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You can only invoke reject() in the lexer when the lexer is of the backtracking persuasion (options.backtrack_lexer = true).\n"+this.showPosition(),{text:"",token:null,line:this.yylineno})},less:function(e){this.unput(this.match.slice(e))},pastInput:function(){var e=this.matched.substr(0,this.matched.length-this.match.length);return(e.length>20?"...":"")+e.substr(-20).replace(/\n/g,"")},upcomingInput:function(){var e=this.match;return e.length<20&&(e+=this._input.substr(0,20-e.length)),(e.substr(0,20)+(e.length>20?"...":"")).replace(/\n/g,"")},showPosition:function(){var e=this.pastInput(),t=new Array(e.length+1).join("-");return e+this.upcomingInput()+"\n"+t+"^"},test_match:function(e,t){var i,a,o;if(this.options.backtrack_lexer&&(o={yylineno:this.yylineno,yylloc:{first_line:this.yylloc.first_line,last_line:this.last_line,first_column:this.yylloc.first_column,last_column:this.yylloc.last_column},yytext:this.yytext,match:this.match,matches:this.matches,matched:this.matched,yyleng:this.yyleng,offset:this.offset,_more:this._more,_input:this._input,yy:this.yy,conditionStack:this.conditionStack.slice(0),done:this.done},this.options.ranges&&(o.yylloc.range=this.yylloc.range.slice(0))),(a=e[0].match(/(?:\r\n?|\n).*/g))&&(this.yylineno+=a.length),this.yylloc={first_line:this.yylloc.last_line,last_line:this.yylineno+1,first_column:this.yylloc.last_column,last_column:a?a[a.length-1].length-a[a.length-1].match(/\r?\n?/)[0].length:this.yylloc.last_column+e[0].length},this.yytext+=e[0],this.match+=e[0],this.matches=e,this.yyleng=this.yytext.length,this.options.ranges&&(this.yylloc.range=[this.offset,this.offset+=this.yyleng]),this._more=!1,this._backtrack=!1,this._input=this._input.slice(e[0].length),this.matched+=e[0],i=this.performAction.call(this,this.yy,this,t,this.conditionStack[this.conditionStack.length-1]),this.done&&this._input&&(this.done=!1),i)return i;if(this._backtrack){for(var n in o)this[n]=o[n];return!1}return!1},next:function(){if(this.done)return this.EOF;var e,t,i,a;this._input||(this.done=!0),this._more||(this.yytext="",this.match="");for(var o=this._currentRules(),n=0;n<o.length;n++)if((i=this._input.match(this.rules[o[n]]))&&(!t||i[0].length>t[0].length)){if(t=i,a=n,this.options.backtrack_lexer){if(!1!==(e=this.test_match(i,o[n])))return e;if(this._backtrack){t=!1;continue}return!1}if(!this.options.flex)break}return t?!1!==(e=this.test_match(t,o[a]))&&e:""===this._input?this.EOF:this.parseError("Lexical error on line "+(this.yylineno+1)+". 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18:return"(";case 17:case 19:return")";case 20:return"[";case 21:return"]";case 22:case 24:return"{";case 23:case 25:return"}";case 26:return"_";case 27:return"|";case 28:return"LEFT|";case 29:return"RIGHT|";case 30:return"!";case 31:return"SIGN";case 32:case 34:case 40:return t.yytext="<=","SIGN";case 33:case 35:case 41:return t.yytext=">=","SIGN";case 36:case 37:case 38:case 39:return t.yytext="<>","SIGN";case 42:case 43:return"FRAC";case 44:return"sqrt";case 45:return"abs";case 46:return"ln";case 47:return"log";case 48:case 49:case 50:case 51:return"TRIG";case 52:return t.yytext="sin","TRIG";case 53:return t.yytext="cos","TRIG";case 54:return t.yytext="tan","TRIG";case 55:return t.yytext="csc","TRIG";case 56:return t.yytext="sec","TRIG";case 57:return t.yytext="cot","TRIG";case 58:return t.yytext="arcsin","TRIG";case 59:return t.yytext="arccos","TRIG";case 60:return t.yytext="arctan","TRIG";case 61:return t.yytext="arccsc","TRIG";case 62:return t.yytext="arcsec","TRIG";case 63:return t.yytext="arccot","TRIG";case 64:case 65:return"TRIGINV";case 66:return t.yytext="sinh","TRIG";case 67:return t.yytext="cosh","TRIG";case 68:case 71:return t.yytext="tanh","TRIG";case 69:return t.yytext="csch","TRIG";case 70:return t.yytext="sech","TRIG";case 72:return"CONST";case 73:case 74:return t.yytext="pi","CONST";case 75:case 78:return"VAR";case 76:case 77:return t.yytext="theta","VAR";case 79:case 80:return t.yytext="phi","VAR";case 81:return e.symbolLexer(t.yytext);case 82:return"EOF";case 83:return"INVALID";case 84:console.log(t.yytext)}},rules:[/^(?:\s+)/,/^(?:\\space)/,/^(?:\\ )/,/^(?:[0-9]+\.?)/,/^(?:([0-9]+)?\.[0-9]+)/,/^(?:\*\*)/,/^(?:\*)/,/^(?:\\cdot|\xb7)/,/^(?:\\times|\xd7)/,/^(?:\\ast)/,/^(?:\/)/,/^(?:\\div|\xf7)/,/^(?:-)/,/^(?:\u2212)/,/^(?:\+)/,/^(?:\^)/,/^(?:\()/,/^(?:\))/,/^(?:\\left\()/,/^(?:\\right\))/,/^(?:\[)/,/^(?:\])/,/^(?:\{)/,/^(?:\})/,/^(?:\\left\{)/,/^(?:\\right\})/,/^(?:_)/,/^(?:\|)/,/^(?:\\left\|)/,/^(?:\\right\|)/,/^(?:\!)/,/^(?:<=|>=|<>|<|>|=)/,/^(?:\\le)/,/^(?:\\ge)/,/^(?:\\leq)/,/^(?:\\geq)/,/^(?:=\/=)/,/^(?:\\ne)/,/^(?:\\neq)/,/^(?:\u2260)/,/^(?:\u2264)/,/^(?:\u2265)/,/^(?:\\frac)/,/^(?:\\dfrac)/,/^(?:sqrt|\\sqrt)/,/^(?:abs|\\abs)/,/^(?:ln|\\ln)/,/^(?:log|\\log)/,/^(?:sin|cos|tan)/,/^(?:csc|sec|cot)/,/^(?:sinh|cosh|tanh)/,/^(?:csch|sech|coth)/,/^(?:\\sin)/,/^(?:\\cos)/,/^(?:\\tan)/,/^(?:\\csc)/,/^(?:\\sec)/,/^(?:\\cot)/,/^(?:\\arcsin)/,/^(?:\\arccos)/,/^(?:\\arctan)/,/^(?:\\arccsc)/,/^(?:\\arcsec)/,/^(?:\\arccot)/,/^(?:arcsin|arccos|arctan)/,/^(?:arccsc|arcsec|arccot)/,/^(?:\\sinh)/,/^(?:\\cosh)/,/^(?:\\tanh)/,/^(?:\\csch)/,/^(?:\\sech)/,/^(?:\\coth)/,/^(?:pi)/,/^(?:\u03c0)/,/^(?:\\pi)/,/^(?:theta)/,/^(?:\u03b8)/,/^(?:\\theta)/,/^(?:phi)/,/^(?:\u03c6)/,/^(?:\\phi)/,/^(?:[a-zA-Z])/,/^(?:$)/,/^(?:.)/,/^(?:.)/],conditions:{INITIAL:{rules:[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84],inclusive:!0}}};function B(){this.yy={}}return O.lexer=S,B.prototype=O,O.Parser=B,new B}();e.parser=t}(o),function(e){var t=function(){var e=function(e,t,i,a){for(i=i||{},a=e.length;a--;i[e[a]]=t);return i},t=[1,11],i=[1,9],a=[8,17],o=[6,11],n=[6,11,13,17],r={trace:function(){},yy:{},symbols_:{error:2,unitvalue:3,magnitude:4,unit:5,EOF:6,float:7,POW:8,int:9,multatoms:10,DIV:11,expatom:12,MUL:13,atom:14,"^":15,nat:16,ATOM:17,FLOAT:18,NAT:19,NEG:20,$accept:0,$end:1},terminals_:{2:"error",6:"EOF",8:"POW",11:"DIV",13:"MUL",15:"^",17:"ATOM",18:"FLOAT",19:"NAT",20:"NEG"},productions_:[0,[3,3],[3,2],[4,3],[4,1],[5,3],[5,1],[10,3],[10,2],[10,1],[12,3],[12,1],[14,1],[7,1],[7,1],[16,1],[9,2],[9,1]],performAction:function(e,t,i,a,o,n,r){var s=n.length-1;switch(o){case 1:return{type:"unitMagnitude",magnitude:n[s-2],unit:n[s-1]};case 2:return{type:"unitStandalone",unit:n[s-1]};case 3:this.$=n[s-2]+"e"+n[s];break;case 4:case 13:case 14:case 15:case 17:this.$=n[s];break;case 5:this.$={num:n[s-2],denom:n[s]};break;case 6:this.$={num:n[s],denom:null};break;case 7:this.$=[n[s-2]].concat(n[s]);break;case 8:this.$=[n[s-1]].concat(n[s]);break;case 9:this.$=[n[s]];break;case 10:this.$={name:n[s-2],pow:n[s]};break;case 11:this.$={name:n[s],pow:1};break;case 12:this.$=e;break;case 16:this.$="-"+n[s]}},table:[{3:1,4:2,5:3,7:4,10:5,12:8,14:10,16:7,17:t,18:[1,6],19:i},{1:[3]},{5:12,10:5,12:8,14:10,17:t},{6:[1,13]},{8:[1,14],17:[2,4]},{6:[2,6],11:[1,15]},e(a,[2,13]),e(a,[2,14]),e(o,[2,9],{12:8,14:10,10:17,13:[1,16],17:t}),e([6,8,11,13,17],[2,15]),e(n,[2,11],{15:[1,18]}),e([6,11,13,15,17],[2,12]),{6:[1,19]},{1:[2,2]},{9:20,19:[1,22],20:[1,21]},{10:23,12:8,14:10,17:t},{10:24,12:8,14:10,17:t},e(o,[2,8]),{16:25,19:i},{1:[2,1]},{17:[2,3]},{19:[1,26]},{17:[2,17]},{6:[2,5]},e(o,[2,7]),e(n,[2,10]),{17:[2,16]}],defaultActions:{13:[2,2],19:[2,1],20:[2,3],22:[2,17],23:[2,5],26:[2,16]},parseError:function(e,t){if(!t.recoverable)throw new Error(e);this.trace(e)},parse:function(e){var t=this,i=[0],a=[null],o=[],n=this.table,r="",s=0,l=0,c=0,p=2,h=1,$=o.slice.call(arguments,1),d=Object.create(this.lexer),u={yy:{}};for(var m in this.yy)Object.prototype.hasOwnProperty.call(this.yy,m)&&(u.yy[m]=this.yy[m]);d.setInput(e,u.yy),u.yy.lexer=d,u.yy.parser=this,"undefined"===typeof d.yylloc&&(d.yylloc={});var b=d.yylloc;o.push(b);var y=d.options&&d.options.ranges;function f(){var e;return"number"!==typeof(e=d.lex()||h)&&(e=t.symbols_[e]||e),e}"function"===typeof u.yy.parseError?this.parseError=u.yy.parseError:this.parseError=Object.getPrototypeOf(this).parseError;for(var g,x,v,T,A,w,C,O,S,B={};;){if(v=i[i.length-1],this.defaultActions[v]?T=this.defaultActions[v]:(null!=g&&"undefined"!==typeof g||(g=f()),T=n[v]&&n[v][g]),"undefined"===typeof T||!T.length||!T[0]){var _="";for(w in S=[],n[v])this.terminals_[w]&&w>p&&S.push("'"+this.terminals_[w]+"'");_=d.showPosition?"Parse error on line "+(s+1)+":\n"+d.showPosition()+"\nExpecting "+S.join(", ")+", got '"+(this.terminals_[g]||g)+"'":"Parse error on line "+(s+1)+": Unexpected "+(g===h?"end of input":"'"+(this.terminals_[g]||g)+"'"),this.parseError(_,{text:d.match,token:this.terminals_[g]||g,line:d.yylineno,loc:b,expected:S})}if(T[0]instanceof Array&&T.length>1)throw new Error("Parse Error: multiple actions possible at state: "+v+", token: "+g);switch(T[0]){case 1:i.push(g),a.push(d.yytext),o.push(d.yylloc),i.push(T[1]),g=null,x?(g=x,x=null):(l=d.yyleng,r=d.yytext,s=d.yylineno,b=d.yylloc,c>0&&c--);break;case 2:if(C=this.productions_[T[1]][1],B.$=a[a.length-C],B._$={first_line:o[o.length-(C||1)].first_line,last_line:o[o.length-1].last_line,first_column:o[o.length-(C||1)].first_column,last_column:o[o.length-1].last_column},y&&(B._$.range=[o[o.length-(C||1)].range[0],o[o.length-1].range[1]]),"undefined"!==typeof(A=this.performAction.apply(B,[r,l,s,u.yy,T[1],a,o].concat($))))return A;C&&(i=i.slice(0,-1*C*2),a=a.slice(0,-1*C),o=o.slice(0,-1*C)),i.push(this.productions_[T[1]][0]),a.push(B.$),o.push(B._$),O=n[i[i.length-2]][i[i.length-1]],i.push(O);break;case 3:return!0}}return!0}},s={EOF:1,parseError:function(e,t){if(!this.yy.parser)throw new Error(e);this.yy.parser.parseError(e,t)},setInput:function(e,t){return this.yy=t||this.yy||{},this._input=e,this._more=this._backtrack=this.done=!1,this.yylineno=this.yyleng=0,this.yytext=this.matched=this.match="",this.conditionStack=["INITIAL"],this.yylloc={first_line:1,first_column:0,last_line:1,last_column:0},this.options.ranges&&(this.yylloc.range=[0,0]),this.offset=0,this},input:function(){var e=this._input[0];return this.yytext+=e,this.yyleng++,this.offset++,this.match+=e,this.matched+=e,e.match(/(?:\r\n?|\n).*/g)?(this.yylineno++,this.yylloc.last_line++):this.yylloc.last_column++,this.options.ranges&&this.yylloc.range[1]++,this._input=this._input.slice(1),e},unput:function(e){var t=e.length,i=e.split(/(?:\r\n?|\n)/g);this._input=e+this._input,this.yytext=this.yytext.substr(0,this.yytext.length-t),this.offset-=t;var a=this.match.split(/(?:\r\n?|\n)/g);this.match=this.match.substr(0,this.match.length-1),this.matched=this.matched.substr(0,this.matched.length-1),i.length-1&&(this.yylineno-=i.length-1);var o=this.yylloc.range;return this.yylloc={first_line:this.yylloc.first_line,last_line:this.yylineno+1,first_column:this.yylloc.first_column,last_column:i?(i.length===a.length?this.yylloc.first_column:0)+a[a.length-i.length].length-i[0].length:this.yylloc.first_column-t},this.options.ranges&&(this.yylloc.range=[o[0],o[0]+this.yyleng-t]),this.yyleng=this.yytext.length,this},more:function(){return this._more=!0,this},reject:function(){return this.options.backtrack_lexer?(this._backtrack=!0,this):this.parseError("Lexical error on line "+(this.yylineno+1)+". 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o=t.exp;t.exp=void 0,t=new p(t,o)}return a=t instanceof p?new p(t.base,c.handleNegative(t.exp,"divide")):new p(t,g.Div),e instanceof v&&1===e.n?[a]:[e,a]};if(e instanceof c){var d=h(a.last(e.terms),t);return new c(a.initial(e.terms).concat(d))}return new c(d=h(e,t)).flatten()},fold:function(e){if(e instanceof c){var t=a.find(a.initial(e.terms),(function(e){return(e instanceof $||e instanceof h)&&e.hints.open})),i=a.indexOf(e.terms,t);if(t){var o,n=a.last(e.terms);if(!(t.hints.parens||n.hints.parens||n.has($)||n.has(h)))return o=t instanceof $?$.create([t.type,t.exp],c.createOrAppend(t.arg,n).fold()):h.create(t.base,c.createOrAppend(t.power,n).fold()),0===i?o:new c(e.terms.slice(0,i).concat(o)).fold();t.hints.open=!1}var r=e.partition()[0].terms,s=function(e){return e.n>0},l=function(e){return-1===e.n&&e.hints.negate};if(r.length>1&&a.some(r,l)&&a.some(r,s)&&a.every(r,(function(e){return s(e)||l(e)}))){var p=a.indexOf(e.terms,a.find(e.terms,l)),d=a.indexOf(e.terms,a.find(e.terms,s));if(p<d)return e.replace(d,e.terms[d].negate()).remove(p)}}return e}}),p.prototype=new r,a.extend(p.prototype,{func:p,args:function(){return[this.base,this.exp]},eval:function(e,t){var i=this.base.eval(e,t),a=this.exp.eval(e,t);if(i<0){var o=this.exp.simplify();if(o instanceof T){var n=o.n,r=(n-n.toFixed()).toString().length-2,s=Math.pow(10,r);o=new x(n*s,s).simplify()}if(o instanceof x)if(Math.abs(o.d)%2===1)return(Math.abs(o.n)%2===1?-1:1)*Math.pow(-1*i,a)}return Math.pow(i,a)},getUnits:function(){return this.base.getUnits().map(function(e){return{unit:e.unit,pow:e.pow*this.exp.n}}.bind(this))},codegen:function(){return"Math.pow("+this.base.codegen()+", "+this.exp.codegen()+")"},print:function(){var e=this.base.print();return(this.base instanceof s||this.base instanceof p)&&(e="("+e+")"),e+"^("+this.exp.print()+")"},tex:function(){if(this.isDivide())return"\\frac{1}{"+this.asDivide().tex()+"}";if(this.isRoot())return 1!==this.exp.n&&i("Node marked with hint 'root' does not have exponent of form 1/x."),2===this.exp.d?"\\sqrt{"+this.base.tex()+"}":"\\sqrt["+this.exp.d+"]{"+this.base.tex()+"}";if(this.base instanceof $&&!this.base.isInverse()&&this.exp instanceof g&&this.exp.isSimple()&&this.exp.eval()>=0){var e=this.base.tex({split:!0});return e[0]+"^{"+this.exp.tex()+"}"+e[1]}var t=this.base.tex();return this.base instanceof s||this.base instanceof p||this.base instanceof g&&!this.base.isSimple()?t="("+t+")":(this.base instanceof $||this.base instanceof h)&&(t="["+t+"]"),t+"^{"+this.exp.tex()+"}"},needsExplicitMul:function(){return!this.isRoot()&&this.base.needsExplicitMul()},expand:function(){var e=this.recurse("expand");if(e.base instanceof c)return new c(a.map(e.base.terms,(function(t){return new p(t,e.exp)}))).expand();if(e.base instanceof l&&e.exp instanceof v&&e.exp.abs().eval()>1){for(var t=e.exp.eval()>0,i=e.exp.abs().eval(),o=function(e){return t?e:new p(e,g.Div)},n={1:e.base},r=2;r<=i;r*=2){var s=new c(n[r/2],n[r/2]);n[r]=s.expand().collect()}if(a.has(n,i))return o(n[i]);var h=a.map(i.toString(2).split(""),(function(e,t,i){return Number(e)*Math.pow(2,i.length-t-1)}));return h=a.without(h,0),o(s=new c(a.pick(n,h)).expand().collect())}return e.exp instanceof l?new c(a.map(e.exp.terms,(function(t){return new p(e.base,t).expand()}))).expand():e},factor:function(){var e=this.recurse("factor");return e.base instanceof c?new c(a.map(e.base.terms,(function(t){return t instanceof v&&e.exp.equals(g.Div)?new x(1,t.n):new p(t,e.exp)}))):e},collect:function(e){if(this.base instanceof p)return new p(this.base.base,n=c.createOrAppend(this.base.exp,this.exp)).collect(e);var t=this.recurse("collect",e),i=function(e){return e instanceof h&&e.base.equals(t.base)};if(t.exp instanceof g&&0===t.exp.eval())return g.One;if(t.exp instanceof g&&1===t.exp.eval())return t.base;if(i(t.exp))return t.exp.power;if(t.exp instanceof c&&a.any(t.exp.terms,i)){var o=a.find(t.exp.terms,i);return new p(o.power,n=t.exp.remove(o).flatten()).collect(e)}if(t.base instanceof g&&t.exp instanceof g){if(e&&e.preciseFloats){var n=t.exp.asRational(),r=t.base.getDecimalPlaces();if(new p(t.base,new x(1,n.d)).collect().getDecimalPlaces()>r){var s=new p(t.base,new v(n.n)).collect();return new p(s,new x(1,n.d))}}return t.base.raiseToThe(t.exp,e)}return t},isDivide:function(){var e=function(e){return e instanceof g&&e.hints.divide};return e(this.exp)||this.exp instanceof c&&a.any(this.exp.terms,e)},asDivide:function(){if(this.exp instanceof g){if(-1===this.exp.eval())return this.base;var e=this.exp.negate();return e.hints=a.clone(this.exp.hints),e.hints.divide=!1,new p(this.base,e)}if(this.exp instanceof c)return new p(this.base,this.exp.factorOut());i("called asDivide() on an Expr that wasn't a Num or Mul")},isRoot:function(){return this.exp instanceof x&&this.exp.hints.root},isSquaredTrig:function(){return this.base instanceof $&&!this.base.isInverse()&&this.exp instanceof g&&2===this.exp.eval()},getDenominator:function(){if(this.exp instanceof g&&-1===this.exp.eval())return c.createOrAppend(this.base,this.base.getDenominator()).flatten();if(this.exp.isNegative()){var e=new p(this.base,c.handleNegative(this.exp).collect());return c.createOrAppend(e,e.collect().getDenominator()).flatten()}return this.base instanceof g?new p(this.base.getDenominator(),this.exp).collect():g.One},findGCD:function(e){var t,i;if(e instanceof p?(t=e.base,i=e.exp):(t=e,i=g.One),this.base.equals(t)){if(this.exp.equals(i))return this;if(this.exp instanceof g&&i instanceof g)return new p(this.base,g.min(this.exp,i)).collect();if(this.exp instanceof g||i instanceof g)return g.One;var a=this.exp.asMul().partition(),o=i.asMul().partition();if(a[1].equals(o[1]))return new p(t,new c(g.min(a[0].reduce(),o[0].reduce()),a[1].flatten()).flatten()).collect()}return g.One},isPositive:function(){if(this.base.isPositive())return!0;var e=this.exp.simplify();return e instanceof v&&e.eval()%2===0},asPositiveFactor:function(){if(this.isPositive())return this;var e=this.exp.simplify();if(e instanceof v){var t=e.eval();if(t>2)return new p(this.base,new v(t-1));if(t<-2)return new p(this.base,new v(t+1))}return g.One}}),a.extend(p,{sqrt:function(e){return new p(e,g.Sqrt)},nthroot:function(e,t){return new p(e,c.fold(c.handleDivide(new v(1),t)).addHint("root"))}}),h.prototype=new r,a.extend(h.prototype,{func:h,args:function(){return[this.base,this.power]},eval:function(e,t){return Math.log(this.power.eval(e,t))/Math.log(this.base.eval(e,t))},codegen:function(){return"(Math.log("+this.power.codegen()+") / Math.log("+this.base.codegen()+"))"},print:function(){var e="("+this.power.print()+")";return this.isNatural()?"ln"+e:"log_("+this.base.print()+") "+e},tex:function(){var e="("+this.power.tex()+")";return this.isNatural()?"\\ln"+e:"\\log_{"+this.base.tex()+"}"+e},collect:function(e){var t=this.recurse("collect",e);return t.power instanceof g&&1===t.power.eval()?g.Zero:t.base.equals(t.power)?g.One:t.power instanceof p&&t.power.base.equals(t.base)?t.power.exp:t},expand:function(){var e=this.recurse("expand");return e.power instanceof c?new l(a.map(e.power.terms,(function(t){return new h(e.base,t).expand()}))):e.power instanceof p?new c(e.power.exp,new h(e.base,e.power.base).expand()).flatten():e.isNatural()?e:c.handleDivide(new h(f.e,e.power),new h(f.e,e.base))},hints:a.extend(h.prototype.hints,{open:!1}),isPositive:function(){var e=this.collect();return e.base instanceof g&&e.power instanceof g&&this.eval()>0},needsExplicitMul:function(){return!1},isNatural:function(){return this.base.equals(f.e)}}),a.extend(h,{natural:function(){return f.e},common:function(){return g.Ten},create:function(e,t){var i=new h(e,t);return t.hints.parens||(i=i.addHint("open")),i}}),$.prototype=new r,a.extend($.prototype,{func:$,args:function(){return[this.type,this.arg]},functions:{sin:{eval:Math.sin,codegen:"Math.sin((",tex:"\\sin",expand:function(){return this}},cos:{eval:Math.cos,codegen:"Math.cos((",tex:"\\cos",expand:function(){return this}},tan:{eval:Math.tan,codegen:"Math.tan((",tex:"\\tan",expand:function(){return c.handleDivide($.sin(this.arg),$.cos(this.arg))}},csc:{eval:function(e){return 1/Math.sin(e)},codegen:"(1/Math.sin(",tex:"\\csc",expand:function(){return c.handleDivide(g.One,$.sin(this.arg))}},sec:{eval:function(e){return 1/Math.cos(e)},codegen:"(1/Math.cos(",tex:"\\sec",expand:function(){return c.handleDivide(g.One,$.cos(this.arg))}},cot:{eval:function(e){return 1/Math.tan(e)},codegen:"(1/Math.tan(",tex:"\\cot",expand:function(){return c.handleDivide($.cos(this.arg),$.sin(this.arg))}},arcsin:{eval:Math.asin,codegen:"Math.asin((",tex:"\\arcsin"},arccos:{eval:Math.acos,codegen:"Math.acos((",tex:"\\arccos"},arctan:{eval:Math.atan,codegen:"Math.atan((",tex:"\\arctan"},arccsc:{eval:function(e){return Math.asin(1/e)},codegen:"Math.asin(1/(",tex:"\\operatorname{arccsc}"},arcsec:{eval:function(e){return Math.acos(1/e)},codegen:"Math.acos(1/(",tex:"\\operatorname{arcsec}"},arccot:{eval:function(e){return Math.atan(1/e)},codegen:"Math.atan(1/(",tex:"\\operatorname{arccot}"},sinh:{eval:function(e){return(Math.exp(e)-Math.exp(-e))/2},codegen:function(e){return"((Math.exp("+e+") - Math.exp(-("+e+"))) / 2)"},tex:"\\sinh",expand:function(){return this}},cosh:{eval:function(e){return(Math.exp(e)+Math.exp(-e))/2},codegen:function(e){return"((Math.exp("+e+") + Math.exp(-("+e+"))) / 2)"},tex:"\\cosh",expand:function(){return this}},tanh:{eval:function(e){return(Math.exp(e)-Math.exp(-e))/(Math.exp(e)+Math.exp(-e))},codegen:function(e){return"((Math.exp("+e+") - Math.exp(-("+e+"))) / (Math.exp("+e+") + Math.exp(-("+e+"))))"},tex:"\\tanh",expand:function(){return c.handleDivide($.sinh(this.arg),$.cosh(this.arg))}},csch:{eval:function(e){return 2/(Math.exp(e)-Math.exp(-e))},codegen:function(e){return"(2 / (Math.exp("+e+") - Math.exp(-("+e+"))))"},tex:"\\csch",expand:function(){return c.handleDivide(g.One,$.sinh(this.arg))}},sech:{eval:function(e){return 2/(Math.exp(e)+Math.exp(-e))},codegen:function(e){return"(2 / (Math.exp("+e+") + Math.exp(-("+e+"))))"},tex:"\\sech",expand:function(){return c.handleDivide(g.One,$.cosh(this.arg))}},coth:{eval:function(e){return(Math.exp(e)+Math.exp(-e))/(Math.exp(e)-Math.exp(-e))},codegen:function(e){return"((Math.exp("+e+") + Math.exp(-("+e+"))) / (Math.exp("+e+") - Math.exp(-("+e+"))))"},tex:"\\coth",expand:function(){return c.handleDivide($.cosh(this.arg),$.sinh(this.arg))}}},isEven:function(){return a.contains(["cos","sec"],this.type)},isInverse:function(){return 0===this.type.indexOf("arc")},isBasic:function(){return a.contains(["sin","cos"],this.type)},eval:function(e,t){return(0,this.functions[this.type].eval)(this.arg.eval(e,t))},codegen:function(){var e=this.functions[this.type].codegen;if("function"===typeof e)return e(this.arg.codegen());if("string"===typeof e)return e+this.arg.codegen()+"))";throw new Error("codegen not implemented for "+this.type)},print:function(){return this.type+"("+this.arg.print()+")"},tex:function(e){var t=this.functions[this.type].tex,i="("+this.arg.tex()+")";return e&&e.split?[t,i]:t+i},hints:a.extend($.prototype.hints,{open:!1}),isPositive:function(){return this.collect().arg instanceof g&&this.eval()>0},completeParse:function(){if(this.exp){var e=new p(this,this.exp);return this.exp=void 0,e}return this},needsExplicitMul:function(){return!1},expand:function(){var e=this.recurse("expand");if(e.isInverse())return e;var t=e.functions[e.type].expand;return a.bind(t,e)()},collect:function(e){var t,i=this.recurse("collect",e);return!i.isInverse()&&i.arg.isNegative()?(t=i.arg instanceof g?i.arg.abs():c.handleDivide(i.arg,g.Neg).collect(e),i.isEven()?new $(i.type,t):new c(g.Neg,new $(i.type,t))):i}}),a.extend($,{create:function(e,t){var i=e[0],a=e[1];a&&a.equals(g.Neg)&&(i="arc"+i,a=void 0);var o=new $(i,t);return t.hints.parens||(o=o.addHint("open")),a&&(o.exp=a),o},sin:function(e){return new $("sin",e)},cos:function(e){return new $("cos",e)},sinh:function(e){return new $("sinh",e)},cosh:function(e){return new $("cosh",e)}}),d.prototype=new r,a.extend(d.prototype,{func:d,args:function(){return[this.arg]},eval:function(e,t){return Math.abs(this.arg.eval(e,t))},codegen:function(){return"Math.abs("+this.arg.codegen()+")"},print:function(){return"abs("+this.arg.print()+")"},tex:function(){return"\\left|"+this.arg.tex()+"\\right|"},collect:function(e){var t=this.recurse("collect",e);if(t.arg.isPositive())return t.arg;if(t.arg instanceof g)return t.arg.abs();if(t.arg instanceof c){var i=a.groupBy(t.arg.terms,(function(e){return e.isPositive()?"positive":e instanceof g?"number":"other"})),o=i.positive.concat(a.invoke(i.number,"abs"));return i.other.length&&o.push(new d(new c(i.other).flatten())),new c(o).flatten()}return t},expand:function(){var e=this.recurse("expand");return e.arg instanceof c?new c(a.map(e.arg.terms,(function(e){return new d(e)}))):e},isPositive:function(){return!0}}),u.prototype=new r,a.extend(u.prototype,{func:u,args:function(){return[this.left,this.type,this.right]},needsExplicitMul:function(){return!1},print:function(){return this.left.print()+this.type+this.right.print()},signs:{"=":" = ","<":" < ",">":" > ","<>":" \\ne ","<=":" \\le ",">=":" \\ge "},tex:function(){return this.left.tex()+this.signs[this.type]+this.right.tex()},normalize:function(){var e=this.recurse("normalize");return a.contains([">",">="],e.type)?new u(e.right,e.type.replace(">","<"),e.left):e},asExpr:function(e){var t=function(e){return e instanceof g&&e.isSimple()&&0===e.eval()},i=[];this.left instanceof l?i=a.clone(this.left.terms):t(this.left)||(i=[this.left]),this.right instanceof l?i=i.concat(this.right.negate().terms):t(this.right)||i.push(this.right.negate());var o=!this.isEquality();i=a.invoke(i,"collect",{preciseFloats:!0});for(var n=0;n<i.length;n++){var r=i[n].getDenominator();o&&!r.isPositive()&&(r=r.asPositiveFactor()),r.equals(g.One)||(i=a.map(i,(function(e){return c.createOrAppend(e,r).simplify({once:!0,preciseFloats:!0})})))}var s=new l(i).flatten();return e?s:this.divideThrough(s)},divideThrough:function(e){var t=!this.isEquality(),i=e.simplify({once:!0}),o=i.factor({keepNegative:t});if(!(o instanceof c))return e;var n=o.terms,r=a.groupBy(n,(function(e){return e instanceof l})),s=r[!0]||[],h=r[!1]||[];if(s.length&&this.isEquality())return new c(s).flatten();var $=h;s.length||($=a.reject($,(function(e){return!!e.getVars().length}))),t&&($=a.invoke($,"asPositiveFactor")),$=a.reject($,(function(e){return e.equals(g.One)})),$=a.map($,(function(e){return new p(e,g.Div)}));var d=new c(n.concat($)).collect();return d.equals(o)?i:d},isEquality:function(){return a.contains(["=","<>"],this.type)},compare:function(e){if(!(e instanceof u))return!1;var t=this.normalize(),i=e.normalize();if(t.type!==i.type)return!1;var a=t.divideThrough(t.asExpr(!0).collect()),o=i.divideThrough(i.asExpr(!0).collect());return t.isEquality()?a.compare(o)||a.compare(c.handleNegative(o)):a.compare(o)},sameForm:function(e){var t=this.normalize(),i=e.normalize(),a=t.left.sameForm(i.left)&&t.right.sameForm(i.right);return t.isEquality()?a||t.left.sameForm(i.right)&&t.right.sameForm(i.left):a},isSimplified:function(){var e=this.asExpr(!0),t=this.divideThrough(e).simplify();return e.equals(t)&&this.left.isSimplified()&&this.right.isSimplified()}}),a.extend(u.prototype,{solveLinearEquationForVariable:function(e){var t=this.asExpr();if(!t.is(l)||2!==t.terms.length)throw new Error("Can only handle linear equations of the form a + bx (= 0)");var i,o,n;return(n=t.terms[0]).has(y)&&a.contains(n.getVars(),e.symbol)?(i=c.handleNegative(t.terms[1]),o=c.handleDivide(t.terms[0],e)):(i=c.handleNegative(t.terms[0]),o=c.handleDivide(t.terms[1],e)),c.handleDivide(i,o).simplify()}}),m.prototype=new r,a.extend(m.prototype,{needsExplicitMul:function(){return!1},findGCD:function(e){return e instanceof m||e instanceof g?this.equals(e)?this:g.One:e.findGCD(this)}}),b.prototype=new m,a.extend(b.prototype,{func:b,args:function(){return[this.symbol,this.arg]},print:function(){return this.symbol+"("+this.arg.print()+")"},tex:function(){return this.symbol+"("+this.arg.tex()+")"},eval:function(t,i){var o=this.arg,n=t[this.symbol],r=a.extend(a.clone(t),{x:o.eval(t,i)}),s=e.parse(n,i);return s.parsed?s.expr.eval(r,i):s},codegen:function(){return'vars["'+this.symbol+'"]('+this.arg.codegen()+")"},getUnits:function(){return this.arg.getUnits()},getVars:function(e){return e?this.arg.getVars():a.union(this.arg.getVars(),[this.symbol]).sort()},getConsts:function(){return this.arg.getConsts()}}),y.prototype=new m,a.extend(y.prototype,{func:y,args:function(){return[this.symbol,this.subscript]},exprArgs:function(){return[]},recurse:function(){return this},print:function(){var e="";return this.subscript&&(e="_("+this.subscript.print()+")"),this.symbol+e},prettyPrint:function(){var e=this.subscript;return e&&(e instanceof g||e instanceof m)?this.symbol+"_"+e.print():this.print()},tex:function(){var e="";return this.subscript&&(e="_{"+this.subscript.tex()+"}"),(this.symbol.length>1?"\\":"")+this.symbol+e},repr:function(){return"Var("+this.print()+")"},eval:function(e,t){return e[this.prettyPrint()]},codegen:function(){return'vars["'+this.prettyPrint()+'"]'},getVars:function(){return[this.prettyPrint()]},isPositive:function(){return!1}}),f.prototype=new m,a.extend(f.prototype,{func:f,args:function(){return[this.symbol]},recurse:function(){return this},eval:function(e,t){return"pi"===this.symbol?Math.PI:"e"===this.symbol?Math.E:void 0},codegen:function(){return"pi"===this.symbol?"Math.PI":"e"===this.symbol?"Math.E":void 0},print:function(){return this.symbol},tex:function(){return"pi"===this.symbol?"\\pi ":"e"===this.symbol?"e":void 0},isPositive:function(){return this.eval()>0},abs:function(){return this.eval()>0?this:c.handleNegative(this)},getConsts:function(){return[this.print()]}}),f.e=new f("e"),f.pi=new f("pi"),g.prototype=new r,a.extend(g.prototype,{repr:function(){return this.print()},strip:function(){return this.abs()},recurse:function(){return this},codegen:function(){return this.print()},add:t,mul:t,negate:t,isSubtract:function(){return 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this},n.prototype.write=function(e){this.block[this.offset]|=(255&e)<<this.shift,this.shift?this.shift-=8:(this.offset++,this.shift=24),16===this.offset&&this.processBlock()},n.prototype.digest=function(e){this.write(128),(this.offset>14||14===this.offset&&this.shift<24)&&this.processBlock(),this.offset=14,this.shift=24,this.write(0),this.write(0),this.write(this.totalLength>0xffffffffff?this.totalLength/1099511627776:0),this.write(this.totalLength>4294967295?this.totalLength/4294967296:0);for(var t=24;t>=0;t-=8)this.write(this.totalLength>>t);var i=new a(20),o=new DataView(i.buffer);return o.setUint32(0,this.h0,!1),o.setUint32(4,this.h1,!1),o.setUint32(8,this.h2,!1),o.setUint32(12,this.h3,!1),o.setUint32(16,this.h4,!1),e?i.toString(e):i},n.prototype.processBlock=function(){for(var e=16;e<80;e++){var t=this.block[e-3]^this.block[e-8]^this.block[e-14]^this.block[e-16];this.block[e]=t<<1|t>>>31}var i,a,o=this.h0,n=this.h1,r=this.h2,s=this.h3,l=this.h4;for(e=0;e<80;e++){e<20?(i=s^n&(r^s),a=1518500249):e<40?(i=n^r^s,a=1859775393):e<60?(i=n&r|s&(n|r),a=2400959708):(i=n^r^s,a=3395469782);var c=(o<<5|o>>>27)+i+l+a+(0|this.block[e]);l=s,s=r,r=n<<30|n>>>2,n=o,o=c}for(this.h0=this.h0+o|0,this.h1=this.h1+n|0,this.h2=this.h2+r|0,this.h3=this.h3+s|0,this.h4=this.h4+l|0,this.offset=0,e=0;e<16;e++)this.block[e]=0}},97467:function(e,t,i){var a=i(19778).lW,o=i(79488),n=64,r=new Uint32Array([1116352408,1899447441,3049323471,3921009573,961987163,1508970993,2453635748,2870763221,3624381080,310598401,607225278,1426881987,1925078388,2162078206,2614888103,3248222580,3835390401,4022224774,264347078,604807628,770255983,1249150122,1555081692,1996064986,2554220882,2821834349,2952996808,3210313671,3336571891,3584528711,113926993,338241895,666307205,773529912,1294757372,1396182291,1695183700,1986661051,2177026350,2456956037,2730485921,2820302411,3259730800,3345764771,3516065817,3600352804,4094571909,275423344,430227734,506948616,659060556,883997877,958139571,1322822218,1537002063,1747873779,1955562222,2024104815,2227730452,2361852424,2428436474,2756734187,3204031479,3329325298]),s=Math.pow(2,53)-1;function l(){this.state=[1779033703,3144134277,1013904242,2773480762,1359893119,2600822924,528734635,1541459225],this.temp=new Int32Array(64),this.buffer=new Uint8Array(64),this.bufferLength=0,this.bytesHashed=0,this.finished=!1}e.exports=l,l.BLOCK_SIZE=n,l.prototype.update=function(e){if(this.finished)throw new Error("Attempted to update an already finished hash.");if(o.isEmptyData(e))return this;var t=0,i=(e=o.convertToBuffer(e)).byteLength;if(this.bytesHashed+=i,8*this.bytesHashed>s)throw new Error("Cannot hash more than 2^53 - 1 bits");for(;i>0;)this.buffer[this.bufferLength++]=e[t++],i--,this.bufferLength===n&&(this.hashBuffer(),this.bufferLength=0);return this},l.prototype.digest=function(e){if(!this.finished){var t=8*this.bytesHashed,i=new DataView(this.buffer.buffer,this.buffer.byteOffset,this.buffer.byteLength),o=this.bufferLength;if(i.setUint8(this.bufferLength++,128),o%n>=56){for(var r=this.bufferLength;r<n;r++)i.setUint8(r,0);this.hashBuffer(),this.bufferLength=0}for(r=this.bufferLength;r<56;r++)i.setUint8(r,0);i.setUint32(56,Math.floor(t/4294967296),!0),i.setUint32(60,t),this.hashBuffer(),this.finished=!0}var s=new a(32);for(r=0;r<8;r++)s[4*r]=this.state[r]>>>24&255,s[4*r+1]=this.state[r]>>>16&255,s[4*r+2]=this.state[r]>>>8&255,s[4*r+3]=this.state[r]>>>0&255;return e?s.toString(e):s},l.prototype.hashBuffer=function(){for(var e=this.buffer,t=this.state,i=t[0],a=t[1],o=t[2],s=t[3],l=t[4],c=t[5],p=t[6],h=t[7],$=0;$<n;$++){if($<16)this.temp[$]=(255&e[4*$])<<24|(255&e[4*$+1])<<16|(255&e[4*$+2])<<8|255&e[4*$+3];else{var d=this.temp[$-2],u=(d>>>17|d<<15)^(d>>>19|d<<13)^d>>>10,m=((d=this.temp[$-15])>>>7|d<<25)^(d>>>18|d<<14)^d>>>3;this.temp[$]=(u+this.temp[$-7]|0)+(m+this.temp[$-16]|0)}var b=(((l>>>6|l<<26)^(l>>>11|l<<21)^(l>>>25|l<<7))+(l&c^~l&p)|0)+(h+(r[$]+this.temp[$]|0)|0)|0,y=((i>>>2|i<<30)^(i>>>13|i<<19)^(i>>>22|i<<10))+(i&a^i&o^a&o)|0;h=p,p=c,c=l,l=s+b|0,s=o,o=a,a=i,i=b+y|0}t[0]+=i,t[1]+=a,t[2]+=o,t[3]+=s,t[4]+=l,t[5]+=c,t[6]+=p,t[7]+=h}},544:function(e,t,i){var a=i(23657);a.crypto.lib=i(82623),a.Buffer=i(19778).lW,a.url=i(12114),a.querystring=i(30863),a.realClock=i(35067),a.environment="js",a.createEventStream=i(28962).createEventStream,a.isBrowser=function(){return!0},a.isNode=function(){return!1};var o=i(8468);if(e.exports=o,i(73916),i(28732),i(14524),i(79382),i(19589),i(89065),i(78437),o.XML.Parser=i(68918),i(76311),"undefined"===typeof n)var n={browser:!0}},65473:function(e,t,i){var a=i(8468),o=a.util.url,n=a.util.crypto.lib,r=a.util.base64.encode,s=a.util.inherit,l=function(e){var t={"+":"-","=":"_","/":"~"};return e.replace(/[\+=\/]/g,(function(e){return t[e]}))},c=function(e,t){var i=n.createSign("RSA-SHA1");return i.write(e),l(i.sign(t,"base64"))},p=function(e,t,i,a){var o=JSON.stringify({Statement:[{Resource:e,Condition:{DateLessThan:{"AWS:EpochTime":t}}}]});return{Expires:t,"Key-Pair-Id":i,Signature:c(o.toString(),a)}},h=function(e,t,i){return e=e.replace(/\s/gm,""),{Policy:l(r(e)),"Key-Pair-Id":t,Signature:c(e,i)}},$=function(e){var t=e.split("://");if(t.length<2)throw new Error("Invalid URL.");return t[0].replace("*","")},d=function(e){var t=o.parse(e);return t.path.replace(/^\//,"")+(t.hash||"")},u=function(e,t){if(!t||"function"!==typeof t)throw e;t(e)},m=function(e,t){if(!t||"function"!==typeof t)return e;t(null,e)};a.CloudFront.Signer=s({constructor:function(e,t){if(void 0===e||void 0===t)throw new Error("A key pair ID and private key are required");this.keyPairId=e,this.privateKey=t},getSignedCookie:function(e,t){var i="policy"in e?h(e.policy,this.keyPairId,this.privateKey):p(e.url,e.expires,this.keyPairId,this.privateKey),a={};for(var o in i)Object.prototype.hasOwnProperty.call(i,o)&&(a["CloudFront-"+o]=i[o]);return m(a,t)},getSignedUrl:function(e,t){try{var i=function(e){switch($(e)){case"http":case"https":return e;case"rtmp":return d(e);default:throw new Error("Invalid URI scheme. Scheme must be one of http, https, or rtmp")}}(e.url)}catch(l){return u(l,t)}var a=o.parse(e.url,!0),n=Object.prototype.hasOwnProperty.call(e,"policy")?h(e.policy,this.keyPairId,this.privateKey):p(i,e.expires,this.keyPairId,this.privateKey);for(var r in a.search=null,n)Object.prototype.hasOwnProperty.call(n,r)&&(a.query[r]=n[r]);try{var s="rtmp"===$(e.url)?d(o.format(a)):o.format(a)}catch(l){return u(l,t)}return m(s,t)}}),e.exports=a.CloudFront.Signer},69070:function(e,t,i){var a,o=i(8468);i(73916),i(28732),o.Config=o.util.inherit({constructor:function(e){void 0===e&&(e={}),e=this.extractCredentials(e),o.util.each.call(this,this.keys,(function(t,i){this.set(t,e[t],i)}))},getCredentials:function(e){var t=this;function i(i){e(i,i?null:t.credentials)}function a(e,t){return new o.util.error(t||new Error,{code:"CredentialsError",message:e,name:"CredentialsError"})}t.credentials?"function"===typeof t.credentials.get?t.credentials.get((function(e){e&&(e=a("Could not load credentials from "+t.credentials.constructor.name,e)),i(e)})):function(){var e=null;t.credentials.accessKeyId&&t.credentials.secretAccessKey||(e=a("Missing credentials")),i(e)}():t.credentialProvider?t.credentialProvider.resolve((function(e,o){e&&(e=a("Could not load credentials from any providers",e)),t.credentials=o,i(e)})):i(a("No credentials to load"))},getToken:function(e){var t=this;function i(i){e(i,i?null:t.token)}function a(e,t){return new o.util.error(t||new Error,{code:"TokenError",message:e,name:"TokenError"})}t.token?"function"===typeof t.token.get?t.token.get((function(e){e&&(e=a("Could not load token from "+t.token.constructor.name,e)),i(e)})):function(){var e=null;t.token.token||(e=a("Missing token")),i(e)}():t.tokenProvider?t.tokenProvider.resolve((function(e,o){e&&(e=a("Could not load token from any providers",e)),t.token=o,i(e)})):i(a("No token to load"))},update:function(e,t){t=t||!1,e=this.extractCredentials(e),o.util.each.call(this,e,(function(e,i){(t||Object.prototype.hasOwnProperty.call(this.keys,e)||o.Service.hasService(e))&&this.set(e,i)}))},loadFromPath:function(e){this.clear();var t=JSON.parse(o.util.readFileSync(e)),i=new o.FileSystemCredentials(e),a=new o.CredentialProviderChain;return a.providers.unshift(i),a.resolve((function(e,i){if(e)throw e;t.credentials=i})),this.constructor(t),this},clear:function(){o.util.each.call(this,this.keys,(function(e){delete this[e]})),this.set("credentials",void 0),this.set("credentialProvider",void 0)},set:function(e,t,i){void 0===t?(void 0===i&&(i=this.keys[e]),this[e]="function"===typeof i?i.call(this):i):"httpOptions"===e&&this[e]?this[e]=o.util.merge(this[e],t):this[e]=t},keys:{credentials:null,credentialProvider:null,region:null,logger:null,apiVersions:{},apiVersion:null,endpoint:void 0,httpOptions:{timeout:12e4},maxRetries:void 0,maxRedirects:10,paramValidation:!0,sslEnabled:!0,s3ForcePathStyle:!1,s3BucketEndpoint:!1,s3DisableBodySigning:!0,s3UsEast1RegionalEndpoint:"legacy",s3UseArnRegion:void 0,computeChecksums:!0,convertResponseTypes:!0,correctClockSkew:!1,customUserAgent:null,dynamoDbCrc32:!0,systemClockOffset:0,signatureVersion:null,signatureCache:!0,retryDelayOptions:{},useAccelerateEndpoint:!1,clientSideMonitoring:!1,endpointDiscoveryEnabled:void 0,endpointCacheSize:1e3,hostPrefixEnabled:!0,stsRegionalEndpoints:"legacy",useFipsEndpoint:!1,useDualstackEndpoint:!1,token:null},extractCredentials:function(e){return e.accessKeyId&&e.secretAccessKey&&((e=o.util.copy(e)).credentials=new o.Credentials(e)),e},setPromisesDependency:function(e){a=e,null===e&&"function"===typeof Promise&&(a=Promise);var t=[o.Request,o.Credentials,o.CredentialProviderChain];o.S3&&(t.push(o.S3),o.S3.ManagedUpload&&t.push(o.S3.ManagedUpload)),o.util.addPromises(t,a)},getPromisesDependency:function(){return a}}),o.config=new o.Config},21980:function(e,t,i){var a=i(8468);function o(e,t){if("string"===typeof e){if(["legacy","regional"].indexOf(e.toLowerCase())>=0)return e.toLowerCase();throw a.util.error(new Error,t)}}e.exports=function(e,t){var i;if((e=e||{})[t.clientConfig]&&(i=o(e[t.clientConfig],{code:"InvalidConfiguration",message:'invalid "'+t.clientConfig+'" configuration. Expect "legacy"  or "regional". Got "'+e[t.clientConfig]+'".'})))return i;if(!a.util.isNode())return i;if(Object.prototype.hasOwnProperty.call({NODE_ENV:"production",PUBLIC_URL:".",WDS_SOCKET_HOST:void 0,WDS_SOCKET_PATH:void 0,WDS_SOCKET_PORT:void 0,FAST_REFRESH:!0,REACT_APP_BUILD_TYPE:"development"},t.env)&&(i=o({NODE_ENV:"production",PUBLIC_URL:".",WDS_SOCKET_HOST:void 0,WDS_SOCKET_PATH:void 0,WDS_SOCKET_PORT:void 0,FAST_REFRESH:!0,REACT_APP_BUILD_TYPE:"development"}[t.env],{code:"InvalidEnvironmentalVariable",message:"invalid "+t.env+' environmental variable. Expect "legacy"  or "regional". Got "'+{NODE_ENV:"production",PUBLIC_URL:".",WDS_SOCKET_HOST:void 0,WDS_SOCKET_PATH:void 0,WDS_SOCKET_PORT:void 0,FAST_REFRESH:!0,REACT_APP_BUILD_TYPE:"development"}[t.env]+'".'})))return i;var n={};try{n=a.util.getProfilesFromSharedConfig(a.util.iniLoader)[{NODE_ENV:"production",PUBLIC_URL:".",WDS_SOCKET_HOST:void 0,WDS_SOCKET_PATH:void 0,WDS_SOCKET_PORT:void 0,FAST_REFRESH:!0,REACT_APP_BUILD_TYPE:"development"}.AWS_PROFILE||a.util.defaultProfile]}catch(r){}return n&&Object.prototype.hasOwnProperty.call(n,t.sharedConfig)&&(i=o(n[t.sharedConfig],{code:"InvalidConfiguration",message:"invalid "+t.sharedConfig+' profile config. Expect "legacy"  or "regional". Got "'+n[t.sharedConfig]+'".'})),i}},8468:function(e,t,i){var a={util:i(23657)};({}).toString(),e.exports=a,a.util.update(a,{VERSION:"2.1692.0",Signers:{},Protocol:{Json:i(89864),Query:i(66588),Rest:i(19701),RestJson:i(12085),RestXml:i(61622)},XML:{Builder:i(98302),Parser:null},JSON:{Builder:i(48212),Parser:i(26815)},Model:{Api:i(65565),Operation:i(91970),Shape:i(3825),Paginator:i(8594),ResourceWaiter:i(15545)},apiLoader:i(44185),EndpointCache:i(50038).$}),i(78451),i(75724),i(69070),i(78168),i(18663),i(33845),i(10767),i(90630),i(85232),i(70778),i(45663),a.events=new a.SequentialExecutor,a.util.memoizedProperty(a,"endpointCache",(function(){return new a.EndpointCache(a.config.endpointCacheSize)}),!0)},73916:function(e,t,i){var a=i(8468);a.Credentials=a.util.inherit({constructor:function(){if(a.util.hideProperties(this,["secretAccessKey"]),this.expired=!1,this.expireTime=null,this.refreshCallbacks=[],1===arguments.length&&"object"===typeof arguments[0]){var e=arguments[0].credentials||arguments[0];this.accessKeyId=e.accessKeyId,this.secretAccessKey=e.secretAccessKey,this.sessionToken=e.sessionToken}else this.accessKeyId=arguments[0],this.secretAccessKey=arguments[1],this.sessionToken=arguments[2]},expiryWindow:15,needsRefresh:function(){var e=a.util.date.getDate().getTime(),t=new Date(e+1e3*this.expiryWindow);return!!(this.expireTime&&t>this.expireTime)||(this.expired||!this.accessKeyId||!this.secretAccessKey)},get:function(e){var t=this;this.needsRefresh()?this.refresh((function(i){i||(t.expired=!1),e&&e(i)})):e&&e()},refresh:function(e){this.expired=!1,e()},coalesceRefresh:function(e,t){var i=this;1===i.refreshCallbacks.push(e)&&i.load((function(e){a.util.arrayEach(i.refreshCallbacks,(function(i){t?i(e):a.util.defer((function(){i(e)}))})),i.refreshCallbacks.length=0}))},load:function(e){e()}}),a.Credentials.addPromisesToClass=function(e){this.prototype.getPromise=a.util.promisifyMethod("get",e),this.prototype.refreshPromise=a.util.promisifyMethod("refresh",e)},a.Credentials.deletePromisesFromClass=function(){delete this.prototype.getPromise,delete this.prototype.refreshPromise},a.util.addPromises(a.Credentials)},79382:function(e,t,i){var a=i(8468),o=i(62362);a.ChainableTemporaryCredentials=a.util.inherit(a.Credentials,{constructor:function(e){a.Credentials.call(this),e=e||{},this.errorCode="ChainableTemporaryCredentialsProviderFailure",this.expired=!0,this.tokenCodeFn=null;var t=a.util.copy(e.params)||{};if(t.RoleArn&&(t.RoleSessionName=t.RoleSessionName||"temporary-credentials"),t.SerialNumber){if(!e.tokenCodeFn||"function"!==typeof e.tokenCodeFn)throw new a.util.error(new Error("tokenCodeFn must be a function when params.SerialNumber is given"),{code:this.errorCode});this.tokenCodeFn=e.tokenCodeFn}var i=a.util.merge({params:t,credentials:e.masterCredentials||a.config.credentials},e.stsConfig||{});this.service=new o(i)},refresh:function(e){this.coalesceRefresh(e||a.util.fn.callback)},load:function(e){var t=this,i=t.service.config.params.RoleArn?"assumeRole":"getSessionToken";this.getTokenCode((function(a,o){var n={};a?e(a):(o&&(n.TokenCode=o),t.service[i](n,(function(i,a){i||t.service.credentialsFrom(a,t),e(i)})))}))},getTokenCode:function(e){var t=this;this.tokenCodeFn?this.tokenCodeFn(this.service.config.params.SerialNumber,(function(i,o){if(i){var n=i;return i instanceof Error&&(n=i.message),void e(a.util.error(new Error("Error fetching MFA token: "+n),{code:t.errorCode}))}e(null,o)})):e(null)}})},89065:function(e,t,i){var a=i(8468),o=i(49055),n=i(62362);a.CognitoIdentityCredentials=a.util.inherit(a.Credentials,{localStorageKey:{id:"aws.cognito.identity-id.",providers:"aws.cognito.identity-providers."},constructor:function(e,t){a.Credentials.call(this),this.expired=!0,this.params=e,this.data=null,this._identityId=null,this._clientConfig=a.util.copy(t||{}),this.loadCachedId();var i=this;Object.defineProperty(this,"identityId",{get:function(){return i.loadCachedId(),i._identityId||i.params.IdentityId},set:function(e){i._identityId=e}})},refresh:function(e){this.coalesceRefresh(e||a.util.fn.callback)},load:function(e){var t=this;t.createClients(),t.data=null,t._identityId=null,t.getId((function(i){i?(t.clearIdOnNotAuthorized(i),e(i)):t.params.RoleArn?t.getCredentialsFromSTS(e):t.getCredentialsForIdentity(e)}))},clearCachedId:function(){this._identityId=null,delete this.params.IdentityId;var e=this.params.IdentityPoolId,t=this.params.LoginId||"";delete this.storage[this.localStorageKey.id+e+t],delete this.storage[this.localStorageKey.providers+e+t]},clearIdOnNotAuthorized:function(e){"NotAuthorizedException"==e.code&&this.clearCachedId()},getId:function(e){var t=this;if("string"===typeof t.params.IdentityId)return e(null,t.params.IdentityId);t.cognito.getId((function(i,a){!i&&a.IdentityId?(t.params.IdentityId=a.IdentityId,e(null,a.IdentityId)):e(i)}))},loadCredentials:function(e,t){e&&t&&(t.expired=!1,t.accessKeyId=e.Credentials.AccessKeyId,t.secretAccessKey=e.Credentials.SecretKey,t.sessionToken=e.Credentials.SessionToken,t.expireTime=e.Credentials.Expiration)},getCredentialsForIdentity:function(e){var t=this;t.cognito.getCredentialsForIdentity((function(i,a){i?t.clearIdOnNotAuthorized(i):(t.cacheId(a),t.data=a,t.loadCredentials(t.data,t)),e(i)}))},getCredentialsFromSTS:function(e){var t=this;t.cognito.getOpenIdToken((function(i,a){i?(t.clearIdOnNotAuthorized(i),e(i)):(t.cacheId(a),t.params.WebIdentityToken=a.Token,t.webIdentityCredentials.refresh((function(i){i||(t.data=t.webIdentityCredentials.data,t.sts.credentialsFrom(t.data,t)),e(i)})))}))},loadCachedId:function(){var e=this;if(a.util.isBrowser()&&!e.params.IdentityId){var t=e.getStorage("id");if(t&&e.params.Logins){var i=Object.keys(e.params.Logins);0!==(e.getStorage("providers")||"").split(",").filter((function(e){return-1!==i.indexOf(e)})).length&&(e.params.IdentityId=t)}else t&&(e.params.IdentityId=t)}},createClients:function(){var e=this._clientConfig;if(this.webIdentityCredentials=this.webIdentityCredentials||new a.WebIdentityCredentials(this.params,e),!this.cognito){var t=a.util.merge({},e);t.params=this.params,this.cognito=new o(t)}this.sts=this.sts||new n(e)},cacheId:function(e){this._identityId=e.IdentityId,this.params.IdentityId=this._identityId,a.util.isBrowser()&&(this.setStorage("id",e.IdentityId),this.params.Logins&&this.setStorage("providers",Object.keys(this.params.Logins).join(",")))},getStorage:function(e){return this.storage[this.localStorageKey[e]+this.params.IdentityPoolId+(this.params.LoginId||"")]},setStorage:function(e,t){try{this.storage[this.localStorageKey[e]+this.params.IdentityPoolId+(this.params.LoginId||"")]=t}catch(i){}},storage:function(){try{var e=a.util.isBrowser()&&null!==window.localStorage&&"object"===typeof window.localStorage?window.localStorage:{};return e["aws.test-storage"]="foobar",delete e["aws.test-storage"],e}catch(t){return{}}}()})},28732:function(e,t,i){var a=i(8468);a.CredentialProviderChain=a.util.inherit(a.Credentials,{constructor:function(e){this.providers=e||a.CredentialProviderChain.defaultProviders.slice(0),this.resolveCallbacks=[]},resolve:function(e){var t=this;if(0===t.providers.length)return e(new Error("No providers")),t;if(1===t.resolveCallbacks.push(e)){var i=0,o=t.providers.slice(0);!function e(n,r){if(!n&&r||i===o.length)return a.util.arrayEach(t.resolveCallbacks,(function(e){e(n,r)})),void(t.resolveCallbacks.length=0);var s=o[i++];(r="function"===typeof s?s.call():s).get?r.get((function(t){e(t,t?null:r)})):e(null,r)}()}return t}}),a.CredentialProviderChain.defaultProviders=[],a.CredentialProviderChain.addPromisesToClass=function(e){this.prototype.resolvePromise=a.util.promisifyMethod("resolve",e)},a.CredentialProviderChain.deletePromisesFromClass=function(){delete this.prototype.resolvePromise},a.util.addPromises(a.CredentialProviderChain)},78437:function(e,t,i){var a=i(8468),o=i(62362);a.SAMLCredentials=a.util.inherit(a.Credentials,{constructor:function(e){a.Credentials.call(this),this.expired=!0,this.params=e},refresh:function(e){this.coalesceRefresh(e||a.util.fn.callback)},load:function(e){var t=this;t.createClients(),t.service.assumeRoleWithSAML((function(i,a){i||t.service.credentialsFrom(a,t),e(i)}))},createClients:function(){this.service=this.service||new o({params:this.params})}})},14524:function(e,t,i){var a=i(8468),o=i(62362);a.TemporaryCredentials=a.util.inherit(a.Credentials,{constructor:function(e,t){a.Credentials.call(this),this.loadMasterCredentials(t),this.expired=!0,this.params=e||{},this.params.RoleArn&&(this.params.RoleSessionName=this.params.RoleSessionName||"temporary-credentials")},refresh:function(e){this.coalesceRefresh(e||a.util.fn.callback)},load:function(e){var t=this;t.createClients(),t.masterCredentials.get((function(){t.service.config.credentials=t.masterCredentials,(t.params.RoleArn?t.service.assumeRole:t.service.getSessionToken).call(t.service,(function(i,a){i||t.service.credentialsFrom(a,t),e(i)}))}))},loadMasterCredentials:function(e){for(this.masterCredentials=e||a.config.credentials;this.masterCredentials.masterCredentials;)this.masterCredentials=this.masterCredentials.masterCredentials;"function"!==typeof this.masterCredentials.get&&(this.masterCredentials=new a.Credentials(this.masterCredentials))},createClients:function(){this.service=this.service||new o({params:this.params})}})},19589:function(e,t,i){var a=i(8468),o=i(62362);a.WebIdentityCredentials=a.util.inherit(a.Credentials,{constructor:function(e,t){a.Credentials.call(this),this.expired=!0,this.params=e,this.params.RoleSessionName=this.params.RoleSessionName||"web-identity",this.data=null,this._clientConfig=a.util.copy(t||{})},refresh:function(e){this.coalesceRefresh(e||a.util.fn.callback)},load:function(e){var t=this;t.createClients(),t.service.assumeRoleWithWebIdentity((function(i,a){t.data=null,i||(t.data=a,t.service.credentialsFrom(a,t)),e(i)}))},createClients:function(){if(!this.service){var e=a.util.merge({},this._clientConfig);e.params=this.params,this.service=new o(e)}}})},24265:function(e,t,i){var a=i(8468),o=i(23657),n=["AWS_ENABLE_ENDPOINT_DISCOVERY","AWS_ENDPOINT_DISCOVERY_ENABLED"];function r(e){var t=e.service,i=t.api||{},a=(i.operations,{});return t.config.region&&(a.region=t.config.region),i.serviceId&&(a.serviceId=i.serviceId),t.config.credentials.accessKeyId&&(a.accessKeyId=t.config.credentials.accessKeyId),a}function s(e,t,i){i&&void 0!==t&&null!==t&&"structure"===i.type&&i.required&&i.required.length>0&&o.arrayEach(i.required,(function(a){var o=i.members[a];if(!0===o.endpointDiscoveryId){var n=o.isLocationName?o.name:a;e[n]=String(t[a])}else s(e,t[a],o)}))}function l(e,t){var i={};return s(i,e.params,t),i}function c(e){var t=e.service,i=t.api,n=i.operations?i.operations[e.operation]:void 0,s=l(e,n?n.input:void 0),c=r(e);Object.keys(s).length>0&&(c=o.update(c,s),n&&(c.operation=n.name));var p=a.endpointCache.get(c);if(!p||1!==p.length||""!==p[0].Address)if(p&&p.length>0)e.httpRequest.updateEndpoint(p[0].Address);else{var h=t.makeRequest(i.endpointOperation,{Operation:n.name,Identifiers:s});$(h),h.removeListener("validate",a.EventListeners.Core.VALIDATE_PARAMETERS),h.removeListener("retry",a.EventListeners.Core.RETRY_CHECK),a.endpointCache.put(c,[{Address:"",CachePeriodInMinutes:1}]),h.send((function(e,t){t&&t.Endpoints?a.endpointCache.put(c,t.Endpoints):e&&a.endpointCache.put(c,[{Address:"",CachePeriodInMinutes:1}])}))}}var p={};function h(e,t){var i=e.service,n=i.api,s=n.operations?n.operations[e.operation]:void 0,c=s?s.input:void 0,h=l(e,c),d=r(e);Object.keys(h).length>0&&(d=o.update(d,h),s&&(d.operation=s.name));var u=a.EndpointCache.getKeyString(d),m=a.endpointCache.get(u);if(m&&1===m.length&&""===m[0].Address)return p[u]||(p[u]=[]),void p[u].push({request:e,callback:t});if(m&&m.length>0)e.httpRequest.updateEndpoint(m[0].Address),t();else{var b=i.makeRequest(n.endpointOperation,{Operation:s.name,Identifiers:h});b.removeListener("validate",a.EventListeners.Core.VALIDATE_PARAMETERS),$(b),a.endpointCache.put(u,[{Address:"",CachePeriodInMinutes:60}]),b.send((function(i,n){if(i){if(e.response.error=o.error(i,{retryable:!1}),a.endpointCache.remove(d),p[u]){var r=p[u];o.arrayEach(r,(function(e){e.request.response.error=o.error(i,{retryable:!1}),e.callback()})),delete p[u]}}else if(n&&(a.endpointCache.put(u,n.Endpoints),e.httpRequest.updateEndpoint(n.Endpoints[0].Address),p[u])){r=p[u];o.arrayEach(r,(function(e){e.request.httpRequest.updateEndpoint(n.Endpoints[0].Address),e.callback()})),delete p[u]}t()}))}}function $(e){var t=e.service.api.apiVersion;t&&!e.httpRequest.headers["x-amz-api-version"]&&(e.httpRequest.headers["x-amz-api-version"]=t)}function d(e){var t=e.error,i=e.httpResponse;if(t&&("InvalidEndpointException"===t.code||421===i.statusCode)){var n=e.request,s=n.service.api.operations||{},c=l(n,s[n.operation]?s[n.operation].input:void 0),p=r(n);Object.keys(c).length>0&&(p=o.update(p,c),s[n.operation]&&(p.operation=s[n.operation].name)),a.endpointCache.remove(p)}}function u(e){return["false","0"].indexOf(e)>=0}e.exports={discoverEndpoint:function(e,t){var i=e.service||{};if(function(e){if(e._originalConfig&&e._originalConfig.endpoint&&!0===e._originalConfig.endpointDiscoveryEnabled)throw o.error(new Error,{code:"ConfigurationException",message:"Custom endpoint is supplied; endpointDiscoveryEnabled must not be true."});var t=a.config[e.serviceIdentifier]||{};return Boolean(a.config.endpoint||t.endpoint||e._originalConfig&&e._originalConfig.endpoint)}(i)||e.isPresigned())return t();var r=(i.api.operations||{})[e.operation],s=r?r.endpointDiscoveryRequired:"NULL",l=function(e){var t=e.service||{};if(void 0!==t.config.endpointDiscoveryEnabled)return t.config.endpointDiscoveryEnabled;if(!o.isBrowser()){for(var i=0;i<n.length;i++){var r=n[i];if(Object.prototype.hasOwnProperty.call({NODE_ENV:"production",PUBLIC_URL:".",WDS_SOCKET_HOST:void 0,WDS_SOCKET_PATH:void 0,WDS_SOCKET_PORT:void 0,FAST_REFRESH:!0,REACT_APP_BUILD_TYPE:"development"},r)){if(""==={NODE_ENV:"production",PUBLIC_URL:".",WDS_SOCKET_HOST:void 0,WDS_SOCKET_PATH:void 0,WDS_SOCKET_PORT:void 0,FAST_REFRESH:!0,REACT_APP_BUILD_TYPE:"development"}[r]||void 0==={NODE_ENV:"production",PUBLIC_URL:".",WDS_SOCKET_HOST:void 0,WDS_SOCKET_PATH:void 0,WDS_SOCKET_PORT:void 0,FAST_REFRESH:!0,REACT_APP_BUILD_TYPE:"development"}[r])throw o.error(new Error,{code:"ConfigurationException",message:"environmental variable "+r+" cannot be set to nothing"});return!u({NODE_ENV:"production",PUBLIC_URL:".",WDS_SOCKET_HOST:void 0,WDS_SOCKET_PATH:void 0,WDS_SOCKET_PORT:void 0,FAST_REFRESH:!0,REACT_APP_BUILD_TYPE:"development"}[r])}}var s={};try{s=a.util.iniLoader?a.util.iniLoader.loadFrom({isConfig:!0,filename:{NODE_ENV:"production",PUBLIC_URL:".",WDS_SOCKET_HOST:void 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"+i.api.className+"."+e.operation+"() requires it. Please check your configurations."}),t();break}e.addNamedListener("INVALIDATE_CACHED_ENDPOINTS","extractError",d),h(e,t);break;default:t()}},requiredDiscoverEndpoint:h,optionalDiscoverEndpoint:c,marshallCustomIdentifiers:l,getCacheKey:r,invalidateCachedEndpoint:d}},8547:function(e,t,i){var a=i(8468),o=a.util,n=i(86337).typeOf,r=i(84218),s=i(31622);function l(e,t){return t?new s(e):Number(e)}function c(e,t){var i={M:{}};for(var o in e){var n=a.DynamoDB.Converter.input(e[o],t);void 0!==n&&(i.M[o]=n)}return i}a.DynamoDB.Converter={input:function e(t,i){i=i||{};var o=n(t);return"Object"===o?c(t,i):"Array"===o?function(e,t){for(var i={L:[]},o=0;o<e.length;o++)i.L.push(a.DynamoDB.Converter.input(e[o],t));return i}(t,i):"Set"===o?function(e,t){t=t||{};var i=e.values;if(t.convertEmptyValues&&0===(i=function(e){var t=[];if({String:!0,Binary:!0,Number:!1}[e.type]){for(var i=0;i<e.values.length;i++)0!==e.values[i].length&&t.push(e.values[i]);return t}return e.values}(e)).length)return 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l(h,i.wrapNumbers);if("B"===p)return o.buffer.toBuffer(h);if("BOOL"===p)return"true"===h||"TRUE"===h||!0===h;if("NULL"===p)return null}},unmarshall:function(e,t){return a.DynamoDB.Converter.output({M:e},t)}},e.exports=a.DynamoDB.Converter},55098:function(e,t,i){var a=i(8468),o=i(74952),n=i(84218);a.DynamoDB.DocumentClient=a.util.inherit({constructor:function(e){var t=this;t.options=e||{},t.configure(t.options)},configure:function(e){var t=this;t.service=e.service,t.bindServiceObject(e),t.attrValue=e.attrValue=t.service.api.operations.putItem.input.members.Item.value.shape},bindServiceObject:function(e){var t=this;if(e=e||{},t.service){var i=a.util.copy(t.service.config);t.service=new t.service.constructor.__super__(i),t.service.config.params=a.util.merge(t.service.config.params||{},e.params)}else t.service=new a.DynamoDB(e)},makeServiceRequest:function(e,t,i){var a=this,o=a.service[e](t);return a.setupRequest(o),a.setupResponse(o),"function"===typeof i&&o.send(i),o},serviceClientOperationsMap:{batchGet:"batchGetItem",batchWrite:"batchWriteItem",delete:"deleteItem",get:"getItem",put:"putItem",query:"query",scan:"scan",update:"updateItem",transactGet:"transactGetItems",transactWrite:"transactWriteItems"},batchGet:function(e,t){var i=this.serviceClientOperationsMap.batchGet;return this.makeServiceRequest(i,e,t)},batchWrite:function(e,t){var i=this.serviceClientOperationsMap.batchWrite;return this.makeServiceRequest(i,e,t)},delete:function(e,t){var i=this.serviceClientOperationsMap.delete;return this.makeServiceRequest(i,e,t)},get:function(e,t){var i=this.serviceClientOperationsMap.get;return this.makeServiceRequest(i,e,t)},put:function(e,t){var i=this.serviceClientOperationsMap.put;return this.makeServiceRequest(i,e,t)},update:function(e,t){var i=this.serviceClientOperationsMap.update;return this.makeServiceRequest(i,e,t)},scan:function(e,t){var i=this.serviceClientOperationsMap.scan;return this.makeServiceRequest(i,e,t)},query:function(e,t){var i=this.serviceClientOperationsMap.query;return this.makeServiceRequest(i,e,t)},transactWrite:function(e,t){var i=this.serviceClientOperationsMap.transactWrite;return this.makeServiceRequest(i,e,t)},transactGet:function(e,t){var i=this.serviceClientOperationsMap.transactGet;return this.makeServiceRequest(i,e,t)},createSet:function(e,t){return new n(e,t=t||{})},getTranslator:function(){return new o(this.options)},setupRequest:function(e){var t=this.getTranslator(),i=e.operation,o=e.service.api.operations[i].input;e._events.validate.unshift((function(e){e.rawParams=a.util.copy(e.params),e.params=t.translateInput(e.rawParams,o)}))},setupResponse:function(e){var t=this,i=t.getTranslator(),o=t.service.api.operations[e.operation].output;e.on("extractData",(function(e){e.data=i.translateOutput(e.data,o)})),e.response.nextPage=function(e){var i,o=this,n=o.request,r=n.service,s=n.operation;try{i=r.paginationConfig(s,!0)}catch(h){o.error=h}if(!o.hasNextPage()){if(e)e(o.error,null);else if(o.error)throw o.error;return null}var l=a.util.copy(n.rawParams);if(o.nextPageTokens){var c=i.inputToken;"string"===typeof c&&(c=[c]);for(var p=0;p<c.length;p++)l[c[p]]=o.nextPageTokens[p];return t[s](l,e)}return e?e(null,null):null}}}),e.exports=a.DynamoDB.DocumentClient},31622:function(e,t,i){var a=i(8468).util.inherit({constructor:function(e){this.wrapperName="NumberValue",this.value=e.toString()},toJSON:function(){return this.toNumber()},toNumber:function(){return Number(this.value)},toString:function(){return this.value}});e.exports=a},84218:function(e,t,i){var a=i(8468).util,o=i(86337).typeOf,n={String:"String",Number:"Number",NumberValue:"Number",Binary:"Binary"},r=a.inherit({constructor:function(e,t){t=t||{},this.wrapperName="Set",this.initialize(e,t.validate)},initialize:function(e,t){var i=this;i.values=[].concat(e),i.detectType(),t&&i.validate()},detectType:function(){if(this.type=n[o(this.values[0])],!this.type)throw a.error(new Error,{code:"InvalidSetType",message:"Sets can contain string, number, or binary values"})},validate:function(){for(var e=this,t=e.values.length,i=e.values,r=0;r<t;r++)if(n[o(i[r])]!==e.type)throw a.error(new Error,{code:"InvalidType",message:e.type+" Set contains "+o(i[r])+" value"})},toJSON:function(){return this.values}});e.exports=r},74952:function(e,t,i){var a=i(8468).util,o=i(8547),n=function(e){e=e||{},this.attrValue=e.attrValue,this.convertEmptyValues=Boolean(e.convertEmptyValues),this.wrapNumbers=Boolean(e.wrapNumbers)};n.prototype.translateInput=function(e,t){return this.mode="input",this.translate(e,t)},n.prototype.translateOutput=function(e,t){return this.mode="output",this.translate(e,t)},n.prototype.translate=function(e,t){var i=this;if(t&&void 0!==e){if(t.shape===i.attrValue)return o[i.mode](e,{convertEmptyValues:i.convertEmptyValues,wrapNumbers:i.wrapNumbers});switch(t.type){case"structure":return i.translateStructure(e,t);case"map":return i.translateMap(e,t);case"list":return i.translateList(e,t);default:return i.translateScalar(e,t)}}},n.prototype.translateStructure=function(e,t){var i=this;if(null!=e){var o={};return a.each(e,(function(e,a){var n=t.members[e];if(n){var r=i.translate(a,n);void 0!==r&&(o[e]=r)}})),o}},n.prototype.translateList=function(e,t){var i=this;if(null!=e){var o=[];return a.arrayEach(e,(function(e){var a=i.translate(e,t.member);void 0===a?o.push(null):o.push(a)})),o}},n.prototype.translateMap=function(e,t){var i=this;if(null!=e){var o={};return a.each(e,(function(e,a){var n=i.translate(a,t.value);o[e]=void 0===n?null:n})),o}},n.prototype.translateScalar=function(e,t){return t.toType(e)},e.exports=n},86337:function(e,t,i){var a=i(8468).util;function o(e){var t=["Buffer","File","Blob","ArrayBuffer","DataView","Int8Array","Uint8Array","Uint8ClampedArray","Int16Array","Uint16Array","Int32Array","Uint32Array","Float32Array","Float64Array"];if(a.isNode()){var i=a.stream.Stream;if(a.Buffer.isBuffer(e)||e instanceof i)return!0}for(var o=0;o<t.length;o++)if(void 0!==e&&e.constructor){if(a.isType(e,t[o]))return!0;if(a.typeName(e.constructor)===t[o])return!0}return!1}e.exports={typeOf:function(e){return null===e&&"object"===typeof e?"null":void 0!==e&&o(e)?"Binary":void 0!==e&&e.constructor?e.wrapperName||a.typeName(e.constructor):void 0!==e&&"object"===typeof e?"Object":"undefined"},isBinary:o}},28962:function(e,t,i){var a=i(62155).eventMessageChunker,o=i(53523).parseEvent;e.exports={createEventStream:function(e,t,i){for(var n=a(e),r=[],s=0;s<n.length;s++)r.push(o(t,n[s],i));return r}}},62155:function(e){e.exports={eventMessageChunker:function(e){for(var t=[],i=0;i<e.length;){var a=e.readInt32BE(i),o=e.slice(i,a+i);i+=a,t.push(o)}return t}}},82553:function(e,t,i){var a=i(8468).util,o=a.buffer.toBuffer;function n(e){if(8!==e.length)throw new Error("Int64 buffers must be exactly 8 bytes");a.Buffer.isBuffer(e)||(e=o(e)),this.bytes=e}function r(e){for(var t=0;t<8;t++)e[t]^=255;for(t=7;t>-1&&(e[t]++,0===e[t]);t--);}n.fromNumber=function(e){if(e>0x8000000000000000||e<-0x8000000000000000)throw new Error(e+" is too large (or, if negative, too small) to represent as an Int64");for(var t=new Uint8Array(8),i=7,a=Math.abs(Math.round(e));i>-1&&a>0;i--,a/=256)t[i]=a;return e<0&&r(t),new n(t)},n.prototype.valueOf=function(){var e=this.bytes.slice(0),t=128&e[0];return t&&r(e),parseInt(e.toString("hex"),16)*(t?-1:1)},n.prototype.toString=function(){return String(this.valueOf())},e.exports={Int64:n}},53523:function(e,t,i){var a=i(72408).parseMessage;e.exports={parseEvent:function(e,t,i){var o=a(t),n=o.headers[":message-type"];if(n){if("error"===n.value)throw function(e){var t=e.headers[":error-code"],i=e.headers[":error-message"],a=new Error(i.value||i);return a.code=a.name=t.value||t,a}(o);if("event"!==n.value)return}var r=o.headers[":event-type"],s=i.members[r.value];if(s){var l={},c=s.eventPayloadMemberName;if(c){var p=s.members[c];"binary"===p.type?l[c]=o.body:l[c]=e.parse(o.body.toString(),p)}for(var h=s.eventHeaderMemberNames,$=0;$<h.length;$++){var d=h[$];o.headers[d]&&(l[d]=s.members[d].toType(o.headers[d].value))}var u={};return u[r.value]=l,u}}}},72408:function(e,t,i){var a=i(82553).Int64,o=i(59184).splitMessage,n="boolean";function r(e){for(var t={},i=0;i<e.length;){var o=e.readUInt8(i++),r=e.slice(i,i+o).toString();switch(i+=o,e.readUInt8(i++)){case 0:t[r]={type:n,value:!0};break;case 1:t[r]={type:n,value:!1};break;case 2:t[r]={type:"byte",value:e.readInt8(i++)};break;case 3:t[r]={type:"short",value:e.readInt16BE(i)},i+=2;break;case 4:t[r]={type:"integer",value:e.readInt32BE(i)},i+=4;break;case 5:t[r]={type:"long",value:new a(e.slice(i,i+8))},i+=8;break;case 6:var s=e.readUInt16BE(i);i+=2,t[r]={type:"binary",value:e.slice(i,i+s)},i+=s;break;case 7:var l=e.readUInt16BE(i);i+=2,t[r]={type:"string",value:e.slice(i,i+l).toString()},i+=l;break;case 8:t[r]={type:"timestamp",value:new Date(new a(e.slice(i,i+8)).valueOf())},i+=8;break;case 9:var c=e.slice(i,i+16).toString("hex");i+=16,t[r]={type:"uuid",value:c.substr(0,8)+"-"+c.substr(8,4)+"-"+c.substr(12,4)+"-"+c.substr(16,4)+"-"+c.substr(20)};break;default:throw new Error("Unrecognized header type tag")}}return t}e.exports={parseMessage:function(e){var t=o(e);return{headers:r(t.headers),body:t.body}}}},59184:function(e,t,i){var a=i(8468).util,o=a.buffer.toBuffer;e.exports={splitMessage:function(e){if(a.Buffer.isBuffer(e)||(e=o(e)),e.length<16)throw new Error("Provided message too short to accommodate event stream message overhead");if(e.length!==e.readUInt32BE(0))throw new Error("Reported message length does not match received message length");var t=e.readUInt32BE(8);if(t!==a.crypto.crc32(e.slice(0,8)))throw new Error("The prelude checksum specified in the message ("+t+") does not match the calculated CRC32 checksum.");var i=e.readUInt32BE(e.length-4);if(i!==a.crypto.crc32(e.slice(0,e.length-4)))throw new Error("The message checksum did not match the expected value of "+i);var n=12+e.readUInt32BE(4);return{headers:e.slice(12,n),body:e.slice(n,e.length-4)}}}},18663:function(e,t,i){var a=i(8468),o=i(78451),n=i(24265).discoverEndpoint;function r(e){if(!e.service.api.operations)return"";var t=e.service.api.operations[e.operation];return t?t.authtype:""}function s(e){var t=e.service;return t.config.signatureVersion?t.config.signatureVersion:t.api.signatureVersion?t.api.signatureVersion:r(e)}a.EventListeners={Core:{}},a.EventListeners={Core:(new o).addNamedListeners((function(e,t){t("VALIDATE_CREDENTIALS","validate",(function(e,t){if(!e.service.api.signatureVersion&&!e.service.config.signatureVersion)return t();"bearer"!==s(e)?e.service.config.getCredentials((function(i){i&&(e.response.error=a.util.error(i,{code:"CredentialsError",message:"Missing credentials in config, if using AWS_CONFIG_FILE, set AWS_SDK_LOAD_CONFIG=1"})),t()})):e.service.config.getToken((function(i){i&&(e.response.error=a.util.error(i,{code:"TokenError"})),t()}))})),e("VALIDATE_REGION","validate",(function(e){if(!e.service.isGlobalEndpoint){var t=new RegExp(/^([a-zA-Z0-9]|[a-zA-Z0-9][a-zA-Z0-9-]{0,61}[a-zA-Z0-9])$/);e.service.config.region?t.test(e.service.config.region)||(e.response.error=a.util.error(new Error,{code:"ConfigError",message:"Invalid region in config"})):e.response.error=a.util.error(new Error,{code:"ConfigError",message:"Missing region in config"})}})),e("BUILD_IDEMPOTENCY_TOKENS","validate",(function(e){if(e.service.api.operations){var t=e.service.api.operations[e.operation];if(t){var i=t.idempotentMembers;if(i.length){for(var o=a.util.copy(e.params),n=0,r=i.length;n<r;n++)o[i[n]]||(o[i[n]]=a.util.uuid.v4());e.params=o}}}})),e("VALIDATE_PARAMETERS","validate",(function(e){if(e.service.api.operations){var t=e.service.api.operations[e.operation].input,i=e.service.config.paramValidation;new a.ParamValidator(i).validate(t,e.params)}})),e("COMPUTE_CHECKSUM","afterBuild",(function(e){if(e.service.api.operations){var t=e.service.api.operations[e.operation];if(t){var i=e.httpRequest.body,o=i&&(a.util.Buffer.isBuffer(i)||"string"===typeof i),n=e.httpRequest.headers;if(t.httpChecksumRequired&&e.service.config.computeChecksums&&o&&!n["Content-MD5"]){var r=a.util.crypto.md5(i,"base64");n["Content-MD5"]=r}}}})),t("COMPUTE_SHA256","afterBuild",(function(e,t){if(e.haltHandlersOnError(),e.service.api.operations){var i=e.service.api.operations[e.operation],o=i?i.authtype:"";if(!e.service.api.signatureVersion&&!o&&!e.service.config.signatureVersion)return t();if(e.service.getSignerClass(e)===a.Signers.V4){var n=e.httpRequest.body||"";if(o.indexOf("unsigned-body")>=0)return e.httpRequest.headers["X-Amz-Content-Sha256"]="UNSIGNED-PAYLOAD",t();a.util.computeSha256(n,(function(i,a){i?t(i):(e.httpRequest.headers["X-Amz-Content-Sha256"]=a,t())}))}else t()}})),e("SET_CONTENT_LENGTH","afterBuild",(function(e){var t=r(e),i=a.util.getRequestPayloadShape(e);if(void 0===e.httpRequest.headers["Content-Length"])try{var o=a.util.string.byteLength(e.httpRequest.body);e.httpRequest.headers["Content-Length"]=o}catch(n){if(i&&i.isStreaming){if(i.requiresLength)throw n;if(t.indexOf("unsigned-body")>=0)return void(e.httpRequest.headers["Transfer-Encoding"]="chunked");throw n}throw n}})),e("SET_HTTP_HOST","afterBuild",(function(e){e.httpRequest.headers.Host=e.httpRequest.endpoint.host})),e("SET_TRACE_ID","afterBuild",(function(e){var t="X-Amzn-Trace-Id";if(a.util.isNode()&&!Object.hasOwnProperty.call(e.httpRequest.headers,t)){var i={NODE_ENV:"production",PUBLIC_URL:".",WDS_SOCKET_HOST:void 0,WDS_SOCKET_PATH:void 0,WDS_SOCKET_PORT:void 0,FAST_REFRESH:!0,REACT_APP_BUILD_TYPE:"development"}.AWS_LAMBDA_FUNCTION_NAME,o={NODE_ENV:"production",PUBLIC_URL:".",WDS_SOCKET_HOST:void 0,WDS_SOCKET_PATH:void 0,WDS_SOCKET_PORT:void 0,FAST_REFRESH:!0,REACT_APP_BUILD_TYPE:"development"}._X_AMZN_TRACE_ID;"string"===typeof i&&i.length>0&&"string"===typeof o&&o.length>0&&(e.httpRequest.headers[t]=o)}})),e("RESTART","restart",(function(){var e=this.response.error;e&&e.retryable&&(this.httpRequest=new a.HttpRequest(this.service.endpoint,this.service.region),this.response.retryCount<this.service.config.maxRetries?this.response.retryCount++:this.response.error=null)}));t("DISCOVER_ENDPOINT","sign",n,!0),t("SIGN","sign",(function(e,t){var i=e.service,a=s(e);if(!a||0===a.length)return t();"bearer"===a?i.config.getToken((function(a,o){if(a)return e.response.error=a,t();try{new(i.getSignerClass(e))(e.httpRequest).addAuthorization(o)}catch(n){e.response.error=n}t()})):i.config.getCredentials((function(a,o){if(a)return e.response.error=a,t();try{var n=i.getSkewCorrectedDate(),r=i.getSignerClass(e),s=(e.service.api.operations||{})[e.operation],l=new r(e.httpRequest,i.getSigningName(e),{signatureCache:i.config.signatureCache,operation:s,signatureVersion:i.api.signatureVersion});l.setServiceClientId(i._clientId),delete e.httpRequest.headers.Authorization,delete e.httpRequest.headers.Date,delete e.httpRequest.headers["X-Amz-Date"],l.addAuthorization(o,n),e.signedAt=n}catch(c){e.response.error=c}t()}))})),e("VALIDATE_RESPONSE","validateResponse",(function(e){this.service.successfulResponse(e,this)?(e.data={},e.error=null):(e.data=null,e.error=a.util.error(new Error,{code:"UnknownError",message:"An unknown error occurred."}))})),e("ERROR","error",(function(e,t){if(t.request.service.api.awsQueryCompatible){var i=t.httpResponse.headers,a=i?i["x-amzn-query-error"]:void 0;a&&a.includes(";")&&(t.error.code=a.split(";")[0])}}),!0),t("SEND","send",(function(e,t){function i(i){e.httpResponse.stream=i;var o=e.request.httpRequest.stream,n=e.request.service,r=n.api,s=e.request.operation,l=r.operations[s]||{};i.on("headers",(function(o,r,s){if(e.request.emit("httpHeaders",[o,r,e,s]),!e.httpResponse.streaming)if(2===a.HttpClient.streamsApiVersion){if(l.hasEventOutput&&n.successfulResponse(e))return e.request.emit("httpDone"),void t();i.on("readable",(function(){var t=i.read();null!==t&&e.request.emit("httpData",[t,e])}))}else i.on("data",(function(t){e.request.emit("httpData",[t,e])}))})),i.on("end",(function(){if(!o||!o.didCallback){if(2===a.HttpClient.streamsApiVersion&&l.hasEventOutput&&n.successfulResponse(e))return;e.request.emit("httpDone"),t()}}))}function o(i){if("RequestAbortedError"!==i.code){var o="TimeoutError"===i.code?i.code:"NetworkingError";i=a.util.error(i,{code:o,region:e.request.httpRequest.region,hostname:e.request.httpRequest.endpoint.hostname,retryable:!0})}e.error=i,e.request.emit("httpError",[e.error,e],(function(){t()}))}function n(){var t,n=a.HttpClient.getInstance(),r=e.request.service.config.httpOptions||{};try{var s=n.handleRequest(e.request.httpRequest,r,i,o);(t=s).on("sendProgress",(function(t){e.request.emit("httpUploadProgress",[t,e])})),t.on("receiveProgress",(function(t){e.request.emit("httpDownloadProgress",[t,e])}))}catch(l){o(l)}}e.httpResponse._abortCallback=t,e.error=null,e.data=null,(e.request.service.getSkewCorrectedDate()-this.signedAt)/1e3>=600?this.emit("sign",[this],(function(e){e?t(e):n()})):n()})),e("HTTP_HEADERS","httpHeaders",(function(e,t,i,o){i.httpResponse.statusCode=e,i.httpResponse.statusMessage=o,i.httpResponse.headers=t,i.httpResponse.body=a.util.buffer.toBuffer(""),i.httpResponse.buffers=[],i.httpResponse.numBytes=0;var n=t.date||t.Date,r=i.request.service;if(n){var s=Date.parse(n);r.config.correctClockSkew&&r.isClockSkewed(s)&&r.applyClockOffset(s)}})),e("HTTP_DATA","httpData",(function(e,t){if(e){if(a.util.isNode()){t.httpResponse.numBytes+=e.length;var i=t.httpResponse.headers["content-length"],o={loaded:t.httpResponse.numBytes,total:i};t.request.emit("httpDownloadProgress",[o,t])}t.httpResponse.buffers.push(a.util.buffer.toBuffer(e))}})),e("HTTP_DONE","httpDone",(function(e){if(e.httpResponse.buffers&&e.httpResponse.buffers.length>0){var t=a.util.buffer.concat(e.httpResponse.buffers);e.httpResponse.body=t}delete e.httpResponse.numBytes,delete e.httpResponse.buffers})),e("FINALIZE_ERROR","retry",(function(e){e.httpResponse.statusCode&&(e.error.statusCode=e.httpResponse.statusCode,void 0===e.error.retryable&&(e.error.retryable=this.service.retryableError(e.error,this)))})),e("INVALIDATE_CREDENTIALS","retry",(function(e){if(e.error)switch(e.error.code){case"RequestExpired":case"ExpiredTokenException":case"ExpiredToken":e.error.retryable=!0,e.request.service.config.credentials.expired=!0}})),e("EXPIRED_SIGNATURE","retry",(function(e){var t=e.error;t&&"string"===typeof t.code&&"string"===typeof t.message&&t.code.match(/Signature/)&&t.message.match(/expired/)&&(e.error.retryable=!0)})),e("CLOCK_SKEWED","retry",(function(e){e.error&&this.service.clockSkewError(e.error)&&this.service.config.correctClockSkew&&(e.error.retryable=!0)})),e("REDIRECT","retry",(function(e){e.error&&e.error.statusCode>=300&&e.error.statusCode<400&&e.httpResponse.headers.location&&(this.httpRequest.endpoint=new a.Endpoint(e.httpResponse.headers.location),this.httpRequest.headers.Host=this.httpRequest.endpoint.host,this.httpRequest.path=this.httpRequest.endpoint.path,e.error.redirect=!0,e.error.retryable=!0)})),e("RETRY_CHECK","retry",(function(e){e.error&&(e.error.redirect&&e.redirectCount<e.maxRedirects?e.error.retryDelay=0:e.retryCount<e.maxRetries&&(e.error.retryDelay=this.service.retryDelays(e.retryCount,e.error)||0))})),t("RESET_RETRY_STATE","afterRetry",(function(e,t){var i,a=!1;e.error&&(i=e.error.retryDelay||0,e.error.retryable&&e.retryCount<e.maxRetries?(e.retryCount++,a=!0):e.error.redirect&&e.redirectCount<e.maxRedirects&&(e.redirectCount++,a=!0)),a&&i>=0?(e.error=null,setTimeout(t,i)):t()}))})),CorePost:(new o).addNamedListeners((function(e){e("EXTRACT_REQUEST_ID","extractData",a.util.extractRequestId),e("EXTRACT_REQUEST_ID","extractError",a.util.extractRequestId),e("ENOTFOUND_ERROR","httpError",(function(e){if("NetworkingError"===e.code&&function(e){return"ENOTFOUND"===e.errno||"number"===typeof e.errno&&"function"===typeof a.util.getSystemErrorName&&["EAI_NONAME","EAI_NODATA"].indexOf(a.util.getSystemErrorName(e.errno)>=0)}(e)){var t="Inaccessible host: `"+e.hostname+"' at port `"+e.port+"'. This service may not be available in the `"+e.region+"' region.";this.response.error=a.util.error(new Error(t),{code:"UnknownEndpoint",region:e.region,hostname:e.hostname,retryable:!0,originalError:e})}}))})),Logger:(new o).addNamedListeners((function(e){e("LOG_REQUEST","complete",(function(e){var t=e.request,o=t.service.config.logger;if(o){var n=function(){var n=(e.request.service.getSkewCorrectedDate().getTime()-t.startTime.getTime())/1e3,s=!!o.isTTY,l=e.httpResponse.statusCode,c=t.params;t.service.api.operations&&t.service.api.operations[t.operation]&&t.service.api.operations[t.operation].input&&(c=r(t.service.api.operations[t.operation].input,t.params));var p=i(49639).inspect(c,!0,null),h="";return s&&(h+="\x1b[33m"),h+="[AWS "+t.service.serviceIdentifier+" "+l,h+=" "+n.toString()+"s "+e.retryCount+" retries]",s&&(h+="\x1b[0;1m"),h+=" "+a.util.string.lowerFirst(t.operation),h+="("+p+")",s&&(h+="\x1b[0m"),h}();"function"===typeof o.log?o.log(n):"function"===typeof o.write&&o.write(n+"\n")}function r(e,t){if(!t)return t;if(e.isSensitive)return"***SensitiveInformation***";switch(e.type){case"structure":var i={};return a.util.each(t,(function(t,a){Object.prototype.hasOwnProperty.call(e.members,t)?i[t]=r(e.members[t],a):i[t]=a})),i;case"list":var o=[];return a.util.arrayEach(t,(function(t,i){o.push(r(e.member,t))})),o;case"map":var n={};return a.util.each(t,(function(t,i){n[t]=r(e.value,i)})),n;default:return t}}}))})),Json:(new o).addNamedListeners((function(e){var t=i(89864);e("BUILD","build",t.buildRequest),e("EXTRACT_DATA","extractData",t.extractData),e("EXTRACT_ERROR","extractError",t.extractError)})),Rest:(new o).addNamedListeners((function(e){var t=i(19701);e("BUILD","build",t.buildRequest),e("EXTRACT_DATA","extractData",t.extractData),e("EXTRACT_ERROR","extractError",t.extractError)})),RestJson:(new o).addNamedListeners((function(e){var t=i(12085);e("BUILD","build",t.buildRequest),e("EXTRACT_DATA","extractData",t.extractData),e("EXTRACT_ERROR","extractError",t.extractError),e("UNSET_CONTENT_LENGTH","afterBuild",t.unsetContentLength)})),RestXml:(new o).addNamedListeners((function(e){var t=i(61622);e("BUILD","build",t.buildRequest),e("EXTRACT_DATA","extractData",t.extractData),e("EXTRACT_ERROR","extractError",t.extractError)})),Query:(new o).addNamedListeners((function(e){var t=i(66588);e("BUILD","build",t.buildRequest),e("EXTRACT_DATA","extractData",t.extractData),e("EXTRACT_ERROR","extractError",t.extractError)}))}},78168:function(e,t,i){var a=i(8468),o=a.util.inherit;a.Endpoint=o({constructor:function(e,t){if(a.util.hideProperties(this,["slashes","auth","hash","search","query"]),"undefined"===typeof e||null===e)throw new Error("Invalid endpoint: "+e);if("string"!==typeof e)return a.util.copy(e);e.match(/^http/)||(e=((t&&void 0!==t.sslEnabled?t.sslEnabled:a.config.sslEnabled)?"https":"http")+"://"+e);a.util.update(this,a.util.urlParse(e)),this.port?this.port=parseInt(this.port,10):this.port="https:"===this.protocol?443:80}}),a.HttpRequest=o({constructor:function(e,t){e=new a.Endpoint(e),this.method="POST",this.path=e.path||"/",this.headers={},this.body="",this.endpoint=e,this.region=t,this._userAgent="",this.setUserAgent()},setUserAgent:function(){this._userAgent=this.headers[this.getUserAgentHeaderName()]=a.util.userAgent()},getUserAgentHeaderName:function(){return(a.util.isBrowser()?"X-Amz-":"")+"User-Agent"},appendToUserAgent:function(e){"string"===typeof e&&e&&(this._userAgent+=" "+e),this.headers[this.getUserAgentHeaderName()]=this._userAgent},getUserAgent:function(){return this._userAgent},pathname:function(){return this.path.split("?",1)[0]},search:function(){var e=this.path.split("?",2)[1];return e?(e=a.util.queryStringParse(e),a.util.queryParamsToString(e)):""},updateEndpoint:function(e){var t=new a.Endpoint(e);this.endpoint=t,this.path=t.path||"/",this.headers.Host&&(this.headers.Host=t.host)}}),a.HttpResponse=o({constructor:function(){this.statusCode=void 0,this.headers={},this.body=void 0,this.streaming=!1,this.stream=null},createUnbufferedStream:function(){return this.streaming=!0,this.stream}}),a.HttpClient=o({}),a.HttpClient.getInstance=function(){return void 0===this.singleton&&(this.singleton=new this),this.singleton}},76311:function(e,t,i){var a=i(8468),o=i(81173).EventEmitter;i(78168),a.XHRClient=a.util.inherit({handleRequest:function(e,t,i,n){var r=this,s=e.endpoint,l=new o,c=s.protocol+"//"+s.hostname;80!==s.port&&443!==s.port&&(c+=":"+s.port),c+=e.path;var p=new XMLHttpRequest,h=!1;e.stream=p,p.addEventListener("readystatechange",(function(){try{if(0===p.status)return}catch(e){return}this.readyState>=this.HEADERS_RECEIVED&&!h&&(l.statusCode=p.status,l.headers=r.parseHeaders(p.getAllResponseHeaders()),l.emit("headers",l.statusCode,l.headers,p.statusText),h=!0),this.readyState===this.DONE&&r.finishRequest(p,l)}),!1),p.upload.addEventListener("progress",(function(e){l.emit("sendProgress",e)})),p.addEventListener("progress",(function(e){l.emit("receiveProgress",e)}),!1),p.addEventListener("timeout",(function(){n(a.util.error(new Error("Timeout"),{code:"TimeoutError"}))}),!1),p.addEventListener("error",(function(){n(a.util.error(new Error("Network Failure"),{code:"NetworkingError"}))}),!1),p.addEventListener("abort",(function(){n(a.util.error(new Error("Request aborted"),{code:"RequestAbortedError"}))}),!1),i(l),p.open(e.method,c,!1!==t.xhrAsync),a.util.each(e.headers,(function(e,t){"Content-Length"!==e&&"User-Agent"!==e&&"Host"!==e&&p.setRequestHeader(e,t)})),t.timeout&&!1!==t.xhrAsync&&(p.timeout=t.timeout),t.xhrWithCredentials&&(p.withCredentials=!0);try{p.responseType="arraybuffer"}catch($){}try{e.body?p.send(e.body):p.send()}catch(d){if(!e.body||"object"!==typeof e.body.buffer)throw d;p.send(e.body.buffer)}return l},parseHeaders:function(e){var t={};return a.util.arrayEach(e.split(/\r?\n/),(function(e){var i=e.split(":",1)[0],a=e.substring(i.length+2);i.length>0&&(t[i.toLowerCase()]=a)})),t},finishRequest:function(e,t){var i;if("arraybuffer"===e.responseType&&e.response){var o=e.response;i=new a.util.Buffer(o.byteLength);for(var n=new Uint8Array(o),r=0;r<i.length;++r)i[r]=n[r]}try{i||"string"!==typeof e.responseText||(i=new a.util.Buffer(e.responseText))}catch(s){}i&&t.emit("data",i),t.emit("end")}}),a.HttpClient.prototype=a.XHRClient.prototype,a.HttpClient.streamsApiVersion=1},48212:function(e,t,i){var a=i(23657);function o(){}function n(e,t){if(t&&void 0!==e&&null!==e)switch(t.type){case"structure":return function(e,t){if(t.isDocument)return e;var i={};return a.each(e,(function(e,a){var o=t.members[e];if(o){if("body"!==o.location)return;var r=o.isLocationName?o.name:e,s=n(a,o);void 0!==s&&(i[r]=s)}})),i}(e,t);case"map":return function(e,t){var i={};return a.each(e,(function(e,a){var o=n(a,t.value);void 0!==o&&(i[e]=o)})),i}(e,t);case"list":return function(e,t){var i=[];return a.arrayEach(e,(function(e){var a=n(e,t.member);void 0!==a&&i.push(a)})),i}(e,t);default:return function(e,t){return t.toWireFormat(e)}(e,t)}}o.prototype.build=function(e,t){return JSON.stringify(n(e,t))},e.exports=o},26815:function(e,t,i){var a=i(23657);function o(){}function n(e,t){if(t&&void 0!==e)switch(t.type){case"structure":return function(e,t){if(null==e)return;if(t.isDocument)return e;var i={},o=t.members,r=t.api&&t.api.awsQueryCompatible;return a.each(o,(function(t,a){var o=a.isLocationName?a.name:t;if(Object.prototype.hasOwnProperty.call(e,o)){var s=n(e[o],a);void 0!==s&&(i[t]=s)}else r&&a.defaultValue&&"list"===a.type&&(i[t]="function"===typeof a.defaultValue?a.defaultValue():a.defaultValue)})),i}(e,t);case"map":return function(e,t){if(null==e)return;var i={};return a.each(e,(function(e,a){var o=n(a,t.value);i[e]=void 0===o?null:o})),i}(e,t);case"list":return function(e,t){if(null==e)return;var i=[];return a.arrayEach(e,(function(e){var a=n(e,t.member);void 0===a?i.push(null):i.push(a)})),i}(e,t);default:return function(e,t){return t.toType(e)}(e,t)}}o.prototype.parse=function(e,t){return n(JSON.parse(e),t)},e.exports=o},45663:function(e){var t=["The AWS SDK for JavaScript (v2) is in maintenance mode."," SDK releases are limited to address critical bug fixes and 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o(e,i,t)}),c.string.lowerFirst,(function(e,t){!0===t.endpointoperation&&p(i,"endpointOperation",c.string.lowerFirst(e)),t.endpointdiscovery&&!i.hasRequiredEndpointDiscovery&&p(i,"hasRequiredEndpointDiscovery",!0===t.endpointdiscovery.required)}))),p(this,"shapes",new a(e.shapes,t,(function(e,i){return n.create(i,t)}))),p(this,"paginators",new a(e.paginators,t,(function(e,i){return new r(e,i,t)}))),p(this,"waiters",new a(e.waiters,t,(function(e,i){return new s(e,i,t)}),c.string.lowerFirst)),t.documentation&&(p(this,"documentation",e.documentation),p(this,"documentationUrl",e.documentationUrl)),p(this,"awsQueryCompatible",e.metadata.awsQueryCompatible)}},1242:function(e,t,i){var a=i(23657).memoizedProperty;function o(e,t,i,o){a(this,o(e),(function(){return i(e,t)}))}e.exports=function(e,t,i,a,n){for(var r in a=a||String,e)Object.prototype.hasOwnProperty.call(e,r)&&(o.call(this,r,e[r],i,a),n&&n(r,e[r]))}},91970:function(e,t,i){var a=i(3825),o=i(23657),n=o.property,r=o.memoizedProperty;e.exports=function(e,t,i){var o=this;i=i||{},n(this,"name",t.name||e),n(this,"api",i.api,!1),t.http=t.http||{},n(this,"endpoint",t.endpoint),n(this,"httpMethod",t.http.method||"POST"),n(this,"httpPath",t.http.requestUri||"/"),n(this,"authtype",t.authtype||""),n(this,"endpointDiscoveryRequired",t.endpointdiscovery?t.endpointdiscovery.required?"REQUIRED":"OPTIONAL":"NULL");var s=t.httpChecksumRequired||t.httpChecksum&&t.httpChecksum.requestChecksumRequired;n(this,"httpChecksumRequired",s,!1),r(this,"input",(function(){return t.input?a.create(t.input,i):new a.create({type:"structure"},i)})),r(this,"output",(function(){return t.output?a.create(t.output,i):new a.create({type:"structure"},i)})),r(this,"errors",(function(){var e=[];if(!t.errors)return null;for(var o=0;o<t.errors.length;o++)e.push(a.create(t.errors[o],i));return e})),r(this,"paginator",(function(){return i.api.paginators[e]})),i.documentation&&(n(this,"documentation",t.documentation),n(this,"documentationUrl",t.documentationUrl)),r(this,"idempotentMembers",(function(){var e=[],t=o.input,i=t.members;if(!t.members)return e;for(var a in i)i.hasOwnProperty(a)&&!0===i[a].isIdempotent&&e.push(a);return e})),r(this,"hasEventOutput",(function(){return function(e){var t=e.members,i=e.payload;if(!e.members)return!1;if(i){return t[i].isEventStream}for(var a in t)if(!t.hasOwnProperty(a)&&!0===t[a].isEventStream)return!0;return!1}(o.output)}))}},8594:function(e,t,i){var a=i(23657).property;e.exports=function(e,t){a(this,"inputToken",t.input_token),a(this,"limitKey",t.limit_key),a(this,"moreResults",t.more_results),a(this,"outputToken",t.output_token),a(this,"resultKey",t.result_key)}},15545:function(e,t,i){var a=i(23657),o=a.property;e.exports=function(e,t,i){i=i||{},o(this,"name",e),o(this,"api",i.api,!1),t.operation&&o(this,"operation",a.string.lowerFirst(t.operation));var n=this;["type","description","delay","maxAttempts","acceptors"].forEach((function(e){var i=t[e];i&&o(n,e,i)}))}},3825:function(e,t,i){var a=i(1242),o=i(23657);function n(e,t,i){null!==i&&void 0!==i&&o.property.apply(this,arguments)}function r(e,t){e.constructor.prototype[t]||o.memoizedProperty.apply(this,arguments)}function 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a})))),e.required&&(n(this,"required",e.required),n(this,"isRequired",(function(t){if(!o){o={};for(var i=0;i<e.required.length;i++)o[e.required[i]]=!0}return o[t]}),!1,!0)),n(this,"resultWrapper",e.resultWrapper||null),e.payload&&n(this,"payload",e.payload),"string"===typeof e.xmlNamespace?n(this,"xmlNamespaceUri",e.xmlNamespace):"object"===typeof e.xmlNamespace&&(n(this,"xmlNamespacePrefix",e.xmlNamespace.prefix),n(this,"xmlNamespaceUri",e.xmlNamespace.uri))}function p(e,t){var i=this,a=!this.isShape;if(l.apply(this,arguments),a&&n(this,"defaultValue",(function(){return[]})),e.member&&r(this,"member",(function(){return s.create(e.member,t)})),this.flattened){var o=this.name;r(this,"name",(function(){return i.member.name||o}))}}function h(e,t){var i=!this.isShape;l.apply(this,arguments),i&&(n(this,"defaultValue",(function(){return{}})),n(this,"key",s.create({type:"string"},t)),n(this,"value",s.create({type:"string"},t))),e.key&&r(this,"key",(function(){return 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0===e?null:"true"===e}}s.normalizedTypes={character:"string",double:"float",long:"integer",short:"integer",biginteger:"integer",bigdecimal:"float",blob:"binary"},s.types={structure:c,list:p,map:h,boolean:m,timestamp:function(e){var t=this;if(s.apply(this,arguments),e.timestampFormat)n(this,"timestampFormat",e.timestampFormat);else if(t.isTimestampFormatSet&&this.timestampFormat)n(this,"timestampFormat",this.timestampFormat);else if("header"===this.location)n(this,"timestampFormat","rfc822");else if("querystring"===this.location)n(this,"timestampFormat","iso8601");else if(this.api)switch(this.api.protocol){case"json":case"rest-json":n(this,"timestampFormat","unixTimestamp");break;case"rest-xml":case"query":case"ec2":n(this,"timestampFormat","iso8601")}this.toType=function(e){return null===e||void 0===e?null:"function"===typeof e.toUTCString?e:"string"===typeof e||"number"===typeof e?o.date.parseTimestamp(e):null},this.toWireFormat=function(e){return o.date.format(e,t.timestampFormat)}},float:function(){s.apply(this,arguments),this.toType=function(e){return null===e||void 0===e?null:parseFloat(e)},this.toWireFormat=this.toType},integer:function(){s.apply(this,arguments),this.toType=function(e){return null===e||void 0===e?null:parseInt(e,10)},this.toWireFormat=this.toType},string:$,base64:u,binary:d},s.resolve=function(e,t){if(e.shape){var i=t.api.shapes[e.shape];if(!i)throw new Error("Cannot find shape reference: "+e.shape);return i}return null},s.create=function(e,t,i){if(e.isShape)return e;var a=s.resolve(e,t);if(a){var o=Object.keys(e);t.documentation||(o=o.filter((function(e){return!e.match(/documentation/)})));var n=function(){a.constructor.call(this,e,t,i)};return n.prototype=a,new n}e.type||(e.members?e.type="structure":e.member?e.type="list":e.key?e.type="map":e.type="string");var r=e.type;if(s.normalizedTypes[e.type]&&(e.type=s.normalizedTypes[e.type]),s.types[e.type])return new s.types[e.type](e,t,i);throw new Error("Unrecognized shape type: "+r)},s.shapes={StructureShape:c,ListShape:p,MapShape:h,StringShape:$,BooleanShape:m,Base64Shape:u},e.exports=s},70778:function(e,t,i){var a=i(8468);a.ParamValidator=a.util.inherit({constructor:function(e){!0!==e&&void 0!==e||(e={min:!0}),this.validation=e},validate:function(e,t,i){if(this.errors=[],this.validateMember(e,t||{},i||"params"),this.errors.length>1){var o=this.errors.join("\n* ");throw o="There were "+this.errors.length+" validation errors:\n* "+o,a.util.error(new Error(o),{code:"MultipleValidationErrors",errors:this.errors})}if(1===this.errors.length)throw this.errors[0];return!0},fail:function(e,t){this.errors.push(a.util.error(new Error(t),{code:e}))},validateStructure:function(e,t,i){if(e.isDocument)return!0;var a;this.validateType(t,i,["object"],"structure");for(var o=0;e.required&&o<e.required.length;o++){var n=t[a=e.required[o]];void 0!==n&&null!==n||this.fail("MissingRequiredParameter","Missing required key '"+a+"' in "+i)}for(a in t)if(Object.prototype.hasOwnProperty.call(t,a)){var r=t[a],s=e.members[a];if(void 0!==s){var l=[i,a].join(".");this.validateMember(s,r,l)}else void 0!==r&&null!==r&&this.fail("UnexpectedParameter","Unexpected key '"+a+"' found in "+i)}return!0},validateMember:function(e,t,i){switch(e.type){case"structure":return this.validateStructure(e,t,i);case"list":return this.validateList(e,t,i);case"map":return this.validateMap(e,t,i);default:return this.validateScalar(e,t,i)}},validateList:function(e,t,i){if(this.validateType(t,i,[Array])){this.validateRange(e,t.length,i,"list member count");for(var a=0;a<t.length;a++)this.validateMember(e.member,t[a],i+"["+a+"]")}},validateMap:function(e,t,i){if(this.validateType(t,i,["object"],"map")){var a=0;for(var o in t)Object.prototype.hasOwnProperty.call(t,o)&&(this.validateMember(e.key,o,i+"[key='"+o+"']"),this.validateMember(e.value,t[o],i+"['"+o+"']"),a++);this.validateRange(e,a,i,"map member count")}},validateScalar:function(e,t,i){switch(e.type){case null:case void 0:case"string":return this.validateString(e,t,i);case"base64":case"binary":return this.validatePayload(t,i);case"integer":case"float":return this.validateNumber(e,t,i);case"boolean":return this.validateType(t,i,["boolean"]);case"timestamp":return this.validateType(t,i,[Date,/^\d{4}-\d{2}-\d{2}T\d{2}:\d{2}:\d{2}(\.\d+)?Z$/,"number"],"Date object, ISO-8601 string, or a UNIX timestamp");default:return this.fail("UnkownType","Unhandled type "+e.type+" for "+i)}},validateString:function(e,t,i){var a=["string"];e.isJsonValue&&(a=a.concat(["number","object","boolean"])),null!==t&&this.validateType(t,i,a)&&(this.validateEnum(e,t,i),this.validateRange(e,t.length,i,"string length"),this.validatePattern(e,t,i),this.validateUri(e,t,i))},validateUri:function(e,t,i){"uri"===e.location&&0===t.length&&this.fail("UriParameterError",'Expected uri parameter to have length >= 1, but found "'+t+'" for '+i)},validatePattern:function(e,t,i){this.validation.pattern&&void 0!==e.pattern&&(new RegExp(e.pattern).test(t)||this.fail("PatternMatchError",'Provided value "'+t+'" does not match regex pattern /'+e.pattern+"/ for "+i))},validateRange:function(e,t,i,a){this.validation.min&&void 0!==e.min&&t<e.min&&this.fail("MinRangeError","Expected "+a+" >= "+e.min+", but found "+t+" for "+i),this.validation.max&&void 0!==e.max&&t>e.max&&this.fail("MaxRangeError","Expected "+a+" <= "+e.max+", but found "+t+" for "+i)},validateEnum:function(e,t,i){this.validation.enum&&void 0!==e.enum&&-1===e.enum.indexOf(t)&&this.fail("EnumError","Found string value of "+t+", but expected "+e.enum.join("|")+" for "+i)},validateType:function(e,t,i,o){if(null===e||void 0===e)return!1;for(var n=!1,r=0;r<i.length;r++){if("string"===typeof i[r]){if(typeof e===i[r])return!0}else if(i[r]instanceof RegExp){if((e||"").toString().match(i[r]))return!0}else{if(e instanceof i[r])return!0;if(a.util.isType(e,i[r]))return!0;o||n||(i=i.slice()),i[r]=a.util.typeName(i[r])}n=!0}var s=o;s||(s=i.join(", ").replace(/,([^,]+)$/,", or$1"));var l=s.match(/^[aeiou]/i)?"n":"";return this.fail("InvalidParameterType","Expected "+t+" to be a"+l+" "+s),!1},validateNumber:function(e,t,i){if(null!==t&&void 0!==t){if("string"===typeof t){var a=parseFloat(t);a.toString()===t&&(t=a)}this.validateType(t,i,["number"])&&this.validateRange(e,t,i,"numeric value")}},validatePayload:function(e,t){if(null!==e&&void 0!==e&&"string"!==typeof e&&(!e||"number"!==typeof e.byteLength)){if(a.util.isNode()){var i=a.util.stream.Stream;if(a.util.Buffer.isBuffer(e)||e instanceof i)return}else if(void 0!==typeof Blob&&e instanceof Blob)return;var o=["Buffer","Stream","File","Blob","ArrayBuffer","DataView"];if(e)for(var n=0;n<o.length;n++){if(a.util.isType(e,o[n]))return;if(a.util.typeName(e.constructor)===o[n])return}this.fail("InvalidParameterType","Expected "+t+" to be a string, Buffer, Stream, Blob, or typed array object")}}})},84796:function(e,t,i){var a=i(8468),o=a.Protocol.Rest;a.Polly.Presigner=a.util.inherit({constructor:function(e){e=e||{},this.options=e,this.service=e.service,this.bindServiceObject(e),this._operations={}},bindServiceObject:function(e){if(e=e||{},this.service){var t=a.util.copy(this.service.config);this.service=new this.service.constructor.__super__(t),this.service.config.params=a.util.merge(this.service.config.params||{},e.params)}else this.service=new a.Polly(e)},modifyInputMembers:function(e){var t=a.util.copy(e);return t.members=a.util.copy(e.members),a.util.each(e.members,(function(e,i){t.members[e]=a.util.copy(i),i.location&&"body"!==i.location||(t.members[e].location="querystring",t.members[e].locationName=e)})),t},convertPostToGet:function(e){e.httpRequest.method="GET";var t=e.service.api.operations[e.operation],i=this._operations[e.operation];i||(this._operations[e.operation]=i=this.modifyInputMembers(t.input));var a=o.generateURI(e.httpRequest.endpoint.path,t.httpPath,i,e.params);e.httpRequest.path=a,e.httpRequest.body="",delete e.httpRequest.headers["Content-Length"],delete e.httpRequest.headers["Content-Type"]},getSynthesizeSpeechUrl:function(e,t,i){var a=this,o=this.service.makeRequest("synthesizeSpeech",e);return o.removeAllListeners("build"),o.on("build",(function(e){a.convertPostToGet(e)})),o.presign(t,i)}})},85363:function(e,t,i){var a=i(23657),o=i(8468);e.exports={populateHostPrefix:function(e){if(!e.service.config.hostPrefixEnabled)return e;var t=e.service.api.operations[e.operation];if(function(e){var t=e.service.api,i=t.operations[e.operation],o=t.endpointOperation&&t.endpointOperation===a.string.lowerFirst(i.name);return"NULL"!==i.endpointDiscoveryRequired||!0===o}(e))return e;if(t.endpoint&&t.endpoint.hostPrefix){var i=function(e,t,i){return a.each(i.members,(function(i,o){if(!0===o.hostLabel){if("string"!==typeof t[i]||""===t[i])throw a.error(new Error,{message:"Parameter "+i+" should be a non-empty string.",code:"InvalidParameter"});var n=new RegExp("\\{"+i+"\\}","g");e=e.replace(n,t[i])}})),e}(t.endpoint.hostPrefix,e.params,t.input);!function(e,t){e.host&&(e.host=t+e.host);e.hostname&&(e.hostname=t+e.hostname)}(e.httpRequest.endpoint,i),function(e){var t=e.split("."),i=/^[a-zA-Z0-9]{1}$|^[a-zA-Z0-9][a-zA-Z0-9\-]*[a-zA-Z0-9]$/;a.arrayEach(t,(function(e){if(!e.length||e.length<1||e.length>63)throw a.error(new Error,{code:"ValidationError",message:"Hostname label length should be between 1 to 63 characters, inclusive."});if(!i.test(e))throw o.util.error(new Error,{code:"ValidationError",message:e+" is not hostname compatible."})}))}(e.httpRequest.endpoint.hostname)}return e}}},89864:function(e,t,i){var a=i(23657),o=i(48212),n=i(26815),r=i(85363).populateHostPrefix;e.exports={buildRequest:function(e){var t=e.httpRequest,i=e.service.api,a=i.targetPrefix+"."+i.operations[e.operation].name,n=i.jsonVersion||"1.0",s=i.operations[e.operation].input,l=new o;1===n&&(n="1.0"),i.awsQueryCompatible&&(t.params||(t.params={}),Object.assign(t.params,e.params)),t.body=l.build(e.params||{},s),t.headers["Content-Type"]="application/x-amz-json-"+n,t.headers["X-Amz-Target"]=a,r(e)},extractError:function(e){var t={},i=e.httpResponse;if(t.code=i.headers["x-amzn-errortype"]||"UnknownError","string"===typeof t.code&&(t.code=t.code.split(":")[0]),i.body.length>0)try{var o=JSON.parse(i.body.toString()),n=o.__type||o.code||o.Code;for(var r in n&&(t.code=n.split("#").pop()),"RequestEntityTooLarge"===t.code?t.message="Request body must be less than 1 MB":t.message=o.message||o.Message||null,o||{})"code"!==r&&"message"!==r&&(t["["+r+"]"]="See error."+r+" for details.",Object.defineProperty(t,r,{value:o[r],enumerable:!1,writable:!0}))}catch(o){t.statusCode=i.statusCode,t.message=i.statusMessage}else t.statusCode=i.statusCode,t.message=i.statusCode.toString();e.error=a.error(new Error,t)},extractData:function(e){var t=e.httpResponse.body.toString()||"{}";if(!1===e.request.service.config.convertResponseTypes)e.data=JSON.parse(t);else{var i=e.request.service.api.operations[e.request.operation].output||{},a=new n;e.data=a.parse(t,i)}}}},66588:function(e,t,i){var a=i(8468),o=i(23657),n=i(31625),r=i(3825),s=i(85363).populateHostPrefix;e.exports={buildRequest:function(e){var t=e.service.api.operations[e.operation],i=e.httpRequest;i.headers["Content-Type"]="application/x-www-form-urlencoded; charset=utf-8",i.params={Version:e.service.api.apiVersion,Action:t.name},(new n).serialize(e.params,t.input,(function(e,t){i.params[e]=t})),i.body=o.queryParamsToString(i.params),s(e)},extractError:function(e){var t,i=e.httpResponse.body.toString();if(i.match("<UnknownOperationException"))t={Code:"UnknownOperation",Message:"Unknown operation "+e.request.operation};else try{t=(new a.XML.Parser).parse(i)}catch(n){t={Code:e.httpResponse.statusCode,Message:e.httpResponse.statusMessage}}t.requestId&&!e.requestId&&(e.requestId=t.requestId),t.Errors&&(t=t.Errors),t.Error&&(t=t.Error),t.Code?e.error=o.error(new Error,{code:t.Code,message:t.Message}):e.error=o.error(new Error,{code:e.httpResponse.statusCode,message:null})},extractData:function(e){var t=e.request,i=t.service.api.operations[t.operation].output||{},n=i;if(n.resultWrapper){var s=r.create({type:"structure"});s.members[n.resultWrapper]=i,s.memberNames=[n.resultWrapper],o.property(i,"name",i.resultWrapper),i=s}var l=new a.XML.Parser;if(i&&i.members&&!i.members._XAMZRequestId){var c=r.create({type:"string"},{api:{protocol:"query"}},"requestId");i.members._XAMZRequestId=c}var p=l.parse(e.httpResponse.body.toString(),i);e.requestId=p._XAMZRequestId||p.requestId,p._XAMZRequestId&&delete p._XAMZRequestId,n.resultWrapper&&p[n.resultWrapper]&&(o.update(p,p[n.resultWrapper]),delete p[n.resultWrapper]),e.data=p}}},19701:function(e,t,i){var a=i(23657),o=i(85363).populateHostPrefix;function n(e,t,i,o){var n=[e,t].join("/");n=n.replace(/\/+/g,"/");var r={},s=!1;if(a.each(i.members,(function(e,t){var i=o[e];if(null!==i&&void 0!==i)if("uri"===t.location){var l=new RegExp("\\{"+t.name+"(\\+)?\\}");n=n.replace(l,(function(e,t){return(t?a.uriEscapePath:a.uriEscape)(String(i))}))}else"querystring"===t.location&&(s=!0,"list"===t.type?r[t.name]=i.map((function(e){return a.uriEscape(t.member.toWireFormat(e).toString())})):"map"===t.type?a.each(i,(function(e,t){Array.isArray(t)?r[e]=t.map((function(e){return a.uriEscape(String(e))})):r[e]=a.uriEscape(String(t))})):r[t.name]=a.uriEscape(t.toWireFormat(i).toString()))})),s){n+=n.indexOf("?")>=0?"&":"?";var l=[];a.arrayEach(Object.keys(r).sort(),(function(e){Array.isArray(r[e])||(r[e]=[r[e]]);for(var t=0;t<r[e].length;t++)l.push(a.uriEscape(String(e))+"="+r[e][t])})),n+=l.join("&")}return n}e.exports={buildRequest:function(e){!function(e){e.httpRequest.method=e.service.api.operations[e.operation].httpMethod}(e),function(e){var t=e.service.api.operations[e.operation],i=t.input,a=n(e.httpRequest.endpoint.path,t.httpPath,i,e.params);e.httpRequest.path=a}(e),function(e){var t=e.service.api.operations[e.operation];a.each(t.input.members,(function(t,i){var o=e.params[t];null!==o&&void 0!==o&&("headers"===i.location&&"map"===i.type?a.each(o,(function(t,a){e.httpRequest.headers[i.name+t]=a})):"header"===i.location&&(o=i.toWireFormat(o).toString(),i.isJsonValue&&(o=a.base64.encode(o)),e.httpRequest.headers[i.name]=o))}))}(e),o(e)},extractError:function(){},extractData:function(e){var t=e.request,i={},o=e.httpResponse,n=t.service.api.operations[t.operation].output,r={};a.each(o.headers,(function(e,t){r[e.toLowerCase()]=t})),a.each(n.members,(function(e,t){var n=(t.name||e).toLowerCase();if("headers"===t.location&&"map"===t.type){i[e]={};var s=t.isLocationName?t.name:"",l=new RegExp("^"+s+"(.+)","i");a.each(o.headers,(function(t,a){var o=t.match(l);null!==o&&(i[e][o[1]]=a)}))}else if("header"===t.location){if(void 0!==r[n]){var c=t.isJsonValue?a.base64.decode(r[n]):r[n];i[e]=t.toType(c)}}else"statusCode"===t.location&&(i[e]=parseInt(o.statusCode,10))})),e.data=i},generateURI:n}},12085:function(e,t,i){var a=i(8468),o=i(23657),n=i(19701),r=i(89864),s=i(48212),l=i(26815),c=["GET","HEAD","DELETE"];function p(e,t){if(!e.httpRequest.headers["Content-Type"]){var i=t?"binary/octet-stream":"application/json";e.httpRequest.headers["Content-Type"]=i}}e.exports={buildRequest:function(e){n.buildRequest(e),c.indexOf(e.httpRequest.method)<0&&function(e){var t=new s,i=e.service.api.operations[e.operation].input;if(i.payload){var a,o=i.members[i.payload];a=e.params[i.payload],"structure"===o.type?(e.httpRequest.body=t.build(a||{},o),p(e)):void 0!==a&&(e.httpRequest.body=a,("binary"===o.type||o.isStreaming)&&p(e,!0))}else e.httpRequest.body=t.build(e.params,i),p(e)}(e)},extractError:function(e){r.extractError(e)},extractData:function(e){n.extractData(e);var t=e.request,i=t.service.api.operations[t.operation],s=t.service.api.operations[t.operation].output||{};if(i.hasEventOutput,s.payload){var c=s.members[s.payload],p=e.httpResponse.body;if(c.isEventStream)h=new l,e.data[s.payload]=o.createEventStream(2===a.HttpClient.streamsApiVersion?e.httpResponse.stream:p,h,c);else if("structure"===c.type||"list"===c.type){var h=new l;e.data[s.payload]=h.parse(p,c)}else"binary"===c.type||c.isStreaming?e.data[s.payload]=p:e.data[s.payload]=c.toType(p)}else{var $=e.data;r.extractData(e),e.data=o.merge($,e.data)}},unsetContentLength:function(e){void 0===o.getRequestPayloadShape(e)&&c.indexOf(e.httpRequest.method)>=0&&delete e.httpRequest.headers["Content-Length"]}}},61622:function(e,t,i){var a=i(8468),o=i(23657),n=i(19701);e.exports={buildRequest:function(e){n.buildRequest(e),["GET","HEAD"].indexOf(e.httpRequest.method)<0&&function(e){var t=e.service.api.operations[e.operation].input,i=new a.XML.Builder,n=e.params,r=t.payload;if(r){var s=t.members[r];if(void 0===(n=n[r]))return;if("structure"===s.type){var l=s.name;e.httpRequest.body=i.toXML(n,s,l,!0)}else e.httpRequest.body=n}else e.httpRequest.body=i.toXML(n,t,t.name||t.shape||o.string.upperFirst(e.operation)+"Request")}(e)},extractError:function(e){var t;n.extractError(e);try{t=(new a.XML.Parser).parse(e.httpResponse.body.toString())}catch(i){t={Code:e.httpResponse.statusCode,Message:e.httpResponse.statusMessage}}t.Errors&&(t=t.Errors),t.Error&&(t=t.Error),t.Code?e.error=o.error(new Error,{code:t.Code,message:t.Message}):e.error=o.error(new Error,{code:e.httpResponse.statusCode,message:null})},extractData:function(e){var t;n.extractData(e);var i=e.request,r=e.httpResponse.body,s=i.service.api.operations[i.operation],l=s.output,c=(s.hasEventOutput,l.payload);if(c){var p=l.members[c];p.isEventStream?(t=new a.XML.Parser,e.data[c]=o.createEventStream(2===a.HttpClient.streamsApiVersion?e.httpResponse.stream:e.httpResponse.body,t,p)):"structure"===p.type?(t=new a.XML.Parser,e.data[c]=t.parse(r.toString(),p)):"binary"===p.type||p.isStreaming?e.data[c]=r:e.data[c]=p.toType(r)}else if(r.length>0){var h=(t=new a.XML.Parser).parse(r.toString(),l);o.update(e.data,h)}}}},31625:function(e,t,i){var a=i(23657);function o(){}function n(e){return e.isQueryName||"ec2"!==e.api.protocol?e.name:e.name[0].toUpperCase()+e.name.substr(1)}function r(e,t,i,o){a.each(i.members,(function(i,a){var r=t[i];if(null!==r&&void 0!==r){var l=n(a);s(l=e?e+"."+l:l,r,a,o)}}))}function s(e,t,i,o){null!==t&&void 0!==t&&("structure"===i.type?r(e,t,i,o):"list"===i.type?function(e,t,i,o){var r=i.member||{};0!==t.length?a.arrayEach(t,(function(t,a){var l="."+(a+1);if("ec2"===i.api.protocol)l+="";else if(i.flattened){if(r.name){var c=e.split(".");c.pop(),c.push(n(r)),e=c.join(".")}}else l="."+(r.name?r.name:"member")+l;s(e+l,t,r,o)})):"ec2"!==i.api.protocol&&o.call(this,e,null)}(e,t,i,o):"map"===i.type?function(e,t,i,o){var n=1;a.each(t,(function(t,a){var r=(i.flattened?".":".entry.")+n+++".",l=r+(i.key.name||"key"),c=r+(i.value.name||"value");s(e+l,t,i.key,o),s(e+c,a,i.value,o)}))}(e,t,i,o):o(e,i.toWireFormat(t).toString()))}o.prototype.serialize=function(e,t,i){r("",e,t,i)},e.exports=o},42418:function(e,t,i){var a=i(8468),o=null,n={signatureVersion:"v4",signingName:"rds-db",operations:{}},r={region:"string",hostname:"string",port:"number",username:"string"};a.RDS.Signer=a.util.inherit({constructor:function(e){this.options=e||{}},convertUrlToAuthToken:function(e){var t="https://";if(0===e.indexOf(t))return e.substring(t.length)},getAuthToken:function(e,t){"function"===typeof e&&void 0===t&&(t=e,e={});var i=this,r="function"===typeof t;e=a.util.merge(this.options,e);var s=this.validateAuthTokenOptions(e);if(!0!==s){if(r)return t(s,null);throw s}var l={region:e.region,endpoint:new a.Endpoint(e.hostname+":"+e.port),paramValidation:!1,signatureVersion:"v4"};e.credentials&&(l.credentials=e.credentials),(o=new a.Service(l)).api=n;var c=o.makeRequest();if(this.modifyRequestForAuthToken(c,e),!r){var p=c.presign(900);return this.convertUrlToAuthToken(p)}c.presign(900,(function(e,a){a&&(a=i.convertUrlToAuthToken(a)),t(e,a)}))},modifyRequestForAuthToken:function(e,t){e.on("build",e.buildAsGet),e.httpRequest.body=a.util.queryParamsToString({Action:"connect",DBUser:t.username})},validateAuthTokenOptions:function(e){var t="";for(var i in e=e||{},r)Object.prototype.hasOwnProperty.call(r,i)&&typeof e[i]!==r[i]&&(t+="option '"+i+"' should have been type '"+r[i]+"', was '"+typeof e[i]+"'.\n");return!t.length||a.util.error(new Error,{code:"InvalidParameter",message:t})}})},35067:function(e){e.exports={now:function(){return"undefined"!==typeof performance&&"function"===typeof performance.now?performance.now():Date.now()}}},88918:function(e){e.exports={isFipsRegion:function(e){return"string"===typeof e&&(e.startsWith("fips-")||e.endsWith("-fips"))},isGlobalRegion:function(e){return"string"===typeof e&&["aws-global","aws-us-gov-global"].includes(e)},getRealRegion:function(e){return["fips-aws-global","aws-fips","aws-global"].includes(e)?"us-east-1":["fips-aws-us-gov-global","aws-us-gov-global"].includes(e)?"us-gov-west-1":e.replace(/fips-(dkr-|prod-)?|-fips/,"")}}},37139:function(e,t,i){var a=i(23657),o=i(80738);function n(e,t){a.each(t,(function(t,i){"globalEndpoint"!==t&&(void 0!==e.config[t]&&null!==e.config[t]||(e.config[t]=i))}))}e.exports={configureEndpoint:function(e){for(var t=function(e){var t=e.config.region,i=function(e){if(!e)return null;var t=e.split("-");return t.length<3?null:t.slice(0,t.length-2).join("-")+"-*"}(t),a=e.api.endpointPrefix;return[[t,a],[i,a],[t,"*"],[i,"*"],["*",a],[t,"internal-*"],["*","*"]].map((function(e){return e[0]&&e[1]?e.join("/"):null}))}(e),i=e.config.useFipsEndpoint,a=e.config.useDualstackEndpoint,r=0;r<t.length;r++){var s=t[r];if(s){var l=i?a?o.dualstackFipsRules:o.fipsRules:a?o.dualstackRules:o.rules;if(Object.prototype.hasOwnProperty.call(l,s)){var c=l[s];"string"===typeof c&&(c=o.patterns[c]),e.isGlobalEndpoint=!!c.globalEndpoint,c.signingRegion&&(e.signingRegion=c.signingRegion),c.signatureVersion||(c.signatureVersion="v4");var p="bearer"===(e.api&&e.api.signatureVersion);return void n(e,Object.assign({},c,{signatureVersion:p?"bearer":c.signatureVersion}))}}}},getEndpointSuffix:function(e){for(var t={"^(us|eu|ap|sa|ca|me)\\-\\w+\\-\\d+$":"amazonaws.com","^cn\\-\\w+\\-\\d+$":"amazonaws.com.cn","^us\\-gov\\-\\w+\\-\\d+$":"amazonaws.com","^us\\-iso\\-\\w+\\-\\d+$":"c2s.ic.gov","^us\\-isob\\-\\w+\\-\\d+$":"sc2s.sgov.gov","^eu\\-isoe\\-west\\-1$":"cloud.adc-e.uk","^us\\-isof\\-\\w+\\-\\d+$":"csp.hci.ic.gov"},i=Object.keys(t),a=0;a<i.length;a++){var o=RegExp(i[a]),n=t[i[a]];if(o.test(e))return n}return"amazonaws.com"}}},33845:function(e,t,i){var a=i(8468),o=i(41496),n=a.util.inherit,r=a.util.domain,s=i(24247),l={success:1,error:1,complete:1};var c=new o;c.setupStates=function(){var e=function(e,t){var i=this;i._haltHandlersOnError=!1,i.emit(i._asm.currentState,(function(e){if(e)if(a=i,Object.prototype.hasOwnProperty.call(l,a._asm.currentState)){if(!(r&&i.domain instanceof r.Domain))throw e;e.domainEmitter=i,e.domain=i.domain,e.domainThrown=!1,i.domain.emit("error",e)}else i.response.error=e,t(e);else t(i.response.error);var a}))};this.addState("validate","build","error",e),this.addState("build","afterBuild","restart",e),this.addState("afterBuild","sign","restart",e),this.addState("sign","send","retry",e),this.addState("retry","afterRetry","afterRetry",e),this.addState("afterRetry","sign","error",e),this.addState("send","validateResponse","retry",e),this.addState("validateResponse","extractData","extractError",e),this.addState("extractError","extractData","retry",e),this.addState("extractData","success","retry",e),this.addState("restart","build","error",e),this.addState("success","complete","complete",e),this.addState("error","complete","complete",e),this.addState("complete",null,null,e)},c.setupStates(),a.Request=n({constructor:function(e,t,i){var n=e.endpoint,s=e.config.region,l=e.config.customUserAgent;e.signingRegion?s=e.signingRegion:e.isGlobalEndpoint&&(s="us-east-1"),this.domain=r&&r.active,this.service=e,this.operation=t,this.params=i||{},this.httpRequest=new a.HttpRequest(n,s),this.httpRequest.appendToUserAgent(l),this.startTime=e.getSkewCorrectedDate(),this.response=new a.Response(this),this._asm=new o(c.states,"validate"),this._haltHandlersOnError=!1,a.SequentialExecutor.call(this),this.emit=this.emitEvent},send:function(e){return e&&(this.httpRequest.appendToUserAgent("callback"),this.on("complete",(function(t){e.call(t,t.error,t.data)}))),this.runTo(),this.response},build:function(e){return this.runTo("send",e)},runTo:function(e,t){return this._asm.runTo(e,t,this),this},abort:function(){return this.removeAllListeners("validateResponse"),this.removeAllListeners("extractError"),this.on("validateResponse",(function(e){e.error=a.util.error(new Error("Request aborted by user"),{code:"RequestAbortedError",retryable:!1})})),this.httpRequest.stream&&!this.httpRequest.stream.didCallback&&(this.httpRequest.stream.abort(),this.httpRequest._abortCallback?this.httpRequest._abortCallback():this.removeAllListeners("send")),this},eachPage:function(e){e=a.util.fn.makeAsync(e,3),this.on("complete",(function t(i){e.call(i,i.error,i.data,(function(o){!1!==o&&(i.hasNextPage()?i.nextPage().on("complete",t).send():e.call(i,null,null,a.util.fn.noop))}))})).send()},eachItem:function(e){var t=this;this.eachPage((function(i,o){if(i)return e(i,null);if(null===o)return e(null,null);var n=t.service.paginationConfig(t.operation).resultKey;Array.isArray(n)&&(n=n[0]);var r=s.search(o,n),l=!0;return a.util.arrayEach(r,(function(t){if(!1===(l=e(null,t)))return a.util.abort})),l}))},isPageable:function(){return!!this.service.paginationConfig(this.operation)},createReadStream:function(){var e=a.util.stream,t=this,i=null;return 2===a.HttpClient.streamsApiVersion?(i=new e.PassThrough,process.nextTick((function(){t.send()}))):((i=new e.Stream).readable=!0,i.sent=!1,i.on("newListener",(function(e){i.sent||"data"!==e||(i.sent=!0,process.nextTick((function(){t.send()})))}))),this.on("error",(function(e){i.emit("error",e)})),this.on("httpHeaders",(function(o,n,r){if(o<300){t.removeListener("httpData",a.EventListeners.Core.HTTP_DATA),t.removeListener("httpError",a.EventListeners.Core.HTTP_ERROR),t.on("httpError",(function(e){r.error=e,r.error.retryable=!1}));var s,l=!1;if("HEAD"!==t.httpRequest.method&&(s=parseInt(n["content-length"],10)),void 0!==s&&!isNaN(s)&&s>=0){l=!0;var c=0}var p=function(){l&&c!==s?i.emit("error",a.util.error(new Error("Stream content length mismatch. Received "+c+" of "+s+" bytes."),{code:"StreamContentLengthMismatch"})):2===a.HttpClient.streamsApiVersion?i.end():i.emit("end")},h=r.httpResponse.createUnbufferedStream();if(2===a.HttpClient.streamsApiVersion)if(l){var $=new e.PassThrough;$._write=function(t){return t&&t.length&&(c+=t.length),e.PassThrough.prototype._write.apply(this,arguments)},$.on("end",p),i.on("error",(function(e){l=!1,h.unpipe($),$.emit("end"),$.end()})),h.pipe($).pipe(i,{end:!1})}else h.pipe(i);else l&&h.on("data",(function(e){e&&e.length&&(c+=e.length)})),h.on("data",(function(e){i.emit("data",e)})),h.on("end",p);h.on("error",(function(e){l=!1,i.emit("error",e)}))}})),i},emitEvent:function(e,t,i){"function"===typeof t&&(i=t,t=null),i||(i=function(){}),t||(t=this.eventParameters(e,this.response)),a.SequentialExecutor.prototype.emit.call(this,e,t,(function(e){e&&(this.response.error=e),i.call(this,e)}))},eventParameters:function(e){switch(e){case"restart":case"validate":case"sign":case"build":case"afterValidate":case"afterBuild":return[this];case"error":return[this.response.error,this.response];default:return[this.response]}},presign:function(e,t){return t||"function"!==typeof e||(t=e,e=null),(new a.Signers.Presign).sign(this.toGet(),e,t)},isPresigned:function(){return Object.prototype.hasOwnProperty.call(this.httpRequest.headers,"presigned-expires")},toUnauthenticated:function(){return this._unAuthenticated=!0,this.removeListener("validate",a.EventListeners.Core.VALIDATE_CREDENTIALS),this.removeListener("sign",a.EventListeners.Core.SIGN),this},toGet:function(){return"query"!==this.service.api.protocol&&"ec2"!==this.service.api.protocol||(this.removeListener("build",this.buildAsGet),this.addListener("build",this.buildAsGet)),this},buildAsGet:function(e){e.httpRequest.method="GET",e.httpRequest.path=e.service.endpoint.path+"?"+e.httpRequest.body,e.httpRequest.body="",delete e.httpRequest.headers["Content-Length"],delete e.httpRequest.headers["Content-Type"]},haltHandlersOnError:function(){this._haltHandlersOnError=!0}}),a.Request.addPromisesToClass=function(e){this.prototype.promise=function(){var t=this;return this.httpRequest.appendToUserAgent("promise"),new e((function(e,i){t.on("complete",(function(t){t.error?i(t.error):e(Object.defineProperty(t.data||{},"$response",{value:t}))})),t.runTo()}))}},a.Request.deletePromisesFromClass=function(){delete this.prototype.promise},a.util.addPromises(a.Request),a.util.mixin(a.Request,a.SequentialExecutor)},90630:function(e,t,i){var a=i(8468),o=a.util.inherit,n=i(24247);function r(e){var t=e.request._waiter,i=t.config.acceptors,a=!1,o="retry";i.forEach((function(i){if(!a){var n=t.matchers[i.matcher];n&&n(e,i.expected,i.argument)&&(a=!0,o=i.state)}})),!a&&e.error&&(o="failure"),"success"===o?t.setSuccess(e):t.setError(e,"retry"===o)}a.ResourceWaiter=o({constructor:function(e,t){this.service=e,this.state=t,this.loadWaiterConfig(this.state)},service:null,state:null,config:null,matchers:{path:function(e,t,i){try{var a=n.search(e.data,i)}catch(o){return!1}return n.strictDeepEqual(a,t)},pathAll:function(e,t,i){try{var a=n.search(e.data,i)}catch(s){return!1}Array.isArray(a)||(a=[a]);var o=a.length;if(!o)return!1;for(var r=0;r<o;r++)if(!n.strictDeepEqual(a[r],t))return!1;return!0},pathAny:function(e,t,i){try{var a=n.search(e.data,i)}catch(s){return!1}Array.isArray(a)||(a=[a]);for(var o=a.length,r=0;r<o;r++)if(n.strictDeepEqual(a[r],t))return!0;return!1},status:function(e,t){var i=e.httpResponse.statusCode;return"number"===typeof i&&i===t},error:function(e,t){return"string"===typeof t&&e.error?t===e.error.code:t===!!e.error}},listeners:(new a.SequentialExecutor).addNamedListeners((function(e){e("RETRY_CHECK","retry",(function(e){var t=e.request._waiter;e.error&&"ResourceNotReady"===e.error.code&&(e.error.retryDelay=1e3*(t.config.delay||0))})),e("CHECK_OUTPUT","extractData",r),e("CHECK_ERROR","extractError",r)})),wait:function(e,t){"function"===typeof e&&(t=e,e=void 0),e&&e.$waiter&&("number"===typeof(e=a.util.copy(e)).$waiter.delay&&(this.config.delay=e.$waiter.delay),"number"===typeof e.$waiter.maxAttempts&&(this.config.maxAttempts=e.$waiter.maxAttempts),delete e.$waiter);var i=this.service.makeRequest(this.config.operation,e);return i._waiter=this,i.response.maxRetries=this.config.maxAttempts,i.addListeners(this.listeners),t&&i.send(t),i},setSuccess:function(e){e.error=null,e.data=e.data||{},e.request.removeAllListeners("extractData")},setError:function(e,t){e.data=null,e.error=a.util.error(e.error||new Error,{code:"ResourceNotReady",message:"Resource is not in the state "+this.state,retryable:t})},loadWaiterConfig:function(e){if(!this.service.api.waiters[e])throw new a.util.error(new Error,{code:"StateNotFoundError",message:"State "+e+" not found."});this.config=a.util.copy(this.service.api.waiters[e])}})},10767:function(e,t,i){var a=i(8468),o=a.util.inherit,n=i(24247);a.Response=o({constructor:function(e){this.request=e,this.data=null,this.error=null,this.retryCount=0,this.redirectCount=0,this.httpResponse=new a.HttpResponse,e&&(this.maxRetries=e.service.numRetries(),this.maxRedirects=e.service.config.maxRedirects)},nextPage:function(e){var t,i=this.request.service,o=this.request.operation;try{t=i.paginationConfig(o,!0)}catch(l){this.error=l}if(!this.hasNextPage()){if(e)e(this.error,null);else if(this.error)throw this.error;return null}var n=a.util.copy(this.request.params);if(this.nextPageTokens){var r=t.inputToken;"string"===typeof r&&(r=[r]);for(var s=0;s<r.length;s++)n[r[s]]=this.nextPageTokens[s];return i.makeRequest(this.request.operation,n,e)}return e?e(null,null):null},hasNextPage:function(){return this.cacheNextPageTokens(),!!this.nextPageTokens||void 0===this.nextPageTokens&&void 0},cacheNextPageTokens:function(){if(Object.prototype.hasOwnProperty.call(this,"nextPageTokens"))return this.nextPageTokens;this.nextPageTokens=void 0;var e=this.request.service.paginationConfig(this.request.operation);if(!e)return this.nextPageTokens;if(this.nextPageTokens=null,e.moreResults&&!n.search(this.data,e.moreResults))return this.nextPageTokens;var t=e.outputToken;return"string"===typeof t&&(t=[t]),a.util.arrayEach.call(this,t,(function(e){var t=n.search(this.data,e);t&&(this.nextPageTokens=this.nextPageTokens||[],this.nextPageTokens.push(t))})),this.nextPageTokens}})},51470:function(e,t,i){var a=i(8468),o=a.util.string.byteLength,n=a.util.Buffer;a.S3.ManagedUpload=a.util.inherit({constructor:function(e){var t=this;a.SequentialExecutor.call(t),t.body=null,t.sliceFn=null,t.callback=null,t.parts={},t.completeInfo=[],t.fillQueue=function(){t.callback(new Error("Unsupported body payload "+typeof t.body))},t.configure(e)},configure:function(e){if(e=e||{},this.partSize=this.minPartSize,e.queueSize&&(this.queueSize=e.queueSize),e.partSize&&(this.partSize=e.partSize),e.leavePartsOnError&&(this.leavePartsOnError=!0),e.tags){if(!Array.isArray(e.tags))throw new Error("Tags must be specified as an array; "+typeof e.tags+" provided.");this.tags=e.tags}if(this.partSize<this.minPartSize)throw new Error("partSize must be greater than "+this.minPartSize);this.service=e.service,this.bindServiceObject(e.params),this.validateBody(),this.adjustTotalBytes()},leavePartsOnError:!1,queueSize:4,partSize:null,minPartSize:5242880,maxTotalParts:1e4,send:function(e){var t=this;t.failed=!1,t.callback=e||function(e){if(e)throw e};var i=!0;if(t.sliceFn)t.fillQueue=t.fillBuffer;else if(a.util.isNode()){var o=a.util.stream.Stream;t.body instanceof o&&(i=!1,t.fillQueue=t.fillStream,t.partBuffers=[],t.body.on("error",(function(e){t.cleanup(e)})).on("readable",(function(){t.fillQueue()})).on("end",(function(){t.isDoneChunking=!0,t.numParts=t.totalPartNumbers,t.fillQueue.call(t),t.isDoneChunking&&t.totalPartNumbers>=1&&t.doneParts===t.numParts&&t.finishMultiPart()})))}i&&t.fillQueue.call(t)},abort:function(){var e=this;!0===e.isDoneChunking&&1===e.totalPartNumbers&&e.singlePart?e.singlePart.abort():e.cleanup(a.util.error(new Error("Request aborted by user"),{code:"RequestAbortedError",retryable:!1}))},validateBody:function(){var e=this;if(e.body=e.service.config.params.Body,"string"===typeof e.body)e.body=a.util.buffer.toBuffer(e.body);else if(!e.body)throw new Error("params.Body is required");e.sliceFn=a.util.arraySliceFn(e.body)},bindServiceObject:function(e){e=e||{};var t=this;if(t.service){var i=t.service,o=a.util.copy(i.config);o.signatureVersion=i.getSignatureVersion(),t.service=new i.constructor.__super__(o),t.service.config.params=a.util.merge(t.service.config.params||{},e),Object.defineProperty(t.service,"_originalConfig",{get:function(){return i._originalConfig},enumerable:!1,configurable:!0})}else t.service=new a.S3({params:e})},adjustTotalBytes:function(){var e=this;try{e.totalBytes=o(e.body)}catch(i){}if(e.totalBytes){var t=Math.ceil(e.totalBytes/e.maxTotalParts);t>e.partSize&&(e.partSize=t)}else e.totalBytes=void 0},isDoneChunking:!1,partPos:0,totalChunkedBytes:0,totalUploadedBytes:0,totalBytes:void 0,numParts:0,totalPartNumbers:0,activeParts:0,doneParts:0,parts:null,completeInfo:null,failed:!1,multipartReq:null,partBuffers:null,partBufferLength:0,fillBuffer:function(){var e=this,t=o(e.body);if(0===t)return e.isDoneChunking=!0,e.numParts=1,void e.nextChunk(e.body);for(;e.activeParts<e.queueSize&&e.partPos<t;){var i=Math.min(e.partPos+e.partSize,t),a=e.sliceFn.call(e.body,e.partPos,i);e.partPos+=e.partSize,(o(a)<e.partSize||e.partPos===t)&&(e.isDoneChunking=!0,e.numParts=e.totalPartNumbers+1),e.nextChunk(a)}},fillStream:function(){var e=this;if(!(e.activeParts>=e.queueSize)){var t=e.body.read(e.partSize-e.partBufferLength)||e.body.read();if(t&&(e.partBuffers.push(t),e.partBufferLength+=t.length,e.totalChunkedBytes+=t.length),e.partBufferLength>=e.partSize){var i=1===e.partBuffers.length?e.partBuffers[0]:n.concat(e.partBuffers);if(e.partBuffers=[],e.partBufferLength=0,i.length>e.partSize){var a=i.slice(e.partSize);e.partBuffers.push(a),e.partBufferLength+=a.length,i=i.slice(0,e.partSize)}e.nextChunk(i)}e.isDoneChunking&&!e.isDoneSending&&(i=1===e.partBuffers.length?e.partBuffers[0]:n.concat(e.partBuffers),e.partBuffers=[],e.partBufferLength=0,e.totalBytes=e.totalChunkedBytes,e.isDoneSending=!0,(0===e.numParts||i.length>0)&&(e.numParts++,e.nextChunk(i))),e.body.read(0)}},nextChunk:function(e){var t=this;if(t.failed)return null;var i=++t.totalPartNumbers;if(t.isDoneChunking&&1===i){var o={Body:e};this.tags&&(o.Tagging=this.getTaggingHeader());var n=t.service.putObject(o);return n._managedUpload=t,n.on("httpUploadProgress",t.progress).send(t.finishSinglePart),t.singlePart=n,null}if(t.service.config.params.ContentMD5){var r=a.util.error(new Error("The Content-MD5 you specified is invalid for multi-part uploads."),{code:"InvalidDigest",retryable:!1});return t.cleanup(r),null}if(t.completeInfo[i]&&null!==t.completeInfo[i].ETag)return null;t.activeParts++,t.service.config.params.UploadId?t.uploadPart(e,i):t.multipartReq?t.queueChunks(e,i):(t.multipartReq=t.service.createMultipartUpload(),t.multipartReq.on("success",(function(e){t.service.config.params.UploadId=e.data.UploadId,t.multipartReq=null})),t.queueChunks(e,i),t.multipartReq.on("error",(function(e){t.cleanup(e)})),t.multipartReq.send())},getTaggingHeader:function(){for(var e=[],t=0;t<this.tags.length;t++)e.push(a.util.uriEscape(this.tags[t].Key)+"="+a.util.uriEscape(this.tags[t].Value));return e.join("&")},uploadPart:function(e,t){var i=this,o={Body:e,ContentLength:a.util.string.byteLength(e),PartNumber:t},n={ETag:null,PartNumber:t};i.completeInfo[t]=n;var r=i.service.uploadPart(o);i.parts[t]=r,r._lastUploadedBytes=0,r._managedUpload=i,r.on("httpUploadProgress",i.progress),r.send((function(e,r){if(delete i.parts[o.PartNumber],i.activeParts--,!e&&(!r||!r.ETag)){var s="No access to ETag property on response.";a.util.isBrowser()&&(s+=" Check CORS configuration to expose ETag header."),e=a.util.error(new Error(s),{code:"ETagMissing",retryable:!1})}return e?i.cleanup(e):i.completeInfo[t]&&null!==i.completeInfo[t].ETag?null:(n.ETag=r.ETag,i.doneParts++,void(i.isDoneChunking&&i.doneParts===i.totalPartNumbers?i.finishMultiPart():i.fillQueue.call(i)))}))},queueChunks:function(e,t){var i=this;i.multipartReq.on("success",(function(){i.uploadPart(e,t)}))},cleanup:function(e){var t=this;t.failed||("function"===typeof t.body.removeAllListeners&&"function"===typeof t.body.resume&&(t.body.removeAllListeners("readable"),t.body.removeAllListeners("end"),t.body.resume()),t.multipartReq&&(t.multipartReq.removeAllListeners("success"),t.multipartReq.removeAllListeners("error"),t.multipartReq.removeAllListeners("complete"),delete t.multipartReq),t.service.config.params.UploadId&&!t.leavePartsOnError?t.service.abortMultipartUpload().send():t.leavePartsOnError&&(t.isDoneChunking=!1),a.util.each(t.parts,(function(e,t){t.removeAllListeners("complete"),t.abort()})),t.activeParts=0,t.partPos=0,t.numParts=0,t.totalPartNumbers=0,t.parts={},t.failed=!0,t.callback(e))},finishMultiPart:function(){var e=this,t={MultipartUpload:{Parts:e.completeInfo.slice(1)}};e.service.completeMultipartUpload(t,(function(t,i){if(t)return e.cleanup(t);if(i&&"string"===typeof i.Location&&(i.Location=i.Location.replace(/%2F/g,"/")),Array.isArray(e.tags)){for(var a=0;a<e.tags.length;a++)e.tags[a].Value=String(e.tags[a].Value);e.service.putObjectTagging({Tagging:{TagSet:e.tags}},(function(t,a){t?e.callback(t):e.callback(t,i)}))}else e.callback(t,i)}))},finishSinglePart:function(e,t){var i=this.request._managedUpload,a=this.request.httpRequest,o=a.endpoint;if(e)return i.callback(e);t.Location=[o.protocol,"//",o.host,a.path].join(""),t.key=this.request.params.Key,t.Key=this.request.params.Key,t.Bucket=this.request.params.Bucket,i.callback(e,t)},progress:function(e){var t=this._managedUpload;"putObject"===this.operation?(e.part=1,e.key=this.params.Key):(t.totalUploadedBytes+=e.loaded-this._lastUploadedBytes,this._lastUploadedBytes=e.loaded,e={loaded:t.totalUploadedBytes,total:t.totalBytes,part:this.params.PartNumber,key:this.params.Key}),t.emit("httpUploadProgress",[e])}}),a.util.mixin(a.S3.ManagedUpload,a.SequentialExecutor),a.S3.ManagedUpload.addPromisesToClass=function(e){this.prototype.promise=a.util.promisifyMethod("send",e)},a.S3.ManagedUpload.deletePromisesFromClass=function(){delete this.prototype.promise},a.util.addPromises(a.S3.ManagedUpload),e.exports=a.S3.ManagedUpload},78451:function(e,t,i){var a=i(8468);a.SequentialExecutor=a.util.inherit({constructor:function(){this._events={}},listeners:function(e){return this._events[e]?this._events[e].slice(0):[]},on:function(e,t,i){return this._events[e]?i?this._events[e].unshift(t):this._events[e].push(t):this._events[e]=[t],this},onAsync:function(e,t,i){return t._isAsync=!0,this.on(e,t,i)},removeListener:function(e,t){var i=this._events[e];if(i){for(var a=i.length,o=-1,n=0;n<a;++n)i[n]===t&&(o=n);o>-1&&i.splice(o,1)}return this},removeAllListeners:function(e){return e?delete this._events[e]:this._events={},this},emit:function(e,t,i){i||(i=function(){});var a=this.listeners(e),o=a.length;return this.callListeners(a,t,i),o>0},callListeners:function(e,t,i,o){var n=this,r=o||null;function s(o){if(o&&(r=a.util.error(r||new Error,o),n._haltHandlersOnError))return i.call(n,r);n.callListeners(e,t,i,r)}for(;e.length>0;){var l=e.shift();if(l._isAsync)return void l.apply(n,t.concat([s]));try{l.apply(n,t)}catch(c){r=a.util.error(r||new Error,c)}if(r&&n._haltHandlersOnError)return void i.call(n,r)}i.call(n,r)},addListeners:function(e){var t=this;return e._events&&(e=e._events),a.util.each(e,(function(e,i){"function"===typeof i&&(i=[i]),a.util.arrayEach(i,(function(i){t.on(e,i)}))})),t},addNamedListener:function(e,t,i,a){return this[e]=i,this.addListener(t,i,a),this},addNamedAsyncListener:function(e,t,i,a){return i._isAsync=!0,this.addNamedListener(e,t,i,a)},addNamedListeners:function(e){var t=this;return e((function(){t.addNamedListener.apply(t,arguments)}),(function(){t.addNamedAsyncListener.apply(t,arguments)})),this}}),a.SequentialExecutor.prototype.addListener=a.SequentialExecutor.prototype.on,e.exports=a.SequentialExecutor},75724:function(e,t,i){var a=i(8468),o=i(65565),n=i(37139),r=a.util.inherit,s=0,l=i(88918);a.Service=r({constructor:function(e){if(!this.loadServiceClass)throw a.util.error(new Error,"Service must be constructed with `new' operator");if(e){if(e.region){var t=e.region;l.isFipsRegion(t)&&(e.region=l.getRealRegion(t),e.useFipsEndpoint=!0),l.isGlobalRegion(t)&&(e.region=l.getRealRegion(t))}"boolean"===typeof e.useDualstack&&"boolean"!==typeof e.useDualstackEndpoint&&(e.useDualstackEndpoint=e.useDualstack)}var i=this.loadServiceClass(e||{});if(i){var o=a.util.copy(e),n=new i(e);return Object.defineProperty(n,"_originalConfig",{get:function(){return o},enumerable:!1,configurable:!0}),n._clientId=++s,n}this.initialize(e)},initialize:function(e){var t=a.config[this.serviceIdentifier];if(this.config=new a.Config(a.config),t&&this.config.update(t,!0),e&&this.config.update(e,!0),this.validateService(),this.config.endpoint||n.configureEndpoint(this),this.config.endpoint=this.endpointFromTemplate(this.config.endpoint),this.setEndpoint(this.config.endpoint),a.SequentialExecutor.call(this),a.Service.addDefaultMonitoringListeners(this),(this.config.clientSideMonitoring||a.Service._clientSideMonitoring)&&this.publisher){var i=this.publisher;this.addNamedListener("PUBLISH_API_CALL","apiCall",(function(e){process.nextTick((function(){i.eventHandler(e)}))})),this.addNamedListener("PUBLISH_API_ATTEMPT","apiCallAttempt",(function(e){process.nextTick((function(){i.eventHandler(e)}))}))}},validateService:function(){},loadServiceClass:function(e){var t=e;if(a.util.isEmpty(this.api)){if(t.apiConfig)return a.Service.defineServiceApi(this.constructor,t.apiConfig);if(this.constructor.services){(t=new a.Config(a.config)).update(e,!0);var i=t.apiVersions[this.constructor.serviceIdentifier];return i=i||t.apiVersion,this.getLatestServiceClass(i)}return null}return null},getLatestServiceClass:function(e){return e=this.getLatestServiceVersion(e),null===this.constructor.services[e]&&a.Service.defineServiceApi(this.constructor,e),this.constructor.services[e]},getLatestServiceVersion:function(e){if(!this.constructor.services||0===this.constructor.services.length)throw new Error("No services defined on "+this.constructor.serviceIdentifier);if(e?a.util.isType(e,Date)&&(e=a.util.date.iso8601(e).split("T")[0]):e="latest",Object.hasOwnProperty(this.constructor.services,e))return e;for(var t=Object.keys(this.constructor.services).sort(),i=null,o=t.length-1;o>=0;o--)if("*"!==t[o][t[o].length-1]&&(i=t[o]),t[o].substr(0,10)<=e)return i;throw new Error("Could not find "+this.constructor.serviceIdentifier+" API to satisfy version constraint `"+e+"'")},api:{},defaultRetryCount:3,customizeRequests:function(e){if(e){if("function"!==typeof e)throw new Error("Invalid callback type '"+typeof e+"' provided in customizeRequests");this.customRequestHandler=e}else this.customRequestHandler=null},makeRequest:function(e,t,i){if("function"===typeof t&&(i=t,t=null),t=t||{},this.config.params){var o=this.api.operations[e];o&&(t=a.util.copy(t),a.util.each(this.config.params,(function(e,i){o.input.members[e]&&(void 0!==t[e]&&null!==t[e]||(t[e]=i))})))}var n=new a.Request(this,e,t);return this.addAllRequestListeners(n),this.attachMonitoringEmitter(n),i&&n.send(i),n},makeUnauthenticatedRequest:function(e,t,i){"function"===typeof t&&(i=t,t={});var a=this.makeRequest(e,t).toUnauthenticated();return i?a.send(i):a},waitFor:function(e,t,i){return new a.ResourceWaiter(this,e).wait(t,i)},addAllRequestListeners:function(e){for(var t=[a.events,a.EventListeners.Core,this.serviceInterface(),a.EventListeners.CorePost],i=0;i<t.length;i++)t[i]&&e.addListeners(t[i]);this.config.paramValidation||e.removeListener("validate",a.EventListeners.Core.VALIDATE_PARAMETERS),this.config.logger&&e.addListeners(a.EventListeners.Logger),this.setupRequestListeners(e),"function"===typeof this.constructor.prototype.customRequestHandler&&this.constructor.prototype.customRequestHandler(e),Object.prototype.hasOwnProperty.call(this,"customRequestHandler")&&"function"===typeof this.customRequestHandler&&this.customRequestHandler(e)},apiCallEvent:function(e){var t=e.service.api.operations[e.operation],i={Type:"ApiCall",Api:t?t.name:e.operation,Version:1,Service:e.service.api.serviceId||e.service.api.endpointPrefix,Region:e.httpRequest.region,MaxRetriesExceeded:0,UserAgent:e.httpRequest.getUserAgent()},a=e.response;if(a.httpResponse.statusCode&&(i.FinalHttpStatusCode=a.httpResponse.statusCode),a.error){var o=a.error;a.httpResponse.statusCode>299?(o.code&&(i.FinalAwsException=o.code),o.message&&(i.FinalAwsExceptionMessage=o.message)):((o.code||o.name)&&(i.FinalSdkException=o.code||o.name),o.message&&(i.FinalSdkExceptionMessage=o.message))}return i},apiAttemptEvent:function(e){var t=e.service.api.operations[e.operation],i={Type:"ApiCallAttempt",Api:t?t.name:e.operation,Version:1,Service:e.service.api.serviceId||e.service.api.endpointPrefix,Fqdn:e.httpRequest.endpoint.hostname,UserAgent:e.httpRequest.getUserAgent()},a=e.response;return a.httpResponse.statusCode&&(i.HttpStatusCode=a.httpResponse.statusCode),!e._unAuthenticated&&e.service.config.credentials&&e.service.config.credentials.accessKeyId&&(i.AccessKey=e.service.config.credentials.accessKeyId),a.httpResponse.headers?(e.httpRequest.headers["x-amz-security-token"]&&(i.SessionToken=e.httpRequest.headers["x-amz-security-token"]),a.httpResponse.headers["x-amzn-requestid"]&&(i.XAmznRequestId=a.httpResponse.headers["x-amzn-requestid"]),a.httpResponse.headers["x-amz-request-id"]&&(i.XAmzRequestId=a.httpResponse.headers["x-amz-request-id"]),a.httpResponse.headers["x-amz-id-2"]&&(i.XAmzId2=a.httpResponse.headers["x-amz-id-2"]),i):i},attemptFailEvent:function(e){var t=this.apiAttemptEvent(e),i=e.response,a=i.error;return i.httpResponse.statusCode>299?(a.code&&(t.AwsException=a.code),a.message&&(t.AwsExceptionMessage=a.message)):((a.code||a.name)&&(t.SdkException=a.code||a.name),a.message&&(t.SdkExceptionMessage=a.message)),t},attachMonitoringEmitter:function(e){var t,i,o,n,r,s,l=0,c=this;e.on("validate",(function(){n=a.util.realClock.now(),s=Date.now()}),true),e.on("sign",(function(){i=a.util.realClock.now(),t=Date.now(),r=e.httpRequest.region,l++}),true),e.on("validateResponse",(function(){o=Math.round(a.util.realClock.now()-i)})),e.addNamedListener("API_CALL_ATTEMPT","success",(function(){var i=c.apiAttemptEvent(e);i.Timestamp=t,i.AttemptLatency=o>=0?o:0,i.Region=r,c.emit("apiCallAttempt",[i])})),e.addNamedListener("API_CALL_ATTEMPT_RETRY","retry",(function(){var n=c.attemptFailEvent(e);n.Timestamp=t,o=o||Math.round(a.util.realClock.now()-i),n.AttemptLatency=o>=0?o:0,n.Region=r,c.emit("apiCallAttempt",[n])})),e.addNamedListener("API_CALL","complete",(function(){var t=c.apiCallEvent(e);if(t.AttemptCount=l,!(t.AttemptCount<=0)){t.Timestamp=s;var i=Math.round(a.util.realClock.now()-n);t.Latency=i>=0?i:0;var o=e.response;o.error&&o.error.retryable&&"number"===typeof o.retryCount&&"number"===typeof o.maxRetries&&o.retryCount>=o.maxRetries&&(t.MaxRetriesExceeded=1),c.emit("apiCall",[t])}}))},setupRequestListeners:function(e){},getSigningName:function(){return this.api.signingName||this.api.endpointPrefix},getSignerClass:function(e){var t,i=null,o="";e&&(o=(i=(e.service.api.operations||{})[e.operation]||null)?i.authtype:"");return t=this.config.signatureVersion?this.config.signatureVersion:"v4"===o||"v4-unsigned-body"===o?"v4":"bearer"===o?"bearer":this.api.signatureVersion,a.Signers.RequestSigner.getVersion(t)},serviceInterface:function(){switch(this.api.protocol){case"ec2":case"query":return a.EventListeners.Query;case"json":return a.EventListeners.Json;case"rest-json":return a.EventListeners.RestJson;case"rest-xml":return a.EventListeners.RestXml}if(this.api.protocol)throw new Error("Invalid service `protocol' "+this.api.protocol+" in API config")},successfulResponse:function(e){return e.httpResponse.statusCode<300},numRetries:function(){return void 0!==this.config.maxRetries?this.config.maxRetries:this.defaultRetryCount},retryDelays:function(e,t){return a.util.calculateRetryDelay(e,this.config.retryDelayOptions,t)},retryableError:function(e){return!!this.timeoutError(e)||(!!this.networkingError(e)||(!!this.expiredCredentialsError(e)||(!!this.throttledError(e)||e.statusCode>=500)))},networkingError:function(e){return"NetworkingError"===e.code},timeoutError:function(e){return"TimeoutError"===e.code},expiredCredentialsError:function(e){return"ExpiredTokenException"===e.code},clockSkewError:function(e){switch(e.code){case"RequestTimeTooSkewed":case"RequestExpired":case"InvalidSignatureException":case"SignatureDoesNotMatch":case"AuthFailure":case"RequestInTheFuture":return!0;default:return!1}},getSkewCorrectedDate:function(){return new Date(Date.now()+this.config.systemClockOffset)},applyClockOffset:function(e){e&&(this.config.systemClockOffset=e-Date.now())},isClockSkewed:function(e){if(e)return Math.abs(this.getSkewCorrectedDate().getTime()-e)>=3e5},throttledError:function(e){if(429===e.statusCode)return!0;switch(e.code){case"ProvisionedThroughputExceededException":case"Throttling":case"ThrottlingException":case"RequestLimitExceeded":case"RequestThrottled":case"RequestThrottledException":case"TooManyRequestsException":case"TransactionInProgressException":case"EC2ThrottledException":return!0;default:return!1}},endpointFromTemplate:function(e){if("string"!==typeof e)return e;var t=e;return t=(t=(t=t.replace(/\{service\}/g,this.api.endpointPrefix)).replace(/\{region\}/g,this.config.region)).replace(/\{scheme\}/g,this.config.sslEnabled?"https":"http")},setEndpoint:function(e){this.endpoint=new a.Endpoint(e,this.config)},paginationConfig:function(e,t){var i=this.api.operations[e].paginator;if(!i){if(t){var o=new Error;throw a.util.error(o,"No pagination configuration for "+e)}return null}return i}}),a.util.update(a.Service,{defineMethods:function(e){a.util.each(e.prototype.api.operations,(function(t){e.prototype[t]||("none"===e.prototype.api.operations[t].authtype?e.prototype[t]=function(e,i){return this.makeUnauthenticatedRequest(t,e,i)}:e.prototype[t]=function(e,i){return this.makeRequest(t,e,i)})}))},defineService:function(e,t,i){a.Service._serviceMap[e]=!0,Array.isArray(t)||(i=t,t=[]);var o=r(a.Service,i||{});if("string"===typeof e){a.Service.addVersions(o,t);var n=o.serviceIdentifier||e;o.serviceIdentifier=n}else o.prototype.api=e,a.Service.defineMethods(o);if(a.SequentialExecutor.call(this.prototype),!this.prototype.publisher&&a.util.clientSideMonitoring){var s=a.util.clientSideMonitoring.Publisher,l=(0,a.util.clientSideMonitoring.configProvider)();this.prototype.publisher=new s(l),l.enabled&&(a.Service._clientSideMonitoring=!0)}return a.SequentialExecutor.call(o.prototype),a.Service.addDefaultMonitoringListeners(o.prototype),o},addVersions:function(e,t){Array.isArray(t)||(t=[t]),e.services=e.services||{};for(var i=0;i<t.length;i++)void 0===e.services[t[i]]&&(e.services[t[i]]=null);e.apiVersions=Object.keys(e.services).sort()},defineServiceApi:function(e,t,i){var n=r(e,{serviceIdentifier:e.serviceIdentifier});function s(t){t.isApi?n.prototype.api=t:n.prototype.api=new o(t,{serviceIdentifier:e.serviceIdentifier})}if("string"===typeof t){if(i)s(i);else try{s(a.apiLoader(e.serviceIdentifier,t))}catch(l){throw a.util.error(l,{message:"Could not find API configuration "+e.serviceIdentifier+"-"+t})}Object.prototype.hasOwnProperty.call(e.services,t)||(e.apiVersions=e.apiVersions.concat(t).sort()),e.services[t]=n}else s(t);return a.Service.defineMethods(n),n},hasService:function(e){return Object.prototype.hasOwnProperty.call(a.Service._serviceMap,e)},addDefaultMonitoringListeners:function(e){e.addNamedListener("MONITOR_EVENTS_BUBBLE","apiCallAttempt",(function(t){var i=Object.getPrototypeOf(e);i._events&&i.emit("apiCallAttempt",[t])})),e.addNamedListener("CALL_EVENTS_BUBBLE","apiCall",(function(t){var i=Object.getPrototypeOf(e);i._events&&i.emit("apiCall",[t])}))},_serviceMap:{}}),a.util.mixin(a.Service,a.SequentialExecutor),e.exports=a.Service},32193:function(e,t,i){var a=i(8468);a.util.update(a.APIGateway.prototype,{setAcceptHeader:function(e){var t=e.httpRequest;t.headers.Accept||(t.headers.Accept="application/json")},setupRequestListeners:function(e){(e.addListener("build",this.setAcceptHeader),"getExport"===e.operation)&&("swagger"===(e.params||{}).exportType&&e.addListener("extractData",a.util.convertPayloadToString))}})},42768:function(e,t,i){var a=i(8468);i(65473),a.util.update(a.CloudFront.prototype,{setupRequestListeners:function(e){e.addListener("extractData",a.util.hoistPayloadMember)}})},89785:function(e,t,i){var a=i(8468);i(55098),a.util.update(a.DynamoDB.prototype,{setupRequestListeners:function(e){e.service.config.dynamoDbCrc32&&(e.removeListener("extractData",a.EventListeners.Json.EXTRACT_DATA),e.addListener("extractData",this.checkCrc32),e.addListener("extractData",a.EventListeners.Json.EXTRACT_DATA))},checkCrc32:function(e){if(!e.httpResponse.streaming&&!e.request.service.crc32IsValid(e))throw e.data=null,e.error=a.util.error(new Error,{code:"CRC32CheckFailed",message:"CRC32 integrity check failed",retryable:!0}),e.request.haltHandlersOnError(),e.error},crc32IsValid:function(e){var t=e.httpResponse.headers["x-amz-crc32"];return!t||parseInt(t,10)===a.util.crypto.crc32(e.httpResponse.body)},defaultRetryCount:10,retryDelays:function(e,t){var i=a.util.copy(this.config.retryDelayOptions);return"number"!==typeof i.base&&(i.base=50),a.util.calculateRetryDelay(e,i,t)}})},89422:function(e,t,i){var a=i(8468);a.util.update(a.EC2.prototype,{setupRequestListeners:function(e){e.removeListener("extractError",a.EventListeners.Query.EXTRACT_ERROR),e.addListener("extractError",this.extractError),"copySnapshot"===e.operation&&e.onAsync("validate",this.buildCopySnapshotPresignedUrl)},buildCopySnapshotPresignedUrl:function(e,t){if(e.params.PresignedUrl||e._subRequest)return t();e.params=a.util.copy(e.params),e.params.DestinationRegion=e.service.config.region;var i=a.util.copy(e.service.config);delete i.endpoint,i.region=e.params.SourceRegion;var o=new e.service.constructor(i)[e.operation](e.params);o._subRequest=!0,o.presign((function(i,a){i?t(i):(e.params.PresignedUrl=a,t())}))},extractError:function(e){var t=e.httpResponse,i=(new a.XML.Parser).parse(t.body.toString()||"");i.Errors?e.error=a.util.error(new Error,{code:i.Errors.Error.Code,message:i.Errors.Error.Message}):e.error=a.util.error(new Error,{code:t.statusCode,message:null}),e.error.requestId=i.RequestID||null}})},17414:function(e,t,i){var a=i(8468),o=["deleteThingShadow","getThingShadow","updateThingShadow"];a.util.update(a.IotData.prototype,{validateService:function(){if(!this.config.endpoint||this.config.endpoint.indexOf("{")>=0){throw a.util.error(new Error,{name:"InvalidEndpoint",message:"AWS.IotData requires an explicit `endpoint' configuration option."})}},setupRequestListeners:function(e){e.addListener("validateResponse",this.validateResponseBody),o.indexOf(e.operation)>-1&&e.addListener("extractData",a.util.convertPayloadToString)},validateResponseBody:function(e){var t=(e.httpResponse.body.toString()||"{}").trim();t&&"{"===t.charAt(0)||(e.httpResponse.body="")}})},85447:function(e,t,i){var a=i(8468);a.util.update(a.Lambda.prototype,{setupRequestListeners:function(e){"invoke"===e.operation&&e.addListener("extractData",a.util.convertPayloadToString)}})},50566:function(e,t,i){var a=i(8468);a.util.update(a.MachineLearning.prototype,{setupRequestListeners:function(e){"predict"===e.operation&&e.addListener("build",this.buildEndpoint)},buildEndpoint:function(e){var t=e.params.PredictEndpoint;t&&(e.httpRequest.endpoint=new a.Endpoint(t))}})},92540:function(e,t,i){i(84796)},99875:function(e,t,i){var a=i(8468),o=i(11483);i(42418);var n=["copyDBSnapshot","createDBInstanceReadReplica","createDBCluster","copyDBClusterSnapshot","startDBInstanceAutomatedBackupsReplication"];a.util.update(a.RDS.prototype,{setupRequestListeners:function(e){o.setupRequestListeners(this,e,n)}})},11483:function(e,t,i){var a=i(8468),o={setupRequestListeners:function(e,t,i){if(-1!==i.indexOf(t.operation)&&t.params.SourceRegion)if(t.params=a.util.copy(t.params),t.params.PreSignedUrl||t.params.SourceRegion===e.config.region)delete t.params.SourceRegion;else{var n=!!e.config.paramValidation;n&&t.removeListener("validate",a.EventListeners.Core.VALIDATE_PARAMETERS),t.onAsync("validate",o.buildCrossRegionPresignedUrl),n&&t.addListener("validate",a.EventListeners.Core.VALIDATE_PARAMETERS)}},buildCrossRegionPresignedUrl:function(e,t){var i=a.util.copy(e.service.config);i.region=e.params.SourceRegion,delete e.params.SourceRegion,delete i.endpoint,delete i.params,i.signatureVersion="v4";var o=e.service.config.region,n=new e.service.constructor(i)[e.operation](a.util.copy(e.params));n.on("build",(function(e){var t=e.httpRequest;t.params.DestinationRegion=o,t.body=a.util.queryParamsToString(t.params)})),n.presign((function(i,a){i?t(i):(e.params.PreSignedUrl=a,t())}))}};e.exports=o},12449:function(e,t,i){var a=i(8468);a.util.update(a.Route53.prototype,{setupRequestListeners:function(e){e.on("build",this.sanitizeUrl)},sanitizeUrl:function(e){var t=e.httpRequest.path;e.httpRequest.path=t.replace(/\/%2F\w+%2F/,"/")},retryableError:function(e){return"PriorRequestNotComplete"===e.code&&400===e.statusCode||a.Service.prototype.retryableError.call(this,e)}})},57700:function(e,t,i){var a=i(8468),o=i(94966),n=i(21980),r=i(22605),s=i(37139);i(51470);var l={completeMultipartUpload:!0,copyObject:!0,uploadPartCopy:!0},c=["AuthorizationHeaderMalformed","BadRequest","PermanentRedirect",301],p="s3-object-lambda";a.util.update(a.S3.prototype,{getSignatureVersion:function(e){var t=this.api.signatureVersion,i=this._originalConfig?this._originalConfig.signatureVersion:null,a=this.config.signatureVersion,o=!!e&&e.isPresigned();return i?i="v2"===i?"s3":i:(!0!==o?t="v4":a&&(t=a),t)},getSigningName:function(e){if(e&&"writeGetObjectResponse"===e.operation)return p;var t=a.Service.prototype.getSigningName;return e&&e._parsedArn&&e._parsedArn.service?e._parsedArn.service:t.call(this)},getSignerClass:function(e){var t=this.getSignatureVersion(e);return a.Signers.RequestSigner.getVersion(t)},validateService:function(){var e,t=[];if(this.config.region||(this.config.region="us-east-1"),!this.config.endpoint&&this.config.s3BucketEndpoint&&t.push("An endpoint must be provided when configuring `s3BucketEndpoint` to true."),1===t.length?e=t[0]:t.length>1&&(e="Multiple configuration errors:\n"+t.join("\n")),e)throw a.util.error(new Error,{name:"InvalidEndpoint",message:e})},shouldDisableBodySigning:function(e){var t=this.getSignerClass();return!0===this.config.s3DisableBodySigning&&t===a.Signers.V4&&"https:"===e.httpRequest.endpoint.protocol},setupRequestListeners:function(e){e.addListener("validateResponse",this.setExpiresString);if(e.addListener("validate",this.validateScheme),e.addListener("validate",this.validateBucketName,true),e.addListener("validate",this.optInUsEast1RegionalEndpoint,true),e.removeListener("validate",a.EventListeners.Core.VALIDATE_REGION),e.addListener("build",this.addContentType),e.addListener("build",this.computeContentMd5),e.addListener("build",this.computeSseCustomerKeyMd5),e.addListener("build",this.populateURI),e.addListener("afterBuild",this.addExpect100Continue),e.addListener("extractError",this.extractError),e.addListener("extractData",a.util.hoistPayloadMember),e.addListener("extractData",this.extractData),e.addListener("extractData",this.extractErrorFrom200Response),e.addListener("beforePresign",this.prepareSignedUrl),this.shouldDisableBodySigning(e)&&(e.removeListener("afterBuild",a.EventListeners.Core.COMPUTE_SHA256),e.addListener("afterBuild",this.disableBodySigning)),"createBucket"!==e.operation&&r.isArnInParam(e,"Bucket"))return e._parsedArn=a.util.ARN.parse(e.params.Bucket),e.removeListener("validate",this.validateBucketName),e.removeListener("build",this.populateURI),"s3"===e._parsedArn.service?(e.addListener("validate",r.validateS3AccessPointArn),e.addListener("validate",this.validateArnResourceType),e.addListener("validate",this.validateArnRegion)):"s3-outposts"===e._parsedArn.service&&(e.addListener("validate",r.validateOutpostsAccessPointArn),e.addListener("validate",r.validateOutpostsArn),e.addListener("validate",r.validateArnRegion)),e.addListener("validate",r.validateArnAccount),e.addListener("validate",r.validateArnService),e.addListener("build",this.populateUriFromAccessPointArn),void e.addListener("build",r.validatePopulateUriFromArn);e.addListener("validate",this.validateBucketEndpoint),e.addListener("validate",this.correctBucketRegionFromCache),e.onAsync("extractError",this.requestBucketRegion),a.util.isBrowser()&&e.onAsync("retry",this.reqRegionForNetworkingError)},validateScheme:function(e){var t=e.params,i=e.httpRequest.endpoint.protocol;if((t.SSECustomerKey||t.CopySourceSSECustomerKey)&&"https:"!==i){throw a.util.error(new Error,{code:"ConfigError",message:"Cannot send SSE keys over HTTP. Set 'sslEnabled'to 'true' in your configuration"})}},validateBucketEndpoint:function(e){if(!e.params.Bucket&&e.service.config.s3BucketEndpoint){throw a.util.error(new Error,{code:"ConfigError",message:"Cannot send requests to root API with `s3BucketEndpoint` set."})}},validateArnRegion:function(e){r.validateArnRegion(e,{allowFipsEndpoint:!0})},validateArnResourceType:function(e){var t=e._parsedArn.resource;if(0!==t.indexOf("accesspoint:")&&0!==t.indexOf("accesspoint/"))throw a.util.error(new Error,{code:"InvalidARN",message:"ARN resource should begin with 'accesspoint/'"})},validateBucketName:function(e){var t=e.service.getSignatureVersion(e),i=e.params&&e.params.Bucket,o=e.params&&e.params.Key,n=i&&i.indexOf("/");if(i&&n>=0)if("string"===typeof o&&n>0){e.params=a.util.copy(e.params);var r=i.substr(n+1)||"";e.params.Key=r+"/"+o,e.params.Bucket=i.substr(0,n)}else if("v4"===t){var s="Bucket names cannot contain forward slashes. Bucket: "+i;throw a.util.error(new Error,{code:"InvalidBucket",message:s})}},isValidAccelerateOperation:function(e){return-1===["createBucket","deleteBucket","listBuckets"].indexOf(e)},optInUsEast1RegionalEndpoint:function(e){var t=e.service,i=t.config;if(i.s3UsEast1RegionalEndpoint=n(t._originalConfig,{env:"AWS_S3_US_EAST_1_REGIONAL_ENDPOINT",sharedConfig:"s3_us_east_1_regional_endpoint",clientConfig:"s3UsEast1RegionalEndpoint"}),!(t._originalConfig||{}).endpoint&&"us-east-1"===e.httpRequest.region&&"regional"===i.s3UsEast1RegionalEndpoint&&e.httpRequest.endpoint.hostname.indexOf("s3.amazonaws.com")>=0){var a=i.endpoint.indexOf(".amazonaws.com"),o=i.endpoint.substring(0,a)+".us-east-1"+i.endpoint.substring(a);e.httpRequest.updateEndpoint(o)}},populateURI:function(e){var t=e.httpRequest,i=e.params.Bucket,a=e.service,o=t.endpoint;if(i&&!a.pathStyleBucketName(i)){a.config.useAccelerateEndpoint&&a.isValidAccelerateOperation(e.operation)?a.config.useDualstackEndpoint?o.hostname=i+".s3-accelerate.dualstack.amazonaws.com":o.hostname=i+".s3-accelerate.amazonaws.com":a.config.s3BucketEndpoint||(o.hostname=i+"."+o.hostname);var n=o.port;o.host=80!==n&&443!==n?o.hostname+":"+o.port:o.hostname,t.virtualHostedBucket=i,a.removeVirtualHostedBucketFromPath(e)}},removeVirtualHostedBucketFromPath:function(e){var t=e.httpRequest,i=t.virtualHostedBucket;if(i&&t.path){if(e.params&&e.params.Key){var o="/"+a.util.uriEscapePath(e.params.Key);if(0===t.path.indexOf(o)&&(t.path.length===o.length||"?"===t.path[o.length]))return}t.path=t.path.replace(new RegExp("/"+i),""),"/"!==t.path[0]&&(t.path="/"+t.path)}},populateUriFromAccessPointArn:function(e){var t=e._parsedArn,i="s3-outposts"===t.service,o="s3-object-lambda"===t.service,n=i?"."+t.outpostId:"",r=i?"s3-outposts":"s3-accesspoint",l=!i&&e.service.config.useFipsEndpoint?"-fips":"",c=!i&&e.service.config.useDualstackEndpoint?".dualstack":"",p=e.httpRequest.endpoint,h=s.getEndpointSuffix(t.region),$=e.service.config.s3UseArnRegion;if(p.hostname=[t.accessPoint+"-"+t.accountId+n,r+l+c,$?t.region:e.service.config.region,h].join("."),o){r="s3-object-lambda";var d=t.resource.split("/")[1];l=e.service.config.useFipsEndpoint?"-fips":"";p.hostname=[d+"-"+t.accountId,r+l,$?t.region:e.service.config.region,h].join(".")}p.host=p.hostname;var u=a.util.uriEscape(e.params.Bucket),m=e.httpRequest.path;e.httpRequest.path=m.replace(new RegExp("/"+u),""),"/"!==e.httpRequest.path[0]&&(e.httpRequest.path="/"+e.httpRequest.path),e.httpRequest.region=t.region},addExpect100Continue:function(e){var t=e.httpRequest.headers["Content-Length"];a.util.isNode()&&(t>=1048576||e.params.Body instanceof a.util.stream.Stream)&&(e.httpRequest.headers.Expect="100-continue")},addContentType:function(e){var t=e.httpRequest;if("GET"!==t.method&&"HEAD"!==t.method){t.headers["Content-Type"]||(t.headers["Content-Type"]="application/octet-stream");var i=t.headers["Content-Type"];if(a.util.isBrowser())if("string"!==typeof t.body||i.match(/;\s*charset=/)){t.headers["Content-Type"]=i.replace(/(;\s*charset=)(.+)$/,(function(e,t,i){return t+i.toUpperCase()}))}else{t.headers["Content-Type"]+="; charset=UTF-8"}}else delete t.headers["Content-Type"]},willComputeChecksums:function(e){var t=e.service.api.operations[e.operation].input.members,i=e.httpRequest.body,o=e.service.config.computeChecksums&&t.ContentMD5&&!e.params.ContentMD5&&i&&(a.util.Buffer.isBuffer(e.httpRequest.body)||"string"===typeof e.httpRequest.body);return!(!o||!e.service.shouldDisableBodySigning(e)||e.isPresigned())||!(!o||"s3"!==this.getSignatureVersion(e)||!e.isPresigned())},computeContentMd5:function(e){if(e.service.willComputeChecksums(e)){var t=a.util.crypto.md5(e.httpRequest.body,"base64");e.httpRequest.headers["Content-MD5"]=t}},computeSseCustomerKeyMd5:function(e){a.util.each({SSECustomerKey:"x-amz-server-side-encryption-customer-key-MD5",CopySourceSSECustomerKey:"x-amz-copy-source-server-side-encryption-customer-key-MD5"},(function(t,i){if(e.params[t]){var o=a.util.crypto.md5(e.params[t],"base64");e.httpRequest.headers[i]=o}}))},pathStyleBucketName:function(e){return!!this.config.s3ForcePathStyle||!this.config.s3BucketEndpoint&&(!r.dnsCompatibleBucketName(e)||!(!this.config.sslEnabled||!e.match(/\./)))},extractErrorFrom200Response:function(e){var t=this.service?this.service:this;if(t.is200Error(e)||l[e.request.operation]){var i=e.httpResponse,o=i.body&&i.body.toString()||"";if(o&&o.indexOf("</Error>")===o.length-8)throw e.data=null,t.extractError(e),e.error.is200Error=!0,e.error;if(!i.body||!o.match(/<[\w_]/))throw e.data=null,a.util.error(new Error,{code:"InternalError",message:"S3 aborted request"})}},is200Error:function(e){if(200!==(e&&e.httpResponse&&e.httpResponse.statusCode))return!1;try{for(var t=e.request,i=t.service.api.operations[t.operation].output.members,a=Object.keys(i),o=0;o<a.length;++o){var n=i[a[o]];if("binary"===n.type&&n.isStreaming)return!1}var r=e.httpResponse.body;if(r&&void 0!==r.byteLength&&(r.byteLength<15||r.byteLength>3e3))return!1;if(!r)return!1;var s=r.toString();if(s.indexOf("</Error>")===s.length-8)return!0}catch(l){return!1}return!1},retryableError:function(e,t){return!!(e.is200Error||l[t.operation]&&200===e.statusCode)||(!t._requestRegionForBucket||!t.service.bucketRegionCache[t._requestRegionForBucket])&&(!(!e||"RequestTimeout"!==e.code)||(e&&-1!=c.indexOf(e.code)&&e.region&&e.region!=t.httpRequest.region?(t.httpRequest.region=e.region,301===e.statusCode&&t.service.updateReqBucketRegion(t),!0):a.Service.prototype.retryableError.call(this,e,t)))},updateReqBucketRegion:function(e,t){var i=e.httpRequest;if("string"===typeof t&&t.length&&(i.region=t),i.endpoint.host.match(/s3(?!-accelerate).*\.amazonaws\.com$/)){var o=e.service,n=o.config,r=n.s3BucketEndpoint;r&&delete n.s3BucketEndpoint;var s=a.util.copy(n);delete s.endpoint,s.region=i.region,i.endpoint=new a.S3(s).endpoint,o.populateURI(e),n.s3BucketEndpoint=r,i.headers.Host=i.endpoint.host,"validate"===e._asm.currentState&&(e.removeListener("build",o.populateURI),e.addListener("build",o.removeVirtualHostedBucketFromPath))}},extractData:function(e){var t=e.request;if("getBucketLocation"===t.operation){var i=e.httpResponse.body.toString().match(/>(.+)<\/Location/);delete e.data._,e.data.LocationConstraint=i?i[1]:""}var a=t.params.Bucket||null;if("deleteBucket"!==t.operation||"string"!==typeof a||e.error){var o=(e.httpResponse.headers||{})["x-amz-bucket-region"]||null;if(!o&&"createBucket"===t.operation&&!e.error){var n=t.params.CreateBucketConfiguration;o=n?"EU"===n.LocationConstraint?"eu-west-1":n.LocationConstraint:"us-east-1"}o&&a&&o!==t.service.bucketRegionCache[a]&&(t.service.bucketRegionCache[a]=o)}else t.service.clearBucketRegionCache(a);t.service.extractRequestIds(e)},extractError:function(e){var t,i={304:"NotModified",403:"Forbidden",400:"BadRequest",404:"NotFound"},o=e.request,n=e.httpResponse.statusCode,r=e.httpResponse.body||"",s=(e.httpResponse.headers||{})["x-amz-bucket-region"]||null,l=o.params.Bucket||null,c=o.service.bucketRegionCache;if(s&&l&&s!==c[l]&&(c[l]=s),i[n]&&0===r.length)l&&!s&&(t=c[l]||null)!==o.httpRequest.region&&(s=t),e.error=a.util.error(new Error,{code:i[n],message:null,region:s});else{var p=(new a.XML.Parser).parse(r.toString());p.Region&&!s?(s=p.Region,l&&s!==c[l]&&(c[l]=s)):!l||s||p.Region||(t=c[l]||null)!==o.httpRequest.region&&(s=t),e.error=a.util.error(new Error,{code:p.Code||n,message:p.Message||null,region:s})}o.service.extractRequestIds(e)},requestBucketRegion:function(e,t){var i=e.error,o=e.request,n=o.params.Bucket||null;if(!i||!n||i.region||"listObjects"===o.operation||a.util.isNode()&&"headBucket"===o.operation||400===i.statusCode&&"headObject"!==o.operation||-1===c.indexOf(i.code))return t();var r=a.util.isNode()?"headBucket":"listObjects",s={Bucket:n};"listObjects"===r&&(s.MaxKeys=0);var l=o.service[r](s);l._requestRegionForBucket=n,l.send((function(){var e=o.service.bucketRegionCache[n]||null;i.region=e,t()}))},reqRegionForNetworkingError:function(e,t){if(!a.util.isBrowser())return t();var i=e.error,o=e.request,n=o.params.Bucket;if(!i||"NetworkingError"!==i.code||!n||"us-east-1"===o.httpRequest.region)return t();var s=o.service,l=s.bucketRegionCache,c=l[n]||null;if(c&&c!==o.httpRequest.region)s.updateReqBucketRegion(o,c),t();else if(r.dnsCompatibleBucketName(n))if(o.httpRequest.virtualHostedBucket){var p=s.listObjects({Bucket:n,MaxKeys:0});s.updateReqBucketRegion(p,"us-east-1"),p._requestRegionForBucket=n,p.send((function(){var e=s.bucketRegionCache[n]||null;e&&e!==o.httpRequest.region&&s.updateReqBucketRegion(o,e),t()}))}else t();else s.updateReqBucketRegion(o,"us-east-1"),"us-east-1"!==l[n]&&(l[n]="us-east-1"),t()},bucketRegionCache:{},clearBucketRegionCache:function(e){var t=this.bucketRegionCache;e?"string"===typeof e&&(e=[e]):e=Object.keys(t);for(var i=0;i<e.length;i++)delete t[e[i]];return t},correctBucketRegionFromCache:function(e){var t=e.params.Bucket||null;if(t){var i=e.service,a=e.httpRequest.region,o=i.bucketRegionCache[t];o&&o!==a&&i.updateReqBucketRegion(e,o)}},extractRequestIds:function(e){var t=e.httpResponse.headers?e.httpResponse.headers["x-amz-id-2"]:null,i=e.httpResponse.headers?e.httpResponse.headers["x-amz-cf-id"]:null;e.extendedRequestId=t,e.cfId=i,e.error&&(e.error.requestId=e.requestId||null,e.error.extendedRequestId=t,e.error.cfId=i)},getSignedUrl:function(e,t,i){var o=(t=a.util.copy(t||{})).Expires||900;if("number"!==typeof o)throw a.util.error(new Error,{code:"InvalidParameterException",message:"The expiration must be a number, received "+typeof o});delete t.Expires;var n=this.makeRequest(e,t);if(!i)return n.presign(o,i);a.util.defer((function(){n.presign(o,i)}))},createPresignedPost:function(e,t){"function"===typeof e&&void 0===t&&(t=e,e=null),e=a.util.copy(e||{});var i=this.config.params||{},o=e.Bucket||i.Bucket,n=this,r=this.config,s=a.util.copy(this.endpoint);function l(){return{url:a.util.urlFormat(s),fields:n.preparePostFields(r.credentials,r.region,o,e.Fields,e.Conditions,e.Expires)}}if(r.s3BucketEndpoint||(s.pathname="/"+o),!t)return l();r.getCredentials((function(e){if(e)t(e);else try{t(null,l())}catch(e){t(e)}}))},preparePostFields:function(e,t,i,n,r,s){var l=this.getSkewCorrectedDate();if(!e||!t||!i)throw new Error("Unable to create a POST object policy without a bucket, region, and credentials");n=a.util.copy(n||{}),r=(r||[]).slice(0),s=s||3600;var c=a.util.date.iso8601(l).replace(/[:\-]|\.\d{3}/g,""),p=c.substr(0,8),h=o.createScope(p,t,"s3"),$=e.accessKeyId+"/"+h;for(var d in n.bucket=i,n["X-Amz-Algorithm"]="AWS4-HMAC-SHA256",n["X-Amz-Credential"]=$,n["X-Amz-Date"]=c,e.sessionToken&&(n["X-Amz-Security-Token"]=e.sessionToken),n)if(n.hasOwnProperty(d)){var u={};u[d]=n[d],r.push(u)}return n.Policy=this.preparePostPolicy(new Date(l.valueOf()+1e3*s),r),n["X-Amz-Signature"]=a.util.crypto.hmac(o.getSigningKey(e,p,t,"s3",!0),n.Policy,"hex"),n},preparePostPolicy:function(e,t){return a.util.base64.encode(JSON.stringify({expiration:a.util.date.iso8601(e),conditions:t}))},prepareSignedUrl:function(e){e.addListener("validate",e.service.noPresignedContentLength),e.removeListener("build",e.service.addContentType),e.params.Body?e.addListener("afterBuild",a.EventListeners.Core.COMPUTE_SHA256):e.removeListener("build",e.service.computeContentMd5)},disableBodySigning:function(e){var t=e.httpRequest.headers;Object.prototype.hasOwnProperty.call(t,"presigned-expires")||(t["X-Amz-Content-Sha256"]="UNSIGNED-PAYLOAD")},noPresignedContentLength:function(e){if(void 0!==e.params.ContentLength)throw a.util.error(new Error,{code:"UnexpectedParameter",message:"ContentLength is not supported in pre-signed URLs."})},createBucket:function(e,t){"function"!==typeof e&&e||(t=t||e,e={});var i=this.endpoint.hostname,o=a.util.copy(e);return"us-east-1"===this.config.region||i===this.api.globalEndpoint||e.CreateBucketConfiguration||(o.CreateBucketConfiguration={LocationConstraint:this.config.region}),this.makeRequest("createBucket",o,t)},writeGetObjectResponse:function(e,t){var i=this.makeRequest("writeGetObjectResponse",a.util.copy(e),t),o=this.endpoint.hostname;return o=-1!==o.indexOf(this.config.region)?o.replace("s3.","s3-object-lambda."):o.replace("s3.","s3-object-lambda."+this.config.region+"."),i.httpRequest.endpoint=new a.Endpoint(o,this.config),i},upload:function(e,t,i){"function"===typeof t&&void 0===i&&(i=t,t=null),t=t||{},t=a.util.merge(t||{},{service:this,params:e});var o=new a.S3.ManagedUpload(t);return"function"===typeof i&&o.send(i),o},setExpiresString:function(e){e&&e.httpResponse&&e.httpResponse.headers&&"expires"in e.httpResponse.headers&&(e.httpResponse.headers.expiresstring=e.httpResponse.headers.expires);try{e&&e.httpResponse&&e.httpResponse.headers&&"expires"in e.httpResponse.headers&&a.util.date.parseTimestamp(e.httpResponse.headers.expires)}catch(t){console.log("AWS SDK","(warning)",t),delete e.httpResponse.headers.expires}}}),a.S3.addPromisesToClass=function(e){this.prototype.getSignedUrlPromise=a.util.promisifyMethod("getSignedUrl",e)},a.S3.deletePromisesFromClass=function(){delete this.prototype.getSignedUrlPromise},a.util.addPromises(a.S3)},22605:function(e,t,i){var a=i(8468),o=i(37139),n={isArnInParam:function(e,t){var i=((e.service.api.operations[e.operation]||{}).input||{}).members||{};return!(!e.params[t]||!i[t])&&a.util.ARN.validate(e.params[t])},validateArnService:function(e){var t=e._parsedArn;if("s3"!==t.service&&"s3-outposts"!==t.service&&"s3-object-lambda"!==t.service)throw a.util.error(new Error,{code:"InvalidARN",message:"expect 's3' or 's3-outposts' or 's3-object-lambda' in ARN service component"})},validateArnAccount:function(e){var t=e._parsedArn;if(!/[0-9]{12}/.exec(t.accountId))throw a.util.error(new Error,{code:"InvalidARN",message:'ARN accountID does not match regex "[0-9]{12}"'})},validateS3AccessPointArn:function(e){var t=e._parsedArn,i=t.resource["accesspoint".length];if(2!==t.resource.split(i).length)throw a.util.error(new Error,{code:"InvalidARN",message:"Access Point ARN should have one resource accesspoint/{accesspointName}"});var o=t.resource.split(i)[1],r=o+"-"+t.accountId;if(!n.dnsCompatibleBucketName(r)||r.match(/\./))throw a.util.error(new Error,{code:"InvalidARN",message:"Access point resource in ARN is not DNS compatible. Got "+o});e._parsedArn.accessPoint=o},validateOutpostsArn:function(e){var t=e._parsedArn;if(0!==t.resource.indexOf("outpost:")&&0!==t.resource.indexOf("outpost/"))throw a.util.error(new Error,{code:"InvalidARN",message:"ARN resource should begin with 'outpost/'"});var i=t.resource["outpost".length],o=t.resource.split(i)[1];if(!new RegExp(/^([a-zA-Z0-9]|[a-zA-Z0-9][a-zA-Z0-9-]{0,61}[a-zA-Z0-9])$/).test(o))throw a.util.error(new Error,{code:"InvalidARN",message:"Outpost resource in ARN is not DNS compatible. Got "+o});e._parsedArn.outpostId=o},validateOutpostsAccessPointArn:function(e){var t=e._parsedArn,i=t.resource["outpost".length];if(4!==t.resource.split(i).length)throw a.util.error(new Error,{code:"InvalidARN",message:"Outposts ARN should have two resources outpost/{outpostId}/accesspoint/{accesspointName}"});var o=t.resource.split(i)[3],r=o+"-"+t.accountId;if(!n.dnsCompatibleBucketName(r)||r.match(/\./))throw a.util.error(new Error,{code:"InvalidARN",message:"Access point resource in ARN is not DNS compatible. Got "+o});e._parsedArn.accessPoint=o},validateArnRegion:function(e,t){void 0===t&&(t={});var i=n.loadUseArnRegionConfig(e),r=e._parsedArn.region,s=e.service.config.region,l=e.service.config.useFipsEndpoint,c=t.allowFipsEndpoint||!1;if(!r){var p="ARN region is empty";throw"s3"===e._parsedArn.service&&(p+="\nYou may want to use multi-regional ARN. The feature is not supported in current SDK. You should consider switching to V3(https://github.com/aws/aws-sdk-js-v3)."),a.util.error(new Error,{code:"InvalidARN",message:p})}if(l&&!c)throw a.util.error(new Error,{code:"InvalidConfiguration",message:"ARN endpoint is not compatible with FIPS region"});if(r.indexOf("fips")>=0)throw a.util.error(new Error,{code:"InvalidConfiguration",message:"FIPS region not allowed in ARN"});if(!i&&r!==s)throw a.util.error(new Error,{code:"InvalidConfiguration",message:"Configured region conflicts with access point region"});if(i&&o.getEndpointSuffix(r)!==o.getEndpointSuffix(s))throw a.util.error(new Error,{code:"InvalidConfiguration",message:"Configured region and access point region not in same partition"});if(e.service.config.useAccelerateEndpoint)throw a.util.error(new Error,{code:"InvalidConfiguration",message:"useAccelerateEndpoint config is not supported with access point ARN"});if("s3-outposts"===e._parsedArn.service&&e.service.config.useDualstackEndpoint)throw a.util.error(new Error,{code:"InvalidConfiguration",message:"Dualstack is not supported with outposts access point ARN"})},loadUseArnRegionConfig:function(e){var t="AWS_S3_USE_ARN_REGION",i="s3_use_arn_region",o=!0,n=e.service._originalConfig||{};if(void 0!==e.service.config.s3UseArnRegion)return e.service.config.s3UseArnRegion;if(void 0!==n.s3UseArnRegion)o=!0===n.s3UseArnRegion;else if(a.util.isNode())if({NODE_ENV:"production",PUBLIC_URL:".",WDS_SOCKET_HOST:void 0,WDS_SOCKET_PATH:void 0,WDS_SOCKET_PORT:void 0,FAST_REFRESH:!0,REACT_APP_BUILD_TYPE:"development"}[t]){var r={NODE_ENV:"production",PUBLIC_URL:".",WDS_SOCKET_HOST:void 0,WDS_SOCKET_PATH:void 0,WDS_SOCKET_PORT:void 0,FAST_REFRESH:!0,REACT_APP_BUILD_TYPE:"development"}[t].trim().toLowerCase();if(["false","true"].indexOf(r)<0)throw a.util.error(new Error,{code:"InvalidConfiguration",message:t+" only accepts true or false. Got "+{NODE_ENV:"production",PUBLIC_URL:".",WDS_SOCKET_HOST:void 0,WDS_SOCKET_PATH:void 0,WDS_SOCKET_PORT:void 0,FAST_REFRESH:!0,REACT_APP_BUILD_TYPE:"development"}[t],retryable:!1});o="true"===r}else{var s={};try{s=a.util.getProfilesFromSharedConfig(a.util.iniLoader)[{NODE_ENV:"production",PUBLIC_URL:".",WDS_SOCKET_HOST:void 0,WDS_SOCKET_PATH:void 0,WDS_SOCKET_PORT:void 0,FAST_REFRESH:!0,REACT_APP_BUILD_TYPE:"development"}.AWS_PROFILE||a.util.defaultProfile]}catch(l){}if(s[i]){if(["false","true"].indexOf(s[i].trim().toLowerCase())<0)throw a.util.error(new Error,{code:"InvalidConfiguration",message:i+" only accepts true or false. Got "+s[i],retryable:!1});o="true"===s[i].trim().toLowerCase()}}return e.service.config.s3UseArnRegion=o,o},validatePopulateUriFromArn:function(e){if(e.service._originalConfig&&e.service._originalConfig.endpoint)throw a.util.error(new Error,{code:"InvalidConfiguration",message:"Custom endpoint is not compatible with access point ARN"});if(e.service.config.s3ForcePathStyle)throw a.util.error(new Error,{code:"InvalidConfiguration",message:"Cannot construct path-style endpoint with access point"})},dnsCompatibleBucketName:function(e){var t=e,i=new RegExp(/^[a-z0-9][a-z0-9\.\-]{1,61}[a-z0-9]$/),a=new RegExp(/(\d+\.){3}\d+/),o=new RegExp(/\.\./);return!(!t.match(i)||t.match(a)||t.match(o))}};e.exports=n},5387:function(e,t,i){var a=i(8468);a.util.update(a.SQS.prototype,{setupRequestListeners:function(e){e.addListener("build",this.buildEndpoint),e.service.config.computeChecksums&&("sendMessage"===e.operation?e.addListener("extractData",this.verifySendMessageChecksum):"sendMessageBatch"===e.operation?e.addListener("extractData",this.verifySendMessageBatchChecksum):"receiveMessage"===e.operation&&e.addListener("extractData",this.verifyReceiveMessageChecksum))},verifySendMessageChecksum:function(e){if(e.data){var t=e.data.MD5OfMessageBody,i=this.params.MessageBody,a=this.service.calculateChecksum(i);if(a!==t){var o='Got "'+e.data.MD5OfMessageBody+'", expecting "'+a+'".';this.service.throwInvalidChecksumError(e,[e.data.MessageId],o)}}},verifySendMessageBatchChecksum:function(e){if(e.data){var t=this.service,i={},o=[],n=[];a.util.arrayEach(e.data.Successful,(function(e){i[e.Id]=e})),a.util.arrayEach(this.params.Entries,(function(e){if(i[e.Id]){var a=i[e.Id].MD5OfMessageBody,r=e.MessageBody;t.isChecksumValid(a,r)||(o.push(e.Id),n.push(i[e.Id].MessageId))}})),o.length>0&&t.throwInvalidChecksumError(e,n,"Invalid messages: "+o.join(", "))}},verifyReceiveMessageChecksum:function(e){if(e.data){var t=this.service,i=[];a.util.arrayEach(e.data.Messages,(function(e){var a=e.MD5OfBody,o=e.Body;t.isChecksumValid(a,o)||i.push(e.MessageId)})),i.length>0&&t.throwInvalidChecksumError(e,i,"Invalid messages: "+i.join(", "))}},throwInvalidChecksumError:function(e,t,i){e.error=a.util.error(new Error,{retryable:!0,code:"InvalidChecksum",messageIds:t,message:e.request.operation+" returned an invalid MD5 response. "+i})},isChecksumValid:function(e,t){return this.calculateChecksum(t)===e},calculateChecksum:function(e){return a.util.crypto.md5(e,"hex")},buildEndpoint:function(e){var t=e.httpRequest.params.QueueUrl;if(t){e.httpRequest.endpoint=new a.Endpoint(t);var i=e.httpRequest.endpoint.host.match(/^sqs\.(.+?)\./);i&&(e.httpRequest.region=i[1])}}})},28177:function(e,t,i){var a=i(8468),o=i(21980);a.util.update(a.STS.prototype,{credentialsFrom:function(e,t){return e?(t||(t=new a.TemporaryCredentials),t.expired=!1,t.accessKeyId=e.Credentials.AccessKeyId,t.secretAccessKey=e.Credentials.SecretAccessKey,t.sessionToken=e.Credentials.SessionToken,t.expireTime=e.Credentials.Expiration,t):null},assumeRoleWithWebIdentity:function(e,t){return this.makeUnauthenticatedRequest("assumeRoleWithWebIdentity",e,t)},assumeRoleWithSAML:function(e,t){return this.makeUnauthenticatedRequest("assumeRoleWithSAML",e,t)},setupRequestListeners:function(e){e.addListener("validate",this.optInRegionalEndpoint,!0)},optInRegionalEndpoint:function(e){var t=e.service,i=t.config;if(i.stsRegionalEndpoints=o(t._originalConfig,{env:"AWS_STS_REGIONAL_ENDPOINTS",sharedConfig:"sts_regional_endpoints",clientConfig:"stsRegionalEndpoints"}),"regional"===i.stsRegionalEndpoints&&t.isGlobalEndpoint){if(!i.region)throw a.util.error(new Error,{code:"ConfigError",message:"Missing region in config"});var n=i.endpoint.indexOf(".amazonaws.com"),r=i.endpoint.substring(0,n)+"."+i.region+i.endpoint.substring(n);e.httpRequest.updateEndpoint(r),e.httpRequest.region=i.region}}})},11261:function(e,t,i){var a=i(8468);a.Signers.Bearer=a.util.inherit(a.Signers.RequestSigner,{constructor:function(e){a.Signers.RequestSigner.call(this,e)},addAuthorization:function(e){this.request.headers.Authorization="Bearer "+e.token}})},10062:function(e,t,i){var a=i(8468),o=a.util.inherit,n="presigned-expires";function r(e){var t=e.httpRequest.headers[n],i=e.service.getSignerClass(e);if(delete e.httpRequest.headers["User-Agent"],delete e.httpRequest.headers["X-Amz-User-Agent"],i===a.Signers.V4){if(t>604800){throw a.util.error(new Error,{code:"InvalidExpiryTime",message:"Presigning does not support expiry time greater than a week with SigV4 signing.",retryable:!1})}e.httpRequest.headers[n]=t}else{if(i!==a.Signers.S3)throw a.util.error(new Error,{message:"Presigning only supports S3 or SigV4 signing.",code:"UnsupportedSigner",retryable:!1});var o=e.service?e.service.getSkewCorrectedDate():a.util.date.getDate();e.httpRequest.headers[n]=parseInt(a.util.date.unixTimestamp(o)+t,10).toString()}}function s(e){var t=e.httpRequest.endpoint,i=a.util.urlParse(e.httpRequest.path),o={};i.search&&(o=a.util.queryStringParse(i.search.substr(1)));var r=e.httpRequest.headers.Authorization.split(" 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a=i(8468),o=a.util.inherit;a.Signers.S3=o(a.Signers.RequestSigner,{subResources:{acl:1,accelerate:1,analytics:1,cors:1,lifecycle:1,delete:1,inventory:1,location:1,logging:1,metrics:1,notification:1,partNumber:1,policy:1,requestPayment:1,replication:1,restore:1,tagging:1,torrent:1,uploadId:1,uploads:1,versionId:1,versioning:1,versions:1,website:1},responseHeaders:{"response-content-type":1,"response-content-language":1,"response-expires":1,"response-cache-control":1,"response-content-disposition":1,"response-content-encoding":1},addAuthorization:function(e,t){this.request.headers["presigned-expires"]||(this.request.headers["X-Amz-Date"]=a.util.date.rfc822(t)),e.sessionToken&&(this.request.headers["x-amz-security-token"]=e.sessionToken);var i=this.sign(e.secretAccessKey,this.stringToSign()),o="AWS "+e.accessKeyId+":"+i;this.request.headers.Authorization=o},stringToSign:function(){var e=this.request,t=[];t.push(e.method),t.push(e.headers["Content-MD5"]||""),t.push(e.headers["Content-Type"]||""),t.push(e.headers["presigned-expires"]||"");var i=this.canonicalizedAmzHeaders();return i&&t.push(i),t.push(this.canonicalizedResource()),t.join("\n")},canonicalizedAmzHeaders:function(){var e=[];a.util.each(this.request.headers,(function(t){t.match(/^x-amz-/i)&&e.push(t)})),e.sort((function(e,t){return e.toLowerCase()<t.toLowerCase()?-1:1}));var t=[];return a.util.arrayEach.call(this,e,(function(e){t.push(e.toLowerCase()+":"+String(this.request.headers[e]))})),t.join("\n")},canonicalizedResource:function(){var e=this.request,t=e.path.split("?"),i=t[0],o=t[1],n="";if(e.virtualHostedBucket&&(n+="/"+e.virtualHostedBucket),n+=i,o){var r=[];a.util.arrayEach.call(this,o.split("&"),(function(e){var t=e.split("=")[0],i=e.split("=")[1];if(this.subResources[t]||this.responseHeaders[t]){var a={name:t};void 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a.util.crypto.hmac(e.secretAccessKey,this.stringToSign(),"base64")},stringToSign:function(){var e=[];return e.push(this.request.method),e.push(this.request.endpoint.host.toLowerCase()),e.push(this.request.pathname()),e.push(a.util.queryParamsToString(this.request.params)),e.join("\n")}}),e.exports=a.Signers.V2},1469:function(e,t,i){var a=i(8468),o=a.util.inherit;a.Signers.V3=o(a.Signers.RequestSigner,{addAuthorization:function(e,t){var i=a.util.date.rfc822(t);this.request.headers["X-Amz-Date"]=i,e.sessionToken&&(this.request.headers["x-amz-security-token"]=e.sessionToken),this.request.headers["X-Amzn-Authorization"]=this.authorization(e,i)},authorization:function(e){return"AWS3 AWSAccessKeyId="+e.accessKeyId+",Algorithm=HmacSHA256,SignedHeaders="+this.signedHeaders()+",Signature="+this.signature(e)},signedHeaders:function(){var e=[];return a.util.arrayEach(this.headersToSign(),(function(t){e.push(t.toLowerCase())})),e.sort().join(";")},canonicalHeaders:function(){var e=this.request.headers,t=[];return a.util.arrayEach(this.headersToSign(),(function(i){t.push(i.toLowerCase().trim()+":"+String(e[i]).trim())})),t.sort().join("\n")+"\n"},headersToSign:function(){var e=[];return a.util.each(this.request.headers,(function(t){("Host"===t||"Content-Encoding"===t||t.match(/^X-Amz/i))&&e.push(t)})),e},signature:function(e){return a.util.crypto.hmac(e.secretAccessKey,this.stringToSign(),"base64")},stringToSign:function(){var e=[];return e.push(this.request.method),e.push("/"),e.push(""),e.push(this.canonicalHeaders()),e.push(this.request.body),a.util.crypto.sha256(e.join("\n"))}}),e.exports=a.Signers.V3},98686:function(e,t,i){var a=i(8468),o=a.util.inherit;i(1469),a.Signers.V3Https=o(a.Signers.V3,{authorization:function(e){return"AWS3-HTTPS AWSAccessKeyId="+e.accessKeyId+",Algorithm=HmacSHA256,Signature="+this.signature(e)},stringToSign:function(){return this.request.headers["X-Amz-Date"]}}),e.exports=a.Signers.V3Https},64355:function(e,t,i){var a=i(8468),o=i(94966),n=a.util.inherit,r="presigned-expires";a.Signers.V4=n(a.Signers.RequestSigner,{constructor:function(e,t,i){a.Signers.RequestSigner.call(this,e),this.serviceName=t,i=i||{},this.signatureCache="boolean"!==typeof i.signatureCache||i.signatureCache,this.operation=i.operation,this.signatureVersion=i.signatureVersion},algorithm:"AWS4-HMAC-SHA256",addAuthorization:function(e,t){var i=a.util.date.iso8601(t).replace(/[:\-]|\.\d{3}/g,"");this.isPresigned()?this.updateForPresigned(e,i):this.addHeaders(e,i),this.request.headers.Authorization=this.authorization(e,i)},addHeaders:function(e,t){this.request.headers["X-Amz-Date"]=t,e.sessionToken&&(this.request.headers["x-amz-security-token"]=e.sessionToken)},updateForPresigned:function(e,t){var i=this.credentialString(t),o={"X-Amz-Date":t,"X-Amz-Algorithm":this.algorithm,"X-Amz-Credential":e.accessKeyId+"/"+i,"X-Amz-Expires":this.request.headers[r],"X-Amz-SignedHeaders":this.signedHeaders()};e.sessionToken&&(o["X-Amz-Security-Token"]=e.sessionToken),this.request.headers["Content-Type"]&&(o["Content-Type"]=this.request.headers["Content-Type"]),this.request.headers["Content-MD5"]&&(o["Content-MD5"]=this.request.headers["Content-MD5"]),this.request.headers["Cache-Control"]&&(o["Cache-Control"]=this.request.headers["Cache-Control"]),a.util.each.call(this,this.request.headers,(function(e,t){if(e!==r&&this.isSignableHeader(e)){var i=e.toLowerCase();0===i.indexOf("x-amz-meta-")?o[i]=t:0===i.indexOf("x-amz-")&&(o[e]=t)}}));var n=this.request.path.indexOf("?")>=0?"&":"?";this.request.path+=n+a.util.queryParamsToString(o)},authorization:function(e,t){var i=[],a=this.credentialString(t);return i.push(this.algorithm+" Credential="+e.accessKeyId+"/"+a),i.push("SignedHeaders="+this.signedHeaders()),i.push("Signature="+this.signature(e,t)),i.join(", ")},signature:function(e,t){var i=o.getSigningKey(e,t.substr(0,8),this.request.region,this.serviceName,this.signatureCache);return a.util.crypto.hmac(i,this.stringToSign(t),"hex")},stringToSign:function(e){var t=[];return t.push("AWS4-HMAC-SHA256"),t.push(e),t.push(this.credentialString(e)),t.push(this.hexEncodedHash(this.canonicalString())),t.join("\n")},canonicalString:function(){var e=[],t=this.request.pathname();return"s3"!==this.serviceName&&"s3v4"!==this.signatureVersion&&(t=a.util.uriEscapePath(t)),e.push(this.request.method),e.push(t),e.push(this.request.search()),e.push(this.canonicalHeaders()+"\n"),e.push(this.signedHeaders()),e.push(this.hexEncodedBodyHash()),e.join("\n")},canonicalHeaders:function(){var e=[];a.util.each.call(this,this.request.headers,(function(t,i){e.push([t,i])})),e.sort((function(e,t){return e[0].toLowerCase()<t[0].toLowerCase()?-1:1}));var t=[];return a.util.arrayEach.call(this,e,(function(e){var i=e[0].toLowerCase();if(this.isSignableHeader(i)){var o=e[1];if("undefined"===typeof o||null===o||"function"!==typeof o.toString)throw a.util.error(new Error("Header "+i+" contains invalid value"),{code:"InvalidHeader"});t.push(i+":"+this.canonicalHeaderValues(o.toString()))}})),t.join("\n")},canonicalHeaderValues:function(e){return e.replace(/\s+/g," ").replace(/^\s+|\s+$/g,"")},signedHeaders:function(){var e=[];return a.util.each.call(this,this.request.headers,(function(t){t=t.toLowerCase(),this.isSignableHeader(t)&&e.push(t)})),e.sort().join(";")},credentialString:function(e){return o.createScope(e.substr(0,8),this.request.region,this.serviceName)},hexEncodedHash:function(e){return a.util.crypto.sha256(e,"hex")},hexEncodedBodyHash:function(){var e=this.request;return this.isPresigned()&&["s3","s3-object-lambda"].indexOf(this.serviceName)>-1&&!e.body?"UNSIGNED-PAYLOAD":e.headers["X-Amz-Content-Sha256"]?e.headers["X-Amz-Content-Sha256"]:this.hexEncodedHash(this.request.body||"")},unsignableHeaders:["authorization","content-type","content-length","user-agent",r,"expect","x-amzn-trace-id"],isSignableHeader:function(e){return 0===e.toLowerCase().indexOf("x-amz-")||this.unsignableHeaders.indexOf(e)<0},isPresigned:function(){return!!this.request.headers[r]}}),e.exports=a.Signers.V4},94966:function(e,t,i){var a=i(8468),o={},n=[],r="aws4_request";e.exports={createScope:function(e,t,i){return[e.substr(0,8),t,i,r].join("/")},getSigningKey:function(e,t,i,s,l){var c=[a.util.crypto.hmac(e.secretAccessKey,e.accessKeyId,"base64"),t,i,s].join("_");if((l=!1!==l)&&c in o)return o[c];var p=a.util.crypto.hmac("AWS4"+e.secretAccessKey,t,"buffer"),h=a.util.crypto.hmac(p,i,"buffer"),$=a.util.crypto.hmac(h,s,"buffer"),d=a.util.crypto.hmac($,r,"buffer");return l&&(o[c]=d,n.push(c),n.length>50&&delete o[n.shift()]),d},emptyCache:function(){o={},n=[]}}},41496:function(e){function t(e,t){this.currentState=t||null,this.states=e||{}}t.prototype.runTo=function(e,t,i,a){"function"===typeof e&&(a=i,i=t,t=e,e=null);var o=this,n=o.states[o.currentState];n.fn.call(i||o,a,(function(a){if(a){if(!n.fail)return t?t.call(i,a):null;o.currentState=n.fail}else{if(!n.accept)return t?t.call(i):null;o.currentState=n.accept}if(o.currentState===e)return t?t.call(i,a):null;o.runTo(e,t,i,a)}))},t.prototype.addState=function(e,t,i,a){return"function"===typeof t?(a=t,t=null,i=null):"function"===typeof i&&(a=i,i=null),this.currentState||(this.currentState=e),this.states[e]={accept:t,fail:i,fn:a},this},e.exports=t},23657:function(e,t,i){var a,o={environment:"nodejs",engine:function(){if(o.isBrowser()&&"undefined"!==typeof navigator)return navigator.userAgent;var e=process.platform+"/"+process.version;return{NODE_ENV:"production",PUBLIC_URL:".",WDS_SOCKET_HOST:void 0,WDS_SOCKET_PATH:void 0,WDS_SOCKET_PORT:void 0,FAST_REFRESH:!0,REACT_APP_BUILD_TYPE:"development"}.AWS_EXECUTION_ENV&&(e+=" exec-env/"+{NODE_ENV:"production",PUBLIC_URL:".",WDS_SOCKET_HOST:void 0,WDS_SOCKET_PATH:void 0,WDS_SOCKET_PORT:void 0,FAST_REFRESH:!0,REACT_APP_BUILD_TYPE:"development"}.AWS_EXECUTION_ENV),e},userAgent:function(){var e=o.environment,t="aws-sdk-"+e+"/"+i(8468).VERSION;return"nodejs"===e&&(t+=" "+o.engine()),t},uriEscape:function(e){var t=encodeURIComponent(e);return t=(t=t.replace(/[^A-Za-z0-9_.~\-%]+/g,escape)).replace(/[*]/g,(function(e){return"%"+e.charCodeAt(0).toString(16).toUpperCase()}))},uriEscapePath:function(e){var t=[];return o.arrayEach(e.split("/"),(function(e){t.push(o.uriEscape(e))})),t.join("/")},urlParse:function(e){return o.url.parse(e)},urlFormat:function(e){return o.url.format(e)},queryStringParse:function(e){return o.querystring.parse(e)},queryParamsToString:function(e){var t=[],i=o.uriEscape,a=Object.keys(e).sort();return o.arrayEach(a,(function(a){var n=e[a],r=i(a),s=r+"=";if(Array.isArray(n)){var l=[];o.arrayEach(n,(function(e){l.push(i(e))})),s=r+"="+l.sort().join("&"+r+"=")}else void 0!==n&&null!==n&&(s=r+"="+i(n));t.push(s)})),t.join("&")},readFileSync:function(e){return o.isBrowser()?null:i(28022).readFileSync(e,"utf-8")},base64:{encode:function(e){if("number"===typeof e)throw o.error(new Error("Cannot base64 encode number "+e));return null===e||"undefined"===typeof e?e:o.buffer.toBuffer(e).toString("base64")},decode:function(e){if("number"===typeof e)throw o.error(new Error("Cannot base64 decode number "+e));return null===e||"undefined"===typeof e?e:o.buffer.toBuffer(e,"base64")}},buffer:{toBuffer:function(e,t){return"function"===typeof o.Buffer.from&&o.Buffer.from!==Uint8Array.from?o.Buffer.from(e,t):new o.Buffer(e,t)},alloc:function(e,t,i){if("number"!==typeof e)throw new Error("size passed to alloc must be a number.");if("function"===typeof o.Buffer.alloc)return o.Buffer.alloc(e,t,i);var a=new 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l(e,t){var i=t>>24&255,a=t>>16&255,o=t>>8&255,n=255&t,r=e.sbox[0][i]+e.sbox[1][a];return r^=e.sbox[2][o],r+=e.sbox[3][n]}function c(e,t,i){for(var o,n=t,r=i,s=0;s<a;++s)o=n^=e.pbox[s],n=r=l(e,n)^r,r=o;return o=n,n=r,r=o,r^=e.pbox[a],{left:n^=e.pbox[a+1],right:r}}function p(e,t,i){for(var o,n=t,r=i,s=a+1;s>1;--s)o=n^=e.pbox[s],n=r=l(e,n)^r,r=o;return o=n,n=r,r=o,r^=e.pbox[1],{left:n^=e.pbox[0],right:r}}function h(e,t,i){for(var n=0;n<4;n++){e.sbox[n]=[];for(var s=0;s<256;s++)e.sbox[n][s]=r[n][s]}for(var l=0,p=0;p<a+2;p++)e.pbox[p]=o[p]^t[l],++l>=i&&(l=0);for(var h=0,$=0,d=0,u=0;u<a+2;u+=2)h=(d=c(e,h,$)).left,$=d.right,e.pbox[u]=h,e.pbox[u+1]=$;for(var m=0;m<4;m++)for(var b=0;b<256;b+=2)h=(d=c(e,h,$)).left,$=d.right,e.sbox[m][b]=h,e.sbox[m][b+1]=$;return!0}var $=i.Blowfish=t.extend({_doReset:function(){if(this._keyPriorReset!==this._key){var e=this._keyPriorReset=this._key,t=e.words,i=e.sigBytes/4;h(s,t,i)}},encryptBlock:function(e,t){var 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e(e){return"string"==typeof e?g:y}return function(t){return{encrypt:function(i,a,o){return e(a).encrypt(t,i,a,o)},decrypt:function(i,a,o){return e(a).decrypt(t,i,a,o)}}}}()}),h=(i.StreamCipher=p.extend({_doFinalize:function(){return this._process(!0)},blockSize:1}),t.mode={}),$=i.BlockCipherMode=a.extend({createEncryptor:function(e,t){return this.Encryptor.create(e,t)},createDecryptor:function(e,t){return this.Decryptor.create(e,t)},init:function(e,t){this._cipher=e,this._iv=t}}),d=h.CBC=function(){var t=$.extend();function i(t,i,a){var o,n=this._iv;n?(o=n,this._iv=e):o=this._prevBlock;for(var r=0;r<a;r++)t[i+r]^=o[r]}return t.Encryptor=t.extend({processBlock:function(e,t){var a=this._cipher,o=a.blockSize;i.call(this,e,t,o),a.encryptBlock(e,t),this._prevBlock=e.slice(t,t+o)}}),t.Decryptor=t.extend({processBlock:function(e,t){var a=this._cipher,o=a.blockSize,n=e.slice(t,t+o);a.decryptBlock(e,t),i.call(this,e,t,o),this._prevBlock=n}}),t}(),u=(t.pad={}).Pkcs7={pad:function(e,t){for(var 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r=(t[n>>>2]>>>24-n%4*8&255)<<16|(t[n+1>>>2]>>>24-(n+1)%4*8&255)<<8|t[n+2>>>2]>>>24-(n+2)%4*8&255,s=0;s<4&&n+.75*s<i;s++)o.push(a.charAt(r>>>6*(3-s)&63));var l=a.charAt(64);if(l)for(;o.length%4;)o.push(l);return o.join("")},parse:function(e){var t=e.length,a=this._map,o=this._reverseMap;if(!o){o=this._reverseMap=[];for(var n=0;n<a.length;n++)o[a.charCodeAt(n)]=n}var r=a.charAt(64);if(r){var s=e.indexOf(r);-1!==s&&(t=s)}return i(e,t,o)},_map:"ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789+/="}}(),o.enc.Base64)}()},4728:function(e,t,i){!function(t,a){var o;e.exports=(o=i(68926),function(){var e=o,t=e.lib.WordArray;function i(e,i,a){for(var o=[],n=0,r=0;r<i;r++)if(r%4){var s=a[e.charCodeAt(r-1)]<<r%4*2|a[e.charCodeAt(r)]>>>6-r%4*2;o[n>>>2]|=s<<24-n%4*8,n++}return t.create(o,n)}e.enc.Base64url={stringify:function(e,t){void 0===t&&(t=!0);var i=e.words,a=e.sigBytes,o=t?this._safe_map:this._map;e.clamp();for(var n=[],r=0;r<a;r+=3)for(var s=(i[r>>>2]>>>24-r%4*8&255)<<16|(i[r+1>>>2]>>>24-(r+1)%4*8&255)<<8|i[r+2>>>2]>>>24-(r+2)%4*8&255,l=0;l<4&&r+.75*l<a;l++)n.push(o.charAt(s>>>6*(3-l)&63));var c=o.charAt(64);if(c)for(;n.length%4;)n.push(c);return n.join("")},parse:function(e,t){void 0===t&&(t=!0);var a=e.length,o=t?this._safe_map:this._map,n=this._reverseMap;if(!n){n=this._reverseMap=[];for(var r=0;r<o.length;r++)n[o.charCodeAt(r)]=r}var s=o.charAt(64);if(s){var l=e.indexOf(s);-1!==l&&(a=l)}return i(e,a,n)},_map:"ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789+/=",_safe_map:"ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789-_"}}(),o.enc.Base64url)}()},91551:function(e,t,i){!function(t,a){var o;e.exports=(o=i(68926),function(){var e=o,t=e.lib.WordArray,i=e.enc;function a(e){return e<<8&4278255360|e>>>8&16711935}i.Utf16=i.Utf16BE={stringify:function(e){for(var t=e.words,i=e.sigBytes,a=[],o=0;o<i;o+=2){var n=t[o>>>2]>>>16-o%4*8&65535;a.push(String.fromCharCode(n))}return 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i=e.byteLength,a=[],o=0;o<i;o++)a[o>>>2]|=e[o]<<24-o%4*8;t.call(this,a,i)}else t.apply(this,arguments)};i.prototype=e}}(),o.lib.WordArray)}()},93074:function(e,t,i){!function(t,a){var o;e.exports=(o=i(68926),function(e){var t=o,i=t.lib,a=i.WordArray,n=i.Hasher,r=t.algo,s=[];!function(){for(var t=0;t<64;t++)s[t]=4294967296*e.abs(e.sin(t+1))|0}();var l=r.MD5=n.extend({_doReset:function(){this._hash=new a.init([1732584193,4023233417,2562383102,271733878])},_doProcessBlock:function(e,t){for(var i=0;i<16;i++){var a=t+i,o=e[a];e[a]=16711935&(o<<8|o>>>24)|4278255360&(o<<24|o>>>8)}var n=this._hash.words,r=e[t+0],l=e[t+1],d=e[t+2],u=e[t+3],m=e[t+4],b=e[t+5],y=e[t+6],f=e[t+7],g=e[t+8],x=e[t+9],v=e[t+10],T=e[t+11],A=e[t+12],w=e[t+13],C=e[t+14],O=e[t+15],S=n[0],B=n[1],_=n[2],z=n[3];S=c(S,B,_,z,r,7,s[0]),z=c(z,S,B,_,l,12,s[1]),_=c(_,z,S,B,d,17,s[2]),B=c(B,_,z,S,u,22,s[3]),S=c(S,B,_,z,m,7,s[4]),z=c(z,S,B,_,b,12,s[5]),_=c(_,z,S,B,y,17,s[6]),B=c(B,_,z,S,f,22,s[7]),S=c(S,B,_,z,g,7,s[8]),z=c(z,S,B,_,x,12,s[9]),_=c(_,z,S,B,v,17,s[10]),B=c(B,_,z,S,T,22,s[11]),S=c(S,B,_,z,A,7,s[12]),z=c(z,S,B,_,w,12,s[13]),_=c(_,z,S,B,C,17,s[14]),S=p(S,B=c(B,_,z,S,O,22,s[15]),_,z,l,5,s[16]),z=p(z,S,B,_,y,9,s[17]),_=p(_,z,S,B,T,14,s[18]),B=p(B,_,z,S,r,20,s[19]),S=p(S,B,_,z,b,5,s[20]),z=p(z,S,B,_,v,9,s[21]),_=p(_,z,S,B,O,14,s[22]),B=p(B,_,z,S,m,20,s[23]),S=p(S,B,_,z,x,5,s[24]),z=p(z,S,B,_,C,9,s[25]),_=p(_,z,S,B,u,14,s[26]),B=p(B,_,z,S,g,20,s[27]),S=p(S,B,_,z,w,5,s[28]),z=p(z,S,B,_,d,9,s[29]),_=p(_,z,S,B,f,14,s[30]),S=h(S,B=p(B,_,z,S,A,20,s[31]),_,z,b,4,s[32]),z=h(z,S,B,_,g,11,s[33]),_=h(_,z,S,B,T,16,s[34]),B=h(B,_,z,S,C,23,s[35]),S=h(S,B,_,z,l,4,s[36]),z=h(z,S,B,_,m,11,s[37]),_=h(_,z,S,B,f,16,s[38]),B=h(B,_,z,S,v,23,s[39]),S=h(S,B,_,z,w,4,s[40]),z=h(z,S,B,_,r,11,s[41]),_=h(_,z,S,B,u,16,s[42]),B=h(B,_,z,S,y,23,s[43]),S=h(S,B,_,z,x,4,s[44]),z=h(z,S,B,_,A,11,s[45]),_=h(_,z,S,B,O,16,s[46]),S=$(S,B=h(B,_,z,S,d,23,s[47]),_,z,r,6,s[48]),z=$(z,S,B,_,f,10,s[49]),_=$(_,z,S,B,C,15,s[50]),B=$(B,_,z,S,b,21,s[51]),S=$(S,B,_,z,A,6,s[52]),z=$(z,S,B,_,u,10,s[53]),_=$(_,z,S,B,v,15,s[54]),B=$(B,_,z,S,l,21,s[55]),S=$(S,B,_,z,g,6,s[56]),z=$(z,S,B,_,O,10,s[57]),_=$(_,z,S,B,y,15,s[58]),B=$(B,_,z,S,w,21,s[59]),S=$(S,B,_,z,m,6,s[60]),z=$(z,S,B,_,T,10,s[61]),_=$(_,z,S,B,d,15,s[62]),B=$(B,_,z,S,x,21,s[63]),n[0]=n[0]+S|0,n[1]=n[1]+B|0,n[2]=n[2]+_|0,n[3]=n[3]+z|0},_doFinalize:function(){var t=this._data,i=t.words,a=8*this._nDataBytes,o=8*t.sigBytes;i[o>>>5]|=128<<24-o%32;var n=e.floor(a/4294967296),r=a;i[15+(o+64>>>9<<4)]=16711935&(n<<8|n>>>24)|4278255360&(n<<24|n>>>8),i[14+(o+64>>>9<<4)]=16711935&(r<<8|r>>>24)|4278255360&(r<<24|r>>>8),t.sigBytes=4*(i.length+1),this._process();for(var s=this._hash,l=s.words,c=0;c<4;c++){var p=l[c];l[c]=16711935&(p<<8|p>>>24)|4278255360&(p<<24|p>>>8)}return s},clone:function(){var e=n.clone.call(this);return e._hash=this._hash.clone(),e}});function c(e,t,i,a,o,n,r){var s=e+(t&i|~t&a)+o+r;return(s<<n|s>>>32-n)+t}function p(e,t,i,a,o,n,r){var s=e+(t&a|i&~a)+o+r;return(s<<n|s>>>32-n)+t}function h(e,t,i,a,o,n,r){var s=e+(t^i^a)+o+r;return(s<<n|s>>>32-n)+t}function $(e,t,i,a,o,n,r){var s=e+(i^(t|~a))+o+r;return(s<<n|s>>>32-n)+t}t.MD5=n._createHelper(l),t.HmacMD5=n._createHmacHelper(l)}(Math),o.MD5)}()},25152:function(e,t,i){!function(t,a,o){var n;e.exports=(n=i(68926),i(33650),n.mode.CFB=function(){var e=n.lib.BlockCipherMode.extend();function t(e,t,i,a){var o,n=this._iv;n?(o=n.slice(0),this._iv=void 0):o=this._prevBlock,a.encryptBlock(o,0);for(var r=0;r<i;r++)e[t+r]^=o[r]}return e.Encryptor=e.extend({processBlock:function(e,i){var a=this._cipher,o=a.blockSize;t.call(this,e,i,o,a),this._prevBlock=e.slice(i,i+o)}}),e.Decryptor=e.extend({processBlock:function(e,i){var a=this._cipher,o=a.blockSize,n=e.slice(i,i+o);t.call(this,e,i,o,a),this._prevBlock=n}}),e}(),n.mode.CFB)}()},87223:function(e,t,i){!function(t,a,o){var n;e.exports=(n=i(68926),i(33650),n.mode.CTRGladman=function(){var e=n.lib.BlockCipherMode.extend();function t(e){if(255===(e>>24&255)){var t=e>>16&255,i=e>>8&255,a=255&e;255===t?(t=0,255===i?(i=0,255===a?a=0:++a):++i):++t,e=0,e+=t<<16,e+=i<<8,e+=a}else e+=1<<24;return e}function i(e){return 0===(e[0]=t(e[0]))&&(e[1]=t(e[1])),e}var a=e.Encryptor=e.extend({processBlock:function(e,t){var a=this._cipher,o=a.blockSize,n=this._iv,r=this._counter;n&&(r=this._counter=n.slice(0),this._iv=void 0),i(r);var s=r.slice(0);a.encryptBlock(s,0);for(var l=0;l<o;l++)e[t+l]^=s[l]}});return e.Decryptor=a,e}(),n.mode.CTRGladman)}()},2992:function(e,t,i){!function(t,a,o){var n;e.exports=(n=i(68926),i(33650),n.mode.CTR=function(){var e=n.lib.BlockCipherMode.extend(),t=e.Encryptor=e.extend({processBlock:function(e,t){var i=this._cipher,a=i.blockSize,o=this._iv,n=this._counter;o&&(n=this._counter=o.slice(0),this._iv=void 0);var r=n.slice(0);i.encryptBlock(r,0),n[a-1]=n[a-1]+1|0;for(var s=0;s<a;s++)e[t+s]^=r[s]}});return e.Decryptor=t,e}(),n.mode.CTR)}()},70130:function(e,t,i){!function(t,a,o){var n;e.exports=(n=i(68926),i(33650),n.mode.ECB=function(){var e=n.lib.BlockCipherMode.extend();return e.Encryptor=e.extend({processBlock:function(e,t){this._cipher.encryptBlock(e,t)}}),e.Decryptor=e.extend({processBlock:function(e,t){this._cipher.decryptBlock(e,t)}}),e}(),n.mode.ECB)}()},15858:function(e,t,i){!function(t,a,o){var n;e.exports=(n=i(68926),i(33650),n.mode.OFB=function(){var e=n.lib.BlockCipherMode.extend(),t=e.Encryptor=e.extend({processBlock:function(e,t){var i=this._cipher,a=i.blockSize,o=this._iv,n=this._keystream;o&&(n=this._keystream=o.slice(0),this._iv=void 0),i.encryptBlock(n,0);for(var r=0;r<a;r++)e[t+r]^=n[r]}});return e.Decryptor=t,e}(),n.mode.OFB)}()},12409:function(e,t,i){!function(t,a,o){var n;e.exports=(n=i(68926),i(33650),n.pad.AnsiX923={pad:function(e,t){var i=e.sigBytes,a=4*t,o=a-i%a,n=i+o-1;e.clamp(),e.words[n>>>2]|=o<<24-n%4*8,e.sigBytes+=o},unpad:function(e){var t=255&e.words[e.sigBytes-1>>>2];e.sigBytes-=t}},n.pad.Ansix923)}()},24823:function(e,t,i){!function(t,a,o){var n;e.exports=(n=i(68926),i(33650),n.pad.Iso10126={pad:function(e,t){var i=4*t,a=i-e.sigBytes%i;e.concat(n.lib.WordArray.random(a-1)).concat(n.lib.WordArray.create([a<<24],1))},unpad:function(e){var t=255&e.words[e.sigBytes-1>>>2];e.sigBytes-=t}},n.pad.Iso10126)}()},36644:function(e,t,i){!function(t,a,o){var n;e.exports=(n=i(68926),i(33650),n.pad.Iso97971={pad:function(e,t){e.concat(n.lib.WordArray.create([2147483648],1)),n.pad.ZeroPadding.pad(e,t)},unpad:function(e){n.pad.ZeroPadding.unpad(e),e.sigBytes--}},n.pad.Iso97971)}()},28413:function(e,t,i){!function(t,a,o){var n;e.exports=(n=i(68926),i(33650),n.pad.NoPadding={pad:function(){},unpad:function(){}},n.pad.NoPadding)}()},51181:function(e,t,i){!function(t,a,o){var n;e.exports=(n=i(68926),i(33650),n.pad.ZeroPadding={pad:function(e,t){var i=4*t;e.clamp(),e.sigBytes+=i-(e.sigBytes%i||i)},unpad:function(e){var t=e.words,i=e.sigBytes-1;for(i=e.sigBytes-1;i>=0;i--)if(t[i>>>2]>>>24-i%4*8&255){e.sigBytes=i+1;break}}},n.pad.ZeroPadding)}()},93822:function(e,t,i){!function(t,a,o){var n;e.exports=(n=i(68926),i(29517),i(45086),function(){var e=n,t=e.lib,i=t.Base,a=t.WordArray,o=e.algo,r=o.SHA256,s=o.HMAC,l=o.PBKDF2=i.extend({cfg:i.extend({keySize:4,hasher:r,iterations:25e4}),init:function(e){this.cfg=this.cfg.extend(e)},compute:function(e,t){for(var i=this.cfg,o=s.create(i.hasher,e),n=a.create(),r=a.create([1]),l=n.words,c=r.words,p=i.keySize,h=i.iterations;l.length<p;){var $=o.update(t).finalize(r);o.reset();for(var d=$.words,u=d.length,m=$,b=1;b<h;b++){m=o.finalize(m),o.reset();for(var y=m.words,f=0;f<u;f++)d[f]^=y[f]}n.concat($),c[0]++}return n.sigBytes=4*p,n}});e.PBKDF2=function(e,t,i){return l.create(i).compute(e,t)}}(),n.PBKDF2)}()},42130:function(e,t,i){!function(t,a,o){var n;e.exports=(n=i(68926),i(53713),i(93074),i(68228),i(33650),function(){var e=n,t=e.lib.StreamCipher,i=e.algo,a=[],o=[],r=[],s=i.RabbitLegacy=t.extend({_doReset:function(){var e=this._key.words,t=this.cfg.iv,i=this._X=[e[0],e[3]<<16|e[2]>>>16,e[1],e[0]<<16|e[3]>>>16,e[2],e[1]<<16|e[0]>>>16,e[3],e[2]<<16|e[1]>>>16],a=this._C=[e[2]<<16|e[2]>>>16,4294901760&e[0]|65535&e[1],e[3]<<16|e[3]>>>16,4294901760&e[1]|65535&e[2],e[0]<<16|e[0]>>>16,4294901760&e[2]|65535&e[3],e[1]<<16|e[1]>>>16,4294901760&e[3]|65535&e[0]];this._b=0;for(var o=0;o<4;o++)l.call(this);for(o=0;o<8;o++)a[o]^=i[o+4&7];if(t){var n=t.words,r=n[0],s=n[1],c=16711935&(r<<8|r>>>24)|4278255360&(r<<24|r>>>8),p=16711935&(s<<8|s>>>24)|4278255360&(s<<24|s>>>8),h=c>>>16|4294901760&p,$=p<<16|65535&c;for(a[0]^=c,a[1]^=h,a[2]^=p,a[3]^=$,a[4]^=c,a[5]^=h,a[6]^=p,a[7]^=$,o=0;o<4;o++)l.call(this)}},_doProcessBlock:function(e,t){var i=this._X;l.call(this),a[0]=i[0]^i[5]>>>16^i[3]<<16,a[1]=i[2]^i[7]>>>16^i[5]<<16,a[2]=i[4]^i[1]>>>16^i[7]<<16,a[3]=i[6]^i[3]>>>16^i[1]<<16;for(var o=0;o<4;o++)a[o]=16711935&(a[o]<<8|a[o]>>>24)|4278255360&(a[o]<<24|a[o]>>>8),e[t+o]^=a[o]},blockSize:4,ivSize:2});function l(){for(var e=this._X,t=this._C,i=0;i<8;i++)o[i]=t[i];for(t[0]=t[0]+1295307597+this._b|0,t[1]=t[1]+3545052371+(t[0]>>>0<o[0]>>>0?1:0)|0,t[2]=t[2]+886263092+(t[1]>>>0<o[1]>>>0?1:0)|0,t[3]=t[3]+1295307597+(t[2]>>>0<o[2]>>>0?1:0)|0,t[4]=t[4]+3545052371+(t[3]>>>0<o[3]>>>0?1:0)|0,t[5]=t[5]+886263092+(t[4]>>>0<o[4]>>>0?1:0)|0,t[6]=t[6]+1295307597+(t[5]>>>0<o[5]>>>0?1:0)|0,t[7]=t[7]+3545052371+(t[6]>>>0<o[6]>>>0?1:0)|0,this._b=t[7]>>>0<o[7]>>>0?1:0,i=0;i<8;i++){var a=e[i]+t[i],n=65535&a,s=a>>>16,l=((n*n>>>17)+n*s>>>15)+s*s,c=((4294901760&a)*a|0)+((65535&a)*a|0);r[i]=l^c}e[0]=r[0]+(r[7]<<16|r[7]>>>16)+(r[6]<<16|r[6]>>>16)|0,e[1]=r[1]+(r[0]<<8|r[0]>>>24)+r[7]|0,e[2]=r[2]+(r[1]<<16|r[1]>>>16)+(r[0]<<16|r[0]>>>16)|0,e[3]=r[3]+(r[2]<<8|r[2]>>>24)+r[1]|0,e[4]=r[4]+(r[3]<<16|r[3]>>>16)+(r[2]<<16|r[2]>>>16)|0,e[5]=r[5]+(r[4]<<8|r[4]>>>24)+r[3]|0,e[6]=r[6]+(r[5]<<16|r[5]>>>16)+(r[4]<<16|r[4]>>>16)|0,e[7]=r[7]+(r[6]<<8|r[6]>>>24)+r[5]|0}e.RabbitLegacy=t._createHelper(s)}(),n.RabbitLegacy)}()},42639:function(e,t,i){!function(t,a,o){var n;e.exports=(n=i(68926),i(53713),i(93074),i(68228),i(33650),function(){var e=n,t=e.lib.StreamCipher,i=e.algo,a=[],o=[],r=[],s=i.Rabbit=t.extend({_doReset:function(){for(var e=this._key.words,t=this.cfg.iv,i=0;i<4;i++)e[i]=16711935&(e[i]<<8|e[i]>>>24)|4278255360&(e[i]<<24|e[i]>>>8);var a=this._X=[e[0],e[3]<<16|e[2]>>>16,e[1],e[0]<<16|e[3]>>>16,e[2],e[1]<<16|e[0]>>>16,e[3],e[2]<<16|e[1]>>>16],o=this._C=[e[2]<<16|e[2]>>>16,4294901760&e[0]|65535&e[1],e[3]<<16|e[3]>>>16,4294901760&e[1]|65535&e[2],e[0]<<16|e[0]>>>16,4294901760&e[2]|65535&e[3],e[1]<<16|e[1]>>>16,4294901760&e[3]|65535&e[0]];for(this._b=0,i=0;i<4;i++)l.call(this);for(i=0;i<8;i++)o[i]^=a[i+4&7];if(t){var n=t.words,r=n[0],s=n[1],c=16711935&(r<<8|r>>>24)|4278255360&(r<<24|r>>>8),p=16711935&(s<<8|s>>>24)|4278255360&(s<<24|s>>>8),h=c>>>16|4294901760&p,$=p<<16|65535&c;for(o[0]^=c,o[1]^=h,o[2]^=p,o[3]^=$,o[4]^=c,o[5]^=h,o[6]^=p,o[7]^=$,i=0;i<4;i++)l.call(this)}},_doProcessBlock:function(e,t){var i=this._X;l.call(this),a[0]=i[0]^i[5]>>>16^i[3]<<16,a[1]=i[2]^i[7]>>>16^i[5]<<16,a[2]=i[4]^i[1]>>>16^i[7]<<16,a[3]=i[6]^i[3]>>>16^i[1]<<16;for(var o=0;o<4;o++)a[o]=16711935&(a[o]<<8|a[o]>>>24)|4278255360&(a[o]<<24|a[o]>>>8),e[t+o]^=a[o]},blockSize:4,ivSize:2});function l(){for(var e=this._X,t=this._C,i=0;i<8;i++)o[i]=t[i];for(t[0]=t[0]+1295307597+this._b|0,t[1]=t[1]+3545052371+(t[0]>>>0<o[0]>>>0?1:0)|0,t[2]=t[2]+886263092+(t[1]>>>0<o[1]>>>0?1:0)|0,t[3]=t[3]+1295307597+(t[2]>>>0<o[2]>>>0?1:0)|0,t[4]=t[4]+3545052371+(t[3]>>>0<o[3]>>>0?1:0)|0,t[5]=t[5]+886263092+(t[4]>>>0<o[4]>>>0?1:0)|0,t[6]=t[6]+1295307597+(t[5]>>>0<o[5]>>>0?1:0)|0,t[7]=t[7]+3545052371+(t[6]>>>0<o[6]>>>0?1:0)|0,this._b=t[7]>>>0<o[7]>>>0?1:0,i=0;i<8;i++){var a=e[i]+t[i],n=65535&a,s=a>>>16,l=((n*n>>>17)+n*s>>>15)+s*s,c=((4294901760&a)*a|0)+((65535&a)*a|0);r[i]=l^c}e[0]=r[0]+(r[7]<<16|r[7]>>>16)+(r[6]<<16|r[6]>>>16)|0,e[1]=r[1]+(r[0]<<8|r[0]>>>24)+r[7]|0,e[2]=r[2]+(r[1]<<16|r[1]>>>16)+(r[0]<<16|r[0]>>>16)|0,e[3]=r[3]+(r[2]<<8|r[2]>>>24)+r[1]|0,e[4]=r[4]+(r[3]<<16|r[3]>>>16)+(r[2]<<16|r[2]>>>16)|0,e[5]=r[5]+(r[4]<<8|r[4]>>>24)+r[3]|0,e[6]=r[6]+(r[5]<<16|r[5]>>>16)+(r[4]<<16|r[4]>>>16)|0,e[7]=r[7]+(r[6]<<8|r[6]>>>24)+r[5]|0}e.Rabbit=t._createHelper(s)}(),n.Rabbit)}()},1325:function(e,t,i){!function(t,a,o){var n;e.exports=(n=i(68926),i(53713),i(93074),i(68228),i(33650),function(){var e=n,t=e.lib.StreamCipher,i=e.algo,a=i.RC4=t.extend({_doReset:function(){for(var e=this._key,t=e.words,i=e.sigBytes,a=this._S=[],o=0;o<256;o++)a[o]=o;o=0;for(var n=0;o<256;o++){var r=o%i,s=t[r>>>2]>>>24-r%4*8&255;n=(n+a[o]+s)%256;var l=a[o];a[o]=a[n],a[n]=l}this._i=this._j=0},_doProcessBlock:function(e,t){e[t]^=o.call(this)},keySize:8,ivSize:0});function o(){for(var e=this._S,t=this._i,i=this._j,a=0,o=0;o<4;o++){i=(i+e[t=(t+1)%256])%256;var n=e[t];e[t]=e[i],e[i]=n,a|=e[(e[t]+e[i])%256]<<24-8*o}return this._i=t,this._j=i,a}e.RC4=t._createHelper(a);var r=i.RC4Drop=a.extend({cfg:a.cfg.extend({drop:192}),_doReset:function(){a._doReset.call(this);for(var e=this.cfg.drop;e>0;e--)o.call(this)}});e.RC4Drop=t._createHelper(r)}(),n.RC4)}()},70077:function(e,t,i){!function(t,a){var o;e.exports=(o=i(68926),function(e){var t=o,i=t.lib,a=i.WordArray,n=i.Hasher,r=t.algo,s=a.create([0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,7,4,13,1,10,6,15,3,12,0,9,5,2,14,11,8,3,10,14,4,9,15,8,1,2,7,0,6,13,11,5,12,1,9,11,10,0,8,12,4,13,3,7,15,14,5,6,2,4,0,5,9,7,12,2,10,14,1,3,8,11,6,15,13]),l=a.create([5,14,7,0,9,2,11,4,13,6,15,8,1,10,3,12,6,11,3,7,0,13,5,10,14,15,8,12,4,9,1,2,15,5,1,3,7,14,6,9,11,8,12,2,10,0,4,13,8,6,4,1,3,11,15,0,5,12,2,13,9,7,10,14,12,15,10,4,1,5,8,7,6,2,13,14,0,3,9,11]),c=a.create([11,14,15,12,5,8,7,9,11,13,14,15,6,7,9,8,7,6,8,13,11,9,7,15,7,12,15,9,11,7,13,12,11,13,6,7,14,9,13,15,14,8,13,6,5,12,7,5,11,12,14,15,14,15,9,8,9,14,5,6,8,6,5,12,9,15,5,11,6,8,13,12,5,12,13,14,11,8,5,6]),p=a.create([8,9,9,11,13,15,15,5,7,7,8,11,14,14,12,6,9,13,15,7,12,8,9,11,7,7,12,7,6,15,13,11,9,7,15,11,8,6,6,14,12,13,5,14,13,13,7,5,15,5,8,11,14,14,6,14,6,9,12,9,12,5,15,8,8,5,12,9,12,5,14,6,8,13,6,5,15,13,11,11]),h=a.create([0,1518500249,1859775393,2400959708,2840853838]),$=a.create([1352829926,1548603684,1836072691,2053994217,0]),d=r.RIPEMD160=n.extend({_doReset:function(){this._hash=a.create([1732584193,4023233417,2562383102,271733878,3285377520])},_doProcessBlock:function(e,t){for(var i=0;i<16;i++){var a=t+i,o=e[a];e[a]=16711935&(o<<8|o>>>24)|4278255360&(o<<24|o>>>8)}var n,r,d,x,v,T,A,w,C,O,S,B=this._hash.words,_=h.words,z=$.words,q=s.words,I=l.words,k=c.words,N=p.words;for(T=n=B[0],A=r=B[1],w=d=B[2],C=x=B[3],O=v=B[4],i=0;i<80;i+=1)S=n+e[t+q[i]]|0,S+=i<16?u(r,d,x)+_[0]:i<32?m(r,d,x)+_[1]:i<48?b(r,d,x)+_[2]:i<64?y(r,d,x)+_[3]:f(r,d,x)+_[4],S=(S=g(S|=0,k[i]))+v|0,n=v,v=x,x=g(d,10),d=r,r=S,S=T+e[t+I[i]]|0,S+=i<16?f(A,w,C)+z[0]:i<32?y(A,w,C)+z[1]:i<48?b(A,w,C)+z[2]:i<64?m(A,w,C)+z[3]:u(A,w,C)+z[4],S=(S=g(S|=0,N[i]))+O|0,T=O,O=C,C=g(w,10),w=A,A=S;S=B[1]+d+C|0,B[1]=B[2]+x+O|0,B[2]=B[3]+v+T|0,B[3]=B[4]+n+A|0,B[4]=B[0]+r+w|0,B[0]=S},_doFinalize:function(){var e=this._data,t=e.words,i=8*this._nDataBytes,a=8*e.sigBytes;t[a>>>5]|=128<<24-a%32,t[14+(a+64>>>9<<4)]=16711935&(i<<8|i>>>24)|4278255360&(i<<24|i>>>8),e.sigBytes=4*(t.length+1),this._process();for(var o=this._hash,n=o.words,r=0;r<5;r++){var s=n[r];n[r]=16711935&(s<<8|s>>>24)|4278255360&(s<<24|s>>>8)}return o},clone:function(){var e=n.clone.call(this);return e._hash=this._hash.clone(),e}});function u(e,t,i){return e^t^i}function m(e,t,i){return e&t|~e&i}function b(e,t,i){return(e|~t)^i}function y(e,t,i){return e&i|t&~i}function f(e,t,i){return e^(t|~i)}function g(e,t){return e<<t|e>>>32-t}t.RIPEMD160=n._createHelper(d),t.HmacRIPEMD160=n._createHmacHelper(d)}(Math),o.RIPEMD160)}()},25590:function(e,t,i){!function(t,a){var o;e.exports=(o=i(68926),function(){var e=o,t=e.lib,i=t.WordArray,a=t.Hasher,n=e.algo,r=[],s=n.SHA1=a.extend({_doReset:function(){this._hash=new i.init([1732584193,4023233417,2562383102,271733878,3285377520])},_doProcessBlock:function(e,t){for(var i=this._hash.words,a=i[0],o=i[1],n=i[2],s=i[3],l=i[4],c=0;c<80;c++){if(c<16)r[c]=0|e[t+c];else{var p=r[c-3]^r[c-8]^r[c-14]^r[c-16];r[c]=p<<1|p>>>31}var h=(a<<5|a>>>27)+l+r[c];h+=c<20?1518500249+(o&n|~o&s):c<40?1859775393+(o^n^s):c<60?(o&n|o&s|n&s)-1894007588:(o^n^s)-899497514,l=s,s=n,n=o<<30|o>>>2,o=a,a=h}i[0]=i[0]+a|0,i[1]=i[1]+o|0,i[2]=i[2]+n|0,i[3]=i[3]+s|0,i[4]=i[4]+l|0},_doFinalize:function(){var e=this._data,t=e.words,i=8*this._nDataBytes,a=8*e.sigBytes;return t[a>>>5]|=128<<24-a%32,t[14+(a+64>>>9<<4)]=Math.floor(i/4294967296),t[15+(a+64>>>9<<4)]=i,e.sigBytes=4*t.length,this._process(),this._hash},clone:function(){var e=a.clone.call(this);return e._hash=this._hash.clone(),e}});e.SHA1=a._createHelper(s),e.HmacSHA1=a._createHmacHelper(s)}(),o.SHA1)}()},43183:function(e,t,i){!function(t,a,o){var n;e.exports=(n=i(68926),i(29517),function(){var e=n,t=e.lib.WordArray,i=e.algo,a=i.SHA256,o=i.SHA224=a.extend({_doReset:function(){this._hash=new t.init([3238371032,914150663,812702999,4144912697,4290775857,1750603025,1694076839,3204075428])},_doFinalize:function(){var e=a._doFinalize.call(this);return e.sigBytes-=4,e}});e.SHA224=a._createHelper(o),e.HmacSHA224=a._createHmacHelper(o)}(),n.SHA224)}()},29517:function(e,t,i){!function(t,a){var o;e.exports=(o=i(68926),function(e){var t=o,i=t.lib,a=i.WordArray,n=i.Hasher,r=t.algo,s=[],l=[];!function(){function t(t){for(var i=e.sqrt(t),a=2;a<=i;a++)if(!(t%a))return!1;return!0}function i(e){return 4294967296*(e-(0|e))|0}for(var a=2,o=0;o<64;)t(a)&&(o<8&&(s[o]=i(e.pow(a,.5))),l[o]=i(e.pow(a,1/3)),o++),a++}();var c=[],p=r.SHA256=n.extend({_doReset:function(){this._hash=new a.init(s.slice(0))},_doProcessBlock:function(e,t){for(var i=this._hash.words,a=i[0],o=i[1],n=i[2],r=i[3],s=i[4],p=i[5],h=i[6],$=i[7],d=0;d<64;d++){if(d<16)c[d]=0|e[t+d];else{var u=c[d-15],m=(u<<25|u>>>7)^(u<<14|u>>>18)^u>>>3,b=c[d-2],y=(b<<15|b>>>17)^(b<<13|b>>>19)^b>>>10;c[d]=m+c[d-7]+y+c[d-16]}var f=a&o^a&n^o&n,g=(a<<30|a>>>2)^(a<<19|a>>>13)^(a<<10|a>>>22),x=$+((s<<26|s>>>6)^(s<<21|s>>>11)^(s<<7|s>>>25))+(s&p^~s&h)+l[d]+c[d];$=h,h=p,p=s,s=r+x|0,r=n,n=o,o=a,a=x+(g+f)|0}i[0]=i[0]+a|0,i[1]=i[1]+o|0,i[2]=i[2]+n|0,i[3]=i[3]+r|0,i[4]=i[4]+s|0,i[5]=i[5]+p|0,i[6]=i[6]+h|0,i[7]=i[7]+$|0},_doFinalize:function(){var t=this._data,i=t.words,a=8*this._nDataBytes,o=8*t.sigBytes;return i[o>>>5]|=128<<24-o%32,i[14+(o+64>>>9<<4)]=e.floor(a/4294967296),i[15+(o+64>>>9<<4)]=a,t.sigBytes=4*i.length,this._process(),this._hash},clone:function(){var e=n.clone.call(this);return e._hash=this._hash.clone(),e}});t.SHA256=n._createHelper(p),t.HmacSHA256=n._createHmacHelper(p)}(Math),o.SHA256)}()},26853:function(e,t,i){!function(t,a,o){var n;e.exports=(n=i(68926),i(73646),function(e){var t=n,i=t.lib,a=i.WordArray,o=i.Hasher,r=t.x64.Word,s=t.algo,l=[],c=[],p=[];!function(){for(var e=1,t=0,i=0;i<24;i++){l[e+5*t]=(i+1)*(i+2)/2%64;var a=(2*e+3*t)%5;e=t%5,t=a}for(e=0;e<5;e++)for(t=0;t<5;t++)c[e+5*t]=t+(2*e+3*t)%5*5;for(var o=1,n=0;n<24;n++){for(var s=0,h=0,$=0;$<7;$++){if(1&o){var d=(1<<$)-1;d<32?h^=1<<d:s^=1<<d-32}128&o?o=o<<1^113:o<<=1}p[n]=r.create(s,h)}}();var h=[];!function(){for(var e=0;e<25;e++)h[e]=r.create()}();var $=s.SHA3=o.extend({cfg:o.cfg.extend({outputLength:512}),_doReset:function(){for(var e=this._state=[],t=0;t<25;t++)e[t]=new r.init;this.blockSize=(1600-2*this.cfg.outputLength)/32},_doProcessBlock:function(e,t){for(var i=this._state,a=this.blockSize/2,o=0;o<a;o++){var n=e[t+2*o],r=e[t+2*o+1];n=16711935&(n<<8|n>>>24)|4278255360&(n<<24|n>>>8),r=16711935&(r<<8|r>>>24)|4278255360&(r<<24|r>>>8),(B=i[o]).high^=r,B.low^=n}for(var s=0;s<24;s++){for(var $=0;$<5;$++){for(var d=0,u=0,m=0;m<5;m++)d^=(B=i[$+5*m]).high,u^=B.low;var b=h[$];b.high=d,b.low=u}for($=0;$<5;$++){var y=h[($+4)%5],f=h[($+1)%5],g=f.high,x=f.low;for(d=y.high^(g<<1|x>>>31),u=y.low^(x<<1|g>>>31),m=0;m<5;m++)(B=i[$+5*m]).high^=d,B.low^=u}for(var v=1;v<25;v++){var T=(B=i[v]).high,A=B.low,w=l[v];w<32?(d=T<<w|A>>>32-w,u=A<<w|T>>>32-w):(d=A<<w-32|T>>>64-w,u=T<<w-32|A>>>64-w);var C=h[c[v]];C.high=d,C.low=u}var O=h[0],S=i[0];for(O.high=S.high,O.low=S.low,$=0;$<5;$++)for(m=0;m<5;m++){var B=i[v=$+5*m],_=h[v],z=h[($+1)%5+5*m],q=h[($+2)%5+5*m];B.high=_.high^~z.high&q.high,B.low=_.low^~z.low&q.low}B=i[0];var I=p[s];B.high^=I.high,B.low^=I.low}},_doFinalize:function(){var t=this._data,i=t.words,o=(this._nDataBytes,8*t.sigBytes),n=32*this.blockSize;i[o>>>5]|=1<<24-o%32,i[(e.ceil((o+1)/n)*n>>>5)-1]|=128,t.sigBytes=4*i.length,this._process();for(var r=this._state,s=this.cfg.outputLength/8,l=s/8,c=[],p=0;p<l;p++){var h=r[p],$=h.high,d=h.low;$=16711935&($<<8|$>>>24)|4278255360&($<<24|$>>>8),d=16711935&(d<<8|d>>>24)|4278255360&(d<<24|d>>>8),c.push(d),c.push($)}return new a.init(c,s)},clone:function(){for(var e=o.clone.call(this),t=e._state=this._state.slice(0),i=0;i<25;i++)t[i]=t[i].clone();return e}});t.SHA3=o._createHelper($),t.HmacSHA3=o._createHmacHelper($)}(Math),n.SHA3)}()},36319:function(e,t,i){!function(t,a,o){var n;e.exports=(n=i(68926),i(73646),i(24345),function(){var e=n,t=e.x64,i=t.Word,a=t.WordArray,o=e.algo,r=o.SHA512,s=o.SHA384=r.extend({_doReset:function(){this._hash=new a.init([new i.init(3418070365,3238371032),new i.init(1654270250,914150663),new i.init(2438529370,812702999),new i.init(355462360,4144912697),new i.init(1731405415,4290775857),new i.init(2394180231,1750603025),new i.init(3675008525,1694076839),new i.init(1203062813,3204075428)])},_doFinalize:function(){var e=r._doFinalize.call(this);return e.sigBytes-=16,e}});e.SHA384=r._createHelper(s),e.HmacSHA384=r._createHmacHelper(s)}(),n.SHA384)}()},24345:function(e,t,i){!function(t,a,o){var n;e.exports=(n=i(68926),i(73646),function(){var e=n,t=e.lib.Hasher,i=e.x64,a=i.Word,o=i.WordArray,r=e.algo;function s(){return a.create.apply(a,arguments)}var l=[s(1116352408,3609767458),s(1899447441,602891725),s(3049323471,3964484399),s(3921009573,2173295548),s(961987163,4081628472),s(1508970993,3053834265),s(2453635748,2937671579),s(2870763221,3664609560),s(3624381080,2734883394),s(310598401,1164996542),s(607225278,1323610764),s(1426881987,3590304994),s(1925078388,4068182383),s(2162078206,991336113),s(2614888103,633803317),s(3248222580,3479774868),s(3835390401,2666613458),s(4022224774,944711139),s(264347078,2341262773),s(604807628,2007800933),s(770255983,1495990901),s(1249150122,1856431235),s(1555081692,3175218132),s(1996064986,2198950837),s(2554220882,3999719339),s(2821834349,766784016),s(2952996808,2566594879),s(3210313671,3203337956),s(3336571891,1034457026),s(3584528711,2466948901),s(113926993,3758326383),s(338241895,168717936),s(666307205,1188179964),s(773529912,1546045734),s(1294757372,1522805485),s(1396182291,2643833823),s(1695183700,2343527390),s(1986661051,1014477480),s(2177026350,1206759142),s(2456956037,344077627),s(2730485921,1290863460),s(2820302411,3158454273),s(3259730800,3505952657),s(3345764771,106217008),s(3516065817,3606008344),s(3600352804,1432725776),s(4094571909,1467031594),s(275423344,851169720),s(430227734,3100823752),s(506948616,1363258195),s(659060556,3750685593),s(883997877,3785050280),s(958139571,3318307427),s(1322822218,3812723403),s(1537002063,2003034995),s(1747873779,3602036899),s(1955562222,1575990012),s(2024104815,1125592928),s(2227730452,2716904306),s(2361852424,442776044),s(2428436474,593698344),s(2756734187,3733110249),s(3204031479,2999351573),s(3329325298,3815920427),s(3391569614,3928383900),s(3515267271,566280711),s(3940187606,3454069534),s(4118630271,4000239992),s(116418474,1914138554),s(174292421,2731055270),s(289380356,3203993006),s(460393269,320620315),s(685471733,587496836),s(852142971,1086792851),s(1017036298,365543100),s(1126000580,2618297676),s(1288033470,3409855158),s(1501505948,4234509866),s(1607167915,987167468),s(1816402316,1246189591)],c=[];!function(){for(var e=0;e<80;e++)c[e]=s()}();var p=r.SHA512=t.extend({_doReset:function(){this._hash=new o.init([new a.init(1779033703,4089235720),new a.init(3144134277,2227873595),new a.init(1013904242,4271175723),new a.init(2773480762,1595750129),new a.init(1359893119,2917565137),new a.init(2600822924,725511199),new a.init(528734635,4215389547),new a.init(1541459225,327033209)])},_doProcessBlock:function(e,t){for(var i=this._hash.words,a=i[0],o=i[1],n=i[2],r=i[3],s=i[4],p=i[5],h=i[6],$=i[7],d=a.high,u=a.low,m=o.high,b=o.low,y=n.high,f=n.low,g=r.high,x=r.low,v=s.high,T=s.low,A=p.high,w=p.low,C=h.high,O=h.low,S=$.high,B=$.low,_=d,z=u,q=m,I=b,k=y,N=f,D=g,Y=x,P=v,E=T,M=A,R=w,L=C,F=O,W=S,G=B,V=0;V<80;V++){var U,H,j=c[V];if(V<16)H=j.high=0|e[t+2*V],U=j.low=0|e[t+2*V+1];else{var Q=c[V-15],K=Q.high,Z=Q.low,J=(K>>>1|Z<<31)^(K>>>8|Z<<24)^K>>>7,X=(Z>>>1|K<<31)^(Z>>>8|K<<24)^(Z>>>7|K<<25),ee=c[V-2],te=ee.high,ie=ee.low,ae=(te>>>19|ie<<13)^(te<<3|ie>>>29)^te>>>6,oe=(ie>>>19|te<<13)^(ie<<3|te>>>29)^(ie>>>6|te<<26),ne=c[V-7],re=ne.high,se=ne.low,le=c[V-16],ce=le.high,pe=le.low;H=(H=(H=J+re+((U=X+se)>>>0<X>>>0?1:0))+ae+((U+=oe)>>>0<oe>>>0?1:0))+ce+((U+=pe)>>>0<pe>>>0?1:0),j.high=H,j.low=U}var he,$e=P&M^~P&L,de=E&R^~E&F,ue=_&q^_&k^q&k,me=z&I^z&N^I&N,be=(_>>>28|z<<4)^(_<<30|z>>>2)^(_<<25|z>>>7),ye=(z>>>28|_<<4)^(z<<30|_>>>2)^(z<<25|_>>>7),fe=(P>>>14|E<<18)^(P>>>18|E<<14)^(P<<23|E>>>9),ge=(E>>>14|P<<18)^(E>>>18|P<<14)^(E<<23|P>>>9),xe=l[V],ve=xe.high,Te=xe.low,Ae=W+fe+((he=G+ge)>>>0<G>>>0?1:0),we=ye+me;W=L,G=F,L=M,F=R,M=P,R=E,P=D+(Ae=(Ae=(Ae=Ae+$e+((he+=de)>>>0<de>>>0?1:0))+ve+((he+=Te)>>>0<Te>>>0?1:0))+H+((he+=U)>>>0<U>>>0?1:0))+((E=Y+he|0)>>>0<Y>>>0?1:0)|0,D=k,Y=N,k=q,N=I,q=_,I=z,_=Ae+(be+ue+(we>>>0<ye>>>0?1:0))+((z=he+we|0)>>>0<he>>>0?1:0)|0}u=a.low=u+z,a.high=d+_+(u>>>0<z>>>0?1:0),b=o.low=b+I,o.high=m+q+(b>>>0<I>>>0?1:0),f=n.low=f+N,n.high=y+k+(f>>>0<N>>>0?1:0),x=r.low=x+Y,r.high=g+D+(x>>>0<Y>>>0?1:0),T=s.low=T+E,s.high=v+P+(T>>>0<E>>>0?1:0),w=p.low=w+R,p.high=A+M+(w>>>0<R>>>0?1:0),O=h.low=O+F,h.high=C+L+(O>>>0<F>>>0?1:0),B=$.low=B+G,$.high=S+W+(B>>>0<G>>>0?1:0)},_doFinalize:function(){var e=this._data,t=e.words,i=8*this._nDataBytes,a=8*e.sigBytes;return t[a>>>5]|=128<<24-a%32,t[30+(a+128>>>10<<5)]=Math.floor(i/4294967296),t[31+(a+128>>>10<<5)]=i,e.sigBytes=4*t.length,this._process(),this._hash.toX32()},clone:function(){var e=t.clone.call(this);return e._hash=this._hash.clone(),e},blockSize:32});e.SHA512=t._createHelper(p),e.HmacSHA512=t._createHmacHelper(p)}(),n.SHA512)}()},42550:function(e,t,i){!function(t,a,o){var n;e.exports=(n=i(68926),i(53713),i(93074),i(68228),i(33650),function(){var e=n,t=e.lib,i=t.WordArray,a=t.BlockCipher,o=e.algo,r=[57,49,41,33,25,17,9,1,58,50,42,34,26,18,10,2,59,51,43,35,27,19,11,3,60,52,44,36,63,55,47,39,31,23,15,7,62,54,46,38,30,22,14,6,61,53,45,37,29,21,13,5,28,20,12,4],s=[14,17,11,24,1,5,3,28,15,6,21,10,23,19,12,4,26,8,16,7,27,20,13,2,41,52,31,37,47,55,30,40,51,45,33,48,44,49,39,56,34,53,46,42,50,36,29,32],l=[1,2,4,6,8,10,12,14,15,17,19,21,23,25,27,28],c=[{0:8421888,268435456:32768,536870912:8421378,805306368:2,1073741824:512,1342177280:8421890,1610612736:8389122,1879048192:8388608,2147483648:514,2415919104:8389120,2684354560:33280,2952790016:8421376,3221225472:32770,3489660928:8388610,3758096384:0,4026531840:33282,134217728:0,402653184:8421890,671088640:33282,939524096:32768,1207959552:8421888,1476395008:512,1744830464:8421378,2013265920:2,2281701376:8389120,2550136832:33280,2818572288:8421376,3087007744:8389122,3355443200:8388610,3623878656:32770,3892314112:514,4160749568:8388608,1:32768,268435457:2,536870913:8421888,805306369:8388608,1073741825:8421378,1342177281:33280,1610612737:512,1879048193:8389122,2147483649:8421890,2415919105:8421376,2684354561:8388610,2952790017:33282,3221225473:514,3489660929:8389120,3758096385:32770,4026531841:0,134217729:8421890,402653185:8421376,671088641:8388608,939524097:512,1207959553:32768,1476395009:8388610,1744830465:2,2013265921:33282,2281701377:32770,2550136833:8389122,2818572289:514,3087007745:8421888,3355443201:8389120,3623878657:0,3892314113:33280,4160749569:8421378},{0:1074282512,16777216:16384,33554432:524288,50331648:1074266128,67108864:1073741840,83886080:1074282496,100663296:1073758208,117440512:16,134217728:540672,150994944:1073758224,167772160:1073741824,184549376:540688,201326592:524304,218103808:0,234881024:16400,251658240:1074266112,8388608:1073758208,25165824:540688,41943040:16,58720256:1073758224,75497472:1074282512,92274688:1073741824,109051904:524288,125829120:1074266128,142606336:524304,159383552:0,176160768:16384,192937984:1074266112,209715200:1073741840,226492416:540672,243269632:1074282496,260046848:16400,268435456:0,285212672:1074266128,301989888:1073758224,318767104:1074282496,335544320:1074266112,352321536:16,369098752:540688,385875968:16384,402653184:16400,419430400:524288,436207616:524304,452984832:1073741840,469762048:540672,486539264:1073758208,503316480:1073741824,520093696:1074282512,276824064:540688,293601280:524288,310378496:1074266112,327155712:16384,343932928:1073758208,360710144:1074282512,377487360:16,394264576:1073741824,411041792:1074282496,427819008:1073741840,444596224:1073758224,461373440:524304,478150656:0,494927872:16400,511705088:1074266128,528482304:540672},{0:260,1048576:0,2097152:67109120,3145728:65796,4194304:65540,5242880:67108868,6291456:67174660,7340032:67174400,8388608:67108864,9437184:67174656,10485760:65792,11534336:67174404,12582912:67109124,13631488:65536,14680064:4,15728640:256,524288:67174656,1572864:67174404,2621440:0,3670016:67109120,4718592:67108868,5767168:65536,6815744:65540,7864320:260,8912896:4,9961472:256,11010048:67174400,12058624:65796,13107200:65792,14155776:67109124,15204352:67174660,16252928:67108864,16777216:67174656,17825792:65540,18874368:65536,19922944:67109120,20971520:256,22020096:67174660,23068672:67108868,24117248:0,25165824:67109124,26214400:67108864,27262976:4,28311552:65792,29360128:67174400,30408704:260,31457280:65796,32505856:67174404,17301504:67108864,18350080:260,19398656:67174656,20447232:0,21495808:65540,22544384:67109120,23592960:256,24641536:67174404,25690112:65536,26738688:67174660,27787264:65796,28835840:67108868,29884416:67109124,30932992:67174400,31981568:4,33030144:65792},{0:2151682048,65536:2147487808,131072:4198464,196608:2151677952,262144:0,327680:4198400,393216:2147483712,458752:4194368,524288:2147483648,589824:4194304,655360:64,720896:2147487744,786432:2151678016,851968:4160,917504:4096,983040:2151682112,32768:2147487808,98304:64,163840:2151678016,229376:2147487744,294912:4198400,360448:2151682112,425984:0,491520:2151677952,557056:4096,622592:2151682048,688128:4194304,753664:4160,819200:2147483648,884736:4194368,950272:4198464,1015808:2147483712,1048576:4194368,1114112:4198400,1179648:2147483712,1245184:0,1310720:4160,1376256:2151678016,1441792:2151682048,1507328:2147487808,1572864:2151682112,1638400:2147483648,1703936:2151677952,1769472:4198464,1835008:2147487744,1900544:4194304,1966080:64,2031616:4096,1081344:2151677952,1146880:2151682112,1212416:0,1277952:4198400,1343488:4194368,1409024:2147483648,1474560:2147487808,1540096:64,1605632:2147483712,1671168:4096,1736704:2147487744,1802240:2151678016,1867776:4160,1933312:2151682048,1998848:4194304,2064384:4198464},{0:128,4096:17039360,8192:262144,12288:536870912,16384:537133184,20480:16777344,24576:553648256,28672:262272,32768:16777216,36864:537133056,40960:536871040,45056:553910400,49152:553910272,53248:0,57344:17039488,61440:553648128,2048:17039488,6144:553648256,10240:128,14336:17039360,18432:262144,22528:537133184,26624:553910272,30720:536870912,34816:537133056,38912:0,43008:553910400,47104:16777344,51200:536871040,55296:553648128,59392:16777216,63488:262272,65536:262144,69632:128,73728:536870912,77824:553648256,81920:16777344,86016:553910272,90112:537133184,94208:16777216,98304:553910400,102400:553648128,106496:17039360,110592:537133056,114688:262272,118784:536871040,122880:0,126976:17039488,67584:553648256,71680:16777216,75776:17039360,79872:537133184,83968:536870912,88064:17039488,92160:128,96256:553910272,100352:262272,104448:553910400,108544:0,112640:553648128,116736:16777344,120832:262144,124928:537133056,129024:536871040},{0:268435464,256:8192,512:270532608,768:270540808,1024:268443648,1280:2097152,1536:2097160,1792:268435456,2048:0,2304:268443656,2560:2105344,2816:8,3072:270532616,3328:2105352,3584:8200,3840:270540800,128:270532608,384:270540808,640:8,896:2097152,1152:2105352,1408:268435464,1664:268443648,1920:8200,2176:2097160,2432:8192,2688:268443656,2944:270532616,3200:0,3456:270540800,3712:2105344,3968:268435456,4096:268443648,4352:270532616,4608:270540808,4864:8200,5120:2097152,5376:268435456,5632:268435464,5888:2105344,6144:2105352,6400:0,6656:8,6912:270532608,7168:8192,7424:268443656,7680:270540800,7936:2097160,4224:8,4480:2105344,4736:2097152,4992:268435464,5248:268443648,5504:8200,5760:270540808,6016:270532608,6272:270540800,6528:270532616,6784:8192,7040:2105352,7296:2097160,7552:0,7808:268435456,8064:268443656},{0:1048576,16:33555457,32:1024,48:1049601,64:34604033,80:0,96:1,112:34603009,128:33555456,144:1048577,160:33554433,176:34604032,192:34603008,208:1025,224:1049600,240:33554432,8:34603009,24:0,40:33555457,56:34604032,72:1048576,88:33554433,104:33554432,120:1025,136:1049601,152:33555456,168:34603008,184:1048577,200:1024,216:34604033,232:1,248:1049600,256:33554432,272:1048576,288:33555457,304:34603009,320:1048577,336:33555456,352:34604032,368:1049601,384:1025,400:34604033,416:1049600,432:1,448:0,464:34603008,480:33554433,496:1024,264:1049600,280:33555457,296:34603009,312:1,328:33554432,344:1048576,360:1025,376:34604032,392:33554433,408:34603008,424:0,440:34604033,456:1049601,472:1024,488:33555456,504:1048577},{0:134219808,1:131072,2:134217728,3:32,4:131104,5:134350880,6:134350848,7:2048,8:134348800,9:134219776,10:133120,11:134348832,12:2080,13:0,14:134217760,15:133152,2147483648:2048,2147483649:134350880,2147483650:134219808,2147483651:134217728,2147483652:134348800,2147483653:133120,2147483654:133152,2147483655:32,2147483656:134217760,2147483657:2080,2147483658:131104,2147483659:134350848,2147483660:0,2147483661:134348832,2147483662:134219776,2147483663:131072,16:133152,17:134350848,18:32,19:2048,20:134219776,21:134217760,22:134348832,23:131072,24:0,25:131104,26:134348800,27:134219808,28:134350880,29:133120,30:2080,31:134217728,2147483664:131072,2147483665:2048,2147483666:134348832,2147483667:133152,2147483668:32,2147483669:134348800,2147483670:134217728,2147483671:134219808,2147483672:134350880,2147483673:134217760,2147483674:134219776,2147483675:0,2147483676:133120,2147483677:2080,2147483678:131104,2147483679:134350848}],p=[4160749569,528482304,33030144,2064384,129024,8064,504,2147483679],h=o.DES=a.extend({_doReset:function(){for(var e=this._key.words,t=[],i=0;i<56;i++){var a=r[i]-1;t[i]=e[a>>>5]>>>31-a%32&1}for(var o=this._subKeys=[],n=0;n<16;n++){var c=o[n]=[],p=l[n];for(i=0;i<24;i++)c[i/6|0]|=t[(s[i]-1+p)%28]<<31-i%6,c[4+(i/6|0)]|=t[28+(s[i+24]-1+p)%28]<<31-i%6;for(c[0]=c[0]<<1|c[0]>>>31,i=1;i<7;i++)c[i]=c[i]>>>4*(i-1)+3;c[7]=c[7]<<5|c[7]>>>27}var h=this._invSubKeys=[];for(i=0;i<16;i++)h[i]=o[15-i]},encryptBlock:function(e,t){this._doCryptBlock(e,t,this._subKeys)},decryptBlock:function(e,t){this._doCryptBlock(e,t,this._invSubKeys)},_doCryptBlock:function(e,t,i){this._lBlock=e[t],this._rBlock=e[t+1],$.call(this,4,252645135),$.call(this,16,65535),d.call(this,2,858993459),d.call(this,8,16711935),$.call(this,1,1431655765);for(var a=0;a<16;a++){for(var o=i[a],n=this._lBlock,r=this._rBlock,s=0,l=0;l<8;l++)s|=c[l][((r^o[l])&p[l])>>>0];this._lBlock=r,this._rBlock=n^s}var h=this._lBlock;this._lBlock=this._rBlock,this._rBlock=h,$.call(this,1,1431655765),d.call(this,8,16711935),d.call(this,2,858993459),$.call(this,16,65535),$.call(this,4,252645135),e[t]=this._lBlock,e[t+1]=this._rBlock},keySize:2,ivSize:2,blockSize:2});function $(e,t){var i=(this._lBlock>>>e^this._rBlock)&t;this._rBlock^=i,this._lBlock^=i<<e}function d(e,t){var i=(this._rBlock>>>e^this._lBlock)&t;this._lBlock^=i,this._rBlock^=i<<e}e.DES=a._createHelper(h);var u=o.TripleDES=a.extend({_doReset:function(){var e=this._key.words;if(2!==e.length&&4!==e.length&&e.length<6)throw new Error("Invalid key length - 3DES requires the key length to be 64, 128, 192 or >192.");var t=e.slice(0,2),a=e.length<4?e.slice(0,2):e.slice(2,4),o=e.length<6?e.slice(0,2):e.slice(4,6);this._des1=h.createEncryptor(i.create(t)),this._des2=h.createEncryptor(i.create(a)),this._des3=h.createEncryptor(i.create(o))},encryptBlock:function(e,t){this._des1.encryptBlock(e,t),this._des2.decryptBlock(e,t),this._des3.encryptBlock(e,t)},decryptBlock:function(e,t){this._des3.decryptBlock(e,t),this._des2.encryptBlock(e,t),this._des1.decryptBlock(e,t)},keySize:6,ivSize:2,blockSize:2});e.TripleDES=a._createHelper(u)}(),n.TripleDES)}()},73646:function(e,t,i){!function(t,a){var o;e.exports=(o=i(68926),function(e){var t=o,i=t.lib,a=i.Base,n=i.WordArray,r=t.x64={};r.Word=a.extend({init:function(e,t){this.high=e,this.low=t}}),r.WordArray=a.extend({init:function(t,i){t=this.words=t||[],this.sigBytes=i!=e?i:8*t.length},toX32:function(){for(var e=this.words,t=e.length,i=[],a=0;a<t;a++){var o=e[a];i.push(o.high),i.push(o.low)}return n.create(i,this.sigBytes)},clone:function(){for(var 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How many boys were in the study group?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a01e792probsolve10a-h1","type":"hint","dependencies":[],"title":"Identify the question","text":"What are we looking for","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve10a-h2","type":"hint","dependencies":["a01e792probsolve10a-h1"],"title":"Identify the question","text":"We are looking for the number of boys in the group","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve10a-h3","type":"hint","dependencies":["a01e792probsolve10a-h2"],"title":"Name","text":"Let\'s assign a variable to represent the number of boys, such as \\"b\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve10a-h4","type":"hint","dependencies":["a01e792probsolve10a-h3"],"title":"Rewrite","text":"Now, let\'s rewrite the question to combine all the important information into one sentence.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve10a-h5","type":"hint","dependencies":["a01e792probsolve10a-h4"],"title":"Rewrite","text":"We can rewrite it like \\"There are $$11$$ girls which is $$3$$ more than twice the number of boys\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve10a-h6","type":"hint","dependencies":["a01e792probsolve10a-h5"],"title":"Translate","text":"Now, use the rewritten question to make an equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve10a-h7","type":"hint","dependencies":["a01e792probsolve10a-h6"],"title":"Translate","text":"We can rewrite it to be $$11=3+2b$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve10a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a01e792probsolve10a-h7"],"title":"Solve","text":"What number can we subtract from both sides?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve10a-h9","type":"hint","dependencies":["a01e792probsolve10a-h8"],"title":"Simplify","text":"Now, we can simplify the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve10a-h10","type":"hint","dependencies":["a01e792probsolve10a-h9"],"title":"Simplify","text":"Finally, we can solve the equation $$8=2b$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve10a-h11","type":"hint","dependencies":["a01e792probsolve10a-h10"],"title":"Simplify","text":"We can divide $$2$$ from both sides of the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a01e792probsolve11","title":"Solve Number Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Use a Problem-Solving Strategy","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a01e792probsolve11a","stepAnswer":["$$19$$"],"problemType":"TextBox","stepTitle":"The difference of a number and six is $$13$$. Find the number.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$19$$","hints":{"DefaultPathway":[{"id":"a01e792probsolve11a-h1","type":"hint","dependencies":[],"title":"Name","text":"Identify a variable for which you are going to represent the number we are looking for (n)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve11a-h2","type":"hint","dependencies":["a01e792probsolve11a-h1"],"title":"Translate","text":"Let\'s rewrite the statement into an equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve11a-h3","type":"hint","dependencies":["a01e792probsolve11a-h2"],"title":"Translate","text":"We would get $$n-6=13$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve11a-h4","type":"hint","dependencies":["a01e792probsolve11a-h3"],"title":"Solve","text":"Solve the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a01e792probsolve11a-h4"],"title":"Solve","text":"What number do we have to add to both sides to be left with only $$n$$ on the left side?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a01e792probsolve12","title":"Use a Problem-Solving Strategy for Word Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Use a Problem-Solving Strategy","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a01e792probsolve12a","stepAnswer":["$$180$$"],"problemType":"TextBox","stepTitle":"Joaquin bought a bookcase on sale for $120, which was two-thirds of the original price. What was the original price of the bookcase?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$180$$","hints":{"DefaultPathway":[{"id":"a01e792probsolve12a-h1","type":"hint","dependencies":[],"title":"Identify and Name","text":"Let\'s first identify what we want to find, and then give it a variable name","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve12a-h2","type":"hint","dependencies":["a01e792probsolve12a-h1"],"title":"Identify and Name","text":"We know that we are trying to find the original price and we can name it \\"n\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve12a-h3","type":"hint","dependencies":["a01e792probsolve12a-h2"],"title":"Rewrite","text":"Let\'s now rewrite the question into a simple sentence which sounds like an equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve12a-h4","type":"hint","dependencies":["a01e792probsolve12a-h3"],"title":"Rewrite","text":"We would get \\"120 is two thirds of the original price\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve12a-h5","type":"hint","dependencies":["a01e792probsolve12a-h4"],"title":"Translate","text":"Let\'s translate the rewritten sentence into a real equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve12a-h6","type":"hint","dependencies":["a01e792probsolve12a-h5"],"title":"Translate","text":"We get $$120=\\\\frac{2}{3} n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve12a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{3}$$"],"dependencies":["a01e792probsolve12a-h6"],"title":"Solve","text":"What number can we divide from both sides to isolate the variable?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve12a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$180$$"],"dependencies":["a01e792probsolve12a-h7"],"title":"Solve","text":"What is $$\\\\frac{120}{\\\\frac{2}{3}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve12a-h9","type":"hint","dependencies":["a01e792probsolve12a-h8"],"title":"Solve","text":"When dividing a number by a fraction we are actually multiplying by the reciprocal. This makes the equation $$120\\\\frac{3}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a01e792probsolve13","title":"Use a Problem-Solving Strategy for Word Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Use a Problem-Solving Strategy","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a01e792probsolve13a","stepAnswer":["$$40$$"],"problemType":"TextBox","stepTitle":"Two-fifths of the songs in Mariel\u2019s playlist are country. If there are $$16$$ country songs, what is the total number of songs in the playlist?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$40$$","hints":{"DefaultPathway":[{"id":"a01e792probsolve13a-h1","type":"hint","dependencies":[],"title":"Identify and Name","text":"Let\'s first identify what we want to find, and then give it a variable name","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve13a-h2","type":"hint","dependencies":["a01e792probsolve13a-h1"],"title":"Identify and Name","text":"We want to find the total number of songs and we can name it \\"n\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve13a-h3","type":"hint","dependencies":["a01e792probsolve13a-h2"],"title":"Rewrite","text":"Let\'s now rewrite the question into a simple sentence which sounds like an equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve13a-h4","type":"hint","dependencies":["a01e792probsolve13a-h3"],"title":"Rewrite","text":"It would be: two fifths of the total songs are country songs.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve13a-h5","type":"hint","dependencies":["a01e792probsolve13a-h4"],"title":"Translate","text":"Let\'s translate the rewritten sentence into a real equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve13a-h6","type":"hint","dependencies":["a01e792probsolve13a-h5"],"title":"Translate","text":"Now we would get \\"(2/5)*n=16\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve13a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{5}$$"],"dependencies":["a01e792probsolve13a-h6"],"title":"Solve","text":"What number can we divide from both sides to isolate the variable?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve13a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$40$$"],"dependencies":["a01e792probsolve13a-h7"],"title":"Solve","text":"What is $$\\\\frac{16}{\\\\frac{2}{5}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve13a-h9","type":"hint","dependencies":["a01e792probsolve13a-h8"],"title":"Solve","text":"When dividing a number by a fraction we are actually multiplying by the reciprocal. This makes the equation $$16\\\\frac{5}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a01e792probsolve14","title":"Use a Problem-Solving Strategy for Word Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Use a Problem-Solving Strategy","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a01e792probsolve14a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"Guillermo bought textbooks and notebooks at the bookstore. The number of textbooks was $$3$$ more than twice the number of notebooks. He bought $$7$$ textbooks. How many notebooks did he buy?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a01e792probsolve14a-h1","type":"hint","dependencies":[],"title":"Identify and Name","text":"Let\'s first identify what we want to find, and then give it a variable name","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve14a-h2","type":"hint","dependencies":["a01e792probsolve14a-h1"],"title":"Identify and Name","text":"We want to find the total number of notebooks and we can name it \\"n\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve14a-h3","type":"hint","dependencies":["a01e792probsolve14a-h2"],"title":"Rewrite","text":"Let\'s now rewrite the question into a simple sentence which sounds like an equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve14a-h4","type":"hint","dependencies":["a01e792probsolve14a-h3"],"title":"Rewrite","text":"It would be: The number of textbooks (7) is $$3$$ more than twice the number of notebooks","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve14a-h5","type":"hint","dependencies":["a01e792probsolve14a-h4"],"title":"Translate","text":"Let\'s translate the rewritten sentence into a real equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve14a-h6","type":"hint","dependencies":["a01e792probsolve14a-h5"],"title":"Translate","text":"Now, we would get $$7=3+2n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve14a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a01e792probsolve14a-h6"],"title":"Solve","text":"What number can we subtract from both sides to isolate the variable?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve14a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a01e792probsolve14a-h7"],"title":"Solve","text":"Now, we get $$4=2n$$. What is $$n$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve14a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a01e792probsolve14a-h8"],"title":"Simplify","text":"What number can we divide from both sides to isolate $$n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a01e792probsolve15","title":"Use a Problem-Solving Strategy for Word Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Use a Problem-Solving Strategy","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a01e792probsolve15a","stepAnswer":["$$7$$"],"problemType":"TextBox","stepTitle":"Gerry worked Sudoku puzzles and crossword puzzles this week. The number of Sudoku puzzles he completed is eight more than twice the number of crossword puzzles. He completed $$22$$ Sudoku puzzles. How many crossword puzzles did he do?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7$$","hints":{"DefaultPathway":[{"id":"a01e792probsolve15a-h1","type":"hint","dependencies":[],"title":"Identify and Name","text":"Let\'s first identify what we want to find, and then give it a variable name","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve15a-h2","type":"hint","dependencies":["a01e792probsolve15a-h1"],"title":"Identify and Name","text":"We want to find the total number of crossword puzzles and we can name it \\"c\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve15a-h3","type":"hint","dependencies":["a01e792probsolve15a-h2"],"title":"Rewrite","text":"Let\'s now rewrite the question into a simple sentence which sounds like an equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve15a-h4","type":"hint","dependencies":["a01e792probsolve15a-h3"],"title":"Rewrite","text":"It would be: The total number of puzzles (22) is $$8$$ more than twice the number of crossword puzzles.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve15a-h5","type":"hint","dependencies":["a01e792probsolve15a-h4"],"title":"Translate","text":"Let\'s translate the rewritten sentence into a real equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve15a-h6","type":"hint","dependencies":["a01e792probsolve15a-h5"],"title":"Translate","text":"Now, we would get $$22=8+2n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve15a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a01e792probsolve15a-h6"],"title":"Solve","text":"What number can we subtract from both sides to isolate the variable?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve15a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a01e792probsolve15a-h7"],"title":"Solve","text":"Now, we get $$(22-8)=2n$$ which is also $$14=2n$$. What is $$n$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve15a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a01e792probsolve15a-h8"],"title":"Simplify","text":"What number can we divide from both sides to isolate $$n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a01e792probsolve16","title":"Solve Number Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Use a Problem-Solving Strategy","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a01e792probsolve16a","stepAnswer":["$$25$$"],"problemType":"TextBox","stepTitle":"The difference of a number and eight is $$17$$. Find the number.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$25$$","hints":{"DefaultPathway":[{"id":"a01e792probsolve16a-h1","type":"hint","dependencies":[],"title":"Name","text":"Identify a variable for which you are going to represent the number we are looking for (n)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve16a-h2","type":"hint","dependencies":["a01e792probsolve16a-h1"],"title":"Translate","text":"Let\'s rewrite the statement into an equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve16a-h3","type":"hint","dependencies":["a01e792probsolve16a-h2"],"title":"Translate","text":"We would get $$n-8=17$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve16a-h4","type":"hint","dependencies":["a01e792probsolve16a-h3"],"title":"Solve","text":"Solve the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve16a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a01e792probsolve16a-h4"],"title":"Solve","text":"What number do we have to add to both sides to be left with only $$n$$ on the left side?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a01e792probsolve17","title":"Solving Number Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Use a Problem-Solving Strategy","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a01e792probsolve17a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"The difference of a number and eleven is $$-7$$. Find the number.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a01e792probsolve17a-h1","type":"hint","dependencies":[],"title":"Use a Variable","text":"Let the number be $$x$$. Set up an equation and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve17a-h2","type":"hint","dependencies":["a01e792probsolve17a-h1"],"title":"Setting up an Equation","text":"The difference of a number and $$11$$ can be represented as $$x-11$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve17a-h3","type":"hint","dependencies":["a01e792probsolve17a-h2"],"title":"The Equation","text":"The equation can be set up as $$x-11=-7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve17a-h4","type":"hint","dependencies":["a01e792probsolve17a-h3"],"title":"Answer","text":"The answer is that $$x=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a01e792probsolve18","title":"Solve Number Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Use a Problem-Solving Strategy","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a01e792probsolve18a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"The sum of twice a number and seven is $$15$$. Find the number.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a01e792probsolve18a-h1","type":"hint","dependencies":[],"title":"Use a Variable","text":"Let the number be $$x$$. Set up an equation and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve18a-h2","type":"hint","dependencies":["a01e792probsolve18a-h1"],"title":"Setting up an Equation","text":"The sum of twice a number and $$7$$ can be represented as $$2x+7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve18a-h3","type":"hint","dependencies":["a01e792probsolve18a-h2"],"title":"The Equation","text":"The equation can be set up as $$2x+7=15$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve18a-h4","type":"hint","dependencies":["a01e792probsolve18a-h3"],"title":"Answer","text":"The answer is that $$x=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a01e792probsolve19","title":"Solve Number Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Use a Problem-Solving Strategy","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a01e792probsolve19a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"The sum of four times a number and two is $$14$$. Find the number.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a01e792probsolve19a-h1","type":"hint","dependencies":[],"title":"Use a Variable","text":"Let the number be $$x$$. Set up an equation and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve19a-h2","type":"hint","dependencies":["a01e792probsolve19a-h1"],"title":"Setting up an Equation","text":"The sum of four times a number and $$2$$ can be represented as $$4x+2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve19a-h3","type":"hint","dependencies":["a01e792probsolve19a-h2"],"title":"The Equation","text":"The equation can be set up as $$4x+2=14$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve19a-h4","type":"hint","dependencies":["a01e792probsolve19a-h3"],"title":"Answer","text":"The answer is that $$x=3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a01e792probsolve2","title":"Solve Number Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Use a Problem-Solving Strategy","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a01e792probsolve2a","stepAnswer":["26,28,30"],"problemType":"MultipleChoice","stepTitle":"Find three consecutive even integers whose sum is $$84$$","stepBody":"","answerType":"string","variabilization":{},"choices":["20,22,24","22,24,28","26,28,30","27,28,29"],"hints":{"DefaultPathway":[{"id":"a01e792probsolve2a-h1","type":"hint","dependencies":[],"title":"Setting up the equation","text":"The first step is to rewrite the statement as a mathematical equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve2a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["n,n+2,n+4"],"dependencies":["a01e792probsolve2a-h1"],"title":"Expressing the numbers in terms of variables","text":"How can we express three consective even integers?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["n,n+1,n+2","n,n+3,n+5","n,n+2,n+4"]},{"id":"a01e792probsolve2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a01e792probsolve2a-h2"],"title":"Difference between two consecutive even numbers.","text":"What is $$6-4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a01e792probsolve2a-h3"],"title":"Difference between two consecutive even numbers.","text":"What is $$98-96$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve2a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a01e792probsolve2a-h4"],"title":"Generalization about the difference between two consecutive even numbers","text":"Based on the examples above, what is the difference between any two consecutive even numbers?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$84$$"],"dependencies":["a01e792probsolve2a-h5"],"title":"Total sum","text":"What should the numbers add up to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve2a-h7","type":"hint","dependencies":["a01e792probsolve2a-h6"],"title":"Solve for $$n$$","text":"The next step is to solve for $$n$$ and find the three consecutive even numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a01e792probsolve20","title":"Solve Number Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Use a Problem-Solving Strategy","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a01e792probsolve20a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"The sum of three times a number and seven is $$25$$. Find the number.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"a01e792probsolve20a-h1","type":"hint","dependencies":[],"title":"Use a Variable","text":"Let the number be $$x$$. Set up an equation and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve20a-h2","type":"hint","dependencies":["a01e792probsolve20a-h1"],"title":"Setting up an Equation","text":"The sum of three times a number and $$7$$ can be represented as $$3x+7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve20a-h3","type":"hint","dependencies":["a01e792probsolve20a-h2"],"title":"The Equation","text":"The equation can be set up as $$3x+7=25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve20a-h4","type":"hint","dependencies":["a01e792probsolve20a-h3"],"title":"Answer","text":"The answer is that $$x=6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a01e792probsolve21","title":"Solve Number Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Use a Problem-Solving Strategy","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a01e792probsolve21a","stepAnswer":["8,13"],"problemType":"TextBox","stepTitle":"One number is five more than another. The sum of the numbers is $$21$$. Find the numbers.","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a01e792probsolve21a-h1","type":"hint","dependencies":[],"title":"Use a Variable","text":"Let the first number be $$x$$. Write the second number in relation to the first.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve21a-h2","type":"hint","dependencies":["a01e792probsolve21a-h1"],"title":"The Second Number","text":"The second number can be written as $$x+5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve21a-h3","type":"hint","dependencies":["a01e792probsolve21a-h2"],"title":"Setting up an Equation","text":"The sum of the two numbers can be written as $$x+x+5$$, or $$2x+5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve21a-h4","type":"hint","dependencies":["a01e792probsolve21a-h3"],"title":"The Equation","text":"The equation can be set up as $$2x+5=21$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve21a-h5","type":"hint","dependencies":["a01e792probsolve21a-h4"],"title":"Answer","text":"The two numbers are $$8$$ and $$13$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a01e792probsolve22","title":"Solve Number Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Use a Problem-Solving Strategy","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a01e792probsolve22a","stepAnswer":["9,15"],"problemType":"TextBox","stepTitle":"One number is six more than another. The sum of the numbers is twenty-four. Find the numbers.","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a01e792probsolve22a-h1","type":"hint","dependencies":[],"title":"Use a Variable","text":"Let the first number be $$x$$. Write the second number in relation to the first.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve22a-h2","type":"hint","dependencies":["a01e792probsolve22a-h1"],"title":"The Second Number","text":"The second number can be written as $$x+6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve22a-h3","type":"hint","dependencies":["a01e792probsolve22a-h2"],"title":"Setting up an Equation","text":"The sum of the two numbers can be written as $$x+x+6$$, or $$2x+6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve22a-h4","type":"hint","dependencies":["a01e792probsolve22a-h3"],"title":"The Equation","text":"The equation can be set up as $$2x+6=24$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve22a-h5","type":"hint","dependencies":["a01e792probsolve22a-h4"],"title":"Answer","text":"The two numbers are $$9$$ and $$15$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a01e792probsolve23","title":"Solve Number Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Use a Problem-Solving Strategy","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a01e792probsolve23a","stepAnswer":["27,31"],"problemType":"TextBox","stepTitle":"The sum of two numbers is fifty-eight. One number is four more than the other. Find the numbers.","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a01e792probsolve23a-h1","type":"hint","dependencies":[],"title":"Use a Variable","text":"Let the first number be $$x$$. Write the second number in relation to the first.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve23a-h2","type":"hint","dependencies":["a01e792probsolve23a-h1"],"title":"The Second Number","text":"The second number can be written as $$x+4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve23a-h3","type":"hint","dependencies":["a01e792probsolve23a-h2"],"title":"Setting up an Equation","text":"The sum of the two numbers can be written as $$x+x+4$$, or $$2x+4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve23a-h4","type":"hint","dependencies":["a01e792probsolve23a-h3"],"title":"The Equation","text":"The equation can be set up as $$2x+4=58$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve23a-h5","type":"hint","dependencies":["a01e792probsolve23a-h4"],"title":"Answer","text":"The two numbers are $$27$$ and $$31$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a01e792probsolve24","title":"Solve Number Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Use a Problem-Solving Strategy","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a01e792probsolve24a","stepAnswer":["-5,-9"],"problemType":"TextBox","stepTitle":"The sum of two numbers is negative fourteen. One number is four less than the other. Find the numbers.","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a01e792probsolve24a-h1","type":"hint","dependencies":[],"title":"Use a Variable","text":"Let the first number be $$x$$. Write the second number in relation to the first.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve24a-h2","type":"hint","dependencies":["a01e792probsolve24a-h1"],"title":"The Second Number","text":"The second number can be written as $$x-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve24a-h3","type":"hint","dependencies":["a01e792probsolve24a-h2"],"title":"Setting up an Equation","text":"The sum of the two numbers can be written as $$x+x-4$$, or $$2x-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve24a-h4","type":"hint","dependencies":["a01e792probsolve24a-h3"],"title":"The Equation","text":"The equation can be set up as $$2x-4=-14$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve24a-h5","type":"hint","dependencies":["a01e792probsolve24a-h4"],"title":"Answer","text":"The two numbers are $$-5$$ and $$-9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a01e792probsolve25","title":"Solve Number Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Use a Problem-Solving Strategy","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a01e792probsolve25a","stepAnswer":["4,-3"],"problemType":"TextBox","stepTitle":"One number is ten more than twice another. Their sum is one. Find the numbers.","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a01e792probsolve25a-h1","type":"hint","dependencies":[],"title":"Define variables","text":"Give the first unknown number a variable (like x) and set the second number interms of $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve25a-h2","type":"hint","dependencies":["a01e792probsolve25a-h1"],"title":"Setting an equation","text":"Based on the question if $$x$$ is the first number then $$2x+10$$ would be the second number. Now create an equation representing the sum of the two numbers which equals $$1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve25a-h3","type":"hint","dependencies":["a01e792probsolve25a-h2"],"title":"Seperating constants","text":"The equation equals $$x+2x+10=1$$. Now we will seperate the constant numbers to one side. Remeber if you subtract from one side you are also subtracting from the other side","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve25a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":["a01e792probsolve25a-h3"],"title":"Seperating constants","text":"What is $$1-10$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve25a-h5","type":"hint","dependencies":["a01e792probsolve25a-h4"],"title":"Seperating variable","text":"The equation equals $$x+2x+10=1$$. Now we will seperate the variables to one side. Remeber if you subtract from one side you are also subtracting from the other side","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve25a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x$$"],"dependencies":["a01e792probsolve25a-h5"],"title":"Seperating variable","text":"what is $$2x+1x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve25a-h7","type":"hint","dependencies":["a01e792probsolve25a-h6"],"title":"Find $$x$$","text":"Now that we have seperated $$x$$ and constants your equations should be $$3x=-9$$. Now find what is the value of $$x$$ by dividing both sides by $$3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve25a-h8","type":"hint","dependencies":["a01e792probsolve25a-h7"],"title":"Number $$2$$","text":"Now that you have found the first number which is $$x$$, plug that number back into the equation for the second number to get the value for the second number","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a01e792probsolve26","title":"Solve Number Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Use a Problem-Solving Strategy","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a01e792probsolve26a","stepAnswer":["-4,0"],"problemType":"TextBox","stepTitle":"One number is eight more than twice another. Their sum is negative four. Find the numbers","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a01e792probsolve26a-h1","type":"hint","dependencies":[],"title":"Define variables","text":"Give the first unknown number a variable (like x) and set the second number interms of $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve26a-h2","type":"hint","dependencies":["a01e792probsolve26a-h1"],"title":"Setting an equation","text":"Based on the question if $$x$$ is the first number then $$2x+8$$ would be the second number. Now create an equation representing the sum of the two numbers which equals $$-4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve26a-h3","type":"hint","dependencies":["a01e792probsolve26a-h2"],"title":"Seperating constants","text":"The equation equals $$x+2x+8=-4$$. Now we will seperate the constant numbers to one side. Remeber if you subtract from one side you are also subtracting from the other side","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve26a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-12$$"],"dependencies":["a01e792probsolve26a-h3"],"title":"Seperating constants","text":"what is $$-4-8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve26a-h5","type":"hint","dependencies":["a01e792probsolve26a-h4"],"title":"Seperating variable","text":"The equation equals $$x+2x+8=-4$$. Now we will seperate the variable to one side. Remeber if you subtract from one side you are also subtracting from the other side","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve26a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x$$"],"dependencies":["a01e792probsolve26a-h5"],"title":"Seperating variable","text":"what is $$2x+1x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve26a-h7","type":"hint","dependencies":["a01e792probsolve26a-h6"],"title":"Find $$x$$","text":"Now that we have seperated $$x$$ and constants your equations should be $$3x=-12$$. Now find what is the value of $$x$$ by dividing both sides by $$3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve26a-h8","type":"hint","dependencies":["a01e792probsolve26a-h7"],"title":"Number $$2$$","text":"Now that you have found the first number which is $$x$$, plug that number back into the equation for the second number to get the value for the second number","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a01e792probsolve27","title":"Solve Number Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Use a Problem-Solving Strategy","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a01e792probsolve27a","stepAnswer":["-2,-3"],"problemType":"TextBox","stepTitle":"One number is three more than three times another. Their sum is $$-5$$. Find the numbers.","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a01e792probsolve27a-h1","type":"hint","dependencies":[],"title":"Define variables","text":"Give the first unknown number a variable (like x) and set the second number interms of $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve27a-h2","type":"hint","dependencies":["a01e792probsolve27a-h1"],"title":"Setting an equation","text":"Based on the question if $$x$$ is the first number then $$3x+3$$ would be the second number. Now create an equation representing the sum of the two numbers which equals $$-5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve27a-h3","type":"hint","dependencies":["a01e792probsolve27a-h2"],"title":"Seperating constants","text":"The equation equals $$x+3x+3=-5$$. Now we will seperate the constant numbers to one side. Remeber if you subtract from one side you are also subtracting from the other side","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve27a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-8$$"],"dependencies":["a01e792probsolve27a-h3"],"title":"Seperating constants","text":"what is $$-5-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve27a-h5","type":"hint","dependencies":["a01e792probsolve27a-h4"],"title":"Seperating variable","text":"The equation equals $$x+3x+3=-5$$. Now we will seperate the variables to one side. Remeber if you subtract from one side you are also subtracting from the other side","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve27a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4x$$"],"dependencies":["a01e792probsolve27a-h5"],"title":"Seperating variable","text":"what is $$3x+1x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve27a-h7","type":"hint","dependencies":["a01e792probsolve27a-h6"],"title":"Find $$x$$","text":"Now that we have seperated $$x$$ and constants your equations should be $$4x=-8$$. Now find what is the value of $$x$$ by dividing both sides by $$4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve27a-h8","type":"hint","dependencies":["a01e792probsolve27a-h7"],"title":"Number $$2$$","text":"Now that you have found the first number which is $$x$$, plug that number back into the equation for the second number to get the value for the second number","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a01e792probsolve28","title":"Solve Number Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Use a Problem-Solving Strategy","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a01e792probsolve28a","stepAnswer":["23,24"],"problemType":"TextBox","stepTitle":"The sum of two consecutive integers is $$47$$. Find the numbers.","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a01e792probsolve28a-h1","type":"hint","dependencies":[],"title":"Define variables","text":"Give the first unknown number a variable (like x) and set the second number interms of $$x$$. Since the problem says consecutive numbers second number would be $$x+1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve28a-h2","type":"hint","dependencies":["a01e792probsolve28a-h1"],"title":"Setting an equation","text":"Based on the question the two numbers are $$x$$ and $$x+1$$ which add up to $$47$$. Try to write an equation based on what you know","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve28a-h3","type":"hint","dependencies":["a01e792probsolve28a-h2"],"title":"Seperate Constants","text":"The equation equals $$x+x+1=47$$. Now we will seperate the constant numbers to one side. Remeber if you subtract from one side you are also subtracting from the other side","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve28a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$46$$"],"dependencies":["a01e792probsolve28a-h3"],"title":"Seperating constants","text":"what is $$47-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve28a-h5","type":"hint","dependencies":["a01e792probsolve28a-h4"],"title":"Seperating variable","text":"The equation equals $$x+x+1=47$$. Now we will seperate the variables to one side. Remeber if you subtract from one side you are also subtracting from the other side","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve28a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x$$"],"dependencies":["a01e792probsolve28a-h5"],"title":"Seperating variable","text":"what is $$x+x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve28a-h7","type":"hint","dependencies":["a01e792probsolve28a-h6"],"title":"Find $$x$$","text":"Now that we have seperated $$x$$ and constants your equations should be $$2x=46$$. Now find what is the value of $$x$$ by dividing both sides by $$2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve28a-h8","type":"hint","dependencies":["a01e792probsolve28a-h7"],"title":"Number $$2$$","text":"Now that you have found the first number which is $$x$$, add one to it to get the second number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a01e792probsolve29","title":"Solve Number Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Use a Problem-Solving Strategy","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a01e792probsolve29a","stepAnswer":["47,48"],"problemType":"TextBox","stepTitle":"The sum of two consecutive integers is $$95$$. Find the numbers.","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a01e792probsolve29a-h1","type":"hint","dependencies":[],"title":"Define variables","text":"Give the first unknown number a variable (like x) and set the second number interms of $$x$$. Since the problem says consecutive numbers second number would be $$x+1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve29a-h2","type":"hint","dependencies":["a01e792probsolve29a-h1"],"title":"Setting an equation","text":"Based on the question the two numbers are $$x$$ and $$x+1$$ which add up to $$95$$. Try to write an equation based on what you know","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve29a-h3","type":"hint","dependencies":["a01e792probsolve29a-h2"],"title":"Seperate Constants","text":"The equation equals $$x+x+1=47$$. Now we will seperate the constant numbers to one side. Remeber if you subtract from one side you are also subtracting from the other side","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve29a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$94$$"],"dependencies":["a01e792probsolve29a-h3"],"title":"Seperating constants","text":"what is $$95-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve29a-h5","type":"hint","dependencies":["a01e792probsolve29a-h4"],"title":"Seperating variable","text":"The equation equals $$x+x+1=95$$. Now we will seperate the variables to one side. Remeber if you subtract from one side you are also subtracting from the other side","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve29a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x$$"],"dependencies":["a01e792probsolve29a-h5"],"title":"Seperating variable","text":"what is $$x+x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve29a-h7","type":"hint","dependencies":["a01e792probsolve29a-h6"],"title":"Find $$x$$","text":"Now that we have seperated $$x$$ and constants your equations should be $$2x=94$$. Now find what is the value of $$x$$ by dividing both sides by $$2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve29a-h8","type":"hint","dependencies":["a01e792probsolve29a-h7"],"title":"Number $$2$$","text":"Now that you have found the first number which is $$x$$, add one to it to get the second number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a01e792probsolve3","title":"Solve number problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Use a Problem-Solving Strategy","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a01e792probsolve3a","stepAnswer":["$42,000"],"problemType":"MultipleChoice","stepTitle":"A married couple together earns $110,000 a year. The wife earns $16,000 less than twice what her husband earns. What does the husband earn?","stepBody":"","answerType":"string","variabilization":{},"choices":["$44,000","$38,000","$42,000","$36,000"],"hints":{"DefaultPathway":[{"id":"a01e792probsolve3a-h1","type":"hint","dependencies":[],"title":"Translate words to expressions","text":"The first step is to express the amount the husband and wife earns as a mathematical expressions","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve3a-h2","type":"hint","dependencies":["a01e792probsolve3a-h1"],"title":"Define the variable for husband","text":"Let $$h$$ represent the amount husband earns","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2h$$"],"dependencies":["a01e792probsolve3a-h2"],"title":"Write wife\'s earnings in terms of the husband\'s earnings","text":"What is twice the amount the husband earns?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2h-16000$$"],"dependencies":["a01e792probsolve3a-h3"],"title":"Write wife\'s earnings in terms of the husband\'s earnings","text":"What is $$16000$$ less than the previous answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2h-16000$$"],"dependencies":["a01e792probsolve3a-h4"],"title":"Write wife\'s earnings in terms of the husband\'s earnings","text":"How much does the wife earn?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve3a-h6","type":"hint","dependencies":["a01e792probsolve3a-h5"],"title":"Translate the problem into an equation","text":"The next step is to express their total earnings as a mathematical equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve3a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$110000$$"],"dependencies":["a01e792probsolve3a-h6"],"title":"Total sum of earnings","text":"How much does the couple together earn per year? Write the answer without the dollar sign","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve3a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$h+2h-16000=110000$$"],"dependencies":["a01e792probsolve3a-h7"],"title":"Writing the equation","text":"How can we express \\"husband\'s $$earning+wife\'s$$ $$earning=110000\\"$$ mathematically?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve3a-h9","type":"hint","dependencies":["a01e792probsolve3a-h8"],"title":"Solving the equation","text":"The last step is to solve for $$h$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve3a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Add $$16000$$, divide by $$2$$"],"dependencies":["a01e792probsolve3a-h9"],"title":"Solving the equation","text":"What should we do to both sides of the equation in order to isolate $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Divide by $$2$$, add $$16000$$","Add $$16000$$, divide by $$2$$","Subtract $$110000$$","none of the above"]}]}}]},{"id":"a01e792probsolve30","title":"Solve Number Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Use a Problem-Solving Strategy","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a01e792probsolve30a","stepAnswer":["-16,-15"],"problemType":"TextBox","stepTitle":"The sum of two consecutive integers is $$-31$$. Find the numbers.","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a01e792probsolve30a-h1","type":"hint","dependencies":[],"title":"Define variables","text":"Give the first unknown number a variable (like x) and set the second number interms of $$x$$. Since the problem says consecutive numbers second number would be $$x+1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve30a-h2","type":"hint","dependencies":["a01e792probsolve30a-h1"],"title":"Setting an equation","text":"Based on the question the two numbers are $$x$$ and $$x+1$$ which add up to $$-31$$. Try to write an equation based on what you know","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve30a-h3","type":"hint","dependencies":["a01e792probsolve30a-h2"],"title":"Seperate Constants","text":"The equation equals $$x+x+1=-31$$. Now we will seperate the constant numbers to one side. Remeber if you subtract from one side you are also subtracting from the other side","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve30a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-32$$"],"dependencies":["a01e792probsolve30a-h3"],"title":"Seperating constants","text":"what is $$-31-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve30a-h5","type":"hint","dependencies":["a01e792probsolve30a-h4"],"title":"Seperating variable","text":"The equation equals $$x+x+1=-31$$. Now we will seperate the variables to one side. Remeber if you subtract from one side you are also subtracting from the other side","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve30a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x$$"],"dependencies":["a01e792probsolve30a-h5"],"title":"Seperating variable","text":"what is $$x+x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve30a-h7","type":"hint","dependencies":["a01e792probsolve30a-h6"],"title":"Find $$x$$","text":"Now that we have seperated $$x$$ and constants your equations should be $$2x=-32$$. Now find what is the value of $$x$$ by dividing both sides by $$2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve30a-h8","type":"hint","dependencies":["a01e792probsolve30a-h7"],"title":"Number $$2$$","text":"Now that you have found the first number which is $$x$$, add one to it to get the second number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a01e792probsolve4","title":"Solve Number Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Use a Problem-Solving Strategy","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a01e792probsolve4a","stepAnswer":["$$-33, -32, -31$$"],"problemType":"MultipleChoice","stepTitle":"Find three consecutive integers whose sum is $$-96$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$-33, -32, -31$$","$$-33, -34, -35$$","$$-32, -33, -34$$"],"hints":{"DefaultPathway":[{"id":"a01e792probsolve4a-h1","type":"hint","dependencies":[],"title":"Set of the math equation","text":"The first step is to express the question as a mathematical expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve4a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["n,n+1,n+2"],"dependencies":["a01e792probsolve4a-h1"],"title":"Consecutive numbers","text":"The problem states that the integers are consecutive. How can we express that condition?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["n,n+1,n+2","n,n+3,n+5","n,n+2,n+4"]},{"id":"a01e792probsolve4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-96$$"],"dependencies":["a01e792probsolve4a-h2"],"title":"Total sum","text":"What should the numbers add up to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve4a-h4","type":"hint","dependencies":["a01e792probsolve4a-h3"],"title":"The next step is to solve the math equation","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a01e792probsolve5","title":"Solve Number 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$$n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-13$$"],"dependencies":["a01e792probsolve5a-h4"],"title":"Solving for $$n$$","text":"What is the value of $$n$$ in the equation $$3n+3=-36$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve5a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-12$$"],"dependencies":["a01e792probsolve5a-h5"],"title":"Finding the consecutive numbers","text":"What is $$n+1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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$$102$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["30,32,34","31,32,33","32,34,36","33,34,45"],"hints":{"DefaultPathway":[{"id":"a01e792probsolve6a-h1","type":"hint","dependencies":[],"title":"Translate the problem into an equation","text":"The first step is to translate the question into a mathematical equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve6a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["n,n+2,n+4"],"dependencies":["a01e792probsolve6a-h1"],"title":"Writing consecutive numbers","text":"Which of the following represent consecutive numbers in terms of $$n$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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$$3n+6=102$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve6a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$34$$"],"dependencies":["a01e792probsolve6a-h5"],"title":"Finding the consecutive numbers","text":"What is $$n+2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve6a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$36$$"],"dependencies":["a01e792probsolve6a-h6"],"title":"Finding the consecutive numbers","text":"What is $$n+4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a01e792probsolve7","title":"Solving Number Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Use a Problem-Solving Strategy","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a01e792probsolve7a","stepAnswer":["$$-10, -8, -6$$"],"problemType":"MultipleChoice","stepTitle":"Find three consecutive even integers whose sum is $$-24$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$-10, -8, -6$$","$$-10, -12, -14$$","10,12,14","6,8,10"],"hints":{"DefaultPathway":[{"id":"a01e792probsolve7a-h1","type":"hint","dependencies":[],"title":"Translate the problem into an equation","text":"The first step is to translate the question into a mathematical equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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$$n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve7a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-10$$"],"dependencies":["a01e792probsolve7a-h4"],"title":"Solving for $$n$$","text":"What is the value of $$n$$ in the equation $$3n+6=102$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve7a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-8$$"],"dependencies":["a01e792probsolve7a-h5"],"title":"Finding the consecutive numbers","text":"What is $$n+2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve7a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6$$"],"dependencies":["a01e792probsolve7a-h6"],"title":"Finding the consecutive numbers","text":"What is $$n+4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a01e792probsolve8","title":"Solve Number Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Use a Problem-Solving Strategy","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a01e792probsolve8a","stepAnswer":["$5,000"],"problemType":"MultipleChoice","stepTitle":"According to the National Automobile Dealers Association, the average cost of a car in $$2014$$ was $28,500. This was $1,500 less than $$6$$ times the cost in $$1975$$. What was the average cost of a car in 1975?","stepBody":"","answerType":"string","variabilization":{},"choices":["$4,000","$5,000","$6,000","$7,000"],"hints":{"DefaultPathway":[{"id":"a01e792probsolve8a-h1","type":"hint","dependencies":[],"title":"Translate words to expressions","text":"The first step is to translate the relationship between the $$2014$$ and $$1975$$ cost in terms of a mathematical equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve8a-h2","type":"hint","dependencies":["a01e792probsolve8a-h1"],"title":"Define the variable for the $$1975$$ cost","text":"Let $$x$$ $$=$$ average cost of car in $$1975$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6x$$"],"dependencies":["a01e792probsolve8a-h2"],"title":"Write the $$2014$$ cost in terms of the $$1975$$ cost","text":"What is six times the amount the husband earns?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6h-1500$$"],"dependencies":["a01e792probsolve8a-h3"],"title":"Write the $$2014$$ cost in terms of the $$1975$$ cost","text":"What is $$1500$$ less than the previous answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve8a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$28500$$"],"dependencies":["a01e792probsolve8a-h4"],"title":"Write the $$2014$$ cost in terms of the $$1975$$ cost","text":"What should $$6x-1500$$ be equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve8a-h6","type":"hint","dependencies":["a01e792probsolve8a-h5"],"title":"Solving the equation","text":"The last step is to solve for $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve8a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Add $$1500$$, divide by $$6$$"],"dependencies":["a01e792probsolve8a-h6"],"title":"Solving the equation","text":"What should we do to both sides of the equation in order to isolate $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Divide by $$6$$, add $$1500$$","Add $$1500$$, divide by $$6$$","Subtract $$28500$$","none of the above"]}]}}]},{"id":"a01e792probsolve9","title":"Use a Problem-Solving Strategy for Word Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Use a Problem-Solving Strategy","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a01e792probsolve9a","stepAnswer":["$$36$$"],"problemType":"TextBox","stepTitle":"Pilar bought a purse on sale for $18, which is one-half of the original price. What was the original price of the purse?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$36$$","hints":{"DefaultPathway":[{"id":"a01e792probsolve9a-h1","type":"hint","dependencies":[],"title":"Rewrite","text":"Rewrite the question in a manner that sounds like an equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve9a-h2","type":"hint","dependencies":["a01e792probsolve9a-h1"],"title":"Rewrite","text":"We can rewrite it as: $$18$$ is one half of the original price","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve9a-h3","type":"hint","dependencies":["a01e792probsolve9a-h2"],"title":"Translate","text":"Use the rewritten sentence, we have to translate the world problem into to an equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve9a-h4","type":"hint","dependencies":["a01e792probsolve9a-h3"],"title":"Translate","text":"After translating the problem into an equation we get: $$18=\\\\frac{1}{2} p$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve9a-h5","type":"hint","dependencies":["a01e792probsolve9a-h4"],"title":"Solve","text":"Now, solve for $$p$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve9a-h6","type":"hint","dependencies":["a01e792probsolve9a-h5"],"title":"Solve","text":"First, we can multiply both sides by $$2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a01e792probsolve9a-h7","type":"hint","dependencies":["a01e792probsolve9a-h6"],"title":"Simplify","text":"Now, we can simply the expression to find $$p!$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a02e810proportions1","title":"Solving for Variables in Proportions","body":"Find the value of $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Solve Proportion and Similar Figure Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a02e810proportions1a","stepAnswer":["$$36$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{x}{63}=\\\\frac{4}{7}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$36$$","hints":{"DefaultPathway":[{"id":"a02e810proportions1a-h1","type":"hint","dependencies":[],"title":"Identifying the Solution","text":"To isolate $$x$$, multiply both sides by the LCD, in this case, $$63$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions1a-h2","type":"hint","dependencies":["a02e810proportions1a-h1"],"title":"Simplifying the Equation","text":"Simplify both sides by reducing and eliminating the denominater under $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions1a-h3","type":"hint","dependencies":["a02e810proportions1a-h2"],"title":"Answer","text":"The answer is $$36$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a02e810proportions10","title":"Solving for Variables in Proportions","body":"Find the value of the variable.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Solve Proportion and Similar Figure Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a02e810proportions10a","stepAnswer":["$$60$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{p+12}{9}=\\\\frac{p-12}{6}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$60$$","hints":{"DefaultPathway":[{"id":"a02e810proportions10a-h1","type":"hint","dependencies":[],"title":"Identifying the Solution","text":"Find the LCD of the two denominators.To isolate $$y$$, multiply both sides by the LCD.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions10a-h2","type":"hint","dependencies":["a02e810proportions10a-h1"],"title":"Simplifying the Equation","text":"Simplify both sides by reducing and eliminating the denominater under the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions10a-h3","type":"hint","dependencies":["a02e810proportions10a-h2"],"title":"Divide","text":"Divide on both sides to isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions10a-h4","type":"hint","dependencies":["a02e810proportions10a-h3"],"title":"Answer","text":"The answer is $$60$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a02e810proportions11","title":"Solving for Variables in Proportions","body":"Find the value of the variable.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Solve Proportion and Similar Figure Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a02e810proportions11a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2x+15}{9}=\\\\frac{7x+3}{15}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"a02e810proportions11a-h1","type":"hint","dependencies":[],"title":"Identifying the Solution","text":"Find the LCD of the two denominators.To isolate the variable, multiply both sides by the LCD.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions11a-h2","type":"hint","dependencies":["a02e810proportions11a-h1"],"title":"Simplifying the Equation","text":"Simplify both sides by reducing and eliminating the denominater under the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions11a-h3","type":"hint","dependencies":["a02e810proportions11a-h2"],"title":"Divide","text":"Divide on both sides to isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions11a-h4","type":"hint","dependencies":["a02e810proportions11a-h3"],"title":"Answer","text":"The answer is $$6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a02e810proportions12","title":"Solving for Variables in Proportions","body":"Find the value of the variable.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Solve Proportion and Similar Figure Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a02e810proportions12a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2x+15}{9}=\\\\frac{7x+3}{15}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"a02e810proportions12a-h1","type":"hint","dependencies":[],"title":"Identifying the Solution","text":"Find the LCD of the two denominators.To isolate the variable, multiply both sides by the LCD.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions12a-h2","type":"hint","dependencies":["a02e810proportions12a-h1"],"title":"Simplifying the Equation","text":"Simplify both sides by reducing and eliminating the denominater under the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions12a-h3","type":"hint","dependencies":["a02e810proportions12a-h2"],"title":"Divide","text":"Divide on both sides to isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions12a-h4","type":"hint","dependencies":["a02e810proportions12a-h3"],"title":"Answer","text":"The answer is $$6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a02e810proportions13","title":"Solving for Variables in Proportions","body":"Find the value of the variable.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Solve Proportion and Similar Figure Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a02e810proportions13a","stepAnswer":["$$49$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{x}{56}=\\\\frac{7}{8}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$49$$","hints":{"DefaultPathway":[{"id":"a02e810proportions13a-h1","type":"hint","dependencies":[],"title":"Identifying the Solution","text":"To isolate $$x$$, multiply both sides by $$56$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions13a-h2","type":"hint","dependencies":["a02e810proportions13a-h1"],"title":"Simplifying the Equation","text":"Simplify both sides by reducing.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions13a-h3","type":"hint","dependencies":["a02e810proportions13a-h2"],"title":"Answer","text":"The answer is $$49$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a02e810proportions14","title":"Solving for Variables in Proportions","body":"Find the value of the variable.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Solve Proportion and Similar Figure Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a02e810proportions14a","stepAnswer":["$$7$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{49}{63}=\\\\frac{z}{9}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7$$","hints":{"DefaultPathway":[{"id":"a02e810proportions14a-h1","type":"hint","dependencies":[],"title":"Identifying the Solution","text":"To isolate $$z$$, multiply both sides by $$9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions14a-h2","type":"hint","dependencies":["a02e810proportions14a-h1"],"title":"Simplifying the Equation","text":"Simplify both sides by reducing.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions14a-h3","type":"hint","dependencies":["a02e810proportions14a-h2"],"title":"Answer","text":"The answer is $$7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a02e810proportions15","title":"Solving for Variables in Proportions","body":"Find the value of the variable.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Solve Proportion and Similar Figure Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a02e810proportions15a","stepAnswer":["$$448$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{56}{72}=\\\\frac{y}{9}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$448$$","hints":{"DefaultPathway":[{"id":"a02e810proportions15a-h1","type":"hint","dependencies":[],"title":"Identifying the Solution","text":"To isolate $$y$$, multiply both sides by $$9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions15a-h2","type":"hint","dependencies":["a02e810proportions15a-h1"],"title":"Simplifying the Equation","text":"Simplify both sides by reducing.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions15a-h3","type":"hint","dependencies":["a02e810proportions15a-h2"],"title":"Answer","text":"The answer is $$448$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a02e810proportions16","title":"Solve Proportion and Similar Figure Applications","body":"Solve the following exercise:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Solve Proportion and Similar Figure Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a02e810proportions16a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2.8}{k}=\\\\frac{2.1}{1.5}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a02e810proportions16a-h1","type":"hint","dependencies":[],"title":"Cross Multiply","text":"Cross multiply by taking the numerator of the first expression and multiplying it with the denominator of the second experssion. Then cross multiply the remaining numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions16a-h2","type":"hint","dependencies":["a02e810proportions16a-h1"],"title":"Reformatting the Expression","text":"Create a new expression by setting the first cross multiplication product with the second resulting in $$2.1k=4.2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions16a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a02e810proportions16a-h2"],"title":"Solving for the Variable","text":"Isolate the variable and solve. What is the value of the variable?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a02e810proportions17","title":"Solve Proportion and Similar Figure Applications","body":"Solve the following exercise:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Solve Proportion and Similar Figure Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a02e810proportions17a","stepAnswer":["$$16$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{a}{a+12}=\\\\frac{4}{7}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16$$","hints":{"DefaultPathway":[{"id":"a02e810proportions17a-h1","type":"hint","dependencies":[],"title":"Cross Multiply","text":"Cross multiply by taking the numerator of the first expression and multiplying it with the denominator of the second experssion. Then cross multiply the remaining numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions17a-h2","type":"hint","dependencies":["a02e810proportions17a-h1"],"title":"Reformatting the Expression","text":"Create a new expression by setting the first cross multiplication product with the second resulting in $$7a=4a+48$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions17a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["a02e810proportions17a-h2"],"title":"Solving for the Variable","text":"Isolate the variable and solve. What is the value of the variable?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a02e810proportions18","title":"Solve Proportion and Similar Figure Applications","body":"Solve the following exercise:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Solve Proportion and Similar Figure Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a02e810proportions18a","stepAnswer":["$$63$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{b}{b-16}=\\\\frac{11}{9}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$63$$","hints":{"DefaultPathway":[{"id":"a02e810proportions18a-h1","type":"hint","dependencies":[],"title":"Cross Multiply","text":"Cross multiply by taking the numerator of the first expression and multiplying it with the denominator of the second experssion. Then cross multiply the remaining numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions18a-h2","type":"hint","dependencies":["a02e810proportions18a-h1"],"title":"Reformatting the Expression","text":"Create a new expression by setting the first cross multiplication product with the second resulting in $$9b=11b-126$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions18a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$63$$"],"dependencies":["a02e810proportions18a-h2"],"title":"Solving for the Variable","text":"Isolate the variable and solve. What is the value of the variable?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a02e810proportions19","title":"Solve Proportion and Similar Figure Applications","body":"Solve the following exercise:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Solve Proportion and Similar Figure Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a02e810proportions19a","stepAnswer":["$$43$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{c}{c-104}=\\\\left(-\\\\frac{5}{8}\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$43$$","hints":{"DefaultPathway":[{"id":"a02e810proportions19a-h1","type":"hint","dependencies":[],"title":"Cross Multiply","text":"Cross multiply by taking the numerator of the first expression and multiplying it with the denominator of the second experssion. Then cross multiply the remaining numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions19a-h2","type":"hint","dependencies":["a02e810proportions19a-h1"],"title":"Reformatting the Expression","text":"Create a new expression by setting the first cross multiplication product with the second resulting in $$8c=\\\\left(-5c\\\\right)+520$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions19a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$43$$"],"dependencies":["a02e810proportions19a-h2"],"title":"Solving for the Variable","text":"Isolate the variable and solve. What is the value of the variable?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a02e810proportions2","title":"Solving for Variables in Proportions","body":"Find the value of the variable.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Solve Proportion and Similar Figure Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a02e810proportions2a","stepAnswer":["$$77$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{n}{84}=\\\\frac{11}{12}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$77$$","hints":{"DefaultPathway":[{"id":"a02e810proportions2a-h1","type":"hint","dependencies":[],"title":"Identifying the Solution","text":"To isolate $$n$$, multiply both sides by the LCD, in this case, $$84$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions2a-h2","type":"hint","dependencies":["a02e810proportions2a-h1"],"title":"Simplifying the Equation","text":"Simplify both sides by reducing and eliminating the denominater under $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions2a-h3","type":"hint","dependencies":["a02e810proportions2a-h2"],"title":"Answer","text":"The answer is $$77$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a02e810proportions20","title":"Solve Proportion and Similar Figure Applications","body":"Solve the following exercise:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Solve Proportion and Similar Figure Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a02e810proportions20a","stepAnswer":["$$39$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{d}{d-48}=\\\\left(-\\\\frac{13}{3}\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$39$$","hints":{"DefaultPathway":[{"id":"a02e810proportions20a-h1","type":"hint","dependencies":[],"title":"Cross Multiply","text":"Cross multiply by taking the numerator of the first expression and multiplying it with the denominator of the second experssion. Then cross multiply the remaining numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions20a-h2","type":"hint","dependencies":["a02e810proportions20a-h1"],"title":"Reformatting the Expression","text":"Create a new expression by setting the first cross multiplication product with the second resulting in $$3d=\\\\left(-13d\\\\right)+624$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions20a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$39$$"],"dependencies":["a02e810proportions20a-h2"],"title":"Solving for the Variable","text":"Isolate the variable and solve. What is the value of the variable?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a02e810proportions21","title":"Solve Proportion and Similar Figure Applications","body":"Solve the following exercise:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Solve Proportion and Similar Figure Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a02e810proportions21a","stepAnswer":["$$60$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{m+90}{25}=\\\\frac{m+30}{15}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$60$$","hints":{"DefaultPathway":[{"id":"a02e810proportions21a-h1","type":"hint","dependencies":[],"title":"Cross Multiply","text":"Cross multiply by taking the numerator of the first expression and multiplying it with the denominator of the second experssion. Then cross multiply the remaining numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions21a-h2","type":"hint","dependencies":["a02e810proportions21a-h1"],"title":"Reformatting the Expression","text":"Create a new expression by setting the first cross multiplication product with the second resulting in $$15m+1350=25m+750$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions21a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$60$$"],"dependencies":["a02e810proportions21a-h2"],"title":"Solving for the Variable","text":"Isolate the variable and solve. What is the value of the variable?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a02e810proportions22","title":"Solve Proportion and Similar Figure Applications","body":"Solve the following exercise:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Solve Proportion and Similar Figure Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a02e810proportions22a","stepAnswer":["$$10$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{n+10}{4}=\\\\frac{40-n}{6}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10$$","hints":{"DefaultPathway":[{"id":"a02e810proportions22a-h1","type":"hint","dependencies":[],"title":"Cross Multiply","text":"Cross multiply by taking the numerator of the first expression and multiplying it with the denominator of the second experssion. Then cross multiply the remaining numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions22a-h2","type":"hint","dependencies":["a02e810proportions22a-h1"],"title":"Reformatting the Expression","text":"Create a new expression by setting the first cross multiplication product with the second resulting in $$6n+60=160-4n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions22a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a02e810proportions22a-h2"],"title":"Solving for the Variable","text":"Isolate the variable and solve. What is the value of the variable?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a02e810proportions23","title":"Solve Proportion and Similar Figure Applications","body":"Solve the following exercise:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Solve Proportion and Similar Figure Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a02e810proportions23a","stepAnswer":["$$30$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2p+4}{8}=\\\\frac{p+18}{6}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$30$$","hints":{"DefaultPathway":[{"id":"a02e810proportions23a-h1","type":"hint","dependencies":[],"title":"Cross Multiply","text":"Cross multiply by taking the numerator of the first expression and multiplying it with the denominator of the second experssion. Then cross multiply the remaining numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions23a-h2","type":"hint","dependencies":["a02e810proportions23a-h1"],"title":"Reformatting the Expression","text":"Create a new expression by setting the first cross multiplication product with the second resulting in $$12p+24=8p+144$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions23a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a02e810proportions23a-h2"],"title":"Solving for the Variable","text":"Isolate the variable and solve. What is the value of the variable?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a02e810proportions24","title":"Solve Proportion and Similar Figure Applications","body":"Solve the following exercise:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Solve Proportion and Similar Figure Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a02e810proportions24a","stepAnswer":["$$\\\\frac{11}{7}$$"],"problemType":"TextBox","stepTitle":"((q-2)/2)=((2*q)-7)/18)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{11}{7}$$","hints":{"DefaultPathway":[{"id":"a02e810proportions24a-h1","type":"hint","dependencies":[],"title":"Cross Multiply","text":"Cross multiply by taking the numerator of the first expression and multiplying it with the denominator of the second experssion. Then cross multiply the remaining numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions24a-h2","type":"hint","dependencies":["a02e810proportions24a-h1"],"title":"Reformatting the Expression","text":"Create a new expression by setting the first cross multiplication product with the second resulting in $$18q-36=4q-14$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions24a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a02e810proportions24a-h2"],"title":"Solving for the Variable","text":"Isolate the variable and solve. What is the value of the variable?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a02e810proportions25","title":"Solve Proportion and Similar Figure Applications","body":"Solve the following exercise:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Solve Proportion and Similar Figure Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a02e810proportions25a","stepAnswer":["$$9$$"],"problemType":"TextBox","stepTitle":"Pediatricians prescribe $$5$$ milliliters (ml) of acetaminophen for every $$25$$ pounds of a child\u2019s weight. How many milliliters of acetaminophen will the doctor prescribe for Jocelyn, who weighs $$45$$ pounds?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9$$","hints":{"DefaultPathway":[{"id":"a02e810proportions25a-h1","type":"hint","dependencies":[],"title":"Setting Up the Proportion","text":"The two main variables in this problem are millimeters to pounds. Using the chilid\'s weight as one proportion and Jocelyn\'s as the other one we can set up the resultinig proportion: $$\\\\frac{5}{25}=\\\\frac{x}{45}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions25a-h2","type":"hint","dependencies":["a02e810proportions25a-h1"],"title":"Cross Multiply","text":"Cross multiply by taking the numerator of the first expression and multiplying it with the denominator of the second experssion. Then cross multiply the remaining numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions25a-h3","type":"hint","dependencies":["a02e810proportions25a-h2"],"title":"Reformatting the Expression","text":"Create a new expression by setting the first cross multiplication product with the second resulting in $$25x=225$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions25a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a02e810proportions25a-h3"],"title":"Solving for the Variable","text":"Isolate the variable and solve. What is the value of the variable?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a02e810proportions26","title":"Solve Proportion and Similar Figure Applications","body":"Solve the following exercise:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Solve Proportion and Similar Figure Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a02e810proportions26a","stepAnswer":["$$90$$"],"problemType":"TextBox","stepTitle":"Brianna, who weighs $$6$$ kg, just received her shots and needs a pain killer. The pain killer is prescribed for children at $$15$$ milligrams (mg) for every $$1$$ kilogram (kg) of the child\u2019s weight. How many milligrams will the doctor prescribe?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$90$$","hints":{"DefaultPathway":[{"id":"a02e810proportions26a-h1","type":"hint","dependencies":[],"title":"Setting Up the Proportion","text":"The two main variables in this problem is weight of the child to weight of the pain killers. Using the standard pain killer prescription ratio as one proportion and Brianna\'s as the other one we can set up the resultinig proportion: $$\\\\frac{1}{15}=\\\\frac{6}{x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions26a-h2","type":"hint","dependencies":["a02e810proportions26a-h1"],"title":"Cross Multiply","text":"Cross multiply by taking the numerator of the first expression and multiplying it with the denominator of the second experssion. Then cross multiply the remaining numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions26a-h3","type":"hint","dependencies":["a02e810proportions26a-h2"],"title":"Reformatting the Expression","text":"Create a new expression by setting the first cross multiplication product with the second resulting in $$x=90$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions26a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$90$$"],"dependencies":["a02e810proportions26a-h3"],"title":"Solving for the Variable","text":"What is the value of the variable?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a02e810proportions27","title":"Solve Proportion and Similar Figure Applications","body":"Solve the following exercise:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Solve Proportion and Similar Figure Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a02e810proportions27a","stepAnswer":["$$325$$"],"problemType":"TextBox","stepTitle":"A veterinarian prescribed Sunny, a $$65$$ pound dog, an antibacterial medicine in case an infection emerges after her teeth were cleaned. If the dosage is $$5$$ mg for every pound, how much medicine was Sunny given?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$325$$","hints":{"DefaultPathway":[{"id":"a02e810proportions27a-h1","type":"hint","dependencies":[],"title":"Setting Up the Proportion","text":"The two main variables in this problem is weight of the dog to weight of the antibacterial medicine. Using the standard antibacterial medicine ratio as one proportion and Sunny\'s as the other one we can set up the resultinig proportion: $$\\\\frac{65}{x}=\\\\frac{1}{5}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions27a-h2","type":"hint","dependencies":["a02e810proportions27a-h1"],"title":"Cross Multiply","text":"Cross multiply by taking the numerator of the first expression and multiplying it with the denominator of the second experssion. Then cross multiply the remaining numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions27a-h3","type":"hint","dependencies":["a02e810proportions27a-h2"],"title":"Reformatting the Expression","text":"Create a new expression by setting the first cross multiplication product with the second resulting in $$x=325$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions27a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$325$$"],"dependencies":["a02e810proportions27a-h3"],"title":"Solving for the Variable","text":"What is the value of the variable?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a02e810proportions28","title":"Solve Proportion and Similar Figure Applications","body":"Solve the following exercise:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Solve Proportion and Similar Figure Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a02e810proportions28a","stepAnswer":["$$23.4$$"],"problemType":"TextBox","stepTitle":"Belle, a $$13$$ pound cat, is suffering from joint pain. How much medicine should the veterinarian prescribe if the dosage is $$1.8$$ mg per pound?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$23.4$$","hints":{"DefaultPathway":[{"id":"a02e810proportions28a-h1","type":"hint","dependencies":[],"title":"Setting Up the Proportion","text":"The two main variables in this problem is weight of the cat to weight of the medicine. Using the standard medicine ratio as one proportion and Belle\'s as the other one we can set up the resultinig proportion: $$\\\\frac{13}{x}=\\\\frac{1}{1.8}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions28a-h2","type":"hint","dependencies":["a02e810proportions28a-h1"],"title":"Cross Multiply","text":"Cross multiply by taking the numerator of the first expression and multiplying it with the denominator of the second experssion. Then cross multiply the remaining numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions28a-h3","type":"hint","dependencies":["a02e810proportions28a-h2"],"title":"Reformatting the Expression","text":"Create a new expression by setting the first cross multiplication product with the second resulting in $$x=325$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions28a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$325$$"],"dependencies":["a02e810proportions28a-h3"],"title":"Solving for the Variable","text":"What is the value of the variable?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a02e810proportions29","title":"Solve Proportion and Similar Figure Applications","body":"Solve the following exercise:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Solve Proportion and Similar Figure Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a02e810proportions29a","stepAnswer":["$$159$$"],"problemType":"TextBox","stepTitle":"A new energy drink advertises $$106$$ calories for $$8$$ ounces. How many calories are in $$12$$ ounces of the drink?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$159$$","hints":{"DefaultPathway":[{"id":"a02e810proportions29a-h1","type":"hint","dependencies":[],"title":"Setting Up the Proportion","text":"The two main variables in this problem is energy drink calories to weight of energy drink. Using the advertised ratio as one proportion and the $$12$$ ounces as the other one we can set up the resultinig proportion: $$\\\\frac{106}{8}=\\\\frac{x}{12}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions29a-h2","type":"hint","dependencies":["a02e810proportions29a-h1"],"title":"Cross Multiply","text":"Cross multiply by taking the numerator of the first expression and multiplying it with the denominator of the second experssion. Then cross multiply the remaining numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions29a-h3","type":"hint","dependencies":["a02e810proportions29a-h2"],"title":"Reformatting the Expression","text":"Create a new expression by setting the first cross multiplication product with the second resulting in $$8x=1272$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions29a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$159$$"],"dependencies":["a02e810proportions29a-h3"],"title":"Solving for the Variable","text":"What is the value of the variable?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a02e810proportions3","title":"Solving for Variables in Proportions","body":"Find the value of the variable.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Solve Proportion and Similar Figure Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a02e810proportions3a","stepAnswer":["$$104$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{y}{96}=\\\\frac{13}{12}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$104$$","hints":{"DefaultPathway":[{"id":"a02e810proportions3a-h1","type":"hint","dependencies":[],"title":"Identifying the Solution","text":"To isolate $$y$$, multiply both sides by the LCD, in this case, $$96$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions3a-h2","type":"hint","dependencies":["a02e810proportions3a-h1"],"title":"Simplifying the Equation","text":"Simplify both sides by reducing and eliminating the denominater under $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a02e810proportions30","title":"Solve Proportion and Similar Figure Applications","body":"Solve the following exercise:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Solve Proportion and Similar Figure Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a02e810proportions30a","stepAnswer":["$$400$$"],"problemType":"TextBox","stepTitle":"One $$12$$ ounce can of soda has $$150$$ calories. If Josiah drinks the big $$32$$ ounce size from the local mini-mart, how many calories does he get?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$400$$","hints":{"DefaultPathway":[{"id":"a02e810proportions30a-h1","type":"hint","dependencies":[],"title":"Setting Up the Proportion","text":"The two main variables in this problem is the soda calories to weight of the soda. Using the ratio of one soda can as one proportion and the amount Josiah drank as the other one we can set up the resultinig proportion: $$\\\\frac{12}{150}=\\\\frac{32}{x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions30a-h2","type":"hint","dependencies":["a02e810proportions30a-h1"],"title":"Cross Multiply","text":"Cross multiply by taking the numerator of the first expression and multiplying it with the denominator of the second experssion. Then cross multiply the remaining numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions30a-h3","type":"hint","dependencies":["a02e810proportions30a-h2"],"title":"Reformatting the Expression","text":"Create a new expression by setting the first cross multiplication product with the second resulting in $$12x=4800$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions30a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$400$$"],"dependencies":["a02e810proportions30a-h3"],"title":"Solving for the Variable","text":"What is the value of the variable?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a02e810proportions4","title":"Solving for Variables in Proportions","body":"Find the value of the variable.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Solve Proportion and Similar Figure Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a02e810proportions4a","stepAnswer":["$$64$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{144}{a}=\\\\frac{9}{4}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$64$$","hints":{"DefaultPathway":[{"id":"a02e810proportions4a-h1","type":"hint","dependencies":[],"title":"Identifying the Solution","text":"To isolate a, multiply both sides by the LCD, in this case, 4a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions4a-h2","type":"hint","dependencies":["a02e810proportions4a-h1"],"title":"Simplifying the Equation","text":"Simplify both sides by reducing and eliminating the denominater under $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions4a-h3","type":"hint","dependencies":["a02e810proportions4a-h2"],"title":"Divide","text":"Divide by $$9$$ on both sides to isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a02e810proportions5","title":"Solving for Variables in Proportions","body":"Find the value of the variable.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Solve Proportion and Similar Figure Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a02e810proportions5a","stepAnswer":["$$65$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{91}{b}=\\\\frac{7}{5}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$65$$","hints":{"DefaultPathway":[{"id":"a02e810proportions5a-h1","type":"hint","dependencies":[],"title":"Identifying the Solution","text":"To isolate $$b$$, multiply both sides by the LCD, in this case, $$5b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions5a-h2","type":"hint","dependencies":["a02e810proportions5a-h1"],"title":"Simplifying the Equation","text":"Simplify both sides by reducing and eliminating the denominater under $$b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions5a-h3","type":"hint","dependencies":["a02e810proportions5a-h2"],"title":"Divide","text":"Divide by $$7$$ on both sides to isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions5a-h4","type":"hint","dependencies":["a02e810proportions5a-h3"],"title":"Answer","text":"The answer is $$65$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a02e810proportions6","title":"Solving for Variables in Proportions","body":"Find the value of the variable.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Solve Proportion and Similar Figure Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a02e810proportions6a","stepAnswer":["$$24$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{39}{c}=\\\\frac{13}{8}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$24$$","hints":{"DefaultPathway":[{"id":"a02e810proportions6a-h1","type":"hint","dependencies":[],"title":"Identifying the Solution","text":"To isolate $$b$$, multiply both sides by the LCD, in this case, 8c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions6a-h2","type":"hint","dependencies":["a02e810proportions6a-h1"],"title":"Simplifying the Equation","text":"Simplify both sides by reducing and eliminating the denominater under c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions6a-h3","type":"hint","dependencies":["a02e810proportions6a-h2"],"title":"Divide","text":"Divide by $$13$$ on both sides to isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions6a-h4","type":"hint","dependencies":["a02e810proportions6a-h3"],"title":"Answer","text":"The answer is $$24$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a02e810proportions7","title":"Solving for Variables in Proportions","body":"Find the value of the variable.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Solve Proportion and Similar Figure Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a02e810proportions7a","stepAnswer":["$$35$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{n}{n+14}=\\\\frac{5}{7}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$35$$","hints":{"DefaultPathway":[{"id":"a02e810proportions7a-h1","type":"hint","dependencies":[],"title":"Identifying the Solution","text":"To isolate $$n$$, multiply both sides by the LCD. Find the LCD by multiplying the two denominators of the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions7a-h2","type":"hint","dependencies":["a02e810proportions7a-h1"],"title":"Simplifying the Equation","text":"Simplify both sides by reducing and eliminating the denominater under the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions7a-h3","type":"hint","dependencies":["a02e810proportions7a-h2"],"title":"Divide","text":"Divide on both sides to isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions7a-h4","type":"hint","dependencies":["a02e810proportions7a-h3"],"title":"Answer","text":"The answer is $$35$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a02e810proportions8","title":"Solving for Variables in Proportions","body":"Find the value of the variable.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Solve Proportion and Similar Figure Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a02e810proportions8a","stepAnswer":["$$33$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{y}{y+55}=\\\\frac{3}{8}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$33$$","hints":{"DefaultPathway":[{"id":"a02e810proportions8a-h1","type":"hint","dependencies":[],"title":"Identifying the Solution","text":"To isolate $$y$$, multiply both sides by the LCD. Find the LCD by multiplying the two denominators of the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions8a-h2","type":"hint","dependencies":["a02e810proportions8a-h1"],"title":"Simplifying the Equation","text":"Simplify both sides by reducing and eliminating the denominater under the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions8a-h3","type":"hint","dependencies":["a02e810proportions8a-h2"],"title":"Divide","text":"Divide on both sides to isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions8a-h4","type":"hint","dependencies":["a02e810proportions8a-h3"],"title":"Answer","text":"The answer is $$33$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a02e810proportions9","title":"Solving for Variables in Proportions","body":"Find the value of the variable.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Solve Proportion and Similar Figure Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a02e810proportions9a","stepAnswer":["$$14$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{z}{z-84}=\\\\frac{-1}{5}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$14$$","hints":{"DefaultPathway":[{"id":"a02e810proportions9a-h1","type":"hint","dependencies":[],"title":"Identifying the Solution","text":"To isolate $$z$$, multiply both sides by the LCD. Find the LCD by multiplying the two denominators of the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions9a-h2","type":"hint","dependencies":["a02e810proportions9a-h1"],"title":"Simplifying the Equation","text":"Simplify both sides by reducing and eliminating the denominater under the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions9a-h3","type":"hint","dependencies":["a02e810proportions9a-h2"],"title":"Divide","text":"Divide on both sides to isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a02e810proportions9a-h4","type":"hint","dependencies":["a02e810proportions9a-h3"],"title":"Answer","text":"The answer is $$14$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0351f0parabola1","title":"Writing the Equation of a Parabola in Standard Form Given its Focus and Directrix","body":"Write the equation for the parabola.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 The Parabola","courseName":"OpenStax: College Algebra","steps":[{"id":"a0351f0parabola1a","stepAnswer":["$$y^2=-2x$$"],"problemType":"TextBox","stepTitle":"The parabola has focus $$(-12,0)$$ and directrix $$x=12$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y^2=-2x$$","hints":{"DefaultPathway":[{"id":"a0351f0parabola1a-h1","type":"hint","dependencies":[],"title":"Find form","text":"The focus has the form (p,0), so the equation will have the form $$y^2=4px$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0351f0parabola1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a0351f0parabola1a-h1"],"title":"Find $$4p$$","text":"Now, solve for $$4p$$. This can be done by finding $$p$$ and multiplying by $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a0351f0parabola1a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a0351f0parabola1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y^2=-2x$$"],"dependencies":["a0351f0parabola1a-h2"],"title":"Substitute","text":"Finally, plug $$4p$$ into the $$y^2=4px$$ equation. What is the final equation?(do not simplify your answer)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a0351f0parabola1a-h3-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$y^2=-2x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a0351f0parabola1a-h4","type":"hint","dependencies":["a0351f0parabola1a-h3"],"title":"Answer","text":"Therefore, the final answer is $$y^2=-2x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0351f0parabola10","title":"Interpretation of a Parabola","body":"Answer the following multiple choice question.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 The Parabola","courseName":"OpenStax: College Algebra","steps":[{"id":"a0351f0parabola10a","stepAnswer":["It increases."],"problemType":"MultipleChoice","stepTitle":"As the graph of a parabola becomes wider, what will happen to the distance between the focus and directrix?","stepBody":"","answerType":"string","variabilization":{},"choices":["It increases.","It decreases."],"hints":{"DefaultPathway":[{"id":"a0351f0parabola10a-h1","type":"hint","dependencies":[],"title":"As the graph of the parabola widens, the locus distances increases, resulting in an increase in the distance between the focus and the directrix.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0351f0parabola11","title":"Writing the Standard Form of a Parabola","body":"Determine whether the given equation is a parabola. If so, rewrite the equation in standard form. If it is not a parabola, enter \\"Not a parabola\\" as your answer.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 The Parabola","courseName":"OpenStax: College Algebra","steps":[{"id":"a0351f0parabola11a","stepAnswer":["Not a parabola"],"problemType":"TextBox","stepTitle":"$$y^2=4-x^2$$","stepBody":"If it is not a parabola, enter \\"Not a parabola\\" as your answer.","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a0351f0parabola11a-h1","type":"hint","dependencies":[],"title":"Determine if parabola","text":"If both the $$x$$ and the $$y$$ values or an equation are squared, the equation is not a parabola.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0351f0parabola11a-h2","type":"hint","dependencies":["a0351f0parabola11a-h1"],"title":"Answer","text":"Therefore, since both the $$x$$ and $$y$$ values are squared, the equation is not a parabola.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0351f0parabola12","title":"Writing the Standard Form of a Parabola","body":"Determine whether the given equation is a parabola. If so, rewrite the equation in standard form. If it is not a parabola, enter \\"Not a parabola\\" as your answer.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 The Parabola","courseName":"OpenStax: College Algebra","steps":[{"id":"a0351f0parabola12a","stepAnswer":["$$y=4\\\\left(1\\\\right) x^2$$"],"problemType":"TextBox","stepTitle":"$$y=4x^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=4\\\\left(1\\\\right) x^2$$","hints":{"DefaultPathway":[{"id":"a0351f0parabola12a-h1","type":"hint","dependencies":[],"title":"Determine if parabola","text":"If both the $$x$$ and the $$y$$ values or an equation are squared, the equation is not a parabola.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0351f0parabola12a-h2","type":"hint","dependencies":["a0351f0parabola12a-h1"],"title":"Specify $$p$$","text":"P in this case is $$1$$. Since the form of the parabola is $$y=4{px}^2$$, the equation should specify $$p$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0351f0parabola12a-h3","type":"hint","dependencies":["a0351f0parabola12a-h2"],"title":"Answer","text":"The answer is $$y=4\\\\left(1\\\\right) x^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0351f0parabola13","title":"Writing the Standard Form of a Parabola","body":"Determine whether the given equation is a parabola. If so, rewrite the equation in standard form. If it is not a parabola, enter \\"Not a parabola\\" as your answer.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 The Parabola","courseName":"OpenStax: College Algebra","steps":[{"id":"a0351f0parabola13a","stepAnswer":["Not a parabola"],"problemType":"TextBox","stepTitle":"$$3x^2-6y^2=12$$","stepBody":"If it is not a parabola, enter \\"Not a parabola\\" as your answer.","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a0351f0parabola13a-h1","type":"hint","dependencies":[],"title":"Determine if parabola","text":"If both the $$x$$ and the $$y$$ values or an equation are squared, the equation is not a parabola.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0351f0parabola13a-h2","type":"hint","dependencies":["a0351f0parabola13a-h1"],"title":"Answer","text":"Therefore, since both the $$x$$ and $$y$$ values are squared, the equation is not a parabola.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0351f0parabola14","title":"Writing the Standard Form of a Parabola","body":"Determine whether the given equation is a parabola. If so, rewrite the equation in standard form. If it is not a parabola, enter \\"Not a parabola\\" as your answer.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 The Parabola","courseName":"OpenStax: College Algebra","steps":[{"id":"a0351f0parabola14a","stepAnswer":["Not a Parabola"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(y-3\\\\right)}^2=8(x-2)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$y=4\\\\left(1\\\\right) x^2$$","$$y=4\\\\left(1\\\\right) x^3$$","Not a Parabola"],"hints":{"DefaultPathway":[{"id":"a0351f0parabola14a-h1","type":"hint","dependencies":[],"title":"Determine if parabola","text":"If both the $$x$$ and the $$y$$ values or an equation are squared, the equation is not a parabola.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0351f0parabola14a-h2","type":"hint","dependencies":["a0351f0parabola14a-h1"],"title":"Answer","text":"Therefore, the equation is not a parabola.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0351f0parabola15","title":"Interpreting a Parabola","body":"Answer the following question regarding the equation of a parabola.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 The Parabola","courseName":"OpenStax: College Algebra","steps":[{"id":"a0351f0parabola15a","stepAnswer":["The graph will open down."],"problemType":"MultipleChoice","stepTitle":"If the equation of a parabola is written in standard form and $$p$$ is negative and the directrix is a horizontal line, then what can we conclude about its graph?","stepBody":"","answerType":"string","variabilization":{},"choices":["The graph will open down.","The graph will open up."],"hints":{"DefaultPathway":[{"id":"a0351f0parabola15a-h1","type":"hint","dependencies":[],"title":"Plug in points","text":"You can determine the answer to this by plugging in points. Create an equation and plug in points to see where the graph is going.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0351f0parabola15a-h2","type":"hint","dependencies":["a0351f0parabola15a-h1"],"title":"Create an equation","text":"The equation can be $$y=-4\\\\left(1\\\\right) x^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0351f0parabola15a-h3","type":"hint","dependencies":["a0351f0parabola15a-h2"],"title":"Substitute","text":"Substitute $$x=0$$, $$-1$$, and $$1$$ into the equation. Where does the graph go? Is it opening down or up?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0351f0parabola15a-h4","type":"hint","dependencies":["a0351f0parabola15a-h3"],"title":"Answer","text":"The equation opens down.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0351f0parabola2","title":"Writing the Equation of a Parabola in Standard Form Given its Focus and Directrix","body":"Write the equation for the parabola.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 The Parabola","courseName":"OpenStax: College Algebra","steps":[{"id":"a0351f0parabola2a","stepAnswer":["$$x^2=14y$$"],"problemType":"TextBox","stepTitle":"The parabola has focus $$(0,\\\\frac{7}{2})$$ and directrix $$y=\\\\frac{-7}{2}$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^2=14y$$","hints":{"DefaultPathway":[{"id":"a0351f0parabola2a-h1","type":"hint","dependencies":[],"title":"Find form","text":"The focus has the form (0,p), so the equation will have the form $$x^2=4py$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0351f0parabola2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$14$$"],"dependencies":["a0351f0parabola2a-h1"],"title":"Find $$4p$$","text":"Now, solve for $$4p$$. This can be done by finding $$p$$ and multiplying by $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a0351f0parabola2a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$14$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a0351f0parabola2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2=14y$$"],"dependencies":["a0351f0parabola2a-h2"],"title":"Substitute","text":"Finally, plug $$4p$$ into the $$x^2=4py$$ equation. What is the final equation?(do not simplify your answer)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a0351f0parabola2a-h3-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$x^2=14y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a0351f0parabola2a-h4","type":"hint","dependencies":["a0351f0parabola2a-h3"],"title":"Answer","text":"Therefore, the final answer is $$x^2=14y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0351f0parabola3","title":"Find components of the graph","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 The Parabola","courseName":"OpenStax: College Algebra","steps":[{"id":"a0351f0parabola3a","stepAnswer":["$$(6,0)$$"],"problemType":"MultipleChoice","stepTitle":"Identify the focus of the graph: $$y^2=24x$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(6,0)$$","choices":["$$(6,0)$$","$$(5,0)$$","$$(4,0)$$"],"hints":{"DefaultPathway":[{"id":"a0351f0parabola3a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":[],"title":"Find $$p$$","text":"Find $$p$$ by using the form $$y^2=4px$$. Solve for $$p$$. What is it?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a0351f0parabola3a-h1-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a0351f0parabola3a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(6,0)$$"],"dependencies":["a0351f0parabola3a-h1"],"title":"Use (p,0)","text":"The coordinates of the focuse are (p,0). Use this to find the focus.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(6,0)$$","$$(5,0)$$","$$(4,0)$$"],"subHints":[{"id":"a0351f0parabola3a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$(6,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a0351f0parabola3a-h3","type":"hint","dependencies":["a0351f0parabola3a-h2"],"title":"Answer","text":"Therefore, the answer is $$(6,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0351f0parabola4","title":"Find Components of the Graph","body":"Identify the directrix of the graph of the following parabola.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 The Parabola","courseName":"OpenStax: College Algebra","steps":[{"id":"a0351f0parabola4a","stepAnswer":["$$x=-6$$"],"problemType":"TextBox","stepTitle":"$$y^2=24x$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x=-6$$","hints":{"DefaultPathway":[{"id":"a0351f0parabola4a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":[],"title":"Find $$p$$","text":"Find $$p$$ by using the form $$y^2=4px$$. Solve for $$p$$. What is it?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a0351f0parabola4a-h1-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a0351f0parabola4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=-6$$"],"dependencies":["a0351f0parabola4a-h1"],"title":"Use $$x=-p$$","text":"The equation of the directrix is $$x=-p$$. Use this to find the directrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a0351f0parabola4a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$x=-6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a0351f0parabola4a-h3","type":"hint","dependencies":["a0351f0parabola4a-h2"],"title":"Answer","text":"Therefore, the answer is $$x=-6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0351f0parabola5","title":"Find Components of the Graph","body":"Identify the endpoints of the latus rectum of the graph.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 The Parabola","courseName":"OpenStax: College Algebra","steps":[{"id":"a0351f0parabola5a","stepAnswer":["$$(6,12)$$ and $$(6,-12)$$"],"problemType":"MultipleChoice","stepTitle":"$$y^2=24x$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(6,12)$$ and $$(6,-12)$$","choices":["$$(6,12)$$ and $$(6,-12)$$","$$(6,11)$$ and $$(6,-11)$$","$$(6,10)$$ and $$(6,-10)$$"],"hints":{"DefaultPathway":[{"id":"a0351f0parabola5a-h1","type":"hint","dependencies":[],"title":"Use focus","text":"The endpoints of the latus rectum have the same x-coordinate at the focus.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0351f0parabola5a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(6,12)$$ and $$(6,-12)$$"],"dependencies":["a0351f0parabola5a-h1"],"title":"Plug in","text":"Plug $$x$$ $$=6$$ into the original equation to find the $$y$$ coordinates of the endpoints of the equation. Submit your answer as $$2$$ coordinate pairs.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(6,12)$$ and $$(6,-12)$$","$$(6,11)$$ and $$(6,-11)$$","$$(6,10)$$ and $$(6,-10)$$"],"subHints":[{"id":"a0351f0parabola5a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$(6,12)$$ and $$(6,-12)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a0351f0parabola5a-h3","type":"hint","dependencies":["a0351f0parabola5a-h2"],"title":"Answer","text":"Therefore, the answer is $$(6,12)$$ and $$(6,-12)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0351f0parabola6","title":"Find Components of the Graph","body":"Identify the vertex of the graph:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 The Parabola","courseName":"OpenStax: College Algebra","steps":[{"id":"a0351f0parabola6a","stepAnswer":["(-3,1)"],"problemType":"TextBox","stepTitle":"$${\\\\left(y-1\\\\right)}^2=-\\\\operatorname{16}\\\\left(x+3\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-3,1)$$","hints":{"DefaultPathway":[{"id":"a0351f0parabola6a-h1","type":"hint","dependencies":[],"title":"Know standard form","text":"The standard form of this equation is $${\\\\left(y-k\\\\right)}^2=4p(x-h)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0351f0parabola6a-h2","type":"hint","dependencies":["a0351f0parabola6a-h1"],"title":"Vertex","text":"The vertex of this type of equation is (h,k).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0351f0parabola6a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(-3,1)$$"],"dependencies":["a0351f0parabola6a-h2"],"title":"Identify","text":"Now, using the standard form and your knowledge of the vertex, what is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(-3,1)$$","$$(-2,1)$$"],"subHints":[{"id":"a0351f0parabola6a-h3-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$(-3,1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a0351f0parabola6a-h4","type":"hint","dependencies":["a0351f0parabola6a-h3"],"title":"Answer","text":"Therefore, the answer is $$(-3,1)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0351f0parabola7","title":"Find Components of the Graph","body":"Identify the axis of symmetry of the graph:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 The Parabola","courseName":"OpenStax: College Algebra","steps":[{"id":"a0351f0parabola7a","stepAnswer":["$$y=1$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(y-1\\\\right)}^2=-\\\\operatorname{16}\\\\left(x+3\\\\right)$$.","stepBody":"Enter your answer in the form $$y=(number)$$.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=1$$","hints":{"DefaultPathway":[{"id":"a0351f0parabola7a-h1","type":"hint","dependencies":[],"title":"Use Vertex","text":"The axis of symmetry of the equation is $$y=k$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0351f0parabola7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a0351f0parabola7a-h1"],"title":"Find k","text":"What is k in this equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a0351f0parabola7a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a0351f0parabola7a-h3","type":"hint","dependencies":["a0351f0parabola7a-h2"],"title":"Answer","text":"Therefore, the axis of symmetry is $$y=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0351f0parabola8","title":"Find Components of the Graph","body":"Identify the directrix of the graph:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 The Parabola","courseName":"OpenStax: College Algebra","steps":[{"id":"a0351f0parabola8a","stepAnswer":["$$x=1$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(y-1\\\\right)}^2=-\\\\operatorname{16}\\\\left(x+3\\\\right)$$","stepBody":"Enter your answer in the form $$x=(number)$$.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x=1$$","hints":{"DefaultPathway":[{"id":"a0351f0parabola8a-h1","type":"hint","dependencies":[],"title":"Know equation for directrix","text":"The equation for a directrix in standard form is $$x=h-p$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0351f0parabola8a-h2","type":"hint","dependencies":["a0351f0parabola8a-h1"],"title":"Find $$h$$ and $$p$$","text":"$$h$$ in this case is $$-3$$, while $$p$$ is $$-4$$. This can be found in the original equation, which is in the form: $${\\\\left(y-k\\\\right)}^2=4p(x-h)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0351f0parabola8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=1$$"],"dependencies":["a0351f0parabola8a-h2"],"title":"Plug in","text":"Given $$h$$ and $$p$$, what is the directrix?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a0351f0parabola8a-h3-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$x=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a0351f0parabola8a-h4","type":"hint","dependencies":["a0351f0parabola8a-h3"],"title":"Answer","text":"Therefore, the directrix is $$x=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0351f0parabola9","title":"Find Components of the Graph","body":"Identify the endpoints of the latus rectum of the graph:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 The Parabola","courseName":"OpenStax: College Algebra","steps":[{"id":"a0351f0parabola9a","stepAnswer":["$$(-7,-7)$$ and $$(-7,9)$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(y-1\\\\right)}^2=-\\\\operatorname{16}\\\\left(x+3\\\\right)$$","stepBody":"Enter your answer in the form: (a,b)(c,d).","answerType":"string","variabilization":{},"answerLatex":"$$(-7,-7)$$ and $$(-7,9)$$","choices":["$$(-7,-7)$$ and $$(-7,9)$$","$$(-7,-7)$$ and $$(-7,9)$$"],"hints":{"DefaultPathway":[{"id":"a0351f0parabola9a-h1","type":"hint","dependencies":[],"title":"Know form","text":"The form of the endpoints of the latus rectum is: (h+p,k+2p),(h+p,k-2p).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0351f0parabola9a-h2","type":"hint","dependencies":["a0351f0parabola9a-h1"],"title":"Plug in","text":"Plug in the values for the endpoints, using the the proper $$h$$, k, and $$p$$ values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0351f0parabola9a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(-7,-7)$$ and $$(-7,9)$$"],"dependencies":["a0351f0parabola9a-h2"],"title":"Endpoints","text":"What are the endpoints? Use the given formula.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(-7,-7)$$ and $$(-7,9)$$","$$(-7,-7)$$ and $$(-7,9)$$"],"subHints":[{"id":"a0351f0parabola9a-h3-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$(-7,7)$$ and $$(-7,9)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a0351f0parabola9a-h4","type":"hint","dependencies":["a0351f0parabola9a-h3"],"title":"Answer","text":"Therefore, the endpoints are $$(-7,7)$$ and $$(-7,9)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0351f0theparabola11","title":"Finding Parabola Characteristics","body":"Find the vertex, focus, and directrix (in order, separated by a comma) of the parabola:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 The Parabola","courseName":"OpenStax: College Algebra","steps":[{"id":"a0351f0theparabola11a","stepAnswer":["(0,0),(1/32,0),x=-1/32"],"problemType":"MultipleChoice","stepTitle":"$$x=8y^2$$","stepBody":"","answerType":"string","variabilization":{},"choices":["(0,0),(1/32,0),x=-1/32","(0,1),(1/31,0),x=-1/32"],"hints":{"DefaultPathway":[{"id":"a0351f0theparabola11a-h1","type":"hint","dependencies":[],"title":"Writing in Standard Form","text":"We simply have to divide both sides by $$8$$ to get $$\\\\frac{x}{8}=y^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0351f0theparabola11a-h2","type":"hint","dependencies":["a0351f0theparabola11a-h1"],"title":"Finding The Characteristics","text":"We know from the equation that the vertex is $$(0,0)$$, and that $$p=\\\\frac{1}{32}$$. This means that that the directrix is $$x=\\\\frac{-1}{32}$$ and the focus is $$(\\\\frac{1}{32},0)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0351f0theparabola12","title":"Finding Parabola Characteristics","body":"Find the vertex, focus, and directrix (in order, separated by a comma) of the parabola:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 The Parabola","courseName":"OpenStax: College Algebra","steps":[{"id":"a0351f0theparabola12a","stepAnswer":["$$(0,0),(0,1),y=-1$$"],"problemType":"MultipleChoice","stepTitle":"$$y=\\\\frac{x^2}{4}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,0),(0,1),y=-1$$","choices":["$$(0,3),(0,1),y=-1$$","$$(0,1),(0,2),y=-1$$","$$(0,0),(0,1),y=-1$$"],"hints":{"DefaultPathway":[{"id":"a0351f0theparabola12a-h1","type":"hint","dependencies":[],"title":"Writing in Standard Form","text":"We just have to multiply by $$4$$ on both sides to get $$4y=x^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0351f0theparabola12a-h2","type":"hint","dependencies":["a0351f0theparabola12a-h1"],"title":"Finding The Characteristics","text":"From the equation, we can see that $$p=1$$, and the vertex is $$(0,0)$$. Because $$p=1$$, we know that the focus is $$(0,1)$$ and the directrix is $$y=-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0351f0theparabola13","title":"Finding Parabola Characteristics","body":"Find the vertex, focus, and directrix (in order, separated by a comma) of the parabola:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 The Parabola","courseName":"OpenStax: College Algebra","steps":[{"id":"a0351f0theparabola13a","stepAnswer":["(0,0),(0,-1/16),y=1/16"],"problemType":"MultipleChoice","stepTitle":"$$y=-4x^2$$","stepBody":"","answerType":"string","variabilization":{},"choices":["(0,0),(0,-1/16),y=1/16","(4,0),(0,-1/16),y=1/16"],"hints":{"DefaultPathway":[{"id":"a0351f0theparabola13a-h1","type":"hint","dependencies":[],"title":"Writing in Standard Form","text":"We can divide both sides by $$-4$$ to put the equation in standard form. We now have $$\\\\frac{-y}{4}=x^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0351f0theparabola13a-h2","type":"hint","dependencies":["a0351f0theparabola13a-h1"],"title":"Finding The Characteristics","text":"From the equation, we can see that the vertex is $$(0,0)$$. Because we know that $$p=\\\\frac{-1}{16}$$, we know that the focus is $$(0,\\\\frac{-1}{16})$$ and the directrix is $$y=\\\\frac{1}{16}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0351f0theparabola14","title":"Finding Parabola Characteristics","body":"Find the vertex, focus, and directrix (in order, separated by a comma) of the parabola:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 The Parabola","courseName":"OpenStax: College Algebra","steps":[{"id":"a0351f0theparabola14a","stepAnswer":["$$(0,0),(2,0),x=-2$$"],"problemType":"MultipleChoice","stepTitle":"$$x=\\\\frac{y^2}{8}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,0),(2,0),x=-2$$","choices":["$$(0,0),(2,0),x=-2$$","$$(1,0),(2,3),x=-2$$"],"hints":{"DefaultPathway":[{"id":"a0351f0theparabola14a-h1","type":"hint","dependencies":[],"title":"Writing in Standard Form","text":"To put the parabola in standard form, we can simply multiply both sides by $$8$$ to get $$8x=y^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0351f0theparabola14a-h2","type":"hint","dependencies":["a0351f0theparabola14a-h1"],"title":"Finding The Characteristics","text":"From the equation, we know that the vertex is $$(0,0)$$ and that $$p=2$$. This means that the focus is $$(2,0)$$ and the directrix is $$x=-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0351f0theparabola15","title":"Finding Parabola Characteristics","body":"Find the vertex, focus, and directrix (in order, separated by a comma) of the parabola:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 The Parabola","courseName":"OpenStax: College Algebra","steps":[{"id":"a0351f0theparabola15a","stepAnswer":["(0,0),(1/144,0),x=-1/144"],"problemType":"MultipleChoice","stepTitle":"$$x=36y^2$$","stepBody":"","answerType":"string","variabilization":{},"choices":["(0,0),(1/144,0),x=-1/144","(0,0),(1/144,0),x=-1/140"],"hints":{"DefaultPathway":[{"id":"a0351f0theparabola15a-h1","type":"hint","dependencies":[],"title":"Writing in Standard Form","text":"We can divide both sides by $$36$$ to get $$\\\\frac{x}{36}=y^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0351f0theparabola15a-h2","type":"hint","dependencies":["a0351f0theparabola15a-h1"],"title":"Finding The Characteristics","text":"From the equation, we know that $$p=\\\\frac{1}{144}$$. This means that the vertex is $$(0,0)$$, the focus is $$(\\\\frac{1}{144},0)$$ and the directrix is $$x=\\\\frac{-1}{144}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0351f0theparabola16","title":"Finding Parabola Characteristics","body":"Find the vertex, focus, and directrix (in order, separated by a comma) of the parabola:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 The Parabola","courseName":"OpenStax: College Algebra","steps":[{"id":"a0351f0theparabola16a","stepAnswer":["$$(0,0),(9,0),x=-9$$"],"problemType":"MultipleChoice","stepTitle":"$$x=\\\\frac{y^2}{36}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,0),(9,0),x=-9$$","choices":["$$(0,0),(9,0),x=-9$$","$$(1,0),(9,0),x=-9$$","$$(2,0),(7,0),x=-9$$"],"hints":{"DefaultPathway":[{"id":"a0351f0theparabola16a-h1","type":"hint","dependencies":[],"title":"Writing in Standard Form","text":"We can write this parabola in standard form by multiplying both sides by $$36$$ to get $$36x=y^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0351f0theparabola16a-h2","type":"hint","dependencies":["a0351f0theparabola16a-h1"],"title":"Finding The Characteristics","text":"From the equation, we know that the vertex is $$(0,0)$$ and that $$p=9$$. This means that the focus is $$(9,0)$$ and the directrix is $$x=-9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0351f0theparabola17","title":"Finding Parabola Characteristics","body":"Find the vertex, focus, and directrix (in order, separated by a comma) of the parabola:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 The Parabola","courseName":"OpenStax: College Algebra","steps":[{"id":"a0351f0theparabola17a","stepAnswer":["$$(1,1),(1,2),y=0$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(x-1\\\\right)}^2=4(y-1)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(1,1),(1,2),y=0$$","choices":["$$(1,5),(1,1),y=0$$","$$(1,3),(1,2),y=0$$","$$(1,1),(1,2),y=0$$"],"hints":{"DefaultPathway":[{"id":"a0351f0theparabola17a-h1","type":"hint","dependencies":[],"title":"Finding The Characteristics","text":"Since this parabola is already in standard form, we know that the vertex is $$(1,1)$$ and that $$p=1$$. This means that the focus is $$(1,2)$$ and the directrix is $$y=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0351f0theparabola18","title":"Finding Parabola Characteristics","body":"Find the vertex, focus, and directrix (in order, separated by a comma) of the parabola:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 The Parabola","courseName":"OpenStax: College Algebra","steps":[{"id":"a0351f0theparabola18a","stepAnswer":["(-4,2),(-4+1/5,2),x=-4-1/5"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(y-2\\\\right)}^2=\\\\frac{4}{5\\\\left(x+4\\\\right)}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["(-4,2),(-4+1/5,2),x=-4-1/5","(-4,1),(-4+3/5,2),x=-4-1/5"],"hints":{"DefaultPathway":[{"id":"a0351f0theparabola18a-h1","type":"hint","dependencies":[],"title":"Finding The Characteristics","text":"Since this parabola is already in standard form, we know that the vertex is $$(-4,2)$$ and that $$p=\\\\frac{1}{5}$$. This means that the focus is $$(-4+\\\\frac{1}{5},2)$$ and the directrix is $$x=-4-\\\\frac{1}{5}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0351f0theparabola19","title":"Finding Parabola Characteristics","body":"Find the vertex, focus, and directrix (in order, separated by a comma) of the parabola:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 The Parabola","courseName":"OpenStax: College Algebra","steps":[{"id":"a0351f0theparabola19a","stepAnswer":["(-3,4),(-5/2,4),x=-7/2"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(y-4\\\\right)}^2=2\\\\left(x+3\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["(-3,4),(-5/3,4),x=-7/2","(-3,4),(-5/2,4),x=-7/2","(-3,4),(-5/2,4),x=-7/3"],"hints":{"DefaultPathway":[{"id":"a0351f0theparabola19a-h1","type":"hint","dependencies":[],"title":"Finding The Characteristics","text":"Since this parabola is already in standard form, we know that the vertex is $$(-3,4)$$ and that $$p=\\\\frac{1}{2}$$. This means that the focus is $$(-2.5, 4)$$ and the directrix is $$x=-3.5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0351f0theparabola20","title":"Finding Parabola Characteristics","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 The Parabola","courseName":"OpenStax: College Algebra","steps":[{"id":"a0351f0theparabola20a","stepAnswer":["$$(-1,4),(-1,4.5),y=3.5$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(x+1\\\\right)}^2=2\\\\left(y+4\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-1,4),(-1,4.5),y=3.5$$","choices":["$$(-1,4),(-1,4.5),y=3.5$$","$$(-1,4),(-1,4.5),y=4.5$$","$$(-1,4),(-1,4.5),y=5.5$$"],"hints":{"DefaultPathway":[{"id":"a0351f0theparabola20a-h1","type":"hint","dependencies":[],"title":"Finding The Characteristics","text":"Since this parabola is already in standard form, we know that the vertex is $$(-1,4)$$ and $$p=\\\\frac{1}{2}$$. This means that the focus is $$(-1, 4.5)$$ and the directrix is $$y=3.5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0351f0theparabola21","title":"Finding Parabola Characteristics","body":"Find the vertex, focus, and directrix (in order, separated by a comma) of the parabola:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 The Parabola","courseName":"OpenStax: College Algebra","steps":[{"id":"a0351f0theparabola21a","stepAnswer":["$$(-4,-1),(-4,5),y=-7$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(x+4\\\\right)}^2=\\\\operatorname{24}\\\\left(y+1\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-4,-1),(-4,5),y=-7$$","choices":["$$(-4,-1),(-4,5),y=-2$$","$$(-4,-1),(-4,5),y=-3$$","$$(-4,-1),(-4,5),y=-7$$"],"hints":{"DefaultPathway":[{"id":"a0351f0theparabola21a-h1","type":"hint","dependencies":[],"title":"Finding The Characteristics","text":"Since this parabola is already in standard form, we know that the vertex is $$(-4,-1)$$ and that $$p=6$$. This means that the focus is $$(-4,5)$$ and that the directrix is $$y=-7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0351f0theparabola22","title":"Finding Parabola Characteristics","body":"Find the vertex, focus, and directrix (in order, separated by a comma) of the parabola:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 The Parabola","courseName":"OpenStax: College Algebra","steps":[{"id":"a0351f0theparabola22a","stepAnswer":["$$(-4,-4),(-4,0),x=8$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(y+4\\\\right)}^2=\\\\operatorname{16}\\\\left(x+4\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-4,-4),(-4,0),x=8$$","choices":["$$(-4,-4),(-4,0),x=8$$","$$(-4,-2),(-4,0),x=4$$"],"hints":{"DefaultPathway":[{"id":"a0351f0theparabola22a-h1","type":"hint","dependencies":[],"title":"Finding The Characteristics","text":"Since this parabola is already in standard form, we know that the vertex is $$(-4,-4)$$ and $$p=4$$. This means that the focus is $$(-4,0)$$ and that the directrix is $$x=-8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0351f0theparabola31","title":"Finding Parabola Characteristics","body":"Find the focus and directrix (in order, separated by a comma) of the parabola:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 The Parabola","courseName":"OpenStax: College Algebra","steps":[{"id":"a0351f0theparabola31a","stepAnswer":["$$(2,0),x=-2$$"],"problemType":"MultipleChoice","stepTitle":"$$x=\\\\frac{1}{8} y^2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(2,0),x=-2$$","choices":["$$(5,0),x=-2$$","$$(1,0),x=-2$$","$$(3,0),x=-2$$","$$(2,0),x=-2$$"],"hints":{"DefaultPathway":[{"id":"a0351f0theparabola31a-h1","type":"hint","dependencies":[],"title":"Finding The Characteristics","text":"We must first put this parabola into standard form by multiplying both sides by $$8$$ to get $$8x=y^2$$. From this, we know that $$p=2$$ and the vertex is $$(0,0)$$. This means that the focus is $$(2,0)$$ and that the directrix is $$x=-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a046eedintegrals1","title":"Finding an Antiderivative of an Exponential Function\\\\n","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.6 Integrals Involving Exponential and Logarithmic Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a046eedintegrals1a","stepAnswer":["$$-\\\\left(e^{\\\\left(-x\\\\right)}\\\\right)+C$$"],"problemType":"MultipleChoice","stepTitle":"Find the antiderivative of the exponential function $$e^{-x}$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-\\\\left(e^{\\\\left(-x\\\\right)}\\\\right)+C$$","choices":["$$-\\\\left(e^{\\\\left(-x\\\\right)}\\\\right)+C$$","$$-\\\\left(e^x\\\\right)+C$$","$$-\\\\left(e^u\\\\right)+C$$"],"hints":{"DefaultPathway":[{"id":"a046eedintegrals1a-h1","type":"hint","dependencies":[],"title":"Using u-substitution","text":"Setting $$u=-x$$, and then $$du=-1dx$$. Multiply the du equation by $$-1$$, so now you have $$dx=-du$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals1a-h2","type":"hint","dependencies":["a046eedintegrals1a-h1"],"title":"Setting the Integral","text":"$$\\\\int e^{-x} \\\\,dx=-\\\\int e^x \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals1a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-e**u)+C"],"dependencies":["a046eedintegrals1a-h2"],"title":"Find the integral","text":"What is the result of the integration?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals1a-h4","type":"hint","dependencies":["a046eedintegrals1a-h3"],"title":"Back-subsitute","text":"As we set $$u=-x$$ at the beginning, we then obtain $$-\\\\left(e^{\\\\left(-x\\\\right)}\\\\right)+C$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a046eedintegrals10","title":"CHECKPOINT $$5.38$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.6 Integrals Involving Exponential and Logarithmic Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a046eedintegrals10a","stepAnswer":["$$ln|x+2|+C$$"],"problemType":"MultipleChoice","stepTitle":"Find the antiderivative of $$\\\\frac{1}{x+2}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$ln|x+2|+C$$","choices":["$$+C$$","ln","$$\\\\ln(x+2)+C$$","$$ln|x+2|+C$$","$$x+2$$"],"hints":{"DefaultPathway":[{"id":"a046eedintegrals10a-h1","type":"hint","dependencies":[],"title":"Using u-substitution","text":"Starting by setting $$u=x+2$$ then $$du=dx$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals10a-h2","type":"hint","dependencies":["a046eedintegrals10a-h1"],"title":"Find the integral","text":"$$\\\\int \\\\frac{1}{u} \\\\,du=ln|u|+C$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals10a-h3","type":"hint","dependencies":["a046eedintegrals10a-h2"],"title":"Back-subsitute","text":"Plug $$x+2$$ back into u to obtain $$ln|x+2|+C$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a046eedintegrals11","title":"Finding an Antiderivative of a Rational Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.6 Integrals Involving Exponential and Logarithmic Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a046eedintegrals11a","stepAnswer":["$$\\\\frac{1}{2} ln|x^4+3x^2|+C$$"],"problemType":"MultipleChoice","stepTitle":"Find the antiderivative of $$\\\\frac{2x^3+3x}{x^4+3x^2}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{1}{2} ln|x^4+3x^2|+C$$","choices":["$$\\\\frac{1}{2} ln$$","$$\\\\frac{1}{2} ln$$","$$\\\\frac{1}{2} \\\\ln(x^4+3x^2)+C$$","$$\\\\frac{1}{2} ln|x^4+3x^2|+C$$","$$x^4+3x^2$$","$$x^4+3x^2$$","$$+C$$"],"hints":{"DefaultPathway":[{"id":"a046eedintegrals11a-h1","type":"hint","dependencies":[],"title":"Rewriting","text":"Rewritting the expression will make it more convenient to use u-substitution. This can be rewritten as $$\\\\int \\\\left(2x^3+3x\\\\right) {\\\\left(x^4+3x^2\\\\right)}^{-1} \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals11a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(x**4+3*x**2)**(-1)"],"dependencies":["a046eedintegrals11a-h1"],"title":"U-substitution","text":"What should we set u equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals11a-h3","type":"hint","dependencies":["a046eedintegrals11a-h2"],"title":"U-substitution","text":"As we set $$u=x^4+3x^2$$ then $$du=\\\\left(4x^3+6x\\\\right) dx$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals11a-h4","type":"hint","dependencies":["a046eedintegrals11a-h3"],"title":"Factoring","text":"Alter du by factoring out the $$2$$ to obtain $$du=2\\\\left(2x^3+3x\\\\right) dx$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals11a-h5","type":"hint","dependencies":["a046eedintegrals11a-h4"],"title":"Dividing by $$2$$","text":"Dividing both side by $$2$$ to obtain $$\\\\frac{du}{2}=\\\\left(2x^3+3x\\\\right) dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals11a-h6","type":"hint","dependencies":["a046eedintegrals11a-h5"],"title":"Rewriting","text":"Rewite an integrand in u: $$(1/2)*\\\\int u^{\\\\left(-1\\\\right)} \\\\,du$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals11a-h7","type":"hint","dependencies":["a046eedintegrals11a-h6"],"title":"Find the integral","text":"$$(1/2)*\\\\int u^{\\\\left(-1\\\\right)} \\\\,du=\\\\frac{1}{2} ln|u|+C$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals11a-h8","type":"hint","dependencies":["a046eedintegrals11a-h7"],"title":"Back-subsitute","text":"$$\\\\frac{1}{2} ln|x^4+3x^2|+C$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a046eedintegrals12","title":"Evaluating a Definite Integral","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.6 Integrals Involving Exponential and Logarithmic Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a046eedintegrals12a","stepAnswer":["ln2"],"problemType":"TextBox","stepTitle":"Find the definite integral of $$\\\\int_{0}^{\\\\frac{\\\\pi}{2}} \\\\frac{sinx}{1+cosx} \\\\,dx$$","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a046eedintegrals12a-h1","type":"hint","dependencies":[],"title":"U-substitution","text":"An ideal strategy is to select \\"u\\" as a function and its derivative that appears within the integral.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals12a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$1+cosx$$"],"dependencies":["a046eedintegrals12a-h1"],"title":"U-substitution","text":"What should we set u equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$1+cosx$$","sinx"],"subHints":[{"id":"a046eedintegrals12a-h2-s1","type":"hint","dependencies":[],"title":"U-substitution","text":"The reason why we want to set $$u=1+cosx$$ is because that expression is on the denominator and we know that $$\\\\frac{d}{\\\\operatorname{dx}\\\\left(1+cosx\\\\right)}=-sinx$$. So that $$du=-sinx$$ or $$-du=sinx$$ which is an expression on the numerator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"a046eedintegrals12a-h3","type":"hint","dependencies":["a046eedintegrals12a-h2"],"title":"Using u-substitution","text":"Bringing the negative sign outside the integral sign, the problem now reads $$-\\\\int u^{\\\\left(-1\\\\right)} \\\\,du$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals12a-h4","type":"hint","dependencies":["a046eedintegrals12a-h3"],"title":"Change the limits","text":"After setting f(x) to f(u), you need to change the limits of integration because the variable of integration has changed from $$x$$ to u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals12a-h5","type":"hint","dependencies":["a046eedintegrals12a-h4"],"title":"Change the limits","text":"As $$u=1+cosx$$, we will have upper and lower limits $$x=1$$ and $$x=2$$ respectively.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals12a-h6","type":"hint","dependencies":["a046eedintegrals12a-h5"],"title":"Interchange the limits","text":"Notice that now the limits begin with the larger number, meaning we must multiply the whole integral by $$-1$$ and interchange the limits.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals12a-h7","type":"hint","dependencies":["a046eedintegrals12a-h6"],"title":"Rewrite the integral in term of u","text":"$$-\\\\int_{2}^{1} u^{\\\\left(-1\\\\right)} \\\\,du=\\\\int_{1}^{2} u^{\\\\left(-1\\\\right)} \\\\,du$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals12a-h8","type":"hint","dependencies":["a046eedintegrals12a-h7"],"title":"Find the integral","text":"$$\\\\int_{1}^{2} u^{\\\\left(-1\\\\right)} \\\\,du=ln|u|$$ with the limit goes from $$u=1$$ to $$u=2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals12a-h9","type":"hint","dependencies":["a046eedintegrals12a-h8"],"title":"Evaluate the integral","text":"$$ln2-ln1=ln2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a046eedintegrals13","title":"In the following exercises, compute each indefinite integral.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.6 Integrals Involving Exponential and Logarithmic Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a046eedintegrals13a","stepAnswer":["$$e^{2x}+C$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\int e^2 x \\\\,dx$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$e^{2x}+C$$","choices":["$$e^{2x}+C$$","$$e^u+C$$"],"hints":{"DefaultPathway":[{"id":"a046eedintegrals13a-h1","type":"hint","dependencies":[],"title":"U-substitution","text":"Set $$u=2x$$ then $$du=2dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals13a-h2","type":"hint","dependencies":["a046eedintegrals13a-h1"],"title":"Rewrite the integral in term of u","text":"$$\\\\int e^u \\\\,du$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals13a-h3","type":"hint","dependencies":["a046eedintegrals13a-h2"],"title":"Find the integral","text":"$$e^u+C$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals13a-h4","type":"hint","dependencies":["a046eedintegrals13a-h3"],"title":"Back-subsitute","text":"$$e^{2x}+C$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a046eedintegrals14","title":"In the following exercises, compute each indefinite integral.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.6 Integrals Involving Exponential and Logarithmic Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a046eedintegrals14a","stepAnswer":["$$\\\\frac{1}{2} ln|x|+C$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\int \\\\frac{2}{x} \\\\,dx$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{1}{2} ln|x|+C$$","choices":["$$\\\\frac{1}{2} ln$$","$$\\\\frac{1}{2} \\\\ln(x)+C$$","$$\\\\frac{1}{2} ln|x|+C$$","$$+C$$","$$x$$"],"hints":{"DefaultPathway":[{"id":"a046eedintegrals14a-h1","type":"hint","dependencies":[],"title":"Rewriting","text":"As a property of integral, the constant $$2$$ can be moved out of the integrand which left $$(1/2)*\\\\int \\\\frac{1}{x} \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals14a-h2","type":"hint","dependencies":["a046eedintegrals14a-h1"],"title":"Find the integral","text":"$$\\\\frac{1}{2} ln|x|+C$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a046eedintegrals15","title":"In the following exercises, find each indefinite integral by using appropriate substitutions.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.6 Integrals Involving Exponential and Logarithmic Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a046eedintegrals15a","stepAnswer":["$$\\\\frac{-1}{\\\\operatorname{lnabs}\\\\left(x\\\\right)}+C$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\int \\\\frac{1}{{\\\\left(x lnx\\\\right)}^2} \\\\,dx$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{-1}{\\\\operatorname{lnabs}\\\\left(x\\\\right)}+C$$","choices":["$$\\\\frac{-1}{\\\\operatorname{lnabs}\\\\left(x\\\\right)}+C$$","$$\\\\frac{1}{\\\\operatorname{lnabs}\\\\left(x\\\\right)}+C$$"],"hints":{"DefaultPathway":[{"id":"a046eedintegrals15a-h1","type":"hint","dependencies":[],"title":"U-substitution","text":"Set $$u=lnx$$ then $$du=\\\\frac{1}{x} dx$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals15a-h2","type":"hint","dependencies":["a046eedintegrals15a-h1"],"title":"Rewrite the integral in term of u","text":"$$\\\\int u^{\\\\left(-2\\\\right)} \\\\,du$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals15a-h3","type":"hint","dependencies":["a046eedintegrals15a-h2"],"title":"Find the integral","text":"$$\\\\int u^{\\\\left(-2\\\\right)} \\\\,du=-\\\\left(u^{\\\\left(-1\\\\right)}\\\\right)+C$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals15a-h4","type":"hint","dependencies":["a046eedintegrals15a-h3"],"title":"Back-substitute","text":"$$\\\\frac{-1}{ln}|x|+C$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a046eedintegrals2","title":"Square Root of an Exponential Function","body":"Find the antiderivative of the exponential function","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.6 Integrals Involving Exponential and Logarithmic Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a046eedintegrals2a","stepAnswer":["$$\\\\frac{2}{3} {\\\\left(1+e^x\\\\right)}^{\\\\frac{3}{2}}+C$$"],"problemType":"MultipleChoice","stepTitle":"$$e^x \\\\sqrt{1+e^x}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{2}{3} {\\\\left(1+e^x\\\\right)}^{\\\\frac{3}{2}}+C$$","choices":["$$\\\\frac{2}{3} {\\\\left(1+e^x\\\\right)}^{\\\\frac{3}{2}}+C$$","$${\\\\left(1+e^x\\\\right)}^{\\\\frac{3}{2}}+C$$"],"hints":{"DefaultPathway":[{"id":"a046eedintegrals2a-h1","type":"hint","dependencies":[],"title":"Rewrite the problem","text":"First rewrite the problem using a rational exponent: $$\\\\int e^x \\\\sqrt{1+e^x} \\\\,dx=\\\\int \\\\frac{e^x {\\\\left(1+e^x\\\\right)}^1}{2} \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals2a-h2","type":"hint","dependencies":["a046eedintegrals2a-h1"],"title":"Using u-substitution","text":"Set $$u=1+e^x$$ then $$du=e^x dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals2a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["/int{u**1/2,u}"],"dependencies":["a046eedintegrals2a-h2"],"title":"Using u-substitution","text":"What is the expression of the integral after using the u-substitution?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals2a-h4","type":"hint","dependencies":["a046eedintegrals2a-h3"],"title":"Find the integral","text":"$$\\\\int \\\\frac{u^1}{2} \\\\,du=\\\\frac{\\\\frac{u^3}{2}}{\\\\frac{3}{2}}+C=\\\\frac{2}{3} u^{\\\\frac{3}{2}}+C$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals2a-h5","type":"hint","dependencies":["a046eedintegrals2a-h4"],"title":"Back-subsitute","text":"The final step is to convert u back into the original variable $$\\\\frac{2}{3} {\\\\left(1+e^x\\\\right)}^{\\\\frac{3}{2}}+C$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a046eedintegrals3","title":"Using Substitution with an Exponential Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.6 Integrals Involving Exponential and Logarithmic Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a046eedintegrals3a","stepAnswer":["$$\\\\frac{1}{2} e^{2x^3}+C$$"],"problemType":"MultipleChoice","stepTitle":"Use substitution to evaluate the indefinite integral /int{(3*(x**2)*e**2*x**3,x}","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{1}{2} e^{2x^3}+C$$","choices":["$$\\\\frac{1}{2} e^{2x^3}+C$$","$$\\\\frac{1}{2} e^u+C$$","$$\\\\frac{1}{2} e^{2x^3}$$"],"hints":{"DefaultPathway":[{"id":"a046eedintegrals3a-h1","type":"hint","dependencies":[],"title":"Choosing u","text":"We need to choose an appropriate u to perform substitution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals3a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2x^3$$"],"dependencies":["a046eedintegrals3a-h1"],"title":"Choosing u","text":"Which function should we set as \'u\'?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$2x^3$$","$$2x$$","$$3x^2$$","$$e^2 x^3$$"]},{"id":"a046eedintegrals3a-h3","type":"hint","dependencies":["a046eedintegrals3a-h2"],"title":"Find the derivative of u","text":"Since $$u=2x^3$$ then $$du=6x^2 dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals3a-h4","type":"hint","dependencies":["a046eedintegrals3a-h3"],"title":"Adjusting the constant coefficient","text":"As we\'ve determined that du $$=$$ $$6x^2$$ dx, and the original function incorporates a factor of $$3x^2$$ (rather than 6x**2), we need to make a minor adjustment to the constant multiplier to align it with the function we intend to substitute.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals3a-h5","type":"hint","dependencies":["a046eedintegrals3a-h4"],"title":"Adjusting the constant coefficient","text":"We can easily see that $$3$$ is $$\\\\frac{1}{2}$$ of $$6$$ so we will divide both side of the expression by $$2$$ and obtain $$\\\\frac{du}{2}=3x^2 dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals3a-h6","type":"hint","dependencies":["a046eedintegrals3a-h5"],"title":"Using u-substitution","text":"$$\\\\int 3x^2 e^2 x^3 \\\\,dx=(1/2)*\\\\int e^u \\\\,du$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals3a-h7","type":"hint","dependencies":["a046eedintegrals3a-h6"],"title":"Find the integral","text":"Integrate the expression in u and then substitute the original expression in $$x$$ back into the u integral: $$(1/2)*\\\\int e^u \\\\,du=\\\\frac{1}{2} e^u+C$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals3a-h8","type":"hint","dependencies":["a046eedintegrals3a-h7"],"title":"Back-subsitute","text":"The final step is to convert u back into the original variable $$\\\\frac{1}{2} e^{2x^3}+C$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a046eedintegrals4","title":"Evaluating a Definite Integral Involving an Exponential Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.6 Integrals Involving Exponential and Logarithmic Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a046eedintegrals4a","stepAnswer":["$$-\\\\left(e^{\\\\left(-1\\\\right)}\\\\right)+1$$"],"problemType":"MultipleChoice","stepTitle":"Evaluate the definite integral $$\\\\int_{1}^{2} e^1-x \\\\,dx$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-\\\\left(e^{\\\\left(-1\\\\right)}\\\\right)+1$$","choices":["$$-\\\\left(e^{\\\\left(-1\\\\right)}\\\\right)+1$$","$$-\\\\left(e^{\\\\left(-1\\\\right)}\\\\right)$$"],"hints":{"DefaultPathway":[{"id":"a046eedintegrals4a-h1","type":"hint","dependencies":[],"title":"Using u-substitution","text":"Let $$u=1-x$$ so $$du=-1dx$$ or $$-du=dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals4a-h2","type":"hint","dependencies":["a046eedintegrals4a-h1"],"title":"Change the limits","text":"After setting f(x) to f(u), you need to change the limits of integration because the variable of integration has changed from $$x$$ to u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a046eedintegrals4a-h2"],"title":"New lower limit","text":"What is the new lower limit in term of u?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"a046eedintegrals4a-h3-s1","type":"hint","dependencies":[],"title":"New lower limit","text":"Using $$u=1-x$$ in the first hint, we have $$u=1-(1)=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"a046eedintegrals4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a046eedintegrals4a-h3"],"title":"New upper limit","text":"What is the new upper limit in term of u?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"a046eedintegrals4a-h4-s1","type":"hint","dependencies":[],"title":"New upper limit","text":"Using $$u=1-x$$ in the first hint, we have $$u=1-(2)=-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"a046eedintegrals4a-h5","type":"hint","dependencies":["a046eedintegrals4a-h4"],"title":"New integral expression","text":"The new integral with a variable of u will be $$-\\\\int e^u \\\\,du$$ as the limit goes from $$u=0$$ to $$u=-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals4a-h6","type":"hint","dependencies":["a046eedintegrals4a-h5"],"title":"Find the integral","text":"$$\\\\int e^u \\\\,du=e^u$$ as the limit goes from $$u=-1$$ to $$u=0$$. Note that the lower limit always has to be less than the upper limit. To change the order of limits, we multply the integral with $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals4a-h7","type":"hint","dependencies":["a046eedintegrals4a-h6"],"title":"Evaluate the integral","text":"$$e^0-e^{\\\\left(-1\\\\right)}=-\\\\left(e^{\\\\left(-1\\\\right)}\\\\right)+1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a046eedintegrals5","title":"CHECKPOINT $$5.34$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.6 Integrals Involving Exponential and Logarithmic Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a046eedintegrals5a","stepAnswer":["$$e^4-1$$"],"problemType":"MultipleChoice","stepTitle":"Evaluate $$\\\\int_{1}^{2} e^{2x} \\\\,dx$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$e^4-1$$","choices":["$$e^4-1$$","$$e^4$$"],"hints":{"DefaultPathway":[{"id":"a046eedintegrals5a-h1","type":"hint","dependencies":[],"title":"Using u-substitution","text":"Let $$u=2x$$ so $$du=2dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals5a-h2","type":"hint","dependencies":["a046eedintegrals5a-h1"],"title":"Change the limits","text":"After setting f(x) to f(u), you need to change the limits of integration because the variable of integration has changed from $$x$$ to u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a046eedintegrals5a-h2"],"title":"New lower limit","text":"What is the new lower limit in term of u?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"a046eedintegrals5a-h3-s1","type":"hint","dependencies":[],"title":"New lower limit","text":"Using $$u=2x$$ in the first hint, we have $$u=2\\\\times0=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"a046eedintegrals5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a046eedintegrals5a-h3"],"title":"New upper limit","text":"What is the new upper limit in term of u?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"a046eedintegrals5a-h4-s1","type":"hint","dependencies":[],"title":"New upper limit","text":"Using $$u=2x$$ in the first hint, we have $$u=2\\\\times2=4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"a046eedintegrals5a-h5","type":"hint","dependencies":["a046eedintegrals5a-h4"],"title":"New integral expression","text":"The new integral with a variable of u will be $$\\\\int e^u \\\\,du$$ as the limit goes from $$u=0$$ to $$u=4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals5a-h6","type":"hint","dependencies":["a046eedintegrals5a-h5"],"title":"Find the integral","text":"$$\\\\int e^u \\\\,du=e^u$$ as the limit goes from $$u=0$$ to $$u=4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals5a-h7","type":"hint","dependencies":["a046eedintegrals5a-h6"],"title":"Evaluate the integral","text":"$$e^4-e^0=e^4-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a046eedintegrals6","title":"Growth of Bacteria in a Culture","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.6 Integrals Involving Exponential and Logarithmic Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a046eedintegrals6a","stepAnswer":["$$17.282$$"],"problemType":"TextBox","stepTitle":"Suppose the rate of growth of bacteria in a Petri dish is given by $$q(t)=3t$$, where $$t$$ is given in hours and q(t) is given in thousands of bacteria per hour. If a culture starts with 10,000 bacteria, find a function Q(t) that gives the number of bacteria in the Petri dish at any time $$t$$. How many bacteria are in the dish after $$2$$ hours?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$17.282$$","hints":{"DefaultPathway":[{"id":"a046eedintegrals6a-h1","type":"hint","dependencies":[],"title":"Express the integral","text":"The first step is to convert the problem from words into mathematical expression. Since $$t$$ is an independent variable in this situation, we have to take the integral with respect to $$t$$ instead of $$x$$ as usual. We have $$\\\\int 3^t \\\\,dt$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals6a-h2","type":"hint","dependencies":["a046eedintegrals6a-h1"],"title":"Find the integral","text":"$$\\\\int 3^t \\\\,dt=\\\\frac{3^t}{\\\\ln(3)}+C$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals6a-h3","type":"hint","dependencies":["a046eedintegrals6a-h2"],"title":"At $$t=0$$","text":"This is a time when the culture starts, we have $$Q(0)=10=\\\\frac{3^t}{\\\\ln(3)}+C$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9.09$$"],"dependencies":["a046eedintegrals6a-h3"],"title":"Find C","text":"Using basic algebra, what will C be? (Rounded to $$2$$ decimal places)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals6a-h5","type":"hint","dependencies":["a046eedintegrals6a-h4"],"title":"At $$t=2$$","text":"The previous step gave us the expression $$Q(t)=\\\\frac{3^t}{\\\\ln(3)}+9.09$$. We now find the ammount of bacteria as they increase in $$2$$ hours by replace $$t$$ with $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals6a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$17.282$$"],"dependencies":["a046eedintegrals6a-h5"],"title":"Find Q(t) after $$2$$ hours","text":"What is the ammount of bacteria after $$2$$ hours? (Rounded to $$3$$ decimal places)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals6a-h7","type":"hint","dependencies":["a046eedintegrals6a-h6"],"title":"Find Q(t) after $$2$$ hours","text":"$$\\\\frac{3^2}{\\\\ln(3)}+9.09=17.282$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a046eedintegrals7","title":"Fruit Fly Population Growth","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.6 Integrals Involving Exponential and Logarithmic Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a046eedintegrals7a","stepAnswer":["$$122$$"],"problemType":"TextBox","stepTitle":"Suppose a population of fruit flies increases at a rate of $$g(t)=2e^{0.02t}$$ in flies per day. If the initial population of fruit flies is $$100$$ flies, how many flies are in the population after $$10$$ days?\\\\n","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$122$$","hints":{"DefaultPathway":[{"id":"a046eedintegrals7a-h1","type":"hint","dependencies":[],"title":"Net Change Theorem","text":"We can start by applying net change theorem then use u-substitution to solve for the integral.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals7a-h2","type":"hint","dependencies":["a046eedintegrals7a-h1"],"title":"Applying Net Change Theorem","text":"$$g(10)-g(0)=\\\\int_{0}^{10} 2e^{0.02} t \\\\,dt$$. Note that $$g(0)=100$$ is given","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals7a-h3","type":"hint","dependencies":["a046eedintegrals7a-h2"],"title":"Solve for g(10)","text":"Add both sides by g(0) or $$100$$ to obtain $$g(10)=100+\\\\int_{0}^{10} 2e^{0.02} t \\\\,dt$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals7a-h4","type":"hint","dependencies":["a046eedintegrals7a-h3"],"title":"Compute the integral","text":"$$100+\\\\frac{2}{0.02} e^{0.02} t$$ with the limit goes from $$t=0$$ to $$t=10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals7a-h5","type":"hint","dependencies":["a046eedintegrals7a-h4"],"title":"Evaluate the integral","text":"$$100+100e^{0.2}-100=122$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a046eedintegrals8","title":"Evaluating a Definite Integral Using Substitution","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.6 Integrals Involving Exponential and Logarithmic Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a046eedintegrals8a","stepAnswer":["$$e-e^{\\\\frac{1}{2}}$$"],"problemType":"TextBox","stepTitle":"Evaluate the definite integral using substitution: $$\\\\int_{1}^{2} \\\\frac{\\\\frac{e^1}{x}}{x^2} \\\\,dx$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$e-e^{\\\\frac{1}{2}}$$","hints":{"DefaultPathway":[{"id":"a046eedintegrals8a-h1","type":"hint","dependencies":[],"title":"Rewriting","text":"This problem requires some rewriting to simplify applying the properties. First, rewrite the exponent on e as a power of $$x$$, then bring the $$x^2$$ in the denominator up to the numerator using a negative exponent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals8a-h2","type":"hint","dependencies":["a046eedintegrals8a-h1"],"title":"Rewriting","text":"We have $$\\\\int_{1}^{2} \\\\frac{e^{\\\\frac{1}{x}}}{x^2} \\\\,dx=\\\\int_{1}^{2} e^{x^{\\\\left(-1\\\\right)}} x^{\\\\left(-2\\\\right)} \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals8a-h3","type":"hint","dependencies":["a046eedintegrals8a-h2"],"title":"Using u-substitution","text":"Let $$u=x^{\\\\left(-1\\\\right)}$$, the exponent on e then $$du=\\\\left(-x^{\\\\left(-2\\\\right)}\\\\right) dx$$ or $$-du=x^{\\\\left(-2\\\\right)} dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals8a-h4","type":"hint","dependencies":["a046eedintegrals8a-h3"],"title":"Using u-substitution","text":"Bringing the negative sign outside the integral sign, the problem now reads $$-\\\\int e^u \\\\,dx$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals8a-h5","type":"hint","dependencies":["a046eedintegrals8a-h4"],"title":"Change the limits","text":"After setting f(x) to f(u), you need to change the limits of integration because the variable of integration has changed from $$x$$ to u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals8a-h6","type":"hint","dependencies":["a046eedintegrals8a-h5"],"title":"Change the limits","text":"With $$u=x^{\\\\left(-1\\\\right)}$$, we will have upper and lower limits $$x=1$$ and $$x=\\\\frac{1}{2}$$ respectively.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals8a-h7","type":"hint","dependencies":["a046eedintegrals8a-h6"],"title":"Interchange the limits","text":"Notice that now the limits begin with the larger number, meaning we must multiply by $$-1$$ and interchange the limits.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals8a-h8","type":"hint","dependencies":["a046eedintegrals8a-h7"],"title":"Find the limit","text":"$$\\\\int_{\\\\frac{1}{2}}^{1} e^u \\\\,du=e^u$$ with the limit goes from $$u=\\\\frac{1}{2}$$ to $$u=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals8a-h9","type":"hint","dependencies":["a046eedintegrals8a-h8"],"title":"Evaluate the limit","text":"$$e-e^{\\\\frac{1}{2}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a046eedintegrals9","title":"Finding an Antiderivative Involving lnx","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.6 Integrals Involving Exponential and Logarithmic Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a046eedintegrals9a","stepAnswer":["$$3ln|x-10|+C$$"],"problemType":"MultipleChoice","stepTitle":"Find the antiderivative of the function $$\\\\frac{3}{x-10}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$3ln|x-10|+C$$","choices":["$$+C$$","$$3ln$$","$$3ln$$","$$3\\\\ln(x-10)+C$$","$$3ln|x-10|+C$$","$$x-10$$","$$x-10$$"],"hints":{"DefaultPathway":[{"id":"a046eedintegrals9a-h1","type":"hint","dependencies":[],"title":"Factoring","text":"First factor the $$3$$ outside the integral symbol. Then use the $$u^{\\\\left(-1\\\\right)}$$ rule. Thus, $$\\\\int \\\\frac{3}{x-10} \\\\,dx=3*\\\\int \\\\frac{1}{x-10} \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals9a-h2","type":"hint","dependencies":["a046eedintegrals9a-h1"],"title":"U-substitution","text":"Set $$u=x-10$$ then $$du=dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals9a-h3","type":"hint","dependencies":["a046eedintegrals9a-h2"],"title":"Find the integral","text":"$$3*\\\\int \\\\frac{1}{u} \\\\,du=3ln|u|+C$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a046eedintegrals9a-h4","type":"hint","dependencies":["a046eedintegrals9a-h3"],"title":"Back-subsitute","text":"As we set $$u=x-10$$ at the beginning, we then obtain $$3ln|x-10|+C$$ with $$x$$ is different from $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a06269afundamental1","title":"Finding the Average Value of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.3 The Fundamental Theorem of Calculus","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a06269afundamental1a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"Find the average value of the function $$f(x)=8-2x$$ over the interval [0,4].","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a06269afundamental1a-h1","type":"hint","dependencies":[],"title":"Area Under","text":"First, find the area under the function in the first quadrant that is bounded by the $$x-$$ and $$y-axes$$. You can see this shape in the figure.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a06269afundamental1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["a06269afundamental1a-h1"],"title":"Area Under","text":"What is $$A=4\\\\frac{1}{2}\\\\times8=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a06269afundamental1a-h3","type":"hint","dependencies":["a06269afundamental1a-h2"],"title":"Average Value","text":"Next, find the average value by multiplying the area by $$\\\\frac{1}{b-a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a06269afundamental1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a06269afundamental1a-h3"],"title":"Average Value","text":"What is $$16\\\\frac{1}{4-0}=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a06269afundamental1a-h5","type":"hint","dependencies":["a06269afundamental1a-h4"],"title":"c","text":"Lastly, set the average value to f(c) and solve for c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a06269afundamental1a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a06269afundamental1a-h5"],"title":"c","text":"Given $$8-2c=4$$, what is $$c=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a06269afundamental14","title":"Fundamental Theorem of Calculus $$Pt.1$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.3 The Fundamental Theorem of Calculus","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a06269afundamental14a","stepAnswer":["$$e^cos^$\\\\left(x\\\\righ$$"],"problemType":"TextBox","stepTitle":"What is the derivative with respect to $$x$$ for $$\\\\int_{1}^{x} e^cos^$\\\\left(t\\\\righ \\\\,dt$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$e^cos^$\\\\left(x\\\\righ$$","hints":{"DefaultPathway":[{"id":"a06269afundamental14a-h1","type":"hint","dependencies":[],"title":"Derivative","text":"Use the first part of the fundamental theorem of calculus to find the derivative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a06269afundamental14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$e^cos^$\\\\left(x\\\\righ$$"],"dependencies":["a06269afundamental14a-h1"],"title":"Derivative","text":"Because a is a constant and $$b=x$$, you can substitute $$t$$ with $$x$$. In other words, plugging in $$x$$ into $$e^cos^$\\\\left(t\\\\righ$$ becomes $$___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a06269afundamental15","title":"Evaluation Theorem","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.3 The Fundamental Theorem of Calculus","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a06269afundamental15a","stepAnswer":["$$e^x+e$$"],"problemType":"TextBox","stepTitle":"Express $$\\\\int_{1}^{x} e^t \\\\,dt$$ as a function F(x).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$e^x+e$$","hints":{"DefaultPathway":[{"id":"a06269afundamental15a-h1","type":"hint","dependencies":[],"title":"Second Fundamental Theorem","text":"Remember that the evaluation theorem is also known as the second part of the fundamental theorem of calculus.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a06269afundamental15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$e^t$$"],"dependencies":["a06269afundamental15a-h1"],"title":"Antiderivative","text":"What is the antiderivative of $$e^t=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a06269afundamental15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$e^x+e$$"],"dependencies":["a06269afundamental15a-h1"],"title":"$$F(b)-F(a)$$","text":"What is $$F(b)-F(a)=___$$? In this case, what is $$e^x-e^1=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a06269afundamental2","title":"Finding the Point Where a Function Takes on Its Average Value","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.3 The Fundamental Theorem of Calculus","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a06269afundamental2a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"Given $$/int{x**2$$, 0,3,x}, find c such that f(c) equals the average value of $$f(x)=x^2$$ over [0,3].","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a06269afundamental2a-h1","type":"hint","dependencies":[],"title":"Area Under","text":"First, find the area under the function in the first quadrant that is bounded by the $$x-$$ and $$y-axes$$. Integrate f(x) between the limits of a and $$b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a06269afundamental2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a06269afundamental2a-h1"],"title":"Area Under","text":"What is $$\\\\frac{3^3}{3}-\\\\frac{0^3}{3}=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a06269afundamental2a-h3","type":"hint","dependencies":["a06269afundamental2a-h2"],"title":"Average Value","text":"Next, find the average value by multiplying the area by $$\\\\frac{1}{b-a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a06269afundamental2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a06269afundamental2a-h3"],"title":"Average Value","text":"What is $$9\\\\frac{1}{3-0}=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a06269afundamental2a-h5","type":"hint","dependencies":["a06269afundamental2a-h4"],"title":"Average Value","text":"Lastly, set the average value to f(c) and solve for c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a06269afundamental2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a06269afundamental2a-h5"],"title":"Average Value","text":"Given the interval of [0,3} and $$c^2=3$$, what is $$c=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a06269afundamental3","title":"Using the Fundamental Theorem and the Chain Rule to Calculate Derivatives","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.3 The Fundamental Theorem of Calculus","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a06269afundamental3a","stepAnswer":["$$\\\\fracsin^2\\\\left(\\\\sqrt{x}\\\\right)\\\\sqrt{x}}$$"],"problemType":"TextBox","stepTitle":"Let $$F(x)=\\\\int_{1}^{\\\\sqrt{x}} sint \\\\,dt$$. FInd F\'(x).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\fracsin^2\\\\left(\\\\sqrt{x}\\\\right)\\\\sqrt{x}}$$","hints":{"DefaultPathway":[{"id":"a06269afundamental3a-h1","type":"hint","dependencies":[],"title":"u(x)","text":"First, let $$u(x)=\\\\sqrt{x}$$ so that way we have $$F(x)=/iint{sin(t),a,u(x),t)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a06269afundamental3a-h2","type":"hint","dependencies":["a06269afundamental3a-h1"],"title":"Fundamental Theorem and chain rule","text":"Then, using the Fundamental theorem of Calculus and the chain rule, find F\'(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a06269afundamental7","title":"A Roller-Skating Race","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.3 The Fundamental Theorem of Calculus","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a06269afundamental7a","stepAnswer":["Kathy"],"problemType":"MultipleChoice","stepTitle":"James and Kathy are racing on roller skates. They race along a long, straight truck, and whoever has gone the farthest after $$5$$ sec wins a prize. If James can skate at a velocity of $$f(t)=2t+5$$ $$\\\\frac{ft}{sec}$$ and Kathy can skate at a velocity of $$g(t)=cos\\\\left(\\\\frac{pit}{2}\\\\right)+10$$ $$\\\\frac{ft}{sec}$$, who is going to win the race?","stepBody":"","answerType":"string","variabilization":{},"choices":["James","Kathy","neither"],"hints":{"DefaultPathway":[{"id":"a06269afundamental7a-h1","type":"hint","dependencies":[],"title":"Integrate both functions over the interval [0,5] and see which value is bigger.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a06269afundamental7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$50$$"],"dependencies":["a06269afundamental7a-h1"],"title":"For James, solve the integral $$\\\\int_{0}^{5} 2t+5 \\\\,dt$$. Round to the first decimal place if necessary.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a06269afundamental7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$50.6$$"],"dependencies":["a06269afundamental7a-h1"],"title":"For Kathy, solve the integral $$\\\\int_{0}^{5} cos\\\\left(\\\\frac{pit}{2}\\\\right)+10 \\\\,dt$$. Round to the first decimal place if necessary.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a06269afundamental8","title":"A Roller-Skating Rematch","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.3 The Fundamental Theorem of Calculus","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a06269afundamental8a","stepAnswer":["Kathy"],"problemType":"MultipleChoice","stepTitle":"James and Kathy have a rematch for their race on roller skates. Again, they race along a long, straight truck, and whoever has gone the farthest after 3-sec wins. Their speeds are the same as last time with James\'s velocity at $$f(t)=2t+5$$ $$\\\\frac{ft}{sec}$$ and Kathy\'s velocity of $$g(t)=cos\\\\left(\\\\frac{pit}{2}\\\\right)+10$$ $$\\\\frac{ft}{sec}$$. Who is going to win the race this time around?","stepBody":"","answerType":"string","variabilization":{},"choices":["James","Kathy","neither"],"hints":{"DefaultPathway":[{"id":"a06269afundamental8a-h1","type":"hint","dependencies":[],"title":"Integrate both functions over the interval [0,3] and see which value is bigger.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a06269afundamental8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$24$$"],"dependencies":["a06269afundamental8a-h1"],"title":"For James, solve the integral $$\\\\int_{0}^{3} 2t+5 \\\\,dt$$. Round to the first decimal place if necessary.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a06269afundamental8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$28.4$$"],"dependencies":["a06269afundamental8a-h1"],"title":"For Kathy, solve the integral $$\\\\int_{0}^{3} cos\\\\left(\\\\frac{pit}{2}\\\\right)+10 \\\\,dt$$. Round to the first decimal place if necessary.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a082a6bGraLiIneq1","title":"Finding the Value of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Graph Linear Inequalities in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a082a6bGraLiIneq1a","stepAnswer":["$$26$$"],"problemType":"TextBox","stepTitle":"Evalute f(3) if $$f(x)=2x^2+3x-1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$26$$","hints":{"DefaultPathway":[{"id":"a082a6bGraLiIneq1a-h1","type":"hint","dependencies":[],"title":"Substituting the Value for $$x$$","text":"We must substitute $$3$$ for $$x$$. $$f(3)={2\\\\left(3\\\\right)}^2+3\\\\times3-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a082a6bGraLiIneq1a-h2","type":"hint","dependencies":["a082a6bGraLiIneq1a-h1"],"title":"Simplifying","text":"$$f(3)=18+9-1=26$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a082a6bGraLiIneq10","title":"Finding the Value of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Graph Linear Inequalities in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a082a6bGraLiIneq10a","stepAnswer":["$$3h^2-5$$"],"problemType":"TextBox","stepTitle":"Evaluate $$f{\\\\left(h^2\\\\right)}$$ if $$f(x)=3x-5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3h^2-5$$","hints":{"DefaultPathway":[{"id":"a082a6bGraLiIneq10a-h1","type":"hint","dependencies":[],"title":"Substituting the Value for $$x$$","text":"Substitute $$h^2$$ for $$x$$. $$f{\\\\left(h^2\\\\right)}=3h^2-5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a082a6bGraLiIneq10a-h2","type":"hint","dependencies":["a082a6bGraLiIneq10a-h1"],"title":"Simplifying","text":"This function cannot be simplified anymore.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a082a6bGraLiIneq11","title":"Finding the Value of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Graph Linear Inequalities in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a082a6bGraLiIneq11a","stepAnswer":["$$3x+1$$"],"problemType":"TextBox","stepTitle":"Evaluate $$f{\\\\left(x+2\\\\right)}$$ if $$f(x)=3x-5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3x+1$$","hints":{"DefaultPathway":[{"id":"a082a6bGraLiIneq11a-h1","type":"hint","dependencies":[],"title":"Substituting the Value for $$x$$","text":"Substituting $$x+2$$ for $$x$$. $$f{\\\\left(x+2\\\\right)}=3\\\\left(x+2\\\\right)-5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a082a6bGraLiIneq11a-h2","type":"hint","dependencies":["a082a6bGraLiIneq11a-h1"],"title":"Simplifying","text":"$$f{\\\\left(x+2\\\\right)}=3x+6-5=3x+1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a082a6bGraLiIneq12","title":"Finding the Value of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Graph Linear Inequalities in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a082a6bGraLiIneq12a","stepAnswer":["$$3x-4$$"],"problemType":"TextBox","stepTitle":"Evalute $$f{\\\\left(x\\\\right)}+f{\\\\left(2\\\\right)}$$ if $$f(x)=3x-5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3x-4$$","hints":{"DefaultPathway":[{"id":"a082a6bGraLiIneq12a-h1","type":"hint","dependencies":[],"title":"Substituting the Value for $$x$$","text":"Substitute $$2$$ for $$x$$. $$f(2)=3(2)-5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a082a6bGraLiIneq12a-h2","type":"hint","dependencies":["a082a6bGraLiIneq12a-h1"],"title":"Simplifying","text":"$$f{\\\\left(x\\\\right)}+f{\\\\left(2\\\\right)}=3x-5+3\\\\left(2\\\\right)-5=3x-5+6-5=3x-4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a082a6bGraLiIneq13","title":"Finding the Value of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Graph Linear Inequalities in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a082a6bGraLiIneq13a","stepAnswer":["$$4m^2-7$$"],"problemType":"TextBox","stepTitle":"Find $$f{\\\\left(m^2\\\\right)}$$ if $$f(x)=4x-7$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4m^2-7$$","hints":{"DefaultPathway":[{"id":"a082a6bGraLiIneq13a-h1","type":"hint","dependencies":[],"title":"Substituting the Value for $$x$$","text":"Substitute $$m^2$$ for $$x$$. $$f{\\\\left(m^2\\\\right)}=4m^2-7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a082a6bGraLiIneq13a-h2","type":"hint","dependencies":["a082a6bGraLiIneq13a-h1"],"title":"Simplifying","text":"This function cannot be simplified anymore.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a082a6bGraLiIneq14","title":"Finding the Value of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Graph Linear Inequalities in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a082a6bGraLiIneq14a","stepAnswer":["$$4x-19$$"],"problemType":"TextBox","stepTitle":"Find $$f(x-3)$$ if $$f(x)=4x-7$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4x-19$$","hints":{"DefaultPathway":[{"id":"a082a6bGraLiIneq14a-h1","type":"hint","dependencies":[],"title":"Substituting the Value for $$x$$","text":"Substitute $$x-3$$ for $$x$$. $$f(x-3)=4(x-3)-7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a082a6bGraLiIneq14a-h2","type":"hint","dependencies":["a082a6bGraLiIneq14a-h1"],"title":"Simplifying","text":"$$f(x-3)=4x-12-7=4x-19$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a082a6bGraLiIneq15","title":"Finding the Value of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Graph Linear Inequalities in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a082a6bGraLiIneq15a","stepAnswer":["$$4x-12$$"],"problemType":"TextBox","stepTitle":"Find $$f(x)-f(3)$$ if $$f(x)=4x-7$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4x-12$$","hints":{"DefaultPathway":[{"id":"a082a6bGraLiIneq15a-h1","type":"hint","dependencies":[],"title":"Substituting the Value for $$x$$","text":"Substitute $$3$$ for $$x$$. $$f(3)=4(3)-7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a082a6bGraLiIneq15a-h2","type":"hint","dependencies":["a082a6bGraLiIneq15a-h1"],"title":"Simplifying","text":"$$f(x)-f(3)=(4x-7)-(4(3)-7)=4x-7-(12-7)=4x-7-5=4x-12$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a082a6bGraLiIneq2","title":"Finding the Value of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Graph Linear Inequalities in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a082a6bGraLiIneq2a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"Evalute $$f(-2)$$ if $$f(x)=2x^2+3x-1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a082a6bGraLiIneq2a-h1","type":"hint","dependencies":[],"title":"Substituting the Value for $$x$$","text":"We must substitute $$-2$$ for $$x$$. f(-2)=2(-2)**2+3*(-2)-1","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a082a6bGraLiIneq2a-h2","type":"hint","dependencies":["a082a6bGraLiIneq2a-h1"],"title":"Simplifying","text":"$$f(-2)=8-6-1=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a082a6bGraLiIneq3","title":"Finding the Value of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Graph Linear Inequalities in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a082a6bGraLiIneq3a","stepAnswer":["$$2a^2+3a-1$$"],"problemType":"TextBox","stepTitle":"Evalute f(a) if $$f(x)=2x^2+3x-1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2a^2+3a-1$$","hints":{"DefaultPathway":[{"id":"a082a6bGraLiIneq3a-h1","type":"hint","dependencies":[],"title":"Substituting the Value for $$x$$","text":"Substitute a for $$x$$. $$f(t)=2a^2+3a-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a082a6bGraLiIneq3a-h2","type":"hint","dependencies":["a082a6bGraLiIneq3a-h1"],"title":"Simplifying","text":"This function cannot be simplified anymore.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a082a6bGraLiIneq4","title":"Finding the Value of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Graph Linear Inequalities in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a082a6bGraLiIneq4a","stepAnswer":["$$22$$"],"problemType":"TextBox","stepTitle":"Evaluate f(3) if $$f(x)=3x^2-2x+1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$22$$","hints":{"DefaultPathway":[{"id":"a082a6bGraLiIneq4a-h1","type":"hint","dependencies":[],"title":"Substituting the Value for $$x$$","text":"Substitute $$3$$ for $$x$$. $$f(3)={3\\\\left(3\\\\right)}^2-2\\\\left(3\\\\right)+1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a082a6bGraLiIneq4a-h2","type":"hint","dependencies":["a082a6bGraLiIneq4a-h1"],"title":"Simplifying","text":"$$f(3)=27-6+1=22$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a082a6bGraLiIneq5","title":"Finding the Value of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Graph Linear Inequalities in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a082a6bGraLiIneq5a","stepAnswer":["$$7$$"],"problemType":"TextBox","stepTitle":"Evaluate $$f(-1)$$ if $$f(x)=3x^2-2x+1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7$$","hints":{"DefaultPathway":[{"id":"a082a6bGraLiIneq5a-h1","type":"hint","dependencies":[],"title":"Substituting the Value for $$x$$","text":"Substitute $$-1$$ for $$x$$. f(-1)=3(-1)**2-2(-1)+2","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a082a6bGraLiIneq5a-h2","type":"hint","dependencies":["a082a6bGraLiIneq5a-h1"],"title":"Simplifying","text":"$$f(-1)=3+2+2=7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a082a6bGraLiIneq6","title":"Finding the Value of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Graph Linear Inequalities in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a082a6bGraLiIneq6a","stepAnswer":["$$3t^2-2t+2$$"],"problemType":"TextBox","stepTitle":"Evaluate f(t) if $$f(x)=3x^2-2x+1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3t^2-2t+2$$","hints":{"DefaultPathway":[{"id":"a082a6bGraLiIneq6a-h1","type":"hint","dependencies":[],"title":"Substituting the Value for $$x$$","text":"Substitute $$t$$ for $$x$$. $$f(t)=3t^2-2t+2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a082a6bGraLiIneq6a-h2","type":"hint","dependencies":["a082a6bGraLiIneq6a-h1"],"title":"Simplifying","text":"This function cannot be simplified anymore.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a082a6bGraLiIneq7","title":"Finding the Value of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Graph Linear Inequalities in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a082a6bGraLiIneq7a","stepAnswer":["$$19$$"],"problemType":"TextBox","stepTitle":"Evaluate f(2) if $$f(x)=2x^2+4x-3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$19$$","hints":{"DefaultPathway":[{"id":"a082a6bGraLiIneq7a-h1","type":"hint","dependencies":[],"title":"Substituting the Value for $$x$$","text":"Substitute $$2$$ for $$x$$. $$f(2)={2\\\\left(2\\\\right)}^2+4\\\\left(2\\\\right)+3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a082a6bGraLiIneq7a-h2","type":"hint","dependencies":["a082a6bGraLiIneq7a-h1"],"title":"Simplifying","text":"$$f(2)=8+8+3=19$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a082a6bGraLiIneq8","title":"Finding the Value of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Graph Linear Inequalities in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a082a6bGraLiIneq8a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"Evaluate $$f(-3)$$ if $$f(x)=2x^2+4x-3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a082a6bGraLiIneq8a-h1","type":"hint","dependencies":[],"title":"Substituting the Value for $$x$$","text":"Substitute $$-3$$ for $$x$$. f(-3)=2(-3)**2+4(-3)-3","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a082a6bGraLiIneq8a-h2","type":"hint","dependencies":["a082a6bGraLiIneq8a-h1"],"title":"Simplifying","text":"$$f(-3)=18-12-3=3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a082a6bGraLiIneq9","title":"Finding the Value of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Graph Linear Inequalities in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a082a6bGraLiIneq9a","stepAnswer":["$$2h^2+4h-3$$"],"problemType":"TextBox","stepTitle":"Evaluate f(h) if $$f(x)=2x^2+4x-3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2h^2+4h-3$$","hints":{"DefaultPathway":[{"id":"a082a6bGraLiIneq9a-h1","type":"hint","dependencies":[],"title":"Substituting the Value 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Write the equation in slope-intercept form.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Find the Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a08a388line1a","stepAnswer":["$$y=4x+1$$"],"problemType":"MultipleChoice","stepTitle":"Slope $$=$$ $$4$$ and y-intercept $$=$$ $$(0,1)$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=4x+1$$","choices":["$$y=x+4$$","$$y=4x-1$$","$$y=x-4$$","$$y=4x+1$$"],"hints":{"DefaultPathway":[{"id":"a08a388line1a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":[],"title":"Identify the Slope","text":"What is the value of the slope given in the problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line1a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(0,1)$$"],"dependencies":["a08a388line1a-h1"],"title":"Identify the y-intercept","text":"What is the coordinate of the y-intercept?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(0,1)$$","$$(0,6)$$","$$(-1,0)$$","$$(0,4)$$"]},{"id":"a08a388line1a-h3","type":"hint","dependencies":["a08a388line1a-h2"],"title":"Point Slope Form","text":"Substitute the values into the point-slope form, $$y-y_1=m\\\\left(x-x_1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line1a-h4","type":"hint","dependencies":["a08a388line1a-h3"],"title":"Slope Intercept Form","text":"Simplify the equation and write it in slope intercept form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a08a388line10","title":"Find an Equation of the Line Given the Slope and a Point","body":"In the following exercises, find the equation of a line with given slope and containing the given point. Write the equation in slope-intercept form.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Find the Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a08a388line10a","stepAnswer":["$$y=\\\\left(-\\\\frac{3x}{5}\\\\right)+1$$"],"problemType":"MultipleChoice","stepTitle":"$$m=\\\\frac{-3}{5}$$, point $$(10,-5)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=\\\\left(-\\\\frac{3x}{5}\\\\right)+1$$","choices":["$$y=\\\\left(-\\\\frac{3x}{5}\\\\right)+1$$","$$y=\\\\frac{x}{5}-10$$","$$y=\\\\left(-\\\\frac{3x}{5}\\\\right)-5$$","$$y=\\\\left(-\\\\frac{3x}{5}\\\\right)+6$$"],"hints":{"DefaultPathway":[{"id":"a08a388line10a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-3}{5}$$"],"dependencies":[],"title":"Identify the Slope","text":"What is the value of the slope given in the problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line10a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(10,-5)$$"],"dependencies":["a08a388line10a-h1"],"title":"Identify the Point","text":"What is the coordinate of the given point?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(5,10)$$","$$(10,-5)$$","$$(\\\\frac{-3}{5},10)$$","$$(5,\\\\frac{-3}{5})$$"]},{"id":"a08a388line10a-h3","type":"hint","dependencies":["a08a388line10a-h2"],"title":"Point Slope Form","text":"Substitute the values into the point-slope form, $$y-y_1=m\\\\left(x-x_1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line10a-h4","type":"hint","dependencies":["a08a388line10a-h3"],"title":"Slope Intercept Form","text":"Simplify the equation and write it in slope intercept form","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a08a388line11","title":"Find an Equation of the Line Given Two Points","body":"In the following exercises, find the equation of a line containing the given points. Write the equation in slope-intercept form.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Find the Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a08a388line11a","stepAnswer":["$$y=-4x+13$$"],"problemType":"MultipleChoice","stepTitle":"$$(3,1)$$ and $$(2,5)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=-4x+13$$","choices":["$$y=-4x+5$$","$$y=3x+1$$","$$y=-4x+13$$","$$y=2x-5$$"],"hints":{"DefaultPathway":[{"id":"a08a388line11a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":[],"title":"Finding the Slope","text":"What is the slope of the line through the two points using $$m=\\\\frac{y_2-y_1}{x_2-x_1}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line11a-h2","type":"hint","dependencies":["a08a388line11a-h1"],"title":"Point Slope Form","text":"Choose one of the two points to plug into the point slope form $$y-y_1=m\\\\left(x-x_1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line11a-h3","type":"hint","dependencies":["a08a388line11a-h2"],"title":"Slope Intercept Form","text":"Simplify the equation and write it in slope intercept form","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a08a388line12","title":"Find an Equation of the Line Given Two Points","body":"In the following exercises, find the equation of a line containing the given points. Write the equation in slope-intercept form.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Find the Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a08a388line12a","stepAnswer":["$$y=x+5$$"],"problemType":"MultipleChoice","stepTitle":"$$(2,7)$$ and $$(3,8)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=x+5$$","choices":["$$y=2x-5$$","$$y=x+5$$","$$y=3x-2$$","$$y=x-5$$"],"hints":{"DefaultPathway":[{"id":"a08a388line12a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Finding the Slope","text":"What is the slope of the line through the two points using $$m=\\\\frac{y_2-y_1}{x_2-x_1}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line12a-h2","type":"hint","dependencies":["a08a388line12a-h1"],"title":"Point Slope Form","text":"Choose one of the two points to plug into the point slope form $$y-y_1=m\\\\left(x-x_1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line12a-h3","type":"hint","dependencies":["a08a388line12a-h2"],"title":"Slope Intercept Form","text":"Simplify the equation and write it in slope intercept form","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a08a388line13","title":"Find an Equation of the Line Given Two Points","body":"In the following exercises, find the equation of a line containing the given points. Write the equation in slope-intercept form.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Find the Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a08a388line13a","stepAnswer":["$$y=\\\\left(-\\\\frac{x}{3}\\\\right)-\\\\frac{14}{3}$$"],"problemType":"MultipleChoice","stepTitle":"$$(-5,-3)$$ and $$(4,-6)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=\\\\left(-\\\\frac{x}{3}\\\\right)-\\\\frac{14}{3}$$","choices":["$$y=\\\\left(-\\\\frac{x}{3}\\\\right)-\\\\frac{14}{3}$$","$$y=\\\\left(-\\\\frac{12x}{3}\\\\right)-3$$","$$y=\\\\left(-\\\\frac{x}{3}\\\\right)-12$$","$$y=-3x+2$$"],"hints":{"DefaultPathway":[{"id":"a08a388line13a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{3}$$"],"dependencies":[],"title":"Finding the Slope","text":"What is the slope of the line through the two points using $$m=\\\\frac{y_2-y_1}{x_2-x_1}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line13a-h2","type":"hint","dependencies":["a08a388line13a-h1"],"title":"Point Slope Form","text":"Choose one of the two points to plug into the point slope form $$y-y_1=m\\\\left(x-x_1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line13a-h3","type":"hint","dependencies":["a08a388line13a-h2"],"title":"Slope Intercept Form","text":"Simplify the equation and write it in slope intercept form","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a08a388line14","title":"Find an Equation of the Line Given Two Points","body":"In the following exercises, find the equation of a line containing the given points. Write the equation in slope-intercept form.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Find the Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a08a388line14a","stepAnswer":["$$y=7x+22$$"],"problemType":"MultipleChoice","stepTitle":"$$(-2,8)$$ and $$(-4,-6)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=7x+22$$","choices":["$$y=7x+8$$","$$y=6x-2$$","$$y=-4x+2$$","$$y=7x+22$$"],"hints":{"DefaultPathway":[{"id":"a08a388line14a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":[],"title":"Finding the Slope","text":"What is the slope of the line through the two points using $$m=\\\\frac{y_2-y_1}{x_2-x_1}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line14a-h2","type":"hint","dependencies":["a08a388line14a-h1"],"title":"Point Slope Form","text":"Choose one of the two points to plug into the point slope form $$y-y_1=m\\\\left(x-x_1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line14a-h3","type":"hint","dependencies":["a08a388line14a-h2"],"title":"Slope Intercept Form","text":"Simplify the equation and write it in slope intercept form","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a08a388line15","title":"Find an Equation of the Line Given Two Points","body":"In the following exercises, find the equation of a line containing the given points. Write the equation in slope-intercept form.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Find the Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a08a388line15a","stepAnswer":["$$y=\\\\left(-\\\\frac{6x}{7}\\\\right)+\\\\frac{4}{7}$$"],"problemType":"MultipleChoice","stepTitle":"$$(3,-2)$$ and $$(-4,4)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=\\\\left(-\\\\frac{6x}{7}\\\\right)+\\\\frac{4}{7}$$","choices":["$$y=\\\\left(-\\\\frac{6x}{7}\\\\right)+\\\\frac{4}{7}$$","$$y=\\\\frac{4x}{6}+7$$","$$y=\\\\left(-\\\\frac{6x}{7}\\\\right)+2$$","$$y=\\\\frac{4x}{7}-\\\\frac{2}{3}$$"],"hints":{"DefaultPathway":[{"id":"a08a388line15a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-6}{7}$$"],"dependencies":[],"title":"Finding the Slope","text":"What is the slope of the line through the two points using $$m=\\\\frac{y_2-y_1}{x_2-x_1}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line15a-h2","type":"hint","dependencies":["a08a388line15a-h1"],"title":"Point Slope Form","text":"Choose one of the two points to plug into the point slope form $$y-y_1=m\\\\left(x-x_1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line15a-h3","type":"hint","dependencies":["a08a388line15a-h2"],"title":"Slope Intercept Form","text":"Simplify the equation and write it in slope intercept form","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a08a388line16","title":"Finding Equations of Lines","body":"Find the expression of a graph from the slope and y-intercept $$(0,-1)$$","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Find the Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a08a388line16a","stepAnswer":["$$y=-7x-1$$"],"problemType":"MultipleChoice","stepTitle":"Find an equation of a line with slope $$-7$$ and y-intercept $$(0,-1)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=-7x-1$$","choices":["$$y=-7x+1$$","$$y=-7x-1$$","$$y=7x-1$$","$$y=7x+1$$","$$y=-1x-7$$"],"hints":{"DefaultPathway":[{"id":"a08a388line16a-h1","type":"hint","dependencies":[],"title":"Equation format","text":"The equation format is $$y=mx+b$$ where $$m$$ is the slope and $$b$$ is the y-intercept.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line16a-h2","type":"hint","dependencies":["a08a388line16a-h1"],"title":"Slope","text":"Substitute the slope value into the equation. You should be left with $$y=-7x+b$$. You can eliminate options that don\'t have $$-7$$ as a slope","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line16a-h3","type":"hint","dependencies":["a08a388line16a-h2"],"title":"Y-intercept","text":"Substitute the y-value into the equation. You should be left with $$y=-7x+\\\\left(-1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a08a388line17","title":"Finding Equations of Lines","body":"Find the expression of a graph from the slope and y-intercept $$(0,4)$$","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Find the Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a08a388line17a","stepAnswer":["$$y=\\\\frac{2}{5} x+4$$"],"problemType":"MultipleChoice","stepTitle":"Find an equation of a line with slope $$\\\\frac{2}{5}$$ and y-intercept $$(0,4)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=\\\\frac{2}{5} x+4$$","choices":["$$y=\\\\frac{-5}{2} x+4$$","$$y=\\\\frac{-2}{5} x+4$$","$$y=\\\\frac{2}{5} x+4$$","$$y=\\\\frac{2}{5} x-4$$","$$y=\\\\frac{-2}{5} x-4$$"],"hints":{"DefaultPathway":[{"id":"a08a388line17a-h1","type":"hint","dependencies":[],"title":"Equation format","text":"The equation format is $$y=mx+b$$ where $$m$$ is the slope and $$b$$ is the y-intercept.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line17a-h2","type":"hint","dependencies":["a08a388line17a-h1"],"title":"Slope","text":"Substitute the slope value into the equation. You should be left with $$y=\\\\frac{2}{5} x+b$$. You can eliminate options that don\'t have $$\\\\frac{2}{5}$$ as a slope","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line17a-h3","type":"hint","dependencies":["a08a388line17a-h2"],"title":"Y-intercept","text":"Substitute the y-value into the equation. You should be left with $$y=\\\\frac{2}{5} x+4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a08a388line18","title":"Finding Equations of Lines","body":"Find the expression of a graph from the slope and y-intercept $$(0,-3)$$","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Find the Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a08a388line18a","stepAnswer":["$$y=-x-3$$"],"problemType":"MultipleChoice","stepTitle":"Find an equation of a line with slope $$-1$$ and y-intercept $$(0,-3)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=-x-3$$","choices":["$$y=-x-3$$","$$y=x-3$$","$$y=x+3$$","$$y=-3x-1$$","$$y=3x-1$$"],"hints":{"DefaultPathway":[{"id":"a08a388line18a-h1","type":"hint","dependencies":[],"title":"Equation format","text":"The equation format is $$y=mx+b$$ where $$m$$ is the slope and $$b$$ is the y-intercept.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line18a-h2","type":"hint","dependencies":["a08a388line18a-h1"],"title":"Slope","text":"Substitute the slope value into the equation. You should be left with $$y=-1x+b$$. You can eliminate options that don\'t have $$-1$$ as a slope","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line18a-h3","type":"hint","dependencies":["a08a388line18a-h2"],"title":"Y-intercept","text":"Substitute the y-value into the equation. You should be left with $$y=-x+\\\\left(-3\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a08a388line19","title":"Finding Equations of Lines","body":"Find the equation of the line passing through the point (3,-2)\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Find the Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a08a388line19a","stepAnswer":["$$y=\\\\frac{2}{3} x-4$$"],"problemType":"MultipleChoice","stepTitle":"Find the equation of the line shown.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=\\\\frac{2}{3} x-4$$","choices":["$$y=3x-4$$","$$y=\\\\frac{3}{2} x+4$$","$$y=\\\\frac{2}{3} x+4$$","$$y=\\\\frac{2}{3} x-4$$","$$y=\\\\frac{-2}{3} x-4$$"],"hints":{"DefaultPathway":[{"id":"a08a388line19a-h1","type":"hint","dependencies":[],"title":"Relevant points on the graph","text":"Find the y-intercept of the line and another point on located on the line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line19a-h2","type":"hint","dependencies":["a08a388line19a-h1"],"title":"Y-intercept and Point","text":"The y-intercept is $$(0,-4)$$ and the hints will use $$(3,-2)$$ as another point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line19a-h3","type":"hint","dependencies":["a08a388line19a-h2"],"title":"Slope","text":"Find the slope by counting the rise and run. The slope is calculated like this: $$\\\\frac{y2-y1}{x2-x1}$$ where x1 and y1 are the values of one and x2 and y2 are the valus of the second point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line19a-h4","type":"hint","dependencies":["a08a388line19a-h3"],"title":"Slope Value","text":"The slope of the line is $$\\\\frac{2}{3}$$. Elimate multiple choice options without this slope.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line19a-h5","type":"hint","dependencies":["a08a388line19a-h4"],"title":"Substitute","text":"Substitute the values into $$y=mx+b$$, where $$m$$ is the calculated slope and $$b$$ is the y-value of the y-intercept.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a08a388line2","title":"Find an Equation of the Line Given the Slope and $$y-Intercept$$","body":"In the following exercises, find the equation of a line with given slope and y-intercept. Write the equation in slope-intercept form.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Find the Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a08a388line2a","stepAnswer":["$$y=3x+5$$"],"problemType":"MultipleChoice","stepTitle":"Slope $$3$$ and y-intercept $$(0,5)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=3x+5$$","choices":["$$y=3x+5$$","$$y=3x-5$$","$$y=5x-3$$","$$y=5x+3$$"],"hints":{"DefaultPathway":[{"id":"a08a388line2a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":[],"title":"Identify the Slope","text":"What is the value of the slope given in the problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line2a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(0,5)$$"],"dependencies":["a08a388line2a-h1"],"title":"Identify the y-intercept","text":"What is the coordinate of the y-intercept?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(0,2)$$","$$(0,5)$$","$$(5,0)$$","$$(-5,0)$$"]},{"id":"a08a388line2a-h3","type":"hint","dependencies":["a08a388line2a-h2"],"title":"Point Slope Form","text":"Substitute the values into the point-slope form, $$y-y_1=m\\\\left(x-x_1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line2a-h4","type":"hint","dependencies":["a08a388line2a-h3"],"title":"Slope Intercept Form","text":"Simplify the equation and write it in slope intercept form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a08a388line20","title":"Finding Equations of Lines","body":"Find the equation of the line passing through the point (5,4)\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Find the Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a08a388line20a","stepAnswer":["$$y=\\\\frac{3}{5} x+1$$"],"problemType":"MultipleChoice","stepTitle":"Find the equation of the line shown.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=\\\\frac{3}{5} x+1$$","choices":["$$y=\\\\frac{5}{3} x+1$$","$$y=\\\\frac{3}{5} x-1$$","$$y=5x+1$$","$$y=3x+1$$","$$y=\\\\frac{3}{5} x+1$$"],"hints":{"DefaultPathway":[{"id":"a08a388line20a-h1","type":"hint","dependencies":[],"title":"Relevant points on the graph","text":"Find the y-intercept of the line and another point on located on the line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line20a-h2","type":"hint","dependencies":["a08a388line20a-h1"],"title":"Y-intercept and Point","text":"The y-intercept is $$(0,1)$$ and the hints will use $$(5,4)$$ as another point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line20a-h3","type":"hint","dependencies":["a08a388line20a-h2"],"title":"Slope","text":"Find the slope by counting the rise and run. The slope is calculated like this: $$\\\\frac{y2-y1}{x2-x1}$$ where x1 and y1 are the values of one and x2 and y2 are the valus of the second point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line20a-h4","type":"hint","dependencies":["a08a388line20a-h3"],"title":"Slope Value","text":"The slope of the line is $$\\\\frac{3}{5}$$. Elimate multiple choice options without this slope.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line20a-h5","type":"hint","dependencies":["a08a388line20a-h4"],"title":"Substitute","text":"Substitute the values into $$y=mx+b$$, where $$m$$ is the calculated slope and $$b$$ is the y-value of the y-intercept.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a08a388line21","title":"Finding Equations of Lines","body":"Find the equation of the line passing through the point (3,-1)\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Find the Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a08a388line21a","stepAnswer":["$$y=\\\\frac{4}{3} x-5$$"],"problemType":"MultipleChoice","stepTitle":"Find the equation of the line shown.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=\\\\frac{4}{3} x-5$$","choices":["$$y=\\\\frac{-4}{3} x-5$$","$$y=\\\\frac{4}{3} x-5$$","$$y=\\\\frac{4}{3} x+5$$","$$y=\\\\frac{3}{4} x-5$$","$$y=\\\\frac{3}{4} x+5$$"],"hints":{"DefaultPathway":[{"id":"a08a388line21a-h1","type":"hint","dependencies":[],"title":"Relevant points on the graph","text":"Find the y-intercept of the line and another point on located on the line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line21a-h2","type":"hint","dependencies":["a08a388line21a-h1"],"title":"Y-intercept and Point","text":"The y-intercept is $$(0,-5)$$ and the hints will use $$(3,-1)$$ as another point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line21a-h3","type":"hint","dependencies":["a08a388line21a-h2"],"title":"Slope","text":"Find the slope by counting the rise and run. The slope is calculated like this: $$\\\\frac{y2-y1}{x2-x1}$$ where x1 and y1 are the values of one and x2 and y2 are the valus of the second point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line21a-h4","type":"hint","dependencies":["a08a388line21a-h3"],"title":"Slope Value","text":"The slope of the line is $$\\\\frac{4}{3}$$. Elimate multiple choice options without this slope.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line21a-h5","type":"hint","dependencies":["a08a388line21a-h4"],"title":"Substitute","text":"Substitute the values into $$y=mx+b$$, where $$m$$ is the calculated slope and $$b$$ is the y-value of the y-intercept.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a08a388line22","title":"Find an Equation of a Line Given the Slope and a Point","body":"Find the equation of the line given its slope and one of its points.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Find the Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a08a388line22a","stepAnswer":["$$y=\\\\frac{2}{5} x-1$$"],"problemType":"MultipleChoice","stepTitle":"Find an equation of a line with slope $$m=\\\\frac{2}{5}$$ that contains the point $$(10,3)$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=\\\\frac{2}{5} x-1$$","choices":["$$y=\\\\frac{2}{5} x+1$$","$$y=\\\\frac{2}{5} x-1$$","$$y=\\\\frac{-2}{5} x-1$$","$$y=\\\\frac{5}{2} x-1$$","$$y=\\\\frac{-5}{2} x-1$$"],"hints":{"DefaultPathway":[{"id":"a08a388line22a-h1","type":"hint","dependencies":[],"title":"Identify the slope","text":"The slope if given. $$m=\\\\frac{2}{5}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line22a-h2","type":"hint","dependencies":["a08a388line22a-h1"],"title":"Identify the point","text":"The point is given. $$(10,3)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line22a-h3","type":"hint","dependencies":["a08a388line22a-h2"],"title":"Substitute","text":"Substitute the values into the point-slope form $$y-y1=m(x-x1)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line22a-h4","type":"hint","dependencies":["a08a388line22a-h3"],"title":"Simplify","text":"Simplify the expression $$y-3=\\\\frac{2}{5} x-4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a08a388line23","title":"Find an Equation of a Line Given the Slope and a Point","body":"Find the equation of the line given its slope and one of its points.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Find the Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a08a388line23a","stepAnswer":["$$y=\\\\frac{5}{6} x-2$$"],"problemType":"MultipleChoice","stepTitle":"Find an equation of a line with slope $$m=\\\\frac{5}{6}$$ that contains the point $$(6,3)$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=\\\\frac{5}{6} x-2$$","choices":["$$y=\\\\frac{5}{6} x-2$$","$$y=\\\\frac{6}{5} x-2$$","$$y=\\\\frac{-5}{6} x+2$$","$$y=\\\\frac{6}{5} x-2$$","$$y=\\\\frac{5}{6} x+2$$"],"hints":{"DefaultPathway":[{"id":"a08a388line23a-h1","type":"hint","dependencies":[],"title":"Identify the slope","text":"The slope if given. $$m=\\\\frac{5}{6}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line23a-h2","type":"hint","dependencies":["a08a388line23a-h1"],"title":"Identify the point","text":"The point is given. $$(6,3)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line23a-h3","type":"hint","dependencies":["a08a388line23a-h2"],"title":"Substitute","text":"Substitute the values into the point-slope form $$y-y1=m(x-x1)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line23a-h4","type":"hint","dependencies":["a08a388line23a-h3"],"title":"Simplify","text":"Simplify the expression $$y-3=\\\\frac{5}{6} x-5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a08a388line24","title":"Find an Equation of a Line Given the Slope and a Point","body":"Find the equation of the line given its slope and one of its points.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Find the Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a08a388line24a","stepAnswer":["$$y=\\\\frac{-1}{3} x-2$$"],"problemType":"MultipleChoice","stepTitle":"Find an equation of a line with slope $$m=\\\\frac{-1}{3}$$ that contains the point $$(6,-4)$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=\\\\frac{-1}{3} x-2$$","choices":["$$y=\\\\frac{1}{3} x-2$$","$$y=-3x-2$$","$$y=\\\\frac{-1}{3} x+2$$","$$y=3x-2$$","$$y=\\\\frac{-1}{3} x-2$$"],"hints":{"DefaultPathway":[{"id":"a08a388line24a-h1","type":"hint","dependencies":[],"title":"Identify the slope","text":"The slope if given. $$m=\\\\frac{-1}{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line24a-h2","type":"hint","dependencies":["a08a388line24a-h1"],"title":"Identify the point","text":"The point is given. $$(6,-4)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line24a-h3","type":"hint","dependencies":["a08a388line24a-h2"],"title":"Substitute","text":"Substitute the values into the point-slope form $$y-y1=m(x-x1)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line24a-h4","type":"hint","dependencies":["a08a388line24a-h3"],"title":"Simplify","text":"Simplify the expression $$y-(-4)=\\\\frac{-1}{3\\\\left(x-6\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a08a388line25","title":"Finding Equations of Horizontal Lines","body":"Using the definition of a horizontal line and one point on the horizontal line, find its equation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Find the Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a08a388line25a","stepAnswer":["$$y=2$$"],"problemType":"MultipleChoice","stepTitle":"Find an equation of a horizontal line that contains the point $$(-1,2)$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=2$$","choices":["$$y=2$$","$$y=0$$","$$y=-2$$","$$y=-1$$","$$y=1$$"],"hints":{"DefaultPathway":[{"id":"a08a388line25a-h1","type":"hint","dependencies":[],"title":"Identify the slope","text":"The slope of a horizontal line is $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line25a-h2","type":"hint","dependencies":["a08a388line25a-h1"],"title":"Identify the point","text":"The point is given. $$(-1,2)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line25a-h3","type":"hint","dependencies":["a08a388line25a-h2"],"title":"Substitute","text":"Substitute the values into the point-slope form $$y-y1=m(x-x1)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line25a-h4","type":"hint","dependencies":["a08a388line25a-h3"],"title":"Simplify","text":"Simplify the expression $$y-(2)=0\\\\left(x+1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a08a388line26","title":"Find an Equation of a Line Given Two Points","body":"Using the two given points, find the equation of the line.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Find the Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a08a388line26a","stepAnswer":["$$y=-x+9$$"],"problemType":"MultipleChoice","stepTitle":"Find an equation of a line that contains the points $$(5,4)$$ and $$(3,6)$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=-x+9$$","choices":["$$y=x+9$$","$$y=x-9$$","$$y=-x+9$$","$$y=-x-9$$","$$y=-9x+9$$"],"hints":{"DefaultPathway":[{"id":"a08a388line26a-h1","type":"hint","dependencies":[],"title":"Slope","text":"Find the slope by counting the rise and run. The slope is calculated like this: $$\\\\frac{y2-y1}{x2-x1}$$ where x1 and y1 are the values of one and x2 and y2 are the valus of the second point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line26a-h2","type":"hint","dependencies":["a08a388line26a-h1"],"title":"Slope Value","text":"The slope of the line is $$-1$$. Elimate multiple choice options without this slope.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line26a-h3","type":"hint","dependencies":["a08a388line26a-h2"],"title":"Choose one point","text":"Choose either point. For the hints, we will use $$(5,4)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line26a-h4","type":"hint","dependencies":["a08a388line26a-h3"],"title":"Substitute","text":"Substitute the values into the point-slope form $$y-y1=m(x-x1)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line26a-h5","type":"hint","dependencies":["a08a388line26a-h4"],"title":"Simplify","text":"Simplify the expression $$y-(4)=-1(x-5)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a08a388line27","title":"Finding Equations of Parallel Lines","body":"Find the equation of the line parallel to the given line.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Find the Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a08a388line27a","stepAnswer":["$$y=2x+5$$"],"problemType":"MultipleChoice","stepTitle":"Find an equation of a line parallel to $$y=2x-3$$ that contains the point $$(-2,1)$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=2x+5$$","choices":["$$y=-2x+5$$","$$y=2x+5$$","$$y=\\\\frac{-1}{2} x+5$$","$$y=2x-5$$","$$y=-2x-5$$"],"hints":{"DefaultPathway":[{"id":"a08a388line27a-h1","type":"hint","dependencies":[],"title":"Slope","text":"Find the slope of the given line. The line is in slope-intercept form, $$y=2x-3$$. The slope is $$m=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line27a-h2","type":"hint","dependencies":["a08a388line27a-h1"],"title":"Slope Value","text":"Parallel lines have the same slope. Slope is $$m=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line27a-h3","type":"hint","dependencies":["a08a388line27a-h2"],"title":"Identify the point","text":"The given point is $$(-2,1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line27a-h4","type":"hint","dependencies":["a08a388line27a-h3"],"title":"Substitute","text":"Substitute the values into the point-slope form $$y-y1=m(x-x1)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line27a-h5","type":"hint","dependencies":["a08a388line27a-h4"],"title":"Simplify","text":"Simplify the expression $$y-(1)=2\\\\left(x+2\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a08a388line28","title":"Finding Equations of Perpendicular Lines","body":"Find the equation of the line perpendicular to the given line.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Find the Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a08a388line28a","stepAnswer":["$$y=\\\\frac{-1}{2} x$$"],"problemType":"MultipleChoice","stepTitle":"Find an equation of a line perpendicular to $$y=2x-3$$ that contains the point $$(-2,1)$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=\\\\frac{-1}{2} x$$","choices":["$$y=\\\\frac{-1}{2} x$$","$$y=2x$$","$$y=\\\\frac{1}{2} x$$","$$y=-2x$$","$$y=-x$$"],"hints":{"DefaultPathway":[{"id":"a08a388line28a-h1","type":"hint","dependencies":[],"title":"Slope","text":"Find the slope of the given line. The line is in slope-intercept form, $$y=2x-3$$. The slope is $$m=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line28a-h2","type":"hint","dependencies":["a08a388line28a-h1"],"title":"Slope Value","text":"The slopes of perpendicular lines are negative reciprocals. Slope is $$m=\\\\frac{-1}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line28a-h3","type":"hint","dependencies":["a08a388line28a-h2"],"title":"Identify the point","text":"The given point is $$(-2,1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line28a-h4","type":"hint","dependencies":["a08a388line28a-h3"],"title":"Substitute","text":"Substitute the values into the point-slope form $$y-y1=m(x-x1)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line28a-h5","type":"hint","dependencies":["a08a388line28a-h4"],"title":"Simplify","text":"Simplify the expression $$y-(1)=\\\\frac{-1}{2\\\\left(x+2\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a08a388line29","title":"Finding Equations of Perpendicular Lines","body":"Find the equation of the line perpendicular to the given line.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Find the Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a08a388line29a","stepAnswer":["$$y=-2$$"],"problemType":"MultipleChoice","stepTitle":"Find an equation of a line perpendicular to $$x=5$$ that contains the point $$(3,-2)$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=-2$$","choices":["$$y=-2$$","$$y=2$$","$$y=-2x$$","$$y=3x$$","$$y=3$$"],"hints":{"DefaultPathway":[{"id":"a08a388line29a-h1","type":"hint","dependencies":[],"title":"Identify the point","text":"The given point is $$(3,-2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line29a-h2","type":"hint","dependencies":["a08a388line29a-h1"],"title":"Slope","text":"Identify the slope of the perpendicular line. $$m=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line29a-h3","type":"hint","dependencies":["a08a388line29a-h2"],"title":"Substitute","text":"Substitute the values into the point-slope form $$y-y1=m(x-x1)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line29a-h4","type":"hint","dependencies":["a08a388line29a-h3"],"title":"Simplify","text":"Simplify the expression $$y-(-2)=0(x-3)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a08a388line3","title":"Find an Equation of the Line Given the Slope and $$y-Intercept$$","body":"In the following exercises, find the equation of a line with given slope and y-intercept. Write the equation in slope-intercept form.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Find the Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a08a388line3a","stepAnswer":["$$y=6x-4$$"],"problemType":"MultipleChoice","stepTitle":"Slope $$6$$ and y-intercept $$(0,-4)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=6x-4$$","choices":["$$y=6x+4$$","$$y=4x+1$$","$$y=6x-4$$","$$y=2x-1$$"],"hints":{"DefaultPathway":[{"id":"a08a388line3a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":[],"title":"Identify the Slope","text":"What is the value of the slope given in the problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line3a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(0,-4)$$"],"dependencies":["a08a388line3a-h1"],"title":"Identify the y-intercept","text":"What is the coordinate of the y-intercept?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(0,-4)$$","$$(0,4)$$","$$(-4,0)$$","$$(4,0)$$"]},{"id":"a08a388line3a-h3","type":"hint","dependencies":["a08a388line3a-h2"],"title":"Point Slope Form","text":"Substitute the values into the point-slope form, $$y-y_1=m\\\\left(x-x_1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line3a-h4","type":"hint","dependencies":["a08a388line3a-h3"],"title":"Slope Intercept Form","text":"Simplify the equation and write it in slope intercept form","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a08a388line30","title":"Finding Equations of Perpendicular Lines","body":"Find the equation of the line perpendicular to the given line.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Find the Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a08a388line30a","stepAnswer":["$$x=-4$$"],"problemType":"MultipleChoice","stepTitle":"Find an equation of a line perpendicular to $$y=-4$$ that contains the point $$(-4,2)$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=-4$$","choices":["$$x=-4$$","$$y=-4$$","$$x=2$$","$$y=2$$","$$y=4$$"],"hints":{"DefaultPathway":[{"id":"a08a388line30a-h1","type":"hint","dependencies":[],"title":"Identify the point","text":"The given point is $$(-4,2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line30a-h2","type":"hint","dependencies":["a08a388line30a-h1"],"title":"Slope","text":"The line $$y=-4$$ is a horizontal line. Any line perpendicular to it must be vertical, in the form $$x=a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a08a388line30a-h3","type":"hint","dependencies":["a08a388line30a-h2"],"title":"Equation","text":"Since the perpendicular line is vertical and passes through $$(-4,2)$$, every point on it has an x-coordinate of $$-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a08a388line4","title":"Find an Equation of the Line Given the Slope and $$y-Intercept$$","body":"In the following exercises, find the equation of a line with given slope and y-intercept. 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Otherwise, use a paranthesis (\\"(\\" or \\")\\").","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal1a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a090e21alginequal1a-h10"],"title":"Interval Notation","text":"Is the lower bound $$6$$ included as a valid value of \'x\' in $$6<x$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"a090e21alginequal1a-h12","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a090e21alginequal1a-h10"],"title":"Interval Notation","text":"Is the upper bound $$\\\\infty$$ included as a valid value of \'x\' in $$6<x$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"],"subHints":[{"id":"a090e21alginequal1a-h12-s1","type":"hint","dependencies":[],"title":"Infinity in Interval Notation","text":"As $$\\\\infty$$ is not a number, it cannot be included as part of a valid value or bound for \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""}]}]}}]},{"id":"a090e21alginequal10","title":"Algebra with Inequalities: Part B","body":"These problems are generally harder, often highlighting an important subtlety. 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$$0$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal10a-h2","type":"hint","dependencies":["a090e21alginequal10a-h1"],"title":"Bounds of a Nth Root","text":"Since $$\\\\sqrt{1-3x}$$ cannot be negative, then $$1-3x$$ $$\\\\geq$$ $$0$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal10a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1}{3}$$ $$\\\\geq$$ $$x$$"],"dependencies":["a090e21alginequal10a-h2"],"title":"Bounds of a Nth Root","text":"Simplify the inequality: $$1-3x$$ $$\\\\geq$$ $$0$$.","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{1}{3}$$ $$\\\\geq$$ $$x$$","$$\\\\frac{-1}{3}$$ $$\\\\geq$$ $$x$$","$$\\\\frac{1}{3}$$ $$\\\\leq$$ $$x$$","$$\\\\frac{-1}{3}$$ $$\\\\leq$$ $$x$$"],"subHints":[{"id":"a090e21alginequal10a-h3-s1","type":"hint","dependencies":[],"title":"Bounds of $$1-3x$$","text":"Rearrange the inequality so that the \'x\'s are separated from the constants: $$1$$ $$\\\\geq$$ $$3x$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal10a-h3-s2","type":"hint","dependencies":["a090e21alginequal10a-h3-s1"],"title":"Bounds of $$1-3x$$","text":"Divide $$3$$ from both sides of $$1$$ $$\\\\geq$$ $$3x$$ to isolate \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a090e21alginequal10a-h4","type":"hint","dependencies":["a090e21alginequal10a-h3"],"title":"Bounds of a Nth Root","text":"Since $$\\\\sqrt{2+5x}$$ cannot be negative, then $$2+5x$$ $$\\\\geq$$ $$0$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal10a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x$$ $$\\\\geq$$ $$\\\\frac{-2}{5}$$"],"dependencies":["a090e21alginequal10a-h4"],"title":"Bounds of a Nth Root","text":"Simplify the inequality: $$2+5x$$ $$\\\\geq$$ $$0$$.","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$x$$ $$\\\\geq$$ $$\\\\frac{-2}{5}$$","$$x$$ $$\\\\geq$$ $$\\\\frac{2}{5}$$","$$x$$ $$\\\\leq$$ $$\\\\frac{-2}{5}$$","$$x$$ $$\\\\leq$$ $$\\\\frac{2}{5}$$"],"subHints":[{"id":"a090e21alginequal10a-h5-s3","type":"hint","dependencies":[],"title":"Bounds of $$2+5x$$","text":"Rearrange the inequality so that the \'x\'s are separated from the constants: $$5x$$ $$\\\\geq$$ $$-2$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal10a-h5-s4","type":"hint","dependencies":["a090e21alginequal10a-h5-s3"],"title":"Bounds of $$2+5x$$","text":"Divide $$5$$ from both sides of $$5x$$ $$\\\\geq$$ $$-2$$ to isolate \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a090e21alginequal10a-h6","type":"hint","dependencies":["a090e21alginequal10a-h3","a090e21alginequal10a-h5"],"title":"Bounds of a Nth Root","text":"$$x$$ must be between or equal to $$\\\\frac{-2}{5}$$ and $$\\\\frac{1}{3}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal10a-h7","type":"hint","dependencies":["a090e21alginequal10a-h6"],"title":"Removing the Square Root","text":"For some nth root $$b$$, $$\\\\sqrt[b]{a}>\\\\sqrt[b]{c}$$ is the same as $$a>c$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal10a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1-3x$$"],"dependencies":["a090e21alginequal10a-h7"],"title":"Removing the Square Root","text":"What is $$\\\\sqrt{1-3x}$$ written without the square root?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal10a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2+5x$$"],"dependencies":["a090e21alginequal10a-h7"],"title":"Removing the Square Root","text":"What is $$\\\\sqrt{2+5x}$$ written without the square root?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal10a-h10","type":"hint","dependencies":["a090e21alginequal10a-h8","a090e21alginequal10a-h9"],"title":"Simplifying the Inequality","text":"Rearrange the inequality such that all the \'x\'s are on one side and the constants are on the other.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal10a-h11","type":"hint","dependencies":["a090e21alginequal10a-h10"],"title":"Simplifying the Inequality","text":"Add $$3x$$ from the left and subtract $$2$$ from the right to get $$1-2>5x+3x$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal10a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a090e21alginequal10a-h11"],"title":"Simplifying the Inequality","text":"What is $$1-2$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal10a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8x$$"],"dependencies":["a090e21alginequal10a-h11"],"title":"Simplifying the Inequality","text":"What is $$5x+3x$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal10a-h14","type":"hint","dependencies":["a090e21alginequal10a-h12","a090e21alginequal10a-h13"],"title":"Simplifying the Inequality","text":"Divide $$8$$ from both sides of $$-1>8x$$ to isolate \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal10a-h15","type":"hint","dependencies":["a090e21alginequal10a-h14"],"title":"Interval Notation","text":"The inequality $$\\\\frac{-1}{8}>x$$ can be written in interval notation by specifying the lower bound first, followed by the upper bound.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal10a-h16","type":"hint","dependencies":["a090e21alginequal10a-h15"],"title":"Interval Notation","text":"Remember that $$x$$ is also bounded between or equal to $$\\\\frac{-2}{5}$$ and $$\\\\frac{1}{3}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal10a-h17","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{-2}{5}$$"],"dependencies":["a090e21alginequal10a-h16"],"title":"Interval Notation","text":"What is the lower bound of the inequality $$\\\\frac{-1}{8}>x$$, with the additional constraints $$\\\\frac{-2}{5}$$ $$\\\\leq$$ $$x$$ $$\\\\leq$$ $$\\\\frac{1}{3}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$-\\\\infty$$","$$\\\\frac{-1}{8}$$","$$\\\\frac{-2}{5}$$","$$\\\\frac{1}{3}$$"]},{"id":"a090e21alginequal10a-h18","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{-1}{8}$$"],"dependencies":["a090e21alginequal10a-h16"],"title":"Interval Notation","text":"What is the upper bound of the inequality $$\\\\frac{-1}{8}>x$$, with the additional constraints $$\\\\frac{-2}{5}$$ $$\\\\leq$$ $$x$$ $$\\\\leq$$ $$\\\\frac{1}{3}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\infty$$","$$\\\\frac{-1}{8}$$","$$\\\\frac{-2}{5}$$","$$\\\\frac{1}{3}$$"],"subHints":[{"id":"a090e21alginequal10a-h18-s5","type":"hint","dependencies":[],"title":"Determining the Upper Bound","text":"$$x$$ is upper bounded by $$x$$ $$\\\\leq$$ $$\\\\frac{1}{3}$$ and $$x<\\\\frac{-1}{8}$$. The intersection between the bounds should be chosen.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal10a-h18-s6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{-1}{8}$$"],"dependencies":["a090e21alginequal10a-h18-s5"],"title":"Determining the Upper Bound","text":"Which number is smaller? $$\\\\frac{1}{3}$$ or $$\\\\frac{-1}{8}$$","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{1}{3}$$","$$\\\\frac{-1}{8}$$"]}]},{"id":"a090e21alginequal10a-h19","type":"hint","dependencies":["a090e21alginequal10a-h17","a090e21alginequal10a-h18"],"title":"Interval Notation","text":"If the bound itself should be included as a valid answer of \'x\', meaning it can be equal to that value, then a bracket (\\"[\\" or \\"]\\") should be used. Otherwise, use a paranthesis (\\"(\\" or \\")\\").","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal10a-h20","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a090e21alginequal10a-h19"],"title":"Interval Notation","text":"Is the lower bound $$\\\\frac{-2}{5}$$ included as a valid value of \'x\' in $$\\\\frac{-2}{5}$$ $$\\\\leq$$ $$x<\\\\frac{-1}{8}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"a090e21alginequal10a-h21","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a090e21alginequal10a-h19"],"title":"Interval Notation","text":"Is the upper bound $$\\\\frac{-1}{8}$$ included as a valid value of \'x\' in $$\\\\frac{-2}{5}$$ $$\\\\leq$$ $$x<\\\\frac{-1}{8}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]}]}}]},{"id":"a090e21alginequal102","title":"Algebra with Inequalities: Part B","body":"These problems are generally harder, often highlighting an important subtlety. Determine which numbers \'x\' satisfy the following conditions. Express your answer in interval notation.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Algebra with Inequalities","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a090e21alginequal102a","stepAnswer":["$$(\\\\frac{9}{2},\\\\frac{16}{3})$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{3}{5}<\\\\frac{1}{2x-9}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(\\\\frac{9}{2},\\\\frac{16}{3})$$","choices":["$$(\\\\frac{9}{2},\\\\frac{16}{3})$$","$$(\\\\frac{9}{2},\\\\infty)$$","$$(-\\\\infty,\\\\frac{9}{2})$$","$$(-\\\\infty,\\\\frac{16}{3})$$"],"hints":{"DefaultPathway":[{"id":"a090e21alginequal102a-h1","type":"hint","dependencies":[],"title":"Bounds of a Fraction","text":"Since $$\\\\frac{3}{5}>0$$, that means $$\\\\frac{1}{2x-9}>0$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal102a-h2","type":"hint","dependencies":["a090e21alginequal102a-h1"],"title":"Bounds of a Fraction","text":"A fraction that whose bound is positive for some value \'a\' $$\\\\frac{1}{a}>0$$ can be rewritten as $$a>0$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal102a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x-9$$"],"dependencies":["a090e21alginequal102a-h2"],"title":"Bounds of a Fraction","text":"What is $$\\\\frac{1}{2x-9}$$ rewritten without the fraction?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal102a-h4","type":"hint","dependencies":["a090e21alginequal102a-h3"],"title":"Bounds of a Fraction","text":"Rearrange the inequality so that the \'x\'s are separated from the constants: $$2x>9$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal102a-h5","type":"hint","dependencies":["a090e21alginequal102a-h4"],"title":"Bounds of a Fraction","text":"Divide $$2$$ from both sides of $$2x>9$$ to isolate \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal102a-h6","type":"hint","dependencies":["a090e21alginequal102a-h5"],"title":"Bounds of a Fraction","text":"$$x$$ must be greater than $$\\\\frac{9}{2}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal102a-h7","type":"hint","dependencies":["a090e21alginequal102a-h6"],"title":"Simplifying the Inequality","text":"For b,d>0, $$\\\\frac{a}{b}<\\\\frac{c}{d}$$ is the same as $$a c<b d$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal102a-h8","type":"hint","dependencies":["a090e21alginequal102a-h7"],"title":"Simplifying the Inequality","text":"Simplify the inequality by multiplying each side by the denominators.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal102a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6x-27$$"],"dependencies":["a090e21alginequal102a-h8"],"title":"Simplifying the Inequality","text":"What is $$3\\\\left(2x-9\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal102a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a090e21alginequal102a-h8"],"title":"Simplifying the Inequality","text":"What is $$1\\\\times5$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal102a-h11","type":"hint","dependencies":["a090e21alginequal102a-h9","a090e21alginequal102a-h10"],"title":"Simplifying the Inequality","text":"Rearrange the inequality so that the \'x\'s are separated from the constants.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal102a-h12","type":"hint","dependencies":["a090e21alginequal102a-h11"],"title":"Simplifying the Inequality","text":"Add $$27$$ from the left side to get $$6x<5+27$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal102a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$32$$"],"dependencies":["a090e21alginequal102a-h12"],"title":"Simplifying the Inequality","text":"What is $$5+27$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal102a-h14","type":"hint","dependencies":["a090e21alginequal102a-h13"],"title":"Simplifying the Inequality","text":"Divide $$6$$ from both sides of $$6x<32$$ to isolate \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal102a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{16}{3}$$"],"dependencies":["a090e21alginequal102a-h14"],"title":"Simplifying the Inequality","text":"What is $$\\\\frac{32}{6}$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal102a-h16","type":"hint","dependencies":["a090e21alginequal102a-h15"],"title":"Interval Notation","text":"The inequality $$x<\\\\frac{16}{3}$$ can be written in interval notation by specifying the lower bound first, followed by the upper bound.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal102a-h17","type":"hint","dependencies":["a090e21alginequal102a-h16"],"title":"Interval Notation","text":"Remember that $$x$$ is also bounded greater than $$\\\\frac{9}{2}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal102a-h18","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{9}{2}$$"],"dependencies":["a090e21alginequal102a-h17"],"title":"Interval Notation","text":"What is the lower bound of the inequality $$x<\\\\frac{16}{3}$$, with the additional constraints $$x>\\\\frac{9}{2}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$-\\\\infty$$","$$\\\\frac{9}{2}$$","$$\\\\frac{16}{3}$$"]},{"id":"a090e21alginequal102a-h19","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{16}{3}$$"],"dependencies":["a090e21alginequal102a-h17"],"title":"Interval Notation","text":"What is the upper bound of the inequality $$x<\\\\frac{16}{3}$$, with the additional constraints $$x>\\\\frac{9}{2}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\infty$$","$$\\\\frac{9}{2}$$","$$\\\\frac{16}{3}$$"]},{"id":"a090e21alginequal102a-h20","type":"hint","dependencies":["a090e21alginequal102a-h18","a090e21alginequal102a-h19"],"title":"Interval Notation","text":"If the bound itself should be included as a valid answer of \'x\', meaning it can be equal to that value, then a bracket (\\"[\\" or \\"]\\") should be used. Otherwise, use a paranthesis (\\"(\\" or \\")\\").","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal102a-h21","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a090e21alginequal102a-h20"],"title":"Interval Notation","text":"Is the lower bound $$\\\\frac{9}{2}$$ included as a valid value of \'x\' in $$\\\\frac{9}{2}<x<\\\\frac{16}{3}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"a090e21alginequal102a-h22","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a090e21alginequal102a-h20"],"title":"Interval Notation","text":"Is the upper bound $$\\\\frac{16}{3}$$ included as a valid value of \'x\' in $$\\\\frac{9}{2}<x<\\\\frac{16}{3}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]}]}}]},{"id":"a090e21alginequal103","title":"Algebra with Inequalities: Part B","body":"These problems are generally harder, often highlighting an important subtlety. Determine which numbers \'x\' satisfy the following conditions. Express your answer in interval notation.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Algebra with Inequalities","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a090e21alginequal103a","stepAnswer":["$$(-1,\\\\frac{1}{2})$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\log_{5}\\\\left(\\\\frac{1-2x}{3}\\\\right)<0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-1,\\\\frac{1}{2})$$","choices":["$$(-1,\\\\frac{1}{2})$$","$$(-\\\\infty,\\\\frac{1}{2})$$","$$(1,\\\\infty)$$","$$(-\\\\infty,1)$$"],"hints":{"DefaultPathway":[{"id":"a090e21alginequal103a-h1","type":"hint","dependencies":[],"title":"Bounds of a Logarithm","text":"The logarithm of some value \'b\' cannot be negative or zero, so $$b>0$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal103a-h2","type":"hint","dependencies":["a090e21alginequal103a-h1"],"title":"Bounds of a Logarithm","text":"Since $$\\\\log_{5}\\\\left(\\\\frac{1-2x}{3}\\\\right)<0$$ cannot be negative or zero, then $$\\\\frac{1-2x}{3}>0$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal103a-h3","type":"hint","dependencies":["a090e21alginequal103a-h2"],"title":"Bounds of a Logarithm","text":"For $$b>0$$, $$\\\\frac{a}{b}>0$$ is the same as $$a>0$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal103a-h4","type":"hint","dependencies":["a090e21alginequal103a-h3"],"title":"Bounds of a Logarithm","text":"Simplify the inequality by multiplying each side by the denominator of the left hand side.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal103a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a090e21alginequal103a-h4"],"title":"Bounds of a Logarithm","text":"What is $$0\\\\times3$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal103a-h6","type":"hint","dependencies":["a090e21alginequal103a-h5"],"title":"Bounds of a Logarithm","text":"Rearrange the inequality so that the \'x\'s are separated from the constants.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal103a-h7","type":"hint","dependencies":["a090e21alginequal103a-h6"],"title":"Bounds of a Logarithm","text":"Add $$2x$$ from the left side to get $$1>0+2x$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal103a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x$$"],"dependencies":["a090e21alginequal103a-h7"],"title":"Bounds of a Logarithm","text":"What is $$0+2x$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal103a-h9","type":"hint","dependencies":["a090e21alginequal103a-h8"],"title":"Bounds of a Logarithm","text":"Divide $$2$$ from both sides of $$1>2x$$ to isolate \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal103a-h10","type":"hint","dependencies":["a090e21alginequal103a-h9"],"title":"Bounds of a Logarithm","text":"$$x$$ must be less than $$\\\\frac{1}{2}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal103a-h11","type":"hint","dependencies":["a090e21alginequal103a-h10"],"title":"Simplifying the Inequality","text":"For $$b>1$$, $$a<c$$ is the same as $$b^a<b^c$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal103a-h12","type":"hint","dependencies":["a090e21alginequal103a-h11"],"title":"Simplifying the Inequality","text":"Exponentiate both sides by $$5$$ to remove the logarithm: $$5**{\\\\log_{5}\\\\left(\\\\frac{1-2x}{3}\\\\right)}<5**0$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal103a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1-2x}{3}$$"],"dependencies":["a090e21alginequal103a-h12"],"title":"Simplifying the Inequality","text":"What is $$5**{\\\\log_{5}{\\\\left(\\\\frac{1-2x}{3}\\\\right)}}$$ ?","variabilization":{},"oer":"https://OATutor.io","license":"","subHints":[{"id":"a090e21alginequal103a-h13-s1","type":"hint","dependencies":[],"title":"Logarithms and Exponents","text":"For some $$b>1$$, $$b**(\\\\log_{b}\\\\left(x\\\\right))=x$$","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a090e21alginequal103a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a090e21alginequal103a-h12"],"title":"Simplifying the Inequality","text":"What is $$5^0$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal103a-h15","type":"hint","dependencies":["a090e21alginequal103a-h13","a090e21alginequal103a-h14"],"title":"Simplifying the Inequality","text":"For $$b>0$$, $$\\\\frac{a}{b}<c$$ is the same as $$a<b c$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal103a-h16","type":"hint","dependencies":["a090e21alginequal103a-h15"],"title":"Simplifying the Inequality","text":"Multiply both sides by $$3$$ to remove the denominators: $$1-2x<1\\\\times3$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal103a-h17","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a090e21alginequal103a-h16"],"title":"Simplifying the Inequality","text":"What is $$1\\\\times3$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal103a-h18","type":"hint","dependencies":["a090e21alginequal103a-h17"],"title":"Simplifying the Inequality","text":"Rearrange the inequality so that the \'x\'s are separated from the constants.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal103a-h19","type":"hint","dependencies":["a090e21alginequal103a-h18"],"title":"Simplifying the Inequality","text":"Add $$2x$$ from the left side and subtract $$3$$ from the right side to get $$1-3>0+2x$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal103a-h20","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a090e21alginequal103a-h19"],"title":"Simplifying the Inequality","text":"What is $$1-3$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal103a-h21","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x$$"],"dependencies":["a090e21alginequal103a-h19"],"title":"Simplifying the Inequality","text":"What is $$0+2x$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal103a-h22","type":"hint","dependencies":["a090e21alginequal103a-h20","a090e21alginequal103a-h21"],"title":"Simplifying the Inequality","text":"Divide $$2$$ from both sides of $$-2<2x$$ to isolate \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal103a-h23","type":"hint","dependencies":["a090e21alginequal103a-h22"],"title":"Interval Notation","text":"The inequality $$-1<x$$ can be written in interval notation by specifying the lower bound first, followed by the upper bound.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal103a-h24","type":"hint","dependencies":["a090e21alginequal103a-h23"],"title":"Interval Notation","text":"Remember that $$x$$ is also bounded less than $$\\\\frac{1}{2}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal103a-h25","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-1$$"],"dependencies":["a090e21alginequal103a-h24"],"title":"Interval Notation","text":"What is the lower bound of the inequality $$-1<x$$, with the additional constraints $$\\\\frac{1}{2}>x$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$-\\\\infty$$","$$-1$$","$$\\\\frac{1}{2}$$"]},{"id":"a090e21alginequal103a-h26","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a090e21alginequal103a-h24"],"title":"Interval Notation","text":"What is the upper bound of the inequality $$-1<x$$, with the additional constraints $$\\\\frac{1}{2}>x$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\infty$$","$$-1$$","$$\\\\frac{1}{2}$$"]},{"id":"a090e21alginequal103a-h27","type":"hint","dependencies":["a090e21alginequal103a-h25","a090e21alginequal103a-h26"],"title":"Interval Notation","text":"If the bound itself should be included as a valid answer of \'x\', meaning it can be equal to that value, then a bracket (\\"[\\" or \\"]\\") should be used. Otherwise, use a paranthesis (\\"(\\" or \\")\\").","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal103a-h28","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a090e21alginequal103a-h27"],"title":"Interval Notation","text":"Is the lower bound $$-1$$ included as a valid value of \'x\' in $$-1<x<\\\\frac{1}{2}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"a090e21alginequal103a-h29","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a090e21alginequal103a-h27"],"title":"Interval Notation","text":"Is the upper bound $$\\\\frac{1}{2}$$ included as a valid value of \'x\' in $$-1<x<\\\\frac{1}{2}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]}]}}]},{"id":"a090e21alginequal11","title":"Algebra with Inequalities: Part C","body":"These questions are challenging, requiring mastery of each concept and their interrelations. Express your answer in interval notation.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Algebra with Inequalities","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a090e21alginequal11a","stepAnswer":["$$(-\\\\infty,0)$$ $$\\\\cup$$ $$(4,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"Determine which numbers \'x\' satisfy the following condition: $$\\\\frac{2x+1}{x}-\\\\frac{2x-7}{x-4}<0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,0)$$ $$\\\\cup$$ $$(4,\\\\infty)$$","choices":["$$(-\\\\infty,0)$$ $$\\\\cup$$ $$(4,\\\\infty)$$","$$(0,4)$$","$$(-\\\\infty,-4)$$ $$\\\\cup$$ $$(0,\\\\infty)$$","$$(-4,0)$$"],"hints":{"DefaultPathway":[{"id":"a090e21alginequal11a-h1","type":"hint","dependencies":[],"title":"Bounds of the Denominator","text":"No value should be divided by zero, so the denominator of the fractions cannot equal zero.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a090e21alginequal11a-h1"],"title":"Bounds of the Denominator","text":"For what value of \'x\' is $$\\\\frac{2x+1}{x}$$ divisible by 0?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal11a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a090e21alginequal11a-h1"],"title":"Bounds of the Denominator","text":"For what value of \'x\' is $$\\\\frac{-\\\\left(2x-7\\\\right)}{x-4}$$ divisible by 0?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal11a-h4","type":"hint","dependencies":["a090e21alginequal11a-h2","a090e21alginequal11a-h3"],"title":"Combining the Fractions","text":"For b,d $$ \\\\neq 0$$, $$\\\\frac{a}{b}-\\\\frac{c}{d}$$ is the same as $$\\\\frac{a d-b c}{b} d$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal11a-h5","type":"hint","dependencies":["a090e21alginequal11a-h4"],"title":"Combining the Fractions","text":"Multiply both fractions by the other fraction\'s denominator to combine.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal11a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x \\\\left(x-4\\\\right)$$"],"dependencies":["a090e21alginequal11a-h5"],"title":"Combining the Fractions","text":"What is the least common denominator of the fractions?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal11a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x^2-7x-4$$"],"dependencies":["a090e21alginequal11a-h6"],"title":"Combining the Fractions","text":"What is $$\\\\left(2x+1\\\\right) \\\\left(x-4\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal11a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2x^2+7x$$"],"dependencies":["a090e21alginequal11a-h6"],"title":"Combining the Fractions","text":"What is $$-\\\\left(2x-7\\\\right) x$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal11a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["a090e21alginequal11a-h7","a090e21alginequal11a-h8"],"title":"Combining the Fractions","text":"What is $$2x^2-7x-4-2x^2+7x$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal11a-h10","type":"hint","dependencies":["a090e21alginequal11a-h9"],"title":"Simplifying the Inequality","text":"Given the inequality $$\\\\frac{-4}{x \\\\left(x-4\\\\right)}<0$$ and $$x$$ $$ \\\\neq 0, 4$$, isolate \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal11a-h11","type":"hint","dependencies":["a090e21alginequal11a-h10"],"title":"Simplifying the Inequality","text":"For a,b $$ \\\\neq 0$$, $$\\\\frac{-a}{b}<0$$ is the same as $$\\\\frac{1}{b}>0$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal11a-h12","type":"hint","dependencies":["a090e21alginequal11a-h11"],"title":"Simplifying the Inequality","text":"Divide $$-4$$ on both sides and flip the inequality sign: $$\\\\frac{1}{x \\\\left(x-4\\\\right)}>0$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal11a-h13","type":"hint","dependencies":["a090e21alginequal11a-h12"],"title":"Simplifying the Inequality","text":"For $$a>0$$, $$\\\\frac{1}{a}>0$$ is the same as $$a>0$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal11a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x \\\\left(x-4\\\\right)$$"],"dependencies":["a090e21alginequal11a-h13"],"title":"Simplifying the Inequality","text":"What is $$\\\\frac{1}{x \\\\left(x-4\\\\right)}$$ rewritten without the fraction?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal11a-h15","type":"hint","dependencies":["a090e21alginequal11a-h14"],"title":"Simplifying the Inequality","text":"For some a,b $$ \\\\neq 0$$, $$a b>0$$ is the same as (a>0 AND b>0) OR (a<0 AND b<0).","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal11a-h16","type":"hint","dependencies":["a090e21alginequal11a-h15"],"title":"Simplifying the Inequality","text":"Split the inequality into four inequalities and solve for \'x\': (x>0 AND x-4>0) OR (x<0 AND x-4<0).","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal11a-h17","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a090e21alginequal11a-h16"],"title":"Simplifying the Inequality","text":"Simplify $$x-4>0$$. Only input the number after the $$\'>\'$$ symbol.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal11a-h18","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a090e21alginequal11a-h16"],"title":"Simplifying the Inequality","text":"Simplify $$x-4<0$$. Only input the number after the $$\'<\'$$ symbol.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal11a-h19","type":"hint","dependencies":["a090e21alginequal11a-h17","a090e21alginequal11a-h18"],"title":"Interval Notation","text":"Since the two inequalities are separated by an OR statement, they will both appear in the final result. First, simplify the ANDs of the two statements.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal11a-h20","type":"hint","dependencies":["a090e21alginequal11a-h19"],"title":"Interval Notation: Part A","text":"The inequality $$x>0$$ AND $$x>4$$ can be written in interval notation by specifying the lower bound first, followed by the upper bound.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal11a-h21","type":"hint","dependencies":["a090e21alginequal11a-h20"],"title":"Interval Notation: Part A","text":"Remember that $$x$$ cannot be 0,4.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal11a-h22","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$4$$"],"dependencies":["a090e21alginequal11a-h21"],"title":"Interval Notation: Part A","text":"What is the lower bound of the inequality $$x>0$$ AND $$x>4$$, with the additional constraints $$x$$ $$ \\\\neq 0, 4$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$-\\\\infty$$","$$0$$","$$4$$"]},{"id":"a090e21alginequal11a-h23","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\infty$$"],"dependencies":["a090e21alginequal11a-h21"],"title":"Interval Notation: Part A","text":"What is the upper bound of the inequality $$x>0$$ AND $$x>4$$, with the additional constraints $$x$$ $$ \\\\neq 0, 4$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\infty$$","$$0$$","$$4$$"]},{"id":"a090e21alginequal11a-h24","type":"hint","dependencies":["a090e21alginequal11a-h22","a090e21alginequal11a-h23"],"title":"Interval Notation: Part A","text":"If the bound itself should be included as a valid answer of \'x\', meaning it can be equal to that value, then a bracket (\\"[\\" or \\"]\\") should be used. Otherwise, use a paranthesis (\\"(\\" or \\")\\").","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal11a-h25","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a090e21alginequal11a-h24"],"title":"Interval Notation: Part A","text":"Is the lower bound $$4$$ included as a valid value of \'x\' in $$4<x$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"a090e21alginequal11a-h26","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a090e21alginequal11a-h24"],"title":"Interval Notation: Part A","text":"Is the upper bound $$\\\\infty$$ included as a valid value of \'x\' in $$4<x$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"],"subHints":[{"id":"a090e21alginequal11a-h26-s1","type":"hint","dependencies":[],"title":"Interval Notation: Part A","text":"As $$\\\\infty$$ is not a number, it cannot be included as part of a valid value or bound for \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a090e21alginequal11a-h27","type":"hint","dependencies":["a090e21alginequal11a-h25","a090e21alginequal11a-h26"],"title":"Interval Notation: Part B","text":"The inequality $$x<0$$ AND $$x<4$$ can be written in interval notation by specifying the lower bound first, followed by the upper bound.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal11a-h28","type":"hint","dependencies":["a090e21alginequal11a-h27"],"title":"Interval Notation: Part B","text":"Remember that $$x$$ $$ \\\\neq 0, 4$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal11a-h29","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-\\\\infty$$"],"dependencies":["a090e21alginequal11a-h28"],"title":"Interval Notation: Part B","text":"What is the lower bound of the inequality $$x<0$$ AND $$x<4$$, with the additional constraints $$x$$ $$ \\\\neq 0, 4$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$-\\\\infty$$","$$0$$","$$4$$"]},{"id":"a090e21alginequal11a-h30","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$0$$"],"dependencies":["a090e21alginequal11a-h28"],"title":"Interval Notation: Part B","text":"What is the upper bound of the inequality $$x<0$$ AND $$x<4$$, with the additional constraints $$x$$ $$ \\\\neq 0, 4$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\infty$$","$$0$$","$$4$$"]},{"id":"a090e21alginequal11a-h31","type":"hint","dependencies":["a090e21alginequal11a-h29","a090e21alginequal11a-h30"],"title":"Interval Notation: Part B","text":"If the bound itself should be included as a valid answer of \'x\', meaning it can be equal to that value, then a bracket (\\"[\\" or \\"]\\") should be used. Otherwise, use a paranthesis (\\"(\\" or \\")\\").","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal11a-h32","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a090e21alginequal11a-h31"],"title":"Interval Notation: Part B","text":"Is the lower bound $$-\\\\infty$$ included as a valid value of \'x\' in $$x<0$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"],"subHints":[{"id":"a090e21alginequal11a-h32-s1","type":"hint","dependencies":[],"title":"Interval Notation: Part B","text":"As $$-\\\\infty$$ is not a number, it cannot be included as part of a valid value or bound for \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a090e21alginequal11a-h33","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a090e21alginequal11a-h31"],"title":"Interval Notation: Part B","text":"Is the upper bound $$0$$ included as a valid value of \'x\' in $$x<0$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"a090e21alginequal11a-h34","type":"hint","dependencies":["a090e21alginequal11a-h32","a090e21alginequal11a-h33"],"title":"Interval Notation","text":"The bounds for the two equations, $$(-\\\\infty,0)$$ and $$(4,\\\\infty)$$, can be ORed together using \' $$\\\\cup$$ \'.","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a090e21alginequal12","title":"Algebra with Inequalities: Part C","body":"These questions are challenging, requiring mastery of each concept and their interrelations. Express your answer in interval notation.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Algebra with Inequalities","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a090e21alginequal12a","stepAnswer":["$$(-\\\\infty,-3)$$ $$\\\\cup$$ $$(\\\\frac{-1}{2},\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"Determine all real numbers \'x\' such that the following expression is defined: $$\\\\log_{2}\\\\left(\\\\frac{1+2x}{x+3}\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,-3)$$ $$\\\\cup$$ $$(\\\\frac{-1}{2},\\\\infty)$$","choices":["$$(-\\\\infty,-3)$$ $$\\\\cup$$ $$(\\\\frac{-1}{2},\\\\infty)$$","$$(-3,\\\\frac{-1}{2})$$","$$(-\\\\infty,-3)$$ $$\\\\cup$$ $$(\\\\frac{1}{2},\\\\infty)$$","$$(-\\\\infty,\\\\frac{1}{2})$$ $$\\\\cup$$ $$(3,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"a090e21alginequal12a-h1","type":"hint","dependencies":[],"title":"Bounds of a Logarithm","text":"The logarithm of some value \'b\' cannot be negative or zero, so $$b>0$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal12a-h2","type":"hint","dependencies":["a090e21alginequal12a-h1"],"title":"Bounds of a Logarithm","text":"Since $$\\\\log_{2}\\\\left(\\\\frac{1+2x}{x+3}\\\\right)$$ cannot be negative or zero, then $$\\\\frac{1+2x}{x+3}>0$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal12a-h3","type":"hint","dependencies":["a090e21alginequal12a-h2"],"title":"When is a Fraction Positive?","text":"Since $$\\\\frac{1+2x}{x+3}>0$$, the fraction is positive and must be considered in the inequality.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal12a-h4","type":"hint","dependencies":["a090e21alginequal12a-h3"],"title":"When is a Fraction Positive?","text":"For some values a,b where $$b$$ $$ \\\\neq 0$$, the fraction $$\\\\frac{a}{b}$$ is positive when (a>0 AND b>0) OR (a<0 AND b<0).","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal12a-h5","type":"hint","dependencies":["a090e21alginequal12a-h4"],"title":"When is a Fraction Positive?","text":"The fraction $$\\\\frac{1+2x}{x+3}$$ is positive when (1+2*x>0 AND x+3>0) OR (1+2*x<0 AND x+3<0).","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal12a-h6","type":"hint","dependencies":["a090e21alginequal12a-h5"],"title":"Simplifying the Inequality","text":"Since the two inequalities are separated by an OR statement, they will both appear in the final result. First, simplify the ANDs of the two statements.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal12a-h7","type":"hint","dependencies":["a090e21alginequal12a-h6"],"title":"Simplifying the Inequality: Part A","text":"Simplify $$1+2x>0$$ AND $$x+3>0$$ by isolating \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal12a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{2}$$"],"dependencies":["a090e21alginequal12a-h7"],"title":"Simplifying the Inequality: Part A","text":"Simplify $$1+2x>0$$. Only input the number after the $$\'>\'$$ symbol.","variabilization":{},"oer":"https://OATutor.io","license":"","subHints":[{"id":"a090e21alginequal12a-h8-s1","type":"hint","dependencies":[],"title":"Simplifying the Inequality: $$1+2x>0$$","text":"Rearrange the inequality so that the \'x\'s are separated from the constants.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal12a-h8-s2","type":"hint","dependencies":["a090e21alginequal12a-h8-s1"],"title":"Simplifying the Inequality: $$1+2x>0$$","text":"Subtract $$1$$ from the left to get $$2x>0-1$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal12a-h8-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a090e21alginequal12a-h8-s2"],"title":"Simplifying the Inequality: $$1+2x>0$$","text":"What is $$0-1$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal12a-h8-s4","type":"hint","dependencies":["a090e21alginequal12a-h8-s3"],"title":"Simplifying the Inequality: $$1+2x>0$$","text":"Divide $$2$$ from both sides of $$2x>-1$$ to isolate \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a090e21alginequal12a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a090e21alginequal12a-h7"],"title":"Simplifying the Inequality: Part A","text":"Simplify $$x+3>0$$. Only input the number after the $$\'>\'$$ symbol.","variabilization":{},"oer":"https://OATutor.io","license":"","subHints":[{"id":"a090e21alginequal12a-h9-s5","type":"hint","dependencies":[],"title":"Simplifying the Inequality: $$x+3>0$$","text":"Rearrange the inequality so that the \'x\'s are separated from the constants.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal12a-h9-s6","type":"hint","dependencies":["a090e21alginequal12a-h9-s5"],"title":"Simplifying the Inequality: $$x+3>0$$","text":"Subtract $$3$$ from the left to get $$x>0-3$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal12a-h9-s7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a090e21alginequal12a-h9-s6"],"title":"Simplifying the Inequality: $$x+3>0$$","text":"What is $$0-3$$?","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a090e21alginequal12a-h10","type":"hint","dependencies":["a090e21alginequal12a-h8","a090e21alginequal12a-h9"],"title":"Simplifying the Inequality: Part B","text":"Simplify $$1+2x<0$$ AND $$x+3<0$$ by isolating \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal12a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{2}$$"],"dependencies":["a090e21alginequal12a-h10"],"title":"Simplifying the Inequality: Part B","text":"Simplify $$1+2x<0$$. Only input the number after the $$\'<\'$$ symbol.","variabilization":{},"oer":"https://OATutor.io","license":"","subHints":[{"id":"a090e21alginequal12a-h11-s8","type":"hint","dependencies":[],"title":"Simplifying the Inequality: $$1+2x<0$$","text":"Rearrange the inequality so that the \'x\'s are separated from the constants.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal12a-h11-s9","type":"hint","dependencies":["a090e21alginequal12a-h11-s8"],"title":"Simplifying the Inequality: $$1+2x<0$$","text":"Subtract $$1$$ from the left to get $$2x<0-1$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal12a-h11-s10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a090e21alginequal12a-h11-s9"],"title":"Simplifying the Inequality: $$1+2x<0$$","text":"What is $$0-1$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal12a-h11-s11","type":"hint","dependencies":["a090e21alginequal12a-h11-s10"],"title":"Simplifying the Inequality: $$1+2x<0$$","text":"Divide $$2$$ from both sides of $$2x<-1$$ to isolate \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a090e21alginequal12a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a090e21alginequal12a-h10"],"title":"Simplifying the Inequality: Part B","text":"Simplify $$x+3<0$$. Only input the number after the $$\'<\'$$ symbol.","variabilization":{},"oer":"https://OATutor.io","license":"","subHints":[{"id":"a090e21alginequal12a-h12-s12","type":"hint","dependencies":[],"title":"Simplifying the Inequality: $$x+3<0$$","text":"Rearrange the inequality so that the \'x\'s are separated from the constants.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal12a-h12-s13","type":"hint","dependencies":["a090e21alginequal12a-h12-s12"],"title":"Simplifying the Inequality: $$x+3<0$$","text":"Subtract $$3$$ from the left to get $$x<0-3$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal12a-h12-s14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a090e21alginequal12a-h12-s13"],"title":"Simplifying the Inequality: $$x+3<0$$","text":"What is $$0-3$$?","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a090e21alginequal12a-h13","type":"hint","dependencies":["a090e21alginequal12a-h11","a090e21alginequal12a-h12"],"title":"Interval Notation","text":"The inequality has been simplified to (x>-1/2 AND x>-3) OR (x<-1/2 AND x<-3).","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal12a-h14","type":"hint","dependencies":["a090e21alginequal12a-h13"],"title":"Interval Notation","text":"Since the two inequalities are separated by an OR statement, they will both appear in the final result. First, simplify the ANDs of the two statements.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal12a-h15","type":"hint","dependencies":["a090e21alginequal12a-h14"],"title":"Interval Notation: Part A","text":"The inequality $$x>\\\\frac{-1}{2}$$ AND $$x>-3$$ can be written in interval notation by specifying the lower bound first, followed by the upper bound.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal12a-h16","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{-1}{2}$$"],"dependencies":["a090e21alginequal12a-h15"],"title":"Interval Notation: Part A","text":"What is the lower bound of the inequality $$x>\\\\frac{-1}{2}$$ AND $$x>-3$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$-\\\\infty$$","$$\\\\frac{-1}{2}$$","$$-3$$"]},{"id":"a090e21alginequal12a-h17","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\infty$$"],"dependencies":["a090e21alginequal12a-h15"],"title":"Interval Notation: Part A","text":"What is the upper bound of the inequality $$x>\\\\frac{-1}{2}$$ AND $$x>-3$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\infty$$","$$\\\\frac{-1}{2}$$","$$-3$$"]},{"id":"a090e21alginequal12a-h18","type":"hint","dependencies":["a090e21alginequal12a-h16","a090e21alginequal12a-h17"],"title":"Interval Notation: Part A","text":"If the bound itself should be included as a valid answer of \'x\', meaning it can be equal to that value, then a bracket (\\"[\\" or \\"]\\") should be used. Otherwise, use a paranthesis (\\"(\\" or \\")\\").","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal12a-h19","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a090e21alginequal12a-h18"],"title":"Interval Notation: Part A","text":"Is the lower bound $$\\\\frac{-1}{2}$$ included as a valid value of \'x\' in $$\\\\frac{-1}{2}<x$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"a090e21alginequal12a-h20","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a090e21alginequal12a-h18"],"title":"Interval Notation: Part A","text":"Is the upper bound $$\\\\infty$$ included as a valid value of \'x\' in $$\\\\frac{-1}{2}<x$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"],"subHints":[{"id":"a090e21alginequal12a-h20-s1","type":"hint","dependencies":[],"title":"Interval Notation: Part A","text":"As $$\\\\infty$$ is not a number, it cannot be included as part of a valid value or bound for \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a090e21alginequal12a-h21","type":"hint","dependencies":["a090e21alginequal12a-h19","a090e21alginequal12a-h20"],"title":"Interval Notation: Part B","text":"The inequality $$x<\\\\frac{-1}{2}$$ AND $$x<-3$$ can be written in interval notation by specifying the lower bound first, followed by the upper bound.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal12a-h22","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-\\\\infty$$"],"dependencies":["a090e21alginequal12a-h21"],"title":"Interval Notation: Part B","text":"What is the lower bound of the inequality $$x<\\\\frac{-1}{2}$$ AND $$x<-3$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$-\\\\infty$$","$$\\\\frac{-1}{2}$$","$$-3$$"]},{"id":"a090e21alginequal12a-h23","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-3$$"],"dependencies":["a090e21alginequal12a-h21"],"title":"Interval Notation: Part B","text":"What is the upper bound of the inequality $$x<\\\\frac{-1}{2}$$ AND $$x<-3$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\infty$$","$$\\\\frac{-1}{2}$$","$$-3$$"]},{"id":"a090e21alginequal12a-h24","type":"hint","dependencies":["a090e21alginequal12a-h22","a090e21alginequal12a-h23"],"title":"Interval Notation: Part B","text":"If the bound itself should be included as a valid answer of \'x\', meaning it can be equal to that value, then a bracket (\\"[\\" or \\"]\\") should be used. Otherwise, use a paranthesis (\\"(\\" or \\")\\").","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal12a-h25","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a090e21alginequal12a-h24"],"title":"Interval Notation: Part B","text":"Is the lower bound $$-\\\\infty$$ included as a valid value of \'x\' in $$x<-3$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"],"subHints":[{"id":"a090e21alginequal12a-h25-s1","type":"hint","dependencies":[],"title":"Interval Notation: Part B","text":"As $$-\\\\infty$$ is not a number, it cannot be included as part of a valid value or bound for \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a090e21alginequal12a-h26","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a090e21alginequal12a-h24"],"title":"Interval Notation: Part B","text":"Is the upper bound $$-3$$ included as a valid value of \'x\' in $$x<-3$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"a090e21alginequal12a-h27","type":"hint","dependencies":["a090e21alginequal12a-h25","a090e21alginequal12a-h26"],"title":"Interval Notation","text":"The bounds for the two equations, $$(-\\\\infty,-3)$$ and $$(\\\\frac{-1}{2},\\\\infty)$$, can be ORed together using \' $$\\\\cup$$ \'.","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a090e21alginequal2","title":"Algebra with Inequalities: Part A","body":"These questions test your knowledge of the core concepts. Determine which numbers \'x\' satisfy the following conditions. Express your answer in interval notation.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Algebra with Inequalities","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a090e21alginequal2a","stepAnswer":["$$[\\\\frac{1}{5},\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$2^{x+1}$$ $$\\\\geq$$ $$4^{1-2x}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[\\\\frac{1}{5},\\\\infty)$$","choices":["$$[\\\\frac{1}{5},\\\\infty)$$","$$(0,\\\\infty)$$","$$(-\\\\infty,1]$$","$$[\\\\frac{3}{4},\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"a090e21alginequal2a-h1","type":"hint","dependencies":[],"title":"Same Base","text":"Rewrite the inequality such that both exponents have the same base.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a090e21alginequal2a-h1"],"title":"Same Base","text":"$$4=2^x$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal2a-h3","type":"hint","dependencies":["a090e21alginequal2a-h2"],"title":"Same Base","text":"Rewrite $$4^{1-2x}$$ with a base of $$2$$. 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Otherwise, use a paranthesis (\\"(\\" or \\")\\").","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal2a-h18","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a090e21alginequal2a-h17"],"title":"Interval Notation","text":"Is the lower bound $$\\\\frac{1}{5}$$ included as a valid value of \'x\' in $$x$$ $$\\\\geq$$ $$\\\\frac{1}{5}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"a090e21alginequal2a-h19","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a090e21alginequal2a-h17"],"title":"Interval Notation","text":"Is the upper bound $$\\\\infty$$ included as a valid value of \'x\' in $$x$$ $$\\\\geq$$ $$\\\\frac{1}{5}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"],"subHints":[{"id":"a090e21alginequal2a-h19-s1","type":"hint","dependencies":[],"title":"Infinity in Interval Notation","text":"As $$\\\\infty$$ is not a number, it cannot be included as part of a valid value or bound for \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""}]}]}}]},{"id":"a090e21alginequal3","title":"Algebra with Inequalities: Part A","body":"These questions test your knowledge of the core concepts. 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Otherwise, use a paranthesis (\\"(\\" or \\")\\").","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a090e21alginequal3a-h12","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a090e21alginequal3a-h11"],"title":"Interval Notation","text":"Is the lower bound $$-\\\\infty$$ included as a valid value of \'x\' in $$1>x$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"],"subHints":[{"id":"a090e21alginequal3a-h12-s1","type":"hint","dependencies":[],"title":"Infinity in Interval Notation","text":"As $$\\\\infty$$ is not a number, it cannot be included as part of a valid value or bound for \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a090e21alginequal3a-h13","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a090e21alginequal3a-h11"],"title":"Interval Notation","text":"Is the upper bound $$\\\\infty$$ included as a valid value of \'x\' in $$1>x$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]}]}}]},{"id":"a0a04b1divmonomial1","title":"Using the Quotient Property for Exponents","body":"Use the Quotient Property for exponents to simplify the following expressions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.5 Divide Monomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a0a04b1divmonomial1a","stepAnswer":["$$x^2$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{x^9}{x^7}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^2$$","hints":{"DefaultPathway":[{"id":"a0a04b1divmonomial1a-h1","type":"hint","dependencies":[],"title":"Definition of the Quotient Property for Exponents","text":"If $$m>n$$, then $$\\\\frac{a^m}{a^n}$$ is equal to $$a^{m-n}$$. 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Otherwise, if $$m<n$$, then $$\\\\frac{a^m}{a^n}=\\\\frac{1}{a^{n-m}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a0a04b1divmonomial9b","stepAnswer":["$$w^4$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{w^{13}}{w^9}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$w^4$$","hints":{"DefaultPathway":[{"id":"a0a04b1divmonomial9b-h1","type":"hint","dependencies":[],"title":"Definition of the Quotient Property for Exponents","text":"If $$m>n$$, then $$\\\\frac{a^m}{a^n}$$ is equal to $$a^{m-n}$$. Otherwise, if $$m<n$$, then $$\\\\frac{a^m}{a^n}=\\\\frac{1}{a^{n-m}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ca101exe1","title":"Factor Trinomials of the Form x2 + bx + c","body":"Factor the Trinomial (Give your answer with the larger value first. eg. $$\\\\left(x+2\\\\right) \\\\left(x+1\\\\right)$$ $$2$$ is greater than 1):","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Factor Trinomials with Leading Coefficient 1","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a0ca101exe1a","stepAnswer":["$$\\\\left(u+100\\\\right) \\\\left(u+1\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$u^2+101u+100$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(u+100\\\\right) \\\\left(u+1\\\\right)$$","hints":{"DefaultPathway":[{"id":"a0ca101exe1a-h1","type":"hint","dependencies":[],"title":"Find Two Factors","text":"In the form $${ax}^2+bx+c$$, find $$2$$ numbers that multiply to c and add up to $$b$$ (Note: This only works when $$a=1)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe1a-h2","type":"hint","dependencies":["a0ca101exe1a-h1"],"title":"Two Factors","text":"The two factors that match these rules are $$100$$ and $$1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe1a-h3","type":"hint","dependencies":["a0ca101exe1a-h2"],"title":"Plug into factored form","text":"The factored form is $$\\\\left(x+y\\\\right) \\\\left(x+z\\\\right)$$. The value of $$x$$ can vary depending on the equation (could be u, $$y$$, f, etc) and the values of $$y$$ and $$z$$ are the two factors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ca101exe10","title":"Factor Trinomials of the Form x2 + bx + c","body":"Factor the Trinomial (Give your answer with the larger value first. eg. $$\\\\left(x+2\\\\right) \\\\left(x+1\\\\right)$$ $$2$$ is greater than 1):","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Factor Trinomials with Leading Coefficient 1","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a0ca101exe10a","stepAnswer":["$$\\\\left(y+1\\\\right) \\\\left(y-7\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$y^2-6y-7$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(y+1\\\\right) \\\\left(y-7\\\\right)$$","hints":{"DefaultPathway":[{"id":"a0ca101exe10a-h1","type":"hint","dependencies":[],"title":"Find Two Factors","text":"In the form $${ax}^2+bx+c$$, find $$2$$ numbers that multiply to c and add up to $$b$$ (Note: This only works when $$a=1)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe10a-h2","type":"hint","dependencies":["a0ca101exe10a-h1"],"title":"Two Factors","text":"The two factors that match these rules are $$1$$ and $$-7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe10a-h3","type":"hint","dependencies":["a0ca101exe10a-h2"],"title":"Plug into factored form","text":"The factored form is $$\\\\left(x+y\\\\right) \\\\left(x+z\\\\right)$$. The value of $$x$$ can vary depending on the equation (could be u, $$y$$, f, etc) and the values of $$y$$ and $$z$$ are the two factored numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ca101exe11","title":"Factor Trinomials of the Form x2 + bx + c","body":"Factor the Trinomial (Give your answer with the larger value first. eg. $$\\\\left(x+2\\\\right) \\\\left(x+1\\\\right)$$ $$2$$ is greater than 1):","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Factor Trinomials with Leading Coefficient 1","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a0ca101exe11a","stepAnswer":["$$\\\\left(y+1\\\\right) \\\\left(y-3\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$v^2-2v-3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(y+1\\\\right) \\\\left(y-3\\\\right)$$","hints":{"DefaultPathway":[{"id":"a0ca101exe11a-h1","type":"hint","dependencies":[],"title":"Find Two Factors","text":"In the form $${ax}^2+bx+c$$, find $$2$$ numbers that multiply to c and add up to $$b$$ (Note: This only works when $$a=1)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe11a-h2","type":"hint","dependencies":["a0ca101exe11a-h1"],"title":"Two Factors","text":"The two factors that match these rules are $$1$$ and $$-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe11a-h3","type":"hint","dependencies":["a0ca101exe11a-h2"],"title":"Plug into factored form","text":"The factored form is $$\\\\left(x+y\\\\right) \\\\left(x+z\\\\right)$$. The value of $$x$$ can vary depending on the equation (could be u, $$y$$, f, etc) and the values of $$y$$ and $$z$$ are the two factored numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ca101exe12","title":"Factor Trinomials of the Form x2 + bx + c","body":"Factor the Trinomial (Give your answer with the larger value first. eg. $$\\\\left(x+2\\\\right) \\\\left(x+1\\\\right)$$ $$2$$ is greater than 1):","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Factor Trinomials with Leading Coefficient 1","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a0ca101exe12a","stepAnswer":["$$\\\\left(x+5\\\\right) \\\\left(x-6\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$x^2-x-12$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(x+5\\\\right) \\\\left(x-6\\\\right)$$","hints":{"DefaultPathway":[{"id":"a0ca101exe12a-h1","type":"hint","dependencies":[],"title":"Find Two Factors","text":"In the form $${ax}^2+bx+c$$, find $$2$$ numbers that multiply to c and add up to $$b$$ (Note: This only works when $$a=1)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe12a-h2","type":"hint","dependencies":["a0ca101exe12a-h1"],"title":"Two Factors","text":"The two factors that match these rules are $$5$$ and $$-6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe12a-h3","type":"hint","dependencies":["a0ca101exe12a-h2"],"title":"Plug into factored form","text":"The factored form is $$\\\\left(x+y\\\\right) \\\\left(x+z\\\\right)$$. The value of $$x$$ can vary depending on the equation (could be u, $$y$$, f, etc) and the values of $$y$$ and $$z$$ are the two factored numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ca101exe13","title":"Factor Trinomials of the Form x2 + bx + c","body":"Factor the Trinomial (Give your answer with the larger value first. eg. $$\\\\left(x+2\\\\right) \\\\left(x+1\\\\right)$$ $$2$$ is greater than 1):","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Factor Trinomials with Leading Coefficient 1","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a0ca101exe13a","stepAnswer":["$$\\\\left(r+2\\\\right) \\\\left(r-4\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$r^2-2r-8$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(r+2\\\\right) \\\\left(r-4\\\\right)$$","hints":{"DefaultPathway":[{"id":"a0ca101exe13a-h1","type":"hint","dependencies":[],"title":"Find Two Factors","text":"In the form $${ax}^2+bx+c$$, find $$2$$ numbers that multiply to c and add up to $$b$$ (Note: This only works when $$a=1)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe13a-h2","type":"hint","dependencies":["a0ca101exe13a-h1"],"title":"Two Factors","text":"The two factors that match these rules are $$2$$ and $$-4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe13a-h3","type":"hint","dependencies":["a0ca101exe13a-h2"],"title":"Plug into factored form","text":"The factored form is $$\\\\left(x+y\\\\right) \\\\left(x+z\\\\right)$$. The value of $$x$$ can vary depending on the equation (could be u, $$y$$, f, etc) and the values of $$y$$ and $$z$$ are the two factored numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ca101exe14","title":"Factor Trinomials of the Form x2 + bx + c","body":"Factor the Trinomial (Give your answer with the larger value first. eg. $$\\\\left(x+2\\\\right) \\\\left(x+1\\\\right)$$ $$2$$ is greater than 1):","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Factor Trinomials with Leading Coefficient 1","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a0ca101exe14a","stepAnswer":["$$\\\\left(a+4\\\\right) \\\\left(a-7\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$a^2-3a-28$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(a+4\\\\right) \\\\left(a-7\\\\right)$$","hints":{"DefaultPathway":[{"id":"a0ca101exe14a-h1","type":"hint","dependencies":[],"title":"Find Two Factors","text":"In the form $${ax}^2+bx+c$$, find $$2$$ numbers that multiply to c and add up to $$b$$ (Note: This only works when $$a=1)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe14a-h2","type":"hint","dependencies":["a0ca101exe14a-h1"],"title":"Two Factors","text":"The two factors that match these rules are $$4$$ and $$-7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe14a-h3","type":"hint","dependencies":["a0ca101exe14a-h2"],"title":"Plug into factored form","text":"The factored form is $$\\\\left(x+y\\\\right) \\\\left(x+z\\\\right)$$. The value of $$x$$ can vary depending on the equation (could be u, $$y$$, f, etc) and the values of $$y$$ and $$z$$ are the two factored numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ca101exe15","title":"Factor Trinomials of the Form x2 + bx + c","body":"Factor the Trinomial (Give your answer with the larger value first. eg. $$\\\\left(x+2\\\\right) \\\\left(x+1\\\\right)$$ $$2$$ is greater than 1):","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Factor Trinomials with Leading Coefficient 1","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a0ca101exe15a","stepAnswer":["$$\\\\left(b+2\\\\right) \\\\left(b-15\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$b^2-13b-30$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(b+2\\\\right) \\\\left(b-15\\\\right)$$","hints":{"DefaultPathway":[{"id":"a0ca101exe15a-h1","type":"hint","dependencies":[],"title":"Find Two Factors","text":"In the form $${ax}^2+bx+c$$, find $$2$$ numbers that multiply to c and add up to $$b$$ (Note: This only works when $$a=1)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe15a-h2","type":"hint","dependencies":["a0ca101exe15a-h1"],"title":"Two Factors","text":"The two factors that match these rules are $$2$$ and $$-15$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe15a-h3","type":"hint","dependencies":["a0ca101exe15a-h2"],"title":"Plug into factored form","text":"The factored form is $$\\\\left(x+y\\\\right) \\\\left(x+z\\\\right)$$. The value of $$x$$ can vary depending on the equation (could be u, $$y$$, f, etc) and the values of $$y$$ and $$z$$ are the two factored numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ca101exe16","title":"Factoring Trinomials","body":"Factor the trinomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Factor Trinomials with Leading Coefficient 1","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a0ca101exe16a","stepAnswer":["$$\\\\left(x+1\\\\right) \\\\left(x+3\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor the following trinomial: $$x^2+4x+3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(x+1\\\\right) \\\\left(x+3\\\\right)$$","hints":{"DefaultPathway":[{"id":"a0ca101exe16a-h1","type":"hint","dependencies":[],"title":"Factoring $$x^2+bx+c$$","text":"When factoring trinomials of the form $$x^2+bx+c$$, we get an answer $$\\\\left(x+k\\\\right) \\\\left(x+h\\\\right)$$, where k and $$h$$ sum to $$b$$ and multiply to c","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe16a-h2","type":"hint","dependencies":["a0ca101exe16a-h1"],"title":"Factoring the Equation with the Given Rules","text":"To factor into the form $$\\\\left(x+h\\\\right) \\\\left(x+k\\\\right)$$, we could take $$k=3$$ and $$h=1$$, because they sum to $$b=4$$ and multiply to $$c=3$$. We have $$\\\\left(x+1\\\\right) \\\\left(x+3\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ca101exe17","title":"Factoring Trinomials","body":"Factor the trinomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Factor Trinomials with Leading Coefficient 1","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a0ca101exe17a","stepAnswer":["$$\\\\left(y+7\\\\right) \\\\left(y+1\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor the following trinomial: $$y^2+8y+7$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(y+7\\\\right) \\\\left(y+1\\\\right)$$","hints":{"DefaultPathway":[{"id":"a0ca101exe17a-h1","type":"hint","dependencies":[],"title":"Factoring $$x^2+bx+c$$","text":"When factoring trinomials of the form $$x^2+bx+c$$, we get an answer $$\\\\left(x+k\\\\right) \\\\left(x+h\\\\right)$$, where k and $$h$$ sum to $$b$$ and multiply to c","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe17a-h2","type":"hint","dependencies":["a0ca101exe17a-h1"],"title":"Factoring the Equation with the Given Rules","text":"To factor into the form $$\\\\left(y+h\\\\right) \\\\left(y+k\\\\right)$$, we could take $$h=7$$ and $$k=1$$, because they sum to $$b=8$$ and multiply to $$c=7$$. We have $$\\\\left(y+7\\\\right) \\\\left(y+1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ca101exe18","title":"Factoring Trinomials","body":"Factor the trinomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Factor Trinomials with Leading Coefficient 1","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a0ca101exe18a","stepAnswer":["$$\\\\left(m+11\\\\right) \\\\left(m+1\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor the following trinomial: $$m^2+12m+11$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(m+11\\\\right) \\\\left(m+1\\\\right)$$","hints":{"DefaultPathway":[{"id":"a0ca101exe18a-h1","type":"hint","dependencies":[],"title":"Factoring $$x^2+bx+c$$","text":"When factoring trinomials of the form $$x^2+bx+c$$, we get an answer $$\\\\left(x+k\\\\right) \\\\left(x+h\\\\right)$$, where k and $$h$$ sum to $$b$$ and multiply to c","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe18a-h2","type":"hint","dependencies":["a0ca101exe18a-h1"],"title":"Factoring the Equation with the Given Rules","text":"To factor into the form $$\\\\left(m+h\\\\right) \\\\left(m+k\\\\right)$$, we could take $$k=11$$ and $$h=1$$, because they sum to $$b=12$$ and multiply to $$c=11$$. We have $$\\\\left(m+11\\\\right) \\\\left(m+1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ca101exe19","title":"Factoring Trinomials","body":"Factor the trinomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Factor Trinomials with Leading Coefficient 1","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a0ca101exe19a","stepAnswer":["$$\\\\left(b+13\\\\right) \\\\left(b+1\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor the following trinomial: $$b^2+14b+13$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(b+13\\\\right) \\\\left(b+1\\\\right)$$","hints":{"DefaultPathway":[{"id":"a0ca101exe19a-h1","type":"hint","dependencies":[],"title":"Factoring $$x^2+bx+c$$","text":"When factoring trinomials of the form $$x^2+bx+c$$, we get an answer $$\\\\left(x+k\\\\right) \\\\left(x+h\\\\right)$$, where k and $$h$$ sum to $$b$$ and multiply to c","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe19a-h2","type":"hint","dependencies":["a0ca101exe19a-h1"],"title":"Factoring the Equation with the Given Rules","text":"To factor into the form $$\\\\left(b+h\\\\right) \\\\left(b+k\\\\right)$$, we could take $$k=13$$ and $$h=1$$, because they sum to $$b=14$$ and multiply to $$c=13$$. We have $$\\\\left(b+13\\\\right) \\\\left(b+1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ca101exe2","title":"Factor Trinomials of the Form x2 + bx + c","body":"Factor the Trinomial (Give your answer with the larger value first. eg. $$\\\\left(x+2\\\\right) \\\\left(x+1\\\\right)$$ $$2$$ is greater than 1):","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Factor Trinomials with Leading Coefficient 1","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a0ca101exe2a","stepAnswer":["$$(x-2)(x-6)$$"],"problemType":"TextBox","stepTitle":"$$x^2-8x+12$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$(x-2)(x-6)$$","hints":{"DefaultPathway":[{"id":"a0ca101exe2a-h1","type":"hint","dependencies":[],"title":"Find Two Factors","text":"In the form $${ax}^2+bx+c$$, find $$2$$ numbers that multiply to c and add up to $$b$$ (Note: This only works when $$a=1)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe2a-h2","type":"hint","dependencies":["a0ca101exe2a-h1"],"title":"Two Factors","text":"The two factors that match these rules are $$-2$$ and $$-6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe2a-h3","type":"hint","dependencies":["a0ca101exe2a-h2"],"title":"Plug into factored form","text":"The factored form is $$\\\\left(x+y\\\\right) \\\\left(x+z\\\\right)$$. The value of $$x$$ can vary depending on the equation (could be u, $$y$$, f, etc) and the values of $$y$$ and $$z$$ are the two factored numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ca101exe20","title":"Factoring Trinomials","body":"Factor the trinomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Factor Trinomials with Leading Coefficient 1","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a0ca101exe20a","stepAnswer":["$$\\\\left(a+5\\\\right) \\\\left(a+4\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor the following trinomial: $$a^2+9a+20$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(a+5\\\\right) \\\\left(a+4\\\\right)$$","hints":{"DefaultPathway":[{"id":"a0ca101exe20a-h1","type":"hint","dependencies":[],"title":"Factoring $$x^2+bx+c$$","text":"When factoring trinomials of the form $$x^2+bx+c$$, we get an answer $$\\\\left(x+k\\\\right) \\\\left(x+h\\\\right)$$, where k and $$h$$ sum to $$b$$ and multiply to c","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe20a-h2","type":"hint","dependencies":["a0ca101exe20a-h1"],"title":"Factoring the Equation with the Given Rules","text":"To factor into the form $$\\\\left(a+h\\\\right) \\\\left(a+k\\\\right)$$, we could take $$k=5$$ and $$h=4$$, because they sum to $$b=9$$ and multiply to $$c=20$$. We have $$\\\\left(a+5\\\\right) \\\\left(a+4\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ca101exe21","title":"Factoring Trinomials","body":"Factor the trinomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Factor Trinomials with Leading Coefficient 1","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a0ca101exe21a","stepAnswer":["$$\\\\left(m+4\\\\right) \\\\left(m+3\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor the following trinomial: $$m^2+7m+12$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(m+4\\\\right) \\\\left(m+3\\\\right)$$","hints":{"DefaultPathway":[{"id":"a0ca101exe21a-h1","type":"hint","dependencies":[],"title":"Factoring $$x^2+bx+c$$","text":"When factoring trinomials of the form $$x^2+bx+c$$, we get an answer $$\\\\left(x+k\\\\right) \\\\left(x+h\\\\right)$$, where k and $$h$$ sum to $$b$$ and multiply to c","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe21a-h2","type":"hint","dependencies":["a0ca101exe21a-h1"],"title":"Factoring the Equation with the Given Rules","text":"To factor into the form $$\\\\left(m+h\\\\right) \\\\left(m+k\\\\right)$$, we could take $$k=4$$ and $$h=3$$, because they sum to $$b=7$$ and multiply to $$c=12$$. We have $$\\\\left(m+4\\\\right) \\\\left(m+3\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ca101exe22","title":"Factoring Trinomials","body":"Factor the trinomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Factor Trinomials with Leading Coefficient 1","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a0ca101exe22a","stepAnswer":["$$\\\\left(p+6\\\\right) \\\\left(p+5\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor the following trinomial: $$p^2+11p+30$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(p+6\\\\right) \\\\left(p+5\\\\right)$$","hints":{"DefaultPathway":[{"id":"a0ca101exe22a-h1","type":"hint","dependencies":[],"title":"Factoring $$x^2+bx+c$$","text":"When factoring trinomials of the form $$x^2+bx+c$$, we get an answer $$\\\\left(x+k\\\\right) \\\\left(x+h\\\\right)$$, where k and $$h$$ sum to $$b$$ and multiply to c","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe22a-h2","type":"hint","dependencies":["a0ca101exe22a-h1"],"title":"Factoring the Equation with the Given Rules","text":"To factor into the form $$\\\\left(p+h\\\\right) \\\\left(p+k\\\\right)$$, we could take $$k=6$$ and $$h=5$$, because they sum to $$b=11$$ and multiply to $$c=30$$. We have $$\\\\left(p+6\\\\right) \\\\left(p+5\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ca101exe23","title":"Factoring Trinomials","body":"Factor the trinomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Factor Trinomials with Leading Coefficient 1","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a0ca101exe23a","stepAnswer":["$$\\\\left(x+7\\\\right) \\\\left(x+3\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor the following trinomial: $$x^2+10x+21$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(x+7\\\\right) \\\\left(x+3\\\\right)$$","hints":{"DefaultPathway":[{"id":"a0ca101exe23a-h1","type":"hint","dependencies":[],"title":"Factoring $$x^2+bx+c$$","text":"When factoring trinomials of the form $$x^2+bx+c$$, we get an answer $$\\\\left(x+k\\\\right) \\\\left(x+h\\\\right)$$, where k and $$h$$ sum to $$b$$ and multiply to c","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe23a-h2","type":"hint","dependencies":["a0ca101exe23a-h1"],"title":"Factoring the Equation with the Given Rules","text":"To factor into the form $$\\\\left(x+h\\\\right) \\\\left(x+k\\\\right)$$, we could take $$k=7$$ and $$h=3$$, because they sum to $$b=10$$ and multiply to $$c=21$$ We have $$\\\\left(x+7\\\\right) \\\\left(x+3\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ca101exe24","title":"Factoring Trinomials","body":"Factor the trinomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Factor Trinomials with Leading Coefficient 1","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a0ca101exe24a","stepAnswer":["$$\\\\left(x+16\\\\right) \\\\left(x+3\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor the following trinomial: $$x^2+19x+48$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(x+16\\\\right) \\\\left(x+3\\\\right)$$","hints":{"DefaultPathway":[{"id":"a0ca101exe24a-h1","type":"hint","dependencies":[],"title":"Factoring $$x^2+bx+c$$","text":"When factoring trinomials of the form $$x^2+bx+c$$, we get an answer $$\\\\left(x+k\\\\right) \\\\left(x+h\\\\right)$$, where k and $$h$$ sum to $$b$$ and multiply to c","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe24a-h2","type":"hint","dependencies":["a0ca101exe24a-h1"],"title":"Factoring the Equation with the Given Rules","text":"To factor into the form $$\\\\left(x+h\\\\right) \\\\left(x+k\\\\right)$$, we could take $$k=16$$ and $$h=3$$, because they sum to $$b=19$$ and multiply to $$c=48$$. We have $$\\\\left(x+16\\\\right) \\\\left(x+3\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ca101exe25","title":"Factoring Trinomials","body":"Factor the trinomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Factor Trinomials with Leading Coefficient 1","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a0ca101exe25a","stepAnswer":["$$\\\\left(x+6\\\\right) \\\\left(x+8\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor the following trinomial: $$x^2+14b+48$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(x+6\\\\right) \\\\left(x+8\\\\right)$$","hints":{"DefaultPathway":[{"id":"a0ca101exe25a-h1","type":"hint","dependencies":[],"title":"Factoring $$x^2+bx+c$$","text":"When factoring trinomials of the form $$x^2+bx+c$$, we get an answer $$\\\\left(x+k\\\\right) \\\\left(x+h\\\\right)$$, where k and $$h$$ sum to $$b$$ and multiply to c","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe25a-h2","type":"hint","dependencies":["a0ca101exe25a-h1"],"title":"Factoring the Equation with the Given Rules","text":"To factor into the form $$\\\\left(x+h\\\\right) \\\\left(x+k\\\\right)$$, we could take $$k=6$$ and $$h=8$$, because they sum to $$b=14$$ and multiply to $$c=48$$. We have $$\\\\left(x+6\\\\right) \\\\left(x+8\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ca101exe26","title":"Factoring Trinomials","body":"Factor the trinomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Factor Trinomials with Leading Coefficient 1","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a0ca101exe26a","stepAnswer":["$$\\\\left(x+5\\\\right) \\\\left(x+20\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor the following trinomial: $$x^2+25x+100$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(x+5\\\\right) \\\\left(x+20\\\\right)$$","hints":{"DefaultPathway":[{"id":"a0ca101exe26a-h1","type":"hint","dependencies":[],"title":"Factoring $$x^2+bx+c$$","text":"When factoring trinomials of the form $$x^2+bx+c$$, we get an answer $$\\\\left(x+k\\\\right) \\\\left(x+h\\\\right)$$, where k and $$h$$ sum to $$b$$ and multiply to c","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe26a-h2","type":"hint","dependencies":["a0ca101exe26a-h1"],"title":"Factoring the Equation with the Given Rules","text":"To factor into the form $$\\\\left(x+h\\\\right) \\\\left(x+k\\\\right)$$, we could take $$k=20$$ and $$h=5$$, because they sum to $$b=25$$ and multiply to $$c=100$$. We have $$\\\\left(x+5\\\\right) \\\\left(x+20\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ca101exe27","title":"Factoring Trinomials","body":"Factor the trinomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Factor Trinomials with Leading Coefficient 1","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a0ca101exe27a","stepAnswer":["$$\\\\left(x-5\\\\right) \\\\left(x+1\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor the following trinomial: $$x^2+25x+100$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(x-5\\\\right) \\\\left(x+1\\\\right)$$","hints":{"DefaultPathway":[{"id":"a0ca101exe27a-h1","type":"hint","dependencies":[],"title":"Factoring $$x^2+bx+c$$","text":"When factoring trinomials of the form $$x^2+bx+c$$, we get an answer $$\\\\left(x+k\\\\right) \\\\left(x+h\\\\right)$$, where k and $$h$$ sum to $$b$$ and multiply to c","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe27a-h2","type":"hint","dependencies":["a0ca101exe27a-h1"],"title":"Factoring the Equation with the Given Rules","text":"To factor into the form $$\\\\left(x+h\\\\right) \\\\left(x+k\\\\right)$$, we could take $$k=-5$$ and $$h=1$$, because they sum to $$b=-4$$ and multiply to $$c=-5$$ We have $$\\\\left(x-5\\\\right) \\\\left(x+1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ca101exe28","title":"Factoring Trinomials","body":"Factor the trinomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Factor Trinomials with Leading Coefficient 1","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a0ca101exe28a","stepAnswer":["$$\\\\left(x-6\\\\right) \\\\left(x+2\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor the following trinomial: $$x^2-4x-12$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(x-6\\\\right) \\\\left(x+2\\\\right)$$","hints":{"DefaultPathway":[{"id":"a0ca101exe28a-h1","type":"hint","dependencies":[],"title":"Factoring $$x^2+bx+c$$","text":"When factoring trinomials of the form $$x^2+bx+c$$, we get an answer $$\\\\left(x+k\\\\right) \\\\left(x+h\\\\right)$$, where k and $$h$$ sum to $$b$$ and multiply to c","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe28a-h2","type":"hint","dependencies":["a0ca101exe28a-h1"],"title":"Factoring the Equation with the Given Rules","text":"To factor into the form $$\\\\left(x+h\\\\right) \\\\left(x+k\\\\right)$$, we could take $$k=-6$$ and $$h=2$$, because they sum to $$b=-4$$ and multiply to $$c=-12$$ We have $$\\\\left(x-6\\\\right) \\\\left(x+2\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ca101exe29","title":"Factoring Trinomials","body":"Factor the trinomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Factor Trinomials with Leading Coefficient 1","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a0ca101exe29a","stepAnswer":["$$\\\\left(x-5\\\\right) \\\\left(x+4\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor the following trinomial: $$x^2-x-20$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(x-5\\\\right) \\\\left(x+4\\\\right)$$","hints":{"DefaultPathway":[{"id":"a0ca101exe29a-h1","type":"hint","dependencies":[],"title":"Factoring $$x^2+bx+c$$","text":"When factoring trinomials of the form $$x^2+bx+c$$, we get an answer $$\\\\left(x+k\\\\right) \\\\left(x+h\\\\right)$$, where k and $$h$$ sum to $$b$$ and multiply to c","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe29a-h2","type":"hint","dependencies":["a0ca101exe29a-h1"],"title":"Factoring the Equation with the Given Rules","text":"To factor into the form $$\\\\left(x+h\\\\right) \\\\left(x+k\\\\right)$$, we could take $$k=-5$$ and $$h=4$$, because they sum to $$b=-1$$ and multiply to $$c=-20$$ We have $$\\\\left(x-5\\\\right) \\\\left(x+4\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ca101exe3","title":"Factor Trinomials of the Form x2 + bx + c","body":"Factor the Trinomial (Give your answer with the larger value first. eg. $$\\\\left(x+2\\\\right) \\\\left(x+1\\\\right)$$ $$2$$ is greater than 1):","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Factor Trinomials with Leading Coefficient 1","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a0ca101exe3a","stepAnswer":["$$(q-4)(q-9)$$"],"problemType":"TextBox","stepTitle":"$$q^2-13q+36$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$(q-4)(q-9)$$","hints":{"DefaultPathway":[{"id":"a0ca101exe3a-h1","type":"hint","dependencies":[],"title":"Find Two Factors","text":"In the form $${ax}^2+bx+c$$, find $$2$$ numbers that multiply to c and add up to $$b$$ (Note: This only works when $$a=1)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe3a-h2","type":"hint","dependencies":["a0ca101exe3a-h1"],"title":"Two Factors","text":"The two factors that match these rules are $$-4$$ and $$-9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe3a-h3","type":"hint","dependencies":["a0ca101exe3a-h2"],"title":"Plug into factored form","text":"The factored form is $$\\\\left(x+y\\\\right) \\\\left(x+z\\\\right)$$. The value of $$x$$ can vary depending on the equation (could be u, $$y$$, f, etc) and the values of $$y$$ and $$z$$ are the two factored numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ca101exe30","title":"Factoring Trinomials","body":"Factor the trinomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Factor Trinomials with Leading Coefficient 1","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a0ca101exe30a","stepAnswer":["$$\\\\left(x-5\\\\right) \\\\left(h+3\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor the following trinomial: $$x^2-2x-15$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(x-5\\\\right) \\\\left(h+3\\\\right)$$","hints":{"DefaultPathway":[{"id":"a0ca101exe30a-h1","type":"hint","dependencies":[],"title":"Factoring $$x^2+bx+c$$","text":"When factoring trinomials of the form $$x^2+bx+c$$, we get an answer $$\\\\left(x+k\\\\right) \\\\left(x+h\\\\right)$$, where k and $$h$$ sum to $$b$$ and multiply to c","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe30a-h2","type":"hint","dependencies":["a0ca101exe30a-h1"],"title":"Factoring the Equation with the Given Rules","text":"To factor into the form $$\\\\left(x+h\\\\right) \\\\left(x+k\\\\right)$$, we could take $$k=-5$$, $$h=3$$, because they sum to $$b=-2$$ and multiply to $$c=-15$$ We have $$\\\\left(x-5\\\\right) \\\\left(h+3\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ca101exe4","title":"Factor Trinomials of the Form x2 + bx + c","body":"Factor the Trinomial (Give your answer with the larger value first. eg. $$\\\\left(x+2\\\\right) \\\\left(x+1\\\\right)$$ $$2$$ is greater than 1):","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Factor Trinomials with Leading Coefficient 1","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a0ca101exe4a","stepAnswer":["$$(y-3)(y-15)$$"],"problemType":"TextBox","stepTitle":"$$y^2-18y+45$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$(y-3)(y-15)$$","hints":{"DefaultPathway":[{"id":"a0ca101exe4a-h1","type":"hint","dependencies":[],"title":"Find Two Factors","text":"In the form $${ax}^2+bx+c$$, find $$2$$ numbers that multiply to c and add up to $$b$$ (Note: This only works when $$a=1)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe4a-h2","type":"hint","dependencies":["a0ca101exe4a-h1"],"title":"Two Factors","text":"The two factors that match these rules are $$-3$$ and $$-15$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe4a-h3","type":"hint","dependencies":["a0ca101exe4a-h2"],"title":"Plug into factored form","text":"The factored form is $$\\\\left(x+y\\\\right) \\\\left(x+z\\\\right)$$. The value of $$x$$ can vary depending on the equation (could be u, $$y$$, f, etc) and the values of $$y$$ and $$z$$ are the two factored numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ca101exe5","title":"Factor Trinomials of the Form x2 + bx + c","body":"Factor the Trinomial (Give your answer with the larger value first. eg. $$\\\\left(x+2\\\\right) \\\\left(x+1\\\\right)$$ $$2$$ is greater than 1):","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Factor Trinomials with Leading Coefficient 1","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a0ca101exe5a","stepAnswer":["$$(m-3)(m-10)$$"],"problemType":"TextBox","stepTitle":"$$m^2-13m+30$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$(m-3)(m-10)$$","hints":{"DefaultPathway":[{"id":"a0ca101exe5a-h1","type":"hint","dependencies":[],"title":"Find Two Factors","text":"In the form $${ax}^2+bx+c$$, find $$2$$ numbers that multiply to c and add up to $$b$$ (Note: This only works when $$a=1)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe5a-h2","type":"hint","dependencies":["a0ca101exe5a-h1"],"title":"Two Factors","text":"The two factors that match these rules are $$-3$$ and $$-10$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe5a-h3","type":"hint","dependencies":["a0ca101exe5a-h2"],"title":"Plug into factored form","text":"The factored form is $$\\\\left(x+y\\\\right) \\\\left(x+z\\\\right)$$. The value of $$x$$ can vary depending on the equation (could be u, $$y$$, f, etc) and the values of $$y$$ and $$z$$ are the two factored numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ca101exe6","title":"Factor Trinomials of the Form x2 + bx + c","body":"Factor the Trinomial (Give your answer with the larger value first. eg. $$\\\\left(x+2\\\\right) \\\\left(x+1\\\\right)$$ $$2$$ is greater than 1):","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Factor Trinomials with Leading Coefficient 1","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a0ca101exe6a","stepAnswer":["$$(x-1)(x-7)$$"],"problemType":"TextBox","stepTitle":"$$x^2-8x+7$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$(x-1)(x-7)$$","hints":{"DefaultPathway":[{"id":"a0ca101exe6a-h1","type":"hint","dependencies":[],"title":"Find Two Factors","text":"In the form $${ax}^2+bx+c$$, find $$2$$ numbers that multiply to c and add up to $$b$$ (Note: This only works when $$a=1)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe6a-h2","type":"hint","dependencies":["a0ca101exe6a-h1"],"title":"Two Factors","text":"The two factors that match these rules are $$-1$$ and $$-7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe6a-h3","type":"hint","dependencies":["a0ca101exe6a-h2"],"title":"Plug into factored form","text":"The factored form is $$\\\\left(x+y\\\\right) \\\\left(x+z\\\\right)$$. The value of $$x$$ can vary depending on the equation (could be u, $$y$$, f, etc) and the values of $$y$$ and $$z$$ are the two factored numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ca101exe7","title":"Factor Trinomials of the Form x2 + bx + c","body":"Factor the Trinomial (Give your answer with the larger value first. eg. $$\\\\left(x+2\\\\right) \\\\left(x+1\\\\right)$$ $$2$$ is greater than 1):","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Factor Trinomials with Leading Coefficient 1","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a0ca101exe7a","stepAnswer":["$$(y-2)(y-3)$$"],"problemType":"TextBox","stepTitle":"$$y^2-5y+6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$(y-2)(y-3)$$","hints":{"DefaultPathway":[{"id":"a0ca101exe7a-h1","type":"hint","dependencies":[],"title":"Find Two Factors","text":"In the form $${ax}^2+bx+c$$, find $$2$$ numbers that multiply to c and add up to $$b$$ (Note: This only works when $$a=1)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe7a-h2","type":"hint","dependencies":["a0ca101exe7a-h1"],"title":"Two Factors","text":"The two factors that match these rules are $$-2$$ and $$-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe7a-h3","type":"hint","dependencies":["a0ca101exe7a-h2"],"title":"Plug into factored form","text":"The factored form is $$\\\\left(x+y\\\\right) \\\\left(x+z\\\\right)$$. The value of $$x$$ can vary depending on the equation (could be u, $$y$$, f, etc) and the values of $$y$$ and $$z$$ are the two factored numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ca101exe8","title":"Factor Trinomials of the Form x2 + bx + c","body":"Factor the Trinomial (Give your answer with the larger value first. eg. $$\\\\left(x+2\\\\right) \\\\left(x+1\\\\right)$$ $$2$$ is greater than 1):","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Factor Trinomials with Leading Coefficient 1","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a0ca101exe8a","stepAnswer":["$$\\\\left(p+6\\\\right) \\\\left(p-1\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$p^2+5p-6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(p+6\\\\right) \\\\left(p-1\\\\right)$$","hints":{"DefaultPathway":[{"id":"a0ca101exe8a-h1","type":"hint","dependencies":[],"title":"Find Two Factors","text":"In the form $${ax}^2+bx+c$$, find $$2$$ numbers that multiply to c and add up to $$b$$ (Note: This only works when $$a=1)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe8a-h2","type":"hint","dependencies":["a0ca101exe8a-h1"],"title":"Two Factors","text":"The two factors that match these rules are $$6$$ and $$-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe8a-h3","type":"hint","dependencies":["a0ca101exe8a-h2"],"title":"Plug into factored form","text":"The factored form is $$\\\\left(x+y\\\\right) \\\\left(x+z\\\\right)$$. The value of $$x$$ can vary depending on the equation (could be u, $$y$$, f, etc) and the values of $$y$$ and $$z$$ are the two factored numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ca101exe9","title":"Factor Trinomials of the Form x2 + bx + c","body":"Factor the Trinomial (Give your answer with the larger value first. eg. $$\\\\left(x+2\\\\right) \\\\left(x+1\\\\right)$$ $$2$$ is greater than 1):","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Factor Trinomials with Leading Coefficient 1","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a0ca101exe9a","stepAnswer":["$$\\\\left(p+7\\\\right) \\\\left(p-1\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$n^2+6n-7$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(p+7\\\\right) \\\\left(p-1\\\\right)$$","hints":{"DefaultPathway":[{"id":"a0ca101exe9a-h1","type":"hint","dependencies":[],"title":"Find Two Factors","text":"In the form $${ax}^2+bx+c$$, find $$2$$ numbers that multiply to c and add up to $$b$$ (Note: This only works when $$a=1)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe9a-h2","type":"hint","dependencies":["a0ca101exe9a-h1"],"title":"Two Factors","text":"The two factors that match these rules are $$7$$ and $$-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ca101exe9a-h3","type":"hint","dependencies":["a0ca101exe9a-h2"],"title":"Plug into factored form","text":"The factored form is $$\\\\left(x+y\\\\right) \\\\left(x+z\\\\right)$$. The value of $$x$$ can vary depending on the equation (could be u, $$y$$, f, etc) and the values of $$y$$ and $$z$$ are the two factored numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0cc26bpoly1","title":"Adding Polynomials","body":"Find the sum of the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a0cc26bpoly1a","stepAnswer":["$$4x^3+20x^2+4x-1$$"],"problemType":"TextBox","stepTitle":"$$12x^2+9x-21+4x^3+8x^2-5x+20$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4x^3+20x^2+4x-1$$","hints":{"DefaultPathway":[{"id":"a0cc26bpoly1a-h1","type":"hint","dependencies":[],"title":"Combining $$x^3$$ Terms","text":"The first step is to combine the terms containing $$x^3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4x^3$$"],"dependencies":["a0cc26bpoly1a-h1"],"title":"Combining $$x^3$$ Terms","text":"How can we combine the $$x^3$$ coefficients?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly1a-h3","type":"hint","dependencies":["a0cc26bpoly1a-h2"],"title":"Combining $$x^3$$ Terms","text":"Since our only $$x^3$$ terms is $$4x^3$$, this is the most simplified version of $$x^3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly1a-h4","type":"hint","dependencies":["a0cc26bpoly1a-h3"],"title":"Combining $$x^2$$ Terms","text":"Next, we can combine the terms with $$x^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly1a-h5","type":"hint","dependencies":["a0cc26bpoly1a-h4"],"title":"Combining $$x^2$$ Terms","text":"How can we simplify the $$2x$$ coefficients?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly1a-h6","type":"hint","dependencies":["a0cc26bpoly1a-h5"],"title":"Combining $$x^2$$ Terms","text":"Our $$x^2$$ terms are $$12x^2$$ and $$8x^2$$. We can simplify the $$x^2$$ terms by adding the coefficients.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly1a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20x^2$$"],"dependencies":["a0cc26bpoly1a-h6"],"title":"Combining $$x^2$$ Terms","text":"What is $$12x^2+8x^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly1a-h8","type":"hint","dependencies":["a0cc26bpoly1a-h7"],"title":"Combining $$x$$ Terms","text":"Now we can simplify the $$x$$ terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly1a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4x$$"],"dependencies":["a0cc26bpoly1a-h8"],"title":"Combining $$x$$ Terms","text":"What is the simplified version of the $$x$$ terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly1a-h10","type":"hint","dependencies":["a0cc26bpoly1a-h9"],"title":"Combining $$x$$ Terms","text":"The $$x$$ terms are $$9x$$ and $$-5x$$. We can simplify the $$x$$ terms by adding the coefficients.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly1a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4x$$"],"dependencies":["a0cc26bpoly1a-h10"],"title":"Combining $$x$$ Terms","text":"What is $$9x-5x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly1a-h12","type":"hint","dependencies":["a0cc26bpoly1a-h11"],"title":"Adding the Constants","text":"The last part we need to simplify is the constants. The constants in this expression are $$-21$$ and $$20$$. By adding these integers, we will have the simplified term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly1a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a0cc26bpoly1a-h12"],"title":"Adding the Constants","text":"What is $$-21+20$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly1a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4x^3+20x^2+4x-1$$"],"dependencies":["a0cc26bpoly1a-h13"],"title":"Simplified Expression","text":"How do we format the new expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly1a-h15","type":"hint","dependencies":["a0cc26bpoly1a-h14"],"title":"Simplified Expression","text":"Since there are no more terms to simplify, we can write the expression as the sum of the simplified terms: $$4x^3+20x^2+4x-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0cc26bpoly10","title":"Expanding the Perfect Squares","body":"Expand the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a0cc26bpoly10a","stepAnswer":["$$16x^2-8x+1$$"],"problemType":"TextBox","stepTitle":"Expand $${\\\\left(4x-1\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16x^2-8x+1$$","hints":{"DefaultPathway":[{"id":"a0cc26bpoly10a-h1","type":"hint","dependencies":[],"title":"Form of the expression","text":"This expression is in the form of $${\\\\left(a-b\\\\right)}^2$$. $${\\\\left(a-b\\\\right)}^2=a^2-2ab+b^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16x^2-8x+1$$"],"dependencies":["a0cc26bpoly10a-h1"],"title":"Substituting","text":"Let $$4x=a$$ and let $$1=b$$. What is $${\\\\left(4x-1\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0cc26bpoly11","title":"Multiplying Binomials Resulting in a Difference of Squares","body":"Expand the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a0cc26bpoly11a","stepAnswer":["$$81x^2-16$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(9x+4\\\\right) \\\\left(9x-4\\\\right)$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$81x^2-16$$","choices":["$$81x^2-16$$","Expanding the expression"],"hints":{"DefaultPathway":[{"id":"a0cc26bpoly11a-h1","type":"hint","dependencies":[],"title":"Difference of Squares","text":"This expression is in the form of $$\\\\left(a+b\\\\right) \\\\left(a-b\\\\right)$$. $$\\\\left(a+b\\\\right) \\\\left(a-b\\\\right)$$ $$=$$ $$a^2-b^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$81x^2-16$$"],"dependencies":["a0cc26bpoly11a-h1"],"title":"Substituting","text":"Let a $$=$$ $$9x$$ and $$b$$ $$=$$ $$4$$. What is $$\\\\left(9x+4\\\\right) \\\\left(9x-4\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0cc26bpoly12","title":"Multiplying Binomials Resulting in a Difference of Squares","body":"Expand the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a0cc26bpoly12a","stepAnswer":["$$3x^2-2xy+17x-8y+20$$"],"problemType":"TextBox","stepTitle":"$$\\\\operatorname{Multiply}\\\\left(x+4\\\\right) \\\\left(3x-2y+5\\\\right)$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3x^2-2xy+17x-8y+20$$","hints":{"DefaultPathway":[{"id":"a0cc26bpoly12a-h1","type":"hint","dependencies":[],"title":"Distributive Property","text":"The first step is to use the distributive property. $$\\\\left(x+4\\\\right) \\\\left(3x-2y+5\\\\right)$$ $$=$$ $$x\\\\left(3x-2y+5\\\\right)+4\\\\left(3x-2y+5\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly12a-h2","type":"hint","dependencies":["a0cc26bpoly12a-h1"],"title":"Multiply","text":"The next step is to multiply those terms. $$x\\\\left(3x-2y+5\\\\right)+4\\\\left(3x-2y+5\\\\right)$$ $$=$$ $$3x^2-2xy+5x+12x-8y+20$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly12a-h3","type":"hint","dependencies":["a0cc26bpoly12a-h2"],"title":"Combining Like Terms","text":"After you multiply, in order to simplify, you need to combine like terms. $$3x^2-2xy+5x+12x-8y+20$$ $$=$$ $$3x^2-2xy+5x+12x-8y+20$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly12a-h4","type":"hint","dependencies":["a0cc26bpoly12a-h3"],"title":"Simplifying","text":"Finally, you need to simplify. $$3x^2-2xy+5x+12x-8y+20$$ $$=$$ $$3x^2-2xy+17x-8y+20$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0cc26bpoly13","title":"Multiplying Binomials Resulting in a Difference of Squares","body":"Expand the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a0cc26bpoly13a","stepAnswer":["$$6x^2+21xy-29x-7y+9$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(3x-1\\\\right) \\\\left(2x+7y-9\\\\right)$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6x^2+21xy-29x-7y+9$$","hints":{"DefaultPathway":[{"id":"a0cc26bpoly13a-h1","type":"hint","dependencies":[],"title":"Distributive Property","text":"The first step is to use the distributive property. $$\\\\left(3x-1\\\\right) \\\\left(2x+7y-9\\\\right)$$ $$=$$ $$3x\\\\left(2x+7y-9\\\\right)-1\\\\left(2x+7y-9\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly13a-h2","type":"hint","dependencies":["a0cc26bpoly13a-h1"],"title":"Multiply","text":"The next step is to multiply those terms. $$3x\\\\left(2x+7y-9\\\\right)-1\\\\left(2x+7y-9\\\\right)=$$ $$6x^2+21x-27x-2x-7y+9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly13a-h3","type":"hint","dependencies":["a0cc26bpoly13a-h2"],"title":"Combining Like Terms","text":"After you multiply, in order to simplify, you need to combine like terms. $$6x^2+21xy-27x-2x-7y+9$$ $$=$$ $$6x^2+21xy-27x+2x-7y+9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly13a-h4","type":"hint","dependencies":["a0cc26bpoly13a-h3"],"title":"Simplifying","text":"Finally, you need to simplify. $$6x^2+21xy-27x+2x-7y+9$$ $$=$$ $$6x^2+21xy-29x-7y+9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0cc26bpoly14","title":"Adding and Subtracting Polynomials","body":"Simplify the expression by adding or subtracting.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a0cc26bpoly14a","stepAnswer":["$$4x^2+3x+19$$"],"problemType":"TextBox","stepTitle":"$$12x^2+3x-8x^2-19$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4x^2+3x+19$$","hints":{"DefaultPathway":[{"id":"a0cc26bpoly14a-h1","type":"hint","dependencies":[],"title":"Distributing the negative sign","text":"The first step is to distribute the negative sign to remove parentheses. $$12x^2+3x-8x^2-19$$ $$=$$ $$12x^2+3x-8x^2+19$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly14a-h2","type":"hint","dependencies":["a0cc26bpoly14a-h1"],"title":"Grouping like terms","text":"The next step is to group like terms. $$12x^2+3x-8x^2+19$$ $$=$$ $$12x^2-8x^2+3x+19$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly14a-h3","type":"hint","dependencies":["a0cc26bpoly14a-h2"],"title":"Combining and simplifying","text":"Finally, simplify the expression. $$12x^2-8x^2+3x+19$$ $$=$$ $$4x^2+3x+19$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0cc26bpoly15","title":"Adding and Subtracting Polynomials","body":"Simplify the expression by adding or subtracting.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a0cc26bpoly15a","stepAnswer":["$$3w^2+30w+21$$"],"problemType":"TextBox","stepTitle":"$$6w^2+24w+24-3w^2-6w+3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3w^2+30w+21$$","hints":{"DefaultPathway":[{"id":"a0cc26bpoly15a-h1","type":"hint","dependencies":[],"title":"Distributing the negative sign","text":"The first step is to distribute the negative sign to remove parentheses. $$6w^2+24w+24-3w^2-6w+3$$ $$=$$ $$6w^2+24w+24-3w^2+6w-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly15a-h2","type":"hint","dependencies":["a0cc26bpoly15a-h1"],"title":"Grouping like terms","text":"The next step is to group like terms. $$6w^2+24w+24-3w^2+6w-3$$ $$=$$ $$6w^2-3w^2+24w+6w+24-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly15a-h3","type":"hint","dependencies":["a0cc26bpoly15a-h2"],"title":"Combining and simplifying","text":"Finally, simplify the expression. $$6w^2-3w^2+24w+6w+24-3$$ $$=$$ $$3w^2+30w+21$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0cc26bpoly16","title":"Adding and Subtracting Polynomials","body":"Simplify the expression by adding or subtracting.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a0cc26bpoly16a","stepAnswer":["$$11b^4-9b^3+12b^2-7b+8$$"],"problemType":"TextBox","stepTitle":"Find the sum or difference. $$11b^4-6b^3+18b^2-4b+8-3b^3+6b^2+3b$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$11b^4-9b^3+12b^2-7b+8$$","hints":{"DefaultPathway":[{"id":"a0cc26bpoly16a-h1","type":"hint","dependencies":[],"title":"Subtracing Polynomials","text":"In this problem you are subtracting one expression from another, so use the distributive property to distribute the negative sign.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly16a-h2","type":"hint","dependencies":["a0cc26bpoly16a-h1"],"title":"Group Like Terms","text":"Find all the $$b^4$$ terms and $$\\\\frac{add}{subtract}$$ the coefficients.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly16a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11$$"],"dependencies":["a0cc26bpoly16a-h2"],"title":"Group Like Terms","text":"What is the coefficient that preceeds $$b^4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly16a-h4","type":"hint","dependencies":["a0cc26bpoly16a-h3"],"title":"Group Like Terms","text":"Find all the $$b^3$$ terms and $$\\\\frac{add}{subtract}$$ the coefficients.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly16a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":["a0cc26bpoly16a-h4"],"title":"Group Like Terms","text":"What is the coefficient that preceeds $$b^3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly16a-h6","type":"hint","dependencies":["a0cc26bpoly16a-h5"],"title":"Group Like Terms","text":"Find all the $$b^2$$ terms and $$\\\\frac{add}{subtract}$$ the coefficients.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly16a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a0cc26bpoly16a-h6"],"title":"Group Like Terms","text":"What is the coefficient that preceeds $$b^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly16a-h8","type":"hint","dependencies":["a0cc26bpoly16a-h7"],"title":"Group Like Terms","text":"Find all the $$b$$ terms and $$\\\\frac{add}{subtract}$$ the coefficients.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly16a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-7$$"],"dependencies":["a0cc26bpoly16a-h8"],"title":"Group Like Terms","text":"What is the coefficient that preceeds $$b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly16a-h10","type":"hint","dependencies":["a0cc26bpoly16a-h9"],"title":"Group Like Terms","text":"Find all the number terms and $$\\\\frac{add}{subtract}$$ them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly16a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a0cc26bpoly16a-h10"],"title":"Group Like Terms","text":"What is the number?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly16a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11b^4-9b^3+12b^2-7b+8$$"],"dependencies":["a0cc26bpoly16a-h11"],"title":"Writing Expressions","text":"Write an expression with all of the combined and simplified terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0cc26bpoly17","title":"Multiplying Polynomials","body":"Find the product of the binomials.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a0cc26bpoly17a","stepAnswer":["$$24x^2-4x-8$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(4x+2\\\\right) \\\\left(6x-4\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$24x^2-4x-8$$","hints":{"DefaultPathway":[{"id":"a0cc26bpoly17a-h1","type":"hint","dependencies":[],"title":"FOIL Method","text":"Use the FOIL (first, outside, inside, last) method to multiply the binomials.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly17a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$24x^2$$"],"dependencies":["a0cc26bpoly17a-h1"],"title":"FOIL Method","text":"What is the product of the first two terms of the binomials?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly17a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-16x$$"],"dependencies":["a0cc26bpoly17a-h2"],"title":"FOIL Method","text":"What is the product of the outside two terms of the binomials?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly17a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12x$$"],"dependencies":["a0cc26bpoly17a-h3"],"title":"FOIL Method","text":"What is the product of the inside two terms of the binomials?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly17a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-8$$"],"dependencies":["a0cc26bpoly17a-h4"],"title":"FOIL Method","text":"What is the product of the last two terms of the binomials?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly17a-h6","type":"hint","dependencies":["a0cc26bpoly17a-h5"],"title":"Combine Like Terms","text":"Add the products and combine like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0cc26bpoly18","title":"Multiplying Polynomials","body":"Find the product of the binomials.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a0cc26bpoly18a","stepAnswer":["$$24b^4-48b^2+24$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(6b^2-6\\\\right) \\\\left(4b^2-4\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$24b^4-48b^2+24$$","hints":{"DefaultPathway":[{"id":"a0cc26bpoly18a-h1","type":"hint","dependencies":[],"title":"FOIL Method","text":"Use the FOIL (first, outside, inside, last) method to multiply the binomials.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$24b^4$$"],"dependencies":["a0cc26bpoly18a-h1"],"title":"FOIL Method","text":"What is the product of the first two terms of the binomials?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly18a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-24b^2$$"],"dependencies":["a0cc26bpoly18a-h2"],"title":"FOIL Method","text":"What is the product of the outside two terms of the binomials?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly18a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-24b^2$$"],"dependencies":["a0cc26bpoly18a-h3"],"title":"FOIL Method","text":"What is the product of the inside two terms of the binomials?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly18a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$24$$"],"dependencies":["a0cc26bpoly18a-h4"],"title":"FOIL Method","text":"What is the product of the last two terms of the binomials?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly18a-h6","type":"hint","dependencies":["a0cc26bpoly18a-h5"],"title":"Combine Like Terms","text":"Add the products and combine like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0cc26bpoly19","title":"Multiplying Polynomials","body":"Find the product of the binomials.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a0cc26bpoly19a","stepAnswer":["$$99v^2-202v+99$$"],"problemType":"TextBox","stepTitle":"$$(9v-11)(11v-9)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$99v^2-202v+99$$","hints":{"DefaultPathway":[{"id":"a0cc26bpoly19a-h1","type":"hint","dependencies":[],"title":"FOIL Method","text":"Use the FOIL (first, outside, inside, last) method to multiply the binomials.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$99v^2$$"],"dependencies":["a0cc26bpoly19a-h1"],"title":"FOIL Method","text":"What is the product of the first two terms of the binomials?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly19a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["-81v"],"dependencies":["a0cc26bpoly19a-h2"],"title":"FOIL Method","text":"What is the product of the outside two terms of the binomials?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly19a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["-121v"],"dependencies":["a0cc26bpoly19a-h3"],"title":"FOIL Method","text":"What is the product of the inside two terms of the binomials?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly19a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$99$$"],"dependencies":["a0cc26bpoly19a-h4"],"title":"FOIL Method","text":"What is the product of the last two terms of the binomials?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly19a-h6","type":"hint","dependencies":["a0cc26bpoly19a-h5"],"title":"Combine Like Terms","text":"Add the products and combine like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0cc26bpoly2","title":"Adding Polynomials","body":"Find the sum of the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a0cc26bpoly2a","stepAnswer":["$$2x^3+7x^2-4x-3$$"],"problemType":"TextBox","stepTitle":"$$2x^3+5x^2-x+1+2x^2-3x-4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2x^3+7x^2-4x-3$$","hints":{"DefaultPathway":[{"id":"a0cc26bpoly2a-h1","type":"hint","dependencies":[],"title":"Combining $$x^3$$ Terms","text":"The first step is to simplify the $$x^3$$ term. However, since, $$2x^3$$ is the only $$x^3$$ term in the expression, this term stays the same.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly2a-h2","type":"hint","dependencies":["a0cc26bpoly2a-h1"],"title":"Combining $$x^2$$ Terms","text":"The next step is to simplify the $$x^2$$ term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly2a-h3","type":"hint","dependencies":["a0cc26bpoly2a-h2"],"title":"Combining $$x^2$$ Terms","text":"We simplify the $$x^2$$ term by combining the coefficients of the values with $$x^2$$. The coefficients with $$x^2$$ are $$5x^2$$ and $$2x^2$$. By adding these values, we simplify the $$x^2$$ term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7x^2$$"],"dependencies":["a0cc26bpoly2a-h3"],"title":"Combining $$x^2$$ Terms","text":"What is $$5x^2+2x^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly2a-h5","type":"hint","dependencies":["a0cc26bpoly2a-h4"],"title":"Combining $$x$$ Terms","text":"Now we need to combine the terms with $$x$$ by adding $$-x$$ and $$-3x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4x$$"],"dependencies":["a0cc26bpoly2a-h5"],"title":"Combining $$x$$ Terms","text":"What is $$-x+\\\\left(-3x\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a0cc26bpoly2a-h6-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":[],"title":"Combining Constants","text":"Finish simplifying the expression by adding the constants. What is $$1-4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a0cc26bpoly2a-h7","type":"hint","dependencies":["a0cc26bpoly2a-h6"],"title":"Simplified Expression","text":"We can finally rewrite the expression as a sum of the simplified terms: $$2x^3+7x^2-4x-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0cc26bpoly20","title":"Multiplying Polynomials","body":"Find the product of the binomials.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a0cc26bpoly20a","stepAnswer":["$$8n^3-4n^2+72n-36$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(8n-4\\\\right) \\\\left(n^2+9\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8n^3-4n^2+72n-36$$","hints":{"DefaultPathway":[{"id":"a0cc26bpoly20a-h1","type":"hint","dependencies":[],"title":"FOIL Method","text":"Use the FOIL (first, outside, inside, last) method to multiply the binomials.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8n^3$$"],"dependencies":["a0cc26bpoly20a-h1"],"title":"FOIL Method","text":"What is the product of the first two terms of the binomials?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly20a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$72n$$"],"dependencies":["a0cc26bpoly20a-h2"],"title":"FOIL Method","text":"What is the product of the outside two terms of the binomials?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly20a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4n^2$$"],"dependencies":["a0cc26bpoly20a-h3"],"title":"FOIL Method","text":"What is the product of the inside two terms of the binomials?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly20a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-36$$"],"dependencies":["a0cc26bpoly20a-h4"],"title":"FOIL Method","text":"What is the product of the last two terms of the binomials?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly20a-h6","type":"hint","dependencies":["a0cc26bpoly20a-h5"],"title":"Combine Like Terms","text":"Add the products and combine like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a0cc26bpoly20b","stepAnswer":["$$9y^2-42y+49$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(3y-7\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9y^2-42y+49$$","hints":{"DefaultPathway":[{"id":"a0cc26bpoly20b-h1","type":"hint","dependencies":[],"title":"Perfect Square Trinomial Formula","text":"When a binomial is squared, the result is a perfect square trinomial. Use the formula: $$x^2+2ax+a^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly20b-h2","type":"hint","dependencies":["a0cc26bpoly20b-h1"],"title":"Perfect Square Trinomial Formula","text":"Square the first term of the binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly20b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9y^2$$"],"dependencies":["a0cc26bpoly20b-h2"],"title":"Perfect Square Trinomial Formula","text":"What is the product?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly20b-h4","type":"hint","dependencies":["a0cc26bpoly20b-h3"],"title":"Perfect Square Trinomial Formula","text":"Square the last term of the binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly20b-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$49$$"],"dependencies":["a0cc26bpoly20b-h4"],"title":"Perfect Square Trinomial Formula","text":"What is the product?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly20b-h6","type":"hint","dependencies":["a0cc26bpoly20b-h5"],"title":"Perfect Square Trinomial Formula","text":"For the middle term of the trinomial, double the product of the two terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly20b-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-42$$"],"dependencies":["a0cc26bpoly20b-h6"],"title":"Perfect Square Trinomial Formula","text":"What is the product?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly20b-h8","type":"hint","dependencies":["a0cc26bpoly20b-h7"],"title":"Combine Like Terms","text":"Add and simplify.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0cc26bpoly21","title":"Expanding the expression","body":"Expand the binomial.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a0cc26bpoly21a","stepAnswer":["$$16p^2+72p+81$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(4p+9\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16p^2+72p+81$$","hints":{"DefaultPathway":[{"id":"a0cc26bpoly21a-h1","type":"hint","dependencies":[],"title":"Perfect Square Trinomial Formula","text":"When a binomial is squared, the result is a perfect square trinomial. Use the formula: $$x^2$$ + 2ax + $$a^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly21a-h2","type":"hint","dependencies":["a0cc26bpoly21a-h1"],"title":"Perfect Square Trinomial Formula","text":"Square the first term of the binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly21a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16p^2$$"],"dependencies":["a0cc26bpoly21a-h2"],"title":"Perfect Square Trinomial Formula","text":"What is the product?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly21a-h4","type":"hint","dependencies":["a0cc26bpoly21a-h3"],"title":"Perfect Square Trinomial Formula","text":"Square the last term of the binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly21a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$81$$"],"dependencies":["a0cc26bpoly21a-h4"],"title":"Perfect Square Trinomial Formula","text":"What is the product?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly21a-h6","type":"hint","dependencies":["a0cc26bpoly21a-h5"],"title":"Perfect Square Trinomial Formula","text":"For the middle term of the trinomial, double the product of the two terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly21a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$72p$$"],"dependencies":["a0cc26bpoly21a-h6"],"title":"Perfect Square Trinomial Formula","text":"What is the product?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly21a-h8","type":"hint","dependencies":["a0cc26bpoly21a-h7"],"title":"Combine Like Terms","text":"Add and simplify.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0cc26bpoly22","title":"Expanding the expression","body":"Expand the binomial.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a0cc26bpoly22a","stepAnswer":["$$9y^2-36y+36$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(3y-6\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9y^2-36y+36$$","hints":{"DefaultPathway":[{"id":"a0cc26bpoly22a-h1","type":"hint","dependencies":[],"title":"Perfect Square Trinomial Formula","text":"When a binomial is squared, the result is a perfect square trinomial. Use the formula: $$x^2+2ax+a^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly22a-h2","type":"hint","dependencies":["a0cc26bpoly22a-h1"],"title":"Perfect Square Trinomial Formula","text":"Square the first term of the binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly22a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9y^2$$"],"dependencies":["a0cc26bpoly22a-h2"],"title":"Perfect Square Trinomial Formula","text":"What is the product?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly22a-h4","type":"hint","dependencies":["a0cc26bpoly22a-h3"],"title":"Perfect Square Trinomial Formula","text":"Square the last term of the binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly22a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$36$$"],"dependencies":["a0cc26bpoly22a-h4"],"title":"Perfect Square Trinomial Formula","text":"What is the product?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly22a-h6","type":"hint","dependencies":["a0cc26bpoly22a-h5"],"title":"Perfect Square Trinomial Formula","text":"For the middle term of the trinomial, double the product of the two terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly22a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-36y$$"],"dependencies":["a0cc26bpoly22a-h6"],"title":"Perfect Square Trinomial Formula","text":"What is the product?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly22a-h8","type":"hint","dependencies":["a0cc26bpoly22a-h7"],"title":"Combine Like Terms","text":"Add and simplify.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0cc26bpoly23","title":"Multiplying Binomials Resulting in a Difference of Squares","body":"Expand the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a0cc26bpoly23a","stepAnswer":["$$4x^2-49$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(2x+7\\\\right) \\\\left(2x-7\\\\right)$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4x^2-49$$","hints":{"DefaultPathway":[{"id":"a0cc26bpoly23a-h1","type":"hint","dependencies":[],"title":"Difference of Squares","text":"This expression is in the form of $$\\\\left(a+b\\\\right) \\\\left(a-b\\\\right)$$. $$\\\\left(a+b\\\\right) \\\\left(a-b\\\\right)$$ $$=$$ $$a^2-b^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly23a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4x^2-49$$"],"dependencies":["a0cc26bpoly23a-h1"],"title":"Substituting","text":"Let $$a=2x$$ and $$b=7$$. What is $$\\\\left(2x+7\\\\right) \\\\left(2x-7\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0cc26bpoly24","title":"Multiplying Polynomials","body":"Expand the polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a0cc26bpoly24a","stepAnswer":["$$16t^4+4t^3-32t^2-t+7$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(4t^2+t-7\\\\right) \\\\left(4t^2-1\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16t^4+4t^3-32t^2-t+7$$","hints":{"DefaultPathway":[{"id":"a0cc26bpoly24a-h1","type":"hint","dependencies":[],"title":"Distributive Property Explanation","text":"The distributive property is defined as when you multiply a number by a sum or difference, you have to multiply each term of the sum or difference by that number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly24a-h2","type":"hint","dependencies":["a0cc26bpoly24a-h1"],"title":"Splitting Sums","text":"Split the $$\\\\frac{sum}{difference}$$ that has the least amount of terms into its individual terms, to use the distributive property with.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly24a-h3","type":"hint","dependencies":["a0cc26bpoly24a-h2"],"title":"Splitting Sums","text":"In this case, we will split the $$4t^2-1$$ into $$4t^2$$ and $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly24a-h4","type":"hint","dependencies":["a0cc26bpoly24a-h3"],"title":"Multiplying Individual Terms","text":"Multiply each term from the split difference to the other $$\\\\frac{sum}{difference}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly24a-h5","type":"hint","dependencies":["a0cc26bpoly24a-h4"],"title":"Multiplying Individual Terms","text":"In this case, we will multiple $$4t^2$$ by $$4t^2+t-7$$ and $$-1$$ by $$4t^2+t-7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly24a-h6","type":"hint","dependencies":["a0cc26bpoly24a-h5"],"title":"Adding Terms","text":"Add all the terms generated after both multiplications and simplify like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly24a-h7","type":"hint","dependencies":["a0cc26bpoly24a-h6"],"title":"Adding Terms","text":"In this case, we will add $$16t^4$$, $$4t^3$$, $$-28t^2$$, $$-4t^2$$, $$-t$$, and $$7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly24a-h8","type":"hint","dependencies":["a0cc26bpoly24a-h7"],"title":"Simplification","text":"Simplify like terms in the sum. Like terms are terms of the same degree.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly24a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16t^4+4t^3-32t^2-t+7$$"],"dependencies":["a0cc26bpoly24a-h8"],"title":"Simplification","text":"After simplification, what is the final polynomial result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0cc26bpoly25","title":"Multiplying Polynomials","body":"Expand the polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a0cc26bpoly25a","stepAnswer":["$$y^3-6y^2-y+18$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(y-2\\\\right) \\\\left(y^2-4y-9\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y^3-6y^2-y+18$$","hints":{"DefaultPathway":[{"id":"a0cc26bpoly25a-h1","type":"hint","dependencies":[],"title":"Distributive Property Explanation","text":"The distributive property is defined as when you multiply a number by a sum or difference, you have to multiply each term of the sum or difference by that number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly25a-h2","type":"hint","dependencies":["a0cc26bpoly25a-h1"],"title":"Splitting Sums","text":"Split the $$\\\\frac{sum}{difference}$$ that has the least amount of terms into its individual terms, to use the distributive property with.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly25a-h3","type":"hint","dependencies":["a0cc26bpoly25a-h2"],"title":"Splitting Sums","text":"In this case, we will split the $$(y-2)$$ into $$y$$ and $$-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly25a-h4","type":"hint","dependencies":["a0cc26bpoly25a-h3"],"title":"Multiplying Individual Terms","text":"Multiply each term from the split difference to the other $$\\\\frac{sum}{difference}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly25a-h5","type":"hint","dependencies":["a0cc26bpoly25a-h4"],"title":"Multiplying Individual Terms","text":"In this case, we will multiple $$y$$ by $$y^2-4y-9$$ and $$-2$$ by $$y^2-4y-9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly25a-h6","type":"hint","dependencies":["a0cc26bpoly25a-h5"],"title":"Adding Terms","text":"Add all the terms generated after both multiplications and simplify like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly25a-h7","type":"hint","dependencies":["a0cc26bpoly25a-h6"],"title":"Adding Terms","text":"In this case, we will add $$y^3$$, $$-4y^2$$, $$-9y$$, $$-2y^2$$, $$8y$$, and $$18$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly25a-h8","type":"hint","dependencies":["a0cc26bpoly25a-h7"],"title":"Simplification","text":"Simplify like terms in the sum. Like terms are terms of the same degree.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly25a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y^3-6y^2-y+18$$"],"dependencies":["a0cc26bpoly25a-h8"],"title":"Simplification","text":"After simplification, what is the final polynomial result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0cc26bpoly26","title":"Multiplying Polynomials","body":"Expand the expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a0cc26bpoly26a","stepAnswer":["$$a^2-b^2$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(a+b\\\\right) \\\\left(a-b\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$a^2-b^2$$","hints":{"DefaultPathway":[{"id":"a0cc26bpoly26a-h1","type":"hint","dependencies":[],"title":"Distributive Property Explanation","text":"The distributive property is defined as when you multiply a number by a sum or difference, you have to multiply each term of the sum or difference by that number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly26a-h2","type":"hint","dependencies":["a0cc26bpoly26a-h1"],"title":"Splitting Sums","text":"Split one of the $$\\\\frac{sums}{differences}$$ into its individual terms, to use the distributive property with.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly26a-h3","type":"hint","dependencies":["a0cc26bpoly26a-h2"],"title":"Splitting Sums","text":"in this case we will split $$a+b$$ into a and $$b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly26a-h4","type":"hint","dependencies":["a0cc26bpoly26a-h3"],"title":"Multiplying Individual Terms","text":"Multiply each term from the split difference to the other $$\\\\frac{sum}{difference}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly26a-h5","type":"hint","dependencies":["a0cc26bpoly26a-h4"],"title":"Multiplying Individual Terms","text":"In this case, we will multiple a by $$(a-b)$$ and $$b$$ by $$(a-b)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly26a-h6","type":"hint","dependencies":["a0cc26bpoly26a-h5"],"title":"Adding Terms","text":"Add all the terms generated after both multiplications and simplify like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly26a-h7","type":"hint","dependencies":["a0cc26bpoly26a-h6"],"title":"Adding Terms","text":"In this case, we will add $$a^2$$, -ab, ab, and $$-\\\\left(b^2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly26a-h8","type":"hint","dependencies":["a0cc26bpoly26a-h7"],"title":"Simplification","text":"Simplify like terms in the sum. Like terms are terms of the same degree.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly26a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$a^2-b^2$$"],"dependencies":["a0cc26bpoly26a-h8"],"title":"Simplification","text":"After simplification, what is the final polynomial result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0cc26bpoly27","title":"Multiplying Polynomials","body":"Expand the polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a0cc26bpoly27a","stepAnswer":["$$4t^2+x^2+4t-5tx-x$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(4t-x\\\\right) \\\\left(t-x+1\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4t^2+x^2+4t-5tx-x$$","hints":{"DefaultPathway":[{"id":"a0cc26bpoly27a-h1","type":"hint","dependencies":[],"title":"Distributive Property Explanation","text":"The distributive property is defined as when you multiply a number by a sum or difference, you have to multiply each term of the sum or difference by that number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly27a-h2","type":"hint","dependencies":["a0cc26bpoly27a-h1"],"title":"Splitting Sums","text":"Split the $$\\\\frac{sum}{difference}$$ that has the least amount of terms into its individual terms, to use the distributive property with.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly27a-h3","type":"hint","dependencies":["a0cc26bpoly27a-h2"],"title":"Splitting Sums","text":"In this case, we will split the $$(4t-x)$$ into $$4t$$ and $$-x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly27a-h4","type":"hint","dependencies":["a0cc26bpoly27a-h3"],"title":"Multiplying Individual Terms","text":"Multiply each term from the split difference to the other $$\\\\frac{sum}{difference}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly27a-h5","type":"hint","dependencies":["a0cc26bpoly27a-h4"],"title":"Multiplying Individual Terms","text":"In this case, we will multiple $$4t$$ by $$t-x+1$$ and $$-x$$ by $$t-x+1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly27a-h6","type":"hint","dependencies":["a0cc26bpoly27a-h5"],"title":"Adding Terms","text":"Add all the terms generated after both multiplications and simplify like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly27a-h7","type":"hint","dependencies":["a0cc26bpoly27a-h6"],"title":"Adding Terms","text":"In this case, we will add $$4t^2$$, -4tx, $$4t$$, -tx, $$x^2$$, and $$-x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly27a-h8","type":"hint","dependencies":["a0cc26bpoly27a-h7"],"title":"Simplification","text":"Simplify like terms in the sum. Like terms are terms of the same degree.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly27a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4t^2+x^2+4t-5tx-x$$"],"dependencies":["a0cc26bpoly27a-h8"],"title":"Simplification","text":"After simplification, what is the final polynomial result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0cc26bpoly28","title":"Multiplying Polynomials","body":"Expand the expression","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a0cc26bpoly28a","stepAnswer":["$$24r^2+22rd-7d^2$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(4r-d\\\\right) \\\\left(6r+7d\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$24r^2+22rd-7d^2$$","hints":{"DefaultPathway":[{"id":"a0cc26bpoly28a-h1","type":"hint","dependencies":[],"title":"Distributive Property Explanation","text":"The distributive property is defined as when you multiply a number by a sum or difference, you have to multiply each term of the sum or difference by that number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly28a-h2","type":"hint","dependencies":["a0cc26bpoly28a-h1"],"title":"Splitting Sums","text":"Split one of the $$\\\\frac{sums}{differences}$$ into its individual terms, to use the distributive property with.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly28a-h3","type":"hint","dependencies":["a0cc26bpoly28a-h2"],"title":"Splitting Sums","text":"In this case, we will split the $$6r+7d$$ into $$6r$$ and $$7d$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly28a-h4","type":"hint","dependencies":["a0cc26bpoly28a-h3"],"title":"Multiplying Individual Terms","text":"Multiply each term from the split difference to the other $$\\\\frac{sum}{difference}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly28a-h5","type":"hint","dependencies":["a0cc26bpoly28a-h4"],"title":"Multiplying Individual Terms","text":"In this case, we will multiple $$6r$$ by $$(4r-d)$$ and $$7d$$ by $$(4r-d)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly28a-h6","type":"hint","dependencies":["a0cc26bpoly28a-h5"],"title":"Adding Terms","text":"Add all the terms generated after both multiplications and simplify like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly28a-h7","type":"hint","dependencies":["a0cc26bpoly28a-h6"],"title":"Adding Terms","text":"In this case, we will add $$24r^2$$, -6rd, 28rd, and $$-7d^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly28a-h8","type":"hint","dependencies":["a0cc26bpoly28a-h7"],"title":"Simplification","text":"Simplify like terms in the sum. Like terms are terms of the same degree.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly28a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$24r^2+22rd-7d^2$$"],"dependencies":["a0cc26bpoly28a-h8"],"title":"Simplification","text":"After simplification, what is the final polynomial result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0cc26bpoly29","title":"Real-World Polynomials","body":"A developer wants to purchase a plot of land to build a house. The area of the plot can be described by the following expression: $$\\\\left(4x+1\\\\right) \\\\left(8x-3\\\\right)$$ where $$x$$ is measured in meters. Multiply the binomials to find the area of the plot in standard form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a0cc26bpoly29a","stepAnswer":["$$32x^2-4x-3$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(4x+1\\\\right) \\\\left(8x-3\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$32x^2-4x-3$$","hints":{"DefaultPathway":[{"id":"a0cc26bpoly29a-h1","type":"hint","dependencies":[],"title":"Distributive Property Explanation","text":"The distributive property is defined as when you multiply a number by a sum or difference, you have to multiply each term of the sum or difference by that number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly29a-h2","type":"hint","dependencies":["a0cc26bpoly29a-h1"],"title":"Splitting Sums","text":"Split one of the $$\\\\frac{sums}{differences}$$ into its individual terms, to use the distributive property with.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly29a-h3","type":"hint","dependencies":["a0cc26bpoly29a-h2"],"title":"Splitting Sums","text":"In this case, we will split the $$4x+1$$ into $$4x$$ and $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly29a-h4","type":"hint","dependencies":["a0cc26bpoly29a-h3"],"title":"Multiplying Individual Terms","text":"Multiply each term from the split difference to the other $$\\\\frac{sum}{difference}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly29a-h5","type":"hint","dependencies":["a0cc26bpoly29a-h4"],"title":"Multiplying Individual Terms","text":"In this case, we will multiple $$4x$$ by $$(8x-3)$$ and $$1$$ by $$(8x-3)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly29a-h6","type":"hint","dependencies":["a0cc26bpoly29a-h5"],"title":"Adding Terms","text":"Add all the terms generated after both multiplications and simplify like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly29a-h7","type":"hint","dependencies":["a0cc26bpoly29a-h6"],"title":"Adding Terms","text":"In this case, we will add $$32x^2$$, $$-12x$$, $$8x$$, and $$-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly29a-h8","type":"hint","dependencies":["a0cc26bpoly29a-h7"],"title":"Simplification","text":"Simplify like terms in the sum. Like terms are terms of the same degree.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly29a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$32x^2-4x-3$$"],"dependencies":["a0cc26bpoly29a-h8"],"title":"Simplification","text":"After simplification, what is the final polynomial result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0cc26bpoly3","title":"Subtracting Polynomials","body":"Find the difference of the following expression","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a0cc26bpoly3a","stepAnswer":["$$7x^4-5x^3+x^2+3x-1$$"],"problemType":"TextBox","stepTitle":"$$7x^4-x^2+6x+1-5x^3-2x^2+3x+2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7x^4-5x^3+x^2+3x-1$$","hints":{"DefaultPathway":[{"id":"a0cc26bpoly3a-h1","type":"hint","dependencies":[],"title":"Distribute the Negative Sign","text":"The first step is to distribute the negative sign to make the second expression negative. This will make the second part of the problem $$-5x^3+2x^3-3x-2$$. Now we can combine like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly3a-h2","type":"hint","dependencies":["a0cc26bpoly3a-h1"],"title":"Grouping Like Terms","text":"$$7x^4$$, $$-5x^2$$, and $$x^2$$ are all the only terms being multiplied by their respective variables, so these cannot be simplified further. However, we can simplify the $$x$$ and constant terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x$$"],"dependencies":["a0cc26bpoly3a-h2"],"title":"Combining $$x$$ Terms","text":"$$6x$$ and $$-3x$$ can be added together to simplify the $$x$$ term. What is $$6x-3x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a0cc26bpoly3a-h3"],"title":"Combining Constants","text":"To get the constant term, we have to add $$1$$ and $$-2$$. What is $$1+\\\\left(-2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly3a-h5","type":"hint","dependencies":["a0cc26bpoly3a-h4"],"title":"Simplified Expression","text":"Since there are no more terms to simplify, we can write the expression as the sum of the simplified terms: $$7x^4$$ $$-5x^3+x^2+3x-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0cc26bpoly30","title":"Polynomial Extensions","body":"Perform the given operations on the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a0cc26bpoly30a","stepAnswer":["$$32t^3-100t^2+40t+38$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(4t-7\\\\right)}^2 \\\\left(2t+1\\\\right)-4t^2+2t+11$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$32t^3-100t^2+40t+38$$","hints":{"DefaultPathway":[{"id":"a0cc26bpoly30a-h1","type":"hint","dependencies":[],"title":"Problem Breakdown","text":"To solve this problem, we will break it up into two parts- the multiplication and the subtraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly30a-h2","type":"hint","dependencies":["a0cc26bpoly30a-h1"],"title":"Distributive Property Explanation","text":"The distributive property is defined as when you multiply a number by a sum or difference, you have to multiply each term of the sum or difference by that number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly30a-h3","type":"hint","dependencies":["a0cc26bpoly30a-h2"],"title":"Solving Products","text":"Solve the most difficult product in the expression in the beginning.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly30a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16t^2-56t+49$$"],"dependencies":["a0cc26bpoly30a-h3"],"title":"Solving Products","text":"In this case, we will solve $${\\\\left(4t-7\\\\right)}^2$$ first. What is it equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly30a-h5","type":"hint","dependencies":["a0cc26bpoly30a-h4"],"title":"Solving Products","text":"Use FOIL (and the distributive property) to multiple by $$4t$$ by $$4t-7$$ and $$-7$$ by $$4t-7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly30a-h6","type":"hint","dependencies":["a0cc26bpoly30a-h5"],"title":"Solving Products","text":"Multiply $$16t^2-56t+49$$ by $$2t+1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly30a-h7","type":"hint","dependencies":["a0cc26bpoly30a-h6"],"title":"Splitting Sums","text":"Split one of the $$\\\\frac{sums}{differences}$$ into its individual terms, to use the distributive property with.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly30a-h8","type":"hint","dependencies":["a0cc26bpoly30a-h7"],"title":"Splitting Sums","text":"In this case, we will split the $$4x+1$$ into $$4x$$ and $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly30a-h9","type":"hint","dependencies":["a0cc26bpoly30a-h8"],"title":"Multiplying Individual Terms","text":"Multiply each term from the split difference to the other $$\\\\frac{sum}{difference}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly30a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$32t^3-100t^2+40t+38$$"],"dependencies":["a0cc26bpoly30a-h9"],"title":"Simplication","text":"Simplify like terms in the sum. Like terms are terms of the same degree. After simplification, what is the final polynomial result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0cc26bpoly31","title":"Polynomial Extensions","body":"Perform the given operations on the expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a0cc26bpoly31a","stepAnswer":["$$a^4+4a^3 c-16a c^3-16c^4$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(a^2+4ac+4c^2\\\\right) \\\\left(a^2-4c^2\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$a^4+4a^3 c-16a c^3-16c^4$$","hints":{"DefaultPathway":[{"id":"a0cc26bpoly31a-h1","type":"hint","dependencies":[],"title":"Distributive Property Explanation","text":"The distributive property is defined as when you multiply a number by a sum or difference, you have to multiply each term of of the sum or difference by that number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly31a-h2","type":"hint","dependencies":["a0cc26bpoly31a-h1"],"title":"Splitting Sums","text":"Split the $$\\\\frac{sum}{difference}$$ that has the least amount of terms into its individual terms, to use the distributive property with.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly31a-h3","type":"hint","dependencies":["a0cc26bpoly31a-h2"],"title":"Splitting Sums","text":"In this case, we will split the $$a^2-4c^2$$ into $$a^2$$ and $$-4c^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly31a-h4","type":"hint","dependencies":["a0cc26bpoly31a-h3"],"title":"Multiplying Individual Terms","text":"Multiply each term from the split difference to the other $$\\\\frac{sum}{difference}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly31a-h5","type":"hint","dependencies":["a0cc26bpoly31a-h4"],"title":"Multiplying Individual Terms","text":"In this case, we will multiple $$a^2$$ by $$a^2+4ac+4c^2$$ and $$-4c^2$$ by $$a^2+4ac+4c^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly31a-h6","type":"hint","dependencies":["a0cc26bpoly31a-h5"],"title":"Adding Terms","text":"Add all the terms generated after both multiplications and simplify like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly31a-h7","type":"hint","dependencies":["a0cc26bpoly31a-h6"],"title":"Adding Terms","text":"In this case, we will add $$a^4$$, $$4a^3 c$$, $$4a^2 c^2$$, $$-4a^2 c^2$$, $$-4{ac}^3$$, and $$-16c^4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly31a-h8","type":"hint","dependencies":["a0cc26bpoly31a-h7"],"title":"Simplification","text":"Simplify like terms in the sum. Like terms are terms of the same degree.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly31a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$a^4+4a^3 c-16a c^3-16c^4$$"],"dependencies":["a0cc26bpoly31a-h8"],"title":"Simplification","text":"After simplification, what is the final polynomial result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0cc26bpoly4","title":"Subtracting Polynomials","body":"Find the difference of the following expression","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a0cc26bpoly4a","stepAnswer":["$$-11x^3-x^2+7x$$ $$-9$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(-7x^3-7x^2+6x-2\\\\right)-4x^3-6x^2-x+7$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-11x^3-x^2+7x$$ $$-9$$","hints":{"DefaultPathway":[{"id":"a0cc26bpoly4a-h1","type":"hint","dependencies":[],"title":"Distribute the Negative Sign","text":"The first step is to distribute the negative sign to make the second expression negative. This will make the second part of the problem $$-4x^3+6x^3+x-7$$. Now we can combine like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly4a-h2","type":"hint","dependencies":["a0cc26bpoly4a-h1"],"title":"Combining $$x^3$$ Terms","text":"$$-7x^3$$ and $$-4x^3$$ are the only terms being multiplied by $$x^3$$, we can add these constants to simplify the $$x^3$$ term. $$-7+\\\\left(-4\\\\right)=-11$$, so the $$x^3$$ term becomes $$-11x^3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly4a-h3","type":"hint","dependencies":["a0cc26bpoly4a-h2"],"title":"Combining $$x^2$$ Terms","text":"Since $$-7x^2$$ and $$6x^2$$ share $$x^2$$, these coefficients can be added to simplify the $$x^2$$ term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-\\\\left(x^2\\\\right)$$"],"dependencies":["a0cc26bpoly4a-h3"],"title":"Combining $$x^2$$ Terms","text":"What is $$-7x^2+6x^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly4a-h5","type":"hint","dependencies":["a0cc26bpoly4a-h4"],"title":"$$x$$ Term","text":"$$6x$$ is the only term being multiplied by $$x$$, so it cannot be simplified further. So, this term stays the same.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly4a-h6","type":"hint","dependencies":["a0cc26bpoly4a-h5"],"title":"Combining Constants","text":"The last term we need to simplify is the constant term. The constants in this expression are $$-2$$ and $$-7$$. By adding these, we can get the simplified constant.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly4a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":["a0cc26bpoly4a-h6"],"title":"Combining Constants","text":"What is $$-2+\\\\left(-7\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly4a-h8","type":"hint","dependencies":["a0cc26bpoly4a-h7"],"title":"Simplified Expression","text":"Since there are no more terms to simplify, we can write the expression as the sum of the simplified terms: $$-11x^3-x^2+7x-9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0cc26bpoly5","title":"Multiplying Polynomials Using the Distributive Property","body":"Find the product of the following expression","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a0cc26bpoly5a","stepAnswer":["$$6x^3+x^2+7x+4$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(2x+1\\\\right) \\\\left(3x^2-x+4\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6x^3+x^2+7x+4$$","hints":{"DefaultPathway":[{"id":"a0cc26bpoly5a-h1","type":"hint","dependencies":[],"title":"Distributive Property","text":"The distributive property states that sum of the factor times each term in the sum is the product of a factor times a sum. So, we can rewrite the equation as the sum of the product of the first term in the first parentheses times the second parentheses plus the second term in the first parentheses time the second parentheses: $$2x\\\\left(3x^2-x-\\\\left(+4\\\\right)\\\\right)+1\\\\left(3x^2-x+4\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly5a-h2","type":"hint","dependencies":["a0cc26bpoly5a-h1"],"title":"Multiplication","text":"The next step is to distribute the $$2x$$. We do not need to distribute the $$1$$ in the second expression since anything multiplied by $$1$$ is itself. Remember: multiply the coefficients and add the exponents.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6x^3-2x^2+8x$$"],"dependencies":["a0cc26bpoly5a-h1"],"title":"Multiplication","text":"What is $$2x\\\\left(3x^2-x+4\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly5a-h4","type":"hint","dependencies":["a0cc26bpoly5a-h3"],"title":"Combine Like Terms","text":"Now we can combine like terms to simplify the expression. The $$x^3$$ and constant terms are in their simplest forms since there are no other like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly5a-h5","type":"hint","dependencies":["a0cc26bpoly5a-h4"],"title":"Combining $$x^2$$ Terms","text":"Since $$-2x^2$$ and $$3x^2$$ share $$x^2$$, these coefficients can be added to simplify the $$x^2$$ term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly5a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2$$"],"dependencies":["a0cc26bpoly5a-h4"],"title":"Combining $$x^2$$ Terms","text":"What is $$-2x^2+3x^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly5a-h7","type":"hint","dependencies":["a0cc26bpoly5a-h5","a0cc26bpoly5a-h6"],"title":"Combining $$x$$ Terms","text":"$$8x$$ and $$-x$$ are the only terms with $$x$$, so we can combine these terms by adding the coefficients","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly5a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7x$$"],"dependencies":["a0cc26bpoly5a-h7"],"title":"Combining $$x$$ Terms","text":"What is $$8x-x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly5a-h9","type":"hint","dependencies":["a0cc26bpoly5a-h8"],"title":"Simplified Expression","text":"Since there are no more terms to simplify, we can write the expression as the sum of the simplified terms: $$6x^3+x^2+7x+4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0cc26bpoly6","title":"Multiplying Polynomials Using the Distributive Property","body":"Find the product of the following expression","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a0cc26bpoly6a","stepAnswer":["$$3x^4-10x^3-8x^2+21x+14$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(3x+2\\\\right) \\\\left(x^3-4x^2+7\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3x^4-10x^3-8x^2+21x+14$$","hints":{"DefaultPathway":[{"id":"a0cc26bpoly6a-h1","type":"hint","dependencies":[],"title":"Distributive Property","text":"The distributive property states that sum of the factor times each term in the sum is the product of a factor times a sum. So, we can rewrite the equation as the sum of the product of the first term in the first parentheses times the second parentheses plus the second term in the first parentheses time the second parentheses: $$3x\\\\left(x^3-4x^2+7\\\\right)+2\\\\left(x^3-4x^2+7\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly6a-h2","type":"hint","dependencies":["a0cc26bpoly6a-h1"],"title":"Distributing the $$3x$$","text":"The next step is to distribute the $$3x$$. Remember: multiply the coefficients and add the exponents.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x^4-12x^3+21x$$"],"dependencies":["a0cc26bpoly6a-h1"],"title":"Distributing the $$3x$$","text":"What is $$3x\\\\left(x^3-4x^2+7\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly6a-h4","type":"hint","dependencies":["a0cc26bpoly6a-h2","a0cc26bpoly6a-h3"],"title":"Distributing the $$2$$","text":"The next step is to distribute the $$3x$$. Remember: multiply the coefficients and add the exponents.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x^3-8x^2+14$$"],"dependencies":["a0cc26bpoly6a-h4"],"title":"Distributing the $$2$$","text":"What is $$2\\\\left(x^3-4x^2+7\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly6a-h6","type":"hint","dependencies":["a0cc26bpoly6a-h5"],"title":"Combine Like Terms","text":"Now we can combine like terms to simplify the expression. The $$x^4$$, $$x^2$$, $$x$$, and constant terms are in their simplest forms since there are no other like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly6a-h7","type":"hint","dependencies":["a0cc26bpoly6a-h6"],"title":"Combining $$x^3$$ Terms","text":"Since $$-12x^3$$ and $$2x^3$$ share $$x^2$$, these coefficients can be added to simplify the $$x^3$$ term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly6a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-10x^3$$"],"dependencies":["a0cc26bpoly6a-h7"],"title":"Combining $$x^3$$ Terms","text":"What is $$-12x^3+2x^3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly6a-h9","type":"hint","dependencies":["a0cc26bpoly6a-h8"],"title":"Simplified Expression","text":"Since there are no more terms to simplify, we can write the expression as the sum of the simplified terms: $$3x^4-10x^3-8x^2+21x+14$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0cc26bpoly7","title":"Using FOIL to Multiply Binomials","body":"Use FOIL to find the product.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a0cc26bpoly7a","stepAnswer":["$$6x^2-48x-54$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(2x-18\\\\right) \\\\left(3x+3\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6x^2-48x-54$$","hints":{"DefaultPathway":[{"id":"a0cc26bpoly7a-h1","type":"hint","dependencies":[],"title":"First Terms","text":"First, find the product of the two first terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6x^2$$"],"dependencies":["a0cc26bpoly7a-h1"],"title":"First Terms","text":"What is $$2x\\\\times3 x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly7a-h3","type":"hint","dependencies":["a0cc26bpoly7a-h2"],"title":"Outside Terms","text":"Now, find the product of the two outside terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6x$$"],"dependencies":["a0cc26bpoly7a-h3"],"title":"Outside Terms","text":"What is $$2x\\\\times3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly7a-h5","type":"hint","dependencies":["a0cc26bpoly7a-h4"],"title":"Inside Terms","text":"Next, find the product of the two inside terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly7a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-54x$$"],"dependencies":["a0cc26bpoly7a-h5"],"title":"Inside Terms","text":"What is $$-18\\\\times3 x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly7a-h7","type":"hint","dependencies":["a0cc26bpoly7a-h6"],"title":"Last Terms","text":"Finally, find the product of the two last terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly7a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-54$$"],"dependencies":["a0cc26bpoly7a-h7"],"title":"Last Terms","text":"What is $$-18\\\\times3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly7a-h9","type":"hint","dependencies":["a0cc26bpoly7a-h8"],"title":"Combine Like Terms","text":"Now that we have the terms foiled out, we can combine like terms. The $$x^2$$ and constant terms cannot be simplified further, but the coeffecients of the $$x$$ terms can be added together to simplify the $$x$$ value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly7a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-48x$$"],"dependencies":["a0cc26bpoly7a-h9"],"title":"Combine Like Terms","text":"What is $$6x-54x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly7a-h11","type":"hint","dependencies":["a0cc26bpoly7a-h10"],"title":"Simplified Expression","text":"The last step is to write out the expression as a sum of all of the values: $$6x^2-48x-54$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0cc26bpoly8","title":"Using FOIL to Multiply Binomials","body":"Use FOIL to find the product.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a0cc26bpoly8a","stepAnswer":["$$3x^2+16x-35$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(x+7\\\\right) \\\\left(3x-5\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3x^2+16x-35$$","hints":{"DefaultPathway":[{"id":"a0cc26bpoly8a-h1","type":"hint","dependencies":[],"title":"First Terms","text":"First, find the product of the two first terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x^2$$"],"dependencies":["a0cc26bpoly8a-h1"],"title":"First Terms","text":"What is $$3x x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly8a-h3","type":"hint","dependencies":["a0cc26bpoly8a-h2"],"title":"Outside Terms","text":"Now, find the product of the two outside terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5x$$"],"dependencies":["a0cc26bpoly8a-h3"],"title":"Outside Terms","text":"What is $$x \\\\left(-5\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly8a-h5","type":"hint","dependencies":["a0cc26bpoly8a-h4"],"title":"Inside Terms","text":"Next, find the product of the two inside terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly8a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$21x$$"],"dependencies":["a0cc26bpoly8a-h5"],"title":"Inside Terms","text":"What is $$7\\\\times3 x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly8a-h7","type":"hint","dependencies":["a0cc26bpoly8a-h6"],"title":"Last Terms","text":"Finally, find the product of the two last terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly8a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-35$$"],"dependencies":["a0cc26bpoly8a-h7"],"title":"Last Terms","text":"What is $$7\\\\left(-5\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly8a-h9","type":"hint","dependencies":["a0cc26bpoly8a-h8"],"title":"Combine Like Terms","text":"Now that we have the terms foiled out, we can combine like terms. The $$x^2$$ and constant terms cannot be simplified further, but the coeffecients of the $$x$$ terms can be added together to simplify the $$x$$ value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly8a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16x$$"],"dependencies":["a0cc26bpoly8a-h9"],"title":"Combine Like Terms","text":"What is $$-5x+21x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly8a-h11","type":"hint","dependencies":["a0cc26bpoly8a-h10"],"title":"Simplified Expression","text":"The last step is to write out the expression as a sum of all of the values: $$3x^2+16x-35$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0cc26bpoly9","title":"Expanding Perfect Squares","body":"Expand the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a0cc26bpoly9a","stepAnswer":["$$9x^2-48x+64$$."],"problemType":"TextBox","stepTitle":"Expand $${\\\\left(3x-8\\\\right)}^2$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9x^2-48x+64$$.","hints":{"DefaultPathway":[{"id":"a0cc26bpoly9a-h1","type":"hint","dependencies":[],"title":"Form of the expression","text":"This expression is in the form of $${\\\\left(a-b\\\\right)}^2$$. $${\\\\left(a-b\\\\right)}^2=a^2-2ab+b^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0cc26bpoly9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9x^2-48x+64$$."],"dependencies":["a0cc26bpoly9a-h1"],"title":"Substituting","text":"Let $$3x=a$$ and let $$8=b$$. What is $${\\\\left(3x-8\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0d19dcInd1","title":"Sampling With or Without Replacement","body":"Determine whether or not the following situation samples with or without replacement.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Independent and Mutually Exclusive Events","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0d19dcInd1a","stepAnswer":["With Replacement"],"problemType":"MultipleChoice","stepTitle":"You have a fair, well-shuffled deck of $$52$$ cards. It consists of four suits. The suits are clubs, diamonds, hearts and spades. There are $$13$$ cards in each suit consisting of $$1$$, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$, $$9$$, $$10$$, J (jack), Q (queen), K (king) of that suit. Suppose you select three cards, The first card you pick out of the $$52$$ cards is the Q of spades. You put this card back, reshuffle the cards and pick a second card from the 52-card deck. It is the ten of clubs. You put this card back, reshuffle the cards and pick a third card from the 52-card deck. This time, the card is the Q of spades again. Your picks are {Q of spades, ten of clubs, Q of spades}.","stepBody":"","answerType":"string","variabilization":{},"choices":["With Replacement","Without Replacement"],"hints":{"DefaultPathway":[{"id":"a0d19dcInd1a-h1","type":"hint","dependencies":[],"title":"Definition of With Replacement","text":"If each member of a population is replaced after it is picked, then that member has the possibility of being chosen more than once. When sampling is done with replacement, then events are considered to be independent, meaning the result of the first pick will not change the probabilities for the second pick.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0d19dcInd1a-h2","type":"hint","dependencies":["a0d19dcInd1a-h1"],"title":"Definition of Without Replacement","text":"When sampling is done without replacement, each member of a population may be chosen only once. In this case, the probabilities for the second pick are affected by the result of the first pick. The events are considered to be dependent or not independent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0d19dcInd10","title":"Mutually Exclusive Events","body":"Determine the probability of the following situation:","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Independent and Mutually Exclusive Events","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0d19dcInd10a","stepAnswer":["$$0.75$$"],"problemType":"TextBox","stepTitle":"Flip two fair coins. What is the probability that we have at most one tail?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.75$$","hints":{"DefaultPathway":[{"id":"a0d19dcInd10a-h1","type":"hint","dependencies":[],"title":"Calculating Probabilities","text":"Find the probability we get heads on one flip.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0d19dcInd10a-h2","type":"hint","dependencies":["a0d19dcInd10a-h1"],"title":"Calculating Probabilities","text":"This problem is equivalent to the situation that we don\'t get {HH}. What is the probability that we get heads in two flips?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0d19dcInd11","title":"Mutually Exclusive Events","body":"Determine the probability of the following situation:","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Independent and Mutually Exclusive Events","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0d19dcInd11a","stepAnswer":["$$0.5$$"],"problemType":"TextBox","stepTitle":"Flip two fair coins. What is the probability that the two faces of our coins are the same? {HH, TT}","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.5$$","hints":{"DefaultPathway":[{"id":"a0d19dcInd11a-h1","type":"hint","dependencies":[],"title":"Calculating Probabilities","text":"Find the probability that we get {HH}.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0d19dcInd11a-h2","type":"hint","dependencies":["a0d19dcInd11a-h1"],"title":"Calculating Probabilities","text":"Find the probability that we get {TT},","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0d19dcInd12","title":"Mutually Exclusive Events","body":"Determine the probability of the following situation:","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Independent and Mutually Exclusive Events","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0d19dcInd12a","stepAnswer":["$$0.5$$"],"problemType":"TextBox","stepTitle":"Flip two fair coins. What is the probability that we get a heads on the first flip, followed by a heads or tails on the second flip?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.5$$","hints":{"DefaultPathway":[{"id":"a0d19dcInd12a-h1","type":"hint","dependencies":[],"title":"Calculating Probabilities","text":"Find the probability we get heads on one flip.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0d19dcInd12a-h2","type":"hint","dependencies":["a0d19dcInd12a-h1"],"title":"Calculating Probabilities","text":"Find the probability we get heads or tails on a single flip.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0d19dcInd13","title":"Mutually Exclusive Events","body":"Determine the probability of the following situation:","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Independent and Mutually Exclusive Events","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0d19dcInd13a","stepAnswer":["$$0.25$$"],"problemType":"TextBox","stepTitle":"Flip two fair coins. What is the probability we get at least one tail, and both of our faces are the same?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.25$$","hints":{"DefaultPathway":[{"id":"a0d19dcInd13a-h1","type":"hint","dependencies":[],"title":"Calculating Probabilities","text":"Find the probability that we get tails on a single flip.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0d19dcInd13a-h2","type":"hint","dependencies":["a0d19dcInd13a-h1"],"title":"Calculating Probabilities","text":"Find the probability that we get two tails on two flips.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0d19dcInd14","title":"Mutually Exclusive Events","body":"Determine the probability of the following situation:","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Independent and Mutually Exclusive Events","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0d19dcInd14a","stepAnswer":["$$0.25$$"],"problemType":"TextBox","stepTitle":"Flip two fair coins. What is the probability that we get two tails?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.25$$","hints":{"DefaultPathway":[{"id":"a0d19dcInd14a-h1","type":"hint","dependencies":[],"title":"Calculating Probabilities","text":"Find the probability we get one tails; use this to find the probability of $$2$$ tails.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0d19dcInd15","title":"Mutually Exclusive Events","body":"Determine if the following situation is mutually exclusive or not:","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Independent and Mutually Exclusive Events","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0d19dcInd15a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Flip two fair coins. Is the situation that we get at most one tail and both of our faces are the same mutually exclusive?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a0d19dcInd15a-h1","type":"hint","dependencies":[],"title":"Mutually Exclusive Definition","text":"A and B are mutually exclusive events if they cannot occur at the same time. This means that A and B do not share any outcomes and P(A AND B) $$=$$ $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0d19dcInd15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.25$$"],"dependencies":["a0d19dcInd15a-h1"],"title":"Calculating Probabilities","text":"What is the probability of our situation occurring?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0d19dcInd16","title":"Mutually Exclusive Events","body":"Determine if the following situation is mutually exclusive or not:","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Independent and Mutually Exclusive Events","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0d19dcInd16a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Flip two fair coins. Is the situation that we get two tails and heads on the first flip followed by heads or tails on the second flip mutually exclusive?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a0d19dcInd16a-h1","type":"hint","dependencies":[],"title":"Mutually Exclusive Definition","text":"A and B are mutually exclusive events if they cannot occur at the same time. This means that A and B do not share any outcomes and P(A AND B) $$=$$ $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0d19dcInd16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a0d19dcInd16a-h1"],"title":"Calculating Probabilities","text":"What is the probability of our situation occurring?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0d19dcInd17","title":"Independent or Not?","body":"Determine if events A and B are independent:","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Independent and Mutually Exclusive Events","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0d19dcInd17a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Let event A $$=$$ learning Spanish. Let event B $$=$$ learning German. Then A AND B $$=$$ learning Spanish and German. Suppose P(A) $$=$$ $$0.4$$ and P(B) $$=$$ $$0.2$$. P(A AND B) $$=$$ $$0.08$$. Are events A and B independent?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a0d19dcInd17a-h1","type":"hint","dependencies":[],"title":"Independent Definition","text":"We can determine whether or not two events are independent by seeing if P(A AND B) $$=$$ P(A)P(B).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0d19dcInd18","title":"Independent or Not?","body":"Determine if events A and B are independent:","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Independent and Mutually Exclusive Events","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0d19dcInd18a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Let event A $$=$$ learning Spanish. Let event B $$=$$ learning German. Then A AND B $$=$$ learning Spanish and German. Suppose P(A) $$=$$ $$0.5$$ and P(B) $$=$$ $$0.5$$. P(A AND B) $$=$$ $$0.20$$. Are events A and B independent?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a0d19dcInd18a-h1","type":"hint","dependencies":[],"title":"Independent Definition","text":"We can determine whether or not two events are independent by seeing if P(A AND B) $$=$$ P(A)P(B).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0d19dcInd19","title":"Independent or Not?","body":"Determine if events A and B are independent:","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Independent and Mutually Exclusive Events","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0d19dcInd19a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Let event A $$=$$ Playing Football. Let event B $$=$$ Playing American Football. Then A AND B $$=$$ Playing Football and American Football. Suppose P(A) $$=$$ $$0.1$$ and P(B) $$=$$ $$0.2$$. P(A AND B) $$=$$ $$0.02$$. Are events A and B independent?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a0d19dcInd19a-h1","type":"hint","dependencies":[],"title":"Independent Definition","text":"We can determine whether or not two events are independent by seeing if P(A AND B) $$=$$ P(A)P(B).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0d19dcInd2","title":"Sampling With or Without Replacement","body":"Determine whether or not the following situation samples with or without replacement.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Independent and Mutually Exclusive Events","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0d19dcInd2a","stepAnswer":["With Replacement"],"problemType":"MultipleChoice","stepTitle":"You have a fair, well-shuffled deck of $$52$$ cards. It consists of four suits. The suits are clubs, diamonds, hearts and spades. There are $$13$$ cards in each suit consisting of $$1$$, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$, $$9$$, $$10$$, J (jack), Q (queen), K (king) of that suit. Three cards are picked at random. Suppose you know that the picked cards are Q of spades, K of hearts and Q of spades. Can you decide if the sampling was with or without replacement?","stepBody":"","answerType":"string","variabilization":{},"choices":["With Replacement","Without Replacement"],"hints":{"DefaultPathway":[{"id":"a0d19dcInd2a-h1","type":"hint","dependencies":[],"title":"Definition of With Replacement","text":"If each member of a population is replaced after it is picked, then that member has the possibility of being chosen more than once. When sampling is done with replacement, then events are considered to be independent, meaning the result of the first pick will not change the probabilities for the second pick.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0d19dcInd2a-h2","type":"hint","dependencies":["a0d19dcInd2a-h1"],"title":"Definition of Without Replacement","text":"When sampling is done without replacement, each member of a population may be chosen only once. In this case, the probabilities for the second pick are affected by the result of the first pick. The events are considered to be dependent or not independent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0d19dcInd20","title":"Independent or Not?","body":"Determine if events A and B are independent:","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Independent and Mutually Exclusive Events","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0d19dcInd20a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Let event A $$=$$ eating Lunch. Let event B $$=$$ eating Dinner. Then A AND B $$=$$ learning Spanish and German. Suppose P(A) $$=$$ $$0.5$$ and P(B) $$=$$ $$0.5$$. P(A AND B) $$=$$ $$0.1$$. Are events A and B independent?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a0d19dcInd20a-h1","type":"hint","dependencies":[],"title":"Independent Definition","text":"We can determine whether or not two events are independent by seeing if P(A AND B) $$=$$ P(A)P(B).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0d19dcInd21","title":"Mutually Exclusive or Not?","body":"Determine if events A and B are mutually exclusive:","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Independent and Mutually Exclusive Events","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0d19dcInd21a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Let event A $$=$$ eating Lunch. Let event B $$=$$ eating Dinner. Then A AND B $$=$$ learning Spanish and German. Suppose P(A) $$=$$ $$0.5$$ and P(B) $$=$$ $$0.5$$. P(A AND B) $$=$$ $$0$$. Are events A and B mutually exclusive?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a0d19dcInd21a-h1","type":"hint","dependencies":[],"title":"Mutually Exclusive Definition","text":"A and B are mutually exclusive if P(A and B) $$=$$ $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0d19dcInd22","title":"Mutually Exclusive or Not?","body":"Determine if events A and B are mutually exclusive:","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Independent and Mutually Exclusive Events","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0d19dcInd22a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Let event A $$=$$ eating Lunch. Let event B $$=$$ eating Dinner. Then A AND B $$=$$ learning Spanish and German. Suppose P(A) $$=$$ $$0.5$$ and P(B) $$=$$ $$0.5$$. P(A AND B) $$=$$ $$0.1$$. Are events A and B mutually exclusive?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a0d19dcInd22a-h1","type":"hint","dependencies":[],"title":"Mutually Exclusive Definition","text":"A and B are mutually exclusive if P(A and B) $$=$$ $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0d19dcInd23","title":"Independent or Not?","body":"Determine if events A and B are independent:","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Independent and Mutually Exclusive Events","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0d19dcInd23a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"In a particular college class, 60% of the students are female. Fifty percent of all students in the class have long hair. Forty-five percent of the students are female and have long hair. Of the female students, 75% have long hair. Let F be the event that a student is female. Let L be the event that a student has long hair. One student is picked randomly. Are the events of being female and having long hair independent?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a0d19dcInd23a-h1","type":"hint","dependencies":[],"title":"Independent Definition","text":"We can determine whether or not two events are independent by seeing if P(A AND B) $$=$$ P(A)P(B).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0d19dcInd24","title":"Probabilities","body":"Determine the probability of the following situation:","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Independent and Mutually Exclusive Events","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0d19dcInd24a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"Mark is deciding which route to take to work. His choices are I $$=$$ the Interstate and F $$=$$ Fifth Street. He knows that P(I) $$=$$ $$.44$$, P(F) $$=$$ $$.56$$, and P(I and F) $$=$$ $$0$$ [since he can only take one route to work]. What is the probability of P(I or F)?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a0d19dcInd24a-h1","type":"hint","dependencies":[],"title":"Mutually Exclusive","text":"It is important to note that since P(I and F) $$=$$ $$0$$, then this means that I and F are mutually exclusive. This is by the definition of being mutually exclusive itself.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0d19dcInd24a-h2","type":"hint","dependencies":["a0d19dcInd24a-h1"],"title":"Mutually Exclusive Rules","text":"If two events are mutually exclusive, lets say I and F, then it follows that P(I or F) $$=$$ P(I) + P(F).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0d19dcInd25","title":"Probabilities","body":"Determine the probability of the following situation:","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Independent and Mutually Exclusive Events","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0d19dcInd25a","stepAnswer":["$$0.3$$"],"problemType":"TextBox","stepTitle":"Let event C $$=$$ taking an English class. Let event D $$=$$ taking a speech class. Suppose P(C) $$=$$ $$0.75$$, P(D) $$=$$ $$0.3$$, P(C|D) $$=$$ $$0.75$$ and P(C AND D) $$=$$ $$0.225$$. What is P(D|C)?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.3$$","hints":{"DefaultPathway":[{"id":"a0d19dcInd25a-h1","type":"hint","dependencies":[],"title":"Given Probabilities","text":"When we see the notation P(D|C), this is essentially asking us \\"What is the probability that D happens, given that C already happened?\\".","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0d19dcInd25a-h2","type":"hint","dependencies":["a0d19dcInd25a-h1"],"title":"Given Probabilities","text":"We can find P(D|C) through this formula: P(D|C) $$=$$ P(D and C)/P(C).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0d19dcInd3","title":"Sampling With or Without Replacement","body":"Determine whether or not the following situation samples with or without replacement.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Independent and Mutually Exclusive Events","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0d19dcInd3a","stepAnswer":["Without Replacement"],"problemType":"MultipleChoice","stepTitle":"You have a fair, well-shuffled deck of $$52$$ cards. It consists of four suits. The suits are clubs, diamonds, hearts and spades. There are $$13$$ cards in each suit consisting of $$1$$, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$, $$9$$, $$10$$, J (jack), Q (queen), K (king) of that suit. Suppose you pick three cards without replacement. The first card you pick out of the $$52$$ cards is the K of hearts. You put this card aside and pick the second card from the $$51$$ cards remaining in the deck. It is the three of diamonds. You put this card aside and pick the third card from the remaining $$50$$ cards in the deck. The third card is the J of spades. Your picks are {K of hearts, three of diamonds, J of spades}.","stepBody":"","answerType":"string","variabilization":{},"choices":["With Replacement","Without Replacement"],"hints":{"DefaultPathway":[{"id":"a0d19dcInd3a-h1","type":"hint","dependencies":[],"title":"Definition of With Replacement","text":"If each member of a population is replaced after it is picked, then that member has the possibility of being chosen more than once. When sampling is done with replacement, then events are considered to be independent, meaning the result of the first pick will not change the probabilities for the second pick.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0d19dcInd3a-h2","type":"hint","dependencies":["a0d19dcInd3a-h1"],"title":"Definition of Without Replacement","text":"When sampling is done without replacement, each member of a population may be chosen only once. In this case, the probabilities for the second pick are affected by the result of the first pick. The events are considered to be dependent or not independent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0d19dcInd4","title":"Sampling With or Without Replacement","body":"Determine whether or not the following situation samples with or without replacement.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Independent and Mutually Exclusive Events","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0d19dcInd4a","stepAnswer":["With Replacement"],"problemType":"MultipleChoice","stepTitle":"You have a fair, well-shuffled deck of $$52$$ cards. It consists of four suits. The suits are clubs, diamonds, hearts, and spades. There are $$13$$ cards in each suit consisting of $$1$$, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$, $$9$$, $$10$$, J (jack), Q (queen), and K (king) of that suit. S $$=$$ spades, H $$=$$ Hearts, D $$=$$ Diamonds, C $$=$$ Clubs. Suppose you pick four cards and put each card back before you pick the next card. Your cards are KH, 7D, 6D, KH.","stepBody":"","answerType":"string","variabilization":{},"choices":["With Replacement","Without Replacement"],"hints":{"DefaultPathway":[{"id":"a0d19dcInd4a-h1","type":"hint","dependencies":[],"title":"Definition of With Replacement","text":"If each member of a population is replaced after it is picked, then that member has the possibility of being chosen more than once. When sampling is done with replacement, then events are considered to be independent, meaning the result of the first pick will not change the probabilities for the second pick.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0d19dcInd4a-h2","type":"hint","dependencies":["a0d19dcInd4a-h1"],"title":"Definition of Without Replacement","text":"When sampling is done without replacement, each member of a population may be chosen only once. In this case, the probabilities for the second pick are affected by the result of the first pick. The events are considered to be dependent or not independent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0d19dcInd5","title":"Sampling With or Without Replacement","body":"Determine whether or not the following situation samples with or without replacement.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Independent and Mutually Exclusive Events","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0d19dcInd5a","stepAnswer":["Without Replacement"],"problemType":"MultipleChoice","stepTitle":"You have a fair, well-shuffled deck of $$52$$ cards. It consists of four suits. The suits are clubs, diamonds, hearts, and spades. There are $$13$$ cards in each suit consisting of $$1$$, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$, $$9$$, $$10$$, J (jack), Q (queen), and K (king) of that suit. S $$=$$ spades, H $$=$$ Hearts, D $$=$$ Diamonds, C $$=$$ Clubs. Suppose you pick four cards and put each card back before you pick the next card. Your cards are KH, 7D, 6D, KH.","stepBody":"","answerType":"string","variabilization":{},"choices":["With Replacement","Without Replacement"],"hints":{"DefaultPathway":[{"id":"a0d19dcInd5a-h1","type":"hint","dependencies":[],"title":"Definition of With Replacement","text":"If each member of a population is replaced after it is picked, then that member has the possibility of being chosen more than once. When sampling is done with replacement, then events are considered to be independent, meaning the result of the first pick will not change the probabilities for the second pick.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0d19dcInd5a-h2","type":"hint","dependencies":["a0d19dcInd5a-h1"],"title":"Definition of Without Replacement","text":"When sampling is done without replacement, each member of a population may be chosen only once. In this case, the probabilities for the second pick are affected by the result of the first pick. The events are considered to be dependent or not independent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0d19dcInd6","title":"Sampling With or Without Replacement","body":"You have a fair, well-shuffled deck of $$52$$ cards. It consists of four suits. The suits are clubs, diamonds, hearts, and spades. There are $$13$$ cards in each suit consisting of $$1$$, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$, $$9$$, $$10$$, J (jack), Q (queen), and K (king) of that suit. S $$=$$ spades, H $$=$$ Hearts, D $$=$$ Diamonds, C $$=$$ Clubs. Suppose that you sample four cards without replacement. Is the following card sample possible?","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Independent and Mutually Exclusive Events","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0d19dcInd6a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"QS, 1D, 1C, QD","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a0d19dcInd6a-h1","type":"hint","dependencies":[],"title":"Definition of With Replacement","text":"If each member of a population is replaced after it is picked, then that member has the possibility of being chosen more than once. When sampling is done with replacement, then events are considered to be independent, meaning the result of the first pick will not change the probabilities for the second pick.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0d19dcInd6a-h2","type":"hint","dependencies":["a0d19dcInd6a-h1"],"title":"Definition of Without Replacement","text":"When sampling is done without replacement, each member of a population may be chosen only once. In this case, the probabilities for the second pick are affected by the result of the first pick. The events are considered to be dependent or not independent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0d19dcInd7","title":"Sampling With or Without Replacement","body":"You have a fair, well-shuffled deck of $$52$$ cards. It consists of four suits. The suits are clubs, diamonds, hearts, and spades. There are $$13$$ cards in each suit consisting of $$1$$, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$, $$9$$, $$10$$, J (jack), Q (queen), and K (king) of that suit. S $$=$$ spades, H $$=$$ Hearts, D $$=$$ Diamonds, C $$=$$ Clubs. Suppose that you sample four cards without replacement. Is the following card sample possible?","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Independent and Mutually Exclusive Events","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0d19dcInd7a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"KH, 7D, 6D, KH","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a0d19dcInd7a-h1","type":"hint","dependencies":[],"title":"Definition of With Replacement","text":"If each member of a population is replaced after it is picked, then that member has the possibility of being chosen more than once. When sampling is done with replacement, then events are considered to be independent, meaning the result of the first pick will not change the probabilities for the second pick.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0d19dcInd7a-h2","type":"hint","dependencies":["a0d19dcInd7a-h1"],"title":"Definition of Without Replacement","text":"When sampling is done without replacement, each member of a population may be chosen only once. In this case, the probabilities for the second pick are affected by the result of the first pick. The events are considered to be dependent or not independent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0d19dcInd8","title":"Sampling With or Without Replacement","body":"You have a fair, well-shuffled deck of $$52$$ cards. It consists of four suits. The suits are clubs, diamonds, hearts, and spades. There are $$13$$ cards in each suit consisting of $$1$$, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$, $$9$$, $$10$$, J (jack), Q (queen), and K (king) of that suit. S $$=$$ spades, H $$=$$ Hearts, D $$=$$ Diamonds, C $$=$$ Clubs. Suppose that you sample four cards with replacement. Is the following card sample possible?","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Independent and Mutually Exclusive Events","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0d19dcInd8a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"KH, KH, KH, KH","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a0d19dcInd8a-h1","type":"hint","dependencies":[],"title":"Definition of With Replacement","text":"If each member of a population is replaced after it is picked, then that member has the possibility of being chosen more than once. When sampling is done with replacement, then events are considered to be independent, meaning the result of the first pick will not change the probabilities for the second pick.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0d19dcInd8a-h2","type":"hint","dependencies":["a0d19dcInd8a-h1"],"title":"Definition of Without Replacement","text":"When sampling is done without replacement, each member of a population may be chosen only once. In this case, the probabilities for the second pick are affected by the result of the first pick. The events are considered to be dependent or not independent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0d19dcInd9","title":"Mutually Exclusive Events","body":"Determine the probability of the following situation:","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Independent and Mutually Exclusive Events","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0d19dcInd9a","stepAnswer":["$$0.75$$"],"problemType":"TextBox","stepTitle":"Draw two cards from a standard 52-card deck with replacement. Find the probability of getting at least one black card.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.75$$","hints":{"DefaultPathway":[{"id":"a0d19dcInd9a-h1","type":"hint","dependencies":[],"title":"Negation","text":"We can find the solution to this problem by finding the probability that we get no black cards when we draw one card.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0d19dcInd9a-h2","type":"hint","dependencies":["a0d19dcInd9a-h1"],"title":"Negation","text":"Now what is the probability that when we draw two cards, neither of them are black? (This is the same probability that we draw two red cards)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0dd0e7corepoly1","title":"Core Functions: Constant, Linear and Quadratic: Part A","body":"These questions test your knowledge of the core concepts. Suppose f is a function whose graph is a straight line passing through the points $$(-3,2)$$ and $$(5,-2)$$.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Core Functions: Constant, Linear, and Quadratic","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a0dd0e7corepoly1a","stepAnswer":["$$\\\\frac{-1}{2}$$"],"problemType":"TextBox","stepTitle":"What is the slope of the straight line?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-1}{2}$$","hints":{"DefaultPathway":[{"id":"a0dd0e7corepoly1a-h1","type":"hint","dependencies":[],"title":"Slope of a Line","text":"The slope of a line can be determined using two points $$(x_1,y_1)$$, $$(x_2,y_2)$$ by subtracting the y-values over the x-values: $$\\\\frac{y_2-y_1}{x_2-x_1}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly1a-h2","type":"hint","dependencies":["a0dd0e7corepoly1a-h1"],"title":"Slope of a Line","text":"Substituting the points $$(-3,2)$$, $$(5,-2)$$, the slope is equal to $$\\\\frac{\\\\left(-2-2\\\\right)}{5-\\\\left(-3\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["a0dd0e7corepoly1a-h2"],"title":"Slope of a Line","text":"What is $$-2-2$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a0dd0e7corepoly1a-h2"],"title":"Slope of a Line","text":"What is $$5-(-3)$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly1a-h5","type":"hint","dependencies":["a0dd0e7corepoly1a-h3","a0dd0e7corepoly1a-h4"],"title":"Slope of a Line","text":"$$\\\\frac{-4}{8}$$ can be simplified to get the slope of the line.","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a0dd0e7corepoly10","title":"Core Functions: Constant, Linear and Quadratic: Part A","body":"These questions test your knowledge of the core concepts. Suppose f is the function $$f(x)=-3x^2+6x+7$$.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Core Functions: Constant, Linear, and Quadratic","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a0dd0e7corepoly10a","stepAnswer":["$$(-\\\\infty,10]$$"],"problemType":"MultipleChoice","stepTitle":"By completing the squre, determine the range of the function.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,10]$$","choices":["$$(-\\\\infty,10]$$","$$(-\\\\infty,7]$$","$$(-\\\\infty,-10]$$","$$(-\\\\infty,-7]$$"],"hints":{"DefaultPathway":[{"id":"a0dd0e7corepoly10a-h1","type":"hint","dependencies":[],"title":"Completing the Square","text":"To complete the square of a quadratic function $$f(x)=a x^2+b x+c$$, first you need to find the greatest common multiple between a,b such that $$f(x)=d \\\\left(\\\\frac{a}{d} x^2+\\\\frac{b}{d} x\\\\right)+c$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a0dd0e7corepoly10a-h1"],"title":"Completing the Square","text":"What is the greatest common multiple of $$3$$ and 6?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly10a-h3","type":"hint","dependencies":["a0dd0e7corepoly10a-h2"],"title":"Completing the Square","text":"Pull out $$3$$ and the negative from the function: $$-3\\\\left(x^2-2x\\\\right)+7$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly10a-h4","type":"hint","dependencies":["a0dd0e7corepoly10a-h3"],"title":"Completing the Square","text":"To find the value to add to $$f(x)=a \\\\left(x^2+b x\\\\right)+c$$, you must add $${\\\\left(\\\\frac{b}{2}\\\\right)}^2$$ within the parantheses. To do so, you must multiply the result by a.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a0dd0e7corepoly10a-h4"],"title":"Completing the Square","text":"What is $${\\\\left(\\\\frac{2}{2}\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly10a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a0dd0e7corepoly10a-h5"],"title":"Completing the Square","text":"What is $$1\\\\left(-3\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly10a-h7","type":"hint","dependencies":["a0dd0e7corepoly10a-h6"],"title":"Completing the Square","text":"Add $$3$$ and subtract $$3$$ to simplify the equation: $$-3\\\\left(x^2-2x\\\\right)-3+3+7$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly10a-h8","type":"hint","dependencies":["a0dd0e7corepoly10a-h7"],"title":"Completing the Square","text":"Move the $$-3$$ within the paranetheses by dividing by three and simplify the constant: $$-3\\\\left(x^2-2x+1\\\\right)+3+7$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly10a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a0dd0e7corepoly10a-h8"],"title":"Completing the Square","text":"What is $$3+7$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly10a-h10","type":"hint","dependencies":["a0dd0e7corepoly10a-h9"],"title":"Completing the Square","text":"For some quadratic $$f(x)=x^2+2b x+b^2$$, it can be simplified to $${\\\\left(x+b\\\\right)}^2$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly10a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${\\\\left(x-1\\\\right)}^2$$"],"dependencies":["a0dd0e7corepoly10a-h10"],"title":"Completing the Square","text":"What is $$x^2-2x+1$$ simplified?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$${\\\\left(x-1\\\\right)}^2$$","$${\\\\left(x+1\\\\right)}^2$$","$${\\\\left(x-2\\\\right)}^2$$","$${\\\\left(x+2\\\\right)}^2$$"]},{"id":"a0dd0e7corepoly10a-h12","type":"hint","dependencies":["a0dd0e7corepoly10a-h11"],"title":"Finding the Range","text":"The range of the function $$f(x)=-3{\\\\left(x-1\\\\right)}^2+10$$ can be found by breaking the function into parts.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly10a-h13","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$[0,\\\\infty)$$"],"dependencies":["a0dd0e7corepoly10a-h12"],"title":"Finding the Range","text":"What is the range of $${\\\\left(x-1\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$[0,\\\\infty)$$","$$[1,\\\\infty)$$","$$[-1,\\\\infty)$$","$$(-\\\\infty,\\\\infty)$$"],"subHints":[{"id":"a0dd0e7corepoly10a-h13-s1","type":"hint","dependencies":[],"title":"Range of $${\\\\left(x+a\\\\right)}^2$$","text":"The range of $${\\\\left(x+a\\\\right)}^2$$ will be $$[0,\\\\infty)$$. a will only shift the equation left and right, changing the domain but not the range.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a0dd0e7corepoly10a-h14","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(-\\\\infty,0]$$"],"dependencies":["a0dd0e7corepoly10a-h13"],"title":"Finding the Range","text":"What is the range of $$-3{\\\\left(x-1\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$[0,\\\\infty)$$","$$(-\\\\infty,0]$$","$$(-\\\\infty,1]$$","$$[1,\\\\infty)$$"],"subHints":[{"id":"a0dd0e7corepoly10a-h14-s1","type":"hint","dependencies":[],"title":"Range of $$-f(x)$$","text":"The range [a,b] for the function f(x) will be $$[-b,-a]$$ for the function $$-f(x)$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a0dd0e7corepoly10a-h15","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(-\\\\infty,10]$$"],"dependencies":["a0dd0e7corepoly10a-h14"],"title":"Finding the Range","text":"What is the range of $$-3{\\\\left(x-1\\\\right)}^2+10$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$(-\\\\infty,10]$$","$$(-\\\\infty,0]$$","$$(-\\\\infty,-10]$$","$$[0,\\\\infty)$$"],"subHints":[{"id":"a0dd0e7corepoly10a-h15-s1","type":"hint","dependencies":[],"title":"Range of $$f{\\\\left(x\\\\right)}+c$$","text":"The range [a,b] for the function f(x) will be [a+c, b+c] for the function $$f{\\\\left(x\\\\right)}+c$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]}]}}]},{"id":"a0dd0e7corepoly100","title":"Core Functions: Constant, Linear and Quadratic: Part B","body":"These problems are generally harder, often highlighting an important subtlety. Every quadratic function $$f(x)=a x^2+b x+c$$ (where a not equal to 0) can be built by applying a composition of transformations to the most basic parabola, $$y=x^2$$. Let\'s explore this with another specific example, $$f(x)=3x^2+12x+7$$.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Core Functions: Constant, Linear, and Quadratic","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a0dd0e7corepoly100a","stepAnswer":["$$f(x)=3{\\\\left(x+2\\\\right)}^2-5$$"],"problemType":"MultipleChoice","stepTitle":"Complete the square to write f in the form $$f(x)=A {\\\\left(x+D\\\\right)}^2+B$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$f(x)=3{\\\\left(x+2\\\\right)}^2-5$$","choices":["$$f(x)=3{\\\\left(x+2\\\\right)}^2-5$$","$$f(x)=3{\\\\left(x-2\\\\right)}^2-5$$","$$f(x)=3{\\\\left(x-2\\\\right)}^2+19$$","$$f(x)=3{\\\\left(x+2\\\\right)}^2+19$$"],"hints":{"DefaultPathway":[{"id":"a0dd0e7corepoly100a-h1","type":"hint","dependencies":[],"title":"Completing the Square","text":"To complete the square of a quadratic function $$f(x)=a x^2+b x+c$$, first you need to find the greatest common multiple between a,b such that $$f(x)=d \\\\left(\\\\frac{a}{d} x^2+\\\\frac{b}{d} x\\\\right)+c$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly100a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a0dd0e7corepoly100a-h1"],"title":"Completing the Square","text":"What is the greatest common multiple of $$3$$ and 12?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly100a-h3","type":"hint","dependencies":["a0dd0e7corepoly100a-h2"],"title":"Completing the Square","text":"Pull out $$3$$ from the function: $$3\\\\left(x^2+4x\\\\right)+7$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly100a-h4","type":"hint","dependencies":["a0dd0e7corepoly100a-h3"],"title":"Completing the Square","text":"To find the value to add to $$f(x)=a \\\\left(x^2+b x\\\\right)+c$$, you must add $${\\\\left(\\\\frac{b}{2}\\\\right)}^2$$ within the parantheses. To do so, you must multiply the result by a.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly100a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a0dd0e7corepoly100a-h4"],"title":"Completing the Square","text":"What is $${\\\\left(\\\\frac{4}{2}\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly100a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a0dd0e7corepoly100a-h5"],"title":"Completing the Square","text":"What is $$4\\\\times3$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly100a-h7","type":"hint","dependencies":["a0dd0e7corepoly100a-h6"],"title":"Completing the Square","text":"Add $$3$$ and subtract $$3$$ to simplify the equation: $$3\\\\left(x^2+4x\\\\right)+12-12+7$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly100a-h8","type":"hint","dependencies":["a0dd0e7corepoly100a-h7"],"title":"Completing the Square","text":"Move the $$12$$ within the paranetheses by dividing by three and simplify the constant: $$3\\\\left(x^2+4x+4\\\\right)-12+7$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly100a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["a0dd0e7corepoly100a-h8"],"title":"Completing the Square","text":"What is $$-12+7$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly100a-h10","type":"hint","dependencies":["a0dd0e7corepoly100a-h9"],"title":"Completing the Square","text":"For some quadratic $$f(x)=x^2+2b x+b^2$$, it can be simplified to $${\\\\left(x+b\\\\right)}^2$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly100a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${\\\\left(x+2\\\\right)}^2$$"],"dependencies":["a0dd0e7corepoly100a-h10"],"title":"Completing the Square","text":"What is $$x^2+4x+4$$ simplified?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$${\\\\left(x-4\\\\right)}^2$$","$${\\\\left(x+4\\\\right)}^2$$","$${\\\\left(x-2\\\\right)}^2$$","$${\\\\left(x+2\\\\right)}^2$$"]}]}}]},{"id":"a0dd0e7corepoly101","title":"Core Functions: Constant, Linear and Quadratic: Part B","body":"These problems are generally harder, often highlighting an important subtlety.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Core Functions: Constant, Linear, and Quadratic","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a0dd0e7corepoly101a","stepAnswer":["$$f(x)=-\\\\left(\\\\frac{1}{2}\\\\right) x+4$$"],"problemType":"MultipleChoice","stepTitle":"Two straight lines are said to be perpendicular if they intersect at a right-angle. It is a fact that this happens if and only if the product of their slopes equals $$-1$$ (assuming neither is vertical). Using this fact, find the formula for the linear function f, whose graph is perpendicular tot he graph of $$g(x)=2x-1$$ and intersects it when $$y=3$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$f(x)=-\\\\left(\\\\frac{1}{2}\\\\right) x+4$$","choices":["$$f(x)=-\\\\left(\\\\frac{1}{2}\\\\right) x+4$$","$$f(x)=-2x+7$$","$$f(x)=\\\\frac{1}{2} x+2$$","$$f(x)=3x-3$$"],"hints":{"DefaultPathway":[{"id":"a0dd0e7corepoly101a-h1","type":"hint","dependencies":[],"title":"Finding the Slope","text":"The slope of the inverse function is when the slope of $$g{\\\\left(x\\\\right)} m=-1$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly101a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a0dd0e7corepoly101a-h1"],"title":"Finding the Slope","text":"What is the slope of $$g(x)=2x-1$$?","variabilization":{},"oer":"https://OATutor.io","license":"","subHints":[{"id":"a0dd0e7corepoly101a-h2-s1","type":"hint","dependencies":[],"title":"Slope of g(x)","text":"The slope of a function $$f(x)=m x+b$$ is $$m$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a0dd0e7corepoly101a-h3","type":"hint","dependencies":["a0dd0e7corepoly101a-h2"],"title":"Finding the Slope","text":"Divide by $$2$$ on both sides to isolate \'m\': $$m=\\\\frac{-1}{2}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly101a-h4","type":"hint","dependencies":["a0dd0e7corepoly101a-h3"],"title":"Finding a Point","text":"The point used to get the perpendicular equation can be down by subtituting the intersection $$y=3$$ into $$g(x)=2x-1$$ to find $$x$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly101a-h5","type":"hint","dependencies":["a0dd0e7corepoly101a-h4"],"title":"Finding a Point","text":"Add one to both sides: $$3+1=2x$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly101a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a0dd0e7corepoly101a-h5"],"title":"Finding a Point","text":"What is $$3+1$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly101a-h7","type":"hint","dependencies":["a0dd0e7corepoly101a-h6"],"title":"Finding a Point","text":"Divide by $$2$$ on both sides: $$\\\\frac{4}{2}=x$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly101a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a0dd0e7corepoly101a-h7"],"title":"Finding a Point","text":"What is $$\\\\frac{4}{2}$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly101a-h9","type":"hint","dependencies":["a0dd0e7corepoly101a-h8"],"title":"Finding a Point","text":"Since $$x=2$$, the point on the graph of g(x) is $$(2,3)$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly101a-h10","type":"hint","dependencies":["a0dd0e7corepoly101a-h9"],"title":"Point-Slope Form","text":"The point slope formula can be used to get the function of a line through the given point: $$y-y_1=m \\\\left(x-x_1\\\\right)$$ where $$m$$ is the slope and $$(x_1,y_1)$$ is the point.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly101a-h11","type":"hint","dependencies":["a0dd0e7corepoly101a-h10"],"title":"Point-Slope Form","text":"Substituing $$m=\\\\frac{-1}{2}$$ and $$(2,3)$$ for the point, the formula is equal to $$y-3=-\\\\left(\\\\frac{1}{2}\\\\right) \\\\left(x-2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly101a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a0dd0e7corepoly101a-h11"],"title":"Point-Slope Form","text":"Distribute $$\\\\frac{-1}{2}$$ to $$x-2$$. What is $$-\\\\left(\\\\frac{1}{2}\\\\right) \\\\left(-2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly101a-h13","type":"hint","dependencies":["a0dd0e7corepoly101a-h12"],"title":"Point-Slope Form","text":"Add $$3$$ to both sides: $$y=-\\\\left(\\\\frac{1}{2}\\\\right) x+1+3$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly101a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a0dd0e7corepoly101a-h13"],"title":"Point-Slope Form","text":"What is $$1+3$$?","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a0dd0e7corepoly1012","title":"Core Functions: Constant, Linear and Quadratic: Part B","body":"These problems are generally harder, often highlighting an important subtlety.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Core Functions: Constant, Linear, and Quadratic","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a0dd0e7corepoly1012a","stepAnswer":["f(0), $$f(-4)$$"],"problemType":"MultipleChoice","stepTitle":"Choose two points that show why the function $$f(x)=2x^2+8x-3$$ does not admit an inverse.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"f(0), $$f(-4)$$","choices":["f(0), $$f(-4)$$","$$f(-2)$$, f(0)","$$f(-1)$$, $$f(-5)$$","f(3), f(7)"],"hints":{"DefaultPathway":[{"id":"a0dd0e7corepoly1012a-h1","type":"hint","dependencies":[],"title":"Understanding the Inverse","text":"A function f can only have an inverse if for every value $$y$$, there is one $$x$$ in its domain. This is also called a one-to-one function.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly1012a-h2","type":"hint","dependencies":["a0dd0e7corepoly1012a-h1"],"title":"Understanding the Inverse","text":"Try plugging in the values to see if two different $$x$$ values get the same $$y$$ value. If they do, then the function cannot be an inverse.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly1012a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a0dd0e7corepoly1012a-h2"],"title":"Solving Functions","text":"What is $$f(-5)=2{\\\\left(-5\\\\right)}^2+8\\\\left(-5\\\\right)-3$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly1012a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a0dd0e7corepoly1012a-h2"],"title":"Solving Functions","text":"What is $$f(-4)=2{\\\\left(-4\\\\right)}^2+8\\\\left(-4\\\\right)-3$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly1012a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-11$$"],"dependencies":["a0dd0e7corepoly1012a-h2"],"title":"Solving Functions","text":"What is $$f(-2)=2{\\\\left(-2\\\\right)}^2+8\\\\left(-2\\\\right)-3$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly1012a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":["a0dd0e7corepoly1012a-h2"],"title":"Solving Functions","text":"What is $$f(-1)=2{\\\\left(-1\\\\right)}^2+8\\\\left(-1\\\\right)-3$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly1012a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a0dd0e7corepoly1012a-h2"],"title":"Solving Functions","text":"What is $$f(0)=2\\\\times0^2+8\\\\times0-3$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly1012a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$39$$"],"dependencies":["a0dd0e7corepoly1012a-h2"],"title":"Solving Functions","text":"What is $$f(3)=2\\\\times3^2+8\\\\times3-3$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly1012a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$151$$"],"dependencies":["a0dd0e7corepoly1012a-h2"],"title":"Solving Functions","text":"What is $$f(7)=2\\\\times7^2+8\\\\times7-3$$?","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a0dd0e7corepoly10b","title":"Core Functions: Constant, Linear and Quadratic: Part A","body":"These questions test your knowledge of the core concepts. Suppose f is the function $$f(x)=-3x^2+6x+7$$.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Core Functions: Constant, Linear, and Quadratic","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a0dd0e7corepoly10ba","stepAnswer":["$$(-\\\\infty,1]$$"],"problemType":"MultipleChoice","stepTitle":"On what interval is the function increasing?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,1]$$","choices":["$$(-\\\\infty,1]$$","$$(-\\\\infty,-1]$$","$$[1,\\\\infty)$$","$$[-1,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"a0dd0e7corepoly10ba-h1","type":"hint","dependencies":[],"title":"Finding the Interval","text":"For $$a<0$$, $$a {\\\\left(x-D\\\\right)}^2+B$$ increases on $$(-\\\\infty,D]$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]}},{"id":"a0dd0e7corepoly10bb","stepAnswer":["$$[1,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"On what interval is the function decreasing?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[1,\\\\infty)$$","choices":["$$(-\\\\infty,1]$$","$$(-\\\\infty,-1]$$","$$[1,\\\\infty)$$","$$[-1,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"a0dd0e7corepoly10bb-h1","type":"hint","dependencies":[],"title":"Finding the Interval","text":"For $$a<0$$, $$a {\\\\left(x-D\\\\right)}^2+B$$ decreases on $$[D,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a0dd0e7corepoly11","title":"Core Functions: Constant, Linear and Quadratic: Part A","body":"These questions test your knowledge of the core concepts.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Core Functions: Constant, Linear, and Quadratic","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a0dd0e7corepoly11a","stepAnswer":["$$A=-1$$"],"problemType":"MultipleChoice","stepTitle":"Find all values of A, such that the graphs of $$f(x)=x^2$$ and $$g(x)=2x+A$$ have a single intersection.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$A=-1$$","choices":["$$A=-1$$","$$A=1$$, $$1$$ $$\\\\pm$$ $$\\\\sqrt{3}$$","$$A=1$$ $$\\\\pm$$ $$\\\\sqrt{3}$$","No possible values"],"hints":{"DefaultPathway":[{"id":"a0dd0e7corepoly11a-h1","type":"hint","dependencies":[],"title":"Solving the Equation","text":"To find the intersection between two functions, you can set them equal to each other: $$x^2=2x+A$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly11a-h2","type":"hint","dependencies":["a0dd0e7corepoly11a-h1"],"title":"Solving the Equation","text":"Move $$2x$$ to the left side by subtracting the right side: $$x^2-2x=A$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly11a-h3","type":"hint","dependencies":["a0dd0e7corepoly11a-h2"],"title":"Solving the Equation","text":"Complete the square for $$x^2-2x$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly11a-h4","type":"hint","dependencies":["a0dd0e7corepoly11a-h3"],"title":"Solving the Equation","text":"To find the value to add to $$x^2+b x$$, you must add $${\\\\left(\\\\frac{b}{2}\\\\right)}^2$$ within the parantheses.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a0dd0e7corepoly11a-h4"],"title":"Solving the Equation","text":"What is $${\\\\left(-\\\\frac{2}{2}\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly11a-h6","type":"hint","dependencies":["a0dd0e7corepoly11a-h5"],"title":"Solving the Equation","text":"Add $$1$$ to both sides to simplify the equation: $$x^2-2x+1=A+1$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly11a-h7","type":"hint","dependencies":["a0dd0e7corepoly11a-h6"],"title":"Solving the Equation","text":"For some quadratic $$f(x)=x^2+2b x+b^2$$, it can be simplified to $${\\\\left(x+b\\\\right)}^2$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly11a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${\\\\left(x-1\\\\right)}^2$$"],"dependencies":["a0dd0e7corepoly11a-h7"],"title":"Solving the Equation","text":"What is $$x^2-2x+1$$ simplified?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$${\\\\left(x-1\\\\right)}^2$$","$${\\\\left(x+1\\\\right)}^2$$","$${\\\\left(x-2\\\\right)}^2$$","$${\\\\left(x+2\\\\right)}^2$$"]},{"id":"a0dd0e7corepoly11a-h9","type":"hint","dependencies":["a0dd0e7corepoly11a-h8"],"title":"Finding One Intersection Point","text":"Given $${\\\\left(x-1\\\\right)}^2=A+1$$, $$A+1$$ could either be $$A+1>0$$, $$A+1<0$$, $$A+1=0$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly11a-h10","type":"hint","dependencies":["a0dd0e7corepoly11a-h9"],"title":"Positive Consant","text":"When $$A+1>0$$, then it is possible to take the square root where $$x-1=$$ $$\\\\pm$$ $$\\\\sqrt{A+1}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly11a-h11","type":"hint","dependencies":["a0dd0e7corepoly11a-h10"],"title":"Positive Consant","text":"Add one to both sides to simplify: $$x=1$$ $$\\\\pm$$ $$\\\\sqrt{A+1}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly11a-h12","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a0dd0e7corepoly11a-h11"],"title":"Positive Consant","text":"Is there exactly one value that would serve as the intersection?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"a0dd0e7corepoly11a-h13","type":"hint","dependencies":["a0dd0e7corepoly11a-h12"],"title":"Negative Consant","text":"When $$A+1<0$$, then it is impossible for $$x$$ to intersect with the line as $${\\\\left(x-1\\\\right)}^2$$ cannot be negative.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly11a-h14","type":"hint","dependencies":["a0dd0e7corepoly11a-h13"],"title":"Zero Constant","text":"When $$A+1=0$$, then it is possible to take the square root where $$x-1=0$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly11a-h15","type":"hint","dependencies":["a0dd0e7corepoly11a-h14"],"title":"Zero Constant","text":"Add $$-1$$ to both sides to simplify: $$x=0-1$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly11a-h16","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a0dd0e7corepoly11a-h15"],"title":"Zero Constant","text":"What is $$0-1$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly11a-h17","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a0dd0e7corepoly11a-h16"],"title":"Zero Constant","text":"Is there exactly one value that would serve as the intersection?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"a0dd0e7corepoly11a-h18","type":"hint","dependencies":["a0dd0e7corepoly11a-h17"],"title":"Finding A","text":"Since there is exactly one intersection when $$A+1=0$$, A can be solved for to get the result.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly11a-h19","type":"hint","dependencies":["a0dd0e7corepoly11a-h18"],"title":"Finding A","text":"Add $$-1$$ to both sides to simplify: $$A=0-1$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly11a-h20","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a0dd0e7corepoly11a-h19"],"title":"Finding A","text":"What is $$0-1$$?","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a0dd0e7corepoly110","title":"Core Functions: Constant, Linear and Quadratic: Part C","body":"These questions are challenging, requiring mastery of each concept and their interrelations.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Core Functions: Constant, Linear, and Quadratic","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a0dd0e7corepoly110a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"A quadratic function f has a graph with vertex $$(2,-1)$$. Is it possible that the graph has x-intercepts $$-1$$ and 4?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a0dd0e7corepoly110a-h1","type":"hint","dependencies":[],"title":"Symetmetry of Quadratics","text":"Quadratics have the same output left and right of the vertex. So, if there is a vertex V, then $$f(v-c)=f{\\\\left(v+c\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly110a-h2","type":"hint","dependencies":["a0dd0e7corepoly110a-h1"],"title":"Symetmetry of Quadratics","text":"Since $$V=2$$, then $$f(2-c)=f{\\\\left(2+c\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly110a-h3","type":"hint","dependencies":["a0dd0e7corepoly110a-h2"],"title":"Solving the Offset","text":"Set the value of $$x$$ to be one of the supposed x-intercepts: $$2+c=-1$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly110a-h4","type":"hint","dependencies":["a0dd0e7corepoly110a-h3"],"title":"Solving the Offset","text":"Subtract $$2$$ from both sides: $$c=-1-2$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly110a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a0dd0e7corepoly110a-h4"],"title":"Solving the Offset","text":"What is $$-1-2$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly110a-h6","type":"hint","dependencies":["a0dd0e7corepoly110a-h5"],"title":"Finding the Other X-Intercept","text":"Since c is $$-3$$, the other x-intercept can be found by using the opposite equation: $$f{\\\\left(2+3\\\\right)}=f(-1)$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly110a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a0dd0e7corepoly110a-h6"],"title":"Finding the Other X-Intercept","text":"What is $$2+3$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly110a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a0dd0e7corepoly110a-h7"],"title":"Finding the Other X-Intercept","text":"Is $$5$$ equal to the other x-intercept of 4?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]}]}}]},{"id":"a0dd0e7corepoly111","title":"Core Functions: Constant, Linear and Quadratic: Part C","body":"These questions are challenging, requiring mastery of each concept and their interrelations.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Core Functions: Constant, Linear, and Quadratic","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a0dd0e7corepoly111a","stepAnswer":["$$[-10,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"Calculate the range of the following function $$f(x)=x^4-6x^2-1$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[-10,\\\\infty)$$","choices":["$$[-10,\\\\infty)$$","$$[-1,\\\\infty)$$","$$(-\\\\infty,10]$$","$$(-\\\\infty,1]$$"],"hints":{"DefaultPathway":[{"id":"a0dd0e7corepoly111a-h1","type":"hint","dependencies":[],"title":"Rewriting the Expression","text":"The equation can be rewritten in the form of a quadratic by subtituting $$a=x^2$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly111a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$a^2-6a-1$$"],"dependencies":["a0dd0e7corepoly111a-h1"],"title":"Rewriting the Expression","text":"What equation represents $$f(x)=x^4-6x^2-1$$ in terms of a?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$a^2-6a-1$$","$$a^4-6a-1$$","$$a^2-6a^2-1$$","$$a^4-6a^2-1$$"]},{"id":"a0dd0e7corepoly111a-h3","type":"hint","dependencies":["a0dd0e7corepoly111a-h2"],"title":"Completing the Square","text":"To find the value to add to $$f(x)=x^2+b x+c$$, you must add $${\\\\left(\\\\frac{b}{2}\\\\right)}^2$$ within the parantheses.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly111a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a0dd0e7corepoly111a-h3"],"title":"Completing the Square","text":"What is $${\\\\left(-\\\\frac{6}{2}\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly111a-h5","type":"hint","dependencies":["a0dd0e7corepoly111a-h4"],"title":"Completing the Square","text":"Add $$9$$ and subtract $$9$$ to simplify the equation: $$a^2-6a+9-9-1$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly111a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-10$$"],"dependencies":["a0dd0e7corepoly111a-h5"],"title":"Completing the Square","text":"What is $$-9-1$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly111a-h7","type":"hint","dependencies":["a0dd0e7corepoly111a-h6"],"title":"Completing the Square","text":"For some quadratic $$f(x)=x^2+2b x+b^2$$, it can be simplified to $${\\\\left(x+b\\\\right)}^2$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly111a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${\\\\left(a-3\\\\right)}^2$$"],"dependencies":["a0dd0e7corepoly111a-h7"],"title":"Completing the Square","text":"What is $$a^2-6a+9$$ simplified?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$${\\\\left(a-3\\\\right)}^2$$","$${\\\\left(a-6\\\\right)}^2$$","$${\\\\left(a+3\\\\right)}^2$$","$${\\\\left(a+6\\\\right)}^2$$"]},{"id":"a0dd0e7corepoly111a-h9","type":"hint","dependencies":["a0dd0e7corepoly111a-h8"],"title":"Rewriting the Expression","text":"Subtitute $$x^2$$ back in for the final result: $$f(x)={\\\\left(x^2-3\\\\right)}^2-10$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly111a-h10","type":"hint","dependencies":["a0dd0e7corepoly111a-h9"],"title":"Finding the Range","text":"The range of the function $$f(x)={\\\\left(x^2-3\\\\right)}^2-10$$ can be found by breaking the function into parts.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly111a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$[0,\\\\infty)$$"],"dependencies":["a0dd0e7corepoly111a-h10"],"title":"Finding the Range","text":"What is the range of $${\\\\left(x^2-3\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$[0,\\\\infty)$$","$$[1,\\\\infty)$$","$$[-1,\\\\infty)$$","$$(-\\\\infty,\\\\infty)$$"],"subHints":[{"id":"a0dd0e7corepoly111a-h11-s1","type":"hint","dependencies":[],"title":"Range of $${\\\\left(x+a\\\\right)}^2$$","text":"The range of $${\\\\left(x+a\\\\right)}^2$$ will be $$[0,\\\\infty)$$. a will only shift the equation left and right, changing the domain but not the range.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a0dd0e7corepoly111a-h12","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$[-10,\\\\infty)$$"],"dependencies":["a0dd0e7corepoly111a-h11"],"title":"Finding the Range","text":"What is the range of $${\\\\left(x^2-3\\\\right)}^2-10$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$(-\\\\infty,10]$$","$$[-10,\\\\infty)$$","$$(-\\\\infty,-10]$$","$$[10,\\\\infty)$$"],"subHints":[{"id":"a0dd0e7corepoly111a-h12-s1","type":"hint","dependencies":[],"title":"Range of $$f{\\\\left(x\\\\right)}+c$$","text":"The range [a,b] for the function f(x) will be [a+c, b+c] for the function $$f{\\\\left(x\\\\right)}+c$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]}]}}]},{"id":"a0dd0e7corepoly112","title":"Core Functions: Constant, Linear and Quadratic: Part C","body":"These questions are challenging, requiring mastery of each concept and their interrelations.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Core Functions: Constant, Linear, and Quadratic","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a0dd0e7corepoly112a","stepAnswer":["$$(-\\\\infty,-1)$$ $$\\\\cup$$ $$(-1,\\\\frac{-\\\\sqrt{2}}{2})$$ $$\\\\cup$$ $$(\\\\frac{-\\\\sqrt{2}}{2},\\\\frac{\\\\sqrt{2}}{2})$$ $$\\\\cup$$ $$(\\\\frac{\\\\sqrt{2}}{2},1)$$ $$\\\\cup$$ $$(1,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"Calculate the domain of the following function $$f(x)=\\\\frac{1}{2x^4-3x^2+1}$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,-1)$$ $$\\\\cup$$ $$(-1,\\\\frac{-\\\\sqrt{2}}{2})$$ $$\\\\cup$$ $$(\\\\frac{-\\\\sqrt{2}}{2},\\\\frac{\\\\sqrt{2}}{2})$$ $$\\\\cup$$ $$(\\\\frac{\\\\sqrt{2}}{2},1)$$ $$\\\\cup$$ $$(1,\\\\infty)$$","choices":["$$(-\\\\infty,-1)$$ $$\\\\cup$$ $$(-1,\\\\frac{-\\\\sqrt{2}}{2})$$ $$\\\\cup$$ $$(\\\\frac{-\\\\sqrt{2}}{2},\\\\frac{\\\\sqrt{2}}{2})$$ $$\\\\cup$$ $$(\\\\frac{\\\\sqrt{2}}{2},1)$$ $$\\\\cup$$ $$(1,\\\\infty)$$","$$(-\\\\infty,-1)$$ $$\\\\cup$$ $$(-1,\\\\frac{-1}{2})$$ $$\\\\cup$$ $$(\\\\frac{-1}{2},\\\\frac{1}{2})$$ $$\\\\cup$$ $$(\\\\frac{1}{2},1)$$ $$\\\\cup$$ $$(1,\\\\infty)$$","$$(-\\\\infty,\\\\frac{1}{2})$$ $$\\\\cup$$ $$(\\\\frac{1}{2},1)$$ $$\\\\cup$$ $$(1,\\\\infty)$$","$$(-\\\\infty,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"a0dd0e7corepoly112a-h1","type":"hint","dependencies":[],"title":"Bounds of a Function","text":"A function is undefined when for some $$\\\\frac{1}{x}$$, $$x$$ not equal to $$0$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly112a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2x^4-3x^2+1 \\\\neq 0$$"],"dependencies":["a0dd0e7corepoly112a-h1"],"title":"Bounds of a Function","text":"Since $$f(x)=\\\\frac{1}{2x^4-3x^2+1}$$, what inequality represents the restriction on a function\'s domain?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$2x^4-3x^2+1 \\\\neq 0$$","$$\\\\frac{1}{2x^4-3x^2+1} \\\\neq 0$$"]},{"id":"a0dd0e7corepoly112a-h3","type":"hint","dependencies":["a0dd0e7corepoly112a-h2"],"title":"Rewriting the Expression","text":"The equation can be rewritten in the form of a quadratic by subtituting $$w=x^2$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly112a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2w^2-3w+1$$"],"dependencies":["a0dd0e7corepoly112a-h3"],"title":"Rewriting the Expression","text":"What equation represents $$f(x)=2x^4-3x^2+1$$ in terms of w?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$2w^2-3w+1$$","$$2w^4-3w+1$$","$$2w^2-3w^2+1$$","$$2w^4-3w^2+1$$"]},{"id":"a0dd0e7corepoly112a-h5","type":"hint","dependencies":["a0dd0e7corepoly112a-h4"],"title":"Finding the x-intercepts","text":"The x-intercepts (when $$x=0)$$ can be found via $$x=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4a c}\\\\right)}{2} a$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly112a-h6","type":"hint","dependencies":["a0dd0e7corepoly112a-h5"],"title":"Finding the x-intercepts","text":"Substitute the values within $$2w^2-3w+1$$ to determine the x-intercepts: $$w=2\\\\frac{3\\\\pm \\\\sqrt{{\\\\left(-3\\\\right)}^2-4\\\\times2\\\\times1}}{2}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly112a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a0dd0e7corepoly112a-h6"],"title":"Finding the x-intercepts","text":"What is $${\\\\left(-3\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly112a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a0dd0e7corepoly112a-h6"],"title":"Finding the x-intercepts","text":"What is $$4\\\\times2\\\\times1$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly112a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a0dd0e7corepoly112a-h6"],"title":"Finding the x-intercepts","text":"What is $$2\\\\times2$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly112a-h10","type":"hint","dependencies":["a0dd0e7corepoly112a-h7","a0dd0e7corepoly112a-h8","a0dd0e7corepoly112a-h9"],"title":"Finding the x-intercepts","text":"So far, $$w=\\\\frac{3\\\\pm\\\\sqrt{9-8}}{4}.$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly112a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a0dd0e7corepoly112a-h10"],"title":"Finding the x-intercepts","text":"What is $$9-8$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly112a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a0dd0e7corepoly112a-h11"],"title":"Finding the x-intercepts","text":"What is $$\\\\sqrt{1}$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly112a-h13","type":"hint","dependencies":["a0dd0e7corepoly112a-h12"],"title":"Finding the x-intercepts","text":"Split w into two parts to solve the different intercepts: $$\\\\frac{3+1}{4}$$ and $$\\\\frac{3-1}{4}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly112a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a0dd0e7corepoly112a-h13"],"title":"Finding the x-intercepts","text":"What is $$\\\\frac{3+1}{4}$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly112a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a0dd0e7corepoly112a-h13"],"title":"Finding the x-intercepts","text":"What is $$\\\\frac{3-1}{4}$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly112a-h16","type":"hint","dependencies":["a0dd0e7corepoly112a-h14","a0dd0e7corepoly112a-h15"],"title":"Finding the x-intercepts","text":"Since w=1,1/2, substitute back in $$x^2$$ to find the answer in terms of $$x$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly112a-h17","type":"hint","dependencies":["a0dd0e7corepoly112a-h16"],"title":"Finding the x-intercepts","text":"Take the square root of both sides to get the result: $$x=$$ $$\\\\pm$$ $$\\\\sqrt{1}$$, $$x=$$ $$\\\\pm$$ $$\\\\sqrt{\\\\frac{1}{2}}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly112a-h18","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\pm$$ $$1$$"],"dependencies":["a0dd0e7corepoly112a-h17"],"title":"Finding the x-intercepts","text":"Simplify $$x=$$ $$\\\\pm$$ $$\\\\sqrt{1}$$.","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\pm$$ $$1$$","$$1$$","$$-1$$","$$\\\\sqrt{1}$$"]},{"id":"a0dd0e7corepoly112a-h19","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\pm$$ $$\\\\frac{\\\\sqrt{2}}{2}$$"],"dependencies":["a0dd0e7corepoly112a-h17"],"title":"Finding the x-intercepts","text":"Simplify $$x=$$ $$\\\\pm$$ $$\\\\sqrt{\\\\frac{1}{2}}$$.","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\pm$$ $$\\\\frac{\\\\sqrt{2}}{2}$$","$$\\\\frac{-\\\\sqrt{2}}{2}$$","$$\\\\frac{\\\\sqrt{2}}{2}$$","$$\\\\sqrt{\\\\frac{1}{2}}$$"]},{"id":"a0dd0e7corepoly112a-h20","type":"hint","dependencies":["a0dd0e7corepoly112a-h18","a0dd0e7corepoly112a-h19"],"title":"Finding the Bounds","text":"The domain in interval notation can be determined since $$x$$ is not equal to $$1,-1,\\\\frac{\\\\sqrt{2}}{2},-\\\\frac{\\\\sqrt{2}}{2}.$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly112a-h21","type":"hint","dependencies":["a0dd0e7corepoly112a-h20"],"title":"Finding the Bounds","text":"For some value c between [a,b] where $$x$$ not equal to c, the interval would be [a,c) $$\\\\cup$$ (c,b].","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly112a-h22","type":"hint","dependencies":["a0dd0e7corepoly112a-h21"],"title":"Finding the Bounds","text":"If there are multiple values $$x$$ cannot equal, say c and $$d$$ where $$c>d$$, the same process can be applied to the interval [a,b]: [a,d) $$\\\\cup$$ (d,c) $$\\\\cup$$ (c,b]","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a0dd0e7corepoly1b","title":"Core Functions: Constant, Linear and Quadratic: Part A","body":"These questions test your knowledge of the core concepts. Suppose f is a function whose graph is a straight line passing through the points $$(-3,2)$$ and $$(5,-2)$$.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Core Functions: Constant, Linear, and Quadratic","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a0dd0e7corepoly1ba","stepAnswer":["$$f(x)=-\\\\left(\\\\frac{1}{2}\\\\right) x+\\\\frac{1}{2}$$"],"problemType":"MultipleChoice","stepTitle":"Find an explicit formula for f(x).","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$f(x)=-\\\\left(\\\\frac{1}{2}\\\\right) x+\\\\frac{1}{2}$$","choices":["$$f(x)=-\\\\left(\\\\frac{1}{2}\\\\right) x+\\\\frac{1}{2}$$","$$f(x)=2x-8$$","$$f(x)=-\\\\left(\\\\frac{1}{2}\\\\right) x-\\\\frac{1}{2}$$","$$f(x)=2x+8$$"],"hints":{"DefaultPathway":[{"id":"a0dd0e7corepoly1ba-h1","type":"hint","dependencies":[],"title":"Slope of a Line","text":"The slope of a line can be determined using two points $$(x_1,y_1)$$, $$(x_2,y_2)$$ by subtracting the y-values over the x-values: $$\\\\frac{y_2-y_1}{x_2-x_1}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly1ba-h2","type":"hint","dependencies":["a0dd0e7corepoly1ba-h1"],"title":"Slope of a Line","text":"Substituting the points $$(-3,2)$$, $$(5,-2)$$, the slope is equal to $$\\\\frac{\\\\left(-2-2\\\\right)}{5-\\\\left(-3\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly1ba-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["a0dd0e7corepoly1ba-h2"],"title":"Slope of a Line","text":"What is $$-2-2$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly1ba-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a0dd0e7corepoly1ba-h2"],"title":"Slope of a Line","text":"What is $$5-(-3)$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly1ba-h5","type":"hint","dependencies":["a0dd0e7corepoly1ba-h3","a0dd0e7corepoly1ba-h4"],"title":"Slope of a Line","text":"$$\\\\frac{-4}{8}$$ can be simplified to get the slope of the line: $$\\\\frac{-1}{2}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly1ba-h6","type":"hint","dependencies":["a0dd0e7corepoly1ba-h5"],"title":"Point-Slope Form","text":"The point slope formula can be used to get the function of a line through the given point: $$y-y_1=m \\\\left(x-x_1\\\\right)$$ where $$m$$ is the slope and $$(x_1,y_1)$$ is the point.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly1ba-h7","type":"hint","dependencies":["a0dd0e7corepoly1ba-h6"],"title":"Point-Slope Form","text":"Substituing $$m=\\\\frac{-1}{2}$$ and $$(-3,2)$$ for the point, the formula is equal to $$y-2=-\\\\left(\\\\frac{1}{2}\\\\right) \\\\left(x+3\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly1ba-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-3}{2}$$"],"dependencies":["a0dd0e7corepoly1ba-h7"],"title":"Point-Slope Form","text":"Distribute $$\\\\frac{-1}{2}$$ to $$x+3$$. What is $$3-\\\\left(\\\\frac{1}{2}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly1ba-h9","type":"hint","dependencies":["a0dd0e7corepoly1ba-h8"],"title":"Point-Slope Form","text":"Add $$2$$ from the left to get $$y=-\\\\left(\\\\frac{1}{2}\\\\right) x-\\\\frac{3}{2}+2$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly1ba-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a0dd0e7corepoly1ba-h9"],"title":"Point-Slope Form","text":"What is $$\\\\frac{-3}{2}+2$$?","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a0dd0e7corepoly2","title":"Core Functions: Constant, Linear and Quadratic: Part A","body":"These questions test your knowledge of the core concepts. Suppose f is a function whose graph is a straight line passing through the points $$(-3,2)$$ and $$(5,-2)$$.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Core Functions: Constant, Linear, and Quadratic","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a0dd0e7corepoly2a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"What is the x-intercept?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a0dd0e7corepoly2a-h1","type":"hint","dependencies":[],"title":"Finding the x-intercept","text":"The x-intercept can be found by plugging in $$y=0$$ for the formula and solving for $$x$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly2a-h2","type":"hint","dependencies":["a0dd0e7corepoly2a-h1"],"title":"Finding the x-intercept","text":"Plugging in $$y=0$$ means $$0=-\\\\left(\\\\frac{1}{2}\\\\right) x+\\\\frac{1}{2}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly2a-h3","type":"hint","dependencies":["a0dd0e7corepoly2a-h2"],"title":"Finding the x-intercept","text":"Subtract $$\\\\frac{1}{2}$$ from the right to get $$\\\\frac{-1}{2}=-\\\\left(\\\\frac{1}{2}\\\\right) x$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly2a-h4","type":"hint","dependencies":["a0dd0e7corepoly2a-h3"],"title":"Finding the x-intercept","text":"Divide $$\\\\frac{-1}{2}$$ on both sides to isolate \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly2a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a0dd0e7corepoly2a-h4"],"title":"Finding the x-intercept","text":"What is $$\\\\frac{\\\\left(-\\\\frac{1}{2}\\\\right)}{\\\\left(-\\\\frac{1}{2}\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a0dd0e7corepoly211","title":"Core Functions: Constant, Linear and Quadratic: Part B","body":"These problems are generally harder, often highlighting an important subtlety. Every quadratic function $$f(x)=a x^2+b x+c$$ (where a not equal to 0) can be built by applying a composition of transformations to the most basic parabola, $$y=x^2$$. Let\'s explore this with another specific example, $$f(x)=3x^2+12x+7$$.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Core Functions: Constant, Linear, and Quadratic","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a0dd0e7corepoly211a","stepAnswer":["Translate left by $$2$$. Stretch vertically by $$3$$. Translate down by $$5$$"],"problemType":"MultipleChoice","stepTitle":"Give a sequence of three basic transformations whose composition transforms the graph of $$y=x^2$$ into $$y=f(x)$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Translate left by $$2$$. Stretch vertically by $$3$$. Translate down by $$5$$","choices":["Translate left by $$2$$. Stretch vertically by $$3$$. Translate down by $$5$$","Translate right by $$2$$. Stretch vertically by $$3$$. Translate up by $$7$$","Translate left by $$4$$. Compress vertically by $$3$$. Translate up by $$5$$","Translate right by $$4$$. Compress vertically by $$3$$. Translate down by $$7$$"],"hints":{"DefaultPathway":[{"id":"a0dd0e7corepoly211a-h1","type":"hint","dependencies":[],"title":"Understanding $$f(x-a)$$","text":"For some value a, $$f(x-a)$$ means that f(x) is translated to the right by a units. Similarly, $$f{\\\\left(x+a\\\\right)}$$ means that f(x) is translated to the left by a units.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly211a-h2","type":"hint","dependencies":["a0dd0e7corepoly211a-h1"],"title":"Understanding $$f(x-a)$$","text":"Since a is $$-2$$, $$f{\\\\left(x+2\\\\right)}$$ means that the graph is translated left by $$2$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly211a-h3","type":"hint","dependencies":["a0dd0e7corepoly211a-h2"],"title":"Understanding $$a f{\\\\left(x\\\\right)}$$","text":"For some value a not equal to $$0$$, $$a f{\\\\left(x\\\\right)}$$ means that f(x) is stretched vertically by a. Similarly, $$\\\\frac{f{\\\\left(x\\\\right)}}{a}$$ means that f(x) is compressed vertically by a.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly211a-h4","type":"hint","dependencies":["a0dd0e7corepoly211a-h3"],"title":"Understanding $$a f{\\\\left(x\\\\right)}$$","text":"Since a is $$3$$, $$3f{\\\\left(x+2\\\\right)}$$ means that the graph is stretched vertically by $$3$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly211a-h5","type":"hint","dependencies":["a0dd0e7corepoly211a-h4"],"title":"Understanding $$f{\\\\left(x\\\\right)}+a$$","text":"For some value a, $$f{\\\\left(x\\\\right)}+a$$ means that f(x) is translated up by a units. Similarly, $$f(x)-a$$ means that f(x) is translated down by a units.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a0dd0e7corepoly211a-h6","type":"hint","dependencies":["a0dd0e7corepoly211a-h5"],"title":"Understanding $$f{\\\\left(x\\\\right)}+a$$","text":"Since a is $$-5$$, $$3f{\\\\left(x+2\\\\right)}-5$$ means that the graph is translated down by $$5$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a0ec58fterminology1","title":"Understanding Probability Terminology","body":"The sample space S is the whole numbers starting at one and less than $$20$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Terminology","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0ec58fterminology1a","stepAnswer":["{1, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$, $$9$$, $$10$$, $$11$$, $$12$$, $$13$$, $$14$$, $$15$$, $$16$$, $$17$$, $$18$$, 19}"],"problemType":"MultipleChoice","stepTitle":"What is the sample space, S?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"{1, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$, $$9$$, $$10$$, $$11$$, $$12$$, $$13$$, $$14$$, $$15$$, $$16$$, $$17$$, $$18$$, 19}","choices":["{1, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$, $$9$$, $$10$$, $$11$$, $$12$$, $$13$$, $$14$$, $$15$$, $$16$$, $$17$$, $$18$$, 19}","{1, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$, $$9$$, $$10$$, $$11$$, $$12$$, $$13$$, $$14$$, $$15$$, $$16$$, $$17$$, $$18$$, $$19$$, 20}","{0, $$1$$, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$, $$9$$, $$10$$, $$11$$, $$12$$, $$13$$, $$14$$, $$15$$, $$16$$, $$17$$, $$18$$, 19}","{0, $$1$$, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$, $$9$$, $$10$$, $$11$$, $$12$$, $$13$$, $$14$$, $$15$$, $$16$$, $$17$$, $$18$$, $$19$$, 20}"],"hints":{"DefaultPathway":[{"id":"a0ec58fterminology1a-h1","type":"hint","dependencies":[],"title":"Definition of Sample Space","text":"The sample space of an experiment is the set of all possible outcomes. In this scenario, we can represent each outcome as one of each of the whole numbers that starts at one and is less than $$20$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology1a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["{1, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$, $$9$$, $$10$$, $$11$$, $$12$$, $$13$$, $$14$$, $$15$$, $$16$$, $$17$$, $$18$$, 19}"],"dependencies":["a0ec58fterminology1a-h1"],"title":"Determining the Sample Space","text":"Knowing the definition of a sample space, what then, is the sample space of whole numbers starting at one and less than 20? What is the list of numbers that starts at one and all of which are less than 20?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["{1, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$, $$9$$, $$10$$, $$11$$, $$12$$, $$13$$, $$14$$, $$15$$, $$16$$, $$17$$, $$18$$, 19}","{1, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$, $$9$$, $$10$$, $$11$$, $$12$$, $$13$$, $$14$$, $$15$$, $$16$$, $$17$$, $$18$$, $$19$$, 20}","{0, $$1$$, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$, $$9$$, $$10$$, $$11$$, $$12$$, $$13$$, $$14$$, $$15$$, $$16$$, $$17$$, $$18$$, 19}","{0, $$1$$, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$, $$9$$, $$10$$, $$11$$, $$12$$, $$13$$, $$14$$, $$15$$, $$16$$, $$17$$, $$18$$, $$19$$, 20}"]}]}},{"id":"a0ec58fterminology1b","stepAnswer":["{2, $$4$$, $$6$$, $$8$$, $$10$$, $$12$$, $$14$$, $$16$$, 18}"],"problemType":"MultipleChoice","stepTitle":"Let event A $$=$$ the even numbers and event B $$=$$ numbers greater than $$13$$. What is the list of events that best represents the combination of outcomes in A?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"{2, $$4$$, $$6$$, $$8$$, $$10$$, $$12$$, $$14$$, $$16$$, 18}","choices":["{2, $$4$$, $$6$$, $$8$$, $$10$$, $$12$$, $$14$$, $$16$$, 18}","{14, $$15$$, $$16$$, $$17$$, $$18$$, 19}","{13, $$14$$, $$15$$, $$16$$, $$17$$, $$18$$, 19}","{2, $$4$$, $$6$$, $$8$$, $$10$$, $$12$$, $$14$$, $$16$$, $$18$$, 20}"],"hints":{"DefaultPathway":[{"id":"a0ec58fterminology1b-h3","type":"hint","dependencies":["a0ec58fterminology1a-h2"],"title":"Determining Events","text":"The list of outcomes for an event A is all the possible outcomes that occur under that event. In this scenario, we can represent each outcome as one of each of the whole numbers that starts at one and is less than $$20$$ AND is even.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology1b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["{2, $$4$$, $$6$$, $$8$$, $$10$$, $$12$$, $$14$$, $$16$$, 18}"],"dependencies":["a0ec58fterminology1b-h3"],"title":"Determining the Outcomes","text":"Knowing the definition of an event, what then, is the list of possible outcomes of whole numbers starting at one and less than $$20$$ that are even? What are the outcomes that together represent A?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["{2, $$4$$, $$6$$, $$8$$, $$10$$, $$12$$, $$14$$, $$16$$, 18}","{14, $$15$$, $$16$$, $$17$$, $$18$$, 19}","{13, $$14$$, $$15$$, $$16$$, $$17$$, $$18$$, 19}","{2, $$4$$, $$6$$, $$8$$, $$10$$, $$12$$, $$14$$, $$16$$, $$18$$, 20}"]}]}},{"id":"a0ec58fterminology1c","stepAnswer":["{14, $$15$$, $$16$$, $$17$$, $$18$$, 19}"],"problemType":"MultipleChoice","stepTitle":"Let event A $$=$$ the even numbers and event B $$=$$ numbers greater than $$13$$. What is the list of events that best represents the combination of outcomes in B?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"{14, $$15$$, $$16$$, $$17$$, $$18$$, 19}","choices":["{2, $$4$$, $$6$$, $$8$$, $$10$$, $$12$$, $$14$$, $$16$$, 18}","{14, $$15$$, $$16$$, $$17$$, $$18$$, 19}","{13, $$14$$, $$15$$, $$16$$, $$17$$, $$18$$, 19}","{2, $$4$$, $$6$$, $$8$$, $$10$$, $$12$$, $$14$$, $$16$$, $$18$$, 20}"],"hints":{"DefaultPathway":[{"id":"a0ec58fterminology1c-h5","type":"hint","dependencies":["a0ec58fterminology1b-h4"],"title":"Determining Events","text":"The list of outcomes for an event B is all the possible outcomes that occur under that event. In this scenario, we can represent each outcome as one of each of the whole numbers that starts at one and is less than $$20$$ AND is greater than $$13$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology1c-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["{14, $$15$$, $$16$$, $$17$$, $$18$$, 19}"],"dependencies":["a0ec58fterminology1c-h5"],"title":"Determining the Outcomes","text":"Knowing the definition of an event, what then, is the list of possible outcomes of whole numbers starting at one and less than $$20$$ that are greater than 13? What are the outcomes that together represent B?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["{2, $$4$$, $$6$$, $$8$$, $$10$$, $$12$$, $$14$$, $$16$$, 18}","{14, $$15$$, $$16$$, $$17$$, $$18$$, 19}","{13, $$14$$, $$15$$, $$16$$, $$17$$, $$18$$, 19}","{2, $$4$$, $$6$$, $$8$$, $$10$$, $$12$$, $$14$$, $$16$$, $$18$$, 20}"]}]}},{"id":"a0ec58fterminology1d","stepAnswer":["$$\\\\frac{9}{19}$$"],"problemType":"TextBox","stepTitle":"What is the probability of event A occurring, P(A)?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{9}{19}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology1d-h7","type":"hint","dependencies":["a0ec58fterminology1c-h6"],"title":"Definition of Probability","text":"The probability of any outcome in the sample space occuring is the long-term relative frequency of that oucome. Effectively, to find the probability of an event, P(event), we need to find the size of that event (how many outcomes are in it) and divide that by the size of the sample space (how many outcomes are in it).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology1d-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a0ec58fterminology1d-h7"],"title":"Size of A","text":"What is the size of A? Effectively, how many numbers are from one to $$19$$ are even?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology1d-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$19$$"],"dependencies":["a0ec58fterminology1d-h8"],"title":"Size of Sample Space","text":"What is the size of the sample space? Effectively, how many total whole numbers are there between $$1$$ and $$19$$ inclusive?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology1d-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{9}{19}$$"],"dependencies":["a0ec58fterminology1d-h9"],"title":"Determining P(A)","text":"What is P(A), or what is the size of A divided by the size of the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a0ec58fterminology1e","stepAnswer":["$$\\\\frac{6}{19}$$"],"problemType":"TextBox","stepTitle":"What is the probability of event B occurring, P(B)?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{6}{19}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology1e-h11","type":"hint","dependencies":["a0ec58fterminology1d-h10"],"title":"Definition of Probability","text":"The probability of any outcome in the sample space occuring is the long-term relative frequency of that oucome. Effectively, to find the probability of an event, P(event), we need to find the size of that event (how many outcomes are in it) and divide that by the size of the sample space (how many outcomes are in it).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology1e-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a0ec58fterminology1e-h11"],"title":"Size of B","text":"What is the size of B? Effectively, how many numbers are from one to $$19$$ are greater than 13?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology1e-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$19$$"],"dependencies":["a0ec58fterminology1e-h12"],"title":"Size of Sample Space","text":"What is the size of the sample space? Effectively, how many total whole numbers are there between $$1$$ and $$19$$ inclusive?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology1e-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{6}{19}$$"],"dependencies":["a0ec58fterminology1e-h13"],"title":"Determining P(B)","text":"What is P(B), or what is the size of B divided by the size of the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a0ec58fterminology1f","stepAnswer":["$$\\\\frac{3}{19}$$"],"problemType":"TextBox","stepTitle":"What is P(A AND B)?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{19}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology1f-h15","type":"hint","dependencies":["a0ec58fterminology1e-h14"],"title":"Determine the Outcomes","text":"To determine the probability of an event occurring, we want to first determine what the possible outcomes that represent the space are.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology1f-h16","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["{14, $$16$$, 18}"],"dependencies":["a0ec58fterminology1f-h15"],"title":"Determining A AND B","text":"What is the list of outcomes that exist in A AND B? Effectively, which numbers between $$1$$ and $$19$$ inclusive are both even AND greater than 13?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["{14, $$16$$, 18}","{15, $$17$$, 19}","{2, $$4$$, $$6$$, $$8$$, $$10$$, $$12$$, $$14$$, $$15$$, $$16$$, $$17$$, $$18$$, 19}","{2, $$4$$, $$6$$, $$8$$, $$10$$, $$12$$, $$14$$, $$16$$, 18}"]},{"id":"a0ec58fterminology1f-h17","type":"hint","dependencies":["a0ec58fterminology1f-h16"],"title":"Definition of Probability","text":"The probability of any outcome in the sample space occuring is the long-term relative frequency of that oucome. Effectively, to find the probability of an event, P(event), we need to find the size of that event (how many outcomes are in it) and divide that by the size of the sample space (how many outcomes are in it).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology1f-h18","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a0ec58fterminology1f-h17"],"title":"Size of A AND B","text":"What is the size of A AND B? Effectively, how many numbers are from one to $$19$$ are even AND greater than 13?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology1f-h19","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$19$$"],"dependencies":["a0ec58fterminology1f-h18"],"title":"Size of Sample Space","text":"What is the size of the sample space? Effectively, how many total whole numbers are there between $$1$$ and $$19$$ inclusive?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology1f-h20","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{19}$$"],"dependencies":["a0ec58fterminology1f-h19"],"title":"Determining P(A AND B)","text":"What is P(A AND B), or what is the size of A AND B divided by the size of the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a0ec58fterminology1g","stepAnswer":["$$\\\\frac{12}{19}$$"],"problemType":"TextBox","stepTitle":"What is P(A OR B)?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{12}{19}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology1g-h15","type":"hint","dependencies":["a0ec58fterminology1e-h14"],"title":"Determine the Outcomes","text":"To determine the probability of an event occurring, we want to first determine what the possible outcomes that represent the space are.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology1g-h16","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["{2, $$4$$, $$6$$, $$8$$, $$10$$, $$12$$, $$14$$, $$15$$, $$16$$, $$17$$, $$18$$, 19}"],"dependencies":["a0ec58fterminology1g-h15"],"title":"Determining A OR B","text":"What is the list of outcomes that exist in A OR B? Effectively, which numbers between $$1$$ and $$19$$ inclusive are at least one of even OR greater than 13?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["{14, $$16$$, 18}","{15, $$17$$, 19}","{2, $$4$$, $$6$$, $$8$$, $$10$$, $$12$$, $$14$$, $$15$$, $$16$$, $$17$$, $$18$$, 19}","{2, $$4$$, $$6$$, $$8$$, $$10$$, $$12$$, $$14$$, $$16$$, 18}"]},{"id":"a0ec58fterminology1g-h17","type":"hint","dependencies":["a0ec58fterminology1g-h16"],"title":"Definition of Probability","text":"The probability of any outcome in the sample space occuring is the long-term relative frequency of that oucome. Effectively, to find the probability of an event, P(event), we need to find the size of that event (how many outcomes are in it) and divide that by the size of the sample space (how many outcomes are in it).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology1g-h18","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a0ec58fterminology1g-h17"],"title":"Size of A OR B","text":"What is the size of A OR B? Effectively, how many numbers are from one to $$19$$ are even OR greater than 13?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology1g-h19","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$19$$"],"dependencies":["a0ec58fterminology1g-h18"],"title":"Size of Sample Space","text":"What is the size of the sample space? Effectively, how many total whole numbers are there between $$1$$ and $$19$$ inclusive?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology1g-h20","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{12}{19}$$"],"dependencies":["a0ec58fterminology1g-h19"],"title":"Determining P(A OR B)","text":"What is P(A OR B), or what is the size of A OR B divided by the size of the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a0ec58fterminology1h","stepAnswer":["$$\\\\frac{10}{19}$$"],"problemType":"TextBox","stepTitle":"What is P(A\')?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{10}{19}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology1h-h21","type":"hint","dependencies":["a0ec58fterminology1g-h20"],"title":"Definition of Complement","text":"The component of an event is denoted as A\'. A\' has all the outcomes that are not in A. Therefore, we note that $$P\\\\left(A\\\\right)+P\\\\left(A\'\\\\right)=1$$ so $$P(A\')=1-P(A)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology1h-h22","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{9}{19}$$"],"dependencies":["a0ec58fterminology1h-h21"],"title":"Remembering P(A)","text":"What is P(A), which you solved for above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology1h-h23","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{10}{19}$$"],"dependencies":["a0ec58fterminology1h-h22"],"title":"Determining P(A\')","text":"What is P(A\')? What is $$1-P(A)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a0ec58fterminology1i","stepAnswer":["$$\\\\frac{3}{6}$$"],"problemType":"TextBox","stepTitle":"What is P(A|B)?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{6}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology1i-h24","type":"hint","dependencies":["a0ec58fterminology1h-h23"],"title":"Definition of Conditional Probability","text":"The conditional probability of A given B is written as P(A|B). P(A|B) is the probability that the event A occurs given that the event B has already occurred. We note that the formula is $$P(A|B)=(P(A$$ AND B))/(P(B)).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology1i-h25","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{19}$$"],"dependencies":["a0ec58fterminology1i-h24"],"title":"Remembering P(A AND B)","text":"What is P(A AND B), which you solved for above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology1i-h26","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{6}{19}$$"],"dependencies":["a0ec58fterminology1i-h25"],"title":"Remembering P(B)","text":"What is P(B), which you solved for above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology1i-h27","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{6}$$"],"dependencies":["a0ec58fterminology1i-h26"],"title":"Determining P(A|B)","text":"What is P(A|B), or in other words, P(A AND B) divided by P(B)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a0ec58fterminology1j","stepAnswer":["$$\\\\frac{3}{9}$$"],"problemType":"TextBox","stepTitle":"What is P(B|A)?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{9}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology1j-h24","type":"hint","dependencies":["a0ec58fterminology1h-h23"],"title":"Definition of Conditional Probability","text":"The conditional probability of B given A is written as P(B|A). P(B|A) is the probability that the event B occurs given that the event A has already occurred. We note that the formula is $$P(B|A)=(P(B$$ AND A))/(P(A)).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology1j-h25","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{19}$$"],"dependencies":["a0ec58fterminology1j-h24"],"title":"Remembering P(A AND B)","text":"What is P(A AND B), which you solved for above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology1j-h26","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{9}{19}$$"],"dependencies":["a0ec58fterminology1j-h25"],"title":"Remembering P(A)","text":"What is P(A), which you solved for above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology1j-h27","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{9}$$"],"dependencies":["a0ec58fterminology1j-h26"],"title":"Determining P(B|A)","text":"What is P(B|A), or in other words, P(A AND B) divided by P(A)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ec58fterminology10","title":"Determining Probability Numerically","body":"A jar of $$150$$ jelly beans contains $$22$$ red jelly beans, $$38$$ yellow, $$20$$ green, $$28$$ purple, $$26$$ blue, and the rest are orange. One jelly bean is chosen from the box at random. Let G $$=$$ the event of getting a green jelly bean.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Terminology","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0ec58fterminology10a","stepAnswer":["$$\\\\frac{20}{150}$$"],"problemType":"TextBox","stepTitle":"Find P(G).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{20}{150}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology10a-h1","type":"hint","dependencies":[],"title":"Definition of Probability","text":"The probability of any outcome in the sample space occuring is the long-term relative frequency of that oucome. Effectively, to find the probability of an event, P(event), we need to find the size of that event (how many outcomes are in it) and divide that by the size of the sample space (how many outcomes are in it).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a0ec58fterminology10a-h1"],"title":"Size of G","text":"What is the size of G? Effectively, how many jelly beans are green in the sample space?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$150$$"],"dependencies":["a0ec58fterminology10a-h2"],"title":"Size of Sample Space","text":"What is the size of the sample space? Effectively, how many total jelly beans were there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{20}{150}$$"],"dependencies":["a0ec58fterminology10a-h3"],"title":"Determining P(G)","text":"What is P(G), or what is the size of G divided by the size of the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ec58fterminology11","title":"Determining Probability Numerically","body":"A jar of $$150$$ jelly beans contains $$22$$ red jelly beans, $$38$$ yellow, $$20$$ green, $$28$$ purple, $$26$$ blue, and the rest are orange. One jelly bean is chosen from the box at random. Let P $$=$$ the event of getting a purple jelly bean.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Terminology","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0ec58fterminology11a","stepAnswer":["$$\\\\frac{28}{150}$$"],"problemType":"TextBox","stepTitle":"Find P(P).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{28}{150}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology11a-h1","type":"hint","dependencies":[],"title":"Definition of Probability","text":"The probability of any outcome in the sample space occuring is the long-term relative frequency of that oucome. Effectively, to find the probability of an event, P(event), we need to find the size of that event (how many outcomes are in it) and divide that by the size of the sample space (how many outcomes are in it).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$28$$"],"dependencies":["a0ec58fterminology11a-h1"],"title":"Size of P","text":"What is the size of P? Effectively, how many jelly beans are purple in the sample space?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology11a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$150$$"],"dependencies":["a0ec58fterminology11a-h2"],"title":"Size of Sample Space","text":"What is the size of the sample space? Effectively, how many total jelly beans were there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{28}{150}$$"],"dependencies":["a0ec58fterminology11a-h3"],"title":"Determining P(P)","text":"What is P(P), or what is the size of P divided by the size of the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ec58fterminology12","title":"Determining Probability Numerically","body":"A jar of $$150$$ jelly beans contains $$22$$ red jelly beans, $$38$$ yellow, $$20$$ green, $$28$$ purple, $$26$$ blue, and the rest are orange. One jelly bean is chosen from the box at random. Let R $$=$$ the event of getting a red jelly bean.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Terminology","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0ec58fterminology12a","stepAnswer":["$$\\\\frac{22}{150}$$"],"problemType":"TextBox","stepTitle":"Find P(R).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{22}{150}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology12a-h1","type":"hint","dependencies":[],"title":"Definition of Probability","text":"The probability of any outcome in the sample space occuring is the long-term relative frequency of that oucome. Effectively, to find the probability of an event, P(event), we need to find the size of that event (how many outcomes are in it) and divide that by the size of the sample space (how many outcomes are in it).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$22$$"],"dependencies":["a0ec58fterminology12a-h1"],"title":"Size of R","text":"What is the size of R? Effectively, how many jelly beans are red in the sample space?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology12a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$150$$"],"dependencies":["a0ec58fterminology12a-h2"],"title":"Size of Sample Space","text":"What is the size of the sample space? Effectively, how many total jelly beans were there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{22}{150}$$"],"dependencies":["a0ec58fterminology12a-h3"],"title":"Determining P(R)","text":"What is P(R), or what is the size of R divided by the size of the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ec58fterminology13","title":"Determining Probability Numerically","body":"A jar of $$150$$ jelly beans contains $$22$$ red jelly beans, $$38$$ yellow, $$20$$ green, $$28$$ purple, $$26$$ blue, and the rest are orange. One jelly bean is chosen from the box at random. Let Y $$=$$ the event of getting a yellow jelly bean.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Terminology","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0ec58fterminology13a","stepAnswer":["$$\\\\frac{38}{150}$$"],"problemType":"TextBox","stepTitle":"Find P(Y).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{38}{150}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology13a-h1","type":"hint","dependencies":[],"title":"Definition of Probability","text":"The probability of any outcome in the sample space occuring is the long-term relative frequency of that oucome. Effectively, to find the probability of an event, P(event), we need to find the size of that event (how many outcomes are in it) and divide that by the size of the sample space (how many outcomes are in it).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$38$$"],"dependencies":["a0ec58fterminology13a-h1"],"title":"Size of Y","text":"What is the size of Y? Effectively, how many jelly beans are yellow in the sample space?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology13a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$150$$"],"dependencies":["a0ec58fterminology13a-h2"],"title":"Size of Sample Space","text":"What is the size of the sample space? Effectively, how many total jelly beans were there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{38}{150}$$"],"dependencies":["a0ec58fterminology13a-h3"],"title":"Determining P(Y)","text":"What is P(Y), or what is the size of Y divided by the size of the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ec58fterminology14","title":"Determining Probability Numerically","body":"A jar of $$150$$ jelly beans contains $$22$$ red jelly beans, $$38$$ yellow, $$20$$ green, $$28$$ purple, $$26$$ blue, and the rest are orange. One jelly bean is chosen from the box at random. Let O $$=$$ the event of getting an orange jelly bean.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Terminology","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0ec58fterminology14a","stepAnswer":["$$\\\\frac{16}{150}$$"],"problemType":"TextBox","stepTitle":"Find P(O).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{16}{150}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology14a-h1","type":"hint","dependencies":[],"title":"Definition of Probability","text":"The probability of any outcome in the sample space occuring is the long-term relative frequency of that oucome. Effectively, to find the probability of an event, P(event), we need to find the size of that event (how many outcomes are in it) and divide that by the size of the sample space (how many outcomes are in it).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["a0ec58fterminology14a-h1"],"title":"Size of O","text":"What is the size of O? Effectively, how many jelly beans are orange in the sample space?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a0ec58fterminology14a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":[],"title":"Size of O","text":"We also know that the rest of the jelly beans (the ones that are not red, yellow, green, purple, or blue) are orange. Therefore, what is the number of orange jelly beans?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology14a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":[],"title":"Size of O","text":"What is $$150-22-38-20-28-27$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a0ec58fterminology14a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$150$$"],"dependencies":["a0ec58fterminology14a-h2"],"title":"Size of Sample Space","text":"What is the size of the sample space? Effectively, how many total jelly beans were there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology14a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{16}{150}$$"],"dependencies":["a0ec58fterminology14a-h3"],"title":"Determining P(O)","text":"What is P(O), or what is the size of O divided by the size of the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ec58fterminology15","title":"Determining Probability Numerically","body":"There are $$23$$ countries in North America, $$12$$ countries in South America, $$47$$ countries in Europe, $$44$$ countries in Asia, $$54$$ countries in Africa, and $$14$$ countries in Oceania (Pacific Ocean region). Let A $$=$$ the event that a country is in Asia.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Terminology","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0ec58fterminology15a","stepAnswer":["$$\\\\frac{44}{194}$$"],"problemType":"TextBox","stepTitle":"Find P(A).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{44}{194}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology15a-h1","type":"hint","dependencies":[],"title":"Definition of Probability","text":"The probability of any outcome in the sample space occuring is the long-term relative frequency of that oucome. Effectively, to find the probability of an event, P(event), we need to find the size of that event (how many outcomes are in it) and divide that by the size of the sample space (how many outcomes are in it).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$44$$"],"dependencies":["a0ec58fterminology15a-h1"],"title":"Size of A","text":"What is the size of A? Effectively, how many countries are in Asia from the sample space?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$194$$"],"dependencies":["a0ec58fterminology15a-h2"],"title":"Size of Sample Space","text":"What is the size of the sample space? Effectively, how many total countries were there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a0ec58fterminology15a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$194$$"],"dependencies":[],"title":"Size of Sample Space","text":"What is $$23+12+47+44+54+14$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a0ec58fterminology15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{44}{194}$$"],"dependencies":["a0ec58fterminology15a-h3"],"title":"Determining P(A)","text":"What is P(A), or what is the size of A divided by the size of the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ec58fterminology16","title":"Determining Probability Numerically","body":"There are $$23$$ countries in North America, $$12$$ countries in South America, $$47$$ countries in Europe, $$44$$ countries in Asia, $$54$$ countries in Africa, and $$14$$ countries in Oceania (Pacific Ocean region). Let E $$=$$ the event that a country is in Europe.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Terminology","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0ec58fterminology16a","stepAnswer":["$$\\\\frac{47}{194}$$"],"problemType":"TextBox","stepTitle":"Find P(E).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{47}{194}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology16a-h1","type":"hint","dependencies":[],"title":"Definition of Probability","text":"The probability of any outcome in the sample space occuring is the long-term relative frequency of that oucome. Effectively, to find the probability of an event, P(event), we need to find the size of that event (how many outcomes are in it) and divide that by the size of the sample space (how many outcomes are in it).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$47$$"],"dependencies":["a0ec58fterminology16a-h1"],"title":"Size of E","text":"What is the size of E? Effectively, how many countries are in Europe from the sample space?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology16a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$194$$"],"dependencies":["a0ec58fterminology16a-h2"],"title":"Size of Sample Space","text":"What is the size of the sample space? Effectively, how many total countries were there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a0ec58fterminology16a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$194$$"],"dependencies":[],"title":"Size of Sample Space","text":"What is $$23+12+47+44+54+14$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a0ec58fterminology16a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{47}{194}$$"],"dependencies":["a0ec58fterminology16a-h3"],"title":"Determining P(E)","text":"What is P(E), or what is the size of E divided by the size of the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ec58fterminology17","title":"Determining Probability Numerically","body":"There are $$23$$ countries in North America, $$12$$ countries in South America, $$47$$ countries in Europe, $$44$$ countries in Asia, $$54$$ countries in Africa, and $$14$$ countries in Oceania (Pacific Ocean region). Let A $$=$$ the event that a country is in Africa.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Terminology","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0ec58fterminology17a","stepAnswer":["$$\\\\frac{54}{194}$$"],"problemType":"TextBox","stepTitle":"Find P(A).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{54}{194}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology17a-h1","type":"hint","dependencies":[],"title":"Definition of Probability","text":"The probability of any outcome in the sample space occuring is the long-term relative frequency of that oucome. Effectively, to find the probability of an event, P(event), we need to find the size of that event (how many outcomes are in it) and divide that by the size of the sample space (how many outcomes are in it).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology17a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$54$$"],"dependencies":["a0ec58fterminology17a-h1"],"title":"Size of A","text":"What is the size of A? Effectively, how many countries are in Asia from the sample space?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology17a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$194$$"],"dependencies":["a0ec58fterminology17a-h2"],"title":"Size of Sample Space","text":"What is the size of the sample space? Effectively, how many total countries were there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a0ec58fterminology17a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$194$$"],"dependencies":[],"title":"Size of Sample Space","text":"What is $$23+12+47+44+54+14$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a0ec58fterminology17a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{54}{194}$$"],"dependencies":["a0ec58fterminology17a-h3"],"title":"Determining P(A)","text":"What is P(A), or what is the size of A divided by the size of the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ec58fterminology18","title":"Determining Probability Numerically","body":"There are $$23$$ countries in North America, $$12$$ countries in South America, $$47$$ countries in Europe, $$44$$ countries in Asia, $$54$$ countries in Africa, and $$14$$ countries in Oceania (Pacific Ocean region). Let N $$=$$ the event that a country is in North America.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Terminology","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0ec58fterminology18a","stepAnswer":["$$\\\\frac{23}{194}$$"],"problemType":"TextBox","stepTitle":"Find P(N).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{23}{194}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology18a-h1","type":"hint","dependencies":[],"title":"Definition of Probability","text":"The probability of any outcome in the sample space occuring is the long-term relative frequency of that oucome. Effectively, to find the probability of an event, P(event), we need to find the size of that event (how many outcomes are in it) and divide that by the size of the sample space (how many outcomes are in it).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$23$$"],"dependencies":["a0ec58fterminology18a-h1"],"title":"Size of N","text":"What is the size of N? Effectively, how many countries are in North America from the sample space?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology18a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$194$$"],"dependencies":["a0ec58fterminology18a-h2"],"title":"Size of Sample Space","text":"What is the size of the sample space? Effectively, how many total countries were there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a0ec58fterminology18a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$194$$"],"dependencies":[],"title":"Size of Sample Space","text":"What is $$23+12+47+44+54+14$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a0ec58fterminology18a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{23}{194}$$"],"dependencies":["a0ec58fterminology18a-h3"],"title":"Determining P(N)","text":"What is P(N), or what is the size of N divided by the size of the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ec58fterminology19","title":"Determining Probability Numerically","body":"There are $$23$$ countries in North America, $$12$$ countries in South America, $$47$$ countries in Europe, $$44$$ countries in Asia, $$54$$ countries in Africa, and $$14$$ countries in Oceania (Pacific Ocean region). Let O $$=$$ the event that a country is in Oceania.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Terminology","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0ec58fterminology19a","stepAnswer":["$$\\\\frac{14}{194}$$"],"problemType":"TextBox","stepTitle":"Find P(O).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{14}{194}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology19a-h1","type":"hint","dependencies":[],"title":"Definition of Probability","text":"The probability of any outcome in the sample space occuring is the long-term relative frequency of that oucome. Effectively, to find the probability of an event, P(event), we need to find the size of that event (how many outcomes are in it) and divide that by the size of the sample space (how many outcomes are in it).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$14$$"],"dependencies":["a0ec58fterminology19a-h1"],"title":"Size of O","text":"What is the size of O? Effectively, how many countries are in Oceania (Pacific Ocean region) from the sample space?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology19a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$194$$"],"dependencies":["a0ec58fterminology19a-h2"],"title":"Size of Sample Space","text":"What is the size of the sample space? Effectively, how many total countries were there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a0ec58fterminology19a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$194$$"],"dependencies":[],"title":"Size of Sample Space","text":"What is $$23+12+47+44+54+14$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a0ec58fterminology19a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{14}{194}$$"],"dependencies":["a0ec58fterminology19a-h3"],"title":"Determining P(O)","text":"What is P(O), or what is the size of O divided by the size of the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ec58fterminology2","title":"Understanding Probability Terminology","body":"The sample space S is all the ordered pairs of two whole numbers, the first from one to three and the second from one to four. Example: $$(1,4)$$ is one such ordered pair.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Terminology","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0ec58fterminology2a","stepAnswer":["{$$(1,1)$$, $$(1,2)$$, $$(1,3)$$, $$(1,4)$$, $$(2,1)$$, $$(2,2)$$, $$(2,3)$$, $$(2,4)$$, $$(3,1)$$, $$(3,2)$$, $$(3,3)$$, $$(3,4)$$}"],"problemType":"MultipleChoice","stepTitle":"What is the sample space, S?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"{$$(1,1)$$, $$(1,2)$$, $$(1,3)$$, $$(1,4)$$, $$(2,1)$$, $$(2,2)$$, $$(2,3)$$, $$(2,4)$$, $$(3,1)$$, $$(3,2)$$, $$(3,3)$$, $$(3,4)$$}","choices":["{$$(1,1)$$, $$(1,2)$$, $$(1,3)$$, $$(1,4)$$, $$(2,1)$$, $$(2,2)$$, $$(2,3)$$, $$(2,4)$$, $$(3,1)$$, $$(3,2)$$, $$(3,3)$$, $$(3,4)$$}","{$$(1,1)$$, $$(1,2)$$, $$(1,3)$$, $$(1,4)$$, $$(2,1)$$, $$(2,2)$$, $$(2,3)$$, $$(2,4)$$}","{$$(1,1)$$, $$(1,2)$$, $$(1,3)$$, $$(2,1)$$, $$(2,2)$$, $$(2,3)$$, $$(3,1)$$, $$(3,2)$$, $$(3,3)$$, $$(4,1)$$, $$(4,2)$$, $$(4,3)$$}","{$$(1,1)$$, $$(2,2)$$, $$(3,3)$$}"],"hints":{"DefaultPathway":[{"id":"a0ec58fterminology2a-h1","type":"hint","dependencies":[],"title":"Definition of Sample Space","text":"The sample space of an experiment is the set of all possible outcomes. In this scenario, we can represent each outcome as an ordered pairs of two whole numbers, the first of which ranges from one to three and the second of the pair ranges from one to four.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology2a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["{$$(1,1)$$, $$(1,2)$$, $$(1,3)$$, $$(1,4)$$, $$(2,1)$$, $$(2,2)$$, $$(2,3)$$, $$(2,4)$$, $$(3,1)$$, $$(3,2)$$, $$(3,3)$$, $$(3,4)$$}"],"dependencies":["a0ec58fterminology2a-h1"],"title":"Determining the Sample Space","text":"Knowing the definition of a sample space, what then, is the sample space of ordered pairs of two whole numbers when the first of the pair can range from one to three and the second of the pair can range from one to four?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["{$$(1,1)$$, $$(1,2)$$, $$(1,3)$$, $$(1,4)$$, $$(2,1)$$, $$(2,2)$$, $$(2,3)$$, $$(2,4)$$, $$(3,1)$$, $$(3,2)$$, $$(3,3)$$, $$(3,4)$$}","{$$(1,1)$$, $$(1,2)$$, $$(1,3)$$, $$(1,4)$$, $$(2,1)$$, $$(2,2)$$, $$(2,3)$$, $$(2,4)$$}","{$$(1,1)$$, $$(1,2)$$, $$(1,3)$$, $$(2,1)$$, $$(2,2)$$, $$(2,3)$$, $$(3,1)$$, $$(3,2)$$, $$(3,3)$$, $$(4,1)$$, $$(4,2)$$, $$(4,3)$$}","{$$(1,1)$$, $$(2,2)$$, $$(3,3)$$}"]}]}},{"id":"a0ec58fterminology2b","stepAnswer":["{$$(1,1)$$, $$(1,3)$$, $$(2,2)$$, $$(2,4)$$, $$(3,1)$$, $$(3,3)$$}"],"problemType":"MultipleChoice","stepTitle":"Let event A $$=$$ the sum is even and event B $$=$$ the first number is prime. What is the list of events that best represents the combination of outcomes in A?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"{$$(1,1)$$, $$(1,3)$$, $$(2,2)$$, $$(2,4)$$, $$(3,1)$$, $$(3,3)$$}","choices":["{$$(1,1)$$, $$(1,3)$$, $$(2,2)$$, $$(2,4)$$, $$(3,1)$$, $$(3,3)$$}","{$$(1,2)$$, $$(1,4)$$, $$(2,1)$$, $$(2,3)$$, $$(3,2)$$, $$(3,4)$$}","{$$(1,1)$$, $$(1,2)$$, $$(1,3)$$, $$(1,4)$$, $$(2,1)$$, $$(2,2)$$, $$(2,3)$$, $$(2,4)$$, $$(3,1)$$, $$(3,2)$$, $$(3,3)$$, $$(3,4)$$}","{$$(2,1)$$, $$(2,2)$$, $$(2,3)$$, $$(2,4)$$, $$(3,1)$$, $$(3,2)$$, $$(3,3)$$, $$(3,4)$$}"],"hints":{"DefaultPathway":[{"id":"a0ec58fterminology2b-h3","type":"hint","dependencies":["a0ec58fterminology2a-h2"],"title":"Determining Events","text":"The list of outcomes for an event A is all the possible outcomes that occur under that event. In this scenario, we can represent each outcome as each pair of ordered whole numbers such that the sum between them is even.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology2b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["{$$(1,1)$$, $$(1,3)$$, $$(2,2)$$, $$(2,4)$$, $$(3,1)$$, $$(3,3)$$}"],"dependencies":["a0ec58fterminology2b-h3"],"title":"Determining the Outcomes","text":"Knowing the definition of an event, what then, is the list of possible outcomes of ordered pairs of whole numbers such that the sum between the pair is even? What are the outcomes that together represent A?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["{$$(1,1)$$, $$(1,3)$$, $$(2,2)$$, $$(2,4)$$, $$(3,1)$$, $$(3,3)$$}","{$$(1,2)$$, $$(1,4)$$, $$(2,1)$$, $$(2,3)$$, $$(3,2)$$, $$(3,4)$$}","{$$(1,1)$$, $$(1,2)$$, $$(1,3)$$, $$(1,4)$$, $$(2,1)$$, $$(2,2)$$, $$(2,3)$$, $$(2,4)$$, $$(3,1)$$, $$(3,2)$$, $$(3,3)$$, $$(3,4)$$}","{$$(2,1)$$, $$(2,2)$$, $$(2,3)$$, $$(2,4)$$, $$(3,1)$$, $$(3,2)$$, $$(3,3)$$, $$(3,4)$$}"]}]}},{"id":"a0ec58fterminology2c","stepAnswer":["{$$(2,1)$$, $$(2,2)$$, $$(2,3)$$, $$(2,4)$$, $$(3,1)$$, $$(3,2)$$, $$(3,3)$$, $$(3,4)$$}"],"problemType":"MultipleChoice","stepTitle":"Let event A $$=$$ the sum is even and event B $$=$$ the first number is prime. What is the list of events that best represents the combination of outcomes in B?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"{$$(2,1)$$, $$(2,2)$$, $$(2,3)$$, $$(2,4)$$, $$(3,1)$$, $$(3,2)$$, $$(3,3)$$, $$(3,4)$$}","choices":["{$$(1,1)$$, $$(1,3)$$, $$(2,2)$$, $$(2,4)$$, $$(3,1)$$, $$(3,3)$$}","{$$(1,2)$$, $$(1,4)$$, $$(2,1)$$, $$(2,3)$$, $$(3,2)$$, $$(3,4)$$}","{$$(1,1)$$, $$(1,2)$$, $$(1,3)$$, $$(1,4)$$, $$(2,1)$$, $$(2,2)$$, $$(2,3)$$, $$(2,4)$$, $$(3,1)$$, $$(3,2)$$, $$(3,3)$$, $$(3,4)$$}","{$$(2,1)$$, $$(2,2)$$, $$(2,3)$$, $$(2,4)$$, $$(3,1)$$, $$(3,2)$$, $$(3,3)$$, $$(3,4)$$}"],"hints":{"DefaultPathway":[{"id":"a0ec58fterminology2c-h5","type":"hint","dependencies":["a0ec58fterminology2b-h4"],"title":"Determining Events","text":"The list of outcomes for an event B is all the possible outcomes that occur under that event. In this scenario, we can represent each outcome as one of each of the ordered pairs such that the first number in the pair is prime. Please note also that $$1$$ is not a prime number. Prime numbers are greater than $$1$$ such that their only divisors are $$1$$ and itself.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology2c-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["{$$(2,1)$$, $$(2,2)$$, $$(2,3)$$, $$(2,4)$$, $$(3,1)$$, $$(3,2)$$, $$(3,3)$$, $$(3,4)$$}"],"dependencies":["a0ec58fterminology2c-h5"],"title":"Determining the Outcomes","text":"Knowing the definition of an event, what then, is the list of possible outcomes in the sample space of ordered pairs where the first number in the ordered pair is prime? What are the outcomes that together represent B?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["{$$(1,1)$$, $$(1,3)$$, $$(2,2)$$, $$(2,4)$$, $$(3,1)$$, $$(3,3)$$}","{$$(1,2)$$, $$(1,4)$$, $$(2,1)$$, $$(2,3)$$, $$(3,2)$$, $$(3,4)$$}","{$$(1,1)$$, $$(1,2)$$, $$(1,3)$$, $$(1,4)$$, $$(2,1)$$, $$(2,2)$$, $$(2,3)$$, $$(2,4)$$, $$(3,1)$$, $$(3,2)$$, $$(3,3)$$, $$(3,4)$$}","{$$(2,1)$$, $$(2,2)$$, $$(2,3)$$, $$(2,4)$$, $$(3,1)$$, $$(3,2)$$, $$(3,3)$$, $$(3,4)$$}"]}]}},{"id":"a0ec58fterminology2d","stepAnswer":["$$\\\\frac{1}{2}$$"],"problemType":"TextBox","stepTitle":"What is the probability of event A occurring, P(A)?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{2}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology2d-h7","type":"hint","dependencies":["a0ec58fterminology2c-h6"],"title":"Definition of Probability","text":"The probability of any outcome in the sample space occuring is the long-term relative frequency of that oucome. Effectively, to find the probability of an event, P(event), we need to find the size of that event (how many outcomes are in it) and divide that by the size of the sample space (how many outcomes are in it).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology2d-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a0ec58fterminology2d-h7"],"title":"Size of A","text":"What is the size of A? Effectively, how many ordered pairs are in the event A?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology2d-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a0ec58fterminology2d-h8"],"title":"Size of Sample Space","text":"What is the size of the sample space? Effectively, how many total ordered pairs were there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology2d-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{6}{12}$$"],"dependencies":["a0ec58fterminology2d-h9"],"title":"Determining P(A)","text":"What is P(A), or what is the size of A divided by the size of the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a0ec58fterminology2e","stepAnswer":["$$\\\\frac{6}{12}$$"],"problemType":"TextBox","stepTitle":"What is the probability of event B occurring, P(B)?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{6}{12}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology2e-h11","type":"hint","dependencies":["a0ec58fterminology2d-h10"],"title":"Definition of Probability","text":"The probability of any outcome in the sample space occuring is the long-term relative frequency of that oucome. Effectively, to find the probability of an event, P(event), we need to find the size of that event (how many outcomes are in it) and divide that by the size of the sample space (how many outcomes are in it).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology2e-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a0ec58fterminology2e-h11"],"title":"Size of B","text":"What is the size of B? Effectively, how many ordered pairs are in the event B?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology2e-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a0ec58fterminology2e-h12"],"title":"Size of Sample Space","text":"What is the size of the sample space? Effectively, how many total ordered pairs were there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology2e-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{6}{12}$$"],"dependencies":["a0ec58fterminology2e-h13"],"title":"Determining P(B)","text":"What is P(B), or what is the size of B divided by the size of the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a0ec58fterminology2f","stepAnswer":["$$\\\\frac{4}{12}$$"],"problemType":"TextBox","stepTitle":"What is P(A AND B)?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{4}{12}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology2f-h15","type":"hint","dependencies":["a0ec58fterminology2e-h14"],"title":"Determine the Outcomes","text":"To determine the probability of an event occurring, we want to first determine what the possible outcomes that represent the space are.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology2f-h16","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["{$$(2,2)$$, $$(2,4)$$, $$(3,1)$$, $$(3,3)$$}"],"dependencies":["a0ec58fterminology2f-h15"],"title":"Determining A AND B","text":"What is the list of outcomes that exist in A AND B? Effectively, which ordered pairs have a sum that is even AND the first number is prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["{$$(1,1)$$, $$(1,3)$$, $$(2,1)$$, $$(2,2)$$, $$(2,3)$$, $$(2,4)$$, $$(3,1)$$, $$(3,2)$$, $$(3,3)$$, $$(3,4)$$, $$(4,2)$$, $$(4,4)$$}","{$$(2,2)$$, $$(2,4)$$, $$(3,1)$$, $$(3,3)$$}","{$$(1,1)$$, $$(1,3)$$, $$(4,2)$$, $$(4,4)$$}","{$$(2,1)$$, $$(2,3)$$, $$(3,2)$$, $$(3,4)$$}"]},{"id":"a0ec58fterminology2f-h17","type":"hint","dependencies":["a0ec58fterminology2f-h16"],"title":"Definition of Probability","text":"The probability of any outcome in the sample space occuring is the long-term relative frequency of that oucome. Effectively, to find the probability of an event, P(event), we need to find the size of that event (how many outcomes are in it) and divide that by the size of the sample space (how many outcomes are in it).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology2f-h18","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a0ec58fterminology2f-h17"],"title":"Size of A AND B","text":"What is the size of A AND B? Effectively, how ordered pairs have a sum that is even AND the first number in the pair is prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology2f-h19","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a0ec58fterminology2f-h18"],"title":"Size of Sample Space","text":"What is the size of the sample space? Effectively, how many total ordered pairs were there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology2f-h20","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{4}{12}$$"],"dependencies":["a0ec58fterminology2f-h19"],"title":"Determining P(A AND B)","text":"What is P(A AND B), or what is the size of A AND B divided by the size of the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a0ec58fterminology2g","stepAnswer":["$$\\\\frac{10}{12}$$"],"problemType":"TextBox","stepTitle":"What is P(A OR B)?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{10}{12}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology2g-h15","type":"hint","dependencies":["a0ec58fterminology2e-h14"],"title":"Determine the Outcomes","text":"To determine the probability of an event occurring, we want to first determine what the possible outcomes that represent the space are.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology2g-h16","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["{$$(1,1)$$, $$(1,3)$$, $$(2,1)$$, $$(2,2)$$, $$(2,3)$$, $$(2,4)$$, $$(3,1)$$, $$(3,2)$$, $$(3,3)$$, $$(3,4)$$, $$(4,2)$$, $$(4,4)$$}"],"dependencies":["a0ec58fterminology2g-h15"],"title":"Determining A OR B","text":"What is the list of outcomes that exist in A OR B? Effectively, which ordered pairs have a sum that is even OR the first number is prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["{$$(1,1)$$, $$(1,3)$$, $$(2,1)$$, $$(2,2)$$, $$(2,3)$$, $$(2,4)$$, $$(3,1)$$, $$(3,2)$$, $$(3,3)$$, $$(3,4)$$, $$(4,2)$$, $$(4,4)$$}","{$$(2,2)$$, $$(2,4)$$, $$(3,1)$$, $$(3,3)$$}","{$$(1,1)$$, $$(1,3)$$, $$(4,2)$$, $$(4,4)$$}","{$$(2,1)$$, $$(2,3)$$, $$(3,2)$$, $$(3,4)$$}"]},{"id":"a0ec58fterminology2g-h17","type":"hint","dependencies":["a0ec58fterminology2g-h16"],"title":"Definition of Probability","text":"The probability of any outcome in the sample space occuring is the long-term relative frequency of that oucome. Effectively, to find the probability of an event, P(event), we need to find the size of that event (how many outcomes are in it) and divide that by the size of the sample space (how many outcomes are in it).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology2g-h18","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a0ec58fterminology2g-h17"],"title":"Size of A OR B","text":"What is the size of A OR B? Effectively, how ordered pairs have a sum that is even OR the first number in the pair is prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology2g-h19","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a0ec58fterminology2g-h18"],"title":"Size of Sample Space","text":"What is the size of the sample space? Effectively, how many total ordered pairs were there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology2g-h20","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{10}{12}$$"],"dependencies":["a0ec58fterminology2g-h19"],"title":"Determining P(A OR B)","text":"What is P(A OR B), or what is the size of A OR B divided by the size of the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a0ec58fterminology2h","stepAnswer":["$$\\\\frac{1}{2}$$"],"problemType":"TextBox","stepTitle":"What is P(A\')?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{2}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology2h-h21","type":"hint","dependencies":["a0ec58fterminology2g-h20"],"title":"Definition of Complement","text":"The component of an event is denoted as A\'. A\' has all the outcomes that are not in A. Therefore, we note that $$P\\\\left(A\\\\right)+P\\\\left(A\'\\\\right)=1$$ so $$P(A\')=1-P(A)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology2h-h22","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a0ec58fterminology2h-h21"],"title":"Remembering P(A)","text":"What is P(A), which you solved for above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology2h-h23","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a0ec58fterminology2h-h22"],"title":"Determining P(A\')","text":"What is P(A\')? What is $$1-P(A)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a0ec58fterminology2i","stepAnswer":["$$\\\\frac{8}{12}$$"],"problemType":"TextBox","stepTitle":"What is P(A|B)?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{8}{12}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology2i-h24","type":"hint","dependencies":["a0ec58fterminology2h-h23"],"title":"Definition of Conditional Probability","text":"The conditional probability of A given B is written as P(A|B). P(A|B) is the probability that the event A occurs given that the event B has already occurred. We note that the formula is $$P(A|B)=(P(A$$ AND B))/(P(B)).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology2i-h25","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{4}{12}$$"],"dependencies":["a0ec58fterminology2i-h24"],"title":"Remembering P(A AND B)","text":"What is P(A AND B), which you solved for above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology2i-h26","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a0ec58fterminology2i-h25"],"title":"Remembering P(B)","text":"What is P(B), which you solved for above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology2i-h27","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{8}{12}$$"],"dependencies":["a0ec58fterminology2i-h26"],"title":"Determining P(A|B)","text":"What is P(A|B), or in other words, P(A AND B) divided by P(B)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a0ec58fterminology2j","stepAnswer":["$$\\\\frac{8}{12}$$"],"problemType":"TextBox","stepTitle":"What is P(B|A)?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{8}{12}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology2j-h24","type":"hint","dependencies":["a0ec58fterminology2h-h23"],"title":"Definition of Conditional Probability","text":"The conditional probability of B given A is written as P(B|A). P(B|A) is the probability that the event B occurs given that the event A has already occurred. We note that the formula is $$P(B|A)=(P(B$$ AND A))/(P(A)).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology2j-h25","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{4}{12}$$"],"dependencies":["a0ec58fterminology2j-h24"],"title":"Remembering P(A AND B)","text":"What is P(A AND B), which you solved for above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology2j-h26","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a0ec58fterminology2j-h25"],"title":"Remembering P(A)","text":"What is P(A), which you solved for above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology2j-h27","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{8}{12}$$"],"dependencies":["a0ec58fterminology2j-h26"],"title":"Determining P(B|A)","text":"What is P(B|A), or in other words, P(A AND B) divided by P(A)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ec58fterminology20","title":"Determining Probability Numerically","body":"There are $$23$$ countries in North America, $$12$$ countries in South America, $$47$$ countries in Europe, $$44$$ countries in Asia, $$54$$ countries in Africa, and $$14$$ countries in Oceania (Pacific Ocean region). Let S $$=$$ the event that a country is in South America.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Terminology","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0ec58fterminology20a","stepAnswer":["$$\\\\frac{12}{194}$$"],"problemType":"TextBox","stepTitle":"Find P(S).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{12}{194}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology20a-h1","type":"hint","dependencies":[],"title":"Definition of Probability","text":"The probability of any outcome in the sample space occuring is the long-term relative frequency of that oucome. Effectively, to find the probability of an event, P(event), we need to find the size of that event (how many outcomes are in it) and divide that by the size of the sample space (how many outcomes are in it).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a0ec58fterminology20a-h1"],"title":"Size of S","text":"What is the size of S? Effectively, how many countries are in South America from the sample space?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology20a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$194$$"],"dependencies":["a0ec58fterminology20a-h2"],"title":"Size of Sample Space","text":"What is the size of the sample space? Effectively, how many total countries were there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a0ec58fterminology20a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$194$$"],"dependencies":[],"title":"Size of Sample Space","text":"What is $$23+12+47+44+54+14$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a0ec58fterminology20a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{12}{194}$$"],"dependencies":["a0ec58fterminology20a-h3"],"title":"Determining P(S)","text":"What is P(S), or what is the size of S divided by the size of the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ec58fterminology21","title":"Determining Standard Probabilities","body":"A standard deck of cards has $$52$$ cards with $$4$$ suits, each with $$13$$ cards: spades, hearts, diamonds, and clubs. Spades and clubs are black cards. Hearts and diamonds are red cards.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Terminology","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0ec58fterminology21a","stepAnswer":["$$\\\\frac{1}{2}$$"],"problemType":"TextBox","stepTitle":"What is the probability of drawing a red card in the standard deck of $$52$$ cards?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{2}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology21a-h1","type":"hint","dependencies":[],"title":"Definition of Probability","text":"The probability of any outcome in the sample space occuring is the long-term relative frequency of that oucome. Effectively, to find the probability of an event, P(event), we need to find the size of that event (how many outcomes are in it) and divide that by the size of the sample space (how many outcomes are in it).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology21a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$26$$"],"dependencies":["a0ec58fterminology21a-h1"],"title":"Number of Red Cards","text":"How many red cards are there in total in the deck?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a0ec58fterminology21a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$26$$"],"dependencies":[],"title":"Number of Red Cards","text":"We know that hearts and diamonds are red suits, each with $$13$$ cards. Therefore, how many red cards are there in total (sum up the two red suits)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a0ec58fterminology21a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$52$$"],"dependencies":["a0ec58fterminology21a-h2"],"title":"Number of Total Cards","text":"How many total cards is in a standard deck?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology21a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{26}{52}$$"],"dependencies":["a0ec58fterminology21a-h3"],"title":"Determining Probability of Red Card","text":"What is the probability of drawing a red card in a standard deck of $$52$$ cards, or what is the number of red cards divided by the total number of cards?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ec58fterminology22","title":"Determining Standard Probabilities","body":"A standard deck of cards has $$52$$ cards with $$4$$ suits, each with $$13$$ cards: spades, hearts, diamonds, and clubs. Spades and clubs are black cards. Hearts and diamonds are red cards.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Terminology","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0ec58fterminology22a","stepAnswer":["$$\\\\frac{13}{52}$$"],"problemType":"TextBox","stepTitle":"What is the probability of drawing a club in the standard deck of $$52$$ cards?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{13}{52}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology22a-h1","type":"hint","dependencies":[],"title":"Definition of Probability","text":"The probability of any outcome in the sample space occuring is the long-term relative frequency of that oucome. Effectively, to find the probability of an event, P(event), we need to find the size of that event (how many outcomes are in it) and divide that by the size of the sample space (how many outcomes are in it).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology22a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$13$$"],"dependencies":["a0ec58fterminology22a-h1"],"title":"Number of Clubs","text":"How many clubs are there in total in the deck?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology22a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$52$$"],"dependencies":["a0ec58fterminology22a-h2"],"title":"Number of Total Cards","text":"How many total cards is in a standard deck?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology22a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{13}{52}$$"],"dependencies":["a0ec58fterminology22a-h3"],"title":"Determining Probability of Club","text":"What is the probability of drawing a red card in a standard deck of $$52$$ cards, or what is the number of red cards divided by the total number of cards?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ec58fterminology23","title":"Determining Standard Probabilities","body":"A fair, six-sided die has six faces, each of which has a certain number of dots, one through six. When you roll a fair, six-sided die, one of the faces shows on top and that is the \\"outcome\\" of that roll.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Terminology","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0ec58fterminology23a","stepAnswer":["$$\\\\frac{1}{2}$$"],"problemType":"TextBox","stepTitle":"What is the probability of rolling an even number of dots with a fair, six-sided die numbered one through six?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{2}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology23a-h1","type":"hint","dependencies":[],"title":"Definition of Probability","text":"The probability of any outcome in the sample space occuring is the long-term relative frequency of that oucome. Effectively, to find the probability of an event, P(event), we need to find the size of that event (how many outcomes are in it) and divide that by the size of the sample space (how many outcomes are in it).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology23a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a0ec58fterminology23a-h1"],"title":"Number of Faces with Even Number of Dots","text":"How many faces of a fair, six-sided die has an even number of dots? In other words, how many numbers in the range one through six inclusive, are even?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology23a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a0ec58fterminology23a-h2"],"title":"Number of Total Faces","text":"How many faces does a fair, six-sided die have?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology23a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{6}$$"],"dependencies":["a0ec58fterminology23a-h3"],"title":"Determining Probability of Even Number of Dots","text":"What is the probability of rolling an even number of dots with a fair, six-sided die? In other words, what is the number of faces with an even number of dots divided by the total number of faces on the die?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ec58fterminology24","title":"Determining Standard Probabilities","body":"A fair, six-sided die has six faces, each of which has a certain number of dots, one through six. When you roll a fair, six-sided die, one of the faces shows on top and that is the \\"outcome\\" of that roll.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Terminology","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0ec58fterminology24a","stepAnswer":["$$\\\\frac{1}{2}$$"],"problemType":"TextBox","stepTitle":"What is the probability of rolling a prime number of dots with a fair, six-sided die numbered one through six?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{2}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology24a-h1","type":"hint","dependencies":[],"title":"Definition of Probability","text":"The probability of any outcome in the sample space occuring is the long-term relative frequency of that oucome. Effectively, to find the probability of an event, P(event), we need to find the size of that event (how many outcomes are in it) and divide that by the size of the sample space (how many outcomes are in it).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology24a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a0ec58fterminology24a-h1"],"title":"Number of Faces with Prime Number of Dots","text":"How many faces of a fair, six-sided die has a prime number of dots? In other words, how many numbers in the range one through six inclusive, are prime? Please note that prime numbers are greater than $$1$$ such that the only divisors of that number are $$1$$ and itself.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a0ec58fterminology24a-h2-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2$$, $$3$$, $$2005$$"],"dependencies":[],"title":"Determining the Prime Numbers","text":"What is the list of prime numbers from one through six inclusive?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2$$, $$3$$, $$5$$","$$1$$, $$2$$, $$3$$, $$5$$","$$1$$, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$","$$2$$, $$3$$, $$4$$, $$5$$, $$6$$"]},{"id":"a0ec58fterminology24a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":[],"title":"Number of Faces with Prime Number of Dots","text":"Now, how many numbers are in the list of the prime numbers from one through six inclusive? How many numbers are in the list you just answered?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a0ec58fterminology24a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a0ec58fterminology24a-h2"],"title":"Number of Total Faces","text":"How many faces does a fair, six-sided die have?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology24a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{6}$$"],"dependencies":["a0ec58fterminology24a-h3"],"title":"Determining Probability of Prime Number of Dots","text":"What is the probability of rolling a prime number of dots with a fair, six-sided die? In other words, what is the number of faces with a prime number of dots divided by the total number of faces on the die?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ec58fterminology25","title":"Solving for Probability Visually","body":"You see a game at a local fair. You have to throw a dart at a color wheel. Each section on the color weel is equal in area, shown in the image.\\\\n- Let B $$=$$ the event of landing on blue.\\\\n- Let R $$=$$ the event of landing on red.\\\\n- Let G $$=$$ the event of landing on green.\\\\n- Let Y $$=$$ the event of landing on yellow.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Terminology","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0ec58fterminology25a","stepAnswer":["$$\\\\frac{1}{8}$$"],"problemType":"TextBox","stepTitle":"If you land on Y, you get the biggest prize. Find P(Y).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{8}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology25a-h1","type":"hint","dependencies":[],"title":"Definition of Probability","text":"The probability of any outcome in the sample space occuring is the long-term relative frequency of that oucome. Effectively, to find the probability of an event, P(event), we need to find the size of that event (how many outcomes are in it) and divide that by the size of the sample space (how many outcomes are in it).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology25a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a0ec58fterminology25a-h1"],"title":"Number of Yellow Sections","text":"How many sections on the color wheel are yellow?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology25a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a0ec58fterminology25a-h2"],"title":"Number of Total Sections","text":"How many sections on the color wheel are there in total?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology25a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{8}$$"],"dependencies":["a0ec58fterminology25a-h3"],"title":"Determining P(Y)","text":"What is P(Y), effectively, the number of yellow sections on the wheel divided by the total number of sections on the wheel?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a0ec58fterminology25b","stepAnswer":["$$\\\\frac{4}{8}$$"],"problemType":"TextBox","stepTitle":"If you land on red, you don\'t get a prize. What is P(R)?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{4}{8}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology25b-h5","type":"hint","dependencies":["a0ec58fterminology25a-h4"],"title":"Definition of Probability","text":"The probability of any outcome in the sample space occuring is the long-term relative frequency of that oucome. Effectively, to find the probability of an event, P(event), we need to find the size of that event (how many outcomes are in it) and divide that by the size of the sample space (how many outcomes are in it).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology25b-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a0ec58fterminology25b-h5"],"title":"Number of Red Sections","text":"How many sections on the color wheel are red?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology25b-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a0ec58fterminology25b-h6"],"title":"Number of Total Sections","text":"How many sections on the color wheel are there in total?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology25b-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{4}{8}$$"],"dependencies":["a0ec58fterminology25b-h7"],"title":"Determining P(R)","text":"What is P(R), effectively, the number of red sections on the wheel divided by the total number of sections on the wheel?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ec58fterminology3","title":"Determining Probability Symbolically","body":"In a particular college class, there are male and female students. Some students have long hair and some students have short hair. Write the symbols for the probabilities of the events for each step. (Note that you cannot find numerical answers here. You were not given enough information ot find any probability values yet; concentrate on understanding the symbols.)\\\\n- Let F be the event that a student is female.\\\\n- Let M be the event that a student is male.\\\\n- Let S be the event that a student has short hair.\\\\n- Let L be the event that a student has long hair.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Terminology","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0ec58fterminology3a","stepAnswer":["$$P(L\')=P(S)$$"],"problemType":"MultipleChoice","stepTitle":"What is the probability that a student does not have long hair?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$P(L\')=P(S)$$","choices":["$$P(L\')=P(S)$$","$$P(S\')=P(L)$$","$$P(F\')=P(M)$$","$$P(M\')=P(F)$$"],"hints":{"DefaultPathway":[{"id":"a0ec58fterminology3a-h1","type":"hint","dependencies":[],"title":"Definition of a Complement","text":"The component of an event is denoted as A\'. A\' has all the outcomes that are not in A. Therefore, we note that $$P\\\\left(A\\\\right)+P\\\\left(A\'\\\\right)=1$$ so $$P(A\')=1-P(A)$$. Therefore, we need to find the probability that a student does NOT have a long hair. This would be the complement of a student having long hair. Think about what this means logically: if a student does not have long hair, then what length of hair do they have?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology3a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$P(L\')=P(S)$$"],"dependencies":["a0ec58fterminology3a-h1"],"title":"Determining Given Probability","text":"What is the probability that a student does not have long hair?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$P(L\')=P(S)$$","$$P(S\')=P(L)$$","$$P(F\')=P(M)$$","$$P(M\')=P(F)$$"]}]}},{"id":"a0ec58fterminology3b","stepAnswer":["P(M OR S)"],"problemType":"MultipleChoice","stepTitle":"What is the probability that a student is male or has short hair?","stepBody":"","answerType":"string","variabilization":{},"choices":["P(M OR S)","P(M AND S)","P(M","S)","P(S","M)"],"hints":{"DefaultPathway":[{"id":"a0ec58fterminology3b-h3","type":"hint","dependencies":["a0ec58fterminology3a-h2"],"title":"Definition of OR","text":"Knowing that we\'re determining the probability of one event occuring OR another occuring, we can use OR symbolically.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology3b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["P(M OR S)"],"dependencies":["a0ec58fterminology3b-h3"],"title":"Determining Given Probability","text":"What is the probability then, that a student is male OR has short hair? We know how to represent both these events symbolically using letters.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["P(M OR S)","P(M AND S)","P(M","S)","P(S","M)"]}]}},{"id":"a0ec58fterminology3c","stepAnswer":["P(F AND L)"],"problemType":"MultipleChoice","stepTitle":"What is the probability that a student is female and has long hair?","stepBody":"","answerType":"string","variabilization":{},"choices":["P(F AND L)","P(F OR L)","P(F","L)","P(L","F)"],"hints":{"DefaultPathway":[{"id":"a0ec58fterminology3c-h5","type":"hint","dependencies":["a0ec58fterminology3b-h4"],"title":"Definition of AND","text":"Knowing that we\'re determining the probability of one event occuring AND another occuring, we can use AND symbolically.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology3c-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["P(F AND L)"],"dependencies":["a0ec58fterminology3c-h5"],"title":"Determining Given Probability","text":"What is the probability then, that a student is female AND has long hair? We know how to represent both these events symbolically using letters.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["P(F AND L)","P(F OR L)","P(F","L)","P(L","F)"]}]}},{"id":"a0ec58fterminology3d","stepAnswer":["P(M|L)"],"problemType":"MultipleChoice","stepTitle":"What is the probability thta a student is male, given that the student has long hair?","stepBody":"","answerType":"string","variabilization":{},"choices":["L)","M)","P(L","P(M","P(M AND L)","P(M OR L)","P(M|L)"],"hints":{"DefaultPathway":[{"id":"a0ec58fterminology3d-h7","type":"hint","dependencies":["a0ec58fterminology3c-h6"],"title":"Definitiong of Conditional Probability","text":"Since we were given the word GIVEN in the problem, we note that we\'re working with conditional probability. Conditional probability essentially, written as P(A|B) is the probability that event A occurs given that event B already occurs.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology3d-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["P(M|L)"],"dependencies":["a0ec58fterminology3d-h7"],"title":"Determining Given Probability","text":"What is the probability that a student is male, given that the student has long hair?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["P(M","L)","P(L","M)","P(M AND L)","P(M OR L)"]}]}},{"id":"a0ec58fterminology3e","stepAnswer":["P(L|M)"],"problemType":"MultipleChoice","stepTitle":"What is the probability that a student has long hair, given that the student is male?","stepBody":"","answerType":"string","variabilization":{},"choices":["L)","M)","P(L","P(L|M)","P(M","P(M AND L)","P(M OR L)"],"hints":{"DefaultPathway":[{"id":"a0ec58fterminology3e-h7","type":"hint","dependencies":["a0ec58fterminology3c-h6"],"title":"Definitiong of Conditional Probability","text":"Since we were given the word GIVEN in the problem, we note that we\'re working with conditional probability. Conditional probability essentially, written as P(A|B) is the probability that event A occurs given that event B already occurs.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology3e-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["P(L|M)"],"dependencies":["a0ec58fterminology3e-h7"],"title":"Determining Given Probability","text":"What is the probability that a student has long hair, given that the student is male?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["P(M","L)","P(L","M)","P(M AND L)","P(M OR L)"]}]}}]},{"id":"a0ec58fterminology4","title":"Determining Probability Symbolically","body":"In a particular college class, there are male and female students. Some students have long hair and some students have short hair. Write the symbols for the probabilities of the events for each step. (Note that you cannot find numerical answers here. You were not given enough information ot find any probability values yet; concentrate on understanding the symbols.)\\\\n- Let F be the event that a student is female.\\\\n- Let M be the event that a student is male.\\\\n- Let S be the event that a student has short hair.\\\\n- Let L be the event that a student has long hair.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Terminology","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0ec58fterminology4a","stepAnswer":["P(S|F)"],"problemType":"MultipleChoice","stepTitle":"Of all the female students, what is the probability that a student has short hair?","stepBody":"","answerType":"string","variabilization":{},"choices":["F)","P(F","P(S","P(S AND F)","P(S OR F)","P(S|F)","S)"],"hints":{"DefaultPathway":[{"id":"a0ec58fterminology4a-h1","type":"hint","dependencies":[],"title":"Definitiong of Conditional Probability","text":"Although we were not given the word GIVEN in this problem, we note that we are picking out of the subset of students that are female. We are finding how many female students have short hair, which can be reworded as the number of students with short hair given they are female. Therefore, we are working with conditional probability. Conditional probability essentially, written as P(A|B) is the probability that event A occurs given that event B already occurs.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology4a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["P(S|F)"],"dependencies":["a0ec58fterminology4a-h1"],"title":"Determining Given Probability","text":"What is the probability that a student has short hair, given that they are female?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["P(S","F)","P(F","S)","P(S AND F)","P(S OR F)"]}]}},{"id":"a0ec58fterminology4b","stepAnswer":["P(F|L)"],"problemType":"MultipleChoice","stepTitle":"Of all students with long hair, the probability that a student is female.","stepBody":"","answerType":"string","variabilization":{},"choices":["F)","L)","P(F","P(F AND L)","P(F OR L)","P(F|L)","P(L"],"hints":{"DefaultPathway":[{"id":"a0ec58fterminology4b-h3","type":"hint","dependencies":["a0ec58fterminology4a-h2"],"title":"Definitiong of Conditional Probability","text":"Although we were not given the word GIVEN in this problem, we note that we are picking out of the subset of students that have long hair. We are finding how many students with long hair are female, which can be reworded as the number of students that are female given that they have long hair. Therefore, we are working with conditional probability. Conditional probability essentially, written as P(A|B) is the probability that event A occurs given that event B already occurs.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology4b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["P(F|L)"],"dependencies":["a0ec58fterminology4b-h3"],"title":"Determining Given Probability","text":"What is the probability that a student is female, given that they have long hair?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["P(F","L)","P(L","F)","P(F AND L)","P(F OR L)"]}]}},{"id":"a0ec58fterminology4c","stepAnswer":["P(F OR L)"],"problemType":"MultipleChoice","stepTitle":"What is the probability that a student is female or has long hair?","stepBody":"","answerType":"string","variabilization":{},"choices":["P(F OR L)","P(F AND L)","P(F","L)","P(L","F)"],"hints":{"DefaultPathway":[{"id":"a0ec58fterminology4c-h5","type":"hint","dependencies":["a0ec58fterminology4b-h4"],"title":"Definition of OR","text":"Knowing that we\'re determining the probability of one event occuring OR another occuring, we can use OR symbolically.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology4c-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["P(F OR L)"],"dependencies":["a0ec58fterminology4c-h5"],"title":"Determining Given Probability","text":"What is the probability then, that a student is female OR has long hair? We know how to represent both these events symbolically using letters.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["P(F OR L)","P(F AND L)","P(F","L)","P(L","F)"]}]}},{"id":"a0ec58fterminology4d","stepAnswer":["P(M AND S)"],"problemType":"MultipleChoice","stepTitle":"What is the probability that a randomly selected student is a male student with short hair?","stepBody":"","answerType":"string","variabilization":{},"choices":["P(M AND S)","P(M OR S)","P(M","S)","P(S","M)"],"hints":{"DefaultPathway":[{"id":"a0ec58fterminology4d-h7","type":"hint","dependencies":["a0ec58fterminology4c-h6"],"title":"Definition of AND","text":"Although this doesn\'t say AND, we note that we want a student who is male with short hair, which we can rewrite as a male student and a student with short hair. Knowing that we\'re determining the probability of one event occuring AND another occuring, we can use AND symbolically.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology4d-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["P(M AND S)"],"dependencies":["a0ec58fterminology4d-h7"],"title":"Determining Given Probability","text":"What is the probability then, that a student is male AND has short hair? We know how to represent both these events symbolically using letters.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["P(M AND S)","P(M OR S)","P(M","S)","P(S","M)"]}]}},{"id":"a0ec58fterminology4e","stepAnswer":["P(F)"],"problemType":"MultipleChoice","stepTitle":"What is the probability that a student is female?","stepBody":"","answerType":"string","variabilization":{},"choices":["P(F)","P(M)","P(S)","P(L)"],"hints":{"DefaultPathway":[{"id":"a0ec58fterminology4e-h9","type":"hint","dependencies":["a0ec58fterminology4d-h8"],"title":"Definition of Probability","text":"We want to find the probability that a student selected is female. We also know how to symbolically represent the event that a student is female as F.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology4e-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["P(F)"],"dependencies":["a0ec58fterminology4e-h9"],"title":"Determining Given Probability","text":"What is the probability that a student is female?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["P(F)","P(M)","P(S)","P(L)"]}]}}]},{"id":"a0ec58fterminology5","title":"Determining Probability Numerically","body":"A box is filled with several party favors. It consists $$12$$ hats, $$15$$ noisemakers, ten finger traps, and five bags of confetti. One party favor is chosen from the box at random. Let H $$=$$ the event of getting a hat.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Terminology","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0ec58fterminology5a","stepAnswer":["$$\\\\frac{12}{42}$$"],"problemType":"TextBox","stepTitle":"Find P(H).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{12}{42}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology5a-h1","type":"hint","dependencies":[],"title":"Definition of Probability","text":"The probability of any outcome in the sample space occuring is the long-term relative frequency of that oucome. Effectively, to find the probability of an event, P(event), we need to find the size of that event (how many outcomes are in it) and divide that by the size of the sample space (how many outcomes are in it).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a0ec58fterminology5a-h1"],"title":"Size of H","text":"What is the size of H? Effectively, how many hats are in the sample space?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$42$$"],"dependencies":["a0ec58fterminology5a-h2"],"title":"Size of Sample Space","text":"What is the size of the sample space? Effectively, how many total party favors were there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a0ec58fterminology5a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$42$$"],"dependencies":[],"title":"Size of Sample Space","text":"What is $$12+15+10+5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a0ec58fterminology5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{12}{42}$$"],"dependencies":["a0ec58fterminology5a-h3"],"title":"Determining P(H)","text":"What is P(H), or what is the size of H divided by the size of the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ec58fterminology6","title":"Determining Probability Numerically","body":"A box is filled with several party favors. It consists $$12$$ hats, $$15$$ noisemakers, ten finger traps, and five bags of confetti. One party favor is chosen from the box at random. Let N $$=$$ the event of getting a noisemaker.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Terminology","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0ec58fterminology6a","stepAnswer":["$$\\\\frac{15}{42}$$"],"problemType":"TextBox","stepTitle":"Find P(N).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{15}{42}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology6a-h1","type":"hint","dependencies":[],"title":"Definition of Probability","text":"The probability of any outcome in the sample space occuring is the long-term relative frequency of that oucome. Effectively, to find the probability of an event, P(event), we need to find the size of that event (how many outcomes are in it) and divide that by the size of the sample space (how many outcomes are in it).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["a0ec58fterminology6a-h1"],"title":"Size of N","text":"What is the size of N? Effectively, how many noisemakers are in the sample space?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$42$$"],"dependencies":["a0ec58fterminology6a-h2"],"title":"Size of Sample Space","text":"What is the size of the sample space? Effectively, how many total party favors were there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a0ec58fterminology6a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$42$$"],"dependencies":[],"title":"Size of Sample Space","text":"What is $$12+15+10+5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a0ec58fterminology6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{15}{42}$$"],"dependencies":["a0ec58fterminology6a-h3"],"title":"Determining P(N)","text":"What is P(N), or what is the size of N divided by the size of the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ec58fterminology7","title":"Determining Probability Numerically","body":"A box is filled with several party favors. It consists $$12$$ hats, $$15$$ noisemakers, ten finger traps, and five bags of confetti. One party favor is chosen from the box at random. Let F $$=$$ the event of getting a finger trap.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Terminology","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0ec58fterminology7a","stepAnswer":["$$\\\\frac{10}{42}$$"],"problemType":"TextBox","stepTitle":"Find P(F).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{10}{42}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology7a-h1","type":"hint","dependencies":[],"title":"Definition of Probability","text":"The probability of any outcome in the sample space occuring is the long-term relative frequency of that oucome. Effectively, to find the probability of an event, P(event), we need to find the size of that event (how many outcomes are in it) and divide that by the size of the sample space (how many outcomes are in it).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a0ec58fterminology7a-h1"],"title":"Size of F","text":"What is the size of F? Effectively, how many finger traps are in the sample space?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$42$$"],"dependencies":["a0ec58fterminology7a-h2"],"title":"Size of Sample Space","text":"What is the size of the sample space? Effectively, how many total party favors were there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a0ec58fterminology7a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$42$$"],"dependencies":[],"title":"Size of Sample Space","text":"What is $$12+15+10+5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a0ec58fterminology7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{10}{42}$$"],"dependencies":["a0ec58fterminology7a-h3"],"title":"Determining P(F)","text":"What is P(F), or what is the size of F divided by the size of the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ec58fterminology8","title":"Determining Probability Numerically","body":"A box is filled with several party favors. It consists $$12$$ hats, $$15$$ noisemakers, ten finger traps, and five bags of confetti. One party favor is chosen from the box at random. Let C $$=$$ the event of getting a bag of confetti.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Terminology","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0ec58fterminology8a","stepAnswer":["$$\\\\frac{5}{42}$$"],"problemType":"TextBox","stepTitle":"Find P(C).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5}{42}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology8a-h1","type":"hint","dependencies":[],"title":"Definition of Probability","text":"The probability of any outcome in the sample space occuring is the long-term relative frequency of that oucome. Effectively, to find the probability of an event, P(event), we need to find the size of that event (how many outcomes are in it) and divide that by the size of the sample space (how many outcomes are in it).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a0ec58fterminology8a-h1"],"title":"Size of C","text":"What is the size of C? Effectively, how many bags of confetti are in the sample space?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$42$$"],"dependencies":["a0ec58fterminology8a-h2"],"title":"Size of Sample Space","text":"What is the size of the sample space? Effectively, how many total party favors were there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a0ec58fterminology8a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$42$$"],"dependencies":[],"title":"Size of Sample Space","text":"What is $$12+15+10+5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a0ec58fterminology8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{42}$$"],"dependencies":["a0ec58fterminology8a-h3"],"title":"Determining P(C)","text":"What is P(C), or what is the size of C divided by the size of the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0ec58fterminology9","title":"Determining Probability Numerically","body":"A jar of $$150$$ jelly beans contains $$22$$ red jelly beans, $$38$$ yellow, $$20$$ green, $$28$$ purple, $$26$$ blue, and the rest are orange. One jelly bean is chosen from the box at random. Let B $$=$$ the event of getting a blue jelly bean.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Terminology","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a0ec58fterminology9a","stepAnswer":["$$\\\\frac{26}{150}$$"],"problemType":"TextBox","stepTitle":"Find P(B).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{26}{150}$$","hints":{"DefaultPathway":[{"id":"a0ec58fterminology9a-h1","type":"hint","dependencies":[],"title":"Definition of Probability","text":"The probability of any outcome in the sample space occuring is the long-term relative frequency of that oucome. Effectively, to find the probability of an event, P(event), we need to find the size of that event (how many outcomes are in it) and divide that by the size of the sample space (how many outcomes are in it).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$26$$"],"dependencies":["a0ec58fterminology9a-h1"],"title":"Size of B","text":"What is the size of B? Effectively, how many jelly beans are blue in the sample space?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$150$$"],"dependencies":["a0ec58fterminology9a-h2"],"title":"Size of Sample Space","text":"What is the size of the sample space? Effectively, how many total jelly beans were there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0ec58fterminology9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{26}{150}$$"],"dependencies":["a0ec58fterminology9a-h3"],"title":"Determining P(B)","text":"What is P(B), or what is the size of B divided by the size of the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0f69c4conic1","title":"Identifying a Conic Given the Polar Form","body":"For each of the following equations, identify the conic with focus at the origin, the directrix, and the eccentricity.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Conic Sections in Polar Coordinates","courseName":"OpenStax: College Algebra","steps":[{"id":"a0f69c4conic1a","stepAnswer":["Ellipse; Eccentricity, $$e=\\\\frac{2}{3}$$ Directrix, $$y=3$$"],"problemType":"MultipleChoice","stepTitle":"$$r=\\\\frac{6}{3+2sin\\\\left(\\\\theta\\\\right)}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Ellipse; Eccentricity, $$e=\\\\frac{2}{3}$$ Directrix, $$y=3$$","choices":["Ellipse; Eccentricity, $$e=\\\\frac{2}{3}$$ Directrix, $$y=3$$","Parabola; Eccentricity, $$e=1;$$ Directrix, $$y=\\\\frac{-7}{2}$$","Hyperbola; Eccentricity, $$e=\\\\frac{5}{4}$$ Directrix, $$x=\\\\frac{12}{5}$$"],"hints":{"DefaultPathway":[{"id":"a0f69c4conic1a-h1","type":"hint","dependencies":[],"title":"Polar Equation of Conic","text":"For a conic with a focus at the origin, if the directrix is $$x=p$$ or $$x=-p$$, where $$p$$ is a positive real number, and the eccentricity is a positive real number e, the conic has a polar equation\\\\n$$r=\\\\frac{e p}{1+e cos\\\\left(\\\\theta\\\\right)}$$ or $$r=\\\\frac{e p}{1-e cos\\\\left(\\\\theta\\\\right)}$$\\\\nFor a conic with a focus at the origin, if the directrix is $$y=p$$ or $$y=-p$$, where $$p$$ is a positive real number, and the eccentricity is a positive real number e, the conic has a polar equation\\\\n$$r=\\\\frac{e p}{1+e sin\\\\left(\\\\theta\\\\right)}$$ or $$r=\\\\frac{e p}{1-e sin\\\\left(\\\\theta\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["a0f69c4conic1a-h1"],"title":"Standard Form","text":"We want to multiply the numerator and denominator by the reciprocal of the constant of the original equation, $$\\\\frac{1}{c}$$, where c is the constant so that we can change the equation to the standard polar form. What is the reciprocal that we want to multiply?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{1+\\\\frac{2}{3} sin\\\\left(\\\\theta\\\\right)}$$"],"dependencies":["a0f69c4conic1a-h2"],"title":"Standard Form","text":"After multiplying by the reciprocal of the constant, what is the polar equation now?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{3}$$"],"dependencies":["a0f69c4conic1a-h3"],"title":"Eccentricity","text":"In the standard polar form, the coefficient in front of $$sin\\\\left(\\\\theta\\\\right)$$ or $$cos\\\\left(\\\\theta\\\\right)$$ is the eccentricity. What is the eccentricity?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic1a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y$$"],"dependencies":["a0f69c4conic1a-h4"],"title":"Directrix","text":"Since $$sin\\\\left(\\\\theta\\\\right)$$ is in the denominator, is the directrix along the $$y$$ or $$x$$ axis?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x$$","$$y$$"]},{"id":"a0f69c4conic1a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a0f69c4conic1a-h5"],"title":"Directrix","text":"The numerator is the product of the eccentricity and the directrix. Now that we know the eccentricity and numerator, what is the directrix?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic1a-h7","type":"hint","dependencies":["a0f69c4conic1a-h6"],"title":"Type of Conic","text":"For a conic with eccentricity e,\\\\n- if $$0 \\\\leq e<1$$, the conic is ellipse\\\\n- if $$e=1$$. the conic is a parabola\\\\n- if $$e>1$$, the conic is a hyperbola","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic1a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Ellipse"],"dependencies":["a0f69c4conic1a-h7"],"title":"Type of Conic","text":"Given that we have found the eccentricity $$e=\\\\frac{2}{3}$$, what type of conic is this?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Ellipse","Parabola","Hyperbola"]}]}},{"id":"a0f69c4conic1b","stepAnswer":["Hyperbola; Eccentricity, $$e=\\\\frac{5}{4}$$ Directrix, $$x=\\\\frac{12}{5}$$"],"problemType":"MultipleChoice","stepTitle":"$$r=\\\\frac{12}{4+5cos\\\\left(\\\\theta\\\\right)}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Hyperbola; Eccentricity, $$e=\\\\frac{5}{4}$$ Directrix, $$x=\\\\frac{12}{5}$$","choices":["Ellipse; Eccentricity, $$e=\\\\frac{2}{3}$$ Directrix, $$y=3$$","Parabola; Eccentricity, $$e=1;$$ Directrix, $$y=\\\\frac{-7}{2}$$","Hyperbola; Eccentricity, $$e=\\\\frac{5}{4}$$ Directrix, $$x=\\\\frac{12}{5}$$"],"hints":{"DefaultPathway":[{"id":"a0f69c4conic1b-h1","type":"hint","dependencies":[],"title":"Polar Equation of Conic","text":"For a conic with a focus at the origin, if the directrix is $$x=p$$ or $$x=-p$$, where $$p$$ is a positive real number, and the eccentricity is a positive real number e, the conic has a polar equation\\\\n$$r=\\\\frac{e p}{1+e cos\\\\left(\\\\theta\\\\right)}$$ or $$r=\\\\frac{e p}{1-e cos\\\\left(\\\\theta\\\\right)}$$\\\\nFor a conic with a focus at the origin, if the directrix is $$y=p$$ or $$y=-p$$, where $$p$$ is a positive real number, and the eccentricity is a positive real number e, the conic has a polar equation\\\\n$$r=\\\\frac{e p}{1+e sin\\\\left(\\\\theta\\\\right)}$$ or $$r=\\\\frac{e p}{1-e sin\\\\left(\\\\theta\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic1b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4}$$"],"dependencies":["a0f69c4conic1b-h1"],"title":"Standard Form","text":"We want to multiply the numerator and denominator by the reciprocal of the constant of the original equation, $$\\\\frac{1}{c}$$, where c is the constant so that we can change the equation to the standard polar form. What is the reciprocal that we want to multiply?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic1b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{1+\\\\frac{5}{4} cos\\\\left(\\\\theta\\\\right)}$$"],"dependencies":["a0f69c4conic1b-h2"],"title":"Standard Form","text":"After multiplying by the reciprocal of the constant, what is the polar equation now?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic1b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{4}$$"],"dependencies":["a0f69c4conic1b-h3"],"title":"Eccentricity","text":"In the standard polar form, the coefficient in front of $$sin\\\\left(\\\\theta\\\\right)$$ or $$cos\\\\left(\\\\theta\\\\right)$$ is the eccentricity. What is the eccentricity?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic1b-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x$$"],"dependencies":["a0f69c4conic1b-h4"],"title":"Directrix","text":"Since $$sin\\\\left(\\\\theta\\\\right)$$ is in the denominator, is the directrix along the $$y$$ or $$x$$ axis?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x$$","$$y$$"]},{"id":"a0f69c4conic1b-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{12}{5}$$"],"dependencies":["a0f69c4conic1b-h5"],"title":"Directrix","text":"The numerator is the product of the eccentricity and the directrix. Now that we know the eccentricity and numerator, what is the directrix?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic1b-h7","type":"hint","dependencies":["a0f69c4conic1b-h6"],"title":"Type of Conic","text":"For a conic with eccentricity e,\\\\n- if $$0 \\\\leq e<1$$, the conic is ellipse\\\\n- if $$e=1$$. the conic is a parabola\\\\n- if $$e>1$$, the conic is a hyperbola","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic1b-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Hyperbola"],"dependencies":["a0f69c4conic1b-h7"],"title":"Type of Conic","text":"Given that we have found the eccentricity $$e=\\\\frac{5}{4}$$, what type of conic is this?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Ellipse","Parabola","Hyperbola"]}]}},{"id":"a0f69c4conic1c","stepAnswer":["Parabola; Eccentricity, $$e=1;$$ Directrix, $$y=\\\\frac{-7}{2}$$"],"problemType":"MultipleChoice","stepTitle":"$$r=\\\\frac{7}{2-2sin\\\\left(\\\\theta\\\\right)}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Parabola; Eccentricity, $$e=1;$$ Directrix, $$y=\\\\frac{-7}{2}$$","choices":["Ellipse; Eccentricity, $$e=\\\\frac{2}{3}$$ Directrix, $$y=3$$","Parabola; Eccentricity, $$e=1;$$ Directrix, $$y=\\\\frac{-7}{2}$$","Hyperbola; Eccentricity, $$e=\\\\frac{5}{4}$$ Directrix, $$x=\\\\frac{12}{5}$$"],"hints":{"DefaultPathway":[{"id":"a0f69c4conic1c-h1","type":"hint","dependencies":[],"title":"Polar Equation of Conic","text":"For a conic with a focus at the origin, if the directrix is $$x=p$$ or $$x=-p$$, where $$p$$ is a positive real number, and the eccentricity is a positive real number e, the conic has a polar equation\\\\n$$r=\\\\frac{e p}{1+e cos\\\\left(\\\\theta\\\\right)}$$ or $$r=\\\\frac{e p}{1-e cos\\\\left(\\\\theta\\\\right)}$$\\\\nFor a conic with a focus at the origin, if the directrix is $$y=p$$ or $$y=-p$$, where $$p$$ is a positive real number, and the eccentricity is a positive real number e, the conic has a polar equation\\\\n$$r=\\\\frac{e p}{1+e sin\\\\left(\\\\theta\\\\right)}$$ or $$r=\\\\frac{e p}{1-e sin\\\\left(\\\\theta\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic1c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a0f69c4conic1c-h1"],"title":"Standard Form","text":"We want to multiply the numerator and denominator by the reciprocal of the constant of the original equation, $$\\\\frac{1}{c}$$, where c is the constant so that we can change the equation to the standard polar form. What is the reciprocal that we want to multiply?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic1c-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\frac{7}{2}}{1-sin\\\\left(\\\\theta\\\\right)}$$"],"dependencies":["a0f69c4conic1c-h2"],"title":"Standard Form","text":"After multiplying by the reciprocal of the constant, what is the polar equation now?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic1c-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a0f69c4conic1c-h3"],"title":"Eccentricity","text":"In the standard polar form, the coefficient in front of $$sin\\\\left(\\\\theta\\\\right)$$ or $$cos\\\\left(\\\\theta\\\\right)$$ is the eccentricity. What is the eccentricity?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic1c-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y$$"],"dependencies":["a0f69c4conic1c-h4"],"title":"Directrix","text":"Since $$sin\\\\left(\\\\theta\\\\right)$$ is in the denominator, is the directrix along the $$y$$ or $$x$$ axis?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x$$","$$y$$"]},{"id":"a0f69c4conic1c-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-7}{2}$$"],"dependencies":["a0f69c4conic1c-h5"],"title":"Directrix","text":"The numerator is the product of the eccentricity and the directrix. Now that we know the eccentricity and numerator, what is the directrix? (Recall that the directrix, $$p$$, follows the sign of the coefficient of the $$sin\\\\left(\\\\theta\\\\right)$$ or cos(\\\\theta))","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic1c-h7","type":"hint","dependencies":["a0f69c4conic1c-h6"],"title":"Type of Conic","text":"For a conic with eccentricity e,\\\\n- if $$0 \\\\leq e<1$$, the conic is ellipse\\\\n- if $$e=1$$. the conic is a parabola\\\\n- if $$e>1$$, the conic is a hyperbola","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic1c-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Parabola"],"dependencies":["a0f69c4conic1c-h7"],"title":"Type of Conic","text":"Given that we have found the eccentricity $$e=1$$, what type of conic is this?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Ellipse","Parabola","Hyperbola"]}]}}]},{"id":"a0f69c4conic10","title":"Identify a Conic Given the Polar Form","body":"$$r=\\\\frac{8}{4-3cos\\\\left(\\\\theta\\\\right)}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Conic Sections in Polar Coordinates","courseName":"OpenStax: College Algebra","steps":[{"id":"a0f69c4conic10a","stepAnswer":["$$\\\\frac{3}{4}$$"],"problemType":"TextBox","stepTitle":"Give the Eccentricity","stepBody":"Identify the eccentricity","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{4}$$","hints":{"DefaultPathway":[{"id":"a0f69c4conic10a-h1","type":"hint","dependencies":[],"title":"Standard Form of Conic","text":"Rewrite the equation in standard form which has a $$1$$ as the constant in the denominator. Achieve standard form by multiplying the numerator and denominator by the reciprocal of the constant of the original equation, $$\\\\frac{1}{4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{4}$$"],"dependencies":["a0f69c4conic10a-h1"],"title":"Identify Eccentricity","text":"Given the standard form is $$r=\\\\frac{ep}{1\\\\pm ecostheta}$$, identify the eccentricity (e).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a0f69c4conic10b","stepAnswer":["$$\\\\frac{8}{3}$$"],"problemType":"TextBox","stepTitle":"Give the Directrix","stepBody":"Identiy the directrix","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{8}{3}$$","hints":{"DefaultPathway":[{"id":"a0f69c4conic10b-h1","type":"hint","dependencies":[],"title":"Format of Directrix","text":"Since cos\u03b8 is in the denominator, the directrix is $$x=p$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic10b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{8}{3}$$"],"dependencies":["a0f69c4conic10b-h1"],"title":"Identify Directrix","text":"Comparing to standard form, $$e=\\\\frac{3}{4}$$. Therefore, from the numerator solve for the directrix (p).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a0f69c4conic10c","stepAnswer":["Ellipse"],"problemType":"MultipleChoice","stepTitle":"Identify the Conic","stepBody":"What type of conic does the polar equation represent?","answerType":"string","variabilization":{},"choices":["Parabola","Hyperbola","Ellipse"],"hints":{"DefaultPathway":[{"id":"a0f69c4conic10c-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Ellipse"],"dependencies":[],"title":"Types of Conic","text":"Since $$e<1$$, identify the conic.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Parabola","Hyperbola","Ellipse"]}]}}]},{"id":"a0f69c4conic11","title":"Identify a Conic Given the Polar Form","body":"$$r=\\\\frac{5}{1+2sintheta}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Conic Sections in Polar Coordinates","courseName":"OpenStax: College Algebra","steps":[{"id":"a0f69c4conic11a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"Give the Eccentricity","stepBody":"Identify the eccentricity","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a0f69c4conic11a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Standard Form of Conic","text":"Since the equation is already in standard form r=(ep)/(1~(esin\u03b8), identify the eccentricity (e).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a0f69c4conic11b","stepAnswer":["$$\\\\frac{5}{2}$$"],"problemType":"TextBox","stepTitle":"Give the Directrix","stepBody":"Identiy the directrix","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5}{2}$$","hints":{"DefaultPathway":[{"id":"a0f69c4conic11b-h1","type":"hint","dependencies":[],"title":"Format of Directrix","text":"Since sin\u03b8 is in the denominator, the directrix is $$y=p$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic11b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{2}$$"],"dependencies":["a0f69c4conic11b-h1"],"title":"Identify Directrix","text":"Comparing to standard form, $$e=2$$. Therefore, from the numerator solve for the directrix (p).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a0f69c4conic11c","stepAnswer":["Hyperbola"],"problemType":"MultipleChoice","stepTitle":"Identify the Conic","stepBody":"What type of conic does the polar equation represent?","answerType":"string","variabilization":{},"choices":["Parabola","Hyperbola","Ellipse"],"hints":{"DefaultPathway":[{"id":"a0f69c4conic11c-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Hyperbola"],"dependencies":[],"title":"Types of Conic","text":"Since $$e>1$$, identify the conic.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Parabola","Hyperbola","Ellipse"]}]}}]},{"id":"a0f69c4conic12","title":"Identify a Conic Given the Polar Form","body":"$$r=\\\\frac{16}{4+3sintheta}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Conic Sections in Polar Coordinates","courseName":"OpenStax: College Algebra","steps":[{"id":"a0f69c4conic12a","stepAnswer":["$$\\\\frac{3}{4}$$"],"problemType":"TextBox","stepTitle":"Give the Eccentricity","stepBody":"Identify the eccentricity","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{4}$$","hints":{"DefaultPathway":[{"id":"a0f69c4conic12a-h1","type":"hint","dependencies":[],"title":"Standard Form of Conic","text":"Rewrite the equation in standard form which has a $$1$$ as the constant in the denominator. Achieve standard form by multiplying the numerator and denominator by the reciprocal of the constant of the original equation, $$\\\\frac{1}{4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{4}$$"],"dependencies":["a0f69c4conic12a-h1"],"title":"Identify Eccentricity","text":"Given the standard form is $$r=\\\\frac{ep}{1\\\\pm esintheta}$$, identify the eccentricity (e).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a0f69c4conic12b","stepAnswer":["$$\\\\frac{16}{3}$$"],"problemType":"TextBox","stepTitle":"Give the Directrix","stepBody":"Identiy the directrix","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{16}{3}$$","hints":{"DefaultPathway":[{"id":"a0f69c4conic12b-h1","type":"hint","dependencies":[],"title":"Format of Directrix","text":"Since sin\u03b8 is in the denominator, the directrix is $$y=p$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic12b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{16}{3}$$"],"dependencies":["a0f69c4conic12b-h1"],"title":"Identify Directrix","text":"Comparing to standard form, $$e=\\\\frac{3}{4}$$. Therefore, from the numerator solve for the directrix (p).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a0f69c4conic12c","stepAnswer":["Ellipse"],"problemType":"MultipleChoice","stepTitle":"Identify the Conic","stepBody":"What type of conic does the polar equation represent?","answerType":"string","variabilization":{},"choices":["Parabola","Hyperbola","Ellipse"],"hints":{"DefaultPathway":[{"id":"a0f69c4conic12c-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Ellipse"],"dependencies":[],"title":"Types of Conic","text":"Since $$e<1$$, identify the conic.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Parabola","Hyperbola","Ellipse"]}]}}]},{"id":"a0f69c4conic13","title":"Identify a Conic Given the Polar Form","body":"$$r=\\\\frac{3}{10+10costheta}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Conic Sections in Polar Coordinates","courseName":"OpenStax: College Algebra","steps":[{"id":"a0f69c4conic13a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"Give the Eccentricity","stepBody":"Identify the eccentricity","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a0f69c4conic13a-h1","type":"hint","dependencies":[],"title":"Standard Form of Conic","text":"Rewrite the equation in standard form which has a $$1$$ as the constant in the denominator. Achieve standard form by multiplying the numerator and denominator by the reciprocal of the constant of the original equation, $$\\\\frac{1}{10}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a0f69c4conic13a-h1"],"title":"Identify Eccentricity","text":"Given the standard form is $$r=\\\\frac{ep}{1\\\\pm ecostheta}$$, identify the eccentricity (e).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a0f69c4conic13b","stepAnswer":["$$\\\\frac{3}{10}$$"],"problemType":"TextBox","stepTitle":"Give the Directrix","stepBody":"Identiy the directrix","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{10}$$","hints":{"DefaultPathway":[{"id":"a0f69c4conic13b-h1","type":"hint","dependencies":[],"title":"Format of Directrix","text":"Since cos\u03b8 is in the denominator, the directrix is $$x=p$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic13b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{10}$$"],"dependencies":["a0f69c4conic13b-h1"],"title":"Identify Directrix","text":"Comparing to standard form, $$e=1$$. Therefore, from the numerator solve for the directrix (p).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a0f69c4conic13c","stepAnswer":["Parabola"],"problemType":"MultipleChoice","stepTitle":"Identify the Conic","stepBody":"What type of conic does the polar equation represent?","answerType":"string","variabilization":{},"choices":["Parabola","Hyperbola","Ellipse"],"hints":{"DefaultPathway":[{"id":"a0f69c4conic13c-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Parabola"],"dependencies":[],"title":"Types of Conic","text":"Since $$e=1$$, identify the conic.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Parabola","Hyperbola","Ellipse"]}]}}]},{"id":"a0f69c4conic14","title":"Identify a Conic Given the Polar Form","body":"$$r=\\\\frac{2}{1-costheta}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Conic Sections in Polar Coordinates","courseName":"OpenStax: College Algebra","steps":[{"id":"a0f69c4conic14a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"Give the Eccentricity","stepBody":"Identify the eccentricity","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a0f69c4conic14a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Standard Form of Conic","text":"Since the equation is already in standard form r=(ep)/(1~(ecos\u03b8), identify the eccentricity (e).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a0f69c4conic14b","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"Give the Directrix","stepBody":"Identiy the directrix","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a0f69c4conic14b-h1","type":"hint","dependencies":[],"title":"Format of Directrix","text":"Since cos\u03b8 is in the denominator, the directrix is $$x=p$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic14b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a0f69c4conic14b-h1"],"title":"Identify Directrix","text":"Comparing to standard form, $$e=1$$. Therefore, from the numerator solve for the directrix (p).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a0f69c4conic14c","stepAnswer":["Parabola"],"problemType":"MultipleChoice","stepTitle":"Identify the Conic","stepBody":"What type of conic does the polar equation represent?","answerType":"string","variabilization":{},"choices":["Parabola","Hyperbola","Ellipse"],"hints":{"DefaultPathway":[{"id":"a0f69c4conic14c-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Parabola"],"dependencies":[],"title":"Types of Conic","text":"Since $$e=1$$, identify the conic.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Parabola","Hyperbola","Ellipse"]}]}}]},{"id":"a0f69c4conic15","title":"Converting Polar Equations to Rectangular Equations","body":"Convert the equation to rectangular form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Conic Sections in Polar Coordinates","courseName":"OpenStax: College Algebra","steps":[{"id":"a0f69c4conic15a","stepAnswer":["$$\\\\sqrt{x^2+y^2}+3y=4$$"],"problemType":"TextBox","stepTitle":"Convert $$r=\\\\frac{4}{1+3sin\\\\left(\\\\theta\\\\right)}$$ to rectangular form.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\sqrt{x^2+y^2}+3y=4$$","hints":{"DefaultPathway":[{"id":"a0f69c4conic15a-h1","type":"hint","dependencies":[],"title":"Making Equivalent Polar to Cartesian Substitutions","text":"$$r=\\\\sqrt{x^2+y^2}$$, $$x=rcos(theta)$$, $$y=rsin(theta)$$. Making these substitutions, we get $$\\\\sqrt{x^2+y^2}+3y=4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0f69c4conic16","title":"Converting Polar Equations to Rectangular Equations","body":"Convert the equation to rectangular form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Conic Sections in Polar Coordinates","courseName":"OpenStax: College Algebra","steps":[{"id":"a0f69c4conic16a","stepAnswer":["$$5\\\\sqrt{x^2+y^2}-3y=2$$"],"problemType":"TextBox","stepTitle":"Convert $$r=\\\\frac{2}{5-3sin\\\\left(\\\\theta\\\\right)}$$ to rectangular form.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5\\\\sqrt{x^2+y^2}-3y=2$$","hints":{"DefaultPathway":[{"id":"a0f69c4conic16a-h1","type":"hint","dependencies":[],"title":"Making Equivalent Polar to Cartesian Substitutions","text":"Convert the equation to rectangular form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0f69c4conic17","title":"Converting Polar Equations to Rectangular Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Conic Sections in Polar Coordinates","courseName":"OpenStax: College Algebra","steps":[{"id":"a0f69c4conic17a","stepAnswer":["$$3\\\\sqrt{x^2+y^2}-2x=8$$"],"problemType":"TextBox","stepTitle":"Convert $$r=\\\\frac{8}{3-2cos\\\\left(\\\\theta\\\\right)}$$ to rectangular form.","stepBody":"Convert the equation to rectangular form.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3\\\\sqrt{x^2+y^2}-2x=8$$","hints":{"DefaultPathway":[{"id":"a0f69c4conic17a-h1","type":"hint","dependencies":[],"title":"Making Equivalent Polar to Cartesian Substitutions","text":"$$r=\\\\sqrt{x^2+y^2}$$, $$x=rcos(theta)$$, $$y=rsin(theta)$$. Making these substitutions, we get $$3\\\\sqrt{x^2+y^2}-2x=8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0f69c4conic18","title":"Converting Polar Equations to Rectangular Equations","body":"Convert the equation to rectangular form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Conic Sections in Polar Coordinates","courseName":"OpenStax: College Algebra","steps":[{"id":"a0f69c4conic18a","stepAnswer":["$$2\\\\sqrt{x^2+y^2}+5x=3$$"],"problemType":"TextBox","stepTitle":"Convert $$r=\\\\frac{3}{2+5cos\\\\left(\\\\theta\\\\right)}$$ to rectangular form.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2\\\\sqrt{x^2+y^2}+5x=3$$","hints":{"DefaultPathway":[{"id":"a0f69c4conic18a-h1","type":"hint","dependencies":[],"title":"Making Equivalent Polar to Cartesian Substitutions","text":"Convert the equation to rectangular form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0f69c4conic19","title":"Converting Polar Equations to Rectangular Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Conic Sections in Polar Coordinates","courseName":"OpenStax: College Algebra","steps":[{"id":"a0f69c4conic19a","stepAnswer":["$$2\\\\sqrt{x^2+y^2}+2y=4$$"],"problemType":"TextBox","stepTitle":"Convert $$r=\\\\frac{4}{2+2sin\\\\left(\\\\theta\\\\right)}$$ to rectangular form.","stepBody":"Convert the equation to rectangular form.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2\\\\sqrt{x^2+y^2}+2y=4$$","hints":{"DefaultPathway":[{"id":"a0f69c4conic19a-h1","type":"hint","dependencies":[],"title":"Making Equivalent Polar to Cartesian Substitutions","text":"$$r=\\\\sqrt{x^2+y^2}$$, $$x=r cos\\\\left(\\\\theta\\\\right)$$, $$y=r sin\\\\left(\\\\theta\\\\right)$$. Making these substitutions, we get $$2\\\\sqrt{x^2+y^2}+2y=4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0f69c4conic2","title":"Identifying a Conic Given the Polar Form","body":"For each of the following equations, identify the conic with focus at the origin, the directrix, and the eccentricity.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Conic Sections in Polar Coordinates","courseName":"OpenStax: College Algebra","steps":[{"id":"a0f69c4conic2a","stepAnswer":["Ellipse; Eccentricity, $$e=\\\\frac{1}{3}$$ Directrix, $$x=-2$$"],"problemType":"MultipleChoice","stepTitle":"$$r=\\\\frac{2}{3-cos\\\\left(\\\\theta\\\\right)}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Ellipse; Eccentricity, $$e=\\\\frac{1}{3}$$ Directrix, $$x=-2$$","choices":["Ellipse; Eccentricity, $$e=\\\\frac{1}{3}$$ Directrix, $$x=-2$$","Parabola; Eccentricity, $$e=1;$$ Directrix, $$x=\\\\frac{-2}{3}$$","Hyperbola; Eccentricity, $$e=\\\\frac{5}{3}$$ Directrix, $$x=-2$$"],"hints":{"DefaultPathway":[{"id":"a0f69c4conic2a-h1","type":"hint","dependencies":[],"title":"Polar Equation of Conic","text":"For a conic with a focus at the origin, if the directrix is $$x=p$$ or $$x=-p$$, where $$p$$ is a positive real number, and the eccentricity is a positive real number e, the conic has a polar equation\\\\n$$r=\\\\frac{e p}{1+e cos\\\\left(\\\\theta\\\\right)}$$ or $$r=\\\\frac{e p}{1-e cos\\\\left(\\\\theta\\\\right)}$$\\\\nFor a conic with a focus at the origin, if the directrix is $$y=p$$ or $$y=-p$$, where $$p$$ is a positive real number, and the eccentricity is a positive real number e, the conic has a polar equation\\\\n$$r=\\\\frac{e p}{1+e sin\\\\left(\\\\theta\\\\right)}$$ or $$r=\\\\frac{e p}{1-e sin\\\\left(\\\\theta\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["a0f69c4conic2a-h1"],"title":"Standard Form","text":"We want to multiply the numerator and denominator by the reciprocal of the constant of the original equation, $$\\\\frac{1}{c}$$, where c is the constant so that we can change the equation to the standard polar form. What is the reciprocal that we want to multiply?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\frac{2}{3}}{1-\\\\frac{1}{3} cos\\\\left(\\\\theta\\\\right)}$$"],"dependencies":["a0f69c4conic2a-h2"],"title":"Standard Form","text":"After multiplying by the reciprocal of the constant, what is the polar equation now?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["a0f69c4conic2a-h3"],"title":"Eccentricity","text":"In the standard polar form, the coefficient in front of $$sin\\\\left(\\\\theta\\\\right)$$ or $$cos\\\\left(\\\\theta\\\\right)$$ is the eccentricity. What is the eccentricity?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic2a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x$$"],"dependencies":["a0f69c4conic2a-h4"],"title":"Directrix","text":"Since $$sin\\\\left(\\\\theta\\\\right)$$ is in the denominator, is the directrix along the $$y$$ or $$x$$ axis?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x$$","$$y$$"]},{"id":"a0f69c4conic2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a0f69c4conic2a-h5"],"title":"Directrix","text":"The numerator is the product of the eccentricity and the directrix. Now that we know the eccentricity and numerator, what is the directrix? (Recall that the directrix, $$p$$, follows the sign of the coefficient of the $$sin\\\\left(\\\\theta\\\\right)$$ or cos(\\\\theta))","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic2a-h7","type":"hint","dependencies":["a0f69c4conic2a-h6"],"title":"Type of Conic","text":"For a conic with eccentricity e,\\\\n- if $$0 \\\\leq e<1$$, the conic is ellipse\\\\n- if $$e=1$$. the conic is a parabola\\\\n- if $$e>1$$, the conic is a hyperbola","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic2a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Ellipse"],"dependencies":["a0f69c4conic2a-h7"],"title":"Type of Conic","text":"Given that we have found the eccentricity $$e=\\\\frac{1}{3}$$, what type of conic is this?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Ellipse","Parabola","Hyperbola"]}]}}]},{"id":"a0f69c4conic20","title":"Converting Polar Equations to Rectangular Equations","body":"Convert the equation to rectangular form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Conic Sections in Polar Coordinates","courseName":"OpenStax: College Algebra","steps":[{"id":"a0f69c4conic20a","stepAnswer":["$$8\\\\sqrt{x^2+y^2}-8x=3$$"],"problemType":"TextBox","stepTitle":"Convert $$r=\\\\frac{3}{8-8cos\\\\left(\\\\theta\\\\right)}$$ to rectangular form.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8\\\\sqrt{x^2+y^2}-8x=3$$","hints":{"DefaultPathway":[{"id":"a0f69c4conic20a-h1","type":"hint","dependencies":[],"title":"Making Equivalent Polar to Cartesian Substitutions","text":"Convert the equation to rectangular form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0f69c4conic21","title":"Converting Polar Equations to Rectangular Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Conic Sections in Polar Coordinates","courseName":"OpenStax: College Algebra","steps":[{"id":"a0f69c4conic21a","stepAnswer":["$$6\\\\sqrt{x^2+y^2}+7x=2$$"],"problemType":"TextBox","stepTitle":"Convert $$r=\\\\frac{2}{6+7cos\\\\left(\\\\theta\\\\right)}$$ to rectangular form.","stepBody":"Convert the equation to rectangular form.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6\\\\sqrt{x^2+y^2}+7x=2$$","hints":{"DefaultPathway":[{"id":"a0f69c4conic21a-h1","type":"hint","dependencies":[],"title":"Making Equivalent Polar to Cartesian Substitutions","text":"$$r=\\\\sqrt{x^2+y^2}$$, $$x=rcos(theta)$$, $$y=rsin(theta)$$. Making these substitutions, we get $$6\\\\sqrt{x^2+y^2}+7x=2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0f69c4conic3","title":"Finding the Polar Form of a Vertical Conic Given a Focus at the Origin and the Eccentricity and Directrix","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Conic Sections in Polar Coordinates","courseName":"OpenStax: College Algebra","steps":[{"id":"a0f69c4conic3a","stepAnswer":["$$r=\\\\frac{6}{1-3sin\\\\left(\\\\theta\\\\right)}$$"],"problemType":"MultipleChoice","stepTitle":"Find the polar form of the conic given a focus at the origin, $$e=3$$ and directrix $$y=-2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$r=\\\\frac{6}{1-3sin\\\\left(\\\\theta\\\\right)}$$","choices":["$$r=\\\\frac{6}{1-3sin\\\\left(\\\\theta\\\\right)}$$","$$r=\\\\frac{-6}{1+3sin\\\\left(\\\\theta\\\\right)}$$","$$r=\\\\frac{6}{1-3cos\\\\left(\\\\theta\\\\right)}$$"],"hints":{"DefaultPathway":[{"id":"a0f69c4conic3a-h1","type":"hint","dependencies":[],"title":"Polar Equation of Conic","text":"For a conic with a focus at the origin, if the directrix is $$x=p$$ or $$x=-p$$, where $$p$$ is a positive real number, and the eccentricity is a positive real number e, the conic has a polar equation\\\\n$$r=\\\\frac{e p}{1+e cos\\\\left(\\\\theta\\\\right)}$$ or $$r=\\\\frac{e p}{1-e cos\\\\left(\\\\theta\\\\right)}$$\\\\nFor a conic with a focus at the origin, if the directrix is $$y=p$$ or $$y=-p$$, where $$p$$ is a positive real number, and the eccentricity is a positive real number e, the conic has a polar equation\\\\n$$r=\\\\frac{e p}{1+e sin\\\\left(\\\\theta\\\\right)}$$ or $$r=\\\\frac{e p}{1-e sin\\\\left(\\\\theta\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic3a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Sine"],"dependencies":["a0f69c4conic3a-h1"],"title":"Directrix","text":"The directrix is $$y=-p$$. Which trigonometric function is in the denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Sine","Cosine"]},{"id":"a0f69c4conic3a-h3","type":"hint","dependencies":["a0f69c4conic3a-h2"],"title":"Polar Equation","text":"Our polar equation takes the form $$r=\\\\frac{e p}{1-e sin\\\\left(\\\\theta\\\\right)}$$ as identified by the directrix $$y=-p$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a0f69c4conic3a-h3"],"title":"Numerator","text":"The numerator is the product of the eccentricity and the absolute of the directrix, |p|. What is the numerator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a0f69c4conic3a-h4"],"title":"Eccentricity","text":"The eccentricity is the magnitude of the coefficient of the trigonometric function in the denominator. Thus, what is the coefficient of the trigonometric function?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic3a-h6","type":"hint","dependencies":["a0f69c4conic3a-h5"],"title":"Polar Form","text":"Substitute the values that were found into the polar equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0f69c4conic4","title":"Finding the Polar Form of a Horizontal Conic Given a Focus at the Origin and the Eccentricity and Directrix","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Conic Sections in Polar Coordinates","courseName":"OpenStax: College Algebra","steps":[{"id":"a0f69c4conic4a","stepAnswer":["$$r=\\\\frac{12}{5+3cos\\\\left(\\\\theta\\\\right)}$$"],"problemType":"MultipleChoice","stepTitle":"Find the polar form of the conic given a focus at the origin, $$e=\\\\frac{3}{5}$$ and directrix $$x=4$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$r=\\\\frac{12}{5+3cos\\\\left(\\\\theta\\\\right)}$$","choices":["$$r=\\\\frac{12}{5+3cos\\\\left(\\\\theta\\\\right)}$$","$$r=\\\\frac{12}{5-3cos\\\\left(\\\\theta\\\\right)}$$","$$r=\\\\frac{-12}{5+3sin\\\\left(\\\\theta\\\\right)}$$"],"hints":{"DefaultPathway":[{"id":"a0f69c4conic4a-h1","type":"hint","dependencies":[],"title":"Polar Equation of Conic","text":"For a conic with a focus at the origin, if the directrix is $$x=p$$ or $$x=-p$$, where $$p$$ is a positive real number, and the eccentricity is a positive real number e, the conic has a polar equation\\\\n$$r=\\\\frac{e p}{1+e cos\\\\left(\\\\theta\\\\right)}$$ or $$r=\\\\frac{e p}{1-e cos\\\\left(\\\\theta\\\\right)}$$\\\\nFor a conic with a focus at the origin, if the directrix is $$y=p$$ or $$y=-p$$, where $$p$$ is a positive real number, and the eccentricity is a positive real number e, the conic has a polar equation\\\\n$$r=\\\\frac{e p}{1+e sin\\\\left(\\\\theta\\\\right)}$$ or $$r=\\\\frac{e p}{1-e sin\\\\left(\\\\theta\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic4a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Cosine"],"dependencies":["a0f69c4conic4a-h1"],"title":"Directrix","text":"The directrix is $$x=p$$. Which trigonometric function is in the denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Sine","Cosine"]},{"id":"a0f69c4conic4a-h3","type":"hint","dependencies":["a0f69c4conic4a-h2"],"title":"Polar Equation","text":"Our polar equation takes the form $$r=\\\\frac{e p}{1+e cos\\\\left(\\\\theta\\\\right)}$$ as identified by the directrix $$x=p$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{12}{5}$$"],"dependencies":["a0f69c4conic4a-h3"],"title":"Numerator","text":"The numerator is the product of the eccentricity and the absolute of the directrix, |p|. What is the numerator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic4a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{5}$$"],"dependencies":["a0f69c4conic4a-h4"],"title":"Eccentricity","text":"The eccentricity is the magnitude of the coefficient of the trigonometric function in the denominator. Thus, what is the coefficient of the trigonometric function?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic4a-h6","type":"hint","dependencies":["a0f69c4conic4a-h5"],"title":"Polar Form","text":"Substitute the values that were found to obtain the polar form of the conic. We can multiply by $$5$$ to the numerator and denominator to remove the fractions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0f69c4conic5","title":"Finding the Polar Form of a Horizontal Conic Given a Focus at the Origin and the Eccentricity and Directrix","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Conic Sections in Polar Coordinates","courseName":"OpenStax: College Algebra","steps":[{"id":"a0f69c4conic5a","stepAnswer":["$$r=\\\\frac{1}{1-cos\\\\left(\\\\theta\\\\right)}$$"],"problemType":"MultipleChoice","stepTitle":"Find the polar form of the conic given a focus at the origin, $$e=1$$ and directrix $$x=-1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$r=\\\\frac{1}{1-cos\\\\left(\\\\theta\\\\right)}$$","choices":["$$r=\\\\frac{1}{1-cos\\\\left(\\\\theta\\\\right)}$$","$$r=\\\\frac{1}{1+cos\\\\left(\\\\theta\\\\right)}$$","$$r=\\\\frac{1}{1-sin\\\\left(\\\\theta\\\\right)}$$"],"hints":{"DefaultPathway":[{"id":"a0f69c4conic5a-h1","type":"hint","dependencies":[],"title":"Polar Equation of Conic","text":"For a conic with a focus at the origin, if the directrix is $$x=p$$ or $$x=-p$$, where $$p$$ is a positive real number, and the eccentricity is a positive real number e, the conic has a polar equation\\\\n$$r=\\\\frac{e p}{1+e cos\\\\left(\\\\theta\\\\right)}$$ or $$r=\\\\frac{e p}{1-e cos\\\\left(\\\\theta\\\\right)}$$\\\\nFor a conic with a focus at the origin, if the directrix is $$y=p$$ or $$y=-p$$, where $$p$$ is a positive real number, and the eccentricity is a positive real number e, the conic has a polar equation\\\\n$$r=\\\\frac{e p}{1+e sin\\\\left(\\\\theta\\\\right)}$$ or $$r=\\\\frac{e p}{1-e sin\\\\left(\\\\theta\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic5a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Cosine"],"dependencies":["a0f69c4conic5a-h1"],"title":"Directrix","text":"The directrix is $$x=-p$$. Which trigonometric function is in the denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Sine","Cosine"]},{"id":"a0f69c4conic5a-h3","type":"hint","dependencies":["a0f69c4conic5a-h2"],"title":"Polar Equation","text":"Our polar equation takes the form $$r=\\\\frac{e p}{1-e cos\\\\left(\\\\theta\\\\right)}$$ as identified by the directrix $$x=-p$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a0f69c4conic5a-h3"],"title":"Numerator","text":"The numerator is the product of the eccentricity and the absolute of the directrix, |p|. What is the numerator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a0f69c4conic5a-h4"],"title":"Eccentricity","text":"The eccentricity is the magnitude of the coefficient of the trigonometric function in the denominator. The sign of the coefficient follows the sign of the directrix, $$x=-1$$. Thus, what is the coefficient of the trigonometric function?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic5a-h6","type":"hint","dependencies":["a0f69c4conic5a-h5"],"title":"Polar Form","text":"Substitute the values that were found to obtain the polar form of the conic.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0f69c4conic6","title":"Converting a Conic in Polar Form to Rectangular Form","body":"Convert the following conic to rectangular form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Conic Sections in Polar Coordinates","courseName":"OpenStax: College Algebra","steps":[{"id":"a0f69c4conic6a","stepAnswer":["$$25x^2-10y=1$$"],"problemType":"MultipleChoice","stepTitle":"$$r=\\\\frac{1}{5-5sin\\\\left(\\\\theta\\\\right)}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$25x^2-10y=1$$","choices":["$$25x^2-10y=1$$","$$25y^2-10x=1$$","$$25x^2-10y^2=1$$"],"hints":{"DefaultPathway":[{"id":"a0f69c4conic6a-h1","type":"hint","dependencies":[],"title":"Identities","text":"Useful identities to switch from polar form to rectangular form are\\\\n$$r=\\\\sqrt{x^2+y^2}$$\\\\n$$x=r cos\\\\left(\\\\theta\\\\right)$$\\\\n$$y=r sin\\\\left(\\\\theta\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic6a-h2","type":"hint","dependencies":["a0f69c4conic6a-h1"],"title":"Eliminate the fraction","text":"We can eliminate the fraction by multiplying the denominator across both sides of the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic6a-h3","type":"hint","dependencies":["a0f69c4conic6a-h2"],"title":"Distribute","text":"We can distribute the $$r$$ that was initially on the LHS into the $$5-5sin\\\\left(\\\\theta\\\\right)$$ that was multiplied to the LHS.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic6a-h4","type":"hint","dependencies":["a0f69c4conic6a-h3"],"title":"Isolate $$5r$$","text":"We want to isolate the $$r$$ terms that are not associated multiplied to a trigonometric function. We do so by adding $$5r sin\\\\left(\\\\theta\\\\right)$$ across both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic6a-h5","type":"hint","dependencies":["a0f69c4conic6a-h4"],"title":"Square Both Sides","text":"Square both sides of the equation. The goal is to obtain the $$r^2$$ so that we can utilize the identity $$r=\\\\sqrt{x^2+y^2}$$ without the square root.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic6a-h6","type":"hint","dependencies":["a0f69c4conic6a-h5"],"title":"Substitution of Identities","text":"Substitute identities $$r=\\\\sqrt{x^2+y^2}$$ and $$y=r sin\\\\left(\\\\theta\\\\right)$$ into the current equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic6a-h7","type":"hint","dependencies":["a0f69c4conic6a-h6"],"title":"Distribute and use FOIL","text":"We want to expand out the equation so that we can later rearrange the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic6a-h8","type":"hint","dependencies":["a0f69c4conic6a-h7"],"title":"Rearrange Terms and Set Constant to $$1$$","text":"We force out the standard rectangular form by rearranging all the terms with variables $$x$$ and $$y$$ to the LHS and ensuring the constant is $$1$$ on the RHS.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0f69c4conic7","title":"Converting a Conic in Polar Form to Rectangular Form","body":"Convert the following conic to rectangular form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Conic Sections in Polar Coordinates","courseName":"OpenStax: College Algebra","steps":[{"id":"a0f69c4conic7a","stepAnswer":["$$\\\\frac{{\\\\left(x-\\\\frac{4}{3}\\\\right)}^2}{\\\\frac{4}{9}}-\\\\frac{y^2}{\\\\frac{4}{3}}=1$$"],"problemType":"MultipleChoice","stepTitle":"$$r=\\\\frac{2}{1+2cos\\\\left(\\\\theta\\\\right)}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{{\\\\left(x-\\\\frac{4}{3}\\\\right)}^2}{\\\\frac{4}{9}}-\\\\frac{y^2}{\\\\frac{4}{3}}=1$$","choices":["$$\\\\frac{{\\\\left(x-\\\\frac{4}{3}\\\\right)}^2}{\\\\frac{4}{9}}-\\\\frac{y^2}{\\\\frac{4}{3}}=1$$","$$\\\\frac{{\\\\left(y-\\\\frac{4}{3}\\\\right)}^2}{\\\\frac{4}{9}}-\\\\frac{x^2}{\\\\frac{4}{3}}=1$$","$$\\\\frac{{\\\\left(x-\\\\frac{4}{3}\\\\right)}^2}{\\\\frac{4}{3}}-\\\\frac{y^2}{\\\\frac{4}{9}}=1$$"],"hints":{"DefaultPathway":[{"id":"a0f69c4conic7a-h1","type":"hint","dependencies":[],"title":"Identities","text":"Useful identities to switch from polar form to rectangular form are\\\\n$$r=\\\\sqrt{x^2+y^2}$$\\\\n$$x=r cos\\\\left(\\\\theta\\\\right)$$\\\\n$$y=r sin\\\\left(\\\\theta\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic7a-h2","type":"hint","dependencies":["a0f69c4conic7a-h1"],"title":"Eliminate the fraction","text":"We can eliminate the fraction by multiplying the denominator across both sides of the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic7a-h3","type":"hint","dependencies":["a0f69c4conic7a-h2"],"title":"Distribute","text":"We can distribute the $$r$$ that was initially on the LHS into the (1+2*cos(\\\\theta))) that was multiplied to the LHS.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic7a-h4","type":"hint","dependencies":["a0f69c4conic7a-h3"],"title":"Isolate $$r$$","text":"We want to isolate the $$r$$ terms that are not associated multiplied to a trigonometric function. We do so by subtracting $$2r cos\\\\left(\\\\theta\\\\right)$$ across both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic7a-h5","type":"hint","dependencies":["a0f69c4conic7a-h4"],"title":"Square Both Sides","text":"Square both sides of the equation. The goal is to obtain the $$r^2$$ so that we can utilize the identity $$r=\\\\sqrt{x^2+y^2}$$ without the square root.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic7a-h6","type":"hint","dependencies":["a0f69c4conic7a-h5"],"title":"Substitution of Identities","text":"Substitute identities $$r=\\\\sqrt{x^2+y^2}$$ and $$x=r cos\\\\left(\\\\theta\\\\right)$$ into the current equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic7a-h7","type":"hint","dependencies":["a0f69c4conic7a-h6"],"title":"Distribute and use FOIL","text":"We want to expand out the equation so that we can later rearrange the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic7a-h8","type":"hint","dependencies":["a0f69c4conic7a-h7"],"title":"Combining Like Terms","text":"We notice that there are multiple terms involving $$x$$. We want to convert to the standard rectangular form, thus we would want to combine all like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic7a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3{\\\\left(x-\\\\frac{4}{3}\\\\right)}^2-\\\\frac{4}{3}$$"],"dependencies":["a0f69c4conic7a-h8"],"title":"Completing the Square","text":"Notice that we have $$y^2=3x^2-8x+4$$ or something similar if you have rearranged differently. We would like to complete the square on the RHS. For $$a x^2+b x+c$$, we can complete the square as $$a {\\\\left(x+d\\\\right)}^2+e$$ where $$d=\\\\frac{b}{2a}$$ and $$e=c-\\\\frac{b^2}{4a}$$. Completing the square on $$3x^2-8x+4$$, what expression do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic7a-h10","type":"hint","dependencies":["a0f69c4conic7a-h9"],"title":"Rearrange Terms and Set Constant to $$1$$","text":"We rearrange so that the constant is on one side and the variables are on the other. Lastly, we divide by constant, $$\\\\frac{4}{3}$$, across both side to set the constant to $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic7a-h11","type":"hint","dependencies":["a0f69c4conic7a-h10"],"title":"Standard Form","text":"If there is a constant coefficient in the numerator, we shift the multiplier to the denominator by multiplying by the reciprocal of the constant. For e.g., a(x-h)**/b will be divided by $$\\\\frac{\\\\frac{1}{a}}{\\\\frac{1}{a}}$$ so that it becomes $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{\\\\frac{b}{a}}$$. This will allow us to obtain the standard form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0f69c4conic8","title":"Identify a Conic Given the Polar Form","body":"$$r=\\\\frac{3}{4-4sintheta}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Conic Sections in Polar Coordinates","courseName":"OpenStax: College Algebra","steps":[{"id":"a0f69c4conic8a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"Give the Eccentricity","stepBody":"Identify the eccentricity","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a0f69c4conic8a-h1","type":"hint","dependencies":[],"title":"Standard Form of Conic","text":"Rewrite the equation in standard form which has a $$1$$ as the constant in the denominator. Achieve standard form by multiplying the numerator and denominator by the reciprocal of the constant of the original equation, $$\\\\frac{1}{4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a0f69c4conic8a-h1"],"title":"Identify Eccentricity","text":"Given the standard form is $$r=\\\\frac{ep}{1\\\\pm esintheta}$$, identify the eccentricity (e).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a0f69c4conic8b","stepAnswer":["$$\\\\frac{3}{4}$$"],"problemType":"TextBox","stepTitle":"Give the Directrix","stepBody":"Identiy the directrix","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{4}$$","hints":{"DefaultPathway":[{"id":"a0f69c4conic8b-h1","type":"hint","dependencies":[],"title":"Format of Directrix","text":"Since sin\u03b8 is in the denominator, the directrix is $$y=p$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic8b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{4}$$"],"dependencies":["a0f69c4conic8b-h1"],"title":"Identify Directrix","text":"Comparing to standard form, $$e=1$$. Therefore, from the numerator solve for the directrix (p).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a0f69c4conic8c","stepAnswer":["Parabola"],"problemType":"MultipleChoice","stepTitle":"Identify the Conic","stepBody":"What type of conic does the polar equation represent?","answerType":"string","variabilization":{},"choices":["Parabola","Hyperbola","Ellipse"],"hints":{"DefaultPathway":[{"id":"a0f69c4conic8c-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Parabola"],"dependencies":[],"title":"Types of Conic","text":"Since $$e=1$$, identify the conic.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Parabola","Hyperbola","Ellipse"]}]}}]},{"id":"a0f69c4conic9","title":"Identify a Conic Given the Polar Form","body":"$$r=\\\\frac{6}{1-2sintheta}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Conic Sections in Polar Coordinates","courseName":"OpenStax: College Algebra","steps":[{"id":"a0f69c4conic9a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"Give the Eccentricity","stepBody":"Identify the eccentricity","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a0f69c4conic9a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Standard Form of Conic","text":"Since the equation is already in standard form r=(ep)/(1~(ecos\u03b8), identify the eccentricity (e).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a0f69c4conic9b","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"Give the Directrix","stepBody":"Identiy the directrix","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a0f69c4conic9b-h1","type":"hint","dependencies":[],"title":"Format of Directrix","text":"Since cos\u03b8 is in the denominator, the directrix is $$x=p$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conic9b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a0f69c4conic9b-h1"],"title":"Identify Directrix","text":"Comparing to standard form, $$e=2$$. Therefore, from the numerator solve for the directrix (p).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a0f69c4conic9c","stepAnswer":["Hyperbola"],"problemType":"MultipleChoice","stepTitle":"Identify the Conic","stepBody":"What type of conic does the polar equation represent?","answerType":"string","variabilization":{},"choices":["Parabola","Hyperbola","Ellipse"],"hints":{"DefaultPathway":[{"id":"a0f69c4conic9c-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Hyperbola"],"dependencies":[],"title":"Types of Conic","text":"Since $$e>1$$, identify the conic.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Parabola","Hyperbola","Ellipse"]}]}}]},{"id":"a0f69c4conics1","title":"Determine parts of the graph","body":"Find the focus of the ellipse.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Conic Sections in Polar Coordinates","courseName":"OpenStax: College Algebra","steps":[{"id":"a0f69c4conics1a","stepAnswer":["$$(0,0)$$"],"problemType":"MultipleChoice","stepTitle":"$$r=\\\\frac{5}{2+costheta}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,0)$$","choices":["$$(0,0)$$","$$(1,0)$$","$$(1,1)$$"],"hints":{"DefaultPathway":[{"id":"a0f69c4conics1a-h1","type":"hint","dependencies":[],"title":"Standard form","text":"The standard form of a conic with directrix $$x=\\\\pm p$$ is r=(ep)/(1\xb1ecos\u03b8). The standard form of a conic with directrix $$y=\\\\pm p$$ is r=(ep)/(1\xb1esin\u03b8).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conics1a-h2","type":"hint","dependencies":["a0f69c4conics1a-h1"],"title":"Multiply","text":"Multiply the numerator and denominator by the reciprocal of the constant in the denominator to rewrite the equation in standard form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conics1a-h3","type":"hint","dependencies":["a0f69c4conics1a-h2"],"title":"Identify","text":"Identify the eccentricity e as the coefficient of the trigonometric function in the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conics1a-h4","type":"hint","dependencies":["a0f69c4conics1a-h3"],"title":"Determine","text":"Because $$e>1$$, the focus is at the origin.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0f69c4conics2","title":"Determine parts of the graph","body":"Find the focus of the hyperbola.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Conic Sections in Polar Coordinates","courseName":"OpenStax: College Algebra","steps":[{"id":"a0f69c4conics2a","stepAnswer":["$$(0,0)$$"],"problemType":"MultipleChoice","stepTitle":"$$r=\\\\frac{8}{4-5costheta}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,0)$$","choices":["$$(0,0)$$","$$(1,0)$$","$$(1,1)$$"],"hints":{"DefaultPathway":[{"id":"a0f69c4conics2a-h1","type":"hint","dependencies":[],"title":"Standard form","text":"The standard form of a conic with directrix $$x=\\\\pm p$$ is r=(ep)/(1\xb1ecos\u03b8). The standard form of a conic with directrix $$y=\\\\pm p$$ is r=(ep)/(1\xb1esin\u03b8).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conics2a-h2","type":"hint","dependencies":["a0f69c4conics2a-h1"],"title":"Multiply","text":"Multiply the numerator and denominator by the reciprocal of the constant in the denominator to rewrite the equation in standard form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conics2a-h3","type":"hint","dependencies":["a0f69c4conics2a-h2"],"title":"Identify","text":"Identify the eccentricity e as the coefficient of the trigonometric function in the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conics2a-h4","type":"hint","dependencies":["a0f69c4conics2a-h3"],"title":"Determine","text":"Because $$e>1$$, the focus is at the origin.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0f69c4conics3","title":"Determine parts of the graph","body":"Find the directrix of the hyperbola.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Conic Sections in Polar Coordinates","courseName":"OpenStax: College Algebra","steps":[{"id":"a0f69c4conics3a","stepAnswer":["$$x=\\\\frac{-8}{5}$$"],"problemType":"TextBox","stepTitle":"$$r=\\\\frac{8}{4-5costheta}$$","stepBody":"Enter your answer in the form: $$x=a$$ or $$y=a$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x=\\\\frac{-8}{5}$$","hints":{"DefaultPathway":[{"id":"a0f69c4conics3a-h1","type":"hint","dependencies":[],"title":"Standard form","text":"The standard form of a conic with directrix $$x=\\\\pm p$$ is r=(ep)/(1\xb1ecos\u03b8). The standard form of a conic with directrix $$y=\\\\pm p$$ is r=(ep)/(1\xb1esin\u03b8).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conics3a-h2","type":"hint","dependencies":["a0f69c4conics3a-h1"],"title":"Multiply","text":"Multiply the numerator and denominator by the reciprocal of the constant in the denominator to rewrite the equation in standard form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conics3a-h3","type":"hint","dependencies":["a0f69c4conics3a-h2"],"title":"Identify","text":"Identify the eccentricity e as the coefficient of the trigonometric function in the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conics3a-h4","type":"hint","dependencies":["a0f69c4conics3a-h3"],"title":"Determine","text":"Since cos is in the denominator, and there is a subtraction sign in the denominator, the directrix is in the form $$x=-p$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conics3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{8}{5}$$"],"dependencies":["a0f69c4conics3a-h4"],"title":"Set ep","text":"Set ep equal to the value in the numerator. Plug in e, and solve for $$p$$. What is $$p$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a0f69c4conics3a-h5-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$\\\\frac{8}{5}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a0f69c4conics3a-h6","type":"hint","dependencies":["a0f69c4conics3a-h5"],"title":"Answer","text":"Therefore, the directrix is $$x=\\\\frac{-8}{5}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0f69c4conics4","title":"Determine parts of the graph","body":"Find the directrix of the parabola.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Conic Sections in Polar Coordinates","courseName":"OpenStax: College Algebra","steps":[{"id":"a0f69c4conics4a","stepAnswer":["$$y=-2$$"],"problemType":"TextBox","stepTitle":"$$r=\\\\frac{2}{1-sintheta}$$","stepBody":"Enter your answer in the form: $$x=a$$ or $$y=a$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=-2$$","hints":{"DefaultPathway":[{"id":"a0f69c4conics4a-h1","type":"hint","dependencies":[],"title":"Standard form","text":"The standard form of a conic with directrix $$x=\\\\pm p$$ is r=(ep)/(1\xb1ecos\u03b8). The standard form of a conic with directrix $$y=\\\\pm p$$ is r=(ep)/(1\xb1esin\u03b8).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conics4a-h2","type":"hint","dependencies":["a0f69c4conics4a-h1"],"title":"Multiply","text":"Multiply the numerator and denominator by the reciprocal of the constant in the denominator to rewrite the equation in standard form(if needed).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conics4a-h3","type":"hint","dependencies":["a0f69c4conics4a-h2"],"title":"Identify","text":"Identify the eccentricity e as the coefficient of the trigonometric function in the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conics4a-h4","type":"hint","dependencies":["a0f69c4conics4a-h3"],"title":"Determine","text":"Since sin is in the denominator, and there is a subtraction sign in the denominator, the directrix is in the form $$y=-p$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conics4a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a0f69c4conics4a-h4"],"title":"Set ep","text":"Set ep equal to the value in the numerator. Plug in e, and solve for $$p$$. What is $$p$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a0f69c4conics4a-h5-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a0f69c4conics4a-h6","type":"hint","dependencies":["a0f69c4conics4a-h5"],"title":"Answer","text":"Therefore, the directrix is $$y=-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0f69c4conics5","title":"Determine parts of the graph","body":"Find the directrix of the parabola.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Conic Sections in Polar Coordinates","courseName":"OpenStax: College Algebra","steps":[{"id":"a0f69c4conics5a","stepAnswer":["$$x=5$$"],"problemType":"TextBox","stepTitle":"$$r\\\\left(1+costheta\\\\right)=5$$","stepBody":"Enter your answer in the form: $$x=a$$ or $$y=a$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x=5$$","hints":{"DefaultPathway":[{"id":"a0f69c4conics5a-h1","type":"hint","dependencies":[],"title":"Standard form","text":"The standard form of a conic with directrix $$x=\\\\pm p$$ is r=(ep)/(1\xb1ecos\u03b8). The standard form of a conic with directrix $$y=\\\\pm p$$ is r=(ep)/(1\xb1esin\u03b8).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conics5a-h2","type":"hint","dependencies":["a0f69c4conics5a-h1"],"title":"Multiply","text":"Multiply the numerator and denominator by the reciprocal of the constant in the denominator to rewrite the equation in standard form(if needed).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conics5a-h3","type":"hint","dependencies":["a0f69c4conics5a-h2"],"title":"Identify","text":"Identify the eccentricity e as the coefficient of the trigonometric function in the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conics5a-h4","type":"hint","dependencies":["a0f69c4conics5a-h3"],"title":"Determine","text":"Since cos is in the denominator, and there is an addition sign in the denominator, the directrix is in the form $$x=p$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conics5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a0f69c4conics5a-h4"],"title":"Set ep","text":"Set ep equal to the value in the numerator. Plug in e, and solve for $$p$$. What is $$p$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a0f69c4conics5a-h5-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a0f69c4conics5a-h6","type":"hint","dependencies":["a0f69c4conics5a-h5"],"title":"Answer","text":"Therefore, the directrix is $$x=5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0f69c4conics6","title":"Determine parts of the graph","body":"Find the focus of the parabola.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Conic Sections in Polar Coordinates","courseName":"OpenStax: College Algebra","steps":[{"id":"a0f69c4conics6a","stepAnswer":["$$(0,0)$$"],"problemType":"MultipleChoice","stepTitle":"$$r=\\\\frac{2}{1-sintheta}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,0)$$","choices":["$$(0,0)$$","$$(1,0)$$","$$(1,1)$$"],"hints":{"DefaultPathway":[{"id":"a0f69c4conics6a-h1","type":"hint","dependencies":[],"title":"Standard form","text":"The standard form of a conic with directrix $$x=\\\\pm p$$ is r=(ep)/(1\xb1ecos\u03b8). The standard form of a conic with directrix $$y=\\\\pm p$$ is r=(ep)/(1\xb1esin\u03b8).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conics6a-h2","type":"hint","dependencies":["a0f69c4conics6a-h1"],"title":"Multiply","text":"Multiply the numerator and denominator by the reciprocal of the constant in the denominator to rewrite the equation in standard form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conics6a-h3","type":"hint","dependencies":["a0f69c4conics6a-h2"],"title":"Identify","text":"Identify the eccentricity e as the coefficient of the trigonometric function in the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conics6a-h4","type":"hint","dependencies":["a0f69c4conics6a-h3"],"title":"Determine","text":"Because $$e>1$$, the focus is at the origin.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a0f69c4conics7","title":"Determine parts of the graph","body":"Find the focus of the parabola.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Conic Sections in Polar Coordinates","courseName":"OpenStax: College Algebra","steps":[{"id":"a0f69c4conics7a","stepAnswer":["$$(0,0)$$"],"problemType":"MultipleChoice","stepTitle":"$$r\\\\left(1+costheta\\\\right)=5$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,0)$$","choices":["$$(0,0)$$","$$(1,0)$$","$$(1,1)$$"],"hints":{"DefaultPathway":[{"id":"a0f69c4conics7a-h1","type":"hint","dependencies":[],"title":"Standard form","text":"The standard form of a conic with directrix $$x=\\\\pm p$$ is r=(ep)/(1\xb1ecos\u03b8). The standard form of a conic with directrix $$y=\\\\pm p$$ is r=(ep)/(1\xb1esin\u03b8).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conics7a-h2","type":"hint","dependencies":["a0f69c4conics7a-h1"],"title":"Multiply","text":"Multiply the numerator and denominator by the reciprocal of the constant in the denominator to rewrite the equation in standard form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conics7a-h3","type":"hint","dependencies":["a0f69c4conics7a-h2"],"title":"Identify","text":"Identify the eccentricity e as the coefficient of the trigonometric function in the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a0f69c4conics7a-h4","type":"hint","dependencies":["a0f69c4conics7a-h3"],"title":"Determine","text":"Because $$e>1$$, the focus is at the origin.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a104861trifactor1","title":"Factor Perfect Square Trinomials","body":"Factor the perfect square trinomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Factor Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a104861trifactor1a","stepAnswer":["$${\\\\left(3x+2\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$9x^2+12x+4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(3x+2\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"a104861trifactor1a-h1","type":"hint","dependencies":[],"title":"Perfect Square Trinomials Pattern","text":"The trinomial fits the perfect square trinomials pattern $$a^2+2a b+b^2$$ because the first and last terms are perfect squares and the middle term matches $$2a b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor1a-h2","type":"hint","dependencies":["a104861trifactor1a-h1"],"title":"Square of the Binomial","text":"Take the perfect squares of the first and last term and write is as a binomial so you will get $${\\\\left(3x+2\\\\right)}^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor1a-h3","type":"hint","dependencies":["a104861trifactor1a-h2"],"title":"Checking Binomial","text":"Check if the squared binomial multiplies out into the original trinomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a104861trifactor10","title":"Factor Special Products","body":"Factor the trinomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Factor Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a104861trifactor10a","stepAnswer":["$$\\\\left(9x+5\\\\right) \\\\left(x+5\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$9x^2+50x+25$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(9x+5\\\\right) \\\\left(x+5\\\\right)$$","hints":{"DefaultPathway":[{"id":"a104861trifactor10a-h1","type":"hint","dependencies":[],"title":"Perfect Square Trinomials Pattern","text":"The first and last terms are perfect squares but the middle term does not match $$2a b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor10a-h2","type":"hint","dependencies":["a104861trifactor10a-h1"],"title":"Alternative Method","text":"Factor using the \\"ac\\" method. Varibles \\"a\\" and \\"c\\" multiply out to $$225$$ and by experimenting with different combinations of the middle term we can see that $$5\\\\times45=225$$ which comes from $$5+45=50$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor10a-h3","type":"hint","dependencies":["a104861trifactor10a-h2"],"title":"Splitting the Middle Term","text":"Split the middle term into the identified numbers before: $$9x^2+5x+45x+25$$. Then factor by grouping: $$x \\\\left(9x+5\\\\right)+5\\\\left(9x+5\\\\right)$$ which results in $$\\\\left(9x+5\\\\right) \\\\left(x+5\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a104861trifactor11","title":"Factor Special Products","body":"Factor the trinomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Factor Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a104861trifactor11a","stepAnswer":["$$\\\\left(8r+3\\\\right) \\\\left(2r+3\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$16r^2+30r+9$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(8r+3\\\\right) \\\\left(2r+3\\\\right)$$","hints":{"DefaultPathway":[{"id":"a104861trifactor11a-h1","type":"hint","dependencies":[],"title":"Perfect Square Trinomials Pattern","text":"The first and last terms are perfect squares but the middle term does not match $$2a b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor11a-h2","type":"hint","dependencies":["a104861trifactor11a-h1"],"title":"Alternative Method","text":"Factor using the \\"ac\\" method. Varibles \\"a\\" and \\"c\\" multiply out to $$144$$ and by experimenting with different combinations of the middle term we can see that $$24\\\\times6=144$$ which comes from $$24+6=144$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor11a-h3","type":"hint","dependencies":["a104861trifactor11a-h2"],"title":"Splitting the Middle Term","text":"Split the middle term into the identified numbers before: $$16r^2+24r+6r+9$$. Then factor by grouping: $$8r \\\\left(2r+3\\\\right)+3\\\\left(2r+3\\\\right)$$ which results in $$\\\\left(8r+3\\\\right) \\\\left(2r+3\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a104861trifactor12","title":"Factor Perfect Square Trinomials","body":"Factor the perfect square trinomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Factor Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a104861trifactor12a","stepAnswer":["$${\\\\left(4y+3\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$16y^2+24y+9$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(4y+3\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"a104861trifactor12a-h1","type":"hint","dependencies":[],"title":"Perfect Square Trinomials Pattern","text":"The trinomial fits the perfect square trinomials pattern $$a^2+2a b+b^2$$ because the first and last terms are perfect squares and the middle term matches $$2a b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor12a-h2","type":"hint","dependencies":["a104861trifactor12a-h1"],"title":"Square of the Binomial","text":"Take the perfect squares of the first and last term and write is as a binomial so you will get $${\\\\left(4y+3\\\\right)}^2$$. Notice how it\'s now subtraction since the original middle term was negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor12a-h3","type":"hint","dependencies":["a104861trifactor12a-h2"],"title":"Checking Binomial","text":"Check if the squared binomial multiplies out into the original trinomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a104861trifactor13","title":"Factor Perfect Square Trinomials","body":"Factor the perfect square trinomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Factor Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a104861trifactor13a","stepAnswer":["$${\\\\left(5v+2\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$25v^2+20v+4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(5v+2\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"a104861trifactor13a-h1","type":"hint","dependencies":[],"title":"Perfect Square Trinomials Pattern","text":"The trinomial fits the perfect square trinomials pattern $$a^2+2a b+b^2$$ because the first and last terms are perfect squares and the middle term matches $$2a b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor13a-h2","type":"hint","dependencies":["a104861trifactor13a-h1"],"title":"Square of the Binomial","text":"Take the perfect squares of the first and last term and write is as a binomial so you will get $${\\\\left(5v+2\\\\right)}^2$$. Notice how it\'s now subtraction since the original middle term was negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor13a-h3","type":"hint","dependencies":["a104861trifactor13a-h2"],"title":"Checking Binomial","text":"Check if the squared binomial multiplies out into the original trinomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a104861trifactor14","title":"Factor Perfect Square Trinomials","body":"Factor the perfect square trinomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Factor Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a104861trifactor14a","stepAnswer":["$${\\\\left(6s+7\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$36s^2+84s+49$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(6s+7\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"a104861trifactor14a-h1","type":"hint","dependencies":[],"title":"Perfect Square Trinomials Pattern","text":"The trinomial fits the perfect square trinomials pattern $$a^2+2a b+b^2$$ because the first and last terms are perfect squares and the middle term matches $$2a b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor14a-h2","type":"hint","dependencies":["a104861trifactor14a-h1"],"title":"Square of the Binomial","text":"Take the perfect squares of the first and last term and write is as a binomial so you will get $${\\\\left(6s+7\\\\right)}^2$$. Notice how it\'s now subtraction since the original middle term was negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor14a-h3","type":"hint","dependencies":["a104861trifactor14a-h2"],"title":"Checking Binomial","text":"Check if the squared binomial multiplies out into the original trinomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a104861trifactor15","title":"Factor Perfect Square Trinomials","body":"Factor the perfect square trinomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Factor Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a104861trifactor15a","stepAnswer":["$${\\\\left(7s+11\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$49s^2+154s+121$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(7s+11\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"a104861trifactor15a-h1","type":"hint","dependencies":[],"title":"Perfect Square Trinomials Pattern","text":"The trinomial fits the perfect square trinomials pattern $$a^2+2a b+b^2$$ because the first and last terms are perfect squares and the middle term matches $$2a b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor15a-h2","type":"hint","dependencies":["a104861trifactor15a-h1"],"title":"Square of the Binomial","text":"Take the perfect squares of the first and last term and write is as a binomial so you will get $${\\\\left(7s+11\\\\right)}^2$$. Notice how it\'s now subtraction since the original middle term was negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor15a-h3","type":"hint","dependencies":["a104861trifactor15a-h2"],"title":"Checking Binomial","text":"Check if the squared binomial multiplies out into the original trinomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a104861trifactor16","title":"Factor Perfect Square Trinomials","body":"Factor.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Factor Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a104861trifactor16a","stepAnswer":["$${\\\\left(4y+3\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$16y^2+24y+9$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(4y+3\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"a104861trifactor16a-h1","type":"hint","dependencies":[],"title":"Perfect Square","text":"This equation is a perfect square","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor16a-h2","type":"hint","dependencies":["a104861trifactor16a-h1"],"title":"Factors","text":"In this equation $${ax}^2+bx+c$$ the square root of a and c are $$4$$ and $$3$$ respectively.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor16a-h3","type":"hint","dependencies":["a104861trifactor16a-h2"],"title":"Plug in","text":"Plug into the perfect square simplified form $${\\\\left(nx+m\\\\right)}^2$$ where $$n$$ is the square root of a and $$m$$ is the square root of c","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a104861trifactor17","title":"Factor Perfect Square Trinomials","body":"Factor.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Factor Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a104861trifactor17a","stepAnswer":["$${\\\\left(5v+2\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$25v^2+20v+4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(5v+2\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"a104861trifactor17a-h1","type":"hint","dependencies":[],"title":"Perfect Square","text":"This equation is a perfect square","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor17a-h2","type":"hint","dependencies":["a104861trifactor17a-h1"],"title":"Factors","text":"In this equation $${ax}^2+bx+c$$ the square root of a and c are $$5$$ and $$2$$ respectively.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor17a-h3","type":"hint","dependencies":["a104861trifactor17a-h2"],"title":"Plug in","text":"Plug into the perfect square simplified form $${\\\\left(nx+m\\\\right)}^2$$ where $$n$$ is the square root of a and $$m$$ is the square root of c","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a104861trifactor18","title":"Factor Perfect Square Trinomials","body":"Factor.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Factor Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a104861trifactor18a","stepAnswer":["$${\\\\left(6s+7\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$36s^2+84s+49$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(6s+7\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"a104861trifactor18a-h1","type":"hint","dependencies":[],"title":"Perfect Square","text":"This equation is a perfect square","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor18a-h2","type":"hint","dependencies":["a104861trifactor18a-h1"],"title":"Factors","text":"In this equation $${ax}^2+bx+c$$ the square root of a and c are $$6$$ and $$7$$ respectively.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor18a-h3","type":"hint","dependencies":["a104861trifactor18a-h2"],"title":"Plug in","text":"Plug into the perfect square simplified form $${\\\\left(nx+m\\\\right)}^2$$ where $$n$$ is the square root of a and $$m$$ is the square root of c","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a104861trifactor19","title":"Factor Perfect Square Trinomials","body":"Factor.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Factor Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a104861trifactor19a","stepAnswer":["$${\\\\left(7s+11\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$49s^2+154s+121$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(7s+11\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"a104861trifactor19a-h1","type":"hint","dependencies":[],"title":"Perfect Square","text":"This equation is a perfect square","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor19a-h2","type":"hint","dependencies":["a104861trifactor19a-h1"],"title":"Factors","text":"In this equation $${ax}^2+bx+c$$ the square root of a and c are $$7$$ and $$11$$ respectively.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor19a-h3","type":"hint","dependencies":["a104861trifactor19a-h2"],"title":"Plug in","text":"Plug into the perfect square simplified form $${\\\\left(nx+m\\\\right)}^2$$ where $$n$$ is the square root of a and $$m$$ is the square root of c","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a104861trifactor2","title":"Factor Perfect Square Trinomials","body":"Factor the perfect square trinomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Factor Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a104861trifactor2a","stepAnswer":["$${\\\\left(2x+3\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$4x^2+12x+9$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(2x+3\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"a104861trifactor2a-h1","type":"hint","dependencies":[],"title":"Perfect Square Trinomials Pattern","text":"The trinomial fits the perfect square trinomials pattern $$a^2+2a b+b^2$$ because the first and last terms are perfect squares and the middle term matches $$2a b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor2a-h2","type":"hint","dependencies":["a104861trifactor2a-h1"],"title":"Square of the Binomial","text":"Take the perfect squares of the first and last term and write is as a binomial so you will get $${\\\\left(2x+3\\\\right)}^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor2a-h3","type":"hint","dependencies":["a104861trifactor2a-h2"],"title":"Checking Binomial","text":"Check if the squared binomial multiplies out into the original trinomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a104861trifactor20","title":"Factor Perfect Square Trinomials","body":"Factor.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Factor Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a104861trifactor20a","stepAnswer":["$${\\\\left(10x-1\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$100x^2-20x+1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(10x-1\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"a104861trifactor20a-h1","type":"hint","dependencies":[],"title":"Perfect Square","text":"This equation is a perfect square","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor20a-h2","type":"hint","dependencies":["a104861trifactor20a-h1"],"title":"Factors","text":"In this equation $${ax}^2+bx+c$$ the square root of a and c are $$10$$ and $$-1$$ respectively.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor20a-h3","type":"hint","dependencies":["a104861trifactor20a-h2"],"title":"Plug in","text":"Plug into the perfect square simplified form $${\\\\left(nx+m\\\\right)}^2$$ where $$n$$ is the square root of a and $$m$$ is the square root of c","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a104861trifactor21","title":"Factor Perfect Square Trinomials","body":"Factor.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Factor Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a104861trifactor21a","stepAnswer":["$${\\\\left(8x-1\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$64m^2-16m+1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(8x-1\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"a104861trifactor21a-h1","type":"hint","dependencies":[],"title":"Perfect Square","text":"This equation is a perfect square","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor21a-h2","type":"hint","dependencies":["a104861trifactor21a-h1"],"title":"Factors","text":"In this equation $${ax}^2+bx+c$$ the square root of a and c are $$8$$ and $$-1$$ respectively.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor21a-h3","type":"hint","dependencies":["a104861trifactor21a-h2"],"title":"Plug in","text":"Plug into the perfect square simplified form $${\\\\left(nx+m\\\\right)}^2$$ where $$n$$ is the square root of a and $$m$$ is the square root of c","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a104861trifactor22","title":"Factor Perfect Square Trinomials","body":"Factor(Give your answer with the larger value first. eg. $$\\\\left(x+2\\\\right) \\\\left(x+1\\\\right)$$ $$2$$ is greater than 1).","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Factor Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a104861trifactor22a","stepAnswer":["$$\\\\left(5n+4\\\\right) \\\\left(5n+1\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$25n^2+25n+4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(5n+4\\\\right) \\\\left(5n+1\\\\right)$$","hints":{"DefaultPathway":[{"id":"a104861trifactor22a-h1","type":"hint","dependencies":[],"title":"Factor","text":"In this equation $${ax}^2+bx+c$$ the square root of a is $$5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor22a-h2","type":"hint","dependencies":["a104861trifactor22a-h1"],"title":"Factor Pt2.","text":"In the form $$\\\\left(px+o\\\\right) \\\\left(mx+n\\\\right)$$. To solve for o and $$n$$, they must multiply to the constant in the original equation and $$pn+om$$ is equal to $$b$$ in $${ax}^2+bx+c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor22a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["4,1"],"dependencies":["a104861trifactor22a-h2"],"title":"Solve","text":"What are the values of o and $$n$$? (Give answer with coma inbetween with the larger number first Ex. 4,2)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor22a-h4","type":"hint","dependencies":[],"title":"Plug in","text":"Plug in the calculated values into the simplified form","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a104861trifactor23","title":"Factor Perfect Square Trinomials","body":"Factor.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Factor Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a104861trifactor23a","stepAnswer":["$${\\\\left(10y-1\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$100y^2-20y+1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(10y-1\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"a104861trifactor23a-h1","type":"hint","dependencies":[],"title":"Perfect Square","text":"This equation is a perfect square","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor23a-h2","type":"hint","dependencies":["a104861trifactor23a-h1"],"title":"Factors","text":"In this equation $${ax}^2+bx+c$$ the square root of a and c are $$10$$ and $$-1$$ respectively.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor23a-h3","type":"hint","dependencies":["a104861trifactor23a-h2"],"title":"Plug in","text":"Plug into the perfect square simplified form $${\\\\left(nx+m\\\\right)}^2$$ where $$n$$ is the square root of a and $$m$$ is the square root of c","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a104861trifactor24","title":"Factor Perfect Square Trinomials","body":"Factor.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Factor Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a104861trifactor24a","stepAnswer":["$${\\\\left(8m-1\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$64m^2-16m+1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(8m-1\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"a104861trifactor24a-h1","type":"hint","dependencies":[],"title":"Perfect Square","text":"This equation is a perfect square","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor24a-h2","type":"hint","dependencies":["a104861trifactor24a-h1"],"title":"Factors","text":"In this equation $${ax}^2+bx+c$$ the square root of a and c are $$8$$ and $$-1$$ respectively.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor24a-h3","type":"hint","dependencies":["a104861trifactor24a-h2"],"title":"Plug in","text":"Plug into the perfect square simplified form $${\\\\left(nx+m\\\\right)}^2$$ where $$n$$ is the square root of a and $$m$$ is the square root of c","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a104861trifactor25","title":"Factor Perfect Square Trinomials","body":"Factor.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Factor Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a104861trifactor25a","stepAnswer":["$$10{\\\\left(x+4\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$10k^2+80k+160$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10{\\\\left(x+4\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"a104861trifactor25a-h1","type":"hint","dependencies":[],"title":"Perfect Square","text":"This equation is a perfect square","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor25a-h2","type":"hint","dependencies":["a104861trifactor25a-h1"],"title":"Common Factor","text":"Take out the $$10$$ to make $$\\\\operatorname{10}\\\\left(k^2+8k+16\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor25a-h3","type":"hint","dependencies":["a104861trifactor25a-h2"],"title":"Factors","text":"In this equation $$z\\\\left({ax}^2+bx+c\\\\right)$$ the square root of a and c are $$1$$ and $$4$$ respectively.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor25a-h4","type":"hint","dependencies":["a104861trifactor25a-h3"],"title":"Plug in","text":"Plug into the perfect square simplified form $${z\\\\left(nx+m\\\\right)}^2$$ where $$n$$ is the square root of a and $$m$$ is the square root of c","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a104861trifactor26","title":"Factor Perfect Square Trinomials","body":"Factor.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Factor Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a104861trifactor26a","stepAnswer":["$$4{\\\\left(4x-3\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$64x^2-96x+36$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4{\\\\left(4x-3\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"a104861trifactor26a-h1","type":"hint","dependencies":[],"title":"Perfect Square","text":"This equation is a perfect square","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor26a-h2","type":"hint","dependencies":["a104861trifactor26a-h1"],"title":"Common Factor","text":"Take out the $$4$$ to make $$4\\\\left(16x^2-24x+9\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor26a-h3","type":"hint","dependencies":["a104861trifactor26a-h2"],"title":"Factors","text":"In this equation $$z\\\\left({ax}^2+bx+c\\\\right)$$ the square root of a and c are $$4$$ and $$-3$$ respectively.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor26a-h4","type":"hint","dependencies":["a104861trifactor26a-h3"],"title":"Plug in","text":"Plug into the perfect square simplified form $${z\\\\left(nx+m\\\\right)}^2$$ where $$n$$ is the square root of a and $$m$$ is the square root of c","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a104861trifactor27","title":"Factor Perfect Square Trinomials","body":"Factor.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Factor Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a104861trifactor27a","stepAnswer":["$$10p {\\\\left(3p+5q\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$90p^3+300p^2 q+250{pq}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10p {\\\\left(3p+5q\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"a104861trifactor27a-h1","type":"hint","dependencies":[],"title":"Perfect Square","text":"This equation is a perfect square","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor27a-h2","type":"hint","dependencies":["a104861trifactor27a-h1"],"title":"Common Factor","text":"Take out the 3u to make $$10p\\\\left(9p^2+30pq+25q^2\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor27a-h3","type":"hint","dependencies":["a104861trifactor27a-h2"],"title":"Factors","text":"In this equation $$z\\\\left({ax}^2+bx+c\\\\right)$$ the square root of a and c are $$3$$ and 5q respectively.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor27a-h4","type":"hint","dependencies":["a104861trifactor27a-h3"],"title":"Plug in","text":"Plug into the perfect square simplified form $${z\\\\left(nx+m\\\\right)}^2$$ where $$n$$ is the square root of a and $$m$$ is the square root of c","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a104861trifactor28","title":"Factor Perfect Square Trinomials","body":"Factor(Give your answer with the larger value first. eg. $$\\\\left(x+2\\\\right) \\\\left(x+1\\\\right)$$ $$2$$ is greater than 1).","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Factor Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a104861trifactor28a","stepAnswer":["$$\\\\left(x+4\\\\right) \\\\left(x-4\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$x^2-16$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(x+4\\\\right) \\\\left(x-4\\\\right)$$","hints":{"DefaultPathway":[{"id":"a104861trifactor28a-h1","type":"hint","dependencies":[],"title":"Rule","text":"If bx is $$0$$ and c is negative, then the simplified form will be $$\\\\left(nx+m\\\\right) \\\\left(nx-m\\\\right)$$. $$n$$ is the square root of a and $$m$$ is the square root of -c","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor28a-h2","type":"hint","dependencies":["a104861trifactor28a-h1"],"title":"Factors","text":"In this equation $${ax}^2+bx+c$$ the square root of a and -c are $$1$$ and $$4$$ respectfully","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor28a-h3","type":"hint","dependencies":["a104861trifactor28a-h2"],"title":"Plug in","text":"Plug into the perfect square simplified form $$\\\\left(nx+m\\\\right) \\\\left(nx-m\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a104861trifactor29","title":"Factor Perfect Square Trinomials","body":"Factor(Give your answer with the larger value first. eg. $$\\\\left(x+2\\\\right) \\\\left(x+1\\\\right)$$ $$2$$ is greater than 1).","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Factor Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a104861trifactor29a","stepAnswer":["$$\\\\left(n+3\\\\right) \\\\left(n-3\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$n^2-9$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(n+3\\\\right) \\\\left(n-3\\\\right)$$","hints":{"DefaultPathway":[{"id":"a104861trifactor29a-h1","type":"hint","dependencies":[],"title":"Rule","text":"If bx is $$0$$ and c is negative, then the simplified form will be $$\\\\left(nx+m\\\\right) \\\\left(nx-m\\\\right)$$. $$n$$ is the square root of a and $$m$$ is the square root of -c","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor29a-h2","type":"hint","dependencies":["a104861trifactor29a-h1"],"title":"Factors","text":"In this equation $${ax}^2+bx+c$$ the square root of a and -c are $$1$$ and $$3$$ respectfully","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor29a-h3","type":"hint","dependencies":["a104861trifactor29a-h2"],"title":"Plug in","text":"Plug into the perfect square simplified form $$\\\\left(nx+m\\\\right) \\\\left(nx-m\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a104861trifactor3","title":"Factor Perfect Square Trinomials","body":"Factor the perfect square trinomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Factor Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a104861trifactor3a","stepAnswer":["$${\\\\left(3y+4\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$9y^2+24y+16$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(3y+4\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"a104861trifactor3a-h1","type":"hint","dependencies":[],"title":"Perfect Square Trinomials Pattern","text":"The trinomial fits the perfect square trinomials pattern $$a^2+2a b+b^2$$ because the first and last terms are perfect squares and the middle term matches $$2a b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor3a-h2","type":"hint","dependencies":["a104861trifactor3a-h1"],"title":"Square of the Binomial","text":"Take the perfect squares of the first and last term and write is as a binomial so you will get $${\\\\left(3y+4\\\\right)}^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor3a-h3","type":"hint","dependencies":["a104861trifactor3a-h2"],"title":"Checking Binomial","text":"Check if the squared binomial multiplies out into the original trinomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a104861trifactor30","title":"Factor Perfect Square Trinomials","body":"Factor(Give your answer with the larger value first. eg. $$\\\\left(x+2\\\\right) \\\\left(x+1\\\\right)$$ $$2$$ is greater than 1).","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Factor Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a104861trifactor30a","stepAnswer":["$$\\\\left(5v+1\\\\right) \\\\left(5v-1\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$25v^2-1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(5v+1\\\\right) \\\\left(5v-1\\\\right)$$","hints":{"DefaultPathway":[{"id":"a104861trifactor30a-h1","type":"hint","dependencies":[],"title":"Rule","text":"If bx is $$0$$ and c is negative, then the simplified form will be $$\\\\left(nx+m\\\\right) \\\\left(nx-m\\\\right)$$. $$n$$ is the square root of a and $$m$$ is the square root of -c","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor30a-h2","type":"hint","dependencies":["a104861trifactor30a-h1"],"title":"Factors","text":"In this equation $${ax}^2+bx+c$$ the square root of a and -c are $$5$$ and $$1$$ respectfully","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor30a-h3","type":"hint","dependencies":["a104861trifactor30a-h2"],"title":"Plug in","text":"Plug into the perfect square simplified form $$\\\\left(nx+m\\\\right) \\\\left(nx-m\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a104861trifactor4","title":"Factor Perfect Square Trinomials","body":"Factor the perfect square trinomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Factor Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a104861trifactor4a","stepAnswer":["$${\\\\left(9y-4\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$81y^2-72y+16$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(9y-4\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"a104861trifactor4a-h1","type":"hint","dependencies":[],"title":"Perfect Square Trinomials Pattern","text":"The trinomial fits the perfect square trinomials pattern $$a^2+2a b+b^2$$ because the first and last terms are perfect squares and the middle term matches $$2a b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor4a-h2","type":"hint","dependencies":["a104861trifactor4a-h1"],"title":"Square of the Binomial","text":"Take the perfect squares of the first and last term and write is as a binomial so you will get $${\\\\left(9y-4\\\\right)}^2$$. Notice how it\'s now subtraction since the original middle term was negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor4a-h3","type":"hint","dependencies":["a104861trifactor4a-h2"],"title":"Checking Binomial","text":"Check if the squared binomial multiplies out into the original trinomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a104861trifactor5","title":"Factor Perfect Square Trinomials","body":"Factor the perfect square trinomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Factor Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a104861trifactor5a","stepAnswer":["$${\\\\left(8y-5\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$64y^2-80y+25$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(8y-5\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"a104861trifactor5a-h1","type":"hint","dependencies":[],"title":"Perfect Square Trinomials Pattern","text":"The trinomial fits the perfect square trinomials pattern $$a^2+2a b+b^2$$ because the first and last terms are perfect squares and the middle term matches $$2a b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor5a-h2","type":"hint","dependencies":["a104861trifactor5a-h1"],"title":"Square of the Binomial","text":"Take the perfect squares of the first and last term and write is as a binomial so you will get $${\\\\left(8y-5\\\\right)}^2$$. Notice how it\'s now subtraction since the original middle term was negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor5a-h3","type":"hint","dependencies":["a104861trifactor5a-h2"],"title":"Checking Binomial","text":"Check if the squared binomial multiplies out into the original trinomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a104861trifactor6","title":"Factor Perfect Square Trinomials","body":"Factor the perfect square trinomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Factor Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a104861trifactor6a","stepAnswer":["$${\\\\left(4z-9\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$16z^2-72z+81$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(4z-9\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"a104861trifactor6a-h1","type":"hint","dependencies":[],"title":"Perfect Square Trinomials Pattern","text":"The trinomial fits the perfect square trinomials pattern $$a^2+2a b+b^2$$ because the first and last terms are perfect squares and the middle term matches $$2a b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor6a-h2","type":"hint","dependencies":["a104861trifactor6a-h1"],"title":"Square of the Binomial","text":"Take the perfect squares of the first and last term and write is as a binomial so you will get $${\\\\left(4z-9\\\\right)}^2$$. Notice how it\'s now subtraction since the original middle term was negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor6a-h3","type":"hint","dependencies":["a104861trifactor6a-h2"],"title":"Checking Binomial","text":"Check if the squared binomial multiplies out into the original trinomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a104861trifactor7","title":"Factor Perfect Square Trinomials","body":"Factor the perfect square trinomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Factor Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a104861trifactor7a","stepAnswer":["$${\\\\left(6x+7y\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$36x^2+84x y+49y^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(6x+7y\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"a104861trifactor7a-h1","type":"hint","dependencies":[],"title":"Perfect Square Trinomials Pattern","text":"The trinomial fits the perfect square trinomials pattern $$a^2+2a b+b^2$$ because the first and last terms are perfect squares and the middle term matches $$2a b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor7a-h2","type":"hint","dependencies":["a104861trifactor7a-h1"],"title":"Square of the Binomial","text":"Take the perfect squares of the first and last term and write is as a binomial so you will get $${\\\\left(6x+7y\\\\right)}^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor7a-h3","type":"hint","dependencies":["a104861trifactor7a-h2"],"title":"Checking Binomial","text":"Check if the squared binomial multiplies out into the original trinomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a104861trifactor8","title":"Factor Perfect Square Trinomials","body":"Factor the perfect square trinomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Factor Special Products","courseName":"OpenStax: Elementary 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$${\\\\left(7x+6y\\\\right)}^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor8a-h3","type":"hint","dependencies":["a104861trifactor8a-h2"],"title":"Checking Binomial","text":"Check if the squared binomial multiplies out into the original trinomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a104861trifactor9","title":"Factor Perfect Square Trinomials","body":"Factor the perfect square trinomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Factor Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a104861trifactor9a","stepAnswer":["$${\\\\left(8m+7n\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$64m^2+112m n+49n^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(8m+7n\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"a104861trifactor9a-h1","type":"hint","dependencies":[],"title":"Perfect Square Trinomials Pattern","text":"The trinomial fits the perfect square trinomials pattern $$a^2+2a b+b^2$$ because the first and last terms are perfect squares and the middle term matches $$2a b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor9a-h2","type":"hint","dependencies":["a104861trifactor9a-h1"],"title":"Square of the Binomial","text":"Take the perfect squares of the first and last term and write is as a binomial so you will get $${\\\\left(8m+7n\\\\right)}^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a104861trifactor9a-h3","type":"hint","dependencies":["a104861trifactor9a-h2"],"title":"Checking Binomial","text":"Check if the squared binomial multiplies out into the original trinomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a10b60arealnumbers1","title":"Simplify Expressions with Higher Roots","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Higher Roots","courseName":"OpenStax: Elementary 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4.0>"}]}},{"id":"a10b60arealnumbers1b","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[4]{81}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[]}},{"id":"a10b60arealnumbers1c","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[5]{32}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[]}}]},{"id":"a10b60arealnumbers10","title":"Simplifying Exponents","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Higher Roots","courseName":"OpenStax: Elementary 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4.0>"},{"id":"a10b60arealnumbers10a-h3","type":"hint","dependencies":[],"title":"Simplifying Odd Roots","text":"When taking the nth root of some integer $$x$$ $$ \\\\geq $$ $$2$$, if $$n$$ is odd, the answer is $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a10b60arealnumbers13","title":"Add and Subtract Higher Roots","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Higher Roots","courseName":"OpenStax: Elementary 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Radicals","text":"For two like radicals, $$a \\\\sqrt[n]{x}-b \\\\sqrt[n]{x}$$ is equivalent to $$\\\\left(a-b\\\\right) \\\\sqrt[n]{x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a10b60arealnumbers16","title":"Simplifying Expressions with Higher Roots","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Higher Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a10b60arealnumbers16a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[3]{216}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"a10b60arealnumbers16a-h1","type":"hint","dependencies":[],"title":"Question Simplification","text":"What number to the power of $$3$$ is 216?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a10b60arealnumbers16a-h2","type":"hint","dependencies":["a10b60arealnumbers16a-h1"],"title":"Finding the Root","text":"$$6^3=216$$, so our answer is $$6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a10b60arealnumbers17","title":"Simplifying Expressions with Higher Roots","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Higher Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a10b60arealnumbers17a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[3]{27}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a10b60arealnumbers17a-h1","type":"hint","dependencies":[],"title":"Question Simplification","text":"What number to the power of $$3$$ is 27?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a10b60arealnumbers17a-h2","type":"hint","dependencies":["a10b60arealnumbers17a-h1"],"title":"Finding the Root","text":"$$3^3=27$$, so our answer is $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a10b60arealnumbers18","title":"Simplifying Expressions with Higher Roots","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Higher Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a10b60arealnumbers18a","stepAnswer":["$$8$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[3]{512}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8$$","hints":{"DefaultPathway":[{"id":"a10b60arealnumbers18a-h1","type":"hint","dependencies":[],"title":"Question Simplification","text":"What number to the power of $$3$$ is 512?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a10b60arealnumbers18a-h2","type":"hint","dependencies":["a10b60arealnumbers18a-h1"],"title":"Finding the Root","text":"$$8^3=512$$, so our answer is $$8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a10b60arealnumbers19","title":"Simplifying Expressions with Higher Roots","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Higher Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a10b60arealnumbers19a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[3]{125}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"a10b60arealnumbers19a-h1","type":"hint","dependencies":[],"title":"Question Simplification","text":"What number to the power of $$3$$ is 125?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a10b60arealnumbers19a-h2","type":"hint","dependencies":["a10b60arealnumbers19a-h1"],"title":"Finding the Root","text":"$$5^3=125$$, so our answer is $$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a10b60arealnumbers2","title":"Simplify Expressions with Higher Roots","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Higher Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a10b60arealnumbers2a","stepAnswer":["$$10$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[3]{1000}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10$$","hints":{"DefaultPathway":[]}},{"id":"a10b60arealnumbers2b","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[4]{16}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[]}},{"id":"a10b60arealnumbers2c","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[5]{32}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[]}}]},{"id":"a10b60arealnumbers20","title":"Simplifying Expressions with Higher Roots","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Higher Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a10b60arealnumbers20a","stepAnswer":["$$-2$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[3]{-8}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-2$$","hints":{"DefaultPathway":[{"id":"a10b60arealnumbers20a-h1","type":"hint","dependencies":[],"title":"Question Simplification","text":"What number to the power of $$3$$ is -8?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a10b60arealnumbers20a-h2","type":"hint","dependencies":["a10b60arealnumbers20a-h1"],"title":"Finding the Root","text":"$${\\\\left(-2\\\\right)}^3=-8$$, so our answer is $$-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a10b60arealnumbers21","title":"Simplifying Expressions with Higher Roots","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Higher Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a10b60arealnumbers21a","stepAnswer":["$$-4$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[3]{-64}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-4$$","hints":{"DefaultPathway":[{"id":"a10b60arealnumbers21a-h1","type":"hint","dependencies":[],"title":"Question Simplification","text":"What number to the power of $$3$$ is -64?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a10b60arealnumbers21a-h2","type":"hint","dependencies":["a10b60arealnumbers21a-h1"],"title":"Finding the Root","text":"$${\\\\left(-4\\\\right)}^3=-64$$, so our answer is $$-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a10b60arealnumbers22","title":"Simplifying Expressions with Higher Roots","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Higher Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a10b60arealnumbers22a","stepAnswer":["$$-3$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[3]{-125}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-3$$","hints":{"DefaultPathway":[{"id":"a10b60arealnumbers22a-h1","type":"hint","dependencies":[],"title":"Question Simplification","text":"What number to the power of $$3$$ is -125?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a10b60arealnumbers22a-h2","type":"hint","dependencies":["a10b60arealnumbers22a-h1"],"title":"Finding the Root","text":"$${\\\\left(-3\\\\right)}^3=-125$$, so our answer is $$-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a10b60arealnumbers23","title":"Simplifying Expressions with Higher Roots","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Higher Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a10b60arealnumbers23a","stepAnswer":["$$-8$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[3]{-512}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-8$$","hints":{"DefaultPathway":[{"id":"a10b60arealnumbers23a-h1","type":"hint","dependencies":[],"title":"Question Simplification","text":"What number to the power of $$3$$ is -512?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a10b60arealnumbers23a-h2","type":"hint","dependencies":["a10b60arealnumbers23a-h1"],"title":"Finding the Root","text":"$${\\\\left(-8\\\\right)}^3=-512$$, so our answer is $$-8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a10b60arealnumbers24","title":"Simplifying Expressions with Higher Roots","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Higher Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a10b60arealnumbers24a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[4]{256}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a10b60arealnumbers24a-h1","type":"hint","dependencies":[],"title":"Question Simplification","text":"What number to the power of $$4$$ is 256?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a10b60arealnumbers24a-h2","type":"hint","dependencies":["a10b60arealnumbers24a-h1"],"title":"Finding the Root","text":"$$4^4=256$$, so our answer is $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a10b60arealnumbers25","title":"Simplifying Expressions with Higher Roots","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Higher Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a10b60arealnumbers25a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[4]{16}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a10b60arealnumbers25a-h1","type":"hint","dependencies":[],"title":"Question Simplification","text":"What number to the power of $$4$$ is 16?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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$$-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a10b60arealnumbers29","title":"Simplifying Expressions with Higher Roots","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Higher Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a10b60arealnumbers29a","stepAnswer":["$$-2$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[4]{-16}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-2$$","hints":{"DefaultPathway":[{"id":"a10b60arealnumbers29a-h1","type":"hint","dependencies":[],"title":"Question Simplification","text":"What number to the power of $$4$$ is -16?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a10b60arealnumbers29a-h2","type":"hint","dependencies":["a10b60arealnumbers29a-h1"],"title":"Finding the Root","text":"$${\\\\left(-2\\\\right)}^4=-16$$, so our answer is $$-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a10b60arealnumbers3","title":"Simplify Expressions with Higher Roots","body":"Simplify. If the result is not a real number, type \\"Complex\\"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Higher Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a10b60arealnumbers3a","stepAnswer":["$$-5$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[3]{-125}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-5$$","hints":{"DefaultPathway":[{"id":"a10b60arealnumbers3a-h1","type":"hint","dependencies":[],"title":"Account for the Negative","text":"When raising a negative number to an exponent, the sign (positive or negative) of your answer will change every time you increment or decrement the exponent","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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roots.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Higher Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a10b60arealnumbers4a","stepAnswer":["$$|x|$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{x^2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$|x|$$","hints":{"DefaultPathway":[{"id":"a10b60arealnumbers4a-h1","type":"hint","dependencies":[],"title":"Simplifying Even Roots","text":"When taking the nth root of some integer $$x$$ $$ \\\\geq $$ $$2$$, if $$n$$ is even, the answer is $$|x|$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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Then, that $$x$$ is the answer to the question","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a10b60arealnumbers5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$64p^6$$"],"dependencies":["a10b60arealnumbers5a-h1"],"title":"Rewriting as an Exponent","text":"What is $${\\\\left(4p\\\\right)}^3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a10b60arealnumbers5b","stepAnswer":["$$2\\\\left(|q^3|\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[4]{16q^{12}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2\\\\left(|q^3|\\\\right)$$","hints":{"DefaultPathway":[{"id":"a10b60arealnumbers5b-h1","type":"hint","dependencies":[],"title":"Transform the Expression into Exponent Form","text":"To make this easier, find some value \'x\' such that $$x^4$$ is equal to $$16q^{12}$$. Then, $$|x|$$ is the answer to the question","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a10b60arealnumbers6","title":"Use the Product Property to Simplify Expressions with Higher Roots","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Higher Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a10b60arealnumbers6a","stepAnswer":["$$x \\\\sqrt[3]{x}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[3]{x^4}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x \\\\sqrt[3]{x}$$","hints":{"DefaultPathway":[{"id":"a10b60arealnumbers6a-h1","type":"hint","dependencies":[],"title":"Product Property of nth Roots","text":"$$\\\\sqrt[n]{a b}$$ is equal to $$\\\\sqrt[n]{a} \\\\sqrt[n]{b}$$ as long as a and $$b$$ are integers $$ \\\\geq $$ $$2$$ and their roots are real numbers","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a10b60arealnumbers6a-h2","type":"hint","dependencies":["a10b60arealnumbers6a-h1"],"title":"Splitting Root Expressions","text":"$$\\\\sqrt[3]{x^4}$$ is equivalent to $$\\\\sqrt[3]{x^3} \\\\sqrt[3]{x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a10b60arealnumbers6a-h3","type":"hint","dependencies":[],"title":"How to Use the Product Property","text":"Since $$3$$ is odd, remember that $$\\\\sqrt[3]{x^3}$$ is just $$x$$. Is there a way to split the root in the problem into two such that one of the new roots is in the form $$\\\\sqrt[3]{x^3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a10b60arealnumbers6b","stepAnswer":["$$|x| \\\\sqrt[4]{x^3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[4]{x^7}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$|x| \\\\sqrt[4]{x^3}$$","hints":{"DefaultPathway":[{"id":"a10b60arealnumbers6b-h1","type":"hint","dependencies":[],"title":"Simplifying Even Roots","text":"When taking the nth root of some integer $$x$$ $$ \\\\geq $$ $$2$$, if $$n$$ is even, the answer is $$|x|$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a10b60arealnumbers7","title":"Use the Product Property to Simplify Expressions with Higher Roots","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Higher Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a10b60arealnumbers7a","stepAnswer":["$$2\\\\sqrt[3]{2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[3]{16}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2\\\\sqrt[3]{2}$$","hints":{"DefaultPathway":[{"id":"a10b60arealnumbers7a-h1","type":"hint","dependencies":[],"title":"Rewriting a Number","text":"$$16$$ is equal to $$2^4$$ which is equal to $$2^3 2^1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a10b60arealnumbers7b","stepAnswer":["$$|3| \\\\sqrt[4]{3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[4]{243}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$|3| \\\\sqrt[4]{3}$$","hints":{"DefaultPathway":[{"id":"a10b60arealnumbers7b-h1","type":"hint","dependencies":[],"title":"Rewriting a Number","text":"$$243$$ is equal to $$3^5$$ which is equal to $$3^4 3^1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a10b60arealnumbers7b-h2","type":"hint","dependencies":[],"title":"Simplifying Even Roots","text":"When taking the nth root of some integer $$x$$ $$ \\\\geq $$ $$2$$, if $$n$$ is even, the answer is $$|x|$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a10b60arealnumbers8","title":"Use the Product Property to Simplify Expressions with Higher Roots","body":"Simplify. If the result is not a real number, type \\"Complex\\"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Higher Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a10b60arealnumbers8a","stepAnswer":["$$-3$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[3]{-27}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-3$$","hints":{"DefaultPathway":[{"id":"a10b60arealnumbers8a-h1","type":"hint","dependencies":[],"title":"Normal Root","text":"What $$x$$ exists such that $$x^3$$ is -27?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a10b60arealnumbers9a-h3","type":"hint","dependencies":[],"title":"Simplifying Odd Roots","text":"When taking the nth root of some integer $$x$$ $$ \\\\geq $$ $$2$$, if $$n$$ is odd, the answer is $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a10b60arealnumbers9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["a"],"dependencies":["a10b60arealnumbers9a-h3"],"title":"Simplifying the Root","text":"What is $$\\\\sqrt[3]{a^3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a10b60arealnumbers9b","stepAnswer":["$$a^2$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[4]{\\\\frac{a^{10}}{a^2}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$a^2$$","hints":{"DefaultPathway":[{"id":"a10b60arealnumbers9b-h1","type":"hint","dependencies":[],"title":"Dividing Exponents","text":"If two exponent expressions in a quotient have the same base (ex: (x**a)/(x**b)) then this is the same as $$x^{a-b}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a10b60arealnumbers9b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$a^8$$"],"dependencies":["a10b60arealnumbers9b-h1"],"title":"SImplifying the Quotient","text":"What does $$\\\\frac{a^{10}}{a^2}$$ simplify to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a10b60arealnumbers9b-h3","type":"hint","dependencies":[],"title":"Rewriting the Expression in the Root","text":"We can rewrite the expression under the root such that it is some $$x^4$$, such that $$x^4$$ is equal to whatever $$\\\\frac{a^{10}}{a^2}$$ simplifies to. Remember that we do this because $$\\\\sqrt[n]{x^n}$$ is just $$|x|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a10b60arealnumbers9b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$a^8$$"],"dependencies":["a10b60arealnumbers9b-h3"],"title":"Rewriting the Expression in the Root","text":"What is $${\\\\left(a^2\\\\right)}^4$$ equivalent to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a10b60arealnumbers9b-h5","type":"hint","dependencies":[],"title":"Absolute Value of a Square","text":"Because the square of all numbers is positive, $$|x^2|$$ can be rewritten as $$x^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1268beother1","title":"Polynomial Equations","body":"Solve the following polynomial equation by grouping or factoring.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Other Types of Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a1268beother1a","stepAnswer":["$$-1, 0, 2$$"],"problemType":"MultipleChoice","stepTitle":"$$x^3+2x^2-x-2=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$-2, -1, 1$$","$$-1, 0, 2$$","$$-2, 1, 1$$","1,1,2"],"hints":{"DefaultPathway":[{"id":"a1268beother1a-h1","type":"hint","dependencies":[],"title":"Grouping","text":"This polynomial consists of $$4$$ terms, so we will solve by grouping. Factor the first $$2$$ terms and then factor the last $$2$$ terms. If the factors in the parantheses are identical, we can continue the process and solve, unless more factoring is suggested.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2$$"],"dependencies":["a1268beother1a-h1"],"title":"Factoring","text":"What can you factor out of the first $$2$$ terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a1268beother1a-h2"],"title":"Factoring","text":"What can you factor out of the last $$2$$ terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother1a-h4","type":"hint","dependencies":["a1268beother1a-h3"],"title":"Factoring","text":"Combine the common expressions and add the factors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother1a-h5","type":"hint","dependencies":["a1268beother1a-h4"],"title":"Factoring","text":"The expression can be rewritten as $$\\\\left(x^2-1\\\\right) \\\\left(x+2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother1a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2-1$$"],"dependencies":["a1268beother1a-h5"],"title":"Factoring","text":"You can factor one of the expressions again. Which one?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother1a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x-1\\\\right) \\\\left(x+1\\\\right) \\\\left(x+2\\\\right)$$"],"dependencies":["a1268beother1a-h6"],"title":"Factoring","text":"What is the expression after factoring $$x^2-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother1a-h8","type":"hint","dependencies":["a1268beother1a-h7"],"title":"Zero-Product Property","text":"Use the Zero-Product property to solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother1a-h9","type":"hint","dependencies":["a1268beother1a-h8"],"title":"Zero-Product Property","text":"Solve for $$x$$ when $$(x-1)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother1a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a1268beother1a-h9"],"title":"Zero-Product Property","text":"What is the solution for $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother1a-h11","type":"hint","dependencies":["a1268beother1a-h10"],"title":"Zero-Product Property","text":"Solve for $$x$$ when $$x+1=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother1a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a1268beother1a-h11"],"title":"Zero-Product Property","text":"What is the solution for $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother1a-h13","type":"hint","dependencies":["a1268beother1a-h12"],"title":"Zero-Product Property","text":"Solve for $$x$$ when $$x+2=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother1a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a1268beother1a-h13"],"title":"Zero-Product Property","text":"What is the solution for $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother1a-h15","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-2, -1, 1$$"],"dependencies":["a1268beother1a-h14"],"title":"Solution","text":"What are the $$3$$ solutions?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$-2, -1, 1$$","$$-1, 1, 2$$","$$-2, 1, 1$$","1,1,2"]}]}}]},{"id":"a1268beother10","title":"Solve the equation","body":"Solve the equation involving absolute value.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Other Types of Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a1268beother10a","stepAnswer":["$$3, -2$$"],"problemType":"MultipleChoice","stepTitle":"$$|2x-1|-7=-2$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$3, -2$$","$$-2, 3$$","$$-1, 3$$","$$1, -3$$"],"hints":{"DefaultPathway":[{"id":"a1268beother10a-h1","type":"hint","dependencies":[],"title":"Adding $$7$$ to Both Sides","text":"The first step is to add $$7$$ to both sides of the equation: $$ans(2x-1)=5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother10a-h2","type":"hint","dependencies":["a1268beother10a-h1"],"title":"Creating Two Equations","text":"Create two equations setting $$2x-1$$ equal to $$5$$ and $$-5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a1268beother10a-h2"],"title":"Solving $$2x-1=5$$","text":"What is $$x$$ when $$2x-1=5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a1268beother10a-h3-s1","type":"hint","dependencies":[],"title":"Solving $$2x-1=5$$","text":"To solve $$2x-1=5$$, start by adding $$1$$ to both sides of the equation: $$2x=6$$. Then, divide both sides by $$2$$ to get $$x=3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a1268beother10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a1268beother10a-h2"],"title":"Solving $$2x-1=-5$$","text":"What is $$x$$ when $$2x-1=-5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a1268beother10a-h4-s1","type":"hint","dependencies":[],"title":"Solving $$2x-1=-5$$","text":"For $$2x-1=-5$$, add $$1$$ to both sides of the equation: $$2x=-4$$. Then, divide both sides by $$2$$ to get $$x=-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a1268beother10a-h5","type":"hint","dependencies":["a1268beother10a-h3","a1268beother10a-h4"],"title":"Final Answer","text":"So, the two values of $$x$$ that would satisfy $$|2x-1|-7=-2$$ are $$3$$ and $$-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1268beother11","title":"Solve the equation","body":"Solve the equation involving absolute value.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Other Types of Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a1268beother11a","stepAnswer":["$$-5$$"],"problemType":"TextBox","stepTitle":"$$|x+5|=0$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-5$$","hints":{"DefaultPathway":[{"id":"a1268beother11a-h1","type":"hint","dependencies":[],"title":"Creating Two Equations","text":"Since $$0=-0$$, we actually keep the expression the way it is and take away the absolute value! Our new expression reads $$x+5=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["a1268beother11a-h1"],"title":"Solving $$x+5=0$$","text":"What is $$x$$ when $$x+5=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a1268beother11a-h2-s1","type":"hint","dependencies":[],"title":"Solving $$x+5=0$$","text":"To solve $$x+5=0$$, subtract $$5$$ from both sides of the equation: $$x=-5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a1268beother11a-h3","type":"hint","dependencies":["a1268beother11a-h2"],"title":"Final Answer","text":"So, the only value of $$x$$ that would satisfy $$|x+5|=0$$ is $$-5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1268beother12","title":"Solve the equation","body":"Solve the equation by identifying the quadratic form. Use a substitute variable and find all real solutions by factoring.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Other Types of Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a1268beother12a","stepAnswer":["$$1, -1, 3, -3$$"],"problemType":"MultipleChoice","stepTitle":"$$x^4-10x^2+9=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["1,3","$$-1, -3$$","$$1, -1, 3, -3$$","$$1, -1, 3, -3, 0$$"],"hints":{"DefaultPathway":[{"id":"a1268beother12a-h1","type":"hint","dependencies":[],"title":"Substitute Variable","text":"Let\'s start by setting a variable $$y$$ equal to $$x^2$$. Now we can substitute $$y$$ into the equation and solve like a normal quadradic: $$y^2-10y+9=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother12a-h2","type":"hint","dependencies":["a1268beother12a-h1"],"title":"Factor","text":"Here, we can factor the quadratic. Since $$-1$$ and $$-9$$ multiply to $$9$$ and add to $$-10$$, we factor the quadratic as $$(y-1)(y-9)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother12a-h3","type":"hint","dependencies":["a1268beother12a-h1","a1268beother12a-h2"],"title":"$$x^2=y$$","text":"Now, because we need to find the answer in terms of $$x$$, we need to substitute $$x^2$$ back in for $$y$$. So we must solve for $$x^2=1$$ and $$x^2=9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother12a-h4","type":"hint","dependencies":["a1268beother12a-h3"],"title":"Answer","text":"The answers are the $$\\\\pm \\\\sqrt{1}$$ and $$\\\\pm \\\\sqrt{9}$$, or $$1$$, $$-1$$, $$3$$, and $$-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1268beother13","title":"Solve the equation","body":"Solve the equation by identifying the quadratic form. Use a substitute variable and find all real solutions by factoring.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Other Types of Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a1268beother13a","stepAnswer":["$$2, -2$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(x^2-1\\\\right)}^2+x^2-1-12=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$2, -2$$","$$2, 0, -2$$","$$4, 2, -2, -4$$","$$-4, 3$$"],"hints":{"DefaultPathway":[{"id":"a1268beother13a-h1","type":"hint","dependencies":[],"title":"Substitute Variable","text":"Let\'s start by setting a variable $$y$$ equal to $$x^2-1$$. Now we can substitute $$y$$ into the equation and solve like a normal quadradic: $$y^2+y-12=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother13a-h2","type":"hint","dependencies":["a1268beother13a-h1"],"title":"Factor","text":"Here, we can factor the quadratic. Since $$4$$ and $$-3$$ multiply to $$-12$$ and add to $$1$$, we factor the quadratic as $$\\\\left(y+4\\\\right) \\\\left(y-3\\\\right)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother13a-h3","type":"hint","dependencies":["a1268beother13a-h2"],"title":"$$x^2-1=y$$","text":"Now, because we need to find the answer in terms of $$x$$, we need to substitute $$x^2-1$$ back in for $$y$$. So we must solve for $$x^2-1=-4$$ and $$x^2-1=3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother13a-h4","type":"hint","dependencies":["a1268beother13a-h3"],"title":"Solve First Equation","text":"To solve $$x^2-1=-4$$, we start by adding $$1$$ to both sides, then square rooting both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother13a-h5","type":"hint","dependencies":["a1268beother13a-h4"],"title":"Solve First Equation","text":"Since $$-4+1=-3$$ is a negative number, the square root of it is unreal. So we will ignore this solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother13a-h6","type":"hint","dependencies":["a1268beother13a-h3"],"title":"Solve Second Equation","text":"To solve $$x^2-1=3$$, we start by adding $$1$$ to both sides, then square rooting both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother13a-h7","type":"hint","dependencies":["a1268beother13a-h6"],"title":"Solve Second Equation","text":"$$\\\\sqrt{3+1}=2$$, $$-2$$. So our solutions for $$x$$ are 2,-2.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1268beother14","title":"Solve the equation","body":"Solve the equation by identifying the quadratic form. Use a substitute variable and find all real solutions by factoring.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Other Types of Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a1268beother14a","stepAnswer":["$$8, -2$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(x+1\\\\right)}^2-8\\\\left(x+1\\\\right)-9=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$8, -2$$","$$2, -8$$","$$4, -4$$","$$3, -6$$"],"hints":{"DefaultPathway":[{"id":"a1268beother14a-h1","type":"hint","dependencies":[],"title":"Substitute Variable","text":"Let\'s start by setting a variable $$y$$ equal to $$x+1$$. Now we can substitute $$y$$ into the equation and solve like a normal quadradic: $$y^2-8y-9=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother14a-h2","type":"hint","dependencies":["a1268beother14a-h1"],"title":"Factor","text":"Here, we can factor the quadratic. Since $$-9$$ and $$1$$ multiply to $$-9$$ and add to $$-8$$, we factor the quadratic as $$\\\\left(y-9\\\\right) \\\\left(y+1\\\\right)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother14a-h3","type":"hint","dependencies":["a1268beother14a-h2"],"title":"$$x+1=y$$","text":"Now, because we need to find the answer in terms of $$x$$, we need to substitute $$x+1$$ back in for $$y$$. So we must solve for $$x+1=9$$ and $$x+1=-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother14a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a1268beother14a-h3"],"title":"Solve First Equation","text":"Solve $$x+1=9$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother14a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a1268beother14a-h4"],"title":"Solve Second Equation","text":"Solve $$x+1=-1$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother14a-h6","type":"hint","dependencies":["a1268beother14a-h4","a1268beother14a-h5"],"title":"Final Answer","text":"So our final answer is $$x=8$$ and $$x=-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1268beother15","title":"Solve the equation by identifying the quadratic form.","body":"Use a substitute variable and find all real solutions by factoring.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Other Types of Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a1268beother15a","stepAnswer":["5,1"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(x-3\\\\right)}^2-4=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["4,2","4,5","5,1","4,6"],"hints":{"DefaultPathway":[{"id":"a1268beother15a-h1","type":"hint","dependencies":[],"title":"Substitute Variable","text":"Let\'s start by setting a variable $$y$$ equal to $$x-3$$. Now we can substitute $$y$$ into the equation and solve like a normal quadradic: $$y^2-4=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother15a-h2","type":"hint","dependencies":["a1268beother15a-h1"],"title":"Factor","text":"Here, we can factor the quadratic as $$\\\\left(y+2\\\\right) \\\\left(y-2\\\\right)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother15a-h3","type":"hint","dependencies":["a1268beother15a-h2"],"title":"$$x-3=y$$","text":"Now, because we need to find the answer in terms of $$x$$, we need to substitute $$x-3$$ back in for $$y$$. So we must solve for $$x-3=2$$ and $$x-3=-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a1268beother15a-h3"],"title":"Solve First Equation","text":"Solve $$x-3=2$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother15a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a1268beother15a-h4"],"title":"Solve Second Equation","text":"Solve $$x-3=-2$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother15a-h6","type":"hint","dependencies":["a1268beother15a-h4","a1268beother15a-h5"],"title":"Final Answer","text":"So our final answer is $$x=5$$ and $$x=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1268beother16","title":"Solve a Polynomial by Grouping","body":"Solve the polynomial by grouping.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Other Types of Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a1268beother16a","stepAnswer":["$$x=-3, 1$$"],"problemType":"MultipleChoice","stepTitle":"$$x^3+x^2-9x-9=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=-3, 1$$","choices":["$$x=-3, 1$$","$$x=3, -1$$","$$x=3, 1$$","$$x=2, -1$$"],"hints":{"DefaultPathway":[{"id":"a1268beother16a-h1","type":"hint","dependencies":[],"title":"How to Group","text":"Grouping requires factoring the first two terms and then factoring the last two terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^{2\\\\left(x+1\\\\right)}$$"],"dependencies":["a1268beother16a-h1"],"title":"Factoring the First Two Terms","text":"What is the factored form of $$x^3+x^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother16a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9\\\\left(x+1\\\\right)$$"],"dependencies":["a1268beother16a-h1"],"title":"Factoring the Last Two Terms","text":"What is the factored form of $$-9x-9$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother16a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["yes"],"dependencies":["a1268beother16a-h2","a1268beother16a-h3"],"title":"Common Factor of First and Second Groups","text":"Are the factors in the parenthese identical?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["yes","no"]},{"id":"a1268beother16a-h5","type":"hint","dependencies":["a1268beother16a-h4"],"title":"Last Step","text":"If the factors in the parenthesis of the first and second groups are identical, the polynomial can be factored by grouping. For example, $$a\\\\left(c+d\\\\right)+b\\\\left(c+d\\\\right)=\\\\left(a+b\\\\right) \\\\left(c+d\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1268beother17","title":"Solving an Equation with One Radical","body":"Solve the following equation for $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Other Types of Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a1268beother17a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{15-2x}=x$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a1268beother17a-h1","type":"hint","dependencies":[],"title":"First Step","text":"The first step is to square both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother17a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15-2x=x^2$$"],"dependencies":["a1268beother17a-h1"],"title":"Result of Squaring Both Sides","text":"What does the equation turn into just after you have squared both sides?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother17a-h3","type":"hint","dependencies":["a1268beother17a-h2"],"title":"Solving a Quadratic Equation","text":"We see that the resulting equation is quadratic. Set the equation up in qudaratic format, $${ax}^2+bx+c=0$$, and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother17a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x=-5, 3$$"],"dependencies":["a1268beother17a-h3"],"title":"Proposed Solutions of the Quadratic Equation","text":"What are the solutions for $$x$$ from the quadratic equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x=5, 3$$","$$x=3, 1$$","$$x=-5, 3$$","$$x=-3, 1$$"]},{"id":"a1268beother17a-h5","type":"hint","dependencies":["a1268beother17a-h4"],"title":"Checking for Extraneous Solutions","text":"Despite the fact that the solutions work in the quadratic equation, they might not work when subsituted for $$x$$ in the original equation. Next, check each $$x$$ value to see it it fits the original equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother17a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["no"],"dependencies":["a1268beother17a-h5"],"title":"scaffold","text":"When $$x=-5$$, does $$\\\\sqrt{15-2x}=x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["yes","no"]},{"id":"a1268beother17a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["yes"],"dependencies":["a1268beother17a-h5"],"title":"scaffold","text":"When $$x=3$$, does $$\\\\sqrt{15-2x}=x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["yes","no"]},{"id":"a1268beother17a-h8","type":"hint","dependencies":["a1268beother17a-h6","a1268beother17a-h7"],"title":"hint","text":"If an $$x$$ value does not work in the original equation, then it is extraneous and not a solution of the original equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1268beother18","title":"Solving a Radical Equation Containing Two Radicals","body":"Solve the following equation for $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Other Types of Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a1268beother18a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{2x+3}+\\\\sqrt{x-2}=4$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a1268beother18a-h1","type":"hint","dependencies":[],"title":"Isolating One Radical","text":"The first step is to isolate one radical, which can be accomplished by subtracting $$\\\\sqrt{x-2}$$ from both sides of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother18a-h2","type":"hint","dependencies":["a1268beother18a-h1"],"title":"Squaring Both Sides of the Equation","text":"Next, after subtracting $$\\\\sqrt{x-2}$$ from both sides of the equation, square both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother18a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2x+3=4^2-2\\\\left(4\\\\right) \\\\sqrt{x-2}+{\\\\sqrt{x-2}}^2$$"],"dependencies":["a1268beother18a-h2"],"title":"Equation after Squaring Both Sides","text":"After squaring both sides and expanding the right side, what is the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2x+3=4^2-2\\\\left(4\\\\right) \\\\sqrt{x-2}+{\\\\sqrt{x-2}}^2$$","$$\\\\sqrt{2x+3}=4^2-2\\\\left(4\\\\right) \\\\sqrt{x-2}+{\\\\sqrt{x-2}}^2$$","$$2x+3={\\\\sqrt{x-2}}^2$$"]},{"id":"a1268beother18a-h4","type":"hint","dependencies":["a1268beother18a-h3"],"title":"Isolating the Remaining Radical","text":"Next, isolate the radical on the right side by moving all other terms to the left.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother18a-h5","type":"hint","dependencies":["a1268beother18a-h4"],"title":"Eliminating the Radical","text":"Then, eliminate the remaining radical on the right side by squaring both sides again.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother18a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x^2-22x+121=64x-128$$"],"dependencies":["a1268beother18a-h5"],"title":"Result after Eliminating the Radical","text":"What is the equation after the radical has been eliminated?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x^2-22x+121=64x-128$$","$$x^2=36x+48$$","$$x^2-8x+16=64x-128$$"]},{"id":"a1268beother18a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x=3, 83$$"],"dependencies":["a1268beother18a-h6"],"title":"Solving the New Quadratic Equation","text":"Solve for $$x$$ from the new quadratic equation. What $$x$$ values make the new equation 0?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x=2, 24$$","$$x=3, 83$$","$$x=4, 84$$","$$x=12, 81$$"]},{"id":"a1268beother18a-h8","type":"hint","dependencies":["a1268beother18a-h7"],"title":"Checking for Extraneous Solutions","text":"Plug in the answers to the quadratic equation to the original problem to see if they are valid solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1268beother19","title":"Solving Absolute Value Equations","body":"Solve the following absolute value equations for $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Other Types of Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a1268beother19a","stepAnswer":["$$\\\\frac{2}{3}-2$$"],"problemType":"MultipleChoice","stepTitle":"$$|6x+4|=8$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{2}{3}-2$$","choices":["$$\\\\frac{2}{3}-2$$","$$3, -2$$","$$2, -1$$"],"hints":{"DefaultPathway":[{"id":"a1268beother19a-h1","type":"hint","dependencies":[],"title":"Writing Two Equations","text":"The first step is to rewrite the absolute value equation into two equations, $$6x+4=8$$ and $$6x+4=-8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{3}$$"],"dependencies":["a1268beother19a-h1"],"title":"Solving Equation One","text":"$$6x+4=8$$, $$x=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother19a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a1268beother19a-h1"],"title":"Solving Equation Two","text":"$$6x+4=-8$$, $$x=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother19a-h4","type":"hint","dependencies":["a1268beother19a-h2","a1268beother19a-h3"],"title":"Final Answer","text":"The answers to the first and second equations are the answer to the original absolute value equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a1268beother19b","stepAnswer":["no solution"],"problemType":"MultipleChoice","stepTitle":"$$|3x+4|=-9$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$x=3, 2$$","$$x=3, 5$$","$$x=\\\\frac{5}{3}$$","no solution"],"hints":{"DefaultPathway":[{"id":"a1268beother19b-h1","type":"hint","dependencies":[],"title":"Requirement of Absolute Value","text":"Absolute Value is always positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother19b-h2","type":"hint","dependencies":["a1268beother19b-h1"],"title":"Classifying $$-9$$","text":"$$-9$$ is a negative number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother19b-h3","type":"hint","dependencies":["a1268beother19b-h1","a1268beother19b-h2"],"title":"Answer","text":"There is no solution if an expression inside absolute value is equated to a negative number, because absolute values cannot be negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a1268beother19c","stepAnswer":["$$5-\\\\frac{5}{3}$$"],"problemType":"MultipleChoice","stepTitle":"$$|3x-5|-4=6$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$5-\\\\frac{5}{3}$$","choices":["-5,5/3","$$5-\\\\frac{5}{3}$$","$$3, -5$$"],"hints":{"DefaultPathway":[{"id":"a1268beother19c-h1","type":"hint","dependencies":[],"title":"Isolating the Absolute Value","text":"First, isolate the absolute value by adding $$4$$ to both sides of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother19c-h2","type":"hint","dependencies":["a1268beother19c-h1"],"title":"Writing Two Equations","text":"Then, rewrite the absolute value equation into two equations, $$3x-5=10$$ and $$3x-5=-10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother19c-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a1268beother19c-h2"],"title":"Solving Equation One","text":"$$3x-5=10$$, $$x=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother19c-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-5}{3}$$"],"dependencies":["a1268beother19c-h2"],"title":"Solving Equation Two","text":"$$3x-5=-10$$, $$x=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother19c-h5","type":"hint","dependencies":["a1268beother19c-h3","a1268beother19c-h4"],"title":"Final Answer","text":"The answers to the first and second equations are the answer to the original absolute value equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a1268beother19d","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"$$|-5x+10|=0$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a1268beother19d-h1","type":"hint","dependencies":[],"title":"Absolute Value of $$0$$","text":"When the absolute value of an expression is equal to $$0$$, the absolute value sign can be removed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother19d-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a1268beother19d-h1"],"title":"Solving the Equation","text":"$$-5x+10=0$$, $$x=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1268beother2","title":"Polynomial Equations","body":"Solve the following polynomial equation by grouping or factoring.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Other Types of Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a1268beother2a","stepAnswer":["0,3/2,-3/2"],"problemType":"MultipleChoice","stepTitle":"$$4y^3-9y=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["0,2/3,-2/3","0,3/2,-3/2"],"hints":{"DefaultPathway":[{"id":"a1268beother2a-h1","type":"hint","dependencies":[],"title":"Factoring","text":"Find the greatest common factor between the two terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y$$"],"dependencies":["a1268beother2a-h1"],"title":"Factoring","text":"What is it?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother2a-h3","type":"hint","dependencies":["a1268beother2a-h2"],"title":"Factoring","text":"Factor out $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4y^2-9$$"],"dependencies":["a1268beother2a-h3"],"title":"Factoring","text":"What expression is left?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother2a-h5","type":"hint","dependencies":["a1268beother2a-h4"],"title":"Zero-Product Property","text":"Use the Zero-Product property to solve for $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother2a-h6","type":"hint","dependencies":["a1268beother2a-h5"],"title":"Zero-Product Property","text":"Solve for $$y$$ when $$y=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother2a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a1268beother2a-h6"],"title":"Zero-Product Property","text":"What is the solution for $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother2a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{3}{2}-\\\\frac{3}{2}$$"],"dependencies":["a1268beother2a-h7"],"title":"Zero-Product Property","text":"Solve for $$y$$ when $$4y^{-9}=0$$. What is $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{2}{3}-\\\\frac{2}{3}$$","$$\\\\frac{3}{2}-\\\\frac{3}{2}$$"]},{"id":"a1268beother2a-s1","type":"hint","dependencies":[],"title":"Solving Equations","text":"Add $$9$$ to both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a1268beother2a-h8-s2","type":"hint","dependencies":["a1268beother2a-s1"],"title":"Solving Equations","text":"Divide both sides by $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother2a-h8-s3","type":"hint","dependencies":["a1268beother2a-h8-s2"],"title":"Solving Equations","text":"Take the square root of both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother2a-h8-s4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{3}{2}$$, $$\\\\frac{-3}{2}$$"],"dependencies":["a1268beother2a-h8-s3"],"title":"Solving Equations","text":"What is the square root of $$\\\\frac{4}{9}$$? (Remember that it can be positive or negative).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{2}{3}-\\\\frac{2}{3}$$","$$\\\\frac{3}{2}-\\\\frac{3}{2}$$"]},{"id":"a1268beother2a-h8-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{3}{2}$$, $$\\\\frac{-3}{2}$$"],"dependencies":["a1268beother2a-h8-s4"],"title":"Solving Equations","text":"Solve for $$y$$. What is $$y$$ equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{2}{3}-\\\\frac{2}{3}$$","$$\\\\frac{3}{2}-\\\\frac{3}{2}$$"]}]},{"id":"a1268beother2a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["0,3/2,-3/2"],"dependencies":["a1268beother2a-h8-h1"],"title":"Solving Polynomial Equations","text":"What are the $$3$$ solutions of the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["0,2/3,-2/3","0,3/2,-3/2"]}]}}]},{"id":"a1268beother20","title":"Solving an Equation With One Radical","body":"Solve the following equation for $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Other Types of Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a1268beother20a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{x+3}=3x-1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a1268beother20a-h1","type":"hint","dependencies":[],"title":"First Step","text":"The first step is to square both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+3={\\\\left(3x-1\\\\right)}^2$$"],"dependencies":["a1268beother20a-h1"],"title":"Result of Squaring Both Sides","text":"What does the equation turn into just after you have squared both sides?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother20a-h3","type":"hint","dependencies":["a1268beother20a-h2"],"title":"Solving a Quadratic Equation","text":"We see that the resulting equation is quadratic. Set the equation up in qudaratic format, $${ax}^2+bx+c=0$$, and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother20a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["x=-2/9,1"],"dependencies":["a1268beother20a-h3"],"title":"Proposed Solutions of the Quadratic Equation","text":"What are the solutions for $$x$$ from the quadratic equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x=1$$","x=3/5,2","x=-2/9,1","$$x=2$$"]},{"id":"a1268beother20a-h5","type":"hint","dependencies":["a1268beother20a-h4"],"title":"Checking for Extraneous Solutions","text":"Despite the fact that the solutions work in the quadratic equation, they might not work when subsituted for $$x$$ in the original equation. Next, check each $$x$$ value to see it it fits the original equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother20a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["no"],"dependencies":["a1268beother20a-h5"],"title":"scaffold","text":"When $$x=\\\\frac{-2}{9}$$, does $$\\\\sqrt{x+3}=3x-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["yes","no"]},{"id":"a1268beother20a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["yes"],"dependencies":["a1268beother20a-h5"],"title":"scaffold","text":"When $$x=1$$, does $$\\\\sqrt{x+3}=3x-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["yes","no"]},{"id":"a1268beother20a-h8","type":"hint","dependencies":["a1268beother20a-h6","a1268beother20a-h7"],"title":"hint","text":"If an $$x$$ value does not work in the original equation, then it is extraneous and not a solution of the original equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1268beother21","title":"Solving an Equation With One Radical","body":"Solve the following equation, an equation with two radicals, for $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Other Types of Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a1268beother21a","stepAnswer":["$$-2$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{3x+7}+\\\\sqrt{x+2}=1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-2$$","hints":{"DefaultPathway":[{"id":"a1268beother21a-h1","type":"hint","dependencies":[],"title":"Isolating One Radical","text":"The first step is to isolate one radical, which can be accomplished by subtracting $$\\\\sqrt{x+2}$$ from both sides of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother21a-h2","type":"hint","dependencies":["a1268beother21a-h1"],"title":"Squaring Both Sides of the Equation","text":"Next, after subtracting $$\\\\sqrt{x+2}$$ from both sides of the equation, square both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother21a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3x+7=1-2\\\\sqrt{x+2}+x+2$$"],"dependencies":["a1268beother21a-h2"],"title":"Equation after Squaring Both Sides","text":"After squaring both sides and expanding the right side, what is the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$3x+7=1-2\\\\sqrt{x+2}+x+2$$","$$3x+7+3=\\\\sqrt{x+2}$$","$$3x+7=4-2\\\\sqrt{x+2}+x+2$$","$$3x+7=1-2\\\\sqrt{x+2}$$"]},{"id":"a1268beother21a-h4","type":"hint","dependencies":["a1268beother21a-h3"],"title":"Isolating the Remaining Radical","text":"Next, isolate the radical on the right side by moving all other terms to the left.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother21a-h5","type":"hint","dependencies":["a1268beother21a-h4"],"title":"Eliminating the Radical","text":"Then, eliminate the remaining radical on the right side by squaring both sides again.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother21a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$4x^2+12x+8=0$$"],"dependencies":["a1268beother21a-h5"],"title":"Result after Eliminating the Radical","text":"What is the quadratic equation (set to 0) after the radical has been eliminated?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$4x^2+12x+8=0$$","$$3x^2+11x+4=0$$","$$x^2+2x+1$$"]},{"id":"a1268beother21a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x=-2, -1$$"],"dependencies":["a1268beother21a-h6"],"title":"Solving the New Quadratic Equation","text":"Solve for $$x$$ from the new quadratic equation. What $$x$$ values make the new equation 0?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x=-2, -1$$","$$x=3, -3$$","$$x=1, 2$$","$$x=1, -2$$"]},{"id":"a1268beother21a-h8","type":"hint","dependencies":[],"title":"Checking for Extraneous Solutions","text":"Plug in the answers to the quadratic equation to the original problem to see if they are valid solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1268beother22","title":"Solving Absolute Value Equations","body":"Solve the following equation for $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Other Types of Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a1268beother22a","stepAnswer":["x=-1,3/2"],"problemType":"MultipleChoice","stepTitle":"$$|1-4x|+8=13$$","stepBody":"","answerType":"string","variabilization":{},"choices":["x=-1,3/2","$$x=-1$$","$$x=-1, 3$$","x=1,2/3"],"hints":{"DefaultPathway":[{"id":"a1268beother22a-h1","type":"hint","dependencies":[],"title":"Isolating the Absolute Value","text":"First, isolate the absolute value by subtracting $$8$$ from both sides of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother22a-h2","type":"hint","dependencies":["a1268beother22a-h1"],"title":"Writing Two Equations","text":"Then, rewrite the absolute value equation into two equations, $$1-4x=5$$ and $$1-4x=-5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother22a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a1268beother22a-h2"],"title":"Solving Equation One","text":"$$1-4x=5$$, $$x=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother22a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{2}$$"],"dependencies":["a1268beother22a-h2"],"title":"Solving Equation Two","text":"$$1-4x=-5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother22a-h5","type":"hint","dependencies":["a1268beother22a-h3","a1268beother22a-h4"],"title":"Final Answer","text":"The answers to the first and second equations are the answer to the original absolute value equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1268beother23","title":"Solving Equations in Quadratic Form","body":"Solve the following equation using subsitution.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Other Types of Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a1268beother23a","stepAnswer":["$$3, -3$$"],"problemType":"MultipleChoice","stepTitle":"$$x^4-8x^2-9=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$3, -3$$","$$3$$","$$3, -3, 1$$","$$3, -3, 1, -1$$"],"hints":{"DefaultPathway":[{"id":"a1268beother23a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["yes"],"dependencies":[],"title":"Evaluating Criteria For Subsitution","text":"Is the exponent of the leading term, $$x^4$$, double the exponent of the second term, $$x^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["yes","no"]},{"id":"a1268beother23a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$u^2-8u-9$$"],"dependencies":["a1268beother23a-h1"],"title":"Using Subsitution","text":"Since the equation fits the criteria for subsitution, let $$u=x^2$$. What is the expression on the left side in terms of u?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother23a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$u=9, -1$$"],"dependencies":["a1268beother23a-h2"],"title":"$$u^2-8u-9=0$$, $$u=$$?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$u=9, -1$$","$$u=-9, 1$$","$$u=9, 1$$","$$u=-9, -1$$"]},{"id":"a1268beother23a-h4","type":"hint","dependencies":["a1268beother23a-h3"],"title":"Subsituting Back In","text":"The next step if to subsitute u back into $$x^2$$, and then solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother23a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x=\\\\pm 3$$"],"dependencies":["a1268beother23a-h4"],"title":"Solving For $$x$$","text":"$$u=9$$, $$x=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x=\\\\pm 3$$","$$x=3$$","$$x=-3$$","$$x=9$$"]},{"id":"a1268beother23a-h6","type":"hint","dependencies":["a1268beother23a-h4"],"title":"Solving For $$x$$","text":"$$u=-1$$, $$x=\\\\sqrt{-1}=i$$, which is not a real number. So we ignore it here.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1268beother3","title":"Polynomial Equations","body":"Solve the following polynomial equation by grouping or factoring.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Other Types of Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a1268beother3a","stepAnswer":["$$1, -1$$"],"problemType":"MultipleChoice","stepTitle":"$$m^3+m^2-m-1=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$0, 1, -1$$","$$1, -1$$","$$1$$","0,2"],"hints":{"DefaultPathway":[{"id":"a1268beother3a-h1","type":"hint","dependencies":[],"title":"Grouping","text":"This polynomial consists of $$4$$ terms, so we will solve by grouping. Factor the first $$2$$ terms and then factor the last $$2$$ terms. If the factors in the parantheses are identical, we can continue the process and solve, unless more factoring is suggested.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$m^2$$"],"dependencies":["a1268beother3a-h1"],"title":"Factoring","text":"What can you factor out of the first $$2$$ terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a1268beother3a-h2"],"title":"Factoring","text":"What can you factor out of the last $$2$$ terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother3a-h4","type":"hint","dependencies":["a1268beother3a-h3"],"title":"Factoring","text":"Combine the common expressions and add the factors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother3a-h5","type":"hint","dependencies":["a1268beother3a-h4"],"title":"Factoring","text":"The expression can be rewritten as $$\\\\left(m^2-1\\\\right) \\\\left(m+1\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother3a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$m^2-1$$"],"dependencies":["a1268beother3a-h5"],"title":"Factoring","text":"You can factor one of the expressions again. Which one?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother3a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(m+1\\\\right) \\\\left(m-1\\\\right) \\\\left(m+1\\\\right)$$"],"dependencies":["a1268beother3a-h6"],"title":"Factoring","text":"What is the expression after factoring $$m^2-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother3a-h8","type":"hint","dependencies":["a1268beother3a-h7"],"title":"Zero-Product Property","text":"Use the Zero-Product property to solve for $$m$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother3a-h9","type":"hint","dependencies":["a1268beother3a-h8"],"title":"Zero-Product Property","text":"Solve for $$m$$ when $$m+1=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother3a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a1268beother3a-h9"],"title":"Zero-Product Property","text":"What is the solution for $$m$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother3a-h11","type":"hint","dependencies":["a1268beother3a-h10"],"title":"Zero-Product Property","text":"Solve for $$m$$ when $$m-1=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother3a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a1268beother3a-h11"],"title":"Zero-Product Property","text":"What is the solution for $$m$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother3a-h13","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$1, -1$$"],"dependencies":["a1268beother3a-h12"],"title":"Solution","text":"What are the solutions?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$0, 1, -1$$","$$1, -1$$","$$1$$","0,2"]}]}}]},{"id":"a1268beother4","title":"Evaluating a Number Raised to a Rational Exponent","body":"Evaluate the expression","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Other Types of Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a1268beother4a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"$$8^{\\\\frac{2}{3}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a1268beother4a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Cube Root","text":"What is $$8^{\\\\frac{1}{3}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother4a-h2","type":"hint","dependencies":["a1268beother4a-h1"],"title":"Simplify","text":"With this information, you can rewrite the original expression as $${\\\\left(8^{\\\\frac{1}{3}}\\\\right)}^2$$ to simplify this problem.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother4a-h3","type":"hint","dependencies":["a1268beother4a-h2"],"title":"Simplify","text":"Simplify the expression by substituting the value for $$8^{\\\\frac{1}{3}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a1268beother4b","stepAnswer":["$$\\\\frac{1}{4}$$"],"problemType":"TextBox","stepTitle":"Evaluate $${64}^{\\\\left(-\\\\frac{1}{3}\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{4}$$","hints":{"DefaultPathway":[{"id":"a1268beother4b-h1","type":"hint","dependencies":[],"title":"Separate","text":"Separate the exponent to $$-1$$ and $$\\\\frac{1}{3}$$, so $${64}^{\\\\left(-\\\\frac{1}{3}\\\\right)}={\\\\left({64}^{\\\\frac{1}{3}}\\\\right)}^{\\\\left(-1\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother4b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a1268beother4b-h1"],"title":"Simplify","text":"What is the cube root of 64?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother4b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4}$$"],"dependencies":["a1268beother4b-h2"],"title":"Substitute","text":"Substitute the value for the cube root of $$64$$ and simplify the expression. What is $$4^{\\\\left(-1\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1268beother5","title":"Solve the Equation Including a Variable Raised to a Rational Exponent","body":"Solve the equation in which a variable is raised to a rational exponent:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Other Types of Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a1268beother5a","stepAnswer":["$$16$$"],"problemType":"TextBox","stepTitle":"$$x^{\\\\frac{5}{4}}=32$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16$$","hints":{"DefaultPathway":[{"id":"a1268beother5a-h1","type":"hint","dependencies":[],"title":"Reciprocal","text":"Raise both sides to the power of $$\\\\frac{4}{5}$$ (reciprocal of the exponent on the left side)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother5a-h2","type":"hint","dependencies":["a1268beother5a-h1"],"title":"Simplify","text":"Simplify the expression by cancelling the left hand side exponent out","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a1268beother5a-h2"],"title":"Evaluate Right Hand Side","text":"What is $${32}^{\\\\frac{1}{5}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother5a-h4","type":"hint","dependencies":["a1268beother5a-h3"],"title":"Simplify","text":"With this information, you can rewirte the original expression as $${\\\\left({32}^{\\\\frac{1}{5}}\\\\right)}^4$$ to simplify this problem.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["a1268beother5a-h4"],"title":"Simplify","text":"Simplify the expression by substituting the value for $${32}^{\\\\frac{1}{5}}$$. What is the value?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a1268beother5b","stepAnswer":["$$25$$"],"problemType":"TextBox","stepTitle":"$$x^{\\\\frac{3}{2}}=125$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$25$$","hints":{"DefaultPathway":[{"id":"a1268beother5b-h1","type":"hint","dependencies":[],"title":"Reciprocal","text":"Raise both sides to the power of 2/3(reciprocal of the exponent on the left side)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother5b-h2","type":"hint","dependencies":["a1268beother5b-h1"],"title":"Simplify","text":"Simplify the expression by cancelling the left hand side exponent out","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother5b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a1268beother5b-h2"],"title":"Cube Root","text":"What is $${125}^{\\\\frac{1}{3}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother5b-h4","type":"hint","dependencies":["a1268beother5b-h3"],"title":"Simplify","text":"With this information, We can separate the original expression as $${\\\\left({125}^{\\\\frac{1}{3}}\\\\right)}^2$$ to simplify this problem.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother5b-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["a1268beother5b-h4"],"title":"Simplify","text":"Simplify the expression by substituting the value for $${32}^{\\\\frac{1}{5}}$$. What is the value?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1268beother6","title":"Solving the equation","body":"Solving an Equation Involving Rational Exponents and Factoring","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Other Types of Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a1268beother6a","stepAnswer":["0,1/81"],"problemType":"MultipleChoice","stepTitle":"$$3x^{\\\\frac{3}{4}}=x^{\\\\frac{1}{2}}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["0,81","$$0, -81$$","0,1/81","$$0-\\\\frac{1}{81}$$"],"hints":{"DefaultPathway":[{"id":"a1268beother6a-h1","type":"hint","dependencies":[],"title":"Subtract","text":"Subract $$x^{\\\\frac{1}{2}}$$ from both sides, and we get the equation $$3x^{\\\\frac{3}{4}}-x^{\\\\frac{1}{2}}=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother6a-h2","type":"hint","dependencies":["a1268beother6a-h1"],"title":"Rewrite","text":"Rewrite $$x^{\\\\frac{1}{2}}$$ as $$x^{\\\\frac{2}{4}}$$, and we get $$3x^{\\\\frac{3}{4}}-x^{\\\\frac{2}{4}}=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother6a-h3","type":"hint","dependencies":["a1268beother6a-h2"],"title":"Factor","text":"Factor out $$x^{\\\\frac{2}{4}}$$, and we get x**(2/4)(3x**(1/4) - $$1)=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother6a-h4","type":"hint","dependencies":["a1268beother6a-h3"],"title":"Zero Product Property","text":"This states that either $$x^{\\\\frac{2}{4}}=0$$ or (3x**(1/4) - $$1)=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a1268beother6a-h4"],"title":"First Solution","text":"Set $$x^{\\\\frac{2}{4}}=0$$ and simplify. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother6a-h6","type":"hint","dependencies":["a1268beother6a-h5"],"title":"Next Solution","text":"Set $$3x^{\\\\frac{1}{4}}-1=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother6a-h7","type":"hint","dependencies":["a1268beother6a-h6"],"title":"Add","text":"Add $$1$$ to both sides, and we get $$3x^{\\\\frac{1}{4}}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother6a-h8","type":"hint","dependencies":["a1268beother6a-h7"],"title":"Divide","text":"Divide both sides by $$3$$, and we get $$x^{\\\\frac{1}{4}}=\\\\frac{1}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother6a-h9","type":"hint","dependencies":["a1268beother6a-h8"],"title":"Reciprocal","text":"Raise both sides to the power of 4(reciprocal of the exponent on the left side), and we get $$x={\\\\left(\\\\frac{1}{3}\\\\right)}^4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother6a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{81}$$"],"dependencies":["a1268beother6a-h9"],"title":"Simplify","text":"Simplify the expression. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a1268beother6b","stepAnswer":["$$-1$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(x+5\\\\right)}^{\\\\frac{3}{2}}=8$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1$$","hints":{"DefaultPathway":[{"id":"a1268beother6b-h1","type":"hint","dependencies":[],"title":"Reciprocal","text":"Raise both sides to the power of 2/3(reciprocal of the exponent on the left side), so we get $${\\\\left({\\\\left(x+5\\\\right)}^{\\\\frac{3}{2}}\\\\right)}^{\\\\frac{2}{3}}=8^{\\\\frac{2}{3}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother6b-h2","type":"hint","dependencies":["a1268beother6b-h1"],"title":"Simplify","text":"Simplify the expression by cancelling the left hand side exponent out, so we get $$x+5=8^{\\\\frac{2}{3}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother6b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a1268beother6b-h2"],"title":"Cube Root","text":"What is $$8^{\\\\frac{1}{3}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother6b-h4","type":"hint","dependencies":["a1268beother6b-h3"],"title":"Simplify","text":"We can rewrite the equation as $$x+5={\\\\left(\\\\frac{8^1}{3}\\\\right)}^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother6b-h5","type":"hint","dependencies":["a1268beother6b-h4"],"title":"Simplify","text":"Simplify the expression by substituting the value for $$8^{\\\\frac{2}{3}}$$, so we get $$x+5=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother6b-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a1268beother6b-h5"],"title":"Subtract","text":"Subtract $$5$$ from both sides. What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1268beother7","title":"Solving a Polynomial by Factoring","body":"Solve the polynomial by factoring:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Other Types of Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a1268beother7a","stepAnswer":["$$0, 4, -4$$"],"problemType":"MultipleChoice","stepTitle":"$$5x^4=80x^2$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$0, 16, -16$$","$$16, -16$$","$$0, 4, -4$$","$$4, -4$$"],"hints":{"DefaultPathway":[{"id":"a1268beother7a-h1","type":"hint","dependencies":[],"title":"Subtract","text":"Subtract $$80x^2$$ from both sides, so we get $$5x^4-80x^2=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother7a-h2","type":"hint","dependencies":["a1268beother7a-h1"],"title":"GCF","text":"Factor out the GCF $$5x^2$$, so we get $$5x^{2\\\\left(x^2-16\\\\right)}=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother7a-h3","type":"hint","dependencies":["a1268beother7a-h2"],"title":"Zero Product Property","text":"This states that either $$5x^2=0$$ or $$x^2-16=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a1268beother7a-h3"],"title":"First Solution","text":"Set $$5x^2=0$$ and simplify. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother7a-h5","type":"hint","dependencies":["a1268beother7a-h4"],"title":"Next Solution","text":"Set $$x^2-16=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother7a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x+4\\\\right) \\\\left(x-4\\\\right)$$"],"dependencies":["a1268beother7a-h5"],"title":"Factor","text":"Factor $$x^2-16$$ as difference of squares. What is it?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother7a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$4, -4$$"],"dependencies":["a1268beother7a-h6"],"title":"Solutions","text":"What two values make the expression equal 0?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$4, -4$$","$$2, -2$$","$$16, -16$$"]}]}},{"id":"a1268beother7b","stepAnswer":["0,-1/2,1/2"],"problemType":"MultipleChoice","stepTitle":"$$12x^4=3x^2$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["-1/2,1/2","0,-1/2,1/2","$$2, -2$$","$$0, 2, -2$$"],"hints":{"DefaultPathway":[{"id":"a1268beother7b-h1","type":"hint","dependencies":[],"title":"Subtract","text":"Subtract $$3x^2$$ from both sides, so we get $$12x\\\\times4-3x^2=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother7b-h2","type":"hint","dependencies":["a1268beother7b-h1"],"title":"GCF","text":"Factor out the GCF $$3x^2$$, and we get $$3x^{2\\\\left(4x^2-1\\\\right)}=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother7b-h3","type":"hint","dependencies":["a1268beother7b-h2"],"title":"Zero Product Property","text":"This states that either $$3x^2=0$$ or $$4x^2-1=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother7b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a1268beother7b-h3"],"title":"First Solution","text":"Set $$3x^2=0$$ and simplify. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother7b-h5","type":"hint","dependencies":["a1268beother7b-h4"],"title":"Next Solution","text":"Set $$4x^2-1=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother7b-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(2x+1\\\\right) \\\\left(2x-1\\\\right)$$"],"dependencies":["a1268beother7b-h5"],"title":"Factor","text":"Factor $$4x^2-1$$ as difference of squares. What expression do we get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother7b-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1}{2}-\\\\frac{1}{2}$$"],"dependencies":["a1268beother7b-h6"],"title":"Solutions","text":"What two values make the expression equal 0?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{1}{2}-\\\\frac{1}{2}$$","$$2, -2$$"]}]}}]},{"id":"a1268beother8","title":"Solving the equation","body":"Solve the equation involving absolute value.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Other Types of Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a1268beother8a","stepAnswer":["-4/3,4"],"problemType":"MultipleChoice","stepTitle":"$$|3x-4|=8$$","stepBody":"","answerType":"string","variabilization":{},"choices":["4,3","6,3/4","-4/3,4","4/3,5"],"hints":{"DefaultPathway":[{"id":"a1268beother8a-h1","type":"hint","dependencies":[],"title":"Creating Two Equations","text":"Create two equations setting $$3x-4$$ equal to $$8$$ and $$-8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a1268beother8a-h1"],"title":"Solving $$3x-4=8$$","text":"What is $$x$$ when $$3x-4=8$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a1268beother8a-h2-s1","type":"hint","dependencies":[],"title":"Solving $$3x-4=8$$","text":"To solve $$3x-4=8$$, start by adding $$4$$ to both sides of the equation: $$3x=12$$. Then, divide both sides by $$3$$ to get $$x=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a1268beother8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-4}{3}$$"],"dependencies":["a1268beother8a-h1"],"title":"Solving $$3x-4=-8$$","text":"What is $$x$$ when $$3x-4=-8$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a1268beother8a-h3-s1","type":"hint","dependencies":[],"title":"Solving $$3x-4=-8$$","text":"For $$3x-4=-8$$, add $$4$$ to both sides of the equation: $$3x=-4$$. Then, divide both sides by $$3$$ to get $$x=\\\\frac{-4}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a1268beother8a-h4","type":"hint","dependencies":["a1268beother8a-h2","a1268beother8a-h3"],"title":"Final Answer","text":"So, the two values of $$x$$ that would satisfy $$|3x-4|=8$$ are $$\\\\frac{-4}{3}$$ and $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1268beother9","title":"Solve the equation","body":"Solve the equation involving absolute value.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Other Types of Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a1268beother9a","stepAnswer":["-5/4,7/4"],"problemType":"MultipleChoice","stepTitle":"$$|1-4x|-1=5$$","stepBody":"","answerType":"string","variabilization":{},"choices":["-7/4,5/4","-5/6,7/6","-5/4,7/4","$$\\\\frac{-5}{4}-\\\\frac{7}{4}$$"],"hints":{"DefaultPathway":[{"id":"a1268beother9a-h1","type":"hint","dependencies":[],"title":"Adding $$1$$ to Both Sides","text":"The first step is to add $$1$$ to both sides of the equation: $$|1-4x|=6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother9a-h2","type":"hint","dependencies":["a1268beother9a-h1"],"title":"Creating Two Equations","text":"Create two equations setting $$1-4x$$ equal to $$6$$ and $$-6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1268beother9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-5}{4}$$"],"dependencies":["a1268beother9a-h2"],"title":"Solving $$1-4x=6$$","text":"What is $$x$$ when $$1-4x=6$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a1268beother9a-h3-s1","type":"hint","dependencies":[],"title":"Solving $$1-4x=6$$","text":"To solve $$1-4x=6$$, start by subtracting $$1$$ from both sides of the equation: $$-4x=5$$. Then, divide both sides by $$-4$$ to get $$x=\\\\frac{-5}{4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a1268beother9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{7}{4}$$"],"dependencies":["a1268beother9a-h2"],"title":"Solving $$1-4x=-6$$","text":"What is $$x$$ when $$1-4x=-6$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a1268beother9a-h4-s1","type":"hint","dependencies":[],"title":"Solving $$1-4x=-6$$","text":"For $$1-4x=-6$$, subtract $$1$$ from both sides of the equation: $$-4x=-7$$. Then, divide both sides by $$-4$$ to get $$x=\\\\frac{7}{4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a1268beother9a-h5","type":"hint","dependencies":["a1268beother9a-h3","a1268beother9a-h4"],"title":"Final Answer","text":"So, the only values of $$x$$ that would satisfy $$|1-4x|-1=5$$ are $$\\\\frac{-5}{4}$$ and $$\\\\frac{7}{4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1279cdpowers1","title":"Simplify Expressions Using the Product Property for Exponents","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Use Multiplication Properties of Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1279cdpowers1a","stepAnswer":["$$d^9$$"],"problemType":"TextBox","stepTitle":"$$d^3 d^6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$d^9$$","hints":{"DefaultPathway":[{"id":"a1279cdpowers1a-h1","type":"hint","dependencies":[],"title":"Product Property for Exponents","text":"The product property: If a is a real number, and $$m$$ and $$n$$ are counting numbers, then: $$a^m a^n=a^{m+n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers1a-h2","type":"hint","dependencies":["a1279cdpowers1a-h1"],"title":"Bases","text":"Since the bases are the same here, you can use the product property.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers1a-h3","type":"hint","dependencies":["a1279cdpowers1a-h2"],"title":"Add","text":"Add the powers of the two exponents together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers1a-h4","type":"hint","dependencies":["a1279cdpowers1a-h3"],"title":"Answer","text":"Therefore, the answer is $$d^9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1279cdpowers10","title":"Simplify Expressions Using Power Properties of Exponents","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Use Multiplication Properties of Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1279cdpowers10a","stepAnswer":["$$48y^4$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(2y\\\\right)}^3 6y$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$48y^4$$","hints":{"DefaultPathway":[{"id":"a1279cdpowers10a-h1","type":"hint","dependencies":[],"title":"Power property","text":"If a is a real number, and $$m$$ and $$n$$ are whole numbers, then $${\\\\left(a^m\\\\right)}^n=a^{m n}$$. To raise a power to a power, multiply the exponents.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers10a-h2","type":"hint","dependencies":["a1279cdpowers10a-h1"],"title":"Use the power property","text":"Using the power property to simplify the expression results in $$8y^3\\\\times6 y$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers10a-h3","type":"hint","dependencies":["a1279cdpowers10a-h2"],"title":"Product Property for Exponents","text":"Now it is possible to simplify the expression by combining terms. Simply add the powers of the term that is being multiplied, and multiply the bases together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers10a-h4","type":"hint","dependencies":["a1279cdpowers10a-h3"],"title":"Answer","text":"Therefore, the answer is $$48y^4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1279cdpowers11","title":"Simplify Expressions Using Power Properties of Exponents","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Use Multiplication Properties of Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1279cdpowers11a","stepAnswer":["$$-18y^{11}$$"],"problemType":"TextBox","stepTitle":"$$6y^7 \\\\left(-3y^4\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-18y^{11}$$","hints":{"DefaultPathway":[{"id":"a1279cdpowers11a-h1","type":"hint","dependencies":[],"title":"Product Property for Exponents","text":"The product property: If a is a real number, and $$m$$ and $$n$$ are counting numbers, then: $$a^m a^n=a^{m+n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers11a-h2","type":"hint","dependencies":["a1279cdpowers11a-h1"],"title":"Product Property","text":"Now it is possible to simplify the expression by combining terms. Simply add the powers of the term that is being multiplied, and multiply the bases together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers11a-h3","type":"hint","dependencies":["a1279cdpowers11a-h2"],"title":"Answer","text":"Therefore, the answer is $$-18y^{11}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1279cdpowers12","title":"Simplify Expressions Using Power Properties of Exponents","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Use Multiplication Properties of Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1279cdpowers12a","stepAnswer":["$$72u^7$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(-8u^6\\\\right) \\\\left(-9u\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$72u^7$$","hints":{"DefaultPathway":[{"id":"a1279cdpowers12a-h1","type":"hint","dependencies":[],"title":"Product Property for Exponents","text":"The product property: If a is a real number, and $$m$$ and $$n$$ are counting numbers, then: $$a^m a^n=a^{m+n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers12a-h2","type":"hint","dependencies":["a1279cdpowers12a-h1"],"title":"Product Property","text":"Now it is possible to simplify the expression by combining terms. Simply add the powers of the term that is being multiplied, and multiply the bases together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers12a-h3","type":"hint","dependencies":["a1279cdpowers12a-h2"],"title":"Answer","text":"Therefore, the answer is $$72u^7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1279cdpowers13","title":"Simplify Expressions Using Power Properties of Exponents","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Use Multiplication Properties of Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1279cdpowers13a","stepAnswer":["$$4f^{11}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{1}{5} f^8 20f^3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4f^{11}$$","hints":{"DefaultPathway":[{"id":"a1279cdpowers13a-h1","type":"hint","dependencies":[],"title":"Product Property for Exponents","text":"The product property: If a is a real number, and $$m$$ and $$n$$ are counting numbers, then: $$a^m a^n=a^{m+n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers13a-h2","type":"hint","dependencies":["a1279cdpowers13a-h1"],"title":"Product Property","text":"Now it is possible to simplify the expression by combining terms. Simply add the powers of the term that is being multiplied, and multiply the bases together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers13a-h3","type":"hint","dependencies":["a1279cdpowers13a-h2"],"title":"Answer","text":"Therefore, the answer is $$4f^{11}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1279cdpowers14","title":"Simplify Expressions Using Power Properties of Exponents","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Use Multiplication Properties of Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1279cdpowers14a","stepAnswer":["$$36a^5 b^7$$"],"problemType":"TextBox","stepTitle":"$$4a^3 b 9a^2 b^6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$36a^5 b^7$$","hints":{"DefaultPathway":[{"id":"a1279cdpowers14a-h1","type":"hint","dependencies":[],"title":"Product Property for Exponents","text":"The product property: If a is a real number, and $$m$$ and $$n$$ are counting numbers, then: $$a^m a^n=a^{m+n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers14a-h2","type":"hint","dependencies":["a1279cdpowers14a-h1"],"title":"Product Property","text":"Now it is possible to simplify the expression by combining terms. Simply add the powers of the term that is being multiplied, and multiply the bases together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers14a-h3","type":"hint","dependencies":["a1279cdpowers14a-h2"],"title":"Answer","text":"Therefore, the answer is $$36a^5 b^7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1279cdpowers15","title":"Simplify Expressions Using Power Properties of Exponents","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Use Multiplication Properties of Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1279cdpowers15a","stepAnswer":["$$8r^2 s^5$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{4}{7} {rs}^2 14{rs}^3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8r^2 s^5$$","hints":{"DefaultPathway":[{"id":"a1279cdpowers15a-h1","type":"hint","dependencies":[],"title":"Product Property for Exponents","text":"The product property: If a is a real number, and $$m$$ and $$n$$ are counting numbers, then: $$a^m a^n=a^{m+n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers15a-h2","type":"hint","dependencies":["a1279cdpowers15a-h1"],"title":"Product Property","text":"Now it is possible to simplify the expression by combining terms. Simply add the powers of the term that is being multiplied, and multiply the bases together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers15a-h3","type":"hint","dependencies":["a1279cdpowers15a-h2"],"title":"Answer","text":"Therefore, the answer is $$8r^2 s^5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1279cdpowers16","title":"Simplify Expressions Using the Product Property for Exponents","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Use Multiplication Properties of Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1279cdpowers16a","stepAnswer":["$$y^{11}$$"],"problemType":"TextBox","stepTitle":"$$y^5 y^6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y^{11}$$","hints":{"DefaultPathway":[{"id":"a1279cdpowers16a-h1","type":"hint","dependencies":[],"title":"Identifying Exponent Property","text":"Use the product property, $$a^m a^n$$ $$=$$ $$a^{m+n}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y^{11}$$"],"dependencies":["a1279cdpowers16a-h1"],"title":"Simplifying the Exponent","text":"What does the exponent evalute to once simplified?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1279cdpowers17","title":"Simplify Expressions Using the Product Property for Exponents","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Use Multiplication Properties of Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1279cdpowers17a","stepAnswer":["$$b^{17}$$"],"problemType":"TextBox","stepTitle":"$$b^9 b^8$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$b^{17}$$","hints":{"DefaultPathway":[{"id":"a1279cdpowers17a-h1","type":"hint","dependencies":[],"title":"Identifying Exponent Property","text":"Use the product property, $$a^m a^n$$ $$=$$ $$a^{m+n}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers17a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$b^{17}$$"],"dependencies":["a1279cdpowers17a-h1"],"title":"Simplifying the Exponent","text":"What does the exponent evalute to once simplified?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1279cdpowers18","title":"Simplify Expressions Using the Product Property for Exponents","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Use Multiplication Properties of Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1279cdpowers18a","stepAnswer":["$$d^{11}$$"],"problemType":"TextBox","stepTitle":"$$d^4 d^5 d^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$d^{11}$$","hints":{"DefaultPathway":[{"id":"a1279cdpowers18a-h1","type":"hint","dependencies":[],"title":"Identifying Exponent Property","text":"Since the bases are the same, we can use the product property to add the exponents.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$d^{11}$$"],"dependencies":["a1279cdpowers18a-h1"],"title":"Simplifying the Exponent","text":"What does the exponent evalute to once simplified?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1279cdpowers19","title":"Simplify Expressions Using the Product Property for Exponents","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Use Multiplication Properties of Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1279cdpowers19a","stepAnswer":["$$x^{18}$$"],"problemType":"TextBox","stepTitle":"$$x^6 x^4 x^8$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^{18}$$","hints":{"DefaultPathway":[{"id":"a1279cdpowers19a-h1","type":"hint","dependencies":[],"title":"Identifying Exponent Property","text":"Since the bases are the same, we can use the product property to add the exponents.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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\\\\left(-7y^4\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-35y^{11}$$","hints":{"DefaultPathway":[{"id":"a1279cdpowers28a-h1","type":"hint","dependencies":[],"title":"Identifying Applicable Property","text":"Use the Commutative Property to rearrange the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers28a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-35y^{11}$$"],"dependencies":["a1279cdpowers28a-h1"],"title":"Multiplying the Terms","text":"What do the terms multiply to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1279cdpowers29","title":"Multiply Monomials","body":"Multiply the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Use Multiplication Properties of Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1279cdpowers29a","stepAnswer":["$$54b^9$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(-6b^4\\\\right) \\\\left(-9b^5\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$54b^9$$","hints":{"DefaultPathway":[{"id":"a1279cdpowers29a-h1","type":"hint","dependencies":[],"title":"Identifying Applicable Property","text":"Use the Commutative Property to rearrange the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers29a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$54b^9$$"],"dependencies":["a1279cdpowers29a-h1"],"title":"Multiplying the Terms","text":"What do the terms multiply to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1279cdpowers3","title":"Simplify Expressions Using the Product Property for Exponents","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Use Multiplication Properties of Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1279cdpowers3a","stepAnswer":["$$n^{31}$$"],"problemType":"TextBox","stepTitle":"$$n^{19} n^{12}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$n^{31}$$","hints":{"DefaultPathway":[{"id":"a1279cdpowers3a-h1","type":"hint","dependencies":[],"title":"Product Property for Exponents","text":"The product property: If a is a real number, and $$m$$ and $$n$$ are counting numbers, then: $$a^m a^n=a^{m+n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers3a-h2","type":"hint","dependencies":["a1279cdpowers3a-h1"],"title":"Bases","text":"Since the bases are the same here, you can use the product property.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers3a-h3","type":"hint","dependencies":["a1279cdpowers3a-h2"],"title":"Add","text":"Add the powers of the two exponents together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers3a-h4","type":"hint","dependencies":["a1279cdpowers3a-h3"],"title":"Answer","text":"Therefore, the answer is $$n^{31}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1279cdpowers30","title":"Multiply Monomials","body":"Multiply the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Use Multiplication Properties of Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1279cdpowers30a","stepAnswer":["$$10x^4 y^3$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{5}{6} x^3 y 12x y^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10x^4 y^3$$","hints":{"DefaultPathway":[{"id":"a1279cdpowers30a-h1","type":"hint","dependencies":[],"title":"Identifying Applicable Property","text":"Use the Commutative Property to rearrange the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers30a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10x^4 y^3$$"],"dependencies":["a1279cdpowers30a-h1"],"title":"Multiplying the Terms","text":"What do the terms multiply to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1279cdpowers4","title":"Simplify Expressions Using the Product Property for Exponents","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Use Multiplication Properties of Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1279cdpowers4a","stepAnswer":["$$q^{42}$$"],"problemType":"TextBox","stepTitle":"$$q^{27} q^{15}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$q^{42}$$","hints":{"DefaultPathway":[{"id":"a1279cdpowers4a-h1","type":"hint","dependencies":[],"title":"Product Property for Exponents","text":"The product property: If a is a real number, and $$m$$ and $$n$$ are counting numbers, then: $$a^m a^n=a^{m+n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers4a-h2","type":"hint","dependencies":["a1279cdpowers4a-h1"],"title":"Bases","text":"Since the bases are the same here, you can use the product property.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers4a-h3","type":"hint","dependencies":["a1279cdpowers4a-h2"],"title":"Add","text":"Add the powers of the two exponents together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers4a-h4","type":"hint","dependencies":["a1279cdpowers4a-h3"],"title":"Answer","text":"Therefore, the answer is $$q^{42}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1279cdpowers5","title":"Simplify Expressions Using the Product Property for Exponents","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Use Multiplication Properties of Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1279cdpowers5a","stepAnswer":["$$w^6$$"],"problemType":"TextBox","stepTitle":"$$w^5 w$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$w^6$$","hints":{"DefaultPathway":[{"id":"a1279cdpowers5a-h1","type":"hint","dependencies":[],"title":"Product Property for Exponents","text":"The product property: If a is a real number, and $$m$$ and $$n$$ are counting numbers, then: $$a^m a^n=a^{m+n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers5a-h2","type":"hint","dependencies":["a1279cdpowers5a-h1"],"title":"Bases","text":"Since the bases are the same here, you can use the product property.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers5a-h3","type":"hint","dependencies":["a1279cdpowers5a-h2"],"title":"Add","text":"Add the powers of the two exponents together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers5a-h4","type":"hint","dependencies":["a1279cdpowers5a-h3"],"title":"Answer","text":"Therefore, the answer is $$w^6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1279cdpowers6","title":"Simplify Expressions Using Power Properties of Exponents","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Use Multiplication Properties of Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1279cdpowers6a","stepAnswer":["$$y^{14}$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(y^2\\\\right)}^4 {\\\\left(y^3\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y^{14}$$","hints":{"DefaultPathway":[{"id":"a1279cdpowers6a-h1","type":"hint","dependencies":[],"title":"Power property","text":"If a is a real number, and $$m$$ and $$n$$ are whole numbers, then $${\\\\left(a^m\\\\right)}^n=a^{m n}$$. To raise a power to a power, multiply the exponents.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers6a-h2","type":"hint","dependencies":["a1279cdpowers6a-h1"],"title":"Use the power property","text":"Using the power property to simplify the expression results in $$y^8 y^6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers6a-h3","type":"hint","dependencies":["a1279cdpowers6a-h2"],"title":"Product Property for Exponents","text":"Now, since the bases are the same, it is possible to simplify the expression by combining terms. Simply add the powers of the term that is being multiplied.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers6a-h4","type":"hint","dependencies":["a1279cdpowers6a-h3"],"title":"Answer","text":"Therefore, the answer is $$y^{14}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1279cdpowers7","title":"Simplify Expressions Using Power Properties of Exponents","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Use Multiplication Properties of Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1279cdpowers7a","stepAnswer":["$$w^{22}$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(w^4\\\\right)}^3 {\\\\left(w^5\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$w^{22}$$","hints":{"DefaultPathway":[{"id":"a1279cdpowers7a-h1","type":"hint","dependencies":[],"title":"Power property","text":"If a is a real number, and $$m$$ and $$n$$ are whole numbers, then $${\\\\left(a^m\\\\right)}^n=a^{m n}$$. To raise a power to a power, multiply the exponents.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers7a-h2","type":"hint","dependencies":["a1279cdpowers7a-h1"],"title":"Use the power property","text":"Using the power property to simplify the expression results in $$w^{12} w^{10}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers7a-h3","type":"hint","dependencies":["a1279cdpowers7a-h2"],"title":"Product Property for Exponents","text":"Now, since the bases are the same, it is possible to simplify the expression by combining terms. Simply add the powers of the term that is being multiplied.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers7a-h4","type":"hint","dependencies":["a1279cdpowers7a-h3"],"title":"Answer","text":"Therefore, the answer is $$w^{22}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1279cdpowers8","title":"Simplify Expressions Using Power Properties of Exponents","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Use Multiplication Properties of Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1279cdpowers8a","stepAnswer":["$$-1000q^6 p^{12}$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(-10q^2 p^4\\\\right)}^3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1000q^6 p^{12}$$","hints":{"DefaultPathway":[{"id":"a1279cdpowers8a-h1","type":"hint","dependencies":[],"title":"Power property","text":"If a is a real number, and $$m$$ and $$n$$ are whole numbers, then $${\\\\left(a^m\\\\right)}^n=a^{m n}$$. To raise a power to a power, multiply the exponents.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers8a-h2","type":"hint","dependencies":["a1279cdpowers8a-h1"],"title":"Exponents with numbers","text":"Raise each variable and number to the power of $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers8a-h3","type":"hint","dependencies":["a1279cdpowers8a-h2"],"title":"Answer","text":"Using the power property to simplify the powers in the variables results in $$-1000q^6 p^{12}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1279cdpowers9","title":"Simplify Expressions Using Power Properties of Exponents","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Use Multiplication Properties of Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1279cdpowers9a","stepAnswer":["$$1125t^8$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(5t^2\\\\right)}^3 {\\\\left(3t\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1125t^8$$","hints":{"DefaultPathway":[{"id":"a1279cdpowers9a-h1","type":"hint","dependencies":[],"title":"Power property","text":"If a is a real number, and $$m$$ and $$n$$ are whole numbers, then $${\\\\left(a^m\\\\right)}^n=a^{m n}$$. To raise a power to a power, multiply the exponents.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers9a-h2","type":"hint","dependencies":["a1279cdpowers9a-h1"],"title":"Use the power property","text":"Using the power property to simplify the expression results in $$125t^6\\\\times9 t^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers9a-h3","type":"hint","dependencies":["a1279cdpowers9a-h2"],"title":"Product Property for Exponents","text":"Now it is possible to simplify the expression by combining terms. Simply add the powers of the term that is being multiplied, and multiply the bases together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1279cdpowers9a-h4","type":"hint","dependencies":["a1279cdpowers9a-h3"],"title":"Answer","text":"Therefore, the answer is $$1125t^8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a137dddgre1","title":"Greatest Common Factor","body":"Find the greatest common factor.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a137dddgre1a","stepAnswer":["$$18$$"],"problemType":"TextBox","stepTitle":"$$54$$, $$36$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$18$$","hints":{"DefaultPathway":[{"id":"a137dddgre1a-h1","type":"hint","dependencies":[],"title":"Factor into Primes","text":"Factor each coefficient into primes. A prime number is a counting number greater than $$1$$, whose only factors are $$1$$ and itself. The first few prime numbers are: $$2$$, $$3$$, $$5$$, $$7$$, $$11$$, $$13$$, etc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre1a-h2","type":"hint","dependencies":["a137dddgre1a-h1"],"title":"Factor First Number","text":"$$54=9\\\\times6$$\\\\n$$54=3\\\\times3 2\\\\times3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre1a-h3","type":"hint","dependencies":["a137dddgre1a-h2"],"title":"Factor Second Number","text":"$$36=6\\\\times6$$\\\\n$$36=2\\\\times3 2\\\\times3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre1a-h4","type":"hint","dependencies":["a137dddgre1a-h2","a137dddgre1a-h3"],"title":"Identify Common Factors in each Column","text":"$$54=3\\\\times3 2\\\\times3$$\\\\n$$36=2\\\\times3 2\\\\times3$$\\\\n$$2$$, $$3$$, and $$3$$ are shared by both numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre1a-h5","type":"hint","dependencies":["a137dddgre1a-h4"],"title":"Multiply Common Factors","text":"Bring down the $$2$$, $$3$$, and $$3$$, and then multiply.\\\\n$$GCF=2\\\\times3\\\\times3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre1a-h6","type":"hint","dependencies":["a137dddgre1a-h5"],"title":"Multiply Common Factors","text":"$$GCF=18$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a137dddgre10","title":"Greatest Common Factor","body":"Find the greatest common factor.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a137dddgre10a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"$$5b$$, $$30$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"a137dddgre10a-h1","type":"hint","dependencies":[],"title":"Factor into Primes","text":"Factor each coefficient into primes and write the variables with exponents in expanded form. A prime number is a counting number greater than $$1$$, whose only factors are $$1$$ and itself. The first few prime numbers are: $$2$$, $$3$$, $$5$$, $$7$$, $$11$$, $$13$$, etc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre10a-h2","type":"hint","dependencies":["a137dddgre10a-h1"],"title":"Factor First Expression","text":"$$5b=5b$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre10a-h3","type":"hint","dependencies":["a137dddgre10a-h2"],"title":"Factor Second Expression","text":"$$30=3\\\\times5\\\\times2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre10a-h4","type":"hint","dependencies":["a137dddgre10a-h2","a137dddgre10a-h3"],"title":"Identify Common Factors in each Column","text":"$$5b=5b$$\\\\n$$30=3\\\\times5\\\\times2$$\\\\n$$5$$ are shared by both expressions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre10a-h5","type":"hint","dependencies":["a137dddgre10a-h4"],"title":"Multiply Common Factors","text":"$$GCF=5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a137dddgre11","title":"Greatest Common Factor","body":"Find the greatest common factor.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a137dddgre11a","stepAnswer":["$$x$$"],"problemType":"TextBox","stepTitle":"$$3x$$, $$10x^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x$$","hints":{"DefaultPathway":[{"id":"a137dddgre11a-h1","type":"hint","dependencies":[],"title":"Factor into Primes","text":"Factor each coefficient into primes and write the variables with exponents in expanded form. A prime number is a counting number greater than $$1$$, whose only factors are $$1$$ and itself. The first few prime numbers are: $$2$$, $$3$$, $$5$$, $$7$$, $$11$$, $$13$$, etc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre11a-h2","type":"hint","dependencies":["a137dddgre11a-h1"],"title":"Factor First Expression","text":"$$3x=3x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre11a-h3","type":"hint","dependencies":["a137dddgre11a-h2"],"title":"Factor Second Expression","text":"$$10x^2=5\\\\times2 x x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre11a-h4","type":"hint","dependencies":["a137dddgre11a-h2","a137dddgre11a-h3"],"title":"Identify Common Factors in each Column","text":"$$3x=3x$$\\\\n$$10x^2=5\\\\times2 x x$$\\\\n$$x$$ are shared by both expressions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre11a-h5","type":"hint","dependencies":["a137dddgre11a-h4"],"title":"Multiply Common Factors","text":"$$GCF=x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a137dddgre12","title":"Greatest Common Factor","body":"Find the greatest common factor.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a137dddgre12a","stepAnswer":["$$7b$$"],"problemType":"TextBox","stepTitle":"$$21b^2$$, $$14b$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7b$$","hints":{"DefaultPathway":[{"id":"a137dddgre12a-h1","type":"hint","dependencies":[],"title":"Factor into Primes","text":"Factor each coefficient into primes and write the variables with exponents in expanded form. A prime number is a counting number greater than $$1$$, whose only factors are $$1$$ and itself. The first few prime numbers are: $$2$$, $$3$$, $$5$$, $$7$$, $$11$$, $$13$$, etc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre12a-h2","type":"hint","dependencies":["a137dddgre12a-h1"],"title":"Factor First Expression","text":"$$21b^2=7\\\\times3 b b$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre12a-h3","type":"hint","dependencies":["a137dddgre12a-h2"],"title":"Factor Second Expression","text":"$$14b=7\\\\times2 b$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre12a-h4","type":"hint","dependencies":["a137dddgre12a-h2","a137dddgre12a-h3"],"title":"Identify Common Factors in each Column","text":"$$21b^2=7\\\\times3 b b$$\\\\n$$14b=7\\\\times2 b$$\\\\n$$7$$ and $$b$$ are shared by both expressions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre12a-h5","type":"hint","dependencies":["a137dddgre12a-h4"],"title":"Multiply Common Factors","text":"Bring down the $$7$$ and $$b$$, and then multiply.\\\\n$$GCF=7b$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre12a-h6","type":"hint","dependencies":["a137dddgre12a-h5"],"title":"Multiply Common Factors","text":"$$GCF=7b$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a137dddgre13","title":"Greatest Common Factor","body":"Find the greatest common factor.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a137dddgre13a","stepAnswer":["$$8w^2$$"],"problemType":"TextBox","stepTitle":"$$8w^2$$, $$24w^3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8w^2$$","hints":{"DefaultPathway":[{"id":"a137dddgre13a-h1","type":"hint","dependencies":[],"title":"Factor into Primes","text":"Factor each coefficient into primes and write the variables with exponents in expanded form. A prime number is a counting number greater than $$1$$, whose only factors are $$1$$ and itself. The first few prime numbers are: $$2$$, $$3$$, $$5$$, $$7$$, $$11$$, $$13$$, etc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre13a-h2","type":"hint","dependencies":["a137dddgre13a-h1"],"title":"Factor First Expression","text":"$$8w^2=2\\\\times2\\\\times2 w w$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre13a-h3","type":"hint","dependencies":["a137dddgre13a-h2"],"title":"Factor Second Expression","text":"$$24w^3=2\\\\times2\\\\times3\\\\times2 w w w$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre13a-h4","type":"hint","dependencies":["a137dddgre13a-h2","a137dddgre13a-h3"],"title":"Identify Common Factors in each Column","text":"$$8w^2=2\\\\times2\\\\times2 w w$$\\\\n$$24w^3=2\\\\times2\\\\times3\\\\times2 w w w$$\\\\n$$2$$, $$2$$, $$2$$, w, and w are shared by both expressions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre13a-h5","type":"hint","dependencies":["a137dddgre13a-h4"],"title":"Multiply Common Factors","text":"Bring down the $$2$$, $$2$$, $$2$$, w, and w, and then multiply.\\\\n$$GCF=2\\\\times2\\\\times2 w w$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre13a-h6","type":"hint","dependencies":["a137dddgre13a-h5"],"title":"Multiply Common Factors","text":"$$GCF=8w^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a137dddgre14","title":"Greatest Common Factor","body":"Find the greatest common factor.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a137dddgre14a","stepAnswer":["$$6x^2$$"],"problemType":"TextBox","stepTitle":"$$30x^2$$, $$18x^3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6x^2$$","hints":{"DefaultPathway":[{"id":"a137dddgre14a-h1","type":"hint","dependencies":[],"title":"Factor into Primes","text":"Factor each coefficient into primes and write the variables with exponents in expanded form. A prime number is a counting number greater than $$1$$, whose only factors are $$1$$ and itself. The first few prime numbers are: $$2$$, $$3$$, $$5$$, $$7$$, $$11$$, $$13$$, etc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre14a-h2","type":"hint","dependencies":["a137dddgre14a-h1"],"title":"Factor First Expression","text":"$$30x^2=5\\\\times2\\\\times3 x x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre14a-h3","type":"hint","dependencies":["a137dddgre14a-h2"],"title":"Factor Second Expression","text":"$$18x^3=3\\\\times3\\\\times2 x x x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre14a-h4","type":"hint","dependencies":["a137dddgre14a-h2","a137dddgre14a-h3"],"title":"Identify Common Factors in each Column","text":"$$30x^2=5\\\\times2\\\\times3 x x$$\\\\n$$18x^3=3\\\\times3\\\\times2 x x x$$\\\\n$$2$$, $$3$$, $$x$$, and $$x$$ are shared by both expressions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre14a-h5","type":"hint","dependencies":["a137dddgre14a-h4"],"title":"Multiply Common Factors","text":"Bring down the $$2$$, $$3$$, $$x$$, and $$x$$, and then multiply.\\\\n$$GCF=2\\\\times3 x x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre14a-h6","type":"hint","dependencies":["a137dddgre14a-h5"],"title":"Multiply Common Factors","text":"$$GCF=6x^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a137dddgre15","title":"Greatest Common Factor","body":"Find the greatest common factor.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a137dddgre15a","stepAnswer":["$$2p q$$"],"problemType":"TextBox","stepTitle":"$$10p^3 q$$, $$12p q^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2p q$$","hints":{"DefaultPathway":[{"id":"a137dddgre15a-h1","type":"hint","dependencies":[],"title":"Factor into Primes","text":"Factor each coefficient into primes and write the variables with exponents in expanded form. A prime number is a counting number greater than $$1$$, whose only factors are $$1$$ and itself. The first few prime numbers are: $$2$$, $$3$$, $$5$$, $$7$$, $$11$$, $$13$$, etc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre15a-h2","type":"hint","dependencies":["a137dddgre15a-h1"],"title":"Factor First Expression","text":"$$10p^3 q=5\\\\times2 p p p q$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre15a-h3","type":"hint","dependencies":["a137dddgre15a-h2"],"title":"Factor Second Expression","text":"$$12p q^2=2\\\\times2\\\\times3 p q q$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre15a-h4","type":"hint","dependencies":["a137dddgre15a-h2","a137dddgre15a-h3"],"title":"Identify Common Factors in each Column","text":"$$10p^3 q=5\\\\times2 p p p q$$\\\\n$$12p q^2=2\\\\times2\\\\times3 p q q$$\\\\n$$2$$, $$p$$, and q are shared by both expressions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre15a-h5","type":"hint","dependencies":["a137dddgre15a-h4"],"title":"Multiply Common Factors","text":"Bring down the $$2$$, $$p$$, and q, and then multiply.\\\\n$$GCF=2p q$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre15a-h6","type":"hint","dependencies":["a137dddgre15a-h5"],"title":"Multiply Common Factors","text":"$$GCF=2p q$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a137dddgre16","title":"Greatest Common Factor","body":"Find the greatest common factor.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a137dddgre16a","stepAnswer":["$$2a b^2$$"],"problemType":"TextBox","stepTitle":"$$8a^2 b^3$$, $$10a b^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2a b^2$$","hints":{"DefaultPathway":[{"id":"a137dddgre16a-h1","type":"hint","dependencies":[],"title":"Factor into Primes","text":"Factor each coefficient into primes and write the variables with exponents in expanded form. A prime number is a counting number greater than $$1$$, whose only factors are $$1$$ and itself. The first few prime numbers are: $$2$$, $$3$$, $$5$$, $$7$$, $$11$$, $$13$$, etc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre16a-h2","type":"hint","dependencies":["a137dddgre16a-h1"],"title":"Factor First Expression","text":"$$8a^2 b^3=2\\\\times2\\\\times2 a a b b b$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre16a-h3","type":"hint","dependencies":["a137dddgre16a-h2"],"title":"Factor Second Expression","text":"$$10a b^2=5\\\\times2 a b b$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre16a-h4","type":"hint","dependencies":["a137dddgre16a-h2","a137dddgre16a-h3"],"title":"Identify Common Factors in each Column","text":"$$8a^2 b^3=2\\\\times2\\\\times2 a a b b b$$\\\\n$$10a b^2=5\\\\times2 a b b$$\\\\n$$2$$, a, $$b$$, and $$b$$ are shared by both expressions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre16a-h5","type":"hint","dependencies":["a137dddgre16a-h4"],"title":"Multiply Common Factors","text":"Bring down the $$2$$, a, $$b$$, and $$b$$, and then multiply.\\\\n$$GCF=2a b b$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre16a-h6","type":"hint","dependencies":["a137dddgre16a-h5"],"title":"Multiply Common Factors","text":"$$GCF=2a b^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a137dddgre17","title":"Greatest Common Factor","body":"Find the greatest common factor.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a137dddgre17a","stepAnswer":["$$6m^2 n^3$$"],"problemType":"TextBox","stepTitle":"$$12m^2 n^3$$, $$30m^5 n^3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6m^2 n^3$$","hints":{"DefaultPathway":[{"id":"a137dddgre17a-h1","type":"hint","dependencies":[],"title":"Factor into Primes","text":"Factor each coefficient into primes and write the variables with exponents in expanded form. A prime number is a counting number greater than $$1$$, whose only factors are $$1$$ and itself. The first few prime numbers are: $$2$$, $$3$$, $$5$$, $$7$$, $$11$$, $$13$$, etc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre17a-h2","type":"hint","dependencies":["a137dddgre17a-h1"],"title":"Factor First Expression","text":"$$12m^2 n^3=3\\\\times2\\\\times2 m m n n n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre17a-h3","type":"hint","dependencies":["a137dddgre17a-h2"],"title":"Factor Second Expression","text":"$$30m^5 n^3=5\\\\times3\\\\times2 m m m m m n n n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre17a-h4","type":"hint","dependencies":["a137dddgre17a-h2","a137dddgre17a-h3"],"title":"Identify Common Factors in each Column","text":"$$12m^2 n^3=3\\\\times2\\\\times2 m m n n n$$\\\\n$$30m^5 n^3=5\\\\times3\\\\times2 m m m m m n n n$$\\\\n$$3$$, $$2$$, $$m$$, $$m$$, $$n$$, $$n$$, and $$n$$ are shared by both expressions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre17a-h5","type":"hint","dependencies":["a137dddgre17a-h4"],"title":"Multiply Common Factors","text":"Bring down the $$3$$, $$2$$, $$m$$, $$m$$, $$n$$, $$n$$, and $$n$$, and then multiply.\\\\n$$GCF=3\\\\times2 m m n n n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre17a-h6","type":"hint","dependencies":["a137dddgre17a-h5"],"title":"Multiply Common Factors","text":"$$GCF=6m^2 n^3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a137dddgre18","title":"Greatest Common Factor","body":"Find the greatest common factor.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a137dddgre18a","stepAnswer":["$$14x^2 y^4$$"],"problemType":"TextBox","stepTitle":"$$28x^2 y^4$$, $$42x^4 y^4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$14x^2 y^4$$","hints":{"DefaultPathway":[{"id":"a137dddgre18a-h1","type":"hint","dependencies":[],"title":"Factor into Primes","text":"Factor each coefficient into primes and write the variables with exponents in expanded form. 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A prime number is a counting number greater than $$1$$, whose only factors are $$1$$ and itself. The first few prime numbers are: $$2$$, $$3$$, $$5$$, $$7$$, $$11$$, $$13$$, etc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre2a-h2","type":"hint","dependencies":["a137dddgre2a-h1"],"title":"Factor First Number","text":"$$8=4\\\\times2$$\\\\n$$8=2\\\\times2\\\\times2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre2a-h3","type":"hint","dependencies":["a137dddgre2a-h2"],"title":"Factor Second Number","text":"$$18=9\\\\times2$$\\\\n$$18=3\\\\times3\\\\times2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre2a-h4","type":"hint","dependencies":["a137dddgre2a-h2","a137dddgre2a-h3"],"title":"Identify Common Factors in each Column","text":"$$8=2\\\\times2\\\\times2$$\\\\n$$18=3\\\\times3\\\\times2$$\\\\n$$2$$ is shared by both numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre2a-h5","type":"hint","dependencies":["a137dddgre2a-h4"],"title":"Multiply Common Factors","text":"$$GCF=2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a137dddgre20","title":"Greatest Common Factor","body":"Find the greatest common factor.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a137dddgre20a","stepAnswer":["$$4y$$"],"problemType":"TextBox","stepTitle":"$$20y^3$$, $$28y^2$$, $$40y$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4y$$","hints":{"DefaultPathway":[{"id":"a137dddgre20a-h1","type":"hint","dependencies":[],"title":"Factor into Primes","text":"Factor each coefficient into primes and write the variables with exponents in expanded form. A prime number is a counting number greater than $$1$$, whose only factors are $$1$$ and itself. The first few prime numbers are: $$2$$, $$3$$, $$5$$, $$7$$, $$11$$, $$13$$, etc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre20a-h2","type":"hint","dependencies":["a137dddgre20a-h1"],"title":"Factor First Expression","text":"$$20y^3=5\\\\times2\\\\times2 y y y$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre20a-h3","type":"hint","dependencies":["a137dddgre20a-h2"],"title":"Factor Second Expression","text":"$$28y^2=7\\\\times2\\\\times2 y y$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre20a-h4","type":"hint","dependencies":["a137dddgre20a-h3"],"title":"Factor Third Expression","text":"$$40y=5\\\\times2\\\\times2\\\\times2 y$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre20a-h5","type":"hint","dependencies":["a137dddgre20a-h2","a137dddgre20a-h3","a137dddgre20a-h4"],"title":"Identify Common Factors in each Column","text":"$$20y^3=5\\\\times2\\\\times2 y y y$$\\\\n$$28y^2=7\\\\times2\\\\times2 y y$$\\\\n$$40y=5\\\\times2\\\\times2\\\\times2 y$$\\\\n$$2$$, $$2$$, and $$y$$ are shared by both expressions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre20a-h6","type":"hint","dependencies":["a137dddgre20a-h5"],"title":"Multiply Common Factors","text":"Bring down the $$2$$, $$2$$, and $$y$$, and then multiply.\\\\n$$GCF=2\\\\times2 y$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre20a-h7","type":"hint","dependencies":["a137dddgre20a-h6"],"title":"Multiply Common Factors","text":"$$GCF=4y$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a137dddgre21","title":"Greatest Common Factor","body":"Find the greatest common factor.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a137dddgre21a","stepAnswer":["$$5x^3$$"],"problemType":"TextBox","stepTitle":"$$35x^3$$, $$10x^4$$, $$5x^5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5x^3$$","hints":{"DefaultPathway":[{"id":"a137dddgre21a-h1","type":"hint","dependencies":[],"title":"Factor into Primes","text":"Factor each coefficient into primes and write the variables with exponents in expanded form. 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The first few prime numbers are: $$2$$, $$3$$, $$5$$, $$7$$, $$11$$, $$13$$, etc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre21a-h2","type":"hint","dependencies":["a137dddgre21a-h1"],"title":"Factor First Expression","text":"$$35x^3=5\\\\times7 x x x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre21a-h3","type":"hint","dependencies":["a137dddgre21a-h2"],"title":"Factor Second Expression","text":"$$10x^4=5\\\\times2 x x x x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre21a-h4","type":"hint","dependencies":["a137dddgre21a-h3"],"title":"Factor Third Expression","text":"$$5x^5=5x x x x x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre21a-h5","type":"hint","dependencies":["a137dddgre21a-h2","a137dddgre21a-h3","a137dddgre21a-h4"],"title":"Identify Common Factors in each Column","text":"$$35x^3=5\\\\times7 x x x$$\\\\n$$10x^4=5\\\\times2 x x x x$$\\\\n$$5x^5=5x x x x x$$\\\\n$$5$$, $$x$$, $$x$$, and $$x$$ are shared by both expressions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre21a-h6","type":"hint","dependencies":["a137dddgre21a-h5"],"title":"Multiply Common Factors","text":"Bring down the $$5$$, $$x$$, $$x$$, and $$x$$, and then multiply.\\\\n$$GCF=5x x x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre21a-h7","type":"hint","dependencies":["a137dddgre21a-h6"],"title":"Multiply Common Factors","text":"$$GCF=5x^3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a137dddgre22","title":"Greatest Common Factor","body":"Find the greatest common factor.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a137dddgre22a","stepAnswer":["$$p^2$$"],"problemType":"TextBox","stepTitle":"$$27p^2$$, $$45p^3$$, $$9p^4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$p^2$$","hints":{"DefaultPathway":[{"id":"a137dddgre22a-h1","type":"hint","dependencies":[],"title":"Factor into Primes","text":"Factor each coefficient into primes and write the variables with exponents in expanded form. A prime number is a counting number greater than $$1$$, whose only factors are $$1$$ and itself. The first few prime numbers are: $$2$$, $$3$$, $$5$$, $$7$$, $$11$$, $$13$$, etc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre22a-h2","type":"hint","dependencies":["a137dddgre22a-h1"],"title":"Factor First Expression","text":"$$27p^2=3\\\\times3\\\\times3 p p$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre22a-h3","type":"hint","dependencies":["a137dddgre22a-h2"],"title":"Factor Second Expression","text":"$$45p^3=5\\\\times7 p p p$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre22a-h4","type":"hint","dependencies":["a137dddgre22a-h3"],"title":"Factor Third Expression","text":"$$9p^4=3\\\\times3 p p p p$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre22a-h5","type":"hint","dependencies":["a137dddgre22a-h2","a137dddgre22a-h3","a137dddgre22a-h4"],"title":"Identify Common Factors in each Column","text":"$$27p^2=3\\\\times3\\\\times3 p p$$\\\\n$$45p^3=5\\\\times7 p p p$$\\\\n$$9p^4=3\\\\times3 p p p p$$\\\\n$$p$$ and $$p$$ are shared by both expressions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre22a-h6","type":"hint","dependencies":["a137dddgre22a-h5"],"title":"Multiply Common Factors","text":"Bring down the $$p$$ and $$p$$, and then multiply.\\\\n$$GCF=p p$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre22a-h7","type":"hint","dependencies":["a137dddgre22a-h6"],"title":"Multiply Common Factors","text":"$$GCF=p^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a137dddgre23","title":"Greatest Common Factor","body":"Factor the greatest common factor from each polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a137dddgre23a","stepAnswer":["$$4\\\\left(x+3\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$4x+12$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4\\\\left(x+3\\\\right)$$","hints":{"DefaultPathway":[{"id":"a137dddgre23a-h1","type":"hint","dependencies":[],"title":"GCF of Terms","text":"Find the greatest common factor of all the terms of the polynomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre23a-h2","type":"hint","dependencies":["a137dddgre23a-h1"],"title":"Factor into Primes","text":"Factor $$4x$$ and $$12$$ into primes.\\\\n$$4x=2\\\\times2 x$$\\\\n$$12=2\\\\times2\\\\times3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre23a-h3","type":"hint","dependencies":["a137dddgre23a-h2"],"title":"Multiply Common Factors","text":"Multiply the terms shared by both expressions.\\\\n$$GCF=2\\\\times2$$\\\\n$$GCF=4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre23a-h4","type":"hint","dependencies":["a137dddgre23a-h3"],"title":"Write each Term as a Product using GCF","text":"Rewrite $$4x$$ and $$12$$ as products of their GCF, $$4$$.\\\\n$$4x+12$$\\\\n$$4x+4\\\\times3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre23a-h5","type":"hint","dependencies":["a137dddgre23a-h4"],"title":"Reverse Distributive Property","text":"Use the reverse Distributive Property to factor the expression.\\\\n$$4\\\\left(x+3\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a137dddgre24","title":"Greatest Common Factor","body":"Factor the greatest common factor from each polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Elementary 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greatest common factor from each polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a137dddgre29a","stepAnswer":["$$7\\\\left(2p+5\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$14p+35$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7\\\\left(2p+5\\\\right)$$","hints":{"DefaultPathway":[{"id":"a137dddgre29a-h1","type":"hint","dependencies":[],"title":"GCF of Terms","text":"Find the greatest common factor of all the terms of the polynomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre29a-h2","type":"hint","dependencies":["a137dddgre29a-h1"],"title":"Factor into Primes","text":"Factor $$14p$$ and $$35$$ into primes.\\\\n$$14p=2\\\\times7 p$$\\\\n$$35=5\\\\times7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre29a-h3","type":"hint","dependencies":["a137dddgre29a-h2"],"title":"Multiply Common Factors","text":"Multiply the terms shared by both expressions.\\\\n$$GCF=7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre29a-h4","type":"hint","dependencies":["a137dddgre29a-h3"],"title":"Write each Term as a Product using GCF","text":"Rewrite $$14p$$ and $$35$$ as products of their GCF, $$7$$.\\\\n$$14p+35$$\\\\n$$7\\\\times2 p+7\\\\times5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre29a-h5","type":"hint","dependencies":["a137dddgre29a-h4"],"title":"Reverse Distributive 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The first few prime numbers are: $$2$$, $$3$$, $$5$$, $$7$$, $$11$$, $$13$$, etc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre4a-h2","type":"hint","dependencies":["a137dddgre4a-h1"],"title":"Factor First Number","text":"$$72=8\\\\times9$$\\\\n$$72=4\\\\times2 3\\\\times3$$\\\\n$$72=2\\\\times2\\\\times2 3\\\\times3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre4a-h3","type":"hint","dependencies":["a137dddgre4a-h2"],"title":"Factor Second Number","text":"$$162=2\\\\times81$$\\\\n$$40=2\\\\times9\\\\times9$$\\\\n$$40=2\\\\times3\\\\times3\\\\times3\\\\times3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre4a-h4","type":"hint","dependencies":["a137dddgre4a-h2","a137dddgre4a-h3"],"title":"Identify Common Factors in each Column","text":"$$72=2\\\\times2\\\\times2 3\\\\times3$$\\\\n$$40=2\\\\times3\\\\times3\\\\times3\\\\times3$$\\\\n$$2$$, $$3$$, and $$3$$ are shared by both numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre4a-h5","type":"hint","dependencies":["a137dddgre4a-h4"],"title":"Multiply Common Factors","text":"Bring down the $$2$$, $$3$$, and $$3$$, and then multiply.\\\\n$$GCF=2\\\\times3\\\\times3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre4a-h6","type":"hint","dependencies":["a137dddgre4a-h5"],"title":"Multiply Common Factors","text":"$$GCF=18$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a137dddgre5","title":"Greatest Common Factor","body":"Find the greatest common factor.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a137dddgre5a","stepAnswer":["$$25$$"],"problemType":"TextBox","stepTitle":"$$150$$, $$275$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$25$$","hints":{"DefaultPathway":[{"id":"a137dddgre5a-h1","type":"hint","dependencies":[],"title":"Factor into Primes","text":"Factor each coefficient into primes. A prime number is a counting number greater than $$1$$, whose only factors are $$1$$ and itself. The first few prime numbers are: $$2$$, $$3$$, $$5$$, $$7$$, $$11$$, $$13$$, etc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre5a-h2","type":"hint","dependencies":["a137dddgre5a-h1"],"title":"Factor First Number","text":"$$150=15\\\\times10$$\\\\n$$150=3\\\\times5 5\\\\times2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre5a-h3","type":"hint","dependencies":["a137dddgre5a-h2"],"title":"Factor Second Number","text":"$$275=25\\\\times11$$\\\\n$$275=5\\\\times5\\\\times11$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre5a-h4","type":"hint","dependencies":["a137dddgre5a-h2","a137dddgre5a-h3"],"title":"Identify Common Factors in each Column","text":"$$150=3\\\\times5 5\\\\times2$$\\\\n$$275=5\\\\times5\\\\times11$$\\\\n$$5$$ and $$5$$ are shared by both numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre5a-h5","type":"hint","dependencies":["a137dddgre5a-h4"],"title":"Multiply Common Factors","text":"Bring down the $$5$$ and $$5$$, and then multiply.\\\\n$$GCF=5\\\\times5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre5a-h6","type":"hint","dependencies":["a137dddgre5a-h5"],"title":"Multiply Common Factors","text":"$$GCF=25$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a137dddgre6","title":"Greatest Common Factor","body":"Find the greatest common factor.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a137dddgre6a","stepAnswer":["$$16$$"],"problemType":"TextBox","stepTitle":"$$48$$, $$80$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16$$","hints":{"DefaultPathway":[{"id":"a137dddgre6a-h1","type":"hint","dependencies":[],"title":"Factor into Primes","text":"Factor each coefficient into primes. A prime number is a counting number greater than $$1$$, whose only factors are $$1$$ and itself. The first few prime numbers are: $$2$$, $$3$$, $$5$$, $$7$$, $$11$$, $$13$$, etc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre6a-h2","type":"hint","dependencies":["a137dddgre6a-h1"],"title":"Factor First Number","text":"$$48=8\\\\times6$$\\\\n$$48=4\\\\times2 3\\\\times2$$\\\\n$$48=2\\\\times2\\\\times2 3\\\\times2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre6a-h3","type":"hint","dependencies":["a137dddgre6a-h2"],"title":"Factor Second Number","text":"$$80=8\\\\times10$$\\\\n$$80=4\\\\times2 5\\\\times2$$\\\\n$$80=2\\\\times2\\\\times2 5\\\\times2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre6a-h4","type":"hint","dependencies":["a137dddgre6a-h2","a137dddgre6a-h3"],"title":"Identify Common Factors in each Column","text":"$$48=2\\\\times2\\\\times2 3\\\\times2$$\\\\n$$80=2\\\\times2\\\\times2 5\\\\times2$$\\\\n$$2$$, $$2$$, $$2$$, and $$2$$ are shared by both numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre6a-h5","type":"hint","dependencies":["a137dddgre6a-h4"],"title":"Multiply Common Factors","text":"Bring down the $$2$$, $$2$$, $$2$$, and $$2$$, and then multiply.\\\\n$$GCF=2\\\\times2\\\\times2\\\\times2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre6a-h6","type":"hint","dependencies":["a137dddgre6a-h5"],"title":"Multiply Common Factors","text":"$$GCF=16$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a137dddgre7","title":"Greatest Common Factor","body":"Find the greatest common factor.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a137dddgre7a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"$$18$$, $$40$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a137dddgre7a-h1","type":"hint","dependencies":[],"title":"Factor into Primes","text":"Factor each coefficient into primes. A prime number is a counting number greater than $$1$$, whose only factors are $$1$$ and itself. The first few prime numbers are: $$2$$, $$3$$, $$5$$, $$7$$, $$11$$, $$13$$, etc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre7a-h2","type":"hint","dependencies":["a137dddgre7a-h1"],"title":"Factor First Number","text":"$$18=9\\\\times2$$\\\\n$$18=3\\\\times3\\\\times2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre7a-h3","type":"hint","dependencies":["a137dddgre7a-h2"],"title":"Factor Second Number","text":"$$40=4\\\\times10$$\\\\n$$40=2\\\\times2 5\\\\times2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre7a-h4","type":"hint","dependencies":["a137dddgre7a-h2","a137dddgre7a-h3"],"title":"Identify Common Factors in each Column","text":"$$18=3\\\\times3\\\\times2$$\\\\n$$40=2\\\\times2 5\\\\times2$$\\\\n$$2$$ is shared by both numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre7a-h5","type":"hint","dependencies":["a137dddgre7a-h4"],"title":"Multiply Common Factors","text":"$$GCF=2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a137dddgre8","title":"Greatest Common Factor","body":"Find the greatest common factor.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a137dddgre8a","stepAnswer":["$$9x^3$$"],"problemType":"TextBox","stepTitle":"$$27x^3$$, $$18x^4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9x^3$$","hints":{"DefaultPathway":[{"id":"a137dddgre8a-h1","type":"hint","dependencies":[],"title":"Factor into Primes","text":"Factor each coefficient into primes and write the variables with exponents in expanded form. A prime number is a counting number greater than $$1$$, whose only factors are $$1$$ and itself. The first few prime numbers are: $$2$$, $$3$$, $$5$$, $$7$$, $$11$$, $$13$$, etc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre8a-h2","type":"hint","dependencies":["a137dddgre8a-h1"],"title":"Factor First Expression","text":"$$27x^3=3\\\\times3\\\\times3 x x x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre8a-h3","type":"hint","dependencies":["a137dddgre8a-h2"],"title":"Factor Second Expression","text":"$$18x^4=2\\\\times3\\\\times3 x x x x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre8a-h4","type":"hint","dependencies":["a137dddgre8a-h2","a137dddgre8a-h3"],"title":"Identify Common Factors in each Column","text":"$$27x^3=3\\\\times3\\\\times3 x x x$$\\\\n$$18x^4=2\\\\times3\\\\times3 x x x x$$\\\\n$$3$$, $$3$$, $$x$$, $$x$$ and $$x$$ are shared by both expressions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre8a-h5","type":"hint","dependencies":["a137dddgre8a-h4"],"title":"Multiply Common Factors","text":"Bring down the $$3$$, $$3$$, $$x$$, $$x$$ and $$x$$, and then multiply.\\\\n$$GCF=3\\\\times3 x x x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre8a-h6","type":"hint","dependencies":["a137dddgre8a-h5"],"title":"Multiply Common Factors","text":"$$GCF=9x^3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a137dddgre9","title":"Greatest Common Factor","body":"Find the greatest common factor.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a137dddgre9a","stepAnswer":["$$10$$"],"problemType":"TextBox","stepTitle":"$$10a$$, $$50$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10$$","hints":{"DefaultPathway":[{"id":"a137dddgre9a-h1","type":"hint","dependencies":[],"title":"Factor into Primes","text":"Factor each coefficient into primes and write the variables with exponents in expanded form. A prime number is a counting number greater than $$1$$, whose only factors are $$1$$ and itself. The first few prime numbers are: $$2$$, $$3$$, $$5$$, $$7$$, $$11$$, $$13$$, etc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre9a-h2","type":"hint","dependencies":["a137dddgre9a-h1"],"title":"Factor First Expression","text":"$$10a=5\\\\times2 a$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre9a-h3","type":"hint","dependencies":["a137dddgre9a-h2"],"title":"Factor Second Expression","text":"$$50=5\\\\times5\\\\times2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre9a-h4","type":"hint","dependencies":["a137dddgre9a-h2","a137dddgre9a-h3"],"title":"Identify Common Factors in each Column","text":"$$10a=5\\\\times2 a$$\\\\n$$50=5\\\\times5\\\\times2$$\\\\n$$5$$ and $$2$$ are shared by both expressions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre9a-h5","type":"hint","dependencies":["a137dddgre9a-h4"],"title":"Multiply Common Factors","text":"Bring down the $$5$$ and $$2$$, and then multiply.\\\\n$$GCF=5\\\\times2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a137dddgre9a-h6","type":"hint","dependencies":["a137dddgre9a-h5"],"title":"Multiply Common Factors","text":"$$GCF=10$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a14ffbcmodeling1","title":"Solving a Direct Variation Problem","body":"The quantity $$y$$ varies directly with the cube of $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.8 Modeling Using Variation","courseName":"OpenStax: College Algebra","steps":[{"id":"a14ffbcmodeling1a","stepAnswer":["$$675$$"],"problemType":"TextBox","stepTitle":"If $$y=25$$ when $$x=2$$, find $$y$$ when $$x$$ is $$6$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$675$$","hints":{"DefaultPathway":[{"id":"a14ffbcmodeling1a-h1","type":"hint","dependencies":[],"title":"Find General Formula","text":"The first step is to identify what general formula can be used for direction variation with a cube.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{25}{8}$$"],"dependencies":["a14ffbcmodeling1a-h1"],"title":"Solve for Constant","text":"Given $$y=25$$ and $$x=2$$, what is the value of constant k from the general formula?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling1a-h3","type":"hint","dependencies":["a14ffbcmodeling1a-h2"],"title":"Find Specific Formula","text":"Use the constant to write an equation that represents the relationship.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$675$$"],"dependencies":["a14ffbcmodeling1a-h3"],"title":"Substitution","text":"What is $$y$$ when $$x=6$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a14ffbcmodeling11","title":"Inverse Variation","body":"Write an expression for $$y$$ that describes the relationship of the given variables.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.8 Modeling Using Variation","courseName":"OpenStax: College Algebra","steps":[{"id":"a14ffbcmodeling11a","stepAnswer":["$$\\\\frac{40}{x^3}$$"],"problemType":"TextBox","stepTitle":"$$y$$ varies inversely as the cube of $$x$$ and when $$x=2$$, $$y=5$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{40}{x^3}$$","hints":{"DefaultPathway":[{"id":"a14ffbcmodeling11a-h1","type":"hint","dependencies":[],"title":"Inverse Variation","text":"If $$x$$ and $$y$$ are related by an equation of the form $$y=\\\\frac{k}{x^n}$$ where k is a nonzero constant, then we say that $$y$$ varies inversely with the nth power of $$x$$. In inversely proportional relationships, or inverse variations, there is a constant multiple $$k=x^n y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling11a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y=\\\\frac{k}{x^3}$$"],"dependencies":["a14ffbcmodeling11a-h1"],"title":"General Formula","text":"What is the general formula for inverse variation of $$y$$ with a cube of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y=k x^3$$","$$y=k \\\\sqrt[3]{x}$$","$$y=\\\\frac{k}{x^3}$$"]},{"id":"a14ffbcmodeling11a-h3","type":"hint","dependencies":["a14ffbcmodeling11a-h2"],"title":"Determining the Constant of Variation, k.","text":"Make k the subject and substitute values to solve for k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^3$$"],"dependencies":["a14ffbcmodeling11a-h3"],"title":"Making k the Subject","text":"What variable do you multiply on both sides to isolate k?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$40$$"],"dependencies":["a14ffbcmodeling11a-h4"],"title":"Substitution","text":"What is k equals to after substituting $$x=2$$ and $$y=5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling11a-h6","type":"hint","dependencies":["a14ffbcmodeling11a-h5"],"title":"Equation","text":"Now that you\'ve found k, substitute k back into the equation that describes the relationship between the variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a14ffbcmodeling12","title":"Inverse Variation","body":"Write an expression for $$y$$ that describes the relationship of the given variables.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.8 Modeling Using Variation","courseName":"OpenStax: College Algebra","steps":[{"id":"a14ffbcmodeling12a","stepAnswer":["$$\\\\frac{81}{x^4}$$"],"problemType":"TextBox","stepTitle":"$$y$$ varies inversely as the fourth power of $$x$$ and when $$x=3$$, $$y=1$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{81}{x^4}$$","hints":{"DefaultPathway":[{"id":"a14ffbcmodeling12a-h1","type":"hint","dependencies":[],"title":"Inverse Variation","text":"If $$x$$ and $$y$$ are related by an equation of the form $$y=\\\\frac{k}{x^n}$$ where k is a nonzero constant, then we say that $$y$$ varies inversely with the nth power of $$x$$. In inversely proportional relationships, or inverse variations, there is a constant multiple $$k=x^n y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling12a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y=\\\\frac{k}{x^4}$$"],"dependencies":["a14ffbcmodeling12a-h1"],"title":"General Formula","text":"What is the general formula for inverse variation of $$y$$ with the fourth power of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y=k x^4$$","$$y=\\\\frac{k}{\\\\sqrt[4]{x}}$$","$$y=\\\\frac{k}{x^4}$$"]},{"id":"a14ffbcmodeling12a-h3","type":"hint","dependencies":["a14ffbcmodeling12a-h2"],"title":"Determining the Constant of Variation, k.","text":"Make k the subject and substitute values to solve for k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^4$$"],"dependencies":["a14ffbcmodeling12a-h3"],"title":"Making k the Subject","text":"What variable do you multiply on both sides to isolate k?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling12a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$81$$"],"dependencies":["a14ffbcmodeling12a-h4"],"title":"Substitution","text":"What is k equals to after substituting $$x=3$$ and $$y=1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling12a-h6","type":"hint","dependencies":["a14ffbcmodeling12a-h5"],"title":"Equation","text":"Now that you\'ve found k, substitute k back into the equation that describes the relationship between the variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a14ffbcmodeling14","title":"Joint Variation","body":"Write an expression for $$y$$ that describes the relationship of the given variables.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.8 Modeling Using Variation","courseName":"OpenStax: College Algebra","steps":[{"id":"a14ffbcmodeling14a","stepAnswer":["$$10x \\\\sqrt{z}$$"],"problemType":"TextBox","stepTitle":"$$y$$ varies jointly as $$x$$ and the square root of $$z$$ and when $$x=2$$ and $$z=25$$, then $$y=100$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10x \\\\sqrt{z}$$","hints":{"DefaultPathway":[{"id":"a14ffbcmodeling14a-h1","type":"hint","dependencies":[],"title":"Joint Variation","text":"Joint variation occurs when a variable varies directly or inversely with multiple variables. For instance, if $$x$$ varies directly with both $$y$$ and $$z$$, we have $$x=k y z$$. If $$x$$ varies directly with $$y$$ and inversely with $$z$$, we have $$x=\\\\frac{k y}{z}$$. Notice that we only use one constant in a joint variation equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling14a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y=k x \\\\sqrt{z}$$"],"dependencies":["a14ffbcmodeling14a-h1"],"title":"General Formula","text":"What is the general formula for $$y$$ that varies directly with $$x$$ and the square root of $$z$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y=k x \\\\sqrt{z}$$","$$y=k x z^2$$","$$y=\\\\frac{k}{x \\\\sqrt{z}}$$","$$y=\\\\frac{k z}{x}$$"]},{"id":"a14ffbcmodeling14a-h3","type":"hint","dependencies":["a14ffbcmodeling14a-h2"],"title":"Determining the Constant of Variation, k.","text":"Make k the subject and substitute values to solve for k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling14a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x \\\\sqrt{z}$$"],"dependencies":["a14ffbcmodeling14a-h3"],"title":"Making k the Subject","text":"What variables do you divide on both sides to isolate k?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling14a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a14ffbcmodeling14a-h4"],"title":"Substitution","text":"What is k equals to after substituting $$x=2$$, $$y=100$$ and $$z=25$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling14a-h6","type":"hint","dependencies":["a14ffbcmodeling14a-h5"],"title":"Equation","text":"Now that you\'ve found k, substitute k back into the equation that describes the relationship between the variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a14ffbcmodeling15","title":"Joint Variation","body":"Write an expression for $$y$$ that describes the relationship of the given variables.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.8 Modeling Using Variation","courseName":"OpenStax: College Algebra","steps":[{"id":"a14ffbcmodeling15a","stepAnswer":["$$x^2 z^3 \\\\sqrt{w}$$"],"problemType":"TextBox","stepTitle":"$$y$$ varies jointly as the square of $$x$$, the cube of $$z$$ and the square root of w. When $$x=1$$, $$z=2$$, and $$w=36$$, then $$y=48$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^2 z^3 \\\\sqrt{w}$$","hints":{"DefaultPathway":[{"id":"a14ffbcmodeling15a-h1","type":"hint","dependencies":[],"title":"Joint Variation","text":"Joint variation occurs when a variable varies directly or inversely with multiple variables. For instance, if $$x$$ varies directly with both $$y$$ and $$z$$, we have $$x=k y z$$. If $$x$$ varies directly with $$y$$ and inversely with $$z$$, we have $$x=\\\\frac{k y}{z}$$. Notice that we only use one constant in a joint variation equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling15a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y=k x^2 z^3 \\\\sqrt{w}$$"],"dependencies":["a14ffbcmodeling15a-h1"],"title":"General Formula","text":"What is the general formula for $$y$$ that varies directly with the square of $$x$$, the cube of $$z$$ and the square root of w?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y=k x^2 z^3 \\\\sqrt{w}$$","$$y=\\\\frac{k x^3 z^2}{w^2}$$","$$y=\\\\frac{k x^3}{z^2 w^2}$$","$$y=k x^3 z^2 \\\\sqrt{w}$$"]},{"id":"a14ffbcmodeling15a-h3","type":"hint","dependencies":["a14ffbcmodeling15a-h2"],"title":"Determining the Constant of Variation, k.","text":"Make k the subject and substitute values to solve for k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2 z^3 \\\\sqrt{w}$$"],"dependencies":["a14ffbcmodeling15a-h3"],"title":"Making k the Subject","text":"What variables do you divide on both sides to isolate k?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling15a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a14ffbcmodeling15a-h4"],"title":"Substitution","text":"What is k equals to after substituting $$w=36$$, $$x=1$$, $$y=48$$ and $$z=2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling15a-h6","type":"hint","dependencies":["a14ffbcmodeling15a-h5"],"title":"Equation","text":"Now that you\'ve found k, substitute k back into the equation that describes the relationship between the variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a14ffbcmodeling16","title":"Joint Variation","body":"Write an expression for $$y$$ that describes the relationship of the given variables.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.8 Modeling Using Variation","courseName":"OpenStax: College Algebra","steps":[{"id":"a14ffbcmodeling16a","stepAnswer":["$$\\\\frac{9x^2 \\\\sqrt{z}}{w^3}$$"],"problemType":"TextBox","stepTitle":"$$y$$ varies jointly as the square of $$x$$ and the square root of $$z$$, and inversely as the cube of w. When $$x=3$$, $$z=4$$, and $$w=3$$, then $$y=6$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{9x^2 \\\\sqrt{z}}{w^3}$$","hints":{"DefaultPathway":[{"id":"a14ffbcmodeling16a-h1","type":"hint","dependencies":[],"title":"Joint Variation","text":"Joint variation occurs when a variable varies directly or inversely with multiple variables. For instance, if $$x$$ varies directly with both $$y$$ and $$z$$, we have $$x=k y z$$. If $$x$$ varies directly with $$y$$ and inversely with $$z$$, we have $$x=\\\\frac{k y}{z}$$. Notice that we only use one constant in a joint variation equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling16a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y=\\\\frac{k x^2 \\\\sqrt{z}}{w^3}$$"],"dependencies":["a14ffbcmodeling16a-h1"],"title":"General Formula","text":"What is the general formula for $$y$$ that varies directly with the square of $$x$$, the square root of $$z$$, and inversely with the cube of w?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y=\\\\frac{k x^2 \\\\sqrt{z}}{w^3}$$","$$y=\\\\frac{k x^3 z^2}{w^2}$$","$$y=\\\\frac{k x^2}{\\\\sqrt{z} w^3}$$","$$y=k x^3 z^2 \\\\sqrt{w}$$"]},{"id":"a14ffbcmodeling16a-h3","type":"hint","dependencies":["a14ffbcmodeling16a-h2"],"title":"Determining the Constant of Variation, k.","text":"Make k the subject and substitute values to solve for k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling16a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{x^2 \\\\sqrt{z}}{w^3}$$"],"dependencies":["a14ffbcmodeling16a-h3"],"title":"Making k the Subject","text":"What variables do you divide on both sides to isolate k?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling16a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a14ffbcmodeling16a-h4"],"title":"Substitution","text":"What is k equals to after substituting $$w=3$$, $$x=3$$, $$y=6$$ and $$z=4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling16a-h6","type":"hint","dependencies":["a14ffbcmodeling16a-h5"],"title":"Equation","text":"Now that you\'ve found k, substitute k back into the equation that describes the relationship between the variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a14ffbcmodeling17","title":"Joint Variation","body":"Write an expression for $$y$$ that describes the relationship of the given variables.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.8 Modeling Using Variation","courseName":"OpenStax: College Algebra","steps":[{"id":"a14ffbcmodeling17a","stepAnswer":["$$\\\\frac{40x z}{\\\\sqrt{w} t^2}$$"],"problemType":"TextBox","stepTitle":"$$y$$ varies jointly as $$x$$ and $$z$$ and inversely as the square root of w and the square of $$t$$ . When $$x=3$$, $$z=1$$, $$w=25$$, and $$t=2$$, then $$y=6$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{40x z}{\\\\sqrt{w} t^2}$$","hints":{"DefaultPathway":[{"id":"a14ffbcmodeling17a-h1","type":"hint","dependencies":[],"title":"Joint Variation","text":"Joint variation occurs when a variable varies directly or inversely with multiple variables. For instance, if $$x$$ varies directly with both $$y$$ and $$z$$, we have $$x=k y z$$. If $$x$$ varies directly with $$y$$ and inversely with $$z$$, we have $$x=\\\\frac{k y}{z}$$. Notice that we only use one constant in a joint variation equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling17a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y=\\\\frac{k x z}{\\\\sqrt{w} t^2}$$"],"dependencies":["a14ffbcmodeling17a-h1"],"title":"General Formula","text":"What is the general formula for $$y$$ that varies directly with $$x$$ and $$z$$, and inversely with the square root of w and the square of $$t$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y=\\\\frac{k x z}{\\\\sqrt{w} t^2}$$","$$y=\\\\frac{k \\\\sqrt{w} t^2}{x z}$$","$$y=\\\\frac{k x t^2}{\\\\sqrt{z} w}$$","$$y=k x z \\\\sqrt{w} t^2$$"]},{"id":"a14ffbcmodeling17a-h3","type":"hint","dependencies":["a14ffbcmodeling17a-h2"],"title":"Determining the Constant of 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$$y=8$$ and $$x=3$$, what is the value of constant k from the general formula?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling6a-h3","type":"hint","dependencies":["a14ffbcmodeling6a-h2"],"title":"Find Specific Formula","text":"Use the constant to write an equation that represents the relationship.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{9}{2}$$"],"dependencies":["a14ffbcmodeling6a-h3"],"title":"Using Subsitution","text":"What is $$y$$ when $$x=4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a14ffbcmodeling7","title":"Solving Problems Involving Joint Variation","body":"A quantity $$x$$ varies directly with the square of $$y$$ and inversely with $$z$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.8 Modeling Using Variation","courseName":"OpenStax: College Algebra","steps":[{"id":"a14ffbcmodeling7a","stepAnswer":["$$20$$"],"problemType":"TextBox","stepTitle":"If $$x=40$$ when $$y=4$$ and $$z=2$$, find $$x$$ when $$y=10$$ and $$z=25$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$20$$","hints":{"DefaultPathway":[{"id":"a14ffbcmodeling7a-h1","type":"hint","dependencies":[],"title":"Find General Formula","text":"The first step is to identify what general formula can be used for direction variation with a square of $$y$$ and inverse variation with $$z$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a14ffbcmodeling7a-h1"],"title":"Solve for Constant","text":"Given $$y=4$$, $$x=40$$, and $$z=2$$, solve for the constant k by plugging those values in.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling7a-h3","type":"hint","dependencies":["a14ffbcmodeling7a-h2"],"title":"Find Specific Formula","text":"Use the constant to write an equation that represents the relationship.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a14ffbcmodeling7a-h3"],"title":"Using Subsitution","text":"Plug in $$y=10$$ and $$z=25$$ to solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a14ffbcmodeling8","title":"Direct Variation","body":"Write an expression for $$y$$ that describes the relationship of the given variables.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.8 Modeling Using Variation","courseName":"OpenStax: College Algebra","steps":[{"id":"a14ffbcmodeling8a","stepAnswer":["$$5x^2$$"],"problemType":"TextBox","stepTitle":"$$y$$ varies directly as the square of $$x$$ and when $$x=4$$, $$y=80$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5x^2$$","hints":{"DefaultPathway":[{"id":"a14ffbcmodeling8a-h1","type":"hint","dependencies":[],"title":"Direct Variation","text":"If $$x$$ and $$y$$ are related by an equation of the form $$y=k x^n$$, then we say that the relationship is direct variation and $$y$$ varies directly with, or is proportional to, the nth power of $$x$$. In direct variation relationships, there is a nonzero constant ratio $$k=\\\\frac{y}{x^n}$$, where k is called the constant of variation, which help defines the relationship between the variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling8a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y=k x^2$$"],"dependencies":["a14ffbcmodeling8a-h1"],"title":"General Formula","text":"What is the general formula for direct variation of $$y$$ with a square of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y=k x^2$$","$$y=\\\\frac{k}{x^2}$$","$$y=k x$$"]},{"id":"a14ffbcmodeling8a-h3","type":"hint","dependencies":["a14ffbcmodeling8a-h2"],"title":"Determining the Constant of Variation, k.","text":"Make k the subject and substitute values to solve for k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2$$"],"dependencies":["a14ffbcmodeling8a-h3"],"title":"Making k the Subject","text":"What variable do you divide on both sides to isolate k?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling8a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a14ffbcmodeling8a-h4"],"title":"Substitution","text":"What is k equals to after substituting $$x=4$$ and $$y=80$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling8a-h6","type":"hint","dependencies":["a14ffbcmodeling8a-h5"],"title":"Equation","text":"Now that you\'ve found k, substitute k back into the equation that describes the relationship between the variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a14ffbcmodeling9","title":"Direct Variation","body":"Write an expression for $$y$$ that describes the relationship of the given variables.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.8 Modeling Using Variation","courseName":"OpenStax: College Algebra","steps":[{"id":"a14ffbcmodeling9a","stepAnswer":["$$4\\\\sqrt{x}$$"],"problemType":"TextBox","stepTitle":"$$y$$ varies directly as the square root of $$x$$ and when $$x=36$$, $$y=24$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4\\\\sqrt{x}$$","hints":{"DefaultPathway":[{"id":"a14ffbcmodeling9a-h1","type":"hint","dependencies":[],"title":"Direct Variation","text":"If $$x$$ and $$y$$ are related by an equation of the form $$y=k x^n$$, then we say that the relationship is direct variation and $$y$$ varies directly with, or is proportional to, the nth power of $$x$$. In direct variation relationships, there is a nonzero constant ratio $$k=\\\\frac{y}{x^n}$$, where k is called the constant of variation, which help defines the relationship between the variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling9a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y=k \\\\sqrt{x}$$"],"dependencies":["a14ffbcmodeling9a-h1"],"title":"General Formula","text":"What is the general formula for direct variation of $$y$$ with a square root of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y=k x^2$$","$$y=k \\\\sqrt{x}$$","$$y=\\\\frac{k}{\\\\sqrt{x}}$$"]},{"id":"a14ffbcmodeling9a-h3","type":"hint","dependencies":["a14ffbcmodeling9a-h2"],"title":"Determining the Constant of Variation, k.","text":"Make k the subject and substitute values to solve for k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{x}$$"],"dependencies":["a14ffbcmodeling9a-h3"],"title":"Making k the Subject","text":"What variable do you divide on both sides to isolate k?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling9a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a14ffbcmodeling9a-h4"],"title":"Substitution","text":"What is k equals to after substituting $$x=36$$ and $$y=24$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a14ffbcmodeling9a-h6","type":"hint","dependencies":["a14ffbcmodeling9a-h5"],"title":"Equation","text":"Now that you\'ve found k, substitute k back into the equation that describes the relationship between the variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a150a865.3anniversary1","title":"Exponential distributions","body":"The amount of time spouses shop for anniversary cards can be modeled by an exponential distribution with the average amount of time equal to eight minutes.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 The Exponential Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a150a865.3anniversary1a","stepAnswer":["$$f(x)=0.125e^{-0.125 x}$$"],"problemType":"MultipleChoice","stepTitle":"Which of the following distribution statements matches the model of the exponential distribution with the average amount of eight minutes?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$f(x)=0.125e^{-0.125 x}$$","choices":["$$f(x)=0.125e^{-0.125 x}$$","$$f(x)=0.15e^{0.8x}$$","$$f(x)=8e^{\\\\left(-8x\\\\right)}$$","$$f(x)=8e^{-0.15 x}$$"],"hints":{"DefaultPathway":[{"id":"a150a865.3anniversary1a-h1","type":"hint","dependencies":[],"title":"Exponential distributions","text":"Use the equation $$f(x)=m e^{\\\\left(-m x\\\\right)}$$, where $$m$$ is the decay parameter, to get the representative distribution of the amount of time spouses shop for anniversary cards.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3anniversary1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{8}$$"],"dependencies":["a150a865.3anniversary1a-h1"],"title":"Exponential distributions","text":"What is the decay parameter $$m$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3anniversary1a-h3","type":"hint","dependencies":["a150a865.3anniversary1a-h2"],"title":"Exponential distributions","text":"The decay parameter is computed by 1/(average random variable value). In our case, the average is $$8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3anniversary1a-h4","type":"hint","dependencies":["a150a865.3anniversary1a-h3"],"title":"Exponential distributions","text":"Plug in the obtained decay parameter into the distiribution formula given earlier into $$m$$ to get the representative exponential distribution.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a150a865.3commuters1","title":"Suppose that the distance, in miles, that people are willing to commute to work is an exponential random variable with a decay parameter $$\\\\frac{1}{20}$$. Let $$X=the$$ distance people are willing to commute in miles.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 The Exponential Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a150a865.3commuters1a","stepAnswer":["$$\\\\frac{1}{20}$$"],"problemType":"TextBox","stepTitle":"What is $$m$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{20}$$","hints":{"DefaultPathway":[{"id":"a150a865.3commuters1a-h1","type":"hint","dependencies":[],"title":"Exponential Distribution","text":"$$m$$ is denoted as the decay parameter where $$1$$ is divided by the random variable average.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a150a865.3commuters1b","stepAnswer":["$$20$$"],"problemType":"TextBox","stepTitle":"What is \u03bc?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$20$$","hints":{"DefaultPathway":[{"id":"a150a865.3commuters1b-h1","type":"hint","dependencies":[],"title":"The mean","text":"\u03bc is the population average.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a150a865.3commuters1c","stepAnswer":["$$0.2865$$"],"problemType":"TextBox","stepTitle":"What is the probability that a person is willing to commute more than $$25$$ miles? Round to $$4$$ decimal points.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.2865$$","hints":{"DefaultPathway":[{"id":"a150a865.3commuters1c-h1","type":"hint","dependencies":[],"title":"Exponential Distribution","text":"We need to find $$P\\\\left(X>25\\\\right)$$ with given information $$m=\\\\frac{1}{20}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3commuters1c-h2","type":"hint","dependencies":["a150a865.3commuters1c-h1"],"title":"Exponential Distribution","text":"Since $$P\\\\left(X<x\\\\right)$$ $$=$$ $$1-e^{\\\\left(-m x\\\\right)}$$, then we can use $$P\\\\left(X>x\\\\right)=1-1-e^{\\\\left(-m x\\\\right)}$$ $$=$$ $$e^{\\\\left(-m x\\\\right)}$$. Remember that $$m$$ is our decay parameter respective to the problem that we are facing.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3commuters1c-h3","type":"hint","dependencies":["a150a865.3commuters1c-h2"],"title":"Exponential Distribution","text":"$$P\\\\left(X>25\\\\right)=e^{25\\\\left(-\\\\frac{1}{20}\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a150a865.3computer1","title":"Exponential Distribution","body":"On the average, a certain computer part lasts ten years. The length of time the computer part lasts is exponentially distributed.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 The Exponential Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a150a865.3computer1a","stepAnswer":["$$0.4966$$"],"problemType":"TextBox","stepTitle":"What is the probability that a computer part lasts more than $$7$$ years? Round to the nearest $$4$$ decimals.\\\\n","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.4966$$","hints":{"DefaultPathway":[{"id":"a150a865.3computer1a-h1","type":"hint","dependencies":[],"title":"Exponential Distribution","text":"To do any calculations, you must know $$m$$, the decay parameter. $$m=\\\\frac{1}{mean}$$. It is given that the mean amount of years that the certain computer lasts is ten years.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3computer1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{10}$$"],"dependencies":["a150a865.3computer1a-h1"],"title":"Exponential Distribution","text":"What is the decay parameter value $$m$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3computer1a-h1","type":"hint","dependencies":[],"title":"Exponential Distribution","text":"$$\\\\frac{1}{mean}=decay$$ parameter. The mean is $$10$$ years.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3computer1a-h1","type":"hint","dependencies":["a150a865.3computer1a-h2"],"title":"Finding Probability","text":"Find the $$P\\\\left(X>7\\\\right)=1-P\\\\left(X<7\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3computer1a-h2","type":"hint","dependencies":["a150a865.3computer1a-h1"],"title":"Finding Probability","text":"Since $$P\\\\left(X<x\\\\right)$$ $$=$$ $$1-e^{\\\\left(-m x\\\\right)}$$, then we can use $$P\\\\left(X>x\\\\right)=1-1-e^{\\\\left(-m x\\\\right)}$$ $$=$$ $$e^{\\\\left(-m x\\\\right)}$$. Remember that $$m$$ is our decay parameter respective to the problem that we are facing.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3computer1a-h3","type":"hint","dependencies":["a150a865.3computer1a-h2"],"title":"Finding Probability","text":"$$P\\\\left(X>7\\\\right)=e^{7\\\\left(-0.1\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a150a865.3computer1b","stepAnswer":["$$50$$"],"problemType":"TextBox","stepTitle":"On the average, how many years would five computer parts last if they are used one after another? (you use one computer until it is no longer useable, then you go on to use the second one, so on so forth, how long until the last computer is used up?)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$50$$","hints":{"DefaultPathway":[{"id":"a150a865.3computer1b-h1","type":"hint","dependencies":[],"title":"Exponential Distribution","text":"On the average, one computer part lasts ten years. Make use of this known average to compute how on average you would last with $$5$$ computers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3computer1b-h2","type":"hint","dependencies":["a150a865.3computer1b-h1"],"title":"Interpretation","text":"Five computer parts, if they are used one right after the other would last, on the average, five computers times ten years.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a150a865.3computer1c","stepAnswer":["$$7$$"],"problemType":"TextBox","stepTitle":"What is the probability that a computer part lasts between nine and $$11$$ years? Answer in percentage form and round to the nearest percentage","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7$$","hints":{"DefaultPathway":[{"id":"a150a865.3computer1c-h1","type":"hint","dependencies":[],"title":"Finding Probability","text":"First, find out how to get the probability that a computer lasts $$9$$ years and $$11$$ years separately. We know that $$m=0.1$$, and using the formula to find the probability at certain years, the probability of $$x$$ years is equal to $$e^{-0.1 x}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3computer1c-h2","type":"hint","dependencies":["a150a865.3computer1c-h1"],"title":"Alternative","text":"We can alternatively use a PDF function to get the $$P\\\\left(X>9\\\\right)$$ and $$P\\\\left(X>11\\\\right)$$. Afterwards, we can subtract the probabilities from each other and set the value to positive since probability is always positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3computer1c-h3","type":"hint","dependencies":["a150a865.3computer1c-h2"],"title":"Calculations","text":"$$P\\\\left(9<X<11\\\\right)=|P\\\\left(X<9\\\\right)-P\\\\left(X<11\\\\right)|$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a150a865.3highway1","title":"Exponential Distribution","body":"Suppose that on a certain stretch of highway, cars pass at an average rate of five cars per minute. Assume that the duration of time between successive cars follows the exponential distribution.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 The Exponential Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a150a865.3highway1a","stepAnswer":["$$12$$"],"problemType":"TextBox","stepTitle":"On average, how many seconds elapse between two successive cars","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12$$","hints":{"DefaultPathway":[{"id":"a150a865.3highway1a-h1","type":"hint","dependencies":[],"title":"Interpretation","text":"Five cars pass by per minute, we want to find the seconds passed between each car on average","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3highway1a-h2","type":"hint","dependencies":["a150a865.3highway1a-h1"],"title":"Interpretation","text":"There are five cars that pass in a minute, so to find the seconds per car passed, we divide the minute in seconds into five: $$60$$ seconds divided by $$5$$ cars","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a150a865.3highway1b","stepAnswer":["$$84$$"],"problemType":"TextBox","stepTitle":"After a car passes by, how many seconds on average will it take for another seven cars to pass by?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$84$$","hints":{"DefaultPathway":[{"id":"a150a865.3highway1b-h1","type":"hint","dependencies":[],"title":"Exponential Distribution","text":"We know that it takes $$12$$ seconds on average for each car to pass by, if there were seven cars that passed by, how many seconds on average has elapsed between the first and seventh car?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3highway1b-h2","type":"hint","dependencies":["a150a865.3highway1b-h1"],"title":"Exponential Distribution","text":"Multiply the average seconds that a car passes by by the number off pass by successions","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a150a865.3highway1c","stepAnswer":["$$0.8111$$"],"problemType":"TextBox","stepTitle":"Find the probability that after a car passes by, the next car will pass within the next $$20$$ seconds. Round to the nearest $$4$$ decimals.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.8111$$","hints":{"DefaultPathway":[{"id":"a150a865.3highway1c-h1","type":"hint","dependencies":[],"title":"Interpretation","text":"We want to find the probability when a car passes by within a $$20$$ second interval. Therefore, any car may pass by within these $$20$$ seconds. Find $$P\\\\left(X<20\\\\right)$$, where X is the time in seconds between successive cars.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3highway1c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{12}$$"],"dependencies":["a150a865.3highway1c-h1"],"title":"Exponential Distribution","text":"Find the probability in respect to a distribution modeled after seconds. Decay parameter $$m=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3highway1c-h3","type":"hint","dependencies":["a150a865.3highway1c-h2"],"title":"Exponential Distribution","text":"$$m$$ is denoted as the decay parameter where $$1$$ is divided by the random variable average.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3highway1c-h4","type":"hint","dependencies":["a150a865.3highway1c-h3"],"title":"Exponential Distribution","text":"Note that the decay parameter must be in seconds. We know that it takes $$5$$ cars in a minute to pass by on average, and earlier calculated that there are on average12 seconds between each car. Use the $$12$$ seconds as the average.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3highway1c-h5","type":"hint","dependencies":["a150a865.3highway1c-h4"],"title":"Exponential Distribution","text":"$$P\\\\left(X<x\\\\right)$$ $$=$$ $$1-e^{\\\\left(-m x\\\\right)}$$, where $$x$$ is the desired conditional random variable cut off. We are finding $$P\\\\left(X<20\\\\right)$$ because within the $$20$$ second interval, there may be a car.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a150a865.3intro1","title":"Exponential Distributions","body":"The amount of time spouses shop for anniversary cards can be modeled by an exponential distribution with the average amount of time equal to eight minutes.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 The Exponential Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a150a865.3intro1a","stepAnswer":["$$\\\\frac{1}{8} e^{\\\\left(-\\\\frac{1}{8}\\\\right) x}$$"],"problemType":"MultipleChoice","stepTitle":"Write the probability density function. $$y=$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{1}{8} e^{\\\\left(-\\\\frac{1}{8}\\\\right) x}$$","choices":["$$\\\\frac{1}{8} e^{\\\\left(-\\\\frac{1}{8}\\\\right) x}$$","$$8e^{\\\\left(-8\\\\right)} x$$","$$8e^{\\\\left(-8\\\\right)} x$$","$$64e^{\\\\left(-64\\\\right)} x$$"],"hints":{"DefaultPathway":[{"id":"a150a865.3intro1a-h1","type":"hint","dependencies":[],"title":"Exponential Distributions","text":"This is an exponential distribution with a given average. Use $$y=m e^{\\\\left(-m\\\\right) x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3intro1a-h2","type":"hint","dependencies":["a150a865.3intro1a-h1"],"title":"Exponential Distributions","text":"In $$y=m e^{\\\\left(-m\\\\right) x}$$, $$m$$, the decay parameter is equal to $$\\\\frac{1}{mean}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3intro1a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1}{8}$$"],"dependencies":["a150a865.3intro1a-h2"],"title":"$$\\\\frac{1}{mean}=$$?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{1}{8}$$","$$\\\\frac{8}{1}$$","$$\\\\frac{8}{2}$$","$$\\\\frac{6}{6}$$"]}]}}]},{"id":"a150a865.3intro2","title":"Exponential Distributions","body":"Let X $$=$$ amount of time (in minutes) a postal clerk spends with his or her customer. The time is known to have an exponential distribution with the average amount of time equal to four minutes.\\\\n##figure3.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 The Exponential Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a150a865.3intro2a","stepAnswer":["$$0.0814$$"],"problemType":"TextBox","stepTitle":"Find the probability that a clerk spends four to five minutes with a randomly selected customer. (round to four decimal places)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.0814$$","hints":{"DefaultPathway":[{"id":"a150a865.3intro2a-h1","type":"hint","dependencies":[],"title":"Exponential Distributions","text":"We want to find find $$P\\\\left(4<x<5\\\\right)$$. Use the CDF function on a calculator to find the area between $$x=4$$ and $$x=5$$\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3intro2a-h2","type":"hint","dependencies":["a150a865.3intro2a-h1"],"title":"Exponential Distributions","text":"You can find the area to the left of $$x=4$$ and find the area to the right of $$x=5$$ to get the complementary probability of $$P\\\\left(4<X<5\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3intro2a-h3","type":"hint","dependencies":["a150a865.3intro2a-h2"],"title":"Exponential Distributions","text":"A simple calculation of can also be done: The probability that a postal clerk spends four to five minutes with a randomly selected customer is P(4 < X < 5) $$=$$ P(<P(X < 4) $$=$$ $$0.7135$$ - $$0.6321$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a150a865.3intro2b","stepAnswer":["$$2.8$$ minutes"],"problemType":"MultipleChoice","stepTitle":"Half of all customers are finished within how long? (Find the 50th percentile).","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2.8$$ minutes","choices":["$$2.8$$ minutes","$$3$$ minutes","$$5.6$$ minutes","$$0.5$$ minutes"],"hints":{"DefaultPathway":[{"id":"a150a865.3intro2b-h1","type":"hint","dependencies":[],"title":"Exponential Distributions","text":"Find the minutes cutoff when the probability of both sides of the exponential distribution is equal to $$0.5$$. The diagram shown has the value k, the 50th percentile.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3intro2b-h2","type":"hint","dependencies":["a150a865.3intro2b-h1"],"title":"Exponential Distributions","text":"You may use $$P\\\\left(X<k\\\\right)$$ and $$P\\\\left(X<k\\\\right)=1-e^{-0.25 k}$$ to find k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3intro2b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.5$$"],"dependencies":["a150a865.3intro2b-h2"],"title":"Exponential Distributions","text":"What is the value of $$P\\\\left(X<k\\\\right)$$ in this case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3intro2b-h4","type":"hint","dependencies":["a150a865.3intro2b-h3"],"title":"Exponential Distributions","text":"Great, we can now set up the equation $$0.5=1-e^{-0.25 k}$$ and solve for k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3intro2b-h5","type":"hint","dependencies":["a150a865.3intro2b-h4"],"title":"Natural Logarithms","text":"Remember that ln(e) can be used to cancel out natural e\'s","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a150a865.3intro2c","stepAnswer":["mean"],"problemType":"MultipleChoice","stepTitle":"Which is larger, the mean or median?","stepBody":"","answerType":"string","variabilization":{},"choices":["mean","median"],"hints":{"DefaultPathway":[{"id":"a150a865.3intro2c-h1","type":"hint","dependencies":[],"title":"Mean and Median","text":"We are given the average minutes that customers took, $$4$$ minutes. Compare it to the 50th percentile minutes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3intro2c-h2","type":"hint","dependencies":["a150a865.3intro2c-h1"],"title":"Mean and Median","text":"Mean $$=$$ $$4$$ minutes, 50th percentile or median $$=$$ $$2.8$$ minutes","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a150a865.3phone1","title":"Exponential Distribution","body":"Suppose that the length of a phone call, in minutes, is an exponential random variable with decay parameter $$\\\\frac{1}{12}$$","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 The Exponential Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a150a865.3phone1a","stepAnswer":["$$0.6592$$"],"problemType":"TextBox","stepTitle":"If another person arrives at a public telephone just before you, find the probability that you will have to wait more than five minutes. Let X be the length of a phone call in minutes. Round to the nearest four decimal places.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.6592$$","hints":{"DefaultPathway":[{"id":"a150a865.3phone1a-h1","type":"hint","dependencies":[],"title":"Exponential Distribution","text":"Using the decay parameter $$\\\\frac{1}{12}$$, we know that the average elapsed time of a ohone call in munutes is $$12$$ minutes on an exponential distribution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3phone1a-h2","type":"hint","dependencies":["a150a865.3phone1a-h1"],"title":"PDF Function","text":"We may use a PDF function and set the decay parameter to $$\\\\frac{1}{12}$$ and use $$P\\\\left(X>x\\\\right)$$, where $$x$$ is the probability cutoff.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3phone1a-h3","type":"hint","dependencies":["a150a865.3phone1a-h2"],"title":"Calculating using $$x$$","text":"Set $$x$$ to $$5$$ minutes since we want to find the area under the exponential distribution curve when X is greater than $$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a150a865.3police1","title":"Exponential Distribution","body":"At a police station in a large city, calls come in at an average rate of four calls per minute. Assume that the time that elapses from one call to the next has the exponential distribution. Take note that we are concerned only with the rate at which calls come in, and we are ignoring the time spent on the phone. We must also assume that the times spent between calls are independent. This means that a particularly long delay between two calls does not mean that there will be a shorter waiting period for the next call. We may then deduce that the total number of calls received during a time period has the Poisson distribution.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 The Exponential Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a150a865.3police1a","stepAnswer":["$$0.4866$$"],"problemType":"TextBox","stepTitle":"Find the average time between two successive calls in minutes.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.4866$$","hints":{"DefaultPathway":[{"id":"a150a865.3police1a-h1","type":"hint","dependencies":[],"title":"Average","text":"On average four calls occur per minutes. Sixty seconds divided by four callers would yield $$15$$ seconds per caller on average.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3police1a-h2","type":"hint","dependencies":["a150a865.3police1a-h1"],"title":"Average","text":"$$m$$ must be in minutes, we can convert to minutes by dividing $$15$$ seconds by $$60$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a150a865.3police1b","stepAnswer":["$$0.4866$$"],"problemType":"TextBox","stepTitle":"Find the probability that after a call is received, the next call occurs in less than $$10$$ seconds. Round to the nearest four decimal places.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.4866$$","hints":{"DefaultPathway":[{"id":"a150a865.3police1b-h1","type":"hint","dependencies":[],"title":"Exponential Distribution","text":"Let T equal the time elapsed between calls. From part a, the mean is found to be $$0.25$$, so the decay parameter is $$1$$ divided by the mean, $$0.25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3police1b-h2","type":"hint","dependencies":["a150a865.3police1b-h1"],"title":"Exponential Distribution","text":"The cumulative distribution function is $$P\\\\left(T<t\\\\right)$$ $$=$$ $$1-e^{\\\\left(-4t\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3police1b-h3","type":"hint","dependencies":["a150a865.3police1b-h2"],"title":"Exponential Distribution","text":"We want to find the area under the exponential distiributiion function where T is less than the $$t=\\\\frac{1}{6}$$ minutes boundary.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3police1b-h4","type":"hint","dependencies":["a150a865.3police1b-h3"],"title":"Exponential Distribution","text":"The probability that the next call occurs in less than $$10$$ seconds (where $$10$$ seconds is equal to $$\\\\frac{1}{6}$$ minutes) is $$P\\\\left(T<\\\\frac{1}{6}\\\\right)=1-e^{\\\\left(-4t\\\\right)}$$, where $$t$$ is $$\\\\frac{1}{6}$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a150a865.3repair1","title":"Exponential Distribution","body":"The cost of all maintenance for a car during its first year is approximately exponentially distributed with a mean of $150.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 The Exponential Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a150a865.3repair1a","stepAnswer":["Cost of maintaining a car within one year of purchasing it."],"problemType":"MultipleChoice","stepTitle":"Define the random variable X.","stepBody":"","answerType":"string","variabilization":{},"choices":["Cost of maintaining a car within one year of purchasing it.","Cost of hiring a car repairer that costs lest than $150.","Cost of the average expendings within the first year of purchasing a new car."],"hints":{"DefaultPathway":[{"id":"a150a865.3repair1a-h1","type":"hint","dependencies":[],"title":"Random Variable","text":"The costs of maintenance within the first year are collected. These values are used to build the exponential distribution and yield the mean of $150","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a150a865.3repair1b","stepAnswer":["$$150$$"],"problemType":"TextBox","stepTitle":"Determine the value of the mean in dollars.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$150$$","hints":{"DefaultPathway":[{"id":"a150a865.3repair1b-h1","type":"hint","dependencies":[],"title":"We are given an exponential distribution of car maintenance costs that has an average of $150.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a150a865.3repair1c","stepAnswer":["$$150$$"],"problemType":"TextBox","stepTitle":"Determine the value of the standard deviation.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$150$$","hints":{"DefaultPathway":[{"id":"a150a865.3repair1c-h1","type":"hint","dependencies":[],"title":"Recall that the standard deviation in exponential distribution is equal to the mean of the distribution.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a150a865.3retire1","title":"Exponential Distribution","body":"The time (in years) after reaching age $$60$$ that it takes an individual to retire is approximately exponentially distributed with a mean of about five years. Suppose we randomly pick one retired individual. We are interested in the time after age $$60$$ to retirement.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 The Exponential Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a150a865.3retire1a","stepAnswer":["The year(s) it takes after reaching past the age $$60$$ that it takes a person to retire"],"problemType":"MultipleChoice","stepTitle":"Define the random variable X.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"The year(s) it takes after reaching past the age $$60$$ that it takes a person to retire","choices":["The year(s) it takes after reaching past the age $$60$$ that it takes a person to retire","The number of retired people in the $$60$$ nearest senior centers","The $$60$$ years needed to retire in the US compared to China","The number of years it takes to retire within a five year interval of retirement."],"hints":{"DefaultPathway":[{"id":"a150a865.3retire1a-h1","type":"hint","dependencies":[],"title":"Random Variables","text":"We are interested in the years it takes AFTER reaching the age $$60$$ for someone to retire.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a150a865.3retire1b","stepAnswer":["discrete"],"problemType":"MultipleChoice","stepTitle":"Is X continuous or discrete?","stepBody":"","answerType":"string","variabilization":{},"choices":["discrete","continuous","neither"],"hints":{"DefaultPathway":[{"id":"a150a865.3retire1b-h1","type":"hint","dependencies":[],"title":"Data types","text":"Since we are interested in the amount of years it takes for an individual to retire after the age of $$60$$, we are specifically organizing data to be separated by years.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a150a865.3retire1c","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"What is the mean in years?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"a150a865.3retire1c-h1","type":"hint","dependencies":[],"title":"Mean","text":"We are given that the data collected can be exponentially distributed with an average of about five years after reaching the age $$60$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a150a865.3retire1d","stepAnswer":["$$18$$"],"problemType":"TextBox","stepTitle":"In a room full of $$1000$$ people over the agae $$80$$, how many do you expect will not have retired yet? Round to the nearest whole person.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$18$$","hints":{"DefaultPathway":[{"id":"a150a865.3retire1d-h1","type":"hint","dependencies":[],"title":"Exponential Distribution","text":"We can set our decay parameter and use a PDF function to find $$P\\\\left(X>x\\\\right)$$ where X is greater than a bounded value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3retire1d-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a150a865.3retire1d-h1"],"title":"Before using $$P\\\\left(X>x\\\\right)$$ function, we need to find what $$x$$ will be. What is $$x$$? Remember the random variable definition.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3retire1d-h3","type":"hint","dependencies":["a150a865.3retire1d-h2"],"title":"Exponential Distribution","text":"Since we are looking at a population aged $$80$$ years old or older, we can subtract $$60$$ from $$80$$ to get $$20$$ as our chosen bound value $$x$$ when solving $$P\\\\left(X>x\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a150a865.3runner1","title":"Exponential Distribution","body":"On average, a pair of running shoes can last $$18$$ months if used every day. The length of time running shoes last is exponentially distributed.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 The Exponential Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a150a865.3runner1a","stepAnswer":["$$0.4346$$"],"problemType":"TextBox","stepTitle":"What is the probability that a pair of running shoes last more than $$15$$ months? Round to the nearest four decimals.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.4346$$","hints":{"DefaultPathway":[{"id":"a150a865.3runner1a-h1","type":"hint","dependencies":[],"title":"Exponential Distribution","text":"We know that the average pair of running shoes last $$18$$ months on an exponential distribution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3runner1a-h2","type":"hint","dependencies":["a150a865.3runner1a-h1"],"title":"Exponential Distribution","text":"The decay parameter is computed by 1/(average random variable value). In our case, the average is $$18$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3runner1a-h3","type":"hint","dependencies":["a150a865.3runner1a-h2"],"title":"Exponential Distribution","text":"Using a PDF function, we can set the appropiate decay parameter and set $$P\\\\left(X>x\\\\right)$$, where $$x$$ is the cutoff boundary for the probability.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3runner1a-h4","type":"hint","dependencies":["a150a865.3runner1a-h3"],"title":"Exponential Distribution","text":"We can set $$x$$ equal to $$15$$ to represent the $$15$$ months and the $$P\\\\left(X>x\\\\right)$$ PDF will yield the area under the curves past $$X>15$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3runner1a-h5","type":"hint","dependencies":["a150a865.3runner1a-h4"],"title":"Lasting Shoes","text":"On average, how long would six pairs of running shoes last if they are used one after the other? Eighty percent of running shoes last at most how long if used every day?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a150a865.3runner1b","stepAnswer":["$$108$$"],"problemType":"TextBox","stepTitle":"On average, how many months would six pairs of running shoes last if they are used one after the other everyday?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$108$$","hints":{"DefaultPathway":[{"id":"a150a865.3runner1b-h1","type":"hint","dependencies":[],"title":"Average","text":"We know that the average amount of time that a running shoe lasts if used everyday lasts $$18$$ months. We can use that value to predict the average amount of time needed to use up six shoes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3runner1b-h2","type":"hint","dependencies":["a150a865.3runner1b-h1"],"title":"Average","text":"We can multiply $$18$$ months and $$6$$ shoes to get the average amount of time to use up $$6$$ running shoes back to back everyday","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a150a865.3runningshoes1","title":"Exponential distribution","body":"On average, a pair of running shoes can last $$18$$ months if used every day. The length of time running shoes last is exponentially distributed.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 The Exponential Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a150a865.3runningshoes1a","stepAnswer":["$$0.43$$"],"problemType":"TextBox","stepTitle":"What is the probability that a pair of running shoes last more than $$15$$ months? Round to the nearest $$2$$ decimals","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.43$$","hints":{"DefaultPathway":[{"id":"a150a865.3runningshoes1a-h1","type":"hint","dependencies":[],"title":"Exponential distribution","text":"You may use a calculator exponential PDF function where $$m$$ can be set to the decay parameter given by $$\\\\frac{1}{18}$$, and finding the $$P\\\\left(X>15\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3runningshoes1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{18}$$"],"dependencies":["a150a865.3runningshoes1a-h1"],"title":"Exponential distribution","text":"Decay parameter $$m=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3runningshoes1a-h3","type":"hint","dependencies":[],"title":"Exponential distribution","text":"The given average is $$18$$ days, divide $$1$$ by $$18$$ to get the decay parameter","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a150a865.3runningshoes1b","stepAnswer":["$$108$$"],"problemType":"TextBox","stepTitle":"How long would six pairs of running shoes last if they are used on after the other? Round to the nearest month.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$108$$","hints":{"DefaultPathway":[{"id":"a150a865.3runningshoes1b-h1","type":"hint","dependencies":[],"title":"Exponential distribution","text":"On the average, one pair of running shoes lasts $$18$$ months if they are used every day. Multiply that average by the number of pairs that you want to calculate how long will last. Remember, a pair is two individual shoes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a150a865.3store1","title":"Exponential Distribution","body":"The time spent waiting between events is often modeled using the exponential distribution. For example, suppose that an average of $$30$$ customers per hour arrive at a store and the time between arrivals is exponentially distributed.\\\\n","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 The Exponential Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a150a865.3store1a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"On average, how many customers do we expect to see every two minutes?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a150a865.3store1a-h1","type":"hint","dependencies":[],"title":"Average","text":"Since we expect $$30$$ customers to arrive per hour (60 minutes), we expect on average one customer to arrive every two minutes on average.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a150a865.3store1b","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"When the store first opens, how many minutes on average does it take for three customers to arrive?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"a150a865.3store1b-h1","type":"hint","dependencies":[],"title":"Average","text":"We know that a customer arrives every two minutes. We can use that to estimate how many minutes it will take for three customers to arrive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3store1b-h2","type":"hint","dependencies":["a150a865.3store1b-h1"],"title":"Average","text":"Since one customer arrives every two minutes on average, it will take six minutes on average for three customers to arrive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a150a865.3store1c","stepAnswer":["$$0.3935$$"],"problemType":"TextBox","stepTitle":"After a customer arrives, find the probability that it takes less than one minute for the next customer to arrive. Round to four decimal points.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.3935$$","hints":{"DefaultPathway":[{"id":"a150a865.3store1c-h1","type":"hint","dependencies":[],"title":"Exponential Distribution","text":"By part a, we know that the average is $$2$$ minutes between each customer. $$\\\\frac{1}{mean}=decay$$ parameter. The mean is $$2$$ minutes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3store1c-h2","type":"hint","dependencies":["a150a865.3store1c-h1"],"title":"Exponential Distribution","text":"$$P\\\\left(X<x\\\\right)$$ $$=$$ $$1-e^{\\\\left(-m x\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3store1c-h3","type":"hint","dependencies":["a150a865.3store1c-h2"],"title":"Exponential Distribution","text":"The cumulative distribution function is $$P\\\\left(X<x\\\\right)=1-e^{-0.5 x}$$, we are finding the area under the curve where X is less than $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3store1c-h4","type":"hint","dependencies":["a150a865.3store1c-h3"],"title":"Exponential Distribution","text":"The expression to compute the probability is $$P\\\\left(X<1\\\\right)=1-e^{1-0.5}$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a150a865.3traveler1","title":"Exponential Distribution","body":"The number of days ahead travelers purchase their airline tickets can be modeled by an exponential distribution with the average amount of time equal to $$15$$ days.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 The Exponential Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a150a865.3traveler1a","stepAnswer":["$$0.487$$"],"problemType":"TextBox","stepTitle":"Find the probability that a traveler will purchase a ticket fewer than ten days in advance.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.487$$","hints":{"DefaultPathway":[{"id":"a150a865.3traveler1a-h1","type":"hint","dependencies":[],"title":"Mean","text":"To do any calculations, you must know $$m$$, the decay parameter. $$m=\\\\frac{1}{mean}$$. It is given that the mean amount of days that travelers purchase their airline tickets is equal to $$15$$ days.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3traveler1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{15}$$"],"dependencies":["a150a865.3traveler1a-h1"],"title":"Value of $$m$$","text":"What is the value of $$m$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3traveler1a-h3","type":"hint","dependencies":["a150a865.3traveler1a-h2"],"title":"Exponential Distribution","text":"The probability desnsity function is $$f(x)=m e^{\\\\left(-m x\\\\right)}$$, where e is the natural number $$e=2.7182818..$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3traveler1a-h4","type":"hint","dependencies":["a150a865.3traveler1a-h3"],"title":"Exponential Distribution","text":"To find the certain probability in an exponential distribution, set the random vairable $$x$$ to the desired probability condition: For example $$f(5)=m e^{\\\\left(-5m\\\\right)}$$ will yield the probability that a random trial yields $$x=5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3traveler1a-h5","type":"hint","dependencies":["a150a865.3traveler1a-h4"],"title":"Exponential Distribution","text":"We are trying to find the probability that travelers purchase their tickets fewer than $$10$$ days in advance, we must find $$P\\\\left(x<10\\\\right)$$, so we may do $$1-P(x=1)-P(x=2)-...-P(x=10)$$, using $$f(x)=m e^{\\\\left(-m x\\\\right)}$$, but to do it faster, we may use the CDF function to find the total area of the values that $$x<10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a150a865.3traveler1a-h6","type":"hint","dependencies":["a150a865.3traveler1a-h5"],"title":"Exponential Distribution","text":"Using CDF of X < $$10$$ and rounding to the nearest $$3$$ decimals, we get $$0.067$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a150a865.3traveler1b","stepAnswer":["$$10$$"],"problemType":"TextBox","stepTitle":"Exponential Distribution","stepBody":"How many days do half of all travelers wait? (round to the nearest day)","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10$$","hints":{"DefaultPathway":[{"id":"a150a865.3traveler1b-h1","type":"hint","dependencies":[],"title":"Exponential Distribution","text":"We want to find the corresponding $$x$$ days value that 50% of travelers wait. One way of doing so would be using a PDF function on a calculator and inputting $$0.5$$ probability to yield the X value cutoff value for half of the exponential distribution curve.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a15666breview1","title":"Review of Functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.1 Review of Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a15666breview1a","stepAnswer":["Domain: Integers from $$-3$$ to $$3$$, Range: [0,9], Is a function"],"problemType":"MultipleChoice","stepTitle":"Determine the domain and the range of each relation, and state whether the relation is a function. 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any undefined values, such as when the denominator is equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a15666breview16a-h3","type":"hint","dependencies":["a15666breview16a-h2"],"title":"Range","text":"What\'s the maximum and minimum of the function as it approaches the edges of the domain?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a15666breview17","title":"Review of Functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.1 Review of Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a15666breview17a","stepAnswer":["Zero: $$(-1,0)$$, Domain: $$[-2,\\\\infty)$$, $$Range:-1$$ to $$\\\\infty$$"],"problemType":"MultipleChoice","stepTitle":"Find the domain, range, and all $$zeros/intercepts$$. 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$$+\\\\infty$$"],"hints":{"DefaultPathway":[{"id":"a15666breview19a-h1","type":"hint","dependencies":[],"title":"Zeroes","text":"Think about what values would cause the function to be equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a15666breview19a-h2","type":"hint","dependencies":["a15666breview19a-h1"],"title":"Domain","text":"When determining domain, solve for any undefined values, such as when the denominator is equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a15666breview19a-h3","type":"hint","dependencies":["a15666breview19a-h2"],"title":"Range","text":"What\'s the maximum and minimum of the function as it approaches the edges of the domain?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a15666breview2","title":"Review of Functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 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Then, find the result of the function","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a15666breview9","title":"Review of Functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.1 Review of Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a15666breview9a","stepAnswer":["a. DNE, $$b$$. $$2$$, c. $$\\\\frac{2}{3}$$, $$d$$. $$\\\\frac{-2}{x}$$, e. $$\\\\frac{2}{a}$$, f. $$\\\\frac{2}{a+h}$$"],"problemType":"MultipleChoice","stepTitle":"Find the values for $$f(x)=\\\\frac{2}{x}$$, if they exist, then simplify. a.$$f(0)$$, b.$$f(1)$$, c.$$f(-x)$$, d.$$f(a)$$, f.$$f(a+h)$$.","stepBody":"$$f(x)=\\\\frac{2}{x}$$","answerType":"string","variabilization":{},"answerLatex":"a. DNE, $$b$$. $$2$$, c. $$\\\\frac{2}{3}$$, $$d$$. $$\\\\frac{-2}{x}$$, e. $$\\\\frac{2}{a}$$, f. $$\\\\frac{2}{a+h}$$","choices":["a. $$0$$, $$b$$. $$2$$, c. $$\\\\frac{2}{3}$$, $$d$$. $$\\\\frac{-2}{x}$$, e. $$\\\\frac{2}{a}$$, f. $$\\\\frac{2}{a+h}$$","a. $$2$$, $$b$$. $$2$$, c. $$\\\\frac{2}{3}$$, $$d$$. $$\\\\frac{-2}{x}$$, e. $$\\\\frac{2}{a}$$, f. $$\\\\frac{-2}{a+h}$$","a. DNE, $$b$$. $$2$$, c. $$\\\\frac{2}{3}$$, $$d$$. $$\\\\frac{-2}{x}$$, e. $$\\\\frac{2}{a}$$, f. $$\\\\frac{2}{a+h}$$"],"hints":{"DefaultPathway":[{"id":"a15666breview9a-h1","type":"hint","dependencies":[],"title":"Plugging in Values","text":"Plug in the required values into the function, such as $$0$$ and $$1$$. If the result is not undefined, then simplify.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a15cd07anov1","title":"Diet Plans","body":"Three different diet plans are to be tested for mean weight loss. The entries in the table are the weight losses for the different plans. The one-way ANOVA results are shown in Table 13.2.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"13.2 The F Distribution and the F-Ratio","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a15cd07anov1a","stepAnswer":["$$0.3769$$"],"problemType":"TextBox","stepTitle":"Use the table to construct the hypothesis test. 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($\\\\SS_{between}$)","text":"$\\\\SS_{between}$ $$=$$ $\\\\sum[\\\\frac{(\\\\sum s_j)**2}{n_j}] - \\\\frac{(\\\\sum s_j)**2}{n}$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.2458$$"],"dependencies":["a15cd07anov1a-h3"],"title":"$\\\\frac{s_1 **2}{4} + \\\\frac{s_2 ** 2}{3} + \\\\frac{s_3 **3}{3} - $$\\\\frac{(16.5$$ + $$15$$ + 15.5)**2}{10}$","text":"Round to the fourth place after the decimal.","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov1a-h5","type":"hint","dependencies":["a15cd07anov1a-h4"],"title":"Compute the Sum of Squares representing variation within samples due to chance. ($\\\\SS_{within})","text":"$\\\\SS_{within} $$=$$ \\\\SS_{total} - \\\\SS_{between}$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov1a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$23.1$$"],"dependencies":["a15cd07anov1a-h5"],"title":"Compute $\\\\SS_{total} $$=$$ \\\\sum $$x^2$$ - \\\\frac{(\\\\sum x) ** 2}{n}$","text":"$(5**2 + $${4.5}^2$$ + $$4^2$$ + $$3^2$$ + $${3.5}^2$$ + $$7^2$$ + $${4.5}^2$$ + $$8^2$$ + $$4^2$$ + 3.5**2) - \\\\frac{(5 + $$4.5$$ + $$4$$ + $$3$$ + $$3.5$$ + $$7$$ + $$4.5$$ + $$8$$ + $$4$$ + 3.5)**2}{4+3+3} $","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov1a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20.8542$$"],"dependencies":["a15cd07anov1a-h6"],"title":"$\\\\SS_{total} $$=$$ \\\\sum $$x^2$$ - \\\\frac{(\\\\sum x) ** 2}{n}$","text":"$$23.1$$ - $$2.2458$$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov1a-h8","type":"hint","dependencies":["a15cd07anov1a-h7"],"title":"What is the Degree of Freedom of Factor (Between)?","text":"$df_{numerator}$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov1a-h9","type":"hint","dependencies":["a15cd07anov1a-h8"],"title":"What is the Degree of Freedom of Error (within)?","text":"$df_{denominator}$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov1a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a15cd07anov1a-h9"],"title":"$df_{denominator} $$=$$ $$n$$ - k$","text":"$$10$$ total data - $$3$$ groups","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov1a-h11","type":"hint","dependencies":["a15cd07anov1a-h10"],"title":"Compute the Mean Square of Factor (Between)","text":"Round to the fourth place after the decimal.","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov1a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.1229$$"],"dependencies":["a15cd07anov1a-h11"],"title":"MS(Factor) $$=$$ SS(Factor) / (k - 1)","text":"$$2.2458$$ / $$2$$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov1a-h13","type":"hint","dependencies":["a15cd07anov1a-h12"],"title":"Compute the Mean Square of Error (Within)","text":"Round to the fourth place after the decimal.","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov1a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.9792$$"],"dependencies":["a15cd07anov1a-h13"],"title":"MS(Error) $$=$$ SS(Error) / (n - k)","text":"$$20.8542$$ / $$7$$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov1a-h15","type":"hint","dependencies":["a15cd07anov1a-h14"],"title":"Compute the F-statistics","text":"Round to the fourth place after the decimal.","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov1a-h16","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.3769$$"],"dependencies":["a15cd07anov1a-h15"],"title":"F $$=$$ MS(Factor) / MS(Error)","text":"$$1.1229$$ / $$2.9792$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"a15cd07anov2","title":"Tomato Plants Treatment","body":"As part of an experiment to see how different types of soil cover would affect slicing tomato production, Marist College students grew tomato plants under different soil cover conditions. Groups of three plants each had one of the following treatments: bare soil, a commercial ground cover, black plastic, straw, compost. All plants grew under the same conditions and were the same variety. Students recorded the weight (in grams) of tomatoes produced by each of the $$n$$ $$=$$ $$15$$ plants.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"13.2 The F Distribution and the F-Ratio","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a15cd07anov2a","stepAnswer":["$$2.1396$$"],"problemType":"TextBox","stepTitle":"Use the table to construct the hypothesis test. What is the F-statistics?","stepBody":"Round to the fourth place after the decimal.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2.1396$$","hints":{"DefaultPathway":[{"id":"a15cd07anov2a-h1","type":"hint","dependencies":[],"title":"What is the sum of the values in the jth group?","text":"Sum of value in each column.","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10537$$"],"dependencies":["a15cd07anov2a-h1"],"title":"Sum of value for group n1.","text":"Add up the value in column n1.","variabilization":{},"oer":"","license":"","subHints":[{"id":"a15cd07anov2a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16512$$"],"dependencies":[],"title":"Sum of value for group n2.","text":"Add up the value in column n2.","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov2a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18973$$"],"dependencies":[],"title":"Sum of value for group n3.","text":"Add up the value in column n3.","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov2a-h2-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$23412$$"],"dependencies":[],"title":"Sum of value for group n4.","text":"Add up the value in column n4.","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov2a-h2-s4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$22772$$"],"dependencies":[],"title":"Sum of value for group n5.","text":"Add up the value in column n5.","variabilization":{},"oer":"","license":""}]},{"id":"a15cd07anov2a-h3","type":"hint","dependencies":["a15cd07anov2a-h2"],"title":"Compute the Sum of Squares representing variation among the different samples. ($\\\\SS_{between}$)","text":"$\\\\SS_{between}$ $$=$$ $\\\\sum[\\\\frac{(\\\\sum s_j)**2}{n_j}] - \\\\frac{(\\\\sum s_j)**2}{n}$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$36648560.93$$"],"dependencies":["a15cd07anov2a-h3"],"title":"$\\\\frac{s_1 **2}{3} + \\\\frac{s_2 ** 2}{3} + \\\\frac{s_3 **2}{3} + \\\\frac{s_4 **2}{3} + \\\\frac{s_5 **2}{3} - \\\\frac{(10537 + $$16512$$ + $$18973$$ + 23412+ 22772)**2}{15}$","text":"Round to the fourth place after the decimal.","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov2a-h5","type":"hint","dependencies":["a15cd07anov2a-h4"],"title":"Compute the Sum of Squares representing variation within samples due to chance. ($\\\\SS_{within})","text":"$\\\\SS_{within} $$=$$ \\\\SS_{total} - \\\\SS_{between}$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$57095286.93$$"],"dependencies":["a15cd07anov2a-h5"],"title":"Compute $\\\\SS_{total} $$=$$ \\\\sum $$x^2$$ - \\\\frac{(\\\\sum x) ** 2}{n}$","text":"$(2625**2 + $${2997}^2$$ + $${4915}^2$$ + $${5348}^2$$ + $${5682}^2$$ + $${5482}^2$$ + $${6583}^2$$ + $${8560}^2$$ + $${3830}^2$$ + $${7285}^2$$ + $${6897}^2$$ + $${9230}^2$$ + $${6277}^2$$ + $${7818}^2$$ + 8677**2) - \\\\frac{(2625 + $$2997$$ + $$4915$$ + $$5348$$ + $$5682$$ + $$5482$$ + $$6583$$ + $$8560$$ + $$3830$$ + $$7285$$ + $$6897$$ + $$9230$$ + $$6277$$ + $$7818$$ + 8677)**2}{3+3+3+3+3} $","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov2a-h7","type":"hint","dependencies":["a15cd07anov2a-h6"],"title":"What is the Degree of Freedom of Factor (Between)?","text":"$df_{numerator}$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov2a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a15cd07anov2a-h7"],"title":"$df_{numerator} $$=$$ k - 1$","text":"$$5$$ groups - $$1$$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov2a-h9","type":"hint","dependencies":["a15cd07anov2a-h8"],"title":"What is the Degree of Freedom of Error (within)?","text":"$df_{denominator}$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov2a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a15cd07anov2a-h9"],"title":"$df_{denominator} $$=$$ $$n$$ - k$","text":"$$15$$ total data - $$5$$ groups","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov2a-h11","type":"hint","dependencies":["a15cd07anov2a-h10"],"title":"Compute the Mean Square of Factor (Between)","text":"Round to the fourth place after the decimal.","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov2a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12216186.98$$"],"dependencies":["a15cd07anov2a-h11"],"title":"MS(Factor) $$=$$ SS(Factor) / (k - 1)","text":"$$36648560.9333$$ / $$3$$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov2a-h13","type":"hint","dependencies":["a15cd07anov2a-h12"],"title":"Compute the Mean Square of Error (Within)","text":"Round to the third place after the decimal.","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov2a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5709528.693$$"],"dependencies":["a15cd07anov2a-h13"],"title":"MS(Error) $$=$$ SS(Error) / (n - k)","text":"$$57095286.93$$ / $$10$$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov2a-h15","type":"hint","dependencies":["a15cd07anov2a-h14"],"title":"Compute the F-statistics","text":"Round to the fourth place after the decimal.","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov2a-h16","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.1396$$"],"dependencies":["a15cd07anov2a-h15"],"title":"F $$=$$ MS(Factor) / MS(Error)","text":"$$12216186.98$$ / $$5709528.693$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"a15cd07anov3","title":"Driver License","body":"Use the following information to answer the next three exercises. Suppose a group is interested in determining whether teenagers obtain their drivers licenses at approximately the same average age across the country. Suppose that the following data are randomly collected from five teenagers in each region of the country. The numbers represent the age at which teenagers obtained their drivers licenses.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"13.2 The F Distribution and the F-Ratio","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a15cd07anov3a","stepAnswer":["$$1.9855$$"],"problemType":"TextBox","stepTitle":"Use the table to construct the hypothesis test. What is the F-statistics?","stepBody":"Round to the fourth place after the decimal.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1.9855$$","hints":{"DefaultPathway":[{"id":"a15cd07anov3a-h1","type":"hint","dependencies":[],"title":"What is the sum of the values in the jth group?","text":"Sum of value in each column.","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$65.3$$"],"dependencies":["a15cd07anov3a-h1"],"title":"Sum of value for group Northeast.","text":"Add up the value in column Northeast.","variabilization":{},"oer":"","license":"","subHints":[{"id":"a15cd07anov3a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$66$$"],"dependencies":[],"title":"Sum of value for group South.","text":"Add up the value in column South.","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov3a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$65.6$$"],"dependencies":[],"title":"Sum of value for group West.","text":"Add up the value in column West.","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov3a-h2-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$65.7$$"],"dependencies":[],"title":"Sum of value for group Central.","text":"Add up the value in column Central.","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov3a-h2-s4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$67.7$$"],"dependencies":[],"title":"Sum of value for group East.","text":"Add up the value in column East.","variabilization":{},"oer":"","license":""}]},{"id":"a15cd07anov3a-h3","type":"hint","dependencies":["a15cd07anov3a-h2"],"title":"Compute the Sum of Squares representing variation among the different samples. ($\\\\SS_{between}$)","text":"$\\\\SS_{between}$ $$=$$ $\\\\sum[\\\\frac{(\\\\sum s_j)**2}{n_j}] - \\\\frac{(\\\\sum s_j)**2}{n}$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.903$$"],"dependencies":["a15cd07anov3a-h3"],"title":"$\\\\frac{s_1 **2}{4} + \\\\frac{s_2 ** 2}{4} + \\\\frac{s_3 **2}{4} + \\\\frac{s_4 **2}{4} + \\\\frac{s_5 **2}{4} - $$\\\\frac{(65.3$$ + $$66$$ + $$65.6$$ + 65.7+ 67.7)**2}{20}$","text":"Round to the third place after the decimal.","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov3a-h5","type":"hint","dependencies":["a15cd07anov3a-h4"],"title":"Compute the Sum of Squares representing variation within samples due to chance. ($\\\\SS_{within})","text":"$\\\\SS_{within} $$=$$ \\\\SS_{total} - \\\\SS_{between}$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov3a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.7055$$"],"dependencies":["a15cd07anov3a-h5"],"title":"Compute $\\\\SS_{total} $$=$$ \\\\sum $$x^2$$ - \\\\frac{(\\\\sum x) ** 2}{n}$","text":"$(16.3**2 + $${16.1}^2$$ + $${16.4}^2$$ + $${16.5}^2$$ + $${16.9}^2$$ + $${16.5}^2$$ + $${16.4}^2$$ + $${16.2}^2$$ + $${16.4}^2$$ + $${16.5}^2$$ + $${16.6}^2$$ + $${16.1}^2$$ + $${16.2}^2$$ + $${16.6}^2$$ + $${16.5}^2$$ + $${16.4}^2$$ + $${17.1}^2$$ + $${17.2}^2$$ + $${16.6}^2$$ + 16.8**2) - $$\\\\frac{(16.3$$ + $$16.1$$ + $$16.4$$ + $$16.5$$ + $$16.9$$ + $$16.5$$ + $$16.4$$ + $$16.2$$ + $$16.5$$ + $$16.4$$ + $$16.6$$ + $$16.1$$ + $$16.2$$ + $$16.6$$ + $$16.5$$ + $$16.4$$ + $$17.1$$ + $$17.2$$ + $$16.6$$ + 16.8)**2}{4+4+4+4+4} $","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov3a-h7","type":"hint","dependencies":["a15cd07anov3a-h6"],"title":"What is the Degree of Freedom of Factor (Between)?","text":"$df_{numerator}$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov3a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a15cd07anov3a-h7"],"title":"$df_{numerator} $$=$$ k - 1$","text":"$$5$$ groups - $$1$$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov3a-h9","type":"hint","dependencies":["a15cd07anov3a-h8"],"title":"What is the Degree of Freedom of Error (within)?","text":"$df_{denominator}$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov3a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["a15cd07anov3a-h9"],"title":"$df_{denominator} $$=$$ $$n$$ - k$","text":"$$20$$ total data - $$5$$ groups","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov3a-h11","type":"hint","dependencies":["a15cd07anov3a-h10"],"title":"Compute the Mean Square of Factor (Between)","text":"Round to the fifith place after the decimal.","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov3a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.22575$$"],"dependencies":["a15cd07anov3a-h11"],"title":"MS(Factor) $$=$$ SS(Factor) / (k - 1)","text":"$$0.903$$ / $$4$$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov3a-h13","type":"hint","dependencies":["a15cd07anov3a-h12"],"title":"Compute the Mean Square of Error (Within)","text":"Round to the third place after the decimal.","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov3a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1137$$"],"dependencies":["a15cd07anov3a-h13"],"title":"MS(Error) $$=$$ SS(Error) / (n - k)","text":"$$1.7055$$ / $$15$$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov3a-h15","type":"hint","dependencies":["a15cd07anov3a-h14"],"title":"Compute the F-statistics","text":"Round to the fourth place after the decimal.","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov3a-h16","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.9855$$"],"dependencies":["a15cd07anov3a-h15"],"title":"F $$=$$ MS(Factor) / MS(Error)","text":"$$0.22575$$ / $$0.1137$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"a15cd07anov4","title":"Average Weight","body":"Use the following information to answer the next eight exercises. Groups of men from three different areas of the country are to be tested for mean weight. The entries in Table $$13.13$$ are the weights for the different groups.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"13.2 The F Distribution and the F-Ratio","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a15cd07anov4a","stepAnswer":["$$2.3045$$"],"problemType":"TextBox","stepTitle":"Use the table to construct the hypothesis test. What is the F-statistics?","stepBody":"Round to the fourth place after the decimal.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2.3045$$","hints":{"DefaultPathway":[{"id":"a15cd07anov4a-h1","type":"hint","dependencies":[],"title":"What is the sum of the values in the jth group?","text":"Sum of value in each column.","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1017$$"],"dependencies":["a15cd07anov4a-h1"],"title":"Sum of value for group1.","text":"Add up the value in column $$1$$.","variabilization":{},"oer":"","license":"","subHints":[{"id":"a15cd07anov4a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1137$$"],"dependencies":[],"title":"Sum of value for group $$2$$.","text":"Add up the value in column $$2$$.","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov4a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$915$$"],"dependencies":[],"title":"Sum of value for group $$3$$.","text":"Add up the value in column $$3$$.","variabilization":{},"oer":"","license":""}]},{"id":"a15cd07anov4a-h3","type":"hint","dependencies":["a15cd07anov4a-h2"],"title":"Compute the Sum of Squares representing variation among the different samples. ($\\\\SS_{between}$)","text":"$\\\\SS_{between}$ $$=$$ $\\\\sum[\\\\frac{(\\\\sum s_j)**2}{n_j}] - \\\\frac{(\\\\sum s_j)**2}{n}$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4939.2$$"],"dependencies":["a15cd07anov4a-h3"],"title":"$\\\\frac{s_1 **2}{5} + \\\\frac{s_2 ** 2}{5} + \\\\frac{s_3 **2}{5} - \\\\frac{(1017 + $$1137$$ + 915)**2}{15}$","text":"Round to the oneth place after the decimal.","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov4a-h5","type":"hint","dependencies":["a15cd07anov4a-h4"],"title":"Compute the Sum of Squares representing variation within samples due to chance. ($\\\\SS_{within})","text":"$\\\\SS_{within} $$=$$ \\\\SS_{total} - \\\\SS_{between}$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov4a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12859.6$$"],"dependencies":["a15cd07anov4a-h5"],"title":"Compute $\\\\SS_{total} $$=$$ \\\\sum $$x^2$$ - \\\\frac{(\\\\sum x) ** 2}{n}$","text":"$(216**2 + $${198}^2$$ + $${240}^2$$ + $${187}^2$$ + $${176}^2$$ + $${202}^2$$ + $${213}^2$$ + $${284}^2$$ + $${228}^2$$ + $${210}^2$$ + $${170}^2$$ + $${165}^2$$ + $${182}^2$$ + $${297}^2$$ + 201**2) - \\\\frac{(216 + $$198$$ + $$240$$ + $$187$$ + $$176$$ + $$202$$ + $$213$$ + $$284$$ + $$228$$ + $$210$$ + $$170$$ + $$165$$ + $$182$$ + $$197$$ + 201)**2}{5+5+5} $","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov4a-h7","type":"hint","dependencies":["a15cd07anov4a-h6"],"title":"What is the Degree of Freedom of Factor (Between)?","text":"$df_{numerator}$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov4a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a15cd07anov4a-h7"],"title":"$df_{numerator} $$=$$ k - 1$","text":"$$3$$ groups - $$1$$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov4a-h9","type":"hint","dependencies":["a15cd07anov4a-h8"],"title":"What is the Degree of Freedom of Error (within)?","text":"$df_{denominator}$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov4a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a15cd07anov4a-h9"],"title":"$df_{denominator} $$=$$ $$n$$ - k$","text":"$$15$$ total data - $$3$$ groups","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov4a-h11","type":"hint","dependencies":["a15cd07anov4a-h10"],"title":"Compute the Mean Square of Factor (Between)","text":"Round to the fourth place after the decimal.","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov4a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2469.6$$"],"dependencies":["a15cd07anov4a-h11"],"title":"MS(Factor) $$=$$ SS(Factor) / (k - 1)","text":"$$4939.2$$ / $$2$$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov4a-h13","type":"hint","dependencies":["a15cd07anov4a-h12"],"title":"Compute the Mean Square of Error (Within)","text":"Round to the third place after the decimal.","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov4a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1071.63$$"],"dependencies":["a15cd07anov4a-h13"],"title":"MS(Error) $$=$$ SS(Error) / (n - k)","text":"$$12859.6$$ / $$12$$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov4a-h15","type":"hint","dependencies":["a15cd07anov4a-h14"],"title":"Compute the F-statistics","text":"Round to the fourth place after the decimal.","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov4a-h16","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.3045$$"],"dependencies":["a15cd07anov4a-h15"],"title":"F $$=$$ MS(Factor) / MS(Error)","text":"$$2469.6$$ / $$1071.63$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"a15cd07anov5","title":"Soccer Teams","body":"Use the following information to answer the next eight exercises. Girls from four different soccer teams are to be tested for mean goals scored per game. The entries in the table are the goals per game for the different teams. The one-way ANOVA results are shown in Table 13.14.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"13.2 The F Distribution and the F-Ratio","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a15cd07anov5a","stepAnswer":["$$3.5258$$"],"problemType":"TextBox","stepTitle":"Use the table to construct the hypothesis test. What is the F-statistics?","stepBody":"Round to the fourth place after the decimal.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3.5258$$","hints":{"DefaultPathway":[{"id":"a15cd07anov5a-h1","type":"hint","dependencies":[],"title":"What is the sum of the values in the jth group?","text":"Sum of value in each column.","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a15cd07anov5a-h1"],"title":"Sum of value for group $$1$$.","text":"Add up the value in column $$1$$.","variabilization":{},"oer":"","license":"","subHints":[{"id":"a15cd07anov5a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":[],"title":"Sum of value for group $$2$$.","text":"Add up the value in column $$2$$.","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov5a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Sum of value for group $$3$$.","text":"Add up the value in column $$3$$.","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov5a-h2-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":[],"title":"Sum of value for group $$4$$.","text":"Add up the value in column $$4$$.","variabilization":{},"oer":"","license":""}]},{"id":"a15cd07anov5a-h3","type":"hint","dependencies":["a15cd07anov5a-h2"],"title":"Compute the Sum of Squares representing variation among the different samples. ($\\\\SS_{between}$)","text":"$\\\\SS_{between}$ $$=$$ $\\\\sum[\\\\frac{(\\\\sum s_j)**2}{n_j}] - \\\\frac{(\\\\sum s_j)**2}{n}$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25.75$$"],"dependencies":["a15cd07anov5a-h3"],"title":"$\\\\frac{s_1 **2}{5} + \\\\frac{s_2 ** 2}{5} + \\\\frac{s_3 **2}{5} + \\\\frac{s_4 **2}{5} - \\\\frac{(8 + $$15$$ + $$2$$ + 16)**2}{20}$","text":"Round to the fourth place after the decimal.","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov5a-h5","type":"hint","dependencies":["a15cd07anov5a-h4"],"title":"Compute the Sum of Squares representing variation within samples due to chance. ($\\\\SS_{within})","text":"$\\\\SS_{within} $$=$$ \\\\SS_{total} - \\\\SS_{between}$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov5a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$38.95$$"],"dependencies":["a15cd07anov5a-h5"],"title":"Compute $\\\\SS_{total} $$=$$ \\\\sum $$x^2$$ - \\\\frac{(\\\\sum x) ** 2}{n}$","text":"$(1**2 + $$2^2$$ + $$0^2$$ + $$3^2$$ + $$2^2$$ + $$2^2$$ + $$3^2$$ + $$2^2$$ + $$4^2$$ + $$4^2$$ + $$0^2$$ + $$1^2$$ + $$1^2$$ + $$0^2$$ + $$0^2$$ + $$3^2$$ + $$4^2$$ + $$4^2$$ + $$3^2$$ + 2**2) - \\\\frac{(1 + $$2$$ + $$0$$ + $$3$$ + $$2$$ + $$2$$ + $$3$$ + $$2$$ + $$4$$ + $$4$$ + $$0$$ + $$1$$ + $$1$$ + $$0$$ + $$0$$ + $$3$$ + $$4$$ + $$4$$ + $$3$$ + 2)**2}{5+5+5+5} $","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov5a-h7","type":"hint","dependencies":["a15cd07anov5a-h6"],"title":"What is the Degree of Freedom of Factor (Between)?","text":"$df_{numerator}$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov5a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a15cd07anov5a-h7"],"title":"$df_{numerator} $$=$$ k - 1$","text":"$$4$$ groups - $$1$$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov5a-h9","type":"hint","dependencies":["a15cd07anov5a-h8"],"title":"What is the Degree of Freedom of Error (within)?","text":"$df_{denominator}$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov5a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["a15cd07anov5a-h9"],"title":"$df_{denominator} $$=$$ $$n$$ - k$","text":"$$20$$ total data - $$4$$ groups","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov5a-h11","type":"hint","dependencies":["a15cd07anov5a-h10"],"title":"Compute the Mean Square of Factor (Between)","text":"Round to the fourth place after the decimal.","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov5a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8.5833$$"],"dependencies":["a15cd07anov5a-h11"],"title":"MS(Factor) $$=$$ SS(Factor) / (k - 1)","text":"$$25.75$$ / $$3$$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov5a-h13","type":"hint","dependencies":["a15cd07anov5a-h12"],"title":"Compute the Mean Square of Error (Within)","text":"Round to the third place after the decimal.","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov5a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.4344$$"],"dependencies":["a15cd07anov5a-h13"],"title":"MS(Error) $$=$$ SS(Error) / (n - k)","text":"$$38.95$$ / $$16$$","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov5a-h15","type":"hint","dependencies":["a15cd07anov5a-h14"],"title":"Compute the F-statistics","text":"Round to the fourth place after the decimal.","variabilization":{},"oer":"","license":""},{"id":"a15cd07anov5a-h16","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3.5258$$"],"dependencies":["a15cd07anov5a-h15"],"title":"F $$=$$ MS(Factor) / MS(Error)","text":"$$8.5833$$ / $$2.4344$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"a161552dividing1","title":"Dividing Polynomial","body":"In the following exercises, divide the monomials.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a161552dividing1a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{15r^4 s^9}{15r^4 s^9}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a161552dividing1a-h1","type":"hint","dependencies":[],"title":"Use the Identity property of mutiplication","text":"For any nonzero real number $$n$$, $$\\\\frac{n}{n}=1$$. Let $$n=15r^4 s^9$$. we get $$\\\\frac{15r^4 s^9}{15r^4 s^9}=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552dividing10","title":"Dividing Polynomial","body":"In the following exercises, divide each polynomial by the monomials.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a161552dividing10a","stepAnswer":["$$\\\\left(-5y\\\\right)-3+\\\\frac{1}{4y}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{20y^2+12y-1}{\\\\left(-4y\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(-5y\\\\right)-3+\\\\frac{1}{4y}$$","hints":{"DefaultPathway":[{"id":"a161552dividing10a-h1","type":"hint","dependencies":[],"title":"Use distributive property","text":"Use distributive property to rewrite the expression $$\\\\frac{20y^2+12y-1}{\\\\left(-4y\\\\right)}=\\\\frac{20y^2}{\\\\left(-4y\\\\right)}+\\\\frac{12y}{\\\\left(-4y\\\\right)}+\\\\frac{\\\\left(-1\\\\right)}{\\\\left(-4y\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing10a-h2","type":"hint","dependencies":["a161552dividing10a-h1"],"title":"Use distributive property","text":"As a reminder, the distributive law states $$\\\\left(a+b+d\\\\right) c=a c+b c+d c$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing10a-h3","type":"hint","dependencies":["a161552dividing10a-h2"],"title":"Use distributive property","text":"in this case, we can see divided by $$\\\\left(-4y\\\\right)$$ same as multiply by $$\\\\frac{1}{\\\\left(-4y\\\\right)}$$, $$a=20y^2$$ , $$b=12y$$, $$d=$$ $$-1$$, $$c=\\\\frac{1}{\\\\left(-4y\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5y$$"],"dependencies":["a161552dividing10a-h3"],"title":"Simplify each term in expression","text":"$$\\\\frac{\\\\operatorname{Simplify}\\\\left(20y^2\\\\right)}{\\\\left(-4y\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a161552dividing10a-h4"],"title":"Simplify each term in expression","text":"Simplify $$\\\\frac{12y}{\\\\left(-4y\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing10a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4y}$$"],"dependencies":["a161552dividing10a-h5"],"title":"Simplify each term in expression","text":"Simplify $$\\\\frac{\\\\left(-1\\\\right)}{\\\\left(-4y\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing10a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4y}$$"],"dependencies":["a161552dividing10a-h6"],"title":"Simplify $$\\\\frac{\\\\left(-1\\\\right)}{\\\\left(-4y\\\\right)}$$","text":"the only common factor in both numerator and denominator is $$-1$$, so the only thing we can do to simplify this term is divided both numerator and denominator by $$-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing10a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(-2x\\\\right)-1+\\\\frac{4}{5x}$$"],"dependencies":["a161552dividing10a-h7"],"title":"Combine the simplified terms","text":"As our last step, we can put the simplified terms back to the expression $$\\\\frac{20y^2}{\\\\left(-4y\\\\right)}+\\\\frac{12y}{\\\\left(-4y\\\\right)}+\\\\frac{\\\\left(-1\\\\right)}{\\\\left(-4y\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividing10a-h8-s1","type":"hint","dependencies":[],"title":"Combine the simplified terms","text":"in the expression $$\\\\frac{20y^2}{\\\\left(-4y\\\\right)}+\\\\frac{12y}{\\\\left(-4y\\\\right)}+\\\\frac{\\\\left(-1\\\\right)}{\\\\left(-4y\\\\right)}$$, replace $$\\\\frac{20y^2}{\\\\left(-4y\\\\right)}$$ with-5*y, replace $$\\\\frac{12y}{\\\\left(-4y\\\\right)}$$ with $$-3$$, and replace $$\\\\frac{\\\\left(-1\\\\right)}{\\\\left(-4y\\\\right)}$$ with $$\\\\frac{1}{4y}$$, then we yield a new simplified expression","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a161552dividing3","title":"Dividing Polynomial","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a161552dividing3a","stepAnswer":["$$\\\\frac{2b^3}{\\\\left(-3a^5\\\\right)}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{18a^4 b^8}{\\\\left(-27a^9 b^5\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2b^3}{\\\\left(-3a^5\\\\right)}$$","hints":{"DefaultPathway":[{"id":"a161552dividing3a-h1","type":"hint","dependencies":[],"title":"Rearrange expression and group like terms","text":"Use the associative and commutative property of multiplication to group like terms together","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$a^9$$"],"dependencies":["a161552dividing3a-h1"],"title":"Rearrange expression and group like terms","text":"In this expression, what is the like term of $$a^4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing3a-h3","type":"hint","dependencies":["a161552dividing3a-h2"],"title":"Rearrange expression and group like terms","text":"$$18$$ and $$-27$$ are consider as like terms because they are both real numbers. $$a^4$$ and $$a^9$$ are consider like terms because they are different power of a. Similar for $$b^8$$ and $$b^5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing3a-h4","type":"hint","dependencies":["a161552dividing3a-h3"],"title":"Simplify each group","text":"We can rewrite the given expression in a new way by grouping the like terms using parenthese, $$\\\\frac{18a^4 b^8}{\\\\left(-27a^9 b^5\\\\right)}=\\\\frac{18}{-27} \\\\frac{a^4}{a^9} \\\\frac{b^8}{b^5}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-2}{3}$$"],"dependencies":["a161552dividing3a-h4"],"title":"Simplify each group","text":"Simplify $$\\\\frac{18}{-27}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing3a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{a^5}$$"],"dependencies":["a161552dividing3a-h5"],"title":"Simplify each group","text":"Simplify $$\\\\frac{a^4}{a^9}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing3a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$b^3$$"],"dependencies":["a161552dividing3a-h6"],"title":"Simplify each group","text":"Simplify $$\\\\frac{b^8}{b^5}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividing3a-h7-s1","type":"hint","dependencies":[],"title":"Rules for simplify power","text":"Given a real number $$x$$, and m,n integers, $$\\\\frac{a^m}{a^n}=a^{m-n}$$ and $$a^m a^n=a^{m+n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a161552dividing3a-h8","type":"hint","dependencies":["a161552dividing3a-h7"],"title":"Multiply each group together","text":"Then, replace the each group with its simplified form we get $$\\\\left(-\\\\frac{2}{3}\\\\right) \\\\frac{1}{a^5} b^3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing3a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2b^3}{\\\\left(-3a^5\\\\right)}$$"],"dependencies":["a161552dividing3a-h8"],"title":"Fraction mutiplication Rule","text":"The final step is take $$\\\\left(-\\\\frac{2}{3}\\\\right) \\\\frac{1}{a^5} b^3$$ and turn it into fraction form using fraction multiplication.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividing3a-h9-s1","type":"hint","dependencies":[],"title":"Fraction mutiplication Rule","text":"As a reminder, given fractions $$\\\\frac{a}{b}$$ and $$\\\\frac{d}{c}$$ where a, $$d$$ are real number, and $$b$$, c are non zero real number, $$\\\\frac{a}{b} \\\\frac{d}{c}=\\\\frac{a d}{b c}$$. we can write $$b^3$$ as $$\\\\frac{b^3}{1}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a161552dividing4","title":"Dividing Polynomial","body":"In the following exercises, divide the monomials.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a161552dividing4a","stepAnswer":["$$\\\\frac{3y^3}{\\\\left(-4x^3\\\\right)}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{45x^5 x^9}{\\\\left(-60x^8 x^6\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3y^3}{\\\\left(-4x^3\\\\right)}$$","hints":{"DefaultPathway":[{"id":"a161552dividing4a-h1","type":"hint","dependencies":[],"title":"Rearrange expression and group like terms","text":"use associative and commutative property of multiplication to group like terms together","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^8$$"],"dependencies":["a161552dividing4a-h1"],"title":"Rearrange expression and group like terms","text":"In this expression, what is the like term of $$x^5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing4a-h3","type":"hint","dependencies":["a161552dividing4a-h2"],"title":"Rearrange expression and group like terms","text":"$$45$$ and $$-60$$ are consider as like terms because they are both real numbers. $$x^5$$ and $$x^8$$ are consider like terms because they are different power of a. Similar for $$y^9$$ and $$y^6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing4a-h4","type":"hint","dependencies":["a161552dividing4a-h3"],"title":"Simplify each group","text":"We can rewrite the given expression in a new way by grouping the like terms using parenthese, $$\\\\frac{45x^5 x^9}{\\\\left(-60x^8 x^6\\\\right)}=\\\\frac{45}{-60} \\\\frac{x^5}{x^8} \\\\frac{y^9}{y^6}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing4a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-3}{4}$$"],"dependencies":["a161552dividing4a-h4"],"title":"Simplify each group","text":"Simplify $$\\\\frac{45}{-60}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing4a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{x^3}$$"],"dependencies":["a161552dividing4a-h5"],"title":"Simplify each group","text":"Simplify $$\\\\frac{x^5}{x^8}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing4a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y^3$$"],"dependencies":["a161552dividing4a-h6"],"title":"Simplify each group","text":"Simplify $$\\\\frac{y^9}{y^6}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividing4a-h7-s1","type":"hint","dependencies":[],"title":"Rules for simplify power","text":"Given a real number $$x$$, and m,n integers, $$\\\\frac{a^m}{a^n}=a^{m-n}$$ and $$a^m a^n=a^{m+n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a161552dividing4a-h8","type":"hint","dependencies":["a161552dividing4a-h7"],"title":"Multiply each group together","text":"Then, replace the each group with its simplified form we get $$\\\\left(-\\\\frac{3}{4}\\\\right) \\\\frac{1}{x^3} y^3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing4a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3y^3}{\\\\left(-4x^3\\\\right)}$$"],"dependencies":["a161552dividing4a-h8"],"title":"Fraction mutiplication Rule","text":"The final step is take $$\\\\left(-\\\\frac{3}{4}\\\\right) \\\\frac{1}{x^3} y^3$$ and turn it into fraction form using fraction multiplication.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividing4a-h9-s1","type":"hint","dependencies":[],"title":"Fraction mutiplication Rule","text":"As a reminder, given fractions $$\\\\frac{a}{b}$$ and $$\\\\frac{d}{c}$$ where a, $$d$$ are real number, and $$b$$, c are non zero real number, $$\\\\frac{a}{b} \\\\frac{d}{c}=\\\\frac{a d}{b c}$$. we can write $$y^3$$ as $$\\\\frac{y^3}{1}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a161552dividing5","title":"Dividing Polynomial","body":"In the following exercises, divide each polynomial by the monomials.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a161552dividing5a","stepAnswer":["$$3n^3+2n^2$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{9n^4+6n^3}{3n}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3n^3+2n^2$$","hints":{"DefaultPathway":[{"id":"a161552dividing5a-h1","type":"hint","dependencies":[],"title":"Use distributive property","text":"Use distributive property to rewrite the expression $$\\\\frac{9n^4+6n^3}{3n}=\\\\frac{9n^4}{3n}+\\\\frac{6n^3}{3n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing5a-h2","type":"hint","dependencies":["a161552dividing5a-h1"],"title":"Use distributive property","text":"As a reminder, the distributive law states $$\\\\left(a+b\\\\right) c=a c+b c$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing5a-h3","type":"hint","dependencies":["a161552dividing5a-h2"],"title":"Use distributive property","text":"in this case, we can see divided by $$3n$$ as multiply by $$\\\\frac{1}{3n}$$, $$a=9n^4$$ , $$b=6n^3$$, $$c=\\\\frac{1}{3n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3n^3$$"],"dependencies":["a161552dividing5a-h3"],"title":"Simplify each term in expression","text":"Simplify $$\\\\frac{9n^4}{3n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2n^2$$"],"dependencies":["a161552dividing5a-h4"],"title":"Simplify each term in expression","text":"Simplify $$\\\\frac{6n^3}{3n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividing5a-h5-s1","type":"hint","dependencies":[],"title":"Simplify each term in expression","text":"As a reminder, the power rule states for any integer $$m$$, $$n$$ and real number a, $$\\\\frac{a^m}{a^n}=a^{m-n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a161552dividing5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3n^3+2n^2$$"],"dependencies":["a161552dividing5a-h2","a161552dividing5a-h3"],"title":"Combine the simplified terms","text":"As our last step, we can put the simplified terms back to the expression $$\\\\frac{9n^4}{3n}+\\\\frac{6n^3}{3n}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividing5a-h4-s1","type":"hint","dependencies":[],"title":"Combine the simplified terms","text":"$$\\\\frac{9n^4}{3n}$$ simplify into $$3n^3$$ $$\\\\frac{6n^3}{3n}$$ simplify into $$2n^2$$ in the expression $$\\\\frac{9n^4}{3n}+\\\\frac{6n^3}{3n}$$, replace $$\\\\frac{9n^4}{3n}$$ with $$3n^3$$ and $$\\\\frac{\\\\operatorname{replace}\\\\left(6n^3\\\\right)}{3n}$$ with $$2n^2$$, then we yield a new expression. What is the new expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a161552dividing6","title":"Dividing Polynomial","body":"In the following exercises, divide each polynomial by the monomials.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a161552dividing6a","stepAnswer":["$$4x^2+3x$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{8x^3+6x^2}{2x}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4x^2+3x$$","hints":{"DefaultPathway":[{"id":"a161552dividing6a-h1","type":"hint","dependencies":[],"title":"Use distributive property","text":"Use distributive property to rewrite the expression $$\\\\frac{8x^3+6x^2}{2x}=\\\\frac{8x^3}{2x}+\\\\frac{6x^2}{2x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing6a-h2","type":"hint","dependencies":["a161552dividing6a-h1"],"title":"Use distributive property","text":"As a reminder, the distributive law states $$\\\\left(a+b\\\\right) c=a c+b c$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing6a-h3","type":"hint","dependencies":["a161552dividing6a-h2"],"title":"Use distributive property","text":"in this case, we can see divided by $$2x$$ as multiply by $$\\\\frac{1}{2x}$$, $$a=8x^3$$ , $$b=6x^2$$, $$c=\\\\frac{1}{2x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4x^2$$"],"dependencies":["a161552dividing6a-h3"],"title":"Simplify each term in expression","text":"Simplify $$\\\\frac{8x^3}{2x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x$$"],"dependencies":["a161552dividing6a-h4"],"title":"Simplify each term in expression","text":"Simplify $$\\\\frac{6x^2}{2x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing6a-h6","type":"hint","dependencies":["a161552dividing6a-h5"],"title":"Simplify each term in expression","text":"As a reminder, the power rule states for any integer $$m$$, $$n$$ and real number a, $$\\\\frac{a^m}{a^n}=a^{m-n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing6a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4x^2+3x$$"],"dependencies":["a161552dividing6a-h6","a161552dividing6a-h5"],"title":"Combine the simplified terms","text":"As our last step, we can put the simplified terms back to the $$\\\\operatorname{expression}\\\\left(\\\\frac{8x^3}{2x}\\\\right)+\\\\frac{6x^2}{2x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividing6a-h7-s1","type":"hint","dependencies":[],"title":"Combine the simplified terms","text":"in the expression $$\\\\frac{8x^3}{2x}+\\\\frac{6x^2}{2x}$$, replace $$\\\\frac{8x^3}{2x}$$ with $$4x^2$$, and $$\\\\frac{\\\\operatorname{replace}\\\\left(6x^2\\\\right)}{2x}$$ with $$3x$$ then we yield a new expression","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a161552dividing7","title":"Dividing Polynomial","body":"In the following exercises, divide each polynomial by the monomials.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a161552dividing7a","stepAnswer":["$$\\\\left(-9m^2\\\\right)+6m$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{63m^4-42m^3}{\\\\left(-7m^2\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(-9m^2\\\\right)+6m$$","hints":{"DefaultPathway":[{"id":"a161552dividing7a-h1","type":"hint","dependencies":[],"title":"Use distributive property","text":"Use distributive property to rewrite the expression $$\\\\frac{63m^4-42m^3}{\\\\left(-7m^2\\\\right)}=\\\\frac{63m^4}{\\\\left(-7m^2\\\\right)}-\\\\frac{42m^3}{\\\\left(-7m^2\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing7a-h2","type":"hint","dependencies":["a161552dividing7a-h1"],"title":"Use distributive property","text":"As a reminder, the distributive law states $$\\\\left(a-b\\\\right) c=a c-b c$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing7a-h3","type":"hint","dependencies":["a161552dividing7a-h2"],"title":"Use distributive property","text":"in this case, we can see divided by $$\\\\left(-7m^2\\\\right)$$ as multiply by $$\\\\frac{1}{\\\\left(-7m^2\\\\right)}$$, $$a=63m^4$$ , $$b=42m^3$$, $$c=\\\\frac{1}{\\\\left(-7m^2\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9m^2$$"],"dependencies":["a161552dividing7a-h3"],"title":"Simplify each term in expression","text":"$$\\\\operatorname{Simplify}\\\\left(\\\\frac{63m^4}{\\\\left(-7m^2\\\\right)}\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing7a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6m$$"],"dependencies":["a161552dividing7a-h4"],"title":"Simplify each term in expression","text":"Simplify $$\\\\frac{42m^3}{\\\\left(-7m^2\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing7a-h6","type":"hint","dependencies":["a161552dividing7a-h5"],"title":"Simplify each term in expression","text":"As a reminder, the power rule states for any integer $$m$$, $$n$$ and real number a, $$\\\\frac{a^m}{a^n}=a^{m-n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing7a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(-9m^2\\\\right)+6m$$"],"dependencies":["a161552dividing7a-h6","a161552dividing7a-h5"],"title":"Combine the simplified terms","text":"As our last step, we can put the simplified terms back to the $$\\\\operatorname{expression}\\\\left(\\\\frac{63m^4}{\\\\left(-7m^2\\\\right)}\\\\right)-\\\\frac{42m^3}{\\\\left(-7m^2\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividing7a-h7-s1","type":"hint","dependencies":[],"title":"Combine the simplified terms","text":"in the expression $$\\\\frac{63m^4}{\\\\left(-7m^2\\\\right)}-\\\\frac{42m^3}{\\\\left(-7m^2\\\\right)}$$, replace $$\\\\frac{63m^4}{\\\\left(-7m^2\\\\right)}$$ with $$-9m^2$$, and replace $$\\\\frac{42m^3}{\\\\left(-7m^2\\\\right)}$$ with $$-6m$$ then we yield a new expression","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a161552dividing8","title":"Dividing Polynomial","body":"In the following exercises, divide each polynomial by the monomials.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a161552dividing8a","stepAnswer":["$$\\\\left(-6y^2\\\\right)+3y$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{48y^4-24y^3}{\\\\left(-8y^2\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(-6y^2\\\\right)+3y$$","hints":{"DefaultPathway":[{"id":"a161552dividing8a-h1","type":"hint","dependencies":[],"title":"Use distributive property","text":"Use distributive property to rewrite the expression $$\\\\frac{48y^4-24y^3}{\\\\left(-8y^2\\\\right)}=\\\\frac{48y^4}{\\\\left(-8y^2\\\\right)}-\\\\frac{24y^3}{\\\\left(-8y^2\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing8a-h2","type":"hint","dependencies":["a161552dividing8a-h1"],"title":"Use distributive property","text":"As a reminder, the distributive law states $$\\\\left(a-b\\\\right) c=a c-b c$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing8a-h3","type":"hint","dependencies":["a161552dividing8a-h2"],"title":"Use distributive property","text":"in this case, we can see divided by $$\\\\left(-8y^2\\\\right)$$ same as multiply by $$\\\\frac{1}{\\\\left(-8y^2\\\\right)}$$, $$a=48y^4$$ , $$b=24y^3$$, $$c=\\\\frac{1}{\\\\left(-8y^2\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6y^2$$"],"dependencies":["a161552dividing8a-h3"],"title":"Simplify each term in expression","text":"Simplify $$\\\\frac{48y^4}{\\\\left(-8y^2\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing8a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3y$$"],"dependencies":["a161552dividing8a-h4"],"title":"Simplify each term in expression","text":"Simplify $$\\\\frac{24y^3}{\\\\left(-8y^2\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing8a-h6","type":"hint","dependencies":["a161552dividing8a-h5"],"title":"Simplify each term in expression","text":"As a reminder, the power rule states for any integer $$m$$, $$n$$ and real number a, $$\\\\frac{a^m}{a^n}=a^{m-n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing8a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(-6y^2\\\\right)+3y$$"],"dependencies":["a161552dividing8a-h6","a161552dividing8a-h5"],"title":"Combine the simplified terms","text":"As our last step, we can put the simplified terms back to the expression $$\\\\frac{48y^4}{\\\\left(-8y^2\\\\right)}-\\\\frac{24y^3}{\\\\left(-8y^2\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividing8a-h7-s1","type":"hint","dependencies":[],"title":"Combine the simplified terms","text":"in the expression $$\\\\frac{48y^4}{\\\\left(-8y^2\\\\right)}-\\\\frac{24y^3}{\\\\left(-8y^2\\\\right)}$$, replace $$\\\\frac{48y^4}{\\\\left(-8y^2\\\\right)}$$ with $$-6y^2$$, and replace $$\\\\frac{24y^3}{\\\\left(-8y^2\\\\right)}$$ with $$-3y$$ then we yield a new expression","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a161552dividing9","title":"Dividing Polynomial","body":"In the following exercises, divide each polynomial by the monomials.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a161552dividing9a","stepAnswer":["$$\\\\left(-2x\\\\right)-1+\\\\frac{4}{5x}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{10x^2+5x-4}{\\\\left(-5x\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(-2x\\\\right)-1+\\\\frac{4}{5x}$$","hints":{"DefaultPathway":[{"id":"a161552dividing9a-h1","type":"hint","dependencies":[],"title":"Use distributive property","text":"Use distributive property to rewrite the expression $$\\\\frac{10x^2+5x-4}{\\\\left(-5x\\\\right)}=\\\\frac{10x^2}{\\\\left(-5x\\\\right)}+\\\\frac{5x}{\\\\left(-5x\\\\right)}+\\\\frac{\\\\left(-4\\\\right)}{\\\\left(-5x\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing9a-h2","type":"hint","dependencies":["a161552dividing9a-h1"],"title":"Use distributive property","text":"As a reminder, the distributive law states $$\\\\left(a+b+d\\\\right) c=a c+b c+d c$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing9a-h3","type":"hint","dependencies":["a161552dividing9a-h2"],"title":"Use distributive property","text":"in this case, we can see divided by $$\\\\left(-5x\\\\right)$$ same as multiply by $$\\\\frac{1}{\\\\left(-5x\\\\right)}$$, $$a=10x^2$$ , $$b=5x$$, $$d=$$ $$-4$$, $$c=\\\\frac{1}{\\\\left(-5x\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2x$$"],"dependencies":["a161552dividing9a-h3"],"title":"Simplify each term in expression","text":"Simplify $$\\\\frac{10x^2}{\\\\left(-5x\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing9a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a161552dividing9a-h4"],"title":"Simplify each term in expression","text":"Simplify $$\\\\frac{5x}{\\\\left(-5x\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing9a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{4}{5x}$$"],"dependencies":["a161552dividing9a-h5"],"title":"Simplify each term in expression","text":"Simplify $$\\\\frac{\\\\left(-4\\\\right)}{\\\\left(-5x\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing9a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{4}{5x}$$"],"dependencies":["a161552dividing9a-h6"],"title":"Simplify $$\\\\frac{\\\\left(-4\\\\right)}{\\\\left(-5x\\\\right)}$$","text":"the only common factor in both numerator and denominator is $$-1$$, so the only thing we can do to simplify this term is divided both numerator and denominator by $$-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividing9a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(-2x\\\\right)-1+\\\\frac{4}{5x}$$"],"dependencies":["a161552dividing9a-h7","a161552dividing9a-h6","a161552dividing9a-h5"],"title":"Combine the simplified terms","text":"As our last step, we can put the simplified terms back to the expression $$\\\\frac{10x^2}{\\\\left(-5x\\\\right)}+\\\\frac{5x}{\\\\left(-5x\\\\right)}+\\\\frac{\\\\left(-4\\\\right)}{\\\\left(-5x\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividing9a-h5-s1","type":"hint","dependencies":[],"title":"Combine the simplified terms","text":"in the expression $$\\\\frac{10x^2}{\\\\left(-5x\\\\right)}+\\\\frac{5x}{\\\\left(-5x\\\\right)}+\\\\frac{\\\\left(-4\\\\right)}{\\\\left(-5x\\\\right)}$$, replace $$\\\\frac{10x^2}{\\\\left(-5x\\\\right)}$$ with $$-2x$$, replace $$\\\\frac{\\\\left(-4\\\\right)}{\\\\left(-5x\\\\right)}$$ with $$\\\\frac{4}{5x}$$, and replace $$\\\\frac{5x}{\\\\left(-5x\\\\right)}$$ with $$-1$$, then we yield a new expression","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a161552dividingpoly1","title":"Dividing Polynomials","body":"Divide each polynomial by the binomial.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a161552dividingpoly1a","stepAnswer":["$$x+3-\\\\frac{8}{x+8}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{x^2+11x+16}{x+8}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x+3-\\\\frac{8}{x+8}$$","hints":{"DefaultPathway":[{"id":"a161552dividingpoly1a-h1","type":"hint","dependencies":[],"title":"Division","text":"Divide the leading term in the dividend by the leading term of the divisor. $$\\\\frac{x^2}{x}=x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly1a-h2","type":"hint","dependencies":["a161552dividingpoly1a-h1"],"title":"Multiplication","text":"Multiply the result by the divisor. $$x \\\\left(x+8\\\\right)=x^2+8x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly1a-h3","type":"hint","dependencies":["a161552dividingpoly1a-h2"],"title":"Subtraction","text":"Subtract the divident from the result. $$x^2+11x+16-x^2+8x=3x+16$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly1a-h4","type":"hint","dependencies":["a161552dividingpoly1a-h3"],"title":"Division","text":"Divide the leading term of the remainder. $$\\\\frac{3x}{x}=3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly1a-h5","type":"hint","dependencies":["a161552dividingpoly1a-h4"],"title":"Multiplication","text":"Multiply the result by the divisor. $$3\\\\left(x+8\\\\right)=3x+24$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly1a-h6","type":"hint","dependencies":["a161552dividingpoly1a-h5"],"title":"Subtraction","text":"Subtract the remainder from the new result. $$3x+16-3x+24=-8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly1a-h7","type":"hint","dependencies":["a161552dividingpoly1a-h6"],"title":"Remainder","text":"Since the degree of the remainder is less than the divisor, our last term is $$\\\\frac{-8}{x+8}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly1a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+3-\\\\frac{8}{x+8}$$"],"dependencies":["a161552dividingpoly1a-h7"],"title":"Division","text":"What is our final result after combining all the terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly1a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+3-\\\\frac{8}{x+8}$$"],"dependencies":["a161552dividingpoly1a-h8"],"title":"Division","text":"$$x+3-\\\\frac{8}{x+8}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552dividingpoly10","title":"Dividing Polynomials","body":"Divide each polynomial function by the binomial function and solve for a given $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a161552dividingpoly10a","stepAnswer":["$$\\\\frac{1}{3}$$"],"problemType":"TextBox","stepTitle":"$$f(x)=x^2-3x+2$$, $$g(x)=x+3$$ Find $$\\\\frac{f}{g}$$ when $$x=3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{3}$$","hints":{"DefaultPathway":[{"id":"a161552dividingpoly10a-h1","type":"hint","dependencies":[],"title":"Rewriting the equation","text":"Remember to fill in missing coefficients with placeholder 0\'s if possible to make the equation easier to read and work with.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a161552dividingpoly10a-h1"],"title":"Division","text":"What is the largest degree in the polynomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly10a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Division","text":"What is the largest degree in the binomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a161552dividingpoly10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x$$"],"dependencies":["a161552dividingpoly10a-h2"],"title":"Multiplication","text":"What can we multiply $$x$$ by to get $$x^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly10a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2+3x$$"],"dependencies":[],"title":"Multiplication","text":"What is $$x \\\\left(x+3\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a161552dividingpoly10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(-6x\\\\right)+2$$"],"dependencies":["a161552dividingpoly10a-h3"],"title":"Subtraction","text":"What is $$x^2-3x+2-x^2+3x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Division","text":"What is the largest degree in the remainder?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly10a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6$$"],"dependencies":["a161552dividingpoly10a-h5"],"title":"Multiplication","text":"What can we multiply $$x$$ by to get $$\\\\left(-6x\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly10a-h6-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(-6x\\\\right)-18$$"],"dependencies":[],"title":"Multiplication","text":"What is $$-6\\\\left(x+3\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a161552dividingpoly10a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a161552dividingpoly10a-h6"],"title":"Subtraction","text":"What is $$\\\\left(-6x\\\\right)+2-\\\\left(-6x\\\\right)-18$$? $$\\\\left(-6x\\\\right)+2$$ is the remainder from the previous subtraction step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly10a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{20}{x+3}$$"],"dependencies":["a161552dividingpoly10a-h7"],"title":"Remainder","text":"What is the remainder? If there is no remainder enter $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly10a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x-6+\\\\frac{20}{x+3}$$"],"dependencies":["a161552dividingpoly10a-h8"],"title":"Combining","text":"Combine all the terms that we used to multiply the divisor by into a single expression along with the divisor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly10a-h9-s1","type":"hint","dependencies":[],"title":"Combining","text":"Reminder: The terms we had were, $$x$$, $$-6$$, $$\\\\frac{20}{x+3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a161552dividingpoly10a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":[],"title":"Plugging it in","text":"What is $$x-6+\\\\frac{20}{x}+3$$ when $$x=3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552dividingpoly2","title":"Dividing Polynomials","body":"Divide each polynomial by the binomial.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a161552dividingpoly2a","stepAnswer":["$$3x^2-2x+2+\\\\frac{2}{x+1}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{3x^3+x^2+4}{x+1}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3x^2-2x+2+\\\\frac{2}{x+1}$$","hints":{"DefaultPathway":[{"id":"a161552dividingpoly2a-h1","type":"hint","dependencies":[],"title":"Rewriting the equation","text":"Remember to fill in missing coefficients with placeholder 0\'s if possible to make the equation easier to read and work with.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a161552dividingpoly2a-h1"],"title":"Division","text":"What is the largest degree in the polynomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly2a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Division","text":"What is the largest degree in the binomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a161552dividingpoly2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x^2$$"],"dependencies":["a161552dividingpoly2a-h2"],"title":"Multiplication","text":"What can we multiply $$x$$ by to get $$3x^3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly2a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x^3+3x^2$$"],"dependencies":[],"title":"Multiplication","text":"What is $$3x^2 \\\\left(x+1\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a161552dividingpoly2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(-2x^2\\\\right)+4$$"],"dependencies":["a161552dividingpoly2a-h3"],"title":"Subtraction","text":"What is $$3x^3+x^2+4-3x^3+3x^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly2a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Division","text":"What is the largest degree in the remainder?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2x$$"],"dependencies":["a161552dividingpoly2a-h5"],"title":"Multiplication","text":"What can we multiply $$x$$ by to get $$\\\\left(-2x^2\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly2a-h6-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2x^2-2x$$"],"dependencies":[],"title":"Multiplication","text":"What is $$\\\\left(-2x\\\\right) \\\\left(x+1\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a161552dividingpoly2a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x+4$$"],"dependencies":["a161552dividingpoly2a-h6"],"title":"Subtraction","text":"What is $$\\\\left(-2x^2+4\\\\right)-\\\\left(-2x^2-2x\\\\right)$$ (-2*(x**2)+4 is the remainder from the previous subtraction step.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly2a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a161552dividingpoly2a-h7"],"title":"Division","text":"What is the largest degree in the remainder?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly2a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Multiplication","text":"What can we multiply $$x$$ by to get $$2x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly2a-h9-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x+2$$"],"dependencies":[],"title":"Multiplication","text":"What is $$2\\\\left(x+1\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a161552dividingpoly2a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a161552dividingpoly2a-h9"],"title":"Subtraction","text":"What is $$2x+4-2x+2$$ ((2*x)+4 is the remainder from the previous subtraction step.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly2a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{x+1}$$"],"dependencies":["a161552dividingpoly2a-h10"],"title":"Remainder","text":"What is the remainder? If there is no remainder enter $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly2a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x^2-2x+2+\\\\frac{2}{x+1}$$"],"dependencies":["a161552dividingpoly2a-h11"],"title":"Combining","text":"Combine all the terms that we used to multiply the divisor by into a single expression along with the divisor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly2a-h12-s1","type":"hint","dependencies":[],"title":"Combining","text":"Reminder: The terms we had were, $$3x^2$$, $$-2x$$, $$2$$, and $$\\\\frac{2}{x+1}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a161552dividingpoly3","title":"Dividing Polynomials","body":"Divide each polynomial by the binomial.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a161552dividingpoly3a","stepAnswer":["$$2x^2-6x+8+\\\\frac{4}{x+3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2x^3-10x+28}{x+3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2x^2-6x+8+\\\\frac{4}{x+3}$$","hints":{"DefaultPathway":[{"id":"a161552dividingpoly3a-h1","type":"hint","dependencies":[],"title":"Rewriting the equation","text":"Remember to fill in missing coefficients with placeholder 0\'s if possible to make the equation easier to read and work with.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a161552dividingpoly3a-h1"],"title":"Division","text":"What is the largest degree in the polynomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly3a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Division","text":"What is the largest degree in the binomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a161552dividingpoly3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x^2$$"],"dependencies":["a161552dividingpoly3a-h2"],"title":"Multiplication","text":"What can we multiply $$x$$ by to get $$2x^3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly3a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x^3+6x^2$$"],"dependencies":[],"title":"Multiplication","text":"What is $$2x^2 \\\\left(x+3\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a161552dividingpoly3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(-6x^2\\\\right)-10x+28$$"],"dependencies":["a161552dividingpoly3a-h3"],"title":"Subtraction","text":"What is $$2x^3-10x+28-2x^3+6x^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Division","text":"What is the largest degree in the remainder?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly3a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6x$$"],"dependencies":["a161552dividingpoly3a-h5"],"title":"Multiplication","text":"What can we multiply $$x$$ by to get $$\\\\left(-6x^2\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly3a-h6-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(-6x^2\\\\right)-18x$$"],"dependencies":[],"title":"Multiplication","text":"What is $$-6x \\\\left(x+3\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a161552dividingpoly3a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8x+28$$"],"dependencies":["a161552dividingpoly3a-h6"],"title":"Subtraction","text":"What is $$\\\\left(-6x^2\\\\right)-10x+28-\\\\left(-6x^2\\\\right)-18x$$ ((-6*(x**2))-(10*x)+28 is the remainder from the previous subtraction step.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly3a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a161552dividingpoly3a-h7"],"title":"Division","text":"What is the largest degree in the remainder?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly3a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":[],"title":"Multiplication","text":"What can we multiply $$x$$ by to get $$8x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly3a-h9-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8x+24$$"],"dependencies":[],"title":"Multiplication","text":"What is $$8\\\\left(x+3\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a161552dividingpoly3a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a161552dividingpoly3a-h9"],"title":"Subtraction","text":"What is $$8x+28-8x+24$$? (8*x+28 is the remainder from the previous subtraction step.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly3a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{4}{x+3}$$"],"dependencies":["a161552dividingpoly3a-h10"],"title":"Remainder","text":"What is the remainder? If there is no remainder enter $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly3a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x^2-6x+8+\\\\frac{4}{x+3}$$"],"dependencies":["a161552dividingpoly3a-h11"],"title":"Combining","text":"Combine all the terms that we used to multiply the divisor by into a single expression along with the divisor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly3a-h12-s1","type":"hint","dependencies":[],"title":"Combining","text":"Reminder: The terms we had were, $$2x^2$$, $$-6x$$, $$8$$, $$\\\\frac{4}{x+3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a161552dividingpoly4","title":"Dividing Polynomials","body":"Divide each polynomial by the binomial.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a161552dividingpoly4a","stepAnswer":["$$x^2-x+1$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{x^3+1}{x+1}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^2-x+1$$","hints":{"DefaultPathway":[{"id":"a161552dividingpoly4a-h1","type":"hint","dependencies":[],"title":"Rewriting the equation","text":"Remember to fill in missing coefficients with placeholder 0\'s if possible to make the equation easier to read and work with.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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$$x^3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly4a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^3+x^2$$"],"dependencies":[],"title":"Multiplication","text":"What is (x**2))*(x+1)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a161552dividingpoly4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(-x^2\\\\right)+1$$"],"dependencies":["a161552dividingpoly4a-h3"],"title":"Subtraction","text":"What is $$x^3+1-x^3+x^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly4a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Division","text":"What is the largest degree in the remainder?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly4a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-x$$"],"dependencies":["a161552dividingpoly4a-h5"],"title":"Multiplication","text":"What can we multiply $$x$$ by to get $$-\\\\left(x^2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly4a-h6-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(-x^2\\\\right)-x$$"],"dependencies":[],"title":"Multiplication","text":"What is $$-x \\\\left(x+1\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a161552dividingpoly4a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+1$$"],"dependencies":["a161552dividingpoly4a-h6"],"title":"Subtraction","text":"What is $$\\\\left(-x^2\\\\right)+1-\\\\left(-x^2\\\\right)-x$$? ((-x**2)+1 is the remainder from the previous subtraction step.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly4a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a161552dividingpoly4a-h7"],"title":"Division","text":"What is the largest degree in the remainder?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly4a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Multiplication","text":"What can we multiply $$x$$ by to get $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly4a-h9-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+1$$"],"dependencies":[],"title":"Multiplication","text":"What is $$1\\\\left(x+1\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a161552dividingpoly4a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a161552dividingpoly4a-h9"],"title":"Subtraction","text":"What is $$x+1-x+1$$ (x+1 is the remainder from the previous subtraction step.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly4a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a161552dividingpoly4a-h10"],"title":"Remainder","text":"What is the remainder? If there is no remainder enter $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly4a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2-x+1$$"],"dependencies":["a161552dividingpoly4a-h11"],"title":"Combining","text":"Combine all the terms that we used to multiply the divisor by into a single expression along with the divisor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly4a-h12-s1","type":"hint","dependencies":[],"title":"Combining","text":"Reminder: The terms we had were, $$x^2$$, $$-x$$, $$1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a161552dividingpoly5","title":"Dividing Polynomials","body":"Divide each polynomial by the binomial.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a161552dividingpoly5a","stepAnswer":["$$x^2-10x+100$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{x^3+1000}{x+10}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^2-10x+100$$","hints":{"DefaultPathway":[{"id":"a161552dividingpoly5a-h1","type":"hint","dependencies":[],"title":"Rewriting the equation","text":"Remember to fill in missing coefficients with placeholder 0\'s if possible to make the equation easier to read and work with.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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$$x^3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly5a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^3+10x^2$$"],"dependencies":[],"title":"Multiplication","text":"What is (x**2))*(x+10)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a161552dividingpoly5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(-10x^2\\\\right)+1000$$"],"dependencies":["a161552dividingpoly5a-h3"],"title":"Subtraction","text":"What is $$x^3+10-x^3+10x^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Division","text":"What is the largest degree in the remainder?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly5a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-10x$$"],"dependencies":["a161552dividingpoly5a-h5"],"title":"Multiplication","text":"What can we multiply $$x$$ by to get $$-10x^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly5a-h6-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-10x^2-100x$$"],"dependencies":[],"title":"Multiplication","text":"What is $$-10x \\\\left(x+10\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a161552dividingpoly5a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$100x+1000$$"],"dependencies":["a161552dividingpoly5a-h6"],"title":"Subtraction","text":"What is $$\\\\left(-10x^2\\\\right)+1000--\\\\left(10x^2-100x\\\\right)$$? $$\\\\left(-10x^2\\\\right)+1000$$ is the remainder from the previous subtraction step.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly5a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a161552dividingpoly5a-h7"],"title":"Division","text":"What is the largest degree in the remainder?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly5a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$100$$"],"dependencies":[],"title":"Multiplication","text":"What can we multiply $$x$$ by to get $$100x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly5a-h9-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$100x+1000$$"],"dependencies":[],"title":"Multiplication","text":"What is $$100\\\\left(x+10\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a161552dividingpoly5a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a161552dividingpoly5a-h9"],"title":"Subtraction","text":"What is $$100x+1000-100x+1000$$? ((100*x)+1000 is the remainder from the last previous step.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly5a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a161552dividingpoly5a-h10"],"title":"Remainder","text":"What is the remainder? If there is no remainder enter $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly5a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2-10x+100$$"],"dependencies":["a161552dividingpoly5a-h11"],"title":"Combining","text":"Combine all the terms that we used to multiply the divisor by into a single expression along with the divisor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly5a-h12-s1","type":"hint","dependencies":[],"title":"Combining","text":"Reminder: The terms we had were, $$x^2$$, $$-10x$$, $$100$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a161552dividingpoly6","title":"Dividing Polynomials","body":"Divide each polynomial by the binomial.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a161552dividingpoly6a","stepAnswer":["$$16x^2+12x+9$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{64x^3-27}{4x-3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16x^2+12x+9$$","hints":{"DefaultPathway":[{"id":"a161552dividingpoly6a-h1","type":"hint","dependencies":[],"title":"Rewriting the equation","text":"Remember to fill in missing coefficients with placeholder 0\'s if possible to make the equation easier to read and work with.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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$$64x^3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly6a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$64x^3-48x^2$$"],"dependencies":[],"title":"Multiplication","text":"What is $$16x^2 \\\\left(4x-3\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a161552dividingpoly6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$48x^2-27$$"],"dependencies":["a161552dividingpoly6a-h3"],"title":"Subtraction","text":"What is $$64x^3-27-64x^3-48x^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Division","text":"What is the largest degree in the remainder?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly6a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12x$$"],"dependencies":["a161552dividingpoly6a-h5"],"title":"Multiplication","text":"What can we multiply $$4x$$ by to get $$48x^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly6a-h6-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$48x^2-36x$$"],"dependencies":[],"title":"Multiplication","text":"What is $$12x \\\\left(4x-3\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a161552dividingpoly6a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$36x-27$$"],"dependencies":["a161552dividingpoly6a-h6"],"title":"Subtraction","text":"What is $$48x^2-27-48x^2-36x$$? ((48*(x**2)-27) is the remainder from the previous subtraction step.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly6a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a161552dividingpoly6a-h7"],"title":"Division","text":"What is the largest degree in the remainder?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly6a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":[],"title":"Multiplication","text":"What can we multiply $$4x$$ by to get $$36x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly6a-h9-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$36x-27$$"],"dependencies":[],"title":"Multiplication","text":"What is $$9\\\\left(4x-3\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a161552dividingpoly6a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a161552dividingpoly6a-h9"],"title":"Subtraction","text":"What is $$36x-27-36x-27$$? (((36*x)-27) is the remainder from the previous subtraction step.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly6a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a161552dividingpoly6a-h10"],"title":"Remainder","text":"What is the remainder? If there is no remainder enter $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly6a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16x^2+12x+9$$"],"dependencies":["a161552dividingpoly6a-h11"],"title":"Combining","text":"Combine all the terms that we used to multiply the divisor by into a single expression along with the divisor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly6a-h12-s1","type":"hint","dependencies":[],"title":"Combining","text":"Reminder: The terms we had were, $$16x^2$$, $$12x$$, $$9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a161552dividingpoly7","title":"Dividing Polynomials","body":"Divide each polynomial by the binomial.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a161552dividingpoly7a","stepAnswer":["$$25x^2+20x+16$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{125x^3-64}{5x-4}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$25x^2+20x+16$$","hints":{"DefaultPathway":[{"id":"a161552dividingpoly7a-h1","type":"hint","dependencies":[],"title":"Rewriting the equation","text":"Remember to fill in missing coefficients with placeholder 0\'s if possible to make the equation easier to read and work with.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a161552dividingpoly7a-h1"],"title":"Division","text":"What is the largest degree in the polynomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly7a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Division","text":"What is the largest degree in the binomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a161552dividingpoly7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25x^2$$"],"dependencies":["a161552dividingpoly7a-h2"],"title":"Multiplication","text":"What can we multiply $$5x$$ by to get $$125x^3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly7a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$125x^3-100x^2$$"],"dependencies":[],"title":"Multiplication","text":"What is $$25x^2 \\\\left(5x-4\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a161552dividingpoly7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$100x^2-64$$"],"dependencies":["a161552dividingpoly7a-h3"],"title":"Subtraction","text":"What is $$125x^3-64-125x^3-100x^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly7a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Division","text":"What is the largest degree in the remainder?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly7a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20x$$"],"dependencies":["a161552dividingpoly7a-h5"],"title":"Multiplication","text":"What can we multiply $$5x$$ by to get $$100x^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly7a-h6-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$100x^2-80x$$"],"dependencies":[],"title":"Multiplication","text":"What is $$20x \\\\left(5x-4\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a161552dividingpoly7a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$80x-64$$"],"dependencies":["a161552dividingpoly7a-h6"],"title":"Subtraction","text":"What is $$100x^2-64-100x^2-80x$$? (((100*(x**2))-64) is the remainder from the previous subtraction step.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly7a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a161552dividingpoly7a-h7"],"title":"Division","text":"What is the largest degree in the remainder?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly7a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":[],"title":"Multiplication","text":"What can we multiply $$5x$$ by to get $$80x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly7a-h9-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$80x-64$$"],"dependencies":[],"title":"Multiplication","text":"What is $$16\\\\left(5x-4\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a161552dividingpoly7a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a161552dividingpoly7a-h9"],"title":"Subtraction","text":"What is $$80x-64-80x-64$$? ((80*x)-64 is the remainder from the previous subtraction step.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly7a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a161552dividingpoly7a-h10"],"title":"Remainder","text":"What is the remainder? If there is no remainder enter $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly7a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25x^2+20x+16$$"],"dependencies":["a161552dividingpoly7a-h11"],"title":"Combining","text":"Combine all the terms that we used to multiply the divisor by into a single expression along with the divisor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly7a-h12-s1","type":"hint","dependencies":[],"title":"Combining","text":"Reminder: The terms we had were, $$25x^2$$, $$20x$$, $$16$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a161552dividingpoly8","title":"Dividing Polynomials","body":"Divide each polynomial function by the binomial function and solve for a given $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a161552dividingpoly8a","stepAnswer":["$$-10$$"],"problemType":"TextBox","stepTitle":"$$f(x)=x^2-13x+36$$ $$g(x)=x-4$$ Find $$\\\\frac{f}{g}$$ when $$x=-1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-10$$","hints":{"DefaultPathway":[{"id":"a161552dividingpoly8a-h1","type":"hint","dependencies":[],"title":"Rewriting the equation","text":"Remember to fill in missing coefficients with placeholder 0\'s if possible to make the equation easier to read and work with.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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$$x^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly8a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2-4x$$"],"dependencies":[],"title":"Multiplication","text":"What is $$x \\\\left(x-4\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a161552dividingpoly8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(-9x\\\\right)+36$$"],"dependencies":["a161552dividingpoly8a-h3"],"title":"Subtraction","text":"What is $$x^2-13x+36-x^2-4x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly8a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Division","text":"What is the largest degree in the remainder?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly8a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":["a161552dividingpoly8a-h5"],"title":"Multiplication","text":"What can we multiply $$x$$ by to get $$\\\\left(-9x\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly8a-h6-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(-9x\\\\right)+36$$"],"dependencies":[],"title":"Multiplication","text":"What is $$-9\\\\left(x-4\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a161552dividingpoly8a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a161552dividingpoly8a-h6"],"title":"Subtraction","text":"What is $$\\\\left(-9x\\\\right)+36-\\\\left(-9x\\\\right)+36$$? ((-9*x)+36 is the remainder from the previous subtraction step.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly8a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a161552dividingpoly8a-h7"],"title":"Remainder","text":"What is the remainder? If there is no remainder enter $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly8a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x-9$$"],"dependencies":["a161552dividingpoly8a-h8"],"title":"Combining","text":"Combine all the terms that we used to multiply the divisor by into a single expression along with the divisor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly8a-h9-s1","type":"hint","dependencies":[],"title":"Combining","text":"Reminder: The terms we had were, $$x$$, $$-9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a161552dividingpoly8a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-10$$"],"dependencies":[],"title":"Plugging it in","text":"What is $$x-9$$ when $$x=-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552dividingpoly9","title":"Dividing Polynomials","body":"Divide each polynomial function by the binomial function and solve for a given $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a161552dividingpoly9a","stepAnswer":["$$-11$$"],"problemType":"TextBox","stepTitle":"$$f(x)=x^2-15x+54$$, $$g(x)=x-9$$ Find $$\\\\frac{f}{g}$$ when $$x=-5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-11$$","hints":{"DefaultPathway":[{"id":"a161552dividingpoly9a-h1","type":"hint","dependencies":[],"title":"Rewriting the equation","text":"Remember to fill in missing coefficients with placeholder 0\'s if possible to make the equation easier to read and work with.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a161552dividingpoly9a-h1"],"title":"Division","text":"What is the largest degree in the polynomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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\\\\left(x-9\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a161552dividingpoly9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(-6x\\\\right)+54$$"],"dependencies":["a161552dividingpoly9a-h3"],"title":"Subtraction","text":"What is $$x^2-15x+54-x^2-9x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly9a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Division","text":"What is the largest degree in the remainder?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly9a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6$$"],"dependencies":["a161552dividingpoly9a-h5"],"title":"Multiplication","text":"What can we multiply $$x$$ by to get $$\\\\left(-6x\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly9a-h6-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(-6x\\\\right)+54$$"],"dependencies":[],"title":"Multiplication","text":"What is $$-6\\\\left(x-9\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a161552dividingpoly9a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a161552dividingpoly9a-h6"],"title":"Subtraction","text":"What is $$\\\\left(-6x\\\\right)+54-\\\\left(-6x\\\\right)+54$$? ((-6*x)+54 is the remainder from the previous subtraction step.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly9a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a161552dividingpoly9a-h7"],"title":"Remainder","text":"What is the remainder? If there is no remainder enter $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552dividingpoly9a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x-6$$"],"dependencies":["a161552dividingpoly9a-h8"],"title":"Combining","text":"Combine all the terms that we used to multiply the divisor by into a single expression along with the divisor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552dividingpoly9a-h9-s1","type":"hint","dependencies":[],"title":"Combining","text":"Reminder: The terms we had were, $$x$$, $$-6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a161552dividingpoly9a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-11$$"],"dependencies":[],"title":"Plugging it in","text":"What is $$x-6$$ when $$x=-5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552divpoly1","title":"Using Long Division to Divide a Second-Degree Polynomial","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a161552divpoly1a","stepAnswer":["$$5x-2$$"],"problemType":"TextBox","stepTitle":"Divide $$5x^2+3x-2$$ by $$x+1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5x-2$$","hints":{"DefaultPathway":[{"id":"a161552divpoly1a-h1","type":"hint","dependencies":[],"title":"Dividing the Expression","text":"Remember, we must start by dividing the first term, $$5x^2$$, by $$x$$. This gives us $$5x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly1a-h2","type":"hint","dependencies":["a161552divpoly1a-h1"],"title":"Multiplying Expressions","text":"Now, we must multiply everything in the divisor by $$5x$$ and subtract from the original polynomial. This means we subtract $$5x^2+5x$$. This cycle of \\"division\\" and \\"multiplication\\" will continue until the divident is of a lesser degree than the divisor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly1a-h3","type":"hint","dependencies":["a161552divpoly1a-h2"],"title":"Full Solution","text":"If you\'re stuck, check your work with the full solution below. The answer is $$5x-2$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552divpoly10","title":"Dividing Polynomials Exercise #1","body":"Use long division to divide. Specify the quotient and the remainder.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a161552divpoly10a","stepAnswer":["Quotient: $$x+6$$, Remainder: $$\\\\frac{5}{x-1}$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{x^2+5x-1}{x-1}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Quotient: $$x+6$$, Remainder: $$\\\\frac{5}{x-1}$$","choices":["Quotient: $$x+6$$, Remainder: $$\\\\frac{5}{x-1}$$","Quotient: $$x+3$$, Remainder: $$\\\\frac{2}{x-1}$$","Quotient: $$x+4$$, Remainder: $$\\\\frac{5}{x-1}$$"],"hints":{"DefaultPathway":[{"id":"a161552divpoly10a-h1","type":"hint","dependencies":[],"title":"Long Division Process","text":"First, determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor. Second, multiply the answer by the divisor and write it below the like terms of the dividend. Third, subtract the bottom binomial from the top binomial. Fourth, bring down the next term of the dividend. Next, repeat steps $$1-4$$ until reaching the last term of the dividend. Finally, if the remainder is non-zero, express as a fraction using the divisor as the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly10a-h2","type":"hint","dependencies":["a161552divpoly10a-h1"],"title":"Long Division Example","text":"The attached image shows an example of long division, where the polynomial $$6x^3+11x^2-31x+15$$ is divided by $$3x-2$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552divpoly11","title":"Dividing Polynomials Exercise #2","body":"Use long division to divide. Specify the quotient and the remainder.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a161552divpoly11a","stepAnswer":["Quotient: $$2x+1$$, Remainder: $$0$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{2x^2-9x-5}{x-5}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Quotient: $$2x+1$$, Remainder: $$0$$","choices":["Quotient: $$2x+1$$, Remainder: $$1$$","Quotient: $$3x+1$$, Remainder: $$1$$","Quotient: $$2x+1$$, Remainder: $$0$$"],"hints":{"DefaultPathway":[{"id":"a161552divpoly11a-h1","type":"hint","dependencies":[],"title":"Long Division Process","text":"First, determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor. Second, multiply the answer by the divisor and write it below the like terms of the dividend. Third, subtract the bottom binomial from the top binomial. Fourth, bring down the next term of the dividend. Next, repeat steps $$1-4$$ until reaching the last term of the dividend. Finally, if the remainder is non-zero, express as a fraction using the divisor as the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly11a-h2","type":"hint","dependencies":["a161552divpoly11a-h1"],"title":"Long Division Example","text":"The attached image shows an example of long division, where the polynomial $$6x^3+11x^2-31x+15$$ is divided by $$3x-2$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552divpoly12","title":"Dividing Polynomials Exercise #3","body":"Use long division to divide. Specify the quotient and the remainder.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a161552divpoly12a","stepAnswer":["Quotient: $$3x+2$$, Remainder: $$0$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{3x^2+23x+14}{x+7}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Quotient: $$3x+2$$, Remainder: $$0$$","choices":["Quotient: $$5x+2$$, Remainder: $$2$$","Quotient: $$3x+2$$, Remainder: $$0$$","Quotient: $$x+2$$, Remainder: $$0$$"],"hints":{"DefaultPathway":[{"id":"a161552divpoly12a-h1","type":"hint","dependencies":[],"title":"Long Division Process","text":"First, determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor. Second, multiply the answer by the divisor and write it below the like terms of the dividend. Third, subtract the bottom binomial from the top binomial. Fourth, bring down the next term of the dividend. Next, repeat steps $$1-4$$ until reaching the last term of the dividend. Finally, if the remainder is non-zero, express as a fraction using the divisor as the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly12a-h2","type":"hint","dependencies":["a161552divpoly12a-h1"],"title":"Long Division Example","text":"The attached image shows an example of long division, where the polynomial $$6x^3+11x^2-31x+15$$ is divided by $$3x-2$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552divpoly13","title":"Dividing Polynomials Exercise #4","body":"Use long division to divide. Specify the quotient and the remainder.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a161552divpoly13a","stepAnswer":["Quotient: $$x-3$$, Remainder: $$\\\\frac{12}{4x+2}$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{4x^2-10x+6}{4x+2}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Quotient: $$x-3$$, Remainder: $$\\\\frac{12}{4x+2}$$","choices":["Quotient: $$x-3$$, Remainder: $$\\\\frac{12}{4x+2}$$","Quotient: $$x-3$$, Remainder: $$\\\\frac{2}{4x+2}$$","Quotient: $$x-2$$, Remainder: $$\\\\frac{12}{4x+2}$$"],"hints":{"DefaultPathway":[{"id":"a161552divpoly13a-h1","type":"hint","dependencies":[],"title":"Long Division Process","text":"First, determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor. Second, multiply the answer by the divisor and write it below the like terms of the dividend. Third, subtract the bottom binomial from the top binomial. Fourth, bring down the next term of the dividend. Next, repeat steps $$1-4$$ until reaching the last term of the dividend. Finally, if the remainder is non-zero, express as a fraction using the divisor as the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly13a-h2","type":"hint","dependencies":["a161552divpoly13a-h1"],"title":"Long Division Example","text":"The attached image shows an example of long division, where the polynomial $$6x^3+11x^2-31x+15$$ is divided by $$3x-2$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552divpoly14","title":"Dividing Polynomials Exercise #5","body":"Use long division to divide. Specify the quotient and the remainder.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a161552divpoly14a","stepAnswer":["Quotient: $$x-5$$, Remainder: $$0$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{6x^2-25x-25}{6x+5}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Quotient: $$x-5$$, Remainder: $$0$$","choices":["Quotient: $$x-3$$, Remainder: $$0$$","Quotient: $$x-5$$, Remainder: $$1$$","Quotient: $$x-5$$, Remainder: $$0$$"],"hints":{"DefaultPathway":[{"id":"a161552divpoly14a-h1","type":"hint","dependencies":[],"title":"Long Division Process","text":"First, determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor. Second, multiply the answer by the divisor and write it below the like terms of the dividend. Third, subtract the bottom binomial from the top binomial. Fourth, bring down the next term of the dividend. Next, repeat steps $$1-4$$ until reaching the last term of the dividend. Finally, if the remainder is non-zero, express as a fraction using the divisor as the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly14a-h2","type":"hint","dependencies":["a161552divpoly14a-h1"],"title":"Long Division Example","text":"The attached image shows an example of long division, where the polynomial $$6x^3+11x^2-31x+15$$ is divided by $$3x-2$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552divpoly15","title":"Dividing Polynomials Exercise #6","body":"Use long division to divide. Specify the quotient and the remainder.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a161552divpoly15a","stepAnswer":["Quotient: $$-x+1$$, Remainder: $$\\\\frac{-2}{x+1}$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{\\\\left(-x^2-1\\\\right)}{x+1}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Quotient: $$-x+1$$, Remainder: $$\\\\frac{-2}{x+1}$$","choices":["Quotient: $$-x+1$$, Remainder: $$\\\\frac{2}{x+1}$$","Quotient: $$x+1$$, Remainder: $$\\\\frac{-1}{x+1}$$","Quotient: $$-x+1$$, Remainder: $$\\\\frac{-2}{x+1}$$"],"hints":{"DefaultPathway":[{"id":"a161552divpoly15a-h1","type":"hint","dependencies":[],"title":"Long Division Process","text":"First, determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor. Second, multiply the answer by the divisor and write it below the like terms of the dividend. Third, subtract the bottom binomial from the top binomial. Fourth, bring down the next term of the dividend. Next, repeat steps $$1-4$$ until reaching the last term of the dividend. Finally, if the remainder is non-zero, express as a fraction using the divisor as the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly15a-h2","type":"hint","dependencies":["a161552divpoly15a-h1"],"title":"Long Division Example","text":"The attached image shows an example of long division, where the polynomial $$6x^3+11x^2-31x+15$$ is divided by $$3x-2$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552divpoly16","title":"Dividing Polynomials Exercise #7","body":"Use long division to divide. Specify the quotient and the remainder.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a161552divpoly16a","stepAnswer":["Quotient: $$2x-7$$, Remainder: $$\\\\frac{16}{x+2}$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{2x^2-3x+2}{x+2}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Quotient: $$2x-7$$, Remainder: $$\\\\frac{16}{x+2}$$","choices":["Quotient: $$2x-7$$, Remainder: $$\\\\frac{16}{x+2}$$","Quotient: $$x-7$$, Remainder: $$\\\\frac{16}{x+2}$$","Quotient: $$2x-7$$, Remainder: $$\\\\frac{6}{x+2}$$"],"hints":{"DefaultPathway":[{"id":"a161552divpoly16a-h1","type":"hint","dependencies":[],"title":"Long Division Process","text":"First, determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor. Second, multiply the answer by the divisor and write it below the like terms of the dividend. Third, subtract the bottom binomial from the top binomial. Fourth, bring down the next term of the dividend. Next, repeat steps $$1-4$$ until reaching the last term of the dividend. Finally, if the remainder is non-zero, express as a fraction using the divisor as the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly16a-h2","type":"hint","dependencies":["a161552divpoly16a-h1"],"title":"Long Division Example","text":"The attached image shows an example of long division, where the polynomial $$6x^3+11x^2-31x+15$$ is divided by $$3x-2$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552divpoly17","title":"Dividing Polynomials Exercise #8","body":"Use long division to divide. Specify the quotient and the remainder.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a161552divpoly17a","stepAnswer":["Quotient: $$x^2+5x+25$$, Remainder: $$\\\\frac{-1}{x-5}$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{x^3-126}{x-5}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Quotient: $$x^2+5x+25$$, Remainder: $$\\\\frac{-1}{x-5}$$","choices":["Quotient: $$x^2+5x+25$$, Remainder: $$\\\\frac{5}{x-5}$$","Quotient: $$x^2+5x+25$$, Remainder: $$\\\\frac{-1}{x-5}$$","Quotient: $$x^2+5x+25$$, Remainder: $$\\\\frac{-3}{x-5}$$"],"hints":{"DefaultPathway":[{"id":"a161552divpoly17a-h1","type":"hint","dependencies":[],"title":"Long Division Process","text":"First, determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor. Second, multiply the answer by the divisor and write it below the like terms of the dividend. Third, subtract the bottom binomial from the top binomial. Fourth, bring down the next term of the dividend. Next, repeat steps $$1-4$$ until reaching the last term of the dividend. Finally, if the remainder is non-zero, express as a fraction using the divisor as the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly17a-h2","type":"hint","dependencies":["a161552divpoly17a-h1"],"title":"Long Division Example","text":"The attached image shows an example of long division, where the polynomial $$6x^3+11x^2-31x+15$$ is divided by $$3x-2$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552divpoly18","title":"Dividing Polynomials Exercise #9","body":"Use long division to divide. Specify the quotient and the remainder.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a161552divpoly18a","stepAnswer":["Quotient: $$x-2$$, Remainder: $$\\\\frac{6}{3x+1}$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{3x^2-5x+4}{3x+1}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Quotient: $$x-2$$, Remainder: $$\\\\frac{6}{3x+1}$$","choices":["Quotient: $$3x-2$$, Remainder: $$\\\\frac{6}{3x+1}$$","Quotient: $$x-2$$, Remainder: $$\\\\frac{2}{3x+1}$$","Quotient: $$x-2$$, Remainder: $$\\\\frac{6}{3x+1}$$"],"hints":{"DefaultPathway":[{"id":"a161552divpoly18a-h1","type":"hint","dependencies":[],"title":"Long Division Process","text":"First, determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor. Second, multiply the answer by the divisor and write it below the like terms of the dividend. Third, subtract the bottom binomial from the top binomial. Fourth, bring down the next term of the dividend. Next, repeat steps $$1-4$$ until reaching the last term of the dividend. Finally, if the remainder is non-zero, express as a fraction using the divisor as the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly18a-h2","type":"hint","dependencies":["a161552divpoly18a-h1"],"title":"Long Division Example","text":"The attached image shows an example of long division, where the polynomial $$6x^3+11x^2-31x+15$$ is divided by $$3x-2$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552divpoly19","title":"Dividing Polynomials Exercise #10","body":"Use long division to divide. Specify the quotient and the remainder.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a161552divpoly19a","stepAnswer":["Quotient: $$x^2-x+3$$, Remainder: $$0$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{x^3-3x^2+5x-6}{x-2}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Quotient: $$x^2-x+3$$, Remainder: $$0$$","choices":["Quotient: $$x^2-x+3$$, Remainder: $$0$$","Quotient: $$x^2-2x+3$$, Remainder: $$0$$","Quotient: $$x^2-5x+3$$, Remainder: $$1$$"],"hints":{"DefaultPathway":[{"id":"a161552divpoly19a-h1","type":"hint","dependencies":[],"title":"Long Division Process","text":"First, determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor. Second, multiply the answer by the divisor and write it below the like terms of the dividend. Third, subtract the bottom binomial from the top binomial. Fourth, bring down the next term of the dividend. Next, repeat steps $$1-4$$ until reaching the last term of the dividend. Finally, if the remainder is non-zero, express as a fraction using the divisor as the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly19a-h2","type":"hint","dependencies":["a161552divpoly19a-h1"],"title":"Long Division Example","text":"The attached image shows an example of long division, where the polynomial $$6x^3+11x^2-31x+15$$ is divided by $$3x-2$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552divpoly2","title":"Using Long Division to Divide a Third-Degree Polynomial","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a161552divpoly2a","stepAnswer":["$$2x^2+5x-7+\\\\frac{1}{3x-2}$$"],"problemType":"TextBox","stepTitle":"$$6x^3+11x^2-31x+15$$ by $$3x-2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2x^2+5x-7+\\\\frac{1}{3x-2}$$","hints":{"DefaultPathway":[{"id":"a161552divpoly2a-h1","type":"hint","dependencies":[],"title":"Dividing Terms","text":"Remember, we must start by dividing the first term, $$6x^3$$, by $$3x$$. This gives us $$2x^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly2a-h2","type":"hint","dependencies":["a161552divpoly2a-h1"],"title":"Multiplying Expressions","text":"Now, we must multiply everything in the divisor by $$2x^2$$ and subtract from the original polynomial. This means we subtract $$6x^3-4x^2$$. This cycle of \\"division\\" and \\"multiplication\\" will continue until the divident is of a lesser degree than the divisor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly2a-h3","type":"hint","dependencies":["a161552divpoly2a-h2"],"title":"Remainder","text":"Sometimes, we will be left with a remainder and will not be able to divide any further (when the dividend has a lower degree than the divisor). When this occurs, we write out the answer and add it to the $$\\\\frac{remainder}{divisor}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly2a-h4","type":"hint","dependencies":["a161552divpoly2a-h3"],"title":"Full Solution","text":"If you\'re stuck, check your work with the full solution below.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552divpoly20","title":"Dividing Polynomials Exercise #11","body":"Use long division to divide. Specify the quotient and the remainder.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a161552divpoly20a","stepAnswer":["Quotient: $$2x^2-3x+5$$, Remainder: $$0$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{2x^3+3x^2-4x+15}{x+3}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Quotient: $$2x^2-3x+5$$, Remainder: $$0$$","choices":["Quotient: $$2x^2-3x+5$$, Remainder: $$0$$","Quotient: $$2x^2-3x+5$$, Remainder: $$\\\\frac{1}{x+3}$$","Quotient: $$2x^2-3x+5$$, Remainder: $$1$$"],"hints":{"DefaultPathway":[{"id":"a161552divpoly20a-h1","type":"hint","dependencies":[],"title":"Long Division Process","text":"First, determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor. Second, multiply the answer by the divisor and write it below the like terms of the dividend. Third, subtract the bottom binomial from the top binomial. Fourth, bring down the next term of the dividend. Next, repeat steps $$1-4$$ until reaching the last term of the dividend. Finally, if the remainder is non-zero, express as a fraction using the divisor as the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly20a-h2","type":"hint","dependencies":["a161552divpoly20a-h1"],"title":"Long Division Example","text":"The attached image shows an example of long division, where the polynomial $$6x^3+11x^2-31x+15$$ is divided by $$3x-2$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552divpoly21","title":"Dividing Polynomials using synthetic division","body":"For the following exercises, use synthetic division to find the quotient and the remainder. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.) Final answer should be in the form: quotient + $$\\\\frac{remainder}{divisor}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a161552divpoly21a","stepAnswer":["$$3x^2-11x+34$$ - $$\\\\frac{106}{x+3}$$"],"problemType":"TextBox","stepTitle":"$$3x^3-2x^2+x-4$$ / $$x+3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3x^2-11x+34$$ - $$\\\\frac{106}{x+3}$$","hints":{"DefaultPathway":[{"id":"a161552divpoly21a-h1","type":"hint","dependencies":[],"title":"Setting up Synthetic Division","text":"Write the coefficients of the divident inside an upside down division symbol and the zero of the $$x+3$$ expression on the left.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly21a-h2","type":"hint","dependencies":["a161552divpoly21a-h1"],"title":"Starting the Procedure","text":"Bring the first coefficient of the divident below the upside down division symbol.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly21a-h3","type":"hint","dependencies":["a161552divpoly21a-h2"],"title":"Finding a Product","text":"Multiply the value carried down with the zero and write the product under the coefficient in the next row","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly21a-h4","type":"hint","dependencies":["a161552divpoly21a-h3"],"title":"Adding Values","text":"Add the values of the product and zero","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly21a-h5","type":"hint","dependencies":["a161552divpoly21a-h4"],"title":"Multiplying Values","text":"Multiply the added value with the zero and write the product under the coefficient in the next row","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly21a-h6","type":"hint","dependencies":["a161552divpoly21a-h5"],"title":"When do you stop?","text":"Stop when you reach the last column and there are no coefficients to the right of you.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly21a-h7","type":"hint","dependencies":["a161552divpoly21a-h6"],"title":"Remainders and Quotients","text":"The value of the quotient is all the values outside of the upside down divison symbol except the last one, which is the remainder.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552divpoly22","title":"Dividing Polynomials using synthetic division","body":"For the following exercises, use synthetic division to find the quotient and the remainder. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.) Final answer should be in the form: quotient + $$\\\\frac{remainder}{divisor}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a161552divpoly22a","stepAnswer":["$$2x^2+2x+1+\\\\frac{10}{x-4}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2x^3-6x^2-7x+6}{x-4}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2x^2+2x+1+\\\\frac{10}{x-4}$$","hints":{"DefaultPathway":[{"id":"a161552divpoly22a-h1","type":"hint","dependencies":[],"title":"Setting up Synthetic Division","text":"Write the coefficients of the divident inside an upside down division symbol and the zero of the $$(x-4)$$ expression on the left.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly22a-h2","type":"hint","dependencies":["a161552divpoly22a-h1"],"title":"Starting the Procedure","text":"Bring the first coefficient of the divident below the upside down division symbol.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly22a-h3","type":"hint","dependencies":["a161552divpoly22a-h2"],"title":"Finding a Product","text":"Multiply the value carried down with the zero and write the product under the coefficient in the next row","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly22a-h4","type":"hint","dependencies":["a161552divpoly22a-h3"],"title":"Adding Values","text":"Add the values of the product and zero","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly22a-h5","type":"hint","dependencies":["a161552divpoly22a-h4"],"title":"Multiplying Values","text":"Multiply the added value with the zero and write the product under the coefficient in the next row","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly22a-h6","type":"hint","dependencies":["a161552divpoly22a-h5"],"title":"When do you stop?","text":"Stop when you reach the last column and there are no coefficients to the right of you.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly22a-h7","type":"hint","dependencies":["a161552divpoly22a-h6"],"title":"Remainders and Quotients","text":"The value of the quotient is all the values outside of the upside down divison symbol except the last one, which is the remainder.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552divpoly23","title":"Dividing Polynomials using synthetic division","body":"For the following exercises, use synthetic division to find the quotient and the remainder. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.) Final answer should be in the form: quotient + $$\\\\frac{remainder}{divisor}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a161552divpoly23a","stepAnswer":["$$6x^2-16x+9$$ - $$\\\\frac{24}{x+1}$$"],"problemType":"TextBox","stepTitle":"$$6x^3-10x^2-7x-15$$ / $$x+1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6x^2-16x+9$$ - $$\\\\frac{24}{x+1}$$","hints":{"DefaultPathway":[{"id":"a161552divpoly23a-h1","type":"hint","dependencies":[],"title":"Setting up Synthetic Division","text":"Write the coefficients of the divident inside an upside down division symbol and the zero of the $$x+1$$ expression on the left.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly23a-h2","type":"hint","dependencies":["a161552divpoly23a-h1"],"title":"Starting the Procedure","text":"Bring the first coefficient of the divident below the upside down division symbol.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly23a-h3","type":"hint","dependencies":["a161552divpoly23a-h2"],"title":"Finding a Product","text":"Multiply the value carried down with the zero and write the product under the coefficient in the next row","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly23a-h4","type":"hint","dependencies":["a161552divpoly23a-h3"],"title":"Adding Values","text":"Add the values of the product and zero","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly23a-h5","type":"hint","dependencies":["a161552divpoly23a-h4"],"title":"Multiplying Values","text":"Multiply the added value with the zero and write the product under the coefficient in the next row","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly23a-h6","type":"hint","dependencies":["a161552divpoly23a-h5"],"title":"When do you stop?","text":"Stop when you reach the last column and there are no coefficients to the right of you.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly23a-h7","type":"hint","dependencies":["a161552divpoly23a-h6"],"title":"Remainders and Quotients","text":"The value of the quotient is all the values outside of the upside down divison symbol except the last one, which is the remainder.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552divpoly24","title":"Dividing Polynomials using synthetic division","body":"For the following exercises, use synthetic division to find the quotient and the remainder. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.) Final answer should be in the form: quotient + $$\\\\frac{remainder}{divisor}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a161552divpoly24a","stepAnswer":["$$2x\\\\times2-7x+1-\\\\frac{2}{2x+1}$$"],"problemType":"TextBox","stepTitle":"$$4x^3-12x^2-5x-1$$ / $$2x+1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2x\\\\times2-7x+1-\\\\frac{2}{2x+1}$$","hints":{"DefaultPathway":[{"id":"a161552divpoly24a-h1","type":"hint","dependencies":[],"title":"Setting up Synthetic Division","text":"Write the coefficients of the divident inside an upside down division symbol and the zero of the $$2x+1$$ expression on the left.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly24a-h2","type":"hint","dependencies":["a161552divpoly24a-h1"],"title":"Starting the Procedure","text":"Bring the first coefficient of the divident below the upside down division symbol.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly24a-h3","type":"hint","dependencies":["a161552divpoly24a-h2"],"title":"Finding a Product","text":"Multiply the value carried down with the zero and write the product under the coefficient in the next row","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly24a-h4","type":"hint","dependencies":["a161552divpoly24a-h3"],"title":"Adding Values","text":"Add the values of the product and zero","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly24a-h5","type":"hint","dependencies":["a161552divpoly24a-h4"],"title":"Multiplying Values","text":"Multiply the added value with the zero and write the product under the coefficient in the next row","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly24a-h6","type":"hint","dependencies":["a161552divpoly24a-h5"],"title":"When do you stop?","text":"Stop when you reach the last column and there are no coefficients to the right of you.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly24a-h7","type":"hint","dependencies":["a161552divpoly24a-h6"],"title":"Remainders and Quotients","text":"The value of the quotient is all the values outside of the upside down divison symbol except the last one, which is the remainder.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552divpoly25","title":"Dividing Polynomials using synthetic division","body":"For the following exercises, use synthetic division to find the quotient and the remainder. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.) Final answer should be in the form: quotient + $$\\\\frac{remainder}{divisor}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a161552divpoly25a","stepAnswer":["$$3x^2-2x+\\\\frac{16}{3}+\\\\frac{31}{3\\\\left(3x-1\\\\right)}$$"],"problemType":"TextBox","stepTitle":"$$9x^3-9x^2+18x+5$$ / $$(3x-1)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3x^2-2x+\\\\frac{16}{3}+\\\\frac{31}{3\\\\left(3x-1\\\\right)}$$","hints":{"DefaultPathway":[{"id":"a161552divpoly25a-h1","type":"hint","dependencies":[],"title":"Setting up Synthetic Division","text":"Write the coefficients of the divident inside an upside down division symbol and the zero of the $$(3x-1)$$ expression on the left.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly25a-h2","type":"hint","dependencies":["a161552divpoly25a-h1"],"title":"Starting the Procedure","text":"Bring the first coefficient of the divident below the upside down division symbol.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly25a-h3","type":"hint","dependencies":["a161552divpoly25a-h2"],"title":"Finding a Product","text":"Multiply the value carried down with the zero and write the product under the coefficient in the next row","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly25a-h4","type":"hint","dependencies":["a161552divpoly25a-h3"],"title":"Adding Values","text":"Add the values of the product and zero","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly25a-h5","type":"hint","dependencies":["a161552divpoly25a-h4"],"title":"Multiplying Values","text":"Multiply the added value with the zero and write the product under the coefficient in the next row","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly25a-h6","type":"hint","dependencies":["a161552divpoly25a-h5"],"title":"When do you stop?","text":"Stop when you reach the last column and there are no coefficients to the right of you.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly25a-h7","type":"hint","dependencies":["a161552divpoly25a-h6"],"title":"Remainders and Quotients","text":"The value of the quotient is all the values outside of the upside down divison symbol except the last one, which is the remainder.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552divpoly26","title":"Dividing Polynomials using synthetic division","body":"For the following exercises, use synthetic division to find the quotient and the remainder. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.) Final answer should be in the form: quotient + $$\\\\frac{remainder}{divisor}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a161552divpoly26a","stepAnswer":["$$3x^2-11x+34-\\\\frac{106}{x+3}$$"],"problemType":"TextBox","stepTitle":"$$3x^3-2x^2+x-4$$ / $$x+3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3x^2-11x+34-\\\\frac{106}{x+3}$$","hints":{"DefaultPathway":[{"id":"a161552divpoly26a-h1","type":"hint","dependencies":[],"title":"Setting up Synthetic Division","text":"Write the coefficients of the divident inside an upside down division symbol and the zero of the $$x+3$$ expression on the left.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly26a-h2","type":"hint","dependencies":["a161552divpoly26a-h1"],"title":"Starting the Procedure","text":"Bring the first coefficient of the divident below the upside down division symbol.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly26a-h3","type":"hint","dependencies":["a161552divpoly26a-h2"],"title":"Finding a Product","text":"Multiply the value carried down with the zero and write the product under the coefficient in the next row","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly26a-h4","type":"hint","dependencies":["a161552divpoly26a-h3"],"title":"Adding Values","text":"Add the values of the product and zero","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly26a-h5","type":"hint","dependencies":["a161552divpoly26a-h4"],"title":"Multiplying Values","text":"Multiply the added value with the zero and write the product under the coefficient in the next row","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly26a-h6","type":"hint","dependencies":["a161552divpoly26a-h5"],"title":"When do you stop?","text":"Stop when you reach the last column and there are no coefficients to the right of you.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly26a-h7","type":"hint","dependencies":["a161552divpoly26a-h6"],"title":"Remainders and Quotients","text":"The value of the quotient is all the values outside of the upside down divison symbol except the last one, which is the remainder.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552divpoly27","title":"Dividing Polynomials using synthetic division","body":"For the following exercises, use synthetic division to find the quotient and the remainder. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.) Final answer should be in the form: quotient + $$\\\\frac{remainder}{divisor}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a161552divpoly27a","stepAnswer":["$$-3x^2-4x-6$$ - $$\\\\frac{22}{2x-3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(-6x^3+x^2\\\\right)-4$$ / $$(2x-3)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-3x^2-4x-6$$ - $$\\\\frac{22}{2x-3}$$","hints":{"DefaultPathway":[{"id":"a161552divpoly27a-h1","type":"hint","dependencies":[],"title":"Setting up Synthetic Division","text":"Write the coefficients of the divident inside an upside down division symbol and the zero of the $$(2x-3)$$ expression on the left.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly27a-h2","type":"hint","dependencies":["a161552divpoly27a-h1"],"title":"Starting the Procedure","text":"Bring the first coefficient of the divident below the upside down division symbol.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly27a-h3","type":"hint","dependencies":["a161552divpoly27a-h2"],"title":"Finding a Product","text":"Multiply the value carried down with the zero and write the product under the coefficient in the next row","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly27a-h4","type":"hint","dependencies":["a161552divpoly27a-h3"],"title":"Adding Values","text":"Add the values of the product and zero","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly27a-h5","type":"hint","dependencies":["a161552divpoly27a-h4"],"title":"Multiplying Values","text":"Multiply the added value with the zero and write the product under the coefficient in the next row","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly27a-h6","type":"hint","dependencies":["a161552divpoly27a-h5"],"title":"When do you stop?","text":"Stop when you reach the last column and there are no coefficients to the right of you.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly27a-h7","type":"hint","dependencies":["a161552divpoly27a-h6"],"title":"Remainders and Quotients","text":"The value of the quotient is all the values outside of the upside down divison symbol except the last one, which is the remainder.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552divpoly28","title":"Dividing Polynomials using synthetic division","body":"For the following exercises, use synthetic division to find the quotient and the remainder. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.) Final answer should be in the form: quotient + $$\\\\frac{remainder}{divisor}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a161552divpoly28a","stepAnswer":["$$x^2+5x+1$$"],"problemType":"TextBox","stepTitle":"$$2x\\\\times83+7x\\\\times82-13x-3$$ / $$(2x-3)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^2+5x+1$$","hints":{"DefaultPathway":[{"id":"a161552divpoly28a-h1","type":"hint","dependencies":[],"title":"Setting up Synthetic Division","text":"Write the coefficients of the divident inside an upside down division symbol and the zero of the $$(2x-3)$$ expression on the left.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly28a-h2","type":"hint","dependencies":["a161552divpoly28a-h1"],"title":"Starting the Procedure","text":"Bring the first coefficient of the divident below the upside down division symbol.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly28a-h3","type":"hint","dependencies":["a161552divpoly28a-h2"],"title":"Finding a Product","text":"Multiply the value carried down with the zero and write the product under the coefficient in the next row","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly28a-h4","type":"hint","dependencies":["a161552divpoly28a-h3"],"title":"Adding Values","text":"Add the values of the product and zero","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly28a-h5","type":"hint","dependencies":["a161552divpoly28a-h4"],"title":"Multiplying Values","text":"Multiply the added value with the zero and write the product under the coefficient in the next row","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly28a-h6","type":"hint","dependencies":["a161552divpoly28a-h5"],"title":"When do you stop?","text":"Stop when you reach the last column and there are no coefficients to the right of you.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly28a-h7","type":"hint","dependencies":["a161552divpoly28a-h6"],"title":"Remainders and Quotients","text":"The value of the quotient is all the values outside of the upside down divison symbol except the last one, which is the remainder.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552divpoly29","title":"Dividing Polynomials using synthetic division","body":"For the following exercises, use synthetic division to find the quotient and the remainder. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.) Final answer should be in the form: quotient + $$\\\\frac{remainder}{divisor}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a161552divpoly29a","stepAnswer":["$$3x^2-11x+24$$ - $$\\\\frac{45}{x+2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{3x^3-5x^2+2x+3}{x+2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3x^2-11x+24$$ - $$\\\\frac{45}{x+2}$$","hints":{"DefaultPathway":[{"id":"a161552divpoly29a-h1","type":"hint","dependencies":[],"title":"Setting up Synthetic Division","text":"Write the coefficients of the divident inside an upside down division symbol and the zero of the $$x+2$$ expression on the left.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly29a-h2","type":"hint","dependencies":["a161552divpoly29a-h1"],"title":"Starting the Procedure","text":"Bring the first coefficient of the divident below the upside down division symbol.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly29a-h3","type":"hint","dependencies":["a161552divpoly29a-h2"],"title":"Finding a Product","text":"Multiply the value carried down with the zero and write the product under the coefficient in the next row","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly29a-h4","type":"hint","dependencies":["a161552divpoly29a-h3"],"title":"Adding Values","text":"Add the values of the product and zero","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly29a-h5","type":"hint","dependencies":["a161552divpoly29a-h4"],"title":"Multiplying Values","text":"Multiply the added value with the zero and write the product under the coefficient in the next row","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly29a-h6","type":"hint","dependencies":["a161552divpoly29a-h5"],"title":"When do you stop?","text":"Stop when you reach the last column and there are no coefficients to the right of you.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly29a-h7","type":"hint","dependencies":["a161552divpoly29a-h6"],"title":"Remainders and Quotients","text":"The value of the quotient is all the values outside of the upside down divison symbol except the last one, which is the remainder.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552divpoly3","title":"Using Synthetic Division to Divide a Second-Degree Polynomial","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a161552divpoly3a","stepAnswer":["$$5x+12$$"],"problemType":"TextBox","stepTitle":"Use synthetic division to divide $$5x^2-3x-36$$ by $$x-3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5x+12$$","hints":{"DefaultPathway":[{"id":"a161552divpoly3a-h1","type":"hint","dependencies":[],"title":"Settting Up","text":"Begin by setting up the coefficients. Write k and the coefficients.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly3a-h2","type":"hint","dependencies":["a161552divpoly3a-h1"],"title":"Beginning the Division","text":"Bring down the lead coefficient. Multiply the lead coefficient by k.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly3a-h3","type":"hint","dependencies":["a161552divpoly3a-h2"],"title":"Finishing the Division","text":"Continue by adding the numbers in the second column. Multiply the resulting number by k. Write the result in the next column. Then add the numbers in the third column.\\\\n##figure3.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly3a-h4","type":"hint","dependencies":["a161552divpoly3a-h3"],"title":"Answer","text":"Our result is $$5x+12$$. There is no remainder.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552divpoly30","title":"Dividing Polynomials using synthetic division","body":"For the following exercises, use synthetic division to find the quotient and the remainder. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.) Final answer should be in the form: quotient + $$\\\\frac{remainder}{divisor}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a161552divpoly30a","stepAnswer":["$$4x^2-13x+26-\\\\frac{39}{x+2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{4x^3-5x^2+13}{x+2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4x^2-13x+26-\\\\frac{39}{x+2}$$","hints":{"DefaultPathway":[{"id":"a161552divpoly30a-h1","type":"hint","dependencies":[],"title":"Setting up Synthetic Division","text":"Write the coefficients of the divident inside an upside down division symbol and the zero of the $$x+2$$ expression on the left.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly30a-h2","type":"hint","dependencies":["a161552divpoly30a-h1"],"title":"Starting the Procedure","text":"Bring the first coefficient of the divident below the upside down division symbol.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly30a-h3","type":"hint","dependencies":["a161552divpoly30a-h2"],"title":"Finding a Product","text":"Multiply the value carried down with the zero and write the product under the coefficient in the next row","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly30a-h4","type":"hint","dependencies":["a161552divpoly30a-h3"],"title":"Adding Values","text":"Add the values of the product and zero","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly30a-h5","type":"hint","dependencies":["a161552divpoly30a-h4"],"title":"Multiplying Values","text":"Multiply the added value with the zero and write the product under the coefficient in the next row","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly30a-h6","type":"hint","dependencies":["a161552divpoly30a-h5"],"title":"When do you stop?","text":"Stop when you reach the last column and there are no coefficients to the right of you.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly30a-h7","type":"hint","dependencies":["a161552divpoly30a-h6"],"title":"Remainders and Quotients","text":"The value of the quotient is all the values outside of the upside down divison symbol except the last one, which is the remainder.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552divpoly4","title":"Using Synthetic Division to Divide a Third-Degree Polynomial","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a161552divpoly4a","stepAnswer":["$$4x^2+2x-10$$"],"problemType":"TextBox","stepTitle":"Use synthetic division to divide $$4x^3+10x^2-6x-20$$ by $$x+2$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4x^2+2x-10$$","hints":{"DefaultPathway":[{"id":"a161552divpoly4a-h1","type":"hint","dependencies":[],"title":"Settting Up","text":"Begin by setting up the coefficients. Write k and the coefficients.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly4a-h2","type":"hint","dependencies":["a161552divpoly4a-h1"],"title":"Beginning the Division","text":"Bring down the lead coefficient. Multiply the lead coefficient by k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly4a-h3","type":"hint","dependencies":["a161552divpoly4a-h2"],"title":"Full Solution","text":"If you\'re stuck, check your work with the full solution below.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly4a-h4","type":"hint","dependencies":["a161552divpoly4a-h3"],"title":"Final Answer","text":"Our result is $$4x^2+2x-10$$. There is no remainder.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552divpoly5","title":"Using Synthetic Division to Divide a Fourth-Degree Polynomial","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a161552divpoly5a","stepAnswer":["$$-9x^3+x^2+8x+8+\\\\frac{2}{x-1}$$"],"problemType":"TextBox","stepTitle":"Use synthetic division to divide $$-9x^4+10x^3+7x^2-6$$ by $$x-1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-9x^3+x^2+8x+8+\\\\frac{2}{x-1}$$","hints":{"DefaultPathway":[{"id":"a161552divpoly5a-h1","type":"hint","dependencies":[],"title":"Settting Up","text":"Begin by setting up the coefficients. Write k and the coefficients.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly5a-h2","type":"hint","dependencies":["a161552divpoly5a-h1"],"title":"Beginning the Division","text":"Bring down the lead coefficient. Multiply the lead coefficient by k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly5a-h3","type":"hint","dependencies":["a161552divpoly5a-h2"],"title":"Full Solution","text":"If you\'re stuck, check your work with the full solution below.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552divpoly6","title":"Using Polynomial Division in an Application Problem","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"a161552divpoly6a","stepAnswer":["$$x^2+x+9$$"],"problemType":"TextBox","stepTitle":"The volume of a rectangular solid is given by the polynomial $$3x^4-3x^3-33x^2+54x$$. The length of the solid is given by $$3x$$ and the width is given by $$x-2$$. Find the height, $$h$$, of the solid.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^2+x+9$$","hints":{"DefaultPathway":[{"id":"a161552divpoly6a-h1","type":"hint","dependencies":[],"title":"Creating a Diagram","text":"There are a few ways to approach this problem. We need to divide the expression for the volume of the solid by the expressions for the length and width. Let us create a sketch.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly6a-h2","type":"hint","dependencies":["a161552divpoly6a-h1"],"title":"Creating an Equation","text":"We can now write an equation by substituting the known values into the formula for the volume of a rectangular solid. $$V=l w h$$. $$3x^4-3x^3-33x^2+54x=3x \\\\left(x-2\\\\right) h$$\\\\n","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly6a-h3","type":"hint","dependencies":["a161552divpoly6a-h2"],"title":"Solving for H","text":"To solve for $$h$$, we can first divide both sides by $$3x$$. We now have $$(x-2)h=x^3-x^2-11x+18$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552divpoly6a-h4","type":"hint","dependencies":["a161552divpoly6a-h3"],"title":"Synthetic Division","text":"We now solve for $$h$$ by using synthetic division as shown.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552poly1","title":"Dividing Monomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a161552poly1a","stepAnswer":["$$-\\\\left(\\\\frac{9a}{b^2}\\\\right)$$"],"problemType":"TextBox","stepTitle":"Find the quotient: $$\\\\frac{54a^2 b^3}{\\\\left(-6a b^5\\\\right)}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-\\\\left(\\\\frac{9a}{b^2}\\\\right)$$","hints":{"DefaultPathway":[{"id":"a161552poly1a-h1","type":"hint","dependencies":[],"title":"Decompose","text":"Use Fraction Multiplication: $$\\\\frac{54}{-6} \\\\frac{a^2}{a} \\\\frac{b^3}{b^5}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552poly1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-\\\\left(\\\\frac{9a}{b^2}\\\\right)$$"],"dependencies":["a161552poly1a-h1"],"title":"Simplify","text":"What is $$\\\\frac{54}{-6} \\\\frac{a^2}{a} \\\\frac{b^3}{b^5}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552poly1a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\frac{54}{-6}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552poly1a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["a"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\frac{a^2}{a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552poly1a-h2-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{b^2}$$"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\frac{b^3}{b^5}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a161552poly10","title":"Factor Theorem","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a161552poly10a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Is $$x-4$$ is a factor of $$f(x)=x^3-64$$?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a161552poly10a-h1","type":"hint","dependencies":[],"title":"Factor Theorem","text":"The Factor Theorem states that if a polynomial function f(x) is divided by $$x-c$$, then the remainder is f(c).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552poly10a-h2","type":"hint","dependencies":["a161552poly10a-h1"],"title":"Factor Theorem","text":"The Factor Theorem tells us that $$x-4$$ is a factor of $$f(x)=x^3-64$$ if $$f(4)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552poly10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a161552poly10a-h2"],"title":"Solve","text":"What is f(4)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552poly2","title":"Division of a Polynomial by a Monomial","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a161552poly2a","stepAnswer":["$$-6x^2+12y$$"],"problemType":"TextBox","stepTitle":"Find the quotient: $$\\\\frac{18x^3 y-36x y^2}{\\\\left(-3x y\\\\right)}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-6x^2+12y$$","hints":{"DefaultPathway":[{"id":"a161552poly2a-h1","type":"hint","dependencies":[],"title":"Decompose","text":"Divide each term by the divisor. Be careful with the signs: $$\\\\frac{18x^3 y}{\\\\left(-3x y\\\\right)}-\\\\frac{36x y^2}{\\\\left(-3x y\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552poly2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6x^2$$"],"dependencies":["a161552poly2a-h1"],"title":"Simplify","text":"What is $$\\\\frac{18x^3 y}{\\\\left(-3x y\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552poly2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-12y$$"],"dependencies":["a161552poly2a-h2"],"title":"Simplify","text":"What is $$\\\\frac{36x y^2}{\\\\left(-3x y\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552poly2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6x^2+12y$$"],"dependencies":["a161552poly2a-h3"],"title":"Simplify","text":"What is $$\\\\left(-6x^2\\\\right)-\\\\left(-12y\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552poly3","title":"Divide Polynomials Using Long Division","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a161552poly3a","stepAnswer":["$$x+4$$"],"problemType":"TextBox","stepTitle":"Find the quotient: $$\\\\frac{x^2+9x+20}{x+5}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x+4$$","hints":{"DefaultPathway":[{"id":"a161552poly3a-h1","type":"hint","dependencies":[],"title":"Long Division","text":"Write it as a long division problem.\\\\nBe sure the dividend is in standard form.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552poly3a-h2","type":"hint","dependencies":["a161552poly3a-h1"],"title":"Long Division","text":"Divide $$x^2$$ by $$x$$. It may help to ask yourself, \u201cWhat do I need\\\\nto multiply $$x$$ by to get x**2?\u201d\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552poly3a-h3","type":"hint","dependencies":["a161552poly3a-h2"],"title":"Long Division","text":"Put the answer, $$x$$, in the quotient over the $$x$$ term.\\\\nMultiply $$x$$ times $$x+5$$. Line up the like terms under the dividend.\\\\n##figure3.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552poly3a-h4","type":"hint","dependencies":["a161552poly3a-h3"],"title":"Long Division","text":"Subtract $$x^2+5x$$ from $$x^2+9x$$.\\\\nYou may find it easier to change the signs and then add.\\\\nThen bring down the last term, $$20$$.\\\\n##figure4.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552poly3a-h5","type":"hint","dependencies":["a161552poly3a-h4"],"title":"Long Division","text":"Divide $$4x$$ by $$x$$. It may help to ask yourself, \u201cWhat do I\\\\nneed to multiply $$x$$ by to get $$4x$$ ?\u201d\\\\nPut the answer, $$4$$ , in the quotient over the constant term.\\\\n##figure5.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552poly3a-h6","type":"hint","dependencies":["a161552poly3a-h5"],"title":"Long Division","text":"Multiply $$4$$ times $$x+5$$.\\\\n##figure6.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552poly3a-h7","type":"hint","dependencies":["a161552poly3a-h6"],"title":"Long Division","text":"Subtract $$4x+20$$ from $$4x+20$$. We get $$0$$ so there is no remainder.\\\\n##figure7.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a161552poly4","title":"Dividing Monomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a161552poly4a","stepAnswer":["$$\\\\frac{-9}{a^5 b}$$"],"problemType":"TextBox","stepTitle":"Find the quotient: $$\\\\frac{\\\\left(-72a^7 b^3\\\\right)}{8a^{12} b^4}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-9}{a^5 b}$$","hints":{"DefaultPathway":[{"id":"a161552poly4a-h1","type":"hint","dependencies":[],"title":"Decompose","text":"Use Fraction Multiplication: $$\\\\left(-\\\\frac{72}{8}\\\\right) \\\\frac{a^7}{a^{12}} \\\\frac{b^3}{b^4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552poly4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-9}{a^5 b}$$"],"dependencies":["a161552poly4a-h1"],"title":"Simplify","text":"What is $$\\\\left(-\\\\frac{72}{8}\\\\right) \\\\frac{a^7}{a^{12}} \\\\frac{b^3}{b^4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552poly4a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\left(-\\\\frac{72}{8}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552poly4a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{a^5}$$"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\frac{a^7}{a^{12}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552poly4a-h2-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{b}$$"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\frac{b^3}{b^4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a161552poly5","title":"Dividing Monomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a161552poly5a","stepAnswer":["$$\\\\frac{\\\\left(-9b\\\\right)}{a^4}$$"],"problemType":"TextBox","stepTitle":"Find the quotient: $$\\\\frac{\\\\left(-63a^8 b^3\\\\right)}{7a^{12} b^2}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{\\\\left(-9b\\\\right)}{a^4}$$","hints":{"DefaultPathway":[{"id":"a161552poly5a-h1","type":"hint","dependencies":[],"title":"Decompose","text":"Use Fraction Multiplication: $$\\\\left(-\\\\frac{63}{7}\\\\right) \\\\frac{a^8}{a^{12}} \\\\frac{b^3}{b^2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552poly5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\left(-9b\\\\right)}{a^4}$$"],"dependencies":["a161552poly5a-h1"],"title":"Simplify","text":"What is $$\\\\left(-\\\\frac{63}{7}\\\\right) \\\\frac{a^8}{a^{12}} \\\\frac{b^3}{b^2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552poly5a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\left(-\\\\frac{63}{7}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552poly5a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{a^4}$$"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\frac{a^8}{a^{12}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552poly5a-h2-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$b$$"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\frac{b^3}{b^2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a161552poly6","title":"Dividing Monomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a161552poly6a","stepAnswer":["$$\\\\frac{2b^6}{3a^4}$$"],"problemType":"TextBox","stepTitle":"Find the quotient: $$\\\\frac{14a^7 b^{12}}{21a^{11} b^6}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2b^6}{3a^4}$$","hints":{"DefaultPathway":[{"id":"a161552poly6a-h1","type":"hint","dependencies":[],"title":"Decompose","text":"Use Fraction Multiplication: $$\\\\frac{14}{21} \\\\frac{a^7}{a^{11}} \\\\frac{b^{12}}{b^6}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552poly6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2b^6}{3a^4}$$"],"dependencies":["a161552poly6a-h1"],"title":"Simplify","text":"What is $$\\\\frac{14}{21}$$ * $$\\\\frac{a^7}{a^{11}}$$ * $$\\\\frac{b^{12}}{b^6}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552poly6a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{3}$$"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\frac{14}{21}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552poly6a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{a^4}$$"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\frac{a^7}{a^{11}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552poly6a-h2-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$b^6$$"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\frac{b^{12}}{b^6}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a161552poly7","title":"Dividing Monomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a161552poly7a","stepAnswer":["$$\\\\frac{4b^2}{7a^4}$$"],"problemType":"TextBox","stepTitle":"Find the quotient: $$\\\\frac{28a^5 b^{14}}{49a^9 b^{12}}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{4b^2}{7a^4}$$","hints":{"DefaultPathway":[{"id":"a161552poly7a-h1","type":"hint","dependencies":[],"title":"Decompose","text":"Use Fraction Multiplication: $$\\\\frac{28}{49} \\\\frac{a^5}{a^9} \\\\frac{b^{14}}{b^{12}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552poly7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{4b^2}{7a^4}$$"],"dependencies":["a161552poly7a-h1"],"title":"Simplify","text":"What is $$\\\\frac{28}{49}$$ * $$\\\\frac{a^5}{a^9}$$ * $$\\\\frac{b^{14}}{b^{12}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552poly7a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{4}{7}$$"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\frac{28}{49}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a161552poly8a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{8}$$"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\frac{30}{48}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552poly8a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{a^5}$$"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\frac{a^5}{a^{10}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a161552poly8a-h2-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{b^3}$$"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\frac{b^{11}}{b^{14}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a161552poly9","title":"Division of a Polynomial by a Monomial","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Dividing Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a161552poly9a","stepAnswer":["$$-4x+2y$$"],"problemType":"TextBox","stepTitle":"Find the quotient: $$\\\\frac{32x^2 y-16x y^2}{\\\\left(-8x y\\\\right)}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-4x+2y$$","hints":{"DefaultPathway":[{"id":"a161552poly9a-h1","type":"hint","dependencies":[],"title":"Decompose","text":"Divide each term by the divisor. 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y^2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{6x y}$$","hints":{"DefaultPathway":[{"id":"a171b3arationals13a-h1","type":"hint","dependencies":[],"title":"Common Factors","text":"Identify the common factor of the numerator and denomiator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a171b3arationals13a-h2","type":"hint","dependencies":["a171b3arationals13a-h1"],"title":"Rewrite the Fraction","text":"Simplify the fraction by diving the numerator and denominator by the common factor, $$3x y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a171b3arationals13a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{6x y}$$"],"dependencies":["a171b3arationals13a-h2"],"title":"Simplifed Fraction","text":"What is the simplified form of the fraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a171b3arationals14","title":"Simplify Rational Expressions","body":"Simplify the following fraction:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 Simplify Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a171b3arationals14a","stepAnswer":["$$\\\\frac{x}{3y}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{4x^2 y}{12x y^2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{x}{3y}$$","hints":{"DefaultPathway":[{"id":"a171b3arationals14a-h1","type":"hint","dependencies":[],"title":"Common Factors","text":"Identify the common factor of the numerator and denomiator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a171b3arationals14a-h2","type":"hint","dependencies":["a171b3arationals14a-h1"],"title":"Rewrite the Fraction","text":"Simplify the fraction by diving the numerator and denominator by the common factor, $$4x y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a171b3arationals14a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{x}{3y}$$"],"dependencies":["a171b3arationals14a-h2"],"title":"Simplifed Fraction","text":"What is the simplified form of the fraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a171b3arationals15","title":"Simplify Rational Expressions","body":"Simplify the following 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numerator and denominator by the common factor, $$2x y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a171b3arationals15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{8x}{y}$$"],"dependencies":["a171b3arationals15a-h2"],"title":"Simplifed Fraction","text":"What is the simplified form of the fraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a171b3arationals15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a171b3arationals15a-h2"],"title":"Solve","text":"What is the value of the variable when $$z=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a171b3arationals16","title":"Determine the 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If there are multiple answers, put a coma in between each in ascending order. Ex. 2,3","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 Simplify Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a171b3arationals16a","stepAnswer":["$$\\\\frac{5}{6}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{4p-1}{6p-5}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5}{6}$$","hints":{"DefaultPathway":[{"id":"a171b3arationals16a-h1","type":"hint","dependencies":[],"title":"Set Denominator to $$0$$","text":"Let\'s focus on the denominator of the expression and set it equal to zero","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a171b3arationals16a-h2","type":"hint","dependencies":["a171b3arationals16a-h1"],"title":"Solve","text":"Solve for the variable when the denominator is equal to $$0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a171b3arationals16a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{6}$$"],"dependencies":["a171b3arationals16a-h2"],"title":"Solve","text":"What is the value of the variable when $$6p-5=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a171b3arationals17","title":"Determine the Values for Which a Rational Expression is Undefined","body":"When is the expression undefined? If there are multiple answers, put a coma in between each in ascending order. Ex. 2,3","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 Simplify Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a171b3arationals17a","stepAnswer":["-2,4"],"problemType":"TextBox","stepTitle":"$$\\\\frac{n-3}{n^2+2n-8}$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a171b3arationals17a-h1","type":"hint","dependencies":[],"title":"Set Denominator to $$0$$","text":"Let\'s focus on the denominator of the expression and set it equal to zero","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a171b3arationals17a-h2","type":"hint","dependencies":["a171b3arationals17a-h1"],"title":"Solve","text":"Solve for the variable when the denominator is equal to $$0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a171b3arationals17a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["-2,4"],"dependencies":["a171b3arationals17a-h2"],"title":"Solve","text":"What is the value of the variable when $$n^2+2n-8=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a171b3arationals18","title":"Determine the Values for Which a Rational Expression is Undefined","body":"When is the expression undefined? If there are multiple answers, put a coma in between each in ascending order. 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Determine the value(s) for which the rational expression is undefined:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 Simplify Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a171b3arationals4a","stepAnswer":["$$-1$$ or $$-3$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{a+10}{a^2+4a+3}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-1$$ or $$-3$$","choices":["$$4$$ or $$2$$","$$-1$$ or $$-3$$","$$-10$$","$$-1$$ or $$-4$$"],"hints":{"DefaultPathway":[{"id":"a171b3arationals4a-h1","type":"hint","dependencies":[],"title":"Dissecting the Denominator","text":"For a fraction, the expression would be undefined if the denominator was zero. 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In order to solve this we must factor out the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a171b3arationals4a-h3","type":"hint","dependencies":["a171b3arationals4a-h2"],"title":"Factored Equation","text":"The factored form of the equation is $$\\\\left(a+1\\\\right) \\\\left(a+3\\\\right)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a171b3arationals4a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-1$$ or $$-3$$"],"dependencies":["a171b3arationals4a-h3"],"title":"Solving for the Value","text":"What are the value(s) for \\"a\\" that make the factored equation true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$-1$$ or $$-3$$","$$5$$ or $$6$$","$$-5$$ or $$2$$","$$1$$ or $$-6$$"]}]}}]},{"id":"a171b3arationals5","title":"Simplify Rational Expressions","body":"Determine the value(s) for which the rational expression is undefined:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 Simplify Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a171b3arationals5a","stepAnswer":["$$\\\\frac{-2}{3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{y-1}{3y+2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-2}{3}$$","hints":{"DefaultPathway":[{"id":"a171b3arationals5a-h1","type":"hint","dependencies":[],"title":"Dissecting the Denominator","text":"For a fraction, the expression would be undefined if the denominator was zero. 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Elementary Algebra","steps":[{"id":"a171b3arationals6a","stepAnswer":["$$\\\\frac{-3}{5}$$"],"problemType":"TextBox","stepTitle":"$$x=0$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-3}{5}$$","hints":{"DefaultPathway":[{"id":"a171b3arationals6a-h1","type":"hint","dependencies":[],"title":"Substituting Variable Into Expression","text":"Substitute $$0$$ for x: $$\\\\frac{2\\\\times0+3}{3\\\\times0-5}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a171b3arationals6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-3}{5}$$"],"dependencies":["a171b3arationals6a-h1"],"title":"Simplifying Expression","text":"What is the result once the expression is simplified?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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If the answer is undefined, enter UND.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 Simplify Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a171b3arationals9a","stepAnswer":["$$\\\\frac{7}{\\\\left(-4\\\\right)}$$"],"problemType":"TextBox","stepTitle":"$$x=0$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{7}{\\\\left(-4\\\\right)}$$","hints":{"DefaultPathway":[{"id":"a171b3arationals9a-h1","type":"hint","dependencies":[],"title":"Substituting Variable Into Expression","text":"Substitute $$0$$ for x: $$\\\\frac{0^2+8\\\\times0+7}{0^2-4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a171b3arationals9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{7}{\\\\left(-4\\\\right)}$$"],"dependencies":["a171b3arationals9a-h1"],"title":"Simplifying Expression","text":"What is the result once the expression is simplified?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a171b3arationals9b","stepAnswer":["UND"],"problemType":"TextBox","stepTitle":"$$x=2$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a171b3arationals9b-h1","type":"hint","dependencies":[],"title":"Substituting Variable Into Expression","text":"Substitute $$2$$ for x: $$\\\\frac{2^2+8\\\\times2+7}{2^2-4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a171b3arationals9b-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["UND"],"dependencies":["a171b3arationals9b-h1"],"title":"Simplifying Expression","text":"What is the result once the expression is simplified?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a171b3arationals9c","stepAnswer":["$$-8$$"],"problemType":"TextBox","stepTitle":"$$x=-1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-8$$","hints":{"DefaultPathway":[{"id":"a171b3arationals9c-h1","type":"hint","dependencies":[],"title":"Substituting Variable Into Expression","text":"Substitute $$-1$$ for x: $$\\\\frac{{\\\\left(-1\\\\right)}^2+8\\\\times2+7}{{\\\\left(-1\\\\right)}^2-4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a171b3arationals9c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-8$$"],"dependencies":["a171b3arationals9c-h1"],"title":"Simplifying Expression","text":"What is the result once the expression is simplified?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18455cprediction1","title":"Prediction","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Testing the Significance of the Correlation Coefficient","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a18455cprediction1a","stepAnswer":["$250,120"],"problemType":"MultipleChoice","stepTitle":"An electronics retailer used regression to find a simple model to predict sales growth in the first quarter of the new year (January through March). The model is good for $$90$$ days, where $$x$$ is the day. The model can be written as follows: \u0177 $$=$$ $$101.32$$ + $$2.48x$$ where \u0177 is in thousands of dollars. What would you predict the sales to be on day 60?","stepBody":"","answerType":"string","variabilization":{},"choices":["$250,120","$255,120","$250,000","$250,620","None of the above"],"hints":{"DefaultPathway":[{"id":"a18455cprediction1a-h1","type":"hint","dependencies":[],"title":"Breaking down the question","text":"Use the given equation and substitute $$x$$ $$=$$ $$60$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18455cprediction10","title":"Prediction","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Testing the Significance of the Correlation Coefficient","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a18455cprediction10a","stepAnswer":["\u0177 $$=$$ $$35.5818045$$ $$-$$ $$0.19182491x$$"],"problemType":"MultipleChoice","stepTitle":"Recently, the annual number of driver deaths per 100,000 for the selected age groups is shown in the table. Calculate the least squares $$(best-fit)$$ line. Put the equation in the form of: \u0177 $$=$$ a + bx","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"\u0177 $$=$$ $$35.5818045$$ - $$0.19182491x$$","choices":["\u0177 $$=$$ $$35.5818045$$ $$-$$ $$0.19182491x$$","\u0177 $$=$$ $$35.5818045$$ $$-$$ $$1.19182491x$$","\u0177 $$=$$ $$37.5818045$$ $$-$$ $$0.19182491x$$","\u0177 $$=$$ $$35.5818045$$ $$-$$ $$0.69182491x$$"],"hints":{"DefaultPathway":[{"id":"a18455cprediction10a-h1","type":"hint","dependencies":[],"title":"Breaking down the question","text":"Plot the given data points or substitute into the $$y=mx+c$$ equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18455cprediction11","title":"Prediction","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Testing the Significance of the Correlation Coefficient","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a18455cprediction11a","stepAnswer":["$$r$$ $$=$$ $$-0.57874$$, Not significant"],"problemType":"MultipleChoice","stepTitle":"Recently, the annual number of driver deaths per 100,000 for the selected age groups is shown in the table. Find the correlation coefficient. Is it significant?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$r$$ $$=$$ $$-0.57874$$, Not significant","choices":["$$r$$ $$=$$ $$-0.57874$$, Significant","$$r$$ $$=$$ $$-0.57874$$, Not significant","$$r$$ $$=$$ $$-0.77874$$, Significant","$$r$$ $$=$$ $$-0.57874$$, Not significant"],"hints":{"DefaultPathway":[{"id":"a18455cprediction11a-h1","type":"hint","dependencies":[],"title":"Significance","text":"For four df and alpha $$=$$ $$0.05$$, the LinRegTTest gives $$p-value$$ $$=$$ $$0.2288$$ so we do not reject the null hypothesis;","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18455cprediction11a-h2","type":"hint","dependencies":["a18455cprediction11a-h1"],"title":"Explanation","text":"Since we do not reject the null hypothesis, there is not a significant linear relationship between deaths and age.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18455cprediction12","title":"Prediction","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Testing the Significance of the Correlation Coefficient","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a18455cprediction12a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Recently, the annual number of driver deaths per 100,000 for the selected age groups is shown in the table. Based on the given data, is there a linear relationship between age of a driver and driver fatality rate?","stepBody":"","answerType":"string","variabilization":{},"choices":["No","Yes"],"hints":{"DefaultPathway":[{"id":"a18455cprediction12a-h1","type":"hint","dependencies":[],"title":"Relationship","text":"$$p-value$$ is greater than $$0.05$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18455cprediction12a-h2","type":"hint","dependencies":["a18455cprediction12a-h1"],"title":"Explanation","text":"Since $$p-value$$ is more than $$0.05$$, there is not a linear relationship between the two variables,","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18455cprediction2","title":"Prediction","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Testing the Significance of the Correlation Coefficient","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a18455cprediction2a","stepAnswer":["1,326 acres"],"problemType":"MultipleChoice","stepTitle":"Use the following information to answer the next three exercises. A landscaping company is hired to mow the grass for several large properties. The total area of the properties combined is 1,345 acres. The rate at which one person can mow is as follows: \u0177 $$=$$ $$1350$$ - $$1.2x$$ where $$x$$ is the number of hours and \u0177 represents the number of acres left to mow. How many acres will be left to mow after $$20$$ hours of work?","stepBody":"","answerType":"string","variabilization":{},"choices":["1,326 acres","1,300 acres","1,226 acres","1,350 acres","None of the above"],"hints":{"DefaultPathway":[{"id":"a18455cprediction2a-h1","type":"hint","dependencies":[],"title":"Breaking down the question","text":"Use the given equation and substitute $$x$$ $$=$$ $$20$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18455cprediction3","title":"Prediction","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Testing the Significance of the Correlation Coefficient","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a18455cprediction3a","stepAnswer":["1,125 hours, or when $$x$$ $$=$$ 1,125"],"problemType":"MultipleChoice","stepTitle":"Use the following information to answer the next three exercises. A landscaping company is hired to mow the grass for several large properties. The total area of the properties combined is 1,345 acres. The rate at which one person can mow is as follows: \u0177 $$=$$ $$1350$$ - $$1.2x$$ where $$x$$ is the number of hours and \u0177 represents the number of acres left to mow. How many hours will it take to mow all of the lawns? (When is \u0177 $$=$$ 0?)","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"1,125 hours, or when $$x$$ $$=$$ 1,125","choices":["1,125 hours, or when $$x$$ $$=$$ 1,125","1,200 hours, or when $$x$$ $$=$$ 1,200","1,100 hours, or when $$x$$ $$=$$ 1,100","1,225 hours, or when $$x$$ $$=$$ 1,225"],"hints":{"DefaultPathway":[{"id":"a18455cprediction3a-h1","type":"hint","dependencies":[],"title":"Breaking down the question","text":"Substitute \u0177 $$=$$ $$0$$ and solve for $$x$$. We end up getting $$x$$ $$=$$ 1,125","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18455cprediction4","title":"Prediction","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Testing the Significance of the Correlation Coefficient","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a18455cprediction4a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"This table contains real data for the first two decades of flu cases reporting. Enter your data into your calculator or computer. Should the pre-1981 data be included or not?","stepBody":"","answerType":"string","variabilization":{},"choices":["No","Yes"],"hints":{"DefaultPathway":[{"id":"a18455cprediction4a-h1","type":"hint","dependencies":[],"title":"Older data","text":"We don\u2019t know if the pre-1981 data was collected from a single year.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18455cprediction4a-h2","type":"hint","dependencies":["a18455cprediction4a-h1"],"title":"Explanation","text":"So we don\u2019t have an accurate $$x$$ value for this figure.Regression equation: \u0177 (#Flu Cases) $$=$$ -3,448,225 + $$1749.777$$ (year)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18455cprediction5","title":"Prediction","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Testing the Significance of the Correlation Coefficient","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a18455cprediction5a","stepAnswer":["2,552"],"problemType":"MultipleChoice","stepTitle":"This table contains real data for the first two decades of flu cases reporting. When $$x$$ $$=$$ $$1985$$, \u0177 $$=$$ $$___$$ .","stepBody":"","answerType":"string","variabilization":{},"choices":["2,552","2,500","2,452","2,700"],"hints":{"DefaultPathway":[{"id":"a18455cprediction5a-h1","type":"hint","dependencies":[],"title":"Breaking down the question","text":"Use the given equation and substitute $$x$$ $$=$$ $$1985$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18455cprediction6","title":"Prediction","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Testing the Significance of the Correlation Coefficient","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a18455cprediction6a","stepAnswer":["34,275"],"problemType":"MultipleChoice","stepTitle":"This table contains real data for the first two decades of flu cases reporting. When $$x$$ $$=$$ $$1990$$, \u0177 $$=$$ $$___$$ .","stepBody":"","answerType":"string","variabilization":{},"choices":["34,275","34,200","34,150","34,000"],"hints":{"DefaultPathway":[{"id":"a18455cprediction6a-h1","type":"hint","dependencies":[],"title":"Breaking down the question","text":"Use the given equation and substitute $$x$$ $$=$$ $$1990$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18455cprediction7","title":"Prediction","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Testing the Significance of the Correlation Coefficient","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a18455cprediction7a","stepAnswer":["$$-725$$, Makes Sense"],"problemType":"MultipleChoice","stepTitle":"This table contains real data for the first two decades of flu cases reporting. When $$x$$ $$=$$ $$1970$$, \u0177 $$=$$ $$___$$ . Also choose whether this answer makes sense or doesn\'t?","stepBody":"Let subscripts $$1$$ $$=$$ Gamma, $$2$$ $$=$$ Zeta","answerType":"string","variabilization":{},"answerLatex":"$$-725$$, Makes Sense","choices":["$$-725$$, Makes Sense","$$-725$$, Doesn\'t make Sense","$$-700$$, Makes Sense","$$-700$$, Doesn\'t make Sense"],"hints":{"DefaultPathway":[{"id":"a18455cprediction7a-h1","type":"hint","dependencies":[],"title":"Does it make sense","text":"The range of $$x$$ values was $$1981$$ to 2002; the year $$1970$$ is not in this range.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18455cprediction7a-h2","type":"hint","dependencies":["a18455cprediction7a-h1"],"title":"Explanation","text":"The regression equation does not apply, because predicting for the year $$1970$$ is extrapolation, which requires a different process. Also, a negative number does not make sense in this context, where we are predicting flu cases diagnosed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18455cprediction8","title":"Prediction","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Testing the Significance of the Correlation Coefficient","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a18455cprediction8a","stepAnswer":["\u0177 $$=$$ $$-3, 448, 225$$ + $$1750x$$"],"problemType":"MultipleChoice","stepTitle":"This table contains real data for the first two decades of flu cases reporting. What is the equation of the regression line for this data?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"\u0177 $$=$$ -3,448,225 + $$1750x$$","choices":["\u0177 $$=$$ $$-3, 448, 225$$ + $$1750x$$","\u0177 $$=$$ $$-3, 448, 225$$ + $$1700x$$","\u0177 $$=$$ $$-3, 448, 000$$ + $$1750x$$","\u0177 $$=$$ $$-3, 450, 225$$ + $$1750x$$"],"hints":{"DefaultPathway":[{"id":"a18455cprediction8a-h1","type":"hint","dependencies":[],"title":"Breaking down the question","text":"The data support that the melting point for Alloy Zeta is different from the melting point of Alloy Gamma.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18455cprediction9","title":"Prediction","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Testing the Significance of the Correlation Coefficient","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a18455cprediction9a","stepAnswer":["$$17.5$$, $$22$$, $$29.5$$, $$44.5$$, $$64.5$$, $$80$$"],"problemType":"MultipleChoice","stepTitle":"Recently, the annual number of driver deaths per 100,000 for the selected age groups is shown in the table. For each age group, pick the midpoint of the interval for the $$x$$ value. (For the 75+ group, use 80.)","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$17.5$$, $$22$$, $$29.5$$, $$44.5$$, $$64.5$$, $$80$$","choices":["$$17.5$$, $$22$$, $$29.5$$, $$44.5$$, $$64.5$$, $$80$$","$$18$$, $$23$$, $$29.5$$, $$44.5$$, $$64.5$$, $$80$$","$$18$$, $$22$$, $$29.5$$, $$45$$, $$65$$, $$80$$"],"hints":{"DefaultPathway":[{"id":"a18455cprediction9a-h1","type":"hint","dependencies":[],"title":"Breaking down the question","text":"Choose the midpoint of the interval for each age range","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a184b1aQuadratic1","title":"Quadratic Formula","body":"Solve the following equation with quadratic formula.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a184b1aQuadratic1a","stepAnswer":["$$x=\\\\frac{1}{2}$$ or $$-5$$"],"problemType":"MultipleChoice","stepTitle":"$$2x^2+9x-5=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=\\\\frac{1}{2}$$ or $$-5$$","choices":["$$x=\\\\frac{1}{2}$$ or $$-5$$","$$x=1$$ or $$5$$","$$x=2$$ or $$\\\\frac{5}{2}$$"],"hints":{"DefaultPathway":[{"id":"a184b1aQuadratic1a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The quadratic formula is in the form of $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$ and $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. Determine the value of a, $$b$$, and c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic1a-h2","type":"hint","dependencies":[],"title":"Using the Quadratic Formula","text":"Plug the values into quadratic formula. A polynomial will typically be into the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":[],"title":"Substitute into the + version","text":"What is $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":[],"title":"Substitute into the - version","text":"What is $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a184b1aQuadratic10","title":"Quadratic Formula","body":"Solve the following equation with quadratic formula.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a184b1aQuadratic10a","stepAnswer":["$$p=\\\\frac{\\\\left(-1+\\\\sqrt{26} i\\\\right)}{3}$$ or $$\\\\frac{\\\\left(-1-\\\\sqrt{26} i\\\\right)}{3}$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(3p\\\\right)}^2+2p+9=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$p=\\\\frac{\\\\left(-1+\\\\sqrt{26} i\\\\right)}{3}$$ or $$\\\\frac{\\\\left(-1-\\\\sqrt{26} i\\\\right)}{3}$$","choices":["$$p=\\\\frac{\\\\left(-1+2\\\\sqrt{6} i\\\\right)}{3}$$ or $$\\\\frac{\\\\left(-1-2\\\\sqrt{6} i\\\\right)}{3}$$","$$p=\\\\frac{\\\\left(-1+\\\\sqrt{13} i\\\\right)}{3}$$ or $$\\\\frac{\\\\left(-1-\\\\sqrt{13} i\\\\right)}{3}$$","$$p=\\\\frac{\\\\left(-1+\\\\sqrt{26} i\\\\right)}{3}$$ or $$\\\\frac{\\\\left(-1-\\\\sqrt{26} i\\\\right)}{3}$$","$$p=\\\\frac{\\\\left(-1+\\\\sqrt{26} i\\\\right)}{3}$$ or $$\\\\frac{\\\\frac{\\\\left(-1-\\\\sqrt{26} i\\\\right)}{3}}{5}$$"],"hints":{"DefaultPathway":[{"id":"a184b1aQuadratic10a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The quadratic formula is in the form of $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$ and $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. Determine the value of a, $$b$$, and c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic10a-h2","type":"hint","dependencies":[],"title":"Using the Quadratic Formula","text":"Plug the values into quadratic formula. A polynomial will typically be into the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\left(-1+\\\\sqrt{26}\\\\right)}{3}$$"],"dependencies":[],"title":"Substitute into the + version","text":"What is $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\left(-1-\\\\sqrt{26} i\\\\right)}{3}$$"],"dependencies":[],"title":"Substitute into the - version","text":"What is $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a184b1aQuadratic11","title":"Quadratic Formula","body":"Solve the following equation with quadratic formula.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a184b1aQuadratic11a","stepAnswer":["$$x=\\\\frac{5}{2}$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(4x\\\\right)}^2-20x=-25$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=\\\\frac{5}{2}$$","choices":["$$x=\\\\frac{5}{2}$$","$$x=\\\\frac{-5}{2}$$ or $$\\\\frac{5}{2}$$","$$x=-5$$ or $$5$$"],"hints":{"DefaultPathway":[{"id":"a184b1aQuadratic11a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The quadratic formula is in the form of $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$ and $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. Determine the value of a, $$b$$, and c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic11a-h2","type":"hint","dependencies":[],"title":"Using the Quadratic Formula","text":"Plug the values into quadratic formula. A polynomial will typically be into the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic11a-h3","type":"hint","dependencies":[],"title":"Creating the Polynomial","text":"Add $$25$$ on both sides to get the equation in the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{2}$$"],"dependencies":[],"title":"Substitute into the + version","text":"What is $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{2}$$"],"dependencies":[],"title":"Substitute into the - version","text":"What is $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a184b1aQuadratic12","title":"Quadratic Formula","body":"Solve the following equation with quadratic formula.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a184b1aQuadratic12a","stepAnswer":["$$r=\\\\frac{-5}{2}$$"],"problemType":"MultipleChoice","stepTitle":"$$r^2+10r+25=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$r=\\\\frac{-5}{2}$$","choices":["$$r=\\\\frac{5}{2}$$","$$r=\\\\frac{-5}{2}$$ or $$\\\\frac{5}{2}$$","$$r=\\\\frac{-5}{2}$$"],"hints":{"DefaultPathway":[{"id":"a184b1aQuadratic12a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The quadratic formula is in the form of $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$ and $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. Determine the value of a, $$b$$, and c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic12a-h2","type":"hint","dependencies":[],"title":"Using the Quadratic Formula","text":"Plug the values into quadratic formula. A polynomial will typically be into the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic12a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-5}{2}$$"],"dependencies":[],"title":"Substitute into the + version","text":"What is $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-5}{2}$$"],"dependencies":[],"title":"Substitute into the - version","text":"What is $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a184b1aQuadratic13","title":"Quadratic Formula","body":"Solve the following equation with quadratic formula.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a184b1aQuadratic13a","stepAnswer":["$$r=\\\\frac{4}{5}$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(25t\\\\right)}^2-40t=-16$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$r=\\\\frac{4}{5}$$","choices":["$$r=\\\\frac{4}{5}$$","$$r=\\\\frac{-4}{5}$$ or $$\\\\frac{4}{5}$$","$$r=\\\\frac{-4}{5}$$"],"hints":{"DefaultPathway":[{"id":"a184b1aQuadratic13a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The quadratic formula is in the form of $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$ and $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. 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A polynomial will typically be into the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic13a-h3","type":"hint","dependencies":[],"title":"Creating the Polynomial","text":"Add $$16$$ on both sides to get the equation in the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{4}{5}$$"],"dependencies":[],"title":"Substitute into the + version","text":"What is $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic13a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{4}{5}$$"],"dependencies":[],"title":"Substitute into the - version","text":"What is $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a184b1aQuadratic14","title":"Quadratic Formula","body":"Determine the number of solutions to the following quadratic equations","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a184b1aQuadratic14a","stepAnswer":["$$2$$ real solutions"],"problemType":"MultipleChoice","stepTitle":"a) $${\\\\left(3x\\\\right)}^2+7x-9=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2$$ real solutions","choices":["$$2$$ real solutions","$$1$$ real solution","$$2$$ complex solutions"],"hints":{"DefaultPathway":[{"id":"a184b1aQuadratic14a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The quadratic formula is in the form of $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$ and $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. Determine the value of a, $$b$$, and c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic14a-h2","type":"hint","dependencies":[],"title":"Using the Quadratic Formula","text":"Plug the values into quadratic formula. A polynomial will typically be into the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic14a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$157$$"],"dependencies":[],"title":"Substitute","text":"What is $$b^2-4ac$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a184b1aQuadratic14b","stepAnswer":["$$2$$ complex solutions"],"problemType":"MultipleChoice","stepTitle":"b) $${\\\\left(5n\\\\right)}^2+n+4=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2$$ complex solutions","choices":["$$2$$ real solutions","$$1$$ real solution","$$2$$ complex solutions"],"hints":{"DefaultPathway":[{"id":"a184b1aQuadratic14b-h1","type":"hint","dependencies":[],"title":"Principle","text":"The quadratic formula is in the form of $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$ and $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. 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Determine the value of a, $$b$$, and c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic14c-h2","type":"hint","dependencies":[],"title":"Using the Quadratic Formula","text":"Plug the values into quadratic formula. A polynomial will typically be into the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic14c-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":[],"title":"Substitute","text":"What is $$b^2-4ac$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a184b1aQuadratic15","title":"Quadratic Formula","body":"Determine the number of solutions to the following quadratic equations","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a184b1aQuadratic15a","stepAnswer":["$$2$$ complex solutions"],"problemType":"MultipleChoice","stepTitle":"a) $${\\\\left(8m\\\\right)}^2-3m+6=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2$$ complex solutions","choices":["$$2$$ real solutions","$$1$$ real solution","$$2$$ complex solutions"],"hints":{"DefaultPathway":[{"id":"a184b1aQuadratic15a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The quadratic formula is in the form of $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$ and $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. Determine the value of a, $$b$$, and c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic15a-h2","type":"hint","dependencies":[],"title":"Using the Quadratic Formula","text":"Plug the values into quadratic formula. A polynomial will typically be into the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-183$$"],"dependencies":[],"title":"Substitute","text":"What is $$b^2-4ac$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a184b1aQuadratic15b","stepAnswer":["$$2$$ real solutions"],"problemType":"MultipleChoice","stepTitle":"b) $${\\\\left(5z\\\\right)}^2+6z-2=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2$$ real solutions","choices":["$$2$$ real solutions","$$1$$ real solution","$$2$$ complex solutions"],"hints":{"DefaultPathway":[{"id":"a184b1aQuadratic15b-h1","type":"hint","dependencies":[],"title":"Principle","text":"The quadratic formula is in the form of $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$ and $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. Determine the value of a, $$b$$, and c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic15b-h2","type":"hint","dependencies":[],"title":"Using the Quadratic Formula","text":"Plug the values into quadratic formula. A polynomial will typically be into the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic15b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$76$$"],"dependencies":[],"title":"Substitute","text":"What is $$b^2-4ac$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a184b1aQuadratic15c","stepAnswer":["$$1$$ real solution"],"problemType":"MultipleChoice","stepTitle":"b) $${\\\\left(9w\\\\right)}^2+24w+16=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$1$$ real solution","choices":["$$2$$ real solutions","$$1$$ real solution","$$2$$ complex solutions"],"hints":{"DefaultPathway":[{"id":"a184b1aQuadratic15c-h1","type":"hint","dependencies":[],"title":"Principle","text":"The quadratic formula is in the form of $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$ and $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. Determine the value of a, $$b$$, and c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic15c-h2","type":"hint","dependencies":[],"title":"Using the Quadratic Formula","text":"Plug the values into quadratic formula. A polynomial will typically be into the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic15c-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":[],"title":"Substitute","text":"What is $$b^2-4ac$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a184b1aQuadratic16","title":"Quadratic Formula","body":"Solve the following equation with quadratic formula.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a184b1aQuadratic16a","stepAnswer":["$$m=\\\\frac{3}{4}$$ or $$-1$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(4m\\\\right)}^2+m-3=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$m=\\\\frac{3}{4}$$ or $$-1$$","choices":["$$m=\\\\frac{1}{2}$$ or $$-1$$","$$m=\\\\frac{3}{4}$$ or $$-1$$","$$m=2$$ or $$-1$$"],"hints":{"DefaultPathway":[{"id":"a184b1aQuadratic16a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The quadratic formula is in the form of $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$ and $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. Determine the value of a, $$b$$, and c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic16a-h2","type":"hint","dependencies":[],"title":"Using the Quadratic Formula","text":"Plug the values into quadratic formula. A polynomial will typically be into the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic16a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{4}$$"],"dependencies":[],"title":"Substitute into the + version","text":"What is $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic16a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":[],"title":"Substitute into the - version","text":"What is $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a184b1aQuadratic17","title":"Quadratic Formula","body":"Solve the following equation with quadratic formula.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a184b1aQuadratic17a","stepAnswer":["$$n=\\\\frac{5}{4}$$ or $$1$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(4n\\\\right)}^2-9n+5=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$n=\\\\frac{5}{4}$$ or $$1$$","choices":["$$n=\\\\frac{5}{4}$$ or $$-1$$","$$n=\\\\frac{5}{4}$$ or $$1$$","$$n=\\\\frac{5}{4}$$ or $$-1$$"],"hints":{"DefaultPathway":[{"id":"a184b1aQuadratic17a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The quadratic formula is in the form of $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$ and $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. Determine the value of a, $$b$$, and c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic17a-h2","type":"hint","dependencies":[],"title":"Using the Quadratic Formula","text":"Plug the values into quadratic formula. A polynomial will typically be into the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic17a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{4}$$"],"dependencies":[],"title":"Substitute into the + version","text":"What is $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic17a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Substitute into the - version","text":"What is $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a184b1aQuadratic18","title":"Quadratic Formula","body":"Solve the following equation with quadratic formula.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a184b1aQuadratic18a","stepAnswer":["$$p=\\\\frac{1}{2}$$ or $$3$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(2p\\\\right)}^2-7p+3=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$p=\\\\frac{1}{2}$$ or $$3$$","choices":["$$p=\\\\frac{1}{2}$$ or $$-3$$","$$p=\\\\frac{1}{2}$$ or $$3$$","$$p=\\\\frac{-1}{2}$$ or $$-3$$"],"hints":{"DefaultPathway":[{"id":"a184b1aQuadratic18a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The quadratic formula is in the form of $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$ and $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. Determine the value of a, $$b$$, and c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic18a-h2","type":"hint","dependencies":[],"title":"Using the Quadratic Formula","text":"Plug the values into quadratic formula. A polynomial will typically be into the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic18a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":[],"title":"Substitute into the + version","text":"What is $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic18a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":[],"title":"Substitute into the - version","text":"What is $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a184b1aQuadratic19","title":"Quadratic Formula","body":"Solve the following equation with quadratic formula.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a184b1aQuadratic19a","stepAnswer":["$$q=-6$$ or $$3$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(3q\\\\right)}^2+8q-3=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$q=-6$$ or $$3$$","choices":["$$q=6$$ or $$-3$$","$$q=-6$$ or $$3$$","$$q=-6$$ or $$-3$$"],"hints":{"DefaultPathway":[{"id":"a184b1aQuadratic19a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The quadratic formula is in the form of $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$ and $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. Determine the value of a, $$b$$, and c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic19a-h2","type":"hint","dependencies":[],"title":"Using the Quadratic Formula","text":"Plug the values into quadratic formula. A polynomial will typically be into the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic19a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":[],"title":"Substitute into the + version","text":"What is $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic19a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6$$"],"dependencies":[],"title":"Substitute into the - version","text":"What is $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a184b1aQuadratic2","title":"Quadratic Formula","body":"Solve the following equation with quadratic formula.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a184b1aQuadratic2a","stepAnswer":["$$y=1$$ or $$\\\\frac{2}{3}$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(3y\\\\right)}^2-5y+2=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=1$$ or $$\\\\frac{2}{3}$$","choices":["$$y=\\\\frac{1}{2}$$ or $$\\\\frac{-2}{3}$$","$$y=1$$ or $$\\\\frac{2}{3}$$","$$y=2$$ or $$\\\\frac{1}{3}$$"],"hints":{"DefaultPathway":[{"id":"a184b1aQuadratic2a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The quadratic formula is in the form of $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$ and $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. Determine the value of a, $$b$$, and c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic2a-h2","type":"hint","dependencies":[],"title":"Using the Quadratic Formula","text":"Plug the values into quadratic formula. A polynomial will typically be into the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Substitute into the + version","text":"What is $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{3}$$"],"dependencies":[],"title":"Substitute into the - version","text":"What is $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a184b1aQuadratic20","title":"Quadratic Formula","body":"Solve the following equation with quadratic formula.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a184b1aQuadratic20a","stepAnswer":["$$r=11$$ or $$-3$$"],"problemType":"MultipleChoice","stepTitle":"$$r^2-8r=33$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$r=11$$ or $$-3$$","choices":["$$r=11$$ or $$-3$$","$$r=11$$ or $$3$$","$$r=-11$$ or $$-3$$"],"hints":{"DefaultPathway":[{"id":"a184b1aQuadratic20a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The quadratic formula is in the form of $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$ and $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. Determine the value of a, $$b$$, and c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic20a-h2","type":"hint","dependencies":[],"title":"Using the Quadratic Formula","text":"Plug the values into quadratic formula. A polynomial will typically be into the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic20a-h3","type":"hint","dependencies":[],"title":"Creating the Polynomial","text":"Subtract $$33$$ on both sides to get the equation in the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic20a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":[],"title":"Substitute into the + version","text":"What is $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic20a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6$$"],"dependencies":[],"title":"Substitute into the - version","text":"What is $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a184b1aQuadratic21","title":"Quadratic Formula","body":"Solve the following equation with quadratic formula.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a184b1aQuadratic21a","stepAnswer":["$$t=-5$$ or $$-8$$"],"problemType":"MultipleChoice","stepTitle":"$$t^2+13t=-40$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$t=-5$$ or $$-8$$","choices":["$$t=5$$ or $$-8$$","$$t=-5$$ or $$8$$","$$t=-5$$ or $$-8$$"],"hints":{"DefaultPathway":[{"id":"a184b1aQuadratic21a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The quadratic formula is in the form of $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$ and $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. Determine the value of a, $$b$$, and c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic21a-h2","type":"hint","dependencies":[],"title":"Using the Quadratic Formula","text":"Plug the values into quadratic formula. A polynomial will typically be into the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic21a-h3","type":"hint","dependencies":[],"title":"Creating the Polynomial","text":"Add $$40$$ on both sides to get the equation in the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic21a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":[],"title":"Substitute into the + version","text":"What is $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic21a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-8$$"],"dependencies":[],"title":"Substitute into the - version","text":"What is $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a184b1aQuadratic22","title":"Quadratic Formula","body":"Solve the following equation with quadratic formula.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a184b1aQuadratic22a","stepAnswer":["$$u=\\\\frac{\\\\left(-7+\\\\sqrt{73}\\\\right)}{6}$$ or $$\\\\frac{\\\\left(-7-\\\\sqrt{73}\\\\right)}{6}$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(3u\\\\right)}^2+7u-2=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$u=\\\\frac{\\\\left(-7+\\\\sqrt{73}\\\\right)}{6}$$ or $$\\\\frac{\\\\left(-7-\\\\sqrt{73}\\\\right)}{6}$$","choices":["$$u=\\\\frac{\\\\left(-7+\\\\sqrt{3}\\\\right)}{2}$$ or $$\\\\frac{\\\\left(-7-\\\\sqrt{3}\\\\right)}{2}$$","$$u=\\\\frac{\\\\left(-7+\\\\sqrt{73}\\\\right)}{2}$$ or $$\\\\frac{\\\\left(-7-\\\\sqrt{73}\\\\right)}{2}$$","$$u=\\\\frac{\\\\left(-7+\\\\sqrt{73}\\\\right)}{6}$$ or $$\\\\frac{\\\\left(-7-\\\\sqrt{73}\\\\right)}{6}$$"],"hints":{"DefaultPathway":[{"id":"a184b1aQuadratic22a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The quadratic formula is in the form of $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$ and $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. Determine the value of a, $$b$$, and c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic22a-h2","type":"hint","dependencies":[],"title":"Using the Quadratic Formula","text":"Plug the values into quadratic formula. A polynomial will typically be into the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic22a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\left(-7+\\\\sqrt{73}\\\\right)}{6}$$"],"dependencies":[],"title":"Substitute into the + version","text":"What is $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic22a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\left(-7-\\\\sqrt{73}\\\\right)}{6}$$"],"dependencies":[],"title":"Substitute into the - version","text":"What is $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a184b1aQuadratic23","title":"Quadratic Formula","body":"Solve the following equation with quadratic formula.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a184b1aQuadratic23a","stepAnswer":["$$p=\\\\frac{\\\\left(-4+\\\\sqrt{6}\\\\right)}{2}$$ or $$\\\\frac{\\\\left(-4-\\\\sqrt{6}\\\\right)}{2}$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(2p\\\\right)}^2+8p+5=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$p=\\\\frac{\\\\left(-4+\\\\sqrt{6}\\\\right)}{2}$$ or $$\\\\frac{\\\\left(-4-\\\\sqrt{6}\\\\right)}{2}$$","choices":["$$p=\\\\frac{\\\\left(-4+\\\\sqrt{6}\\\\right)}{2}$$ or $$\\\\frac{\\\\left(-4-\\\\sqrt{6}\\\\right)}{2}$$","$$p=\\\\frac{\\\\left(-4+\\\\sqrt{3}\\\\right)}{2}$$ or $$\\\\frac{\\\\left(-4-\\\\sqrt{3}\\\\right)}{2}$$","$$u=\\\\frac{\\\\left(-4+\\\\sqrt{2}\\\\right)}{2}$$ or $$\\\\frac{\\\\left(-4-\\\\sqrt{2}\\\\right)}{2}$$"],"hints":{"DefaultPathway":[{"id":"a184b1aQuadratic23a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The quadratic formula is in the form of $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$ and $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. Determine the value of a, $$b$$, and c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic23a-h2","type":"hint","dependencies":[],"title":"Using the Quadratic Formula","text":"Plug the values into quadratic formula. A polynomial will typically be into the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic23a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\left(-4+\\\\sqrt{6}\\\\right)}{2}$$"],"dependencies":[],"title":"Substitute into the + version","text":"What is $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic23a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\left(-4-\\\\sqrt{6}\\\\right)}{2}$$"],"dependencies":[],"title":"Substitute into the - version","text":"What is $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a184b1aQuadratic24","title":"Quadratic Formula","body":"Solve the following equation with quadratic formula.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a184b1aQuadratic24a","stepAnswer":["$$a=\\\\frac{3+\\\\sqrt{3}}{2}$$ or $$\\\\frac{3-\\\\sqrt{3}}{2}$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(2a\\\\right)}^2-6a+3=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$a=\\\\frac{3+\\\\sqrt{3}}{2}$$ or $$\\\\frac{3-\\\\sqrt{3}}{2}$$","choices":["$$a=\\\\frac{\\\\left(-3+\\\\sqrt{6}\\\\right)}{2}$$ or $$\\\\frac{\\\\left(-3-\\\\sqrt{6}\\\\right)}{2}$$","$$a=\\\\frac{\\\\left(-3+\\\\sqrt{3}\\\\right)}{2}$$ or $$\\\\frac{\\\\left(-3-\\\\sqrt{3}\\\\right)}{2}$$","$$a=\\\\frac{3+\\\\sqrt{3}}{2}$$ or $$\\\\frac{3-\\\\sqrt{3}}{2}$$"],"hints":{"DefaultPathway":[{"id":"a184b1aQuadratic24a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The quadratic formula is in the form of $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$ and $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. Determine the value of a, $$b$$, and c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic24a-h2","type":"hint","dependencies":[],"title":"Using the Quadratic Formula","text":"Plug the values into quadratic formula. A polynomial will typically be into the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic24a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3+\\\\sqrt{3}}{2}$$"],"dependencies":[],"title":"Substitute into the + version","text":"What is $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic24a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3-\\\\sqrt{3}}{2}$$"],"dependencies":[],"title":"Substitute into the - version","text":"What is $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a184b1aQuadratic25","title":"Quadratic Formula","body":"Solve the following equation with quadratic formula.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a184b1aQuadratic25a","stepAnswer":["$$b=\\\\frac{\\\\left(-1+\\\\sqrt{21}\\\\right)}{5}$$ or $$\\\\frac{\\\\left(-1-\\\\sqrt{21}\\\\right)}{5}$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(5b\\\\right)}^2+2b-4=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$b=\\\\frac{\\\\left(-1+\\\\sqrt{21}\\\\right)}{5}$$ or $$\\\\frac{\\\\left(-1-\\\\sqrt{21}\\\\right)}{5}$$","choices":["$$b=\\\\frac{\\\\left(-1+\\\\sqrt{21}\\\\right)}{5}$$ or $$\\\\frac{\\\\left(-1-\\\\sqrt{21}\\\\right)}{5}$$","$$b=\\\\frac{1+\\\\sqrt{21}}{5}$$ or $$\\\\frac{1-\\\\sqrt{21}}{5}$$","$$b=\\\\frac{1+\\\\sqrt{7}}{2}$$ or $$\\\\frac{1-\\\\sqrt{7}}{2}$$"],"hints":{"DefaultPathway":[{"id":"a184b1aQuadratic25a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The quadratic formula is in the form of $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$ and $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. 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A polynomial will typically be into the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic25a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\left(-1+\\\\sqrt{21}\\\\right)}{5}$$"],"dependencies":[],"title":"Substitute into the + version","text":"What is $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic25a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\left(-1-\\\\sqrt{21}\\\\right)}{5}$$"],"dependencies":[],"title":"Substitute into the - version","text":"What is $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a184b1aQuadratic26","title":"Quadratic Formula","body":"Solve the following equation with quadratic formula.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a184b1aQuadratic26a","stepAnswer":["$$x=-4+2\\\\sqrt{5}$$ or $$--4-2\\\\sqrt{5}$$"],"problemType":"MultipleChoice","stepTitle":"$$x^2+8x-4=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=-4+2\\\\sqrt{5}$$ or $$--4-2\\\\sqrt{5}$$","choices":["$$x=-2+2\\\\sqrt{5}$$ or $$-2-2\\\\sqrt{5}$$","$$x=-4+2\\\\sqrt{5}$$ or $$--4-2\\\\sqrt{5}$$","$$x=-4+2\\\\sqrt{5}$$ or $$-4-2\\\\sqrt{5}$$","$$x=-4+\\\\sqrt{5}$$ or $$-4-\\\\sqrt{5}$$"],"hints":{"DefaultPathway":[{"id":"a184b1aQuadratic26a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The quadratic formula is in the form of $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$ and $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. 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A polynomial will typically be into the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic26a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4+2\\\\sqrt{5}$$"],"dependencies":[],"title":"Substitute into the + version","text":"What is $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic26a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4-2\\\\sqrt{5}$$"],"dependencies":[],"title":"Substitute into the - version","text":"What is $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a184b1aQuadratic27","title":"Quadratic Formula","body":"Solve the following equation with quadratic formula.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a184b1aQuadratic27a","stepAnswer":["$$y=-2+2\\\\sqrt{2}$$ or $$--2-2\\\\sqrt{2}$$"],"problemType":"MultipleChoice","stepTitle":"$$y^2+4y-4=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=-2+2\\\\sqrt{2}$$ or $$--2-2\\\\sqrt{2}$$","choices":["$$y=-1+2\\\\sqrt{2}$$ or $$-1-2\\\\sqrt{2}$$","$$y=-2+2\\\\sqrt{2}$$ or $$--2-2\\\\sqrt{2}$$","$$y=-2+2\\\\sqrt{2}$$ or $$-2-2\\\\sqrt{2}$$","$$y=-2+\\\\sqrt{2}$$ or $$-2-\\\\sqrt{2}$$"],"hints":{"DefaultPathway":[{"id":"a184b1aQuadratic27a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The quadratic formula is in the form of $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$ and $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. Determine the value of a, $$b$$, and c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic27a-h2","type":"hint","dependencies":[],"title":"Using the Quadratic Formula","text":"Plug the values into quadratic formula. A polynomial will typically be into the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic27a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2+2\\\\sqrt{2}$$"],"dependencies":[],"title":"Substitute into the + version","text":"What is $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic27a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2-2\\\\sqrt{2}$$"],"dependencies":[],"title":"Substitute into the - version","text":"What is $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a184b1aQuadratic28","title":"Quadratic Formula","body":"Solve the following equation with quadratic formula.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a184b1aQuadratic28a","stepAnswer":["$$y=\\\\frac{1}{3}$$ or $$-2$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(3y\\\\right)}^2+5y-2=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=\\\\frac{1}{3}$$ or $$-2$$","choices":["$$y=\\\\frac{1}{3}$$ or $$-2$$","$$y=\\\\frac{2}{3}$$ or $$-2$$","$$y=\\\\frac{1}{3}$$ or $$2$$"],"hints":{"DefaultPathway":[{"id":"a184b1aQuadratic28a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The quadratic formula is in the form of $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$ and $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. Determine the value of a, $$b$$, and c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic28a-h2","type":"hint","dependencies":[],"title":"Using the Quadratic Formula","text":"Plug the values into quadratic formula. A polynomial will typically be into the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic28a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":[],"title":"Substitute into the + version","text":"What is $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic28a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":[],"title":"Substitute into the - version","text":"What is $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a184b1aQuadratic29","title":"Quadratic Formula","body":"Solve the following equation with quadratic formula.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a184b1aQuadratic29a","stepAnswer":["$$y=\\\\frac{5}{3}$$ or $$-2$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(6x\\\\right)}^2+2x-20=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=\\\\frac{5}{3}$$ or $$-2$$","choices":["$$y=\\\\frac{5}{3}$$ or $$-2$$","$$y=\\\\frac{-5}{3}$$ or $$-2$$","$$y=\\\\frac{5}{3}$$ or $$2$$"],"hints":{"DefaultPathway":[{"id":"a184b1aQuadratic29a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The quadratic formula is in the form of $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$ and $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. Determine the value of a, $$b$$, and c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic29a-h2","type":"hint","dependencies":[],"title":"Using the Quadratic Formula","text":"Plug the values into quadratic formula. A polynomial will typically be into the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic29a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{3}$$"],"dependencies":[],"title":"Substitute into the + version","text":"What is $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic29a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":[],"title":"Substitute into the - version","text":"What is $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a184b1aQuadratic3","title":"Quadratic Formula","body":"Solve the following equation with quadratic formula.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a184b1aQuadratic3a","stepAnswer":["$$z=1$$ or $$\\\\frac{-3}{2}$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(4z\\\\right)}^2+2z-6=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$z=1$$ or $$\\\\frac{-3}{2}$$","choices":["$$z=1$$ or $$\\\\frac{-2}{3}$$","$$z=1$$ or $$\\\\frac{3}{2}$$","$$z=1$$ or $$\\\\frac{-3}{2}$$"],"hints":{"DefaultPathway":[{"id":"a184b1aQuadratic3a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The quadratic formula is in the form of $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$ and $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. Determine the value of a, $$b$$, and c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic3a-h2","type":"hint","dependencies":[],"title":"Using the Quadratic Formula","text":"Plug the values into quadratic formula. A polynomial will typically be into the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Substitute into the + version","text":"What is $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-3}{2}$$"],"dependencies":[],"title":"Substitute into the - version","text":"What is $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a184b1aQuadratic30","title":"Quadratic Formula","body":"Solve the following equation with quadratic formula.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a184b1aQuadratic30a","stepAnswer":["$$c=\\\\frac{-3}{4}$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(16c\\\\right)}^2+24c+9=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$c=\\\\frac{-3}{4}$$","choices":["$$c=\\\\frac{3}{4}$$ or $$\\\\frac{-3}{4}$$","$$c=\\\\frac{-3}{4}$$","$$c=\\\\frac{3}{4}$$"],"hints":{"DefaultPathway":[{"id":"a184b1aQuadratic30a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The quadratic formula is in the form of $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$ and $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. Determine the value of a, $$b$$, and c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic30a-h2","type":"hint","dependencies":[],"title":"Using the Quadratic Formula","text":"Plug the values into quadratic formula. A polynomial will typically be into the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic30a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-3}{4}$$"],"dependencies":[],"title":"Substitute into the + version","text":"What is $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic30a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-3}{4}$$"],"dependencies":[],"title":"Substitute into the - version","text":"What is $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a184b1aQuadratic4","title":"Quadratic Formula","body":"Solve the following equation with quadratic formula.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a184b1aQuadratic4a","stepAnswer":["$$x=1$$ or $$5$$"],"problemType":"MultipleChoice","stepTitle":"$$x^2-6x=-5$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=1$$ or $$5$$","choices":["$$x=1$$ or $$5$$","$$x=-1$$ or $$5$$","$$x=1$$ or $$-5$$"],"hints":{"DefaultPathway":[{"id":"a184b1aQuadratic4a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The quadratic formula is in the form of $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$ and $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. Determine the value of a, $$b$$, and c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic4a-h2","type":"hint","dependencies":[],"title":"Using the Quadratic Formula","text":"Plug the values into quadratic formula. A polynomial will typically be into the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic4a-h3","type":"hint","dependencies":[],"title":"Creating the Polynomial","text":"Add $$5$$ on both sides to get the equation in the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Substitute into the + version","text":"What is $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic4a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":[],"title":"Substitute into the - version","text":"What is $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a184b1aQuadratic5","title":"Quadratic Formula","body":"Solve the following equation with quadratic formula.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a184b1aQuadratic5a","stepAnswer":["$$a=-3$$ or $$5$$"],"problemType":"MultipleChoice","stepTitle":"$$a^2-2a=15$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$a=-3$$ or $$5$$","choices":["$$a=1$$ or $$5$$","$$a=-2$$ or $$5$$","$$a=-3$$ or $$5$$"],"hints":{"DefaultPathway":[{"id":"a184b1aQuadratic5a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The quadratic formula is in the form of $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$ and $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. Determine the value of a, $$b$$, and c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic5a-h2","type":"hint","dependencies":[],"title":"Using the Quadratic Formula","text":"Plug the values into quadratic formula. A polynomial will typically be into the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic5a-h3","type":"hint","dependencies":[],"title":"Creating the Polynomial","text":"Subtract $$15$$ on both sides to get the equation in the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":[],"title":"Substitute into the + version","text":"What is $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":[],"title":"Substitute into the - version","text":"What is $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a184b1aQuadratic6","title":"Quadratic Formula","body":"Solve the following equation with quadratic formula.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a184b1aQuadratic6a","stepAnswer":["$$b=-4$$ or $$-6$$"],"problemType":"MultipleChoice","stepTitle":"$$b^2+24=\\\\left(-10b\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$b=-4$$ or $$-6$$","choices":["$$b=-4$$ or $$6$$","$$b=-4$$ or $$-6$$","$$b=4$$ or $$-6$$"],"hints":{"DefaultPathway":[{"id":"a184b1aQuadratic6a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The quadratic formula is in the form of $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$ and $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. Determine the value of a, $$b$$, and c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic6a-h2","type":"hint","dependencies":[],"title":"Using the Quadratic Formula","text":"Plug the values into quadratic formula. A polynomial will typically be into the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic6a-h3","type":"hint","dependencies":[],"title":"Creating the Polynomial","text":"Add $$10b$$ on both sides to get the equation in the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":[],"title":"Substitute into the + version","text":"What is $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6$$"],"dependencies":[],"title":"Substitute into the - version","text":"What is $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a184b1aQuadratic7","title":"Quadratic Formula","body":"Solve the following equation with quadratic formula.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a184b1aQuadratic7a","stepAnswer":["$$x=\\\\frac{\\\\left(-5+\\\\sqrt{3}\\\\right)}{2}$$ or $$\\\\frac{\\\\left(-5-\\\\sqrt{3}\\\\right)}{2}$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(2x\\\\right)}^2+10x+11=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=\\\\frac{\\\\left(-5+\\\\sqrt{3}\\\\right)}{2}$$ or $$\\\\frac{\\\\left(-5-\\\\sqrt{3}\\\\right)}{2}$$","choices":["$$x=\\\\frac{\\\\left(-5+\\\\sqrt{3}\\\\right)}{2}$$ or $$\\\\frac{\\\\left(-5-\\\\sqrt{3}\\\\right)}{2}$$","$$x=\\\\frac{\\\\left(-3+\\\\sqrt{5}\\\\right)}{2}$$ or $$\\\\frac{\\\\left(-3-\\\\sqrt{5}\\\\right)}{2}$$","$$x=\\\\frac{\\\\left(-5+\\\\sqrt{3}\\\\right)}{2}$$ or $$\\\\frac{\\\\left(-3-\\\\sqrt{5}\\\\right)}{2}$$"],"hints":{"DefaultPathway":[{"id":"a184b1aQuadratic7a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The quadratic formula is in the form of $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$ and $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. Determine the value of a, $$b$$, and c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic7a-h2","type":"hint","dependencies":[],"title":"Using the Quadratic Formula","text":"Plug the values into quadratic formula. 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Determine the value of a, $$b$$, and c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic8a-h2","type":"hint","dependencies":[],"title":"Using the Quadratic Formula","text":"Plug the values into quadratic formula. A polynomial will typically be into the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\left(-6+\\\\sqrt{15}\\\\right)}{3}$$"],"dependencies":[],"title":"Substitute into the + version","text":"What is $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\left(-6-\\\\sqrt{15}\\\\right)}{3}$$"],"dependencies":[],"title":"Substitute into the - version","text":"What is $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a184b1aQuadratic9","title":"Quadratic Formula","body":"Solve the following equation with quadratic formula.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a184b1aQuadratic9a","stepAnswer":["$$n=\\\\frac{\\\\left(-2+2\\\\sqrt{6}\\\\right)}{5}$$ or $$\\\\frac{\\\\left(-2-2\\\\sqrt{6}\\\\right)}{5}$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(5n\\\\right)}^2+4n-4=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$n=\\\\frac{\\\\left(-2+2\\\\sqrt{6}\\\\right)}{5}$$ or $$\\\\frac{\\\\left(-2-2\\\\sqrt{6}\\\\right)}{5}$$","choices":["$$n=\\\\frac{\\\\left(-2+\\\\sqrt{6}\\\\right)}{5}$$ or $$\\\\frac{\\\\left(-2-\\\\sqrt{6}\\\\right)}{5}$$","$$n=\\\\frac{\\\\left(-2+2\\\\sqrt{6}\\\\right)}{5}$$ or $$\\\\frac{\\\\left(-2-2\\\\sqrt{6}\\\\right)}{5}$$","$$n=\\\\frac{\\\\left(-2+\\\\sqrt{3}\\\\right)}{5}$$ or $$\\\\frac{\\\\left(-2-\\\\sqrt{3}\\\\right)}{5}$$"],"hints":{"DefaultPathway":[{"id":"a184b1aQuadratic9a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The quadratic formula is in the form of $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$ and $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. Determine the value of a, $$b$$, and c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic9a-h2","type":"hint","dependencies":[],"title":"Using the Quadratic Formula","text":"Plug the values into quadratic formula. A polynomial will typically be into the form of $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\left(-2+\\\\sqrt{6}\\\\right)}{5}$$"],"dependencies":[],"title":"Substitute into the + version","text":"What is $$\\\\frac{\\\\left(-b+\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a184b1aQuadratic9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\left(-2-\\\\sqrt{6}\\\\right)}{5}$$"],"dependencies":[],"title":"Substitute into the - version","text":"What is $$\\\\frac{\\\\left(-b-\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18a5a6slope1","title":"Calculating the Slope of a Line","body":"Use the image to answer the question.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4  Understand Slope of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a18a5a6slope1a","stepAnswer":["$$\\\\frac{3}{4}$$"],"problemType":"TextBox","stepTitle":"What is the slope of the line on the geoboard shown?","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{4}$$","hints":{"DefaultPathway":[{"id":"a18a5a6slope1a-h1","type":"hint","dependencies":[],"title":"Definition of the Slope of a Line","text":"The slope of a line is $$m=\\\\frac{rise}{run}$$. The rise measures the vertical change and the run measures the horizontal change between two points on the line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18a5a6slope10","title":"Calculating the Slope of a Line From a Graph","body":"Use the image to answer the question.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4  Understand Slope of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a18a5a6slope10a","stepAnswer":["$$\\\\frac{-2}{3}$$"],"problemType":"TextBox","stepTitle":"What is the slope of the line on the graph shown?","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-2}{3}$$","hints":{"DefaultPathway":[{"id":"a18a5a6slope10a-h1","type":"hint","dependencies":[],"title":"Definition of the Slope of a Line","text":"The slope of a line is $$m=\\\\frac{rise}{run}$$. The rise measures the vertical change and the run measures the horizontal change between two points on the line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18a5a6slope10a-h2","type":"hint","dependencies":["a18a5a6slope10a-h1"],"title":"Choosing Two Points","text":"We locate two points on the line whose coordinates are integers. We then start with the point on the left and sketch a right triangle, so we can count the rise and run.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18a5a6slope11","title":"Calculating the Slope of a Line From a Graph","body":"Use the image to answer the question.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4  Understand Slope of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a18a5a6slope11a","stepAnswer":["$$\\\\frac{-4}{3}$$"],"problemType":"TextBox","stepTitle":"What is the slope of the line on the graph shown?","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-4}{3}$$","hints":{"DefaultPathway":[{"id":"a18a5a6slope11a-h1","type":"hint","dependencies":[],"title":"Definition of the Slope of a Line","text":"The slope of a line is $$m=\\\\frac{rise}{run}$$. The rise measures the vertical change and the run measures the horizontal change between two points on the line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18a5a6slope11a-h2","type":"hint","dependencies":["a18a5a6slope11a-h1"],"title":"Choosing Two Points","text":"We locate two points on the line whose coordinates are integers. We then start with the point on the left and sketch a right triangle, so we can count the rise and run.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18a5a6slope12","title":"Calculating the Slope of a Line From a Graph","body":"Use the image to answer the question.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4  Understand Slope of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a18a5a6slope12a","stepAnswer":["$$\\\\frac{-2}{3}$$"],"problemType":"TextBox","stepTitle":"What is the slope of the line on the graph shown?","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-2}{3}$$","hints":{"DefaultPathway":[{"id":"a18a5a6slope12a-h1","type":"hint","dependencies":[],"title":"Definition of the Slope of a Line","text":"The slope of a line is $$m=\\\\frac{rise}{run}$$. The rise measures the vertical change and the run measures the horizontal change between two points on the line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18a5a6slope12a-h2","type":"hint","dependencies":["a18a5a6slope12a-h1"],"title":"Choosing Two Points","text":"We locate two points on the line whose coordinates are integers. We then start with the point on the left and sketch a right triangle, so we can count the rise and run.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18a5a6slope13","title":"Calculating the Slope of a Line From a Graph","body":"Use the image to answer the question.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4  Understand Slope of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a18a5a6slope13a","stepAnswer":["$$\\\\frac{3}{5}$$"],"problemType":"TextBox","stepTitle":"What is the slope of the line on the graph shown?","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{5}$$","hints":{"DefaultPathway":[{"id":"a18a5a6slope13a-h1","type":"hint","dependencies":[],"title":"Definition of the Slope of a Line","text":"The slope of a line is $$m=\\\\frac{rise}{run}$$. The rise measures the vertical change and the run measures the horizontal change between two points on the line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18a5a6slope13a-h2","type":"hint","dependencies":["a18a5a6slope13a-h1"],"title":"Choosing Two Points","text":"We locate two points on the line whose coordinates are integers. 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Effectively, what is $$\\\\frac{128}{64}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric1a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes, the series is geometric. The common ratio is $$r=2$$."],"dependencies":["a18d102geometric1a-h6"],"title":"Determining Commonality in Ratios","text":"Are the ratios between consecutive terms common?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes, the series is geometric. The common ratio is $$r=2$$.","Yes, the series is geometric. The common ratio is $$r=4$$.","Yes, the series is geometric. 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Effectively, what is $$\\\\frac{6}{3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric10a-h3-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Ratio Between $$6$$ and $$12$$","text":"What is the common ratio between $$6$$ and 12? Effectively, what is $$\\\\frac{12}{6}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric10a-h3-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Ratio Between $$12$$ and $$24$$","text":"What is the common ratio between $$12$$ and 24? Effectively, what is $$\\\\frac{24}{12}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric10a-h3-s4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Ratio Between $$24$$ and $$48$$","text":"What is the common ratio between $$24$$ and 48? Effectively, what is $$\\\\frac{48}{24}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric10a-h3-s5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Ratio Between $$48$$ and $$96$$","text":"What is the common ratio between $$48$$ and 96? Effectively, what is $$\\\\frac{96}{48}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric10a-h3-s6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Determining Common Ratio","text":"What is the common ratio between consecutive terms? What is $$r$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a18d102geometric10a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$a_n=3\\\\times2^{n-1}$$"],"dependencies":["a18d102geometric10a-h3"],"title":"Writing the Formula","text":"What is the correct formula for the general term using the terms just determined?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$a_n=3\\\\times2^{n-1}$$","$$a_n=2\\\\times3^{n-1}$$","$$a_n=3\\\\times2^n$$","$$a_n=2\\\\times3^n$$"]}]}}]},{"id":"a18d102geometric11","title":"Determining the General Term","body":"Determine the general term for the sequence below.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.3 Geometric Sequences and Series","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a18d102geometric11a","stepAnswer":["$$a_n=6\\\\times3^{n-1}$$"],"problemType":"MultipleChoice","stepTitle":"$$6$$, $$18$$, $$54$$, $$162$$, $$486$$, $$1458$$, ...","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$a_n=6\\\\times3^{n-1}$$","choices":["$$a_n=6\\\\times3^{n-1}$$","$$a_n=3\\\\times6^{n-1}$$","$$a_n=6\\\\times3^n$$","$$a_n=3\\\\times6 n$$"],"hints":{"DefaultPathway":[{"id":"a18d102geometric11a-h1","type":"hint","dependencies":[],"title":"Formula for General Term","text":"To find the nth term, $$a_n$$, we use the formula with $$a_1$$ being the starting term and $$r$$ being the common ratio that $$a_n=a_1 r^{n-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a18d102geometric11a-h1"],"title":"Determine $$a_1$$","text":"What is $$a_1$$ in the formula above? 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Effectively, what is $$\\\\frac{162}{54}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric11a-h3-s4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":[],"title":"Ratio Between $$162$$ and $$486$$","text":"What is the common ratio between $$162$$ and 486? Effectively, what is $$\\\\frac{486}{162}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric11a-h3-s5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":[],"title":"Ratio Between $$486$$ and $$1458$$","text":"What is the common ratio between $$486$$ and 1458? 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What is $$r$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a18d102geometric11a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$a_n=6\\\\times3^{n-1}$$"],"dependencies":["a18d102geometric11a-h3"],"title":"Writing the Formula","text":"What is the correct formula for the general term using the terms just determined?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$a_n=6\\\\times3^{n-1}$$","$$a_n=3\\\\times6^{n-1}$$","$$a_n=6\\\\times3^n$$","$$a_n=3\\\\times6 n$$"]}]}}]},{"id":"a18d102geometric12","title":"Determining the General Term","body":"Determine the general term for the sequence below.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.3 Geometric Sequences and Series","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a18d102geometric12a","stepAnswer":["$$a_n=7\\\\times2^{n-1}$$"],"problemType":"MultipleChoice","stepTitle":"$$7$$, $$14$$, $$28$$, $$56$$, $$112$$, $$224$$, ...","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$a_n=7\\\\times2^{n-1}$$","choices":["$$a_n=7\\\\times2^{n-1}$$","$$a_n=2\\\\times7^{n-1}$$","$$a_n=7\\\\times2^n$$","$$a_n=2\\\\times7 n$$"],"hints":{"DefaultPathway":[{"id":"a18d102geometric12a-h1","type":"hint","dependencies":[],"title":"Formula for General Term","text":"To find the nth term, $$a_n$$, we use the formula with $$a_1$$ being the starting term and $$r$$ being the common ratio that $$a_n=a_1 r^{n-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a18d102geometric12a-h1"],"title":"Determine $$a_1$$","text":"What is $$a_1$$ in the formula above? 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Effectively, what is $$\\\\frac{56}{28}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric12a-h3-s4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Ratio Between $$56$$ and $$112$$","text":"What is the common ratio between $$56$$ and 112? Effectively, what is $$\\\\frac{112}{56}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric12a-h3-s5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Ratio Between $$112$$ and $$224$$","text":"What is the common ratio between $$112$$ and 224? Effectively, what is $$\\\\frac{224}{112}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric12a-h3-s6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Determining Common Ratio","text":"What is the common ratio between consecutive terms? What is $$r$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a18d102geometric12a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$a_n=7\\\\times2^{n-1}$$"],"dependencies":["a18d102geometric12a-h3"],"title":"Writing the Formula","text":"What is the correct formula for the general term using the terms just determined?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$a_n=7\\\\times2^{n-1}$$","$$a_n=2\\\\times7^{n-1}$$","$$a_n=7\\\\times2^n$$","$$a_n=2\\\\times7 n$$"]}]}}]},{"id":"a18d102geometric13","title":"Find the Sum of the First $$n$$ Terms","body":"Find the sum of the first $$20$$ terms of the following geometric sequence.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.3 Geometric Sequences and Series","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a18d102geometric13a","stepAnswer":["$$7340025$$"],"problemType":"TextBox","stepTitle":"$$7$$, $$14$$, $$28$$, $$56$$, $$112$$, $$224$$, ...","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7340025$$","hints":{"DefaultPathway":[{"id":"a18d102geometric13a-h1","type":"hint","dependencies":[],"title":"Sum of the First $$n$$ Terms Formula","text":"The sum, $$S_n$$, of the first $$n$$ terms of a geometric sequence is $$S_n=\\\\frac{a_1 \\\\left(1-r^n\\\\right)}{1-r}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a18d102geometric13a-h1"],"title":"Determine $$a_1$$","text":"What is $$a_1$$ in the formula above? Essentially, what is the starting term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric13a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a18d102geometric13a-h2"],"title":"Determine $$r$$","text":"What is $$r$$? What is the common ratio?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a18d102geometric13a-h3"],"title":"Determine $$n$$","text":"What is $$n$$? How many terms of the sequence do we want to sum?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric13a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7340025$$"],"dependencies":["a18d102geometric13a-h4"],"title":"Solve for the Summation","text":"Plug in the values into the equation for $$S_n$$ and solve. What is the sum of the first $$20$$ terms of the geometric sequence?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a18d102geometric13a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7340025$$"],"dependencies":[],"title":"Solve for the Summation","text":"What is $$S_{20}$$? Effectively, what is $$\\\\frac{7\\\\left(1-2^{20}\\\\right)}{1-2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a18d102geometric14","title":"Find the Sum of the First $$n$$ Terms","body":"Find the sum of the first $$20$$ terms of the following geometric sequence.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.3 Geometric Sequences and Series","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a18d102geometric14a","stepAnswer":["$$3145725$$"],"problemType":"TextBox","stepTitle":"$$3$$, $$6$$, $$12$$, $$24$$, $$48$$, $$96$$, ...","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3145725$$","hints":{"DefaultPathway":[{"id":"a18d102geometric14a-h1","type":"hint","dependencies":[],"title":"Sum of the First $$n$$ Terms Formula","text":"The sum, $$S_n$$, of the first $$n$$ terms of a geometric sequence is $$S_n=\\\\frac{a_1 \\\\left(1-r^n\\\\right)}{1-r}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a18d102geometric14a-h1"],"title":"Determine $$a_1$$","text":"What is $$a_1$$ in the formula above? Essentially, what is the starting term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric14a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a18d102geometric14a-h2"],"title":"Determine $$r$$","text":"What is $$r$$? What is the common ratio?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric14a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a18d102geometric14a-h3"],"title":"Determine $$n$$","text":"What is $$n$$? How many terms of the sequence do we want to sum?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric14a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3145725$$"],"dependencies":["a18d102geometric14a-h4"],"title":"Solve for the Summation","text":"Plug in the values into the equation for $$S_n$$ and solve. What is the sum of the first $$20$$ terms of the geometric sequence?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a18d102geometric14a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3145725$$"],"dependencies":[],"title":"Solve for the Summation","text":"What is $$S_{20}$$? Effectively, what is $$\\\\frac{3\\\\left(1-2^{20}\\\\right)}{1-2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a18d102geometric15","title":"Find the Sum of the First $$n$$ Terms","body":"Find the sum of the first $$20$$ terms of the following geometric sequence.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.3 Geometric Sequences and Series","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a18d102geometric15a","stepAnswer":["$$10460353200$$"],"problemType":"TextBox","stepTitle":"$$6$$, $$18$$, $$54$$, $$162$$, $$486$$, $$1458$$, ...","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10460353200$$","hints":{"DefaultPathway":[{"id":"a18d102geometric15a-h1","type":"hint","dependencies":[],"title":"Sum of the First $$n$$ Terms Formula","text":"The sum, $$S_n$$, of the first $$n$$ terms of a geometric sequence is $$S_n=\\\\frac{a_1 \\\\left(1-r^n\\\\right)}{1-r}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a18d102geometric15a-h1"],"title":"Determine $$a_1$$","text":"What is $$a_1$$ in the formula above? Essentially, what is the starting term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a18d102geometric15a-h2"],"title":"Determine $$r$$","text":"What is $$r$$? What is the common ratio?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a18d102geometric15a-h3"],"title":"Determine $$n$$","text":"What is $$n$$? How many terms of the sequence do we want to sum?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric15a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10460353200$$"],"dependencies":["a18d102geometric15a-h4"],"title":"Solve for the Summation","text":"Plug in the values into the equation for $$S_n$$ and solve. What is the sum of the first $$20$$ terms of the geometric sequence?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a18d102geometric15a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10460353200$$"],"dependencies":[],"title":"Solve for the Summation","text":"What is $$S_{20}$$? Effectively, what is $$\\\\frac{6\\\\left(1-3^{20}\\\\right)}{1-3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a18d102geometric16","title":"Find the Sum of the First $$n$$ Terms","body":"Find the sum below.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.3 Geometric Sequences and Series","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a18d102geometric16a","stepAnswer":["$$43046718$$"],"problemType":"TextBox","stepTitle":"sum{i\\\\=1}{15}{2*(3)**i}","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$43046718$$","hints":{"DefaultPathway":[{"id":"a18d102geometric16a-h1","type":"hint","dependencies":[],"title":"Sum of the First $$n$$ Terms Formula","text":"The sum, $$S_n$$, of the first $$n$$ terms of a geometric sequence is $$S_n=\\\\frac{a_1 \\\\left(1-r^n\\\\right)}{1-r}$$. We note that the form that is given is sum{i\\\\=1}{15}{2*(3)**i} and this is written in the form sum{i\\\\=1}{k}{a*(r)**i} where we\'re summing the first k terms where $$r$$ is the common ratio and $$a r$$ is the starting value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a18d102geometric16a-h1"],"title":"Determine $$a_1$$","text":"What is $$a_1$$? As in, if you plug in $$i=1$$, what is $$2\\\\times3^i$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a18d102geometric16a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":[],"title":"Determine $$a_1$$","text":"What is $$2\\\\times3^1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a18d102geometric16a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a18d102geometric16a-h2"],"title":"Determine $$r$$","text":"What is $$r$$? What is the common ratio? This is usually the base of the exponent in the summation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric16a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["a18d102geometric16a-h3"],"title":"Determine $$n$$","text":"What is $$n$$? As in, how many values are in the summation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric16a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$43046718$$"],"dependencies":["a18d102geometric16a-h4"],"title":"Solve for the Summation","text":"Plug in the values into the equation for $$S_n$$ and solve. What is the sum of the first $$15$$ terms of the geometric sequence?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a18d102geometric16a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$43046718$$"],"dependencies":[],"title":"Solve for the Summation","text":"What is $$\\\\frac{6\\\\left(1-3^{15}\\\\right)}{1-3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a18d102geometric17","title":"Find the Sum of the First $$n$$ Terms","body":"Find the sum below.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.3 Geometric Sequences and Series","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a18d102geometric17a","stepAnswer":["$$393204$$"],"problemType":"TextBox","stepTitle":"sum{i\\\\=1}{15}{6*(2)**i}","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$393204$$","hints":{"DefaultPathway":[{"id":"a18d102geometric17a-h1","type":"hint","dependencies":[],"title":"Sum of the First $$n$$ Terms Formula","text":"The sum, $$S_n$$, of the first $$n$$ terms of a geometric sequence is $$S_n=\\\\frac{a_1 \\\\left(1-r^n\\\\right)}{1-r}$$. We note that the form that is given is sum{i\\\\=1}{15}{6*(2)**i} and this is written in the form sum{i\\\\=1}{k}{a*(r)**i} where we\'re summing the first k terms where $$r$$ is the common ratio and $$a r$$ is the starting value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric17a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a18d102geometric17a-h1"],"title":"Determine $$a_1$$","text":"What is $$a_1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a18d102geometric17a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":[],"title":"Determine $$a_1$$","text":"What is $$6\\\\times2^1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a18d102geometric17a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a18d102geometric17a-h2"],"title":"Determine $$r$$","text":"What is $$r$$? What is the common ratio? This is usually the base of the exponent in the summation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric17a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["a18d102geometric17a-h3"],"title":"Determine $$n$$","text":"What is $$n$$? As in, how many values are in the summation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric17a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$393204$$"],"dependencies":["a18d102geometric17a-h4"],"title":"Solve for the Summation","text":"Plug in the values into the equation for $$S_n$$ and solve. What is the sum of the first $$15$$ terms of the geometric sequence?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a18d102geometric17a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$393204$$"],"dependencies":[],"title":"Solve for the Summation","text":"What is $$\\\\frac{12\\\\left(1-2^{15}\\\\right)}{1-2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a18d102geometric18","title":"Find the Sum of the First $$n$$ Terms","body":"Find the sum below.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.3 Geometric Sequences and Series","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a18d102geometric18a","stepAnswer":["$$10230$$"],"problemType":"TextBox","stepTitle":"sum{i\\\\=1}{10}{5*(2)**i}","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10230$$","hints":{"DefaultPathway":[{"id":"a18d102geometric18a-h1","type":"hint","dependencies":[],"title":"Sum of the First $$n$$ Terms Formula","text":"The sum, $$S_n$$, of the first $$n$$ terms of a geometric sequence is $$S_n=\\\\frac{a_1 \\\\left(1-r^n\\\\right)}{1-r}$$. We note that the form that is given is sum{i\\\\=1}{10}{5*(2)**i} and this is written in the form sum{i\\\\=1}{k}{a*(r)**i} where we\'re summing the first k terms where $$r$$ is the common ratio and $$a r$$ is the starting value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a18d102geometric18a-h1"],"title":"Determine $$a_1$$","text":"What is $$a_1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a18d102geometric18a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":[],"title":"Determine $$a_1$$","text":"What is $$5\\\\times2^1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a18d102geometric18a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a18d102geometric18a-h2"],"title":"Determine $$r$$","text":"What is $$r$$? What is the common ratio? This is usually the base of the exponent in the summation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric18a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a18d102geometric18a-h3"],"title":"Determine $$n$$","text":"What is $$n$$? As in, how many values are in the summation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric18a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10230$$"],"dependencies":["a18d102geometric18a-h4"],"title":"Solve for the Summation","text":"Plug in the values into the equation for $$S_n$$ and solve. What is the sum of the first $$10$$ terms of the geometric sequence?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a18d102geometric18a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10230$$"],"dependencies":[],"title":"Solve for the Summation","text":"What is $$\\\\frac{10\\\\left(1-2^{10}\\\\right)}{1-2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a18d102geometric19","title":"Finding Sums of Infinite Geometric Series","body":"Find the sum of the infinite geometric series listed below.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.3 Geometric Sequences and Series","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a18d102geometric19a","stepAnswer":["$$81$$"],"problemType":"TextBox","stepTitle":"54+18+6+2+2/3+2/9+...","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$81$$","hints":{"DefaultPathway":[{"id":"a18d102geometric19a-h1","type":"hint","dependencies":[],"title":"Formula for an Infinite Geometric Series","text":"For an infinite geometric series whose first term is $$a_1$$ and common ratio $$r$$, if |r|<1, then the sum is $$S=\\\\frac{a_1}{1-r}$$. If |r|>1, then the infinite geometric series does not have a sum and diverges.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$54$$"],"dependencies":["a18d102geometric19a-h1"],"title":"Determine $$a_1$$","text":"What is $$a_1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric19a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["a18d102geometric19a-h2"],"title":"Determine $$r$$","text":"What is $$r$$? What is $$\\\\frac{18}{54}$$? What is $$\\\\frac{6}{18}$$? What is $$\\\\frac{2}{6}$$? These should all be the same number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric19a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a18d102geometric19a-h3"],"title":"Determining if there is a Sum","text":"Is |r|<1? We can only find a sum if |r|<1.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a18d102geometric19a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$81$$"],"dependencies":["a18d102geometric19a-h4"],"title":"Using the Formula","text":"Substitute $$a_1$$ and $$r$$ into the summation formula. What is the sum?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a18d102geometric19a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$81$$"],"dependencies":[],"title":"Using the Formula","text":"What is $$\\\\frac{54}{1-\\\\frac{1}{3}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a18d102geometric2","title":"Determine the Common Ratio","body":"Determine if each sequence is geometric. If so, indicate the common ratio.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.3 Geometric Sequences and Series","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a18d102geometric2a","stepAnswer":["Yes, the series is geometric. The common ratio is $$r=3$$."],"problemType":"MultipleChoice","stepTitle":"$$7$$, $$21$$, $$63$$, $$189$$, $$567$$, $$1701$$, ...","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Yes, the series is geometric. The common ratio is $$r=3$$.","choices":["Yes, the series is geometric. The common ratio is $$r=3$$.","Yes, the series is geometric. The common ratio is $$r=4$$.","Yes, the series is geometric. The common ratio is $$r=\\\\frac{1}{4}$$.","No, the series is not geometric."],"hints":{"DefaultPathway":[{"id":"a18d102geometric2a-h1","type":"hint","dependencies":[],"title":"Find Ratios of Consecutive Terms","text":"To determine if a sequence is geometric, we find the ratio of the consecutive terms shown.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a18d102geometric2a-h1"],"title":"Ratio Between $$7$$ and $$21$$","text":"What is the common ratio between $$7$$ and 21? 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Effectively, what is $$\\\\frac{189}{63}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric2a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a18d102geometric2a-h4"],"title":"Ratio Between $$189$$ and $$567$$","text":"What is the common ratio between $$189$$ and 567? Effectively, what is $$\\\\frac{567}{189}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a18d102geometric2a-h5"],"title":"Ratio Between $$567$$ and $$1701$$","text":"What is the common ratio between $$567$$ and 1701? Effectively, what is $$\\\\frac{1701}{567}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric2a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes, the series is geometric. The common ratio is $$r=3$$."],"dependencies":["a18d102geometric2a-h6"],"title":"Determining Commonality in Ratios","text":"Are the ratios between consecutive terms common?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes, the series is geometric. The common ratio is $$r=3$$.","Yes, the series is geometric. The common ratio is $$r=4$$.","Yes, the series is geometric. The common ratio is $$r=\\\\frac{1}{4}$$.","No, the series is not geometric."]}]}},{"id":"a18d102geometric2b","stepAnswer":["Yes, the series is geometric. The common ratio is $$r=\\\\frac{1}{4}$$."],"problemType":"MultipleChoice","stepTitle":"$$64$$, $$16$$, $$4$$, $$1$$, $$\\\\frac{1}{4}$$, $$\\\\frac{1}{16}$$, ...","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Yes, the series is geometric. The common ratio is $$r=\\\\frac{1}{4}$$.","choices":["Yes, the series is geometric. The common ratio is $$r=3$$.","Yes, the series is geometric. The common ratio is $$r=4$$.","Yes, the series is geometric. The common ratio is $$r=\\\\frac{1}{4}$$.","No, the series is not geometric."],"hints":{"DefaultPathway":[{"id":"a18d102geometric2b-h8","type":"hint","dependencies":["a18d102geometric2a-h7"],"title":"Find Ratios of Consecutive Terms","text":"To determine if a sequence is geometric, we find the ratio of the consecutive terms shown.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric2b-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4}$$"],"dependencies":["a18d102geometric2b-h8"],"title":"Ratio Between $$64$$ and $$16$$","text":"What is the common ratio between $$64$$ and 16? 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Effectively, what is $$\\\\frac{1}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric2b-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4}$$"],"dependencies":["a18d102geometric2b-h11"],"title":"Ratio Between $$1$$ and $$\\\\frac{1}{4}$$","text":"What is the common ratio between $$1$$ and $$\\\\frac{1}{4}$$? Effectively, what is $$\\\\frac{\\\\frac{1}{4}}{1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric2b-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4}$$"],"dependencies":["a18d102geometric2b-h12"],"title":"Ratio Between $$\\\\frac{1}{4}$$ and $$\\\\frac{1}{16}$$","text":"What is the common ratio between $$\\\\frac{1}{4}$$ and $$\\\\frac{1}{16}$$? Effectively, what is $$\\\\frac{\\\\frac{1}{4}}{\\\\frac{1}{16}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric2b-h14","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes, the series is geometric. The common ratio is $$r=\\\\frac{1}{4}$$."],"dependencies":["a18d102geometric2b-h13"],"title":"Determining Commonality in Ratios","text":"Are the ratios between consecutive terms common?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes, the series is geometric. The common ratio is $$r=3$$.","Yes, the series is geometric. The common ratio is $$r=4$$.","Yes, the series is geometric. The common ratio is $$r=\\\\frac{1}{4}$$.","No, the series is not geometric."]}]}},{"id":"a18d102geometric2c","stepAnswer":["No, the series is not geometric."],"problemType":"MultipleChoice","stepTitle":"$$2$$, $$4$$, $$12$$, $$48$$, $$240$$, $$1440$$, ...","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes, the series is geometric. The common ratio is $$r=3$$.","Yes, the series is geometric. The common ratio is $$r=4$$.","Yes, the series is geometric. The common ratio is $$r=\\\\frac{1}{4}$$.","No, the series is not geometric."],"hints":{"DefaultPathway":[{"id":"a18d102geometric2c-h15","type":"hint","dependencies":["a18d102geometric2b-h14"],"title":"Find Ratios of Consecutive Terms","text":"To determine if a sequence is geometric, we find the ratio of the consecutive terms shown.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric2c-h16","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a18d102geometric2c-h15"],"title":"Ratio Between $$2$$ and $$4$$","text":"What is the common ratio between $$2$$ and 4? Effectively, what is $$\\\\frac{4}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric2c-h17","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a18d102geometric2c-h16"],"title":"Ratio Between $$4$$ and $$12$$","text":"What is the common ratio between $$4$$ and 12? Effectively, what is $$\\\\frac{12}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric2c-h18","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a18d102geometric2c-h17"],"title":"Ratio Between $$12$$ and $$48$$","text":"What is the common ratio between $$12$$ and 48? Effectively, what is $$\\\\frac{48}{12}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric2c-h19","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a18d102geometric2c-h18"],"title":"Ratio Between $$240$$ and $$48$$","text":"What is the common ratio between $$48$$ and 240? Effectively, what is $$\\\\frac{240}{48}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric2c-h20","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a18d102geometric2c-h19"],"title":"Ratio Between $$240$$ and $$1440$$","text":"What is the common ratio between $$240$$ and 1440? Effectively, what is $$\\\\frac{1440}{240}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric2c-h21","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No, the series is not geometric."],"dependencies":["a18d102geometric2c-h20"],"title":"Determining Commonality in Ratios","text":"Are the ratios between consecutive terms common?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes, the series is geometric. The common ratio is $$r=3$$.","Yes, the series is geometric. The common ratio is $$r=4$$.","Yes, the series is geometric. The common ratio is $$r=\\\\frac{1}{4}$$.","No, the series is not geometric."]}]}}]},{"id":"a18d102geometric20","title":"Finding Sums of Infinite Geometric Series","body":"Find the sum of the infinite geometric series listed below.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.3 Geometric Sequences and Series","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a18d102geometric20a","stepAnswer":["$$96$$"],"problemType":"TextBox","stepTitle":"48+24+12+6+3+3/2+...","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$96$$","hints":{"DefaultPathway":[{"id":"a18d102geometric20a-h1","type":"hint","dependencies":[],"title":"Formula for an Infinite Geometric Series","text":"For an infinite geometric series whose first term is $$a_1$$ and common ratio $$r$$, if |r|<1, then the sum is $$S=\\\\frac{a_1}{1-r}$$. If |r|>1, then the infinite geometric series does not have a sum and diverges.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$48$$"],"dependencies":["a18d102geometric20a-h1"],"title":"Determine $$a_1$$","text":"What is $$a_1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric20a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a18d102geometric20a-h2"],"title":"Determine $$r$$","text":"What is $$r$$? What is $$\\\\frac{24}{48}$$? What is $$\\\\frac{12}{24}$$? What is $$\\\\frac{6}{12}$$? These should all be the same number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric20a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a18d102geometric20a-h3"],"title":"Determining if there is a Sum","text":"Is |r|<1? We can only find a sum if |r|<1.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a18d102geometric20a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$96$$"],"dependencies":["a18d102geometric20a-h4"],"title":"Using the Formula","text":"Substitute $$a_1$$ and $$r$$ into the summation formula. What is the sum?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a18d102geometric20a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$96$$"],"dependencies":[],"title":"Using the Formula","text":"What is $$\\\\frac{48}{1-\\\\frac{1}{2}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a18d102geometric21","title":"Finding Sums of Infinite Geometric Series","body":"Find the sum of the infinite geometric series listed below.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.3 Geometric Sequences and Series","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a18d102geometric21a","stepAnswer":["$$\\\\frac{256}{3}$$"],"problemType":"TextBox","stepTitle":"64+16+4+1+1/4+1/16+... Do not round your answer.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{256}{3}$$","hints":{"DefaultPathway":[{"id":"a18d102geometric21a-h1","type":"hint","dependencies":[],"title":"Formula for an Infinite Geometric Series","text":"For an infinite geometric series whose first term is $$a_1$$ and common ratio $$r$$, if |r|<1, then the sum is $$S=\\\\frac{a_1}{1-r}$$. If |r|>1, then the infinite geometric series does not have a sum and diverges.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric21a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$64$$"],"dependencies":["a18d102geometric21a-h1"],"title":"Determine $$a_1$$","text":"What is $$a_1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric21a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4}$$"],"dependencies":["a18d102geometric21a-h2"],"title":"Determine $$r$$","text":"What is $$r$$? What is $$\\\\frac{16}{64}$$? What is $$\\\\frac{4}{16}$$? What is $$\\\\frac{1}{4}$$? These should all be the same number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric21a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a18d102geometric21a-h3"],"title":"Determining if there is a Sum","text":"Is |r|<1? We can only find a sum if |r|<1.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a18d102geometric21a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{256}{3}$$"],"dependencies":["a18d102geometric21a-h4"],"title":"Using the Formula","text":"Substitute $$a_1$$ and $$r$$ into the summation formula. What is the sum? Do not round your answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a18d102geometric21a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{256}{3}$$"],"dependencies":[],"title":"Using the Formula","text":"What is $$\\\\frac{64}{1-\\\\frac{1}{4}}$$? Do not round your answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a18d102geometric22","title":"Apply Geometric Sequences and Series in the Real World","body":"The government has decided to give a $1,000 tax rebate to each household in order to stimulate the economy. The government statistics say that each household will spend 80% of the rebate in goods and services. The businesses and individuals who benefitted from that 80% will then spend 80% of what they received and so on. The result is called the multiplier effect.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.3 Geometric Sequences and Series","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a18d102geometric22a","stepAnswer":["$$5000$$"],"problemType":"TextBox","stepTitle":"What is the total effect of the rebate on the economy?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5000$$","hints":{"DefaultPathway":[{"id":"a18d102geometric22a-h1","type":"hint","dependencies":[],"title":"Formula for an Infinite Geometric Series","text":"We can represent this as an infinite geometric series. Every time money goes into the economy, 80% of it is spent and is then in the economy to be spent. Therefore, we have an infinite geometric series 1000+1000(0.8)+1000(0.8)**2+... For an infinite geometric series whose first term is $$a_1$$ and common ratio $$r$$, if |r|<1, then the sum is $$S=\\\\frac{a_1}{1-r}$$. If |r|>1, then the infinite geometric series does not have a sum and diverges.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric22a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1000$$"],"dependencies":["a18d102geometric22a-h1"],"title":"Determine $$a_1$$","text":"What is $$a_1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric22a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.8$$"],"dependencies":["a18d102geometric22a-h2"],"title":"Determine $$r$$","text":"What is $$r$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric22a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a18d102geometric22a-h3"],"title":"Determining if there is a Sum","text":"Is |r|<1? We can only find a sum if |r|<1.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a18d102geometric22a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5000$$"],"dependencies":["a18d102geometric22a-h4"],"title":"Using the Formula","text":"Substitute $$a_1$$ and $$r$$ into the summation formula. What is the sum? This will be the overall total effect of the rebate on the economy.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a18d102geometric22a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5000$$"],"dependencies":[],"title":"Using the Formula","text":"What is $$\\\\frac{1000}{1-0.8}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a18d102geometric23","title":"Apply Geometric Sequences and Series in the Real World","body":"New parents decide to invest $100 per month into an annuity for their baby daughter. The account will pay 5% interest per year which is compounded monthly.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.3 Geometric Sequences and Series","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a18d102geometric23a","stepAnswer":["$$34920.2$$"],"problemType":"TextBox","stepTitle":"How much will be in the child\'s account at her eighteenth birthday?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$34920.2$$","hints":{"DefaultPathway":[{"id":"a18d102geometric23a-h1","type":"hint","dependencies":[],"title":"Formula for Value of an Annuity","text":"For a principal, P, invested at the end of a compounding period, with an interest rate, $$r$$, which is compounded $$n$$ times in a year, the new balance A, after $$t$$ years, is $$A_t=\\\\frac{P \\\\left({\\\\left(1+\\\\frac{r}{n}\\\\right)}^{nt}-1\\\\right)}{\\\\frac{r}{n}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric23a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$100$$"],"dependencies":["a18d102geometric23a-h1"],"title":"Determine P","text":"What is P? What is the amount invested each month?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric23a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.05$$"],"dependencies":["a18d102geometric23a-h2"],"title":"Determine $$r$$","text":"What is $$r$$? What is the annual interest rate, in decimal form?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric23a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a18d102geometric23a-h3"],"title":"Determine $$n$$","text":"What is $$n$$? What is the number of times the deposit will be made and the interest compounded each year? How many months are in a year, essentially?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric23a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18$$"],"dependencies":["a18d102geometric23a-h4"],"title":"Determine $$t$$","text":"What is $$t$$? What is the number of years that will pass?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric23a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$34920.2$$"],"dependencies":["a18d102geometric23a-h5"],"title":"Use the Formula for Value of an Annuity","text":"Plug in P, $$r$$, $$n$$, and $$t$$ into the summation formula to get $$A_t$$, the amount of money in the account after $$18$$ years.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric23a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$34920.2$$"],"dependencies":["a18d102geometric23a-h6"],"title":"Use the Formula for Value of an Annuity","text":"What is $$\\\\frac{100\\\\left({\\\\left(1+\\\\frac{0.05}{12}\\\\right)}^{12\\\\times18}-1\\\\right)}{\\\\frac{0.05}{12}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18d102geometric24","title":"Determine the Common Ratio","body":"Determine if each sequence is geometric. If so, indicate the common ratio.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.3 Geometric Sequences and Series","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a18d102geometric24a","stepAnswer":["Yes, the series is geometric. The common ratio is $$r=4$$."],"problemType":"MultipleChoice","stepTitle":"$$3$$, $$12$$, $$48$$, $$192$$, $$768$$, $$3072$$, ...","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Yes, the series is geometric. The common ratio is $$r=4$$.","choices":["Yes, the series is geometric. The common ratio is $$r=2$$.","Yes, the series is geometric. The common ratio is $$r=4$$.","Yes, the series is geometric. The common ratio is $$r=\\\\frac{1}{3}$$.","No, the series is not geometric."],"hints":{"DefaultPathway":[{"id":"a18d102geometric24a-h1","type":"hint","dependencies":[],"title":"Find Ratios of Consecutive Terms","text":"To determine if a sequence is geometric, we find the ratio of the consecutive terms shown.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric24a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a18d102geometric24a-h1"],"title":"Ratio Between $$3$$ and $$12$$","text":"What is the common ratio between $$3$$ and 12? Effectively, what is $$\\\\frac{12}{3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric24a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a18d102geometric24a-h2"],"title":"Ratio Between $$12$$ and $$48$$","text":"What is the common ratio between $$12$$ and 48? Effectively, what is $$\\\\frac{48}{12}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric24a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a18d102geometric24a-h3"],"title":"Ratio Between $$48$$ and $$192$$","text":"What is the common ratio between $$48$$ and 192? Effectively, what is $$\\\\frac{192}{48}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric24a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a18d102geometric24a-h4"],"title":"Ratio Between $$192$$ and $$768$$","text":"What is the common ratio between $$192$$ and 768? Effectively, what is $$\\\\frac{768}{192}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric24a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a18d102geometric24a-h5"],"title":"Ratio Between $$768$$ and $$3072$$","text":"What is the common ratio between $$768$$ and 3072? Effectively, what is $$\\\\frac{3072}{768}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric24a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes, the series is geometric. The common ratio is $$r=4$$."],"dependencies":["a18d102geometric24a-h6"],"title":"Determining Commonality in Ratios","text":"Are the ratios between consecutive terms common?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes, the series is geometric. The common ratio is $$r=2$$.","Yes, the series is geometric. The common ratio is $$r=4$$.","Yes, the series is geometric. The common ratio is $$r=\\\\frac{1}{3}$$.","No, the series is not geometric."]}]}}]},{"id":"a18d102geometric25","title":"Determine the Common Ratio","body":"Determine if each sequence is geometric. If so, indicate the common ratio.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.3 Geometric Sequences and Series","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a18d102geometric25a","stepAnswer":["Yes, the series is geometric. The common ratio is $$r=\\\\frac{1}{2}$$."],"problemType":"MultipleChoice","stepTitle":"$$72$$, $$36$$, $$18$$, $$9$$, $$\\\\frac{9}{2}$$, $$\\\\frac{9}{4}$$, ...","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Yes, the series is geometric. The common ratio is $$r=\\\\frac{1}{2}$$.","choices":["Yes, the series is geometric. The common ratio is $$r=2$$.","Yes, the series is geometric. The common ratio is $$r=4$$.","Yes, the series is geometric. The common ratio is $$r=\\\\frac{1}{2}$$.","No, the series is not geometric."],"hints":{"DefaultPathway":[{"id":"a18d102geometric25a-h1","type":"hint","dependencies":[],"title":"Find Ratios of Consecutive Terms","text":"To determine if a sequence is geometric, we find the ratio of the consecutive terms shown.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric25a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a18d102geometric25a-h1"],"title":"Ratio Between $$72$$ and $$36$$","text":"What is the common ratio between $$72$$ and 36? Effectively, what is $$\\\\frac{36}{72}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric25a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a18d102geometric25a-h2"],"title":"Ratio Between $$36$$ and $$18$$","text":"What is the common ratio between $$36$$ and 18? Effectively, what is $$\\\\frac{18}{36}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric25a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a18d102geometric25a-h3"],"title":"Ratio Between $$18$$ and $$9$$","text":"What is the common ratio between $$18$$ and 9? Effectively, what is $$\\\\frac{9}{18}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric25a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a18d102geometric25a-h4"],"title":"Ratio Between $$9$$ and $$\\\\frac{9}{2}$$","text":"What is the common ratio between $$9$$ and $$\\\\frac{9}{2}$$? Effectively, what is $$\\\\frac{\\\\frac{9}{2}}{9}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric25a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a18d102geometric25a-h5"],"title":"Ratio Between $$\\\\frac{9}{2}$$ and $$\\\\frac{9}{4}$$","text":"What is the common ratio between $$\\\\frac{9}{2}$$ and $$\\\\frac{9}{4}$$? Effectively, what is $$\\\\frac{\\\\frac{9}{4}}{\\\\frac{9}{2}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric25a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes, the series is geometric. The common ratio is $$r=\\\\frac{1}{2}$$."],"dependencies":["a18d102geometric25a-h6"],"title":"Determining Commonality in Ratios","text":"Are the ratios between consecutive terms common?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes, the series is geometric. The common ratio is $$r=2$$.","Yes, the series is geometric. The common ratio is $$r=4$$.","Yes, the series is geometric. The common ratio is $$r=\\\\frac{1}{2}$$.","No, the series is not geometric."]}]}}]},{"id":"a18d102geometric26","title":"Writing Terms of a Sequence","body":"Determine the first five terms of the sequence defined below.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.3 Geometric Sequences and Series","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a18d102geometric26a","stepAnswer":["$$4$$, $$12$$, $$36$$, $$108$$, $$324$$, ..."],"problemType":"MultipleChoice","stepTitle":"The first term is $$a_1=4$$ and the common ratio is $$r=3$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$4$$, $$12$$, $$36$$, $$108$$, $$324$$, ...","choices":["$$4$$, $$12$$, $$36$$, $$108$$, $$324$$, ...","$$4$$, $$-12$$, $$36$$, $$-108$$, $$324$$, ...","$$4$$, $$\\\\frac{4}{3}$$, $$\\\\frac{4}{9}$$, $$\\\\frac{4}{27}$$, $$\\\\frac{4}{81}$$, ...","$$4$$, $$\\\\frac{-4}{3}$$, $$\\\\frac{4}{9}$$, $$\\\\frac{-4}{27}$$, $$\\\\frac{4}{81}$$, ..."],"hints":{"DefaultPathway":[{"id":"a18d102geometric26a-h1","type":"hint","dependencies":[],"title":"Multiply by Common Ratio","text":"To get the sequence, start with the first term and multiply it by the common ratio. Then we continue to multiply that result by the common ratio to get the next term, and so on.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric26a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a18d102geometric26a-h1"],"title":"Multiply $$4$$ by the Common Ratio","text":"What is $$4\\\\times3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric26a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$36$$"],"dependencies":["a18d102geometric26a-h2"],"title":"Multiply $$12$$ by the Common Ratio","text":"What is $$12\\\\times3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric26a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$108$$"],"dependencies":["a18d102geometric26a-h3"],"title":"Multiply $$36$$ by the Common Ratio","text":"What is $$36\\\\times3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric26a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$324$$"],"dependencies":["a18d102geometric26a-h4"],"title":"Multiply $$108$$ by the Common Ratio","text":"What is $$108\\\\times3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18d102geometric27","title":"Writing Terms of a Sequence","body":"Determine the first five terms of the sequence defined below.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.3 Geometric Sequences and Series","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a18d102geometric27a","stepAnswer":["$$-4$$, $$8$$, $$-16$$, $$32$$, $$-64$$, ..."],"problemType":"MultipleChoice","stepTitle":"The first term is $$a_1=-4$$ and the common ratio is $$r=-2$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-4$$, $$8$$, $$-16$$, $$32$$, $$-64$$, ...","choices":["$$-4$$, $$8$$, $$-16$$, $$32$$, $$-64$$, ...","$$4$$, $$-8$$, $$16$$, $$-32$$, $$64$$, ...","$$4$$, $$8$$, $$16$$, $$32$$, $$64$$, ...","$$-4$$, $$-8$$, $$-16$$, $$-32$$, $$-64$$, ..."],"hints":{"DefaultPathway":[{"id":"a18d102geometric27a-h1","type":"hint","dependencies":[],"title":"Multiply by Common Ratio","text":"To get the sequence, start with the first term and multiply it by the common ratio. 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Essentially, which term of the sequence do we want?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric28a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a18d102geometric28a-h2"],"title":"Determine $$a_1$$","text":"What is $$a_1$$ in the formula above? Essentially, what is the starting term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric28a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a18d102geometric28a-h3"],"title":"Determine $$r$$","text":"What is $$r$$? What is the common ratio?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric28a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$472392$$"],"dependencies":["a18d102geometric28a-h4"],"title":"Plug in the Values into Formula","text":"Plug in the values determined above into the formula to find the Xth term and calculate the values. What is the eleventh term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a18d102geometric28a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$472392$$"],"dependencies":[],"title":"Plug in the Values into Formula","text":"What is $$8\\\\times3^{11-1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a18d102geometric29","title":"Find the Sum of the First $$n$$ Terms","body":"Find the sum below.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.3 Geometric Sequences and Series","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a18d102geometric29a","stepAnswer":["$$65534$$"],"problemType":"TextBox","stepTitle":"sum{i\\\\=1}{15}{(2)**i}","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$65534$$","hints":{"DefaultPathway":[{"id":"a18d102geometric29a-h1","type":"hint","dependencies":[],"title":"Sum of the First $$n$$ Terms Formula","text":"The sum, $$S_n$$, of the first $$n$$ terms of a geometric sequence is $$S_n=\\\\frac{a_1 \\\\left(1-r^n\\\\right)}{1-r}$$. We note that the form that is given is sum{i\\\\=1}{15}{(2)**i} and this is written in the form sum{i\\\\=1}{k}{a*(r)**i} where we\'re summing the first k terms where $$r$$ is the common ratio and $$a r$$ is the starting value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric29a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a18d102geometric29a-h1"],"title":"Determine $$a_1$$","text":"What is $$a_1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a18d102geometric29a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Determine $$a_1$$","text":"What is $$2^1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a18d102geometric29a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a18d102geometric29a-h2"],"title":"Determine $$r$$","text":"What is $$r$$? What is the common ratio? This is usually the base of the exponent in the summation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric29a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["a18d102geometric29a-h3"],"title":"Determine $$n$$","text":"What is $$n$$? As in, how many values are in the summation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric29a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$65534$$"],"dependencies":["a18d102geometric29a-h4"],"title":"Solve for the Summation","text":"Plug in the values into the equation for $$S_n$$ and solve. What is the sum of the first $$15$$ terms of the geometric sequence?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a18d102geometric29a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$65534$$"],"dependencies":[],"title":"Solve for the Summation","text":"What is $$\\\\frac{2\\\\left(1-2^{15}\\\\right)}{1-2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a18d102geometric3","title":"Determine the Common Ratio","body":"Determine if each sequence is geometric. If so, indicate the common ratio.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.3 Geometric Sequences and Series","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a18d102geometric3a","stepAnswer":["No, the series is not geometric."],"problemType":"MultipleChoice","stepTitle":"$$-150$$, $$-30$$, $$-15$$, $$-5$$, $$\\\\frac{-5}{2}$$, $$0$$, ...","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes, the series is geometric. The common ratio is $$r=2$$.","Yes, the series is geometric. The common ratio is $$r=\\\\frac{1}{2}$$.","Yes, the series is geometric. The common ratio is $$r=\\\\frac{1}{4}$$.","No, the series is not geometric."],"hints":{"DefaultPathway":[{"id":"a18d102geometric3a-h1","type":"hint","dependencies":[],"title":"Find Ratios of Consecutive Terms","text":"To determine if a sequence is geometric, we find the ratio of the consecutive terms shown.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{5}$$"],"dependencies":["a18d102geometric3a-h1"],"title":"Ratio Between $$-150$$ and $$-30$$","text":"What is the common ratio between $$-150$$ and -30? Effectively, what is $$\\\\frac{-30}{-150}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a18d102geometric3a-h2"],"title":"Ratio Between $$-30$$ and $$-15$$","text":"What is the common ratio between $$-30$$ and -15? Effectively, what is $$\\\\frac{-15}{-30}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["a18d102geometric3a-h3"],"title":"Ratio Between $$-15$$ and $$-5$$","text":"What is the common ratio between $$-15$$ and -5? Effectively, what is $$\\\\frac{-5}{-15}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a18d102geometric3a-h4"],"title":"Ratio Between $$-5$$ and $$\\\\frac{-5}{2}$$","text":"What is the common ratio between $$-5$$ and $$\\\\frac{-5}{2}$$? Effectively, what is $$\\\\frac{\\\\left(-\\\\frac{5}{2}\\\\right)}{\\\\left(-5\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric3a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a18d102geometric3a-h5"],"title":"Ratio Between $$\\\\frac{-5}{2}$$ and $$0$$","text":"What is the common ratio between $$\\\\frac{-5}{2}$$ and 0? Effectively, what is $$\\\\frac{\\\\frac{0}{-5}}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric3a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No, the series is not geometric."],"dependencies":["a18d102geometric3a-h6"],"title":"Determining Commonality in Ratios","text":"Are the ratios between consecutive terms common?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes, the series is geometric. The common ratio is $$r=2$$.","Yes, the series is geometric. The common ratio is $$r=\\\\frac{1}{2}$$.","Yes, the series is geometric. The common ratio is $$r=\\\\frac{1}{4}$$.","No, the series is not geometric."]}]}},{"id":"a18d102geometric3b","stepAnswer":["Yes, the series is geometric. The common ratio is $$r=2$$."],"problemType":"MultipleChoice","stepTitle":"$$5$$, $$10$$, $$20$$, $$40$$, $$80$$, $$160$$, ...","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Yes, the series is geometric. The common ratio is $$r=2$$.","choices":["Yes, the series is geometric. The common ratio is $$r=2$$.","Yes, the series is geometric. The common ratio is $$r=\\\\frac{1}{2}$$.","Yes, the series is geometric. The common ratio is $$r=\\\\frac{1}{4}$$.","No, the series is not geometric."],"hints":{"DefaultPathway":[{"id":"a18d102geometric3b-h8","type":"hint","dependencies":["a18d102geometric3a-h7"],"title":"Find Ratios of Consecutive Terms","text":"To determine if a sequence is geometric, we find the ratio of the consecutive terms shown.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric3b-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a18d102geometric3b-h8"],"title":"Ratio Between $$5$$ and $$10$$","text":"What is the common ratio between $$5$$ and 10? Effectively, what is $$\\\\frac{10}{5}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric3b-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a18d102geometric3b-h9"],"title":"Ratio Between $$10$$ and $$20$$","text":"What is the common ratio between $$10$$ and 20? Effectively, what is $$\\\\frac{20}{10}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric3b-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a18d102geometric3b-h10"],"title":"Ratio Between $$20$$ and $$40$$","text":"What is the common ratio between $$20$$ and 40? Effectively, what is $$\\\\frac{40}{20}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric3b-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a18d102geometric3b-h11"],"title":"Ratio Between $$40$$ and $$80$$","text":"What is the common ratio between $$40$$ and 80? Effectively, what is $$\\\\frac{80}{40}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric3b-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a18d102geometric3b-h12"],"title":"Ratio Between $$80$$ and $$160$$","text":"What is the common ratio between $$80$$ and 160? Effectively, what is $$\\\\frac{160}{80}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric3b-h14","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes, the series is geometric. The common ratio is $$r=2$$."],"dependencies":["a18d102geometric3b-h13"],"title":"Determining Commonality in Ratios","text":"Are the ratios between consecutive terms common?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes, the series is geometric. The common ratio is $$r=2$$.","Yes, the series is geometric. The common ratio is $$r=\\\\frac{1}{2}$$.","Yes, the series is geometric. The common ratio is $$r=\\\\frac{1}{4}$$.","No, the series is not geometric."]}]}},{"id":"a18d102geometric3c","stepAnswer":["Yes, the series is geometric. The common ratio is $$r=\\\\frac{1}{2}$$."],"problemType":"MultipleChoice","stepTitle":"$$8$$, $$4$$, $$2$$, $$1$$, $$\\\\frac{1}{2}$$, $$\\\\frac{1}{4}$$, ...","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Yes, the series is geometric. The common ratio is $$r=\\\\frac{1}{2}$$.","choices":["Yes, the series is geometric. The common ratio is $$r=2$$.","Yes, the series is geometric. The common ratio is $$r=\\\\frac{1}{2}$$.","Yes, the series is geometric. The common ratio is $$r=\\\\frac{1}{4}$$.","No, the series is not geometric."],"hints":{"DefaultPathway":[{"id":"a18d102geometric3c-h15","type":"hint","dependencies":["a18d102geometric3b-h14"],"title":"Find Ratios of Consecutive Terms","text":"To determine if a sequence is geometric, we find the ratio of the consecutive terms shown.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric3c-h16","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a18d102geometric3c-h15"],"title":"Ratio Between $$8$$ and $$4$$","text":"What is the common ratio between $$8$$ and 4? Effectively, what is $$\\\\frac{4}{8}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric3c-h17","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a18d102geometric3c-h16"],"title":"Ratio Between $$4$$ and $$2$$","text":"What is the common ratio between $$4$$ and 2? Effectively, what is $$\\\\frac{2}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric3c-h18","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a18d102geometric3c-h17"],"title":"Ratio Between $$2$$ and $$1$$","text":"What is the common ratio between $$2$$ and 1? Effectively, what is $$\\\\frac{1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric3c-h19","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a18d102geometric3c-h18"],"title":"Ratio Between $$1$$ and $$\\\\frac{1}{2}$$","text":"What is the common ratio between $$1$$ and $$\\\\frac{1}{2}$$? Effectively, what is $$\\\\frac{\\\\frac{1}{2}}{1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric3c-h20","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a18d102geometric3c-h19"],"title":"Ratio Between $$\\\\frac{1}{2}$$ and $$\\\\frac{1}{4}$$","text":"What is the common ratio between $$\\\\frac{1}{2}$$ and $$\\\\frac{1}{4}$$? Effectively, what is $$\\\\frac{\\\\frac{1}{4}}{\\\\frac{1}{2}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric3c-h21","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes, the series is geometric. The common ratio is $$r=\\\\frac{1}{2}$$."],"dependencies":["a18d102geometric3c-h20"],"title":"Determining Commonality in Ratios","text":"Are the ratios between consecutive terms common?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes, the series is geometric. The common ratio is $$r=2$$.","Yes, the series is geometric. The common ratio is $$r=\\\\frac{1}{2}$$.","Yes, the series is geometric. The common ratio is $$r=\\\\frac{1}{4}$$.","No, the series is not geometric."]}]}}]},{"id":"a18d102geometric30","title":"Representation of Repeating Decimals","body":"Write the repeating decimal below as a fraction.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.3 Geometric Sequences and Series","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a18d102geometric30a","stepAnswer":["$$\\\\frac{5}{9}$$"],"problemType":"TextBox","stepTitle":"The value $$0.55555..$$. (repeating $$5$$ after the decimal).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5}{9}$$","hints":{"DefaultPathway":[{"id":"a18d102geometric30a-h1","type":"hint","dependencies":[],"title":"Represent Repeating Decimal with Summation","text":"We can rewrite $$0.5$$ as $$0.5555..$$. with repeating 5s forever. We can represent this summation as $$0.5+0.05+0.005+0.0005+...$$, which is an infinite sum. Now, we use this infinite geometric series and find $$a_1$$ and $$r$$ to find the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric30a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1$$"],"dependencies":["a18d102geometric30a-h1"],"title":"Determine $$r$$","text":"What is the common ratio between the consecutive terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a18d102geometric30a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1$$"],"dependencies":[],"title":"Ratio Between $$0.5$$ and $$0.05$$","text":"What is the common ratio between $$0.5$$ and $$0.05$$? Effectively, what is $$\\\\frac{0.05}{0.5}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric30a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1$$"],"dependencies":[],"title":"Ratio Between $$0.05$$ and $$0.005$$","text":"What is the common ratio between $$0.05$$ and $$0.005$$? Effectively, what is $$\\\\frac{0.005}{0.05}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric30a-h2-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1$$"],"dependencies":[],"title":"Ratio Between $$0.005$$ and $$0.0005$$","text":"What is the common ratio between $$0.005$$ and $$0.0005$$? Effectively, what is $$\\\\frac{0.0005}{0.005}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric30a-h2-s4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1$$"],"dependencies":[],"title":"Ratio Between $$0.0005$$ and $$0.00005$$","text":"What is the common ratio between $$0.0005$$ and $$0.00005$$? Effectively, what is $$\\\\frac{5e-05}{0.0005}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric30a-h2-s5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1$$"],"dependencies":[],"title":"Ratio Between $$0.00005$$ and $$0.000005$$","text":"What is the common ratio between $$0.00005$$ and $$0.000005$$? Effectively, what is $$\\\\frac{5e-06}{0.5}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a18d102geometric30a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.5$$"],"dependencies":["a18d102geometric30a-h2"],"title":"Determine $$a_1$$","text":"What is $$a_1$$? What is the first term in the series?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric30a-h4","type":"hint","dependencies":["a18d102geometric30a-h3"],"title":"Formula for Summation of an Infinite Geometric Series","text":"The sum, S, of an infinite geometric series with starting term $$a_1$$ and common ratio $$r$$ given |r|<1 is $$S=\\\\frac{a_1}{1-r}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric30a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a18d102geometric30a-h4"],"title":"Determining if there is a Sum","text":"Is |r|<1? We can only find a sum if |r|<1.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a18d102geometric30a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{9}$$"],"dependencies":["a18d102geometric30a-h5"],"title":"Using the Formula","text":"Substitute $$a_1$$ and $$r$$ into the summation formula. What is the sum? Leave this in fraction form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a18d102geometric30a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{9}$$"],"dependencies":[],"title":"Using the Formula","text":"What is $$\\\\frac{0.5}{1-0.1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a18d102geometric4","title":"Writing Terms of a Sequence","body":"Determine the first five terms of the sequence defined below.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.3 Geometric Sequences and Series","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a18d102geometric4a","stepAnswer":["$$3$$, $$-6$$, $$12$$, $$-24$$, $$48$$, ..."],"problemType":"MultipleChoice","stepTitle":"The first term is $$3$$ and the common ratio is $$r=-2$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$3$$, $$-6$$, $$12$$, $$-24$$, $$48$$, ...","choices":["$$3$$, $$-6$$, $$12$$, $$-24$$, $$48$$, ...","$$3$$, $$6$$, $$12$$, $$24$$, $$48$$, ...","$$3$$, $$\\\\frac{3}{2}$$, $$\\\\frac{3}{4}$$, $$\\\\frac{3}{8}$$, $$\\\\frac{3}{16}$$, ...","$$3$$, $$\\\\frac{-3}{2}$$, $$\\\\frac{3}{4}$$, $$\\\\frac{-3}{8}$$, $$3-16$$"],"hints":{"DefaultPathway":[{"id":"a18d102geometric4a-h1","type":"hint","dependencies":[],"title":"Multiply by Common Ratio","text":"To get the sequence, start with the first term and multiply it by the common ratio. Then we continue to multiply that result by the common ratio to get the next term, and so on.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6$$"],"dependencies":["a18d102geometric4a-h1"],"title":"Multiply $$3$$ by the Common Ratio","text":"What is $$3\\\\left(-2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a18d102geometric4a-h2"],"title":"Multiply $$-6$$ by the Common Ratio","text":"What is $$\\\\left(-6\\\\right) \\\\left(-2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-24$$"],"dependencies":["a18d102geometric4a-h3"],"title":"Multiply $$12$$ by the Common Ratio","text":"What is $$12\\\\left(-2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric4a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$48$$"],"dependencies":["a18d102geometric4a-h4"],"title":"Multiply $$-24$$ by the Common Ratio","text":"What is $$\\\\left(-24\\\\right) \\\\left(-2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18d102geometric5","title":"Writing Terms of a Sequence","body":"Determine the first five terms of the sequence defined below.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.3 Geometric Sequences and Series","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a18d102geometric5a","stepAnswer":["$$7$$, $$-21$$, $$63$$, $$-189$$, $$567$$, ..."],"problemType":"MultipleChoice","stepTitle":"The first term is $$7$$ and the common ratio is $$r=-3$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$7$$, $$-21$$, $$63$$, $$-189$$, $$567$$, ...","choices":["$$7$$, $$-21$$, $$63$$, $$-189$$, $$567$$, ...","$$7$$, $$21$$, $$63$$, $$189$$, $$567$$, ...","$$7$$, $$\\\\frac{7}{3}$$, $$\\\\frac{7}{9}$$, $$\\\\frac{7}{27}$$, $$\\\\frac{7}{81}$$, ...","$$7$$, $$\\\\frac{-7}{3}$$, $$\\\\frac{7}{9}$$, $$\\\\frac{-7}{27}$$, $$\\\\frac{7}{81}$$, ..."],"hints":{"DefaultPathway":[{"id":"a18d102geometric5a-h1","type":"hint","dependencies":[],"title":"Multiply by Common Ratio","text":"To get the sequence, start with the first term and multiply it by the common ratio. Then we continue to multiply that result by the common ratio to get the next term, and so on.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-21$$"],"dependencies":["a18d102geometric5a-h1"],"title":"Multiply $$7$$ by the Common Ratio","text":"What is $$7\\\\left(-3\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$63$$"],"dependencies":["a18d102geometric5a-h2"],"title":"Multiply $$-21$$ by the Common Ratio","text":"What is $$\\\\left(-21\\\\right) \\\\left(-3\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-189$$"],"dependencies":["a18d102geometric5a-h3"],"title":"Multiply $$63$$ by the Common Ratio","text":"What is $$63\\\\left(-3\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$567$$"],"dependencies":["a18d102geometric5a-h4"],"title":"Multiply $$-189$$ by the Common Ratio","text":"What is $$\\\\left(-189\\\\right) \\\\left(-3\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18d102geometric6","title":"Writing Terms of a Sequence","body":"Determine the first five terms of the sequence defined below.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.3 Geometric Sequences and Series","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a18d102geometric6a","stepAnswer":["$$6$$, $$-24$$, $$96$$, $$-384$$, $$1536$$, ..."],"problemType":"MultipleChoice","stepTitle":"The first term is $$6$$ and the common ratio is $$r=-4$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$6$$, $$-24$$, $$96$$, $$-384$$, $$1536$$, ...","choices":["$$6$$, $$-24$$, $$96$$, $$-384$$, $$1536$$, ...","$$6$$, $$24$$, $$96$$, $$384$$, $$1536$$, ...","$$6$$, $$\\\\frac{3}{2}$$, $$\\\\frac{3}{8}$$, $$\\\\frac{3}{32}$$, $$\\\\frac{3}{128}$$, ...","$$6$$, $$\\\\frac{-3}{2}$$, $$\\\\frac{3}{8}$$, $$\\\\frac{-3}{32}$$, $$\\\\frac{3}{128}$$, ..."],"hints":{"DefaultPathway":[{"id":"a18d102geometric6a-h1","type":"hint","dependencies":[],"title":"Multiply by Common Ratio","text":"To get the sequence, start with the first term and multiply it by the common ratio. Then we continue to multiply that result by the common ratio to get the next term, and so on.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-24$$"],"dependencies":["a18d102geometric6a-h1"],"title":"Multiply $$6$$ by the Common Ratio","text":"What is $$6\\\\left(-4\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$96$$"],"dependencies":["a18d102geometric6a-h2"],"title":"Multiply $$-24$$ by the Common Ratio","text":"What is $$\\\\left(-24\\\\right) \\\\left(-4\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-384$$"],"dependencies":["a18d102geometric6a-h3"],"title":"Multiply $$96$$ by the Common Ratio","text":"What is $$96\\\\left(-4\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1536$$"],"dependencies":["a18d102geometric6a-h4"],"title":"Multiply $$-384$$ by the Common Ratio","text":"What is $$\\\\left(-384\\\\right) \\\\left(-4\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18d102geometric7","title":"Finding the Xth Term of a Sequence","body":"Find the fourteenth term of the following sequence.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.3 Geometric Sequences and Series","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a18d102geometric7a","stepAnswer":["$$\\\\frac{1}{128}$$"],"problemType":"TextBox","stepTitle":"The first term is $$64$$ and the common ratio is $$r=\\\\frac{1}{2}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{128}$$","hints":{"DefaultPathway":[{"id":"a18d102geometric7a-h1","type":"hint","dependencies":[],"title":"Formula to Find the Xth Term","text":"To find the Xth term, $$a_x$$, we use the formula with $$a_1$$ being the starting term and $$r$$ being the common ratio that $$a_x=a_1 r^{n-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$14$$"],"dependencies":["a18d102geometric7a-h1"],"title":"Determine $$x$$","text":"What is $$x$$ in the formula above? Essentially, which term of the sequence do we want?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$64$$"],"dependencies":["a18d102geometric7a-h2"],"title":"Determine $$a_1$$","text":"What is $$a_1$$ in the formula above? Essentially, what is the starting term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a18d102geometric7a-h3"],"title":"Determine $$r$$","text":"What is $$r$$? What is the common ratio?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric7a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{128}$$"],"dependencies":["a18d102geometric7a-h4"],"title":"Plug in the Values into Formula","text":"Plug in the values determined above into the formula to find the Xth term and calculate the values. What is the fourteenth term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a18d102geometric7a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{128}$$"],"dependencies":[],"title":"Plug in the Values into Formula","text":"What is $$64{\\\\left(\\\\frac{1}{2}\\\\right)}^{14-1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a18d102geometric8","title":"Finding the Xth Term of a Sequence","body":"Find the thirteenth term of the following sequence.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.3 Geometric Sequences and Series","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a18d102geometric8a","stepAnswer":["$$\\\\frac{1}{6561}$$"],"problemType":"TextBox","stepTitle":"The first term is $$81$$ and the common ratio is $$r=\\\\frac{1}{3}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{6561}$$","hints":{"DefaultPathway":[{"id":"a18d102geometric8a-h1","type":"hint","dependencies":[],"title":"Formula to Find the Xth Term","text":"To find the Xth term, $$a_x$$, we use the formula with $$a_1$$ being the starting term and $$r$$ being the common ratio that $$a_x=a_1 r^{n-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$13$$"],"dependencies":["a18d102geometric8a-h1"],"title":"Determine $$x$$","text":"What is $$x$$ in the formula above? Essentially, which term of the sequence do we want?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$81$$"],"dependencies":["a18d102geometric8a-h2"],"title":"Determine $$a_1$$","text":"What is $$a_1$$ in the formula above? Essentially, what is the starting term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["a18d102geometric8a-h3"],"title":"Determine $$r$$","text":"What is $$r$$? What is the common ratio?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric8a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{6561}$$"],"dependencies":["a18d102geometric8a-h4"],"title":"Plug in the Values into Formula","text":"Plug in the values determined above into the formula to find the Xth term and calculate the values. What is the thirteenth term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a18d102geometric8a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{6561}$$"],"dependencies":[],"title":"Plug in the Values into Formula","text":"What is $$81{\\\\left(\\\\frac{1}{3}\\\\right)}^{13-1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a18d102geometric9","title":"Finding the Xth Term of a Sequence","body":"Find the twelfth term of the following sequence.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.3 Geometric Sequences and Series","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a18d102geometric9a","stepAnswer":["$$\\\\frac{1}{16384}$$"],"problemType":"TextBox","stepTitle":"The first term is $$256$$ and the common ratio is $$r=\\\\frac{1}{4}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{16384}$$","hints":{"DefaultPathway":[{"id":"a18d102geometric9a-h1","type":"hint","dependencies":[],"title":"Formula to Find the Xth Term","text":"To find the Xth term, $$a_x$$, we use the formula with $$a_1$$ being the starting term and $$r$$ being the common ratio that $$a_x=a_1 r^{n-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a18d102geometric9a-h1"],"title":"Determine $$x$$","text":"What is $$x$$ in the formula above? Essentially, which term of the sequence do we want?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$256$$"],"dependencies":["a18d102geometric9a-h2"],"title":"Determine $$a_1$$","text":"What is $$a_1$$ in the formula above? Essentially, what is the starting term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4}$$"],"dependencies":["a18d102geometric9a-h3"],"title":"Determine $$r$$","text":"What is $$r$$? What is the common ratio?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18d102geometric9a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{16384}$$"],"dependencies":["a18d102geometric9a-h4"],"title":"Plug in the Values into Formula","text":"Plug in the values determined above into the formula to find the Xth term and calculate the values. What is the twelfth term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a18d102geometric9a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{16384}$$"],"dependencies":[],"title":"Plug in the Values into Formula","text":"What is $$256{\\\\left(\\\\frac{1}{4}\\\\right)}^{12-1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a18dde9partfrac1","title":"Decomposing a Rational Function with Distinct Linear Factors","body":"Decompose the given rational expression with distinct linear factors.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Partial Fractions","courseName":"OpenStax: College Algebra","steps":[{"id":"a18dde9partfrac1a","stepAnswer":["$$\\\\frac{2}{x+2}+\\\\frac{1}{x-1}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{3x}{\\\\left(x+2\\\\right) \\\\left(x-1\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2}{x+2}+\\\\frac{1}{x-1}$$","hints":{"DefaultPathway":[{"id":"a18dde9partfrac1a-h1","type":"hint","dependencies":[],"title":"Using the definition of partial fraction decomposition","text":"The first step is to separate the denominator factors and give each numerator a symbolic label, like A, B, or C.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac1a-h2","type":"hint","dependencies":["a18dde9partfrac1a-h1"],"title":"Removing denominator","text":"The next step is to multiply both sides of the equation by the common denominator to eliminate the fractions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac1a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3x=A\\\\left(x-1\\\\right)+B\\\\left(x+2\\\\right)$$"],"dependencies":["a18dde9partfrac1a-h2"],"title":"Setting up the system of equations","text":"What is the resulting equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$3x=A\\\\left(x-1\\\\right)+B\\\\left(x+2\\\\right)$$","$$3x=A\\\\left(x-3\\\\right)+B\\\\left(x+1\\\\right)$$","$$3x=A\\\\left(x+1\\\\right)+B\\\\left(x+2\\\\right)$$"]},{"id":"a18dde9partfrac1a-h4","type":"hint","dependencies":["a18dde9partfrac1a-h3"],"title":"Simplifying the equation","text":"The next step is to expand the right side of the equation and combine like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac1a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3x=\\\\left(A+B\\\\right) x-A+2B$$"],"dependencies":["a18dde9partfrac1a-h4"],"title":"Simplifying the equation","text":"What is the simplified equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$3x=\\\\left(A-B\\\\right) x-A+2B$$","$$3x=\\\\left(A+B\\\\right) x-A+2B$$","$$3x=\\\\left(A+1\\\\right) x-A+2B$$"]},{"id":"a18dde9partfrac1a-h6","type":"hint","dependencies":["a18dde9partfrac1a-h5"],"title":"Setting up the system of equations","text":"Set up a system of equations associating corresponding coefficients.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac1a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a18dde9partfrac1a-h6"],"title":"Equating the coefficients to the terms","text":"What coefficient does $$A+B$$ correspond to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac1a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a18dde9partfrac1a-h7"],"title":"Equating the coefficients to the terms","text":"What coefficient does $$-A+2B$$ correspond to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18dde9partfrac10","title":"Decomposing a Rational Function with Distinct Linear Factors","body":"Decompose the given rational expression with distinct linear factors: $$\\\\frac{4x-1}{x^2-x-6}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Partial Fractions","courseName":"OpenStax: College Algebra","steps":[{"id":"a18dde9partfrac10a","stepAnswer":["$$\\\\frac{8}{x-3}-\\\\frac{5}{x-2}$$"],"problemType":"TextBox","stepTitle":"","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{8}{x-3}-\\\\frac{5}{x-2}$$","hints":{"DefaultPathway":[{"id":"a18dde9partfrac10a-h1","type":"hint","dependencies":[],"title":"Factoring the denominator","text":"The first step is to factor the denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac10a-h2","type":"hint","dependencies":["a18dde9partfrac10a-h1"],"title":"Using the definition of partial fraction decomposition","text":"The first step is to separate the denominator factors and give each numerator a symbolic label, like A, B, or C.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac10a-h3","type":"hint","dependencies":["a18dde9partfrac10a-h2"],"title":"Removing denominator","text":"The next step is to multiply both sides of the equation by the common denominator to eliminate the fractions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac10a-h4","type":"hint","dependencies":["a18dde9partfrac10a-h3"],"title":"Simplifying the equation","text":"The next step is to expand the right side of the equation and combine like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac10a-h5","type":"hint","dependencies":["a18dde9partfrac10a-h4"],"title":"Setting up the Systems of Equations","text":"Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18dde9partfrac11","title":"Decomposing with Repeated Linear Factors","body":"Decompose the given rational expression with repeated linear factors.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Partial Fractions","courseName":"OpenStax: College Algebra","steps":[{"id":"a18dde9partfrac11a","stepAnswer":["$$\\\\frac{1}{x-2}+\\\\frac{2}{{\\\\left(x-2\\\\right)}^2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{x}{{\\\\left(x-2\\\\right)}^2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{x-2}+\\\\frac{2}{{\\\\left(x-2\\\\right)}^2}$$","hints":{"DefaultPathway":[{"id":"a18dde9partfrac11a-h1","type":"hint","dependencies":[],"title":"Representing the numerators symbolically","text":"The first step is to use variables like A, B, C, etc. for the numerators and account for the increasing powers of the denominators","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac11a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(x-2)$$, $${\\\\left(x-2\\\\right)}^2$$"],"dependencies":["a18dde9partfrac11a-h1"],"title":"Denominators","text":"What would be the denominators of the decomposed factors?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(x-2)$$, $${\\\\left(x-2\\\\right)}^2$$","$$x+2$$, $${\\\\left(x+2\\\\right)}^2$$","$$(x-2)$$, $$x+2$$"]},{"id":"a18dde9partfrac11a-h3","type":"hint","dependencies":["a18dde9partfrac11a-h2"],"title":"Using the definition of partial fraction decomposition","text":"For each of the denominators in the decomposed form, give each numerator a symbolic label, like A, B, or C.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac11a-h4","type":"hint","dependencies":["a18dde9partfrac11a-h3"],"title":"Remove the denominator","text":"Multiply both sides by the common denominator. On the right side of the equation, expand and collect like terms. The resulting equation should be $$-\\\\left(x^2\\\\right)+2x+4=\\\\left(A+B\\\\right) x^2+\\\\left(-4A-2B+C\\\\right) x+4A$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac11a-h5","type":"hint","dependencies":["a18dde9partfrac11a-h4"],"title":"Finding the resulting equation","text":"Expand the right side of the equation and collect like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac11a-h6","type":"hint","dependencies":["a18dde9partfrac11a-h5"],"title":"Setting up the system of equations","text":"Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18dde9partfrac14","title":"Decomposing with Repeated Linear Factors","body":"Decompose the given rational expression with repeated linear factors.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Partial Fractions","courseName":"OpenStax: College Algebra","steps":[{"id":"a18dde9partfrac14a","stepAnswer":["$$\\\\frac{4}{x}-\\\\frac{3}{2\\\\left(x+1\\\\right)}+\\\\frac{7}{2{\\\\left(3x+2\\\\right)}^2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{5x^2+20x+8}{2{x\\\\left(x+1\\\\right)}^2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{4}{x}-\\\\frac{3}{2\\\\left(x+1\\\\right)}+\\\\frac{7}{2{\\\\left(3x+2\\\\right)}^2}$$","hints":{"DefaultPathway":[{"id":"a18dde9partfrac14a-h1","type":"hint","dependencies":[],"title":"Representing the numerators symbolically","text":"The first step is to use variables like A, B, C, etc. for the numerators and account for the increasing powers of the denominators","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac14a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["x, 2(x+1), 2(x+1)**2"],"dependencies":["a18dde9partfrac14a-h1"],"title":"Denominators","text":"What would be the denominators of the decomposed factors?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac14a-h3","type":"hint","dependencies":["a18dde9partfrac14a-h2"],"title":"Using the definition of partial fraction decomposition","text":"For each of the denominators in the decomposed form, give each numerator a symbolic label, like A, B, or C.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac14a-h4","type":"hint","dependencies":["a18dde9partfrac14a-h3"],"title":"Remove the denominator","text":"Multiply both sides by the common denominator. On the right side of the equation, expand and collect like terms. The resulting equation should be $$-\\\\left(x^2\\\\right)+2x+4=\\\\left(A+B\\\\right) x^2+\\\\left(-4A-2B+C\\\\right) x+4A$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac14a-h5","type":"hint","dependencies":["a18dde9partfrac14a-h4"],"title":"Finding the resulting equation","text":"Expand the right side of the equation and collect like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac14a-h6","type":"hint","dependencies":["a18dde9partfrac14a-h5"],"title":"Setting up the system of equations","text":"Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18dde9partfrac15","title":"Decomposing with Repeated Linear Factors","body":"Decompose the given rational expression with repeated linear factors.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Partial Fractions","courseName":"OpenStax: College Algebra","steps":[{"id":"a18dde9partfrac15a","stepAnswer":["$$\\\\frac{4}{x}+\\\\frac{2}{x^2}-\\\\frac{3}{3x+2}+\\\\frac{7}{2{\\\\left(3x+2\\\\right)}^2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{54x^3+127x^2+80x+16}{2x^{{2\\\\left(3x+2\\\\right)}^2}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{4}{x}+\\\\frac{2}{x^2}-\\\\frac{3}{3x+2}+\\\\frac{7}{2{\\\\left(3x+2\\\\right)}^2}$$","hints":{"DefaultPathway":[{"id":"a18dde9partfrac15a-h1","type":"hint","dependencies":[],"title":"Representing the numerators symbolically","text":"The first step is to use variables like A, B, C, etc. for the numerators and account for the increasing powers of the denominators","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac15a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["x, x**2, 3x+2, 2(3x+2)**2"],"dependencies":["a18dde9partfrac15a-h1"],"title":"Denominators","text":"What would be the denominators of the decomposed factors?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac15a-h3","type":"hint","dependencies":["a18dde9partfrac15a-h2"],"title":"Using the definition of partial fraction decomposition","text":"For each of the denominators in the decomposed form, give each numerator a symbolic label, like A, B, or C.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac15a-h4","type":"hint","dependencies":["a18dde9partfrac15a-h3"],"title":"Remove the denominator","text":"Multiply both sides by the common denominator. On the right side of the equation, expand and collect like terms. The resulting equation should be $$-\\\\left(x^2\\\\right)+2x+4=\\\\left(A+B\\\\right) x^2+\\\\left(-4A-2B+C\\\\right) x+4A$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac15a-h5","type":"hint","dependencies":["a18dde9partfrac15a-h4"],"title":"Finding the resulting equation","text":"Expand the right side of the equation and collect like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac15a-h6","type":"hint","dependencies":["a18dde9partfrac15a-h5"],"title":"Setting up the system of equations","text":"Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18dde9partfrac17","title":"Decomposing a Rational Function with Distinct Linear Factors","body":"Find the decomposition of the parital fraction for the nonrepeating linear factors:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Partial Fractions","courseName":"OpenStax: College Algebra","steps":[{"id":"a18dde9partfrac17a","stepAnswer":["$$\\\\frac{-2}{x+4}+\\\\frac{7}{x-6}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{5x+16}{x^2+10x+24}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-2}{x+4}+\\\\frac{7}{x-6}$$","hints":{"DefaultPathway":[{"id":"a18dde9partfrac17a-h1","type":"hint","dependencies":[],"title":"Factoring the denominator","text":"The first step is to factor the denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac17a-h2","type":"hint","dependencies":["a18dde9partfrac17a-h1"],"title":"Using the definition of partial fraction decomposition","text":"Separate the denominator factors and give each numerator a symbolic label, like A, B, or C.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac17a-h3","type":"hint","dependencies":["a18dde9partfrac17a-h2"],"title":"Removing denominator","text":"The next step is to multiply both sides of the equation by the common denominator to eliminate the fractions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac17a-h4","type":"hint","dependencies":["a18dde9partfrac17a-h3"],"title":"Simplifying the equation","text":"The next step is to expand the right side of the equation and combine like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac17a-h5","type":"hint","dependencies":["a18dde9partfrac17a-h4"],"title":"Setting up the Systems of Equations","text":"Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18dde9partfrac18","title":"Decomposing a Rational Function with Distinct Linear Factors","body":"Find the decomposition of the parital fraction for the nonrepeating linear factors:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Partial Fractions","courseName":"OpenStax: College Algebra","steps":[{"id":"a18dde9partfrac18a","stepAnswer":["$$\\\\frac{2}{x+4}-\\\\frac{3}{x-6}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\left(-x-24\\\\right)}{x^2-2x-24}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2}{x+4}-\\\\frac{3}{x-6}$$","hints":{"DefaultPathway":[{"id":"a18dde9partfrac18a-h1","type":"hint","dependencies":[],"title":"Factoring the denominator","text":"The first step is to factor the denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac18a-h2","type":"hint","dependencies":["a18dde9partfrac18a-h1"],"title":"Using the definition of partial fraction decomposition","text":"Separate the denominator factors and give each numerator a symbolic label, like A, B, or C.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac18a-h3","type":"hint","dependencies":["a18dde9partfrac18a-h2"],"title":"Removing denominator","text":"The next step is to multiply both sides of the equation by the common denominator to eliminate the fractions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac18a-h4","type":"hint","dependencies":["a18dde9partfrac18a-h3"],"title":"Simplifying the equation","text":"The next step is to expand the right side of the equation and combine like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac18a-h5","type":"hint","dependencies":["a18dde9partfrac18a-h4"],"title":"Setting up the Systems of Equations","text":"Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18dde9partfrac19","title":"Decomposing a Rational Function with Distinct Linear Factors","body":"Find the decomposition of the parital fraction for the nonrepeating linear factors:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Partial Fractions","courseName":"OpenStax: College Algebra","steps":[{"id":"a18dde9partfrac19a","stepAnswer":["$$\\\\frac{-1}{3x+5}+\\\\frac{1}{2x+5}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{x}{6x^2+25x+25}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-1}{3x+5}+\\\\frac{1}{2x+5}$$","hints":{"DefaultPathway":[{"id":"a18dde9partfrac19a-h1","type":"hint","dependencies":[],"title":"Factoring the denominator","text":"The first step is to factor the denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac19a-h2","type":"hint","dependencies":["a18dde9partfrac19a-h1"],"title":"Using the definition of partial fraction decomposition","text":"Separate the denominator factors and give each numerator a symbolic label, like A, B, or C.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac19a-h3","type":"hint","dependencies":["a18dde9partfrac19a-h2"],"title":"Removing denominator","text":"The next step is to multiply both sides of the equation by the common denominator to eliminate the fractions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac19a-h4","type":"hint","dependencies":["a18dde9partfrac19a-h3"],"title":"Simplifying the equation","text":"The next step is to expand the right side of the equation and combine like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac19a-h5","type":"hint","dependencies":["a18dde9partfrac19a-h4"],"title":"Setting up the Systems of Equations","text":"Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18dde9partfrac2","title":"Decomposing with Repeated Linear Factors","body":"Decompose the following rational expression (with repeated linear factors.)","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Partial Fractions","courseName":"OpenStax: College Algebra","steps":[{"id":"a18dde9partfrac2a","stepAnswer":["$$\\\\frac{1}{x}-\\\\frac{2}{x-2}+\\\\frac{2}{{\\\\left(x-2\\\\right)}^2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\left(-x^2+2x+4\\\\right)}{x^3-4x^2+4x}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{x}-\\\\frac{2}{x-2}+\\\\frac{2}{{\\\\left(x-2\\\\right)}^2}$$","hints":{"DefaultPathway":[{"id":"a18dde9partfrac2a-h1","type":"hint","dependencies":[],"title":"Factor the denominator","text":"The first step is to factor the denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac2a-h2","type":"hint","dependencies":["a18dde9partfrac2a-h1"],"title":"Decomposition with repeated linear factors","text":"To allow for the repeated factor of $$(x-2)$$, the decomposition will include the denominators $$x$$, $$(x-2)$$, and $${\\\\left(x-2\\\\right)}^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac2a-h3","type":"hint","dependencies":["a18dde9partfrac2a-h2"],"title":"Using the definition of partial fraction decomposition","text":"For each of the denominators in the decomposed form, give each numerator a symbolic label, like A, B, or C.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac2a-h4","type":"hint","dependencies":["a18dde9partfrac2a-h3"],"title":"Remove the denominator","text":"Multiply both sides by the common denominator. On the right side of the equation, expand and collect like terms. The resulting equation should be $$-\\\\left(x^2\\\\right)+2x+4=\\\\left(A+B\\\\right) x^2+\\\\left(-4A-2B+C\\\\right) x+4A$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac2a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{x}-\\\\frac{2}{x-2}+\\\\frac{2}{{\\\\left(x-2\\\\right)}^2}$$"],"dependencies":["a18dde9partfrac2a-h4"],"title":"Resulting equation","text":"What is the resulting equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac2a-h6","type":"hint","dependencies":["a18dde9partfrac2a-h5"],"title":"Setting up the System of Equations","text":"Compare the coefficients of both sides, which will give rise to a system of equations in three variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac2a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a18dde9partfrac2a-h6"],"title":"Comparing Coefficients","text":"What coefficient does $$A+B$$ correspond to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac2a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a18dde9partfrac2a-h7"],"title":"Comparing Coefficients","text":"What coefficient does $$-4A-2B+C$$ correspond to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac2a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a18dde9partfrac2a-h8"],"title":"Comparing Coefficients","text":"What coefficient does 4A correspond to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18dde9partfrac20","title":"Decomposing a Fraction with Repeating Linear Factors","body":"Decompose the partial fraction for the repeating linear factors:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Partial Fractions","courseName":"OpenStax: College Algebra","steps":[{"id":"a18dde9partfrac20a","stepAnswer":["$$\\\\frac{-24}{{\\\\left(x+4\\\\right)}^2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\left(-5-19\\\\right)}{{\\\\left(x+4\\\\right)}^2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-24}{{\\\\left(x+4\\\\right)}^2}$$","hints":{"DefaultPathway":[{"id":"a18dde9partfrac20a-h1","type":"hint","dependencies":[],"title":"Representing the numerators symbolically","text":"The first step is to use variables like A, B, C, etc. for the numerators and account for the increasing powers of the denominators","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac20a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x+4$$, $${\\\\left(x+4\\\\right)}^2$$"],"dependencies":["a18dde9partfrac20a-h1"],"title":"Denominators","text":"What would be the denominators of the decomposed factors?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x+4$$, $${\\\\left(x+4\\\\right)}^2$$","$$x$$, $$x+2$$, $$x+2$$","$$(x-4)$$, $$x+4$$"]},{"id":"a18dde9partfrac20a-h3","type":"hint","dependencies":["a18dde9partfrac20a-h2"],"title":"Using the definition of partial fraction decomposition","text":"For each of the denominators in the decomposed form, give each numerator a symbolic label, like A, B, or C.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac20a-h4","type":"hint","dependencies":["a18dde9partfrac20a-h3"],"title":"Remove the denominator","text":"Multiply both sides by the common denominator. On the right side of the equation, expand and collect like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac20a-h5","type":"hint","dependencies":["a18dde9partfrac20a-h4"],"title":"Finding the resulting equation","text":"Expand the right side of the equation and collect like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac20a-h6","type":"hint","dependencies":["a18dde9partfrac20a-h5"],"title":"Setting up the system of equations","text":"Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18dde9partfrac21","title":"Decomposing a Fraction with Repeating Linear Factors","body":"Decompose the partial fraction for the repeating linear factors:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Partial Fractions","courseName":"OpenStax: College Algebra","steps":[{"id":"a18dde9partfrac21a","stepAnswer":["$$\\\\frac{7}{x+3}-\\\\frac{7}{{\\\\left(x+3\\\\right)}^2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{7x+14}{{\\\\left(x+3\\\\right)}^2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{7}{x+3}-\\\\frac{7}{{\\\\left(x+3\\\\right)}^2}$$","hints":{"DefaultPathway":[{"id":"a18dde9partfrac21a-h1","type":"hint","dependencies":[],"title":"Representing the numerators symbolically","text":"The first step is to use variables like A, B, C, etc. for the numerators and account for the increasing powers of the denominators","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac21a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x+3$$, $${\\\\left(x+3\\\\right)}^2$$"],"dependencies":["a18dde9partfrac21a-h1"],"title":"Denominators","text":"What would be the denominators of the decomposed factors?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x+3$$, $$x+3$$","$$x+3$$, $${\\\\left(x+3\\\\right)}^2$$","$$x+2$$, $$x+1$$"]},{"id":"a18dde9partfrac21a-h3","type":"hint","dependencies":["a18dde9partfrac21a-h2"],"title":"Using the definition of partial fraction decomposition","text":"For each of the denominators in the decomposed form, give each numerator a symbolic label, like A, B, or C.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac21a-h4","type":"hint","dependencies":["a18dde9partfrac21a-h3"],"title":"Remove the denominator","text":"Multiply both sides by the common denominator. On the right side of the equation, expand and collect like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac21a-h5","type":"hint","dependencies":["a18dde9partfrac21a-h4"],"title":"Finding the resulting equation","text":"Expand the right side of the equation and collect like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac21a-h6","type":"hint","dependencies":["a18dde9partfrac21a-h5"],"title":"Setting up the system of equations","text":"Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18dde9partfrac22","title":"Decomposing a Fraction with Repeating Linear Factors","body":"Decompose the partial fraction for the repeating linear factors:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Partial Fractions","courseName":"OpenStax: College Algebra","steps":[{"id":"a18dde9partfrac22a","stepAnswer":["$$\\\\frac{-4}{6x-7}-\\\\frac{55}{{\\\\left(6x-7\\\\right)}^2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\left(-24x-27\\\\right)}{{\\\\left(6x-7\\\\right)}^2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-4}{6x-7}-\\\\frac{55}{{\\\\left(6x-7\\\\right)}^2}$$","hints":{"DefaultPathway":[{"id":"a18dde9partfrac22a-h1","type":"hint","dependencies":[],"title":"Representing the numerators symbolically","text":"The first step is to use variables like A, B, C, etc. for the numerators and account for the increasing powers of the denominators","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac22a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(6x-7)$$, $${\\\\left(6x-7\\\\right)}^2$$"],"dependencies":["a18dde9partfrac22a-h1"],"title":"Denominators","text":"What would be the denominators of the decomposed factors?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(6x-7)$$, $${\\\\left(6x-7\\\\right)}^2$$","$$(6x-7)$$, $$6x+7$$","$$6x$$, $$-7$$"]},{"id":"a18dde9partfrac22a-h3","type":"hint","dependencies":["a18dde9partfrac22a-h2"],"title":"Using the definition of partial fraction decomposition","text":"For each of the denominators in the decomposed form, give each numerator a symbolic label, like A, B, or C.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac22a-h4","type":"hint","dependencies":["a18dde9partfrac22a-h3"],"title":"Remove the denominator","text":"Multiply both sides by the common denominator. On the right side of the equation, expand and collect like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac22a-h5","type":"hint","dependencies":["a18dde9partfrac22a-h4"],"title":"Finding the resulting equation","text":"Expand the right side of the equation and collect like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac22a-h6","type":"hint","dependencies":["a18dde9partfrac22a-h5"],"title":"Setting up the system of equations","text":"Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18dde9partfrac23","title":"Decomposing a Fraction with Repeating Linear Factors","body":"Decompose the partial fraction for the repeating linear factors:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Partial Fractions","courseName":"OpenStax: College Algebra","steps":[{"id":"a18dde9partfrac23a","stepAnswer":["$$\\\\frac{5}{2\\\\left(x+3\\\\right)}-\\\\frac{1}{2{\\\\left(x+3\\\\right)}^2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{5x+14}{2x^2+12x+18}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5}{2\\\\left(x+3\\\\right)}-\\\\frac{1}{2{\\\\left(x+3\\\\right)}^2}$$","hints":{"DefaultPathway":[{"id":"a18dde9partfrac23a-h1","type":"hint","dependencies":[],"title":"Factoring the Denominator","text":"The first step is to factor the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac23a-h2","type":"hint","dependencies":["a18dde9partfrac23a-h1"],"title":"Representing the numerators symbolically","text":"Use variables like A, B, C, etc. for the numerators and account for the increasing powers of the denominators","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac23a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x+3$$, $${\\\\left(x+3\\\\right)}^2$$"],"dependencies":["a18dde9partfrac23a-h2"],"title":"Denominators","text":"What would be the denominators of the decomposed factors?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x+3$$, $${\\\\left(x+3\\\\right)}^2$$","$$(x-3),(x-3)2**$$","$$x\\\\left(x+3\\\\right)$$"]},{"id":"a18dde9partfrac23a-h4","type":"hint","dependencies":["a18dde9partfrac23a-h3"],"title":"Using the definition of partial fraction decomposition","text":"For each of the denominators in the decomposed form, give each numerator a symbolic label, like A, B, or C.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac23a-h5","type":"hint","dependencies":["a18dde9partfrac23a-h4"],"title":"Remove the denominator","text":"Multiply both sides by the common denominator. On the right side of the equation, expand and collect like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac23a-h6","type":"hint","dependencies":["a18dde9partfrac23a-h5"],"title":"Finding the resulting equation","text":"Expand the right side of the equation and collect like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac23a-h7","type":"hint","dependencies":["a18dde9partfrac23a-h6"],"title":"Setting up the system of equations","text":"Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18dde9partfrac24","title":"Decomposing a Fraction with Repeating Linear Factors","body":"Decompose the partial fraction for the repeating linear factors:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Partial Fractions","courseName":"OpenStax: College Algebra","steps":[{"id":"a18dde9partfrac24a","stepAnswer":["$$\\\\frac{1}{5} x-\\\\frac{1}{3\\\\left(3x-5\\\\right)}+\\\\frac{20}{3{\\\\left(3x+5\\\\right)}^2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{4x^2+55x+25}{5{x\\\\left(3x+5\\\\right)}^2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{5} x-\\\\frac{1}{3\\\\left(3x-5\\\\right)}+\\\\frac{20}{3{\\\\left(3x+5\\\\right)}^2}$$","hints":{"DefaultPathway":[{"id":"a18dde9partfrac24a-h1","type":"hint","dependencies":[],"title":"Representing the numerators symbolically","text":"Use variables like A, B, C, etc. for the numerators and account for the increasing powers of the denominators","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac24a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["x,(3x+5),(3x+5)**2"],"dependencies":["a18dde9partfrac24a-h1"],"title":"Denominators","text":"What would be the denominators of the decomposed factors?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["x,(3x+5),(3x+5)**2","$$x$$, $$3x+5$$","None of the above"]},{"id":"a18dde9partfrac24a-h3","type":"hint","dependencies":["a18dde9partfrac24a-h2"],"title":"Using the definition of partial fraction decomposition","text":"For each of the denominators in the decomposed form, give each numerator a symbolic label, like A, B, or C. For the term with the quadratic denominator, express the numerator as a linear expression like $$Ax+B$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac24a-h4","type":"hint","dependencies":["a18dde9partfrac24a-h3"],"title":"Remove the denominator","text":"Multiply both sides by the common denominator. On the right side of the equation, expand and collect like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac24a-h5","type":"hint","dependencies":["a18dde9partfrac24a-h4"],"title":"Finding the resulting equation","text":"Expand the right side of the equation and collect like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac24a-h6","type":"hint","dependencies":["a18dde9partfrac24a-h5"],"title":"Setting up the system of equations","text":"Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18dde9partfrac25","title":"Decomposing a Fraction with Repeating Linear Factors","body":"Decompose the partial fraction for the repeating linear factors:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Partial Fractions","courseName":"OpenStax: College Algebra","steps":[{"id":"a18dde9partfrac25a","stepAnswer":["$$\\\\frac{-354}{169x}+\\\\frac{72}{13x^2}+\\\\frac{523x+2467}{169\\\\left(x^2+12x+26\\\\right)}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{x^3-5x^2+12x+144}{x^{2\\\\left(x^2+12x+26\\\\right)}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-354}{169x}+\\\\frac{72}{13x^2}+\\\\frac{523x+2467}{169\\\\left(x^2+12x+26\\\\right)}$$","hints":{"DefaultPathway":[{"id":"a18dde9partfrac25a-h1","type":"hint","dependencies":[],"title":"Using the definition of partial fraction decomposition","text":"For each of the denominators in the decomposed form, give each numerator a symbolic label, like A, B, or C. For the term with the quadratic denominator, express the numerator as a linear expression like $$Ax+B$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac25a-h2","type":"hint","dependencies":["a18dde9partfrac25a-h1"],"title":"Remove the denominator","text":"Multiply both sides by the common denominator. On the right side of the equation, expand and collect like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac25a-h3","type":"hint","dependencies":["a18dde9partfrac25a-h2"],"title":"Finding the resulting equation","text":"Expand the right side of the equation and collect like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac25a-h4","type":"hint","dependencies":["a18dde9partfrac25a-h3"],"title":"Setting up the system of equations","text":"Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18dde9partfrac26","title":"Decomposing a Fraction with Nonrepeating Quadratic Factors","body":"Find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Partial Fractions","courseName":"OpenStax: College Algebra","steps":[{"id":"a18dde9partfrac26a","stepAnswer":["$$\\\\frac{2}{x-1}+\\\\frac{2x-1}{x^2+6x+11}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{4x^2+9x+23}{\\\\left(x-1\\\\right) \\\\left(x^2+6x+11\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2}{x-1}+\\\\frac{2x-1}{x^2+6x+11}$$","hints":{"DefaultPathway":[{"id":"a18dde9partfrac26a-h1","type":"hint","dependencies":[],"title":"Representing the numerators symbolically","text":"The first step is to use variables like A, B, C, etc. for the constant numerators and $$Ax+B$$ for the numeratores of each quadratic factor in the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac26a-h2","type":"hint","dependencies":["a18dde9partfrac26a-h1"],"title":"Using the definition of partial fraction decomposition","text":"For each of the denominators in the decomposed form, give each numerator a symbolic label, like A, B, or C.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac26a-h3","type":"hint","dependencies":["a18dde9partfrac26a-h2"],"title":"Remove the denominator","text":"Multiply both sides by the common denominator. On the right side of the equation, expand and collect like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac26a-h4","type":"hint","dependencies":["a18dde9partfrac26a-h3"],"title":"Finding the resulting equation","text":"Expand the right side of the equation and collect like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac26a-h5","type":"hint","dependencies":["a18dde9partfrac26a-h4"],"title":"Setting up the system of equations","text":"Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18dde9partfrac27","title":"Decomposing a Fraction with Nonrepeating Quadratic Factors","body":"Find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Partial Fractions","courseName":"OpenStax: College Algebra","steps":[{"id":"a18dde9partfrac27a","stepAnswer":["$$\\\\frac{1}{6\\\\left(x+1\\\\right)}+\\\\frac{5x+8}{6\\\\left(x^2+5x-2\\\\right)}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{x^2+3x+1}{\\\\left(x+1\\\\right) \\\\left(x^2+5x-2\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{6\\\\left(x+1\\\\right)}+\\\\frac{5x+8}{6\\\\left(x^2+5x-2\\\\right)}$$","hints":{"DefaultPathway":[{"id":"a18dde9partfrac27a-h1","type":"hint","dependencies":[],"title":"Representing the numerators symbolically","text":"The first step is to use variables like A, B, C, etc. for the constant numerators and $$Ax+B$$ for the numeratores of each quadratic factor in the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac27a-h2","type":"hint","dependencies":["a18dde9partfrac27a-h1"],"title":"Using the definition of partial fraction decomposition","text":"For each of the denominators in the decomposed form, give each numerator a symbolic label, like A, B, or C.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac27a-h3","type":"hint","dependencies":["a18dde9partfrac27a-h2"],"title":"Remove the denominator","text":"Multiply both sides by the common denominator. On the right side of the equation, expand and collect like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac27a-h4","type":"hint","dependencies":["a18dde9partfrac27a-h3"],"title":"Finding the resulting equation","text":"Expand the right side of the equation and collect like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac27a-h5","type":"hint","dependencies":["a18dde9partfrac27a-h4"],"title":"Setting up the system of equations","text":"Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18dde9partfrac28","title":"Decomposing a Fraction with Nonrepeating Quadratic Factors","body":"Find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Partial Fractions","courseName":"OpenStax: College Algebra","steps":[{"id":"a18dde9partfrac28a","stepAnswer":["$$\\\\frac{-20}{3\\\\left(x+5\\\\right)}+\\\\frac{32x-20}{3\\\\left(x^2+7x-5\\\\right)}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{4x^2}{\\\\left(x+5\\\\right) \\\\left(x^2+7x-5\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-20}{3\\\\left(x+5\\\\right)}+\\\\frac{32x-20}{3\\\\left(x^2+7x-5\\\\right)}$$","hints":{"DefaultPathway":[{"id":"a18dde9partfrac28a-h1","type":"hint","dependencies":[],"title":"Representing the numerators symbolically","text":"The first step is to use variables like A, B, C, etc. for the constant numerators and $$Ax+B$$ for the numeratores of each quadratic factor in the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac28a-h2","type":"hint","dependencies":["a18dde9partfrac28a-h1"],"title":"Using the definition of partial fraction decomposition","text":"For each of the denominators in the decomposed form, give each numerator a symbolic label, like A, B, or C.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac28a-h3","type":"hint","dependencies":["a18dde9partfrac28a-h2"],"title":"Remove the denominator","text":"Multiply both sides by the common denominator. On the right side of the equation, expand and collect like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac28a-h4","type":"hint","dependencies":["a18dde9partfrac28a-h3"],"title":"Finding the resulting equation","text":"Expand the right side of the equation and collect like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac28a-h5","type":"hint","dependencies":["a18dde9partfrac28a-h4"],"title":"Setting up the system of equations","text":"Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18dde9partfrac29","title":"Decomposing a Fraction with Nonrepeating Quadratic Factors","body":"Find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Partial Fractions","courseName":"OpenStax: College Algebra","steps":[{"id":"a18dde9partfrac29a","stepAnswer":["$$\\\\left(-\\\\frac{5}{x+2}\\\\right)+\\\\frac{8}{x^2-2x+4}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\left(-5x^2+18x-4\\\\right)}{x^3+8}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(-\\\\frac{5}{x+2}\\\\right)+\\\\frac{8}{x^2-2x+4}$$","hints":{"DefaultPathway":[{"id":"a18dde9partfrac29a-h1","type":"hint","dependencies":[],"title":"Representing the numerators symbolically","text":"The first step is to use variables like A, B, C, etc. for the constant numerators and $$Ax+B$$ for the numeratores of each quadratic factor in the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac29a-h2","type":"hint","dependencies":["a18dde9partfrac29a-h1"],"title":"Using the definition of partial fraction decomposition","text":"For each of the denominators in the decomposed form, give each numerator a symbolic label, like A, B, or C.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac29a-h3","type":"hint","dependencies":["a18dde9partfrac29a-h2"],"title":"Remove the denominator","text":"Multiply both sides by the common denominator. On the right side of the equation, expand and collect like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac29a-h4","type":"hint","dependencies":["a18dde9partfrac29a-h3"],"title":"Finding the resulting equation","text":"Expand the right side of the equation and collect like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac29a-h5","type":"hint","dependencies":["a18dde9partfrac29a-h4"],"title":"Setting up the system of equations","text":"Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18dde9partfrac3","title":"Decomposing polynomial fractions when the denominator contains a nonrepeated irreducible quadratic factor","body":"Find the partial fraction decomposition of the given expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Partial Fractions","courseName":"OpenStax: College Algebra","steps":[{"id":"a18dde9partfrac3a","stepAnswer":["$$\\\\frac{2}{x+3}+\\\\frac{6x-8}{x^2+x+2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{8x^2+12x-20}{\\\\left(x+3\\\\right) \\\\left(x^2+x+2\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2}{x+3}+\\\\frac{6x-8}{x^2+x+2}$$","hints":{"DefaultPathway":[{"id":"a18dde9partfrac3a-h1","type":"hint","dependencies":[],"title":"Using the definition of partial fraction decomposition","text":"Since the denominator contains one linear factor and one irreducible quadratic factor, the numerator of the linear factor will be a constant (A), and the numerator of the quadratic factor will be a linear expression (e.g. Bx+C)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac3a-h2","type":"hint","dependencies":["a18dde9partfrac3a-h1"],"title":"Removing the denominator","text":"Multiply both sides of the equation by the common denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac3a-h3","type":"hint","dependencies":["a18dde9partfrac3a-h2"],"title":"Finding the resulting equation","text":"Expand the right side and combine like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac3a-h4","type":"hint","dependencies":["a18dde9partfrac3a-h3"],"title":"Resulting equation","text":"The resulting equation is: $$8x^2+12x-20=\\\\left(A+B\\\\right) x^2+\\\\left(A+3B+C\\\\right) x+2A+3C$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac3a-h5","type":"hint","dependencies":["a18dde9partfrac3a-h4"],"title":"Setting up the system of equations","text":"Compare the coefficients of both sides, which will give rise to a system of equations in three variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac3a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a18dde9partfrac3a-h5"],"title":"Comparing Coefficients","text":"What coefficient does $$A+B$$ correspond to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac3a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a18dde9partfrac3a-h6"],"title":"Comparing Coefficients","text":"What coefficient does $$A+3B+C$$ correspond to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac3a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-20$$"],"dependencies":["a18dde9partfrac3a-h7"],"title":"Comparing Coefficients","text":"What coefficient does $$2A+3C$$ correspond to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18dde9partfrac30","title":"Decomposing a Fraction with Nonrepeating Quadratic Factors","body":"Find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Partial Fractions","courseName":"OpenStax: College Algebra","steps":[{"id":"a18dde9partfrac30a","stepAnswer":["$$\\\\frac{1}{x-5}-\\\\frac{3}{x^2+5x+25}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{x^2+2x+40}{x^3-125}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{x-5}-\\\\frac{3}{x^2+5x+25}$$","hints":{"DefaultPathway":[{"id":"a18dde9partfrac30a-h1","type":"hint","dependencies":[],"title":"Representing the numerators symbolically","text":"The first step is to use variables like A, B, C, etc. for the constant numerators and $$Ax+B$$ for the numeratores of each quadratic factor in the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac30a-h2","type":"hint","dependencies":["a18dde9partfrac30a-h1"],"title":"Using the definition of partial fraction decomposition","text":"For each of the denominators in the decomposed form, give each numerator a symbolic label, like A, B, or C.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac30a-h3","type":"hint","dependencies":["a18dde9partfrac30a-h2"],"title":"Remove the denominator","text":"Multiply both sides by the common denominator. On the right side of the equation, expand and collect like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac30a-h4","type":"hint","dependencies":["a18dde9partfrac30a-h3"],"title":"Finding the resulting equation","text":"Expand the right side of the equation and collect like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac30a-h5","type":"hint","dependencies":["a18dde9partfrac30a-h4"],"title":"Setting up the system of equations","text":"Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18dde9partfrac4","title":"Decomposing a Rational Function with Distinct Linear Factors","body":"Decompose the given rational expression with distinct linear factors.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Partial Fractions","courseName":"OpenStax: College Algebra","steps":[{"id":"a18dde9partfrac4a","stepAnswer":["$$\\\\frac{8}{x+3}-\\\\frac{5}{x-8}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{3x-79}{x^2-5x-24}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{8}{x+3}-\\\\frac{5}{x-8}$$","hints":{"DefaultPathway":[{"id":"a18dde9partfrac4a-h1","type":"hint","dependencies":[],"title":"Factoring the denominator","text":"The first step is to factor the denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac4a-h2","type":"hint","dependencies":["a18dde9partfrac4a-h1"],"title":"Using the definition of partial fraction decomposition","text":"The first step is to separate the denominator factors and give each numerator a symbolic label, like A, B, or C.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac4a-h3","type":"hint","dependencies":["a18dde9partfrac4a-h2"],"title":"Removing denominator","text":"The next step is to multiply both sides of the equation by the common denominator to eliminate the fractions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac4a-h4","type":"hint","dependencies":["a18dde9partfrac4a-h3"],"title":"Simplifying the equation","text":"The next step is to expand the right side of the equation and combine like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac4a-h5","type":"hint","dependencies":["a18dde9partfrac4a-h4"],"title":"Setting up the Systems of Equations","text":"Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18dde9partfrac5","title":"Decomposing a Rational Function with Distinct Linear Factors","body":"Decompose the given rational expression with distinct linear factors: $$\\\\frac{10x+47}{x^2+7x+10}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Partial Fractions","courseName":"OpenStax: College Algebra","steps":[{"id":"a18dde9partfrac5a","stepAnswer":["$$\\\\frac{1}{x+5}+\\\\frac{9}{x+2}$$"],"problemType":"TextBox","stepTitle":"","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{x+5}+\\\\frac{9}{x+2}$$","hints":{"DefaultPathway":[{"id":"a18dde9partfrac5a-h1","type":"hint","dependencies":[],"title":"Factoring the denominator","text":"The first step is to factor the denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac5a-h2","type":"hint","dependencies":["a18dde9partfrac5a-h1"],"title":"Using the definition of partial fraction decomposition","text":"The first step is to separate the denominator factors and give each numerator a symbolic label, like A, B, or C.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac5a-h3","type":"hint","dependencies":["a18dde9partfrac5a-h2"],"title":"Removing denominator","text":"The next step is to multiply both sides of the equation by the common denominator to eliminate the fractions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac5a-h4","type":"hint","dependencies":["a18dde9partfrac5a-h3"],"title":"Simplifying the equation","text":"The next step is to expand the right side of the equation and combine like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac5a-h5","type":"hint","dependencies":["a18dde9partfrac5a-h4"],"title":"Setting up the Systems of Equations","text":"Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18dde9partfrac6","title":"Decomposing a Rational Function with Distinct Linear Factors","body":"Decompose the given rational expression with distinct linear factors: $$\\\\frac{32x-11}{20x^2-13x+2}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Partial Fractions","courseName":"OpenStax: College Algebra","steps":[{"id":"a18dde9partfrac6a","stepAnswer":["$$\\\\frac{3}{5x-2}+\\\\frac{4}{4x-1}$$"],"problemType":"TextBox","stepTitle":"","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{5x-2}+\\\\frac{4}{4x-1}$$","hints":{"DefaultPathway":[{"id":"a18dde9partfrac6a-h1","type":"hint","dependencies":[],"title":"Factoring the denominator","text":"The first step is to factor the denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac6a-h2","type":"hint","dependencies":["a18dde9partfrac6a-h1"],"title":"Using the definition of partial fraction decomposition","text":"The first step is to separate the denominator factors and give each numerator a symbolic label, like A, B, or C.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac6a-h3","type":"hint","dependencies":["a18dde9partfrac6a-h2"],"title":"Removing denominator","text":"The next step is to multiply both sides of the equation by the common denominator to eliminate the fractions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac6a-h4","type":"hint","dependencies":["a18dde9partfrac6a-h3"],"title":"Simplifying the equation","text":"The next step is to expand the right side of the equation and combine like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac6a-h5","type":"hint","dependencies":["a18dde9partfrac6a-h4"],"title":"Setting up the Systems of Equations","text":"Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18dde9partfrac7","title":"Decomposing a Rational Function with Distinct Linear Factors","body":"Decompose the given rational expression with distinct linear factors: $$\\\\frac{5x}{x^2-9}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Partial Fractions","courseName":"OpenStax: College Algebra","steps":[{"id":"a18dde9partfrac7a","stepAnswer":["$$\\\\frac{5}{2\\\\left(x+3\\\\right)}+\\\\frac{5}{2\\\\left(x-3\\\\right)}$$"],"problemType":"TextBox","stepTitle":"","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5}{2\\\\left(x+3\\\\right)}+\\\\frac{5}{2\\\\left(x-3\\\\right)}$$","hints":{"DefaultPathway":[{"id":"a18dde9partfrac7a-h1","type":"hint","dependencies":[],"title":"Factoring the denominator","text":"The first step is to factor the denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac7a-h2","type":"hint","dependencies":["a18dde9partfrac7a-h1"],"title":"Using the definition of partial fraction decomposition","text":"The first step is to separate the denominator factors and give each numerator a symbolic label, like A, B, or C.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac7a-h3","type":"hint","dependencies":["a18dde9partfrac7a-h2"],"title":"Removing denominator","text":"The next step is to multiply both sides of the equation by the common denominator to eliminate the fractions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac7a-h4","type":"hint","dependencies":["a18dde9partfrac7a-h3"],"title":"Simplifying the equation","text":"The next step is to expand the right side of the equation and combine like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac7a-h5","type":"hint","dependencies":["a18dde9partfrac7a-h4"],"title":"Setting up the Systems of Equations","text":"Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18dde9partfrac8","title":"Decomposing a Rational Function with Distinct Linear Factors","body":"Decompose the given rational expression with distinct linear factors: $$\\\\frac{6x}{x^2-4}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Partial Fractions","courseName":"OpenStax: College Algebra","steps":[{"id":"a18dde9partfrac8a","stepAnswer":["$$\\\\frac{3}{x+2}+\\\\frac{3}{x-2}$$"],"problemType":"TextBox","stepTitle":"","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{x+2}+\\\\frac{3}{x-2}$$","hints":{"DefaultPathway":[{"id":"a18dde9partfrac8a-h1","type":"hint","dependencies":[],"title":"Factoring the denominator","text":"The first step is to factor the denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac8a-h2","type":"hint","dependencies":["a18dde9partfrac8a-h1"],"title":"Using the definition of partial fraction decomposition","text":"The first step is to separate the denominator factors and give each numerator a symbolic label, like A, B, or C.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac8a-h3","type":"hint","dependencies":["a18dde9partfrac8a-h2"],"title":"Removing denominator","text":"The next step is to multiply both sides of the equation by the common denominator to eliminate the fractions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac8a-h4","type":"hint","dependencies":["a18dde9partfrac8a-h3"],"title":"Simplifying the equation","text":"The next step is to expand the right side of the equation and combine like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac8a-h5","type":"hint","dependencies":["a18dde9partfrac8a-h4"],"title":"Setting up the Systems of Equations","text":"Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18dde9partfrac9","title":"Decomposing a Rational Function with Distinct Linear Factors","body":"Decompose the given rational expression with distinct linear factors: $$\\\\frac{4x-1}{x^2-x-6}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Partial Fractions","courseName":"OpenStax: College Algebra","steps":[{"id":"a18dde9partfrac9a","stepAnswer":["$$\\\\frac{9}{5\\\\left(x+2\\\\right)}+\\\\frac{11}{5\\\\left(x-3\\\\right)}$$"],"problemType":"TextBox","stepTitle":"","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{9}{5\\\\left(x+2\\\\right)}+\\\\frac{11}{5\\\\left(x-3\\\\right)}$$","hints":{"DefaultPathway":[{"id":"a18dde9partfrac9a-h1","type":"hint","dependencies":[],"title":"Factoring the denominator","text":"The first step is to factor the denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac9a-h2","type":"hint","dependencies":["a18dde9partfrac9a-h1"],"title":"Using the definition of partial fraction decomposition","text":"The first step is to separate the denominator factors and give each numerator a symbolic label, like A, B, or C.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac9a-h3","type":"hint","dependencies":["a18dde9partfrac9a-h2"],"title":"Removing denominator","text":"The next step is to multiply both sides of the equation by the common denominator to eliminate the fractions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac9a-h4","type":"hint","dependencies":["a18dde9partfrac9a-h3"],"title":"Simplifying the equation","text":"The next step is to expand the right side of the equation and combine like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfrac9a-h5","type":"hint","dependencies":["a18dde9partfrac9a-h4"],"title":"Setting up the Systems of Equations","text":"Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18dde9partfract12","title":"Decomposing with Repeated Linear Factors","body":"Decompose the given rational expression with repeated linear factors.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Partial Fractions","courseName":"OpenStax: College Algebra","steps":[{"id":"a18dde9partfract12a","stepAnswer":["$$-\\\\left(\\\\frac{6}{4x+5}\\\\right)+\\\\frac{3}{{\\\\left(4x+5\\\\right)}^2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\left(-24x-27\\\\right)}{{\\\\left(6x-7\\\\right)}^2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-\\\\left(\\\\frac{6}{4x+5}\\\\right)+\\\\frac{3}{{\\\\left(4x+5\\\\right)}^2}$$","hints":{"DefaultPathway":[{"id":"a18dde9partfract12a-h1","type":"hint","dependencies":[],"title":"Representing the numerators symbolically","text":"The first step is to use variables like A, B, C, etc. for the numerators and account for the increasing powers of the denominators","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfract12a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(6x-7), (6x-7)**2"],"dependencies":["a18dde9partfract12a-h1"],"title":"Denominators","text":"What would be the denominators of the decomposed factors?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfract12a-h3","type":"hint","dependencies":["a18dde9partfract12a-h2"],"title":"Using the definition of partial fraction decomposition","text":"For each of the denominators in the decomposed form, give each numerator a symbolic label, like A, B, or C.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfract12a-h4","type":"hint","dependencies":["a18dde9partfract12a-h3"],"title":"Remove the denominator","text":"Multiply both sides by the common denominator. On the right side of the equation, expand and collect like terms. The resulting equation should be $$-\\\\left(x^2\\\\right)+2x+4=\\\\left(A+B\\\\right) x^2+\\\\left(-4A-2B+C\\\\right) x+4A$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfract12a-h5","type":"hint","dependencies":["a18dde9partfract12a-h4"],"title":"Finding the resulting equation","text":"Expand the right side of the equation and collect like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfract12a-h6","type":"hint","dependencies":["a18dde9partfract12a-h5"],"title":"Setting up the system of equations","text":"Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a18dde9partfract13","title":"Decomposing with Repeated Linear Factors","body":"Decompose the given rational expression with repeated linear factors.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Partial Fractions","courseName":"OpenStax: College Algebra","steps":[{"id":"a18dde9partfract13a","stepAnswer":["$$-\\\\left(\\\\frac{1}{x-7}\\\\right)-\\\\frac{2}{{\\\\left(x-7\\\\right)}^2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{5-x}{{\\\\left(x-7\\\\right)}^2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-\\\\left(\\\\frac{1}{x-7}\\\\right)-\\\\frac{2}{{\\\\left(x-7\\\\right)}^2}$$","hints":{"DefaultPathway":[{"id":"a18dde9partfract13a-h1","type":"hint","dependencies":[],"title":"Representing the numerators symbolically","text":"The first step is to use variables like A, B, C, etc. for the numerators and account for the increasing powers of the denominators","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfract13a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(x-7),(x-7)**2"],"dependencies":["a18dde9partfract13a-h1"],"title":"Denominators","text":"What would be the denominators of the decomposed factors?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfract13a-h3","type":"hint","dependencies":["a18dde9partfract13a-h2"],"title":"Using the definition of partial fraction decomposition","text":"For each of the denominators in the decomposed form, give each numerator a symbolic label, like A, B, or C.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfract13a-h4","type":"hint","dependencies":["a18dde9partfract13a-h3"],"title":"Remove the denominator","text":"Multiply both sides by the common denominator. On the right side of the equation, expand and collect like terms. The resulting equation should be $$-\\\\left(x^2\\\\right)+2x+4=\\\\left(A+B\\\\right) x^2+\\\\left(-4A-2B+C\\\\right) x+4A$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfract13a-h5","type":"hint","dependencies":["a18dde9partfract13a-h4"],"title":"Finding the resulting equation","text":"Expand the right side of the equation and collect like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a18dde9partfract13a-h6","type":"hint","dependencies":["a18dde9partfract13a-h5"],"title":"Setting up the system of equations","text":"Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a1ee1measure1","title":"How to Make Unit Conversions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.10 Systems of Measurement","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1a1ee1measure1a","stepAnswer":["$$5.5$$"],"problemType":"TextBox","stepTitle":"MaryAnne is $$66$$ inches tall. Convert her height into feet.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5.5$$","hints":{"DefaultPathway":[{"id":"a1a1ee1measure1a-h1","type":"hint","dependencies":[],"title":"Convert","text":"Multiply the measurement to be converted by 1; write $$1$$ as a fraction relating the units given and the units needed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a1a1ee1measure1a-h1"],"title":"Convert","text":"How many inches are in a foot?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure1a-h3","type":"hint","dependencies":["a1a1ee1measure1a-h2"],"title":"Multiply","text":"Multiply $$66$$ inches by $$1$$ $$\\\\frac{foot}{12}$$ inches (inches should be in the denominator so that the inches will divide out).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5.5$$"],"dependencies":["a1a1ee1measure1a-h3"],"title":"Multiply","text":"What do we get after the multiplication?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a1ee1measure10","title":"Use Mixed Units of Measurement in the Metric System","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.10 Systems of Measurement","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1a1ee1measure10a","stepAnswer":["$$0.75$$"],"problemType":"TextBox","stepTitle":"Ryland is $$1.6$$ meters tall. His younger brother is $$85$$ centimeters tall. How much taller is Ryland than his younger brother (in meters)?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.75$$","hints":{"DefaultPathway":[{"id":"a1a1ee1measure10a-h1","type":"hint","dependencies":[],"title":"Convert","text":"We can convert both measurements to either centimeters or meters. Since meters is the larger unit, we will subtract the lengths in meters.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$100$$"],"dependencies":["a1a1ee1measure10a-h1"],"title":"Convert","text":"How many centimeters are in one meter?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.85$$"],"dependencies":["a1a1ee1measure10a-h2"],"title":"Convert","text":"We convert $$85$$ centimeters to meters by moving the decimal $$2$$ places to the left. What do we get after the conversion?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure10a-h4","type":"hint","dependencies":["a1a1ee1measure10a-h3"],"title":"Subtract","text":"We now want to find the difference between $$1.60$$ and $$0.85$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.75$$"],"dependencies":["a1a1ee1measure10a-h4"],"title":"Subtract","text":"What do we get after the subtraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a1ee1measure11","title":"Use Mixed Units of Measurement in the Metric System","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.10 Systems of Measurement","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1a1ee1measure11a","stepAnswer":["$$0.45$$"],"problemType":"TextBox","stepTitle":"Dena\u2019s recipe for lentil soup calls for $$150$$ milliliters of olive oil. Dena wants to triple the recipe. How many liters of olive oil will she need?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.45$$","hints":{"DefaultPathway":[{"id":"a1a1ee1measure11a-h1","type":"hint","dependencies":[],"title":"Triple","text":"We will find the amount of olive oil in millileters then convert to liters.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$450$$"],"dependencies":["a1a1ee1measure11a-h1"],"title":"Triple","text":"What do we get after tripling $$150$$ mL (in mL)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure11a-h3","type":"hint","dependencies":["a1a1ee1measure11a-h2"],"title":"Convert","text":"Multiply by $$1$$, writing $$1$$ as a fraction relating mL to L.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1000$$"],"dependencies":["a1a1ee1measure11a-h3"],"title":"Convert","text":"How many mL are in one liter?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure11a-h5","type":"hint","dependencies":["a1a1ee1measure11a-h4"],"title":"Multiply","text":"Multiply $$450$$ mL by $$1$$ $$\\\\frac{L}{1000}$$ mL (mL should be in the denominator so that mL will divide out).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure11a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.45$$"],"dependencies":["a1a1ee1measure11a-h5"],"title":"Multiply","text":"What do we get after the multiplication?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a1ee1measure12","title":"Convert Between the U.S. and the Metric Systems of Measurement","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.10 Systems of Measurement","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1a1ee1measure12a","stepAnswer":["$$16.7$$"],"problemType":"TextBox","stepTitle":"Lee\u2019s water bottle holds $$500$$ mL of water. How many ounces are in the bottle? Round to the nearest tenth of an ounce.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16.7$$","hints":{"DefaultPathway":[{"id":"a1a1ee1measure12a-h1","type":"hint","dependencies":[],"title":"Convert","text":"Multiply by a unit conversion factor relating mL and ounces.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$30$$"],"dependencies":["a1a1ee1measure12a-h1"],"title":"Convert","text":"How many mL are in one ounce (rounded to a digit)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure12a-h3","type":"hint","dependencies":["a1a1ee1measure12a-h2"],"title":"Multiply","text":"Multiply $$500$$ mL by $$1$$ $$\\\\frac{ounce}{30}$$ mL (mL should be in the denominator so that mL will divide out).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16.7$$"],"dependencies":["a1a1ee1measure12a-h3"],"title":"Multiply","text":"What do we get after the multiplication?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a1ee1measure13","title":"Convert Between the U.S. and the Metric Systems of Measurement","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.10 Systems of Measurement","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1a1ee1measure13a","stepAnswer":["$$62$$"],"problemType":"TextBox","stepTitle":"Soleil was on a road trip and saw a sign that said the next rest stop was in $$100$$ kilometers. How many miles until the next rest stop? Round to the nearest integer.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$62$$","hints":{"DefaultPathway":[{"id":"a1a1ee1measure13a-h1","type":"hint","dependencies":[],"title":"Convert","text":"Multiply by a unit conversion factor relating km and mi.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.61$$"],"dependencies":["a1a1ee1measure13a-h1"],"title":"Convert","text":"How many km are in one mile?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure13a-h3","type":"hint","dependencies":["a1a1ee1measure13a-h2"],"title":"Multiply","text":"Multiply $$100$$ km by $$1$$ $$\\\\frac{mile}{1.61}$$ km (km should be in the denominator so that km will divide out).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$62$$"],"dependencies":["a1a1ee1measure13a-h3"],"title":"Multiply","text":"What do we get after the multiplication?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a1ee1measure14","title":"Convert between Fahrenheit and Celsius Temperatures","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.10 Systems of Measurement","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1a1ee1measure14a","stepAnswer":["$$10$$"],"problemType":"TextBox","stepTitle":"Convert $$50$$ degrees Fahrenheit into degrees Celsius.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10$$","hints":{"DefaultPathway":[{"id":"a1a1ee1measure14a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We will substitute $$50$$ degrees Fahrenheit into the formula, $$C=\\\\frac{5}{9} \\\\left(F-32\\\\right)$$, to find C.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure14a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$C=\\\\frac{5}{9} \\\\left(50-32\\\\right)$$"],"dependencies":["a1a1ee1measure14a-h1"],"title":"Substitute","text":"What do we get after the substitution?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$C=\\\\frac{9}{5} \\\\left(50-32\\\\right)$$","$$C=\\\\frac{5}{9} \\\\left(50-32\\\\right)$$","$$C=\\\\frac{5}{9} \\\\left(50+32\\\\right)$$"]},{"id":"a1a1ee1measure14a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a1a1ee1measure14a-h2"],"title":"Simplify","text":"What do we get for C after simplifying the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a1ee1measure15","title":"Convert between Fahrenheit and Celsius Temperatures","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.10 Systems of Measurement","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1a1ee1measure15a","stepAnswer":["$$68$$"],"problemType":"TextBox","stepTitle":"While visiting Paris, Woody saw the temperature was $$20$$ degrees Celsius. Convert the temperature into degrees Fahrenheit.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$68$$","hints":{"DefaultPathway":[{"id":"a1a1ee1measure15a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We will substitute $$20$$ degrees Celsius into the formula, $$F=\\\\frac{9}{5} C+32$$, to find F.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure15a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$F=20\\\\frac{9}{5}+32$$"],"dependencies":["a1a1ee1measure15a-h1"],"title":"Substitute","text":"What do we get after the substitution?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$F=20\\\\frac{9}{5}-32$$","$$F=20\\\\frac{5}{9}+32$$","$$F=20\\\\frac{9}{5}+32$$"]},{"id":"a1a1ee1measure15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$68$$"],"dependencies":["a1a1ee1measure15a-h2"],"title":"Simplify","text":"What do we get for F after simplifying the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a1ee1measure16","title":"In the following exercise, convert the units.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.10 Systems of Measurement","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1a1ee1measure16a","stepAnswer":["$$72$$"],"problemType":"TextBox","stepTitle":"A park bench is $$6$$ feet long. Convert the length to inches.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$72$$","hints":{"DefaultPathway":[{"id":"a1a1ee1measure16a-h1","type":"hint","dependencies":[],"title":"Multiply by $$1$$","text":"The first step is to multiply the measurement to be converted by 1; write $$1$$ as a fraction relating the units given and the units needed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure16a-h2","type":"hint","dependencies":["a1a1ee1measure16a-h1"],"title":"Relating Feet to Inches","text":"The conversion from feet to inches is $$1$$ $$foot=12$$ inches.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure16a-h3","type":"hint","dependencies":["a1a1ee1measure16a-h2"],"title":"Writing $$1$$ as a Fraction","text":"Using the relation between feet and inches, we can rewrite the $$1$$ we multiply in the first step as (12 inches)/(1 feet), so we get the expression $$6$$ feet*(12 inches)/(1 feet).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure16a-h4","type":"hint","dependencies":["a1a1ee1measure16a-h3"],"title":"Simplifying","text":"Feet divides out, so we get the expression 6*(12 inches)/1.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure16a-h5","type":"hint","dependencies":["a1a1ee1measure16a-h4"],"title":"Multiplying","text":"Multiply the expression, we get the final answer $$72$$ inches","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a1ee1measure17","title":"In the following exercise, convert the units.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.10 Systems of Measurement","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1a1ee1measure17a","stepAnswer":["$$7920$$"],"problemType":"TextBox","stepTitle":"Ulises lives $$1.5$$ miles from school. Convert the distance to feet.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7920$$","hints":{"DefaultPathway":[{"id":"a1a1ee1measure17a-h1","type":"hint","dependencies":[],"title":"Multiply by $$1$$","text":"The first step is to multiply the measurement to be converted by 1; write $$1$$ as a fraction relating the units given and the units needed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure17a-h2","type":"hint","dependencies":["a1a1ee1measure17a-h1"],"title":"Relating Miles to Feet","text":"The conversion from miles to feet is $$1$$ $$mile=5280$$ feet.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure17a-h3","type":"hint","dependencies":["a1a1ee1measure17a-h2"],"title":"Writing $$1$$ as a Fraction","text":"Using the relation between miles and feet, we can rewrite the $$1$$ we multiply in the first step as (5280 feet)/(1 mile), so we get the expression $$(1.5$$ miles)*(5280 feet)/(1 mile).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure17a-h4","type":"hint","dependencies":["a1a1ee1measure17a-h3"],"title":"Simplifying","text":"Mile divides out, so we get the expression (1.5)*(5280 feet)/1.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure17a-h5","type":"hint","dependencies":["a1a1ee1measure17a-h4"],"title":"Multiplying","text":"Multiply the expression, we get the final answer $$7920$$ feet.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a1ee1measure18","title":"In the following exercise, convert the units.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.10 Systems of Measurement","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1a1ee1measure18a","stepAnswer":["$$5400$$"],"problemType":"TextBox","stepTitle":"Rocco waited $$1.5$$ hours for his appointment. Convert the time to seconds.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5400$$","hints":{"DefaultPathway":[{"id":"a1a1ee1measure18a-h1","type":"hint","dependencies":[],"title":"Multiply by $$1$$","text":"The first step is to multiply the measurement to be converted by 1; write $$1$$ as a fraction relating the units given and the units needed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure18a-h2","type":"hint","dependencies":["a1a1ee1measure18a-h1"],"title":"Relating Hours to Seconds","text":"The conversion from hours to seconds is $$1$$ $$hour=3600$$ seconds.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure18a-h3","type":"hint","dependencies":["a1a1ee1measure18a-h2"],"title":"Writing $$1$$ as a Fraction","text":"Using the relation between hours and seconds, we can rewrite the $$1$$ we multiply in the first step as (3600 seconds)/(1 hour), so we get the expression $$(1.5$$ hours)*(3600 seconds)/(1 hour).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure18a-h4","type":"hint","dependencies":["a1a1ee1measure18a-h3"],"title":"Simplifying","text":"Hour divides out, so we get the expression (1.5)*(3600 seconds)/1.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure18a-h5","type":"hint","dependencies":["a1a1ee1measure18a-h4"],"title":"Multiplying","text":"Multiply the expression, we get the final answer $$5400$$ seconds.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a1ee1measure19","title":"In the following exercise, convert the units.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.10 Systems of Measurement","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1a1ee1measure19a","stepAnswer":["$$76$$"],"problemType":"TextBox","stepTitle":"Jon is $$6$$ feet $$4$$ inches tall. Convert his height to inches.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$76$$","hints":{"DefaultPathway":[{"id":"a1a1ee1measure19a-h1","type":"hint","dependencies":[],"title":"Separating Feet and Inches","text":"To convert $$6$$ feet $$4$$ inches into inches, we can convert $$6$$ feet to inches first, and add $$4$$ to that value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure19a-h2","type":"hint","dependencies":["a1a1ee1measure19a-h1"],"title":"Multiply by $$1$$","text":"To convert $$6$$ feet into inches, the first step is to multiply the measurement to be converted by 1; write $$1$$ as a fraction relating the units given and the units needed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure19a-h3","type":"hint","dependencies":["a1a1ee1measure19a-h2"],"title":"Relating Feet to Inches","text":"The conversion from feet to inches is $$1$$ $$foot=12$$ inches.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure19a-h4","type":"hint","dependencies":["a1a1ee1measure19a-h3"],"title":"Writing $$1$$ as a Fraction","text":"Using the relation between hours and seconds, we can rewrite the $$1$$ we multiply in the first step as (12 inches)/(1 foot), so we get the expression (6 feet)*(12 inches)/(1 foot).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure19a-h5","type":"hint","dependencies":["a1a1ee1measure19a-h4"],"title":"Simplifying","text":"Feet (foot) divides out, so we get the expression (6)*(12 inches)/1.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure19a-h6","type":"hint","dependencies":["a1a1ee1measure19a-h5"],"title":"Multiplying","text":"Multiply the expression, we get $$6$$ $$feet=72$$ inches.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure19a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$76$$"],"dependencies":["a1a1ee1measure19a-h6"],"title":"Addition","text":"The last step is to add the additional $$4$$ inches to the converted result. What is $$72+4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a1ee1measure2","title":"How to Make Unit Conversions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.10 Systems of Measurement","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1a1ee1measure2a","stepAnswer":["$$6400$$"],"problemType":"TextBox","stepTitle":"Ndula, an elephant at the San Diego Safari Park, weighs almost $$3.2$$ tons. Convert her weight to pounds.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6400$$","hints":{"DefaultPathway":[{"id":"a1a1ee1measure2a-h1","type":"hint","dependencies":[],"title":"Convert","text":"Multiply the measurement to be converted by 1; write $$1$$ as a fraction relating the units given and the units needed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2000$$"],"dependencies":["a1a1ee1measure2a-h1"],"title":"Convert","text":"How many pounds are in a ton?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure2a-h3","type":"hint","dependencies":["a1a1ee1measure2a-h2"],"title":"Multiply","text":"Multiply $$3.2$$ tons by $$2000$$ $$\\\\frac{pounds}{1}$$ ton (tons should be in the denominator so that the tons will divide out).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6400$$"],"dependencies":["a1a1ee1measure2a-h3"],"title":"Multiply","text":"What do we get after the multiplication?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a1ee1measure20","title":"In the following exercise, solve.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.10 Systems of Measurement","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1a1ee1measure20a","stepAnswer":["$$8$$ lb. $$13$$ oz."],"problemType":"MultipleChoice","stepTitle":"Eli caught three fish. The weights of the fish were $$2$$ pounds $$4$$ ounces, $$1$$ pound $$11$$ ounces, and $$4$$ pounds $$14$$ ounces. What was the total weight of the three fish?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$8$$ lb. $$13$$ oz.","choices":["$$7$$ lb. $$13$$ oz.","7lb. $$3$$ oz.","$$8$$ lb. $$13$$ oz","$$8$$ lb. $$13$$ oz.","$$8$$ lb. $$3$$ oz"],"hints":{"DefaultPathway":[{"id":"a1a1ee1measure20a-h1","type":"hint","dependencies":[],"title":"Separating Terms","text":"To find the total weight of the three fish, we can first add the pounds and the ounces separately.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a1a1ee1measure20a-h1"],"title":"Adding Pounds","text":"We want to add the pounds together. What is $$2+1+4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure20a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$29$$"],"dependencies":["a1a1ee1measure20a-h1"],"title":"Adding Ounces","text":"We want to add the ounces together. What is $$4+11+14$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure20a-h4","type":"hint","dependencies":["a1a1ee1measure20a-h2","a1a1ee1measure20a-h3"],"title":"Converting Ounces","text":"The next step is to convert the total ounces into pounds and ounces, so that we can later add this pounds value to the previous sum of pounds.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure20a-h5","type":"hint","dependencies":["a1a1ee1measure20a-h4"],"title":"Relation between Pounds and Ounces","text":"The relation between pounds and ounces is $$1$$ pound $$(lb.)=16$$ ounces (0z.).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure20a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["1lb. $$13$$ oz."],"dependencies":["a1a1ee1measure20a-h5"],"title":"Converting Ounces to Pounds and Ounces","text":"What is $$29$$ ounces in pounds and ounces?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["2lb.","1lb. $$13$$ oz.","$$29$$ oz.","$$1$$ lb. $$3$$ oz."]},{"id":"a1a1ee1measure20a-h7","type":"hint","dependencies":["a1a1ee1measure20a-h6"],"title":"Final Answer","text":"Therefore, the final answer becomes $$7$$ lb.+1 lb. $$+13$$ $$oz.=8$$ lb. $$13$$ oz.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a1ee1measure21","title":"In the following exercise, solve.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.10 Systems of Measurement","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1a1ee1measure21a","stepAnswer":["$$3.05$$"],"problemType":"TextBox","stepTitle":"One day Anya kept track of the number of minutes she spent driving. She recorded $$45$$, $$10$$, $$8$$, $$65$$, $$20$$, and $$35$$. How many hours did Anya spend driving?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3.05$$","hints":{"DefaultPathway":[{"id":"a1a1ee1measure21a-h1","type":"hint","dependencies":[],"title":"Approach","text":"For this problem, our approach is to add up all the minutes first and then convert the sum into hours.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure21a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$183$$"],"dependencies":["a1a1ee1measure21a-h1"],"title":"Adding Up the Minutes","text":"We start by adding up the minutes. What is $$45+10+8+65+20+35$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure21a-h3","type":"hint","dependencies":["a1a1ee1measure21a-h2"],"title":"Converting Minutes to Hours","text":"Then, we convert $$183$$ minutes into hours.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure21a-h4","type":"hint","dependencies":["a1a1ee1measure21a-h3"],"title":"Relation between Minutes and Hours","text":"The relation between minutes and hours is $$60$$ $$min=1$$ hr.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure21a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3.05$$"],"dependencies":["a1a1ee1measure21a-h4"],"title":"Applying the Conversion","text":"What is $$183$$ minutes in hours?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure21a-h6","type":"hint","dependencies":["a1a1ee1measure21a-h5"],"title":"Applying the Conversion","text":"To convert $$183$$ minutes into hours, we use the expression $$183$$ $$min=(183$$ min)*(1 hr)/(60 min)=(183)*(1 hr)/(60)=3.05 hr. The first equality holds because $$1$$ $$hr=60$$ min, and the second equation holds because we can cancel the min.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a1ee1measure22","title":"In the following exercise, solve.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.10 Systems of Measurement","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1a1ee1measure22a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"Leilani wants to make $$8$$ placemats. For each placemat she needs $$18$$ inches of fabric. How many yards of fabric will she need for the $$8$$ placemats?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a1a1ee1measure22a-h1","type":"hint","dependencies":[],"title":"Approach","text":"To approach this problem, we can find the total length of fabric needed in inches first, and then convert this sum into yards.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure22a-h2","type":"hint","dependencies":["a1a1ee1measure22a-h1"],"title":"Finding Answer in Inches","text":"To get the total length of fabric in inches, all we need to do is multiply $$18$$ by $$8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure22a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$144$$"],"dependencies":["a1a1ee1measure22a-h2"],"title":"Finding Answer in Inches","text":"What is $$18\\\\times8$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure22a-h4","type":"hint","dependencies":["a1a1ee1measure22a-h3"],"title":"Converting Inches to Yard","text":"The next step is to convert $$144$$ inches into yards. But since we don\'t know the conversion between inches and yards right away, we will need to convert inches into feet first, and then convert feet into yards.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure22a-h5","type":"hint","dependencies":["a1a1ee1measure22a-h4"],"title":"Relation between Inches and Feet","text":"The relation between inches and feet is $$12$$ $$inches=1$$ foot.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure22a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a1a1ee1measure22a-h5"],"title":"Converting Inches to Feet","text":"What is $$144$$ inches in feet?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure22a-h7","type":"hint","dependencies":["a1a1ee1measure22a-h6"],"title":"Applying the Conversion","text":"To convert $$144$$ inches into feet, we use the expression $$144$$ $$inches=(144$$ inches)*(1 foot)/(12 inches)=(144)*(1 foot)/(12)=12 feet. The first equality holds because $$1$$ $$foot=12$$ inches, and the second equation holds because we can cancel the inches.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure22a-h8","type":"hint","dependencies":["a1a1ee1measure22a-h7"],"title":"Relation between Feet and Yards","text":"The relation between feet and yards is $$3$$ $$feet=1$$ yard.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure22a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a1a1ee1measure22a-h8"],"title":"Converting Feet to Yard","text":"What is $$12$$ feet in yards?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure22a-h10","type":"hint","dependencies":["a1a1ee1measure22a-h9"],"title":"Applying the Conversion","text":"To convert $$12$$ feet into yards, we use the expression $$12$$ $$feet=(12$$ feet)*(1 yard)/(3 feet)=(12)*(1 yard)/(3)=4 yards. The first equality holds because $$1$$ $$yard=3$$ feet, and the second equation holds because we can cancel the feet.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a1ee1measure23","title":"In the following exercise, convert the units.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.10 Systems of Measurement","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1a1ee1measure23a","stepAnswer":["$$3.072$$"],"problemType":"TextBox","stepTitle":"Mount Whitney is 3,072 meters tall. Convert the height to kilometers.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3.072$$","hints":{"DefaultPathway":[{"id":"a1a1ee1measure23a-h1","type":"hint","dependencies":[],"title":"Multiply by $$1$$","text":"The first step is to multiply the measurement to be converted by 1; write $$1$$ as a fraction relating the units given and the units needed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure23a-h2","type":"hint","dependencies":["a1a1ee1measure23a-h1"],"title":"Relating $$m$$ to km","text":"The conversion from meters to kilometers is 1,000 $$m=1$$ km.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure23a-h3","type":"hint","dependencies":["a1a1ee1measure23a-h2"],"title":"Writing $$1$$ as a Fraction","text":"Using the relation between km and $$m$$, we can rewrite the $$1$$ we multiply in the first step as (1 km)/(1000 m), so we get the expression (3072 m)*(1 km)/(1000 m).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure23a-h4","type":"hint","dependencies":["a1a1ee1measure23a-h3"],"title":"Simplifying","text":"$$m$$ divides out, so we get the expression (3072)*(1 km)/1000.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure23a-h5","type":"hint","dependencies":["a1a1ee1measure23a-h4"],"title":"Dividing","text":"Dividing the expression, we get the final answer $$3.072$$ km.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a1ee1measure24","title":"In the following exercise, solve.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.10 Systems of Measurement","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1a1ee1measure24a","stepAnswer":["$$91$$"],"problemType":"TextBox","stepTitle":"Matthias is $$1.8$$ meters tall. His son is $$89$$ centimeters tall. How much taller is Matthias than his son in centimeters?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$91$$","hints":{"DefaultPathway":[{"id":"a1a1ee1measure24a-h1","type":"hint","dependencies":[],"title":"Approach","text":"Since the problem wants the answer in centimeters, we will convert all measurements into centimeters before proceeding with arithmetics.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure24a-h2","type":"hint","dependencies":["a1a1ee1measure24a-h1"],"title":"Relation between $$m$$ and cm","text":"The relation between meters and centimeters is $$1$$ $$m=100$$ cm.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure24a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$180$$"],"dependencies":["a1a1ee1measure24a-h2"],"title":"Converting Dad\'s Height","text":"We start by converting Matthias\' height into centimeters. What is $$1.8$$ $$m$$ in cm?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure24a-h4","type":"hint","dependencies":["a1a1ee1measure24a-h3"],"title":"Applying the Conversion","text":"To convert $$1.8$$ $$m$$ into cm, we use the expression $$1.8$$ $$m=(1.8$$ m)*(100 cm)/(1 m)=(1.8)*(100 cm)/(1)=180 cm. The first equality holds because $$100$$ $$cm=1$$ $$m$$, and the second equation holds because we can cancel the $$m$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure24a-h5","type":"hint","dependencies":["a1a1ee1measure24a-h4"],"title":"Final Answer","text":"Now that we have both Matthias and his son\'s heights in centimeters, to find the difference between their heights in centimeters, we can simply subtract the the two numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure24a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$91$$"],"dependencies":["a1a1ee1measure24a-h5"],"title":"Final Answer","text":"What is $$180-89$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a1ee1measure25","title":"In the following exercise, solve.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.10 Systems of Measurement","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1a1ee1measure25a","stepAnswer":["$$855$$"],"problemType":"TextBox","stepTitle":"A typical dove weighs $$345$$ grams. A typical duck weighs $$1.2$$ kilograms. What is the difference, in grams, of the weights of a duck and a dove?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$855$$","hints":{"DefaultPathway":[{"id":"a1a1ee1measure25a-h1","type":"hint","dependencies":[],"title":"Approach","text":"Since the problem wants the answer in grams, we will convert all measurements into grams before proceeding with arithmetics.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure25a-h2","type":"hint","dependencies":["a1a1ee1measure25a-h1"],"title":"Relation between kg and g","text":"The relation between grams and kilograms is $$1000$$ $$g=1$$ kg.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure25a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1200$$"],"dependencies":["a1a1ee1measure25a-h2"],"title":"Converting the Weight of Duck","text":"We start by converting the weight of the duck into grams. What is $$1.2$$ kg in g?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure25a-h4","type":"hint","dependencies":["a1a1ee1measure25a-h3"],"title":"Applying the Conversion","text":"To convert $$1.2$$ kg into g, we use the expression $$1.2$$ $$kg=(1.2$$ kg)*(1000 g)/(1 kg)=(1.2)*(1000 g)/(1)=1200 g. The first equality holds because $$1000$$ $$g=1$$ kg, and the second equation holds because we can cancel the kg.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure25a-h5","type":"hint","dependencies":["a1a1ee1measure25a-h4"],"title":"Final Answer","text":"Now that we have both the dove\'s weight and the duck\'s weight in grams, to find the difference between their weights, we can simply subtract the the two numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure25a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$855$$"],"dependencies":["a1a1ee1measure25a-h5"],"title":"Final Answer","text":"What is $$1200-345$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a1ee1measure26","title":"In the following exercises, make the unit conversions. Round to the nearest tenth.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.10 Systems of Measurement","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1a1ee1measure26a","stepAnswer":["$$190.5$$"],"problemType":"TextBox","stepTitle":"Bill is $$75$$ inches tall. Convert his height to centimeters.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$190.5$$","hints":{"DefaultPathway":[{"id":"a1a1ee1measure26a-h1","type":"hint","dependencies":[],"title":"Multiply by $$1$$","text":"The first step is to multiply the measurement to be converted by 1; write $$1$$ as a fraction relating the units given and the units needed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure26a-h2","type":"hint","dependencies":["a1a1ee1measure26a-h1"],"title":"Relating in. to cm","text":"The conversion from inches to centimeters is $$1$$ $$in.=2.54$$ cm.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure26a-h3","type":"hint","dependencies":["a1a1ee1measure26a-h2"],"title":"Writing $$1$$ as a Fraction","text":"Using the relation between in. and cm, we can rewrite the $$1$$ we multiply in the first step as $$(2.54$$ cm)/(1 in.), so we get the expression (75 in.)*(2.54 cm)/(1 in.).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure26a-h4","type":"hint","dependencies":["a1a1ee1measure26a-h3"],"title":"Simplifying","text":"in. divides out, so we get the expression (75)*(2.54 cm)/(1).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure26a-h5","type":"hint","dependencies":["a1a1ee1measure26a-h4"],"title":"Multiplying","text":"Multiplying the expression, we get the final answer $$190.5$$ cm.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a1ee1measure27","title":"In the following exercises, make the unit conversions. Round to the nearest tenth.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.10 Systems of Measurement","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1a1ee1measure27a","stepAnswer":["$$44$$"],"problemType":"TextBox","stepTitle":"Dawn\u2019s suitcase weighed $$20$$ kilograms. Convert the weight to pounds.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$44$$","hints":{"DefaultPathway":[{"id":"a1a1ee1measure27a-h1","type":"hint","dependencies":[],"title":"Multiply by $$1$$","text":"The first step is to multiply the measurement to be converted by 1; write $$1$$ as a fraction relating the units given and the units needed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure27a-h2","type":"hint","dependencies":["a1a1ee1measure27a-h1"],"title":"Relating kg to lb.","text":"The conversion from kilograms to pounds is $$1$$ $$kg=2.2$$ lb.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure27a-h3","type":"hint","dependencies":["a1a1ee1measure27a-h2"],"title":"Writing $$1$$ as a Fraction","text":"Using the relation between kg and lb., we can rewrite the $$1$$ we multiply in the first step as $$(2.2$$ lb)/(1 kg), so we get the expression (20 kg)*(2.2 lb)/(1 kg).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure27a-h4","type":"hint","dependencies":["a1a1ee1measure27a-h3"],"title":"Simplifying","text":"kg divides out, so we get the expression (20)*(2.2 lb)/(1).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure27a-h5","type":"hint","dependencies":["a1a1ee1measure27a-h4"],"title":"Multiplying","text":"Multiplying the expression, we get the final answer $$44$$ lb.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a1ee1measure28","title":"In the following exercises, make the unit conversions. Round to the nearest tenth.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.10 Systems of Measurement","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1a1ee1measure28a","stepAnswer":["$$53.2$$"],"problemType":"TextBox","stepTitle":"Ozzie put $$14$$ gallons of gas in his truck. Convert the volume to liters.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$53.2$$","hints":{"DefaultPathway":[{"id":"a1a1ee1measure28a-h1","type":"hint","dependencies":[],"title":"Relating gal to L","text":"We don\'t know the conversion between gallons and liters right away, but we do know the conversion from gallons to quart and the conversion from quart to liters.Therefore, we can convert the measurement into quarts first, then convert it into liters.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure28a-h2","type":"hint","dependencies":["a1a1ee1measure28a-h1"],"title":"Relating gal to qt.","text":"The conversion from gallons to quarts is $$1$$ $$gal=4$$ qt.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure28a-h3","type":"hint","dependencies":["a1a1ee1measure28a-h2"],"title":"Multiply by $$1$$","text":"The first step is to multiply the measurement to be converted by 1; write $$1$$ as a fraction relating the units given and the units needed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure28a-h4","type":"hint","dependencies":["a1a1ee1measure28a-h3"],"title":"Writing $$1$$ as a Fraction","text":"Using the relation between gal and qt., we can rewrite the $$1$$ we multiply in the last step as (4 qt.)/(1 gal), so we get the expression (14 gal)*(4 qt.)/(1 gal).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure28a-h5","type":"hint","dependencies":["a1a1ee1measure28a-h4"],"title":"Simplifying","text":"gal divides out, so we get the expression (14)*(4 qt.)/(1).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure28a-h6","type":"hint","dependencies":["a1a1ee1measure28a-h5"],"title":"Multiplying","text":"Multiplying the expression, we get $$14$$ $$gal=56$$ qt.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure28a-h7","type":"hint","dependencies":["a1a1ee1measure28a-h6"],"title":"Converting qt. to L","text":"Our next step is to convert $$56$$ qt. into liters.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure28a-h8","type":"hint","dependencies":["a1a1ee1measure28a-h7"],"title":"Relating qt. to liters","text":"The conversion from quarts to liters is $$1$$ $$qt.=0.95$$ L.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure28a-h9","type":"hint","dependencies":["a1a1ee1measure28a-h8"],"title":"Multiply by $$1$$","text":"The first step is to multiply the measurement to be converted by 1; write $$1$$ as a fraction relating the units given and the units needed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure28a-h10","type":"hint","dependencies":["a1a1ee1measure28a-h9"],"title":"Writing $$1$$ as a Fraction","text":"Using the relation between qt. and L, we can rewrite the $$1$$ we multiply in the last step as $$(0.95$$ L)/(1 qt.), so we get the expression (56 qt.)*(0.95 L)/(1 qt.).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure28a-h11","type":"hint","dependencies":["a1a1ee1measure28a-h10"],"title":"Simplifying","text":"qt. divides out, so we get the expression (56)*(0.95 L)/(1).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure28a-h12","type":"hint","dependencies":["a1a1ee1measure28a-h11"],"title":"Multiplying","text":"Multiplying the expression, we get the final answer $$53.2$$ L.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a1ee1measure29","title":"In the following exercises, convert the Fahrenheit temperatures to degrees Celsius. Round to the nearest tenth.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.10 Systems of Measurement","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1a1ee1measure29a","stepAnswer":["$$30$$"],"problemType":"TextBox","stepTitle":"$$86$$ degrees Fahrenheit","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$30$$","hints":{"DefaultPathway":[{"id":"a1a1ee1measure29a-h1","type":"hint","dependencies":[],"title":"Conversion Formula","text":"Recall that the formula converting Fahrenheit into Celsius is $$C=\\\\frac{5}{9} \\\\left(F-32\\\\right)$$, where C is the temperature in Celsius and F is the temperature in Fahrenheit.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure29a-h2","type":"hint","dependencies":["a1a1ee1measure29a-h1"],"title":"Applying the Conversion","text":"To use the formula, all we need to do is substitute $$86$$ for the F, so we get the equation $$C=\\\\frac{5}{9} \\\\left(86-32\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure29a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$30$$"],"dependencies":["a1a1ee1measure29a-h2"],"title":"Evaluating the Expression","text":"What is $$\\\\frac{5}{9} \\\\left(86-32\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a1a1ee1measure29a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$54$$"],"dependencies":[],"title":"Evaluating the Expression","text":"To evaluate $$\\\\frac{5}{9} \\\\left(86-32\\\\right)$$, we evaluate what\'s inside the parenthesis first. What is $$86-32$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure29a-h3-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$30$$"],"dependencies":["a1a1ee1measure29a-h3-s1"],"title":"Evaluating the Expression","text":"Then, we evaluate $$54\\\\frac{5}{9}$$. What is this value?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a1a1ee1measure3","title":"How to Make Unit Conversions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.10 Systems of Measurement","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1a1ee1measure3a","stepAnswer":["$$90720$$"],"problemType":"TextBox","stepTitle":"Juliet is going with her family to their summer home. She will be away from her boyfriend for $$9$$ weeks. Convert the time to minutes.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$90720$$","hints":{"DefaultPathway":[{"id":"a1a1ee1measure3a-h1","type":"hint","dependencies":[],"title":"Convert","text":"To convert weeks into minutes we will convert weeks into days, days into hours, and then hours into minutes. To do this we will multiply by conversion factors of $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a1a1ee1measure3a-h1"],"title":"Convert","text":"How many days are in a week?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$24$$"],"dependencies":["a1a1ee1measure3a-h2"],"title":"Convert","text":"How many hours are in a day?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$60$$"],"dependencies":["a1a1ee1measure3a-h3"],"title":"Convert","text":"How many minutes are in an hour?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure3a-h5","type":"hint","dependencies":["a1a1ee1measure3a-h4"],"title":"Multiply","text":"Multiply $$9$$ weeks by (7 $$\\\\frac{days}{1}$$ week)*(24 $$\\\\frac{hours}{1}$$ day)*(60 $$\\\\frac{minutes}{1}$$ hour)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure3a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$90720$$"],"dependencies":["a1a1ee1measure3a-h5"],"title":"Multiply","text":"What do we get after the multiplication?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a1ee1measure30","title":"In the following exercises, convert the Celsius temperatures to degrees Fahrenheit. Round to the nearest tenth.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.10 Systems of Measurement","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1a1ee1measure30a","stepAnswer":["$$41$$"],"problemType":"TextBox","stepTitle":"$$5$$ degrees Celsius","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$41$$","hints":{"DefaultPathway":[{"id":"a1a1ee1measure30a-h1","type":"hint","dependencies":[],"title":"Conversion Formula","text":"Recall that the formula converting Celsius into Fahrenheit is $$F=\\\\frac{9}{5} C+32$$, where C is the temperature in Celsius and F is the temperature in Fahrenheit.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure30a-h2","type":"hint","dependencies":["a1a1ee1measure30a-h1"],"title":"Applying the Conversion","text":"To use the formula, all we need to do is substitute $$5$$ for the C, so we get the equation $$F=5\\\\frac{9}{5}+32$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure30a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$41$$"],"dependencies":["a1a1ee1measure30a-h2"],"title":"Evaluating the Expression","text":"What is $$5\\\\frac{9}{5}+32$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a1a1ee1measure30a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":[],"title":"Evaluating the Expression","text":"To evaluate $$5\\\\frac{9}{5}+32$$, we evaluate the multiplication first. What is $$5\\\\frac{9}{5}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure30a-h3-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$41$$"],"dependencies":["a1a1ee1measure30a-h3-s1"],"title":"Evaluating the Expression","text":"Then, we evaluate $$9+32$$. What is this value?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a1a1ee1measure4","title":"How to Make Unit Conversions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.10 Systems of Measurement","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1a1ee1measure4a","stepAnswer":["$$128$$"],"problemType":"TextBox","stepTitle":"How many ounces are in $$1$$ gallon?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$128$$","hints":{"DefaultPathway":[{"id":"a1a1ee1measure4a-h1","type":"hint","dependencies":[],"title":"Convert","text":"We will convert gallons to ounces by multiplying by several conversion factors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a1a1ee1measure4a-h1"],"title":"Convert","text":"How many quarts are in a gallon?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a1a1ee1measure4a-h2"],"title":"Convert","text":"How many pints are in a quart?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a1a1ee1measure4a-h3"],"title":"Convert","text":"How many cups are in a pint?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure4a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a1a1ee1measure4a-h4"],"title":"Convert","text":"How many ounces are in a cup?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure4a-h6","type":"hint","dependencies":["a1a1ee1measure4a-h5"],"title":"Multiply","text":"Multiply $$1$$ gallon by (4 $$\\\\frac{quarts}{1}$$ gallon)*(2 $$\\\\frac{pints}{1}$$ quart)*(2 $$\\\\frac{cups}{1}$$ pint)*(8 $$\\\\frac{ounces}{1}$$ cup)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure4a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$128$$"],"dependencies":["a1a1ee1measure4a-h6"],"title":"Multiply","text":"What do we get after the multiplication?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a1ee1measure5","title":"Use Mixed Units of Measurement in the U.S. System","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.10 Systems of Measurement","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1a1ee1measure5a","stepAnswer":["$$3$$ pounds, $$6$$ ounces"],"problemType":"MultipleChoice","stepTitle":"Seymour bought three steaks for a barbecue. Their weights were $$14$$ ounces, $$1$$ pound $$2$$ ounces and $$1$$ pound $$6$$ ounces. How many total pounds of steak did he buy?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$3$$ pounds, $$6$$ ounces","choices":["$$3$$ pounds, $$6$$ ounces","$$3$$ pounds, $$3$$ ounces","$$2$$ pounds, $$6$$ ounces","$$3$$ pounds"],"hints":{"DefaultPathway":[{"id":"a1a1ee1measure5a-h1","type":"hint","dependencies":[],"title":"Add","text":"We will add the weights of the steaks to find the total weight of the steaks. We first add the ounces, then add the pounds.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure5a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2$$ pounds, $$22$$ ounces"],"dependencies":["a1a1ee1measure5a-h1"],"title":"Add","text":"What do we get after the addition?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2$$ pounds, $$20$$ ounces","$$2$$ pounds","$$2$$ pounds, $$22$$ ounces"]},{"id":"a1a1ee1measure5a-h3","type":"hint","dependencies":["a1a1ee1measure5a-h2"],"title":"Convert","text":"We then need to convert $$22$$ ounces to pounds and ounces.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure5a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$1$$ pound, $$6$$ ounces"],"dependencies":["a1a1ee1measure5a-h3"],"title":"Convert","text":"What do we get after the conversion?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$1$$ pound, $$8$$ ounces","$$1$$ pound, $$3$$ ounces","$$1$$ pound, $$6$$ ounces"]},{"id":"a1a1ee1measure5a-h5","type":"hint","dependencies":["a1a1ee1measure5a-h4"],"title":"Add","text":"We then need to add $$2$$ pounds with $$1$$ pound, $$6$$ ounces.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure5a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3$$ pounds, $$6$$ ounces"],"dependencies":["a1a1ee1measure5a-h5"],"title":"Add","text":"What do we get after the addition?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$3$$ pounds, $$6$$ ounces","$$3$$ pounds, $$3$$ ounces","$$2$$ pounds, $$6$$ ounces","$$3$$ pounds"]}]}}]},{"id":"a1a1ee1measure6","title":"Use Mixed Units of Measurement in the U.S. System","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.10 Systems of Measurement","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1a1ee1measure6a","stepAnswer":["$$25$$ feet, $$4$$ inches"],"problemType":"MultipleChoice","stepTitle":"Anthony bought four planks of wood that were each $$6$$ feet $$4$$ inches long. What is the total length of the wood he purchased?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$25$$ feet, $$4$$ inches","choices":["$$24$$ feet, $$3$$ inches","$$25$$ feet, $$4$$ inches","$$25$$ feet, $$8$$ inches","$$25$$ feet"],"hints":{"DefaultPathway":[{"id":"a1a1ee1measure6a-h1","type":"hint","dependencies":[],"title":"Multiply","text":"We will multiply the length of one plank to find the total length.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure6a-h2","type":"hint","dependencies":["a1a1ee1measure6a-h1"],"title":"Multiply","text":"Multiply the inches and then the feet.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure6a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$24$$ feet, $$16$$ inches"],"dependencies":["a1a1ee1measure6a-h2"],"title":"Multiply","text":"What do we get after multiplying the length of the plank by 4?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$24$$ feet","$$24$$ feet, $$16$$ inches","$$24$$ feet, $$18$$ inches"]},{"id":"a1a1ee1measure6a-h4","type":"hint","dependencies":["a1a1ee1measure6a-h3"],"title":"Convert","text":"We then need to convert the $$16$$ inches to feet.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure6a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$1$$ foot, $$4$$ inches"],"dependencies":["a1a1ee1measure6a-h4"],"title":"Convert","text":"What do we get after the conversion?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$1$$ foot, $$4$$ inches","$$1$$ foot, $$6$$ inches","$$1$$ foot, $$8$$ inches"]},{"id":"a1a1ee1measure6a-h6","type":"hint","dependencies":["a1a1ee1measure6a-h5"],"title":"Add","text":"We then need to add $$24$$ feet with $$1$$ foot, $$4$$ inches.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure6a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$25$$ feet, $$4$$ inches"],"dependencies":["a1a1ee1measure6a-h6"],"title":"Add","text":"What do we get after the addition?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$24$$ feet, $$3$$ inches","$$25$$ feet, $$4$$ inches","$$25$$ feet, $$8$$ inches","$$25$$ feet"]}]}}]},{"id":"a1a1ee1measure7","title":"Make Unit Conversions in the Metric System","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.10 Systems of Measurement","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1a1ee1measure7a","stepAnswer":["$$10000$$"],"problemType":"TextBox","stepTitle":"Nick ran a 10K race. How many meters did he run?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10000$$","hints":{"DefaultPathway":[{"id":"a1a1ee1measure7a-h1","type":"hint","dependencies":[],"title":"Identity property of multiplication","text":"We will convert kilometers to meters using the identity property of multiplication.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure7a-h2","type":"hint","dependencies":["a1a1ee1measure7a-h1"],"title":"Convert","text":"Multiply the measurement to be converted by 1; write $$1$$ as a fraction relating the units given and the units needed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1000$$"],"dependencies":["a1a1ee1measure7a-h2"],"title":"Convert","text":"How many meters are in one kilometer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure7a-h4","type":"hint","dependencies":["a1a1ee1measure7a-h3"],"title":"Multiply","text":"Multiply 10K by $$1000$$ $$\\\\frac{meters}{1}$$ kilometer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure7a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10000$$"],"dependencies":["a1a1ee1measure7a-h4"],"title":"Multiply","text":"What do we get after the multiplication?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a1ee1measure8","title":"Make Unit Conversions in the Metric System","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.10 Systems of Measurement","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1a1ee1measure8a","stepAnswer":["$$3.2$$"],"problemType":"TextBox","stepTitle":"Eleanor\u2019s newborn baby weighed 3,200 grams. How many kilograms did the baby weigh?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3.2$$","hints":{"DefaultPathway":[{"id":"a1a1ee1measure8a-h1","type":"hint","dependencies":[],"title":"Convert","text":"Multiply the measurement to be converted by 1; write $$1$$ as a fraction relating the units given and the units needed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1000$$"],"dependencies":["a1a1ee1measure8a-h1"],"title":"Convert","text":"How many grams are in one kilogram?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure8a-h3","type":"hint","dependencies":["a1a1ee1measure8a-h2"],"title":"Multiply","text":"Multiply $$3200$$ grams by $$1$$ $$\\\\frac{kg}{1000}$$ grams (grams should be in the denominator so that the grams will divide out).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3.2$$"],"dependencies":["a1a1ee1measure8a-h3"],"title":"Multiply","text":"What do we get after the multiplication?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a1ee1measure9","title":"Make Unit Conversions in the Metric System","body":"Convert:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.10 Systems of Measurement","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1a1ee1measure9a","stepAnswer":["$$0.35$$"],"problemType":"TextBox","stepTitle":"$$350$$ L to kiloliters","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.35$$","hints":{"DefaultPathway":[{"id":"a1a1ee1measure9a-h1","type":"hint","dependencies":[],"title":"Convert","text":"Multiply by $$1$$, writing $$1$$ as a fraction relating liters to kiloliters.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1000$$"],"dependencies":["a1a1ee1measure9a-h1"],"title":"Convert","text":"How many liters are in one kiloliter?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure9a-h3","type":"hint","dependencies":["a1a1ee1measure9a-h2"],"title":"Multiply","text":"Multiply $$350$$ L by $$1$$ $$\\\\frac{kiloliter}{1000}$$ L (L should be in the denominator so that L will divide out).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a1ee1measure9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.35$$"],"dependencies":["a1a1ee1measure9a-h3"],"title":"Multiply","text":"What do we get after the multiplication?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a538fpairs1","title":"Determine Whether an Ordered Pair is a Solution of a System of Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Solve Systems of Linear Equations with Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1a538fpairs1a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Determine whether $$(3,1)$$ is a solution to the system: $$2x-6y=0, 3x-4y=5$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No","Undefined"],"hints":{"DefaultPathway":[{"id":"a1a538fpairs1a-h1","type":"hint","dependencies":[],"title":"How to Verify a Solution","text":"If a point is a solution to a system, then it satisfies both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fpairs1a-h2","type":"hint","dependencies":["a1a538fpairs1a-h1"],"title":"Plugging in the Point","text":"Let us plug in the point. $$2(3)-6(1)=0$$, $$0=0$$. The first equation is satisfied. $$3(3)-4(1)=5$$, $$5=5$$. The second equation is satisfied. The point is a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a538fpairs10","title":"Solve a System of Equations by Elimination","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Solve Systems of Linear Equations with Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1a538fpairs10a","stepAnswer":["$$(4,5)$$"],"problemType":"MultipleChoice","stepTitle":"Solve the systems of equations: 6x-5y=-1,2x+y=13","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(4,5)$$","choices":["$$(-4,5)$$","$$(-4,-5)$$","$$(4,5)$$"],"hints":{"DefaultPathway":[{"id":"a1a538fpairs10a-h1","type":"hint","dependencies":[],"title":"Condensing into One Equation","text":"We can multiply the second equation by $$5$$ and then add to get $$16x=64$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fpairs10a-h2","type":"hint","dependencies":["a1a538fpairs10a-h1"],"title":"Solving the Equation","text":"$$16x=64$$ can be simplified to $$x=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fpairs10a-h3","type":"hint","dependencies":["a1a538fpairs10a-h2"],"title":"Substituting for the Other Variable","text":"Plugging $$x=4$$ into the first equation, we get that $$2\\\\left(4\\\\right)+y=13$$, which means that $$y=5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a538fpairs11","title":"Solve a System of Equations by Elimination","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Solve Systems of Linear Equations with Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1a538fpairs11a","stepAnswer":["$$(6,1)$$"],"problemType":"MultipleChoice","stepTitle":"Solve the systems of equations: $$2x-5y=7, 3x-y=17$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(6,1)$$","choices":["$$(-6,-1)$$","$$(-6,1)$$","$$(1,6)$$","$$(6,1)$$"],"hints":{"DefaultPathway":[{"id":"a1a538fpairs11a-h1","type":"hint","dependencies":[],"title":"Condensing into One Equation","text":"We can multiply the second equation by $$-5$$ and then add to get $$-13x=-78$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fpairs11a-h2","type":"hint","dependencies":["a1a538fpairs11a-h1"],"title":"Solving the Equation","text":"$$-13x=-78$$ simplifies to $$x=6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fpairs11a-h3","type":"hint","dependencies":["a1a538fpairs11a-h2"],"title":"Substituting for the Other Variable","text":"Plugging $$x=4$$ into the second equation, we get that $$3(6)-y=17$$, which means that $$y=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a538fpairs12","title":"Solve a System of Equations by Elimination","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Solve Systems of Linear Equations with Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1a538fpairs12a","stepAnswer":["$$(7,12)$$"],"problemType":"MultipleChoice","stepTitle":"Solve the systems of equations: $$5x-3y=-1, 2x-y=2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(7,12)$$","choices":["$$(-7,12)$$","$$(-7,-12)$$","$$(7,12)$$"],"hints":{"DefaultPathway":[{"id":"a1a538fpairs12a-h1","type":"hint","dependencies":[],"title":"Condensing into One Equation","text":"We can multiply the second equation by $$-3$$ and then add to get $$-x=-7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fpairs12a-h2","type":"hint","dependencies":["a1a538fpairs12a-h1"],"title":"Solving the Equation","text":"$$-x=-7$$ simplifies to $$x=7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fpairs12a-h3","type":"hint","dependencies":["a1a538fpairs12a-h2"],"title":"Substituting for the Other Variable","text":"Plugging $$x=7$$ into the second equation, we get that $$2(7)-y=2$$, which means that $$y=12$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a538fpairs13","title":"Solve a System of Equations by Elimination","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Solve Systems of Linear Equations with Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1a538fpairs13a","stepAnswer":["$$(2,3)$$"],"problemType":"MultipleChoice","stepTitle":"Solve the systems of equations: 3x-5y=-9,6x+2y=16","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(2,3)$$","choices":["$$(-2,-3)$$","$$(-2,3)$$","$$(2,3)$$","$$(2,3)$$"],"hints":{"DefaultPathway":[{"id":"a1a538fpairs13a-h1","type":"hint","dependencies":[],"title":"Condensing into One Equation","text":"We can multiply the second equation by $$5$$ and the first equation by $$2$$ and add to get $$31x=62$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fpairs13a-h2","type":"hint","dependencies":["a1a538fpairs13a-h1"],"title":"Solving the Equation","text":"$$31x=62$$ simplifies to $$x=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fpairs13a-h3","type":"hint","dependencies":["a1a538fpairs13a-h2"],"title":"Substituting for the Other Variable","text":"Plugging $$x=2$$ into the second equation, we get that $$5\\\\left(2\\\\right)+2y=16$$, which means that $$y=3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a538fpairs14","title":"Solve a System of Equations by Elimination","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Solve Systems of Linear Equations with Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1a538fpairs14a","stepAnswer":["$$(-3,-5)$$"],"problemType":"MultipleChoice","stepTitle":"Solve the systems of equations: 4x-3y=3,2x+5y=-31","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-3,-5)$$","choices":["$$(-3,-5)$$","$$(-3,-5)$$","$$(3,5)$$"],"hints":{"DefaultPathway":[{"id":"a1a538fpairs14a-h1","type":"hint","dependencies":[],"title":"Condensing into One Equation","text":"We can multiply the second equation by $$-2$$ and add to get $$-13y=65$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fpairs14a-h2","type":"hint","dependencies":["a1a538fpairs14a-h1"],"title":"Solving the Equation","text":"$$-13y=65$$ simplifies to $$y=-5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fpairs14a-h3","type":"hint","dependencies":["a1a538fpairs14a-h2"],"title":"Substituting for the Other Variable","text":"Plugging $$y=-5$$ into the second equation, we get that 2x+5(-5)=-31, which means that $$x=-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a538fpairs2","title":"Determine Whether an Ordered Pair is a Solution of a System of Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Solve Systems of Linear Equations with Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1a538fpairs2a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Determine whether $$(-3,4)$$ is a solution to the system: $$2x-6y=0, 3x-4y=5$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No","Undefined"],"hints":{"DefaultPathway":[{"id":"a1a538fpairs2a-h1","type":"hint","dependencies":[],"title":"How to Verify a Solution","text":"If a point is a solution to a system, then it satisfies both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fpairs2a-h2","type":"hint","dependencies":["a1a538fpairs2a-h1"],"title":"Plugging in the Point","text":"Let us plug in the point. $$2(-3)-6(4)=0$$, $$-30=0$$. The first equation is not satisfied, so the point is not a solution. $$3(3)-4(1)=5$$, $$5=5$$. The second equation is satisfied. The point is a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a538fpairs3","title":"Determine Whether an Ordered Pair is a Solution of a System of Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Solve Systems of Linear Equations with Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1a538fpairs3a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Determine whether $$(-5,-7)$$ is a solution to the system: $$-3x+y=8-x+2y=-9$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No","Undefined"],"hints":{"DefaultPathway":[{"id":"a1a538fpairs3a-h1","type":"hint","dependencies":[],"title":"How to Verify a Solution","text":"If a point is a solution to a system, then it satisfies both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fpairs3a-h2","type":"hint","dependencies":["a1a538fpairs3a-h1"],"title":"Plugging in the Point","text":"Let us plug in the point. $$(-3)(-5)-7=8$$, $$8=8$$. The first equation is satisfied. -(-5)F+2(-7)=-9, $$-9=-9$$. The second equation is satisfied. The point is a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a538fpairs4","title":"Determine Whether an Ordered Pair is a Solution of a System of Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Solve Systems of Linear Equations with Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1a538fpairs4a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Determine whether $$(-5,7)$$ is a solution to the system: $$-3x+y=8-x+2y=-9$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No","Undefined"],"hints":{"DefaultPathway":[{"id":"a1a538fpairs4a-h1","type":"hint","dependencies":[],"title":"How to Verify a Solution","text":"If a point is a solution to a system, then it satisfies both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fpairs4a-h2","type":"hint","dependencies":["a1a538fpairs4a-h1"],"title":"Plugging in the Point","text":"Let us plug in the point. $$\\\\left(-3\\\\right) \\\\left(-5\\\\right)+7=-8$$. The first equation is not satisfied, so the point is not a solution. -(-5)F+2(-7)=-9, $$-9=-9$$. The second equation is satisfied. The point is a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a538fpairs5","title":"Determine Whether an Ordered Pair is a Solution of a System of Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Solve Systems of Linear Equations with Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1a538fpairs5a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Determine whether $$(\\\\frac{8}{7},\\\\frac{6}{7})$$ is a solution to the system: $$x+y=2, y=\\\\frac{3}{4} x$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No","Undefined"],"hints":{"DefaultPathway":[{"id":"a1a538fpairs5a-h1","type":"hint","dependencies":[],"title":"How to Verify a Solution","text":"If a point is a solution to a system, then it satisfies both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fpairs5a-h2","type":"hint","dependencies":["a1a538fpairs5a-h1"],"title":"Plugging in the Point","text":"Let us plug in the point. $$\\\\frac{8}{7}+\\\\frac{6}{7}=2$$. The first equation is satisfied. $$\\\\frac{6}{7}=\\\\frac{3}{4} \\\\frac{8}{7}$$. The second equation is satisfied. The point is a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a538fpairs6","title":"Determine Whether an Ordered Pair is a Solution of a System of Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Solve Systems of Linear Equations with Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1a538fpairs6a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Determine whether $$(1,\\\\frac{3}{4})$$ is a solution to the system: $$x+y=2, y=\\\\frac{3}{4} x$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No","Undefined"],"hints":{"DefaultPathway":[{"id":"a1a538fpairs6a-h1","type":"hint","dependencies":[],"title":"How to Verify a Solution","text":"If a point is a solution to a system, then it satisfies both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fpairs6a-h2","type":"hint","dependencies":["a1a538fpairs6a-h1"],"title":"Plugging in the Point","text":"Let us plug in the point. $$1+\\\\frac{3}{4}=2$$. The first equation is not satisfied, so the point is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a538fpairs7","title":"Determine Whether an Ordered Pair is a Solution of a System of Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Solve Systems of Linear Equations with Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1a538fpairs7a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Determine whether $$(-6,2)$$ is a solution to the system: $$2x+3y=6, y=\\\\frac{2}{3} x+2$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No","Undefined"],"hints":{"DefaultPathway":[{"id":"a1a538fpairs7a-h1","type":"hint","dependencies":[],"title":"How to Verify a Solution","text":"If a point is a solution to a system, then it satisfies both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fpairs7a-h2","type":"hint","dependencies":["a1a538fpairs7a-h1"],"title":"Plugging in the Point","text":"Let us plug in the point. 2(-6)+3(2)=6. The first equation is not satisfied, so the point is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a538fpairs8","title":"Determine Whether an Ordered Pair is a Solution of a System of Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Solve Systems of Linear Equations with Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1a538fpairs8a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Determine whether $$(-3,4)$$ is a solution to the system: $$2x+3y=6, y=\\\\frac{2}{3} x+2$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No","Undefined"],"hints":{"DefaultPathway":[{"id":"a1a538fpairs8a-h1","type":"hint","dependencies":[],"title":"How to Verify a Solution","text":"If a point is a solution to a system, then it satisfies both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fpairs8a-h2","type":"hint","dependencies":["a1a538fpairs8a-h1"],"title":"Plugging in the Point","text":"Let us plug in the point. 2(-3)+3(4)=6. The first equation is satisfied. $$4=\\\\frac{2}{3} \\\\left(-3\\\\right)+2$$. The second equation is not satisfied, so the point is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a538fpairs9","title":"Solve a System of Equations by Elimination","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Solve Systems of Linear Equations with Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1a538fpairs9a","stepAnswer":["$$(-2,6)$$"],"problemType":"MultipleChoice","stepTitle":"Solve the systems of equations: $$5x+2y=2, -3x-y=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-2,6)$$","choices":["$$(-2,6)$$","$$(-2,-6)$$","$$(2,6)$$"],"hints":{"DefaultPathway":[{"id":"a1a538fpairs9a-h1","type":"hint","dependencies":[],"title":"Condensing into One Equation","text":"We can multiply the second equation by $$2$$ and then add the two to get $$-x=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fpairs9a-h2","type":"hint","dependencies":["a1a538fpairs9a-h1"],"title":"Solving the Equation","text":"$$-x=2$$ can be simplified to $$x=-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fpairs9a-h3","type":"hint","dependencies":["a1a538fpairs9a-h2"],"title":"Substituting for the Other Variable","text":"Plugging $$x=-2$$ into the first equation, we get that 5(-2)+2y=2, which means that $$2y=12$$, $$y=6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a538fsystems1","title":"Determine whether each ordered pair is a solution to the system.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Solve Systems of Linear Equations with Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1a538fsystems1a","stepAnswer":["The ordered pair is not a solution to the system."],"problemType":"MultipleChoice","stepTitle":"$$x-y=-1$$ and $$2x-y=-5$$","stepBody":"$$(-2,-1)$$","answerType":"string","variabilization":{},"choices":["The ordered pair is a solution to the system.","The ordered pair is not a solution to the system."],"hints":{"DefaultPathway":[{"id":"a1a538fsystems1a-h1","type":"hint","dependencies":[],"title":"Plug in","text":"Plug in the values from the ordered pair into the system of equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fsystems1a-h2","type":"hint","dependencies":["a1a538fsystems1a-h1"],"title":"Check sides","text":"The equation is true if both sides equal each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fsystems1a-h3","type":"hint","dependencies":["a1a538fsystems1a-h2"],"title":"Simplify","text":"Simplify the equations. Check if the simplified form to see if it makes the equation true.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fsystems1a-h4","type":"hint","dependencies":["a1a538fsystems1a-h3"],"title":"Answer","text":"The ordered pair is not a solution to the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a538fsystems10","title":"Determine whether the system of equations is intersecting, parallel, or coincident.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Solve Systems of Linear Equations with Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1a538fsystems10a","stepAnswer":["The system is intersecting."],"problemType":"MultipleChoice","stepTitle":"$$3x+2y=2$$ and $$2x+y=1$$","stepBody":"","answerType":"string","variabilization":{},"choices":["The system is intersecting.","The system is parallel.","The system is coincident."],"hints":{"DefaultPathway":[{"id":"a1a538fsystems10a-h1","type":"hint","dependencies":[],"title":"Slope Intercept form","text":"Put both equations into slope intercept form. This will allow us to compare their characteristics.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fsystems10a-h2","type":"hint","dependencies":["a1a538fsystems10a-h1"],"title":"Extract slopes and intercepts","text":"Determine the slopes and intercepts of each line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fsystems10a-h3","type":"hint","dependencies":["a1a538fsystems10a-h2"],"title":"Interpret","text":"If the slopes and intercepts of both lines are the same, the system is coincident. If the slopes are the same, but the intercepts are not, the lines are parallel. If neither are the same, the system intersects.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fsystems10a-h4","type":"hint","dependencies":["a1a538fsystems10a-h3"],"title":"Answer","text":"The system is intersecting.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a538fsystems2","title":"Determine whether each ordered pair is a solution to the system.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Solve Systems of Linear Equations with Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1a538fsystems2a","stepAnswer":["The ordered pair is a solution to the system."],"problemType":"MultipleChoice","stepTitle":"$$x-y=-1$$ and $$2x-y=-5$$","stepBody":"$$(-4,-3)$$","answerType":"string","variabilization":{},"choices":["The ordered pair is a solution to the system.","The ordered pair is not a solution to the system."],"hints":{"DefaultPathway":[{"id":"a1a538fsystems2a-h1","type":"hint","dependencies":[],"title":"Plug in","text":"Plug in the values from the ordered pair into the system of equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fsystems2a-h2","type":"hint","dependencies":["a1a538fsystems2a-h1"],"title":"Check sides","text":"The equation is true if both sides equal each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fsystems2a-h3","type":"hint","dependencies":["a1a538fsystems2a-h2"],"title":"Simplify","text":"Simplify the equations. Check if the simplified form to see if it makes the equation true.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fsystems2a-h4","type":"hint","dependencies":["a1a538fsystems2a-h3"],"title":"Answer","text":"The ordered pair is a solution to the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a538fsystems3","title":"Determine whether each ordered pair is a solution to the system.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Solve Systems of Linear Equations with Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1a538fsystems3a","stepAnswer":["The ordered pair is a solution to the system."],"problemType":"MultipleChoice","stepTitle":"$$3x+y=0$$ and $$x+2y=-5$$","stepBody":"$$(1,-3)$$","answerType":"string","variabilization":{},"choices":["The ordered pair is a solution to the system.","The ordered pair is not a solution to the system."],"hints":{"DefaultPathway":[{"id":"a1a538fsystems3a-h1","type":"hint","dependencies":[],"title":"Plug in","text":"Plug in the values from the ordered pair into the system of equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fsystems3a-h2","type":"hint","dependencies":["a1a538fsystems3a-h1"],"title":"Check sides","text":"The equation is true if both sides equal each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fsystems3a-h3","type":"hint","dependencies":["a1a538fsystems3a-h2"],"title":"Simplify","text":"Simplify the equations. Check if the simplified form to see if it makes the equation true.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fsystems3a-h4","type":"hint","dependencies":["a1a538fsystems3a-h3"],"title":"Answer","text":"The ordered pair is a solution to the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a538fsystems4","title":"Determine whether each ordered pair is a solution to the system.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Solve Systems of Linear Equations with Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1a538fsystems4a","stepAnswer":["The ordered pair is not a solution to the system."],"problemType":"MultipleChoice","stepTitle":"$$3x+y=0$$ and $$x+2y=-5$$","stepBody":"$$(0,0)$$","answerType":"string","variabilization":{},"choices":["The ordered pair is a solution to the system.","The ordered pair is not a solution to the system."],"hints":{"DefaultPathway":[{"id":"a1a538fsystems4a-h1","type":"hint","dependencies":[],"title":"Plug in","text":"Plug in the values from the ordered pair into the system of equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fsystems4a-h2","type":"hint","dependencies":["a1a538fsystems4a-h1"],"title":"Check sides","text":"The equation is true if both sides equal each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fsystems4a-h3","type":"hint","dependencies":["a1a538fsystems4a-h2"],"title":"Simplify","text":"Simplify the equations. Check if the simplified form to see if it makes the equation true.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fsystems4a-h4","type":"hint","dependencies":["a1a538fsystems4a-h3"],"title":"Answer","text":"The ordered pair is not a solution to the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a538fsystems5","title":"Determine whether each ordered pair is a solution to the system.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Solve Systems of Linear Equations with Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1a538fsystems5a","stepAnswer":["The ordered pair is not a solution to the system."],"problemType":"MultipleChoice","stepTitle":"$$x-3y=-8$$ and $$-3x-y=4$$","stepBody":"$$(2,-2)$$","answerType":"string","variabilization":{},"choices":["The ordered pair is a solution to the system.","The ordered pair is not a solution to the system."],"hints":{"DefaultPathway":[{"id":"a1a538fsystems5a-h1","type":"hint","dependencies":[],"title":"Plug in","text":"Plug in the values from the ordered pair into the system of equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fsystems5a-h2","type":"hint","dependencies":["a1a538fsystems5a-h1"],"title":"Check sides","text":"The equation is true if both sides equal each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fsystems5a-h3","type":"hint","dependencies":["a1a538fsystems5a-h2"],"title":"Simplify","text":"Simplify the equations. Check if the simplified form to see if it makes the equation true.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fsystems5a-h4","type":"hint","dependencies":["a1a538fsystems5a-h3"],"title":"Answer","text":"The ordered pair is not a solution to the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a538fsystems6","title":"Determine whether each ordered pair is a solution to the system.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Solve Systems of Linear Equations with Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1a538fsystems6a","stepAnswer":["The ordered pair is a solution to the system."],"problemType":"MultipleChoice","stepTitle":"$$x-3y=-8$$ and $$-3x-y=4$$","stepBody":"$$(-2,2)$$","answerType":"string","variabilization":{},"choices":["The ordered pair is a solution to the system.","The ordered pair is not a solution to the system."],"hints":{"DefaultPathway":[{"id":"a1a538fsystems6a-h1","type":"hint","dependencies":[],"title":"Plug in","text":"Plug in the values from the ordered pair into the system of equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fsystems6a-h2","type":"hint","dependencies":["a1a538fsystems6a-h1"],"title":"Check sides","text":"The equation is true if both sides equal each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fsystems6a-h3","type":"hint","dependencies":["a1a538fsystems6a-h2"],"title":"Simplify","text":"Simplify the equations. Check if the simplified form to see if it makes the equation true.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fsystems6a-h4","type":"hint","dependencies":["a1a538fsystems6a-h3"],"title":"Answer","text":"The ordered pair is a solution to the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a538fsystems7","title":"Determine whether the system of equations is intersecting, parallel, or coincident.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Solve Systems of Linear Equations with Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1a538fsystems7a","stepAnswer":["The system is parallel."],"problemType":"MultipleChoice","stepTitle":"$$y=3x-1$$ and $$6x-2y=12$$","stepBody":"","answerType":"string","variabilization":{},"choices":["The system is intersecting.","The system is parallel.","The system is coincident."],"hints":{"DefaultPathway":[{"id":"a1a538fsystems7a-h1","type":"hint","dependencies":[],"title":"Slope Intercept form","text":"Put both equations into slope intercept form. This will allow us to compare their characteristics.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fsystems7a-h2","type":"hint","dependencies":["a1a538fsystems7a-h1"],"title":"Extract slopes and intercepts","text":"Determine the slopes and intercepts of each line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fsystems7a-h3","type":"hint","dependencies":["a1a538fsystems7a-h2"],"title":"Interpret","text":"If the slopes and intercepts of both lines are the same, the system is coincident. If the slopes are the same, but the intercepts are not, the lines are parallel. If neither are the same, the system intersects.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fsystems7a-h4","type":"hint","dependencies":["a1a538fsystems7a-h3"],"title":"Answer","text":"The system is parallel.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a538fsystems8","title":"Determine whether the system of equations is intersecting, parallel, or coincident.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Solve Systems of Linear Equations with Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1a538fsystems8a","stepAnswer":["The system is intersecting."],"problemType":"MultipleChoice","stepTitle":"$$2x+y=-3$$ and $$x-5y=5$$","stepBody":"","answerType":"string","variabilization":{},"choices":["The system is intersecting.","The system is parallel.","The system is coincident."],"hints":{"DefaultPathway":[{"id":"a1a538fsystems8a-h1","type":"hint","dependencies":[],"title":"Slope Intercept form","text":"Put both equations into slope intercept form. This will allow us to compare their characteristics.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fsystems8a-h2","type":"hint","dependencies":["a1a538fsystems8a-h1"],"title":"Extract slopes and intercepts","text":"Determine the slopes and intercepts of each line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fsystems8a-h3","type":"hint","dependencies":["a1a538fsystems8a-h2"],"title":"Interpret","text":"If the slopes and intercepts of both lines are the same, the system is coincident. If the slopes are the same, but the intercepts are not, the lines are parallel. If neither are the same, the system intersects.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fsystems8a-h4","type":"hint","dependencies":["a1a538fsystems8a-h3"],"title":"Answer","text":"The system is intersecting.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a538fsystems9","title":"Determine whether the system of equations is intersecting, parallel, or coincident.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Solve Systems of Linear Equations with Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1a538fsystems9a","stepAnswer":["The system is parallel."],"problemType":"MultipleChoice","stepTitle":"$$y=-2x-4$$ and $$4x+2y=9$$","stepBody":"","answerType":"string","variabilization":{},"choices":["The system is intersecting.","The system is parallel.","The system is coincident."],"hints":{"DefaultPathway":[{"id":"a1a538fsystems9a-h1","type":"hint","dependencies":[],"title":"Slope Intercept form","text":"Put both equations into slope intercept form. This will allow us to compare their characteristics.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fsystems9a-h2","type":"hint","dependencies":["a1a538fsystems9a-h1"],"title":"Extract slopes and intercepts","text":"Determine the slopes and intercepts of each line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fsystems9a-h3","type":"hint","dependencies":["a1a538fsystems9a-h2"],"title":"Interpret","text":"If the slopes and intercepts of both lines are the same, the system is coincident. If the slopes are the same, but the intercepts are not, the lines are parallel. If neither are the same, the system intersects.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a538fsystems9a-h4","type":"hint","dependencies":["a1a538fsystems9a-h3"],"title":"Answer","text":"The system is parallel.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a9374graphExp1","title":"Solve Exponential Equations","body":"Solve the exponential equation for $$x$$","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Evaluate and Graph Exponential Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1a9374graphExp1a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"$$3^{2x-5}=27$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a1a9374graphExp1a-h1","type":"hint","dependencies":[],"title":"Same Base","text":"Write both sides of the equation with the same base.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3^3$$"],"dependencies":["a1a9374graphExp1a-h1"],"title":"Same Base","text":"What is $$27$$ written with base 3? (** written as **)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp1a-h3","type":"hint","dependencies":["a1a9374graphExp1a-h2"],"title":"Analyzing Same Base Exponents","text":"When bases are equal in an equation, exponents can be equaled to each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp1a-h4","type":"hint","dependencies":["a1a9374graphExp1a-h3"],"title":"Analyzing Same Base Exponents","text":"Solve for $$x$$, $$2x-5=3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp1a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a1a9374graphExp1a-h4"],"title":"Solving for $$x$$","text":"What is $$x$$ after isolating?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a9374graphExp10","title":"Solve Exponential Equations","body":"Use the properties of exponents to solve for $$x$$. (If multiple answers, enter the numbers with a comma in between, no spaces, larger number first)","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Evaluate and Graph Exponential Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1a9374graphExp10a","stepAnswer":["$$-1$$"],"problemType":"TextBox","stepTitle":"$$2^{x^2+2x}=\\\\frac{1}{2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1$$","hints":{"DefaultPathway":[{"id":"a1a9374graphExp10a-h1","type":"hint","dependencies":[],"title":"Same Base","text":"Write both sides of the equation with the same base.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2^{-1}$$"],"dependencies":["a1a9374graphExp10a-h1"],"title":"Same Base","text":"What is $$\\\\frac{1}{2}$$ written with base 2? (** written as **)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp10a-h3","type":"hint","dependencies":["a1a9374graphExp10a-h2"],"title":"Analyzing Same Base Exponents","text":"When bases are equal in an equation, exponents can be equaled to each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp10a-h4","type":"hint","dependencies":["a1a9374graphExp10a-h3"],"title":"Analyzing Same Base Exponents","text":"Solve for $$x$$, $$x^2+2x=-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a1a9374graphExp10a-h4"],"title":"Solving for $$x$$","text":"What is $$x$$ after isolating?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a9374graphExp11","title":"Graph Exponential Functions","body":"Choose the correct graph of the following exponential function:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Evaluate and Graph Exponential Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1a9374graphExp11a","stepAnswer":["Graph B"],"problemType":"MultipleChoice","stepTitle":"$$2^x$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Graph A","Graph B","Graph C","Graph D"],"hints":{"DefaultPathway":[{"id":"a1a9374graphExp11a-h1","type":"hint","dependencies":[],"title":"Determining Points of the Graph","text":"We will use point plotting to determine which graph is correct. It will be easier to start with values of $$y$$ to get values of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4}$$"],"dependencies":["a1a9374graphExp11a-h1"],"title":"Determining $$y$$ when $$x=-2$$","text":"We know $$y=2^x$$ so what is $$y$$ when $$x=-2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp11a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a1a9374graphExp11a-h2"],"title":"Determining $$y$$ when $$x=0$$","text":"We know $$y=2^x$$ so what is $$y$$ when $$x=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a1a9374graphExp11a-h3"],"title":"Determining $$y$$ when $$x=1$$","text":"We know $$y=2^x$$ so what is $$y$$ when $$x=1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a1a9374graphExp11a-h4"],"title":"Determining $$y$$ when $$x=2$$","text":"We know $$y=2^x$$ so what is $$y$$ when $$x=2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp11a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph B"],"dependencies":["a1a9374graphExp11a-h5"],"title":"Determining the Correct Graph","text":"Which of the graphs fits each of the plot points we found with $$x$$ being the horizontal axis and $$y$$ being the vertical axis?\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph B","Graph C","Graph D"]}]}}]},{"id":"a1a9374graphExp12","title":"Graph Exponential Functions","body":"Choose the correct graph of the following exponential function:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Evaluate and Graph Exponential Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1a9374graphExp12a","stepAnswer":["Graph A"],"problemType":"MultipleChoice","stepTitle":"$$3^x$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Graph A","Graph B","Graph C","Graph D"],"hints":{"DefaultPathway":[{"id":"a1a9374graphExp12a-h1","type":"hint","dependencies":[],"title":"Determining Points of the Graph","text":"We will use point plotting to determine which graph is correct. It will be easier to start with values of $$y$$ to get values of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{9}$$"],"dependencies":["a1a9374graphExp12a-h1"],"title":"Determining $$y$$ when $$x=-2$$","text":"We know $$y=3^x$$ so what is $$y$$ when $$x=-2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp12a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a1a9374graphExp12a-h2"],"title":"Determining $$y$$ when $$x=0$$","text":"We know $$y=3^x$$ so what is $$y$$ when $$x=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a1a9374graphExp12a-h3"],"title":"Determining $$y$$ when $$x=1$$","text":"We know $$y=3^x$$ so what is $$y$$ when $$x=1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp12a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a1a9374graphExp12a-h4"],"title":"Determining $$y$$ when $$x=2$$","text":"We know $$y=3^x$$ so what is $$y$$ when $$x=2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp12a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph A"],"dependencies":["a1a9374graphExp12a-h5"],"title":"Determining the Correct Graph","text":"Which of the graphs fits each of the plot points we found with $$x$$ being the horizontal axis and $$y$$ being the vertical axis?\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph B","Graph C","Graph D"]}]}}]},{"id":"a1a9374graphExp13","title":"Graph Exponential Functions","body":"Choose the correct graph of the following exponential function:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Evaluate and Graph Exponential Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1a9374graphExp13a","stepAnswer":["Graph C"],"problemType":"MultipleChoice","stepTitle":"$$2^x-2$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Graph A","Graph B","Graph C","Graph D"],"hints":{"DefaultPathway":[{"id":"a1a9374graphExp13a-h1","type":"hint","dependencies":[],"title":"Determining Points of the Graph","text":"We will use point plotting to determine which graph is correct. It will be easier to start with values of $$y$$ to get values of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-7}{4}$$"],"dependencies":["a1a9374graphExp13a-h1"],"title":"Determining $$y$$ when $$x=-2$$","text":"We know $$y=2^x-2$$ so what is $$y$$ when $$x=-2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp13a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a1a9374graphExp13a-h2"],"title":"Determining $$y$$ when $$x=0$$","text":"We know $$y=2^x-2$$ so what is $$y$$ when $$x=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a1a9374graphExp13a-h3"],"title":"Determining $$y$$ when $$x=1$$","text":"We know $$y=2^x-2$$ so what is $$y$$ when $$x=1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp13a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a1a9374graphExp13a-h4"],"title":"Determining $$y$$ when $$x=2$$","text":"We know $$y=2^x-2$$ so what is $$y$$ when $$x=2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp13a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph C"],"dependencies":["a1a9374graphExp13a-h5"],"title":"Determining the Correct Graph","text":"Which of the graphs fits each of the plot points we found with $$x$$ being the horizontal axis and $$y$$ being the vertical axis?\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph B","Graph C","Graph D"]}]}}]},{"id":"a1a9374graphExp14","title":"Solve Exponential Equations","body":"Solve the exponential equation for $$x$$. (If multiple answers, enter the numbers with a comma in between, no spaces, larger number first)","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Evaluate and Graph Exponential Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1a9374graphExp14a","stepAnswer":["$$-1$$"],"problemType":"TextBox","stepTitle":"$$4^{x+3}=16$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1$$","hints":{"DefaultPathway":[{"id":"a1a9374graphExp14a-h1","type":"hint","dependencies":[],"title":"Same Base","text":"Write both sides of the equation with the same base.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4^2$$"],"dependencies":["a1a9374graphExp14a-h1"],"title":"Same Base","text":"What is $$16$$ written with base 4? (** written as **)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp14a-h3","type":"hint","dependencies":["a1a9374graphExp14a-h2"],"title":"Analyzing Same Base Exponents","text":"When bases are equal in an equation, exponents can be equaled to each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp14a-h4","type":"hint","dependencies":["a1a9374graphExp14a-h3"],"title":"Analyzing Same Base Exponents","text":"Solve for $$x$$, $$x+3=2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp14a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a1a9374graphExp14a-h4"],"title":"Solving for $$x$$","text":"What is $$x$$ after isolating?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a9374graphExp15","title":"Solve Exponential Equations","body":"Solve the exponential equation for $$x$$. (If multiple answers, enter the numbers with a comma in between, no spaces, larger number first)","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Evaluate and Graph Exponential Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1a9374graphExp15a","stepAnswer":["2,-2"],"problemType":"TextBox","stepTitle":"$$3^{x^2}=81$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a1a9374graphExp15a-h1","type":"hint","dependencies":[],"title":"Same Base","text":"Write both sides of the equation with the same base.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3^4$$"],"dependencies":["a1a9374graphExp15a-h1"],"title":"Same Base","text":"What is $$81$$ written with base 3? (** written as **)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp15a-h3","type":"hint","dependencies":["a1a9374graphExp15a-h2"],"title":"Analyzing Same Base Exponents","text":"When bases are equal in an equation, exponents can be equaled to each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp15a-h4","type":"hint","dependencies":["a1a9374graphExp15a-h3"],"title":"Analyzing Same Base Exponents","text":"$$x^2=4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp15a-h5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["2,-2"],"dependencies":["a1a9374graphExp15a-h4"],"title":"Solving for $$x$$","text":"What is $$x$$ after isolating?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a9374graphExp2","title":"Solve Exponential Equations","body":"Solve the Exponential Equation for $$x$$","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Evaluate and Graph Exponential Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1a9374graphExp2a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"$$3^{3x-2}=81$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a1a9374graphExp2a-h1","type":"hint","dependencies":[],"title":"Same Base","text":"Write both sides of the equation with the same base.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3^4$$"],"dependencies":["a1a9374graphExp2a-h1"],"title":"Same Base","text":"What is $$81$$ written with base 3? 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(If multiple answers, enter the numbers with a comma in between, no spaces, larger number first)","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Evaluate and Graph Exponential Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1a9374graphExp4a","stepAnswer":["3,-1"],"problemType":"TextBox","stepTitle":"$$\\\\frac{e^{x^2}}{e^3}=e^{2x}$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a1a9374graphExp4a-h1","type":"hint","dependencies":[],"title":"Division Property of Exponents","text":"When dividing same base exponents: $$\\\\frac{a^m}{a^n}=a^{m-n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp4a-h2","type":"hint","dependencies":["a1a9374graphExp4a-h1"],"title":"Division Property of Exponents","text":"The left side of the equation equals $$e^{x^2-3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp4a-h3","type":"hint","dependencies":["a1a9374graphExp4a-h2"],"title":"Analyzing Same Base Exponents","text":"When bases are equal in an equation, exponents can be equaled to each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp4a-h4","type":"hint","dependencies":["a1a9374graphExp4a-h3"],"title":"Analyzing Same Base Exponents","text":"Solve for $$x$$, $$x^2-3=2x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp4a-h5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["3,-1"],"dependencies":["a1a9374graphExp4a-h4"],"title":"Solving for $$x$$","text":"What is $$x$$ after isolating?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a9374graphExp5","title":"Solve Exponential Equations","body":"Use the properties of exponents to solve for $$x$$. (If multiple answers, enter the numbers with a comma in between, no spaces, larger number first)","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Evaluate and Graph Exponential Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1a9374graphExp5a","stepAnswer":["2,-1"],"problemType":"TextBox","stepTitle":"$$\\\\frac{e^{x^2}}{e^x}=e^2$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a1a9374graphExp5a-h1","type":"hint","dependencies":[],"title":"Division Property of Exponents","text":"When dividing same base exponents: $$\\\\frac{a^m}{a^n}=a^{m-n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp5a-h2","type":"hint","dependencies":["a1a9374graphExp5a-h1"],"title":"Division Property of Exponents","text":"The left side of the equation equals $$e^{x^2-x}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp5a-h3","type":"hint","dependencies":["a1a9374graphExp5a-h2"],"title":"Analyzing Same Base Exponents","text":"When bases are equal in an equation, exponents can be equaled to each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp5a-h4","type":"hint","dependencies":["a1a9374graphExp5a-h3"],"title":"Analyzing Same Base Exponents","text":"Solve for $$x$$, $$x^2-x=2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1a9374graphExp5a-h5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["2,-1"],"dependencies":["a1a9374graphExp5a-h4"],"title":"Solving for $$x$$","text":"What is $$x$$ after isolating?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1a9374graphExp6","title":"Solve Exponential Equations","body":"Use the properties of exponents to solve for $$x$$. 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over the interval [1,4], find the area of region R.","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{57}{4}$$","hints":{"DefaultPathway":[{"id":"a1cc0dcareas1a-h1","type":"hint","dependencies":[],"title":"Identify the curves","text":"First, you need to identify the two curves that define the region whose area you want to find. In this case, f(x) is the upper curve and g(x) is the lower curve","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas1a-h2","type":"hint","dependencies":["a1cc0dcareas1a-h1"],"title":"Set up the integral","text":"$$A=\\\\int (f(x)-g(x) \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas1a-h3","type":"hint","dependencies":["a1cc0dcareas1a-h2"],"title":"Set up the integral","text":"With $$f(x)=x+4$$ and $$g(x)=3-\\\\frac{x}{2}$$ over the given integral [1,4], we can complete the integral expression as $$A=\\\\int_{1}^{4} x+4-3-\\\\frac{x}{2} \\\\,dx=\\\\int_{1}^{4} \\\\frac{3x}{2}+1 \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas1a-h4","type":"hint","dependencies":["a1cc0dcareas1a-h3"],"title":"Find the integral","text":"$$\\\\int_{1}^{4} \\\\frac{3x}{2}+1 \\\\,dx=\\\\frac{3x^2}{4}+x$$ as the limits go from $$x=1$$ to $$x=4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas1a-h5","type":"hint","dependencies":["a1cc0dcareas1a-h4"],"title":"Evaluate","text":"$$16-\\\\frac{7}{4}=\\\\frac{57}{4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a1cc0dcareas10","title":"For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x-axis.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.1 Areas between Curves","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a1cc0dcareas10a","stepAnswer":["$$243$$"],"problemType":"TextBox","stepTitle":"$$y=x^2$$ and $$y=-\\\\left(x^2\\\\right)+18x$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$243$$","hints":{"DefaultPathway":[{"id":"a1cc0dcareas10a-h1","type":"hint","dependencies":[],"title":"Find the limits","text":"In order to set up the definite integrals of the $$2$$ functions, we have to find the limits where they intersect by setting them equal to each other and solve for $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas10a-h2","type":"hint","dependencies":["a1cc0dcareas10a-h1"],"title":"Find the limits","text":"$$x^2=-\\\\left(x^2\\\\right)+18x$$ then $$x=0$$ and $$x=9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas10a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y=-\\\\left(x^2\\\\right)+18x$$"],"dependencies":["a1cc0dcareas10a-h2"],"title":"Define the right graph","text":"What is the upper graph in this problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$y=-\\\\left(x^2\\\\right)+18x$$","$$y=x^2$$"]},{"id":"a1cc0dcareas10a-h4","type":"hint","dependencies":["a1cc0dcareas10a-h3"],"title":"Define the upper graph","text":"The area of A is given by $$\\\\int_{a}^{b} |f{\\\\left(x\\\\right)}-g{\\\\left(x\\\\right)}| \\\\,dx$$, as f(x) is an upper graph and g(x) is the lower one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas10a-h5","type":"hint","dependencies":["a1cc0dcareas10a-h4"],"title":"Set up the integral","text":"$$\\\\int_{0}^{9} -\\\\left(x^2\\\\right)+18x-x^2 \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas10a-h6","type":"hint","dependencies":["a1cc0dcareas10a-h5"],"title":"Compute the integral","text":"$$-\\\\left(\\\\frac{2x^2}{3}\\\\right)+\\\\frac{18x^2}{2}$$ as the limits go from $$x=0$$ to $$x=9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas10a-h7","type":"hint","dependencies":["a1cc0dcareas10a-h6"],"title":"Evaluate","text":"$$-\\\\left(\\\\frac{2\\\\times9^3}{3}\\\\right)+9\\\\times9^2=243$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a1cc0dcareas11","title":"For the following exercises, graph the equations and shade the area of the region between the curves. If necessary, break the region into sub-regions to determine its entire area.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.1 Areas between Curves","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a1cc0dcareas11a","stepAnswer":["$$\\\\frac{5}{2}$$"],"problemType":"TextBox","stepTitle":"$$y=x^3$$ and $$y=x^2-2x$$ over $$x=[-1,1]$$","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5}{2}$$","hints":{"DefaultPathway":[{"id":"a1cc0dcareas11a-h1","type":"hint","dependencies":[],"title":"Seperate the interval","text":"In this problem, the area of the region between $$2$$ curves can be broken into $$2$$ sub-regions as [-1,0] and [0,1].","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{19}{12}$$"],"dependencies":["a1cc0dcareas11a-h1"],"title":"Area from $$-1$$ to $$0$$","text":"What is the area of the interval from $$-1$$ to 0?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas11a-h3","type":"hint","dependencies":["a1cc0dcareas11a-h2"],"title":"Area from $$-1$$ to $$0$$","text":"/int{x**2-(2*x)-x**3,-1,0,x}=0-(((-1)**3)/3)-(-1)**2-(-1)**4)/4)=19/12","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a1cc0dcareas11a-h3"],"title":"Area from $$0$$ to $$1$$","text":"What is the area of the interval from $$0$$ to 1?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas11a-h5","type":"hint","dependencies":["a1cc0dcareas11a-h4"],"title":"Area from $$0$$ to $$1$$","text":"$$\\\\int_{0}^{1} x^3-x^2+2x \\\\,dx=\\\\frac{1^4}{4}-\\\\frac{1^3}{3}+1^2=\\\\frac{11}{12}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas11a-h6","type":"hint","dependencies":["a1cc0dcareas11a-h5"],"title":"The area of the whole region","text":"$$\\\\frac{19}{12}+\\\\frac{11}{12}=\\\\frac{5}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a1cc0dcareas2","title":"Finding the Area of a Region between Two Curves $$2$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.1 Areas between Curves","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a1cc0dcareas2a","stepAnswer":["$$3x-\\\\frac{x^3}{12}+\\\\frac{x^2}{2}$$ as the limits go from $$x=-2$$ to $$x=6$$"],"problemType":"MultipleChoice","stepTitle":"If R is the region bounded above by the graph of the function $$f(x)=9-{\\\\left(\\\\frac{x}{2}\\\\right)}^2$$ and below by the graph of the function $$g(x)=6-x$$, fin the area of region R.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$3x-\\\\frac{x^3}{12}+\\\\frac{x^2}{2}$$ as the limits go from $$x=-2$$ to $$x=6$$","choices":["$$3x-\\\\frac{x^3}{12}+\\\\frac{x^2}{2}$$ as the limits go from $$x=-2$$ to $$x=6$$","$$3x-\\\\frac{x^3}{12}+\\\\frac{x^2}{2}$$ as the limits go from $$x=-2$$ to $$y=6$$"],"hints":{"DefaultPathway":[{"id":"a1cc0dcareas2a-h1","type":"hint","dependencies":[],"title":"Determine the limits of integration","text":"You\'ll need to find the x-values where the two curves intersect by setting the given $$2$$ functions equal to each other $$f(x)=g(x)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas2a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$[-2,6]$$"],"dependencies":[],"title":"Determine the limits of integration","text":"What are the limits of integration?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$[-2,6]$$","$$[-5.5, 5.5]$$","[0,10]"]},{"id":"a1cc0dcareas2a-s1","type":"hint","dependencies":[],"title":"Determine the limits of integration","text":"As $$f(x)=g(x)$$, we then have $$9-{\\\\left(\\\\frac{x}{2}\\\\right)}^2=6-x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas2a-s2","type":"hint","dependencies":[],"title":"Solve for $$x$$","text":"$$9-\\\\frac{x}{4}=6-x$$ then $$\\\\left(x-6\\\\right) \\\\left(x+2\\\\right)=0$$. The graphs of functions intersect when $$x=6$$ and $$x=-2$$ so we want to integrate from $$-2$$ to $$6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas2a-h3","type":"hint","dependencies":["a1cc0dcareas2a-h2"],"title":"Set up the integral","text":"$$\\\\int_{-2}^{6} 9-{\\\\left(\\\\frac{x}{2}\\\\right)}^2-6-x \\\\,dx=\\\\int_{-2}^{6} 3-\\\\frac{x^2}{4}+x \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas2a-h4","type":"hint","dependencies":["a1cc0dcareas2a-h3"],"title":"Evaluate","text":"$$3x-\\\\frac{x^3}{12}+\\\\frac{x^2}{2}$$ as the limits go from $$x=-2$$ to $$x=6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas2a-h5","type":"hint","dependencies":["a1cc0dcareas2a-h4"],"title":"Conclusion","text":"The area of the region is $$\\\\frac{64}{3}$$ $${unit}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a1cc0dcareas3","title":"If R is the region bounded above by the graph of the function $$f(x)=sinx$$ and below by the graph of the function $$g(x)=cosx$$ over the interval [0,pi], find the area of region R.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.1 Areas between Curves","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a1cc0dcareas3a","stepAnswer":["$$\\\\sqrt{2}+1$$"],"problemType":"TextBox","stepTitle":"Identify the curves","stepBody":"First, you need to identify the two curves that define the region whose area you want to find. In this case, f(x) is the upper curve and g(x) is the lower curve. Note that from $$0$$ to $$\\\\frac{\\\\pi}{4}$$, the graph g(x) is an upper curve. But from $$\\\\frac{\\\\pi}{4}$$ to pi, f(x) is an upper curve and g(x) is a lower curve.##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\sqrt{2}+1$$","hints":{"DefaultPathway":[{"id":"a1cc0dcareas3a-h1","type":"hint","dependencies":[],"title":"Set up the integral","text":"$$A=\\\\int_{a}^{b} |f(x)-g(x)| \\\\,dx=\\\\int_{0}^{\\\\frac{\\\\pi}{4}} cosx-sinx \\\\,dx+\\\\int_{\\\\frac{\\\\pi}{4}}^{pi} sinx-cosx \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas3a-h2","type":"hint","dependencies":["a1cc0dcareas3a-h1"],"title":"Integrate from $$0$$ to $$\\\\frac{\\\\pi}{4}$$","text":"$$\\\\int_{0}^{\\\\frac{\\\\pi}{4}} cosx-sinx \\\\,dx=sinx+cox$$ with $$x$$ goes from $$0$$ to $$\\\\frac{\\\\pi}{4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{2}-1$$"],"dependencies":["a1cc0dcareas3a-h2"],"title":"Evaluate","text":"What is the Area of a region bounded from $$x=0$$ to $$x=\\\\frac{\\\\pi}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas3a-h4","type":"hint","dependencies":["a1cc0dcareas3a-h3"],"title":"Integrate from $$\\\\frac{\\\\pi}{4}$$ to pi","text":"$$\\\\int_{0}^{\\\\frac{\\\\pi}{4}} -cosx-sinx \\\\,dx=sinx+cox$$ with $$x$$ goes from $$\\\\frac{\\\\pi}{4}$$ to pi","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{2}+1$$"],"dependencies":["a1cc0dcareas3a-h4"],"title":"Evaluate","text":"What is the Area of a region bounded from $$x=0$$ to $$x=\\\\frac{\\\\pi}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas3a-h6","type":"hint","dependencies":["a1cc0dcareas3a-h5"],"title":"The area of the whole region","text":"$$\\\\sqrt{2}-1+\\\\sqrt{2}+1=2\\\\sqrt{2}$$ $${units}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a1cc0dcareas4","title":"Finding the Area of a Complex Region","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.1 Areas between Curves","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a1cc0dcareas4a","stepAnswer":["$$\\\\frac{5}{6}$$"],"problemType":"TextBox","stepTitle":"Consider the region depicted in Figure $$6.7$$. Find the area of R.","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5}{6}$$","hints":{"DefaultPathway":[{"id":"a1cc0dcareas4a-h1","type":"hint","dependencies":[],"title":"Find the intersection of $$2$$ functions","text":"Set $$f(x)=g(x)$$ and solve for $$x$$. After solving for $$x$$, we obtain $$x=1$$ where the graphs intersect.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas4a-h2","type":"hint","dependencies":["a1cc0dcareas4a-h1"],"title":"Seperate the interval","text":"Since the Area from $$0$$ to $$1$$ and the Area from $$1$$ to $$2$$ are not the same, we have to integrate each of them seperately.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["a1cc0dcareas4a-h2"],"title":"Area from $$0$$ to $$1$$","text":"What is the area of the interval from $$0$$ to 1?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"a1cc0dcareas4a-h3-s1","type":"hint","dependencies":[],"title":"Area from $$0$$ to $$1$$","text":"$$\\\\int_{0}^{1} x^2 \\\\,dx=\\\\frac{1^3}{3}-0=\\\\frac{1}{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"a1cc0dcareas4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a1cc0dcareas4a-h3"],"title":"Area from $$1$$ to $$2$$","text":"What is the area of the interval from $$1$$ to 2?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"a1cc0dcareas4a-h4-s1","type":"hint","dependencies":[],"title":"Area from $$1$$ to $$2$$","text":"$$\\\\int_{1}^{2} 2-x \\\\,dx=2\\\\times2-\\\\frac{2^2}{2}-2\\\\times1-\\\\frac{1^2}{2}=\\\\frac{1}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"a1cc0dcareas4a-h5","type":"hint","dependencies":["a1cc0dcareas4a-h4"],"title":"Adding the areas","text":"Adding these areas together, we obtain $$A=\\\\frac{1}{3}+\\\\frac{1}{2}=\\\\frac{5}{6}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a1cc0dcareas5","title":"Integrating with Respect to $$y$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.1 Areas between Curves","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a1cc0dcareas5a","stepAnswer":["$$\\\\frac{5}{6}$$"],"problemType":"TextBox","stepTitle":"Let R be the region depicted in Figure $$6.10$$. Find the area of R by integrating with respect to $$y$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5}{6}$$","hints":{"DefaultPathway":[{"id":"a1cc0dcareas5a-h1","type":"hint","dependencies":[],"title":"Express the function","text":"We must first express the graphs as functions of $$y$$. The curve on the left can be represented by the function $$x=\\\\sqrt{y}$$. The curve on the right can be represented by the funcgion $$x=2-y$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas5a-h2","type":"hint","dependencies":["a1cc0dcareas5a-h1"],"title":"Change the limits","text":"Now we have to determine the limits of integration. The region is bounded below by the x-axis, so the lower limit of integration is $$y=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas5a-h3","type":"hint","dependencies":["a1cc0dcareas5a-h2"],"title":"Change the limits","text":"The region is bounded below by the x-axis, so the lower limit of integration is $$y=0$$. The upper limit of integration is determined by the point where the two graphs intersect, which is the point $$(1,1)$$ so the upper limit of integration is $$y=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas5a-h4","type":"hint","dependencies":["a1cc0dcareas5a-h3"],"title":"Find the area","text":"The area of the region $$A=\\\\int_{0}^{1} 2-y-\\\\sqrt{y} \\\\,dy$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas5a-h5","type":"hint","dependencies":["a1cc0dcareas5a-h4"],"title":"Compute the integral","text":"$$2y-\\\\frac{y^2}{2}-\\\\frac{\\\\frac{2}{3} y^3}{2}$$ with the limits go from $$y=0$$ to $$y=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas5a-h6","type":"hint","dependencies":["a1cc0dcareas5a-h5"],"title":"Evaluate","text":"$$2\\\\times1-\\\\frac{1^2}{2}-\\\\frac{\\\\frac{2}{3} 1^3}{2}-2\\\\times0-\\\\frac{0^2}{2}-\\\\frac{\\\\frac{2}{3} 0^3}{2}=\\\\frac{5}{6}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a1cc0dcareas6","title":"For the following exercises, determine the area of the region between the two curves in the given figure by integrating over the x-axis.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.1 Areas between Curves","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a1cc0dcareas6a","stepAnswer":["$$\\\\frac{32}{3}$$"],"problemType":"TextBox","stepTitle":"$$y=x^2-3$$ and $$y=1$$","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{32}{3}$$","hints":{"DefaultPathway":[{"id":"a1cc0dcareas6a-h1","type":"hint","dependencies":[],"title":"Find the limits","text":"In order to set up the definite integrals of the $$2$$ functions, we have to find the limits where they intersect by setting them equal to each other and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas6a-h2","type":"hint","dependencies":["a1cc0dcareas6a-h1"],"title":"Find the limits","text":"$$x^2-3=1$$ then $$x=2$$ and $$x=-2$$. So we obtain $$x=-2$$ as a lower limit and $$x=2$$ as an upper limit.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas6a-h3","type":"hint","dependencies":["a1cc0dcareas6a-h2"],"title":"Define the upper graph","text":"The area of A is given by $$int{|f(x)-g(x)|, a, b, x$$,} as f(x) is an upper graph and g(x) is the lower one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas6a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["g(x)"],"dependencies":["a1cc0dcareas6a-h3"],"title":"Define the upper graph","text":"What is the upper graph in this problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["g(x)","f(x)"]},{"id":"a1cc0dcareas6a-h5","type":"hint","dependencies":["a1cc0dcareas6a-h4"],"title":"Set up the integral","text":"$$\\\\int_{-2}^{2} 1-x^2+3 \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas6a-h6","type":"hint","dependencies":["a1cc0dcareas6a-h5"],"title":"Compute the integral","text":"$$4x-\\\\frac{x^3}{3}$$ as the limtis go from $$x=-2$$ to $$x=2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas6a-h7","type":"hint","dependencies":["a1cc0dcareas6a-h6"],"title":"Evaluate","text":"$$4\\\\times2-\\\\frac{2^3}{3}-4\\\\left(-2\\\\right)-\\\\frac{{\\\\left(-2\\\\right)}^3}{3}=\\\\frac{32}{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a1cc0dcareas7","title":"For the following exercises, split the region between the two curves into two smaller regions, then determine the area by integrating over the x-axis. Note that you will have two integrals to solve.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.1 Areas between Curves","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a1cc0dcareas7a","stepAnswer":["$$\\\\frac{13}{12}$$"],"problemType":"TextBox","stepTitle":"$$y=x^3$$ and $$y=x^2+x$$","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{13}{12}$$","hints":{"DefaultPathway":[{"id":"a1cc0dcareas7a-h1","type":"hint","dependencies":[],"title":"Find the limits","text":"In order to set up the definite integrals of the $$2$$ functions, we have to find the limits where they intersect by setting them equal to each other and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas7a-h2","type":"hint","dependencies":["a1cc0dcareas7a-h1"],"title":"Find the limits","text":"$$x^3=x^2+x$$ then $$x=0$$, $$x=\\\\frac{1-\\\\sqrt{5}}{2}$$ and $$x=\\\\frac{1+\\\\sqrt{5}}{2}$$. So we obtain $$x=\\\\frac{1-\\\\sqrt{5}}{2}$$ as a lower limit and $$x=0$$ as an upper limit for the region on the left side of the y-axis. The second limit for the region on the right side of the y-axis has a lower limit as $$x=0$$ and an upper limit as $$x=\\\\frac{1+\\\\sqrt{5}}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas7a-h3","type":"hint","dependencies":["a1cc0dcareas7a-h2"],"title":"Define the upper graph","text":"The area of A is given by int{abs(f(x)-g(x)),a,b,x}, as f(x) is an upper graph and g(x) is the lower one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas7a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["f(x)"],"dependencies":["a1cc0dcareas7a-h3"],"title":"Define the upper graph","text":"What is the upper graph for the region on the left side of y-axis?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["g(x)","f(x)"]},{"id":"a1cc0dcareas7a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["g(x)"],"dependencies":["a1cc0dcareas7a-h4"],"title":"Define the upper graph","text":"What is the upper graph for the region on the right side of y-axis?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["g(x)","f(x)"]},{"id":"a1cc0dcareas7a-h6","type":"hint","dependencies":["a1cc0dcareas7a-h5"],"title":"Set up the integral","text":"$$\\\\int_{\\\\frac{1-\\\\sqrt{5}}{2}}^{0} x^3-x^2-x \\\\,dx+\\\\int_{0}^{\\\\frac{1+\\\\sqrt{5}}{2}} x^2+x-x^3 \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas7a-h7","type":"hint","dependencies":["a1cc0dcareas7a-h6"],"title":"Evaluate","text":"$$\\\\frac{13-5\\\\sqrt{5}}{24}+\\\\frac{13+5\\\\sqrt{5}}{24}=\\\\frac{13}{12}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a1cc0dcareas8","title":"For the following exercises, determine the area of the region between the two curves by integrating over the y-axis","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.1 Areas between Curves","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a1cc0dcareas8a","stepAnswer":["$$36$$"],"problemType":"TextBox","stepTitle":"$$x=y^2$$ and $$x=9$$","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$36$$","hints":{"DefaultPathway":[{"id":"a1cc0dcareas8a-h1","type":"hint","dependencies":[],"title":"Find the limits","text":"In order to set up the definite integrals of the $$2$$ functions, we have to find the limits where they intersect by setting them equal to each other and solve for $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas8a-h2","type":"hint","dependencies":["a1cc0dcareas8a-h1"],"title":"Find the limits","text":"$$y^2=9$$ then $$y=-3$$ and $$y=3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas8a-h3","type":"hint","dependencies":["a1cc0dcareas8a-h2"],"title":"Define the right graph","text":"The area of A is given by $$\\\\int_{c}^{d} |u\\\\left(y\\\\right)-v\\\\left(y\\\\right)| \\\\,dy$$, as u(y) is an upper graph and v(y)is the lower one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas8a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y^2$$"],"dependencies":["a1cc0dcareas8a-h3"],"title":"Define the right graph","text":"What is the right graph in this problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$y^2$$","$$9$$"]},{"id":"a1cc0dcareas8a-h5","type":"hint","dependencies":["a1cc0dcareas8a-h4"],"title":"Set up the integral","text":"$$\\\\int_{-3}^{3} 9-y^2 \\\\,dy$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas8a-h6","type":"hint","dependencies":["a1cc0dcareas8a-h5"],"title":"Compute the integral","text":"$$9y-\\\\frac{y^3}{3}$$ as the limits go from $$y=-3$$ to $$y=3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas8a-h7","type":"hint","dependencies":["a1cc0dcareas8a-h6"],"title":"Evaluate","text":"$$9\\\\times3-\\\\frac{3^3}{3}-9\\\\left(-3\\\\right)-\\\\frac{{\\\\left(-3\\\\right)}^3}{3}=36$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a1cc0dcareas9","title":"For the following exercises, determine the area of the region between the two curves by integrating over the y-axis","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.1 Areas between Curves","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a1cc0dcareas9a","stepAnswer":["$$\\\\frac{1}{6}$$"],"problemType":"TextBox","stepTitle":"$$y=x$$ and $$x=y^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{6}$$","hints":{"DefaultPathway":[{"id":"a1cc0dcareas9a-h1","type":"hint","dependencies":[],"title":"Find the limits","text":"In order to set up the definite integrals of the $$2$$ functions, we have to find the limits where they intersect by setting them equal to each other and solve for $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas9a-h2","type":"hint","dependencies":["a1cc0dcareas9a-h1"],"title":"Find the limits","text":"$$y^2=y$$ then $$y=0$$ and $$y=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas9a-h3","type":"hint","dependencies":["a1cc0dcareas9a-h2"],"title":"Define the right graph","text":"The area of A is given by $$\\\\int_{c}^{d} |u\\\\left(y\\\\right)-v\\\\left(y\\\\right)| \\\\,dy$$, as u(y) is an upper graph and v(y)is the lower one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas9a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y$$"],"dependencies":["a1cc0dcareas9a-h3"],"title":"Define the right graph","text":"What is the right graph in this problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$y$$","$$y^2$$"]},{"id":"a1cc0dcareas9a-h5","type":"hint","dependencies":["a1cc0dcareas9a-h4"],"title":"Set up the integral","text":"$$\\\\int_{0}^{1} y-y^2 \\\\,dy$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas9a-h6","type":"hint","dependencies":["a1cc0dcareas9a-h5"],"title":"Compute the integral","text":"$$\\\\frac{y^2}{2}-\\\\frac{y^3}{3}$$ as the limits go from $$y=0$$ to $$y=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a1cc0dcareas9a-h7","type":"hint","dependencies":["a1cc0dcareas9a-h6"],"title":"Evaluate","text":"(1**2)/2)-(1**3)/3=1/6","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a1def42clt1","title":"Sample Mean Probabilty","body":"An unknown distribution has a mean of $$90$$ and a standard deviation of $$15$$. Samples of size $$n$$ $$=$$ $$25$$ are randomly drawn from the population.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 The Central Limit Theorem for Sample Means (Averages)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a1def42clt1a","stepAnswer":["$$0.6997$$"],"problemType":"TextBox","stepTitle":"Find the probabilty that the sample mean is between $$85$$ and $$92$$.","stepBody":"Round answers to four decimal places.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.6997$$","hints":{"DefaultPathway":[{"id":"a1def42clt1a-h1","type":"hint","dependencies":[],"title":"Using a calculator","text":"Using normalcdf","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt1a-h2","type":"hint","dependencies":["a1def42clt1a-h1"],"title":"First parameter","text":"Lower Value","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$85$$"],"dependencies":["a1def42clt1a-h2"],"title":"What is the lower value?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt1a-h4","type":"hint","dependencies":["a1def42clt1a-h3"],"title":"Second parameter","text":"Upper Value","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt1a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$92$$"],"dependencies":["a1def42clt1a-h4"],"title":"What is the upper value?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt1a-h6","type":"hint","dependencies":["a1def42clt1a-h5"],"title":"Third parameter","text":"Mean","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt1a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$90$$"],"dependencies":["a1def42clt1a-h6"],"title":"Mean","text":"What is the mean?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt1a-h8","type":"hint","dependencies":["a1def42clt1a-h7"],"title":"Fourth parameter","text":"Standard error of the mean","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt1a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{15}{\\\\sqrt{25}}$$"],"dependencies":["a1def42clt1a-h8"],"title":"What is the standard error of the mean?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1def42clt10","title":"Carrying Change","body":"Previously, De Anza statistics students estimated that the amount of change daytime statistics students carry is distributed with a mean of $$\\\\$0.88$$ and standard deviation of $$0.1$$. Suppose that we randomly pick $$25$$ daytime statistics students.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 The Central Limit Theorem for Sample Means (Averages)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a1def42clt10a","stepAnswer":["$$0.6731$$"],"problemType":"TextBox","stepTitle":"Find the probability that an individual had between $$\\\\$0.80$$ and $$\\\\$1.00$$.","stepBody":"Round to four decimal places.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.6731$$","hints":{"DefaultPathway":[{"id":"a1def42clt10a-h1","type":"hint","dependencies":[],"title":"Using a calculator","text":"Using normalcdf","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt10a-h2","type":"hint","dependencies":["a1def42clt10a-h1"],"title":"First parameter","text":"Lower Value","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.8$$"],"dependencies":["a1def42clt10a-h2"],"title":"What is the lower value?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt10a-h4","type":"hint","dependencies":["a1def42clt10a-h3"],"title":"Second parameter","text":"Upper Value","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a1def42clt10a-h4"],"title":"What is the upper value?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt10a-h6","type":"hint","dependencies":["a1def42clt10a-h5"],"title":"Third parameter","text":"Mean","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt10a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.88$$"],"dependencies":["a1def42clt10a-h6"],"title":"What is the mean?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt10a-h8","type":"hint","dependencies":["a1def42clt10a-h7"],"title":"Fourth parameter","text":"Standard Deviation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt10a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1$$"],"dependencies":["a1def42clt10a-h8"],"title":"What is the standard deviation?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1def42clt11","title":"Tax Forms","body":"According to the Internal Revenue Service, the average length of time for an individual to complete (keep records for, learn, prepare, copy, assemble, and send) IRS Form $$1040$$ is $$10.53$$ hours. Let us assume that the standard deviation is two hours. Suppose we randomly sample $$36$$ taxpayers.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 The Central Limit Theorem for Sample Means (Averages)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a1def42clt11a","stepAnswer":["$$0.3733$$"],"problemType":"TextBox","stepTitle":"Find the probability that the sample mean is between $$10$$ hours and $$12$$ hours","stepBody":"Round to four decimal places.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.3733$$","hints":{"DefaultPathway":[{"id":"a1def42clt11a-h1","type":"hint","dependencies":[],"title":"Using a calculator","text":"Using normalcdf","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt11a-h2","type":"hint","dependencies":["a1def42clt11a-h1"],"title":"First parameter","text":"Lower Value","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt11a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a1def42clt11a-h2"],"title":"What is the lower value?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt11a-h4","type":"hint","dependencies":["a1def42clt11a-h3"],"title":"Second parameter","text":"Upper Value","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a1def42clt11a-h4"],"title":"What is the upper value?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt11a-h6","type":"hint","dependencies":["a1def42clt11a-h5"],"title":"Third parameter","text":"Mean","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt11a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10.53$$"],"dependencies":["a1def42clt11a-h6"],"title":"What is the mean?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt11a-h8","type":"hint","dependencies":["a1def42clt11a-h7"],"title":"Fourth parameter","text":"Standard error of the mean","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt11a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{\\\\sqrt{36}}$$"],"dependencies":["a1def42clt11a-h8"],"title":"What is the standard error of the mean?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1def42clt12","title":"Marathon Runner","body":"Suppose that a category of world-class runners are known to run a marathon (26 miles) in an average of $$145$$ minutes with a standard deviation of $$14$$ minutes. Consider $$49$$ of the races","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 The Central Limit Theorem for Sample Means (Averages)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a1def42clt12a","stepAnswer":["$$0.6247$$"],"problemType":"TextBox","stepTitle":"Find the probability that the runners will average between $$142$$ and $$146$$ minutes in these $$49$$ marathons","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.6247$$","hints":{"DefaultPathway":[{"id":"a1def42clt12a-h1","type":"hint","dependencies":[],"title":"Using a calculator","text":"Using normalcdf","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt12a-h2","type":"hint","dependencies":["a1def42clt12a-h1"],"title":"First parameter","text":"Lower Value","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt12a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$142$$"],"dependencies":["a1def42clt12a-h2"],"title":"What is the lower value?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt12a-h4","type":"hint","dependencies":["a1def42clt12a-h3"],"title":"Second parameter","text":"Upper Value","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt12a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$146$$"],"dependencies":["a1def42clt12a-h4"],"title":"What is the upper value?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt12a-h6","type":"hint","dependencies":["a1def42clt12a-h5"],"title":"Third parameter","text":"Mean","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt12a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$145$$"],"dependencies":["a1def42clt12a-h6"],"title":"What is the mean?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt12a-h8","type":"hint","dependencies":["a1def42clt12a-h7"],"title":"Fourth parameter","text":"Standard error of the mean","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt12a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{14}{\\\\sqrt{49}}$$"],"dependencies":["a1def42clt12a-h8"],"title":"What is the standard error of the mean?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1def42clt13","title":"Wealth Distribution","body":"The distribution of income in some Third World countries is considered wedge shaped (many very poor people, very few middle income people, and even fewer wealthy people). Suppose we pick a country with a wedge shaped distribution. Let the average salary be $2,000 per year with a standard deviation of $8,000. We randomly survey 1,000 residents of that country.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 The Central Limit Theorem for Sample Means (Averages)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a1def42clt13a","stepAnswer":["$$0.1537$$"],"problemType":"TextBox","stepTitle":"Find the probabilty that the average is between $2000 and $2100","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.1537$$","hints":{"DefaultPathway":[{"id":"a1def42clt13a-h1","type":"hint","dependencies":[],"title":"Using a calculator","text":"Using normalcdf","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt13a-h2","type":"hint","dependencies":["a1def42clt13a-h1"],"title":"First parameter","text":"Lower Value","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt13a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2000$$"],"dependencies":["a1def42clt13a-h2"],"title":"What is the lower value?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt13a-h4","type":"hint","dependencies":["a1def42clt13a-h3"],"title":"Second parameter","text":"Upper Value","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt13a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2100$$"],"dependencies":["a1def42clt13a-h4"],"title":"What is the upper value?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt13a-h6","type":"hint","dependencies":["a1def42clt13a-h5"],"title":"Third parameter","text":"Mean","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt13a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2000$$"],"dependencies":["a1def42clt13a-h6"],"title":"What is the mean?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt13a-h8","type":"hint","dependencies":["a1def42clt13a-h7"],"title":"Fourth parameter","text":"Standard error of the mean","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt13a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{8000}{\\\\sqrt{1000}}$$"],"dependencies":["a1def42clt13a-h8"],"title":"What is the standard error of the mean?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1def42clt14","title":"Gas Prices","body":"The cost of unleaded gasoline in the Bay Area once followed a distribution with a mean of $$\\\\$4.59$$ and a standard deviation of $$\\\\$0.10$$. Sixteen gas stations from the Bay Area are randomly chosen.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 The Central Limit Theorem for Sample Means (Averages)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a1def42clt14a","stepAnswer":["$$0.6803$$"],"problemType":"TextBox","stepTitle":"What is the probabilty the sample mean was between $$\\\\$4.50$$ and $$\\\\$4.70$$?","stepBody":"Round to four decimal places","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.6803$$","hints":{"DefaultPathway":[{"id":"a1def42clt14a-h1","type":"hint","dependencies":[],"title":"Using a calculator","text":"Using normalcdf","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt14a-h2","type":"hint","dependencies":["a1def42clt14a-h1"],"title":"First parameter","text":"Lower Value","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt14a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4.5$$"],"dependencies":["a1def42clt14a-h2"],"title":"What is the lower value?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt14a-h4","type":"hint","dependencies":["a1def42clt14a-h3"],"title":"Second parameter","text":"Upper Value","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt14a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4.7$$"],"dependencies":["a1def42clt14a-h4"],"title":"What is the upper value?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt14a-h6","type":"hint","dependencies":["a1def42clt14a-h5"],"title":"Third parameter","text":"Mean","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt14a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4.59$$"],"dependencies":["a1def42clt14a-h6"],"title":"What is the mean?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt14a-h8","type":"hint","dependencies":["a1def42clt14a-h7"],"title":"Fourth parameter","text":"Standard Deviation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt14a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1$$"],"dependencies":["a1def42clt14a-h8"],"title":"What is the standard deviation?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1def42clt15","title":"Calorie Consumption","body":"The percent of fat calories that a person in America consumes each day is normally distributed with a mean of about $$36$$ and a standard deviation of about ten. Suppose that $$16$$ individuals are randomly chosen.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 The Central Limit Theorem for Sample Means (Averages)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a1def42clt15a","stepAnswer":["$$0.999$$"],"problemType":"TextBox","stepTitle":"What is the probabilty the sample mean is above 5?","stepBody":"Round to four decimal places","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.999$$","hints":{"DefaultPathway":[{"id":"a1def42clt15a-h1","type":"hint","dependencies":[],"title":"Using a calculator","text":"Using normalcdf","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt15a-h2","type":"hint","dependencies":["a1def42clt15a-h1"],"title":"First parameter","text":"Lower Value","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a1def42clt15a-h2"],"title":"What is the lower value?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt15a-h4","type":"hint","dependencies":["a1def42clt15a-h3"],"title":"Second parameter","text":"Upper Value","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt15a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${10}^{99}$$"],"dependencies":["a1def42clt15a-h4"],"title":"What is the upper value?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt15a-h6","type":"hint","dependencies":["a1def42clt15a-h5"],"title":"Third parameter","text":"Mean","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt15a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$36$$"],"dependencies":["a1def42clt15a-h6"],"title":"What is the mean?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt15a-h8","type":"hint","dependencies":["a1def42clt15a-h7"],"title":"Fourth parameter","text":"Standard Deviation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt15a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a1def42clt15a-h8"],"title":"What is the standard deviation?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1def42clt2","title":"Sample Mean Probabilty","body":"An unknown distribution has a mean of $$90$$ and a standard deviation of $$15$$. Samples of size $$n$$ $$=$$ $$25$$ are randomly drawn from the population.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 The Central Limit Theorem for Sample Means (Averages)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a1def42clt2a","stepAnswer":["$$96$$"],"problemType":"TextBox","stepTitle":"Find the value that is two standard deviations above the expected value, $$90$$, of the sample mean.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$96$$","hints":{"DefaultPathway":[{"id":"a1def42clt2a-h1","type":"hint","dependencies":[],"title":"Use the formula","text":"mean * (# of sdevs) * standard error of the mean","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1def42clt3","title":"Soccer Game","body":"The length of time, in hours, it takes an \\"over 40\\" group of people to play one soccer match is normally distributed with a mean of two hours and a standard deviation of $$0.5$$ hours. A sample of size $$n$$ $$=$$ $$50$$ is drawn randomly from the population.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 The Central Limit Theorem for Sample Means (Averages)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a1def42clt3a","stepAnswer":["$$0.9977$$"],"problemType":"TextBox","stepTitle":"Find the probability that the sample mean is between $$1.8$$ hours and $$2.3$$ hours.","stepBody":"Round answers to four decimal places.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.9977$$","hints":{"DefaultPathway":[{"id":"a1def42clt3a-h1","type":"hint","dependencies":[],"title":"Using a calculator","text":"Using normalcdf","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt3a-h2","type":"hint","dependencies":["a1def42clt3a-h1"],"title":"First parameter","text":"Lower Value","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.8$$"],"dependencies":["a1def42clt3a-h2"],"title":"What is the lower value?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt3a-h4","type":"hint","dependencies":["a1def42clt3a-h3"],"title":"Second parameter","text":"Upper Value","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.3$$"],"dependencies":["a1def42clt3a-h4"],"title":"What is the upper value?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt3a-h6","type":"hint","dependencies":["a1def42clt3a-h5"],"title":"Third parameter","text":"Mean","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt3a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a1def42clt3a-h6"],"title":"What is the mean?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt3a-h8","type":"hint","dependencies":["a1def42clt3a-h7"],"title":"Fourth parameter","text":"Standard error of the mean","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt3a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{0.5}{\\\\sqrt{50}}$$"],"dependencies":["a1def42clt3a-h8"],"title":"What is the standard error of the mean?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1def42clt4","title":"Tablet Users","body":"In a recent study reported Oct. $$29$$, $$2012$$ on the Flurry Blog, the mean age of tablet users is $$34$$ years. Suppose the standard deviation is $$15$$ years. Take a sample of size $$n$$ $$=$$ $$100$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 The Central Limit Theorem for Sample Means (Averages)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a1def42clt4a","stepAnswer":["$$\\\\frac{15}{11}$$"],"problemType":"TextBox","stepTitle":"What is the sample standard deviation for the ages of tablet users?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{15}{11}$$","hints":{"DefaultPathway":[{"id":"a1def42clt4a-h1","type":"hint","dependencies":[],"title":"Sample standard deviation formula","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1def42clt5","title":"Tablet Users","body":"In a recent study reported Oct. $$29$$, $$2012$$ on the Flurry Blog, the mean age of tablet users is $$34$$ years. Suppose the standard deviation is $$15$$ years. Take a sample of size $$n$$ $$=$$ $$100$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 The Central Limit Theorem for Sample Means (Averages)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a1def42clt5a","stepAnswer":["$$0.9962$$"],"problemType":"TextBox","stepTitle":"Find the probability that the sample mean age is more than $$30$$ years (the reported mean age of tablet users in this particular study).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.9962$$","hints":{"DefaultPathway":[{"id":"a1def42clt5a-h1","type":"hint","dependencies":[],"title":"Using a calculator","text":"Using normalcdf","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt5a-h2","type":"hint","dependencies":["a1def42clt5a-h1"],"title":"First parameter","text":"Lower Value","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$30$$"],"dependencies":["a1def42clt5a-h2"],"title":"What is the lower value?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt5a-h4","type":"hint","dependencies":["a1def42clt5a-h3"],"title":"Second parameter","text":"Upper Value","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${10}^{99}$$"],"dependencies":["a1def42clt5a-h4"],"title":"What is the upper value?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt5a-h6","type":"hint","dependencies":["a1def42clt5a-h5"],"title":"Third parameter","text":"Mean","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt5a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$34$$"],"dependencies":["a1def42clt5a-h6"],"title":"What is the mean?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt5a-h8","type":"hint","dependencies":["a1def42clt5a-h7"],"title":"Fourth parameter","text":"Standard error of the mean","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt5a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.5$$"],"dependencies":["a1def42clt5a-h8"],"title":"What is the standard error of the mean?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1def42clt6","title":"Tablet Users","body":"In a recent study reported Oct. $$29$$, $$2012$$ on the Flurry Blog, the mean age of tablet users is $$34$$ years. Suppose the standard deviation is $$15$$ years. Take a sample of size $$n$$ $$=$$ $$100$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 The Central Limit Theorem for Sample Means (Averages)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a1def42clt6a","stepAnswer":["$$36.5$$"],"problemType":"TextBox","stepTitle":"Find the 95th percentile for the sample mean age (to one decimal place).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$36.5$$","hints":{"DefaultPathway":[{"id":"a1def42clt6a-h1","type":"hint","dependencies":[],"title":"Using a Calculator (TI-83, 83+, $$84$$, 84+ Calculator)","text":"Using invNorm","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1def42clt7","title":"App Engagement","body":"The mean number of minutes for app engagement by a tablet user is $$8.2$$ minutes. Suppose the standard deviation is one minute. Take a sample of $$60$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 The Central Limit Theorem for Sample Means (Averages)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a1def42clt7a","stepAnswer":["$$1.5$$"],"problemType":"TextBox","stepTitle":"What is the sample standard deviation for the number of app engaged minutes of tablet users?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1.5$$","hints":{"DefaultPathway":[{"id":"a1def42clt7a-h1","type":"hint","dependencies":[],"title":"Sample standard deviation formula","text":"sdev / $$\\\\sqrt{n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1def42clt8","title":"App Engagement","body":"The mean number of minutes for app engagement by a tablet user is $$8.2$$ minutes. Suppose the standard deviation is one minute. Take a sample of $$60$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 The Central Limit Theorem for Sample Means (Averages)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a1def42clt8a","stepAnswer":["$$8.37$$"],"problemType":"TextBox","stepTitle":"Find the 90th percentile for the sample mean time for app engagement for a tablet user.","stepBody":"Round to two decimal places","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8.37$$","hints":{"DefaultPathway":[{"id":"a1def42clt8a-h1","type":"hint","dependencies":[],"title":"Using a Calculator (TI-83, 83+, $$84$$, 84+ Calculator)","text":"Using invNorm","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8.37$$"],"dependencies":["a1def42clt8a-h1"],"title":"invNorm(0.90,8.2,1/sqrt(60))","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1def42clt9","title":"App Engagement","body":"The mean number of minutes for app engagement by a tablet user is $$8.2$$ minutes. Suppose the standard deviation is one minute. Take a sample of $$60$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 The Central Limit Theorem for Sample Means (Averages)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a1def42clt9a","stepAnswer":["$$0.9293$$"],"problemType":"TextBox","stepTitle":"Find the probability that the sample mean is between eight minutes and $$8.5$$ minutes.","stepBody":"Round to four decimal places","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.9293$$","hints":{"DefaultPathway":[{"id":"a1def42clt9a-h1","type":"hint","dependencies":[],"title":"Using a calculator","text":"Using normalcdf","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt9a-h2","type":"hint","dependencies":["a1def42clt9a-h1"],"title":"First parameter","text":"Lower Value","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a1def42clt9a-h2"],"title":"What is the lower value?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt9a-h4","type":"hint","dependencies":["a1def42clt9a-h3"],"title":"Second parameter","text":"Upper Value","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt9a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8.5$$"],"dependencies":["a1def42clt9a-h4"],"title":"What is the upper value?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt9a-h6","type":"hint","dependencies":["a1def42clt9a-h5"],"title":"Third parameter","text":"Mean","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt9a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8.2$$"],"dependencies":["a1def42clt9a-h6"],"title":"What is the mean?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt9a-h8","type":"hint","dependencies":["a1def42clt9a-h7"],"title":"Fourth parameter","text":"Standard error of the mean","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1def42clt9a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{\\\\sqrt{60}}$$"],"dependencies":["a1def42clt9a-h8"],"title":"What is the standard error of the mean?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1ee7a8quadratics1","title":"Solve the quadratic","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Solve Quadratic Equations in Quadratic Form","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1ee7a8quadratics1a","stepAnswer":["$$x=\\\\pm \\\\sqrt{3}$$, $$x=\\\\pm 2$$"],"problemType":"MultipleChoice","stepTitle":"$$x^4-{\\\\left(7x\\\\right)}^2+12=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=\\\\pm \\\\sqrt{3}$$, $$x=\\\\pm 2$$","choices":["$$x=\\\\pm \\\\sqrt{3}$$, $$x=\\\\pm 2$$","$$x=\\\\pm \\\\sqrt{5}$$, $$x=\\\\pm \\\\sqrt{3}$$","$$x=\\\\pm \\\\sqrt{2}$$, $$x=\\\\pm 1$$","$$x=\\\\pm \\\\sqrt{7}$$, $$x=\\\\pm \\\\sqrt{5}$$"],"hints":{"DefaultPathway":[{"id":"a1ee7a8quadratics1a-h1","type":"hint","dependencies":[],"title":"Set a placeholder variable","text":"Since $${\\\\left(x^2\\\\right)}^2=x^4$$, we can let $$u=x^2$$. Substitute $$x^2$$ for u throughout the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics1a-h2","type":"hint","dependencies":["a1ee7a8quadratics1a-h1"],"title":"Factor","text":"Now, factor the equation so that it is in the form (au~b)(cu~d).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics1a-h3","type":"hint","dependencies":["a1ee7a8quadratics1a-h2"],"title":"Use the Zero Product Property","text":"Solve for the variable u by setting each of the two terms equal to zero. Then, once you have found a value for u, plug in $$x^2$$ for u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics1a-h4","type":"hint","dependencies":["a1ee7a8quadratics1a-h3"],"title":"Solve for X","text":"Solve for $$x$$ by taking the square root of both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics1a-h5","type":"hint","dependencies":["a1ee7a8quadratics1a-h4"],"title":"Check","text":"Check all $$4$$ $$x$$ values(including both positive and negative values) you just solved for by plugging them back into the original equation. If the value is correct, the equation will be equal on both sides after solving.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1ee7a8quadratics10","title":"Solve the quadratic","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Solve Quadratic Equations in Quadratic Form","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1ee7a8quadratics10a","stepAnswer":["$$x=7$$, $$x=-8$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(x+2\\\\right)}^2-3\\\\left(x+2\\\\right)-54=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=7$$, $$x=-8$$","choices":["$$x=7$$, $$x=-8$$","$$x=3$$, $$x=-4$$","$$x=-7$$, $$x=1$$","$$x=5$$, $$x=6$$"],"hints":{"DefaultPathway":[{"id":"a1ee7a8quadratics10a-h1","type":"hint","dependencies":[],"title":"Set a placeholder variable","text":"You can use a placeholder variable to simplify this problem. Set $$u=x+2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics10a-h2","type":"hint","dependencies":["a1ee7a8quadratics10a-h1"],"title":"Factor","text":"Now, factor the equation so that it is in the form (au~b)(cu~d).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics10a-h3","type":"hint","dependencies":["a1ee7a8quadratics10a-h2"],"title":"Use the Zero Product Property","text":"Solve for the variable u by setting each of the two terms equal to zero. Then, once you have found a value for u, plug in the term we substituted for u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics10a-h4","type":"hint","dependencies":["a1ee7a8quadratics10a-h3"],"title":"Solve","text":"Solve for $$x$$ by setting the equation equal to zero and solving.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics10a-h5","type":"hint","dependencies":["a1ee7a8quadratics10a-h4"],"title":"Check","text":"Check all $$x$$ values(including both positive and negative values) you just solved for by plugging them back into the original equation. If the value is correct, the equation will be equal on both sides after solving.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1ee7a8quadratics11","title":"Solve the quadratic","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Solve Quadratic Equations in Quadratic Form","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1ee7a8quadratics11a","stepAnswer":["$$y=\\\\left(-\\\\frac{5}{3}\\\\right)$$, $$y=0$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(3y+2\\\\right)}^2+3y+2-6=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=\\\\left(-\\\\frac{5}{3}\\\\right)$$, $$y=0$$","choices":["$$y=\\\\left(-\\\\frac{5}{3}\\\\right)$$, $$y=0$$","$$y=\\\\frac{4}{3}$$, $$y=5$$","$$y=\\\\frac{1}{3}$$, $$y=-2$$","$$y=3$$, $$y=2$$"],"hints":{"DefaultPathway":[{"id":"a1ee7a8quadratics11a-h1","type":"hint","dependencies":[],"title":"Set a placeholder variable","text":"You can use a placeholder variable to simplify this problem. Set $$u=3y+2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics11a-h2","type":"hint","dependencies":["a1ee7a8quadratics11a-h1"],"title":"Factor","text":"Now, factor the equation so that it is in the form (au~b)(cu~d).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics11a-h3","type":"hint","dependencies":["a1ee7a8quadratics11a-h2"],"title":"Use the Zero Product Property","text":"Solve for the variable u by setting each of the two terms equal to zero. Then, once you have found a value for u, plug in the term we substituted for u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics11a-h4","type":"hint","dependencies":["a1ee7a8quadratics11a-h3"],"title":"Solve","text":"Solve for $$y$$ by setting the equation equal to zero and solving.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics11a-h5","type":"hint","dependencies":["a1ee7a8quadratics11a-h4"],"title":"Check","text":"Check all $$y$$ values(including positive and negative values) you just solved for by plugging them back into the original equation. If the value is correct, the equation will be equal on both sides after solving.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1ee7a8quadratics12","title":"Solve the quadratic","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Solve Quadratic Equations in Quadratic Form","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1ee7a8quadratics12a","stepAnswer":["$$y=\\\\left(-\\\\frac{6}{5}\\\\right)$$, $$y=1$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(5y-1\\\\right)}^2+3\\\\left(5y-1\\\\right)-28=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=\\\\left(-\\\\frac{6}{5}\\\\right)$$, $$y=1$$","choices":["$$y=\\\\left(-\\\\frac{6}{5}\\\\right)$$, $$y=1$$","$$y=\\\\frac{4}{3}$$, $$y=5$$","$$y=\\\\frac{1}{3}$$, $$y=-2$$","$$y=3$$, $$y=2$$"],"hints":{"DefaultPathway":[{"id":"a1ee7a8quadratics12a-h1","type":"hint","dependencies":[],"title":"Set a placeholder variable","text":"You can use a placeholder variable to simplify this problem. Set $$u=(5y-1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics12a-h2","type":"hint","dependencies":["a1ee7a8quadratics12a-h1"],"title":"Factor","text":"Now, factor the equation so that it is in the form (au~b)(cu~d).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics12a-h3","type":"hint","dependencies":["a1ee7a8quadratics12a-h2"],"title":"Use the Zero Product Property","text":"Solve for the variable u by setting each of the two terms equal to zero. Then, once you have found a value for u, plug in the term we substituted for u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics12a-h4","type":"hint","dependencies":["a1ee7a8quadratics12a-h3"],"title":"Solve","text":"Solve for $$y$$ by setting the equation equal to zero and solving.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics12a-h5","type":"hint","dependencies":["a1ee7a8quadratics12a-h4"],"title":"Check","text":"Check all $$y$$ values(including positive and negative values) you just solved for by plugging them back into the original equation. If the value is correct, the equation will be equal on both sides after solving.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1ee7a8quadratics13","title":"Solve the quadratic","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Solve Quadratic Equations in Quadratic Form","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1ee7a8quadratics13a","stepAnswer":["$$x=0$$, $$x=\\\\pm \\\\sqrt{3}$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(x^2+1\\\\right)}^2-5\\\\left(x^2+1\\\\right)+4=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=0$$, $$x=\\\\pm \\\\sqrt{3}$$","choices":["$$x=\\\\pm 7$$, $$x=-8$$","$$x=0$$, $$x=\\\\pm \\\\sqrt{3}$$","$$x=-7$$, $$x=1$$","$$x=5$$, $$x=\\\\pm 6$$"],"hints":{"DefaultPathway":[{"id":"a1ee7a8quadratics13a-h1","type":"hint","dependencies":[],"title":"Set a placeholder variable","text":"You can use a placeholder variable to simplify this problem. Set $$u=x^2+1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics13a-h2","type":"hint","dependencies":["a1ee7a8quadratics13a-h1"],"title":"Factor","text":"Now, factor the equation so that it is in the form (au~b)(cu~d).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics13a-h3","type":"hint","dependencies":["a1ee7a8quadratics13a-h2"],"title":"Use the Zero Product Property","text":"Solve for the variable u by setting each of the two terms equal to zero. Then, once you have found a value for u, plug in the term we substituted for u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics13a-h4","type":"hint","dependencies":["a1ee7a8quadratics13a-h3"],"title":"Solve","text":"Solve by isolating $$x$$ and taking the square root of both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics13a-h5","type":"hint","dependencies":["a1ee7a8quadratics13a-h4"],"title":"Check","text":"Check all $$x$$ values(including both positive and negative values) you just solved for by plugging them back into the original equation. If the value is correct, the equation will be equal on both sides after solving.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1ee7a8quadratics14","title":"Solve the quadratic","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Solve Quadratic Equations in Quadratic Form","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1ee7a8quadratics14a","stepAnswer":["$$x=\\\\pm \\\\sqrt{7}$$, $$x=\\\\pm \\\\sqrt{5}$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(x^2-4\\\\right)}^2-4\\\\left(x^2-4\\\\right)+3=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=\\\\pm \\\\sqrt{7}$$, $$x=\\\\pm \\\\sqrt{5}$$","choices":["$$x=\\\\pm 7$$, $$x=-8$$","$$x=0$$, $$x=\\\\pm \\\\sqrt{3}$$","$$x=\\\\pm \\\\sqrt{7}$$, $$x=\\\\pm \\\\sqrt{5}$$","$$x=5$$, $$x=\\\\pm 6$$"],"hints":{"DefaultPathway":[{"id":"a1ee7a8quadratics14a-h1","type":"hint","dependencies":[],"title":"Set a placeholder variable","text":"You can use a placeholder variable to simplify this problem. Set $$u=x^2-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics14a-h2","type":"hint","dependencies":["a1ee7a8quadratics14a-h1"],"title":"Factor","text":"Now, factor the equation so that it is in the form (au~b)(cu~d).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics14a-h3","type":"hint","dependencies":["a1ee7a8quadratics14a-h2"],"title":"Use the Zero Product Property","text":"Solve for the variable u by setting each of the two terms equal to zero. Then, once you have found a value for u, plug in the term we substituted for u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics14a-h4","type":"hint","dependencies":["a1ee7a8quadratics14a-h3"],"title":"Solve","text":"Solve by isolating $$x$$ and taking the square root of both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics14a-h5","type":"hint","dependencies":["a1ee7a8quadratics14a-h4"],"title":"Check","text":"Check all $$x$$ values(including both positive and negative values) you just solved for by plugging them back into the original equation. If the value is correct, the equation will be equal on both sides after solving.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1ee7a8quadratics15","title":"Solve the quadratic","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Solve Quadratic Equations in Quadratic Form","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1ee7a8quadratics15a","stepAnswer":["$$x=\\\\pm \\\\left(\\\\frac{\\\\sqrt{22}}{2}\\\\right)$$, $$x=\\\\pm \\\\sqrt{7}$$"],"problemType":"MultipleChoice","stepTitle":"$$2{\\\\left(x^2-5\\\\right)}^2-5\\\\left(x^2-5\\\\right)+2=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=\\\\pm \\\\left(\\\\frac{\\\\sqrt{22}}{2}\\\\right)$$, $$x=\\\\pm \\\\sqrt{7}$$","choices":["$$x=\\\\pm 7$$, $$x=-8$$","$$x=0$$, $$x=\\\\pm \\\\sqrt{3}$$","$$x=\\\\pm \\\\sqrt{7}$$, $$x=\\\\pm \\\\sqrt{5}$$","$$x=\\\\pm \\\\left(\\\\frac{\\\\sqrt{22}}{2}\\\\right)$$, $$x=\\\\pm \\\\sqrt{7}$$"],"hints":{"DefaultPathway":[{"id":"a1ee7a8quadratics15a-h1","type":"hint","dependencies":[],"title":"Set a placeholder variable","text":"You can use a placeholder variable to simplify this problem. Set $$u=x^2-5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics15a-h2","type":"hint","dependencies":["a1ee7a8quadratics15a-h1"],"title":"Factor","text":"Now, factor the equation so that it is in the form (au~b)(cu~d).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics15a-h3","type":"hint","dependencies":["a1ee7a8quadratics15a-h2"],"title":"Use the Zero Product Property","text":"Solve for the variable u by setting each of the two terms equal to zero. Then, once you have found a value for u, plug in the term we substituted for u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics15a-h4","type":"hint","dependencies":["a1ee7a8quadratics15a-h3"],"title":"Solve","text":"Solve by isolating $$x$$ and taking the square root of both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics15a-h5","type":"hint","dependencies":["a1ee7a8quadratics15a-h4"],"title":"Check","text":"Check all $$x$$ values(including both positive and negative values) you just solved for by plugging them back into the original equation. If the value is correct, the equation will be equal on both sides after solving.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1ee7a8quadratics16","title":"Solve the quadratic","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Solve Quadratic Equations in Quadratic Form","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1ee7a8quadratics16a","stepAnswer":["$$x=\\\\pm \\\\left(\\\\frac{\\\\sqrt{13}}{2}\\\\right)$$, $$x=\\\\pm \\\\sqrt{7}$$"],"problemType":"MultipleChoice","stepTitle":"$$2{\\\\left(x^2-5\\\\right)}^2-7\\\\left(x^2-5\\\\right)+6=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=\\\\pm \\\\left(\\\\frac{\\\\sqrt{13}}{2}\\\\right)$$, $$x=\\\\pm \\\\sqrt{7}$$","choices":["$$x=\\\\pm \\\\left(\\\\frac{\\\\sqrt{13}}{2}\\\\right)$$, $$x=\\\\pm \\\\sqrt{7}$$","$$x=0$$, $$x=\\\\pm \\\\sqrt{3}$$","$$x=\\\\pm \\\\sqrt{7}$$, $$x=\\\\pm \\\\sqrt{5}$$","$$x=\\\\pm \\\\left(\\\\frac{\\\\sqrt{22}}{2}\\\\right)$$, $$x=\\\\pm \\\\sqrt{7}$$"],"hints":{"DefaultPathway":[{"id":"a1ee7a8quadratics16a-h1","type":"hint","dependencies":[],"title":"Set a placeholder variable","text":"You can use a placeholder variable to simplify this problem. Set $$u=x^2-5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics16a-h2","type":"hint","dependencies":["a1ee7a8quadratics16a-h1"],"title":"Factor","text":"Now, factor the equation so that it is in the form (au~b)(cu~d).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics16a-h3","type":"hint","dependencies":["a1ee7a8quadratics16a-h2"],"title":"Use the Zero Product Property","text":"Solve for the variable u by setting each of the two terms equal to zero. Then, once you have found a value for u, plug in the term we substituted for u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics16a-h4","type":"hint","dependencies":["a1ee7a8quadratics16a-h3"],"title":"Solve","text":"Solve by isolating $$x$$ and taking the square root of both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics16a-h5","type":"hint","dependencies":["a1ee7a8quadratics16a-h4"],"title":"Check","text":"Check all $$x$$ values(including both positive and negative values) you just solved for by plugging them back into the original equation. If the value is correct, the equation will be equal on both sides after solving.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1ee7a8quadratics17","title":"Solve the quadratic","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Solve Quadratic Equations in Quadratic Form","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1ee7a8quadratics17a","stepAnswer":["$$x=25$$"],"problemType":"MultipleChoice","stepTitle":"$$x-\\\\sqrt{x}-20=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=25$$","choices":["$$x=\\\\pm \\\\left(\\\\frac{\\\\sqrt{13}}{2}\\\\right)$$, $$x=\\\\pm \\\\sqrt{7}$$","$$x=25$$","$$x=\\\\pm \\\\sqrt{7}$$, $$x=\\\\pm \\\\sqrt{5}$$","$$x=\\\\pm \\\\left(\\\\frac{\\\\sqrt{22}}{2}\\\\right)$$, $$x=\\\\pm \\\\sqrt{7}$$"],"hints":{"DefaultPathway":[{"id":"a1ee7a8quadratics17a-h1","type":"hint","dependencies":[],"title":"Set a placeholder variable","text":"You can use a placeholder variable to simplify this problem. Set $$u=\\\\sqrt{x}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics17a-h2","type":"hint","dependencies":["a1ee7a8quadratics17a-h1"],"title":"Factor","text":"Now, factor the equation so that it is in the form (au~b)(cu~d).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics17a-h3","type":"hint","dependencies":["a1ee7a8quadratics17a-h2"],"title":"Use the Zero Product Property","text":"Solve for the variable u by setting each of the two terms equal to zero. Then, once you have found a value for u, plug in the term we substituted for u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics17a-h4","type":"hint","dependencies":["a1ee7a8quadratics17a-h3"],"title":"Solve","text":"Solve by isolating $$x$$ and squaring both sides to eliminate the radical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics17a-h5","type":"hint","dependencies":["a1ee7a8quadratics17a-h4"],"title":"Check","text":"Check all $$x$$ values(including both positive and negative values) you just solved for by plugging them back into the original equation. If the value is correct, the equation will be equal on both sides after solving.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1ee7a8quadratics18","title":"Solve the quadratic","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Solve Quadratic Equations in Quadratic Form","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1ee7a8quadratics18a","stepAnswer":["$$x=25$$, $$x=9$$"],"problemType":"MultipleChoice","stepTitle":"$$x-8\\\\sqrt{x}+15=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=25$$, $$x=9$$","choices":["$$x=16$$, $$x=9$$","$$x=25$$","$$x=\\\\pm \\\\sqrt{7}$$, $$x=\\\\pm \\\\sqrt{5}$$","$$x=25$$, $$x=9$$"],"hints":{"DefaultPathway":[{"id":"a1ee7a8quadratics18a-h1","type":"hint","dependencies":[],"title":"Set a placeholder variable","text":"You can use a placeholder variable to simplify this problem. Set $$u=\\\\sqrt{x}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics18a-h2","type":"hint","dependencies":["a1ee7a8quadratics18a-h1"],"title":"Factor","text":"Now, factor the equation so that it is in the form (au~b)(cu~d).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics18a-h3","type":"hint","dependencies":["a1ee7a8quadratics18a-h2"],"title":"Use the Zero Product Property","text":"Solve for the variable u by setting each of the two terms equal to zero. Then, once you have found a value for u, plug in the term we substituted for u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics18a-h4","type":"hint","dependencies":["a1ee7a8quadratics18a-h3"],"title":"Solve","text":"Solve by isolating $$x$$ and squaring both sides to eliminate the radical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics18a-h5","type":"hint","dependencies":["a1ee7a8quadratics18a-h4"],"title":"Check","text":"Check all $$x$$ values(including both positive and negative values) you just solved for by plugging them back into the original equation. If the value is correct, the equation will be equal on both sides after solving.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1ee7a8quadratics19","title":"Solve the quadratic","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Solve Quadratic Equations in Quadratic Form","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1ee7a8quadratics19a","stepAnswer":["$$x=4$$"],"problemType":"MultipleChoice","stepTitle":"$$x+6\\\\sqrt{x}-16=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=4$$","choices":["$$x=16$$, $$x=9$$","$$x=25$$","$$x=4$$","$$x=25$$, $$x=9$$"],"hints":{"DefaultPathway":[{"id":"a1ee7a8quadratics19a-h1","type":"hint","dependencies":[],"title":"Set a placeholder variable","text":"You can use a placeholder variable to simplify this problem. Set $$u=\\\\sqrt{x}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics19a-h2","type":"hint","dependencies":["a1ee7a8quadratics19a-h1"],"title":"Factor","text":"Now, factor the equation so that it is in the form (au~b)(cu~d).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics19a-h3","type":"hint","dependencies":["a1ee7a8quadratics19a-h2"],"title":"Use the Zero Product Property","text":"Solve for the variable u by setting each of the two terms equal to zero. Then, once you have found a value for u, plug in the term we substituted for u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics19a-h4","type":"hint","dependencies":["a1ee7a8quadratics19a-h3"],"title":"Solve","text":"Solve by isolating $$x$$ and squaring both sides to eliminate the radical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics19a-h5","type":"hint","dependencies":["a1ee7a8quadratics19a-h4"],"title":"Check","text":"Check all $$x$$ values(including both positive and negative values) you just solved for by plugging them back into the original equation. If the value is correct, the equation will be equal on both sides after solving.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1ee7a8quadratics2","title":"Solve the quadratic","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Solve Quadratic Equations in Quadratic Form","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1ee7a8quadratics2a","stepAnswer":["$$x=\\\\pm \\\\sqrt{3}$$, $$x=\\\\sqrt{6}$$"],"problemType":"MultipleChoice","stepTitle":"(x**4)-((9*x**2)+18=0","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=\\\\pm \\\\sqrt{3}$$, $$x=\\\\sqrt{6}$$","choices":["$$x=\\\\pm \\\\sqrt{3}$$, $$x=\\\\pm 2$$","$$x=\\\\pm \\\\sqrt{3}$$, $$x=\\\\sqrt{6}$$","$$x=\\\\pm \\\\sqrt{2}$$, $$x=\\\\pm 1$$","$$x=\\\\pm \\\\sqrt{7}$$, $$x=\\\\pm \\\\sqrt{5}$$"],"hints":{"DefaultPathway":[{"id":"a1ee7a8quadratics2a-h1","type":"hint","dependencies":[],"title":"Set a placeholder variable","text":"Since $${\\\\left(x^2\\\\right)}^2=x^4$$, we can let $$u=x^2$$. Substitute $$x^2$$ for u throughout the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics2a-h2","type":"hint","dependencies":["a1ee7a8quadratics2a-h1"],"title":"Factor","text":"Now, factor the equation so that it is in the form (au~b)(cu~d).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics2a-h3","type":"hint","dependencies":["a1ee7a8quadratics2a-h2"],"title":"Use the Zero Product Property","text":"Solve for the variable u by setting each of the two terms equal to zero. Then, once you have found a value for u, plug in $$x^2$$ for u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics2a-h4","type":"hint","dependencies":["a1ee7a8quadratics2a-h3"],"title":"Solve for X","text":"Solve for $$x$$ by taking the square root of both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics2a-h5","type":"hint","dependencies":["a1ee7a8quadratics2a-h4"],"title":"Check","text":"Check all $$4$$ $$x$$ values(including both positive and negative values) you just solved for by plugging them back into the original equation. If the value is correct, the equation will be equal on both sides after solving. Remove any solutions which do not satisfy the original equation from your final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1ee7a8quadratics20","title":"Solve the quadratic","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Solve Quadratic Equations in Quadratic Form","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1ee7a8quadratics20a","stepAnswer":["$$x=4$$"],"problemType":"MultipleChoice","stepTitle":"$$x+6\\\\sqrt{x}-16=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=4$$","choices":["$$x=16$$, $$x=9$$","$$x=25$$","$$x=4$$","$$x=25$$, $$x=9$$"],"hints":{"DefaultPathway":[{"id":"a1ee7a8quadratics20a-h1","type":"hint","dependencies":[],"title":"Set a placeholder variable","text":"You can use a placeholder variable to simplify this problem. Set $$u=\\\\sqrt{x}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics20a-h2","type":"hint","dependencies":["a1ee7a8quadratics20a-h1"],"title":"Factor","text":"Now, factor the equation so that it is in the form (au~b)(cu~d).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics20a-h3","type":"hint","dependencies":["a1ee7a8quadratics20a-h2"],"title":"Use the Zero Product Property","text":"Solve for the variable u by setting each of the two terms equal to zero. Then, once you have found a value for u, plug in the term we substituted for u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics20a-h4","type":"hint","dependencies":["a1ee7a8quadratics20a-h3"],"title":"Solve","text":"Solve by isolating $$x$$ and squaring both sides to eliminate the radical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics20a-h5","type":"hint","dependencies":["a1ee7a8quadratics20a-h4"],"title":"Check","text":"Check all $$x$$ values(including both positive and negative values) you just solved for by plugging them back into the original equation. If the value is correct, the equation will be equal on both sides after solving.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1ee7a8quadratics21","title":"Solve the quadratic","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Solve Quadratic Equations in Quadratic Form","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1ee7a8quadratics21a","stepAnswer":["$$x=9$$"],"problemType":"MultipleChoice","stepTitle":"$$x+4\\\\sqrt{x}-21=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=9$$","choices":["$$x=16$$, $$x=9$$","$$x=25$$","$$x=9$$","$$x=25$$, $$x=9$$"],"hints":{"DefaultPathway":[{"id":"a1ee7a8quadratics21a-h1","type":"hint","dependencies":[],"title":"Set a placeholder variable","text":"You can use a placeholder variable to simplify this problem. Set $$u=\\\\sqrt{x}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics21a-h2","type":"hint","dependencies":["a1ee7a8quadratics21a-h1"],"title":"Factor","text":"Now, factor the equation so that it is in the form (au~b)(cu~d).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics21a-h3","type":"hint","dependencies":["a1ee7a8quadratics21a-h2"],"title":"Use the Zero Product Property","text":"Solve for the variable u by setting each of the two terms equal to zero. Then, once you have found a value for u, plug in the term we substituted for u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics21a-h4","type":"hint","dependencies":["a1ee7a8quadratics21a-h3"],"title":"Solve","text":"Solve by isolating $$x$$ and squaring both sides to eliminate the radical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics21a-h5","type":"hint","dependencies":["a1ee7a8quadratics21a-h4"],"title":"Check","text":"Check all $$x$$ values(including both positive and negative values) you just solved for by plugging them back into the original equation. If the value is correct, the equation will be equal on both sides after solving.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1ee7a8quadratics22","title":"Solve the quadratic","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Solve Quadratic Equations in Quadratic Form","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1ee7a8quadratics22a","stepAnswer":["$$x=\\\\frac{1}{4}$$"],"problemType":"MultipleChoice","stepTitle":"$$6x+\\\\sqrt{x}-2=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=\\\\frac{1}{4}$$","choices":["$$x=\\\\frac{1}{4}$$","$$x=25$$","$$x=9$$","$$x=25$$, $$x=9$$"],"hints":{"DefaultPathway":[{"id":"a1ee7a8quadratics22a-h1","type":"hint","dependencies":[],"title":"Set a placeholder variable","text":"You can use a placeholder variable to simplify this problem. Set $$u=\\\\sqrt{x}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics22a-h2","type":"hint","dependencies":["a1ee7a8quadratics22a-h1"],"title":"Factor","text":"Now, factor the equation so that it is in the form (au~b)(cu~d).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics22a-h3","type":"hint","dependencies":["a1ee7a8quadratics22a-h2"],"title":"Use the Zero Product Property","text":"Solve for the variable u by setting each of the two terms equal to zero. Then, once you have found a value for u, plug in the term we substituted for u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics22a-h4","type":"hint","dependencies":["a1ee7a8quadratics22a-h3"],"title":"Solve","text":"Solve by isolating $$x$$ and squaring both sides to eliminate the radical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics22a-h5","type":"hint","dependencies":["a1ee7a8quadratics22a-h4"],"title":"Check","text":"Check all $$x$$ values(including both positive and negative values) you just solved for by plugging them back into the original equation. If the value is correct, the equation will be equal on both sides after solving.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1ee7a8quadratics23","title":"Solve the quadratic","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Solve Quadratic Equations in Quadratic Form","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1ee7a8quadratics23a","stepAnswer":["$$x=\\\\frac{1}{9}$$"],"problemType":"MultipleChoice","stepTitle":"$$6x+\\\\sqrt{x}-1=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=\\\\frac{1}{9}$$","choices":["$$x=\\\\frac{1}{4}$$","$$x=25$$","$$x=\\\\frac{1}{9}$$","$$x=25$$, $$x=9$$"],"hints":{"DefaultPathway":[{"id":"a1ee7a8quadratics23a-h1","type":"hint","dependencies":[],"title":"Set a placeholder variable","text":"You can use a placeholder variable to simplify this problem. Set $$u=\\\\sqrt{x}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics23a-h2","type":"hint","dependencies":["a1ee7a8quadratics23a-h1"],"title":"Factor","text":"Now, factor the equation so that it is in the form (au~b)(cu~d).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics23a-h3","type":"hint","dependencies":["a1ee7a8quadratics23a-h2"],"title":"Use the Zero Product Property","text":"Solve for the variable u by setting each of the two terms equal to zero. Then, once you have found a value for u, plug in the term we substituted for u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics23a-h4","type":"hint","dependencies":["a1ee7a8quadratics23a-h3"],"title":"Solve","text":"Solve by isolating $$x$$ and squaring both sides to eliminate the radical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics23a-h5","type":"hint","dependencies":["a1ee7a8quadratics23a-h4"],"title":"Check","text":"Check all $$x$$ values(including both positive and negative values) you just solved for by plugging them back into the original equation. If the value is correct, the equation will be equal on both sides after solving.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1ee7a8quadratics24","title":"Solve the quadratic","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Solve Quadratic Equations in Quadratic Form","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1ee7a8quadratics24a","stepAnswer":["$$x=\\\\frac{1}{25}$$, $$x=\\\\frac{9}{4}$$"],"problemType":"MultipleChoice","stepTitle":"$$10x-17\\\\sqrt{x}+3=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=\\\\frac{1}{25}$$, $$x=\\\\frac{9}{4}$$","choices":["$$x=\\\\frac{1}{4}$$","$$x=\\\\frac{1}{25}$$, $$x=\\\\frac{9}{4}$$","$$x=\\\\frac{1}{9}$$","$$x=25$$, $$x=9$$"],"hints":{"DefaultPathway":[{"id":"a1ee7a8quadratics24a-h1","type":"hint","dependencies":[],"title":"Set a placeholder variable","text":"You can use a placeholder variable to simplify this problem. Set $$u=\\\\sqrt{x}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics24a-h2","type":"hint","dependencies":["a1ee7a8quadratics24a-h1"],"title":"Factor","text":"Now, factor the equation so that it is in the form (au~b)(cu~d).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics24a-h3","type":"hint","dependencies":["a1ee7a8quadratics24a-h2"],"title":"Use the Zero Product Property","text":"Solve for the variable u by setting each of the two terms equal to zero. Then, once you have found a value for u, plug in the term we substituted for u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics24a-h4","type":"hint","dependencies":["a1ee7a8quadratics24a-h3"],"title":"Solve","text":"Solve by isolating $$x$$ and squaring both sides to eliminate the radical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics24a-h5","type":"hint","dependencies":["a1ee7a8quadratics24a-h4"],"title":"Check","text":"Check all $$x$$ values(including both positive and negative values) you just solved for by plugging them back into the original equation. If the value is correct, the equation will be equal on both sides after solving.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1ee7a8quadratics25","title":"Solve the quadratic","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Solve Quadratic Equations in Quadratic Form","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1ee7a8quadratics25a","stepAnswer":["$$x=\\\\frac{1}{9}$$"],"problemType":"MultipleChoice","stepTitle":"$$12x+5\\\\sqrt{x}-3=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=\\\\frac{1}{9}$$","choices":["$$x=\\\\frac{1}{4}$$","$$x=\\\\frac{1}{25}$$, $$x=\\\\frac{9}{4}$$","$$x=\\\\frac{1}{9}$$","$$x=25$$, $$x=9$$"],"hints":{"DefaultPathway":[{"id":"a1ee7a8quadratics25a-h1","type":"hint","dependencies":[],"title":"Set a placeholder variable","text":"You can use a placeholder variable to simplify this problem. Set $$u=\\\\sqrt{x}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics25a-h2","type":"hint","dependencies":["a1ee7a8quadratics25a-h1"],"title":"Factor","text":"Now, factor the equation so that it is in the form (au~b)(cu~d).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics25a-h3","type":"hint","dependencies":["a1ee7a8quadratics25a-h2"],"title":"Use the Zero Product Property","text":"Solve for the variable u by setting each of the two terms equal to zero. Then, once you have found a value for u, plug in the term we substituted for u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics25a-h4","type":"hint","dependencies":["a1ee7a8quadratics25a-h3"],"title":"Solve","text":"Solve by isolating $$x$$ and squaring both sides to eliminate the radical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics25a-h5","type":"hint","dependencies":["a1ee7a8quadratics25a-h4"],"title":"Check","text":"Check all $$x$$ values(including both positive and negative values) you just solved for by plugging them back into the original equation. If the value is correct, the equation will be equal on both sides after solving.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1ee7a8quadratics26","title":"Solve the quadratic","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Solve Quadratic Equations in Quadratic Form","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1ee7a8quadratics26a","stepAnswer":["$$x=-1$$, $$x=-512$$"],"problemType":"MultipleChoice","stepTitle":"$$x^{\\\\frac{2}{3}}+{\\\\left(9x\\\\right)}^{\\\\frac{1}{3}}+8=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=-1$$, $$x=-512$$","choices":["$$x=7$$, $$x=-8$$","$$x=3$$, $$x=-4$$","$$x=-7$$, $$x=1$$","$$x=-1$$, $$x=-512$$"],"hints":{"DefaultPathway":[{"id":"a1ee7a8quadratics26a-h1","type":"hint","dependencies":[],"title":"Set a placeholder variable","text":"You can use a placeholder variable to simplify this problem. Set $$u=x^{\\\\frac{1}{3}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics26a-h2","type":"hint","dependencies":["a1ee7a8quadratics26a-h1"],"title":"Factor","text":"Now, factor the equation so that it is in the form (au~b)(cu~d).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics26a-h3","type":"hint","dependencies":["a1ee7a8quadratics26a-h2"],"title":"Use the Zero Product Property","text":"Solve for the variable u by setting each of the two terms equal to zero. Then, once you have found a value for u, plug in the term we substituted for u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics26a-h4","type":"hint","dependencies":["a1ee7a8quadratics26a-h3"],"title":"Solve","text":"Solve by isolating $$x$$ and exponentiating both sides by the power of $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics26a-h5","type":"hint","dependencies":["a1ee7a8quadratics26a-h4"],"title":"Check","text":"Check all $$x$$ values(including both positive and negative values) you just solved for by plugging them back into the original equation. If the value is correct, the equation will be equal on both sides after solving.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1ee7a8quadratics27","title":"Solve the quadratic","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Solve Quadratic Equations in Quadratic Form","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1ee7a8quadratics27a","stepAnswer":["$$x=-64$$, $$x=343$$"],"problemType":"MultipleChoice","stepTitle":"$$x^{\\\\frac{2}{3}}-{\\\\left(3x\\\\right)}^{\\\\frac{1}{3}}=28$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=-64$$, $$x=343$$","choices":["$$x=-64$$, $$x=343$$","$$x=3$$, $$x=-4$$","$$x=-7$$, $$x=1$$","$$x=-1$$, $$x=-512$$"],"hints":{"DefaultPathway":[{"id":"a1ee7a8quadratics27a-h1","type":"hint","dependencies":[],"title":"Set a placeholder variable","text":"You can use a placeholder variable to simplify this problem. Set $$u=x^{\\\\frac{1}{3}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics27a-h2","type":"hint","dependencies":["a1ee7a8quadratics27a-h1"],"title":"Factor","text":"Now, move everything to one side so the entire equation is set equal to $$0$$ and factor the equation so that it is in the form (au~b)(cu~d).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics27a-h3","type":"hint","dependencies":["a1ee7a8quadratics27a-h2"],"title":"Use the Zero Product Property","text":"Solve for the variable u by setting each of the two terms equal to zero. Then, once you have found a value for u, plug in the term we substituted for u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics27a-h4","type":"hint","dependencies":["a1ee7a8quadratics27a-h3"],"title":"Solve","text":"Solve by isolating $$x$$ and exponentiating both sides by the power of $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics27a-h5","type":"hint","dependencies":["a1ee7a8quadratics27a-h4"],"title":"Check","text":"Check all $$x$$ values(including both positive and negative values) you just solved for by plugging them back into the original equation. If the value is correct, the equation will be equal on both sides after solving.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1ee7a8quadratics28","title":"Solve the quadratic","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Solve Quadratic Equations in Quadratic Form","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1ee7a8quadratics28a","stepAnswer":["$$x=8$$, $$x=-216$$"],"problemType":"MultipleChoice","stepTitle":"$$x^{\\\\frac{2}{3}}+{\\\\left(4x\\\\right)}^{\\\\frac{1}{3}}=12$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=8$$, $$x=-216$$","choices":["$$x=-64$$, $$x=343$$","$$x=8$$, $$x=-216$$","$$x=-7$$, $$x=1$$","$$x=-1$$, $$x=-512$$"],"hints":{"DefaultPathway":[{"id":"a1ee7a8quadratics28a-h1","type":"hint","dependencies":[],"title":"Set a placeholder variable","text":"You can use a placeholder variable to simplify this problem. Set $$u=x^{\\\\frac{1}{3}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics28a-h2","type":"hint","dependencies":["a1ee7a8quadratics28a-h1"],"title":"Factor","text":"Now, move everything to one side so the entire equation is set equal to $$0$$ and factor the equation so that it is in the form (au~b)(cu~d).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics28a-h3","type":"hint","dependencies":["a1ee7a8quadratics28a-h2"],"title":"Use the Zero Product Property","text":"Solve for the variable u by setting each of the two terms equal to zero. Then, once you have found a value for u, plug in the term we substituted for u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics28a-h4","type":"hint","dependencies":["a1ee7a8quadratics28a-h3"],"title":"Solve","text":"Solve by isolating $$x$$ and exponentiating both sides by the power of $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics28a-h5","type":"hint","dependencies":["a1ee7a8quadratics28a-h4"],"title":"Check","text":"Check all $$x$$ values(including both positive and negative values) you just solved for by plugging them back into the original equation. If the value is correct, the equation will be equal on both sides after solving.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1ee7a8quadratics29","title":"Solve the quadratic","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Solve Quadratic Equations in Quadratic Form","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1ee7a8quadratics29a","stepAnswer":["$$x=125$$, $$x=216$$"],"problemType":"MultipleChoice","stepTitle":"$$x^{\\\\frac{2}{3}}-{\\\\left(11x\\\\right)}^{\\\\frac{1}{3}}+30=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=125$$, $$x=216$$","choices":["$$x=-64$$, $$x=343$$","$$x=8$$, $$x=-216$$","$$x=125$$, $$x=216$$","$$x=-1$$, $$x=-512$$"],"hints":{"DefaultPathway":[{"id":"a1ee7a8quadratics29a-h1","type":"hint","dependencies":[],"title":"Set a placeholder variable","text":"You can use a placeholder variable to simplify this problem. Set $$u=x^{\\\\frac{1}{3}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics29a-h2","type":"hint","dependencies":["a1ee7a8quadratics29a-h1"],"title":"Factor","text":"Now, move everything to one side so the entire equation is set equal to $$0$$ and factor the equation so that it is in the form (au~b)(cu~d).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics29a-h3","type":"hint","dependencies":["a1ee7a8quadratics29a-h2"],"title":"Use the Zero Product Property","text":"Solve for the variable u by setting each of the two terms equal to zero. Then, once you have found a value for u, plug in the term we substituted for u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics29a-h4","type":"hint","dependencies":["a1ee7a8quadratics29a-h3"],"title":"Solve","text":"Solve by isolating $$x$$ and exponentiating both sides by the power of $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics29a-h5","type":"hint","dependencies":["a1ee7a8quadratics29a-h4"],"title":"Check","text":"Check all $$x$$ values(including both positive and negative values) you just solved for by plugging them back into the original equation. If the value is correct, the equation will be equal on both sides after solving.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1ee7a8quadratics3","title":"Solve the quadratic","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Solve Quadratic Equations in Quadratic Form","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1ee7a8quadratics3a","stepAnswer":["$$x=\\\\pm \\\\sqrt{15}$$, $$x=\\\\pm \\\\sqrt{2} i$$"],"problemType":"MultipleChoice","stepTitle":"$$x^4-{\\\\left(13x\\\\right)}^2-30=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=\\\\pm \\\\sqrt{15}$$, $$x=\\\\pm \\\\sqrt{2} i$$","choices":["$$x=\\\\pm \\\\sqrt{3}$$, $$x=\\\\pm 2$$","$$x=\\\\pm \\\\sqrt{3}$$, $$x=\\\\sqrt{6}$$","$$x=\\\\pm \\\\sqrt{15}$$, $$x=\\\\pm \\\\sqrt{2} i$$","$$x=\\\\pm \\\\sqrt{7}$$, $$x=\\\\pm \\\\sqrt{5}$$"],"hints":{"DefaultPathway":[{"id":"a1ee7a8quadratics3a-h1","type":"hint","dependencies":[],"title":"Set a placeholder variable","text":"Since $${\\\\left(x^2\\\\right)}^2=x^4$$, we can let $$u=x^2$$. Substitute $$x^2$$ for u throughout the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics3a-h2","type":"hint","dependencies":["a1ee7a8quadratics3a-h1"],"title":"Factor","text":"Now, factor the equation so that it is in the form (au~b)(cu~d).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics3a-h3","type":"hint","dependencies":["a1ee7a8quadratics3a-h2"],"title":"Use the Zero Product Property","text":"Solve for the variable u by setting each of the two terms equal to zero. Then, once you have found a value for u, plug in $$x^2$$ for u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics3a-h4","type":"hint","dependencies":["a1ee7a8quadratics3a-h3"],"title":"Solve","text":"Solve for $$x$$ by taking the square root of both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics3a-h5","type":"hint","dependencies":["a1ee7a8quadratics3a-h4"],"title":"Check","text":"Check all $$4$$ $$x$$ values(including both positive and negative values) you just solved for by plugging them back into the original equation. If the value is correct, the equation will be equal on both sides after solving. Remove any solutions which do not satisfy the original equation from your final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1ee7a8quadratics30","title":"Solve the quadratic","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Solve Quadratic Equations in Quadratic Form","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1ee7a8quadratics30a","stepAnswer":["$$x=\\\\frac{27}{8}$$, $$x=\\\\frac{-64}{27}$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(6x\\\\right)}^{\\\\frac{2}{3}}-x^{\\\\frac{1}{3}}=12$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=\\\\frac{27}{8}$$, $$x=\\\\frac{-64}{27}$$","choices":["$$x=-64$$, $$x=343$$","$$x=8$$, $$x=-216$$","$$x=\\\\frac{27}{8}$$, $$x=\\\\frac{-64}{27}$$","$$x=-1$$, $$x=-512$$"],"hints":{"DefaultPathway":[{"id":"a1ee7a8quadratics30a-h1","type":"hint","dependencies":[],"title":"Set a placeholder variable","text":"You can use a placeholder variable to simplify this problem. Set $$u=x^{\\\\frac{1}{3}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics30a-h2","type":"hint","dependencies":["a1ee7a8quadratics30a-h1"],"title":"Factor","text":"Now, move everything to one side so the entire equation is set equal to $$0$$ and factor the equation so that it is in the form (au~b)(cu~d).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics30a-h3","type":"hint","dependencies":["a1ee7a8quadratics30a-h2"],"title":"Use the Zero Product Property","text":"Solve for the variable u by setting each of the two terms equal to zero. Then, once you have found a value for u, plug in the term we substituted for u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics30a-h4","type":"hint","dependencies":["a1ee7a8quadratics30a-h3"],"title":"Solve","text":"Solve by isolating $$x$$ and exponentiating both sides by the power of $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics30a-h5","type":"hint","dependencies":["a1ee7a8quadratics30a-h4"],"title":"Check","text":"Check all $$x$$ values(including both positive and negative values) you just solved for by plugging them back into the original equation. If the value is correct, the equation will be equal on both sides after solving.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1ee7a8quadratics4","title":"Solve the quadratic","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Solve Quadratic Equations in Quadratic Form","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1ee7a8quadratics4a","stepAnswer":["$$x=\\\\pm \\\\sqrt{2}$$, $$x=\\\\pm 3 i$$"],"problemType":"MultipleChoice","stepTitle":"$$x^4+{\\\\left(5x\\\\right)}^2-36=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=\\\\pm \\\\sqrt{2}$$, $$x=\\\\pm 3 i$$","choices":["$$x=\\\\pm \\\\sqrt{3}$$, $$x=\\\\pm 2$$","$$x=\\\\pm \\\\sqrt{3}$$, $$x=\\\\sqrt{6}$$","$$x=\\\\pm \\\\sqrt{15}$$, $$x=\\\\pm \\\\sqrt{2} i$$","$$x=\\\\pm \\\\sqrt{2}$$, $$x=\\\\pm 3 i$$"],"hints":{"DefaultPathway":[{"id":"a1ee7a8quadratics4a-h1","type":"hint","dependencies":[],"title":"Set a placeholder variable","text":"Since $${\\\\left(x^2\\\\right)}^2=x^4$$, we can let $$u=x^2$$. Substitute $$x^2$$ for u throughout the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics4a-h2","type":"hint","dependencies":["a1ee7a8quadratics4a-h1"],"title":"Factor","text":"Now, factor the equation so that it is in the form (au~b)(cu~d).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics4a-h3","type":"hint","dependencies":["a1ee7a8quadratics4a-h2"],"title":"Use the Zero Product Property","text":"Solve for the variable u by setting each of the two terms equal to zero. Then, once you have found a value for u, plug in $$x^2$$ for u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics4a-h4","type":"hint","dependencies":["a1ee7a8quadratics4a-h3"],"title":"Solve","text":"Solve for $$x$$ by taking the square root of both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics4a-h5","type":"hint","dependencies":["a1ee7a8quadratics4a-h4"],"title":"Check","text":"Check all $$4$$ $$x$$ values(including both positive and negative values) you just solved for by plugging them back into the original equation. If the value is correct, the equation will be equal on both sides after solving. Remove any solutions which do not satisfy the original equation from your final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1ee7a8quadratics5","title":"Solve the quadratic","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Solve Quadratic Equations in Quadratic Form","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1ee7a8quadratics5a","stepAnswer":["$$x=\\\\pm 1$$, $$x=\\\\pm \\\\left(\\\\frac{\\\\sqrt{6}}{2}\\\\right)$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(2x\\\\right)}^4-{\\\\left(5x\\\\right)}^2+3=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=\\\\pm 1$$, $$x=\\\\pm \\\\left(\\\\frac{\\\\sqrt{6}}{2}\\\\right)$$","choices":["$$x=\\\\pm 1$$, $$x=\\\\pm \\\\left(\\\\frac{\\\\sqrt{6}}{2}\\\\right)$$","$$x=\\\\pm \\\\sqrt{3}$$, $$x=\\\\sqrt{6}$$","$$x=\\\\pm \\\\sqrt{15}$$, $$x=\\\\pm \\\\sqrt{2} i$$","$$x=\\\\pm \\\\sqrt{2}$$, $$x=\\\\pm 3 i$$"],"hints":{"DefaultPathway":[{"id":"a1ee7a8quadratics5a-h1","type":"hint","dependencies":[],"title":"Set a placeholder variable","text":"Since $${\\\\left(x^2\\\\right)}^2=x^4$$, we can let $$u=x^2$$. Substitute $$x^2$$ for u throughout the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics5a-h2","type":"hint","dependencies":["a1ee7a8quadratics5a-h1"],"title":"Factor","text":"Now, factor the equation so that it is in the form (au~b)(cu~d).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics5a-h3","type":"hint","dependencies":["a1ee7a8quadratics5a-h2"],"title":"Use the Zero Product Property","text":"Solve for the variable u by setting each of the two terms equal to zero. Then, once you have found a value for u, plug in $$x^2$$ for u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics5a-h4","type":"hint","dependencies":["a1ee7a8quadratics5a-h3"],"title":"Solve","text":"Solve for $$x$$ by taking the square root of both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics5a-h5","type":"hint","dependencies":["a1ee7a8quadratics5a-h4"],"title":"Check","text":"Check all $$4$$ $$x$$ values(including both positive and negative values) you just solved for by plugging them back into the original equation. If the value is correct, the equation will be equal on both sides after solving. Remove any solutions which do not satisfy the original equation from your final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1ee7a8quadratics6","title":"Solve the quadratic","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Solve Quadratic Equations in Quadratic Form","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1ee7a8quadratics6a","stepAnswer":["$$x=\\\\frac{\\\\pm 1}{2}$$, $$x=\\\\pm 1$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(4x\\\\right)}^4-{\\\\left(5x\\\\right)}^2+1=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=\\\\frac{\\\\pm 1}{2}$$, $$x=\\\\pm 1$$","choices":["$$x=\\\\pm 1$$, $$x=\\\\pm \\\\left(\\\\frac{\\\\sqrt{6}}{2}\\\\right)$$","$$x=\\\\frac{\\\\pm 1}{2}$$, $$x=\\\\pm 1$$","$$x=\\\\pm \\\\sqrt{15}$$, $$x=\\\\pm \\\\sqrt{2} i$$","$$x=\\\\pm \\\\sqrt{2}$$, $$x=\\\\pm 3 i$$"],"hints":{"DefaultPathway":[{"id":"a1ee7a8quadratics6a-h1","type":"hint","dependencies":[],"title":"Set a placeholder variable","text":"Since $${\\\\left(x^2\\\\right)}^2=x^4$$, we can let $$u=x^2$$. Substitute $$x^2$$ for u throughout the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics6a-h2","type":"hint","dependencies":["a1ee7a8quadratics6a-h1"],"title":"Factor","text":"Now, factor the equation so that it is in the form (au~b)(cu~d).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics6a-h3","type":"hint","dependencies":["a1ee7a8quadratics6a-h2"],"title":"Use the Zero Product Property","text":"Solve for the variable u by setting each of the two terms equal to zero. Then, once you have found a value for u, plug in $$x^2$$ for u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics6a-h4","type":"hint","dependencies":["a1ee7a8quadratics6a-h3"],"title":"Solve","text":"Solve for $$x$$ by taking the square root of both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics6a-h5","type":"hint","dependencies":["a1ee7a8quadratics6a-h4"],"title":"Check","text":"Check all $$4$$ $$x$$ values(including both positive and negative values) you just solved for by plugging them back into the original equation. If the value is correct, the equation will be equal on both sides after solving.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1ee7a8quadratics7","title":"Solve the quadratic","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Solve Quadratic Equations in Quadratic Form","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1ee7a8quadratics7a","stepAnswer":["$$x=\\\\pm \\\\sqrt{3}$$, $$x=\\\\pm \\\\left(\\\\frac{\\\\sqrt{2}}{2}\\\\right)$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(2x\\\\right)}^4-{\\\\left(7x\\\\right)}^2+3=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=\\\\pm \\\\sqrt{3}$$, $$x=\\\\pm \\\\left(\\\\frac{\\\\sqrt{2}}{2}\\\\right)$$","choices":["$$x=\\\\pm 1$$, $$x=\\\\pm \\\\left(\\\\frac{\\\\sqrt{6}}{2}\\\\right)$$","$$x=\\\\frac{\\\\pm 1}{2}$$, $$x=\\\\pm 1$$","$$x=\\\\pm \\\\sqrt{3}$$, $$x=\\\\pm \\\\left(\\\\frac{\\\\sqrt{2}}{2}\\\\right)$$","$$x=\\\\pm \\\\sqrt{2}$$, $$x=\\\\pm 3 i$$"],"hints":{"DefaultPathway":[{"id":"a1ee7a8quadratics7a-h1","type":"hint","dependencies":[],"title":"Set a placeholder variable","text":"Since $${\\\\left(x^2\\\\right)}^2=x^4$$, we can let $$u=x^2$$. Substitute $$x^2$$ for u throughout the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics7a-h2","type":"hint","dependencies":["a1ee7a8quadratics7a-h1"],"title":"Factor","text":"Now, factor the equation so that it is in the form (au~b)(cu~d).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics7a-h3","type":"hint","dependencies":["a1ee7a8quadratics7a-h2"],"title":"Use the Zero Product Property","text":"Solve for the variable u by setting each of the two terms equal to zero. Then, once you have found a value for u, plug in $$x^2$$ for u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics7a-h4","type":"hint","dependencies":["a1ee7a8quadratics7a-h3"],"title":"Solve","text":"Solve for $$x$$ by taking the square root of both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics7a-h5","type":"hint","dependencies":["a1ee7a8quadratics7a-h4"],"title":"Check","text":"Check all $$4$$ $$x$$ values(including both positive and negative values) you just solved for by plugging them back into the original equation. If the value is correct, the equation will be equal on both sides after solving.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1ee7a8quadratics8","title":"Solve the quadratic","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Solve Quadratic Equations in Quadratic Form","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1ee7a8quadratics8a","stepAnswer":["$$x=\\\\pm \\\\sqrt{\\\\frac{2}{3}}$$, $$x=\\\\pm 2$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(3x\\\\right)}^4-{\\\\left(14x\\\\right)}^2+8=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=\\\\pm \\\\sqrt{\\\\frac{2}{3}}$$, $$x=\\\\pm 2$$","choices":["$$x=\\\\pm 1$$, $$x=\\\\pm \\\\left(\\\\frac{\\\\sqrt{6}}{2}\\\\right)$$","$$x=\\\\frac{\\\\pm 1}{2}$$, $$x=\\\\pm 1$$","$$x=\\\\pm \\\\sqrt{3}$$, $$x=\\\\pm \\\\left(\\\\frac{\\\\sqrt{2}}{2}\\\\right)$$","$$x=\\\\pm \\\\sqrt{\\\\frac{2}{3}}$$, $$x=\\\\pm 2$$"],"hints":{"DefaultPathway":[{"id":"a1ee7a8quadratics8a-h1","type":"hint","dependencies":[],"title":"Set a placeholder variable","text":"Since $${\\\\left(x^2\\\\right)}^2=x^4$$, we can let $$u=x^2$$. Substitute $$x^2$$ for u throughout the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics8a-h2","type":"hint","dependencies":["a1ee7a8quadratics8a-h1"],"title":"Factor","text":"Now, factor the equation so that it is in the form (au~b)(cu~d).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics8a-h3","type":"hint","dependencies":["a1ee7a8quadratics8a-h2"],"title":"Use the Zero Product Property","text":"Solve for the variable u by setting each of the two terms equal to zero. Then, once you have found a value for u, plug in $$x^2$$ for u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics8a-h4","type":"hint","dependencies":["a1ee7a8quadratics8a-h3"],"title":"Solve","text":"Solve for $$x$$ by taking the square root of both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics8a-h5","type":"hint","dependencies":["a1ee7a8quadratics8a-h4"],"title":"Check","text":"Check all $$4$$ $$x$$ values(including both positive and negative values) you just solved for by plugging them back into the original equation. If the value is correct, the equation will be equal on both sides after solving.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1ee7a8quadratics9","title":"Solve the quadratic","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Solve Quadratic Equations in Quadratic Form","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1ee7a8quadratics9a","stepAnswer":["$$x=-1$$, $$x=12$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(x-3\\\\right)}^2-5\\\\left(x-3\\\\right)-36=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=-1$$, $$x=12$$","choices":["$$x=-1$$, $$x=12$$","$$x=3$$, $$x=-4$$","$$x=-7$$, $$x=1$$","$$x=5$$, $$x=6$$"],"hints":{"DefaultPathway":[{"id":"a1ee7a8quadratics9a-h1","type":"hint","dependencies":[],"title":"Set a placeholder variable","text":"You can use a placeholder variable to simplify this problem. Set $$u=(x-3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics9a-h2","type":"hint","dependencies":["a1ee7a8quadratics9a-h1"],"title":"Factor","text":"Now, factor the equation so that it is in the form (au~b)(cu~d).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1ee7a8quadratics9a-h3","type":"hint","dependencies":["a1ee7a8quadratics9a-h2"],"title":"Use the Zero Product Property","text":"Solve for the variable u by setting each of the two terms equal to zero. 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$$\\\\frac{4\\\\times2}{{\\\\left(-2\\\\right)}^2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences19b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a1f0162sequences19b-h3"],"title":"Calculate the denominator","text":"What is $${\\\\left(-2\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences19b-h5","type":"hint","dependencies":["a1f0162sequences19b-h4"],"title":"Putting It Together","text":"Put the numerator and denominator together to create a fraction","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a1f0162sequences19c","stepAnswer":["$$\\\\frac{-12}{8}$$"],"problemType":"TextBox","stepTitle":"Write the third 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Begin with $$n=1$$ to find the first term, a1. To find the second term, a2, use all $$n$$ terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences2a-h2","type":"hint","dependencies":["a1f0162sequences2a-h1"],"title":"Finding All Needed Values","text":"Continue in the same manner until you have identified all $$n$$ terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences2a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["1,2,9,14,81,66,729,256"],"dependencies":["a1f0162sequences2a-h2"],"title":"First $$8$$ Terms of Sequence","text":"What are the first $$8$$ terms of the sequence?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["1,2,9,14,81,66,729,256","2,5,9,16,28,39,53,68","2,4,6,8,10,12,14,16","4,7,22,48,69,84,96,105"]}]}}]},{"id":"a1f0162sequences20","title":"Writing an Explicit Formula for the nth Term of a Sequence","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Sequences and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a1f0162sequences20a","stepAnswer":["$$-\\\\left({\\\\left(-9\\\\right)}^n\\\\right)$$"],"problemType":"TextBox","stepTitle":"Write an explicit formula for the nth term of the sequence. {9, $$-81$$, $$729$$, $$-6561$$, $$59049$$, ...}","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-\\\\left({\\\\left(-9\\\\right)}^n\\\\right)$$","hints":{"DefaultPathway":[{"id":"a1f0162sequences20a-h1","type":"hint","dependencies":[],"title":"Alternating Terms","text":"The terms alternate between positive and negative. Use $${\\\\left(-1\\\\right)}^n$$ to make the terms alternate.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences20a-h2","type":"hint","dependencies":["a1f0162sequences20a-h1"],"title":"Pattern Among Terms","text":"Notice that the absolute values of terms increase by a multiple of $$9$$. $$9$$ is multiplied by $$9$$ to make $$81$$, $$81$$ is multiplied by $$9$$ to make $$729$$ etc. Use $$9^n$$ to represent this.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences20a-h3","type":"hint","dependencies":["a1f0162sequences20a-h2"],"title":"Putting it together","text":"Multiply your two results together to get your answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f0162sequences21","title":"Writing the Terms of a Sequence Defined by a Recursive Formula","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Sequences and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a1f0162sequences21a","stepAnswer":["$$9$$"],"problemType":"TextBox","stepTitle":"Write the first term of the sequence defined by the recursive formula. $$a_1=9$$, $$a_n=3\\\\left(a_n-1\\\\right)-20$$ for $$a_n>1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9$$","hints":{"DefaultPathway":[{"id":"a1f0162sequences21a-h1","type":"hint","dependencies":[],"title":"Indicated Answer","text":"The first term $$a_1$$ is given in the formula.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a1f0162sequences21b","stepAnswer":["$$7$$"],"problemType":"TextBox","stepTitle":"Write the second term of the sequence defined by the recursive formula. $$a_1=9$$, $$a_n=3a_n-20$$ for $$a_n>1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7$$","hints":{"DefaultPathway":[{"id":"a1f0162sequences21b-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Replace the term $$a_n-1$$ with the value of $$a_1$$. As indicated in the formula, $$a_1=9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences21b-h2","type":"hint","dependencies":["a1f0162sequences21b-h1"],"title":"Calculation","text":"Calculate the expression $$a_2=3\\\\times9-20$$ for the answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a1f0162sequences21c","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"Write the third term of the sequence defined by the recursive formula. $$a_1=9$$, $$a_n=3a_n-20$$ for $$a_n>1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a1f0162sequences21c-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Replace the term $$a_n-1$$ with the value of $$a_2$$. As indicated in the previous problem, $$a_2=7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences21c-h2","type":"hint","dependencies":["a1f0162sequences21c-h1"],"title":"Calculation","text":"Calculate the expression $$a_3=3\\\\times7-20$$ for the answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a1f0162sequences21d","stepAnswer":["$$-17$$"],"problemType":"TextBox","stepTitle":"Write the fourth term of the sequence defined by the recursive formula. $$a_1=9$$, $$a_n=3a_n-20$$ for $$a_n>1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-17$$","hints":{"DefaultPathway":[{"id":"a1f0162sequences21d-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Replace the term $$a_n-1$$ with the value of $$a_3$$. As indicated in the previous problem, $$a_3=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences21d-h2","type":"hint","dependencies":["a1f0162sequences21d-h1"],"title":"Calculation","text":"Calculate the expression $$a_4=3\\\\times1-20$$ for the answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a1f0162sequences21e","stepAnswer":["$$-71$$"],"problemType":"TextBox","stepTitle":"Write the fifth term of the sequence defined by the recursive formula. $$a_1=9$$, $$a_n=3a_n-20$$ for $$a_n>1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-71$$","hints":{"DefaultPathway":[{"id":"a1f0162sequences21e-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Replace the term $$a_n-1$$ with the value of $$a_4$$. As indicated in the previous problem, $$a_4=-17$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences21e-h2","type":"hint","dependencies":["a1f0162sequences21e-h1"],"title":"Calculation","text":"Calculate the expression $$a_5=3\\\\left(-17\\\\right)-20$$ for the answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f0162sequences22","title":"Writing the Terms of a Sequence Using Factorials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Sequences and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a1f0162sequences22a","stepAnswer":["$$\\\\frac{5}{6}$$"],"problemType":"TextBox","stepTitle":"Write the first term of the sequence defined by the explicit formula a_n=(5n)/((n+2)!)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5}{6}$$","hints":{"DefaultPathway":[{"id":"a1f0162sequences22a-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Substitute $$n=1$$ into the formula","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences22a-h2","type":"hint","dependencies":["a1f0162sequences22a-h1"],"title":"Simplification","text":"Calculate the expression a_1=(5*1)/((1+2)!)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences22a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a1f0162sequences22a-h2"],"title":"Calculate the numerator","text":"What is $$5\\\\times1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences22a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a1f0162sequences22a-h3"],"title":"Calculate the denominator","text":"((1+2)!) is equal to 3! What is $$3\\\\times2\\\\times1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences22a-h5","type":"hint","dependencies":["a1f0162sequences22a-h4"],"title":"Putting It Together","text":"Put the numerator and denominator together to create a fraction","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a1f0162sequences22b","stepAnswer":["$$\\\\frac{5}{12}$$"],"problemType":"TextBox","stepTitle":"Write the second term of the sequence defined by the explicit formula a_n=(5n)/((n+2)!)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5}{12}$$","hints":{"DefaultPathway":[{"id":"a1f0162sequences22b-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Substitute $$n=2$$ into the formula","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences22b-h2","type":"hint","dependencies":["a1f0162sequences22b-h1"],"title":"Simplification","text":"Calculate the expression a_2=(5*2)/((2+2)!)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences22b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a1f0162sequences22b-h2"],"title":"Calculate the numerator","text":"What is $$5\\\\times2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences22b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$24$$"],"dependencies":["a1f0162sequences22b-h3"],"title":"Calculate the denominator","text":"((2+2)!) is equal to 4! What is $$4\\\\times3\\\\times2\\\\times1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences22b-h5","type":"hint","dependencies":["a1f0162sequences22b-h4"],"title":"Putting It Together","text":"Put the numerator and denominator together to create a fraction","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a1f0162sequences22c","stepAnswer":["$$\\\\frac{1}{8}$$"],"problemType":"TextBox","stepTitle":"Write the third term of the sequence defined by the explicit formula a_n=(5n)/((n+2)!)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{8}$$","hints":{"DefaultPathway":[{"id":"a1f0162sequences22c-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Substitute $$n=3$$ into the formula","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences22c-h2","type":"hint","dependencies":["a1f0162sequences22c-h1"],"title":"Simplification","text":"Calculate the expression a_3=(5*3)/((3+2)!)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences22c-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["a1f0162sequences22c-h2"],"title":"Calculate the numerator","text":"What is $$5\\\\times3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences22c-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$120$$"],"dependencies":["a1f0162sequences22c-h3"],"title":"Calculate the denominator","text":"((3+2)!) is equal to 5! What is $$5\\\\times4\\\\times3\\\\times2\\\\times1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences22c-h5","type":"hint","dependencies":["a1f0162sequences22c-h4"],"title":"Putting It Together","text":"Put the numerator and denominator together to create a fraction","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f0162sequences23","title":"Practice Writing the Terms of a Sequence Defined by a Formula","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Sequences and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a1f0162sequences23a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"Write the first term of the sequence defined by the explicit formula $$a_n=2^n-2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a1f0162sequences23a-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Substitute $$n=1$$ into the formula","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences23a-h2","type":"hint","dependencies":["a1f0162sequences23a-h1"],"title":"Simplification","text":"Calculate the expression $$2^1-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a1f0162sequences23b","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"Write the second term of the sequence defined by the explicit formula $$a_n=2^n-2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a1f0162sequences23b-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Substitute $$n=2$$ into the formula","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences23b-h2","type":"hint","dependencies":["a1f0162sequences23b-h1"],"title":"Simplification","text":"Calculate the expression $$2^2-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a1f0162sequences23c","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"Write the third term of the sequence defined by the explicit formula $$a_n=2^n-2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"a1f0162sequences23c-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Substitute $$n=3$$ into the formula","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences23c-h2","type":"hint","dependencies":["a1f0162sequences23c-h1"],"title":"Simplification","text":"Calculate the expression $$2^3-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f0162sequences24","title":"More Practice Writing the Terms of a Sequence Defined by a Formula","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Sequences and Their 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4.0>"}]}},{"id":"a1f0162sequences24b","stepAnswer":["$$\\\\frac{-16}{3}$$"],"problemType":"TextBox","stepTitle":"Write the second term of the sequence defined by the explicit formula $$a_n=\\\\frac{\\\\left(-16\\\\right)}{n+1}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-16}{3}$$","hints":{"DefaultPathway":[{"id":"a1f0162sequences24b-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Substitute $$n=2$$ into the formula","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences24b-h2","type":"hint","dependencies":["a1f0162sequences24b-h1"],"title":"Simplification","text":"Calculate the expression $$\\\\frac{\\\\left(-16\\\\right)}{2+1}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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first eight terms of the piecewise sequence.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["2,1,2/3,2/4,2/5,2/6,2/7,2/8","9,25,49,81,121,169,225,289","2,25,2/3,81,2/5,2/6,2/7,289","2,10,2/3,25,2/5,64,2/7,81"],"hints":{"DefaultPathway":[{"id":"a1f0162sequences4a-h1","type":"hint","dependencies":[],"title":"Substituting N into Formula","text":"Substitute each value of $$n$$ into the formula. Begin with $$n=1$$ to find the first term, a1. To find the second term, a2, use all $$n$$ terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences4a-h2","type":"hint","dependencies":["a1f0162sequences4a-h1"],"title":"Finding All Needed Values","text":"Continue in the same manner until you have identified all $$n$$ terms.","variabilization":{},"oer":"https://openstax.org/","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences4a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["2,25,2/3,81,2/5,2/6,2/7,289"],"dependencies":["a1f0162sequences4a-h2"],"title":"First $$8$$ Terms of Sequence","text":"What are the first $$8$$ terms of the sequence?","variabilization":{},"oer":"https://openstax.org/","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["2,1,2/3,2/4,2/5,2/6,2/7,2/8","9,25,49,81,121,169,225,289","2,25,2/3,81,2/5,2/6,2/7,289","2,10,2/3,25,2/5,64,2/7,81"]}]}}]},{"id":"a1f0162sequences5","title":"Finding First N Terms of a Sequence","body":"","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"9.1 Sequences and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a1f0162sequences5a","stepAnswer":["$$-0.6, -3, -15, -20, -375, -80, -9375, -320$$"],"problemType":"MultipleChoice","stepTitle":"For the following exercises, write the first eight terms of the piecewise sequence.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$-0.6, -3, -15, -20, -375, -80, -9375, -320$$","choices":["$$-0.6, -3, -15, -20, -375, -80, -9375, -320$$","$$-0.5, -4, -20, -2, -52, -1257, -81, -51$$","$$-7, -0.2, -19, -29, -0.9, -12, -42, -90$$","$$-12, -8, -0.6, -0.2, -122, -23, -48, -52$$"],"hints":{"DefaultPathway":[{"id":"a1f0162sequences5a-h1","type":"hint","dependencies":[],"title":"Substituting N into Formula","text":"Substitute each value of $$n$$ into the formula. Begin with $$n=1$$ to find the first term, a1. To find the second term, a2, use all $$n$$ terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences5a-h2","type":"hint","dependencies":["a1f0162sequences5a-h1"],"title":"Finding All Needed Values","text":"Continue in the same manner until you have identified all $$n$$ terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences5a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-0.6, -3, -15, -20, -375, -80, -9375, -320$$"],"dependencies":["a1f0162sequences5a-h2"],"title":"First $$8$$ Terms of Sequence","text":"What are the first $$8$$ terms of the sequence?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$-0.6, -3, -15, -20, -375, -80, -9375, -320$$","$$-0.5, -4, -20, -2, -52, -1257, -81, -51$$","$$-7, -0.2, -19, -29, -0.9, -12, -42, -90$$","$$-12, -8, -0.6, -0.2, -122, -23, -48, -52$$"]}]}}]},{"id":"a1f0162sequences6","title":"Finding First N Terms of a Sequence","body":"","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"9.1 Sequences and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a1f0162sequences6a","stepAnswer":["-4,8,28,14/4,23/4,38/4,188,248"],"problemType":"MultipleChoice","stepTitle":"For the following exercises, write the first eight terms of the piecewise sequence.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["-4,-8,-28,14/4,-23/4,38/4,-188,248","-4,8,28,14/4,23/4,38/4,188,248","$$12, -36, 42, -5, 24, -6, 52, -12$$","$$-8, -2, 3, 12, 26, -7, 42, -9$$"],"hints":{"DefaultPathway":[{"id":"a1f0162sequences6a-h1","type":"hint","dependencies":[],"title":"Substituting N into Formula","text":"Substitute each value of $$n$$ into the formula. Begin with $$n=1$$ to find the first term, a1. To find the second term, a2, use all $$n$$ terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences6a-h2","type":"hint","dependencies":["a1f0162sequences6a-h1"],"title":"Finding All Needed Values","text":"Continue in the same manner until you have identified all $$n$$ terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences6a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["-4,8,28,14/4,23/4,38/4,188,248"],"dependencies":["a1f0162sequences6a-h2"],"title":"First $$8$$ Terms of Sequence","text":"What are the first $$8$$ terms of the sequence?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["-4,-8,-28,14/4,-23/4,38/4,-188,248","-4,8,28,14/4,23/4,38/4,188,248","$$12, -36, 42, -5, 24, -6, 52, -12$$","$$-8, -2, 3, 12, 26, -7, 42, -9$$"]}]}}]},{"id":"a1f0162sequences7","title":"Finding Explicit Formula","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Sequences and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a1f0162sequences7a","stepAnswer":["$$\\\\frac{2^n}{2n}$$"],"problemType":"MultipleChoice","stepTitle":"For the following exercise, write an explicit formula for each sequence.","stepBody":"1,1,4/3,2,16/5,...","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{2^n}{2n}$$","choices":["$$\\\\frac{2^n}{2n}$$","$$4n^2$$","$$\\\\frac{2n}{2^n}$$","$$5n+1$$"],"hints":{"DefaultPathway":[{"id":"a1f0162sequences7a-h1","type":"hint","dependencies":[],"title":"Pattern Among the Signs","text":"Terms are all positive","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences7a-h2","type":"hint","dependencies":["a1f0162sequences7a-h1"],"title":"Write in Explicit Form","text":"Write a formula for $$a_n$$ in terms of $$n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences7a-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{2^n}{2n}$$"],"dependencies":["a1f0162sequences7a-h2"],"title":"Selecting Formula","text":"What is the explicit formula?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{2^n}{2n}$$","$$4n^2$$","$$\\\\frac{2n}{2^n}$$","$$5n+1$$"]}]}}]},{"id":"a1f0162sequences8","title":"Finding Explicit Formula","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Sequences and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a1f0162sequences8a","stepAnswer":["$${\\\\left(-\\\\frac{1}{2}\\\\right)}^{n-1}$$"],"problemType":"MultipleChoice","stepTitle":"For the following exercise, write an explicit formula for each sequence.","stepBody":"1,-1/2,1/4,-1/8,1/16","answerType":"string","variabilization":{},"answerLatex":"$${\\\\left(-\\\\frac{1}{2}\\\\right)}^{n-1}$$","choices":["$${\\\\left(-\\\\frac{1}{4}\\\\right)}^n$$","$${\\\\left(\\\\frac{1}{2}\\\\right)}^{n-1}$$","$${\\\\left(-\\\\frac{1}{2}\\\\right)}^{n-1}$$","$$2^{n-1}$$"],"hints":{"DefaultPathway":[{"id":"a1f0162sequences8a-h1","type":"hint","dependencies":[],"title":"Pattern Among the Signs","text":"Terms are alternating between positive and negative","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences8a-h2","type":"hint","dependencies":["a1f0162sequences8a-h1"],"title":"Pattern Among the Terms","text":"Numerator is always $$1$$ but denominator is increasing in doubles","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences8a-h3","type":"hint","dependencies":["a1f0162sequences8a-h2"],"title":"Write in Explicit Form","text":"Write a formula for $$a_n$$ in terms of $$n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences8a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["1,-1/2,1/4,-1/8,1/16"],"dependencies":["a1f0162sequences8a-h3"],"title":"Selecting Formula","text":"What is the explicit formula?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["1,-1/2,1/4,-1/8,1/16","1,-1/2,3/4,-1/8,3/16"]}]}}]},{"id":"a1f0162sequences9","title":"Finding First N Terms of a Sequence","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Sequences and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a1f0162sequences9a","stepAnswer":["$$3, -9, 27, -81, 243$$"],"problemType":"MultipleChoice","stepTitle":"For the following exercises, write the first fove terms of the sequence.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["$$3, -9, 27, -81, 243$$","3,9,27,81,243","$$0, -3, 9, -27, 81$$","0,3,9,27,81"],"hints":{"DefaultPathway":[{"id":"a1f0162sequences9a-h1","type":"hint","dependencies":[],"title":"Identifying Given Values","text":"Identify the initial term, a1, which is given as part of the formula. This is the first term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences9a-h2","type":"hint","dependencies":["a1f0162sequences9a-h1"],"title":"Finding Next Terms","text":"To find the second term, a2, substitute the initial term into the formula for an-1. Solve. Repeat until you have solved for the 5th term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f0162sequences9a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3, -9, 27, -81, 243$$"],"dependencies":["a1f0162sequences9a-h2"],"title":"First $$5$$ Terms of Sequence","text":"What are the first $$5$$ terms of the sequence?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$3, -9, 27, -81, 243$$","3,9,27,81,243","$$0, -3, 9, -27, 81$$","0,3,9,27,81"]}]}}]},{"id":"a1f32dfFormula1","title":"Jamal\'s Bicycle Ride","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve a Formula for a Specific Variable ","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f32dfFormula1a","stepAnswer":["$$42$$"],"problemType":"TextBox","stepTitle":"Jamal rides his bike at a uniform rate of $$12$$ miles per hour for $$3.5$$ hours. What distance has he traveled?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$42$$","hints":{"DefaultPathway":[{"id":"a1f32dfFormula1a-h1","type":"hint","dependencies":[],"title":"Read and Understand the Problem","text":"The question tells us that Jamal rides his bike at the speed of $$12$$ miles per hour which means that every hour he is riding $$12$$ miles.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula1a-h2","type":"hint","dependencies":["a1f32dfFormula1a-h1"],"title":"Identify the Unknown","text":"We are looking for the total distance traveled and we can name it \\"d\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula1a-h3","type":"hint","dependencies":["a1f32dfFormula1a-h2"],"title":"Translate","text":"We can use the formula $$distance=rate time$$ --\x3e $$d=rt$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula1a-h4","type":"hint","dependencies":["a1f32dfFormula1a-h3"],"title":"Substitute","text":"Since we are given that the rate is $$12$$ mph and the time is $$3.5$$ hours, we can substitute $$r=12$$, $$t=3.5$$ and get $$d=12\\\\times3.5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula1a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$42$$"],"dependencies":["a1f32dfFormula1a-h4"],"title":"Evaluate","text":"Evaluate $$12\\\\times3.5$$ to find the distance in miles. What is the distance?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f32dfFormula10","title":"Solve the formula $$A=\\\\frac{1}{2} bh$$ for h:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve a Formula for a Specific Variable ","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f32dfFormula10a","stepAnswer":["$$20$$"],"problemType":"TextBox","stepTitle":"When $$A=170$$ and $$b=17$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$20$$","hints":{"DefaultPathway":[{"id":"a1f32dfFormula10a-h1","type":"hint","dependencies":[],"title":"Identify the Unknown","text":"The formula is $$A=0.5bh$$, and we want to find $$h$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula10a-h2","type":"hint","dependencies":["a1f32dfFormula10a-h1"],"title":"Substitute","text":"Since we know that $$A=170$$ and $$b=17$$, we can substitute these values into the formula and get $$170=0.5(17)(h)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula10a-h3","type":"hint","dependencies":["a1f32dfFormula10a-h2"],"title":"Simplify","text":"Simplifying the equation, we get $$170=(8.5)(h)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a1f32dfFormula10a-h3"],"title":"Solve","text":"To solve for $$h$$, we can divide both sides of the equation by $$8.5$$. What is $$h$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a1f32dfFormula10b","stepAnswer":["$$h=\\\\frac{2A}{b}$$"],"problemType":"MultipleChoice","stepTitle":"In general","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$h=\\\\frac{2A}{b}$$","choices":["$$h=2Ab$$","$$h=\\\\frac{A}{2} b$$","$$h=\\\\frac{b}{2} A$$","$$h=\\\\frac{2A}{b}$$"],"hints":{"DefaultPathway":[{"id":"a1f32dfFormula10b-h1","type":"hint","dependencies":[],"title":"Identify the Unknown","text":"The formula is $$A=0.5bh$$, and we want to find $$h$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula10b-h2","type":"hint","dependencies":["a1f32dfFormula10b-h1"],"title":"Isolate","text":"To solve for $$h$$, we can first multiply both sides by $$2$$ to get rid of the fraction. Now, we the equation becomes $$2A=bh$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula10b-h3","type":"hint","dependencies":["a1f32dfFormula10b-h2"],"title":"Solve","text":"We can then divide both sides by $$b$$ to isolate $$h$$. Therefore, we get $$h=\\\\frac{2A}{b}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f32dfFormula11","title":"$$I=Prt$$","body":"The formula $$I=Prt$$ is used to calculate simple interest, I, for a principal, P, invested at rate, $$r$$, for $$t$$ years. Solve the formula $$I=Prt$$ to find the principal, P:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve a Formula for a Specific Variable ","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f32dfFormula11a","stepAnswer":["$$20000$$"],"problemType":"TextBox","stepTitle":"When $$I=\\\\$5, 600$$, $$r=4\\\\%$$, $$t=7$$ years","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$20000$$","hints":{"DefaultPathway":[{"id":"a1f32dfFormula11a-h1","type":"hint","dependencies":[],"title":"Identify the Unknown","text":"The formula is $$I=Prt$$, and we want to find P.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula11a-h2","type":"hint","dependencies":["a1f32dfFormula11a-h1"],"title":"Substitute","text":"Since we know that $$I=\\\\$5, 600$$, $$r=4\\\\%$$ $$(0.04)$$, and $$t=7$$ years, we can substitute these values into the formula and get $$5600=P(0.04)(7)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula11a-h3","type":"hint","dependencies":["a1f32dfFormula11a-h2"],"title":"Simplify","text":"Simplifying the equation, we get $$5600=0.28P$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20000$$"],"dependencies":["a1f32dfFormula11a-h3"],"title":"Solve","text":"To solve for P, we can divide both sides of the equation by $$0.28$$. What is P?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a1f32dfFormula11b","stepAnswer":["$$P=\\\\frac{I}{rt}$$"],"problemType":"MultipleChoice","stepTitle":"In general","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$P=\\\\frac{I}{rt}$$","choices":["$$P=\\\\frac{I}{rt}$$","$$P=Irt$$","$$P=\\\\frac{rt}{I}$$"],"hints":{"DefaultPathway":[{"id":"a1f32dfFormula11b-h1","type":"hint","dependencies":[],"title":"Write the Formula","text":"The formula is $$I=Prt$$, and we want to find P.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula11b-h2","type":"hint","dependencies":["a1f32dfFormula11b-h1"],"title":"Isolate","text":"To solve for P, we can divide both sides by rt to isolate P, so we get $$P=\\\\frac{I}{rt}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f32dfFormula12","title":"$$I=Prt$$","body":"The formula $$I=Prt$$ is used to calculate simple interest, I, for a principal, P, invested at rate, $$r$$, for $$t$$ years. Solve the formula $$I=Prt$$ to find the principal, P:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve a Formula for a Specific Variable ","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f32dfFormula12a","stepAnswer":["$$12000$$"],"problemType":"TextBox","stepTitle":"When $$I=\\\\$2, 160$$, $$r=6\\\\%$$, $$t=3$$ years","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12000$$","hints":{"DefaultPathway":[{"id":"a1f32dfFormula12a-h1","type":"hint","dependencies":[],"title":"Identify the Unknown","text":"The formula is $$I=Prt$$, and we want to find P.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula12a-h2","type":"hint","dependencies":["a1f32dfFormula12a-h1"],"title":"Substitute","text":"Since we know that $$I=\\\\$2, 160$$, $$r=6\\\\%$$ $$(0.06)$$, and $$t=3$$ years, we can substitute these values into the formula and get $$2160=P(0.06)(3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula12a-h3","type":"hint","dependencies":["a1f32dfFormula12a-h2"],"title":"Simplify","text":"Simplifying the equation, we get $$2160=0.18P$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12000$$"],"dependencies":["a1f32dfFormula12a-h3"],"title":"Solve","text":"To solve for P, we can divide both sides of the equation by $$0.18$$. What is P?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a1f32dfFormula12b","stepAnswer":["$$P=\\\\frac{I}{rt}$$"],"problemType":"MultipleChoice","stepTitle":"In general","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$P=\\\\frac{I}{rt}$$","choices":["$$P=\\\\frac{I}{rt}$$","$$P=Irt$$","$$P=\\\\frac{rt}{I}$$"],"hints":{"DefaultPathway":[{"id":"a1f32dfFormula12b-h1","type":"hint","dependencies":[],"title":"Write the Formula","text":"The formula is $$I=Prt$$, and we want to find P.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula12b-h2","type":"hint","dependencies":["a1f32dfFormula12b-h1"],"title":"Isolate","text":"To solve for P, we can divide both sides by rt to isolate P, so we get $$P=\\\\frac{I}{rt}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f32dfFormula13","title":"Solve the formula $$3x+2y=18$$ for y:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve a Formula for a Specific Variable ","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f32dfFormula13a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"When $$x=4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a1f32dfFormula13a-h1","type":"hint","dependencies":[],"title":"Identify the Unknown","text":"The formula is $$3x+2y=18$$, and we want to find $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula13a-h2","type":"hint","dependencies":["a1f32dfFormula13a-h1"],"title":"Substitute","text":"Since we know that $$x=4$$, we can substitute $$4$$ for $$x$$ into the equation and get $$3\\\\left(4\\\\right)+2y=18$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula13a-h3","type":"hint","dependencies":["a1f32dfFormula13a-h2"],"title":"Simplify","text":"Simplifying the equation, we get $$12+2y=18$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula13a-h4","type":"hint","dependencies":["a1f32dfFormula13a-h3"],"title":"Solve","text":"To solve the equation, we can start by subtracting $$12$$ from both sides, which gives us the equation $$2y=6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula13a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a1f32dfFormula13a-h4"],"title":"Isolate","text":"Now, we can divide both sides by $$2$$ to isolate $$y$$. What is $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a1f32dfFormula13b","stepAnswer":["$$y=\\\\frac{18-3x}{2}$$"],"problemType":"MultipleChoice","stepTitle":"In general","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=\\\\frac{18-3x}{2}$$","choices":["$$y=2\\\\left(18+3x\\\\right)$$","$$y=2\\\\left(18-3x\\\\right)$$","$$y=\\\\frac{18-3x}{2}$$","$$y=\\\\frac{18+3x}{2}$$"],"hints":{"DefaultPathway":[{"id":"a1f32dfFormula13b-h1","type":"hint","dependencies":[],"title":"Identify the Unknown","text":"The formula is $$3x+2y=18$$, and we want to find $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula13b-h2","type":"hint","dependencies":["a1f32dfFormula13b-h1"],"title":"Isolate","text":"To isolate $$y$$, the first step is to subtract $$3x$$ from both sides to get $$2y=18-3x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula13b-h3","type":"hint","dependencies":["a1f32dfFormula13b-h2"],"title":"Isolate","text":"Then, we can divide both sides of the equation by $$2$$ and get $$y=\\\\frac{18-3x}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f32dfFormula14","title":"Solve the formula $$3x+4y=10$$ for y:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve a Formula for a Specific Variable ","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f32dfFormula14a","stepAnswer":["$$-1$$"],"problemType":"TextBox","stepTitle":"When $$x=\\\\frac{14}{3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1$$","hints":{"DefaultPathway":[{"id":"a1f32dfFormula14a-h1","type":"hint","dependencies":[],"title":"Identify the Unknown","text":"The formula is $$3x+4y=10$$, and we want to find $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula14a-h2","type":"hint","dependencies":["a1f32dfFormula14a-h1"],"title":"Substitute","text":"Since we know that $$x=\\\\frac{14}{3}$$, we can substitute $$\\\\frac{14}{3}$$ for $$x$$ into the equation and get $$3\\\\left(\\\\frac{14}{3}\\\\right)+4y=10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula14a-h3","type":"hint","dependencies":["a1f32dfFormula14a-h2"],"title":"Simplify","text":"Simplifying the equation, we get $$14+4y=10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula14a-h4","type":"hint","dependencies":["a1f32dfFormula14a-h3"],"title":"Solve","text":"To solve the equation, we can start by subtracting $$14$$ from both sides, which gives us the equation $$4y=-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula14a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a1f32dfFormula14a-h4"],"title":"Isolate","text":"Now, we can divide both sides by $$4$$ to isolate $$y$$. What is $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a1f32dfFormula14b","stepAnswer":["$$y=\\\\frac{10-3x}{4}$$"],"problemType":"MultipleChoice","stepTitle":"In general","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=\\\\frac{10-3x}{4}$$","choices":["$$y=4\\\\left(10+3x\\\\right)$$","$$y=4\\\\left(10-3x\\\\right)$$","$$y=\\\\frac{10-3x}{4}$$","$$y=\\\\frac{10+3x}{4}$$"],"hints":{"DefaultPathway":[{"id":"a1f32dfFormula14b-h1","type":"hint","dependencies":[],"title":"Identify the Unknown","text":"The formula is $$3x+4y=10$$, and we want to find $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula14b-h2","type":"hint","dependencies":["a1f32dfFormula14b-h1"],"title":"Isolate","text":"To isolate $$y$$, the first step is to subtract $$3x$$ from both sides to get $$4y=10-3x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula14b-h3","type":"hint","dependencies":["a1f32dfFormula14b-h2"],"title":"Isolate","text":"Then, we can divide both sides of the equation by $$4$$ and get $$y=\\\\frac{10-3x}{4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f32dfFormula15","title":"$$P=a+b+c$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve a Formula for a Specific Variable ","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f32dfFormula15a","stepAnswer":["$$b=P-a-c$$"],"problemType":"MultipleChoice","stepTitle":"Solve the formula $$P=a+b+c$$ for $$b$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$b=P-a-c$$","choices":["$$b=P$$","$$b=P+a+c$$","$$b=P-a-c$$"],"hints":{"DefaultPathway":[{"id":"a1f32dfFormula15a-h1","type":"hint","dependencies":[],"title":"Write the Formula","text":"The formula is $$P=a+b+c$$, and we want to find $$b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula15a-h2","type":"hint","dependencies":["a1f32dfFormula15a-h1"],"title":"Isolate","text":"To isolate $$b$$, we can subtract a and c from both sides to get $$P-a-c=b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f32dfFormula16","title":"Solve a Formula for a Specific Variable","body":"In the following exercises, use the formula $$d=rt$$.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve a Formula for a Specific Variable ","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f32dfFormula16a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"Solve for $$t$$ when $$d=350$$ and $$r=70$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"a1f32dfFormula16a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Since we know $$d=350$$ and $$r=70$$, we can substitute $$350$$ for $$d$$ and $$70$$ for $$r$$ into the equation, which gives us $$350=70t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula16a-h2","type":"hint","dependencies":["a1f32dfFormula16a-h1"],"title":"Isolate","text":"To isolate $$t$$, we can divide both sides by $$70$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a1f32dfFormula16b","stepAnswer":["$$t=\\\\frac{d}{r}$$"],"problemType":"MultipleChoice","stepTitle":"Solve for $$t$$ in general","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$t=\\\\frac{d}{r}$$","choices":["$$t=\\\\frac{d}{r}$$","$$t=\\\\frac{r}{d}$$","$$t=dr$$"],"hints":{"DefaultPathway":[{"id":"a1f32dfFormula16b-h1","type":"hint","dependencies":[],"title":"Isolate","text":"To isolate $$t$$, we can divide both sides of the equation by $$r$$, which gives us $$t=\\\\frac{d}{r}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f32dfFormula17","title":"Solve a Formula for a Specific Variable","body":"In the following exercises, use the formula $$d=rt$$.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve a Formula for a Specific Variable ","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f32dfFormula17a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"Solve for $$t$$ when $$d=240$$ and $$r=60$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a1f32dfFormula17a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Since we know $$d=240$$ and $$r=60$$, we can substitute $$240$$ for $$d$$ and $$60$$ for $$r$$ into the equation, which gives us $$240=60t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula17a-h2","type":"hint","dependencies":["a1f32dfFormula17a-h1"],"title":"Isolate","text":"To isolate $$t$$, we can divide both sides by $$60$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a1f32dfFormula17b","stepAnswer":["$$t=\\\\frac{d}{r}$$"],"problemType":"MultipleChoice","stepTitle":"Solve for $$t$$ in general","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$t=\\\\frac{d}{r}$$","choices":["$$t=\\\\frac{d}{r}$$","$$t=\\\\frac{r}{d}$$","$$t=dr$$"],"hints":{"DefaultPathway":[{"id":"a1f32dfFormula17b-h1","type":"hint","dependencies":[],"title":"Isolate","text":"To isolate $$t$$, we can divide both sides of the equation by $$r$$, which gives us $$t=\\\\frac{d}{r}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f32dfFormula18","title":"Solve a Formula for a Specific Variable","body":"In the following exercises, use the formula $$d=rt$$.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve a Formula for a Specific Variable ","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f32dfFormula18a","stepAnswer":["$$8.5$$"],"problemType":"TextBox","stepTitle":"Solve for $$t$$ when $$d=510$$ and $$r=60$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8.5$$","hints":{"DefaultPathway":[{"id":"a1f32dfFormula18a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Since we know $$d=510$$ and $$r=60$$, we can substitute $$510$$ for $$d$$ and $$60$$ for $$r$$ into the equation, which gives us $$510=60t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula18a-h2","type":"hint","dependencies":["a1f32dfFormula18a-h1"],"title":"Isolate","text":"To isolate $$t$$, we can divide both sides by $$60$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a1f32dfFormula18b","stepAnswer":["$$t=\\\\frac{d}{r}$$"],"problemType":"MultipleChoice","stepTitle":"Solve for $$t$$ in general","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$t=\\\\frac{d}{r}$$","choices":["$$t=\\\\frac{d}{r}$$","$$t=\\\\frac{r}{d}$$","$$t=dr$$"],"hints":{"DefaultPathway":[{"id":"a1f32dfFormula18b-h1","type":"hint","dependencies":[],"title":"Isolate","text":"To isolate $$t$$, we can divide both sides of the equation by $$r$$, which gives us $$t=\\\\frac{d}{r}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f32dfFormula19","title":"Solve a Formula for a Specific Variable","body":"In the following exercises, use the formula $$d=rt$$.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve a Formula for a Specific Variable ","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f32dfFormula19a","stepAnswer":["$$64$$"],"problemType":"TextBox","stepTitle":"Solve for $$r$$ when $$d=204$$ and $$t=3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$64$$","hints":{"DefaultPathway":[{"id":"a1f32dfFormula19a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Since we know $$d=204$$ and $$t=3$$, we can substitute $$204$$ for $$d$$ and $$3$$ for $$t$$ into the equation, which gives us $$204=3r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula19a-h2","type":"hint","dependencies":["a1f32dfFormula19a-h1"],"title":"Isolate","text":"To isolate $$r$$, we can divide both sides by $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a1f32dfFormula19b","stepAnswer":["$$r=\\\\frac{d}{t}$$"],"problemType":"MultipleChoice","stepTitle":"Solve for $$r$$ in general","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$r=\\\\frac{d}{t}$$","choices":["$$r=\\\\frac{d}{t}$$","$$r=\\\\frac{t}{d}$$","$$r=dt$$"],"hints":{"DefaultPathway":[{"id":"a1f32dfFormula19b-h1","type":"hint","dependencies":[],"title":"Isolate","text":"To isolate $$r$$, we can divide both sides of the equation by $$t$$, which gives us $$r=\\\\frac{d}{t}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f32dfFormula2","title":"Lindsay\'s Distance","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve a Formula for a Specific Variable ","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f32dfFormula2a","stepAnswer":["$$330$$"],"problemType":"TextBox","stepTitle":"Lindsay drove for $$5.5$$ hours at $$60$$ miles per hour. How much distance did she travel?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$330$$","hints":{"DefaultPathway":[{"id":"a1f32dfFormula2a-h1","type":"hint","dependencies":[],"title":"Identify the Unknown","text":"We are looking for the total distance traveled and we can name it \\"d\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula2a-h2","type":"hint","dependencies":["a1f32dfFormula2a-h1"],"title":"Translate","text":"We can use the formula $$distance=rate time$$ --\x3e $$d=rt$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula2a-h3","type":"hint","dependencies":["a1f32dfFormula2a-h2"],"title":"Substitute","text":"Since we are given that the rate is $$60$$ mph and the time is $$5.5$$ hours, we can substitute $$r=60$$, $$t=5.5$$ and get $$d=60\\\\times5.5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$330$$"],"dependencies":["a1f32dfFormula2a-h3"],"title":"Solve","text":"Solve $$60\\\\times5.5$$ to find the distance in miles. What is the distance?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f32dfFormula20","title":"Solve a Formula for a Specific Variable","body":"In the following exercises, use the formula $$d=rt$$.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve a Formula for a Specific Variable ","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f32dfFormula20a","stepAnswer":["$$70$$"],"problemType":"TextBox","stepTitle":"Solve for $$r$$ when $$d=420$$ and $$t=6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$70$$","hints":{"DefaultPathway":[{"id":"a1f32dfFormula20a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Since we know $$d=420$$ and $$t=6$$, we can substitute $$420$$ for $$d$$ and $$6$$ for $$t$$ into the equation, which gives us $$240=6r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula20a-h2","type":"hint","dependencies":["a1f32dfFormula20a-h1"],"title":"Isolate","text":"To isolate $$r$$, we can divide both sides by $$6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a1f32dfFormula20b","stepAnswer":["$$r=\\\\frac{d}{t}$$"],"problemType":"MultipleChoice","stepTitle":"Solve for $$r$$ in general","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$r=\\\\frac{d}{t}$$","choices":["$$r=\\\\frac{d}{t}$$","$$r=\\\\frac{t}{d}$$","$$r=dt$$"],"hints":{"DefaultPathway":[{"id":"a1f32dfFormula20b-h1","type":"hint","dependencies":[],"title":"Isolate","text":"To isolate $$r$$, we can divide both sides of the equation by $$t$$, which gives us $$r=\\\\frac{d}{t}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f32dfFormula21","title":"Solve a Formula for a Specific Variable","body":"In the following exercises, use the formula $$d=rt$$.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve a Formula for a Specific Variable ","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f32dfFormula21a","stepAnswer":["$$64$$"],"problemType":"TextBox","stepTitle":"Solve for $$r$$ when $$d=160$$ and $$t=2.5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$64$$","hints":{"DefaultPathway":[{"id":"a1f32dfFormula21a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Since we know $$d=160$$ and $$t=2.5$$, we can substitute $$160$$ for $$d$$ and $$2.5$$ for $$t$$ into the equation, which gives us $$160=2.5r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula21a-h2","type":"hint","dependencies":["a1f32dfFormula21a-h1"],"title":"Isolate","text":"To isolate $$r$$, we can divide both sides by $$2.5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a1f32dfFormula21b","stepAnswer":["$$r=\\\\frac{d}{t}$$"],"problemType":"MultipleChoice","stepTitle":"Solve for $$r$$ in general","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$r=\\\\frac{d}{t}$$","choices":["$$r=\\\\frac{d}{t}$$","$$r=\\\\frac{t}{d}$$","$$r=dt$$"],"hints":{"DefaultPathway":[{"id":"a1f32dfFormula21b-h1","type":"hint","dependencies":[],"title":"Isolate","text":"To isolate $$r$$, we can divide both sides of the equation by $$t$$, which gives us $$r=\\\\frac{d}{t}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f32dfFormula22","title":"Solve a Formula for a Specific Variable","body":"In the following exercises, use the formula $$d=rt$$.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve a Formula for a Specific Variable ","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f32dfFormula22a","stepAnswer":["$$40$$"],"problemType":"TextBox","stepTitle":"Solve for $$r$$ when $$d=180$$ and $$t=4.5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$40$$","hints":{"DefaultPathway":[{"id":"a1f32dfFormula22a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Since we know $$d=180$$ and $$t=4.5$$, we can substitute $$180$$ for $$d$$ and $$4.5$$ for $$t$$ into the equation, which gives us $$180=4.5r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula22a-h2","type":"hint","dependencies":["a1f32dfFormula22a-h1"],"title":"Isolate","text":"To isolate $$r$$, we can divide both sides by $$4.5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a1f32dfFormula22b","stepAnswer":["$$r=\\\\frac{d}{t}$$"],"problemType":"MultipleChoice","stepTitle":"Solve for $$r$$ in general","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$r=\\\\frac{d}{t}$$","choices":["$$r=\\\\frac{d}{t}$$","$$r=\\\\frac{t}{d}$$","$$r=dt$$"],"hints":{"DefaultPathway":[{"id":"a1f32dfFormula22b-h1","type":"hint","dependencies":[],"title":"Isolate","text":"To isolate $$r$$, we can divide both sides of the equation by $$t$$, which gives us $$r=\\\\frac{d}{t}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f32dfFormula23","title":"In the following exercises, solve.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve a Formula for a Specific Variable ","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f32dfFormula23a","stepAnswer":["$$b=90-a$$"],"problemType":"MultipleChoice","stepTitle":"Solve $$a+b=90$$ for $$b$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$b=90-a$$","choices":["$$b=90a$$","$$b=90+a$$","$$b=90-a$$"],"hints":{"DefaultPathway":[{"id":"a1f32dfFormula23a-h1","type":"hint","dependencies":[],"title":"Isolate","text":"To isolate $$b$$, we can subtract a from both sides of the equation, which gives us $$b=90-a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f32dfFormula24","title":"In the following exercises, solve.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve a Formula for a Specific Variable ","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f32dfFormula24a","stepAnswer":["$$a=90-b$$"],"problemType":"MultipleChoice","stepTitle":"Solve $$a+b=90$$ for a","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$a=90-b$$","choices":["$$a=90b$$","$$a=90-b$$","$$a=90+b$$"],"hints":{"DefaultPathway":[{"id":"a1f32dfFormula24a-h1","type":"hint","dependencies":[],"title":"Isolate","text":"To isolate a, we can subtract $$b$$ from both sides of the equation, which gives us $$a=90-b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f32dfFormula25","title":"In the following exercises, solve.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve a Formula for a Specific Variable ","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f32dfFormula25a","stepAnswer":["$$y=15-8x$$"],"problemType":"MultipleChoice","stepTitle":"Solve the formula $$8x+y=15$$ for $$y$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=15-8x$$","choices":["$$y=15+8x$$","$$y=15-8x$$","$$y=8x-15$$"],"hints":{"DefaultPathway":[{"id":"a1f32dfFormula25a-h1","type":"hint","dependencies":[],"title":"Isolate","text":"To isolate $$y$$, we can subtract $$8x$$ from both sides of the equation, which gives us $$y=15-8x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f32dfFormula26","title":"In the following exercises, solve.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve a Formula for a Specific Variable ","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f32dfFormula26a","stepAnswer":["$$y=13-9x$$"],"problemType":"MultipleChoice","stepTitle":"Solve the formula $$9x+y=13$$ for $$y$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=13-9x$$","choices":["$$y=13+9x$$","$$y=13-9x$$","$$y=9x-13$$"],"hints":{"DefaultPathway":[{"id":"a1f32dfFormula26a-h1","type":"hint","dependencies":[],"title":"Isolate","text":"To isolate $$y$$, we can subtract $$9x$$ from both sides of the equation, which gives us $$y=13-9x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f32dfFormula27","title":"In the following exercises, solve.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve a Formula for a Specific Variable ","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f32dfFormula27a","stepAnswer":["$$y=-6+4x$$"],"problemType":"MultipleChoice","stepTitle":"Solve the formula $$-4x+y=-6$$ for $$y$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=-6+4x$$","choices":["$$y=-6-4x$$","$$y=-6+4x$$","$$y=-4x+6$$"],"hints":{"DefaultPathway":[{"id":"a1f32dfFormula27a-h1","type":"hint","dependencies":[],"title":"Isolate","text":"To isolate $$y$$, we can add $$4x$$ to both sides of the equation, which gives us $$y=-6+4x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f32dfFormula28","title":"In the following exercises, solve.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve a Formula for a Specific Variable ","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f32dfFormula28a","stepAnswer":["$$y=-1+5x$$"],"problemType":"MultipleChoice","stepTitle":"Solve the formula $$-5x+y=-1$$ for $$y$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=-1+5x$$","choices":["$$y=-1+5x$$","$$y=-1-5x$$","$$y=5x$$"],"hints":{"DefaultPathway":[{"id":"a1f32dfFormula28a-h1","type":"hint","dependencies":[],"title":"Isolate","text":"To isolate $$y$$, we can add $$5x$$ to both sides of the equation, which gives us $$y=-1+5x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f32dfFormula29","title":"In the following exercises, solve.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve a Formula for a Specific Variable ","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f32dfFormula29a","stepAnswer":["$$d=\\\\frac{C}{\\\\pi}$$"],"problemType":"MultipleChoice","stepTitle":"Solve the formula $$C=\\\\pi d$$ for $$d$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$d=\\\\frac{C}{\\\\pi}$$","choices":["$$d=\\\\frac{C}{\\\\pi}$$","$$d=C \\\\pi$$","$$d=\\\\frac{\\\\pi}{C}$$"],"hints":{"DefaultPathway":[{"id":"a1f32dfFormula29a-h1","type":"hint","dependencies":[],"title":"Isolate","text":"To isolate $$d$$, we can divide both sides of the equation by pi, which gives us $$d=\\\\frac{C}{\\\\pi}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f32dfFormula3","title":"Trinh\'s Distance","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve a Formula for a Specific Variable ","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f32dfFormula3a","stepAnswer":["$$7$$"],"problemType":"TextBox","stepTitle":"Trinh walked for $$\\\\frac{7}{3}$$ hours at $$3$$ miles per hour. How far did she walk?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7$$","hints":{"DefaultPathway":[{"id":"a1f32dfFormula3a-h1","type":"hint","dependencies":[],"title":"Identify the Unknown","text":"We are looking for the total distance traveled and we can name it \\"d\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula3a-h2","type":"hint","dependencies":["a1f32dfFormula3a-h1"],"title":"Translate","text":"We can use the formula $$distance=rate time$$ --\x3e $$d=rt$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula3a-h3","type":"hint","dependencies":["a1f32dfFormula3a-h2"],"title":"Substitute","text":"Since we are given that the rate is $$3$$ mph and the time is $$\\\\frac{7}{3}$$ hours, we can $$substituter=3$$, $$t=\\\\frac{7}{3}$$ and get $$d=3\\\\frac{7}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a1f32dfFormula3a-h3"],"title":"Solve","text":"Solve $$\\\\frac{3\\\\times7}{3}$$ to find the distance in miles. What is the distance?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f32dfFormula30","title":"In the following exercises, solve.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve a Formula for a Specific Variable ","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f32dfFormula30a","stepAnswer":["$$pi=\\\\frac{C}{d}$$"],"problemType":"MultipleChoice","stepTitle":"Solve the formula $$C=\\\\pi d$$ for pi","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$pi=\\\\frac{C}{d}$$","choices":["$$pi=\\\\frac{C}{d}$$","$$pi=Cd$$","$$pi=\\\\frac{d}{C}$$"],"hints":{"DefaultPathway":[{"id":"a1f32dfFormula30a-h1","type":"hint","dependencies":[],"title":"Isolate","text":"To isolate pi, we can divide both sides of the equation by $$d$$, which gives us $$pi=\\\\frac{C}{d}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f32dfFormula31","title":"In the following exercises, solve.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve a Formula for a Specific Variable ","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f32dfFormula31a","stepAnswer":["$$L=\\\\frac{V}{WH}$$"],"problemType":"MultipleChoice","stepTitle":"Solve the formula $$V=LWH$$ for L .","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$L=\\\\frac{V}{WH}$$","choices":["$$L=\\\\frac{V}{W}$$","$$L=\\\\frac{V}{WH}$$","$$L=VWH$$","$$L=\\\\frac{WH}{V}$$"],"hints":{"DefaultPathway":[{"id":"a1f32dfFormula31a-h1","type":"hint","dependencies":[],"title":"Isolate","text":"To isolate L, we can divide both sides of the equation by WH, which gives us $$L=\\\\frac{V}{WH}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f32dfFormula4","title":"Rey\'s Driving Time","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve a Formula for a Specific Variable ","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f32dfFormula4a","stepAnswer":["$$8$$"],"problemType":"TextBox","stepTitle":"Rey is planning to drive from his house in San Diego to visit his grandmother in Sacramento, a distance of $$520$$ miles. If he can drive at a steady rate of $$65$$ miles per hour, how many hours will the trip take?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8$$","hints":{"DefaultPathway":[{"id":"a1f32dfFormula4a-h1","type":"hint","dependencies":[],"title":"Identify the Unknown","text":"We are looking for the total time that the journey took, and we can name it \\"h\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula4a-h2","type":"hint","dependencies":["a1f32dfFormula4a-h1"],"title":"Translate","text":"We can use the formula $$distance=rate time$$ --\x3e $$d=rt$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula4a-h3","type":"hint","dependencies":["a1f32dfFormula4a-h2"],"title":"Substitute","text":"Since we are given that the distance is $$520$$ miles and the rate is $$65$$ mph, we can substitute $$d=520$$, $$r=65$$ and get $$520=65t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a1f32dfFormula4a-h3"],"title":"Solve","text":"Solve for $$t$$ in $$520=65t$$. (hint: divide both sides by 65.) What is $$t$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f32dfFormula5","title":"Lee\'s Driving Time","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve a Formula for a Specific Variable ","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f32dfFormula5a","stepAnswer":["$$11$$"],"problemType":"TextBox","stepTitle":"Lee wants to drive from Phoenix to his brother\u2019s apartment in San Francisco, a distance of $$770$$ miles. If he drives at a steady rate of $$70$$ miles per hour, how many hours will the trip take?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$11$$","hints":{"DefaultPathway":[{"id":"a1f32dfFormula5a-h1","type":"hint","dependencies":[],"title":"Identify the Unknown","text":"We are looking for the total time that the journey took, and we can name it \\"h\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula5a-h2","type":"hint","dependencies":["a1f32dfFormula5a-h1"],"title":"Translate","text":"We can use the formula $$distance=rate time$$ --\x3e $$d=rt$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula5a-h3","type":"hint","dependencies":["a1f32dfFormula5a-h2"],"title":"Substitute","text":"Since we are given that the distance is $$770$$ miles and the rate is $$70$$ mph, we can substitute $$d=770$$, $$r=70$$ and get $$770=70t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11$$"],"dependencies":["a1f32dfFormula5a-h3"],"title":"Solve","text":"Solve for $$t$$ in the given equation (hint: divide both sides by 70). What is $$t$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f32dfFormula6","title":"Yesenia\'s Needed Speed","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve a Formula for a Specific Variable ","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f32dfFormula6a","stepAnswer":["$$56$$"],"problemType":"TextBox","stepTitle":"Yesenia is $$168$$ miles from Chicago. If she needs to be in Chicago in $$3$$ hours, at what rate does she need to drive?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$56$$","hints":{"DefaultPathway":[{"id":"a1f32dfFormula6a-h1","type":"hint","dependencies":[],"title":"Identify the Unknown","text":"We are looking for the the rate of the journey, and we can name it \\"r\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula6a-h2","type":"hint","dependencies":["a1f32dfFormula6a-h1"],"title":"Translate","text":"We can use the formula $$distance=rate time$$ --\x3e $$d=rt$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula6a-h3","type":"hint","dependencies":["a1f32dfFormula6a-h2"],"title":"Substitute","text":"Since we are given that the distance is $$168$$ miles and the time is $$3$$ hours, we can substitute $$d=168$$, $$t=3$$ and get $$168=3r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$56$$"],"dependencies":["a1f32dfFormula6a-h3"],"title":"Solve","text":"Solve for $$h$$ in the given equation (hint: divide both sides by 3). What is $$h$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f32dfFormula7","title":"Solve the formula $$d=rt$$ for t:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve a Formula for a Specific Variable ","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f32dfFormula7a","stepAnswer":["$$8$$"],"problemType":"TextBox","stepTitle":"When $$d=520$$ and $$r=65$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8$$","hints":{"DefaultPathway":[{"id":"a1f32dfFormula7a-h1","type":"hint","dependencies":[],"title":"Identify the Unknown","text":"The formula is $$d=rt$$, and we want to find $$t$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula7a-h2","type":"hint","dependencies":["a1f32dfFormula7a-h1"],"title":"Substitute","text":"Since we know that $$d=520$$, $$r=65$$ we can substitute these values into the formula and get $$520=65t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a1f32dfFormula7a-h2"],"title":"Solve","text":"To solve for $$t$$, we can divide both sides of the equation by $$65$$. What is $$t$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a1f32dfFormula7b","stepAnswer":["$$t=\\\\frac{d}{r}$$"],"problemType":"MultipleChoice","stepTitle":"in general","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$t=\\\\frac{d}{r}$$","choices":["$$t=\\\\frac{d}{r}$$","$$t=dr$$","$$t=\\\\frac{r}{d}$$"],"hints":{"DefaultPathway":[{"id":"a1f32dfFormula7b-h1","type":"hint","dependencies":[],"title":"Identify the Unknown","text":"The formula is $$d=rt$$, and we want to find $$t$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula7b-h2","type":"hint","dependencies":["a1f32dfFormula7b-h1"],"title":"Isolate","text":"To solve for $$r$$, we can divide both sides by $$r$$ to isolate $$t$$, so we get $$t=\\\\frac{d}{r}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f32dfFormula8","title":"Solve the formula $$d=rt$$ for r:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve a Formula for a Specific Variable ","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f32dfFormula8a","stepAnswer":["$$45$$"],"problemType":"TextBox","stepTitle":"When $$d=180$$ and $$t=4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$45$$","hints":{"DefaultPathway":[{"id":"a1f32dfFormula8a-h1","type":"hint","dependencies":[],"title":"Identify the Unknown","text":"The formula is $$d=rt$$, and we want to find $$r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula8a-h2","type":"hint","dependencies":["a1f32dfFormula8a-h1"],"title":"Substitute","text":"Since we know that $$d=180$$, $$t=12$$ we can substitute these values into the formula and get $$180=12r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$45$$"],"dependencies":["a1f32dfFormula8a-h2"],"title":"Solve","text":"To solve for $$r$$, we can divide both sides of the equation by $$4$$. What is $$r$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a1f32dfFormula8b","stepAnswer":["$$r=\\\\frac{d}{t}$$"],"problemType":"MultipleChoice","stepTitle":"in general","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$r=\\\\frac{d}{t}$$","choices":["$$r=\\\\frac{d}{t}$$","$$r=dt$$","$$r=\\\\frac{t}{d}$$"],"hints":{"DefaultPathway":[{"id":"a1f32dfFormula8b-h1","type":"hint","dependencies":[],"title":"Identify the Unknown","text":"The formula is $$d=rt$$, and we want to find $$r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula8b-h2","type":"hint","dependencies":["a1f32dfFormula8b-h1"],"title":"Isolate","text":"To solve for $$r$$, we can divide both sides by $$t$$ to isolate $$r$$, so we get $$r=\\\\frac{d}{t}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f32dfFormula9","title":"Solve the formula $$A=\\\\frac{1}{2} bh$$ for h:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve a Formula for a Specific Variable ","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f32dfFormula9a","stepAnswer":["$$12$$"],"problemType":"TextBox","stepTitle":"When $$A=90$$ and $$b=15$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12$$","hints":{"DefaultPathway":[{"id":"a1f32dfFormula9a-h1","type":"hint","dependencies":[],"title":"Identify the Unknown","text":"The formula is $$A=0.5bh$$, and we want to find $$h$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula9a-h2","type":"hint","dependencies":["a1f32dfFormula9a-h1"],"title":"Substitute","text":"Since we know that $$A=90$$ and $$b=15$$, we can substitute these values into the formula and get $$90=0.5(15)(h)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula9a-h3","type":"hint","dependencies":["a1f32dfFormula9a-h2"],"title":"Simplify","text":"Simplifying the equation, we get $$90=(7.5)(h)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a1f32dfFormula9a-h3"],"title":"Solve","text":"To solve for $$h$$, we can divide both sides of the equation by $$7.5$$. What is $$h$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a1f32dfFormula9b","stepAnswer":["$$h=\\\\frac{2A}{b}$$"],"problemType":"MultipleChoice","stepTitle":"In general","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$h=\\\\frac{2A}{b}$$","choices":["$$h=2Ab$$","$$h=\\\\frac{A}{2} b$$","$$h=\\\\frac{b}{2} A$$","$$h=\\\\frac{2A}{b}$$"],"hints":{"DefaultPathway":[{"id":"a1f32dfFormula9b-h1","type":"hint","dependencies":[],"title":"Identify the Unknown","text":"The formula is $$A=0.5bh$$, and we want to find $$h$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula9b-h2","type":"hint","dependencies":["a1f32dfFormula9b-h1"],"title":"Isolate","text":"To solve for $$h$$, we can first multiply both sides by $$2$$ to get rid of the fraction. Now, we the equation becomes $$2A=bh$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f32dfFormula9b-h3","type":"hint","dependencies":["a1f32dfFormula9b-h2"],"title":"Solve","text":"We can then divide both sides by $$b$$ to isolate $$h$$. Therefore, we get $$h=\\\\frac{2A}{b}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f74d6coordinates1","title":"Identify Points in Quadrants","body":"Plot the following point and identify the quadrant in which it is located.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Use the Rectangular Coordinate System","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f74d6coordinates1a","stepAnswer":["Quadrant $$2$$"],"problemType":"MultipleChoice","stepTitle":"Plot $$(-5,4)$$ in the rectangular coordinate system and identify the quadrant in which the point is located.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Quadrant $$2$$","choices":["Quadrant $$1$$","Quadrant $$2$$","Quadrant $$3$$","Quadrant $$4$$"],"hints":{"DefaultPathway":[{"id":"a1f74d6coordinates1a-h1","type":"hint","dependencies":[],"title":"Know Quadrants","text":"The quadrants are the $$4$$ regions of the graph, divided by the axes. The first quadrant is the top right region. The quadrants then continue counter clockwise in order, such that the top left region is the second quadrant.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates1a-h2","type":"hint","dependencies":["a1f74d6coordinates1a-h1"],"title":"Use Negatives","text":"If the $$x$$ coordinate is negative, the point lies to the left of the $$y$$ axis. If the $$y$$ coordinate is negative, the point lies below the $$x$$ axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates1a-h3","type":"hint","dependencies":["a1f74d6coordinates1a-h2"],"title":"Know Where the Points Lie","text":"Since $$x$$ is negative, and $$y$$ is positive, the point is in quadrant $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f74d6coordinates10","title":"Verify Ordered Pairs as Solutions","body":"Determine if the following point is a valid solution to the equation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Use the Rectangular Coordinate System","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f74d6coordinates10a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Determine if $$(2,0)$$ is a solution to the equation $$2x+3y=6$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a1f74d6coordinates10a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute the $$x$$ and $$y$$ values of the ordered pair into the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates10a-h2","type":"hint","dependencies":["a1f74d6coordinates10a-h1"],"title":"Simplify","text":"Simplify each side such that their is a single number on both sides of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates10a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a1f74d6coordinates10a-h2"],"title":"Interpret","text":"If both sides of the equation equal each other, the ordered pair is a solution. Is the ordered pair a solution in this case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a1f74d6coordinates10a-h4","type":"hint","dependencies":["a1f74d6coordinates10a-h3"],"title":"Answer","text":"Therefore, the ordered pair is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f74d6coordinates11","title":"Finding Solutions to the Equation","body":"Find the solution to the equation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Use the Rectangular Coordinate System","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f74d6coordinates11a","stepAnswer":["$$(0,-5)$$"],"problemType":"MultipleChoice","stepTitle":"Given $$x=0$$, find a solution for the equation $$5x-4y=20$$.","stepBody":"Enter your answer as a coordinate pair.","answerType":"string","variabilization":{},"answerLatex":"$$(0,-5)$$","choices":["$$(0,-5)$$","$$(0,-4)$$","$$(0,-3)$$"],"hints":{"DefaultPathway":[{"id":"a1f74d6coordinates11a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Plug $$x=0$$ into the equation. The new equation should be $$0-4y=20$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates11a-h2","type":"hint","dependencies":["a1f74d6coordinates11a-h1"],"title":"Solve","text":"Solve the equation for $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates11a-h3","type":"hint","dependencies":["a1f74d6coordinates11a-h2"],"title":"Answer","text":"$$y=-5$$, so the coordinate pair for the solution to the equation is $$(0,-5)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f74d6coordinates12","title":"Finding Solutions to the Equation","body":"Find the solution to the equation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Use the Rectangular Coordinate System","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f74d6coordinates12a","stepAnswer":["$$(4,0)$$"],"problemType":"MultipleChoice","stepTitle":"Given $$y=0$$, find a solution for the equation $$5x-4y=20$$.","stepBody":"Enter your answer as a coordinate pair.","answerType":"string","variabilization":{},"answerLatex":"$$(4,0)$$","choices":["$$(3,0)$$","$$(4,0)$$","$$(5,0)$$"],"hints":{"DefaultPathway":[{"id":"a1f74d6coordinates12a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Plug $$y=0$$ into the equation. The new equation should be $$5x-0=20$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates12a-h2","type":"hint","dependencies":["a1f74d6coordinates12a-h1"],"title":"Solve","text":"Solve the equation for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates12a-h3","type":"hint","dependencies":["a1f74d6coordinates12a-h2"],"title":"Answer","text":"$$x=4$$, so the coordinate pair for the solution to the equation is $$(4,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f74d6coordinates13","title":"Finding Solutions to the Equation","body":"Find the solution to the equation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Use the Rectangular Coordinate System","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f74d6coordinates13a","stepAnswer":["$$(8,5)$$"],"problemType":"MultipleChoice","stepTitle":"Given $$y=5$$, find a solution for the equation $$5x-4y=20$$.","stepBody":"Enter your answer as a coordinate pair.","answerType":"string","variabilization":{},"answerLatex":"$$(8,5)$$","choices":["$$(0,5)$$","$$(4,5)$$","$$(8,5)$$"],"hints":{"DefaultPathway":[{"id":"a1f74d6coordinates13a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Plug $$y=5$$ into the equation. The new equation should be $$5x-20=20$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates13a-h2","type":"hint","dependencies":["a1f74d6coordinates13a-h1"],"title":"Solve","text":"Solve the equation for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates13a-h3","type":"hint","dependencies":["a1f74d6coordinates13a-h2"],"title":"Answer","text":"$$x=8$$, so the coordinate pair for the solution to the equation is $$(8,5)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f74d6coordinates14","title":"Finding Solutions to the Equation","body":"Find the solution to the equation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Use the Rectangular Coordinate System","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f74d6coordinates14a","stepAnswer":["$$(-5,6)$$"],"problemType":"MultipleChoice","stepTitle":"Given $$x=-5$$, find a solution for the equation $$2x-5y=20$$","stepBody":"Enter your answer as a coordinate pair.","answerType":"string","variabilization":{},"answerLatex":"$$(-5,6)$$","choices":["$$(-5,6)$$","$$(-5,2)$$","$$(-5,3)$$"],"hints":{"DefaultPathway":[{"id":"a1f74d6coordinates14a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Plug $$x=-5$$ into the equation. The new equation should be $$-10-5y=20$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates14a-h2","type":"hint","dependencies":["a1f74d6coordinates14a-h1"],"title":"Solve","text":"Solve the equation for $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates14a-h3","type":"hint","dependencies":["a1f74d6coordinates14a-h2"],"title":"Answer","text":"$$y=-6$$, so the coordinate pair for the solution to the equation is $$(-5,-6)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f74d6coordinates15","title":"Finding Solutions to the Equation","body":"Find the solution to the equation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Use the Rectangular Coordinate System","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f74d6coordinates15a","stepAnswer":["$$(0,-4)$$"],"problemType":"MultipleChoice","stepTitle":"Given $$x=0$$, find a solution for the equation $$2x-5y=20$$","stepBody":"Enter your answer as a coordinate pair.","answerType":"string","variabilization":{},"answerLatex":"$$(0,-4)$$","choices":["$$(0,-2)$$","$$(0,-4)$$","$$(0,-1)$$"],"hints":{"DefaultPathway":[{"id":"a1f74d6coordinates15a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Plug $$x=0$$ into the equation. The new equation should be $$0-5y=20$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates15a-h2","type":"hint","dependencies":["a1f74d6coordinates15a-h1"],"title":"Solve","text":"Solve the equation for $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates15a-h3","type":"hint","dependencies":["a1f74d6coordinates15a-h2"],"title":"Answer","text":"$$y=-4$$, so the coordinate pair for the solution to the equation is $$(0,-4)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f74d6coordinates16","title":"Verify Solutions to an Equation in Two Variables","body":"In the following exercises, which ordered pairs are solutions to the given equations?","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Use the Rectangular Coordinate System","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f74d6coordinates16a","stepAnswer":["$$(4,0)$$ and $$(2,-3)$$"],"problemType":"MultipleChoice","stepTitle":"$$3x-2y=12$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(4,0)$$ and $$(2,-3)$$","choices":["$$(4,0)$$ and $$(2,-3)$$","$$(1,6)$$ and $$(2,-3)$$","$$(4,0)$$ and $$(1,6)$$"],"hints":{"DefaultPathway":[{"id":"a1f74d6coordinates16a-h1","type":"hint","dependencies":[],"title":"Understanding Coordinate Format","text":"The ordered pairs are given in coordinate format with (x,y). The $$x$$ value will be plugged into the $$x$$ variable of the given equation and the same for the $$y$$ value in the $$y$$ variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates16a-h2","type":"hint","dependencies":["a1f74d6coordinates16a-h1"],"title":"Plugging in Ordered Pairs","text":"In order to check which ordered pairs are solutions, plug in each option into the given equation to check if it outputs the right answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates16a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a1f74d6coordinates16a-h2"],"title":"Plugging in $$(4,0)$$","text":"What does the ordered pair $$(4,0)$$ output when plugged into the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates16a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a1f74d6coordinates16a-h3"],"title":"Plugging in $$(2,-3)$$","text":"What does the ordered pair $$(2,-3)$$ output when plugged into the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates16a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":["a1f74d6coordinates16a-h4"],"title":"Plugging in $$(1,6)$$","text":"What does the ordered pair $$(1,6)$$ output when plugged into the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates16a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(4,0)$$ and $$(2,-3)$$"],"dependencies":["a1f74d6coordinates16a-h5"],"title":"Identifying Correct Ordered Pairs","text":"Which ordered pairs satisfy the solution to the given equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(4,0)$$ and $$(2,-3)$$","$$(1,6)$$ and $$(2,-3)$$","$$(4,0)$$ and $$(1,6)$$"]}]}}]},{"id":"a1f74d6coordinates17","title":"Verify Solutions to an Equation in Two Variables","body":"In the following exercises, which ordered pairs are solutions to the given equations?","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Use the Rectangular Coordinate System","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f74d6coordinates17a","stepAnswer":["$$(-1,-1)$$ and $$(\\\\frac{1}{2},5)$$"],"problemType":"MultipleChoice","stepTitle":"$$y=4x+3$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-1,-1)$$ and $$(\\\\frac{1}{2},5)$$","choices":["$$(4,3)$$ and $$(-1,-1)$$","$$(\\\\frac{1}{2},5)$$ and $$(4,3)$$","$$(-1,-1)$$ and $$(\\\\frac{1}{2},5)$$"],"hints":{"DefaultPathway":[{"id":"a1f74d6coordinates17a-h1","type":"hint","dependencies":[],"title":"Understanding Coordinate Format","text":"The ordered pairs are given in coordinate format with (x,y). The $$x$$ value will be plugged into the $$x$$ variable of the given equation and the same for the $$y$$ value in the $$y$$ variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates17a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y-4x=3$$"],"dependencies":["a1f74d6coordinates17a-h1"],"title":"Rearranging Equation","text":"What will the equation look like when rearranged so that all variables are on one side?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates17a-h3","type":"hint","dependencies":["a1f74d6coordinates17a-h2"],"title":"Plugging in Ordered Pairs","text":"In order to check which ordered pairs are solutions, plug in each option into the rearranged equation to check if it outputs the right answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates17a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-8$$"],"dependencies":["a1f74d6coordinates17a-h3"],"title":"Plugging in $$(4,3)$$","text":"What does the ordered pair $$(4,3)$$ output when plugged into the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates17a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a1f74d6coordinates17a-h4"],"title":"Plugging in $$(-1,-1)$$","text":"What does the ordered pair $$(-1,-1)$$ output when plugged into the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates17a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a1f74d6coordinates17a-h5"],"title":"Plugging in $$(\\\\frac{1}{2},5)$$","text":"What does the ordered pair $$(\\\\frac{1}{2},5)$$ output when plugged into the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates17a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(-1,-1)$$ and $$(\\\\frac{1}{2},5)$$"],"dependencies":["a1f74d6coordinates17a-h6"],"title":"Identifying Correct Ordered Pairs","text":"Which ordered pairs satisfy the solution to the given equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(4,3)$$ and $$(-1,-1)$$","$$(\\\\frac{1}{2},5)$$ and $$(4,3)$$","$$(-1,-1)$$ and $$(\\\\frac{1}{2},5)$$"]}]}}]},{"id":"a1f74d6coordinates18","title":"Verify Solutions to an Equation in Two Variables","body":"In the following exercises, which ordered pairs are solutions to the given equations?","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Use the Rectangular Coordinate System","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f74d6coordinates18a","stepAnswer":["$$(0,-5)$$ and $$(\\\\frac{1}{2},-4)$$"],"problemType":"MultipleChoice","stepTitle":"$$y=2x-5$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,-5)$$ and $$(\\\\frac{1}{2},-4)$$","choices":["$$(0,-5)$$ and $$(2,1)$$","$$(0,-5)$$ and $$(\\\\frac{1}{2},-4)$$","$$(\\\\frac{1}{2},-4)$$ and $$(2,1)$$"],"hints":{"DefaultPathway":[{"id":"a1f74d6coordinates18a-h1","type":"hint","dependencies":[],"title":"Understanding Coordinate Format","text":"The ordered pairs are given in coordinate format with (x,y). The $$x$$ value will be plugged into the $$x$$ variable of the given equation and the same for the $$y$$ value in the $$y$$ variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y-2x=-5$$"],"dependencies":["a1f74d6coordinates18a-h1"],"title":"Rearranging Equation","text":"What will the equation look like when rearranged so that all variables are on one side?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates18a-h3","type":"hint","dependencies":["a1f74d6coordinates18a-h2"],"title":"Plugging in Ordered Pairs","text":"In order to check which ordered pairs are solutions, plug in each option into the rearranged equation to check if it outputs the right answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates18a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["a1f74d6coordinates18a-h3"],"title":"Plugging in $$(0,-5)$$","text":"What does the ordered pair $$(0,-5)$$ output when plugged into the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates18a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a1f74d6coordinates18a-h4"],"title":"Plugging in $$(2,1)$$","text":"What does the ordered pair $$(2,1)$$ output when plugged into the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates18a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["a1f74d6coordinates18a-h5"],"title":"Plugging in $$(\\\\frac{1}{2},-4)$$","text":"What does the ordered pair $$(\\\\frac{1}{2},-4)$$ output when plugged into the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates18a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(0,-5)$$ and $$(\\\\frac{1}{2},-4)$$"],"dependencies":["a1f74d6coordinates18a-h6"],"title":"Identifying Correct Ordered Pairs","text":"Which ordered pairs satisfy the solution to the given equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(0,-5)$$ and $$(2,1)$$","$$(0,-5)$$ and $$(\\\\frac{1}{2},-4)$$","$$(\\\\frac{1}{2},-4)$$ and $$(2,1)$$"]}]}}]},{"id":"a1f74d6coordinates19","title":"Verify Solutions to an Equation in Two Variables","body":"In the following exercises, which ordered pairs are solutions to the given equations?","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Use the Rectangular Coordinate System","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f74d6coordinates19a","stepAnswer":["$$(2,0)$$ and $$(-6,-4)$$"],"problemType":"MultipleChoice","stepTitle":"$$y=\\\\frac{x}{2}-1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(2,0)$$ and $$(-6,-4)$$","choices":["$$(2,0)$$ and $$(-6,-4)$$","$$(-4,-1)$$ and $$(2,0)$$","$$(-6,-4)$$ and $$(-4,1)$$"],"hints":{"DefaultPathway":[{"id":"a1f74d6coordinates19a-h1","type":"hint","dependencies":[],"title":"Understanding Coordinate Format","text":"The ordered pairs are given in coordinate format with (x,y). The $$x$$ value will be plugged into the $$x$$ variable of the given equation and the same for the $$y$$ value in the $$y$$ variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y-\\\\frac{x}{2}=-1$$"],"dependencies":["a1f74d6coordinates19a-h1"],"title":"Rearranging Equation","text":"What will the equation look like when rearranged so that all variables are on one side?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates19a-h3","type":"hint","dependencies":["a1f74d6coordinates19a-h2"],"title":"Plugging in Ordered Pairs","text":"In order to check which ordered pairs are solutions, plug in each option into the rearranged equation to check if it outputs the right answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates19a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a1f74d6coordinates19a-h3"],"title":"Plugging in $$(2,0)$$","text":"What does the ordered pair $$(2,0)$$ output when plugged into the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates19a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a1f74d6coordinates19a-h4"],"title":"Plugging in $$(-6,-4)$$","text":"What does the ordered pair $$(-6,-4)$$ output when plugged into the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates19a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a1f74d6coordinates19a-h5"],"title":"Plugging in $$(-4,-1)$$","text":"What does the ordered pair $$(-4,-1)$$ output when plugged into the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates19a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(2,0)$$ and $$(-6,-4)$$"],"dependencies":["a1f74d6coordinates19a-h6"],"title":"Identifying Correct Ordered Pairs","text":"Which ordered pairs satisfy the solution to the given equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(2,0)$$ and $$(-6,-4)$$","$$(-4,-1)$$ and $$(2,0)$$","$$(-6,-4)$$ and $$(-4,1)$$"]}]}}]},{"id":"a1f74d6coordinates2","title":"Identify Points in Quadrants","body":"Plot the following point and identify the quadrant in which it is located.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Use the Rectangular Coordinate System","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f74d6coordinates2a","stepAnswer":["Quadrant $$3$$"],"problemType":"MultipleChoice","stepTitle":"Plot $$(-3,-4)$$ in the rectangular coordinate system and identify the quadrant in which the point is located.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Quadrant $$3$$","choices":["Quadrant $$1$$","Quadrant $$2$$","Quadrant $$3$$","Quadrant $$4$$"],"hints":{"DefaultPathway":[{"id":"a1f74d6coordinates2a-h1","type":"hint","dependencies":[],"title":"Know Quadrants","text":"The quadrants are the $$4$$ regions of the graph, divided by the axes. The first quadrant is the top right region. The quadrants then continue counter clockwise in order, such that the top left region is the second quadrant.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates2a-h2","type":"hint","dependencies":["a1f74d6coordinates2a-h1"],"title":"Use Negatives","text":"If the $$x$$ coordinate is negative, the point lies to the left of the $$y$$ axis. If the $$y$$ coordinate is negative, the point lies below the $$x$$ axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates2a-h3","type":"hint","dependencies":["a1f74d6coordinates2a-h2"],"title":"Know Where the Points Lie","text":"Since $$x$$ is negative, and $$y$$ is negative, the point is in quadrant $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f74d6coordinates20","title":"Verify Solutions to an Equation in Two Variables","body":"In the following exercises, which ordered pairs are solutions to the given equations?","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Use the Rectangular Coordinate System","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f74d6coordinates20a","stepAnswer":["$$(-3,0)$$ and $$(9,4)$$ and $$(-6,-1)$$"],"problemType":"MultipleChoice","stepTitle":"$$y=\\\\frac{x}{3}+1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-3,0)$$ and $$(9,4)$$ and $$(-6,-1)$$","choices":["$$(-3,0)$$ and $$(9,4)$$","$$(9,4)(-6,-1)$$","$$(-6,-1)$$ and $$(-3,0)$$","$$(-3,0)$$ and $$(9,4)$$ and $$(-6,-1)$$"],"hints":{"DefaultPathway":[{"id":"a1f74d6coordinates20a-h1","type":"hint","dependencies":[],"title":"Understanding Coordinate Format","text":"The ordered pairs are given in coordinate format with (x,y). 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$$y$$?","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-3$$","hints":{"DefaultPathway":[{"id":"a1f74d6coordinates28b-h1","type":"hint","dependencies":[],"title":"Plugging in x-value","text":"After plugging in $$x=3$$, use PEMDAS to determine how to solve the equation for $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates28b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a1f74d6coordinates28b-h1"],"title":"Solving for $$y$$","text":"What is the value of $$y$$ when $$x=3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a1f74d6coordinates28c","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"$$y=\\\\left(-\\\\frac{2x}{3}\\\\right)-1$$","stepBody":"If $$x=-3$$, what is $$y$$?","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a1f74d6coordinates28c-h1","type":"hint","dependencies":[],"title":"Plugging in x-value","text":"After plugging in $$x=-3$$, use PEMDAS to determine how to solve the equation for $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates28c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a1f74d6coordinates28c-h1"],"title":"Solving for $$y$$","text":"What is the value of $$y$$ when $$x=-3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f74d6coordinates29","title":"Find Solutions to a Linear Equation","body":"In the following exercise, find the other variable given the equation and the value of another 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The first quadrant is the top right region. The quadrants then continue counter clockwise in order, such that the top left region is the second quadrant.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates3a-h2","type":"hint","dependencies":["a1f74d6coordinates3a-h1"],"title":"Use Negatives","text":"If the $$x$$ coordinate is negative, the point lies to the left of the $$y$$ axis. If the $$y$$ coordinate is negative, the point lies below the $$x$$ axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates3a-h3","type":"hint","dependencies":["a1f74d6coordinates3a-h2"],"title":"Know Where the Points Lie","text":"Since $$x$$ is positive, and $$y$$ is negative, the point is in quadrant $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f74d6coordinates30","title":"Find Solutions to a Linear Equation","body":"In the following exercise, find the other variable given the equation and the value of another variable.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Use the Rectangular Coordinate 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If the $$y$$ coordinate is negative, the point lies below the $$x$$ axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates4a-h3","type":"hint","dependencies":["a1f74d6coordinates4a-h2"],"title":"Know Where the Points Lie","text":"Since $$x$$ is negative, and $$y$$ is positive, the point is in quadrant $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f74d6coordinates5","title":"Identify Points in Quadrants","body":"Plot the following point and identify the quadrant in which it is located.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Use the Rectangular Coordinate System","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f74d6coordinates5a","stepAnswer":["Quadrant $$1$$"],"problemType":"MultipleChoice","stepTitle":"Plot $$(3,\\\\frac{5}{2})$$ in the rectangular coordinate system and identify the quadrant in which the point is located.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Quadrant $$1$$","choices":["Quadrant $$1$$","Quadrant $$2$$","Quadrant $$3$$","Quadrant $$4$$"],"hints":{"DefaultPathway":[{"id":"a1f74d6coordinates5a-h1","type":"hint","dependencies":[],"title":"Know Quadrants","text":"The quadrants are the $$4$$ regions of the graph, divided by the axes. 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If the $$y$$ coordinate is negative, the point lies below the $$x$$ axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates5a-h3","type":"hint","dependencies":["a1f74d6coordinates5a-h2"],"title":"Know Where the Points Lie","text":"Since $$x$$ is positive, and $$y$$ is positive, the point is in quadrant $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f74d6coordinates6","title":"Verify Ordered Pairs as Solutions","body":"Determine if the following point is a valid solution to the equation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Use the Rectangular Coordinate System","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f74d6coordinates6a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Determine if $$(0,2)$$ is a solution to the equation $$x+4y=8$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a1f74d6coordinates6a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute the $$x$$ and $$y$$ values of the ordered pair into the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates6a-h2","type":"hint","dependencies":["a1f74d6coordinates6a-h1"],"title":"Simplify","text":"Simplify each side such that their is a single number on both sides of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates6a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a1f74d6coordinates6a-h2"],"title":"Interpret","text":"If both sides of the equation equal each other, the ordered pair is a solution. 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Is the ordered pair a solution in this case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a1f74d6coordinates7a-h4","type":"hint","dependencies":["a1f74d6coordinates7a-h3"],"title":"Answer","text":"Therefore, the ordered pair is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f74d6coordinates8","title":"Verify Ordered Pairs as Solutions","body":"Determine if the following point is a valid solution to the equation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Use the Rectangular Coordinate System","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a1f74d6coordinates8a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Determine if $$(-4,3)$$ is a solution to the equation $$x+4y=8$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a1f74d6coordinates8a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute the $$x$$ and $$y$$ values of the ordered pair into the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates8a-h2","type":"hint","dependencies":["a1f74d6coordinates8a-h1"],"title":"Simplify","text":"Simplify each side such that their is a single number on both sides of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a1f74d6coordinates8a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a1f74d6coordinates8a-h2"],"title":"Interpret","text":"If both sides of the equation equal each other, the ordered pair is a solution. 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a1f9370SolvingFormul16","title":"Solve the Formula for the given variable","body":"Solve the formula to get an expression for L.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.3 Solve a Formula for a Specific Variable","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1f9370SolvingFormul16a","stepAnswer":["$$L=\\\\frac{V}{W H}$$"],"problemType":"TextBox","stepTitle":"$$V=LWH$$ for L","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$L=\\\\frac{V}{W H}$$","hints":{"DefaultPathway":[{"id":"a1f9370SolvingFormul16a-h1","type":"hint","dependencies":[],"title":"Divide","text":"Divide on both sides by WH in order to isolate L.","variabilization":{},"oer":"https://OATutor.io 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4.0>"}]}}]},{"id":"a1f9370SolvingFormul9","title":"Solve a Formula for a Specific Variable","body":"Solve the formula to get an expression for $$y$$.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.3 Solve a Formula for a Specific Variable","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a1f9370SolvingFormul9a","stepAnswer":["$$y=\\\\frac{15-8x}{7}$$"],"problemType":"TextBox","stepTitle":"$$8x+7y=15$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=\\\\frac{15-8x}{7}$$","hints":{"DefaultPathway":[{"id":"a1f9370SolvingFormul9a-h1","type":"hint","dependencies":[],"title":"Isolating the Variable","text":"Simplify the equation by isolating the requested variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad1","title":"Solving Quadratic Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad1a","stepAnswer":["2/3, -1/2"],"problemType":"TextBox","stepTitle":"$$2+z=6z^2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{2}{3}$$, $$\\\\frac{-1}{2}$$","hints":{"DefaultPathway":[{"id":"a20771equad1a-h1","type":"hint","dependencies":[],"title":"Completing the Square","text":"To complete the square, we need to add a term to make the entire expression of the form $$a^2+2ab+b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad1a-h2","type":"hint","dependencies":["a20771equad1a-h1"],"title":"Completing the Square","text":"In this case, we want to add a term to $$6z^2-z$$ to make it a perfect square. Assume $$6z^2$$ is the $$a^2$$ term and $$-z$$ is the 2ab term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{24}$$"],"dependencies":["a20771equad1a-h2"],"title":"Completing the Square","text":"What term should we add to both sides of the equation to make this true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad1a-h4","type":"hint","dependencies":["a20771equad1a-h3"],"title":"Simplification","text":"Now that we added $$\\\\frac{1}{24}$$ to both sides, we get that $${\\\\left(\\\\sqrt{6} x-\\\\frac{\\\\sqrt{6}}{12}\\\\right)}^2=2+\\\\frac{1}{24}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad1a-h5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["2/3, -1/2"],"dependencies":["a20771equad1a-h4"],"title":"Simplification","text":"By taking the square root of both sides, what two answers do we get for $$z$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad10","title":"Solving a Polynomial of Higher Degree by Factoring","body":"Solve the equation by factoring.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad10a","stepAnswer":["0, -1, -10"],"problemType":"TextBox","stepTitle":"$$x^3+11x^2+10x=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$0$$, $$-1$$, $$-10$$","hints":{"DefaultPathway":[{"id":"a20771equad10a-h1","type":"hint","dependencies":[],"title":"Factoring Out Terms","text":"Since all of the terms inclue $$x$$, we can factor it out of the equation: $$x \\\\left(x^2+11x+10\\\\right)=0$$,","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad10a-h2","type":"hint","dependencies":["a20771equad10a-h1"],"title":"Grouping","text":"Now we can factor by grouping. $$1\\\\times10=10$$, so we should look for two numbers that multiply to $$10$$ and add to $$11$$. $$10$$ and $$1$$ satisfy this.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x+1\\\\right) \\\\left(x+10\\\\right)$$"],"dependencies":["a20771equad10a-h2"],"title":"Grouping","text":"What is the factored expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad10a-h4","type":"hint","dependencies":["a20771equad10a-h3"],"title":"Grouping","text":"We can rewrite the expression as $$x^2+x+10x+10$$ or $$x^2+x+10x+10$$. Let\'s factor $$x^2+x$$. $$x$$ goes into both expressions, so let\'s rewrite this as $$x \\\\left(x+1\\\\right)$$. Now, factor $$10x+10$$. $$10$$ goes into both terms, so it is rewritten as $$10\\\\left(x+1\\\\right)$$. Because both $$x \\\\left(x+1\\\\right)$$ and $$10\\\\left(x+1\\\\right)$$ are multiplied by $$x+1$$, we can use the distributive property to rewrite the expression as $$\\\\left(x+1\\\\right) \\\\left(x+10\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad10a-h5","type":"hint","dependencies":["a20771equad10a-h4"],"title":"Zero Product Property","text":"Now, we can bring back the $$x$$, making our equation $$x \\\\left(x+1\\\\right) \\\\left(x+10\\\\right)$$. Using the Zero Product Property, we can set all the terms equal to zero and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad10a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a20771equad10a-h5"],"title":"Zero Product Property","text":"What what value of $$x$$ makes $$x=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad10a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a20771equad10a-h6"],"title":"Zero Product Property","text":"What what value of $$x$$ makes $$x+1=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad10a-h8","type":"hint","dependencies":["a20771equad10a-h7"],"title":"Zero Product Property","text":"Subtract both sides of the equation by $$1$$ to get $$x=-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad10a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-10$$"],"dependencies":["a20771equad10a-h8"],"title":"Zero Product Property","text":"What what value of $$x$$ makes $$x+10=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad10a-h10","type":"hint","dependencies":["a20771equad10a-h9"],"title":"Zero Product Property","text":"Subtract $$10$$ from both sides of the equation to get $$x=-10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad10a-h11","type":"hint","dependencies":["a20771equad10a-h10"],"title":"Final Answer","text":"So, our factors are $$x=0$$, $$-1$$, $$-10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad11","title":"Solving a Quadratic Equation Using the Square Root Property","body":"Solve the quadratic equation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad11a","stepAnswer":["(2*sqrt(2)), (-2*sqrt(2))"],"problemType":"TextBox","stepTitle":"$$x^2=8$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2\\\\sqrt{2}$$, $$\\\\left(-2\\\\sqrt{2}\\\\right)$$","hints":{"DefaultPathway":[{"id":"a20771equad11a-h1","type":"hint","dependencies":[],"title":"Square Root","text":"Take the +/- square root of both sides: $$x=\\\\pm \\\\sqrt{8}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad11a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(2*sqrt(2)), (-2*sqrt(2))"],"dependencies":["a20771equad11a-h1"],"title":"Square Root","text":"What is $$\\\\pm \\\\sqrt{8}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad11a-h3","type":"hint","dependencies":["a20771equad11a-h2"],"title":"Square Root","text":"$$\\\\sqrt{4}=2$$, and $$\\\\frac{8}{4}=2$$, so $$\\\\sqrt{8}=2\\\\sqrt{2}$$. Remember to take the +/- values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad12","title":"Solving a Quadratic Equation Using the Square Root Property","body":"Solve the quadratic equation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad12a","stepAnswer":["sqrt(6)/2, -sqrt(6)/2"],"problemType":"TextBox","stepTitle":"$$4x^2+1=7$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{\\\\sqrt{6}}{2}$$, $$\\\\frac{-\\\\sqrt{6}}{2}$$","hints":{"DefaultPathway":[{"id":"a20771equad12a-h1","type":"hint","dependencies":[],"title":"Simplify the Expression","text":"To isolate the variable, first subtract both sides by one to get $$4x^2=6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad12a-h2","type":"hint","dependencies":["a20771equad12a-h1"],"title":"Simplify the Expression","text":"Next, divide both sides by 4: $$x^2=\\\\frac{6}{4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad12a-h3","type":"hint","dependencies":["a20771equad12a-h2"],"title":"Square Root","text":"Take the +/- square root of both sides: $$x=\\\\pm \\\\sqrt{\\\\frac{6}{4}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad12a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["sqrt(6)/2, -sqrt(6)/2"],"dependencies":["a20771equad12a-h3"],"title":"Square Root","text":"What is $$\\\\pm \\\\sqrt{\\\\frac{6}{4}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad12a-h5","type":"hint","dependencies":["a20771equad12a-h4"],"title":"Square Root","text":"$$\\\\sqrt{6}$$ is in its simplest form, so we can leave it as it is. $$\\\\sqrt{4}=2$$, so the final answer is $$\\\\frac{\\\\pm \\\\sqrt{6}}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad13","title":"Solving a Quadratic Equation Using the Square Root Property","body":"Solve the quadratic equation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad13a","stepAnswer":["sqrt(5)+4, -sqrt(5)+4"],"problemType":"TextBox","stepTitle":"$${3\\\\left(x-4\\\\right)}^2=15$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\sqrt{5}+4$$, $$-\\\\sqrt{5}+4$$","hints":{"DefaultPathway":[{"id":"a20771equad13a-h1","type":"hint","dependencies":[],"title":"Simplify the Expression","text":"To isolate the variable, first divide both sides by $$3$$ to get $${\\\\left(x-4\\\\right)}^2=5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad13a-h2","type":"hint","dependencies":["a20771equad13a-h1"],"title":"Square Root","text":"Take the +/- square root of both sides: $$x-4=\\\\pm \\\\sqrt{5}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad13a-h3","type":"hint","dependencies":["a20771equad13a-h2"],"title":"Isolating $$x$$","text":"Add $$4$$ to both sides of the equation: $$x=\\\\pm \\\\sqrt{5}+4$$. This is our final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad14","title":"Solving a Quadratic by Completing the Square","body":"Solve the quadratic equation by completing the square.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad14a","stepAnswer":["(3+sqrt(29))/2, (3-sqrt(29))/2"],"problemType":"TextBox","stepTitle":"$$x^2-3x-5=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{3+\\\\sqrt{29}}{2}$$, $$\\\\frac{3-\\\\sqrt{29}}{2}$$","hints":{"DefaultPathway":[{"id":"a20771equad14a-h1","type":"hint","dependencies":[],"title":"Isolating the Variable","text":"First, we need to move the constant term to the other side by adding both sides by 5: $$x^2-3x=5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad14a-h2","type":"hint","dependencies":["a20771equad14a-h1"],"title":"Completing the Square","text":"Now, we can take our $$b$$ term and find the value we need to complete the square. $$\\\\frac{\\\\left(-3\\\\right)}{2}=\\\\left(-\\\\frac{3}{2}\\\\right)$$. Then, $${\\\\left(-\\\\frac{3}{2}\\\\right)}^2=\\\\frac{9}{4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad14a-h3","type":"hint","dependencies":["a20771equad14a-h2"],"title":"Completing the Square","text":"Here, we can add $$\\\\frac{9}{4}$$ to both sides of the equation: $$x^2-3x+\\\\frac{9}{4}=5+\\\\frac{9}{4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad14a-h4","type":"hint","dependencies":["a20771equad14a-h3"],"title":"Factoring","text":"Finally, we can simplify and factor: $${\\\\left(x-\\\\frac{3}{2}\\\\right)}^2=\\\\frac{29}{4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad14a-h5","type":"hint","dependencies":["a20771equad14a-h4"],"title":"Square Root","text":"We now take the square root of both sides: $$x-\\\\frac{3}{2}=\\\\left(+plusminus\\\\right)+\\\\frac{\\\\sqrt{29}}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad14a-h6","type":"hint","dependencies":["a20771equad14a-h5"],"title":"Simplify the Expression","text":"The last step is to add $$\\\\frac{3}{2}$$ to both sides, so the answers are $$\\\\frac{3+\\\\sqrt{29}}{2}$$ and $$\\\\frac{3-\\\\sqrt{29}}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad15","title":"Solving a Quadratic by Completing the Square","body":"Solve the quadratic equation by completing the square.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad15a","stepAnswer":["(3+sqrt(22)), (3\u2212sqrt(22))"],"problemType":"TextBox","stepTitle":"$$2x-6x=13$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$3+\\\\sqrt{22}$$, $$3-\\\\sqrt{22}$$","hints":{"DefaultPathway":[{"id":"a20771equad15a-h1","type":"hint","dependencies":[],"title":"Completing the Square","text":"We need to take our $$b$$ term and find the value we need to complete the square. $$\\\\frac{\\\\left(-6\\\\right)}{2}=\\\\left(-\\\\frac{6}{2}\\\\right)$$. Then, $${\\\\left(-\\\\frac{6}{2}\\\\right)}^2=\\\\frac{36}{4}$$, or $$9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad15a-h2","type":"hint","dependencies":["a20771equad15a-h1"],"title":"Completing the Square","text":"Here, we can add $$9$$ to both sides of the equation: $$x^2-6x+9=13+9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad15a-h3","type":"hint","dependencies":["a20771equad15a-h2"],"title":"Factoring","text":"Finally, we can simplify and factor: $${\\\\left(x-3\\\\right)}^2=22$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad15a-h4","type":"hint","dependencies":["a20771equad15a-h3"],"title":"Square Root","text":"We now take the square root of both sides: $$(x-(3))=\\\\left(+plusminus\\\\right)+\\\\sqrt{22}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad15a-h5","type":"hint","dependencies":["a20771equad15a-h4"],"title":"Simplify the Expression","text":"The last step is to add $$3$$ to both sides, so the answers are $$3+\\\\sqrt{22}$$ and $$3-\\\\sqrt{22}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad16","title":"Solve the Quadratic Equation Using the Quadratic Formula","body":"Solve the quadratic equation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad16a","stepAnswer":["(-5+sqrt(21))/2, (-5-sqrt(21))/2"],"problemType":"TextBox","stepTitle":"$$x^2+5x+1=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{\\\\left(-5+\\\\sqrt{21}\\\\right)}{2}$$, $$\\\\frac{\\\\left(-5-\\\\sqrt{21}\\\\right)}{2}$$","hints":{"DefaultPathway":[{"id":"a20771equad16a-h1","type":"hint","dependencies":[],"title":"Naming Varaible","text":"First, we need to identify our variables: $$a=1$$, $$b=5$$, and $$c=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad16a-h2","type":"hint","dependencies":["a20771equad16a-h1"],"title":"Quadratic Formula","text":"Now we can plug our variables into the Quadratic Formula: $$\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4ac}\\\\right)}{2} a$$ becomes $$\\\\frac{\\\\left(-5\\\\pm \\\\sqrt{5^2-4\\\\times1\\\\times1}\\\\right)}{2\\\\times1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad16a-h3","type":"hint","dependencies":["a20771equad16a-h2"],"title":"Smplifying","text":"This simplies to our final answers: $$\\\\frac{\\\\left(-5+\\\\sqrt{21}\\\\right)}{2}$$ and $$\\\\frac{\\\\left(-5-\\\\sqrt{21}\\\\right)}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad17","title":"Solving a Quadratic Equation with the Quadratic Formula","body":"Use the quadratic formula to solve the equation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad17a","stepAnswer":["$$\\\\frac{\\\\left(-1\\\\pm i \\\\sqrt{7}\\\\right)}{2}$$"],"problemType":"MultipleChoice","stepTitle":"$$x^2+x+2=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{\\\\left(-1\\\\pm i \\\\sqrt{7}\\\\right)}{2}$$","choices":["$$\\\\frac{1\\\\pm i \\\\sqrt{7}}{2}$$","$$\\\\frac{\\\\left(-1\\\\pm i \\\\sqrt{7}\\\\right)}{2}$$","$$\\\\frac{\\\\left(-1\\\\pm i \\\\sqrt{6}\\\\right)}{4}$$","$$\\\\frac{1\\\\pm i \\\\sqrt{7}}{4}$$"],"hints":{"DefaultPathway":[{"id":"a20771equad17a-h1","type":"hint","dependencies":[],"title":"Finding the Coefficients","text":"The first step is to identify the coefficient of each term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad17a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a20771equad17a-h1"],"title":"Coefficient of $$x^2$$","text":"What is the coefficient of $$x^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad17a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a20771equad17a-h2"],"title":"Coefficient of $$x$$","text":"What is the coefficient of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad17a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a20771equad17a-h3"],"title":"Coefficient of $$2$$","text":"What is the coefficient of 2?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad17a-h5","type":"hint","dependencies":["a20771equad17a-h4"],"title":"Using the Quadratic Formula","text":"Next, subsitute the coefficients of each term into the quadratic formula.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad17a-h6","type":"hint","dependencies":["a20771equad17a-h5"],"title":"Quadratic Formula Definition","text":"The quadratic formula is $$x=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4ac}\\\\right)}{2a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad18","title":"Using the Discriminant to Find the Nature of the Solutions to a Quadratic Equation","body":"Using the discriminant, classify the nature of the following quadratic equations.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College 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quadratic formula: $$b^2-4ac$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad18a-h3","type":"hint","dependencies":["a20771equad18a-h2"],"title":"Interpreting Discriminant Values","text":"If the discriminant is $$0$$, there is one rational solution (double solution.) If the discriminant is greater than $$0$$, if has two rational values if it is a perfect square, and two irrational values if it is not. If the discriminant is less than $$0$$, it has two complex solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad18a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["a20771equad18a-h3"],"title":"Identity of $$b^2$$","text":"Using the equation, which is given in the form $${ax}^2+bx+c=0$$, what is $$b^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad18a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["a20771equad18a-h4"],"title":"Identity of 4ac","text":"Using the equation, which is given in the form $${ax}^2+bx+c=0$$, what is 4ac?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a20771equad18b","stepAnswer":["two rational solutions"],"problemType":"MultipleChoice","stepTitle":"$$8x^2+14x+3=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["one rational solution (double solution)","two rational solutions","two irrational solutions","two complex solutions"],"hints":{"DefaultPathway":[{"id":"a20771equad18b-h1","type":"hint","dependencies":[],"title":"Nature of a Quadratic Equation Definition","text":"The nature of a quadratic equation is whether the solutions are real of complex numbers, and how many solutions of each type to expect.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad18b-h2","type":"hint","dependencies":["a20771equad18b-h1"],"title":"Discriminant Definition","text":"For $${ax}^2+bx+c=0$$, where a, $$b$$, and c are real numbers, the discriminant is the expression under the radical in the quadratic formula: $$b^2-4ac$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad18b-h3","type":"hint","dependencies":["a20771equad18b-h2"],"title":"Interpreting Discriminant Values","text":"If the discriminant is $$0$$, there is one rational solution (double solution.) If the discriminant is greater than $$0$$, if has two rational values if it is a perfect square, and two irrational values if it is not. If the discriminant is less than $$0$$, it has two complex solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad18b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$196$$"],"dependencies":["a20771equad18b-h1","a20771equad18b-h2","a20771equad18b-h3"],"title":"Identity of $$b^2$$","text":"Using the equation, which is given in the form $${ax}^2+bx+c=0$$, what is $$b^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad18b-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$96$$"],"dependencies":["a20771equad18b-h1","a20771equad18b-h2","a20771equad18b-h3"],"title":"Identity of 4ac","text":"Using the equation, which is given in the form $${ax}^2+bx+c=0$$, what is 4ac?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad18b-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$100$$"],"dependencies":["a20771equad18b-h4","a20771equad18b-h5"],"title":"Value of the Discriminant","text":"What is the value of the discriminant, $$b^2-4ac$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a20771equad18c","stepAnswer":["two irrational solutions"],"problemType":"MultipleChoice","stepTitle":"$$3x^2-5x-2=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["one rational solution (double solution)","two rational solutions","two irrational solutions","two complex solutions"],"hints":{"DefaultPathway":[{"id":"a20771equad18c-h1","type":"hint","dependencies":[],"title":"Nature of a Quadratic Equation Definition","text":"The nature of a quadratic equation is whether the solutions are real of complex numbers, and how many solutions of each type to expect.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad18c-h2","type":"hint","dependencies":["a20771equad18c-h1"],"title":"Discriminant Definition","text":"For $${ax}^2+bx+c=0$$, where a, $$b$$, and c are real numbers, the discriminant is the expression under the radical in the quadratic formula: $$b^2-4ac$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad18c-h3","type":"hint","dependencies":["a20771equad18c-h2"],"title":"Interpreting Discriminant Values","text":"If the discriminant is $$0$$, there is one rational solution (double solution.) If the discriminant is greater than $$0$$, if has two rational values if it is a perfect square, and two irrational values if it is not. If the discriminant is less than $$0$$, it has two complex solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad18c-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["a20771equad18c-h1","a20771equad18c-h2","a20771equad18c-h3"],"title":"Identity of $$b^2$$","text":"Using the equation, which is given in the form $${ax}^2+bx+c=0$$, what is $$b^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad18c-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-48$$"],"dependencies":["a20771equad18c-h1","a20771equad18c-h2","a20771equad18c-h3"],"title":"Identity of 4ac","text":"Using the equation, which is given in the form $${ax}^2+bx+c=0$$, what is 4ac?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad18c-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$73$$"],"dependencies":["a20771equad18c-h4","a20771equad18c-h5"],"title":"Value of the Discriminant","text":"What is the value of the discriminant, $$b^2-4ac$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a20771equad18d","stepAnswer":["two complex solutions"],"problemType":"MultipleChoice","stepTitle":"$$3x^2-10x+15=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["one rational solution (double solution)","two rational solutions","two irrational solutions","two complex solutions"],"hints":{"DefaultPathway":[{"id":"a20771equad18d-h1","type":"hint","dependencies":[],"title":"Nature of a Quadratic Equation Definition","text":"The nature of a quadratic equation is whether the solutions are real of complex numbers, and how many solutions of each type to expect.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad18d-h2","type":"hint","dependencies":["a20771equad18d-h1"],"title":"Discriminant Definition","text":"For $${ax}^2+bx+c=0$$, where a, $$b$$, and c are real numbers, the discriminant is the expression under the radical in the quadratic formula: $$b^2-4ac$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad18d-h3","type":"hint","dependencies":["a20771equad18d-h2"],"title":"Interpreting Discriminant Values","text":"If the discriminant is $$0$$, there is one rational solution (double solution.) If the discriminant is greater than $$0$$, if has two rational values if it is a perfect square, and two irrational values if it is not. If the discriminant is less than $$0$$, it has two complex solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad18d-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$100$$"],"dependencies":["a20771equad18d-h1","a20771equad18d-h2","a20771equad18d-h3"],"title":"Identity of $$b^2$$","text":"Using the equation, which is given in the form $${ax}^2+bx+c=0$$, what is $$b^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad18d-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$180$$"],"dependencies":["a20771equad18d-h1","a20771equad18d-h2","a20771equad18d-h3"],"title":"Identity of 4ac","text":"Using the equation, which is given in the form $${ax}^2+bx+c=0$$, what is 4ac?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad18d-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-80$$"],"dependencies":["a20771equad18d-h4","a20771equad18d-h5"],"title":"Value of the Discriminant","text":"What is the value of the discriminant, $$b^2-4ac$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad19","title":"Finding the Length of the Missing Side of a Right Triangle","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad19a","stepAnswer":["$$\\\\sqrt{128}$$"],"problemType":"MultipleChoice","stepTitle":"Find the length of the missing side of the right triangle in the diagram.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$\\\\sqrt{128}$$","choices":["$$\\\\sqrt{128}$$","$$8$$","$$\\\\sqrt{138}$$","$$\\\\sqrt{148}$$"],"hints":{"DefaultPathway":[{"id":"a20771equad19a-h1","type":"hint","dependencies":[],"title":"Given Measurements","text":"The diagram gives the measurements of the hypotenuse and one of the legs.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad19a-h2","type":"hint","dependencies":["a20771equad19a-h1"],"title":"Pythagorean Theorem Definition","text":"The Pythagorean Theorem is given as $$a^2+b^2=c^2$$, where a and $$b$$ refer to the legs of a right triangle adjacent to the $$90$$ degree angle, and c refers to the hypotenuse.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad2","title":"Solving Quadratic Equations","body":"For the following exercises, solve the quadratic equation by completing the square. Show each step:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad2a","stepAnswer":["(3+sqrt(17)/4, (3-sqrt(17)/4"],"problemType":"TextBox","stepTitle":"$$2x^2-3x-1=0$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a20771equad2a-h1","type":"hint","dependencies":[],"title":"Completing the Square","text":"To complete the square, we need to add a term to make the entire expression of the form $$a^2+2ab+b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad2a-h2","type":"hint","dependencies":["a20771equad2a-h1"],"title":"Completing the Square","text":"In this case, we want to add a term to $$2x^2-3x$$ to make it a perfect square. Assume $$2x^2$$ is the $$a^2$$ term and $$-3x$$ is the 2ab term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{9}{8}$$"],"dependencies":["a20771equad2a-h2"],"title":"Completing the Square","text":"What term should we add to both sides of the equation to make this true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad2a-h4","type":"hint","dependencies":["a20771equad2a-h3"],"title":"Simplification","text":"Now that we added $$\\\\frac{9}{8}$$ to both sides, we get that $${\\\\left(\\\\sqrt{2} x-\\\\frac{3\\\\sqrt{2}}{4}\\\\right)}^2=1+\\\\frac{9}{8}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad2a-h5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(3+sqrt(17)/4, (3-sqrt(17)/4"],"dependencies":["a20771equad2a-h4"],"title":"Simplification","text":"By taking the square root of both sides, what two answers do we get for $$z$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad20","title":"Try It: Solving the Quadratic Equation with the Quadratic Formula","body":"Solve the quadratic equation using the quadratic formula.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad20a","stepAnswer":["x=-1/3,2/3"],"problemType":"MultipleChoice","stepTitle":"$$9x^2+3x-2=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["x=-1/3,2/3","$$x=-1, 2$$","x=-1/4,2/5","$$x=-3, 3$$"],"hints":{"DefaultPathway":[{"id":"a20771equad20a-h1","type":"hint","dependencies":[],"title":"Quadratic Formula Definition","text":"The quadratic formula to find the roots of an equation $${ax}^2+bx+c=0$$ is $$x=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4ac}\\\\right)}{2} a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a20771equad20a-h1"],"title":"Identifying a","text":"What is a in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad20a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a20771equad20a-h1"],"title":"Identifying $$b$$","text":"What is $$b$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad20a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a20771equad20a-h1"],"title":"Identifying c","text":"What is c in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad21","title":"Try It: Finding the Length of the Missing Side of a Right Triangle","body":"Use the Pythagorean Theorem to solve the following problem:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad21a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"Leg a measures $$4$$ units, leg $$b$$ measures $$3$$ units. Find the length of the hypotenuse.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"a20771equad21a-h1","type":"hint","dependencies":[],"title":"Pythagorean Theorem Definition","text":"The Pythagorean Theorem is given as $$a^2+b^2=c^2$$, where a and $$b$$ refer to the legs of a right triangle adjacent to the $$90$$ degree angle, and c refers to the hypotenuse.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad21a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["a20771equad21a-h1"],"title":"Calculating $$a^2$$","text":"What is $$a^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad21a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a20771equad21a-h1"],"title":"Calculating $$b^2$$","text":"What is $$b^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad21a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["a20771equad21a-h2","a20771equad21a-h3"],"title":"Calculating $$c^2$$","text":"$$c^2$$ is equal to the sum of $$a^2+b^2$$. What is $$c^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad21a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a20771equad21a-h4"],"title":"Calculating c","text":"c is equal to $$\\\\sqrt{c^2}$$. What is c?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad22","title":"Solve the quadratic equation by factoring","body":"Solve the quadratic equation by factoring.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad22a","stepAnswer":["$$x=3, 6$$"],"problemType":"MultipleChoice","stepTitle":"$$x^2-9x+18=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=3, 6$$","choices":["$$x=2, 9$$","$$x=-2, -9$$","$$x=-3, -6$$","$$x=3, 6$$"],"hints":{"DefaultPathway":[{"id":"a20771equad22a-h1","type":"hint","dependencies":[],"title":"Factoring a Quadratic Equation","text":"To factor a quadratic expression $$x^2+bx+c$$, the first step is to find two numbers, $$p$$ and q, for which $$p q=c$$ and $$p+q=b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad22a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18$$"],"dependencies":["a20771equad22a-h1"],"title":"Product of Factors","text":"What is the product of $$\\\\left(-3\\\\right) \\\\left(-6\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad22a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":["a20771equad22a-h1"],"title":"Sum of Factors","text":"What is the sum of $$\\\\left(-3\\\\right)+\\\\left(-6\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad22a-h4","type":"hint","dependencies":["a20771equad22a-h1","a20771equad22a-h2","a20771equad22a-h3"],"title":"Rewriting the Expression","text":"Then, rewrite the equation as $$\\\\left(x+p\\\\right) \\\\left(x+q\\\\right)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad22a-h5","type":"hint","dependencies":["a20771equad22a-h4"],"title":"Answers of a Quadratic Equation","text":"The answers of a quadratic equation are its roots, the $$p$$ and q values that make the expression $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad22a-h6","type":"hint","dependencies":["a20771equad22a-h5"],"title":"Significance of $$p$$ and q","text":"When $$x=-p$$ and -q, $$\\\\left(x+p\\\\right) \\\\left(x+q\\\\right)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad23","title":"Solve the quadratic equation by factoring.","body":"Solve the quadratic equation by factoring:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad23a","stepAnswer":["$$x=\\\\frac{-5}{2}$$, $$x=\\\\frac{-1}{3}$$"],"problemType":"MultipleChoice","stepTitle":"$$6x^2+17x+5=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=\\\\frac{-5}{2}$$, $$x=\\\\frac{-1}{3}$$","choices":["$$x=\\\\frac{-5}{2}$$, $$x=\\\\frac{-1}{3}$$","$$x=-5$$, $$x=-1$$","$$x=\\\\frac{-5}{6}$$, $$x=-1$$","$$x=\\\\frac{-5}{2}$$, $$x=-2$$"],"hints":{"DefaultPathway":[{"id":"a20771equad23a-h1","type":"hint","dependencies":[],"title":"Factoring a Quadratic Equation","text":"To factor a quadratic expression $${ax}^2+bx+c=0$$, the first step is to divide both sides by a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad23a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x^2+\\\\frac{17}{6} x+\\\\frac{5}{6}=0$$"],"dependencies":["a20771equad23a-h1"],"title":"Dividing Both Sides of the Equation","text":"After dividing both sides by a, what does the equation turn into?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x^2+17x+\\\\frac{5}{6}=0$$","$$x^2+\\\\frac{17}{6} x+\\\\frac{5}{6}=0$$","$$x^2+\\\\frac{17}{6} x+5=0$$"]},{"id":"a20771equad23a-h3","type":"hint","dependencies":["a20771equad23a-h2"],"title":"Factoring a Quadratic Equation","text":"The second step is to find two numbers, $$p$$ and q, for which $$p q=c$$ and $$p+q=b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad23a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{6}$$"],"dependencies":["a20771equad23a-h3"],"title":"Product of Factors","text":"What is the product of $$\\\\frac{5}{2} \\\\frac{1}{3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad23a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{17}{6}$$"],"dependencies":["a20771equad23a-h3"],"title":"Sum of Factors","text":"What is the sum of $$\\\\frac{5}{2}+\\\\frac{1}{3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad23a-h6","type":"hint","dependencies":["a20771equad23a-h3","a20771equad23a-h4","a20771equad23a-h5"],"title":"Rewriting the Expression","text":"Then, rewrite the equation as $$\\\\left(x+p\\\\right) \\\\left(x+q\\\\right)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad23a-h7","type":"hint","dependencies":["a20771equad23a-h6"],"title":"Answers of a Quadratic Equation","text":"The answers of a quadratic equation are its roots, the $$p$$ and q values that make the expression $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad23a-h8","type":"hint","dependencies":["a20771equad23a-h7"],"title":"Significance of $$p$$ and q","text":"When $$x=-p$$ and -q, $$\\\\left(x+p\\\\right) \\\\left(x+q\\\\right)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad24","title":"Solve the quadratic equation by factoring","body":"Solve the quadratic equation by factoring.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad24a","stepAnswer":["x=-sqrt(17/3),sqrt(17/3)"],"problemType":"MultipleChoice","stepTitle":"$$3x^2-17=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["x=-sqrt(17/3),sqrt(17/3)","x=-sqrt(6),sqrt(6)","x=-sqrt(17),sqrt(17)","x=-sqrt(3),sqrt(3)"],"hints":{"DefaultPathway":[{"id":"a20771equad24a-h1","type":"hint","dependencies":[],"title":"Factoring a Quadratic Equation","text":"To factor a quadratic expression $${ax}^2-c=0$$, the first step is to divide both sides by a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad24a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x^2-\\\\frac{17}{3}=0$$"],"dependencies":["a20771equad24a-h1"],"title":"Dividing Both Sides of the Equation","text":"After dividing both sides by a, what does the equation turn into?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x^2-\\\\frac{17}{3}=0$$","$$3x^2-\\\\frac{17}{3}=0$$","$$x^2-17=0$$","$$x^2-6=0$$"]},{"id":"a20771equad24a-h3","type":"hint","dependencies":["a20771equad24a-h2"],"title":"Difference of Squares","text":"The expression then turns into a difference of squares, $$x^2-d^2$$, that can be factored as $$\\\\left(x+d\\\\right) \\\\left(x-d\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad24a-h4","type":"hint","dependencies":["a20771equad24a-h3"],"title":"Answers of a Quadratic Equation","text":"The answers of a quadratic equation are its roots, the $$x$$ values that make the expression $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad24a-h5","type":"hint","dependencies":["a20771equad24a-h4"],"title":"Significance of Quadratic Roots","text":"When $$x=-d$$ or $$x=d$$, $$\\\\left(x+d\\\\right) \\\\left(x-d\\\\right)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad25","title":"Solving Quadratics","body":"Solving Quadratics with a Leading Coefficient of $$1$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad25a","stepAnswer":["2, -3"],"problemType":"TextBox","stepTitle":"$$x^2+x-6=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2$$, $$-3$$","hints":{"DefaultPathway":[{"id":"a20771equad25a-h1","type":"hint","dependencies":[],"title":"Factoring","text":"Look for two numbers whose product equals $$-6$$ and whose sum equals $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad25a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["3, -2"],"dependencies":["a20771equad25a-h1"],"title":"Factoring","text":"Look at the possible factors of $$-6$$. Which pair of factors is equal 1?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad25a-h3","type":"hint","dependencies":["a20771equad25a-h2"],"title":"Factoring","text":"The factors will be $$\\\\left(x-2\\\\right) \\\\left(x+3\\\\right)=0$$. Solve the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad25a-h4","type":"hint","dependencies":["a20771equad25a-h3"],"title":"Zero-Product Property","text":"Use the zero-product property. Set each factor equal to zero and solve.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad25a-h5","type":"hint","dependencies":["a20771equad25a-h4"],"title":"Zero-Product Property","text":"Solve $$(x-2)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad25a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a20771equad25a-h5"],"title":"Solving Equations","text":"What is $$x$$ equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad25a-h7","type":"hint","dependencies":["a20771equad25a-h4"],"title":"Zero-Product Property","text":"Solve $$x+3=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad25a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a20771equad25a-h7"],"title":"Solving Equations","text":"What is $$x$$ equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad25a-h9","type":"hint","dependencies":["a20771equad25a-h6","a20771equad25a-h8"],"title":"X-Intercepts","text":"The solutions are the x-intercepts of $$y=x^2+x-6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad26","title":"Solving Quadratics","body":"Solving Quadratics with a Leading Coefficient of $$1$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad26a","stepAnswer":["6, -1"],"problemType":"TextBox","stepTitle":"$$x^2-5x-6=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$6$$, $$-1$$","hints":{"DefaultPathway":[{"id":"a20771equad26a-h1","type":"hint","dependencies":[],"title":"Factoring","text":"Look for two numbers whose product equals $$-6$$ and whose sum equals $$-5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad26a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["-6, 1"],"dependencies":["a20771equad26a-h1"],"title":"Factoring","text":"Look at the possible factors of $$-6$$. Which pair of factors is equal -5?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad26a-h3","type":"hint","dependencies":["a20771equad26a-h2"],"title":"Factoring","text":"The factors will be $$\\\\left(x-6\\\\right) \\\\left(x+5\\\\right)=0$$. Solve the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad26a-h4","type":"hint","dependencies":["a20771equad26a-h3"],"title":"Zero-Product Property","text":"Use the zero-product property. Set each factor equal to zero and solve.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad26a-h5","type":"hint","dependencies":["a20771equad26a-h4"],"title":"Zero-Product Property","text":"Solve $$(x-6)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad26a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a20771equad26a-h5"],"title":"Solving Equations","text":"What is $$x$$ equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad26a-h7","type":"hint","dependencies":["a20771equad26a-h4"],"title":"Zero-Product Property","text":"Solve $$x+1=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad26a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a20771equad26a-h7"],"title":"Solving Equations","text":"What is $$x$$ equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad26a-h9","type":"hint","dependencies":["a20771equad26a-h6","a20771equad26a-h8"],"title":"X-Intercepts","text":"The solutions are the x-intercepts of $$y=x^2-5x-6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad27","title":"Solving Quadratics","body":"Solve the Quadratic Equation by Factoring","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad27a","stepAnswer":["-3, -5"],"problemType":"TextBox","stepTitle":"$$x^2+8x+15=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-3$$, $$-5$$","hints":{"DefaultPathway":[{"id":"a20771equad27a-h1","type":"hint","dependencies":[],"title":"Factoring","text":"Look for two numbers whose product equals $$15$$ and whose sum equals $$15$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad27a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["3, 5"],"dependencies":["a20771equad27a-h1"],"title":"Factoring","text":"Look at the possible factors of $$15$$. Which pair of factors is equal 8?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad27a-h3","type":"hint","dependencies":["a20771equad27a-h2"],"title":"Factoring","text":"The factors will be $$\\\\left(x+5\\\\right) \\\\left(x+3\\\\right)=0$$. Solve the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad27a-h4","type":"hint","dependencies":["a20771equad27a-h3"],"title":"Zero-Product Property","text":"Use the zero-product property. Set each factor equal to zero and solve.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad27a-h5","type":"hint","dependencies":["a20771equad27a-h4"],"title":"Zero-Product Property","text":"Solve $$x+5=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad27a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["a20771equad27a-h5"],"title":"Solving Equations","text":"What is $$x$$ equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad27a-h7","type":"hint","dependencies":["a20771equad27a-h4"],"title":"Zero-Product Property","text":"Solve $$x+3=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad27a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a20771equad27a-h7"],"title":"Solving Equations","text":"What is $$x$$ equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad27a-h9","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["-3, -5"],"dependencies":["a20771equad27a-h6","a20771equad27a-h8"],"title":"X-Intercepts","text":"What are the solutions?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad28","title":"Solving Quadratics","body":"Solve the Quadratic Equation by Factoring","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad28a","stepAnswer":["7, -3"],"problemType":"TextBox","stepTitle":"$$x^2-4x-21=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$7$$, $$-3$$","hints":{"DefaultPathway":[{"id":"a20771equad28a-h1","type":"hint","dependencies":[],"title":"Factoring","text":"Look for two numbers whose product equals $$-21$$ and whose sum equals $$-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad28a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["-7, 3"],"dependencies":["a20771equad28a-h1"],"title":"Factoring","text":"Look at the possible factors of $$-21$$. Which pair of factors is equal -4?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad28a-h3","type":"hint","dependencies":["a20771equad28a-h2"],"title":"Factoring","text":"The factors will be $$\\\\left(x-7\\\\right) \\\\left(x+3\\\\right)=0$$. Solve the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad28a-h4","type":"hint","dependencies":["a20771equad28a-h3"],"title":"Zero-Product Property","text":"Use the zero-product property. Set each factor equal to zero and solve.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad28a-h5","type":"hint","dependencies":["a20771equad28a-h4"],"title":"Zero-Product Property","text":"Solve $$(x-7)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad28a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a20771equad28a-h5"],"title":"Solving Equations","text":"What is $$x$$ equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad28a-h7","type":"hint","dependencies":["a20771equad28a-h4"],"title":"Zero-Product Property","text":"Solve $$x+3=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad28a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a20771equad28a-h7"],"title":"Solving Equations","text":"What is $$x$$ equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad28a-h9","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["3, -3"],"dependencies":["a20771equad28a-h6","a20771equad28a-h8"],"title":"X-Intercepts","text":"What are the solutions?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad29","title":"Solve Quadratic Equation","body":"Using the Zero-Product Property to Solve a Quadratic Equation Written as the Difference of Squares","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad29a","stepAnswer":["3, -3"],"problemType":"TextBox","stepTitle":"$$x^2-9=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$3$$, $$-3$$","hints":{"DefaultPathway":[{"id":"a20771equad29a-h1","type":"hint","dependencies":[],"title":"Difference of Squares","text":"Write the two factors by taking the square root of each term, using a minus sign as the operator in one factor and a plus sign as the operator in the other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad29a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(x-3)$$ and $$x+3$$"],"dependencies":["a20771equad29a-h1"],"title":"Difference of Squares","text":"What are the factors?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(x-6)$$ and $$x+3$$","$$(x-3)$$ and $$x+3$$","$$(x-3)$$ and $$x+9$$"]},{"id":"a20771equad29a-h3","type":"hint","dependencies":["a20771equad29a-h2"],"title":"Difference of Squares","text":"Use the zero-factor property to solve each factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad29a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a20771equad29a-h3"],"title":"Zero-Factor Property","text":"What is the solution of $$(x-3)=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad29a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a20771equad29a-h3"],"title":"Zero-Factor Property","text":"What is the solution of $$x+3=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad29a-h6","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["3, -3"],"dependencies":["a20771equad29a-h4","a20771equad29a-h5"],"title":"Zero-Factor Property","text":"What are the solutions?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad3","title":"Solving Quadratic Equations","body":"For the following exercises, determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad3a","stepAnswer":["not real"],"problemType":"MultipleChoice","stepTitle":"$$x^2+4x+7=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["not real","$$1$$ real","$$2$$ real"],"hints":{"DefaultPathway":[{"id":"a20771equad3a-h1","type":"hint","dependencies":[],"title":"Discriminant","text":"To find the discriminant of a quadratic equation, we calculate $$b^2-4ac$$. If that value is greater than $$0$$ it has $$2$$ real solutions, if it is smaller than $$0$$ it has no real solutions, and if it is equal to $$0$$ it has $$1$$ real solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-12$$"],"dependencies":["a20771equad3a-h1"],"title":"Discriminant","text":"In this case, what is our discriminant?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad3a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["not real"],"dependencies":["a20771equad3a-h2"],"title":"Discriminant","text":"Since this value is less than $$0$$, is this equation\'s solutions real or not real?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["not real","$$1$$ real","$$2$$ real"]}]}}]},{"id":"a20771equad30","title":"Solve Quadratic Equation","body":"Using the Zero-Product Property to Solve a Quadratic Equation Written as the Difference of Squares","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad30a","stepAnswer":["5, -5"],"problemType":"TextBox","stepTitle":"$$x^2-25=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$5$$, $$-5$$","hints":{"DefaultPathway":[{"id":"a20771equad30a-h1","type":"hint","dependencies":[],"title":"Difference of Squares","text":"Write the two factors by taking the square root of each term, using a minus sign as the operator in one factor and a plus sign as the operator in the other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad30a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(x-5)$$ and $$x+5$$"],"dependencies":["a20771equad30a-h1"],"title":"Difference of Squares","text":"What are the factors?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(x-5)$$ and $$(x-10)$$","$$x+5$$ and $$x+5$$","$$(x-5)$$ and $$x+5$$"]},{"id":"a20771equad30a-h3","type":"hint","dependencies":["a20771equad30a-h2"],"title":"Difference of Squares","text":"Use the zero-factor property to solve each factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad30a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a20771equad30a-h3"],"title":"Zero-Factor Property","text":"What is the solution of $$(x-5)=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad30a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["a20771equad30a-h3"],"title":"Finding Factors","text":"What is the solution of $$x+5=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad30a-h6","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["5, -5"],"dependencies":["a20771equad30a-h4","a20771equad30a-h5"],"title":"Zero-Factor Property","text":"What are the solutions?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad31","title":"Solve Quadratic Equation","body":"Solving a Quadratic Equation Using Grouping (please enter your answer as $$x$$, y).","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad31a","stepAnswer":["-3/4, -3"],"problemType":"TextBox","stepTitle":"$$4x^2+15x+9=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{-3}{4}$$, $$-3$$","hints":{"DefaultPathway":[{"id":"a20771equad31a-h1","type":"hint","dependencies":[],"title":"Finding the LCM","text":"Multiply a and c (4 and 9).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad31a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$36$$"],"dependencies":["a20771equad31a-h1"],"title":"Finding the LCM","text":"What is $$4\\\\times9$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad31a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["12, 3"],"dependencies":["a20771equad31a-h2"],"title":"Finding Factors","text":"List the factors of $$36$$. Which pair of factors equals 15?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad31a-h4","type":"hint","dependencies":["a20771equad31a-h3"],"title":"Finding Factors","text":"Separate $$15x$$ into $$12x$$ and $$3x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad31a-h5","type":"hint","dependencies":["a20771equad31a-h4"],"title":"Finding Factors","text":"Rewrite quadratic equation as $$4x^2+3x+12x+9=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad31a-h6","type":"hint","dependencies":["a20771equad31a-h5"],"title":"Finding Factors","text":"Find a common factor between the first two terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad31a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x$$"],"dependencies":["a20771equad31a-h6"],"title":"Finding Factors","text":"What is the common term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad31a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x\\\\left(4x+3\\\\right)$$"],"dependencies":["a20771equad31a-h7"],"title":"Finding Factors","text":"Factor out the common term. What are the first two terms now?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad31a-h9","type":"hint","dependencies":["a20771equad31a-h8"],"title":"Finding Factors","text":"Find a common factor between the last two terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad31a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a20771equad31a-h9"],"title":"Finding Factors","text":"What is the common term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad31a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3\\\\left(4x+3\\\\right)$$"],"dependencies":["a20771equad31a-h10"],"title":"Finding Factors","text":"Factor out the common term. What are the first two terms now?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad31a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4x+3$$"],"dependencies":["a20771equad31a-h11"],"title":"Finding Factors","text":"Factor out the common expression. What is the common expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad31a-h13","type":"hint","dependencies":["a20771equad31a-h12"],"title":"Finding Factors","text":"Rewrite the quadratic equation as $$\\\\left(4x+3\\\\right) \\\\left(x+3\\\\right)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad31a-h14","type":"hint","dependencies":["a20771equad31a-h13"],"title":"Finding Factors","text":"Use zero-product property to find the solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad31a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-3}{4}$$"],"dependencies":["a20771equad31a-h14"],"title":"Finding Factors","text":"What is the solution of $$4x+3=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad31a-h16","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a20771equad31a-h15"],"title":"Finding Factors","text":"What is the solution of $$x+3=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad31a-h17","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["-3/4, -3"],"dependencies":["a20771equad31a-h16"],"title":"Finding Factors","text":"What are the solutions? (Please format as: $$x$$ and y)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad32","title":"Solve Quadratic Equation","body":"Solving a Quadratic Equation Using Grouping","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad32a","stepAnswer":["-1/4, -2/3"],"problemType":"TextBox","stepTitle":"$$12x^2+11x+2=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{-1}{4}$$, $$\\\\frac{-2}{3}$$","hints":{"DefaultPathway":[{"id":"a20771equad32a-h1","type":"hint","dependencies":[],"title":"Finding the LCM","text":"Multiply a and c (12 and 2).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad32a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$24$$"],"dependencies":["a20771equad32a-h1"],"title":"Finding the LCM","text":"What is $$12\\\\times2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad32a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["3, 8"],"dependencies":["a20771equad32a-h2"],"title":"Finding Factors","text":"List the factors of $$24$$. Which pair of factors equals 11?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad32a-h4","type":"hint","dependencies":["a20771equad32a-h3"],"title":"Finding Factors","text":"Separate $$11x$$ into $$3x$$ and $$8x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad32a-h5","type":"hint","dependencies":["a20771equad32a-h4"],"title":"Finding Factors","text":"Rewrite quadratic equation as $$12x^2+3x+8x+2=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad32a-h6","type":"hint","dependencies":["a20771equad32a-h5"],"title":"Finding Factors","text":"Find a common factor between the first two terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad32a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x$$"],"dependencies":["a20771equad32a-h6"],"title":"Finding Factors","text":"What is the common term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad32a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x \\\\left(4x+1\\\\right)$$"],"dependencies":["a20771equad32a-h7"],"title":"Finding Factors","text":"Factor out the common term. What are the first two terms now?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad32a-h9","type":"hint","dependencies":["a20771equad32a-h8"],"title":"Finding Factors","text":"Find a common factor between the last two terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad32a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a20771equad32a-h9"],"title":"Finding Factors","text":"What is the common term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad32a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2\\\\left(4x+1\\\\right)$$"],"dependencies":["a20771equad32a-h10"],"title":"Finding Factors","text":"Factor out the common term. What are the first two terms now?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad32a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4x+1$$"],"dependencies":["a20771equad32a-h11"],"title":"Finding Factors","text":"Factor out the common expression. What is the common expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad32a-h13","type":"hint","dependencies":["a20771equad32a-h12"],"title":"Finding Factors","text":"Rewrite the quadratic equation as $$\\\\left(4x+1\\\\right) \\\\left(3x+2\\\\right)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad32a-h14","type":"hint","dependencies":["a20771equad32a-h13"],"title":"Finding Factors","text":"Use zero-product property to find the solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad32a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{4}$$"],"dependencies":["a20771equad32a-h14"],"title":"Finding Factors","text":"What is the solution of $$4x+1=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad32a-h16","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-2}{3}$$"],"dependencies":["a20771equad32a-h15"],"title":"Finding Factors","text":"What is the solution of $$3x+2=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad32a-h17","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["-1/4, -2/3"],"dependencies":["a20771equad32a-h16"],"title":"Finding Factors","text":"What are the solutions? (please format as: $$x$$, y)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad33","title":"Solve Quadratics by Factoring","body":"For the following exercises, solve the quadratic equation by factoring.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad33a","stepAnswer":["-3/2, 3/2"],"problemType":"TextBox","stepTitle":"$$4x^2=9$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{-3}{2}$$, $$\\\\frac{3}{2}$$","hints":{"DefaultPathway":[{"id":"a20771equad33a-h1","type":"hint","dependencies":[],"title":"Simplify","text":"Move the term on the right hand side to the left hand side to make the right hand side $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad33a-h2","type":"hint","dependencies":["a20771equad33a-h1"],"title":"Difference of Squares","text":"Use difference of squares to rearrange the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad33a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\left(2x-3\\\\right) \\\\left(2x+3\\\\right)=0$$"],"dependencies":["a20771equad33a-h2"],"title":"Difference of Squares","text":"What do we get after rearranging the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\left(2x-3\\\\right) \\\\left(2x+3\\\\right)=0$$","$$\\\\left(x+3\\\\right) \\\\left(2x+3\\\\right)=0$$","$$\\\\left(2x-1\\\\right) \\\\left(2x+9\\\\right)=0$$"]},{"id":"a20771equad33a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["-3/2, 3/2"],"dependencies":["a20771equad33a-h3"],"title":"Factors","text":"What are the two numbers that make the expression $$0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad34","title":"Solve Quadratics by Factoring","body":"For the following exercises, solve the quadratic equation by factoring.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad34a","stepAnswer":["-3, 2"],"problemType":"TextBox","stepTitle":"$$5x^2=5x+30$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-3$$, $$2$$","hints":{"DefaultPathway":[{"id":"a20771equad34a-h1","type":"hint","dependencies":[],"title":"Simplify","text":"Move all the terms from the right hand side to the ledt hand side to make the right hand side equal to $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad34a-h2","type":"hint","dependencies":["a20771equad34a-h1"],"title":"Divide","text":"Divide the left hand side by $$5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad34a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x^2-x-6=0$$"],"dependencies":["a20771equad34a-h2"],"title":"Divide","text":"What do we get after the 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4.0>"},{"id":"a20771equad35a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x\\\\left(7x+3\\\\right)=0$$"],"dependencies":["a20771equad35a-h2"],"title":"Factor","text":"What do we get after the factoring?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x\\\\left(7x^2+3\\\\right)=0$$","$$x(7x-3)=0$$","$$x\\\\left(7x+3\\\\right)=0$$"]},{"id":"a20771equad35a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["0, -3/7"],"dependencies":["a20771equad35a-h3"],"title":"Factors","text":"Which factors will make the left side zero $$(x=0$$ & $$7x$$ - $$3$$ $$=0)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad36","title":"Solve Quadratics by Square Root","body":"For the following exercises, solve the quadratic equation by using the square root property.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad36a","stepAnswer":["-6, 6"],"problemType":"TextBox","stepTitle":"$$x^2=36$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-6$$, $$6$$","hints":{"DefaultPathway":[{"id":"a20771equad36a-h1","type":"hint","dependencies":[],"title":"Square Root","text":"Take the square root of both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad36a-h2","type":"hint","dependencies":["a20771equad36a-h1"],"title":"Positive and Negative","text":"Remember that taking the square root of $$36$$ results in one positive number and one negative number!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad37","title":"Solve Quadratics by Square Root","body":"For the following exercises, solve the quadratic equation by using the square root property.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad37a","stepAnswer":["6, -4"],"problemType":"TextBox","stepTitle":"$${\\\\left(x-1\\\\right)}^2=25$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$6$$, $$-4$$","hints":{"DefaultPathway":[{"id":"a20771equad37a-h1","type":"hint","dependencies":[],"title":"Square root","text":"Take the square root of both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad37a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x-1=\\\\sqrt{25}$$"],"dependencies":["a20771equad37a-h1"],"title":"Square root","text":"What do we get after taking the square root of both sides?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x+1=\\\\sqrt{25}$$","$$x-1=\\\\sqrt{25}$$","$$x-1=\\\\sqrt{20}$$"]},{"id":"a20771equad37a-h3","type":"hint","dependencies":["a20771equad37a-h2"],"title":"Careful!","text":"Remember to be careful about the square root of 36! (remember positve and negative numbers)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad37a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["-5, 5"],"dependencies":["a20771equad37a-h3"],"title":"Square root","text":"What is the square root of 25?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad37a-h5","type":"hint","dependencies":["a20771equad37a-h4"],"title":"Set Equal!","text":"set $$x-1$$ equal to both values and solve for $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad38","title":"Solve Quadratics by Square Root","body":"For the following exercises, solve the quadratic equation by using the square root property.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad38a","stepAnswer":["1, -2"],"problemType":"TextBox","stepTitle":"$${\\\\left(2x+1\\\\right)}^2=9$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$1$$, $$-2$$","hints":{"DefaultPathway":[{"id":"a20771equad38a-h1","type":"hint","dependencies":[],"title":"Square root","text":"Take the square root of both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad38a-h2","type":"hint","dependencies":["a20771equad38a-h1"],"title":"Careful!","text":"Remember to be careful about the square root of 9! (remember positve and negative numbers)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad38a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["3, -3"],"dependencies":["a20771equad38a-h2"],"title":"Square root","text":"What is the square root of 9?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad38a-h4","type":"hint","dependencies":["a20771equad38a-h3"],"title":"Set Equal!","text":"Set $$2x+1$$ to be equal to both values and solve for $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad39","title":"Solve Quadratics by Completing the Square","body":"For the following exercises, solve the quadratic equation by completing the square. Show each step","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad39a","stepAnswer":["11, -2"],"problemType":"TextBox","stepTitle":"$$x^2-9x-22=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$11$$, $$-2$$","hints":{"DefaultPathway":[{"id":"a20771equad39a-h1","type":"hint","dependencies":[],"title":"Simplify","text":"Move the constant term(22) to the right side by adding $$22$$ to both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad39a-h2","type":"hint","dependencies":["a20771equad39a-h1"],"title":"Add","text":"Add $${\\\\left(\\\\frac{9}{2}\\\\right)}^2$$ to both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad39a-h3","type":"hint","dependencies":["a20771equad39a-h2"],"title":"Simplify","text":"Now simplify the expression $$x^2-9x+\\\\frac{81}{4}=22+\\\\frac{81}{4}$$ as a factor squared","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad39a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${\\\\left(x-\\\\frac{9}{2}\\\\right)}^2=22+\\\\frac{81}{4}$$"],"dependencies":["a20771equad39a-h3"],"title":"Simplify","text":"What do we get after rewriting the left-hand side?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$${\\\\left(x-\\\\frac{9}{2}\\\\right)}^2=22+\\\\frac{81}{4}$$","$${\\\\left(x+\\\\frac{9}{3}\\\\right)}^2=22+\\\\frac{81}{4}$$","$${\\\\left(x-\\\\frac{3}{2}\\\\right)}^2=22+\\\\frac{81}{4}$$"]},{"id":"a20771equad39a-h5","type":"hint","dependencies":["a20771equad39a-h4"],"title":"Simplify","text":"Calculate the right hand side $$22+\\\\frac{81}{16}$$, which is equal to $$\\\\frac{169}{4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad39a-h6","type":"hint","dependencies":["a20771equad39a-h5"],"title":"Square Root","text":"Take the square root of both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad39a-h7","type":"hint","dependencies":["a20771equad39a-h6"],"title":"Add","text":"Add $$\\\\frac{9}{2}$$ to both values","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad4","title":"Solving Quadratic Equations","body":"For the following exercises, determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad4a","stepAnswer":["$$1$$ real"],"problemType":"MultipleChoice","stepTitle":"$$9x^2-30x+25=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$1$$ real","choices":["not real","$$1$$ real","$$2$$ real"],"hints":{"DefaultPathway":[{"id":"a20771equad4a-h1","type":"hint","dependencies":[],"title":"Discriminant","text":"To find the discriminant, of a quadratic equation, we calculate $$b^2-4ac$$. If that value is greater than $$0$$ it has $$2$$ real solutions, if it is smaller than $$0$$ it has no real solutions, and if it is equal to $$0$$ it has $$1$$ real solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a20771equad4a-h1"],"title":"Discriminant","text":"In this case, what is our discriminant?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad4a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$1$$ real"],"dependencies":["a20771equad4a-h2"],"title":"Discriminant","text":"Since this value is equal to $$0$$, is this equation\'s solutions real or not real? If it is real, how many solutions does it have?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["not real","$$1$$ real","$$2$$ real"]}]}}]},{"id":"a20771equad40","title":"Solve Quadratics by Completing the Square","body":"For the following exercises, solve the quadratic equation by completing the square.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad40a","stepAnswer":["5, 1"],"problemType":"TextBox","stepTitle":"$$x^2-6x=13$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$5$$, $$1$$","hints":{"DefaultPathway":[{"id":"a20771equad40a-h1","type":"hint","dependencies":[],"title":"Simplify","text":"Add $${\\\\left(\\\\frac{6}{2}\\\\right)}^2$$ to both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad40a-h2","type":"hint","dependencies":["a20771equad40a-h1"],"title":"Simplify","text":"Rewrite as a factor of squares","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad40a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${\\\\left(x-3\\\\right)}^2=4$$"],"dependencies":["a20771equad40a-h2"],"title":"Simplify","text":"What do we get after rewriting the left-hand side?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$${\\\\left(x-3\\\\right)}^2=4$$","$${\\\\left(x+3\\\\right)}^2=6$$","$${\\\\left(x-6\\\\right)}^2=6$$"]},{"id":"a20771equad40a-h4","type":"hint","dependencies":["a20771equad40a-h3"],"title":"Square Root","text":"Take the square root of both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad40a-h5","type":"hint","dependencies":["a20771equad40a-h4"],"title":"Add","text":"Add $$3$$ to both sides and solve for $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad5","title":"Solving Quadratic Equations","body":"For the following exercises, determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad5a","stepAnswer":["$$2$$ real"],"problemType":"MultipleChoice","stepTitle":"$$6x^2-x-2=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2$$ real","choices":["not real","$$1$$ real","$$2$$ real"],"hints":{"DefaultPathway":[{"id":"a20771equad5a-h1","type":"hint","dependencies":[],"title":"Discriminant","text":"To find the discriminant, of a quadratic equation, we calculate $$b^2-4ac$$. If that value is greater than $$0$$ it has $$2$$ real solutions, if it is smaller than $$0$$ it has no real solutions, and if it is equal to $$0$$ it has $$1$$ real solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$49$$"],"dependencies":["a20771equad5a-h1"],"title":"Discriminant","text":"In this case, what is our discriminant?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad5a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2$$ real"],"dependencies":["a20771equad5a-h2"],"title":"Discriminant","text":"Since this value is greater than $$0$$, is this equation\'s solutions real or not real? If it is real, how many solutions does it have?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["not real","$$1$$ real","$$2$$ real"]}]}}]},{"id":"a20771equad6","title":"Solving Quadratic Equations","body":"For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No Real Solution.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad6a","stepAnswer":["(-1+sqrt(17))/2, (-1-sqrt(17))/2"],"problemType":"TextBox","stepTitle":"$$x^2+x=4$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{\\\\left(-1+\\\\sqrt{17}\\\\right)}{2}$$, $$\\\\frac{\\\\left(-1-\\\\sqrt{17}\\\\right)}{2}$$","hints":{"DefaultPathway":[{"id":"a20771equad6a-h1","type":"hint","dependencies":[],"title":"Discriminant","text":"To find the discriminant, of a quadratic equation, we calculate $$b^2-4ac$$. If that value is greater than $$0$$ it has $$2$$ real solutions, if it is smaller than $$0$$ it has no real solutions, and if it is equal to $$0$$ it has $$1$$ real solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$17$$"],"dependencies":["a20771equad6a-h1"],"title":"Discriminant","text":"In this case, what is our discriminant?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad6a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{\\\\left(-1+\\\\sqrt{17}\\\\right)}{2}$$, $$\\\\frac{\\\\left(-1-\\\\sqrt{17}\\\\right)}{2}$$"],"dependencies":["a20771equad6a-h2"],"title":"Discriminant","text":"Since this value is greater than $$0$$, this equation has $$2$$ solutions. Using the rest of the formula (-b+sqrt(discriminant)/2a and -b-sqrt(discriminant)/2a) what are our two solutions?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{\\\\left(-1+\\\\sqrt{17}\\\\right)}{2}$$, $$\\\\frac{\\\\left(-1-\\\\sqrt{17}\\\\right)}{2}$$","$$\\\\frac{\\\\left(-1+\\\\sqrt{17}\\\\right)}{2}$$, $$\\\\frac{\\\\left(-1+\\\\sqrt{17}\\\\right)}{2}$$","$$\\\\frac{\\\\left(-1-\\\\sqrt{17}\\\\right)}{2}$$, $$\\\\frac{\\\\left(-1-\\\\sqrt{17}\\\\right)}{2}$$"]}]}}]},{"id":"a20771equad7","title":"Solving Quadratic Equations","body":"For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No Real Solution.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad7a","stepAnswer":["(5+sqrt(13))/6, (5-sqrt(13))/6"],"problemType":"TextBox","stepTitle":"$$3x^2-5x+1=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{5+\\\\sqrt{13}}{6}$$, $$\\\\frac{5-\\\\sqrt{13}}{6}$$","hints":{"DefaultPathway":[{"id":"a20771equad7a-h1","type":"hint","dependencies":[],"title":"Discriminant","text":"To find the discriminant, of a quadratic equation, we calculate $$b^2-4ac$$. If that value is greater than $$0$$ it has $$2$$ real solutions, if it is smaller than $$0$$ it has no real solutions, and if it is equal to $$0$$ it has $$1$$ real solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$13$$"],"dependencies":["a20771equad7a-h1"],"title":"Discriminant","text":"In this case, what is our discriminant?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad7a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(5+sqrt(13))/6, (5-sqrt(13))/6"],"dependencies":["a20771equad7a-h2"],"title":"Discriminant","text":"Since this value is greater than $$0$$, this equation has $$2$$ solutions. Using the rest of the formula (-b+sqrt(discriminant)/2a and -b-sqrt(discriminant)/2a) what are our two solutions?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad8","title":"Solving Quadratic Equations","body":"For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No Real Solution.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad8a","stepAnswer":["(-1+sqrt(17))/8, (-1-sqrt(17))/8"],"problemType":"TextBox","stepTitle":"$$4+\\\\frac{1}{x}-\\\\frac{1}{x^2}=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{\\\\left(-1+\\\\sqrt{17}\\\\right)}{8}$$, $$\\\\frac{\\\\left(-1-\\\\sqrt{17}\\\\right)}{8}$$","hints":{"DefaultPathway":[{"id":"a20771equad8a-h1","type":"hint","dependencies":[],"title":"Multiplication","text":"In this case, we will multiple both sides by $$x^2$$ to get rid of the fractions. This gives us $$4x^2+x-1=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad8a-h2","type":"hint","dependencies":["a20771equad8a-h1"],"title":"Discriminant","text":"To find the discriminant, of a quadratic equation, we calculate $$b^2-4ac$$. If that value is greater than $$0$$ it has $$2$$ real solutions, if it is smaller than $$0$$ it has no real solutions, and if it is equal to $$0$$ it has $$1$$ real solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$17$$"],"dependencies":["a20771equad8a-h2"],"title":"Discriminant","text":"In this case, what is our discriminant?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad8a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-1+sqrt(17))/8, (-1-sqrt(17))/8"],"dependencies":["a20771equad8a-h3"],"title":"Discriminant","text":"Since this value is greater than $$0$$, this equation has $$2$$ solutions. Using the rest of the formula (-b+sqrt(discriminant)/2a and -b-sqrt(discriminant)/2a) what are our two solutions?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a20771equad9","title":"Solving a Polynomial of Higher Degree by Factoring","body":"Solve the equation by factoring.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Quadratic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a20771equad9a","stepAnswer":["0, -2/3, -1"],"problemType":"TextBox","stepTitle":"$$-3x^3-5x^2-2x=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$0$$, $$\\\\frac{-2}{3}$$, $$-1$$","hints":{"DefaultPathway":[{"id":"a20771equad9a-h1","type":"hint","dependencies":[],"title":"Factoring Out Terms","text":"Since all of the terms inclue $$-x$$, we can factor it out of the equation: $$-x \\\\left(3x^2+5x+2\\\\right)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad9a-h2","type":"hint","dependencies":["a20771equad9a-h1"],"title":"Grouping","text":"Now we can factor by grouping. $$2\\\\times3=6$$, so we should look for two numbers that multiply to $$6$$ and add to $$5$$. $$2$$ and $$3$$ satisfy this.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x+1\\\\right) \\\\left(3x+2\\\\right)$$"],"dependencies":["a20771equad9a-h2"],"title":"Grouping","text":"What is the factored expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad9a-h4","type":"hint","dependencies":["a20771equad9a-h3"],"title":"Grouping","text":"We can rewrite the expression as $$3x^2+3x+2x+2$$ or $$3x^2+3x+2x+2$$. Let\'s factor $$3x^2+3x$$. $$3x$$ goes into both expressions, so let\'s rewrite this as $$3x \\\\left(x+1\\\\right)$$. Now, factor $$2x+2$$. $$2$$ goes into both terms, so it is rewritten as $$2\\\\left(x+1\\\\right)$$. Because both $$3x \\\\left(x+1\\\\right)$$ and $$2\\\\left(x+1\\\\right)$$ are multiplied by $$x+1$$, we can use the distributive property to rewrite the expression as $$\\\\left(x+1\\\\right) \\\\left(3x+2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad9a-h5","type":"hint","dependencies":["a20771equad9a-h4"],"title":"Zero Product Property","text":"Now, we can bring back the $$-x$$, making our equation $$-x \\\\left(x+1\\\\right) \\\\left(3x+2\\\\right)$$. Using the Zero Product Property, we can set all the terms equal to zero and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad9a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a20771equad9a-h5"],"title":"Zero Product Property","text":"What what value of $$x$$ makes $$-x=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad9a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a20771equad9a-h6"],"title":"Zero Product Property","text":"What what value of $$x$$ makes $$x+1=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad9a-h8","type":"hint","dependencies":["a20771equad9a-h7"],"title":"Zero Product Property","text":"Subtract $$1$$ from both sides of the equation to get $$x=-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad9a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-2}{3}$$"],"dependencies":["a20771equad9a-h8"],"title":"Zero Product Property","text":"What what value of $$x$$ makes $$3x+2=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad9a-h10","type":"hint","dependencies":["a20771equad9a-h9"],"title":"Zero Product Property","text":"Subtract $$2$$ from both sides of the equation, then divide by three to get $$x=\\\\frac{-2}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a20771equad9a-h11","type":"hint","dependencies":["a20771equad9a-h10"],"title":"Final Answer","text":"So, our factors are $$x=0$$, $$\\\\frac{-2}{3}$$, $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a23945econtingency1","title":"Contingency Tables","body":"Suppose a study of speeding violations and drivers who use cell phones produced the following fictional data: Of $$305$$ people who use their phone while driving, $$25$$ have gotten a ticket in the past year, while the other $$280$$ people did not recieve a ticket. Of $$450$$ people who don\'t use their phone while driving, $$45$$ have gotten a ticket in the past year, while the other $$405$$ have not recieved a ticket.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Contingency Tables","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a23945econtingency1a","stepAnswer":["$$\\\\frac{305}{755}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a randomly selected driver is a cell phone user?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{305}{755}$$","hints":{"DefaultPathway":[{"id":"a23945econtingency1a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of this situation, we first have to find out the total number of occurrences in which someone is driving. AKA, we want to know how many drivers there are.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$755$$"],"dependencies":["a23945econtingency1a-h1"],"title":"Total Drivers","text":"How many total drivers are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency1a-h3","type":"hint","dependencies":["a23945econtingency1a-h2"],"title":"Probability Rules","text":"After getting the total number of drivers, we just need to find the total number of people who fulfill our original requirement. AKA, how many people use their phone while driving.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$305$$"],"dependencies":["a23945econtingency1a-h3"],"title":"Total Phone Users","text":"How many drivers are there that use their phone whilst driving?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency1a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{305}{755}$$"],"dependencies":["a23945econtingency1a-h4"],"title":"Answer","text":"Now that we have our proportions set up, we can divide the \\"phone\\" drivers by the number of total drivers to get our final probability. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a23945econtingency10","title":"Contingency Tables","body":"Suppose a study of athletes stretching habits was recently released to the public. In this study, $$800$$ people were examined. Out of $$350$$ athletes who stretch before exercise, $$55$$ have gotten injured in the previous year while $$295$$ have not. Out of $$450$$ athletes who do not stretch before exercise, $$231$$ have been injured in the year prior while $$219$$ have not.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Contingency Tables","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a23945econtingency10a","stepAnswer":["$$\\\\frac{55}{800}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a randoly selected athlete stretches before exercising and has gotten injured in the past year?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{55}{800}$$","hints":{"DefaultPathway":[{"id":"a23945econtingency10a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of this situation, we have to first identify how many athletes there are.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$800$$"],"dependencies":["a23945econtingency10a-h1"],"title":"Athlete Count","text":"How many athletes are there in this study?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency10a-h3","type":"hint","dependencies":["a23945econtingency10a-h2"],"title":"Probability Rules","text":"To find the probability of this situation, we can divide the number of athletes who stretch and have gotten injured by the total number of athletes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$55$$"],"dependencies":["a23945econtingency10a-h3"],"title":"Injured and Stretch Count","text":"How many athletes are there that have gotten injured and also stretch before exercising?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{55}{800}$$"],"dependencies":["a23945econtingency10a-h4"],"title":"Answer","text":"Given the information in the hints, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a23945econtingency11","title":"Contingency Tables","body":"Suppose there exists a sample of $$2000$$ people who drink something during their lunch. Of $$1200$$ people who drank water during lunch, $$200$$ did so with a disposeable bottle while the other $$1000$$ did so with a reuseable bottle. Of the $$800$$ people who drank something else, $$700$$ drank out of a disposeable cup, while only $$100$$ drank out of a reuseable cup.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Contingency Tables","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a23945econtingency11a","stepAnswer":["$$\\\\frac{3}{5}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a randomly selected person is drinking water?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{5}$$","hints":{"DefaultPathway":[{"id":"a23945econtingency11a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of this situation, we need to first figure out how many total people are in our situation, then we must figure out the total number of people that \\"could\\" be in our situation (Occurences vs Total Events).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1200$$"],"dependencies":["a23945econtingency11a-h1"],"title":"Number of Water Drinkers","text":"How many people drink water in this sample?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency11a-h3","type":"hint","dependencies":["a23945econtingency11a-h2"],"title":"Probability Rules","text":"Now that we have the total number of people who are already in our situation, we just need to find out the total number of people who \\"could\\" be in our situation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2000$$"],"dependencies":["a23945econtingency11a-h3"],"title":"Number of People","text":"How many people are in this study?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{5}$$"],"dependencies":["a23945econtingency11a-h4"],"title":"Answer","text":"Now that we have the total number of people who drink water and the total number of people in the study, what is our answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a23945econtingency12","title":"Contingency Tables","body":"Suppose there exists a sample of $$2000$$ people who drink something during their lunch. Of $$1200$$ people who drank water during lunch, $$200$$ did so with a disposeable bottle while the other $$1000$$ did so with a reuseable bottle. Of the $$800$$ people who drank something else, $$700$$ drank out of a disposeable cup, while only $$100$$ drank out of a reuseable cup.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Contingency Tables","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a23945econtingency12a","stepAnswer":["$$\\\\frac{2}{5}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a randomly selected person is not drinking water?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2}{5}$$","hints":{"DefaultPathway":[{"id":"a23945econtingency12a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of this situation, we need to first figure out how many total people are in our situation, then we must figure out the total number of people that \\"could\\" be in our situation (Occurences vs Total Events).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$800$$"],"dependencies":["a23945econtingency12a-h1"],"title":"Number of Water Drinkers","text":"How many people are not drinking water in this sample?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency12a-h3","type":"hint","dependencies":["a23945econtingency12a-h2"],"title":"Probability Rules","text":"Now that we have the total number of people who are already in our situation, we just need to find out the total number of people who \\"could\\" be in our situation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2000$$"],"dependencies":["a23945econtingency12a-h3"],"title":"Number of People","text":"How many people are in this study?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency12a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{5}$$"],"dependencies":["a23945econtingency12a-h4"],"title":"Answer","text":"Now that we have the total number of people who don\'t drink water and the total number of people in the study, what is our answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a23945econtingency13","title":"Contingency Tables","body":"Suppose there exists a sample of $$2000$$ people who drink something during their lunch. Of $$1200$$ people who drank water during lunch, $$200$$ did so with a disposeable bottle while the other $$1000$$ did so with a reuseable bottle. Of the $$800$$ people who drank something else, $$700$$ drank out of a disposeable cup, while only $$100$$ drank out of a reuseable cup.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Contingency Tables","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a23945econtingency13a","stepAnswer":["$$\\\\frac{11}{20}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a randomly selected person is drinking out of a reuseable cup?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{11}{20}$$","hints":{"DefaultPathway":[{"id":"a23945econtingency13a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of this situation, we need to first figure out how many total people are in our situation, then we must figure out the total number of people that \\"could\\" be in our situation (Occurences vs Total Events).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1100$$"],"dependencies":["a23945econtingency13a-h1"],"title":"Number of Reuseable Cups","text":"How many people are drinking out of a reuseable cup?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency13a-h3","type":"hint","dependencies":["a23945econtingency13a-h2"],"title":"Probability Rules","text":"Now that we have the total number of people who are already in our situation, we just need to find out the total number of people who \\"could\\" be in our situation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2000$$"],"dependencies":["a23945econtingency13a-h3"],"title":"Number of People","text":"How many people are in this study?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency13a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{11}{20}$$"],"dependencies":["a23945econtingency13a-h4"],"title":"Answer","text":"Now that we have the total number of people in the study along with how many people use reuseable bottles, what is our answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a23945econtingency14","title":"Contingency Tables","body":"Suppose there exists a sample of $$2000$$ people who drink something during their lunch. Of $$1200$$ people who drank water during lunch, $$200$$ did so with a disposeable bottle while the other $$1000$$ did so with a reuseable bottle. Of the $$800$$ people who drank something else, $$700$$ drank out of a disposeable cup, while only $$100$$ drank out of a reuseable cup.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Contingency Tables","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a23945econtingency14a","stepAnswer":["$$\\\\frac{3}{20}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a randomly selected person drank from a disposeable water bottle or another fluid in a reuseable cup?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{20}$$","hints":{"DefaultPathway":[{"id":"a23945econtingency14a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of this situation, we need to first figure out how many total people are in our situation, then we must figure out the total number of people that \\"could\\" be in our situation (Occurences vs Total Events).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$300$$"],"dependencies":["a23945econtingency14a-h1"],"title":"Number in Situation","text":"How many people either drank from a disposeable water bottle or another fluid in a reuseable cup?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency14a-h3","type":"hint","dependencies":["a23945econtingency14a-h2"],"title":"Probability Rules","text":"Now that we have the total number of people who are already in our situation, we just need to find out the total number of people who \\"could\\" be in our situation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency14a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2000$$"],"dependencies":["a23945econtingency14a-h3"],"title":"Number of People","text":"How many people are in this study?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency14a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{20}$$"],"dependencies":["a23945econtingency14a-h4"],"title":"Answer","text":"Now that we have the total number of people in the study along with the number of people who drank from a disposeable water bottle or another fluid in a reuseable cup, what is our answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a23945econtingency15","title":"Contingency Tables","body":"Suppose there exists a sample of $$2000$$ people who drink something during their lunch. Of $$1200$$ people who drank water during lunch, $$200$$ did so with a disposeable bottle while the other $$1000$$ did so with a reuseable bottle. Of the $$800$$ people who drank something else, $$700$$ drank out of a disposeable cup, while only $$100$$ drank out of a reuseable cup.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Contingency Tables","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a23945econtingency15a","stepAnswer":["$$\\\\frac{1}{11}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a randomly selected person is not drinking water given that they are using a reuseable cup?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{11}$$","hints":{"DefaultPathway":[{"id":"a23945econtingency15a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of this situation, we need to first figure out how many total people are in our situation, then we must figure out the total number of people that \\"could\\" be in our situation (Occurences vs Total Events).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$100$$"],"dependencies":["a23945econtingency15a-h1"],"title":"Number in Situation","text":"What is the number of people who are not drinking water given that they are using a reuseable cup?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency15a-h3","type":"hint","dependencies":["a23945econtingency15a-h2"],"title":"Probability Rules","text":"Now that we have the total number of people who are already in our situation, we just need to find out the total number of people who \\"could\\" be in our situation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1100$$"],"dependencies":["a23945econtingency15a-h3"],"title":"Number of People","text":"How many people are in this study and using reuseable cups?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency15a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{11}$$"],"dependencies":["a23945econtingency15a-h4"],"title":"Answer","text":"Now that we have the total number of people who don\'t drink water out of reuseable cups and the number of people who use reuseable cups, what is our answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a23945econtingency16","title":"Contingency Tables","body":"Suppose we take a simple random sample of $$402$$ jazz artists in NYC. Of $$301$$ people that primarily practice at night, $$252$$ of them play Guitar, $$12$$ play Piano, and $$37$$ play the Saxophone. Of $$101$$ people that primarily practice during the day, $$48$$ play Guitar, $$30$$ play Piano, and $$23$$ play the Saxophone.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Contingency Tables","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a23945econtingency16a","stepAnswer":["$$\\\\frac{300}{401}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a randomly selected artist plays the Guitar?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{300}{401}$$","hints":{"DefaultPathway":[{"id":"a23945econtingency16a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of this situation, we need to first figure out how many total people are in our situation, then we must figure out the total number of people that \\"could\\" be in our situation (Occurences vs Total Events).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$300$$"],"dependencies":["a23945econtingency16a-h1"],"title":"Number of Guitars","text":"How many people play the guitar?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency16a-h3","type":"hint","dependencies":["a23945econtingency16a-h2"],"title":"Probability Rules","text":"Now that we have the total number of people who are already in our situation, we just need to find out the total number of people who \\"could\\" be in our situation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency16a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$401$$"],"dependencies":["a23945econtingency16a-h3"],"title":"Number of People","text":"How many people are in this study?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency16a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{300}{401}$$"],"dependencies":["a23945econtingency16a-h4"],"title":"Answer","text":"Now that we have the total number of guitarists and people in the study, what is our answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a23945econtingency17","title":"Contingency Tables","body":"Suppose we take a simple random sample of $$402$$ jazz artists in NYC. Of $$301$$ people that primarily practice at night, $$252$$ of them play Guitar, $$12$$ play Piano, and $$37$$ play the Saxophone. Of $$101$$ people that primarily practice during the day, $$48$$ play Guitar, $$30$$ play Piano, and $$23$$ play the Saxophone.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Contingency Tables","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a23945econtingency17a","stepAnswer":["$$\\\\frac{301}{401}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a randomly selected artist plays during the night?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{301}{401}$$","hints":{"DefaultPathway":[{"id":"a23945econtingency17a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of this situation, we need to first figure out how many total people are in our situation, then we must figure out the total number of people that \\"could\\" be in our situation (Occurences vs Total Events).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency17a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$301$$"],"dependencies":["a23945econtingency17a-h1"],"title":"Number of Night Practicers","text":"How many people practice at night?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency17a-h3","type":"hint","dependencies":["a23945econtingency17a-h2"],"title":"Probability Rules","text":"Now that we have the total number of people who are already in our situation, we just need to find out the total number of people who \\"could\\" be in our situation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency17a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$401$$"],"dependencies":["a23945econtingency17a-h3"],"title":"Number of People","text":"How many people are in this study?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency17a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{301}{401}$$"],"dependencies":["a23945econtingency17a-h4"],"title":"Answer","text":"Now that we have the total number of people who play at night and people in the study, what is our answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a23945econtingency18","title":"Contingency Tables","body":"Suppose we take a simple random sample of $$402$$ jazz artists in NYC. Of $$301$$ people that primarily practice at night, $$252$$ of them play Guitar, $$12$$ play Piano, and $$37$$ play the Saxophone. Of $$101$$ people that primarily practice during the day, $$48$$ play Guitar, $$30$$ play Piano, and $$23$$ play the Saxophone.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Contingency Tables","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a23945econtingency18a","stepAnswer":["$$\\\\frac{275}{401}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a randomly selected artist plays Guitar at Night or plays Saxophone during the Day?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{275}{401}$$","hints":{"DefaultPathway":[{"id":"a23945econtingency18a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of this situation, we need to first figure out how many total people are in our situation, then we must figure out the total number of people that \\"could\\" be in our situation (Occurences vs Total Events).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$275$$"],"dependencies":["a23945econtingency18a-h1"],"title":"Number in Situation","text":"How many people play Guitar at Night or play Saxophone during the Day?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency18a-h3","type":"hint","dependencies":["a23945econtingency18a-h2"],"title":"Probability Rules","text":"Now that we have the total number of people who are already in our situation, we just need to find out the total number of people who \\"could\\" be in our situation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency18a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$401$$"],"dependencies":["a23945econtingency18a-h3"],"title":"Number of People","text":"How many people are in this study?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency18a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{275}{401}$$"],"dependencies":["a23945econtingency18a-h4"],"title":"Answer","text":"Now that we have the total number of people who play Guitar at Night or play Saxophone during the Day and people in the study, what is our answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a23945econtingency19","title":"Contingency Tables","body":"Suppose we take a simple random sample of $$402$$ jazz artists in NYC. Of $$301$$ people that primarily practice at night, $$252$$ of them play Guitar, $$12$$ play Piano, and $$37$$ play the Saxophone. Of $$101$$ people that primarily practice during the day, $$48$$ play Guitar, $$30$$ play Piano, and $$23$$ play the Saxophone.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Contingency Tables","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a23945econtingency19a","stepAnswer":["$$\\\\frac{12}{301}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a random artist plays Piano given that they only play during the Night?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{12}{301}$$","hints":{"DefaultPathway":[{"id":"a23945econtingency19a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of this situation, we need to first figure out how many total people are in our situation, then we must figure out the total number of people that \\"could\\" be in our situation (Occurences vs Total Events).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a23945econtingency19a-h1"],"title":"Number in Situation","text":"How many people play Piano at Night?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency19a-h3","type":"hint","dependencies":["a23945econtingency19a-h2"],"title":"Probability Rules","text":"Now that we have the total number of people who are already in our situation, we just need to find out the total number of people who \\"could\\" be in our situation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency19a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$301$$"],"dependencies":["a23945econtingency19a-h3"],"title":"Number of People","text":"How many people only play during the night?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency19a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{12}{301}$$"],"dependencies":["a23945econtingency19a-h4"],"title":"Answer","text":"Now that we have the total number of people who play Piano at Night and people who play at Night, what is our answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a23945econtingency2","title":"Contingency Tables","body":"Suppose a study of speeding violations and drivers who use cell phones produced the following fictional data: Of $$305$$ people who use their phone while driving, $$25$$ have gotten a ticket in the past year, while the other $$280$$ people did not recieve a ticket. Of $$450$$ people who don\'t use their phone while driving, $$45$$ have gotten a ticket in the past year, while the other $$405$$ have not recieved a ticket.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Contingency Tables","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a23945econtingency2a","stepAnswer":["$$\\\\frac{685}{755}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a randomly selected driver has had no violation in the past year?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{685}{755}$$","hints":{"DefaultPathway":[{"id":"a23945econtingency2a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of this situation, we first have to find out the total number of occurrences in which someone is driving. AKA, we want to know how many drivers there are.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$755$$"],"dependencies":["a23945econtingency2a-h1"],"title":"Total Drivers","text":"How many total drivers are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency2a-h3","type":"hint","dependencies":["a23945econtingency2a-h2"],"title":"Probability Rules","text":"After getting the total number of drivers, we just need to find the total number of people who fulfill our original requirement. AKA, how many people have not had a ticket in the past year.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$685$$"],"dependencies":["a23945econtingency2a-h3"],"title":"Total Phone Users","text":"How many drivers have not gotten a ticket in the past year?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency2a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{685}{755}$$"],"dependencies":["a23945econtingency2a-h4"],"title":"Answer","text":"Now that we have our proportions set up, we can divide the \\"safe\\" drivers by the number of total drivers to get our final probability. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a23945econtingency20","title":"Contingency Tables","body":"Suppose we take a simple random sample of $$402$$ jazz artists in NYC. Of $$301$$ people that primarily practice at night, $$252$$ of them play Guitar, $$12$$ play Piano, and $$37$$ play the Saxophone. Of $$101$$ people that primarily practice during the day, $$48$$ play Guitar, $$30$$ play Piano, and $$23$$ play the Saxophone.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Contingency Tables","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a23945econtingency20a","stepAnswer":["$$\\\\frac{23}{60}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a randomly selected Saxophone player practices during the Day?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{23}{60}$$","hints":{"DefaultPathway":[{"id":"a23945econtingency20a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of this situation, we need to first figure out how many total people are in our situation, then we must figure out the total number of people that \\"could\\" be in our situation (Occurences vs Total Events).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$23$$"],"dependencies":["a23945econtingency20a-h1"],"title":"Number in Situation","text":"What is the number of Saxophonists who play during the day?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency20a-h3","type":"hint","dependencies":["a23945econtingency20a-h2"],"title":"Probability Rules","text":"Now that we have the total number of people who are already in our situation, we just need to find out the total number of people who \\"could\\" be in our situation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency20a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$23$$"],"dependencies":["a23945econtingency20a-h3"],"title":"Number of People","text":"How many people play the saxophone?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency20a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{23}{60}$$"],"dependencies":["a23945econtingency20a-h4"],"title":"Answer","text":"Now that we have the total number of people who play Sax during the day and people who play Sax, what is our answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a23945econtingency21","title":"Contingency Tables","body":"Suppose we obtain data from $$405$$ people who play \\"games\\" at a High School. Of the $$121$$ that prefer games played outside, $$10$$ like to play alone, while the other $$111$$ play with others. Of the $$354$$ people that prefer games played inside, $$245$$ liked to play alone, while the other $$109$$ played with others.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Contingency Tables","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a23945econtingency21a","stepAnswer":["$$\\\\frac{10}{475}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a randomly selected student plays alone and outside?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{10}{475}$$","hints":{"DefaultPathway":[{"id":"a23945econtingency21a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of this situation, we need to first figure out how many total people are in our situation, then we must figure out the total number of people that \\"could\\" be in our situation (Occurences vs Total Events).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency21a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a23945econtingency21a-h1"],"title":"Number in Situation","text":"How many students play alone and outside?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency21a-h3","type":"hint","dependencies":["a23945econtingency21a-h2"],"title":"Probability Rules","text":"Now that we have the total number of people who are already in our situation, we just need to find out the total number of people who \\"could\\" be in our situation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency21a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$475$$"],"dependencies":["a23945econtingency21a-h3"],"title":"Number of Students","text":"How many students are in the sample?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency21a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{10}{475}$$"],"dependencies":["a23945econtingency21a-h4"],"title":"Answer","text":"Now that we have the total number of students and the number of students who play alone and outside, what is our answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a23945econtingency22","title":"Contingency Tables","body":"Suppose we obtain data from $$405$$ people who play \\"games\\" at a High School. Of the $$121$$ that prefer games played outside, $$10$$ like to play alone, while the other $$111$$ play with others. Of the $$354$$ people that prefer games played inside, $$245$$ liked to play alone, while the other $$109$$ played with others.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Contingency Tables","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a23945econtingency22a","stepAnswer":["$$\\\\frac{220}{475}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a randomly selected student plays with other people?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{220}{475}$$","hints":{"DefaultPathway":[{"id":"a23945econtingency22a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of this situation, we need to first figure out how many total people are in our situation, then we must figure out the total number of people that \\"could\\" be in our situation (Occurences vs Total Events).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency22a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$220$$"],"dependencies":["a23945econtingency22a-h1"],"title":"Number in Situation","text":"How many students play with others?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency22a-h3","type":"hint","dependencies":["a23945econtingency22a-h2"],"title":"Probability Rules","text":"Now that we have the total number of people who are already in our situation, we just need to find out the total number of people who \\"could\\" be in our situation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency22a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$475$$"],"dependencies":["a23945econtingency22a-h3"],"title":"Number of Students","text":"How many students are in the sample?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency22a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{220}{475}$$"],"dependencies":["a23945econtingency22a-h4"],"title":"Answer","text":"Now that we have the total number of students and the number of students who play with others, what is our answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a23945econtingency23","title":"Contingency Tables","body":"Suppose we obtain data from $$405$$ people who play \\"games\\" at a High School. Of the $$121$$ that prefer games played outside, $$10$$ like to play alone, while the other $$111$$ play with others. Of the $$354$$ people that prefer games played inside, $$245$$ liked to play alone, while the other $$109$$ played with others.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Contingency Tables","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a23945econtingency23a","stepAnswer":["$$\\\\frac{121}{475}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a randomly selected student prefers to play outside?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{121}{475}$$","hints":{"DefaultPathway":[{"id":"a23945econtingency23a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of this situation, we need to first figure out how many total people are in our situation, then we must figure out the total number of people that \\"could\\" be in our situation (Occurences vs Total Events).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency23a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$121$$"],"dependencies":["a23945econtingency23a-h1"],"title":"Number in Situation","text":"How many students prefer to play outside?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency23a-h3","type":"hint","dependencies":["a23945econtingency23a-h2"],"title":"Probability Rules","text":"Now that we have the total number of people who are already in our situation, we just need to find out the total number of people who \\"could\\" be in our situation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency23a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$475$$"],"dependencies":["a23945econtingency23a-h3"],"title":"Number of Students","text":"How many students are in the sample?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency23a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{121}{475}$$"],"dependencies":["a23945econtingency23a-h4"],"title":"Answer","text":"Now that we have the total number of students and the number of students who play outside, what is our answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a23945econtingency24","title":"Contingency Tables","body":"Suppose we obtain data from $$405$$ people who play \\"games\\" at a High School. Of the $$121$$ that prefer games played outside, $$10$$ like to play alone, while the other $$111$$ play with others. Of the $$354$$ people that prefer games played inside, $$245$$ liked to play alone, while the other $$109$$ played with others.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Contingency Tables","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a23945econtingency24a","stepAnswer":["$$\\\\frac{111}{220}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a randomly selected student plays outside given that they play with friends?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{111}{220}$$","hints":{"DefaultPathway":[{"id":"a23945econtingency24a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of this situation, we need to first figure out how many total people are in our situation, then we must figure out the total number of people that \\"could\\" be in our situation (Occurences vs Total Events).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency24a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$111$$"],"dependencies":["a23945econtingency24a-h1"],"title":"Number in Situation","text":"How many students play outside with friends?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency24a-h3","type":"hint","dependencies":["a23945econtingency24a-h2"],"title":"Probability Rules","text":"Now that we have the total number of people who are already in our situation, we just need to find out the total number of people who \\"could\\" be in our situation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency24a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$220$$"],"dependencies":["a23945econtingency24a-h3"],"title":"Number of Students With Friends","text":"How many students play with friends in the sample?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency24a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{111}{220}$$"],"dependencies":["a23945econtingency24a-h4"],"title":"Answer","text":"Now that we have the total number of students who play with friends and the number of students who play with outside with friends, what is our answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a23945econtingency25","title":"Contingency Tables","body":"Suppose we obtain data from $$405$$ people who play \\"games\\" at a High School. Of the $$121$$ that prefer games played outside, $$10$$ like to play alone, while the other $$111$$ play with others. Of the $$354$$ people that prefer games played inside, $$245$$ liked to play alone, while the other $$109$$ played with others.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Contingency Tables","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a23945econtingency25a","stepAnswer":["$$\\\\frac{121}{475}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a randomly selected student plays alone and outside or with friends outside?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{121}{475}$$","hints":{"DefaultPathway":[{"id":"a23945econtingency25a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of this situation, we need to first figure out how many total people are in our situation, then we must figure out the total number of people that \\"could\\" be in our situation (Occurences vs Total Events).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency25a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$121$$"],"dependencies":["a23945econtingency25a-h1"],"title":"Number in Situation","text":"How many students play alone and outside or with friends outside?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency25a-h3","type":"hint","dependencies":["a23945econtingency25a-h2"],"title":"Probability Rules","text":"Now that we have the total number of people who are already in our situation, we just need to find out the total number of people who \\"could\\" be in our situation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency25a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$475$$"],"dependencies":["a23945econtingency25a-h3"],"title":"Number of Students","text":"How many students are in the sample?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency25a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{121}{475}$$"],"dependencies":["a23945econtingency25a-h4"],"title":"Answer","text":"Now that we have the total number of students and the number of students who play alone and outside or with friends outside, what is our answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a23945econtingency3","title":"Contingency Tables","body":"Suppose a study of speeding violations and drivers who use cell phones produced the following fictional data: Of $$305$$ people who use their phone while driving, $$25$$ have gotten a ticket in the past year, while the other $$280$$ people did not recieve a ticket. Of $$450$$ people who don\'t use their phone while driving, $$45$$ have gotten a ticket in the past year, while the other $$405$$ have not recieved a ticket.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Contingency Tables","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a23945econtingency3a","stepAnswer":["$$\\\\frac{280}{755}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a randomly selected driver is a cell phone user and has had no tickets in the past year?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{280}{755}$$","hints":{"DefaultPathway":[{"id":"a23945econtingency3a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of this situation, we first have to find out the total number of occurrences in which someone is driving. AKA, we want to know how many drivers there are.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$755$$"],"dependencies":["a23945econtingency3a-h1"],"title":"Total Drivers","text":"How many total drivers are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency3a-h3","type":"hint","dependencies":["a23945econtingency3a-h2"],"title":"Probability Rules","text":"After getting the total number of drivers, we just need to find the total number of people who fulfill our original requirement. AKA, how many people are phone users and have not gotten a ticket recently.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$280$$"],"dependencies":["a23945econtingency3a-h3"],"title":"Total Phone Users","text":"How many drivers are there that use their phone whilst driving and have not gotten a ticket in the past year?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{280}{755}$$"],"dependencies":["a23945econtingency3a-h4"],"title":"Answer","text":"Now that we have our proportions set up, we can divide the \\"phone\\" drivers by the number of total drivers to get our final probability. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a23945econtingency4","title":"Contingency Tables","body":"Suppose a study of speeding violations and drivers who use cell phones produced the following fictional data: Of $$305$$ people who use their phone while driving, $$25$$ have gotten a ticket in the past year, while the other $$280$$ people did not recieve a ticket. Of $$450$$ people who don\'t use their phone while driving, $$45$$ have gotten a ticket in the past year, while the other $$405$$ have not recieved a ticket.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Contingency Tables","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a23945econtingency4a","stepAnswer":["$$\\\\frac{710}{755}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a randomly selected driver is a cell phone user or has had no violation in the last year?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{710}{755}$$","hints":{"DefaultPathway":[{"id":"a23945econtingency4a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of this situation, we first have to find out the total number of occurrences in which someone is driving. AKA, we want to know how many drivers there are.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$755$$"],"dependencies":["a23945econtingency4a-h1"],"title":"Total Drivers","text":"How many total drivers are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency4a-h3","type":"hint","dependencies":["a23945econtingency4a-h2"],"title":"Probability Rules","text":"After getting the total number of drivers, we just need to find the total number of people who fulfill our original requirement. AKA, how many people use their phone while driving or have had no tickets in the past year.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$710$$"],"dependencies":["a23945econtingency4a-h3"],"title":"Total Phone Users","text":"How many drivers are there that use their phone whilst driving or have had no violations in the past year? HINT: Inclusion Exclusion Principle","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency4a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{710}{755}$$"],"dependencies":["a23945econtingency4a-h4"],"title":"Answer","text":"Now that we have our proportions set up, we can divide the \\"phone\\" or \\"safe\\" drivers by the number of total drivers to get our final probability. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a23945econtingency5","title":"Contingency Tables","body":"Suppose a study of speeding violations and drivers who use cell phones produced the following fictional data: Of $$305$$ people who use their phone while driving, $$25$$ have gotten a ticket in the past year, while the other $$280$$ people did not recieve a ticket. Of $$450$$ people who don\'t use their phone while driving, $$45$$ have gotten a ticket in the past year, while the other $$405$$ have not recieved a ticket.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Contingency Tables","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a23945econtingency5a","stepAnswer":["$$\\\\frac{25}{70}$$"],"problemType":"TextBox","stepTitle":"What is the probability that given a random driver, they are a cell phone user GIVEN that the driver had a violation in the last year.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{25}{70}$$","hints":{"DefaultPathway":[{"id":"a23945econtingency5a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of this situation, we first have to find out the total number of occurrences in which someone is driving given that they have had a violation in the past year.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$70$$"],"dependencies":["a23945econtingency5a-h1"],"title":"Total Drivers","text":"How many total drivers with tickets in the past year are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency5a-h3","type":"hint","dependencies":["a23945econtingency5a-h2"],"title":"Probability Rules","text":"After getting the total number of drivers, we just need to find the total number of people who fulfill our original requirement. AKA, how many people are cell phone users in the \\"ticket\\" subgroup.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["a23945econtingency5a-h3"],"title":"Total Phone Users","text":"How many drivers are there that use their phone whilst driving given that they have gotten a traffic ticket?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{25}{70}$$"],"dependencies":["a23945econtingency5a-h4"],"title":"Answer","text":"Now that we have our proportions set up, we can divide the \\"phone\\" drivers by the number of total drivers with tickets to get our final probability. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a23945econtingency6","title":"Contingency Tables","body":"Suppose a study of speeding violations and drivers who use cell phones produced the following fictional data: Of $$305$$ people who use their phone while driving, $$25$$ have gotten a ticket in the past year, while the other $$280$$ people did not recieve a ticket. Of $$450$$ people who don\'t use their phone while driving, $$45$$ have gotten a ticket in the past year, while the other $$405$$ have not recieved a ticket.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Contingency Tables","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a23945econtingency6a","stepAnswer":["$$\\\\frac{405}{450}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a randomly selected driver had no violation last year GIVEN that the driver was not a cell phone user?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{405}{450}$$","hints":{"DefaultPathway":[{"id":"a23945econtingency6a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of this situation, we first have to find out the total number of occurrences in which someone is driving. AKA, we want to know how many non-cell phone drivers there are.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$450$$"],"dependencies":["a23945econtingency6a-h1"],"title":"Total Drivers","text":"How many total non-cell phone drivers are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency6a-h3","type":"hint","dependencies":["a23945econtingency6a-h2"],"title":"Probability Rules","text":"After getting the total number of drivers, we just need to find the total number of people who fulfill our original requirement. AKA, how many people have had no tickets in the past year.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$405$$"],"dependencies":["a23945econtingency6a-h3"],"title":"Total Phone Users","text":"How many drivers have had no tickets in the past year given that they are not a cell phone user?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{405}{450}$$"],"dependencies":["a23945econtingency6a-h4"],"title":"Answer","text":"Now that we have our proportions set up, we can divide the \\"non-phone\\" drivers by the number of total drivers to get our final probability. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a23945econtingency7","title":"Contingency Tables","body":"Suppose a study of athletes stretching habits was recently released to the public. In this study, $$800$$ people were examined. Out of $$350$$ athletes who stretch before exercise, $$55$$ have gotten injured in the previous year while $$295$$ have not. Out of $$450$$ athletes who do not stretch before exercise, $$231$$ have been injured in the year prior while $$219$$ have not.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Contingency Tables","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a23945econtingency7a","stepAnswer":["$$\\\\frac{350}{800}$$"],"problemType":"TextBox","stepTitle":"What is the probability that an athlete stretches before exercise?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{350}{800}$$","hints":{"DefaultPathway":[{"id":"a23945econtingency7a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of this situation, we have to first identify how many athletes there are.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$800$$"],"dependencies":["a23945econtingency7a-h1"],"title":"Athlete Count","text":"How many athletes are there in this study?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency7a-h3","type":"hint","dependencies":["a23945econtingency7a-h2"],"title":"Probability Rules","text":"To find the probability of this situation, we can divide how many athletes stretch before exercising by the total number of athletes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$350$$"],"dependencies":["a23945econtingency7a-h3"],"title":"Stretch Count","text":"How many athletes are there that stretch before they workout?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency7a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{350}{800}$$"],"dependencies":["a23945econtingency7a-h4"],"title":"Answer","text":"Given the information in the hints, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a23945econtingency8","title":"Contingency Tables","body":"Suppose a study of athletes stretching habits was recently released to the public. In this study, $$800$$ people were examined. Out of $$350$$ athletes who stretch before exercise, $$55$$ have gotten injured in the previous year while $$295$$ have not. Out of $$450$$ athletes who do not stretch before exercise, $$231$$ have been injured in the year prior while $$219$$ have not.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Contingency Tables","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a23945econtingency8a","stepAnswer":["$$\\\\frac{295}{514}$$"],"problemType":"TextBox","stepTitle":"What is the probability that an athlete stretches before exercising given that they have had no injuries in the past year?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{295}{514}$$","hints":{"DefaultPathway":[{"id":"a23945econtingency8a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of this situation, we first need to find out the total number of athletes that have had no injuries in the past year.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$514$$"],"dependencies":["a23945econtingency8a-h1"],"title":"Injury Count","text":"How many athletes have not gotten injured in the past year?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency8a-h3","type":"hint","dependencies":["a23945econtingency8a-h2"],"title":"Probability Rules","text":"To find the probability of this situation, we can divide the total number of athletes who stretch and have not gotten injured by the total number of athletes who have not gotten injured.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$295$$"],"dependencies":["a23945econtingency8a-h3"],"title":"Stretch and Injury Count","text":"How many athletes stretch and have not gotten injured?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency8a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{295}{514}$$"],"dependencies":["a23945econtingency8a-h4"],"title":"Answer","text":"Given the information in the hints, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a23945econtingency9","title":"Contingency Tables","body":"Suppose a study of athletes stretching habits was recently released to the public. In this study, $$800$$ people were examined. Out of $$350$$ athletes who stretch before exercise, $$55$$ have gotten injured in the previous year while $$295$$ have not. Out of $$450$$ athletes who do not stretch before exercise, $$231$$ have been injured in the year prior while $$219$$ have not.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Contingency Tables","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a23945econtingency9a","stepAnswer":["$$\\\\frac{514}{800}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a randomly selected athlete has had no injuries in the past year?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{514}{800}$$","hints":{"DefaultPathway":[{"id":"a23945econtingency9a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of this situation, we have to first identify how many athletes there are.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$800$$"],"dependencies":["a23945econtingency9a-h1"],"title":"Athlete Count","text":"How many athletes are there in this study?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency9a-h3","type":"hint","dependencies":["a23945econtingency9a-h2"],"title":"Probability Rules","text":"To find the probability of this situation, we can divide the number of non-injured athletes by the total number of athletes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$514$$"],"dependencies":["a23945econtingency9a-h3"],"title":"Non-Injured Count","text":"How many athletes are there in the study that haven\'t gotten injured in the past year?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a23945econtingency9a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{514}{800}$$"],"dependencies":["a23945econtingency9a-h4"],"title":"Answer","text":"Given the information in the hints, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a251a50multiplication1","title":"Multiply the binomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Multiply Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a251a50multiplication1a","stepAnswer":["$$y^2-8y+12$$"],"problemType":"TextBox","stepTitle":"$$(y-6)(y-2)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y^2-8y+12$$","hints":{"DefaultPathway":[{"id":"a251a50multiplication1a-h1","type":"hint","dependencies":[],"title":"Multiply first terms","text":"Use FOIL. The acronym is First, Outer, Inner, Last. Multiply the first terms of each binomial","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication1a-h2","type":"hint","dependencies":["a251a50multiplication1a-h1"],"title":"Multiply outer terms","text":"Multiply the two outer terms of the binomial, which is the first term of the first binomial, and the second term of the second binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication1a-h3","type":"hint","dependencies":["a251a50multiplication1a-h2"],"title":"Multiply inner terms","text":"Multiply the two inner terms, which are the second term of the first binomial, and the first term of the second binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication1a-h4","type":"hint","dependencies":["a251a50multiplication1a-h3"],"title":"Multiply last terms","text":"Multiply the last terms of the two binomials together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication1a-h5","type":"hint","dependencies":["a251a50multiplication1a-h4"],"title":"Add it all","text":"Now, add all the terms together, combining like terms where possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a251a50multiplication10","title":"Multiply the binomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Multiply Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a251a50multiplication10a","stepAnswer":["$$y^3+3y^2-4y-12$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(y^2-4\\\\right) \\\\left(y+3\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y^3+3y^2-4y-12$$","hints":{"DefaultPathway":[{"id":"a251a50multiplication10a-h1","type":"hint","dependencies":[],"title":"Multiply first terms","text":"Use FOIL. The acronym is First, Outer, Inner, Last. Multiply the first terms of each binomial","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication10a-h2","type":"hint","dependencies":["a251a50multiplication10a-h1"],"title":"Multiply outer terms","text":"Multiply the two outer terms of the binomial, which is the first term of the first binomial, and the second term of the second binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication10a-h3","type":"hint","dependencies":["a251a50multiplication10a-h2"],"title":"Multiply inner terms","text":"Multiply the two inner terms, which are the second term of the first binomial, and the first term of the second binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication10a-h4","type":"hint","dependencies":["a251a50multiplication10a-h3"],"title":"Multiply last terms","text":"Multiply the last terms of the two binomials together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication10a-h5","type":"hint","dependencies":["a251a50multiplication10a-h4"],"title":"Add it all","text":"Now, add all the terms together, combining like terms where possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a251a50multiplication11","title":"Multiply the binomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Multiply Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a251a50multiplication11a","stepAnswer":["$$10a^2 b^2+13ab-3$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(5ab-1\\\\right) \\\\left(2ab+3\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10a^2 b^2+13ab-3$$","hints":{"DefaultPathway":[{"id":"a251a50multiplication11a-h1","type":"hint","dependencies":[],"title":"Multiply first terms","text":"Use FOIL. The acronym is First, Outer, Inner, Last. Multiply the first terms of each binomial","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication11a-h2","type":"hint","dependencies":["a251a50multiplication11a-h1"],"title":"Multiply outer terms","text":"Multiply the two outer terms of the binomial, which is the first term of the first binomial, and the second term of the second binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication11a-h3","type":"hint","dependencies":["a251a50multiplication11a-h2"],"title":"Multiply inner terms","text":"Multiply the two inner terms, which are the second term of the first binomial, and the first term of the second binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication11a-h4","type":"hint","dependencies":["a251a50multiplication11a-h3"],"title":"Multiply last terms","text":"Multiply the last terms of the two binomials together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication11a-h5","type":"hint","dependencies":["a251a50multiplication11a-h4"],"title":"Add it all","text":"Now, add all the terms together, combining like terms where possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a251a50multiplication12","title":"Multiply the binomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Multiply Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a251a50multiplication12a","stepAnswer":["$$6x^2 y^2+13xy+6$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(2xy+3\\\\right) \\\\left(3xy+2\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6x^2 y^2+13xy+6$$","hints":{"DefaultPathway":[{"id":"a251a50multiplication12a-h1","type":"hint","dependencies":[],"title":"Multiply first terms","text":"Use FOIL. The acronym is First, Outer, Inner, Last. Multiply the first terms of each binomial","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication12a-h2","type":"hint","dependencies":["a251a50multiplication12a-h1"],"title":"Multiply outer terms","text":"Multiply the two outer terms of the binomial, which is the first term of the first binomial, and the second term of the second binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication12a-h3","type":"hint","dependencies":["a251a50multiplication12a-h2"],"title":"Multiply inner terms","text":"Multiply the two inner terms, which are the second term of the first binomial, and the first term of the second binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication12a-h4","type":"hint","dependencies":["a251a50multiplication12a-h3"],"title":"Multiply last terms","text":"Multiply the last terms of the two binomials together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication12a-h5","type":"hint","dependencies":["a251a50multiplication12a-h4"],"title":"Add it all","text":"Now, add all the terms together, combining like terms where possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a251a50multiplication13","title":"Multiply the binomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Multiply Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a251a50multiplication13a","stepAnswer":["$$x^4+3x^2-40$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(x^2+8\\\\right) \\\\left(x^2-5\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^4+3x^2-40$$","hints":{"DefaultPathway":[{"id":"a251a50multiplication13a-h1","type":"hint","dependencies":[],"title":"Multiply first terms","text":"Use FOIL. The acronym is First, Outer, Inner, Last. Multiply the first terms of each binomial","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication13a-h2","type":"hint","dependencies":["a251a50multiplication13a-h1"],"title":"Multiply outer terms","text":"Multiply the two outer terms of the binomial, which is the first term of the first binomial, and the second term of the second binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication13a-h3","type":"hint","dependencies":["a251a50multiplication13a-h2"],"title":"Multiply inner terms","text":"Multiply the two inner terms, which are the second term of the first binomial, and the first term of the second binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication13a-h4","type":"hint","dependencies":["a251a50multiplication13a-h3"],"title":"Multiply last terms","text":"Multiply the last terms of the two binomials together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication13a-h5","type":"hint","dependencies":["a251a50multiplication13a-h4"],"title":"Add it all","text":"Now, add all the terms together, combining like terms where possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a251a50multiplication14","title":"Multiply the binomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Multiply Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a251a50multiplication14a","stepAnswer":["$$y^4-11y^2+28$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(y^2-7\\\\right) \\\\left(y^2-4\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y^4-11y^2+28$$","hints":{"DefaultPathway":[{"id":"a251a50multiplication14a-h1","type":"hint","dependencies":[],"title":"Multiply first terms","text":"Use FOIL. The acronym is First, Outer, Inner, Last. Multiply the first terms of each binomial","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication14a-h2","type":"hint","dependencies":["a251a50multiplication14a-h1"],"title":"Multiply outer terms","text":"Multiply the two outer terms of the binomial, which is the first term of the first binomial, and the second term of the second binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication14a-h3","type":"hint","dependencies":["a251a50multiplication14a-h2"],"title":"Multiply inner terms","text":"Multiply the two inner terms, which are the second term of the first binomial, and the first term of the second binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication14a-h4","type":"hint","dependencies":["a251a50multiplication14a-h3"],"title":"Multiply last terms","text":"Multiply the last terms of the two binomials together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication14a-h5","type":"hint","dependencies":["a251a50multiplication14a-h4"],"title":"Add it all","text":"Now, add all the terms together, combining like terms where possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a251a50multiplication15","title":"Multiply the binomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Multiply Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a251a50multiplication15a","stepAnswer":["$$9\\\\left(r^2\\\\right) s^2-33rs+28$$"],"problemType":"TextBox","stepTitle":"$$(3rs-7)(3rs-4)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9\\\\left(r^2\\\\right) s^2-33rs+28$$","hints":{"DefaultPathway":[{"id":"a251a50multiplication15a-h1","type":"hint","dependencies":[],"title":"Multiply first terms","text":"Use FOIL. The acronym is First, Outer, Inner, Last. Multiply the first terms of each binomial","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication15a-h2","type":"hint","dependencies":["a251a50multiplication15a-h1"],"title":"Multiply outer terms","text":"Multiply the two outer terms of the binomial, which is the first term of the first binomial, and the second term of the second binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication15a-h3","type":"hint","dependencies":["a251a50multiplication15a-h2"],"title":"Multiply inner terms","text":"Multiply the two inner terms, which are the second term of the first binomial, and the first term of the second binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication15a-h4","type":"hint","dependencies":["a251a50multiplication15a-h3"],"title":"Multiply last terms","text":"Multiply the last terms of the two binomials together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication15a-h5","type":"hint","dependencies":["a251a50multiplication15a-h4"],"title":"Add it all","text":"Now, add all the terms together, combining like terms where possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a251a50multiplication2","title":"Multiply the binomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Multiply Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a251a50multiplication2a","stepAnswer":["$$x^2+11x+24$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(x+8\\\\right) \\\\left(x+3\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^2+11x+24$$","hints":{"DefaultPathway":[{"id":"a251a50multiplication2a-h1","type":"hint","dependencies":[],"title":"Multiply first terms","text":"Use FOIL. The acronym is First, Outer, Inner, Last. Multiply the first terms of each binomial","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication2a-h2","type":"hint","dependencies":["a251a50multiplication2a-h1"],"title":"Multiply outer terms","text":"Multiply the two outer terms of the binomial, which is the first term of the first binomial, and the second term of the second binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication2a-h3","type":"hint","dependencies":["a251a50multiplication2a-h2"],"title":"Multiply inner terms","text":"Multiply the two inner terms, which are the second term of the first binomial, and the first term of the second binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication2a-h4","type":"hint","dependencies":["a251a50multiplication2a-h3"],"title":"Multiply last terms","text":"Multiply the last terms of the two binomials together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication2a-h5","type":"hint","dependencies":["a251a50multiplication2a-h4"],"title":"Add it all","text":"Now, add all the terms together, combining like terms where possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a251a50multiplication3","title":"Multiply the binomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Multiply Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a251a50multiplication3a","stepAnswer":["$$20t^2-88t-9$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(2t-9\\\\right) \\\\left(10t+1\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$20t^2-88t-9$$","hints":{"DefaultPathway":[{"id":"a251a50multiplication3a-h1","type":"hint","dependencies":[],"title":"Multiply first terms","text":"Use FOIL. The acronym is First, Outer, Inner, Last. Multiply the first terms of each binomial","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication3a-h2","type":"hint","dependencies":["a251a50multiplication3a-h1"],"title":"Multiply outer terms","text":"Multiply the two outer terms of the binomial, which is the first term of the first binomial, and the second term of the second binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication3a-h3","type":"hint","dependencies":["a251a50multiplication3a-h2"],"title":"Multiply inner terms","text":"Multiply the two inner terms, which are the second term of the first binomial, and the first term of the second binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication3a-h4","type":"hint","dependencies":["a251a50multiplication3a-h3"],"title":"Multiply last terms","text":"Multiply the last terms of the two binomials together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication3a-h5","type":"hint","dependencies":["a251a50multiplication3a-h4"],"title":"Add it all","text":"Now, add all the terms together, combining like terms where possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a251a50multiplication4","title":"Multiply the binomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Multiply Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a251a50multiplication4a","stepAnswer":["$$6p^2+11p+5$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(6p+5\\\\right) \\\\left(p+1\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6p^2+11p+5$$","hints":{"DefaultPathway":[{"id":"a251a50multiplication4a-h1","type":"hint","dependencies":[],"title":"Multiply first terms","text":"Use FOIL. The acronym is First, Outer, Inner, Last. Multiply the first terms of each binomial","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication4a-h2","type":"hint","dependencies":["a251a50multiplication4a-h1"],"title":"Multiply outer terms","text":"Multiply the two outer terms of the binomial, which is the first term of the first binomial, and the second term of the second binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication4a-h3","type":"hint","dependencies":["a251a50multiplication4a-h2"],"title":"Multiply inner terms","text":"Multiply the two inner terms, which are the second term of the first binomial, and the first term of the second binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication4a-h4","type":"hint","dependencies":["a251a50multiplication4a-h3"],"title":"Multiply last terms","text":"Multiply the last terms of the two binomials together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication4a-h5","type":"hint","dependencies":["a251a50multiplication4a-h4"],"title":"Add it all","text":"Now, add all the terms together, combining like terms where possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a251a50multiplication5","title":"Multiply the binomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Multiply Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a251a50multiplication5a","stepAnswer":["$$q^2+3q-40$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(q-5\\\\right) \\\\left(q+8\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$q^2+3q-40$$","hints":{"DefaultPathway":[{"id":"a251a50multiplication5a-h1","type":"hint","dependencies":[],"title":"Multiply first terms","text":"Use FOIL. The acronym is First, Outer, Inner, Last. Multiply the first terms of each binomial","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication5a-h2","type":"hint","dependencies":["a251a50multiplication5a-h1"],"title":"Multiply outer terms","text":"Multiply the two outer terms of the binomial, which is the first term of the first binomial, and the second term of the second binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication5a-h3","type":"hint","dependencies":["a251a50multiplication5a-h2"],"title":"Multiply inner terms","text":"Multiply the two inner terms, which are the second term of the first binomial, and the first term of the second binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication5a-h4","type":"hint","dependencies":["a251a50multiplication5a-h3"],"title":"Multiply last terms","text":"Multiply the last terms of the two binomials together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication5a-h5","type":"hint","dependencies":["a251a50multiplication5a-h4"],"title":"Add it all","text":"Now, add all the terms together, combining like terms where possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a251a50multiplication6","title":"Multiply the binomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Multiply Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a251a50multiplication6a","stepAnswer":["$$m^2+7m-44$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(m+11\\\\right) \\\\left(m-4\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$m^2+7m-44$$","hints":{"DefaultPathway":[{"id":"a251a50multiplication6a-h1","type":"hint","dependencies":[],"title":"Multiply first terms","text":"Use FOIL. The acronym is First, Outer, Inner, Last. Multiply the first terms of each binomial","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication6a-h2","type":"hint","dependencies":["a251a50multiplication6a-h1"],"title":"Multiply outer terms","text":"Multiply the two outer terms of the binomial, which is the first term of the first binomial, and the second term of the second binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication6a-h3","type":"hint","dependencies":["a251a50multiplication6a-h2"],"title":"Multiply inner terms","text":"Multiply the two inner terms, which are the second term of the first binomial, and the first term of the second binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication6a-h4","type":"hint","dependencies":["a251a50multiplication6a-h3"],"title":"Multiply last terms","text":"Multiply the last terms of the two binomials together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication6a-h5","type":"hint","dependencies":["a251a50multiplication6a-h4"],"title":"Add it all","text":"Now, add all the terms together, combining like terms where possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a251a50multiplication7","title":"Multiply the binomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Multiply Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a251a50multiplication7a","stepAnswer":["$$7m^2-20m-3$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(7m+1\\\\right) \\\\left(m-3\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7m^2-20m-3$$","hints":{"DefaultPathway":[{"id":"a251a50multiplication7a-h1","type":"hint","dependencies":[],"title":"Multiply first terms","text":"Use FOIL. The acronym is First, Outer, Inner, Last. Multiply the first terms of each binomial","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication7a-h2","type":"hint","dependencies":["a251a50multiplication7a-h1"],"title":"Multiply outer terms","text":"Multiply the two outer terms of the binomial, which is the first term of the first binomial, and the second term of the second binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication7a-h3","type":"hint","dependencies":["a251a50multiplication7a-h2"],"title":"Multiply inner terms","text":"Multiply the two inner terms, which are the second term of the first binomial, and the first term of the second binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication7a-h4","type":"hint","dependencies":["a251a50multiplication7a-h3"],"title":"Multiply last terms","text":"Multiply the last terms of the two binomials together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50multiplication7a-h5","type":"hint","dependencies":["a251a50multiplication7a-h4"],"title":"Add it all","text":"Now, add all the terms together, combining like terms where possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a251a50multiplication8","title":"Multiply the binomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Multiply Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a251a50multiplication8a","stepAnswer":["$$33r^2-85r-8$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(3r-8\\\\right) \\\\left(11r+1\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$33r^2-85r-8$$","hints":{"DefaultPathway":[{"id":"a251a50multiplication8a-h1","type":"hint","dependencies":[],"title":"Multiply first terms","text":"Use FOIL. The acronym is First, Outer, Inner, Last. 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The acronym is First, Outer, Inner, Last. 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variables can add together","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a251a50MultiplyPoly10b","stepAnswer":["$$8a^2 b^2+12ab-20$$"],"problemType":"TextBox","stepTitle":"b) $$\\\\left(2ab+5\\\\right) \\\\left(4ab-4\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8a^2 b^2+12ab-20$$","hints":{"DefaultPathway":[{"id":"a251a50MultiplyPoly10b-h1","type":"hint","dependencies":[],"title":"Principle","text":"Use the FOIL method for multiplication","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50MultiplyPoly10b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8a^2 b^2$$"],"dependencies":["a251a50MultiplyPoly10b-h1"],"title":"Multiplication","text":"What is the first term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50MultiplyPoly10b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["-8ab"],"dependencies":["a251a50MultiplyPoly10b-h2"],"title":"Multiplication","text":"What is the outer term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50MultiplyPoly10b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["20ab"],"dependencies":["a251a50MultiplyPoly10b-h3"],"title":"Multiplication","text":"What is the inner term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50MultiplyPoly10b-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-20$$"],"dependencies":["a251a50MultiplyPoly10b-h4"],"title":"Multiplication","text":"What is the last term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a251a50MultiplyPoly10b-h6","type":"hint","dependencies":["a251a50MultiplyPoly10b-h5"],"title":"Principle","text":"The terms with same degree of variables can add together","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a251a50MultiplyPoly11","title":"Multiply a Polynomial by a Polynomial","body":"Multiply the following polynomials","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Multiply Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a251a50MultiplyPoly11a","stepAnswer":["$$2b^3+b^2-7b+24$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(b+3\\\\right) 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What is the value of $$\\\\frac{x-3}{x+4}$$ after plugging in the value $$x=-5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational1a-h10","type":"hint","dependencies":["a276c42SolveRational1a-h9"],"title":"Test a Value in Each Interval.","text":"Since $$8$$ is a positive numer greater than $$0$$, we get the sign of quotient is positive in the interval $$(-\\\\infty,-4)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational1a-h11","type":"hint","dependencies":["a276c42SolveRational1a-h10"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-4,3)$$, take the point $$x=0$$ within the interval and plug it into the original expression $$\\\\frac{x-3}{x+4}$$. We get $$\\\\frac{-3}{4}$$ after plugging in $$x=0$$ into the quotient $$\\\\frac{x-3}{x+4}$$. $$\\\\frac{-3}{4}$$ is a negative number, so we can mark the quotient negative in the interval $$(-4,3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational1a-h12","type":"hint","dependencies":["a276c42SolveRational1a-h11"],"title":"Test a Value in Each Interval.","text":"For the interval $$(3,\\\\infty)$$, take the point $$x=4$$ within the interval and plug it into the original expression $$\\\\frac{x-3}{x+4}$$. We get $$\\\\frac{1}{8}$$ after plugging in $$x=4$$ into the quotient $$\\\\frac{x-3}{x+4}$$. $$\\\\frac{1}{8}$$ is a positive number, so we can mark the quotient positive in the interval $$(3,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational1a-h13","type":"hint","dependencies":["a276c42SolveRational1a-h12"],"title":"Determine the Intervals Where the Inequality is Correct.","text":"We want the quotient to be greater than or equal to zero, so the numbers in the intervals $$(-\\\\infty,-4)$$ and $$(3,\\\\infty)$$ are solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational1a-h14","type":"hint","dependencies":["a276c42SolveRational1a-h13"],"title":"Determine the Critical Points Where the Inequality is Correct","text":"The critical point $$x=-4$$ makes the denominator $$0$$, so it must be excluded from the solution and we mark it with a parenthesis.\\\\nThe critical point $$x=3$$ makes the whole rational expression $$0$$. The inequality requires that the rational expression be greater than or equal to $$0$$. So, $$3$$ is part of the solution and we will mark it with a bracket.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational1a-h15","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-inf,-4)U[3,inf)"],"dependencies":["a276c42SolveRational1a-h14"],"title":"Write the Final Answer in the interval Form","text":"We know that $$(-\\\\infty,4)$$, $$(3,\\\\infty)$$ and $$x=3$$ are the answer, please use \\"[ ]\\". \\"( )\\" and U to write it in a mathematical form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a276c42SolveRational10","title":"Solve Rational Inequality","body":"In the following exercises, solve each rational inequality and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6  Solve Rational Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a276c42SolveRational10a","stepAnswer":["(-inf,-3)U(6,inf)"],"problemType":"TextBox","stepTitle":"$$\\\\frac{6x}{x-6}>2$$","stepBody":"Please enter your answer as (x,y). Use U to indicate the union of two intervals, if applicable. If your answer is the interval from $$x$$ to $$\\\\infty$$, please enter it as $$(x,\\\\infty)$$. If your answer includes the endpoint of the interval, please use [ ] to indicate the inclusion of endpoints. For instance, [x,y) refers to the interval including endpoint $$x$$ and excluding endpoint $$y$$.","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,-3) \\\\cup (6,\\\\infty)$$","hints":{"DefaultPathway":[{"id":"a276c42SolveRational10a-h1","type":"hint","dependencies":[],"title":"Put The Equation In The Correct Form with Zero By Itself On One Side","text":"Write the inequality as one quotient on the left and zero on the right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational10a-h2","type":"hint","dependencies":["a276c42SolveRational10a-h1"],"title":"Put The Equation In The Correct Form with Zero By Itself On One Side","text":"By subtracting $$2$$ to get zero on the right, we get $$\\\\frac{6x}{x-6}-2>0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational10a-h3","type":"hint","dependencies":["a276c42SolveRational10a-h2"],"title":"Rewrite $$2$$ As a Fraction Using the LCD.","text":"LCD for $$\\\\frac{6x}{x-6}$$ and $$2$$ is $$(x-6)$$, we can rewrite $$2$$ as $$\\\\frac{2\\\\left(x-6\\\\right)}{x-6}$$. We can rewrite $$\\\\frac{6x}{x-6}-2$$ as $$\\\\frac{6x}{x-6}-\\\\frac{2\\\\left(x-6\\\\right)}{x-6}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational10a-h4","type":"hint","dependencies":["a276c42SolveRational10a-h3"],"title":"Rewrite $$2$$ As a Fraction Using the LCD.","text":"By subtracting the numerators and placing the difference over the common denominator, we can rewrite ((6*x)/(x-6))-(2(x-6)/(x-6))) as $$\\\\frac{6x-2\\\\left(x-6\\\\right)}{x-6}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational10a-h5","type":"hint","dependencies":["a276c42SolveRational10a-h4"],"title":"Simplify $$\\\\frac{6x-2\\\\left(x-6\\\\right)}{x-6}$$","text":"$$\\\\frac{6x-2\\\\left(x-6\\\\right)}{x-6}=\\\\frac{4x+12}{x-6}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational10a-h6","type":"hint","dependencies":["a276c42SolveRational10a-h5"],"title":"Find The Critical Points of Equation","text":"Determine the critical points\u2014the points where the rational expression will be zero or undefined.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational10a-h7","type":"hint","dependencies":["a276c42SolveRational10a-h6"],"title":"Find The Critical Points of Equation","text":"When is the equation undefine? Note that the equation is undefined when the denominator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational10a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a276c42SolveRational10a-h7"],"title":"Find The Critical Points of Equation","text":"Solve the equation $$x-6=0$$. What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational10a-h9","type":"hint","dependencies":["a276c42SolveRational10a-h8"],"title":"Find The Critical Points of Equation","text":"When does the equation equal zero? Note that the equation is undefined when the numerator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational10a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a276c42SolveRational10a-h9"],"title":"Find The Critical Points of Equation","text":"Solve the equation $$4x+12=0$$. What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational10a-h11","type":"hint","dependencies":["a276c42SolveRational10a-h10"],"title":"Use the Critical Points to Divide the Number Line into Intervals.","text":"Given that the $$x=-3$$ and $$x=6$$ are the two critical points, we can divide the number line into three intervals, namely, $$(-\\\\infty,-3)$$, $$(-3,6)$$ and $$(6,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational10a-h12","type":"hint","dependencies":["a276c42SolveRational10a-h11"],"title":"Test a Value in Each Interval.","text":"To find the sign of each factor in an interval, we choose any point in that interval and use it as a test point. Any point in the interval will give the expression the same sign, so we can choose any point in the interval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational10a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{5}$$"],"dependencies":["a276c42SolveRational10a-h12"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-\\\\infty,-3)$$, take the point $$x=-4$$ within the interval and plug it into the quotient $$\\\\frac{4x+12}{x-6}$$. What is the value of $$\\\\frac{4x+2}{x-2}$$ after plugging in the value $$x=-4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational10a-h14","type":"hint","dependencies":["a276c42SolveRational10a-h13"],"title":"Test a Value in Each Interval.","text":"Since $$\\\\frac{2}{5}$$ is a positive numer greater than $$0$$, we get the sign of quotient is positive in the interval $$(-\\\\infty,-3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational10a-h15","type":"hint","dependencies":["a276c42SolveRational10a-h14"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-3,6)$$, take the point $$x=0$$ within the interval and plug it into the original expression $$\\\\frac{4x+12}{x-6}$$. We get $$-2$$ after plugging in $$x=0$$ into the quotient $$\\\\frac{4x+12}{x-6}$$. $$-2$$ is a negative number, so we can mark the quotient negative in the interval $$(-3,6)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational10a-h16","type":"hint","dependencies":["a276c42SolveRational10a-h15"],"title":"Test a Value in Each Interval.","text":"For the interval $$(6,\\\\infty)$$, take the point $$x=7$$ within the interval and plug it into the original expression $$\\\\frac{4x+12}{x-6}$$. We get $$40$$ after plugging in $$x=7$$ into the quotient $$\\\\frac{4x+12}{x-6}$$. $$40$$ is a positive number, so we can mark the quotient positive in the interval $$(6,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational10a-h17","type":"hint","dependencies":["a276c42SolveRational10a-h16"],"title":"Determine the Intervals Where the Inequality is Correct.","text":"We want the quotient to be greater than zero, so the numbers in the interval $$(-\\\\infty,-3)$$ or $$(6,\\\\infty)$$ are the solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational10a-h18","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-inf,-3)U(6,inf)"],"dependencies":["a276c42SolveRational10a-h17"],"title":"Write the Final Answer in the interval Form","text":"We know that $$(-\\\\infty,-3)$$ and $$(6,\\\\infty)$$ are the answer, please use \\"[ ]\\". \\"( )\\" and U to write it in a mathematical form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a276c42SolveRational11","title":"Solve Rational Inequality","body":"In the following exercises, solve each rational inequality and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6  Solve Rational Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a276c42SolveRational11a","stepAnswer":["(-inf,-8)U(4,inf)"],"problemType":"TextBox","stepTitle":"$$\\\\frac{3x}{x-4}>2$$","stepBody":"Please enter your answer as (x,y). Use U to indicate the union of two intervals, if applicable. If your answer is the interval from $$x$$ to $$\\\\infty$$, please enter it as $$(x,\\\\infty)$$. If your answer includes the endpoint of the interval, please use [ ] to indicate the inclusion of endpoints. For instance, [x,y) refers to the interval including endpoint $$x$$ and excluding endpoint $$y$$.","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,-8) \\\\cup (4,\\\\infty)$$","hints":{"DefaultPathway":[{"id":"a276c42SolveRational11a-h1","type":"hint","dependencies":[],"title":"Put The Equation In The Correct Form with Zero By Itself On One Side","text":"Write the inequality as one quotient on the left and zero on the right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational11a-h2","type":"hint","dependencies":["a276c42SolveRational11a-h1"],"title":"Put The Equation In The Correct Form with Zero By Itself On One Side","text":"By subtracting $$2$$ to get zero on the right, we get $$\\\\frac{3x}{x-4}-2>0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational11a-h3","type":"hint","dependencies":["a276c42SolveRational11a-h2"],"title":"Rewrite $$2$$ As a Fraction Using the LCD.","text":"LCD for $$\\\\frac{3x}{x-4}$$ and $$2$$ is $$(x-4)$$, we can rewrite $$2$$ as $$\\\\frac{2\\\\left(x-4\\\\right)}{x-4}$$. We can rewrite $$\\\\frac{3x}{x-4}-2$$ as $$\\\\frac{3x}{x-4}-\\\\frac{2\\\\left(x-4\\\\right)}{x-4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational11a-h4","type":"hint","dependencies":["a276c42SolveRational11a-h3"],"title":"Rewrite $$2$$ As a Fraction Using the LCD.","text":"By subtracting the numerators and placing the difference over the common denominator, we can rewrite $$\\\\frac{3x}{x-4}-\\\\frac{2\\\\left(x-4\\\\right)}{x-4}$$ as $$\\\\frac{3x-2\\\\left(x-4\\\\right)}{x-4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational11a-h5","type":"hint","dependencies":["a276c42SolveRational11a-h4"],"title":"Simplify $$\\\\frac{3x-2\\\\left(x-4\\\\right)}{x-4}$$","text":"$$\\\\frac{3x-2\\\\left(x-4\\\\right)}{x-4}=\\\\frac{x+8}{x-4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational11a-h6","type":"hint","dependencies":["a276c42SolveRational11a-h5"],"title":"Find The Critical Points of Equation","text":"Determine the critical points\u2014the points where the rational expression will be zero or undefined.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational11a-h7","type":"hint","dependencies":["a276c42SolveRational11a-h6"],"title":"Find The Critical Points of Equation","text":"When is the equation undefine? Note that the equation is undefined when the denominator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational11a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a276c42SolveRational11a-h7"],"title":"Find The Critical Points of Equation","text":"Solve the equation $$x-4=0$$. What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational11a-h9","type":"hint","dependencies":["a276c42SolveRational11a-h8"],"title":"Find The Critical Points of Equation","text":"When does the equation equal zero? Note that the equation is undefined when the numerator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational11a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-8$$"],"dependencies":["a276c42SolveRational11a-h9"],"title":"Find The Critical Points of Equation","text":"Solve the equation $$x+8=0$$. What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational11a-h11","type":"hint","dependencies":["a276c42SolveRational11a-h10"],"title":"Use the Critical Points to Divide the Number Line into Intervals.","text":"Given that the $$x=-8$$ and $$x=4$$ are the two critical points, we can divide the number line into three intervals, namely, $$(-\\\\infty,-8)$$, $$(-8,4)$$ and $$(4,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational11a-h12","type":"hint","dependencies":["a276c42SolveRational11a-h11"],"title":"Test a Value in Each Interval.","text":"To find the sign of each factor in an interval, we choose any point in that interval and use it as a test point. Any point in the interval will give the expression the same sign, so we can choose any point in the interval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational11a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{13}$$"],"dependencies":["a276c42SolveRational11a-h12"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-\\\\infty,-8)$$, take the point $$x=-9$$ within the interval and plug it into the quotient $$\\\\frac{x+8}{x-4}$$. What is the value $$\\\\frac{\\\\operatorname{of}\\\\left(x+8\\\\right)}{x-4}$$ after plugging in the value $$x=-9$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational11a-h14","type":"hint","dependencies":["a276c42SolveRational11a-h13"],"title":"Test a Value in Each Interval.","text":"Since $$\\\\frac{1}{13}$$ is a positive numer greater than $$0$$, we get the sign of quotient is positive in the interval $$(-\\\\infty,-8)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational11a-h15","type":"hint","dependencies":["a276c42SolveRational11a-h14"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-8,4)$$, take the point $$x=0$$ within the interval and plug it into the original expression $$\\\\frac{x+8}{x-4}$$. We get $$-2$$ after plugging in $$x=0$$ into the quotient $$\\\\frac{x+8}{x-4}$$. $$-2$$ is a negative number, so we can mark the quotient negative in the interval $$(-8,4)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational11a-h16","type":"hint","dependencies":["a276c42SolveRational11a-h15"],"title":"Test a Value in Each Interval.","text":"For the interval $$(4,\\\\infty)$$, take the point $$x=5$$ within the interval and plug it into the original expression $$\\\\frac{x+8}{x-4}$$. We get $$13$$ after plugging in $$x=5$$ into the quotient $$\\\\frac{x+8}{x-4}$$. $$13$$ is a positive number, so we can mark the quotient positive in the interval $$(4,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational11a-h17","type":"hint","dependencies":["a276c42SolveRational11a-h16"],"title":"Determine the Intervals Where the Inequality is Correct.","text":"We want the quotient to be greater than zero, so the numbers in the interval $$(-\\\\infty,-8)$$ or $$(4,\\\\infty)$$ are the solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational11a-h18","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-inf,-8)U(4,inf)"],"dependencies":["a276c42SolveRational11a-h17"],"title":"Write the Final Answer in the interval Form","text":"We know that $$(-\\\\infty,-8)$$ and $$(4,\\\\infty)$$ are the answer, please use \\"[ ]\\". \\"( )\\" and U to write it in a mathematical form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a276c42SolveRational12","title":"Solve Rational Inequality","body":"In the following exercises, solve each rational inequality and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6  Solve Rational Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a276c42SolveRational12a","stepAnswer":["[-9,6)"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2x+3}{x-6} \\\\leq 1$$","stepBody":"Please enter your answer as (x,y). Use U to indicate the union of two intervals, if applicable. If your answer is the interval from $$x$$ to $$\\\\infty$$, please enter it as $$(x,\\\\infty)$$. If your answer includes the endpoint of the interval, please use [ ] to indicate the inclusion of endpoints. For instance, [x,y) refers to the interval including endpoint $$x$$ and excluding endpoint $$y$$.","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a276c42SolveRational12a-h1","type":"hint","dependencies":[],"title":"Put The Equation In The Correct Form with Zero By Itself On One Side","text":"Write the inequality as one quotient on the left and zero on the right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational12a-h2","type":"hint","dependencies":["a276c42SolveRational12a-h1"],"title":"Put The Equation In The Correct Form with Zero By Itself On One Side","text":"By subtracting $$1$$ to get zero on the right, we get $$\\\\frac{2x+3}{x-6}-1 \\\\leq 0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational12a-h3","type":"hint","dependencies":["a276c42SolveRational12a-h2"],"title":"Rewrite $$1$$ As a Fraction Using the LCD.","text":"LCD for $$\\\\frac{2x+3}{x-6}$$ and $$1$$ is $$(x-6)$$, we can rewrite $$1$$ as $$\\\\frac{x-6}{x-6}$$. We can rewrite $$\\\\frac{2x+3}{x-6}-1$$ as $$\\\\frac{2x+3}{x-6}-\\\\frac{x-6}{x-6}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational12a-h4","type":"hint","dependencies":["a276c42SolveRational12a-h3"],"title":"Rewrite $$1$$ As a Fraction Using the LCD.","text":"By subtracting the numerators and placing the difference over the common denominator, we can rewrite $$\\\\frac{2x+3}{x-6}-\\\\frac{x-6}{x-6}$$ as $$\\\\frac{2x+3-x-6}{x-6}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational12a-h5","type":"hint","dependencies":["a276c42SolveRational12a-h4"],"title":"Simplify $$\\\\frac{2x+3-x-6}{x-6}$$","text":"$$\\\\frac{2x+3-x-6}{x-6}=\\\\frac{x+9}{x-6}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational12a-h6","type":"hint","dependencies":["a276c42SolveRational12a-h5"],"title":"Find The Critical Points of Equation","text":"Determine the critical points\u2014the points where the rational expression will be zero or undefined.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational12a-h7","type":"hint","dependencies":["a276c42SolveRational12a-h6"],"title":"Find The Critical Points of Equation","text":"When is the equation undefine? Note that the equation is undefined when the denominator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational12a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a276c42SolveRational12a-h7"],"title":"Find The Critical Points of Equation","text":"Solve the equation $$x-6=0$$. What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational12a-h9","type":"hint","dependencies":["a276c42SolveRational12a-h8"],"title":"Find The Critical Points of Equation","text":"When does the equation equal zero? Note that the equation is undefined when the numerator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational12a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":["a276c42SolveRational12a-h9"],"title":"Find The Critical Points of Equation","text":"Solve the equation $$x+9=0$$. What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational12a-h11","type":"hint","dependencies":["a276c42SolveRational12a-h10"],"title":"Use the Critical Points to Divide the Number Line into Intervals.","text":"Given that the $$x=-9$$ and $$x=6$$ are the two critical points, we can divide the number line into three intervals, namely, $$(-\\\\infty,-9)$$, $$(-9,6)$$ and $$(6,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational12a-h12","type":"hint","dependencies":["a276c42SolveRational12a-h11"],"title":"Test a Value in Each Interval.","text":"To find the sign of each factor in an interval, we choose any point in that interval and use it as a test point. Any point in the interval will give the expression the same sign, so we can choose any point in the interval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational12a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{16}$$"],"dependencies":["a276c42SolveRational12a-h12"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-\\\\infty,-9)$$, take the point $$x=-10$$ within the interval and plug it into the quotient $$\\\\frac{x+9}{x-6}$$. What is the value of $$\\\\frac{x+9}{x-6}$$ after plugging in the value $$x=-10$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational12a-h14","type":"hint","dependencies":["a276c42SolveRational12a-h13"],"title":"Test a Value in Each Interval.","text":"Since $$\\\\frac{1}{16}$$ is a positive numer greater than $$0$$, we get the sign of quotient is positive in the interval $$(-\\\\infty,-9)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational12a-h15","type":"hint","dependencies":["a276c42SolveRational12a-h14"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-9,6)$$, take the point $$x=0$$ within the interval and plug it into the original expression $$\\\\frac{x+9}{x-6}$$. We get $$\\\\frac{-3}{2}$$ after plugging in $$x=0$$ into the quotient $$\\\\frac{x+9}{x-6}$$. $$\\\\frac{-3}{2}$$ is a negative number, so we can mark the quotient negative in the interval $$(-9,6)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational12a-h16","type":"hint","dependencies":["a276c42SolveRational12a-h15"],"title":"Test a Value in Each Interval.","text":"For the interval $$(6,\\\\infty)$$, take the point $$x=7$$ within the interval and plug it into the original expression $$\\\\frac{x+9}{x-6}$$. We get $$16$$ after plugging in $$x=7$$ into the quotient $$\\\\frac{x+9}{x-6}$$. $$16$$ is a positive number, so we can mark the quotient positive in the interval $$(6,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational12a-h17","type":"hint","dependencies":["a276c42SolveRational12a-h16"],"title":"Determine the Intervals Where the Inequality is Correct.","text":"We want the quotient to be less than and equal to zero, so the numbers in the interval $$(-9,6)$$ are the solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational12a-h18","type":"hint","dependencies":["a276c42SolveRational12a-h17"],"title":"Determine the Intervals Where the Inequality is Correct.","text":"We want the quotient to be less than or equal to zero, so the numbers in the interval $$(-9,6)$$ are the solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational12a-h19","type":"hint","dependencies":["a276c42SolveRational12a-h18"],"title":"Determine the Critical Points Where the Inequality is Correct","text":"The critical point $$x=6$$ makes the denominator $$0$$, so it must be excluded from the solution and we mark it with a parenthesis.\\\\nThe critical point $$x=-9$$ makes the whole rational expression $$0$$. The inequality requires that the rational expression be greater than or equal to $$0$$. So, $$-9$$ is part of the solution and we will mark it with a bracket.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational12a-h20","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["[-9,6)"],"dependencies":["a276c42SolveRational12a-h19"],"title":"Write the Final Answer in the interval Form","text":"We know that $$(-9,6)$$ and $$x=-9$$ are the answer, please use \\"[ ]\\". \\"( )\\" and U to write it in a mathematical form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a276c42SolveRational13","title":"Solve Rational Inequality","body":"In the following exercises, solve each rational inequality and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6  Solve Rational Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a276c42SolveRational13a","stepAnswer":["[-1,4)"],"problemType":"TextBox","stepTitle":"$$\\\\frac{4x-1}{x-4} \\\\leq 1$$","stepBody":"Please enter your answer as (x,y). Use U to indicate the union of two intervals, if applicable. If your answer is the interval from $$x$$ to $$\\\\infty$$, please enter it as $$(x,\\\\infty)$$. If your answer includes the endpoint of the interval, please use [ ] to indicate the inclusion of endpoints. For instance, [x,y) refers to the interval including endpoint $$x$$ and excluding endpoint $$y$$.","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a276c42SolveRational13a-h1","type":"hint","dependencies":[],"title":"Put The Equation In The Correct Form with Zero By Itself On One Side","text":"Write the inequality as one quotient on the left and zero on the right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational13a-h2","type":"hint","dependencies":["a276c42SolveRational13a-h1"],"title":"Put The Equation In The Correct Form with Zero By Itself On One Side","text":"By subtracting $$1$$ to get zero on the right, we get $$\\\\frac{4x-1}{x-4}-1 \\\\leq 0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational13a-h3","type":"hint","dependencies":["a276c42SolveRational13a-h2"],"title":"Rewrite $$1$$ As a Fraction Using the LCD.","text":"LCD for $$\\\\frac{4x-1}{x-4}$$ and $$1$$ is $$(x-4)$$, we can rewrite $$1$$ as $$\\\\frac{x-4}{x-4}$$. We can rewrite $$\\\\frac{4x-1}{x-4}-1$$ as $$\\\\frac{4x-1}{x-4}-\\\\frac{x-4}{x-4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational13a-h4","type":"hint","dependencies":["a276c42SolveRational13a-h3"],"title":"Rewrite $$1$$ As a Fraction Using the LCD.","text":"By subtracting the numerators and placing the difference over the common denominator, we can rewrite $$\\\\frac{4x-1}{x-4}-\\\\frac{x-4}{x-4}$$ as $$\\\\frac{4x-1-x-4}{x-4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational13a-h5","type":"hint","dependencies":["a276c42SolveRational13a-h4"],"title":"Simplify $$\\\\frac{4x-1-x-4}{x-4}$$","text":"$$\\\\frac{4x-1-x-4}{x-4}=\\\\frac{3x+3}{x-4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational13a-h6","type":"hint","dependencies":["a276c42SolveRational13a-h5"],"title":"Find The Critical Points of Equation","text":"Determine the critical points\u2014the points where the rational expression will be zero or undefined.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational13a-h7","type":"hint","dependencies":["a276c42SolveRational13a-h6"],"title":"Find The Critical Points of Equation","text":"When is the equation undefine? Note that the equation is undefined when the denominator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational13a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a276c42SolveRational13a-h7"],"title":"Find The Critical Points of Equation","text":"Solve the equation $$x-4=0$$. What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational13a-h9","type":"hint","dependencies":["a276c42SolveRational13a-h8"],"title":"Find The Critical Points of Equation","text":"When does the equation equal zero? Note that the equation is undefined when the numerator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational13a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a276c42SolveRational13a-h9"],"title":"Find The Critical Points of Equation","text":"Solve the equation $$3x+3=0$$. What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational13a-h11","type":"hint","dependencies":["a276c42SolveRational13a-h10"],"title":"Use the Critical Points to Divide the Number Line into Intervals.","text":"Given that the $$x=-1$$ and $$x=4$$ are the two critical points, we can divide the number line into three intervals, namely, $$(-\\\\infty,-1)$$, $$(-1,4)$$ and $$(4,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational13a-h12","type":"hint","dependencies":["a276c42SolveRational13a-h11"],"title":"Test a Value in Each Interval.","text":"To find the sign of each factor in an interval, we choose any point in that interval and use it as a test point. Any point in the interval will give the expression the same sign, so we can choose any point in the interval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational13a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a276c42SolveRational13a-h12"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-\\\\infty,-1)$$, take the point $$x=-2$$ within the interval and plug it into the $$\\\\frac{\\\\operatorname{quotient}\\\\left(3x+3\\\\right)}{x-4}$$. What is the value of $$\\\\frac{3x+3}{x-4}$$ after plugging in the value $$x=-2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational13a-h14","type":"hint","dependencies":["a276c42SolveRational13a-h13"],"title":"Test a Value in Each Interval.","text":"Since $$\\\\frac{1}{2}$$ is a positive numer greater than $$0$$, we get the sign of quotient is positive in the interval $$(-\\\\infty,-1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational13a-h15","type":"hint","dependencies":["a276c42SolveRational13a-h14"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-1,4)$$, take the point $$x=0$$ within the interval and plug it into the original expression $$\\\\frac{3x+3}{x-4}$$. We get $$\\\\frac{-3}{4}$$ after plugging in $$x=0$$ into the quotient $$\\\\frac{3x+3}{x-4}$$. $$\\\\frac{-3}{4}$$ is a negative number, so we can mark the quotient negative in the interval $$(-1,4)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational13a-h16","type":"hint","dependencies":["a276c42SolveRational13a-h15"],"title":"Test a Value in Each Interval.","text":"For the interval $$(4,\\\\infty)$$, take the point $$x=5$$ within the interval and plug it into the original expression $$\\\\frac{3x+3}{x-4}$$. We get $$18$$ after plugging in $$x=5$$ into the quotient $$\\\\frac{3x+3}{x-4}$$. $$18$$ is a positive number, so we can mark the quotient positive in the interval $$(4,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational13a-h17","type":"hint","dependencies":["a276c42SolveRational13a-h16"],"title":"Determine the Intervals Where the Inequality is Correct.","text":"We want the quotient to be less than and equal to zero, so the numbers in the interval $$(-1,4)$$ are the solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational13a-h18","type":"hint","dependencies":["a276c42SolveRational13a-h17"],"title":"Determine the Intervals Where the Inequality is Correct.","text":"We want the quotient to be less than or equal to zero, so the numbers in the interval $$(-1,4)$$ are the solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational13a-h19","type":"hint","dependencies":["a276c42SolveRational13a-h18"],"title":"Determine the Critical Points Where the Inequality is Correct","text":"The critical point $$x=4$$ makes the denominator $$0$$, so it must be excluded from the solution and we mark it with a parenthesis.\\\\nThe critical point $$x=-1$$ makes the whole rational expression $$0$$. The inequality requires that the rational expression be greater than or equal to $$0$$. So, $$-9$$ is part of the solution and we will mark it with a bracket.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational13a-h20","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["[-1,4)"],"dependencies":["a276c42SolveRational13a-h19"],"title":"Write the Final Answer in the interval Form","text":"We know that $$(-1,4)$$ and $$x=-1$$ are the answer, please use \\"[ ]\\". \\"( )\\" and U to write it in a mathematical form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a276c42SolveRational14","title":"Solve Rational Inequality","body":"In the following exercises, solve each rational inequality and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6  Solve Rational Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a276c42SolveRational14a","stepAnswer":["(-inf,-6]U(4,inf)"],"problemType":"TextBox","stepTitle":"$$\\\\frac{3x-2}{x-4} \\\\geq 2$$","stepBody":"Please enter your answer as (x,y). Use U to indicate the union of two intervals, if applicable. If your answer is the interval from $$x$$ to $$\\\\infty$$, please enter it as $$(x,\\\\infty)$$. If your answer includes the endpoint of the interval, please use [ ] to indicate the inclusion of endpoints. For instance, [x,y) refers to the interval including endpoint $$x$$ and excluding endpoint $$y$$.","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,-6] \\\\cup (4,\\\\infty)$$","hints":{"DefaultPathway":[{"id":"a276c42SolveRational14a-h1","type":"hint","dependencies":[],"title":"Put The Equation In The Correct Form with Zero By Itself On One Side","text":"Write the inequality as one quotient on the left and zero on the right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational14a-h2","type":"hint","dependencies":["a276c42SolveRational14a-h1"],"title":"Put The Equation In The Correct Form with Zero By Itself On One Side","text":"By subtracting $$2$$ to get zero on the right, we get $$\\\\frac{3x-2}{x-4}-2 \\\\geq 0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational14a-h3","type":"hint","dependencies":["a276c42SolveRational14a-h2"],"title":"Rewrite $$2$$ As a Fraction Using the LCD.","text":"LCD for $$\\\\frac{3x-2}{x-4}$$ and $$2$$ is $$(x-4)$$, we can rewrite $$2$$ as $$\\\\frac{2\\\\left(x-4\\\\right)}{x-4}$$. We can rewrite $$\\\\frac{3x-2}{x-4}-2$$ as $$\\\\frac{3x-2}{x-4}-\\\\frac{2\\\\left(x-4\\\\right)}{x-4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational14a-h4","type":"hint","dependencies":["a276c42SolveRational14a-h3"],"title":"Rewrite $$2$$ As a Fraction Using the LCD.","text":"By subtracting the numerators and placing the difference over the common denominator, we can rewrite $$\\\\frac{3x-2}{x-4}-\\\\frac{2\\\\left(x-4\\\\right)}{x-4}$$ as $$\\\\frac{3x-2-2\\\\left(x-4\\\\right)}{x-4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational14a-h5","type":"hint","dependencies":["a276c42SolveRational14a-h4"],"title":"Simplify $$\\\\frac{3x-2-2\\\\left(x-4\\\\right)}{x-4}$$","text":"$$\\\\frac{3x-2-2\\\\left(x-4\\\\right)}{x-4}=\\\\frac{x+6}{x-4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational14a-h6","type":"hint","dependencies":["a276c42SolveRational14a-h5"],"title":"Find The Critical Points of Equation","text":"Determine the critical points\u2014the points where the rational expression will be zero or undefined.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational14a-h7","type":"hint","dependencies":["a276c42SolveRational14a-h6"],"title":"Find The Critical Points of Equation","text":"When is the equation undefine? Note that the equation is undefined when the denominator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational14a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a276c42SolveRational14a-h7"],"title":"Find The Critical Points of Equation","text":"Solve the equation $$x-4=0$$. What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational14a-h9","type":"hint","dependencies":["a276c42SolveRational14a-h8"],"title":"Find The Critical Points of Equation","text":"When does the equation equal zero? Note that the equation is undefined when the numerator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational14a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6$$"],"dependencies":["a276c42SolveRational14a-h9"],"title":"Find The Critical Points of Equation","text":"Solve the equation $$x+6=0$$. What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational14a-h11","type":"hint","dependencies":["a276c42SolveRational14a-h10"],"title":"Use the Critical Points to Divide the Number Line into Intervals.","text":"Given that the $$x=-6$$ and $$x=4$$ are the two critical points, we can divide the number line into three intervals, namely, $$(-\\\\infty,-6)$$, $$(-6,4)$$ and $$(4,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational14a-h12","type":"hint","dependencies":["a276c42SolveRational14a-h11"],"title":"Test a Value in Each Interval.","text":"To find the sign of each factor in an interval, we choose any point in that interval and use it as a test point. Any point in the interval will give the expression the same sign, so we can choose any point in the interval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational14a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{11}$$"],"dependencies":["a276c42SolveRational14a-h12"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-\\\\infty,-6)$$, take the point $$x=-7$$ within the interval and plug it into the $$\\\\frac{\\\\operatorname{quotient}\\\\left(x+6\\\\right)}{x-4}$$. What is the value of $$\\\\frac{x+6}{x-4}$$ after plugging in the value $$x=-7$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational14a-h14","type":"hint","dependencies":["a276c42SolveRational14a-h13"],"title":"Test a Value in Each Interval.","text":"Since $$\\\\frac{1}{11}$$ is a positive numer greater than $$0$$, we get the sign of quotient is positive in the interval $$(-\\\\infty,-6)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational14a-h15","type":"hint","dependencies":["a276c42SolveRational14a-h14"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-6,4)$$, take the point $$x=0$$ within the interval and plug it into the original expression $$\\\\frac{x+6}{x-4}$$. We get $$\\\\frac{-3}{2}$$ after plugging in $$x=0$$ into the quotient $$\\\\frac{x+6}{x-4}$$. $$\\\\frac{-3}{2}$$ is a negative number, so we can mark the quotient negative in the interval $$(-6,4)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational14a-h16","type":"hint","dependencies":["a276c42SolveRational14a-h15"],"title":"Test a Value in Each Interval.","text":"For the interval $$(4,\\\\infty)$$, take the point $$x=5$$ within the interval and plug it into the original expression $$\\\\frac{x+6}{x-4}$$. We get $$18$$ after plugging in $$x=5$$ into the quotient $$\\\\frac{x+6}{x-4}$$. $$11$$ is a positive number, so we can mark the quotient positive in the interval $$(4,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational14a-h17","type":"hint","dependencies":["a276c42SolveRational14a-h16"],"title":"Determine the Intervals Where the Inequality is Correct.","text":"We want the quotient to be greater than and equal to zero, so the numbers in the intervals $$(-\\\\infty,-6)$$ and $$(4,\\\\infty)$$ are the solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational14a-h18","type":"hint","dependencies":["a276c42SolveRational14a-h17"],"title":"Determine the Intervals Where the Inequality is Correct.","text":"We want the quotient to be less than or equal to zero, so the numbers in the interval $$(-1,4)$$ are the solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational14a-h19","type":"hint","dependencies":["a276c42SolveRational14a-h18"],"title":"Determine the Critical Points Where the Inequality is Correct","text":"The critical point $$x=4$$ makes the denominator $$0$$, so it must be excluded from the solution and we mark it with a parenthesis.\\\\nThe critical point $$x=-6$$ makes the whole rational expression $$0$$. The inequality requires that the rational expression be greater than or equal to $$0$$. So, $$-6$$ is part of the solution and we will mark it with a bracket.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational14a-h20","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-inf,-6]U(4,inf)"],"dependencies":["a276c42SolveRational14a-h19"],"title":"Write the Final Answer in the interval Form","text":"We know that $$(-\\\\infty,-6)$$ or $$(4,\\\\infty)$$ and $$x=-6$$ are the answer, please use \\"[ ]\\". \\"( )\\" and U to write it in a mathematical form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a276c42SolveRational15","title":"Solve Rational Inequality","body":"In the following exercises, solve each rational inequality and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6  Solve Rational Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a276c42SolveRational15a","stepAnswer":["(-inf,-3/2]U(3,inf)"],"problemType":"TextBox","stepTitle":"$$\\\\frac{4x-3}{x-3} \\\\geq 2$$","stepBody":"Please enter your answer as (x,y). Use U to indicate the union of two intervals, if applicable. If your answer is the interval from $$x$$ to $$\\\\infty$$, please enter it as $$(x,\\\\infty)$$. If your answer includes the endpoint of the interval, please use [ ] to indicate the inclusion of endpoints. For instance, [x,y) refers to the interval including endpoint $$x$$ and excluding endpoint $$y$$.","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\frac{-3}{2}] \\\\cup (3,\\\\infty)$$","hints":{"DefaultPathway":[{"id":"a276c42SolveRational15a-h1","type":"hint","dependencies":[],"title":"Put The Equation In The Correct Form with Zero By Itself On One Side","text":"Write the inequality as one quotient on the left and zero on the right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational15a-h2","type":"hint","dependencies":["a276c42SolveRational15a-h1"],"title":"Put The Equation In The Correct Form with Zero By Itself On One Side","text":"By subtracting $$2$$ to get zero on the right, we get $$\\\\frac{4x-3}{x-3}-2 \\\\geq 0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational15a-h3","type":"hint","dependencies":["a276c42SolveRational15a-h2"],"title":"Rewrite $$2$$ As a Fraction Using the LCD.","text":"LCD for(4*x-3)/(x-3) and $$2$$ is $$(x-3)$$, we can rewrite $$2$$ as $$\\\\frac{2\\\\left(x-3\\\\right)}{x-3}$$. We can rewrite (4*x-3)/(x-3))-2 as $$\\\\frac{4x-3}{x-3}-\\\\frac{2\\\\left(x-3\\\\right)}{x-3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational15a-h4","type":"hint","dependencies":["a276c42SolveRational15a-h3"],"title":"Rewrite $$2$$ As a Fraction Using the LCD.","text":"By subtracting the numerators and placing the difference over the common denominator, we can rewrite $$\\\\frac{4x-3}{x-3}-\\\\frac{2\\\\left(x-3\\\\right)}{x-3}$$ as $$\\\\frac{4x-3-2\\\\left(x-3\\\\right)}{x-3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational15a-h5","type":"hint","dependencies":["a276c42SolveRational15a-h4"],"title":"Simplify $$\\\\frac{4x-3-2\\\\left(x-3\\\\right)}{x-3}$$","text":"$$\\\\frac{4x-3-2\\\\left(x-3\\\\right)}{x-3}=\\\\frac{2x+3}{x-3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational15a-h6","type":"hint","dependencies":["a276c42SolveRational15a-h5"],"title":"Find The Critical Points of Equation","text":"Determine the critical points\u2014the points where the rational expression will be zero or undefined.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational15a-h7","type":"hint","dependencies":["a276c42SolveRational15a-h6"],"title":"Find The Critical Points of Equation","text":"When is the equation undefine? Note that the equation is undefined when the denominator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational15a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a276c42SolveRational15a-h7"],"title":"Find The Critical Points of Equation","text":"Solve the equation $$x-3=0$$. What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational15a-h9","type":"hint","dependencies":["a276c42SolveRational15a-h8"],"title":"Find The Critical Points of Equation","text":"When does the equation equal zero? Note that the equation is undefined when the numerator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational15a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-3}{2}$$"],"dependencies":["a276c42SolveRational15a-h9"],"title":"Find The Critical Points of Equation","text":"Solve the equation $$2x+3=0$$. What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational15a-h11","type":"hint","dependencies":["a276c42SolveRational15a-h10"],"title":"Use the Critical Points to Divide the Number Line into Intervals.","text":"Given that the $$x=\\\\frac{-3}{2}$$ and $$x=3$$ are the two critical points, we can divide the number line into three intervals, namely, $$(-\\\\infty,\\\\frac{-3}{2})$$, $$(\\\\frac{-3}{2},3)$$ and $$(3,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational15a-h12","type":"hint","dependencies":["a276c42SolveRational15a-h11"],"title":"Test a Value in Each Interval.","text":"To find the sign of each factor in an interval, we choose any point in that interval and use it as a test point. Any point in the interval will give the expression the same sign, so we can choose any point in the interval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational15a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{5}$$"],"dependencies":["a276c42SolveRational15a-h12"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-\\\\infty,\\\\frac{-3}{2})$$, take the point $$x=-2$$ within the interval and plug it into the quotient $$\\\\frac{2x+3}{x-3}$$. What is the value of $$\\\\frac{2x+3}{x-3}$$ after plugging in the value $$x=-2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational15a-h14","type":"hint","dependencies":["a276c42SolveRational15a-h13"],"title":"Test a Value in Each Interval.","text":"Since $$\\\\frac{1}{5}$$ is a positive numer greater than $$0$$, we get the sign of quotient is positive in the interval $$(-\\\\infty,\\\\frac{-3}{2})$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational15a-h15","type":"hint","dependencies":["a276c42SolveRational15a-h14"],"title":"Test a Value in Each Interval.","text":"For the interval $$(\\\\frac{-3}{2},3)$$, take the point $$x=0$$ within the interval and plug it into the original expression $$\\\\frac{2x+3}{x-3}$$. We get $$-1$$ after plugging in $$x=0$$ into the quotient $$\\\\frac{2x+3}{x-3}$$. $$-1$$ is a negative number, so we can mark the quotient negative in the interval $$(\\\\frac{-3}{2},3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational15a-h16","type":"hint","dependencies":["a276c42SolveRational15a-h15"],"title":"Test a Value in Each Interval.","text":"For the interval $$(3,\\\\infty)$$, take the point $$x=4$$ within the interval and plug it into the original expression $$\\\\frac{2x+3}{x-3}$$. We get $$11$$ after plugging in $$x=4$$ into the quotient $$\\\\frac{2x+3}{x-3}$$. $$11$$ is a positive number, so we can mark the quotient positive in the interval $$(3,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational15a-h17","type":"hint","dependencies":["a276c42SolveRational15a-h16"],"title":"Determine the Intervals Where the Inequality is Correct.","text":"We want the quotient to be greater than and equal to zero, so the numbers in the intervals $$(-\\\\infty,-6)$$ or $$(4,\\\\infty)$$ are the solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational15a-h18","type":"hint","dependencies":["a276c42SolveRational15a-h17"],"title":"Determine the Intervals Where the Inequality is Correct.","text":"We want the quotient to be greater than or equal to zero, so the numbers in the intervals $$(-\\\\infty,\\\\frac{-3}{2})$$ and $$(3,\\\\infty)$$ are the solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational15a-h19","type":"hint","dependencies":["a276c42SolveRational15a-h18"],"title":"Determine the Critical Points Where the Inequality is Correct","text":"The critical point $$x=3$$ makes the denominator $$0$$, so it must be excluded from the solution and we mark it with a parenthesis.\\\\nThe critical point $$x=\\\\frac{-3}{2}$$ makes the whole rational expression $$0$$. The inequality requires that the rational expression be greater than or equal to $$0$$. So, $$\\\\frac{-3}{2}$$ is part of the solution and we will mark it with a bracket.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational15a-h20","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-inf,-3/2]U(3,inf)"],"dependencies":["a276c42SolveRational15a-h19"],"title":"Write the Final Answer in the interval Form","text":"We know that $$(-\\\\infty,\\\\frac{-3}{2})$$ and $$(3,\\\\infty)$$ and $$x=\\\\frac{-3}{2}$$ are the answer, please use \\"[ ]\\". \\"( )\\" and U to write it in a mathematical form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a276c42SolveRational16","title":"Solve Rational Inequality","body":"In the following exercises, solve each rational inequality and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6  Solve Rational Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a276c42SolveRational16a","stepAnswer":["(-inf,-4)U(-3,inf)"],"problemType":"TextBox","stepTitle":"$$\\\\frac{1}{x^2+7x+12}>0$$","stepBody":"Please enter your answer as (x,y). Use U to indicate the union of two intervals, if applicable. If your answer is the interval from $$x$$ to $$\\\\infty$$, please enter it as $$(x,\\\\infty)$$. If your answer includes the endpoint of the interval, please use [ ] to indicate the inclusion of endpoints. For instance, [x,y) refers to the interval including endpoint $$x$$ and excluding endpoint $$y$$.","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,-4) \\\\cup (-3,\\\\infty)$$","hints":{"DefaultPathway":[{"id":"a276c42SolveRational16a-h1","type":"hint","dependencies":[],"title":"Put The Equation In The Correct Form with Zero By Itself On One Side","text":"Write the inequality as one quotient on the left and zero on the right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x+3\\\\right) \\\\left(x+4\\\\right)$$"],"dependencies":["a276c42SolveRational16a-h1"],"title":"Factor the denominator","text":"Factor $$x^2+7x+12$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational16a-h3","type":"hint","dependencies":["a276c42SolveRational16a-h2"],"title":"Find The Critical Points of Equation","text":"The quotient is $$0$$ when the numerator is $$0$$. Since the numerator is always $$1$$, the quotient cannot be $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational16a-h4","type":"hint","dependencies":["a276c42SolveRational16a-h3"],"title":"Find The Critical Points of Equation","text":"The quotient will be undefined when the denominator is zero. $$\\\\left(x+3\\\\right) \\\\left(x+4\\\\right)=0$$ when $$x=-3$$, $$x=-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational16a-h5","type":"hint","dependencies":["a276c42SolveRational16a-h4"],"title":"Use the Critical Points to Divide the Number Line into Intervals.","text":"Given that the $$x=-3$$ and $$x=-4$$ are the two critical points, we can divide the number line into three intervals, namely, $$(-\\\\infty,-4)$$, $$(-4,-3)$$ and $$(-3,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational16a-h6","type":"hint","dependencies":["a276c42SolveRational16a-h5"],"title":"Test a Value in Each Interval.","text":"To find the sign of each factor in an interval, we choose any point in that interval and use it as a test point. Any point in the interval will give the expression the same sign, so we can choose any point in the interval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational16a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a276c42SolveRational16a-h6"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-\\\\infty,-4)$$, take the point $$x=-5$$ within the interval and plug it into the quotient $$\\\\frac{1}{\\\\left(x+3\\\\right) \\\\left(x+4\\\\right)}$$. What is the value of $$\\\\frac{1}{\\\\left(x+3\\\\right) \\\\left(x+4\\\\right)}$$ after plugging in the value $$x=-5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational16a-h8","type":"hint","dependencies":["a276c42SolveRational16a-h7"],"title":"Test a Value in Each Interval.","text":"Since $$\\\\frac{1}{2}$$ is a positive numer greater than $$0$$, we get the sign of quotient is positive in the interval $$(-\\\\infty,-4)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational16a-h9","type":"hint","dependencies":["a276c42SolveRational16a-h8"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-4,-3)$$, take the point $$x=-3.5$$ within the interval and plug it into the expression $$\\\\frac{1}{\\\\left(x+3\\\\right) \\\\left(x+4\\\\right)}$$ . We get $$-4$$ after plugging in $$x=-3.5$$ into the quotient $$\\\\frac{1}{\\\\left(x+3\\\\right) \\\\left(x+4\\\\right)}$$. $$-4$$ is a negative number, so we can mark the quotient negative in the interval $$(-4,-3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational16a-h10","type":"hint","dependencies":["a276c42SolveRational16a-h9"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-3,\\\\infty)$$, take the point $$x=0$$ within the interval and plug it into the original expression $$\\\\frac{1}{\\\\left(x+3\\\\right) \\\\left(x+4\\\\right)}$$. We get $$\\\\frac{1}{12}$$ after plugging in $$x=0$$ into the quotient $$\\\\frac{1}{\\\\left(x+3\\\\right) \\\\left(x+4\\\\right)}$$. $$\\\\frac{1}{12}$$ is a positive number, so we can mark the quotient positive in the interval $$(-3,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational16a-h11","type":"hint","dependencies":["a276c42SolveRational16a-h10"],"title":"Determine the Intervals Where the Inequality is Correct.","text":"We want the quotient to be greater than zero, so the numbers in the intervals $$(-\\\\infty,-4)$$ and $$(-3,\\\\infty)$$ are the solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational16a-h12","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-inf,-4)U(-3,inf)"],"dependencies":["a276c42SolveRational16a-h11"],"title":"Write the Final Answer in the interval Form","text":"We know that $$(-\\\\infty,-4)$$ and $$(-3,\\\\infty)$$ are the answer, please use \\"[ ]\\". \\"( )\\" and U to write it in a mathematical form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a276c42SolveRational17","title":"Solve Rational Inequality","body":"In the following exercises, solve each rational inequality and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6  Solve Rational Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a276c42SolveRational17a","stepAnswer":["(-inf,-2)U(6,inf)"],"problemType":"TextBox","stepTitle":"$$\\\\frac{1}{x^2-4x-12}>0$$","stepBody":"Please enter your answer as (x,y). Use U to indicate the union of two intervals, if applicable. If your answer is the interval from $$x$$ to $$\\\\infty$$, please enter it as $$(x,\\\\infty)$$. If your answer includes the endpoint of the interval, please use [ ] to indicate the inclusion of endpoints. For instance, [x,y) refers to the interval including endpoint $$x$$ and excluding endpoint $$y$$.","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,-2) \\\\cup (6,\\\\infty)$$","hints":{"DefaultPathway":[{"id":"a276c42SolveRational17a-h1","type":"hint","dependencies":[],"title":"Put The Equation In The Correct Form with Zero By Itself On One Side","text":"Write the inequality as one quotient on the left and zero on the right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational17a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x-6\\\\right) \\\\left(x+2\\\\right)$$"],"dependencies":["a276c42SolveRational17a-h1"],"title":"Factor the denominator","text":"Factor $$x^2-4x-12$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational17a-h3","type":"hint","dependencies":["a276c42SolveRational17a-h2"],"title":"Find The Critical Points of Equation","text":"The quotient is $$0$$ when the numerator is $$0$$. Since the numerator is always $$1$$, the quotient cannot be $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational17a-h4","type":"hint","dependencies":["a276c42SolveRational17a-h3"],"title":"Find The Critical Points of Equation","text":"The quotient will be undefined when the denominator is zero. $$\\\\left(x-6\\\\right) \\\\left(x+2\\\\right)=0$$ when $$x=-2$$, $$x=6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational17a-h5","type":"hint","dependencies":["a276c42SolveRational17a-h4"],"title":"Use the Critical Points to Divide the Number Line into Intervals.","text":"Given that the $$x=-2$$ and $$x=6$$ are the two critical points, we can divide the number line into three intervals, namely, $$(-\\\\infty,-2)$$, $$(-2,6)$$ and $$(6,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational17a-h6","type":"hint","dependencies":["a276c42SolveRational17a-h5"],"title":"Test a Value in Each Interval.","text":"To find the sign of each factor in an interval, we choose any point in that interval and use it as a test point. Any point in the interval will give the expression the same sign, so we can choose any point in the interval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational17a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{9}$$"],"dependencies":["a276c42SolveRational17a-h6"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-\\\\infty,-2)$$, take the point $$x=-3$$ within the interval and plug it into the quotient $$\\\\frac{1}{\\\\left(x-6\\\\right) \\\\left(x+2\\\\right)}$$. What is the value of $$\\\\frac{1}{\\\\left(x-6\\\\right) \\\\left(x+2\\\\right)}$$ after plugging in the value $$x=-3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational17a-h8","type":"hint","dependencies":["a276c42SolveRational17a-h7"],"title":"Test a Value in Each Interval.","text":"Since $$\\\\frac{1}{9}$$ is a positive numer greater than $$0$$, we get the sign of quotient is positive in the interval $$(-\\\\infty,-2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational17a-h9","type":"hint","dependencies":["a276c42SolveRational17a-h8"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-2,6)$$, take the point $$x=0$$ within the interval and plug it into the expression $$\\\\frac{1}{\\\\left(x-6\\\\right) \\\\left(x+2\\\\right)}$$. We get $$\\\\frac{-1}{12}$$ after plugging in $$x=0$$ into the quotient $$\\\\frac{1}{\\\\left(x-6\\\\right) \\\\left(x+2\\\\right)}$$. $$\\\\frac{-1}{12}$$ is a negative number, so we can mark the quotient negative in the interval $$(-2,6)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational17a-h10","type":"hint","dependencies":["a276c42SolveRational17a-h9"],"title":"Test a Value in Each Interval.","text":"For the interval $$(6,\\\\infty)$$, take the point $$x=7$$ within the interval and plug it into the original expression $$\\\\frac{1}{\\\\left(x-6\\\\right) \\\\left(x+2\\\\right)}$$. We get $$\\\\frac{1}{9}$$ after plugging in $$x=7$$ into the $$\\\\frac{quotient1}{\\\\left(x-6\\\\right) \\\\left(x+2\\\\right)}$$. $$\\\\frac{1}{9}$$ is a positive number, so we can mark the quotient positive in the interval $$(6,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational17a-h11","type":"hint","dependencies":["a276c42SolveRational17a-h10"],"title":"Determine the Intervals Where the Inequality is Correct.","text":"We want the quotient to be greater than zero, so the numbers in the intervals $$(-\\\\infty,-2)$$ and $$(6,\\\\infty)$$ are the solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational17a-h12","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-inf,-2)U(6,inf)"],"dependencies":["a276c42SolveRational17a-h11"],"title":"Write the Final Answer in the interval Form","text":"We know that $$(-\\\\infty,-2)$$ and $$(6,\\\\infty)$$ are the answer, please use \\"[ ]\\". \\"( )\\" and U to write it in a mathematical form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a276c42SolveRational18","title":"Solve Rational Inequality","body":"In the following exercises, solve each rational inequality and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6  Solve Rational Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a276c42SolveRational18a","stepAnswer":["(1,4)"],"problemType":"TextBox","stepTitle":"$$\\\\frac{3}{x^2-5x+4}<0$$","stepBody":"Please enter your answer as (x,y). Use U to indicate the union of two intervals, if applicable. If your answer is the interval from $$x$$ to $$\\\\infty$$, please enter it as $$(x,\\\\infty)$$. If your answer includes the endpoint of the interval, please use [ ] to indicate the inclusion of endpoints. For instance, [x,y) refers to the interval including endpoint $$x$$ and excluding endpoint $$y$$.","answerType":"string","variabilization":{},"answerLatex":"$$(1,4)$$","hints":{"DefaultPathway":[{"id":"a276c42SolveRational18a-h1","type":"hint","dependencies":[],"title":"Put The Equation In The Correct Form with Zero By Itself On One Side","text":"Write the inequality as one quotient on the left and zero on the right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x-1\\\\right) \\\\left(x-4\\\\right)$$"],"dependencies":["a276c42SolveRational18a-h1"],"title":"Factor the denominator","text":"Factor $$x^2-5x+4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational18a-h3","type":"hint","dependencies":["a276c42SolveRational18a-h2"],"title":"Find The Critical Points of Equation","text":"The quotient is $$0$$ when the numerator is $$0$$. Since the numerator is always $$3$$, the quotient cannot be $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational18a-h4","type":"hint","dependencies":["a276c42SolveRational18a-h3"],"title":"Find The Critical Points of Equation","text":"The quotient will be undefined when the denominator is zero. $$\\\\left(x-1\\\\right) \\\\left(x-4\\\\right)=0$$ when $$x=1$$, $$x=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational18a-h5","type":"hint","dependencies":["a276c42SolveRational18a-h4"],"title":"Use the Critical Points to Divide the Number Line into Intervals.","text":"Given that the $$x=1$$ and $$x=4$$ are the two critical points, we can divide the number line into three intervals, namely, $$(-\\\\infty,1)$$, $$(1,4)$$ and $$(4,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational18a-h6","type":"hint","dependencies":["a276c42SolveRational18a-h5"],"title":"Test a Value in Each Interval.","text":"To find the sign of each factor in an interval, we choose any point in that interval and use it as a test point. Any point in the interval will give the expression the same sign, so we can choose any point in the interval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational18a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4}$$"],"dependencies":["a276c42SolveRational18a-h6"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-\\\\infty,1)$$, take the point $$x=0$$ within the interval and plug it into the quotient $$\\\\frac{1}{\\\\left(x-1\\\\right) \\\\left(x-4\\\\right)}$$. What is the value of $$\\\\frac{1}{\\\\left(x-1\\\\right) \\\\left(x-4\\\\right)}$$ after plugging in the value $$x=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational18a-h8","type":"hint","dependencies":["a276c42SolveRational18a-h7"],"title":"Test a Value in Each Interval.","text":"Since $$\\\\frac{1}{4}$$ is a positive numer greater than $$0$$, we get the sign of quotient is positive in the interval $$(-\\\\infty,1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational18a-h9","type":"hint","dependencies":["a276c42SolveRational18a-h8"],"title":"Test a Value in Each Interval.","text":"For the interval $$(1,4)$$, take the point $$x=2$$ within the interval and plug it into the expression $$\\\\frac{1}{\\\\left(x-1\\\\right) \\\\left(x-4\\\\right)}$$. We get $$\\\\frac{-1}{2}$$ after plugging in $$x=2$$ into the $$\\\\frac{quotient1}{\\\\left(x-1\\\\right) \\\\left(x-4\\\\right)}$$. $$\\\\frac{-1}{2}$$ is a negative number, so we can mark the quotient negative in the interval $$(1,4)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational18a-h10","type":"hint","dependencies":["a276c42SolveRational18a-h9"],"title":"Test a Value in Each Interval.","text":"For the interval $$(4,\\\\infty)$$, take the point $$x=5$$ within the interval and plug it into the original expression $$\\\\frac{1}{\\\\left(x-1\\\\right) \\\\left(x-4\\\\right)}$$. We get $$\\\\frac{1}{4}$$ after plugging in $$x=5$$ into the quotient $$\\\\frac{1}{\\\\left(x-1\\\\right) \\\\left(x-4\\\\right)}$$. $$\\\\frac{1}{4}$$ is a positive number, so we can mark the quotient positive in the interval $$(4,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational18a-h11","type":"hint","dependencies":["a276c42SolveRational18a-h10"],"title":"Determine the Intervals Where the Inequality is Correct.","text":"We want the quotient to be less than zero, so the numbers in the intervals $$(1,4)$$ are the solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a276c42SolveRational19","title":"Solve Rational Inequality","body":"In the following exercises, solve each rational inequality and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6  Solve Rational Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a276c42SolveRational19a","stepAnswer":["(-4,-3)"],"problemType":"TextBox","stepTitle":"$$\\\\frac{4}{x^2+7x+12}<0$$","stepBody":"Please enter your answer as (x,y). Use U to indicate the union of two intervals, if applicable. If your answer is the interval from $$x$$ to $$\\\\infty$$, please enter it as $$(x,\\\\infty)$$. If your answer includes the endpoint of the interval, please use [ ] to indicate the inclusion of endpoints. For instance, [x,y) refers to the interval including endpoint $$x$$ and excluding endpoint $$y$$.","answerType":"string","variabilization":{},"answerLatex":"$$(-4,-3)$$","hints":{"DefaultPathway":[{"id":"a276c42SolveRational19a-h1","type":"hint","dependencies":[],"title":"Put The Equation In The Correct Form with Zero By Itself On One Side","text":"Write the inequality as one quotient on the left and zero on the right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x+3\\\\right) \\\\left(x+4\\\\right)$$"],"dependencies":["a276c42SolveRational19a-h1"],"title":"Factor the denominator","text":"Factor $$x^2+7x+12$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational19a-h3","type":"hint","dependencies":["a276c42SolveRational19a-h2"],"title":"Find The Critical Points of Equation","text":"The quotient is $$0$$ when the numerator is $$0$$. Since the numerator is always $$4$$, the quotient cannot be $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational19a-h4","type":"hint","dependencies":["a276c42SolveRational19a-h3"],"title":"Find The Critical Points of Equation","text":"The quotient will be undefined when the denominator is zero. $$\\\\left(x+3\\\\right) \\\\left(x+4\\\\right)=0$$ when $$x=-3$$, $$x=-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational19a-h5","type":"hint","dependencies":["a276c42SolveRational19a-h4"],"title":"Use the Critical Points to Divide the Number Line into Intervals.","text":"Given that the $$x=-3$$ and $$x=-4$$ are the two critical points, we can divide the number line into three intervals, namely, $$(-\\\\infty,-4)$$, $$(-4,-3)$$ and $$(-3,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational19a-h6","type":"hint","dependencies":["a276c42SolveRational19a-h5"],"title":"Test a Value in Each Interval.","text":"To find the sign of each factor in an interval, we choose any point in that interval and use it as a test point. Any point in the interval will give the expression the same sign, so we can choose any point in the interval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational19a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a276c42SolveRational19a-h6"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-\\\\infty,-4)$$, take the point $$x=-5$$ within the interval and plug it into the quotient $$\\\\frac{4}{\\\\left(x+3\\\\right) \\\\left(x+4\\\\right)}$$. What is the value of $$\\\\frac{4}{\\\\left(x+3\\\\right) \\\\left(x+4\\\\right)}$$ after plugging in the value $$x=-5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational19a-h8","type":"hint","dependencies":["a276c42SolveRational19a-h7"],"title":"Test a Value in Each Interval.","text":"Since $$2$$ is a positive numer greater than $$0$$, we get the sign of quotient is positive in the interval $$(-\\\\infty,-4)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational19a-h9","type":"hint","dependencies":["a276c42SolveRational19a-h8"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-4,-3)$$, take the point $$x=-3.5$$ within the interval and plug it into the expression $$\\\\frac{4}{\\\\left(x+3\\\\right) \\\\left(x+4\\\\right)}$$. We get $$-16$$ after plugging in $$x=-3.5$$ into the quotient $$\\\\frac{4}{\\\\left(x+3\\\\right) \\\\left(x+4\\\\right)}$$. $$-16$$ is a negative number, so we can mark the quotient negative in the interval $$(-4,-3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational19a-h10","type":"hint","dependencies":["a276c42SolveRational19a-h9"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-3,\\\\infty)$$, take the point $$x=0$$ within the interval and plug it into the original expression $$\\\\frac{4}{\\\\left(x+3\\\\right) \\\\left(x+4\\\\right)}$$. We get $$\\\\frac{1}{3}$$ after plugging in $$x=5$$ into the quotient $$\\\\frac{1}{\\\\left(x-1\\\\right) \\\\left(x-4\\\\right)}$$. $$\\\\frac{1}{3}$$ is a positive number, so we can mark the quotient positive in the interval $$(-3,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational19a-h11","type":"hint","dependencies":["a276c42SolveRational19a-h10"],"title":"Determine the Intervals Where the Inequality is Correct.","text":"We want the quotient to be less than zero, so the numbers in the intervals $$(-4,-3)$$ are the solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a276c42SolveRational2","title":"Solve Rational Inequality","body":"In the following exercises, solve each rational inequality and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6  Solve Rational Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a276c42SolveRational2a","stepAnswer":["(-inf,-6]U(5,inf)"],"problemType":"TextBox","stepTitle":"$$\\\\frac{x+6}{x-5} \\\\geq 0$$","stepBody":"Please enter your answer as (x,y). Use U to indicate the union of two intervals, if applicable. If your answer is the interval from $$x$$ to $$\\\\infty$$, please enter it as $$(x,\\\\infty)$$. If your answer includes the endpoint of the interval, please use [ ] to indicate the inclusion of endpoints. For instance, [x,y) refers to the interval including endpoint $$x$$ and excluding endpoint $$y$$.","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,-6] \\\\cup (5,\\\\infty)$$","hints":{"DefaultPathway":[{"id":"a276c42SolveRational2a-h1","type":"hint","dependencies":[],"title":"Put The Equation In The Correct Form with Zero By Itself On One Side","text":"Write the inequality as one quotient on the left and zero on the right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational2a-h2","type":"hint","dependencies":["a276c42SolveRational2a-h1"],"title":"Find The Critical Points of Equation","text":"Determine the critical points\u2014the points where the rational expression will be zero or undefined.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational2a-h3","type":"hint","dependencies":["a276c42SolveRational2a-h2"],"title":"Find The Critical Points of Equation","text":"When is the equation undefine? Note that the equation is undefined when the denominator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a276c42SolveRational2a-h3"],"title":"Find The Critical Points of Equation","text":"Solve the equation $$x-5=0$$. What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational2a-h5","type":"hint","dependencies":["a276c42SolveRational2a-h3"],"title":"Find The Critical Points of Equation","text":"When does the equation equal zero? Note that the equation is undefined when the numerator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6$$"],"dependencies":["a276c42SolveRational2a-h3"],"title":"Find The Critical Points of Equation","text":"Solve the equation $$x+6=0$$. What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational2a-h7","type":"hint","dependencies":["a276c42SolveRational2a-h4","a276c42SolveRational2a-h5","a276c42SolveRational2a-h6"],"title":"Use the Critical Points to Divide the Number Line into Intervals.","text":"Given that the $$x=-6$$ and $$x=5$$ are the two critical points, we can divide the number line into three intervals, namely, $$(-\\\\infty,-6)$$, $$(-6,5)$$ and $$(5,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational2a-h8","type":"hint","dependencies":["a276c42SolveRational2a-h7"],"title":"Test a Value in Each Interval.","text":"To find the sign of each factor in an interval, we choose any point in that interval and use it as a test point. Any point in the interval will give the expression the same sign, so we can choose any point in the interval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational2a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{12}$$"],"dependencies":["a276c42SolveRational2a-h8"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-\\\\infty,-6)$$, take the point $$x=-7$$ within the interval and plug it into the original expression. What is the value of $$\\\\frac{x+6}{x-5}$$ after plugging in the value $$x=-7$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational2a-h10","type":"hint","dependencies":["a276c42SolveRational2a-h9"],"title":"Test a Value in Each Interval.","text":"Since $$\\\\frac{1}{12}$$ is a positive numer greater than $$0$$, we get the sign of quotient is positive in the interval $$(-\\\\infty,-6)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational2a-h11","type":"hint","dependencies":["a276c42SolveRational2a-h10"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-6,5)$$, take the point $$x=0$$ within the interval and plug it into the original $$\\\\frac{\\\\operatorname{expression}\\\\left(x+6\\\\right)}{x-5}$$. We get $$\\\\frac{-6}{5}$$ after plugging in $$x=0$$ into the quotient $$\\\\frac{x+6}{x-5}$$. $$\\\\frac{-6}{5}$$ is a negative number, so we can mark the quotient negative in the interval $$(-6,5)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational2a-h12","type":"hint","dependencies":["a276c42SolveRational2a-h11"],"title":"Test a Value in Each Interval.","text":"For the interval $$(5,\\\\infty)$$, take the point $$x=6$$ within the interval and plug it into the original expression $$\\\\frac{x+6}{x-5}$$. We get $$12$$ after plugging in $$x=6$$ into the quotient $$\\\\frac{x+6}{x-5}$$. $$12$$ is a positive number, so we can mark the quotient positive in the interval $$(5,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational2a-h13","type":"hint","dependencies":["a276c42SolveRational2a-h12"],"title":"Determine the Intervals Where the Inequality is Correct.","text":"We want the quotient to be greater than or equal to zero, so the numbers in the intervals $$(-\\\\infty,-6)$$, and $$(5,\\\\infty)$$ are solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational2a-h14","type":"hint","dependencies":["a276c42SolveRational2a-h13"],"title":"Determine the Critical Points Where the Inequality is Correct","text":"The critical point $$x=5$$ makes the denominator $$0$$, so it must be excluded from the solution and we mark it with a parenthesis.\\\\nThe critical point $$x=-6$$ makes the whole rational expression $$0$$. The inequality requires that the rational expression be greater than or equal to $$0$$. So, $$-6$$ is part of the solution and we will mark it with a bracket.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational2a-h15","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-inf,-6]U(5,inf)"],"dependencies":["a276c42SolveRational2a-h14"],"title":"Write the Final Answer in the interval Form","text":"We know that $$(-\\\\infty,-6)$$, $$(5,\\\\infty)$$ and $$x=-6$$ are the answer, please use \\"[ ]\\". \\"( )\\" and U to write it in a mathematical form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a276c42SolveRational20","title":"Solve Rational Inequality","body":"In the following exercises, solve each rational inequality and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6  Solve Rational Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a276c42SolveRational20a","stepAnswer":["(-inf,-3)U(5/2,inf)"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2}{2x^2+x-15} \\\\geq 0$$","stepBody":"Please enter your answer as (x,y). Use U to indicate the union of two intervals, if applicable. If your answer is the interval from $$x$$ to $$\\\\infty$$, please enter it as $$(x,\\\\infty)$$. If your answer includes the endpoint of the interval, please use [ ] to indicate the inclusion of endpoints. For instance, [x,y) refers to the interval including endpoint $$x$$ and excluding endpoint $$y$$.","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,-3) \\\\cup (\\\\frac{5}{2},\\\\infty)$$","hints":{"DefaultPathway":[{"id":"a276c42SolveRational20a-h1","type":"hint","dependencies":[],"title":"Put The Equation In The Correct Form with Zero By Itself On One Side","text":"Write the inequality as one quotient on the left and zero on the right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(2x-5\\\\right) \\\\left(x+3\\\\right)$$"],"dependencies":["a276c42SolveRational20a-h1"],"title":"Factor the denominator","text":"Factor $$2x^2+x-15$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational20a-h3","type":"hint","dependencies":["a276c42SolveRational20a-h2"],"title":"Find The Critical Points of Equation","text":"The quotient is $$0$$ when the numerator is $$0$$. Since the numerator is always $$2$$, the quotient cannot be $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational20a-h4","type":"hint","dependencies":["a276c42SolveRational20a-h3"],"title":"Find The Critical Points of Equation","text":"The quotient will be undefined when the denominator is zero. $$\\\\left(2x-5\\\\right) \\\\left(x+3\\\\right)=0$$ when $$x=-3$$, $$x=\\\\frac{5}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational20a-h5","type":"hint","dependencies":["a276c42SolveRational20a-h4"],"title":"Use the Critical Points to Divide the Number Line into Intervals.","text":"Given that the $$x=-3$$ and $$x=\\\\frac{5}{2}$$ are the two critical points, we can divide the number line into three intervals, namely, $$(-\\\\infty,-3)$$, $$(-3,\\\\frac{5}{2})$$ and $$(\\\\frac{5}{2},\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational20a-h6","type":"hint","dependencies":["a276c42SolveRational20a-h5"],"title":"Test a Value in Each Interval.","text":"To find the sign of each factor in an interval, we choose any point in that interval and use it as a test point. Any point in the interval will give the expression the same sign, so we can choose any point in the interval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational20a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{13}$$"],"dependencies":["a276c42SolveRational20a-h6"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-\\\\infty,-3)$$, take the point $$x=-4$$ within the interval and plug it into the quotient $$\\\\frac{2}{\\\\left(2x-5\\\\right) \\\\left(x+3\\\\right)}$$. What is the value of $$\\\\frac{2}{\\\\left(2x-5\\\\right) \\\\left(x+3\\\\right)}$$ after plugging in the value $$x=-4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational20a-h8","type":"hint","dependencies":["a276c42SolveRational20a-h7"],"title":"Test a Value in Each Interval.","text":"Since $$\\\\frac{2}{13}$$ is a positive numer greater than $$0$$, we get the sign of quotient is positive in the interval $$(-\\\\infty,-3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational20a-h9","type":"hint","dependencies":["a276c42SolveRational20a-h8"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-3,\\\\frac{5}{2})$$, take the point $$x=0$$ within the interval and plug it into the expression $$\\\\frac{2}{\\\\left(2x-5\\\\right) \\\\left(x+3\\\\right)}$$. We get $$\\\\frac{-2}{15}$$ after plugging in $$x=0$$ into the quotient $$\\\\frac{2}{\\\\left(2x-5\\\\right) \\\\left(x+3\\\\right)}$$. $$\\\\frac{-2}{15}$$ is a negative number, so we can mark the quotient negative in the interval $$(-3,\\\\frac{5}{2})$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational20a-h10","type":"hint","dependencies":["a276c42SolveRational20a-h9"],"title":"Test a Value in Each Interval.","text":"For the interval $$(\\\\frac{5}{2},\\\\infty)$$, take the point $$x=3$$ within the interval and plug it into the original expression $$\\\\frac{2}{\\\\left(2x-5\\\\right) \\\\left(x+3\\\\right)}$$. We get $$\\\\frac{1}{3}$$ after plugging in $$x=3$$ into the quotient $$\\\\frac{2}{\\\\left(2x-5\\\\right) \\\\left(x+3\\\\right)}$$. $$\\\\frac{1}{3}$$ is a positive number, so we can mark the quotient positive in the interval $$(\\\\frac{5}{2},\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational20a-h11","type":"hint","dependencies":["a276c42SolveRational20a-h10"],"title":"Determine the Intervals Where the Inequality is Correct.","text":"We want the quotient to be greater than and equal to zero, since the quotient can never equal to $$0$$, so the numbers in the intervals $$(-\\\\infty,-3)$$ and $$(\\\\frac{5}{2},\\\\infty)$$ are the solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational20a-h12","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-inf,-3)U(5/2,inf)"],"dependencies":["a276c42SolveRational20a-h11"],"title":"Write the Final Answer in the interval Form","text":"We know that $$(-\\\\infty,-3)$$ and $$(\\\\frac{5}{2},\\\\infty)$$ are the answer, please use \\"[ ]\\". \\"( )\\" and U to write it in a mathematical form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a276c42SolveRational21","title":"Solve Rational Inequality","body":"In the following exercises, solve each rational inequality and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6  Solve Rational Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a276c42SolveRational21a","stepAnswer":["(-inf,-1)U(5/3,inf)"],"problemType":"TextBox","stepTitle":"$$\\\\frac{6}{3x^2-2x-5} \\\\geq 0$$","stepBody":"Please enter your answer as (x,y). Use U to indicate the union of two intervals, if applicable. If your answer is the interval from $$x$$ to $$\\\\infty$$, please enter it as $$(x,\\\\infty)$$. If your answer includes the endpoint of the interval, please use [ ] to indicate the inclusion of endpoints. For instance, [x,y) refers to the interval including endpoint $$x$$ and excluding endpoint $$y$$.","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,-1) \\\\cup (\\\\frac{5}{3},\\\\infty)$$","hints":{"DefaultPathway":[{"id":"a276c42SolveRational21a-h1","type":"hint","dependencies":[],"title":"Put The Equation In The Correct Form with Zero By Itself On One Side","text":"Write the inequality as one quotient on the left and zero on the right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational21a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(3x-5\\\\right) \\\\left(x+1\\\\right)$$"],"dependencies":["a276c42SolveRational21a-h1"],"title":"Factor the denominator","text":"Factor $$3x^2-2x-5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational21a-h3","type":"hint","dependencies":["a276c42SolveRational21a-h2"],"title":"Find The Critical Points of Equation","text":"The quotient is $$0$$ when the numerator is $$0$$. Since the numerator is always $$6$$, the quotient cannot be $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational21a-h4","type":"hint","dependencies":["a276c42SolveRational21a-h3"],"title":"Find The Critical Points of Equation","text":"The quotient will be undefined when the denominator is zero. $$\\\\left(3x-5\\\\right) \\\\left(x+1\\\\right)=0$$ when $$x=-1$$, $$x=\\\\frac{5}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational21a-h5","type":"hint","dependencies":["a276c42SolveRational21a-h4"],"title":"Use the Critical Points to Divide the Number Line into Intervals.","text":"Given that the $$x=-1$$ and $$x=\\\\frac{5}{3}$$ are the two critical points, we can divide the number line into three intervals, namely, $$(-\\\\infty,-1)$$, $$(-1,\\\\frac{5}{3})$$ and $$(\\\\frac{5}{3},\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational21a-h6","type":"hint","dependencies":["a276c42SolveRational21a-h5"],"title":"Test a Value in Each Interval.","text":"To find the sign of each factor in an interval, we choose any point in that interval and use it as a test point. Any point in the interval will give the expression the same sign, so we can choose any point in the interval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational21a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{6}{11}$$"],"dependencies":["a276c42SolveRational21a-h6"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-\\\\infty,-1)$$, take the point $$x=-2$$ within the interval and plug it into the quotient $$\\\\frac{6}{\\\\left(3x-5\\\\right) \\\\left(x+1\\\\right)}$$. What is the value of $$\\\\frac{6}{\\\\left(3x-5\\\\right) \\\\left(x+1\\\\right)}$$ after plugging in the value $$x=-2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational21a-h8","type":"hint","dependencies":["a276c42SolveRational21a-h7"],"title":"Test a Value in Each Interval.","text":"Since $$\\\\frac{6}{11}$$ is a positive numer greater than $$0$$, we get the sign of quotient is positive in the interval $$(-\\\\infty,-1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational21a-h9","type":"hint","dependencies":["a276c42SolveRational21a-h8"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-1,\\\\frac{5}{3})$$, take the point $$x=0$$ within the interval and plug it into the expression $$\\\\frac{6}{\\\\left(3x-5\\\\right) \\\\left(x+1\\\\right)}$$. We get $$\\\\frac{-6}{5}$$ after plugging in $$x=0$$ into the quotient $$\\\\frac{6}{\\\\left(3x-5\\\\right) \\\\left(x+1\\\\right)}$$. $$\\\\frac{-6}{5}$$ is a negative number, so we can mark the quotient negative in the interval $$(-1,\\\\frac{5}{3})$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational21a-h10","type":"hint","dependencies":["a276c42SolveRational21a-h9"],"title":"Test a Value in Each Interval.","text":"For the interval $$(\\\\frac{5}{3},\\\\infty)$$, take the point $$x=2$$ within the interval and plug it into the original expression $$\\\\frac{6}{\\\\left(3x-5\\\\right) \\\\left(x+1\\\\right)}$$. We get $$2$$ after plugging in $$x=2$$ into the quotient $$\\\\frac{6}{\\\\left(3x-5\\\\right) \\\\left(x+1\\\\right)}$$. $$2$$ is a positive number, so we can mark the quotient positive in the interval $$(\\\\frac{5}{3},\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational21a-h11","type":"hint","dependencies":["a276c42SolveRational21a-h10"],"title":"Determine the Intervals Where the Inequality is Correct.","text":"We want the quotient to be greater than and equal to zero, since the quotient can never equal to $$0$$, so the numbers in the intervals $$(-\\\\infty,-1)$$ and $$(\\\\frac{5}{3},\\\\infty)$$ are the solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational21a-h12","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-inf,-1)U(5/3,inf)"],"dependencies":["a276c42SolveRational21a-h11"],"title":"Write the Final Answer in the interval Form","text":"We know that $$(-\\\\infty,-1)$$ and $$(\\\\frac{5}{3},\\\\infty)$$ are the answer, please use \\"[ ]\\". \\"( )\\" and U to write it in a mathematical form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a276c42SolveRational22","title":"Solve Rational Inequality","body":"In the following exercises, solve each rational inequality and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6  Solve Rational Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a276c42SolveRational22a","stepAnswer":["(-inf,2/3)U(3/2,inf)"],"problemType":"TextBox","stepTitle":"$$\\\\frac{-2}{6x^2-13x+6} \\\\leq 0$$","stepBody":"Please enter your answer as (x,y). Use U to indicate the union of two intervals, if applicable. If your answer is the interval from $$x$$ to $$\\\\infty$$, please enter it as $$(x,\\\\infty)$$. If your answer includes the endpoint of the interval, please use [ ] to indicate the inclusion of endpoints. For instance, [x,y) refers to the interval including endpoint $$x$$ and excluding endpoint $$y$$.","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\frac{2}{3}) \\\\cup (\\\\frac{3}{2},\\\\infty)$$","hints":{"DefaultPathway":[{"id":"a276c42SolveRational22a-h1","type":"hint","dependencies":[],"title":"Put The Equation In The Correct Form with Zero By Itself On One Side","text":"Write the inequality as one quotient on the left and zero on the right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational22a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(2x-3\\\\right) \\\\left(3x-2\\\\right)$$"],"dependencies":["a276c42SolveRational22a-h1"],"title":"Factor the denominator","text":"Factor $$6x^2-13x+6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational22a-h3","type":"hint","dependencies":["a276c42SolveRational22a-h2"],"title":"Find The Critical Points of Equation","text":"The quotient is $$0$$ when the numerator is $$0$$. Since the numerator is always $$-2$$, the quotient cannot be $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational22a-h4","type":"hint","dependencies":["a276c42SolveRational22a-h3"],"title":"Find The Critical Points of Equation","text":"The quotient will be undefined when the denominator is zero. $$\\\\left(2x-3\\\\right) \\\\left(3x-2\\\\right)=0$$ when $$x=\\\\frac{3}{2}$$, $$x=\\\\frac{2}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational22a-h5","type":"hint","dependencies":["a276c42SolveRational22a-h4"],"title":"Use the Critical Points to Divide the Number Line into Intervals.","text":"Given that the $$x=\\\\frac{2}{3}$$ and $$x=\\\\frac{3}{2}$$ are the two critical points, we can divide the number line into three intervals, namely, $$(-\\\\infty,\\\\frac{2}{3})$$, $$(\\\\frac{2}{3},\\\\frac{3}{2})$$ and $$(\\\\frac{3}{2},\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational22a-h6","type":"hint","dependencies":["a276c42SolveRational22a-h5"],"title":"Test a Value in Each Interval.","text":"To find the sign of each factor in an interval, we choose any point in that interval and use it as a test point. Any point in the interval will give the expression the same sign, so we can choose any point in the interval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational22a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{3}$$"],"dependencies":["a276c42SolveRational22a-h6"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-\\\\infty,\\\\frac{2}{3})$$, take the point $$x=0$$ within the interval and plug it into the quotient $$\\\\frac{-2}{\\\\left(2x-3\\\\right) \\\\left(3x-2\\\\right)}$$. What is the value of $$\\\\frac{-2}{\\\\left(2x-3\\\\right) \\\\left(3x-2\\\\right)}$$ after plugging in the value $$x=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational22a-h8","type":"hint","dependencies":["a276c42SolveRational22a-h7"],"title":"Test a Value in Each Interval.","text":"Since $$\\\\frac{-1}{3}$$ is a negative numer less than $$0$$, we get the sign of quotient is negative in the interval $$(-\\\\infty,\\\\frac{2}{3})$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational22a-h9","type":"hint","dependencies":["a276c42SolveRational22a-h8"],"title":"Test a Value in Each Interval.","text":"For the interval $$(\\\\frac{2}{3},\\\\frac{3}{2})$$, take the point $$x=1$$ within the interval and plug it into the expression $$\\\\frac{-2}{\\\\left(2x-3\\\\right) \\\\left(3x-2\\\\right)}$$. We get $$2$$ after plugging in $$x=1$$ into the quotient $$\\\\frac{-2}{\\\\left(2x-3\\\\right) \\\\left(3x-2\\\\right)}$$. $$2$$ is a positive number, so we can mark the quotient negative in the interval $$(\\\\frac{2}{3},\\\\frac{3}{2})$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational22a-h10","type":"hint","dependencies":["a276c42SolveRational22a-h9"],"title":"Test a Value in Each Interval.","text":"For the interval $$(\\\\frac{3}{2},\\\\infty)$$, take the point $$x=2$$ within the interval and plug it into the original expression $$\\\\frac{-2}{\\\\left(2x-3\\\\right) \\\\left(3x-2\\\\right)}$$. We get $$-2$$ after plugging in $$x=2$$ into the quotient $$\\\\frac{6}{\\\\left(3x-5\\\\right) \\\\left(x+1\\\\right)}$$. $$-2$$ is a negative number, so we can mark the quotient negative in the interval $$(\\\\frac{3}{2},\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational22a-h11","type":"hint","dependencies":["a276c42SolveRational22a-h10"],"title":"Determine the Intervals Where the Inequality is Correct.","text":"We want the quotient to be less than and equal to zero, since the quotient can never equal to $$0$$, so the numbers in the intervals $$(-\\\\infty,\\\\frac{2}{3})$$ and $$(\\\\frac{3}{2},\\\\infty)$$ are the solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational22a-h12","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-inf,2/3)U(3/2,inf)"],"dependencies":["a276c42SolveRational22a-h11"],"title":"Write the Final Answer in the interval Form","text":"We know that $$(-\\\\infty,\\\\frac{2}{3})$$ and $$(\\\\frac{3}{2},\\\\infty)$$ are the answer, please use \\"[ ]\\". \\"( )\\" and U to write it in a mathematical form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a276c42SolveRational23","title":"Solve Rational Inequality","body":"In the following exercises, solve each rational inequality and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6  Solve Rational Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a276c42SolveRational23a","stepAnswer":["(-inf,-3/2)U(2/5,inf)"],"problemType":"TextBox","stepTitle":"$$\\\\frac{-1}{10x^2+11x-6}$$","stepBody":"Please enter your answer as (x,y). Use U to indicate the union of two intervals, if applicable. If your answer is the interval from $$x$$ to $$\\\\infty$$, please enter it as $$(x,\\\\infty)$$. If your answer includes the endpoint of the interval, please use [ ] to indicate the inclusion of endpoints. For instance, [x,y) refers to the interval including endpoint $$x$$ and excluding endpoint $$y$$.","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\frac{-3}{2}) \\\\cup (\\\\frac{2}{5},\\\\infty)$$","hints":{"DefaultPathway":[{"id":"a276c42SolveRational23a-h1","type":"hint","dependencies":[],"title":"Put The Equation In The Correct Form with Zero By Itself On One Side","text":"Write the inequality as one quotient on the left and zero on the right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational23a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(5x-2\\\\right) \\\\left(2x+3\\\\right)$$"],"dependencies":["a276c42SolveRational23a-h1"],"title":"Factor the denominator","text":"Factor $$10x^2+11x-6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational23a-h3","type":"hint","dependencies":["a276c42SolveRational23a-h2"],"title":"Find The Critical Points of Equation","text":"The quotient is $$0$$ when the numerator is $$0$$. Since the numerator is always $$-1$$, the quotient cannot be $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational23a-h4","type":"hint","dependencies":["a276c42SolveRational23a-h3"],"title":"Find The Critical Points of Equation","text":"The quotient will be undefined when the denominator is zero. $$\\\\left(5x-2\\\\right) \\\\left(2x+3\\\\right)=0$$ when $$x=\\\\frac{2}{5}$$, $$x=\\\\frac{-3}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational23a-h5","type":"hint","dependencies":["a276c42SolveRational23a-h4"],"title":"Use the Critical Points to Divide the Number Line into Intervals.","text":"Given that the $$x=\\\\frac{-3}{2}$$ and $$x=\\\\frac{2}{5}$$ are the two critical points, we can divide the number line into three intervals, namely, $$(-\\\\infty,\\\\frac{-3}{2})$$, $$(\\\\frac{-3}{2},\\\\frac{2}{5})$$ and $$(\\\\frac{2}{5},\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational23a-h6","type":"hint","dependencies":["a276c42SolveRational23a-h5"],"title":"Test a Value in Each Interval.","text":"To find the sign of each factor in an interval, we choose any point in that interval and use it as a test point. Any point in the interval will give the expression the same sign, so we can choose any point in the interval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational23a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{12}$$"],"dependencies":["a276c42SolveRational23a-h6"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-\\\\infty,\\\\frac{-3}{2})$$, take the point $$x=-2$$ within the interval and plug it into the quotient $$\\\\frac{-1}{\\\\left(5x-2\\\\right) \\\\left(2x+3\\\\right)}$$. What is the value of $$\\\\frac{-1}{\\\\left(5x-2\\\\right) \\\\left(2x+3\\\\right)}$$ after plugging in the value $$x=-2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational23a-h8","type":"hint","dependencies":["a276c42SolveRational23a-h7"],"title":"Test a Value in Each Interval.","text":"Since $$\\\\frac{-1}{12}$$ is a negative numer less than $$0$$, we get the sign of quotient is negative in the interval $$(-\\\\infty,\\\\frac{-3}{2})$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational23a-h9","type":"hint","dependencies":["a276c42SolveRational23a-h8"],"title":"Test a Value in Each Interval.","text":"For the interval $$(\\\\frac{-3}{2},\\\\frac{2}{5})$$, take the point $$x=0$$ within the interval and plug it into the expression $$\\\\frac{-1}{\\\\left(5x-2\\\\right) \\\\left(2x+3\\\\right)}$$. We get $$\\\\frac{1}{6}$$ after plugging in $$x=0$$ into the quotient $$\\\\frac{-1}{\\\\left(5x-2\\\\right) \\\\left(2x+3\\\\right)}$$. 1/6is a positive number, so we can mark the quotient negative in the interval $$(\\\\frac{-3}{2},\\\\frac{2}{5})$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational23a-h10","type":"hint","dependencies":["a276c42SolveRational23a-h9"],"title":"Test a Value in Each Interval.","text":"For the interval $$(\\\\frac{2}{5},\\\\infty)$$, take the point $$x=1$$ within the interval and plug it into the original expression $$\\\\frac{-1}{\\\\left(5x-2\\\\right) \\\\left(2x+3\\\\right)}$$. We get $$\\\\frac{-1}{15}$$ after plugging in $$x=1$$ into the quotient $$\\\\frac{-1}{\\\\left(5x-2\\\\right) \\\\left(2x+3\\\\right)}$$. $$\\\\frac{-1}{15}$$ is a negative number, so we can mark the quotient negative in the interval $$(\\\\frac{2}{5},\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational23a-h11","type":"hint","dependencies":["a276c42SolveRational23a-h10"],"title":"Determine the Intervals Where the Inequality is Correct.","text":"We want the quotient to be less than and equal to zero, since the quotient can never equal to $$0$$, so the numbers in the intervals $$(-\\\\infty,\\\\frac{-3}{2})$$ and $$(\\\\frac{2}{5},\\\\infty)$$ are the solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational23a-h12","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-inf,-3/2)U(2/5,inf)"],"dependencies":["a276c42SolveRational23a-h11"],"title":"Write the Final Answer in the interval Form","text":"We know that $$(-\\\\infty,\\\\frac{-3}{2})$$ and $$(\\\\frac{2}{5},\\\\infty)$$ are the answer, please use \\"[ ]\\". \\"( )\\" and U to write it in a mathematical form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational23a-h12","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-inf,-5)U(5,inf)"],"dependencies":["a276c42SolveRational23a-h11"],"title":"Write the Final Answer in the interval Form","text":"We know that $$(-\\\\infty,-5)$$ and $$(5,\\\\infty)$$ are the answer, please use \\"[ ]\\". \\"( )\\" and U to write it in a mathematical form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a276c42SolveRational24","title":"Solve Rational Inequality","body":"In the following exercises, solve each rational inequality and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6  Solve Rational Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a276c42SolveRational24a","stepAnswer":["(-4,4)"],"problemType":"TextBox","stepTitle":"$$\\\\frac{1}{x^2-16}$$","stepBody":"Please enter your answer as (x,y). Use U to indicate the union of two intervals, if applicable. If your answer is the interval from $$x$$ to $$\\\\infty$$, please enter it as $$(x,\\\\infty)$$. If your answer includes the endpoint of the interval, please use [ ] to indicate the inclusion of endpoints. For instance, [x,y) refers to the interval including endpoint $$x$$ and excluding endpoint $$y$$.","answerType":"string","variabilization":{},"answerLatex":"$$(-4,4)$$","hints":{"DefaultPathway":[{"id":"a276c42SolveRational24a-h1","type":"hint","dependencies":[],"title":"Put The Equation In The Correct Form with Zero By Itself On One Side","text":"Write the inequality as one quotient on the left and zero on the right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational24a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x+4\\\\right) \\\\left(x-4\\\\right)$$"],"dependencies":["a276c42SolveRational24a-h1"],"title":"Factor the denominator","text":"Factor $$x^2-16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational24a-h3","type":"hint","dependencies":["a276c42SolveRational24a-h2"],"title":"Find The Critical Points of Equation","text":"The quotient is $$0$$ when the numerator is $$0$$. Since the numerator is always $$1$$, the quotient cannot be $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational24a-h4","type":"hint","dependencies":["a276c42SolveRational24a-h3"],"title":"Find The Critical Points of Equation","text":"The quotient will be undefined when the denominator is zero. $$\\\\left(x+4\\\\right) \\\\left(x-4\\\\right)=0$$ when $$x=-4$$, $$x=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational24a-h5","type":"hint","dependencies":["a276c42SolveRational24a-h4"],"title":"Use the Critical Points to Divide the Number Line into Intervals.","text":"Given that the $$x=-4$$ and $$x=4$$ are the two critical points, we can divide the number line into three intervals, namely, $$(-\\\\infty,-4)$$, $$(-4,4)$$ and $$(4,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational24a-h6","type":"hint","dependencies":["a276c42SolveRational24a-h5"],"title":"Test a Value in Each Interval.","text":"To find the sign of each factor in an interval, we choose any point in that interval and use it as a test point. Any point in the interval will give the expression the same sign, so we can choose any point in the interval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational24a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{9}$$"],"dependencies":["a276c42SolveRational24a-h6"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-\\\\infty,-4)$$, take the point $$x=-5$$ within the interval and plug it into the quotient $$\\\\frac{1}{\\\\left(x+4\\\\right) \\\\left(x-4\\\\right)}$$. What is the value of $$\\\\frac{1}{\\\\left(x+4\\\\right) \\\\left(x-4\\\\right)}$$ after plugging in the value $$x=-5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational24a-h8","type":"hint","dependencies":["a276c42SolveRational24a-h7"],"title":"Test a Value in Each Interval.","text":"Since $$\\\\frac{1}{9}$$ is a positive numer greater than $$0$$, we get the sign of quotient is positive in the interval $$(-\\\\infty,-4)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational24a-h9","type":"hint","dependencies":["a276c42SolveRational24a-h8"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-4,4)$$, take the point $$x=0$$ within the interval and plug it into the expression $$\\\\frac{1}{\\\\left(x+4\\\\right) \\\\left(x-4\\\\right)}$$. We get $$\\\\frac{-1}{16}$$ after plugging in $$x=0$$ into the quotient $$\\\\frac{1}{\\\\left(x+4\\\\right) \\\\left(x-4\\\\right)}$$. $$\\\\frac{-1}{16}$$ is a negative number, so we can mark the quotient negative in the interval $$(-4,4)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational24a-h10","type":"hint","dependencies":["a276c42SolveRational24a-h9"],"title":"Test a Value in Each Interval.","text":"For the interval $$(4,\\\\infty)$$, take the point $$x=5$$ within the interval and plug it into the original expression $$\\\\frac{1}{\\\\left(x+4\\\\right) \\\\left(x-4\\\\right)}$$. We get $$\\\\frac{1}{9}$$ after plugging in $$x=5$$ into the quotient $$\\\\frac{1}{\\\\left(x+4\\\\right) \\\\left(x-4\\\\right)}$$. $$\\\\frac{1}{9}$$ is a positive number, so we can mark the quotient positive in the interval $$(4,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational24a-h11","type":"hint","dependencies":["a276c42SolveRational24a-h10"],"title":"Determine the Intervals Where the Inequality is Correct.","text":"We want the quotient to be less than zero, so the numbers in the intervals $$(-4,4)$$ are the solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a276c42SolveRational25","title":"Solve Rational Inequality","body":"In the following exercises, solve each rational inequality and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6  Solve Rational Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a276c42SolveRational25a","stepAnswer":["(-inf,-5)U(5,inf)"],"problemType":"TextBox","stepTitle":"$$\\\\frac{4}{x^2-25}>0$$","stepBody":"Please enter your answer as (x,y). Use U to indicate the union of two intervals, if applicable. If your answer is the interval from $$x$$ to $$\\\\infty$$, please enter it as $$(x,\\\\infty)$$. If your answer includes the endpoint of the interval, please use [ ] to indicate the inclusion of endpoints. For instance, [x,y) refers to the interval including endpoint $$x$$ and excluding endpoint $$y$$.","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,-5) \\\\cup (5,\\\\infty)$$","hints":{"DefaultPathway":[{"id":"a276c42SolveRational25a-h1","type":"hint","dependencies":[],"title":"Put The Equation In The Correct Form with Zero By Itself On One Side","text":"Write the inequality as one quotient on the left and zero on the right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational25a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x+5\\\\right) \\\\left(x-5\\\\right)$$"],"dependencies":["a276c42SolveRational25a-h1"],"title":"Factor the denominator","text":"Factor $$x^2-25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational25a-h3","type":"hint","dependencies":["a276c42SolveRational25a-h2"],"title":"Find The Critical Points of Equation","text":"The quotient is $$0$$ when the numerator is $$0$$. Since the numerator is always $$4$$, the quotient cannot be $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational25a-h4","type":"hint","dependencies":["a276c42SolveRational25a-h3"],"title":"Find The Critical Points of Equation","text":"The quotient will be undefined when the denominator is zero. $$\\\\left(x+5\\\\right) \\\\left(x-5\\\\right)=0$$ when $$x=-5$$, $$x=5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational25a-h5","type":"hint","dependencies":["a276c42SolveRational25a-h4"],"title":"Use the Critical Points to Divide the Number Line into Intervals.","text":"Given that the $$x=-5$$ and $$x=5$$ are the two critical points, we can divide the number line into three intervals, namely, $$(-\\\\infty,-5)$$, $$(-5,5)$$ and $$(5,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational25a-h6","type":"hint","dependencies":["a276c42SolveRational25a-h5"],"title":"Test a Value in Each Interval.","text":"To find the sign of each factor in an interval, we choose any point in that interval and use it as a test point. Any point in the interval will give the expression the same sign, so we can choose any point in the interval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational25a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{4}{11}$$"],"dependencies":["a276c42SolveRational25a-h6"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-\\\\infty,-5)$$, take the point $$x=-6$$ within the interval and plug it into the quotient $$\\\\frac{4}{\\\\left(x+5\\\\right) \\\\left(x-5\\\\right)}$$. What is the value of $$\\\\frac{4}{\\\\left(x+5\\\\right) \\\\left(x-5\\\\right)}$$ after plugging in the value $$x=-6$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational25a-h8","type":"hint","dependencies":["a276c42SolveRational25a-h7"],"title":"Test a Value in Each Interval.","text":"Since $$\\\\frac{4}{11}$$ is a positive numer greater than $$0$$, we get the sign of quotient is positive in the interval $$(-\\\\infty,-5)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational25a-h9","type":"hint","dependencies":["a276c42SolveRational25a-h8"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-5,5)$$, take the point $$x=0$$ within the interval and plug it into the expression $$\\\\frac{4}{\\\\left(x+5\\\\right) \\\\left(x-5\\\\right)}$$. We get $$\\\\frac{-4}{25}$$ after plugging in $$x=0$$ into the quotient $$\\\\frac{4}{\\\\left(x+5\\\\right) \\\\left(x-5\\\\right)}$$. $$\\\\frac{-4}{25}$$ is a negative number, so we can mark the quotient negative in the interval $$(-5,5)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational25a-h10","type":"hint","dependencies":["a276c42SolveRational25a-h9"],"title":"Test a Value in Each Interval.","text":"For the interval $$(5,\\\\infty)$$, take the point $$x=6$$ within the interval and plug it into the original expression $$\\\\frac{4}{\\\\left(x+5\\\\right) \\\\left(x-5\\\\right)}$$. We get $$\\\\frac{4}{11}$$ after plugging in $$x=6$$ into the quotient $$\\\\frac{4}{\\\\left(x+5\\\\right) \\\\left(x-5\\\\right)}$$. $$\\\\frac{4}{11}$$ is a positive number, so we can mark the quotient positive in the interval $$(5,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational25a-h11","type":"hint","dependencies":["a276c42SolveRational25a-h10"],"title":"Determine the Intervals Where the Inequality is Correct.","text":"We want the quotient to be greater than zero, so the numbers in the intervals $$(-\\\\infty,-5)$$ and $$(5,\\\\infty)$$. are the solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a276c42SolveRational3","title":"Solve Rational Inequality","body":"In the following exercises, solve each rational inequality and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6  Solve Rational Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a276c42SolveRational3a","stepAnswer":["(-2,4]"],"problemType":"TextBox","stepTitle":"$$\\\\frac{x-4}{x+2} \\\\leq 0$$","stepBody":"Please enter your answer as (x,y). Use U to indicate the union of two intervals, if applicable. If your answer is the interval from $$x$$ to $$\\\\infty$$, please enter it as $$(x,\\\\infty)$$. If your answer includes the endpoint of the interval, please use [ ] to indicate the inclusion of endpoints. For instance, [x,y) refers to the interval including endpoint $$x$$ and excluding endpoint $$y$$.","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a276c42SolveRational3a-h1","type":"hint","dependencies":[],"title":"Put The Equation In The Correct Form with Zero By Itself On One Side","text":"Write the inequality as one quotient on the left and zero on the right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational3a-h2","type":"hint","dependencies":["a276c42SolveRational3a-h1"],"title":"Find The Critical Points of Equation","text":"Determine the critical points\u2014the points where the rational expression will be zero or undefined.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational3a-h3","type":"hint","dependencies":["a276c42SolveRational3a-h2"],"title":"Find The Critical Points of Equation","text":"When is the equation undefine? Note that the equation is undefined when the denominator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a276c42SolveRational3a-h3"],"title":"Find The Critical Points of Equation","text":"Solve the equation $$x+2=0$$. What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational3a-h5","type":"hint","dependencies":["a276c42SolveRational3a-h3"],"title":"Find The Critical Points of Equation","text":"When does the equation equal zero? Note that the equation is undefined when the numerator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational3a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a276c42SolveRational3a-h3"],"title":"Find The Critical Points of Equation","text":"Solve the equation $$x-4=0$$. What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational3a-h7","type":"hint","dependencies":["a276c42SolveRational3a-h4","a276c42SolveRational3a-h5","a276c42SolveRational3a-h6"],"title":"Use the Critical Points to Divide the Number Line into Intervals.","text":"Given that the $$x=-2$$ and $$x=4$$ are the two critical points, we can divide the number line into three intervals, namely, $$(-\\\\infty,-2)$$, $$(-2,4)$$ and $$(4,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational3a-h8","type":"hint","dependencies":["a276c42SolveRational3a-h7"],"title":"Test a Value in Each Interval.","text":"To find the sign of each factor in an interval, we choose any point in that interval and use it as a test point. Any point in the interval will give the expression the same sign, so we can choose any point in the interval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational3a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a276c42SolveRational3a-h8"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-\\\\infty,-2)$$, take the point $$x=-3$$ within the interval and plug it into the original expression. What is the value of $$\\\\frac{x-4}{x+2}$$ after plugging in the value $$x=-3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational3a-h10","type":"hint","dependencies":["a276c42SolveRational3a-h9"],"title":"Test a Value in Each Interval.","text":"Since $$7$$ is a positive numer greater than $$0$$, we get the sign of quotient is positive in the interval $$(-\\\\infty,-2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational3a-h11","type":"hint","dependencies":["a276c42SolveRational3a-h10"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-2,4)$$, take the point $$x=0$$ within the interval and plug it into the original expression $$\\\\frac{x-4}{x+2}$$. We get $$-2$$ after plugging in $$x=0$$ into the quotient $$\\\\frac{x-4}{x+2}$$. $$-2$$ is a negative number, so we can mark the quotient negative in the interval $$(-2,4)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational3a-h12","type":"hint","dependencies":["a276c42SolveRational3a-h11"],"title":"Test a Value in Each Interval.","text":"For the interval $$(5,\\\\infty)$$, take the point $$x=6$$ within the interval and plug it into the original expression $$\\\\frac{x+6}{x-5}$$. We get $$12$$ after plugging in $$x=6$$ into the quotient $$\\\\frac{x+6}{x-5}$$. $$12$$ is a positive number, so we can mark the quotient positive in the interval $$(5,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational3a-h13","type":"hint","dependencies":["a276c42SolveRational3a-h12"],"title":"Determine the Intervals Where the Inequality is Correct.","text":"We want the quotient to be less than or equal to zero, so the numbers in the interval $$(-2,4)$$ are the solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational3a-h14","type":"hint","dependencies":["a276c42SolveRational3a-h13"],"title":"Determine the Critical Points Where the Inequality is Correct","text":"The critical point $$x=-2$$ makes the denominator $$0$$, so it must be excluded from the solution and we mark it with a parenthesis.\\\\nThe critical point $$x=4$$ makes the whole rational expression $$0$$. The inequality requires that the rational expression be greater than or equal to $$0$$. So, $$4$$ is part of the solution and we will mark it with a bracket.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational3a-h15","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-2,4]"],"dependencies":["a276c42SolveRational3a-h14"],"title":"Write the Final Answer in the interval Form","text":"We know that $$(-2,4)$$ and $$x=4$$ are the answer, please use \\"[ ]\\". \\"( )\\" and U to write it in a mathematical form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a276c42SolveRational4","title":"Solve Rational Inequality","body":"In the following exercises, solve each rational inequality and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6  Solve Rational Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a276c42SolveRational4a","stepAnswer":["(-inf,1)U(7,inf)"],"problemType":"TextBox","stepTitle":"$$\\\\frac{x-7}{x-1}>0$$","stepBody":"Please enter your answer as (x,y). Use U to indicate the union of two intervals, if applicable. If your answer is the interval from $$x$$ to $$\\\\infty$$, please enter it as $$(x,\\\\infty)$$. If your answer includes the endpoint of the interval, please use [ ] to indicate the inclusion of endpoints. For instance, [x,y) refers to the interval including endpoint $$x$$ and excluding endpoint $$y$$.","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,1) \\\\cup (7,\\\\infty)$$","hints":{"DefaultPathway":[{"id":"a276c42SolveRational4a-h1","type":"hint","dependencies":[],"title":"Put The Equation In The Correct Form with Zero By Itself On One Side","text":"Write the inequality as one quotient on the left and zero on the right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational4a-h2","type":"hint","dependencies":["a276c42SolveRational4a-h1"],"title":"Find The Critical Points of Equation","text":"Determine the critical points\u2014the points where the rational expression will be zero or undefined.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational4a-h3","type":"hint","dependencies":["a276c42SolveRational4a-h2"],"title":"Find The Critical Points of Equation","text":"When is the equation undefine? Note that the equation is undefined when the denominator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a276c42SolveRational4a-h3"],"title":"Find The Critical Points of Equation","text":"Solve the equation $$x-1=0$$. What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational4a-h5","type":"hint","dependencies":["a276c42SolveRational4a-h3"],"title":"Find The Critical Points of Equation","text":"When does the equation equal zero? Note that the equation is undefined when the numerator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational4a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a276c42SolveRational4a-h3"],"title":"Find The Critical Points of Equation","text":"Solve the equation $$x-7=0$$. What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational4a-h7","type":"hint","dependencies":["a276c42SolveRational4a-h4","a276c42SolveRational4a-h5","a276c42SolveRational4a-h6"],"title":"Use the Critical Points to Divide the Number Line into Intervals.","text":"Given that the $$x=-2$$ and $$x=4$$ are the two critical points, we can divide the number line into three intervals, namely, $$(-\\\\infty,-2)$$, $$(-2,4)$$ and $$(4,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational4a-h8","type":"hint","dependencies":["a276c42SolveRational4a-h7"],"title":"Test a Value in Each Interval.","text":"To find the sign of each factor in an interval, we choose any point in that interval and use it as a test point. Any point in the interval will give the expression the same sign, so we can choose any point in the interval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational4a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a276c42SolveRational4a-h8"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-\\\\infty,1)$$, take the point $$x=0$$ within the interval and plug it into the original expression. What is the value $$\\\\frac{\\\\operatorname{of}\\\\left(x-7\\\\right)}{x-1}$$ after plugging in the value $$x=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational4a-h10","type":"hint","dependencies":["a276c42SolveRational4a-h9"],"title":"Test a Value in Each Interval.","text":"Since $$7$$ is a positive numer greater than $$0$$, we get the sign of quotient is positive in the interval $$(-\\\\infty,1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational4a-h11","type":"hint","dependencies":["a276c42SolveRational4a-h10"],"title":"Test a Value in Each Interval.","text":"For the interval $$(1,7)$$, take the point $$x=2$$ within the interval and plug it into the original expression $$\\\\frac{x-7}{x-1}$$. We get $$-5$$ after plugging in $$x=2$$ into the quotient $$\\\\frac{x-7}{x-1}$$. $$-5$$ is a negative number, so we can mark the quotient negative in the interval $$(1,7)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational4a-h12","type":"hint","dependencies":["a276c42SolveRational4a-h11"],"title":"Test a Value in Each Interval.","text":"For the interval $$(5,\\\\infty)$$, take the point $$x=6$$ within the interval and plug it into the original expression $$\\\\frac{x+6}{x-5}$$. We get $$12$$ after plugging in $$x=6$$ into the quotient $$\\\\frac{x+6}{x-5}$$. $$12$$ is a positive number, so we can mark the quotient positive in the interval $$(5,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational4a-h13","type":"hint","dependencies":["a276c42SolveRational4a-h12"],"title":"Determine the Intervals Where the Inequality is Correct.","text":"We want the quotient to be greater than zero, so the numbers in the interval $$(1,7)$$ are the solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a276c42SolveRational5","title":"Solve Rational Inequality","body":"In the following exercises, solve each rational inequality and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6  Solve Rational Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a276c42SolveRational5a","stepAnswer":["(-inf,-8)U(-3,inf)"],"problemType":"TextBox","stepTitle":"$$\\\\frac{x+8}{x+3}>0$$","stepBody":"Please enter your answer as (x,y). Use U to indicate the union of two intervals, if applicable. If your answer is the interval from $$x$$ to $$\\\\infty$$, please enter it as $$(x,\\\\infty)$$. If your answer includes the endpoint of the interval, please use [ ] to indicate the inclusion of endpoints. For instance, [x,y) refers to the interval including endpoint $$x$$ and excluding endpoint $$y$$.","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,-8) \\\\cup (-3,\\\\infty)$$","hints":{"DefaultPathway":[{"id":"a276c42SolveRational5a-h1","type":"hint","dependencies":[],"title":"Put The Equation In The Correct Form with Zero By Itself On One Side","text":"Write the inequality as one quotient on the left and zero on the right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational5a-h2","type":"hint","dependencies":["a276c42SolveRational5a-h1"],"title":"Find The Critical Points of Equation","text":"Determine the critical points\u2014the points where the rational expression will be zero or undefined.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational5a-h3","type":"hint","dependencies":["a276c42SolveRational5a-h2"],"title":"Find The Critical Points of Equation","text":"When is the equation undefine? Note that the equation is undefined when the denominator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a276c42SolveRational5a-h3"],"title":"Find The Critical Points of Equation","text":"Solve the equation $$x+3=0$$. What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational5a-h5","type":"hint","dependencies":["a276c42SolveRational5a-h3"],"title":"Find The Critical Points of Equation","text":"When does the equation equal zero? Note that the equation is undefined when the numerator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational5a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-8$$"],"dependencies":["a276c42SolveRational5a-h3"],"title":"Find The Critical Points of Equation","text":"Solve the equation $$x+8=0$$. What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational5a-h7","type":"hint","dependencies":["a276c42SolveRational5a-h4","a276c42SolveRational5a-h5","a276c42SolveRational5a-h6"],"title":"Use the Critical Points to Divide the Number Line into Intervals.","text":"Given that the $$x=-3$$ and $$x=-8$$ are the two critical points, we can divide the number line into three intervals, namely, $$(-\\\\infty,-8)$$, $$(-8,-3)$$ and $$(-3,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational5a-h8","type":"hint","dependencies":["a276c42SolveRational5a-h7"],"title":"Test a Value in Each Interval.","text":"To find the sign of each factor in an interval, we choose any point in that interval and use it as a test point. Any point in the interval will give the expression the same sign, so we can choose any point in the interval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational5a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{6}$$"],"dependencies":["a276c42SolveRational5a-h8"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-\\\\infty,-8)$$, take the point $$x=-9$$ within the interval and plug it into the original expression. What is the value of $$\\\\frac{x+8}{x+3}$$ after plugging in the value $$x=-9$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational5a-h10","type":"hint","dependencies":["a276c42SolveRational5a-h9"],"title":"Test a Value in Each Interval.","text":"Since $$\\\\frac{1}{6}$$ is a positive numer greater than $$0$$, we get the sign of quotient is positive in the interval $$(-\\\\infty,-8)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational5a-h11","type":"hint","dependencies":["a276c42SolveRational5a-h10"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-8,-3)$$, take the point $$x=-6$$ within the interval and plug it into the original expression $$\\\\frac{x+8}{x+3}$$. We get $$\\\\frac{-2}{3}$$ after plugging in $$x=-6$$ into the quotient $$\\\\frac{x+8}{x+3}$$. $$-5$$ is a negative number, so we can mark the quotient negative in the interval $$(-8,-3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational5a-h12","type":"hint","dependencies":["a276c42SolveRational5a-h11"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-3,\\\\infty)$$, take the point $$x=0$$ within the interval and plug it into the original expression $$\\\\frac{x+8}{x+3}$$. We get $$\\\\frac{8}{3}$$ after plugging in $$x=0$$ into the quotient $$\\\\frac{x+8}{x+3}$$. $$\\\\frac{8}{3}$$ is a positive number, so we can mark the quotient positive in the interval $$(-3,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational5a-h13","type":"hint","dependencies":["a276c42SolveRational5a-h12"],"title":"Determine the Intervals Where the Inequality is Correct.","text":"We want the quotient to be greater than zero, so the numbers in the interval $$(-\\\\infty,-8) \\\\cup (-3,\\\\infty)$$ are the solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a276c42SolveRational6","title":"Solve Rational Inequality","body":"In the following exercises, solve each rational inequality and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6  Solve Rational Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a276c42SolveRational6a","stepAnswer":["(-5,6)"],"problemType":"TextBox","stepTitle":"$$\\\\frac{x-6}{x+5}<0$$","stepBody":"Please enter your answer as (x,y). Use U to indicate the union of two intervals, if applicable. If your answer is the interval from $$x$$ to $$\\\\infty$$, please enter it as $$(x,\\\\infty)$$. If your answer includes the endpoint of the interval, please use [ ] to indicate the inclusion of endpoints. For instance, [x,y) refers to the interval including endpoint $$x$$ and excluding endpoint $$y$$.","answerType":"string","variabilization":{},"answerLatex":"$$(-5,6)$$","hints":{"DefaultPathway":[{"id":"a276c42SolveRational6a-h1","type":"hint","dependencies":[],"title":"Put The Equation In The Correct Form with Zero By Itself On One Side","text":"Write the inequality as one quotient on the left and zero on the right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational6a-h2","type":"hint","dependencies":["a276c42SolveRational6a-h1"],"title":"Find The Critical Points of Equation","text":"Determine the critical points\u2014the points where the rational expression will be zero or undefined.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational6a-h3","type":"hint","dependencies":["a276c42SolveRational6a-h2"],"title":"Find The Critical Points of Equation","text":"When is the equation undefine? Note that the equation is undefined when the denominator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["a276c42SolveRational6a-h3"],"title":"Find The Critical Points of Equation","text":"Solve the equation $$x+5=0$$. What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational6a-h5","type":"hint","dependencies":["a276c42SolveRational6a-h3"],"title":"Find The Critical Points of Equation","text":"When does the equation equal zero? Note that the equation is undefined when the numerator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational6a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a276c42SolveRational6a-h3"],"title":"Find The Critical Points of Equation","text":"Solve the equation $$x-6=0$$. What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational6a-h7","type":"hint","dependencies":["a276c42SolveRational6a-h4","a276c42SolveRational6a-h5","a276c42SolveRational6a-h6"],"title":"Use the Critical Points to Divide the Number Line into Intervals.","text":"Given that the $$x=-5$$ and $$x=6$$ are the two critical points, we can divide the number line into three intervals, namely, $$(-\\\\infty,-5)$$, $$(-5,6)$$ and $$(6,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational6a-h8","type":"hint","dependencies":["a276c42SolveRational6a-h7"],"title":"Test a Value in Each Interval.","text":"To find the sign of each factor in an interval, we choose any point in that interval and use it as a test point. Any point in the interval will give the expression the same sign, so we can choose any point in the interval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational6a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a276c42SolveRational6a-h8"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-\\\\infty,-5)$$, take the point $$x=-6$$ within the interval and plug it into the original expression. What is the value of $$\\\\frac{x-6}{x+5}$$ after plugging in the value $$x=-6$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational6a-h10","type":"hint","dependencies":["a276c42SolveRational6a-h9"],"title":"Test a Value in Each Interval.","text":"Since $$12$$ is a positive numer greater than $$0$$, we get the sign of quotient is positive in the interval $$(-\\\\infty,-5)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational6a-h11","type":"hint","dependencies":["a276c42SolveRational6a-h10"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-5,6)$$, take the point $$x=0$$ within the interval and plug it into the original expression $$\\\\frac{x-6}{x+5}$$. We get $$\\\\frac{-6}{5}$$ after plugging in $$x=0$$ into the quotient $$\\\\frac{x-6}{x+5}$$. $$\\\\frac{-6}{5}$$ is a negative number, so we can mark the quotient negative in the interval $$(-5,6)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational6a-h12","type":"hint","dependencies":["a276c42SolveRational6a-h11"],"title":"Test a Value in Each Interval.","text":"For the interval $$(6,\\\\infty)$$, take the point $$x=7$$ within the interval and plug it into the original expression $$\\\\frac{x-6}{x+5}$$. We get $$\\\\frac{1}{12}$$ after plugging in $$x=7$$ into the quotient $$\\\\frac{x-6}{x+5}$$. $$\\\\frac{1}{12}$$ is a positive number, so we can mark the quotient positive in the interval $$(6,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational6a-h13","type":"hint","dependencies":["a276c42SolveRational6a-h12"],"title":"Determine the Intervals Where the Inequality is Correct.","text":"We want the quotient to be less than zero, so the numbers in the interval $$(-5,6)$$ are the solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a276c42SolveRational7","title":"Solve Rational Inequality","body":"In the following exercises, solve each rational inequality and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6  Solve Rational Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a276c42SolveRational7a","stepAnswer":["(-5,2)"],"problemType":"TextBox","stepTitle":"$$\\\\frac{x+5}{x-2}<0$$","stepBody":"Please enter your answer as (x,y). Use U to indicate the union of two intervals, if applicable. If your answer is the interval from $$x$$ to $$\\\\infty$$, please enter it as $$(x,\\\\infty)$$. If your answer includes the endpoint of the interval, please use [ ] to indicate the inclusion of endpoints. For instance, [x,y) refers to the interval including endpoint $$x$$ and excluding endpoint $$y$$.","answerType":"string","variabilization":{},"answerLatex":"$$(-5,2)$$","hints":{"DefaultPathway":[{"id":"a276c42SolveRational7a-h1","type":"hint","dependencies":[],"title":"Put The Equation In The Correct Form with Zero By Itself On One Side","text":"Write the inequality as one quotient on the left and zero on the right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational7a-h2","type":"hint","dependencies":["a276c42SolveRational7a-h1"],"title":"Find The Critical Points of Equation","text":"Determine the critical points\u2014the points where the rational expression will be zero or undefined.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational7a-h3","type":"hint","dependencies":["a276c42SolveRational7a-h2"],"title":"Find The Critical Points of Equation","text":"When is the equation undefine? Note that the equation is undefined when the denominator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a276c42SolveRational7a-h3"],"title":"Find The Critical Points of Equation","text":"Solve the equation $$x-2=0$$. What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational7a-h5","type":"hint","dependencies":["a276c42SolveRational7a-h3"],"title":"Find The Critical Points of Equation","text":"When does the equation equal zero? Note that the equation is undefined when the numerator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational7a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["a276c42SolveRational7a-h3"],"title":"Find The Critical Points of Equation","text":"Solve the equation $$x+5=0$$. What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational7a-h7","type":"hint","dependencies":["a276c42SolveRational7a-h4","a276c42SolveRational7a-h5","a276c42SolveRational7a-h6"],"title":"Use the Critical Points to Divide the Number Line into Intervals.","text":"Given that the $$x=-5$$ and $$x=2$$ are the two critical points, we can divide the number line into three intervals, namely, $$(-\\\\infty,-5)$$, $$(-5,2)$$ and $$(2,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational7a-h8","type":"hint","dependencies":["a276c42SolveRational7a-h7"],"title":"Test a Value in Each Interval.","text":"To find the sign of each factor in an interval, we choose any point in that interval and use it as a test point. Any point in the interval will give the expression the same sign, so we can choose any point in the interval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational7a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{8}$$"],"dependencies":["a276c42SolveRational7a-h8"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-\\\\infty,-5)$$, take the point $$x=-6$$ within the interval and plug it into the original expression. What is the value of $$\\\\frac{x-6}{x+5}$$ after plugging in the value $$x=-6$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational7a-h10","type":"hint","dependencies":["a276c42SolveRational7a-h9"],"title":"Test a Value in Each Interval.","text":"Since $$\\\\frac{1}{8}$$ is a positive numer greater than $$0$$, we get the sign of quotient is positive in the interval $$(-\\\\infty,-5)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational7a-h11","type":"hint","dependencies":["a276c42SolveRational7a-h10"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-5,2)$$, take the point $$x=0$$ within the interval and plug it into the original expression $$\\\\frac{x+5}{x-2}$$. We get $$\\\\frac{-5}{2}$$ after plugging in $$x=0$$ into the quotient $$\\\\frac{x+5}{x-2}$$. $$\\\\frac{-5}{2}$$ is a negative number, so we can mark the quotient negative in the interval $$(-5,2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational7a-h12","type":"hint","dependencies":["a276c42SolveRational7a-h11"],"title":"Test a Value in Each Interval.","text":"For the interval $$(2,\\\\infty)$$, take the point $$x=3$$ within the interval and plug it into the original expression $$\\\\frac{x+5}{x-2}$$. We get $$8$$ after plugging in $$x=3$$ into the quotient $$\\\\frac{x+5}{x-2}$$. $$8$$ is a positive number, so we can mark the quotient positive in the interval $$(2,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational7a-h13","type":"hint","dependencies":["a276c42SolveRational7a-h12"],"title":"Determine the Intervals Where the Inequality is Correct.","text":"We want the quotient to be less than zero, so the numbers in the interval $$(-5,2)$$ are the solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a276c42SolveRational8","title":"Solve Rational Inequality","body":"In the following exercises, solve each rational inequality and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6  Solve Rational Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a276c42SolveRational8a","stepAnswer":["(-5/2,5)"],"problemType":"TextBox","stepTitle":"$$\\\\frac{3x}{x-5}<1$$","stepBody":"Please enter your answer as (x,y). Use U to indicate the union of two intervals, if applicable. If your answer is the interval from $$x$$ to $$\\\\infty$$, please enter it as $$(x,\\\\infty)$$. If your answer includes the endpoint of the interval, please use [ ] to indicate the inclusion of endpoints. For instance, [x,y) refers to the interval including endpoint $$x$$ and excluding endpoint $$y$$.","answerType":"string","variabilization":{},"answerLatex":"$$(\\\\frac{-5}{2},5)$$","hints":{"DefaultPathway":[{"id":"a276c42SolveRational8a-h1","type":"hint","dependencies":[],"title":"Put The Equation In The Correct Form with Zero By Itself On One Side","text":"Write the inequality as one quotient on the left and zero on the right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational8a-h2","type":"hint","dependencies":["a276c42SolveRational8a-h1"],"title":"Put The Equation In The Correct Form with Zero By Itself On One Side","text":"By subtracting $$1$$ to get zero on the right, we get $$\\\\frac{3x}{x-5}-1<0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational8a-h3","type":"hint","dependencies":["a276c42SolveRational8a-h2"],"title":"Rewrite $$1$$ As a Fraction Using the LCD.","text":"LCD for $$\\\\frac{3x}{x-5}$$ and $$1$$ is $$(x-5)$$, we can rewrite $$1$$ as $$\\\\frac{x-5}{x-5}$$. We can rewrite (3*x)/(x-5)-1as $$\\\\frac{3x}{x-5}-\\\\frac{x-5}{x-5}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational8a-h4","type":"hint","dependencies":["a276c42SolveRational8a-h3"],"title":"Rewrite $$1$$ As a Fraction Using the LCD.","text":"By subtracting the numerators and placing the difference over the common denominator, we can rewrite ((3*x)/(x-5))-((x-5)/(x-5) as $$\\\\frac{3x-x-5}{x-5}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational8a-h5","type":"hint","dependencies":["a276c42SolveRational8a-h4"],"title":"Simplify $$\\\\frac{3x-x-5}{x-5}$$","text":"$$\\\\frac{3x-x-5}{x-5}=\\\\frac{2x+5}{x-5}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational8a-h6","type":"hint","dependencies":["a276c42SolveRational8a-h5"],"title":"Find The Critical Points of Equation","text":"Determine the critical points\u2014the points where the rational expression will be zero or undefined.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational8a-h7","type":"hint","dependencies":["a276c42SolveRational8a-h6"],"title":"Find The Critical Points of Equation","text":"When is the equation undefine? Note that the equation is undefined when the denominator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational8a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a276c42SolveRational8a-h7"],"title":"Find The Critical Points of Equation","text":"Solve the equation $$x-5=0$$. What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational8a-h9","type":"hint","dependencies":["a276c42SolveRational8a-h8"],"title":"Find The Critical Points of Equation","text":"When does the equation equal zero? Note that the equation is undefined when the numerator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational8a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-5}{2}$$"],"dependencies":["a276c42SolveRational8a-h9"],"title":"Find The Critical Points of Equation","text":"Solve the equation $$2x+5=0$$. What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational8a-h11","type":"hint","dependencies":["a276c42SolveRational8a-h10"],"title":"Use the Critical Points to Divide the Number Line into Intervals.","text":"Given that the $$x=\\\\frac{-5}{2}$$ and $$x=5$$ are the two critical points, we can divide the number line into three intervals, namely, $$(-\\\\infty,\\\\frac{-5}{2})$$, $$(\\\\frac{-5}{2},5)$$ and $$(5,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational8a-h12","type":"hint","dependencies":["a276c42SolveRational8a-h11"],"title":"Test a Value in Each Interval.","text":"To find the sign of each factor in an interval, we choose any point in that interval and use it as a test point. Any point in the interval will give the expression the same sign, so we can choose any point in the interval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational8a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{8}$$"],"dependencies":["a276c42SolveRational8a-h12"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-\\\\infty,\\\\frac{-5}{2})$$, take the point $$x=-3$$ within the interval and plug it into the quotient $$\\\\frac{2x+5}{x-5}$$. What is the value $$\\\\frac{\\\\operatorname{of}\\\\left(2x+5\\\\right)}{x-5}$$ after plugging in the value $$x=-3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational8a-h14","type":"hint","dependencies":["a276c42SolveRational8a-h13"],"title":"Test a Value in Each Interval.","text":"Since $$\\\\frac{1}{8}$$ is a positive numer greater than $$0$$, we get the sign of quotient is positive in the interval $$(-\\\\infty,\\\\frac{-5}{2})$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational8a-h15","type":"hint","dependencies":["a276c42SolveRational8a-h14"],"title":"Test a Value in Each Interval.","text":"For the interval $$(\\\\frac{-5}{2},5)$$, take the point $$x=0$$ within the interval and plug it into the original expression $$\\\\frac{2x+5}{x-5}$$. We get $$--1$$ after plugging in $$x=0$$ into the quotient $$\\\\frac{2x+5}{x-5}$$. $$-1$$ is a negative number, so we can mark the quotient negative in the interval $$(\\\\frac{-5}{2},5)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational8a-h16","type":"hint","dependencies":["a276c42SolveRational8a-h15"],"title":"Test a Value in Each Interval.","text":"For the interval $$(5,\\\\infty)$$, take the point $$x=6$$ within the interval and plug it into the original expression $$\\\\frac{2x+5}{x-5}$$. We get $$17$$ after plugging in $$x=6$$ into the quotient $$\\\\frac{2x+5}{x-5}$$. $$17$$ is a positive number, so we can mark the quotient positive in the interval $$(5,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational8a-h17","type":"hint","dependencies":["a276c42SolveRational8a-h16"],"title":"Determine the Intervals Where the Inequality is Correct.","text":"We want the quotient to be less than zero, so the numbers in the interval $$(\\\\frac{-5}{2},5)$$ are the solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a276c42SolveRational9","title":"Solve Rational Inequality","body":"In the following exercises, solve each rational inequality and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6  Solve Rational Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a276c42SolveRational9a","stepAnswer":["(-1/2,2)"],"problemType":"TextBox","stepTitle":"$$\\\\frac{5x}{x-2}<1$$","stepBody":"Please enter your answer as (x,y). Use U to indicate the union of two intervals, if applicable. If your answer is the interval from $$x$$ to $$\\\\infty$$, please enter it as $$(x,\\\\infty)$$. If your answer includes the endpoint of the interval, please use [ ] to indicate the inclusion of endpoints. For instance, [x,y) refers to the interval including endpoint $$x$$ and excluding endpoint $$y$$.","answerType":"string","variabilization":{},"answerLatex":"$$(\\\\frac{-1}{2},2)$$","hints":{"DefaultPathway":[{"id":"a276c42SolveRational9a-h1","type":"hint","dependencies":[],"title":"Put The Equation In The Correct Form with Zero By Itself On One Side","text":"Write the inequality as one quotient on the left and zero on the right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational9a-h2","type":"hint","dependencies":["a276c42SolveRational9a-h1"],"title":"Put The Equation In The Correct Form with Zero By Itself On One Side","text":"By subtracting $$1$$ to get zero on the right, we get $$\\\\frac{5x}{x-2}-1<0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational9a-h3","type":"hint","dependencies":["a276c42SolveRational9a-h2"],"title":"Rewrite $$1$$ As a Fraction Using the LCD.","text":"LCD for $$\\\\frac{5x}{x-2}$$ and $$1$$ is $$(x-2)$$, we can rewrite $$1$$ as $$\\\\frac{x-2}{x-2}$$. We can rewrite (5*x)/(x-2)-1as $$\\\\frac{5x}{x-2}-\\\\frac{x-2}{x-2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational9a-h4","type":"hint","dependencies":["a276c42SolveRational9a-h3"],"title":"Rewrite $$1$$ As a Fraction Using the LCD.","text":"By subtracting the numerators and placing the difference over the common denominator, we can rewrite $$\\\\frac{5x}{x-2}-\\\\frac{x-2}{x-2}$$ as $$\\\\frac{5x-x-2}{x-2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational9a-h5","type":"hint","dependencies":["a276c42SolveRational9a-h4"],"title":"Simplify $$\\\\frac{5x-x-2}{x-2}$$","text":"$$\\\\frac{5x-x-2}{x-2}=\\\\frac{4x+2}{x-2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational9a-h6","type":"hint","dependencies":["a276c42SolveRational9a-h5"],"title":"Find The Critical Points of Equation","text":"Determine the critical points\u2014the points where the rational expression will be zero or undefined.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational9a-h7","type":"hint","dependencies":["a276c42SolveRational9a-h6"],"title":"Find The Critical Points of Equation","text":"When is the equation undefine? Note that the equation is undefined when the denominator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational9a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a276c42SolveRational9a-h7"],"title":"Find The Critical Points of Equation","text":"Solve the equation $$x-2=0$$. What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational9a-h9","type":"hint","dependencies":["a276c42SolveRational9a-h8"],"title":"Find The Critical Points of Equation","text":"When does the equation equal zero? Note that the equation is undefined when the numerator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational9a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{2}$$"],"dependencies":["a276c42SolveRational9a-h9"],"title":"Find The Critical Points of Equation","text":"Solve the equation $$4x+2=0$$. What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational9a-h11","type":"hint","dependencies":["a276c42SolveRational9a-h10"],"title":"Use the Critical Points to Divide the Number Line into Intervals.","text":"Given that the $$x=\\\\frac{-1}{2}$$ and $$x=2$$ are the two critical points, we can divide the number line into three intervals, namely, $$(-\\\\infty,\\\\frac{-1}{2})$$, $$(\\\\frac{-1}{2},2)$$ and $$(2,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational9a-h12","type":"hint","dependencies":["a276c42SolveRational9a-h11"],"title":"Test a Value in Each Interval.","text":"To find the sign of each factor in an interval, we choose any point in that interval and use it as a test point. Any point in the interval will give the expression the same sign, so we can choose any point in the interval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational9a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{2}$$"],"dependencies":["a276c42SolveRational9a-h12"],"title":"Test a Value in Each Interval.","text":"For the interval $$(-\\\\infty,\\\\frac{-1}{2})$$, take the point $$x=-2$$ within the interval and plug it into the quotient $$\\\\frac{4x+2}{x-2}$$. What is the value of $$\\\\frac{4x+2}{x-2}$$ after plugging in the value $$x=-2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational9a-h14","type":"hint","dependencies":["a276c42SolveRational9a-h13"],"title":"Test a Value in Each Interval.","text":"Since $$\\\\frac{3}{2}$$ is a positive numer greater than $$0$$, we get the sign of quotient is positive in the interval $$(-\\\\infty,\\\\frac{-1}{2})$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational9a-h15","type":"hint","dependencies":["a276c42SolveRational9a-h14"],"title":"Test a Value in Each Interval.","text":"For the interval $$(\\\\frac{-1}{2},2)$$, take the point $$x=0$$ within the interval and plug it into the original expression $$\\\\frac{4x+2}{x-2}$$. We get $$--1$$ after plugging in $$x=0$$ into the quotient $$\\\\frac{4x+2}{x-2}$$. $$-1$$ is a negative number, so we can mark the quotient negative in the interval $$(\\\\frac{-1}{2},2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational9a-h16","type":"hint","dependencies":["a276c42SolveRational9a-h15"],"title":"Test a Value in Each Interval.","text":"For the interval $$(2,\\\\infty)$$, take the point $$x=3$$ within the interval and plug it into the original expression $$\\\\frac{4x+2}{x-2}$$. We get $$14$$ after plugging in $$x=3$$ into the quotient $$\\\\frac{4x+2}{x-2}$$. $$14$$ is a positive number, so we can mark the quotient positive in the interval $$(2,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a276c42SolveRational9a-h17","type":"hint","dependencies":["a276c42SolveRational9a-h16"],"title":"Determine the Intervals Where the Inequality is Correct.","text":"We want the quotient to be less than zero, so the numbers in the interval $$(\\\\frac{-1}{2},2)$$ are the solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28448cslope1","title":"Slope and y-intercept","body":"Use the graph to find the slope and y-intercept of the line.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Use the Slope-Intercept Form of an Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28448cslope1a","stepAnswer":["slope $$m=2$$ and $$y-intercept$$ $$(0,1)$$"],"problemType":"MultipleChoice","stepTitle":"$$y=2x+1$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"slope $$m=2$$ and y-intercept $$(0,1)$$","choices":["slope $$m=2$$ and $$y-intercept$$ $$(0,1)$$","slope $$m=\\\\frac{1}{2}$$ and $$y-intercept$$ $$(0,1)$$","slope $$m=2$$ and $$y-intercept$$ $$(1,0)$$"],"hints":{"DefaultPathway":[{"id":"a28448cslope1a-h1","type":"hint","dependencies":[],"title":"Slope","text":"To find the slope of the line, we need to choose two points on the line. We\u2019ll use the points $$(0,1)$$ and $$(1,3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope1a-h2","type":"hint","dependencies":["a28448cslope1a-h1"],"title":"Slope","text":"Find the rise and run using the formula $$m=\\\\frac{rise}{run}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope1a-h3","type":"hint","dependencies":["a28448cslope1a-h2"],"title":"Slope","text":"From those two points, there is a rise of $$2$$ units and a run of $$1$$ unit; therefore $$m=\\\\frac{2}{1}=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope1a-h4","type":"hint","dependencies":["a28448cslope1a-h3"],"title":"y-intercept","text":"Find the y-intercept of the line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope1a-h5","type":"hint","dependencies":["a28448cslope1a-h4"],"title":"y-intercept","text":"When $$x=0$$, $$y=1$$. Therefore, the y-intercept is the point $$(0,1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope1a-h6","type":"hint","dependencies":["a28448cslope1a-h5"],"title":"Slope and y-intercept","text":"The slope is $$m=2$$ and the y-intercept is $$(0,1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28448cslope10","title":"Slope and y-intercept","body":"Identify the slope and y-intercept of the line.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Use the Slope-Intercept Form of an Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28448cslope10a","stepAnswer":["$$-4;(0,8)$$"],"problemType":"MultipleChoice","stepTitle":"$$4x+y=8$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-4;(0,8)$$","choices":["$$4;(0,8)$$","$$-4;(0,8)$$","$$-4;(8,0)$$"],"hints":{"DefaultPathway":[{"id":"a28448cslope10a-h1","type":"hint","dependencies":[],"title":"Solve for $$y$$","text":"Isolate $$y$$ to one side: $$4x+y=8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope10a-h2","type":"hint","dependencies":["a28448cslope10a-h1"],"title":"Solve for $$y$$","text":"Substract $$4x$$ from each side: $$y=-4x+8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope10a-h3","type":"hint","dependencies":["a28448cslope10a-h2"],"title":"Slope-Intercept Form of an Equation of a Line","text":"We compare our equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=-4x+8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["a28448cslope10a-h3"],"title":"Identify the slope.","text":"What is $$m$$ in our given equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a28448cslope10a-h4"],"title":"Identify $$b$$.","text":"What is $$b$$ in our given equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope10a-h6","type":"hint","dependencies":["a28448cslope10a-h5"],"title":"Identify the y-intercept.","text":"The y-intercept is (0,b). Since $$b$$ is $$8$$, the y-intercept would be $$(0,8)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope10a-h7","type":"hint","dependencies":["a28448cslope10a-h4","a28448cslope10a-h6"],"title":"Slope and y-intercept","text":"The slope is $$-4$$ and the y-intercept is $$(0,8)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28448cslope11","title":"Slope and y-intercept","body":"Identify the slope and y-intercept of the line.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Use the Slope-Intercept Form of an Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28448cslope11a","stepAnswer":["$$-1/2;(0,3)$$"],"problemType":"MultipleChoice","stepTitle":"$$x+2y=6$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-1/2;(0,3)$$","choices":["$$-1/2;(3,0)$$","$$3;(0,\\\\frac{-1}{2})$$","$$-1/2;(0,3)$$"],"hints":{"DefaultPathway":[{"id":"a28448cslope11a-h1","type":"hint","dependencies":[],"title":"Solve for $$y$$","text":"Isolate $$y$$ to one side: $$x+2y=6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope11a-h2","type":"hint","dependencies":["a28448cslope11a-h1"],"title":"Solve for $$y$$","text":"Substract $$x$$ from each side: $$2y=-x+6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope11a-h3","type":"hint","dependencies":["a28448cslope11a-h2"],"title":"Solve for $$y$$","text":"Divide both sides by 2: $$\\\\frac{2y}{2}=\\\\frac{\\\\left(-x+6\\\\right)}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope11a-h4","type":"hint","dependencies":["a28448cslope11a-h3"],"title":"Solve for $$y$$","text":"Remember $$\\\\frac{a+b}{c}=\\\\frac{a}{c}+\\\\frac{b}{c}$$: $$\\\\frac{2y}{2}=\\\\frac{-x}{2}+\\\\frac{6}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope11a-h5","type":"hint","dependencies":["a28448cslope11a-h4"],"title":"Solve for $$y$$","text":"Simplify: $$y=\\\\left(-\\\\frac{1}{2}\\\\right) x+3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope11a-h6","type":"hint","dependencies":["a28448cslope11a-h5"],"title":"Slope-Intercept Form of an Equation of a Line","text":"We compare our equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=\\\\left(-\\\\frac{1}{2}\\\\right) x+3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope11a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{2}$$"],"dependencies":["a28448cslope11a-h6"],"title":"Identify the slope.","text":"What is $$m$$ in our given equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope11a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a28448cslope11a-h7"],"title":"Identify $$b$$.","text":"What is $$b$$ in our given equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope11a-h9","type":"hint","dependencies":["a28448cslope11a-h8"],"title":"Identify the y-intercept.","text":"The y-intercept is (0,b). Since $$b$$ is $$3$$, the y-intercept would be $$(0,3)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope11a-h10","type":"hint","dependencies":["a28448cslope11a-h7","a28448cslope11a-h9"],"title":"Slope and y-intercept","text":"The slope is $$\\\\frac{-1}{2}$$ and the y-intercept is $$(0,3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28448cslope12","title":"Slope and y-intercept","body":"Identify the slope and y-intercept of the line.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Use the Slope-Intercept Form of an Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28448cslope12a","stepAnswer":["$$-3/2;(0,3)$$"],"problemType":"MultipleChoice","stepTitle":"$$6x+4y=12$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-3/2;(0,3)$$","choices":["$$-3/2;(3,0)$$","$$3;(0,\\\\frac{-3}{2})$$","$$-3/2;(0,3)$$"],"hints":{"DefaultPathway":[{"id":"a28448cslope12a-h1","type":"hint","dependencies":[],"title":"Solve for $$y$$","text":"Isolate $$y$$ to one side: $$6x+4y=12$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope12a-h2","type":"hint","dependencies":["a28448cslope12a-h1"],"title":"Solve for $$y$$","text":"Substract $$6x$$ from each side: $$4y=-6x+12$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope12a-h3","type":"hint","dependencies":["a28448cslope12a-h2"],"title":"Solve for $$y$$","text":"Divide both sides by 4: $$\\\\frac{4y}{4}=\\\\frac{\\\\left(-6x+12\\\\right)}{4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope12a-h4","type":"hint","dependencies":["a28448cslope12a-h3"],"title":"Solve for $$y$$","text":"Remember $$\\\\frac{a+b}{c}=\\\\frac{a}{c}+\\\\frac{b}{c}$$: $$\\\\frac{4y}{4}=\\\\frac{-6x}{4}+\\\\frac{12}{4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope12a-h5","type":"hint","dependencies":["a28448cslope12a-h4"],"title":"Solve for $$y$$","text":"Simplify: $$y=\\\\left(-\\\\frac{3}{2}\\\\right) x+3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope12a-h6","type":"hint","dependencies":["a28448cslope12a-h5"],"title":"Slope-Intercept Form of an Equation of a Line","text":"We compare our equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=\\\\left(-\\\\frac{3}{2}\\\\right) x+3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope12a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-3}{2}$$"],"dependencies":["a28448cslope12a-h6"],"title":"Identify the slope.","text":"What is $$m$$ in our given equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope12a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a28448cslope12a-h7"],"title":"Identify $$b$$.","text":"What is $$b$$ in our given equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope12a-h9","type":"hint","dependencies":["a28448cslope12a-h8"],"title":"Identify the y-intercept.","text":"The y-intercept is (0,b). Since $$b$$ is $$3$$, the y-intercept would be $$(0,3)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope12a-h10","type":"hint","dependencies":["a28448cslope12a-h7","a28448cslope12a-h9"],"title":"Slope and y-intercept","text":"The slope is $$\\\\frac{-3}{2}$$ and the y-intercept is $$(0,3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28448cslope13","title":"Slope and y-intercept","body":"Identify the slope and y-intercept of the line.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Use the Slope-Intercept Form of an Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28448cslope13a","stepAnswer":["$$7/3;(0,-3)$$"],"problemType":"MultipleChoice","stepTitle":"$$7x-3y=9$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$7/3;(0,-3)$$","choices":["$$7/3;(3,0)$$","$$3;(0,\\\\frac{7}{3})$$","$$7/3;(0,-3)$$"],"hints":{"DefaultPathway":[{"id":"a28448cslope13a-h1","type":"hint","dependencies":[],"title":"Solve for $$y$$","text":"Isolate $$y$$ to one side: $$7x-3y=9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope13a-h2","type":"hint","dependencies":["a28448cslope13a-h1"],"title":"Solve for $$y$$","text":"Substract $$7x$$ from each side: $$-3y=-7x+9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope13a-h3","type":"hint","dependencies":["a28448cslope13a-h2"],"title":"Solve for $$y$$","text":"Divide both sides by -3: $$\\\\frac{\\\\left(-3y\\\\right)}{-3}=\\\\frac{\\\\left(-7x+9\\\\right)}{-3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope13a-h4","type":"hint","dependencies":["a28448cslope13a-h3"],"title":"Solve for $$y$$","text":"Remember $$\\\\frac{a+b}{c}=\\\\frac{a}{c}+\\\\frac{b}{c}$$: $$\\\\frac{\\\\left(-3y\\\\right)}{-3}=\\\\frac{-7x}{-3}+\\\\frac{9}{-3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope13a-h5","type":"hint","dependencies":["a28448cslope13a-h4"],"title":"Solve for $$y$$","text":"Simplify: $$y=\\\\frac{7}{3} x-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope13a-h6","type":"hint","dependencies":["a28448cslope13a-h5"],"title":"Slope-Intercept Form of an Equation of a Line","text":"We compare our equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=\\\\frac{7}{3} x-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope13a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{7}{3}$$"],"dependencies":["a28448cslope13a-h6"],"title":"Identify the slope.","text":"What is $$m$$ in our given equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope13a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a28448cslope13a-h7"],"title":"Identify $$b$$.","text":"What is $$b$$ in our given equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope13a-h9","type":"hint","dependencies":["a28448cslope13a-h8"],"title":"Identify the y-intercept.","text":"The y-intercept is (0,b). Since $$b$$ is $$-3$$, the y-intercept would be $$(0,-3)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope13a-h10","type":"hint","dependencies":["a28448cslope13a-h7","a28448cslope13a-h9"],"title":"Slope and y-intercept","text":"The slope is $$\\\\frac{7}{3}$$ and the y-intercept is $$(0,-3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28448cslope14","title":"Parallel Lines","body":"Use slopes and y-intercepts to determine if the lines are parallel.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Use the Slope-Intercept Form of an Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28448cslope14a","stepAnswer":["Parallel"],"problemType":"MultipleChoice","stepTitle":"$$3x-2y=6$$, $$y=\\\\frac{3}{2} x+1$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Parallel","Not Parallel"],"hints":{"DefaultPathway":[{"id":"a28448cslope14a-h1","type":"hint","dependencies":[],"title":"Solve the first equation for $$y$$","text":"$$3x-2y=6$$\\\\n$$-2y=-3x+6$$\\\\n$$\\\\frac{\\\\left(-2y\\\\right)}{-2}=\\\\frac{\\\\left(-3x+6\\\\right)}{-2}$$\\\\n$$y=\\\\frac{3}{2} x-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope14a-h2","type":"hint","dependencies":["a28448cslope14a-h1"],"title":"Slope-Intercept Form","text":"Both equations are now in slope-intercept form: $$y=\\\\frac{3}{2} x-3$$, $$y=\\\\frac{3}{2} x+1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope14a-h3","type":"hint","dependencies":["a28448cslope14a-h2"],"title":"Slope and y-intercept","text":"Identify the slope and y-intercept of both lines.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope14a-h4","type":"hint","dependencies":["a28448cslope14a-h3"],"title":"Slope-Intercept Form of an Equation of the First Line","text":"Compare the first equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=\\\\frac{3}{2} x-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope14a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{2}$$"],"dependencies":["a28448cslope14a-h4"],"title":"Identify the Slope of First Line","text":"What is $$m$$ in the first equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope14a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(0,-3)$$"],"dependencies":["a28448cslope14a-h5"],"title":"Identify the y-intercept of First Line","text":"What is the y-intercept in the first equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(0,-3)$$","$$(0,3)$$","$$(-3,0)$$"]},{"id":"a28448cslope14a-h7","type":"hint","dependencies":["a28448cslope14a-h5","a28448cslope14a-h6"],"title":"Slope and y-intercept of First Line","text":"The slope of the first equation is $$\\\\frac{3}{2}$$ and the y-intercept is $$(0,-3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope14a-h8","type":"hint","dependencies":["a28448cslope14a-h7"],"title":"Slope-Intercept Form of an Equation of Second Line","text":"Compare the second equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=\\\\frac{3}{2} x+1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope14a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{2}$$"],"dependencies":["a28448cslope14a-h8"],"title":"Identify the Slope of Second Line","text":"What is $$m$$ in the second equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope14a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(0,1)$$"],"dependencies":["a28448cslope14a-h9"],"title":"Identify the y-intercept of Second Line","text":"What is the y-intercept in the second equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(0,-1)$$","$$(0,1)$$","$$(1,0)$$"]},{"id":"a28448cslope14a-h11","type":"hint","dependencies":["a28448cslope14a-h9","a28448cslope14a-h10"],"title":"Slope and y-intercept of Second Line","text":"The slope of the second equation is $$\\\\frac{3}{2}$$ and the y-intercept is $$(0,1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope14a-h12","type":"hint","dependencies":["a28448cslope14a-h7","a28448cslope14a-h11"],"title":"Parallel Lines","text":"The lines have the same slope and different y-intercepts and so they are parallel.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28448cslope15","title":"Parallel Lines","body":"Use slopes and y-intercepts to determine if the lines are parallel.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Use the Slope-Intercept Form of an Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28448cslope15a","stepAnswer":["Parallel"],"problemType":"MultipleChoice","stepTitle":"$$y=\\\\frac{3}{4} x-3$$, $$3x-4y=-2$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Parallel","Not Parallel"],"hints":{"DefaultPathway":[{"id":"a28448cslope15a-h1","type":"hint","dependencies":[],"title":"Solve the second equation for $$y$$","text":"$$3x-4y=-2$$\\\\n$$-4y=-3x-2$$\\\\n$$\\\\frac{\\\\left(-4y\\\\right)}{-4}=\\\\frac{\\\\left(-3x-2\\\\right)}{-4}$$\\\\n$$y=\\\\frac{3}{4} x+\\\\frac{1}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope15a-h2","type":"hint","dependencies":["a28448cslope15a-h1"],"title":"Slope-Intercept Form","text":"Both equations are now in slope-intercept form: $$y=\\\\frac{3}{4} x-3$$, $$y=\\\\frac{3}{4} x+\\\\frac{1}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope15a-h3","type":"hint","dependencies":["a28448cslope15a-h2"],"title":"Slope and y-intercept","text":"Identify the slope and y-intercept of both lines.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope15a-h4","type":"hint","dependencies":["a28448cslope15a-h3"],"title":"Slope-Intercept Form of an Equation of the First Line","text":"Compare the first equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=\\\\frac{3}{4} x-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope15a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{4}$$"],"dependencies":["a28448cslope15a-h4"],"title":"Identify the Slope of First Line","text":"What is $$m$$ in the first equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope15a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(0,-3)$$"],"dependencies":["a28448cslope15a-h5"],"title":"Identify the y-intercept of First Line","text":"What is the y-intercept in the first equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(0,-3)$$","$$(0,3)$$","$$(3,0)$$"]},{"id":"a28448cslope15a-h7","type":"hint","dependencies":["a28448cslope15a-h5","a28448cslope15a-h6"],"title":"Slope and y-intercept of First Line","text":"The slope of the first equation is $$\\\\frac{3}{4}$$ and the y-intercept is $$(0,-3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope15a-h8","type":"hint","dependencies":["a28448cslope15a-h7"],"title":"Slope-Intercept Form of an Equation of Second Line","text":"Compare the second equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=\\\\frac{3}{4} x+\\\\frac{1}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope15a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{4}$$"],"dependencies":["a28448cslope15a-h8"],"title":"Identify the Slope of Second Line","text":"What is $$m$$ in the second equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope15a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(0,\\\\frac{1}{2})$$"],"dependencies":["a28448cslope15a-h9"],"title":"Identify the y-intercept of Second Line","text":"What is the y-intercept in the second equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(0,\\\\frac{1}{2})$$","$$(0,\\\\frac{-1}{2})$$","$$(\\\\frac{1}{2},0)$$"]},{"id":"a28448cslope15a-h11","type":"hint","dependencies":["a28448cslope15a-h9","a28448cslope15a-h10"],"title":"Slope and y-intercept of Second Line","text":"The slope of the second equation is $$\\\\frac{3}{4}$$ and the y-intercept is $$(0,\\\\frac{1}{2})$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope15a-h12","type":"hint","dependencies":["a28448cslope15a-h7","a28448cslope15a-h11"],"title":"Parallel Lines","text":"The lines have the same slope and different y-intercepts and so they are parallel.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28448cslope16","title":"Parallel Lines","body":"Use slopes and y-intercepts to determine if the lines are parallel.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Use the Slope-Intercept Form of an Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28448cslope16a","stepAnswer":["Parallel"],"problemType":"MultipleChoice","stepTitle":"$$y=\\\\frac{2}{3} x-1$$, $$2x-3y=-2$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Parallel","Not Parallel"],"hints":{"DefaultPathway":[{"id":"a28448cslope16a-h1","type":"hint","dependencies":[],"title":"Solve the second equation for $$y$$","text":"$$2x-3y=-2$$\\\\n$$-3y=-2x-2$$\\\\n$$\\\\frac{\\\\left(-3y\\\\right)}{-3}=\\\\frac{\\\\left(-2x-2\\\\right)}{-3}$$\\\\n$$y=\\\\frac{2}{3} x+\\\\frac{2}{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope16a-h2","type":"hint","dependencies":["a28448cslope16a-h1"],"title":"Slope-Intercept Form","text":"Both equations are now in slope-intercept form: $$y=\\\\frac{2}{3} x-1$$, $$y=\\\\frac{2}{3} x+\\\\frac{2}{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope16a-h3","type":"hint","dependencies":["a28448cslope16a-h2"],"title":"Slope and y-intercept","text":"Identify the slope and y-intercept of both lines.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope16a-h4","type":"hint","dependencies":["a28448cslope16a-h3"],"title":"Slope-Intercept Form of an Equation of the First Line","text":"Compare the first equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=\\\\frac{2}{3} x-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope16a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{3}$$"],"dependencies":["a28448cslope16a-h4"],"title":"Identify the Slope of First Line","text":"What is $$m$$ in the first equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope16a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(0,-1)$$"],"dependencies":["a28448cslope16a-h5"],"title":"Identify the y-intercept of First Line","text":"What is the y-intercept in the first equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(0,1)$$","$$(0,-1)$$","$$(1,0)$$"]},{"id":"a28448cslope16a-h7","type":"hint","dependencies":["a28448cslope16a-h5","a28448cslope16a-h6"],"title":"Slope and y-intercept of First Line","text":"The slope of the first equation is $$\\\\frac{2}{3}$$ and the y-intercept is $$(0,-1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope16a-h8","type":"hint","dependencies":["a28448cslope16a-h7"],"title":"Slope-Intercept Form of an Equation of Second Line","text":"Compare the second equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=\\\\frac{2}{3} x+\\\\frac{2}{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope16a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{3}$$"],"dependencies":["a28448cslope16a-h8"],"title":"Identify the Slope of Second Line","text":"What is $$m$$ in the second equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope16a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(0,\\\\frac{2}{3})$$"],"dependencies":["a28448cslope16a-h9"],"title":"Identify the y-intercept of Second Line","text":"What is the y-intercept in the second equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(0,\\\\frac{2}{3})$$","$$(0,\\\\frac{-2}{3})$$","$$(\\\\frac{2}{3},0)$$"]},{"id":"a28448cslope16a-h11","type":"hint","dependencies":["a28448cslope16a-h9","a28448cslope16a-h10"],"title":"Slope and y-intercept of Second Line","text":"The slope of the second equation is $$\\\\frac{2}{3}$$ and the y-intercept is $$(0,\\\\frac{2}{3})$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope16a-h12","type":"hint","dependencies":["a28448cslope16a-h7","a28448cslope16a-h11"],"title":"Parallel Lines","text":"The lines have the same slope and different y-intercepts and so they are parallel.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28448cslope17","title":"Parallel Lines","body":"Use slopes and y-intercepts to determine if the lines are parallel.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Use the Slope-Intercept Form of an Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28448cslope17a","stepAnswer":["Parallel"],"problemType":"MultipleChoice","stepTitle":"$$2x-5y=-3$$, $$y=\\\\frac{2}{5} x+1$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Parallel","Not Parallel"],"hints":{"DefaultPathway":[{"id":"a28448cslope17a-h1","type":"hint","dependencies":[],"title":"Solve the first equation for $$y$$","text":"$$2x-5y=-3$$\\\\n$$-5y=-2x-3$$\\\\n$$\\\\frac{\\\\left(-5y\\\\right)}{-5}=\\\\frac{\\\\left(-2x-3\\\\right)}{-5}$$\\\\n$$y=\\\\frac{2}{5} x+\\\\frac{3}{5}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope17a-h2","type":"hint","dependencies":["a28448cslope17a-h1"],"title":"Slope-Intercept Form","text":"Both equations are now in slope-intercept form: $$y=\\\\frac{2}{5} x+\\\\frac{3}{5}$$, $$y=\\\\frac{2}{5} x+1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope17a-h3","type":"hint","dependencies":["a28448cslope17a-h2"],"title":"Slope and y-intercept","text":"Identify the slope and y-intercept of both lines.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope17a-h4","type":"hint","dependencies":["a28448cslope17a-h3"],"title":"Slope-Intercept Form of an Equation of the First Line","text":"Compare the first equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=\\\\frac{2}{5} x+\\\\frac{3}{5}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope17a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{5}$$"],"dependencies":["a28448cslope17a-h4"],"title":"Identify the Slope of First Line","text":"What is $$m$$ in the first equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope17a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(0,\\\\frac{3}{5})$$"],"dependencies":["a28448cslope17a-h5"],"title":"Identify the y-intercept of First Line","text":"What is the y-intercept in the first equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(0,\\\\frac{-3}{5})$$","$$(0,\\\\frac{3}{5})$$","$$(\\\\frac{-3}{5},0)$$"]},{"id":"a28448cslope17a-h7","type":"hint","dependencies":["a28448cslope17a-h5","a28448cslope17a-h6"],"title":"Slope and y-intercept of First Line","text":"The slope of the first equation is $$\\\\frac{2}{5}$$ and the y-intercept is $$(0,\\\\frac{3}{5})$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope17a-h8","type":"hint","dependencies":["a28448cslope17a-h7"],"title":"Slope-Intercept Form of an Equation of Second Line","text":"Compare the second equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=\\\\frac{2}{5} x+1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope17a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{5}$$"],"dependencies":["a28448cslope17a-h8"],"title":"Identify the Slope of Second Line","text":"What is $$m$$ in the second equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope17a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(0,1)$$"],"dependencies":["a28448cslope17a-h9"],"title":"Identify the y-intercept of Second Line","text":"What is the y-intercept in the second equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(0,-1)$$","$$(0,1)$$","$$(1,0)$$"]},{"id":"a28448cslope17a-h11","type":"hint","dependencies":["a28448cslope17a-h9","a28448cslope17a-h10"],"title":"Slope and y-intercept of Second Line","text":"The slope of the second equation is $$\\\\frac{2}{5}$$ and the y-intercept is $$(0,1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope17a-h12","type":"hint","dependencies":["a28448cslope17a-h7","a28448cslope17a-h11"],"title":"Parallel Lines","text":"The lines have the same slope and different y-intercepts and so they are parallel.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28448cslope18","title":"Parallel Lines","body":"Use slopes and y-intercepts to determine if the lines are parallel.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Use the Slope-Intercept Form of an Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28448cslope18a","stepAnswer":["Not Parallel"],"problemType":"MultipleChoice","stepTitle":"$$6x-3y=9$$, $$2x-y=3$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Parallel","Not Parallel"],"hints":{"DefaultPathway":[{"id":"a28448cslope18a-h1","type":"hint","dependencies":[],"title":"Solve the first equation for $$y$$","text":"$$6x-3y=9$$\\\\n$$-3y=-6x+9$$\\\\n$$\\\\frac{\\\\left(-3y\\\\right)}{-3}=\\\\frac{\\\\left(-6x+9\\\\right)}{-3}$$\\\\n$$y=2x-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope18a-h2","type":"hint","dependencies":["a28448cslope18a-h1"],"title":"Solve the second equation for $$y$$","text":"$$2x-y=3$$\\\\n$$-y=-2x+3$$\\\\n$$y=2x-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope18a-h3","type":"hint","dependencies":["a28448cslope18a-h2"],"title":"Parallel Lines","text":"Since the lines have the same equation, they are the same line. Therefore, they cannot be parallel.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28448cslope19","title":"Parallel Lines","body":"Use slopes and y-intercepts to determine if the lines are parallel.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Use the Slope-Intercept Form of an Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28448cslope19a","stepAnswer":["Not Parallel"],"problemType":"MultipleChoice","stepTitle":"$$5x-2y=11$$, $$5x-y=7$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Parallel","Not Parallel"],"hints":{"DefaultPathway":[{"id":"a28448cslope19a-h1","type":"hint","dependencies":[],"title":"Solve the first equation for $$y$$","text":"$$5x-2y=11$$\\\\n$$-2y=-5x+11$$\\\\n$$\\\\frac{\\\\left(-2y\\\\right)}{-2}=\\\\frac{\\\\left(-5x+11\\\\right)}{-2}$$\\\\n$$y=\\\\frac{5}{2} x-\\\\frac{11}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope19a-h2","type":"hint","dependencies":["a28448cslope19a-h1"],"title":"Slope-Intercept Form of an Equation of Second Line","text":"Compare the second equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=\\\\frac{5}{2} x-\\\\frac{11}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope19a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{2}$$"],"dependencies":["a28448cslope19a-h2"],"title":"Identify the Slope of First Line","text":"What is $$m$$ in the first equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope19a-h4","type":"hint","dependencies":["a28448cslope19a-h3"],"title":"Solve the second equation for $$y$$","text":"$$5x-y=7$$\\\\n$$-y=-5x+7$$\\\\n$$y=5x-7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope19a-h5","type":"hint","dependencies":["a28448cslope19a-h4"],"title":"Slope-Intercept Form of an Equation of Second Line","text":"Compare the second equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=5x-7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope19a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a28448cslope19a-h5"],"title":"Identify the Slope of Second Line","text":"What is $$m$$ in the second equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope19a-h7","type":"hint","dependencies":["a28448cslope19a-h3","a28448cslope19a-h6"],"title":"Parallel Lines","text":"The lines have different slopes, therefore they are not parallel.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28448cslope2","title":"Slope and y-intercept","body":"Use the graph to find the slope and y-intercept of the line.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Use the Slope-Intercept Form of an Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28448cslope2a","stepAnswer":["slope $$m=3$$ and $$y-intercept$$ $$(0,-5)$$"],"problemType":"MultipleChoice","stepTitle":"$$y=3x-5$$\\\\n","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"slope $$m=3$$ and y-intercept $$(0,-5)$$","choices":["slope $$m=\\\\frac{1}{3}$$ and $$y-intercept$$ $$(0,-5)$$","slope $$m=3$$ and $$y-intercept$$ $$(0,-5)$$","slope $$m=3$$ and $$y-intercept$$ $$(-5,0)$$"],"hints":{"DefaultPathway":[{"id":"a28448cslope2a-h1","type":"hint","dependencies":[],"title":"Slope","text":"To find the slope of the line, we need to choose two points on the line. We\u2019ll use the points $$(2,1)$$ and $$(3,4)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope2a-h2","type":"hint","dependencies":["a28448cslope2a-h1"],"title":"Slope","text":"Find the rise and run using the formula $$m=\\\\frac{rise}{run}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope2a-h3","type":"hint","dependencies":["a28448cslope2a-h2"],"title":"Slope","text":"From those two points, there is a rise of $$3$$ units and a run of $$1$$ unit; therefore $$m=\\\\frac{3}{1}=3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope2a-h4","type":"hint","dependencies":["a28448cslope2a-h3"],"title":"y-intercept","text":"Find the y-intercept of the line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope2a-h5","type":"hint","dependencies":["a28448cslope2a-h4"],"title":"y-intercept","text":"When $$x=0$$, $$y=-5$$. Therefore, the y-intercept is the point $$(0,-5)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope2a-h6","type":"hint","dependencies":["a28448cslope2a-h5"],"title":"Slope and y-intercept","text":"The slope is $$m=3$$ and the y-intercept is $$(0,-5)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28448cslope20","title":"Parallel Lines","body":"Use slopes and y-intercepts to determine if the lines are parallel.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Use the Slope-Intercept Form of an Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28448cslope20a","stepAnswer":["Parallel"],"problemType":"MultipleChoice","stepTitle":"$$y=-4$$, $$y=3$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Parallel","Not Parallel"],"hints":{"DefaultPathway":[{"id":"a28448cslope20a-h1","type":"hint","dependencies":[],"title":"Slope-Intercept Form","text":"Write each equation in slope-intercept form. Since there is no $$x$$ term we write $$0x$$:\\\\n$$y=0x-4$$, $$y=0x+3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope20a-h2","type":"hint","dependencies":["a28448cslope20a-h1"],"title":"Slope","text":"Since $$m=0$$ for both equation, the lines have the same slope.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope20a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(0,-4)$$"],"dependencies":["a28448cslope20a-h2"],"title":"Identify the y-intercept of First Line","text":"What is the y-intercept for $$y=0x-4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(0,-4)$$","$$(0,4)$$","$$(4,0)$$"]},{"id":"a28448cslope20a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(0,3)$$"],"dependencies":["a28448cslope20a-h3"],"title":"Identify the y-intercept of Second Line","text":"What is the y-intercept for $$y=0x+3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(0,3)$$","$$(0,-3)$$","$$(3,0)$$"]},{"id":"a28448cslope20a-h5","type":"hint","dependencies":["a28448cslope20a-h4"],"title":"Parallel Lines","text":"The lines have the same slope and different y-intercepts and so they are parallel.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28448cslope21","title":"Parallel Lines","body":"Use slopes and y-intercepts to determine if the lines are parallel.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Use the Slope-Intercept Form of an Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28448cslope21a","stepAnswer":["Parallel"],"problemType":"MultipleChoice","stepTitle":"$$y=2$$, $$y=6$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Parallel","Not Parallel"],"hints":{"DefaultPathway":[{"id":"a28448cslope21a-h1","type":"hint","dependencies":[],"title":"Horizontal Lines","text":"Horizontal lines are equations where $$y$$ equals a single constant.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope21a-h2","type":"hint","dependencies":["a28448cslope21a-h1"],"title":"Slope","text":"Slopes of horizontal lines are always $$0$$. Therefore, both lines have the same slope of $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope21a-h3","type":"hint","dependencies":["a28448cslope21a-h2"],"title":"Identify the y-intercept","text":"Since the horizontal lines cross the y-axis at $$y=2$$ and at $$y=6$$, we know the y-intercepts are $$(0,2)$$ and $$(0,6)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope21a-h4","type":"hint","dependencies":["a28448cslope21a-h3"],"title":"Parallel Lines","text":"The lines have the same slope and different y-intercepts and so they are parallel.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28448cslope22","title":"Parallel Lines","body":"Use slopes and y-intercepts to determine if the lines are parallel.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Use the Slope-Intercept Form of an Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28448cslope22a","stepAnswer":["Parallel"],"problemType":"MultipleChoice","stepTitle":"$$x=-2$$, $$x=-5$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Parallel","Not Parallel"],"hints":{"DefaultPathway":[{"id":"a28448cslope22a-h1","type":"hint","dependencies":[],"title":"Slope-Intercept Form","text":"Since there is no $$y$$, the equations cannot be put in slope-intercept form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope22a-h2","type":"hint","dependencies":["a28448cslope22a-h1"],"title":"Vertical Lines","text":"But we recognize them as equations of vertical lines. Vertical lines are equations where $$x$$ equals a single constant.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope22a-h3","type":"hint","dependencies":["a28448cslope22a-h2"],"title":"Identify the x-intercept","text":"Since the vertical lines cross the x-axis at $$x=-2$$ and at $$x=-5$$, we know the x-intercepts are $$(-2,0)$$ and $$(-5,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope22a-h4","type":"hint","dependencies":["a28448cslope22a-h3"],"title":"Parallel Lines","text":"Since their x-intercepts are different, the vertical lines are parallel.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28448cslope23","title":"Perpendicular Lines","body":"Use slopes to determine if the lines are perpendicular.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Use the Slope-Intercept Form of an Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28448cslope23a","stepAnswer":["Perpendicular"],"problemType":"MultipleChoice","stepTitle":"$$y=-5x-4$$, $$x-5y=5$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Perpendicular","Not Perpendicular"],"hints":{"DefaultPathway":[{"id":"a28448cslope23a-h1","type":"hint","dependencies":[],"title":"Solve the second equation for $$y$$","text":"$$x-5y=5$$\\\\n$$-5y=-x+5$$\\\\n$$\\\\frac{\\\\left(-5y\\\\right)}{-5}=\\\\frac{\\\\left(-x+5\\\\right)}{-5}$$\\\\n$$y=\\\\frac{1}{5} x-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope23a-h2","type":"hint","dependencies":["a28448cslope23a-h1"],"title":"Slope-Intercept Form","text":"Both equations are now in slope-intercept form: $$y=-5x-4$$, $$y=\\\\frac{1}{5} x-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope23a-h3","type":"hint","dependencies":["a28448cslope23a-h2"],"title":"Slope","text":"Identify the slope of both lines.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope23a-h4","type":"hint","dependencies":["a28448cslope23a-h3"],"title":"Slope-Intercept Form of an Equation of the First Line","text":"Compare the first equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=-5x-4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope23a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["a28448cslope23a-h4"],"title":"Identify the Slope of First Line","text":"What is $$m$$ in the first equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope23a-h6","type":"hint","dependencies":["a28448cslope23a-h5"],"title":"Slope-Intercept Form of an Equation of Second Line","text":"Compare the second equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=\\\\frac{1}{5} x-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope23a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{5}$$"],"dependencies":["a28448cslope23a-h6"],"title":"Identify the Slope of Second Line","text":"What is $$m$$ in the second equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope23a-h8","type":"hint","dependencies":["a28448cslope23a-h7"],"title":"Perpendicular Lines","text":"Perpendicular lines have slopes that are negative reciprocals of each other. We check by multiplying the slopes and see if it equals $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope23a-h9","type":"hint","dependencies":["a28448cslope23a-h5","a28448cslope23a-h7","a28448cslope23a-h8"],"title":"Checking if perpendicular","text":"$$m_1 m_2$$\\\\n$$-5\\\\frac{1}{5}=-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope23a-h10","type":"hint","dependencies":["a28448cslope23a-h9"],"title":"Perpendicular Lines","text":"The slopes are negative reciprocals of each other, so the lines are perpendicular.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28448cslope24","title":"Perpendicular Lines","body":"Use slopes to determine if the lines are perpendicular.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Use the Slope-Intercept Form of an Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28448cslope24a","stepAnswer":["Not Perpendicular"],"problemType":"MultipleChoice","stepTitle":"$$7x+2y=3$$, $$2x+7y=5$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Perpendicular","Not Perpendicular"],"hints":{"DefaultPathway":[{"id":"a28448cslope24a-h1","type":"hint","dependencies":[],"title":"Solve the first equation for $$y$$","text":"$$7x+2y=3$$\\\\n$$2y=-7x+3$$\\\\n$$\\\\frac{2y}{2}=\\\\frac{\\\\left(-7x+3\\\\right)}{2}$$\\\\n$$y=\\\\frac{-7}{2} x+\\\\frac{3}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope24a-h2","type":"hint","dependencies":["a28448cslope24a-h1"],"title":"Slope-Intercept Form of an Equation of Second Line","text":"Compare the second equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=\\\\frac{-7}{2} x+\\\\frac{3}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope24a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-7}{2}$$"],"dependencies":["a28448cslope24a-h2"],"title":"Identify the Slope of First Line","text":"What is $$m$$ in the first equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope24a-h4","type":"hint","dependencies":["a28448cslope24a-h3"],"title":"Solve the second equation for $$y$$","text":"$$2x+7y=5$$\\\\n$$7y=-2x+5$$\\\\n$$\\\\frac{7y}{7}=\\\\frac{\\\\left(-2x+5\\\\right)}{7}$$\\\\n$$y=\\\\frac{-2}{7} x+\\\\frac{5}{7}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope24a-h5","type":"hint","dependencies":["a28448cslope24a-h4"],"title":"Slope-Intercept Form of an Equation of Second Line","text":"Compare the second equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=\\\\frac{-2}{7} x+\\\\frac{5}{7}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope24a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-2}{7}$$"],"dependencies":["a28448cslope24a-h5"],"title":"Identify the Slope of Second Line","text":"What is $$m$$ in the second equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope24a-h7","type":"hint","dependencies":["a28448cslope24a-h6"],"title":"Perpendicular Lines","text":"Perpendicular lines have slopes that are negative reciprocals of each other. We check by multiplying the slopes and see if it equals $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope24a-h8","type":"hint","dependencies":["a28448cslope24a-h3","a28448cslope24a-h6","a28448cslope24a-h7"],"title":"Checking if perpendicular","text":"$$m_1 m_2$$\\\\n$$\\\\frac{-2\\\\frac{-7}{2}}{7}=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope24a-h9","type":"hint","dependencies":["a28448cslope24a-h8"],"title":"Perpendicular Lines","text":"The slopes are reciprocals of each other, but they have the same sign. Since they are not negative reciprocals, the lines are not perpendicular.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28448cslope25","title":"Perpendicular Lines","body":"Use slopes to determine if the lines are perpendicular.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Use the Slope-Intercept Form of an Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28448cslope25a","stepAnswer":["Perpendicular"],"problemType":"MultipleChoice","stepTitle":"$$x-4y=8$$, $$4x+y=2$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Perpendicular","Not Perpendicular"],"hints":{"DefaultPathway":[{"id":"a28448cslope25a-h1","type":"hint","dependencies":[],"title":"Solve the first equation for $$y$$","text":"$$x-4y=8$$\\\\n$$-4y=-x+8$$\\\\n$$\\\\frac{\\\\left(-4y\\\\right)}{-4}=\\\\frac{\\\\left(-x+8\\\\right)}{-4}$$\\\\n$$y=\\\\frac{1}{4} x-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope25a-h2","type":"hint","dependencies":["a28448cslope25a-h1"],"title":"Slope-Intercept Form of an Equation of Second Line","text":"Compare the second equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=\\\\frac{1}{4} x-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope25a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4}$$"],"dependencies":["a28448cslope25a-h2"],"title":"Identify the Slope of First Line","text":"What is $$m$$ in the first equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope25a-h4","type":"hint","dependencies":["a28448cslope25a-h3"],"title":"Solve the second equation for $$y$$","text":"$$4x+y=2$$\\\\n$$y=-4x+2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope25a-h5","type":"hint","dependencies":["a28448cslope25a-h4"],"title":"Slope-Intercept Form of an Equation of Second Line","text":"Compare the second equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=-4x+2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope25a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["a28448cslope25a-h5"],"title":"Identify the Slope of Second Line","text":"What is $$m$$ in the second equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope25a-h7","type":"hint","dependencies":["a28448cslope25a-h6"],"title":"Perpendicular Lines","text":"Perpendicular lines have slopes that are negative reciprocals of each other. We check by multiplying the slopes and see if it equals $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope25a-h8","type":"hint","dependencies":["a28448cslope25a-h3","a28448cslope25a-h6","a28448cslope25a-h7"],"title":"Checking if perpendicular","text":"$$m_1 m_2$$\\\\n$$-4\\\\frac{1}{4}=-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope25a-h9","type":"hint","dependencies":["a28448cslope25a-h8"],"title":"Perpendicular Lines","text":"The slopes are negative reciprocals of each other, so the lines are perpendicular.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28448cslope26","title":"Perpendicular Lines","body":"Use slopes to determine if the lines are perpendicular.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Use the Slope-Intercept Form of an Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28448cslope26a","stepAnswer":["Perpendicular"],"problemType":"MultipleChoice","stepTitle":"$$2x+3y=5$$, $$3x-2y=7$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Perpendicular","Not Perpendicular"],"hints":{"DefaultPathway":[{"id":"a28448cslope26a-h1","type":"hint","dependencies":[],"title":"Solve the first equation for $$y$$","text":"$$2x+3y=5$$\\\\n$$3y=-2x+5$$\\\\n$$\\\\frac{3y}{3}=\\\\frac{\\\\left(-2x+5\\\\right)}{3}$$\\\\n$$y=\\\\frac{-2}{3} x+\\\\frac{5}{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope26a-h2","type":"hint","dependencies":["a28448cslope26a-h1"],"title":"Slope-Intercept Form of an Equation of Second Line","text":"Compare the second equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=\\\\frac{-2}{3} x+\\\\frac{5}{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope26a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-2}{3}$$"],"dependencies":["a28448cslope26a-h2"],"title":"Identify the Slope of First Line","text":"What is $$m$$ in the first equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope26a-h4","type":"hint","dependencies":["a28448cslope26a-h3"],"title":"Solve the second equation for $$y$$","text":"$$3x-2y=7$$\\\\n$$-2y=-3x+7$$\\\\n$$\\\\frac{\\\\left(-2y\\\\right)}{-2}=\\\\frac{\\\\left(-3x+7\\\\right)}{-2}$$\\\\n$$y=\\\\frac{3}{2} x-\\\\frac{7}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope26a-h5","type":"hint","dependencies":["a28448cslope26a-h4"],"title":"Slope-Intercept Form of an Equation of Second Line","text":"Compare the second equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=\\\\frac{3}{2} x-\\\\frac{7}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope26a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{2}$$"],"dependencies":["a28448cslope26a-h5"],"title":"Identify the Slope of Second Line","text":"What is $$m$$ in the second equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope26a-h7","type":"hint","dependencies":["a28448cslope26a-h6"],"title":"Perpendicular Lines","text":"Perpendicular lines have slopes that are negative reciprocals of each other. We check by multiplying the slopes and see if it equals $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope26a-h8","type":"hint","dependencies":["a28448cslope26a-h3","a28448cslope26a-h6","a28448cslope26a-h7"],"title":"Checking if perpendicular","text":"$$m_1 m_2$$\\\\n$$\\\\frac{3\\\\frac{-2}{3}}{2}=-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope26a-h9","type":"hint","dependencies":["a28448cslope26a-h8"],"title":"Perpendicular Lines","text":"The slopes are negative reciprocals of each other, so the lines are perpendicular.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28448cslope27","title":"Perpendicular Lines","body":"Use slopes to determine if the lines are perpendicular.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Use the Slope-Intercept Form of an Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28448cslope27a","stepAnswer":["Not Perpendicular"],"problemType":"MultipleChoice","stepTitle":"$$3x-4y=8$$, $$4x-3y=6$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Perpendicular","Not Perpendicular"],"hints":{"DefaultPathway":[{"id":"a28448cslope27a-h1","type":"hint","dependencies":[],"title":"Solve the first equation for $$y$$","text":"$$3x-4y=8$$\\\\n$$-4y=-3x+8$$\\\\n$$\\\\frac{\\\\left(-4y\\\\right)}{-4}=\\\\frac{\\\\left(-3x+8\\\\right)}{-4}$$\\\\n$$y=\\\\frac{3}{4} x-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope27a-h2","type":"hint","dependencies":["a28448cslope27a-h1"],"title":"Slope-Intercept Form of an Equation of Second Line","text":"Compare the second equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=\\\\frac{3}{4} x-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope27a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{4}$$"],"dependencies":["a28448cslope27a-h2"],"title":"Identify the Slope of First Line","text":"What is $$m$$ in the first equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope27a-h4","type":"hint","dependencies":["a28448cslope27a-h3"],"title":"Solve the second equation for $$y$$","text":"$$4x-3y=6$$\\\\n$$-3y=-4x+6$$\\\\n$$\\\\frac{\\\\left(-3y\\\\right)}{-3}=\\\\frac{\\\\left(-4x+6\\\\right)}{-3}$$\\\\n$$y=\\\\frac{4}{3} x-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope27a-h5","type":"hint","dependencies":["a28448cslope27a-h4"],"title":"Slope-Intercept Form of an Equation of Second Line","text":"Compare the second equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=\\\\frac{4}{3} x-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope27a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{4}{3}$$"],"dependencies":["a28448cslope27a-h5"],"title":"Identify the Slope of Second Line","text":"What is $$m$$ in the second equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope27a-h7","type":"hint","dependencies":["a28448cslope27a-h6"],"title":"Perpendicular Lines","text":"Perpendicular lines have slopes that are negative reciprocals of each other. We check by multiplying the slopes and see if it equals $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope27a-h8","type":"hint","dependencies":["a28448cslope27a-h3","a28448cslope27a-h6","a28448cslope27a-h7"],"title":"Checking if perpendicular","text":"$$m_1 m_2$$\\\\n$$\\\\frac{4\\\\frac{3}{4}}{3}=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope27a-h9","type":"hint","dependencies":["a28448cslope27a-h8"],"title":"Perpendicular Lines","text":"The slopes are reciprocals of each other, but they have the same sign. Since they are not negative reciprocals, the lines are not perpendicular.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28448cslope28","title":"Perpendicular Lines","body":"Use slopes to determine if the lines are perpendicular.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Use the Slope-Intercept Form of an Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28448cslope28a","stepAnswer":["Not Perpendicular"],"problemType":"MultipleChoice","stepTitle":"$$2x+4y=3$$, $$6x+3y=2$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Perpendicular","Not Perpendicular"],"hints":{"DefaultPathway":[{"id":"a28448cslope28a-h1","type":"hint","dependencies":[],"title":"Solve the first equation for $$y$$","text":"$$2x+4y=3$$\\\\n$$4y=-2x+3$$\\\\n$$\\\\frac{4y}{4}=\\\\frac{\\\\left(-2x+3\\\\right)}{4}$$\\\\n$$y=\\\\frac{-1}{2} x+\\\\frac{3}{4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope28a-h2","type":"hint","dependencies":["a28448cslope28a-h1"],"title":"Slope-Intercept Form of an Equation of Second Line","text":"Compare the second equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=\\\\frac{-1}{2} x+\\\\frac{3}{4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope28a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{2}$$"],"dependencies":["a28448cslope28a-h2"],"title":"Identify the Slope of First Line","text":"What is $$m$$ in the first equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope28a-h4","type":"hint","dependencies":["a28448cslope28a-h3"],"title":"Solve the second equation for $$y$$","text":"$$6x+3y=2$$\\\\n$$3y=-6x+2$$\\\\n$$\\\\frac{3y}{3}=\\\\frac{\\\\left(-6x+2\\\\right)}{3}$$\\\\n$$y=-2x+\\\\frac{2}{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope28a-h5","type":"hint","dependencies":["a28448cslope28a-h4"],"title":"Slope-Intercept Form of an Equation of Second Line","text":"Compare the second equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=-2x+\\\\frac{2}{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope28a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a28448cslope28a-h5"],"title":"Identify the Slope of Second Line","text":"What is $$m$$ in the second equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope28a-h7","type":"hint","dependencies":["a28448cslope28a-h6"],"title":"Perpendicular Lines","text":"Perpendicular lines have slopes that are negative reciprocals of each other. We check by multiplying the slopes and see if it equals $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope28a-h8","type":"hint","dependencies":["a28448cslope28a-h3","a28448cslope28a-h6","a28448cslope28a-h7"],"title":"Checking if perpendicular","text":"$$m_1 m_2$$\\\\n$$-2\\\\frac{-1}{2}=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope28a-h9","type":"hint","dependencies":["a28448cslope28a-h8"],"title":"Perpendicular Lines","text":"The slopes are reciprocals of each other, but they have the same sign. Since they are not negative reciprocals, the lines are not perpendicular.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28448cslope29","title":"Perpendicular Lines","body":"Use slopes to determine if the lines are perpendicular.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Use the Slope-Intercept Form of an Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28448cslope29a","stepAnswer":["Perpendicular"],"problemType":"MultipleChoice","stepTitle":"$$2x-6y=4$$, $$12x+4y=9$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Perpendicular","Not Perpendicular"],"hints":{"DefaultPathway":[{"id":"a28448cslope29a-h1","type":"hint","dependencies":[],"title":"Solve the first equation for $$y$$","text":"$$2x-6y=4$$\\\\n$$-6y=-2x+4$$\\\\n$$\\\\frac{\\\\left(-6y\\\\right)}{-6}=\\\\frac{\\\\left(-2x+4\\\\right)}{-6}$$\\\\n$$y=\\\\frac{1}{3} x-\\\\frac{2}{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope29a-h2","type":"hint","dependencies":["a28448cslope29a-h1"],"title":"Slope-Intercept Form of an Equation of Second Line","text":"Compare the second equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=\\\\frac{1}{3} x-\\\\frac{2}{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope29a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["a28448cslope29a-h2"],"title":"Identify the Slope of First Line","text":"What is $$m$$ in the first equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope29a-h4","type":"hint","dependencies":["a28448cslope29a-h3"],"title":"Solve the second equation for $$y$$","text":"$$12x+4y=9$$\\\\n$$4y=-12x+9$$\\\\n$$\\\\frac{4y}{4}=\\\\frac{\\\\left(-12x+9\\\\right)}{4}$$\\\\n$$y=-3x+\\\\frac{9}{4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope29a-h5","type":"hint","dependencies":["a28448cslope29a-h4"],"title":"Slope-Intercept Form of an Equation of Second Line","text":"Compare the second equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=-3x+\\\\frac{9}{4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope29a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a28448cslope29a-h5"],"title":"Identify the Slope of Second Line","text":"What is $$m$$ in the second equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope29a-h7","type":"hint","dependencies":["a28448cslope29a-h6"],"title":"Perpendicular Lines","text":"Perpendicular lines have slopes that are negative reciprocals of each other. We check by multiplying the slopes and see if it equals $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope29a-h8","type":"hint","dependencies":["a28448cslope29a-h3","a28448cslope29a-h6","a28448cslope29a-h7"],"title":"Checking if perpendicular","text":"$$m_1 m_2$$\\\\n$$-3\\\\frac{1}{3}=-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope29a-h9","type":"hint","dependencies":["a28448cslope29a-h8"],"title":"Perpendicular Lines","text":"The slopes are negative reciprocals of each other, so the lines are perpendicular.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28448cslope3","title":"Slope and y-intercept","body":"Use the graph to find the slope and y-intercept of the line.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Use the Slope-Intercept Form of an Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28448cslope3a","stepAnswer":["slope $$m=4$$ and $$y-intercept$$ $$(0,-2)$$"],"problemType":"MultipleChoice","stepTitle":"$$y=4x-2$$\\\\n","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"slope $$m=4$$ and y-intercept $$(0,-2)$$","choices":["slope $$m=\\\\frac{1}{4}$$ and $$y-intercept$$ $$(0,-2)$$","slope $$m=4$$ and $$y-intercept$$ $$(-2,0)$$","slope $$m=4$$ and $$y-intercept$$ $$(0,-2)$$"],"hints":{"DefaultPathway":[{"id":"a28448cslope3a-h1","type":"hint","dependencies":[],"title":"Slope","text":"To find the slope of the line, we need to choose two points on the line. We\u2019ll use the points $$(0,-2)$$ and $$(1,2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope3a-h2","type":"hint","dependencies":["a28448cslope3a-h1"],"title":"Slope","text":"Find the rise and run using the formula $$m=\\\\frac{rise}{run}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope3a-h3","type":"hint","dependencies":["a28448cslope3a-h2"],"title":"Slope","text":"From those two points, there is a rise of $$4$$ units and a run of $$1$$ unit; therefore $$m=\\\\frac{4}{1}=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope3a-h4","type":"hint","dependencies":["a28448cslope3a-h3"],"title":"y-intercept","text":"Find the y-intercept of the line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope3a-h5","type":"hint","dependencies":["a28448cslope3a-h4"],"title":"y-intercept","text":"When $$x=0$$, $$y=-2$$. Therefore, the y-intercept is the point $$(0,-2)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope3a-h6","type":"hint","dependencies":["a28448cslope3a-h5"],"title":"Slope and y-intercept","text":"The slope is $$m=4$$ and the y-intercept is $$(0,-2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28448cslope30","title":"Perpendicular Lines","body":"Use slopes to determine if the lines are perpendicular.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Use the Slope-Intercept Form of an Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28448cslope30a","stepAnswer":["Perpendicular"],"problemType":"MultipleChoice","stepTitle":"$$8x-2y=7$$, $$3x+12y=9$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Perpendicular","Not Perpendicular"],"hints":{"DefaultPathway":[{"id":"a28448cslope30a-h1","type":"hint","dependencies":[],"title":"Solve the first equation for $$y$$","text":"$$8x-2y=7$$\\\\n$$-2y=-8x+7$$\\\\n$$\\\\frac{\\\\left(-2y\\\\right)}{-2}=\\\\frac{\\\\left(-8x+7\\\\right)}{-2}$$\\\\n$$y=4x-\\\\frac{7}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope30a-h2","type":"hint","dependencies":["a28448cslope30a-h1"],"title":"Slope-Intercept Form of an Equation of Second Line","text":"Compare the second equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=4x-\\\\frac{7}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope30a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a28448cslope30a-h2"],"title":"Identify the Slope of First Line","text":"What is $$m$$ in the first equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope30a-h4","type":"hint","dependencies":["a28448cslope30a-h3"],"title":"Solve the second equation for $$y$$","text":"$$3x+12y=9$$\\\\n$$12y=-3x+9$$\\\\n$$\\\\frac{12y}{12}=\\\\frac{\\\\left(-3x+9\\\\right)}{12}$$\\\\n$$y=\\\\frac{-1}{4} x+\\\\frac{3}{4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope30a-h5","type":"hint","dependencies":["a28448cslope30a-h4"],"title":"Slope-Intercept Form of an Equation of Second Line","text":"Compare the second equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=\\\\frac{-1}{4} x+\\\\frac{3}{4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope30a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{4}$$"],"dependencies":["a28448cslope30a-h5"],"title":"Identify the Slope of Second Line","text":"What is $$m$$ in the second equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope30a-h7","type":"hint","dependencies":["a28448cslope30a-h6"],"title":"Perpendicular Lines","text":"Perpendicular lines have slopes that are negative reciprocals of each other. We check by multiplying the slopes and see if it equals $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope30a-h8","type":"hint","dependencies":["a28448cslope30a-h7"],"title":"Checking if perpendicular","text":"$$m_1 m_2$$\\\\n$$\\\\frac{4\\\\times-1}{4}=-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope30a-h9","type":"hint","dependencies":["a28448cslope30a-h8"],"title":"Perpendicular Lines","text":"The slopes are negative reciprocals of each other, so the lines are perpendicular.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28448cslope4","title":"Slope and y-intercept","body":"Use the graph to find the slope and y-intercept of the line.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Use the Slope-Intercept Form of an Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28448cslope4a","stepAnswer":["slope $$m=-1$$ and $$y-intercept$$ $$(0,4)$$"],"problemType":"MultipleChoice","stepTitle":"$$y=-x+4$$\\\\n","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"slope $$m=-1$$ and y-intercept $$(0,4)$$","choices":["slope $$m=-1$$ and $$y-intercept$$ $$(0,4)$$","slope $$m=1$$ and $$y-intercept$$ $$(4,0)$$","slope $$m=-1$$ and $$y-intercept$$ $$(4,0)$$"],"hints":{"DefaultPathway":[{"id":"a28448cslope4a-h1","type":"hint","dependencies":[],"title":"Slope","text":"To find the slope of the line, we need to choose two points on the line. We\u2019ll use the points $$(0,4)$$ and $$(1,3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope4a-h2","type":"hint","dependencies":["a28448cslope4a-h1"],"title":"Slope","text":"Find the rise and run using the formula $$m=\\\\frac{rise}{run}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope4a-h3","type":"hint","dependencies":["a28448cslope4a-h2"],"title":"Slope","text":"From those two points, there is a rise of $$1$$ unit down and a run of $$1$$ unit; therefore $$m=\\\\frac{-1}{1}=-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope4a-h4","type":"hint","dependencies":["a28448cslope4a-h3"],"title":"y-intercept","text":"Find the y-intercept of the line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope4a-h5","type":"hint","dependencies":["a28448cslope4a-h4"],"title":"y-intercept","text":"When $$x=0$$, $$y=4$$. Therefore, the y-intercept is the point $$(0,4)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope4a-h6","type":"hint","dependencies":["a28448cslope4a-h5"],"title":"Slope and y-intercept","text":"The slope is $$m=-1$$ and the y-intercept is $$(0,4)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28448cslope5","title":"Slope and y-intercept","body":"Identify the slope and y-intercept of the line.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Use the Slope-Intercept Form of an Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28448cslope5a","stepAnswer":["$$-3;(0,5)$$"],"problemType":"MultipleChoice","stepTitle":"$$y=-3x+5$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-3;(0,5)$$","choices":["$$-3;(0,5)$$","$$-3;(5,0)$$","$$-3;5$$"],"hints":{"DefaultPathway":[{"id":"a28448cslope5a-h1","type":"hint","dependencies":[],"title":"Slope-Intercept Form of an Equation of a Line","text":"We compare our equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=-3x+5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a28448cslope5a-h1"],"title":"Identify the slope.","text":"What is $$m$$ in our given equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a28448cslope5a-h2"],"title":"Identify $$b$$.","text":"What is $$b$$ in our given equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope5a-h4","type":"hint","dependencies":["a28448cslope5a-h3"],"title":"Identify the y-intercept.","text":"The y-intercept is (0,b). Since $$b$$ is $$5$$, the y-intercept would be $$(0,5)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope5a-h5","type":"hint","dependencies":["a28448cslope5a-h2","a28448cslope5a-h4"],"title":"Slope and y-intercept","text":"The slope is $$-3$$ and the y-intercept is $$(0,5)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28448cslope6","title":"Slope and y-intercept","body":"Identify the slope and y-intercept of the line.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Use the Slope-Intercept Form of an Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28448cslope6a","stepAnswer":["$$-7;(0,3)$$"],"problemType":"MultipleChoice","stepTitle":"$$y=-7x+3$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-7;(0,3)$$","choices":["$$-7;(3,0)$$","$$-7;(0,3)$$","$$-7;3$$"],"hints":{"DefaultPathway":[{"id":"a28448cslope6a-h1","type":"hint","dependencies":[],"title":"Slope-Intercept Form of an Equation of a Line","text":"We compare our equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=-7x+3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-7$$"],"dependencies":["a28448cslope6a-h1"],"title":"Identify the slope.","text":"What is $$m$$ in our given equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a28448cslope6a-h2"],"title":"Identify $$b$$.","text":"What is $$b$$ in our given equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope6a-h4","type":"hint","dependencies":["a28448cslope6a-h3"],"title":"Identify the y-intercept.","text":"The y-intercept is (0,b). Since $$b$$ is $$3$$, the y-intercept would be $$(0,3)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope6a-h5","type":"hint","dependencies":["a28448cslope6a-h2","a28448cslope6a-h4"],"title":"Slope and y-intercept","text":"The slope is $$-7$$ and the y-intercept is $$(0,3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28448cslope7","title":"Slope and y-intercept","body":"Identify the slope and y-intercept of the line.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Use the Slope-Intercept Form of an Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28448cslope7a","stepAnswer":["$$-9;(0,7)$$"],"problemType":"MultipleChoice","stepTitle":"$$y=-9x+7$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-9;(0,7)$$","choices":["$$-9;(0,7)$$","$$-9;(7,0)$$","$$7;(0,-9)$$"],"hints":{"DefaultPathway":[{"id":"a28448cslope7a-h1","type":"hint","dependencies":[],"title":"Slope-Intercept Form of an Equation of a Line","text":"We compare our equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=-9x+7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":["a28448cslope7a-h1"],"title":"Identify the slope.","text":"What is $$m$$ in our given equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a28448cslope7a-h2"],"title":"Identify $$b$$.","text":"What is $$b$$ in our given equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope7a-h4","type":"hint","dependencies":["a28448cslope7a-h3"],"title":"Identify the y-intercept.","text":"The y-intercept is (0,b). Since $$b$$ is $$7$$, the y-intercept would be $$(0,7)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope7a-h5","type":"hint","dependencies":["a28448cslope7a-h2","a28448cslope7a-h4"],"title":"Slope and y-intercept","text":"The slope is $$-9$$ and the y-intercept is $$(0,7)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28448cslope8","title":"Slope and y-intercept","body":"Identify the slope and y-intercept of the line.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Use the Slope-Intercept Form of an Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28448cslope8a","stepAnswer":["$$4;(0,-10)$$"],"problemType":"MultipleChoice","stepTitle":"$$y=4x-10$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$4;(0,-10)$$","choices":["$$-4;(0,10)$$","$$4;(10,0)$$","$$4;(0,-10)$$"],"hints":{"DefaultPathway":[{"id":"a28448cslope8a-h1","type":"hint","dependencies":[],"title":"Slope-Intercept Form of an Equation of a Line","text":"We compare our equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=4x-10$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a28448cslope8a-h1"],"title":"Identify the slope.","text":"What is $$m$$ in our given equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-10$$"],"dependencies":["a28448cslope8a-h2"],"title":"Identify $$b$$.","text":"What is $$b$$ in our given equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope8a-h4","type":"hint","dependencies":["a28448cslope8a-h3"],"title":"Identify the y-intercept.","text":"The y-intercept is (0,b). Since $$b$$ is $$-10$$, the y-intercept would be $$(0,-10)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope8a-h5","type":"hint","dependencies":["a28448cslope8a-h2","a28448cslope8a-h4"],"title":"Slope and y-intercept","text":"The slope is $$4$$ and the y-intercept is $$(0,-10)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28448cslope9","title":"Slope and y-intercept","body":"Identify the slope and y-intercept of the line.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Use the Slope-Intercept Form of an Equation of a Line","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28448cslope9a","stepAnswer":["$$-3;(0,5)$$"],"problemType":"MultipleChoice","stepTitle":"$$3x+y=5$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-3;(0,5)$$","choices":["$$-3;(0,5)$$","$$-3;(5,0)$$","$$3;(0,5)$$"],"hints":{"DefaultPathway":[{"id":"a28448cslope9a-h1","type":"hint","dependencies":[],"title":"Solve for $$y$$","text":"Isolate $$y$$ to one side: $$3x+y=5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope9a-h2","type":"hint","dependencies":["a28448cslope9a-h1"],"title":"Solve for $$y$$","text":"Substract $$3x$$ from each side: $$y=-3x+5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope9a-h3","type":"hint","dependencies":["a28448cslope9a-h2"],"title":"Slope-Intercept Form of an Equation of a Line","text":"We compare our equation to the slope-intercept form of the equation.\\\\n$$y=m x+b$$\\\\n$$y=-3x+5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a28448cslope9a-h3"],"title":"Identify the slope.","text":"What is $$m$$ in our given equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope9a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a28448cslope9a-h4"],"title":"Identify $$b$$.","text":"What is $$b$$ in our given equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope9a-h6","type":"hint","dependencies":["a28448cslope9a-h5"],"title":"Identify the y-intercept.","text":"The y-intercept is (0,b). Since $$b$$ is $$5$$, the y-intercept would be $$(0,5)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28448cslope9a-h7","type":"hint","dependencies":["a28448cslope9a-h4","a28448cslope9a-h6"],"title":"Slope and y-intercept","text":"The slope is $$-3$$ and the y-intercept is $$(0,5)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28722cappquad1","title":"Solve Applications of the Quadratic Formula","body":"Find the number(s) that satisfy the given conditions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Solve Applications Modeled by Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28722cappquad1a","stepAnswer":["$$13$$, $$15$$, and $$-13$$, $$-15$$"],"problemType":"MultipleChoice","stepTitle":"The product of two consecutive odd integers is $$195$$. Find the integers.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$13$$, $$15$$, and $$-13$$, $$-15$$","choices":["$$13$$, $$15$$, and $$-13$$, $$-15$$","$$15$$, $$17$$, and $$-15$$, $$-17$$","$$17$$, $$19$$, and $$-17$$, $$-19$$","$$19$$, $$21$$, and $$-19$$, $$-21$$"],"hints":{"DefaultPathway":[{"id":"a28722cappquad1a-h1","type":"hint","dependencies":[],"title":"Writing the Problem Algebraically","text":"The problem can be rewritten as $$n \\\\left(n+2\\\\right)$$ equals $$195$$, find $$n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad1a-h2","type":"hint","dependencies":["a28722cappquad1a-h1"],"title":"Use the Quadratic Formula","text":"To easily find $$n$$, rewrite the problem in the form $$a n^2$$ + bn + c $$=$$ $$0$$, where a, $$b$$, and c are constants","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$n^2$$ + $$2n$$ $$-195$$ $$=$$ $$0$$"],"dependencies":["a28722cappquad1a-h2"],"title":"Use the Quadratic Formula","text":"What is $$n \\\\left(n+2\\\\right)=195$$ in the form given above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad1a-h4","type":"hint","dependencies":["a28722cappquad1a-h3"],"title":"Quadratic Formula","text":"If you have an equation in the form $$a n^2$$ + bn + c $$=$$ $$0$$, $$n$$ $$=$$ (-b~sqrt(b**2 -4ac))/(2a)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad1a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a28722cappquad1a-h4"],"title":"Quadratic Formula","text":"What is the value of a if we rewrite the problem above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad1a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a28722cappquad1a-h4"],"title":"Quadratic Formula","text":"What is the value of $$b$$ if we rewrite the problem above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad1a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-195$$"],"dependencies":["a28722cappquad1a-h4"],"title":"Quadratic Formula","text":"What is the value of c if we rewrite the problem above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad1a-h8","type":"hint","dependencies":["a28722cappquad1a-h7"],"title":"Multiple Answers","text":"The quadratic formula can yield two different answers, based on the positive or negative value of the square root. These can both be answers","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad1a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$13$$, $$-15$$"],"dependencies":["a28722cappquad1a-h8"],"title":"Multiple Answers","text":"What are the two possible values of $$n$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$13$$, $$-15$$","$$15$$, $$-17$$","$$17$$, $$-19$$","$$19$$, $$-21$$"]}]}}]},{"id":"a28722cappquad10","title":"An Application of the Quadratic Formula","body":"Solve for the number that satisfies the given conditions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Solve Applications Modeled by Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28722cappquad10a","stepAnswer":["$$24$$"],"problemType":"TextBox","stepTitle":"The product of two positive, consecutive even numbers is $$624$$. Find the lower number.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$24$$","hints":{"DefaultPathway":[{"id":"a28722cappquad10a-h1","type":"hint","dependencies":[],"title":"Creating an Equation","text":"If we let $$n$$ be an even number, then we know that $$n\\\\left(n+2\\\\right)=624$$. We can simplify this to get $$n^2+2n-624=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad10a-h2","type":"hint","dependencies":["a28722cappquad10a-h1"],"title":"Solving the Equation","text":"Using the quadratic formula, we get $$n=24, -26$$. We can remove the negative solution. So, the two consecutive odd numbers are $$24$$ and $$26$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28722cappquad11","title":"An Application of the Quadratic Formula","body":"Solve for the number that satisfies the given conditions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Solve Applications Modeled by Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28722cappquad11a","stepAnswer":["$$31$$"],"problemType":"TextBox","stepTitle":"The product of two positive, consecutive odd numbers is $$1023$$. Find the lower number.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$31$$","hints":{"DefaultPathway":[{"id":"a28722cappquad11a-h1","type":"hint","dependencies":[],"title":"Creating an Equation","text":"If we let $$n$$ be an odd number, then we know that $$n\\\\left(n+2\\\\right)=1023$$. We can simplify this to get $$n^2+2n-1023=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad11a-h2","type":"hint","dependencies":["a28722cappquad11a-h1"],"title":"Solving the Equation","text":"Using the quadratic formula, we get $$n=31, -33$$. We can remove the negative solution. So, the two consecutive odd numbers are $$31$$ and $$33$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28722cappquad12","title":"An Application of the Quadratic Formula","body":"Solve for the number that satisfies the given conditions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Solve Applications Modeled by Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28722cappquad12a","stepAnswer":["$$21$$"],"problemType":"TextBox","stepTitle":"The product of two positive, consecutive odd numbers is $$483$$. Find the lower number.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$21$$","hints":{"DefaultPathway":[{"id":"a28722cappquad12a-h1","type":"hint","dependencies":[],"title":"Creating an Equation","text":"If we let $$n$$ be an odd number, then we know that $$n\\\\left(n+2\\\\right)=483$$. We can simplify this to get $$n^2+2n-483=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad12a-h2","type":"hint","dependencies":["a28722cappquad12a-h1"],"title":"Solving the Equation","text":"Using the quadratic formula, we get $$n=21, -23$$. We can remove the negative solution. So, the two consecutive odd numbers are $$21$$ and $$23$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28722cappquad13","title":"An Application of the Quadratic Formula","body":"Solve for the number that satisfies the given conditions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Solve Applications Modeled by Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28722cappquad13a","stepAnswer":["$$22$$"],"problemType":"TextBox","stepTitle":"The product of two positive, consecutive even numbers is $$528$$. Find the lower number.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$22$$","hints":{"DefaultPathway":[{"id":"a28722cappquad13a-h1","type":"hint","dependencies":[],"title":"Creating an Equation","text":"If we let $$n$$ be an even number, then we know that $$n\\\\left(n+2\\\\right)=528$$. We can simplify this to get $$n^2+2n-528=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad13a-h2","type":"hint","dependencies":["a28722cappquad13a-h1"],"title":"Solving the Equation","text":"Using the quadratic formula, we get $$n=22, -24$$. We can remove the negative solution. So, the two consecutive odd numbers are $$22$$ and $$24$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28722cappquad14","title":"An Application of the Quadratic Formula","body":"Solve for the number that satisfies the given conditions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Solve Applications Modeled by Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28722cappquad14a","stepAnswer":["$$18$$"],"problemType":"TextBox","stepTitle":"A triangle with area $$45$$ square inches has a height that is two less than four times the width. Find the height of the triangle.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$18$$","hints":{"DefaultPathway":[{"id":"a28722cappquad14a-h1","type":"hint","dependencies":[],"title":"Creating an Equation","text":"Let w represent the width. We know that the area of a triangle is $$\\\\frac{1}{2}$$ * height * width. 4w-2 is the height, while w is the width. So, $$area=\\\\frac{1}{2\\\\left(4w-2\\\\right)} w$$. The area is $$45$$, so our final equation is $$2w^2-w-45=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad14a-h2","type":"hint","dependencies":["a28722cappquad14a-h1"],"title":"Solving the Equation","text":"Using the quadratic formula, we get $$w=5, w=\\\\frac{-9}{2}$$. We can remove the negative solution. So, the width is $$5$$, which means that the height is $$18$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28722cappquad15","title":"Solve Applications of the Quadratic Formula","body":"Find the correct dimension.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Solve Applications Modeled by Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28722cappquad15a","stepAnswer":["$$22$$"],"problemType":"TextBox","stepTitle":"The width of a triangle is six more than twice the height. The area of the triangle is $$88$$ square yards. Find the width of the triangle.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$22$$","hints":{"DefaultPathway":[{"id":"a28722cappquad15a-h1","type":"hint","dependencies":[],"title":"Identify what we are looking for.","text":"We are looking for the height and width","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad15a-h2","type":"hint","dependencies":["a28722cappquad15a-h1"],"title":"Name what we are looking for.","text":"Let $$h$$ $$=$$ the height of the triangle, $$6+2h$$ $$=$$ the width of the triangle","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad15a-h3","type":"hint","dependencies":["a28722cappquad15a-h2"],"title":"Translate","text":"We know the area. Write the formula for the area of a triangle $$A=\\\\frac{1}{2} b h$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad15a-h4","type":"hint","dependencies":["a28722cappquad15a-h3"],"title":"Solve the equation. Substitue in the values","text":"$$\\\\frac{1}{2} \\\\left(6+2h\\\\right) h=88$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad15a-h5","type":"hint","dependencies":["a28722cappquad15a-h4"],"title":"Distribute","text":"$$3h+h^2$$ $$=$$ $$88$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad15a-h6","type":"hint","dependencies":["a28722cappquad15a-h5"],"title":"Rewrite in standard form.","text":"$$h^2+3h-88=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad15a-h7","type":"hint","dependencies":["a28722cappquad15a-h6"],"title":"Solve the equation using the Quadratic Formula.","text":"Idenfity the a, $$b$$, c value: $$a=1$$, $$b=3$$, $$c=-88$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad15a-h8","type":"hint","dependencies":["a28722cappquad15a-h7"],"title":"Solve the equation using the Quadratic Formula, $$h=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4a c}\\\\right)}{2a}$$.","text":"$$h=\\\\frac{\\\\left(-3\\\\pm 19\\\\right)}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad15a-h9","type":"hint","dependencies":["a28722cappquad15a-h8"],"title":"Rewrite to show two solutions.","text":"$$h=-11$$, $$h=8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad15a-h10","type":"hint","dependencies":["a28722cappquad15a-h9"],"title":"Since $$h$$ is the height of a triangle, it must be positive","text":"$$h=8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad15a-h11","type":"hint","dependencies":["a28722cappquad15a-h10"],"title":"Solve for width.","text":"width $$=$$ $$6+2h$$ $$=$$ $$22$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28722cappquad16","title":"Solve Applications of the Quadratic Formula","body":"Find the correct dimension.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Solve Applications Modeled by Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28722cappquad16a","stepAnswer":["$$1.7$$, $$3$$, $$3.4$$"],"problemType":"MultipleChoice","stepTitle":"The hypotenuse of a right triangle is twice the length of one of its legs. The length of the other leg is three feet. Find the lengths of the three sides of the triangle. Round to the nearest tenth.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$1.7$$, $$3$$, $$3.4$$","choices":["$$1.1$$, $$2.6$$, $$3.1$$","$$1.7$$, $$3$$, $$3.4$$","$$1.4$$, $$2.8$$, $$3.2$$","$$1.6$$, $$3$$, $$3.3$$"],"hints":{"DefaultPathway":[{"id":"a28722cappquad16a-h1","type":"hint","dependencies":[],"title":"Identify what we are looking for.","text":"We are looking for the length of the three sides of the triangle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad16a-h2","type":"hint","dependencies":["a28722cappquad16a-h1"],"title":"Name what we are looking for.","text":"Let $$x$$ $$=$$ length of one side of the triangle, $$2x=length$$ of the hypotenuse.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad16a-h3","type":"hint","dependencies":["a28722cappquad16a-h2"],"title":"Translate into an equation.","text":"We can use the Pythagorean Theorem to solve for x: $$3^2$$ + $$x^2$$ $$=$$ $${\\\\left(2x\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad16a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2$$ $$=$$ $$3$$"],"dependencies":["a28722cappquad16a-h3"],"title":"Simplify","text":"Simplify the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad16a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=\\\\pm \\\\sqrt{3}$$"],"dependencies":["a28722cappquad16a-h4"],"title":"Solve for $$x$$","text":"Use the Square Root Property.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad16a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.7$$"],"dependencies":["a28722cappquad16a-h5"],"title":"Rounding","text":"Round number to neareat tenth.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad16a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3.4$$"],"dependencies":["a28722cappquad16a-h6"],"title":"Solve for hypotenuse","text":"Solve for $$2x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28722cappquad17","title":"Solve Applications of the Quadratic Formula","body":"Find the correct dimension.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Solve Applications Modeled by Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28722cappquad17a","stepAnswer":["$$3.2$$, $$9.5$$, $$10$$"],"problemType":"MultipleChoice","stepTitle":"The hypotenuse of a right triangle is $$10$$ cm long. One of the triangle\u2019s legs is three times the length of the other leg. Find the lengths of the three sides of the triangle in cm. Round to the nearest tenth.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$3.2$$, $$9.5$$, $$10$$","choices":["$$3.2$$, $$9.5$$, $$10$$","$$2.2$$, $$6.6$$, $$10$$","$$2.8$$, $$7.5$$, $$10$$","$$3.5$$, $$10$$, $$10$$"],"hints":{"DefaultPathway":[{"id":"a28722cappquad17a-h1","type":"hint","dependencies":[],"title":"Identify what we are looking for.","text":"We are looking for the length of the three sides of the triangle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad17a-h2","type":"hint","dependencies":["a28722cappquad17a-h1"],"title":"Name what we are looking for.","text":"Let $$x$$ $$=$$ length of one side of the triangle, $$3x=length$$ of the other side..","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad17a-h3","type":"hint","dependencies":["a28722cappquad17a-h2"],"title":"Translate into an equation.","text":"We can use the Pythagorean Theorem to solve for x: $$x^2$$ $$+{\\\\left(3x\\\\right)}^2$$ $$=$$ $${10}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad17a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2=10$$"],"dependencies":["a28722cappquad17a-h3"],"title":"Simplify","text":"Simplify the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad17a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=\\\\pm \\\\sqrt{10}$$"],"dependencies":["a28722cappquad17a-h4"],"title":"Solve for $$x$$","text":"Use the Square Root Property.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad17a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x=3\\\\left(+plusminus\\\\right)+\\\\sqrt{10}$$"],"dependencies":["a28722cappquad17a-h5"],"title":"Solve for","text":"Solve for the other side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad17a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3.2$$"],"dependencies":["a28722cappquad17a-h6"],"title":"Rounding","text":"Round $$x$$ to the nearest tenth.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad17a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9.5$$"],"dependencies":["a28722cappquad17a-h7"],"title":"Rounding","text":"Round $$3x$$ to the nearest tenth.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28722cappquad18","title":"Solve Applications of the Quadratic Formula","body":"Find the correct dimension.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Solve Applications Modeled by Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28722cappquad18a","stepAnswer":["$$7.3$$"],"problemType":"TextBox","stepTitle":"A farmer plans to fence off sections of a rectangular corral. The diagonal distance from one corner of the corral to the opposite corner is five yards longer than the width of the corral. The length of the corral is three times the width. Find the length of the diagonal of the corral. Round to the nearest tenth.","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7.3$$","hints":{"DefaultPathway":[{"id":"a28722cappquad18a-h1","type":"hint","dependencies":[],"title":"Identify what we are looking for.","text":"We are looking for the lengh of the diagonal of the corral, which is the hypotenuse of a triangle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad18a-h2","type":"hint","dependencies":["a28722cappquad18a-h1"],"title":"Name what we are looking for.","text":"As we can see in the picture, the side of the triangle is w, $$3w$$, and $$w+5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad18a-h3","type":"hint","dependencies":["a28722cappquad18a-h2"],"title":"Translate into an equation.","text":"We can use the Pythagorean Theorem to solve for w: $$w^2+{\\\\left(3w\\\\right)}^2={\\\\left(w+5\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad18a-h4","type":"hint","dependencies":["a28722cappquad18a-h3"],"title":"Rewrite in standard form.","text":"$$10w^2-10w-25=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad18a-h5","type":"hint","dependencies":["a28722cappquad18a-h4"],"title":"Solve the equation using the Quadratic Formula.","text":"Idenfity the a, $$b$$, c value: $$a=10$$, $$b=-10$$, $$c=-25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad18a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$w=\\\\frac{10\\\\pm \\\\sqrt{1100}}{20}$$"],"dependencies":["a28722cappquad18a-h5"],"title":"Solve the equation.","text":"We could solve the equation using the Quadratic Formula, $$h=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4a c}\\\\right)}{2a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad18a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.3$$"],"dependencies":["a28722cappquad18a-h6"],"title":"Rounding","text":"Round to the nearest tenth.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad18a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7.3$$"],"dependencies":["a28722cappquad18a-h7"],"title":"Solve for $$w+5$$","text":"Solve for $$w+5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28722cappquad19","title":"Solve Applications of the Quadratic Formula","body":"Find the correct dimension.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Solve Applications Modeled by Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28722cappquad19a","stepAnswer":["$$7.2$$"],"problemType":"TextBox","stepTitle":"Nautical flags are used to represent letters of the alphabet. The flag for the letter O consists of a yellow right triangle and a red right triangle which are sewn together along their hypotenuse to form a square. The adjoining side of the two triangles is three inches longer than a side of the flag. Find the length of the side of the flag.","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7.2$$","hints":{"DefaultPathway":[{"id":"a28722cappquad19a-h1","type":"hint","dependencies":[],"title":"Identify what we are looking for.","text":"We are looking for the length of the side of the flag, which is s.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad19a-h2","type":"hint","dependencies":["a28722cappquad19a-h1"],"title":"Name what we are looking for.","text":"As we can see in the picture, the sides are: s, s, $$s+3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad19a-h3","type":"hint","dependencies":["a28722cappquad19a-h2"],"title":"Translate into an equation.","text":"We can use the Pythagorean Theorem to solve for s: $$s^2$$ $$+s^2$$ $$=$$ $${\\\\left(s+3\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad19a-h4","type":"hint","dependencies":["a28722cappquad19a-h3"],"title":"Rewrite in standard form.","text":"$$s^2-6s-9=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad19a-h5","type":"hint","dependencies":["a28722cappquad19a-h4"],"title":"Solve the equation using the Quadratic Formula.","text":"Idenfity the a, $$b$$, c value: $$a=1$$, $$b=-6$$, $$c=-9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad19a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$s=\\\\frac{6\\\\pm \\\\sqrt{72}}{2}$$"],"dependencies":["a28722cappquad19a-h5"],"title":"Solve the equation.","text":"We could solve the equation using the Quadratic Formula, $$h=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4a c}\\\\right)}{2a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad19a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7.2$$"],"dependencies":["a28722cappquad19a-h6"],"title":"Rounding","text":"Round to nearest tenth","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28722cappquad2","title":"Solve Applications of the Quadratic Formula","body":"Find the number(s) that satisfy the given conditions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Solve Applications Modeled by Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28722cappquad2a","stepAnswer":["$$9$$, $$11$$ and $$-9$$, $$-11$$"],"problemType":"MultipleChoice","stepTitle":"The product of two consecutive odd integers is $$99$$. Find the integers.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$9$$, $$11$$ and $$-9$$, $$-11$$","choices":["$$11$$, $$13$$, and $$-11$$, $$-13$$","$$13$$, $$15$$, and $$-13$$, $$-15$$","$$7$$, $$9$$, and $$-7$$, $$-9$$","$$9$$, $$11$$ and $$-9$$, $$-11$$","$$9$$, $$11$$, and $$-9$$, $$-11$$"],"hints":{"DefaultPathway":[{"id":"a28722cappquad2a-h1","type":"hint","dependencies":[],"title":"Writing the Problem Algebraically","text":"The problem can be rewritten as $$n \\\\left(n+2\\\\right)$$ equals $$99$$, find $$n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad2a-h2","type":"hint","dependencies":[],"title":"Use the Quadratic Formula","text":"To easily find $$n$$, rewrite the problem in the form $${an}^2$$ + bn + c $$=$$ $$0$$, where a, $$b$$, and c are constants","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$n^2$$ + $$2n$$ - $$99=$$ $$0$$"],"dependencies":["a28722cappquad2a-h2"],"title":"Use the Quadratic Formula","text":"What is $$n \\\\left(n+2\\\\right)=99$$ in the form given above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad2a-h4","type":"hint","dependencies":["a28722cappquad2a-h3"],"title":"Quadratic Formula","text":"If you have an equation in the form $${an}^2$$ + bn + c $$=$$ $$0$$, $$n$$ $$=$$ (-b~sqrt(b**2 -4ac))/(2a)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad2a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a28722cappquad2a-h4"],"title":"Quadratic Formula","text":"What is the value of a if we rewrite the problem above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a28722cappquad2a-h4"],"title":"Quadratic Formula","text":"What is the value of $$b$$ if we rewrite the problem above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad2a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-99$$"],"dependencies":["a28722cappquad2a-h4"],"title":"Quadratic Formula","text":"What is the value of c if we rewrite the problem above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad2a-h8","type":"hint","dependencies":[],"title":"Multiple Answers","text":"The quadratic formula can yield two different answers, based on the positive or negative value of the square root. These can both be answers","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad2a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$9$$, $$-11$$"],"dependencies":["a28722cappquad2a-h8"],"title":"Multiple Answers","text":"What are the two possible values of $$n$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$7$$, $$-9$$","$$9$$, $$-11$$","$$11$$, $$-13$$","$$13$$, $$-15$$"]}]}}]},{"id":"a28722cappquad20","title":"Solve Applications of the Quadratic Formula","body":"Find the correct dimension.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Solve Applications Modeled by Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28722cappquad20a","stepAnswer":["$$35$$"],"problemType":"TextBox","stepTitle":"The length of a rectangular driveway is five feet more than three times the width. The area is $$350$$ square feet. 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Its width that is six less than twice the length. What is the width of the lawn in yards?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$14$$","hints":{"DefaultPathway":[{"id":"a28722cappquad21a-h1","type":"hint","dependencies":[],"title":"Identify what we are looking for.","text":"We are looking for the width of the lawn.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad21a-h2","type":"hint","dependencies":["a28722cappquad21a-h1"],"title":"Name what we are looking for.","text":"Let $$x=length$$ of the lawn, $$2x-6=width$$ of the lawn.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad21a-h3","type":"hint","dependencies":["a28722cappquad21a-h2"],"title":"Translate into an equation.","text":"We know the area, and we could use the formula for the area of a rectangle to solve for x: $$\\\\left(2x-6\\\\right) x=140$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad21a-h4","type":"hint","dependencies":["a28722cappquad21a-h3"],"title":"Rewrite in standard form.","text":"$$2x^2-6x-140=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad21a-h5","type":"hint","dependencies":["a28722cappquad21a-h4"],"title":"Solve the equation using the Quadratic Formula.","text":"Idenfity the a, $$b$$, c value: $$a=2$$, $$b=-6$$, $$c=-140$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad21a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=\\\\frac{6\\\\pm \\\\sqrt{1156}}{4}$$"],"dependencies":["a28722cappquad21a-h5"],"title":"Solve the equation.","text":"We could solve the equation using the Quadratic Formula, $$h=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4a c}\\\\right)}{2a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad21a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a28722cappquad21a-h6"],"title":"Solve for $$x$$","text":"Simplify the equation and solve for $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad21a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$14$$"],"dependencies":["a28722cappquad21a-h7"],"title":"Solve for $$2x-6$$","text":"Using the value of $$x$$ to solve for $$2x-6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28722cappquad3","title":"Solve Applications of the Quadratic Formula","body":"Find the number(s) that satisfy the given conditions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Solve Applications Modeled by Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28722cappquad3a","stepAnswer":["$$24$$"],"problemType":"TextBox","stepTitle":"An architect is designing the entryway of a restaurant. She wants to put a triangular window above the doorway. Due to energy restrictions, the window can have an area of $$120$$ square feet and the architect wants the width to be $$4$$ feet more than twice the height. 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$$120$$ $$=$$ $$0$$"],"dependencies":["a28722cappquad3a-h6"],"title":"Quadratic Formula","text":"What is $$h^2$$ $$\\\\left(+2\\\\right) h$$ $$=$$ $$120$$ written in the form above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad3a-h8","type":"hint","dependencies":["a28722cappquad3a-h7"],"title":"Quadratic Formula","text":"If you have an equation in the form $$a h^2$$ + bh+ c $$=$$ $$0$$, $$h$$ $$=$$ (-b~sqrt(b**2 -4ac))/(2a)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad3a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a28722cappquad3a-h8"],"title":"Quadratic Formula","text":"What is the value of a if we rewrite the problem above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad3a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a28722cappquad3a-h8"],"title":"Quadratic Formula","text":"What is the value of $$b$$ if we rewrite the problem above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad3a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-120$$"],"dependencies":["a28722cappquad3a-h8"],"title":"Quadratic Formula","text":"What is the value of c if we rewrite the problem above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad3a-h12","type":"hint","dependencies":["a28722cappquad3a-h11"],"title":"Multiple Answers","text":"The quadratic formula can yield two different answers, based on the positive or negative value of the square root, BUT we must choose the positive answer since the height of something can\'t be negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad3a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a28722cappquad3a-h12"],"title":"Multiple Answers","text":"What is the positive value of $$h$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28722cappquad4","title":"Solve Applications of the Quadratic Formula","body":"Find the number(s) that satisfy the given conditions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Solve Applications Modeled by Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28722cappquad4a","stepAnswer":["$$20$$"],"problemType":"TextBox","stepTitle":"If a triangle that has an area of $$110$$ square feet has a height that is two feet less than twice the width, what is its height in feet?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$20$$","hints":{"DefaultPathway":[{"id":"a28722cappquad4a-h1","type":"hint","dependencies":[],"title":"Area of a Triangle","text":"The area of a triangle is $$110$$ $$=$$ $$0.5h w$$, where $$h$$ and w respectively represent the height and width of the triangle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad4a-h2","type":"hint","dependencies":["a28722cappquad4a-h1"],"title":"Rewrite the Height in Terms of w","text":"Find $$h$$ in terms of w","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2w$$ - $$2$$"],"dependencies":["a28722cappquad4a-h2"],"title":"Rewrite the Height in Terms of w","text":"What is $$h$$ in terms of w?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad4a-h4","type":"hint","dependencies":["a28722cappquad4a-h3"],"title":"Rewriting the Problem Algebraically","text":"Substitute the value of $$h$$ with its equivalent in terms of w, in the area formula $$110$$ $$=$$ $$0.5h w$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad4a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["0.5*(2w**2 - 2w)"],"dependencies":["a28722cappquad4a-h4"],"title":"Rewriting the Problem Algebraically","text":"What is the area of this triangle in terms of $$h$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad4a-h6","type":"hint","dependencies":["a28722cappquad4a-h5"],"title":"Quadratic Formula","text":"Now that the area formula can be rewritten in terms of w, we can rewrite it in the form $${aw}^2$$ + bw + c $$=$$ $$0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad4a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$w^2$$ - 2w - $$110$$ $$=$$ $$0$$"],"dependencies":["a28722cappquad4a-h6"],"title":"Quadratic Formula","text":"What is $$w^2$$ - 2w $$=$$ $$110$$ written in the form above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad4a-h8","type":"hint","dependencies":["a28722cappquad4a-h7"],"title":"Quadratic Formula","text":"If you have an equation in the form $${ah}^2$$ + $$b h$$ + c $$=$$ $$0$$, $$h$$ $$=$$ (-b~sqrt(b**2 -4ac))/(2a)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad4a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a28722cappquad4a-h8"],"title":"Quadratic Formula","text":"What is the value of a if we rewrite the problem above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad4a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a28722cappquad4a-h8"],"title":"Quadratic Formula","text":"What is the value of $$b$$ if we rewrite the problem above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad4a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-110$$"],"dependencies":["a28722cappquad4a-h8"],"title":"Quadratic Formula","text":"What is the value of c if we rewrite the problem above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad4a-h12","type":"hint","dependencies":["a28722cappquad4a-h11"],"title":"Multiple Answers","text":"The quadratic formula can yield two different answers, based on the positive or negative value of the square root, BUT we must choose the positive answer since the width of something can\'t be negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad4a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11$$"],"dependencies":["a28722cappquad4a-h12"],"title":"Multiple Answers","text":"What is the positive value of w?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28722cappquad5","title":"Solve Applications of the Quadratic Formula","body":"Find the number(s) that satisfy the given conditions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Solve Applications Modeled by Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28722cappquad5a","stepAnswer":["$$(6.3$$, $$19.0)$$"],"problemType":"MultipleChoice","stepTitle":"The sun casts a shadow from a flag pole. The height of the flag pole is three times the length of its shadow. The distance between the end of the shadow and the top of the flag pole is $$20$$ feet. Find the length of the shadow and the length of the flag pole. Round to the nearest tenth of a foot.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(6.3$$, $$19.0)$$","choices":["$$(5.5$$, $$16.5)$$","$$(5.8$$, $$17.4)$$","$$(6.3$$, $$19.0)$$","$$(6.0$$, $$18.0)$$"],"hints":{"DefaultPathway":[{"id":"a28722cappquad5a-h1","type":"hint","dependencies":[],"title":"Writing the Problem Algebraically","text":"Use the Pythagorean Theorem to rewrite the problem like $$a^2$$ + $$b^2$$ $$=$$ $$c^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a28722cappquad5a-h1"],"title":"Writing the Problem Algebraically","text":"What is the value of c in this scenario?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad5a-h3","type":"hint","dependencies":["a28722cappquad5a-h2"],"title":"Rewrite the Problem in Terms of One Unknown","text":"Currently, there are two unknowns, height of the flag pole (h) and length of the shadow (l). WIth the information given, we can write $$h$$ in terms of l","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3l$$"],"dependencies":["a28722cappquad5a-h3"],"title":"Rewrite the Problem in Terms of One Unknown","text":"What is $$h$$ in terms of l?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad5a-h5","type":"hint","dependencies":["a28722cappquad5a-h4"],"title":"Simplify","text":"Now that you have $$h$$ in terms of l, you can plug that value into the original Pythagorean Theorem equation $$h^2$$ + $$l^2$$ $$=$$ $${20}^2$$, then do the squares and solve","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad5a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9l^2$$"],"dependencies":["a28722cappquad5a-h5"],"title":"Simplify","text":"What is $${\\\\left(3l\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad5a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10l^2$$"],"dependencies":["a28722cappquad5a-h6"],"title":"Simplify","text":"What is $$9l^2$$ + $$l^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad5a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$400$$"],"dependencies":["a28722cappquad5a-h7"],"title":"Simplify","text":"What is $${20}^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad5a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{40}$$"],"dependencies":["a28722cappquad5a-h8"],"title":"Simplify","text":"What is l?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad5a-h10","type":"hint","dependencies":["a28722cappquad5a-h9"],"title":"Solving for $$h$$","text":"Now that we have a value of l, we can plug that value back into the equation for $$h$$ in terms of l, to get $$h$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad5a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$19$$"],"dependencies":["a28722cappquad5a-h10"],"title":"Solving for $$h$$","text":"What is $$h$$ (rounded to the nearest tenth)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28722cappquad6","title":"Solve Applications of the Quadratic Formula","body":"Find the number(s) that satisfy the given conditions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Solve Applications Modeled by Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28722cappquad6a","stepAnswer":["$$(18.1$$, $$11.0)$$"],"problemType":"MultipleChoice","stepTitle":"The length of a $$200$$ square foot rectangular vegetable garden is four feet less than twice the width. Find the length and width of the garden. Round to the nearest tenth of a foot.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(18.1$$, $$11.0)$$","choices":["$$(18.1$$, $$11.0)$$","$$(20.1$$, $$12.0)$$","$$(22.0$$","$$12.5)$$","$$(22.5$$","$$13.0)$$"],"hints":{"DefaultPathway":[{"id":"a28722cappquad6a-h1","type":"hint","dependencies":[],"title":"Writing the Problem Algebraically","text":"Use the formula for the area of a rectangle to rewrite the problem: $$200$$ $$=$$ $$L W$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad6a-h2","type":"hint","dependencies":["a28722cappquad6a-h1"],"title":"Rewrite the Problem in Terms of One Unknown","text":"Currently, there are two unknowns in the area formula: L and W. Since we know how L relates to W, we can find an equation for L in terms of W.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2W$$ - $$4$$"],"dependencies":["a28722cappquad6a-h2"],"title":"Rewrite the Problem in Terms of One Unknown","text":"What is L in terms of W?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad6a-h4","type":"hint","dependencies":["a28722cappquad6a-h3"],"title":"Simplify","text":"Now that you have L in terms of W, you can rewrite the original area equation, $$200$$ $$=$$ $$L W$$ in terms of W","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2W^2$$ - $$4W$$"],"dependencies":["a28722cappquad6a-h4"],"title":"Simplify","text":"What is W*(2*W - 4)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad6a-h6","type":"hint","dependencies":["a28722cappquad6a-h5"],"title":"Quadratic Formula","text":"We can rewrite this equation in the format $$a W^2$$ + $$b W$$ + c $$=$$ $$0$$ to use the quadratic formula","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad6a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a28722cappquad6a-h6"],"title":"Quadratic Formula","text":"What is the value of a if we rewrite the problem above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad6a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["a28722cappquad6a-h6"],"title":"Quadratic Formula","text":"What is the value of $$b$$ if we rewrite the problem above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad6a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-200$$"],"dependencies":["a28722cappquad6a-h6"],"title":"Quadratic Formula","text":"What is the value of c if we rewrite the problem above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad6a-h10","type":"hint","dependencies":["a28722cappquad6a-h6"],"title":"Quadratic Formula","text":"If you have an equation in the form $$a W^2$$ + $$b W$$ + c $$=$$ $$0$$, W $$=$$ (-b~sqrt(b**2 -4ac))/(2a)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad6a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11.05$$"],"dependencies":["a28722cappquad6a-h10"],"title":"Quadratic Formula","text":"What POSITIVE value does the quadratic formula yield, rounded to $$2$$ decimal places?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad6a-h12","type":"hint","dependencies":["a28722cappquad6a-h11"],"title":"Solving for L","text":"Now that you have a value for W, you can plug in into the equation for L in terms of W to get L","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad6a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18.1$$"],"dependencies":["a28722cappquad6a-h12"],"title":"Solving for L","text":"What is L?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28722cappquad7","title":"Projectile Motion","body":"Find the number(s) that satisfy the given conditions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Solve Applications Modeled by Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28722cappquad7a","stepAnswer":["$$3.6$$"],"problemType":"TextBox","stepTitle":"A firework is shot upwards with initial velocity $$130$$ feet per second. How many seconds will it take to FIRST reach a height of $$260$$ feet? Round to the nearest tenth of a second.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3.6$$","hints":{"DefaultPathway":[{"id":"a28722cappquad7a-h1","type":"hint","dependencies":[],"title":"Writing the Problem Algebraically","text":"Substitute the values for $$h$$ and $$t$$ into the projectile motion formula $$h$$ $$=$$ $$-16t^2$$ + $$vo t$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$130$$"],"dependencies":["a28722cappquad7a-h1"],"title":"Writing the Problem Algebraically","text":"What is vo, the initial velocity, in $$\\\\frac{ft}{s}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$260$$"],"dependencies":["a28722cappquad7a-h2"],"title":"Writing the Problem Algebraically","text":"What is $$h$$, the height to reach, in ft?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad7a-h4","type":"hint","dependencies":["a28722cappquad7a-h3"],"title":"Quadratic Formula","text":"We can rewrite this equation in the format $$a t^2$$ + $$b t$$ + c $$=$$ $$0$$ to use the quadratic formula","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad7a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-16$$"],"dependencies":["a28722cappquad7a-h4"],"title":"Quadratic Formula","text":"What is the value of a if we rewrite the problem above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad7a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$130$$"],"dependencies":["a28722cappquad7a-h4"],"title":"Quadratic Formula","text":"What is the value of $$b$$ if we rewrite the problem above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad7a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-260$$"],"dependencies":["a28722cappquad7a-h4"],"title":"Quadratic Formula","text":"What is the value of c if we rewrite the problem above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad7a-h8","type":"hint","dependencies":["a28722cappquad7a-h7"],"title":"Quadratic Formula","text":"If you have an equation in the form $$a t^2$$ + $$b t$$ + c $$=$$ $$0$$, $$t$$ $$=$$ (-b~sqrt(b**2 -4ac))/(2a)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a28722cappquad7a-h9","type":"hint","dependencies":["a28722cappquad7a-h8"],"title":"Quadratic Formula","text":"The quadratic formula may yield two positive answers. If this is the case, choose the lower value because we want to find the time it takes to initially reach $$260$$ ft","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28722cappquad8","title":"An Application of the Quadratic Formula","body":"Solve for the number that satisfies the given conditions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Solve Applications Modeled by Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28722cappquad8a","stepAnswer":["$$15$$"],"problemType":"TextBox","stepTitle":"The product of two positive, consecutive odd numbers is $$255$$. 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So, the two consecutive odd numbers are $$15$$ and $$17$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a28722cappquad9","title":"An Application of the Quadratic Formula","body":"Solve for the number that satisfies the given conditions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Solve Applications Modeled by Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a28722cappquad9a","stepAnswer":["$$18$$"],"problemType":"TextBox","stepTitle":"The product of two positive, consecutive even numbers is $$360$$. 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So, the two consecutive odd numbers are $$18$$ and $$20$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a298b40expectedvalue1","title":"Practice with Expected Value","body":"A men\'s soccer team plays soccer zero, one, or two days a week. The probability that they play zero days is $$0.2$$, the probability that they play one day is $$0.5$$, and the probability that they play two days is $$0.3$$. Let the random variable X $$=$$ the number of days the men\'s soccer team plays soccer per week. X takes the values $$0$$, $$1$$, $$2$$. In the table, we have rows for values of X, $$P(X=x)$$, and x*P(X=x).\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Mean or Expected Value and Standard Deviation","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a298b40expectedvalue1a","stepAnswer":["$$1.1$$"],"problemType":"TextBox","stepTitle":"Find the long-term average or expected value of the number of days per week the men\'s soccer team plays soccer based on the table provided.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1.1$$","hints":{"DefaultPathway":[{"id":"a298b40expectedvalue1a-h1","type":"hint","dependencies":[],"title":"How to Calculate Expected Value","text":"Expected value, or the long-term average or mean can be calculated as the overall summation of each individual possible value of the random variable multiplied by the probability in the sample space of that outcome. Therefore, if we have a random variable X, the expected value of X can be calculated as the summation of all possible outcomes $$x$$ by the probability of $$x$$, $$x P\\\\left(x\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.1$$"],"dependencies":["a298b40expectedvalue1a-h1"],"title":"Expected Value as Summation","text":"What is the expected value, or the summation of all the rows of $$x P\\\\left(x\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a298b40expectedvalue1a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.1$$"],"dependencies":[],"title":"Expected Value as Summation","text":"What is $$0+0.5+0.6$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a298b40expectedvalue10","title":"Practice with Random Variables","body":"Suppose you play a game with a spinner. You play each game by spinning the spinner once. $$P(red)=\\\\frac{2}{5}$$, $$P(blue)=\\\\frac{2}{5}$$, and $$P(green)=\\\\frac{1}{5}$$. If you land on red, you pay $10. If you land on blue, you don\'t pay or win anything. If you land on green, you win $10. We want to find how much we expect to earn in the process.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Mean or Expected Value and Standard Deviation","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a298b40expectedvalue10a","stepAnswer":["$$X=amount$$ of profit"],"problemType":"MultipleChoice","stepTitle":"Define a random variable X.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$X=amount$$ of profit","choices":["$$X=amount$$ of profit","$$X=number$$ of spins","$$X=if$$ you win","$$X=if$$ you lose"],"hints":{"DefaultPathway":[{"id":"a298b40expectedvalue10a-h1","type":"hint","dependencies":[],"title":"Defining Discrete Random Variables","text":"Remember that when we define discrete random variables, we want variables to be countable (5 marbles, $$2$$ heads, $5).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue10a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$X=amount$$ of profit OR $$X=number$$ of spins"],"dependencies":["a298b40expectedvalue10a-h1"],"title":"Determining Countable Random Variables","text":"Which pair of random variables listed is countable?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$X=amount$$ of profit OR $$X=number$$ of spins","$$X=if$$ you win OR $$X=if$$ you lose"]},{"id":"a298b40expectedvalue10a-h3","type":"hint","dependencies":["a298b40expectedvalue10a-h2"],"title":"Determining Aligned Random Variables","text":"Now, we have limited down to either the amount of profit or the number of spins, both of which are applicable to our situation. However, note that we want to determine if we\'ll come out ahead (gain profit).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue10a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$X=amount$$ of profit"],"dependencies":["a298b40expectedvalue10a-h3"],"title":"Determining Aligned Random Variables","text":"Which random variable seems more aligned with what we want to solve for?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$X=amount$$ of profit","$$X=number$$ of spins"]}]}}]},{"id":"a298b40expectedvalue11","title":"Practice with Expected Value Tables","body":"Suppose you play a game with a spinner. You play each game by spinning the spinner once. $$P(red)=\\\\frac{2}{5}$$, $$P(blue)=\\\\frac{2}{5}$$, and $$P(green)=\\\\frac{1}{5}$$. If you land on red, you pay $10. If you land on blue, you don\'t pay or win anything. If you land on green, you win $10. We want to find how much we expect to earn in the process. Let\'s complete the expected value table provided.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Mean or Expected Value and Standard Deviation","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a298b40expectedvalue11a","stepAnswer":["$$-10$$"],"problemType":"TextBox","stepTitle":"What is the value for $$x$$ for red?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-10$$","hints":{"DefaultPathway":[{"id":"a298b40expectedvalue11a-h1","type":"hint","dependencies":[],"title":"Look At the Table","text":"Notice that the table has $$10$$ listed for the $$x$$ value for green. This is due to the fact that if you spin a green, you win $10. Therefore, red must align with what happens if you spin a red.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-10$$"],"dependencies":["a298b40expectedvalue11a-h1"],"title":"Determining Red\'s $$x$$","text":"What is the $$x$$ value for red? Reminder that red is associated with pay, which is a negative number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a298b40expectedvalue11b","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"What is the value for $$x$$ for blue?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a298b40expectedvalue11b-h3","type":"hint","dependencies":["a298b40expectedvalue11a-h2"],"title":"Look At the Table","text":"Notice that the table has $$10$$ listed for the $$x$$ value for green. This is due to the fact that if you spin a green, you win $10. Therefore, blue must align with what happens if you spin a blue.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue11b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a298b40expectedvalue11b-h3"],"title":"Determining Blue\'s $$x$$","text":"What is the $$x$$ value for blue?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a298b40expectedvalue11c","stepAnswer":["$$\\\\frac{2}{5}$$"],"problemType":"TextBox","stepTitle":"What is the P(x) value for red?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2}{5}$$","hints":{"DefaultPathway":[{"id":"a298b40expectedvalue11c-h5","type":"hint","dependencies":["a298b40expectedvalue11b-h4"],"title":"Look At the Table","text":"Notice that the table has $$\\\\frac{2}{5}$$ listed for the P(x) value for blue. This is due to the fact that there is a probability of $$\\\\frac{2}{5}$$ of spinning a blue since $$P(blue)=\\\\frac{2}{5}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue11c-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{5}$$"],"dependencies":["a298b40expectedvalue11c-h5"],"title":"Determining Red\'s P(x)","text":"What is the value for P(x) for red, P(red)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a298b40expectedvalue11d","stepAnswer":["$$\\\\frac{1}{5}$$"],"problemType":"TextBox","stepTitle":"What is the P(x) value for green?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{5}$$","hints":{"DefaultPathway":[{"id":"a298b40expectedvalue11d-h7","type":"hint","dependencies":["a298b40expectedvalue11c-h6"],"title":"Look At the Table","text":"Notice that the table has $$\\\\frac{2}{5}$$ listed for the P(x) value for blue. This is due to the fact that there is a probability of $$\\\\frac{2}{5}$$ of spinning a blue since $$P(blue)=\\\\frac{2}{5}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue11d-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{5}$$"],"dependencies":["a298b40expectedvalue11d-h7"],"title":"Determining Green\'s P(x)","text":"What is the value for P(x) for green, P(geen)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a298b40expectedvalue11e","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"What is the value for $$x P\\\\left(x\\\\right)$$ for blue?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a298b40expectedvalue11e-h9","type":"hint","dependencies":["a298b40expectedvalue11d-h8"],"title":"Definition of $$x P\\\\left(x\\\\right)$$","text":"Now, we want to find $$x P\\\\left(x\\\\right)$$. In the provided table or previous steps of this question we found $$x$$ and P(x) for blue. Now, let\'s multiply the two together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue11e-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a298b40expectedvalue11e-h9"],"title":"Determining Blue\'s $$x P\\\\left(x\\\\right)$$","text":"What is the value for $$x P\\\\left(x\\\\right)$$ for blue?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a298b40expectedvalue11f","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"What is the value for $$x P\\\\left(x\\\\right)$$ for green?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a298b40expectedvalue11f-h11","type":"hint","dependencies":["a298b40expectedvalue11e-h10"],"title":"Definition of $$x P\\\\left(x\\\\right)$$","text":"Now, we want to find $$x P\\\\left(x\\\\right)$$. In the provided table or previous steps of this question we found $$x$$ and P(x) for green. Now, let\'s multiply the two together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue11f-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a298b40expectedvalue11f-h11"],"title":"Determining Green\'s $$x P\\\\left(x\\\\right)$$","text":"What is the value for $$x P\\\\left(x\\\\right)$$ for green?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a298b40expectedvalue12","title":"Practice with Expected Value","body":"Suppose you play a game with a spinner. You play each game by spinning the spinner once. $$P(red)=\\\\frac{2}{5}$$, $$P(blue)=\\\\frac{2}{5}$$, and $$P(green)=\\\\frac{1}{5}$$. If you land on red, you pay $10. If you land on blue, you don\'t pay or win anything. If you land on green, you win $10. We want to find how much we expect to earn in the process.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Mean or Expected Value and Standard Deviation","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a298b40expectedvalue12a","stepAnswer":["$$\\\\frac{-8}{6}$$"],"problemType":"TextBox","stepTitle":"What is the expected value for your profit?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-8}{6}$$","hints":{"DefaultPathway":[{"id":"a298b40expectedvalue12a-h1","type":"hint","dependencies":[],"title":"How to Calculate Expected Value","text":"Expected value, or the long-term average or mean can be calculated as the overall summation of each individual possible value of the random variable multiplied by the probability in the sample space of that outcome. Therefore, if we have a random variable X, the expected value of X can be calculated as the summation of all possible outcomes $$x$$ by the probability of $$x$$, $$x P\\\\left(x\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-8}{6}$$"],"dependencies":["a298b40expectedvalue12a-h1"],"title":"Expected Value as Summation","text":"What is the expected value, or the summation of all the rows of $$x P\\\\left(x\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a298b40expectedvalue12a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-8}{6}$$"],"dependencies":[],"title":"Expected Value as Summation","text":"What is $$\\\\frac{-20}{6}+0+2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a298b40expectedvalue13","title":"Practice with Expected Value","body":"Toss a fair, six-sided die twice. Let X $$=$$ the number of faces that show an even number. Provided is an expected value table.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Mean or Expected Value and Standard Deviation","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a298b40expectedvalue13a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"What is the expected value for X?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a298b40expectedvalue13a-h1","type":"hint","dependencies":[],"title":"How to Calculate Expected Value","text":"Expected value, or the long-term average or mean can be calculated as the overall summation of each individual possible value of the random variable multiplied by the probability in the sample space of that outcome. Therefore, if we have a random variable X, the expected value of X can be calculated as the summation of all possible outcomes $$x$$ by the probability of $$x$$, $$x P\\\\left(x\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a298b40expectedvalue13a-h1"],"title":"Expected Value as Summation","text":"What is the expected value, or the summation of all the rows of $$x P\\\\left(x\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a298b40expectedvalue13a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Expected Value as Summation","text":"What is $$0+\\\\frac{18}{36}+\\\\frac{18}{36}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a298b40expectedvalue14","title":"Practice with Expected Value","body":"Toss a fair, six-sided die twice. Let X $$=$$ the number of faces that show an even number. Provided is an expected value table.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Mean or Expected Value and Standard Deviation","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a298b40expectedvalue14a","stepAnswer":["$$0.707$$"],"problemType":"TextBox","stepTitle":"What is the standard deviation for X? Round to the nearest thousandth.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.707$$","hints":{"DefaultPathway":[{"id":"a298b40expectedvalue14a-h1","type":"hint","dependencies":[],"title":"How to Calculate Standard Deviation","text":"The standard deviation of a PDF (probability density function) is the square root of variance. The value for variance is in the fourth column of the provided table since variance is the summation of all those rows, so we need to calculate standard deviation from that.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.707$$"],"dependencies":["a298b40expectedvalue14a-h1"],"title":"Standard Deviation as the Square Root of Variance","text":"What is the standard deviation given that standard deviation is the square root of variance and variance is the summation of the values in the fourth column? Round this value to the nearest thousandth.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a298b40expectedvalue14a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{18}{36}$$"],"dependencies":[],"title":"Finding Variance","text":"What is the variance? What is the summation of all the values in the fourth column? Keep in fractional form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue14a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.707$$"],"dependencies":[],"title":"Finding Standard Deviation","text":"Based on the variance found, standard deviation is the square root of variance. Therefore, what is the square root of the variance you found, rounded to the nearest thousandth?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a298b40expectedvalue15","title":"Real World Applications of Expected Value","body":"On May $$11$$, $$2013$$ at 9:30 PM, the probability that moderate seismic activity (one moderate earthquake) would occur in the next $$48$$ hours in Iran was about $$21.42\\\\%$$. Suppose you make a bet that a moderate earthquake will occur in Iran during this period. If you win the bet, you win $50. If you lose the bet, you pay $20. Let X $$=$$ the amount of profit from a bet.\\\\nP(win) $$=$$ P(one moderate earthquake will occur) $$=$$ $$21.42\\\\%$$\\\\n\\\\nP(loss) $$=$$ P(one moderate earthquake will not occur) $$=$$ 100% - 21.42%\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Mean or Expected Value and Standard Deviation","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a298b40expectedvalue15a","stepAnswer":["$$-5.006$$"],"problemType":"TextBox","stepTitle":"Determine the expected value for your profit.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-5.006$$","hints":{"DefaultPathway":[{"id":"a298b40expectedvalue15a-h1","type":"hint","dependencies":[],"title":"How to Calculate Expected Value","text":"Expected value, or the long-term average or mean can be calculated as the overall summation of each individual possible value of the random variable multiplied by the probability in the sample space of that outcome. Therefore, if we have a random variable X, the expected value of X can be calculated as the summation of all possible outcomes $$x$$ by the probability of $$x$$, $$x P\\\\left(x\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5.006$$"],"dependencies":["a298b40expectedvalue15a-h1"],"title":"Expected Value as Summation","text":"What is the expected value, or the summation of all the rows of $$x P\\\\left(x\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a298b40expectedvalue15a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5.006$$"],"dependencies":[],"title":"Expected Value as Summation","text":"What is $$10.71-15.716$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a298b40expectedvalue16","title":"Expected Value Warm Up","body":"Using the expected table provided in the questions below, we will complete it together.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Mean or Expected Value and Standard Deviation","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a298b40expectedvalue16a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"What is the value for $$x P\\\\left(x\\\\right)$$ for $$x=0$$ and $$P(x)=0.2$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a298b40expectedvalue16a-h1","type":"hint","dependencies":[],"title":"Definition of $$x P\\\\left(x\\\\right)$$","text":"Now, we want to find $$x P\\\\left(x\\\\right)$$. In the table, we have the provided values for $$x P\\\\left(x\\\\right)$$ for $$x=0$$ and $$P(x)=0.2$$. Now, let\'s multiply the two together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a298b40expectedvalue16a-h1"],"title":"Determining $$x=0\'s$$ $$x P\\\\left(x\\\\right)$$","text":"What is the value for $$x P\\\\left(x\\\\right)$$ for $$x=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a298b40expectedvalue16b","stepAnswer":["$$0.2$$"],"problemType":"TextBox","stepTitle":"What is the value for $$x P\\\\left(x\\\\right)$$ for $$x=1and$$ $$P(x)=0.2$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.2$$","hints":{"DefaultPathway":[{"id":"a298b40expectedvalue16b-h3","type":"hint","dependencies":["a298b40expectedvalue16a-h2"],"title":"Definition of $$x P\\\\left(x\\\\right)$$","text":"Now, we want to find $$x P\\\\left(x\\\\right)$$. In the table, we have the provided values for $$x P\\\\left(x\\\\right)$$ for $$x=1$$ and $$P(x)=0.2$$. Now, let\'s multiply the two together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue16b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.2$$"],"dependencies":["a298b40expectedvalue16b-h3"],"title":"Determining $$x=1\'s$$ $$x P\\\\left(x\\\\right)$$","text":"What is the value for $$x P\\\\left(x\\\\right)$$ for $$x=1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a298b40expectedvalue16c","stepAnswer":["$$0.8$$"],"problemType":"TextBox","stepTitle":"What is the value for $$x P\\\\left(x\\\\right)$$ for $$x=2$$ and $$P(x)=0.4$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.8$$","hints":{"DefaultPathway":[{"id":"a298b40expectedvalue16c-h5","type":"hint","dependencies":["a298b40expectedvalue16b-h4"],"title":"Definition of $$x P\\\\left(x\\\\right)$$","text":"Now, we want to find $$x P\\\\left(x\\\\right)$$. In the table, we have the provided values for $$x P\\\\left(x\\\\right)$$ for $$x=2$$ and $$P(x)=0.4$$. Now, let\'s multiply the two together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue16c-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.8$$"],"dependencies":["a298b40expectedvalue16c-h5"],"title":"Determining $$x=2\'s$$ $$x P\\\\left(x\\\\right)$$","text":"What is the value for $$x P\\\\left(x\\\\right)$$ for $$x=2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a298b40expectedvalue16d","stepAnswer":["$$0.6$$"],"problemType":"TextBox","stepTitle":"What is the value for $$x P\\\\left(x\\\\right)$$ for $$x=3$$ and $$P(x)=0.2$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.6$$","hints":{"DefaultPathway":[{"id":"a298b40expectedvalue16d-h7","type":"hint","dependencies":["a298b40expectedvalue16a-h1"],"title":"Definition of $$x P\\\\left(x\\\\right)$$","text":"Now, we want to find $$x P\\\\left(x\\\\right)$$. In the table, we have the provided values for $$x P\\\\left(x\\\\right)$$ for $$x=3$$ and $$P(x)=0.2$$. Now, let\'s multiply the two together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue16d-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.6$$"],"dependencies":["a298b40expectedvalue16d-h7"],"title":"Determining $$x=3\'s$$ $$x P\\\\left(x\\\\right)$$","text":"What is the value for $$x P\\\\left(x\\\\right)$$ for $$x=3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a298b40expectedvalue16e","stepAnswer":["$$1.6$$"],"problemType":"TextBox","stepTitle":"Determine the expected value.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1.6$$","hints":{"DefaultPathway":[{"id":"a298b40expectedvalue16e-h9","type":"hint","dependencies":["a298b40expectedvalue16d-h8"],"title":"How to Calculate Expected Value","text":"Expected value, or the long-term average or mean can be calculated as the overall summation of each individual possible value of the random variable multiplied by the probability in the sample space of that outcome. Therefore, if we have a random variable X, the expected value of X can be calculated as the summation of all possible outcomes $$x$$ by the probability of $$x$$, $$x P\\\\left(x\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue16e-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.6$$"],"dependencies":["a298b40expectedvalue16e-h9"],"title":"Expected Value as Summation","text":"What is the expected value, or the summation of all the $$x P\\\\left(x\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a298b40expectedvalue16e-h10-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.6$$"],"dependencies":[],"title":"Expected Value as Summation","text":"What is $$0+0.2+0.8+0.6$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a298b40expectedvalue17","title":"Practice with Expected Value","body":"Use the expected table provided in the question below.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Mean or Expected Value and Standard Deviation","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a298b40expectedvalue17a","stepAnswer":["$$5.4$$"],"problemType":"TextBox","stepTitle":"Find the expected value from the expected value table.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5.4$$","hints":{"DefaultPathway":[{"id":"a298b40expectedvalue17a-h1","type":"hint","dependencies":[],"title":"How to Calculate Expected Value","text":"Expected value, or the long-term average or mean can be calculated as the overall summation of each individual possible value of the random variable multiplied by the probability in the sample space of that outcome. Therefore, if we have a random variable X, the expected value of X can be calculated as the summation of all possible outcomes $$x$$ by the probability of $$x$$, $$x P\\\\left(x\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue17a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5.4$$"],"dependencies":["a298b40expectedvalue17a-h1"],"title":"Expected Value as Summation","text":"What is the expected value, or the summation of all the $$x P\\\\left(x\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a298b40expectedvalue17a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5.4$$"],"dependencies":[],"title":"Expected Value as Summation","text":"What is $$0.2+1.2+2.4+1.6$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a298b40expectedvalue18","title":"Practice with Standard Deviation","body":"Use the table provided in the following question.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Mean or Expected Value and Standard Deviation","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a298b40expectedvalue18a","stepAnswer":["$$1.8$$"],"problemType":"TextBox","stepTitle":"Calculate the standard deviation.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1.8$$","hints":{"DefaultPathway":[{"id":"a298b40expectedvalue18a-h1","type":"hint","dependencies":[],"title":"How to Calculate Standard Deviation","text":"The standard deviation of a PDF (probability density function) is the square root of variance. The value for variance is in the fourth column of the provided table since variance is the summation of all those rows, so we need to calculate standard deviation from that.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.8$$"],"dependencies":["a298b40expectedvalue18a-h1"],"title":"Standard Deviation as the Square Root of Variance","text":"What is the standard deviation given that standard deviation is the square root of variance and variance is the summation of the values in the fourth column?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a298b40expectedvalue18a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3.24$$"],"dependencies":[],"title":"Finding Variance","text":"What is the variance? What is the summation of all the values in the fourth column?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue18a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.8$$"],"dependencies":[],"title":"Finding Standard Deviation","text":"Based on the variance found, standard deviation is the square root of variance. Therefore, what is the square root of the variance you found?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a298b40expectedvalue19","title":"Real World Applications of Probability","body":"A physics professor wants to know what percent of physics majors will spend the next several years doing post-graduate research. He has the following probability distribution.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Mean or Expected Value and Standard Deviation","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a298b40expectedvalue19a","stepAnswer":["$$0.15$$"],"problemType":"TextBox","stepTitle":"Find the probability that a physics major will do post-graduate research for four years, $$P(x=4)$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.15$$","hints":{"DefaultPathway":[{"id":"a298b40expectedvalue19a-h1","type":"hint","dependencies":[],"title":"Total Probability","text":"By the definition of probability, the summation of the probabilities of all possible outcomes in a sample space must add up to $$1$$. We know that the possible outcomes here are $$1$$, $$2$$, $$3$$, $$4$$, $$5$$, and $$6$$. Therefore, their probabilities, P(1), P(2), P(3), P(4), P(5), and P(6) must add up to one. We can use subtraction from $$1$$ to calculate $$P(x=4)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.15$$"],"dependencies":["a298b40expectedvalue19a-h1"],"title":"Determining $$P(x=4)$$","text":"What is P(x=4)=1-[P(x=1)+P(x=2)+P(x=3)+P(x=5)+P(x=6)]?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a298b40expectedvalue2","title":"Practice with Expected Value","body":"Let the random variable X $$=$$ the number of times a newborn baby\'s crying wakes its mother after midnight. Below, we\'ve listed $$x$$, the possible outcomes for X, P(x), $$x P\\\\left(x\\\\right)$$, and (x-\\\\mu)**2*P(x).\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Mean or Expected Value and Standard Deviation","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a298b40expectedvalue2a","stepAnswer":["$$2.1$$"],"problemType":"TextBox","stepTitle":"Find the expected value of the number of times a newborn baby\'s crying wakes its mother after midnight. This expected value represents the expected number of times per week a newborn baby\'s crying wakes its mother after midnight.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2.1$$","hints":{"DefaultPathway":[{"id":"a298b40expectedvalue2a-h1","type":"hint","dependencies":[],"title":"How to Calculate Expected Value","text":"Expected value, or the long-term average or mean can be calculated as the overall summation of each individual possible value of the random variable multiplied by the probability in the sample space of that outcome. Therefore, if we have a random variable X, the expected value of X can be calculated as the summation of all possible outcomes $$x$$ by the probability of $$x$$, $$x P\\\\left(x\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.1$$"],"dependencies":["a298b40expectedvalue2a-h1"],"title":"Expected Value as Summation","text":"What is the expected value, or the summation of all the rows of $$x P\\\\left(x\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a298b40expectedvalue2a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.1$$"],"dependencies":[],"title":"Expected Value as Summation","text":"What is $$0+\\\\frac{11}{50}+\\\\frac{46}{50}+\\\\frac{27}{50}+\\\\frac{16}{50}+\\\\frac{5}{50}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a298b40expectedvalue20","title":"Real World Applications of Probability","body":"A physics professor wants to know what percent of physics majors will spend the next several years doing post-graduate research. He has the following probability distribution.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Mean or Expected Value and Standard Deviation","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a298b40expectedvalue20a","stepAnswer":["$$2.6$$"],"problemType":"TextBox","stepTitle":"On average, how many years would you expect a physics major to spend doing post graduate research?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2.6$$","hints":{"DefaultPathway":[{"id":"a298b40expectedvalue20a-h1","type":"hint","dependencies":[],"title":"How to Calculate Expected Value","text":"Expected value, or the long-term average or mean can be calculated as the overall summation of each individual possible value of the random variable multiplied by the probability in the sample space of that outcome. 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What does $$P\\\\left(red\\\\right)+P\\\\left(blue\\\\right)$$ sum up to? Is this applicable to all probability distributions?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a298b40expectedvalue22","title":"Expected Value with Real World Applications","body":"You are playing a game by drawing a card from a standard deck and replacing it. If the card is a face card, you win $30. If it is not a face card, you pay $2. There are $$12$$ face cards in a deck of $$52$$ cards.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Mean or Expected Value and Standard Deviation","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a298b40expectedvalue22a","stepAnswer":["$$5.38$$"],"problemType":"TextBox","stepTitle":"What is the expected value of playing the game?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5.38$$","hints":{"DefaultPathway":[{"id":"a298b40expectedvalue22a-h1","type":"hint","dependencies":[],"title":"How to Calculate Expected Value","text":"Expected value, or the long-term average or mean can be calculated as the overall summation of each individual possible value of the random variable multiplied by the probability in the sample space of that outcome. 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The probability distribution for DVD rentals per customer at Video To Go is given in the following table. There is a five-video limit per customer at this store, so nobody ever rents more than five DVDs.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Mean or Expected Value and Standard Deviation","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a298b40expectedvalue24a","stepAnswer":["$$0.12$$"],"problemType":"TextBox","stepTitle":"Find the probability that a customer rents three DVDs.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.12$$","hints":{"DefaultPathway":[{"id":"a298b40expectedvalue24a-h1","type":"hint","dependencies":[],"title":"Total Probability","text":"By the definition of probability, the summation of the probabilities of all possible outcomes in a sample space must add up to $$1$$. 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We can use subtraction from $$1$$ to calculate $$P(x=3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue24a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.12$$"],"dependencies":["a298b40expectedvalue24a-h1"],"title":"Determining $$P(x=4)$$","text":"What is P(x=3)=1-[P(x=0)+P(x=1)+P(x=2)+P(x=4)+P(x=5)]?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a298b40expectedvalue24b","stepAnswer":["$$0.11$$"],"problemType":"TextBox","stepTitle":"Find the probability a customer rents at least four DVDs.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.11$$","hints":{"DefaultPathway":[{"id":"a298b40expectedvalue24b-h3","type":"hint","dependencies":["a298b40expectedvalue24a-h2"],"title":"Definition of AT LEAST","text":"When we want to determine the probability that a customer rents at least four DVDs, we know that we need to calculate the probability of getting four OR MORE DVDs. 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What are the possible outcomes?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$0$$, $$5$$, $$25$$, $$100$$","$$5$$, $$25$$, $$100$$","$$0$$, $$5$$","$$0$$, $$5$$, $$10$$, $$25$$, $$100$$"]},{"id":"a298b40expectedvalue25a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.969$$"],"dependencies":["a298b40expectedvalue25a-h2"],"title":"Determining Probability of $0","text":"Determine the probability of earning $0 in profit. This means that you do not earn any money at all.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a298b40expectedvalue25a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.969$$"],"dependencies":[],"title":"Determining Probability of $0","text":"We know that there are $$250$$ people who get $5, $$50$$ who get $25, and $$10$$ that get $100. That means there are $$10, 000-250-25-10=969$$ left to get $0. What is the probability of getting $0?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a298b40expectedvalue25a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.025$$"],"dependencies":["a298b40expectedvalue25a-h3"],"title":"Determining Probability of $5","text":"Determine the probability of earning $5 in profit.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue25a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.005$$"],"dependencies":["a298b40expectedvalue25a-h4"],"title":"Determining Probability of $10","text":"Determine the probability of earning $10 in profit.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue25a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.001$$"],"dependencies":["a298b40expectedvalue25a-h5"],"title":"Determining Probability of $100","text":"Determine the probability of earning $100 in profit.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue25a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a298b40expectedvalue25a-h6"],"title":"$$x P\\\\left(x\\\\right)$$ for $$x=0$$","text":"Determine $$x P\\\\left(x\\\\right)$$ for $$x=\\\\$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a298b40expectedvalue25a-h7-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":[],"title":"$$x P\\\\left(x\\\\right)$$ for $$x=0$$","text":"We know that $$x=0$$ and $$P(x)=0.969$$. What is $$x P\\\\left(x\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a298b40expectedvalue25a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.125$$"],"dependencies":["a298b40expectedvalue25a-h7"],"title":"$$x P\\\\left(x\\\\right)$$ for $$x=5$$","text":"Determine $$x P\\\\left(x\\\\right)$$ for $$x=-\\\\$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue25a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.125$$"],"dependencies":["a298b40expectedvalue25a-h8"],"title":"$$x P\\\\left(x\\\\right)$$ for $$x=25$$","text":"Determine $$x P\\\\left(x\\\\right)$$ for $$x=-\\\\$25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue25a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1$$"],"dependencies":["a298b40expectedvalue25a-h9"],"title":"$$x P\\\\left(x\\\\right)$$ for $$x=100$$","text":"Determine $$x P\\\\left(x\\\\right)$$ for $$x=-\\\\$100$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue25a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.35$$"],"dependencies":["a298b40expectedvalue25a-h10"],"title":"Summation of $$x P\\\\left(x\\\\right)$$","text":"To find the expected value, sum up the answers for $$x P\\\\left(x\\\\right)$$ for $$x=\\\\$0$$, $$x=\\\\$5$$, $$x=\\\\$25$$, and $$x=\\\\$100$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a298b40expectedvalue3","title":"Practice with Standard Deviation","body":"Let the random variable X $$=$$ the number of times a newborn baby\'s crying wakes its mother after midnight. Below, we\'ve listed $$x$$, the possible outcomes for X, P(x), $$x P\\\\left(x\\\\right)$$, and (x-\\\\mu)**2*P(x).\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Mean or Expected Value and Standard Deviation","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a298b40expectedvalue3a","stepAnswer":["$$1.025$$"],"problemType":"TextBox","stepTitle":"Calculate the standard deviation of X, rounded to the nearest thousandth.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1.025$$","hints":{"DefaultPathway":[{"id":"a298b40expectedvalue3a-h1","type":"hint","dependencies":[],"title":"How to Calculate Standard Deviation","text":"The standard deviation of a PDF (probability density function) is the square root of variance. The value for variance is in the fourth column of the provided table since variance is the summation of all those rows, so we need to calculate standard deviation from that.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.025$$"],"dependencies":["a298b40expectedvalue3a-h1"],"title":"Standard Deviation as the Square Root of Variance","text":"What is the standard deviation given that standard deviation is the square root of variance and variance is the summation of the values in the fourth column? Round this value to the nearest thousandth.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a298b40expectedvalue3a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.05$$"],"dependencies":[],"title":"Finding Variance","text":"What is the variance? What is the summation of all the values in the fourth column? Round to the nearest hundredth.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue3a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.025$$"],"dependencies":[],"title":"Finding Standard Deviation","text":"Based on the variance found, standard deviation is the square root of variance. Therefore, what is the square root of the variance you found, rounded to the nearest thousandth?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a298b40expectedvalue4","title":"Practice with Expected Value","body":"A hospital researcher is interested in the number of times the average post-op patient will ring the nurse during a 12-hour shift. For a random sample of $$50$$ patients, the following information was obtained.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Mean or Expected Value and Standard Deviation","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a298b40expectedvalue4a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"What is the expected value?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a298b40expectedvalue4a-h1","type":"hint","dependencies":[],"title":"How to Calculate Expected Value","text":"Expected value, or the long-term average or mean can be calculated as the overall summation of each individual possible value of the random variable multiplied by the probability in the sample space of that outcome. Therefore, if we have a random variable X, the expected value of X can be calculated as the summation of all possible outcomes $$x$$ by the probability of $$x$$, $$x P\\\\left(x\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a298b40expectedvalue4a-h1"],"title":"Expected Value as Summation","text":"What is the expected value, or the summation of all the rows of $$x P\\\\left(x\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a298b40expectedvalue4a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Expected Value as Summation","text":"What is $$\\\\frac{4}{50}+\\\\frac{8}{50}+\\\\frac{16}{50}+\\\\frac{14}{50}+\\\\frac{6}{50}+\\\\frac{2}{50}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a298b40expectedvalue5","title":"Practice with Expected Value","body":"Suppose you play a game of chance in which five numbers are chosen from $$0$$, $$1$$, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$, $$9$$. A computer randomly selects five numbers from zero to nine with replacement. You pay $2 to play and could profit $100,000 if you match all five numbers in order (you get your $2 back plus $100,000). The expected value table below is provided. We can calculate the probabilities by calculating (1)((10)**(-5)=0.00001 for profit and $$1-0.00001=0.99999$$ for loss.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Mean or Expected Value and Standard Deviation","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a298b40expectedvalue5a","stepAnswer":["$$-1$$"],"problemType":"TextBox","stepTitle":"What is the expected profit of playing the game, rounded to the nearest dollar?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1$$","hints":{"DefaultPathway":[{"id":"a298b40expectedvalue5a-h1","type":"hint","dependencies":[],"title":"How to Calculate Expected Value","text":"Expected value, or the long-term average or mean can be calculated as the overall summation of each individual possible value of the random variable multiplied by the probability in the sample space of that outcome. Therefore, if we have a random variable X, the expected value of X can be calculated as the summation of all possible outcomes $$x$$ by the probability of $$x$$, $$x P\\\\left(x\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a298b40expectedvalue5a-h1"],"title":"Expected Value as Summation","text":"What is the expected value, or the summation of all the rows of $$x P\\\\left(x\\\\right)$$, rounded to the nearest ones place (nearest dollar)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a298b40expectedvalue5a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":[],"title":"Expected Value as Summation","text":"What is $$-1.99998+1$$, rounded to the nearest dollar?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a298b40expectedvalue6","title":"Expected Value without Tables","body":"You are playing a game of chance in whcih four cards are drawn from a standard deck of $$52$$ cards. You guess the suit of each card before it is drawn. The cards are replaced i the deck on each draw. You pay $1 to play. If you guess the right suit every time, you get your money back and $256.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Mean or Expected Value and Standard Deviation","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a298b40expectedvalue6a","stepAnswer":["$$\\\\frac{1}{256}$$"],"problemType":"TextBox","stepTitle":"What is your expected profit of playing the game over the long term? Keep it in fractional form.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{256}$$","hints":{"DefaultPathway":[{"id":"a298b40expectedvalue6a-h1","type":"hint","dependencies":[],"title":"How to Calculate Expected Value","text":"Expected value, or the long-term average or mean can be calculated as the overall summation of each individual possible value of the random variable multiplied by the probability in the sample space of that outcome. Therefore, if we have a random variable X, the expected value of X can be calculated as the summation of all possible outcomes $$x$$ by the probability of $$x$$, $$x P\\\\left(x\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue6a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-1$$, $$256$$"],"dependencies":["a298b40expectedvalue6a-h1"],"title":"Determining Possible Outcomes","text":"First, we need to determine the possible outcomes (how much profit) you can earn from the game. What are the possible outcomes?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$-1$$, $$256$$","$$0$$, $$256$$","$$0$$, $$255$$","$$-1$$, $$255$$"]},{"id":"a298b40expectedvalue6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{256}$$"],"dependencies":["a298b40expectedvalue6a-h2"],"title":"Determining Probability of $256","text":"Determine the probability of earning $256 in profit. This means that for all four cards, you guess their suit correctly.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a298b40expectedvalue6a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{256}$$"],"dependencies":[],"title":"Determining Probability of $256","text":"The probability of guessing the suit correctly is $$\\\\frac{1}{4}$$ since there are $$4$$ suits, each with an even number of cards. Then, we need to guess correctly $$4$$ times. Therefore, we have to take $$\\\\frac{1}{4}$$ to the power of $$4$$. What is $$\\\\frac{1}{4}$$ to the power of 4?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a298b40expectedvalue6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{255}{256}$$"],"dependencies":["a298b40expectedvalue6a-h3"],"title":"Determining Probability of -$1","text":"Determine the probability of earning -$1 in profit. This means that you did NOT get all four suits correct. Hint: this is the complement rule applied to the previous step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a298b40expectedvalue6a-h4"],"title":"$$x P\\\\left(x\\\\right)$$ for $$x=256$$","text":"Determine $$x P\\\\left(x\\\\right)$$ for $$x=\\\\$256$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue6a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-255}{256}$$"],"dependencies":["a298b40expectedvalue6a-h5"],"title":"$$x P\\\\left(x\\\\right)$$ for $$x=-1$$","text":"Determine $$x P\\\\left(x\\\\right)$$ for $$x=-\\\\$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue6a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{256}$$"],"dependencies":["a298b40expectedvalue6a-h6"],"title":"Summation of $$x P\\\\left(x\\\\right)$$","text":"To find the expected value, sum up the answers for $$x P\\\\left(x\\\\right)$$ for $$x=\\\\$256$$ and $$x=-\\\\$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a298b40expectedvalue7","title":"Practice with Random Variables","body":"Suppose you play a game with a biased coin. You play each game by tossing the coin once. $$P(heads)=\\\\frac{2}{3}$$ and $$P(tails)=\\\\frac{1}{3}$$. If you toss a head, you pay $6. If you toss a tail, you win $10. We want to determine whether we\'ll come out ahead (gain a profit).","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Mean or Expected Value and Standard Deviation","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a298b40expectedvalue7a","stepAnswer":["$$X=amount$$ of profit"],"problemType":"MultipleChoice","stepTitle":"Define a random variable X.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$X=amount$$ of profit","choices":["$$X=amount$$ of profit","$$X=number$$ of heads","$$X=if$$ you win","$$X=if$$ you lose"],"hints":{"DefaultPathway":[{"id":"a298b40expectedvalue7a-h1","type":"hint","dependencies":[],"title":"Defining Discrete Random Variables","text":"Remember that when we define discrete random variables, we want variables to be countable (5 marbles, $$2$$ heads, $5).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue7a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$X=amount$$ of profit OR $$X=number$$ of heads"],"dependencies":["a298b40expectedvalue7a-h1"],"title":"Determining Countable Random Variables","text":"Which pair of random variables listed is countable?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$X=amount$$ of profit OR $$X=number$$ of heads","$$X=if$$ you win OR $$X=if$$ you lose"]},{"id":"a298b40expectedvalue7a-h3","type":"hint","dependencies":["a298b40expectedvalue7a-h2"],"title":"Determining Aligned Random Variables","text":"Now, we have limited down to either the amount of profit or the number of heads, both of which are applicable to our situation. However, note that we want to determine if we\'ll come out ahead (gain profit).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue7a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$X=amount$$ of profit"],"dependencies":["a298b40expectedvalue7a-h3"],"title":"Determining Aligned Random Variables","text":"Which random variable seems more aligned with what we want to solve for?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$X=amount$$ of profit","$$X=number$$ of heads"]}]}}]},{"id":"a298b40expectedvalue8","title":"Practice with Expected Value Tables","body":"Suppose you play a game with a biased coin. You play each game by tossing the coin once. $$P(heads)=\\\\frac{2}{3}$$ and $$P(tails)=\\\\frac{1}{3}$$. If you toss a head, you pay $6. If you toss a tail, you win $10. We want to determine whether we\'ll come out ahead (gain a profit). Let\'s complete the expected value table provided.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Mean or Expected Value and Standard Deviation","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a298b40expectedvalue8a","stepAnswer":["$$-6$$"],"problemType":"TextBox","stepTitle":"What is the value for $$x$$ for lose?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-6$$","hints":{"DefaultPathway":[{"id":"a298b40expectedvalue8a-h1","type":"hint","dependencies":[],"title":"Look At the Table","text":"Notice that the table has $$10$$ listed for the $$x$$ value for win. This is due to the fact that if you toss a tail, you win $10. Therefore, lose must align with what happens if you toss a head.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6$$"],"dependencies":["a298b40expectedvalue8a-h1"],"title":"Determining Lose\'s $$x$$","text":"What is the $$x$$ value for lose? Reminder that lose is associated with a negative number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a298b40expectedvalue8b","stepAnswer":["$$\\\\frac{2}{3}$$"],"problemType":"TextBox","stepTitle":"What is the P(x) value for lose?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2}{3}$$","hints":{"DefaultPathway":[{"id":"a298b40expectedvalue8b-h3","type":"hint","dependencies":["a298b40expectedvalue8a-h2"],"title":"Definition of Total Probability","text":"Since P(x) represents the probability and the total probability must sum up to one, we can add up P(x) for all values of $$x$$ (-6 and 10) to get $$1$$. Since we already know $$P(10)=\\\\frac{1}{3}$$, we know that $$P(Lose)=P(-6)=1-P(10)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue8b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{3}$$"],"dependencies":["a298b40expectedvalue8b-h3"],"title":"Determining Lose\'s P(x)","text":"What is the value for P(x) for lose?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a298b40expectedvalue8c","stepAnswer":["$$\\\\frac{-12}{3}$$"],"problemType":"TextBox","stepTitle":"What is the value for $$x P\\\\left(x\\\\right)$$ for lose?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-12}{3}$$","hints":{"DefaultPathway":[{"id":"a298b40expectedvalue8c-h5","type":"hint","dependencies":["a298b40expectedvalue8b-h4"],"title":"Definition of $$x P\\\\left(x\\\\right)$$","text":"Now, we want to find $$x P\\\\left(x\\\\right)$$. In the first step of this question we found $$x$$ for lose and in the second step we found P(x) for lose. Now, let\'s multiply the two answers together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue8c-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-12}{3}$$"],"dependencies":["a298b40expectedvalue8c-h5"],"title":"Determining Lose\'s $$x P\\\\left(x\\\\right)$$","text":"What is the value for $$x P\\\\left(x\\\\right)$$ for lose?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a298b40expectedvalue9","title":"Practice with Expected Value","body":"Suppose you play a game with a biased coin. You play each game by tossing the coin once. $$P(heads)=\\\\frac{2}{3}$$ and $$P(tails)=\\\\frac{1}{3}$$. If you toss a head, you pay $6. If you toss a tail, you win $10. We want to determine whether we\'ll come out ahead (gain a profit).\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Mean or Expected Value and Standard Deviation","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a298b40expectedvalue9a","stepAnswer":["$$\\\\frac{-2}{3}$$"],"problemType":"TextBox","stepTitle":"What is the expected value?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-2}{3}$$","hints":{"DefaultPathway":[{"id":"a298b40expectedvalue9a-h1","type":"hint","dependencies":[],"title":"How to Calculate Expected Value","text":"Expected value, or the long-term average or mean can be calculated as the overall summation of each individual possible value of the random variable multiplied by the probability in the sample space of that outcome. Therefore, if we have a random variable X, the expected value of X can be calculated as the summation of all possible outcomes $$x$$ by the probability of $$x$$, $$x P\\\\left(x\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a298b40expectedvalue9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-2}{3}$$"],"dependencies":["a298b40expectedvalue9a-h1"],"title":"Expected Value as Summation","text":"What is the expected value, or the summation of all the rows of $$x P\\\\left(x\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a298b40expectedvalue9a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-2}{3}$$"],"dependencies":[],"title":"Expected Value as Summation","text":"What is $$\\\\frac{10}{3}+\\\\left(-\\\\frac{12}{3}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a29c32b4.4darts1","title":"","body":"You throw darts at a board until you hit the center area. Your probability of hitting the center area is $$p$$ $$=$$ $$0.17$$. You want to find the probability that it takes eight throws until you hit the center.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Geometric Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a29c32b4.4darts1a","stepAnswer":["1,2,3,...inf"],"problemType":"MultipleChoice","stepTitle":"What values does random variable X take on?","stepBody":"","answerType":"string","variabilization":{},"choices":["1,2,3,4,5,6,7,8","1,2,3,...inf","$$8$$"],"hints":{"DefaultPathway":[{"id":"a29c32b4.4darts1a-h1","type":"hint","dependencies":[],"title":"Interpretation","text":"The probability that you hit the center after $$8$$ throws can be shown as $$P(X=8)$$, what are the other possible values that can take place of 8? Can you find the limits of X?","variabilization":{},"oer":"","license":""},{"id":"a29c32b4.4darts1a-h2","type":"hint","dependencies":["a29c32b4.4darts1a-h1"],"title":"Interpretation","text":"As long that the probability of hitting the center each throw is higher than $$0$$, then you can just ignore the probability value of $$0.17$$ and focus on the possible values of X.","variabilization":{},"oer":"","license":""},{"id":"a29c32b4.4darts1a-h3","type":"hint","dependencies":["a29c32b4.4darts1a-h2"],"title":"Random Variable of a Geometric Distribution","text":"Theoretically, the number of trials in a geometric experiment can be infinite.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a29c32b4.4defective1","title":"Probability","body":"Assume that the probability of a defective computer component is $$0.02$$. Components are randomly selected.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Geometric Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a29c32b4.4defective1a","stepAnswer":["$$0.0177$$"],"problemType":"MultipleChoice","stepTitle":"Find the probability that the first defect is caused by the seventh component tested.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$0.0177$$","choices":["$$0.0177$$","$$0.0200$$","$$0.2000$$","$$0.1400$$"],"hints":{"DefaultPathway":[{"id":"a29c32b4.4defective1a-h1","type":"hint","dependencies":[],"title":"Geometric Distribution","text":"To find the probability, you can use a geometric PDF distribution function on a graphing calculator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a29c32b4.4defective1a-h2","type":"hint","dependencies":["a29c32b4.4defective1a-h1"],"title":"Geometric Distribution","text":"When using the calculator, set the $$p$$ to $$0.02$$, and the X value to $$7$$, since we want to find the probability that the FIRST defect is caused by the seventh component tested.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a29c32b4.4defective1b","stepAnswer":["$$0.1319$$"],"problemType":"MultipleChoice","stepTitle":"What is the probability that the first defect found is found in fewer than $$7$$ consecutive trials? (round to $$4$$ decimal points)","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$0.1319$$","choices":["$$0.1319$$","$$-0.8681$$","$$0.1500$$","$$0.0177$$"],"hints":{"DefaultPathway":[{"id":"a29c32b4.4defective1b-h1","type":"hint","dependencies":[],"title":"Interpretation","text":"This probability problem asks to find the probability that the first component is less than or equal to the seventh component tested.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a29c32b4.4defective1b-h2","type":"hint","dependencies":["a29c32b4.4defective1b-h1"],"title":"Probability","text":"You are trying to find the sum of the probabilities from one to seven components. You can either manually use Geometric PDF function for X equal 1,2,3,4,5,6, and $$7$$, then add the $$7$$ values calculated, or the more convenient way would be using Geometric CDF and setting X to $$7$$, because doing so does the same sum of other former tedious option described.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a29c32b4.4defective1b-h3","type":"hint","dependencies":["a29c32b4.4defective1b-h2"],"title":"Visualizing Probability Distributions","text":"The graph of X ~ $$G(0.02)$$ is shown. You must find the sum of $$X=0, 1, 2, 3, ..., 7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a29c32b4.4defective1b-h4","type":"hint","dependencies":["a29c32b4.4defective1b-h3"],"title":"","text":"Geometric PDF of $$X=7$$ gets only the value of the $$X=7$$ bar, whereas Geometric CDF of $$X=7$$ gets ALL the values of $$X=7$$ and below. You want to find all the values of $$X=7$$ and below.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a29c32b4.4game1","title":"Proability","body":"Suppose a game has two outcomes, win or lose. You repeatedly play that game until you lose. The\\\\nprobability of losing is $$p$$ $$=$$ $$0.57$$.\\\\nIf we let X $$=$$ the number of games you play until you lose (includes the losing game), then X is a geometric random variable. All three characteristics are met. Each game you play is a Bernoulli trial, either win or lose.\\\\nYou would need to play at least one game before you stop. X takes on the values $$1$$, $$2$$, $$3$$, . (could go on\\\\nindefinitely). Since we are measuring the number of games you play until you lose, we define a success as\\\\nlosing a game and a failure as winning a game. The probability of a success $$p$$ $$=$$ $$.57$$ and the probability of a failure q $$=$$ $$1$$ - $$p$$ $$=$$ $$1$$ - $$0.57$$ $$=$$ $$0.43$$. Both $$p$$ and q remain the same from game to game.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Geometric Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a29c32b4.4game1a","stepAnswer":["$$P(X=5)$$"],"problemType":"MultipleChoice","stepTitle":"If we want to find the probability that it takes five games until you lose, which of the following choices would be represented as the probability?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$P(X=5)$$","choices":["$$P(X=5)$$","P(X)","$$X=5$$","P(5X)"],"hints":{"DefaultPathway":[{"id":"a29c32b4.4game1a-h1","type":"hint","dependencies":[],"title":"Probability","text":"P(X) means the probability of X. In this case, X is the number of games you play until you lose. In order to specify the probability on a certain number of games that it takes until you lose, enter $$P(X=a)$$, where a is the number of games that you play until you lose.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a29c32b4.4instructor1","title":"Probability","body":"An instructor feels that 15% of students get below a C on their final exam. She decides to look at final\\\\nexams (selected randomly and replaced in the pile after reading) until she finds one that shows a grade\\\\nbelow a C. We want to know the probability that the instructor will have to examine at least ten exams\\\\nuntil she finds one with a grade below a C.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Geometric Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a29c32b4.4instructor1a","stepAnswer":["$$P(X \\\\leq 10)$$"],"problemType":"MultipleChoice","stepTitle":"Probability","stepBody":"What is the probability question stated mathematically?","answerType":"string","variabilization":{},"answerLatex":"$$P(X \\\\leq 10)$$","choices":["$$P(X \\\\leq 10)$$","$$P(X \\\\geq 15)$$","$$P(X \\\\leq 150)$$","$$P(X \\\\leq 100)$$"],"hints":{"DefaultPathway":[{"id":"a29c32b4.4instructor1a-h1","type":"hint","dependencies":[],"title":"Random Variable of a Geometric Distribution","text":"X should be the random variable value of the specific probability that we want in the probability statement.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a29c32b4.4instructor1a-h2","type":"hint","dependencies":["a29c32b4.4instructor1a-h1"],"title":"Random Variable of a Geometric Distribution","text":"Since we are concerned with the number of papers that are be examined until the teacher finds a C grade, the number of papers that need to be examined until the teacher finds a C grade is the random variable X.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a29c32b4.4Jeremiah1","title":"Random Variables","body":"Jeremiah has basketball practice two days a week. Ninety percent of the time, he attends both practices. Eight percent of the time, he attends one practice. Two percent of the time, he does not attend either practice.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Geometric Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a29c32b4.4Jeremiah1a","stepAnswer":["Days that Jeremiah practices basketball in a week"],"problemType":"MultipleChoice","stepTitle":"What is the random variable X?","stepBody":"","answerType":"string","variabilization":{},"choices":["Days that Jeremiah practices basketball in a week","Days that Jeremiah practices basketball in a year","Jeremiah\'s teammates","Average number of basketball players that practice two times a day"],"hints":{"DefaultPathway":[{"id":"a29c32b4.4Jeremiah1a-h1","type":"hint","dependencies":[],"title":"Random Variables","text":"The given probabilities are different from one another, but each has their own similar characteristic.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a29c32b4.4Jeremiah1a-h2","type":"hint","dependencies":["a29c32b4.4Jeremiah1a-h1"],"title":"Random Variables","text":"Notice that each of the probabilities are concerning the number of days that Jeremiah practices.basketball in a week.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a29c32b4.4miata1","title":"","body":"A consumer looking to buy a used red Miata car will call dealerships until she finds a dealership that carries the car. She estimates the probability that any independent dealership will have the car will be 28%. We are interested in the number of dealerships she must call.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Geometric Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a29c32b4.4miata1a","stepAnswer":["Number of dealerships that need to be called until one contains a used red Miata car"],"problemType":"MultipleChoice","stepTitle":"In words, define the random variable X","stepBody":"","answerType":"string","variabilization":{},"choices":["Number of dealerships that need to be called until one contains a used red Miata car","Number of dealerships that need to be called until one contains three or more used red Miata cars","Number of dealerships that contains a used red Miata car","Number of consumers with red cars"],"hints":{"DefaultPathway":[{"id":"a29c32b4.4miata1a-h1","type":"hint","dependencies":[],"title":"Random Variable of a Geometric Distribution","text":"There is a chance that the first dealership consulted doesn\'t contain a red Miata car.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a29c32b4.4miata1b","stepAnswer":["$$1$$ to $$\\\\infty$$"],"problemType":"MultipleChoice","stepTitle":"List the values that X may take on","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$1$$ to $$\\\\infty$$","choices":["$$1$$ to $$\\\\infty$$","$$0$$ to $$\\\\infty$$","$$1$$ to $$28$$","$$0$$ to $$28$$"],"hints":{"DefaultPathway":[]}}]},{"id":"a29c32b4.4nearby1","title":"Geometric Distributions","body":"Suppose that you are looking for a student at your college who lives within five miles of you. You know that\\\\n55% of the 25,000 students do live within five miles of you. You randomly contact students from the college\\\\nuntil one says he or she lives within five miles of you. What is the probability that you need to contact four\\\\npeople?\\\\nThis is a geometric problem because you may have a number of failures before you have the one success you desire. Also, the probability of a success stays the same each time you ask a student if he or she lives within\\\\nfive miles of you. There is no definite number of trials (number of times you ask a student).","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Geometric Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a29c32b4.4nearby1a","stepAnswer":["students, until one"],"problemType":"MultipleChoice","stepTitle":"Let the random variable X $$=$$ the number of $$_{}$$ you must ask $$_{}$$ says yes.","stepBody":"","answerType":"string","variabilization":{},"choices":["colleges, until","colleges, when $$twenty-five$$ thousand","students, until","students, until five","students, until one"],"hints":{"DefaultPathway":[{"id":"a29c32b4.4nearby1a-h1","type":"hint","dependencies":[],"title":"Interpretation","text":"We are concerned with the probability of approaaching a student at your college that lives within five miles of you.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a29c32b4.4nearby1b","stepAnswer":["1,2,3,...,n number of students"],"problemType":"MultipleChoice","stepTitle":"What values does X take on?","stepBody":"","answerType":"string","variabilization":{},"choices":["1,2,3,...,n number of students","1,2,3,4,5","0,1,2,3,...,n number of students","0,1,2,3,4,5"],"hints":{"DefaultPathway":[{"id":"a29c32b4.4nearby1b-h1","type":"hint","dependencies":[],"title":"Random Variable of a Geometric Distribution","text":"Since the colleague must be contacted by the student, the number of possible values must be at least $$1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a29c32b4.4nearby1b-h2","type":"hint","dependencies":["a29c32b4.4nearby1b-h1"],"title":"Random Variable of a Geometric Distribution","text":"Remember that each contact is guaranteed to be a student, so there is a limit to the possible values of X","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a29c32b4.4nearby1c","stepAnswer":["$$0.55$$"],"problemType":"TextBox","stepTitle":"What is p?(p is the probability of success, while q is the complementary probability of failure)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.55$$","hints":{"DefaultPathway":[{"id":"a29c32b4.4nearby1c-h1","type":"hint","dependencies":[],"title":"Probability","text":"The probability that 25,000 students do live within $$5$$ miles of you should be the $$p$$ value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a29c32b4.4nearby1c-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["55%"],"dependencies":["a29c32b4.4nearby1c-h1"],"title":"What is p?(p is the probability of success, while q is the complementary probability of failure)","text":"What is the probability value discussed in the previous hint?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["55%","65%","45%","20%"]},{"id":"a29c32b4.4nearby1c-h3","type":"hint","dependencies":["a29c32b4.4nearby1c-h2"],"title":"Interpretation","text":"If we know that 55% of 25,000 students do live within $$5$$ miles of you, then that is the probability value discussed in the last hint as it is follows the same description.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a29c32b4.4nearby1c-h4","type":"hint","dependencies":["a29c32b4.4nearby1c-h3"],"title":"Interpretation","text":"The probability is 55%","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a29c32b4.4nearby1c-h5","type":"hint","dependencies":["a29c32b4.4nearby1c-h4"],"title":"Conversion","text":"To convert a percentage back to numeric form, divide by $$100$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a29c32b4.4nearby1d","stepAnswer":["$$P(X=4)$$"],"problemType":"MultipleChoice","stepTitle":"Interpretation","stepBody":"The probability question is $$P\\\\left(_{}\\\\right)$$.","answerType":"string","variabilization":{},"answerLatex":"$$P(X=4)$$","choices":["$$P(X=4)$$","$$P(X=55)$$","$$P(X=25000)$$","$$P(X=40)$$"],"hints":{"DefaultPathway":[{"id":"a29c32b4.4nearby1d-h1","type":"hint","dependencies":[],"title":"Random Variable of a Geometric Distribution","text":"The random variable X is the number of college students contacted until one is found to be within five miles of you. The probability question that was provided was \\" What is the probability that you need to contact four people?\\", so X must be $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a29c32b4.4pancrea1","title":"Geometric","body":"The lifetime risk of developing pancreatic cancer is about one in $$78$$ $$(1.28\\\\%)$$. Let X $$=$$ the number of people you ask until one says he or she has pancreatic cancer. Then X is a discrete random variable with a geometric distribution: $$X\\\\pm G \\\\frac{1}{78}$$ or $$X\\\\pm 0.0128G$$","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Geometric Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a29c32b4.4pancrea1a","stepAnswer":["$$0.0114$$"],"problemType":"TextBox","stepTitle":"What is the probability of that you ask ten people before one says he or she has pancreatic cancer?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.0114$$","hints":{"DefaultPathway":[{"id":"a29c32b4.4pancrea1a-h1","type":"hint","dependencies":[],"title":"Interpretation","text":"Find $$P(X=10)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a29c32b4.4pancrea1a-h2","type":"hint","dependencies":["a29c32b4.4pancrea1a-h1"],"title":"Geometric distribution notation","text":"Remember that $$X\\\\pm G p$$ can be read as \\"X is a random variable with a geometric distribution.\\" The parameter is p; $$p$$ $$=$$ the probability of a\\\\nsuccess for each trial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a29c32b4.4pancrea1a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$10$$"],"dependencies":["a29c32b4.4pancrea1a-h2"],"title":"Geometric distribution","text":"To find $$P(X=10)$$, you can use a calculator function $$geometpdf(0.0128$$, _) to find the probability.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$0.0128$$","$$10$$","$$78$$","$$\\\\frac{1}{78}$$"]}]}},{"id":"a29c32b4.4pancrea1b","stepAnswer":["$$0.01$$"],"problemType":"TextBox","stepTitle":"What is the probability that you must ask $$20$$ people?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.01$$","hints":{"DefaultPathway":[{"id":"a29c32b4.4pancrea1b-h1","type":"hint","dependencies":[],"title":"Geometric distribution notation","text":"Remember that $$X\\\\pm G p$$ can be read as \\"X is a random variable with a geometric distribution.\\" The parameter is p; $$p$$ $$=$$ the probability of a\\\\nsuccess for each trial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a29c32b4.4pancrea1b-h2","type":"hint","dependencies":["a29c32b4.4pancrea1b-h1"],"title":"Geometric distribution","text":"To find $$P(X=20)$$, you can use a calculator function $$geometpdf(0.0128$$, _) to find the probability.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a29c32b4.4pancrea1b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$20$$"],"dependencies":["a29c32b4.4pancrea1b-h2"],"title":"Geometric distribution","text":"To find $$P(X=10)$$, you can use a calculator function $$geometpdf(0.0128$$, _) to find the probability.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$10$$","$$20$$","$$30$$","$$78$$","$$\\\\frac{1}{78}$$"]}]}},{"id":"a29c32b4.4pancrea1c","stepAnswer":["$$78$$"],"problemType":"TextBox","stepTitle":"Find the mean of X","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$78$$","hints":{"DefaultPathway":[{"id":"a29c32b4.4pancrea1c-h1","type":"hint","dependencies":[],"title":"Geometric distribution","text":"In a geometric distribution, the formula for the mean is $$mean=\\\\frac{1}{p}$$, where $$p$$ is the probability of success for each trial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a29c32b4.4pancrea1c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$78$$"],"dependencies":["a29c32b4.4pancrea1c-h1"],"title":"Geometric distribution","text":"$$\\\\frac{1}{p}=mean;p=0.0128;$$ $$mean=\\\\frac{1}{0.0128}=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a29c32b4.4pancrea1d","stepAnswer":["$$\\\\sqrt{\\\\frac{1-0.0128}{{0.0128}^2}}$$"],"problemType":"TextBox","stepTitle":"What is the standard deviation of X?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\sqrt{\\\\frac{1-0.0128}{{0.0128}^2}}$$","hints":{"DefaultPathway":[{"id":"a29c32b4.4pancrea1d-h1","type":"hint","dependencies":[],"title":"Geometric distribution","text":"The formula for standard deviation in a geometric distribution is $$\\\\sqrt{\\\\frac{1-p}{p^2}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a29c32b4.4pancrea1d-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.0128$$"],"dependencies":["a29c32b4.4pancrea1d-h1"],"title":"Geometric distribution","text":"What is $$p$$ if we were to follow the formula shown in the last step?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a29c32b4.4pancrea1d-h3","type":"hint","dependencies":["a29c32b4.4pancrea1d-h2"],"title":"Geometric distribution","text":"Plug in $$p=0.0128$$ into the given equation to get the standard deviation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a29c32b4.4singer1","title":"Interpretation","body":"Ellen has music practice three days a week. She practices for all of the three days 85% of the time, two days 8% of the time, one day 4% of the time, and no days 3% of the time.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Geometric Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a29c32b4.4singer1a","stepAnswer":["0,1,2,3"],"problemType":"MultipleChoice","stepTitle":"One week is selected at random. What values does random variable X take on?","stepBody":"","answerType":"string","variabilization":{},"choices":["0,1,2,3","1,2,3","0,1,2","1,3"],"hints":{"DefaultPathway":[{"id":"a29c32b4.4singer1a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["0,1,2,3"],"dependencies":[],"title":"Random Variable","text":"Ellen can have practice three, two, one, or even no practice in a week. What are the possible amount of days that she practices?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["0,1,2,3","1,2,3","0,1,2","1,3"]}]}}]},{"id":"a29c32b4.4social1","title":"","body":"According to a recent Pew Research poll, 75% of millenials (people born between $$1981$$ and 1995) have a profile on a social networking site. Let X $$=$$ the number of millenials you ask until you find a person without a profile on a social networking site.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Geometric Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a29c32b4.4social1a","stepAnswer":["$$X\\\\pm 0.25G$$"],"problemType":"MultipleChoice","stepTitle":"Describe the distribution of X.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$X\\\\pm 0.25G$$","choices":["$$X\\\\pm 0.25G$$","$$X\\\\pm 0.75G$$","normal distribution","$$X\\\\pm 0.1875G$$"],"hints":{"DefaultPathway":[{"id":"a29c32b4.4social1a-h1","type":"hint","dependencies":[],"title":"Geometric Distribution Notation","text":"Given $$X\\\\pm G Z$$, Remember that the Z value is the probability of the success of the random variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a29c32b4.4social1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.25$$"],"dependencies":["a29c32b4.4social1a-h1"],"title":"Probability","text":"The probability of success of finding a millenial user that doesn\'t have a social network profile is the random variable X. What is the probability for each trial that you select a random user?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a29c32b4.4social1a-h3","type":"hint","dependencies":["a29c32b4.4social1a-h2"],"title":"Interpretation","text":"There is a 75% chance of selecting a user from the population of millenials from the PEW research study that has a social networking profile, therefore the complementary probability of 25% is the probability of selecting a user from the same population that doesn\'t have a social networking profile online.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a29c32b4.4springs1","title":"Random Variable","body":"In one of its Spring catalogs, L.L. Bean\xae advertised footwear on $$29$$ of its $$192$$ catalog pages. Suppose we randomly survey $$20$$ pages. We are interested in the number of pages that advertise footwear. Each page may be picked more than once.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Geometric Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a29c32b4.4springs1a","stepAnswer":["the number of pages that advertise footwear"],"problemType":"MultipleChoice","stepTitle":"In words, define the random variable X.","stepBody":"","answerType":"string","variabilization":{},"choices":["the number of pages that advertise footwear","the number of footwear companies","the number of Spring catalogue catalog pages","the number of advertisements of companies"],"hints":{"DefaultPathway":[{"id":"a29c32b4.4springs1a-h1","type":"hint","dependencies":[],"title":"Random Variable","text":"The random variable is typically the subject that is stated to be of interest","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a29c32b4.4springs1b","stepAnswer":["$$0$$ to $$20$$"],"problemType":"MultipleChoice","stepTitle":"List the values that X may take on.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$0$$ to $$20$$","choices":["$$0$$ to $$20$$","$$1$$ to $$20$$","$$1$$ to $$10$$","$$0$$ to $$10$$"],"hints":{"DefaultPathway":[{"id":"a29c32b4.4springs1b-h1","type":"hint","dependencies":[],"title":"Random Variable","text":"Our sample consists of $$20$$ pages, so the max random variable value is $$20$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a29c32b4.4springs1b-h2","type":"hint","dependencies":["a29c32b4.4springs1b-h1"],"title":"Random Variable","text":"Of the total $$192$$ catalog pages population, our sample could very well have none, so the lowest random variable value could be $$0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a29c32b4.4springs1c","stepAnswer":["$$3.02$$"],"problemType":"MultipleChoice","stepTitle":"How many pages do you expect to advertise footwear on them?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$3.02$$","choices":["$$3.02$$","$$0.15$$","$$0.13$$","$$2.7$$"],"hints":{"DefaultPathway":[{"id":"a29c32b4.4springs1c-h1","type":"hint","dependencies":[],"title":"Probability","text":"Find the probability that you will get an advertisement of footwear","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a29c32b4.4springs1c-h2","type":"hint","dependencies":["a29c32b4.4springs1c-h1"],"title":"Probability","text":"The probability is $$\\\\frac{29}{192}$$, now you can multiply it by the number of pages in the sample to get the expected number of pages.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a29c32b4.4springs1d","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Is it highly probable that all twenty will advertise footwear on them?","stepBody":"","answerType":"string","variabilization":{},"choices":["No","Yes"],"hints":{"DefaultPathway":[{"id":"a29c32b4.4springs1d-h1","type":"hint","dependencies":[],"title":"Probability","text":"The probability of getiting a footwear advertisement is $$\\\\frac{29}{192}$$, which is around 15%. Let\'s say that highly probable is higher than 50%.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a29c32b4.4springs1d-h2","type":"hint","dependencies":["a29c32b4.4springs1d-h1"],"title":"Probability Distribution Function","text":"We can use Geometric PDF function on the calculator to find the probability that there are $$20$$ foot wear advertisements in our sample. Set $$p$$ to the probability that the advertisement is a footwear advertisement, and X to our desired random variable value, $$20$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a29c32b4.4superbowl1","title":"Random Variable","body":"Suppose that the probability that an adult in America will watch the Super Bowl is 40%. Each person is\\\\nconsidered independent. We are interested in the number of adults in America we must survey until we find one\\\\nwho will watch the Super Bowl.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Geometric Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a29c32b4.4superbowl1a","stepAnswer":["the number of adults in America who are surveyed until one says he or she will watch the Super Bowl"],"problemType":"MultipleChoice","stepTitle":"Define the random variable X.","stepBody":"","answerType":"string","variabilization":{},"choices":["the number of adults in America who are surveyed until one says he or she will watch the Super Bowl","The number of Patriots fans","the probability that an adult in America will watch the Super Bowl"],"hints":{"DefaultPathway":[{"id":"a29c32b4.4superbowl1a-h1","type":"hint","dependencies":[],"title":"Random Variable","text":"Consider the random variable to be the subject matter of interest in the problem.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a29c32b4.4ucla1","title":"Same Sex Marriage","body":"The Higher Education Research Institute at UCLA collected data from 203,967 incoming first-time, full-time freshmen from $$270$$ four-year colleges and universities in the U.S. $$71.3\\\\%$$ of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly select freshman from the study until you find one who replies \u201cyes.\u201d You are interested in the number of freshmen you must ask.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Geometric Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a29c32b4.4ucla1a","stepAnswer":["the number of freshmen selected from the study until one replied \\"yes\\" that $$same-sex$$ couples should have the right to legal marital status.|"],"problemType":"MultipleChoice","stepTitle":"Random Variable","stepBody":"In words, define the random variable X","answerType":"string","variabilization":{},"choices":["the number of freshmen selected from the study until one replied \\"no\\" that $$same-sex$$ couples should have the right to legal marital status.","the number of freshmen selected from the study until one replied \\"yes\\" that $$same-sex$$ couples should have the right to legal marital status.","the number of freshmen selected from the study until one replied \\"yes\\" that $$same-sex$$ couples should have the right to legal marital status.|","the number of same sex couple freshmen selected from the study","the number of same sex couples"],"hints":{"DefaultPathway":[{"id":"a29c32b4.4ucla1a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Geometric distribution"],"dependencies":[],"title":"Distribution and Experiment type","text":"We are trying to find a participant until one says yes, what type of distribution is this?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Geometric distribution","Binomial distribution","Hypergeometric distribution","Exponential Distribution Curve"]},{"id":"a29c32b4.4ucla1a-h2","type":"hint","dependencies":["a29c32b4.4ucla1a-h1"],"title":"Distribution and Experiment type","text":"Since there isn\'t a fixed number of trials and there are only success and failure options, this is a Geometric Distribution","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a29c32b4.4ucla1a-h3","type":"hint","dependencies":["a29c32b4.4ucla1a-h2"],"title":"Random Variable","text":"Our population parameter metric is based of students that replied \\"yes\\" to the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a29c32b4.4ucla1b","stepAnswer":["$$0.713$$"],"problemType":"TextBox","stepTitle":"Probability Distribution Function","stepBody":"$$P(X=1)$$ $$=$$?","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.713$$","hints":{"DefaultPathway":[{"id":"a29c32b4.4ucla1b-h1","type":"hint","dependencies":[],"title":"Use the probability distribution function formula P(X) $$=$$ $$p {\\\\left(1-p\\\\right)}^{x-1}$$","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2a280bgaussian1","title":"Writing the Augmented Matrix for a System of Equations","body":"Write the augmented matrix for the given system of equations.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Solving Systems with Gaussian Elimination","courseName":"OpenStax: College Algebra","steps":[{"id":"a2a280bgaussian1a","stepAnswer":["$$\\\\begin{bmatrix} 1 & 2 & -1 & 3 \\\\\\\\ 2 & -1 & 2 & 6 \\\\\\\\ 1 & -3 & 3 & 4 \\\\end{bmatrix}$$"],"problemType":"MultipleChoice","stepTitle":"$$x+2y-z=3$$ $$2x-y+2z=6$$ $$x-3y+3z=4$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 1 & 2 & -1 & 3 \\\\\\\\ 2 & -1 & 2 & 6 \\\\\\\\ 1 & -3 & 3 & 4 \\\\end{bmatrix}$$","choices":["$$\\\\begin{bmatrix} 1 & 2 & -1 & 3 \\\\\\\\ 2 & -1 & 2 & 6 \\\\\\\\ 1 & -3 & 3 & 4 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} 1 & 2 & -1 & 3 \\\\\\\\ 2 & -1 & 2 & 6 \\\\\\\\ 1 & -3 & 3 & 1 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} 1 & 2 & -1 & 3 \\\\\\\\ 2 & -1 & 21 & 6 \\\\\\\\ 1 & 10 & 3 & 4 \\\\end{bmatrix}$$","None of the above"],"hints":{"DefaultPathway":[{"id":"a2a280bgaussian1a-h1","type":"hint","dependencies":[],"title":"Definition","text":"The augmented matrix displays the coefficients of the variables, and an additional column for the constants.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2a280bgaussian1a-h2","type":"hint","dependencies":["a2a280bgaussian1a-h1"],"title":"Write","text":"The first row has the values 1,2,-1,3.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2a280bgaussian1a-h3","type":"hint","dependencies":["a2a280bgaussian1a-h2"],"title":"Answer","text":"The answer is $$\\\\begin{bmatrix} 1 & 2 & -1 & 3 \\\\\\\\ 2 & -1 & 2 & 6 \\\\\\\\ 1 & -3 & 3 & 4 \\\\end{bmatrix}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2a280bgaussian11","title":"Solving Systems of Equations with Matrices Using a Calculator","body":"Solve the system of equations.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Solving Systems with Gaussian Elimination","courseName":"OpenStax: College Algebra","steps":[{"id":"a2a280bgaussian11a","stepAnswer":["$$\\\\frac{61}{187}-\\\\frac{92}{187}-\\\\frac{24}{187}$$"],"problemType":"MultipleChoice","stepTitle":"$$5x+3y+9z=-1$$ $$-2x+3y-z=-2$$ $$-x-4y+5z=1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{61}{187}-\\\\frac{92}{187}-\\\\frac{24}{187}$$","choices":["$$\\\\frac{61}{187}-\\\\frac{92}{187}-\\\\frac{24}{187}$$","$$(61, -92, -24)$$","No Solution"],"hints":{"DefaultPathway":[{"id":"a2a280bgaussian11a-h1","type":"hint","dependencies":[],"title":"Augumented Matrix","text":"First, write the system of equations as an augumented matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2a280bgaussian11a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\begin{bmatrix} 5 & 3 & 9 & -1 \\\\\\\\ -2 & 3 & -1 & -2 \\\\\\\\ -1 & -4 & 5 & -1 \\\\end{bmatrix}$$"],"dependencies":["a2a280bgaussian11a-h1"],"title":"Augumented Matrix","text":"How do you write the system as an augumented matrix?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\begin{bmatrix} 5 & 3 & 9 & -1 \\\\\\\\ -2 & 3 & -1 & -2 \\\\\\\\ 1 & 4 & 5 & 1 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} 5 & 3 & 9 & -1 \\\\\\\\ -2 & 3 & -1 & -2 \\\\\\\\ -1 & -4 & 5 & -1 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} 1 & 9 & 3 & -1 \\\\\\\\ -2 & 3 & -1 & -2 \\\\\\\\ -1 & -4 & 5 & -1 \\\\end{bmatrix}$$","None of the Above"]},{"id":"a2a280bgaussian11a-h3","type":"hint","dependencies":["a2a280bgaussian11a-h2"],"title":"Use a calculator","text":"On the matrix page of the calculator, enter the augmented matrix above as the matrix variable [A].","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2a280bgaussian11a-h4","type":"hint","dependencies":["a2a280bgaussian11a-h3"],"title":"Use a calculator","text":"Use the ref( function in the calculator, calling up the matrix variable [A]. ref([A])","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2a280bgaussian11a-h5","type":"hint","dependencies":["a2a280bgaussian11a-h4"],"title":"Translate","text":"Using the matrix the calculator outputed, transform the matrix to a system, using spaces to seperate the equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2a280bgaussian11a-h6","type":"hint","dependencies":["a2a280bgaussian11a-h5"],"title":"Answer","text":"The answer is $$x+\\\\frac{3}{5} y+\\\\frac{9}{5} z=\\\\frac{1}{5}$$ $$y+\\\\frac{13}{21} z=-\\\\left(\\\\frac{4}{7}\\\\right)$$ $$z=-\\\\left(\\\\frac{24}{187}\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2a280bgaussian11a-h7","type":"hint","dependencies":["a2a280bgaussian11a-h6"],"title":"Back-substitute","text":"Use back substituition to solve the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2a280bgaussian11a-h8","type":"hint","dependencies":["a2a280bgaussian11a-h7"],"title":"Answer","text":"Therefore, the answer to the system of equations is $$\\\\frac{61}{187}-\\\\frac{92}{187}-\\\\frac{24}{187}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2a280bgaussian12","title":"Solving a 2\xd72 System by Gaussian Elimination","body":"Solve the given system by Gaussian elimination. Write the answer as a coordinate pair.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Solving Systems with Gaussian Elimination","courseName":"OpenStax: College Algebra","steps":[{"id":"a2a280bgaussian12a","stepAnswer":["(2,1)"],"problemType":"TextBox","stepTitle":"$$4x+3y=11$$ $$x-3y=-1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(2,1)$$","hints":{"DefaultPathway":[{"id":"a2a280bgaussian12a-h1","type":"hint","dependencies":[],"title":"Augumented Matrix","text":"First, write the system of equations as an augumented matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2a280bgaussian12a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\begin{bmatrix} 4 & 3 & 11 \\\\\\\\ 1 & -3 & -1 \\\\end{bmatrix}$$"],"dependencies":["a2a280bgaussian12a-h1"],"title":"Augumented Matrix","text":"How do you write the system as an augumented matrix?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\begin{bmatrix} 4 & 3 & 11 \\\\\\\\ 1 & -3 & -1 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} 4 & 1 & 11 \\\\\\\\ 1 & -3 & -1 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} 14 & 3 & 11 \\\\\\\\ 1 & 3 & -1 \\\\end{bmatrix}$$","No Solution"]},{"id":"a2a280bgaussian12a-h3","type":"hint","dependencies":["a2a280bgaussian12a-h2"],"title":"Obtain row-echelon form","text":"Now, perform row operations to get row-echelon form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2a280bgaussian12a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\begin{bmatrix} 1 & 0 & 2 \\\\\\\\ 0 & 1 & 1 \\\\end{bmatrix}$$"],"dependencies":["a2a280bgaussian12a-h3"],"title":"Obtain row-echelon form","text":"What is the matrix in row-echelon form?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\begin{bmatrix} 1 & 0 & 2 \\\\\\\\ 0 & 10 & 1 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} 1 & 0 & 2 \\\\\\\\ 0 & 1 & 1 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} 0 & 0 & 2 \\\\\\\\ 0 & 1 & 1 \\\\end{bmatrix}$$","None of the above"],"subHints":[{"id":"a2a280bgaussian12a-h4-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$\\\\begin{bmatrix} 1 & 0 & 2 \\\\\\\\ 0 & 1 & 1 \\\\end{bmatrix}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a2a280bgaussian12a-h5","type":"hint","dependencies":["a2a280bgaussian12a-h4"],"title":"Form","text":"Now, you have the appropriate form to find $$x$$ and $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2a280bgaussian12a-h6","type":"hint","dependencies":["a2a280bgaussian12a-h5"],"title":"Answer","text":"The answer is $$(2,1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2a280bgaussian13","title":"Solving Systems with Gaussian Elimination","body":"For the following exercises, write the linear system from the augmented matrix.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Solving Systems with Gaussian Elimination","courseName":"OpenStax: College Algebra","steps":[{"id":"a2a280bgaussian13a","stepAnswer":["$$-2x+5y=5, 6x-18y=26$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\begin{bmatrix} -2 & 5 & 5 \\\\\\\\ 6 & -18 & 26 \\\\end{bmatrix}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-2x+5y=5, 6x-18y=26$$","choices":["$$-2x+5y=5, 6x-18y=26$$","$$-2x+5y=5, 6x-18y=2$$","$$-2x+5y=5, x-8y=26$$","None of the Above"],"hints":{"DefaultPathway":[{"id":"a2a280bgaussian13a-h1","type":"hint","dependencies":[],"title":"Number of Variables","text":"Identify the number of variables present. $$3$$ columns indicates $$2$$ variables (x and y).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2a280bgaussian13a-h2","type":"hint","dependencies":[],"title":"Aligning Variables","text":"Aligns numbers in same column with the same variable. Last column to the right is the output.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2a280bgaussian14","title":"Solving Systems with Gaussian Elimination","body":"For the following exercises, write the linear system from the augmented matrix.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Solving Systems with Gaussian Elimination","courseName":"OpenStax: College Algebra","steps":[{"id":"a2a280bgaussian14a","stepAnswer":["3x+4y=10,10x+17y=439"],"problemType":"MultipleChoice","stepTitle":"$$\\\\begin{bmatrix} 3 & 4 & 10 \\\\\\\\ 10 & 17 & 439 \\\\end{bmatrix}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["3x+4y=10,10x+17y=439","3x+4y=10,10x+17y=43","3x+4y=10,10x+y=439","None of the Above"],"hints":{"DefaultPathway":[{"id":"a2a280bgaussian14a-h1","type":"hint","dependencies":[],"title":"Number of Variables","text":"Identify the number of variables present. $$3$$ columns indicates $$2$$ variables (x and y).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2a280bgaussian14a-h2","type":"hint","dependencies":[],"title":"Aligning Variables","text":"Aligns numbers in same column with the same variable. Last column to the right is the output.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2a280bgaussian15","title":"Solving Systems with Gaussian Elimination","body":"For the following exercises, write the linear system from the augmented matrix.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Solving Systems with Gaussian Elimination","courseName":"OpenStax: College Algebra","steps":[{"id":"a2a280bgaussian15a","stepAnswer":["3x+2y=3,-1x-9y+4z=-1,8x+5y+7z=8"],"problemType":"MultipleChoice","stepTitle":"$$\\\\begin{bmatrix} 3 & 2 & 0 & 3 \\\\\\\\ -1 & -9 & 4 & -1 \\\\\\\\ 8 & 5 & 7 & 8 \\\\end{bmatrix}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["3x+2y=3,-1x-9y+4z=-1,8x+5y+7z=8","3x+2y=3,-1x-9y+4z=1,8x+5y+7z=10","5x+2y=3,-1x-9y+4z=-1,8x+5y+7z=8","None of the Above"],"hints":{"DefaultPathway":[{"id":"a2a280bgaussian15a-h1","type":"hint","dependencies":[],"title":"Number of Variables","text":"Identify the number of variables present. $$4$$ columns indicates $$3$$ variables (x,y,z).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2a280bgaussian15a-h2","type":"hint","dependencies":[],"title":"Aligning Variables","text":"Aligns numbers in same column with the same variable. Last column to the right is the output.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2a280bgaussian16","title":"Solving Systems with Gaussian Elimination","body":"For the following exercises, write the linear system from the augmented matrix.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Solving Systems with Gaussian Elimination","courseName":"OpenStax: College Algebra","steps":[{"id":"a2a280bgaussian16a","stepAnswer":["$$8x+29y+z=43-x+7y+5z=38, 3z=10$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\begin{bmatrix} 8 & 29 & 1 & 43 \\\\\\\\ -1 & 7 & 5 & 38 \\\\\\\\ 0 & 0 & 3 & 10 \\\\end{bmatrix}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$8x+29y+z=43-x+7y+5z=38, 3z=10$$","choices":["$$8x+2y+z=43-x+7y+5z=38, 3z=1$$","$$8x+29y+z=43-x+7y+5z=38, 3z=10$$","$$-8x-29y+z=43-x-7y+5z=38, 3z=10$$","None of the Above"],"hints":{"DefaultPathway":[{"id":"a2a280bgaussian16a-h1","type":"hint","dependencies":[],"title":"Number of Variables","text":"Identify the number of variables present. $$4$$ columns indicates $$3$$ variables (x,y,z).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2a280bgaussian16a-h2","type":"hint","dependencies":[],"title":"Aligning Variables","text":"Aligns numbers in same column with the same variable. Last column to the right is the output.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2a280bgaussian17","title":"Solving Systems with Gaussian Elimination","body":"For the following exercises, write the linear system from the augmented matrix.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Solving Systems with Gaussian Elimination","courseName":"OpenStax: College Algebra","steps":[{"id":"a2a280bgaussian17a","stepAnswer":["4x+5y-2z=12,y+58z=2,8x+7y-3z=-5"],"problemType":"MultipleChoice","stepTitle":"$$\\\\begin{bmatrix} 4 & 5 & -2 & 12 \\\\\\\\ 0 & 1 & 58 & 2 \\\\\\\\ 8 & 7 & -3 & -5 \\\\end{bmatrix}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["4x+5y-2z=12,y+58z=2,8x+7y-3z=-5","4x+5y-2z=12,y+8z=2,8x+7y-3z=5","x-5y-2z=12,y+58z=2,8x+7y-3z=5","None of the Above"],"hints":{"DefaultPathway":[{"id":"a2a280bgaussian17a-h1","type":"hint","dependencies":[],"title":"Number of Variables","text":"Identify the number of variables present. $$4$$ columns indicates $$3$$ variables (x,y,z).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2a280bgaussian17a-h2","type":"hint","dependencies":[],"title":"Aligning Variables","text":"Aligns numbers in same column with the same variable. Last column to the right is the output.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2a280bgaussian18","title":"Solving Systems with Gaussian Elimination","body":"For the following exercises, write the augmented matrix for the linear system.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Solving Systems with Gaussian Elimination","courseName":"OpenStax: College Algebra","steps":[{"id":"a2a280bgaussian18a","stepAnswer":["$$\\\\begin{bmatrix} 8 & -37 & 8 \\\\\\\\ 2 & 12 & 3 \\\\end{bmatrix}$$"],"problemType":"MultipleChoice","stepTitle":"8x-37y=8,2x+12y=3","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 8 & -37 & 8 \\\\\\\\ 2 & 12 & 3 \\\\end{bmatrix}$$","choices":["$$\\\\begin{bmatrix} 8 & -37 & 8 \\\\\\\\ 2 & 12 & 3 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} 8 & 7 & 8 \\\\\\\\ 2 & 12 & 3 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} -37 & 8 & 8 \\\\\\\\ 12 & 2 & 3 \\\\end{bmatrix}$$","None of the Above"],"hints":{"DefaultPathway":[{"id":"a2a280bgaussian18a-h1","type":"hint","dependencies":[],"title":"Matrix Columns","text":"Identify the number of variables present. Add the number of variables by $$1$$ to determine number of columns in matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2a280bgaussian18a-h2","type":"hint","dependencies":[],"title":"Matrix Rows","text":"The number of equations will serve as the number of rows in the matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2a280bgaussian19","title":"Solving Systems with Gaussian Elimination","body":"For the following exercises, write the augmented matrix for the linear system.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Solving Systems with Gaussian Elimination","courseName":"OpenStax: College Algebra","steps":[{"id":"a2a280bgaussian19a","stepAnswer":["$$\\\\begin{bmatrix} 0 & 16 & 4 \\\\\\\\ 9 & -1 & 2 \\\\end{bmatrix}$$"],"problemType":"MultipleChoice","stepTitle":"$$16y=4, 9x-y=2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 0 & 16 & 4 \\\\\\\\ 9 & -1 & 2 \\\\end{bmatrix}$$","choices":["$$\\\\begin{bmatrix} 4 & 16 & 0 \\\\\\\\ 9 & -1 & 2 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} 0 & 16 & 4 \\\\\\\\ -9 & -1 & 2 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} 0 & 16 & 4 \\\\\\\\ 9 & -1 & 2 \\\\end{bmatrix}$$","None of the above"],"hints":{"DefaultPathway":[{"id":"a2a280bgaussian19a-h1","type":"hint","dependencies":[],"title":"Matrix Columns","text":"Identify the number of variables present. Add the number of variables by $$1$$ to determine number of columns in matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2a280bgaussian19a-h2","type":"hint","dependencies":[],"title":"Matrix Rows","text":"The number of equations will serve as the number of rows in the matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2a280bgaussian2","title":"Writing the Augmented Matrix for a System of Equations","body":"Write the augmented matrix of the given system of equations.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Solving Systems with Gaussian Elimination","courseName":"OpenStax: College Algebra","steps":[{"id":"a2a280bgaussian2a","stepAnswer":["$$\\\\begin{bmatrix} 4 & -3 & 11 \\\\\\\\ 3 & 2 & 4 \\\\end{bmatrix}$$"],"problemType":"MultipleChoice","stepTitle":"4x-3y=11,3x+2y=4","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 4 & -3 & 11 \\\\\\\\ 3 & 2 & 4 \\\\end{bmatrix}$$","choices":["$$\\\\begin{bmatrix} 4 & -3 & 11 \\\\\\\\ 3 & 1 & 4 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} 4 & -3 & 11 \\\\\\\\ 3 & 2 & 4 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} 4 & 3 & -11 \\\\\\\\ 3 & 2 & 4 \\\\end{bmatrix}$$","None of the above"],"hints":{"DefaultPathway":[{"id":"a2a280bgaussian2a-h1","type":"hint","dependencies":[],"title":"Definition","text":"The augmented matrix displays the coefficients of the variables, and an additional column for the constants.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2a280bgaussian2a-h2","type":"hint","dependencies":["a2a280bgaussian2a-h1"],"title":"Write","text":"The first row has the values 4,-3,11.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2a280bgaussian2a-h3","type":"hint","dependencies":["a2a280bgaussian2a-h2"],"title":"Answer","text":"The answer is $$\\\\begin{bmatrix} 4 & -3 & 11 \\\\\\\\ 3 & 2 & 4 \\\\end{bmatrix}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2a280bgaussian20","title":"Solving Systems with Gaussian Elimination","body":"For the following exercises, write the augmented matrix for the linear system.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Solving Systems with Gaussian Elimination","courseName":"OpenStax: College Algebra","steps":[{"id":"a2a280bgaussian20a","stepAnswer":["$$\\\\begin{bmatrix} 3 & 2 & 10 & 3 \\\\\\\\ -6 & 2 & 5 & 13 \\\\\\\\ 4 & 0 & 1 & 18 \\\\end{bmatrix}$$"],"problemType":"MultipleChoice","stepTitle":"3x+2y+10z=3,-6x+2y+5z=13,4x+z=18","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 3 & 2 & 10 & 3 \\\\\\\\ -6 & 2 & 5 & 13 \\\\\\\\ 4 & 0 & 1 & 18 \\\\end{bmatrix}$$","choices":["$$\\\\begin{bmatrix} 3 & 2 & 3 & 3 \\\\\\\\ -6 & 2 & 5 & 13 \\\\\\\\ 4 & 0 & 1 & 18 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} 3 & 2 & 10 & 3 \\\\\\\\ -6 & 2 & 5 & 13 \\\\\\\\ 4 & 0 & 1 & 18 \\\\end{bmatrix}$$","None of the Above"],"hints":{"DefaultPathway":[{"id":"a2a280bgaussian20a-h1","type":"hint","dependencies":[],"title":"Matrix Columns","text":"Identify the number of variables present. Add the number of variables by $$1$$ to determine number of columns in matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2a280bgaussian20a-h2","type":"hint","dependencies":[],"title":"Matrix Rows","text":"The number of equations will serve as the number of rows in the matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2a280bgaussian21","title":"Solving Systems with Gaussian Elimination","body":"For the following exercises, write the augmented matrix for the linear system.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Solving Systems with Gaussian Elimination","courseName":"OpenStax: College Algebra","steps":[{"id":"a2a280bgaussian21a","stepAnswer":["$$\\\\begin{bmatrix} 1 & 5 & 8 & 19 \\\\\\\\ 12 & 3 & 0 & 4 \\\\\\\\ 3 & 4 & 9 & -7 \\\\end{bmatrix}$$"],"problemType":"MultipleChoice","stepTitle":"x+5y+8z=19,12x+3y=4,3x+4y+9z=-7","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 1 & 5 & 8 & 19 \\\\\\\\ 12 & 3 & 0 & 4 \\\\\\\\ 3 & 4 & 9 & -7 \\\\end{bmatrix}$$","choices":["$$\\\\begin{bmatrix} 1 & 5 & 8 & 19 \\\\\\\\ 12 & 3 & 0 & 4 \\\\\\\\ 3 & 4 & 9 & -7 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} 1 & 5 & 8 & 19 \\\\\\\\ 12 & 3 & 0 & 4 \\\\\\\\ 3 & -4 & 9 & 7 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} 19 & 5 & 8 & 1 \\\\\\\\ 12 & 3 & 0 & 4 \\\\\\\\ 3 & 4 & 9 & -7 \\\\end{bmatrix}$$","None of the Above"],"hints":{"DefaultPathway":[{"id":"a2a280bgaussian21a-h1","type":"hint","dependencies":[],"title":"Matrix Columns","text":"Identify the number of variables present. Add the number of variables by $$1$$ to determine number of columns in matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2a280bgaussian21a-h2","type":"hint","dependencies":[],"title":"Matrix Rows","text":"The number of equations will serve as the number of rows in the matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2a280bgaussian22","title":"Solving Systems with Gaussian Elimination","body":"For the following exercises, write the augmented matrix for the linear system.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Solving Systems with Gaussian Elimination","courseName":"OpenStax: College Algebra","steps":[{"id":"a2a280bgaussian22a","stepAnswer":["$$\\\\begin{bmatrix} 6 & 12 & 16 & 4 \\\\\\\\ 19 & -5 & 3 & -9 \\\\\\\\ 1 & 0 & 2 & -8 \\\\end{bmatrix}$$"],"problemType":"MultipleChoice","stepTitle":"6x+12y+16z=4,19x-5y+3z=-9,x+2y=-8","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 6 & 12 & 16 & 4 \\\\\\\\ 19 & -5 & 3 & -9 \\\\\\\\ 1 & 0 & 2 & -8 \\\\end{bmatrix}$$","choices":["$$\\\\begin{bmatrix} 6 & 12 & 16 & 4 \\\\\\\\ 19 & -5 & 3 & -9 \\\\\\\\ 1 & 0 & -2 & 8 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} 6 & 12 & 16 & 4 \\\\\\\\ 19 & -5 & 3 & -9 \\\\\\\\ 1 & 0 & 2 & -8 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} -6 & 12 & 16 & 4 \\\\\\\\ 19 & -5 & 3 & -9 \\\\\\\\ 1 & 0 & 2 & -8 \\\\end{bmatrix}$$","None of the Above"],"hints":{"DefaultPathway":[{"id":"a2a280bgaussian22a-h1","type":"hint","dependencies":[],"title":"Matrix Columns","text":"Identify the number of variables present. Add the number of variables by $$1$$ to determine number of columns in matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2a280bgaussian22a-h2","type":"hint","dependencies":[],"title":"Matrix Rows","text":"The number of equations will serve as the number of rows in the matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2a280bgaussian3","title":"Writing a System of Equations from an Augmented Matrix Form","body":"Find the system of equations from the augmented matrix when the variables are $$x$$, $$y$$, and $$z$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Solving Systems with Gaussian Elimination","courseName":"OpenStax: College Algebra","steps":[{"id":"a2a280bgaussian3a","stepAnswer":["$$x-3y-5z=-2$$, $$2x-5y-4z=5$$, $$-3x+5y+4z=6$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\begin{bmatrix} 1 & -3 & -5 & -2 \\\\\\\\ 2 & -5 & -4 & 5 \\\\\\\\ -3 & 5 & 4 & 6 \\\\end{bmatrix}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x-3y-5z=-2$$, $$2x-5y-4z=5$$, $$-3x+5y+4z=6$$","choices":["$$x-3y-5z=-2$$, $$2x-5y-4z=5$$, $$-3x+5y+4z=-6$$","$$x-3y-5z=-2$$, $$2x-5y-4z=5$$, $$-3x+5y+4z=6$$","$$3x-y+5z=-2$$, $$2x-5y-4z=5$$, $$-3x+5y+4z=6$$","None of the above"],"hints":{"DefaultPathway":[{"id":"a2a280bgaussian3a-h1","type":"hint","dependencies":[],"title":"Rows","text":"Rows represent a single equation. There are $$3$$ rows, so you know there are $$3$$ equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2a280bgaussian3a-h2","type":"hint","dependencies":["a2a280bgaussian3a-h1"],"title":"Coefficients","text":"The numbers in the matrix represent coefficients of variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2a280bgaussian3a-h3","type":"hint","dependencies":["a2a280bgaussian3a-h2"],"title":"Answer","text":"The answer is $$x-3y-5z=-2$$ $$2x-5y-4z=5$$ $$-3x+5y+4z=6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2a280bgaussian4","title":"Writing a System of Equations from an Augmented Matrix Form","body":"Find the system of equations from the augmented matrix when the variables are $$x$$, $$y$$, and $$z$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Solving Systems with Gaussian Elimination","courseName":"OpenStax: College Algebra","steps":[{"id":"a2a280bgaussian4a","stepAnswer":["$$x-y+z=5$$, $$2x-y+3z=1$$, $$y+z=-9$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\begin{bmatrix} 1 & -1 & 1 & 5 \\\\\\\\ 2 & -1 & 3 & 1 \\\\\\\\ 0 & 1 & 1 & -9 \\\\end{bmatrix}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x-y+z=5$$, $$2x-y+3z=1$$, $$y+z=-9$$","choices":["$$x-y+z=5$$, $$2x-y+3z=1$$, $$y+z=-9$$","$$x-y+z=5$$, $$2x-y+3z=1$$, $$y+z=9$$","$$x-y+z=5$$, $$2x+3z=1$$, $$y+z=-9$$","None of the above"],"hints":{"DefaultPathway":[{"id":"a2a280bgaussian4a-h1","type":"hint","dependencies":[],"title":"Rows","text":"Rows represent a single equation. There are $$3$$ rows, so you know there are $$3$$ equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2a280bgaussian4a-h2","type":"hint","dependencies":["a2a280bgaussian4a-h1"],"title":"Coefficients","text":"The numbers in the matrix represent coefficients of variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2a280bgaussian4a-h3","type":"hint","dependencies":["a2a280bgaussian4a-h2"],"title":"Answer","text":"The answer is $$x-y+z=5$$ $$2x-y+3z=1$$ $$y+z=-9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2a280bgaussian9","title":"Solving a System of Linear Equations Using Matrices","body":"Solve the system of linear equations using matrices.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Solving Systems with Gaussian Elimination","courseName":"OpenStax: College Algebra","steps":[{"id":"a2a280bgaussian9a","stepAnswer":["$$(4, -3, 1)$$"],"problemType":"MultipleChoice","stepTitle":"$$x-y+z=8$$ $$2x+3y-z=-2$$ $$3x-2y-9z=9$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(4, -3, 1)$$","choices":["$$(4, -3, 1)$$","(4,3,1)","$$(-4, -3, -1)$$","None of the Above"],"hints":{"DefaultPathway":[{"id":"a2a280bgaussian9a-h1","type":"hint","dependencies":[],"title":"Augumented Matrix","text":"First, write the system of equations as an augumented matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2a280bgaussian9a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\begin{bmatrix} 1 & -1 & 1 & 8 \\\\\\\\ 2 & 3 & -1 & -2 \\\\\\\\ 3 & -2 & -9 & 9 \\\\end{bmatrix}$$"],"dependencies":["a2a280bgaussian9a-h1"],"title":"Augumented Matrix","text":"How do you write the system as an augumented matrix?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\begin{bmatrix} 1 & -1 & 1 & 10 \\\\\\\\ 2 & 3 & -1 & -2 \\\\\\\\ 5 & -2 & 9 & 9 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} 1 & -1 & 1 & 8 \\\\\\\\ 2 & 3 & -1 & -2 \\\\\\\\ 3 & -2 & -9 & 9 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} 1 & 1 & 1 & 4 \\\\\\\\ 2 & 3 & 1 & -2 \\\\\\\\ 3 & -2 & -9 & 9 \\\\end{bmatrix}$$","None of the Above"],"subHints":[{"id":"a2a280bgaussian9a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$\\\\begin{bmatrix} 1 & -1 & 1 & 8 \\\\\\\\ 2 & 3 & -1 & -2 \\\\\\\\ 3 & -2 & -9 & 9 \\\\end{bmatrix}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a2a280bgaussian9a-h3","type":"hint","dependencies":["a2a280bgaussian9a-h2"],"title":"Obtain row-echelon form","text":"Now, perform row operations to obtain row-echelon form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2a280bgaussian9a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\begin{bmatrix} 1 & -1 & 1 & 8 \\\\\\\\ 0 & 1 & -12 & -15 \\\\\\\\ 0 & 0 & 1 & 1 \\\\end{bmatrix}$$"],"dependencies":["a2a280bgaussian9a-h3"],"title":"Obtain row-echelon form","text":"What is the new matrix in row-echelon form?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\begin{bmatrix} 1 & -1 & 1 & 8 \\\\\\\\ 0 & 0 & -1 & -5 \\\\\\\\ 0 & 0 & 1 & 1 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} 1 & -1 & 1 & 8 \\\\\\\\ 0 & 1 & -12 & -15 \\\\\\\\ 1 & 1 & 1 & 1 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} 1 & -1 & 1 & 8 \\\\\\\\ 0 & 1 & -12 & -15 \\\\\\\\ 0 & 0 & 1 & 1 \\\\end{bmatrix}$$","None of the above"],"subHints":[{"id":"a2a280bgaussian9a-h4-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$\\\\begin{bmatrix} 1 & -1 & 1 & 8 \\\\\\\\ 0 & 1 & -12 & -15 \\\\\\\\ 0 & 0 & 1 & 1 \\\\end{bmatrix}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a2a280bgaussian9a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x-y+z=8$$ $$y-12z=-15$$ $$z=1$$"],"dependencies":["a2a280bgaussian9a-h4"],"title":"Convert to equations","text":"What is the system of equations after converting it from the matrix?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a2a280bgaussian9a-h5-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$x-y+z=8$$ $$y-12z=-15$$ $$z=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a2a280bgaussian9a-h6","type":"hint","dependencies":["a2a280bgaussian9a-h5"],"title":"Back substitute","text":"Now, back substitute the variables into the system of equations to obtain the final solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2a280bgaussian9a-h7","type":"hint","dependencies":["a2a280bgaussian9a-h6"],"title":"Answer","text":"Therefore, the final solution is $$(4, -3, 1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2a5cd1alginequalabs1","title":"Inequalities and Absolute Values: Part A","body":"These questions test your knowledge of the core concepts. Express each of the following statements as a mathematical equation or inequality.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Inequalities and Absolute Values","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a2a5cd1alginequalabs1a","stepAnswer":["$$-1<x$$ $$\\\\leq$$ $$3$$"],"problemType":"MultipleChoice","stepTitle":"A number $$x$$ is less than or equal to $$3$$ and greater than $$-1$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-1<x$$ $$\\\\leq$$ $$3$$","choices":["$$-1<x$$ $$\\\\leq$$ $$3$$","$$-1<x<3$$","$$x<-1$$ or $$x$$ $$\\\\geq$$ $$3$$","$$x<-1$$ or $$x>3$$"],"hints":{"DefaultPathway":[{"id":"a2a5cd1alginequalabs1a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\leq$$"],"dependencies":[],"title":"Inequality of \'less than or equal to 3\'","text":"What symbol does \'less than or equal to\' refer to?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["<",">","$$\\\\leq$$","$$\\\\geq$$"]},{"id":"a2a5cd1alginequalabs1a-h2","type":"hint","dependencies":["a2a5cd1alginequalabs1a-h1"],"title":"Inequality of \'less than or equal to 3\'","text":"The sentence \'a is less than or equal to b\' means that a $$\\\\leq$$ $$b$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs1a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x$$ $$\\\\leq$$ $$3$$"],"dependencies":["a2a5cd1alginequalabs1a-h2"],"title":"Inequality of \'less than or equal to 3\'","text":"What inequality represents \'x\' is less than or equal to 3\'?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$x$$ $$\\\\leq$$ $$3$$","$$3$$ $$\\\\leq$$ $$x$$"]},{"id":"a2a5cd1alginequalabs1a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":[">"],"dependencies":["a2a5cd1alginequalabs1a-h3"],"title":"Inequality of \'greater than -1\'","text":"What symbol does \'greater than\' refer to?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["<",">","$$\\\\leq$$","$$\\\\geq$$"]},{"id":"a2a5cd1alginequalabs1a-h5","type":"hint","dependencies":["a2a5cd1alginequalabs1a-h4"],"title":"Inequality of \'greater than -1\'","text":"The sentence \'a is greater than b\' means that $$a>b$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs1a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x>-1$$"],"dependencies":["a2a5cd1alginequalabs1a-h5"],"title":"Inequality of \'greater than -1\'","text":"What inequality represents \'x is greater than -1\'?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$x>-1$$","$$-1>x$$"]},{"id":"a2a5cd1alginequalabs1a-h7","type":"hint","dependencies":["a2a5cd1alginequalabs1a-h6"],"title":"ANDing Two Inequalities","text":"When ANDing two inequalities together, this means that both inequalities must be true for the value of \'x\' to be valid.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs1a-h8","type":"hint","dependencies":["a2a5cd1alginequalabs1a-h7"],"title":"ANDing Two Inequalities","text":"For some values a,b where $$a<x$$ and $$x$$ $$\\\\leq$$ $$b$$, these two inequalities can be combined to say $$a<x$$ $$\\\\leq$$ $$b$$","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a2a5cd1alginequalabs10","title":"Inequalities and Absolute Values: Part A","body":"These questions test your knowledge of the core concepts. Describe the set of all numbers \'x\' satisfying each inequality using interval notation.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Inequalities and Absolute Values","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a2a5cd1alginequalabs10a","stepAnswer":["$$[-2,8]$$"],"problemType":"MultipleChoice","stepTitle":"$$|x-3|$$ $$\\\\leq$$ $$5$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$[-2,8]$$","$$(-2,8)$$","$$(-\\\\infty,-2)$$ $$\\\\cup$$ $$(8,\\\\infty)$$","$$(-\\\\infty,-2]$$ $$\\\\cup$$ $$[8,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"a2a5cd1alginequalabs10a-h1","type":"hint","dependencies":[],"title":"Expanding the Inequality","text":"For some numbers a and $$b$$ $$\\\\geq$$ $$0$$ where $$|a|$$ $$\\\\leq$$ $$b$$, this is the same as a $$\\\\leq$$ $$b$$ AND a $$\\\\geq$$ $$-b$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs10a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x-3$$ $$\\\\leq$$ $$5$$ and $$x-3$$ $$\\\\geq$$ $$-5$$"],"dependencies":["a2a5cd1alginequalabs10a-h1"],"title":"Expanding the Inequality","text":"What is $$|x-3|$$ $$\\\\leq$$ $$5$$ rewritten without the absolute value sign?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$x-3$$ $$\\\\leq$$ $$5$$ and $$x-3$$ $$\\\\geq$$ $$-5$$","$$x-3$$ $$\\\\leq$$ $$5$$ and $$x-3$$ $$\\\\geq$$ $$5$$","$$x-3$$ $$\\\\leq$$ $$5$$ and $$x-3$$ $$\\\\leq$$ $$-5$$","$$x-3$$ $$\\\\leq$$ $$5$$"]},{"id":"a2a5cd1alginequalabs10a-h3","type":"hint","dependencies":["a2a5cd1alginequalabs10a-h2"],"title":"Simplifying the Inequality: Part A","text":"Rearrange the inequality such that all the \'x\'s are on one side and the constants are on the other.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs10a-h4","type":"hint","dependencies":["a2a5cd1alginequalabs10a-h3"],"title":"Simplifying the Inequality: Part A","text":"Add $$3$$ from the left to get $$x$$ $$\\\\leq$$ $$5+3$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a2a5cd1alginequalabs10a-h4"],"title":"Simplifying the Inequality: Part A","text":"What is $$5+3$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs10a-h6","type":"hint","dependencies":["a2a5cd1alginequalabs10a-h5"],"title":"Simplifying the Inequality: Part B","text":"Rearrange the inequality such that all the \'x\'s are on one side and the constants are on the other.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs10a-h7","type":"hint","dependencies":["a2a5cd1alginequalabs10a-h6"],"title":"Simplifying the Inequality: Part B","text":"Add $$3$$ from the left to get $$x$$ $$\\\\geq$$ $$-5+3$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs10a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a2a5cd1alginequalabs10a-h7"],"title":"Simplifying the Inequality: Part B","text":"What is $$-5+3$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs10a-h9","type":"hint","dependencies":["a2a5cd1alginequalabs10a-h8"],"title":"ANDing the Distance","text":"Since the distance is less than a value, the inequalities provide the upper and lower bounds of the value by ANDing them together.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs10a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-2$$"],"dependencies":["a2a5cd1alginequalabs10a-h9"],"title":"Interval Notation","text":"What is the lower bound of the inequality $$x$$ $$\\\\leq$$ $$8$$ and $$x$$ $$\\\\geq$$ -2?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$-\\\\infty$$","$$8$$","$$-2$$"]},{"id":"a2a5cd1alginequalabs10a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$8$$"],"dependencies":["a2a5cd1alginequalabs10a-h9"],"title":"Interval Notation","text":"What is the upper bound of the inequality $$x$$ $$\\\\leq$$ $$8$$ and $$x$$ $$\\\\geq$$ -2?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\infty$$","$$8$$","$$-2$$"]},{"id":"a2a5cd1alginequalabs10a-h12","type":"hint","dependencies":["a2a5cd1alginequalabs10a-h10","a2a5cd1alginequalabs10a-h11"],"title":"Interval Notation","text":"If the bound itself should be included as a valid answer of \'x\', meaning it can be equal to that value, then a bracket (\\"[\\" or \\"]\\") should be used. Otherwise, use a paranthesis (\\"(\\" or \\")\\").","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs10a-h13","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a2a5cd1alginequalabs10a-h12"],"title":"Interval Notation","text":"Is the lower bound $$-2$$ included as a valid value of \'x\' in $$x$$ $$\\\\leq$$ $$8$$ and $$x$$ $$\\\\geq$$ -2?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"a2a5cd1alginequalabs10a-h14","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a2a5cd1alginequalabs10a-h12"],"title":"Interval Notation","text":"Is the upper bound $$8$$ included as a valid value of \'x\' in $$x$$ $$\\\\leq$$ $$8$$ and $$x$$ $$\\\\geq$$ -2?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]}]}},{"id":"a2a5cd1alginequalabs10b","stepAnswer":["$$(-\\\\infty,-9)$$ $$\\\\cup$$ $$(-1,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$|x+5|>4$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,-9)$$ $$\\\\cup$$ $$(-1,\\\\infty)$$","choices":["$$(-\\\\infty,-9)$$ $$\\\\cup$$ $$(-1,\\\\infty)$$","$$(-\\\\infty,-9]$$ $$\\\\cup$$ $$[-1,\\\\infty)$$","$$(-9,-1)$$","$$[-9,-1]$$"],"hints":{"DefaultPathway":[{"id":"a2a5cd1alginequalabs10b-h1","type":"hint","dependencies":[],"title":"Expanding the Inequality","text":"For some numbers a and $$b$$ $$\\\\geq$$ $$0$$ where $$|a|>b$$, this is the same as $$a>b$$ OR $$a<-b$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs10b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x+5>4$$ or $$x+5<-4$$"],"dependencies":["a2a5cd1alginequalabs10b-h1"],"title":"Expanding the Inequality","text":"What is $$|x+5|>4$$ rewritten without the absolute value sign?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$x+5>4$$ or $$x+5<-4$$","$$x+5>4$$ or $$x+5>-4$$","$$x+5>4$$ or $$x+5<4$$","$$x+5>4$$"]},{"id":"a2a5cd1alginequalabs10b-h3","type":"hint","dependencies":["a2a5cd1alginequalabs10b-h2"],"title":"Simplifying the Inequality: Part A","text":"Rearrange the inequality such that all the \'x\'s are on one side and the constants are on the other.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs10b-h4","type":"hint","dependencies":["a2a5cd1alginequalabs10b-h3"],"title":"Simplifying the Inequality: Part A","text":"Subtract $$5$$ from the left to get $$x>4-5$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs10b-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a2a5cd1alginequalabs10b-h4"],"title":"Simplifying the Inequality: Part A","text":"What is $$4-5$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs10b-h6","type":"hint","dependencies":["a2a5cd1alginequalabs10b-h5"],"title":"Simplifying the Inequality: Part B","text":"Rearrange the inequality such that all the \'x\'s are on one side and the constants are on the other.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs10b-h7","type":"hint","dependencies":["a2a5cd1alginequalabs10b-h6"],"title":"Simplifying the Inequality: Part B","text":"Subtract $$5$$ from the left to get $$x<-4-5$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs10b-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":["a2a5cd1alginequalabs10b-h7"],"title":"Simplifying the Inequality: Part B","text":"What is $$-4-5$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs10b-h9","type":"hint","dependencies":["a2a5cd1alginequalabs10b-h8"],"title":"ORing the Distance","text":"Since the distance is greater than a value, the value outside of the distance are considered valid, ORing the two inequalities together.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs10b-h10","type":"hint","dependencies":["a2a5cd1alginequalabs10b-h9"],"title":"Interval Notation","text":"Since the two inequalities are separated by an OR statement, they will both appear in the final result.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs10b-h11","type":"hint","dependencies":["a2a5cd1alginequalabs10b-h10"],"title":"Interval Notation: Part A","text":"The inequality $$x>-1$$ can be written in interval notation by specifying the lower bound first, followed by the upper bound.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs10b-h12","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-1$$"],"dependencies":["a2a5cd1alginequalabs10b-h11"],"title":"Interval Notation: Part A","text":"What is the lower bound of the inequality $$x>-1$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$-\\\\infty$$","$$-1$$"]},{"id":"a2a5cd1alginequalabs10b-h13","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\infty$$"],"dependencies":["a2a5cd1alginequalabs10b-h11"],"title":"Interval Notation: Part A","text":"What is the upper bound of the inequality $$x>-1$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\infty$$","$$-1$$"]},{"id":"a2a5cd1alginequalabs10b-h14","type":"hint","dependencies":["a2a5cd1alginequalabs10b-h12","a2a5cd1alginequalabs10b-h13"],"title":"Interval Notation: Part A","text":"If the bound itself should be included as a valid answer of \'x\', meaning it can be equal to that value, then a bracket (\\"[\\" or \\"]\\") should be used. Otherwise, use a paranthesis (\\"(\\" or \\")\\").","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs10b-h15","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a2a5cd1alginequalabs10b-h14"],"title":"Interval Notation: Part A","text":"Is the lower bound $$-1$$ included as a valid value of \'x\' in $$x>-1$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"a2a5cd1alginequalabs10b-h16","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a2a5cd1alginequalabs10b-h14"],"title":"Interval Notation: Part A","text":"Is the upper bound $$\\\\infty$$ included as a valid value of \'x\' in $$x>-1$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"],"subHints":[{"id":"a2a5cd1alginequalabs10b-h16-s1","type":"hint","dependencies":[],"title":"Interval Notation: Part A","text":"As $$\\\\infty$$ is not a number, it cannot be included as part of a valid value or bound for \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a2a5cd1alginequalabs10b-h17","type":"hint","dependencies":["a2a5cd1alginequalabs10b-h15","a2a5cd1alginequalabs10b-h16"],"title":"Interval Notation: Part B","text":"The inequality $$x<-9$$ can be written in interval notation by specifying the lower bound first, followed by the upper bound.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs10b-h18","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-\\\\infty$$"],"dependencies":["a2a5cd1alginequalabs10b-h17"],"title":"Interval Notation: Part B","text":"What is the lower bound of the inequality $$x<-9$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$-\\\\infty$$","$$-9$$"]},{"id":"a2a5cd1alginequalabs10b-h19","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-9$$"],"dependencies":["a2a5cd1alginequalabs10b-h17"],"title":"Interval Notation: Part B","text":"What is the upper bound of the inequality $$x<-9$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\infty$$","$$-9$$"]},{"id":"a2a5cd1alginequalabs10b-h20","type":"hint","dependencies":["a2a5cd1alginequalabs10b-h18","a2a5cd1alginequalabs10b-h19"],"title":"Interval Notation: Part B","text":"If the bound itself should be included as a valid answer of \'x\', meaning it can be equal to that value, then a bracket (\\"[\\" or \\"]\\") should be used. Otherwise, use a paranthesis (\\"(\\" or \\")\\").","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs10b-h21","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a2a5cd1alginequalabs10b-h20"],"title":"Interval Notation: Part B","text":"Is the lower bound $$-\\\\infty$$ included as a valid value of \'x\' in $$x<-9$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"],"subHints":[{"id":"a2a5cd1alginequalabs10b-h21-s2","type":"hint","dependencies":[],"title":"Interval Notation: Part B","text":"As $$-\\\\infty$$ is not a number, it cannot be included as part of a valid value or bound for \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a2a5cd1alginequalabs10b-h22","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a2a5cd1alginequalabs10b-h20"],"title":"Interval Notation: Part B","text":"Is the upper bound $$-9$$ included as a valid value of \'x\' in $$x<-9$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"a2a5cd1alginequalabs10b-h23","type":"hint","dependencies":["a2a5cd1alginequalabs10b-h21","a2a5cd1alginequalabs10b-h22"],"title":"Interval Notation","text":"The bounds for the two equations, $$(-\\\\infty,-9)$$ and $$(-1,\\\\infty)$$, can be ORed together using \' $$$$ $$\\\\cup$$ $$$$ \'.","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a2a5cd1alginequalabs100","title":"Inequalities and Absolute Values: Part B","body":"These problems are harder, often highlighting an important subtlety.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Inequalities and Absolute Values","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a2a5cd1alginequalabs100a","stepAnswer":["$$A=3$$, $$B=6$$"],"problemType":"MultipleChoice","stepTitle":"Find constants \'A\' and \'B\' such that the number \'x\' is contained in $$(-3,9)$$ if and only if $$|x-A|<B$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$A=3$$, $$B=6$$","choices":["$$A=3$$, $$B=6$$","$$A=-3$$, $$B=9$$","$$A=9$$, $$B=6$$","No solution"],"hints":{"DefaultPathway":[{"id":"a2a5cd1alginequalabs100a-h1","type":"hint","dependencies":[],"title":"Expanding the Inequality","text":"For some numbers a and $$b$$ $$\\\\geq$$ $$0$$ where $$|a|<b$$, this is the same as $$a<b$$ AND $$a>-b$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs100a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x-A<B$$ and $$x-A>-B$$"],"dependencies":["a2a5cd1alginequalabs100a-h1"],"title":"Expanding the Inequality","text":"What is $$|x-A|<B$$ rewritten without the absolute value sign?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$x-A<B$$ and $$x-A>-B$$","$$x-A<B$$ and $$x-A<-B$$","$$x-A<B$$ and $$x-A>B$$","$$x-A<B$$"]},{"id":"a2a5cd1alginequalabs100a-h3","type":"hint","dependencies":["a2a5cd1alginequalabs100a-h2"],"title":"Simplifying the Inequality: Part A","text":"Rearrange the inequality such that all the \'x\'s are on one side and the constants are on the other.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs100a-h4","type":"hint","dependencies":["a2a5cd1alginequalabs100a-h3"],"title":"Simplifying the Inequality: Part A","text":"Add A from the left to get $$x<A+B$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs100a-h5","type":"hint","dependencies":["a2a5cd1alginequalabs100a-h4"],"title":"Simplifying the Inequality: Part B","text":"Rearrange the inequality such that all the \'x\'s are on one side and the constants are on the other.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs100a-h6","type":"hint","dependencies":["a2a5cd1alginequalabs100a-h5"],"title":"Simplifying the Inequality: Part B","text":"Add A from the left to get $$x>A-B$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs100a-h7","type":"hint","dependencies":["a2a5cd1alginequalabs100a-h6"],"title":"ANDing the Distance","text":"Since the distance is less than a value, the inequalities provide the upper and lower bounds of the value by ANDing them together.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs100a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$A-B$$"],"dependencies":["a2a5cd1alginequalabs100a-h7"],"title":"Interval Notation","text":"What is the lower bound of the inequality $$x<A+B$$ and $$x>A-B$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$A-B$$","$$A+B$$","A","B"]},{"id":"a2a5cd1alginequalabs100a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$A+B$$"],"dependencies":["a2a5cd1alginequalabs100a-h7"],"title":"Interval Notation","text":"What is the upper bound of the inequality $$x<A+B$$ and $$x>A-B$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$A-B$$","$$A+B$$","A","B"]},{"id":"a2a5cd1alginequalabs100a-h10","type":"hint","dependencies":["a2a5cd1alginequalabs100a-h8","a2a5cd1alginequalabs100a-h9"],"title":"Interval Notation","text":"If the bound itself should be included as a valid answer of \'x\', meaning it can be equal to that value, then a bracket (\\"[\\" or \\"]\\") should be used. Otherwise, use a paranthesis (\\"(\\" or \\")\\").","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs100a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a2a5cd1alginequalabs100a-h10"],"title":"Interval Notation","text":"Is the lower bound A-B included as a valid value of \'x\' in $$x<A+B$$ and $$x>A-B$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"a2a5cd1alginequalabs100a-h12","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a2a5cd1alginequalabs100a-h10"],"title":"Interval Notation","text":"Is the upper bound $$A+B$$ included as a valid value of \'x\' in $$x<A+B$$ and $$x>A-B$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"a2a5cd1alginequalabs100a-h13","type":"hint","dependencies":["a2a5cd1alginequalabs100a-h11","a2a5cd1alginequalabs100a-h12"],"title":"Solving for A and B","text":"Since $$x$$ is contained in $$(-3,9)$$ if and only if $$x$$ is in $$(A-B,A+B)$$, A and B can be found by solving the system of linear equations.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs100a-h14","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$A-B=-3$$ and $$A+B=9$$"],"dependencies":["a2a5cd1alginequalabs100a-h13"],"title":"Solving for A and B","text":"What system of linear equations can be extrapolated from $$(-3,9)$$ and (A-B, A+B)?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$A-B=-3$$ and $$A+B=9$$","$$A-B=9$$ and $$A+B=-3$$"]},{"id":"a2a5cd1alginequalabs100a-h15","type":"hint","dependencies":["a2a5cd1alginequalabs100a-h14"],"title":"Solving for A and B","text":"The two equations can be added together to eliminate B from the equation to solve for A.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs100a-h16","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2A=6$$"],"dependencies":["a2a5cd1alginequalabs100a-h15"],"title":"Solving for A and B","text":"What equation is left when you add $$A-B=-3$$ and $$A+B=9$$ together?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$2A=6$$","$$A=6$$","$$2A=12$$","$$A=12$$"]},{"id":"a2a5cd1alginequalabs100a-h17","type":"hint","dependencies":["a2a5cd1alginequalabs100a-h16"],"title":"Solving for A and B","text":"Divide $$2$$ from both sides of $$2A=6$$ to isolate \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs100a-h18","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a2a5cd1alginequalabs100a-h17"],"title":"Solving for A and B","text":"What is $$\\\\frac{6}{2}$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs100a-h19","type":"hint","dependencies":["a2a5cd1alginequalabs100a-h18"],"title":"Solving for A and B","text":"Since $$A=3$$, substitute A into $$A+B=9$$ and solve for B: $$3+B=9$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs100a-h20","type":"hint","dependencies":["a2a5cd1alginequalabs100a-h19"],"title":"Solving for A and B","text":"Rearrange the inequality such that all the \'x\'s are on one side and the constants are on the other.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs100a-h21","type":"hint","dependencies":["a2a5cd1alginequalabs100a-h20"],"title":"Solving for A and B","text":"Subtract $$3$$ from the left to get $$B=9-3$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs100a-h22","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a2a5cd1alginequalabs100a-h21"],"title":"Solving for A and B","text":"What is $$9-3$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs100a-h23","type":"hint","dependencies":["a2a5cd1alginequalabs100a-h22"],"title":"Validating the Assumption","text":"b\' had an assumption that $$b$$ $$\\\\geq$$ $$0$$ such that $$|a|<b$$ is the same as $$a<b$$ AND $$a>-b$$. Since B is equivalent to \'b\', B $$\\\\geq$$ $$0$$ for the answer to be correct.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs100a-h24","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a2a5cd1alginequalabs100a-h23"],"title":"Validating the Assumption","text":"Is B $$\\\\geq$$ $$0$$ when $$B=6$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]}]}}]},{"id":"a2a5cd1alginequalabs1002","title":"Inequalities and Absolute Values: Part B","body":"These problems are harder, often highlighting an important subtlety.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Inequalities and Absolute Values","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a2a5cd1alginequalabs1002a","stepAnswer":["No solution"],"problemType":"MultipleChoice","stepTitle":"Determine which positive numbers \'x\' satisfy the condition $$|x+7|<6$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["No solution","$$0<x<13$$","$$x=1$$","All positive numbers"],"hints":{"DefaultPathway":[{"id":"a2a5cd1alginequalabs1002a-h1","type":"hint","dependencies":[],"title":"Expanding the Inequality","text":"For some numbers a and $$b$$ $$\\\\geq$$ $$0$$ where $$|a|<b$$, this is the same as $$a<b$$ AND $$a>-b$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs1002a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x+7<6$$ and $$x+7>-6$$"],"dependencies":["a2a5cd1alginequalabs1002a-h1"],"title":"Expanding the Inequality","text":"What is $$|x+7|<6$$ rewritten without the absolute value sign?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$x+7<6$$ and $$x+7>-6$$","$$x+7<6$$ and $$x+7<-6$$","$$x+7<6$$ and 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$$6-7$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs1002a-h6","type":"hint","dependencies":["a2a5cd1alginequalabs1002a-h5"],"title":"Simplifying the Inequality: Part B","text":"Rearrange the inequality such that all the \'x\'s are on one side and the constants are on the other.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs1002a-h7","type":"hint","dependencies":["a2a5cd1alginequalabs1002a-h6"],"title":"Simplifying the Inequality: Part B","text":"Subtract $$7$$ from the left to get $$x>-6-7$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs1002a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-13$$"],"dependencies":["a2a5cd1alginequalabs1002a-h7"],"title":"Simplifying the Inequality: Part B","text":"What is $$-6-7$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs1002a-h9","type":"hint","dependencies":["a2a5cd1alginequalabs1002a-h8"],"title":"ANDing the Distance","text":"Since the distance is less than a value, the inequalities provide the upper and lower bounds of the value by ANDing them together.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs1002a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-13$$"],"dependencies":["a2a5cd1alginequalabs1002a-h9"],"title":"Determining the Bounds","text":"What is the lower bound of the inequality $$x<-1$$ and $$x>-13$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$-\\\\infty$$","$$-1$$","$$-13$$"]},{"id":"a2a5cd1alginequalabs1002a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-1$$"],"dependencies":["a2a5cd1alginequalabs1002a-h9"],"title":"Determining the Bounds","text":"What is the upper bound of the inequality $$x<-1$$ and $$x>-13$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\infty$$","$$-1$$","$$-13$$"]},{"id":"a2a5cd1alginequalabs1002a-h12","type":"hint","dependencies":["a2a5cd1alginequalabs1002a-h10","a2a5cd1alginequalabs1002a-h11"],"title":"Positive Numbers","text":"Since the question asks for all positive numbers \'x\', $$x>0$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs1002a-h13","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a2a5cd1alginequalabs1002a-h12"],"title":"Positive Numbers","text":"Are any values between $$-13<x<-1$$ positive?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]}]}}]},{"id":"a2a5cd1alginequalabs101","title":"Inequalities and Absolute Values: Part C","body":"These questions are challenging, requiring mastery of each concept and their interrelations. 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$$2-10<x$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs101a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-8$$"],"dependencies":["a2a5cd1alginequalabs101a-h4"],"title":"Simplifying $$|2-x|<10$$: Part A","text":"What is $$2-10$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs101a-h6","type":"hint","dependencies":["a2a5cd1alginequalabs101a-h5"],"title":"Simplifying $$|2-x|<10$$: Part B","text":"Rearrange the inequality such that all the \'x\'s are on one side and the constants are on the other.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs101a-h7","type":"hint","dependencies":["a2a5cd1alginequalabs101a-h6"],"title":"Simplifying $$|2-x|<10$$: Part B","text":"Add $$x$$ from the left and add $$10$$ from the right to get $$2+10>x$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs101a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a2a5cd1alginequalabs101a-h7"],"title":"Simplifying $$|2-x|<10$$: Part B","text":"What is $$2+10$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs101a-h9","type":"hint","dependencies":["a2a5cd1alginequalabs101a-h8"],"title":"Inequalities of $$|2-x|<10$$","text":"Since the distance is less than a value, the inequalities provide the upper and lower bounds of the value by ANDing them together: $$-8<x<12$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs101a-h10","type":"hint","dependencies":["a2a5cd1alginequalabs101a-h9"],"title":"Expanding $$|x+4|$$ $$\\\\geq$$ $$5$$","text":"For some numbers a and $$b$$ $$\\\\geq$$ $$0$$ where $$|a|$$ $$\\\\geq$$ $$b$$, this is the same as a $$\\\\geq$$ $$b$$ OR a $$\\\\leq$$ $$-b$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs101a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x+4$$ $$\\\\geq$$ $$5$$ or $$x+4$$ $$\\\\leq$$ $$-5$$"],"dependencies":["a2a5cd1alginequalabs101a-h10"],"title":"Expanding $$|x+4|$$ $$\\\\geq$$ $$5$$","text":"What is $$|x+4|$$ $$\\\\geq$$ $$5$$ rewritten without the absolute value sign?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$x+4$$ $$\\\\geq$$ $$5$$ or $$x+4$$ $$\\\\leq$$ $$-5$$","$$x+4$$ $$\\\\geq$$ $$5$$ or $$x+4$$ $$\\\\leq$$ $$5$$","$$x+4$$ $$\\\\geq$$ $$5$$ or $$x+4$$ $$\\\\geq$$ $$-5$$","$$x+4$$ $$\\\\geq$$ $$5$$"]},{"id":"a2a5cd1alginequalabs101a-h12","type":"hint","dependencies":["a2a5cd1alginequalabs101a-h11"],"title":"Simplifying $$|x+4|$$ $$\\\\geq$$ 5: Part A","text":"Rearrange the inequality such that all the \'x\'s are on one side and the constants are on the other.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs101a-h13","type":"hint","dependencies":["a2a5cd1alginequalabs101a-h12"],"title":"Simplifying $$|x+4|$$ $$\\\\geq$$ 5: Part A","text":"Subtract $$4$$ from the left to get $$x$$ $$\\\\geq$$ $$5-4$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs101a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a2a5cd1alginequalabs101a-h13"],"title":"Simplifying $$|x+4|$$ $$\\\\geq$$ 5: Part A","text":"What is $$5-4$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs101a-h15","type":"hint","dependencies":["a2a5cd1alginequalabs101a-h14"],"title":"Simplifying $$|x+4|$$ $$\\\\geq$$ 5: Part B","text":"Rearrange the inequality such that all the \'x\'s are on one side and the constants are on the other.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs101a-h16","type":"hint","dependencies":["a2a5cd1alginequalabs101a-h15"],"title":"Simplifying $$|x+4|$$ $$\\\\geq$$ 5: Part B","text":"Subtract $$4$$ from the left to get $$x$$ $$\\\\leq$$ $$-5-4$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs101a-h17","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":["a2a5cd1alginequalabs101a-h16"],"title":"Simplifying $$|x+4|$$ $$\\\\geq$$ 5: Part B","text":"What is $$-5-4$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs101a-h18","type":"hint","dependencies":["a2a5cd1alginequalabs101a-h17"],"title":"Inequalities of $$|x+4|$$ $$\\\\geq$$ $$5$$","text":"Since the distance is greater than a value, the value outside of the distance are considered valid, ORing the two inequalities together: $$x$$ $$\\\\geq$$ $$1$$ or $$x$$ $$\\\\leq$$ $$-9$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs101a-h19","type":"hint","dependencies":["a2a5cd1alginequalabs101a-h18"],"title":"Interval Notation","text":"Since the two inequalities are separated by an AND statement, the result will contain the intersection or overlap of $$-8<x<12$$ AND $$x$$ $$\\\\leq$$ $$-9$$ or $$x$$ $$\\\\geq$$ $$1$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs101a-h20","type":"hint","dependencies":["a2a5cd1alginequalabs101a-h19"],"title":"Interval Notation: Part A","text":"The inequality $$-8<x<12$$ AND $$x$$ $$\\\\leq$$ $$-9$$ or $$x$$ $$\\\\geq$$ $$1$$ can be written in interval notation by specifying the lower bound first, followed by the upper bound.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs101a-h21","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$1$$"],"dependencies":["a2a5cd1alginequalabs101a-h20"],"title":"Interval Notation","text":"What is the lower bound that overlaps both $$-8<x<12$$ AND $$x$$ $$\\\\leq$$ $$-9$$ or $$x$$ $$\\\\geq$$ 1?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$-\\\\infty$$","$$-8$$","$$-9$$","$$1$$"]},{"id":"a2a5cd1alginequalabs101a-h22","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$12$$"],"dependencies":["a2a5cd1alginequalabs101a-h20"],"title":"Interval Notation","text":"What is the upper bound that overlaps both $$-8<x<12$$ AND $$x$$ $$\\\\leq$$ $$-9$$ or $$x$$ $$\\\\geq$$ 1?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\infty$$","$$-8$$","$$12$$","$$1$$"]},{"id":"a2a5cd1alginequalabs101a-h23","type":"hint","dependencies":["a2a5cd1alginequalabs101a-h21","a2a5cd1alginequalabs101a-h22"],"title":"Interval Notation","text":"If the bound itself should be included as a valid answer of \'x\', meaning it can be equal to that value, then a bracket (\\"[\\" or \\"]\\") should be used. Otherwise, use a paranthesis (\\"(\\" or \\")\\").","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs101a-h24","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a2a5cd1alginequalabs101a-h23"],"title":"Interval Notation","text":"Is the lower bound $$1$$ included as a valid value of \'x\' in $$-8<x<12$$ AND $$x$$ $$\\\\leq$$ $$-9$$ or $$x$$ $$\\\\geq$$ 1?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"a2a5cd1alginequalabs101a-h25","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a2a5cd1alginequalabs101a-h23"],"title":"Interval Notation","text":"Is the upper bound $$12$$ included as a valid value of \'x\' in $$-8<x<12$$ AND $$x$$ $$\\\\leq$$ $$-9$$ or $$x$$ $$\\\\geq$$ 1?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]}]}}]},{"id":"a2a5cd1alginequalabs102","title":"Inequalities and Absolute Values: Part C","body":"These questions are challenging, requiring mastery of each concept and their interrelations. Determine which numbers \'x\' satisfy the following condition. Express you answer in interval notation.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Inequalities and Absolute Values","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a2a5cd1alginequalabs102a","stepAnswer":["$$(-9,8)$$"],"problemType":"MultipleChoice","stepTitle":"$$2<x<8$$ or $$|3+x|<6$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-9,8)$$","choices":["$$(-9,8)$$","$$(2,8)$$","$$(-9,3)$$","$$(2,3)$$"],"hints":{"DefaultPathway":[{"id":"a2a5cd1alginequalabs102a-h1","type":"hint","dependencies":[],"title":"Expanding $$|3+x|<6$$","text":"For some numbers a and $$b$$ $$\\\\geq$$ $$0$$ where $$|a|<b$$, this is the same as $$a<b$$ AND $$a>-b$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs102a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3+x<6$$ and $$3+x>-6$$"],"dependencies":["a2a5cd1alginequalabs102a-h1"],"title":"Expanding $$|3+x|<6$$","text":"What is $$|3+x|<6$$ rewritten without the absolute value sign?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$3+x<6$$ and $$3+x>-6$$","$$3+x<6$$ and $$3+x>6$$","$$3+x<6$$ and $$3+x<-6$$","$$3+x<6$$"]},{"id":"a2a5cd1alginequalabs102a-h3","type":"hint","dependencies":["a2a5cd1alginequalabs102a-h2"],"title":"Simplifying $$|3+x|<6$$: Part A","text":"Rearrange the inequality such that all the \'x\'s are on one side and the constants are on the other.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs102a-h4","type":"hint","dependencies":["a2a5cd1alginequalabs102a-h3"],"title":"Simplifying $$|3+x|<6$$: Part A","text":"Subtract $$3$$ from the left to get $$x<6-3$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs102a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a2a5cd1alginequalabs102a-h4"],"title":"Simplifying $$|3+x|<6$$: Part A","text":"What is $$6-3$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs102a-h6","type":"hint","dependencies":["a2a5cd1alginequalabs102a-h5"],"title":"Simplifying $$|3+x|<6$$: Part B","text":"Rearrange the inequality such that all the \'x\'s are on one side and the constants are on the other.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs102a-h7","type":"hint","dependencies":["a2a5cd1alginequalabs102a-h6"],"title":"Simplifying $$|3+x|<6$$: Part B","text":"Subtract $$3$$ from the left to get $$x>-6-3$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs102a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":["a2a5cd1alginequalabs102a-h7"],"title":"Simplifying $$|3+x|<6$$: Part B","text":"What is $$-6-3$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs102a-h9","type":"hint","dependencies":["a2a5cd1alginequalabs102a-h8"],"title":"Inequalities of $$|3+x|<6$$","text":"Since the distance is less than a value, the inequalities provide the upper and lower bounds of the value by ANDing them together: $$-9<x<3$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs102a-h10","type":"hint","dependencies":["a2a5cd1alginequalabs102a-h9"],"title":"Interval Notation","text":"Since the two inequalities are separated by an OR statement, the result will contain values within either $$2<x<8$$ OR $$-9<x<3$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs102a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a2a5cd1alginequalabs102a-h10"],"title":"Interval Notation","text":"Is there an overlap between $$2<x<8$$ and $$-9<x<3$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"a2a5cd1alginequalabs102a-h12","type":"hint","dependencies":["a2a5cd1alginequalabs102a-h11"],"title":"Interval Notation","text":"Since there is an overlap between $$2<x<8$$ and $$-9<x<3$$, the inequalities can be combined to one inequality using the lowest number as the lower bound and the higest number as the higher bound.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs102a-h13","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-9$$"],"dependencies":["a2a5cd1alginequalabs102a-h12"],"title":"Interval Notation","text":"What is the lower bound of $$2<x<8$$ OR $$-9<x<3$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$-\\\\infty$$","$$-9$$","$$2$$","$$3$$"]},{"id":"a2a5cd1alginequalabs102a-h14","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$8$$"],"dependencies":["a2a5cd1alginequalabs102a-h12"],"title":"Interval Notation","text":"What is the upper bound of $$2<x<8$$ OR $$-9<x<3$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\infty$$","$$8$$","$$2$$","$$3$$"]},{"id":"a2a5cd1alginequalabs102a-h15","type":"hint","dependencies":["a2a5cd1alginequalabs102a-h13","a2a5cd1alginequalabs102a-h14"],"title":"Interval Notation","text":"If the bound itself should be included as a valid answer of \'x\', meaning it can be equal to that value, then a bracket (\\"[\\" or \\"]\\") should be used. Otherwise, use a paranthesis (\\"(\\" or \\")\\").","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs102a-h16","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a2a5cd1alginequalabs102a-h15"],"title":"Interval Notation","text":"Is the lower bound $$-9$$ included as a valid value of \'x\' in $$2<x<8$$ OR $$-9<x<3$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"a2a5cd1alginequalabs102a-h17","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a2a5cd1alginequalabs102a-h15"],"title":"Interval Notation","text":"Is the upper bound $$8$$ included as a valid value of \'x\' in $$2<x<8$$ OR $$-9<x<3$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]}]}}]},{"id":"a2a5cd1alginequalabs11","title":"Inequalities and Absolute Values: Part B","body":"These problems are harder, often highlighting an important subtlety. Determine which numbers $$x$$ satisfy each inequality. Express your answers in interval notation.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Inequalities and Absolute Values","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a2a5cd1alginequalabs11a","stepAnswer":["$$(-\\\\infty,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$|x-8|$$ $$\\\\geq$$ $$0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\infty)$$","choices":["$$(-\\\\infty,\\\\infty)$$","$$(-\\\\infty,8)$$ $$\\\\cup$$ $$(8,\\\\infty)$$","[8]","No solution"],"hints":{"DefaultPathway":[{"id":"a2a5cd1alginequalabs11a-h1","type":"hint","dependencies":[],"title":"Properties of Absolute Values","text":"For any value a, $$|a|$$ $$\\\\geq$$ $$0$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs11a-h2","type":"hint","dependencies":["a2a5cd1alginequalabs11a-h1"],"title":"Properties of Absolute Values","text":"Because the inequality takes the absolute value of $$x-8$$, $$|x-8|$$ will always be greater than or equal to $$0$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]}},{"id":"a2a5cd1alginequalabs11b","stepAnswer":["$$(-\\\\infty,8)$$ $$\\\\cup$$ $$(8,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$|x-8|>0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,8)$$ $$\\\\cup$$ $$(8,\\\\infty)$$","choices":["$$(-\\\\infty,\\\\infty)$$","$$(-\\\\infty,8)$$ $$\\\\cup$$ $$(8,\\\\infty)$$","[8]","No solution"],"hints":{"DefaultPathway":[{"id":"a2a5cd1alginequalabs11b-h1","type":"hint","dependencies":[],"title":"Expanding the Inequality","text":"For some numbers a and $$b$$ $$\\\\geq$$ $$0$$ where $$|a|>b$$, this is the same as $$a>b$$ OR $$a<-b$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs11b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x-8>0$$ or $$x-8<0$$"],"dependencies":["a2a5cd1alginequalabs11b-h1"],"title":"Expanding the Inequality","text":"What is $$|x-8|>0$$ rewritten without the absolute value sign?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$x-8>0$$ or $$x-8<0$$","$$x-8>0$$","$$x-8<0$$"]},{"id":"a2a5cd1alginequalabs11b-h3","type":"hint","dependencies":["a2a5cd1alginequalabs11b-h2"],"title":"Simplifying the Inequality: Part A","text":"Rearrange the inequality such that all the \'x\'s are on one side and the constants are on the other.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs11b-h4","type":"hint","dependencies":["a2a5cd1alginequalabs11b-h3"],"title":"Simplifying the Inequality: Part A","text":"Add $$8$$ from the left to get $$x>0+8$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs11b-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a2a5cd1alginequalabs11b-h4"],"title":"Simplifying the Inequality: Part A","text":"What is $$0+8$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs11b-h6","type":"hint","dependencies":["a2a5cd1alginequalabs11b-h5"],"title":"Simplifying the Inequality: Part B","text":"Rearrange the inequality such that all the \'x\'s are on one side and the constants are on the other.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs11b-h7","type":"hint","dependencies":["a2a5cd1alginequalabs11b-h6"],"title":"Simplifying the Inequality: Part B","text":"Add $$8$$ from the left to get $$x<0+8$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs11b-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a2a5cd1alginequalabs11b-h7"],"title":"Simplifying the Inequality: Part B","text":"What is $$0+8$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs11b-h9","type":"hint","dependencies":["a2a5cd1alginequalabs11b-h8"],"title":"ORing the Distance","text":"Since the distance is greater than a value, the value outside of the distance are considered valid, ORing the two inequalities together.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs11b-h10","type":"hint","dependencies":["a2a5cd1alginequalabs11b-h9"],"title":"Interval Notation","text":"Since the two inequalities are separated by an OR statement, they will both appear in the final result.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs11b-h11","type":"hint","dependencies":["a2a5cd1alginequalabs11b-h10"],"title":"Interval Notation: Part A","text":"The inequality $$x>8$$ can be written in interval notation by specifying the lower bound first, followed by the upper bound.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs11b-h12","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$8$$"],"dependencies":["a2a5cd1alginequalabs11b-h11"],"title":"Interval Notation: Part A","text":"What is the lower bound of the inequality $$x>8$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$-\\\\infty$$","$$8$$"]},{"id":"a2a5cd1alginequalabs11b-h13","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\infty$$"],"dependencies":["a2a5cd1alginequalabs11b-h11"],"title":"Interval Notation: Part A","text":"What is the upper bound of the inequality $$x>8$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\infty$$","$$8$$"]},{"id":"a2a5cd1alginequalabs11b-h14","type":"hint","dependencies":["a2a5cd1alginequalabs11b-h12","a2a5cd1alginequalabs11b-h13"],"title":"Interval Notation: Part A","text":"If the bound itself should be included as a valid answer of \'x\', meaning it can be equal to that value, then a bracket (\\"[\\" or \\"]\\") should be used. Otherwise, use a paranthesis (\\"(\\" or \\")\\").","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs11b-h15","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a2a5cd1alginequalabs11b-h14"],"title":"Interval Notation: Part A","text":"Is the lower bound $$8$$ included as a valid value of \'x\' in $$x>8$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"a2a5cd1alginequalabs11b-h16","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a2a5cd1alginequalabs11b-h14"],"title":"Interval Notation: Part A","text":"Is the upper bound $$\\\\infty$$ included as a valid value of \'x\' in $$x>8$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"],"subHints":[{"id":"a2a5cd1alginequalabs11b-h16-s1","type":"hint","dependencies":[],"title":"Interval Notation: Part A","text":"As $$\\\\infty$$ is not a number, it cannot be included as part of a valid value or bound for \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a2a5cd1alginequalabs11b-h17","type":"hint","dependencies":["a2a5cd1alginequalabs11b-h15","a2a5cd1alginequalabs11b-h16"],"title":"Interval Notation: Part B","text":"The inequality $$x<8$$ can be written in interval notation by specifying the lower bound first, followed by the upper bound.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs11b-h18","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-\\\\infty$$"],"dependencies":["a2a5cd1alginequalabs11b-h17"],"title":"Interval Notation: Part B","text":"What is the lower bound of the inequality $$x<8$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$-\\\\infty$$","$$8$$"]},{"id":"a2a5cd1alginequalabs11b-h19","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$8$$"],"dependencies":["a2a5cd1alginequalabs11b-h17"],"title":"Interval Notation: Part B","text":"What is the upper bound of the inequality $$x<8$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\infty$$","$$8$$"]},{"id":"a2a5cd1alginequalabs11b-h20","type":"hint","dependencies":["a2a5cd1alginequalabs11b-h18","a2a5cd1alginequalabs11b-h19"],"title":"Interval Notation: Part B","text":"If the bound itself should be included as a valid answer of \'x\', meaning it can be equal to that value, then a bracket (\\"[\\" or \\"]\\") should be used. Otherwise, use a paranthesis (\\"(\\" or \\")\\").","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs11b-h21","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a2a5cd1alginequalabs11b-h20"],"title":"Interval Notation: Part B","text":"Is the lower bound $$-\\\\infty$$ included as a valid value of \'x\' in $$x<8$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"],"subHints":[{"id":"a2a5cd1alginequalabs11b-h21-s1","type":"hint","dependencies":[],"title":"Interval Notation: Part B","text":"As $$-\\\\infty$$ is not a number, it cannot be included as part of a valid value or bound for \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a2a5cd1alginequalabs11b-h22","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a2a5cd1alginequalabs11b-h20"],"title":"Interval Notation: Part B","text":"Is the upper bound $$8$$ included as a valid value of \'x\' in $$x<8$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"a2a5cd1alginequalabs11b-h23","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a2a5cd1alginequalabs11b-h21","a2a5cd1alginequalabs11b-h22"],"title":"Interval Notation","text":"Can $$(-\\\\infty,8)$$ and $$(8,\\\\infty)$$ be combined into one interval?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"a2a5cd1alginequalabs11b-h24","type":"hint","dependencies":["a2a5cd1alginequalabs11b-h23"],"title":"Interval Notation","text":"The bounds for the two equations, $$(-\\\\infty,8)$$ and $$(8,\\\\infty)$$, can be ORed together using \' $$\\\\cup$$ \'.","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a2a5cd1alginequalabs11b","title":"Inequalities and Absolute Values: Part B","body":"These problems are harder, often highlighting an important subtlety. Determine which numbers $$x$$ satisfy each inequality. Express your answers in interval notation.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Inequalities and Absolute Values","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a2a5cd1alginequalabs11ba","stepAnswer":["No solution"],"problemType":"MultipleChoice","stepTitle":"$$|2-x|<0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$(-\\\\infty,\\\\infty)$$","$$(-\\\\infty,-2)$$ $$\\\\cup$$ $$(-2,\\\\infty)$$","$$(-\\\\infty,2)$$ $$\\\\cup$$ $$(2,\\\\infty)$$","No solution"],"hints":{"DefaultPathway":[{"id":"a2a5cd1alginequalabs11ba-h1","type":"hint","dependencies":[],"title":"Properties of Absolute Values","text":"For any value a, $$|a|$$ $$\\\\geq$$ $$0$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs11ba-h2","type":"hint","dependencies":["a2a5cd1alginequalabs11ba-h1"],"title":"Properties of Absolute Values","text":"Because the inequality takes the absolute value of $$2-x$$, $$|2-x|$$ will always be greater than or equal to $$0$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs11ba-h3","type":"hint","dependencies":["a2a5cd1alginequalabs11ba-h2"],"title":"Properties of Absolute Values","text":"Because $$|2-x|$$ will always be greater than or equal to $$0$$, there will never be some value \'x\' which makes $$|2-x|$$ less than $$0$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a2a5cd1alginequalabs2","title":"Inequalities and Absolute Values: Part A","body":"These questions test your knowledge of the core concepts. Express each of the following statements as a mathematical equation or inequality.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Inequalities and Absolute Values","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a2a5cd1alginequalabs2a","stepAnswer":["$$-6$$ $$\\\\leq$$ $$x$$ $$\\\\leq$$ $$14$$"],"problemType":"MultipleChoice","stepTitle":"The distance between the number $$2x$$ and $$8$$ is at most $$20$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-6$$ $$\\\\leq$$ $$x$$ $$\\\\leq$$ $$14$$","choices":["$$-6$$ $$\\\\leq$$ $$x$$ $$\\\\leq$$ $$14$$","$$-6<x<14$$","$$x$$ $$\\\\leq$$ $$-6$$ or $$x$$ $$\\\\geq$$ $$14$$","$$x<-6$$ or $$x>14$$"],"hints":{"DefaultPathway":[{"id":"a2a5cd1alginequalabs2a-h1","type":"hint","dependencies":[],"title":"Understanding Distance","text":"Distance\' is used to refer to the absolute value between two numbers a,b: $$|a-b|$$ since distance does not care about whether the value is positive or negative. For example, four units away from zero could either refer to $$4$$ or $$-4$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs2a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$|2x-8|$$"],"dependencies":["a2a5cd1alginequalabs2a-h1"],"title":"Understanding Distance","text":"What is the distance between $$2x$$ and $$8$$ written as an absolute value?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$|2x-8|$$","$$|2x+8|$$"]},{"id":"a2a5cd1alginequalabs2a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\leq$$"],"dependencies":["a2a5cd1alginequalabs2a-h2"],"title":"Inequality of \'at most 20\'","text":"What symbol does \'at most\' refer to?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["<",">","$$\\\\leq$$","$$\\\\geq$$"],"subHints":[{"id":"a2a5cd1alginequalabs2a-h3-s1","type":"hint","dependencies":[],"title":"at most 20\' as a Symbol","text":"The sentence \'a is at most b\' means a is less than or equal to $$b$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a2a5cd1alginequalabs2a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$|2x-8|$$ $$\\\\leq$$ $$20$$"],"dependencies":["a2a5cd1alginequalabs2a-h3"],"title":"Inequality of \'at most 20\'","text":"What inequality represents \'the distance between the number $$2x$$ and $$8$$ is at most 20\'?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$|2x-8|$$ $$\\\\leq$$ $$20$$","$$20$$ $$\\\\leq$$ $$|2x-8|$$"]},{"id":"a2a5cd1alginequalabs2a-h5","type":"hint","dependencies":["a2a5cd1alginequalabs2a-h4"],"title":"Expanding the Inequality","text":"For some numbers a and $$b$$ $$\\\\geq$$ $$0$$ where $$|a|$$ $$\\\\leq$$ $$b$$, this is the same as a $$\\\\leq$$ $$b$$ AND a $$\\\\geq$$ $$-b$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs2a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2x-8$$ $$\\\\leq$$ $$20$$ and $$2x-8$$ $$\\\\geq$$ $$-20$$"],"dependencies":["a2a5cd1alginequalabs2a-h5"],"title":"Expanding the Inequality","text":"What is $$|2x-8|$$ $$\\\\leq$$ $$20$$ rewritten without the absolute value sign?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$2x-8$$ $$\\\\leq$$ $$20$$ and $$2x-8$$ $$\\\\geq$$ $$-20$$","$$2x-8$$ $$\\\\leq$$ $$20$$ and $$2x-8$$ $$\\\\geq$$ $$20$$","$$2x-8$$ $$\\\\leq$$ $$20$$ and $$2x-8$$ $$\\\\leq$$ $$-20$$","$$2x-8$$ $$\\\\leq$$ $$20$$"]},{"id":"a2a5cd1alginequalabs2a-h7","type":"hint","dependencies":["a2a5cd1alginequalabs2a-h6"],"title":"Simplifying the Inequality: Part A","text":"Rearrange the inequality such that all the \'x\'s are on one side and the constants are on the other.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs2a-h8","type":"hint","dependencies":["a2a5cd1alginequalabs2a-h7"],"title":"Simplifying the Inequality: Part A","text":"Add $$8$$ from the left to get $$2x$$ $$\\\\leq$$ $$20+8$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs2a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$28$$"],"dependencies":["a2a5cd1alginequalabs2a-h8"],"title":"Simplifying the Inequality: Part A","text":"What is $$20+8$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs2a-h10","type":"hint","dependencies":["a2a5cd1alginequalabs2a-h9"],"title":"Simplifying the Inequality: Part A","text":"Divide $$2$$ from both sides of $$2x$$ $$\\\\leq$$ $$28$$ to isolate \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs2a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$14$$"],"dependencies":["a2a5cd1alginequalabs2a-h10"],"title":"Simplifying the Inequality: Part A","text":"What is $$\\\\frac{28}{2}$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs2a-h12","type":"hint","dependencies":["a2a5cd1alginequalabs2a-h11"],"title":"Simplifying the Inequality: Part B","text":"Rearrange the inequality such that all the \'x\'s are on one side and the constants are on the other.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs2a-h13","type":"hint","dependencies":["a2a5cd1alginequalabs2a-h12"],"title":"Simplifying the Inequality: Part B","text":"Add $$8$$ from the left to get $$2x$$ $$\\\\geq$$ $$-20+8$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs2a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-12$$"],"dependencies":["a2a5cd1alginequalabs2a-h13"],"title":"Simplifying the Inequality: Part B","text":"What is $$-20+8$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs2a-h15","type":"hint","dependencies":["a2a5cd1alginequalabs2a-h14"],"title":"Simplifying the Inequality: Part B","text":"Divide $$2$$ from both sides of $$2x$$ $$\\\\geq$$ $$-12$$ to isolate \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs2a-h16","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6$$"],"dependencies":["a2a5cd1alginequalabs2a-h15"],"title":"Simplifying the Inequality: Part B","text":"What is $$\\\\frac{-12}{2}$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs2a-h17","type":"hint","dependencies":["a2a5cd1alginequalabs2a-h16"],"title":"ANDing the Distance","text":"Since the distance is less than a value, the inequalities provide the upper and lower bounds of the value by ANDing them together.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs2a-h18","type":"hint","dependencies":["a2a5cd1alginequalabs2a-h17"],"title":"ANDing the Distance","text":"For some values a,b where a $$\\\\leq$$ $$x$$ and $$x$$ $$\\\\leq$$ $$b$$, these two inequalities can be combined to say a $$\\\\leq$$ $$x$$ $$\\\\leq$$ $$b$$","variabilization":{},"oer":"https://OATutor.io","license":""}]}},{"id":"a2a5cd1alginequalabs2b","stepAnswer":["$$x>59$$ or $$x<25$$"],"problemType":"MultipleChoice","stepTitle":"The distance between $$42$$ and $$x$$ is greater than 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For example, four units away from zero could either refer to $$4$$ or $$-4$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs2b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$|42-x|$$"],"dependencies":["a2a5cd1alginequalabs2b-h1"],"title":"Understanding Distance","text":"What is the distance between $$42$$ and $$x$$ written as an absolute value?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$|42-x|$$","$$|42+x|$$"]},{"id":"a2a5cd1alginequalabs2b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":[">"],"dependencies":["a2a5cd1alginequalabs2b-h2"],"title":"Inequality of \'greater than 17\'","text":"What symbol does \'greater than\' refer to?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["<",">","$$\\\\leq$$","$$\\\\geq$$"]},{"id":"a2a5cd1alginequalabs2b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$|42-x|>17$$"],"dependencies":["a2a5cd1alginequalabs2b-h3"],"title":"Inequality of \'greater than 17\'","text":"What inequality represents \'the distance between $$42$$ and $$x$$ is greater than 17\'?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$|42-x|>17$$","$$17>|42-x|$$"]},{"id":"a2a5cd1alginequalabs2b-h5","type":"hint","dependencies":["a2a5cd1alginequalabs2b-h4"],"title":"Expanding the Inequality","text":"For some numbers a and $$b$$ $$\\\\geq$$ $$0$$ where $$|a|>b$$, this is the same as $$a>b$$ OR $$a<-b$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs2b-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$42-x>17$$ or $$42-x<-17$$"],"dependencies":["a2a5cd1alginequalabs2b-h5"],"title":"Expanding the Inequality","text":"What is $$|42-x|>17$$ rewritten without the absolute value sign?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$42-x>17$$ or $$42-x<-17$$","$$42-x>17$$ or $$42-x<17$$","$$42-x>17$$ or $$42-x>-17$$","$$42-x>17$$"]},{"id":"a2a5cd1alginequalabs2b-h7","type":"hint","dependencies":["a2a5cd1alginequalabs2b-h6"],"title":"Simplifying the Inequality: Part A","text":"Rearrange the inequality such that all the \'x\'s are on one side and the constants are on the other.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs2b-h8","type":"hint","dependencies":["a2a5cd1alginequalabs2b-h7"],"title":"Simplifying the Inequality: Part A","text":"Subtract $$17$$ from the left and add $$x$$ from the right to get $$42-17>x$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs2b-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["a2a5cd1alginequalabs2b-h8"],"title":"Simplifying the Inequality: Part A","text":"What is $$42-17$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs2b-h10","type":"hint","dependencies":["a2a5cd1alginequalabs2b-h9"],"title":"Simplifying the Inequality: Part B","text":"Rearrange the inequality such that all the \'x\'s are on one side and the constants are on the other.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs2b-h11","type":"hint","dependencies":["a2a5cd1alginequalabs2b-h10"],"title":"Simplifying the Inequality: Part B","text":"Add $$17$$ from the left and add $$x$$ from the right to get $$42+17<x$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs2b-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$59$$"],"dependencies":["a2a5cd1alginequalabs2b-h11"],"title":"Simplifying the Inequality: Part B","text":"What is $$42+17$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a2a5cd1alginequalabs2b-h13","type":"hint","dependencies":["a2a5cd1alginequalabs2b-h12"],"title":"ORing the Distance","text":"Since the distance is greater than a value, the value outside of the distance are considered valid, ORing the two inequalities together.","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a2c7986stat11","title":"Types of Sampling","body":"The instructor takes her sample by gathering data on five randomly selected students from each Lake Tahoe Community College math class.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Frequency, Frequency Tables, and Levels of Measurement","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a2c7986stat11a","stepAnswer":["Stratified Sampling"],"problemType":"MultipleChoice","stepTitle":"The type of sampling she used is:","stepBody":"","answerType":"string","variabilization":{},"choices":["Cluster sampling","Convenience sampling","Simple random sampling","Stratified Sampling","Stratified sampling"],"hints":{"DefaultPathway":[{"id":"a2c7986stat11a-h1","type":"hint","dependencies":[],"title":"Divide Up the Population","text":"There are two types of sampling that divide the population into groups of people: stratified and cluster. The difference between the two are that stratified sampling takes a proportionate number of subjects from each stratum to include into the sample while clustering usually means only a few clusters are selected to represent the entire population.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2c7986stat11a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Stratified"],"dependencies":["a2c7986stat11a-h1"],"title":"Stratified versus Cluster","text":"Does the instructor want to pick students from each group that they\'ve created based on Lake Tahoe Community College math classes? Or do they only want one or two groups to be represented? If the instructor wants each Lake Tahoe Community College math class to be included in the sample select stratified. If the instructor only wants one or two groups to be represented in full, select cluster.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Stratified","Cluster"]}]}}]},{"id":"a2c7986stat12","title":"Determining Percentages from Tables","body":"Using the table, determine the percentages desired.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Frequency, Frequency Tables, and Levels of Measurement","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a2c7986stat12a","stepAnswer":["$$23$$"],"problemType":"TextBox","stepTitle":"From the table provided, find the percentage of heights that are less than $$65.95$$ inches.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$23$$","hints":{"DefaultPathway":[{"id":"a2c7986stat12a-h1","type":"hint","dependencies":[],"title":"Figure Out the Rows to Use","text":"First, determine which rows from the table to look at. Since we want heights that are less than $$65.95$$ inches, we note that we want to only look at the first three rows because everything past that will have a larger height.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2c7986stat12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$23$$"],"dependencies":["a2c7986stat12a-h1"],"title":"Add Up the Data from Chosen Rows","text":"Now, knowing the frequency of each row, we can add up the first three rows. What is the sum?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a2c7986stat12a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$23$$"],"dependencies":[],"title":"Add Up the Data from Chosen Rows","text":"What is $$5+3+15$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a2c7986stat12a-h3","type":"hint","dependencies":["a2c7986stat12a-h2"],"title":"Definition of Percentages","text":"The definition of a percentage is the proportion out of 100%. Since we know that there are $$23$$ players out of a total of $$100$$ players (the total of the frequency column) that are less than $$65.95$$ inches tall, we can determine the percentage of players that is less than $$65.95$$ inches tall.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2c7986stat12a-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$23$$"],"dependencies":["a2c7986stat12a-h3"],"title":"Find the Overall Percentage of Heights","text":"What is $$23$$ / $$100$$ written as a percentage?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2c7986stat13","title":"Determining Percentages from Tables","body":"Using the table, determine the percentages desired.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Frequency, Frequency Tables, and Levels of Measurement","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a2c7986stat13a","stepAnswer":["$$18$$"],"problemType":"TextBox","stepTitle":"From the table provided, find the percentage of heights that fall between $$61.95$$ and $$65.96$$ inches.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$18$$","hints":{"DefaultPathway":[{"id":"a2c7986stat13a-h1","type":"hint","dependencies":[],"title":"Figure Out the Rows to Use","text":"First, determine which rows from the table to look at. Since we want heights that fall between $$61.95$$ and $$65.95$$, we note that we want to only look at the second and third rows because everything in a later that will have a larger height and everything in an earlier row will have a smaller height.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2c7986stat13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18$$"],"dependencies":["a2c7986stat13a-h1"],"title":"Add Up the Relative Frequencies of the Chosen Rows","text":"Now, knowing which rows to access, we can actually look at the relative frequencies and add them together to get the percentage of players in that range. What is the sum of the relative frequencies, written as a percentage (multiply proportions by 100)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a2c7986stat13a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18$$"],"dependencies":[],"title":"Add Up the Relative Frequencies of the Chosen Rows","text":"What is $$100\\\\left(0.03+0.15\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a2c7986stat14","title":"$$\\\\frac{Mean}{Average}$$","body":"The table shows the commute time by state for workers at least $$16$$ years old who are not working at home.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Frequency, Frequency Tables, and Levels of Measurement","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a2c7986stat14a","stepAnswer":["$$23.46$$"],"problemType":"TextBox","stepTitle":"How much time does it take to travel to work on average? Find the mean travel time, and round off the answer properly.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$23.46$$","hints":{"DefaultPathway":[{"id":"a2c7986stat14a-h1","type":"hint","dependencies":[],"title":"Definition of an Average","text":"To find the $$\\\\frac{average}{mean}$$ of a data set, sum up all the values and then divide this sum by the number of elements.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2c7986stat14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1173.1$$"],"dependencies":["a2c7986stat14a-h1"],"title":"Sum All Values from the Table","text":"What is the sum of all of the values in the table?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a2c7986stat14a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$225.2$$"],"dependencies":[],"title":"Sum All Values from Row $$1$$","text":"We can go row by row. What is the sum of all the values in row 1?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2c7986stat14a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$230.6$$"],"dependencies":[],"title":"Sum All Values from Row $$2$$","text":"What is the sum of all of the values in the row 2?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2c7986stat14a-h2-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$247.9$$"],"dependencies":[],"title":"Sum All Values from Row $$3$$","text":"What is the sum of all the values in row 3?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2c7986stat14a-h2-s4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$225.2$$"],"dependencies":[],"title":"Sum All Values from Row $$4$$","text":"What is the sum of all the values in row 4?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2c7986stat14a-h2-s5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$244.2$$"],"dependencies":[],"title":"Sum All Values from Row $$5$$","text":"What is the sum of all the values in row 5?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2c7986stat14a-h2-s6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1173.1$$"],"dependencies":[],"title":"Total Sum of All Values from the Table","text":"What is the sum of all the rows?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2c7986stat14a-h2-s7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1173.1$$"],"dependencies":[],"title":"Total Sum of All Values from the Table","text":"What is $$225.2+230.6+247.9+225.2+244.2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a2c7986stat14a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$50$$"],"dependencies":["a2c7986stat14a-h2"],"title":"Finding the Total Number of Items in the Table","text":"What is the total number of items? In this case, how many workers were surveyed?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2c7986stat14a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$23.46$$"],"dependencies":["a2c7986stat14a-h3"],"title":"Determining Average from Total Sum and Total Number of Items","text":"What is the average time it takes to travel to work? In other words, what is the total number of minutes divided by the number of workers surveyed?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a2c7986stat14a-h4-s8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$23.46$$"],"dependencies":[],"title":"Determining Average from Total Sum and Total Number of Items","text":"What is $$\\\\frac{1173.1}{50}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a2c7986stat15","title":"Relative Frequency","body":"The table contains data on hurricanes that have made direct hits on the U.S. between $$1851$$ and $$2004$$. A hurricane is given a strength category rating based on the minimum wind speed generated by the storm.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Frequency, Frequency Tables, and Levels of Measurement","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a2c7986stat15a","stepAnswer":["$$0.0659$$"],"problemType":"MultipleChoice","stepTitle":"What is the relative frequency of direct hits that were category $$4$$ hurricanes?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$0.0659$$","choices":["$$0.0768$$","$$0.0659$$","$$0.2601$$","Not enough information to calculate"],"hints":{"DefaultPathway":[{"id":"a2c7986stat15a-h1","type":"hint","dependencies":[],"title":"Definition of Relative Frequency","text":"The relative frequency is essentially the proportion of that specific row. In this case, relative frequency is calculated as the number of direct hits in that specific category divided by the total number of hurricanes (273).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2c7986stat15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18$$"],"dependencies":["a2c7986stat15a-h1"],"title":"Determining the Frequency of Category $$4$$ Hurricanes","text":"What is the frequency of the number of direct hits in Category 4?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2c7986stat15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$273$$"],"dependencies":["a2c7986stat15a-h2"],"title":"Finding the Overall Total Number of Direct Hits","text":"What is the total number of direct hits?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2c7986stat15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.0659$$"],"dependencies":["a2c7986stat15a-h3"],"title":"Calculate Relative Frequency from Frequency of Category $$4$$ and Overall Total Hits","text":"What is the relative frequency of direct hits Category $$4$$ hurricanes? Note that relative frequency is calculated as frequency of category divided by total.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2c7986stat15a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.0659$$"],"dependencies":["a2c7986stat15a-h4"],"title":"Calculate Relative Frequency from Frequency of Category $$4$$ and Overall Total Hits","text":"What is $$\\\\frac{18}{273}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a2c7986stat15b","stepAnswer":["$$0.9231$$"],"problemType":"MultipleChoice","stepTitle":"What is the relative frequency of direct hits that were AT MOST a category $$3$$ storm?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$0.9231$$","choices":["$$0.3480$$","$$0.9231$$","$$0.2601$$","$$0.3370$$"],"hints":{"DefaultPathway":[{"id":"a2c7986stat15b-h6","type":"hint","dependencies":["a2c7986stat15a-h5"],"title":"Relationship Between Relative and Cumulative Frequency","text":"The cumulative frequency column accumulates the total relative frequency until that point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2c7986stat15b-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.9231$$"],"dependencies":["a2c7986stat15b-h6"],"title":"Finding Cumulative Frequency up Until Category $$3$$ Hurricanes","text":"To get the relative frequency of direct hits that were AT MOST a category $$3$$ storm, we can add up the cumulative frequency for Category $$2$$ (as it includes the relative frequencies for Category $$1$$ and Category 2) with the relative frequency of Category $$3$$. This will give the cumulative frequency up until Category $$3$$ as desired. What is that value?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a2c7986stat15b-h7-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.663$$"],"dependencies":[],"title":"Cumulative Frequency up Until Category $$2$$","text":"What is the cumulative frequency up until a Category $$2$$ hurricane?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2c7986stat15b-h7-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.2601$$"],"dependencies":[],"title":"Relative Frequency of Category $$3$$","text":"What is the relative frequency for a Category $$3$$ hurricane?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a2c7986stat15b-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.9231$$"],"dependencies":["a2c7986stat15b-h7"],"title":"Sum to Find Relative Frequency of Hits","text":"What is the sum of the cumulative frequency for Category $$2$$ and the relative frequency for Category 3?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a2c7986stat15b-h8-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.9231$$"],"dependencies":[],"title":"Sum to Find Relative Frequency of Hits","text":"What is $$0.663+0.2601$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a2d77b8chainrule1","title":"Chain Rule Exercises","body":"Given $$y=f(u)$$ and $$u=g(x)$$, find $$\\\\frac{dy}{dx}$$ in terms of $$x!$$","variabilization":{},"oer":"https://openstax.org/books/college-physics-2e/pages/3-problems-exercises <OpenStax: College Physics 2e>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 The Chain Rule","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a2d77b8chainrule1a","stepAnswer":["$$12x$$"],"problemType":"MultipleChoice","stepTitle":"$$y=3u-6$$, $$u=2x^2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$12x$$","choices":["$$12x$$","$$4x$$","$$16x$$","$$8x$$"],"hints":{"DefaultPathway":[{"id":"a2d77b8chainrule1a-h1","type":"hint","dependencies":[],"title":"The Chain Rule","text":"Remember, the chain rule in Leibniz\'s notation tells us that $$\\\\frac{dy}{dx}=\\\\frac{dy}{du} \\\\frac{du}{dx}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d77b8chainrule1a-h2","type":"hint","dependencies":["a2d77b8chainrule1a-h1"],"title":"Find the Derivative","text":"First, try finding the derivative of $$y$$ in terms of u. Then, find the derivative of u in terms of $$x$$. Multiplying the two should yield the correct answer!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d77b8chainrule1a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3$$"],"dependencies":["a2d77b8chainrule1a-h2"],"title":"Find $$\\\\frac{dy}{du}$$","text":"What is $$\\\\frac{dy}{du}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$3$$","$$\\\\frac{3}{2} u^2$$","3u","$$3u^2$$"]},{"id":"a2d77b8chainrule1a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$4x$$"],"dependencies":["a2d77b8chainrule1a-h3"],"title":"Find $$\\\\frac{du}{dx}$$","text":"What is $$\\\\frac{du}{dx}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$4x$$","$$\\\\frac{2}{3} x^3$$","$$4x^2$$","$$2x^3$$"]},{"id":"a2d77b8chainrule1a-h5","type":"hint","dependencies":["a2d77b8chainrule1a-h4"],"title":"Plug in","text":"Don\'t forget to plug in what we have for u in terms of $$x$$ to give our final answer in terms of $$x!$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2d77b8chainrule10","title":"$$y={cot}^{2\\\\left(x\\\\right)}$$","body":"","variabilization":{},"oer":"https://openstax.org/books/college-physics-2e/pages/3-problems-exercises <OpenStax: College Physics 2e>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 The Chain Rule","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a2d77b8chainrule10a","stepAnswer":["$$-2\\\\operatorname{cot}\\\\left(x\\\\right) {\\\\operatorname{csc}\\\\left(x\\\\right)}^2$$"],"problemType":"MultipleChoice","stepTitle":"Find $$\\\\frac{dy}{dx}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-2\\\\operatorname{cot}\\\\left(x\\\\right) {\\\\operatorname{csc}\\\\left(x\\\\right)}^2$$","choices":["$$-2\\\\operatorname{cot}\\\\left(x\\\\right) {\\\\operatorname{csc}\\\\left(x\\\\right)}^2$$","$$\\\\operatorname{cot}\\\\left(x\\\\right) {\\\\operatorname{csc}\\\\left(x\\\\right)}^2$$","$$-2\\\\operatorname{cot}\\\\left(x\\\\right) \\\\operatorname{csc}\\\\left(x\\\\right)$$","$$\\\\operatorname{cot}\\\\left(x\\\\right) \\\\operatorname{csc}\\\\left(x\\\\right)$$"],"hints":{"DefaultPathway":[{"id":"a2d77b8chainrule10a-h1","type":"hint","dependencies":[],"title":"The Chain Rule","text":"Remember, the chain rule in Leibniz\'s notation tells us that $$\\\\frac{dy}{dx}=\\\\frac{dy}{du} \\\\frac{du}{dx}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d77b8chainrule10a-h2","type":"hint","dependencies":["a2d77b8chainrule10a-h1"],"title":"Decompose the Function","text":"First we need to decompose our function $$y$$ into the form $$y=f(u)$$ and $$u=g(x)$$, try finding what f(u) and g(x) should be in this case!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d77b8chainrule10a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$u^2$$"],"dependencies":["a2d77b8chainrule10a-h2"],"title":"Find f(u)","text":"What is f(u)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$u^2$$","cot(u)","$$u^3$$","$$\\\\sqrt{u}$$"]},{"id":"a2d77b8chainrule10a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["cot(x)"],"dependencies":["a2d77b8chainrule10a-h3"],"title":"Find g(x)","text":"What is g(x)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["cot(x)","$$x^2$$","$$x^3$$","$$\\\\sqrt{x}$$"]},{"id":"a2d77b8chainrule10a-h5","type":"hint","dependencies":["a2d77b8chainrule10a-h4"],"title":"Find the Derivative","text":"Next, we take the derivative of f(u) in terms of u, $$\\\\frac{dy}{du}$$, and the derivative of g(x) in terms of $$x$$, $$\\\\frac{du}{dx}$$. Multiplying these two expressions together yields our final answer, dy/dx!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d77b8chainrule10a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["2u"],"dependencies":["a2d77b8chainrule10a-h5"],"title":"Find $$\\\\frac{dy}{du}$$","text":"What is $$\\\\frac{dy}{du}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["2u","$$\\\\frac{2}{3} u^3$$","$$u^2$$","$$2$$"]},{"id":"a2d77b8chainrule10a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-\\\\left({\\\\operatorname{csc}\\\\left(x\\\\right)}^2\\\\right)$$"],"dependencies":["a2d77b8chainrule10a-h6"],"title":"Find $$\\\\frac{du}{dx}$$","text":"What is $$\\\\frac{du}{dx}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$${\\\\operatorname{sec}\\\\left(x\\\\right)}^2$$","$$\\\\operatorname{sec}\\\\left(x\\\\right) tan\\\\left(x\\\\right)$$","$$-\\\\left({\\\\operatorname{csc}\\\\left(x\\\\right)}^2\\\\right)$$","$$-\\\\operatorname{csc}\\\\left(x\\\\right) \\\\operatorname{cot}\\\\left(x\\\\right)$$"]},{"id":"a2d77b8chainrule10a-h8","type":"hint","dependencies":["a2d77b8chainrule10a-h7"],"title":"Plug in","text":"Finally, we only need to multiply these two expressions to find our final answer. Don\'t forget to plug in what we have for u in terms of $$x$$ into our final answer!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2d77b8chainrule11","title":"$$y={\\\\left(5-2x\\\\right)}^{\\\\left(-2\\\\right)}$$","body":"","variabilization":{},"oer":"https://openstax.org/books/college-physics-2e/pages/3-problems-exercises <OpenStax: College Physics 2e>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 The Chain Rule","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a2d77b8chainrule11a","stepAnswer":["$$4{\\\\left(5-2x\\\\right)}^{\\\\left(-3\\\\right)}$$"],"problemType":"MultipleChoice","stepTitle":"Find $$\\\\frac{dy}{dx}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$4{\\\\left(5-2x\\\\right)}^{\\\\left(-3\\\\right)}$$","choices":["$$4{\\\\left(5-2x\\\\right)}^{\\\\left(-3\\\\right)}$$","$$4u^{\\\\left(-3\\\\right)}$$","$${\\\\left(5-2x\\\\right)}^{\\\\left(-3\\\\right)}$$","$$4{\\\\left(5-2x\\\\right)}^{\\\\left(-2\\\\right)}$$"],"hints":{"DefaultPathway":[{"id":"a2d77b8chainrule11a-h1","type":"hint","dependencies":[],"title":"The Chain Rule","text":"Remember, the chain rule in Leibniz\'s notation tells us that $$\\\\frac{dy}{dx}=\\\\frac{dy}{du} \\\\frac{du}{dx}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2d77b8chainrule12","title":"$$y={\\\\left(2x^3-x^2+6x+1\\\\right)}^3$$","body":"","variabilization":{},"oer":"https://openstax.org/books/college-physics-2e/pages/3-problems-exercises <OpenStax: College Physics 2e>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 The Chain Rule","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a2d77b8chainrule12a","stepAnswer":["$$\\\\left(18x^2-6x+18\\\\right) {\\\\left(2x^3-x^2+6x+1\\\\right)}^2$$"],"problemType":"MultipleChoice","stepTitle":"Find $$\\\\frac{dy}{dx}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\left(18x^2-6x+18\\\\right) {\\\\left(2x^3-x^2+6x+1\\\\right)}^2$$","choices":["$$\\\\left(18x^2-6x+18\\\\right) {\\\\left(2x^3-x^2+6x+1\\\\right)}^2$$","$$6{\\\\left(2x^3-x^2+6x+1\\\\right)}^2$$","$$\\\\left(3x^2-x+3\\\\right) {\\\\left(2x^3-x^2+6x+1\\\\right)}^2$$","$$\\\\left(18x^2-6x+18\\\\right) \\\\left(2x^3-x^2+6x+1\\\\right)$$"],"hints":{"DefaultPathway":[{"id":"a2d77b8chainrule12a-h1","type":"hint","dependencies":[],"title":"The Chain Rule","text":"Remember, the chain rule in Leibniz\'s notation tells us that $$\\\\frac{dy}{dx}=\\\\frac{dy}{du} \\\\frac{du}{dx}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2d77b8chainrule13","title":"$$y={\\\\left(tan\\\\left(x\\\\right)+sin\\\\left(x\\\\right)\\\\right)}^{\\\\left(-3\\\\right)}$$","body":"","variabilization":{},"oer":"https://openstax.org/books/college-physics-2e/pages/3-problems-exercises <OpenStax: College Physics 2e>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 The Chain Rule","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a2d77b8chainrule13a","stepAnswer":["-3*((sec(x))**2 + cos(x))*(tan(x)+sin(x))**(-4)"],"problemType":"MultipleChoice","stepTitle":"Find $$\\\\frac{dy}{dx}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["-3*((sec(x))**2 + cos(x))*(tan(x)+sin(x))**(-4)","-3*((csc(x))**2 $$-$$ cos(x))*(tan(x)+sin(x))**(-4)","-3*((sec(x))**2 + cos(x))*(tan(x)+sin(x))**(-3)","((sec(x))**2 + cos(x))*(tan(x)+sin(x))**(-4)"],"hints":{"DefaultPathway":[{"id":"a2d77b8chainrule13a-h1","type":"hint","dependencies":[],"title":"The Chain Rule","text":"Remember, the chain rule in Leibniz\'s notation tells us that $$\\\\frac{dy}{dx}=\\\\frac{dy}{du} \\\\frac{du}{dx}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2d77b8chainrule14","title":"$$y=sin(cos(7x))$$","body":"","variabilization":{},"oer":"https://openstax.org/books/college-physics-2e/pages/3-problems-exercises <OpenStax: College Physics 2e>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 The Chain Rule","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a2d77b8chainrule14a","stepAnswer":["$$-7cos\\\\left(cos\\\\left(7x\\\\right)\\\\right) sin\\\\left(7x\\\\right)$$"],"problemType":"MultipleChoice","stepTitle":"Find $$\\\\frac{dy}{dx}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-7cos\\\\left(cos\\\\left(7x\\\\right)\\\\right) sin\\\\left(7x\\\\right)$$","choices":["$$-7cos\\\\left(cos\\\\left(7x\\\\right)\\\\right) sin\\\\left(7x\\\\right)$$","$$-7sin\\\\left(cos\\\\left(7x\\\\right)\\\\right) sin\\\\left(7x\\\\right)$$","$$7cos\\\\left(cos\\\\left(7x\\\\right)\\\\right) cos\\\\left(7x\\\\right)$$","$$cos\\\\left(cos\\\\left(7x\\\\right)\\\\right) sin\\\\left(7x\\\\right)$$"],"hints":{"DefaultPathway":[{"id":"a2d77b8chainrule14a-h1","type":"hint","dependencies":[],"title":"The Chain Rule","text":"Remember, the chain rule in Leibniz\'s notation tells us that $$\\\\frac{dy}{dx}=\\\\frac{dy}{du} \\\\frac{du}{dx}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2d77b8chainrule15","title":"$$y={\\\\operatorname{cot}\\\\left(4x+1\\\\right)}^3$$","body":"","variabilization":{},"oer":"https://openstax.org/books/college-physics-2e/pages/3-problems-exercises <OpenStax: College Physics 2e>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 The Chain Rule","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a2d77b8chainrule15a","stepAnswer":["$$-12{\\\\operatorname{csc}\\\\left(4x+1\\\\right)}^2 {\\\\operatorname{cot}\\\\left(4x+1\\\\right)}^2$$"],"problemType":"MultipleChoice","stepTitle":"Find $$\\\\frac{dy}{dx}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-12{\\\\operatorname{csc}\\\\left(4x+1\\\\right)}^2 {\\\\operatorname{cot}\\\\left(4x+1\\\\right)}^2$$","choices":["$$-12{\\\\operatorname{csc}\\\\left(4x+1\\\\right)}^2 {\\\\operatorname{cot}\\\\left(4x+1\\\\right)}^2$$","$$-12{\\\\operatorname{sec}\\\\left(4x+1\\\\right)}^2 {\\\\operatorname{cot}\\\\left(4x+1\\\\right)}^2$$","$$-12{\\\\operatorname{csc}\\\\left(4x+1\\\\right)}^2 {\\\\operatorname{cot}\\\\left(4x+1\\\\right)}^3$$","$${\\\\operatorname{csc}\\\\left(4x+1\\\\right)}^2 {\\\\operatorname{cot}\\\\left(4x+1\\\\right)}^2$$"],"hints":{"DefaultPathway":[{"id":"a2d77b8chainrule15a-h1","type":"hint","dependencies":[],"title":"The Chain Rule","text":"Remember, the chain rule in Leibniz\'s notation tells us that $$\\\\frac{dy}{dx}=\\\\frac{dy}{du} \\\\frac{du}{dx}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2d77b8chainrule2","title":"Chain Rule Exercises","body":"Given $$y=f(u)$$ and $$u=g(x)$$, find $$\\\\frac{dy}{dx}$$ in terms of $$x!$$","variabilization":{},"oer":"https://openstax.org/books/college-physics-2e/pages/3-problems-exercises <OpenStax: College Physics 2e>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 The Chain Rule","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a2d77b8chainrule2a","stepAnswer":["$$126{\\\\left(7x-4\\\\right)}^2$$"],"problemType":"MultipleChoice","stepTitle":"$$y=6u^3$$, $$u=7x-4$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$126{\\\\left(7x-4\\\\right)}^2$$","choices":["$$126{\\\\left(7x-4\\\\right)}^2$$","$$18{\\\\left(7x-4\\\\right)}^2$$","$${\\\\left(7x-4\\\\right)}^2$$","$$126u^2$$"],"hints":{"DefaultPathway":[{"id":"a2d77b8chainrule2a-h1","type":"hint","dependencies":[],"title":"The Chain Rule","text":"Remember, the chain rule in Leibniz\'s notation tells us that $$\\\\frac{dy}{dx}=\\\\frac{dy}{du} \\\\frac{du}{dx}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d77b8chainrule2a-h2","type":"hint","dependencies":["a2d77b8chainrule2a-h1"],"title":"Find the Derivative","text":"First, try finding the derivative of $$y$$ in terms of u. Then, find the derivative of u in terms of $$x$$. 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Multiplying these two expressions together yields our final answer, dy/dx!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d77b8chainrule7a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3u^2$$"],"dependencies":["a2d77b8chainrule7a-h5"],"title":"Find $$\\\\frac{dy}{du}$$","text":"What is $$\\\\frac{dy}{du}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["u","$$u^2$$","$$3u^2$$","$$\\\\frac{3}{4} u^4$$"]},{"id":"a2d77b8chainrule7a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$6x$$"],"dependencies":["a2d77b8chainrule7a-h6"],"title":"Find $$\\\\frac{du}{dx}$$","text":"What is $$\\\\frac{du}{dx}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$6x$$","$$6x+1$$","$$6x^2$$","$$6x^2+1$$"]},{"id":"a2d77b8chainrule7a-h8","type":"hint","dependencies":["a2d77b8chainrule7a-h7"],"title":"Plug in","text":"Finally, we only need to multiply these two expressions to find our final answer. Don\'t forget to plug in what we have for u in terms of $$x$$ into our final answer!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2d77b8chainrule8","title":"$$y={\\\\left(\\\\frac{x}{7}+\\\\frac{7}{x}\\\\right)}^7$$","body":"","variabilization":{},"oer":"https://openstax.org/books/college-physics-2e/pages/3-problems-exercises <OpenStax: College Physics 2e>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 The Chain Rule","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a2d77b8chainrule8a","stepAnswer":["$$7\\\\left(\\\\frac{1}{7}-\\\\frac{7}{x^2}\\\\right) {\\\\left(\\\\frac{x}{7}+\\\\frac{7}{x}\\\\right)}^6$$"],"problemType":"MultipleChoice","stepTitle":"Find $$\\\\frac{dy}{dx}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$7\\\\left(\\\\frac{1}{7}-\\\\frac{7}{x^2}\\\\right) {\\\\left(\\\\frac{x}{7}+\\\\frac{7}{x}\\\\right)}^6$$","choices":["$$7\\\\left(\\\\frac{1}{7}-\\\\frac{7}{x^2}\\\\right) {\\\\left(\\\\frac{x}{7}+\\\\frac{7}{x}\\\\right)}^6$$","$$\\\\left(\\\\frac{1}{7}-\\\\frac{7}{x^2}\\\\right) {\\\\left(\\\\frac{x}{7}+\\\\frac{7}{x}\\\\right)}^6$$","$$7{\\\\left(\\\\frac{x}{7}+\\\\frac{7}{x}\\\\right)}^6$$","$$7\\\\left(\\\\frac{1}{7}-\\\\frac{7}{x^2}\\\\right) {\\\\left(\\\\frac{x}{7}+\\\\frac{7}{x}\\\\right)}^7$$"],"hints":{"DefaultPathway":[{"id":"a2d77b8chainrule8a-h1","type":"hint","dependencies":[],"title":"The Chain Rule","text":"Remember, the chain rule in Leibniz\'s notation tells us that $$\\\\frac{dy}{dx}=\\\\frac{dy}{du} \\\\frac{du}{dx}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d77b8chainrule8a-h2","type":"hint","dependencies":["a2d77b8chainrule8a-h1"],"title":"Decompose the Function","text":"First we need to decompose our function $$y$$ into the form $$y=f(u)$$ and $$u=g(x)$$, try finding what f(u) and g(x) should be in this case!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d77b8chainrule8a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$u^7$$"],"dependencies":["a2d77b8chainrule8a-h2"],"title":"Find f(u)","text":"What is f(u)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$u^3$$","$$u^2$$","$$u^6$$","$$u^7$$"]},{"id":"a2d77b8chainrule8a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{x}{7}+\\\\frac{7}{x}$$"],"dependencies":["a2d77b8chainrule8a-h3"],"title":"Find g(x)","text":"What is g(x)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{x}{7}+\\\\frac{7}{x}$$","$$7\\\\ln(x)$$","$$\\\\frac{x}{7}$$","$$\\\\frac{7}{x}$$"]},{"id":"a2d77b8chainrule8a-h5","type":"hint","dependencies":["a2d77b8chainrule8a-h4"],"title":"Find the Derivative","text":"Next, we take the derivative of f(u) in terms of u, $$\\\\frac{dy}{du}$$, and the derivative of g(x) in terms of $$x$$, $$\\\\frac{du}{dx}$$. Multiplying these two expressions together yields our final answer, dy/dx!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d77b8chainrule8a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$7u^6$$"],"dependencies":["a2d77b8chainrule8a-h5"],"title":"Find $$\\\\frac{dy}{du}$$","text":"What is $$\\\\frac{dy}{du}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$7u^6$$","$$7u^7$$","$$u^6$$","$$\\\\frac{1}{8} u^8$$"]},{"id":"a2d77b8chainrule8a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1}{7}-\\\\frac{7}{x^2}$$"],"dependencies":["a2d77b8chainrule8a-h6"],"title":"Find $$\\\\frac{du}{dx}$$","text":"What is $$\\\\frac{du}{dx}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$1-\\\\frac{7}{x^2}$$","$$\\\\frac{1}{7}-\\\\frac{1}{x^2}$$","$$1-\\\\frac{1}{x^2}$$","$$\\\\frac{1}{7}-\\\\frac{7}{x^2}$$"]},{"id":"a2d77b8chainrule8a-h8","type":"hint","dependencies":["a2d77b8chainrule8a-h7"],"title":"Plug in","text":"Finally, we only need to multiply these two expressions to find our final answer. Don\'t forget to plug in what we have for u in terms of $$x$$ into our final answer!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2d77b8chainrule9","title":"$$y=\\\\operatorname{csc}\\\\left(x+1\\\\right)$$","body":"","variabilization":{},"oer":"https://openstax.org/books/college-physics-2e/pages/3-problems-exercises <OpenStax: College Physics 2e>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 The Chain Rule","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a2d77b8chainrule9a","stepAnswer":["$$-\\\\operatorname{csc}\\\\left(x+1\\\\right) \\\\operatorname{cot}\\\\left(x+1\\\\right)$$"],"problemType":"MultipleChoice","stepTitle":"Find $$\\\\frac{dy}{dx}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-\\\\operatorname{csc}\\\\left(x+1\\\\right) \\\\operatorname{cot}\\\\left(x+1\\\\right)$$","choices":["$$-\\\\operatorname{csc}\\\\left(x+1\\\\right) \\\\operatorname{cot}\\\\left(x+1\\\\right)$$","$$\\\\operatorname{sec}\\\\left(x+1\\\\right) tan\\\\left(x+1\\\\right)$$","$$-\\\\left({\\\\operatorname{csc}\\\\left(x+1\\\\right)}^2\\\\right)$$","$${\\\\operatorname{sec}\\\\left(x+1\\\\right)}^2$$"],"hints":{"DefaultPathway":[{"id":"a2d77b8chainrule9a-h1","type":"hint","dependencies":[],"title":"The Chain Rule","text":"Remember, the chain rule in Leibniz\'s notation tells us that $$\\\\frac{dy}{dx}=\\\\frac{dy}{du} \\\\frac{du}{dx}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d77b8chainrule9a-h2","type":"hint","dependencies":["a2d77b8chainrule9a-h1"],"title":"Decompose the Function","text":"First we need to decompose our function $$y$$ into the form $$y=f(u)$$ and $$u=g(x)$$, try finding what f(u) and g(x) should be in this case!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d77b8chainrule9a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["csc(u)"],"dependencies":["a2d77b8chainrule9a-h2"],"title":"Find f(u)","text":"What is f(u)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["csc(u)","sec(u)","$$\\\\sqrt{u}$$","$$u^2$$"]},{"id":"a2d77b8chainrule9a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x+1$$"],"dependencies":["a2d77b8chainrule9a-h3"],"title":"Find g(x)","text":"What is g(x)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x+1$$","$$x^2+1$$","$$x^2+x$$","$$x$$"]},{"id":"a2d77b8chainrule9a-h5","type":"hint","dependencies":["a2d77b8chainrule9a-h4"],"title":"Find the Derivative","text":"Next, we take the derivative of f(u) in terms of u, $$\\\\frac{dy}{du}$$, and the derivative of g(x) in terms of $$x$$, $$\\\\frac{du}{dx}$$. Multiplying these two expressions together yields our final answer, dy/dx!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d77b8chainrule9a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-\\\\operatorname{csc}\\\\left(u\\\\right) \\\\operatorname{cot}\\\\left(u\\\\right)$$"],"dependencies":["a2d77b8chainrule9a-h5"],"title":"Find $$\\\\frac{dy}{du}$$","text":"What is $$\\\\frac{dy}{du}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$-\\\\operatorname{csc}\\\\left(u\\\\right) \\\\operatorname{cot}\\\\left(u\\\\right)$$","$$\\\\operatorname{sec}\\\\left(u\\\\right) tan\\\\left(u\\\\right)$$","$$-\\\\left({\\\\operatorname{csc}\\\\left(u\\\\right)}^2\\\\right)$$","$${\\\\operatorname{sec}\\\\left(u\\\\right)}^2$$"]},{"id":"a2d77b8chainrule9a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$1$$"],"dependencies":["a2d77b8chainrule9a-h6"],"title":"Find $$\\\\frac{du}{dx}$$","text":"What is $$\\\\frac{du}{dx}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$1$$","$$x$$","$$x^2$$","$$x^3$$"]},{"id":"a2d77b8chainrule9a-h8","type":"hint","dependencies":["a2d77b8chainrule9a-h7"],"title":"Plug in","text":"Finally, we only need to multiply these two expressions to find our final answer. Don\'t forget to plug in what we have for u in terms of $$x$$ into our final answer!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2d8720LinEqua1","title":"Solving an Equation in One Variable","body":"Solve the following equation:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Linear Equations in One Variable","courseName":"OpenStax: College Algebra","steps":[{"id":"a2d8720LinEqua1a","stepAnswer":["$$x=6$$"],"problemType":"TextBox","stepTitle":"$$2x+7=19$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x=6$$","hints":{"DefaultPathway":[{"id":"a2d8720LinEqua1a-h1","type":"hint","dependencies":[],"title":"Isolation","text":"First we should isolate the variable on one side of the equation by adding, subtracting, multiplying or dividing the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x=12$$"],"dependencies":["a2d8720LinEqua1a-h1"],"title":"Subtraction","text":"What is the result after subtracting $$7$$ from both sides?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua1a-h3","type":"hint","dependencies":["a2d8720LinEqua1a-h2"],"title":"Normalization","text":"When the variable is multiplied by a coefficient in the final stage, multiply both sides of the equation by the reciprocal of the cofficient.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=6$$"],"dependencies":["a2d8720LinEqua1a-h3"],"title":"Multiplication","text":"What is the result after multiplying both sides by $$\\\\frac{1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2d8720LinEqua10","title":"Solving a Rational Equation by Factoring the Denominator","body":"Solve the rational equation:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Linear Equations in One Variable","courseName":"OpenStax: College Algebra","steps":[{"id":"a2d8720LinEqua10a","stepAnswer":["$$x=1$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{-5}{2x}+\\\\frac{3}{4x}=\\\\frac{-7}{4}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x=1$$","hints":{"DefaultPathway":[{"id":"a2d8720LinEqua10a-h1","type":"hint","dependencies":[],"title":"Factoring the denominator","text":"The three denominators in factored form are $$2x=2x$$, $$4x=2\\\\times2 x$$, and $$4=2\\\\times2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua10a-h2","type":"hint","dependencies":["a2d8720LinEqua10a-h1"],"title":"Find LCD","text":"The LCD is the smallest expression that is divisible by each one of the denominators.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4x$$"],"dependencies":["a2d8720LinEqua10a-h2"],"title":"LCD","text":"What is the LCD of this equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-10+3=-7x$$"],"dependencies":["a2d8720LinEqua10a-h3"],"title":"Eliminating","text":"Simplify $$4x \\\\left(-\\\\frac{5}{2x}+\\\\frac{3}{4x}\\\\right)=4x \\\\left(-\\\\frac{4}{7}\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua10a-h5","type":"hint","dependencies":["a2d8720LinEqua10a-h4"],"title":"Solve equation","text":"Then we should solve the linear equation obtained.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua10a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=1$$"],"dependencies":["a2d8720LinEqua10a-h5"],"title":"Linear equation","text":"Solve the linear equation $$-10+3=-7x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2d8720LinEqua11","title":"Solving Rational Equations with a Binomial in the Denominator","body":"Solve the following rational equations and state the excluded values.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Linear Equations in One Variable","courseName":"OpenStax: College Algebra","steps":[{"id":"a2d8720LinEqua11a","stepAnswer":["Excluded values: $$6$$, $$0$$ $$x=15$$"],"problemType":"MultipleChoice","stepTitle":"3/(x - 6) $$=$$ $$\\\\frac{5}{x}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Excluded values: $$6$$, $$0$$ $$x=15$$","choices":["Excluded values: $$6$$, $$0$$ $$x=15$$","Excluded values: $$6$$ $$x=15$$","Excluded values: $$6$$, $$0$$ $$x=3$$","Excluded values: $$6$$ $$x=3$$"],"hints":{"DefaultPathway":[{"id":"a2d8720LinEqua11a-h1","type":"hint","dependencies":[],"title":"Excluded values","text":"The excluded values are those making a denominator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua11a-h2","type":"hint","dependencies":["a2d8720LinEqua11a-h1"],"title":"Denominator","text":"The denominators are $$x-6$$ and $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua11a-h3","type":"hint","dependencies":["a2d8720LinEqua11a-h2"],"title":"Excluded values","text":"The excluded values are $$6$$ and $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua11a-h4","type":"hint","dependencies":["a2d8720LinEqua11a-h3"],"title":"Factoring the denominator","text":"The denominators $$x$$ and $$x-6$$ have nothing in common.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua11a-h5","type":"hint","dependencies":["a2d8720LinEqua11a-h4"],"title":"Find LCD","text":"The LCD is the smallest expression that is divisible by each one of the denominators.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua11a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x(x-6)$$"],"dependencies":["a2d8720LinEqua11a-h5"],"title":"LCD","text":"What is the LCD of this equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua11a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x=5x-30$$"],"dependencies":["a2d8720LinEqua11a-h6"],"title":"Eliminating","text":"Simplify $$x\\\\left(x-6\\\\right) \\\\frac{3}{x-6}=x\\\\left(x-6\\\\right) \\\\frac{5}{x}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua11a-h8","type":"hint","dependencies":["a2d8720LinEqua11a-h7"],"title":"Solve equation","text":"Then we should solve the linear equation obtained.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua11a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["a2d8720LinEqua11a-h8"],"title":"Linear equation","text":"Solve the equation $$3x=5x-30$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a2d8720LinEqua11b","stepAnswer":["Excluded value: $$3$$ $$x=\\\\frac{13}{3}$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{x}{x-3}=\\\\frac{5}{x-3}-\\\\frac{1}{2}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Excluded value: $$3$$ $$x=\\\\frac{13}{3}$$","choices":["Excluded value: $$3$$ $$x=\\\\frac{3}{13}$$","Excluded value: $$2$$ $$x=\\\\frac{3}{13}$$","Excluded value: $$3$$ $$x=\\\\frac{13}{3}$$","Excluded value: $$2$$ $$x=\\\\frac{13}{3}$$"],"hints":{"DefaultPathway":[{"id":"a2d8720LinEqua11b-h1","type":"hint","dependencies":[],"title":"Excluded values","text":"The excluded values are those making a denominator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua11b-h2","type":"hint","dependencies":["a2d8720LinEqua11b-h1"],"title":"Denominator","text":"The denominators are $$x-3$$, $$x-3$$ and $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua11b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a2d8720LinEqua11b-h2"],"title":"Excluded values","text":"State the excluded value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua11b-h4","type":"hint","dependencies":["a2d8720LinEqua11b-h3"],"title":"Factoring the denominator","text":"The three denominators in factored form are $$x-3$$, $$x-3$$, and $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua11b-h5","type":"hint","dependencies":["a2d8720LinEqua11b-h4"],"title":"Find LCD","text":"The LCD is the smallest expression that is divisible by each one of the denominators.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua11b-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2(x-3)$$"],"dependencies":["a2d8720LinEqua11b-h5"],"title":"LCD","text":"What is the LCD of this equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua11b-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x=10-x+3$$"],"dependencies":["a2d8720LinEqua11b-h6"],"title":"Eliminating","text":"Simplify $$2\\\\left(x-3\\\\right) \\\\frac{x}{x-3}=2\\\\left(x-3\\\\right) \\\\left(\\\\frac{5}{x-3}-\\\\frac{1}{2}\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua11b-h8","type":"hint","dependencies":["a2d8720LinEqua11b-h7"],"title":"Solve equation","text":"Then we should solve the linear equation obtained.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua11b-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{13}{3}$$"],"dependencies":["a2d8720LinEqua11b-h8"],"title":"Linear equation","text":"Solve the equation $$2x=10-x+3$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a2d8720LinEqua11c","stepAnswer":["Excluded value: $$2$$ $$x=4$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{x}{x-2}=\\\\frac{5}{x-2}-\\\\frac{1}{2}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Excluded value: $$2$$ $$x=4$$","choices":["Excluded value: $$0$$ $$x=4$$","Excluded value: $$2$$ $$x=4$$","Excluded value: $$0$$ $$x=6$$","Excluded value: $$2$$ $$x=6$$"],"hints":{"DefaultPathway":[{"id":"a2d8720LinEqua11c-h1","type":"hint","dependencies":[],"title":"Excluded values","text":"The excluded values are those making a denominator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua11c-h2","type":"hint","dependencies":["a2d8720LinEqua11c-h1"],"title":"Denominator","text":"The denominators are $$x-2$$, $$x-2$$ and $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua11c-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a2d8720LinEqua11c-h2"],"title":"Excluded values","text":"State the excluded values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua11c-h4","type":"hint","dependencies":["a2d8720LinEqua11c-h3"],"title":"Factoring the denominator","text":"The three denominators in factored form are $$x-2$$ , $$x-2$$, and $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua11c-h5","type":"hint","dependencies":["a2d8720LinEqua11c-h4"],"title":"Find LCD","text":"The LCD is the smallest expression that is divisible by each one of the denominators.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua11c-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2(x-2)$$"],"dependencies":["a2d8720LinEqua11c-h5"],"title":"LCD","text":"What is the LCD of this equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua11c-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x=12-x$$"],"dependencies":["a2d8720LinEqua11c-h6"],"title":"Eliminating","text":"Simplify $$2\\\\left(x-2\\\\right) \\\\frac{x}{x-2}=2\\\\left(x-2\\\\right) \\\\left(\\\\frac{5}{x-2}-\\\\frac{1}{2}\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua11c-h8","type":"hint","dependencies":["a2d8720LinEqua11c-h7"],"title":"Solve equation","text":"Then we should solve the linear equation obtained.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua11c-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a2d8720LinEqua11c-h8"],"title":"Linear equation","text":"Solve the equation $$2x=12-x$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2d8720LinEqua12","title":"Solving Rational Equations with a Binomial in the Denominator","body":"Solve the equation and state the excluded values.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Linear Equations in One Variable","courseName":"OpenStax: College Algebra","steps":[{"id":"a2d8720LinEqua12a","stepAnswer":["Excluded values: $$\\\\frac{-1}{2}$$, $$\\\\frac{-1}{3}$$ $$x=\\\\frac{-7}{17}$$"],"problemType":"MultipleChoice","stepTitle":"Solve $$\\\\frac{-3}{2x+1}=\\\\frac{4}{3x+1}$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Excluded values: $$\\\\frac{-1}{2}$$, $$\\\\frac{-1}{3}$$ $$x=\\\\frac{-7}{17}$$","choices":["Excluded values: $$\\\\frac{1}{2}$$, $$\\\\frac{1}{3}$$ $$x=\\\\frac{-7}{17}$$","Excluded values: $$\\\\frac{-1}{2}$$, $$\\\\frac{-1}{3}$$ $$x=\\\\frac{-7}{17}$$","Excluded values: $$\\\\frac{1}{2}$$, $$\\\\frac{1}{3}$$ $$x=\\\\frac{7}{17}$$","Excluded values: $$\\\\frac{-1}{2}$$, $$\\\\frac{-1}{3}$$ $$x=\\\\frac{7}{17}$$"],"hints":{"DefaultPathway":[{"id":"a2d8720LinEqua12a-h1","type":"hint","dependencies":[],"title":"Excluded values","text":"The excluded values are those making a denominator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua12a-h2","type":"hint","dependencies":["a2d8720LinEqua12a-h1"],"title":"Denominator","text":"The denominators are $$2x+1$$ and $$3x+1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua12a-h3","type":"hint","dependencies":["a2d8720LinEqua12a-h2"],"title":"Excluded values","text":"The excluded values are $$\\\\frac{-1}{2}$$ and $$\\\\frac{-1}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua12a-h4","type":"hint","dependencies":["a2d8720LinEqua12a-h3"],"title":"Factoring the denominator","text":"The two denominators in factored form are $$2x+1$$, and $$3x+1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua12a-h5","type":"hint","dependencies":["a2d8720LinEqua12a-h4"],"title":"Find LCD","text":"The LCD is the smallest expression that is divisible by each one of the denominators.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua12a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(2x+1\\\\right) \\\\left(3x+1\\\\right)$$"],"dependencies":["a2d8720LinEqua12a-h5"],"title":"LCD","text":"What is the LCD of this equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua12a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9x-3=8x+4$$"],"dependencies":["a2d8720LinEqua12a-h6"],"title":"Eliminating","text":"Simplify $$\\\\left(2x+1\\\\right) \\\\left(3x+1\\\\right) \\\\left(-\\\\frac{3}{2x+1}\\\\right)=\\\\left(2x+1\\\\right) \\\\left(3x+1\\\\right) \\\\frac{4}{3x+1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua12a-h8","type":"hint","dependencies":["a2d8720LinEqua12a-h7"],"title":"Solve equation","text":"Then we should solve the linear equation obtained.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua12a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-7}{17}$$"],"dependencies":["a2d8720LinEqua12a-h8"],"title":"Linear equation","text":"Solve the equation $$-9x-3=8x+4$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2d8720LinEqua13","title":"Solving a Rational Equation with Factored Denominators and Stating Excluded Values","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Linear Equations in One Variable","courseName":"OpenStax: College Algebra","steps":[{"id":"a2d8720LinEqua13a","stepAnswer":["Excluded values: $$1$$, $$-1$$ $$x=-3$$"],"problemType":"MultipleChoice","stepTitle":"Solve the rational equation after factoring the denominators: $$\\\\frac{2}{x+1}-\\\\frac{1}{x-1}=\\\\frac{2x}{x^2-1}$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Excluded values: $$1$$, $$-1$$ $$x=-3$$","choices":["Excluded values: $$1$$, $$-1$$ $$x=-3$$","Excluded values: $$1$$ $$x=-3$$","Excluded values: $$1$$, $$-1$$ $$x=3$$","Excluded values: $$1$$ $$x=3$$"],"hints":{"DefaultPathway":[{"id":"a2d8720LinEqua13a-h1","type":"hint","dependencies":[],"title":"Excluded values","text":"The excluded values are those making a denominator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua13a-h2","type":"hint","dependencies":["a2d8720LinEqua13a-h1"],"title":"Denominator","text":"The denominators are $$x+1$$, $$x-1$$ and $$x^2-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua13a-h3","type":"hint","dependencies":["a2d8720LinEqua13a-h2"],"title":"Excluded values","text":"The excluded values are $$1$$ and $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua13a-h4","type":"hint","dependencies":["a2d8720LinEqua13a-h3"],"title":"Factoring the denominator","text":"The three denominators in factored form are $$x+1$$, $$x-1$$ and $$\\\\left(x+1\\\\right) \\\\left(x-1\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua13a-h5","type":"hint","dependencies":["a2d8720LinEqua13a-h4"],"title":"Find LCD","text":"The LCD is the smallest expression that is divisible by each one of the denominators.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua13a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x-1\\\\right) \\\\left(x+1\\\\right)$$"],"dependencies":["a2d8720LinEqua13a-h5"],"title":"LCD","text":"What is the LCD of this equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua13a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x-2-x-1=2x$$"],"dependencies":["a2d8720LinEqua13a-h6"],"title":"Eliminating","text":"Simplify $$\\\\left(x-1\\\\right) \\\\left(x+1\\\\right) \\\\left(\\\\frac{2}{x+1}-\\\\frac{1}{x-1}\\\\right)=\\\\left(x-1\\\\right) \\\\left(x+1\\\\right) \\\\frac{2x}{x^2+1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua13a-h8","type":"hint","dependencies":["a2d8720LinEqua13a-h7"],"title":"Solve equation","text":"Then we should solve the linear equation obtained.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua13a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a2d8720LinEqua13a-h8"],"title":"Linear equation","text":"Solve the equation $$2x-2-x-1=2x$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2d8720LinEqua14","title":"Solving a Rational Equation with Factored Denominators and Stating Excluded Values","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Linear Equations in One Variable","courseName":"OpenStax: College Algebra","steps":[{"id":"a2d8720LinEqua14a","stepAnswer":["$$\\\\frac{1}{3}$$"],"problemType":"TextBox","stepTitle":"Solve the rational equation: $$\\\\frac{2}{x-2}+\\\\frac{1}{x-1}=\\\\frac{1}{x^2-x-2}$$. State the value of $$x$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{3}$$","hints":{"DefaultPathway":[{"id":"a2d8720LinEqua14a-h1","type":"hint","dependencies":[],"title":"Excluded values","text":"The excluded values are those making a denominator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua14a-h2","type":"hint","dependencies":["a2d8720LinEqua14a-h1"],"title":"Denominator","text":"The denominators are $$x-2$$, $$x+1$$ and $$x^2-x-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua14a-h3","type":"hint","dependencies":["a2d8720LinEqua14a-h2"],"title":"Excluded values","text":"The excluded values are $$2$$ and $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua14a-h4","type":"hint","dependencies":["a2d8720LinEqua14a-h3"],"title":"Factoring the denominator","text":"The three denominators in factored form are $$x-2$$, $$x+1$$ and $$\\\\left(x-2\\\\right) \\\\left(x+1\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua14a-h5","type":"hint","dependencies":["a2d8720LinEqua14a-h4"],"title":"Find LCD","text":"The LCD is the smallest expression that is divisible by each one of the denominators.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua14a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x-2\\\\right) \\\\left(x+1\\\\right)$$"],"dependencies":["a2d8720LinEqua14a-h5"],"title":"LCD","text":"What is the LCD of this equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua14a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x+2+x-2=1$$"],"dependencies":["a2d8720LinEqua14a-h6"],"title":"Eliminating","text":"Simplify $$\\\\left(x-2\\\\right) \\\\left(x+1\\\\right) \\\\left(\\\\frac{2}{x-2}+\\\\frac{1}{x+1}\\\\right)=\\\\left(x-2\\\\right) \\\\left(x+1\\\\right) \\\\frac{1}{x^2-x-2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua14a-h8","type":"hint","dependencies":["a2d8720LinEqua14a-h7"],"title":"Solve equation","text":"Then we should solve the linear equation obtained.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua14a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["a2d8720LinEqua14a-h8"],"title":"Linear equation","text":"Solve the equation $$2x+2+x-2=1$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2d8720LinEqua15","title":"For the following exercises, solve the equation for $$x$$. State all x-values that are excluded from the solution set.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Linear Equations in One Variable","courseName":"OpenStax: College Algebra","steps":[{"id":"a2d8720LinEqua15a","stepAnswer":["Excluded values: $$-4$$ $$x=-3$$"],"problemType":"MultipleChoice","stepTitle":"$$2-\\\\frac{3}{x+4}=\\\\frac{x+2}{x+4}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Excluded values: $$-4$$ $$x=-3$$","choices":["Excluded values: $$-4$$ $$x=-3$$","Excluded values: $$4$$ $$x=-3$$","Excluded values: $$-4$$ $$x=1$$","Excluded values: $$4$$ $$x=1$$"],"hints":{"DefaultPathway":[{"id":"a2d8720LinEqua15a-h1","type":"hint","dependencies":[],"title":"Excluded values","text":"The excluded values are those making a denominator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua15a-h2","type":"hint","dependencies":["a2d8720LinEqua15a-h1"],"title":"Denominator","text":"The only denominator is $$x+4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["a2d8720LinEqua15a-h2"],"title":"Excluded values","text":"State the excluded value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua15a-h4","type":"hint","dependencies":["a2d8720LinEqua15a-h3"],"title":"Factoring the denominator","text":"The denominator in factored form is $$x+4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua15a-h5","type":"hint","dependencies":["a2d8720LinEqua15a-h4"],"title":"Find LCD","text":"The LCD is the smallest expression that is divisible by each one of the denominators.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua15a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+4$$"],"dependencies":["a2d8720LinEqua15a-h5"],"title":"LCD","text":"What is the LCD of this equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua15a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x+8-3=x+2$$"],"dependencies":["a2d8720LinEqua15a-h6"],"title":"Eliminating","text":"Simplify $$2\\\\left(x+4\\\\right)-\\\\left(x+4\\\\right) \\\\frac{3}{x+4}=\\\\left(x+4\\\\right) \\\\frac{x+2}{x+4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua15a-h8","type":"hint","dependencies":["a2d8720LinEqua15a-h7"],"title":"Solve equation","text":"Then we should solve the linear equation obtained.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua15a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a2d8720LinEqua15a-h8"],"title":"Linear equation","text":"Solve the equation $$2x+8-3=x+2$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a2d8720LinEqua15b","stepAnswer":["Excluded values: $$1$$ No solution for $$x$$."],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{3x}{x-1}+2=\\\\frac{3}{x-1}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Excluded values: $$1$$ No solution for $$x$$.","choices":["Excluded values: $$-1$$ $$x=1$$","Excluded values: $$1$$ $$x=1$$","Excluded values: $$1$$ No solution for $$x$$."],"hints":{"DefaultPathway":[{"id":"a2d8720LinEqua15b-h1","type":"hint","dependencies":[],"title":"Excluded values","text":"The excluded values are those making a denominator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua15b-h2","type":"hint","dependencies":["a2d8720LinEqua15b-h1"],"title":"Denominator","text":"The only denominator is $$x-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua15b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a2d8720LinEqua15b-h2"],"title":"Excluded values","text":"State the excluded value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua15b-h4","type":"hint","dependencies":["a2d8720LinEqua15b-h3"],"title":"Factoring the denominator","text":"The denominator in factored form is $$x-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua15b-h5","type":"hint","dependencies":["a2d8720LinEqua15b-h4"],"title":"Find LCD","text":"The LCD is the smallest expression that is divisible by each one of the denominators.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua15b-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x-1$$"],"dependencies":["a2d8720LinEqua15b-h5"],"title":"LCD","text":"What is the LCD of this equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua15b-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x+2x-2=3$$"],"dependencies":["a2d8720LinEqua15b-h6"],"title":"Eliminating","text":"Simplify $$\\\\left(x-1\\\\right) \\\\frac{3x}{x-1}+2\\\\left(x-1\\\\right)=\\\\left(x-1\\\\right) \\\\frac{3}{x-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua15b-h8","type":"hint","dependencies":["a2d8720LinEqua15b-h7"],"title":"Solve equation","text":"Then we should solve the linear equation obtained.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua15b-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a2d8720LinEqua15b-h8"],"title":"Linear equation","text":"Solve the equation $$3x+2x-2=3$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua15b-h10","type":"hint","dependencies":["a2d8720LinEqua15b-h9"],"title":"Check the solution","text":"Since $$1$$ is an excluded value, it is not the true solution for the original rational function. As a result, there is no solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a2d8720LinEqua15c","stepAnswer":["Excluded value: $$0$$ $$x=\\\\frac{-5}{2}$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{1}{x}=\\\\frac{1}{5}+\\\\frac{3}{2x}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Excluded value: $$0$$ $$x=\\\\frac{-5}{2}$$","choices":["Excluded value: $$0$$ $$x=\\\\frac{-5}{2}$$","Excluded value: None $$x=\\\\frac{-5}{2}$$","Excluded value: $$0$$ $$x=\\\\frac{-7}{2}$$","Excluded value: None $$x=\\\\frac{-7}{2}$$"],"hints":{"DefaultPathway":[{"id":"a2d8720LinEqua15c-h1","type":"hint","dependencies":[],"title":"Excluded values","text":"The excluded values are those making a denominator equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua15c-h2","type":"hint","dependencies":["a2d8720LinEqua15c-h1"],"title":"Denominator","text":"The denominators are $$x$$ and $$2x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua15c-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a2d8720LinEqua15c-h2"],"title":"Excluded values","text":"State the excluded value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua15c-h4","type":"hint","dependencies":["a2d8720LinEqua15c-h3"],"title":"Factoring the denominator","text":"The denominator in factored form is $$x$$ and $$2x$$ $$=$$ $$2x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua15c-h5","type":"hint","dependencies":["a2d8720LinEqua15c-h4"],"title":"Find LCD","text":"The LCD is the smallest expression that is divisible by each one of the denominators.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua15c-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x$$"],"dependencies":["a2d8720LinEqua15c-h5"],"title":"LCD","text":"What is the LCD of this equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua15c-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2=\\\\frac{2x}{5}+3$$"],"dependencies":["a2d8720LinEqua15c-h6"],"title":"Eliminating","text":"Simplify $$2x\\\\left(\\\\frac{1}{x}\\\\right)=2x\\\\left(\\\\frac{1}{5}\\\\right)+2x\\\\left(\\\\frac{3}{2} x\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua15c-h8","type":"hint","dependencies":["a2d8720LinEqua15c-h7"],"title":"Solve equation","text":"Then we should solve the linear equation obtained.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua15c-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-5}{2}$$"],"dependencies":["a2d8720LinEqua15c-h8"],"title":"Linear equation","text":"Solve the equation $$2=\\\\frac{2x}{5}+3$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2d8720LinEqua16","title":"Finding the Slope of a Line Given Two Points","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Linear Equations in One Variable","courseName":"OpenStax: College Algebra","steps":[{"id":"a2d8720LinEqua16a","stepAnswer":["$$\\\\frac{-4}{7}$$"],"problemType":"TextBox","stepTitle":"Find the slope of a line that passes through the points $$(2,-1)$$ and $$(-5,3)$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-4}{7}$$","hints":{"DefaultPathway":[{"id":"a2d8720LinEqua16a-h1","type":"hint","dependencies":[],"title":"The slope of a line","text":"The slope of a line, $$m$$, represents the change in $$y$$ over the change in $$x$$. Given two points, (x_1, y_1) and (x_2, y_2), the following formula determines the slope of a line containing these points: $$m=\\\\frac{y_2-y_1}{x_2-x_1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua16a-h2","type":"hint","dependencies":["a2d8720LinEqua16a-h1"],"title":"Substitute","text":"Substitute the y-values and the x-values into the formula, and we get $$m=\\\\frac{3-\\\\left(-1\\\\right)}{\\\\left(-5-2\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua16a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a2d8720LinEqua16a-h2"],"title":"Subtraction","text":"what is $$3-(-1)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua16a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-7$$"],"dependencies":["a2d8720LinEqua16a-h3"],"title":"Subtraction","text":"What is $$-5-2$$ ?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua16a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-4}{7}$$"],"dependencies":["a2d8720LinEqua16a-h4"],"title":"Division","text":"What is $$\\\\frac{4}{\\\\left(-7\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2d8720LinEqua17","title":"Finding the Slope of a Line Given Two Points","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Linear Equations in One Variable","courseName":"OpenStax: College Algebra","steps":[{"id":"a2d8720LinEqua17a","stepAnswer":["$$\\\\frac{-2}{3}$$"],"problemType":"TextBox","stepTitle":"Find the slope of a line that passes through the points $$(-2,6)$$ and $$(1,4)$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-2}{3}$$","hints":{"DefaultPathway":[{"id":"a2d8720LinEqua17a-h1","type":"hint","dependencies":[],"title":"The slope of a line","text":"The slope of a line, $$m$$, represents the change in $$y$$ over the change in $$x$$. Given two points, (x_1, y_1) and (x_2, y_2), the following formula determines the slope of a line containing these points: $$m$$ $$=$$ (y_2 - y_1)/(x_2 - x_1).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua17a-h2","type":"hint","dependencies":["a2d8720LinEqua17a-h1"],"title":"Substitute","text":"Substitute the y-values and the x-values into the formula, and we get $$m=\\\\frac{4-6}{1-\\\\left(-2\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua17a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a2d8720LinEqua17a-h2"],"title":"Subtraction","text":"What is $$4-6$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua17a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a2d8720LinEqua17a-h3"],"title":"Subtraction","text":"What is $$1-(-2)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua17a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-2}{3}$$"],"dependencies":["a2d8720LinEqua17a-h4"],"title":"Division","text":"What is $$\\\\frac{\\\\left(-2\\\\right)}{3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2d8720LinEqua18","title":"Identifying the Slope and y-intercept of a Line Given an Equation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Linear Equations in One Variable","courseName":"OpenStax: College Algebra","steps":[{"id":"a2d8720LinEqua18a","stepAnswer":["$$\\\\frac{-3}{4}$$"],"problemType":"TextBox","stepTitle":"Identifying the slope","stepBody":"Identify the slope given the equation $$y=-\\\\left(\\\\frac{3}{4}\\\\right) x-4$$.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-3}{4}$$","hints":{"DefaultPathway":[{"id":"a2d8720LinEqua18a-h1","type":"hint","dependencies":[],"title":"The slope of a line","text":"When the line is in $$y=mx+b$$ form, the coefficient of $$x$$ is the slope of the line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a2d8720LinEqua18b","stepAnswer":["$$-4$$"],"problemType":"TextBox","stepTitle":"Identifying the y-intercept","stepBody":"Identify the y-intercept given the equation $$y=-\\\\left(\\\\frac{3}{4}\\\\right) x-4$$.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-4$$","hints":{"DefaultPathway":[{"id":"a2d8720LinEqua18b-h1","type":"hint","dependencies":[],"title":"The y-intercept of a line","text":"The y-intercept is the point at which the line crosses the y-axis. On the y-axis, $$x$$ $$=$$ $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua18b-h2","type":"hint","dependencies":["a2d8720LinEqua18b-h1"],"title":"Identify the y-intercept","text":"We can always identify the y-intercept when the line is in slope-intercept from, as it will always equal $$b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2d8720LinEqua19","title":"Finding the Equation of a Line Given the Slope and One Point","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Linear Equations in One Variable","courseName":"OpenStax: College Algebra","steps":[{"id":"a2d8720LinEqua19a","stepAnswer":["$$y=-3x+20$$"],"problemType":"TextBox","stepTitle":"Write the equation of the line with slope $$m=-3$$ and passing through the point $$(4,8)$$. Write the final equation in slope-intercept form","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=-3x+20$$","hints":{"DefaultPathway":[{"id":"a2d8720LinEqua19a-h1","type":"hint","dependencies":[],"title":"The point-slope formula","text":"Given one point and the slope, the point-slope formula will lead to the equation of a line: $$y-y_1=m\\\\left(x-x_1\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y-8=-3(x-4)$$"],"dependencies":["a2d8720LinEqua19a-h1"],"title":"Substitute","text":"Write the point-slope formula.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua19a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y-8=-3x+12$$"],"dependencies":["a2d8720LinEqua19a-h2"],"title":"Parentheses","text":"Using the distributive property.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua19a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y=-3x+20$$"],"dependencies":["a2d8720LinEqua19a-h3"],"title":"Simplification","text":"Simplify the equation $$y-8=-3x+12$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2d8720LinEqua2","title":"Solving an Equation in One Variable","body":"Solve the following equation:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Linear Equations in One Variable","courseName":"OpenStax: College Algebra","steps":[{"id":"a2d8720LinEqua2a","stepAnswer":["$$x=-5$$"],"problemType":"TextBox","stepTitle":"$$2x+1=-9$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x=-5$$","hints":{"DefaultPathway":[{"id":"a2d8720LinEqua2a-h1","type":"hint","dependencies":[],"title":"Isolation","text":"First we should isolate the variable on one side of the equation by adding, subtracting, multiplying or dividing the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x=-10$$"],"dependencies":["a2d8720LinEqua2a-h1"],"title":"Subtraction","text":"What is the result after subtracting $$1$$ from both sides?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua2a-h3","type":"hint","dependencies":["a2d8720LinEqua2a-h2"],"title":"Normalization","text":"When the variable is multiplied by a coefficient in the final stage, multiply both sides fo the equation by the reciprocal of the cofficient.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=-5$$"],"dependencies":["a2d8720LinEqua2a-h3"],"title":"Multiplication","text":"What is the result after multiplying both sides by $$\\\\frac{1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2d8720LinEqua20","title":"Finding the Equation of a Line Given the Slope and One Point","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Linear Equations in One Variable","courseName":"OpenStax: College Algebra","steps":[{"id":"a2d8720LinEqua20a","stepAnswer":["$$y=4x-3$$"],"problemType":"TextBox","stepTitle":"Given $$m=4$$, find the equation of the line in slope-intercept form passing through the point $$(2,5)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=4x-3$$","hints":{"DefaultPathway":[{"id":"a2d8720LinEqua20a-h1","type":"hint","dependencies":[],"title":"The point-slope formula","text":"Given one point and the slope, the point-slope formula will lead to the equation of a line: $$y-y_1=m\\\\left(x-x_1\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y-5=4(x-2)$$"],"dependencies":["a2d8720LinEqua20a-h1"],"title":"Substitute","text":"Write the point-slope formula.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua20a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y-5=4x-8$$"],"dependencies":["a2d8720LinEqua20a-h2"],"title":"Parentheses","text":"Using the distributive property.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua20a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y=4x-3$$"],"dependencies":["a2d8720LinEqua20a-h3"],"title":"Simplification","text":"Simplify the equation $$y-5=4x-8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2d8720LinEqua21","title":"Finding the Equation of a Line Passing Through Two Given Points","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Linear Equations in One Variable","courseName":"OpenStax: College Algebra","steps":[{"id":"a2d8720LinEqua21a","stepAnswer":["$$y=\\\\frac{7}{3} x-3$$"],"problemType":"TextBox","stepTitle":"Find the equation of the line passing through the points $$(3,4)$$ and $$(0,-3)$$. Write the final equation in slope-intercept form.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=\\\\frac{7}{3} x-3$$","hints":{"DefaultPathway":[{"id":"a2d8720LinEqua21a-h1","type":"hint","dependencies":[],"title":"Slope","text":"First, we calculate the slope using the slope formula and two points.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua21a-h2","type":"hint","dependencies":["a2d8720LinEqua21a-h1"],"title":"Slope Formula","text":"The slope of a line, $$m$$, represents the change in $$y$$ over the change in $$x$$. Given two points, $$(x_1,y_1)$$ and $$(x_2,y_2)$$, the following formula determines the slope of a line containing these points: $$m=\\\\frac{y_2-y_1}{x_2-x_1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua21a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{7}{3}$$"],"dependencies":["a2d8720LinEqua21a-h2"],"title":"Calculate the slope","text":"What is the slope of the line given two points $$(3,4)$$ and $$(0,-3)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua21a-h4","type":"hint","dependencies":["a2d8720LinEqua21a-h3"],"title":"Write the equation","text":"Next, we use the point-slope formula with the slope of $$\\\\frac{7}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua21a-h5","type":"hint","dependencies":["a2d8720LinEqua21a-h4"],"title":"The point-slope formula","text":"Given one point and the slope, the point-slope formula will lead to the equation of a line: $$y-y_1=m(x$$ -x_1).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua21a-h6","type":"hint","dependencies":["a2d8720LinEqua21a-h5"],"title":"Picking a point","text":"We can pick either $$(3,4)$$ or $$(0,-3)$$ as $$(x_1,y_1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua21a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y-4=\\\\frac{7}{3} \\\\left(x-3\\\\right)$$"],"dependencies":["a2d8720LinEqua21a-h6"],"title":"Substitute","text":"Write the point-slope formula using the point $$(3,4)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua21a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y-4=\\\\frac{7}{3} x-7$$"],"dependencies":["a2d8720LinEqua21a-h7"],"title":"Parentheses","text":"Using the distributive property.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua21a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y=\\\\frac{7}{3} x-3$$"],"dependencies":["a2d8720LinEqua21a-h8"],"title":"Simplification","text":"Simplify the equation $$y-4=\\\\frac{7}{3} x-7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2d8720LinEqua22","title":"Finding the Equation of a Line and Writing It in Standard Form","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Linear Equations in One Variable","courseName":"OpenStax: College Algebra","steps":[{"id":"a2d8720LinEqua22a","stepAnswer":["$$12x+2y=-1$$"],"problemType":"TextBox","stepTitle":"Find the equation of the line with $$m=-6$$ and passing through the point $$(\\\\frac{1}{4},-2)$$. Write the equation in standard form.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12x+2y=-1$$","hints":{"DefaultPathway":[{"id":"a2d8720LinEqua22a-h1","type":"hint","dependencies":[],"title":"Writing in slope-intercept form","text":"We begin using the point-slope formula with the slope $$m$$ and the point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua22a-h2","type":"hint","dependencies":["a2d8720LinEqua22a-h1"],"title":"The point-slope formula","text":"Given one point and the slope, the point-slope formula will lead to the equation of a line: $$y-y_1=m\\\\left(x-x_1\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua22a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y-(-2)=-6\\\\left(x-\\\\frac{1}{4}\\\\right)$$"],"dependencies":["a2d8720LinEqua22a-h2"],"title":"Substitute","text":"Write the point-slope formula.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua22a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y+2=-6x+\\\\frac{3}{2}$$"],"dependencies":["a2d8720LinEqua22a-h3"],"title":"Parentheses","text":"Using the distributive property.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua22a-h5","type":"hint","dependencies":["a2d8720LinEqua22a-h4"],"title":"Eliminating denominators","text":"We should multiply through by $$2$$, as no fractions are permitted in standard form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua22a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2y+4=-12x+3$$"],"dependencies":["a2d8720LinEqua22a-h5"],"title":"Multiplication","text":"What is the result of $$2\\\\left(y+2\\\\right)=2\\\\left(-6x+\\\\frac{3}{2}\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua22a-h7","type":"hint","dependencies":["a2d8720LinEqua22a-h6"],"title":"Moving terms","text":"Both variables should be moved to the left side of the equal sign and the constants should be moved to the right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua22a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12x+2y=-1$$"],"dependencies":["a2d8720LinEqua22a-h7"],"title":"Writing in standard form","text":"Write the equation in standard form after moving terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2d8720LinEqua23","title":"Finding the Equation of a Line and Writing It in Standard Form","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Linear Equations in One Variable","courseName":"OpenStax: College Algebra","steps":[{"id":"a2d8720LinEqua23a","stepAnswer":["$$x+3y=2$$"],"problemType":"TextBox","stepTitle":"Find the equation of the line in standard form with slope $$m=\\\\frac{-1}{3}$$ and passing through the point $$(1,\\\\frac{1}{3})$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x+3y=2$$","hints":{"DefaultPathway":[{"id":"a2d8720LinEqua23a-h1","type":"hint","dependencies":[],"title":"Writing in slope-intercept form","text":"We begin using the point-slope formula with the slope $$m$$ and the point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua23a-h2","type":"hint","dependencies":["a2d8720LinEqua23a-h1"],"title":"The point-slope formula","text":"Given one point and the slope, the point-slope formula will lead to the equation of a line: $$y-y_1=m\\\\left(x-x_1\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua23a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y-\\\\frac{1}{3}=\\\\frac{-1}{3\\\\left(x-1\\\\right)}$$"],"dependencies":["a2d8720LinEqua23a-h2"],"title":"Substitute","text":"Write the point-slope formula.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua23a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y-\\\\frac{1}{3}=\\\\left(-\\\\frac{1}{3}\\\\right) x+\\\\frac{1}{3}$$"],"dependencies":["a2d8720LinEqua23a-h3"],"title":"Parentheses","text":"Using the distributive property.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua23a-h5","type":"hint","dependencies":["a2d8720LinEqua23a-h4"],"title":"Eliminating denominators","text":"We should multiply through by $$3$$, as no fractions are permitted in standard form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua23a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3y-1=-x+1$$"],"dependencies":["a2d8720LinEqua23a-h5"],"title":"Multiplication","text":"What is the result of $$3\\\\left(y-\\\\frac{1}{3}\\\\right)=3\\\\left(\\\\left(-\\\\frac{1}{3}\\\\right) x+\\\\frac{1}{3}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua23a-h7","type":"hint","dependencies":["a2d8720LinEqua23a-h6"],"title":"Moving terms","text":"Both variables should be moved to the left aside of the equal sign and the constants should be moved to the right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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Point","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Linear Equations in One Variable","courseName":"OpenStax: College Algebra","steps":[{"id":"a2d8720LinEqua30a","stepAnswer":["$$y=5x+3$$"],"problemType":"TextBox","stepTitle":"Find the equation of the line parallel to $$5x=7+y$$ and passing through the point $$(-1,-2)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=5x+3$$","hints":{"DefaultPathway":[{"id":"a2d8720LinEqua30a-h1","type":"hint","dependencies":[],"title":"Find the slope","text":"First, we should write the given equation in slope-intercept form to find the slope.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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Since the line is parallel to $$5x=7+y$$, they have the same slope.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua30a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a2d8720LinEqua30a-h3"],"title":"Slope","text":"What is the slope of the target line?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua30a-h5","type":"hint","dependencies":["a2d8720LinEqua30a-h4"],"title":"Point-slope formula","text":"Given one point and the slope, the point-slope formula will lead to the equation of a line: $$y-y_1=m\\\\left(x-x_1\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua30a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y-(-2)=5(x-(-1))$$"],"dependencies":["a2d8720LinEqua30a-h5"],"title":"Substitute","text":"Given the slope $$5$$ and the point $$(-1,-2)$$, what equation can we get using the point-slope formula?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua30a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y=5x+3$$"],"dependencies":["a2d8720LinEqua30a-h6"],"title":"Simplification","text":"What is the point-intercept form of equation $$y-(-2)=5(x-(-1))$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2d8720LinEqua31","title":"Finding the Equation of a Line Perpendicular to a Given Line Passing Through a Given Point","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Linear Equations in One Variable","courseName":"OpenStax: College Algebra","steps":[{"id":"a2d8720LinEqua31a","stepAnswer":["$$y=-\\\\left(\\\\frac{3}{5}\\\\right) x-\\\\frac{7}{5}$$"],"problemType":"TextBox","stepTitle":"Find the equation of the line perpendicular to $$5x-3y+4=0$$ and passing through the point $$(-4,1)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=-\\\\left(\\\\frac{3}{5}\\\\right) x-\\\\frac{7}{5}$$","hints":{"DefaultPathway":[{"id":"a2d8720LinEqua31a-h1","type":"hint","dependencies":[],"title":"Find the slope","text":"First, we should write the given equation in slope-intercept form to find the slope.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua31a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y=\\\\frac{5}{3} x+\\\\frac{4}{3}$$"],"dependencies":["a2d8720LinEqua31a-h1"],"title":"Slope-intercept form","text":"What is the slope-intercept form of equation $$5x-3y+4=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua31a-h3","type":"hint","dependencies":["a2d8720LinEqua31a-h2"],"title":"Relationship between slopes","text":"In order to write the equation of a line, we need to calculate its slope. Since the line is perpendicular to $$5x-3y+4=0$$, the product of their slopes is $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua31a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-3}{5}$$"],"dependencies":["a2d8720LinEqua31a-h3"],"title":"Slope","text":"What is the slope of the target line?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua31a-h5","type":"hint","dependencies":["a2d8720LinEqua31a-h4"],"title":"Point-slope formula","text":"Given one point and the slope, the point-slope formula will lead to the equation of a line: $$y-y_1=m\\\\left(x-x_1\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua31a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y-1=\\\\frac{-3}{5\\\\left(x-\\\\left(-4\\\\right)\\\\right)}$$"],"dependencies":["a2d8720LinEqua31a-h5"],"title":"Substitute","text":"Given the slope $$\\\\frac{-3}{5}$$ and the point $$(-4,1)$$, what equation can we get using the point-slope formula?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua31a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y=-\\\\left(\\\\frac{3}{5}\\\\right) x-\\\\frac{7}{5}$$"],"dependencies":["a2d8720LinEqua31a-h6"],"title":"Simplification","text":"What is the point-intercept form of equation $$y-1=\\\\frac{-3}{5\\\\left(x-\\\\left(-4\\\\right)\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2d8720LinEqua4","title":"Solving an Equation Algebraically When the Variable Appears on Both Sides","body":"Solve the equation in one variable","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Linear Equations in One Variable","courseName":"OpenStax: College Algebra","steps":[{"id":"a2d8720LinEqua4a","stepAnswer":["$$x=-3$$"],"problemType":"TextBox","stepTitle":"$$-2\\\\left(3x-1\\\\right)+x=14-x$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x=-3$$","hints":{"DefaultPathway":[{"id":"a2d8720LinEqua4a-h1","type":"hint","dependencies":[],"title":"Distributive property","text":"First we should apply the distributive property.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6x+2+x=14-x$$"],"dependencies":["a2d8720LinEqua4a-h1"],"title":"Distributive property","text":"What is the result after applying the distributive property?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua4a-h3","type":"hint","dependencies":["a2d8720LinEqua4a-h2"],"title":"Simplification","text":"Place $$x-terms$$ on one side and simplify.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=-3$$"],"dependencies":["a2d8720LinEqua4a-h3"],"title":"Simplification","text":"What is the result after placing all $$x-terms$$ in the left side and multiplying both sides by $$\\\\frac{-1}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2d8720LinEqua5","title":"For the following exercises, solve the equation for $$x$$.","body":"Solve the equation in one variable","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Linear Equations in One Variable","courseName":"OpenStax: College Algebra","steps":[{"id":"a2d8720LinEqua5a","stepAnswer":["$$x=2$$"],"problemType":"TextBox","stepTitle":"$$4x-3=5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x=2$$","hints":{"DefaultPathway":[{"id":"a2d8720LinEqua5a-h1","type":"hint","dependencies":[],"title":"Isolation","text":"First we should isolate the variable on one side of the equation by adding, subtracting, multiplying or dividing the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4x=8$$"],"dependencies":["a2d8720LinEqua5a-h1"],"title":"Addition","text":"What is the result after adding $$3$$ from both sides?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua5a-h3","type":"hint","dependencies":["a2d8720LinEqua5a-h2"],"title":"Normalization","text":"When the variable is multiplied by a coefficient in the final stage, multiply both sides fo the equation by the reciprocal of the cofficient.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=2$$"],"dependencies":["a2d8720LinEqua5a-h3"],"title":"Multiplication","text":"What is the result after multiplying both sides by $$\\\\frac{1}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a2d8720LinEqua5b","stepAnswer":["$$x=\\\\frac{2}{7}$$"],"problemType":"TextBox","stepTitle":"$$12-5\\\\left(x+3\\\\right)=2x-5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x=\\\\frac{2}{7}$$","hints":{"DefaultPathway":[{"id":"a2d8720LinEqua5b-h1","type":"hint","dependencies":[],"title":"Distributive property","text":"First we should apply the distributive property.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua5b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12-5x-15=2x-5$$"],"dependencies":["a2d8720LinEqua5b-h1"],"title":"Distributive property","text":"What is the result after applying the distributive property?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua5b-h3","type":"hint","dependencies":["a2d8720LinEqua5b-h2"],"title":"Simplification","text":"Place $$x-terms$$ on one side and simplify.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua5b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=\\\\frac{2}{7}$$"],"dependencies":["a2d8720LinEqua5b-h3"],"title":"Simplification","text":"What is the result after placing all $$x-terms$$ in the left side and multiplying both sides by $$\\\\frac{-1}{7}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2d8720LinEqua6","title":"Solving a Rational Equation","body":"Solve the rational equation:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Linear Equations in One Variable","courseName":"OpenStax: College Algebra","steps":[{"id":"a2d8720LinEqua6a","stepAnswer":["$$x=\\\\frac{1}{4}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{7}{2x}-\\\\frac{5}{3x}=\\\\frac{22}{3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x=\\\\frac{1}{4}$$","hints":{"DefaultPathway":[{"id":"a2d8720LinEqua6a-h1","type":"hint","dependencies":[],"title":"Find LCD","text":"First, we should eliminate all denominators by multiplying both sides of the equation by the least common denominator (LCD).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6x$$"],"dependencies":["a2d8720LinEqua6a-h1"],"title":"LCD","text":"What is the LCD of this equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$21-10=44x$$"],"dependencies":["a2d8720LinEqua6a-h2"],"title":"Eliminating","text":"Simplify $$6\\\\left(\\\\frac{7}{2x}-\\\\frac{5}{3x}\\\\right) x$$ $$=$$ $$6\\\\frac{22}{3} x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua6a-h4","type":"hint","dependencies":["a2d8720LinEqua6a-h3"],"title":"Solve equation","text":"Then we should solve the linear equation obtained.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=\\\\frac{1}{4}$$"],"dependencies":["a2d8720LinEqua6a-h4"],"title":"Linear equation","text":"Solve the linear equation $$21-10=44x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2d8720LinEqua7","title":"Solving a Rational Equation without Factoring.","body":"Solve the following rational equation:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Linear Equations in One Variable","courseName":"OpenStax: College Algebra","steps":[{"id":"a2d8720LinEqua7a","stepAnswer":["$$x=-1$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2}{x}-\\\\frac{3}{2}=\\\\frac{7}{2x}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x=-1$$","hints":{"DefaultPathway":[{"id":"a2d8720LinEqua7a-h1","type":"hint","dependencies":[],"title":"Find LCD","text":"We have three denominators: $$x$$, $$2$$ and $$2x$$. The product of the first two denominators is equal to the third denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x$$"],"dependencies":["a2d8720LinEqua7a-h1"],"title":"LCD","text":"What is the LCD of this equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4-3x=7$$"],"dependencies":["a2d8720LinEqua7a-h2"],"title":"Eliminating","text":"Simplify $$2x \\\\left(\\\\frac{2}{x}-\\\\frac{3}{2}\\\\right)=2x \\\\frac{7}{2x}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua7a-h4","type":"hint","dependencies":["a2d8720LinEqua7a-h3"],"title":"Solve equation","text":"Then we should solve the linear equation obtained.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua7a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=-1$$"],"dependencies":["a2d8720LinEqua7a-h4"],"title":"Linear equation","text":"Solve the linear equation $$4-3x=7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2d8720LinEqua8","title":"Solve the rational equation","body":"Solve the rational equation:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Linear Equations in One Variable","courseName":"OpenStax: College Algebra","steps":[{"id":"a2d8720LinEqua8a","stepAnswer":["$$x=\\\\frac{10}{3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2}{3x}=\\\\frac{1}{4}-\\\\frac{1}{6x}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x=\\\\frac{10}{3}$$","hints":{"DefaultPathway":[{"id":"a2d8720LinEqua8a-h1","type":"hint","dependencies":[],"title":"Find LCD","text":"We have three denominators: $$3x$$, $$4$$, and $$6x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12x$$"],"dependencies":["a2d8720LinEqua8a-h1"],"title":"LCD","text":"What is the LCD of this equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8=3x-2$$"],"dependencies":["a2d8720LinEqua8a-h2"],"title":"Eliminating","text":"Simplify $$12x \\\\frac{2}{3x}=12x \\\\frac{1}{4}-12x \\\\frac{1}{6x}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua8a-h4","type":"hint","dependencies":["a2d8720LinEqua8a-h3"],"title":"Solve equation","text":"Then we should solve the linear equation obtained.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua8a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=\\\\frac{10}{3}$$"],"dependencies":["a2d8720LinEqua8a-h4"],"title":"Linear equation","text":"Solve the linear equation $$8=3x-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2d8720LinEqua9","title":"Solving a Rational Equation by Factoring the Denominator","body":"Solve the following rational equation:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Linear Equations in One Variable","courseName":"OpenStax: College Algebra","steps":[{"id":"a2d8720LinEqua9a","stepAnswer":["$$x=\\\\frac{35}{2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{1}{x}=\\\\frac{1}{10}-\\\\frac{3}{4x}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x=\\\\frac{35}{2}$$","hints":{"DefaultPathway":[{"id":"a2d8720LinEqua9a-h1","type":"hint","dependencies":[],"title":"Factoring the denominator","text":"The three denominators in factored form are $$x$$, $$10=2\\\\times5$$, and $$4x=2\\\\times2 x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua9a-h2","type":"hint","dependencies":["a2d8720LinEqua9a-h1"],"title":"Find LCD","text":"The LCD is the smallest expression that is divisible by each one of the denominators.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20x$$"],"dependencies":["a2d8720LinEqua9a-h2"],"title":"LCD","text":"What is the LCD of this equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20=2x-15$$"],"dependencies":["a2d8720LinEqua9a-h3"],"title":"Eliminating","text":"Simplify $$20x \\\\frac{1}{x}=20x \\\\left(\\\\frac{1}{10}-\\\\frac{3}{4x}\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua9a-h5","type":"hint","dependencies":["a2d8720LinEqua9a-h4"],"title":"Solve equation","text":"Then we should solve the linear equation obtained.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2d8720LinEqua9a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=\\\\frac{35}{2}$$"],"dependencies":["a2d8720LinEqua9a-h5"],"title":"Linear equation","text":"Solve the linear equation $$20=2x-15$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2ef97dGrphingQuads1","title":"Finding the Axis of Symmetry and Vertex","body":"$$y=\\\\left(-2x^2\\\\right)-8x-3$$: First find the Axis of Symmetry of this equation, then find the vertex. Write the answers as a string: eg. $$\\"x=answer$$, (x,y)\\" without the quotes.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Graphing Quadratic Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a2ef97dGrphingQuads1a","stepAnswer":["x=-2, (-2,5)"],"problemType":"TextBox","stepTitle":"$$y=\\\\left(-2x^2\\\\right)-8x-3$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=-2$$, $$(-2,5)$$","hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads1a-h1","type":"hint","dependencies":[],"title":"Finding the Axis of Symmetry","text":"To find the axis of symmetry, you use this equation: $$x=\\\\left(-\\\\frac{b}{2} a\\\\right)$$ where $$y=a x^2+b x+c$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=-2$$"],"dependencies":["a2ef97dGrphingQuads1a-h1"],"title":"Finding the Axis of Symmetry","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads1a-h3","type":"hint","dependencies":["a2ef97dGrphingQuads1a-h2"],"title":"Finding the Vertex","text":"You find the vertex by plugging in the axis of symmetry as $$x$$ and finding $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads1a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-2,5)"],"dependencies":["a2ef97dGrphingQuads1a-h3"],"title":"Finding the Vertex","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2ef97dGrphingQuads10","title":"Finding Intercepts of a Parabola","body":"Find the $$x$$ intercepts of the parabola.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Graphing Quadratic Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a2ef97dGrphingQuads10a","stepAnswer":["No Real Roots"],"problemType":"MultipleChoice","stepTitle":"$$y=x^2-2x-8$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$1$$ and $$2$$","$$3$$ and $$-3$$","No Real Roots"],"hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads10a-h1","type":"hint","dependencies":[],"title":"Solving for intercepts","text":"Set the entire equation equal to zero","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads10a-h2","type":"hint","dependencies":["a2ef97dGrphingQuads10a-h1"],"title":"Solving for intercepts","text":"Calculate the determinant of the fucntion via the formula $$b^2$$ - 4ac","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads10a-h3","type":"hint","dependencies":["a2ef97dGrphingQuads10a-h2"],"title":"Solving for intercepts","text":"Notice that the determinant is negative. Therefore, there are no real roots.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a2ef97dGrphingQuads10b","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"Finding Intercepts of a Parabola","stepBody":"Find the $$y$$ intercepts of the parabola $$y=x^2-2x-8$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads10b-h1","type":"hint","dependencies":[],"title":"Solving for intercepts","text":"Plug in $$0$$ for $$x$$ and solve the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2ef97dGrphingQuads11","title":"Finding Intercepts of a Parabola","body":"The quadratic equation $$h=16t^2+vt+h$$ models the height of a volleyball hit straight upwards with initial velocity $$176$$ feet per second from a height of $$4$$ feet. $$h$$ is the hiehgt, $$t$$ is the time, and v is the initial velocity.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Graphing Quadratic Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a2ef97dGrphingQuads11a","stepAnswer":["$$\\\\frac{3}{2}$$"],"problemType":"TextBox","stepTitle":"Find the $$x$$ intercepts of the parabola. $$y=4x^2-12x-9$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{2}$$","hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads11a-h1","type":"hint","dependencies":[],"title":"Solving for intercepts","text":"Set the entire equation equal to zero","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads11a-h2","type":"hint","dependencies":["a2ef97dGrphingQuads11a-h1"],"title":"Solving for intercepts","text":"Calculate the determinant of the fucntion via the formula $$b^2$$ - 4ac","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads11a-h3","type":"hint","dependencies":["a2ef97dGrphingQuads11a-h2"],"title":"Solving for intercepts","text":"Notice that the determinant is zero. Therefore, there is only one root.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads11a-h4","type":"hint","dependencies":["a2ef97dGrphingQuads11a-h3"],"title":"Solving for intercepts","text":"Factor the equation by using the perfect square trinomial","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a2ef97dGrphingQuads11b","stepAnswer":["$$9$$"],"problemType":"TextBox","stepTitle":"Finding Intercepts of a Parabola","stepBody":"Find the $$y$$ intercepts of the parabola $$y=x^2-2x-8$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9$$","hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads11b-h1","type":"hint","dependencies":[],"title":"Solving for intercepts","text":"Plug in $$0$$ for $$x$$ and solve the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2ef97dGrphingQuads12","title":"Finding minimum value of a parabola","body":"Find the x-value of minimum value of the quadratic equation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Graphing Quadratic Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a2ef97dGrphingQuads12a","stepAnswer":["$$-1$$"],"problemType":"TextBox","stepTitle":"$$y=x^2+2x-8$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1$$","hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads12a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":[],"title":"Solving for the axis of symmetry","text":"What is the x-coordinate that results from the equation $$x$$ $$=$$ $$\\\\frac{-b}{2a}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a2ef97dGrphingQuads12b","stepAnswer":["$$-9$$"],"problemType":"TextBox","stepTitle":"Find the y-value of the minimum value of the quadratic equation $$y=x^2+2x-8$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-9$$","hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads12b-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":[],"title":"Solving for the axis of symmetry","text":"What is the y-coordinate that results from plugging in the axis of symmetry to the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2ef97dGrphingQuads13","title":"The quadratic equation $$h=16t^2+vt+h_0$$ models the height of a volleyball hit straight upwards with initial velocity $$176$$ feet per second from a height of $$4$$ feet. $$h$$ is the height of the volleyball, $$t$$ is the time, v is the initial velocity, and $$h_0$$ is the initial height.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Graphing Quadratic Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a2ef97dGrphingQuads13a","stepAnswer":["$$5.5$$"],"problemType":"TextBox","stepTitle":"How many seconds will it take the volleyball to reach its maximum height?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5.5$$","hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads13a-h1","type":"hint","dependencies":[],"title":"Axis of Symmetry","text":"Find the axis of symmetry through the formula $$x$$ $$=$$ - $$\\\\frac{b}{2} a$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a2ef97dGrphingQuads13b","stepAnswer":["$$488$$"],"problemType":"TextBox","stepTitle":"Find the maximum height of the volleyball.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$488$$","hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads13b-h1","type":"hint","dependencies":[],"title":"Finding the Vertex of Parabolas","text":"Plug in the axis of symmetry and solve the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2ef97dGrphingQuads2","title":"Finding the $$x$$ and $$y$$ Intercepts","body":"$$y=x^2+7x+6$$: Find the $$y$$ intercept then the $$x$$ intercept. Write the answer as a string: eg. (0,y), (x,0), (x,0). For the $$x$$ intercepts write the smaller $$x$$ intercepts first.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Graphing Quadratic Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a2ef97dGrphingQuads2a","stepAnswer":["(0,6), (-6,0), (-1,0)"],"problemType":"TextBox","stepTitle":"$$y=x^2+7x+6$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,6)$$, $$(-6,0)$$, $$(-1,0)$$","hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads2a-h1","type":"hint","dependencies":[],"title":"Finding the $$y$$ Intercept","text":"To find the $$y$$ intercept plug $$0$$ in for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a2ef97dGrphingQuads2a-h1"],"title":"Finding the $$y$$ Intercept","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads2a-h3","type":"hint","dependencies":["a2ef97dGrphingQuads2a-h2"],"title":"Finding the $$x$$ Intercept","text":"You find the $$x$$ intercept by plugging $$0$$ in for $$y$$ and solving the quadratic equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads2a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["-6, -1"],"dependencies":["a2ef97dGrphingQuads2a-h3"],"title":"Finding the $$x$$ Intercept","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2ef97dGrphingQuads20","title":"Determining Parabola Properties","body":"Determine if the parabola opens up or down.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Graphing Quadratic Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a2ef97dGrphingQuads20a","stepAnswer":["Opens down"],"problemType":"MultipleChoice","stepTitle":"$$y=-2x^2$$ $$-6x-7$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Opens down","Opens up"],"hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads20a-h1","type":"hint","dependencies":[],"title":"Locate \\"a\\"","text":"The quadratic is in the form $${ax}^2+bx+c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads20a-h2","type":"hint","dependencies":["a2ef97dGrphingQuads20a-h1"],"title":"Use \\"a\\"","text":"If a is negative, the parabola will open downwards. If it is positive, the parabola will open upwards.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads20a-h3","type":"hint","dependencies":["a2ef97dGrphingQuads20a-h2"],"title":"Answer","text":"The parabola opens down.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2ef97dGrphingQuads21","title":"Determining Parabola Properties","body":"Determine if the parabola opens up or down.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Graphing Quadratic Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a2ef97dGrphingQuads21a","stepAnswer":["Opens up"],"problemType":"MultipleChoice","stepTitle":"$$y=6x^2+2x+3$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Opens down","Opens up"],"hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads21a-h1","type":"hint","dependencies":[],"title":"Locate \\"a\\"","text":"The quadratic is in the form $${ax}^2+bx+c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads21a-h2","type":"hint","dependencies":["a2ef97dGrphingQuads21a-h1"],"title":"Use \\"a\\"","text":"If a is negative, the parabola will open downwards. If it is positive, the parabola will open upwards.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads21a-h3","type":"hint","dependencies":["a2ef97dGrphingQuads21a-h2"],"title":"Answer","text":"The parabola opens up.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2ef97dGrphingQuads22","title":"Determining Parabola Properties","body":"Determine if the parabola opens up or down.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Graphing Quadratic Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a2ef97dGrphingQuads22a","stepAnswer":["Opens up"],"problemType":"MultipleChoice","stepTitle":"$$y=4x^2+x-4$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Opens down","Opens up"],"hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads22a-h1","type":"hint","dependencies":[],"title":"Locate \\"a\\"","text":"The quadratic is in the form $${ax}^2+bx+c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads22a-h2","type":"hint","dependencies":["a2ef97dGrphingQuads22a-h1"],"title":"Use \\"a\\"","text":"If a is negative, the parabola will open downwards. If it is positive, the parabola will open upwards.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads22a-h3","type":"hint","dependencies":["a2ef97dGrphingQuads22a-h2"],"title":"Answer","text":"The parabola opens up.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2ef97dGrphingQuads23","title":"Determining Parabola Properties","body":"Determine if the parabola opens up or down.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Graphing Quadratic Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a2ef97dGrphingQuads23a","stepAnswer":["Opens down"],"problemType":"MultipleChoice","stepTitle":"$$y=-2x^2$$ $$-6x-7$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Opens down","Opens up"],"hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads23a-h1","type":"hint","dependencies":[],"title":"Locate \\"a\\"","text":"The quadratic is in the form $${ax}^2+bx+c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads23a-h2","type":"hint","dependencies":["a2ef97dGrphingQuads23a-h1"],"title":"Use \\"a\\"","text":"If a is negative, the parabola will open downwards. If it is positive, the parabola will open upwards.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads23a-h3","type":"hint","dependencies":["a2ef97dGrphingQuads23a-h2"],"title":"Answer","text":"The parabola opens down.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2ef97dGrphingQuads24","title":"Determining Parabola Properties","body":"Identify qualities of the parabola.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Graphing Quadratic Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a2ef97dGrphingQuads24a","stepAnswer":["$$x=-4$$"],"problemType":"TextBox","stepTitle":"$$x^2+8x+1$$","stepBody":"Find the axis of symmmetry. Input your answer as $$x=(axis$$ of symmetry).","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x=-4$$","hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads24a-h1","type":"hint","dependencies":[],"title":"Equation for axis of symmetry","text":"The equation for the axis of symmetry of a quadratic is $$x=\\\\frac{-b}{2} a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads24a-h2","type":"hint","dependencies":["a2ef97dGrphingQuads24a-h1"],"title":"Plug in","text":"Plug in the a and $$b$$ values into the axis of symmetry equation. Then simplify to find it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads24a-h3","type":"hint","dependencies":["a2ef97dGrphingQuads24a-h2"],"title":"Answer","text":"The answer is $$x=-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a2ef97dGrphingQuads24b","stepAnswer":["$$(-4,-17)$$"],"problemType":"MultipleChoice","stepTitle":"$$x^2+8x+1$$","stepBody":"Find the vertex.","answerType":"string","variabilization":{},"answerLatex":"$$(-4,-17)$$","choices":["$$(-4,-17)$$","$$(-2,3)$$","$$(-7,12)$$","$$(5,27)$$"],"hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads24b-h1","type":"hint","dependencies":[],"title":"X-coordinate","text":"The x-coordinate of the vertex is the axis of symmetry.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads24b-h2","type":"hint","dependencies":["a2ef97dGrphingQuads24b-h1"],"title":"Y-coordinate","text":"Plug the x-coordinate of the vertex into the quadratic to get the $$y$$ coordinate of the vertex.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads24b-h3","type":"hint","dependencies":["a2ef97dGrphingQuads24b-h2"],"title":"Answer","text":"The vertex is $$(-4,-17)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2ef97dGrphingQuads25","title":"Determining Parabola Properties","body":"Identify qualities of the parabola.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Graphing Quadratic Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a2ef97dGrphingQuads25a","stepAnswer":["$$x=-5$$"],"problemType":"TextBox","stepTitle":"$$x^2+10x+25$$","stepBody":"Find the axis of symmmetry. Input your answer as $$x=(axis$$ of symmetry).","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x=-5$$","hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads25a-h1","type":"hint","dependencies":[],"title":"Equation for axis of symmetry","text":"The equation for the axis of symmetry of a quadratic is $$x=\\\\frac{-b}{2} a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads25a-h2","type":"hint","dependencies":["a2ef97dGrphingQuads25a-h1"],"title":"Plug in","text":"Plug in the a and $$b$$ values into the axis of symmetry equation. Then simplify to find it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads25a-h3","type":"hint","dependencies":["a2ef97dGrphingQuads25a-h2"],"title":"Answer","text":"The answer is $$x=-5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a2ef97dGrphingQuads25b","stepAnswer":["$$(-5,0)$$"],"problemType":"MultipleChoice","stepTitle":"$$x^2+10x+25$$","stepBody":"Find the vertex.","answerType":"string","variabilization":{},"answerLatex":"$$(-5,0)$$","choices":["$$(-5,0)$$","$$(-2,3)$$","$$(-3,2)$$","$$(2,-16)$$"],"hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads25b-h1","type":"hint","dependencies":[],"title":"X-coordinate","text":"The x-coordinate of the vertex is the axis of symmetry.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads25b-h2","type":"hint","dependencies":["a2ef97dGrphingQuads25b-h1"],"title":"Y-coordinate","text":"Plug the x-coordinate of the vertex into the quadratic to get the $$y$$ coordinate of the vertex.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads25b-h3","type":"hint","dependencies":["a2ef97dGrphingQuads25b-h2"],"title":"Answer","text":"The vertex is $$(-5,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2ef97dGrphingQuads26","title":"Determining Parabola Properties","body":"Identify qualities of the parabola.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Graphing Quadratic Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a2ef97dGrphingQuads26a","stepAnswer":["$$x=1$$"],"problemType":"TextBox","stepTitle":"$$-\\\\left(x^2\\\\right)+2x+5$$","stepBody":"Find the axis of symmmetry. Input your answer as $$x=(axis$$ of symmetry).","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x=1$$","hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads26a-h1","type":"hint","dependencies":[],"title":"Equation for axis of symmetry","text":"The equation for the axis of symmetry of a quadratic is $$x=\\\\frac{-b}{2} a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads26a-h2","type":"hint","dependencies":["a2ef97dGrphingQuads26a-h1"],"title":"Plug in","text":"Plug in the a and $$b$$ values into the axis of symmetry equation. Then simplify to find it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads26a-h3","type":"hint","dependencies":["a2ef97dGrphingQuads26a-h2"],"title":"Answer","text":"The answer is $$x=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a2ef97dGrphingQuads26b","stepAnswer":["$$(1,6)$$"],"problemType":"MultipleChoice","stepTitle":"$$x^2+10x+25$$","stepBody":"Find the vertex.","answerType":"string","variabilization":{},"answerLatex":"$$(1,6)$$","choices":["$$(1,6)$$","$$(3,7)$$","$$(6,-4)$$","$$(2,-16)$$"],"hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads26b-h1","type":"hint","dependencies":[],"title":"X-coordinate","text":"The x-coordinate of the vertex is the axis of symmetry.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads26b-h2","type":"hint","dependencies":["a2ef97dGrphingQuads26b-h1"],"title":"Y-coordinate","text":"Plug the x-coordinate of the vertex into the quadratic to get the $$y$$ coordinate of the vertex.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads26b-h3","type":"hint","dependencies":["a2ef97dGrphingQuads26b-h2"],"title":"Answer","text":"The vertex is $$(1,6)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2ef97dGrphingQuads27","title":"Graphing Quadratic Equations in Two Variables","body":"$$y=-\\\\left(x^2\\\\right)-14x-49$$","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Graphing Quadratic Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a2ef97dGrphingQuads27a","stepAnswer":["$$(-7,0)$$"],"problemType":"MultipleChoice","stepTitle":"Find the x-intercept of the given equation.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-7,0)$$","choices":["$$(-7,0)$$","$$(4,0)$$","$$(-2,0)$$","$$(3,0)$$"],"hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads27a-h1","type":"hint","dependencies":[],"title":"Defining an x-intercept","text":"A x-intercept is the point where an equation hits the x-axis. This means the y-coordinate will be $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads27a-h2","type":"hint","dependencies":["a2ef97dGrphingQuads27a-h1"],"title":"Plugging in $$0$$","text":"Given the definition of a x-intercept, plug in $$0$$ for $$y$$ to solve for the x-value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads27a-h3","type":"hint","dependencies":["a2ef97dGrphingQuads27a-h2"],"title":"Factoring a Quadratic Equation","text":"The factored form of the quadratic equation is $${\\\\left(x+7\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads27a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-7$$"],"dependencies":["a2ef97dGrphingQuads27a-h3"],"title":"Solving for $$x$$","text":"What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a2ef97dGrphingQuads27b","stepAnswer":["$$(0,49)$$"],"problemType":"MultipleChoice","stepTitle":"Find the y-intercept of the given equation.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,49)$$","choices":["$$(0,7)$$","$$(0,-5)$$","$$(0,21)$$","$$(0,49)$$"],"hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads27b-h1","type":"hint","dependencies":[],"title":"Defining an y-intercept","text":"A y-intercept is the point where an equation hits the y-axis. This means the x-coordinate will be $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads27b-h2","type":"hint","dependencies":["a2ef97dGrphingQuads27b-h1"],"title":"Plugging in $$0$$","text":"Given the definition of a y-intercept, plug in $$0$$ for $$x$$ to solve for the y-value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads27b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$49$$"],"dependencies":["a2ef97dGrphingQuads27b-h2"],"title":"Solving for $$y$$","text":"What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2ef97dGrphingQuads28","title":"Solve Maximum and Minimum","body":"In the following exercise, find the maximum or minimum value","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Graphing Quadratic Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a2ef97dGrphingQuads28a","stepAnswer":["$$(-0.25, -1.125)$$"],"problemType":"MultipleChoice","stepTitle":"$$y=2x^2+x-1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-0.25, -1.125)$$","choices":["$$(-0.5, 1)$$","$$(-0.25, -1.125)$$","$$(0.75, 2)$$","$$(1, -0.25)$$"],"hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads28a-h1","type":"hint","dependencies":[],"title":"Determining Maximum or Minimum","text":"We can first identify what direction the parabola is facing to determine whether we will be solving for a maximum or minimum. Since the coefficient with the greatest variable power is positive, the parabola will be oriented as a \\"U\\" meaning we will be solving for a minimum.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads28a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-0.25$$"],"dependencies":["a2ef97dGrphingQuads28a-h1"],"title":"Finding the x-coordinate","text":"Solving for the minimum value is the same as finding the vertex of the parabola. To find the x-coordinate of the vertex we can use the equation $$x=\\\\frac{-b}{2a}$$. What is the x-coordinate of the vertex?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads28a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1.125$$"],"dependencies":["a2ef97dGrphingQuads28a-h2"],"title":"Finding the y-coordinate","text":"Given the x-coordinate, we can plug it in to solve for the y-coordinate of the vertex. What is the y-coordinate?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2ef97dGrphingQuads29","title":"Solve Maximum and Minimum","body":"In the following exercise, find the maximum or minimum value","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Graphing Quadratic Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a2ef97dGrphingQuads29a","stepAnswer":["$$(1.5, 4)$$"],"problemType":"MultipleChoice","stepTitle":"$$y=\\\\left(-4x^2\\\\right)+12x-5$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(1.5, 4)$$","choices":["$$(1.5, 4)$$","$$(2,3)$$","$$(3.5, 1)$$","$$(7,2)$$"],"hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads29a-h1","type":"hint","dependencies":[],"title":"Determining Maximum or Minimum","text":"We can first identify what direction the parabola is facing to determine whether we will be solving for a maximum or minimum. Since the coefficient with the greatest variable power is negative, the parabola will be oriented as a upside down \\"U\\" meaning we will be solving for a maximum.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads29a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.5$$"],"dependencies":["a2ef97dGrphingQuads29a-h1"],"title":"Finding the x-coordinate","text":"Solving for the minimum value is the same as finding the vertex of the parabola. To find the x-coordinate of the vertex we can use the equation $$x=\\\\frac{-b}{2a}$$. What is the x-coordinate of the vertex?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads29a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a2ef97dGrphingQuads29a-h2"],"title":"Finding the y-coordinate","text":"Given the x-coordinate, we can plug it in to solve for the y-coordinate of the vertex. What is the y-coordinate?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2ef97dGrphingQuads3","title":"Finding the $$x$$ and $$y$$ Intercepts","body":"$$y=x^2+10x-11$$: Find the $$y$$ intercept then the $$x$$ intercept. Write the answer as a string: eg. (0,y), (x,0), (x,0). For the $$x$$ intercepts write the smaller $$x$$ intercepts first.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Graphing Quadratic Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a2ef97dGrphingQuads3a","stepAnswer":["(0,-11), (-11,0), (1,0)"],"problemType":"TextBox","stepTitle":"$$y=x^2+10x-11$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,-11)$$, $$(-11,0)$$, $$(1,0)$$","hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads3a-h1","type":"hint","dependencies":[],"title":"Finding the $$y$$ Intercept","text":"To find the $$y$$ intercept plug $$0$$ in for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-11$$"],"dependencies":["a2ef97dGrphingQuads3a-h1"],"title":"Finding the $$y$$ Intercept","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads3a-h3","type":"hint","dependencies":["a2ef97dGrphingQuads3a-h2"],"title":"Finding the $$x$$ Intercept","text":"You find the $$x$$ intercept by plugging $$0$$ in for $$y$$ and solving the quadratic equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads3a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["-11, 1"],"dependencies":["a2ef97dGrphingQuads3a-h3"],"title":"Finding the $$x$$ Intercept","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2ef97dGrphingQuads30","title":"Solve Maximum and Minimum","body":"In the following exercise, find the maximum or minimum value","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Graphing Quadratic Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a2ef97dGrphingQuads30a","stepAnswer":["$$(3,6)$$"],"problemType":"MultipleChoice","stepTitle":"$$y=x^2-6x+15$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(3,6)$$","choices":["$$(2,1)$$","$$(3,6)$$","$$(4,9)$$","$$(7,2)$$"],"hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads30a-h1","type":"hint","dependencies":[],"title":"Determining Maximum or Minimum","text":"We can first identify what direction the parabola is facing to determine whether we will be solving for a maximum or minimum. Since the coefficient with the greatest variable power is positive, the parabola will be oriented as a \\"U\\" meaning we will be solving for a minimum.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads30a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a2ef97dGrphingQuads30a-h1"],"title":"Finding the x-coordinate","text":"Solving for the minimum value is the same as finding the vertex of the parabola. To find the x-coordinate of the vertex we can use the equation $$x=\\\\frac{-b}{2a}$$. What is the x-coordinate of the vertex?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads30a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a2ef97dGrphingQuads30a-h2"],"title":"Finding the y-coordinate","text":"Given the x-coordinate, we can plug it in to solve for the y-coordinate of the vertex. What is the y-coordinate?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2ef97dGrphingQuads31","title":"Solve Maximum and Minimum","body":"In the following exercise, find the maximum or minimum value","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Graphing Quadratic Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a2ef97dGrphingQuads31a","stepAnswer":["$$(2,-1)$$"],"problemType":"MultipleChoice","stepTitle":"$$y=-\\\\left(x^2\\\\right)+4x-5$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(2,-1)$$","choices":["$$(0,1)$$","$$(7,-2)$$","$$(2,-1)$$","$$(3,5)$$"],"hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads31a-h1","type":"hint","dependencies":[],"title":"Determining Maximum or Minimum","text":"We can first identify what direction the parabola is facing to determine whether we will be solving for a maximum or minimum. Since the coefficient with the greatest variable power is negative, the parabola will be oriented as a upside down \\"U\\" meaning we will be solving for a maximum.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads31a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a2ef97dGrphingQuads31a-h1"],"title":"Finding the x-coordinate","text":"Solving for the minimum value is the same as finding the vertex of the parabola. To find the x-coordinate of the vertex we can use the equation $$x=\\\\frac{-b}{2a}$$. What is the x-coordinate of the vertex?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads31a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a2ef97dGrphingQuads31a-h2"],"title":"Finding the y-coordinate","text":"Given the x-coordinate, we can plug it in to solve for the y-coordinate of the vertex. What is the y-coordinate?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2ef97dGrphingQuads32","title":"Solve Maximum and Minimum","body":"In the following exercise, find the maximum or minimum value","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Graphing Quadratic Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a2ef97dGrphingQuads32a","stepAnswer":["$$(0,16)$$"],"problemType":"MultipleChoice","stepTitle":"$$y=\\\\left(-9x^2\\\\right)+16$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,16)$$","choices":["$$(0,5)$$","$$(0,10)$$","$$(0,16)$$","$$(0,81)$$"],"hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads32a-h1","type":"hint","dependencies":[],"title":"Determining Maximum or Minimum","text":"We can first identify what direction the parabola is facing to determine whether we will be solving for a maximum or minimum. Since the coefficient with the greatest variable power is negative, the parabola will be oriented as a upside down \\"U\\" meaning we will be solving for a maximum.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads32a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a2ef97dGrphingQuads32a-h1"],"title":"Finding the x-coordinate","text":"Solving for the minimum value is the same as finding the vertex of the parabola. To find the x-coordinate of the vertex we can use the equation $$x=\\\\frac{-b}{2a}$$. What is the x-coordinate of the vertex?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads32a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["a2ef97dGrphingQuads32a-h2"],"title":"Finding the y-coordinate","text":"Given the x-coordinate, we can plug it in to solve for the y-coordinate of the vertex. What is the y-coordinate?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2ef97dGrphingQuads33","title":"Solve Maximum and Minimum","body":"In the following exercise, find the maximum or minimum value","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Graphing Quadratic Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a2ef97dGrphingQuads33a","stepAnswer":["$$(0,-49)$$"],"problemType":"MultipleChoice","stepTitle":"$$y=4x^2-49$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,-49)$$","choices":["$$(0,-16)$$","$$(0,-25)$$","$$(0,-36)$$","$$(0,-49)$$"],"hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads33a-h1","type":"hint","dependencies":[],"title":"Determining Maximum or Minimum","text":"We can first identify what direction the parabola is facing to determine whether we will be solving for a maximum or minimum. Since the coefficient with the greatest variable power is positive, the parabola will be oriented as a \\"U\\" meaning we will be solving for a minimum.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads33a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a2ef97dGrphingQuads33a-h1"],"title":"Finding the x-coordinate","text":"Solving for the minimum value is the same as finding the vertex of the parabola. To find the x-coordinate of the vertex we can use the equation $$x=\\\\frac{-b}{2a}$$. What is the x-coordinate of the vertex?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads33a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-49$$"],"dependencies":["a2ef97dGrphingQuads33a-h2"],"title":"Finding the y-coordinate","text":"Given the x-coordinate, we can plug it in to solve for the y-coordinate of the vertex. What is the y-coordinate?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2ef97dGrphingQuads4","title":"Finding the $$x$$ and $$y$$ Intercepts","body":"$$y=\\\\left(-x^2\\\\right)+8x+19$$: Find the $$y$$ intercept then the $$x$$ intercept. Write the answer as a string: eg. (0,y), (x,0), (x,0). For the $$x$$ intercepts write the smaller $$x$$ intercepts first. If there are no $$x$$ intercepts write DNE for Does Not Exist.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Graphing Quadratic Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a2ef97dGrphingQuads4a","stepAnswer":["(0,-19), DNE"],"problemType":"TextBox","stepTitle":"$$y=\\\\left(-x^2\\\\right)+8x+19$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,-19)$$, DNE","hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads4a-h1","type":"hint","dependencies":[],"title":"Finding the $$y$$ Intercept","text":"To find the $$y$$ intercept plug $$0$$ in for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-19$$"],"dependencies":["a2ef97dGrphingQuads4a-h1"],"title":"Finding the $$y$$ Intercept","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads4a-h3","type":"hint","dependencies":["a2ef97dGrphingQuads4a-h2"],"title":"Finding the $$x$$ Intercept","text":"You find the $$x$$ intercept by plugging $$0$$ in for $$y$$ and solving the quadratic equation. If there is no solution then write DNE for Does Not Exist","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads4a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["DNE"],"dependencies":["a2ef97dGrphingQuads4a-h3"],"title":"Finding the $$x$$ Intercept","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2ef97dGrphingQuads5","title":"Finding the $$x$$ and $$y$$ Intercepts","body":"$$y=x^2+6x+13$$: Find the $$y$$ intercept then the $$x$$ intercept. Write the answer as a string: eg. (0,y), (x,0), (x,0). For the $$x$$ intercepts write the smaller $$x$$ intercepts first. If there are no $$x$$ intercepts write DNE for Does Not Exist.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Graphing Quadratic Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a2ef97dGrphingQuads5a","stepAnswer":["(0,13), DNE"],"problemType":"TextBox","stepTitle":"$$y=x^2+6x+13$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,13)$$, DNE","hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads5a-h1","type":"hint","dependencies":[],"title":"Finding the $$y$$ Intercept","text":"To find the $$y$$ intercept plug $$0$$ in for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$13$$"],"dependencies":["a2ef97dGrphingQuads5a-h1"],"title":"Finding the $$y$$ Intercept","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads5a-h3","type":"hint","dependencies":["a2ef97dGrphingQuads5a-h2"],"title":"Finding the $$x$$ Intercept","text":"You find the $$x$$ intercept by plugging $$0$$ in for $$y$$ and solving the quadratic equation. If there is no solution then write DNE for Does Not Exist","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads5a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["DNE"],"dependencies":["a2ef97dGrphingQuads5a-h3"],"title":"Finding the $$x$$ Intercept","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2ef97dGrphingQuads6","title":"Finding the $$x$$ and $$y$$ Intercepts","body":"$$y=4x^2-20x+25$$: Find the $$y$$ intercept then the $$x$$ intercept. Write the answer as a string: eg. (0,y), (x,0), (x,0). For the $$x$$ intercepts write the smaller $$x$$ intercepts first.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Graphing Quadratic Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a2ef97dGrphingQuads6a","stepAnswer":["(0,25), (2.5,0)"],"problemType":"TextBox","stepTitle":"$$y=4x^2-20x+25$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,25)$$, $$(2.5, 0)$$","hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads6a-h1","type":"hint","dependencies":[],"title":"Finding the $$y$$ Intercept","text":"To find the $$y$$ intercept plug $$0$$ in for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["a2ef97dGrphingQuads6a-h1"],"title":"Finding the $$y$$ Intercept","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads6a-h3","type":"hint","dependencies":["a2ef97dGrphingQuads6a-h2"],"title":"Finding the $$x$$ Intercept","text":"You find the $$x$$ intercept by plugging $$0$$ in for $$y$$ and solving the quadratic equation. If there is no solution then write DNE for Does Not Exist","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.5$$"],"dependencies":["a2ef97dGrphingQuads6a-h3"],"title":"Finding the $$x$$ Intercept","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2ef97dGrphingQuads7","title":"Parabola Orientation","body":"Determine whether each parabola opens upward or downward:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Graphing Quadratic Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a2ef97dGrphingQuads7a","stepAnswer":["Downward"],"problemType":"MultipleChoice","stepTitle":"$$y$$ $$=$$ $$-3x^2+2x-4$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Upward","Downward"],"hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads7a-h1","type":"hint","dependencies":[],"title":"Parabola Orientation","text":"Consider the parabolic form $${ax}^2+bx+c$$. Find the sign of a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a2ef97dGrphingQuads7b","stepAnswer":["Upward"],"problemType":"MultipleChoice","stepTitle":"Determine whether each parabola opens upward or downward:","stepBody":"$$y$$ $$=$$ $$6x^2+7x-9$$","answerType":"string","variabilization":{},"choices":["Upward","Downward"],"hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads7b-h1","type":"hint","dependencies":[],"title":"Parabola Orientation","text":"Consider the parabolic form $${ax}^2+bx+c$$. Find the sign of a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2ef97dGrphingQuads8","title":"Axis of Symmetry and Vertex of a Parabola","body":"Find the axis of symmetry for the following function.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Graphing Quadratic Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a2ef97dGrphingQuads8a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"$$y=3x^2-6x+2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads8a-h1","type":"hint","dependencies":[],"title":"Axis of Symmetry","text":"Remember the formula for the axis of symmetry: $$x$$ $$=$$ $$\\\\frac{-b}{2a}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a2ef97dGrphingQuads8b","stepAnswer":["$$-1$$"],"problemType":"TextBox","stepTitle":"Vertex of Parabolas","stepBody":"Find the axis of symmetry for the function $$y=3x2-6x+2$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1$$","hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads8b-h1","type":"hint","dependencies":[],"title":"Vertex of Parabolas","text":"Remember how to find the vertex of the parabola. Plug in the $$x$$ coordinate achieved from the axis of symmetry to the original parabolic equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a2ef97dGrphingQuads9","title":"Finding Intercepts of a Parabola","body":"Find the $$x$$ intercepts of the parabola.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Graphing Quadratic Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a2ef97dGrphingQuads9a","stepAnswer":["$$4$$ and $$-2$$"],"problemType":"MultipleChoice","stepTitle":"$$y=x^2-2x-8$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$4$$ and $$-2$$","choices":["$$4$$ and $$-2$$","$$2$$ and $$3$$","$$5$$ and $$4$$"],"hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads9a-h1","type":"hint","dependencies":[],"title":"Solving for intercepts","text":"Set the entire equation equal to zero","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads9a-h2","type":"hint","dependencies":["a2ef97dGrphingQuads9a-h1"],"title":"Solving for intercepts","text":"Factor the equation by thinking about numbers that add to $$-2$$, but whose product is negative $$8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a2ef97dGrphingQuads9a-h3","type":"hint","dependencies":["a2ef97dGrphingQuads9a-h2"],"title":"Solving for intercepts","text":"Set each of the factors equal to zero","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a2ef97dGrphingQuads9b","stepAnswer":["$$-8$$"],"problemType":"TextBox","stepTitle":"Finding Intercepts of a Parabola","stepBody":"Find the $$y$$ intercepts of the parabola $$y=x^2-2x-8$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-8$$","hints":{"DefaultPathway":[{"id":"a2ef97dGrphingQuads9b-h1","type":"hint","dependencies":[],"title":"Solving for intercepts","text":"Plug in $$0$$ for $$x$$ and solve the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a320fba11.6airline1","title":"Test of a Single Variance","body":"Suppose an airline claims that its flights are consistently on time with an average delay of at most $$15$$ minutes. It claims that the average delay is so consistent that the variance is no more than $$150$$ minutes. Doubting the consistency part of the claim, a disgruntled traveler calculates the delays for his next $$25$$ flights. The average delay for those $$25$$ flights is $$22$$ minutes with a standard deviation of $$15$$ minutes.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.6 Test of a Single Variance","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a320fba11.6airline1a","stepAnswer":["variance"],"problemType":"MultipleChoice","stepTitle":"Is the traveler disputing the claim about the average or about the variance?","stepBody":"","answerType":"string","variabilization":{},"choices":["variance","average"],"hints":{"DefaultPathway":[{"id":"a320fba11.6airline1a-h1","type":"hint","dependencies":[],"title":"Test of a Single Variance","text":"Notice that the traveler is disputing about the consistency of the average flight times, not the literal average.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a320fba11.6airline1a-h2","type":"hint","dependencies":["a320fba11.6airline1a-h1"],"title":"Test of a Single Variance","text":"Since the consistency of flight times is discussed, the variance is being disputed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a320fba11.6airline1b","stepAnswer":["$$225$$"],"problemType":"TextBox","stepTitle":"A sample standard deviation of $$15$$ minutes is the same as a sample variance of $$___$$ minutes.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$225$$","hints":{"DefaultPathway":[{"id":"a320fba11.6airline1b-h1","type":"hint","dependencies":[],"title":"Sample Standard Deviation","text":"A sample standard deviation is the square root of the variance. Let s denote standard deviation. If s would be the standard deviation, what would be thte variance?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a320fba11.6airline1b-h2","type":"hint","dependencies":["a320fba11.6airline1b-h1"],"title":"Sample Standard Deviation","text":"$$s^2$$ is the variance, therefore the variance can be found by squaring the standard deviation, or the square of $$15$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a320fba11.6airline2","title":"Test of a Single Variance","body":"Suppose an airline claims that its flights are consistently on time with an average delay of at most $$15$$ minutes. It claims that the average delay is so consistent that the variance is no more than $$150$$ minutes. Doubting the consistency part of the claim, a disgruntled traveler calculates the delays for his next $$25$$ flights. The average delay for those $$25$$ flights is $$22$$ minutes with a standard deviation of $$15$$ minutes.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.6 Test of a Single Variance","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a320fba11.6airline2a","stepAnswer":["H_0:s**2<=150; H_a:s**2>150"],"problemType":"MultipleChoice","stepTitle":"State the null and alternative hypotheses. let $$s^2$$ denote the variance.","stepBody":"","answerType":"string","variabilization":{},"choices":["H_0:s**2<=150; H_a:s**2>150","H_0:s=150; H_a:s>150","H_0:s**2=12.24; H_a:s**2>12.24","H_0:s**2<12.24; H_a:s**2>12.24"],"hints":{"DefaultPathway":[{"id":"a320fba11.6airline2a-h1","type":"hint","dependencies":[],"title":"Test of a Single Variance","text":"The traveler is disputing the claim that the variance is $$150$$. The traveler collected their own data to test against the claim, therefore the traveler\'s claim is the alternative claim.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a320fba11.6airline2a-h2","type":"hint","dependencies":["a320fba11.6airline2a-h1"],"title":"Test of a Single Variance","text":"We have to be specific about the hypothesis when constructing tests so that we can get proper conclusions. Sometimes it can be hard to pin point the key alternative hypothesis like in this case because the traveler only \\"doubts\\" the claim which would lead us to believe that the alternative hypothesis would be that the variance isn\'t equal to $$150$$. However, the \\"no more than 150\\" claim by the airline is the most clear message that the null hypothesis is that the variance is less than or equal to $$150$$, therefore we should model our hypotheses off of airline\'s null claim.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a320fba11.6airline2b","stepAnswer":["$$right-tailed$$"],"problemType":"MultipleChoice","stepTitle":"Is this a right-tailed, left-tailed, or two-tailed test?","stepBody":"","answerType":"string","variabilization":{},"choices":["$$left-tailed$$","none of the listed options work","$$right-tailed$$","$$right-tailed$$","$$two-tailed$$"],"hints":{"DefaultPathway":[{"id":"a320fba11.6airline2b-h1","type":"hint","dependencies":[],"title":"Hypothesis Testing","text":"When the alternative claim is that the test statistic is greater than the null value, then the test is a right tailed test.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a320fba11.6airline2c","stepAnswer":["$$24$$"],"problemType":"TextBox","stepTitle":"What is the value of the degrees of freedom?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$24$$","hints":{"DefaultPathway":[{"id":"a320fba11.6airline2c-h1","type":"hint","dependencies":[],"title":"Degrees of freedom is calculated by subtracting one from the sample.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a320fba11.6airline2c-h2","type":"hint","dependencies":["a320fba11.6airline2c-h1"],"title":"Our sample size is $$n=25$$. The degrees of freedom would be $$25-1$$.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a320fba11.6airline2d","stepAnswer":["$$36$$"],"problemType":"MultipleChoice","stepTitle":"Determine the value of the chi-square test statistic.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$36$$","choices":["$$36$$","$$25$$","$$13$$","$$3$$"],"hints":{"DefaultPathway":[{"id":"a320fba11.6airline2d-h1","type":"hint","dependencies":[],"title":"Chi-square Test Statistic","text":"Recall that the test statistic in a test of a single variance is test statistic $$X^2=\\\\frac{\\\\left(n-1\\\\right) s^2}{q^2}$$. Note that $$q^2$$ here denotes the population variance whereas s denotes sample variance.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a320fba11.6airline2d-h2","type":"hint","dependencies":["a320fba11.6airline2d-h1"],"title":"Chi-square Test Statistic","text":"With n=25,q**2=150,s=15, $$X^2=\\\\frac{\\\\left(25-1\\\\right) {15}^2}{150}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a320fba11.6archer","title":"Test of a Single Variance","body":"An archer\u2019s standard deviation for his hits is six (data is measured in distance from the center of the target). An observer claims the standard deviation is less.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.6 Test of a Single Variance","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a320fba11.6archera","stepAnswer":["Test of a single variance"],"problemType":"MultipleChoice","stepTitle":"What type of test should be used?","stepBody":"","answerType":"string","variabilization":{},"choices":["Test of a single variance","Test of a population median population deviation","Test of a sample variance","Test of a population proportion"],"hints":{"DefaultPathway":[{"id":"a320fba11.6archera-h1","type":"hint","dependencies":[],"title":"Hypothesis testing","text":"The statistic being tested is the single standard deviation concerned.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a320fba11.6archera-h2","type":"hint","dependencies":["a320fba11.6archera-h1"],"title":"Hypothesis testing","text":"Since the standard deviation is involved, this is a test of a single variance.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a320fba11.6archerb","stepAnswer":["$$H_0$$: q**2=6**2,H_a:q**2<6**2"],"problemType":"MultipleChoice","stepTitle":"State the null and alternative hypotheses.Let q denote population variance.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$H_0$$: q**2=6**2,H_a:q**2<6**2","choices":["$$H_0$$: q**2=6**2,H_a:q**2<6**2","$$H_0$$: q**2=5**2,H_a:q**2<5**2","$$H_0$$: q**2>3**2,H_a:q**2>3**2","$$H_0$$: q**2>=6**2,H_a:q**2<6**2"],"hints":{"DefaultPathway":[{"id":"a320fba11.6archerb-h1","type":"hint","dependencies":[],"title":"Test of a Single Variance","text":"The observer\'s claim is that the standard deviation is less than six hits, which will be the alternate claim.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a320fba11.6archerb-h2","type":"hint","dependencies":["a320fba11.6archerb-h1"],"title":"Test of a Single Variance","text":"What the archer recorded in this case is assumed to be true as the observer is conducting the test to prove the archer wrong.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a320fba11.6archerb-h3","type":"hint","dependencies":["a320fba11.6archerb-h2"],"title":"Test of a Single Variance","text":"Even though we are given the population standard deviation the test using populaton variance can still follow $$H_0$$: q**2=6**2,H_a:q**2<6**2","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a320fba11.6archerc","stepAnswer":["$$left-tailed$$"],"problemType":"MultipleChoice","stepTitle":"Is this a right-tailed, left-tailed, or two-tailed test?","stepBody":"","answerType":"string","variabilization":{},"choices":["$$left-tailed$$","$$right-tailed$$","$$two-tailed$$","none of the listed options work"],"hints":{"DefaultPathway":[{"id":"a320fba11.6archerc-h1","type":"hint","dependencies":[],"title":"Test of a Single Variance","text":"Think of s as the random variable in this test, where the test of a single variance can be right tailed, left tailed, or two tailed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a320fba11.6archerc-h2","type":"hint","dependencies":["a320fba11.6archerc-h1"],"title":"Test of a Single Variance","text":"The word \'less\' in the claim that the archer\'s hits has less variation is the key determining factor to how we choose our test-we will be using a left tailed test.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a320fba11.6doctor1","title":"Test of a Single Variance","body":"The average waiting time in a doctor\u2019s office varies. The standard deviation of waiting times in a doctor\u2019s office is $$3.4$$ minutes. A random sample of $$30$$ patients in the doctor\u2019s office has a standard deviation of waiting times of $$4.1$$ minutes. One doctor believes the variance of waiting times is greater than originally thought.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.6 Test of a Single Variance","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a320fba11.6doctor1a","stepAnswer":["Test of a single variance"],"problemType":"MultipleChoice","stepTitle":"What type of test should be used?","stepBody":"","answerType":"string","variabilization":{},"choices":["Test of a single variance","Test of a population median population deviation","Test of a sample variance","Test of a population proportion"],"hints":{"DefaultPathway":[{"id":"a320fba11.6doctor1a-h1","type":"hint","dependencies":[],"title":"Choosing tests","text":"The statistic being tested is the single standard deviation concerned.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a320fba11.6doctor1a-h2","type":"hint","dependencies":["a320fba11.6doctor1a-h1"],"title":"Choosing tests","text":"To test variability, use the chi-square test of a single variance. The test may be left-, right-, or two-tailed, and its hypotheses are always expressed in terms of the variance (or standard deviation).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a320fba11.6doctor1a-h3","type":"hint","dependencies":["a320fba11.6doctor1a-h2"],"title":"Test of a Single Variance","text":"The doctor is testing if the variances are different than what was originally thought, so this is a test of a single variance test.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a320fba11.6FCC","title":"Test of a Single Variance","body":"The FCC conducts broadband speed tests to measure how much data per second passes between a consumer\u2019s computer and the internet. As of August $$2012$$, the standard deviation of internet speeds across internet service providers (ISPs) was $$12.2$$ percent. Suppose a sample of $$15$$ ISPs is taken, and the standard deviation is $$13.2$$. An analyst claims that the standard deviation of speeds is more than what was reported.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.6 Test of a Single Variance","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a320fba11.6FCCa","stepAnswer":["$$H_0$$: $$q^2=12.2$$, $$H_a$$: $$q^2>12.2$$"],"problemType":"MultipleChoice","stepTitle":"State the null and alternative hypotheses. Let $$q^2$$ denote standard deviation in percentage.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$H_0$$: $$q^2=12.2$$, $$H_a$$: $$q^2>12.2$$","choices":["$$H_0$$: $$q^2=12.2$$, $$H_a$$: $$q^2>12.2$$","$$H_0$$: $$q^2 \\\\geq 12.2$$, $$H_a$$: $$q^2<12.2$$","$$H_0$$: $$q^2>12.2$$, $$H_a$$: $$q^2>12.2$$","$$H_0$$: $$q^2>12.2$$, $$H_a$$: $$q^2=12.2$$"],"hints":{"DefaultPathway":[{"id":"a320fba11.6FCCa-h1","type":"hint","dependencies":[],"title":"Test of a Single Variance","text":"The analyst\'s test result is higher than the claimed standard deviation. The analyst\'s claim is that the standard deviation is more than $$12.2$$ percent, which will be the alternate claim.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a320fba11.6FCCa-h2","type":"hint","dependencies":["a320fba11.6FCCa-h1"],"title":"Test of a Single Variance","text":"Even though we are given the population standard deviation the test using populaton variance can still follow $$H_0$$: q**2=5**2,H_a:q**2>5**2","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a320fba11.6FCCb","stepAnswer":["$$16.389$$"],"problemType":"TextBox","stepTitle":"Compute the test statistic. Round to the third decimal point.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16.389$$","hints":{"DefaultPathway":[{"id":"a320fba11.6FCCb-h1","type":"hint","dependencies":[],"title":"Test statistic","text":"The test statistic formula is $$X^2=\\\\frac{\\\\left(n-1\\\\right) s^2}{q^2}$$, where $$q^2$$ is population variance.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a320fba11.6FCCb-h2","type":"hint","dependencies":["a320fba11.6FCCb-h1"],"title":"Test statistic","text":"Let $$X^2$$ denote the test statistic, s denote sample standard deviation, and q denote population standard deviation. $$X^2=\\\\frac{\\\\left(n-1\\\\right) s^2}{q^2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a320fba11.6FCCb-h3","type":"hint","dependencies":["a320fba11.6FCCb-h2"],"title":"Test statistic","text":"With $$n=15$$, $$s=13.2$$, and $$q=12.2$$, $$X^2$$ is found to be $$16.389$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a320fba11.6FCCc","stepAnswer":["$$right-tailed$$"],"problemType":"MultipleChoice","stepTitle":"What test type should be used?","stepBody":"","answerType":"string","variabilization":{},"choices":["$$left-tailed$$","$$right-tailed$$","$$two-tailed$$","none of the listed options work"],"hints":{"DefaultPathway":[{"id":"a320fba11.6FCCc-h1","type":"hint","dependencies":[],"title":"Test of a Single Variance","text":"The analyst claims that the standard deviation is greater than what was reported.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a320fba11.6FCCc-h2","type":"hint","dependencies":["a320fba11.6FCCc-h1"],"title":"Test of a Single Variance","text":"The word \\"greater\\" in the claim that the broadband speed variation is greater than what was reported is the key determining factor to how we choose our test. Additionally, since the alternative claim has a \\"greater than\\" sign, we will be using a right tailed test.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a320fba11.6FCCd","stepAnswer":["$$14$$"],"problemType":"TextBox","stepTitle":"Compute the degrees of freedom.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$14$$","hints":{"DefaultPathway":[{"id":"a320fba11.6FCCd-h1","type":"hint","dependencies":[],"title":"Degrees of freedom","text":"The formula for degrees of freedom denoted by df is $$df=n-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a320fba11.6FCCe","stepAnswer":["$$0.2902$$"],"problemType":"TextBox","stepTitle":"What is the $$p-value$$? Test at the $$1$$ percent significance level and round to four decimal points.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.2902$$","hints":{"DefaultPathway":[{"id":"a320fba11.6FCCe-h1","type":"hint","dependencies":[],"title":"P-value","text":"We found that the test statistic is equal to $$16.389$$, so we can now find $$p-value$$ $$P\\\\left(X^2>16.389\\\\right)=0.2902$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a320fba11.6FCCe-h2","type":"hint","dependencies":["a320fba11.6FCCe-h1"],"title":"Test of a Single Variance","text":"Since $$p-value$$ $$0.2902>0.05$$, we fail to reject the null hypothesis and reject the alternate hypothesis by the analyst that the FCC reported the broadband speed standard deviation lower than the dataset really represents.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a320fba11.6heights","title":"Test of a Single Variance","body":"The standard deviation of heights for students in a school is $$0.81$$. A random sample of $$50$$ students is taken, and the standard deviation of heights of the sample is $$0.96$$. A researcher in charge of the study believes the standard deviation of heights for the school is greater than $$0.81$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.6 Test of a Single Variance","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a320fba11.6heightsa","stepAnswer":["$$49$$"],"problemType":"TextBox","stepTitle":"What is the degree of freedom value?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$49$$","hints":{"DefaultPathway":[{"id":"a320fba11.6heightsa-h1","type":"hint","dependencies":[],"title":"Degree of freedom","text":"Degrees of freedom, often denoted as $$d f$$, are used to find critical cutoff values especially when performing inferential statistics.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a320fba11.6heightsa-h2","type":"hint","dependencies":["a320fba11.6heightsa-h1"],"title":"Degree of freedom","text":"Recall that degrees of freedom is k-1, where k is the population size.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a320fba11.6mathinstructor","title":"Test of a Single Variance","body":"Math instructors are not only interested in how their students do on exams, on average, but how the exam scores vary. To many instructors, the variance (or standard deviation) may be more important than the average.\\\\n","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.6 Test of a Single Variance","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a320fba11.6mathinstructora","stepAnswer":["$$H_0$$: q**2=5**2,H_a:q**2>5**2"],"problemType":"MultipleChoice","stepTitle":"Suppose a math instructor believes that the standard deviation for his final exam is five points. One of his best students thinks otherwise. The student claims that the standard deviation is more than five points. If the student were to conduct a hypothesis test, what would the null and alternative hypotheses be? Let q denote population variance.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$H_0$$: q**2=5**2,H_a:q**2>5**2","choices":["$$H_0$$: q<5,H_a:q<=5","$$H_0$$: q**2<=25,H_a:q**2>=25","$$H_0$$: q**2=25,H_a:q**2>25","$$H_0$$: q**2=5**2,H_a:q**2>5**2","$$H_0$$: q**5=25,H_a:q**5>25"],"hints":{"DefaultPathway":[{"id":"a320fba11.6mathinstructora-h1","type":"hint","dependencies":[],"title":"Test of a Single Variance","text":"What the math instructor in this case believes is assumed to be true as the student is conducting the test to prove the instructor wrong.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a320fba11.6mathinstructora-h2","type":"hint","dependencies":["a320fba11.6mathinstructora-h1"],"title":"Test of a Single Variance","text":"The student\'s claim is that the standard deviation is more than five points, which will be the alternate claim.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a320fba11.6mathinstructora-h3","type":"hint","dependencies":["a320fba11.6mathinstructora-h2"],"title":"Test of a Single Variance","text":"Think of s as the random variable in this test, where the test of a single variance can be right tailed, left tailed, or two tailed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a320fba11.6mathinstructora-h4","type":"hint","dependencies":["a320fba11.6mathinstructora-h3"],"title":"Test of a Single Variance","text":"Even though we are given the population standard deviation the test using populaton variance can still follow $$H_0$$: q**2=5**2,H_a:q**2>5**2","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a320fba11.6post","title":"With individual lines at its various windows, a post office finds that the standard deviation for normally distributed waiting times for customers on Friday afternoon is $$7.2$$ minutes. The post office experiments with a single, main waiting line and finds that for a random sample of $$25$$ customers, the waiting times for customers have a standard deviation of $$3.5$$ minutes.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.6 Test of a Single Variance","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a320fba11.6posta","stepAnswer":["At a 5% level of significance, from the data, there is sufficient evidence to conclude that a single line causes a lower variation among the waiting times or with a single line, the customer waiting times vary less than $$7.2$$ minutes."],"problemType":"MultipleChoice","stepTitle":"With a significance level of 5%, test the claim that a single line causes lower variation among waiting times (shorter waiting times) for customers.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"At a 5% level of significance, from the data, there is sufficient evidence to conclude that a single line causes a lower variation among the waiting times or with a single line, the customer waiting times vary less than $$7.2$$ minutes.","choices":["At a 5% level of significance, from the data, there is sufficient evidence to conclude that a single line causes a lower variation among the waiting times or with a single line, the customer waiting times vary less than $$7.2$$ minutes.","At a 5% level of significance, from the data, there is insufficient evidence to conclude that a single line causes a lower variation among the waiting times or with a single line, the customer waiting times vary less than $$7.2$$ minutes.\\\\n\\\\n"],"hints":{"DefaultPathway":[{"id":"a320fba11.6posta-h1","type":"hint","dependencies":[],"title":"Test of a Single Variance","text":"Since the claim is that a single line causes less variation, this is a test of a single variance. The parameter is the population variance $$q^2$$ or the population standard deviation q.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a320fba11.6posta-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$H_0$$: $$q^2={7.2}^2$$, $$H_a$$: $$q^2<{7.2}^2$$"],"dependencies":["a320fba11.6posta-h1"],"title":"Which of the following claims follow the situation? 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(Round to $$2$$ decimal places)","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a320fba11.6posta-h12","type":"hint","dependencies":["a320fba11.6posta-h11"],"title":"From $$n=25$$, $$s=3.5$$, and population standard deviation $$q=7.2$$, the test statistic found is $$5.67$$, so we can now find $$p-value$$ $$P\\\\left(X^2<5.67\\\\right)$$.","text":"\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a320fba11.6posta-h13","type":"hint","dependencies":["a320fba11.6posta-h12"],"title":"P-value","text":"The $$p-value$$ is found to be $$0.000042$$, now compare that to our 5% level of significance to make a conclusion.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a320fba11.6posta-h14","type":"hint","dependencies":["a320fba11.6posta-h13"],"title":"Hypothesis Testing","text":"Since our significance level $$0.05$$ is less than our $$p-value$$ $$0.000042$$, we reject that the poulation variance is equal to $${7.2}^2$$. We can interpret this as in that we do not think the variation in waiting times is $$7.2$$ minutes and rather think that the variation in waiting times is less.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a320fba11.6studentheights","title":"Test of a Single Variance","body":"The standard deviation of heights for students in a school is $$0.81$$. A random sample of $$50$$ students is taken, and the standard deviation of heights of the sample is $$0.96$$. 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a330233CompSq7a-h3","type":"hint","dependencies":["a330233CompSq7a-h2"],"title":"Add $${\\\\left(\\\\frac{1}{2} b\\\\right)}^2$$ to $$x^2+bx$$","text":"$$x^2-4x+4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a330233CompSq7a-h4","type":"hint","dependencies":["a330233CompSq7a-h3"],"title":"Write as a perfect squared binomial","text":"$${\\\\left(x-2\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a330233CompSq8","title":"Complete the Square of a Binomial Expression","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Solve Quadratic Equations by 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The solid of revolution generated by revolving R about the y-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes1a-h3","type":"hint","dependencies":["a33da99volumes1a-h2"],"title":"Find the volume","text":"$$V=\\\\int_{1}^{3} \\\\frac{2\\\\pi x\\\\times1}{x} \\\\,dx=\\\\int_{1}^{3} 2\\\\pi \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes1a-h4","type":"hint","dependencies":["a33da99volumes1a-h3"],"title":"Find the integral","text":"$$2\\\\pi x$$ with the limits going from $$x=1$$ to $$x=3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes1a-h5","type":"hint","dependencies":["a33da99volumes1a-h4"],"title":"Evaluate","text":"$$2\\\\pi\\\\times3-2\\\\pi\\\\times1=4\\\\pi$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a33da99volumes10","title":"Finding Volume","body":"For the following exercises, find the volume generated when the region between the two curves is rotated around the given axis. Use both the shell method and the washer method. Use technology to graph the functions and draw a typical slice by hand.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.3 Volumes of Revolution: Cylindrical Shells","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a33da99volumes10a","stepAnswer":["$$54\\\\pi$$"],"problemType":"TextBox","stepTitle":"Bounded by the curves $$y=3x$$, $$y=0$$, and $$x=3$$ rotated around the y-axis.","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$54\\\\pi$$","hints":{"DefaultPathway":[{"id":"a33da99volumes10a-h1","type":"hint","dependencies":[],"title":"Graphing","text":"First, graph the region and the solid of revolution as shown.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes10a-h2","type":"hint","dependencies":["a33da99volumes10a-h1"],"title":"Shell Method","text":"We set up a rectangle that is parallel to the y-axis and runs over the x-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes10a-h3","type":"hint","dependencies":["a33da99volumes10a-h2"],"title":"Set up the integral","text":"$$2*pi*\\\\int_{0}^{3} 3x x \\\\,dx=2*pi*\\\\int_{0}^{3} 3x^2 \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes10a-h4","type":"hint","dependencies":["a33da99volumes10a-h3"],"title":"Find the integral","text":"$$2\\\\pi \\\\frac{3}{3} x^3$$ as the limits going from $$x=0$$ to $$x=3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes10a-h5","type":"hint","dependencies":["a33da99volumes10a-h4"],"title":"Evaluate","text":"Solve $$2\\\\pi 3^3=54\\\\pi$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a33da99volumes11","title":"Finding Volume","body":"For the following exercises, find the volume generated when the region between the two curves is rotated around the given axis. Use both the shell method and the washer method. Use technology to graph the functions and draw a typical slice by hand.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.3 Volumes of Revolution: Cylindrical Shells","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a33da99volumes11a","stepAnswer":["$$81\\\\pi$$"],"problemType":"TextBox","stepTitle":"Bounded by the curves $$y=3x$$, $$y=0$$, and $$x=3$$ rotated around the x-axis.","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$81\\\\pi$$","hints":{"DefaultPathway":[{"id":"a33da99volumes11a-h1","type":"hint","dependencies":[],"title":"Graphing","text":"First, graph the region and the solid of revolution as shown.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes11a-h2","type":"hint","dependencies":["a33da99volumes11a-h1"],"title":"Shell Method","text":"We set up a rectangle that is perpendicular to the x-axis and runs over the x-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes11a-h3","type":"hint","dependencies":["a33da99volumes11a-h2"],"title":"Set up the integral","text":"$$(pi)*\\\\int_{0}^{3} {\\\\left(3x\\\\right)}^2 \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes11a-h4","type":"hint","dependencies":["a33da99volumes11a-h3"],"title":"Find the integral","text":"$$\\\\frac{9\\\\pi x^3}{3}$$ as the limits going from $$x=0$$ to $$x=3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes11a-h5","type":"hint","dependencies":["a33da99volumes11a-h4"],"title":"Evaluate","text":"Solve $$\\\\frac{9\\\\pi 3^3}{3}=81\\\\pi$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a33da99volumes12","title":"A Region of Revolution Revolved around the Line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.3 Volumes of Revolution: Cylindrical Shells","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a33da99volumes12a","stepAnswer":["$$23\\\\pi$$ /3"],"problemType":"TextBox","stepTitle":"Define R as the region bounded above by the graph of $$f(x)=x$$ and below by the x-axis over the interval [1,2]. 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The solid of revolution generated by revolving R about the line $$x=-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes12a-h3","type":"hint","dependencies":["a33da99volumes12a-h2"],"title":"Radius","text":"the radius of the shell is given by $$x+1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes12a-h4","type":"hint","dependencies":["a33da99volumes12a-h3"],"title":"Find the volume","text":"$$V=\\\\int_{1}^{2} 2\\\\pi \\\\left(x+1\\\\right) f{\\\\left(x\\\\right)} \\\\,dx=\\\\int_{1}^{2} 2\\\\pi \\\\left(x+1\\\\right) x \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes12a-h5","type":"hint","dependencies":["a33da99volumes12a-h4"],"title":"Simplify the equation","text":"You can put $$2\\\\pi$$ in front of the integral, which makes it $$2*pi*(\\\\int_{1}^{2} \\\\left(x+1\\\\right) x \\\\,dx)=2*pi*(/int{x**2+x$$, 1,2,x})","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes12a-h6","type":"hint","dependencies":["a33da99volumes12a-h5"],"title":"Find the integral","text":"$$2\\\\pi \\\\left(\\\\frac{x^3}{3}+\\\\frac{x^2}{2}\\\\right)$$ with the limits going from $$x=1$$ to $$x=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes12a-h7","type":"hint","dependencies":["a33da99volumes12a-h6"],"title":"Evaluate","text":"Solve $$2\\\\pi$$ *(2**3/3+2**2/2) $$-\\\\left(2\\\\pi \\\\left(\\\\frac{1^3}{3}+\\\\frac{1^2}{2}\\\\right)\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a33da99volumes13","title":"Find the volume","body":"Find the volume generated when the region between the two curves is rotated around the given axis. Use technology to graph the functions and draw a typical slice by hand.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.3 Volumes of Revolution: Cylindrical Shells","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a33da99volumes13a","stepAnswer":["$$54\\\\pi$$"],"problemType":"TextBox","stepTitle":"Bounded by the curves $$y=3x$$, $$y=0, and$$ $$x=3$$ rotated around the y-axis.","stepBody":"Find the volume","answerType":"arithmetic","variabilization":{},"answerLatex":"$$54\\\\pi$$","hints":{"DefaultPathway":[{"id":"a33da99volumes13a-h1","type":"hint","dependencies":[],"title":"Graphing region","text":"First we must graph the region R, as shown in the following figure.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes13a-h2","type":"hint","dependencies":["a33da99volumes13a-h1"],"title":"Define the region","text":"Looking at graph, we can find the region is under the line of $$y=3x$$ over the interval [0,3] on x-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes13a-h3","type":"hint","dependencies":["a33da99volumes13a-h2"],"title":"Find the volume","text":"$$V=\\\\int_{0}^{3} 2\\\\pi x y \\\\,dx=\\\\int_{0}^{3} 2\\\\pi x\\\\times3 x \\\\,dx=\\\\int_{0}^{3} 6\\\\pi x^2 \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes13a-h4","type":"hint","dependencies":["a33da99volumes13a-h3"],"title":"Find the integral","text":"So we need to solve $$6\\\\pi$$ times $$\\\\frac{x^3}{3}$$ with the limits going from $$x=0$$ to $$x=3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a33da99volumes14","title":"Find the volume","body":"Find the volume generated when the region between the two curves is rotated around the given axis. Use technology to graph the functions and draw a typical slice by hand.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.3 Volumes of Revolution: Cylindrical Shells","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a33da99volumes14a","stepAnswer":["$$81\\\\pi$$"],"problemType":"TextBox","stepTitle":"Bounded by the curves $$y=3x$$, $$y=0$$, and $$x=3$$ rotated around the x-axis.","stepBody":"Find the volume","answerType":"arithmetic","variabilization":{},"answerLatex":"$$81\\\\pi$$","hints":{"DefaultPathway":[{"id":"a33da99volumes14a-h1","type":"hint","dependencies":[],"title":"Graphing region","text":"First we must graph the region R, as shown in the following figure.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a33da99volumes14a-h1"],"title":"Finding range on y-axis","text":"when $$x=0$$, $$y=3\\\\times0=0$$, then what is $$y$$ when $$x=3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes14a-h3","type":"hint","dependencies":["a33da99volumes14a-h2"],"title":"Define the region","text":"Looking at graph, we can find the region is under the line of $$y=3x$$ over the interval [0,9] on y-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes14a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{y}{3}$$"],"dependencies":["a33da99volumes14a-h3"],"title":"Find g(y)","text":"$$y=3x$$ so $$x=\\\\frac{y}{3}$$. so what is g(y)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes14a-h5","type":"hint","dependencies":["a33da99volumes14a-h4"],"title":"Height","text":"The height is the horizontal distance from $$y=0$$ to $$x=3$$, minus the distance from $$y=0$$ to $$x=\\\\frac{y}{3}$$, which is $$3-\\\\frac{y}{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes14a-h6","type":"hint","dependencies":["a33da99volumes14a-h5"],"title":"Find the volume","text":"$$V=\\\\int_{0}^{9} 2\\\\pi y \\\\left(3-\\\\frac{y}{3}\\\\right) \\\\,dy=\\\\int_{0}^{9} 2\\\\pi \\\\left(3y-\\\\frac{y^2}{3}\\\\right) \\\\,dy$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes14a-h7","type":"hint","dependencies":["a33da99volumes14a-h6"],"title":"Find the integral","text":"So we need to solve $$2\\\\pi$$ times (3*y**2)/2- $$\\\\frac{y^3}{9}$$ with the limits going from $$y=0$$ to $$y=9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a33da99volumes15","title":"Find the volume","body":"Use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the x-axis and are rotated around the y-axis.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.3 Volumes of Revolution: Cylindrical Shells","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a33da99volumes15a","stepAnswer":["$$2\\\\pi$$"],"problemType":"TextBox","stepTitle":"$$y=5x^3$$, $$x=0$$, and $$x=1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2\\\\pi$$","hints":{"DefaultPathway":[{"id":"a33da99volumes15a-h1","type":"hint","dependencies":[],"title":"Define the region","text":"According to the problem note, the region is under the line of $$y=5x^3$$ over the interval [0,1] on x-axis","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes15a-h2","type":"hint","dependencies":["a33da99volumes15a-h1"],"title":"Find the volume","text":"Using the shell, $$V=/int{2*pi*x*$$ (5*(x**3)),0,1,x}=2*pi*(/int{5*(x**4),0,1,x})","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes15a-h3","type":"hint","dependencies":["a33da99volumes15a-h2"],"title":"Find the integral","text":"So we need to solve $$2\\\\pi$$ times $$\\\\frac{5x^5}{5}$$ with limits from $$0$$ to $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a33da99volumes2","title":"The Method of Cylindrical Shells $$2$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.3 Volumes of Revolution: Cylindrical Shells","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a33da99volumes2a","stepAnswer":["$$\\\\frac{8\\\\pi}{3}$$"],"problemType":"TextBox","stepTitle":"Define R as the region above by the graph of $$f(x)=2x-x^2$$ and below by the x-axis over the interval [0,2]. Find the volume of the solid of revolution formed by revolving R around the y-axis.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{8\\\\pi}{3}$$","hints":{"DefaultPathway":[{"id":"a33da99volumes2a-h1","type":"hint","dependencies":[],"title":"Graphing","text":"First we must graph the region R and the associated solid of revolution, as shown in the following figure.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes2a-h2","type":"hint","dependencies":["a33da99volumes2a-h1"],"title":"Define the region","text":"The region R under the graph of $$f(x)=2x-x^2$$ over the interval [0,2]. The solid of revolution generated by revolving R about the y-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes2a-h3","type":"hint","dependencies":["a33da99volumes2a-h2"],"title":"Find the volume","text":"$$V=(2*pi)*\\\\int_{0}^{2} x \\\\left(2x-x^2\\\\right) \\\\,dx=(2*pi)*\\\\int_{0}^{2} 2x^2-x^3 \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes2a-h4","type":"hint","dependencies":["a33da99volumes2a-h3"],"title":"Find the integral","text":"$$2\\\\pi \\\\left(\\\\frac{2x^3}{3}-\\\\frac{x^4}{4}\\\\right)$$ with the limits going from $$x=0$$ to $$x=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes2a-h5","type":"hint","dependencies":["a33da99volumes2a-h4"],"title":"Evaluate","text":"$$2\\\\pi \\\\left(\\\\frac{2\\\\times2^3}{3}-\\\\frac{2^4}{4}\\\\right)-0=\\\\frac{8\\\\pi}{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a33da99volumes3","title":"The Method of Cylindrical Shells for a Solid Revolved around the x-axis","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.3 Volumes of Revolution: Cylindrical Shells","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a33da99volumes3a","stepAnswer":["$$\\\\frac{256\\\\pi}{5}$$"],"problemType":"TextBox","stepTitle":"Define Q as the region bounded on the right by the graph of $$g(y)=2\\\\sqrt{y}$$ and on the left by the y-axis or y\u2208[0,4]. Find the volume of the solid of revolution formed by revolving Q around the x-axis.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{256\\\\pi}{5}$$","hints":{"DefaultPathway":[{"id":"a33da99volumes3a-h1","type":"hint","dependencies":[],"title":"Graphing","text":"First we need to graph the region Q and the associated solid of revolution, as shown in the following figure.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes3a-h2","type":"hint","dependencies":["a33da99volumes3a-h1"],"title":"Define the region","text":"The region Q to the left of the function g(y) over the interval [0,4]. The solid of revolution generated by revolving Q around the x-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes3a-h3","type":"hint","dependencies":["a33da99volumes3a-h2"],"title":"Find the volume","text":"$$V=2*pi*\\\\int_{0}^{4} 2y \\\\sqrt{y} \\\\,dy=4*pi*\\\\int_{0}^{4} y^{\\\\frac{3}{2}} \\\\,dy$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes3a-h4","type":"hint","dependencies":["a33da99volumes3a-h3"],"title":"Find the integral","text":"$$4\\\\pi \\\\frac{2y^{\\\\frac{5}{2}}}{5}$$ as the limits going from $$x=0$$ to $$x=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes3a-h5","type":"hint","dependencies":["a33da99volumes3a-h4"],"title":"Evaluate","text":"$$4\\\\pi \\\\frac{2\\\\times4^{\\\\frac{5}{2}}}{5}-0=\\\\frac{256\\\\pi}{5}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a33da99volumes4","title":"A Region of Revolution Bounded by the Graphs of Two Functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.3 Volumes of Revolution: Cylindrical Shells","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a33da99volumes4a","stepAnswer":["$$\\\\frac{94\\\\pi}{5}$$"],"problemType":"TextBox","stepTitle":"Define R as the region bounded above by the graph of the function $$f(x)=\\\\sqrt{x}$$ and below by the graph of the function. $$g(x)=\\\\frac{1}{x}$$ over the interval [1,4]. Find the volume of the solid of revolution generated by revolving R around the y-axis.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{94\\\\pi}{5}$$","hints":{"DefaultPathway":[{"id":"a33da99volumes4a-h1","type":"hint","dependencies":[],"title":"Graphing","text":"First we must graph the region R and the associated solid of revolution, as shown in the following figure.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes4a-h2","type":"hint","dependencies":["a33da99volumes4a-h1"],"title":"Define the region","text":"The region R between the graph of f(x) and the graph of g(x) over the interval [1,4]. The solid of revolution generated by revolving R around the y-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes4a-h3","type":"hint","dependencies":["a33da99volumes4a-h2"],"title":"Find the height of a shell","text":"Note that the axis of revolution is the y-axis, so the radius of a shell is given simply by $$x$$. We don\u2019t need to make any adjustments to the $$x-term$$ of our integrand. The height of a shell, though, is given by $$f(x)-g(x)$$, so in this case we need to adjust the f(x) term of the integrand.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes4a-h4","type":"hint","dependencies":["a33da99volumes4a-h3"],"title":"Find the volume","text":"$$V=2*pi*\\\\int_{1}^{4} x \\\\left(\\\\sqrt{x}-\\\\frac{1}{x}\\\\right) \\\\,dx=2*pi*\\\\int_{1}^{4} x^{\\\\frac{3}{2}}-1 \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes4a-h5","type":"hint","dependencies":["a33da99volumes4a-h4"],"title":"Find the integral","text":"$$2\\\\pi \\\\frac{2x^{\\\\frac{5}{2}}}{5}-x$$ with the limits going from $$x=1$$ to $$x=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes4a-h6","type":"hint","dependencies":["a33da99volumes4a-h5"],"title":"Evaluate","text":"2*pi*(((2*4**(5/2))/5)-4-(2*pi*(((2*1**(5/2))/5)-1)=94*pi/5","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a33da99volumes5","title":"Selecting the Best Method","body":"For each of the following problems, select the best method to find the volume of a solid of revolution generated by revolving the given region around the x-axis, and set up the integral to find the volume (do not evaluate the integral).","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.3 Volumes of Revolution: Cylindrical Shells","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a33da99volumes5a","stepAnswer":["$$\\\\int_{0}^{1} \\\\pi x^2 \\\\,dx+\\\\int_{1}^{2} \\\\pi {\\\\left(2-x\\\\right)}^2 \\\\,dx$$"],"problemType":"MultipleChoice","stepTitle":"The region bounded by the graph of $$y=x, y=2-x$$, and the x-axis. Give an answer using Disk Method.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\int_{0}^{1} \\\\pi x^2 \\\\,dx+\\\\int_{1}^{2} \\\\pi {\\\\left(2-x\\\\right)}^2 \\\\,dx$$","choices":["$$\\\\int_{0}^{1} \\\\pi x^2 \\\\,dx+\\\\int_{1}^{2} \\\\pi {\\\\left(2-x\\\\right)}^2 \\\\,dx$$","$$\\\\int_{0}^{1} \\\\pi {\\\\left(2-x\\\\right)}^2 \\\\,dx+\\\\int_{1}^{2} \\\\pi x^2 \\\\,dx$$"],"hints":{"DefaultPathway":[{"id":"a33da99volumes5a-h1","type":"hint","dependencies":[],"title":"Sketching","text":"First, sketch the region and the solid of revolution as shown.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes5a-h2","type":"hint","dependencies":["a33da99volumes5a-h1"],"title":"Seperate the region","text":"If we want to integrate with respect to $$x$$, we would have to break the integral into two pieces, because we have different functions bounding the region over [0,1] and [1,2]. In this case, using the disk method,","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes5a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y=x$$"],"dependencies":["a33da99volumes5a-h2"],"title":"Seperate the region","text":"What is the function bounding over [0,1]?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$y=x$$","$$y=2-x$$"]},{"id":"a33da99volumes5a-h4","type":"hint","dependencies":["a33da99volumes5a-h3"],"title":"Disk Method","text":"$$V=\\\\int_{0}^{1} \\\\pi x^2 \\\\,dx+\\\\int_{1}^{2} \\\\pi {\\\\left(2-x\\\\right)}^2 \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes5a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2-2y$$"],"dependencies":["a33da99volumes5a-h4"],"title":"Shell Method","text":"What would the function be if we used the shell method instead?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$2-2x$$","$$2-2y$$","$$y-2+y$$"],"subHints":[{"id":"a33da99volumes5a-h5-s1","type":"hint","dependencies":[],"title":"Shell Method","text":"In order to use Shell Method in this case, we need to use function of $$y$$ to represent the curves. Since we have $$x=y$$ and $$y=2-x$$ then $$x=2-y$$, we will replace $$x$$ in the given functions to get a new function in term of $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"a33da99volumes5a-h6","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["[0,1]"],"dependencies":["a33da99volumes5a-h5"],"title":"Shell Method","text":"What would the limits of the function revolving around the y-axis? Express in Interval Expression","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes5a-h7","type":"hint","dependencies":["a33da99volumes5a-h6"],"title":"Shell Method","text":"The integral of the Shell Method can be set up as $$V=\\\\int_{0}^{1} 2\\\\pi y \\\\left(2-y-y\\\\right) \\\\,dy=\\\\int_{0}^{1} 2\\\\pi y \\\\left(2-2y\\\\right) \\\\,dy$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a33da99volumes6","title":"Selecting the Best Method","body":"For each of the following problems, select the best method to find the volume of a solid of revolution generated by revolving the given region around the x-axis, and set up the integral to find the volume (do not evaluate the integral).","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.3 Volumes of Revolution: Cylindrical Shells","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a33da99volumes6a","stepAnswer":["$$\\\\int_{0}^{4} \\\\pi {\\\\left(4x-x^2\\\\right)}^2 \\\\,dx$$"],"problemType":"MultipleChoice","stepTitle":"The region bounded by the graph of $$y=4x-x^2$$ and the x-axis.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\int_{0}^{4} \\\\pi {\\\\left(4x-x^2\\\\right)}^2 \\\\,dx$$","choices":["$$\\\\int_{0}^{4} \\\\pi {\\\\left(4x-x^2\\\\right)}^2 \\\\,dx$$","$$\\\\int_{0}^{4} \\\\pi 4x-x^2 \\\\,dx$$"],"hints":{"DefaultPathway":[{"id":"a33da99volumes6a-h1","type":"hint","dependencies":[],"title":"Sketching","text":"First, sketch the region and the solid of revolution as shown.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes6a-h2","type":"hint","dependencies":["a33da99volumes6a-h1"],"title":"Disk Method","text":"$$V=\\\\int_{0}^{4} \\\\pi {\\\\left(4x-x^2\\\\right)}^2 \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes6a-h3","type":"hint","dependencies":["a33da99volumes6a-h2"],"title":"Shell Method","text":"Looking at the region, it would be problematic to define a horizontal rectangle; the region is bounded on the left and right by the same function. Therefore, we can dismiss the method of shells. The solid has no cavity in the middle, so we can use the method of disks.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a33da99volumes7","title":"For the following exercises, find the volume generated when the region between the two curves is rotated around the given axis. Use both the shell method and the washer method. Use technology to graph the functions and draw a typical slice by hand.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.3 Volumes of Revolution: Cylindrical Shells","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a33da99volumes7a","stepAnswer":["$$\\\\frac{512\\\\pi}{7}$$"],"problemType":"TextBox","stepTitle":"Bounded by the curves $$y=2x^3$$, $$y=0$$, and $$x=2$$ rotated around the x-axis.","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{512\\\\pi}{7}$$","hints":{"DefaultPathway":[{"id":"a33da99volumes7a-h1","type":"hint","dependencies":[],"title":"Graphing","text":"First, graph the region and the solid of revolution as shown.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes7a-h2","type":"hint","dependencies":["a33da99volumes7a-h1"],"title":"Washer Method","text":"In order to use the Washer Method for a function rotating over x-axis, we need to define the limits of it.The upper limit is given as $$x=2$$. Now we have to define the lower limit.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a33da99volumes7a-h2"],"title":"Define the limits","text":"What is the lower limit?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"a33da99volumes7a-h3-s1","type":"hint","dependencies":[],"title":"Define the limits","text":"Set $$y=0=2x^3$$ and solve for $$x$$. Then $$x=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"a33da99volumes7a-h4","type":"hint","dependencies":["a33da99volumes7a-h3"],"title":"Set up the integral","text":"$$pi*\\\\int_{0}^{2} {\\\\left(2x^3\\\\right)}^2 \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a33da99volumes8","title":"Graphing region","body":"For the following exercises, use technology to graph the region. Determine which method you think would be easiest to use to calculate the volume generated when the function is rotated around the specified axis. Then, use your chosen method to find the volume.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.3 Volumes of Revolution: Cylindrical Shells","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a33da99volumes8a","stepAnswer":["$$\\\\frac{419\\\\pi}{15}$$"],"problemType":"TextBox","stepTitle":"$$y=x^2-2x$$, $$x=2$$ and $$x=4$$ rotated around the x-axis.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{419\\\\pi}{15}$$","hints":{"DefaultPathway":[{"id":"a33da99volumes8a-h1","type":"hint","dependencies":[],"title":"Graphing","text":"First, graph the region and the solid of revolution as shown.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes8a-h2","type":"hint","dependencies":["a33da99volumes8a-h1"],"title":"Washer Method","text":"Set up the integral: $$pi*\\\\int_{2}^{4} x^2-2x \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a33da99volumes9","title":"Graphing Region","body":"For the following exercises, use technology to graph the region. Determine which method you think would be easiest to use to calculate the volume generated when the function is rotated around the specified axis. Then, use your chosen method to find the volume.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.3 Volumes of Revolution: Cylindrical Shells","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a33da99volumes9a","stepAnswer":["$$15.9074$$"],"problemType":"TextBox","stepTitle":"$$x=y^2$$, $$x=y^2-2y+1$$ and $$x=2$$ rotated around the y-axis.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$15.9074$$","hints":{"DefaultPathway":[{"id":"a33da99volumes9a-h1","type":"hint","dependencies":[],"title":"Graphing","text":"First, graph the region and the solid of revolution as shown.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes9a-h2","type":"hint","dependencies":["a33da99volumes9a-h1"],"title":"Choose a method","text":"Note that either Wash and Shell method works in this case. Since it will be more convenient to set up a rectangle that is parallel to the y-axis and run it over the x-axis, we proceed to use Shell Method to calculate the volume.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes9a-h3","type":"hint","dependencies":["a33da99volumes9a-h2"],"title":"Shell Method","text":"The general formula of cylindrical shells is given as $$2*pi*\\\\int_{a}^{b} x f{\\\\left(x\\\\right)} \\\\,dx$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4}$$"],"dependencies":["a33da99volumes9a-h3"],"title":"Define a lower limit","text":"What is a lower limit of the integral?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"a33da99volumes9a-h4-s1","type":"hint","dependencies":[],"title":"A point of intersection","text":"The lower limit is also the point where two graphs intersect each other. To find that point, we set $$y^2=y^2-2y+1$$ and solve for $$x$$. Then $$x=\\\\frac{1}{4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"a33da99volumes9a-h5","type":"hint","dependencies":["a33da99volumes9a-h4"],"title":"Define an upper limit","text":"The upper limit is already given as $$x=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes9a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2\\\\sqrt{x}-1$$"],"dependencies":["a33da99volumes9a-h5"],"title":"Define f(x)","text":"What is the f(x) of the integral?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"a33da99volumes9a-h6-s1","type":"hint","dependencies":[],"title":"Define f(x)","text":"Since we are working on the change over the x-axis or dx, we will need to rewrite both functions in term of $$x$$ instead of $$y$$ as given.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes9a-h6-s2","type":"hint","dependencies":["a33da99volumes9a-h6-s1"],"title":"$$x=y^2$$","text":"To solve for $$y$$, we take the square root of both sides to obtain $$y=\\\\sqrt{x}$$ and $$y=-\\\\sqrt{x}$$. Although we have $$2$$ solutions, we only consider the positive solution since it is the upper function of the shaded region.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes9a-h6-s3","type":"hint","dependencies":["a33da99volumes9a-h6-s2"],"title":"$$x=y^2-2y+1$$","text":"Recall that $$y^2-2y+1={\\\\left(y-1\\\\right)}^2$$ then $$y-1=-\\\\sqrt{x}$$. After solving for $$y$$, we have $$y=1-\\\\sqrt{x}$$. Again in this case we chose $$-\\\\sqrt{x}$$ as an answer to use because it is the lower function of the shaded region.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"a33da99volumes9a-h7","type":"hint","dependencies":["a33da99volumes9a-h6"],"title":"Find f(x)","text":"$$\\\\sqrt{x}-1-\\\\sqrt{x}=2\\\\sqrt{x}-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes9a-h8","type":"hint","dependencies":["a33da99volumes9a-h7"],"title":"Set up the integral","text":"$$2*pi*\\\\int_{\\\\frac{1}{4}}^{2} x \\\\left(2\\\\sqrt{x}-1\\\\right) \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes9a-h9","type":"hint","dependencies":["a33da99volumes9a-h8"],"title":"Find the integral","text":"$$2\\\\pi \\\\left(\\\\frac{4}{5} \\\\frac{x^5}{2}-\\\\frac{x^2}{2}\\\\right)$$ as the limits going from $$x=\\\\frac{1}{4}$$ to $$x=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a33da99volumes9a-h10","type":"hint","dependencies":["a33da99volumes9a-h9"],"title":"Evaluate","text":"$$15.9074$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a342e92limit1","title":"The Limit of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1<OpenStax: Calculus Volume 1>","license":0,"lesson":"2.2 The Limit of a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a342e92limit1a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"Evaluate the following using a table of functional values. If the limit does not exist, write \'DNE\'.","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a342e92limit1a-h1","type":"hint","dependencies":[],"title":"Functional Values Table","text":"The values of $$f(x)=\\\\fracsin^x\\\\left(x\\\\right)}$$ and $$x$$ are listed in Table $$2.2$$. Note: the values in this table were obtained using a calculator and using all the places given in the calculator output.\\\\n##figure2.gif##","variabilization":{},"oer":"","license":""},{"id":"a342e92limit1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a342e92limit1a-h1"],"title":"The Approaching Limit","text":"As we read down each (sin x)/x column, what number are the values in each column approaching?","variabilization":{},"oer":"","license":"","subHints":[{"id":"a342e92limit1a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Graphing the Function","text":"The values of (sin x)/x are as follows: $$0.998334166468$$, $$0.999983333417$$, $$0.999999833333$$, $$0.999999998333$$, $$0.999999998333$$. A calculator or computer-generated graph of f(x) $$=$$ $$\\\\frac{sinx}{x}$$ can also be used to determine the value we are approaching. Around what y-value does the graph intersect?","variabilization":{},"oer":"","license":""}]},{"id":"a342e92limit1a-h3","type":"hint","dependencies":["a342e92limit1a-h2"],"title":"Finding $$\\\\lim_{x\\\\to0} \\\\fracsin^x\\\\left(x\\\\right)}$$","text":"The values are approaching $$1$$. Thus, it is fairly reasonable to conclude that $$\\\\lim_{x\\\\to0} \\\\frac{sinx}{x}=1$$. A calculator or computer-generated graph of f(x) $$=$$ (sin x)/x would be similar to the figure shown, and it confirms our estimate.\\\\n##figure3.gif##","variabilization":{},"oer":"","license":""}]}}]},{"id":"a342e92limit10","title":"The Limit of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1<OpenStax: Calculus Volume 1>","license":0,"lesson":"2.2 The Limit of a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a342e92limit10a","stepAnswer":["$$+\\\\infty$$"],"problemType":"MultipleChoice","stepTitle":"Evaluate /lim{x,-3**-,1/(x+3)**4} using the theorem of infinite limits from positive integers.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$+\\\\infty$$","choices":["$$-\\\\infty$$","$$+\\\\infty$$"],"hints":{"DefaultPathway":[{"id":"a342e92limit10a-h1","type":"hint","dependencies":[],"title":"Infinite Limits from Positive Integers","text":"Use the table to guide your answer.\\\\n##figure1.gif##","variabilization":{},"oer":"","license":""},{"id":"a342e92limit10a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Even"],"dependencies":["a342e92limit10a-h1"],"title":"Positive Even or Odd Integer","text":"Is $$n$$ an even or odd integer?","variabilization":{},"oer":"","license":"","choices":["Even","Odd"],"subHints":[{"id":"a342e92limit10a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":[],"title":"Determining $$n$$","text":"Using the function $$\\\\lim_{x\\\\toa} \\\\frac{1}{{\\\\left(x-a\\\\right)}^n}$$, what is $$n$$ in /lim{x,-3**-,1/(x+3)**4}?","variabilization":{},"oer":"","license":""}]},{"id":"a342e92limit10a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$+\\\\infty$$"],"dependencies":["a342e92limit10a-h2"],"title":"Finding /lim{x,-3**-,1/(x+3)**4}","text":"Given that $$n$$ is a positive even integer, what is the limit?","variabilization":{},"oer":"","license":"","choices":["$$-\\\\infty$$","$$+\\\\infty$$","DNE"]},{"id":"a342e92limit10a-h4","type":"hint","dependencies":["a342e92limit10a-h3"],"title":"Explanation","text":"From the function $$\\\\lim_{x\\\\toa} \\\\frac{1}{{\\\\left(x-a\\\\right)}^n}$$, we know that $$n$$ $$=$$ $$4$$ in /lim{x,-3**-,1/(x+3)**4}. According to the Infinite Limits from Positive Integers Theorem, if $$n$$ is a positive even integer, then $$\\\\lim_{x\\\\toa} \\\\frac{1}{{\\\\left(x-a\\\\right)}^n}=+\\\\infty$$. Since $$4$$ is a positive even integer, this means that /lim{x,-3**-,1/(x+3)**4}=+inf.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a342e92limit11","title":"The Limit of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1<OpenStax: Calculus Volume 1>","license":0,"lesson":"2.2 The Limit of a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a342e92limit11a","stepAnswer":["$$+\\\\infty$$"],"problemType":"MultipleChoice","stepTitle":"Evaluate /lim{x,-3**+,1/(x+3)**4} using the theorem of infinite limits from positive integers.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$+\\\\infty$$","choices":["$$-\\\\infty$$","$$+\\\\infty$$","DNE"],"hints":{"DefaultPathway":[{"id":"a342e92limit11a-h1","type":"hint","dependencies":[],"title":"Infinite Limits from Positive Integers","text":"Use the table to guide your answer.\\\\n##figure1.gif##","variabilization":{},"oer":"","license":""},{"id":"a342e92limit11a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Even"],"dependencies":["a342e92limit11a-h1"],"title":"Positive Even or Odd Integer","text":"Is $$n$$ an even or odd integer?","variabilization":{},"oer":"","license":"","choices":["Even","Odd"],"subHints":[{"id":"a342e92limit11a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":[],"title":"Determining $$n$$","text":"Using the function $$\\\\lim_{x\\\\toa} \\\\frac{1}{{\\\\left(x-a\\\\right)}^n}$$, what is $$n$$ in /lim{x,-3**+,1/(x+3)**4}?","variabilization":{},"oer":"","license":""}]},{"id":"a342e92limit11a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$+\\\\infty$$"],"dependencies":["a342e92limit11a-h2"],"title":"Finding /lim{x,-3**+,1/(x+3)**4}","text":"Given that $$n$$ is a positive even integer, what is the limit?","variabilization":{},"oer":"","license":"","choices":["$$-\\\\infty$$","$$+\\\\infty$$","DNE"]},{"id":"a342e92limit11a-h4","type":"hint","dependencies":["a342e92limit11a-h3"],"title":"Explanation","text":"From the function $$\\\\lim_{x\\\\toa} \\\\frac{1}{{\\\\left(x-a\\\\right)}^n}$$, we know that $$n$$ $$=$$ $$4$$ in /lim{x,-3**+,1/(x+3)**4}. According to the Infinite Limits from Positive Integers Theorem, if $$n$$ is a positive even integer, then $$\\\\lim_{x\\\\toa} \\\\frac{1}{{\\\\left(x-a\\\\right)}^n}=+\\\\infty$$. Since $$4$$ is a positive even integer, this means that /lim{x,-3**+,1/(x+3)**4}=+inf.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a342e92limit12","title":"The Limit of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1<OpenStax: Calculus Volume 1>","license":0,"lesson":"2.2 The Limit of a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a342e92limit12a","stepAnswer":["$$+\\\\infty$$"],"problemType":"MultipleChoice","stepTitle":"Evaluate $$\\\\lim_{x\\\\to-3} \\\\frac{1}{{\\\\left(x+3\\\\right)}^4}$$ using the theorem of infinite limits from positive integers.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$+\\\\infty$$","choices":["$$-\\\\infty$$","$$+\\\\infty$$","DNE"],"hints":{"DefaultPathway":[{"id":"a342e92limit12a-h1","type":"hint","dependencies":[],"title":"Infinite Limits from Positive Integers","text":"Use the table to guide your answer.\\\\n##figure1.gif##","variabilization":{},"oer":"","license":""},{"id":"a342e92limit12a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Even"],"dependencies":["a342e92limit12a-h1"],"title":"Positive Even or Odd Integer","text":"Is $$n$$ an even or odd integer?","variabilization":{},"oer":"","license":"","choices":["Even","Odd"],"subHints":[{"id":"a342e92limit12a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":[],"title":"Determining $$n$$","text":"Using the function $$\\\\lim_{x\\\\toa} \\\\frac{1}{{\\\\left(x-a\\\\right)}^n}$$, what is $$n$$ in $$\\\\lim_{x\\\\to-3} \\\\frac{1}{{\\\\left(x+3\\\\right)}^4}$$?","variabilization":{},"oer":"","license":""}]},{"id":"a342e92limit12a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$+\\\\infty$$"],"dependencies":["a342e92limit12a-h2"],"title":"Finding $$\\\\lim_{x\\\\to-3} \\\\frac{1}{{\\\\left(x+3\\\\right)}^4}$$","text":"Given that $$n$$ is a positive even integer, what is the limit?","variabilization":{},"oer":"","license":"","choices":["$$-\\\\infty$$","$$+\\\\infty$$","DNE"]},{"id":"a342e92limit12a-h4","type":"hint","dependencies":["a342e92limit12a-h3"],"title":"Explanation","text":"From the function $$\\\\lim_{x\\\\toa} \\\\frac{1}{{\\\\left(x-a\\\\right)}^n}$$, we know that $$n$$ $$=$$ $$4$$ in $$\\\\lim_{x\\\\to-3} \\\\frac{1}{{\\\\left(x+3\\\\right)}^4}$$. According to the Infinite Limits from Positive Integers Theorem, if $$n$$ is a positive even integer, then $$\\\\lim_{x\\\\toa} \\\\frac{1}{{\\\\left(x-a\\\\right)}^n}=+\\\\infty$$. Since $$4$$ is a positive even integer, this means that $$\\\\lim_{x\\\\to-3} \\\\frac{1}{{\\\\left(x+3\\\\right)}^4}=+\\\\infty$$.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a342e92limit13","title":"The Limit of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1<OpenStax: Calculus Volume 1>","license":0,"lesson":"2.2 The Limit of a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a342e92limit13a","stepAnswer":["$$-3$$"],"problemType":"TextBox","stepTitle":"Identify any vertical asymptotes of the function $$f(x)=\\\\frac{1}{{\\\\left(x+3\\\\right)}^4}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-3$$","hints":{"DefaultPathway":[{"id":"a342e92limit13a-h1","type":"hint","dependencies":[],"title":"How to Find Vertical Asymptotes","text":"Set the denominator equal to zero and solve for $$x$$.","variabilization":{},"oer":"","license":""},{"id":"a342e92limit13a-h2","type":"hint","dependencies":["a342e92limit13a-h1"],"title":"Identifying the Denominator","text":"The denominator of $$f(x)=\\\\frac{1}{{\\\\left(x+3\\\\right)}^4}$$ is $${\\\\left(x+3\\\\right)}^4$$. Solve for $${\\\\left(x+3\\\\right)}^4=0$$.","variabilization":{},"oer":"","license":""},{"id":"a342e92limit13a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a342e92limit13a-h2"],"title":"Solving for $$x$$","text":"What is $${\\\\left(x+3\\\\right)}^4=0$$?","variabilization":{},"oer":"","license":""},{"id":"a342e92limit13a-h4","type":"hint","dependencies":["a342e92limit13a-h3"],"title":"Taking the Square Root","text":"Take the square root of $$4$$ on both sides of the equation $${\\\\left(x+3\\\\right)}^4=0$$.","variabilization":{},"oer":"","license":""},{"id":"a342e92limit13a-h5","type":"hint","dependencies":["a342e92limit13a-h4"],"title":"Substracting from Both Sides","text":"Now, the simplified equation is $$x+3=0$$. To solve for $$x$$, substract $$3$$ from both sides of the equation.","variabilization":{},"oer":"","license":""},{"id":"a342e92limit13a-h6","type":"hint","dependencies":["a342e92limit13a-h5"],"title":"The Vertical Asymptote","text":"The function $$f(x)=\\\\frac{1}{{\\\\left(x+3\\\\right)}^4}$$ has a vertical asymptote of $$x=-3$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"a342e92limit14","title":"Behavior of a Function at Different Points","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1<OpenStax: Calculus Volume 1>","license":0,"lesson":"2.2 The Limit of a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a342e92limit14a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"Use the graph of f(x) to determine the value of /lim{x,-4**-,f(x)}; /lim{x,-4**+,f(x)}; $$\\\\lim_{x\\\\to-4} f(x);$$ $$f(-4)$$.","stepBody":"/lim{x,-4**-,f(x)} is the value of f(x) as $$x$$ approaches $$-4$$ from the left.\\\\n/lim{x,-4**-+,f(x)} is the value of f(x) as $$x$$ approaches $$-4$$ from the right.\\\\n$$\\\\lim_{x\\\\to-4} f(x)$$ is the value of f(x) as $$x$$ approaches $$-4$$ from both sides.\\\\n$$f(-4)$$ is the value of f(x) when $$x=-4$$.\\\\nLocate $$x=-4$$ on the graph and observe the values of f(x).##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a342e92limit14a-h1","type":"hint","dependencies":[],"title":"Understanding the Functions and Graph","text":"\\\\n##figure2.gif##","variabilization":{},"oer":"","license":""},{"id":"a342e92limit14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a342e92limit14a-h1"],"title":"Finding the Limit","text":"What value is f(x) approaching as $$x$$ approaches -4?","variabilization":{},"oer":"","license":""},{"id":"a342e92limit14a-h3","type":"hint","dependencies":["a342e92limit14a-h2"],"title":"Limit Value","text":"/lim{x,-4**-,f(x)}=0; /lim{x,-4**+,f(x)}=0; $$\\\\lim_{x\\\\to-4} f(x)=0;$$ $$f(-4)=0$$\\\\n##figure3.gif##","variabilization":{},"oer":"","license":""}]}}]},{"id":"a342e92limit15","title":"Behavior of a Function at Different Points","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1<OpenStax: Calculus Volume 1>","license":0,"lesson":"2.2 The Limit of a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a342e92limit15a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"Use the graph of f(x) to determine the value of /lim{x,-2**-,f(x)}; /lim{x,-2**+,f(x)}; $$\\\\lim_{x\\\\to-2} f(x)$$.","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a342e92limit15a-h1","type":"hint","dependencies":[],"title":"Understanding the Functions and Graph","text":"/lim{x,-2**-,f(x)} is the value of f(x) as $$x$$ approaches $$-2$$ from the left.\\\\n/lim{x,-2**+,f(x)} is the value of f(x) as $$x$$ approaches $$-2$$ from the right.\\\\n$$\\\\lim_{x\\\\to-2} f(x)$$ is the value of f(x) as $$x$$ approaches $$-2$$ from both sides.\\\\nLocate $$x=-2$$ on the graph and observe the values of f(x) as it approaches that x-value.\\\\n##figure2.gif##","variabilization":{},"oer":"","license":""},{"id":"a342e92limit15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a342e92limit15a-h1"],"title":"Finding the Limit","text":"What value is f(x) approaching as $$x$$ approaches -2?","variabilization":{},"oer":"","license":""},{"id":"a342e92limit15a-h3","type":"hint","dependencies":["a342e92limit15a-h2"],"title":"Limit Value","text":"/lim{x,1**+,f(x)}=2. Looking at the right side of the function as $$x$$ approaches $$1$$, we can see that the value of f(x) approaches $$3$$. Remember that there is no difference with an open or closed circle because a limit is the value the graph is approaching from the right and left sides. In this case, we are looking solely at the right side of the function.\\\\n##figure3.gif##","variabilization":{},"oer":"","license":""}]}}]},{"id":"a342e92limit16","title":"Behavior of a Function at Different Points","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1<OpenStax: Calculus Volume 1>","license":0,"lesson":"2.2 The Limit of a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a342e92limit16a","stepAnswer":["undefined"],"problemType":"MultipleChoice","stepTitle":"Use the graph of f(x) to determine the value of $$f(-2)$$. If the value is undefined, write \\"undefined\\".","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["defined","undefined"],"hints":{"DefaultPathway":[{"id":"a342e92limit16a-h1","type":"hint","dependencies":[],"title":"Understanding the Function and Graph","text":"$$f(-2)$$ is the value of f(x) when $$x=-2$$. Locate $$x=-2$$ on the graph and observe the value of f(x). Keep in mind that open circles are used for numbers that are less than or greater than (< or >). In other words, it is not a value in the function. Closed circles are used for numbers that are less than or equal to and greater than or equal to $$( \\\\leq $$ or $$ \\\\geq )$$, meaning it is a value in the function.","variabilization":{},"oer":"","license":""},{"id":"a342e92limit16a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["undefined"],"dependencies":["a342e92limit16a-h1"],"title":"Finding the Limit","text":"What value is f(x) when $$x=-2$$?","variabilization":{},"oer":"","license":"","choices":["defined","undefined"]},{"id":"a342e92limit16a-h3","type":"hint","dependencies":["a342e92limit16a-h2"],"title":"Limit Value","text":"Because there is an open circle at $$x=-2$$, this means that the function is undefined at that particular x-value. Since we are asked to find $$f(-2)$$, the value would be undefined. If we were asked to find limits, however, there is no difference with an open or closed circle because a limit is the value the graph is approaching from both the right and left sides.\\\\n##figure2.gif##","variabilization":{},"oer":"","license":""}]}}]},{"id":"a342e92limit17","title":"Behavior of a Function at Different Points","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1<OpenStax: Calculus Volume 1>","license":0,"lesson":"2.2 The Limit of a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a342e92limit17a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"Use the graph of f(x) to determine the value of /lim{x,1**-,f(x)}. If the limit does not exist, write \'DNE\'.","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"a342e92limit17a-h1","type":"hint","dependencies":[],"title":"Understanding the Function and Graph","text":"/lim{x,1**-,f(x)} is the value of f(x) as $$x$$ approaches $$1$$ from the left. Locate $$x$$ $$=$$ $$1$$ on the graph and observe the values of f(x) as it approaches that x-value from the left.","variabilization":{},"oer":"","license":""},{"id":"a342e92limit17a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a342e92limit17a-h1"],"title":"Finding the Limit","text":"What value is f(x) approaching as $$x$$ approaches $$1$$ from the left?","variabilization":{},"oer":"","license":""},{"id":"a342e92limit17a-h3","type":"hint","dependencies":["a342e92limit17a-h2"],"title":"Limit Value","text":"/lim{x,1**-,f(x)}=6. Looking at the left side of the function as $$x$$ approaches $$1$$, we can see that the value of f(x) approaches $$6$$. Remember that there is no difference with an open or closed circle because a limit is the value the graph is approaching from the right and left sides. In this case, we are looking solely at the left side of the function.\\\\n##figure2.gif##","variabilization":{},"oer":"","license":""}]}}]},{"id":"a342e92limit18","title":"Behavior of a Function at Different Points","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1<OpenStax: Calculus Volume 1>","license":0,"lesson":"2.2 The Limit of a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a342e92limit18a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"Use the graph of f(x) to determine the value of /lim{x,1**+,f(x)}. If the limit does not exist, write \'DNE\'.","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a342e92limit18a-h1","type":"hint","dependencies":[],"title":"Understanding the Function and Graph","text":"/lim{x,1**+,f(x)} is the value of f(x) as $$x$$ approaches $$1$$ from the right. Locate $$x$$ $$=$$ $$1$$ on the graph and observe the values of f(x) as it approaches that x-value from the right.","variabilization":{},"oer":"","license":""},{"id":"a342e92limit18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a342e92limit18a-h1"],"title":"Finding the Limit","text":"What value is f(x) approaching as $$x$$ approaches $$1$$ from the right?","variabilization":{},"oer":"","license":""},{"id":"a342e92limit18a-h3","type":"hint","dependencies":["a342e92limit18a-h2"],"title":"Limit Value","text":"/lim{x,1**+,f(x)}=3. Looking at the right side of the function as $$x$$ approaches $$1$$, we can see that the value of f(x) approaches $$3$$. Remember that there is no difference with an open or closed circle because a limit is the value the graph is approaching from the right and left sides. In this case, we are looking solely at the right side of the function.\\\\n##figure2.gif##","variabilization":{},"oer":"","license":""}]}}]},{"id":"a342e92limit19","title":"Behavior of a Function at Different Points","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1<OpenStax: Calculus Volume 1>","license":0,"lesson":"2.2 The Limit of a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a342e92limit19a","stepAnswer":["DNE"],"problemType":"MultipleChoice","stepTitle":"Use the graph of f(x) to determine the value of $$\\\\lim_{x\\\\to1} f(x)$$. If the limit does not exist, write \'DNE\'.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["$$-\\\\infty$$","$$+\\\\infty$$","DNE"],"hints":{"DefaultPathway":[{"id":"a342e92limit19a-h1","type":"hint","dependencies":[],"title":"Understanding the Function and Graph","text":"$$\\\\lim_{x\\\\to1} f(x)$$ is the value of f(x) as $$x$$ approaches $$1$$ from both sides. Locate $$x=1$$ on the graph and observe the values of f(x) as it approaches that x-value from both sides.","variabilization":{},"oer":"","license":""},{"id":"a342e92limit19a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["DNE"],"dependencies":["a342e92limit19a-h1"],"title":"Finding the Limit","text":"What value is f(x) approaching as $$x$$ approaches 1? If the limit does not exist, write \'DNE\'.","variabilization":{},"oer":"","license":"","choices":["$$-\\\\infty$$","$$+\\\\infty$$","DNE"]},{"id":"a342e92limit19a-h3","type":"hint","dependencies":["a342e92limit19a-h2"],"title":"Limit Value","text":"$$\\\\lim_{x\\\\to1} f(x)$$ DNE. There are two different points that f(x) is approaching as $$x$$ approaches 1: $$6$$ (from the left side of the function) and $$3$$ (from the right side of the function). Because the y-values do not approach any one single value, the limit does not exist.\\\\n##figure2.gif##","variabilization":{},"oer":"","license":""}]}}]},{"id":"a342e92limit2","title":"The Limit of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1<OpenStax: Calculus Volume 1>","license":0,"lesson":"2.2 The Limit of a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a342e92limit2a","stepAnswer":["$$0.25$$"],"problemType":"TextBox","stepTitle":"Evaluate the following using a table of functional values. If the limit does not exist, write \'DNE\'.","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.25$$","hints":{"DefaultPathway":[{"id":"a342e92limit2a-h1","type":"hint","dependencies":[],"title":"Functional Values Table","text":"The values of $$f(x)=\\\\frac{\\\\sqrt{x}-2}{x-4}$$ as $$x$$ approaches $$4$$ are listed in Table $$2.3$$. Note: the values in this table were obtained using a calculator and using all the places given in the calculator output.\\\\n##figure2.gif##","variabilization":{},"oer":"","license":""},{"id":"a342e92limit2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.25$$"],"dependencies":["a342e92limit2a-h1"],"title":"The Approaching Limit","text":"As we read down each (sqrt(x) - 2)/(x - 4) column, what number are the values in each column approaching? Round to the nearest hundredths place.","variabilization":{},"oer":"","license":"","subHints":[{"id":"a342e92limit2a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.25$$"],"dependencies":[],"title":"Graphing the Function","text":"The values of (sqrt(x) - 2)/(x - 4) as $$x$$ approaches $$4$$ range from $$0.251582341869$$ to $$0.24999984$$. A calculator or computer-generated graph of f(x) $$=$$ (sqrt(x) - 2)/(x - 4) can also be used to determine the value we are approaching. Around what y-value does the graph intersect?","variabilization":{},"oer":"","license":""}]},{"id":"a342e92limit2a-h3","type":"hint","dependencies":["a342e92limit2a-h2"],"title":"Finding $$/lim{x,4,(sqrt(x)$$ - 2)/(x - 4)}","text":"After inspecting the table, we see that the functional values less than $$4$$ appear to be decreasing toward $$0.25$$ whereas the functional values greater than $$4$$ appear to be increasing toward $$0.25$$. We conclude that $$/lim{x,4,(sqrt(x)$$ - 2)/(x - $$4)=0.25$$. We confirm this estimate using the graph of f(x)=(sqrt(x) - 2)/(x - 4) shown in the following figure.\\\\n##figure3.gif##","variabilization":{},"oer":"","license":""}]}}]},{"id":"a342e92limit20","title":"Behavior of a Function at Different Points","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1<OpenStax: Calculus Volume 1>","license":0,"lesson":"2.2 The Limit of a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a342e92limit20a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"Use the graph of f(x) to determine the value of f(1). If the value is undefined, write \\"undefined\\".","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"a342e92limit20a-h1","type":"hint","dependencies":[],"title":"Understanding the Function and Graph","text":"f(1) is the value of f(x) when $$x=1$$. Locate $$x=1$$ on the graph and observe the value of f(x). Keep in mind that open circles are used for numbers that are less than or greater than (< or >). In other words, it is not a value in the function. Closed circles are used for numbers that are less than or equal to and greater than or equal to $$( \\\\leq $$ or $$ \\\\geq )$$, meaning it is a value in the function.","variabilization":{},"oer":"","license":""},{"id":"a342e92limit20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a342e92limit20a-h1"],"title":"Finding the Limit","text":"What value is f(x) when $$x=1$$?","variabilization":{},"oer":"","license":""},{"id":"a342e92limit20a-h3","type":"hint","dependencies":["a342e92limit20a-h2"],"title":"Limit Value","text":"When $$x=1$$, there are two points that f(x) approaches: one is at a closed circle (from the left) and the other is at an open circle (from the right). Because open circles indicate that the function is undefined at that particular x-value, the f(x) value when $$x=1$$ would be at the closed circle where $$f(x)=6$$. If we were asked to find limits, however, there would be no difference with an open or closed circle because a limit is the value the graph is approaching from both the right and left sides.\\\\n##figure2.gif##","variabilization":{},"oer":"","license":""}]}}]},{"id":"a342e92limit21","title":"Behavior of a Function at Different Points","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1<OpenStax: Calculus Volume 1>","license":0,"lesson":"2.2 The Limit of a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a342e92limit21a","stepAnswer":["$$-\\\\infty$$"],"problemType":"MultipleChoice","stepTitle":"Use the graph of f(x) to determine the value of /lim{x,3**-,f(x)}; /lim{x,3**+,f(x)}; $$\\\\lim_{x\\\\to3} f(x)$$. If the limit is $$\\\\infty$$, write $$-\\\\infty$$ or $$+\\\\infty$$. If the limit does not exist, write \'DNE\'.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$-\\\\infty$$","choices":["$$-\\\\infty$$","$$+\\\\infty$$","DNE"],"hints":{"DefaultPathway":[{"id":"a342e92limit21a-h1","type":"hint","dependencies":[],"title":"Understanding the Functions and Graph","text":"/lim{x,3**-,f(x)} is the value of f(x) as $$x$$ approaches $$3$$ from the left.\\\\n/lim{x,3**+,f(x)} is the value of f(x) as $$x$$ approaches $$3$$ from the right.\\\\n$$\\\\lim_{x\\\\to3} f(x)is$$ the value of f(x) as $$x$$ approaches $$3$$ from both sides.\\\\nLocate $$x=3$$ on the graph and observe the values of f(x) as it approaches that x-value.\\\\n##figure2.gif##","variabilization":{},"oer":"","license":""},{"id":"a342e92limit21a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-\\\\infty$$"],"dependencies":["a342e92limit21a-h1"],"title":"Finding the Limit","text":"What value is f(x) approaching as $$x$$ approaches 3? If it is $$\\\\infty$$, write $$-\\\\infty$$ or $$+\\\\infty$$. If it does not exist, write \'DNE\'.","variabilization":{},"oer":"","license":"","choices":["$$-\\\\infty$$","$$+\\\\infty$$","DNE"]},{"id":"a342e92limit21a-h3","type":"hint","dependencies":["a342e92limit21a-h2"],"title":"Limit Value","text":"/lim{x,3**-,f(x)})=-inf; /lim{x,3**+,f(x)}=-inf; $$\\\\lim_{x\\\\to3} f(x)=-\\\\infty$$. As $$x$$ approaches $$3$$, the values of f(x) continuously go down into the negatives.\\\\n##figure3.gif##","variabilization":{},"oer":"","license":""}]}}]},{"id":"a342e92limit22","title":"Behavior of a Function at Different Points","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1<OpenStax: Calculus Volume 1>","license":0,"lesson":"2.2 The Limit of a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a342e92limit22a","stepAnswer":["undefined"],"problemType":"MultipleChoice","stepTitle":"Use the graph of f(x) to determine the value of f(3). If the value is undefined, write \\"undefined\\".","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["defined","undefined"],"hints":{"DefaultPathway":[{"id":"a342e92limit22a-h1","type":"hint","dependencies":[],"title":"Understanding the Function and Graph","text":"f(3) is the value of f(x) when $$x=3$$. Locate $$x=3$$ on the graph and observe the value of f(x).","variabilization":{},"oer":"","license":""},{"id":"a342e92limit22a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["undefined"],"dependencies":["a342e92limit22a-h1"],"title":"Finding the Limit","text":"What value is f(x) when $$x=3$$? If the value is undefined, write \\"undefined\\".","variabilization":{},"oer":"","license":"","choices":["defined","undefined"]},{"id":"a342e92limit22a-h3","type":"hint","dependencies":["a342e92limit22a-h2"],"title":"Limit Value","text":"As $$x$$ approaches $$3$$, the value of f(x) does not approach any one single value, meaning that f(3) is undefined.\\\\n##figure2.gif##","variabilization":{},"oer":"","license":""}]}}]},{"id":"a342e92limit23","title":"The Limit of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1<OpenStax: Calculus Volume 1>","license":0,"lesson":"2.2 The Limit of a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a342e92limit23a","stepAnswer":["False"],"problemType":"MultipleChoice","stepTitle":"Consider the graph of the function $$y=f(x)$$ shown here. Is the statement /lim{x,-2**+,f(x)}=3 true or false about $$y=f(x)$$?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["True","False"],"hints":{"DefaultPathway":[{"id":"a342e92limit23a-h1","type":"hint","dependencies":[],"title":"Understanding the Function and Graph","text":"/lim{x,-2**+,f(x)} is the value of f(x) as $$x$$ approaches $$-2$$ from the right. Locate $$x$$ $$=$$ $$-2$$ on the graph and observe the values of f(x) from the right.\\\\n##figure2.gif##","variabilization":{},"oer":"","license":""},{"id":"a342e92limit23a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$+\\\\infty$$"],"dependencies":["a342e92limit23a-h1"],"title":"Finding the Limit","text":"What value is f(x) approaching when $$x$$ approaches $$-2$$ from the right? If the limit is $$\\\\infty$$, write $$-\\\\infty$$ or $$+\\\\infty$$. If the limit does not exist, write \'DNE\'.","variabilization":{},"oer":"","license":""},{"id":"a342e92limit23a-h3","type":"hint","dependencies":["a342e92limit23a-h2"],"title":"Limit Value","text":"/lim{x,-2**+,f(x)}=+inf. Looking at the right side of the function as $$x$$ approaches $$-2$$, we can see that the values of f(x) continuously goes up into the positives.\\\\n##figure3.gif##","variabilization":{},"oer":"","license":""}]}}]},{"id":"a342e92limit24","title":"The Limit of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1<OpenStax: Calculus Volume 1>","license":0,"lesson":"2.2 The Limit of a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a342e92limit24a","stepAnswer":["False"],"problemType":"MultipleChoice","stepTitle":"Consider the graph of the function $$y=f(x)$$ shown here. Is the statement $$\\\\lim_{x\\\\to6} f(x)=5$$ true or false about $$y=f(x)$$?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["True","False"],"hints":{"DefaultPathway":[{"id":"a342e92limit24a-h1","type":"hint","dependencies":[],"title":"Understanding the Function and Graph","text":"$$\\\\lim_{x\\\\to6} f(x)$$ is the value of f(x) as $$x$$ approaches $$6$$ from both sides. Locate $$x=6$$ on the graph and observe the value of f(x) from both sides.\\\\n##figure2.gif##","variabilization":{},"oer":"","license":""},{"id":"a342e92limit24a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["DNE"],"dependencies":["a342e92limit24a-h1"],"title":"Finding the Limit","text":"What value is f(x) approaching when $$x$$ approaches $$6$$ from both sides? If the limit is $$\\\\infty$$, write $$-\\\\infty$$ or $$+\\\\infty$$. If the limit does not exist, write \'DNE\'.","variabilization":{},"oer":"","license":"","choices":["$$-\\\\infty$$","$$+\\\\infty$$","DNE"]},{"id":"a342e92limit24a-h3","type":"hint","dependencies":["a342e92limit24a-h2"],"title":"Limit Value","text":"$$\\\\lim_{x\\\\to6} f(x)$$ DNE because /lim{x,6**-,f(x)}=2 and /lim{x,6**+,f(x)}=5.\\\\n##figure3.gif##","variabilization":{},"oer":"","license":""}]}}]},{"id":"a342e92limit25","title":"The Limit of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1<OpenStax: Calculus Volume 1>","license":0,"lesson":"2.2 The Limit of a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a342e92limit25a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"Use the following graph of the function $$y=f(x)$$ to find the value of /lim{x,1**+,f(x)}, if possible. Estimate when necessary.","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a342e92limit25a-h1","type":"hint","dependencies":[],"title":"Understanding the Function and Graph","text":"/lim{x,1**+,f(x)} is the value of f(x) as $$x$$ approaches $$1$$ from the right side. Locate $$x=1$$ on the graph and observe the value of f(x) from the right.\\\\n##figure2.gif##","variabilization":{},"oer":"","license":""},{"id":"a342e92limit25a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a342e92limit25a-h1"],"title":"Finding the Limit","text":"What value is f(x) approaching when $$x$$ approaches $$1$$ from the right side? If the limit is $$\\\\infty$$, write $$-\\\\infty$$ or $$+\\\\infty$$. If the limit does not exist, write \'DNE\'.","variabilization":{},"oer":"","license":""},{"id":"a342e92limit25a-h3","type":"hint","dependencies":["a342e92limit25a-h2"],"title":"Limit Value","text":"/lim{x,1**+,f(x)}=2. Looking at the right side of the function as $$x$$ approaches $$1$$, we can see that the value of f(x) approaches $$2$$. Remember that there is no difference with an open or closed circle because a limit is the value the graph is approaching from the right and left sides. In this case, we are looking solely at the right side of the function.\\\\n##figure3.gif##","variabilization":{},"oer":"","license":""}]}}]},{"id":"a342e92limit3","title":"The Limit of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1<OpenStax: Calculus Volume 1>","license":0,"lesson":"2.2 The Limit of a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a342e92limit3a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"For g(x) shown in the figure, evaluate $$\\\\lim_{x\\\\to-1} g(x)$$. If the limit does not exist, write \'DNE\'.","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a342e92limit3a-h1","type":"hint","dependencies":[],"title":"Understanding the Graph","text":"Look at the g(x) values as the x-values approach $$-1$$ from either side to determine the limit of the function.","variabilization":{},"oer":"","license":""},{"id":"a342e92limit3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a342e92limit3a-h1"],"title":"The Approaching Limit","text":"What number do the g(x) values approach as the x-values approach -1?","variabilization":{},"oer":"","license":""},{"id":"a342e92limit3a-h3","type":"hint","dependencies":["a342e92limit3a-h2"],"title":"Determining $$/lim{x$$ ,-1, g(x)}","text":"Despite the fact that $$g(-1)$$ $$=$$ $$4$$, as the x-values approach $$-1$$ from either side, the g(x) values approach $$3$$. Therefore, $$lim_x$$ \u2192 $$-1$$ g(x) $$=$$ $$3$$. Note that we can determine this limit without even knowing the algebraic expression of the function.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a342e92limit4","title":"The Limit of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1<OpenStax: Calculus Volume 1>","license":0,"lesson":"2.2 The Limit of a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a342e92limit4a","stepAnswer":["DNE"],"problemType":"TextBox","stepTitle":"Evaluate the following using a table of functional values. If the limit does not exist, write \'DNE\'.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a342e92limit4a-h1","type":"hint","dependencies":[],"title":"Functional Values Table","text":"The values of $$f(x)=sin\\\\left(\\\\frac{1}{x}\\\\right)$$ as $$x$$ approaches $$0$$ are listed in Table $$2.5$$. Note: the values in this table were obtained using a calculator and using all the places given in the calculator output.\\\\n##figure2.gif##","variabilization":{},"oer":"","license":""},{"id":"a342e92limit4a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a342e92limit4a-h1"],"title":"The Approaching Limit","text":"As we read down each $$sin\\\\left(\\\\frac{1}{x}\\\\right)$$ column, are the values in each column approaching one single number?","variabilization":{},"oer":"","license":"","choices":["Yes","No"]},{"id":"a342e92limit4a-h3","type":"hint","dependencies":["a342e92limit4a-h2"],"title":"Finding $$\\\\lim_{x\\\\to0} sin\\\\left(\\\\frac{1}{x}\\\\right)$$","text":"After examining the table of functional values, we can see that the y-values do not seem to approach any one single value. It appears the limit does not exist. Before drawing this conclusion, let\u2019s take a more systematic approach. Take the following sequence of x-values approaching 0: $$\\\\frac{2}{\\\\pi}$$, $$\\\\frac{2}{3} \\\\pi$$, $$\\\\frac{2}{5} \\\\pi$$, $$\\\\frac{2}{7} \\\\pi$$, $$\\\\frac{2}{9} \\\\pi$$, $$\\\\frac{2}{11} \\\\pi$$, ... The corresponding y-values are $$1$$, $$-1$$, $$1$$, $$-1$$, $$1$$, $$-1$$, ... At this point we can indeed conclude that $$lim_x$$ \u2192 $$0$$ $$sin\\\\left(\\\\frac{1}{x}\\\\right)$$ does not exist. Thus, we would write $$lim_x$$ \u2192 $$0$$ $$sin\\\\left(\\\\frac{1}{x}\\\\right)$$ DNE. The graph of f(x) $$=$$ $$sin\\\\left(\\\\frac{1}{x}\\\\right)$$ is shown in the figure and gives a clearer picture of the behavior of $$sin\\\\left(\\\\frac{1}{x}\\\\right)$$ as $$x$$ approaches $$0$$. You can see that $$sin\\\\left(\\\\frac{1}{x}\\\\right)$$ oscillates ever more wildly between $$-1$$ and $$1$$ as $$x$$ approaches $$0$$.\\\\n##figure3.gif##","variabilization":{},"oer":"","license":""}]}}]},{"id":"a342e92limit5","title":"The Limit of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1<OpenStax: Calculus Volume 1>","license":0,"lesson":"2.2 The Limit of a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a342e92limit5a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"For the function, evaluate the following limit: /lim{x,2**-,f(x)}.","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a342e92limit5a-h1","type":"hint","dependencies":[],"title":"Functional Values Table","text":"The functional values are shown in the table. Observe that for values of $$x$$ less than $$2$$, we use $$f(x)=x+1$$. The values in this table were obtained using a calculator and using all the places given in the calculator output.\\\\n##figure2.gif##","variabilization":{},"oer":"","license":""},{"id":"a342e92limit5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a342e92limit5a-h1"],"title":"The Approaching Limit","text":"As we read down the $$f(x)=x+1$$ column, what number are the values approaching?","variabilization":{},"oer":"","license":""},{"id":"a342e92limit5a-h3","type":"hint","dependencies":["a342e92limit5a-h2"],"title":"Finding /lim{x,2**-,f(x)}","text":"Based on the table, we can conclude that /lim{x,2**-,f(x)}=3. The figure shows a graph of f(x) and reinforces our conclusion about the limit.\\\\n##figure3.gif##","variabilization":{},"oer":"","license":""}]}}]},{"id":"a342e92limit6","title":"The Limit of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1<OpenStax: Calculus Volume 1>","license":0,"lesson":"2.2 The Limit of a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a342e92limit6a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"For the function, evaluate the following limit: /lim{x,2**+,f(x)}.","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a342e92limit6a-h1","type":"hint","dependencies":[],"title":"Functional Values Table","text":"The functional values are shown in the table. Observe that for values of $$x$$ greater than $$2$$, we use $$f(x)=x^2-4$$. The values in this table were obtained using a calculator and using all the places given in the calculator output.\\\\n##figure2.gif##","variabilization":{},"oer":"","license":""},{"id":"a342e92limit6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a342e92limit6a-h1"],"title":"The Approaching Limit","text":"As we read down the $$f(x)=x^2-4$$ column, what number are the values approaching?","variabilization":{},"oer":"","license":""},{"id":"a342e92limit6a-h3","type":"hint","dependencies":["a342e92limit6a-h2"],"title":"Finding /lim{x,2**+,f(x)}","text":"Based on the table, we can conclude that /lim{x,2**+,f(x)}=0. The figure shows a graph of f(x) and reinforces our conclusion about the limit.\\\\n##figure3.gif##","variabilization":{},"oer":"","license":""}]}}]},{"id":"a342e92limit7","title":"The Limit of a Function","body":"Use a table of functional values and graph $$f(x)=$$ $$\\\\frac{1}{x}$$ to confirm your conclusion.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1<OpenStax: Calculus Volume 1>","license":0,"lesson":"2.2 The Limit of a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a342e92limit7a","stepAnswer":["$$-\\\\infty$$"],"problemType":"MultipleChoice","stepTitle":"Evaluate /lim{x,0**-,1/x} If the limit is $$\\\\infty$$, write $$-\\\\infty$$ or $$+\\\\infty$$. If the limit does not exist, write \'DNE\'.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-\\\\infty$$","choices":["$$-\\\\infty$$","$$+\\\\infty$$"],"hints":{"DefaultPathway":[{"id":"a342e92limit7a-h1","type":"hint","dependencies":[],"title":"Functional Values Table","text":"The functional values are shown in the table. Note: the values in this table were obtained using a calculator and using all the places given in the calculator output.\\\\n##figure1.gif##","variabilization":{},"oer":"","license":""},{"id":"a342e92limit7a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-\\\\infty$$"],"dependencies":["a342e92limit7a-h1"],"title":"The Approaching Limit","text":"As we read down the $$\\\\frac{1}{x}$$ column, what are the values approaching?","variabilization":{},"oer":"","license":"","choices":["$$-\\\\infty$$","$$+\\\\infty$$"]},{"id":"a342e92limit7a-h3","type":"hint","dependencies":["a342e92limit7a-h2"],"title":"Finding /lim{x,0**-,1/x}","text":"The values of $$\\\\frac{1}{x}$$ decrease without bound as $$x$$ approaches $$0$$ from the left. As such, we conclude that /lim{x,0**-,1/x}=-inf. The graph of $$f(x)=\\\\frac{1}{x}$$ in the figure confirms these conclusions.\\\\n##figure2.gif##","variabilization":{},"oer":"","license":""}]}}]},{"id":"a342e92limit8","title":"The Limit of a Function","body":"Use a table of functional values and graph $$f(x)=\\\\frac{1}{x}$$ to confirm your conclusion.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1<OpenStax: Calculus Volume 1>","license":0,"lesson":"2.2 The Limit of a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a342e92limit8a","stepAnswer":["$$+\\\\infty$$"],"problemType":"MultipleChoice","stepTitle":"Evaluate /lim{x,0**+,1/x}. If the limit is $$\\\\infty$$, write $$-\\\\infty$$ or $$+\\\\infty$$. If the limit does not exist, write \'DNE\'.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$+\\\\infty$$","choices":["$$-\\\\infty$$","$$+\\\\infty$$"],"hints":{"DefaultPathway":[{"id":"a342e92limit8a-h1","type":"hint","dependencies":[],"title":"Functional Values Table","text":"The functional values are shown in the table. Note: the values in this table were obtained using a calculator and using all the places given in the calculator output.\\\\n##figure1.gif##","variabilization":{},"oer":"","license":""},{"id":"a342e92limit8a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$+\\\\infty$$"],"dependencies":["a342e92limit8a-h1"],"title":"The Approaching Limit","text":"As we read down the $$\\\\frac{1}{x}$$ column, what are the values approaching?","variabilization":{},"oer":"","license":"","choices":["$$-\\\\infty$$","$$+\\\\infty$$"]},{"id":"a342e92limit8a-h3","type":"hint","dependencies":["a342e92limit8a-h2"],"title":"Finding /lim{x,0**+,1/x}","text":"The values of $$\\\\frac{1}{x}$$ increase without bound as $$x$$ approaches $$0$$ from the right. As such, we conclude that /lim{x,0**+,1/x}=+inf. The graph of $$f(x)=\\\\frac{1}{x}$$ in the figure confirms these conclusions.\\\\n##figure2.gif##","variabilization":{},"oer":"","license":""}]}}]},{"id":"a342e92limit9","title":"The Limit of a Function","body":"Use a table of functional values and graph $$f(x)=\\\\frac{1}{x}$$ to confirm your conclusion.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1<OpenStax: Calculus Volume 1>","license":0,"lesson":"2.2 The Limit of a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a342e92limit9a","stepAnswer":["DNE"],"problemType":"MultipleChoice","stepTitle":"Evaluate $$\\\\lim_{x\\\\to0} \\\\frac{1}{x}$$. If the limit is $$\\\\infty$$, write $$-\\\\infty$$ or $$+\\\\infty$$. If the limit does not exist, write \'DNE\'.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$-\\\\infty$$","$$+\\\\infty$$","DNE"],"hints":{"DefaultPathway":[{"id":"a342e92limit9a-h1","type":"hint","dependencies":[],"title":"Functional Values Table","text":"The functional values are shown in the table. Note: the values in this table were obtained using a calculator and using all the places given in the calculator output.\\\\n##figure1.gif##","variabilization":{},"oer":"","license":""},{"id":"a342e92limit9a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a342e92limit9a-h1"],"title":"The Approaching Limit","text":"As we read down the $$\\\\frac{1}{x}$$ columns, does /lim{x,0**-,1/x} and /lim{x,0**+,1/x} approach the same values?","variabilization":{},"oer":"","license":"","choices":["Yes","No"]},{"id":"a342e92limit9a-h3","type":"hint","dependencies":["a342e92limit9a-h2"],"title":"Finding $$/lim{x$$ ,0,1/x}","text":"Since /lim{x,0**-,1/x}=-inf and /lim{x,0**+,1/x}=+inf have different values, we conclude that $$/lim(x$$ , $$0$$, $$\\\\frac{1}{x}$$ )DNE. The graph of $$f(x)=\\\\frac{1}{x}$$ in the figure confirms these conclusions.\\\\n##figure2.gif##","variabilization":{},"oer":"","license":""}]}}]},{"id":"a343428l\'hopital1","title":"Evaluate each of the following limits by applying L\u2019H\xf4pital\u2019s Rule.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.8 L\u2019H\xf4pital\u2019s Rule","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a343428l\'hopital1a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"$$\\\\lim_{x\\\\to0} \\\\frac{sinx-x}{x^2}=0$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a343428l\'hopital1a-h1","type":"hint","dependencies":[],"title":"Substitute $$x$$ with the value $$x$$ approaches to. If the function results in one of these indeterminate form: $$\\\\frac{0}{0}$$, $$\\\\frac{\\\\infty}{\\\\infty}$$ or $$\\\\frac{-\\\\infty}{\\\\infty}$$, we apply L\u2019H\xf4pital\u2019s Rule.","text":"Substitute $$x$$ with the value $$x$$ approaches to. If the function results in one of these indeterminate form: $$\\\\frac{0}{0}$$, $$\\\\frac{\\\\infty}{\\\\infty}$$ or $$\\\\frac{-\\\\infty}{\\\\infty}$$, we apply L\u2019H\xf4pital\u2019s Rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital1a-h2","type":"hint","dependencies":["a343428l\'hopital1a-h1"],"title":"Substitute the limit","text":"$$\\\\frac{0-0}{0}=\\\\frac{0}{0}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital1a-h3","type":"hint","dependencies":["a343428l\'hopital1a-h2"],"title":"Apply L\u2019H\xf4pital\u2019s Rule","text":"Differentiate the denominator and the numerator $$seperatedly:\\\\lim_{x\\\\to0} \\\\frac{\\\\frac{d}{dx} \\\\left(sinx-x\\\\right)}{\\\\frac{d}{dx} x^2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital1a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{x\\\\to0} \\\\frac{cosx-1}{2x}$$"],"dependencies":["a343428l\'hopital1a-h3"],"title":"Find the derivative","text":"What is the answer of the function after L\u2019H\xf4pital\u2019s Rule is applied?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\lim_{x\\\\to0} \\\\frac{x^2 cosx-2x cosx}{x^4}$$","$$\\\\lim_{x\\\\to0} \\\\frac{cosx}{2x}$$","$$\\\\lim_{x\\\\to0} \\\\frac{\\\\left(-cosx-1\\\\right)}{2x}$$","$$\\\\lim_{x\\\\to0} \\\\frac{cosx-1}{2x}$$"],"subHints":[{"id":"a343428l\'hopital1a-h4-s1","type":"hint","dependencies":[],"title":"Indeterminate forms in answers","text":"If the result is still in an indeterminate form after we apply L\u2019H\xf4pital\u2019s Rule and evaluate the limit, we repeat the process.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"a343428l\'hopital1a-h5","type":"hint","dependencies":["a343428l\'hopital1a-h4"],"title":"Evaluate the limit","text":"$$\\\\fraccos^2\\\\left(0\\\\right)-1\\\\times0}=\\\\frac{1-1}{0}=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital1a-h6","type":"hint","dependencies":["a343428l\'hopital1a-h5"],"title":"Apply L\u2019H\xf4pital\u2019s Rule again","text":"Differentiate the denominator and the numerator $$seperatedly:\\\\lim_{x\\\\to0} \\\\frac{\\\\frac{d}{dx} \\\\left(cosx-1\\\\right)}{\\\\frac{d}{dx} 2x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital1a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{x\\\\to0} \\\\frac{-sinx}{2}$$"],"dependencies":["a343428l\'hopital1a-h6"],"title":"Apply L\u2019H\xf4pital\u2019s Rule again","text":"$$0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\lim_{x\\\\to0} \\\\frac{2x cosx-2cosx}{4x^2}$$","$$\\\\lim_{x\\\\to0} \\\\frac{sinx}{2}$$","$$\\\\lim_{x\\\\to0} \\\\frac{-sinx}{2}$$","$$\\\\lim_{x\\\\to0} \\\\frac{sinx-1}{2}$$"]},{"id":"a343428l\'hopital1a-h8","type":"hint","dependencies":["a343428l\'hopital1a-h7"],"title":"Evaluate the limit","text":"$$\\\\frac{-sin\\\\left(0\\\\right)}{2}=\\\\frac{-0}{2}=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a343428l\'hopital10","title":"For the following exercises, evaluate the limits with either L\u2019H\xf4pital\u2019s rule or previously learned methods.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.8 L\u2019H\xf4pital\u2019s Rule","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a343428l\'hopital10a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"$$\\\\lim_{x\\\\to3} \\\\frac{x^2-9}{x-3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"a343428l\'hopital10a-h1","type":"hint","dependencies":[],"title":"Indeterminate form","text":"Substitute $$x$$ with the value $$x$$ approaches to. If the function results in one of these indeterminate form: $$\\\\frac{0}{0}$$, $$\\\\frac{\\\\infty}{\\\\infty}$$ or $$\\\\frac{-\\\\infty}{\\\\infty}$$, we apply L\u2019H\xf4pital\u2019s Rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital10a-h2","type":"hint","dependencies":["a343428l\'hopital10a-h1"],"title":"Substitute the limit","text":"$$\\\\frac{3^2-9}{3-3}=\\\\frac{0}{0}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital10a-h3","type":"hint","dependencies":["a343428l\'hopital10a-h2"],"title":"Apply L\u2019H\xf4pital\u2019s Rule","text":"Differentiate the denominator and the numerator $$seperatedly:\\\\lim_{x\\\\to3} \\\\frac{\\\\frac{d}{dx} \\\\left(x^2-9\\\\right)}{\\\\frac{d}{dx} \\\\left(x-3\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital10a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{x\\\\to3} \\\\frac{2x}{1}$$"],"dependencies":["a343428l\'hopital10a-h3"],"title":"Find the derivative","text":"What is the answer of the function after L\u2019H\xf4pital\u2019s Rule is applied?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\lim_{x\\\\to3} \\\\frac{2x}{1}$$","$$\\\\lim_{x\\\\to3} \\\\frac{x}{2}$$","$$\\\\lim_{x\\\\to3} \\\\frac{2x \\\\left(x-3\\\\right)-x^2}{{\\\\left(x-3\\\\right)}^2}$$","$$\\\\lim_{x\\\\to3} x$$"]},{"id":"a343428l\'hopital10a-h5","type":"hint","dependencies":["a343428l\'hopital10a-h4"],"title":"Evaluate the limit","text":"$$\\\\frac{2\\\\times3}{1}=6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a343428l\'hopital12","title":"For the following exercises, determine whether you can apply L\u2019H\xf4pital\u2019s Rule directly. Explain why or why not. Then, indicate if there is some way you can alter the limit so you can apply L\u2019H\xf4pital\u2019s Rule.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.8 L\u2019H\xf4pital\u2019s Rule","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a343428l\'hopital12a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"$$\\\\lim_{x\\\\to0} \\\\frac{x^2}{\\\\frac{1}{x}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a343428l\'hopital12a-h1","type":"hint","dependencies":[],"title":"Indeterminate form","text":"Substitute $$x$$ with the value $$x$$ approaches to. If the function results in one of these indeterminate form: $$\\\\frac{0}{0}$$, $$\\\\frac{\\\\infty}{\\\\infty}$$ or $$\\\\frac{-\\\\infty}{\\\\infty}$$, we apply L\u2019H\xf4pital\u2019s Rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital12a-h2","type":"hint","dependencies":["a343428l\'hopital12a-h1"],"title":"Substitute the limit","text":"$$\\\\frac{0^2}{\\\\frac{1}{0}}=\\\\frac{0}{\\\\infty}=\\\\frac{0}{0}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital12a-h3","type":"hint","dependencies":["a343428l\'hopital12a-h2"],"title":"Indeterminate form","text":"Substitution does not give us an indeterminate form therefore we have to look for another approach.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital12a-h4","type":"hint","dependencies":["a343428l\'hopital12a-h3"],"title":"Rearrange the expression","text":"$$\\\\lim_{x\\\\to0} \\\\frac{x^2}{\\\\frac{1}{x}}=\\\\lim_{x\\\\to0} x^3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital12a-h5","type":"hint","dependencies":["a343428l\'hopital12a-h4"],"title":"Evaluate","text":"$$0^3=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a343428l\'hopital13","title":"For the following exercises, determine whether you can apply L\u2019H\xf4pital\u2019s Rule directly. Explain why or why not. Then, indicate if there is some way you can alter the limit so you can apply L\u2019H\xf4pital\u2019s Rule.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.8 L\u2019H\xf4pital\u2019s Rule","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a343428l\'hopital13a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"Evaluate $$\\\\lim_{x\\\\to\\\\infty} x^{\\\\frac{1}{x}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a343428l\'hopital13a-h1","type":"hint","dependencies":[],"title":"Operation","text":"Let $$y=x^{\\\\frac{1}{x}}$$. Using one-to-one property of logarithms, we obtain $$ln(y)=\\\\ln(x^{\\\\frac{1}{x}})$$. According to the Power Properties of Logarithms, the expression can be written as $$ln(y)=\\\\frac{1}{x} \\\\ln(x)=\\\\frac{\\\\ln(x)}{x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital13a-h2","type":"hint","dependencies":["a343428l\'hopital13a-h1"],"title":"Operation","text":"If we immediately evaluate the limit of the given function, the result would be $${\\\\infty}^0$$ which is not the indeterminate form we need in order to apply L\u2019H\xf4pital\u2019s Rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital13a-h3","type":"hint","dependencies":["a343428l\'hopital13a-h2"],"title":"Operation","text":"We set up a new limit based on the function we just created $$\\\\lim_{x\\\\to\\\\infty} \\\\frac{\\\\ln(x)}{x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital13a-h4","type":"hint","dependencies":["a343428l\'hopital13a-h3"],"title":"Indeterminate form","text":"Substitute $$x$$ with the value $$x$$ approaches to. If the function results in one of these indeterminate form: $$\\\\frac{0}{0}$$, $$\\\\frac{\\\\infty}{\\\\infty}$$ or $$\\\\frac{-\\\\infty}{\\\\infty}$$, we apply L\u2019H\xf4pital\u2019s Rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital13a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a343428l\'hopital13a-h4"],"title":"Indeterminate form","text":"Does direct substitution of this limit yield any required indeterminate form?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["Yes","No"],"subHints":[{"id":"a343428l\'hopital13a-h5-s1","type":"hint","dependencies":[],"title":"Substitute the limit","text":"$$\\\\frac{\\\\ln(\\\\infty)}{\\\\infty}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"a343428l\'hopital13a-h6","type":"hint","dependencies":["a343428l\'hopital13a-h3","a343428l\'hopital13a-h4","a343428l\'hopital13a-h5"],"title":"Apply L\u2019H\xf4pital\u2019s Rule","text":"Differentiate the denominator and the numerator seperatedly: $$\\\\lim_{x\\\\to\\\\infty} \\\\frac{\\\\frac{d}{dx} \\\\ln(x)}{\\\\frac{d}{dx} x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital13a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{\\\\frac{1}{x}}{1}$$"],"dependencies":["a343428l\'hopital13a-h6"],"title":"Find the derivative","text":"What is the answer of the function after L\u2019H\xf4pital\u2019s Rule is applied?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{\\\\frac{1}{x}}{1}$$","$$\\\\lim_{x\\\\to\\\\infty} x$$","$$/l\\\\lim_{x\\\\to\\\\infty} \\\\frac{1}{x^2}$$","/lim{x,inf,x**2)}"]},{"id":"a343428l\'hopital13a-h8","type":"hint","dependencies":["a343428l\'hopital13a-h7"],"title":"Evaluate the limit","text":"$$\\\\frac{\\\\frac{1}{\\\\infty}}{1}=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital13a-h9","type":"hint","dependencies":["a343428l\'hopital13a-h8"],"title":"Operation","text":"$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{\\\\ln(x)}{x}=\\\\lim_{x\\\\to\\\\infty} ln(y)=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"a343428l\'hopital13a-h7-s1","type":"hint","dependencies":[],"title":"Operation","text":"We set $$ln(y)=\\\\frac{\\\\ln(x)}{x}$$ in the very first step","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"a343428l\'hopital13a-h10","type":"hint","dependencies":["a343428l\'hopital13a-h9"],"title":"Limit of Logarithm Rule","text":"$$\\\\lim_{x\\\\to\\\\infty} ln(y)=ln(\\\\lim_{x\\\\to\\\\infty} (y))=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital13a-h11","type":"hint","dependencies":["a343428l\'hopital13a-h10"],"title":"Properties of Exponents and Natural Logarithms","text":"$$ln(\\\\lim_{x\\\\to\\\\infty} (y))=\\\\lim_{x\\\\to\\\\infty} (y)=e^0=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital13a-h12","type":"hint","dependencies":["a343428l\'hopital13a-h1","a343428l\'hopital13a-h11"],"title":"Conclusion","text":"$$\\\\lim_{x\\\\to\\\\infty} y=\\\\lim_{x\\\\to\\\\infty} x^{\\\\frac{1}{x}}=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a343428l\'hopital14","title":"Comparing the Growth Rates of ln(x),x**2 and $$e^x$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.8 L\u2019H\xf4pital\u2019s Rule","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a343428l\'hopital14a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"For each of the following pairs of functions, use L\u2019H\xf4pital\u2019s rule to evaluate $$\\\\lim_{x\\\\to\\\\infty} \\\\frac{f{\\\\left(x\\\\right)}}{g{\\\\left(x\\\\right)}}$$.","stepBody":"$$a.f(x)=x^2$$ and $$g(x)=e^x$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a343428l\'hopital14a-h1","type":"hint","dependencies":[],"title":"Comparing the Growth Rates","text":"In order to determine which function grows more quickly than the other as $$x$$ approaches $$infinty$$, we can choose to set up ONE of these two limits $$\\\\lim_{x\\\\to\\\\infty} \\\\frac{f{\\\\left(x\\\\right)}}{g{\\\\left(x\\\\right)}}$$ or $$\\\\lim_{x\\\\to\\\\infty} \\\\frac{g{\\\\left(x\\\\right)}}{f{\\\\left(x\\\\right)}}$$. If evaluating the limits results in $$0$$, the function in the denominator is greater and grows faster than the function on the numerator. If evaluating the limits results in $$\\\\infty$$, the function in the numerator is is greater and grows fastethan the function on the denominator. In this case, let\'s use $$\\\\lim_{x\\\\to\\\\infty} \\\\frac{f{\\\\left(x\\\\right)}}{g{\\\\left(x\\\\right)}}=\\\\lim_{x\\\\to\\\\infty} \\\\frac{x^2}{e^x}$$ .","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital14a-h2","type":"hint","dependencies":["a343428l\'hopital14a-h1"],"title":"Indeterminate form","text":"Substitute $$x$$ with the value $$x$$ approaches to. If the function results in one of these indeterminate form: $$\\\\frac{0}{0}$$, $$\\\\frac{\\\\infty}{\\\\infty}$$ or $$\\\\frac{-\\\\infty}{\\\\infty}$$, we apply L\u2019H\xf4pital\u2019s Rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital14a-h3","type":"hint","dependencies":["a343428l\'hopital14a-h2"],"title":"Substitute the limit","text":"$$\\\\frac{{\\\\infty}^2}{e^{\\\\infty}}=\\\\frac{\\\\infty}{\\\\infty}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital14a-h4","type":"hint","dependencies":["a343428l\'hopital14a-h3"],"title":"Apply L\u2019H\xf4pital\u2019s Rule","text":"Differentiate the denominator and the numerator seperatedly: $$\\\\lim_{x\\\\to\\\\infty} \\\\frac{\\\\frac{d}{dx} x^2}{\\\\frac{d}{dx} e^x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital14a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{2x}{e^x}$$"],"dependencies":["a343428l\'hopital14a-h4"],"title":"Find the derivative","text":"What is the answer of the function after L\u2019H\xf4pital\u2019s Rule is applied?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{2x}{e^x}$$","$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{2x}{{xe}^{x-1}}$$","$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{2x}{x e}$$","$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{2x}{e}$$"]},{"id":"a343428l\'hopital14a-h6","type":"hint","dependencies":["a343428l\'hopital14a-h5"],"title":"Evaluate the limit","text":"$$\\\\frac{2\\\\infty}{e^{\\\\infty}}=\\\\frac{\\\\infty}{\\\\infty}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital14a-h7","type":"hint","dependencies":["a343428l\'hopital14a-h6"],"title":"Indeterminate forms in answers","text":"If the result is still in an indeterminate form after we apply L\u2019H\xf4pital\u2019s Rule and evaluate the limit, we repeat the process.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital14a-h8","type":"hint","dependencies":["a343428l\'hopital14a-h7"],"title":"Indeterminate form","text":"Substitute $$x$$ with the value $$x$$ approaches to. If the function results in one of these indeterminate form: $$\\\\frac{0}{0}$$, $$\\\\frac{\\\\infty}{\\\\infty}$$ or $$\\\\frac{-\\\\infty}{\\\\infty}$$, we apply L\u2019H\xf4pital\u2019s Rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital14a-h9","type":"hint","dependencies":["a343428l\'hopital14a-h8"],"title":"Substitute the limit","text":"$$\\\\frac{2\\\\infty}{e^{\\\\infty}}=\\\\frac{\\\\infty}{\\\\infty}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital14a-h10","type":"hint","dependencies":["a343428l\'hopital14a-h9"],"title":"Apply L\u2019H\xf4pital\u2019s Rule","text":"Differentiate the denominator and the numerator seperatedly: $$\\\\lim_{x\\\\to\\\\infty} \\\\frac{\\\\frac{d}{dx} 2x}{\\\\frac{d}{dx} e^x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital14a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{2}{e^x}$$"],"dependencies":["a343428l\'hopital14a-h10"],"title":"Find the derivative","text":"What is the answer of the function after L\u2019H\xf4pital\u2019s Rule is applied?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{2x}{e^x}$$","$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{2}{{xe}^{x-1}}$$","$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{2}{x e}$$","$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{2}{e^x}$$"]},{"id":"a343428l\'hopital14a-h12","type":"hint","dependencies":["a343428l\'hopital14a-h11"],"title":"Evaluate the limit","text":"$$\\\\frac{2}{e^{\\\\infty}}=\\\\frac{2}{\\\\infty}=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital14a-h13","type":"hint","dependencies":["a343428l\'hopital14a-h12"],"title":"Conclusion","text":"As the result is $$\\\\infty$$, we can conclude that the function in the denominator which is $$e^x$$ is greater and grows faster.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a343428l\'hopital15","title":"For the following exercises, evaluate the limits with either L\u2019H\xf4pital\u2019s rule or previously learned methods.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.8 L\u2019H\xf4pital\u2019s Rule","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a343428l\'hopital15a","stepAnswer":["$$\\\\frac{1}{2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\lim_{x\\\\to0} \\\\frac{e^x-x-1}{x^2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{2}$$","hints":{"DefaultPathway":[{"id":"a343428l\'hopital15a-h1","type":"hint","dependencies":[],"title":"Indeterminate form","text":"Substitute $$x$$ with the value $$x$$ approaches to. If the function results in one of these indeterminate form: $$\\\\frac{0}{0}$$, $$\\\\frac{\\\\infty}{\\\\infty}$$ or $$\\\\frac{-\\\\infty}{\\\\infty}$$, we apply L\u2019H\xf4pital\u2019s Rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital15a-h2","type":"hint","dependencies":["a343428l\'hopital15a-h1"],"title":"Substitute the limit","text":"$$\\\\frac{e^0-0-1}{0^2}=\\\\frac{1-0-1}{0}=\\\\frac{0}{0}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital15a-h3","type":"hint","dependencies":["a343428l\'hopital15a-h2"],"title":"Apply L\u2019H\xf4pital\u2019s Rule","text":"Differentiate the denominator and the numerator seperatedly: $$\\\\lim_{x\\\\to0} \\\\frac{\\\\frac{d}{dx} \\\\left(e^x-x-1\\\\right)}{\\\\frac{d}{dx} x^2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital15a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{x\\\\to0} \\\\frac{e^x-1}{2} x$$"],"dependencies":["a343428l\'hopital15a-h3"],"title":"Find the derivative","text":"What is the answer of the function after L\u2019H\xf4pital\u2019s Rule is applied?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\lim_{x\\\\to0} \\\\frac{e^x}{2} x$$","$$\\\\lim_{x\\\\to0} \\\\frac{x e^x-1}{2} x$$","$$\\\\lim_{x\\\\to0} \\\\frac{e^x-1}{2} x$$","$$\\\\lim_{x\\\\to0} \\\\frac{e^x-1}{2}$$"]},{"id":"a343428l\'hopital15a-h5","type":"hint","dependencies":["a343428l\'hopital15a-h4"],"title":"Evaluate the limit","text":"$$0\\\\frac{e^0-1}{2}=\\\\frac{1-1}{0}=\\\\frac{0}{0}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital15a-h6","type":"hint","dependencies":["a343428l\'hopital15a-h5"],"title":"Indeterminate forms in answers","text":"If the result is still in an indeterminate form after we apply L\u2019H\xf4pital\u2019s Rule and evaluate the limit, we repeat the process.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital15a-h7","type":"hint","dependencies":["a343428l\'hopital15a-h6"],"title":"Apply L\u2019H\xf4pital\u2019s Rule again","text":"Differentiate the denominator and the numerator seperatedly: /lim{x,0,d/dx((e**(x))-1))/d/dx(2*x)}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital15a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{x\\\\to0} \\\\frac{e^x}{2}$$"],"dependencies":["a343428l\'hopital15a-h7"],"title":"Find the derivative","text":"What is the answer of the function after L\u2019H\xf4pital\u2019s Rule is applied?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\lim_{x\\\\to0} \\\\frac{e^x}{2}$$","$$\\\\lim_{x\\\\to0} 2\\\\left(e^x\\\\right)$$","$$\\\\lim_{x\\\\to0} \\\\frac{x e^x}{2}$$","$$\\\\lim_{x\\\\to0} \\\\frac{x e^{x-1}}{2}$$"]},{"id":"a343428l\'hopital15a-h9","type":"hint","dependencies":["a343428l\'hopital15a-h8"],"title":"Evaluate the limit","text":"$$\\\\frac{e^0}{2}=\\\\frac{1}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a343428l\'hopital16","title":"For the following exercises, evaluate the limit.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.8 L\u2019H\xf4pital\u2019s Rule","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a343428l\'hopital16a","stepAnswer":["$$\\\\infty$$"],"problemType":"MultipleChoice","stepTitle":"Evaluate the limit $$\\\\lim_{x\\\\to\\\\infty} \\\\frac{e^x}{x^k}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\infty$$","choices":["$$\\\\infty$$","$$0$$"],"hints":{"DefaultPathway":[{"id":"a343428l\'hopital16a-h1","type":"hint","dependencies":[],"title":"Indeterminate form","text":"Substitute $$x$$ with the value $$x$$ approaches to. If the function results in one of these indeterminate form: $$\\\\frac{0}{0}$$, $$\\\\frac{\\\\infty}{\\\\infty}$$ or $$\\\\frac{-\\\\infty}{\\\\infty}$$, we apply L\u2019H\xf4pital\u2019s Rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital16a-h2","type":"hint","dependencies":["a343428l\'hopital16a-h1"],"title":"Substitute the limit","text":"$$\\\\frac{e^{\\\\infty}}{{\\\\infty}^k}=\\\\frac{\\\\infty}{\\\\infty}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital16a-h3","type":"hint","dependencies":["a343428l\'hopital16a-h2"],"title":"Apply L\u2019H\xf4pital\u2019s Rule","text":"Differentiate the denominator and the numerator seperatedly: $$\\\\lim_{x\\\\to\\\\infty} \\\\frac{\\\\frac{d}{dx} e^x}{\\\\frac{d}{dx} x^k}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital16a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{e^x}{k x^{k-1}}$$"],"dependencies":["a343428l\'hopital16a-h3"],"title":"Find the derivative","text":"What is the answer of the function after L\u2019H\xf4pital\u2019s Rule is applied?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{x e^x}{k}$$","/lim{x,inf,(x*(e**(x)))/(k*x))}","$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{e^x}{k x}$$","$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{e^x}{k x^{k-1}}$$"]},{"id":"a343428l\'hopital16a-h5","type":"hint","dependencies":["a343428l\'hopital16a-h4"],"title":"Rewrite the limit","text":"/lim{x,inf,(e**(x))/(x*(k!))}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"a343428l\'hopital16a-h4-s1","type":"hint","dependencies":[],"title":"Factorial Formula","text":"Take a look at the denominator, if we keep applying the L\u2019H\xf4pital\u2019s Rule $$infinitely$$, we will obtain the general form of factorial: $$x k \\\\left(k-1\\\\right) \\\\neq k \\\\left(k-1\\\\right) \\\\neq k!$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital16a-h4-s2","type":"hint","dependencies":["a343428l\'hopital16a-h4-s1"],"title":"Explanation","text":"No matter how many time we apply the L\u2019H\xf4pital\u2019s Rule, the limit will always have the form $$\\\\frac{\\\\infty}{\\\\infty}$$. The derivative of $$e^x$$ on the numerator will always be the same. Meanwhile, as we apply the L\u2019H\xf4pital\u2019s Rule successively, the function in the denominator will behave: $$k x^{k-1}$$, $$k \\\\left(k-1\\\\right) x^{k-2}$$, k*(k-1)*(k-2)*x**(k-3),...,k!x,k!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"a343428l\'hopital16a-h6","type":"hint","dependencies":["a343428l\'hopital16a-h5"],"title":"Evaluate the limit","text":"(e**(inf))/(k!)=inf/k!=inf","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a343428l\'hopital17","title":"For the following exercises, evaluate the limits with either L\u2019H\xf4pital\u2019s rule or previously learned methods.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.8 L\u2019H\xf4pital\u2019s Rule","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a343428l\'hopital17a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"$$\\\\lim_{x\\\\to0} \\\\frac{tanx}{\\\\sqrt{x}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a343428l\'hopital17a-h1","type":"hint","dependencies":[],"title":"Indeterminate form","text":"Substitute $$x$$ with the value $$x$$ approaches to. If the function results in one of these indeterminate form: $$\\\\frac{0}{0}$$, $$\\\\frac{\\\\infty}{\\\\infty}$$ or $$\\\\frac{-\\\\infty}{\\\\infty}$$, we apply L\u2019H\xf4pital\u2019s Rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital17a-h2","type":"hint","dependencies":["a343428l\'hopital17a-h1"],"title":"Substitute the limit","text":"$$\\\\fractan^\\\\\\\\left(0\\\\right)sqrt{0}}=\\\\frac{0}{0}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital17a-h3","type":"hint","dependencies":["a343428l\'hopital17a-h2"],"title":"Apply L\u2019H\xf4pital\u2019s Rule","text":"Differentiate the denominator and the numerator seperatedly: /lim{x,0**+,((d/dx)*tan(x))/((d/dx)*sqrt(x))}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital17a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["/lim{x,0**+,(sec**2(x))/(1/(2*sqrt(x)))}"],"dependencies":["a343428l\'hopital17a-h3"],"title":"Find the derivative","text":"What is the answer of the function after L\u2019H\xf4pital\u2019s Rule is applied?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["/lim{x,0**+,(sec**2(x))/(1/sqrt(x))}","/lim{x,0**+,sec(x)/(1/sqrt(x))}","/lim{x,0**+,(sec**2(x))/-(1/sqrt(x))}","/lim{x,0**+,(sec**2(x))/(1/(2*sqrt(x)))}"]},{"id":"a343428l\'hopital17a-h5","type":"hint","dependencies":["a343428l\'hopital17a-h4"],"title":"Rearrange the expression","text":"/lim{x,0**+,(sec**2(x))/(1/(2*sqrt(x)))}=/lim{x,0**+,(sec**2(x))*2*sqrt(x)}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital17a-h6","type":"hint","dependencies":["a343428l\'hopital17a-h5"],"title":"Evaluate the limit","text":"$$2{sec}^{2\\\\left(0\\\\right)} \\\\sqrt{0}=0\\\\times2\\\\times0=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a343428l\'hopital19","title":"Evaluate the limit","body":"For the following exercises, evaluate the limit.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.8 L\u2019H\xf4pital\u2019s Rule","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a343428l\'hopital19a","stepAnswer":["$$\\\\frac{-1}{2}$$"],"problemType":"TextBox","stepTitle":"Evaluate $$\\\\lim_{x\\\\to0} \\\\frac{sinx-tan\\\\left(x\\\\right)}{x^3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-1}{2}$$","hints":{"DefaultPathway":[{"id":"a343428l\'hopital19a-h1","type":"hint","dependencies":[],"title":"Indeterminate form","text":"Substitute $$x$$ with the value $$x$$ approaches to. If the function results in one of these indeterminate form: $$\\\\frac{0}{0}$$, $$\\\\frac{\\\\infty}{\\\\infty}$$ or $$\\\\frac{-\\\\infty}{\\\\infty}$$, we apply L\u2019H\xf4pital\u2019s Rule.","variabilization":{},"oer":"","license":""},{"id":"a343428l\'hopital19a-h2","type":"hint","dependencies":["a343428l\'hopital19a-h1"],"title":"Substitute the limit","text":"(sin(0)-tan(0))/(0)**3)}=(0-0)/0=0/0","variabilization":{},"oer":"","license":""},{"id":"a343428l\'hopital19a-h3","type":"hint","dependencies":["a343428l\'hopital19a-h2"],"title":"Apply L\u2019H\xf4pital\u2019s Rule","text":"Differentiate the denominator and the numerator seperatedly: $$\\\\lim_{x\\\\to0} \\\\frac{\\\\frac{d}{dx} \\\\left(sinx-tan\\\\left(x\\\\right)\\\\right)}{\\\\frac{d}{dx} x^3}$$","variabilization":{},"oer":"","license":""},{"id":"a343428l\'hopital19a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{x\\\\to0} \\\\frac{cosx-{sec}^{2\\\\left(x\\\\right)}}{3x^2}$$"],"dependencies":["a343428l\'hopital19a-h3"],"title":"Find the derivative","text":"What is the answer of the function after L\u2019H\xf4pital\u2019s Rule is applied?","variabilization":{},"oer":"","license":"","choices":["$$\\\\lim_{x\\\\to0} \\\\frac{\\\\left(-cosx-{sec}^{2\\\\left(x\\\\right)}\\\\right)}{3x^2}$$","$$\\\\lim_{x\\\\to0} \\\\frac{cosx-\\\\operatorname{sec}\\\\left(x\\\\right)}{3x^2}$$","$$\\\\lim_{x\\\\to0} \\\\frac{cosx+{sec}^{2\\\\left(x\\\\right)}}{3x^2}$$","$$\\\\lim_{x\\\\to0} \\\\frac{cosx-{sec}^{2\\\\left(x\\\\right)}}{3x^2}$$"],"subHints":[{"id":"a343428l\'hopital19a-h4-s1","type":"hint","dependencies":[],"title":"Indeterminate forms in answers","text":"If the result is still in an indeterminate form after we apply L\u2019H\xf4pital\u2019s Rule and evaluate the limit, we repeat the process.","variabilization":{},"oer":"","license":""}]},{"id":"a343428l\'hopital19a-h5","type":"hint","dependencies":["a343428l\'hopital19a-h4"],"title":"Evaluate the limit","text":"$$\\\\fraccos^3\\\\left(0\\\\right)-{sec}^{2\\\\left(0\\\\right)}\\\\times0^2}=\\\\frac{1-1}{0}=\\\\frac{0}{0}$$","variabilization":{},"oer":"","license":""},{"id":"a343428l\'hopital19a-h6","type":"hint","dependencies":["a343428l\'hopital19a-h5"],"title":"Apply L\u2019H\xf4pital\u2019s Rule again","text":"Differentiate the denominator and the numerator seperatedly: $$\\\\lim_{x\\\\to0} \\\\frac{\\\\frac{\\\\frac{d}{\\\\operatorname{dx}\\\\left(cosx-{sec}^{2\\\\left(x\\\\right)}\\\\right)}}{d}}{\\\\operatorname{dx}\\\\left(3x^2\\\\right)}$$","variabilization":{},"oer":"","license":""},{"id":"a343428l\'hopital19a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{x\\\\to0} \\\\frac{\\\\left(-sinx-2{sec}^{2\\\\left(x\\\\right)} tan\\\\left(x\\\\right)\\\\right)}{6x}$$"],"dependencies":["a343428l\'hopital19a-h6"],"title":"Find the derivative","text":"What is the answer of the function after L\u2019H\xf4pital\u2019s Rule is applied?","variabilization":{},"oer":"","license":"","choices":["$$\\\\lim_{x\\\\to0} \\\\frac{sinx-2{sec}^{2\\\\left(x\\\\right)} {tan}^{2\\\\left(x\\\\right)}}{6x}$$","$$\\\\lim_{x\\\\to0} \\\\frac{\\\\left(-sinx-2{sec}^{2\\\\left(x\\\\right)} tan\\\\left(x\\\\right)\\\\right)}{6x}$$","$$\\\\lim_{x\\\\to0} \\\\frac{\\\\left(-sinx+2\\\\operatorname{sec}\\\\left(x\\\\right) tan\\\\left(x\\\\right)\\\\right)}{6x}$$","/lim{x,0,(-sinx-2*tan(x)))/(6*x)}"],"subHints":[{"id":"a343428l\'hopital19a-h7-s1","type":"hint","dependencies":[],"title":"Indeterminate forms in answers","text":"If the result is still in an indeterminate form after we apply L\u2019H\xf4pital\u2019s Rule and evaluate the limit, we repeat the process.","variabilization":{},"oer":"","license":""},{"id":"a343428l\'hopital19a-h7-s2","type":"hint","dependencies":["a343428l\'hopital19a-h7-s1"],"title":"Apply L\u2019H\xf4pital\u2019s Rule multiple times","text":"You can keep applying L\'Hospital\'s Rule as long as there is an indeterminate form in the answers. If the problem is out of the indeterminate forms, you will not be able to apply L\'Hospital\'s Rule anymore.","variabilization":{},"oer":"","license":""}]},{"id":"a343428l\'hopital19a-h8","type":"hint","dependencies":["a343428l\'hopital19a-h7"],"title":"Evaluate the limit","text":"$$\\\\frac{\\\\left(-sin\\\\left(0\\\\right)-2{sec}^{2\\\\left(0\\\\right)} tan\\\\left(0\\\\right)\\\\right)}{6\\\\times0}=\\\\frac{0-2\\\\times0}{0}=\\\\frac{0}{0}$$","variabilization":{},"oer":"","license":""},{"id":"a343428l\'hopital19a-h9","type":"hint","dependencies":["a343428l\'hopital19a-h8"],"title":"Apply L\u2019H\xf4pital\u2019s Rule again","text":"Differentiate the denominator and the numerator seperatedly: $$\\\\lim_{x\\\\to0} \\\\frac{\\\\frac{\\\\frac{d}{dx\\\\left(-sinx-2{sec}^{2\\\\left(x\\\\right)} tan\\\\left(x\\\\right)\\\\right)}}{d}}{\\\\operatorname{dx}\\\\left(6x\\\\right)}$$","variabilization":{},"oer":"","license":""},{"id":"a343428l\'hopital19a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{x\\\\to0} \\\\frac{\\\\left(-cosx-2\\\\left(2{sec}^{2\\\\left(x\\\\right)} {tan}^{2\\\\left(x\\\\right)}+{sec}^{4\\\\left(x\\\\right)}\\\\right)\\\\right)}{6}$$"],"dependencies":["a343428l\'hopital19a-h9"],"title":"Find the derivative","text":"What is the answer of the function after L\u2019H\xf4pital\u2019s Rule is applied?","variabilization":{},"oer":"","license":"","choices":["$$\\\\lim_{x\\\\to0} \\\\frac{\\\\left(-cosx-2\\\\left(\\\\operatorname{sec}\\\\left(x\\\\right) tan\\\\left(x\\\\right)+2{sec}^{4\\\\left(x\\\\right)}\\\\right)\\\\right)}{6}$$","$$\\\\lim_{x\\\\to0} \\\\frac{\\\\left(-cosx-2\\\\left({sec}^{2\\\\left(x\\\\right)} {tan}^{2\\\\left(x\\\\right)}+2{sec}^{4\\\\left(x\\\\right)}\\\\right)\\\\right)}{6}$$","$$\\\\lim_{x\\\\to0} \\\\frac{\\\\left(-cosx-2\\\\left({sec}^{2\\\\left(x\\\\right)} {tan}^{2\\\\left(x\\\\right)}+{sec}^{2\\\\left(x\\\\right)}\\\\right)\\\\right)}{6}$$","$$\\\\lim_{x\\\\to0} \\\\frac{\\\\left(-cosx-2\\\\left(2{sec}^{2\\\\left(x\\\\right)} {tan}^{2\\\\left(x\\\\right)}+{sec}^{4\\\\left(x\\\\right)}\\\\right)\\\\right)}{6}$$"]},{"id":"a343428l\'hopital19a-h11","type":"hint","dependencies":["a343428l\'hopital19a-h10"],"title":"Evaluate the limit","text":"$$\\\\frac{\\\\left(-cosx-2\\\\left(2{sec}^{2\\\\left(x\\\\right)} {tan}^{2\\\\left(x\\\\right)}+{sec}^{4\\\\left(x\\\\right)}\\\\right)\\\\right)}{6}=\\\\frac{\\\\left(-1-2\\\\left(0+1\\\\right)\\\\right)}{6}=\\\\frac{\\\\left(-1-2\\\\right)}{6}=\\\\frac{-1}{2}$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"a343428l\'hopital2","title":"Indeterminate form of Type $$0^0$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.8 L\u2019H\xf4pital\u2019s Rule","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a343428l\'hopital2a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"Evaluate $$\\\\lim_{x\\\\to0} x^x$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a343428l\'hopital2a-h1","type":"hint","dependencies":[],"title":"Operation","text":"Let $$y=x^x$$. Using one-to-one property of logarithms, we obtain $$ln(y)=\\\\ln(x^x)$$. According to the Power Properties of Logarithms, the expression can be written as $$ln(y)=x \\\\ln(x)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital2a-h2","type":"hint","dependencies":["a343428l\'hopital2a-h1"],"title":"Operation","text":"If we immediately evaluate the limit of the given function, the result would be $$0^0$$ which is not the indeterminate form we need in order to apply L\u2019H\xf4pital\u2019s Rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital2a-h3","type":"hint","dependencies":["a343428l\'hopital2a-h2"],"title":"Operation","text":"We set up a new limit based on the function we just created /lim{x,0**+,x*ln(x)}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital2a-h4","type":"hint","dependencies":["a343428l\'hopital2a-h3"],"title":"Indeterminate form","text":"Substitute $$x$$ with the value $$x$$ approaches to. If the function results in one of these indeterminate form: $$\\\\frac{0}{0}$$, $$\\\\frac{\\\\infty}{\\\\infty}$$ or $$\\\\frac{-\\\\infty}{\\\\infty}$$, we apply L\u2019H\xf4pital\u2019s Rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital2a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["/lim{x,0**+,ln(x)/(1/x)}"],"dependencies":["a343428l\'hopital2a-h4"],"title":"Rearrange the expression","text":"The purpose of rearranging an expression is to make it fit the indeterminate form and from there we can apply L\u2019H\xf4pital\u2019s Rule. Using the Negative Exponents Law, how can we rearrange the new expression so that it could fit the indeterminate form?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["/lim{x,0**+,ln(x)/(1/x)}","/lim{x,0**+,x/(1/ln(x))}"],"subHints":[{"id":"a343428l\'hopital2a-h5-s1","type":"hint","dependencies":[],"title":"Rearrange the expression","text":"/lim{x,0**+,ln(x)/x**(-1)}=/lim{x,0**+,ln(x)/(1/x)}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital2a-h5-s2","type":"hint","dependencies":["a343428l\'hopital2a-h5-s1"],"title":"Rearrange the expression","text":"Note that the other way of rearrangement does not give us a proper indeterminate form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"a343428l\'hopital2a-h6","type":"hint","dependencies":["a343428l\'hopital2a-h5"],"title":"Substitute the limit","text":"$$\\\\frac{\\\\ln(0)}{\\\\frac{1}{0}}=\\\\frac{-\\\\infty}{\\\\infty}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital2a-h7","type":"hint","dependencies":["a343428l\'hopital2a-h6"],"title":"Apply L\u2019H\xf4pital\u2019s Rule","text":"Differentiate the denominator and the numerator seperatedly: /lim{x,0**+,d/dx(ln(x))/d/dx(1/x)}=/lim{x,0**+,(1/x)/-(1/x**2)}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital2a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["/lim{x,0**+,(1/x)/-(1/x**2)}"],"dependencies":["a343428l\'hopital2a-h7"],"title":"Find the derivative","text":"What is the answer of the function after L\u2019H\xf4pital\u2019s Rule is applied?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["/lim{x,0**+,x/-(1/x**2)}","/lim{x,0**+,(1/x)/(1/x**2)}","/lim{x,0**+,(x**2/x)/-(1/x**2)}","/lim{x,0**+,(1/x)/-(1/x**2)}"]},{"id":"a343428l\'hopital2a-h9","type":"hint","dependencies":["a343428l\'hopital2a-h8"],"title":"Rearrange the expression","text":"/lim{x,0**+,(1/x)/-(1/x**2)}=/lim{x,0**+,-x}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital2a-h10","type":"hint","dependencies":["a343428l\'hopital2a-h9"],"title":"Evaluate the limit","text":"$$-(0)=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital2a-h11","type":"hint","dependencies":["a343428l\'hopital2a-h1","a343428l\'hopital2a-h10"],"title":"Operation","text":"We set $$ln(y)=x \\\\ln(x)$$ in the very first step therefore we can obtain /lim{x,0**+,x*ln(x)}=/lim{x,0**+,ln(y)}=0.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital2a-h12","type":"hint","dependencies":["a343428l\'hopital2a-h11"],"title":"Limit of Logarithm Rule","text":"/lim{x,0**+,ln(y)}=ln(/lim{x,0**+,(y)})=0","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital2a-h13","type":"hint","dependencies":["a343428l\'hopital2a-h12"],"title":"Properties of Exponents and Natural Logarithms","text":"ln(/lim{x,0**+,(y)})=/lim{x,0**+,(y)}=e**0=1","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital2a-h14","type":"hint","dependencies":["a343428l\'hopital2a-h1","a343428l\'hopital2a-h13"],"title":"Conclusion","text":"$$/lim{x,0,(y)=\\\\lim_{x\\\\to0} x^x=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a343428l\'hopital20","title":"For the following exercises, evaluate the limits with either L\u2019H\xf4pital\u2019s rule or previously learned methods.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.8 L\u2019H\xf4pital\u2019s Rule","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a343428l\'hopital20a","stepAnswer":["$$ln(3)-ln(2)$$"],"problemType":"TextBox","stepTitle":"$$\\\\lim_{x\\\\to0} \\\\frac{3^x-2^x}{x}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$ln(3)-ln(2)$$","hints":{"DefaultPathway":[{"id":"a343428l\'hopital20a-h1","type":"hint","dependencies":[],"title":"Indeterminate form","text":"Substitute $$x$$ with the value $$x$$ approaches to. If the function results in one of these indeterminate form: $$\\\\frac{0}{0}$$, $$\\\\frac{\\\\infty}{\\\\infty}$$ or $$\\\\frac{-\\\\infty}{\\\\infty}$$, we apply L\u2019H\xf4pital\u2019s Rule.","variabilization":{},"oer":"","license":""},{"id":"a343428l\'hopital20a-h2","type":"hint","dependencies":["a343428l\'hopital20a-h1"],"title":"Substitute the limit","text":"$$3^0-\\\\frac{2^0}{0}=\\\\frac{1-1}{0}=\\\\frac{0}{0}$$","variabilization":{},"oer":"","license":""},{"id":"a343428l\'hopital20a-h3","type":"hint","dependencies":["a343428l\'hopital20a-h2"],"title":"Apply L\u2019H\xf4pital\u2019s Rule","text":"Differentiate the denominator and the numerator seperatedly: $$\\\\lim_{x\\\\to0} \\\\frac{\\\\frac{d}{dx} \\\\left(3^x-2^x\\\\right)}{\\\\frac{d}{dx} x}$$","variabilization":{},"oer":"","license":""},{"id":"a343428l\'hopital20a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{x\\\\to0} \\\\frac{\\\\ln(3) 3^x-\\\\ln(2) 2^x}{1}$$"],"dependencies":["a343428l\'hopital20a-h3"],"title":"Find the derivative","text":"What is the answer of the function after L\u2019H\xf4pital\u2019s Rule is applied?","variabilization":{},"oer":"","license":"","choices":["$$\\\\lim_{x\\\\to0} \\\\frac{\\\\ln(3)-\\\\ln(2)}{1}$$","$$\\\\lim_{x\\\\to0} \\\\frac{\\\\ln(3) 3^x-\\\\ln(2) 2^x}{1}$$","$$\\\\lim_{x\\\\to0} \\\\frac{3^x-2^x}{1}$$","$$\\\\lim_{x\\\\to0} \\\\frac{\\\\ln(x) 3^x-\\\\ln(x) 2^x}{1}$$"]},{"id":"a343428l\'hopital20a-h5","type":"hint","dependencies":["a343428l\'hopital20a-h4"],"title":"Evaluate the limit","text":"$$\\\\frac{\\\\ln(3) 3^x-\\\\ln(2) 2^x}{1}=\\\\ln(3) 3^0-\\\\ln(2) 2^0=1\\\\ln(3)-1\\\\ln(2)=ln(3)-ln(2)$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"a343428l\'hopital21","title":"For the following exercises, evaluate the limit.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.8 L\u2019H\xf4pital\u2019s Rule","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a343428l\'hopital21a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"Evaluate $$\\\\lim_{x\\\\to\\\\infty} x sin\\\\left(\\\\frac{1}{x}\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a343428l\'hopital21a-h1","type":"hint","dependencies":[],"title":"Indeterminate forms in answers","text":"If the result is still in an indeterminate form after we apply L\u2019H\xf4pital\u2019s Rule and evaluate the limit, we repeat the process.","variabilization":{},"oer":"","license":""},{"id":"a343428l\'hopital21a-h2","type":"hint","dependencies":["a343428l\'hopital21a-h1"],"title":"Rearrange the expression","text":"$$\\\\lim_{x\\\\to\\\\infty} \\\\fracsin^x\\\\left(\\\\frac{1}{x}\\\\right)^{\\\\left(-1\\\\right)}}=\\\\lim_{x\\\\to\\\\infty} \\\\fracsin^\\\\\\\\left(\\\\frac{1}{x}\\\\right)frac{1}{x}}$$","variabilization":{},"oer":"","license":""},{"id":"a343428l\'hopital21a-h3","type":"hint","dependencies":["a343428l\'hopital21a-h2"],"title":"Substitute the limit","text":"$$\\\\fracsin^\\\\\\\\left(\\\\frac{1}{\\\\infty}\\\\right)frac{1}{\\\\infty}}=\\\\fracsin^0\\\\left(0\\\\right)}=\\\\frac{0}{0}$$","variabilization":{},"oer":"","license":""},{"id":"a343428l\'hopital21a-h4","type":"hint","dependencies":["a343428l\'hopital21a-h3"],"title":"Apply L\u2019H\xf4pital\u2019s Rule again","text":"Differentiate the denominator and the numerator seperatedly: $$\\\\lim_{x\\\\to\\\\infty} \\\\frac{\\\\frac{d}{dx} sin\\\\left(\\\\frac{1}{x}\\\\right)}{\\\\frac{d}{dx} \\\\frac{1}{x}}$$","variabilization":{},"oer":"","license":""},{"id":"a343428l\'hopital21a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["/lim{x,inf,((-1/x**2)*cos(1/x)))/(-1/x**2))}"],"dependencies":["a343428l\'hopital21a-h4"],"title":"Find the derivative","text":"What is the answer of the function after L\u2019H\xf4pital\u2019s Rule is applied?","variabilization":{},"oer":"","license":"","choices":["/lim{x,inf,((-1/x**2)*cos(1/x)))/(-1/x**2))}","/lim{x,inf,((1/x**2)*sin(1/x))/(1/x**2))}","/lim{x,inf,(cos(1/x))/(-1/x**2))}","$$\\\\lim_{x\\\\to\\\\infty} \\\\left(-\\\\frac{1}{x^2}\\\\right) cos\\\\left(\\\\frac{1}{x}\\\\right)$$"],"subHints":[{"id":"a343428l\'hopital21a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["((-1/x**2)*cos(1/x)))"],"dependencies":[],"title":"Chain Rule","text":"What is the derivative of $$sin\\\\left(\\\\frac{1}{x}\\\\right)$$?","variabilization":{},"oer":"","license":""}]},{"id":"a343428l\'hopital21a-h6","type":"hint","dependencies":["a343428l\'hopital21a-h5"],"title":"Simplify","text":"$$\\\\frac{\\\\left(-\\\\frac{1}{x^2}\\\\right) cos\\\\left(\\\\frac{1}{x}\\\\right)}{\\\\left(-\\\\frac{1}{x^2}\\\\right)}=cos\\\\left(\\\\frac{1}{x}\\\\right)$$","variabilization":{},"oer":"","license":""},{"id":"a343428l\'hopital21a-h7","type":"hint","dependencies":["a343428l\'hopital21a-h6"],"title":"Evaluate the limit","text":"$$cos\\\\left(\\\\frac{1}{\\\\infty}\\\\right)=cos(0)=1$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"a343428l\'hopital3","title":"Evaluate","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.8 L\u2019H\xf4pital\u2019s Rule","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a343428l\'hopital3a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{lnx}{5} x$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a343428l\'hopital3a-h1","type":"hint","dependencies":[],"title":"Indeterminate form","text":"Substitute $$x$$ with the value $$x$$ approaches to. If the function results in one of these indeterminate form: $$\\\\frac{0}{0}$$, $$\\\\frac{\\\\infty}{\\\\infty}$$ or $$\\\\frac{-\\\\infty}{\\\\infty}$$, we apply L\u2019H\xf4pital\u2019s Rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital3a-h2","type":"hint","dependencies":["a343428l\'hopital3a-h1"],"title":"Substitute the limit","text":"$$\\\\frac{\\\\ln(\\\\infty)}{5} \\\\infty=\\\\frac{\\\\infty}{\\\\infty}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital3a-h3","type":"hint","dependencies":["a343428l\'hopital3a-h2"],"title":"Apply L\u2019H\xf4pital\u2019s Rule","text":"Differentiate the denominator and the numerator seperatedly: $$\\\\lim_{x\\\\to\\\\infty} \\\\frac{\\\\frac{d}{dx} lnx}{\\\\frac{d}{dx} 5x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital3a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{1}{5x}$$"],"dependencies":["a343428l\'hopital3a-h3"],"title":"Find the derivative","text":"What is the answer of the function after L\u2019H\xf4pital\u2019s Rule is applied?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{1}{5x}$$","$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{5}{x}$$","$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{1}{x}$$","$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{x}{5}$$"]},{"id":"a343428l\'hopital3a-h5","type":"hint","dependencies":["a343428l\'hopital3a-h4"],"title":"Evaluate the limit","text":"$$\\\\frac{1}{5\\\\infty}=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a343428l\'hopital4","title":"Evaluate","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.8 L\u2019H\xf4pital\u2019s Rule","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a343428l\'hopital4a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"$$\\\\lim_{x\\\\to0} \\\\frac{x}{tanx}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a343428l\'hopital4a-h1","type":"hint","dependencies":[],"title":"Indeterminate form","text":"Substitute $$x$$ with the value $$x$$ approaches to. If the function results in one of these indeterminate form: $$\\\\frac{0}{0}$$, $$\\\\frac{\\\\infty}{\\\\infty}$$ or $$\\\\frac{-\\\\infty}{\\\\infty}$$, we apply L\u2019H\xf4pital\u2019s Rule.","variabilization":{},"oer":"","license":""},{"id":"a343428l\'hopital4a-h2","type":"hint","dependencies":["a343428l\'hopital4a-h1"],"title":"Substitute the limit","text":"$$\\\\frac{0}tan^$\\\\left(0\\\\righ=\\\\frac{0}{0}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital4a-h3","type":"hint","dependencies":["a343428l\'hopital4a-h2"],"title":"Apply L\u2019H\xf4pital\u2019s Rule","text":"Differentiate the denominator and the numerator seperatedly: $$\\\\lim_{x\\\\to0} \\\\frac{\\\\frac{d}{dx} x}{\\\\frac{d}{dx} tanx}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital4a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{x\\\\to0} \\\\frac{1}{{sec}^{2\\\\left(x\\\\right)}}$$"],"dependencies":["a343428l\'hopital4a-h3"],"title":"Find the derivative","text":"What is the answer of the function after L\u2019H\xf4pital\u2019s Rule is applied?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\lim_{x\\\\to0} \\\\frac{1}{\\\\operatorname{sec}\\\\left(x\\\\right)}$$","$$\\\\lim_{x\\\\to0} \\\\frac{1}{{sec}^{2\\\\left(x\\\\right)}}$$","$$\\\\lim_{x\\\\to0} \\\\frac{1}{secx tanx}$$","$$\\\\lim_{x\\\\to0} \\\\frac{1}{{cos}^{2\\\\left(x\\\\right)}}$$"]},{"id":"a343428l\'hopital4a-h5","type":"hint","dependencies":["a343428l\'hopital4a-h4"],"title":"Evaluate the limit","text":"$$\\\\frac{1}{{sec}^{2\\\\left(0\\\\right)}}=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a343428l\'hopital5","title":"Evaluate each of the following limits by applying L\u2019H\xf4pital\u2019s rule","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.8 L\u2019H\xf4pital\u2019s Rule","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a343428l\'hopital5a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"$$\\\\lim_{x\\\\to0} \\\\frac{lnx}{cotx}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a343428l\'hopital5a-h1","type":"hint","dependencies":[],"title":"Indeterminate form","text":"Substitute $$x$$ with the value $$x$$ approaches to. If the function results in one of these indeterminate form: $$\\\\frac{0}{0}$$, $$\\\\frac{\\\\infty}{\\\\infty}$$ or $$\\\\frac{-\\\\infty}{\\\\infty}$$, we apply L\u2019H\xf4pital\u2019s Rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital5a-h2","type":"hint","dependencies":["a343428l\'hopital5a-h1"],"title":"Substitute the limit","text":"As $$x$$ approaches $$0$$ from the right side of the $$y$$ axis, the graph of lnx approaches negative infinity and the graph of cotx approaches $$\\\\infty$$. Therefore, /lim{x,0**+,lnx/cotx}=-inf/inf.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital5a-h3","type":"hint","dependencies":["a343428l\'hopital5a-h2"],"title":"Apply L\u2019H\xf4pital\u2019s Rule","text":"Differentiate the denominator and the numerator seperatedly: /lim{x,0**+,d/dx(lnx)/d/dx(cotx)}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital5a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["/lim{x,0**+,1/-(x*csc**2(x))}"],"dependencies":["a343428l\'hopital5a-h3"],"title":"Find the derivative","text":"What is the answer of the function after L\u2019H\xf4pital\u2019s Rule is applied?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["/lim{x,0**+,1/(x*csc**2(x))}","/lim{x,0**+,1/-(x*csc**2(x))}","/lim{x,0**+,x/(-csc**2(x))}","/lim{x,0**+,-csc**2(x)/x}"],"subHints":[{"id":"a343428l\'hopital5a-h4-s1","type":"hint","dependencies":[],"title":"Evaluate the limit","text":"If we plug $$0$$ into $$x$$ in the denominator at this point, the first term is approaching zero meanwhile the second term is approaching a very large number. The expression would be written as: $$\\\\frac{1}{0} \\\\left(-\\\\infty\\\\right)$$. In such a case, we can not make any conclusion yet.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"a343428l\'hopital5a-h5","type":"hint","dependencies":["a343428l\'hopital5a-h4"],"title":"Rearrange the expression","text":"In order to make the expression easier to apply the L\u2019H\xf4pital\u2019s Rule a second time, we need to rearrange the expression using the definition of cscx. We then apply the apply L\u2019H\xf4pital\u2019s Rule and Evaluate the limit","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital5a-h6","type":"hint","dependencies":["a343428l\'hopital5a-h5"],"title":"Apply L\u2019H\xf4pital\u2019s Rule again","text":"Differentiate the denominator and the numerator seperatedly: /lim{x,0**+,d/dx(sin**2(x)/-x)/d/dx(-x)}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital5a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["/lim{x,0**+,2*sinx*cosx/(-1)}"],"dependencies":["a343428l\'hopital5a-h6"],"title":"Apply L\u2019H\xf4pital\u2019s Rule again","text":"What is the answer of the function after the L\u2019H\xf4pital\u2019s Rule and the Chain Rule are applied?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["/lim{x,0**+,(2*sinx*cosx)/(-1)}","/lim{x,0**+,(2*sin**2(x))/(-1)}","/lim{x,0**+,(2*sinx)/(-1)}","lim{x,0**+,(sinx*cos**2(x))/(-1)}"],"subHints":[{"id":"a343428l\'hopital5a-h7-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2sinx cosx$$"],"dependencies":[],"title":"Apply Chain Rule","text":"What is the derivative of $${sin}^{2\\\\left(x\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$2{sin}^{2\\\\left(x\\\\right)}$$","$$2sinx$$","$$2sinx cosx$$","$$sinx {cos}^{2\\\\left(x\\\\right)}$$"]}]},{"id":"a343428l\'hopital5a-h8","type":"hint","dependencies":["a343428l\'hopital5a-h7"],"title":"Evaluate the limit","text":"$$\\\\frac{2sin\\\\left(0\\\\right) cos\\\\left(0\\\\right)}{-1}=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a343428l\'hopital7","title":"Evaluate each of the following limits by applying L\u2019H\xf4pital\u2019s Rule.","body":"For the following exercises, evaluate the limits with either L\u2019H\xf4pital\u2019s Rule or previously learned methods.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.8 L\u2019H\xf4pital\u2019s Rule","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a343428l\'hopital7a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"$$\\\\lim_{x\\\\to0} x \\\\ln(x^4)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a343428l\'hopital7a-h1","type":"hint","dependencies":[],"title":"Indeterminate form","text":"Substitute $$x$$ with the value $$x$$ approaches to. If the function results in one of these indeterminate form: $$\\\\frac{0}{0}$$, $$\\\\frac{\\\\infty}{\\\\infty}$$ or $$\\\\frac{-\\\\infty}{\\\\infty}$$, we apply L\u2019H\xf4pital\u2019s Rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital7a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["/lim{x,0**+,ln(x**4)/(1/x)}"],"dependencies":[],"title":"Rearrange the expression","text":"The purpose of rearranging the expression is to make it fit the indeterminate form and from there we can apply L\u2019H\xf4pital\u2019s Rule. Using the Negative Exponents Law, how can we rearrange the expression so that it could fit the indeterminate form?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["/lim{x,0**+,ln(x**4)/(1/x)}","/lim{x,0**+,x/(1/ln(x**4))}"],"subHints":[{"id":"a343428l\'hopital7a-h2-s1","type":"hint","dependencies":[],"title":"Rearrange the expression","text":"/lim{x,0+,ln(x**4)/x**(-1)}=/lim{x,0+,ln(x**4)/(1/x)}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital7a-h2-s2","type":"hint","dependencies":["a343428l\'hopital7a-h2-s1"],"title":"Rearrange the expression","text":"The other way of rearrangement does not give us a proper indeterminate form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital7a-h2-s3","type":"hint","dependencies":["a343428l\'hopital7a-h2-s1"],"title":"Substitute the limit","text":"$$\\\\frac{\\\\ln(0^4)}{\\\\frac{1}{0}}=\\\\frac{\\\\ln(0)}{\\\\frac{1}{0}}=\\\\frac{-\\\\infty}{\\\\infty}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"a343428l\'hopital7a-h3","type":"hint","dependencies":["a343428l\'hopital7a-h2"],"title":"Apply L\u2019H\xf4pital\u2019s Rule","text":"Differentiate the denominator and the numerator seperatedly: /lim{x,0**+,d/dx(ln(x**4))/d/dx(1/x)}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital7a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["/lim{x,0**+,(4/x)/-(1/x**2)}"],"dependencies":["a343428l\'hopital7a-h3"],"title":"Find the derivative","text":"What is the answer of the function after L\u2019H\xf4pital\u2019s Rule is applied?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["/lim{x,0**+,(4*x)/-(1/x**2)}","/lim{x,0**+,(4/x)/(1/x**2)}","/lim{x,0**+,(4*x**2/x)/-(1/x**2)}","/lim{x,0**+,(4/x)/-(1/x**2)}"],"subHints":[{"id":"a343428l\'hopital7a-h4-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{4}{x}$$"],"dependencies":[],"title":"Chain Rule","text":"What is the derivative of $$\\\\ln(x^4)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\frac{4}{x}$$","$$4x^2$$","$$\\\\frac{1}{x^4}$$","$$4x$$"]},{"id":"a343428l\'hopital7a-h4-s2","type":"hint","dependencies":["a343428l\'hopital7a-h4-s1"],"title":"Chain Rule","text":"$$\\\\frac{d}{\\\\operatorname{dx}\\\\left(\\\\ln(x^4)\\\\right)}=\\\\frac{4x^3}{x^4}=\\\\frac{4}{x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital7a-h4-s3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{-1}{x^2}$$"],"dependencies":[],"title":"Power Rule","text":"What is the derivative of $$\\\\frac{1}{x}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\frac{1}{x^2}$$","$$1$$","$$\\\\frac{-1}{x^2}$$","$$x$$"]}]},{"id":"a343428l\'hopital7a-h5","type":"hint","dependencies":["a343428l\'hopital7a-h4"],"title":"Simplify","text":"/lim{x,0**+,(4/x)/-(1/x**2)}=/lim{x,0**+,-(4*x**2)/x}=/lim{x,0**+,-4*x}=-4*x","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital7a-h6","type":"hint","dependencies":["a343428l\'hopital7a-h5"],"title":"Evaluate the limit","text":"$$-4(0)=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a343428l\'hopital8","title":"For the following exercises, evaluate the limits with either L\u2019H\xf4pital\u2019s Rule or previously learned methods.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.8 L\u2019H\xf4pital\u2019s Rule","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a343428l\'hopital8a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"$$\\\\lim_{x\\\\to\\\\infty} x^2 e^{\\\\left(-x\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a343428l\'hopital8a-h1","type":"hint","dependencies":[],"title":"Indeterminate form","text":"Substitute $$x$$ with the value $$x$$ approaches to. If the function results in one of these indeterminate form: $$\\\\frac{0}{0}$$, $$\\\\frac{\\\\infty}{\\\\infty}$$ or $$\\\\frac{-\\\\infty}{\\\\infty}$$, we apply L\u2019H\xf4pital\u2019s Rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital8a-h2","type":"hint","dependencies":["a343428l\'hopital8a-h1"],"title":"Rearrange the expression","text":"$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{x^2}{e^x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital8a-h3","type":"hint","dependencies":["a343428l\'hopital8a-h2"],"title":"Substitute the limit","text":"$$\\\\frac{{\\\\infty}^2}{e^{\\\\infty}}=\\\\frac{\\\\infty}{\\\\infty}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital8a-h4","type":"hint","dependencies":["a343428l\'hopital8a-h3"],"title":"Apply L\u2019H\xf4pital\u2019s Rule","text":"Differentiate the denominator and the numerator seperatedly: $$\\\\lim_{x\\\\to\\\\infty} \\\\frac{\\\\frac{d}{dx} x^2}{\\\\frac{d}{dx} e^x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital8a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{2x}{e^x}$$"],"dependencies":["a343428l\'hopital8a-h4"],"title":"Find the derivative","text":"What is the answer of the function after L\u2019H\xf4pital\u2019s Rule is applied?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{2x}{e^x}$$","$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{2x e^x-e^x x^2}{e^2} x$$","$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{2x}{e^2} x$$","$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{2x^2}{e^x}$$"]},{"id":"a343428l\'hopital8a-h6","type":"hint","dependencies":["a343428l\'hopital8a-h5"],"title":"Indeterminate forms in answers.","text":"If the result is still in an indeterminate form after we apply L\u2019H\xf4pital\u2019s Rule and evaluate the limit, we repeat the process.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"a343428l\'hopital8a-h5-s1","type":"hint","dependencies":[],"title":"Evaluate the limit","text":"$$\\\\frac{2\\\\infty}{e^{\\\\infty}}=\\\\frac{\\\\infty}{\\\\infty}=\\\\frac{\\\\infty}{\\\\infty}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"a343428l\'hopital8a-h7","type":"hint","dependencies":["a343428l\'hopital8a-h6"],"title":"Apply L\u2019H\xf4pital\u2019s Rule again","text":"Differentiate the denominator and the numerator seperatedly: $$\\\\lim_{x\\\\to\\\\infty} \\\\frac{\\\\frac{d}{dx} 2x}{\\\\frac{d}{dx} e^x}=\\\\lim_{x\\\\to\\\\infty} \\\\frac{2}{e^x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital8a-h8","type":"hint","dependencies":["a343428l\'hopital8a-h7"],"title":"Evaluate the limit","text":"$$\\\\frac{2}{e^{\\\\infty}}=\\\\frac{2}{\\\\infty}=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a343428l\'hopital9","title":"When L\u2019H\xf4pital\u2019s Rule does not apply","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.8 L\u2019H\xf4pital\u2019s Rule","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a343428l\'hopital9a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"Explain why we cannot apply L\u2019H\xf4pital\u2019s Rule to evaluate /lim{x,0**+,cos(x)/x}. Evaluate /lim{x,0**+,cos(x)/x} by other means","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a343428l\'hopital9a-h1","type":"hint","dependencies":[],"title":"Indeterminate form","text":"Substitute $$x$$ with the value $$x$$ approaches to. If the function results in one of these indeterminate form: $$\\\\frac{0}{0}$$, $$\\\\frac{\\\\infty}{\\\\infty}$$ or $$\\\\frac{-\\\\infty}{\\\\infty}$$, we apply L\u2019H\xf4pital\u2019s Rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital9a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a343428l\'hopital9a-h1"],"title":"Indeterminate form","text":"Does direct substitution of this limit yield any required indeterminate form?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["Yes","No"],"subHints":[{"id":"a343428l\'hopital9a-h2-s1","type":"hint","dependencies":[],"title":"Substitute the limit","text":"$$\\\\fraccos^0\\\\left(0\\\\right)}=\\\\frac{1}{0}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"a343428l\'hopital9a-h3","type":"hint","dependencies":["a343428l\'hopital9a-h2"],"title":"Explaination","text":"L\'H\xf4pital\'s rule is a technique to evaluate limits of indeterminate forms: $$\\\\frac{0}{0}$$, $$\\\\frac{\\\\infty}{\\\\infty}$$ or $$\\\\frac{-\\\\infty}{\\\\infty}$$. Since $$\\\\frac{1}{0}$$ is not an indeterminate form, we then have to try another approach. In addition, this problem can be double-checked with a graphing calculator to prove this point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital9a-h4","type":"hint","dependencies":["a343428l\'hopital9a-h3"],"title":"Rearrange the expression","text":"$$\\\\lim_{x\\\\to0} cos\\\\left(x\\\\right) \\\\frac{1}{x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital9a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\infty$$"],"dependencies":["a343428l\'hopital9a-h4"],"title":"Evaluate the limit","text":"What is a value of the limit as $$x$$ approaches $$0$$ from the right side of the origin?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$-\\\\infty$$","$$0$$","$$\\\\infty$$","DNE"],"subHints":[{"id":"a343428l\'hopital9a-h5-s1","type":"hint","dependencies":[],"title":"Evaluate the limit","text":"After rearranging the expression, we directly substitute the value $$x$$ approaching to and obtain: $$cos\\\\left(0\\\\right) \\\\frac{1}{0}=1\\\\infty=\\\\infty$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"a343428l\'hopital9a-h6","type":"hint","dependencies":["a343428l\'hopital9a-h5"],"title":"Assumption","text":"Even if we tried to apply L\u2019H\xf4pital\u2019s Rule ignoring the fact that this limit does not meet L\u2019H\xf4pital\u2019s Rule condition, the answer would still be wrong eventually.We can double check by looking at the graph of the limit.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital9a-h7","type":"hint","dependencies":["a343428l\'hopital9a-h6"],"title":"Apply L\u2019H\xf4pital\u2019s Rule","text":"$$\\\\lim_{x\\\\to0} \\\\frac{\\\\frac{d}{dx} cos\\\\left(x\\\\right)}{\\\\frac{d}{dx} x}=\\\\lim_{x\\\\to0} \\\\frac{-sin\\\\left(x\\\\right)}{1}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a343428l\'hopital9a-h8","type":"hint","dependencies":["a343428l\'hopital9a-h7"],"title":"Evaluate the limit","text":"$$\\\\frac{-sin\\\\left(0\\\\right)}{1}=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a35b0d9quadratic1","title":"Identifying the Characteristics of a Parabola","body":"Use the attached graph.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Quadratic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a35b0d9quadratic1a","stepAnswer":["$$(3,1)$$"],"problemType":"MultipleChoice","stepTitle":"Determine the vertex of the parabola shown.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(3,1)$$","choices":["$$(0,6)$$","$$(3,1)$$","$$(4,2)$$","$$(1,3)$$"],"hints":{"DefaultPathway":[{"id":"a35b0d9quadratic1a-h1","type":"hint","dependencies":[],"title":"Identifying the Vertex","text":"The vertex is the turning point of the graph. We can see that the vertex is at $$(3,1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a35b0d9quadratic1b","stepAnswer":["$$x=3$$"],"problemType":"MultipleChoice","stepTitle":"Determine the axis of symmetry of the parabola shown.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=3$$","choices":["$$y=x$$","$$x=3$$","$$x=0$$"],"hints":{"DefaultPathway":[{"id":"a35b0d9quadratic1b-h1","type":"hint","dependencies":[],"title":"Identifying the Axis of Symmetry","text":"Because this parabola opens upward, the axis of symmetry is the vertical line that intersects the parabola at the vertex. So the axis of symmetry is $$x=3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a35b0d9quadratic1c","stepAnswer":["There are no zeros"],"problemType":"MultipleChoice","stepTitle":"Determine the zeros of the parabola shown.","stepBody":"","answerType":"string","variabilization":{},"choices":["at $$x=0$$","at $$x=0$$ and $$x=6$$","at $$x=6$$","There are no zeros"],"hints":{"DefaultPathway":[{"id":"a35b0d9quadratic1c-h1","type":"hint","dependencies":[],"title":"Identifying the Zeros","text":"This parabola does not cross the $$x-$$ axis, so it has no zeros.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a35b0d9quadratic1d","stepAnswer":["$$(0,7)$$"],"problemType":"MultipleChoice","stepTitle":"Determine the y-intercept of the parabola shown.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,7)$$","choices":["$$(0,7)$$","$$(7,0)$$","$$(0,0)$$","$$(0,1)$$"],"hints":{"DefaultPathway":[{"id":"a35b0d9quadratic1d-h1","type":"hint","dependencies":[],"title":"Identifying the y-intercept","text":"The parabola crosses the $$y-$$ axis at $$(0,7)$$ so this is the y-intercept.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35b0d9quadratic10","title":"Domain and Range of a Quadratic Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Quadratic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a35b0d9quadratic10a","stepAnswer":["$$f(x) \\\\geq \\\\frac{8}{11}-\\\\infty<x<\\\\infty$$"],"problemType":"MultipleChoice","stepTitle":"Find the domain and range of the function $$f(x)={2\\\\left(x-\\\\frac{4}{7}\\\\right)}^2+\\\\frac{8}{11}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$f(x) \\\\geq \\\\frac{8}{11}-\\\\infty<x<\\\\infty$$","choices":["$$f(x) \\\\geq \\\\frac{8}{11}-\\\\infty<x<\\\\infty$$","$$f(x) \\\\geq \\\\frac{6}{11}-\\\\infty<x<\\\\infty$$","$$f(x) \\\\geq \\\\frac{10}{11}-\\\\infty<x<0$$"],"hints":{"DefaultPathway":[{"id":"a35b0d9quadratic10a-h1","type":"hint","dependencies":[],"title":"Finding the Range","text":"Since we know that the quadratic is positive, the minima must be at the vertex. The function is in vertex form, so we know that the range of the function is greater than or equal to $$\\\\frac{8}{11}$$. This can be written as $$f(x) \\\\geq \\\\frac{8}{11}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic10a-h2","type":"hint","dependencies":["a35b0d9quadratic10a-h1"],"title":"Finding the Domain","text":"Quadratic functions always have a domain that consists of all real numbers. This can be written as $$-\\\\infty<x<\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35b0d9quadratic11","title":"Finding the Intercepts of a Quadratic Equation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Quadratic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a35b0d9quadratic11a","stepAnswer":["x=NA, y=13"],"problemType":"TextBox","stepTitle":"Find the $$x-$$ and $$y-$$ intercept values of the of the function $$g(x)=13+x^2-6x$$. Enter the answer in the following format: $$x=$$ $$___$$ , $$y=$$ $$___$$ . If there is no intercept, enter \'NA\'.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=NA$$, $$y=13$$","hints":{"DefaultPathway":[{"id":"a35b0d9quadratic11a-h1","type":"hint","dependencies":[],"title":"Setting up an Equation to Solve for the X-Intercept","text":"We must first set g(x) equal to $$0$$ and solve for $$x$$. The equation we now have is $$0=13+x^2-6x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic11a-h2","type":"hint","dependencies":["a35b0d9quadratic11a-h1"],"title":"Solving the Equation","text":"We\'ll notice that this equation has no real solutions because the discriminant $$b^2-4ac$$ is less than $$0$$. Thus, there is no x-intercept of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic11a-h3","type":"hint","dependencies":["a35b0d9quadratic11a-h2"],"title":"Finding the Y-Intercept","text":"To solve for the y-intercept, we must merely plug in $$0$$ for $$x$$ in the equation. We now have $$g(x)=13$$. Thus, the y-intercept is $$(0,13)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35b0d9quadratic12","title":"Throwing Rocks","body":"A ball is thrown upward from the top of a $$40$$ foot high building at a speed of $$80$$ feet per second. The ball\u2019s height above ground can be modeled by the equation $$h(t)=-16t^2+96t+112$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Quadratic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a35b0d9quadratic12a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"When does the rock reach maximum height?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a35b0d9quadratic12a-h1","type":"hint","dependencies":[],"title":"Finding the X-value of the Vertex","text":"To determine the maximum height of the rock, we must find the x-value of the the vertex. Since we already have the equation, we can use the formula $$\\\\frac{-b}{2} a$$ to find the x-value. $$\\\\frac{-90}{-32}=3$$. So, the ball reaches its maximum height at $$3$$ seconds.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a35b0d9quadratic12b","stepAnswer":["$$256$$"],"problemType":"TextBox","stepTitle":"What is the maximum height of the rock?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$256$$","hints":{"DefaultPathway":[{"id":"a35b0d9quadratic12b-h1","type":"hint","dependencies":[],"title":"Finding the Y-Value of the Vertex","text":"To determine the maximum height of the rock, we must find the y-value of the vertex. Since we already know that the x-value is $$3$$, we simply plug in $$3$$ for $$x$$ into the equation. We now get $$256$$ as the y-value. This means that the maximum height of the ball is $$256$$ feet.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a35b0d9quadratic12c","stepAnswer":["$$7$$"],"problemType":"TextBox","stepTitle":"When does the rock hit the ground?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7$$","hints":{"DefaultPathway":[{"id":"a35b0d9quadratic12c-h1","type":"hint","dependencies":[],"title":"Finding the X-Intercept","text":"To find when the rock hits the ground, we can simply find the x-intercept that is greater than $$0$$. To do this, we must set h(x) equal to as follows: $$0=-16t^2+96t+112$$. After solving for $$t$$, we will get $$7$$. This means that the ball hits the ground after $$7$$ seconds.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35b0d9quadratic13","title":"Rewriting Quadratics in Standard Form and Finding the Vertex","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Quadratic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a35b0d9quadratic13a","stepAnswer":["$$f(x)={\\\\left(x-6\\\\right)}^2-4$$"],"problemType":"TextBox","stepTitle":"Rewrite the quadratic in standard form: $$x^2-12x+32$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$f(x)={\\\\left(x-6\\\\right)}^2-4$$","hints":{"DefaultPathway":[{"id":"a35b0d9quadratic13a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"The standard form of a quadratic is as follows: $$f(x)={a\\\\left(x-h\\\\right)}^2+k$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic13a-h2","type":"hint","dependencies":["a35b0d9quadratic13a-h1"],"title":"Finding $$h$$","text":"We can find $$h$$ by using the formula $$\\\\frac{-b}{2} a$$. $$\\\\frac{12}{2}=6$$. Thus, $$h=6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic13a-h3","type":"hint","dependencies":["a35b0d9quadratic13a-h2"],"title":"Finding k","text":"We can substitue $$x=h$$ into the general form of the quadratic to find k. $$6^2-\\\\operatorname{12}\\\\left(6\\\\right)+32=k=-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic13a-h4","type":"hint","dependencies":["a35b0d9quadratic13a-h3"],"title":"Rewriting into Standard Form","text":"Using the values of $$h$$ and k derived in the previous steps, we can write the equation in standard form. $$f(x)={\\\\left(x-6\\\\right)}^2-4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a35b0d9quadratic13b","stepAnswer":["(6,-4)"],"problemType":"TextBox","stepTitle":"Find the vertex of the equation derived in the previous question.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(6,-4)$$","hints":{"DefaultPathway":[{"id":"a35b0d9quadratic13b-h1","type":"hint","dependencies":[],"title":"Determing the Vertex","text":"The vertex is (h,k). If we use the values of $$h$$ and k from before, we get the vertex to be $$(6,-4)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35b0d9quadratic14","title":"Rewriting Quadratics in Standard Form and Finding the Vertex","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Quadratic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a35b0d9quadratic14a","stepAnswer":["$$f(x)={\\\\left(x+1\\\\right)}^2-4$$"],"problemType":"TextBox","stepTitle":"Rewrite the quadratic in standard form: $$x^2+2x-3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$f(x)={\\\\left(x+1\\\\right)}^2-4$$","hints":{"DefaultPathway":[{"id":"a35b0d9quadratic14a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"The standard form of a quadratic is as follows: $$f(x)={a\\\\left(x-h\\\\right)}^2+k$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic14a-h2","type":"hint","dependencies":["a35b0d9quadratic14a-h1"],"title":"Finding $$h$$","text":"We can find $$h$$ by using the formula $$\\\\frac{-b}{2} a$$. $$\\\\frac{-2}{2}=-1$$. Thus, $$h=-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic14a-h3","type":"hint","dependencies":["a35b0d9quadratic14a-h2"],"title":"Finding k","text":"We can substitue $$x=h$$ into the general form of the quadratic to find k. (-1)**2+2(-1)-3=k=-4.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic14a-h4","type":"hint","dependencies":["a35b0d9quadratic14a-h3"],"title":"Rewriting into Standard Form","text":"Using the values of $$h$$ and k derived in the previous steps, we can write the equation in standard form. $$f(x)={\\\\left(x+1\\\\right)}^2-4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a35b0d9quadratic14b","stepAnswer":["(-1,-4)"],"problemType":"TextBox","stepTitle":"Find the vertex of the equation derived in the previous question.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-1,-4)$$","hints":{"DefaultPathway":[{"id":"a35b0d9quadratic14b-h1","type":"hint","dependencies":[],"title":"Determining the Vertex","text":"The vertex is (h,k). If we use the values of $$h$$ and k from before, we get the vertex to be $$(-1,-4)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35b0d9quadratic15","title":"Rewriting Quadratics in Standard Form and Finding the Vertex","body":"Rewrite the quadratic in standard form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Quadratic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a35b0d9quadratic15a","stepAnswer":["$$f(x)={\\\\left(x-\\\\frac{1}{2}\\\\right)}^2-0.25$$"],"problemType":"TextBox","stepTitle":"$$x^2-x$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$f(x)={\\\\left(x-\\\\frac{1}{2}\\\\right)}^2-0.25$$","hints":{"DefaultPathway":[{"id":"a35b0d9quadratic15a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"The standard form of a quadratic is as follows: $$f(x)={a\\\\left(x-h\\\\right)}^2+k$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic15a-h2","type":"hint","dependencies":["a35b0d9quadratic15a-h1"],"title":"Finding $$h$$","text":"We can find $$h$$ by using the formula $$\\\\frac{-b}{2} a$$. $$\\\\frac{-1}{-2}=\\\\frac{1}{2}$$. Thus, $$h=\\\\frac{1}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic15a-h3","type":"hint","dependencies":["a35b0d9quadratic15a-h2"],"title":"Finding k","text":"We can substitue $$x=h$$ into the general form of the quadratic to find k. $${\\\\left(\\\\frac{1}{2}\\\\right)}^2-\\\\frac{1}{2}=k=-.25$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic15a-h4","type":"hint","dependencies":["a35b0d9quadratic15a-h3"],"title":"Rewriting into Standard Form","text":"Using the values of $$h$$ and k derived in the previous steps, we can write the equation in standard form. $$f(x)={\\\\left(x-\\\\frac{1}{2}\\\\right)}^2-0.25$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a35b0d9quadratic15b","stepAnswer":["(1/2,-.25)"],"problemType":"TextBox","stepTitle":"Find the vertex of the equation derived in the previous question.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{1}{2}-0.25$$","hints":{"DefaultPathway":[{"id":"a35b0d9quadratic15b-h1","type":"hint","dependencies":[],"title":"The vertex is (h,k). If we use the values of $$h$$ and k from before, we get the vertex to be $$(-1,-4)$$","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35b0d9quadratic16","title":"Rewriting Quadratics in Standard Form and Finding the Vertex","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Quadratic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a35b0d9quadratic16a","stepAnswer":["$$f(x)={\\\\left(x+\\\\frac{5}{2}\\\\right)}^2-8.25$$"],"problemType":"TextBox","stepTitle":"Rewrite the quadratic in standard form: $$x^2+5x-2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$f(x)={\\\\left(x+\\\\frac{5}{2}\\\\right)}^2-8.25$$","hints":{"DefaultPathway":[{"id":"a35b0d9quadratic16a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"The standard form of a quadratic is as follows: $$f(x)={a\\\\left(x-h\\\\right)}^2+k$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic16a-h2","type":"hint","dependencies":["a35b0d9quadratic16a-h1"],"title":"Finding $$h$$","text":"We can find $$h$$ by using the formula $$\\\\frac{-b}{2} a$$. $$h=\\\\frac{-5}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic16a-h3","type":"hint","dependencies":["a35b0d9quadratic16a-h2"],"title":"Finding k","text":"We can substitue $$x=h$$ into the general form of the quadratic to find k. (-5/2)**2+5(-5/2)-2=k=-8.25","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic16a-h4","type":"hint","dependencies":["a35b0d9quadratic16a-h3"],"title":"Rewriting into Standard Form","text":"Using the values of $$h$$ and k derived in the previous steps, we can write the equation in standard form. $$f(x)={\\\\left(x+\\\\frac{5}{2}\\\\right)}^2-8.25$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a35b0d9quadratic16b","stepAnswer":["(5/2,-8.25)"],"problemType":"TextBox","stepTitle":"Find the vertex of the equation derived in the previous question.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{5}{2}-8.25$$","hints":{"DefaultPathway":[{"id":"a35b0d9quadratic16b-h1","type":"hint","dependencies":[],"title":"The vertex is (h,k). If we use the values of $$h$$ and k from before, we get the vertex to be $$\\\\frac{5}{2}-8.25$$","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35b0d9quadratic17","title":"Rewriting Quadratics in Standard Form and Finding the Vertex","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Quadratic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a35b0d9quadratic17a","stepAnswer":["$$f(x)=2{\\\\left(x+2\\\\right)}^2-18$$"],"problemType":"TextBox","stepTitle":"Rewrite the quadratic in standard form: $$2x^2+8x-10$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$f(x)=2{\\\\left(x+2\\\\right)}^2-18$$","hints":{"DefaultPathway":[{"id":"a35b0d9quadratic17a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"The standard form of a quadratic is as follows: $$f(x)={a\\\\left(x-h\\\\right)}^2+k$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic17a-h2","type":"hint","dependencies":["a35b0d9quadratic17a-h1"],"title":"Finding $$h$$","text":"We can find $$h$$ by using the formula $$\\\\frac{-b}{2} a$$. $$\\\\frac{-8}{4}=-2$$. $$h=-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic17a-h3","type":"hint","dependencies":["a35b0d9quadratic17a-h2"],"title":"Finding k","text":"We can substitue $$x=h$$ into the general form of the quadratic to find k. 2(-2)**2+8(-2)-10=k=-18","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic17a-h4","type":"hint","dependencies":["a35b0d9quadratic17a-h3"],"title":"Rewriting into Standard Form","text":"Using the values of a, $$h$$, and k derived in the previous steps, we can write the equation in standard form. $$f(x)={2\\\\left(x+2\\\\right)}^2-18$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a35b0d9quadratic17b","stepAnswer":["(-2,-18)"],"problemType":"TextBox","stepTitle":"Find the vertex of the equation derived in the previous question.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-2,-18)$$","hints":{"DefaultPathway":[{"id":"a35b0d9quadratic17b-h1","type":"hint","dependencies":[],"title":"The vertex is (h,k). If we use the values of $$h$$ and k from before, we get the vertex to be $$(-2,-18)$$","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35b0d9quadratic18","title":"Rewriting Quadratics in Standard Form and Finding the Vertex","body":"Rewrite the quadratic in standard form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Quadratic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a35b0d9quadratic18a","stepAnswer":["$$3{\\\\left(x-1\\\\right)}^2-12$$"],"problemType":"TextBox","stepTitle":"$$f(x)=3x^2-6x-9$$. For the rewritten quadratic, $$f(x)=$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3{\\\\left(x-1\\\\right)}^2-12$$","hints":{"DefaultPathway":[{"id":"a35b0d9quadratic18a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"The standard form of a quadratic is as follows: $$f(x)={a\\\\left(x-h\\\\right)}^2+k$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic18a-h2","type":"hint","dependencies":["a35b0d9quadratic18a-h1"],"title":"Finding $$h$$","text":"We can find $$h$$ by using the formula $$\\\\frac{-b}{2} a$$. $$\\\\frac{6}{6}=1$$. $$h=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic18a-h3","type":"hint","dependencies":["a35b0d9quadratic18a-h2"],"title":"Finding k","text":"We can substitue $$x=h$$ into the general form of the quadratic to find k. $${3\\\\left(1\\\\right)}^2-6\\\\left(1\\\\right)-9=k=-6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic18a-h4","type":"hint","dependencies":["a35b0d9quadratic18a-h3"],"title":"Rewriting into Standard Form","text":"Using the values of a, $$h$$, and k derived in the previous steps, we can write the equation in standard form. $$f(x)={3\\\\left(x-1\\\\right)}^2-12$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a35b0d9quadratic18b","stepAnswer":["(1,-12)"],"problemType":"TextBox","stepTitle":"Find the vertex of the equation derived in the previous question.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(1,-12)$$","hints":{"DefaultPathway":[{"id":"a35b0d9quadratic18b-h1","type":"hint","dependencies":[],"title":"The vertex is (h,k). If we use the values of $$h$$ and k from before, we get the vertex to be $$(1,-12)$$","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35b0d9quadratic19","title":"Rewriting Quadratics in Standard Form and Finding the Vertex","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Quadratic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a35b0d9quadratic19a","stepAnswer":["$$f(x)=2{\\\\left(x-1.5\\\\right)}^2-4.5$$"],"problemType":"TextBox","stepTitle":"Rewrite the quadratic in standard form: $$f(x)=2x^2-6x$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$f(x)=2{\\\\left(x-1.5\\\\right)}^2-4.5$$","hints":{"DefaultPathway":[{"id":"a35b0d9quadratic19a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"The standard form of a quadratic is as follows: $$f(x)={a\\\\left(x-h\\\\right)}^2+k$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic19a-h2","type":"hint","dependencies":["a35b0d9quadratic19a-h1"],"title":"Finding $$h$$","text":"We can find $$h$$ by using the formula $$\\\\frac{-b}{2} a$$. $$\\\\frac{6}{4}=\\\\frac{3}{2}$$. $$h=\\\\frac{3}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic19a-h3","type":"hint","dependencies":["a35b0d9quadratic19a-h2"],"title":"Finding k","text":"We can substitue $$x=h$$ into the general form of the quadratic to find k. $${2\\\\left(1.5\\\\right)}^2-6\\\\left(1.5\\\\right)=k=-4.5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic19a-h4","type":"hint","dependencies":["a35b0d9quadratic19a-h3"],"title":"Rewriting into Standard Form","text":"Using the values of a, $$h$$, and k derived in the previous steps, we can write the equation in standard form. $$f(x)={2\\\\left(x-1.5\\\\right)}^2-4.5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a35b0d9quadratic19b","stepAnswer":["(3/2,-4.5)"],"problemType":"TextBox","stepTitle":"Find the vertex of the equation derived in the previous question.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{3}{2}-4.5$$","hints":{"DefaultPathway":[{"id":"a35b0d9quadratic19b-h1","type":"hint","dependencies":[],"title":"The vertex is (h,k). If we use the values of $$h$$ and k from before, we get the vertex to be $$\\\\frac{3}{2}-4.5$$","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35b0d9quadratic2","title":"Writing the Equation of a Quadratic Function from the Graph","body":"Use the graph to answer the question.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Quadratic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a35b0d9quadratic2a","stepAnswer":["$$\\\\frac{1}{2} x^2+2x-1$$"],"problemType":"TextBox","stepTitle":"Write an equation for the quadratic function g as a transformation of $$f(x)=x^2$$, and then expand the formula, and simplify terms to write the equation in general form. What is the equation in general form?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{2} x^2+2x-1$$","hints":{"DefaultPathway":[{"id":"a35b0d9quadratic2a-h1","type":"hint","dependencies":[],"title":"Analyzing the Form of the Graph","text":"We can see the graph of g is the graph of $$f(x)=x^2$$ shifted to the left $$2$$ and down $$3$$, giving a formula in the form $$g(x)={a\\\\left(x-\\\\left(-2\\\\right)\\\\right)}^2-3={a\\\\left(x+2\\\\right)}^2-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic2a-h2","type":"hint","dependencies":["a35b0d9quadratic2a-h1"],"title":"Solving for Stretch Factor","text":"Substituting the coordinates of a point on the curve, such as $$(0,-1)$$, we can solve for the stretch factor. We then get $$a=\\\\frac{1}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic2a-h3","type":"hint","dependencies":["a35b0d9quadratic2a-h2"],"title":"Standard Form of the Polynomial","text":"In standard form, the algebraic model for this graph is $$g(x)=\\\\frac{1}{{2\\\\left(x+2\\\\right)}^2}-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic2a-h4","type":"hint","dependencies":["a35b0d9quadratic2a-h3"],"title":"Writing in General Polynomial Form","text":"To write this in general polynomial form, we can expand the formula and simplify terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35b0d9quadratic20","title":"Quadratic Functions","body":"For the following exercises, determine whether there is a Minimum or Maximum value to each quadratic function. Find the value and the axis of symmetry.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Quadratic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a35b0d9quadratic20a","stepAnswer":["Minimum"],"problemType":"MultipleChoice","stepTitle":"$$y(x)=2x^2+10x+12$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Minimum","Maximum"],"hints":{"DefaultPathway":[{"id":"a35b0d9quadratic20a-h1","type":"hint","dependencies":[],"title":"Dividing the Right Side","text":"Divide the right hand side by a factor of $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic20a-h2","type":"hint","dependencies":["a35b0d9quadratic20a-h1"],"title":"Complete the Square","text":"Complete the square to find the vertex and how the graph opens.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic20a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(x+\\\\frac{5}{2}\\\\right)}^2$$ + $$\\\\frac{37}{2}$$"],"dependencies":["a35b0d9quadratic20a-h2"],"title":"Determining the Equation\'s New Form","text":"What is the equation after completing the square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic20a-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a35b0d9quadratic20a-h3"],"title":"Determining the Coefficient","text":"What is the coefficient of the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic20a-h4","type":"hint","dependencies":["a35b0d9quadratic20a-s1"],"title":"Identifying the Shape","text":"If the coefficient is positive, it opens upwards.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic20a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Minimum"],"dependencies":["a35b0d9quadratic20a-h4"],"title":"Interpreting the Shape","text":"Does this mean that the vertex will be a Minimum or Maximum?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Minimum","Maximum"]}]}}]},{"id":"a35b0d9quadratic21","title":"Quadratic Functions","body":"For the following exercises, determine whether there is a Minimum or Maximum value to each quadratic function. Find the value and the axis of symmetry.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Quadratic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a35b0d9quadratic21a","stepAnswer":["Minimum"],"problemType":"MultipleChoice","stepTitle":"$$y(x)=2x^2-10x+4$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Minimum","Maximum"],"hints":{"DefaultPathway":[{"id":"a35b0d9quadratic21a-h1","type":"hint","dependencies":[],"title":"Dividing the Right Side","text":"Divide the right hand side by a factor of $$2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic21a-h2","type":"hint","dependencies":["a35b0d9quadratic21a-h1"],"title":"Complete the Square","text":"Complete the square to find the vertex and how the graph opens.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic21a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(x-\\\\frac{5}{2}\\\\right)}^2$$ + $$\\\\frac{33}{2}$$"],"dependencies":["a35b0d9quadratic21a-h2"],"title":"Determining the Equation\'s New Form","text":"What is the equation after completing the square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic21a-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a35b0d9quadratic21a-h3"],"title":"Determining the Coefficient","text":"What is the coefficient of the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic21a-h4","type":"hint","dependencies":["a35b0d9quadratic21a-s1"],"title":"Identifying the Shape","text":"If the coefficient is positive, it opens upwards.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic21a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Minimum"],"dependencies":["a35b0d9quadratic21a-h4"],"title":"Interpreting the Shape","text":"Does this mean that the vertex will be a Minimum or Maximum?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Minimum","Maximum"]}]}}]},{"id":"a35b0d9quadratic22","title":"Quadratic Functions","body":"For the following exercises, determine whether there is a Minimum or Maximum value to each quadratic function. Find the value and the axis of symmetry.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Quadratic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a35b0d9quadratic22a","stepAnswer":["Maximum"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=-\\\\left(x^2\\\\right)+4x+3$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Minimum","Maximum"],"hints":{"DefaultPathway":[{"id":"a35b0d9quadratic22a-h1","type":"hint","dependencies":[],"title":"Complete the Square","text":"Complete the square to find the vertex and how the graph opens.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic22a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a35b0d9quadratic22a-h1"],"title":"Determining the Coefficient","text":"What is the coefficient of the right hand expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic22a-h3","type":"hint","dependencies":["a35b0d9quadratic22a-h2"],"title":"Identifying the Shape","text":"If the coefficient is negative, it opens downwards.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic22a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Maximum"],"dependencies":["a35b0d9quadratic22a-h3"],"title":"Interpreting the Shape","text":"Does this mean that the vertex will be a Minimum or Maximum?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Minimum","Maximum"]}]}}]},{"id":"a35b0d9quadratic23","title":"Quadratic Functions","body":"For the following exercises, determine whether there is a Minimum or Maximum value to each quadratic function. 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35b0d9quadratic28","title":"Quadratic Functions","body":"For the following exercises, determine the domain and range of the quadratic function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Quadratic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a35b0d9quadratic28a","stepAnswer":["Domain: $$(-\\\\infty,\\\\infty)$$ Range $$(-\\\\infty,-6)$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=$$ $$-\\\\left({2\\\\left(x-3\\\\right)}^2\\\\right)$$ - $$6$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Domain: $$(-\\\\infty,\\\\infty)$$ Range $$(-\\\\infty,-6)$$","choices":["Domain: $$(-\\\\infty,\\\\infty)$$ Range $$(-\\\\infty,-6)$$","Domain: $$(-\\\\infty,-6)$$ Range $$(2,-6)$$","Domain: 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Algebra","steps":[{"id":"a35b0d9quadratic3a","stepAnswer":["$$2{\\\\left(x-\\\\frac{3}{2}\\\\right)}^2+\\\\frac{5}{2}$$"],"problemType":"TextBox","stepTitle":"Find the vertex of f(x), and rewrite the equation in standard form (vertex form.) $$f(x)=$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2{\\\\left(x-\\\\frac{3}{2}\\\\right)}^2+\\\\frac{5}{2}$$","hints":{"DefaultPathway":[{"id":"a35b0d9quadratic3a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{2}$$"],"dependencies":[],"title":"Finding the Horizontal Coordinate","text":"The horizontal coordinate of the vertex will be at $$h=\\\\frac{-b}{2a}$$. What is $$h$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{2}$$"],"dependencies":["a35b0d9quadratic3a-h1"],"title":"Finding the Vertical Coordinate","text":"The vertical coordinate is at $$k=f(h)$$. What is f(h)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$f(x)=2x^2-6x+7$$"],"dependencies":["a35b0d9quadratic3a-h2"],"title":"Rewriting into Standard Form","text":"Rewriting into standard form, the stretch factor will be the same as the a in the original quadratic. First, find the horizontal coordinate of the vertex. Then find the vertical coordinate of the vertex. Substitute the values into standard form, using the \\"a\\" from the general form. $$f(x)={ax}^2+bx+c$$ with numerical values of a, $$b$$, and c, what is f?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35b0d9quadratic4","title":"Finding the Domain and Range of a Quadratic Function","body":"$$f(x)=-5x^2+9x-1$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Quadratic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a35b0d9quadratic4a","stepAnswer":["all real numbers"],"problemType":"MultipleChoice","stepTitle":"What is the domain of f?","stepBody":"","answerType":"string","variabilization":{},"choices":["$$x>0$$","all real numbers","all real numbers except $$0$$"],"hints":{"DefaultPathway":[{"id":"a35b0d9quadratic4a-h1","type":"hint","dependencies":[],"title":"Domain of Quadratic Functions","text":"As with any quadratic function, the domain is all real numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a35b0d9quadratic4b","stepAnswer":["(-inf., 61/20]"],"problemType":"MultipleChoice","stepTitle":"What is the range of f?","stepBody":"","answerType":"string","variabilization":{},"choices":["(-inf., 61/20]","all real numbers","(61/20, inf.]","(-61/20, 61/20]"],"hints":{"DefaultPathway":[{"id":"a35b0d9quadratic4b-h1","type":"hint","dependencies":[],"title":"Analyzing the Parabola","text":"Because a is negative, the parabola opens downward and has a maximum value. We need to determine the maximum value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic4b-h2","type":"hint","dependencies":["a35b0d9quadratic4b-h1"],"title":"Finding the X-Coordinate of the Vertex","text":"The x-coordinate is equal to $$\\\\frac{-b}{2} a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic4b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{9}{10}$$"],"dependencies":["a35b0d9quadratic4b-h2"],"title":"Determinig the X-Coordinate","text":"What is the x-coordinate?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic4b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{61}{20}$$"],"dependencies":["a35b0d9quadratic4b-h3"],"title":"Determing the Y-Coordinate of the Vertex","text":"The maximum value is the vertex\'s y-coordinate. When $$x=\\\\frac{9}{10}$$, $$f=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35b0d9quadratic5","title":"Finding the Maximum Value of a Quadratic Function","body":"A backyard farmer wants to enclose a rectangular space for a new garden within her fenced backyard. She has purchased $$80$$ feet of wire fencing to enclose three sides, and she will use a section of the backyard fence as the fourth side.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Quadratic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a35b0d9quadratic5a","stepAnswer":["$$-2L^2+80L$$"],"problemType":"TextBox","stepTitle":"Find a formula for the area enclosed by the fence if the sides of fencing perpendicular to the existing fence have length L.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-2L^2+80L$$","hints":{"DefaultPathway":[{"id":"a35b0d9quadratic5a-h1","type":"hint","dependencies":[],"title":"Creating a Diagram","text":"First, use a diagram such as the attached one to record the given information. It is also helpful to introduce a temporary variable, W, to represent the width of the garden and the length of the fence section parallel to the backyard fence.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic5a-h2","type":"hint","dependencies":["a35b0d9quadratic5a-h1"],"title":"Relating Two Variables Through an Equation","text":"We know we have only $$80$$ feet of fence available, and $$L+W+L=80$$, or more simply, $$2L+W=80$$. This allows us to represent the width, W, in terms of L with the equation $$W=80-2L$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic5a-h3","type":"hint","dependencies":["a35b0d9quadratic5a-h2"],"title":"Writing an Equation for Area","text":"We know the area of a rectangle is length multiplied by width, so $$A=LW=L(80-2L)=80L-2L^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic5a-h4","type":"hint","dependencies":["a35b0d9quadratic5a-h3"],"title":"General Form of the Equation","text":"In general form, $$A(L)=-2L^2+80L$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a35b0d9quadratic5b","stepAnswer":["$$20$$ feet $$x$$ $$40$$ feet"],"problemType":"MultipleChoice","stepTitle":"What dimensions should she make her garden to maximize the enclosed area?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$20$$ feet $$x$$ $$40$$ feet","choices":["$$20$$ feet $$x$$ $$10$$ feet","$$20$$ feet $$x$$ $$20$$ feet","$$20$$ feet $$x$$ $$30$$ feet","$$20$$ feet $$x$$ $$40$$ feet"],"hints":{"DefaultPathway":[{"id":"a35b0d9quadratic5b-h1","type":"hint","dependencies":[],"title":"Analyzing the Function","text":"The quadratic has a negative leading coefficient, so the graph will open downward, and the vertex will be the maximum value for the area. In finding the vertex, we must be careful because the equation is not written in standard polynomial form with decreasing powers. This is why we rewrote the function in general form for the last step. Since a is the coefficient of the squared term, $$a=-2, b=80$$, and $$v=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic5b-h2","type":"hint","dependencies":["a35b0d9quadratic5b-h1"],"title":"Finding the Vertex of a Quadratic Function","text":"The x-coordinate is equal to $$\\\\frac{-b}{2} a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic5b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a35b0d9quadratic5b-h2"],"title":"Determining the x-coordinate of the vertex","text":"Since $$b=80$$ and $$a=-2$$, what is the x-coordinate of the vertex?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic5b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$800$$"],"dependencies":["a35b0d9quadratic5b-h3"],"title":"Determining the y-coordinate of the vertex","text":"Knowing that the x-coordinate of the vertex is $$20$$, what is the y-coordinate of the vertex?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic5b-h5","type":"hint","dependencies":["a35b0d9quadratic5b-h4"],"title":"Dimensions","text":"The maximum value of the function is an area of $$800$$ square feet, which occurs when $$L=20$$ feet. When the shorter sides are $$20$$ feet, there is $$40$$ feet of fencing left for the longer side. To maximize the area, she should enclose the garden so the two shorter sides have length $$20$$ feet and the longer side parallel to the existing fence has length $$40$$ feet.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35b0d9quadratic6","title":"Finding Maximum Revenue","body":"The unit price of an item affects its supply and demand. That is, if the unit price goes up, the demand for the item will usually decrease. For example, a local newspaper currently has 84,000 subscribers at a quarterly charge of $30. Market research has suggested that if the owners raise the price to $32, they would lose 5,000 subscribers.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Quadratic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a35b0d9quadratic6a","stepAnswer":["$$79500$$"],"problemType":"TextBox","stepTitle":"Assuming that subscriptions are linearly related to the price, what price should the newspaper charge for a quarterly subscription to maximize their revenue?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$79500$$","hints":{"DefaultPathway":[{"id":"a35b0d9quadratic6a-h1","type":"hint","dependencies":[],"title":"Equation for Revenue","text":"Revenue is the amount of money a company brings in. In this case, the revenue can be found by multiplying the price per subscription times the number of subscribers, or quantity. We can introduce variables, $$p$$ for price per subscription and Q for quantity, giving us the equation $$Revenue=pQ$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic6a-h2","type":"hint","dependencies":["a35b0d9quadratic6a-h1"],"title":"Finding a Relationship Between Variables","text":"Because the number of subscribers changes with the price, we need to find a relationship between the variables. We know that currently $$\ud835\udc5d=30p=30$$ and $$\ud835\udc44=84, 000.Q=84, 000$$. We also know that if the price rises to $32, the newspaper would lose 5,000 subscribers, giving a second pair of values, $$\ud835\udc5d=32p=32$$ and $$\ud835\udc44=79, 000.Q=79, 000$$. From this we can find a linear equation relating the two quantities. The slope will be","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2500$$"],"dependencies":["a35b0d9quadratic6a-h2"],"title":"Equation for Slope","text":"The slope is equal to (79,000-84,000)/(32-30). What is the simplified answer for slope?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic6a-h4","type":"hint","dependencies":["a35b0d9quadratic6a-h3"],"title":"Solving for the Y-Intercept","text":"This tells us the paper will lose 2,500 subscribers for each dollar they raise the price. We can then solve for the y-intercept by subsituting in $$Q=84, 000$$ and $$p=30$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic6a-h5","type":"hint","dependencies":["a35b0d9quadratic6a-h4"],"title":"Rewriting the Revenue Equation","text":"This gives us the linear equation Q=-2,500p+159,000 relating cost and subscribers. We now return to our revenue equation by plugging in the new formula for Q. The new equation is Revenue $$=$$ $$-2500p^2+159000p$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic6a-h6","type":"hint","dependencies":["a35b0d9quadratic6a-h5"],"title":"Use of Finding the Vertex","text":"To find the price that will maximize revenue for the newspaper, we can find the vertex.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic6a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$31.8$$"],"dependencies":["a35b0d9quadratic6a-h6"],"title":"Identifying the $$p-value$$ of the Vertex","text":"What is the first coordinate of the vertex?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic6a-h8","type":"hint","dependencies":["a35b0d9quadratic6a-h7"],"title":"Indentifying Revenue at the Vertex","text":"The model tells us that the maximum revenue will occur if the newspaper charges $$\\\\$31.80$$ for a subscription. To find what the maximum revenue is, we evaluate the revenue function at the vertex.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35b0d9quadratic7","title":"Finding the $$y-$$ and $$x-Intercepts$$ of a Parabola","body":"Find the $$x$$ and $$y$$ intercepts of the following function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Quadratic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a35b0d9quadratic7a","stepAnswer":["$$y$$ intercept: $$(0,-2);$$ $$x$$ intercepts: (1/3, 0), $$(-2,0)$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=3x^2+5x-2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y$$ intercept: $$(0,-2);$$ $$x$$ intercepts: (1/3, 0), $$(-2,0)$$","choices":["$$y$$ intercept: $$(0,-2);$$ $$x$$ intercepts: (1/3, 0), $$(-2,0)$$","$$y$$ intercept: $$(0,-3);$$ $$x$$ intercepts: (1/3, 0), $$(-2,0)$$","$$y$$ intercept: $$(0,-2);$$ $$x$$ intercepts: (1/4, 0), $$(-2,0)$$"],"hints":{"DefaultPathway":[{"id":"a35b0d9quadratic7a-h1","type":"hint","dependencies":[],"title":"Finding the y-intercept","text":"Find the $$y$$ intercept by evaluating f(0).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic7a-h2","type":"hint","dependencies":["a35b0d9quadratic7a-h1"],"title":"Finding the x-intercept","text":"To find the $$x$$ intercept, factor f(x) to find the $$x$$ values when $$f(x)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35b0d9quadratic8","title":"Finding the $$x-Intercepts$$ of a Parabola","body":"Find the $$x-$$ intercepts of the quadratic function","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Quadratic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a35b0d9quadratic8a","stepAnswer":["$$(-1-\\\\sqrt{3},0)$$, $$(-1+\\\\sqrt{3},0)$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=2x^2+4x-4$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-1-\\\\sqrt{3},0)$$, $$(-1+\\\\sqrt{3},0)$$","choices":["$$(-1-\\\\sqrt{3},0)$$, $$(-2+\\\\sqrt{3},0)$$","$$(-1-\\\\sqrt{5},0)$$, $$(-1+\\\\sqrt{5},0)$$","$$(-1-\\\\sqrt{3},0)$$, $$(-1+\\\\sqrt{3},0)$$"],"hints":{"DefaultPathway":[{"id":"a35b0d9quadratic8a-h1","type":"hint","dependencies":[],"title":"Rewriting the Equation in Standard Form","text":"Because the quadratic is not easily factorable in this case, we solve for the intercepts by first rewriting the quadratic in standard form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic8a-h2","type":"hint","dependencies":["a35b0d9quadratic8a-h1"],"title":"Identifying the Standard Form","text":"The standard form is $$f(x)={2\\\\left(x+1\\\\right)}^2-6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic8a-h3","type":"hint","dependencies":["a35b0d9quadratic8a-h2"],"title":"Solving the Zeros","text":"Finally, set $$f(x)=0$$ and solve for the $$x$$ values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35b0d9quadratic9","title":"Try It: Writing an Equation in General and Standard Form","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Quadratic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a35b0d9quadratic9a","stepAnswer":["General form: $$g(x)=x^2-6x+13$$ Standard Form: $$g(x)={\\\\left(x-3\\\\right)}^2+4$$"],"problemType":"MultipleChoice","stepTitle":"$$g(x)=13+x^2-6x$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"General form: $$g(x)=x^2-6x+13$$ Standard Form: $$g(x)={\\\\left(x-3\\\\right)}^2+4$$","choices":["General form: $$g(x)=$$ $${\\\\left(x-3\\\\right)}^2+4$$ Standard Form: $$g(x)=x^2-6x+13$$","General form: $$g(x)=x^2-6x+13$$ Standard Form: $$g(x)={\\\\left(x-3\\\\right)}^2+4$$"],"hints":{"DefaultPathway":[{"id":"a35b0d9quadratic9a-h1","type":"hint","dependencies":[],"title":"Definition of the General Form","text":"The general form has the formula $$g(x)={ax}^2+bx+c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35b0d9quadratic9a-h2","type":"hint","dependencies":["a35b0d9quadratic9a-h1"],"title":"Definition of the Standard Form","text":"The standard form is $$g(x)={a\\\\left(x-h\\\\right)}^2+k$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35ba99cou1","title":"Addition Principle","body":"Assume that there are $$n$$ ways an event A can happen, $$m$$ ways an event B can happen, and that A and B are non-overlapping.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Counting Principles","courseName":"OpenStax: College Algebra","steps":[{"id":"a35ba99cou1a","stepAnswer":["$$m+n$$"],"problemType":"MultipleChoice","stepTitle":"Use the Addition Principle of counting to show how many ways event A or B can occur.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$m+n$$","choices":["$$m+n$$","$$m n$$","$$m+m+n+n$$"],"hints":{"DefaultPathway":[{"id":"a35ba99cou1a-h1","type":"hint","dependencies":[],"title":"Addition Principle","text":"According to the Addition Principle, if one event can occur in $$m$$ ways and a second event with no common outcomes can occur in $$n$$ ways, then the first or second event can occur in $$m+n$$ ways.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou1a-h2","type":"hint","dependencies":["a35ba99cou1a-h1"],"title":"Addition Principle","text":"There are $$m+n$$ ways for either event A or event B to occur.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35ba99cou10","title":"Numeric","body":"Use the Addition Principle or the Multiplication Principle to perform the calculations.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Counting Principles","courseName":"OpenStax: College Algebra","steps":[{"id":"a35ba99cou10a","stepAnswer":["$$12$$"],"problemType":"TextBox","stepTitle":"How many outcomes are possible from tossing a coin and rolling a 6-sided die?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12$$","hints":{"DefaultPathway":[{"id":"a35ba99cou10a-h1","type":"hint","dependencies":[],"title":"Product of the Number of Options","text":"To find the total number of outcomes, find the product of the number of tossing a coin outcomes and the number of rolling a 6-sided die outcomes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou10a-h2","type":"hint","dependencies":["a35ba99cou10a-h1"],"title":"Tossing a Coin Outcomes","text":"There are $$2$$ outcomes in a coin toss: heads or tails.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou10a-h3","type":"hint","dependencies":["a35ba99cou10a-h2"],"title":"Rolling a 6-Sided Die Outcomes","text":"There are $$6$$ outcomes to rolling a 6-sided die because there are $$6$$ sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou10a-h4","type":"hint","dependencies":["a35ba99cou10a-h3"],"title":"Add the Number of Options","text":"# of tossing a coin outcomes+# of rolling a 6-sided die outcomes\\\\n$$2\\\\times6=12$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou10a-h5","type":"hint","dependencies":["a35ba99cou10a-h4"],"title":"Total Ways","text":"There are $$12$$ outcomes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35ba99cou11","title":"Numeric","body":"Use the Addition Principle or the Multiplication Principle to perform the calculations.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Counting Principles","courseName":"OpenStax: College Algebra","steps":[{"id":"a35ba99cou11a","stepAnswer":["$$60$$"],"problemType":"TextBox","stepTitle":"A restaurant offers a breakfast special that includes a breakfast sandwich, a side dish, and a beverage. There are $$3$$ types of breakfast sandwiches, $$4$$ side dish options, and $$5$$ beverage choices. Find the total number of possible breakfast specials.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$60$$","hints":{"DefaultPathway":[{"id":"a35ba99cou11a-h1","type":"hint","dependencies":[],"title":"Product of the Number of Options","text":"To find the total number of outcomes, find the product of the number of breakfast sandwich options, the number of side dish options, and the number of beverage options.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou11a-h2","type":"hint","dependencies":["a35ba99cou11a-h1"],"title":"Number of Breakfast Sandwiches","text":"There are $$3$$ types of breakfast sandwiches.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou11a-h3","type":"hint","dependencies":["a35ba99cou11a-h2"],"title":"Number of Side Dishes","text":"There are $$4$$ side dish options.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou11a-h4","type":"hint","dependencies":["a35ba99cou11a-h3"],"title":"Number of Beverages","text":"There are $$5$$ beverage choices.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou11a-h5","type":"hint","dependencies":["a35ba99cou11a-h4"],"title":"Add the Number of Options","text":"# of breakfast sandwiches options+# of side dish options+# of beverage options\\\\n$$3\\\\times4\\\\times5=60$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou11a-h6","type":"hint","dependencies":["a35ba99cou11a-h5"],"title":"Total Ways","text":"There are $$60$$ possible breakfast specials.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35ba99cou12","title":"Number of Permutations","body":"Finding the Number of Permutations Using the Formula","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Counting Principles","courseName":"OpenStax: College Algebra","steps":[{"id":"a35ba99cou12a","stepAnswer":["$$79833600$$"],"problemType":"TextBox","stepTitle":"A professor is creating an exam of $$9$$ questions from a test bank of $$12$$ questions. How many ways can she select and arrange the questions?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$79833600$$","hints":{"DefaultPathway":[{"id":"a35ba99cou12a-h1","type":"hint","dependencies":[],"title":"Formula for Permutations of $$n$$ Distinct Objects","text":"Substitute $$n=12$$ and $$r=9$$ into the permutation formula: P(n,r)=n!/(n-r)!.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou12a-h2","type":"hint","dependencies":["a35ba99cou12a-h1"],"title":"Substitute","text":"P(12,9)=12!/(12-9)!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou12a-h3","type":"hint","dependencies":["a35ba99cou12a-h2"],"title":"Simplify","text":"P(12,9)=12!/3!=79833600","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou12a-h4","type":"hint","dependencies":["a35ba99cou12a-h3"],"title":"Permutations","text":"There are $$79833600$$ possible permutations of exam questions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35ba99cou13","title":"Number of Permutations","body":"Compute the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Counting Principles","courseName":"OpenStax: College Algebra","steps":[{"id":"a35ba99cou13a","stepAnswer":["$$20$$"],"problemType":"TextBox","stepTitle":"$$P(5,2)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$20$$","hints":{"DefaultPathway":[{"id":"a35ba99cou13a-h1","type":"hint","dependencies":[],"title":"Formula for Permutations of $$n$$ Distinct Objects","text":"Substitute $$n=5$$ and $$r=2$$ into the permutation formula: P(n,r)=n!/(n-r)!.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou13a-h2","type":"hint","dependencies":["a35ba99cou13a-h1"],"title":"Substitute","text":"P(5,2)=5!/(5-2)!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou13a-h3","type":"hint","dependencies":["a35ba99cou13a-h2"],"title":"Simplify","text":"P(5,2)=5!/3!=20","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35ba99cou14","title":"Number of Permutations","body":"Compute the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Counting Principles","courseName":"OpenStax: College Algebra","steps":[{"id":"a35ba99cou14a","stepAnswer":["$$1680$$"],"problemType":"TextBox","stepTitle":"$$P(8,4)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1680$$","hints":{"DefaultPathway":[{"id":"a35ba99cou14a-h1","type":"hint","dependencies":[],"title":"Formula for Permutations of $$n$$ Distinct Objects","text":"Substitute $$n=8$$ and $$r=4$$ into the permutation formula: P(n,r)=n!/(n-r)!.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou14a-h2","type":"hint","dependencies":["a35ba99cou14a-h1"],"title":"Substitute","text":"P(8,4)=8!/(8-4)!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou14a-h3","type":"hint","dependencies":["a35ba99cou14a-h2"],"title":"Simplify","text":"P(5,2)=8!/4!=1680","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35ba99cou15","title":"Number of Permutations","body":"Compute the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Counting Principles","courseName":"OpenStax: College Algebra","steps":[{"id":"a35ba99cou15a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"$$P(3,3)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"a35ba99cou15a-h1","type":"hint","dependencies":[],"title":"Formula for Permutations of $$n$$ Distinct Objects","text":"Substitute $$n=3$$ and $$r=3$$ into the permutation formula: P(n,r)=n!/(n-r)!.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou15a-h2","type":"hint","dependencies":["a35ba99cou15a-h1"],"title":"Substitute","text":"P(3,3)=3!/(3-3)!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou15a-h3","type":"hint","dependencies":["a35ba99cou15a-h2"],"title":"Simplify","text":"P(3,3)=3!/0!=6","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35ba99cou16","title":"Number of Permutations","body":"Compute the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Counting Principles","courseName":"OpenStax: College Algebra","steps":[{"id":"a35ba99cou16a","stepAnswer":["$$60480$$"],"problemType":"TextBox","stepTitle":"$$P(9,6)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$60480$$","hints":{"DefaultPathway":[{"id":"a35ba99cou16a-h1","type":"hint","dependencies":[],"title":"Formula for Permutations of $$n$$ Distinct Objects","text":"Substitute $$n=9$$ and $$r=6$$ into the permutation formula: P(n,r)=n!/(n-r)!.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou16a-h2","type":"hint","dependencies":["a35ba99cou16a-h1"],"title":"Substitute","text":"P(9,6)=9!/(9-6)!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou16a-h3","type":"hint","dependencies":["a35ba99cou16a-h2"],"title":"Simplify","text":"P(9,6)=9!/3!=60480","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35ba99cou17","title":"Number of Permutations","body":"Compute the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Counting Principles","courseName":"OpenStax: College Algebra","steps":[{"id":"a35ba99cou17a","stepAnswer":["$$55440$$"],"problemType":"TextBox","stepTitle":"$$P(11,5)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$55440$$","hints":{"DefaultPathway":[{"id":"a35ba99cou17a-h1","type":"hint","dependencies":[],"title":"Formula for Permutations of $$n$$ Distinct Objects","text":"Substitute $$n=11$$ and $$r=5$$ into the permutation formula: P(n,r)=n!/(n-r)!.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou17a-h2","type":"hint","dependencies":["a35ba99cou17a-h1"],"title":"Substitute","text":"P(11,5)=11!/(11-5)!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou17a-h3","type":"hint","dependencies":["a35ba99cou17a-h2"],"title":"Simplify","text":"P(11,5)=11!/6!=55440","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35ba99cou18","title":"Number of Combinations","body":"A fast food restaurant offers five side dish options. Your meal comes with two side dishes.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Counting Principles","courseName":"OpenStax: College Algebra","steps":[{"id":"a35ba99cou18a","stepAnswer":["$$10$$"],"problemType":"TextBox","stepTitle":"How many ways can you select your side dishes?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10$$","hints":{"DefaultPathway":[{"id":"a35ba99cou18a-h1","type":"hint","dependencies":[],"title":"Define $$n$$ and $$r$$","text":"We want to choose $$2$$ side dishes from $$5$$ options. Therefore $$n=5$$ and $$r=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou18a-h2","type":"hint","dependencies":["a35ba99cou18a-h1"],"title":"Formula for Combinations of $$n$$ Distinct Objects","text":"Substitute $$n=5$$ and $$r=2$$ into the combination formula: C(n,r)=n!/(r!*(n-r)!).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou18a-h3","type":"hint","dependencies":["a35ba99cou18a-h2"],"title":"Substitute","text":"C(5,2)=5!/(2!*(5-2)!)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou18a-h4","type":"hint","dependencies":["a35ba99cou18a-h3"],"title":"Simplify","text":"$$C(5,2)=10$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a35ba99cou18b","stepAnswer":["$$10$$"],"problemType":"TextBox","stepTitle":"How many ways can you select $$3$$ side dishes?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10$$","hints":{"DefaultPathway":[{"id":"a35ba99cou18b-h1","type":"hint","dependencies":[],"title":"Define $$n$$ and $$r$$","text":"We want to choose $$3$$ side dishes from $$5$$ options. Therefore $$n=3$$ and $$r=5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou18b-h2","type":"hint","dependencies":["a35ba99cou18b-h1"],"title":"Formula for Combinations of $$n$$ Distinct Objects","text":"Substitute $$n=5$$ and $$r=3$$ into the combination formula: C(n,r)=n!/(r!*(n-r)!).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou18b-h3","type":"hint","dependencies":["a35ba99cou18b-h2"],"title":"Substitute","text":"C(5,3)=5!/(3!*(5-3)!)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou18b-h4","type":"hint","dependencies":["a35ba99cou18b-h3"],"title":"Simplify","text":"$$C(5,3)=10$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35ba99cou19","title":"Number of Combinations","body":"Compute the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Counting Principles","courseName":"OpenStax: College Algebra","steps":[{"id":"a35ba99cou19a","stepAnswer":["$$56$$"],"problemType":"TextBox","stepTitle":"$$C(8,5)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$56$$","hints":{"DefaultPathway":[{"id":"a35ba99cou19a-h1","type":"hint","dependencies":[],"title":"Formula for Combinations of $$n$$ Distinct Objects","text":"Substitute $$n=8$$ and $$r=5$$ into the combination formula: C(n,r)=n!/(r!*(n-r)!).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou19a-h2","type":"hint","dependencies":["a35ba99cou19a-h1"],"title":"Substitute","text":"C(8,5)=8!/(5!*(8-5)!)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou19a-h3","type":"hint","dependencies":["a35ba99cou19a-h2"],"title":"Simplify","text":"$$C(8,5)=56$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35ba99cou2","title":"Addition Principle","body":"Using the Addition Principle","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Counting Principles","courseName":"OpenStax: College Algebra","steps":[{"id":"a35ba99cou2a","stepAnswer":["$$7$$"],"problemType":"TextBox","stepTitle":"There are $$2$$ vegetarian entr\xe9e options and $$5$$ meat entr\xe9e options on a dinner menu. What is the total number of entr\xe9e options?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7$$","hints":{"DefaultPathway":[{"id":"a35ba99cou2a-h1","type":"hint","dependencies":[],"title":"Add the Number of Options","text":"We can add the number of vegetarian options to the number of meat options to find the total number of entr\xe9e options.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou2a-h2","type":"hint","dependencies":["a35ba99cou2a-h1"],"title":"Add the Number of Options","text":"\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou2a-h3","type":"hint","dependencies":["a35ba99cou2a-h2"],"title":"Total Options","text":"There are $$7$$ total options.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35ba99cou20","title":"Number of Combinations","body":"Compute the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Counting Principles","courseName":"OpenStax: College Algebra","steps":[{"id":"a35ba99cou20a","stepAnswer":["$$495$$"],"problemType":"TextBox","stepTitle":"$$C(12,4)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$495$$","hints":{"DefaultPathway":[{"id":"a35ba99cou20a-h1","type":"hint","dependencies":[],"title":"Formula for Combinations of $$n$$ Distinct Objects","text":"Substitute $$n=12$$ and $$r=4$$ into the combination formula: C(n,r)=n!/(r!*(n-r)!).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou20a-h2","type":"hint","dependencies":["a35ba99cou20a-h1"],"title":"Substitute","text":"C(12,4)=12!/(4!*(12-4)!)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou20a-h3","type":"hint","dependencies":["a35ba99cou20a-h2"],"title":"Simplify","text":"$$C(12,4)=495$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35ba99cou21","title":"Number of Combinations","body":"Compute the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Counting Principles","courseName":"OpenStax: College Algebra","steps":[{"id":"a35ba99cou21a","stepAnswer":["$$2600$$"],"problemType":"TextBox","stepTitle":"$$C(26,3)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2600$$","hints":{"DefaultPathway":[{"id":"a35ba99cou21a-h1","type":"hint","dependencies":[],"title":"Formula for Combinations of $$n$$ Distinct Objects","text":"Substitute $$n=26$$ and $$r=3$$ into the combination formula: C(n,r)=n!/(r!*(n-r)!).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou21a-h2","type":"hint","dependencies":["a35ba99cou21a-h1"],"title":"Substitute","text":"C(26,3)=26!/(3!*(26-3)!)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou21a-h3","type":"hint","dependencies":["a35ba99cou21a-h2"],"title":"Simplify","text":"$$C(26,3)=2600$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35ba99cou22","title":"Number of Combinations","body":"Compute the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Counting Principles","courseName":"OpenStax: College Algebra","steps":[{"id":"a35ba99cou22a","stepAnswer":["$$7$$"],"problemType":"TextBox","stepTitle":"$$C(7,6)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7$$","hints":{"DefaultPathway":[{"id":"a35ba99cou22a-h1","type":"hint","dependencies":[],"title":"Formula for Combinations of $$n$$ Distinct Objects","text":"Substitute $$n=7$$ and $$r=6$$ into the combination formula: C(n,r)=n!/(r!*(n-r)!).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou22a-h2","type":"hint","dependencies":["a35ba99cou22a-h1"],"title":"Substitute","text":"C(7,6)=7!/(6!*(7-6)!)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou22a-h3","type":"hint","dependencies":["a35ba99cou22a-h2"],"title":"Simplify","text":"$$C(7,6)=7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35ba99cou23","title":"Number of Combinations","body":"Compute the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Counting Principles","courseName":"OpenStax: College Algebra","steps":[{"id":"a35ba99cou23a","stepAnswer":["$$120$$"],"problemType":"TextBox","stepTitle":"$$C(10,3)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$120$$","hints":{"DefaultPathway":[{"id":"a35ba99cou23a-h1","type":"hint","dependencies":[],"title":"Formula for Combinations of $$n$$ Distinct Objects","text":"Substitute $$n=10$$ and $$r=3$$ into the combination formula: C(n,r)=n!/(r!*(n-r)!).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou23a-h2","type":"hint","dependencies":["a35ba99cou23a-h1"],"title":"Substitute","text":"C(10,3)=10!/(3!*(10-3)!)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou23a-h3","type":"hint","dependencies":["a35ba99cou23a-h2"],"title":"Simplify","text":"$$C(10,3)=120$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35ba99cou24","title":"Subsets","body":"Find the number of subsets in each given set.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Counting Principles","courseName":"OpenStax: College Algebra","steps":[{"id":"a35ba99cou24a","stepAnswer":["$$16$$"],"problemType":"TextBox","stepTitle":"A restaurant offers butter, cheese, chives, and sour cream as toppings for a baked potato. How many different ways are there to order a potato?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16$$","hints":{"DefaultPathway":[{"id":"a35ba99cou24a-h1","type":"hint","dependencies":[],"title":"Identify $$n$$","text":"We are looking for the number of subsets of a set with $$4$$ objects.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou24a-h2","type":"hint","dependencies":["a35ba99cou24a-h1"],"title":"Formula for the Number of Subsets of a Set","text":"Substitute $$n=4$$ into the formula: $$2^n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou24a-h3","type":"hint","dependencies":["a35ba99cou24a-h2"],"title":"Substitute and Simplify","text":"$$2^4=16$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou24a-h4","type":"hint","dependencies":["a35ba99cou24a-h3"],"title":"Possible Ways","text":"There are $$16$$ possible ways to order a potato.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35ba99cou25","title":"Subsets","body":"Find the number of subsets in each given set.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Counting Principles","courseName":"OpenStax: College Algebra","steps":[{"id":"a35ba99cou25a","stepAnswer":["$$1024$$"],"problemType":"TextBox","stepTitle":"{1,2,3,4,5,6,7,8,9,10}","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1024$$","hints":{"DefaultPathway":[{"id":"a35ba99cou25a-h1","type":"hint","dependencies":[],"title":"Identify $$n$$","text":"We are looking for the number of subsets of a set with $$10$$ distinct numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou25a-h2","type":"hint","dependencies":["a35ba99cou25a-h1"],"title":"Formula for the Number of Subsets of a Set","text":"Substitute $$n=10$$ into the formula: $$2^n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou25a-h3","type":"hint","dependencies":["a35ba99cou25a-h2"],"title":"Substitute and Simplify","text":"$$2^{10}=1024$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35ba99cou26","title":"Subsets","body":"Find the number of subsets in each given set.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Counting Principles","courseName":"OpenStax: College Algebra","steps":[{"id":"a35ba99cou26a","stepAnswer":["$$67108864$$"],"problemType":"TextBox","stepTitle":"{a,b,c,\u2026,z}","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$67108864$$","hints":{"DefaultPathway":[{"id":"a35ba99cou26a-h1","type":"hint","dependencies":[],"title":"Identify $$n$$","text":"We are looking for the number of subsets of a set with $$26$$ distinct letters.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou26a-h2","type":"hint","dependencies":["a35ba99cou26a-h1"],"title":"Formula for the Number of Subsets of a Set","text":"Substitute $$n=26$$ into the formula: $$2^n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou26a-h3","type":"hint","dependencies":["a35ba99cou26a-h2"],"title":"Substitute and Simplify","text":"$$2^{26}=67108864$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35ba99cou27","title":"Subsets","body":"Find the number of subsets in each given set.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Counting Principles","courseName":"OpenStax: College Algebra","steps":[{"id":"a35ba99cou27a","stepAnswer":["$$4096$$"],"problemType":"TextBox","stepTitle":"A set containing $$5$$ distinct numbers, $$4$$ distinct letters, and $$3$$ distinct symbols","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4096$$","hints":{"DefaultPathway":[{"id":"a35ba99cou27a-h1","type":"hint","dependencies":[],"title":"Amount of Distinct Objects","text":"The set contains a total of $$5+4+3=12$$ distinct objects.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou27a-h2","type":"hint","dependencies":["a35ba99cou27a-h1"],"title":"Identify $$n$$","text":"We are looking for the number of subsets of a set with $$12$$ objects.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou27a-h3","type":"hint","dependencies":["a35ba99cou27a-h2"],"title":"Formula for the Number of Subsets of a Set","text":"Substitute $$n=12$$ into the formula: $$2^n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou27a-h4","type":"hint","dependencies":["a35ba99cou27a-h3"],"title":"Substitute and Simplify","text":"$$2^{12}=4096$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35ba99cou28","title":"Subsets","body":"Find the number of subsets in each given set.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Counting Principles","courseName":"OpenStax: College Algebra","steps":[{"id":"a35ba99cou28a","stepAnswer":["$$16384$$"],"problemType":"TextBox","stepTitle":"The set of even numbers from $$2$$ to $$28$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16384$$","hints":{"DefaultPathway":[{"id":"a35ba99cou28a-h1","type":"hint","dependencies":[],"title":"Amount of Distinct Objects","text":"The set contains 2,4,6,8,10,12,14,16,18,20,22,24,26,28.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou28a-h2","type":"hint","dependencies":["a35ba99cou28a-h1"],"title":"Identify $$n$$","text":"We are looking for the number of subsets of a set with $$14$$ distinct even numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou28a-h3","type":"hint","dependencies":["a35ba99cou28a-h2"],"title":"Formula for the Number of Subsets of a Set","text":"Substitute $$n=14$$ into the formula: $$2^n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou28a-h4","type":"hint","dependencies":["a35ba99cou28a-h3"],"title":"Substitute and Simplify","text":"$$2^{14}=16384$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35ba99cou29","title":"Number of Permutations","body":"Finding the Number of Permutations of $$n$$ Non-Distinct Objects","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Counting Principles","courseName":"OpenStax: College Algebra","steps":[{"id":"a35ba99cou29a","stepAnswer":["$$10080$$"],"problemType":"TextBox","stepTitle":"Find the number of rearrangements of the letters in the word DISTINCT.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10080$$","hints":{"DefaultPathway":[{"id":"a35ba99cou29a-h1","type":"hint","dependencies":[],"title":"Identify the elements","text":"There are $$8$$ letters. Both I and T are repeated $$2$$ times.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou29a-h2","type":"hint","dependencies":["a35ba99cou29a-h1"],"title":"Formula for Finding the Number of Permutations of $$n$$ Non-Distinct Objects","text":"Substitute $$n=8$$, $$r_1=2$$, and $$r_2=2$$ into the formula: n!/(r!*r_2!...r_k!)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou29a-h3","type":"hint","dependencies":["a35ba99cou29a-h2"],"title":"Substitute and Simplify","text":"8!/(2!*2!)=10080","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35ba99cou3","title":"Multiplication Principle","body":"Using the Multiplication Principle","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Counting Principles","courseName":"OpenStax: College Algebra","steps":[{"id":"a35ba99cou3a","stepAnswer":["$$16$$"],"problemType":"TextBox","stepTitle":"Diane packed $$2$$ skirts, $$4$$ blouses, and a sweater for her business trip. She will need to choose a skirt and a blouse for each outfit and decide whether to wear the sweater. Use the Multiplication Principle to find the total number of possible outfits.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16$$","hints":{"DefaultPathway":[{"id":"a35ba99cou3a-h1","type":"hint","dependencies":[],"title":"Product of the Number of Options","text":"To find the total number of outfits, find the product of the number of skirt options, the number of blouse options, and the number of sweater options.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou3a-h2","type":"hint","dependencies":["a35ba99cou3a-h1"],"title":"Product of the Number of Options","text":"\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou3a-h3","type":"hint","dependencies":["a35ba99cou3a-h2"],"title":"Total Outfits","text":"There are $$16$$ possible outfits.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35ba99cou30","title":"Number of Permutations","body":"Finding the Number of Permutations of $$n$$ Non-Distinct Objects","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Counting Principles","courseName":"OpenStax: College Algebra","steps":[{"id":"a35ba99cou30a","stepAnswer":["$$840$$"],"problemType":"TextBox","stepTitle":"Find the number of rearrangements of the letters in the word CARRIER.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$840$$","hints":{"DefaultPathway":[{"id":"a35ba99cou30a-h1","type":"hint","dependencies":[],"title":"Identify the elements","text":"There are $$7$$ letters. R is repeated $$3$$ times.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou30a-h2","type":"hint","dependencies":["a35ba99cou30a-h1"],"title":"Formula for Finding the Number of Permutations of $$n$$ Non-Distinct Objects","text":"Substitute $$n=7$$, $$r_1=3$$ into the formula: n!/(r!*r_2!...r_k!)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou30a-h3","type":"hint","dependencies":["a35ba99cou30a-h2"],"title":"Substitute and Simplify","text":"7!/3!=840","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35ba99cou4","title":"Numeric","body":"Use the Addition Principle or the Multiplication Principle to perform the calculations.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Counting Principles","courseName":"OpenStax: College Algebra","steps":[{"id":"a35ba99cou4a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"Let the set A $$=$$ {-5,-3,-1,2,3,4,5,6}. How many ways are there to choose a negative or an even number from A?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"a35ba99cou4a-h1","type":"hint","dependencies":[],"title":"Add the Number of Options","text":"We can add the number of negative number options to the number of even number options to find the total number of options.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou4a-h2","type":"hint","dependencies":["a35ba99cou4a-h1"],"title":"Number of Negatives","text":"There are $$3$$ negative numbers: $$-5$$, $$-3$$, $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou4a-h3","type":"hint","dependencies":["a35ba99cou4a-h2"],"title":"Number of Evens","text":"There are $$3$$ even numbers: $$2$$, $$4$$, $$6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou4a-h4","type":"hint","dependencies":["a35ba99cou4a-h3"],"title":"Add the Number of Options","text":"# of negative number options+# of even number options\\\\n$$3+3=6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou4a-h5","type":"hint","dependencies":["a35ba99cou4a-h4"],"title":"Total Ways","text":"There are $$6$$ ways to choose a negative or an even number from A.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35ba99cou5","title":"Numeric","body":"Use the Addition Principle or the Multiplication Principle to perform the calculations.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Counting Principles","courseName":"OpenStax: College Algebra","steps":[{"id":"a35ba99cou5a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"Let the set B $$=$$ {-23,-16,-7,-2,20,36,48,72}. How many ways are there to choose a positive or an odd number from B?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"a35ba99cou5a-h1","type":"hint","dependencies":[],"title":"Add the Number of Options","text":"We can add the number of positive number options to the number of odd number options to find the total number of options.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou5a-h2","type":"hint","dependencies":["a35ba99cou5a-h1"],"title":"Number of Positives","text":"There are $$4$$ positive numbers: $$20$$, $$36$$, $$48$$, $$72$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou5a-h3","type":"hint","dependencies":["a35ba99cou5a-h2"],"title":"Number of Odds","text":"There are $$2$$ odd numbers: $$-23$$, $$-7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou5a-h4","type":"hint","dependencies":["a35ba99cou5a-h3"],"title":"Add the Number of Options","text":"# of positive number options+# of odd number options\\\\n$$4+2=6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou5a-h5","type":"hint","dependencies":["a35ba99cou5a-h4"],"title":"Total Ways","text":"There are $$6$$ ways to choose a positive or an odd number from B.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35ba99cou6","title":"Numeric","body":"Use the Addition Principle or the Multiplication Principle to perform the calculations.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Counting Principles","courseName":"OpenStax: College Algebra","steps":[{"id":"a35ba99cou6a","stepAnswer":["$$15$$"],"problemType":"TextBox","stepTitle":"How many ways are there to pick a red ace or a club from a standard card playing deck?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$15$$","hints":{"DefaultPathway":[{"id":"a35ba99cou6a-h1","type":"hint","dependencies":[],"title":"Add the Number of Options","text":"We can add the number of red ace options to the number of club options to find the total number of options.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou6a-h2","type":"hint","dependencies":["a35ba99cou6a-h1"],"title":"Number of Red Aces","text":"There are $$2$$ red aces in a deck of cards: an ace of diamonds and an ace of hearts.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou6a-h3","type":"hint","dependencies":["a35ba99cou6a-h2"],"title":"Number of Clubs","text":"There are $$13$$ clubs in a deck of cards, from the ace of clubs to the king of clubs.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou6a-h4","type":"hint","dependencies":["a35ba99cou6a-h3"],"title":"Add the Number of Options","text":"# of red ace options+# of club options\\\\n$$2+13=15$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou6a-h5","type":"hint","dependencies":["a35ba99cou6a-h4"],"title":"Total Ways","text":"There are $$15$$ ways to pick a red ace or a club from a standard card playing deck.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35ba99cou7","title":"Numeric","body":"Use the Addition Principle or the Multiplication Principle to perform the calculations.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Counting Principles","courseName":"OpenStax: College Algebra","steps":[{"id":"a35ba99cou7a","stepAnswer":["$$16$$"],"problemType":"TextBox","stepTitle":"How many ways are there to pick a paint color from $$5$$ shades of green, $$4$$ shades of blue, or $$7$$ shades of yellow?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16$$","hints":{"DefaultPathway":[{"id":"a35ba99cou7a-h1","type":"hint","dependencies":[],"title":"Add the Number of Options","text":"We can add the number of shades of green options to the number of shades of blue options to the number of shades of yellow options to find the total number of options.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou7a-h2","type":"hint","dependencies":["a35ba99cou7a-h1"],"title":"Number of Greens","text":"There are $$5$$ shades of green.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou7a-h3","type":"hint","dependencies":["a35ba99cou7a-h2"],"title":"Number of Blues","text":"There are $$4$$ shades of blue.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou7a-h4","type":"hint","dependencies":["a35ba99cou7a-h3"],"title":"Number of Yellows","text":"There are $$7$$ shades of yellow.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou7a-h5","type":"hint","dependencies":["a35ba99cou7a-h4"],"title":"Add the Number of Options","text":"# of shades of green options+# of shades of blue options+# of shades of yellow options\\\\n$$5+4+7=16$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou7a-h6","type":"hint","dependencies":["a35ba99cou7a-h5"],"title":"Total Ways","text":"There are $$16$$ ways to pick a paint color.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35ba99cou8","title":"Numeric","body":"Use the Addition Principle or the Multiplication Principle to perform the calculations.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Counting Principles","courseName":"OpenStax: College Algebra","steps":[{"id":"a35ba99cou8a","stepAnswer":["$$7$$"],"problemType":"TextBox","stepTitle":"A student is shopping for a new computer. He is deciding among $$3$$ desktop computers and $$4$$ laptop computers. What is the total number of computer options?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7$$","hints":{"DefaultPathway":[{"id":"a35ba99cou8a-h1","type":"hint","dependencies":[],"title":"Add the Number of Options","text":"We can add the number of desktop options to the number of laptop options to find the total number of options.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou8a-h2","type":"hint","dependencies":["a35ba99cou8a-h1"],"title":"Number of Desktops","text":"There are $$3$$ desktop computers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou8a-h3","type":"hint","dependencies":["a35ba99cou8a-h2"],"title":"Number of Laptops","text":"There are $$4$$ laptop computers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou8a-h4","type":"hint","dependencies":["a35ba99cou8a-h3"],"title":"Add the Number of Options","text":"# of desktops options+# of laptop options\\\\n$$3+4=7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou8a-h5","type":"hint","dependencies":["a35ba99cou8a-h4"],"title":"Total Ways","text":"There are $$7$$ computer options.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a35ba99cou9","title":"Numeric","body":"Use the Addition Principle or the Multiplication Principle to perform the calculations.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Counting Principles","courseName":"OpenStax: College Algebra","steps":[{"id":"a35ba99cou9a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"How many outcomes are possible from tossing a pair of coins?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a35ba99cou9a-h1","type":"hint","dependencies":[],"title":"Number of Options","text":"In a coin, there is only $$2$$ sides: heads or tails. We are tossing $$2$$ coins.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a35ba99cou9a-h2","type":"hint","dependencies":["a35ba99cou9a-h1"],"title":"Outcomes","text":"There are $$4$$ outcomes from tossing a pair of coins: HH, HT, TH, TT","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a372017cramer1","title":"Find the determinant.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.8 Solving Systems with Cramer\'s Rule","courseName":"OpenStax: College Algebra","steps":[{"id":"a372017cramer1a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"$$\\\\begin{bmatrix} -1 & 4 & 0 \\\\\\\\ 0 & 2 & 3 \\\\\\\\ 0 & 0 & -3 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"a372017cramer1a-h1","type":"hint","dependencies":[],"title":"Augument","text":"First, augument the matrix with the first two columns. In other words, duplicate the first two columns, and add them to the end of the matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer1a-h2","type":"hint","dependencies":["a372017cramer1a-h1"],"title":"Formula","text":"Now, follow the formula to find the determinant. First, multiply the entries down from the row $$1$$ column one spot, to the row $$3$$ column $$3$$ spot, in a diagonal fashion. Do this for the next two diagonals as well, from the row $$1$$ column $$2$$ spot, to the row $$3$$ column $$4$$ spot, and from the row $$1$$ column $$3$$ spot to the row $$3$$ column $$5$$ spot.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a372017cramer1a-h2"],"title":"Add","text":"Now, add all the previous products together. What do you get as your answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a372017cramer1a-h3-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a372017cramer1a-h4","type":"hint","dependencies":["a372017cramer1a-h3"],"title":"Multiply","text":"Next, multiply the values from the bottom left of the matrix to the location $$3$$ spots to the top in a diagonal. Another way to see this is a diagonal from the bottom to the top. Do this for the next $$2$$ columns.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer1a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a372017cramer1a-h4"],"title":"Subtract","text":"Now, subtract all the values that were just multiplied from the first set of diagonal multiplications(those that you calculated previously).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a372017cramer1a-h5-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a372017cramer1a-h6","type":"hint","dependencies":["a372017cramer1a-h5"],"title":"Answer","text":"Therefore, the determinant is $$6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a372017cramer10","title":"Solve the system of linear equations using Cramer\u2019s Rule.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.8 Solving Systems with Cramer\'s Rule","courseName":"OpenStax: College Algebra","steps":[{"id":"a372017cramer10a","stepAnswer":["(1,1)"],"problemType":"TextBox","stepTitle":"$$2x-3y=-1$$ $$4x+5y=9$$","stepBody":"Enter your solution as a coordinate pair.","answerType":"string","variabilization":{},"answerLatex":"$$(1,1)$$","hints":{"DefaultPathway":[{"id":"a372017cramer10a-h1","type":"hint","dependencies":[],"title":"Use the formula","text":"First, use the formula for Cramer\'s rule to solve for $$x$$ and $$y$$. If using a 2x2 matrix, the formula for it is $$x=(\\\\begin{bmatrix} c1 & b1 \\\\\\\\ c2 & b2 \\\\end{bmatrix})/(\\\\begin{bmatrix} a1 & b1 \\\\\\\\ a2 & b2 \\\\end{bmatrix})$$ and $$y=(\\\\begin{bmatrix} a1 & c1 \\\\\\\\ a2 & c2 \\\\end{bmatrix})/(\\\\begin{bmatrix} a1 & b1 \\\\\\\\ a2 & b2 \\\\end{bmatrix}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a372017cramer10a-h1"],"title":"Determinant","text":"Now, take the determinant of all matrices in the $$x=$$ section and simplify. What does $$x$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a372017cramer10a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a372017cramer10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a372017cramer10a-h2"],"title":"Determinant","text":"Next, take the determinant of all matrices in the $$y=$$ section and simplify. What does $$y$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a372017cramer10a-h3-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a372017cramer10a-h4","type":"hint","dependencies":["a372017cramer10a-h3"],"title":"Answer","text":"Therefore, the answer is $$(1,1)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a372017cramer11","title":"Solve the system of linear equations using Cramer\u2019s Rule.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.8 Solving Systems with Cramer\'s Rule","courseName":"OpenStax: College Algebra","steps":[{"id":"a372017cramer11a","stepAnswer":["(2,2)"],"problemType":"TextBox","stepTitle":"$$5x-4y=2$$ $$-4x+7y=6$$","stepBody":"Enter your solution as a coordinate pair.","answerType":"string","variabilization":{},"answerLatex":"$$(2,2)$$","hints":{"DefaultPathway":[{"id":"a372017cramer11a-h1","type":"hint","dependencies":[],"title":"Use the formula","text":"First, use the formula for Cramer\'s rule to solve for $$x$$ and $$y$$. If using a 2x2 matrix, the formula for it is $$x=(\\\\begin{bmatrix} c1 & b1 \\\\\\\\ c2 & b2 \\\\end{bmatrix})/(\\\\begin{bmatrix} a1 & b1 \\\\\\\\ a2 & b2 \\\\end{bmatrix})$$ and $$y=(\\\\begin{bmatrix} a1 & c1 \\\\\\\\ a2 & c2 \\\\end{bmatrix})/(\\\\begin{bmatrix} a1 & b1 \\\\\\\\ a2 & b2 \\\\end{bmatrix}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a372017cramer11a-h1"],"title":"Determinant","text":"Now, take the determinant of all matrices in the $$x=$$ section and simplify. What does $$x$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a372017cramer11a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a372017cramer11a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a372017cramer11a-h2"],"title":"Determinant","text":"Next, take the determinant of all matrices in the $$y=$$ section and simplify. What does $$y$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a372017cramer11a-h3-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a372017cramer11a-h4","type":"hint","dependencies":["a372017cramer11a-h3"],"title":"Answer","text":"Therefore, the answer is $$(2,2)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a372017cramer12","title":"Solve the system of linear equations using Cramer\u2019s Rule.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.8 Solving Systems with Cramer\'s Rule","courseName":"OpenStax: College Algebra","steps":[{"id":"a372017cramer12a","stepAnswer":["(1/2,1/3)"],"problemType":"TextBox","stepTitle":"$$6x-3y=2$$ $$-8x+9y=-1$$","stepBody":"Enter your solution as a coordinate pair.","answerType":"string","variabilization":{},"answerLatex":"$$(\\\\frac{1}{2},\\\\frac{1}{3})$$","hints":{"DefaultPathway":[{"id":"a372017cramer12a-h1","type":"hint","dependencies":[],"title":"Use the formula","text":"First, use the formula for Cramer\'s rule to solve for $$x$$ and $$y$$. If using a 2x2 matrix, the formula for it is $$x=(\\\\begin{bmatrix} c1 & b1 \\\\\\\\ c2 & b2 \\\\end{bmatrix})/(\\\\begin{bmatrix} a1 & b1 \\\\\\\\ a2 & b2 \\\\end{bmatrix})$$ and $$y=(\\\\begin{bmatrix} a1 & c1 \\\\\\\\ a2 & c2 \\\\end{bmatrix})/(\\\\begin{bmatrix} a1 & b1 \\\\\\\\ a2 & b2 \\\\end{bmatrix}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a372017cramer12a-h1"],"title":"Determinant","text":"Now, take the determinant of all matrices in the $$x=$$ section and simplify. What does $$x$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a372017cramer12a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$\\\\frac{1}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a372017cramer12a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["a372017cramer12a-h2"],"title":"Determinant","text":"Next, take the determinant of all matrices in the $$y=$$ section and simplify. What does $$y$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a372017cramer12a-h3-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$\\\\frac{1}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a372017cramer12a-h4","type":"hint","dependencies":["a372017cramer12a-h3"],"title":"Answer","text":"Therefore, the answer is $$(\\\\frac{1}{2},\\\\frac{1}{3})$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a372017cramer13","title":"Solve the system of linear equations using Cramer\u2019s Rule.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.8 Solving Systems with Cramer\'s Rule","courseName":"OpenStax: College Algebra","steps":[{"id":"a372017cramer13a","stepAnswer":["(3,1)"],"problemType":"TextBox","stepTitle":"$$2x+6y=12$$ $$5x-2y=13$$","stepBody":"Enter your solution as a coordinate pair.","answerType":"string","variabilization":{},"answerLatex":"$$(3,1)$$","hints":{"DefaultPathway":[{"id":"a372017cramer13a-h1","type":"hint","dependencies":[],"title":"Use the formula","text":"First, use the formula for Cramer\'s rule to solve for $$x$$ and $$y$$. If using a 2x2 matrix, the formula for it is $$x=(\\\\begin{bmatrix} c1 & b1 \\\\\\\\ c2 & b2 \\\\end{bmatrix})/(\\\\begin{bmatrix} a1 & b1 \\\\\\\\ a2 & b2 \\\\end{bmatrix})$$ and $$y=(\\\\begin{bmatrix} a1 & c1 \\\\\\\\ a2 & c2 \\\\end{bmatrix})/(\\\\begin{bmatrix} a1 & b1 \\\\\\\\ a2 & b2 \\\\end{bmatrix}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a372017cramer13a-h1"],"title":"Determinant","text":"Now, take the determinant of all matrices in the $$x=$$ section and simplify. What does $$x$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a372017cramer13a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a372017cramer13a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a372017cramer13a-h2"],"title":"Determinant","text":"Next, take the determinant of all matrices in the $$y=$$ section and simplify. What does $$y$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a372017cramer13a-h3-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a372017cramer13a-h4","type":"hint","dependencies":["a372017cramer13a-h3"],"title":"Answer","text":"Therefore, the answer is $$(3,1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a372017cramer14","title":"Solve the system of linear equations using Cramer\u2019s Rule.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.8 Solving Systems with Cramer\'s Rule","courseName":"OpenStax: College Algebra","steps":[{"id":"a372017cramer14a","stepAnswer":["(2,5)"],"problemType":"TextBox","stepTitle":"$$4x+3y=23$$ $$2x-y=-1$$","stepBody":"Enter your solution as a coordinate pair.","answerType":"string","variabilization":{},"answerLatex":"$$(2,5)$$","hints":{"DefaultPathway":[{"id":"a372017cramer14a-h1","type":"hint","dependencies":[],"title":"Use the formula","text":"First, use the formula for Cramer\'s rule to solve for $$x$$ and $$y$$. If using a 2x2 matrix, the formula for it is $$x=(\\\\begin{bmatrix} c1 & b1 \\\\\\\\ c2 & b2 \\\\end{bmatrix})/(\\\\begin{bmatrix} a1 & b1 \\\\\\\\ a2 & b2 \\\\end{bmatrix})$$ and $$y=(\\\\begin{bmatrix} a1 & c1 \\\\\\\\ a2 & c2 \\\\end{bmatrix})/(\\\\begin{bmatrix} a1 & b1 \\\\\\\\ a2 & b2 \\\\end{bmatrix}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a372017cramer14a-h1"],"title":"Determinant","text":"Now, take the determinant of all matrices in the $$x=$$ section and simplify. What does $$x$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a372017cramer14a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a372017cramer14a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a372017cramer14a-h2"],"title":"Determinant","text":"Next, take the determinant of all matrices in the $$y=$$ section and simplify. What does $$y$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a372017cramer14a-h3-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a372017cramer14a-h4","type":"hint","dependencies":["a372017cramer14a-h3"],"title":"Answer","text":"Therefore, the answer is $$(2,5)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a372017cramer15","title":"Solve the system of linear equations using Cramer\u2019s Rule.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.8 Solving Systems with Cramer\'s Rule","courseName":"OpenStax: College Algebra","steps":[{"id":"a372017cramer15a","stepAnswer":["(1/5,0)"],"problemType":"TextBox","stepTitle":"$$10x-6y=2$$ $$-5x+8y=-1$$","stepBody":"Enter your solution as a coordinate pair.","answerType":"string","variabilization":{},"answerLatex":"$$(\\\\frac{1}{5},0)$$","hints":{"DefaultPathway":[{"id":"a372017cramer15a-h1","type":"hint","dependencies":[],"title":"Use the formula","text":"First, use the formula for Cramer\'s rule to solve for $$x$$ and $$y$$. If using a 2x2 matrix, the formula for it is $$x=(\\\\begin{bmatrix} c1 & b1 \\\\\\\\ c2 & b2 \\\\end{bmatrix})/(\\\\begin{bmatrix} a1 & b1 \\\\\\\\ a2 & b2 \\\\end{bmatrix})$$ and $$y=(\\\\begin{bmatrix} a1 & c1 \\\\\\\\ a2 & c2 \\\\end{bmatrix})/(\\\\begin{bmatrix} a1 & b1 \\\\\\\\ a2 & b2 \\\\end{bmatrix}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{5}$$"],"dependencies":["a372017cramer15a-h1"],"title":"Determinant","text":"Now, take the determinant of all matrices in the $$x=$$ section and simplify. What does $$x$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a372017cramer15a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$\\\\frac{1}{5}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a372017cramer15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a372017cramer15a-h2"],"title":"Determinant","text":"Next, take the determinant of all matrices in the $$y=$$ section and simplify. What does $$y$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a372017cramer15a-h3-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a372017cramer15a-h4","type":"hint","dependencies":["a372017cramer15a-h3"],"title":"Answer","text":"Therefore, the answer is $$(\\\\frac{1}{5},0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a372017cramer16","title":"Finding the Determinant of a 2x2 Matrix","body":"Find the determinant of the given matrix","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.8 Solving Systems with Cramer\'s Rule","courseName":"OpenStax: College Algebra","steps":[{"id":"a372017cramer16a","stepAnswer":["$$27$$"],"problemType":"TextBox","stepTitle":"$$A=\\\\begin{bmatrix} 5 & 2 \\\\\\\\ -6 & 3 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$27$$","hints":{"DefaultPathway":[{"id":"a372017cramer16a-h1","type":"hint","dependencies":[],"title":"Determinant of 2x2 Matrix","text":"The determinant of a 2\xd72 matrix, given $$A=\\\\begin{bmatrix} a & b \\\\\\\\ c & d \\\\end{bmatrix}$$\\\\nis defined as $$det(A)=a d-c b$$.\\\\nNotice the change in notation. There are several ways to indicate the determinant, including det(A) and replacing the brackets in a matrix with straight lines, |A|.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$27$$"],"dependencies":["a372017cramer16a-h1"],"title":"Finding the Determinant","text":"Apply the formula that $$det(A)=a d-b c$$, what is the determinant?\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a372017cramer17","title":"Using Cramer\'s Rule to Solve a 2x2 System","body":"Solving the following 2x2 system using Cramer\'s Rule.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.8 Solving Systems with Cramer\'s Rule","courseName":"OpenStax: College Algebra","steps":[{"id":"a372017cramer17a","stepAnswer":["$$(2,-3)$$"],"problemType":"MultipleChoice","stepTitle":"$$12x+3y=15;$$\\\\n$$2x-3y=13$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(2,-3)$$","choices":["$$(2,-3)$$","$$(-3,2)$$"],"hints":{"DefaultPathway":[{"id":"a372017cramer17a-h1","type":"hint","dependencies":[],"title":"Cramer\'s Rule for 2x2 Systems","text":"Cramer\u2019s Rule is a method that uses determinants to solve systems of equations that have the same number of equations as variables.\\\\nConsider a system of two linear equations in two variables.\\\\n$$a_1 x+b_1 y=c_1$$\\\\n$$a_2 x+b_2 y=c_2$$\\\\n\\\\nThe solution using Cramer\'s Rule is given as\\\\n$$x=\\\\frac{D_x}{D}=$$ |(c_1,b_1),(c_2,b_2)|/|(a_1,b_1),(a_2,b_2)|, $$D \\\\neq 0;$$\\\\n$$y=\\\\frac{D_y}{D}=$$ |(a_1,c_1),(a_2,c_2)|/|(a_1,b_1),(a_2,b_2)|, $$D \\\\neq 0;$$\\\\nIf we are solving for $$x$$, the $$x$$ column is replaced with the constant column. If we are solving for $$y$$, the $$y$$ column is replaced with the constant column.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer17a-h2","type":"hint","dependencies":["a372017cramer17a-h1"],"title":"Determinant","text":"The determinant of a 2\xd72 matrix, given $$A=\\\\begin{bmatrix} a & b \\\\\\\\ c & d \\\\end{bmatrix}$$ is defined as $$det(A)=a d-c b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer17a-h3","type":"hint","dependencies":["a372017cramer17a-h2"],"title":"Solve for $$x$$","text":"Using the Cramer\'s Rule, replace the column for $$x$$ with the constant column, then find the determinant of $$D_x$$ and D to solve for $$x=\\\\frac{D_x}{D}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer17a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-84$$"],"dependencies":["a372017cramer17a-h3"],"title":"Solve for $$x$$","text":"Find the determinant $$D_x$$ $$=$$ $$|(15,3),(13,-3)|$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer17a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-42$$"],"dependencies":["a372017cramer17a-h4"],"title":"Solve for $$x$$","text":"Find the determinant D $$=$$ $$|(12,3),(2,-3)|$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer17a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a372017cramer17a-h5"],"title":"Solve for $$x$$","text":"What is $$\\\\frac{D_x}{D}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer17a-h7","type":"hint","dependencies":["a372017cramer17a-h6"],"title":"Solve for $$y$$","text":"Using the Cramer\'s Rule, replace the column for $$y$$ with the constant column, then find the determinant of $$D_y$$ and D to solve for $$y=\\\\frac{D_y}{D}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer17a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$126$$"],"dependencies":["a372017cramer17a-h7"],"title":"Solve for $$y$$","text":"Find the determinant $$D_y$$ $$=$$ $$|(12,15),(2,13)|$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer17a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-42$$"],"dependencies":["a372017cramer17a-h8"],"title":"Solve for $$y$$","text":"Find the determinant D $$=$$ $$|(12,3),(2,-3)|$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer17a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a372017cramer17a-h9"],"title":"Solve for $$y$$","text":"What is $$\\\\frac{D_y}{D}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a372017cramer18","title":"Using Cramer\'s Rule to Solve a 2x2 System","body":"Solving the following 2x2 system using Cramer\'s Rule.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.8 Solving Systems with Cramer\'s Rule","courseName":"OpenStax: College Algebra","steps":[{"id":"a372017cramer18a","stepAnswer":["$$(3,-7)$$"],"problemType":"MultipleChoice","stepTitle":"$$x+2y=-11;$$\\\\n$$-2x+y=-13$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(3,-7)$$","choices":["$$(3,-7)$$","$$(-7,3)$$"],"hints":{"DefaultPathway":[{"id":"a372017cramer18a-h1","type":"hint","dependencies":[],"title":"Cramer\'s Rule for 2x2 Systems","text":"Cramer\u2019s Rule is a method that uses determinants to solve systems of equations that have the same number of equations as variables.\\\\nConsider a system of two linear equations in two variables.\\\\n$$a_1 x+b_1 y=c_1$$\\\\n$$a_2 x+b_2 y=c_2$$\\\\n\\\\nThe solution using Cramer\'s Rule is given as\\\\n$$x=\\\\frac{D_x}{D}=$$ |(c_1,b_1),(c_2,b_2)|/|(a_1,b_1),(a_2,b_2)|, $$D \\\\neq 0;$$\\\\n$$y=\\\\frac{D_y}{D}=$$ |(a_1,c_1),(a_2,c_2)|/|(a_1,b_1),(a_2,b_2)|, $$D \\\\neq 0;$$\\\\nIf we are solving for $$x$$, the $$x$$ column is replaced with the constant column. If we are solving for $$y$$, the $$y$$ column is replaced with the constant column.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer18a-h2","type":"hint","dependencies":["a372017cramer18a-h1"],"title":"Determinant of 2x2 Matrix","text":"The determinant of a 2\xd72 matrix, given $$A=\\\\begin{bmatrix} a & b \\\\\\\\ c & d \\\\end{bmatrix}$$ is defined as $$det(A)=a d-c b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer18a-h3","type":"hint","dependencies":["a372017cramer18a-h2"],"title":"Solve for $$x$$","text":"Using the Cramer\'s Rule, replace the column for $$x$$ with the constant column, then find the determinant of $$D_x$$ and D to solve for $$x=\\\\frac{D_x}{D}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer18a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["a372017cramer18a-h3"],"title":"Solve for $$x$$","text":"Find the determinant $$D_x$$ $$=$$ $$|(-11,2),(-13,1)|$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer18a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a372017cramer18a-h4"],"title":"Solve for $$x$$","text":"Find the determinant D $$=$$ $$|(1,2),(-2,1)|$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer18a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a372017cramer18a-h5"],"title":"Solve for $$x$$","text":"What is $$\\\\frac{D_x}{D}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer18a-h7","type":"hint","dependencies":["a372017cramer18a-h6"],"title":"Solve for $$y$$","text":"Using the Cramer\'s Rule, replace the column for $$y$$ with the constant column, then find the determinant of $$D_y$$ and D to solve for $$y=\\\\frac{D_y}{D}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer18a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-35$$"],"dependencies":["a372017cramer18a-h7"],"title":"Solve for $$y$$","text":"Find the determinant $$D_y$$ $$=$$ $$|(1,-11),(-2,-13)|$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer18a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a372017cramer18a-h8"],"title":"Solve for $$y$$","text":"Find the determinant D $$=$$ $$|(1,2),(-2,1)|$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer18a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-7$$"],"dependencies":["a372017cramer18a-h9"],"title":"Solve for $$y$$","text":"What is $$\\\\frac{D_y}{D}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a372017cramer19","title":"Finding the Determinant of a 3x3 Matrix","body":"Find the determinant of the 3x3 matrix given.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.8 Solving Systems with Cramer\'s Rule","courseName":"OpenStax: College Algebra","steps":[{"id":"a372017cramer19a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"$$A=\\\\begin{bmatrix} 0 & 2 & 1 \\\\\\\\ 3 & -1 & 1 \\\\\\\\ 4 & 0 & 1 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"a372017cramer19a-h1","type":"hint","dependencies":[],"title":"Determinant of 3x3 Matrix","text":"Finding the determinant of a 2\xd72 matrix is straightforward, but finding the determinant of a 3\xd73 matrix is more complicated. One method is to augment the 3\xd73 matrix with a repetition of the first two columns, giving a 3\xd75 matrix. Then we calculate the sum of the products of entries down each of the three diagonals (upper left to lower right), and subtract the products of entries up each of the three diagonals (lower left to upper right). This is more easily understood with a visual and an example.\\\\nFor a matrix $$A=\\\\begin{bmatrix} a_1 & b_1 & c_1 \\\\\\\\ a_2 & b_2 & c_2 \\\\\\\\ a_3 & b_3 & c_3 \\\\end{bmatrix}$$,\\\\n$$1$$. Augment A with the first $$2$$ columns, det(A)=|(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|(a_1,b_1),(a_2,b_2),(a_3,b_3)|\\\\n$$2$$. From upper left to lower right: Multiply the entries down the first diagonal. Add the result to the product of entries down the second diagonal. Add this result to the product of the entries down the third diagonal.\\\\n$$3$$. From lower left to upper right: Subtract the product of entries up the first diagonal. From this result subtract the product of entries up the second diagonal. From this result, subtract the product of entries up the third diagonal.\\\\nThe algebra is as follows: $$|A|=a_1 b_2 c_3+b_1 c_2 a_3+c_1 a_2 b_3-a_3 b_2 c_1-b_3 c_2 a_1-c_3 a_2 b_1$$\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a372017cramer19a-h1"],"title":"Finding the Determinant","text":"Applying the formula $$|A|=a_1 b_2 c_3+b_1 c_2 a_3+c_1 a_2 b_3-a_3 b_2 c_1-b_3 c_2 a_1-c_3 a_2 b_1$$, what is the det(A) for the given matrix?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a372017cramer2","title":"Find the determinant.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.8 Solving Systems with Cramer\'s Rule","courseName":"OpenStax: College Algebra","steps":[{"id":"a372017cramer2a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"$$\\\\begin{bmatrix} 1 & 0 & 1 \\\\\\\\ 0 & 1 & 0 \\\\\\\\ 1 & 0 & 0 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"a372017cramer2a-h1","type":"hint","dependencies":[],"title":"Augument","text":"First, augument the matrix with the first two columns. In other words, duplicate the first two columns, and add them to the end of the matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer2a-h2","type":"hint","dependencies":["a372017cramer2a-h1"],"title":"Formula","text":"Now, follow the formula to find the determinant. First, multiply the entries down from the row $$1$$ column one spot, to the row $$3$$ column $$3$$ spot, in a diagonal fashion. Do this for the next two diagonals as well, from the row $$1$$ column $$2$$ spot, to the row $$3$$ column $$4$$ spot, and from the row $$1$$ column $$3$$ spot to the row $$3$$ column $$5$$ spot.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a372017cramer2a-h2"],"title":"Add","text":"Now, add all the previous products together. What do you get as your answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a372017cramer2a-h3-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a372017cramer2a-h4","type":"hint","dependencies":["a372017cramer2a-h3"],"title":"Multiply","text":"Next, multiply the values from the bottom left of the matrix to the location $$3$$ spots to the top in a diagonal. Another way to see this is a diagonal from the bottom to the top. Do this for the next $$2$$ columns.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer2a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a372017cramer2a-h4"],"title":"Subtract","text":"Now, subtract all the values that were just multiplied from the first set of diagonal multiplications(those that you calculated previously).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a372017cramer2a-h5-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a372017cramer2a-h6","type":"hint","dependencies":["a372017cramer2a-h5"],"title":"Answer","text":"Therefore, the determinant is $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a372017cramer20","title":"Finding the Determinant of a 3x3 Matrix","body":"Find the determinant of the 3x3 matrix given.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.8 Solving Systems with Cramer\'s Rule","courseName":"OpenStax: College Algebra","steps":[{"id":"a372017cramer20a","stepAnswer":["$$-10$$"],"problemType":"TextBox","stepTitle":"$$A=\\\\begin{bmatrix} 1 & -3 & 7 \\\\\\\\ 1 & 1 & 1 \\\\\\\\ 1 & -2 & 3 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-10$$","hints":{"DefaultPathway":[{"id":"a372017cramer20a-h1","type":"hint","dependencies":[],"title":"Determinant of 3x3 Matrix","text":"Finding the determinant of a 2\xd72 matrix is straightforward, but finding the determinant of a 3\xd73 matrix is more complicated. One method is to augment the 3\xd73 matrix with a repetition of the first two columns, giving a 3\xd75 matrix. Then we calculate the sum of the products of entries down each of the three diagonals (upper left to lower right), and subtract the products of entries up each of the three diagonals (lower left to upper right). This is more easily understood with a visual and an example.\\\\nFor a matrix $$A=\\\\begin{bmatrix} a_1 & b_1 & c_1 \\\\\\\\ a_2 & b_2 & c_2 \\\\\\\\ a_3 & b_3 & c_3 \\\\end{bmatrix}$$,\\\\n$$1$$. Augment A with the first $$2$$ columns, det(A)=|(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|(a_1,b_1),(a_2,b_2),(a_3,b_3)|\\\\n$$2$$. From upper left to lower right: Multiply the entries down the first diagonal. Add the result to the product of entries down the second diagonal. Add this result to the product of the entries down the third diagonal.\\\\n$$3$$. From lower left to upper right: Subtract the product of entries up the first diagonal. From this result subtract the product of entries up the second diagonal. From this result, subtract the product of entries up the third diagonal.\\\\nThe algebra is as follows: $$|A|=a_1 b_2 c_3+b_1 c_2 a_3+c_1 a_2 b_3-a_3 b_2 c_1-b_3 c_2 a_1-c_3 a_2 b_1$$\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-10$$"],"dependencies":["a372017cramer20a-h1"],"title":"Finding the Determinant","text":"Applying the formula $$|A|=a_1 b_2 c_3+b_1 c_2 a_3+c_1 a_2 b_3-a_3 b_2 c_1-b_3 c_2 a_1-c_3 a_2 b_1$$, what is the det(A) for the given matrix?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a372017cramer21","title":"Solving a 3x3 System Using Cramer\'s Rule","body":"Find the solution to the given 3x3 system using Cramer\'s Rule.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.8 Solving Systems with Cramer\'s Rule","courseName":"OpenStax: College Algebra","steps":[{"id":"a372017cramer21a","stepAnswer":["$$(1, 3, -2)$$"],"problemType":"MultipleChoice","stepTitle":"$$x+y-z=6;$$\\\\n$$3x-2y+z=-5;$$\\\\n$$x+3y-2z=14$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(1, 3, -2)$$","choices":["$$(1, 3, -2)$$","$$(1, -2, 3)$$","$$(3, 1, -2)$$"],"hints":{"DefaultPathway":[{"id":"a372017cramer21a-h1","type":"hint","dependencies":[],"title":"Cramer\'s Rule for 3x3 Systems","text":"We can apply Cramer\u2019s Rule to solve a system of three equations in three variables. Cramer\u2019s Rule is straightforward, following a pattern consistent with Cramer\u2019s Rule for $$2$$ \xd7 $$2$$ matrices.\\\\nConsider a 3x3 system of equations.\\\\n$$a_1 x+b_1 y+c_1 z=d_1$$\\\\n$$a_2 x+b_2 y+c_2 z=d_2$$\\\\n$$a_3 x+b_3 y+c_3 z=d_3$$\\\\n$$x=\\\\frac{D_x}{D}$$, $$y=\\\\frac{D_y}{D}$$, $$z=\\\\frac{D_z}{D}$$, $$D \\\\neq 0$$\\\\nIf we are writing the determinant $$D_x$$, we replace the $$x$$ column with the constant column. If we are writing the determinant $$D_y$$, we replace the $$y$$ column with the constant column. If we are writing the determinant $$D_z$$, we replace the $$z$$ column with the constant column.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer21a-h2","type":"hint","dependencies":["a372017cramer21a-h1"],"title":"Determinant of 3x3 Matrix","text":"Finding the determinant of a 2\xd72 matrix is straightforward, but finding the determinant of a 3\xd73 matrix is more complicated. One method is to augment the 3\xd73 matrix with a repetition of the first two columns, giving a 3\xd75 matrix. Then we calculate the sum of the products of entries down each of the three diagonals (upper left to lower right), and subtract the products of entries up each of the three diagonals (lower left to upper right). This is more easily understood with a visual and an example.\\\\nFor a matrix $$A=\\\\begin{bmatrix} a_1 & b_1 & c_1 \\\\\\\\ a_2 & b_2 & c_2 \\\\\\\\ a_3 & b_3 & c_3 \\\\end{bmatrix}$$,\\\\n$$1$$. Augment A with the first $$2$$ columns, det(A)=|(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|(a_1,b_1),(a_2,b_2),(a_3,b_3)|\\\\n$$2$$. From upper left to lower right: Multiply the entries down the first diagonal. Add the result to the product of entries down the second diagonal. Add this result to the product of the entries down the third diagonal.\\\\n$$3$$. From lower left to upper right: Subtract the product of entries up the first diagonal. From this result subtract the product of entries up the second diagonal. From this result, subtract the product of entries up the third diagonal.\\\\nThe algebra is as follows: $$|A|=a_1 b_2 c_3+b_1 c_2 a_3+c_1 a_2 b_3-a_3 b_2 c_1-b_3 c_2 a_1-c_3 a_2 b_1$$\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer21a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a372017cramer21a-h2"],"title":"Finding D","text":"Find the determinant $$D=|(1, 1, -1), (3, -2, 1), (1, 3, -2)|$$. We can use the formula for finding determinant of a 3x3 matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer21a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a372017cramer21a-h3"],"title":"Finding $$D_x$$","text":"Recall that we can replace the $$x$$ column with the constant column in the matrix to find the determinant of $$D_x$$. Find the determinant $$D_x=|(6, 1, -1), (-5, -2, 1), (14, 3, -2)|$$. We can use the formula for finding determinant of a 3x3 matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer21a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":["a372017cramer21a-h4"],"title":"Finding $$D_y$$","text":"Recall that we can replace the $$y$$ column with the constant column in the matrix to find the determinant of $$D_y$$. Find the determinant $$D_y=|(1, 6, -1), (3, -5, 1), (1, 14, -2)|$$. We can use the formula for finding determinant of a 3x3 matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer21a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a372017cramer21a-h5"],"title":"Finding $$D_z$$","text":"Recall that we can replace the $$z$$ column with the constant column in the matrix to find the determinant of $$D_z$$. Find the determinant $$D_z=|(1, 1, 6), (3, -2, -5), (1, 3, 14)|$$. We can use the formula for finding determinant of a 3x3 matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer21a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a372017cramer21a-h6"],"title":"Solving for $$x$$","text":"By Cramer\'s Rule, $$x=\\\\frac{D_x}{D}$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer21a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a372017cramer21a-h7"],"title":"Solving for $$y$$","text":"By Cramer\'s Rule, $$y=\\\\frac{D_y}{D}$$. What is $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer21a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a372017cramer21a-h8"],"title":"Solving for $$z$$","text":"By Cramer\'s Rule, $$z=\\\\frac{D_z}{D}$$. What is $$z$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a372017cramer22","title":"Solving a 3x3 System Using Cramer\'s Rule","body":"Find the solution to the given 3x3 system using Cramer\'s Rule.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.8 Solving Systems with Cramer\'s Rule","courseName":"OpenStax: College Algebra","steps":[{"id":"a372017cramer22a","stepAnswer":["(-2,3/5,12/5)"],"problemType":"MultipleChoice","stepTitle":"$$x-3y+7z=13;$$\\\\n$$x+y+z=1;$$\\\\n$$x-2y+3z=4$$","stepBody":"","answerType":"string","variabilization":{},"choices":["(-2,3/5,12/5)","(2,3/5,12/5)","(3/5,-2,12/5)"],"hints":{"DefaultPathway":[{"id":"a372017cramer22a-h1","type":"hint","dependencies":[],"title":"Cramer\'s Rule for 3x3 Systems","text":"We can apply Cramer\u2019s Rule to solve a system of three equations in three variables. Cramer\u2019s Rule is straightforward, following a pattern consistent with Cramer\u2019s Rule for $$2$$ \xd7 $$2$$ matrices.\\\\nConsider a 3x3 system of equations.\\\\n$$a_1 x+b_1 y+c_1 z=d_1$$\\\\n$$a_2 x+b_2 y+c_2 z=d_2$$\\\\n$$a_3 x+b_3 y+c_3 z=d_3$$\\\\n$$x=\\\\frac{D_x}{D}$$, $$y=\\\\frac{D_y}{D}$$, $$z=\\\\frac{D_z}{D}$$, $$D \\\\neq 0$$\\\\nIf we are writing the determinant $$D_x$$, we replace the $$x$$ column with the constant column. If we are writing the determinant $$D_y$$, we replace the $$y$$ column with the constant column. If we are writing the determinant $$D_z$$, we replace the $$z$$ column with the constant column.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer22a-h2","type":"hint","dependencies":["a372017cramer22a-h1"],"title":"Determinant of 3x3 Matrix","text":"Finding the determinant of a 2\xd72 matrix is straightforward, but finding the determinant of a 3\xd73 matrix is more complicated. One method is to augment the 3\xd73 matrix with a repetition of the first two columns, giving a 3\xd75 matrix. Then we calculate the sum of the products of entries down each of the three diagonals (upper left to lower right), and subtract the products of entries up each of the three diagonals (lower left to upper right). This is more easily understood with a visual and an example.\\\\nFor a matrix $$A=\\\\begin{bmatrix} a_1 & b_1 & c_1 \\\\\\\\ a_2 & b_2 & c_2 \\\\\\\\ a_3 & b_3 & c_3 \\\\end{bmatrix}$$,\\\\n$$1$$. Augment A with the first $$2$$ columns, det(A)=|(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|(a_1,b_1),(a_2,b_2),(a_3,b_3)|\\\\n$$2$$. From upper left to lower right: Multiply the entries down the first diagonal. Add the result to the product of entries down the second diagonal. Add this result to the product of the entries down the third diagonal.\\\\n$$3$$. From lower left to upper right: Subtract the product of entries up the first diagonal. From this result subtract the product of entries up the second diagonal. From this result, subtract the product of entries up the third diagonal.\\\\nThe algebra is as follows: $$|A|=a_1 b_2 c_3+b_1 c_2 a_3+c_1 a_2 b_3-a_3 b_2 c_1-b_3 c_2 a_1-c_3 a_2 b_1$$\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer22a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-10$$"],"dependencies":["a372017cramer22a-h2"],"title":"Finding D","text":"Find the determinant $$D=|(1, 1, -1), (3, -2, 1), (1, 3, -2)|$$. We can use the formula for finding determinant of a 3x3 matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer22a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a372017cramer22a-h3"],"title":"Finding $$D_x$$","text":"Recall that we can replace the $$x$$ column with the constant column in the matrix to find the determinant of $$D_x$$. Find the determinant $$D_x=|(6, 1, -1), (-5, -2, 1), (14, 3, -2)|$$. We can use the formula for finding determinant of a 3x3 matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer22a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6$$"],"dependencies":["a372017cramer22a-h4"],"title":"Finding $$D_y$$","text":"Recall that we can replace the $$y$$ column with the constant column in the matrix to find the determinant of $$D_y$$. Find the determinant $$D_y=|(1, 6, -1), (3, -5, 1), (1, 14, -2)|$$. We can use the formula for finding determinant of a 3x3 matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer22a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-24$$"],"dependencies":["a372017cramer22a-h5"],"title":"Finding $$D_z$$","text":"Recall that we can replace the $$z$$ column with the constant column in the matrix to find the determinant of $$D_z$$. Find the determinant $$D_z=|(1, 1, 6), (3, -2, -5), (1, 3, 14)|$$. We can use the formula for finding determinant of a 3x3 matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer22a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a372017cramer22a-h6"],"title":"Solving for $$x$$","text":"By Cramer\'s Rule, $$x=\\\\frac{D_x}{D}$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer22a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{5}$$"],"dependencies":["a372017cramer22a-h7"],"title":"Solving for $$y$$","text":"By Cramer\'s Rule, $$y=\\\\frac{D_y}{D}$$. What is $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer22a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{12}{5}$$"],"dependencies":["a372017cramer22a-h8"],"title":"Solving for $$z$$","text":"By Cramer\'s Rule, $$z=\\\\frac{D_z}{D}$$. What is $$z$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a372017cramer23","title":"Using Cramer\'s Rule to Solve an Inconsistent System","body":"Solve the system using Cramer\'s Rule.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.8 Solving Systems with Cramer\'s Rule","courseName":"OpenStax: College Algebra","steps":[{"id":"a372017cramer23a","stepAnswer":["Inconsistent"],"problemType":"MultipleChoice","stepTitle":"$$3x-2y=4;$$\\\\n$$6x-4y=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$(2,1)$$","$$(4,6)$$","Inconsistent","Infinite Solutions"],"hints":{"DefaultPathway":[{"id":"a372017cramer23a-h1","type":"hint","dependencies":[],"title":"Cramer\'s Rule for 2x2 Systems","text":"Cramer\u2019s Rule is a method that uses determinants to solve systems of equations that have the same number of equations as variables.\\\\nConsider a system of two linear equations in two variables.\\\\n$$a_1 x+b_1 y=c_1$$\\\\n$$a_2 x+b_2 y=c_2$$\\\\n\\\\nThe solution using Cramer\'s Rule is given as\\\\n$$x=\\\\frac{D_x}{D}=$$ |(c_1,b_1),(c_2,b_2)|/|(a_1,b_1),(a_2,b_2)|, $$D \\\\neq 0;$$\\\\n$$y=\\\\frac{D_y}{D}=$$ |(a_1,c_1),(a_2,c_2)|/|(a_1,b_1),(a_2,b_2)|, $$D \\\\neq 0;$$\\\\nIf we are solving for $$x$$, the $$x$$ column is replaced with the constant column. If we are solving for $$y$$, the $$y$$ column is replaced with the constant column.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer23a-h2","type":"hint","dependencies":["a372017cramer23a-h1"],"title":"Determinant of 2x2 Matrix","text":"The determinant of a 2\xd72 matrix, given $$A=\\\\begin{bmatrix} a & b \\\\\\\\ c & d \\\\end{bmatrix}$$ is defined as $$det(A)=a d-c b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer23a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a372017cramer23a-h2"],"title":"Finding D","text":"Find the determinant $$D=|(3,-2),(6,-4)|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer23a-h4","type":"hint","dependencies":["a372017cramer23a-h3"],"title":"Determinant of Zero","text":"A determinant of zero means that either the system has no solution or it has an infinite number of solutions. We use the process of elimination to see which one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer23a-h5","type":"hint","dependencies":["a372017cramer23a-h4"],"title":"Elimination","text":"When there are like terms in both equation, we can add or subtract the common terms so that we are left with only one variable that we can solve.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer23a-h6","type":"hint","dependencies":["a372017cramer23a-h5"],"title":"Elimination","text":"Multiply the first equation by $$-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer23a-h7","type":"hint","dependencies":["a372017cramer23a-h6"],"title":"Elimination","text":"Add the result to the second equation. What do we notice?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer23a-h8","type":"hint","dependencies":["a372017cramer23a-h7"],"title":"Inconsistency","text":"Adding the scaled first equation with the second equation, we obtain the equation $$0=-8$$, which is false. Therefore, the system has no solution. Graphing the system reveals two parallel lines.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a372017cramer24","title":"Using Cramer\'s Rule to Solve a Dependent System","body":"Solve the system with an infinite number of solution.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.8 Solving Systems with Cramer\'s Rule","courseName":"OpenStax: College Algebra","steps":[{"id":"a372017cramer24a","stepAnswer":["Infinite Solutions"],"problemType":"MultipleChoice","stepTitle":"$$x-2y+3z=0;$$\\\\n$$3x+y-2z=0;$$\\\\n$$2x-4y+6z=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$(-1, 1, 1)$$","$$(1, -1, 1)$$","Inconsistent","Infinite Solutions"],"hints":{"DefaultPathway":[{"id":"a372017cramer24a-h1","type":"hint","dependencies":[],"title":"Cramer\'s Rule for 3x3 Systems","text":"We can apply Cramer\u2019s Rule to solve a system of three equations in three variables. Cramer\u2019s Rule is straightforward, following a pattern consistent with Cramer\u2019s Rule for $$2$$ \xd7 $$2$$ matrices.\\\\nConsider a 3x3 system of equations.\\\\n$$a_1 x+b_1 y+c_1 z=d_1$$\\\\n$$a_2 x+b_2 y+c_2 z=d_2$$\\\\n$$a_3 x+b_3 y+c_3 z=d_3$$\\\\n$$x=\\\\frac{D_x}{D}$$, $$y=\\\\frac{D_y}{D}$$, $$z=\\\\frac{D_z}{D}$$, $$D \\\\neq 0$$\\\\nIf we are writing the determinant $$D_x$$, we replace the $$x$$ column with the constant column. If we are writing the determinant $$D_y$$, we replace the $$y$$ column with the constant column. If we are writing the determinant $$D_z$$, we replace the $$z$$ column with the constant column.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer24a-h2","type":"hint","dependencies":["a372017cramer24a-h1"],"title":"Determinant of 3x3 Matrix","text":"Finding the determinant of a 2\xd72 matrix is straightforward, but finding the determinant of a 3\xd73 matrix is more complicated. One method is to augment the 3\xd73 matrix with a repetition of the first two columns, giving a 3\xd75 matrix. Then we calculate the sum of the products of entries down each of the three diagonals (upper left to lower right), and subtract the products of entries up each of the three diagonals (lower left to upper right). This is more easily understood with a visual and an example.\\\\nFor a matrix $$A=\\\\begin{bmatrix} a_1 & b_1 & c_1 \\\\\\\\ a_2 & b_2 & c_2 \\\\\\\\ a_3 & b_3 & c_3 \\\\end{bmatrix}$$,\\\\n$$1$$. Augment A with the first $$2$$ columns, det(A)=|(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|(a_1,b_1),(a_2,b_2),(a_3,b_3)|\\\\n$$2$$. From upper left to lower right: Multiply the entries down the first diagonal. Add the result to the product of entries down the second diagonal. Add this result to the product of the entries down the third diagonal.\\\\n$$3$$. From lower left to upper right: Subtract the product of entries up the first diagonal. From this result subtract the product of entries up the second diagonal. From this result, subtract the product of entries up the third diagonal.\\\\nThe algebra is as follows: $$|A|=a_1 b_2 c_3+b_1 c_2 a_3+c_1 a_2 b_3-a_3 b_2 c_1-b_3 c_2 a_1-c_3 a_2 b_1$$\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer24a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a372017cramer24a-h2"],"title":"Finding D","text":"Find the determinant $$D=|(1, -2, 3), (3, 1, -2), (2, -4, 6)|$$. We can use the formula for finding determinant of a 3x3 matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer24a-h4","type":"hint","dependencies":["a372017cramer24a-h3"],"title":"Determinant of Zero","text":"A determinant of zero means that either the system has no solution or it has an infinite number of solutions. We use the process of elimination to see which one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer24a-h5","type":"hint","dependencies":["a372017cramer24a-h4"],"title":"Eliminating Terms","text":"When there are like terms in both equation, we can add or subtract the common terms so that we are left with only one variable that we can solve.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer24a-h6","type":"hint","dependencies":["a372017cramer24a-h5"],"title":"Eliminating Terms","text":"Multiply the first equation by $$-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer24a-h7","type":"hint","dependencies":["a372017cramer24a-h6"],"title":"Eliminating Terms","text":"Add the result to the third equation. What do we notice?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer24a-h8","type":"hint","dependencies":["a372017cramer24a-h7"],"title":"Interpreting the Solution","text":"Adding the scaled first equation to the third equation, we obtain $$0=0$$, a statement that is always true. This means that the system has an infinite number of solutions. Graphing the system, we see that two of the planes are the same and they both intersect the third plane on a line.\\\\n##figure3.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a372017cramer25","title":"Using Cramer\'s Rule and Determinant Properties to Solve a System","body":"Find the solution to the given 3x3 system. Is it Consistent or Inconsistent?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.8 Solving Systems with Cramer\'s Rule","courseName":"OpenStax: College Algebra","steps":[{"id":"a372017cramer25a","stepAnswer":["Inconsistent"],"problemType":"MultipleChoice","stepTitle":"$$2x+4y+4z=2;$$\\\\n$$3x+7y+7z=-5;$$\\\\n$$x+2y+2z=4$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Consistent","Inconsistent"],"hints":{"DefaultPathway":[{"id":"a372017cramer25a-h1","type":"hint","dependencies":[],"title":"Cramer\'s Rule for 3x3 Systems","text":"We can apply Cramer\u2019s Rule to solve a system of three equations in three variables. Cramer\u2019s Rule is straightforward, following a pattern consistent with Cramer\u2019s Rule for $$2$$ \xd7 $$2$$ matrices.\\\\nConsider a 3x3 system of equations.\\\\n$$a_1 x+b_1 y+c_1 z=d_1$$\\\\n$$a_2 x+b_2 y+c_2 z=d_2$$\\\\n$$a_3 x+b_3 y+c_3 z=d_3$$\\\\n$$x=\\\\frac{D_x}{D}$$, $$y=\\\\frac{D_y}{D}$$, $$z=\\\\frac{D_z}{D}$$, $$D \\\\neq 0$$\\\\nIf we are writing the determinant $$D_x$$, we replace the $$x$$ column with the constant column. If we are writing the determinant $$D_y$$, we replace the $$y$$ column with the constant column. If we are writing the determinant $$D_z$$, we replace the $$z$$ column with the constant column.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer25a-h2","type":"hint","dependencies":["a372017cramer25a-h1"],"title":"Determinant of 3x3 Matrix","text":"Finding the determinant of a 2\xd72 matrix is straightforward, but finding the determinant of a 3\xd73 matrix is more complicated. One method is to augment the 3\xd73 matrix with a repetition of the first two columns, giving a 3\xd75 matrix. Then we calculate the sum of the products of entries down each of the three diagonals (upper left to lower right), and subtract the products of entries up each of the three diagonals (lower left to upper right). This is more easily understood with a visual and an example.\\\\nFor a matrix $$A=\\\\begin{bmatrix} a_1 & b_1 & c_1 \\\\\\\\ a_2 & b_2 & c_2 \\\\\\\\ a_3 & b_3 & c_3 \\\\end{bmatrix}$$,\\\\n$$1$$. Augment A with the first $$2$$ columns, det(A)=|(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|(a_1,b_1),(a_2,b_2),(a_3,b_3)|\\\\n$$2$$. From upper left to lower right: Multiply the entries down the first diagonal. Add the result to the product of entries down the second diagonal. Add this result to the product of the entries down the third diagonal.\\\\n$$3$$. From lower left to upper right: Subtract the product of entries up the first diagonal. From this result subtract the product of entries up the second diagonal. From this result, subtract the product of entries up the third diagonal.\\\\nThe algebra is as follows: $$|A|=a_1 b_2 c_3+b_1 c_2 a_3+c_1 a_2 b_3-a_3 b_2 c_1-b_3 c_2 a_1-c_3 a_2 b_1$$\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer25a-h3","type":"hint","dependencies":["a372017cramer25a-h2"],"title":"Properties of Determinants","text":"$$1$$. If the matrix is in upper triangular form, the determinant equals the product of entries down the main diagonal.\\\\n$$2$$. When two rows are interchanged, the determinant changes sign.\\\\n$$3$$. If either two rows or two columns are identical, the determinant equals zero.\\\\n$$4$$. If a matrix contains either a row of zeros or a column of zeros, the determinant equals zero.\\\\n$$5$$. The determinant of an inverse matrix A-1 is the reciprocal of the determinant of the matrix A.\\\\n$$6$$. If any row or column is multiplied by a constant, the determinant is multiplied by the same factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer25a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a372017cramer25a-h3"],"title":"Finding D","text":"Using Cramer\'s Rule, we have that $$D=|(2, 4, 4), (3, 7, 7), (1, 2, 2)|$$. Observe that the second and third columns are identical. Which property, from $$1$$ to $$6$$, can we use here?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer25a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a372017cramer25a-h4"],"title":"Finding D","text":"What is D?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer25a-h6","type":"hint","dependencies":["a372017cramer25a-h5"],"title":"Determinant of Zero","text":"A determinant of zero means that either the system has no solution or it has an infinite number of solutions. We use the process of elimination to see which one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer25a-h7","type":"hint","dependencies":["a372017cramer25a-h6"],"title":"Eliminating Terms","text":"When there are like terms in both equation, we can add or subtract the common terms so that we are left with only one variable that we can solve.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer25a-h8","type":"hint","dependencies":["a372017cramer25a-h7"],"title":"Eliminating Terms","text":"Multiply the third equation by $$-2$$ and add the result to the first equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer25a-h9","type":"hint","dependencies":["a372017cramer25a-h8"],"title":"Interpreting the Solution","text":"Adding the scaled third equation with the first equation, we obtain the equation $$0=-6$$, which is false. Therefore, the system has no solution..","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a372017cramer26","title":"Finding the Determinant of a 2x2 Matrix","body":"Find the determinant of the given matrix","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.8 Solving Systems with Cramer\'s Rule","courseName":"OpenStax: College Algebra","steps":[{"id":"a372017cramer26a","stepAnswer":["$$7$$"],"problemType":"TextBox","stepTitle":"$$A=\\\\begin{bmatrix} 2 & -5 \\\\\\\\ -1 & 6 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7$$","hints":{"DefaultPathway":[{"id":"a372017cramer26a-h1","type":"hint","dependencies":[],"title":"Determinant of 2x2 Matrix","text":"The determinant of a 2\xd72 matrix, given $$A=\\\\begin{bmatrix} a & b \\\\\\\\ c & d \\\\end{bmatrix}$$\\\\nis defined as $$det(A)=a d-c b$$.\\\\nNotice the change in notation. There are several ways to indicate the determinant, including det(A) and replacing the brackets in a matrix with straight lines, |A|.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer26a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a372017cramer26a-h1"],"title":"Finding the Determinant","text":"Apply the formula that $$det(A)=a d-b c$$, what is the determinant?\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a372017cramer27","title":"Finding the Determinant of a 3x3 Matrix","body":"Find the determinant of the 3x3 matrix given.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.8 Solving Systems with Cramer\'s Rule","courseName":"OpenStax: College Algebra","steps":[{"id":"a372017cramer27a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"$$A=\\\\begin{bmatrix} -1 & 0 & 0 \\\\\\\\ 0 & 1 & 0 \\\\\\\\ 0 & 0 & -3 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a372017cramer27a-h1","type":"hint","dependencies":[],"title":"Determinant of 3x3 Matrix","text":"Finding the determinant of a 2\xd72 matrix is straightforward, but finding the determinant of a 3\xd73 matrix is more complicated. One method is to augment the 3\xd73 matrix with a repetition of the first two columns, giving a 3\xd75 matrix. Then we calculate the sum of the products of entries down each of the three diagonals (upper left to lower right), and subtract the products of entries up each of the three diagonals (lower left to upper right). This is more easily understood with a visual and an example.\\\\nFor a matrix $$A=\\\\begin{bmatrix} a_1 & b_1 & c_1 \\\\\\\\ a_2 & b_2 & c_2 \\\\\\\\ a_3 & b_3 & c_3 \\\\end{bmatrix}$$,\\\\n$$1$$. Augment A with the first $$2$$ columns, det(A)=|(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|(a_1,b_1),(a_2,b_2),(a_3,b_3)|\\\\n$$2$$. From upper left to lower right: Multiply the entries down the first diagonal. Add the result to the product of entries down the second diagonal. Add this result to the product of the entries down the third diagonal.\\\\n$$3$$. From lower left to upper right: Subtract the product of entries up the first diagonal. From this result subtract the product of entries up the second diagonal. From this result, subtract the product of entries up the third diagonal.\\\\nThe algebra is as follows: $$|A|=a_1 b_2 c_3+b_1 c_2 a_3+c_1 a_2 b_3-a_3 b_2 c_1-b_3 c_2 a_1-c_3 a_2 b_1$$\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer27a-h2","type":"hint","dependencies":["a372017cramer27a-h1"],"title":"Properties of Determinants","text":"$$1$$. If the matrix is in upper triangular form, the determinant equals the product of entries down the main diagonal.\\\\n$$2$$. When two rows are interchanged, the determinant changes sign.\\\\n$$3$$. If either two rows or two columns are identical, the determinant equals zero.\\\\n$$4$$. If a matrix contains either a row of zeros or a column of zeros, the determinant equals zero.\\\\n$$5$$. The determinant of an inverse matrix A-1 is the reciprocal of the determinant of the matrix A.\\\\n$$6$$. If any row or column is multiplied by a constant, the determinant is multiplied by the same factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer27a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a372017cramer27a-h2"],"title":"Finding D","text":"We have that $$D=|(-1, 4, 0), (0, 2, -3), (0, 0, -3)|$$. Observe that the matrix is in upper triangular form. Which property, from $$1$$ to $$6$$, can we use here?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer27a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a372017cramer27a-h3"],"title":"Finding D","text":"What is D?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a372017cramer28","title":"Finding the Determinant of a 2x2 Matrix","body":"Find the determinant of the given matrix","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.8 Solving Systems with Cramer\'s Rule","courseName":"OpenStax: College Algebra","steps":[{"id":"a372017cramer28a","stepAnswer":["$$-100$$"],"problemType":"TextBox","stepTitle":"$$A=\\\\begin{bmatrix} 10 & 20 \\\\\\\\ 0 & -10 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-100$$","hints":{"DefaultPathway":[{"id":"a372017cramer28a-h1","type":"hint","dependencies":[],"title":"Determinant of 2x2 Matrix","text":"The determinant of a 2\xd72 matrix, given $$A=\\\\begin{bmatrix} a & b \\\\\\\\ c & d \\\\end{bmatrix}$$\\\\nis defined as $$det(A)=a d-c b$$.\\\\nNotice the change in notation. There are several ways to indicate the determinant, including det(A) and replacing the brackets in a matrix with straight lines, |A|.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer28a-h2","type":"hint","dependencies":["a372017cramer28a-h1"],"title":"Properties of Determinants","text":"$$1$$. If the matrix is in upper triangular form, the determinant equals the product of entries down the main diagonal.\\\\n$$2$$. When two rows are interchanged, the determinant changes sign.\\\\n$$3$$. If either two rows or two columns are identical, the determinant equals zero.\\\\n$$4$$. If a matrix contains either a row of zeros or a column of zeros, the determinant equals zero.\\\\n$$5$$. The determinant of an inverse matrix A-1 is the reciprocal of the determinant of the matrix A.\\\\n$$6$$. If any row or column is multiplied by a constant, the determinant is multiplied by the same factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer28a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a372017cramer28a-h2"],"title":"Finding D","text":"We have that $$D=|(10,20),(0,-10)|$$. Observe that the matrix is in upper triangular form. Which property, from $$1$$ to $$6$$, can we use here?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer28a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-100$$"],"dependencies":["a372017cramer28a-h3"],"title":"Finding D","text":"What is D?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a372017cramer29","title":"Finding the Determinant of a 2x2 Matrix","body":"Find the determinant of the given matrix","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.8 Solving Systems with Cramer\'s Rule","courseName":"OpenStax: College Algebra","steps":[{"id":"a372017cramer29a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"$$A=\\\\begin{bmatrix} 10 & 0.2 \\\\\\\\ 5 & 0.1 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a372017cramer29a-h1","type":"hint","dependencies":[],"title":"Determinant of 2x2 Matrix","text":"The determinant of a 2\xd72 matrix, given $$A=\\\\begin{bmatrix} a & b \\\\\\\\ c & d \\\\end{bmatrix}$$\\\\nis defined as $$det(A)=a d-c b$$.\\\\nNotice the change in notation. There are several ways to indicate the determinant, including det(A) and replacing the brackets in a matrix with straight lines, |A|.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer29a-h2","type":"hint","dependencies":["a372017cramer29a-h1"],"title":"Properties of Determinants","text":"$$1$$. If the matrix is in upper triangular form, the determinant equals the product of entries down the main diagonal.\\\\n$$2$$. When two rows are interchanged, the determinant changes sign.\\\\n$$3$$. If either two rows or two columns are identical, the determinant equals zero.\\\\n$$4$$. If a matrix contains either a row of zeros or a column of zeros, the determinant equals zero.\\\\n$$5$$. The determinant of an inverse matrix A-1 is the reciprocal of the determinant of the matrix A.\\\\n$$6$$. If any row or column is multiplied by a constant, the determinant is multiplied by the same factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer29a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a372017cramer29a-h2"],"title":"Finding D","text":"We have that $$D=|(10, 0.2), (5, 0.1)|$$. A common trick is to multiply a row or column by a constant so that we can utilize other properties later. Which property, from $$1$$ to $$6$$, is that?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer29a-h4","type":"hint","dependencies":["a372017cramer29a-h3"],"title":"Finding D","text":"Using property $$6$$, we multiply the second column by $$50$$ from $$\\\\begin{bmatrix} 0.2 \\\\\\\\ 0.1 \\\\end{bmatrix}$$ to $$\\\\begin{bmatrix} 10 \\\\\\\\ 5 \\\\end{bmatrix}$$. This would mean that the determinant would be multiplied by $$50$$ later on.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer29a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a372017cramer29a-h4"],"title":"Finding D","text":"We now have that $$D=|(10,10),(5,5)|$$. We realize that both columns are identical. Which property, from $$1$$ to $$6$$, can we use here?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer29a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a372017cramer29a-h5"],"title":"Finding D","text":"What is D? We thus observe that by extension of Property $$3$$ and $$6$$, that if $$2$$ rows or columns are a scalar multiple of each other, that the determinant is zero as well.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a372017cramer3","title":"Find the determinant.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.8 Solving Systems with Cramer\'s Rule","courseName":"OpenStax: College Algebra","steps":[{"id":"a372017cramer3a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"$$\\\\begin{bmatrix} 2 & -3 & 1 \\\\\\\\ 3 & -4 & 1 \\\\\\\\ -5 & 6 & 1 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a372017cramer3a-h1","type":"hint","dependencies":[],"title":"Augument","text":"First, augument the matrix with the first two columns. In other words, duplicate the first two columns, and add them to the end of the matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer3a-h2","type":"hint","dependencies":["a372017cramer3a-h1"],"title":"Formula","text":"Now, follow the formula to find the determinant. First, multiply the entries down from the row $$1$$ column one spot, to the row $$3$$ column $$3$$ spot, in a diagonal fashion. Do this for the next two diagonals as well, from the row $$1$$ column $$2$$ spot, to the row $$3$$ column $$4$$ spot, and from the row $$1$$ column $$3$$ spot to the row $$3$$ column $$5$$ spot.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["a372017cramer3a-h2"],"title":"Add","text":"Now, add all the previous products together. What do you get as your answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a372017cramer3a-h3-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a372017cramer3a-h4","type":"hint","dependencies":["a372017cramer3a-h3"],"title":"Multiply","text":"Next, multiply the values from the bottom left of the matrix to the location $$3$$ spots to the top in a diagonal. Another way to see this is a diagonal from the bottom to the top. Do this for the next $$2$$ columns.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a372017cramer3a-h4"],"title":"Subtract","text":"Now, subtract all the values that were just multiplied from the first set of diagonal multiplications(those that you calculated previously).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a372017cramer3a-h5-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a372017cramer3a-h6","type":"hint","dependencies":["a372017cramer3a-h5"],"title":"Answer","text":"Therefore, the determinant is $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a372017cramer30","title":"Finding the Determinant of a 2x2 Matrix","body":"Find the determinant of the given matrix","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.8 Solving Systems with Cramer\'s Rule","courseName":"OpenStax: College Algebra","steps":[{"id":"a372017cramer30a","stepAnswer":["$$-4$$"],"problemType":"TextBox","stepTitle":"$$A=\\\\begin{bmatrix} 1 & 0 \\\\\\\\ 3 & -4 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-4$$","hints":{"DefaultPathway":[{"id":"a372017cramer30a-h1","type":"hint","dependencies":[],"title":"Determinant of 2x2 Matrix","text":"The determinant of a 2\xd72 matrix, given $$A=\\\\begin{bmatrix} a & b \\\\\\\\ c & d \\\\end{bmatrix}$$\\\\nis defined as $$det(A)=a d-c b$$.\\\\nNotice the change in notation. There are several ways to indicate the determinant, including det(A) and replacing the brackets in a matrix with straight lines, |A|.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer30a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["a372017cramer30a-h1"],"title":"Finding the Determinant","text":"Apply the formula that $$det(A)=a d-b c$$, what is the determinant?\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a372017cramer4","title":"Find the determinant.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.8 Solving Systems with Cramer\'s Rule","courseName":"OpenStax: College Algebra","steps":[{"id":"a372017cramer4a","stepAnswer":["$$224$$"],"problemType":"TextBox","stepTitle":"$$\\\\begin{bmatrix} -2 & 1 & 4 \\\\\\\\ -4 & 2 & -8 \\\\\\\\ 2 & -8 & -3 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$224$$","hints":{"DefaultPathway":[{"id":"a372017cramer4a-h1","type":"hint","dependencies":[],"title":"Augument","text":"First, augument the matrix with the first two columns. In other words, duplicate the first two columns, and add them to the end of the matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer4a-h2","type":"hint","dependencies":["a372017cramer4a-h1"],"title":"Formula","text":"Now, follow the formula to find the determinant. First, multiply the entries down from the row $$1$$ column one spot, to the row $$3$$ column $$3$$ spot, in a diagonal fashion. Do this for the next two diagonals as well, from the row $$1$$ column $$2$$ spot, to the row $$3$$ column $$4$$ spot, and from the row $$1$$ column $$3$$ spot to the row $$3$$ column $$5$$ spot.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$124$$"],"dependencies":["a372017cramer4a-h2"],"title":"Add","text":"Now, add all the previous products together. What do you get as your answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a372017cramer4a-h3-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$124$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a372017cramer4a-h4","type":"hint","dependencies":["a372017cramer4a-h3"],"title":"Multiply","text":"Next, multiply the values from the bottom left of the matrix to the location $$3$$ spots to the top in a diagonal. Another way to see this is a diagonal from the bottom to the top. Do this for the next $$2$$ columns.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer4a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$224$$"],"dependencies":["a372017cramer4a-h4"],"title":"Subtract","text":"Now, subtract all the values that were just multiplied from the first set of diagonal multiplications(those that you calculated previously).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a372017cramer4a-h5-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$224$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a372017cramer4a-h6","type":"hint","dependencies":["a372017cramer4a-h5"],"title":"Answer","text":"Therefore, the determinant is $$224$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a372017cramer5","title":"Find the determinant.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.8 Solving Systems with Cramer\'s Rule","courseName":"OpenStax: College Algebra","steps":[{"id":"a372017cramer5a","stepAnswer":["$$-319$$"],"problemType":"TextBox","stepTitle":"$$\\\\begin{bmatrix} 6 & -1 & 2 \\\\\\\\ -4 & -3 & 5 \\\\\\\\ 1 & 9 & -1 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-319$$","hints":{"DefaultPathway":[{"id":"a372017cramer5a-h1","type":"hint","dependencies":[],"title":"Augument","text":"First, augument the matrix with the first two columns. In other words, duplicate the first two columns, and add them to the end of the matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer5a-h2","type":"hint","dependencies":["a372017cramer5a-h1"],"title":"Formula","text":"Now, follow the formula to find the determinant. First, multiply the entries down from the row $$1$$ column one spot, to the row $$3$$ column $$3$$ spot, in a diagonal fashion. Do this for the next two diagonals as well, from the row $$1$$ column $$2$$ spot, to the row $$3$$ column $$4$$ spot, and from the row $$1$$ column $$3$$ spot to the row $$3$$ column $$5$$ spot.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-59$$"],"dependencies":["a372017cramer5a-h2"],"title":"Add","text":"Now, add all the previous products together. What do you get as your answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a372017cramer5a-h3-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$-59$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a372017cramer5a-h4","type":"hint","dependencies":["a372017cramer5a-h3"],"title":"Multiply","text":"Next, multiply the values from the bottom left of the matrix to the location $$3$$ spots to the top in a diagonal. Another way to see this is a diagonal from the bottom to the top. Do this for the next $$2$$ columns.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-319$$"],"dependencies":["a372017cramer5a-h4"],"title":"Subtract","text":"Now, subtract all the values that were just multiplied from the first set of diagonal multiplications(those that you calculated previously).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a372017cramer5a-h5-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$-319$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a372017cramer5a-h6","type":"hint","dependencies":["a372017cramer5a-h5"],"title":"Answer","text":"Therefore, the determinant is $$-319$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a372017cramer6","title":"Find the determinant.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.8 Solving Systems with Cramer\'s Rule","courseName":"OpenStax: College Algebra","steps":[{"id":"a372017cramer6a","stepAnswer":["$$15$$"],"problemType":"TextBox","stepTitle":"$$\\\\begin{bmatrix} 5 & 1 & -1 \\\\\\\\ 2 & 3 & 1 \\\\\\\\ 3 & -6 & 3 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$15$$","hints":{"DefaultPathway":[{"id":"a372017cramer6a-h1","type":"hint","dependencies":[],"title":"Augument","text":"First, augument the matrix with the first two columns. In other words, duplicate the first two columns, and add them to the end of the matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer6a-h2","type":"hint","dependencies":["a372017cramer6a-h1"],"title":"Formula","text":"Now, follow the formula to find the determinant. First, multiply the entries down from the row $$1$$ column one spot, to the row $$3$$ column $$3$$ spot, in a diagonal fashion. Do this for the next two diagonals as well, from the row $$1$$ column $$2$$ spot, to the row $$3$$ column $$4$$ spot, and from the row $$1$$ column $$3$$ spot to the row $$3$$ column $$5$$ spot.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-30$$"],"dependencies":["a372017cramer6a-h2"],"title":"Add","text":"Now, add all the previous products together. What do you get as your answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a372017cramer6a-h3-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$-30$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a372017cramer6a-h4","type":"hint","dependencies":["a372017cramer6a-h3"],"title":"Multiply","text":"Next, multiply the values from the bottom left of the matrix to the location $$3$$ spots to the top in a diagonal. Another way to see this is a diagonal from the bottom to the top. Do this for the next $$2$$ columns.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["a372017cramer6a-h4"],"title":"Subtract","text":"Now, subtract all the values that were just multiplied from the first set of diagonal multiplications(those that you calculated previously).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a372017cramer6a-h5-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$15$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a372017cramer6a-h6","type":"hint","dependencies":["a372017cramer6a-h5"],"title":"Answer","text":"Therefore, the determinant is $$15$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a372017cramer7","title":"Find the determinant.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.8 Solving Systems with Cramer\'s Rule","courseName":"OpenStax: College Algebra","steps":[{"id":"a372017cramer7a","stepAnswer":["$$18.4$$"],"problemType":"TextBox","stepTitle":"$$\\\\begin{bmatrix} 1.1 & 2.-1 \\\\\\\\ -4 & 0 & 0 \\\\\\\\ 4.1 & -0.4 & 2.5 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$18.4$$","hints":{"DefaultPathway":[{"id":"a372017cramer7a-h1","type":"hint","dependencies":[],"title":"Augument","text":"First, augument the matrix with the first two columns. In other words, duplicate the first two columns, and add them to the end of the matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer7a-h2","type":"hint","dependencies":["a372017cramer7a-h1"],"title":"Formula","text":"Now, follow the formula to find the determinant. First, multiply the entries down from the row $$1$$ column one spot, to the row $$3$$ column $$3$$ spot, in a diagonal fashion. Do this for the next two diagonals as well, from the row $$1$$ column $$2$$ spot, to the row $$3$$ column $$4$$ spot, and from the row $$1$$ column $$3$$ spot to the row $$3$$ column $$5$$ spot.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1.6$$"],"dependencies":["a372017cramer7a-h2"],"title":"Add","text":"Now, add all the previous products together. What do you get as your answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a372017cramer7a-h3-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$-1.6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a372017cramer7a-h4","type":"hint","dependencies":["a372017cramer7a-h3"],"title":"Multiply","text":"Next, multiply the values from the bottom left of the matrix to the location $$3$$ spots to the top in a diagonal. Another way to see this is a diagonal from the bottom to the top. Do this for the next $$2$$ columns.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer7a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18.4$$"],"dependencies":["a372017cramer7a-h4"],"title":"Subtract","text":"Now, subtract all the values that were just multiplied from the first set of diagonal multiplications(those that you calculated previously).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a372017cramer7a-h5-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$18.4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a372017cramer7a-h6","type":"hint","dependencies":["a372017cramer7a-h5"],"title":"Answer","text":"Therefore, the determinant is $$18.4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a372017cramer8","title":"Find the determinant.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.8 Solving Systems with Cramer\'s Rule","courseName":"OpenStax: College Algebra","steps":[{"id":"a372017cramer8a","stepAnswer":["$$-17.03$$"],"problemType":"TextBox","stepTitle":"$$\\\\begin{bmatrix} 2 & -1.6 & 3.1 \\\\\\\\ 1.1 & 3 & -8 \\\\\\\\ -9.3 & 0 & 2 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-17.03$$","hints":{"DefaultPathway":[{"id":"a372017cramer8a-h1","type":"hint","dependencies":[],"title":"Augument","text":"First, augument the matrix with the first two columns. In other words, duplicate the first two columns, and add them to the end of the matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer8a-h2","type":"hint","dependencies":["a372017cramer8a-h1"],"title":"Formula","text":"Now, follow the formula to find the determinant. First, multiply the entries down from the row $$1$$ column one spot, to the row $$3$$ column $$3$$ spot, in a diagonal fashion. Do this for the next two diagonals as well, from the row $$1$$ column $$2$$ spot, to the row $$3$$ column $$4$$ spot, and from the row $$1$$ column $$3$$ spot to the row $$3$$ column $$5$$ spot.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$131.04$$"],"dependencies":["a372017cramer8a-h2"],"title":"Add","text":"Now, add all the previous products together. What do you get as your answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a372017cramer8a-h3-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$131.04$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a372017cramer8a-h4","type":"hint","dependencies":["a372017cramer8a-h3"],"title":"Multiply","text":"Next, multiply the values from the bottom left of the matrix to the location $$3$$ spots to the top in a diagonal. Another way to see this is a diagonal from the bottom to the top. Do this for the next $$2$$ columns.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer8a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-17.03$$"],"dependencies":["a372017cramer8a-h4"],"title":"Subtract","text":"Now, subtract all the values that were just multiplied from the first set of diagonal multiplications(those that you calculated previously).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a372017cramer8a-h5-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$-17.03$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a372017cramer8a-h6","type":"hint","dependencies":["a372017cramer8a-h5"],"title":"Answer","text":"Therefore, the determinant is $$-17.03$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a372017cramer9","title":"Find the determinant.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.8 Solving Systems with Cramer\'s Rule","courseName":"OpenStax: College Algebra","steps":[{"id":"a372017cramer9a","stepAnswer":["$$\\\\frac{1}{480}$$"],"problemType":"TextBox","stepTitle":"$$\\\\begin{bmatrix} \\\\frac{-1}{2} & \\\\frac{1}{3} & \\\\frac{1}{4} \\\\\\\\ 1.1 & 3 & -8 \\\\\\\\ -9.3 & 0 & 2 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{480}$$","hints":{"DefaultPathway":[{"id":"a372017cramer9a-h1","type":"hint","dependencies":[],"title":"Augument","text":"First, augument the matrix with the first two columns. In other words, duplicate the first two columns, and add them to the end of the matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer9a-h2","type":"hint","dependencies":["a372017cramer9a-h1"],"title":"Formula","text":"Now, follow the formula to find the determinant. First, multiply the entries down from the row $$1$$ column one spot, to the row $$3$$ column $$3$$ spot, in a diagonal fashion. Do this for the next two diagonals as well, from the row $$1$$ column $$2$$ spot, to the row $$3$$ column $$4$$ spot, and from the row $$1$$ column $$3$$ spot to the row $$3$$ column $$5$$ spot.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{96}$$"],"dependencies":["a372017cramer9a-h2"],"title":"Add","text":"Now, add all the previous products together. What do you get as your answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a372017cramer9a-h3-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$\\\\frac{1}{96}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a372017cramer9a-h4","type":"hint","dependencies":["a372017cramer9a-h3"],"title":"Multiply","text":"Next, multiply the values from the bottom left of the matrix to the location $$3$$ spots to the top in a diagonal. Another way to see this is a diagonal from the bottom to the top. Do this for the next $$2$$ columns.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a372017cramer9a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{480}$$"],"dependencies":["a372017cramer9a-h4"],"title":"Subtract","text":"Now, subtract all the values that were just multiplied from the first set of diagonal multiplications(those that you calculated previously).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a372017cramer9a-h5-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$\\\\frac{1}{480}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a372017cramer9a-h6","type":"hint","dependencies":["a372017cramer9a-h5"],"title":"Answer","text":"Therefore, the determinant is $$\\\\frac{1}{480}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a374ff4inequalities1","title":"Find the Slope of a Line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Slope of a Line","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a374ff4inequalities1a","stepAnswer":["$$\\\\frac{2}{5}$$"],"problemType":"TextBox","stepTitle":"","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2}{5}$$","hints":{"DefaultPathway":[{"id":"a374ff4inequalities1a-h1","type":"hint","dependencies":[],"title":"Find Two Points","text":"Find two points on the graph whose coordinates are integers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities1a-h2","type":"hint","dependencies":["a374ff4inequalities1a-h1"],"title":"Rise","text":"Imagine a right triangle for this line as shown in the graph. Count the rise.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities1a-h3","type":"hint","dependencies":["a374ff4inequalities1a-h2"],"title":"Run","text":"Count the run. It\'s the horizontal distance between the right angle and the line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities1a-h4","type":"hint","dependencies":["a374ff4inequalities1a-h3"],"title":"Use Slope Formula","text":"Use the slope formula: $$m=\\\\frac{rise}{run}$$. Substitute the values of rise and run.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities1a-h5","type":"hint","dependencies":["a374ff4inequalities1a-h4"],"title":"Answer","text":"The answer is $$\\\\frac{2}{5}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a374ff4inequalities10","title":"Find the Slope of a Line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Slope of a Line","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a374ff4inequalities10a","stepAnswer":["$$\\\\frac{-6}{5}$$"],"problemType":"TextBox","stepTitle":"$$(3,6)(8,0)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-6}{5}$$","hints":{"DefaultPathway":[{"id":"a374ff4inequalities10a-h1","type":"hint","dependencies":[],"title":"Use Slope Formula","text":"The slope formula is $$m=\\\\frac{y_2-y_1}{x_2-x_1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities10a-h2","type":"hint","dependencies":["a374ff4inequalities10a-h1"],"title":"Substitue","text":"Substitute the values from the points to the slope formula, choosing one point to be the first point, and the other to be the second.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities10a-h3","type":"hint","dependencies":["a374ff4inequalities10a-h2"],"title":"Simplify","text":"Simplify the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities10a-h4","type":"hint","dependencies":["a374ff4inequalities10a-h3"],"title":"Answer","text":"The answer is $$\\\\frac{-6}{5}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a374ff4inequalities11","title":"Find the Slope of a Line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Slope of a Line","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a374ff4inequalities11a","stepAnswer":["$$-1$$"],"problemType":"TextBox","stepTitle":"$$(-2,4)(3,-1)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1$$","hints":{"DefaultPathway":[{"id":"a374ff4inequalities11a-h1","type":"hint","dependencies":[],"title":"Use Slope Formula","text":"The slope formula is $$m=\\\\frac{y_2-y_1}{x_2-x_1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities11a-h2","type":"hint","dependencies":["a374ff4inequalities11a-h1"],"title":"Substitue","text":"Substitute the values from the points to the slope formula, choosing one point to be the first point, and the other to be the second.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities11a-h3","type":"hint","dependencies":["a374ff4inequalities11a-h2"],"title":"Simplify","text":"Simplify the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities11a-h4","type":"hint","dependencies":["a374ff4inequalities11a-h3"],"title":"Answer","text":"The answer is $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a374ff4inequalities12","title":"Find the Slope of a Line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Slope of a Line","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a374ff4inequalities12a","stepAnswer":["$$\\\\frac{3}{4}$$"],"problemType":"TextBox","stepTitle":"$$(-2,-1)(6,5)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{4}$$","hints":{"DefaultPathway":[{"id":"a374ff4inequalities12a-h1","type":"hint","dependencies":[],"title":"Use Slope Formula","text":"The slope formula is $$m=\\\\frac{y_2-y_1}{x_2-x_1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities12a-h2","type":"hint","dependencies":["a374ff4inequalities12a-h1"],"title":"Substitue","text":"Substitute the values from the points to the slope formula, choosing one point to be the first point, and the other to be the second.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities12a-h3","type":"hint","dependencies":["a374ff4inequalities12a-h2"],"title":"Simplify","text":"Simplify the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities12a-h4","type":"hint","dependencies":["a374ff4inequalities12a-h3"],"title":"Answer","text":"The answer is $$\\\\frac{3}{4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a374ff4inequalities13","title":"Find the Slope of a Line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Slope of a Line","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a374ff4inequalities13a","stepAnswer":["$$-4$$"],"problemType":"TextBox","stepTitle":"$$(3,-6)(2,-2)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-4$$","hints":{"DefaultPathway":[{"id":"a374ff4inequalities13a-h1","type":"hint","dependencies":[],"title":"Use Slope Formula","text":"The slope formula is $$m=\\\\frac{y_2-y_1}{x_2-x_1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities13a-h2","type":"hint","dependencies":["a374ff4inequalities13a-h1"],"title":"Substitue","text":"Substitute the values from the points to the slope formula, choosing one point to be the first point, and the other to be the second.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities13a-h3","type":"hint","dependencies":["a374ff4inequalities13a-h2"],"title":"Simplify","text":"Simplify the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities13a-h4","type":"hint","dependencies":["a374ff4inequalities13a-h3"],"title":"Answer","text":"The answer is $$-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a374ff4inequalities14","title":"Find the Slope of a Line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Slope of a Line","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a374ff4inequalities14a","stepAnswer":["$$-1$$"],"problemType":"TextBox","stepTitle":"$$(-7,4)(3,-6)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1$$","hints":{"DefaultPathway":[{"id":"a374ff4inequalities14a-h1","type":"hint","dependencies":[],"title":"Use Slope Formula","text":"The slope formula is $$m=\\\\frac{y_2-y_1}{x_2-x_1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities14a-h2","type":"hint","dependencies":["a374ff4inequalities14a-h1"],"title":"Substitue","text":"Substitute the values from the points to the slope formula, choosing one point to be the first point, and the other to be the second.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities14a-h3","type":"hint","dependencies":["a374ff4inequalities14a-h2"],"title":"Simplify","text":"Simplify the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities14a-h4","type":"hint","dependencies":["a374ff4inequalities14a-h3"],"title":"Answer","text":"The answer is $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a374ff4inequalities15","title":"Find the Slope of a Line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Slope of a Line","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a374ff4inequalities15a","stepAnswer":["$$\\\\frac{-1}{5}$$"],"problemType":"TextBox","stepTitle":"$$(-3,6)(2,5)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-1}{5}$$","hints":{"DefaultPathway":[{"id":"a374ff4inequalities15a-h1","type":"hint","dependencies":[],"title":"Use Slope Formula","text":"The slope formula is $$m=\\\\frac{y_2-y_1}{x_2-x_1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities15a-h2","type":"hint","dependencies":["a374ff4inequalities15a-h1"],"title":"Substitue","text":"Substitute the values from the points to the slope formula, choosing one point to be the first point, and the other to be the second.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities15a-h3","type":"hint","dependencies":["a374ff4inequalities15a-h2"],"title":"Simplify","text":"Simplify the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities15a-h4","type":"hint","dependencies":["a374ff4inequalities15a-h3"],"title":"Answer","text":"The answer is $$\\\\frac{-1}{5}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a374ff4inequalities16","title":"Finding the Slope of a Line","body":"Analyze the line from the graph.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Slope of a Line","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a374ff4inequalities16a","stepAnswer":["$$\\\\frac{-2}{3}$$"],"problemType":"TextBox","stepTitle":"What is the slope of the line shown? If the slope is undefined, enter \\"und\\" without the quotes.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-2}{3}$$","hints":{"DefaultPathway":[{"id":"a374ff4inequalities16a-h1","type":"hint","dependencies":[],"title":"Analyzing the Line","text":"Choose two points on the line. Then, count the rise and run, which are respectively the amount of $$y$$ units and the amount of $$x$$ units it changes from the point on the left to the point on the right. (Rise is positive if the line goes up and negative if it goes down.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities16a-h2","type":"hint","dependencies":["a374ff4inequalities16a-h1"],"title":"Slope Formula","text":"The slope formula is $$\\\\frac{rise}{run}$$. (Hint: Using the points as the hypotenuse of a small right triangle, count its rise and run. Rise is positive if the line goes up and negative if it goes down. Count the run from left to right, which will give you a positive value.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a374ff4inequalities17","title":"Finding the Slope of a Line","body":"Analyze the line from the graph.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Slope of a Line","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a374ff4inequalities17a","stepAnswer":["$$\\\\frac{-4}{3}$$"],"problemType":"TextBox","stepTitle":"What is the slope of the line shown? If the slope is undefined, enter \\"und\\" without the quotes.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-4}{3}$$","hints":{"DefaultPathway":[{"id":"a374ff4inequalities17a-h1","type":"hint","dependencies":[],"title":"Analyzing the Line","text":"Choose two points on the line. Then, count the rise and run, which are respectively the amount of $$y$$ units and the amount of $$x$$ units it changes from the point on the left to the point on the right. (Rise is positive if the line goes up and negative if it goes down.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities17a-h2","type":"hint","dependencies":["a374ff4inequalities17a-h1"],"title":"Slope Formula","text":"The slope formula is $$\\\\frac{rise}{run}$$. (Hint: Using the points as the hypotenuse of a small right triangle, count its rise and run. Rise is positive if the line goes up and negative if it goes down. Count the run from left to right, which will give you a positive value.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a374ff4inequalities18","title":"Finding the Slope of a Line","body":"Analyze the line from the graph.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Slope of a Line","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a374ff4inequalities18a","stepAnswer":["$$\\\\frac{-3}{5}$$"],"problemType":"TextBox","stepTitle":"What is the slope of the line shown? If the slope is undefined, enter \\"und\\" without the quotes.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-3}{5}$$","hints":{"DefaultPathway":[{"id":"a374ff4inequalities18a-h1","type":"hint","dependencies":[],"title":"Analyzing the Line","text":"Choose two points on the line. Then, count the rise and run, which are respectively the amount of $$y$$ units and the amount of $$x$$ units it changes from the point on the left to the point on the right. (Rise is positive if the line goes up and negative if it goes down.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities18a-h2","type":"hint","dependencies":["a374ff4inequalities18a-h1"],"title":"Slope Formula","text":"The slope formula is $$\\\\frac{rise}{run}$$. (Hint: Using the points as the hypotenuse of a small right triangle, count its rise and run. Rise is positive if the line goes up and negative if it goes down. Count the run from left to right, which will give you a positive value.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a374ff4inequalities19","title":"Finding the Slope of a Line","body":"Find the slope of the line. If the slope is undefined, enter \\"und\\" without the quotes.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Slope of a Line","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a374ff4inequalities19a","stepAnswer":["und"],"problemType":"TextBox","stepTitle":"What is the slope of the line $$x=-4$$?","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a374ff4inequalities19a-h1","type":"hint","dependencies":[],"title":"Vertical Lines","text":"The slopes of vertical lines, $$x=a$$, are undefined.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a374ff4inequalities2","title":"Find the Slope of a Line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Slope of a Line","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a374ff4inequalities2a","stepAnswer":["$$\\\\frac{5}{4}$$"],"problemType":"TextBox","stepTitle":"","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5}{4}$$","hints":{"DefaultPathway":[{"id":"a374ff4inequalities2a-h1","type":"hint","dependencies":[],"title":"Find Two Points","text":"Find two points on the graph whose coordinates are integers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities2a-h2","type":"hint","dependencies":["a374ff4inequalities2a-h1"],"title":"Rise","text":"Imagine a right triangle for this line as shown in the graph. Count the rise.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities2a-h3","type":"hint","dependencies":["a374ff4inequalities2a-h2"],"title":"Run","text":"Count the run. It\'s the horizontal distance between the right angle and the line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities2a-h4","type":"hint","dependencies":["a374ff4inequalities2a-h3"],"title":"Use Slope Formula","text":"Use the slope formula: $$m=\\\\frac{rise}{run}$$. Substitute the values of rise and run.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities2a-h5","type":"hint","dependencies":["a374ff4inequalities2a-h4"],"title":"Answer","text":"The answer is $$\\\\frac{5}{4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a374ff4inequalities20","title":"Finding the Slope of a Line","body":"Find the slope of the line. If the slope is undefined, enter \\"und\\" without the quotes.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Slope of a Line","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a374ff4inequalities20a","stepAnswer":["und"],"problemType":"TextBox","stepTitle":"What is the slope of the line $$x=8$$?","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a374ff4inequalities20a-h1","type":"hint","dependencies":[],"title":"Vertical Lines","text":"The slopes of vertical lines, $$x=a$$, are undefined.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a374ff4inequalities21","title":"Finding the Slope of a Line","body":"Find the slope of the line. If the slope is undefined, enter \\"und\\" without the quotes.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Slope of a Line","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a374ff4inequalities21a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"What is the slope of the line $$y=-5$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a374ff4inequalities21a-h1","type":"hint","dependencies":[],"title":"Vertical Lines","text":"The slopes of horizontal lines, $$y=b$$, are $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a374ff4inequalities22","title":"Finding the Slope of a Line","body":"Find the slope of the line. If the slope is undefined, enter \\"und\\" without the quotes.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Slope of a Line","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a374ff4inequalities22a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"What is the slope of the line $$y=7$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a374ff4inequalities22a-h1","type":"hint","dependencies":[],"title":"Horizontal Lines","text":"The slopes of horizontal lines, $$y=b$$, are $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a374ff4inequalities23","title":"Finding the Slope of a Line","body":"Use the slope formula to find the slope through the two points.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Slope of a Line","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a374ff4inequalities23a","stepAnswer":["$$\\\\frac{-7}{5}$$"],"problemType":"TextBox","stepTitle":"$$(-2,-3)$$ and $$(-7,4)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-7}{5}$$","hints":{"DefaultPathway":[{"id":"a374ff4inequalities23a-h1","type":"hint","dependencies":[],"title":"Slope Formula","text":"The slope of a line, $$m$$, between two points, (x1,y1) and (x2,y2), is $$m=\\\\frac{y2-y1}{x2-x1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a374ff4inequalities24","title":"Finding the Slope of a Line","body":"Use the slope formula to find the slope through the two points.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Slope of a Line","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a374ff4inequalities24a","stepAnswer":["$$-1$$"],"problemType":"TextBox","stepTitle":"$$(-3,4)$$ and $$(2,-1)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1$$","hints":{"DefaultPathway":[{"id":"a374ff4inequalities24a-h1","type":"hint","dependencies":[],"title":"Slope Formula","text":"The slope of a line, $$m$$, between two points, (x1,y1) and (x2,y2), is $$m=\\\\frac{y2-y1}{x2-x1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a374ff4inequalities25","title":"Finding the Slope of a Line","body":"Use the slope formula to find the slope through the two points.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Slope of a Line","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a374ff4inequalities25a","stepAnswer":["$$10$$"],"problemType":"TextBox","stepTitle":"$$(-2,6)$$ and $$(-3,-4)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10$$","hints":{"DefaultPathway":[{"id":"a374ff4inequalities25a-h1","type":"hint","dependencies":[],"title":"Slope Formula","text":"The slope of a line, $$m$$, between two points, (x1,y1) and (x2,y2), is $$m=\\\\frac{y2-y1}{x2-x1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a374ff4inequalities26","title":"Identifying Slope and y-intercept","body":"Identify the slope and y-intercept of the line. Input your answer in the form \\"m,(0,b)\\" without the quotes, where $$m$$ is the slope of the line and (0,b) is the y-intercept of the line. Simplify $$m$$ and $$b$$ if possible.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Slope of a Line","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a374ff4inequalities26a","stepAnswer":["-4/7,(0,-2)"],"problemType":"TextBox","stepTitle":"$$y=\\\\frac{-4}{7} x-2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-4/7,(0,-2)$$","hints":{"DefaultPathway":[{"id":"a374ff4inequalities26a-h1","type":"hint","dependencies":[],"title":"Slope-Intercept Form","text":"The slope-intercept form of an equation of a line with slope $$m$$ and y-intercept, (0,b), is $$y=mx+b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a374ff4inequalities27","title":"Identifying Slope and y-intercept","body":"Identify the slope and y-intercept of the line. Input your answer in the form \\"m,(0,b)\\" without the quotes, where $$m$$ is the slope of the line and (0,b) is the y-intercept of the line. Simplify $$m$$ and $$b$$ if possible.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Slope of a Line","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a374ff4inequalities27a","stepAnswer":["-1/3,(0,3)"],"problemType":"TextBox","stepTitle":"$$x+3y=9$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-1/3,(0,3)$$","hints":{"DefaultPathway":[{"id":"a374ff4inequalities27a-h1","type":"hint","dependencies":[],"title":"Slope-Intercept Form","text":"The slope-intercept form of an equation of a line with slope $$m$$ and y-intercept, (0,b), is $$y=mx+b$$. Rewrite the equation into this format.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities27a-h2","type":"hint","dependencies":["a374ff4inequalities27a-h1"],"title":"Rewritten Form","text":"The rewritten equation is $$y=\\\\left(-\\\\frac{1}{3}\\\\right) x+3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a374ff4inequalities28","title":"Identifying Slope and y-intercept","body":"Identify the slope and y-intercept of the line. Input your answer in the form \\"m,(0,b)\\" without the quotes, where $$m$$ is the slope of the line and (0,b) is the y-intercept of the line. Simplify $$m$$ and $$b$$ if possible.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Slope of a Line","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a374ff4inequalities28a","stepAnswer":["2/5,(0,-1)"],"problemType":"TextBox","stepTitle":"$$y=\\\\frac{2}{5} x-1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2/5,(0,-1)$$","hints":{"DefaultPathway":[{"id":"a374ff4inequalities28a-h1","type":"hint","dependencies":[],"title":"Slope-Intercept Form","text":"The slope-intercept form of an equation of a line with slope $$m$$ and y-intercept, (0,b), is $$y=mx+b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a374ff4inequalities29","title":"Identifying Slope and y-intercept","body":"Identify the slope and y-intercept of the line. Input your answer in the form \\"m,(0,b)\\" without the quotes, where $$m$$ is the slope of the line and (0,b) is the y-intercept of the line. Simplify $$m$$ and $$b$$ if possible.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Slope of a Line","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a374ff4inequalities29a","stepAnswer":["-1/4,(0,2)"],"problemType":"TextBox","stepTitle":"$$x+4y=8$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-1/4,(0,2)$$","hints":{"DefaultPathway":[{"id":"a374ff4inequalities29a-h1","type":"hint","dependencies":[],"title":"Slope-Intercept Form","text":"The slope-intercept form of an equation of a line with slope $$m$$ and y-intercept, (0,b), is $$y=mx+b$$. Rewrite the equation into this format.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities29a-h2","type":"hint","dependencies":["a374ff4inequalities29a-h1"],"title":"Rewritten Form","text":"The rewritten equation is $$y=\\\\left(-\\\\frac{1}{4}\\\\right) x+2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a374ff4inequalities3","title":"Find the Slope of a Line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Slope of a Line","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a374ff4inequalities3a","stepAnswer":["$$\\\\frac{-1}{3}$$"],"problemType":"TextBox","stepTitle":"","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-1}{3}$$","hints":{"DefaultPathway":[{"id":"a374ff4inequalities3a-h1","type":"hint","dependencies":[],"title":"Find Two Points","text":"Find two points on the graph whose coordinates are integers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities3a-h2","type":"hint","dependencies":["a374ff4inequalities3a-h1"],"title":"Rise","text":"Imagine a right triangle for this line as shown in the graph. Count the rise.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities3a-h3","type":"hint","dependencies":["a374ff4inequalities3a-h2"],"title":"Run","text":"Count the run. It\'s the horizontal distance between the right angle and the line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities3a-h4","type":"hint","dependencies":["a374ff4inequalities3a-h3"],"title":"Use Slope Formula","text":"Use the slope formula: $$m=\\\\frac{rise}{run}$$. Substitute the values of rise and run.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities3a-h5","type":"hint","dependencies":["a374ff4inequalities3a-h4"],"title":"Answer","text":"The answer is $$\\\\frac{-1}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a374ff4inequalities30","title":"Identifying Slope and y-intercept","body":"Identify the slope and y-intercept of the line. Input your answer in the form \\"m,(0,b)\\" without the quotes, where $$m$$ is the slope of the line and (0,b) is the y-intercept of the line. Simplify $$m$$ and $$b$$ if possible.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Slope of a Line","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a374ff4inequalities30a","stepAnswer":["-4/3,(0,1)"],"problemType":"TextBox","stepTitle":"$$y=\\\\frac{-4}{3} x+1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-4/3,(0,1)$$","hints":{"DefaultPathway":[{"id":"a374ff4inequalities30a-h1","type":"hint","dependencies":[],"title":"Slope-Intercept Form","text":"The slope-intercept form of an equation of a line with slope $$m$$ and y-intercept, (0,b), is $$y=mx+b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities30a-h2","type":"hint","dependencies":["a374ff4inequalities30a-h1"],"title":"Rewritten Form","text":"The rewritten equation is $$y=\\\\left(-\\\\frac{4}{3}\\\\right) x+1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a374ff4inequalities4","title":"Find the Slope of a Line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Slope of a Line","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a374ff4inequalities4a","stepAnswer":["$$\\\\frac{-5}{2}$$"],"problemType":"TextBox","stepTitle":"","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-5}{2}$$","hints":{"DefaultPathway":[{"id":"a374ff4inequalities4a-h1","type":"hint","dependencies":[],"title":"Find Two Points","text":"Find two points on the graph whose coordinates are integers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities4a-h2","type":"hint","dependencies":["a374ff4inequalities4a-h1"],"title":"Rise","text":"Imagine a right triangle for this line as shown in the graph. Count the rise.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities4a-h3","type":"hint","dependencies":["a374ff4inequalities4a-h2"],"title":"Run","text":"Count the run. It\'s the horizontal distance between the right angle and the line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities4a-h4","type":"hint","dependencies":["a374ff4inequalities4a-h3"],"title":"Use Slope Formula","text":"Use the slope formula: $$m=\\\\frac{rise}{run}$$. Substitute the values of rise and run.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities4a-h5","type":"hint","dependencies":["a374ff4inequalities4a-h4"],"title":"Answer","text":"The answer is $$\\\\frac{-5}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a374ff4inequalities5","title":"Find the Slope of a Line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Slope of a Line","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a374ff4inequalities5a","stepAnswer":["$$\\\\frac{-2}{3}$$"],"problemType":"TextBox","stepTitle":"","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-2}{3}$$","hints":{"DefaultPathway":[{"id":"a374ff4inequalities5a-h1","type":"hint","dependencies":[],"title":"Find Two Points","text":"Find two points on the graph whose coordinates are integers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities5a-h2","type":"hint","dependencies":["a374ff4inequalities5a-h1"],"title":"Rise","text":"Imagine a right triangle for this line as shown in the graph. Count the rise.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities5a-h3","type":"hint","dependencies":["a374ff4inequalities5a-h2"],"title":"Run","text":"Count the run. It\'s the horizontal distance between the right angle and the line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities5a-h4","type":"hint","dependencies":["a374ff4inequalities5a-h3"],"title":"Use Slope Formula","text":"Use the slope formula: $$m=\\\\frac{rise}{run}$$. Substitute the values of rise and run.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities5a-h5","type":"hint","dependencies":["a374ff4inequalities5a-h4"],"title":"Answer","text":"The answer is $$\\\\frac{-2}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a374ff4inequalities6","title":"Find the Slope of a Line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Slope of a Line","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a374ff4inequalities6a","stepAnswer":["$$\\\\frac{-5}{2}$$"],"problemType":"TextBox","stepTitle":"$$(2,5)(4,0)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-5}{2}$$","hints":{"DefaultPathway":[{"id":"a374ff4inequalities6a-h1","type":"hint","dependencies":[],"title":"Use Slope Formula","text":"The slope formula is $$m=\\\\frac{y_2-y_1}{x_2-x_1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities6a-h2","type":"hint","dependencies":["a374ff4inequalities6a-h1"],"title":"Substitue","text":"Substitute the values from the points to the slope formula, choosing one point to be the first point, and the other to be the second.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities6a-h3","type":"hint","dependencies":["a374ff4inequalities6a-h2"],"title":"Simplify","text":"Simplify the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities6a-h4","type":"hint","dependencies":["a374ff4inequalities6a-h3"],"title":"Answer","text":"The answer is $$\\\\frac{-5}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a374ff4inequalities7","title":"Find the Slope of a Line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Slope of a Line","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a374ff4inequalities7a","stepAnswer":["$$\\\\frac{-8}{7}$$"],"problemType":"TextBox","stepTitle":"$$(-3,3)(4,-5)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-8}{7}$$","hints":{"DefaultPathway":[{"id":"a374ff4inequalities7a-h1","type":"hint","dependencies":[],"title":"Use Slope Formula","text":"The slope formula is $$m=\\\\frac{y_2-y_1}{x_2-x_1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities7a-h2","type":"hint","dependencies":["a374ff4inequalities7a-h1"],"title":"Substitue","text":"Substitute the values from the points to the slope formula, choosing one point to be the first point, and the other to be the second.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities7a-h3","type":"hint","dependencies":["a374ff4inequalities7a-h2"],"title":"Simplify","text":"Simplify the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities7a-h4","type":"hint","dependencies":["a374ff4inequalities7a-h3"],"title":"Answer","text":"The answer is $$\\\\frac{-8}{7}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a374ff4inequalities8","title":"Find the Slope of a Line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Slope of a Line","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a374ff4inequalities8a","stepAnswer":["$$\\\\frac{7}{3}$$"],"problemType":"TextBox","stepTitle":"$$(-1,-2)(2,5)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{7}{3}$$","hints":{"DefaultPathway":[{"id":"a374ff4inequalities8a-h1","type":"hint","dependencies":[],"title":"Use Slope Formula","text":"The slope formula is $$m=\\\\frac{y_2-y_1}{x_2-x_1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities8a-h2","type":"hint","dependencies":["a374ff4inequalities8a-h1"],"title":"Substitue","text":"Substitute the values from the points to the slope formula, choosing one point to be the first point, and the other to be the second.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities8a-h3","type":"hint","dependencies":["a374ff4inequalities8a-h2"],"title":"Simplify","text":"Simplify the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities8a-h4","type":"hint","dependencies":["a374ff4inequalities8a-h3"],"title":"Answer","text":"The answer is $$\\\\frac{7}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a374ff4inequalities9","title":"Find the Slope of a Line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Slope of a Line","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a374ff4inequalities9a","stepAnswer":["$$-1$$"],"problemType":"TextBox","stepTitle":"$$(4,-5)(1,-2)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1$$","hints":{"DefaultPathway":[{"id":"a374ff4inequalities9a-h1","type":"hint","dependencies":[],"title":"Use Slope Formula","text":"The slope formula is $$m=\\\\frac{y_2-y_1}{x_2-x_1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities9a-h2","type":"hint","dependencies":["a374ff4inequalities9a-h1"],"title":"Substitue","text":"Substitute the values from the points to the slope formula, choosing one point to be the first point, and the other to be the second.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities9a-h3","type":"hint","dependencies":["a374ff4inequalities9a-h2"],"title":"Simplify","text":"Simplify the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a374ff4inequalities9a-h4","type":"hint","dependencies":["a374ff4inequalities9a-h3"],"title":"Answer","text":"The answer is $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a381217systemeq1","title":"Solve Mixture Applications","body":"Translate to a system of equations and solve","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Solve Mixture Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a381217systemeq1a","stepAnswer":["42a105c"],"problemType":"TextBox","stepTitle":"The box office at a movie theater sold $$147$$ tickets for the evening show, and receipts totaled $1,302. How many $11 adult and how many $8 child tickets were sold? (Answer in the form with the number of a variable followed by the first letter of the variable (adult $$=$$ a): 11a13c (These numbers and variables are not the answers))","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a381217systemeq1a-h1","type":"hint","dependencies":[],"title":"Convert amounts into first equation","text":"Let\'s use \\"a\\" as the number of adult tickets and \\"c\\" as the number of childrens tickets. The total number of tickets combined is $$147$$, therefore we can determine that the first equation is $$a+c=147$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq1a-h2","type":"hint","dependencies":["a381217systemeq1a-h1"],"title":"Convert amoutn into second equation","text":"Because the total amount of money made was $1302 and adult tickets are $11 and childrens $8 we can determine that the second equation is $$11a+8c=1302$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq1a-h3","type":"hint","dependencies":["a381217systemeq1a-h2"],"title":"Isolate Variable","text":"Let\'s use the first equation to isolate c. With the equation, we can subtract a from both sides to get $$c=147-a$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq1a-h4","type":"hint","dependencies":["a381217systemeq1a-h3"],"title":"Plug in","text":"We can plug in $$c=147-a$$ using the second equation to get $$11a+8\\\\left(147-a\\\\right)=1302$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq1a-h5","type":"hint","dependencies":["a381217systemeq1a-h4"],"title":"Solve for a","text":"We can determine that the second equation after plugging in C would be $$3a+1176=1302$$. We now isolate the variable \\"a\\" and get $$3a=126$$. Then, we can divide both sides by $$3$$ to get $$a=42$$. There are $$42$$ adult tickets.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq1a-h6","type":"hint","dependencies":["a381217systemeq1a-h5"],"title":"Solve for c","text":"After solving for \\"a\\", we can plug that value into the first equation to get $$42+c=147$$ which we can determine c to be $$105$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a381217systemeq10","title":"Solve Mixture Applications","body":"Translate to a system of equations and solve","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Solve Mixture Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a381217systemeq10a","stepAnswer":["6m10t"],"problemType":"TextBox","stepTitle":"Tickets for a baseball game are $69 for Main Level seats and $39 for Terrace Level seats. A group of sixteen friends went to the game and spent a total of $804 for the tickets. How many of Main Level and how many Terrace Level tickets did they buy? (Answer in the form with the number of a variable followed by the first letter of the variable (Main $$=$$ $$m$$, Terrace $$=$$ t: 11m13t (These numbers and variables are not the answers))","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a381217systemeq10a-h1","type":"hint","dependencies":[],"title":"Convert amounts into first equation","text":"Let\'s use \\"m\\" as the number of Main Level seats and \\"t\\" as the number of Terrace Level seats. The total number of people combined is $$16$$, therefore we can determine that the first equation is $$m+t=16$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq10a-h2","type":"hint","dependencies":["a381217systemeq10a-h1"],"title":"Convert amoutn into second equation","text":"Because the total amount of money made was $804 and Main Level seats are $69 and Terrace Level seats $39 we can determine that the second equation is $$69m+39t=804$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq10a-h3","type":"hint","dependencies":["a381217systemeq10a-h2"],"title":"Isolate Variable","text":"Let\'s use the first equation to isolate $$m$$. With the equation, we can subtract a from both sides to get $$m=16-t$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq10a-h4","type":"hint","dependencies":["a381217systemeq10a-h3"],"title":"Plug in","text":"We can plug in $$m=16-t$$ using the second equation to get $$\\\\operatorname{69}\\\\left(16-t\\\\right)+39t=804$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq10a-h5","type":"hint","dependencies":["a381217systemeq10a-h4"],"title":"Solve for a","text":"We can determine that the second equation after plugging in would be -30t+1,104=804. We now isolate the variable \\"t\\" and get $$-30t=-300$$. Then, we can divide both sides by $$-30$$ to get $$t=10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq10a-h6","type":"hint","dependencies":["a381217systemeq10a-h5"],"title":"Solve for c","text":"After solving for \\"t\\", we can plug that value into the first equation to get $$m+10=16$$ which we can determine \\"m\\" to be $$6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a381217systemeq11","title":"Solve Mixture Applications","body":"Translate to a system of equations and solve","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Solve Mixture Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a381217systemeq11a","stepAnswer":["125a128c"],"problemType":"TextBox","stepTitle":"Tickets for a dance recital cost $15 for adults and $7 for children. The dance company sold $$253$$ tickets and the total receipts were $2,771. How many adult tickets and how many child tickets were sold? (Answer in the form with the number of a variable followed by the first letter of the variable (adult $$=$$ a): 11a13c (These numbers and variables are not the answers))","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a381217systemeq11a-h1","type":"hint","dependencies":[],"title":"Convert amounts into first equation","text":"Let\'s use \\"a\\" as the number of adult tickets and \\"c\\" as the number of childrens tickets. The total number of tickets combined is $$253$$, therefore we can determine that the first equation is $$a+c=253$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq11a-h2","type":"hint","dependencies":["a381217systemeq11a-h1"],"title":"Convert amoutn into second equation","text":"Because the total amount of money made was $2,771 and adult tickets are $15 and childrens $7 we can determine that the second equation is $$15a+7c=2, 771$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq11a-h3","type":"hint","dependencies":["a381217systemeq11a-h2"],"title":"Isolate Variable","text":"Let\'s use the first equation to isolate c. With the equation, we can subtract a from both sides to get $$c=253-a$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq11a-h4","type":"hint","dependencies":["a381217systemeq11a-h3"],"title":"Plug in","text":"We can plug in $$c=253-a$$ using the second equation to get $$15a+7\\\\left(253-a\\\\right)=2, 771$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq11a-h5","type":"hint","dependencies":["a381217systemeq11a-h4"],"title":"Solve for a","text":"We can determine that the second equation after plugging in \\"c\\" would be 8a+1,771=2,771. We now isolate the variable \\"a\\" and get $$8a=1000$$. Then, we can divide both sides by $$8$$ to get $$a=125$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq11a-h6","type":"hint","dependencies":["a381217systemeq11a-h5"],"title":"Solve for c","text":"After solving for \\"a\\", we can plug that value into the first equation to get $$125+c=253$$ which we can determine \\"c\\" to be $$128$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a381217systemeq12","title":"Solve Mixture Applications","body":"Translate to a system of equations and solve","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Solve Mixture Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a381217systemeq12a","stepAnswer":["92a220c"],"problemType":"TextBox","stepTitle":"Tickets for the community fair cost $12 for adults and $5 dollars for children. On the first day of the fair, $$312$$ tickets were sold for a total of $2,204. How many adult tickets and how many child tickets were sold? (Answer in the form with the number of a variable followed by the first letter of the variable (adult $$=$$ a): 11a13c (These numbers and variables are not the answers))","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a381217systemeq12a-h1","type":"hint","dependencies":[],"title":"Convert amounts into first equation","text":"Let\'s use \\"a\\" as the number of adult tickets and \\"c\\" as the number of childrens tickets. The total number of tickets combined is $$312$$, therefore we can determine that the first equation is $$a+c=312$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq12a-h2","type":"hint","dependencies":["a381217systemeq12a-h1"],"title":"Convert amoutn into second equation","text":"Because the total amount of money made was $2,204 and adult tickets are $12 and childrens $5 we can determine that the second equation is $$12a+5c=2, 204$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq12a-h3","type":"hint","dependencies":["a381217systemeq12a-h2"],"title":"Isolate Variable","text":"Let\'s use the first equation to isolate c. With the equation, we can subtract a from both sides to get $$c=312-a$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq12a-h4","type":"hint","dependencies":["a381217systemeq12a-h3"],"title":"Plug in","text":"We can plug in $$c=312-a$$ using the second equation to get $$12a+5\\\\left(312-a\\\\right)=2, 204$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq12a-h5","type":"hint","dependencies":["a381217systemeq12a-h4"],"title":"Solve for a","text":"We can determine that the second equation after plugging in \\"c\\" would be $$7a+1560=2, 204$$. We now isolate the variable \\"a\\" and get $$7a=644$$. Then, we can divide both sides by $$7$$ to get $$a=92$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq12a-h6","type":"hint","dependencies":["a381217systemeq12a-h5"],"title":"Solve for c","text":"After solving for \\"a\\", we can plug that value into the first equation to get $$92+c=312$$ which we can determine \\"c\\" to be $$220$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a381217systemeq13","title":"Solve Mixture Applications","body":"Translate to a system of equations and solve","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Solve Mixture Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a381217systemeq13a","stepAnswer":["206a347c"],"problemType":"TextBox","stepTitle":"The ticket office at the zoo sold $$553$$ tickets one day. The receipts totaled $3,936. How many $9 adult tickets and how many $6 child tickets were sold? (Answer in the form with the number of a variable followed by the first letter of the variable (adult $$=$$ a): 11a13c (These numbers and variables are not the answers))","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a381217systemeq13a-h1","type":"hint","dependencies":[],"title":"Convert amounts into first equation","text":"Let\'s use \\"a\\" as the number of adult tickets and \\"c\\" as the number of childrens tickets. The total number of tickets combined is $$553$$, therefore we can determine that the first equation is $$a+c=553$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq13a-h2","type":"hint","dependencies":["a381217systemeq13a-h1"],"title":"Convert amoutn into second equation","text":"Because the total amount of money made was $3,936 and adult tickets are $9 and childrens $6 we can determine that the second equation is $$9a+6c=3, 936$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq13a-h3","type":"hint","dependencies":["a381217systemeq13a-h2"],"title":"Isolate Variable","text":"Let\'s use the first equation to isolate c. With the equation, we can subtract a from both sides to get $$c=553-a$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq13a-h4","type":"hint","dependencies":["a381217systemeq13a-h3"],"title":"Plug in","text":"We can plug in $$c=553-a$$ using the second equation to get $$9a+6\\\\left(553-a\\\\right)=3, 936$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq13a-h5","type":"hint","dependencies":["a381217systemeq13a-h4"],"title":"Solve for a","text":"We can determine that the second equation after plugging in \\"c\\" would be 3a+3,318=3,936. We now isolate the variable \\"a\\" and get $$3a=618$$. Then, we can divide both sides by $$3$$ to get $$a=206$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq13a-h6","type":"hint","dependencies":["a381217systemeq13a-h5"],"title":"Solve for c","text":"After solving for \\"a\\", we can plug that value into the first equation to get $$206+c=553$$ which we can determine \\"c\\" to be $$347$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a381217systemeq14","title":"Solve Mixture Applications","body":"Translate to a system of equations and solve","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Solve Mixture Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a381217systemeq14a","stepAnswer":["521a842c"],"problemType":"TextBox","stepTitle":"A science center sold 1,363 tickets on a busy weekend. The receipts totaled $12,146. How many $12 adult tickets and how many $7 child tickets were sold? (Answer in the form with the number of a variable followed by the first letter of the variable (adult $$=$$ a): 11a13c (These numbers and variables are not the answers))","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a381217systemeq14a-h1","type":"hint","dependencies":[],"title":"Convert amounts into first equation","text":"Let\'s use \\"a\\" as the number of adult tickets and \\"c\\" as the number of childrens tickets. The total number of tickets combined is 1,363, therefore we can determine that the first equation is $$a+c=1, 363$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq14a-h2","type":"hint","dependencies":["a381217systemeq14a-h1"],"title":"Convert amoutn into second equation","text":"Because the total amount of money made was $12,146. and adult tickets are $12 and childrens $7 we can determine that the second equation is $$12a+7c=12, 146$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq14a-h3","type":"hint","dependencies":["a381217systemeq14a-h2"],"title":"Isolate Variable","text":"Let\'s use the first equation to isolate c. With the equation, we can subtract a from both sides to get $$c=1, 363-a$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq14a-h4","type":"hint","dependencies":["a381217systemeq14a-h3"],"title":"Plug in","text":"We can plug in $$c=1, 363-a$$ using the second equation to get $$12a+7(1,363-a)=12,146$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq14a-h5","type":"hint","dependencies":["a381217systemeq14a-h4"],"title":"Solve for a","text":"We can determine that the second equation after plugging in \\"c\\" would be 5a+9,541=12,146. We now isolate the variable \\"a\\" and get $$5a=2, 605$$. Then, we can divide both sides by $$5$$ to get $$a=521$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq14a-h6","type":"hint","dependencies":["a381217systemeq14a-h5"],"title":"Solve for c","text":"After solving for \\"a\\", we can plug that value into the first equation to get $$521+c=1, 363$$ which we can determine \\"c\\" to be $$842$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a381217systemeq15","title":"Solve Mixture Applications","body":"Translate to a system of equations and solve","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Solve Mixture Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a381217systemeq15a","stepAnswer":["29q13d"],"problemType":"TextBox","stepTitle":"Matilda has a handful of quarters and dimes, with a total value of $$\\\\$8.55$$. The number of quarters is $$3$$ more than twice the number of dimes. How many dimes and how many quarters does she have? (Answer in the form with the number of a variable followed by the first letter of the variable (quarter $$=$$ q, dime $$=$$ d): 2q13d (These numbers are not the answers))","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a381217systemeq15a-h1","type":"hint","dependencies":[],"title":"Convert amounts into first equation","text":"Let\'s use \\"q\\" as the number of quarters and \\"d\\" as the number of dimes. Because the number of quarters is $$3$$ more than $$2$$ times the number of dims, we can identify the first equation to be $$q=2d+3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq15a-h2","type":"hint","dependencies":["a381217systemeq15a-h1"],"title":"Convert amoutn into second equation","text":"Because the total amount of money is $$\\\\$8.55$$ and quarters are $$\\\\$0.25$$ and dimes are $$\\\\$0.10$$ we can determine that the second equation is $$0.25q+0.1d=8.55$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq15a-h3","type":"hint","dependencies":["a381217systemeq15a-h2"],"title":"Plug in","text":"We can plug in $$q=2d+3$$ using the second equation to get $$\\\\operatorname{0.25}\\\\left(2d+3\\\\right)+0.1d=8.55$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq15a-h4","type":"hint","dependencies":["a381217systemeq15a-h3"],"title":"Solve for a","text":"We can determine that the second equation simplified would be $$0.6d+0.75=8.55$$. We now isolate the variable \\"d\\" and get $$0.6d=7.8$$. Then, we can divide both sides by $$0.6$$ to get $$d=13$$. There are $$13$$ dimes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq15a-h5","type":"hint","dependencies":["a381217systemeq15a-h4"],"title":"Solve for c","text":"After solving for \\"d\\", we can plug that value into the first equation to get $$q=2\\\\left(13\\\\right)+3$$ which we can determine $$n$$ to be $$29$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a381217systemeq16","title":"Solving Systems of Equations in Word Problems","body":"Lucinda had a pocketful of dimes and quarters with a value of $ $$\\\\$6.20$$. The number of dimes is eighteen more than three times the number of quarters.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Solve Mixture Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a381217systemeq16a","stepAnswer":["$$42$$ dimes and $$8$$ quarters"],"problemType":"MultipleChoice","stepTitle":"How many dimes and how many quarters does Lucinda have?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$42$$ dimes and $$8$$ quarters","choices":["$$37$$ dimes and $$6$$ quarters","$$42$$ dimes and $$8$$ quarters","$$28$$ dimes and $$9$$ quarters"],"hints":{"DefaultPathway":[{"id":"a381217systemeq16a-h1","type":"hint","dependencies":[],"title":"Creating a System of Equations From Relevant Variables","text":"The relevant variables here are Lucinda\'s dimes and quarters. Since their total value is $$\\\\$6.20$$ and we know the individual values of dimes and quarters, the first equation is $$0.1d+0.25q=6.20$$, where $$d$$ is dimes and q is quarters. Additionally, the last sentence of the problem tells us that $$3q+18=d$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq16a-h2","type":"hint","dependencies":["a381217systemeq16a-h1"],"title":"How to Use the Systems of Equations","text":"Solve the two equations as a normal system of equations to find the number of dimes and quarters (equal to the values of $$d$$ and q respectively.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a381217systemeq17","title":"Solving Systems of Equations in Word Problems","body":"A cashier has $$30$$ bills, all of which are $10 or $20 bills. The total value of the money is $460.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Solve Mixture Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a381217systemeq17a","stepAnswer":["$$14$$ $10 bills, $$16$$ $20 bills"],"problemType":"MultipleChoice","stepTitle":"How many $10 bills and $20 bills does the cashier have?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$14$$ $10 bills, $$16$$ $20 bills","choices":["$$14$$ $10 bills, $$16$$ $20 bills","$$10$$ $10 bills, $$15$$ $20 bills","$$9$$ $10 bills, $$15$$ $20 bills"],"hints":{"DefaultPathway":[{"id":"a381217systemeq17a-h1","type":"hint","dependencies":[],"title":"Creating a System of Equations From Relevant Variables","text":"The relevant variables here are the $10 and $20 bills. We know that the total value is $460. Using a and $$b$$ to represent the number of $$10$$ and $$20$$ dollar bills respectively, this means that $$10a+20b=460$$. Furthermore, since there are $$30$$ bills in total, $$a+b=30$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq17a-h2","type":"hint","dependencies":["a381217systemeq17a-h1"],"title":"How to Use the Systems of Equations","text":"Solve the two equations as a normal system of equations to find the values of the two variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a381217systemeq18","title":"Solving Systems of Equations in Word Problems","body":"A cashier has $$54$$ bills, all of which are $10 or $20 bills. The total value of the money is $910.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Solve Mixture Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a381217systemeq18a","stepAnswer":["$$17$$ $10 bills, $$37$$ $20 bills"],"problemType":"MultipleChoice","stepTitle":"How many $10 bills and $20 bills does the cashier have?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$17$$ $10 bills, $$37$$ $20 bills","choices":["$$17$$ $10 bills, $$37$$ $20 bills","$$23$$ $10 bills, $$18$$ $20 bills","$$25$$ $10 bills, $$16$$ $20 bills","$$25$$ $10 bills, $$16$$ $20 bills"],"hints":{"DefaultPathway":[{"id":"a381217systemeq18a-h1","type":"hint","dependencies":[],"title":"Creating a System of Equations From Relevant Variables","text":"The relevant variables here are the $10 and $20 bills. We know that the total value is $910. Using a and $$b$$ to represent the number of $$10$$ and $$20$$ dollar bills respectively, this means that $$10a+20b=910$$. Furthermore, since there are $$54$$ bills in total, $$a+b=54$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq18a-h2","type":"hint","dependencies":["a381217systemeq18a-h1"],"title":"How to Use the Systems of Equations","text":"Solve the two equations as a normal system of equations to find the values of the two variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a381217systemeq19","title":"Solving Systems of Equations in Word Problems","body":"Marissa wants to blend candy selling for $$\\\\$1.80$$ per pound with candy costing $$\\\\$1.20$$ per pound to get a mixture that costs her $$\\\\$1.40$$ per pound to make. She wants to make $$90$$ pounds of the candy blend. How many pounds of each type of candy should she use?","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Solve Mixture Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a381217systemeq19a","stepAnswer":["$$30$$ pounds of $$\\\\$1.80$$ candy and $$60$$ pounds of $$\\\\$1.20$$ candy"],"problemType":"MultipleChoice","stepTitle":"How many pounds of each type of candy should she use?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$30$$ pounds of $$\\\\$1.80$$ candy and $$60$$ pounds of $$\\\\$1.20$$ candy","choices":["$$30$$ pounds of $$\\\\$1.80$$ candy and $$60$$ pounds of $$\\\\$1.20$$ candy","$$35$$ pounds of $$\\\\$1.80$$ candy and $$55$$ pounds of $$\\\\$1.20$$ candy","$$20$$ pounds of $$\\\\$1.80$$ candy and $$40$$ pounds of $$\\\\$1.20$$ candy"],"hints":{"DefaultPathway":[{"id":"a381217systemeq19a-h1","type":"hint","dependencies":[],"title":"Creating a System of Equations From Relevant Variables","text":"Using a to represent the number of pounds of $$\\\\$1.80$$ candies and $$b$$ to represent the number of pounds of $$\\\\$1.20$$ candies, we know that $$a+b=90$$ since the total is $$90$$ pounds. Additionally, since we know the $$90$$ pounds cost $$\\\\$1.40$$ per pound, the total cost is $$90\\\\times1.4=126$$. Knowing this total cost, we can get our second equation: $$1.8a+1.2b=126$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq19a-h2","type":"hint","dependencies":["a381217systemeq19a-h1"],"title":"How to Use the Systems of Equations","text":"Solve the two equations as a normal system of equations to find the values of the two variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a381217systemeq2","title":"Solve Mixture Applications","body":"Translate to a system of equations and solve","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Solve Mixture Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a381217systemeq2a","stepAnswer":["19q51n"],"problemType":"TextBox","stepTitle":"Priam has a collection of nickels and quarters, with a total value of $$\\\\$7.30$$. The number of nickels is six less than three times the number of quarters. How many nickels and how many quarters does he have? (Answer in the form with the number of a variable followed by the first letter of the variable (quarter $$=$$ q): 2q13n (These numbers are not the answers))","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a381217systemeq2a-h1","type":"hint","dependencies":[],"title":"Convert amounts into first equation","text":"Let\'s use \\"q\\" as the number of quarters and \\"n\\" as the number of nickels. Because the number of nickels is $$6$$ less than $$3$$ times the number of quarters, we can identify the first equation to be $$n=3q-6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq2a-h2","type":"hint","dependencies":["a381217systemeq2a-h1"],"title":"Convert amoutn into second equation","text":"Because the total amount of money is $$\\\\$7.30$$ and quarters are $$\\\\$0.25$$ and nickels are $$\\\\$0.05$$ we can determine that the second equation is $$0.25q+0.05n=7.3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq2a-h3","type":"hint","dependencies":["a381217systemeq2a-h2"],"title":"Plug in","text":"We can plug in $$n=3q-6$$ using the second equation to get $$0.25q+\\\\operatorname{0.05}\\\\left(3q-6\\\\right)=7.3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq2a-h4","type":"hint","dependencies":["a381217systemeq2a-h3"],"title":"Solve for a","text":"We can determine that the second equation simplified would be $$0.4q-0.3=7.3$$. We now isolate the variable \\"q\\" and get $$0.4q=7.6$$. Then, we can divide both sides by $$0.4$$ to get $$q=19$$. There are $$19$$ quarters.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq2a-h5","type":"hint","dependencies":["a381217systemeq2a-h4"],"title":"Solve for c","text":"After solving for \\"q\\", we can plug that value into the first equation to get $$n=3(19)-6$$ which we can determine $$n$$ to be $$51$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a381217systemeq20","title":"Solving Systems of Equations in Word Problems","body":"How many pounds of nuts selling for $6 per pound and raisins selling for $3 per pound should Kurt combine to obtain $$120$$ pounds of trail mix that cost him $5 per pound?","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Solve Mixture Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a381217systemeq20a","stepAnswer":["$$80$$ pounds of nuts and $$40$$ pounds of raisins"],"problemType":"MultipleChoice","stepTitle":"How many pounds of nuts and raisins should Kurt use?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$80$$ pounds of nuts and $$40$$ pounds of raisins","choices":["$$70$$ pounds of nuts and $$50$$ pounds of raisins","$$80$$ pounds of nuts and $$40$$ pounds of raisins","$$30$$ pounds of nuts and $$16$$ pounds of raisins"],"hints":{"DefaultPathway":[{"id":"a381217systemeq20a-h1","type":"hint","dependencies":[],"title":"Creating a System of Equations From Relevant Variables","text":"Using a to represent the number of pounds of $6 nuts and $$b$$ to represent the number of pounds of $13 raisins, we know that $$a+b=120$$ since the total is $$120$$ pounds. Additionally, since we know the $$120$$ pounds cost $5 per pound, the total cost is $$120\\\\times5=600$$. Knowing this total cost, we can get our second equation: $$6a+3b=600$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq20a-h2","type":"hint","dependencies":["a381217systemeq20a-h1"],"title":"How to Use the Systems of Equations","text":"Solve the two equations as a normal system of equations to find the values of the two variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a381217systemeq21","title":"Solving Systems of Equations in Word Problems","body":"Hannah has to make twenty-five gallons of punch for a potluck. The punch is made of soda and fruit drink. The cost of the soda is $$\\\\$1.79$$ per gallon and the cost of the fruit drink is $$\\\\$2.49$$ per gallon. Hannah\u2019s budget requires that the punch cost $$\\\\$2.21$$ per gallon.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Solve Mixture Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a381217systemeq21a","stepAnswer":["$$10$$ gallons of soda and $$15$$ gallons of fruit drink"],"problemType":"MultipleChoice","stepTitle":"How many gallons of soda and how many gallons of fruit drink does she need?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$10$$ gallons of soda and $$15$$ gallons of fruit drink","choices":["$$10$$ gallons of soda and $$12$$ gallons of fruit drink","$$8$$ gallons of soda and $$15$$ gallons of fruit drink","$$10$$ gallons of soda and $$15$$ gallons of fruit drink"],"hints":{"DefaultPathway":[{"id":"a381217systemeq21a-h1","type":"hint","dependencies":[],"title":"Creating a System of Equations From Relevant Variables","text":"Using a to represent the number of gallons of soda and $$b$$ to represent the number of gallons of the fruit drink, we know that $$a+b=25$$ since the total is $$25$$ gallons. Additionally, since we know the $$25$$ gallons cost $$\\\\$2.21$$ per gallon, the total cost is $$25\\\\times2.21=55.25$$. Knowing this total cost, we can get our second equation: $$1.79a+2.49b=55.25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq21a-h2","type":"hint","dependencies":["a381217systemeq21a-h1"],"title":"How to Use the Systems of Equations","text":"Solve the two equations as a normal system of equations to find the values of the two variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a381217systemeq22","title":"Solving Systems of Equations in Word Problems","body":"Joseph would like to make $$12$$ pounds of a coffee blend at a cost of $$\\\\$6.25$$ per pound. He blends Ground Chicory at $$\\\\$4.40$$ a pound with Jamaican Blue Mountain at $$\\\\$8.84$$ per pound.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Solve Mixture Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a381217systemeq22a","stepAnswer":["$$7$$ pounds of Ground Chicory and $$5$$ pounds of Jamaican Blue Mountain"],"problemType":"MultipleChoice","stepTitle":"How much of each type of coffee should he use?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$7$$ pounds of Ground Chicory and $$5$$ pounds of Jamaican Blue Mountain","choices":["$$6$$ pounds of Ground Chicory and $$5$$ pounds of Jamaican Blue Mountain","$$7$$ pounds of Ground Chicory and $$5$$ pounds of Jamaican Blue Mountain","$$8$$ pounds of Ground Chicory and $$5$$ pounds of Jamaican Blue Mountain"],"hints":{"DefaultPathway":[{"id":"a381217systemeq22a-h1","type":"hint","dependencies":[],"title":"Creating a System of Equations From Relevant Variables","text":"Using a to represent the number of pounds of Ground Chicory and $$b$$ to represent the number of pounds of Jamaican Blue Mountain, we know that $$a+b=12$$ since the total is $$12$$ pounds. Additionally, since we know the $$12$$ pounds cost $$\\\\$6.25$$ per pound, the total cost is $$12\\\\times6.25=\\\\$75$$. Knowing this total cost, we can get our second equation: $$4.4a+8.84b=75$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq22a-h2","type":"hint","dependencies":["a381217systemeq22a-h1"],"title":"How to Use the Systems of Equations","text":"Solve the two equations as a normal system of equations to find the values of the two variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a381217systemeq23","title":"Solving Systems of Equations in Word Problems","body":"Julia and her husband own a coffee shop. They experimented with mixing a City Roast Columbian coffee that cost $$\\\\$7.80$$ per pound with French Roast Columbian coffee that cost $$\\\\$8.10$$ per pound to make a $$20$$ pound blend. Their blend should cost them $$\\\\$7.92$$ per pound.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Solve Mixture Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a381217systemeq23a","stepAnswer":["$$12$$ pounds of City Roast Columbian coffee and $$8$$ pounds of French Roast Columbian coffee"],"problemType":"MultipleChoice","stepTitle":"How much of each type of coffee should they buy?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$12$$ pounds of City Roast Columbian coffee and $$8$$ pounds of French Roast Columbian coffee","choices":["$$10$$ pounds of City Roast Columbian coffee and $$7$$ pounds of French Roat Columbian coffee","$$12$$ pounds of City Roast Columbian coffee and $$8$$ pounds of French Roast Columbian coffee","$$12$$ pounds of City Roast Columbian coffee and $$8$$ pounds of French Roat Columbian coffee","$$13$$ pounds of City Roast Columbian coffee and $$5$$ pounds of French Roat Columbian coffee"],"hints":{"DefaultPathway":[{"id":"a381217systemeq23a-h1","type":"hint","dependencies":[],"title":"Creating a System of Equations From Relevant Variables","text":"Using a to represent the number of pounds of City Roast Columbian and $$b$$ to represent the number of pounds of French Roast Columbian, we know that $$a+b=20$$ since the total is $$20$$ pounds. Additionally, since we know the $$20$$ pounds cost $$\\\\$7.92$$ per pound, the total cost is $$20\\\\times7.92=158.4$$. Knowing this total cost, we can get our second equation: $$7.8a+8.1b=158.4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq23a-h2","type":"hint","dependencies":["a381217systemeq23a-h1"],"title":"How to Use the Systems of Equations","text":"Solve the two equations as a normal system of equations to find the values of the two variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a381217systemeq24","title":"Solving Systems of Equations in Word Problems","body":"Melody wants to sell bags of mixed candy at her lemonade stand. She will mix chocolate pieces that cost $$\\\\$4.89$$ per bag with peanut butter pieces that cost $$\\\\$3.79$$ per bag to get a total of twenty-five bags of mixed candy. Melody wants the bags of mixed candy to cost her $$\\\\$4.23$$ a bag to make.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Solve Mixture Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a381217systemeq24a","stepAnswer":["$$10$$ bags of chocolate pieces and $$15$$ bags of peanut butter pieces"],"problemType":"MultipleChoice","stepTitle":"How many bags of chocolate pieces and how many bags of peanut butter pieces should she use?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$10$$ bags of chocolate pieces and $$15$$ bags of peanut butter pieces","choices":["$$10$$ bags of chocolate pieces and $$15$$ bags of peanut butter pieces","$$12$$ bags of chocolate pieces and $$16$$ bags of peanut butter pieces","$$8$$ bags of chocolate pieces and $$15$$ bags of peanut butter pieces"],"hints":{"DefaultPathway":[{"id":"a381217systemeq24a-h1","type":"hint","dependencies":[],"title":"Creating a System of Equations From Relevant Variables","text":"Using a to represent the number of bags of chocolate pieces and $$b$$ to represent the number of bags of peanut butter pieces, we know that $$a+b=25$$ since the total is $$25$$ bags. Additionally, since we know the $$25$$ bags cost $$\\\\$4.23$$ per bag, the total cost is $$25\\\\times4.23=105.75$$. Knowing this total cost, we can get our second equation: $$4.89a+3.79b=105.75$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq24a-h2","type":"hint","dependencies":["a381217systemeq24a-h1"],"title":"How to Use the Systems of Equations","text":"Solve the two equations as a normal system of equations to find the values of the two variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a381217systemeq25","title":"Solving Systems of Equations in Word Problems","body":"As the treasurer of her daughter\u2019s Girl Scout troop, Laney collected money for some girls and adults to go to a three-day camp. Each girl paid $75 and each adult paid $30. The total amount of money collected for camp was $765. The number of girls is three times the number of adults.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Solve Mixture Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a381217systemeq25a","stepAnswer":["$$9$$ girls and $$3$$ adults"],"problemType":"MultipleChoice","stepTitle":"How many girls and how many adults paid for camp?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$9$$ girls and $$3$$ adults","choices":["$$8$$ girls and $$4$$ adults","$$9$$ girls and $$3$$ adults","$$15$$ girls and $$3$$ adults"],"hints":{"DefaultPathway":[{"id":"a381217systemeq25a-h1","type":"hint","dependencies":[],"title":"Creating a System of Equations From Relevant Variables","text":"Using g to represent the number of girls and a to represent the number of adults, we know $$75g+30a=765$$ from the given information. Furthermore, since the number of girls is $$3$$ times the number of adults, we know that $$g=3a$$. This can be rewritten as $$g-3a=0$$ for our second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq25a-h2","type":"hint","dependencies":["a381217systemeq25a-h1"],"title":"How to Use the Systems of Equations","text":"Solve the two equations as a normal system of equations to find the values of the two variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a381217systemeq3","title":"Solve Mixture Applications","body":"Translate to a system of equations and solve","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Solve Mixture Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a381217systemeq3a","stepAnswer":["16n4c"],"problemType":"TextBox","stepTitle":"Carson wants to make $$20$$ pounds of trail mix using nuts and chocolate chips. His budget requires that the trail mix costs him $$\\\\$7.60$$ per pound. Nuts cost $$\\\\$9.00$$ per pound and chocolate chips cost $$\\\\$2.00$$ per pound. How many pounds of nuts and how many pounds of chocolate chips should he use? (Answer in the form with the number of a variable followed by the first letter of the variable (nuts $$=$$ n): 2n13c (These numbers are not the answers))","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a381217systemeq3a-h1","type":"hint","dependencies":[],"title":"Convert amounts into first equation","text":"Let\'s use \\"n\\" as the amount in pounds of nuts and \\"c\\" as the amount in pounds of chocolate chips. The total weight combined is $$20$$ pounds, therefore we can determine that the first equation is $$n+c=20$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq3a-h2","type":"hint","dependencies":["a381217systemeq3a-h1"],"title":"Convert amoutn into second equation","text":"Because the total amount of money was 20*$7.60 $$=$$ $152 and nuts are $$\\\\$9.00$$ per pound and chocolate chips $2 per pound, we can determine that the second equation is $$9n+2c=152$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq3a-h3","type":"hint","dependencies":["a381217systemeq3a-h2"],"title":"Isolate Variable","text":"Let\'s use the first equation to isolate c. With the equation, we can subtract a from both sides to get $$c=20-n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq3a-h4","type":"hint","dependencies":["a381217systemeq3a-h3"],"title":"Plug in","text":"We can plug in $$c=20-n$$ using the second equation to get $$9n+2\\\\left(20-n\\\\right)=152$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq3a-h5","type":"hint","dependencies":["a381217systemeq3a-h4"],"title":"Solve for a","text":"We can determine that the second equation after plugging in \\"c\\" would be $$7n+40=152$$. We now isolate the variable \\"n\\" and get $$7n=112$$. Then, we can divide both sides by $$7$$ to get $$n=16$$. There are $$16$$ pounds of nuts.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq3a-h6","type":"hint","dependencies":["a381217systemeq3a-h5"],"title":"Solve for c","text":"After solving for \\"n\\", we can plug that value into the first equation to get $$16+c=20$$ which we can determine c to be $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a381217systemeq4","title":"Solve Mixture Applications","body":"Translate to a system of equations and solve","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Solve Mixture Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a381217systemeq4a","stepAnswer":["80t120f"],"problemType":"TextBox","stepTitle":"Sasheena is a lab assistant at her community college. She needs to make $$200$$ milliliters of a 40% solution of sulfuric acid for a lab experiment. The lab has only 25% and 50% solutions in the storeroom. How much should she mix of the 25% and the 50% solutions to make the 40% solution? (Answer in the form with the number of a variable followed by the first letter of the variable (25% solution $$=$$ $$t$$, 50% solution $$=$$ f): 20n30c (These numbers are not the answers))","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a381217systemeq4a-h1","type":"hint","dependencies":[],"title":"Convert amounts into first equation","text":"Let\'s use \\"t\\" as the 25% solution and \\"f\\" as the 50% solution. The total amount of solution combined is $$200$$, therefore we can determine that the first equation is $$t+f=200$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq4a-h2","type":"hint","dependencies":["a381217systemeq4a-h1"],"title":"Convert amoutn into second equation","text":"Because the total amount is 40% of $$200$$ (80) and there are 25% solutions and 50% solutions, we can determine that the second equation is $$0.25t+0.5f=80$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq4a-h3","type":"hint","dependencies":["a381217systemeq4a-h2"],"title":"Isolate Variable","text":"Let\'s use the first equation to isolate $$t$$. With the equation, we can subtract a from both sides to get $$t=200-f$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq4a-h4","type":"hint","dependencies":["a381217systemeq4a-h3"],"title":"Plug in","text":"We can plug in $$t=200-f$$ using the second equation to get $$\\\\operatorname{0.25}\\\\left(200-f\\\\right)+0.5f=80$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq4a-h5","type":"hint","dependencies":["a381217systemeq4a-h4"],"title":"Solve for a","text":"We can determine that the second equation after plugging in $$t$$ would be $$50+0.25f=80$$. We now isolate the variable \\"f\\" and get $$0.25f=30$$. Then, we can divide both sides by $$0.25$$ to get $$f=120$$. There is $$120$$ mililiters of 50% solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq4a-h6","type":"hint","dependencies":["a381217systemeq4a-h5"],"title":"Solve for c","text":"After solving for \\"f\\", we can plug that value into the first equation to get $$t+120=200$$ which we can determine $$t$$ to be $$80$$ mililiters.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a381217systemeq5","title":"Solve Mixture Applications","body":"Translate to a system of equations and solve","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Solve Mixture Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a381217systemeq5a","stepAnswer":["32800s7200b"],"problemType":"TextBox","stepTitle":"Adnan has $40,000 to invest and hopes to earn $$7.1\\\\%$$ interest per year. He will put some of the money into a stock fund that earns 8% per year and the rest into bonds that earns 3% per year. How much money should he put into each fund? (Answer in the form with the number of a variable followed by the first letter of the variable (stocks $$=$$ s, bonds $$=$$ b): 20s30b (These numbers are not the answers))","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a381217systemeq5a-h1","type":"hint","dependencies":[],"title":"Convert amounts into first equation","text":"Let\'s use \\"s\\" as stocks and \\"b\\" as the bonds. The total amount of money combined is $40,000, therefore we can determine that the first equation is $$s+b=40, 000$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq5a-h2","type":"hint","dependencies":["a381217systemeq5a-h1"],"title":"Convert amoutn into second equation","text":"Because the total amount is $$0.071(40,000)$$ and an interest rate of 8% on stocks and 3% on bonds, we can determine that the second equation is $$0.08s+0.03b=2, 840$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq5a-h3","type":"hint","dependencies":["a381217systemeq5a-h2"],"title":"Isolate Variable","text":"Let\'s use the first equation to isolate $$b$$. With the equation, we can subtract a from both sides to get $$b=40, 000-s$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq5a-h4","type":"hint","dependencies":["a381217systemeq5a-h3"],"title":"Plug in","text":"We can plug in $$b=40, 000-s$$ using the second equation to get $$0.08s+0.03(40,000-s)=2,840$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq5a-h5","type":"hint","dependencies":["a381217systemeq5a-h4"],"title":"Solve for a","text":"We can determine that the second equation after plugging in \\"b\\" would be 0.05s+1,200=2,840. We now isolate the variable \\"s\\" and get $$0.05s=1640$$. Then, we can divide both sides by $$0.05$$ to get $$s=32, 800$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq5a-h6","type":"hint","dependencies":["a381217systemeq5a-h5"],"title":"Solve for c","text":"After solving for \\"s\\", we can plug that value into the first equation to get 32,800+b=40,000 which we can determine $$b$$ to be 7,200.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a381217systemeq6","title":"Solve Mixture Applications","body":"Translate to a system of equations and solve","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Solve Mixture Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a381217systemeq6a","stepAnswer":["8670b12870f"],"problemType":"TextBox","stepTitle":"Rosie owes $21,540 on her two student loans. The interest rate on her bank loan is $$10.5\\\\%$$ and the interest rate on the federal loan is $$5.9\\\\%$$. The total amount of interest she paid last year was $$\\\\$1, 669.68$$. What was the principal for each loan? (Answer in the form with the number of a variable followed by the first letter of the variable (bank $$=$$ $$b$$, federal $$=$$ f): 20b30f (These numbers are not the answers))","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a381217systemeq6a-h1","type":"hint","dependencies":[],"title":"Convert amounts into first equation","text":"Let\'s use \\"b\\" as bank loan and \\"f\\" as the federal loan. The total amount of money combined is $21,540, therefore we can determine that the first equation is $$b+f=21, 540$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq6a-h2","type":"hint","dependencies":["a381217systemeq6a-h1"],"title":"Convert amoutn into second equation","text":"Because the total amount is how much she paid $$(\\\\$1, 669.68)$$ and there is an interest rate of $$10.5\\\\%$$ from bank and $$5.9\\\\%$$ from federal, we can determine that the second equation is $$0.105b+0.059f=1, 669.68$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq6a-h3","type":"hint","dependencies":["a381217systemeq6a-h2"],"title":"Isolate Variable","text":"Let\'s use the first equation to isolate $$b$$. With the equation, we can subtract a from both sides to get $$b=21, 540-f$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq6a-h4","type":"hint","dependencies":["a381217systemeq6a-h3"],"title":"Plug in","text":"We can plug in $$b=21, 540-f$$ using the second equation to get $$0.105(21,540-f)f+0.059=1,669.68$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq6a-h5","type":"hint","dependencies":["a381217systemeq6a-h4"],"title":"Solve for a","text":"We can determine that the second equation after plugging in \\"b\\" would be $$2, 261.7-0.046f=1, 669.68$$. We now isolate the variable \\"f\\" and get $$-0.046f=-592.02$$. Then, we can divide both sides by $$-0.046$$ to get $$f=12, 870$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq6a-h6","type":"hint","dependencies":["a381217systemeq6a-h5"],"title":"Solve for c","text":"After solving for \\"f\\", we can plug that value into the first equation to get b+12,870=21,540 which we can determine $$b$$ to be 8,670.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a381217systemeq7","title":"Solve Mixture Applications","body":"Translate to a system of equations and solve","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Solve Mixture Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a381217systemeq7a","stepAnswer":["1120a530c"],"problemType":"TextBox","stepTitle":"Tickets to a Broadway show cost $35 for adults and $15 for children. The total receipts for $$1650$$ tickets at one performance were $47,150. How many adult and how many child tickets were sold? (Answer in the form with the number of a variable followed by the first letter of the variable (adult $$=$$ a): 11a13c (These numbers and variables are not the answers))","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a381217systemeq7a-h1","type":"hint","dependencies":[],"title":"Convert amounts into first equation","text":"Let\'s use \\"a\\" as the number of adult tickets and \\"c\\" as the number of childrens tickets. The total number of tickets combined is $$1650$$, therefore we can determine that the first equation is $$a+c=1, 650$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq7a-h2","type":"hint","dependencies":["a381217systemeq7a-h1"],"title":"Convert amoutn into second equation","text":"Because the total amount of money made was $47,150 and adult tickets are $35 and childrens $15 we can determine that the second equation is $$35a+15c=47, 150$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq7a-h3","type":"hint","dependencies":["a381217systemeq7a-h2"],"title":"Isolate Variable","text":"Let\'s use the first equation to isolate c. With the equation, we can subtract a from both sides to get $$c=1, 650-a$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq7a-h4","type":"hint","dependencies":["a381217systemeq7a-h3"],"title":"Plug in","text":"We can plug in $$c=1, 650-a$$ using the second equation to get $$35a+15(1,650-a)=47,150$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq7a-h5","type":"hint","dependencies":["a381217systemeq7a-h4"],"title":"Solve for a","text":"We can determine that the second equation after plugging in \\"c\\" would be 20a+24,750=47,150. We now isolate the variable \\"a\\" and get $$20a=22, 400$$. Then, we can divide both sides by $$20$$ to get $$a=1, 120$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq7a-h6","type":"hint","dependencies":["a381217systemeq7a-h5"],"title":"Solve for c","text":"After solving for \\"a\\", we can plug that value into the first equation to get 1,120+c=1,650 which we can determine \\"c\\" to be $$530$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a381217systemeq8","title":"Solve Mixture Applications","body":"Translate to a system of equations and solve","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Solve Mixture Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a381217systemeq8a","stepAnswer":["83a217c"],"problemType":"TextBox","stepTitle":"Tickets for a show are $70 for adults and $50 for children. One evening performance had a total of $$300$$ tickets sold and the receipts totaled $17,200. How many adult and how many child tickets were sold? (Answer in the form with the number of a variable followed by the first letter of the variable (adult $$=$$ a): 11a13c (These numbers and variables are not the answers))","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a381217systemeq8a-h1","type":"hint","dependencies":[],"title":"Convert amounts into first equation","text":"Let\'s use \\"a\\" as the number of adult tickets and \\"c\\" as the number of childrens tickets. The total number of tickets combined is $$300$$, therefore we can determine that the first equation is $$a+c=300$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq8a-h2","type":"hint","dependencies":["a381217systemeq8a-h1"],"title":"Convert amoutn into second equation","text":"Because the total amount of money made was $17,200 and adult tickets are $70 and childrens $50 we can determine that the second equation is $$70a+50c=17, 200$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq8a-h3","type":"hint","dependencies":["a381217systemeq8a-h2"],"title":"Isolate Variable","text":"Let\'s use the first equation to isolate c. With the equation, we can subtract a from both sides to get $$c=300-a$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq8a-h4","type":"hint","dependencies":["a381217systemeq8a-h3"],"title":"Plug in","text":"We can plug in $$c=300-a$$ using the second equation to get $$70a+\\\\operatorname{50}\\\\left(300-a\\\\right)=17, 200$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq8a-h5","type":"hint","dependencies":["a381217systemeq8a-h4"],"title":"Solve for a","text":"We can determine that the second equation after plugging in \\"c\\" would be $$20a+600=17, 200$$. We now isolate the variable \\"a\\" and get $$20a=16, 600$$. Then, we can divide both sides by $$20$$ to get $$a=83$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq8a-h6","type":"hint","dependencies":["a381217systemeq8a-h5"],"title":"Solve for c","text":"After solving for \\"a\\", we can plug that value into the first equation to get $$83+c=300$$ which we can determine \\"c\\" to be $$217$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a381217systemeq9","title":"Solve Mixture Applications","body":"Translate to a system of equations and solve","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Solve Mixture Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a381217systemeq9a","stepAnswer":["40a32c"],"problemType":"TextBox","stepTitle":"Tickets for a train cost $10 for children and $22 for adults. Josie paid $1,200 for a total of $$72$$ tickets. How many children\u2019s tickets and how many adult tickets did Josie buy? (Answer in the form with the number of a variable followed by the first letter of the variable (adult $$=$$ a): 11a13c (These numbers and variables are not the answers))","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a381217systemeq9a-h1","type":"hint","dependencies":[],"title":"Convert amounts into first equation","text":"Let\'s use \\"a\\" as the number of adult tickets and \\"c\\" as the number of childrens tickets. The total number of tickets combined is $$72$$, therefore we can determine that the first equation is $$a+c=72$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq9a-h2","type":"hint","dependencies":["a381217systemeq9a-h1"],"title":"Convert amoutn into second equation","text":"Because the total amount of money made was $1,200 and adult tickets are $22 and childrens $10 we can determine that the second equation is $$22a+10c=1, 200$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq9a-h3","type":"hint","dependencies":["a381217systemeq9a-h2"],"title":"Isolate Variable","text":"Let\'s use the first equation to isolate c. With the equation, we can subtract a from both sides to get $$c=72-a$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq9a-h4","type":"hint","dependencies":["a381217systemeq9a-h3"],"title":"Plug in","text":"We can plug in $$c=72-a$$ using the second equation to get $$22a+\\\\operatorname{10}\\\\left(72-a\\\\right)=1, 200$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq9a-h5","type":"hint","dependencies":["a381217systemeq9a-h4"],"title":"Solve for a","text":"We can determine that the second equation after plugging in \\"c\\" would be $$12a+720=1, 200$$. We now isolate the variable \\"a\\" and get $$12a=480$$. Then, we can divide both sides by $$12$$ to get $$a=40$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a381217systemeq9a-h6","type":"hint","dependencies":["a381217systemeq9a-h5"],"title":"Solve for c","text":"After solving for \\"a\\", we can plug that value into the first equation to get $$40+c=72$$ which we can determine \\"c\\" to be $$32$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a382995DiscreteDist1","title":"Playing Card Experiment","body":"Suppose we are conducting the \\"Playing Card Experiment.\\" Essentially, the experimental procedure for empirical data is to pick one card from a deck of shuffled cards. Then, we can record the card. We can repeat this prcoess (about $$10$$ times) and calculate certain probabilites based on the data collected.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Discrete Distribution (Playing Card Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a382995DiscreteDist1a","stepAnswer":["$$\\\\frac{1}{4}$$"],"problemType":"TextBox","stepTitle":"In a single turn, what is the probability that the card is a diamond?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{4}$$","hints":{"DefaultPathway":[{"id":"a382995DiscreteDist1a-h1","type":"hint","dependencies":[],"title":"Probability","text":"To calculate the probability, we divide the number of cards with the desired outcome by the total number of possible outcomes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$52$$"],"dependencies":["a382995DiscreteDist1a-h1"],"title":"Possible Outcomes","text":"How many total cards are there in the deck? This represents the number of possible outcomes for a single turn.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$13$$"],"dependencies":["a382995DiscreteDist1a-h2"],"title":"Desired Outcome","text":"How many cards have a diamond? This is the number of cards with the desired outcome.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4}$$"],"dependencies":["a382995DiscreteDist1a-h3"],"title":"Divide","text":"What is $$\\\\frac{13}{52}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a382995DiscreteDist10","title":"Playing Card Experiment","body":"Suppose we are conducting the \\"Playing Card Experiment.\\" Essentially, the experimental procedure for empirical data is to pick one card from a deck of shuffled cards. Then, we can record the card. We can repeat this prcoess (about $$10$$ times) and calculate certain probabilites based on the data collected.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Discrete Distribution (Playing Card Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a382995DiscreteDist10a","stepAnswer":["No, we are never guaranteed to get exactly the same proportion in our sample as the ideal proportion."],"problemType":"MultipleChoice","stepTitle":"In ten turns, is it guaranteed that we will flip a king exactly the same number of times as the ideal number of times we should flip a king in tens turns?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes, because the ideal proportion represents the proportion we would get for any sample.","No, we are never guaranteed to get exactly the same proportion in our sample as the ideal proportion."],"hints":{"DefaultPathway":[{"id":"a382995DiscreteDist10a-h1","type":"hint","dependencies":[],"title":"Interpret","text":"Since ten turns is a small sample size, it is quite unlikely that we will flip a king exactly the same number of times as the ideal number of times we should flip a king in ten turns. Even in a large enough sample size, though, it is not ever guaranteed that we achieve the same proportion of our desired outcome with the given sample as the ideal proportion.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a382995DiscreteDist11","title":"Playing Card Experiment","body":"Suppose we are conducting the \\"Playing Card Experiment.\\" Essentially, the experimental procedure for empirical data is to pick one card from a deck of shuffled cards. Then, we can record the card. We can repeat this prcoess (about $$10$$ times) and calculate certain probabilites based on the data collected.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Discrete Distribution (Playing Card Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a382995DiscreteDist11a","stepAnswer":["$$\\\\frac{1}{169}$$"],"problemType":"TextBox","stepTitle":"In two turns, what is the probability that both cards are queens?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{169}$$","hints":{"DefaultPathway":[{"id":"a382995DiscreteDist11a-h1","type":"hint","dependencies":[],"title":"Probability","text":"The first step is calculating the probability of flipping a queen in one turn. To calculate the probability, we divide the number of cards with the desired outcome by the total number of possible outcomes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$52$$"],"dependencies":["a382995DiscreteDist11a-h1"],"title":"Possible Outcomes","text":"How many total cards are there in the deck?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist11a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a382995DiscreteDist11a-h2"],"title":"Desired Outcome","text":"How many cards are a queen? This is the number of cards with the desired outcome.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{13}$$"],"dependencies":["a382995DiscreteDist11a-h3"],"title":"Divide","text":"What is $$\\\\frac{4}{52}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist11a-h5","type":"hint","dependencies":["a382995DiscreteDist11a-h4"],"title":"$$2$$ In A Row","text":"To calculate the probability that $$2$$ cards in a row are both queens, we can multiply the individual probabilities together since we need to flip a queen card AND another queen card. The AND keyword hints at the idea that the probabilities need to be multiplied together. Essentially, we would multiply the probability of flipping a red with itself.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist11a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{169}$$"],"dependencies":["a382995DiscreteDist11a-h5"],"title":"Calculation","text":"What is $$\\\\frac{1}{13} \\\\frac{1}{13}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a382995DiscreteDist12","title":"Playing Card Experiment","body":"Suppose we are conducting the \\"Playing Card Experiment.\\" Essentially, the experimental procedure for empirical data is to pick one card from a deck of shuffled cards. Then, we can record the card. We can repeat this prcoess (about $$10$$ times) and calculate certain probabilites based on the data collected.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Discrete Distribution (Playing Card Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a382995DiscreteDist12a","stepAnswer":["$$\\\\frac{1}{26}$$"],"problemType":"TextBox","stepTitle":"What is the probability of selecting a red king in a single turn?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{26}$$","hints":{"DefaultPathway":[{"id":"a382995DiscreteDist12a-h1","type":"hint","dependencies":[],"title":"Probability","text":"To calculate the probability, we divide the number of cards with the desired outcome by the total number of possible outcomes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$52$$"],"dependencies":["a382995DiscreteDist12a-h1"],"title":"Possible Outcomes","text":"How many total cards are there in the deck? This represents the number of possible outcomes for a single turn.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist12a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$26$$"],"dependencies":["a382995DiscreteDist12a-h2"],"title":"Desired Outcome","text":"How many cards are red kings? This is the number of cards with the desired outcome.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{26}$$"],"dependencies":["a382995DiscreteDist12a-h3"],"title":"Divide","text":"What is $$\\\\frac{2}{52}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a382995DiscreteDist13","title":"Playing Card Experiment","body":"Suppose we are conducting the \\"Playing Card Experiment.\\" Essentially, the experimental procedure for empirical data is to pick one card from a deck of shuffled cards. Then, we can record the card. We can repeat this prcoess (about $$10$$ times) and calculate certain probabilites based on the data collected.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Discrete Distribution (Playing Card Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a382995DiscreteDist13a","stepAnswer":["$$\\\\frac{8}{13}$$"],"problemType":"TextBox","stepTitle":"In a single turn, what is the probability of selecting a number card? (Number cards are the cards from $$2-10)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{8}{13}$$","hints":{"DefaultPathway":[{"id":"a382995DiscreteDist13a-h1","type":"hint","dependencies":[],"title":"Probability","text":"To calculate the probability, we divide the number of cards with the desired outcome by the total number of possible outcomes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$52$$"],"dependencies":["a382995DiscreteDist13a-h1"],"title":"Possible Outcomes","text":"How many total cards are there in the deck? This represents the number of possible outcomes for a single turn.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist13a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$32$$"],"dependencies":["a382995DiscreteDist13a-h2"],"title":"Desired Outcome","text":"How many cards are number cards? This is the number of cards with the desired outcome.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{8}{13}$$"],"dependencies":["a382995DiscreteDist13a-h3"],"title":"Divide","text":"What is $$\\\\frac{32}{52}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a382995DiscreteDist14","title":"Playing Card Experiment","body":"Suppose we are conducting the \\"Playing Card Experiment.\\" Essentially, the experimental procedure for empirical data is to pick one card from a deck of shuffled cards. Then, we can record the card. We can repeat this prcoess (about $$10$$ times) and calculate certain probabilites based on the data collected.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Discrete Distribution (Playing Card Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a382995DiscreteDist14a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"In ten turns, what is the ideal number of red cards we should have flipped? (round down to the nearest whole number)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a382995DiscreteDist14a-h1","type":"hint","dependencies":[],"title":"Probability","text":"The first step is calculating the probability of flipping a red card in one turn. 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Then, we can record the card. We can repeat this prcoess (about $$10$$ times) and calculate certain probabilites based on the data collected.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Discrete Distribution (Playing Card Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a382995DiscreteDist15a","stepAnswer":["$$\\\\frac{2}{13}$$"],"problemType":"TextBox","stepTitle":"What is the probability of flipping a king or a queen in a single turn?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2}{13}$$","hints":{"DefaultPathway":[{"id":"a382995DiscreteDist15a-h1","type":"hint","dependencies":[],"title":"Probability","text":"To calculate the probability, we divide the number of cards with the desired outcome by the total number of possible outcomes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$52$$"],"dependencies":["a382995DiscreteDist15a-h1"],"title":"Possible Outcomes","text":"How many total cards are there in the deck? 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We can repeat this prcoess (about $$10$$ times) and calculate certain probabilites based on the data collected.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Discrete Distribution (Playing Card Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a382995DiscreteDist2a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"In ten turns, what should be the ideal number of times we flip a diamond? (Round down to the nearest whole number)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a382995DiscreteDist2a-h1","type":"hint","dependencies":[],"title":"Probability","text":"The first step is calculating the probability of flipping a diamond in one turn. 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This is the number of cards with the desired outcome.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4}$$"],"dependencies":["a382995DiscreteDist2a-h3"],"title":"Divide","text":"What is $$\\\\frac{13}{52}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist2a-h5","type":"hint","dependencies":["a382995DiscreteDist2a-h4"],"title":"Ten Turns","text":"To find the number of times we should flip a diamond in ten turns, we can multiply the probability of flipping a diamond in one turn by the number of turns.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{2}$$"],"dependencies":["a382995DiscreteDist2a-h5"],"title":"Calculate","text":"What is $$10\\\\frac{1}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist2a-h7","type":"hint","dependencies":["a382995DiscreteDist2a-h6"],"title":"Rounding","text":"Round $$\\\\frac{5}{2}$$ down to the nearest whole number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a382995DiscreteDist3","title":"Playing Card Experiment","body":"Suppose we are conducting the \\"Playing Card Experiment.\\" Essentially, the experimental procedure for empirical data is to pick one card from a deck of shuffled cards. Then, we can record the card. We can repeat this prcoess (about $$10$$ times) and calculate certain probabilites based on the data collected.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Discrete Distribution (Playing Card Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a382995DiscreteDist3a","stepAnswer":["No, we are never guaranteed to get exactly the same proportion in our sample as the ideal proportion."],"problemType":"MultipleChoice","stepTitle":"In ten turns, is it guaranteed that we will flip a diamond exactly the same number of times as the ideal number of times we should flip a diamond in tens turns?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes, because the ideal proportion represents the proportion we would get for any sample.","No, we are never guaranteed to get exactly the same proportion in our sample as the ideal proportion."],"hints":{"DefaultPathway":[{"id":"a382995DiscreteDist3a-h1","type":"hint","dependencies":[],"title":"Interpret","text":"Since ten turns is a small sample size, it is quite unlikely that we will flip a diamond exactly the same number of times as the ideal number of times we should flip a diamond in ten turns. Even in a large enough sample size, though, it is not ever guaranteed that we achieve the same proportion of our desired outcome with the given sample as the ideal proportion.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a382995DiscreteDist4","title":"Playing Card Experiment","body":"Suppose we are conducting the \\"Playing Card Experiment.\\" Essentially, the experimental procedure for empirical data is to pick one card from a deck of shuffled cards. Then, we can record the card. We can repeat this prcoess (about $$10$$ times) and calculate certain probabilites based on the data collected.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Discrete Distribution (Playing Card Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a382995DiscreteDist4a","stepAnswer":["$$\\\\frac{1}{16}$$"],"problemType":"TextBox","stepTitle":"In two turns, what is the probability that both cards are spades?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{16}$$","hints":{"DefaultPathway":[{"id":"a382995DiscreteDist4a-h1","type":"hint","dependencies":[],"title":"Probability","text":"The first step is calculating the probability of flipping a spades in one turn. 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This is the number of cards with the desired outcome.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4}$$"],"dependencies":["a382995DiscreteDist4a-h3"],"title":"Divide","text":"What is $$\\\\frac{13}{52}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist4a-h5","type":"hint","dependencies":["a382995DiscreteDist4a-h4"],"title":"$$2$$ In A Row","text":"To calculate the probability that $$2$$ cards in a row are both spades, we can multiply the individual probabilities together since we need to flip a card with spades AND another card with spades. The AND keyword hints at the idea that the probabilities need to be multiplied together. Essentially, we would multiply the probability of flipping a spade with itself.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist4a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{16}$$"],"dependencies":["a382995DiscreteDist4a-h5"],"title":"Calculation","text":"What is $$\\\\frac{1}{4} \\\\frac{1}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a382995DiscreteDist5","title":"Playing Card Experiment","body":"Suppose we are conducting the \\"Playing Card Experiment.\\" Essentially, the experimental procedure for empirical data is to pick one card from a deck of shuffled cards. Then, we can record the card. We can repeat this prcoess (about $$10$$ times) and calculate certain probabilites based on the data collected.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Discrete Distribution (Playing Card Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a382995DiscreteDist5a","stepAnswer":["$$\\\\frac{1}{4}$$"],"problemType":"TextBox","stepTitle":"What is the probability of getting $$2$$ red cards in a row?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{4}$$","hints":{"DefaultPathway":[{"id":"a382995DiscreteDist5a-h1","type":"hint","dependencies":[],"title":"Probability","text":"The first step is calculating the probability of flipping a red card in one turn. 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This is the number of cards with the desired outcome.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a382995DiscreteDist5a-h3"],"title":"Divide","text":"What is $$\\\\frac{26}{52}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist5a-h5","type":"hint","dependencies":["a382995DiscreteDist5a-h4"],"title":"$$2$$ In A Row","text":"To calculate the probability that $$2$$ cards in a row are both red, we can multiply the individual probabilities together since we need to flip a red card AND another red card. The AND keyword hints at the idea that the probabilities need to be multiplied together. Essentially, we would multiply the probability of flipping a red with itself.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist5a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4}$$"],"dependencies":["a382995DiscreteDist5a-h5"],"title":"Calculation","text":"What is $$\\\\frac{1}{2} \\\\frac{1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a382995DiscreteDist6","title":"Playing Card Experiment","body":"Suppose we are conducting the \\"Playing Card Experiment.\\" Essentially, the experimental procedure for empirical data is to pick one card from a deck of shuffled cards. Then, we can record the card. We can repeat this prcoess (about $$10$$ times) and calculate certain probabilites based on the data collected.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Discrete Distribution (Playing Card Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a382995DiscreteDist6a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"In a single turn, what is the probability of picking a red or a black card?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a382995DiscreteDist6a-h1","type":"hint","dependencies":[],"title":"Probability","text":"To calculate the probability, we divide the number of cards with the desired outcome by the total number of possible outcomes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$52$$"],"dependencies":["a382995DiscreteDist6a-h1"],"title":"Possible Outcomes","text":"How many total cards are there in the deck? This represents the number of possible outcomes for a single turn.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$52$$"],"dependencies":["a382995DiscreteDist6a-h2"],"title":"Desired Outcome","text":"How many cards are either black or red? This is the number of cards with the desired outcome.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a382995DiscreteDist6a-h3"],"title":"Divide","text":"What is $$\\\\frac{52}{52}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a382995DiscreteDist7","title":"Playing Card Experiment","body":"Suppose we are conducting the \\"Playing Card Experiment.\\" Essentially, the experimental procedure for empirical data is to pick one card from a deck of shuffled cards. Then, we can record the card. We can repeat this prcoess (about $$10$$ times) and calculate certain probabilites based on the data collected.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Discrete Distribution (Playing Card Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a382995DiscreteDist7a","stepAnswer":["$$\\\\frac{1}{2}$$"],"problemType":"TextBox","stepTitle":"In a single turn, what is the probability of picking a diamond or a spades?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{2}$$","hints":{"DefaultPathway":[{"id":"a382995DiscreteDist7a-h1","type":"hint","dependencies":[],"title":"Probability","text":"To calculate the probability, we divide the number of cards with the desired outcome by the total number of possible outcomes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$52$$"],"dependencies":["a382995DiscreteDist7a-h1"],"title":"Possible Outcomes","text":"How many total cards are there in the deck? This represents the number of possible outcomes for a single turn.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$26$$"],"dependencies":["a382995DiscreteDist7a-h2"],"title":"Desired Outcome","text":"How many cards are either a diamond or a spade? This is the number of cards with the desired outcome.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a382995DiscreteDist7a-h3"],"title":"Divide","text":"What is $$\\\\frac{26}{52}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a382995DiscreteDist8","title":"Playing Card Experiment","body":"Suppose we are conducting the \\"Playing Card Experiment.\\" Essentially, the experimental procedure for empirical data is to pick one card from a deck of shuffled cards. Then, we can record the card. We can repeat this prcoess (about $$10$$ times) and calculate certain probabilites based on the data collected.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Discrete Distribution (Playing Card Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a382995DiscreteDist8a","stepAnswer":["$$\\\\frac{1}{2}$$"],"problemType":"TextBox","stepTitle":"In a single turn, what is the probability that the card is black?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{2}$$","hints":{"DefaultPathway":[{"id":"a382995DiscreteDist8a-h1","type":"hint","dependencies":[],"title":"Probability","text":"To calculate the probability, we divide the number of cards with the desired outcome by the total number of possible outcomes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$52$$"],"dependencies":["a382995DiscreteDist8a-h1"],"title":"Possible Outcomes","text":"How many total cards are there in the deck? This represents the number of possible outcomes for a single turn.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$26$$"],"dependencies":["a382995DiscreteDist8a-h2"],"title":"Desired Outcome","text":"How many cards are black? This is the number of cards with the desired outcome.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a382995DiscreteDist8a-h3"],"title":"Divide","text":"What is $$\\\\frac{26}{52}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a382995DiscreteDist9","title":"Playing Card Experiment","body":"Suppose we are conducting the \\"Playing Card Experiment.\\" Essentially, the experimental procedure for empirical data is to pick one card from a deck of shuffled cards. Then, we can record the card. We can repeat this prcoess (about $$10$$ times) and calculate certain probabilites based on the data collected.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Discrete Distribution (Playing Card Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a382995DiscreteDist9a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"In ten turns, what should be the ideal number of times we flip a king? (round down to the nearest whole number)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a382995DiscreteDist9a-h1","type":"hint","dependencies":[],"title":"Probability","text":"The first step is calculating the probability of flipping a king in one turn. To calculate the probability, we divide the number of cards with the desired outcome by the total number of possible outcomes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$52$$"],"dependencies":["a382995DiscreteDist9a-h1"],"title":"Possible Outcomes","text":"How many total cards are there in the deck?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a382995DiscreteDist9a-h2"],"title":"Desired Outcome","text":"How many cards are a king? This is the number of cards with the desired outcome.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{13}$$"],"dependencies":["a382995DiscreteDist9a-h3"],"title":"Divide","text":"What is $$\\\\frac{4}{52}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist9a-h5","type":"hint","dependencies":["a382995DiscreteDist9a-h4"],"title":"Ten Turns","text":"To find the number of times we should flip a king in ten turns, we can multiply the probability of flipping a king in one turn by the number of turns.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist9a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{10}{13}$$"],"dependencies":["a382995DiscreteDist9a-h5"],"title":"Calculate","text":"What is $$10\\\\frac{1}{13}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a382995DiscreteDist9a-h7","type":"hint","dependencies":["a382995DiscreteDist9a-h6"],"title":"Rounding","text":"Round $$\\\\frac{10}{13}$$ down to the nearest whole number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3837e8graphline1","title":"Interpreting Graphs of Linear Equations","body":"Based on the attached graph, for each ordered pair, determine whether it is a solution to the equation and if it is on the line.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Graph Linear Equations in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3837e8graphline1a","stepAnswer":["It is a solution and is on the line"],"problemType":"MultipleChoice","stepTitle":"$$(0,-3)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["It is a solution and is on the line","It is a solution and is not on the line","It is not a solution and is on the line","It is not a solution and is not on the line"],"hints":{"DefaultPathway":[{"id":"a3837e8graphline1a-h1","type":"hint","dependencies":[],"title":"How Points and Solutions Are Related","text":"Every point on the line is a solution of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a3837e8graphline1b","stepAnswer":["It is a solution and is on the line"],"problemType":"MultipleChoice","stepTitle":"$$(3,3)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["It is a solution and is on the line","It is a solution and is not on the line","It is not a solution and is on the line","It is not a solution and is not on the line"],"hints":{"DefaultPathway":[{"id":"a3837e8graphline1b-h1","type":"hint","dependencies":[],"title":"How Points and Solutions Are Related","text":"Every point on the line is a solution of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a3837e8graphline1c","stepAnswer":["It is not a solution and is not on the line"],"problemType":"MultipleChoice","stepTitle":"$$(2,-3)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["It is a solution and is on the line","It is a solution and is not on the line","It is not a solution and is on the line","It is not a solution and is not on the line"],"hints":{"DefaultPathway":[{"id":"a3837e8graphline1c-h1","type":"hint","dependencies":[],"title":"How Points and Solutions Are Related","text":"Every point on the line is a solution of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a3837e8graphline1d","stepAnswer":["It is a solution and is on the line"],"problemType":"MultipleChoice","stepTitle":"$$(-1,-5)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["It is a solution and is on the line","It is a solution and is not on the line","It is not a solution and is on the line","It is not a solution and is not on the line"],"hints":{"DefaultPathway":[{"id":"a3837e8graphline1d-h1","type":"hint","dependencies":[],"title":"How Points and Solutions Are Related","text":"Every point on the line is a solution of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3837e8graphline10","title":"Graphing Linear Equations by Plotting Points","body":"Graph the equation. From your graph, identify the slope and y-intercept. Input the slope and y-intercept through the format \\"slope,y-intercept\\" without the quotes. Example: $$3,(0,1)$$","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Graph Linear Equations in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3837e8graphline10a","stepAnswer":["1/2,(0,3)"],"problemType":"TextBox","stepTitle":"$$y=\\\\frac{1}{2} x+3$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$1/2,(0,3)$$","hints":{"DefaultPathway":[{"id":"a3837e8graphline10a-h1","type":"hint","dependencies":[],"title":"Finding Points to Graph","text":"Find three points whose coordinates are solutions to the equation. Organize them in a table.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3837e8graphline10a-h2","type":"hint","dependencies":["a3837e8graphline10a-h1"],"title":"Plotting Points","text":"Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3837e8graphline10a-h3","type":"hint","dependencies":["a3837e8graphline10a-h2"],"title":"Graphing the Line","text":"Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3837e8graphline11","title":"Graphing Linear Equations by Plotting Points","body":"Graph the equation. From your graph, identify the slope and y-intercept. Input the slope and y-intercept through the format \\"slope,y-intercept\\" without the quotes. Example: $$3,(0,1)$$","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Graph Linear Equations in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3837e8graphline11a","stepAnswer":["1/3,(0,-1)"],"problemType":"TextBox","stepTitle":"$$y=\\\\frac{1}{3} x-1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$1/3,(0,-1)$$","hints":{"DefaultPathway":[{"id":"a3837e8graphline11a-h1","type":"hint","dependencies":[],"title":"Finding Points to Graph","text":"Find three points whose coordinates are solutions to the equation. Organize them in a table.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3837e8graphline11a-h2","type":"hint","dependencies":["a3837e8graphline11a-h1"],"title":"Plotting Points","text":"Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3837e8graphline11a-h3","type":"hint","dependencies":["a3837e8graphline11a-h2"],"title":"Graphing the Line","text":"Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3837e8graphline12","title":"Graphing Linear Equations by Plotting Points","body":"Graph the equation. From your graph, identify the slope and y-intercept. Input the slope and y-intercept through the format \\"slope,y-intercept\\" without the quotes. Example: $$3,(0,1)$$","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Graph Linear Equations in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3837e8graphline12a","stepAnswer":["1/4,2"],"problemType":"TextBox","stepTitle":"$$y=\\\\frac{1}{4} x+2$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a3837e8graphline12a-h1","type":"hint","dependencies":[],"title":"Finding Points to Graph","text":"Find three points whose coordinates are solutions to the equation. Organize them in a table.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3837e8graphline12a-h2","type":"hint","dependencies":["a3837e8graphline12a-h1"],"title":"Plotting Points","text":"Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3837e8graphline12a-h3","type":"hint","dependencies":["a3837e8graphline12a-h2"],"title":"Graphing the Line","text":"Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3837e8graphline13","title":"Finding $$x$$ and $$y$$ intercepts","body":"Find the $$x$$ and $$y$$ intercepts of the line. Enter your answer in the form \\"x-intercept,y-intercept\\" without the quotes. Example: $$(1,0),(0,1)$$","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Graph Linear Equations in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3837e8graphline13a","stepAnswer":["(4,0),(0,8)"],"problemType":"TextBox","stepTitle":"$$2x+y=8$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(4,0),(0,8)$$","hints":{"DefaultPathway":[{"id":"a3837e8graphline13a-h1","type":"hint","dependencies":[],"title":"Finding x-intercept","text":"Let $$y=0$$ and find the value of $$x$$ to find the x-intercept.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3837e8graphline13a-h2","type":"hint","dependencies":["a3837e8graphline13a-h1"],"title":"Finding y-intercept","text":"Let $$x=0$$ and find the value of $$x$$ to find the x-intercept.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3837e8graphline14","title":"Finding $$x$$ and $$y$$ intercepts","body":"Find the $$x$$ and $$y$$ intercepts of the line. Enter your answer in the form \\"x-intercept,y-intercept\\" without the quotes. Example: $$(1,0),(0,1)$$","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Graph Linear Equations in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3837e8graphline14a","stepAnswer":["(4,0),(0,12)"],"problemType":"TextBox","stepTitle":"$$3x+y=12$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(4,0),(0,12)$$","hints":{"DefaultPathway":[{"id":"a3837e8graphline14a-h1","type":"hint","dependencies":[],"title":"Finding x-intercept","text":"Let $$y=0$$ and find the value of $$x$$ to find the x-intercept.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3837e8graphline14a-h2","type":"hint","dependencies":["a3837e8graphline14a-h1"],"title":"Finding y-intercept","text":"Let $$x=0$$ and find the value of $$x$$ to find the x-intercept.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3837e8graphline15","title":"Finding $$x$$ and $$y$$ intercepts","body":"Find the $$x$$ and $$y$$ intercepts of the line. Enter your answer in the form \\"x-intercept,y-intercept\\" without the quotes. Example: $$(1,0),(0,1)$$","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Graph Linear Equations in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3837e8graphline15a","stepAnswer":["(8,0),(0,2)"],"problemType":"TextBox","stepTitle":"$$x+4y=8$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(8,0),(0,2)$$","hints":{"DefaultPathway":[{"id":"a3837e8graphline15a-h1","type":"hint","dependencies":[],"title":"Finding x-intercept","text":"Let $$y=0$$ and find the value of $$x$$ to find the x-intercept.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3837e8graphline15a-h2","type":"hint","dependencies":["a3837e8graphline15a-h1"],"title":"Finding y-intercept","text":"Let $$x=0$$ and find the value of $$x$$ to find the x-intercept.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3837e8graphline5","title":"Interpreting Graphs of Linear Equations","body":"Based on the attached graph, for each ordered pair, determine whether it is a solution to the equation and if it is on the line.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Graph Linear Equations in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3837e8graphline5a","stepAnswer":["It is a solution and is on the line"],"problemType":"MultipleChoice","stepTitle":"$$(0,-1)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["It is a solution and is on the line","It is a solution and is not on the line","It is not a solution and is on the line","It is not a solution and is not on the line"],"hints":{"DefaultPathway":[{"id":"a3837e8graphline5a-h1","type":"hint","dependencies":[],"title":"How Points and Solutions Are Related","text":"Every point on the line is a solution of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a3837e8graphline5b","stepAnswer":["It is a solution and is on the line"],"problemType":"MultipleChoice","stepTitle":"$$(2,5)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["It is a solution and is on the line","It is a solution and is not on the line","It is not a solution and is on the line","It is not a solution and is not on the line"],"hints":{"DefaultPathway":[{"id":"a3837e8graphline5b-h1","type":"hint","dependencies":[],"title":"How Points and Solutions Are Related","text":"Every point on the line is a solution of the 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the line","It is not a solution and is not on the line"],"hints":{"DefaultPathway":[{"id":"a3837e8graphline6a-h1","type":"hint","dependencies":[],"title":"How Points and Solutions Are Related","text":"Every point on the line is a solution of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a3837e8graphline6b","stepAnswer":["It is not a solution and is not on the line"],"problemType":"MultipleChoice","stepTitle":"$$(-1,-4)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["It is a solution and is on the line","It is a solution and is not on the line","It is not a solution and is on the line","It is not a solution and is not on the line"],"hints":{"DefaultPathway":[{"id":"a3837e8graphline6b-h1","type":"hint","dependencies":[],"title":"How Points and Solutions Are Related","text":"Every point on the line is a solution of the 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From your graph, identify the slope and y-intercept. Input the slope and y-intercept through the format \\"slope,y-intercept\\" without the quotes. Example: $$3,(0,1)$$","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Graph Linear Equations in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3837e8graphline7a","stepAnswer":["2,(0,1)"],"problemType":"TextBox","stepTitle":"$$y=2x+1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2,(0,1)$$","hints":{"DefaultPathway":[{"id":"a3837e8graphline7a-h1","type":"hint","dependencies":[],"title":"Finding Points to Graph","text":"Find three points whose coordinates are solutions to the equation. Organize them in a table.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3837e8graphline7a-h2","type":"hint","dependencies":["a3837e8graphline7a-h1"],"title":"Plotting Points","text":"Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3837e8graphline7a-h3","type":"hint","dependencies":["a3837e8graphline7a-h2"],"title":"Graphing the Line","text":"Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3837e8graphline8","title":"Graphing Linear Equations by Plotting Points","body":"Graph the equation. From your graph, identify the slope and y-intercept. Input the slope and y-intercept through the format \\"slope,y-intercept\\" without the quotes. Example: $$3,(0,1)$$","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Graph Linear Equations in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3837e8graphline8a","stepAnswer":["2,(0,-3)"],"problemType":"TextBox","stepTitle":"$$y=2x-3$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2,(0,-3)$$","hints":{"DefaultPathway":[{"id":"a3837e8graphline8a-h1","type":"hint","dependencies":[],"title":"Finding Points to Graph","text":"Find three points whose coordinates are solutions to the equation. Organize them in a table.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3837e8graphline8a-h2","type":"hint","dependencies":["a3837e8graphline8a-h1"],"title":"Plotting Points","text":"Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3837e8graphline8a-h3","type":"hint","dependencies":["a3837e8graphline8a-h2"],"title":"Graphing the Line","text":"Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3837e8graphline9","title":"Graphing Linear Equations by Plotting Points","body":"Graph the equation. From your graph, identify the slope and y-intercept. Input the slope and y-intercept through the format \\"slope,y-intercept\\" without the quotes. Example: $$3,(0,1)$$","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Graph Linear Equations in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3837e8graphline9a","stepAnswer":["-2,(0,4)"],"problemType":"TextBox","stepTitle":"$$y=-2x+4$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-2,(0,4)$$","hints":{"DefaultPathway":[{"id":"a3837e8graphline9a-h1","type":"hint","dependencies":[],"title":"Finding Points to Graph","text":"Find three points whose coordinates are solutions to the equation. Organize them in a table.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3837e8graphline9a-h2","type":"hint","dependencies":["a3837e8graphline9a-h1"],"title":"Plotting Points","text":"Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3837e8graphline9a-h3","type":"hint","dependencies":["a3837e8graphline9a-h2"],"title":"Graphing the Line","text":"Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3837e8points1","title":"Locating a Point Quadrant","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Graph Linear Equations in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3837e8points1a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"What quadrant is the point $$(-4,2)$$ in? Enter only the numeric value.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a3837e8points1a-h1","type":"hint","dependencies":[],"title":"Overview of Quadrant System","text":"Quadrant I consists points where $$x$$ and $$y$$ are greater than $$0$$. Quadrant II consists points where $$x>0$$ and $$y<0$$. Quadrant III consists points where $$x$$ and $$y$$ are less than $$0$$. Quadrant IV consists points where $$x>0$$ and $$y<0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3837e8points1a-h2","type":"hint","dependencies":["a3837e8points1a-h1"],"title":"Determining Quadrant of $$(-4,2)$$","text":"$$x=-4<0$$ and $$y=2>0$$. Following the details given in the previous step, the point is in quadrant II.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3837e8points10","title":"Determining Whether a Point is On a Line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Graph Linear Equations in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3837e8points10a","stepAnswer":["The point is not on the line."],"problemType":"MultipleChoice","stepTitle":"Determine whether the point $$(1,2)$$ lies on the line $$y=x+2$$. Enter \\"1\\" if it does, and \\"0\\" if it does not.","stepBody":"","answerType":"string","variabilization":{},"choices":["The point is on the line.","The point is not on the line."],"hints":{"DefaultPathway":[{"id":"a3837e8points10a-h1","type":"hint","dependencies":[],"title":"Plugging in X and Y","text":"We must plug in $$x=1$$ and $$y=2$$. Doing this, we get $$2=1+2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3837e8points10a-h2","type":"hint","dependencies":["a3837e8points10a-h1"],"title":"Determining Equality","text":"$$2=1+2$$ can be simplified to $$2=3$$, which is false statement. Thus the point does not lie on the line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3837e8points11","title":"Determining Whether a Point is On a Line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Graph Linear Equations in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3837e8points11a","stepAnswer":["The point is on the line."],"problemType":"MultipleChoice","stepTitle":"Determine whether the point $$(-1,1)$$ lies on the line $$y=x+2$$. Enter \\"1\\" if it does, and \\"0\\" if it does not.","stepBody":"","answerType":"string","variabilization":{},"choices":["The point is on the line.","The point is not on the line."],"hints":{"DefaultPathway":[{"id":"a3837e8points11a-h1","type":"hint","dependencies":[],"title":"Plugging in X and Y","text":"We must plug in $$x=-1$$ and $$y=1$$. Doing this, we get $$1=-1+2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3837e8points11a-h2","type":"hint","dependencies":["a3837e8points11a-h1"],"title":"Determining Equality","text":"$$1=-1+2$$ is simplified to $$1=1$$, which is a false statement. Thus the point does lie on the line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3837e8points12","title":"Determining Whether a Point is On a Line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Graph Linear Equations in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3837e8points12a","stepAnswer":["The point is on the line."],"problemType":"MultipleChoice","stepTitle":"Determine whether the point $$(-3,-1)$$ lies on the line $$y=x+2$$. Enter \\"1\\" if it does, and \\"0\\" if it does not.","stepBody":"","answerType":"string","variabilization":{},"choices":["The point is on the line.","The point is not on the line."],"hints":{"DefaultPathway":[{"id":"a3837e8points12a-h1","type":"hint","dependencies":[],"title":"Plugging in X and Y","text":"We must plug in $$x=-3$$ and $$y=-1$$. Doing this, we get $$-1=-3+2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3837e8points12a-h2","type":"hint","dependencies":["a3837e8points12a-h1"],"title":"Determining Equality","text":"$$-1=-3+2$$ is simplified to $$-1=-1$$, which is a true statement. Thus, the point lies on the line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3837e8points13","title":"Determining Whether a Point is On a Line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Graph Linear Equations in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3837e8points13a","stepAnswer":["The point is on the line."],"problemType":"MultipleChoice","stepTitle":"Determine whether the point $$(0,-4)$$ lies on the line $$y=x-4$$. Enter \\"1\\" if it does, and \\"0\\" if it does not.","stepBody":"","answerType":"string","variabilization":{},"choices":["The point is on the line.","The point is not on the line."],"hints":{"DefaultPathway":[{"id":"a3837e8points13a-h1","type":"hint","dependencies":[],"title":"Plugging in X and Y","text":"We must plug in $$x=0$$ and $$y=-4$$. Doing this, we get $$-4=0-4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3837e8points13a-h2","type":"hint","dependencies":["a3837e8points13a-h1"],"title":"Determining Equality","text":"$$-4=0-4$$ can be simplified to $$-4=-4$$, which is a true statement. Thus, the point lies on the line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3837e8points14","title":"Determining Whether a Point is On a Line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Graph Linear Equations in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3837e8points14a","stepAnswer":["The point is on the line."],"problemType":"MultipleChoice","stepTitle":"Determine whether the point $$(3,-1)$$ lies on the line $$y=x-4$$. Enter \\"1\\" if it does, and \\"0\\" if it does not.","stepBody":"","answerType":"string","variabilization":{},"choices":["The point is on the line.","The point is not on the line."],"hints":{"DefaultPathway":[{"id":"a3837e8points14a-h1","type":"hint","dependencies":[],"title":"Plugging in X and Y","text":"We must plug in $$x=3$$ and $$y=-1$$. Doing this, we get $$-1=3-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3837e8points14a-h2","type":"hint","dependencies":["a3837e8points14a-h1"],"title":"Determining Equality","text":"$$-1=3-4$$ can be simplified to $$-1=-1$$, which is a true statement. Thus, the point lies on the line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3837e8points15","title":"Determining Whether a Point is On a Line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Graph Linear Equations in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3837e8points15a","stepAnswer":["The point is not on the line."],"problemType":"MultipleChoice","stepTitle":"Determine whether the point $$(2,2)$$ lies on the line $$y=x-4$$. Enter \\"1\\" if it does, and \\"0\\" if it does not.","stepBody":"","answerType":"string","variabilization":{},"choices":["The point is on the line.","The point is not on the line."],"hints":{"DefaultPathway":[{"id":"a3837e8points15a-h1","type":"hint","dependencies":[],"title":"Plugging in X and Y","text":"We must plug in $$x=2$$ and $$y=2$$. Doing this, we get $$2=2-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3837e8points15a-h2","type":"hint","dependencies":["a3837e8points15a-h1"],"title":"Determining Equality","text":"$$2=2-4$$ simplifies to $$2=-2$$, which is a false statement. Thus, the point does not lie on the line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3837e8points2","title":"Locating a Point Quadrant","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Graph Linear Equations in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3837e8points2a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"What quadrant is the point $$(-1,-2)$$ in? Enter only the numeric value.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a3837e8points2a-h1","type":"hint","dependencies":[],"title":"Overview of Quadrant System","text":"Quadrant I consists points where $$x$$ and $$y$$ are greater than $$0$$. Quadrant II consists points where $$x>0$$ and $$y<0$$. Quadrant III consists points where $$x$$ and $$y$$ are less than $$0$$. Quadrant IV consists points where $$x>0$$ and $$y<0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3837e8points2a-h2","type":"hint","dependencies":["a3837e8points2a-h1"],"title":"Determining Quadrant of $$(-4,2)$$","text":"$$x=-1<0$$ and $$y=-2<0$$. Following the details given in the previous step, the point is in quadrant III.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3837e8points3","title":"Locating a Point Quadrant","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Graph Linear Equations in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3837e8points3a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"What quadrant is the point $$(3,-5)$$ in? Enter only the numeric value.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a3837e8points3a-h1","type":"hint","dependencies":[],"title":"Overview of Quadrant System","text":"Quadrant I consists points where $$x$$ and $$y$$ are greater than $$0$$. Quadrant II consists points where $$x>0$$ and $$y<0$$. Quadrant III consists points where $$x$$ and $$y$$ are less than $$0$$. Quadrant IV consists points where $$x>0$$ and $$y<0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3837e8points3a-h2","type":"hint","dependencies":["a3837e8points3a-h1"],"title":"Determining Quadrant of $$(-4,2)$$","text":"$$x=3>0$$ and $$y=-5<0$$. Following the details given in the previous step, the point is in quadrant IV.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3837e8points4","title":"Locating a Point Quadrant","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Graph Linear Equations in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3837e8points4a","stepAnswer":["NA"],"problemType":"TextBox","stepTitle":"What quadrant is the point $$(-3,0)$$ in? Enter only the numeric value. If a point is not in a quadrant, enter \\"NA\\".","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a3837e8points4a-h1","type":"hint","dependencies":[],"title":"Overview of Quadrant System","text":"Quadrant I consists points where $$x$$ and $$y$$ are greater than $$0$$. Quadrant II consists points where $$x>0$$ and $$y<0$$. Quadrant III consists points where $$x$$ and $$y$$ are less than $$0$$. Quadrant IV consists points where $$x>0$$ and $$y<0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3837e8points4a-h2","type":"hint","dependencies":["a3837e8points4a-h1"],"title":"Determining Quadrant of $$(-4,2)$$","text":"Since $$y=0$$, we know that the point lies on an axis, not in any quadrant.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3837e8points5","title":"Locating a Point Quadrant","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Graph Linear Equations in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3837e8points5a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"What quadrant is the point $$(-2,-3)$$ in? Enter only the numeric value. If a point is not in a quadrant, enter \\"NA\\".","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a3837e8points5a-h1","type":"hint","dependencies":[],"title":"Overview of Quadrant System","text":"Quadrant I consists points where $$x$$ and $$y$$ are greater than $$0$$. Quadrant II consists points where $$x>0$$ and $$y<0$$. Quadrant III consists points where $$x$$ and $$y$$ are less than $$0$$. Quadrant IV consists points where $$x>0$$ and $$y<0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3837e8points5a-h2","type":"hint","dependencies":["a3837e8points5a-h1"],"title":"Determining Quadrant of $$(-4,2)$$","text":"$$x=-2<0$$ and $$y=-3<0$$. Following the details given in the previous step, the point is in quadrant III.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3837e8points6","title":"Locating a Point Quadrant","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Graph Linear Equations in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3837e8points6a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"What quadrant is the point $$(3,-3)$$ in? Enter only the numeric value. If a point is not in a quadrant, enter \\"NA\\".","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a3837e8points6a-h1","type":"hint","dependencies":[],"title":"Overview of Quadrant System","text":"Quadrant I consists points where $$x$$ and $$y$$ are greater than $$0$$. Quadrant II consists points where $$x>0$$ and $$y<0$$. Quadrant III consists points where $$x$$ and $$y$$ are less than $$0$$. Quadrant IV consists points where $$x>0$$ and $$y<0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3837e8points6a-h2","type":"hint","dependencies":["a3837e8points6a-h1"],"title":"Determining Quadrant of $$(-4,2)$$","text":"$$x=3>0$$ and $$y=-3<0$$. Following the details given in the previous step, the point is in quadrant IV.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3837e8points7","title":"Locating a Point Quadrant","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Graph Linear Equations in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3837e8points7a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"What quadrant is the point $$(-4,1)$$ in? Enter only the numeric value. If a point is not in a quadrant, enter \\"NA\\".","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a3837e8points7a-h1","type":"hint","dependencies":[],"title":"Overview of Quadrant System","text":"Quadrant I consists points where $$x$$ and $$y$$ are greater than $$0$$. Quadrant II consists points where $$x>0$$ and $$y<0$$. Quadrant III consists points where $$x$$ and $$y$$ are less than $$0$$. Quadrant IV consists points where $$x>0$$ and $$y<0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3837e8points7a-h2","type":"hint","dependencies":["a3837e8points7a-h1"],"title":"Determining Quadrant of $$(-4,2)$$","text":"$$x=-4<0$$ and $$y=1>0$$. Following the details given in the previous step, the point is in quadrant II.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3837e8points8","title":"Locating a Point Quadrant","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Graph Linear Equations in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3837e8points8a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"What quadrant is the point $$(4,-2)$$ in? Enter only the numeric value. If a point is not in a quadrant, enter \\"NA\\".","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a3837e8points8a-h1","type":"hint","dependencies":[],"title":"Overview of Quadrant System","text":"Quadrant I consists points where $$x$$ and $$y$$ are greater than $$0$$. Quadrant II consists points where $$x>0$$ and $$y<0$$. Quadrant III consists points where $$x$$ and $$y$$ are less than $$0$$. Quadrant IV consists points where $$x>0$$ and $$y<0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3837e8points8a-h2","type":"hint","dependencies":["a3837e8points8a-h1"],"title":"Determining Quadrant of $$(-4,2)$$","text":"$$x=4>0$$ and $$y=-2<0$$. Following the details given in the previous step, the point is in quadrant IV.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3837e8points9","title":"Determining Whether a Point is On a Line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Graph Linear Equations in Two Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3837e8points9a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"Determine whether the point $$(0,2)$$ lies on the line $$y=x+2$$. Enter \\"1\\" if it does, and \\"0\\" if it does not.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a3837e8points9a-h1","type":"hint","dependencies":[],"title":"Plugging in X and Y","text":"We must plug in $$x=0$$ and $$y=2$$. Doing this, we get $$2=0+2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3837e8points9a-h2","type":"hint","dependencies":["a3837e8points9a-h1"],"title":"Determining Equality","text":"$$2=0+2$$ can be simplified to $$2=2$$, which is a true statement. Thus, the point lies on the line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a38bd54basicrules1","title":"The Addition Rule","body":"Klaus is trying to choose where to go on vacation. His two choices are: A $$=$$ New Zealand and B $$=$$ Alaska. Klaus can only afford one vacation. The probability that he chooses A is $$P(A)=0.6$$ and the probability that he chooses B is $$P(B)=0.35$$. P(A AND $$B)=0$$ since Klaus can only afford to take one vacation.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Two Basic Rules of Probability","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a38bd54basicrules1a","stepAnswer":["$$0.05$$"],"problemType":"TextBox","stepTitle":"What is the probability that Klaus chooses to not go anywhere on vacation?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.05$$","hints":{"DefaultPathway":[{"id":"a38bd54basicrules1a-h1","type":"hint","dependencies":[],"title":"The Addition Rule","text":"We note that P(A AND B) $$=$$ $$0$$. By definition, this means that A and B are mutually exclusive. For two mutually exclusive events, we note that P(A OR $$B)=P\\\\left(A\\\\right)+P\\\\left(B\\\\right)$$. We want to find the probability that Kluas doesn\'t choose A or B. Therefore, we want to find 1-P(A OR $$B)=1-P\\\\left(A\\\\right)+P\\\\left(B\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.05$$"],"dependencies":["a38bd54basicrules1a-h1"],"title":"The Addition Rule Applied","text":"What is the probability that Klaus chooses neither New Zealand nor Alaska?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.05$$"],"dependencies":["a38bd54basicrules1a-h2"],"title":"The Addition Rule Applied","text":"What is $$1-0.6+0.35$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a38bd54basicrules10","title":"Probability Revisited","body":"Studies show that about one woman in seven (approximately $$14.3\\\\%)$$ who live to be $$90$$ will develop breast cancer. Suppose that of those women who develop breast cancer, a test is negative 2% of the time. Also suppose that in the general population of women, the test for breast cancer is negative about 85% of the time. Let B $$=$$ woman develops breast cancer and let N $$=$$ tests negative while P $$=$$ tests positive. Suppose one woman is selected at random.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Two Basic Rules of Probability","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a38bd54basicrules10a","stepAnswer":["$$0.98$$"],"problemType":"TextBox","stepTitle":"Given that a woman develops breast cancer, what is the probability that she tests positive?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.98$$","hints":{"DefaultPathway":[{"id":"a38bd54basicrules10a-h1","type":"hint","dependencies":[],"title":"Complement Rule","text":"We note that we want to find the conditional probability $$P(P|B)=P(not$$ N|B). We can use the complement rule here and note that P(not $$N|B)=1-P(N|B)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.98$$"],"dependencies":["a38bd54basicrules10a-h1"],"title":"Using the Complement Rule","text":"What is $$1-P(N|B)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a38bd54basicrules10b","stepAnswer":["$$0.14014$$"],"problemType":"TextBox","stepTitle":"What is the probability that a woman develops breast cancer and tests positive.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.14014$$","hints":{"DefaultPathway":[{"id":"a38bd54basicrules10b-h2","type":"hint","dependencies":["a38bd54basicrules10a-h1"],"title":"The Multiplication Rule","text":"If B and P are two events defined on a sample space, then P(B AND $$P)=P\\\\left(B\\\\right) P\\\\left(P|B\\\\right)$$. We want to find the probability that B and P both occur: this is the formula to use.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules10b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.143$$"],"dependencies":["a38bd54basicrules10b-h2"],"title":"Applying the Multiplication Rule","text":"What is P(B)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules10b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.98$$"],"dependencies":["a38bd54basicrules10b-h3"],"title":"Applying the Multiplication Rule","text":"What is $$P(P|B)=P(not$$ P|B), which we found above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules10b-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.14014$$"],"dependencies":["a38bd54basicrules10b-h4"],"title":"Applying the Multiplication Rule","text":"What is P(B AND $$P)=P\\\\left(B\\\\right) P\\\\left(P|B\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a38bd54basicrules11","title":"Revisiting the Complement Rule","body":"Studies show that about one woman in seven (approximately $$14.3\\\\%)$$ who live to be $$90$$ will develop breast cancer. Suppose that of those women who develop breast cancer, a test is negative 2% of the time. Also suppose that in the general population of women, the test for breast cancer is negative about 85% of the time. Let B $$=$$ woman develops breast cancer and let N $$=$$ tests negative while P $$=$$ tests positive. Suppose one woman is selected at random.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Two Basic Rules of Probability","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a38bd54basicrules11a","stepAnswer":["$$0.857$$"],"problemType":"TextBox","stepTitle":"What is the probability that a woman does not develop breast cancer?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.857$$","hints":{"DefaultPathway":[{"id":"a38bd54basicrules11a-h1","type":"hint","dependencies":[],"title":"Complement Rule","text":"We note that we want to find the probability that a woman does not develop breast cancer. We can use the complement rule here: the probability that a woman does not develop breast cancer is just $$1-P(B)$$, or one minus the probability that a woman develops breast cancer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.857$$"],"dependencies":["a38bd54basicrules11a-h1"],"title":"Applying the Complement Rule","text":"What is $$1-P(B)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a38bd54basicrules11b","stepAnswer":["$$0.15$$"],"problemType":"TextBox","stepTitle":"What is the probability that a woman tests positive for breast cancer?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.15$$","hints":{"DefaultPathway":[{"id":"a38bd54basicrules11b-h1","type":"hint","dependencies":[],"title":"Complement Rule","text":"We note that we want to find the probability that a woman tests positive for breast cancer. We can use the complement rule here: the probability that a woman from the general population tests positive for breast cancer is just $$1-P(N)$$, or one minus the probability that a woman tests negative cancer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules11b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.15$$"],"dependencies":["a38bd54basicrules11b-h1"],"title":"Applying the Complement Rule","text":"What is $$1-P(N)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a38bd54basicrules12","title":"Applying Rules of Probability","body":"A student goes to the library. Let events B $$=$$ the student checks out a book and D $$=$$ the student checks out a DVD. Suppose that $$P(B)=0.40$$, $$P(D)=0.30$$, and $$P(D|B)=0.50$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Two Basic Rules of Probability","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a38bd54basicrules12a","stepAnswer":["$$0.6$$"],"problemType":"TextBox","stepTitle":"Find P(B\').","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.6$$","hints":{"DefaultPathway":[{"id":"a38bd54basicrules12a-h1","type":"hint","dependencies":[],"title":"Complement Rule","text":"Remember that the notation for finding the complement is the apostrophe (\'). We note that the formula for complements is $$P(B\')=1-P(B)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.6$$"],"dependencies":["a38bd54basicrules12a-h1"],"title":"Applying the Complement Rule","text":"What is P(B\')? What is $$1-P(B)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a38bd54basicrules12b","stepAnswer":["$$0.2$$"],"problemType":"TextBox","stepTitle":"Find P(D AND B).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.2$$","hints":{"DefaultPathway":[{"id":"a38bd54basicrules12b-h1","type":"hint","dependencies":[],"title":"The Multiplication Rule","text":"If B and D are two events defined on a sample space, then P(B AND $$D)=P\\\\left(B\\\\right) P\\\\left(D|B\\\\right)$$. We want to find the probability that B and D both occur: this is the formula to use.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules12b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.4$$"],"dependencies":["a38bd54basicrules12b-h1"],"title":"Applying the Multiplication Rule","text":"What is P(B)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules12b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.5$$"],"dependencies":["a38bd54basicrules12b-h2"],"title":"Applying the Multiplication Rule","text":"What is P(D|B)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules12b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.2$$"],"dependencies":["a38bd54basicrules12b-h3"],"title":"Applying the Multiplication Rule","text":"What is P(B AND $$D)=P\\\\left(B\\\\right) P\\\\left(D|B\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a38bd54basicrules13","title":"Applying Rules of Probability","body":"A student goes to the library. Let events B $$=$$ the student checks out a book and D $$=$$ the student checks out a DVD. Suppose that $$P(B)=0.40$$, $$P(D)=0.30$$, and $$P(D|B)=0.50$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Two Basic Rules of Probability","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a38bd54basicrules13a","stepAnswer":["$$\\\\frac{2}{3}$$"],"problemType":"TextBox","stepTitle":"Find P(B|D).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2}{3}$$","hints":{"DefaultPathway":[{"id":"a38bd54basicrules13a-h1","type":"hint","dependencies":[],"title":"Conditional Probability","text":"We note that $$P(B|D)=P(B$$ AND D)/P(D). However, we also know that $$P(D|B)=P(B$$ AND D)/P(B)=0.5.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.2$$"],"dependencies":["a38bd54basicrules13a-h1"],"title":"Solve for P(B AND D)","text":"Algebraically solve for P(B AND D) using the expression above.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules13a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.2$$"],"dependencies":["a38bd54basicrules13a-h2"],"title":"Solve for P(B AND D)","text":"What is $$0.5\\\\times0.4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{3}$$"],"dependencies":["a38bd54basicrules13a-h3"],"title":"Determining P(B|D)","text":"What is P(B AND D)/P(D)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a38bd54basicrules13b","stepAnswer":["$$1.1$$"],"problemType":"TextBox","stepTitle":"Find P(D AND B\').","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1.1$$","hints":{"DefaultPathway":[{"id":"a38bd54basicrules13b-h1","type":"hint","dependencies":[],"title":"Multiplication Rule","text":"We note that by the multiplication rule, P(D AND $$B\')=P\\\\left(D\\\\right) P\\\\left(B\'|D\\\\right)$$. We also note that $$P\\\\left(B|D\\\\right)+P\\\\left(B\'|D\\\\right)=1$$, so we can use this to our advantage.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules13b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.3$$"],"dependencies":["a38bd54basicrules13b-h1"],"title":"Determining P(D)","text":"What is P(D)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules13b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["a38bd54basicrules13b-h2"],"title":"Determining P(B\'|D)","text":"What is $$P(B\'|D)=1-P(B|D)$$? Hint: we found P(B|D) in the previous step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules13b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{10}$$"],"dependencies":["a38bd54basicrules13b-h3"],"title":"Determining P(D AND B\')","text":"What is P(D AND $$B\')=P\\\\left(D\\\\right) P\\\\left(B\'|D\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a38bd54basicrules14","title":"Applying Probability to Real-World Problems","body":"Forty-eight percent of all Californians registered voters prefer life in prison without parole over the death penalty for a person convicted of first degree murder. Among Latino California registered voters, 55% prefer life in prison without parole over the death penalty for a person convicted of first degree murder. $$37.6\\\\%$$ of all Californians are Latino. In this problem, let C $$=$$ Calofornians (registered voters) preferring life in prison without parole over the death penalty for a person convicted of first degree murder and let L $$=$$ Latino Californians. Suppose that one Californian is randomly selected.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Two Basic Rules of Probability","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a38bd54basicrules14a","stepAnswer":["$$0.48$$"],"problemType":"TextBox","stepTitle":"Find P(C).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.48$$","hints":{"DefaultPathway":[{"id":"a38bd54basicrules14a-h1","type":"hint","dependencies":[],"title":"Converting Percentages to Probabilities","text":"Note that in the problem, we note the percent of all Californians registered voters that prefer life in prison without parole over the death penalty for a person convicted of first degree murder. Also note, this is the definition of C, the event we are seeking the probability for. Probabilities are proportions between $$0$$ and $$1$$ while percentages are between $$0$$ and $$100$$. To get a probability from a percentage, divide by $$100$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.48$$"],"dependencies":["a38bd54basicrules14a-h1"],"title":"Determining the Probability P(C)","text":"What is P(C), or the percentage of all Californians registered voters that prefer life in prison without parole over the death penalty for a person convicted of first degree murder divided by 100?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a38bd54basicrules15","title":"Applying Probability to Real-World Problems","body":"Forty-eight percent of all Californians registered voters prefer life in prison without parole over the death penalty for a person convicted of first degree murder. Among Latino California registered voters, 55% prefer life in prison without parole over the death penalty for a person convicted of first degree murder. $$37.6\\\\%$$ of all Californians are Latino. In this problem, let C $$=$$ Calofornians (registered voters) preferring life in prison without parole over the death penalty for a person convicted of first degree murder and let L $$=$$ Latino Californians. Suppose that one Californian is randomly selected.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Two Basic Rules of Probability","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a38bd54basicrules15a","stepAnswer":["$$0.376$$"],"problemType":"TextBox","stepTitle":"Find P(L).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.376$$","hints":{"DefaultPathway":[{"id":"a38bd54basicrules15a-h1","type":"hint","dependencies":[],"title":"Converting Percentages to Probabilities","text":"Note that in the problem, we note the percent of all Californians that are Latino. Also note, this is the definition of L, the event we are seeking the probability for. Probabilities are proportions between $$0$$ and $$1$$ while percentages are between $$0$$ and $$100$$. To get a probability from a percentage, divide by $$100$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.376$$"],"dependencies":["a38bd54basicrules15a-h1"],"title":"Determining the Probability P(L)","text":"What is P(L), or the percentage of all Californians that are Latino, divided by 100?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a38bd54basicrules16","title":"Applying Probability to Real-World Problems","body":"Forty-eight percent of all Californians registered voters prefer life in prison without parole over the death penalty for a person convicted of first degree murder. Among Latino California registered voters, 55% prefer life in prison without parole over the death penalty for a person convicted of first degree murder. $$37.6\\\\%$$ of all Californians are Latino. In this problem, let C $$=$$ Calofornians (registered voters) preferring life in prison without parole over the death penalty for a person convicted of first degree murder and let L $$=$$ Latino Californians. Suppose that one Californian is randomly selected.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Two Basic Rules of Probability","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a38bd54basicrules16a","stepAnswer":["$$0.55$$"],"problemType":"TextBox","stepTitle":"Find P(C|L).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.55$$","hints":{"DefaultPathway":[{"id":"a38bd54basicrules16a-h1","type":"hint","dependencies":[],"title":"Converting Percentages to Probabilities","text":"Note that in the problem, we note that given a voter is Latino California, the percent that prefer life in prison without parole over the death penalty for a person convicted of first degree murder. Also note, this is the definition of C|L, the event we are seeking the probability for. Probabilities are proportions between $$0$$ and $$1$$ while percentages are between $$0$$ and $$100$$. To get a probability from a percentage, divide by $$100$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.55$$"],"dependencies":["a38bd54basicrules16a-h1"],"title":"Determining the Probability P(C|L)","text":"What is P(C|L), or among Latino Californians, the percentage that prefer life in prison without parole over the death penalty for a person convicted of first degree murder, divided by 100?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a38bd54basicrules17","title":"Applying Probability to Real-World Problems","body":"Forty-eight percent of all Californians registered voters prefer life in prison without parole over the death penalty for a person convicted of first degree murder. Among Latino California registered voters, 55% prefer life in prison without parole over the death penalty for a person convicted of first degree murder. $$37.6\\\\%$$ of all Californians are Latino. In this problem, let C $$=$$ Calofornians (registered voters) preferring life in prison without parole over the death penalty for a person convicted of first degree murder and let L $$=$$ Latino Californians. Suppose that one Californian is randomly selected.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Two Basic Rules of Probability","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a38bd54basicrules17a","stepAnswer":["Given the person chosen is Latino Californian, the person is a registered voter who prefers life in person without parole for a person convicted of first degree murder."],"problemType":"MultipleChoice","stepTitle":"In words, what is C|L?","stepBody":"","answerType":"string","variabilization":{},"choices":["Given the person chosen is Latino Californian, the person is a registered voter who prefers life in person without parole for a person convicted of first degree murder.","The person chosen is a Latino California registered voter who prefers life without parole over the death penalty for a person convicted of first degree murder.","The person chosen is a Californian registered voter who prefers life in prison without parole over the death penalty for a person convicted of first degree murder or a person is a Latino Californian.","The person is neither Latino Californian nor prefers life in prison without parole over the death penalty for a person convicted of first degree murder."],"hints":{"DefaultPathway":[{"id":"a38bd54basicrules17a-h1","type":"hint","dependencies":[],"title":"Identifying Conditional Probability","text":"We note that C|L is a conditional statement, which usually involves words such as \\"given\\", \\"due to\\", or \\"since\\". In this case, we want a statement specifically that allows for the language of given the event L occurs, the event C occurs.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules17a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Given the person chosen is Latino Californian, the person is a registered voter who prefers life in person without parole for a person convicted of first degree murder."],"dependencies":["a38bd54basicrules17a-h1"],"title":"Identifying Conditional Probability","text":"Which statement best aligns with the idea of \\"given the event L occurs, the event C occurs\\"?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Given the person chosen is Latino Californian, the person is a registered voter who prefers life in person without parole for a person convicted of first degree murder.","The person chosen is a Latino California registered voter who prefers life without parole over the death penalty for a person convicted of first degree murder.","The person chosen is a Californian registered voter who prefers life in prison without parole over the death penalty for a person convicted of first degree murder or a person is a Latino Californian.","The person is neither Latino Californian nor prefers life in prison without parole over the death penalty for a person convicted of first degree murder."]}]}}]},{"id":"a38bd54basicrules18","title":"Applying Probability to Real-World Problems","body":"Forty-eight percent of all Californians registered voters prefer life in prison without parole over the death penalty for a person convicted of first degree murder. Among Latino California registered voters, 55% prefer life in prison without parole over the death penalty for a person convicted of first degree murder. $$37.6\\\\%$$ of all Californians are Latino. In this problem, let C $$=$$ Calofornians (registered voters) preferring life in prison without parole over the death penalty for a person convicted of first degree murder and let L $$=$$ Latino Californians. Suppose that one Californian is randomly selected.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Two Basic Rules of Probability","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a38bd54basicrules18a","stepAnswer":["$$0.2068$$"],"problemType":"TextBox","stepTitle":"Find P(L AND C).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.2068$$","hints":{"DefaultPathway":[{"id":"a38bd54basicrules18a-h1","type":"hint","dependencies":[],"title":"The Multiplication Rule","text":"We note that for events C and L, the mulitplication rule states that P(L AND $$C)=P\\\\left(L\\\\right) P\\\\left(C|L\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.376$$"],"dependencies":["a38bd54basicrules18a-h1"],"title":"Applying the Multiplication Rule","text":"What is P(L)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules18a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.55$$"],"dependencies":["a38bd54basicrules18a-h2"],"title":"Applying the Multiplication Rule","text":"What is P(C|L), or the probability that a California registered voter would prefer life in prison without parole over the death penalty for a person convicted of first degree murder given they are Latino Californian?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules18a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.2068$$"],"dependencies":["a38bd54basicrules18a-h3"],"title":"Applying the Multiplication Rule","text":"What is P(L AND $$C)=P\\\\left(L\\\\right) P\\\\left(C|L\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a38bd54basicrules2","title":"Applying the Addition and Multiplication Rules","body":"Carlos plays college soccer. he makes a goal 65% of the time he shoots. Carlos is going to attempt two goals in a row in the next game. A $$=$$ the event Carlos is successful on his first attempt. $$P(A)=0.65$$. B $$=$$ the event Carlos is successful on his second attempt. $$P(B)=0.65$$. Carlos tends to shoot in streaks. The probability that he makes hte second goal GIVEN that he made the first goal is $$0.90$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Two Basic Rules of Probability","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a38bd54basicrules2a","stepAnswer":["$$0.585$$"],"problemType":"TextBox","stepTitle":"What is the probability that he makes both goals?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.585$$","hints":{"DefaultPathway":[{"id":"a38bd54basicrules2a-h1","type":"hint","dependencies":[],"title":"The Multiplication Rule","text":"If A and B are two events defined on a sample space, then P(A AND $$B)=P\\\\left(B\\\\right) P\\\\left(A|B\\\\right)$$. We want to find the probability that A and B both occur: this is the formula to use.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.585$$"],"dependencies":["a38bd54basicrules2a-h1"],"title":"The Multiplication Rule Applied","text":"What is P(A AND $$B)=P(B$$ AND A)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.585$$"],"dependencies":["a38bd54basicrules2a-h2"],"title":"The Multiplication Rule Applied","text":"We know that $$P(B|A)=0.90$$. What is P(B AND $$A)=P\\\\left(A\\\\right) P\\\\left(B|A\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a38bd54basicrules2b","stepAnswer":["$$0.715$$"],"problemType":"TextBox","stepTitle":"What is the probability that Carlos makes either the first goal or the second goal?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.715$$","hints":{"DefaultPathway":[{"id":"a38bd54basicrules2b-h1","type":"hint","dependencies":[],"title":"The Addition Rule","text":"If A and B are defined on a sample space, then P(A OR B)=P(A)+P(B)-P(A AND B). We want to the find the probability that A or B occurs: this is the formula to use.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules2b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.715$$"],"dependencies":["a38bd54basicrules2b-h1"],"title":"The Addition Rule Applied","text":"What is P(A OR B)=P(A)+P(B)-P(A AND B)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules2b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.715$$"],"dependencies":["a38bd54basicrules2b-h2"],"title":"The Addition Rule Applied","text":"What is $$0.65+0.65-0.585$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a38bd54basicrules2c","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Are A and B independent?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a38bd54basicrules2c-h1","type":"hint","dependencies":[],"title":"Definition of Independence","text":"If A and B are independent, then $$P(A|B)=P(A)$$ and P(A AND $$B)=P(A)P(B)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules2c-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a38bd54basicrules2c-h1"],"title":"Application of Independence","text":"Are A and B independent events?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a38bd54basicrules2c-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.585$$"],"dependencies":["a38bd54basicrules2c-h2"],"title":"Application of Independence","text":"What is P(B AND A), solved for above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules2c-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.423$$"],"dependencies":["a38bd54basicrules2c-h3"],"title":"Application of Independence","text":"What is $$P\\\\left(A\\\\right) P\\\\left(B\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules2c-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a38bd54basicrules2c-h4"],"title":"Application of Independence","text":"Is P(A AND B) equal to $$P\\\\left(A\\\\right) P\\\\left(B\\\\right)$$? If yes, then A and B are independent; if no, then A and B are not independent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}},{"id":"a38bd54basicrules2d","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Are A and B mutually exclusive?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a38bd54basicrules2d-h1","type":"hint","dependencies":[],"title":"Definition of Mutual Independence","text":"If A and B are mutually exclusive, then P(A AND $$B)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules2d-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.585$$"],"dependencies":["a38bd54basicrules2d-h1"],"title":"Application of Mutual Independence","text":"What is P(A AND B), which was solved for above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules2d-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a38bd54basicrules2d-h2"],"title":"Application of Mutual Independence","text":"Is P(A AND $$B)=0$$? If yes, then they are mutually exclusive; if not, then they are not mutually exclusive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a38bd54basicrules20","title":"Applying Probability to Real-World Problems","body":"Forty-eight percent of all Californians registered voters prefer life in prison without parole over the death penalty for a person convicted of first degree murder. Among Latino California registered voters, 55% prefer life in prison without parole over the death penalty for a person convicted of first degree murder. $$37.6\\\\%$$ of all Californians are Latino. In this problem, let C $$=$$ Calofornians (registered voters) preferring life in prison without parole over the death penalty for a person convicted of first degree murder and let L $$=$$ Latino Californians. Suppose that one Californian is randomly selected.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Two Basic Rules of Probability","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a38bd54basicrules20a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Are L and C independent events?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a38bd54basicrules20a-h1","type":"hint","dependencies":[],"title":"Definition of Independence","text":"If L and C are independent, then P(C AND $$L)=P\\\\left(C\\\\right) P\\\\left(L\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.2068$$"],"dependencies":["a38bd54basicrules20a-h1"],"title":"Application of Independence","text":"What is P(C AND L)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules20a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.376$$"],"dependencies":["a38bd54basicrules20a-h2"],"title":"Applying the Multiplication Rule","text":"What is P(L)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules20a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.55$$"],"dependencies":["a38bd54basicrules20a-h3"],"title":"Applying the Multiplication Rule","text":"What is P(C|L), or the probability that a California registered voter would prefer life in prison without parole over the death penalty for a person convicted of first degree murder given they are Latino Californian?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules20a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.2068$$"],"dependencies":["a38bd54basicrules20a-h4"],"title":"Applying the Multiplication Rule","text":"What is P(L AND $$C)=P\\\\left(L\\\\right) P\\\\left(C|L\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules20a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.18048$$"],"dependencies":["a38bd54basicrules20a-h5"],"title":"Application of Independence","text":"What is $$P\\\\left(L\\\\right) P\\\\left(C\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules20a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a38bd54basicrules20a-h6"],"title":"Application of Independence","text":"Is P(L AND C) equal to $$P\\\\left(L\\\\right) P\\\\left(C\\\\right)$$? If yes, then L and C are independent; if no, then L and C are not independent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a38bd54basicrules21","title":"Applying Probability to Real-World Problems","body":"Forty-eight percent of all Californians registered voters prefer life in prison without parole over the death penalty for a person convicted of first degree murder. Among Latino California registered voters, 55% prefer life in prison without parole over the death penalty for a person convicted of first degree murder. $$37.6\\\\%$$ of all Californians are Latino. In this problem, let C $$=$$ Calofornians (registered voters) preferring life in prison without parole over the death penalty for a person convicted of first degree murder and let L $$=$$ Latino Californians. Suppose that one Californian is randomly selected.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Two Basic Rules of Probability","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a38bd54basicrules21a","stepAnswer":["$$0.6492$$"],"problemType":"TextBox","stepTitle":"Find P(L OR C).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.6492$$","hints":{"DefaultPathway":[{"id":"a38bd54basicrules21a-h1","type":"hint","dependencies":[],"title":"The Addition Rule","text":"If L and C are defined on a sample space, then P(L OR C)=P(L)+P(C)-P(L AND C). We want to the find the probability that L or C occurs: this is the formula to use.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules21a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.6492$$"],"dependencies":["a38bd54basicrules21a-h1"],"title":"The Addition Rule Applied","text":"What is P(L OR C)=P(L)+P(C)-P(L AND C)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules21a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.376$$"],"dependencies":["a38bd54basicrules21a-h2"],"title":"The Addition Rule Applied","text":"What is P(L)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules21a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.48$$"],"dependencies":["a38bd54basicrules21a-h3"],"title":"The Addition Rule Applied","text":"What is P(C)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules21a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.2068$$"],"dependencies":["a38bd54basicrules21a-h4"],"title":"The Addition Rule Applied","text":"What is P(L AND C)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules21a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.55$$"],"dependencies":["a38bd54basicrules21a-h5"],"title":"Applying the Multiplication Rule","text":"What is P(C|L), or the probability that a California registered voter would prefer life in prison without parole over the death penalty for a person convicted of first degree murder given they are Latino Californian?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules21a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.2068$$"],"dependencies":["a38bd54basicrules21a-h6"],"title":"Applying the Multiplication Rule","text":"What is P(L AND $$C)=P\\\\left(L\\\\right) P\\\\left(C|L\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules21a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.6492$$"],"dependencies":["a38bd54basicrules21a-h7"],"title":"Determining P(L OR C)","text":"What is $$0.376+0.48-0.2068$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a38bd54basicrules23","title":"Applying Probability to Real-World Problems","body":"Forty-eight percent of all Californians registered voters prefer life in prison without parole over the death penalty for a person convicted of first degree murder. Among Latino California registered voters, 55% prefer life in prison without parole over the death penalty for a person convicted of first degree murder. $$37.6\\\\%$$ of all Californians are Latino. In this problem, let C $$=$$ Calofornians (registered voters) preferring life in prison without parole over the death penalty for a person convicted of first degree murder and let L $$=$$ Latino Californians. Suppose that one Californian is randomly selected.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Two Basic Rules of Probability","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a38bd54basicrules23a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Are L and C mutually exclusive events?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a38bd54basicrules23a-h1","type":"hint","dependencies":[],"title":"Definition of Mutually Exclusive","text":"If L and C are mutually exclusive, then P(L AND $$C)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules23a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.2068$$"],"dependencies":["a38bd54basicrules23a-h1"],"title":"Application of Mutual Independence","text":"What is P(L AND C)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules23a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.376$$"],"dependencies":["a38bd54basicrules23a-h2"],"title":"Applying the Multiplication Rule","text":"What is P(L)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules23a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.55$$"],"dependencies":["a38bd54basicrules23a-h3"],"title":"Applying the Multiplication Rule","text":"What is P(C|L), or the probability that a California registered voter would prefer life in prison without parole over the death penalty for a person convicted of first degree murder given they are Latino Californian?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules23a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.2068$$"],"dependencies":["a38bd54basicrules23a-h4"],"title":"Applying the Multiplication Rule","text":"What is P(L AND $$C)=P\\\\left(L\\\\right) P\\\\left(C|L\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules23a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a38bd54basicrules23a-h5"],"title":"Application of Mutual Independence","text":"Is P(A AND $$B)=0$$? If yes, then they are mutually exclusive; if not, then they are not mutually exclusive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a38bd54basicrules24","title":"Applying Probability to Real-World Problems","body":"United Blood Services is a blood bank that serves more than $$500$$ hospitals in $$18$$ states. According to their website, a person with type O blood and a negative Rh factor $$(Rh-)$$ can donate blood to any person with any blood type. Their data show that 43% of people have type O blood and 15% of people have Rh- factor; 52% of people have type O or Rh- factor. Let O $$=$$ a person with type O blood and R $$=$$ a person with a negative Rh factor $$(Rh-)$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Two Basic Rules of Probability","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a38bd54basicrules24a","stepAnswer":["$$0.06$$"],"problemType":"TextBox","stepTitle":"Find the probability that a person has both type O blood and the Rh- factor.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.06$$","hints":{"DefaultPathway":[{"id":"a38bd54basicrules24a-h1","type":"hint","dependencies":[],"title":"The Addition Rule","text":"This question is actually a cool application of the addition rule. We note that the addition rule states that P(O OR R)=P(O)+P(R)-P(O AND R). However, we can algebraically manipulate this to get P(O AND R)=P(O)+P(R)-P(O OR R), which is what we want to solve for.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules24a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.06$$"],"dependencies":["a38bd54basicrules24a-h1"],"title":"The Addition Rule Applied","text":"What is the probability that a person has both type O blood and the Rh- factor, or in other words, what is P(O)+P(R)-P(O OR R)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules24a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.06$$"],"dependencies":["a38bd54basicrules24a-h2"],"title":"The Addition Rule Applied","text":"What is $$0.43+0.15-0.52$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a38bd54basicrules24b","stepAnswer":["$$0.94$$"],"problemType":"TextBox","stepTitle":"Find the probability that a person does NOT have both type O blood and the Rh- factor.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.94$$","hints":{"DefaultPathway":[{"id":"a38bd54basicrules24b-h1","type":"hint","dependencies":[],"title":"Complement Rule","text":"This question is asking for the complement rule. We are essentially asked the probability that a person does not have type O blood and the Rh- factor. We can rewrite this express mathematically as P(NOT(O AND R)) and using complement notation (with the apostrophe), this can be simplified to P((O AND R)\'), or the complement of what we found in the previous step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules24b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.94$$"],"dependencies":["a38bd54basicrules24b-h1"],"title":"Applying the Complement Rule","text":"What is 1-P(NOT(O AND R))?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules24b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.94$$"],"dependencies":["a38bd54basicrules24b-h2"],"title":"Applying the Complement Rule","text":"What is $$1-0.06$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a38bd54basicrules25","title":"Applying Probability to Real-World Problems","body":"In a box of assorted cookies, 36% contain chocolate and 12% contain nuts. In the box, 8% contain both chocolate and nuts. Sean is allergic to both chocolate and nuts. Let C $$=$$ the cookie contains chocolate and let N $$=$$ the chocolate contains nuts.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Two Basic Rules of Probability","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a38bd54basicrules25a","stepAnswer":["$$0.4$$"],"problemType":"TextBox","stepTitle":"Find the probability that a cookie contains chocolate or nuts (he can\'t eat it).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.4$$","hints":{"DefaultPathway":[{"id":"a38bd54basicrules25a-h1","type":"hint","dependencies":[],"title":"The Addition Rule","text":"The addition rule states that P(C OR N)=P(C)+P(N)-P(C AND N). Since we want to find the probability that a cookie contains chocolate or nuts, this works well with us as it uses an OR statement.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules25a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.36$$"],"dependencies":["a38bd54basicrules25a-h1"],"title":"The Addition Rule Applied","text":"What is P(C), the probability that the cookie contians choclate?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules25a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.12$$"],"dependencies":["a38bd54basicrules25a-h2"],"title":"The Addition Rule Applied","text":"What is P(N), the probability the cookie contains nuts?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules25a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.08$$"],"dependencies":["a38bd54basicrules25a-h3"],"title":"The Addition Rule Applied","text":"What is P(C AND N), the probability the cookie contains both chocolate and nuts?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules25a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.4$$"],"dependencies":["a38bd54basicrules25a-h4"],"title":"The Addition Rule Applied","text":"What is P(C OR N)=P(C)+P(N)-P(C AND N)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules25a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.4$$"],"dependencies":["a38bd54basicrules25a-h5"],"title":"The Addition Rule Applied","text":"What is $$0.36+0.12-0.08$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a38bd54basicrules25b","stepAnswer":["$$0.6$$"],"problemType":"TextBox","stepTitle":"Find the probability that a cookie does not contain chocolate or nuts (he can eat it).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.6$$","hints":{"DefaultPathway":[{"id":"a38bd54basicrules25b-h1","type":"hint","dependencies":[],"title":"Complement Rule","text":"This question is asking for the complement rule. We are essentially asked the probability that a cookie does not contain chocolate or nuts, which can be expressed mathematically as P(NOT(C OR N)) and using complement notation (with the apostrophe), this can be simplified to P((C OR N)\'), or the complement of what we found in the previous step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules25b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.6$$"],"dependencies":["a38bd54basicrules25b-h1"],"title":"Applying the Complement Rule","text":"What is 1-P(NOT(C OR N))?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules25b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.6$$"],"dependencies":["a38bd54basicrules25b-h2"],"title":"Applying the Complement Rule","text":"What is $$1-0.4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a38bd54basicrules4","title":"Probability Review","body":"A community swim team has $$150$$ members. Seventy-five of the members are advanced swimmers. Forty-seven of the members are intermediate swimmers. The remainder are novice swimmers. Forty of the advanced swimmers practice four times a week. Thirty of the intermediate swimmers practice four times a week. Ten of the novice swimmers practice four times a week. Suppose one member of the swim team is chosen randomly.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Two Basic Rules of Probability","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a38bd54basicrules4a","stepAnswer":["$$\\\\frac{28}{150}$$"],"problemType":"TextBox","stepTitle":"What is the probability that the member is a novice swimmer?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{28}{150}$$","hints":{"DefaultPathway":[{"id":"a38bd54basicrules4a-h1","type":"hint","dependencies":[],"title":"Determining Probabilities","text":"To find the probability of an event occurring, we find the number of occurrences of that event, and divide that by the total number of occurrences possible in the sample space.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$28$$"],"dependencies":["a38bd54basicrules4a-h1"],"title":"Determining Novice Swimmers","text":"What is the total number of members that are novice swimmers?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$28$$"],"dependencies":["a38bd54basicrules4a-h2"],"title":"Determining Novice Swimmers","text":"What is $$150-75-47$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$150$$"],"dependencies":["a38bd54basicrules4a-h3"],"title":"Determining the Sample Space","text":"What is the total number of swimmers?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules4a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{28}{150}$$"],"dependencies":["a38bd54basicrules4a-h4"],"title":"Determining Probability of Novice","text":"What is the probability that the member is a novice swimmer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a38bd54basicrules4b","stepAnswer":["$$\\\\frac{80}{150}$$"],"problemType":"TextBox","stepTitle":"What is the probability that the member practices $$4$$ times a week?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{80}{150}$$","hints":{"DefaultPathway":[{"id":"a38bd54basicrules4b-h1","type":"hint","dependencies":[],"title":"Determining Probabilities","text":"To find the probability of an event occurring, we find the number of occurrences of that event, and divide that by the total number of occurrences possible in the sample space.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules4b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$80$$"],"dependencies":["a38bd54basicrules4b-h1"],"title":"Determining Members That Practice $$4$$ Times a Week","text":"How many mmebers practice four times a week?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules4b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$150$$"],"dependencies":["a38bd54basicrules4b-h2"],"title":"Determining the Sample Space","text":"What is the total number of swimmers?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules4b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{80}{150}$$"],"dependencies":["a38bd54basicrules4b-h3"],"title":"Determining Probability of Novice","text":"What is the probability that the member practices four times a week?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a38bd54basicrules4c","stepAnswer":["$$\\\\frac{40}{150}$$"],"problemType":"TextBox","stepTitle":"What is the probability that the member is an advanced swimmer and practices four times a week?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{40}{150}$$","hints":{"DefaultPathway":[{"id":"a38bd54basicrules4c-h1","type":"hint","dependencies":[],"title":"Determining Probabilities","text":"To find the probability of an event occurring, we find the number of occurrences of that event, and divide that by the total number of occurrences possible in the sample space.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules4c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$40$$"],"dependencies":["a38bd54basicrules4c-h1"],"title":"Determining Advanced and $$4$$ Times a Week","text":"How many swimmers are advanced and practice four times a week?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules4c-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$150$$"],"dependencies":["a38bd54basicrules4c-h2"],"title":"Determining the Sample Space","text":"What is the total number of swimmers?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules4c-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{40}{150}$$"],"dependencies":["a38bd54basicrules4c-h3"],"title":"Determining Probability of Novice","text":"What is the probability that the member is an advanced swimmer and practices four times a week?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a38bd54basicrules5","title":"Understanding Mutually Exclusive and Independent Events","body":"A community swim team has $$150$$ members. Seventy-five of the members are advanced swimmers. Forty-seven of the members are intermediate swimmers. The remainder are novice swimmers. Forty of the advanced swimmers practice four times a week. Thirty of the intermediate swimmers practice four times a week. Ten of the novice swimmers practice four times a week. Suppose one member of the swim team is chosen randomly.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Two Basic Rules of Probability","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a38bd54basicrules5a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Are being an advanced swimmer and an intermediate swimmer mutually exclusive?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a38bd54basicrules5a-h1","type":"hint","dependencies":[],"title":"Definition of Mutually Exclusive","text":"If A and B are mutually exclusive, then P(A AND $$B)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a38bd54basicrules5a-h1"],"title":"Application of Mutual Independence","text":"If we let A $$=$$ the member is an advanced swimmer and B $$=$$ the member is an intermediate swimmer, what is P(A AND B)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules5a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a38bd54basicrules5a-h2"],"title":"Application of Mutual Independence","text":"Is P(A AND $$B)=0$$? If yes, then they are mutually exclusive; if not, then they are not mutually exclusive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}},{"id":"a38bd54basicrules5b","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Are being a novice swimmer and practicing four times a week independent events?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a38bd54basicrules5b-h1","type":"hint","dependencies":[],"title":"Definition of Independence","text":"If A and B are independent, then $$P(A|B)=P(A)$$ and P(A AND $$B)=P(A)P(B)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules5b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{10}{150}$$"],"dependencies":["a38bd54basicrules5b-h1"],"title":"Application of Independence","text":"If we let A $$=$$ the member is a novice swimmer and B $$=$$ the member practices four times per week, what is P(A AND B)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules5b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2240}{22500}$$"],"dependencies":["a38bd54basicrules5b-h2"],"title":"Application of Independence","text":"What is $$P\\\\left(A\\\\right) P\\\\left(B\\\\right)$$, or the probability that the member is a novice multiplied by the probability that the member practices four times a week?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules5b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a38bd54basicrules5b-h3"],"title":"Application of Independence","text":"Is P(A AND B) equal to $$P\\\\left(A\\\\right) P\\\\left(B\\\\right)$$? If yes, then A and B are independent; if no, then A and B are not independent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a38bd54basicrules6","title":"Probability Revisited","body":"A school has $$200$$ seniors of whom $$140$$ will be going to college next year. Forty will be going directly to work. The remainder are taking a gap year. Fifty of the seniors going to college play sports. Thirty of the seniors going directly work play sports. Five of the seniors taking a gap year play sports.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Two Basic Rules of Probability","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a38bd54basicrules6a","stepAnswer":["$$\\\\frac{20}{200}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a senior is taking a gap year?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{20}{200}$$","hints":{"DefaultPathway":[{"id":"a38bd54basicrules6a-h1","type":"hint","dependencies":[],"title":"Complement Rule","text":"Sometimes, it is easier to determine if an event does not occur. For example, here, we can find the probability that a senior is NOT taking a gap year. Let A $$=$$ a senior who is taking a gap year. Therefore, $$P(A)-1-(not$$ A).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$180$$"],"dependencies":["a38bd54basicrules6a-h1"],"title":"Determining Number of Seniors NOT Taking a Gap Year","text":"How many seniors are not taking a gap year?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a38bd54basicrules6a-h2"],"title":"Determining Number of Seniors Taking a Gap Year","text":"How many seniors are taking a gap year? This would be the number of seniors NOT taking a gap year subtracted from the total number of seniors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$200$$"],"dependencies":["a38bd54basicrules6a-h3"],"title":"Determining Total Seniors","text":"How many seniors are there in total?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{20}{200}$$"],"dependencies":["a38bd54basicrules6a-h4"],"title":"Finding Probability","text":"We note that the probability an event occurs is the number outcomes in that event divided by the total number of outcomes. In this case, we want to know the probability that a senior is taking a gap year. What is the total number of seniors taking a gap year divided by the total number of seniors?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a38bd54basicrules7","title":"Applying Rules of Probability","body":"Felicity attends Modesto JC in Modesto, CA. The probability that Felicity enrolls in a math class is $$0.2$$ and the probability that she enrolls in a speech class is $$0.65$$. The probability that she enrolls in a math class GIVEN that she enrolls in a speech class is $$0.25$$. Let M $$=$$ math class, S $$=$$ speech class, M|S $$=$$ math given speech.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Two Basic Rules of Probability","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a38bd54basicrules7a","stepAnswer":["$$0.1625$$"],"problemType":"TextBox","stepTitle":"What is the probability that Felicity enrolls in math and speech?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.1625$$","hints":{"DefaultPathway":[{"id":"a38bd54basicrules7a-h1","type":"hint","dependencies":[],"title":"The Multiplication Rule","text":"If M and S are two events defined on a sample space, then P(M AND $$S)=P\\\\left(S\\\\right) P\\\\left(M|S\\\\right)$$. We want to find the probability that M and S both occur: this is the formula to use.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1625$$"],"dependencies":["a38bd54basicrules7a-h1"],"title":"The Multiplication Rule Applied","text":"What is P(M AND $$S)=P\\\\left(M|S\\\\right) P\\\\left(S\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1625$$"],"dependencies":["a38bd54basicrules7a-h2"],"title":"The Multiplication Rule Applied","text":"What is $$0.25\\\\times0.65$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a38bd54basicrules7b","stepAnswer":["$$0.6875$$"],"problemType":"TextBox","stepTitle":"What is the probability that Felicity enrolls in math or speech classes?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.6875$$","hints":{"DefaultPathway":[{"id":"a38bd54basicrules7b-h1","type":"hint","dependencies":[],"title":"The Addition Rule","text":"To find if the events M or S occur (one or the other, or both), we want to use the Addition Rule: P(M OR S)=P(M)+P(S)-P(M AND S).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules7b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.6875$$"],"dependencies":["a38bd54basicrules7b-h1"],"title":"The Addition Rule Applied","text":"What is P(M OR S)=P(M)+P(S)-P(M AND S)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules7b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.6875$$"],"dependencies":["a38bd54basicrules7b-h2"],"title":"The Addition Rule Applied","text":"What is $$0.2+0.65-0.1625$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a38bd54basicrules7c","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Are M and S independent?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a38bd54basicrules7c-h1","type":"hint","dependencies":[],"title":"Definition of Independence","text":"Events M and S are independent is $$P(M|S)=P(M)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules7c-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a38bd54basicrules7c-h1"],"title":"Application of Independence","text":"Does P(M|S) equal P(M)? Both are given in the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}},{"id":"a38bd54basicrules7d","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Are M and S mutually exclusive?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a38bd54basicrules7d-h1","type":"hint","dependencies":[],"title":"Definition of Mutually Exclusive","text":"Events M and S are mutually exclusive if P(M AND $$S)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules7d-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a38bd54basicrules7d-h1"],"title":"Application of Mutually Exclusive","text":"Is P(M AND S) equal to 0? We solved for this earlier in the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a38bd54basicrules8","title":"Applying Rules of Probability","body":"A student goes to the library. Let events B $$=$$ the student checks out a book and D $$=$$ the student check out a DVD. Suppose that $$P(B)=0.40$$, $$P(D)=0.30$$ and $$P(D|B)=0.5$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Two Basic Rules of Probability","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a38bd54basicrules8a","stepAnswer":["$$0.2$$"],"problemType":"TextBox","stepTitle":"Find P(B AND D).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.2$$","hints":{"DefaultPathway":[{"id":"a38bd54basicrules8a-h1","type":"hint","dependencies":[],"title":"The Multiplication Rule","text":"If B and D are two events defined on a sample space, then P(B AND $$D)=P\\\\left(B\\\\right) P\\\\left(D|B\\\\right)$$. We want to find the probability that B and D both occur: this is the formula to use.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.2$$"],"dependencies":["a38bd54basicrules8a-h1"],"title":"The Multiplication Rule Applied","text":"What is P(B AND $$D)=P\\\\left(D|B\\\\right) P\\\\left(B\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.2$$"],"dependencies":["a38bd54basicrules8a-h2"],"title":"The Multiplication Rule Applied","text":"What is $$0.5\\\\times0.4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a38bd54basicrules8b","stepAnswer":["$$0.5$$"],"problemType":"TextBox","stepTitle":"Find P(B OR D).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.5$$","hints":{"DefaultPathway":[{"id":"a38bd54basicrules8b-h1","type":"hint","dependencies":[],"title":"The Addition Rule","text":"To find if the events B or D occur (one or the other, or both), we want to use the Addition Rule: P(B OR D)=P(B)+P(D)-P(B AND D).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules8b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.5$$"],"dependencies":["a38bd54basicrules8b-h1"],"title":"The Addition Rule Applied","text":"What is P(B OR D)=P(B)+P(D)-P(B AND D)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules8b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.5$$"],"dependencies":["a38bd54basicrules8b-h2"],"title":"The Addition Rule Applied","text":"What is $$0.4+0.3-0.2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a38bd54basicrules9","title":"The Multiplication Rule","body":"A school has $$200$$ seniors of whom $$140$$ will be going to college next year. Forty will be going directly to work. The remainder are taking a gap year. Fifty of the seniors going to college play sports. Thirty of the seniors going directly work play sports. Five of the seniors taking a gap year play sports.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Two Basic Rules of Probability","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a38bd54basicrules9a","stepAnswer":["$$\\\\frac{50}{200}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a senior is going to college and plays sports?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{50}{200}$$","hints":{"DefaultPathway":[{"id":"a38bd54basicrules9a-h1","type":"hint","dependencies":[],"title":"The Multiplication Rule","text":"If A and B are two events defined on a sample space, then P(A AND $$B)=P\\\\left(A\\\\right) P\\\\left(B|A\\\\right)$$. Let A $$=$$ a senior that is going college and B $$=$$ a senior plays sports. We want to find the probability that A and B both occur: this is the formula to use.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{140}{200}$$"],"dependencies":["a38bd54basicrules9a-h1"],"title":"The Multiplication Rule Applied","text":"What is P(A), the probability that a senior is going to college?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{50}{140}$$"],"dependencies":["a38bd54basicrules9a-h2"],"title":"The Multiplication Rule Applied","text":"What is P(B|A), the probability that a senior plays sports GIVEN that they are going to college?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{50}{200}$$"],"dependencies":["a38bd54basicrules9a-h3"],"title":"Conditional Probability","text":"P(B|A) can be determined as P(A AND B) divided by P(A). We know that $$P(A)=\\\\frac{140}{200}$$. What is P(A AND B), the probability that a senior is going to college and plays sports?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules9a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{50}{140}$$"],"dependencies":["a38bd54basicrules9a-h4"],"title":"Conditional Probability","text":"What is P(B|A), or $$\\\\frac{\\\\frac{50}{200}}{\\\\frac{140}{200}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a38bd54basicrules9a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{50}{200}$$"],"dependencies":["a38bd54basicrules9a-h5"],"title":"The Multiplication Rule Applied","text":"What is P(A AND $$B)=P\\\\left(A\\\\right) P\\\\left(B|A\\\\right)$$, using the values we calculated above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a38da08geo1","title":"Geometry and Algebra","body":"These questions test your knowledge of the core concepts.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Geometry and Algebra","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a38da08geo1a","stepAnswer":["$$2\\\\pi R^2+2\\\\pi h R$$"],"problemType":"MultipleChoice","stepTitle":"By drawing the net of a cylinder, what is the surface area of a cylinder with radius R and height $$h$$? Hint: The net is the cylinder laid flat. It should consist of a rectangle and two circles.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$2\\\\pi R^2+2\\\\pi h R$$","choices":["$$2\\\\pi R^2+2\\\\pi h R$$","$$\\\\pi R^2+2\\\\pi h R$$","$$2\\\\pi R^2+\\\\pi h R$$","$$\\\\pi R^2+\\\\pi h R$$"],"hints":{"DefaultPathway":[{"id":"a38da08geo1a-h1","type":"hint","dependencies":[],"title":"Draw the graph","text":"The area of a circle is $$\\\\pi R^2$$. The area of a rectangle is $$h d=2\\\\pi h R$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a38da08geo1a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2\\\\pi R^2+2\\\\pi h R$$"],"dependencies":["a38da08geo1a-h1"],"title":"See the graph","text":"Following the hint, the area of a cylinder is $$A\\\\left(circle1\\\\right)+A\\\\left(circle2\\\\right)+A\\\\left(rectangle\\\\right)$$. Choose the correct area.","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$2\\\\pi R^2+2\\\\pi h R$$","$$\\\\pi R^2+2\\\\pi h R$$","$$2\\\\pi R^2+\\\\pi h R$$","$$\\\\pi R^2+\\\\pi h R$$"]}]}}]},{"id":"a38da08geo10","title":"Geometry and Algebra","body":"These questions are challenging, requiring mastery of each concept and their interrelations.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Geometry and Algebra","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a38da08geo10a","stepAnswer":["$$\\\\frac{4+\\\\sqrt{15}+2\\\\sqrt{3}+\\\\sqrt{7}}{4}$$"],"problemType":"MultipleChoice","stepTitle":"In the following diagram the curve is a semi-circle of radius $$2$$. Calculate the exact area of the shaded rectangles. Hint: How would you calculate the height of each rectangle? What\u2019s the equation for a circle centered at $$(0,0)$$ of radius 2?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{4+\\\\sqrt{15}+2\\\\sqrt{3}+\\\\sqrt{7}}{4}$$","choices":["$$\\\\frac{4+\\\\sqrt{15}+2\\\\sqrt{3}+\\\\sqrt{7}}{4}$$","$$\\\\frac{1+\\\\sqrt{15}+2\\\\sqrt{3}+\\\\sqrt{7}}{4}$$","$$\\\\frac{4+\\\\sqrt{15}+\\\\sqrt{3}+\\\\sqrt{7}}{4}$$"],"hints":{"DefaultPathway":[{"id":"a38da08geo10a-h1","type":"hint","dependencies":[],"title":"Equation for a circle","text":"Circle centered at $$(0,0)$$ of radius $$2$$ has equation $$x^2+y^2=2^2$$.","variabilization":{},"oer":"","license":""},{"id":"a38da08geo10a-h2","type":"hint","dependencies":["a38da08geo10a-h1"],"title":"Equation for a circle","text":"$$x^2+y^2=2^2$$ is equivalent to $$y=\\\\pm \\\\sqrt{4-x^2}$$ for $$x$$ in [-2,2].","variabilization":{},"oer":"","license":""},{"id":"a38da08geo10a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y=\\\\sqrt{4-x^2}$$"],"dependencies":["a38da08geo10a-h2"],"title":"The shaded region","text":"Which equation fits the upper semicircle region shown in the shaded plot?","variabilization":{},"oer":"","license":"","choices":["$$y=\\\\pm \\\\sqrt{4-x^2}$$","$$y=-\\\\sqrt{4-x^2}$$","$$y=\\\\sqrt{4-x^2}$$"]},{"id":"a38da08geo10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a38da08geo10a-h3"],"title":"Find the height of each rectangle","text":"For the equation $$y=\\\\sqrt{4-x^2}$$, what is $$y$$ when $$x=0$$?","variabilization":{},"oer":"","license":""},{"id":"a38da08geo10a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{\\\\sqrt{15}}{2}$$"],"dependencies":["a38da08geo10a-h4"],"title":"Find the height of each rectangle","text":"For the equation $$y=\\\\sqrt{4-x^2}$$, what is $$y$$ when $$x=0.5$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{\\\\sqrt{15}}{2}$$","$$-\\\\left(\\\\frac{\\\\sqrt{15}}{2}\\\\right)$$"]},{"id":"a38da08geo10a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\sqrt{3}$$"],"dependencies":["a38da08geo10a-h5"],"title":"Find the height of each rectangle","text":"For the equation $$y=\\\\sqrt{4-x^2}$$, what is $$y$$ when $$x=1$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\sqrt{3}$$","$$-\\\\sqrt{3}$$"]},{"id":"a38da08geo10a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{\\\\sqrt{7}}{2}$$"],"dependencies":["a38da08geo10a-h6"],"title":"Find the height of each rectangle","text":"For the equation $$y=\\\\sqrt{4-x^2}$$, what is $$y$$ when $$x=1.5$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{\\\\sqrt{7}}{2}$$","$$-\\\\left(\\\\frac{\\\\sqrt{7}}{2}\\\\right)$$"]},{"id":"a38da08geo10a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a38da08geo10a-h7"],"title":"Find the area of each rectangle","text":"Using the rectangle area formula $$A=length$$ times width, what is the area of R1?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a38da08geo10a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{\\\\sqrt{15}}{4}$$"],"dependencies":["a38da08geo10a-h8"],"title":"Find the area of each rectangle","text":"Using the rectangle area formula $$A=length$$ times width, what is the area of R2?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{\\\\sqrt{15}}{4}$$","$$-\\\\left(\\\\frac{\\\\sqrt{15}}{4}\\\\right)$$"]},{"id":"a38da08geo10a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{\\\\sqrt{3}}{2}$$"],"dependencies":["a38da08geo10a-h9"],"title":"Find the area of each rectangle","text":"Using the rectangle area formula $$A=length$$ times width, what is the area of R3?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{\\\\sqrt{3}}{2}$$","$$-\\\\left(\\\\frac{\\\\sqrt{3}}{2}\\\\right)$$"]},{"id":"a38da08geo10a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{\\\\sqrt{7}}{4}$$"],"dependencies":["a38da08geo10a-h10"],"title":"Find the area of each rectangle","text":"Using the rectangle area formula $$A=length$$ times width, what is the area of R4?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{\\\\sqrt{7}}{4}$$","$$-\\\\left(\\\\frac{\\\\sqrt{7}}{4}\\\\right)$$"]},{"id":"a38da08geo10a-h12","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{4+\\\\sqrt{15}+2\\\\sqrt{3}+\\\\sqrt{7}}{4}$$"],"dependencies":["a38da08geo10a-h11"],"title":"Shaded Rectangles","text":"Calculate the exact area of the shaded rectangles. (What is the total area?)","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{4+\\\\sqrt{15}+2\\\\sqrt{3}+\\\\sqrt{7}}{4}$$","$$\\\\frac{1+\\\\sqrt{15}+2\\\\sqrt{3}+\\\\sqrt{7}}{4}$$","$$\\\\frac{4+\\\\sqrt{15}+\\\\sqrt{3}+\\\\sqrt{7}}{4}$$"]}]}}]},{"id":"a38da08geo2","title":"Geometry and Algebra","body":"These questions test your knowledge of the core concepts.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Geometry and Algebra","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a38da08geo2a","stepAnswer":["$$(\\\\frac{5}{13},\\\\frac{-1}{13})$$"],"problemType":"MultipleChoice","stepTitle":"Determine the point of intersection of the two straights $$2x-3y=1$$ and $$5x-y=2$$.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$(\\\\frac{5}{13},\\\\frac{-1}{13})$$","choices":["$$(\\\\frac{5}{13},\\\\frac{1}{13})$$","$$(\\\\frac{5}{13},\\\\frac{-1}{13})$$","$$(\\\\frac{-5}{13},\\\\frac{-1}{13})$$","$$(\\\\frac{-5}{13},\\\\frac{1}{13})$$"],"hints":{"DefaultPathway":[{"id":"a38da08geo2a-h1","type":"hint","dependencies":[],"title":"Transformation","text":"We can express $$5x-y=2$$ as $$y=5x-2$$.","variabilization":{},"oer":"","license":""},{"id":"a38da08geo2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{13}$$"],"dependencies":["a38da08geo2a-h1"],"title":"Substitution","text":"Substitute $$y=5x-2$$ into $$2x-3y=1$$, what is $$x$$?","variabilization":{},"oer":"","license":"","subHints":[{"id":"a38da08geo2a-h2-s1","type":"hint","dependencies":[],"title":"Substitution","text":"$$2x-3\\\\left(5x-2\\\\right)=1$$. $$2x-15x+6=1$$. $$-13x=-5$$, so $$x=\\\\frac{5}{13}$$.","variabilization":{},"oer":"","license":""}]},{"id":"a38da08geo2a-h3","type":"hint","dependencies":["a38da08geo2a-h2"],"title":"Substitution","text":"We can substitute $$x=\\\\frac{5}{13}$$ in any of the two lines. We choose $$5x-y=2$$ in this case.","variabilization":{},"oer":"","license":""},{"id":"a38da08geo2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{13}$$"],"dependencies":["a38da08geo2a-h3"],"title":"Substitution","text":"What is the value of $$y$$ for the equation $$\\\\frac{5\\\\times5}{13}-y=2$$?","variabilization":{},"oer":"","license":"","subHints":[{"id":"a38da08geo2a-h4-s1","type":"hint","dependencies":[],"title":"Substitution","text":"$$y=\\\\frac{25}{13}-2=\\\\frac{-1}{13}$$","variabilization":{},"oer":"","license":""}]}]}},{"id":"a38da08geo2b","stepAnswer":["No intersection"],"problemType":"MultipleChoice","stepTitle":"Determine the point of intersection of the two straights $$2x-4y=1$$ and $$x-2y=3$$. Explain your result geometrically.","stepBody":"##figure2.gif## ","answerType":"string","variabilization":{},"choices":["We can find intersection","No intersection"],"hints":{"DefaultPathway":[{"id":"a38da08geo2b-h1","type":"hint","dependencies":[],"title":"Transformation","text":"We can express $$x-2y=3$$ as $$x=3+2y$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a38da08geo2b-h2","type":"hint","dependencies":["a38da08geo2b-h1"],"title":"Substitution","text":"Substitude $$x=3+2y$$ into $$2x-4y=1$$. Simplify $$2\\\\left(3+2y\\\\right)-4y=1$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a38da08geo2b-h3","type":"hint","dependencies":["a38da08geo2b-h2"],"title":"LHS","text":"$$2\\\\left(3+2y\\\\right)-4y=6$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a38da08geo2b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a38da08geo2b-h3"],"title":"$$LHS=RHS$$","text":"Does $$6=1$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"a38da08geo2b-h5","type":"hint","dependencies":["a38da08geo2b-h4"],"title":"Conclusion","text":"Since there is no solution to the equation, there is no intersection. Also, from the plot, we can see two lines are parallel","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a38da08geo3","title":"Geometry and Algebra","body":"These questions test your knowledge of the core concepts.\\\\n##figure1.gif","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Geometry and Algebra","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a38da08geo3a","stepAnswer":["$$a b+\\\\frac{\\\\pi}{8} a^2$$"],"problemType":"MultipleChoice","stepTitle":"Consider the following shape S: Express the area of S as a formula involving a and $$b$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$a b+\\\\frac{\\\\pi}{8} a^2$$","choices":["$$a b+\\\\frac{\\\\pi}{8} a^2$$","$$a b+\\\\frac{\\\\pi}{2} a^2$$"],"hints":{"DefaultPathway":[{"id":"a38da08geo3a-h1","type":"hint","dependencies":[],"title":"Graph","text":"S is composed of a rectangle and a semi-circle","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a38da08geo3a-h2","type":"hint","dependencies":["a38da08geo3a-h1"],"title":"Rectangle Area","text":"The area of the rectangle is $$a b$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a38da08geo3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{a}{2}$$"],"dependencies":["a38da08geo3a-h2"],"title":"Radius of the semi-circle","text":"What is the radius of the semi-circle?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a38da08geo3a-h4","type":"hint","dependencies":["a38da08geo3a-h3"],"title":"Semi-circle Area","text":"The area of the semi-circle is $$\\\\frac{\\\\pi r^2}{2}=\\\\frac{\\\\pi {\\\\left(\\\\frac{a}{2}\\\\right)}^2}{2}=\\\\frac{\\\\pi}{8} a^2$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a38da08geo3a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$a b+\\\\frac{\\\\pi}{8} a^2$$"],"dependencies":["a38da08geo3a-h4"],"title":"Total Area","text":"What is the total area?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$a b+\\\\frac{\\\\pi}{8} a^2$$","$$a b+\\\\frac{\\\\pi}{2} a^2$$"]}]}},{"id":"a38da08geo3b","stepAnswer":["$$2b+\\\\left(1+\\\\frac{\\\\pi}{2}\\\\right) a=100$$"],"problemType":"MultipleChoice","stepTitle":"Consider the following shape S: Assume now that the perimeter of S is $$100$$ units. Express this fact as a mathematical equation involving a and $$b$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2b+\\\\left(1+\\\\frac{\\\\pi}{2}\\\\right) a=100$$","choices":["$$2b+\\\\left(1+\\\\frac{\\\\pi}{2}\\\\right) a=100$$","$$b+\\\\left(1+\\\\frac{\\\\pi}{2}\\\\right) a=100$$"],"hints":{"DefaultPathway":[{"id":"a38da08geo3b-h1","type":"hint","dependencies":[],"title":"Perimeter of S","text":"Perimeter of S is $$b+a+b+\\\\pi r=2b+a+\\\\pi \\\\frac{a}{2}$$ since $$r=\\\\frac{a}{2}$$ in the semi-circle.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a38da08geo3b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2b+\\\\left(1+\\\\frac{\\\\pi}{2}\\\\right) a=100$$"],"dependencies":["a38da08geo3b-h1"],"title":"Perimeter of S","text":"Since $$perimeter=100$$, what is the correct mathematical equation?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$2b+\\\\left(1+\\\\frac{\\\\pi}{2}\\\\right) a=100$$","$$b+\\\\left(1+\\\\frac{\\\\pi}{2}\\\\right) a=100$$"]}]}},{"id":"a38da08geo3c","stepAnswer":["$$50a+\\\\frac{\\\\left(-4-\\\\pi\\\\right)}{8} a^2$$"],"problemType":"MultipleChoice","stepTitle":"Consider the following shape S: By solving this equation in $$b$$, find an expression for the area of S purely in terms of a.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$50a+\\\\frac{\\\\left(-4-\\\\pi\\\\right)}{8} a^2$$","choices":["$$50a+\\\\frac{\\\\left(-4-\\\\pi\\\\right)}{8} a^2$$","$$50a+\\\\frac{\\\\left(-4-\\\\pi\\\\right)}{8} a$$","$$a+\\\\frac{\\\\left(-4-\\\\pi\\\\right)}{8} a^2$$"],"hints":{"DefaultPathway":[{"id":"a38da08geo3c-h1","type":"hint","dependencies":[],"title":"Simplification","text":"Simplify $$2b+\\\\left(1+\\\\frac{\\\\pi}{2}\\\\right) a=100$$ to find the relationship between a and $$b$$ first. $$2b=100-\\\\left(1+\\\\frac{\\\\pi}{2}\\\\right) a$$. So $$b=50-\\\\frac{2+\\\\pi}{4} a$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a38da08geo3c-h2","type":"hint","dependencies":["a38da08geo3c-h1"],"title":"Substitution","text":"Substitute $$b=50-\\\\frac{2+\\\\pi}{4} a$$ into $$Area=a b+\\\\frac{\\\\pi}{8} a^2$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a38da08geo3c-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$50a+\\\\frac{\\\\left(-4-\\\\pi\\\\right)}{8} a^2$$"],"dependencies":["a38da08geo3c-h2"],"title":"Substitution","text":"What is the correct expression?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$50a+\\\\frac{\\\\left(-4-\\\\pi\\\\right)}{8} a^2$$","$$50a+\\\\frac{\\\\left(-4-\\\\pi\\\\right)}{8} a$$","$$a+\\\\frac{\\\\left(-4-\\\\pi\\\\right)}{8} a^2$$"],"subHints":[{"id":"a38da08geo3c-h3-s1","type":"hint","dependencies":[],"title":"Substitution","text":"$$a \\\\left(50-\\\\frac{2+\\\\pi}{4} a\\\\right)+\\\\frac{\\\\pi}{8} a^2=50a+\\\\left(\\\\frac{\\\\pi}{8}+\\\\frac{\\\\left(-2-\\\\pi\\\\right)}{4}\\\\right) a^2=50a+\\\\frac{\\\\left(-4-\\\\pi\\\\right)}{8} a^2$$","variabilization":{},"oer":"https://OATutor.io","license":""}]}]}},{"id":"a38da08geo3d","stepAnswer":["$$[0,\\\\frac{200}{2+\\\\pi}]$$"],"problemType":"MultipleChoice","stepTitle":"Consider the following shape S: Notice that this expression makes mathematical sense for any value of a. What values of a are applicable to the problem? Give your answer in interval notation. Hint: Remember, both a and $$b$$ must be non-negative.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[0,\\\\frac{200}{2+\\\\pi}]$$","choices":["$$[0,\\\\frac{400}{4+\\\\pi}]$$","$$[0,\\\\frac{200}{2+\\\\pi}]$$"],"hints":{"DefaultPathway":[{"id":"a38da08geo3d-h1","type":"hint","dependencies":[],"title":"Follow the hint","text":"a and $$b$$ must be non-negative implies that $$a \\\\geq 0$$, $$b \\\\geq 0$$.","variabilization":{},"oer":"","license":""},{"id":"a38da08geo3d-h2","type":"hint","dependencies":["a38da08geo3d-h1"],"title":"Simplification","text":"$$50-\\\\frac{2+\\\\pi}{4} a \\\\geq 0$$","variabilization":{},"oer":"","license":""},{"id":"a38da08geo3d-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$a \\\\leq \\\\frac{200}{2+\\\\pi}$$"],"dependencies":["a38da08geo3d-h2"],"title":"Simplification","text":"What is the range of a?","variabilization":{},"oer":"","license":"","choices":["$$a \\\\leq \\\\frac{200}{2+\\\\pi}$$","$$a \\\\leq \\\\frac{400}{2+\\\\pi}$$"]},{"id":"a38da08geo3d-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$[0,\\\\frac{200}{2+\\\\pi}]$$"],"dependencies":["a38da08geo3d-h3"],"title":"Interval Notation","text":"What is a in interval notation?","variabilization":{},"oer":"","license":"","choices":["$$[0,\\\\frac{400}{4+\\\\pi}]$$","$$[0,\\\\frac{200}{2+\\\\pi}]$$"]}]}}]},{"id":"a38da08geo4","title":"Geometry and Algebra","body":"These questions test your knowledge of the core concepts.\\\\n##figure2.gif","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Geometry and Algebra","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a38da08geo4a","stepAnswer":["$$\\\\frac{13}{8}$$"],"problemType":"TextBox","stepTitle":"The sloping straight line in the following diagram is given by the formula $$2x-6y=-3$$. Determine the area of the shaded region.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{13}{8}$$","hints":{"DefaultPathway":[{"id":"a38da08geo4a-h1","type":"hint","dependencies":[],"title":"Expression","text":"Express $$2x-6y=-3$$ in the form of $$y$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a38da08geo4a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y=\\\\frac{1}{3} x+\\\\frac{1}{2}$$"],"dependencies":["a38da08geo4a-h1"],"title":"Expression","text":"Which option is the correct expression?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$y=\\\\frac{-1}{3} x+\\\\frac{1}{2}$$","$$y=\\\\frac{1}{3} x+\\\\frac{1}{2}$$","$$y=\\\\frac{1}{3} x-\\\\frac{1}{2}$$","$$y=\\\\frac{-1}{3} x-\\\\frac{1}{2}$$"]},{"id":"a38da08geo4a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{5}{6}$$"],"dependencies":["a38da08geo4a-h2"],"title":"Plot","text":"When $$x=1$$ (specified in the plot), what is the corresponding $$y$$ value?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{5}{6}$$","$$1$$"],"subHints":[{"id":"a38da08geo4a-h3-s1","type":"hint","dependencies":[],"title":"Plot","text":"$$y=\\\\frac{1}{3}+\\\\frac{1}{2}=\\\\frac{5}{6}$$","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a38da08geo4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{4}{3}$$"],"dependencies":["a38da08geo4a-h3"],"title":"Plot","text":"When $$x=2.5=\\\\frac{5}{2}$$ (specified in the plot), what is the corresponding $$y$$ value?","variabilization":{},"oer":"https://OATutor.io","license":"","subHints":[{"id":"a38da08geo4a-h4-s1","type":"hint","dependencies":[],"title":"Plot","text":"$$y=\\\\frac{1}{3} \\\\frac{5}{2}+\\\\frac{1}{2}=\\\\frac{5}{6}+\\\\frac{1}{2}=\\\\frac{8}{6}=\\\\frac{4}{3}$$","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a38da08geo4a-h5","type":"hint","dependencies":["a38da08geo4a-h4"],"title":"The other plot","text":"The shaded region (a trapezoid) can be considered as a rectangle and a triangle as shown in the other plot.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a38da08geo4a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{2}$$"],"dependencies":["a38da08geo4a-h5"],"title":"The other plot","text":"What is the length (L) of the rectangle? (the same value as the base (b) of the triangle)","variabilization":{},"oer":"https://OATutor.io","license":"","subHints":[{"id":"a38da08geo4a-h6-s1","type":"hint","dependencies":[],"title":"The other plot","text":"$$length=\\\\frac{5}{2}-1=\\\\frac{3}{2}$$","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a38da08geo4a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{6}$$"],"dependencies":["a38da08geo4a-h6"],"title":"The other plot","text":"What is the height (h1) of the rectangle?","variabilization":{},"oer":"https://OATutor.io","license":"","subHints":[{"id":"a38da08geo4a-h7-s1","type":"hint","dependencies":[],"title":"The other plot","text":"$$h1=\\\\frac{5}{6}-0=\\\\frac{5}{6}$$","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a38da08geo4a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a38da08geo4a-h7"],"title":"The other plot","text":"What is the height (h2) of the triangle?","variabilization":{},"oer":"https://OATutor.io","license":"","subHints":[{"id":"a38da08geo4a-h8-s1","type":"hint","dependencies":[],"title":"The other plot","text":"$$h2=\\\\frac{4}{3}-\\\\frac{5}{6}=\\\\frac{3}{6}=\\\\frac{1}{2}$$","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a38da08geo4a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{8}$$"],"dependencies":["a38da08geo4a-h8"],"title":"Area","text":"What is the area of the triangle?","variabilization":{},"oer":"https://OATutor.io","license":"","subHints":[{"id":"a38da08geo4a-h9-s1","type":"hint","dependencies":[],"title":"Area","text":"$$A=\\\\frac{1}{2} b h2=\\\\frac{1}{2} \\\\frac{3}{2} \\\\frac{1}{2}=\\\\frac{3}{8}$$","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a38da08geo4a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{4}$$"],"dependencies":["a38da08geo4a-h9"],"title":"Area","text":"What is the area of the rectangle?","variabilization":{},"oer":"https://OATutor.io","license":"","subHints":[{"id":"a38da08geo4a-h10-s1","type":"hint","dependencies":[],"title":"Area","text":"$$A=L h1=\\\\frac{3}{2} \\\\frac{5}{6}=\\\\frac{15}{12}=\\\\frac{5}{4}$$","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a38da08geo4a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{13}{8}$$"],"dependencies":["a38da08geo4a-h10"],"title":"Total Area","text":"What is the total area?","variabilization":{},"oer":"https://OATutor.io","license":"","subHints":[{"id":"a38da08geo4a-h11-s1","type":"hint","dependencies":[],"title":"Total Area","text":"$$\\\\frac{3}{8}+\\\\frac{5}{4}=\\\\frac{13}{8}$$","variabilization":{},"oer":"https://OATutor.io","license":""}]}]}}]},{"id":"a38da08geo5","title":"Geometry and Algebra","body":"These problems are generally harder, often highlighting an important subtlety","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Geometry and Algebra","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a38da08geo5a","stepAnswer":["$$w^2 h$$"],"problemType":"MultipleChoice","stepTitle":"A rectangular solid S with a square base, has height $$h$$ and width w. The surface area of S is $$32$$ square units. Express the volume of S (shown in the plot) as a formula involving $$h$$ and w.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$w^2 h$$","choices":["$$w^2 h$$","$$w h$$"],"hints":{"DefaultPathway":[{"id":"a38da08geo5a-h1","type":"hint","dependencies":[],"title":"Volumn Formula","text":"$$Volume=w w h$$","variabilization":{},"oer":"","license":""},{"id":"a38da08geo5a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$w^2 h$$"],"dependencies":["a38da08geo5a-h1"],"title":"Volumn Formula","text":"Which answer represents the volumn?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$w^2 h$$","$$w h$$"]}]}},{"id":"a38da08geo5b","stepAnswer":["$$2w^2+4w h=32$$"],"problemType":"MultipleChoice","stepTitle":"A rectangular solid S with a square base, has height $$h$$ and width w. The surface area of S is $$32$$ square units. Express the fact that the surface area is $$32$$ square units as a mathematical equation involving $$h$$ and w.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2w^2+4w h=32$$","choices":["$$2w^2+4w h=32$$","$$4w+4w h=32$$","$$4w^2+4w h=32$$","$$w^2+w h=32$$"],"hints":{"DefaultPathway":[{"id":"a38da08geo5b-h1","type":"hint","dependencies":[],"title":"Surface Area","text":"The surfact area can be considered as $$2Area$$ of the square $$base+4Area$$ of the Rectangle","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a38da08geo5b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$w^2$$"],"dependencies":["a38da08geo5b-h1"],"title":"Area of square base","text":"What is the area of a square base?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$w^2$$","$$2w$$"]},{"id":"a38da08geo5b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$w h$$"],"dependencies":["a38da08geo5b-h2"],"title":"Area of rectangle","text":"What is the area of a rectangle?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$w h$$","$$h^2$$","$$w^2$$"]},{"id":"a38da08geo5b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2w^2+4w h=32$$"],"dependencies":["a38da08geo5b-h3"],"title":"Expression","text":"Which one is the correct expression?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$2w^2+4w h=32$$","$$4w+4w h=32$$","$$4w^2+4w h=32$$","$$w^2+w h=32$$"]}]}},{"id":"a38da08geo5c","stepAnswer":["(0,4]"],"problemType":"MultipleChoice","stepTitle":"A rectangular solid S with a square base, has height $$h$$ and width w. The surface area of S is $$32$$ square units. By solving this equation in $$h$$, find an expression for the volume purely in terms of w. Describe, in interval notation, the values of w which are applicable to this problem.","stepBody":"","answerType":"string","variabilization":{},"choices":["(0,4]","[0,4]","$$(0,4)$$"],"hints":{"DefaultPathway":[{"id":"a38da08geo5c-h1","type":"hint","dependencies":[],"title":"When $$w=0$$","text":"If $$w=0$$, then $$2w^2+4w h=0 \\\\neq 32$$, so $$w \\\\neq 0$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a38da08geo5c-h2","type":"hint","dependencies":["a38da08geo5c-h1"],"title":"Simplification","text":"$$2w^2+4w h=32$$. $$w \\\\left(w+2h\\\\right)=16$$. $$\\\\frac{16}{w}=w+2h$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a38da08geo5c-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$h=\\\\frac{8}{w}-\\\\frac{w}{2}$$"],"dependencies":["a38da08geo5c-h2"],"title":"Simplification","text":"Express $$\\\\frac{16}{w}=w+2h$$ in the form of $$h$$. What is the result?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$h=\\\\frac{8}{w}+\\\\frac{w}{2}$$","$$h=\\\\frac{8}{w}-\\\\frac{w}{2}$$","$$h=\\\\frac{8}{w}-\\\\frac{2}{w}$$","$$h=\\\\frac{-8}{w}-\\\\frac{w}{2}$$"]},{"id":"a38da08geo5c-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$8w-\\\\frac{w^3}{2}$$"],"dependencies":["a38da08geo5c-h3"],"title":"Volume","text":"Substitute $$h=\\\\frac{8}{w}-\\\\frac{w}{2}$$ to $$V=w^2 h$$. What is the volume expressed in w only?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$8w-\\\\frac{w^3}{2}$$","$$8-\\\\frac{w^2}{2}$$"]},{"id":"a38da08geo5c-h5","type":"hint","dependencies":["a38da08geo5c-h4"],"title":"Range","text":"w must be nonnegative since it represents width, and since $$w \\\\neq 0$$, $$w>0$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a38da08geo5c-h6","type":"hint","dependencies":["a38da08geo5c-h5"],"title":"Range","text":"$$h \\\\geq 0$$, so $$\\\\frac{8}{w}-\\\\frac{w}{2} \\\\geq 0$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a38da08geo5c-h7","type":"hint","dependencies":["a38da08geo5c-h6"],"title":"Simplification","text":"Multiply w on both sides, we get $$8-\\\\frac{w^2}{2} \\\\geq 0$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a38da08geo5c-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$w \\\\leq 4$$"],"dependencies":["a38da08geo5c-h7"],"title":"Simplification","text":"What is the range of w?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$w \\\\leq 4$$","$$w \\\\leq -4$$","$$w \\\\geq 4$$","$$w \\\\geq -4$$"],"subHints":[{"id":"a38da08geo5c-h8-s1","type":"hint","dependencies":[],"title":"Simplification","text":"$$\\\\frac{w^2}{2} \\\\leq 8$$, $$w^2 \\\\leq 16$$. Since w >0, $$w \\\\leq 4$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a38da08geo5c-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["(0,4]"],"dependencies":["a38da08geo5c-h8"],"title":"Simplification","text":"What is $$w \\\\leq 4$$ in interval notation?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["(0,4]","[0,4]","$$(0,4)$$"]}]}},{"id":"a38da08geo5d","stepAnswer":["$$2h^3-2h^2 \\\\sqrt{h^2+16}+16h$$"],"problemType":"MultipleChoice","stepTitle":"A rectangular solid S with a square base, has height $$h$$ and width w. The surface area of S is $$32$$ square units. Would you run into any diculties trying to express the volume purely in terms of $$h$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2h^3-2h^2 \\\\sqrt{h^2+16}+16h$$","choices":["$$2h^3-2h^2 \\\\sqrt{h^2+16}+16h$$","$$2h^3+2h^2 \\\\sqrt{h^2+16}+16h$$","$$2h^3+2h^2 \\\\sqrt{h^2+16}-16h$$"],"hints":{"DefaultPathway":[{"id":"a38da08geo5d-h1","type":"hint","dependencies":[],"title":"Simplification","text":"$$2w^2+4w h=32$$. $$w \\\\left(w+2h\\\\right)=16$$, so $$w^2+2h w-16=0$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a38da08geo5d-h2","type":"hint","dependencies":["a38da08geo5d-h1"],"title":"Simplification","text":"Using the Quadratic Formula, $$w=\\\\frac{\\\\left(-2h\\\\pm \\\\sqrt{4h^2+64}\\\\right)}{2}=\\\\frac{\\\\left(-2h\\\\pm \\\\sqrt{4\\\\left(h^2+16\\\\right)}\\\\right)}{2}=-h\\\\pm \\\\sqrt{h^2+16}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a38da08geo5d-h3","type":"hint","dependencies":["a38da08geo5d-h2"],"title":"Comparison","text":"$$h^2+16>h^2$$ is equivalent to $$\\\\sqrt{h^2+16}>\\\\sqrt{h^2}=h$$. Thus, $$-h-\\\\sqrt{h^2+16}<0$$. Since $$w \\\\geq 0$$, the only solution is $$w=-h+\\\\sqrt{h^2+16}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a38da08geo5d-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2h^3-2h^2 \\\\sqrt{h^2+16}+16h$$"],"dependencies":["a38da08geo5d-h3"],"title":"Volume","text":"Substitute $$w=-h+\\\\sqrt{h^2+16}$$ into $$V=w^2 h$$, what is V expressed in $$h$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$2h^3-2h^2 \\\\sqrt{h^2+16}+16h$$","$$2h^3+2h^2 \\\\sqrt{h^2+16}+16h$$","$$2h^3+2h^2 \\\\sqrt{h^2+16}-16h$$"],"subHints":[{"id":"a38da08geo5d-h4-s1","type":"hint","dependencies":[],"title":"Volumn Formula","text":"$$V=w^2 h=h {\\\\left(-h+\\\\sqrt{h^2+16}\\\\right)}^2=h \\\\left(h^2-2h \\\\sqrt{h^2+16}+h^2+16\\\\right)=2h^3-2h^2 \\\\sqrt{h^2+16}+16h$$","variabilization":{},"oer":"https://OATutor.io","license":""}]}]}}]},{"id":"a38da08geo6","title":"Geometry and Algebra","body":"These problems are generally harder, often highlighting an important subtlety","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Geometry and Algebra","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a38da08geo6a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"On the following diagram, shade the following: (a) The points (x, y) such that $$|y-1| \\\\leq \\\\frac{1}{2}$$. (b) The points (x, y) such that $$|x+2| \\\\leq \\\\frac{1}{2}$$. (c) The points (x, y) such that $$|y-1| \\\\leq \\\\frac{1}{2}$$ and $$|x+2| \\\\leq \\\\frac{1}{2}$$. Does the graph correctly show the information?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a38da08geo6a-h1","type":"hint","dependencies":[],"title":"T or F","text":"See the graph.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a38da08geo6a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a38da08geo6a-h1"],"title":"T or F","text":"Does the graph correctly show the information?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]}]}}]},{"id":"a38da08geo7","title":"Geometry and Algebra","body":"These problems are generally harder, often highlighting an important subtlety","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Geometry and Algebra","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a38da08geo7a","stepAnswer":["$$11.6$$"],"problemType":"MultipleChoice","stepTitle":"A solid shape has volume $$1000$$ $${cm}^3$$. The solid is then scaled until it has volume $$10$$ $${cm}^3$$. If the original solid had surface area $$250$$ $${cm}^2$$, what is the approximate surface area of the scaled solid?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$11.6$$","choices":["$$10$$","$$11.6$$","$$12.5$$"],"hints":{"DefaultPathway":[{"id":"a38da08geo7a-h1","type":"hint","dependencies":[],"title":"Define k","text":"Scaling by factor k multiplies volumes by $$k^3$$ and areas by $$k^2$$.","variabilization":{},"oer":"","license":""},{"id":"a38da08geo7a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${10}^{\\\\left(-\\\\frac{2}{3}\\\\right)}$$"],"dependencies":["a38da08geo7a-h1"],"title":"Define k","text":"$$1000k^3=10$$. What is k?","variabilization":{},"oer":"","license":"","choices":["$${10}^{\\\\left(-\\\\frac{2}{3}\\\\right)}$$","$${10}^{\\\\left(-2\\\\right)}$$"],"subHints":[{"id":"a38da08geo7a-h2-s1","type":"hint","dependencies":[],"title":"Define k","text":"$$k^3=\\\\frac{10}{1000}={10}^{\\\\left(-2\\\\right)}$$, so $$k={10}^{\\\\left(-\\\\frac{2}{3}\\\\right)}$$","variabilization":{},"oer":"","license":""}]},{"id":"a38da08geo7a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${10}^{\\\\left(-\\\\frac{4}{3}\\\\right)}$$"],"dependencies":["a38da08geo7a-h2"],"title":"Computation","text":"What is $$k^2$$?","variabilization":{},"oer":"","license":"","choices":["$${10}^{\\\\left(-\\\\frac{4}{3}\\\\right)}$$","$${10}^{\\\\left(-\\\\frac{3}{4}\\\\right)}$$"],"subHints":[{"id":"a38da08geo7a-h3-s1","type":"hint","dependencies":[],"title":"Computation","text":"$$k^2={10}^{\\\\left(-\\\\frac{4}{3}\\\\right)}$$","variabilization":{},"oer":"","license":""}]},{"id":"a38da08geo7a-h4","type":"hint","dependencies":["a38da08geo7a-h3"],"title":"Computation","text":"New surface $$area=250k^2=250\\\\times {10}^{\\\\left(-\\\\frac{4}{3}\\\\right)}$$.","variabilization":{},"oer":"","license":""},{"id":"a38da08geo7a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$11.6$$"],"dependencies":["a38da08geo7a-h4"],"title":"Computation","text":"What is the approximate value of the new surface area?","variabilization":{},"oer":"","license":"","choices":["$$10$$","$$11.6$$","$$12.5$$"]}]}}]},{"id":"a38da08geo8","title":"Geometry and Algebra","body":"These questions are challenging, requiring mastery of each concept and their interrelations.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Geometry and Algebra","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a38da08geo8a","stepAnswer":["All values in R"],"problemType":"MultipleChoice","stepTitle":"In the following diagram, the curve is given by the equation $$y$$ $$=$$ $$x^2$$. Express the distance between (a, b), a point on the curve, and the point $$(2,1)$$ as an expression purely in terms of a. 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What values of $$b$$ are applicable to the problem? Hint: Look for a relationship between a and $$b$$ using appropriate similar triangles.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["[0,4]","[0,4)","(0,4]","$$(0,4)$$","R"],"hints":{"DefaultPathway":[{"id":"a38da08geo9a-h1","type":"hint","dependencies":[],"title":"From the plot","text":"Note that triangle1 is similar to triangle2 by AAA\\\\n##figure2.gif##","variabilization":{},"oer":"","license":""},{"id":"a38da08geo9a-h2","type":"hint","dependencies":["a38da08geo9a-h1"],"title":"From the plot","text":"Ratio of lengths of similar shapes are equal, so $$\\\\frac{b}{4-b}=\\\\frac{3-a}{a}$$","variabilization":{},"oer":"","license":""},{"id":"a38da08geo9a-h3","type":"hint","dependencies":["a38da08geo9a-h2"],"title":"Simplification","text":"$$\\\\frac{b}{4-b}=\\\\frac{3-a}{a}$$ is equivalent to $$a=\\\\frac{3}{4} \\\\left(4-b\\\\right)$$ by simplification.","variabilization":{},"oer":"","license":""},{"id":"a38da08geo9a-h4","type":"hint","dependencies":["a38da08geo9a-h3"],"title":"Area of rectangle","text":"The area of rectangle is $$a b$$.","variabilization":{},"oer":"","license":""},{"id":"a38da08geo9a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3b-\\\\frac{3}{4} b^2$$"],"dependencies":["a38da08geo9a-h4"],"title":"Area of rectangle","text":"Substitute $$a=\\\\frac{3}{4} \\\\left(4-b\\\\right)$$ into $$a b$$, what is the area expressed in the form of $$b$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$3b-\\\\frac{3}{4} b^2$$","$$\\\\frac{3}{4} b-\\\\frac{3}{4} b^2$$"]},{"id":"a38da08geo9a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a38da08geo9a-h5"],"title":"Range of a, $$b$$","text":"Can a, $$b$$ be negative?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"],"subHints":[{"id":"a38da08geo9a-h6-s1","type":"hint","dependencies":[],"title":"Range of a, $$b$$","text":"The length of a and $$b$$ should be nonnegative, so $$a \\\\geq 0$$ and $$b \\\\geq 0$$","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a38da08geo9a-h7","type":"hint","dependencies":["a38da08geo9a-h6"],"title":"Simplification","text":"$$a \\\\geq 0$$ implies $$\\\\frac{3}{4} \\\\left(4-b\\\\right) \\\\geq 0$$. Simplify this inequality.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a38da08geo9a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$b \\\\leq 4$$"],"dependencies":["a38da08geo9a-h7"],"title":"Simplification","text":"What is the result of the simplification?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$b \\\\leq 4$$","$$b \\\\leq -4$$","$$b \\\\geq 4$$","$$b>-4$$","$$b<4$$"],"subHints":[{"id":"a38da08geo9a-h8-s1","type":"hint","dependencies":[],"title":"Simplification","text":"$$\\\\frac{3}{4} \\\\left(4-b\\\\right) \\\\geq 0$$. $$(4-b) \\\\geq 0$$. So $$b \\\\leq 4$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a38da08geo9a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["[0,4]"],"dependencies":["a38da08geo9a-h8"],"title":"Values of $$b$$","text":"What values of $$b$$ are applicable to the problem? (note $$b \\\\geq 0)$$","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["[0,4]","[0,4)","(0,4]","$$(0,4)$$","R"]}]}}]},{"id":"a391214Sequ1","title":"Write some terms of a sequence.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.1 Sequences","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a391214Sequ1a","stepAnswer":["1, 5, 9, 13, 17"],"problemType":"TextBox","stepTitle":"Write some terms of a sequence.","stepBody":"Write the first five terms of the sequence whose general term is $$an=4n-3$$. 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What should the general term be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a391214Sequ11","title":"Find a general term for the sequence","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.1 Sequences","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a391214Sequ11a","stepAnswer":["an $$=$$ $$3n$$"],"problemType":"TextBox","stepTitle":"Find a general term for the sequence (Note: Answer in $$(an=\\"$$ \\") format.)","stepBody":"Find a general term for the sequence whose first five terms are shown. $$3$$, $$6$$, $$9$$, $$12$$, $$15$$, \u2026 (Please enter your answer in the form \\"an $$=$$ x\\")","answerType":"arithmetic","variabilization":{},"answerLatex":"an $$=$$ $$3n$$","hints":{"DefaultPathway":[{"id":"a391214Sequ11a-h1","type":"hint","dependencies":[],"title":"Find a pattern","text":"We look for a pattern in the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a391214Sequ11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["an $$=$$ $$3n$$"],"dependencies":["a391214Sequ11a-h1"],"title":"Find the result","text":"The numbers are all multiples of $$3$$. What should the general term be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a391214Sequ12","title":"Find a general term for the sequence","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.1 Sequences","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a391214Sequ12a","stepAnswer":["an $$=$$ $$5n$$"],"problemType":"TextBox","stepTitle":"Find a general term for the sequence (Note: Answer in $$(an=\\"$$ \\") format.)","stepBody":"Find a general term for the sequence whose first five terms are shown. $$5$$, $$10$$, $$15$$, $$20$$, $$25$$, \u2026","answerType":"arithmetic","variabilization":{},"answerLatex":"an $$=$$ $$5n$$","hints":{"DefaultPathway":[{"id":"a391214Sequ12a-h1","type":"hint","dependencies":[],"title":"Find a pattern","text":"We look for a pattern in the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a391214Sequ12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["an $$=$$ $$5n$$"],"dependencies":["a391214Sequ12a-h1"],"title":"Find the result","text":"The numbers are all multiples of $$5$$. What should the general term be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a391214Sequ13","title":"Find a general term for the sequence","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.1 Sequences","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a391214Sequ13a","stepAnswer":["an $$=$$ $${\\\\left(-1\\\\right)}^{n+1}$$ * $$2^n$$"],"problemType":"TextBox","stepTitle":"Find a general term for the sequence (Note: Answer in $$(an=\\"$$ \\") format.)","stepBody":"Find a general term for the sequence whose first five terms are shown. $$2$$, $$-4$$, $$8$$, $$-16$$, $$32$$","answerType":"arithmetic","variabilization":{},"answerLatex":"an $$=$$ $${\\\\left(-1\\\\right)}^{n+1}$$ * $$2^n$$","hints":{"DefaultPathway":[{"id":"a391214Sequ13a-h1","type":"hint","dependencies":[],"title":"Find a pattern","text":"We look for a pattern in the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a391214Sequ13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["an $$=$$ $${\\\\left(-1\\\\right)}^{n+1}$$ * $$2^n$$"],"dependencies":["a391214Sequ13a-h1"],"title":"Find the result","text":"The numbers are powers of $$2$$. The signs are alternating, with even $$n$$ negative.. What should the general term be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a391214Sequ14","title":"Find a general term for the sequence","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.1 Sequences","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a391214Sequ14a","stepAnswer":["an $$=$$ $${\\\\left(-1\\\\right)}^n$$ * $$3^n$$"],"problemType":"TextBox","stepTitle":"Find a general term for the sequence (Note: Answer in $$(an=\\"$$ \\") format.)","stepBody":"Find a general term for the sequence whose first five terms are shown. $$-3$$, $$9$$, $$-27$$, $$81$$, $$-243$$, \u2026","answerType":"arithmetic","variabilization":{},"answerLatex":"an $$=$$ $${\\\\left(-1\\\\right)}^n$$ * $$3^n$$","hints":{"DefaultPathway":[{"id":"a391214Sequ14a-h1","type":"hint","dependencies":[],"title":"Find a pattern","text":"We look for a pattern in the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a391214Sequ14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["an $$=$$ $${\\\\left(-1\\\\right)}^n$$ * $$3^n$$"],"dependencies":["a391214Sequ14a-h1"],"title":"Find the result","text":"The numbers are powers of $$3$$. The signs are alternating, with even $$n$$ positive.. What should the general term be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a391214Sequ15","title":"Find a general term for the sequence","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.1 Sequences","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a391214Sequ15a","stepAnswer":["a(n) $$=$$ $${\\\\left(-1\\\\right)}^{n+1}$$ * $$n^2$$"],"problemType":"TextBox","stepTitle":"Find a general term for the sequence (Note: Answer in $$(an=\\"$$ \\") format.)","stepBody":"Find a general term for the sequence whose first five terms are shown. $$1$$, $$-4$$, $$9$$, $$-16$$, $$25$$, \u2026","answerType":"arithmetic","variabilization":{},"answerLatex":"a(n) $$=$$ $${\\\\left(-1\\\\right)}^{n+1}$$ * $$n^2$$","hints":{"DefaultPathway":[{"id":"a391214Sequ15a-h1","type":"hint","dependencies":[],"title":"Find a pattern","text":"We look for a pattern in the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a391214Sequ15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["an $$=$$ $${\\\\left(-1\\\\right)}^{n+1}$$ * $$n^2$$"],"dependencies":["a391214Sequ15a-h1"],"title":"Find the result","text":"The numbers are powers of $$n$$. The signs are alternating, with even $$n$$ negative.. What should the general term be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a391214Sequ16","title":"Find a general term for the sequence","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.1 Sequences","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a391214Sequ16a","stepAnswer":["an $$=$$ $$\\\\frac{1}{3^n}$$"],"problemType":"TextBox","stepTitle":"Find a general term for the sequence (Note: Answer in $$(an=\\"$$ \\") format.)","stepBody":"Find a general term for the sequence whose first five terms are shown. $$\\\\frac{1}{3}$$, $$\\\\frac{1}{9}$$, $$\\\\frac{1}{27}$$, $$\\\\frac{1}{81}$$, $$\\\\frac{1}{243}$$, \u2026","answerType":"arithmetic","variabilization":{},"answerLatex":"an $$=$$ $$\\\\frac{1}{3^n}$$","hints":{"DefaultPathway":[{"id":"a391214Sequ16a-h1","type":"hint","dependencies":[],"title":"Find a pattern","text":"We look for a pattern in the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a391214Sequ16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["an $$=$$ $$\\\\frac{1}{3^n}$$"],"dependencies":["a391214Sequ16a-h1"],"title":"Find the result","text":"The numerators are all $$1$$. The denominators are powers of $$3$$. 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<OATutor>","license":"","choices":["$$-5sin\\\\left(x\\\\right)$$","$$-5cos\\\\left(x\\\\right)$$","$$5sin\\\\left(x\\\\right)$$","$$5cos\\\\left(x\\\\right)$$"]},{"id":"a394625deriv24a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$5cos\\\\left(x\\\\right)$$"],"dependencies":["a394625deriv24a-h1"],"title":"Sub-question","text":"What\'s the derivative of the third derivative?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$-5sin\\\\left(x\\\\right)$$","$$-5cos\\\\left(x\\\\right)$$","$$5sin\\\\left(x\\\\right)$$","$$5cos\\\\left(x\\\\right)$$"]},{"id":"a394625deriv24a-h4","type":"hint","dependencies":["a394625deriv24a-h3"],"title":"The derivative of the $$n-1th$$ derivative is the nth derivative","text":"Conclude your answer from sub-questions","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a394625deriv25","title":"Derivatives of Trigonometric Functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1","license":0,"lesson":"3.5 Derivatives of Trigonometric Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a394625deriv25a","stepAnswer":["$$\\\\frac{\\\\pi}{6}$$, $$\\\\frac{5\\\\pi}{6}$$"],"problemType":"MultipleChoice","stepTitle":"Find all $$x$$ values on the graph of $$f(x)=x-2cos\\\\left(x\\\\right)$$ for $$0<x<2\\\\pi$$ where the tangent line has slope $$2$$.","stepBody":"$$f(x)=x-2cos\\\\left(x\\\\right)$$","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{\\\\pi}{6}$$, $$\\\\frac{5\\\\pi}{6}$$","choices":["$$\\\\frac{\\\\pi}{6}$$, $$\\\\frac{5\\\\pi}{6}$$","$$\\\\frac{5\\\\pi}{6}$$","$$\\\\frac{\\\\pi}{6}$$"],"hints":{"DefaultPathway":[{"id":"a394625deriv25a-h1","type":"hint","dependencies":[],"title":"Breaking down the question","text":"The derivative of the function is the slope of the tangent 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be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a394625deriv25a-h4","type":"hint","dependencies":["a394625deriv25a-h3"],"title":"$$sin(x)=0.5$$ find specific $$x$$","text":"$$x$$ is from $$0$$ to $$2\\\\pi$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a394625deriv25a-h5","type":"hint","dependencies":["a394625deriv25a-h4"],"title":"The derivative of the function is the slope of the tangent line","text":"Conclude your answer from sub-questions","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a394625deriv3","title":"Derivatives of Trigonometric Functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1","license":0,"lesson":"3.5 Derivatives of Trigonometric Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a394625deriv3a","stepAnswer":["$$2x \\\\operatorname{cot}\\\\left(x\\\\right)-x^2 {\\\\operatorname{csc2}\\\\left(x\\\\right)}^2$$"],"problemType":"MultipleChoice","stepTitle":"For the following exercises, find $$\\\\frac{dy}{dx}$$ for the given functions.","stepBody":"$$y=x^2 \\\\operatorname{cot}\\\\left(x\\\\right)$$","answerType":"string","variabilization":{},"answerLatex":"$$2x \\\\operatorname{cot}\\\\left(x\\\\right)-x^2 {\\\\operatorname{csc2}\\\\left(x\\\\right)}^2$$","choices":["$$2x \\\\operatorname{cot}\\\\left(x\\\\right)-x^2 {\\\\operatorname{csc2}\\\\left(x\\\\right)}^2$$","$$2x \\\\operatorname{cot}\\\\left(x\\\\right)$$","$$2x \\\\operatorname{csc}\\\\left(x\\\\right)$$"],"hints":{"DefaultPathway":[{"id":"a394625deriv3a-h1","type":"hint","dependencies":[],"title":"Breaking down the question","text":"We can separate the functions into $$x^2$$ and cot(x)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a394625deriv3a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2x$$"],"dependencies":["a394625deriv3a-h1"],"title":"Sub-question","text":"What\'s the derivative of $$x^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$2x$$","$$x$$","$$2$$"]},{"id":"a394625deriv3a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-\\\\left({\\\\operatorname{csc}\\\\left(x\\\\right)}^2\\\\right)$$"],"dependencies":["a394625deriv3a-h2"],"title":"Sub-question","text":"What\'s the derivative of cot(x)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$-\\\\left({\\\\operatorname{csc}\\\\left(x\\\\right)}^2\\\\right)$$","csc(x)","$${\\\\operatorname{csc}\\\\left(x\\\\right)}^2$$"]},{"id":"a394625deriv3a-h4","type":"hint","dependencies":["a394625deriv3a-h3"],"title":"Apply derivatives\' rule","text":"Apply the derivative rule of composed functions to find the final solution","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a394625deriv4","title":"Derivatives of Trigonometric Functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1","license":0,"lesson":"3.5 Derivatives of Trigonometric Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a394625deriv4a","stepAnswer":["$$1-3x^2 sin\\\\left(x\\\\right)-x^3 cos\\\\left(x\\\\right)$$"],"problemType":"MultipleChoice","stepTitle":"For the following exercises, find $$\\\\frac{dy}{dx}$$ for the given functions.","stepBody":"$$y=x-x^3 sin\\\\left(x\\\\right)$$","answerType":"string","variabilization":{},"answerLatex":"$$1-3x^2 sin\\\\left(x\\\\right)-x^3 cos\\\\left(x\\\\right)$$","choices":["$$1-3x^2 sin\\\\left(x\\\\right)-x^3 cos\\\\left(x\\\\right)$$","$$1-3x sin\\\\left(x\\\\right)$$","$$1-3sin(x)$$","$$1-3x^2 sin\\\\left(x\\\\right)-x^3 cos\\\\left(x\\\\right)$$"],"hints":{"DefaultPathway":[{"id":"a394625deriv4a-h1","type":"hint","dependencies":[],"title":"Breaking down the question","text":"We can separate the hard functions into $$x^3$$ and $$3sin\\\\left(x\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a394625deriv4a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3x^2$$"],"dependencies":["a394625deriv4a-h1"],"title":"Sub-question","text":"What\'s the derivative of $$x^3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$3x^2$$","$$3x$$","$$3$$"]},{"id":"a394625deriv4a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["cos(x)"],"dependencies":["a394625deriv4a-h2"],"title":"Sub-question","text":"What\'s the derivative of sin(x)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["cos(x)","tan(x)","sin(x)"]},{"id":"a394625deriv4a-h4","type":"hint","dependencies":["a394625deriv4a-h3"],"title":"Apply derivatives\' rule","text":"Apply the derivative rule of composed functions to find the final solution","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a394625deriv5","title":"Derivatives of Trigonometric Functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1","license":0,"lesson":"3.5 Derivatives of Trigonometric Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a394625deriv5a","stepAnswer":["$$\\\\frac{x \\\\operatorname{sec}\\\\left(x\\\\right) tan\\\\left(x\\\\right)-\\\\operatorname{sec}\\\\left(x\\\\right)}{x^2}$$"],"problemType":"MultipleChoice","stepTitle":"For the following exercises, find $$\\\\frac{dy}{dx}$$ for the given functions.","stepBody":"$$y=\\\\frac{\\\\operatorname{sec}\\\\left(x\\\\right)}{x}$$","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{x \\\\operatorname{sec}\\\\left(x\\\\right) tan\\\\left(x\\\\right)-\\\\operatorname{sec}\\\\left(x\\\\right)}{x^2}$$","choices":["$$\\\\frac{x \\\\operatorname{sec}\\\\left(x\\\\right) tan\\\\left(x\\\\right)-\\\\operatorname{sec}\\\\left(x\\\\right)}{x^2}$$","sec(x)","$$\\\\operatorname{sec}\\\\left(x\\\\right) tan\\\\left(x\\\\right)$$"],"hints":{"DefaultPathway":[{"id":"a394625deriv5a-h1","type":"hint","dependencies":[],"title":"Breaking down the question","text":"We can separate the functions into sec(x) and $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a394625deriv5a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\operatorname{sec}\\\\left(x\\\\right) tan\\\\left(x\\\\right)$$"],"dependencies":["a394625deriv5a-h1"],"title":"Sub-question","text":"What\'s the derivative of sec(x)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\operatorname{sec}\\\\left(x\\\\right) tan\\\\left(x\\\\right)$$","sec(x)","tan(x)"]},{"id":"a394625deriv5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a394625deriv5a-h2"],"title":"Sub-question","text":"What\'s the derivative of $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a394625deriv5a-h4","type":"hint","dependencies":["a394625deriv5a-h3"],"title":"Apply derivatives\' rule","text":"Apply the quotient derivative rule of composed functions to find the final solution","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a394625deriv6","title":"Derivatives of Trigonometric Functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1","license":0,"lesson":"3.5 Derivatives of Trigonometric Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a394625deriv6a","stepAnswer":["$${\\\\operatorname{sec}\\\\left(x\\\\right)}^2 sin\\\\left(x\\\\right)$$ + $$cos\\\\left(x\\\\right) tan\\\\left(x\\\\right)$$"],"problemType":"MultipleChoice","stepTitle":"For the following exercises, find $$\\\\frac{dy}{dx}$$ for the given functions.","stepBody":"$$y=sin\\\\left(x\\\\right) tan\\\\left(x\\\\right)$$","answerType":"string","variabilization":{},"answerLatex":"$${\\\\operatorname{sec}\\\\left(x\\\\right)}^2 sin\\\\left(x\\\\right)$$ + $$cos\\\\left(x\\\\right) tan\\\\left(x\\\\right)$$","choices":["$${\\\\operatorname{sec}\\\\left(x\\\\right)}^2 sin\\\\left(x\\\\right)$$ + $$cos\\\\left(x\\\\right) tan\\\\left(x\\\\right)$$","tan(x)","$$cos\\\\left(x\\\\right) tan\\\\left(x\\\\right)$$"],"hints":{"DefaultPathway":[{"id":"a394625deriv6a-h1","type":"hint","dependencies":[],"title":"Breaking down the question","text":"We can separate the functions into sin(x) and tan(x)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a394625deriv6a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["cos(x)"],"dependencies":["a394625deriv6a-h1"],"title":"Sub-question","text":"What\'s the derivative of sin(x)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["cos(x)","sin(x)","tan(x)"]},{"id":"a394625deriv6a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${\\\\operatorname{sec}\\\\left(x\\\\right)}^2$$"],"dependencies":["a394625deriv6a-h2"],"title":"Sub-question","text":"What\'s the derivative of tan(x)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$tan^2\\\\left(x\\\\right)$$","csc(x)","$${\\\\operatorname{sec}\\\\left(x\\\\right)}^2$$"]},{"id":"a394625deriv6a-h4","type":"hint","dependencies":["a394625deriv6a-h3"],"title":"Apply derivatives\' rule","text":"Apply the product derivative rule of composed functions to find the final solution","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a394625deriv7","title":"Derivatives of Trigonometric Functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1","license":0,"lesson":"3.5 Derivatives of Trigonometric Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a394625deriv7a","stepAnswer":["$$\\\\left(1-sin\\\\left(x\\\\right)\\\\right) \\\\left(x-sin\\\\left(x\\\\right)-cos\\\\left(x\\\\right)\\\\right)$$"],"problemType":"MultipleChoice","stepTitle":"For the following exercises, find $$\\\\frac{dy}{dx}$$ for the given functions.","stepBody":"$$y=\\\\left(x+cos\\\\left(x\\\\right)\\\\right) \\\\left(1-sin\\\\left(x\\\\right)\\\\right)$$","answerType":"string","variabilization":{},"answerLatex":"$$\\\\left(1-sin\\\\left(x\\\\right)\\\\right) \\\\left(x-sin\\\\left(x\\\\right)-cos\\\\left(x\\\\right)\\\\right)$$","choices":["$$1+cos\\\\left(x\\\\right)$$","$$1-sin(x)$$","$$x cos\\\\left(x\\\\right)$$","$$\\\\left(1-sin\\\\left(x\\\\right)\\\\right) \\\\left(x-sin\\\\left(x\\\\right)-cos\\\\left(x\\\\right)\\\\right)$$"],"hints":{"DefaultPathway":[{"id":"a394625deriv7a-h1","type":"hint","dependencies":[],"title":"Breaking down the question","text":"We can separate the functions into $$x+cos\\\\left(x\\\\right)$$ and $$1-sin(x)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a394625deriv7a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$1-sin(x)$$"],"dependencies":["a394625deriv7a-h1"],"title":"Sub-question","text":"What\'s the derivative of $$x+cos\\\\left(x\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$1-sin(x)$$","cos(x)","sin(x)"]},{"id":"a394625deriv7a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-cos(x)$$"],"dependencies":["a394625deriv7a-h2"],"title":"Sub-question","text":"What\'s the derivative of $$1-sin(x)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["sin(x)","$$-cos(x)$$","$$-sin(x)$$"]},{"id":"a394625deriv7a-h4","type":"hint","dependencies":["a394625deriv7a-h3"],"title":"Apply derivatives\' rule","text":"Apply the product derivative rule of composed functions to find the final solution","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a394625deriv8","title":"Derivatives of Trigonometric Functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1","license":0,"lesson":"3.5 Derivatives of Trigonometric Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a394625deriv8a","stepAnswer":["$$\\\\frac{-\\\\left({\\\\operatorname{sec}\\\\left(x\\\\right)}^2\\\\right) tan\\\\left(x\\\\right)}{{\\\\left(1-\\\\operatorname{sec}\\\\left(x\\\\right)\\\\right)}^2}$$"],"problemType":"MultipleChoice","stepTitle":"For the following exercises, find $$\\\\frac{dy}{dx}$$ for the given functions.","stepBody":"$$y=\\\\fractan^1\\\\left(x\\\\right)-\\\\operatorname{sec}\\\\left(x\\\\right)}$$","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{-\\\\left({\\\\operatorname{sec}\\\\left(x\\\\right)}^2\\\\right) tan\\\\left(x\\\\right)}{{\\\\left(1-\\\\operatorname{sec}\\\\left(x\\\\right)\\\\right)}^2}$$","choices":["sec(x)","$$\\\\fractan^o\\\\left(x\\\\right)s^$\\\\left(x\\\\righ$$","$$\\\\frac{-\\\\left({\\\\operatorname{sec}\\\\left(x\\\\right)}^2\\\\right) tan\\\\left(x\\\\right)}{{\\\\left(1-\\\\operatorname{sec}\\\\left(x\\\\right)\\\\right)}^2}$$"],"hints":{"DefaultPathway":[{"id":"a394625deriv8a-h1","type":"hint","dependencies":[],"title":"Breaking down the question","text":"We can separate the functions into tan(x) and $$1-sec(x)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a394625deriv8a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${\\\\operatorname{sec}\\\\left(x\\\\right)}^2$$"],"dependencies":["a394625deriv8a-h1"],"title":"Sub-question","text":"What\'s the derivative of tan(x)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["sec(x)","csc(x)","$${\\\\operatorname{sec}\\\\left(x\\\\right)}^2$$"]},{"id":"a394625deriv8a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-\\\\operatorname{sec}\\\\left(x\\\\right) tan\\\\left(x\\\\right)$$"],"dependencies":["a394625deriv8a-h2"],"title":"Sub-question","text":"What\'s the derivative of $$1-sec(x)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$-\\\\operatorname{sec}\\\\left(x\\\\right) tan\\\\left(x\\\\right)$$","$$\\\\operatorname{sec}\\\\left(x\\\\right) tan\\\\left(x\\\\right)$$","sec(x)"]},{"id":"a394625deriv8a-h4","type":"hint","dependencies":["a394625deriv8a-h3"],"title":"Apply derivatives\' rule","text":"Apply the quotient derivative rule of composed functions to find the final solution","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a394625deriv9","title":"Derivatives of Trigonometric Functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1","license":0,"lesson":"3.5 Derivatives of Trigonometric Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a394625deriv9a","stepAnswer":["$$\\\\frac{2{\\\\operatorname{csc}\\\\left(x\\\\right)}^2}{{\\\\left(1+\\\\operatorname{cot}\\\\left(x\\\\right)\\\\right)}^2}$$"],"problemType":"MultipleChoice","stepTitle":"For the following exercises, find $$\\\\frac{dy}{dx}$$ for the given functions.","stepBody":"$$y=\\\\frac{1-\\\\operatorname{cot}\\\\left(x\\\\right)}{1+\\\\operatorname{cot}\\\\left(x\\\\right)}$$","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{2{\\\\operatorname{csc}\\\\left(x\\\\right)}^2}{{\\\\left(1+\\\\operatorname{cot}\\\\left(x\\\\right)\\\\right)}^2}$$","choices":["$$\\\\frac{2\\\\operatorname{csc}\\\\left(x\\\\right)}{{\\\\left(1+\\\\operatorname{cot}\\\\left(x\\\\right)\\\\right)}^2}$$","$$\\\\frac{2{\\\\operatorname{csc}\\\\left(x\\\\right)}^2}{1+\\\\operatorname{cot}\\\\left(x\\\\right)}$$","$$\\\\frac{2{\\\\operatorname{csc}\\\\left(x\\\\right)}^2}{{\\\\left(1+\\\\operatorname{cot}\\\\left(x\\\\right)\\\\right)}^2}$$"],"hints":{"DefaultPathway":[{"id":"a394625deriv9a-h1","type":"hint","dependencies":[],"title":"Breaking down the question","text":"We can separate the functions into $$1-cot(x)$$ and $$1+\\\\operatorname{cot}\\\\left(x\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a394625deriv9a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-\\\\left({\\\\operatorname{csc}\\\\left(x\\\\right)}^2\\\\right)$$"],"dependencies":["a394625deriv9a-h1"],"title":"Sub-question","text":"What\'s the derivative of $$1-cot(x)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$-\\\\left({\\\\operatorname{csc}\\\\left(x\\\\right)}^2\\\\right)$$","$${\\\\operatorname{csc}\\\\left(x\\\\right)}^2$$","csc(x)"]},{"id":"a394625deriv9a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${\\\\operatorname{csc}\\\\left(x\\\\right)}^2$$"],"dependencies":["a394625deriv9a-h2"],"title":"Sub-question","text":"What\'s the derivative of $$1+\\\\operatorname{cot}\\\\left(x\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$-\\\\left({\\\\operatorname{csc}\\\\left(x\\\\right)}^2\\\\right)$$","$${\\\\operatorname{csc}\\\\left(x\\\\right)}^2$$","csc(x)"]},{"id":"a394625deriv9a-h4","type":"hint","dependencies":["a394625deriv9a-h3"],"title":"Apply derivatives\' rule","text":"Apply the quotient derivative rule of composed functions to find the final solution","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a39b8a0def1","title":"Finding the Secant","body":"","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"3.1 Defining the Derivative","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a39b8a0def1a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"Find the slope of the secant line between the values $$x_1$$ and $$x_2$$ for $$f(x)=4x+7$$, where $$x_1=2$$ and $$x_2=5$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a39b8a0def1a-h1","type":"hint","dependencies":[],"title":"Secant Line Equation","text":"To find the secant line between two points, use the equation $$\\\\frac{f{\\\\left(x\\\\right)}-f{\\\\left(a\\\\right)}}{x-a}$$.","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def1a-h2","type":"hint","dependencies":["a39b8a0def1a-h1"],"title":"Plugging In","text":"In the previous equation, plug the values of $$x_2$$ and $$x_1$$ into $$x$$ and a respectively.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a39b8a0def10","title":"Finding Slope of the Tangent","body":"","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"3.1 Defining the Derivative","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a39b8a0def10a","stepAnswer":["$$-2$$"],"problemType":"TextBox","stepTitle":"Find the slope of the tangent line $$m=f\'(a)$$, where $$f(x)=\\\\frac{2}{3+x}$$ and $$a=-4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-2$$","hints":{"DefaultPathway":[{"id":"a39b8a0def10a-h1","type":"hint","dependencies":[],"title":"Tangent Line Equation","text":"The tangent line to f(x) at a is the line passing through the point (a,f(a)) having slope $$\\\\lim_{h\\\\to0} \\\\frac{f{\\\\left(a+h\\\\right)}-f{\\\\left(a\\\\right)}}{h}$$.","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def10a-h2","type":"hint","dependencies":["a39b8a0def10a-h1"],"title":"Plugging In","text":"As $$a=2$$ in this equation, plug $$-4$$ into every a.","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def10a-h3","type":"hint","dependencies":["a39b8a0def10a-h2"],"title":"Simplify","text":"Simplify the resulting equation: $$\\\\frac{\\\\frac{2}{h-4+3}-\\\\frac{2}{3-4}}{h}$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"a39b8a0def11","title":"Finding the Derivative","body":"","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"3.1 Defining the Derivative","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a39b8a0def11a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"For the function $$y=f(x)=5x+4$$, find f\u2032(a), where $$a=-1$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"a39b8a0def11a-h1","type":"hint","dependencies":[],"title":"Derivative Equation","text":"The derivative of the function f(x) at a is defined by /lim{x,a}(f(x)-f(a))/(x-a).","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def11a-h2","type":"hint","dependencies":["a39b8a0def11a-h1"],"title":"Plugging In","text":"Plug $$-1$$ into a and expand the equation.","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def11a-h3","type":"hint","dependencies":["a39b8a0def11a-h2"],"title":"Simplify","text":"Simplify the resulting equation: $$\\\\frac{5x+4-5\\\\left(-1\\\\right)+4}{x+1}$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"a39b8a0def12","title":"Finding the Derivative","body":"","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"3.1 Defining the Derivative","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a39b8a0def12a","stepAnswer":["$$13$$"],"problemType":"TextBox","stepTitle":"For the function $$f(x)=x^2+9x$$, find f\u2032(a), where $$a=2$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$13$$","hints":{"DefaultPathway":[{"id":"a39b8a0def12a-h1","type":"hint","dependencies":[],"title":"Derivative Equation","text":"The derivative of the function f(x) at a is defined by /lim{x,a}(f(x)-f(a))/(x-a).","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def12a-h2","type":"hint","dependencies":["a39b8a0def12a-h1"],"title":"Plugging In","text":"Plug $$2$$ into a and expand the equation.","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def12a-h3","type":"hint","dependencies":["a39b8a0def12a-h2"],"title":"Simplify","text":"Simplify the resulting equation: $$\\\\frac{x^2+9x-2^2+9\\\\times2}{x-2}$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"a39b8a0def13","title":"Finding the Derivative","body":"","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"3.1 Defining the Derivative","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a39b8a0def13a","stepAnswer":["$$\\\\frac{1}{4}$$"],"problemType":"TextBox","stepTitle":"For the function $$f(x)=\\\\sqrt{x}$$, find f\u2032(a), where $$a=4$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{4}$$","hints":{"DefaultPathway":[{"id":"a39b8a0def13a-h1","type":"hint","dependencies":[],"title":"Derivative Equation","text":"The derivative of the function f(x) at a is defined by $$\\\\lim_{x\\\\toa} \\\\frac{f{\\\\left(x\\\\right)}-f{\\\\left(a\\\\right)}}{x-a}$$.","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def13a-h2","type":"hint","dependencies":["a39b8a0def13a-h1"],"title":"Plugging In","text":"Plug $$4$$ into a and expand the equation.","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def13a-h3","type":"hint","dependencies":["a39b8a0def13a-h2"],"title":"Simplify","text":"Simplify the resulting equation: $$\\\\frac{\\\\sqrt{x}-\\\\sqrt{4}}{x-4}$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"a39b8a0def14","title":"Finding the Derivative","body":"","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"3.1 Defining the Derivative","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a39b8a0def14a","stepAnswer":["$$\\\\frac{-1}{4}$$"],"problemType":"TextBox","stepTitle":"For the function $$f(x)=\\\\frac{1}{x}$$, find f\u2032(a), where $$a=2$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-1}{4}$$","hints":{"DefaultPathway":[{"id":"a39b8a0def14a-h1","type":"hint","dependencies":[],"title":"Derivative Equation","text":"The derivative of the function f(x) at a is defined by $$\\\\lim_{x\\\\toa} \\\\frac{f{\\\\left(x\\\\right)}-f{\\\\left(a\\\\right)}}{x-a}$$.","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def14a-h2","type":"hint","dependencies":["a39b8a0def14a-h1"],"title":"Plugging In","text":"Plug $$2$$ into a and expand the equation.","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def14a-h3","type":"hint","dependencies":["a39b8a0def14a-h2"],"title":"Simplify","text":"Simplify the resulting equation: $$\\\\frac{\\\\frac{1}{x}-\\\\frac{1}{2}}{x-2}$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"a39b8a0def15","title":"Finding the Derivative","body":"","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"3.1 Defining the 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$$\\\\frac{\\\\frac{1}{x^3}-\\\\frac{1}{1^3}}{x-1}$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"a39b8a0def16","title":"Average Velocity","body":"","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"3.1 Defining the Derivative","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a39b8a0def16a","stepAnswer":["$$\\\\frac{1}{3}$$"],"problemType":"TextBox","stepTitle":"For the following position function $$y=\\\\frac{s}{t}+5$$, find the simplified expression for the average velocity from $$t=2$$\\\\nto $$t=2+h$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{3}$$","hints":{"DefaultPathway":[{"id":"a39b8a0def16a-h1","type":"hint","dependencies":[],"title":"Average Velocity Equation","text":"If s(t) is the position of an object moving along a coordinate axis, the average velocity of the object over a time interval [a,t] is $$V_{avg}=\\\\frac{s\\\\left(t\\\\right)-s\\\\left(a\\\\right)}{t-a}$$","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def16a-h2","type":"hint","dependencies":["a39b8a0def16a-h1"],"title":"Plugging In","text":"In this question, $$t$$ goes from $$2$$ to $$2+h$$. Plug $$2+h$$ in $$t$$ and $$2$$ in a for the previous equation.","variabilization":{},"oer":"","license":""}]}},{"id":"a39b8a0def16b","stepAnswer":["$$\\\\frac{1}{3}$$"],"problemType":"TextBox","stepTitle":"For the previous equation, find the average velocity between $$t=2$$ and $$t=2+h$$, where $$h=0.1$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{3}$$","hints":{"DefaultPathway":[{"id":"a39b8a0def16b-h1","type":"hint","dependencies":[],"title":"Plugging In","text":"Plug $$t=2$$ and $$t=2+h$$ into the equation found in the previous section for the average velocity.","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def16b-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["the same"],"dependencies":["a39b8a0def16b-h1"],"title":"Constants","text":"As there are no variables, is the average velocity the same or different for different time intervals?","variabilization":{},"oer":"","license":"","choices":["the same","different"]}]}},{"id":"a39b8a0def16c","stepAnswer":["$$\\\\frac{1}{3}$$"],"problemType":"TextBox","stepTitle":"Use the answer from a. to estimate the instantaneous velocity at $$t=2$$ seconds.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{3}$$","hints":{"DefaultPathway":[{"id":"a39b8a0def16c-h1","type":"hint","dependencies":[],"title":"Instantaneous Velocity Equation","text":"Instantaneous velocity at a is given by v(a)=lim{t,a,(s(t)-s(a))/(t-a)}","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def16c-h2","type":"hint","dependencies":["a39b8a0def16c-h1"],"title":"Plugging In","text":"Plug in $$a=2$$ into the instantaneous velocity equation.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a39b8a0def17","title":"Average Velocity","body":"","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"3.1 Defining the Derivative","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a39b8a0def17a","stepAnswer":["$$2\\\\left(h^2+6h+12\\\\right)$$\\\\n"],"problemType":"MultipleChoice","stepTitle":"For the following position function $$y=2t^3+3$$, find the simplified expression for the average velocity from $$t=2$$\\\\nto $$t=2+h$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2\\\\left(h^2+6h+12\\\\right)$$\\\\n","choices":["$$2\\\\left(h^2+4h+4\\\\right)$$","$$2\\\\left(h^2+6h+12\\\\right)$$","$$2\\\\left(h^2+6h+12\\\\right)$$\\\\n","$$2\\\\left(h+2\\\\right)$$\\\\n","$$3\\\\left(h^3+6h^2+12h+18\\\\right)$$"],"hints":{"DefaultPathway":[{"id":"a39b8a0def17a-h1","type":"hint","dependencies":[],"title":"Average Velocity Equation","text":"If s(t) is the position of an object moving along a coordinate axis, the average velocity of the object over a time interval [a,t] is $$V_{avg}=\\\\frac{s\\\\left(t\\\\right)-s\\\\left(a\\\\right)}{t-a}$$","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def17a-h2","type":"hint","dependencies":["a39b8a0def17a-h1"],"title":"Plugging In","text":"In this question, $$t$$ goes from $$2$$ to $$2+h$$. Plug $$2+h$$ in $$t$$ and $$2$$ in a for the previous equation.","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def17a-h3","type":"hint","dependencies":["a39b8a0def17a-h2"],"title":"Expanding Powers of Three","text":"The equation for $${\\\\left(a+b\\\\right)}^3$$ is $$a^3+3a^b+{3\\\\left(a b\\\\right)}^2+b^3$$.","variabilization":{},"oer":"","license":""}]}},{"id":"a39b8a0def17b","stepAnswer":["$$25.22$$"],"problemType":"TextBox","stepTitle":"For the previous equation, find the average velocity between $$t=2$$ and $$t=2+h$$, where $$h=0.1$$. 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What is the slope of a horizontal line?","variabilization":{},"oer":"","license":"","choices":["DNE","$$0$$","$$1$$","$$-1$$"]}]}}]},{"id":"a39b8a0def2","title":"Finding the Secant","body":"","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"3.1 Defining the Derivative","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a39b8a0def2a","stepAnswer":["$$8.5$$"],"problemType":"TextBox","stepTitle":"Find the slope of the secant line between the values $$x_1$$ and $$x_2$$ for $$f(x)=x^2+2x+1$$, where $$x_1=3$$ and $$x_2=3.5$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8.5$$","hints":{"DefaultPathway":[{"id":"a39b8a0def2a-h1","type":"hint","dependencies":[],"title":"Secant Line Equation","text":"To find the secant line between two points, use the equation $$\\\\frac{f{\\\\left(x\\\\right)}-f{\\\\left(a\\\\right)}}{x-a}$$.","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def2a-h2","type":"hint","dependencies":["a39b8a0def2a-h1"],"title":"Plugging In","text":"In the previous equation, plug the values of $$x_2$$ and $$x_1$$ into $$x$$ and a respectively.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a39b8a0def20","title":"Limit Definition for Derivatives","body":"","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"3.1 Defining the Derivative","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a39b8a0def20a","stepAnswer":["DNE"],"problemType":"MultipleChoice","stepTitle":"What is the derivative for $$y=x^3$$ at $$x=0$$?","stepBody":"","answerType":"string","variabilization":{},"choices":["DNE","$$1$$","$$8$$","$$-8$$"],"hints":{"DefaultPathway":[{"id":"a39b8a0def20a-h1","type":"hint","dependencies":[],"title":"Limit Definition of the Derivative","text":"The limit defintion of a derivative is $$\\\\lim_{h\\\\to0} \\\\frac{f{\\\\left(x+h\\\\right)}-f{\\\\left(x\\\\right)}}{h}$$.","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def20a-h2","type":"hint","dependencies":["a39b8a0def20a-h1"],"title":"Plugging In","text":"Plug in $$h=0$$ into the previous equation.","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def20a-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["infinity"],"dependencies":["a39b8a0def20a-h2"],"title":"Resulting Limit","text":"What is the value of this limit?","variabilization":{},"oer":"","license":"","choices":["negative infinity","$$-1$$","$$0$$","infinity"]},{"id":"a39b8a0def20a-h3","type":"hint","dependencies":["a39b8a0def20a-h2"],"title":"Undefined Limits","text":"Limits are undefined when they do not approach a finite value or when both sides do not approach the same value.","variabilization":{},"oer":"","license":""}]}},{"id":"a39b8a0def20b","stepAnswer":["DNE"],"problemType":"MultipleChoice","stepTitle":"f(x) is $$1$$ when $$x<1$$, and $$x$$ when $$x \\\\geq 1$$. What is the derivative for f(x) when at $$x=1$$?","stepBody":"","answerType":"string","variabilization":{},"choices":["DNE","$$1$$","$$-1$$"],"hints":{"DefaultPathway":[{"id":"a39b8a0def20b-h1","type":"hint","dependencies":[],"title":"Limit Definition of the Derivative","text":"The limit defintion of a derivative is $$\\\\lim_{h\\\\to0} \\\\frac{f{\\\\left(x+h\\\\right)}-f{\\\\left(x\\\\right)}}{h}$$.","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def20b-h2","type":"hint","dependencies":["a39b8a0def20b-h1"],"title":"Plugging In","text":"Plug in $$h=1$$ into the previous equation, approaching the $$h$$ from 0- and 0+.","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def20b-h3","type":"hint","dependencies":["a39b8a0def20b-h2"],"title":"Undefined Limits","text":"Limits are undefined when they do not approach a finite value or when both sides do not approach the same value.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a39b8a0def21","title":"Average Velocity","body":"","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"3.1 Defining the Derivative","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a39b8a0def21a","stepAnswer":["$$61.0725$$"],"problemType":"TextBox","stepTitle":"The position in feet of a race car along a straight track after $$t$$ seconds is modeled by the function $$s(t)=8t^2-\\\\frac{t^3}{16}$$. Find the average velocity of the vehicle from 4s to $$4.01s$$ to $$4$$ decimal places.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$61.0725$$","hints":{"DefaultPathway":[{"id":"a39b8a0def21a-h1","type":"hint","dependencies":[],"title":"Average Velocity Equation","text":"If s(t) is the position of an object moving along a coordinate axis, the average velocity of the object over a time interval [a,t] is $$V_{avg}=\\\\frac{s\\\\left(t\\\\right)-s\\\\left(a\\\\right)}{t-a}$$","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def21a-h2","type":"hint","dependencies":["a39b8a0def21a-h1"],"title":"Plugging In","text":"In this question, $$t$$ goes from $$4$$ to $$4.01$$. Plug $$4.01s$$ in for $$t$$ and 4s in for a in the previous equation.","variabilization":{},"oer":"","license":""}]}},{"id":"a39b8a0def21b","stepAnswer":["$$61$$"],"problemType":"TextBox","stepTitle":"Draw a conclusion about the instantaneous velocity of the vehicle at $$t=4$$ seconds. From [4, $$4.1]$$, the average velocity is $$61.7244$$ $$\\\\frac{ft}{s}$$. From [4, $$4.001]$$, the average velocity is $$61.0072$$ $$\\\\frac{ft}{s}$$. From [4, $$4.0001]$$, the average velocity is $$61.0007$$ $$\\\\frac{ft}{s}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$61$$","hints":{"DefaultPathway":[{"id":"a39b8a0def21b-h1","type":"hint","dependencies":[],"title":"Approaching Instantaeous Velocity","text":"The value that is being approached by the average velocities as $$t$$ gets closer to $$4$$ seconds is likely the instantaneous velocity when $$t=4$$.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a39b8a0def22","title":"Average Velocity","body":"","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"3.1 Defining the Derivative","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a39b8a0def22a","stepAnswer":["g(t)"],"problemType":"MultipleChoice","stepTitle":"Two vehicles start out traveling side by side along a straight road. Their position functions, shown in the following graph, are given by $$s=f(t)$$ and $$s=g(t)$$, where s is measured in feet and $$t$$ is measured in seconds. Which vehicle is driving faster at $$t=4$$ seconds?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["g(t)","s(t)"],"hints":{"DefaultPathway":[{"id":"a39b8a0def22a-h1","type":"hint","dependencies":[],"title":"Average Velocity Equation","text":"The instantaneous velocity of both vehicles is given by the slope of the tangent line at $$t=4$$.","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def22a-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["g(t)"],"dependencies":["a39b8a0def22a-h1"],"title":"Velocities of Vehicles","text":"Is f(t) or g(t)\'s slope greater when $$t=4$$?","variabilization":{},"oer":"","license":"","choices":["g(t)","s(t)"]}]}},{"id":"a39b8a0def22b","stepAnswer":["same distance"],"problemType":"MultipleChoice","stepTitle":"Which vehicle traveled further at $$4$$ seconds?","stepBody":"","answerType":"string","variabilization":{},"choices":["g(t)","s(t)","same distance"],"hints":{"DefaultPathway":[{"id":"a39b8a0def22b-h1","type":"hint","dependencies":[],"title":"Position","text":"The position of both vehicles is given by s, which is measured in feet.","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def22b-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a39b8a0def22b-h1"],"title":"Position at $$4$$ Seconds","text":"How far did s(t) travel at $$t=4$$?","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def22b-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a39b8a0def22b-h1"],"title":"Position at $$4$$ Seconds","text":"How far did g(t) travel at $$t=4$$?","variabilization":{},"oer":"","license":""}]}}]},{"id":"a39b8a0def23","title":"Graphing Functions","body":"","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"3.1 Defining the Derivative","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a39b8a0def23a","stepAnswer":["$$2.694$$"],"problemType":"MultipleChoice","stepTitle":"For the function $$f(x)=x^3-2x^2-11x+12$$, graph the function on the graphing calculator. Then find a value of a where $$f\'(a)=0$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2.694$$","choices":["$$2.694$$","$$0.122$$","$$5.675$$","$$-1.349$$"],"hints":{"DefaultPathway":[{"id":"a39b8a0def23a-h1","type":"hint","dependencies":[],"title":"Graphing a Function","text":"Use a graphing calculator to graph f(x). Then use the ZOOM function to approximate values of a.","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def23a-h2","type":"hint","dependencies":[],"title":"Zero Derivative","text":"When a tangent line at some $$x$$ on the f(x) is horizontal, $$f\'(x)=0$$.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a39b8a0def24","title":"Graphing Functions","body":"","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"3.1 Defining the Derivative","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a39b8a0def24a","stepAnswer":["$$2.694$$"],"problemType":"MultipleChoice","stepTitle":"For the function $$f(x)=\\\\frac{x^2}{x^2+1}$$ graph the function on the graphing calculator, and find the value of f\'(2).","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2.694$$","choices":["$$2.694$$","$$0.122$$","$$5.675$$","$$-1.349$$"],"hints":{"DefaultPathway":[{"id":"a39b8a0def24a-h1","type":"hint","dependencies":[],"title":"Graphing a Function","text":"Use a graphing calculator to graph f(x). Then use the nDeriv function to approximate values at $$2$$.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a39b8a0def25","title":"Mileage","body":"","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"3.1 Defining the Derivative","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a39b8a0def25a","stepAnswer":["$$\\\\frac{x}{30}$$"],"problemType":"MultipleChoice","stepTitle":"Suppose that N(x) computes the number of gallons of gas used by a vehicle traveling $$x$$ miles. Suppose the vehicle gets $$30$$ mpg. What is a mathematical expression for N(x)?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{x}{30}$$","choices":["$$\\\\frac{x}{30}$$","$$30x$$","$$\\\\frac{30}{x}$$","$$30$$"],"hints":{"DefaultPathway":[{"id":"a39b8a0def25a-h1","type":"hint","dependencies":[],"title":"Graphing a Function","text":"For every thirty miles traveled, $$1$$ gallon of gas is used. In this equation, $$x$$ would be the number of gallons of gas consumed.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a39b8a0def26","title":"Finding Derivative of an Equation","body":"","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"3.1 Defining the Derivative","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a39b8a0def26a","stepAnswer":["$$\\\\frac{1}{30}$$"],"problemType":"TextBox","stepTitle":"$$N(x)=\\\\frac{x}{30}$$ is the number of gallons of gas used by a vehicle after traveling $$x$$ miles. What is N\u2032(100)? This is the gas consumption rate in gallons per mile that the vehicle achieves after having traveled $$100$$ miles.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{30}$$","hints":{"DefaultPathway":[{"id":"a39b8a0def26a-h1","type":"hint","dependencies":[],"title":"Derivative Equation","text":"N\'(100) would be the slope of the tangent line when $$x=100$$.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a39b8a0def3","title":"Finding the Secant","body":"","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"3.1 Defining the Derivative","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a39b8a0def3a","stepAnswer":["$$-0.75$$"],"problemType":"TextBox","stepTitle":"Find the slope of the secant line between the values $$x_1$$ and $$x_2$$ for $$f(x)=\\\\frac{4}{3x-1}$$, where $$x_1=1$$ and $$x_2=3$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-0.75$$","hints":{"DefaultPathway":[{"id":"a39b8a0def3a-h1","type":"hint","dependencies":[],"title":"Secant Line Equation","text":"To find the secant line between two points, use the equation $$\\\\frac{f{\\\\left(x\\\\right)}-f{\\\\left(a\\\\right)}}{x-a}$$.","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def3a-h2","type":"hint","dependencies":["a39b8a0def3a-h1"],"title":"Plugging In","text":"In the previous equation, plug the values of $$x_2$$ and $$x_1$$ into $$x$$ and a respectively.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a39b8a0def4","title":"Finding the Secant","body":"","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"3.1 Defining the Derivative","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a39b8a0def4a","stepAnswer":["$$0.2$$"],"problemType":"TextBox","stepTitle":"Find the slope of the secant line between the values $$x_1$$ and $$x_2$$ for $$f(x)=\\\\sqrt{x}$$, where $$x_1=1$$ and $$x_2=16$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.2$$","hints":{"DefaultPathway":[{"id":"a39b8a0def4a-h1","type":"hint","dependencies":[],"title":"Secant Line Equation","text":"To find the secant line between two points, use the equation $$\\\\frac{f{\\\\left(x\\\\right)}-f{\\\\left(a\\\\right)}}{x-a}$$.","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def4a-h2","type":"hint","dependencies":["a39b8a0def4a-h1"],"title":"Plugging In","text":"In the previous equation, plug the values of $$x_2$$ and $$x_1$$ into $$x$$ and a respectively.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a39b8a0def5","title":"Finding the Secant","body":"","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"3.1 Defining the Derivative","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a39b8a0def5a","stepAnswer":["$$0.25$$"],"problemType":"TextBox","stepTitle":"Find the slope of the secant line between the values $$x_1$$ and $$x_2$$ for $$f(x)=x^{\\\\frac{1}{3}}+1$$, where $$x_1=0$$ and $$x_2=8$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.25$$","hints":{"DefaultPathway":[{"id":"a39b8a0def5a-h1","type":"hint","dependencies":[],"title":"Secant Line Equation","text":"To find the secant line between two points, use the equation $$\\\\frac{f{\\\\left(x\\\\right)}-f{\\\\left(a\\\\right)}}{x-a}$$.","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def5a-h2","type":"hint","dependencies":["a39b8a0def5a-h1"],"title":"Plugging In","text":"In the previous equation, plug the values of $$x_2$$ and $$x_1$$ into $$x$$ and a respectively.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a39b8a0def6","title":"Finding Slope of the Tangent","body":"","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"3.1 Defining the Derivative","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a39b8a0def6a","stepAnswer":["$$-4$$"],"problemType":"TextBox","stepTitle":"Find the slope of the tangent line $$m=f\'(a)$$, where $$f(x)=3-4x$$ and $$a=2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-4$$","hints":{"DefaultPathway":[{"id":"a39b8a0def6a-h1","type":"hint","dependencies":[],"title":"Tangent Line Equation","text":"The tangent line to f(x) at a is the line passing through the point (a,f(a)) having slope $$\\\\lim_{h\\\\to0} \\\\frac{f{\\\\left(a+h\\\\right)}-f{\\\\left(a\\\\right)}}{h}$$.","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def6a-h2","type":"hint","dependencies":["a39b8a0def6a-h1"],"title":"Plugging In","text":"As $$a=2$$ in this equation, plug $$2$$ into every f(a).","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def6a-h3","type":"hint","dependencies":["a39b8a0def6a-h2"],"title":"Simplify","text":"Simplify the resulting equation: $$\\\\frac{3-4\\\\left(2+h\\\\right)-\\\\left(-4\\\\times2\\\\right)}{h}$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"a39b8a0def7","title":"Finding Slope of the Tangent","body":"","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"3.1 Defining the Derivative","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a39b8a0def7a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"Find the slope of the tangent line $$m=f\'(a)$$, where $$f(x)=x^2+x$$ and $$a=1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a39b8a0def7a-h1","type":"hint","dependencies":[],"title":"Tangent Line Equation","text":"The tangent line to f(x) at a is the line passing through the point (a,f(a)) having slope $$\\\\lim_{h\\\\to0} \\\\frac{f{\\\\left(a+h\\\\right)}-f{\\\\left(a\\\\right)}}{h}$$.","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def7a-h2","type":"hint","dependencies":["a39b8a0def7a-h1"],"title":"Plugging In","text":"As $$a=1$$ in this equation, plug $$1$$ into every f(a).","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def7a-h3","type":"hint","dependencies":["a39b8a0def7a-h2"],"title":"Simplify","text":"Simplify the resulting equation: $${\\\\left(1+h\\\\right)}^2+1+h-\\\\frac{1^2-1}{h}$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"a39b8a0def8","title":"Finding Slope of the Tangent","body":"","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"3.1 Defining the Derivative","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a39b8a0def8a","stepAnswer":["$$\\\\frac{-7}{9}$$"],"problemType":"TextBox","stepTitle":"Find the slope of the tangent line $$m=f\'(a)$$, where $$f(x)=\\\\frac{7}{x}$$ and $$a=3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-7}{9}$$","hints":{"DefaultPathway":[{"id":"a39b8a0def8a-h1","type":"hint","dependencies":[],"title":"Tangent Line Equation","text":"The tangent line to f(x) at a is the line passing through the point (a,f(a)) having slope $$\\\\lim_{h\\\\to0} \\\\frac{f{\\\\left(a+h\\\\right)}-f{\\\\left(a\\\\right)}}{h}$$.","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def8a-h2","type":"hint","dependencies":["a39b8a0def8a-h1"],"title":"Plugging In","text":"As $$a=3$$ in this equation, plug $$3+h$$ into f(a).","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def8a-h3","type":"hint","dependencies":["a39b8a0def8a-h2"],"title":"Simplify","text":"Simplify the resulting equation: $$\\\\frac{\\\\frac{7}{3+h}-\\\\frac{7}{3}}{h}$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"a39b8a0def9","title":"Finding Slope of the Tangent","body":"","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"3.1 Defining the Derivative","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a39b8a0def9a","stepAnswer":["$$12$$"],"problemType":"TextBox","stepTitle":"Find the slope of the tangent line $$m=f\'(a)$$, where $$f(x)=2-3x^2$$ and $$a=-2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12$$","hints":{"DefaultPathway":[{"id":"a39b8a0def9a-h1","type":"hint","dependencies":[],"title":"Tangent Line Equation","text":"The tangent line to f(x) at a is the line passing through the point (a,f(a)) having slope $$\\\\lim_{h\\\\to0} \\\\frac{f{\\\\left(a+h\\\\right)}-f{\\\\left(a\\\\right)}}{h}$$.","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def9a-h2","type":"hint","dependencies":["a39b8a0def9a-h1"],"title":"Plugging In","text":"As $$a=-2$$ in this equation, plug $$2$$ into f(a).","variabilization":{},"oer":"","license":""},{"id":"a39b8a0def9a-h3","type":"hint","dependencies":["a39b8a0def9a-h2"],"title":"Simplify","text":"Simplify the resulting equation: $$\\\\frac{2-3{\\\\left(h-2\\\\right)}^2-3-3{\\\\left(-2\\\\right)}^2}{h}$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"a3abd22AddRational1","title":"Rational Expression Addition","body":"Add the following rational expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Add and Subtract Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3abd22AddRational1a","stepAnswer":["$$x+7$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{11x+28}{x+4}+\\\\frac{x^2}{x+4}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x+7$$","hints":{"DefaultPathway":[{"id":"a3abd22AddRational1a-h1","type":"hint","dependencies":[],"title":"Find the Common Denominator","text":"Do all the terms in the expression have common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational1a-h2","type":"hint","dependencies":["a3abd22AddRational1a-h1"],"title":"Find the Common Denominator","text":"Yes, $$\\\\frac{11x+28}{x+4}$$ and $$\\\\frac{x^2}{x+4}$$ shared the same common denominator of $$x+4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational1a-h3","type":"hint","dependencies":["a3abd22AddRational1a-h2"],"title":"Add Numerators","text":"Since all of fractions in the expression have the least common denominator $$x+4$$, we can add the numerators.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2+11x+28$$"],"dependencies":["a3abd22AddRational1a-h3"],"title":"Add Numerators","text":"What is the sum of numerators for $$\\\\frac{11x+28}{x+4}$$ and $$\\\\frac{x^2}{x+4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational1a-h5","type":"hint","dependencies":["a3abd22AddRational1a-h4"],"title":"Factor the Numerator","text":"We can factor the sum of numerator that we yield from step above to check if we can simplify the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational1a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x+4\\\\right) \\\\left(x+7\\\\right)$$"],"dependencies":["a3abd22AddRational1a-h5"],"title":"Factor the Numerator","text":"Factor $$x^2+11x+28$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational1a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x+4\\\\right) \\\\left(x+7\\\\right)$$"],"dependencies":["a3abd22AddRational1a-h6"],"title":"Factor the Numerator","text":"In order to factor $$x^2+11x+28$$, you can use reverse foiling, completing square or quadratic formula. In here we can use reverse-foiling. $$\\\\left(x+a\\\\right) \\\\left(x+b\\\\right)=x^2+\\\\left(a+b\\\\right) x+ab$$. In this case, we get $$a b=28$$ and $$a+b=11$$. Assume both a and $$b$$ are integers, $$a=4$$ and $$b=7$$ is only one possible combination that satisfies $$a b=28$$ and $$a+b=11$$. So you can factor $$x^2+11x+28$$ as?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational1a-h8","type":"hint","dependencies":["a3abd22AddRational1a-h7"],"title":"Rewrite the expression","text":"From above steps, we get the sum of numerators is $$\\\\left(x+4\\\\right) \\\\left(x+7\\\\right)$$ and the denominator is $$x+4$$. So we can rewrite $$\\\\frac{11x+28}{x+4}+\\\\frac{x^2}{x+4}$$ as $$\\\\frac{\\\\left(x+4\\\\right) \\\\left(x+7\\\\right)}{x+4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational1a-h9","type":"hint","dependencies":["a3abd22AddRational1a-h8"],"title":"Simplify Expression by Canceling out Terms","text":"Are there any ways to simplify $$\\\\frac{\\\\left(x+4\\\\right) \\\\left(x+7\\\\right)}{x+4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational1a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+4$$"],"dependencies":["a3abd22AddRational1a-h9"],"title":"Simplify Expression by Canceling out Terms","text":"What is the greatest common factor shared by both numerator and denominator in $$\\\\frac{\\\\left(x+4\\\\right) \\\\left(x+7\\\\right)}{x+4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational1a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+7$$"],"dependencies":["a3abd22AddRational1a-h10"],"title":"Simplify Expression by Canceling out Terms","text":"As the final step, we cancel out $$x-4$$ on both numerator and denominator, what is the simplified form of $$\\\\frac{\\\\left(x+4\\\\right) \\\\left(x+7\\\\right)}{x+4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational1a-h12","type":"hint","dependencies":["a3abd22AddRational1a-h11"],"title":"Final Remark","text":"Note that the expression $$\\\\frac{11x+28}{x+4}+\\\\frac{x^2}{x+4}$$ and $$x+7$$ are different because the domain of $$\\\\frac{11x+28}{x+4}+\\\\frac{x^2}{x+4}$$ does not include $$x=-4$$. The domain of $$x+7$$ includes $$x=-4$$. Strictly speaking, $$\\\\frac{11x+28}{x+4}+\\\\frac{x^2}{x+4}$$ equals $$x+7$$ excluding the domain $$x=-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3abd22AddRational10","title":"AddRationalExpressions","body":"Add the Following Rational Expressions with Unlike Denominators","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Add and Subtract Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3abd22AddRational10a","stepAnswer":["$$\\\\frac{5x-12}{\\\\left(x-3\\\\right) \\\\left(x-2\\\\right)}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{3}{x-3}+\\\\frac{2}{x-2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5x-12}{\\\\left(x-3\\\\right) \\\\left(x-2\\\\right)}$$","hints":{"DefaultPathway":[{"id":"a3abd22AddRational10a-h1","type":"hint","dependencies":[],"title":"Determine if the Expression Have a Common Denominator","text":"The expression does not have a common denominator, so we need to find the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x-3\\\\right) \\\\left(x-2\\\\right)$$"],"dependencies":["a3abd22AddRational10a-h1"],"title":"Find the Common Denominator","text":"Since both $$(x-3)$$ and $$(x-2)$$ are linear factors and they are not multiple of each other. What is the least common denominator for $$\\\\frac{3}{x-3}+\\\\frac{2}{x-2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational10a-h3","type":"hint","dependencies":["a3abd22AddRational10a-h2"],"title":"Rewrite the Expression","text":"Rewrite each rational expression as an equivalent rational expression with the LCD.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3\\\\left(x-2\\\\right)}{\\\\left(x-3\\\\right) \\\\left(x-2\\\\right)}$$"],"dependencies":["a3abd22AddRational10a-h3"],"title":"Rewrite the Expression","text":"What is the equivalent rational expression with the LCD $$\\\\left(x-2\\\\right) \\\\left(x-3\\\\right)$$ for $$\\\\frac{3}{x-3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational10a-h5","type":"hint","dependencies":["a3abd22AddRational10a-h4"],"title":"Rewrite the Expression","text":"Since $$\\\\frac{x-2}{x-2}=1$$ and $$1\\\\frac{3}{x-3}=\\\\frac{3}{x-3}$$ We can rewrite $$\\\\frac{3}{x-3}$$ as $$\\\\frac{3\\\\left(x-2\\\\right)}{\\\\left(x-3\\\\right) \\\\left(x-2\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational10a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2\\\\left(x-3\\\\right)}{\\\\left(x-2\\\\right) \\\\left(x-3\\\\right)}$$"],"dependencies":["a3abd22AddRational10a-h5"],"title":"Rewrite the Expression","text":"What is the equivalent rational expression with the LCD $$\\\\left(x-2\\\\right) \\\\left(x-3\\\\right)$$ for $$\\\\frac{2}{x-2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational10a-h7","type":"hint","dependencies":["a3abd22AddRational10a-h6"],"title":"Add the Numerators for Expression","text":"Add the numerators and place the sum over the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational10a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5x-12$$"],"dependencies":["a3abd22AddRational10a-h7"],"title":"Add the Numerators for Expression","text":"What is the sum of numerators for $$\\\\frac{3\\\\left(x-2\\\\right)}{\\\\left(x-3\\\\right) \\\\left(x-2\\\\right)}+\\\\frac{2\\\\left(x-3\\\\right)}{\\\\left(x-2\\\\right) \\\\left(x-3\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational10a-h9","type":"hint","dependencies":["a3abd22AddRational10a-h8"],"title":"Simplify the Expression","text":"According to the above steps, we can rewrite $$\\\\frac{3}{x-3}+\\\\frac{2}{x-2}$$ as $$\\\\frac{5x-12}{\\\\left(x-3\\\\right) \\\\left(x-2\\\\right)}$$, are there any ways to simplify the new expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational10a-h10","type":"hint","dependencies":["a3abd22AddRational10a-h9"],"title":"Simplify the Expression","text":"The answer of $$\\\\frac{5x-12}{\\\\left(x-3\\\\right) \\\\left(x-2\\\\right)}$$ is simplified because $$5x-12$$ cannot be factored.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3abd22AddRational11","title":"AddRationalExpressions","body":"Add the Following Rational Expressions with Unlike Denominators","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Add and Subtract Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3abd22AddRational11a","stepAnswer":["$$\\\\frac{7x-4}{\\\\left(x+3\\\\right) \\\\left(x-2\\\\right)}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2}{x-2}+\\\\frac{5}{x+3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{7x-4}{\\\\left(x+3\\\\right) \\\\left(x-2\\\\right)}$$","hints":{"DefaultPathway":[{"id":"a3abd22AddRational11a-h1","type":"hint","dependencies":[],"title":"Determine if the Expression Have a Common Denominator","text":"The expression does not have a common denominator, so we need to find the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x+3\\\\right) \\\\left(x-2\\\\right)$$"],"dependencies":["a3abd22AddRational11a-h1"],"title":"Find the Common Denominator","text":"Since both $$x+3$$ and $$(x-2)$$ are linear factors and they are not multiple of each other. What is the least common denominator for $$\\\\frac{5}{x+3}+\\\\frac{2}{x-2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational11a-h3","type":"hint","dependencies":["a3abd22AddRational11a-h2"],"title":"Rewrite the Expression","text":"Rewrite each rational expression as an equivalent rational expression with the LCD.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5\\\\left(x-2\\\\right)}{\\\\left(x+3\\\\right) \\\\left(x-2\\\\right)}$$"],"dependencies":["a3abd22AddRational11a-h3"],"title":"Rewrite the Expression","text":"What is the equivalent rational expression with the LCD $$\\\\left(x-2\\\\right) \\\\left(x+3\\\\right)$$ for $$\\\\frac{5}{x+3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational11a-h5","type":"hint","dependencies":["a3abd22AddRational11a-h4"],"title":"Rewrite the Expression","text":"Since $$\\\\frac{x-2}{x-2}=1$$ and $$1\\\\frac{5}{x+3}=\\\\frac{5}{x+3}$$, we can rewrite $$\\\\frac{5}{x+3}$$ as $$\\\\frac{5\\\\left(x-2\\\\right)}{\\\\left(x+3\\\\right) \\\\left(x-2\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational11a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2\\\\left(x+3\\\\right)}{\\\\left(x-2\\\\right) \\\\left(x+3\\\\right)}$$"],"dependencies":["a3abd22AddRational11a-h5"],"title":"Rewrite the Expression","text":"What is the equivalent rational expression with the LCD $$\\\\left(x-2\\\\right) \\\\left(x+3\\\\right)$$ for $$\\\\frac{2}{x-2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational11a-h7","type":"hint","dependencies":["a3abd22AddRational11a-h6"],"title":"Add the Numerators for Expression","text":"Add the numerators and place the sum over the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational11a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7x-4$$"],"dependencies":["a3abd22AddRational11a-h7"],"title":"Add the Numerators for Expression","text":"What is the sum of numerators for $$\\\\frac{5\\\\left(x-2\\\\right)}{\\\\left(x+3\\\\right) \\\\left(x-2\\\\right)}+\\\\frac{2\\\\left(x+3\\\\right)}{\\\\left(x-2\\\\right) \\\\left(x+3\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational11a-h9","type":"hint","dependencies":["a3abd22AddRational11a-h8"],"title":"Simplify the Expression","text":"According to the above steps, we can rewrite $$\\\\frac{2}{x-2}+\\\\frac{5}{x+3}$$ as $$\\\\frac{7x-4}{\\\\left(x+3\\\\right) \\\\left(x-2\\\\right)}$$, are there any ways to simplify the new expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational11a-h10","type":"hint","dependencies":["a3abd22AddRational11a-h9"],"title":"Simplify the Expression","text":"The answer of $$\\\\frac{7x-4}{\\\\left(x+3\\\\right) \\\\left(x-2\\\\right)}$$ is simplified because $$7x-4$$ cannot be factored.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3abd22AddRational12","title":"AddRationalExpressions","body":"Add the Following Rational Expressions with Unlike Denominators","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Add and Subtract Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3abd22AddRational12a","stepAnswer":["$$\\\\frac{7m+25}{\\\\left(m+3\\\\right) \\\\left(m+4\\\\right)}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{4}{m+3}+\\\\frac{3}{m+4}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{7m+25}{\\\\left(m+3\\\\right) \\\\left(m+4\\\\right)}$$","hints":{"DefaultPathway":[{"id":"a3abd22AddRational12a-h1","type":"hint","dependencies":[],"title":"Determine if the Expression Have a Common Denominator","text":"The expression does not have a common denominator, so we need to find the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(m+3\\\\right) \\\\left(m+4\\\\right)$$"],"dependencies":["a3abd22AddRational12a-h1"],"title":"Find the Common Denominator","text":"Since both $$m+3$$ and $$m+4$$ are linear factors and they are not multiple of each other. What is the least common denominator for $$\\\\frac{4}{m+3}+\\\\frac{3}{m+4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational12a-h3","type":"hint","dependencies":["a3abd22AddRational12a-h2"],"title":"Rewrite the Expression","text":"Rewrite each rational expression as an equivalent rational expression with the LCD.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{4\\\\left(m+4\\\\right)}{\\\\left(m+4\\\\right) \\\\left(m+3\\\\right)}$$"],"dependencies":["a3abd22AddRational12a-h3"],"title":"Rewrite the Expression","text":"What is the equivalent rational expression with the LCD $$\\\\left(m+3\\\\right) \\\\left(m+4\\\\right)$$ for $$\\\\frac{4}{m+3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational12a-h5","type":"hint","dependencies":["a3abd22AddRational12a-h4"],"title":"Rewrite the Expression","text":"Since $$\\\\frac{m+4}{m+4}=1$$ and $$1\\\\frac{4}{m+3}=\\\\frac{4}{m+3}$$, we can rewrite 4/(m+3)) as $$\\\\frac{4\\\\left(m+4\\\\right)}{\\\\left(m+4\\\\right) \\\\left(m+3\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational12a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3\\\\left(m+3\\\\right)}{\\\\left(m+3\\\\right) \\\\left(m+4\\\\right)}$$"],"dependencies":["a3abd22AddRational12a-h5"],"title":"Rewrite the Expression","text":"What is the equivalent rational expression with the LCD $$\\\\left(m+3\\\\right) \\\\left(m+4\\\\right)$$ for $$\\\\frac{3}{m+4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational12a-h7","type":"hint","dependencies":["a3abd22AddRational12a-h6"],"title":"Add the Numerators for Expression","text":"Add the numerators and place the sum over the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational12a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7m+25$$"],"dependencies":["a3abd22AddRational12a-h7"],"title":"Add the Numerators for Expression","text":"What is the sum of numerators for $$\\\\frac{4\\\\left(m+4\\\\right)}{\\\\left(m+4\\\\right) \\\\left(m+3\\\\right)}+\\\\frac{3\\\\left(m+3\\\\right)}{\\\\left(m+3\\\\right) \\\\left(m+4\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational12a-h9","type":"hint","dependencies":["a3abd22AddRational12a-h8"],"title":"Simplify the Expression","text":"According to the above steps, we can rewrite $$\\\\frac{4}{m+3}+\\\\frac{3}{m+4}$$ as $$\\\\frac{7m+25}{\\\\left(m+3\\\\right) \\\\left(m+4\\\\right)}$$, are there any ways to simplify the new expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational12a-h10","type":"hint","dependencies":["a3abd22AddRational12a-h9"],"title":"Simplify the Expression","text":"The answer of $$\\\\frac{7m+25}{\\\\left(m+3\\\\right) \\\\left(m+4\\\\right)}$$ is simplified because $$7m+25$$ cannot be factored.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3abd22AddRational13","title":"Subtract Rational Expressions","body":"Subtract the Following Rational Expressions with Unlike Denominators","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Add and Subtract Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3abd22AddRational13a","stepAnswer":["$$\\\\frac{4}{y+4}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{8y}{y^2-16}-\\\\frac{4}{y-4}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{4}{y+4}$$","hints":{"DefaultPathway":[{"id":"a3abd22AddRational13a-h1","type":"hint","dependencies":[],"title":"Determine if the Expression Have a Common Denominator","text":"The expression does not have a common denominator, so we need to find the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(y-4\\\\right) \\\\left(y+4\\\\right)$$"],"dependencies":["a3abd22AddRational13a-h1"],"title":"Find the Common Denominator","text":"Since the one of the denominator is quadratic, we can try to factor the it. What is the linear factor form of $$y^2-16$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational13a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(y-4\\\\right) \\\\left(y+4\\\\right)$$"],"dependencies":["a3abd22AddRational13a-h2"],"title":"Find the Common Denominator","text":"We can rewrite $$\\\\frac{8y}{y^2-16}$$ as $$\\\\frac{8y}{\\\\left(y-4\\\\right) \\\\left(y+4\\\\right)}$$, what is the least common denominator for $$\\\\frac{8y}{\\\\left(y-4\\\\right) \\\\left(y+4\\\\right)}-\\\\frac{4}{y-4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational13a-h4","type":"hint","dependencies":["a3abd22AddRational13a-h3"],"title":"Rewrite the Expression","text":"Rewrite each rational expression as an equivalent rational expression with the LCD.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational13a-h5","type":"hint","dependencies":["a3abd22AddRational13a-h4"],"title":"Rewrite the Expression","text":"Since $$\\\\frac{8y}{y^2-16}=\\\\frac{8y}{\\\\left(y-4\\\\right) \\\\left(y+4\\\\right)}$$ already has LCD as denominator, we need to find the equivalent rational expression with the LCD $$\\\\left(y+4\\\\right) \\\\left(y-4\\\\right)$$ for $$\\\\frac{4}{y-4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational13a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{4\\\\left(y+4\\\\right)}{\\\\left(y-4\\\\right) \\\\left(y+4\\\\right)}$$"],"dependencies":["a3abd22AddRational13a-h5"],"title":"Rewrite the Expression","text":"Find the equivalent rational expression with the LCD $$\\\\left(y+4\\\\right) \\\\left(y-4\\\\right)$$ for $$\\\\frac{4}{y-4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational13a-h7","type":"hint","dependencies":["a3abd22AddRational13a-h6"],"title":"Subtract the Numerators for Expression","text":"Subtract the numerators and place the difference over the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational13a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4y-16$$"],"dependencies":["a3abd22AddRational13a-h7"],"title":"Subtract the Numerators for Expression","text":"What is the difference of numerators for ((8*y)/((y-4)*(y+4))-((4*(y+4))/((y-4)*(y+4))?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational13a-h9","type":"hint","dependencies":["a3abd22AddRational13a-h8"],"title":"Simplify the Expression","text":"According to the above steps, we can rewrite $$\\\\frac{8y}{y^2-16}-\\\\frac{4}{y-4}$$ as $$\\\\frac{4y-16}{\\\\left(y+4\\\\right) \\\\left(y-4\\\\right)}$$, are there any ways to simplify the new expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational13a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{4}{y+4}$$"],"dependencies":["a3abd22AddRational13a-h9"],"title":"Simplify the Expression","text":"Yes, we can further factor the numerator $$4y-16$$ as $$4\\\\left(y-4\\\\right)$$. So $$\\\\frac{4y-16}{\\\\left(y+4\\\\right) \\\\left(y-4\\\\right)}$$ can be rewritten as $$\\\\frac{4\\\\left(y-4\\\\right)}{\\\\left(y+4\\\\right) \\\\left(y-4\\\\right)}$$. What is the simplified form of $$\\\\frac{4\\\\left(y-4\\\\right)}{\\\\left(y+4\\\\right) \\\\left(y-4\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3abd22AddRational14","title":"Subtract Rational Expressions","body":"Subtract the Following Rational Expressions with Unlike Denominators","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Add and Subtract Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3abd22AddRational14a","stepAnswer":["$$\\\\frac{1}{x-2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2x}{x^2-4}-\\\\frac{1}{x+2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{x-2}$$","hints":{"DefaultPathway":[{"id":"a3abd22AddRational14a-h1","type":"hint","dependencies":[],"title":"Determine if the Expression Have a Common Denominator","text":"The expression does not have a common denominator, so we need to find the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x+2\\\\right) \\\\left(x-2\\\\right)$$"],"dependencies":["a3abd22AddRational14a-h1"],"title":"Find the Common Denominator","text":"Since the one of the denominator is quadratic, we can try to factor the it. What is the linear factor form of $$x^2-4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational14a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x+2\\\\right) \\\\left(x-2\\\\right)$$"],"dependencies":["a3abd22AddRational14a-h2"],"title":"Find the Common Denominator","text":"We can rewrite $$\\\\frac{2x}{x^2-4}$$ as $$\\\\frac{2x}{\\\\left(x+2\\\\right) \\\\left(x-2\\\\right)}$$, what is the least common denominator for $$\\\\frac{2x}{\\\\left(x+2\\\\right) \\\\left(x-2\\\\right)}-\\\\frac{1}{x+2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational14a-h4","type":"hint","dependencies":["a3abd22AddRational14a-h3"],"title":"Rewrite the Expression","text":"Rewrite each rational expression as an equivalent rational expression with the LCD.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational14a-h5","type":"hint","dependencies":["a3abd22AddRational14a-h4"],"title":"Rewrite the Expression","text":"Since $$\\\\frac{2x}{x^2-4}=\\\\frac{2x}{\\\\left(x+2\\\\right) \\\\left(x-2\\\\right)}$$ already has LCD as denominator, we need to find the equivalent rational expression with the LCD $$\\\\left(x+2\\\\right) \\\\left(x-2\\\\right)$$ for $$\\\\frac{1}{x+2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational14a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{x-2}{x+2} \\\\left(x-2\\\\right)$$"],"dependencies":["a3abd22AddRational14a-h5"],"title":"Rewrite the Expression","text":"Find the equivalent rational expression with the LCD $$\\\\left(x+2\\\\right) \\\\left(x-2\\\\right)$$ for $$\\\\frac{1}{x+2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational14a-h7","type":"hint","dependencies":["a3abd22AddRational14a-h6"],"title":"Subtract the Numerators for Expression","text":"Subtract the numerators and place the difference over the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational14a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+2$$"],"dependencies":["a3abd22AddRational14a-h7"],"title":"Subtract the Numerators for Expression","text":"What is the difference of numerators for $$\\\\frac{2x}{\\\\left(x+2\\\\right) \\\\left(x-2\\\\right)}-\\\\frac{x-2}{x+2} \\\\left(x-2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational14a-h9","type":"hint","dependencies":["a3abd22AddRational14a-h8"],"title":"Simplify the Expression","text":"According to the above steps, we can rewrite $$\\\\frac{2x}{x^2-4}-\\\\frac{1}{x+2}$$ as $$\\\\frac{x+2}{\\\\left(x+2\\\\right) \\\\left(x-2\\\\right)}$$, are there any ways to simplify the new expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational14a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{x-2}$$"],"dependencies":["a3abd22AddRational14a-h9"],"title":"Simplify the Expression","text":"Yes, we can there is a common factor of $$x+2$$ in both numerator and denominator of $$\\\\frac{x+2}{x+2} \\\\left(x-2\\\\right)$$. By cancelling out $$x+2$$ in both numerator and denominator, what do we get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3abd22AddRational15","title":"Subtract Rational Expressions","body":"Subtract the Following Rational Expressions with Unlike Denominators","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Add and Subtract Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3abd22AddRational15a","stepAnswer":["$$\\\\frac{-3}{z-3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{3}{z+3}-\\\\frac{6z}{z^2-9}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-3}{z-3}$$","hints":{"DefaultPathway":[{"id":"a3abd22AddRational15a-h1","type":"hint","dependencies":[],"title":"Determine if the Expression Have a Common Denominator","text":"The expression does not have a common denominator, so we need to find the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(z+3\\\\right) \\\\left(z-3\\\\right)$$"],"dependencies":["a3abd22AddRational15a-h1"],"title":"Find the Common Denominator","text":"Since the one of the denominator is quadratic, we can try to factor the it. What is the linear factor form of $$z^2-9$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(z+3\\\\right) \\\\left(z-3\\\\right)$$"],"dependencies":["a3abd22AddRational15a-h2"],"title":"Find the Common Denominator","text":"We can rewrite $$\\\\frac{6z}{z^2-9}$$ as $$\\\\frac{6z}{\\\\left(z+3\\\\right) \\\\left(z-3\\\\right)}$$, what is the least common denominator for $$\\\\frac{3}{z+3}-\\\\frac{6z}{\\\\left(z+3\\\\right) \\\\left(z-3\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational15a-h4","type":"hint","dependencies":["a3abd22AddRational15a-h3"],"title":"Rewrite the Expression","text":"Rewrite each rational expression as an equivalent rational expression with the LCD.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational15a-h5","type":"hint","dependencies":["a3abd22AddRational15a-h4"],"title":"Rewrite the Expression","text":"Since $$\\\\frac{6z}{z^2-9}=\\\\frac{6z}{\\\\left(z+3\\\\right) \\\\left(z-3\\\\right)}$$ already has LCD as denominator, we need to find the equivalent rational expression with the LCD $$\\\\left(z+3\\\\right) \\\\left(z-3\\\\right)$$ for $$\\\\frac{3}{z+3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational15a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3\\\\left(z-3\\\\right)}{\\\\left(z+3\\\\right) \\\\left(z-3\\\\right)}$$"],"dependencies":["a3abd22AddRational15a-h5"],"title":"Rewrite the Expression","text":"Find the equivalent rational expression with the LCD $$\\\\left(z+3\\\\right) \\\\left(z-3\\\\right)$$ for $$\\\\frac{3}{z+3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational15a-h7","type":"hint","dependencies":["a3abd22AddRational15a-h6"],"title":"Subtract the Numerators for Expression","text":"Subtract the numerators and place the difference over the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational15a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3z-9$$"],"dependencies":["a3abd22AddRational15a-h7"],"title":"Subtract the Numerators for Expression","text":"What is the difference of numerators for (3*(z-3))/((z+3)*(z-3)))-((6*z)/((z+3)*(z-3)))?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational15a-h9","type":"hint","dependencies":["a3abd22AddRational15a-h8"],"title":"Simplify the Expression","text":"According to the above steps, we can $$\\\\operatorname{rewrite}\\\\left(\\\\frac{3}{z+3}\\\\right)-\\\\frac{6z}{z^2-9}$$ as $$\\\\frac{\\\\left(-3z-9\\\\right)}{\\\\left(z+3\\\\right) \\\\left(z-3\\\\right)}$$, are there any ways to simplify the new expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational15a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-3}{z-3}$$"],"dependencies":["a3abd22AddRational15a-h9"],"title":"Simplify the Expression","text":"Yes, we can further factor the numerator $$-3z-9$$ as $$-3\\\\left(z+3\\\\right)$$. So $$\\\\frac{\\\\left(-3z-9\\\\right)}{\\\\left(z+3\\\\right) \\\\left(z-3\\\\right)}$$ can be rewritten as $$\\\\frac{\\\\left(-3\\\\left(z+3\\\\right)\\\\right)}{\\\\left(z+3\\\\right) \\\\left(z-3\\\\right)}$$. What is the simplified form of $$\\\\frac{\\\\left(-3\\\\left(z+3\\\\right)\\\\right)}{\\\\left(z+3\\\\right) \\\\left(z-3\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3abd22AddRational2","title":"Rational Expression Addition","body":"Add the following rational expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Add and Subtract Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3abd22AddRational2a","stepAnswer":["$$x+2$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{9x+14}{x+7}+\\\\frac{x^2}{x+7}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x+2$$","hints":{"DefaultPathway":[{"id":"a3abd22AddRational2a-h1","type":"hint","dependencies":[],"title":"Find the Common Denominator","text":"Do all the terms in the expression have common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational2a-h2","type":"hint","dependencies":["a3abd22AddRational2a-h1"],"title":"Find the Common Denominator","text":"Yes, $$\\\\frac{9x+14}{x+7}$$ and $$\\\\frac{x^2}{x+7}$$ shared the same common denominator of $$x+7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational2a-h3","type":"hint","dependencies":["a3abd22AddRational2a-h2"],"title":"Add Numerators","text":"Since all of fractions in the expression have common denominator, we can add the numerators and place the sum over the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2+9x+14$$"],"dependencies":["a3abd22AddRational2a-h3"],"title":"Add Numerators","text":"What is the sum of numerators for $$\\\\frac{9x+14}{x+7}$$ and $$\\\\frac{x^2}{x+7}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational2a-h5","type":"hint","dependencies":["a3abd22AddRational2a-h4"],"title":"Factor the Numerator","text":"We can factor the sum of numerator we yield from step above to check if we can simplify the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x+2\\\\right) \\\\left(x+7\\\\right)$$"],"dependencies":["a3abd22AddRational2a-h5"],"title":"Factor the Numerator","text":"Factor $$x^2+9x+14$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational2a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x+2\\\\right) \\\\left(x+7\\\\right)$$"],"dependencies":["a3abd22AddRational2a-h6"],"title":"Factor the Numerator","text":"In order to $$\\\\operatorname{factor}\\\\left(x^2\\\\right)+9x+14$$, you can use reverse foiling, completing square or quadratic formula. In here we can use reverse-foiling. $$\\\\left(x+a\\\\right) \\\\left(x+b\\\\right)=x^2+\\\\left(a+b\\\\right) x+ab$$. In this case, we get $$a b=14$$ and $$a+b=9$$. Assume both a and $$b$$ are integers, $$a=2$$ and $$b=7$$ is only one possible combination that satisfies $$a b=14$$ and $$a+b=9$$. So you can factor $$x^2+9x+14$$ as ?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational2a-h8","type":"hint","dependencies":["a3abd22AddRational2a-h7"],"title":"Rewrite the expression","text":"From above steps, we get the sum of numerator is $$\\\\left(x+2\\\\right) \\\\left(x+7\\\\right)$$ and the denominator is $$x+7$$. So we can rewrite((9*x+14)/(x+7))+((x**2)/(x+7))as $$\\\\frac{\\\\left(x+2\\\\right) \\\\left(x+7\\\\right)}{x+7}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational2a-h9","type":"hint","dependencies":["a3abd22AddRational2a-h8"],"title":"Simplify Expression by Canceling out Terms","text":"Are there any ways to simplify $$\\\\frac{\\\\left(x+2\\\\right) \\\\left(x+7\\\\right)}{x+7}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational2a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+7$$"],"dependencies":["a3abd22AddRational2a-h9"],"title":"Simplify Expression by Canceling out Terms","text":"What is the greatest common factor shared by the numerator and denominator in $$\\\\frac{\\\\left(x+2\\\\right) \\\\left(x+7\\\\right)}{x+7}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational2a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+2$$"],"dependencies":["a3abd22AddRational2a-h10"],"title":"Simplify Expression by Canceling out Terms","text":"As the final step, we cancel out $$x+7$$ on both numerator and denominator, what is the simplified form of $$\\\\frac{\\\\left(x+2\\\\right) \\\\left(x+7\\\\right)}{x+7}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational2a-h12","type":"hint","dependencies":["a3abd22AddRational2a-h11"],"title":"Final Remark","text":"Note that the $$\\\\operatorname{expression}\\\\left(\\\\frac{9x+14}{x+7}\\\\right)+\\\\frac{x^2}{x+7}$$ and $$x+2$$ are different because the domain of $$\\\\frac{9x+14}{x+7}+\\\\frac{x^2}{x+7}$$ does not include $$x=-7$$. The domain of $$x+2$$ includes $$x=-7$$. Strictly speaking, $$\\\\frac{9x+14}{x+7}+\\\\frac{x^2}{x+7}$$ equals $$x+2$$ excluding the domain $$x=-7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3abd22AddRational3","title":"Rational Expression Addition","body":"Add the following rational expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Add and Subtract Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3abd22AddRational3a","stepAnswer":["$$x+3$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{x^2+8x}{x+5}+\\\\frac{15}{x+5}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x+3$$","hints":{"DefaultPathway":[{"id":"a3abd22AddRational3a-h1","type":"hint","dependencies":[],"title":"Find the Common Denominator","text":"Do all the terms in the expression have common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational3a-h2","type":"hint","dependencies":["a3abd22AddRational3a-h1"],"title":"Find the Common Denominator","text":"Yes, $$\\\\frac{x^2+8x}{x+5}$$ and $$\\\\frac{15}{x+5}$$ shared the same common denominator of $$x+5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational3a-h3","type":"hint","dependencies":["a3abd22AddRational3a-h2"],"title":"Add Numerators","text":"Since all of fractions in the expression have common denominator, we can add the numerators and place the sum over the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2+8x+15$$"],"dependencies":["a3abd22AddRational3a-h3"],"title":"Add Numerators","text":"What is the sum of numerators for $$\\\\frac{x^2+8x}{x+5}$$ and $$\\\\frac{15}{x+5}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational3a-h5","type":"hint","dependencies":["a3abd22AddRational3a-h4"],"title":"Factor the Numerator","text":"We can factor the sum of numerator we yield from step above to check if we can simplify the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational3a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x+2\\\\right) \\\\left(x+7\\\\right)$$"],"dependencies":["a3abd22AddRational3a-h5"],"title":"Factor the Numerator","text":"Factor $$x^2+8x+15$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational3a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x+2\\\\right) \\\\left(x+7\\\\right)$$"],"dependencies":["a3abd22AddRational3a-h6"],"title":"Factor the Numerator","text":"In order to factor $$x^2+8x+15$$, you can use reverse foiling, completing square or quadratic formula. In here we can use reverse-foiling. $$\\\\left(x+a\\\\right) \\\\left(x+b\\\\right)=x^2+\\\\left(a+b\\\\right) x+ab$$. In this case, we get $$a b=15$$ and $$a+b=8$$. Assume both a and $$b$$ are integers, $$a=3$$ and $$b=5$$ is only one possible combination that satisfies $$a b=15$$ and $$a+b=8$$. So you can factor $$x^2+8x+15$$ as ?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational3a-h8","type":"hint","dependencies":["a3abd22AddRational3a-h7"],"title":"Rewrite the expression","text":"From above steps, we get the sum of numerator is $$\\\\left(x+3\\\\right) \\\\left(x+5\\\\right)$$ and the denominator is $$x+5$$. So we can rewrite((x**2+8*x)/(x+5))+((15)/(x+5))as $$\\\\frac{\\\\left(x+3\\\\right) \\\\left(x+5\\\\right)}{x+5}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational3a-h9","type":"hint","dependencies":["a3abd22AddRational3a-h8"],"title":"Simplify expression by Canceling out Terms","text":"Are there any ways to simplify $$\\\\frac{\\\\left(x+3\\\\right) \\\\left(x+5\\\\right)}{x+5}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational3a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+5$$"],"dependencies":["a3abd22AddRational3a-h9"],"title":"Simplify expression by Canceling out Terms","text":"What is the greatest common factor shared by the numerator and denominator in $$\\\\frac{\\\\left(x+3\\\\right) \\\\left(x+5\\\\right)}{x+5}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational3a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+3$$"],"dependencies":["a3abd22AddRational3a-h10"],"title":"Simplify expression by Canceling out Terms","text":"As the final step, we cancel out $$x+5$$ on both numerator and denominator, what is the simplified form of $$\\\\frac{\\\\left(x+3\\\\right) \\\\left(x+5\\\\right)}{x+5}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational3a-h12","type":"hint","dependencies":["a3abd22AddRational3a-h11"],"title":"Final Remark","text":"Note that the expression $$\\\\frac{x^2+8x}{x+5}+\\\\frac{15}{x+5}$$ and $$x+3$$ are different because the domain of ((x**2+8*x)/(x+5))+((15)/(x+5))) does not include $$x=-5$$. The domain of $$x+3$$ does not exclude $$x=-5$$. Strictly $$\\\\operatorname{speaking}\\\\left(\\\\frac{x^2+8x}{x+5}\\\\right)+\\\\frac{15}{x+5}$$ equals $$x+3$$ excluding the domain $$x=-5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3abd22AddRational4","title":"Rational Expression Subtraction","body":"Subtract the following rational expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Add and Subtract Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3abd22AddRational4a","stepAnswer":["$$\\\\frac{x-2}{x+3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{5x^2-7x+3}{x^2-3x-18}-\\\\frac{4x^2+x-9}{x^2-3x-18}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{x-2}{x+3}$$","hints":{"DefaultPathway":[{"id":"a3abd22AddRational4a-h1","type":"hint","dependencies":[],"title":"Common Denominator","text":"Do all the terms in the expression have common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational4a-h2","type":"hint","dependencies":["a3abd22AddRational4a-h1"],"title":"Common Denominator","text":"Yes, $$\\\\frac{5x^2-7x+3}{x^2-3x-18}$$ and $$\\\\frac{4x^2+x-9}{x^2-3x-18}$$ shared the same common denominator of $$x^2-3x-18$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational4a-h3","type":"hint","dependencies":["a3abd22AddRational4a-h2"],"title":"Subtract the Numerators for Expression","text":"Since all of fractions in the expression have common denominator, we can subtract the numerators and place the difference over the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2-8x+12$$"],"dependencies":["a3abd22AddRational4a-h3"],"title":"Subtract the Numerators for Expression","text":"What is the difference of numerators for $$\\\\frac{5x^2-7x+3}{x^2-3x-18}-\\\\frac{4x^2+x-9}{x^2-3x-18}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational4a-h5","type":"hint","dependencies":["a3abd22AddRational4a-h4"],"title":"Factor the Numerator","text":"We can factor the difference of numerator we yield from step above to check if we can simplify the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational4a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x-2\\\\right) \\\\left(x-6\\\\right)$$"],"dependencies":["a3abd22AddRational4a-h5"],"title":"Factor the Numerator","text":"Factor $$x^2-8x+12$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational4a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x-2\\\\right) \\\\left(x-6\\\\right)$$"],"dependencies":["a3abd22AddRational4a-h6"],"title":"Factor the Numerator","text":"In order to factor $$x^2-8x+12$$, you can use reverse foiling, completing square or quadratic formula. In here we can use reverse-foiling. $$\\\\left(x-a\\\\right) \\\\left(x-b\\\\right)=x^2-\\\\left(a+b\\\\right) x+ab$$. In this case, we get $$a b=12$$ and $$-\\\\left(a+b\\\\right)=-8$$. Assume both a and $$b$$ are integers, $$a=2$$ and $$b=6$$ is only one possible combination that satisfies $$a b=12$$ and $$-\\\\left(a+b\\\\right)=-8$$. So you can factor $$x^2-8x+12$$ as ?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational4a-h8","type":"hint","dependencies":["a3abd22AddRational4a-h7"],"title":"Rewrite the expression","text":"From above steps, we get the difference of numerator is $$\\\\left(x-2\\\\right) \\\\left(x-6\\\\right)$$ and the denominator is $$x^2-3x-18$$. So we can rewrite $$\\\\frac{5x^2-7x+3}{x^2-3x-18}-\\\\frac{4x^2+x-9}{x^2-3x-18}$$ as $$\\\\frac{\\\\left(x-2\\\\right) \\\\left(x-6\\\\right)}{x^2-3x-18}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational4a-h9","type":"hint","dependencies":["a3abd22AddRational4a-h8"],"title":"Simplify expression by Canceling out Terms","text":"Are there any ways to simplify $$\\\\frac{\\\\left(x-2\\\\right) \\\\left(x-6\\\\right)}{x^2-3x-18}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational4a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x-6\\\\right) \\\\left(x+3\\\\right)$$"],"dependencies":["a3abd22AddRational4a-h9"],"title":"Factor the Denominator","text":"Factor $$x^2-3x-18$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational4a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x-6$$"],"dependencies":["a3abd22AddRational4a-h10"],"title":"Simplify expression by Canceling out Terms","text":"Put back together, we can rewrite $$\\\\frac{5x^2-7x+3}{x^2-3x-18}-\\\\frac{4x^2+x-9}{x^2-3x-18}$$ as $$\\\\frac{\\\\left(x-2\\\\right) \\\\left(x-6\\\\right)}{\\\\left(x+3\\\\right) \\\\left(x-6\\\\right)}$$. What is the greatest common factor shared by the numerator and denominator in $$\\\\frac{\\\\left(x-2\\\\right) \\\\left(x-6\\\\right)}{\\\\left(x+3\\\\right) \\\\left(x-6\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational4a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{x-2}{x+3}$$"],"dependencies":["a3abd22AddRational4a-h11"],"title":"Simplify expression by Canceling out Terms","text":"As the final step, we cancel out $$x-6$$ on both numerator and denominator, what is the simplified form of $$\\\\frac{\\\\left(x-2\\\\right) \\\\left(x-6\\\\right)}{\\\\left(x+3\\\\right) \\\\left(x-6\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational4a-h13","type":"hint","dependencies":["a3abd22AddRational4a-h12"],"title":"Final Remark","text":"Note that the expression $$\\\\frac{5x^2-7x+3}{x^2-3x-18}-\\\\frac{4x^2+x-9}{x^2-3x-18}$$ and $$\\\\frac{x-2}{x+3}$$ are different because the domain of $$\\\\frac{5x^2-7x+3}{x^2-3x-18}-\\\\frac{4x^2+x-9}{x^2-3x-18}$$ does not include $$x=-3$$ and $$x=6$$. The domain of $$\\\\frac{x-2}{x+3}$$ does not exclude $$x=6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3abd22AddRational5","title":"Rational Expression Subtraction","body":"Subtract the following rational expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Add and Subtract Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3abd22AddRational5a","stepAnswer":["$$\\\\frac{x-11}{x-2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{4x^2-11x+8}{x^2-3x+2}-\\\\frac{3x^2+x-3}{x^2-3x+2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{x-11}{x-2}$$","hints":{"DefaultPathway":[{"id":"a3abd22AddRational5a-h1","type":"hint","dependencies":[],"title":"Find the Common Denominator","text":"Do all the terms in the expression have common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational5a-h2","type":"hint","dependencies":["a3abd22AddRational5a-h1"],"title":"Find the Common Denominator","text":"Yes, $$\\\\frac{4x^2-11x+8}{x^2-3x+2}$$ and $$\\\\frac{3x^2+x-3}{x^2-3x+2}$$ shared the same common denominator of $$x^2-3x+2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational5a-h3","type":"hint","dependencies":["a3abd22AddRational5a-h2"],"title":"Subtract the Numerators for Expression","text":"Since all of fractions in the expression have common denominator, we can subtract the numerators and place the difference over the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2-12x+11$$"],"dependencies":["a3abd22AddRational5a-h3"],"title":"Subtract the Numerators for Expression","text":"What is the difference of the numerators for (((4*x**2)-11*x+8)/(x**2-3*x+2))-((3*x**2+x-3)/(x**2-3*x+2)))?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational5a-h5","type":"hint","dependencies":["a3abd22AddRational5a-h4"],"title":"Factor the Numerator","text":"We can factor the difference of numerator we yield from step above to check if we can simplify the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational5a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x-11\\\\right) \\\\left(x-1\\\\right)$$"],"dependencies":["a3abd22AddRational5a-h5"],"title":"Factor the Numerator","text":"Factor $$x^2-12x+11$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational5a-h7","type":"hint","dependencies":["a3abd22AddRational5a-h6"],"title":"Rewrite the expression","text":"From above steps, we get the difference of numerator is $$\\\\left(x-11\\\\right) \\\\left(x-1\\\\right)$$ and the denominator is $$x^2-3x+2$$. So we can rewrite $$\\\\frac{4x^2-11x+8}{x^2-3x+2}-\\\\frac{3x^2+x-3}{x^2-3x+2}$$ as $$\\\\frac{\\\\left(x-11\\\\right) \\\\left(x-1\\\\right)}{x^2-3x+2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational5a-h8","type":"hint","dependencies":["a3abd22AddRational5a-h7"],"title":"Simplify expression by Canceling out Terms","text":"Are there any ways to simplify $$\\\\frac{\\\\left(x-11\\\\right) \\\\left(x-1\\\\right)}{x^2-3x+2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational5a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x-1\\\\right) \\\\left(x-2\\\\right)$$"],"dependencies":["a3abd22AddRational5a-h8"],"title":"Factor the Denominator","text":"Factor $$x^2-3x+2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational5a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x-1$$"],"dependencies":["a3abd22AddRational5a-h9"],"title":"Simplify Expression By Canceling out Terms","text":"Put back together, we can rewrite $$\\\\frac{4x^2-11x+8}{x^2-3x+2}-\\\\frac{3x^2+x-3}{x^2-3x+2}$$ as $$\\\\frac{\\\\left(x-11\\\\right) \\\\left(x-1\\\\right)}{\\\\left(x-1\\\\right) \\\\left(x-2\\\\right)}$$. What is the greatest common factor shared by the numerator and denominator in $$\\\\frac{\\\\left(x-11\\\\right) \\\\left(x-1\\\\right)}{\\\\left(x-1\\\\right) \\\\left(x-2\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational5a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{x-11}{x-2}$$"],"dependencies":["a3abd22AddRational5a-h10"],"title":"Simplify Expression By Canceling out Terms","text":"As the final step, we cancel out $$x-1$$ on both numerator and denominator, what is the simplified form of ((x-11)*(x-1))/((x-1)*(x-2)))?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational5a-h12","type":"hint","dependencies":["a3abd22AddRational5a-h11"],"title":"Final Remark","text":"Note that the expression (((4*x**2)-11*x+8)/(x**2-3*x+2))-((3*x**2+x-3)/(x**2-3*x+2))) and $$\\\\frac{x-11}{x-2}$$ are different because the domain of (((4*x**2)-11*x+8)/(x**2-3*x+2))-((3*x**2+x-3)/(x**2-3*x+2))) does not include $$x=1$$. The domain of $$\\\\frac{x-2}{x+3}$$ does not exclude $$x=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3abd22AddRational6","title":"Rational Expression Subtraction","body":"Subtract the following rational expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Add and Subtract Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3abd22AddRational6a","stepAnswer":["$$\\\\frac{x-3}{x+9}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{6x^2-x+20}{x^2-81}-\\\\frac{5x^2+11x-7}{x^2-81}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{x-3}{x+9}$$","hints":{"DefaultPathway":[{"id":"a3abd22AddRational6a-h1","type":"hint","dependencies":[],"title":"Find the Common Denominator","text":"Do all the terms in the expression have common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational6a-h2","type":"hint","dependencies":["a3abd22AddRational6a-h1"],"title":"Find the Common Denominator","text":"Yes, $$\\\\frac{6x^2-x+20}{x^2-81}$$ and $$\\\\frac{5x^2+11x-7}{x^2-81}$$ shared the same common denominator of $$x^2-81$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational6a-h3","type":"hint","dependencies":["a3abd22AddRational6a-h2"],"title":"Subtract the Numerators for Expression","text":"Since all of fractions in the expression have common denominator, we can subtract the numerators and place the sum over the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2-12x+27$$"],"dependencies":["a3abd22AddRational6a-h3"],"title":"Subtract the Numerators for Expression","text":"What is the difference of the numerators for $$\\\\frac{6x^2-x+20}{x^2-81}-\\\\frac{5x^2+11x-7}{x^2-81}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational6a-h5","type":"hint","dependencies":["a3abd22AddRational6a-h4"],"title":"Factor the Numerator","text":"We can factor the difference of numerator we yield from step above to check if we can simplify the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational6a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x-3\\\\right) \\\\left(x-9\\\\right)$$"],"dependencies":["a3abd22AddRational6a-h5"],"title":"Factor the Numerator","text":"Factor $$x^2-12x+27$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational6a-h7","type":"hint","dependencies":["a3abd22AddRational6a-h6"],"title":"Rewrite the expression","text":"From above steps, we get the difference of numerator is $$\\\\left(x-3\\\\right) \\\\left(x-9\\\\right)$$ and the denominator is $$x^2-81$$. So we can rewrite $$\\\\frac{6x^2-x+20}{x^2-81}-\\\\frac{5x^2+11x-7}{x^2-81}$$ as $$\\\\frac{\\\\left(x-3\\\\right) \\\\left(x-9\\\\right)}{x^2-81}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational6a-h8","type":"hint","dependencies":["a3abd22AddRational6a-h7"],"title":"Simplify expression by Canceling out Terms","text":"Are there any ways to simplify $$\\\\frac{\\\\left(x-3\\\\right) \\\\left(x-9\\\\right)}{x^2-81}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational6a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x-9\\\\right) \\\\left(x+9\\\\right)$$"],"dependencies":["a3abd22AddRational6a-h8"],"title":"Factor the Denominator","text":"Factor $$x^2-81$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational6a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x-9$$"],"dependencies":["a3abd22AddRational6a-h9"],"title":"Simplify Expression By Canceling out Terms","text":"Put back together, we can rewrite $$\\\\frac{6x^2-x+20}{x^2-81}-\\\\frac{5x^2+11x-7}{x^2-81}$$ as $$\\\\frac{\\\\left(x-3\\\\right) \\\\left(x-9\\\\right)}{\\\\left(x-9\\\\right) \\\\left(x+9\\\\right)}$$. What is the greatest common factor shared by the numerator and denominator in $$\\\\frac{\\\\left(x-3\\\\right) \\\\left(x-9\\\\right)}{\\\\left(x-9\\\\right) \\\\left(x+9\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational6a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{x-3}{x+9}$$"],"dependencies":["a3abd22AddRational6a-h10"],"title":"Simplify Expression By Canceling out Terms","text":"As the final step, we cancel out $$x-9$$ on both numerator and denominator, what is the simplified form of $$\\\\frac{\\\\left(x-3\\\\right) \\\\left(x-9\\\\right)}{\\\\left(x-9\\\\right) \\\\left(x+9\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational6a-h12","type":"hint","dependencies":["a3abd22AddRational6a-h11"],"title":"Final Remark","text":"Note that the $$\\\\operatorname{expression}\\\\left(\\\\frac{6x^2-x+20}{x^2-81}\\\\right)-\\\\frac{5x^2+11x-7}{x^2-81}$$ and $$\\\\frac{x-3}{x+9}$$ are different because the domain of $$\\\\frac{6x^2-x+20}{x^2-81}-\\\\frac{5x^2+11x-7}{x^2-81}$$ does not include $$x=9$$. The domain of $$\\\\frac{x-3}{x+9}$$ does not exclude $$x=9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3abd22AddRational7","title":"Rational Expression Subtraction","body":"Subtract the following rational expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Add and Subtract Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3abd22AddRational7a","stepAnswer":["$$\\\\frac{m-2}{m+1}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{m^2-6m}{m^2-1}-\\\\frac{3m+2}{1-m^2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{m-2}{m+1}$$","hints":{"DefaultPathway":[{"id":"a3abd22AddRational7a-h1","type":"hint","dependencies":[],"title":"Find the Common Denominator","text":"Do all the terms in the expression have common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational7a-h2","type":"hint","dependencies":["a3abd22AddRational7a-h1"],"title":"Find the Common Denominator","text":"No, the denominators of terms in the expression are different. We can find the common denominator for both terms. Notice that $$m^2-1=-1\\\\left(1-m^2\\\\right)$$. So we can rewrite the expression as $$\\\\frac{m^2-6m}{m^2-1}-\\\\left(-\\\\frac{3m+2}{m^2-1}\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational7a-h3","type":"hint","dependencies":["a3abd22AddRational7a-h2"],"title":"Subtract the Numerators for Expression","text":"Since all the terms in the expression have the same denominator, we can directly subtract their numerators.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$m^2-3m+2$$"],"dependencies":["a3abd22AddRational7a-h3"],"title":"Subtract the Numerators for Expression","text":"What is the difference of numerators for $$\\\\frac{m^2-6m}{m^2-1}-\\\\left(-\\\\frac{3m+2}{m^2-1}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational7a-h5","type":"hint","dependencies":["a3abd22AddRational7a-h4"],"title":"Rewrite the expression","text":"We can rewrite $$\\\\frac{m^2-6m}{m^2-1}-\\\\frac{3m+2}{1-m^2}$$ as $$\\\\frac{m^2-3m+2}{m^2-1}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational7a-h6","type":"hint","dependencies":["a3abd22AddRational7a-h5"],"title":"Simplify Expression By Canceling out Terms","text":"What is the greatest common factor shared by the numerator and denominator in $$\\\\frac{\\\\left(x+2\\\\right) \\\\left(x+7\\\\right)}{x+7}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational7a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(m-1\\\\right) \\\\left(m-2\\\\right)$$"],"dependencies":["a3abd22AddRational7a-h6"],"title":"Factor the Numerator","text":"Factor $$m^2-3m+2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational7a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(m-1\\\\right) \\\\left(m+1\\\\right)$$"],"dependencies":["a3abd22AddRational7a-h7"],"title":"Factor the Denominator","text":"Factor $$m^2-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational7a-h9","type":"hint","dependencies":["a3abd22AddRational7a-h8"],"title":"Simplify Expression By Canceling out Terms","text":"We can rewrite $$\\\\frac{m^2-6m}{m^2-1}-\\\\frac{3m+2}{1-m^2}$$ as $$\\\\frac{\\\\left(m-1\\\\right) \\\\left(m-2\\\\right)}{\\\\left(m-1\\\\right) \\\\left(m+1\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational7a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$m-1$$"],"dependencies":["a3abd22AddRational7a-h9"],"title":"Simplify Expression By Canceling out Terms","text":"What is the greatest common factor in numerator and denominator of $$\\\\frac{\\\\left(m-1\\\\right) \\\\left(m-2\\\\right)}{\\\\left(m-1\\\\right) \\\\left(m+1\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational7a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{m-2}{m+1}$$"],"dependencies":["a3abd22AddRational7a-h10"],"title":"Simplify Expression By Canceling out Terms","text":"As the last step, we will cancel out $$m-1$$ from both numerator and denominator, what is the simplified form of $$\\\\frac{\\\\left(m-1\\\\right) \\\\left(m-2\\\\right)}{\\\\left(m-1\\\\right) \\\\left(m+1\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational7a-h12","type":"hint","dependencies":["a3abd22AddRational7a-h11"],"title":"Final Remark","text":"Note that the expression $$\\\\frac{m^2-6m}{m^2-1}-\\\\frac{3m+2}{1-m^2}$$ and $$\\\\frac{m-2}{m+1}$$ are different because the domain of $$\\\\frac{m^2-6m}{m^2-1}-\\\\frac{3m+2}{1-m^2}$$ does not include $$x=1$$ and the domain of $$\\\\frac{m-2}{m+1}$$ does not exclude $$x=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3abd22AddRational8","title":"Rational Expression Subtraction","body":"Subtract the following rational expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Add and Subtract Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3abd22AddRational8a","stepAnswer":["$$\\\\frac{y+3}{y+2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{y^2-5y}{y^2-4}-\\\\frac{6y-6}{4-y^2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{y+3}{y+2}$$","hints":{"DefaultPathway":[{"id":"a3abd22AddRational8a-h1","type":"hint","dependencies":[],"title":"Find the Common Denominator","text":"Do all the terms in the expression have common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational8a-h2","type":"hint","dependencies":["a3abd22AddRational8a-h1"],"title":"Find the Common Denominator","text":"No, the denominators of terms in the expression are different. We can find the common denominator for both terms. Notice that $$y^2-4=-1\\\\left(4-y^2\\\\right)$$. We can multiply $$\\\\frac{\\\\left(-1\\\\right)}{\\\\left(-1\\\\right)}$$ with $$\\\\frac{6y-6}{4-y^2}$$. We can rewrite the expression as $$\\\\frac{y^2-5y}{y^2-4}-\\\\left(-\\\\frac{6y-6}{y^2-4}\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational8a-h3","type":"hint","dependencies":["a3abd22AddRational8a-h2"],"title":"Subtract the Numerators for Expression","text":"Since all the terms in the expression have the same denominator, we can directly subtract their numerators.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y^2+y-6$$"],"dependencies":["a3abd22AddRational8a-h3"],"title":"Subtract the Numerators for Expression","text":"What is the difference of numerators for $$\\\\frac{y^2-5y}{y^2-4}-\\\\left(-\\\\frac{6y-6}{y^2-4}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational8a-h5","type":"hint","dependencies":["a3abd22AddRational8a-h4"],"title":"Rewrite the expression","text":"We can rewrite $$\\\\frac{y^2-5y}{y^2-4}-\\\\frac{6y-6}{4-y^2}$$ as $$\\\\frac{y^2+y-6}{y^2-4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational8a-h6","type":"hint","dependencies":["a3abd22AddRational8a-h5"],"title":"Simplify Expression By Canceling out Terms","text":"We can try to factor the numerator and denominator to see if we can further simplify the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational8a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(y+3\\\\right) \\\\left(y-2\\\\right)$$"],"dependencies":["a3abd22AddRational8a-h6"],"title":"Factor the Numerator","text":"Factor $$y^2+y-6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational8a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(y-2\\\\right) \\\\left(y+2\\\\right)$$"],"dependencies":["a3abd22AddRational8a-h7"],"title":"Factor the Denominator","text":"Factor $$y^2-4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational8a-h9","type":"hint","dependencies":["a3abd22AddRational8a-h8"],"title":"Simplify Expression By Canceling out Terms","text":"We can rewrite $$\\\\frac{y^2-5y}{y^2-4}-\\\\frac{6y-6}{4-y^2}$$ as $$\\\\frac{\\\\left(y+3\\\\right) \\\\left(y-2\\\\right)}{\\\\left(y-2\\\\right) \\\\left(y+2\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational8a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y-2$$"],"dependencies":["a3abd22AddRational8a-h9"],"title":"Simplify Expression By Canceling out Terms","text":"What is the greatest common factor in numerator and denominator of $$\\\\frac{\\\\left(y+3\\\\right) \\\\left(y-2\\\\right)}{\\\\left(y-2\\\\right) \\\\left(y+2\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational8a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{y+3}{y+2}$$"],"dependencies":["a3abd22AddRational8a-h10"],"title":"Simplify Expression By Canceling out Terms","text":"As the last step, we will cancel out $$y-2$$ from both numerator and denominator, what is the simplified form of $$\\\\frac{\\\\left(y+3\\\\right) \\\\left(y-2\\\\right)}{\\\\left(y-2\\\\right) \\\\left(y+2\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational8a-h12","type":"hint","dependencies":["a3abd22AddRational8a-h11"],"title":"Final Remark","text":"Note that the expression ((y**2-5*y)/(y**2-4))-((6*y-6)/(4-y**2))and $$\\\\frac{y+3}{y+2}$$ are different because the domain of $$\\\\frac{y^2-5y}{y^2-4}-\\\\frac{6y-6}{4-y^2}$$ does not include $$x=2$$ and the domain of $$\\\\frac{y+3}{y+2}$$ does not exclude $$x=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3abd22AddRational9","title":"Rational Expression Subtraction","body":"Subtract the following rational expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Add and Subtract Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3abd22AddRational9a","stepAnswer":["$$\\\\frac{3n-2}{n-1}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2n^2+8n-1}{n^2-1}-\\\\frac{n^2-7n-1}{1-n^2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3n-2}{n-1}$$","hints":{"DefaultPathway":[{"id":"a3abd22AddRational9a-h1","type":"hint","dependencies":[],"title":"Find the Common Denominator","text":"Do all the terms in the expression have common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational9a-h2","type":"hint","dependencies":["a3abd22AddRational9a-h1"],"title":"Find the Common Denominator","text":"No, the denominators of terms in the expression are different. We can find the common denominator for both terms. Notice that $$n^2-1=-1\\\\left(1-n^2\\\\right)$$. We can multiply $$\\\\frac{\\\\left(-1\\\\right)}{\\\\left(-1\\\\right)}$$ with $$\\\\frac{n^2-7n-1}{1-n^2}$$. We can rewrite the expression as $$\\\\frac{2n^2+8n-1}{n^2-1}-\\\\left(-\\\\frac{n^2-7n-1}{n^2-1}\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational9a-h3","type":"hint","dependencies":["a3abd22AddRational9a-h2"],"title":"Subtract Numerators","text":"Since all the terms in the expression have the same denominator, we can directly subtract their numerators.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3n^2+n-2$$"],"dependencies":["a3abd22AddRational9a-h3"],"title":"Subtract Numerators","text":"What is the difference of numerators for $$\\\\frac{2n^2+8n-1}{n^2-1}-\\\\left(-\\\\frac{n^2-7n-1}{n^2-1}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational9a-h5","type":"hint","dependencies":["a3abd22AddRational9a-h4"],"title":"Rewrite the expression","text":"We can rewrite $$\\\\frac{2n^2+8n-1}{n^2-1}-\\\\frac{n^2-7n-1}{1-n^2}$$ as $$\\\\frac{3n^2+n-2}{n^2-1}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational9a-h6","type":"hint","dependencies":["a3abd22AddRational9a-h5"],"title":"Simplify Expression By Canceling out Terms","text":"We can try to factor the numerator and denominator to see if we can further simplify the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational9a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(3n-2\\\\right) \\\\left(n+1\\\\right)$$"],"dependencies":["a3abd22AddRational9a-h6"],"title":"Factor the Numerator","text":"Factor $$3n^2+n-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational9a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(n-1\\\\right) \\\\left(n+1\\\\right)$$"],"dependencies":["a3abd22AddRational9a-h7"],"title":"Factor the Denominator","text":"Factor $$n^2-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational9a-h9","type":"hint","dependencies":["a3abd22AddRational9a-h8"],"title":"Simplify Expression By Canceling out Terms","text":"We can rewrite $$\\\\frac{2n^2+8n-1}{n^2-1}-\\\\frac{n^2-7n-1}{1-n^2}$$ as $$\\\\frac{\\\\left(3n-2\\\\right) \\\\left(n+1\\\\right)}{\\\\left(n-1\\\\right) \\\\left(n+1\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational9a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$n+1$$"],"dependencies":["a3abd22AddRational9a-h9"],"title":"Simplify Expression By Canceling out Terms","text":"What is the greatest common factor in numerator and denominator of $$\\\\frac{\\\\left(3n-2\\\\right) \\\\left(n+1\\\\right)}{\\\\left(n-1\\\\right) \\\\left(n+1\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational9a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3n-2}{n-1}$$"],"dependencies":["a3abd22AddRational9a-h10"],"title":"Simplify Expression By Canceling out Terms","text":"As the last step, we will cancel out $$n+1$$ from both numerator and denominator, what is the simplified form of $$\\\\frac{\\\\left(3n-2\\\\right) \\\\left(n+1\\\\right)}{\\\\left(n-1\\\\right) \\\\left(n+1\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22AddRational9a-h12","type":"hint","dependencies":["a3abd22AddRational9a-h11"],"title":"Final Remark","text":"Note that the expression (((2n**2)+8*n-1)/(n**2-1))-((n**2-7*n-1)/(1-n**2))) and $$\\\\frac{3n-2}{n-1}$$ are different because the domain of $$\\\\frac{2n^2+8n-1}{n^2-1}-\\\\frac{n^2-7n-1}{1-n^2}$$ does not include $$x=-1$$ and the domain of $$\\\\frac{3n-2}{n-1}$$ does not exclude $$x=-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3abd22addsubrational1","title":"Adding and Subtracting Rational Expressions","body":"Add the rational expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Add and Subtract Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3abd22addsubrational1a","stepAnswer":["$$\\\\frac{3}{5}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2}{15}+\\\\frac{7}{15}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{5}$$","hints":{"DefaultPathway":[{"id":"a3abd22addsubrational1a-h1","type":"hint","dependencies":[],"title":"Same Denominators","text":"Make sure the fractions have the same denominator before adding them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational1a-h2","type":"hint","dependencies":["a3abd22addsubrational1a-h1"],"title":"Divide Rational Expressions","text":"Since the denominators are the same, put both of the numerators of the rational expression over a single denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational1a-h3","type":"hint","dependencies":["a3abd22addsubrational1a-h2"],"title":"Finding Numerators","text":"Think about what the numerators are.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational1a-h4","type":"hint","dependencies":["a3abd22addsubrational1a-h3"],"title":"Adding and Subtracting the Numerators","text":"Add the numerators together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational1a-h5","type":"hint","dependencies":["a3abd22addsubrational1a-h4"],"title":"Simplify","text":"Can $$\\\\frac{9}{15}$$ be simplified further?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3abd22addsubrational10","title":"Adding and Subtracting Rational Expressions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Add and Subtract Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3abd22addsubrational10a","stepAnswer":["$$5b+6$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{25b^2}{5b-6}-\\\\frac{36}{5b-6}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5b+6$$","hints":{"DefaultPathway":[{"id":"a3abd22addsubrational10a-h1","type":"hint","dependencies":[],"title":"Same Denominators","text":"Make sure the fractions have the same denominator before adding them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational10a-h2","type":"hint","dependencies":["a3abd22addsubrational10a-h1"],"title":"Divide Rational Expressions","text":"Since the denominators are the same, put both of the numerators of the rational expression over a single denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational10a-h3","type":"hint","dependencies":["a3abd22addsubrational10a-h2"],"title":"Finding Numerators","text":"Think about what the numerators are.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational10a-h4","type":"hint","dependencies":["a3abd22addsubrational10a-h3"],"title":"Adding and Subtracting the Numerators","text":"Subtract the numerators from each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational10a-h5","type":"hint","dependencies":["a3abd22addsubrational10a-h4"],"title":"Simplify","text":"Can (25*(b**2))-36)/((5*b)-6) be simplified further?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational10a-h6","type":"hint","dependencies":["a3abd22addsubrational10a-h5"],"title":"Factoring","text":"Try factoring the numerator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational10a-h7","type":"hint","dependencies":["a3abd22addsubrational10a-h6"],"title":"Difference of Squares","text":"Notice that the numerator has a difference of squares.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational10a-h8","type":"hint","dependencies":["a3abd22addsubrational10a-h7"],"title":"Simplify","text":"Can something in the numerator and denominator cancel out?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3abd22addsubrational11","title":"Adding and Subtracting Rational Expressions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Add and Subtract Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3abd22addsubrational11a","stepAnswer":["$$\\\\frac{m-2}{2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{3m^2}{6m-30}-\\\\frac{21m-30}{6m-30}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{m-2}{2}$$","hints":{"DefaultPathway":[{"id":"a3abd22addsubrational11a-h1","type":"hint","dependencies":[],"title":"Same Denominators","text":"Make sure the fractions have the same denominator before adding them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational11a-h2","type":"hint","dependencies":["a3abd22addsubrational11a-h1"],"title":"Divide Rational Expressions","text":"Since the denominators are the same, put both of the numerators of the rational expression over a single denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational11a-h3","type":"hint","dependencies":["a3abd22addsubrational11a-h2"],"title":"Finding Numerators","text":"Think about what the numerators are.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational11a-h4","type":"hint","dependencies":["a3abd22addsubrational11a-h3"],"title":"Adding and Subtracting the Numerators","text":"Subtract the numerators from each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational11a-h5","type":"hint","dependencies":["a3abd22addsubrational11a-h4"],"title":"Simplify","text":"Can $$\\\\frac{3m^2-21m-30}{6m-30}$$ be simplified further?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational11a-h6","type":"hint","dependencies":["a3abd22addsubrational11a-h5"],"title":"Factoring","text":"Try factoring the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational11a-h7","type":"hint","dependencies":["a3abd22addsubrational11a-h6"],"title":"Factoring the Numerator","text":"Notice that the numerator can be factored by $$3$$ and then factored again as a polynomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational11a-h8","type":"hint","dependencies":["a3abd22addsubrational11a-h7"],"title":"Factoring the Denominator","text":"Notice that the denominator can also be factored by $$6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational11a-h9","type":"hint","dependencies":["a3abd22addsubrational11a-h8"],"title":"Simplify","text":"Can something in the numerator and denominator cancel out and simplify?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3abd22addsubrational12","title":"Adding and Subtracting Rational Expressions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Add and Subtract Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3abd22addsubrational12a","stepAnswer":["$$\\\\frac{n-1}{2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2n^2}{4n-32}-\\\\frac{18n-16}{4n-32}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{n-1}{2}$$","hints":{"DefaultPathway":[{"id":"a3abd22addsubrational12a-h1","type":"hint","dependencies":[],"title":"Same Denominators","text":"Make sure the fractions have the same denominator before adding them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational12a-h2","type":"hint","dependencies":["a3abd22addsubrational12a-h1"],"title":"Divide Rational Expressions","text":"Since the denominators are the same, put both of the numerators of the rational expression over a single denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational12a-h3","type":"hint","dependencies":["a3abd22addsubrational12a-h2"],"title":"Finding Numerators","text":"Think about what the numerators are.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational12a-h4","type":"hint","dependencies":["a3abd22addsubrational12a-h3"],"title":"Adding and Subtracting the Numerators","text":"Subtract the numerators from each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational12a-h5","type":"hint","dependencies":["a3abd22addsubrational12a-h4"],"title":"Simplify","text":"Can $$\\\\frac{2n^2-18n-16}{4n-32}$$ be simplified further?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational12a-h6","type":"hint","dependencies":["a3abd22addsubrational12a-h5"],"title":"Factoring","text":"Try factoring the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational12a-h7","type":"hint","dependencies":["a3abd22addsubrational12a-h6"],"title":"Factoring the Numerator","text":"Notice that the numerator can be factored by $$2$$ and then factored again as a polynomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational12a-h8","type":"hint","dependencies":["a3abd22addsubrational12a-h7"],"title":"Factoring the Denominator","text":"Notice that the denominator can also be factored by $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational12a-h9","type":"hint","dependencies":["a3abd22addsubrational12a-h8"],"title":"Simplify","text":"Can something in the numerator and denominator cancel out and simplify?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3abd22addsubrational13","title":"Adding and Subtracting Rational Expressions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Add and Subtract Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3abd22addsubrational13a","stepAnswer":["$$\\\\frac{p+3}{p+5}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{6p^2+3p+4}{p^2+4p-5}-\\\\frac{5p^2+p+7}{p^2+4p-5}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{p+3}{p+5}$$","hints":{"DefaultPathway":[{"id":"a3abd22addsubrational13a-h1","type":"hint","dependencies":[],"title":"Same Denominators","text":"Make sure the fractions have the same denominator before adding them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational13a-h2","type":"hint","dependencies":["a3abd22addsubrational13a-h1"],"title":"Divide Rational Expressions","text":"Since the denominators are the same, put both of the numerators of the rational expression over a single denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational13a-h3","type":"hint","dependencies":["a3abd22addsubrational13a-h2"],"title":"Finding Numerators","text":"Group the numerators by like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational13a-h4","type":"hint","dependencies":["a3abd22addsubrational13a-h3"],"title":"Adding and Subtracting the Numerators","text":"Subtract the numerators from each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational13a-h5","type":"hint","dependencies":["a3abd22addsubrational13a-h4"],"title":"Simplify","text":"Can $$\\\\frac{p^2+2n-3}{p^2+4p-5}$$ be simplified further?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational13a-h6","type":"hint","dependencies":["a3abd22addsubrational13a-h5"],"title":"Factoring","text":"Try factoring the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational13a-h7","type":"hint","dependencies":["a3abd22addsubrational13a-h6"],"title":"Factoring the Numerator","text":"Notice that the numerator can be factored by undoing the FOIL.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational13a-h8","type":"hint","dependencies":["a3abd22addsubrational13a-h7"],"title":"Factoring the Denominator","text":"Notice that the denominator can be factored by undoing the FOIL.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational13a-h9","type":"hint","dependencies":["a3abd22addsubrational13a-h8"],"title":"Simplify","text":"Can something in the numerator and denominator cancel out and simplify?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3abd22addsubrational14","title":"Adding and Subtracting Rational Expressions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Add and Subtract Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3abd22addsubrational14a","stepAnswer":["$$\\\\frac{q-8}{q+4}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{5q^2+3q-9}{q^2+6q+8}-\\\\frac{4q^2+9q+7}{q^2+6p+8}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{q-8}{q+4}$$","hints":{"DefaultPathway":[{"id":"a3abd22addsubrational14a-h1","type":"hint","dependencies":[],"title":"Same Denominators","text":"Make sure the fractions have the same denominator before adding them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational14a-h2","type":"hint","dependencies":["a3abd22addsubrational14a-h1"],"title":"Divide Rational Expressions","text":"Since the denominators are the same, put both of the numerators of the rational expression over a single denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational14a-h3","type":"hint","dependencies":["a3abd22addsubrational14a-h2"],"title":"Finding Numerators","text":"Group the numerators by like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational14a-h4","type":"hint","dependencies":["a3abd22addsubrational14a-h3"],"title":"Adding and Subtracting the Numerators","text":"Subtract the numerators from each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational14a-h5","type":"hint","dependencies":["a3abd22addsubrational14a-h4"],"title":"Simplify","text":"Can $$\\\\frac{q^2-6q-16}{q^2+6q+8}$$ be simplified further?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational14a-h6","type":"hint","dependencies":["a3abd22addsubrational14a-h5"],"title":"Factoring","text":"Try factoring the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational14a-h7","type":"hint","dependencies":["a3abd22addsubrational14a-h6"],"title":"Factoring the Numerator","text":"Notice that the numerator can be factored by undoing the FOIL.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational14a-h8","type":"hint","dependencies":["a3abd22addsubrational14a-h7"],"title":"Factoring the Denominator","text":"Notice that the denominator can be factored by undoing the FOIL.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational14a-h9","type":"hint","dependencies":["a3abd22addsubrational14a-h8"],"title":"Simplify","text":"Can something in the numerator and denominator cancel out and simplify?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3abd22addsubrational15","title":"Adding and Subtracting Rational Expressions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Add and Subtract Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3abd22addsubrational15a","stepAnswer":["$$\\\\frac{r+9}{r+7}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{5r^2+7r-33}{r^2-49}-\\\\frac{4r^2+5r+30}{r^2-49}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{r+9}{r+7}$$","hints":{"DefaultPathway":[{"id":"a3abd22addsubrational15a-h1","type":"hint","dependencies":[],"title":"Same Denominators","text":"Make sure the fractions have the same denominator before adding them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational15a-h2","type":"hint","dependencies":["a3abd22addsubrational15a-h1"],"title":"Divide Rational Expressions","text":"Since the denominators are the same, put both of the numerators of the rational expression over a single denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational15a-h3","type":"hint","dependencies":["a3abd22addsubrational15a-h2"],"title":"Finding Numerators","text":"Group the numerators by like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational15a-h4","type":"hint","dependencies":["a3abd22addsubrational15a-h3"],"title":"Adding and Subtracting the Numerators","text":"Subtract the numerators from each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational15a-h5","type":"hint","dependencies":["a3abd22addsubrational15a-h4"],"title":"Simplify","text":"Can $$\\\\frac{r^2+2r-63}{r^2-49}$$ be simplified further?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational15a-h6","type":"hint","dependencies":["a3abd22addsubrational15a-h5"],"title":"Factoring","text":"Try factoring the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational15a-h7","type":"hint","dependencies":["a3abd22addsubrational15a-h6"],"title":"Factoring the Numerator","text":"Notice that the numerator can be factored by undoing the FOIL.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational15a-h8","type":"hint","dependencies":["a3abd22addsubrational15a-h7"],"title":"Factoring the Denominator","text":"Notice that the denominator is a binomial difference of squares.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational15a-h9","type":"hint","dependencies":["a3abd22addsubrational15a-h8"],"title":"Simplify","text":"Can something in the numerator and denominator cancel out and simplify?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3abd22addsubrational2","title":"Adding and Subtracting Rational Expressions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Add and Subtract Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3abd22addsubrational2a","stepAnswer":["$$\\\\frac{3}{4}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{7}{24}+\\\\frac{11}{24}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{4}$$","hints":{"DefaultPathway":[{"id":"a3abd22addsubrational2a-h1","type":"hint","dependencies":[],"title":"Same Denominators","text":"Make sure the fractions have the same denominator before adding them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational2a-h2","type":"hint","dependencies":["a3abd22addsubrational2a-h1"],"title":"Divide Rational Expressions","text":"Since the denominators are the same, put both of the numerators of the rational expression over a single denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational2a-h3","type":"hint","dependencies":["a3abd22addsubrational2a-h2"],"title":"Finding Numerators","text":"Think about what the numerators are.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational2a-h4","type":"hint","dependencies":["a3abd22addsubrational2a-h3"],"title":"Adding and Subtracting the Numerators","text":"Add the numerators together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational2a-h5","type":"hint","dependencies":["a3abd22addsubrational2a-h4"],"title":"Simplify","text":"Can $$\\\\frac{18}{24}$$ be simplified further?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3abd22addsubrational3","title":"Adding and Subtracting Rational Expressions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Add and Subtract Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3abd22addsubrational3a","stepAnswer":["$$\\\\frac{3x+5}{4x-5}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{3x}{4x-5}+\\\\frac{5}{4x-5}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3x+5}{4x-5}$$","hints":{"DefaultPathway":[{"id":"a3abd22addsubrational3a-h1","type":"hint","dependencies":[],"title":"Same Denominators","text":"Make sure the fractions have the same denominator before adding them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational3a-h2","type":"hint","dependencies":["a3abd22addsubrational3a-h1"],"title":"Divide Rational Expressions","text":"Since the denominators are the same, put both of the numerators of the rational expression over a single denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational3a-h3","type":"hint","dependencies":["a3abd22addsubrational3a-h2"],"title":"Finding Numerators","text":"Think about what the numerators are.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational3a-h4","type":"hint","dependencies":["a3abd22addsubrational3a-h3"],"title":"Adding and Subtracting the Numerators","text":"Add the numerators together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational3a-h5","type":"hint","dependencies":["a3abd22addsubrational3a-h4"],"title":"Simplify","text":"Can $$\\\\frac{3x+5}{4x-5}$$ be simplified further?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3abd22addsubrational4","title":"Adding and Subtracting Rational Expressions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Add and Subtract Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3abd22addsubrational4a","stepAnswer":["$$\\\\frac{7x+4}{2x+y}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{7x}{2x+y}+\\\\frac{4}{2x+y}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{7x+4}{2x+y}$$","hints":{"DefaultPathway":[{"id":"a3abd22addsubrational4a-h1","type":"hint","dependencies":[],"title":"Same Denominators","text":"Make sure the fractions have the same denominator before adding them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational4a-h2","type":"hint","dependencies":["a3abd22addsubrational4a-h1"],"title":"Divide Rational Expressions","text":"Since the denominators are the same, put both of the numerators of the rational expression over a single denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational4a-h3","type":"hint","dependencies":["a3abd22addsubrational4a-h2"],"title":"Finding Numerators","text":"Think about what the numerators are.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational4a-h4","type":"hint","dependencies":["a3abd22addsubrational4a-h3"],"title":"Adding and Subtracting the Numerators","text":"Add the numerators together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational4a-h5","type":"hint","dependencies":["a3abd22addsubrational4a-h4"],"title":"Simplify","text":"Can $$\\\\frac{7x+4}{2x+y}$$ be simplified further?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3abd22addsubrational5","title":"Adding and Subtracting Rational Expressions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Add and Subtract Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3abd22addsubrational5a","stepAnswer":["$$r+8$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2r^2}{2r-1}+\\\\frac{15r-8}{2r+1}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$r+8$$","hints":{"DefaultPathway":[{"id":"a3abd22addsubrational5a-h1","type":"hint","dependencies":[],"title":"Same Denominators","text":"Make sure the fractions have the same denominator before adding them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational5a-h2","type":"hint","dependencies":["a3abd22addsubrational5a-h1"],"title":"Divide Rational Expressions","text":"Since the denominators are the same, put both of the numerators of the rational expression over a single denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational5a-h3","type":"hint","dependencies":["a3abd22addsubrational5a-h2"],"title":"Finding Numerators","text":"Think about what the numerators are.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational5a-h4","type":"hint","dependencies":["a3abd22addsubrational5a-h3"],"title":"Adding and Subtracting the Numerators","text":"Add the numerators together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational5a-h5","type":"hint","dependencies":["a3abd22addsubrational5a-h4"],"title":"Simplify","text":"Can $$\\\\frac{2r^2+15r-8}{2r+1}$$ be simplified further?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational5a-h6","type":"hint","dependencies":["a3abd22addsubrational5a-h5"],"title":"Factoring","text":"Try factoring the numerator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational5a-h7","type":"hint","dependencies":["a3abd22addsubrational5a-h6"],"title":"Simplify","text":"Can something in the numerator and denominator cancel out?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3abd22addsubrational6","title":"Adding and Subtracting Rational Expressions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Add and Subtract Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3abd22addsubrational6a","stepAnswer":["$$s+5$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{3s^2}{3s-2}+\\\\frac{13s-10}{3s-2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$s+5$$","hints":{"DefaultPathway":[{"id":"a3abd22addsubrational6a-h1","type":"hint","dependencies":[],"title":"Same Denominators","text":"Make sure the fractions have the same denominator before adding them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational6a-h2","type":"hint","dependencies":["a3abd22addsubrational6a-h1"],"title":"Divide Rational Expressions","text":"Since the denominators are the same, put both of the numerators of the rational expression over a single denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational6a-h3","type":"hint","dependencies":["a3abd22addsubrational6a-h2"],"title":"Finding Numerators","text":"Think about what the numerators are.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational6a-h4","type":"hint","dependencies":["a3abd22addsubrational6a-h3"],"title":"Adding and Subtracting the Numerators","text":"Add the numerators together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational6a-h5","type":"hint","dependencies":["a3abd22addsubrational6a-h4"],"title":"Simplify","text":"Can $$\\\\frac{3s^2+13s-10}{2r+1}$$ be simplified further?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational6a-h6","type":"hint","dependencies":["a3abd22addsubrational6a-h5"],"title":"Factoring","text":"Try factoring the numerator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational6a-h7","type":"hint","dependencies":["a3abd22addsubrational6a-h6"],"title":"Simplify","text":"Can something in the numerator and denominator cancel out?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3abd22addsubrational7","title":"Adding and Subtracting Rational Expressions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Add and Subtract Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3abd22addsubrational7a","stepAnswer":["$$\\\\frac{2w}{w+4}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2w^2}{3w^2-16}+\\\\frac{8w}{3w^2-16}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2w}{w+4}$$","hints":{"DefaultPathway":[{"id":"a3abd22addsubrational7a-h1","type":"hint","dependencies":[],"title":"Same Denominators","text":"Make sure the fractions have the same denominator before adding them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational7a-h2","type":"hint","dependencies":["a3abd22addsubrational7a-h1"],"title":"Divide Rational Expressions","text":"Since the denominators are the same, put both of the numerators of the rational expression over a single denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational7a-h3","type":"hint","dependencies":["a3abd22addsubrational7a-h2"],"title":"Finding Numerators","text":"Think about what the numerators are.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational7a-h4","type":"hint","dependencies":["a3abd22addsubrational7a-h3"],"title":"Adding and Subtracting the Numerators","text":"Add the numerators together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational7a-h5","type":"hint","dependencies":["a3abd22addsubrational7a-h4"],"title":"Simplify","text":"Can $$\\\\frac{2w^2+8w}{3w^2-16}$$ be simplified further?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational7a-h6","type":"hint","dependencies":["a3abd22addsubrational7a-h5"],"title":"Factoring","text":"Try factoring the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational7a-h7","type":"hint","dependencies":["a3abd22addsubrational7a-h6"],"title":"Difference of Squares","text":"Notice that the denominator has a difference of squares.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational7a-h8","type":"hint","dependencies":["a3abd22addsubrational7a-h7"],"title":"Simplify","text":"Can something in the numerator and denominator cancel out?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3abd22addsubrational8","title":"Adding and Subtracting Rational Expressions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Add and Subtract Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3abd22addsubrational8a","stepAnswer":["$$\\\\frac{7x}{x-3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{7x^2}{x^2-9}+\\\\frac{21x}{x^2-9}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{7x}{x-3}$$","hints":{"DefaultPathway":[{"id":"a3abd22addsubrational8a-h1","type":"hint","dependencies":[],"title":"Same Denominators","text":"Make sure the fractions have the same denominator before adding them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational8a-h2","type":"hint","dependencies":["a3abd22addsubrational8a-h1"],"title":"Divide Rational Expressions","text":"Since the denominators are the same, put both of the numerators of the rational expression over a single denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational8a-h3","type":"hint","dependencies":["a3abd22addsubrational8a-h2"],"title":"Finding Numerators","text":"Think about what the numerators are.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational8a-h4","type":"hint","dependencies":["a3abd22addsubrational8a-h3"],"title":"Adding and Subtracting the Numerators","text":"Add the numerators together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational8a-h5","type":"hint","dependencies":["a3abd22addsubrational8a-h4"],"title":"Simplify","text":"Can $$\\\\frac{7x^2+21x}{x^2-9}$$ be simplified further?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational8a-h6","type":"hint","dependencies":["a3abd22addsubrational8a-h5"],"title":"Factoring","text":"Try factoring the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational8a-h7","type":"hint","dependencies":["a3abd22addsubrational8a-h6"],"title":"Difference of Squares","text":"Notice that the denominator has a difference of squares.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational8a-h8","type":"hint","dependencies":["a3abd22addsubrational8a-h7"],"title":"Simplify","text":"Can something in the numerator and denominator cancel out?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3abd22addsubrational9","title":"Adding and Subtracting Rational Expressions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Add and Subtract Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3abd22addsubrational9a","stepAnswer":["$$3a+7$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{9a^2}{3a-7}-\\\\frac{49}{3a-7}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3a+7$$","hints":{"DefaultPathway":[{"id":"a3abd22addsubrational9a-h1","type":"hint","dependencies":[],"title":"Same Denominators","text":"Make sure the fractions have the same denominator before adding them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational9a-h2","type":"hint","dependencies":["a3abd22addsubrational9a-h1"],"title":"Divide Rational Expressions","text":"Since the denominators are the same, put both of the numerators of the rational expression over a single denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational9a-h3","type":"hint","dependencies":["a3abd22addsubrational9a-h2"],"title":"Finding Numerators","text":"Think about what the numerators are.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational9a-h4","type":"hint","dependencies":["a3abd22addsubrational9a-h3"],"title":"Adding and Subtracting the Numerators","text":"Subtract the numerators from each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational9a-h5","type":"hint","dependencies":["a3abd22addsubrational9a-h4"],"title":"Simplify","text":"Can (9*(a**2))-49)/((3*a)-7) be simplified further?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational9a-h6","type":"hint","dependencies":["a3abd22addsubrational9a-h5"],"title":"Factoring","text":"Try factoring the numerator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational9a-h7","type":"hint","dependencies":["a3abd22addsubrational9a-h6"],"title":"Difference of Squares","text":"Notice that the numerator has a difference of squares.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3abd22addsubrational9a-h8","type":"hint","dependencies":["a3abd22addsubrational9a-h7"],"title":"Simplify","text":"Can something in the numerator and denominator cancel out?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b09a3binomseq1","title":"Writing a Given Term of a Binomial Expansion","body":"Find the tenth term of $${\\\\left(x+2y\\\\right)}^{16}$$ without fully expanding the binomial","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Binomial Theorem","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b09a3binomseq1a","stepAnswer":["$$5857280x^7 y^9$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(x+2y\\\\right)}^{16}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5857280x^7 y^9$$","hints":{"DefaultPathway":[{"id":"a3b09a3binomseq1a-h1","type":"hint","dependencies":[],"title":"Equation","text":"Use the equation $$C(n,r)rx**n-$$ * $$y^r$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq1a-h2","type":"hint","dependencies":["a3b09a3binomseq1a-h1"],"title":"Find $$r$$","text":"Since it is asking for the 10th term, $$r+1=10$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a3b09a3binomseq1a-h2"],"title":"Solve For $$r$$","text":"What is $$r$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["a3b09a3binomseq1a-h3"],"title":"Solve For $$n$$","text":"What is $$n$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq1a-h5","type":"hint","dependencies":["a3b09a3binomseq1a-h4"],"title":"Plug In Values","text":"Plug in all the values of $$n$$, $$r$$, $$x$$, and $$y$$ into the equation and solve.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq1a-h6","type":"hint","dependencies":["a3b09a3binomseq1a-h5"],"title":"Solve C(n,r)","text":"C(n,r) can be converted to the equation (n!)/(r!(n-r)!)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq1a-h7","type":"hint","dependencies":["a3b09a3binomseq1a-h6"],"title":"Plug In Values","text":"Plug in the values of $$n$$ and $$r$$ into the equation that C(n,r) can be converted to","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq1a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5857280$$"],"dependencies":["a3b09a3binomseq1a-h7"],"title":"Solve","text":"Combine all the coefficients of variables into one","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b09a3binomseq10","title":"Finding Binomial Coefficients","body":"For the following exercise, evaluate the binomial coefficient","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Binomial Theorem","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b09a3binomseq10a","stepAnswer":["$$10$$"],"problemType":"TextBox","stepTitle":"$$C(5,3)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10$$","hints":{"DefaultPathway":[{"id":"a3b09a3binomseq10a-h1","type":"hint","dependencies":[],"title":"Equation","text":"Use the equation C(n,r)=(n!)/(r!(n-r)!) to solve","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq10a-h2","type":"hint","dependencies":[],"title":"Plug In Values","text":"Plug in the values to the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$120$$"],"dependencies":[],"title":"Solve For the Numerator","text":"What is 5!?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a3b09a3binomseq10a-h3"],"title":"Solve For the Denominator: Part $$1$$","text":"What is 3!?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a3b09a3binomseq10a-h4"],"title":"Solve For the Denominator: Part $$2$$","text":"What is $$(5-3)!$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq10a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a3b09a3binomseq10a-h5"],"title":"Solve","text":"What is $$\\\\frac{120}{6\\\\times2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b09a3binomseq11","title":"Finding Binomial Coefficients","body":"For the following exercise, evaluate the binomial coefficient","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Binomial Theorem","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b09a3binomseq11a","stepAnswer":["$$36$$"],"problemType":"TextBox","stepTitle":"$$C(9,2)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$36$$","hints":{"DefaultPathway":[{"id":"a3b09a3binomseq11a-h1","type":"hint","dependencies":[],"title":"Equation","text":"Use the equation C(n,r)=(n!)/(r!(n-r)!) to solve","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq11a-h2","type":"hint","dependencies":[],"title":"Plug In Values","text":"Plug in the values to the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq11a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$362880$$"],"dependencies":[],"title":"Solve For the Numerator","text":"What is 9!?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a3b09a3binomseq11a-h3"],"title":"Solve For the Denominator: Part $$1$$","text":"What is 2!?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5040$$"],"dependencies":["a3b09a3binomseq11a-h4"],"title":"Solve For the Denominator: Part $$2$$","text":"What is $$(9-2)!$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq11a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$36$$"],"dependencies":["a3b09a3binomseq11a-h5"],"title":"Solve","text":"What is $$\\\\frac{362880}{2\\\\times5040}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b09a3binomseq12","title":"Finding Binomial Coefficients","body":"For the following exercise, evaluate the binomial coefficient","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Binomial Theorem","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b09a3binomseq12a","stepAnswer":["$$36$$"],"problemType":"TextBox","stepTitle":"$$C(9,7)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$36$$","hints":{"DefaultPathway":[{"id":"a3b09a3binomseq12a-h1","type":"hint","dependencies":[],"title":"Equation","text":"Use the equation C(n,r)=(n!)/(r!(n-r)!) to solve","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq12a-h2","type":"hint","dependencies":[],"title":"Plug In Values","text":"Plug in the values to the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq12a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$362880$$"],"dependencies":[],"title":"Solve For the Numerator","text":"What is 9!?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5040$$"],"dependencies":["a3b09a3binomseq12a-h3"],"title":"Solve For the Denominator: Part $$1$$","text":"What is 7!?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq12a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a3b09a3binomseq12a-h4"],"title":"Solve For the Denominator: Part $$2$$","text":"What is $$(9-7)!$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq12a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$36$$"],"dependencies":["a3b09a3binomseq12a-h5"],"title":"Solve","text":"What is $$\\\\frac{362880}{5040\\\\times2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b09a3binomseq13","title":"Finding Binomial Coefficients","body":"For the following exercise, evaluate the binomial coefficient","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Binomial Theorem","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b09a3binomseq13a","stepAnswer":["$$36$$"],"problemType":"TextBox","stepTitle":"$$C(7,3)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$36$$","hints":{"DefaultPathway":[{"id":"a3b09a3binomseq13a-h1","type":"hint","dependencies":[],"title":"Equation","text":"Use the equation C(n,r)=(n!)/(r!(n-r)!) to solve","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq13a-h2","type":"hint","dependencies":[],"title":"Plug In Values","text":"Plug in the values to the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq13a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5040$$"],"dependencies":[],"title":"Solve For the Numerator","text":"What is 7!?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a3b09a3binomseq13a-h3"],"title":"Solve For the Denominator: Part $$1$$","text":"What is 3!?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq13a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$24$$"],"dependencies":["a3b09a3binomseq13a-h4"],"title":"Solve For the Denominator: Part $$2$$","text":"What is $$(7-3)!$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq13a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$35$$"],"dependencies":["a3b09a3binomseq13a-h5"],"title":"Solve","text":"What is $$\\\\frac{5040}{6\\\\times24}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b09a3binomseq14","title":"Finding Binomial Coefficients","body":"For the following exercise, evaluate the binomial coefficient","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Binomial Theorem","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b09a3binomseq14a","stepAnswer":["$$330$$"],"problemType":"TextBox","stepTitle":"$$C(11,4)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$330$$","hints":{"DefaultPathway":[{"id":"a3b09a3binomseq14a-h1","type":"hint","dependencies":[],"title":"Equation","text":"Use the equation C(n,r)=(n!)/(r!(n-r)!) to solve","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq14a-h2","type":"hint","dependencies":[],"title":"Plug In Values","text":"Plug in the values to the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq14a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$39916800$$"],"dependencies":[],"title":"Solve For the Numerator","text":"What is 11!?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq14a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$24$$"],"dependencies":["a3b09a3binomseq14a-h3"],"title":"Solve For the Denominator: Part $$1$$","text":"What is 4!?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq14a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5040$$"],"dependencies":["a3b09a3binomseq14a-h4"],"title":"Solve For the Denominator: Part $$2$$","text":"What is $$(11-4)!$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq14a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$330$$"],"dependencies":["a3b09a3binomseq14a-h5"],"title":"Solve","text":"What is $$\\\\frac{39916800}{24\\\\times5040}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b09a3binomseq15","title":"Writing a Given Term of a Binomial Expansion","body":"Find the tenth term of $${\\\\left(3x-y\\\\right)}^9$$ without fully expanding the binomial","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Binomial Theorem","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b09a3binomseq15a","stepAnswer":["$$-10206x^4$$ $$y^5$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(3x-y\\\\right)}^9$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-10206x^4$$ $$y^5$$","hints":{"DefaultPathway":[{"id":"a3b09a3binomseq15a-h1","type":"hint","dependencies":[],"title":"Equation","text":"Use the equation $$C(n,r)rx**n-$$ * $$y^r$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq15a-h2","type":"hint","dependencies":["a3b09a3binomseq15a-h1"],"title":"Find $$r$$","text":"Since it is asking for the 10th term, $$r+1=6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a3b09a3binomseq15a-h2"],"title":"Solve For $$r$$","text":"What is $$r$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a3b09a3binomseq15a-h3"],"title":"Solve For $$n$$","text":"What is $$n$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq15a-h5","type":"hint","dependencies":["a3b09a3binomseq15a-h4"],"title":"Plug In Values","text":"Plug in all the values of $$n$$, $$r$$, $$x$$, and $$y$$ into the equation and solve.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq15a-h6","type":"hint","dependencies":["a3b09a3binomseq15a-h5"],"title":"Solve C(n,r)","text":"C(n,r) can be converted to the equation (n!)/(r!(n-r)!)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq15a-h7","type":"hint","dependencies":["a3b09a3binomseq15a-h6"],"title":"Plug In Values","text":"Plug in the values of $$n$$ and $$r$$ into the equation that C(n,r) can be converted to","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq15a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-10206$$"],"dependencies":["a3b09a3binomseq15a-h7"],"title":"Solve","text":"Combine all the coefficients of variables into one","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b09a3binomseq16","title":"Evaluating the Binomial Coefficient","body":"Evaluate the binomial coefficient:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Binomial Theorem","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b09a3binomseq16a","stepAnswer":["$$15$$"],"problemType":"TextBox","stepTitle":"$$(6,2)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$15$$","hints":{"DefaultPathway":[{"id":"a3b09a3binomseq16a-h1","type":"hint","dependencies":[],"title":"Using the Combination Formula","text":"$$(6,2)$$ $$=$$ $$6$$ Choose $$2$$ $$=$$ $$C(6,2)$$ $$=$$ $$15$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b09a3binomseq17","title":"Binomial Expansion","body":"Find the fourth term of the sequence.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Binomial Theorem","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b09a3binomseq17a","stepAnswer":["$$216{xy}^3$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(2x-3y\\\\right)}^4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$216{xy}^3$$","hints":{"DefaultPathway":[{"id":"a3b09a3binomseq17a-h1","type":"hint","dependencies":[],"title":"Binomial Coefficient Formula","text":"The formula to find the nth term is as follows: C(n, $$n-(r-1))$$ * $$a^{n \\\\left(r-1\\\\right)}$$ * $$b^{r-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq17a-h2","type":"hint","dependencies":["a3b09a3binomseq17a-h1"],"title":"Plugging into the Formula","text":"$$r=4$$ and $$n=4$$. Plugging in these values into the formula above gives: $$216{xy}^3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b09a3binomseq18","title":"Binomial Expansion","body":"Find the fourth term:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Binomial Theorem","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b09a3binomseq18a","stepAnswer":["$$-720x^2 y^3$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(3x-2y\\\\right)}^5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-720x^2 y^3$$","hints":{"DefaultPathway":[{"id":"a3b09a3binomseq18a-h1","type":"hint","dependencies":[],"title":"Binomial Coefficient Formula","text":"The formula to find the nth term is as follows: C(n, $$n-(r-1))$$ * $$a^{n \\\\left(r-1\\\\right)}$$ * $$b^{r-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq18a-h2","type":"hint","dependencies":["a3b09a3binomseq18a-h1"],"title":"Plugging into the Formula","text":"$$r=4$$ and $$n=5$$. Plugging in these values into the formula above gives: $$-720x^2 y^3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b09a3binomseq19","title":"Binomial Expansion","body":"Find the third term of the binomial expansion.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Binomial Theorem","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b09a3binomseq19a","stepAnswer":["$$1469664x^5 y^2$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(6x-3y\\\\right)}^7$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1469664x^5 y^2$$","hints":{"DefaultPathway":[{"id":"a3b09a3binomseq19a-h1","type":"hint","dependencies":[],"title":"Binomial Coefficient Formula","text":"The formula to find the nth term is as follows: C(n, $$n-(r-1))$$ * $$a^{n \\\\left(r-1\\\\right)}$$ * $$b^{r-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq19a-h2","type":"hint","dependencies":["a3b09a3binomseq19a-h1"],"title":"Plugging into the Formula","text":"$$r=3$$ and $$n=7$$. Plugging in these values into the formula above gives: $$1469664x^5 y^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b09a3binomseq2","title":"Finding Binomial Coefficients","body":"For the following exercise, evaluate the binomial coefficient","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Binomial Theorem","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b09a3binomseq2a","stepAnswer":["$$15$$"],"problemType":"TextBox","stepTitle":"$$C(6,2)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$15$$","hints":{"DefaultPathway":[{"id":"a3b09a3binomseq2a-h1","type":"hint","dependencies":[],"title":"Equation","text":"Use the equation C(n,r)=(n!)/(r!(n-r)!) to solve","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq2a-h2","type":"hint","dependencies":[],"title":"Plug In Values","text":"Plug in the values to the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$720$$"],"dependencies":[],"title":"Solve For the Numerator","text":"What is 6!?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a3b09a3binomseq2a-h3"],"title":"Solve For the Denominator: Part $$1$$","text":"What is 2!?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq2a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$24$$"],"dependencies":["a3b09a3binomseq2a-h4"],"title":"Solve For the Denominator: Part $$2$$","text":"What is $$(6-2)!$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["a3b09a3binomseq2a-h5"],"title":"Solve","text":"What is $$\\\\frac{720}{24\\\\times2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b09a3binomseq20","title":"Binomial Expansion","body":"Find the eighth term of:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Binomial Theorem","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b09a3binomseq20a","stepAnswer":["$$220812466875000y^7$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(7+5y\\\\right)}^{14}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$220812466875000y^7$$","hints":{"DefaultPathway":[{"id":"a3b09a3binomseq20a-h1","type":"hint","dependencies":[],"title":"Binomial Coefficient Formula","text":"The formula to find the nth term is as follows: C(n, $$n-(r-1))$$ * $$a^{n \\\\left(r-1\\\\right)}$$ * $$b^{r-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq20a-h2","type":"hint","dependencies":["a3b09a3binomseq20a-h1"],"title":"Plugging into the Formula","text":"$$r=8$$, $$n=14$$. Plugging in these values into the formula above gives: $$220812466875000y^7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b09a3binomseq21","title":"Binomial Expansion","body":"Find the eighth term of:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Binomial Theorem","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b09a3binomseq21a","stepAnswer":["$$462a^5 b^6$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(a+b\\\\right)}^{11}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$462a^5 b^6$$","hints":{"DefaultPathway":[{"id":"a3b09a3binomseq21a-h1","type":"hint","dependencies":[],"title":"Binomial Coefficient Formula","text":"The formula to find the nth term is as follows: C(n, $$n-(r-1))$$ * $$a^{n \\\\left(r-1\\\\right)}$$ * $$b^{r-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq21a-h2","type":"hint","dependencies":["a3b09a3binomseq21a-h1"],"title":"Plugging into the Formula","text":"$$r=7$$, $$n=11$$. Plugging in these values into the formula above gives: $$462a^5 b^6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b09a3binomseq22","title":"Binomial Expansion","body":"Find the fifth term of:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Binomial Theorem","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b09a3binomseq22a","stepAnswer":["$$35x^3 y^4$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(x-y\\\\right)}^7$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$35x^3 y^4$$","hints":{"DefaultPathway":[{"id":"a3b09a3binomseq22a-h1","type":"hint","dependencies":[],"title":"Binomial Coefficient Formula","text":"The formula to find the nth term is as follows: C(n, $$n-(r-1))$$ * $$a^{n \\\\left(r-1\\\\right)}$$ * $$b^{r-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq22a-h2","type":"hint","dependencies":["a3b09a3binomseq22a-h1"],"title":"Plugging into the Formula","text":"$$r=5$$, $$n=7$$. Pluggin in these values into the formula above gives: $$35x^3 y^4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b09a3binomseq23","title":"Binomial Expansion","body":"Find the tenth term of:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Binomial Theorem","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b09a3binomseq23a","stepAnswer":["$$-220x^3$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(x-1\\\\right)}^{12}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-220x^3$$","hints":{"DefaultPathway":[{"id":"a3b09a3binomseq23a-h1","type":"hint","dependencies":[],"title":"Binomial Coefficient Formula","text":"The formula to find the nth term is as follows: C(n, $$n-(r-1))$$ * $$a^{n \\\\left(r-1\\\\right)}$$ * $$b^{r-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq23a-h2","type":"hint","dependencies":["a3b09a3binomseq23a-h1"],"title":"Plugging into the Formula","text":"$$r=10$$, $$n=12$$. Plugging in these values into the formula above gives: $$-220x^3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b09a3binomseq24","title":"Binomial Expansion","body":"Find the ninth term of:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Binomial Theorem","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b09a3binomseq24a","stepAnswer":["$$1082565a^3 b^{16}$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(a-3b^2\\\\right)}^{11}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1082565a^3 b^{16}$$","hints":{"DefaultPathway":[{"id":"a3b09a3binomseq24a-h1","type":"hint","dependencies":[],"title":"Binomial Coefficient Formula","text":"The formula to find the nth term is as follows: C(n, $$n-(r-1))$$ * $$a^{n \\\\left(r-1\\\\right)}$$ * $$b^{r-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq24a-h2","type":"hint","dependencies":["a3b09a3binomseq24a-h1"],"title":"Plugging into the Formula","text":"$$r=9$$, $$n=11$$. Plugging in these values into the formula above gives: $$1082565a^3 b^{16}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b09a3binomseq25","title":"Binomial Expansion","body":"Find the fourth term of:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Binomial Theorem","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b09a3binomseq25a","stepAnswer":["$$-15x^{21}$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(x^3-\\\\frac{1}{2}\\\\right)}^{10}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-15x^{21}$$","hints":{"DefaultPathway":[{"id":"a3b09a3binomseq25a-h1","type":"hint","dependencies":[],"title":"Binomial Coefficient Formula","text":"The formula to find the nth term is as follows: C(n, $$n-(r-1))$$ * $$a^{n \\\\left(r-1\\\\right)}$$ * $$b^{r-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq25a-h2","type":"hint","dependencies":["a3b09a3binomseq25a-h1"],"title":"Plugging into the Formula","text":"$$r=4$$, $$n=10$$. Plugging these values into the formula above gives: $$-15x^{21}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b09a3binomseq26","title":"Binomial Expansion","body":"Find the eighth term of:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Binomial Theorem","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b09a3binomseq26a","stepAnswer":["$$\\\\frac{1152y^2}{x^7}$$"],"problemType":"TextBox","stepTitle":"(y/2 + 2/x)**9","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1152y^2}{x^7}$$","hints":{"DefaultPathway":[{"id":"a3b09a3binomseq26a-h1","type":"hint","dependencies":[],"title":"Binomial Coefficient Formula","text":"The formula to find the nth term is as follows: C(n, $$n-(r-1))$$ * $$a^{n \\\\left(r-1\\\\right)}$$ * $$b^{r-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq26a-h2","type":"hint","dependencies":["a3b09a3binomseq26a-h1"],"title":"Plugging into the Formula","text":"$$r=8$$, $$n=9$$. Plugging in these values into the formula above gives: $$\\\\frac{1152y^2}{x^7}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b09a3binomseq27","title":"Binomial Expansion","body":"Expand the following.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Binomial Theorem","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b09a3binomseq27a","stepAnswer":["$$64a^3-48a^2 b+12{ab}^2-b^3$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(4a-b\\\\right)}^3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$64a^3-48a^2 b+12{ab}^2-b^3$$","hints":{"DefaultPathway":[{"id":"a3b09a3binomseq27a-h1","type":"hint","dependencies":[],"title":"Binomial Coefficient Formula","text":"The formula to find the nth term is as follows: C(n, $$n-(r-1))$$ * $$a^{n \\\\left(r-1\\\\right)}$$ * $$b^{r-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq27a-h2","type":"hint","dependencies":["a3b09a3binomseq27a-h1"],"title":"Plugging into the Formula","text":"$$r=1$$, $$2$$, $$3$$, 4; $$n=3$$. Plugging in these values into the formula above gives: $$64a^3-48a^2 b+12{ab}^2-b^3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b09a3binomseq28","title":"Binomial Expansion","body":"Expand the following.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Binomial Theorem","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b09a3binomseq28a","stepAnswer":["$$27a^3+54a^2 b+36{ab}^2+8b^3$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(3a+2b\\\\right)}^3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$27a^3+54a^2 b+36{ab}^2+8b^3$$","hints":{"DefaultPathway":[{"id":"a3b09a3binomseq28a-h1","type":"hint","dependencies":[],"title":"Binomial Coefficient Formula","text":"The formula to find the nth term is as follows: C(n, $$n-(r-1))$$ * $$a^{n \\\\left(r-1\\\\right)}$$ * $$b^{r-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq28a-h2","type":"hint","dependencies":["a3b09a3binomseq28a-h1"],"title":"Plugging into the Formula","text":"$$r=1$$, $$2$$, $$3$$, 4; $$n=3$$. Plugging these values into the formula above gives: $$27a^3+54a^2 b+36{ab}^2+8b^3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b09a3binomseq29","title":"Binomial Expansion","body":"Expand the following.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Binomial Theorem","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b09a3binomseq29a","stepAnswer":["$$1024x^5+2560x^4 y+2560x^3 y^2+1280x^{y^3}+320{xy}^4+32y^4$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(4x+2y\\\\right)}^5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1024x^5+2560x^4 y+2560x^3 y^2+1280x^{y^3}+320{xy}^4+32y^4$$","hints":{"DefaultPathway":[{"id":"a3b09a3binomseq29a-h1","type":"hint","dependencies":[],"title":"Binomial Coefficient Formula","text":"The formula to find the nth term is as follows: C(n, $$n-(r-1))$$ * $$a^{n \\\\left(r-1\\\\right)}$$ * $$b^{r-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq29a-h2","type":"hint","dependencies":["a3b09a3binomseq29a-h1"],"title":"Plugging into the Formula","text":"$$r=1$$, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$. $$n=5$$. Plugging in these values into the formula above gives: $$1024x^5+2560x^4 y+2560x^3 y^2+1280x^{y^3}+320{xy}^4+32y^4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b09a3binomseq3","title":"Finding Binomial Coefficients","body":"For the following exercise, evaluate the binomial coefficient","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Binomial Theorem","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b09a3binomseq3a","stepAnswer":["$$10$$"],"problemType":"TextBox","stepTitle":"$$C(5,3)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10$$","hints":{"DefaultPathway":[{"id":"a3b09a3binomseq3a-h1","type":"hint","dependencies":[],"title":"Equation","text":"Use the equation C(n,r)=(n!)/(r!(n-r)!) to solve","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq3a-h2","type":"hint","dependencies":[],"title":"Plug In Values","text":"Plug in the values to the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$120$$"],"dependencies":[],"title":"Solve For the Numerator","text":"What is 5!?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a3b09a3binomseq3a-h3"],"title":"Solve For the Denominator: Part $$1$$","text":"What is 3!?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a3b09a3binomseq3a-h4"],"title":"Solve For the Denominator: Part $$2$$","text":"What is $$(5-3)!$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq3a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a3b09a3binomseq3a-h5"],"title":"Solve","text":"What is $$\\\\frac{120}{6\\\\times2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b09a3binomseq30","title":"Binomial Expansion","body":"Expand the following.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Binomial Theorem","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b09a3binomseq30a","stepAnswer":["$$1024x^5-3840x^4 yy+5760x^3 y^2-4320x^2 y^3+1620{xy}^4-243y^5$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(4x-3y\\\\right)}^5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1024x^5-3840x^4 yy+5760x^3 y^2-4320x^2 y^3+1620{xy}^4-243y^5$$","hints":{"DefaultPathway":[{"id":"a3b09a3binomseq30a-h1","type":"hint","dependencies":[],"title":"Binomial Coefficient Formula","text":"The formula to find the nth term is as follows: C(n, $$n-(r-1))$$ * $$a^{n \\\\left(r-1\\\\right)}$$ * $$b^{r-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq30a-h2","type":"hint","dependencies":["a3b09a3binomseq30a-h1"],"title":"Plugging into the Formula","text":"$$r=1$$, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$. $$n=5$$\\\\nPlugging these values into the formula above gives: $$1024x^5-3840x^4 yy+5760x^3 y^2-4320x^2 y^3+1620{xy}^4-243y^5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b09a3binomseq4","title":"Finding Binomial Coefficients","body":"For the following exercise, evaluate the binomial coefficient","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Binomial Theorem","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b09a3binomseq4a","stepAnswer":["$$35$$"],"problemType":"TextBox","stepTitle":"$$C(7,4)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$35$$","hints":{"DefaultPathway":[{"id":"a3b09a3binomseq4a-h1","type":"hint","dependencies":[],"title":"Equation","text":"Use the equation C(n,r)=(n!)/(r!(n-r)!) to solve","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq4a-h2","type":"hint","dependencies":[],"title":"Plug In Values","text":"Plug in the values to the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5040$$"],"dependencies":[],"title":"Solve For the Numerator","text":"What is 7!?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$24$$"],"dependencies":["a3b09a3binomseq4a-h3"],"title":"Solve For the Denominator: Part $$1$$","text":"What is 4!?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq4a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a3b09a3binomseq4a-h4"],"title":"Solve For the Denominator: Part $$2$$","text":"What is $$(7-4)!$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq4a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$35$$"],"dependencies":["a3b09a3binomseq4a-h5"],"title":"Solve","text":"What is $$\\\\frac{5040}{24\\\\times6}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b09a3binomseq5","title":"Finding Binomial Coefficients","body":"For the following exercise, evaluate the binomial coefficient","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Binomial Theorem","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b09a3binomseq5a","stepAnswer":["$$36$$"],"problemType":"TextBox","stepTitle":"$$C(9,7)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$36$$","hints":{"DefaultPathway":[{"id":"a3b09a3binomseq5a-h1","type":"hint","dependencies":[],"title":"Equation","text":"Use the equation C(n,r)=(n!)/(r!(n-r)!) to solve","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq5a-h2","type":"hint","dependencies":[],"title":"Plug In Values","text":"Plug in the values to the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$362880$$"],"dependencies":[],"title":"Solve For the Numerator","text":"What is 9!?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5040$$"],"dependencies":["a3b09a3binomseq5a-h3"],"title":"Solve For the Denominator: Part $$1$$","text":"What is 7!?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a3b09a3binomseq5a-h4"],"title":"Solve For the Denominator: Part $$2$$","text":"What is $$(9-7)!$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq5a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$36$$"],"dependencies":["a3b09a3binomseq5a-h5"],"title":"Solve","text":"What is $$\\\\frac{362880}{5040\\\\times2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b09a3binomseq6","title":"Finding Binomial Coefficients","body":"For the following exercise, evaluate the binomial coefficient","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Binomial Theorem","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b09a3binomseq6a","stepAnswer":["$$10$$"],"problemType":"TextBox","stepTitle":"$$C(10,9)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10$$","hints":{"DefaultPathway":[{"id":"a3b09a3binomseq6a-h1","type":"hint","dependencies":[],"title":"Equation","text":"Use the equation C(n,r)=(n!)/(r!(n-r)!) to solve","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq6a-h2","type":"hint","dependencies":[],"title":"Plug In Values","text":"Plug in the values to the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3628800$$"],"dependencies":[],"title":"Solve For the Numerator","text":"What is 10!?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$362880$$"],"dependencies":["a3b09a3binomseq6a-h3"],"title":"Solve For the Denominator: Part $$1$$","text":"What is 9!?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a3b09a3binomseq6a-h4"],"title":"Solve For the Denominator: Part $$2$$","text":"What is $$(10-9)!$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq6a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a3b09a3binomseq6a-h5"],"title":"Solve","text":"What is $$\\\\frac{3628800}{362880\\\\times1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b09a3binomseq7","title":"Finding Binomial Coefficients","body":"For the following exercise, evaluate the binomial coefficient","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Binomial Theorem","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b09a3binomseq7a","stepAnswer":["$$4457400$$"],"problemType":"TextBox","stepTitle":"$$C(25,11)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4457400$$","hints":{"DefaultPathway":[{"id":"a3b09a3binomseq7a-h1","type":"hint","dependencies":[],"title":"Equation","text":"Use the equation C(n,r)=(n!)/(r!(n-r)!) to solve","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq7a-h2","type":"hint","dependencies":[],"title":"Plug In Values","text":"Plug in the values to the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.55E+25$$"],"dependencies":[],"title":"Solve For the Numerator","text":"What is 25!? (Write in the form $$1.22E+2$$ and round to the nearest hundredth)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$39916800$$"],"dependencies":["a3b09a3binomseq7a-h3"],"title":"Solve For the Denominator: Part $$1$$","text":"What is 11!?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq7a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$87178291200$$"],"dependencies":["a3b09a3binomseq7a-h4"],"title":"Solve For the Denominator: Part $$2$$","text":"What is $$(25-11)!$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq7a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4457400$$"],"dependencies":["a3b09a3binomseq7a-h5"],"title":"Solve","text":"What is $$1.55E+\\\\frac{25}{39916800\\\\times87178291200}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b09a3binomseq8","title":"Finding Binomial Coefficients","body":"For the following exercise, evaluate the binomial coefficient","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Binomial Theorem","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b09a3binomseq8a","stepAnswer":["$$12376$$"],"problemType":"TextBox","stepTitle":"$$C(17,6)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12376$$","hints":{"DefaultPathway":[{"id":"a3b09a3binomseq8a-h1","type":"hint","dependencies":[],"title":"Equation","text":"Use the equation C(n,r)=(n!)/(r!(n-r)!) to solve","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq8a-h2","type":"hint","dependencies":[],"title":"Plug In Values","text":"Plug in the values to the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3.56E+14$$"],"dependencies":[],"title":"Solve For the Numerator","text":"What is 17!? (Write in the form $$1.22E+2$$ and round to the nearest hundredth)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$720$$"],"dependencies":["a3b09a3binomseq8a-h3"],"title":"Solve For the Denominator: Part $$1$$","text":"What is 6!?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq8a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$39916800$$"],"dependencies":["a3b09a3binomseq8a-h4"],"title":"Solve For the Denominator: Part $$2$$","text":"What is $$(17-6)!$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq8a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12376$$"],"dependencies":["a3b09a3binomseq8a-h5"],"title":"Solve","text":"What is $$3.56E+\\\\frac{14}{720\\\\times39916800}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b09a3binomseq9","title":"Finding Binomial Coefficients","body":"For the following exercise, evaluate the binomial coefficient","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Binomial Theorem","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b09a3binomseq9a","stepAnswer":["$$200$$"],"problemType":"TextBox","stepTitle":"$$C(200,199)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$200$$","hints":{"DefaultPathway":[{"id":"a3b09a3binomseq9a-h1","type":"hint","dependencies":[],"title":"Equation","text":"Use the equation C(n,r)=(n!)/(r!(n-r)!) to solve","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq9a-h2","type":"hint","dependencies":[],"title":"Plug In Values","text":"Plug in the values to the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1E+0$$"],"dependencies":[],"title":"Solve Denominator Pt1","text":"What is $$(200-199)!$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq9a-h4","type":"hint","dependencies":["a3b09a3binomseq9a-h3"],"title":"Simplify","text":"Because $$(200-199)!$$ is $$1$$, you can imply that the equation is 200!/199!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b09a3binomseq9a-h5","type":"hint","dependencies":["a3b09a3binomseq9a-h4"],"title":"Infer","text":"200!/200 is the same as 199!, so you can infer that 200!/199! is $$200$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b7d40expgraph1","title":"Finding the domain and range","body":"What is the domain and range of $$f(x)={0.25}^x$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Graphs of Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b7d40expgraph1a","stepAnswer":["domain: $$(-\\\\infty,\\\\infty)$$, range: $$(0,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"domain: $$(-\\\\infty,\\\\infty)$$, range: $$(0,\\\\infty)$$","choices":["domain: $$(-\\\\infty,\\\\infty)$$, range: $$(1,\\\\infty)$$","domain: $$(0,\\\\infty)$$, range: $$(-\\\\infty,\\\\infty)$$","domain: $$(1,\\\\infty)$$, range: $$(-\\\\infty,\\\\infty)$$","domain: $$(-\\\\infty,\\\\infty)$$, range: $$(0,\\\\infty)$$","domain: $$(-\\\\infty,\\\\infty)$$, range: $$(0,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"a3b7d40expgraph1a-h1","type":"hint","dependencies":[],"title":"Characteristics of an exponential function","text":"Recall that an exponential function with the form $$f(x)=b^x$$, $$b>0$$, where $$b$$ does not equal $$1$$, has these characteristics:\\\\n\\\\n$$1$$. one-to-one function\\\\n$$2$$. horizontal asymptote: $$y=0$$\\\\n$$3$$. domain: $$(-\\\\infty,\\\\infty)$$\\\\n$$4$$. range: $$(0,\\\\infty)$$\\\\n$$5$$. x-intercept: none\\\\n$$6$$. y-intercept: $$(0,1)$$\\\\n$$7$$. increasing if $$b>1$$\\\\n$$8$$. decreasing if $$b<1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b7d40expgraph10","title":"Finding horizontal asymptotes","body":"What is the horizontal asymptote of $$f(x)=2x-2$$?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Graphs of Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b7d40expgraph10a","stepAnswer":["$$y=0$$"],"problemType":"TextBox","stepTitle":"","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=0$$","hints":{"DefaultPathway":[{"id":"a3b7d40expgraph10a-h1","type":"hint","dependencies":[],"title":"Definition of horizontal asymptote","text":"The horizontal asymptote describes the end behavior of the function. In other words, it describes the value f(x) converges to as $$x$$ approaches infinity and/or negative $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b7d40expgraph10a-h2","type":"hint","dependencies":[],"title":"Finding the horizontal asymptote","text":"Determine the range of the function. It may be helpful to graph the function","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b7d40expgraph10a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(0,inf)"],"dependencies":[],"title":"Finding the range","text":"What is the range of the function? Write the range in interval notation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b7d40expgraph10a-h4","type":"hint","dependencies":[],"title":"Finding the horizontal asymptote","text":"The last step is to determine the value that f(x) converges to as $$x$$ approaches positive or negative $$\\\\infty$$. It may be helpful to check various values of $$x$$ (e.g. $$-100$$, $$-1000$$, $$-10000$$, etc.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b7d40expgraph11","title":"Transforming exponential functions","body":"Let the parent function be f(x) $$=$$ $$4^x$$. Write the function that results from shifting f(x) $$5$$ units up and $$4$$ units to the right.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Graphs of Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b7d40expgraph11a","stepAnswer":["$$4^{x-4}+5$$"],"problemType":"TextBox","stepTitle":"","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4^{x-4}+5$$","hints":{"DefaultPathway":[{"id":"a3b7d40expgraph11a-h1","type":"hint","dependencies":[],"title":"General form of exponential function","text":"In the general form of the function f(x) $$=$$ $$a b^{x+c}+d$$, a vertically $$\\\\frac{stretches}{shrinks}$$ the function, c translates the function to the left or right, and $$d$$ translates the function up or down.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b7d40expgraph11a-h2","type":"hint","dependencies":[],"title":"Vertical translations","text":"The function f(x) $$=$$ $$a b^{x+c}+d$$ is being translated upwards by $$d$$ units.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b7d40expgraph11a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":[],"title":"Identifying $$d$$","text":"What is the value of $$d$$ in our given equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b7d40expgraph11a-h4","type":"hint","dependencies":[],"title":"Horizontal translations","text":"The function f(x) $$=$$ $$a b^{x+c}+d$$ is being translated c units to the left.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b7d40expgraph11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":[],"title":"Identifying c","text":"What is the value of c in our given equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b7d40expgraph12","title":"Solving Exponential Equations","body":"Evaluate the exponential functions for the indicated value of x: g(x) $$=$$ $$13\\\\times7^x-2$$ for g(6).","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Graphs of Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b7d40expgraph12a","stepAnswer":["$$1529435$$"],"problemType":"TextBox","stepTitle":"","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1529435$$","hints":{"DefaultPathway":[{"id":"a3b7d40expgraph12a-h1","type":"hint","dependencies":[],"title":"Rewriting into an equation","text":"The function can be rewritten as $$13\\\\times7^6-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b7d40expgraph13","title":"Solving Exponential Equations","body":"Evaluate the exponential functions for the indicated value of x: $$f(x)={4\\\\left(2\\\\right)}^{x-1}-2$$ for f(5).","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Graphs of Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b7d40expgraph13a","stepAnswer":["$$62$$"],"problemType":"TextBox","stepTitle":"","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$62$$","hints":{"DefaultPathway":[{"id":"a3b7d40expgraph13a-h1","type":"hint","dependencies":[],"title":"Rewriting into an equation","text":"The function can be rewritten as $${4\\\\left(2\\\\right)}^{5-1}-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b7d40expgraph14","title":"Solving Exponential Functions","body":"Evaluate the exponential functions for the indicated value of x: $$h(x)=-\\\\left(\\\\frac{1}{2}\\\\right) {\\\\left(\\\\frac{1}{2}\\\\right)}^x+6$$ for $$h(-7)$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Graphs of Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b7d40expgraph14a","stepAnswer":["$$-58$$"],"problemType":"TextBox","stepTitle":"","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-58$$","hints":{"DefaultPathway":[{"id":"a3b7d40expgraph14a-h1","type":"hint","dependencies":[],"title":"Rewriting into an equation","text":"The function can be rewritten as $$-\\\\left(\\\\frac{1}{2}\\\\right) {\\\\left(\\\\frac{1}{2}\\\\right)}^{\\\\left(-7\\\\right)}+6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b7d40expgraph15","title":"Transformations of exponential functions","body":"The graph of $$f(x)={10}^x$$ is reflected about the x-axis and shifted upward $$7$$ units. What is the equation of the new function, g(x)?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Graphs of Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b7d40expgraph15a","stepAnswer":["$$g(x)=$$ $$-\\\\left({10}^x\\\\right)+7$$"],"problemType":"TextBox","stepTitle":"","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$g(x)=$$ $$-\\\\left({10}^x\\\\right)+7$$","hints":{"DefaultPathway":[{"id":"a3b7d40expgraph15a-h1","type":"hint","dependencies":[],"title":"Vertical translations","text":"For the parent function f(x) $$=$$ $$b^x$$, a vertical translation upwards by $$d$$ units is represented by g(x) $$=$$ $$b^x$$ + $$d$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b7d40expgraph15a-h2","type":"hint","dependencies":[],"title":"Reflection across the x-axis","text":"For the parent function f(x) $$=$$ $$b^x$$, a reflection across the x-axis is represented by g(x) $$=$$ $$-\\\\left(b^x\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b7d40expgraph16","title":"Transformations of Exponential Functions","body":"The graph of $$f(x)={1.68}^x$$ is shifted right $$3.3$$ units, stretched vertically by a factor of $$2.2$$, reflected about the x-axis, and then shifted downward $$3.3$$ units. What is the equation of the new function, g(x)?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Graphs of Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b7d40expgraph16a","stepAnswer":["$$g(x)=-2.2{1.68}^{x-3.3}-3.3$$"],"problemType":"TextBox","stepTitle":"","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$g(x)=-2.2{1.68}^{x-3.3}-3.3$$","hints":{"DefaultPathway":[{"id":"a3b7d40expgraph16a-h1","type":"hint","dependencies":[],"title":"Vertical translations","text":"For the parent function f(x) $$=$$ $$b^x$$, a vertical translation downwards by $$d$$ units is represented by g(x) $$=$$ $$b^x$$ - $$d$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b7d40expgraph16a-h2","type":"hint","dependencies":[],"title":"Reflection across the x-axis","text":"For the parent function f(x) $$=$$ $$b^x$$, a reflection across the x-axis is represented by g(x) $$=$$ $$-\\\\left(b^x\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b7d40expgraph16a-h3","type":"hint","dependencies":[],"title":"Horizontal translations","text":"For the parent function f(x) $$=$$ $$b^x$$, a horizontal translation c units to the right is represented by g(x) $$=$$ $$b^{x-c}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b7d40expgraph17","title":"Transformations of Exponential Functions","body":"The graph of $$f(x)=-\\\\left(\\\\frac{1}{2}\\\\right) {\\\\left(\\\\frac{1}{4}\\\\right)}^{x-2}+4$$ is shifted downward $$4$$ units, and then shifted left $$2$$ units, stretched vertically by a factor of $$4$$, and reflected about the x-axis. What is the equation of the new function, g(x)?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Graphs of Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b7d40expgraph17a","stepAnswer":["$$g(x)=2{\\\\left(\\\\frac{1}{4}\\\\right)}^x$$"],"problemType":"TextBox","stepTitle":"","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$g(x)=2{\\\\left(\\\\frac{1}{4}\\\\right)}^x$$","hints":{"DefaultPathway":[{"id":"a3b7d40expgraph17a-h1","type":"hint","dependencies":[],"title":"Vertical translations","text":"For the parent function f(x) $$=$$ $$b^x$$, a vertical translation downwards by $$d$$ units is represented by g(x) $$=$$ $$b^x$$ - $$d$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b7d40expgraph17a-h2","type":"hint","dependencies":[],"title":"Reflection across the x-axis","text":"For the parent function f(x) $$=$$ $$b^x$$, a reflection across the x-axis is represented by g(x) $$=$$ $$-\\\\left(b^x\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b7d40expgraph17a-h3","type":"hint","dependencies":[],"title":"Horizontal translations","text":"For the parent function f(x) $$=$$ $$b^x$$, a horizontal translation c units to the left is represented by g(x) $$=$$ $$b^{x+c}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b7d40expgraph18","title":"Transformations of Exponential Functions","body":"What is the function, g(x), after reflecting $$f(x)={3\\\\left(\\\\frac{1}{2}\\\\right)}^x$$ about the y-axis?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Graphs of Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b7d40expgraph18a","stepAnswer":["$$g(x)=3{\\\\left(\\\\frac{1}{2}\\\\right)}^{-x}$$"],"problemType":"TextBox","stepTitle":"","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$g(x)=3{\\\\left(\\\\frac{1}{2}\\\\right)}^{-x}$$","hints":{"DefaultPathway":[{"id":"a3b7d40expgraph18a-h1","type":"hint","dependencies":[],"title":"Reflection across the x-axis","text":"For the parent function f(x) $$=$$ $$b^x$$, a reflection across the y-axis is represented by g(x) $$=$$ $$b^{-x}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b7d40expgraph19","title":"Transformations of Exponential Functions","body":"What is the function, g(x), after reflecting $$f(x)=-\\\\left({4\\\\left(2\\\\right)}^x\\\\right)+2$$ about the x-axis?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Graphs of Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b7d40expgraph19a","stepAnswer":["$$g(x)=4\\\\times2^x-2$$"],"problemType":"TextBox","stepTitle":"","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$g(x)=4\\\\times2^x-2$$","hints":{"DefaultPathway":[{"id":"a3b7d40expgraph19a-h1","type":"hint","dependencies":[],"title":"Rewriting the equation","text":"Let $$b$$ $$=$$ $$-\\\\left({4\\\\left(2\\\\right)}^x\\\\right)+2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b7d40expgraph19a-h2","type":"hint","dependencies":[],"title":"Reflections across the x-axis","text":"For the parent function f(x) $$=$$ $$b^x$$, a reflection across the x-axis is represented by g(x) $$=$$ $$-\\\\left(b^x\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b7d40expgraph2","title":"Finding the domain and range of an exponential function","body":"What is the domain and range of $$f(x)=2^{x+1}-3$$?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Graphs of Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b7d40expgraph2a","stepAnswer":["domain: $$(-\\\\infty,\\\\infty)$$, range: $$(-3,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"domain: $$(-\\\\infty,\\\\infty)$$, range: $$(-3,\\\\infty)$$","choices":["domain: $$(-1,\\\\infty)$$, range: $$(3,\\\\infty)$$","domain: $$(-\\\\infty,\\\\infty)$$, range: $$(-3,\\\\infty)$$","domain: $$(3,\\\\infty)$$, range: $$(-1,\\\\infty)$$","domain: $$(-3,\\\\infty)$$, range: $$(-\\\\infty,\\\\infty)$$","domain: $$(-\\\\infty,\\\\infty)$$, range: $$(-3,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"a3b7d40expgraph2a-h1","type":"hint","dependencies":[],"title":"Characteristics of an exponential function","text":"Recall that an exponential function with the form $$f(x)=b^x$$, $$b>0$$, where $$b$$ does not equal $$1$$, has these characteristics:\\\\n\\\\n$$1$$. one-to-one function\\\\n$$2$$. horizontal asymptote: $$y$$ $$=$$ $$0$$\\\\n$$3$$. domain: $$(-\\\\infty,\\\\infty)$$\\\\n$$4$$. range: $$(0,\\\\infty)$$\\\\n$$5$$. x-intercept: none\\\\n$$6$$. y-intercept: $$(0,1)$$\\\\n$$7$$. increasing if $$b$$ > $$1$$\\\\n$$8$$. decreasing if $$b$$ < $$1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b7d40expgraph2a-h2","type":"hint","dependencies":["a3b7d40expgraph2a-h1"],"title":"Identifying the transformations","text":"The next step is to identify how the function is being transformed from the parent function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b7d40expgraph2a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Move all $$x-coordinates$$ $$1$$ unit to the left"],"dependencies":["a3b7d40expgraph2a-h2"],"title":"Translation on the $$x$$","text":"In the equation $$2^{x+1}-3$$, how would you describe the translation on $$x$$ in words?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Move all $$x-coordinates$$ $$1$$ unit to the left","Move all $$y-coordinates$$ $$1$$ unit to the left","Move all $$x-coordinates$$ $$1$$ unit to the right","Move all $$y-coordinates$$ $$1$$ unit to the right"]},{"id":"a3b7d40expgraph2a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Move all $$y-coordinates$$ $$3$$ units down"],"dependencies":["a3b7d40expgraph2a-h3"],"title":"Translation on the $$y$$","text":"In the equation $$2^{x+1}-3$$, how would you describe the translation on $$y$$ in words?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Move all $$x-coordinates$$ $$3$$ units up","Move all $$y-coordinates$$ $$3$$ units up","Move all $$x-coordinates$$ $$3$$ units down","Move all $$y-coordinates$$ $$3$$ units down"]},{"id":"a3b7d40expgraph2a-h5","type":"hint","dependencies":["a3b7d40expgraph2a-h4"],"title":"Finding the domain and range","text":"Transforming the parent function will also change the domain and range. It will be helpful to draw a graph to visualize.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b7d40expgraph20","title":"Transformations of Exponential Functions","body":"What is the function, g(x), after reflecting $$f(x)={3\\\\left(0.75\\\\right)}^x-1$$ about the x-axis?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Graphs of Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b7d40expgraph20a","stepAnswer":["$$g(x)=-3{0.75}^x+1$$"],"problemType":"TextBox","stepTitle":"","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$g(x)=-3{0.75}^x+1$$","hints":{"DefaultPathway":[{"id":"a3b7d40expgraph20a-h1","type":"hint","dependencies":[],"title":"Rewriting the equation","text":"Let $$b$$ $$=$$ $$-\\\\left({4\\\\left(2\\\\right)}^x\\\\right)+2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b7d40expgraph20a-h2","type":"hint","dependencies":[],"title":"Reflections across the x-axis","text":"For the parent function f(x) $$=$$ $$b^x$$, a reflection across the x-axis is represented by g(x) $$=$$ $$-\\\\left(b^x\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b7d40expgraph3","title":"Finding the domain and range of exponential functions","body":"What is the domain and range of $$f(x)={4\\\\left(\\\\frac{1}{2}\\\\right)}^x$$?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Graphs of Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b7d40expgraph3a","stepAnswer":["domain: $$(-\\\\infty,\\\\infty)$$, range: $$(0,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"domain: $$(-\\\\infty,\\\\infty)$$, range: $$(0,\\\\infty)$$","choices":["domain: $$(-\\\\infty,\\\\infty)$$, range: $$(0,\\\\infty)$$","domain: $$(0,\\\\infty)$$, range: $$(-\\\\infty,\\\\infty)$$","domain: $$(-\\\\infty,\\\\infty)$$, range: $$(1,\\\\infty)$$","domain: $$(1,\\\\infty)$$, range: $$(-\\\\infty,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"a3b7d40expgraph3a-h1","type":"hint","dependencies":[],"title":"Characteristics of an exponential function","text":"Recall that an exponential function with the form $$f(x)=b^x$$, $$b>0$$, where $$b$$ does not equal $$1$$, has these characteristics:\\\\n\\\\n$$1$$. one-to-one function\\\\n$$2$$. horizontal asymptote: $$y$$ $$=$$ $$0$$\\\\n$$3$$. domain: $$(-\\\\infty,\\\\infty)$$\\\\n$$4$$. range: $$(0,\\\\infty)$$\\\\n$$5$$. x-intercept: none\\\\n$$6$$. y-intercept: $$(0,1)$$\\\\n$$7$$. increasing if $$b$$ > $$1$$\\\\n$$8$$. decreasing if $$b$$ < $$1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b7d40expgraph3a-h2","type":"hint","dependencies":[],"title":"Stretching and Compressing the Parent Function","text":"For any factor a > $$0$$, the function $$f(x)={a\\\\left(b\\\\right)}^x$$\\\\n- is stretched vertically by a factor of a if |a|>1.\\\\n- is compressed vertically by a factor of a if |a| <1.\\\\n- has a y-intercept of (0,a).\\\\n- has a horizontal asymptote at $$y=0$$, a range of $$(0,\\\\infty)$$, and a domain of $$(-\\\\infty,\\\\infty)$$, which are unchanged from the parent function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b7d40expgraph3a-h3","type":"hint","dependencies":[],"title":"Finding the domain and range","text":"Transforming the parent function will also change the domain and range. It will be helpful to draw a graph to visualize.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b7d40expgraph4","title":"Find the equation of the graph, g(x), that reflects f(x) $$=$$ $${\\\\left(\\\\frac{1}{4}\\\\right)}^x$$ about the x-axis.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Graphs of Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b7d40expgraph4a","stepAnswer":["$$g(x)=-\\\\left({\\\\left(\\\\frac{1}{4}\\\\right)}^x\\\\right)$$"],"problemType":"MultipleChoice","stepTitle":"","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$g(x)=-\\\\left({\\\\left(\\\\frac{1}{4}\\\\right)}^x\\\\right)$$","choices":["$$g(x)=-\\\\left({\\\\left(\\\\frac{1}{4}\\\\right)}^x\\\\right)$$","$$g(x)={\\\\left(\\\\frac{1}{4}\\\\right)}^{\\\\left(-x\\\\right)}$$","$$g(x)={\\\\left(\\\\frac{1}{4}\\\\right)}^x+1$$","$$g(x)=-\\\\left({\\\\left(\\\\frac{1}{4}\\\\right)}^x\\\\right)-1$$"],"hints":{"DefaultPathway":[{"id":"a3b7d40expgraph4a-h1","type":"hint","dependencies":[],"title":"Graphing the function","text":"The first step would be to graph the current function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b7d40expgraph4a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(1,-2)"],"dependencies":["a3b7d40expgraph4a-h1"],"title":"Identifying which coordinate to change","text":"Say that we want to reflect $$(1,2)$$ across the x-axis. What will the reflected coordinate look like?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b7d40expgraph4a-h3","type":"hint","dependencies":["a3b7d40expgraph4a-h2"],"title":"Reflecting exponential functions","text":"The function f(x) $$=$$ $$-\\\\left(b^x\\\\right)$$ reflects the parent function f(x) $$=$$ $$b^x$$ about the x-axis. The function f(x) $$=$$ $$b^{\\\\left(-x\\\\right)}$$ reflects the parent function f(x) $$=$$ $$b^x$$ about the y-axis.\\\\n","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b7d40expgraph6","title":"Transforming exponential functions","body":"The graph of $$f(x)=3^x$$ is reflected about the y-axis and stretched vertically by a factor of $$4.4$$. What is the equation of the new function g(x)?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Graphs of Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b7d40expgraph6a","stepAnswer":["$$g(x)=4.4\\\\times3^{\\\\left(-x\\\\right)}$$"],"problemType":"TextBox","stepTitle":"","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$g(x)=4.4\\\\times3^{\\\\left(-x\\\\right)}$$","choices":["$$g(x)=4.4\\\\times3^{\\\\left(-x\\\\right)}$$","$$g(x)=-4.4\\\\times3^{\\\\left(-x\\\\right)}$$","$$g(x)=4.4-\\\\left(3^{\\\\left(-x\\\\right)}\\\\right)$$"],"hints":{"DefaultPathway":[{"id":"a3b7d40expgraph6a-h1","type":"hint","dependencies":[],"title":"Determining the transformed coordinates","text":"The first step is to identify which coordinates are being changed during each transformation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b7d40expgraph6a-h2","type":"hint","dependencies":[],"title":"Definition of reflection across the y-axis","text":"Recall that reflecting across the y-axis changes the x-coordinate (i.e. $$(-x$$, y)).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b7d40expgraph6a-h3","type":"hint","dependencies":[],"title":"Definition of vertical $$\\\\frac{stretch}{shrink}$$","text":"Recall that vertically stretching the function changes the $$y$$ coordinate (i.e (x, $$4.4y))$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b7d40expgraph6a-h4","type":"hint","dependencies":[],"title":"Mathematical representation of transformations","text":"In the general form of the function f(x) $$=$$ $$a b^{x+c}+d$$, a vertically $$\\\\frac{stretches}{shrinks}$$ the function, c translates the function to the left or right, and $$d$$ translates the function up or down.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b7d40expgraph6a-h5","type":"hint","dependencies":[],"title":"Determing the value of a","text":"What is a?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b7d40expgraph6a-h6","type":"hint","dependencies":[],"title":"Mathematical representation of a reflection across the y-axis","text":"The mathematical representation of a reflection across the y-axis is $$f(x)=b^{\\\\left(-x\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b7d40expgraph7","title":"Finding the domain and range","body":"What is the domain and range of $$f(x)={12}^{\\\\left(-x\\\\right)}$$?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Graphs of Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b7d40expgraph7a","stepAnswer":["domain: $$(-\\\\infty,\\\\infty)$$, range: $$(0,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"domain: $$(-\\\\infty,\\\\infty)$$, range: $$(0,\\\\infty)$$","choices":["domain: $$(-\\\\infty,\\\\infty)$$, range: $$(-\\\\infty,0)$$","domain: $$(-\\\\infty,\\\\infty)$$, range: $$(0,\\\\infty)$$","none of the above","domain: $$(-\\\\infty,\\\\infty)$$, range: $$(0,\\\\infty)$$","domain: $$(0,\\\\infty)$$, range: $$(-\\\\infty,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"a3b7d40expgraph7a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Manipulating exponents","text":"What is $${\\\\left(\\\\frac{1}{2}\\\\right)}^{-1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b7d40expgraph7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2^x$$"],"dependencies":[],"title":"Rewriting the equation","text":"Based on the previous answer, how can the equation be rewritten?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b7d40expgraph7a-h3","type":"hint","dependencies":[],"title":"Characteristics of an exponential function","text":"Recall that an exponential function with the form $$f(x)=b^x$$, $$b>0$$, where $$b$$ does not equal $$1$$, has these characteristics:\\\\n\\\\n$$1$$. one-to-one function\\\\n$$2$$. horizontal asymptote: $$y$$ $$=$$ $$0$$\\\\n$$3$$. domain: $$(-\\\\infty,\\\\infty)$$\\\\n$$4$$. range: $$(0,\\\\infty)$$\\\\n$$5$$. x-intercept: none\\\\n$$6$$. y-intercept: $$(0,1)$$\\\\n$$7$$. increasing if $$b$$ > $$1$$\\\\n$$8$$. decreasing if $$b$$ < $$1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b7d40expgraph8","title":"Finding horizontal asymptotes","body":"What is the horizontal asymptote of $$f(x)=-\\\\left({5\\\\left(4\\\\right)}^x\\\\right)-1$$?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Graphs of Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b7d40expgraph8a","stepAnswer":["$$-1$$"],"problemType":"TextBox","stepTitle":"","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1$$","hints":{"DefaultPathway":[{"id":"a3b7d40expgraph8a-h1","type":"hint","dependencies":[],"title":"Definition of horizontal asymptote","text":"The horizontal asymptote describes the end behavior of the function. In other words, it describes the value f(x) converges to as $$x$$ approaches infinity and/or negative $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b7d40expgraph8a-h2","type":"hint","dependencies":[],"title":"Finding the horizontal asymptote","text":"Determine the range of the function. It may be helpful to graph the function","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b7d40expgraph8a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-inf,-1)"],"dependencies":[],"title":"Finding the range","text":"What is the range of the function?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b7d40expgraph8a-h4","type":"hint","dependencies":[],"title":"Finding the horizontal asymptote","text":"The last step is to determine the value that f(x) converges to as $$x$$ approaches positive or negative $$\\\\infty$$. It may be helpful to check various values of $$x$$ (e.g. $$-100$$, $$-1000$$, $$-10000$$, etc.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3b7d40expgraph9","title":"Finding horizontal asymptotes","body":"What is the horizontal asymptote of $$h(x)=2x+3$$?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Graphs of Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a3b7d40expgraph9a","stepAnswer":["$$y=3$$"],"problemType":"TextBox","stepTitle":"","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=3$$","hints":{"DefaultPathway":[{"id":"a3b7d40expgraph9a-h1","type":"hint","dependencies":[],"title":"Definition of horizontal asymptote","text":"The horizontal asymptote describes the end behavior of the function. In other words, it describes the value f(x) converges to as $$x$$ approaches infinity and/or negative $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b7d40expgraph9a-h2","type":"hint","dependencies":[],"title":"Finding the horizontal asymptote","text":"Determine the range of the function. It may be helpful to graph the function","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b7d40expgraph9a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(3,inf)"],"dependencies":[],"title":"Finding the range","text":"What is the range of the function? Write the range in interval notation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3b7d40expgraph9a-h4","type":"hint","dependencies":[],"title":"Finding the horizontal asymptote","text":"The last step is to determine the value that f(x) converges to as $$x$$ approaches positive or negative $$\\\\infty$$. It may be helpful to check various values of $$x$$ (e.g. $$-100$$, $$-1000$$, $$-10000$$, etc.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c2b68root1","title":"Multiplying Square Roots","body":"Simplify the following square roots.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Multiply Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3c2b68root1a","stepAnswer":["$$48$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{2} \\\\sqrt{6}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$48$$","hints":{"DefaultPathway":[{"id":"a3c2b68root1a-h1","type":"hint","dependencies":[],"title":"Multiplying Each Root","text":"$$\\\\sqrt{2} \\\\sqrt{6}=\\\\sqrt{12}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root1a-h2","type":"hint","dependencies":["a3c2b68root1a-h1"],"title":"Simplifying the Product","text":"We can simplify $$\\\\sqrt{12}$$ to become $$2\\\\sqrt{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root1a-h3","type":"hint","dependencies":["a3c2b68root1a-h2"],"title":"$$4\\\\sqrt{3}\\\\times2 \\\\sqrt{12}$$","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root1a-h4","type":"hint","dependencies":["a3c2b68root1a-h3"],"title":"Multiplying Each Root","text":"$$4\\\\sqrt{3}\\\\times2 \\\\sqrt{12}$$ $$=$$ $$8\\\\sqrt{36}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root1a-h5","type":"hint","dependencies":["a3c2b68root1a-h4"],"title":"Simplifying the Product","text":"$$8\\\\sqrt{36}$$ can be simplified as $$48$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c2b68root10","title":"Multiplying Square Roots","body":"Simplify the following square roots.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Multiply Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3c2b68root10a","stepAnswer":["$$1+9x+6\\\\sqrt{x}$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(1+3\\\\sqrt{x}\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1+9x+6\\\\sqrt{x}$$","hints":{"DefaultPathway":[{"id":"a3c2b68root10a-h1","type":"hint","dependencies":[],"title":"Multiplying Each Root","text":"Recall binomial square: $${\\\\left(a+b\\\\right)}^2=a^2+2ab+b^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root10a-h2","type":"hint","dependencies":["a3c2b68root10a-h1"],"title":"Substitution","text":"Substitute the value in the binomial equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c2b68root11","title":"Multiplying Square Roots","body":"Simplify the following square roots.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Multiply Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3c2b68root11a","stepAnswer":["$$4+25m+20\\\\sqrt{m}$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(2+5\\\\sqrt{m}\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4+25m+20\\\\sqrt{m}$$","hints":{"DefaultPathway":[{"id":"a3c2b68root11a-h1","type":"hint","dependencies":[],"title":"Multiplying Each Root","text":"Recall binomial square: $${\\\\left(a+b\\\\right)}^2=a^2+2ab+b^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root11a-h2","type":"hint","dependencies":["a3c2b68root11a-h1"],"title":"Simplifying the Product","text":"Substitute the value in the binomial equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c2b68root12","title":"Multiplying Square Roots","body":"Simplify the following square roots.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Multiply Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3c2b68root12a","stepAnswer":["$$14$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(4-\\\\sqrt{2}\\\\right) \\\\left(4+\\\\sqrt{2}\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$14$$","hints":{"DefaultPathway":[{"id":"a3c2b68root12a-h1","type":"hint","dependencies":[],"title":"Multiplying Each Root","text":"Recall the formula of Product of Conjugate: $$\\\\left(a+b\\\\right) \\\\left(a-b\\\\right)=a^2-b^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root12a-h2","type":"hint","dependencies":["a3c2b68root12a-h1"],"title":"Simplifying the Product","text":"As $$a=4$$, $$b=\\\\sqrt{2}$$, $$a^2-b^2=14$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c2b68root13","title":"Multiplying Square Roots","body":"Simplify the following square roots.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Multiply Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3c2b68root13a","stepAnswer":["$$13$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(5-2\\\\sqrt{3}\\\\right) \\\\left(5+2\\\\sqrt{3}\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$13$$","hints":{"DefaultPathway":[{"id":"a3c2b68root13a-h1","type":"hint","dependencies":[],"title":"Multiplying Each Root","text":"Recall the formula of Product of Conjugate: $$\\\\left(a+b\\\\right) \\\\left(a-b\\\\right)=a^2-b^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root13a-h2","type":"hint","dependencies":["a3c2b68root13a-h1"],"title":"Simplifying the Product","text":"As $$a=5$$, $$b=2\\\\sqrt{3}$$, $$a^2-b^2=13$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c2b68root14","title":"Multiplying Square Roots","body":"Simplify the following square roots.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Multiply Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3c2b68root14a","stepAnswer":["$$-11$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(3-2\\\\sqrt{5}\\\\right) \\\\left(3+2\\\\sqrt{5}\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-11$$","hints":{"DefaultPathway":[{"id":"a3c2b68root14a-h1","type":"hint","dependencies":[],"title":"Multiplying Each Root","text":"Recall the formula of Product of Conjugate: $$\\\\left(a+b\\\\right) \\\\left(a-b\\\\right)=a^2-b^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root14a-h2","type":"hint","dependencies":["a3c2b68root14a-h1"],"title":"Simplifying the Product","text":"As $$a=3$$, $$b=2\\\\sqrt{5}$$, $$a^2-b^2=-11$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c2b68root15","title":"Multiplying Square Roots","body":"Simplify the following square roots.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Multiply Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3c2b68root15a","stepAnswer":["$$-159$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(4+5\\\\sqrt{7}\\\\right) \\\\left(4-5\\\\sqrt{7}\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-159$$","hints":{"DefaultPathway":[{"id":"a3c2b68root15a-h1","type":"hint","dependencies":[],"title":"Multiplying Each Root","text":"Recall the formula of Product of Conjugate: $$\\\\left(a+b\\\\right) \\\\left(a-b\\\\right)=a^2-b^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root15a-h2","type":"hint","dependencies":["a3c2b68root15a-h1"],"title":"Simplifying the Product","text":"As $$a=4$$, $$b=5\\\\sqrt{7}$$, $$a^2-b^2=-159$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c2b68root16","title":"Multiplying Square Roots","body":"Simplify the following square roots.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Multiply Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3c2b68root16a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{2} \\\\sqrt{8}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a3c2b68root16a-h1","type":"hint","dependencies":[],"title":"Multiplication","text":"Multiplying Each Root","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root16a-h2","type":"hint","dependencies":["a3c2b68root16a-h1"],"title":"Simplify","text":"Simplifying the Product","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c2b68root17","title":"Multiplying Square Roots","body":"Simplify the following square roots.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Multiply Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3c2b68root17a","stepAnswer":["$$18\\\\sqrt{6}$$"],"problemType":"TextBox","stepTitle":"$$3\\\\sqrt{3}\\\\times2 \\\\sqrt{18}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$18\\\\sqrt{6}$$","hints":{"DefaultPathway":[{"id":"a3c2b68root17a-h1","type":"hint","dependencies":[],"title":"Multiplying Each Root","text":"$$3\\\\sqrt{3}\\\\times2 \\\\sqrt{18}=6\\\\sqrt{54}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root17a-h2","type":"hint","dependencies":["a3c2b68root17a-h1"],"title":"Simplifying the Product","text":"We can simplify $$\\\\sqrt{54}$$ to become $$3\\\\sqrt{6}$$. $$6\\\\times3 \\\\sqrt{6}=18\\\\sqrt{6}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c2b68root18","title":"Multiplying Square Roots","body":"Simplify the following square roots.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Multiply Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3c2b68root18a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{6} \\\\sqrt{6}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"a3c2b68root18a-h1","type":"hint","dependencies":[],"title":"Multiplying Each Root","text":"$$\\\\sqrt{6} \\\\sqrt{6}=\\\\sqrt{36}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root18a-h2","type":"hint","dependencies":["a3c2b68root18a-h1"],"title":"Simplifying the Product","text":"$$\\\\sqrt{36}$$ can be simplified to $$6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c2b68root19","title":"Multiplying Square Roots","body":"Simplify the following square roots.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Multiply Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3c2b68root19a","stepAnswer":["$$48$$"],"problemType":"TextBox","stepTitle":"$$3\\\\sqrt{2}\\\\times2 \\\\sqrt{32}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$48$$","hints":{"DefaultPathway":[{"id":"a3c2b68root19a-h1","type":"hint","dependencies":[],"title":"Multiplying Each Root","text":"$$\\\\sqrt{2} \\\\sqrt{32}=\\\\sqrt{64}$$. So, we now have $$6\\\\sqrt{64}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root19a-h2","type":"hint","dependencies":["a3c2b68root19a-h1"],"title":"Simplifying the Product","text":"$$\\\\sqrt{64}=8$$, so we have $$6\\\\times8=48$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c2b68root2","title":"Multiplying Square Roots","body":"Simplify the following square roots.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Multiply Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3c2b68root2a","stepAnswer":["$$12\\\\sqrt{15}$$"],"problemType":"TextBox","stepTitle":"$$3\\\\sqrt{2}\\\\times2 \\\\sqrt{30}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12\\\\sqrt{15}$$","hints":{"DefaultPathway":[{"id":"a3c2b68root2a-h1","type":"hint","dependencies":[],"title":"Multiplying Each Root","text":"$$3\\\\sqrt{2}\\\\times2 \\\\sqrt{30}=6\\\\sqrt{60}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root2a-h2","type":"hint","dependencies":["a3c2b68root2a-h1"],"title":"Simplifying the Product","text":"We can simplify $$6\\\\sqrt{60}$$ to become $$12\\\\sqrt{15}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c2b68root20","title":"Multiplying Square Roots","body":"Simplify the following square roots.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Multiply Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3c2b68root20a","stepAnswer":["$$7\\\\sqrt{2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{7} \\\\sqrt{14}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7\\\\sqrt{2}$$","hints":{"DefaultPathway":[{"id":"a3c2b68root20a-h1","type":"hint","dependencies":[],"title":"Multiplying Each Root","text":"$$\\\\sqrt{7} \\\\sqrt{14}=\\\\sqrt{98}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root20a-h2","type":"hint","dependencies":["a3c2b68root20a-h1"],"title":"Simplifying the Product","text":"$$\\\\sqrt{98}=\\\\sqrt{49\\\\times2}=7\\\\sqrt{2}$$. The simplified form of the above product is $$7\\\\sqrt{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c2b68root21","title":"Multiplying Square Roots","body":"Simplify the following square roots.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Multiply Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3c2b68root21a","stepAnswer":["$$160$$"],"problemType":"TextBox","stepTitle":"$$4\\\\sqrt{8}\\\\times5 \\\\sqrt{8}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$160$$","hints":{"DefaultPathway":[{"id":"a3c2b68root21a-h1","type":"hint","dependencies":[],"title":"Multiplying Each Root","text":"$$\\\\sqrt{8} \\\\sqrt{8}=\\\\sqrt{64}$$. So, we have $$4\\\\times5 \\\\sqrt{64}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root21a-h2","type":"hint","dependencies":["a3c2b68root21a-h1"],"title":"Simplifying the Product","text":"The product simplifies to $$20\\\\sqrt{64}$$, which is $$160$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c2b68root22","title":"Multiplying Square Roots","body":"Simplify the following square roots.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Multiply Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3c2b68root22a","stepAnswer":["$$6\\\\sqrt{2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{6} \\\\sqrt{12}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6\\\\sqrt{2}$$","hints":{"DefaultPathway":[{"id":"a3c2b68root22a-h1","type":"hint","dependencies":[],"title":"Multiplying Each Root","text":"$$\\\\sqrt{6} \\\\sqrt{12}=\\\\sqrt{72}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root22a-h2","type":"hint","dependencies":["a3c2b68root22a-h1"],"title":"Simplifying the Product","text":"$$\\\\sqrt{72}=\\\\sqrt{36\\\\times2}=6\\\\sqrt{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c2b68root23","title":"Multiplying Square Roots","body":"Simplify the following square roots.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Multiply Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3c2b68root23a","stepAnswer":["$$20\\\\sqrt{2}$$"],"problemType":"TextBox","stepTitle":"$$2\\\\sqrt{5}\\\\times2 \\\\sqrt{10}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$20\\\\sqrt{2}$$","hints":{"DefaultPathway":[{"id":"a3c2b68root23a-h1","type":"hint","dependencies":[],"title":"Multiplying Each Root","text":"$$\\\\sqrt{5} \\\\sqrt{10}=\\\\sqrt{50}$$. $$2\\\\times2 \\\\sqrt{50}=4\\\\sqrt{50}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root23a-h2","type":"hint","dependencies":["a3c2b68root23a-h1"],"title":"Simplifying the Product","text":"$$\\\\sqrt{50}=\\\\sqrt{25\\\\times2}=5\\\\sqrt{2}$$. $$4\\\\times5 \\\\sqrt{2}=20\\\\sqrt{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c2b68root24","title":"Multiplying Square Roots","body":"Simplify the following square roots.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Multiply Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3c2b68root24a","stepAnswer":["$$30\\\\sqrt{3}$$"],"problemType":"TextBox","stepTitle":"$$5\\\\sqrt{2}\\\\times3 \\\\sqrt{6}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$30\\\\sqrt{3}$$","hints":{"DefaultPathway":[{"id":"a3c2b68root24a-h1","type":"hint","dependencies":[],"title":"Multiplying Each Root","text":"$$\\\\sqrt{2} \\\\sqrt{6}=\\\\sqrt{12}$$. We now have $$15\\\\sqrt{12}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root24a-h2","type":"hint","dependencies":["a3c2b68root24a-h1"],"title":"Simplifying the Product","text":"$$\\\\sqrt{12}=\\\\sqrt{4\\\\times3}=2\\\\sqrt{3}$$. $$15\\\\times2 \\\\sqrt{3}=30\\\\sqrt{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c2b68root25","title":"Multiplying Square Roots","body":"Simplify the following square roots.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Multiply Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3c2b68root25a","stepAnswer":["$$24\\\\sqrt{2}$$"],"problemType":"TextBox","stepTitle":"$$2\\\\sqrt{3}\\\\times4 \\\\sqrt{6}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$24\\\\sqrt{2}$$","hints":{"DefaultPathway":[{"id":"a3c2b68root25a-h1","type":"hint","dependencies":[],"title":"Multiplying Each Root","text":"$$\\\\sqrt{3} \\\\sqrt{6}=\\\\sqrt{18}$$. $$2\\\\times4 \\\\sqrt{18}=8\\\\sqrt{18}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root25a-h2","type":"hint","dependencies":["a3c2b68root25a-h1"],"title":"Simplifying the Product","text":"$$8\\\\sqrt{18}=8\\\\sqrt{9\\\\times2}=24\\\\sqrt{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c2b68root26","title":"Multiplying Square Roots","body":"Simplify the following square roots.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Multiply Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3c2b68root26a","stepAnswer":["$$-18\\\\sqrt{6}$$"],"problemType":"TextBox","stepTitle":"$$-2\\\\sqrt{3}\\\\times3 \\\\sqrt{18}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-18\\\\sqrt{6}$$","hints":{"DefaultPathway":[{"id":"a3c2b68root26a-h1","type":"hint","dependencies":[],"title":"Multiplying Each Root","text":"$$\\\\sqrt{3} \\\\sqrt{18}=\\\\sqrt{54}$$. $$-2\\\\times3=-6$$. $$-6\\\\sqrt{54}$$ is the product.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root26a-h2","type":"hint","dependencies":["a3c2b68root26a-h1"],"title":"Simplifying the Product","text":"$$-6\\\\sqrt{54}=-6\\\\sqrt{9\\\\times6}=-18\\\\sqrt{6}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c2b68root27","title":"Multiplying Square Roots","body":"Simplify the following square roots.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Multiply Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3c2b68root27a","stepAnswer":["$$-100\\\\sqrt{2}$$"],"problemType":"TextBox","stepTitle":"$$-4\\\\sqrt{5}\\\\times5 \\\\sqrt{10}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-100\\\\sqrt{2}$$","hints":{"DefaultPathway":[{"id":"a3c2b68root27a-h1","type":"hint","dependencies":[],"title":"Multiplying Each Root","text":"$$-4\\\\sqrt{5}\\\\times5 \\\\sqrt{10}=-20\\\\sqrt{50}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root27a-h2","type":"hint","dependencies":["a3c2b68root27a-h1"],"title":"Simplifying the Product","text":"We must simplify the product. $$-20\\\\sqrt{50}=-20\\\\sqrt{25\\\\times2}=-100\\\\sqrt{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c2b68root28","title":"Multiplying Square Roots","body":"Simplify the following square roots.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Multiply Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3c2b68root28a","stepAnswer":["$$-10\\\\sqrt{15}$$"],"problemType":"TextBox","stepTitle":"$$5\\\\sqrt{6} \\\\left(-\\\\sqrt{12}\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-10\\\\sqrt{15}$$","hints":{"DefaultPathway":[{"id":"a3c2b68root28a-h1","type":"hint","dependencies":[],"title":"Multiplying Each Root","text":"$$5\\\\sqrt{6} \\\\left(-1\\\\right) \\\\sqrt{12}=-5\\\\sqrt{60}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root28a-h2","type":"hint","dependencies":["a3c2b68root28a-h1"],"title":"Simplifying the Product","text":"$$-5\\\\sqrt{60}=-5\\\\sqrt{15\\\\times4}=-10\\\\sqrt{15}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c2b68root29","title":"Multiplying Square Roots","body":"Simplify the following square roots.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Multiply Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3c2b68root29a","stepAnswer":["$$-8\\\\sqrt{5}$$"],"problemType":"TextBox","stepTitle":"$$6\\\\sqrt{2} \\\\left(-\\\\sqrt{10}\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-8\\\\sqrt{5}$$","hints":{"DefaultPathway":[{"id":"a3c2b68root29a-h1","type":"hint","dependencies":[],"title":"Multiplying Each Root","text":"$$6\\\\sqrt{2} \\\\left(-\\\\sqrt{10}\\\\right)=-6\\\\sqrt{20}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root29a-h2","type":"hint","dependencies":["a3c2b68root29a-h1"],"title":"Simplifying the Product","text":"$$-6\\\\sqrt{20}=-6\\\\sqrt{4\\\\times5}=-8\\\\sqrt{5}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c2b68root3","title":"Multiplying Square Roots","body":"Simplify the following equation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Multiply Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3c2b68root3a","stepAnswer":["$$27\\\\sqrt{2}$$"],"problemType":"TextBox","stepTitle":"$$3\\\\sqrt{3}\\\\times3 \\\\sqrt{6}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$27\\\\sqrt{2}$$","hints":{"DefaultPathway":[{"id":"a3c2b68root3a-h1","type":"hint","dependencies":[],"title":"Multiplying Each Root","text":"$$3\\\\sqrt{3}\\\\times3 \\\\sqrt{6}=9\\\\sqrt{18}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root3a-h2","type":"hint","dependencies":["a3c2b68root3a-h1"],"title":"Simplifying the Product","text":"$$9\\\\sqrt{18}$$ can be simplified to $$27\\\\sqrt{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c2b68root30","title":"Multiplying Square Roots","body":"Simplify the following square roots.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Multiply Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3c2b68root30a","stepAnswer":["$$27\\\\sqrt{2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(-2\\\\sqrt{7}\\\\right) \\\\left(-2\\\\sqrt{14}\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$27\\\\sqrt{2}$$","hints":{"DefaultPathway":[{"id":"a3c2b68root30a-h1","type":"hint","dependencies":[],"title":"Multiplying Each Root","text":"$$\\\\left(-2\\\\sqrt{7}\\\\right) \\\\left(-2\\\\sqrt{14}\\\\right)=4\\\\sqrt{98}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root30a-h2","type":"hint","dependencies":["a3c2b68root30a-h1"],"title":"Simplifying the Product","text":"$$4\\\\sqrt{98}$$ can be simplified: $$4\\\\sqrt{98}=4\\\\sqrt{47\\\\times2}=27\\\\sqrt{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c2b68root4","title":"Multiplying Square Roots","body":"Simplify the following square roots.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Multiply Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3c2b68root4a","stepAnswer":["$$144x^3 \\\\sqrt{10}$$"],"problemType":"TextBox","stepTitle":"$$6\\\\sqrt{2x^2}\\\\times8 \\\\sqrt{45x^4}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$144x^3 \\\\sqrt{10}$$","hints":{"DefaultPathway":[{"id":"a3c2b68root4a-h1","type":"hint","dependencies":[],"title":"Multiplying Each Root","text":"$$6\\\\sqrt{2x^2}\\\\times8 \\\\sqrt{45x^4}$$ $$=$$ $$48\\\\sqrt{90x^6}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root4a-h2","type":"hint","dependencies":["a3c2b68root4a-h1"],"title":"Simplifying the Product","text":"$$\\\\sqrt{90x^6}=3x^3 \\\\sqrt{10}$$, so we have $$48\\\\times3 x^3 \\\\sqrt{10}=144x^3 \\\\sqrt{10}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c2b68root5","title":"Multiplying Square Roots","body":"Simplify the following square roots.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Multiply Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3c2b68root5a","stepAnswer":["$$180p^2 \\\\sqrt{3}$$"],"problemType":"TextBox","stepTitle":"$$10\\\\sqrt{6p^3} 3\\\\sqrt{18p}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$180p^2 \\\\sqrt{3}$$","hints":{"DefaultPathway":[{"id":"a3c2b68root5a-h1","type":"hint","dependencies":[],"title":"Multiplying Each Root","text":"$$10\\\\sqrt{6p^3}\\\\times3 \\\\sqrt{18p}=30\\\\sqrt{108p^4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root5a-h2","type":"hint","dependencies":["a3c2b68root5a-h1"],"title":"Simplifying the Product","text":"$$\\\\sqrt{108p^4}=6p^2 \\\\sqrt{3}$$. The simplified form of the above product is $$30\\\\times6 p^2 \\\\sqrt{3}$$ $$=$$ $$180p^2 \\\\sqrt{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c2b68root6","title":"Multiplying Square Roots","body":"Simplify the following square roots.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Multiply Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3c2b68root6a","stepAnswer":["a) $$16$$ b) $$25$$"],"problemType":"MultipleChoice","stepTitle":"a) $${\\\\sqrt{16}}^2$$ b) $${\\\\sqrt{-20}}^2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"a) $$16$$ b) $$25$$","choices":["a) $$16$$ b) $$25$$","a) $$4$$ b) $$25$$","a) $$16$$ b) $$125$$","a) $$4$$ b) $$5$$"],"hints":{"DefaultPathway":[{"id":"a3c2b68root6a-h1","type":"hint","dependencies":[],"title":"Multiplying Each Root","text":"$${\\\\sqrt{16}}^2=\\\\sqrt{256}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root6a-h2","type":"hint","dependencies":["a3c2b68root6a-h1"],"title":"Simplifying the Product","text":"$$\\\\sqrt{256}$$ can be simplified to $$16$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root6a-h3","type":"hint","dependencies":[],"title":"Multiplying Each Root","text":"$${\\\\sqrt{-25}}^2=\\\\sqrt{625}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root6a-h4","type":"hint","dependencies":["a3c2b68root6a-h3"],"title":"Simplifying the Product","text":"$$\\\\sqrt{625}$$ can be simplified to $$25$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c2b68root7","title":"Multiplying Square Roots","body":"Simplify the following square roots.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Multiply Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3c2b68root7a","stepAnswer":["a) $$48$$ b) $$54$$"],"problemType":"MultipleChoice","stepTitle":"a) $$2\\\\sqrt{3} 8\\\\sqrt{3}$$ b) $${\\\\left(3\\\\sqrt{6}\\\\right)}^2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"a) $$48$$ b) $$54$$","choices":["a) $$24$$ b) $$54$$","a) $$24$$ b) $$27$$","a) $$48$$ b) $$54$$","a) $$48$$ b) $$27$$"],"hints":{"DefaultPathway":[{"id":"a3c2b68root7a-h1","type":"hint","dependencies":[],"title":"Multiplying Each Root","text":"$$2\\\\sqrt{3}\\\\times8 \\\\sqrt{3}$$ $$=$$ $$16\\\\sqrt{9}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root7a-h2","type":"hint","dependencies":["a3c2b68root7a-h1"],"title":"Simplifying the Product","text":"$$\\\\sqrt{9}$$ can be simplified to $$3$$. $$16\\\\times3=48$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root7a-h3","type":"hint","dependencies":[],"title":"Multiplying Each Root","text":"$${\\\\left(3\\\\sqrt{6}\\\\right)}^2=9\\\\sqrt{36}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root7a-h4","type":"hint","dependencies":["a3c2b68root7a-h3"],"title":"Simplifying the Product","text":"$$\\\\sqrt{36}=6$$ $$9\\\\times6=54$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c2b68root8","title":"Multiplying Square Roots","body":"Simplify the following square roots.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Multiply Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3c2b68root8a","stepAnswer":["$$-3+2\\\\sqrt{6}$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(1+\\\\sqrt{6}\\\\right) \\\\left(3-\\\\sqrt{6}\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-3+2\\\\sqrt{6}$$","hints":{"DefaultPathway":[{"id":"a3c2b68root8a-h1","type":"hint","dependencies":[],"title":"Multiplying Each Root","text":"$$\\\\left(1+\\\\sqrt{6}\\\\right) \\\\left(3-\\\\sqrt{6}\\\\right)=3+3\\\\sqrt{6}-\\\\sqrt{6}-6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root8a-h2","type":"hint","dependencies":["a3c2b68root8a-h1"],"title":"Simplifying the Product","text":"Simplify the answer above to $$-3+2\\\\sqrt{6}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c2b68root9","title":"Multiplying Square Roots","body":"Simplify the following square roots.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Multiply Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3c2b68root9a","stepAnswer":["$$40-14\\\\sqrt{7}$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(3-2\\\\sqrt{7}\\\\right) \\\\left(4-2\\\\sqrt{7}\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$40-14\\\\sqrt{7}$$","hints":{"DefaultPathway":[{"id":"a3c2b68root9a-h1","type":"hint","dependencies":[],"title":"Multiplying Each Root","text":"$$\\\\left(3-2\\\\sqrt{7}\\\\right) \\\\left(4-2\\\\sqrt{7}\\\\right)=12-8\\\\sqrt{7}-6\\\\sqrt{7}+4\\\\sqrt{49}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c2b68root9a-h2","type":"hint","dependencies":["a3c2b68root9a-h1"],"title":"Simplifying the Product","text":"$$4\\\\sqrt{49}$$ can be simplified as $$28$$. The answer above can be simplified as $$40-14\\\\sqrt{7}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c51e5DiscreteD1","title":"Lucky Dice Game","body":"Suppose we are playing the lucky dice game. The rules of the game are as follows: First, each player selects any number from $$1-6$$. Next, each player will roll $$3$$ dice and count how times each dice matched their lucky number. $$1$$ correct guess will result in $$1$$ point. This process will then be repeated as many times are you want (Typically $$3-10$$ times), and the highest scoring player wins.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.8 Discrete Distribution (Lucky Dice Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3c51e5DiscreteD1a","stepAnswer":["$$\\\\frac{1}{6}$$"],"problemType":"TextBox","stepTitle":"What is the probability that if our player only rolls $$1$$ die, they correctly guess the right number?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{6}$$","hints":{"DefaultPathway":[{"id":"a3c51e5DiscreteD1a-h1","type":"hint","dependencies":[],"title":"Calculating Probabilities","text":"The procedure to find the probability of something is as follows: First find the total number of possible outcomes for your situation. Second, find the total number of outcomes where your situation matches the one you want. Finally, divide your number acquired in the second step of this procedure by the first number, and you have the probability of your desired situation occuring.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD1a-h2","type":"hint","dependencies":["a3c51e5DiscreteD1a-h1"],"title":"Calculating Probabilities","text":"To calculate the probability of this situation, we first have to count the total number of outcomes in this situation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a3c51e5DiscreteD1a-h2"],"title":"Dice Faces","text":"Assuming we select a number beforehand, how many different faces can our dice show on $$1$$ roll?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD1a-h4","type":"hint","dependencies":["a3c51e5DiscreteD1a-h3"],"title":"Calculating Probabilities","text":"The next step of calculating this probability is to find the number of instances of our situation occurring.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD1a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a3c51e5DiscreteD1a-h4"],"title":"Correct Dice Face","text":"Out of our $$6$$ rolls, how many of them result in us having the correct number?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD1a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{6}$$"],"dependencies":["a3c51e5DiscreteD1a-h5"],"title":"Answer","text":"Knowing the total number of outcomes and the total number of instances in which our situation occurs, what is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c51e5DiscreteD10","title":"Lucky Dice Game","body":"Suppose we are playing the lucky dice game. The rules of the game are as follows: First, each player selects any number from $$1-6$$. Next, each player will roll $$3$$ dice and count how times each dice matched their lucky number. $$1$$ correct guess will result in $$1$$ point. This process will then be repeated as many times are you want (Typically $$3-10$$ times), and the highest scoring player wins.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.8 Discrete Distribution (Lucky Dice Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3c51e5DiscreteD10a","stepAnswer":["$$0.0694$$"],"problemType":"TextBox","stepTitle":"What is the probability that if our player rolls $$3$$ dice, they have two matches?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.0694$$","hints":{"DefaultPathway":[{"id":"a3c51e5DiscreteD10a-h1","type":"hint","dependencies":[],"title":"Calculating Binomial Probabilities","text":"To find a binomial probability, we can use the equation: (n choose x)*(p**x)*(q**n-x) where $$n=total$$ trials, $$x=number$$ of times our situation occurs, $$p=probability$$ of success, and $$q=probability$$ of failure.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a3c51e5DiscreteD10a-h1"],"title":"Acquiring Variables","text":"How many total trials are there? What is $$n$$ in our equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a3c51e5DiscreteD10a-h2"],"title":"Acquiring Variables","text":"How many times does our situation occur? What is $$x$$ in our equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{6}$$"],"dependencies":["a3c51e5DiscreteD10a-h3"],"title":"Acquiring Variables","text":"What is the probability of success in $$1$$ trial? What is $$p$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{6}$$"],"dependencies":["a3c51e5DiscreteD10a-h4"],"title":"Acquiring Variables","text":"What is the probability of failure in $$1$$ trial? What is q?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD10a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.0694$$"],"dependencies":["a3c51e5DiscreteD10a-h5"],"title":"Answer","text":"With all of our variables set up, what will our final answer be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c51e5DiscreteD11","title":"Lucky Dice Game","body":"Suppose we are playing the lucky dice game. The rules of the game are as follows: First, each player selects any number from $$1-6$$. Next, each player will roll $$3$$ dice and count how times each dice matched their lucky number. $$1$$ correct guess will result in $$1$$ point. This process will then be repeated as many times are you want (Typically $$3-10$$ times), and the highest scoring player wins.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.8 Discrete Distribution (Lucky Dice Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3c51e5DiscreteD11a","stepAnswer":["$$0.0046$$"],"problemType":"TextBox","stepTitle":"What is the probability that if our player rolls $$3$$ dice, they have three matches?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.0046$$","hints":{"DefaultPathway":[{"id":"a3c51e5DiscreteD11a-h1","type":"hint","dependencies":[],"title":"Calculating Binomial Probabilities","text":"To find a binomial probability, we can use the equation: (n choose x)*(p**x)*(q**n-x) where $$n=total$$ trials, $$x=number$$ of times our situation occurs, $$p=probability$$ of success, and $$q=probability$$ of failure.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a3c51e5DiscreteD11a-h1"],"title":"Acquiring Variables","text":"How many total trials are there? What is $$n$$ in our equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD11a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a3c51e5DiscreteD11a-h2"],"title":"Acquiring Variables","text":"How many times does our situation occur? What is $$x$$ in our equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{6}$$"],"dependencies":["a3c51e5DiscreteD11a-h3"],"title":"Acquiring Variables","text":"What is the probability of success in $$1$$ trial? What is $$p$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{6}$$"],"dependencies":["a3c51e5DiscreteD11a-h4"],"title":"Acquiring Variables","text":"What is the probability of failure in $$1$$ trial? What is q?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD11a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.0046$$"],"dependencies":["a3c51e5DiscreteD11a-h5"],"title":"Answer","text":"With all of our variables set up, what will our final answer be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c51e5DiscreteD12","title":"Lucky Dice Game","body":"Suppose we are playing the lucky dice game. The rules of the game are as follows: First, each player selects any number from $$1-6$$. Next, each player will roll $$3$$ dice and count how times each dice matched their lucky number. $$1$$ correct guess will result in $$1$$ point. This process will then be repeated as many times are you want (Typically $$3-10$$ times), and the highest scoring player wins.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.8 Discrete Distribution (Lucky Dice Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3c51e5DiscreteD12a","stepAnswer":["$$\\\\frac{1}{2}$$"],"problemType":"TextBox","stepTitle":"If our player rolls three dice, how many points can we expect them to get?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{2}$$","hints":{"DefaultPathway":[{"id":"a3c51e5DiscreteD12a-h1","type":"hint","dependencies":[],"title":"Calculating Expected Values","text":"To calculate the expected value of something, we can use the equation E[X] $$=$$ sum{i\\\\=0}{...}{p(X_i)*X_i}, where $$p\\\\left(X_i\\\\right)$$ is the probability of $$X_i$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD12a-h2","type":"hint","dependencies":["a3c51e5DiscreteD12a-h1"],"title":"What is X?","text":"We can set $$X_0$$ $$=$$ Getting no points, $$X_1$$ $$=$$ Getting $$1$$ point, $$X_2$$ $$=$$ Getting $$2$$ points, and $$X_3$$ $$=$$ Getting $$3$$ points.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD12a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.5787$$"],"dependencies":["a3c51e5DiscreteD12a-h2"],"title":"Calculating Probabilities","text":"What is the probability that we get no points?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.3472$$"],"dependencies":["a3c51e5DiscreteD12a-h3"],"title":"Calculating Probabilities","text":"What is the probability that we get $$1$$ point?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD12a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.0694$$"],"dependencies":["a3c51e5DiscreteD12a-h4"],"title":"Calculating Probabilities","text":"What is the probability that we get $$2$$ points?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD12a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.0046$$"],"dependencies":["a3c51e5DiscreteD12a-h5"],"title":"Calculating Probabilities","text":"What is the probability that we get $$3$$ points?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD12a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a3c51e5DiscreteD12a-h6"],"title":"Answer","text":"Now that we have all of our variables, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c51e5DiscreteD13","title":"Lucky Dice Game","body":"Suppose we are playing the lucky dice game. The rules of the game are as follows: First, each player selects any number from $$1-6$$. Next, each player will roll $$3$$ dice and count how times each dice matched their lucky number. $$1$$ correct guess will result in $$1$$ point. This process will then be repeated as many times are you want (Typically $$3-10$$ times), and the highest scoring player wins.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.8 Discrete Distribution (Lucky Dice Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3c51e5DiscreteD13a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"If our player rolls three dice and plays the game twice, how many point(s) can we expect them to get?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a3c51e5DiscreteD13a-h1","type":"hint","dependencies":[],"title":"Calculating Expected Values","text":"To calculate the expected value for this problem, we can multiply the expected value for $$1$$ game of lucky dice by two, since we are playing the game twice.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a3c51e5DiscreteD13a-h1"],"title":"Expected Value for $$1$$ game","text":"If our player rolls three dice, how many points can we expect them to get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD13a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a3c51e5DiscreteD13a-h2"],"title":"Answer","text":"Now that we have our equation and expected value for one game, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c51e5DiscreteD14","title":"Lucky Dice Game","body":"Suppose we are playing the lucky dice game. The rules of the game are as follows: First, each player selects any number from $$1-6$$. Next, each player will roll $$3$$ dice and count how times each dice matched their lucky number. $$1$$ correct guess will result in $$1$$ point. This process will then be repeated as many times are you want (Typically $$3-10$$ times), and the highest scoring player wins.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.8 Discrete Distribution (Lucky Dice Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3c51e5DiscreteD14a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"If our player rolls three dice and plays the game ten times, how many point(s) can we expect them to get?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"a3c51e5DiscreteD14a-h1","type":"hint","dependencies":[],"title":"Calculating Expected Values","text":"To calculate the expected value for this problem, we can multiply the expected value for $$1$$ game of lucky dice by ten, since we are playing the game ten times.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a3c51e5DiscreteD14a-h1"],"title":"Expected Value for $$1$$ game","text":"If our player rolls three dice, how many points can we expect them to get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD14a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a3c51e5DiscreteD14a-h2"],"title":"Answer","text":"Now that we have our equation and expected value for one game, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c51e5DiscreteD15","title":"Lucky Dice Game","body":"Suppose we are playing the lucky dice game. The rules of the game are as follows: First, each player selects any number from $$1-6$$. Next, each player will roll $$3$$ dice and count how times each dice matched their lucky number. $$1$$ correct guess will result in $$1$$ point. This process will then be repeated as many times are you want (Typically $$3-10$$ times), and the highest scoring player wins.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.8 Discrete Distribution (Lucky Dice Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3c51e5DiscreteD15a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"If our player rolls three dice and plays the game six times, how many point(s) can we expect them to get?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a3c51e5DiscreteD15a-h1","type":"hint","dependencies":[],"title":"Calculating Expected Values","text":"To calculate the expected value for this problem, we can multiply the expected value for $$1$$ game of lucky dice by six, since we are playing the game six times.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a3c51e5DiscreteD15a-h1"],"title":"Expected Value for $$1$$ game","text":"If our player rolls three dice, how many points can we expect them to get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a3c51e5DiscreteD15a-h2"],"title":"Answer","text":"Now that we have our equation and expected value for one game, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c51e5DiscreteD16","title":"Lucky Dice Game","body":"Suppose we are playing the lucky dice game. The rules of the game are as follows: First, each player selects any number from $$1-6$$. Next, each player will roll $$3$$ dice and count how times each dice matched their lucky number. $$1$$ correct guess will result in $$1$$ point. This process will then be repeated as many times are you want (Typically $$3-10$$ times), and the highest scoring player wins.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.8 Discrete Distribution (Lucky Dice Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3c51e5DiscreteD16a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"If our player rolls two dice and plays the game three times, how many point(s) can we expect them to get?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a3c51e5DiscreteD16a-h1","type":"hint","dependencies":[],"title":"Calculating Expected Values","text":"To calculate the expected value for this problem, we can multiply the expected value for $$1$$ game of lucky dice by three, since we are playing the game three times.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["a3c51e5DiscreteD16a-h1"],"title":"Expected Value for $$1$$ game","text":"If our player rolls two dice, how many points can we expect them to get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD16a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a3c51e5DiscreteD16a-h2"],"title":"Answer","text":"Now that we have our equation and expected value for one game, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c51e5DiscreteD17","title":"Lucky Dice Game","body":"Suppose we are playing the lucky dice game. The rules of the game are as follows: First, each player selects any number from $$1-6$$. Next, each player will roll $$3$$ dice and count how times each dice matched their lucky number. $$1$$ correct guess will result in $$1$$ point. This process will then be repeated as many times are you want (Typically $$3-10$$ times), and the highest scoring player wins.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.8 Discrete Distribution (Lucky Dice Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3c51e5DiscreteD17a","stepAnswer":["$$11$$"],"problemType":"TextBox","stepTitle":"If our player rolls two dice and plays the game $$33$$ times, how many point(s) can we expect them to get?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$11$$","hints":{"DefaultPathway":[{"id":"a3c51e5DiscreteD17a-h1","type":"hint","dependencies":[],"title":"Calculating Expected Values","text":"To calculate the expected value for this problem, we can multiply the expected value for $$1$$ game of lucky dice by $$33$$, since we are playing the game $$33$$ times.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD17a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["a3c51e5DiscreteD17a-h1"],"title":"Expected Value for $$1$$ game","text":"If our player rolls two dice, how many points can we expect them to get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD17a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11$$"],"dependencies":["a3c51e5DiscreteD17a-h2"],"title":"Answer","text":"Now that we have our equation and expected value for one game, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c51e5DiscreteD18","title":"Lucky Dice Game","body":"Suppose we are playing the lucky dice game. The rules of the game are as follows: First, each player selects any number from $$1-6$$. Next, each player will roll $$3$$ dice and count how times each dice matched their lucky number. $$1$$ correct guess will result in $$1$$ point. This process will then be repeated as many times are you want (Typically $$3-10$$ times), and the highest scoring player wins.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.8 Discrete Distribution (Lucky Dice Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3c51e5DiscreteD18a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"If our player rolls two dice and plays the game six times, how many point(s) can we expect them to get?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a3c51e5DiscreteD18a-h1","type":"hint","dependencies":[],"title":"Calculating Expected Values","text":"To calculate the expected value for this problem, we can multiply the expected value for $$1$$ game of lucky dice by six, since we are playing the game six times.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["a3c51e5DiscreteD18a-h1"],"title":"Expected Value for $$1$$ game","text":"If our player rolls two dice, how many points can we expect them to get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD18a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a3c51e5DiscreteD18a-h2"],"title":"Answer","text":"Now that we have our equation and expected value for one game, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c51e5DiscreteD19","title":"Lucky Dice Game","body":"Suppose we are playing the lucky dice game. The rules of the game are as follows: First, each player selects any number from $$1-6$$. Next, each player will roll $$3$$ dice and count how times each dice matched their lucky number. $$1$$ correct guess will result in $$1$$ point. This process will then be repeated as many times are you want (Typically $$3-10$$ times), and the highest scoring player wins.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.8 Discrete Distribution (Lucky Dice Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3c51e5DiscreteD19a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"If our player rolls three dice, about how many games should it take for them to reach two points?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a3c51e5DiscreteD19a-h1","type":"hint","dependencies":[],"title":"Using Expected Values","text":"To calculate how many games it will take to reach two points, we can use the equation: $$2$$ $$=$$ $$E[X] n$$, where E[X] is the expected points from one game and $$n$$ is the amount of games it will take two reach two points. Solving for $$n$$ will give us the correct solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a3c51e5DiscreteD19a-h1"],"title":"Expected Value","text":"If our player rolls three dice, how many points can we expect them to get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD19a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a3c51e5DiscreteD19a-h2"],"title":"Answer","text":"With our equation and variables in hand, we can now solve for $$n$$. What is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c51e5DiscreteD2","title":"Lucky Dice Game","body":"Suppose we are playing the lucky dice game. The rules of the game are as follows: First, each player selects any number from $$1-6$$. Next, each player will roll $$3$$ dice and count how times each dice matched their lucky number. $$1$$ correct guess will result in $$1$$ point. This process will then be repeated as many times are you want (Typically $$3-10$$ times), and the highest scoring player wins.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.8 Discrete Distribution (Lucky Dice Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3c51e5DiscreteD2a","stepAnswer":["$$\\\\frac{5}{6}$$"],"problemType":"TextBox","stepTitle":"What is the probability that if our player only rolls $$1$$ die, they don\'t guess the right number?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5}{6}$$","hints":{"DefaultPathway":[{"id":"a3c51e5DiscreteD2a-h1","type":"hint","dependencies":[],"title":"Calculating Probabilities","text":"The procedure to find the probability of something is as follows: First find the total number of possible outcomes for your situation. Second, find the total number of outcomes where your situation matches the one you want. Finally, divide your number acquired in the second step of this procedure by the first number, and you have the probability of your desired situation occuring.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD2a-h2","type":"hint","dependencies":["a3c51e5DiscreteD2a-h1"],"title":"Calculating Probabilities","text":"To calculate the probability of this situation, we first have to count the total number of outcomes in this situation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a3c51e5DiscreteD2a-h2"],"title":"Dice Faces","text":"Assuming we select a number beforehand, how many different faces can our dice show on $$1$$ roll?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD2a-h4","type":"hint","dependencies":["a3c51e5DiscreteD2a-h3"],"title":"Calculating Probabilities","text":"The next step of calculating this probability is to find the number of instances of our situation occurring.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD2a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a3c51e5DiscreteD2a-h4"],"title":"Correct Dice Face","text":"Out of our $$6$$ rolls, how many of them result in us not having the correct number?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{6}$$"],"dependencies":["a3c51e5DiscreteD2a-h5"],"title":"Answer","text":"Knowing the total number of outcomes and the total number of instances in which our situation occurs, what is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c51e5DiscreteD20","title":"Lucky Dice Game","body":"Suppose we are playing the lucky dice game. The rules of the game are as follows: First, each player selects any number from $$1-6$$. Next, each player will roll $$3$$ dice and count how times each dice matched their lucky number. $$1$$ correct guess will result in $$1$$ point. This process will then be repeated as many times are you want (Typically $$3-10$$ times), and the highest scoring player wins.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.8 Discrete Distribution (Lucky Dice Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3c51e5DiscreteD20a","stepAnswer":["$$20$$"],"problemType":"TextBox","stepTitle":"If our player rolls three dice, about how many games should it take for them to reach ten points?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$20$$","hints":{"DefaultPathway":[{"id":"a3c51e5DiscreteD20a-h1","type":"hint","dependencies":[],"title":"Using Expected Values","text":"To calculate how many games it will take to reach ten points, we can use the equation: $$10$$ $$=$$ $$E[X] n$$, where E[X] is the expected points from one game and $$n$$ is the amount of games it will take two reach ten points. Solving for $$n$$ will give us the correct solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a3c51e5DiscreteD20a-h1"],"title":"Expected Value","text":"If our player rolls three dice, how many points can we expect them to get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD20a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a3c51e5DiscreteD20a-h2"],"title":"Answer","text":"With our equation and variables in hand, we can now solve for $$n$$. What is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c51e5DiscreteD3","title":"Lucky Dice Game","body":"Suppose we are playing the lucky dice game. The rules of the game are as follows: First, each player selects any number from $$1-6$$. Next, each player will roll $$3$$ dice and count how times each dice matched their lucky number. $$1$$ correct guess will result in $$1$$ point. This process will then be repeated as many times are you want (Typically $$3-10$$ times), and the highest scoring player wins.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.8 Discrete Distribution (Lucky Dice Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3c51e5DiscreteD3a","stepAnswer":["$$\\\\frac{1}{6}$$"],"problemType":"TextBox","stepTitle":"If our player rolls only $$1$$ die, how many points can we expect them to get?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{6}$$","hints":{"DefaultPathway":[{"id":"a3c51e5DiscreteD3a-h1","type":"hint","dependencies":[],"title":"Calculating Expected Values","text":"To calculate the expected value of something, we can use the equation E[X] $$=$$ sum{i\\\\=0}{...}{p(X_i)*X_i}, where $$p\\\\left(X_i\\\\right)$$ is the probability of $$X_i$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD3a-h2","type":"hint","dependencies":["a3c51e5DiscreteD3a-h1"],"title":"What is X?","text":"Since $$X=Getting$$ one point, we can set $$X=1$$ in our formula.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{6}$$"],"dependencies":["a3c51e5DiscreteD3a-h2"],"title":"Calculating Probabilities","text":"What is the probability that we get a point? In other words, what is the probability our guess matches a random dice roll?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{6}$$"],"dependencies":["a3c51e5DiscreteD3a-h3"],"title":"Answer","text":"Now that we have all of our variables, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c51e5DiscreteD4","title":"Lucky Dice Game","body":"Suppose we are playing the lucky dice game. The rules of the game are as follows: First, each player selects any number from $$1-6$$. Next, each player will roll $$3$$ dice and count how times each dice matched their lucky number. $$1$$ correct guess will result in $$1$$ point. This process will then be repeated as many times are you want (Typically $$3-10$$ times), and the highest scoring player wins.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.8 Discrete Distribution (Lucky Dice Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3c51e5DiscreteD4a","stepAnswer":["$$0.2777$$"],"problemType":"TextBox","stepTitle":"What is the probability that if our player rolls $$2$$ dice, they have one match?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.2777$$","hints":{"DefaultPathway":[{"id":"a3c51e5DiscreteD4a-h1","type":"hint","dependencies":[],"title":"Calculating Binomial Probabilities","text":"To find a binomial probability, we can use the equation: (n choose x)*(p**x)*(q**n-x) where $$n=total$$ trials, $$x=number$$ of times our situation occurs, $$p=probability$$ of success, and $$q=probability$$ of failure.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a3c51e5DiscreteD4a-h1"],"title":"Acquiring Variables","text":"How many total trials are there? What is $$n$$ in our equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a3c51e5DiscreteD4a-h2"],"title":"Acquiring Variables","text":"How many times does our situation occur? What is $$x$$ in our equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{6}$$"],"dependencies":["a3c51e5DiscreteD4a-h3"],"title":"Acquiring Variables","text":"What is the probability of success in $$1$$ trial? What is $$p$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD4a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{6}$$"],"dependencies":["a3c51e5DiscreteD4a-h4"],"title":"Acquiring Variables","text":"What is the probability of failure in $$1$$ trial? What is q?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD4a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.2777$$"],"dependencies":["a3c51e5DiscreteD4a-h5"],"title":"Answer","text":"With all of our variables set up, what will our final answer be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c51e5DiscreteD5","title":"Lucky Dice Game","body":"Suppose we are playing the lucky dice game. The rules of the game are as follows: First, each player selects any number from $$1-6$$. Next, each player will roll $$3$$ dice and count how times each dice matched their lucky number. $$1$$ correct guess will result in $$1$$ point. This process will then be repeated as many times are you want (Typically $$3-10$$ times), and the highest scoring player wins.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.8 Discrete Distribution (Lucky Dice Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3c51e5DiscreteD5a","stepAnswer":["$$0.6944$$"],"problemType":"TextBox","stepTitle":"What is the probability that if our player rolls $$2$$ dice, they have no matches?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.6944$$","hints":{"DefaultPathway":[{"id":"a3c51e5DiscreteD5a-h1","type":"hint","dependencies":[],"title":"Calculating Binomial Probabilities","text":"To find a binomial probability, we can use the equation: (n choose x)*(p**x)*(q**n-x) where $$n=total$$ trials, $$x=number$$ of times our situation occurs, $$p=probability$$ of success, and $$q=probability$$ of failure.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a3c51e5DiscreteD5a-h1"],"title":"Acquiring Variables","text":"How many total trials are there? What is $$n$$ in our equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a3c51e5DiscreteD5a-h2"],"title":"Acquiring Variables","text":"How many times does our situation occur? What is $$x$$ in our equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{6}$$"],"dependencies":["a3c51e5DiscreteD5a-h3"],"title":"Acquiring Variables","text":"What is the probability of success in $$1$$ trial? What is $$p$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{6}$$"],"dependencies":["a3c51e5DiscreteD5a-h4"],"title":"Acquiring Variables","text":"What is the probability of failure in $$1$$ trial? What is q?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD5a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.6944$$"],"dependencies":["a3c51e5DiscreteD5a-h5"],"title":"Answer","text":"With all of our variables set up, what will our final answer be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c51e5DiscreteD6","title":"Lucky Dice Game","body":"Suppose we are playing the lucky dice game. The rules of the game are as follows: First, each player selects any number from $$1-6$$. Next, each player will roll $$3$$ dice and count how times each dice matched their lucky number. $$1$$ correct guess will result in $$1$$ point. This process will then be repeated as many times are you want (Typically $$3-10$$ times), and the highest scoring player wins.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.8 Discrete Distribution (Lucky Dice Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3c51e5DiscreteD6a","stepAnswer":["$$0.0277$$"],"problemType":"TextBox","stepTitle":"What is the probability that if our player rolls $$2$$ dice, they have one match?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.0277$$","hints":{"DefaultPathway":[{"id":"a3c51e5DiscreteD6a-h1","type":"hint","dependencies":[],"title":"Calculating Binomial Probabilities","text":"To find a binomial probability, we can use the equation: (n choose x)*(p**x)*(q**n-x) where $$n=total$$ trials, $$x=number$$ of times our situation occurs, $$p=probability$$ of success, and $$q=probability$$ of failure.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a3c51e5DiscreteD6a-h1"],"title":"Acquiring Variables","text":"How many total trials are there? What is $$n$$ in our equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a3c51e5DiscreteD6a-h2"],"title":"Acquiring Variables","text":"How many times does our situation occur? What is $$x$$ in our equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{6}$$"],"dependencies":["a3c51e5DiscreteD6a-h3"],"title":"Acquiring Variables","text":"What is the probability of success in $$1$$ trial? What is $$p$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{6}$$"],"dependencies":["a3c51e5DiscreteD6a-h4"],"title":"Acquiring Variables","text":"What is the probability of failure in $$1$$ trial? What is q?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD6a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.0277$$"],"dependencies":["a3c51e5DiscreteD6a-h5"],"title":"Answer","text":"With all of our variables set up, what will our final answer be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c51e5DiscreteD7","title":"Lucky Dice Game","body":"Suppose we are playing the lucky dice game. The rules of the game are as follows: First, each player selects any number from $$1-6$$. Next, each player will roll $$3$$ dice and count how times each dice matched their lucky number. $$1$$ correct guess will result in $$1$$ point. This process will then be repeated as many times are you want (Typically $$3-10$$ times), and the highest scoring player wins.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.8 Discrete Distribution (Lucky Dice Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3c51e5DiscreteD7a","stepAnswer":["$$\\\\frac{1}{3}$$"],"problemType":"TextBox","stepTitle":"If our player rolls two dice, how many points can we expect them to get?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{3}$$","hints":{"DefaultPathway":[{"id":"a3c51e5DiscreteD7a-h1","type":"hint","dependencies":[],"title":"Calculating Expected Values","text":"To calculate the expected value of something, we can use the equation E[X] $$=$$ sum{i\\\\=0}{...}{p(X_i)*X_i}, where $$p\\\\left(X_i\\\\right)$$ is the probability of $$X_i$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD7a-h2","type":"hint","dependencies":["a3c51e5DiscreteD7a-h1"],"title":"What is X?","text":"We can set $$X_0$$ $$=$$ Getting no points, $$X_1$$ $$=$$ Getting $$1$$ point, and $$X_2$$ $$=$$ Getting $$2$$ points.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.6944$$"],"dependencies":["a3c51e5DiscreteD7a-h2"],"title":"Calculating Probabilities","text":"What is the probability that we get no points?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.2777$$"],"dependencies":["a3c51e5DiscreteD7a-h3"],"title":"Calculating Probabilities","text":"What is the probability that we get $$1$$ point?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD7a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.0277$$"],"dependencies":["a3c51e5DiscreteD7a-h4"],"title":"Calculating Probabilities","text":"What is the probability that we get $$2$$ points?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD7a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["a3c51e5DiscreteD7a-h5"],"title":"Answer","text":"Now that we have all of our variables, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c51e5DiscreteD8","title":"Lucky Dice Game","body":"Suppose we are playing the lucky dice game. The rules of the game are as follows: First, each player selects any number from $$1-6$$. Next, each player will roll $$3$$ dice and count how times each dice matched their lucky number. $$1$$ correct guess will result in $$1$$ point. This process will then be repeated as many times are you want (Typically $$3-10$$ times), and the highest scoring player wins.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.8 Discrete Distribution (Lucky Dice Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3c51e5DiscreteD8a","stepAnswer":["$$0.5787$$"],"problemType":"TextBox","stepTitle":"What is the probability that if our player rolls $$3$$ dice, they have no matches?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.5787$$","hints":{"DefaultPathway":[{"id":"a3c51e5DiscreteD8a-h1","type":"hint","dependencies":[],"title":"Calculating Binomial Probabilities","text":"To find a binomial probability, we can use the equation: (n choose x)*(p**x)*(q**n-x) where $$n=total$$ trials, $$x=number$$ of times our situation occurs, $$p=probability$$ of success, and $$q=probability$$ of failure.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a3c51e5DiscreteD8a-h1"],"title":"Acquiring Variables","text":"How many total trials are there? What is $$n$$ in our equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a3c51e5DiscreteD8a-h2"],"title":"Acquiring Variables","text":"How many times does our situation occur? What is $$x$$ in our equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{6}$$"],"dependencies":["a3c51e5DiscreteD8a-h3"],"title":"Acquiring Variables","text":"What is the probability of success in $$1$$ trial? What is $$p$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD8a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{6}$$"],"dependencies":["a3c51e5DiscreteD8a-h4"],"title":"Acquiring Variables","text":"What is the probability of failure in $$1$$ trial? What is q?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD8a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.5787$$"],"dependencies":["a3c51e5DiscreteD8a-h5"],"title":"Answer","text":"With all of our variables set up, what will our final answer be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3c51e5DiscreteD9","title":"Lucky Dice Game","body":"Suppose we are playing the lucky dice game. The rules of the game are as follows: First, each player selects any number from $$1-6$$. Next, each player will roll $$3$$ dice and count how times each dice matched their lucky number. $$1$$ correct guess will result in $$1$$ point. This process will then be repeated as many times are you want (Typically $$3-10$$ times), and the highest scoring player wins.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.8 Discrete Distribution (Lucky Dice Experiment)","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3c51e5DiscreteD9a","stepAnswer":["$$0.3472$$"],"problemType":"TextBox","stepTitle":"What is the probability that if our player rolls $$3$$ dice, they have one match?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.3472$$","hints":{"DefaultPathway":[{"id":"a3c51e5DiscreteD9a-h1","type":"hint","dependencies":[],"title":"Calculating Binomial Probabilities","text":"To find a binomial probability, we can use the equation: (n choose x)*(p**x)*(q**n-x) where $$n=total$$ trials, $$x=number$$ of times our situation occurs, $$p=probability$$ of success, and $$q=probability$$ of failure.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a3c51e5DiscreteD9a-h1"],"title":"Acquiring Variables","text":"How many total trials are there? What is $$n$$ in our equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a3c51e5DiscreteD9a-h2"],"title":"Acquiring Variables","text":"How many times does our situation occur? What is $$x$$ in our equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{6}$$"],"dependencies":["a3c51e5DiscreteD9a-h3"],"title":"Acquiring Variables","text":"What is the probability of success in $$1$$ trial? What is $$p$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD9a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{6}$$"],"dependencies":["a3c51e5DiscreteD9a-h4"],"title":"Acquiring Variables","text":"What is the probability of failure in $$1$$ trial? What is q?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3c51e5DiscreteD9a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.3472$$"],"dependencies":["a3c51e5DiscreteD9a-h5"],"title":"Answer","text":"With all of our variables set up, what will our final answer be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3cce77popu1","title":"Study of Acupuncture","body":"Suppose you do a study of acupuncture to determine how effective it is in relieving pain. You measure sensory rates for $$15$$ subjects with the results given.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 A Single Population Mean using the Student t Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3cce77popu1a","stepAnswer":["(7.30,9.15)"],"problemType":"TextBox","stepTitle":"Use the example data to construct a 95% confidence interval for the mean sensory rate for the population (assumed normal) from which you took the data. $$(8.6$$, $$9.4$$, $$7.9$$, $$6.8$$, $$8.3$$, $$7.3$$, $$9.2$$, $$9.6$$, $$8.7$$, $$11.4$$, $$10.3$$, $$5.4$$, $$8.1$$, $$5.5$$, $$6.9)$$","stepBody":"Round answers to two decimal place. Wrap answer with parenthesis, separete by a comma, no spacing.","answerType":"string","variabilization":{},"answerLatex":"$$(7.30, 9.15)$$","hints":{"DefaultPathway":[{"id":"a3cce77popu1a-h1","type":"hint","dependencies":[],"title":"Find sample statistics","text":"To find the confidence interval, we need to the sample mean \u03bc, the sample standard error s and sample size $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8.2267$$"],"dependencies":["a3cce77popu1a-h1"],"title":"Sample Mean \u03bc","text":"What is the mean of the given data set above? Round answers to four decimal place. Sqrt (variance)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.6722$$"],"dependencies":["a3cce77popu1a-h1"],"title":"Sample Standard Error s","text":"What is the standard deviation of the given data set above? Round answers to four decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["a3cce77popu1a-h1"],"title":"Sample Size $$n$$","text":"What is the size of the sample?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu1a-h5","type":"hint","dependencies":["a3cce77popu1a-h4"],"title":"Area Under the Curve","text":"What are the area to the right and the area to the left?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu1a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.025$$"],"dependencies":["a3cce77popu1a-h5"],"title":"What is the area to the right if using a 95% Confidence Interval?","text":"$$\\\\frac{1-CL}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu1a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.975$$"],"dependencies":["a3cce77popu1a-h5"],"title":"What is the area to the left if using a 95% confidence interval?","text":"$$1$$ - area to the $$right=1-0.025$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu1a-h8","type":"hint","dependencies":["a3cce77popu1a-h7"],"title":"T-Score","text":"To find the confidence interval, we need to compute $$t-score$$, and to find the $$t-score$$ we will need to find the EBM (error bound)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu1a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.14$$"],"dependencies":["a3cce77popu1a-h8"],"title":"$$\\\\frac{t_\u03b1}{2}$$","text":"Using $$invT(0.975$$, df) on the TI-84+ calculator. Round answers to two decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu1a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.924$$"],"dependencies":["a3cce77popu1a-h8"],"title":"EBM","text":"$$EBM=\\\\frac{t_\u03b1}{2} \\\\frac{s}{\\\\sqrt{n}}$$. Round answer to $$3$$ decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu1a-h11","type":"hint","dependencies":["a3cce77popu1a-h10"],"title":"Bounds of the Confidence Interval","text":"What is the lower and upper bound the confidence interval?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu1a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7.3$$"],"dependencies":["a3cce77popu1a-h11"],"title":"Lower Bound","text":"$$x\u0304-EBM=8.2267-0.9240$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu1a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9.15$$"],"dependencies":["a3cce77popu1a-h11"],"title":"Upper Bound","text":"$$x\u0304+EBM=8.2267+0.924$$. Round answers to two decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3cce77popu10","title":"Grams of fat per serving of chocolate chip cookies","body":"Six different national brands of chocolate chip cookies were randomly selected at the supermarket. The grams of fat per serving are as follows: 8; 8; 10; 7; 9; $$9$$. Assume the underlying distribution is approximately normal.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 A Single Population Mean using the Student t Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3cce77popu10a","stepAnswer":["(7.64,9.36)"],"problemType":"TextBox","stepTitle":"Construct a 90% confidence interval for the population mean grams of fat per serving of chocolate chip cookies sold in supermarkets.","stepBody":"Round answers to two decimal place. Wrap answer with parenthesis, separete by a comma, no spacing.","answerType":"string","variabilization":{},"answerLatex":"$$(7.64, 9.36)$$","hints":{"DefaultPathway":[{"id":"a3cce77popu10a-h1","type":"hint","dependencies":[],"title":"Find sample statistics","text":"To find the confidence interval, we need to the sample mean \u03bc, the sample standard error s and sample size $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8.5$$"],"dependencies":["a3cce77popu10a-h1"],"title":"Sample Mean \u03bc","text":"What is the mean of the given data set above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.05$$"],"dependencies":["a3cce77popu10a-h1"],"title":"Sample Standard Error s","text":"What is the standard deviation of the given data set above? 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Round answers to two decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu10a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.86$$"],"dependencies":["a3cce77popu10a-h10"],"title":"EBM","text":"$$EBM=\\\\frac{t_\u03b1}{2} \\\\frac{s}{\\\\sqrt{n}}$$. Round answers to two decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu10a-h13","type":"hint","dependencies":["a3cce77popu10a-h12"],"title":"Bounds of the Confidence Interval","text":"What is the lower and upper bound the confidence interval?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu10a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7.64$$"],"dependencies":["a3cce77popu10a-h13"],"title":"Lower Bound","text":"$$x\u0304-EBM=8.5-0.86$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu10a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9.36$$"],"dependencies":["a3cce77popu10a-h13"],"title":"Upper Bound","text":"$$x\u0304+EBM=8.5+0.86$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3cce77popu11","title":"Time Wasted","body":"Suppose that a committee is studying whether or not there is waste of time in our judicial system. It is interested in the mean amount of time individuals waste at the courthouse waiting to be called for jury duty. The committee randomly surveyed $$81$$ people who recently served as jurors. The sample mean wait time was eight hours with a sample standard deviation of four hours.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 A Single Population Mean using the Student t Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3cce77popu11a","stepAnswer":["(7.12,8.88)"],"problemType":"TextBox","stepTitle":"Construct a 95% confidence interval for the population mean time wasted.","stepBody":"Round answers to two decimal place. Wrap answer with parenthesis, separete by a comma, no spacing.","answerType":"string","variabilization":{},"answerLatex":"$$(7.12, 8.88)$$","hints":{"DefaultPathway":[{"id":"a3cce77popu11a-h1","type":"hint","dependencies":[],"title":"Find sample statistics","text":"To find the confidence interval, we need to the sample mean \u03bc, the sample standard error s and sample size $$n$$.","variabilization":{},"oer":"","license":""},{"id":"a3cce77popu11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a3cce77popu11a-h1"],"title":"Sample Mean \u03bc","text":"What is the mean of the given data set above?","variabilization":{},"oer":"","license":""},{"id":"a3cce77popu11a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a3cce77popu11a-h1"],"title":"Sample Standard Error s","text":"What is the standard deviation of the given data set above?","variabilization":{},"oer":"","license":""},{"id":"a3cce77popu11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$81$$"],"dependencies":["a3cce77popu11a-h1"],"title":"Sample Size $$n$$","text":"What is the size of the sample?","variabilization":{},"oer":"","license":""},{"id":"a3cce77popu11a-h5","type":"hint","dependencies":["a3cce77popu11a-h4"],"title":"Degree of Freedom - df","text":"T-scores follow a Student\'s $$t-distribution$$ with $$n-1$$ degrees of freedom.","variabilization":{},"oer":"","license":""},{"id":"a3cce77popu11a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$80$$"],"dependencies":["a3cce77popu11a-h5"],"title":"Compute degree of freedom","text":"What is $$n-1$$?","variabilization":{},"oer":"","license":""},{"id":"a3cce77popu11a-h7","type":"hint","dependencies":["a3cce77popu11a-h6"],"title":"Area Under the Curve","text":"What are the area to the right and the area to the left?","variabilization":{},"oer":"","license":""},{"id":"a3cce77popu11a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.025$$"],"dependencies":["a3cce77popu11a-h7"],"title":"What is the area to the right if using a 95% Confidence Interval?","text":"$$\\\\frac{1-CL}{2}$$","variabilization":{},"oer":"","license":""},{"id":"a3cce77popu11a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.975$$"],"dependencies":["a3cce77popu11a-h7"],"title":"What is the area to the left if using a 95% confidence interval?","text":"$$1$$ - area to the $$right=1-0.025$$","variabilization":{},"oer":"","license":""},{"id":"a3cce77popu11a-h10","type":"hint","dependencies":["a3cce77popu11a-h9"],"title":"T-Score","text":"To find the confidence interval, we need to compute $$t-score$$, and to find the $$t-score$$ we will need to find the EBM (error bound)","variabilization":{},"oer":"","license":""},{"id":"a3cce77popu11a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.99$$"],"dependencies":["a3cce77popu11a-h10"],"title":"$$\\\\frac{t_\u03b1}{2}$$","text":"Using $$invT(0.975$$, df) on the TI-84+ calculator. Round answers to two decimal place.","variabilization":{},"oer":"","license":""},{"id":"a3cce77popu11a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.88$$"],"dependencies":["a3cce77popu11a-h10"],"title":"EBM","text":"$$EBM=\\\\frac{t_\u03b1}{2} \\\\frac{s}{\\\\sqrt{n}}$$. Round answers to two decimal place.","variabilization":{},"oer":"","license":""},{"id":"a3cce77popu11a-h13","type":"hint","dependencies":["a3cce77popu11a-h12"],"title":"Bounds of the Confidence Interval","text":"What is the lower and upper bound the confidence interval?","variabilization":{},"oer":"","license":""},{"id":"a3cce77popu11a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7.12$$"],"dependencies":["a3cce77popu11a-h13"],"title":"Lower Bound","text":"$$x\u0304-EBM=8-0.88$$","variabilization":{},"oer":"","license":""},{"id":"a3cce77popu11a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8.88$$"],"dependencies":["a3cce77popu11a-h13"],"title":"Upper Bound","text":"$$x\u0304+EBM=8+0.88$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"a3cce77popu12","title":"Training Wheels","body":"Suppose that $$14$$ children, who were learning to ride two-wheel bikes, were surveyed to determine how long they had to use training wheels. It was revealed that they used them an average of six months with a sample standard deviation of three months. Assume that the underlying population distribution is normal.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 A Single Population Mean using the Student t Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3cce77popu12a","stepAnswer":["(4.58,7.42)"],"problemType":"TextBox","stepTitle":"Construct a 99% confidence interval for the population mean length of time using training wheels.","stepBody":"Round answers to two decimal place. Wrap answer with parenthesis, separete by a comma, no spacing.","answerType":"string","variabilization":{},"answerLatex":"$$(4.58, 7.42)$$","hints":{"DefaultPathway":[{"id":"a3cce77popu12a-h1","type":"hint","dependencies":[],"title":"Find sample statistics","text":"To find the confidence interval, we need to the sample mean \u03bc, the sample standard error s and sample size $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a3cce77popu12a-h1"],"title":"Sample Mean \u03bc","text":"What is the mean of the given data set above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu12a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a3cce77popu12a-h1"],"title":"Sample Standard Error s","text":"What is the standard deviation of the given data set above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$14$$"],"dependencies":["a3cce77popu12a-h1"],"title":"Sample Size $$n$$","text":"What is the size of the sample?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu12a-h5","type":"hint","dependencies":["a3cce77popu12a-h4"],"title":"Degree of Freedom - df","text":"T-scores follow a Student\'s $$t-distribution$$ with $$n-1$$ degrees of freedom.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu12a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$13$$"],"dependencies":["a3cce77popu12a-h5"],"title":"Compute degree of freedom","text":"What is $$n-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu12a-h7","type":"hint","dependencies":["a3cce77popu12a-h6"],"title":"Area Under the Curve","text":"What are the area to the right and the area to the left?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu12a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.05$$"],"dependencies":["a3cce77popu12a-h7"],"title":"What is the area to the right if using a 95% Confidence Interval?","text":"$$\\\\frac{1-CL}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu12a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.95$$"],"dependencies":["a3cce77popu12a-h7"],"title":"What is the area to the left if using a 95% confidence interval?","text":"$$1$$ - area to the $$right=1-0.05$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu12a-h10","type":"hint","dependencies":["a3cce77popu12a-h9"],"title":"T-Score","text":"To find the confidence interval, we need to compute $$t-score$$, and to find the $$t-score$$ we will need to find the EBM (error bound)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu12a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.77$$"],"dependencies":["a3cce77popu12a-h10"],"title":"$$\\\\frac{t_\u03b1}{2}$$","text":"Using $$invT(0.95$$, df) on the TI-84+ calculator. Round answers to two decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu12a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.42$$"],"dependencies":["a3cce77popu12a-h10"],"title":"EBM","text":"$$EBM=\\\\frac{t_\u03b1}{2} \\\\frac{s}{\\\\sqrt{n}}$$. Round answers to two decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu12a-h13","type":"hint","dependencies":["a3cce77popu12a-h12"],"title":"Bounds of the Confidence Interval","text":"What is the lower and upper bound the confidence interval?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu12a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4.58$$"],"dependencies":["a3cce77popu12a-h13"],"title":"Lower Bound","text":"$$x\u0304-EBM=6-1.42$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu12a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7.42$$"],"dependencies":["a3cce77popu12a-h13"],"title":"Upper Bound","text":"$$x\u0304+EBM=6+1.42$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3cce77popu13","title":"Forbes Megazine","body":"Forbes magazine published data on the best small firms in $$2012$$. These were firms that had been publicly traded for at least a year, have a stock price of at least $5 per share, and have reported annual revenue between $5 million and $1 billion. The data below shows the ages of the corporate CEOs for a random sample of these firms.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 A Single Population Mean using the Student t Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3cce77popu13a","stepAnswer":["(54.427,58.713)"],"problemType":"TextBox","stepTitle":"Use this sample data to construct a 90% confidence interval for the mean age of CEO\u2019s for these top small firms. Use the Student\'s $$t-distribution$$. (48, $$59$$, $$59$$, $$74$$, $$60$$, $$63$$, $$43$$, $$67$$, $$55$$, $$57$$, $$67$$, $$58$$, $$51$$, $$63$$, $$60$$, $$57$$, $$61$$, $$55$$, $$61$$, $$53$$, $$57$$, $$47$$, $$62$$, $$55$$, $$56$$, $$50$$, $$46$$, $$55$$, $$49$$, 49)","stepBody":"Use the Student\'s $$t-distribution$$. Round answers to three decimal place. Wrap answer with parenthesis, separete by a comma, no spacing.","answerType":"string","variabilization":{},"answerLatex":"$$(54.427, 58.713)$$","hints":{"DefaultPathway":[{"id":"a3cce77popu13a-h1","type":"hint","dependencies":[],"title":"Find sample statistics","text":"To find the confidence interval, we need to the sample mean \u03bc, the sample standard error s and sample size $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$56.57$$"],"dependencies":["a3cce77popu13a-h1"],"title":"Sample Mean \u03bc","text":"What is the mean of the given data set above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu13a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6.91$$"],"dependencies":["a3cce77popu13a-h1"],"title":"Sample Standard Error s","text":"What is the standard deviation of the given data set above? Round answer to two decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$30$$"],"dependencies":["a3cce77popu13a-h1"],"title":"Sample Size $$n$$","text":"What is the size of the sample?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu13a-h5","type":"hint","dependencies":["a3cce77popu13a-h4"],"title":"Degree of Freedom - df","text":"T-scores follow a Student\'s $$t-distribution$$ with $$n-1$$ degrees of freedom.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu13a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$29$$"],"dependencies":["a3cce77popu13a-h5"],"title":"Compute degree of freedom","text":"What is $$n-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu13a-h7","type":"hint","dependencies":["a3cce77popu13a-h6"],"title":"Area Under the Curve","text":"What are the area to the right and the area to the left?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu13a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.05$$"],"dependencies":["a3cce77popu13a-h7"],"title":"What is the area to the right if using a 95% Confidence Interval?","text":"$$\\\\frac{1-CL}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu13a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.95$$"],"dependencies":["a3cce77popu13a-h7"],"title":"What is the area to the left if using a 95% confidence interval?","text":"$$1$$ - area to the $$right=1-0.05$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu13a-h10","type":"hint","dependencies":["a3cce77popu13a-h9"],"title":"T-Score","text":"To find the confidence interval, we need to compute $$t-score$$, and to find the $$t-score$$ we will need to find the EBM (error bound)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu13a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.699$$"],"dependencies":["a3cce77popu13a-h10"],"title":"$$\\\\frac{t_\u03b1}{2}$$","text":"Using $$invT(0.95$$, df) on the TI-84+ calculator. Round answers to three decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu13a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.143$$"],"dependencies":["a3cce77popu13a-h10"],"title":"EBM","text":"$$EBM=\\\\frac{t_\u03b1}{2} \\\\frac{s}{\\\\sqrt{n}}$$. Round answers to three decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu13a-h13","type":"hint","dependencies":["a3cce77popu13a-h12"],"title":"Bounds of the Confidence Interval","text":"What is the lower and upper bound the confidence interval?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu13a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$54.427$$"],"dependencies":["a3cce77popu13a-h13"],"title":"Lower Bound","text":"$$x\u0304-EBM=56.57-2.143$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu13a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$58.713$$"],"dependencies":["a3cce77popu13a-h13"],"title":"Upper Bound","text":"$$x\u0304+EBM=56.57+2.143$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3cce77popu14","title":"Cost of a Used Car","body":"In a recent sample of $$84$$ used car sales costs, the sample mean was $6,425 with a standard deviation of $3,156. Assume the underlying distribution is approximately normal.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 A Single Population Mean using the Student t Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3cce77popu14a","stepAnswer":["(5454.0,7478.0)"],"problemType":"TextBox","stepTitle":"Construct a $$99.7\\\\%$$ confidence interval for the population mean cost of a used car.","stepBody":"Wrap answer with parenthesis, separete by a comma, no spacing. Round answers to one decimal place.","answerType":"string","variabilization":{},"answerLatex":"$$(5454.0, 7478.0)$$","hints":{"DefaultPathway":[{"id":"a3cce77popu14a-h1","type":"hint","dependencies":[],"title":"Find sample statistics","text":"To find the confidence interval, we need to the sample mean \u03bc, the sample standard error s and sample size $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6425$$"],"dependencies":["a3cce77popu14a-h1"],"title":"Sample Mean \u03bc","text":"What is the mean of the given data set above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu14a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3156$$"],"dependencies":["a3cce77popu14a-h1"],"title":"Sample Standard Error s","text":"What is the standard deviation of the given data set above? Round answer to two decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu14a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$84$$"],"dependencies":["a3cce77popu14a-h1"],"title":"Sample Size $$n$$","text":"What is the size of the sample?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu14a-h5","type":"hint","dependencies":["a3cce77popu14a-h4"],"title":"Degree of Freedom - df","text":"T-scores follow a Student\'s $$t-distribution$$ with $$n-1$$ degrees of freedom.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu14a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$83$$"],"dependencies":["a3cce77popu14a-h5"],"title":"Compute degree of freedom","text":"What is $$n-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu14a-h7","type":"hint","dependencies":["a3cce77popu14a-h6"],"title":"Area Under the Curve","text":"What are the area to the right and the area to the left?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu14a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.0015$$"],"dependencies":["a3cce77popu14a-h7"],"title":"What is the area to the right if using a 95% Confidence Interval?","text":"$$\\\\frac{1-CL}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu14a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.9985$$"],"dependencies":["a3cce77popu14a-h7"],"title":"What is the area to the left if using a 95% confidence interval?","text":"$$1$$ - area to the $$right=1-0.0015$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu14a-h10","type":"hint","dependencies":["a3cce77popu14a-h9"],"title":"T-Score","text":"To find the confidence interval, we need to compute $$t-score$$, and to find the $$t-score$$ we will need to find the EBM (error bound)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu14a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3.06$$"],"dependencies":["a3cce77popu14a-h10"],"title":"$$\\\\frac{t_\u03b1}{2}$$","text":"Using $$invT(0.9985$$, df) on the TI-84+ calculator. Round answers to two decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu14a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1053$$"],"dependencies":["a3cce77popu14a-h10"],"title":"EBM","text":"$$EBM=\\\\frac{t_\u03b1}{2} \\\\frac{s}{\\\\sqrt{n}}$$. Round answers to integer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu14a-h13","type":"hint","dependencies":["a3cce77popu14a-h12"],"title":"Bounds of the Confidence Interval","text":"What is the lower and upper bound the confidence interval?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu14a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5454$$"],"dependencies":["a3cce77popu14a-h13"],"title":"Lower Bound","text":"$$x\u0304-EBM=6425-1053$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu14a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7478$$"],"dependencies":["a3cce77popu14a-h13"],"title":"Upper Bound","text":"$$x\u0304+EBM=6425+1053$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3cce77popu15","title":"Worth of Coupons","body":"A survey of the mean number of cents off that coupons give was conducted by randomly surveying one coupon per page from the coupon sections of a recent San Jose Mercury News. The following data were collected: 20\xa2; 75\xa2; 50\xa2; 65\xa2; 30\xa2; 55\xa2; 40\xa2; 40\xa2; 30\xa2; 55\xa2; $$\\\\$1.50;$$ 40\xa2; 65\xa2; 40\xa2. Assume the underlying distribution is approximately normal.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 A Single Population Mean using the Student t Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3cce77popu15a","stepAnswer":["(41.90,69.96)"],"problemType":"TextBox","stepTitle":"Construct a 88% confidence interval for the population mean worth of coupons.","stepBody":"Wrap answer with parenthesis, separete by a comma, no spacing. Round answers to two decimal place.","answerType":"string","variabilization":{},"answerLatex":"$$(41.90, 69.96)$$","hints":{"DefaultPathway":[{"id":"a3cce77popu15a-h1","type":"hint","dependencies":[],"title":"Find sample statistics","text":"To find the confidence interval, we need to the sample mean \u03bc, the sample standard error s and sample size $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$55.93$$"],"dependencies":["a3cce77popu15a-h1"],"title":"Sample Mean \u03bc","text":"What is the mean of the given data set above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$31.63$$"],"dependencies":["a3cce77popu15a-h1"],"title":"Sample Standard Error s","text":"What is the standard deviation of the given data set above? Round answer to two decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$14$$"],"dependencies":["a3cce77popu15a-h1"],"title":"Sample Size $$n$$","text":"What is the size of the sample?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu15a-h5","type":"hint","dependencies":["a3cce77popu15a-h4"],"title":"Degree of Freedom - df","text":"T-scores follow a Student\'s $$t-distribution$$ with $$n-1$$ degrees of freedom.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu15a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$13$$"],"dependencies":["a3cce77popu15a-h5"],"title":"Compute degree of freedom","text":"What is $$n-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu15a-h7","type":"hint","dependencies":["a3cce77popu15a-h6"],"title":"Area Under the Curve","text":"What are the area to the right and the area to the left?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu15a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.06$$"],"dependencies":["a3cce77popu15a-h7"],"title":"What is the area to the right if using a 95% Confidence Interval?","text":"$$\\\\frac{1-CL}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu15a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.94$$"],"dependencies":["a3cce77popu15a-h7"],"title":"What is the area to the left if using a 95% confidence interval?","text":"$$1$$ - area to the $$right=1-0.06$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu15a-h10","type":"hint","dependencies":["a3cce77popu15a-h9"],"title":"T-Score","text":"To find the confidence interval, we need to compute $$t-score$$, and to find the $$t-score$$ we will need to find the EBM (error bound)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu15a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.66$$"],"dependencies":["a3cce77popu15a-h10"],"title":"$$\\\\frac{t_\u03b1}{2}$$","text":"Using $$invT(0.94$$, df) on the TI-84+ calculator. Round answers to two decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu15a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$14.03$$"],"dependencies":["a3cce77popu15a-h10"],"title":"EBM","text":"$$EBM=\\\\frac{t_\u03b1}{2} \\\\frac{s}{\\\\sqrt{n}}$$. Round answers to two decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu15a-h13","type":"hint","dependencies":["a3cce77popu15a-h12"],"title":"Bounds of the Confidence Interval","text":"What is the lower and upper bound the confidence interval?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu15a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$41.9$$"],"dependencies":["a3cce77popu15a-h13"],"title":"Lower Bound","text":"$$x\u0304-EBM=55.93-14.03$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu15a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$69.96$$"],"dependencies":["a3cce77popu15a-h13"],"title":"Upper Bound","text":"$$x\u0304+EBM=55.93+14.03$$. Round answer to two decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3cce77popu2","title":"Study of Hypnotherapy","body":"You do a study of hypnotherapy to determine how effective it is in increasing the number of hours of sleep subjects get each night. You measure hours of sleep for $$12$$ subjects with the following results.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 A Single Population Mean using the Student t Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3cce77popu2a","stepAnswer":["(8.16,9.80)"],"problemType":"TextBox","stepTitle":"Construct a 95% confidence interval for the mean number of hours slept for the population (assumed normal) from which you took the data. $$(8.2$$, $$9.1$$, $$7.7$$, $$8.6$$, $$6.9$$, $$11.2$$, $$10.1$$, $$9.9$$, $$8.9$$, $$9.2$$, $$7.5$$, $$10.5)$$","stepBody":"Round answers to two decimal place. Wrap answer with parenthesis, separete by a comma, no spacing.","answerType":"string","variabilization":{},"answerLatex":"$$(8.16, 9.80)$$","hints":{"DefaultPathway":[{"id":"a3cce77popu2a-h1","type":"hint","dependencies":[],"title":"Find sample statistics","text":"To find the confidence interval, we need to the sample mean \u03bc, the sample standard error s and sample size $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8.9833$$"],"dependencies":["a3cce77popu2a-h1"],"title":"Sample Mean \u03bc","text":"What is the mean of the given data set above? Round answers to four decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.2904$$"],"dependencies":["a3cce77popu2a-h1"],"title":"Sample Standard Error s","text":"What is the standard deviation of the given data set above? Round answers to four decimal place. Square root of the variance.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a3cce77popu2a-h1"],"title":"Sample Size $$n$$","text":"What is the size of the sample?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu2a-h5","type":"hint","dependencies":["a3cce77popu2a-h4"],"title":"Area Under the Curve","text":"What are the area to the right and the area to the left?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.025$$"],"dependencies":["a3cce77popu2a-h5"],"title":"What is the area to the right if using a 95% Confidence Interval?","text":"$$\\\\frac{1-CL}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu2a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.975$$"],"dependencies":["a3cce77popu2a-h5"],"title":"What is the area to the left if using a 95% confidence interval?","text":"$$1$$ - area to the $$right=1-0.025$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu2a-h8","type":"hint","dependencies":["a3cce77popu2a-h7"],"title":"T-Score","text":"To find the confidence interval, we need to compute $$t-score$$, and to find the $$t-score$$ we will need to find the EBM (error bound)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu2a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.2$$"],"dependencies":["a3cce77popu2a-h8"],"title":"$$\\\\frac{t_\u03b1}{2}$$","text":"Using $$invT(0.975$$, df) on the TI-84+ calculator. Round answers to one decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu2a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.8195$$"],"dependencies":["a3cce77popu2a-h8"],"title":"EBM","text":"$$EBM=\\\\frac{t_\u03b1}{2} \\\\frac{s}{\\\\sqrt{n}}$$. Round answers to four decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu2a-h11","type":"hint","dependencies":["a3cce77popu2a-h10"],"title":"Bounds of the Confidence Interval","text":"What is the lower and upper bound the confidence interval?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu2a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8.16$$"],"dependencies":["a3cce77popu2a-h11"],"title":"Lower Bound","text":"$$x\u0304-EBM=8.9833-0.8195$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu2a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9.8$$"],"dependencies":["a3cce77popu2a-h11"],"title":"Upper Bound","text":"$$x\u0304-EBM=8.9833+0.8195$$. Round answers to one decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3cce77popu3","title":"The Human Toxome Project (HTP)","body":"The Human Toxome Project (HTP) is working to understand the scope of industrial pollution in the human body. Industrial chemicals may enter the body through pollution or as ingredients in consumer products. In October $$2008$$, the scientists at HTP tested cord blood samples for $$20$$ newborn infants in the United States. The cord blood of the \\\\\\"In utero/newborn\\\\\\" group was tested for $$430$$ industrial compounds, pollutants, and other chemicals, including chemicals linked to brain and nervous system toxicity, immune system toxicity, and reproductive toxicity, and fertility problems. There are health concerns about the effects of some chemicals on the brain and nervous system. The data shows how many of the targeted chemicals were found in each infant\u2019s cord blood.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 A Single Population Mean using the Student t Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3cce77popu3a","stepAnswer":["(117.412,137.488)"],"problemType":"TextBox","stepTitle":"Use this sample data to construct a 90% confidence interval for the mean number of targeted industrial chemicals to be found in an in infant\u2019s blood. (79, $$145$$, $$147$$, $$160$$, $$116$$, $$100$$, $$159$$, $$151$$, $$156$$, $$126$$, $$137$$, $$83$$, $$156$$, $$94$$, $$121$$, $$144$$, $$123$$, $$114$$, $$139$$, 99)","stepBody":"Round answers to three decimal place. Wrap answer with parenthesis, separete by a comma, no spacing.","answerType":"string","variabilization":{},"answerLatex":"$$(117.412, 137.488)$$","hints":{"DefaultPathway":[{"id":"a3cce77popu3a-h1","type":"hint","dependencies":[],"title":"Find sample statistics","text":"To find the confidence interval, we need to the sample mean \u03bc, the sample standard error s and sample size $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$127.45$$"],"dependencies":["a3cce77popu3a-h1"],"title":"Sample Mean \u03bc","text":"What is the mean of the given data set above? Round answers to two decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25.965$$"],"dependencies":["a3cce77popu3a-h1"],"title":"Sample Standard Error s","text":"What is the standard deviation of the given data set above? Round answers to three decimal place. Square root of the variance.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a3cce77popu3a-h1"],"title":"Sample Size $$n$$","text":"What is the size of the sample?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu3a-h5","type":"hint","dependencies":["a3cce77popu3a-h4"],"title":"Degree of Freedom - df","text":"T-scores follow a Student\'s $$t-distribution$$ with $$n-1$$ degrees of freedom.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu3a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$19$$"],"dependencies":["a3cce77popu3a-h5"],"title":"Compute degree of freedom","text":"What is $$n-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu3a-h7","type":"hint","dependencies":["a3cce77popu3a-h6"],"title":"Area Under the Curve","text":"What are the area to the right and the area to the left?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu3a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.05$$"],"dependencies":["a3cce77popu3a-h7"],"title":"What is the area to the right if using a 95% Confidence Interval?","text":"$$\\\\frac{1-CL}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu3a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.95$$"],"dependencies":["a3cce77popu3a-h7"],"title":"What is the area to the left if using a 95% confidence interval?","text":"$$1$$ - area to the $$right=1-0.05$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu3a-h10","type":"hint","dependencies":["a3cce77popu3a-h9"],"title":"T-Score","text":"To find the confidence interval, we need to compute $$t-score$$, and to find the $$t-score$$ we will need to find the EBM (error bound)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu3a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.729$$"],"dependencies":["a3cce77popu3a-h10"],"title":"$$\\\\frac{t_\u03b1}{2}$$","text":"Using $$invT(0.95$$, df) on the TI-84+ calculator. Round answers to three decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu3a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10.038$$"],"dependencies":["a3cce77popu3a-h10"],"title":"EBM","text":"$$EBM=\\\\frac{t_\u03b1}{2} \\\\frac{s}{\\\\sqrt{n}}$$. Round answers to three decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu3a-h13","type":"hint","dependencies":["a3cce77popu3a-h12"],"title":"Bounds of the Confidence Interval","text":"What is the lower and upper bound the confidence interval?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu3a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$117.412$$"],"dependencies":["a3cce77popu3a-h13"],"title":"Lower Bound","text":"$$x\u0304-EBM=127.45-10.038$$. Round answers to three decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu3a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$137.488$$"],"dependencies":["a3cce77popu3a-h13"],"title":"Upper Bound","text":"$$x\u0304+EBM=127.45+10.038$$. Round answers to three decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3cce77popu4","title":"Hours Spend on Television","body":"A random sample of statistics students were asked to estimate the total number of hours they spend watching television in an average week.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 A Single Population Mean using the Student t Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3cce77popu4a","stepAnswer":["(2.397, 9.869)"],"problemType":"TextBox","stepTitle":"Use this sample data to construct a 98% confidence interval for the mean number of hours statistics students will spend watching television in one week. (0, $$3$$, $$1$$, $$20$$, $$9$$, $$5$$, $$10$$, $$1$$, $$10$$, $$4$$, $$14$$, $$2$$, $$4$$, $$4$$, 5)","stepBody":"Round answers to three decimal place. Wrap answer with parenthesis, separete by a comma, no spacing.","answerType":"string","variabilization":{},"answerLatex":"$$(2.397$$, $$9.869)$$","hints":{"DefaultPathway":[{"id":"a3cce77popu4a-h1","type":"hint","dependencies":[],"title":"Find sample statistics","text":"To find the confidence interval, we need to the sample mean \u03bc, the sample standard error s and sample size $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6.133$$"],"dependencies":["a3cce77popu4a-h1"],"title":"Sample Mean \u03bc","text":"What is the mean of the given data set above? Round answers to three decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5.514$$"],"dependencies":["a3cce77popu4a-h1"],"title":"Sample Standard Error s","text":"What is the standard deviation of the given data set above? $$s=\\\\sqrt{\\\\sigma}$$ Round answers to three decimal place. Square root of the variance.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["a3cce77popu4a-h1"],"title":"Sample Size $$n$$","text":"What is the size of the sample?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu4a-h5","type":"hint","dependencies":["a3cce77popu4a-h4"],"title":"Degree of Freedom - df","text":"T-scores follow a Student\'s $$t-distribution$$ with $$n-1$$ degrees of freedom.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu4a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$14$$"],"dependencies":["a3cce77popu4a-h5"],"title":"Compute degree of freedom","text":"What is $$n-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu4a-h7","type":"hint","dependencies":["a3cce77popu4a-h6"],"title":"Area Under the Curve","text":"What are the area to the right and the area to the left?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu4a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.01$$"],"dependencies":["a3cce77popu4a-h7"],"title":"What is the area to the right if using a 95% Confidence Interval?","text":"$$\\\\frac{1-CL}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu4a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.99$$"],"dependencies":["a3cce77popu4a-h7"],"title":"What is the area to the left if using a 95% confidence interval?","text":"$$1$$ - area to the $$right=1-0.01$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu4a-h10","type":"hint","dependencies":["a3cce77popu4a-h9"],"title":"T-Score","text":"To find the confidence interval, we need to compute $$t-score$$, and to find the $$t-score$$ we will need to find the EBM (error bound)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu4a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.624$$"],"dependencies":["a3cce77popu4a-h10"],"title":"$$\\\\frac{t_\u03b1}{2}$$","text":"Using $$invT(0.99$$, df) on the TI-84+ calculator. Round to three decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu4a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3.736$$"],"dependencies":["a3cce77popu4a-h10"],"title":"EBM","text":"$$EBM=\\\\frac{t_\u03b1}{2} \\\\frac{s}{\\\\sqrt{n}}$$. Round answers to three decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu4a-h13","type":"hint","dependencies":["a3cce77popu4a-h12"],"title":"Bounds of the Confidence Interval","text":"What is the lower and upper bound the confidence interval?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu4a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.397$$"],"dependencies":["a3cce77popu4a-h13"],"title":"Lower Bound","text":"$$x\u0304-EBM=6.133-3.736$$. Round answers to three decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu4a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9.869$$"],"dependencies":["a3cce77popu4a-h13"],"title":"Upper Bound","text":"$$x\u0304+EBM=6.133+3.736$$. Round answers to three decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3cce77popu5","title":"Emergency Room Wait Times","body":"A hospital is trying to cut down on emergency room wait times. It is interested in the amount of time patients must wait before being called back to be examined. An investigation committee randomly surveyed $$70$$ patients. The sample mean was $$1.5$$ hours with a sample standard deviation of $$0.5$$ hours.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 A Single Population Mean using the Student t Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3cce77popu5a","stepAnswer":["(1.331,1.669)"],"problemType":"TextBox","stepTitle":"Construct a 95% confidence interval for the population mean time spent waiting.","stepBody":"Round answers to three decimal place. Wrap answer with parenthesis, separete by a comma, no spacing.","answerType":"string","variabilization":{},"answerLatex":"$$(1.331, 1.669)$$","hints":{"DefaultPathway":[{"id":"a3cce77popu5a-h1","type":"hint","dependencies":[],"title":"Find sample statistics","text":"To find the confidence interval, we need to the sample mean \u03bc, the sample standard error s and sample size $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.5$$"],"dependencies":["a3cce77popu5a-h1"],"title":"Sample Mean \u03bc","text":"What is the mean of the given data set above? Round answers to one decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.7071$$"],"dependencies":["a3cce77popu5a-h1"],"title":"Sample Standard Error s","text":"What is the standard deviation of the given data set above? $$s=\\\\sqrt{\\\\sigma}$$ Round answers to four decimal place. Square root of the variance.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$70$$"],"dependencies":["a3cce77popu5a-h1"],"title":"Sample Size $$n$$","text":"What is the size of the sample?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu5a-h5","type":"hint","dependencies":["a3cce77popu5a-h4"],"title":"Degree of Freedom - df","text":"T-scores follow a Student\'s $$t-distribution$$ with $$n-1$$ degrees of freedom.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu5a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$69$$"],"dependencies":["a3cce77popu5a-h5"],"title":"Compute degree of freedom","text":"What is $$n-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu5a-h7","type":"hint","dependencies":["a3cce77popu5a-h6"],"title":"Area Under the Curve","text":"What are the area to the right and the area to the left?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu5a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.025$$"],"dependencies":["a3cce77popu5a-h7"],"title":"What is the area to the right if using a 95% Confidence Interval?","text":"$$\\\\frac{1-CL}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu5a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.975$$"],"dependencies":["a3cce77popu5a-h7"],"title":"What is the area to the left if using a 95% confidence interval?","text":"$$1$$ - area to the $$right=1-0.025$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu5a-h10","type":"hint","dependencies":["a3cce77popu5a-h9"],"title":"T-Score","text":"To find the confidence interval, we need to compute $$t-score$$, and to find the $$t-score$$ we will need to find the EBM (error bound)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu5a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.995$$"],"dependencies":["a3cce77popu5a-h10"],"title":"$$\\\\frac{t_\u03b1}{2}$$","text":"Using $$invT(0.975$$, df) on the TI-84+ calculator. Round answers to three decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu5a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1686$$"],"dependencies":["a3cce77popu5a-h10"],"title":"EBM","text":"$$EBM=\\\\frac{t_\u03b1}{2} \\\\frac{s}{\\\\sqrt{n}}$$. Round answers to four decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu5a-h13","type":"hint","dependencies":["a3cce77popu5a-h12"],"title":"Bounds of the Confidence Interval","text":"What is the lower and upper bound the confidence interval?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu5a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.331$$"],"dependencies":["a3cce77popu5a-h13"],"title":"Lower Bound","text":"$$x\u0304-EBM=1.5-0.1686$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu5a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.669$$"],"dependencies":["a3cce77popu5a-h13"],"title":"Upper Bound","text":"x\u0304+EBM=1.5+ $$0.1686$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3cce77popu6","title":"Tranquilizer","body":"A pharmaceutical company makes tranquilizers. It is assumed that the distribution for the length of time they last is approximately normal. Researchers in a hospital used the drug on a random sample of nine patients.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 A Single Population Mean using the Student t Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3cce77popu6a","stepAnswer":["(2.270,2.754)"],"problemType":"TextBox","stepTitle":"Construct a 95% confidence interval for the population mean length of time. The effective period of the tranquilizer for each patient (in hours) was as follows: $$2.7;$$ $$2.8;$$ $$3.0;$$ $$2.3;$$ $$2.3;$$ $$2.2;$$ $$2.8;$$ $$2.1;$$ and $$2.4$$.","stepBody":"Round answers to three decimal place. Wrap answer with parenthesis, separete by a comma, no spacing.","answerType":"string","variabilization":{},"answerLatex":"$$(2.270, 2.754)$$","hints":{"DefaultPathway":[{"id":"a3cce77popu6a-h1","type":"hint","dependencies":[],"title":"Find sample statistics","text":"To find the confidence interval, we need to the sample mean \u03bc, the sample standard error s and sample size $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.51$$"],"dependencies":["a3cce77popu6a-h1"],"title":"Sample Mean \u03bc","text":"What is the mean of the given data set above? 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Square root of the variance.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a3cce77popu6a-h1"],"title":"Sample Size $$n$$","text":"What is the size of the sample?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu6a-h5","type":"hint","dependencies":["a3cce77popu6a-h4"],"title":"Degree of Freedom - df","text":"T-scores follow a Student\'s $$t-distribution$$ with $$n-1$$ degrees of freedom.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu6a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a3cce77popu6a-h5"],"title":"Compute degree of freedom","text":"What is $$n-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu6a-h7","type":"hint","dependencies":["a3cce77popu6a-h6"],"title":"Area Under the Curve","text":"What are the area to the right and the area to the left?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu6a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.025$$"],"dependencies":["a3cce77popu6a-h7"],"title":"What is the area to the right if using a 95% Confidence Interval?","text":"$$\\\\frac{1-CL}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu6a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.975$$"],"dependencies":["a3cce77popu6a-h7"],"title":"What is the area to the left if using a 95% confidence interval?","text":"$$1$$ - area to the $$right=1-0.025$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu6a-h10","type":"hint","dependencies":["a3cce77popu6a-h9"],"title":"T-Score","text":"To find the confidence interval, we need to compute $$t-score$$, and to find the $$t-score$$ we will need to find the EBM (error bound)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu6a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.306$$"],"dependencies":["a3cce77popu6a-h10"],"title":"$$\\\\frac{t_\u03b1}{2}$$","text":"Using $$invT(0.975$$, df) on the TI-84+ calculator. Round answers to three decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu6a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.244$$"],"dependencies":["a3cce77popu6a-h10"],"title":"EBM","text":"$$EBM=\\\\frac{t_\u03b1}{2} \\\\frac{s}{\\\\sqrt{n}}$$. Round answers to three decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu6a-h13","type":"hint","dependencies":["a3cce77popu6a-h12"],"title":"Bounds of the Confidence Interval","text":"What is the lower and upper bound the confidence interval?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu6a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.27$$"],"dependencies":["a3cce77popu6a-h13"],"title":"Lower Bound","text":"$$x\u0304-EBM=2.51-0.244$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu6a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.754$$"],"dependencies":["a3cce77popu6a-h13"],"title":"Upper Bound","text":"$$x\u0304+EBM=2.51+0.244$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3cce77popu7","title":"Unoccupied Seats in Airplanes","body":"Unoccupied seats on flights cause airlines to lose revenue. Suppose a large airline wants to estimate its mean number of unoccupied seats per flight over the past year. To accomplish this, the records of $$225$$ flights are randomly selected and the number of unoccupied seats is noted for each of the sampled flights. The sample mean is $$11.6$$ seats and the sample standard deviation is $$4.1$$ seats.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 A Single Population Mean using the Student t Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3cce77popu7a","stepAnswer":["(11.12,12.08)"],"problemType":"TextBox","stepTitle":"Construct a 92% confidence interval for the population mean number of unoccupied seats per flight.","stepBody":"Round answers to two decimal place. Wrap answer with parenthesis, separete by a comma, no spacing.","answerType":"string","variabilization":{},"answerLatex":"$$(11.12, 12.08)$$","hints":{"DefaultPathway":[{"id":"a3cce77popu7a-h1","type":"hint","dependencies":[],"title":"Find sample statistics","text":"To find the confidence interval, we need to the sample mean \u03bc, the sample standard error s and sample size $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11.6$$"],"dependencies":["a3cce77popu7a-h1"],"title":"Sample Mean \u03bc","text":"What is the mean of the given data set above? 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Square root of the variance.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$225$$"],"dependencies":["a3cce77popu7a-h1"],"title":"Sample Size $$n$$","text":"What is the size of the sample?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu7a-h5","type":"hint","dependencies":["a3cce77popu7a-h4"],"title":"Degree of Freedom - df","text":"T-scores follow a Student\'s $$t-distribution$$ with $$n-1$$ degrees of freedom.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu7a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$224$$"],"dependencies":["a3cce77popu7a-h5"],"title":"Compute degree of freedom","text":"What is $$n-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu7a-h7","type":"hint","dependencies":["a3cce77popu7a-h6"],"title":"Area Under the Curve","text":"What are the area to the right and the area to the left?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu7a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.04$$"],"dependencies":["a3cce77popu7a-h7"],"title":"What is the area to the right if using a 95% Confidence Interval?","text":"$$\\\\frac{1-CL}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu7a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.96$$"],"dependencies":["a3cce77popu7a-h7"],"title":"What is the area to the left if using a 95% confidence interval?","text":"$$1$$ - area to the $$right=1-0.04$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu7a-h10","type":"hint","dependencies":["a3cce77popu7a-h9"],"title":"T-Score","text":"To find the confidence interval, we need to compute $$t-score$$, and to find the $$t-score$$ we will need to find the EBM (error bound)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu7a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.759$$"],"dependencies":["a3cce77popu7a-h10"],"title":"$$\\\\frac{t_\u03b1}{2}$$","text":"Using $$invT(0.96$$, df) on the TI-84+ calculator. Round answers to three decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu7a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.481$$"],"dependencies":["a3cce77popu7a-h10"],"title":"EBM","text":"$$EBM=\\\\frac{t_\u03b1}{2} \\\\frac{s}{\\\\sqrt{n}}$$. Round answers to three decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu7a-h13","type":"hint","dependencies":["a3cce77popu7a-h12"],"title":"Bounds of the Confidence Interval","text":"What is the lower and upper bound the confidence interval?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu7a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11.12$$"],"dependencies":["a3cce77popu7a-h13"],"title":"Lower Bound","text":"$$x\u0304-EBM=11.6-0.481$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu7a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12.08$$"],"dependencies":["a3cce77popu7a-h13"],"title":"Upper Bound","text":"$$x\u0304+EBM=11.6+0.481$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3cce77popu8","title":"Amount of Soda Served","body":"A quality control specialist for a restaurant chain takes a random sample of size $$12$$ to check the amount of soda served in the $$16$$ oz. serving size. The sample mean is $$13.30$$ with a sample standard deviation of $$1.55$$. Assume the underlying population is normally distributed.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 A Single Population Mean using the Student t Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3cce77popu8a","stepAnswer":["$$B.(12.32$$, $$14.29)$$"],"problemType":"MultipleChoice","stepTitle":"Find the 95% Confidence Interval for the true population mean for the amount of soda served.","stepBody":"Choose one from the following.","answerType":"string","variabilization":{},"answerLatex":"$$B.(12.32$$, $$14.29)$$","choices":["$$A.(12.42$$, $$14.18)$$","$$B.(12.32$$, $$14.29)$$","$$C.(12.50$$, $$14.10)$$","D.Impossible to determine."],"hints":{"DefaultPathway":[{"id":"a3cce77popu8a-h1","type":"hint","dependencies":[],"title":"Find sample statistics","text":"To find the confidence interval, we need to the sample mean \u03bc, the sample standard error s and sample size $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$13.3$$"],"dependencies":["a3cce77popu8a-h1"],"title":"Sample Mean \u03bc","text":"What is the mean of the given data set above? Round answers to one decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.55$$"],"dependencies":["a3cce77popu8a-h1"],"title":"Sample Standard Error s","text":"What is the standard deviation of the given data set above? Round answers to two decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a3cce77popu8a-h1"],"title":"Sample Size $$n$$","text":"What is the size of the sample?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu8a-h5","type":"hint","dependencies":["a3cce77popu8a-h4"],"title":"Degree of Freedom - df","text":"T-scores follow a Student\'s $$t-distribution$$ with $$n-1$$ degrees of freedom.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu8a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11$$"],"dependencies":["a3cce77popu8a-h5"],"title":"Compute degree of freedom","text":"What is $$n-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu8a-h7","type":"hint","dependencies":["a3cce77popu8a-h6"],"title":"Area Under the Curve","text":"What are the area to the right and the area to the left?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu8a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.025$$"],"dependencies":["a3cce77popu8a-h7"],"title":"What is the area to the right if using a 95% Confidence Interval?","text":"$$\\\\frac{1-CL}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu8a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.975$$"],"dependencies":["a3cce77popu8a-h7"],"title":"What is the area to the left if using a 95% confidence interval?","text":"$$1$$ - area to the $$right=1-0.025$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu8a-h10","type":"hint","dependencies":["a3cce77popu8a-h9"],"title":"T-Score","text":"To find the confidence interval, we need to compute $$t-score$$, and to find the $$t-score$$ we will need to find the EBM (error bound)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu8a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.2$$"],"dependencies":["a3cce77popu8a-h10"],"title":"$$\\\\frac{t_\u03b1}{2}$$","text":"Using $$invT(0.975$$, df) on the TI-84+ calculator. Round answers to one decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu8a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.98$$"],"dependencies":["a3cce77popu8a-h10"],"title":"EBM","text":"$$EBM=\\\\frac{t_\u03b1}{2} \\\\frac{s}{\\\\sqrt{n}}$$. Round answers to two decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu8a-h13","type":"hint","dependencies":["a3cce77popu8a-h12"],"title":"Bounds of the Confidence Interval","text":"What is the lower and upper bound the confidence interval?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu8a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12.32$$"],"dependencies":["a3cce77popu8a-h13"],"title":"Lower Bound","text":"$$x\u0304-EBM=13.3-0.98$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu8a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$14.28$$"],"dependencies":["a3cce77popu8a-h13"],"title":"Upper Bound","text":"$$x\u0304+EBM=13.3+0.98$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3cce77popu9","title":"Enrollment at Community Colleges","body":"A random survey of enrollment at $$35$$ community colleges across the United States yielded the following figures: 6,414; 1,550; 2,109; 9,350; 21,828; 4,300; 5,944; 5,722; 2,825; 2,044; 5,481; 5,200; 5,853; 2,750; 10,012; 6,357; 27,000; 9,414; 7,681; 3,200; 17,500; 9,200; 7,380; 18,314; 6,557; 13,713; 17,768; 7,493; 2,771; 2,861; 1,263; 7,285; 28,165; 5,080; 11,622. Assume the underlying population is normal.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 A Single Population Mean using the Student t Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a3cce77popu9a","stepAnswer":["(6244.0,11014.0)"],"problemType":"TextBox","stepTitle":"Construct a 95% confidence interval for the population mean enrollment at community colleges in the United States.","stepBody":"Round answers to one decimal place. Wrap answer with parenthesis, separete by a comma, no spacing.","answerType":"string","variabilization":{},"answerLatex":"$$(6244.0, 11014.0)$$","hints":{"DefaultPathway":[{"id":"a3cce77popu9a-h1","type":"hint","dependencies":[],"title":"Find sample statistics","text":"To find the confidence interval, we need to the sample mean \u03bc, the sample standard error s and sample size $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8629$$"],"dependencies":["a3cce77popu9a-h1"],"title":"Sample Mean \u03bc","text":"What is the mean of the given data set above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6944$$"],"dependencies":["a3cce77popu9a-h1"],"title":"Sample Standard Error s","text":"What is the standard deviation of the given data set above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$35$$"],"dependencies":["a3cce77popu9a-h1"],"title":"Sample Size $$n$$","text":"What is the size of the sample?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu9a-h5","type":"hint","dependencies":["a3cce77popu9a-h4"],"title":"Degree of Freedom - df","text":"T-scores follow a Student\'s $$t-distribution$$ with $$n-1$$ degrees of freedom.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu9a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$34$$"],"dependencies":["a3cce77popu9a-h5"],"title":"Compute degree of freedom","text":"What is $$n-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu9a-h7","type":"hint","dependencies":["a3cce77popu9a-h6"],"title":"Area Under the Curve","text":"What are the area to the right and the area to the left?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu9a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.025$$"],"dependencies":["a3cce77popu9a-h7"],"title":"What is the area to the right if using a 95% Confidence Interval?","text":"$$\\\\frac{1-CL}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu9a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.975$$"],"dependencies":["a3cce77popu9a-h7"],"title":"What is the area to the left if using a 95% confidence interval?","text":"$$1$$ - area to the $$right=1-0.025$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu9a-h10","type":"hint","dependencies":["a3cce77popu9a-h9"],"title":"T-Score","text":"To find the confidence interval, we need to compute $$t-score$$, and to find the $$t-score$$ we will need to find the EBM (error bound)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu9a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.032$$"],"dependencies":["a3cce77popu9a-h10"],"title":"$$\\\\frac{t_\u03b1}{2}$$","text":"Using $$invT(0.975$$, df) on the TI-84+ calculator. Round answers to three decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu9a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2385$$"],"dependencies":["a3cce77popu9a-h10"],"title":"EBM","text":"$$EBM=\\\\frac{t_\u03b1}{2} \\\\frac{s}{\\\\sqrt{n}}$$. Round answers to integer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu9a-h13","type":"hint","dependencies":["a3cce77popu9a-h12"],"title":"Bounds of the Confidence Interval","text":"What is the lower and upper bound the confidence interval?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu9a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6244$$"],"dependencies":["a3cce77popu9a-h13"],"title":"Lower Bound","text":"$$x\u0304-EBM=8629-2385$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3cce77popu9a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11014$$"],"dependencies":["a3cce77popu9a-h13"],"title":"Upper Bound","text":"$$x\u0304+EBM=8629+2385$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6a35variation1","title":"How to Solve Direct Variation Problems","body":"Solve for the relating equation (Convert all fractions into decimals).","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.9 Use Direct and Inverse Variation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6a35variation1a","stepAnswer":["$$y=2.5x$$"],"problemType":"TextBox","stepTitle":"If $$y$$ varies directly with $$x$$ and $$y=20$$ when $$x=8$$, find the equation that relates $$x$$ and $$y$$ (in the form $$y=?)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=2.5x$$","hints":{"DefaultPathway":[{"id":"a3d6a35variation1a-h1","type":"hint","dependencies":[],"title":"Set up equation","text":"Plug in all the given values in $$y=kx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation1a-h2","type":"hint","dependencies":["a3d6a35variation1a-h1"],"title":"Plugged in","text":"$$20=8k$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation1a-h3","type":"hint","dependencies":["a3d6a35variation1a-h2"],"title":"Divide","text":"Divide both sides by a value that isolates K","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation1a-h4","type":"hint","dependencies":["a3d6a35variation1a-h3"],"title":"Convert to decimal","text":"The equation after dividing both sides by $$8$$ should be $$k=\\\\frac{20}{8}$$. Convert it into an decimal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6a35variation10","title":"How to Solve Direct Variation Problems","body":"Solve for the relating equation (Convert all fractions into decimals).","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.9 Use Direct and Inverse Variation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6a35variation10a","stepAnswer":["$$a=0.46b$$"],"problemType":"TextBox","stepTitle":"If a varies directly as $$b$$ and $$a=6$$ when $$b=13$$, find the equation that relates a and $$b$$ (in the form $$a=?)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$a=0.46b$$","hints":{"DefaultPathway":[{"id":"a3d6a35variation10a-h1","type":"hint","dependencies":[],"title":"Set up equation","text":"Plug in all the given values in $$a=kb$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation10a-h2","type":"hint","dependencies":["a3d6a35variation10a-h1"],"title":"Plugged in","text":"$$6=13k$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation10a-h3","type":"hint","dependencies":["a3d6a35variation10a-h2"],"title":"Divide","text":"Divide both sides by a value that isolates K","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation10a-h4","type":"hint","dependencies":["a3d6a35variation10a-h3"],"title":"Convert to decimal","text":"The equation after dividing both sides by $$13$$ should be $$k=\\\\frac{6}{13}$$. Convert it into an decimal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6a35variation11","title":"How to Solve Direct Variation Problems","body":"Solve for the relating equation (Convert all fractions into decimals).","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.9 Use Direct and Inverse Variation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6a35variation11a","stepAnswer":["$$v=0.67w$$"],"problemType":"TextBox","stepTitle":"If v varies directly as w and $$v=8, when$$ $$w=12$$, find the equation that relates v and w (in the form $$v=?)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$v=0.67w$$","hints":{"DefaultPathway":[{"id":"a3d6a35variation11a-h1","type":"hint","dependencies":[],"title":"Set up equation","text":"Plug in all the given values in $$v=kw$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation11a-h2","type":"hint","dependencies":["a3d6a35variation11a-h1"],"title":"Plugged in","text":"$$8=12w$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation11a-h3","type":"hint","dependencies":["a3d6a35variation11a-h2"],"title":"Divide","text":"Divide both sides by a value that isolates K","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation11a-h4","type":"hint","dependencies":["a3d6a35variation11a-h3"],"title":"Convert to decimal","text":"The equation after dividing both sides by $$12$$ should be $$k=\\\\frac{8}{12}$$. Convert it into an decimal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6a35variation12","title":"How to Solve Inverse Variation Problems","body":"Solve for the relating equation (Convert all fractions into decimals).","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.9 Use Direct and Inverse Variation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6a35variation12a","stepAnswer":["$$y=\\\\frac{160}{x}$$"],"problemType":"TextBox","stepTitle":"If $$y$$ varies inversely with $$x$$ and $$y=20$$ when $$x=8$$ , find the equation that relates $$x$$ and $$y$$ (in the form $$y=?)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=\\\\frac{160}{x}$$","hints":{"DefaultPathway":[{"id":"a3d6a35variation12a-h1","type":"hint","dependencies":[],"title":"Set up equation","text":"Plug in all the given values in $$y=\\\\frac{k}{x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation12a-h2","type":"hint","dependencies":["a3d6a35variation12a-h1"],"title":"Plugged in","text":"$$20=\\\\frac{k}{8}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation12a-h3","type":"hint","dependencies":["a3d6a35variation12a-h2"],"title":"Divide","text":"Multiply both sides by a value that isolates K","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation12a-h4","type":"hint","dependencies":["a3d6a35variation12a-h3"],"title":"Convert to decimal","text":"The equation after multiplying both sides by $$8$$ should be $$k=160$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6a35variation13","title":"How to Solve Inverse Variation Problems","body":"Solve for the relating equation (Convert all fractions into decimals).","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.9 Use Direct and Inverse Variation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6a35variation13a","stepAnswer":["$$p=\\\\frac{360}{q}$$"],"problemType":"TextBox","stepTitle":"If $$p$$ varies inversely with q and $$p=30$$ when $$q=12$$, find the equation that relates $$p$$ and q (in the form $$p=?)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$p=\\\\frac{360}{q}$$","hints":{"DefaultPathway":[{"id":"a3d6a35variation13a-h1","type":"hint","dependencies":[],"title":"Set up equation","text":"Plug in all the given values in $$p=\\\\frac{k}{q}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation13a-h2","type":"hint","dependencies":["a3d6a35variation13a-h1"],"title":"Plugged in","text":"$$30=\\\\frac{k}{12}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation13a-h3","type":"hint","dependencies":["a3d6a35variation13a-h2"],"title":"Divide","text":"Multiply both sides by a value that isolates K","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation13a-h4","type":"hint","dependencies":["a3d6a35variation13a-h3"],"title":"Convert to decimal","text":"The equation after multiplying both sides by $$12$$ should be $$k=360$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6a35variation14","title":"How to Solve Inverse Variation Problems","body":"Solve for the relating equation (Convert all fractions into decimals).","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.9 Use Direct and Inverse Variation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6a35variation14a","stepAnswer":["$$y=\\\\frac{16}{x}$$"],"problemType":"TextBox","stepTitle":"If $$y$$ varies inversely with $$x$$ and $$y=8$$ when $$x=2$$, find the equation that relates $$x$$ and $$y$$ (in the form $$y=?)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=\\\\frac{16}{x}$$","hints":{"DefaultPathway":[{"id":"a3d6a35variation14a-h1","type":"hint","dependencies":[],"title":"Set up equation","text":"Plug in all the given values in $$y=\\\\frac{k}{x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation14a-h2","type":"hint","dependencies":["a3d6a35variation14a-h1"],"title":"Plugged in","text":"$$8=\\\\frac{k}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation14a-h3","type":"hint","dependencies":["a3d6a35variation14a-h2"],"title":"Divide","text":"Multiply both sides by a value that isolates K","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation14a-h4","type":"hint","dependencies":["a3d6a35variation14a-h3"],"title":"Convert to decimal","text":"The equation after multiplying both sides by $$2$$ should be $$k=16$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6a35variation16","title":"Solving Relating Equations","body":"Find the equation that relates $$x$$ and $$y$$.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.9 Use Direct and Inverse Variation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6a35variation16a","stepAnswer":["$$y=\\\\frac{2}{x}$$"],"problemType":"TextBox","stepTitle":"Y varies inversely with $$x$$, and $$y=2$$ when $$x=1$$. Write your answer in the form $$y=$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=\\\\frac{2}{x}$$","hints":{"DefaultPathway":[{"id":"a3d6a35variation16a-h1","type":"hint","dependencies":[],"title":"Use formula","text":"The inverse variation formula is $$y=\\\\frac{k}{x}$$. You can use this to find the relationship between $$x$$ and $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation16a-h2","type":"hint","dependencies":["a3d6a35variation16a-h1"],"title":"Plug In","text":"Plug in $$x$$ and $$y$$ from the point given. Solve for k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation16a-h3","type":"hint","dependencies":["a3d6a35variation16a-h2"],"title":"Simplify","text":"Simplify the equation, finding k and plugging it back into the inverse variation formula.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation16a-h4","type":"hint","dependencies":["a3d6a35variation16a-h3"],"title":"Answer","text":"The answer is $$y=\\\\frac{2}{x}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6a35variation17","title":"Solving Relating Equations","body":"Find the equation that relates $$x$$ and $$y$$.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.9 Use Direct and Inverse Variation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6a35variation17a","stepAnswer":["$$y=\\\\frac{3}{x}$$"],"problemType":"TextBox","stepTitle":"Y varies inversely with $$x$$, and $$y=6$$ when $$x=\\\\frac{1}{2}$$. Write your answer in the form $$y=$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=\\\\frac{3}{x}$$","hints":{"DefaultPathway":[{"id":"a3d6a35variation17a-h1","type":"hint","dependencies":[],"title":"Use formula","text":"The inverse variation formula is $$y=\\\\frac{k}{x}$$. You can use this to find the relationship between $$x$$ and $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation17a-h2","type":"hint","dependencies":["a3d6a35variation17a-h1"],"title":"Plug In","text":"Plug in $$x$$ and $$y$$ from the point given. Solve for k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation17a-h3","type":"hint","dependencies":["a3d6a35variation17a-h2"],"title":"Simplify","text":"Simplify the equation, finding k and plugging it back into the inverse variation formula.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation17a-h4","type":"hint","dependencies":["a3d6a35variation17a-h3"],"title":"Answer","text":"The answer is $$y=\\\\frac{3}{x}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6a35variation18","title":"Solving Relating Equations","body":"Find the equation that relates $$x$$ and $$y$$.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.9 Use Direct and Inverse Variation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6a35variation18a","stepAnswer":["$$y=\\\\frac{4}{x}$$"],"problemType":"TextBox","stepTitle":"Y varies inversely with $$x$$, and $$y=12$$ when $$x=\\\\frac{1}{3}$$. Write your answer in the form $$y=$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=\\\\frac{4}{x}$$","hints":{"DefaultPathway":[{"id":"a3d6a35variation18a-h1","type":"hint","dependencies":[],"title":"Use formula","text":"The inverse variation formula is $$y=\\\\frac{k}{x}$$. You can use this to find the relationship between $$x$$ and $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation18a-h2","type":"hint","dependencies":["a3d6a35variation18a-h1"],"title":"Plug In","text":"Plug in $$x$$ and $$y$$ from the point given. Solve for k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation18a-h3","type":"hint","dependencies":["a3d6a35variation18a-h2"],"title":"Simplify","text":"Simplify the equation, finding k and plugging it back into the inverse variation formula.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation18a-h4","type":"hint","dependencies":["a3d6a35variation18a-h3"],"title":"Answer","text":"The answer is $$y=\\\\frac{4}{x}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6a35variation19","title":"Sally\'s Necklaces","body":"Solve the word problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.9 Use Direct and Inverse Variation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6a35variation19a","stepAnswer":["$$P=10n$$"],"problemType":"TextBox","stepTitle":"The amount of money Sally earns, P, varies directly with the number, $$n$$, of necklaces she sells. When Sally sells $$15$$ necklaces she earns $150. Write the equation that relates P and $$n$$ (in the form $$P=?)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$P=10n$$","hints":{"DefaultPathway":[{"id":"a3d6a35variation19a-h1","type":"hint","dependencies":[],"title":"Use formula","text":"The word problem implies a direct variation formula is needed. This formula is $$P=kn$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation19a-h2","type":"hint","dependencies":["a3d6a35variation19a-h1"],"title":"Find values","text":"Since Sally sells $$15$$ necklaces for $150, we can assign P to be $$150$$ and $$n$$ to be $$15$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation19a-h3","type":"hint","dependencies":["a3d6a35variation19a-h2"],"title":"Plug in","text":"Plug in the values, and solve for k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation19a-h4","type":"hint","dependencies":["a3d6a35variation19a-h3"],"title":"Answer","text":"The answer is $$P=10n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6a35variation2","title":"How to Solve Direct Variation Problems","body":"Solve for the relating equation (Convert all fractions into decimals).","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.9 Use Direct and Inverse Variation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6a35variation2a","stepAnswer":["$$y=0.33x$$"],"problemType":"TextBox","stepTitle":"If $$y$$ varies directly with $$x$$ and $$y=3$$ when $$x=10$$, find the equation that relates $$x$$ and $$y$$ (in the form $$y=?)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=0.33x$$","hints":{"DefaultPathway":[{"id":"a3d6a35variation2a-h1","type":"hint","dependencies":[],"title":"Set up equation","text":"Plug in all the given values in $$y=kx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation2a-h2","type":"hint","dependencies":["a3d6a35variation2a-h1"],"title":"Plugged in","text":"$$3=10k$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation2a-h3","type":"hint","dependencies":["a3d6a35variation2a-h2"],"title":"Divide","text":"Divide both sides by a value that isolates K","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation2a-h4","type":"hint","dependencies":["a3d6a35variation2a-h3"],"title":"Convert to decimal","text":"The equation after dividing both sides by $$10$$ should be $$k=\\\\frac{3}{10}$$. Convert it into an decimal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6a35variation20","title":"Terri\'s Pies","body":"Solve the word problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.9 Use Direct and Inverse Variation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6a35variation20a","stepAnswer":["$$a=4.5p$$"],"problemType":"TextBox","stepTitle":"Terri needs to make some pies for a fundraiser. The number of apples, a, varies directly with number of pies, $$p$$. It takes nine apples to make two pies. Write the equation that relates a and $$p$$ (in the form $$a=?)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$a=4.5p$$","hints":{"DefaultPathway":[{"id":"a3d6a35variation20a-h1","type":"hint","dependencies":[],"title":"Use formula","text":"The word problem implies a direct variation formula is needed. This formula is $$a=kp$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation20a-h2","type":"hint","dependencies":["a3d6a35variation20a-h1"],"title":"Find values","text":"Since Terri needs $$9$$ apples to make $$2$$ pies, we can assign a to be $$9$$ and $$p$$ to be $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation20a-h3","type":"hint","dependencies":["a3d6a35variation20a-h2"],"title":"Plug in","text":"Plug in the values, and solve for k. Then, put the k value into the direct variation equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation20a-h4","type":"hint","dependencies":["a3d6a35variation20a-h3"],"title":"Answer","text":"The answer is $$a=4.5p$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6a35variation21","title":"Jesse\'s Gas","body":"Solve the word problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.9 Use Direct and Inverse Variation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6a35variation21a","stepAnswer":["$$p=3.98g$$"],"problemType":"TextBox","stepTitle":"The price (p) of gas that Jesse purchased varies directly to how many gallons (g) he purchased. He purchased $$10$$ gallons of gas for $$\\\\$39.80$$. Write the equation that relates the price to the number of gallons (in the form $$p=?)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$p=3.98g$$","hints":{"DefaultPathway":[{"id":"a3d6a35variation21a-h1","type":"hint","dependencies":[],"title":"Use formula","text":"The word problem implies a direct variation formula is needed. This formula is $$p=kg$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation21a-h2","type":"hint","dependencies":["a3d6a35variation21a-h1"],"title":"Find values","text":"Since Terri bought $$10$$ gallons of gas for $$39.80$$, we can assign $$p$$ to be $$39.80$$ and g to be $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation21a-h3","type":"hint","dependencies":["a3d6a35variation21a-h2"],"title":"Plug in","text":"Plug in the values, and solve for k. Then, put the k value into the direct variation equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation21a-h4","type":"hint","dependencies":["a3d6a35variation21a-h3"],"title":"Answer","text":"The answer is $$p=3.98g$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6a35variation22","title":"Volume and Mass","body":"Solve the word problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.9 Use Direct and Inverse Variation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6a35variation22a","stepAnswer":["$$m=8v$$"],"problemType":"TextBox","stepTitle":"The mass (m) of a liquid varies directly with its volume (v). A liquid with mass $$16$$ kilograms has a volume of $$2$$ liters. Write the equation that relates the mass to the volume (in the form $$m=?)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$m=8v$$","hints":{"DefaultPathway":[{"id":"a3d6a35variation22a-h1","type":"hint","dependencies":[],"title":"Use formula","text":"The word problem implies a direct variation formula is needed. This formula is $$m=kv$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation22a-h2","type":"hint","dependencies":["a3d6a35variation22a-h1"],"title":"Find values","text":"Since the liquid has a mass of $$16$$ kilos and has a volume of $$2$$ liters, we can assign $$m$$ to be $$16$$ and v to be $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation22a-h3","type":"hint","dependencies":["a3d6a35variation22a-h2"],"title":"Plug in","text":"Plug in the values, and solve for k. Then, put the k value into the direct variation equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation22a-h4","type":"hint","dependencies":["a3d6a35variation22a-h3"],"title":"Answer","text":"The answer is $$m=8v$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6a35variation23","title":"Falling Distance","body":"Solve the word problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.9 Use Direct and Inverse Variation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6a35variation23a","stepAnswer":["$$d=5t^2$$"],"problemType":"TextBox","stepTitle":"The distance (d) an object falls varies directly to the square of the time (t) it falls. A ball falls $$45$$ feet in $$3$$ seconds. Write the equation that relates the distance to the time (in the form $$d=?)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$d=5t^2$$","hints":{"DefaultPathway":[{"id":"a3d6a35variation23a-h1","type":"hint","dependencies":[],"title":"Use formula","text":"The word problem implies a direct variation formula with a square is needed. This formula is $$d=k t^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation23a-h2","type":"hint","dependencies":["a3d6a35variation23a-h1"],"title":"Find values","text":"Since the ball falls $$45$$ feet in $$3$$ seconds, we can assign $$d$$ to be $$45$$ and $$t$$ to be $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation23a-h3","type":"hint","dependencies":["a3d6a35variation23a-h2"],"title":"Plug in","text":"Plug in the values, and solve for k. Then, put the k value into the direct variation equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation23a-h4","type":"hint","dependencies":["a3d6a35variation23a-h3"],"title":"Answer","text":"The answer is $$d=5t^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6a35variation24","title":"Pizza Size","body":"Solve the word problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.9 Use Direct and Inverse Variation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6a35variation24a","stepAnswer":["$$A=3.14r^2$$"],"problemType":"TextBox","stepTitle":"The area of a circle (A) varies directly as the square of the radius (r). A circular pizza with a radius of $$6$$ inches has an area of $$113.04$$ square inches. Write the equation that relates the area to the radius (in the form $$A=?)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$A=3.14r^2$$","hints":{"DefaultPathway":[{"id":"a3d6a35variation24a-h1","type":"hint","dependencies":[],"title":"Use formula","text":"The word problem implies a direct variation formula with a square is needed. This formula is $$d={kt}^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation24a-h2","type":"hint","dependencies":["a3d6a35variation24a-h1"],"title":"Find values","text":"Since the pizza has a radius of $$6$$ inches and an area of $$113.04$$ square inches, we can assign A to be $$113.04$$ and $$r$$ to be $$6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation24a-h3","type":"hint","dependencies":["a3d6a35variation24a-h2"],"title":"Plug in","text":"Plug in the values, and solve for k. Then, put the k value into the direct variation equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation24a-h4","type":"hint","dependencies":["a3d6a35variation24a-h3"],"title":"Answer","text":"The answer is $$A=3.14r^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6a35variation25","title":"Fuel Consumption","body":"Solve the word problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.9 Use Direct and Inverse Variation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6a35variation25a","stepAnswer":["$$g=\\\\frac{92400}{w}$$"],"problemType":"TextBox","stepTitle":"The fuel consumption (g) of a car varies inversely with its weight (w). A Toyota Corolla weighs $$2800$$ pounds and gets $$33$$ mpg on the highway. Write the equation that relates the mpg to the car\u2019s weight (in the form $$g=?)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$g=\\\\frac{92400}{w}$$","hints":{"DefaultPathway":[{"id":"a3d6a35variation25a-h1","type":"hint","dependencies":[],"title":"Use formula","text":"The word problem implies an inverse variation formula. This formula is $$g=\\\\frac{k}{w}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation25a-h2","type":"hint","dependencies":["a3d6a35variation25a-h1"],"title":"Find values","text":"Since the car weighs $$2800$$ pounts and gets $$33$$ mpg, we can assign g to be $$33$$ and w to be $$2800$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation25a-h3","type":"hint","dependencies":["a3d6a35variation25a-h2"],"title":"Plug in","text":"Plug in the values, and solve for k. Then, put the k value into the inverse variation equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation25a-h4","type":"hint","dependencies":["a3d6a35variation25a-h3"],"title":"Answer","text":"The answer is $$g=\\\\frac{92400}{w}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6a35variation26","title":"Janet\'s Basement","body":"Solve the word problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.9 Use Direct and Inverse Variation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6a35variation26a","stepAnswer":["$$t=\\\\frac{1000}{r}$$"],"problemType":"TextBox","stepTitle":"The time (t) required to empty a tank varies inversely as the rate (r) of pumping. It took Janet $$5$$ hours to pump her flooded basement using a pump that was rated at $$200$$ gpm (gallons per minute). Write the equation that relates the number of hours to the pump rate (in the form $$t=?)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$t=\\\\frac{1000}{r}$$","hints":{"DefaultPathway":[{"id":"a3d6a35variation26a-h1","type":"hint","dependencies":[],"title":"Use formula","text":"The word problem implies an inverse variation formula. This formula is $$t=\\\\frac{k}{r}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation26a-h2","type":"hint","dependencies":["a3d6a35variation26a-h1"],"title":"Find values","text":"Since it took Janet $$5$$ hours to pump the basement with a 200gpm pump, we can assign $$t$$ to be $$5$$ and $$r$$ to be $$200$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation26a-h3","type":"hint","dependencies":["a3d6a35variation26a-h2"],"title":"Plug in","text":"Plug in the values, and solve for k. Then, put the k value into the inverse variation equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation26a-h4","type":"hint","dependencies":["a3d6a35variation26a-h3"],"title":"Answer","text":"The answer is $$t=\\\\frac{1000}{r}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6a35variation27","title":"Violin Strings","body":"Solve the word problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.9 Use Direct and Inverse Variation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6a35variation27a","stepAnswer":["$$L=\\\\frac{4400}{f}$$"],"problemType":"TextBox","stepTitle":"On a string instrument, the length (L) of a string varies inversely as the frequency (f) of its vibrations. An 11-inch string on a violin has a frequency of $$400$$ cycles per second. Write the equation that relates the string length to its frequency (in the form $$L=?)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$L=\\\\frac{4400}{f}$$","hints":{"DefaultPathway":[{"id":"a3d6a35variation27a-h1","type":"hint","dependencies":[],"title":"Use formula","text":"The word problem implies an inverse variation formula. This formula is $$L=\\\\frac{k}{f}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation27a-h2","type":"hint","dependencies":["a3d6a35variation27a-h1"],"title":"Find values","text":"Since the string has a frequency of $$400$$ cycles per second when the string is $$11$$ inches long, we can assign L to be $$11$$ and f to be $$400$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation27a-h3","type":"hint","dependencies":["a3d6a35variation27a-h2"],"title":"Plug in","text":"Plug in the values, and solve for k. Then, put the k value into the inverse variation equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation27a-h4","type":"hint","dependencies":["a3d6a35variation27a-h3"],"title":"Answer","text":"The answer is $$L=\\\\frac{4400}{f}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6a35variation28","title":"Brianna\'s Tickets","body":"Solve the word problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.9 Use Direct and Inverse Variation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6a35variation28a","stepAnswer":["$$t=\\\\frac{125}{p}$$"],"problemType":"TextBox","stepTitle":"The number of tickets (t) for a sports fundraiser varies inversely to the price (p) of each ticket. Brianna can buy $$25$$ tickets at $5 each. Write the equation that relates the number of tickets to the price of each ticket (in the form $$t=?)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$t=\\\\frac{125}{p}$$","hints":{"DefaultPathway":[{"id":"a3d6a35variation28a-h1","type":"hint","dependencies":[],"title":"Use formula","text":"The word problem implies an inverse variation formula. This formula is $$t=\\\\frac{k}{p}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation28a-h2","type":"hint","dependencies":["a3d6a35variation28a-h1"],"title":"Find values","text":"Since Brianna can buy $$25$$ tickets at $5 each, we can assign $$t$$ to be $$25$$ and $$p$$ to be $$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation28a-h3","type":"hint","dependencies":["a3d6a35variation28a-h2"],"title":"Plug in","text":"Plug in the values, and solve for k. Then, put the k value into the inverse variation equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation28a-h4","type":"hint","dependencies":["a3d6a35variation28a-h3"],"title":"Answer","text":"The answer is $$t=\\\\frac{125}{p}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6a35variation29","title":"Solving Relating Equations","body":"Solve the problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.9 Use Direct and Inverse Variation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6a35variation29a","stepAnswer":["$$y=\\\\frac{5}{3} x$$"],"problemType":"TextBox","stepTitle":"If $$y$$ varies directly as $$x$$, and $$y=5$$ when $$x=3$$, find the equation that relates $$x$$ and $$y$$ (n the form $$y=?)$$","stepBody":"Find the equation that relates $$x$$ and $$y$$.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=\\\\frac{5}{3} x$$","hints":{"DefaultPathway":[{"id":"a3d6a35variation29a-h1","type":"hint","dependencies":[],"title":"Use formula","text":"The direct variation formula is $$y=kx$$. You can use this to find the relationship between $$x$$ and $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation29a-h2","type":"hint","dependencies":["a3d6a35variation29a-h1"],"title":"Plug In","text":"Plug in $$x$$ and $$y$$ from the point given. Solve for k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation29a-h3","type":"hint","dependencies":["a3d6a35variation29a-h2"],"title":"Simplify","text":"Simplify the equation, finding k and plugging it back into the direct variation formula.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation29a-h4","type":"hint","dependencies":["a3d6a35variation29a-h3"],"title":"Answer","text":"The answer is $$y=\\\\frac{5}{3} x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6a35variation3","title":"How to Solve Direct Variation Problems","body":"Solve for the relating equation (Convert all fractions into decimals).","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.9 Use Direct and Inverse Variation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6a35variation3a","stepAnswer":["$$y=3x$$"],"problemType":"TextBox","stepTitle":"If $$y$$ varies directly with $$x$$ and $$y=12$$ when $$x=4$$, find the equation that relates $$x$$ and $$y$$ (in the form $$y=?)$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=3x$$","hints":{"DefaultPathway":[{"id":"a3d6a35variation3a-h1","type":"hint","dependencies":[],"title":"Set up equation","text":"Plug in all the given values in $$y=kx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation3a-h2","type":"hint","dependencies":["a3d6a35variation3a-h1"],"title":"Plugged in","text":"$$12=4k$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation3a-h3","type":"hint","dependencies":["a3d6a35variation3a-h2"],"title":"Divide","text":"Divide both sides by a value that isolates K","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation3a-h4","type":"hint","dependencies":["a3d6a35variation3a-h3"],"title":"Convert to decimal","text":"The equation after dividing both sides by $$4$$ should be $$k=\\\\frac{12}{4}$$. Convert it into an decimal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6a35variation30","title":"Solving Relating Equations","body":"Find the equation that relates $$x$$ and $$y$$.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.9 Use Direct and Inverse Variation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6a35variation30a","stepAnswer":["$$y=\\\\frac{20}{x}$$"],"problemType":"TextBox","stepTitle":"Y varies inversely with $$x$$, and $$y=5$$ when $$x=4$$. Write your answer in the form $$y=$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=\\\\frac{20}{x}$$","hints":{"DefaultPathway":[{"id":"a3d6a35variation30a-h1","type":"hint","dependencies":[],"title":"Use formula","text":"The inverse variation formula is $$y=\\\\frac{k}{x}$$. You can use this to find the relationship between $$x$$ and $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation30a-h2","type":"hint","dependencies":["a3d6a35variation30a-h1"],"title":"Plug In","text":"Plug in $$x$$ and $$y$$ from the point given. Solve for k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation30a-h3","type":"hint","dependencies":["a3d6a35variation30a-h2"],"title":"Simplify","text":"Simplify the equation, finding k and plugging it back into the inverse variation formula.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation30a-h4","type":"hint","dependencies":["a3d6a35variation30a-h3"],"title":"Answer","text":"The answer is $$y=\\\\frac{20}{x}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6a35variation4","title":"How to Solve Direct Variation Problems","body":"Solve for the relating equation (Convert all fractions into decimals).","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.9 Use Direct and Inverse Variation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6a35variation4a","stepAnswer":["$$y=4.67x$$"],"problemType":"TextBox","stepTitle":"If $$y$$ varies directly with $$x$$ and $$y=14$$ when $$x=3$$, find the equation that relates $$x$$ and $$y$$ (in the form $$y=?)$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=4.67x$$","hints":{"DefaultPathway":[{"id":"a3d6a35variation4a-h1","type":"hint","dependencies":[],"title":"Set up equation","text":"Plug in all the given values in $$y=kx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation4a-h2","type":"hint","dependencies":["a3d6a35variation4a-h1"],"title":"Plugged in","text":"$$14=3k$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation4a-h3","type":"hint","dependencies":["a3d6a35variation4a-h2"],"title":"Divide","text":"Divide both sides by a value that isolates K","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation4a-h4","type":"hint","dependencies":["a3d6a35variation4a-h3"],"title":"Convert to decimal","text":"The equation after dividing both sides by $$3$$ should be $$k=\\\\frac{14}{3}$$. Convert it into an decimal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6a35variation5","title":"How to Solve Direct Variation Problems","body":"Solve for the relating equation (Convert all fractions into decimals).","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.9 Use Direct and Inverse Variation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6a35variation5a","stepAnswer":["$$p=2.5q$$"],"problemType":"TextBox","stepTitle":"If $$p$$ varies directly as q and $$p=5, whenq=2$$, find the equation that relates pandq (in the form $$p=?)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$p=2.5q$$","hints":{"DefaultPathway":[{"id":"a3d6a35variation5a-h1","type":"hint","dependencies":[],"title":"Set up equation","text":"Plug in all the given values in $$p=kq$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation5a-h2","type":"hint","dependencies":["a3d6a35variation5a-h1"],"title":"Plugged in","text":"$$5=2k$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation5a-h3","type":"hint","dependencies":["a3d6a35variation5a-h2"],"title":"Divide","text":"Divide both sides by a value that isolates K","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation5a-h4","type":"hint","dependencies":["a3d6a35variation5a-h3"],"title":"Convert to decimal","text":"The equation after dividing both sides by $$2$$ should be $$k=\\\\frac{5}{2}$$. Convert it into an decimal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6a35variation6","title":"How to Solve Direct Variation Problems","body":"Solve for the relating equation (Convert all fractions into decimals).","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.9 Use Direct and Inverse Variation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6a35variation6a","stepAnswer":["$$v=3w$$"],"problemType":"TextBox","stepTitle":"If v varies directly as w and $$v=24$$ when $$w=8$$, find the equation that relates v and w (in the form $$v=?)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$v=3w$$","hints":{"DefaultPathway":[{"id":"a3d6a35variation6a-h1","type":"hint","dependencies":[],"title":"Set up equation","text":"Plug in all the given values in $$v=kw$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation6a-h2","type":"hint","dependencies":["a3d6a35variation6a-h1"],"title":"Plugged in","text":"$$24=8k$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation6a-h3","type":"hint","dependencies":["a3d6a35variation6a-h2"],"title":"Divide","text":"Divide both sides by a value that isolates K","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation6a-h4","type":"hint","dependencies":["a3d6a35variation6a-h3"],"title":"Convert to decimal","text":"The equation after dividing both sides by $$8$$ should be $$k=\\\\frac{24}{8}$$. Convert it into an decimal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6a35variation7","title":"How to Solve Direct Variation Problems","body":"Solve for the relating equation (Convert all fractions into decimals).","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.9 Use Direct and Inverse Variation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6a35variation7a","stepAnswer":["$$a=4b$$"],"problemType":"TextBox","stepTitle":"If a varies directly as $$b$$ and $$a=16$$ when $$b$$ $$=4$$, find the equation that relates a and $$b$$ (in the form a $$=?)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$a=4b$$","hints":{"DefaultPathway":[{"id":"a3d6a35variation7a-h1","type":"hint","dependencies":[],"title":"Set up equation","text":"Plug in all the given values in $$a=kb$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation7a-h2","type":"hint","dependencies":["a3d6a35variation7a-h1"],"title":"Plugged in","text":"$$16=4k$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation7a-h3","type":"hint","dependencies":["a3d6a35variation7a-h2"],"title":"Divide","text":"Divide both sides by a value that isolates K","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation7a-h4","type":"hint","dependencies":["a3d6a35variation7a-h3"],"title":"Convert to decimal","text":"The equation after dividing both sides by $$4$$ should be $$k=\\\\frac{16}{4}$$. Convert it into an decimal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6a35variation8","title":"How to Solve Direct Variation Problems","body":"Solve for the relating equation (Convert all fractions into decimals).","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.9 Use Direct and Inverse Variation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6a35variation8a","stepAnswer":["$$p=3.2q$$"],"problemType":"TextBox","stepTitle":"If $$p$$ varies directly as q and $$p=9.6$$ when $$q=3$$, find the equation that relates $$p$$ and q (in the form $$p=?)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$p=3.2q$$","hints":{"DefaultPathway":[{"id":"a3d6a35variation8a-h1","type":"hint","dependencies":[],"title":"Set up equation","text":"Plug in all the given values in $$p=kq$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation8a-h2","type":"hint","dependencies":["a3d6a35variation8a-h1"],"title":"Plugged in","text":"$$9.6=3k$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation8a-h3","type":"hint","dependencies":["a3d6a35variation8a-h2"],"title":"Divide","text":"Divide both sides by a value that isolates K","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation8a-h4","type":"hint","dependencies":["a3d6a35variation8a-h3"],"title":"Convert to decimal","text":"The equation after dividing both sides by $$3$$ should be $$k=\\\\frac{9.6}{3}$$. Convert it into an decimal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6a35variation9","title":"How to Solve Direct Variation Problems","body":"Solve for the relating equation (Convert all fractions into decimals).","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.9 Use Direct and Inverse Variation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6a35variation9a","stepAnswer":["$$y=3.1x$$"],"problemType":"TextBox","stepTitle":"If $$y$$ varies directly as $$x$$ and $$y=12.4, whenx=4$$, find the equation that relates xandy (in the form $$y=?)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=3.1x$$","hints":{"DefaultPathway":[{"id":"a3d6a35variation9a-h1","type":"hint","dependencies":[],"title":"Set up equation","text":"Plug in all the given values in $$y=kx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation9a-h2","type":"hint","dependencies":["a3d6a35variation9a-h1"],"title":"Plugged in","text":"$$12.4=4k$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation9a-h3","type":"hint","dependencies":["a3d6a35variation9a-h2"],"title":"Divide","text":"Divide both sides by a value that isolates K","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6a35variation9a-h4","type":"hint","dependencies":["a3d6a35variation9a-h3"],"title":"Convert to decimal","text":"The equation after dividing both sides by $$4$$ should be $$k=\\\\frac{12.4}{4}$$. Convert it into an decimal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6ae2sys1","title":"Solutions of a System of Equations","body":"Determine if the following points are solutions to the given system of equation:\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Solve Systems of Equations by Graphing","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6ae2sys1a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$(-2,-1)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys1a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitue $$x=-2$$ and $$y=-1$$ into both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys1a-h2","type":"hint","dependencies":["a3d6ae2sys1a-h1"],"title":"Substitute into First Equation","text":"$$x-y=-1$$\\\\n$$-2-(-1)=-1$$\\\\n$$-1=-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys1a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a3d6ae2sys1a-h2"],"title":"Substitute into First Equation","text":"Is the equation above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a3d6ae2sys1a-h4","type":"hint","dependencies":["a3d6ae2sys1a-h3"],"title":"Solution to First Equation","text":"Therefore, $$(-2,-1)$$ satisfies the first equation, but it must also safisfy the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys1a-h5","type":"hint","dependencies":["a3d6ae2sys1a-h4"],"title":"Substitute into Second Equation","text":"$$2x-y=-5$$\\\\n$$2\\\\times-2-\\\\left(-1\\\\right)=-5$$\\\\n$$-3=-5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys1a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["FALSE"],"dependencies":["a3d6ae2sys1a-h5"],"title":"Substitute into Second Equation","text":"Is the equation above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a3d6ae2sys1a-h7","type":"hint","dependencies":["a3d6ae2sys1a-h6"],"title":"Solution to Second Equation","text":"Therefore, $$(-2,-1)$$ does not satisfies the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys1a-h8","type":"hint","dependencies":["a3d6ae2sys1a-h7"],"title":"Solutions of a System of Equations","text":"$$(-2,-1)$$ does not make both equations true. $$(-2,-1)$$ is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a3d6ae2sys1b","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$(-4,-3)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys1b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitue $$x=-4$$ and $$y=-3$$ into both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys1b-h2","type":"hint","dependencies":["a3d6ae2sys1b-h1"],"title":"Substitute into First Equation","text":"$$x-y=-1$$\\\\n$$-4-(-3)=-1$$\\\\n$$-1=-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys1b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a3d6ae2sys1b-h2"],"title":"Substitute into First Equation","text":"Is the equation above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a3d6ae2sys1b-h4","type":"hint","dependencies":["a3d6ae2sys1b-h3"],"title":"Solution to First Equation","text":"Therefore, $$(-4,-3)$$ satisfies the first equation, but it must also safisfy the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys1b-h5","type":"hint","dependencies":["a3d6ae2sys1b-h4"],"title":"Substitute into Second Equation","text":"$$2x-y=-5$$\\\\n$$2\\\\times-4-\\\\left(-3\\\\right)=-5$$\\\\n$$-5=-5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys1b-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a3d6ae2sys1b-h5"],"title":"Substitute into Second Equation","text":"Is the equation above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a3d6ae2sys1b-h7","type":"hint","dependencies":["a3d6ae2sys1b-h6"],"title":"Solution to Second Equation","text":"Therefore, $$(-4,-3)$$ does satisfies the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys1b-h8","type":"hint","dependencies":["a3d6ae2sys1b-h7"],"title":"Solutions of a System of Equations","text":"$$(-4,-3)$$ does make both equations true. $$(-4,-3)$$ is a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6ae2sys10","title":"Solutions of a System of Equations","body":"Determine if the following points are solutions to the given system of equation:\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Solve Systems of Equations by Graphing","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6ae2sys10a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$(1,-3)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys10a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitue $$x=1$$ and $$y=-3$$ into both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys10a-h2","type":"hint","dependencies":["a3d6ae2sys10a-h1"],"title":"Substitute into First Equation","text":"$$3x+y=0$$\\\\n$$3\\\\times1-3=0$$\\\\n$$0=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys10a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a3d6ae2sys10a-h2"],"title":"Substitute into First Equation","text":"Is the equation above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a3d6ae2sys10a-h4","type":"hint","dependencies":["a3d6ae2sys10a-h3"],"title":"Solution to First Equation","text":"Therefore, $$(-6,5)$$ satisfies the first equation, but it must also safisfy the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys10a-h5","type":"hint","dependencies":["a3d6ae2sys10a-h4"],"title":"Substitute into Second Equation","text":"$$x+2y=-5$$\\\\n$$1+2\\\\times-3=-5$$\\\\n$$-5=-5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys10a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a3d6ae2sys10a-h5"],"title":"Substitute into Second Equation","text":"Is the equation above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a3d6ae2sys10a-h7","type":"hint","dependencies":["a3d6ae2sys10a-h6"],"title":"Solution to Second Equation","text":"Therefore, $$(1,-3)$$ does satisfies the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys10a-h8","type":"hint","dependencies":["a3d6ae2sys10a-h7"],"title":"Solutions of a System of Equations","text":"$$(1,-3)$$ does make both equations true. $$(1,-3)$$ is a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a3d6ae2sys10b","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$(0,0)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys10b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitue $$x=0$$ and $$y=0$$ into both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys10b-h2","type":"hint","dependencies":["a3d6ae2sys10b-h1"],"title":"Substitute into First Equation","text":"$$3x+y=0$$\\\\n$$3\\\\times0+0=0$$\\\\n$$0=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys10b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a3d6ae2sys10b-h2"],"title":"Substitute into First Equation","text":"Is the equation above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a3d6ae2sys10b-h4","type":"hint","dependencies":["a3d6ae2sys10b-h3"],"title":"Solution to First Equation","text":"Therefore, $$(0,0)$$ satisfies the first equation, but it must also safisfy the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys10b-h5","type":"hint","dependencies":["a3d6ae2sys10b-h4"],"title":"Substitute into Second Equation","text":"$$x+2y=-5$$\\\\n$$0+2\\\\times0=-5$$\\\\n$$0=-5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys10b-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["FALSE"],"dependencies":["a3d6ae2sys10b-h5"],"title":"Substitute into Second Equation","text":"Is the equation above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a3d6ae2sys10b-h7","type":"hint","dependencies":["a3d6ae2sys10b-h6"],"title":"Solution to Second Equation","text":"Therefore, $$(0,0)$$ does not satisfies the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys10b-h8","type":"hint","dependencies":["a3d6ae2sys10b-h7"],"title":"Solutions of a System of Equations","text":"$$(0,0)$$ does not make both equations true. $$(0,0)$$ is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6ae2sys11","title":"Number of Solutions of Linear Systems of Equations","body":"Without graphing, determine the number of solutions and then classify the system of equation:\\\\n##figure2.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Solve Systems of Equations by Graphing","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6ae2sys11a","stepAnswer":["No solution"],"problemType":"MultipleChoice","stepTitle":"Determine Number of Solutions.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$1$$ solution","No solution","Infinitely many"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys11a-h1","type":"hint","dependencies":[],"title":"Compare Slopes and Intercepts","text":"We will compare the slopes and intercepts of the two lines:\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys11a-h2","type":"hint","dependencies":["a3d6ae2sys11a-h1"],"title":"Slope-Intercept Form","text":"The first equation is already in slope-intercept form: $$y=3x-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys11a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a3d6ae2sys11a-h2"],"title":"Identify the Slope","text":"What is $$m$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a3d6ae2sys11a-h3"],"title":"Identify Y-Intercept","text":"What is $$b$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys11a-h5","type":"hint","dependencies":["a3d6ae2sys11a-h4"],"title":"Slope-Intercept Form","text":"Write the second equation in slope-intercept form:\\\\n$$6x-2y=12$$\\\\n$$-2y=-6x+12$$\\\\n$$\\\\frac{-2y}{-2}=\\\\frac{\\\\left(-6x+12\\\\right)}{-2}$$\\\\n$$y=3x-6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys11a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a3d6ae2sys11a-h5"],"title":"Identify the Slope","text":"What is $$m$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys11a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6$$"],"dependencies":["a3d6ae2sys11a-h6"],"title":"Identify Y-Intercept","text":"What is $$b$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys11a-h8","type":"hint","dependencies":["a3d6ae2sys11a-h3","a3d6ae2sys11a-h4","a3d6ae2sys11a-h6","a3d6ae2sys11a-h7"],"title":"Parallel Lines","text":"Since the slopes are the same and y-intercepts are different, the lines are parallel.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys11a-h9","type":"hint","dependencies":["a3d6ae2sys11a-h8"],"title":"Parallel Lines","text":"A system of equations whose graphs are parallel lines has no solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a3d6ae2sys11b","stepAnswer":["Inconsistent & Independent"],"problemType":"MultipleChoice","stepTitle":"Classify the system of equations.","stepBody":"","answerType":"string","variabilization":{},"choices":["Consistent & Independent","Inconsistent & Independent","Consistent & Dependent"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys11b-h1","type":"hint","dependencies":[],"title":"Classifications","text":"A system of equations whose graphs are parallel lines has no solution and is inconsistent and independent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6ae2sys12","title":"Number of Solutions of Linear Systems of Equations","body":"Without graphing, determine the number of solutions and then classify the system of equation:\\\\n##figure2.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Solve Systems of Equations by Graphing","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6ae2sys12a","stepAnswer":["$$1$$ solution"],"problemType":"MultipleChoice","stepTitle":"Determine number of solutions.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$1$$ solution","choices":["$$1$$ solution","No solution","Infinitely many"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys12a-h1","type":"hint","dependencies":[],"title":"Compare Slopes and Intercepts","text":"We will compare the slopes and intercepts of the two lines:\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys12a-h2","type":"hint","dependencies":["a3d6ae2sys12a-h1"],"title":"Slope-Intercept Form","text":"Write the first equation in slope-intercept form:\\\\n$$2x+y=-3$$\\\\n$$y=-2x-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys12a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a3d6ae2sys12a-h2"],"title":"Identify the Slope","text":"What is $$m$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a3d6ae2sys12a-h3"],"title":"Identify Y-Intercept","text":"What is $$b$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys12a-h5","type":"hint","dependencies":["a3d6ae2sys12a-h4"],"title":"Slope-Intercept Form","text":"Write the second equation in slope-intercept form:\\\\n$$x-5y=5$$\\\\n$$-5y=-x+5$$\\\\n$$\\\\frac{-5y}{-5}=\\\\frac{\\\\left(-x+5\\\\right)}{-5}$$\\\\n$$y=\\\\frac{1}{5} x-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys12a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{5}$$"],"dependencies":["a3d6ae2sys12a-h5"],"title":"Identify the Slope","text":"What is $$m$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys12a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a3d6ae2sys12a-h6"],"title":"Identify Y-Intercept","text":"What is $$b$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys12a-h8","type":"hint","dependencies":["a3d6ae2sys12a-h3","a3d6ae2sys12a-h4","a3d6ae2sys12a-h6","a3d6ae2sys12a-h7"],"title":"Intersecting Lines","text":"Since the slopes are different, the lines intersect.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys12a-h9","type":"hint","dependencies":["a3d6ae2sys12a-h8"],"title":"Intersecting Lines","text":"A system of equations whose graphs intersect have $$1$$ solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a3d6ae2sys12b","stepAnswer":["Consistent & Independent"],"problemType":"MultipleChoice","stepTitle":"Classify the system of equations.","stepBody":"","answerType":"string","variabilization":{},"choices":["Consistent & Independent","Inconsistent & Independent","Consistent & Dependent"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys12b-h1","type":"hint","dependencies":[],"title":"Classifications","text":"A system of equations whose graphs are intersect has $$1$$ solution and is consistent and independent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6ae2sys13","title":"Number of Solutions of Linear Systems of Equations","body":"Without graphing, determine the number of solutions and then classify the system of equation:\\\\n##figure2.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Solve Systems of Equations by Graphing","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6ae2sys13a","stepAnswer":["Infinitely many"],"problemType":"MultipleChoice","stepTitle":"Determine number of solutions.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$1$$ solution","No solution","Infinitely many"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys13a-h1","type":"hint","dependencies":[],"title":"Compare Slopes and Intercepts","text":"We will compare the slopes and intercepts of the two lines:\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys13a-h2","type":"hint","dependencies":["a3d6ae2sys13a-h1"],"title":"Slope-Intercept Form","text":"Write the first equation in slope-intercept form:\\\\n$$3x-2y=4$$\\\\n$$-2y=-3x+4$$\\\\n$$\\\\frac{-2y}{-2}=\\\\frac{\\\\left(-3x+4\\\\right)}{-2}$$\\\\n$$y=\\\\frac{3}{2} x-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys13a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{2}$$"],"dependencies":["a3d6ae2sys13a-h2"],"title":"Identify the Slope","text":"What is $$m$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a3d6ae2sys13a-h3"],"title":"Identify Y-Intercept","text":"What is $$b$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys13a-h5","type":"hint","dependencies":["a3d6ae2sys13a-h4"],"title":"Slope-Intercept Form","text":"The second equation is already in slope-intercept form: $$y=\\\\frac{3}{2} x-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys13a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{2}$$"],"dependencies":["a3d6ae2sys13a-h5"],"title":"Identify the Slope","text":"What is $$m$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys13a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a3d6ae2sys13a-h6"],"title":"Identify Y-Intercept","text":"What is $$b$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys13a-h8","type":"hint","dependencies":["a3d6ae2sys13a-h3","a3d6ae2sys13a-h4","a3d6ae2sys13a-h6","a3d6ae2sys13a-h7"],"title":"Same Lines","text":"Since the slopes and y-intercept are the same, the equations have the same line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys13a-h9","type":"hint","dependencies":["a3d6ae2sys13a-h8"],"title":"Same Lines","text":"A system of equations whose graphs are coincident lines has infinitely many solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a3d6ae2sys13b","stepAnswer":["Consistent & Dependent"],"problemType":"MultipleChoice","stepTitle":"Classify the system of equations.","stepBody":"","answerType":"string","variabilization":{},"choices":["Consistent & Independent","Inconsistent & Independent","Consistent & Dependent"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys13b-h1","type":"hint","dependencies":[],"title":"Classifications","text":"A system of equations whose graphs are coincident lines has infinitely many solutions and is consistent and dependent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6ae2sys14","title":"Number of Solutions of Linear Systems of Equations","body":"Without graphing, determine the number of solutions and then classify the system of equation:\\\\n##figure2.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Solve Systems of Equations by Graphing","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6ae2sys14a","stepAnswer":["No solution"],"problemType":"MultipleChoice","stepTitle":"Determine number of solutions.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$1$$ solution","No solution","Infinitely many"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys14a-h1","type":"hint","dependencies":[],"title":"Compare Slopes and Intercepts","text":"We will compare the slopes and intercepts of the two lines:\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys14a-h2","type":"hint","dependencies":["a3d6ae2sys14a-h1"],"title":"Slope-Intercept Form","text":"The first equation is already in slope-intercept form: $$y=\\\\frac{2}{3} x+1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys14a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{3}$$"],"dependencies":["a3d6ae2sys14a-h2"],"title":"Identify the Slope","text":"What is $$m$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys14a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a3d6ae2sys14a-h3"],"title":"Identify Y-Intercept","text":"What is $$b$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys14a-h5","type":"hint","dependencies":["a3d6ae2sys14a-h4"],"title":"Slope-Intercept Form","text":"Write the second equation in slope-intercept form:\\\\n$$-2x+3y=5$$\\\\n$$3y=2x+5$$\\\\n$$\\\\frac{3y}{3}=\\\\frac{2x+5}{3}$$\\\\n$$y=\\\\frac{2}{3} x+\\\\frac{5}{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys14a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{3}$$"],"dependencies":["a3d6ae2sys14a-h5"],"title":"Identify the Slope","text":"What is $$m$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys14a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{3}$$"],"dependencies":["a3d6ae2sys14a-h6"],"title":"Identify Y-Intercept","text":"What is $$b$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys14a-h8","type":"hint","dependencies":["a3d6ae2sys14a-h3","a3d6ae2sys14a-h4","a3d6ae2sys14a-h6","a3d6ae2sys14a-h7"],"title":"Parallel Lines","text":"Since the slopes are the same and y-intercepts are different, the lines are parallel.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys14a-h9","type":"hint","dependencies":["a3d6ae2sys14a-h8"],"title":"Parallel Lines","text":"A system of equations whose graphs are parallel lines has no solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a3d6ae2sys14b","stepAnswer":["Inconsistent & Independent"],"problemType":"MultipleChoice","stepTitle":"Classify the system of equations.","stepBody":"","answerType":"string","variabilization":{},"choices":["Consistent & Independent","Inconsistent & Independent","Consistent & Dependent"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys14b-h1","type":"hint","dependencies":[],"title":"Classifications","text":"A system of equations whose graphs are parallel lines has no solution and is inconsistent and independent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6ae2sys15","title":"Number of Solutions of Linear Systems of Equations","body":"Without graphing, determine the number of solutions and then classify the system of equation:\\\\n##figure2.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Solve Systems of Equations by Graphing","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6ae2sys15a","stepAnswer":["No solution"],"problemType":"MultipleChoice","stepTitle":"Determine number of solutions.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$1$$ solution","No solution","Infinitely many"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys15a-h1","type":"hint","dependencies":[],"title":"Compare Slopes and Intercepts","text":"We will compare the slopes and intercepts of the two lines:\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys15a-h2","type":"hint","dependencies":["a3d6ae2sys15a-h1"],"title":"Slope-Intercept Form","text":"The first equation is already in slope-intercept form: $$y=\\\\frac{1}{3} x+2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["a3d6ae2sys15a-h2"],"title":"Identify the Slope","text":"What is $$m$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a3d6ae2sys15a-h3"],"title":"Identify Y-Intercept","text":"What is $$b$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys15a-h5","type":"hint","dependencies":["a3d6ae2sys15a-h4"],"title":"Slope-Intercept Form","text":"Write the second equation in slope-intercept form:\\\\n$$x-3y=9$$\\\\n$$-3y=-x+9$$\\\\n$$\\\\frac{-3y}{-3}=\\\\frac{\\\\left(-x+9\\\\right)}{-3}$$\\\\n$$y=\\\\frac{1}{3} x-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys15a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["a3d6ae2sys15a-h5"],"title":"Identify the Slope","text":"What is $$m$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys15a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a3d6ae2sys15a-h6"],"title":"Identify Y-Intercept","text":"What is $$b$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys15a-h8","type":"hint","dependencies":["a3d6ae2sys15a-h3","a3d6ae2sys15a-h4","a3d6ae2sys15a-h6","a3d6ae2sys15a-h7"],"title":"Parallel Lines","text":"Since the slopes are the same and y-intercepts are different, the lines are parallel.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys15a-h9","type":"hint","dependencies":["a3d6ae2sys15a-h8"],"title":"Parallel Lines","text":"A system of equations whose graphs are parallel lines has no solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a3d6ae2sys15b","stepAnswer":["Inconsistent & Independent"],"problemType":"MultipleChoice","stepTitle":"Classify the system of equations.","stepBody":"","answerType":"string","variabilization":{},"choices":["Consistent & Independent","Inconsistent & Independent","Consistent & Dependent"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys15b-h1","type":"hint","dependencies":[],"title":"Classifications","text":"A system of equations whose graphs are parallel lines has no solution and is inconsistent and independent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6ae2sys16","title":"Number of Solutions of Linear Systems of Equations","body":"Without graphing, determine the number of solutions and then classify the system of equations: image1\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Solve Systems of Equations by Graphing","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6ae2sys16a","stepAnswer":["No solution"],"problemType":"MultipleChoice","stepTitle":"Determine number of solutions.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$1$$ solution","No solution","Infinitely many"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys16a-h1","type":"hint","dependencies":[],"title":"Compare Slopes and Intercepts","text":"We will compare the slopes and intercepts of the two lines: image1","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys16a-h2","type":"hint","dependencies":["a3d6ae2sys16a-h1"],"title":"Slope-Intercept Form","text":"The first equation is already in slope-intercept form: $$y=-2x+1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys16a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a3d6ae2sys16a-h2"],"title":"Identify the Slope","text":"What is $$m$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys16a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a3d6ae2sys16a-h3"],"title":"Identify y-intercept","text":"What is $$b$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys16a-h5","type":"hint","dependencies":["a3d6ae2sys16a-h4"],"title":"Slope-Intercept Form","text":"Write the second equation in slope-intercept form:\\\\n$$4x+2y=8$$\\\\n$$2y=-4x+8$$\\\\n$$\\\\frac{2y}{2}=\\\\frac{\\\\left(-4x+8\\\\right)}{2}$$\\\\n$$y=-2x+4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys16a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a3d6ae2sys16a-h5"],"title":"Identify the Slope","text":"What is $$m$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys16a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a3d6ae2sys16a-h6"],"title":"Identify y-intercept","text":"What is $$b$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys16a-h8","type":"hint","dependencies":["a3d6ae2sys16a-h3","a3d6ae2sys16a-h4","a3d6ae2sys16a-h6","a3d6ae2sys16a-h7"],"title":"Parallel Lines","text":"Since the slopes are the same and y-intercepts are different, the lines are parallel.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys16a-h9","type":"hint","dependencies":["a3d6ae2sys16a-h8"],"title":"Parallel Lines","text":"A system of equations whose graphs are parallel lines has no solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a3d6ae2sys16b","stepAnswer":["Inconsistent & Independent"],"problemType":"MultipleChoice","stepTitle":"Classify the system of equations.","stepBody":"","answerType":"string","variabilization":{},"choices":["Consistent & Independent","Inconsistent & Independent","Consistent & Dependent"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys16b-h1","type":"hint","dependencies":[],"title":"Classifications","text":"A system of equations whose graphs are parallel lines has no solution and is inconsistent and independent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6ae2sys17","title":"Number of Solutions of Linear Systems of Equations","body":"Without graphing, determine the number of solutions and then classify the system of equations: image1\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Solve Systems of Equations by Graphing","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6ae2sys17a","stepAnswer":["No solution"],"problemType":"MultipleChoice","stepTitle":"Determine number of solutions.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$1$$ solution","No solution","Infinitely many"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys17a-h1","type":"hint","dependencies":[],"title":"Compare Slopes and Intercepts","text":"We will compare the slopes and intercepts of the two lines: image1","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys17a-h2","type":"hint","dependencies":["a3d6ae2sys17a-h1"],"title":"Slope-Intercept Form","text":"The first equation is already in slope-intercept form: $$y=3x+4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys17a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a3d6ae2sys17a-h2"],"title":"Identify the Slope","text":"What is $$m$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys17a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a3d6ae2sys17a-h3"],"title":"Identify y-intercept","text":"What is $$b$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys17a-h5","type":"hint","dependencies":["a3d6ae2sys17a-h4"],"title":"Slope-Intercept Form","text":"Write the second equation in slope-intercept form:\\\\n$$9x-3y=18$$\\\\n$$-3y=-9x+18$$\\\\n$$\\\\frac{-3y}{-3}=\\\\frac{\\\\left(-9x+18\\\\right)}{-3}$$\\\\n$$y=3x-6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys17a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a3d6ae2sys17a-h5"],"title":"Identify the Slope","text":"What is $$m$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys17a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6$$"],"dependencies":["a3d6ae2sys17a-h6"],"title":"Identify y-intercept","text":"What is $$b$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys17a-h8","type":"hint","dependencies":["a3d6ae2sys17a-h3","a3d6ae2sys17a-h4","a3d6ae2sys17a-h6","a3d6ae2sys17a-h7"],"title":"Parallel Lines","text":"Since the slopes are the same and y-intercepts are different, the lines are parallel.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys17a-h9","type":"hint","dependencies":["a3d6ae2sys17a-h8"],"title":"Parallel Lines","text":"A system of equations whose graphs are parallel lines has no solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a3d6ae2sys17b","stepAnswer":["Inconsistent & Independent"],"problemType":"MultipleChoice","stepTitle":"Classify the system of equations.","stepBody":"","answerType":"string","variabilization":{},"choices":["Consistent & Independent","Inconsistent & Independent","Consistent & Dependent"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys17b-h1","type":"hint","dependencies":[],"title":"Classifications","text":"A system of equations whose graphs are parallel lines has no solution and is inconsistent and independent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6ae2sys18","title":"Number of Solutions of Linear Systems of Equations","body":"Without graphing, determine the number of solutions and then classify the system of equations: image1\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Solve Systems of Equations by Graphing","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6ae2sys18a","stepAnswer":["No solution"],"problemType":"MultipleChoice","stepTitle":"Determine number of solutions.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$1$$ solution","No solution","Infinitely many"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys18a-h1","type":"hint","dependencies":[],"title":"Compare Slopes and Intercepts","text":"We will compare the slopes and intercepts of the two lines: image1","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys18a-h2","type":"hint","dependencies":["a3d6ae2sys18a-h1"],"title":"Slope-Intercept Form","text":"The first equation is already in slope-intercept form: $$y=\\\\frac{2}{3} x+1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys18a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{3}$$"],"dependencies":["a3d6ae2sys18a-h2"],"title":"Identify the Slope","text":"What is $$m$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys18a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a3d6ae2sys18a-h3"],"title":"Identify y-intercept","text":"What is $$b$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys18a-h5","type":"hint","dependencies":["a3d6ae2sys18a-h4"],"title":"Slope-Intercept Form","text":"Write the second equation in slope-intercept form:\\\\n$$2x-3y=7$$\\\\n$$-3y=-2x+7$$\\\\n$$\\\\frac{-3y}{-3}=\\\\frac{\\\\left(-2x+7\\\\right)}{-3}$$\\\\n$$y=\\\\frac{2}{3} x-\\\\frac{7}{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys18a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{3}$$"],"dependencies":["a3d6ae2sys18a-h5"],"title":"Identify the Slope","text":"What is $$m$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys18a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-7}{3}$$"],"dependencies":["a3d6ae2sys18a-h6"],"title":"Identify y-intercept","text":"What is $$b$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys18a-h8","type":"hint","dependencies":["a3d6ae2sys18a-h3","a3d6ae2sys18a-h4","a3d6ae2sys18a-h6","a3d6ae2sys18a-h7"],"title":"Parallel Lines","text":"Since the slopes are the same and y-intercepts are different, the lines are parallel.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys18a-h9","type":"hint","dependencies":["a3d6ae2sys18a-h8"],"title":"Parallel Lines","text":"A system of equations whose graphs are parallel lines has no solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a3d6ae2sys18b","stepAnswer":["Inconsistent & Independent"],"problemType":"MultipleChoice","stepTitle":"Classify the system of equations.","stepBody":"","answerType":"string","variabilization":{},"choices":["Consistent & Independent","Inconsistent & Independent","Consistent & Dependent"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys18b-h1","type":"hint","dependencies":[],"title":"Classifications","text":"A system of equations whose graphs are parallel lines has no solution and is inconsistent and independent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6ae2sys19","title":"Number of Solutions of Linear Systems of Equations","body":"Without graphing, determine the number of solutions and then classify the system of equations: image1\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Solve Systems of Equations by Graphing","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6ae2sys19a","stepAnswer":["$$1$$ solution"],"problemType":"MultipleChoice","stepTitle":"Determine number of solutions.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$1$$ solution","choices":["$$1$$ solution","No solution","Infinitely many"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys19a-h1","type":"hint","dependencies":[],"title":"Compare Slopes and Intercepts","text":"We will compare the slopes and intercepts of the two lines: image1","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys19a-h2","type":"hint","dependencies":["a3d6ae2sys19a-h1"],"title":"Slope-Intercept Form","text":"Write the first equation in slope-intercept form:\\\\n$$3x+4y=12$$\\\\n$$4y=-3x+12$$\\\\n$$\\\\frac{4y}{4}=\\\\frac{\\\\left(-3x+12\\\\right)}{4}$$\\\\n$$y=\\\\frac{-3}{4} x+3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys19a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-3}{4}$$"],"dependencies":["a3d6ae2sys19a-h2"],"title":"Identify the Slope","text":"What is $$m$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys19a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a3d6ae2sys19a-h3"],"title":"Identify y-intercept","text":"What is $$b$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys19a-h5","type":"hint","dependencies":["a3d6ae2sys19a-h4"],"title":"Slope-Intercept Form","text":"The second equation is already in slope-intercept form: $$y=-3x-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys19a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a3d6ae2sys19a-h5"],"title":"Identify the Slope","text":"What is $$m$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys19a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a3d6ae2sys19a-h6"],"title":"Identify y-intercept","text":"What is $$b$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys19a-h8","type":"hint","dependencies":["a3d6ae2sys19a-h3","a3d6ae2sys19a-h4","a3d6ae2sys19a-h6","a3d6ae2sys19a-h7"],"title":"Intersecting Lines","text":"Since the slopes are different, the lines intersect.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys19a-h9","type":"hint","dependencies":["a3d6ae2sys19a-h8"],"title":"Intersecting Lines","text":"A system of equations whose graphs are intersect has $$1$$ solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a3d6ae2sys19b","stepAnswer":["Consistent & Independent"],"problemType":"MultipleChoice","stepTitle":"Classify the system of equations.","stepBody":"","answerType":"string","variabilization":{},"choices":["Consistent & Independent","Inconsistent & Independent","Consistent & Dependent"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys19b-h1","type":"hint","dependencies":[],"title":"Classifications","text":"A system of equations whose graphs are intersect has $$1$$ solution and is consistent and independent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6ae2sys2","title":"Solutions of a System of Equations","body":"Determine if the following points are solutions to the given system of equation:\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Solve Systems of Equations by Graphing","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6ae2sys2a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$(3,1)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys2a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitue $$x=3$$ and $$y=1$$ into both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys2a-h2","type":"hint","dependencies":["a3d6ae2sys2a-h1"],"title":"Substitute into First Equation","text":"$$2x-6y=0$$\\\\n$$2\\\\times3-6\\\\times1=0$$\\\\n$$0=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys2a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a3d6ae2sys2a-h2"],"title":"Substitute into First Equation","text":"Is the equation above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a3d6ae2sys2a-h4","type":"hint","dependencies":["a3d6ae2sys2a-h3"],"title":"Solution to First Equation","text":"Therefore, $$(3,1)$$ satisfies the first equation, but it must also safisfy the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys2a-h5","type":"hint","dependencies":["a3d6ae2sys2a-h4"],"title":"Substitute into Second Equation","text":"$$3x-4y=5$$\\\\n$$3\\\\times3-4\\\\times1=5$$\\\\n$$5=5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys2a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a3d6ae2sys2a-h5"],"title":"Substitute into Second Equation","text":"Is the equation above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a3d6ae2sys2a-h7","type":"hint","dependencies":["a3d6ae2sys2a-h6"],"title":"Solution to Second Equation","text":"Therefore, $$(3,1)$$ does satisfies the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys2a-h8","type":"hint","dependencies":["a3d6ae2sys2a-h7"],"title":"Solutions of a System of Equations","text":"$$(3,1)$$ does make both equations true. $$(3,1)$$ is a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a3d6ae2sys2b","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$(-3,4)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys2b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitue $$x=-3$$ and $$y=4$$ into both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys2b-h2","type":"hint","dependencies":["a3d6ae2sys2b-h1"],"title":"Substitute into First Equation","text":"$$2x-6y=0$$\\\\n$$2\\\\times-3-6\\\\times4=0$$\\\\n$$-30=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys2b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["FALSE"],"dependencies":["a3d6ae2sys2b-h2"],"title":"Substitute into First Equation","text":"Is the equation above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a3d6ae2sys2b-h4","type":"hint","dependencies":["a3d6ae2sys2b-h3"],"title":"Solution to First Equation","text":"Therefore, $$(-3,4)$$ does not satisfies the first equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys2b-h5","type":"hint","dependencies":["a3d6ae2sys2b-h4"],"title":"Solutions of a System of Equations","text":"$$(-3,4)$$ does not make both equations true. $$(-3,4)$$ is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6ae2sys20","title":"Number of Solutions of Linear Systems of Equations","body":"Without graphing, determine the number of solutions and then classify the system of equations: image1\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Solve Systems of Equations by Graphing","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6ae2sys20a","stepAnswer":["$$1$$ solution"],"problemType":"MultipleChoice","stepTitle":"Determine number of solutions.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$1$$ solution","choices":["$$1$$ solution","No solution","Infinitely many"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys20a-h1","type":"hint","dependencies":[],"title":"Compare Slopes and Intercepts","text":"We will compare the slopes and intercepts of the two lines: image1","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys20a-h2","type":"hint","dependencies":["a3d6ae2sys20a-h1"],"title":"Slope-Intercept Form","text":"Write the first equation in slope-intercept form:\\\\n$$4x+2y=10$$\\\\n$$2y=-4x+10$$\\\\n$$\\\\frac{2y}{2}=\\\\frac{\\\\left(-4x+10\\\\right)}{2}$$\\\\n$$y=-2x+5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys20a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a3d6ae2sys20a-h2"],"title":"Identify the Slope","text":"What is $$m$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys20a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a3d6ae2sys20a-h3"],"title":"Identify y-intercept","text":"What is $$b$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys20a-h5","type":"hint","dependencies":["a3d6ae2sys20a-h4"],"title":"Slope-Intercept Form","text":"Write the second equation in slope-intercept form:\\\\n$$4x-2y=-6$$\\\\n$$-2y=-4x-6$$\\\\n$$\\\\frac{-2y}{-2}=\\\\frac{\\\\left(-4x-6\\\\right)}{-2}$$\\\\n$$y=2x+3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys20a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a3d6ae2sys20a-h5"],"title":"Identify the Slope","text":"What is $$m$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys20a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a3d6ae2sys20a-h6"],"title":"Identify y-intercept","text":"What is $$b$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys20a-h8","type":"hint","dependencies":["a3d6ae2sys20a-h3","a3d6ae2sys20a-h4","a3d6ae2sys20a-h6","a3d6ae2sys20a-h7"],"title":"Intersecting Lines","text":"Since the slopes are different, the lines intersect.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys20a-h9","type":"hint","dependencies":["a3d6ae2sys20a-h8"],"title":"Intersecting Lines","text":"A system of equations whose graphs are intersect has $$1$$ solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a3d6ae2sys20b","stepAnswer":["Consistent & Independent"],"problemType":"MultipleChoice","stepTitle":"Classify the system of equations.","stepBody":"","answerType":"string","variabilization":{},"choices":["Consistent & Independent","Inconsistent & Independent","Consistent & Dependent"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys20b-h1","type":"hint","dependencies":[],"title":"Classifications","text":"A system of equations whose graphs are intersect has $$1$$ solution and is consistent and independent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6ae2sys21","title":"Number of Solutions of Linear Systems of Equations","body":"Without graphing, determine the number of solutions and then classify the system of equations: image1\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Solve Systems of Equations by Graphing","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6ae2sys21a","stepAnswer":["$$1$$ solution"],"problemType":"MultipleChoice","stepTitle":"Determine number of solutions.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$1$$ solution","choices":["$$1$$ solution","No solution","Infinitely many"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys21a-h1","type":"hint","dependencies":[],"title":"Compare Slopes and Intercepts","text":"We will compare the slopes and intercepts of the two lines: image1","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys21a-h2","type":"hint","dependencies":["a3d6ae2sys21a-h1"],"title":"Slope-Intercept Form","text":"Write the first equation in slope-intercept form:\\\\n$$5x+3y=4$$\\\\n$$3y=-5x+4$$\\\\n$$\\\\frac{3y}{3}=\\\\frac{\\\\left(-5x+4\\\\right)}{3}$$\\\\n$$y=\\\\frac{-5}{3} x+\\\\frac{4}{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys21a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-5}{3}$$"],"dependencies":["a3d6ae2sys21a-h2"],"title":"Identify the Slope","text":"What is $$m$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys21a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{4}{3}$$"],"dependencies":["a3d6ae2sys21a-h3"],"title":"Identify y-intercept","text":"What is $$b$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys21a-h5","type":"hint","dependencies":["a3d6ae2sys21a-h4"],"title":"Slope-Intercept Form","text":"Write the second equation in slope-intercept form:\\\\n$$2x-3y=5$$\\\\n$$-3y=-2x+5$$\\\\n$$\\\\frac{-3y}{-3}=\\\\frac{\\\\left(-2x+5\\\\right)}{-3}$$\\\\n$$y=\\\\frac{2}{3} x-\\\\frac{5}{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys21a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{3}$$"],"dependencies":["a3d6ae2sys21a-h5"],"title":"Identify the Slope","text":"What is $$m$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys21a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-5}{3}$$"],"dependencies":["a3d6ae2sys21a-h6"],"title":"Identify y-intercept","text":"What is $$b$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys21a-h8","type":"hint","dependencies":["a3d6ae2sys21a-h3","a3d6ae2sys21a-h4","a3d6ae2sys21a-h6","a3d6ae2sys21a-h7"],"title":"Intersecting Lines","text":"Since the slopes are different, the lines intersect.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys21a-h9","type":"hint","dependencies":["a3d6ae2sys21a-h8"],"title":"Intersecting Lines","text":"A system of equations whose graphs are intersect has $$1$$ solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a3d6ae2sys21b","stepAnswer":["Consistent & Independent"],"problemType":"MultipleChoice","stepTitle":"Classify the system of equations.","stepBody":"","answerType":"string","variabilization":{},"choices":["Consistent & Independent","Inconsistent & Independent","Consistent & Dependent"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys21b-h1","type":"hint","dependencies":[],"title":"Classifications","text":"A system of equations whose graphs are intersect has $$1$$ solution and is consistent and independent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6ae2sys22","title":"Number of Solutions of Linear Systems of Equations","body":"Without graphing, determine the number of solutions and then classify the system of equations: image1\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Solve Systems of Equations by Graphing","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6ae2sys22a","stepAnswer":["Infinitely many"],"problemType":"MultipleChoice","stepTitle":"Determine number of solutions.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$1$$ solution","No solution","Infinitely many"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys22a-h1","type":"hint","dependencies":[],"title":"Compare Slopes and Intercepts","text":"We will compare the slopes and intercepts of the two lines: image1","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys22a-h2","type":"hint","dependencies":["a3d6ae2sys22a-h1"],"title":"Slope-Intercept Form","text":"The first equation is already in slope-intercept form: $$y=\\\\frac{-1}{2} x+5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys22a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{2}$$"],"dependencies":["a3d6ae2sys22a-h2"],"title":"Identify the Slope","text":"What is $$m$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys22a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a3d6ae2sys22a-h3"],"title":"Identify y-intercept","text":"What is $$b$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys22a-h5","type":"hint","dependencies":["a3d6ae2sys22a-h4"],"title":"Slope-Intercept Form","text":"Write the second equation in slope-intercept form:\\\\n$$x+2y=10$$\\\\n$$2y=-x+10$$\\\\n$$\\\\frac{2y}{2}=\\\\frac{\\\\left(-x+10\\\\right)}{2}$$\\\\n$$y=\\\\frac{-1}{2} x+5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys22a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{2}$$"],"dependencies":["a3d6ae2sys22a-h5"],"title":"Identify the Slope","text":"What is $$m$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys22a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a3d6ae2sys22a-h6"],"title":"Identify y-intercept","text":"What is $$b$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys22a-h8","type":"hint","dependencies":["a3d6ae2sys22a-h3","a3d6ae2sys22a-h4","a3d6ae2sys22a-h6","a3d6ae2sys22a-h7"],"title":"Same Lines","text":"Since the slopes and y-intercept are the same, the lines are the same.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys22a-h9","type":"hint","dependencies":["a3d6ae2sys22a-h8"],"title":"Same Lines","text":"A system of equations whose graphs are coincident lines has infinitely many solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a3d6ae2sys22b","stepAnswer":["Consistent & Dependent"],"problemType":"MultipleChoice","stepTitle":"Classify the system of equations.","stepBody":"","answerType":"string","variabilization":{},"choices":["Consistent & Independent","Inconsistent & Independent","Consistent & Dependent"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys22b-h1","type":"hint","dependencies":[],"title":"Classifications","text":"A system of equations whose graphs are coincident lines has infinitely many solutions and is consistent and dependent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6ae2sys23","title":"Number of Solutions of Linear Systems of Equations","body":"Without graphing, determine the number of solutions and then classify the system of equations: image1\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Solve Systems of Equations by Graphing","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6ae2sys23a","stepAnswer":["Infinitely many"],"problemType":"MultipleChoice","stepTitle":"Determine number of solutions.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$1$$ solution","No solution","Infinitely many"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys23a-h1","type":"hint","dependencies":[],"title":"Compare Slopes and Intercepts","text":"We will compare the slopes and intercepts of the two lines: image1","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys23a-h2","type":"hint","dependencies":["a3d6ae2sys23a-h1"],"title":"Slope-Intercept Form","text":"The first equation is already in slope-intercept form: $$y=x+1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys23a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a3d6ae2sys23a-h2"],"title":"Identify the Slope","text":"What is $$m$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys23a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a3d6ae2sys23a-h3"],"title":"Identify y-intercept","text":"What is $$b$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys23a-h5","type":"hint","dependencies":["a3d6ae2sys23a-h4"],"title":"Slope-Intercept Form","text":"Write the second equation in slope-intercept form:\\\\n$$-x+y=1$$\\\\n$$y=x+1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys23a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a3d6ae2sys23a-h5"],"title":"Identify the Slope","text":"What is $$m$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys23a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a3d6ae2sys23a-h6"],"title":"Identify y-intercept","text":"What is $$b$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys23a-h8","type":"hint","dependencies":["a3d6ae2sys23a-h3","a3d6ae2sys23a-h4","a3d6ae2sys23a-h6","a3d6ae2sys23a-h7"],"title":"Same Lines","text":"Since the slopes and y-intercept are the same, the lines are the same.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys23a-h9","type":"hint","dependencies":["a3d6ae2sys23a-h8"],"title":"Same Lines","text":"A system of equations whose graphs are coincident lines has infinitely many solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a3d6ae2sys23b","stepAnswer":["Consistent & Dependent"],"problemType":"MultipleChoice","stepTitle":"Classify the system of equations.","stepBody":"","answerType":"string","variabilization":{},"choices":["Consistent & Independent","Inconsistent & Independent","Consistent & Dependent"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys23b-h1","type":"hint","dependencies":[],"title":"Classifications","text":"A system of equations whose graphs are coincident lines has infinitely many solutions and is consistent and dependent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6ae2sys24","title":"Number of Solutions of Linear Systems of Equations","body":"Without graphing, determine the number of solutions and then classify the system of equations: image1\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Solve Systems of Equations by Graphing","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6ae2sys24a","stepAnswer":["Infinitely many"],"problemType":"MultipleChoice","stepTitle":"Determine number of solutions.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$1$$ solution","No solution","Infinitely many"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys24a-h1","type":"hint","dependencies":[],"title":"Compare Slopes and Intercepts","text":"We will compare the slopes and intercepts of the two lines: image1","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys24a-h2","type":"hint","dependencies":["a3d6ae2sys24a-h1"],"title":"Slope-Intercept Form","text":"The first equation is already in slope-intercept form: $$y=2x+3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys24a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a3d6ae2sys24a-h2"],"title":"Identify the Slope","text":"What is $$m$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys24a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a3d6ae2sys24a-h3"],"title":"Identify y-intercept","text":"What is $$b$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys24a-h5","type":"hint","dependencies":["a3d6ae2sys24a-h4"],"title":"Slope-Intercept Form","text":"Write the second equation in slope-intercept form:\\\\n$$2x-y=-3$$\\\\n$$-y=-2x-3$$\\\\n$$\\\\frac{-y}{-1}=\\\\frac{\\\\left(-2x-3\\\\right)}{-1}$$\\\\n$$y=2x+3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys24a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a3d6ae2sys24a-h5"],"title":"Identify the Slope","text":"What is $$m$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys24a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a3d6ae2sys24a-h6"],"title":"Identify y-intercept","text":"What is $$b$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys24a-h8","type":"hint","dependencies":["a3d6ae2sys24a-h3","a3d6ae2sys24a-h4","a3d6ae2sys24a-h6","a3d6ae2sys24a-h7"],"title":"Same Lines","text":"Since the slopes and y-intercept are the same, the lines are the same.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys24a-h9","type":"hint","dependencies":["a3d6ae2sys24a-h8"],"title":"Same Lines","text":"A system of equations whose graphs are coincident lines has infinitely many solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a3d6ae2sys24b","stepAnswer":["Consistent & Dependent"],"problemType":"MultipleChoice","stepTitle":"Classify the system of equations.","stepBody":"","answerType":"string","variabilization":{},"choices":["Consistent & Independent","Inconsistent & Independent","Consistent & Dependent"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys24b-h1","type":"hint","dependencies":[],"title":"Classifications","text":"A system of equations whose graphs are coincident lines has infinitely many solutions and is consistent and dependent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6ae2sys25","title":"Number of Solutions of Linear Systems of Equations","body":"Without graphing, determine the number of solutions and then classify the system of equations: image1\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Solve Systems of Equations by Graphing","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6ae2sys25a","stepAnswer":["Infinitely many"],"problemType":"MultipleChoice","stepTitle":"Determine number of solutions.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$1$$ solution","No solution","Infinitely many"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys25a-h1","type":"hint","dependencies":[],"title":"Compare Slopes and Intercepts","text":"We will compare the slopes and intercepts of the two lines: image1","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys25a-h2","type":"hint","dependencies":["a3d6ae2sys25a-h1"],"title":"Slope-Intercept Form","text":"Write the first equation in slope-intercept form:\\\\n$$5x-2y=10$$\\\\n$$-2y=-5x+10$$\\\\n$$\\\\frac{-2y}{-2}=\\\\frac{\\\\left(-5x+10\\\\right)}{-2}$$\\\\n$$y=\\\\frac{5}{2} x-5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys25a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{2}$$"],"dependencies":["a3d6ae2sys25a-h2"],"title":"Identify the Slope","text":"What is $$m$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys25a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["a3d6ae2sys25a-h3"],"title":"Identify y-intercept","text":"What is $$b$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys25a-h5","type":"hint","dependencies":["a3d6ae2sys25a-h4"],"title":"Slope-Intercept Form","text":"The second equation is already in slope-intercept form: $$y=\\\\frac{5}{2} x-5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys25a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{2}$$"],"dependencies":["a3d6ae2sys25a-h5"],"title":"Identify the Slope","text":"What is $$m$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys25a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["a3d6ae2sys25a-h6"],"title":"Identify y-intercept","text":"What is $$b$$ in the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys25a-h8","type":"hint","dependencies":["a3d6ae2sys25a-h3","a3d6ae2sys25a-h4","a3d6ae2sys25a-h6","a3d6ae2sys25a-h7"],"title":"Same Lines","text":"Since the slopes and y-intercept are the same, the lines are the same.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys25a-h9","type":"hint","dependencies":["a3d6ae2sys25a-h8"],"title":"Same Lines","text":"A system of equations whose graphs are coincident lines has infinitely many solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a3d6ae2sys25b","stepAnswer":["Consistent & Dependent"],"problemType":"MultipleChoice","stepTitle":"Classify the system of equations.","stepBody":"","answerType":"string","variabilization":{},"choices":["Consistent & Independent","Inconsistent & Independent","Consistent & Dependent"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys25b-h1","type":"hint","dependencies":[],"title":"Classifications","text":"A system of equations whose graphs are coincident lines has infinitely many solutions and is consistent and dependent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6ae2sys26","title":"Graphing Linear Equations","body":"Solve the system by graphing.\\\\n##figure3.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Solve Systems of Equations by Graphing","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6ae2sys26a","stepAnswer":["$$(4,-1)$$"],"problemType":"MultipleChoice","stepTitle":"image1","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(4,-1)$$","choices":["$$(4,-1)$$","$$(4,1)$$","$$(-1,4)$$"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys26a-h1","type":"hint","dependencies":[],"title":"Graph the First Equation","text":"To graph the first line, write equation in slope-intercept form:\\\\n$$2x+y=7$$\\\\n$$y=-2x+7$$\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys26a-h2","type":"hint","dependencies":["a3d6ae2sys26a-h1"],"title":"Plot the y-intercept","text":"Plot $$(0,7)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys26a-h3","type":"hint","dependencies":["a3d6ae2sys26a-h2"],"title":"Identify the rise and the run","text":"Use the slope formula $$m=\\\\frac{rise}{run}$$:\\\\n$$m=-2$$\\\\n$$\\\\frac{rise}{run}=\\\\frac{-2}{1}$$\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys26a-h4","type":"hint","dependencies":["a3d6ae2sys26a-h3"],"title":"Plot line","text":"Start at $$(0,7)$$ and count the rise and the run. Down $$2$$, right $$1$$. Mark the second point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys26a-h5","type":"hint","dependencies":["a3d6ae2sys26a-h4"],"title":"Plot line","text":"Connect the two points with a line: image2","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys26a-h6","type":"hint","dependencies":["a3d6ae2sys26a-h5"],"title":"Graph the Second Equation","text":"To graph the second line, write equation in slope-intercept form:\\\\n$$x-2y=6$$\\\\n$$-2y=-x+6$$\\\\n$$\\\\frac{-2y}{-2}=\\\\frac{\\\\left(-x+6\\\\right)}{-2}$$\\\\n$$y=\\\\frac{1}{2} x-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys26a-h7","type":"hint","dependencies":["a3d6ae2sys26a-h6"],"title":"Plot the y-intercept","text":"Plot $$(0,-3)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys26a-h8","type":"hint","dependencies":["a3d6ae2sys26a-h7"],"title":"Identify the rise and the run","text":"Use the slope formula $$m=\\\\frac{rise}{run}$$:\\\\n$$m=\\\\frac{1}{2}$$\\\\n$$\\\\frac{rise}{run}=\\\\frac{1}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys26a-h9","type":"hint","dependencies":["a3d6ae2sys26a-h7","a3d6ae2sys26a-h8"],"title":"Plot line","text":"Start at $$(0,-3)$$ and count the rise and the run. Up $$1$$, right $$2$$. Mark the second point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys26a-h10","type":"hint","dependencies":["a3d6ae2sys26a-h9"],"title":"Plot line","text":"Connect the two points with a line: image3","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys26a-h11","type":"hint","dependencies":["a3d6ae2sys26a-h10"],"title":"Determine type of line","text":"The lines intersect.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys26a-h12","type":"hint","dependencies":["a3d6ae2sys26a-h11"],"title":"Solution to system","text":"The lines intersect at $$(4,-1)$$. Therefore the solution is $$(4,-1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6ae2sys27","title":"Graphing Linear Equations","body":"Solve the system by graphing.\\\\n##figure3.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Solve Systems of Equations by Graphing","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6ae2sys27a","stepAnswer":["$$(1,3)$$"],"problemType":"MultipleChoice","stepTitle":"image1","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(1,3)$$","choices":["$$(3,1)$$","$$(1,3)$$","$$(-1,-3)$$"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys27a-h1","type":"hint","dependencies":[],"title":"Find the slope and y-intercept of the first equation","text":"image2\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys27a-h2","type":"hint","dependencies":["a3d6ae2sys27a-h1"],"title":"Find the slope and y-intercept of the second equation","text":"image3","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys27a-h3","type":"hint","dependencies":["a3d6ae2sys27a-h2"],"title":"Graph the two lines","text":"image4\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys27a-h4","type":"hint","dependencies":["a3d6ae2sys27a-h3"],"title":"Determine the point of intersection","text":"The lines intersect at $$(1,3)$$. Therefore the solution is $$(1,3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6ae2sys28","title":"Graphing Linear Equations","body":"Solve the system by graphing\\\\n##figure2.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Solve Systems of Equations by Graphing","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6ae2sys28a","stepAnswer":["$$(-1,2)$$"],"problemType":"MultipleChoice","stepTitle":"image1","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-1,2)$$","choices":["$$(2,1)$$","$$(1,-2)$$","$$(-1,2)$$"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys28a-h1","type":"hint","dependencies":[],"title":"Solve the first equation for $$y$$.","text":"$$3x+y=-1$$\\\\n$$y=-3x-1$$\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys28a-h2","type":"hint","dependencies":["a3d6ae2sys28a-h1"],"title":"Find the slope and y-intercept.","text":"$$m=-3$$\\\\n$$b=-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys28a-h3","type":"hint","dependencies":["a3d6ae2sys28a-h2"],"title":"Solve the second equation for $$y$$.","text":"$$2x+y=0$$\\\\n$$y=-2x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys28a-h4","type":"hint","dependencies":["a3d6ae2sys28a-h3"],"title":"Find the slope and y-intercept.","text":"$$m=-2$$\\\\n$$b=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys28a-h5","type":"hint","dependencies":["a3d6ae2sys28a-h2","a3d6ae2sys28a-h4"],"title":"Graph the lines","text":"image2","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys28a-h6","type":"hint","dependencies":["a3d6ae2sys28a-h5"],"title":"Determine the point of intersection.","text":"The lines intersect at $$(-1,2)$$. Therefore the solution is $$(-1,2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6ae2sys29","title":"Graphing Linear Equations","body":"Solve the system by graphing.\\\\n##figure2.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Solve Systems of Equations by Graphing","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6ae2sys29a","stepAnswer":["$$(3,-1)$$"],"problemType":"MultipleChoice","stepTitle":"image1","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(3,-1)$$","choices":["$$(3,-1)$$","$$(1,-3)$$","$$(-1,3)$$"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys29a-h1","type":"hint","dependencies":[],"title":"Graph using intercepts","text":"We will find the $$x-$$ and y-intercepts of both equations and use them to graph the lines.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys29a-h2","type":"hint","dependencies":["a3d6ae2sys29a-h1"],"title":"x-intercept for first equation","text":"To find the x-intercept, let $$x=0$$ and solve for y:\\\\n$$x+y=2$$\\\\n$$0+y=2$$\\\\n$$y=2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys29a-h3","type":"hint","dependencies":["a3d6ae2sys29a-h2"],"title":"y-intercept for first equation","text":"To find the y-intercept, let $$y=0$$ and solve for x:\\\\n$$x+y=2$$\\\\n$$x+0=2$$\\\\n$$x=2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys29a-h4","type":"hint","dependencies":["a3d6ae2sys29a-h3"],"title":"Intercepts for first equation","text":"For equation $$x+y=2$$, plot points $$(0,2)$$ and $$(2,0)$$ and connect the lines.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys29a-h5","type":"hint","dependencies":["a3d6ae2sys29a-h4"],"title":"x-intercept for second equation","text":"To find the x-intercept, let $$x=0$$ and solve for y:\\\\n$$x-y=4$$\\\\n$$0-y=4$$\\\\n$$-y=4$$\\\\n$$y=-4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys29a-h6","type":"hint","dependencies":["a3d6ae2sys29a-h5"],"title":"y-intercept for second equation","text":"To find the y-intercept, let $$y=0$$ and solve for x:\\\\n$$x-y=4$$\\\\n$$x-0=4$$\\\\n$$x=4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys29a-h7","type":"hint","dependencies":["a3d6ae2sys29a-h6"],"title":"Intercepts for second equation","text":"For equation $$x-y=4$$, plot points $$(0,-4)$$ and $$(4,0)$$ and connect the lines.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys29a-h8","type":"hint","dependencies":["a3d6ae2sys29a-h4","a3d6ae2sys29a-h7"],"title":"Graph the line","text":"image2","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys29a-h9","type":"hint","dependencies":["a3d6ae2sys29a-h8"],"title":"Determine the point of intersection.","text":"The lines intersect at $$(3,-1)$$. Therefore the solution is $$(3,-1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6ae2sys3","title":"Solutions of a System of Equations","body":"Determine if the following points are solutions to the given system of equation:\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Solve Systems of Equations by Graphing","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6ae2sys3a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$(1,2)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys3a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitue $$x=1$$ and $$y=2$$ into both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys3a-h2","type":"hint","dependencies":["a3d6ae2sys3a-h1"],"title":"Substitute into First Equation","text":"$$7x-4y=-1$$\\\\n$$7\\\\times1-4\\\\times2=-1$$\\\\n$$-1=-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys3a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a3d6ae2sys3a-h2"],"title":"Substitute into First Equation","text":"Is the equation above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a3d6ae2sys3a-h4","type":"hint","dependencies":["a3d6ae2sys3a-h3"],"title":"Solution to First Equation","text":"Therefore, $$(1,2)$$ satisfies the first equation, but it must also safisfy the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys3a-h5","type":"hint","dependencies":["a3d6ae2sys3a-h4"],"title":"Substitute into Second Equation","text":"$$-3x-2y=1$$\\\\n$$-3\\\\times1-2\\\\times2=-1$$\\\\n$$-7=-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys3a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["FALSE"],"dependencies":["a3d6ae2sys3a-h5"],"title":"Substitute into Second Equation","text":"Is the equation above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a3d6ae2sys3a-h7","type":"hint","dependencies":["a3d6ae2sys3a-h6"],"title":"Solution to Second Equation","text":"Therefore, $$(3,1)$$ does not satisfies the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys3a-h8","type":"hint","dependencies":["a3d6ae2sys3a-h7"],"title":"Solutions of a System of Equations","text":"$$(1,2)$$ does not make both equations true. $$(1,2)$$ is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a3d6ae2sys3b","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$(1,-2)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys3b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitue $$x=1$$ and $$y=-2$$ into both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys3b-h2","type":"hint","dependencies":["a3d6ae2sys3b-h1"],"title":"Substitute into First Equation","text":"$$7x-4y=-1$$\\\\n$$7\\\\times1-4\\\\times-2=-1$$\\\\n$$15=-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys3b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["FALSE"],"dependencies":["a3d6ae2sys3b-h2"],"title":"Substitute into First Equation","text":"Is the equation above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a3d6ae2sys3b-h4","type":"hint","dependencies":["a3d6ae2sys3b-h3"],"title":"Solution to First Equation","text":"Therefore, $$(1,-2)$$ does not satisfies the first equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys3b-h5","type":"hint","dependencies":["a3d6ae2sys3b-h4"],"title":"Solutions of a System of Equations","text":"$$(1,-2)$$ does not make both equations true. $$(1,-2)$$ is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6ae2sys30","title":"Graphing Linear Equations","body":"Solve the system by graphing\\\\n##figure2.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Solve Systems of Equations by Graphing","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6ae2sys30a","stepAnswer":["$$(-3,6)$$"],"problemType":"MultipleChoice","stepTitle":"Graph the given image","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-3,6)$$","choices":["$$(3,6)$$","$$(6,-3)$$","$$(-3,6)$$"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys30a-h1","type":"hint","dependencies":[],"title":"First Equation","text":"We know the first equation represents a horizontal line whose y-intercept is 6: $$y=6$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys30a-h2","type":"hint","dependencies":["a3d6ae2sys30a-h1"],"title":"Graph using intercepts","text":"We will find the $$x-$$ and y-intercepts of the second equation to graph it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys30a-h3","type":"hint","dependencies":["a3d6ae2sys30a-h2"],"title":"x-intercept for first equation","text":"To find the x-intercept, let $$x=0$$ and solve for y:\\\\n$$2x+3y=12$$\\\\n$$2\\\\times0+3y=12$$\\\\n$$3y=12$$\\\\n$$y=4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys30a-h4","type":"hint","dependencies":["a3d6ae2sys30a-h3"],"title":"y-intercept for first equation","text":"To find the y-intercept, let $$y=0$$ and solve for x:\\\\n$$2x+3y=12$$\\\\n$$2x+3\\\\times0=12$$\\\\n$$2x=12$$\\\\n$$x=6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys30a-h5","type":"hint","dependencies":["a3d6ae2sys30a-h3","a3d6ae2sys30a-h4"],"title":"Intercepts for first equation","text":"For equation $$x+y=2$$, plot points $$(0,4)$$ and $$(6,0)$$ and connect the lines.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys30a-h6","type":"hint","dependencies":["a3d6ae2sys30a-h5"],"title":"Graph the line","text":"Look at the second image.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys30a-h7","type":"hint","dependencies":["a3d6ae2sys30a-h6"],"title":"Determine the point of intersection.","text":"The lines intersect at $$(-3,6)$$. Therefore the solution is $$(-3,6)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6ae2sys4","title":"Solutions of a System of Equations","body":"Determine if the following points are solutions to the given system of equation:\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Solve Systems of Equations by Graphing","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6ae2sys4a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$(4,-3)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys4a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitue $$x=4$$ and $$y=-3$$ into both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys4a-h2","type":"hint","dependencies":["a3d6ae2sys4a-h1"],"title":"Substitute into First Equation","text":"$$2x+y=5$$\\\\n$$2\\\\times4-3=5$$\\\\n$$5=5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys4a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a3d6ae2sys4a-h2"],"title":"Substitute into First Equation","text":"Is the equation above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a3d6ae2sys4a-h4","type":"hint","dependencies":["a3d6ae2sys4a-h3"],"title":"Solution to First Equation","text":"Therefore, $$(4,-3)$$ satisfies the first equation, but it must also safisfy the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys4a-h5","type":"hint","dependencies":["a3d6ae2sys4a-h4"],"title":"Substitute into Second Equation","text":"$$x+y=1$$\\\\n$$4-3=1$$\\\\n$$1=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys4a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a3d6ae2sys4a-h5"],"title":"Substitute into Second Equation","text":"Is the equation above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a3d6ae2sys4a-h7","type":"hint","dependencies":["a3d6ae2sys4a-h6"],"title":"Solution to Second Equation","text":"Therefore, $$(4,-3)$$ does satisfies the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys4a-h8","type":"hint","dependencies":["a3d6ae2sys4a-h7"],"title":"Solutions of a System of Equations","text":"$$(4,-3)$$ does make both equations true. $$(4,-3)$$ is a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a3d6ae2sys4b","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$(2,0)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys4b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitue $$x=2$$ and $$y=0$$ into both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys4b-h2","type":"hint","dependencies":["a3d6ae2sys4b-h1"],"title":"Substitute into First Equation","text":"$$2x+y=5$$\\\\n$$2\\\\times2+0=5$$\\\\n$$4=5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys4b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["FALSE"],"dependencies":["a3d6ae2sys4b-h2"],"title":"Substitute into First Equation","text":"Is the equation above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a3d6ae2sys4b-h4","type":"hint","dependencies":["a3d6ae2sys4b-h3"],"title":"Solution to First Equation","text":"Therefore, $$(2,0)$$ does not satisfies the first equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys4b-h5","type":"hint","dependencies":["a3d6ae2sys4b-h4"],"title":"Solutions of a System of Equations","text":"$$(2,0)$$ does not make both equations true. $$(2,0)$$ is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6ae2sys5","title":"Solutions of a System of Equations","body":"Determine if the following points are solutions to the given system of equation:\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Solve Systems of Equations by Graphing","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6ae2sys5a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$(-5,-7)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys5a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitue $$x=-5$$ and $$y=-7$$ into both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys5a-h2","type":"hint","dependencies":["a3d6ae2sys5a-h1"],"title":"Substitute into First Equation","text":"$$-3x+y=8$$\\\\n$$-3\\\\times-5-7=8$$\\\\n$$8=8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys5a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a3d6ae2sys5a-h2"],"title":"Substitute into First Equation","text":"Is the equation above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a3d6ae2sys5a-h4","type":"hint","dependencies":["a3d6ae2sys5a-h3"],"title":"Solution to First Equation","text":"Therefore, $$(-5,-7)$$ satisfies the first equation, but it must also safisfy the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys5a-h5","type":"hint","dependencies":["a3d6ae2sys5a-h4"],"title":"Substitute into Second Equation","text":"$$-x+2y=-9$$\\\\n$$-\\\\left(-5\\\\right)+2\\\\times-7=-9$$\\\\n$$-9=-9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys5a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a3d6ae2sys5a-h5"],"title":"Substitute into Second Equation","text":"Is the equation above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a3d6ae2sys5a-h7","type":"hint","dependencies":["a3d6ae2sys5a-h6"],"title":"Solution to Second Equation","text":"Therefore, $$(-5,-7)$$ does satisfies the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys5a-h8","type":"hint","dependencies":["a3d6ae2sys5a-h7"],"title":"Solutions of a System of Equations","text":"$$(-5,-7)$$ does make both equations true. $$(-5,-7)$$ is a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a3d6ae2sys5b","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$(-5,7)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys5b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitue $$x=-5$$ and $$y=7$$ into both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys5b-h2","type":"hint","dependencies":["a3d6ae2sys5b-h1"],"title":"Substitute into First Equation","text":"$$-3x+y=8$$\\\\n$$-3\\\\times-5+7=8$$\\\\n$$22=8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys5b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["FALSE"],"dependencies":["a3d6ae2sys5b-h2"],"title":"Substitute into First Equation","text":"Is the equation above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a3d6ae2sys5b-h4","type":"hint","dependencies":["a3d6ae2sys5b-h3"],"title":"Solution to First Equation","text":"Therefore, $$(-5,7)$$ does not satisfies the first equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys5b-h5","type":"hint","dependencies":["a3d6ae2sys5b-h4"],"title":"Solutions of a System of Equations","text":"$$(-5,7)$$ does not make both equations true. $$(-5,7)$$ is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6ae2sys6","title":"Solutions of a System of Equations","body":"Determine if the following points are solutions to the given system of equation:\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Solve Systems of Equations by Graphing","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6ae2sys6a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$(\\\\frac{8}{7},\\\\frac{6}{7})$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys6a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitue $$x=\\\\frac{8}{7}$$ and $$y=\\\\frac{6}{7}$$ into both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys6a-h2","type":"hint","dependencies":["a3d6ae2sys6a-h1"],"title":"Substitute into First Equation","text":"$$x+y=2$$\\\\n$$\\\\frac{8}{7}+\\\\frac{6}{7}=2$$\\\\n$$\\\\frac{14}{7}=2$$\\\\n$$2=2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys6a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a3d6ae2sys6a-h2"],"title":"Substitute into First Equation","text":"Is the equation above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a3d6ae2sys6a-h4","type":"hint","dependencies":["a3d6ae2sys6a-h3"],"title":"Solution to First Equation","text":"Therefore, $$(\\\\frac{8}{7},\\\\frac{6}{7})$$ satisfies the first equation, but it must also safisfy the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys6a-h5","type":"hint","dependencies":["a3d6ae2sys6a-h4"],"title":"Substitute into Second Equation","text":"$$y=\\\\frac{3}{4} x$$\\\\n$$\\\\frac{6}{7}=\\\\frac{8\\\\frac{3}{4}}{7}$$\\\\n$$\\\\frac{6}{7}=\\\\frac{6}{7}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys6a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a3d6ae2sys6a-h5"],"title":"Substitute into Second Equation","text":"Is the equation above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a3d6ae2sys6a-h7","type":"hint","dependencies":["a3d6ae2sys6a-h6"],"title":"Solution to Second Equation","text":"Therefore, $$(\\\\frac{8}{7},\\\\frac{6}{7})$$ does satisfies the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys6a-h8","type":"hint","dependencies":["a3d6ae2sys6a-h7"],"title":"Solutions of a System of Equations","text":"$$(\\\\frac{8}{7},\\\\frac{6}{7})$$ does make both equations true. $$(\\\\frac{8}{7},\\\\frac{6}{7})$$ is a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a3d6ae2sys6b","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$(1,\\\\frac{3}{4})$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys6b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitue $$x=1$$ and $$y=\\\\frac{3}{4}$$ into both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys6b-h2","type":"hint","dependencies":["a3d6ae2sys6b-h1"],"title":"Substitute into First Equation","text":"$$x+y=2$$\\\\n$$1+\\\\frac{3}{4}=2$$\\\\n$$\\\\frac{7}{4}=2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys6b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["FALSE"],"dependencies":["a3d6ae2sys6b-h2"],"title":"Substitute into First Equation","text":"Is the equation above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a3d6ae2sys6b-h4","type":"hint","dependencies":["a3d6ae2sys6b-h3"],"title":"Solution to First Equation","text":"Therefore, $$(1,\\\\frac{3}{4})$$ does not satisfies the first equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys6b-h5","type":"hint","dependencies":["a3d6ae2sys6b-h4"],"title":"Solutions of a System of Equations","text":"$$(1,\\\\frac{3}{4})$$ does not make both equations true. $$(1,\\\\frac{3}{4})$$ is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6ae2sys7","title":"Solutions of a System of Equations","body":"Determine if the following points are solutions to the given system of equation:\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Solve Systems of Equations by Graphing","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6ae2sys7a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$(\\\\frac{5}{7},\\\\frac{2}{7})$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys7a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitue $$x=\\\\frac{5}{7}$$ and $$y=\\\\frac{2}{7}$$ into both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys7a-h2","type":"hint","dependencies":["a3d6ae2sys7a-h1"],"title":"Substitute into First Equation","text":"$$x+y=1$$\\\\n$$\\\\frac{5}{7}+\\\\frac{2}{7}=1$$\\\\n$$\\\\frac{7}{7}=1$$\\\\n$$1=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys7a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a3d6ae2sys7a-h2"],"title":"Substitute into First Equation","text":"Is the equation above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a3d6ae2sys7a-h4","type":"hint","dependencies":["a3d6ae2sys7a-h3"],"title":"Solution to First Equation","text":"Therefore, $$(\\\\frac{5}{7},\\\\frac{2}{7})$$ satisfies the first equation, but it must also safisfy the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys7a-h5","type":"hint","dependencies":["a3d6ae2sys7a-h4"],"title":"Substitute into Second Equation","text":"$$y=\\\\frac{2}{5} x$$\\\\n$$\\\\frac{2}{7}=\\\\frac{5\\\\frac{2}{5}}{7}$$\\\\n$$\\\\frac{2}{7}=\\\\frac{2}{7}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys7a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a3d6ae2sys7a-h5"],"title":"Substitute into Second Equation","text":"Is the equation above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a3d6ae2sys7a-h7","type":"hint","dependencies":["a3d6ae2sys7a-h6"],"title":"Solution to Second Equation","text":"Therefore, $$(\\\\frac{5}{7},\\\\frac{2}{7})$$ does satisfies the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys7a-h8","type":"hint","dependencies":["a3d6ae2sys7a-h7"],"title":"Solutions of a System of Equations","text":"$$(\\\\frac{5}{7},\\\\frac{2}{7})$$ does make both equations true. $$(\\\\frac{5}{7},\\\\frac{2}{7})$$ is a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a3d6ae2sys7b","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$(5,2)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys7b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitue $$x=5$$ and $$y=2$$ into both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys7b-h2","type":"hint","dependencies":["a3d6ae2sys7b-h1"],"title":"Substitute into First Equation","text":"$$x+y=1$$\\\\n$$5+2=1$$\\\\n$$7=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys7b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["FALSE"],"dependencies":["a3d6ae2sys7b-h2"],"title":"Substitute into First Equation","text":"Is the equation above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a3d6ae2sys7b-h4","type":"hint","dependencies":["a3d6ae2sys7b-h3"],"title":"Solution to First Equation","text":"Therefore, $$(5,2)$$ does not satisfies the first equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys7b-h5","type":"hint","dependencies":["a3d6ae2sys7b-h4"],"title":"Solutions of a System of Equations","text":"$$(5,2)$$ does not make both equations true. $$(5,2)$$ is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6ae2sys8","title":"Solutions of a System of Equations","body":"Determine if the following points are solutions to the given system of equation:\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Solve Systems of Equations by Graphing","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6ae2sys8a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$(-10,4)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys8a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitue $$x=-10$$ and $$y=4$$ into both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys8a-h2","type":"hint","dependencies":["a3d6ae2sys8a-h1"],"title":"Substitute into First Equation","text":"$$x+5y=10$$\\\\n$$-10+5\\\\times4=10$$\\\\n$$10=10$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys8a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a3d6ae2sys8a-h2"],"title":"Substitute into First Equation","text":"Is the equation above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a3d6ae2sys8a-h4","type":"hint","dependencies":["a3d6ae2sys8a-h3"],"title":"Solution to First Equation","text":"Therefore, $$(-10,4)$$ satisfies the first equation, but it must also safisfy the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys8a-h5","type":"hint","dependencies":["a3d6ae2sys8a-h4"],"title":"Substitute into Second Equation","text":"$$y=\\\\frac{3}{5} x+1$$\\\\n$$4=-10\\\\frac{3}{5}+1$$\\\\n$$4=-5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys8a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["FALSE"],"dependencies":["a3d6ae2sys8a-h5"],"title":"Substitute into Second Equation","text":"Is the equation above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a3d6ae2sys8a-h7","type":"hint","dependencies":["a3d6ae2sys8a-h6"],"title":"Solution to Second Equation","text":"Therefore, $$(-10,4)$$ does not satisfies the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys8a-h8","type":"hint","dependencies":["a3d6ae2sys8a-h7"],"title":"Solutions of a System of Equations","text":"$$(-10,4)$$ does not make both equations true. $$(-10,4)$$ is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a3d6ae2sys8b","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$(\\\\frac{5}{4},\\\\frac{7}{4})$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys8b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitue $$x=\\\\frac{5}{4}$$ and $$y=\\\\frac{7}{4}$$ into both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys8b-h2","type":"hint","dependencies":["a3d6ae2sys8b-h1"],"title":"Substitute into First Equation","text":"$$x+5y=10$$\\\\n$$\\\\frac{5}{4}+\\\\frac{5\\\\times7}{4}=10$$\\\\n$$\\\\frac{40}{4}=10$$\\\\n$$10=10$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys8b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a3d6ae2sys8b-h2"],"title":"Substitute into First Equation","text":"Is the equation above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a3d6ae2sys8b-h4","type":"hint","dependencies":["a3d6ae2sys8b-h3"],"title":"Solution to First Equation","text":"Therefore, $$(\\\\frac{5}{4},\\\\frac{7}{4})$$ satisfies the first equation, but it must also safisfy the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys8b-h5","type":"hint","dependencies":["a3d6ae2sys8b-h4"],"title":"Substitute into Second Equation","text":"$$y=\\\\frac{3}{5} x+1$$\\\\n$$\\\\frac{7}{4}=\\\\frac{5\\\\frac{3}{5}}{4}+1$$\\\\n$$\\\\frac{7}{4}=\\\\frac{7}{4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys8b-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a3d6ae2sys8b-h5"],"title":"Substitute into Second Equation","text":"Is the equation above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a3d6ae2sys8b-h7","type":"hint","dependencies":["a3d6ae2sys8b-h6"],"title":"Solution to Second Equation","text":"Therefore, $$(\\\\frac{5}{4},\\\\frac{7}{4})$$ does satisfies the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys8b-h8","type":"hint","dependencies":["a3d6ae2sys8b-h7"],"title":"Solutions of a System of Equations","text":"$$(\\\\frac{5}{4},\\\\frac{7}{4})$$ does make both equations true. $$(\\\\frac{5}{4},\\\\frac{7}{4})$$ is a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d6ae2sys9","title":"Solutions of a System of Equations","body":"Determine if the following points are solutions to the given system of equation:\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Solve Systems of Equations by Graphing","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3d6ae2sys9a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$(-6,5)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys9a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitue $$x=-6$$ and $$y=5$$ into both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys9a-h2","type":"hint","dependencies":["a3d6ae2sys9a-h1"],"title":"Substitute into First Equation","text":"$$x+3y=9$$\\\\n$$-6+3\\\\times5=9$$\\\\n$$9=9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys9a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a3d6ae2sys9a-h2"],"title":"Substitute into First Equation","text":"Is the equation above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a3d6ae2sys9a-h4","type":"hint","dependencies":["a3d6ae2sys9a-h3"],"title":"Solution to First Equation","text":"Therefore, $$(-6,5)$$ satisfies the first equation, but it must also safisfy the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys9a-h5","type":"hint","dependencies":["a3d6ae2sys9a-h4"],"title":"Substitute into Second Equation","text":"$$y=\\\\frac{2}{3} x-2$$\\\\n$$5=-6\\\\frac{2}{3}-2$$\\\\n$$5=-6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys9a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["FALSE"],"dependencies":["a3d6ae2sys9a-h5"],"title":"Substitute into Second Equation","text":"Is the equation above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a3d6ae2sys9a-h7","type":"hint","dependencies":["a3d6ae2sys9a-h6"],"title":"Solution to Second Equation","text":"Therefore, $$(-6,5)$$ does not satisfies the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys9a-h8","type":"hint","dependencies":["a3d6ae2sys9a-h7"],"title":"Solutions of a System of Equations","text":"$$(-6,5)$$ does not make both equations true. $$(-6,5)$$ is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a3d6ae2sys9b","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$(5,\\\\frac{4}{3})$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a3d6ae2sys9b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitue $$x=5$$ and $$y=\\\\frac{4}{3}$$ into both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys9b-h2","type":"hint","dependencies":["a3d6ae2sys9b-h1"],"title":"Substitute into First Equation","text":"$$x+3y=9$$\\\\n$$5+\\\\frac{3\\\\times4}{3}=9$$\\\\n$$9=9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys9b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a3d6ae2sys9b-h2"],"title":"Substitute into First Equation","text":"Is the equation above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a3d6ae2sys9b-h4","type":"hint","dependencies":["a3d6ae2sys9b-h3"],"title":"Solution to First Equation","text":"Therefore, $$(5,\\\\frac{4}{3})$$ satisfies the first equation, but it must also safisfy the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys9b-h5","type":"hint","dependencies":["a3d6ae2sys9b-h4"],"title":"Substitute into Second Equation","text":"$$y=\\\\frac{2}{3} x-2$$\\\\n$$\\\\frac{4}{3}=5\\\\frac{2}{3}-2$$\\\\n$$\\\\frac{4}{3}=\\\\frac{10}{3}-\\\\frac{6}{3}$$\\\\n$$\\\\frac{4}{3}=\\\\frac{4}{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys9b-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a3d6ae2sys9b-h5"],"title":"Substitute into Second Equation","text":"Is the equation above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a3d6ae2sys9b-h7","type":"hint","dependencies":["a3d6ae2sys9b-h6"],"title":"Solution to Second Equation","text":"Therefore, $$(5,\\\\frac{4}{3})$$ does satisfies the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d6ae2sys9b-h8","type":"hint","dependencies":["a3d6ae2sys9b-h7"],"title":"Solutions of a System of Equations","text":"$$(5,\\\\frac{4}{3})$$ does make both equations true. $$(5,\\\\frac{4}{3})$$ is a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d9e92Inequality1","title":"Solve Application with Compound Inequalities","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve Compound Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3d9e92Inequality1a","stepAnswer":["21<x<95"],"problemType":"TextBox","stepTitle":"Due to the drought in California, many communities have tiered water rates. There are different rates for Conservation Usage, Normal Usage and Excessive Usage. The usage is measured in the number of hundred cubic feet (hcf) the property owner uses. During the summer, a property owner will pay $$\\\\$24.72$$ plus $$\\\\$1.54$$ per hcf for Normal Usage. The bill for Normal Usage would be between or equal to $$\\\\$57.06$$ and $$\\\\$171.02$$. How many hcf can the owner use if he wants his usage to stay in the normal range?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$21<x<95$$","hints":{"DefaultPathway":[{"id":"a3d9e92Inequality1a-h1","type":"hint","dependencies":[],"title":"Setup","text":"Assume $$x=the$$ number of hcf, the bill $$=$$ $$24.72+1.54x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d9e92Inequality1a-h2","type":"hint","dependencies":[],"title":"Setup","text":"The bill can be set in the frame of maximum and minimum: $$57.06 \\\\leq 24.74+1.54x \\\\leq 171.02$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d9e92Inequality1a-h3","type":"hint","dependencies":[],"title":"Calculation","text":"Solve the inequality by reducing the middle term to $$x$$ while operating on both ends","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d9e92Inequality1a-h4","type":"hint","dependencies":["a3d9e92Inequality1a-h3"],"title":"Calculation","text":"Reduce the in equality to $$ \\\\leq x \\\\leq $$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d9e92Inequality1a-h5","type":"hint","dependencies":["a3d9e92Inequality1a-h4"],"title":"Caluculation","text":"Both $$57.06$$ and $$171.02$$ have to be minused by $$24.72$$ then divided by $$1.54$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d9e92Inequality10","title":"Solving Compound Inequalities with \\"and\\"","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve Compound Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3d9e92Inequality10a","stepAnswer":["[-1, 2)"],"problemType":"TextBox","stepTitle":"Write the solution in interval notation: $$5x-2<8$$ and $$6x+9 \\\\geq 3$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a3d9e92Inequality10a-h1","type":"hint","dependencies":[],"title":"Interval Notation Overview","text":"Open parentheses \\"(\\" and \\")\\" represent excluded bounds, while closed parentheses \\"[\\" and \\"]\\" represent included bounds. If two separate ranges don\'t overlap, then we can use \\"U\\" (union) to combine them into the same expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d9e92Inequality10a-h2","type":"hint","dependencies":["a3d9e92Inequality10a-h1"],"title":"Solving Each Inequality for X","text":"We can solve each inequality. $$5x-2<8$$ becomes $$x<2$$, and $$6x+9 \\\\geq 3$$ becomes $$x \\\\geq -1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d9e92Inequality10a-h3","type":"hint","dependencies":["a3d9e92Inequality10a-h2"],"title":"Creating Our Notation","text":"Using the general guide given in the previous hints, we can write the solution [-1, 2)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d9e92Inequality11","title":"Solving Compound Inequalities with \\"and\\"","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve Compound Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3d9e92Inequality11a","stepAnswer":["[-2, 2)"],"problemType":"TextBox","stepTitle":"Write the solution in interval notation: $$4x-1<7$$ and $$2x+8 \\\\geq 4$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a3d9e92Inequality11a-h1","type":"hint","dependencies":[],"title":"Interval Notation Overview","text":"Open parentheses \\"(\\" and \\")\\" represent excluded bounds, while closed parentheses \\"[\\" and \\"]\\" represent included bounds. If two separate ranges don\'t overlap, then we can use \\"U\\" (union) to combine them into the same expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d9e92Inequality11a-h2","type":"hint","dependencies":["a3d9e92Inequality11a-h1"],"title":"Solving Each Inequality for X","text":"We can solve each inequality. $$4x-1<7$$ becomes $$x<2$$, and $$2x+8 \\\\geq 4$$ becomes $$x \\\\geq -2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d9e92Inequality11a-h3","type":"hint","dependencies":["a3d9e92Inequality11a-h2"],"title":"Creating Our Notation","text":"Using the general guide given in the previous hints, we can write the solution [-2, 2)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d9e92Inequality12","title":"Solving Compound Inequalities with \\"and\\"","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve Compound Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3d9e92Inequality12a","stepAnswer":["[-3, -1]"],"problemType":"TextBox","stepTitle":"Write the solution in interval notation: $$4x+6 \\\\leq 2$$ and $$2x+1 \\\\geq -5$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a3d9e92Inequality12a-h1","type":"hint","dependencies":[],"title":"Interval Notation Overview","text":"Open parentheses \\"(\\" and \\")\\" represent excluded bounds, while closed parentheses \\"[\\" and \\"]\\" represent included bounds. If two separate ranges don\'t overlap, then we can use \\"U\\" (union) to combine them into the same expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d9e92Inequality12a-h2","type":"hint","dependencies":["a3d9e92Inequality12a-h1"],"title":"Solving Each Inequality for X","text":"We can solve each inequality. $$4x+6 \\\\leq 2$$ becomes $$x \\\\leq -1$$, and $$2x+1 \\\\geq -5$$ becomes $$x \\\\geq -3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d9e92Inequality12a-h3","type":"hint","dependencies":["a3d9e92Inequality12a-h2"],"title":"Creating Our Notation","text":"Using the general guide given in the previous hints, we can write the solution [-3, -1]","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d9e92Inequality13","title":"Solving Compound Inequalities with \\"and\\"","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve Compound Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3d9e92Inequality13a","stepAnswer":["[-1, 3/2]"],"problemType":"TextBox","stepTitle":"Write the solution in interval notation: $$4x-2 \\\\leq 4$$ and $$7x-1 \\\\geq -8$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a3d9e92Inequality13a-h1","type":"hint","dependencies":[],"title":"Interval Notation Overview","text":"Open parentheses \\"(\\" and \\")\\" represent excluded bounds, while closed parentheses \\"[\\" and \\"]\\" represent included bounds. If two separate ranges don\'t overlap, then we can use \\"U\\" (union) to combine them into the same expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d9e92Inequality13a-h2","type":"hint","dependencies":["a3d9e92Inequality13a-h1"],"title":"Solving Each Inequality for X","text":"We can solve each inequality. $$4x-2 \\\\leq 4$$ becomes $$x \\\\leq \\\\frac{3}{2}$$, and $$7x-1>-8$$ becomes $$x>-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d9e92Inequality13a-h3","type":"hint","dependencies":["a3d9e92Inequality13a-h2"],"title":"Creating Our Notation","text":"Using the general guide given in the previous hints, we can write the solution as $$[-1,\\\\frac{3}{2}]$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d9e92Inequality14","title":"Solving Compound Inequalities with \\"and\\"","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve Compound Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3d9e92Inequality14a","stepAnswer":["(1, 8)"],"problemType":"TextBox","stepTitle":"Write the solution in interval notation: $$2x-11<5$$ and $$3x-8>-5$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(1,8)$$","hints":{"DefaultPathway":[{"id":"a3d9e92Inequality14a-h1","type":"hint","dependencies":[],"title":"Interval Notation Overview","text":"Open parentheses \\"(\\" and \\")\\" represent excluded bounds, while closed parentheses \\"[\\" and \\"]\\" represent included bounds. If two separate ranges don\'t overlap, then we can use \\"U\\" (union) to combine them into the same expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d9e92Inequality14a-h2","type":"hint","dependencies":["a3d9e92Inequality14a-h1"],"title":"Solving Each Inequality for X","text":"We can solve each inequality. $$2x-11<5$$ becomes $$x<8$$, and $$3x-8>-5$$ becomes $$x>1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d9e92Inequality14a-h3","type":"hint","dependencies":["a3d9e92Inequality14a-h2"],"title":"Creating Our Notation","text":"Using the general guide given in the previous hints, we can write the solution $$(1,8)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d9e92Inequality15","title":"Solving Compound Inequalities with \\"and\\"","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve Compound Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3d9e92Inequality15a","stepAnswer":["(-2, 2)"],"problemType":"TextBox","stepTitle":"Write the solution in interval notation: $$7x-8<6$$ and $$5x+7>-3$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-2,2)$$","hints":{"DefaultPathway":[{"id":"a3d9e92Inequality15a-h1","type":"hint","dependencies":[],"title":"Interval Notation Overview","text":"Open parentheses \\"(\\" and \\")\\" represent excluded bounds, while closed parentheses \\"[\\" and \\"]\\" represent included bounds. If two separate ranges don\'t overlap, then we can use \\"U\\" (union) to combine them into the same expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d9e92Inequality15a-h2","type":"hint","dependencies":["a3d9e92Inequality15a-h1"],"title":"Solving Each Inequality for X","text":"We can solve each inequality. $$7x-8<6$$ becomes $$x<2$$, and $$5x+7>-3$$ becomes $$x>-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d9e92Inequality15a-h3","type":"hint","dependencies":["a3d9e92Inequality15a-h2"],"title":"Creating Our Notation","text":"Using the general guide given in the previous hints, we can write the solution $$(-2,2)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d9e92Inequality2","title":"Solving Compound Inequalities with \\"and\\"","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve Compound Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3d9e92Inequality2a","stepAnswer":["[1, 3)"],"problemType":"TextBox","stepTitle":"Write the solution in interval notation: $$x<3$$ and $$x \\\\geq 1$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a3d9e92Inequality2a-h1","type":"hint","dependencies":[],"title":"Interval Notation Overview","text":"Open parentheses \\"(\\" and \\")\\" represent excluded bounds, while closed parentheses \\"[\\" and \\"]\\" represent included bounds.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d9e92Inequality2a-h2","type":"hint","dependencies":["a3d9e92Inequality2a-h1"],"title":"Creating Our Notation","text":"Using the general guide given in the previous hint, we can write the solution [1, 3)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d9e92Inequality3","title":"Solving Compound Inequalities with \\"and\\"","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve Compound Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3d9e92Inequality3a","stepAnswer":["(-2, 4]"],"problemType":"TextBox","stepTitle":"Write the solution in interval notation: $$x \\\\leq 4$$ and $$x>-2$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a3d9e92Inequality3a-h1","type":"hint","dependencies":[],"title":"Interval Notation Overview","text":"Open parentheses \\"(\\" and \\")\\" represent excluded bounds, while closed parentheses \\"[\\" and \\"]\\" represent included bounds.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d9e92Inequality3a-h2","type":"hint","dependencies":["a3d9e92Inequality3a-h1"],"title":"Creating Our Notation","text":"Using the general guide given in the previous hint, we can write the solution (-2, 4]","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d9e92Inequality4","title":"Solving Compound Inequalities with \\"and\\"","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve Compound Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3d9e92Inequality4a","stepAnswer":["[-4, -1]"],"problemType":"TextBox","stepTitle":"Write the solution in interval notation: $$x \\\\geq -4$$ and $$x \\\\leq -1$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a3d9e92Inequality4a-h1","type":"hint","dependencies":[],"title":"Interval Notation Overview","text":"Open parentheses \\"(\\" and \\")\\" represent excluded bounds, while closed parentheses \\"[\\" and \\"]\\" represent included bounds.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d9e92Inequality4a-h2","type":"hint","dependencies":["a3d9e92Inequality4a-h1"],"title":"Creating Our Notation","text":"Using the general guide given in the previous hint, we can write the solution [-4, -1]","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d9e92Inequality5","title":"Solving Compound Inequalities with \\"and\\"","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve Compound Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3d9e92Inequality5a","stepAnswer":["(-6, -3)"],"problemType":"TextBox","stepTitle":"Write the solution in interval notation: $$x>-6$$ and $$x<-3$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-6,-3)$$","hints":{"DefaultPathway":[{"id":"a3d9e92Inequality5a-h1","type":"hint","dependencies":[],"title":"Interval Notation Overview","text":"Open parentheses \\"(\\" and \\")\\" represent excluded bounds, while closed parentheses \\"[\\" and \\"]\\" represent included bounds.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d9e92Inequality5a-h2","type":"hint","dependencies":["a3d9e92Inequality5a-h1"],"title":"Creating Our Notation","text":"Using the general guide given in the previous hint, we can write the solution $$(-6,-3)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d9e92Inequality6","title":"Solving Compound Inequalities with \\"or\\"","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve Compound Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3d9e92Inequality6a","stepAnswer":["(-inf, -2] U (3, inf)"],"problemType":"TextBox","stepTitle":"Write the solution in interval notation: $$x \\\\leq -2$$ or $$x>3$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a3d9e92Inequality6a-h1","type":"hint","dependencies":[],"title":"Interval Notation Overview","text":"Open parentheses \\"(\\" and \\")\\" represent excluded bounds, while closed parentheses \\"[\\" and \\"]\\" represent included bounds. If two separate ranges don\'t overlap, then we can use \\"U\\" (union) to combine them into the same expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d9e92Inequality6a-h2","type":"hint","dependencies":["a3d9e92Inequality6a-h1"],"title":"Creating Our Notation","text":"Using the general guide given in the previous hint, we can write the solution (-inf, -2] U (3, inf)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d9e92Inequality7","title":"Solving Compound Inequalities with \\"or\\"","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve Compound Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3d9e92Inequality7a","stepAnswer":["(-inf, -4] U (-3, inf)"],"problemType":"TextBox","stepTitle":"Write the solution in interval notation: $$x \\\\leq -4$$ or $$x>-3$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a3d9e92Inequality7a-h1","type":"hint","dependencies":[],"title":"Interval Notation Overview","text":"Open parentheses \\"(\\" and \\")\\" represent excluded bounds, while closed parentheses \\"[\\" and \\"]\\" represent included bounds. If two separate ranges don\'t overlap, then we can use \\"U\\" (union) to combine them into the same expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d9e92Inequality7a-h2","type":"hint","dependencies":["a3d9e92Inequality7a-h1"],"title":"Creating Our Notation","text":"Using the general guide given in the previous hint, we can write the solution (-inf, -4] U (-3, inf)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d9e92Inequality8","title":"Solving Compound Inequalities with \\"or\\"","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve Compound Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3d9e92Inequality8a","stepAnswer":["(-inf, 2) U [5, inf)"],"problemType":"TextBox","stepTitle":"Write the solution in interval notation: $$x<2$$ or $$x \\\\geq 5$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a3d9e92Inequality8a-h1","type":"hint","dependencies":[],"title":"Interval Notation Overview","text":"Open parentheses \\"(\\" and \\")\\" represent excluded bounds, while closed parentheses \\"[\\" and \\"]\\" represent included bounds. If two separate ranges don\'t overlap, then we can use \\"U\\" (union) to combine them into the same expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d9e92Inequality8a-h2","type":"hint","dependencies":["a3d9e92Inequality8a-h1"],"title":"Creating Our Notation","text":"Using the general guide given in the previous hint, we can write the solution (-inf, 2) U [5, inf)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3d9e92Inequality9","title":"Solving Compound Inequalities with \\"or\\"","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Solve Compound Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a3d9e92Inequality9a","stepAnswer":["(-inf,0) U [4, inf)"],"problemType":"TextBox","stepTitle":"Write the solution in interval notation: $$x<0$$ or $$x \\\\geq 4$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,0)$$ U [4, inf)","hints":{"DefaultPathway":[{"id":"a3d9e92Inequality9a-h1","type":"hint","dependencies":[],"title":"Interval Notation Overview","text":"Open parentheses \\"(\\" and \\")\\" represent excluded bounds, while closed parentheses \\"[\\" and \\"]\\" represent included bounds. If two separate ranges don\'t overlap, then we can use \\"U\\" (union) to combine them into the same expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3d9e92Inequality9a-h2","type":"hint","dependencies":["a3d9e92Inequality9a-h1"],"title":"Creating Our Notation","text":"Using the general guide given in the previous hint, we can write the solution $$(-\\\\infty,0)$$ U [4, inf)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3e1067arc1","title":"Calculating the Arc Length of a Function $$x$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.4 Arc Length of a Curve and Surface Area","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a3e1067arc1a","stepAnswer":["$$2.268$$"],"problemType":"TextBox","stepTitle":"Let f(x) $$=$$ $$2x^{\\\\frac{3}{2}}$$. Calculate the arc length of the graph of f(x) over the interval [0,1]. Round the answer to three decimal places.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2.268$$","hints":{"DefaultPathway":[{"id":"a3e1067arc1a-h1","type":"hint","dependencies":[],"title":"Arc length formula","text":"Arc $$Length=\\\\int_{a}^{b} \\\\sqrt{1+{\\\\operatorname{f\'}\\\\left(x\\\\right)}^2} \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc1a-h2","type":"hint","dependencies":["a3e1067arc1a-h1"],"title":"Take the derivative of f(x) and plug it into the integral with the given bounds","text":"Arc $$Length=\\\\int_{0}^{1} \\\\sqrt{1+{\\\\left(3x^{\\\\frac{1}{2}}\\\\right)}^2} \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc1a-h3","type":"hint","dependencies":["a3e1067arc1a-h2"],"title":"Simplify $${\\\\operatorname{f\'}\\\\left(x\\\\right)}^2$$","text":"Arc $$Length=\\\\int_{0}^{1} \\\\sqrt{1+9x} \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc1a-h4","type":"hint","dependencies":["a3e1067arc1a-h3"],"title":"U substitution","text":"Perform u substitution with $$u=1+9x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc1a-h5","type":"hint","dependencies":["a3e1067arc1a-h4"],"title":"Integral after u substitution","text":"Arc $$Length=(1/9)*\\\\int_{1}^{10} \\\\sqrt{u} \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a3e1067arc10","title":"Find the length","body":"Find the length of the functions over the given interval.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.4 Arc Length of a Curve and Surface Area","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a3e1067arc10a","stepAnswer":["$$\\\\frac{3\\\\sqrt{5}}{2}$$"],"problemType":"TextBox","stepTitle":"$$y=-\\\\left(\\\\frac{1}{2}\\\\right) x+25$$ from $$x=1$$ to $$x=4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3\\\\sqrt{5}}{2}$$","hints":{"DefaultPathway":[{"id":"a3e1067arc10a-h1","type":"hint","dependencies":[],"title":"Arc length formula","text":"Arc Length can be found by computing $$\\\\int_{a}^{b} \\\\sqrt{1+{\\\\operatorname{f\'}\\\\left(x\\\\right)}^2} \\\\,dx$$, f\'(x) is derivative of f(x) and interval [a,b]","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc10a-h2","type":"hint","dependencies":["a3e1067arc10a-h1"],"title":"Find the derivative of f(x)","text":"We have $$f(x)=y=-\\\\left(\\\\frac{1}{2}\\\\right) x+25$$, so $$f\'(x)=-\\\\left(\\\\frac{1}{2}\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc10a-h3","type":"hint","dependencies":["a3e1067arc10a-h2"],"title":"Plug in","text":"After we plug f\'(x) and interval, we will have Arc $$Length=\\\\int_{1}^{4} \\\\sqrt{1+{\\\\left(-\\\\frac{1}{2}\\\\right)}^2} \\\\,dx=\\\\int_{1}^{4} \\\\sqrt{\\\\frac{5}{4}} \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc10a-h4","type":"hint","dependencies":["a3e1067arc10a-h3"],"title":"Evaluate the integral","text":"We can take $$\\\\sqrt{\\\\frac{5}{4}}$$ out of integral, which gives us Arc $$Length=\\\\sqrt{\\\\frac{5}{4}}$$ times $$x$$ with limit from $$1$$ to $$4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3\\\\sqrt{5}}{2}$$"],"dependencies":["a3e1067arc10a-h4"],"title":"Evaluate the integral","text":"What is $$\\\\sqrt{\\\\frac{5}{4}} \\\\left(4-1\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a3e1067arc11","title":"Find the length","body":"Find the length of the functions over the given interval.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.4 Arc Length of a Curve and Surface Area","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a3e1067arc11a","stepAnswer":["$$2\\\\sqrt{17}$$"],"problemType":"TextBox","stepTitle":"$$x=4y$$ from $$y=-1$$ to $$y=1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2\\\\sqrt{17}$$","hints":{"DefaultPathway":[{"id":"a3e1067arc11a-h1","type":"hint","dependencies":[],"title":"Arc length formula","text":"Arc Length can be found by computing $$\\\\int_{a}^{b} \\\\sqrt{1+{\\\\operatorname{g\'}\\\\left(y\\\\right)}^2} \\\\,dy$$, g\'(y) is derivative of g(y) and interval [a,b]","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc11a-h2","type":"hint","dependencies":["a3e1067arc11a-h1"],"title":"Find the derivative of g(y)","text":"We have $$g(y)=x=4y$$, so $$g\'(y)=4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc11a-h3","type":"hint","dependencies":["a3e1067arc11a-h2"],"title":"Plug in","text":"After we plug g\'(y) and interval, we will have Arc $$Length=\\\\int_{-1}^{1} \\\\sqrt{1+4^2} \\\\,dy=\\\\int_{-1}^{1} \\\\sqrt{17} \\\\,dy$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc11a-h4","type":"hint","dependencies":["a3e1067arc11a-h3"],"title":"Evaluate the integral","text":"We can take $$\\\\sqrt{17}$$ out of integral, which gives us Arc $$Length=\\\\sqrt{17}$$ times $$y$$ with limit from $$-1$$ to $$1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2\\\\sqrt{17}$$"],"dependencies":["a3e1067arc11a-h4"],"title":"Evaluate the integral","text":"What is $$\\\\sqrt{17} \\\\left(1-\\\\left(-1\\\\right)\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a3e1067arc12","title":"Find the lengths of the functions of $$x$$ over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.4 Arc Length of a Curve and Surface Area","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a3e1067arc12a","stepAnswer":["$$\\\\frac{13\\\\sqrt{13}-8}{27}$$"],"problemType":"TextBox","stepTitle":"$$y=x^{\\\\frac{3}{2}}$$ from $$(0,0)$$ to $$(1,1)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{13\\\\sqrt{13}-8}{27}$$","hints":{"DefaultPathway":[{"id":"a3e1067arc12a-h1","type":"hint","dependencies":[],"title":"Arc length formula","text":"Arc Length can be found by computing $$\\\\int_{a}^{b} \\\\sqrt{1+{\\\\operatorname{f\'}\\\\left(x\\\\right)}^2} \\\\,dx$$, f\'(x) is derivative of f(x) and interval [a,b]","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc12a-h2","type":"hint","dependencies":["a3e1067arc12a-h1"],"title":"Find the derivative of f(x)","text":"We have f(x) $$=y=x^{\\\\frac{3}{2}}$$, so $$f\'(x)=$$ $$\\\\frac{3}{2} x^{\\\\frac{1}{2}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc12a-h3","type":"hint","dependencies":["a3e1067arc12a-h2"],"title":"Find the interval","text":"We need to find the interval on x-axis, which is [0,1]","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc12a-h4","type":"hint","dependencies":["a3e1067arc12a-h3"],"title":"Plug in","text":"After we plug f\'(x) and interval, we will have Arc $$Length=\\\\int_{0}^{1} \\\\sqrt{1+{\\\\left(\\\\frac{3}{2} x^{\\\\frac{1}{2}}\\\\right)}^2} \\\\,dx=\\\\int_{0}^{1} \\\\sqrt{1+\\\\frac{9}{4} x} \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc12a-h5","type":"hint","dependencies":["a3e1067arc12a-h4"],"title":"Evaluate the integral","text":"Use a computer, calculator, or another device to solve the integral.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a3e1067arc13","title":"Find the lengths of the functions of $$y$$ over the given interval. If you cannot evaluate the integral exactly, use technology to approximate it.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.4 Arc Length of a Curve and Surface Area","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a3e1067arc13a","stepAnswer":["$$18\\\\left(4\\\\sqrt{5}+\\\\ln(9+4\\\\sqrt{5})\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$x=\\\\sqrt{y}$$ from $$y=0$$ to $$y=1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$18\\\\left(4\\\\sqrt{5}+\\\\ln(9+4\\\\sqrt{5})\\\\right)$$","hints":{"DefaultPathway":[{"id":"a3e1067arc13a-h1","type":"hint","dependencies":[],"title":"Arc length formula","text":"Arc Length can be found by computing $$\\\\int_{a}^{b} \\\\sqrt{1+{\\\\operatorname{g\'}\\\\left(y\\\\right)}^2} \\\\,dy$$, g\'(y) is derivative of g(y) and interval [a,b]","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc13a-h2","type":"hint","dependencies":["a3e1067arc13a-h1"],"title":"Find the derivative of f(x)","text":"We have $$g(y)=x=\\\\sqrt{y}$$, so $$g\'(y)=\\\\frac{1}{2\\\\sqrt{y}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc13a-h3","type":"hint","dependencies":["a3e1067arc13a-h2"],"title":"Plug in","text":"After we plug g\'(y) and interval, we will have Arc $$Length=\\\\int_{0}^{1} \\\\sqrt{1+{\\\\left(\\\\frac{1}{2\\\\sqrt{y}}\\\\right)}^2} \\\\,dy=\\\\int_{0}^{1} \\\\sqrt{1+\\\\frac{1}{4y}} \\\\,dy$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc13a-h4","type":"hint","dependencies":["a3e1067arc13a-h3"],"title":"Evaluate the integral","text":"Use a computer, calculator, or another device to solve the integral.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a3e1067arc14","title":"Find the surface area","body":"Find the surface area of the volume generated when the following curves revolve around the x-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.4 Arc Length of a Curve and Surface Area","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a3e1067arc14a","stepAnswer":["$$\\\\frac{49\\\\pi}{3}$$"],"problemType":"TextBox","stepTitle":"$$y=\\\\sqrt{x}$$ from $$x=2$$ to $$x=6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{49\\\\pi}{3}$$","hints":{"DefaultPathway":[{"id":"a3e1067arc14a-h1","type":"hint","dependencies":[],"title":"Find the surface area","text":"The surface area can be found with $$(2*pi)*\\\\int_{a}^{b} f{\\\\left(x\\\\right)} \\\\sqrt{1+{\\\\operatorname{f\'}\\\\left(x\\\\right)}^2} \\\\,dx$$, where f\'(x) indicates the derivative of f(x). We need to plug f(x) and intervals into this equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2\\\\sqrt{x}}$$"],"dependencies":["a3e1067arc14a-h1"],"title":"Derivative of f(x)","text":"What is f\'(x)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc14a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4x}$$"],"dependencies":["a3e1067arc14a-h2"],"title":"Simplify $${\\\\operatorname{f\'}\\\\left(x\\\\right)}^2$$","text":"What is $${\\\\operatorname{f\'}\\\\left(x\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc14a-h4","type":"hint","dependencies":["a3e1067arc14a-h3"],"title":"Find the surface area","text":"After plugging f(x) and the interval into the equation, we will have Surface $$Area=(2*pi)*\\\\int_{2}^{6} \\\\sqrt{x} \\\\sqrt{1+\\\\frac{1}{4x}} \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc14a-h5","type":"hint","dependencies":["a3e1067arc14a-h4"],"title":"Evaluate the integral","text":"Use a computer, calculator, or another device to solve the integral.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a3e1067arc15","title":"Find the surface area","body":"Find the surface area of the volume generated when the following curves revolve around the x-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.4 Arc Length of a Curve and Surface Area","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a3e1067arc15a","stepAnswer":["$$70\\\\pi \\\\sqrt{2}$$"],"problemType":"TextBox","stepTitle":"$$y=7x$$ from $$x=-1$$ to $$x=1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$70\\\\pi \\\\sqrt{2}$$","hints":{"DefaultPathway":[{"id":"a3e1067arc15a-h1","type":"hint","dependencies":[],"title":"Find the surface area","text":"The surface area can be found with $$(2*pi)*\\\\int_{a}^{b} f{\\\\left(x\\\\right)} \\\\sqrt{1+{\\\\operatorname{f\'}\\\\left(x\\\\right)}^2} \\\\,dx$$, where f\'(x) indicates the derivative of f(x). We need to plug f(x) and intervals into this equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a3e1067arc15a-h1"],"title":"Derivative of f(x)","text":"What is f\'(x)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$49$$"],"dependencies":["a3e1067arc15a-h2"],"title":"Simplify $${\\\\operatorname{f\'}\\\\left(x\\\\right)}^2$$","text":"What is $${\\\\operatorname{f\'}\\\\left(x\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc15a-h4","type":"hint","dependencies":["a3e1067arc15a-h3"],"title":"Find the surface area","text":"After plugging f(x) and the interval into the equation, we will have Surface $$Area=(2*pi)*\\\\int_{-1}^{1} 7x \\\\sqrt{1+49} \\\\,dx=(2*pi)*\\\\int_{-1}^{1} 7x \\\\sqrt{50} \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc15a-h5","type":"hint","dependencies":["a3e1067arc15a-h4"],"title":"Evaluate the integral","text":"Use a computer, calculator, or another device to solve the integral.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a3e1067arc2","title":"Calculating the Arc Length of a Function $$x$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.4 Arc Length of a Curve and Surface Area","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a3e1067arc2a","stepAnswer":["$$1.657$$"],"problemType":"TextBox","stepTitle":"Let $$f(x)=\\\\frac{4}{3} x^{\\\\frac{3}{2}}$$. Calculate the arc length of the graph of f(x) over the interval [0,1]. Round the answer to three decimal places.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1.657$$","hints":{"DefaultPathway":[{"id":"a3e1067arc2a-h1","type":"hint","dependencies":[],"title":"Arc length formula","text":"Arc $$Length=\\\\int_{a}^{b} \\\\sqrt{1+{\\\\operatorname{f\'}\\\\left(x\\\\right)}^2} \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc2a-h2","type":"hint","dependencies":["a3e1067arc2a-h1"],"title":"Take the derivative of f(x)","text":"$$f\'(x)=\\\\frac{4}{3} \\\\frac{3}{2} x^{\\\\frac{1}{2}}=2x^{\\\\frac{1}{2}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc2a-h3","type":"hint","dependencies":["a3e1067arc2a-h2"],"title":"Plug the derivative into the integral with the given bounds","text":"Arc $$Length=\\\\int_{0}^{1} \\\\sqrt{1+{\\\\left(2x^{\\\\frac{1}{2}}\\\\right)}^2} \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc2a-h4","type":"hint","dependencies":["a3e1067arc2a-h3"],"title":"Simplify $${\\\\operatorname{f\'}\\\\left(x\\\\right)}^2$$ and evaluate the integral","text":"Arc $$Length=\\\\int_{0}^{1} \\\\sqrt{1+{\\\\left(4x\\\\right)}^2} \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a3e1067arc3","title":"Using a Computer or Calculator to Determine the Arc Length of a Function of $$x$$. Round to three decimal places.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.4 Arc Length of a Curve and Surface Area","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a3e1067arc3a","stepAnswer":["$$8.269$$"],"problemType":"TextBox","stepTitle":"Let $$f(x)=x^2$$. Calculate the arc length of the graph of f(x) over the interval [1,3].","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8.269$$","hints":{"DefaultPathway":[{"id":"a3e1067arc3a-h1","type":"hint","dependencies":[],"title":"Take the derivative of f(x) and plug it into the integral with the given bounds","text":"Arc $$Length=\\\\int_{1}^{3} \\\\sqrt{1+{\\\\left(2x\\\\right)}^2} \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc3a-h2","type":"hint","dependencies":["a3e1067arc3a-h1"],"title":"Simplify $${\\\\operatorname{f\'}\\\\left(x\\\\right)}^2$$","text":"Arc $$Length=\\\\int_{1}^{3} \\\\sqrt{1+4x^2} \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc3a-h3","type":"hint","dependencies":["a3e1067arc3a-h2"],"title":"Evaluate the integral","text":"Use a computer, calculator, or another device to solve the integral.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a3e1067arc4","title":"Using a Computer or Calculator to Determine the Arc Length of a Function of $$x$$. Round to three decimal places.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.4 Arc Length of a Curve and Surface Area","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a3e1067arc4a","stepAnswer":["$$3.819$$"],"problemType":"TextBox","stepTitle":"Let $$f(x)=sin(x)$$ Calculate the arc length of the graph of f(x) over the interval [0,pi].","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3.819$$","hints":{"DefaultPathway":[{"id":"a3e1067arc4a-h1","type":"hint","dependencies":[],"title":"Take the derivative of f(x)","text":"$$f\'(x)=cos(x)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc4a-h2","type":"hint","dependencies":["a3e1067arc4a-h1"],"title":"Plug the f\'(x) into the arc length formula with the given bounds.","text":"Arc $$Length=\\\\int_{0}^{pi} \\\\sqrt{1+cos^2\\\\left(x\\\\right)} \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc4a-h3","type":"hint","dependencies":["a3e1067arc4a-h2"],"title":"Evaluate the integral","text":"Use a computer, calculator, or another device to solve the integral.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a3e1067arc5","title":"Calculating the Arc Length of a Function of $$y$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.4 Arc Length of a Curve and Surface Area","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a3e1067arc5a","stepAnswer":["$$21.0277$$"],"problemType":"TextBox","stepTitle":"Let $$g(y)=3y^3$$. Calculate the arc length of the graph of g(y) over the interval [1,2].","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$21.0277$$","hints":{"DefaultPathway":[{"id":"a3e1067arc5a-h1","type":"hint","dependencies":[],"title":"Take the derivative of g(y) and plug it into the integral with the given bounds","text":"Arc $$Length=\\\\int_{1}^{2} \\\\sqrt{1+{\\\\left(9y^2\\\\right)}^2} \\\\,dy$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc5a-h2","type":"hint","dependencies":["a3e1067arc5a-h1"],"title":"Simplify $${\\\\operatorname{g\'}\\\\left(y\\\\right)}^2$$","text":"Arc $$Length=\\\\int_{1}^{2} \\\\sqrt{1+81y^4} \\\\,dy$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc5a-h3","type":"hint","dependencies":["a3e1067arc5a-h2"],"title":"Evaluate the integral","text":"Use a computer, calculator, or another device to solve the integral.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a3e1067arc6","title":"Calculating the Arc Length of a Function of $$y$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.4 Arc Length of a Curve and Surface Area","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a3e1067arc6a","stepAnswer":["$$3.15018$$"],"problemType":"TextBox","stepTitle":"Let $$g(y)=\\\\frac{1}{y}$$. Calculate the arc length of the graph of g(y) over the interval [1,4].","stepBody":"Use a computer or calculator to approximate the value of the integral.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3.15018$$","hints":{"DefaultPathway":[{"id":"a3e1067arc6a-h1","type":"hint","dependencies":[],"title":"Take the derivative of g(y) and plug it into the integral with the given bounds","text":"Arc $$Length=\\\\int_{1}^{4} \\\\sqrt{1+{\\\\left(\\\\frac{1}{y^2}\\\\right)}^2} \\\\,dy$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc6a-h2","type":"hint","dependencies":["a3e1067arc6a-h1"],"title":"Simplify $${\\\\operatorname{g\'}\\\\left(y\\\\right)}^2$$","text":"Arc $$Length=\\\\int_{1}^{4} \\\\sqrt{1+{\\\\left(\\\\frac{1}{y}\\\\right)}^4} \\\\,dy$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc6a-h3","type":"hint","dependencies":["a3e1067arc6a-h2"],"title":"Evaluate the integral","text":"Use a computer, calculator, or another device to solve the integral.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a3e1067arc7","title":"Calculating the Surface Area of a Surface of Revolution $$1$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.4 Arc Length of a Curve and Surface Area","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a3e1067arc7a","stepAnswer":["$$30.846$$"],"problemType":"TextBox","stepTitle":"Let $$f(x)=\\\\sqrt{x}$$ over the interval [0,4]. Find the surface area of the surface generated by revolving the graph of f(x) around the x-axis. Round the answer to three decimal places.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$30.846$$","hints":{"DefaultPathway":[{"id":"a3e1067arc7a-h1","type":"hint","dependencies":[],"title":"Graphing","text":"First, we need to draw a graph of f(x) and the surface of rotation, which are shown in the following figure. Note (a) is the graph of f(x) and (b) is the surface of rotation.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc7a-h2","type":"hint","dependencies":["a3e1067arc7a-h1"],"title":"Find the surface area","text":"The surface area can be found with $$\\\\int_{a}^{b} 2\\\\pi f{\\\\left(x\\\\right)} \\\\sqrt{1+{\\\\operatorname{f\'}\\\\left(x\\\\right)}^2} \\\\,dx$$, where f\'(x) indicates the derivative of f(x). We need to plug f(x) and intervals into this equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2\\\\sqrt{x}}$$"],"dependencies":["a3e1067arc7a-h2"],"title":"Derivative of f(x)","text":"What is f\'(x)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4x}$$"],"dependencies":["a3e1067arc7a-h3"],"title":"Simplify $${\\\\operatorname{f\'}\\\\left(x\\\\right)}^2$$","text":"What is $${\\\\operatorname{f\'}\\\\left(x\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc7a-h5","type":"hint","dependencies":["a3e1067arc7a-h4"],"title":"Find the surface area","text":"After plugging f(x) and the interval into the equation, we will have Surface $$Area=\\\\int_{1}^{4} 2\\\\pi \\\\sqrt{x} \\\\sqrt{1+\\\\frac{1}{4x}} \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc7a-h6","type":"hint","dependencies":["a3e1067arc7a-h5"],"title":"Simpify","text":"After simplified the equation, we will have Surface $$Area=\\\\int_{1}^{4} 2\\\\pi \\\\sqrt{x+\\\\frac{1}{4}} \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc7a-h7","type":"hint","dependencies":["a3e1067arc7a-h6"],"title":"Evaluate the integral","text":"Let $$u=x+\\\\frac{1}{4}$$. Then, $$du=dx$$. When $$x=1$$, $$u=\\\\frac{5}{4}$$, and when $$x=4$$, $$u=\\\\frac{17}{4}$$. This gives us $$\\\\int_{1}^{4} 2\\\\pi \\\\sqrt{x+\\\\frac{1}{4}} \\\\,dx=\\\\int_{\\\\frac{5}{4}}^{\\\\frac{17}{4}} 2\\\\pi \\\\sqrt{u} \\\\,du$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc7a-h8","type":"hint","dependencies":["a3e1067arc7a-h7"],"title":"Evaluate the integral","text":"Use a computer, calculator, or another device to solve the integral.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a3e1067arc8","title":"Calculating the Surface Area of a Surface of Revolution $$2$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.4 Arc Length of a Curve and Surface Area","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a3e1067arc8a","stepAnswer":["$$24.118$$"],"problemType":"TextBox","stepTitle":"Let $$f(x)=y=\\\\sqrt[3]{3x}$$. Consider the portion of the curve where $$0 \\\\leq y \\\\leq 2$$. Find the surface area of the surface generated by revolving the graph of f(x) around the y-axis.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$24.118$$","hints":{"DefaultPathway":[{"id":"a3e1067arc8a-h1","type":"hint","dependencies":[],"title":"Find g(y)","text":"Notice that we are revolving the curve around the y-axis, and the interval is in terms of $$y$$, so we want to rewrite the function as a function of $$y$$. We get $$x=g(y)=\\\\frac{1}{3} y^3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc8a-h2","type":"hint","dependencies":["a3e1067arc8a-h1"],"title":"Graphing","text":"First, we need to draw a graph of g(y) and the surface of rotation, which are shown in the following figure. Note (a) is the graph of g(y) and (b) is the surface of rotation.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc8a-h3","type":"hint","dependencies":["a3e1067arc8a-h2"],"title":"Find the surface area","text":"The surface area can be found with $$\\\\int_{a}^{b} 2\\\\pi g{\\\\left(y\\\\right)} \\\\sqrt{1+{\\\\operatorname{g\'}\\\\left(y\\\\right)}^2} \\\\,dy$$, where g\'(y) indicates the derivative of g(y). We need to plug g(y) and intervals into this equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y^2$$"],"dependencies":["a3e1067arc8a-h3"],"title":"Derivative of g(y)","text":"What is g\'(y)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc8a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y^4$$"],"dependencies":["a3e1067arc8a-h4"],"title":"Simplify $${\\\\operatorname{g\'}\\\\left(y\\\\right)}^2$$","text":"What is $${\\\\operatorname{g\'}\\\\left(y\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc8a-h6","type":"hint","dependencies":["a3e1067arc8a-h5"],"title":"Find the surface area","text":"After plugging g(y) and the interval into the equation, we will have Surface $$Area=\\\\int_{0}^{2} 2\\\\pi \\\\frac{1}{3} y^3 \\\\sqrt{1+y^4} \\\\,dy$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc8a-h7","type":"hint","dependencies":["a3e1067arc8a-h6"],"title":"Simpify","text":"After simplified the equation, we will have Surface $$Area=((2*pi)/3)*(\\\\int_{0}^{2} y^3 \\\\sqrt{1+y^4} \\\\,dy)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc8a-h8","type":"hint","dependencies":["a3e1067arc8a-h7"],"title":"Evaluate the integral","text":"Let $$u=y^4+1$$. Then $$du=4y^3 dy$$. When $$y=0$$, $$u=1$$, and when $$y=2$$, $$u=17$$. Then, $$((2*pi)/3)*(\\\\int_{0}^{2} y^3 \\\\sqrt{1+y^4} \\\\,dy)=((2*pi)/3)*\\\\int_{1}^{17} \\\\frac{1}{4} \\\\sqrt{u} \\\\,du$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc8a-h9","type":"hint","dependencies":["a3e1067arc8a-h8"],"title":"Evaluate the integral","text":"Use a computer, calculator, or another device to solve the integral.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a3e1067arc9","title":"Find the length","body":"Find the length of the functions over the given interval.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.4 Arc Length of a Curve and Surface Area","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a3e1067arc9a","stepAnswer":["$$2\\\\sqrt{26}$$"],"problemType":"TextBox","stepTitle":"$$y=5x$$ from $$x=0$$ to $$x=2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2\\\\sqrt{26}$$","hints":{"DefaultPathway":[{"id":"a3e1067arc9a-h1","type":"hint","dependencies":[],"title":"Arc length formula","text":"Arc Length can be found by computing $$\\\\int_{a}^{b} \\\\sqrt{1+{\\\\operatorname{f\'}\\\\left(x\\\\right)}^2} \\\\,dx$$, f\'(x) is derivative of f(x) and interval [a,b]","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc9a-h2","type":"hint","dependencies":["a3e1067arc9a-h1"],"title":"Find the derivative of f(x)","text":"We have $$f(x)=y=5x$$, so $$f\'(x)=5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc9a-h3","type":"hint","dependencies":["a3e1067arc9a-h2"],"title":"Plug in","text":"After we plug f\'(x) and interval, we will have Arc $$Length=\\\\int_{0}^{2} \\\\sqrt{1+5^2} \\\\,dx=\\\\int_{0}^{2} \\\\sqrt{26} \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc9a-h4","type":"hint","dependencies":["a3e1067arc9a-h3"],"title":"Evaluate the integral","text":"We can take $$\\\\sqrt{26}$$ out of integral, which gives us Arc $$Length=\\\\sqrt{26}$$ times $$x$$ with limit from $$0$$ to $$2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a3e1067arc9a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2\\\\sqrt{26}$$"],"dependencies":["a3e1067arc9a-h4"],"title":"Evaluate the integral","text":"What is $$\\\\sqrt{26} \\\\left(2-0\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a3e5c4cpercent1","title":"Translate and Solve Basic Percent Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Solve Percent Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3e5c4cpercent1a","stepAnswer":["$$31.5$$"],"problemType":"TextBox","stepTitle":"Translate and solve: What number is 35% of 90?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$31.5$$","hints":{"DefaultPathway":[{"id":"a3e5c4cpercent1a-h1","type":"hint","dependencies":[],"title":"Translate into algebra","text":"Let $$n=the$$ number. $$n=0.35\\\\times90$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent1a-h2","type":"hint","dependencies":["a3e5c4cpercent1a-h1"],"title":"Of and \\"Is\\" in math","text":"Of means multiple and \\"is\\" means equals","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent1a-h3","type":"hint","dependencies":["a3e5c4cpercent1a-h2"],"title":"Multiply.","text":"$$n=31.5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3e5c4cpercent10","title":"Find Percent Increase","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Solve Percent Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3e5c4cpercent10a","stepAnswer":["$$8.8$$"],"problemType":"TextBox","stepTitle":"Find the percent increase. (Round to the nearest tenth of a percent.) In $$2011$$, the IRS increased the deductible mileage cost to $$55.5$$ cents from $$51$$ cents.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8.8$$","hints":{"DefaultPathway":[{"id":"a3e5c4cpercent10a-h1","type":"hint","dependencies":[],"title":"Find the Difference","text":"Find the difference between the two numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent10a-h2","type":"hint","dependencies":["a3e5c4cpercent10a-h1"],"title":"The Difference","text":"The difference is $$55.5-51$$, or $$4.5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent10a-h3","type":"hint","dependencies":["a3e5c4cpercent10a-h2"],"title":"Find the Percent","text":"Divide the difference by the original value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent10a-h4","type":"hint","dependencies":["a3e5c4cpercent10a-h3"],"title":"The Percentage","text":"$$4.5$$ divided by $$51$$ is $$0.088$$. Change this to percent form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent10a-h5","type":"hint","dependencies":["a3e5c4cpercent10a-h4"],"title":"The Answer","text":"Multiply by $$100$$ to change to percentage. The percentage is $$8.8\\\\%$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3e5c4cpercent11","title":"Find Percent Increase","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Solve Percent Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3e5c4cpercent11a","stepAnswer":["$$50$$"],"problemType":"TextBox","stepTitle":"Find the percent increase. In $$1995$$, the standard bus fare in Chicago was $$\\\\$1.50$$. In $$2008$$, the standard bus fare was $$\\\\$2.25$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$50$$","hints":{"DefaultPathway":[{"id":"a3e5c4cpercent11a-h1","type":"hint","dependencies":[],"title":"Find the Difference","text":"Find the difference between the two numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent11a-h2","type":"hint","dependencies":["a3e5c4cpercent11a-h1"],"title":"The Difference","text":"The difference is $$2.25-1.5$$, or $$0.75$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent11a-h3","type":"hint","dependencies":["a3e5c4cpercent11a-h2"],"title":"Find the Percent","text":"Divide the difference by the original value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent11a-h4","type":"hint","dependencies":["a3e5c4cpercent11a-h3"],"title":"The Percentage","text":"$$0.75$$ divided by $$1.5$$ is $$0.5$$. Change this to percent form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent11a-h5","type":"hint","dependencies":["a3e5c4cpercent11a-h4"],"title":"The Answer","text":"Multiply by $$100$$ to change to percentage. The percentage is 50%.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3e5c4cpercent12","title":"Find Percent Decrease","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Solve Percent Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3e5c4cpercent12a","stepAnswer":["$$1.9$$"],"problemType":"TextBox","stepTitle":"The average price of a gallon of gas in one city in June $$2014$$ was $$\\\\$3.71$$. The average price in that city in July was $$\\\\$3.64$$. Find the percent decrease.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1.9$$","hints":{"DefaultPathway":[{"id":"a3e5c4cpercent12a-h1","type":"hint","dependencies":[],"title":"Find the Difference","text":"Find the difference between the two numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent12a-h2","type":"hint","dependencies":["a3e5c4cpercent12a-h1"],"title":"The Difference","text":"The difference is $$3.71-3.64$$, or $$0.07$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent12a-h3","type":"hint","dependencies":["a3e5c4cpercent12a-h2"],"title":"Find the Percent","text":"Divide the difference by the original value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent12a-h4","type":"hint","dependencies":["a3e5c4cpercent12a-h3"],"title":"The Percentage","text":"$$0.07$$ divided by $$3.71$$ is $$0.019$$. Change this to percent form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent12a-h5","type":"hint","dependencies":["a3e5c4cpercent12a-h4"],"title":"The Answer","text":"Multiply by $$100$$ to change to percentage. The percentage is $$1.9\\\\%$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3e5c4cpercent13","title":"Find Percent Decrease","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Solve Percent Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3e5c4cpercent13a","stepAnswer":["$$6.3$$"],"problemType":"TextBox","stepTitle":"Find the percent decrease. (Round to the nearest tenth of a percent.) The population of North Dakota was about 672,000 in $$2010$$. The population is projected to be about 630,000 in $$2020$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6.3$$","hints":{"DefaultPathway":[{"id":"a3e5c4cpercent13a-h1","type":"hint","dependencies":[],"title":"Find the Difference","text":"Find the difference between the two numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent13a-h2","type":"hint","dependencies":["a3e5c4cpercent13a-h1"],"title":"The Difference","text":"The difference is $$672000-630000$$, or $$42000$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent13a-h3","type":"hint","dependencies":["a3e5c4cpercent13a-h2"],"title":"Find the Percent","text":"Divide the difference by the original value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent13a-h4","type":"hint","dependencies":["a3e5c4cpercent13a-h3"],"title":"The Percentage","text":"$$42000$$ divided by $$672000$$ is $$0.063$$. Change this to percent form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent13a-h5","type":"hint","dependencies":["a3e5c4cpercent13a-h4"],"title":"The Answer","text":"Multiply by $$100$$ to change to percentage. The percentage is $$6.3\\\\%$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3e5c4cpercent14","title":"Find Percent Decrease","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Solve Percent Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3e5c4cpercent14a","stepAnswer":["$$10$$"],"problemType":"TextBox","stepTitle":"Find the percent decrease. Last year, Sheila\u2019s salary was $42,000. Because of furlough days, this year, her salary was $37,800.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10$$","hints":{"DefaultPathway":[{"id":"a3e5c4cpercent14a-h1","type":"hint","dependencies":[],"title":"Find the Difference","text":"Find the difference between the two numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent14a-h2","type":"hint","dependencies":["a3e5c4cpercent14a-h1"],"title":"The Difference","text":"The difference is $$42000-37800$$, or $$4200$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent14a-h3","type":"hint","dependencies":["a3e5c4cpercent14a-h2"],"title":"Find the Percent","text":"Divide the difference by the original value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent14a-h4","type":"hint","dependencies":["a3e5c4cpercent14a-h3"],"title":"The Percentage","text":"$$4200$$ divided by $$42000$$ is $$0.1$$. Change this to percent form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent14a-h5","type":"hint","dependencies":["a3e5c4cpercent14a-h4"],"title":"The Answer","text":"Multiply by $$100$$ to change to percentage. The percentage is 10%.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3e5c4cpercent15","title":"Solve Applications of Percent","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Solve Percent Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3e5c4cpercent15a","stepAnswer":["$$26.1$$"],"problemType":"TextBox","stepTitle":"Veronica is planning to make muffins from a mix. 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Change this to percentage.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent15a-h3","type":"hint","dependencies":["a3e5c4cpercent15a-h2"],"title":"The Answer","text":"Multiply by $$100$$ to change to percentage. The percentage is $$26.1\\\\%$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3e5c4cpercent16","title":"Solve Applications of Percent","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Solve Percent Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3e5c4cpercent16a","stepAnswer":["$$40$$"],"problemType":"TextBox","stepTitle":"Solve. Round to the nearest whole percent. 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Change this to percentage.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent16a-h3","type":"hint","dependencies":["a3e5c4cpercent16a-h2"],"title":"The Answer","text":"Multiply by $$100$$ to change to percentage. 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In other words, what variable are we solving for?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["P","$$r$$","$$t$$","I"]},{"id":"a3e5c4cpercent22a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$900$$"],"dependencies":["a3e5c4cpercent22a-h2"],"title":"Finding the value for I","text":"How much interest did Jim\'s sister pay? 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Write without the dollar sign.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent22a-h6","type":"hint","dependencies":["a3e5c4cpercent22a-h5"],"title":"Solving for $$r$$","text":"The formula for $$r$$ is $$r=\\\\frac{I}{P t}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3e5c4cpercent23","title":"Solve Simple Interest Applications","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Solve Percent Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3e5c4cpercent23a","stepAnswer":["4%"],"problemType":"MultipleChoice","stepTitle":"Hang borrowed $7,500 from her parents to pay her tuition. In $$5$$ years, she paid them $1,500 interest in addition to the $7,500 she borrowed. What was the rate of interest?","stepBody":"","answerType":"string","variabilization":{},"choices":["$$0.04\\\\%$$","4%","1%","10%"],"hints":{"DefaultPathway":[{"id":"a3e5c4cpercent23a-h1","type":"hint","dependencies":[],"title":"Simple Interest Formula","text":"The formula for simple interest is $$I=Prt$$, with I being interest, P being the initial amount of money invested (also called principal), $$r$$ being the rate, and $$t$$ being the time.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent23a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$r$$"],"dependencies":["a3e5c4cpercent23a-h1"],"title":"Variable in question","text":"What do we want to find? In other words, what variable are we solving for?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["P","$$r$$","$$t$$","I"]},{"id":"a3e5c4cpercent23a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1500$$"],"dependencies":["a3e5c4cpercent23a-h2"],"title":"Finding the value for I","text":"How much interest did Hang pay? Write without the dollar sign.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent23a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a3e5c4cpercent23a-h3"],"title":"Finding the value for $$t$$","text":"How many years are we considering?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent23a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7500$$"],"dependencies":["a3e5c4cpercent23a-h4"],"title":"Finding the value for P","text":"How much money did Hang borrow? Write without the dollar sign.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent23a-h6","type":"hint","dependencies":["a3e5c4cpercent23a-h5"],"title":"Solving for $$r$$","text":"The formula for $$r$$ is $$r=\\\\frac{I}{P t}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3e5c4cpercent3","title":"Translate and Solve Basic Percent Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Solve Percent Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3e5c4cpercent3a","stepAnswer":["$$33$$"],"problemType":"TextBox","stepTitle":"Translate and solve: What number is 55% of 60?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$33$$","hints":{"DefaultPathway":[{"id":"a3e5c4cpercent3a-h1","type":"hint","dependencies":[],"title":"Translate into Algebra","text":"Let $$n=number$$. $$n=0.55$$ * $$60$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent3a-h2","type":"hint","dependencies":["a3e5c4cpercent3a-h1"],"title":"Of and \\"Is\\" in math","text":"Of means multiple and \\"is\\" means equals","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent3a-h3","type":"hint","dependencies":["a3e5c4cpercent3a-h2"],"title":"Multiply.","text":"$$n=33$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3e5c4cpercent4","title":"Translate and Solve Basic Percent Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Solve Percent Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3e5c4cpercent4a","stepAnswer":["$18"],"problemType":"MultipleChoice","stepTitle":"Translate and solve: $$6.5\\\\%$$ of what number is $$\\\\$1.17$$?","stepBody":"","answerType":"string","variabilization":{},"choices":["$16","$18","$17","$15"],"hints":{"DefaultPathway":[{"id":"a3e5c4cpercent4a-h1","type":"hint","dependencies":[],"title":"Translate.","text":"Let $$n=the$$ number. $$0.065n=1.17$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent4a-h2","type":"hint","dependencies":["a3e5c4cpercent4a-h1"],"title":"Multiply","text":"$$0.065n=1.17$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18$$"],"dependencies":["a3e5c4cpercent4a-h2"],"title":"Divide","text":"What do you get for $$n$$ when you divide both sides by $$0.065$$ and simplify?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3e5c4cpercent5","title":"Translate and Solve Basic Percent Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Solve Percent Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3e5c4cpercent5a","stepAnswer":["$26"],"problemType":"MultipleChoice","stepTitle":"Translate and solve: $$7.5\\\\%$$ of what number is $$\\\\$1.95$$?","stepBody":"","answerType":"string","variabilization":{},"choices":["$24","$25","$26","$27"],"hints":{"DefaultPathway":[{"id":"a3e5c4cpercent5a-h1","type":"hint","dependencies":[],"title":"Translate.","text":"Let $$n=the$$ number. $$0.075n=1.95$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent5a-h2","type":"hint","dependencies":["a3e5c4cpercent5a-h1"],"title":"Multiply.","text":"$$0.075n=1.95$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$26$$"],"dependencies":["a3e5c4cpercent5a-h2"],"title":"Divide","text":"What do you get for $$n$$ when you divide both sides by $$0.075$$ and simplify?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3e5c4cpercent6","title":"Translate and Solve Basic Percent Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Solve Percent Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3e5c4cpercent6a","stepAnswer":["$36"],"problemType":"MultipleChoice","stepTitle":"Translate and solve: $$8.5\\\\%$$ of what number is $$\\\\$3.06$$?","stepBody":"","answerType":"string","variabilization":{},"choices":["$33","$34","$35","$36"],"hints":{"DefaultPathway":[{"id":"a3e5c4cpercent6a-h1","type":"hint","dependencies":[],"title":"Translate.","text":"Let $$n=the$$ number. $$0.085n=3.06$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent6a-h2","type":"hint","dependencies":["a3e5c4cpercent6a-h1"],"title":"Multiply","text":"$$0.085n=3.06$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$36$$"],"dependencies":["a3e5c4cpercent6a-h2"],"title":"Divide","text":"What do you get for $$n$$ when you divide both sides by $$0.08$$ $$5$$ and simplify?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3e5c4cpercent7","title":"Translate and Solve Basic Percent Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Solve Percent Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3e5c4cpercent7a","stepAnswer":["150%"],"problemType":"MultipleChoice","stepTitle":"Translate and solve: $$144$$ is what percent of 96?","stepBody":"","answerType":"string","variabilization":{},"choices":["136%","140%","150%","148%"],"hints":{"DefaultPathway":[{"id":"a3e5c4cpercent7a-h1","type":"hint","dependencies":[],"title":"Translate into algebra","text":"Let $$p=the$$ percent. $$144=96p$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent7a-h2","type":"hint","dependencies":["a3e5c4cpercent7a-h1"],"title":"Multiply","text":"$$144=96p$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.5$$"],"dependencies":["a3e5c4cpercent7a-h2"],"title":"Divide","text":"What do you get for $$p$$ when you divide by $$96$$ and simplify?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent7a-h4","type":"hint","dependencies":["a3e5c4cpercent7a-h3"],"title":"Convert to percent.","text":"$$150\\\\%=p$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3e5c4cpercent8","title":"Translate and Solve Basic Percent Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Solve Percent Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3e5c4cpercent8a","stepAnswer":["125%"],"problemType":"MultipleChoice","stepTitle":"Translate and solve: $$110$$ is what percent of 88?","stepBody":"","answerType":"string","variabilization":{},"choices":["95%","110%","115%","125%"],"hints":{"DefaultPathway":[{"id":"a3e5c4cpercent8a-h1","type":"hint","dependencies":[],"title":"Translate into algebra.","text":"Let $$p=the$$ percent. $$110=88p$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent8a-h2","type":"hint","dependencies":["a3e5c4cpercent8a-h1"],"title":"Multiply","text":"$$110=88p$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.25$$"],"dependencies":["a3e5c4cpercent8a-h2"],"title":"Divide","text":"What do you get for $$p$$ when you divide by $$88$$ and simplify?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent8a-h4","type":"hint","dependencies":["a3e5c4cpercent8a-h3"],"title":"Convert to percent.","text":"$$125\\\\%=p$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a3e5c4cpercent9","title":"Find Percent Increase","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Solve Percent Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a3e5c4cpercent9a","stepAnswer":["$$38.5$$"],"problemType":"TextBox","stepTitle":"In $$2011$$, the California governor proposed raising community college fees from $26 a unit to $36 a unit. Find the percent increase. (Round to the nearest tenth of a percent.)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$38.5$$","hints":{"DefaultPathway":[{"id":"a3e5c4cpercent9a-h1","type":"hint","dependencies":[],"title":"Find the Difference","text":"Find the difference between the two numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent9a-h2","type":"hint","dependencies":["a3e5c4cpercent9a-h1"],"title":"The Difference","text":"The difference is $36-$26, or $10.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent9a-h3","type":"hint","dependencies":["a3e5c4cpercent9a-h2"],"title":"Find the Percent","text":"Divide the difference by the original value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent9a-h4","type":"hint","dependencies":["a3e5c4cpercent9a-h3"],"title":"The Percentage","text":"$$10$$ divided by $$26$$ is $$0.385$$. Change this to percent form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a3e5c4cpercent9a-h5","type":"hint","dependencies":["a3e5c4cpercent9a-h4"],"title":"The Answer","text":"Multiply by $$100$$ to change to percentage. The percentage is $$38.4\\\\%$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a41316cmatrices1","title":"Adding and Subtracting Matrices","body":"For the following exercises, use the matrices below and perform the matrix addition or subtraction.\\\\n$$A=\\\\begin{bmatrix} 1 & 3 \\\\\\\\ 0 & 7 \\\\end{bmatrix}$$, $$B=\\\\begin{bmatrix} 2 & 14 \\\\\\\\ 22 & 6 \\\\end{bmatrix}$$, $$C=\\\\begin{bmatrix} 1 & 5 \\\\\\\\ 8 & 92 \\\\\\\\ 12 & 6 \\\\end{bmatrix}$$, $$D=\\\\begin{bmatrix} 10 & 14 \\\\\\\\ 7 & 2 \\\\\\\\ 5 & 61 \\\\end{bmatrix}$$, $$E=\\\\begin{bmatrix} 6 & 12 \\\\\\\\ 14 & 5 \\\\end{bmatrix}$$, $$F=\\\\begin{bmatrix} 0 & 9 \\\\\\\\ 78 & 17 \\\\\\\\ 15 & 4 \\\\end{bmatrix}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Matrices and Matrix Operations","courseName":"OpenStax: College Algebra","steps":[{"id":"a41316cmatrices1a","stepAnswer":["$$\\\\begin{bmatrix} 11 & 19 \\\\\\\\ 15 & 94 \\\\\\\\ 17 & 67 \\\\end{bmatrix}$$"],"problemType":"MultipleChoice","stepTitle":"$$C+D$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 11 & 19 \\\\\\\\ 15 & 94 \\\\\\\\ 17 & 67 \\\\end{bmatrix}$$","choices":["$$\\\\begin{bmatrix} 19 & 11 \\\\\\\\ 15 & 49 \\\\\\\\ 17 & 67 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} 11 & 19 \\\\\\\\ 15 & 94 \\\\\\\\ 17 & 67 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} 11 & 19 \\\\\\\\ 5 & 94 \\\\\\\\ 1 & 6 \\\\end{bmatrix}$$","None of the above"],"hints":{"DefaultPathway":[{"id":"a41316cmatrices1a-h1","type":"hint","dependencies":[],"title":"Adding and Subtracting Matrices","text":"Given matrices A and B of like dimensions, addition and subtraction of A and B will produce matrix C or matrix D of the same dimension.\\\\n$$A+B=C$$ such that $$a_{i,j}+b_{i,j}=c_{i,j}$$\\\\n$$A-B=D$$ such that $$a_{i,j}-b_{i,j}=d_{i,j}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices1a-h2","type":"hint","dependencies":["a41316cmatrices1a-h1"],"title":"Adding Corresponding Entries","text":"Since the dimension of the matrices are the same, we perform matrix addition $$A+B=C$$ such that $$a_{i,j}+b_{i,j}=c_{i,j}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11$$"],"dependencies":["a41316cmatrices1a-h2"],"title":"Adding Corresponding Entries","text":"We will start by adding the top left entry of C, $$c_{1,1}$$, and D, $$d_{1,1}$$. What is $$c_{1,1}+d_{1,1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$19$$"],"dependencies":["a41316cmatrices1a-h3"],"title":"Adding Corresponding Entries","text":"We will next add the top right entry of C, $$c_{1,2}$$, and D, $$d_{1,2}$$. What is $$c_{1,2}+d_{1,2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices1a-h5","type":"hint","dependencies":["a41316cmatrices1a-h4"],"title":"Adding Corresponding Entries","text":"Repeat the same process for each corresponding entries to compute the addition of the two matrices.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a41316cmatrices10","title":"Adding and Subtracting Matrices","body":"Use the matrices below to perform the indicated operation if possible. $$A=\\\\begin{bmatrix} 2 & -5 \\\\\\\\ 6 & 7 \\\\end{bmatrix}$$, $$B=\\\\begin{bmatrix} -9 & 6 \\\\\\\\ -4 & 2 \\\\end{bmatrix}$$, $$C=\\\\begin{bmatrix} 0 & 9 \\\\\\\\ 7 & 1 \\\\end{bmatrix}$$, $$D=\\\\begin{bmatrix} -8 & 7 & -5 \\\\\\\\ 4 & 3 & 2 \\\\\\\\ 0 & 9 & 2 \\\\end{bmatrix}$$, $$E=\\\\begin{bmatrix} 4 & 5 & 3 \\\\\\\\ 7 & -6 & -5 \\\\\\\\ 1 & 0 & 9 \\\\end{bmatrix}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Matrices and Matrix Operations","courseName":"OpenStax: College Algebra","steps":[{"id":"a41316cmatrices10a","stepAnswer":["$$\\\\begin{bmatrix} -7 & -8 \\\\\\\\ -5 & 8 \\\\end{bmatrix}$$"],"problemType":"MultipleChoice","stepTitle":"$$A+B-C$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} -7 & -8 \\\\\\\\ -5 & 8 \\\\end{bmatrix}$$","choices":["$$\\\\begin{bmatrix} -7 & -8 \\\\\\\\ -5 & 8 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} 7 & 8 \\\\\\\\ 5 & -8 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} -7 & -8 \\\\\\\\ 8 & -5 \\\\end{bmatrix}$$","None of the above"],"hints":{"DefaultPathway":[{"id":"a41316cmatrices10a-h1","type":"hint","dependencies":[],"title":"Adding and Subtracting Matrices","text":"Given matrices A and B of like dimensions, addition and subtraction of A and B will produce matrix C or matrix D of the same dimension.\\\\n$$A+B=C$$ such that $$a_{i,j}+b_{i,j}=c_{i,j}$$\\\\n$$A-B=D$$ such that $$a_{i,j}-b_{i,j}=d_{i,j}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices10a-h2","type":"hint","dependencies":["a41316cmatrices10a-h1"],"title":"Adding Corresponding Entries","text":"Since the dimension of the matrices are the same, we perform matrix addition $$A+B=D$$ such that $$a_{i,j}+b_{i,j}=d_{i,j}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-7$$"],"dependencies":["a41316cmatrices10a-h2"],"title":"Adding Corresponding Entries","text":"We will start by adding the top left entry of A, $$a_{1,1}$$, and B, $$b_{1,1}$$. What is $$a_{1,1}+b_{1,1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a41316cmatrices10a-h3"],"title":"Adding Corresponding Entries","text":"We will next add the top right entry of A, $$a_{1,2}$$, and B, $$b_{1,2}$$. What is $$a_{1,2}+b_{1,2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices10a-h5","type":"hint","dependencies":["a41316cmatrices10a-h4"],"title":"Adding Corresponding Entries","text":"Repeat the same process for each corresponding entries to compute the addition of the two matrices, A and B. This would enable us to proceed to perform the subtractoin of matrices","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices10a-h6","type":"hint","dependencies":["a41316cmatrices10a-h5"],"title":"Subtracting Corresponding Entries","text":"Since the dimension of the matrices are the same, we perform matrix subtraction $$D-C=E$$ such that $$d_{i,j}-c_{i,j}=e_{i,j}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices10a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-7$$"],"dependencies":["a41316cmatrices10a-h6"],"title":"Subtracting Corresponding Entries","text":"We will start by subtracting the top left entry of C, $$c_{1,1}$$, from D, $$d_{1,1}$$. What is $$d_{1,1}-c_{1,1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices10a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-8$$"],"dependencies":["a41316cmatrices10a-h7"],"title":"Subtracting Corresponding Entries","text":"We will next subtract the top right entry of C, $$c_{1,2}$$, from D, $$d_{1,2}$$. What is $$d_{1,2}-c_{1,2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices10a-h9","type":"hint","dependencies":["a41316cmatrices10a-h8"],"title":"Subtracting Corresponding Entries","text":"Repeat the same process for each corresponding entries to compute the subtraction between the two matrices.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a41316cmatrices11","title":"Finding the Dimensions of the Given Matrix","body":"Find the dimensions of matrix A.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Matrices and Matrix Operations","courseName":"OpenStax: College Algebra","steps":[{"id":"a41316cmatrices11a","stepAnswer":["3x3"],"problemType":"MultipleChoice","stepTitle":"$$A=\\\\begin{bmatrix} 2 & 1 & 0 \\\\\\\\ 2 & 4 & 7 \\\\\\\\ 3 & 1 & -2 \\\\end{bmatrix}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["3x1","3x2","3x3","1x3"],"hints":{"DefaultPathway":[{"id":"a41316cmatrices11a-h1","type":"hint","dependencies":[],"title":"Dimensions of a Matrix","text":"Matrices are often referred to by their dimensions: mxn indicating $$m$$ rows and $$n$$ columns.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices11a-h2","type":"hint","dependencies":["a41316cmatrices11a-h1"],"title":"Number of Rows and Columns in Matrix A","text":"Matrix A has three rows and three columns.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a41316cmatrices12","title":"Locating Entries of a Matrix","body":"Locate a31 and a22 for Matrix A.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Matrices and Matrix Operations","courseName":"OpenStax: College Algebra","steps":[{"id":"a41316cmatrices12a","stepAnswer":["$$a31=3$$, a22 $$=$$ $$4$$"],"problemType":"MultipleChoice","stepTitle":"$$A=\\\\begin{bmatrix} 2 & 1 & 0 \\\\\\\\ 2 & 4 & 7 \\\\\\\\ 3 & 1 & -2 \\\\end{bmatrix}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$a31=3$$, a22 $$=$$ $$4$$","choices":["$$a31=3$$, a22 $$=$$ $$4$$","a31 $$=$$ $$4$$, a22 $$=$$ $$3$$"],"hints":{"DefaultPathway":[{"id":"a41316cmatrices12a-h1","type":"hint","dependencies":[],"title":"Interpreting Entries","text":"Entry a31 is the number at row $$3$$, column $$1$$, which is $$3$$. The entry a22 is the number at row $$2$$, column $$2$$, which is $$4$$. Remember, the row comes first, then the column.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a41316cmatrices13","title":"Finding the Sum of Two Matrices","body":"Find the sum of matrices A and B.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Matrices and Matrix Operations","courseName":"OpenStax: College Algebra","steps":[{"id":"a41316cmatrices13a","stepAnswer":["$$\\\\begin{bmatrix} 6 & 8 \\\\\\\\ 10 & 12 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"$$A=\\\\begin{bmatrix} 1 & 2 \\\\\\\\ 3 & 4 \\\\end{bmatrix}$$ and $$B=\\\\begin{bmatrix} 5 & 6 \\\\\\\\ 7 & 8 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 6 & 8 \\\\\\\\ 10 & 12 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"a41316cmatrices13a-h1","type":"hint","dependencies":[],"title":"Matrix Addition","text":"Add corresponding entries. Add the entry in row $$1$$, column $$1$$, a11, of matrix A to the entry in row $$1$$, column $$1$$, b11, of B. Continue the pattern until all entries have been added.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a41316cmatrices14","title":"Finding the Sum of Two Matrices","body":"Find the sum of matrices A and B.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Matrices and Matrix Operations","courseName":"OpenStax: College Algebra","steps":[{"id":"a41316cmatrices14a","stepAnswer":["$$\\\\begin{bmatrix} a+e & b+f \\\\\\\\ c+g & d+h \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"$$A=\\\\begin{bmatrix} a & b \\\\\\\\ c & d \\\\end{bmatrix}$$ and $$B=\\\\begin{bmatrix} e & f \\\\\\\\ g & h \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} a+e & b+f \\\\\\\\ c+g & d+h \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"a41316cmatrices14a-h1","type":"hint","dependencies":[],"title":"Matrix Addition","text":"Add corresponding entries. Add the entry in row $$1$$, column $$1$$, a11, of matrix A to the entry in row $$1$$, column $$1$$, b11, of B. Continue the pattern until all entries have been added.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a41316cmatrices15","title":"Finding the Sum of Two Matrices","body":"Find the sum of matrices A and B.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Matrices and Matrix Operations","courseName":"OpenStax: College Algebra","steps":[{"id":"a41316cmatrices15a","stepAnswer":["$$\\\\begin{bmatrix} 9 & 10 \\\\\\\\ 3 & 9 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"$$A=\\\\begin{bmatrix} 4 & 1 \\\\\\\\ 3 & 2 \\\\end{bmatrix}$$ and $$B=\\\\begin{bmatrix} 5 & 9 \\\\\\\\ 0 & 7 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 9 & 10 \\\\\\\\ 3 & 9 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"a41316cmatrices15a-h1","type":"hint","dependencies":[],"title":"Matrix Addition","text":"Add corresponding entries. Add the entry in row $$1$$, column $$1$$, a11, of matrix A to the entry in row $$1$$, column $$1$$, b11, of B. Continue the pattern until all entries have been added.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a41316cmatrices16","title":"Finding the Difference of Two Matrices","body":"Find the difference of A and B.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Matrices and Matrix Operations","courseName":"OpenStax: College Algebra","steps":[{"id":"a41316cmatrices16a","stepAnswer":["$$\\\\begin{bmatrix} -10 & 2 \\\\\\\\ -5 & -3 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"$$A=\\\\begin{bmatrix} -2 & 3 \\\\\\\\ 0 & 1 \\\\end{bmatrix}$$ and $$B=\\\\begin{bmatrix} 8 & 1 \\\\\\\\ 5 & 4 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} -10 & 2 \\\\\\\\ -5 & -3 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"a41316cmatrices16a-h1","type":"hint","dependencies":[],"title":"Matrix Subtraction","text":"Subtract corresponding entries. Subtract the entry in row $$1$$, column $$1$$, a11, of matrix A from the entry in row $$1$$, column $$1$$, b11, of B. Continue the pattern until all entries B have been subtracted from each corresponding entry of A.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a41316cmatrices17","title":"Finding the Sum of Two Matrices","body":"Find the sum of matrices A and B.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Matrices and Matrix Operations","courseName":"OpenStax: College Algebra","steps":[{"id":"a41316cmatrices17a","stepAnswer":["$$\\\\begin{bmatrix} 5 & 4 \\\\\\\\ 2 & 5 \\\\\\\\ -3 & 0 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"$$A=\\\\begin{bmatrix} 2 & 6 \\\\\\\\ 1 & 0 \\\\\\\\ 1 & -3 \\\\end{bmatrix}$$ and $$B=\\\\begin{bmatrix} 3 & -2 \\\\\\\\ 1 & 5 \\\\\\\\ -4 & 3 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 5 & 4 \\\\\\\\ 2 & 5 \\\\\\\\ -3 & 0 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"a41316cmatrices17a-h1","type":"hint","dependencies":[],"title":"Matrix Addition","text":"Add corresponding entries. Add the entry in row $$1$$, column $$1$$, a11, of matrix A to the entry in row $$1$$, column $$1$$, b11, of B. Continue the pattern until all entries have been added.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a41316cmatrices18","title":"Multiplying the Matrix By a Scalar","body":"Multiply matrix A by the scalar $$3$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Matrices and Matrix Operations","courseName":"OpenStax: College Algebra","steps":[{"id":"a41316cmatrices18a","stepAnswer":["$$\\\\begin{bmatrix} 24 & 3 \\\\\\\\ 15 & 12 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"$$A=\\\\begin{bmatrix} 8 & 1 \\\\\\\\ 5 & 4 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 24 & 3 \\\\\\\\ 15 & 12 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"a41316cmatrices18a-h1","type":"hint","dependencies":[],"title":"Multiplying a Matrix by a Scalar","text":"To multiply a matrix A by a scaler C, multiply each entry in A by C.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a41316cmatrices19","title":"Multiplying the Matrix By a Scalar","body":"Given matrix B, find -2B.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Matrices and Matrix Operations","courseName":"OpenStax: College Algebra","steps":[{"id":"a41316cmatrices19a","stepAnswer":["$$\\\\begin{bmatrix} -8 & -2 \\\\\\\\ -6 & -4 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"$$B=\\\\begin{bmatrix} 4 & 1 \\\\\\\\ 3 & 2 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} -8 & -2 \\\\\\\\ -6 & -4 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"a41316cmatrices19a-h1","type":"hint","dependencies":[],"title":"Multiplying a Matrix by a Scalar","text":"To multiply a matrix A by a scaler C, multiply each entry in A by C.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a41316cmatrices2","title":"Adding and Subtracting Matrices","body":"For the following exercises, use the matrices below and perform the matrix addition or subtraction.\\\\n$$A=\\\\begin{bmatrix} 1 & 3 \\\\\\\\ 0 & 7 \\\\end{bmatrix}$$, $$B=\\\\begin{bmatrix} 2 & 14 \\\\\\\\ 22 & 6 \\\\end{bmatrix}$$, $$C=\\\\begin{bmatrix} 1 & 5 \\\\\\\\ 8 & 92 \\\\\\\\ 12 & 6 \\\\end{bmatrix}$$, $$D=\\\\begin{bmatrix} 10 & 14 \\\\\\\\ 7 & 2 \\\\\\\\ 5 & 61 \\\\end{bmatrix}$$, $$E=\\\\begin{bmatrix} 6 & 12 \\\\\\\\ 14 & 5 \\\\end{bmatrix}$$, $$F=\\\\begin{bmatrix} 0 & 9 \\\\\\\\ 78 & 17 \\\\\\\\ 15 & 4 \\\\end{bmatrix}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Matrices and Matrix Operations","courseName":"OpenStax: College Algebra","steps":[{"id":"a41316cmatrices2a","stepAnswer":["$$\\\\begin{bmatrix} -4 & 2 \\\\\\\\ 8 & 1 \\\\end{bmatrix}$$"],"problemType":"MultipleChoice","stepTitle":"B-E","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} -4 & 2 \\\\\\\\ 8 & 1 \\\\end{bmatrix}$$","choices":["$$\\\\begin{bmatrix} -4 & 2 \\\\\\\\ 8 & 1 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} -3 & 2 \\\\\\\\ 4 & 1 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} 4 & 2 \\\\\\\\ -8 & 1 \\\\end{bmatrix}$$","None of the above"],"hints":{"DefaultPathway":[{"id":"a41316cmatrices2a-h1","type":"hint","dependencies":[],"title":"Adding and Subtracting Matrices","text":"Given matrices A and B of like dimensions, addition and subtraction of A and B will produce matrix C or matrix D of the same dimension.\\\\n$$A+B=C$$ such that $$a_{i,j}+b_{i,j}=c_{i,j}$$\\\\n$$A-B=D$$ such that $$a_{i,j}-b_{i,j}=d_{i,j}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices2a-h2","type":"hint","dependencies":["a41316cmatrices2a-h1"],"title":"Subtracting Corresponding Entries","text":"Since the dimension of the matrices are the same, we perform matrix subtraction $$A-B=D$$ such that $$a_{i,j}-b_{i,j}=d_{i,j}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["a41316cmatrices2a-h2"],"title":"Subtracting Corresponding Entries","text":"We will start by subtracting the top left entry of D, $$d_{1,1}$$, from C, $$c_{1,1}$$. What is $$c_{1,1}-d_{1,1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a41316cmatrices2a-h3"],"title":"Subtracting Corresponding Entries","text":"We will next subtract the top right entry of D, $$d_{1,2}$$, from C, $$c_{1,2}$$. What is $$c_{1,2}-d_{1,2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices2a-h5","type":"hint","dependencies":["a41316cmatrices2a-h4"],"title":"Subtracting Corresponding Entries","text":"Repeat the same process for each corresponding entries to compute the subtraction between the two matrices.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a41316cmatrices20","title":"Multiplying the Matrix By a Scalar","body":"Find the sum $$3A+2B$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Matrices and Matrix Operations","courseName":"OpenStax: College Algebra","steps":[{"id":"a41316cmatrices20a","stepAnswer":["$$\\\\begin{bmatrix} 1 & -2 & 2 \\\\\\\\ 0 & -9 & 10 \\\\\\\\ 12 & 11 & -26 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"$$A=\\\\begin{bmatrix} 1 & -2 & 0 \\\\\\\\ 0 & -1 & 2 \\\\\\\\ 4 & 3 & -6 \\\\end{bmatrix}$$ and $$B=\\\\begin{bmatrix} -1 & 2 & 1 \\\\\\\\ 0 & -3 & 2 \\\\\\\\ 0 & 1 & -4 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 1 & -2 & 2 \\\\\\\\ 0 & -9 & 10 \\\\\\\\ 12 & 11 & -26 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"a41316cmatrices20a-h1","type":"hint","dependencies":[],"title":"Order of Steps","text":"First, find 3A, then 2B. Next, add them together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices20a-h2","type":"hint","dependencies":["a41316cmatrices20a-h1"],"title":"Matrix Addition","text":"Add corresponding entries. Add the entry in row $$1$$, column $$1$$, a11, of matrix A to the entry in row $$1$$, column $$1$$, b11, of B. Continue the pattern until all entries have been added.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices20a-h3","type":"hint","dependencies":["a41316cmatrices20a-h2"],"title":"Multiplying a Matrix by a Scalar","text":"To multiply a matrix A by a scaler C, multiply each entry in A by C.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a41316cmatrices22","title":"Matrix Operations","body":"Use the following matrices to answer the questions: $$A=\\\\begin{bmatrix} 2 & -5 \\\\\\\\ 6 & 7 \\\\end{bmatrix}$$, $$B=\\\\begin{bmatrix} -9 & 6 \\\\\\\\ -4 & 2 \\\\end{bmatrix}$$, $$C=\\\\begin{bmatrix} 0 & 9 \\\\\\\\ 7 & 1 \\\\end{bmatrix}$$, $$D=\\\\begin{bmatrix} -8 & 7-5 \\\\\\\\ 4 & 3 & 2 \\\\\\\\ 0 & 9 & 2 \\\\end{bmatrix}$$, $$E=\\\\begin{bmatrix} 4 & 5 & 3 \\\\\\\\ 7 & -6 & -5 \\\\\\\\ 1 & 0 & 9 \\\\end{bmatrix}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Matrices and Matrix Operations","courseName":"OpenStax: College Algebra","steps":[{"id":"a41316cmatrices22a","stepAnswer":["No Solution"],"problemType":"TextBox","stepTitle":"4A + 5D","stepBody":"If the problem has no solution, simply enter \\"No Solution\\" as your answer.","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a41316cmatrices22a-h1","type":"hint","dependencies":[],"title":"Determining Solvability","text":"First, determine if the expression is defined. A matrix operation of addition is defined when the two matrices have the same dimensions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices22a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a41316cmatrices22a-h1"],"title":"Determining Solvability","text":"Do the two matrices you are adding have the same dimensions?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a41316cmatrices22a-h3","type":"hint","dependencies":["a41316cmatrices22a-h2"],"title":"Interpreting Solvability","text":"Therefore, the matrix cannot be added and is undefined with no solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a41316cmatrices25","title":"Matrix Operations","body":"Use the following matrices to answer the questions: $$A=\\\\begin{bmatrix} 2 & -5 \\\\\\\\ 6 & 7 \\\\end{bmatrix}$$, $$B=\\\\begin{bmatrix} -9 & 6 \\\\\\\\ -4 & 2 \\\\end{bmatrix}$$, $$C=\\\\begin{bmatrix} 0 & 9 \\\\\\\\ 7 & 1 \\\\end{bmatrix}$$, $$D=\\\\begin{bmatrix} -8 & 7-5 \\\\\\\\ 4 & 3 & 2 \\\\\\\\ 0 & 9 & 2 \\\\end{bmatrix}$$, $$E=\\\\begin{bmatrix} 4 & 5 & 3 \\\\\\\\ 7 & -6 & -5 \\\\\\\\ 1 & 0 & 9 \\\\end{bmatrix}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Matrices and Matrix Operations","courseName":"OpenStax: College Algebra","steps":[{"id":"a41316cmatrices25a","stepAnswer":["No Solution"],"problemType":"TextBox","stepTitle":"$$C-0.5D$$","stepBody":"If the problem has no solution, simply enter \\"No Solution\\" as your answer.","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a41316cmatrices25a-h1","type":"hint","dependencies":[],"title":"Determining Solvability","text":"First, determine if the expression is defined. A matrix operation of addition is defined when the two matrices have the same dimensions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices25a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a41316cmatrices25a-h1"],"title":"Determining Solvability","text":"Do the two matrices you are working with have the same dimensions?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a41316cmatrices25a-h3","type":"hint","dependencies":["a41316cmatrices25a-h2"],"title":"Interpreting Solvability","text":"Therefore, the matrix cannot be simplified and is undefined with no solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a41316cmatrices3","title":"Adding and Subtracting Matrices","body":"For the following exercises, use the matrices below and perform the matrix addition or subtraction.\\\\n$$A=\\\\begin{bmatrix} 1 & 3 \\\\\\\\ 0 & 7 \\\\end{bmatrix}$$, $$B=\\\\begin{bmatrix} 2 & 14 \\\\\\\\ 22 & 6 \\\\end{bmatrix}$$, $$C=\\\\begin{bmatrix} 1 & 5 \\\\\\\\ 8 & 92 \\\\\\\\ 12 & 6 \\\\end{bmatrix}$$, $$D=\\\\begin{bmatrix} 10 & 14 \\\\\\\\ 7 & 2 \\\\\\\\ 5 & 61 \\\\end{bmatrix}$$, $$E=\\\\begin{bmatrix} 6 & 12 \\\\\\\\ 14 & 5 \\\\end{bmatrix}$$, $$F=\\\\begin{bmatrix} 0 & 9 \\\\\\\\ 78 & 17 \\\\\\\\ 15 & 4 \\\\end{bmatrix}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Matrices and Matrix Operations","courseName":"OpenStax: College Algebra","steps":[{"id":"a41316cmatrices3a","stepAnswer":["$$\\\\begin{bmatrix} 1 & 14 \\\\\\\\ 86 & 109 \\\\\\\\ 27 & 10 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"$$C+F$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 1 & 14 \\\\\\\\ 86 & 109 \\\\\\\\ 27 & 10 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"a41316cmatrices3a-h1","type":"hint","dependencies":[],"title":"Adding and Subtracting Matrices","text":"Given matrices A and B of like dimensions, addition and subtraction of A and B will produce matrix C or matrix D of the same dimension.\\\\n$$A+B=C$$ such that $$a_{i,j}+b_{i,j}=c_{i,j}$$\\\\n$$A-B=D$$ such that $$a_{i,j}-b_{i,j}=d_{i,j}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices3a-h2","type":"hint","dependencies":["a41316cmatrices3a-h1"],"title":"Adding Corresponding Entries","text":"Since the dimension of the matrices are the same, we perform matrix addition $$A+B=C$$ such that $$a_{i,j}+b_{i,j}=c_{i,j}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a41316cmatrices3a-h2"],"title":"Adding Corresponding Entries","text":"We will start by adding the top left entry of C, $$c_{1,1}$$, and D, $$d_{1,1}$$. What is $$c_{1,1}+d_{1,1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$14$$"],"dependencies":["a41316cmatrices3a-h3"],"title":"Adding Corresponding Entries","text":"We will next add the top right entry of C, $$c_{1,2}$$, and D, $$d_{1,2}$$. What is $$c_{1,2}+d_{1,2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices3a-h5","type":"hint","dependencies":["a41316cmatrices3a-h4"],"title":"Adding Corresponding Entries","text":"Repeat the same process for each corresponding entries to compute the addition of the two matrices.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a41316cmatrices30","title":"Matrix Operations","body":"Use the following matrices to answer the questions: $$A=\\\\begin{bmatrix} -10 & 20 \\\\\\\\ 5 & 25 \\\\end{bmatrix}$$, $$B=\\\\begin{bmatrix} 40 & 10 \\\\\\\\ -20 & 30 \\\\end{bmatrix}$$, $$C=\\\\begin{bmatrix} -1 & 0 \\\\\\\\ 0 & -1 \\\\\\\\ 1 & 0 \\\\end{bmatrix}$$. If the problem has no solution, simply enter \\"No Solution\\" as your answer.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Matrices and Matrix Operations","courseName":"OpenStax: College Algebra","steps":[{"id":"a41316cmatrices30a","stepAnswer":["No Solution"],"problemType":"MultipleChoice","stepTitle":"BC","stepBody":"","answerType":"string","variabilization":{},"choices":["$$\\\\begin{bmatrix} 10 & -20 \\\\\\\\ -5 & -25 \\\\\\\\ -10 & 20 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} 11 & -20 \\\\\\\\ -5 & -25 \\\\\\\\ -15 & 20 \\\\end{bmatrix}$$","No Solution"],"hints":{"DefaultPathway":[{"id":"a41316cmatrices30a-h1","type":"hint","dependencies":[],"title":"Detemine if Multipliable","text":"Two matrices are Multipliable(able to be multiplied) if their inner dimensions are the same.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices30a-h2","type":"hint","dependencies":["a41316cmatrices30a-h1"],"title":"Determining If Multipliable","text":"The inner dimensions are not the same, therefore the matrix is not multipliable, and no solution exists.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a41316cmatrices4","title":"Finding Scalar Mulitples of a Matrix","body":"Perform scalar multiplication on the matrix $$C=\\\\begin{bmatrix} 16 & 3 & 7 & 18 \\\\\\\\ 90 & 5 & 3 & 29 \\\\end{bmatrix}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Matrices and Matrix Operations","courseName":"OpenStax: College Algebra","steps":[{"id":"a41316cmatrices4a","stepAnswer":["$$\\\\begin{bmatrix} -64 & -12 & -28 & -72 \\\\\\\\ -360 & -20 & -12 & -116 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"$$-4C$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} -64 & -12 & -28 & -72 \\\\\\\\ -360 & -20 & -12 & -116 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"a41316cmatrices4a-h1","type":"hint","dependencies":[],"title":"Scalar Multiplication","text":"Scalar multiplication involves finding the product of a constant by each entry in the matrix. Given $$A=\\\\begin{bmatrix} a_{1,1} & a_{1,2} \\\\\\\\ a_{2,1} & a_{2,2} \\\\end{bmatrix}$$, the scalar multiple\\\\n$$c A$$ is $$c A=c*\\\\begin{bmatrix} a_{1,1} & a_{1,2} \\\\\\\\ a_{2,1} & a_{2,2} \\\\end{bmatrix}=\\\\begin{bmatrix} c a_{1,1} & c a_{1,2} \\\\\\\\ c a_{2,1} & c a_{2,2} \\\\end{bmatrix}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-64$$"],"dependencies":["a41316cmatrices4a-h1"],"title":"Scalar Multiplication","text":"Multiply each entry in C by scalar $$-4$$. What is the top left entry, $$c_{1,1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-360$$"],"dependencies":["a41316cmatrices4a-h2"],"title":"Scalar Multiplication","text":"What is the bottom left entry, $$c_{2,1}$$, after multiplying by 4?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices4a-h4","type":"hint","dependencies":["a41316cmatrices4a-h3"],"title":"Scalar Multiplication","text":"Repeat the same process for each corresponding entries.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a41316cmatrices5","title":"Finding Scalar Mulitples of a Matrix","body":"Perform scalar multiplication on the matrix $$C=\\\\begin{bmatrix} 16 & 3 & 7 & 18 \\\\\\\\ 90 & 5 & 3 & 29 \\\\end{bmatrix}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Matrices and Matrix Operations","courseName":"OpenStax: College Algebra","steps":[{"id":"a41316cmatrices5a","stepAnswer":["$$\\\\begin{bmatrix} 8 & \\\\frac{3}{2} & \\\\frac{7}{2} & 9 \\\\\\\\ 45 & \\\\frac{5}{2} & \\\\frac{3}{2} & \\\\frac{29}{2} \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{1}{2} C$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 8 & \\\\frac{3}{2} & \\\\frac{7}{2} & 9 \\\\\\\\ 45 & \\\\frac{5}{2} & \\\\frac{3}{2} & \\\\frac{29}{2} \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"a41316cmatrices5a-h1","type":"hint","dependencies":[],"title":"Scalar Multiplication","text":"Scalar multiplication involves finding the product of a constant by each entry in the matrix. Given $$A=\\\\begin{bmatrix} a_{1,1} & a_{1,2} \\\\\\\\ a_{2,1} & a_{2,2} \\\\end{bmatrix}$$, the scalar multiple\\\\n$$c A$$ is $$c A=c*\\\\begin{bmatrix} a_{1,1} & a_{1,2} \\\\\\\\ a_{2,1} & a_{2,2} \\\\end{bmatrix}=\\\\begin{bmatrix} c a_{1,1} & c a_{1,2} \\\\\\\\ c a_{2,1} & c a_{2,2} \\\\end{bmatrix}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a41316cmatrices5a-h1"],"title":"Scalar Multiplication","text":"Multiply each entry in C by scalar $$\\\\frac{1}{2}$$. What is the top left entry, $$c_{1,1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$45$$"],"dependencies":["a41316cmatrices5a-h2"],"title":"Scalar Multiplication","text":"What is the bottom left entry, $$c_{2,1}$$, after multiplying by $$\\\\frac{1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices5a-h4","type":"hint","dependencies":["a41316cmatrices5a-h3"],"title":"Scalar Multiplication","text":"Repeat the same process for each corresponding entries.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a41316cmatrices6","title":"Finding the Product of Two Matrices","body":"Use the matrices below to perform the indicated operation if possible.\\\\n$$A=\\\\begin{bmatrix} -1 & 5 \\\\\\\\ 3 & 2 \\\\end{bmatrix}$$, $$B=\\\\begin{bmatrix} 3 & 6 & 4 \\\\\\\\ -8 & 0 & 12 \\\\end{bmatrix}$$, $$C=\\\\begin{bmatrix} 4 & 10 \\\\\\\\ -2 & 6 \\\\\\\\ 5 & 9 \\\\end{bmatrix}$$, $$D=\\\\begin{bmatrix} 2 & -3 & 12 \\\\\\\\ 9 & 3 & 1 \\\\\\\\ 0 & 8 & -10 \\\\end{bmatrix}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Matrices and Matrix Operations","courseName":"OpenStax: College Algebra","steps":[{"id":"a41316cmatrices6a","stepAnswer":["$$\\\\begin{bmatrix} 20 & 102 \\\\\\\\ 28 & 28 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"BC","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 20 & 102 \\\\\\\\ 28 & 28 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"a41316cmatrices6a-h1","type":"hint","dependencies":[],"title":"Product of Matrices","text":"Finding the product of two matrices is only possible when the inner dimensions are the same, meaning that the number of columns of the first matrix is equal to the number of rows of the second matrix. If A is an m\xd7r matrix and B is an r\xd7n matrix, then the product matrix AB is an m\xd7n matrix. For example, the product AB is possible because the number of columns in A is the same as the number of rows in B. If the inner dimensions do not match, the product is not defined.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices6a-h2","type":"hint","dependencies":["a41316cmatrices6a-h1"],"title":"Product of Matrices:Computing Entries","text":"For a matrix $$AB=C$$, we can obtain entry $$c_{i,j}$$ by multipying the entries in row i of A by column of j in B and adding them. For example, given matrices A and B, where the dimensions of A are 2x3 and the dimensions of B are 3x3, the product of AB will be a 2x3 matrix.\\\\n$$A=/\\\\begin{bmatrix} a_{1,1} & a_{1,2} & a_{1,3} \\\\\\\\ a_{2,1} & a_{2,2} & a_{2,3} \\\\end{bmatrix}$$ and $$B=\\\\begin{bmatrix} b_{1,1} & b_{1,2} & b_{1,3} \\\\\\\\ b_{2,1} & b_{2,2} & b_{2,3} \\\\\\\\ b_{3,1} & b_{3,2} & b_{3,3} \\\\end{bmatrix}$$.\\\\nTo obtain $$c_{1,1}$$ we multiply the first row of A with the first column of B and add. Thus, $$c_{1,1}=a_{1,1} b_{1,1}+a_{1,2} b_{2,1}+a_{1,3} b_{3,1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a41316cmatrices6a-h2"],"title":"Product of Matrices:Computing Entries","text":"We will start by computing the top left entry of BC. We can do so by calculating $$b_{1,1} c_{1,1}+b_{1,2} c_{2,1}+b_{1,3} c_{3,1}$$. What is the value?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$28$$"],"dependencies":["a41316cmatrices6a-h3"],"title":"Product of Matrices:Computing Entries","text":"We will next compute the bottom left entry of BC. We can do so by calculating $$b_{2,1} c_{1,1}+b_{2,2} c_{2,1}+b_{2,3} c_{3,1}$$. What is the value?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices6a-h5","type":"hint","dependencies":["a41316cmatrices6a-h4"],"title":"Product of Matrices:Computing Entries","text":"Repeat the same process for each corresponding entries.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a41316cmatrices7","title":"Finding the Product of Two Matrices","body":"Use the matrices below to perform the indicated operation if possible.\\\\n$$A=\\\\begin{bmatrix} -1 & 5 \\\\\\\\ 3 & 2 \\\\end{bmatrix}$$, $$B=\\\\begin{bmatrix} 3 & 6 & 4 \\\\\\\\ -8 & 0 & 12 \\\\end{bmatrix}$$, $$C=\\\\begin{bmatrix} 4 & 10 \\\\\\\\ -2 & 6 \\\\\\\\ 5 & 9 \\\\end{bmatrix}$$, $$D=\\\\begin{bmatrix} 2 & -3 & 12 \\\\\\\\ 9 & 3 & 1 \\\\\\\\ 0 & 8 & -10 \\\\end{bmatrix}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Matrices and Matrix Operations","courseName":"OpenStax: College Algebra","steps":[{"id":"a41316cmatrices7a","stepAnswer":["$$\\\\begin{bmatrix} 60 & 41 & 2 \\\\\\\\ -16 & 120 & -216 \\\\end{bmatrix}$$"],"problemType":"MultipleChoice","stepTitle":"BD","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 60 & 41 & 2 \\\\\\\\ -16 & 120 & -216 \\\\end{bmatrix}$$","choices":["$$\\\\begin{bmatrix} 64 & 41 & 2 \\\\\\\\ 16 & 120 & -216 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} 60 & 46 & 2 \\\\\\\\ -16 & 10 & -216 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} 60 & 41 & 2 \\\\\\\\ -16 & 120 & -216 \\\\end{bmatrix}$$","None of the above"],"hints":{"DefaultPathway":[{"id":"a41316cmatrices7a-h1","type":"hint","dependencies":[],"title":"Product of Matrices","text":"Finding the product of two matrices is only possible when the inner dimensions are the same, meaning that the number of columns of the first matrix is equal to the number of rows of the second matrix. If A is an m\xd7r matrix and B is an r\xd7n matrix, then the product matrix AB is an m\xd7n matrix. For example, the product AB is possible because the number of columns in A is the same as the number of rows in B. If the inner dimensions do not match, the product is not defined.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices7a-h2","type":"hint","dependencies":["a41316cmatrices7a-h1"],"title":"Product of Matrices:Computing Entries","text":"For a matrix $$AB=C$$, we can obtain entry $$c_{i,j}$$ by multipying the entries in row i of A by column of j in B and adding them. For example, given matrices A and B, where the dimensions of A are 2x3 and the dimensions of B are 3x3, the product of AB will be a 2x3 matrix.\\\\n$$A=/\\\\begin{bmatrix} a_{1,1} & a_{1,2} & a_{1,3} \\\\\\\\ a_{2,1} & a_{2,2} & a_{2,3} \\\\end{bmatrix}$$ and $$B=\\\\begin{bmatrix} b_{1,1} & b_{1,2} & b_{1,3} \\\\\\\\ b_{2,1} & b_{2,2} & b_{2,3} \\\\\\\\ b_{3,1} & b_{3,2} & b_{3,3} \\\\end{bmatrix}$$.\\\\nTo obtain $$c_{1,1}$$ we multiply the first row of A with the first column of B and add. Thus, $$c_{1,1}=a_{1,1} b_{1,1}+a_{1,2} b_{2,1}+a_{1,3} b_{3,1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$60$$"],"dependencies":["a41316cmatrices7a-h2"],"title":"Product of Matrices:Computing Entries","text":"We will start by computing the top left entry of BD. We can do so by calculating $$b_{1,1} d_{1,1}+b_{1,2} d_{2,1}+b_{1,3} d_{3,1}$$. What is the value?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-16$$"],"dependencies":["a41316cmatrices7a-h3"],"title":"Product of Matrices:Computing Entries","text":"We will next compute the bottom left entry of BD. We can do so by calculating $$b_{2,1} d_{1,1}+b_{2,2} d_{2,1}+b_{2,3} d_{3,1}$$. What is the value?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices7a-h5","type":"hint","dependencies":["a41316cmatrices7a-h4"],"title":"Product of Matrices:Computing Entries","text":"Repeat the same process for each corresponding entries.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a41316cmatrices8","title":"Finding the Product of Two Matrices","body":"Use the matrices below to perform the indicated operation if possible.\\\\n$$A=\\\\begin{bmatrix} -1 & 5 \\\\\\\\ 3 & 2 \\\\end{bmatrix}$$, $$B=\\\\begin{bmatrix} 3 & 6 & 4 \\\\\\\\ -8 & 0 & 12 \\\\end{bmatrix}$$, $$C=\\\\begin{bmatrix} 4 & 10 \\\\\\\\ -2 & 6 \\\\\\\\ 5 & 9 \\\\end{bmatrix}$$, $$D=\\\\begin{bmatrix} 2 & -3 & 12 \\\\\\\\ 9 & 3 & 1 \\\\\\\\ 0 & 8 & -10 \\\\end{bmatrix}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Matrices and Matrix Operations","courseName":"OpenStax: College Algebra","steps":[{"id":"a41316cmatrices8a","stepAnswer":["$$\\\\begin{bmatrix} 74 & 110 \\\\\\\\ 35 & 117 \\\\\\\\ -66 & -42 \\\\end{bmatrix}$$"],"problemType":"MultipleChoice","stepTitle":"DC","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 74 & 110 \\\\\\\\ 35 & 117 \\\\\\\\ -66 & -42 \\\\end{bmatrix}$$","choices":["$$\\\\begin{bmatrix} 74 & 110 \\\\\\\\ 35 & 117 \\\\\\\\ -66 & -42 \\\\end{bmatrix}$$"],"hints":{"DefaultPathway":[{"id":"a41316cmatrices8a-h1","type":"hint","dependencies":[],"title":"Product of Matrices","text":"Finding the product of two matrices is only possible when the inner dimensions are the same, meaning that the number of columns of the first matrix is equal to the number of rows of the second matrix. If A is an m\xd7r matrix and B is an r\xd7n matrix, then the product matrix AB is an m\xd7n matrix. For example, the product AB is possible because the number of columns in A is the same as the number of rows in B. If the inner dimensions do not match, the product is not defined.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices8a-h2","type":"hint","dependencies":["a41316cmatrices8a-h1"],"title":"Product of Matrices:Computing Entries","text":"For a matrix $$AB=C$$, we can obtain entry $$c_{i,j}$$ by multipying the entries in row i of A by column of j in B and adding them. For example, given matrices A and B, where the dimensions of A are 2x3 and the dimensions of B are 3x3, the product of AB will be a 2x3 matrix.\\\\n$$A=/\\\\begin{bmatrix} a_{1,1} & a_{1,2} & a_{1,3} \\\\\\\\ a_{2,1} & a_{2,2} & a_{2,3} \\\\end{bmatrix}$$ and $$B=\\\\begin{bmatrix} b_{1,1} & b_{1,2} & b_{1,3} \\\\\\\\ b_{2,1} & b_{2,2} & b_{2,3} \\\\\\\\ b_{3,1} & b_{3,2} & b_{3,3} \\\\end{bmatrix}$$.\\\\nTo obtain $$c_{1,1}$$ we multiply the first row of A with the first column of B and add. Thus, $$c_{1,1}=a_{1,1} b_{1,1}+a_{1,2} b_{2,1}+a_{1,3} b_{3,1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$74$$"],"dependencies":["a41316cmatrices8a-h2"],"title":"Product of Matrices:Computing Entries","text":"We will start by computing the top left entry of DC. We can do so by calculating $$d_{1,1} c_{1,1}+d_{1,2} c_{2,1}+d_{1,3} c_{3,1}$$. What is the value?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$110$$"],"dependencies":["a41316cmatrices8a-h3"],"title":"Product of Matrices:Computing Entries","text":"We will next compute the top right entry of DC, $$dc_{1,2}$$. We can do so by calculating $$d_{1,1} c_{1,2}+d_{1,2} c_{2,2}+d_{1,3} c_{3,2}$$. What is the value?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices8a-h5","type":"hint","dependencies":["a41316cmatrices8a-h4"],"title":"Product of Matrices:Computing Entries","text":"Repeat the same process for each corresponding entries.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a41316cmatrices9","title":"Finding the Product of Two Matrices","body":"Use the matrices below to perform the indicated operation if possible.\\\\n$$A=\\\\begin{bmatrix} -1 & 5 \\\\\\\\ 3 & 2 \\\\end{bmatrix}$$, $$B=\\\\begin{bmatrix} 3 & 6 & 4 \\\\\\\\ -8 & 0 & 12 \\\\end{bmatrix}$$, $$C=\\\\begin{bmatrix} 4 & 10 \\\\\\\\ -2 & 6 \\\\\\\\ 5 & 9 \\\\end{bmatrix}$$, $$D=\\\\begin{bmatrix} 2 & -3 & 12 \\\\\\\\ 9 & 3 & 1 \\\\\\\\ 0 & 8 & -10 \\\\end{bmatrix}$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Matrices and Matrix Operations","courseName":"OpenStax: College Algebra","steps":[{"id":"a41316cmatrices9a","stepAnswer":["$$\\\\begin{bmatrix} -68 & 24 & 136 \\\\\\\\ -54 & -12 & 64 \\\\\\\\ -57 & 30 & 128 \\\\end{bmatrix}$$"],"problemType":"MultipleChoice","stepTitle":"CB","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} -68 & 24 & 136 \\\\\\\\ -54 & -12 & 64 \\\\\\\\ -57 & 30 & 128 \\\\end{bmatrix}$$","choices":["$$\\\\begin{bmatrix} -68 & 4 & 136 \\\\\\\\ -54 & -2 & 64 \\\\\\\\ -53 & 30 & 18 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} -68 & 24 & 136 \\\\\\\\ -54 & -12 & 64 \\\\\\\\ -57 & 30 & 128 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} -68 & 24 & 16 \\\\\\\\ -5 & -12 & 64 \\\\\\\\ -57 & 60 & 128 \\\\end{bmatrix}$$"],"hints":{"DefaultPathway":[{"id":"a41316cmatrices9a-h1","type":"hint","dependencies":[],"title":"Product of Matrices","text":"Finding the product of two matrices is only possible when the inner dimensions are the same, meaning that the number of columns of the first matrix is equal to the number of rows of the second matrix. If A is an m\xd7r matrix and B is an r\xd7n matrix, then the product matrix AB is an m\xd7n matrix. For example, the product AB is possible because the number of columns in A is the same as the number of rows in B. If the inner dimensions do not match, the product is not defined.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices9a-h2","type":"hint","dependencies":["a41316cmatrices9a-h1"],"title":"Product of Matrices:Computing Entries","text":"For a matrix $$AB=C$$, we can obtain entry $$c_{i,j}$$ by multipying the entries in row i of A by column of j in B and adding them. For example, given matrices A and B, where the dimensions of A are 2x3 and the dimensions of B are 3x3, the product of AB will be a 2x3 matrix.\\\\n$$A=/\\\\begin{bmatrix} a_{1,1} & a_{1,2} & a_{1,3} \\\\\\\\ a_{2,1} & a_{2,2} & a_{2,3} \\\\end{bmatrix}$$ and $$B=\\\\begin{bmatrix} b_{1,1} & b_{1,2} & b_{1,3} \\\\\\\\ b_{2,1} & b_{2,2} & b_{2,3} \\\\\\\\ b_{3,1} & b_{3,2} & b_{3,3} \\\\end{bmatrix}$$.\\\\nTo obtain $$c_{1,1}$$ we multiply the first row of A with the first column of B and add. Thus, $$c_{1,1}=a_{1,1} b_{1,1}+a_{1,2} b_{2,1}+a_{1,3} b_{3,1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-68$$"],"dependencies":["a41316cmatrices9a-h2"],"title":"Product of Matrices:Computing Entries","text":"We will start by computing the top left entry of CB. We can do so by calculating $$c_{1,1} b_{1,1}+c_{1,2} b_{2,1}+c_{1,3} b_{3,1}$$. What is the value?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$24$$"],"dependencies":["a41316cmatrices9a-h3"],"title":"Product of Matrices:Computing Entries","text":"We will next compute the middle entry of the first row of CB, $$cb_{1,2}$$. We can do so by calculating $$c_{1,1} b_{1,2}+c_{1,2} b_{2,2}+c_{1,3} b_{3,2}$$. What is the value?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a41316cmatrices9a-h5","type":"hint","dependencies":["a41316cmatrices9a-h4"],"title":"Product of Matrices:Computing Entries","text":"Repeat the same process for each corresponding entries.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a42d06ec1","title":"Core Functions: Exponential and Logarithmic","body":"These questions test your knowledge of the core concepts.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Core Functions: Exponential and Logarithmic","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a42d06ec1a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"On the following graph roughly plot the following graphs: $$y=3^x$$ and $$y=\\\\log_{3}\\\\left(x\\\\right)$$. Is this the correct graph?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a42d06ec1a-h1","type":"hint","dependencies":[],"title":"Special Points","text":"Start by choosing a few $$x$$ values, calculating the corresponding $$y$$ values, and plotting points on the graph.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a42d06ec1a-h1"],"title":"Special Points","text":"For the line $$y=3^x$$, what is the value of $$y$$ when $$x=0$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a42d06ec1a-h2"],"title":"Special Points","text":"For the line $$y=3^x$$, what is the value of $$y$$ when $$x=1$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{9}$$"],"dependencies":["a42d06ec1a-h3"],"title":"Special Points","text":"For the line $$y=3^x$$, what is the value of $$y$$ when $$x=-2$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec1a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a42d06ec1a-h4"],"title":"Special Points","text":"For the line $$y=\\\\log_{3}\\\\left(x\\\\right)$$, what is the value of $$y$$ when $$x=1$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec1a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a42d06ec1a-h5"],"title":"Special Points","text":"For the line $$y=\\\\log_{3}\\\\left(x\\\\right)$$, what is the value of $$y$$ when $$x=3$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec1a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a42d06ec1a-h6"],"title":"Select the graph","text":"Is this the correct graph?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]}]}},{"id":"a42d06ec1b","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Similarly, on the following graph roughly plot the following graphs: $$y={\\\\left(\\\\frac{1}{3}\\\\right)}^x$$ and $$y$$ $$=$$ $$\\\\log_{1/3}\\\\left(x\\\\right)$$. Is this the correct graph?","stepBody":"##figure2.gif## ","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a42d06ec1b-h1","type":"hint","dependencies":[],"title":"Special Points","text":"Start by choosing a few $$x$$ values, calculating the corresponding $$y$$ values, and plotting points on the graph.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec1b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a42d06ec1b-h1"],"title":"Special Points","text":"For the line $$y={\\\\left(\\\\frac{1}{3}\\\\right)}^x$$, what is the value of $$y$$ when $$x=0$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec1b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a42d06ec1b-h2"],"title":"Special Points","text":"For the line $$y={\\\\left(\\\\frac{1}{3}\\\\right)}^x$$, what is the value of $$y$$ when $$x=-1$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec1b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["a42d06ec1b-h3"],"title":"Special Points","text":"For the line $$y={\\\\left(\\\\frac{1}{3}\\\\right)}^x$$, what is the value of $$y$$ when $$x=1$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec1b-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a42d06ec1b-h4"],"title":"Special Points","text":"For the line $$y=\\\\log_{1/3}\\\\left(x\\\\right)$$, what is the value of $$y$$ when $$x=1$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec1b-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a42d06ec1b-h5"],"title":"Special Points","text":"For the line $$y=\\\\log_{1/3}\\\\left(x\\\\right)$$, what is the value of $$y$$ when $$x=3$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec1b-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a42d06ec1b-h6"],"title":"Select the graph","text":"Is this the correct graph?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]}]}}]},{"id":"a42d06ec2","title":"Core Functions: Exponential and Logarithmic","body":"These questions test your knowledge of the core concepts.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Core Functions: Exponential and Logarithmic","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a42d06ec2a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Find an exponential function, whose graph contains the point $$(-3,\\\\frac{1}{125})$$. Does $$f(x)=5^x$$ satisfy the question\'s condition?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a42d06ec2a-h1","type":"hint","dependencies":[],"title":"The general form","text":"The general form of an exponential function is $$y=a b^x$$, where a is the initial value or y-intercept, and $$b$$ is the base of the exponential function.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec2a-h2","type":"hint","dependencies":["a42d06ec2a-h1"],"title":"Use the Given Point","text":"Plug in the given point $$(-3,\\\\frac{1}{125})$$ into the general form. This gives you the equation $$\\\\frac{1}{125}=a b^{\\\\left(-3\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec2a-h3","type":"hint","dependencies":["a42d06ec2a-h2"],"title":"Solve for a and $$b$$","text":"Let $$a=1$$ for simplicity.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a42d06ec2a-h3"],"title":"Solve for a and $$b$$","text":"What is the value of $$b$$ when $$a=1$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec2a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a42d06ec2a-h4"],"title":"Exponential Function","text":"Does $$f(x)=5^x$$ satisfy the question\'s condition?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]}]}},{"id":"a42d06ec2b","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Find a logarithmic function, whose graph contains the point $$(81,4)$$. Does $$f(x)=\\\\log_{3}\\\\left(x\\\\right)$$ satisfy the question\'s condition?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a42d06ec2b-h1","type":"hint","dependencies":[],"title":"The general form","text":"If a is a coefficient, $$b$$ is the base of the logarithm, and $$x$$ is the input, the general form of a logarithmic function is $$a\\\\log_{b}{x}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec2b-h2","type":"hint","dependencies":["a42d06ec2b-h1"],"title":"Use the Given Point","text":"Plug in the given point $$(81,4)$$ into the general form. This gives you the equation $$4=a\\\\log_{b}{81}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec2b-h3","type":"hint","dependencies":["a42d06ec2b-h2"],"title":"Solve for a and $$b$$","text":"Let $$a=1$$ for simplicity.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec2b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a42d06ec2b-h3"],"title":"Solve for a and $$b$$","text":"What is the value of $$b$$ when $$a=1$$? (suppose $$b$$ is positive)","variabilization":{},"oer":"https://OATutor.io","license":"","subHints":[{"id":"a42d06ec2b-h4-s1","type":"hint","dependencies":[],"title":"Solve for a and $$b$$","text":"$$a^4=81$$, $$a^4=3^4$$, so $$a=3$$ is a valid answer.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a42d06ec2b-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a42d06ec2b-h4"],"title":"Exponential Function","text":"Does $$f(x)=\\\\log_{3}\\\\left(x\\\\right)$$ satisfy the question\'s condition?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]}]}}]},{"id":"a42d06ec3","title":"Core Functions: Exponential and Logarithmic","body":"These questions test your knowledge of the core concepts.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Core Functions: Exponential and Logarithmic","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a42d06ec3a","stepAnswer":["$$(\\\\frac{-1}{2},\\\\frac{1}{2})$$"],"problemType":"MultipleChoice","stepTitle":"Calculate the domain of the function: f(x) $$=$$ $$\\\\log_{2}\\\\left(1-4x^2\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(\\\\frac{-1}{2},\\\\frac{1}{2})$$","choices":["$$(\\\\frac{-1}{2},\\\\frac{1}{2})$$","$$(-\\\\infty,\\\\frac{1}{2})$$","$$(\\\\frac{-1}{2},\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"a42d06ec3a-h1","type":"hint","dependencies":[],"title":"The Logarithmic Domain","text":"Recall that the domain of a logarithmic function is determined by the argument inside the logarithm. For $$\\\\log_{2}\\\\left(y\\\\right)$$ to be defined, $$y$$ must be greater than zero.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec3a-h2","type":"hint","dependencies":["a42d06ec3a-h1"],"title":"The Logarithmic Domain","text":"In this case, $$1-4x^2>0$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec3a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{-1}{2}<x<\\\\frac{1}{2}$$"],"dependencies":["a42d06ec3a-h2"],"title":"Solve the inequality","text":"Solve $$1-4x^2>0$$.","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{-1}{2}<x<\\\\frac{1}{2}$$","$$\\\\frac{-1}{2}>x$$","$$x<\\\\frac{1}{2}$$","$$\\\\frac{-1}{2}<x$$","$$x>\\\\frac{1}{2}$$"],"subHints":[{"id":"a42d06ec3a-h3-s1","type":"hint","dependencies":[],"title":"Solve the inequality","text":"$$x^2<\\\\frac{1}{4}$$, so $$\\\\frac{-1}{2}<x<\\\\frac{1}{2}$$","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a42d06ec3a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(\\\\frac{-1}{2},\\\\frac{1}{2})$$"],"dependencies":["a42d06ec3a-h3"],"title":"Domain","text":"What is the domain?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$(\\\\frac{-1}{2},\\\\frac{1}{2})$$","$$(-\\\\infty,\\\\frac{1}{2})$$","$$(\\\\frac{-1}{2},\\\\infty)$$"]}]}}]},{"id":"a42d06ec4","title":"Core Functions: Exponential and Logarithmic","body":"These problems are generally harder, often highlighting an important subtlety","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Core Functions: Exponential and Logarithmic","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a42d06ec4a","stepAnswer":["$$(-\\\\infty,6)$$"],"problemType":"MultipleChoice","stepTitle":"Determine the range of the following function $$f(x)=6-3\\\\times2^{1-x}$$. Hint: What is the range of $$2^x$$? How could you apply elementary transformations to get f? How would the range be affected by each of these transformations?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,6)$$","choices":["$$(-\\\\infty,6)$$","$$(6,\\\\infty)$$","$$(-\\\\infty,0)$$","$$(0,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"a42d06ec4a-h1","type":"hint","dependencies":[],"title":"Range of $$2^x$$","text":"Consider the base function $$g(x)=2^x$$. The range of this exponential function is $$(0,\\\\infty)$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec4a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(0,\\\\infty)$$"],"dependencies":["a42d06ec4a-h1"],"title":"Transformations","text":"What is the range of $$2^{1-x}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$(0,\\\\infty)$$","$$(-\\\\infty,0)$$","$$(-\\\\infty,\\\\infty)$$"]},{"id":"a42d06ec4a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(0,\\\\infty)$$"],"dependencies":["a42d06ec4a-h2"],"title":"Transformations","text":"What is the range of $$3\\\\times2^{1-x}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$(0,\\\\infty)$$","$$(-\\\\infty,0)$$","$$(-\\\\infty,\\\\infty)$$"]},{"id":"a42d06ec4a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(-\\\\infty,0)$$"],"dependencies":["a42d06ec4a-h3"],"title":"Transformations","text":"What is the range of $$-3\\\\times2^{1-x}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$(-\\\\infty,0)$$","$$(0,\\\\infty)$$","$$(-\\\\infty,\\\\infty)$$"]},{"id":"a42d06ec4a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(-\\\\infty,6)$$"],"dependencies":["a42d06ec4a-h4"],"title":"Transformations","text":"What is the range of $$6-3\\\\times2^{1-x}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$(-\\\\infty,6)$$","$$(6,\\\\infty)$$","$$(-\\\\infty,0)$$","$$(0,\\\\infty)$$"]}]}}]},{"id":"a42d06ec5","title":"Core Functions: Exponential and Logarithmic","body":"These problems are generally harder, often highlighting an important subtlety","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Core Functions: Exponential and Logarithmic","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a42d06ec5a","stepAnswer":["$$K=3$$, $$b=2$$"],"problemType":"MultipleChoice","stepTitle":"Let f be a function of the form f(t) $$=$$ $$K b^t$$ where K and $$b$$ are constants with $$b$$ > $$1$$. Any quantity which is modelled by such a function (with $$t$$ representing time) is said to experience natural growth. If $$f(1)=6$$ and $$f(3)=24$$, determine the values of K and $$b$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$K=3$$, $$b=2$$","choices":["$$K=3$$, $$b=2$$","$$K=2$$, $$b=3$$"],"hints":{"DefaultPathway":[{"id":"a42d06ec5a-h1","type":"hint","dependencies":[],"title":"Setting up Equations using Given Information","text":"$$f(1)=6$$ implies $$K b=6$$. $$f(3)=24$$ implies $$K b^3=24$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec5a-h2","type":"hint","dependencies":["a42d06ec5a-h1"],"title":"Solving the Equations","text":"From the two equations, we can get $$\\\\frac{24}{6}=\\\\frac{K b^3}{K b}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a42d06ec5a-h2"],"title":"Solve the value of $$b$$","text":"What is the value of $$b$$?","variabilization":{},"oer":"https://OATutor.io","license":"","subHints":[{"id":"a42d06ec5a-h3-s1","type":"hint","dependencies":[],"title":"Solve the value of $$b$$","text":"$$b^2=4$$. Since $$b>1$$, $$b=2$$","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a42d06ec5a-h4","type":"hint","dependencies":["a42d06ec5a-h3"],"title":"Solve the value of K","text":"Substitute $$b=2$$ into $$K b=6$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a42d06ec5a-h4"],"title":"Solve the value of K","text":"What is the value of K?","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a42d06ec6","title":"Core Functions: Exponential and Logarithmic","body":"These problems are generally harder, often highlighting an important subtlety. Let f be a function of the form $$f(t)=K b^x$$ where K and $$b$$ are constants with $$b$$ > $$0$$. On the following graph there are several plots of the graphs of such functions: \\\\n##figure1.gif","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Core Functions: Exponential and Logarithmic","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a42d06ec6a","stepAnswer":["D,E,F"],"problemType":"MultipleChoice","stepTitle":"Which of the graphs correspond to $$b>1$$?","stepBody":"","answerType":"string","variabilization":{},"choices":["D,E,F","A,B,C"],"hints":{"DefaultPathway":[{"id":"a42d06ec6a-h1","type":"hint","dependencies":[],"title":"Interpretation","text":"$$b>1$$ implies $$b^x$$ is increasing.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec6a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["D,E,F"],"dependencies":["a42d06ec6a-h1"],"title":"Interpretation","text":"Which of the graphs are increasing?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["D,E,F","A,B,C"]}]}},{"id":"a42d06ec6b","stepAnswer":["A,B,C"],"problemType":"MultipleChoice","stepTitle":"Which of the graphs correspond to $$b<1$$?","stepBody":"","answerType":"string","variabilization":{},"choices":["D,E,F","A,B,C"],"hints":{"DefaultPathway":[{"id":"a42d06ec6b-h1","type":"hint","dependencies":[],"title":"Interpretation","text":"$$b<1$$ implies $$b^x$$ is decreasing.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec6b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["A,B,C"],"dependencies":["a42d06ec6b-h1"],"title":"Interpretation","text":"Which of the graphs are decreasing?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["D,E,F","A,B,C"]}]}},{"id":"a42d06ec6c","stepAnswer":["D"],"problemType":"MultipleChoice","stepTitle":"Which graph has the largest value of $$b$$?","stepBody":"","answerType":"string","variabilization":{},"choices":["A","B","C","D","E","F"],"hints":{"DefaultPathway":[{"id":"a42d06ec6c-h1","type":"hint","dependencies":[],"title":"Interpretation","text":"This question is the same way of asking which graph increases the most rapidly.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec6c-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["D"],"dependencies":["a42d06ec6c-h1"],"title":"Select the graph","text":"Which graph increases the most rapidly?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["A","B","C","D","E","F"]}]}}]},{"id":"a42d06ec61","title":"Core Functions: Exponential and Logarithmic","body":"These problems are generally harder, often highlighting an important subtlety. Let f be a function of the form $$f(t)=K b^x$$ where K and $$b$$ are constants with $$b$$ > $$0$$. On the following graph there are several plots of the graphs of such functions: \\\\n##figure1.gif","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Core Functions: Exponential and Logarithmic","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a42d06ec61a","stepAnswer":["C"],"problemType":"MultipleChoice","stepTitle":"Let f be a function of the form $$f(t)=K b^x$$ where K and $$b$$ are constants with $$b$$ > $$0$$. On the following graph there are several plots of the graphs of such functions: which graph corresponds to the largest value of K?","stepBody":"","answerType":"string","variabilization":{},"choices":["A","B","C","D","E","F"],"hints":{"DefaultPathway":[{"id":"a42d06ec61a-h1","type":"hint","dependencies":[],"title":"Interpretation","text":"$$f(0)=K$$. The graph with the highest inercept is the graph with the largest value of K.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec61a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["C"],"dependencies":["a42d06ec61a-h1"],"title":"Select the graph","text":"Which graph has the highest intercept?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["A","B","C","D","E","F"]}]}},{"id":"a42d06ec61b","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Do any two of these graphs correspond to the same value of K?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a42d06ec61b-h1","type":"hint","dependencies":[],"title":"Interpretation","text":"Same y-intercept implies same value of K","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec61b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a42d06ec61b-h1"],"title":"Select the graph","text":"Do any two of these graphs correspond to the same value of K?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"],"subHints":[{"id":"a42d06ec61b-h2-s1","type":"hint","dependencies":[],"title":"Select the graph","text":"A,B and D,E are each pairs with the same value of K.","variabilization":{},"oer":"https://OATutor.io","license":""}]}]}},{"id":"a42d06ec61c","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Do any two of these graphs correspond to the same value of $$b$$?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a42d06ec61c-h1","type":"hint","dependencies":[],"title":"Interpretation","text":"One is a constant multiple of the other means same value of $$b$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec61c-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a42d06ec61c-h1"],"title":"Select the graph","text":"Do any two of these graphs correspond to the same value of $$b$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"],"subHints":[{"id":"a42d06ec61c-h2-s1","type":"hint","dependencies":[],"title":"Select the graph","text":"E,F have the same value of $$b$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]}]}}]},{"id":"a42d06ec7","title":"Core Functions: Exponential and Logarithmic","body":"These questions are challenging, requiring mastery of each concept and their interrelations.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Core Functions: Exponential and Logarithmic","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a42d06ec7a","stepAnswer":["$$b^a$$"],"problemType":"MultipleChoice","stepTitle":"Assume that a quantity P experiences natural growth and is modelled by the function $$f(t)=K b^t$$. Let $$a>0$$ be a fixed constant. Show that over any time interval of length a, the quantity P changes by multiplication by the same fixed constant. What is this constant?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$b^a$$","choices":["$$b^a$$","$$b^t$$","$$K b^a$$","$$K b^t$$"],"hints":{"DefaultPathway":[{"id":"a42d06ec7a-h1","type":"hint","dependencies":[],"title":"Find the constant","text":"Let $$t$$ be arbitrary. To find the constant, we can find the value of $$\\\\frac{f{\\\\left(t+a\\\\right)}}{f{\\\\left(t\\\\right)}}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec7a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$b^a$$"],"dependencies":["a42d06ec7a-h1"],"title":"Find the constant","text":"What is the result of $$\\\\frac{f{\\\\left(t+a\\\\right)}}{f{\\\\left(t\\\\right)}}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$b^a$$","$$b^t$$","$$K b^a$$","$$K b^t$$"]}]}}]},{"id":"a42d06ec8","title":"Core Functions: Exponential and Logarithmic","body":"These questions are challenging, requiring mastery of each concept and their interrelations.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Core Functions: Exponential and Logarithmic","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a42d06ec8a","stepAnswer":["$$(-2,3]$$"],"problemType":"MultipleChoice","stepTitle":"Consider the following function: $$f(x)=\\\\log_{3}\\\\left(\\\\frac{2+x}{5}\\\\right)$$. Find a restriction of the domain of f so that the range becomes $$(-\\\\infty,0]$$. Give your answer in interval notation.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$(-2,3]$$","$$(-2,3)$$","$$(0,\\\\infty)$$","$$(-\\\\infty,0)$$"],"hints":{"DefaultPathway":[{"id":"a42d06ec8a-h1","type":"hint","dependencies":[],"title":"Follow the hint","text":"Start by considering the same question of $$\\\\log_{3}\\\\left(x\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec8a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(-\\\\infty,\\\\infty)$$"],"dependencies":["a42d06ec8a-h1"],"title":"Follow the hint","text":"What is the range of $$\\\\log_{3}\\\\left(x\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$(-\\\\infty,\\\\infty)$$","$$(-\\\\infty,0)$$","$$(0,\\\\infty)$$","$$(-\\\\infty,0]$$","$$[0,\\\\infty)$$"]},{"id":"a42d06ec8a-h3","type":"hint","dependencies":["a42d06ec8a-h2"],"title":"Consider the following case","text":"When $$0<x \\\\leq 1$$, $$\\\\log_{3}\\\\left(x\\\\right) \\\\leq \\\\log_{3}\\\\left(1\\\\right)=0$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec8a-h4","type":"hint","dependencies":["a42d06ec8a-h3"],"title":"Consider the following case","text":"When $$x>1$$, $$\\\\log_{3}\\\\left(x\\\\right)>\\\\log_{3}\\\\left(1\\\\right)=0$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a42d06ec8a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["(0,1]"],"dependencies":["a42d06ec8a-h4"],"title":"Choose a domain","text":"What is the domain of $$\\\\log_{3}\\\\left(x\\\\right)$$ when it has a range of $$(-\\\\infty,0]$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["(0,1]","$$(0,1)$$","$$[1,\\\\infty)$$","$$(1,\\\\infty)$$"]},{"id":"a42d06ec8a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(-2,3]$$"],"dependencies":["a42d06ec8a-h5"],"title":"Solve the question","text":"We have $$\\\\frac{2+x}{5}$$ in (0,1]. What is the range of $$x$$ (the domain of f)?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$(-2,3]$$","$$(-2,3)$$","$$(0,\\\\infty)$$","$$(-\\\\infty,0)$$"],"subHints":[{"id":"a42d06ec8a-h6-s1","type":"hint","dependencies":[],"title":"Solve the question","text":"$$\\\\frac{2+x}{5}$$ in (0,1]. $$2+x$$ in (0,5]. $$x$$ in (-2,3].","variabilization":{},"oer":"https://OATutor.io","license":""}]}]}}]},{"id":"a443311sqroots1","title":"Simplifying Dvided Square Roots","body":"Simplify the square root espression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Divide Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a443311sqroots1a","stepAnswer":["$$\\\\frac{\\\\sqrt{6}}{2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\sqrt{54}}{6}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{\\\\sqrt{6}}{2}$$","hints":{"DefaultPathway":[{"id":"a443311sqroots1a-h1","type":"hint","dependencies":[],"title":"Split","text":"Split the numerator into the product of two square roots, one of which can be simplified to a single number such as $$\\\\sqrt{4}$$ or $$\\\\sqrt{16}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots1a-h2","type":"hint","dependencies":["a443311sqroots1a-h1"],"title":"Simplify","text":"Now, simplify the radical so that there is a product of a constant and a square root.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots1a-h3","type":"hint","dependencies":["a443311sqroots1a-h2"],"title":"Remove common factors","text":"Remove the common factors of the numerator and denominator by dividing them out.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots1a-h4","type":"hint","dependencies":["a443311sqroots1a-h3"],"title":"Answer","text":"The answer is $$\\\\frac{\\\\sqrt{6}}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a443311sqroots10","title":"Simplifying Dvided Square Roots","body":"Simplify the square root espression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Divide Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a443311sqroots10a","stepAnswer":["$$\\\\frac{9x^2}{y^2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\sqrt{162x^{10} y^2}}{\\\\sqrt{2x^6 y^6}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{9x^2}{y^2}$$","hints":{"DefaultPathway":[{"id":"a443311sqroots10a-h1","type":"hint","dependencies":[],"title":"Combine","text":"Combine the radicals into one radical, since $$\\\\frac{\\\\sqrt{a}}{\\\\sqrt{b}}$$ $$=$$ $$\\\\sqrt{\\\\frac{a}{b}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots10a-h2","type":"hint","dependencies":["a443311sqroots10a-h1"],"title":"Simplify","text":"Now, simplify the quotient under the radical as much as possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots10a-h3","type":"hint","dependencies":["a443311sqroots10a-h2"],"title":"Reduce","text":"If necessary, simplify the radical further by splitting it apart and reducing it to a constant (or variable) and another radical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots10a-h4","type":"hint","dependencies":["a443311sqroots10a-h3"],"title":"Answer","text":"The answer is $$\\\\frac{9x^2}{y^2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a443311sqroots11","title":"Simplifying Dvided Square Roots","body":"Simplify the square root espression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Divide Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a443311sqroots11a","stepAnswer":["$$\\\\frac{4\\\\sqrt{3}}{3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{4}{\\\\sqrt{3}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{4\\\\sqrt{3}}{3}$$","hints":{"DefaultPathway":[{"id":"a443311sqroots11a-h1","type":"hint","dependencies":[],"title":"Multiply","text":"Multiply the numerator and the denominator by the radical in the denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots11a-h2","type":"hint","dependencies":["a443311sqroots11a-h1"],"title":"Simplify","text":"Simplify the numerator and the denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots11a-h3","type":"hint","dependencies":["a443311sqroots11a-h2"],"title":"Remove common factors","text":"Remove the common factors of the numerator and denominator by dividing them out.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots11a-h4","type":"hint","dependencies":["a443311sqroots11a-h3"],"title":"Answer","text":"The answer is $$\\\\frac{4\\\\sqrt{3}}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a443311sqroots12","title":"Simplifying Dvided Square Roots","body":"Simplify the square root espression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Divide Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a443311sqroots12a","stepAnswer":["$$\\\\frac{\\\\sqrt{15}}{6}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{\\\\frac{5}{12}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{\\\\sqrt{15}}{6}$$","hints":{"DefaultPathway":[{"id":"a443311sqroots12a-h1","type":"hint","dependencies":[],"title":"Split","text":"Split the radical into a fraction, since $$\\\\sqrt{\\\\frac{a}{b}}$$ $$=$$ $$\\\\frac{\\\\sqrt{a}}{\\\\sqrt{b}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots12a-h2","type":"hint","dependencies":["a443311sqroots12a-h1"],"title":"Rationalize","text":"Rationalize the denominator by multiplying the numerator and denominator by the radical in the denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots12a-h3","type":"hint","dependencies":["a443311sqroots12a-h2"],"title":"Remove common factors","text":"Remove the common factors of the numerator and denominator by dividing them out.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots12a-h4","type":"hint","dependencies":["a443311sqroots12a-h3"],"title":"Answer","text":"The answer is $$\\\\frac{\\\\sqrt{15}}{6}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a443311sqroots13","title":"Simplifying Dvided Square Roots","body":"Simplify the square root espression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Divide Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a443311sqroots13a","stepAnswer":["$$\\\\frac{\\\\sqrt{77}}{14}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{\\\\frac{11}{28}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{\\\\sqrt{77}}{14}$$","hints":{"DefaultPathway":[{"id":"a443311sqroots13a-h1","type":"hint","dependencies":[],"title":"Split","text":"Split the radical into a fraction, since $$\\\\sqrt{\\\\frac{a}{b}}$$ $$=$$ $$\\\\frac{\\\\sqrt{a}}{\\\\sqrt{b}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots13a-h2","type":"hint","dependencies":["a443311sqroots13a-h1"],"title":"Simplify","text":"Simplify the denominator if possible","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots13a-h3","type":"hint","dependencies":["a443311sqroots13a-h2"],"title":"Rationalize","text":"Rationalize the denominator by multiplying the numerator and denominator by the radical in the denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots13a-h4","type":"hint","dependencies":["a443311sqroots13a-h3"],"title":"Answer","text":"The answer is $$\\\\frac{\\\\sqrt{77}}{14}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a443311sqroots14","title":"Simplifying Dvided Square Roots","body":"Simplify the square root espression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Divide Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a443311sqroots14a","stepAnswer":["$$\\\\frac{2\\\\left(4-\\\\sqrt{2}\\\\right)}{7}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{4}{4+\\\\sqrt{2}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2\\\\left(4-\\\\sqrt{2}\\\\right)}{7}$$","hints":{"DefaultPathway":[{"id":"a443311sqroots14a-h1","type":"hint","dependencies":[],"title":"Multiply by the Conjugate","text":"Multiply the numerator and denominator by the congugate of the denominator $$4-\\\\sqrt{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots14a-h2","type":"hint","dependencies":["a443311sqroots14a-h1"],"title":"Simplify","text":"Simplify the denominator as much as possible","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots14a-h3","type":"hint","dependencies":["a443311sqroots14a-h2"],"title":"Remove common factors","text":"Remove the common factors of the numerator and denominator by dividing them out.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots14a-h4","type":"hint","dependencies":["a443311sqroots14a-h3"],"title":"Answer","text":"The answer is (2*(4-sqrt(2))/7.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a443311sqroots15","title":"Simplifying Dvided Square Roots","body":"Simplify the square root espression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Divide Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a443311sqroots15a","stepAnswer":["$$5\\\\left(2+\\\\sqrt{3}\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{5}{2-\\\\sqrt{3}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5\\\\left(2+\\\\sqrt{3}\\\\right)$$","hints":{"DefaultPathway":[{"id":"a443311sqroots15a-h1","type":"hint","dependencies":[],"title":"Multiply by the Conjugate","text":"Multiply the numerator and denominator by the congugate of the denominator $$2-\\\\left(+\\\\sqrt{3}\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots15a-h2","type":"hint","dependencies":["a443311sqroots15a-h1"],"title":"Simplify","text":"Simplify the denominator as much as possible","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots15a-h3","type":"hint","dependencies":["a443311sqroots15a-h2"],"title":"Remove common factors","text":"Remove the common factors of the numerator and denominator by dividing them out.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots15a-h4","type":"hint","dependencies":["a443311sqroots15a-h3"],"title":"Answer","text":"The answer is $$5\\\\left(2+\\\\sqrt{3}\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a443311sqroots16","title":"Simplifying Dvided Square Roots","body":"Simplify the square root espression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Divide Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a443311sqroots16a","stepAnswer":["$$\\\\frac{\\\\sqrt{3}}{2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\sqrt{27}}{6}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{\\\\sqrt{3}}{2}$$","hints":{"DefaultPathway":[{"id":"a443311sqroots16a-h1","type":"hint","dependencies":[],"title":"Split","text":"Split the numerator into the product of two square roots, one of which can be simplified to a single number such as $$\\\\sqrt{4}$$ or $$\\\\sqrt{16}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots16a-h2","type":"hint","dependencies":["a443311sqroots16a-h1"],"title":"Simplify","text":"Now, simplify the radical so that there is a product of a constant and a square root.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots16a-h3","type":"hint","dependencies":["a443311sqroots16a-h2"],"title":"Remove common factors","text":"Remove the common factors of the numerator and denominator by dividing them out.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots16a-h4","type":"hint","dependencies":["a443311sqroots16a-h3"],"title":"Answer","text":"The answer is $$\\\\frac{\\\\sqrt{3}}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a443311sqroots17","title":"Simplifying Dvided Square Roots","body":"Simplify the square root espression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Divide Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a443311sqroots17a","stepAnswer":["$$\\\\frac{2\\\\sqrt{2}}{3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\sqrt{72}}{9}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2\\\\sqrt{2}}{3}$$","hints":{"DefaultPathway":[{"id":"a443311sqroots17a-h1","type":"hint","dependencies":[],"title":"Split","text":"Split the numerator into the product of two square roots, one of which can be simplified to a single number such as $$\\\\sqrt{4}$$ or $$\\\\sqrt{16}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots17a-h2","type":"hint","dependencies":["a443311sqroots17a-h1"],"title":"Simplify","text":"Now, simplify the radical so that there is a product of a constant and a square root.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots17a-h3","type":"hint","dependencies":["a443311sqroots17a-h2"],"title":"Remove common factors","text":"Remove the common factors of the numerator and denominator by dividing them out.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots17a-h4","type":"hint","dependencies":["a443311sqroots17a-h3"],"title":"Answer","text":"The answer is $$\\\\frac{2\\\\sqrt{2}}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a443311sqroots18","title":"Simplifying Dvided Square Roots","body":"Simplify the square root espression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Divide Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a443311sqroots18a","stepAnswer":["$$\\\\frac{1-2\\\\sqrt{2}}{4}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2-\\\\sqrt{32}}{8}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1-2\\\\sqrt{2}}{4}$$","hints":{"DefaultPathway":[{"id":"a443311sqroots18a-h1","type":"hint","dependencies":[],"title":"Split","text":"Split the radical into the product of two square roots, one of which can be simplified to a single number such as $$\\\\sqrt{4}$$ or $$\\\\sqrt{16}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots18a-h2","type":"hint","dependencies":["a443311sqroots18a-h1"],"title":"Simplify","text":"Now, simplify the radical so that there is a product of a constant and a square root.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots18a-h3","type":"hint","dependencies":["a443311sqroots18a-h2"],"title":"Remove common factors","text":"Remove the common factors of the numerator and denominator by dividing them out.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots18a-h4","type":"hint","dependencies":["a443311sqroots18a-h3"],"title":"Answer","text":"The answer is $$\\\\frac{1-2\\\\sqrt{2}}{4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a443311sqroots19","title":"Simplifying Dvided Square Roots","body":"Simplify the square root espression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Divide Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a443311sqroots19a","stepAnswer":["$$\\\\frac{2+\\\\sqrt{5}}{2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{6+\\\\sqrt{45}}{6}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2+\\\\sqrt{5}}{2}$$","hints":{"DefaultPathway":[{"id":"a443311sqroots19a-h1","type":"hint","dependencies":[],"title":"Split","text":"Split the radical into the product of two square roots, one of which can be simplified to a single number such as $$\\\\sqrt{4}$$ or $$\\\\sqrt{16}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots19a-h2","type":"hint","dependencies":["a443311sqroots19a-h1"],"title":"Simplify","text":"Now, simplify the radical so that there is a product of a constant and a square root.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots19a-h3","type":"hint","dependencies":["a443311sqroots19a-h2"],"title":"Remove common factors","text":"Remove the common factors of the numerator and denominator by dividing them out.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots19a-h4","type":"hint","dependencies":["a443311sqroots19a-h3"],"title":"Answer","text":"The answer is (2+*sqrt(5))/2.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a443311sqroots2","title":"Simplifying Dvided Square Roots","body":"Simplify the square root espression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Divide Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a443311sqroots2a","stepAnswer":["$$\\\\frac{\\\\sqrt{2}}{2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\sqrt{32}}{8}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{\\\\sqrt{2}}{2}$$","hints":{"DefaultPathway":[{"id":"a443311sqroots2a-h1","type":"hint","dependencies":[],"title":"Split","text":"Split the numerator into the product of two square roots, one of which can be simplified to a single number such as $$\\\\sqrt{4}$$ or $$\\\\sqrt{16}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots2a-h2","type":"hint","dependencies":["a443311sqroots2a-h1"],"title":"Simplify","text":"Now, simplify the radical so that there is a product of a constant and a square root.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots2a-h3","type":"hint","dependencies":["a443311sqroots2a-h2"],"title":"Remove common factors","text":"Remove the common factors of the numerator and denominator by dividing them out.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots2a-h4","type":"hint","dependencies":["a443311sqroots2a-h3"],"title":"Answer","text":"The answer is $$\\\\frac{\\\\sqrt{2}}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a443311sqroots20","title":"Simplifying Dvided Square Roots","body":"Simplify the square root espression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Divide Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a443311sqroots20a","stepAnswer":["$$\\\\frac{4}{5}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\sqrt{80}}{\\\\sqrt{125}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{4}{5}$$","hints":{"DefaultPathway":[{"id":"a443311sqroots20a-h1","type":"hint","dependencies":[],"title":"Split","text":"Split the radicals into the product of two square roots, one of which can be simplified to a single number such as $$\\\\sqrt{4}$$ or $$\\\\sqrt{16}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots20a-h2","type":"hint","dependencies":["a443311sqroots20a-h1"],"title":"Simplify","text":"Now, simplify the radicals so that there is a product of a constant and a square root.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots20a-h3","type":"hint","dependencies":["a443311sqroots20a-h2"],"title":"Remove common factors","text":"Remove the common factors of the numerator and denominator by dividing them out.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots20a-h4","type":"hint","dependencies":["a443311sqroots20a-h3"],"title":"Answer","text":"The answer is $$\\\\frac{4}{5}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a443311sqroots21","title":"Simplifying Dvided Square Roots","body":"Simplify the square root espression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Divide Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a443311sqroots21a","stepAnswer":["$$\\\\frac{4}{3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\sqrt{128}}{\\\\sqrt{72}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{4}{3}$$","hints":{"DefaultPathway":[{"id":"a443311sqroots21a-h1","type":"hint","dependencies":[],"title":"Split","text":"Split the radicals into the product of two square roots, one of which can be simplified to a single number such as $$\\\\sqrt{4}$$ or $$\\\\sqrt{16}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots21a-h2","type":"hint","dependencies":["a443311sqroots21a-h1"],"title":"Simplify","text":"Now, simplify the radicals so that there is a product of a constant and a square root.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots21a-h3","type":"hint","dependencies":["a443311sqroots21a-h2"],"title":"Remove common factors","text":"Remove the common factors of the numerator and denominator by dividing them out.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots21a-h4","type":"hint","dependencies":["a443311sqroots21a-h3"],"title":"Answer","text":"The answer is $$\\\\frac{4}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a443311sqroots22","title":"Simplifying Dvided Square Roots","body":"Simplify the square root espression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Divide Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a443311sqroots22a","stepAnswer":["$$2x^2$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\sqrt{8x^6}}{\\\\sqrt{2x^2}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2x^2$$","hints":{"DefaultPathway":[{"id":"a443311sqroots22a-h1","type":"hint","dependencies":[],"title":"Split","text":"Split the radicals into the product of two square roots, one of which can be simplified to a single number such as $$\\\\sqrt{4}$$ or $$\\\\sqrt{16}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots22a-h2","type":"hint","dependencies":["a443311sqroots22a-h1"],"title":"Simplify","text":"Now, simplify the radicals so that there is a product of a constant(or variable) and a square root.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots22a-h3","type":"hint","dependencies":["a443311sqroots22a-h2"],"title":"Remove common factors","text":"Remove the common factors of the numerator and denominator by dividing them out.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots22a-h4","type":"hint","dependencies":["a443311sqroots22a-h3"],"title":"Answer","text":"The answer is $$2x^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a443311sqroots23","title":"Simplifying Dvided Square Roots","body":"Simplify the square root espression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Divide Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a443311sqroots23a","stepAnswer":["$$\\\\frac{10m^2}{7}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\sqrt{200m^5}}{\\\\sqrt{98m}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{10m^2}{7}$$","hints":{"DefaultPathway":[{"id":"a443311sqroots23a-h1","type":"hint","dependencies":[],"title":"Split","text":"Split the radicals into the product of two square roots, one of which can be simplified to a single number such as $$\\\\sqrt{4}$$ or $$\\\\sqrt{16}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots23a-h2","type":"hint","dependencies":["a443311sqroots23a-h1"],"title":"Simplify","text":"Now, simplify the radicals so that there is a product of a constant(or variable) and a square root.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots23a-h3","type":"hint","dependencies":["a443311sqroots23a-h2"],"title":"Remove common factors","text":"Remove the common factors of the numerator and denominator by dividing them out.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots23a-h4","type":"hint","dependencies":["a443311sqroots23a-h3"],"title":"Answer","text":"The answer is $$\\\\frac{10m^2}{7}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a443311sqroots24","title":"Simplifying Dvided Square Roots","body":"Simplify the square root espression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Divide Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a443311sqroots24a","stepAnswer":["$$\\\\frac{5r}{6}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\sqrt{75r^3}}{\\\\sqrt{108r}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5r}{6}$$","hints":{"DefaultPathway":[{"id":"a443311sqroots24a-h1","type":"hint","dependencies":[],"title":"Split","text":"Split the radicals into the product of two square roots, one of which can be simplified to a single number such as $$\\\\sqrt{4}$$ or $$\\\\sqrt{16}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots24a-h2","type":"hint","dependencies":["a443311sqroots24a-h1"],"title":"Simplify","text":"Now, simplify the radicals so that there is a product of a constant(or variable) and a square root.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots24a-h3","type":"hint","dependencies":["a443311sqroots24a-h2"],"title":"Remove common factors","text":"Remove the common factors of the numerator and denominator by dividing them out.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots24a-h4","type":"hint","dependencies":["a443311sqroots24a-h3"],"title":"Answer","text":"The answer is $$\\\\frac{5r}{6}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a443311sqroots25","title":"Simplifying Dvided Square Roots","body":"Simplify the square root espression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Divide Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a443311sqroots25a","stepAnswer":["$$\\\\frac{3p \\\\sqrt{102}}{17q^2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\sqrt{108p^5 q^2}}{\\\\sqrt{3p^3 q^6}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3p \\\\sqrt{102}}{17q^2}$$","hints":{"DefaultPathway":[{"id":"a443311sqroots25a-h1","type":"hint","dependencies":[],"title":"Split","text":"Split the radicals into the product of two square roots, one of which can be simplified to a single number such as $$\\\\sqrt{4}$$ or $$\\\\sqrt{16}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots25a-h2","type":"hint","dependencies":["a443311sqroots25a-h1"],"title":"Simplify","text":"Now, simplify the radicals so that there is a product of a constant(or variable) and a square root.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots25a-h3","type":"hint","dependencies":["a443311sqroots25a-h2"],"title":"Remove common factors","text":"Remove the common factors of the numerator and denominator by dividing them out.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots25a-h4","type":"hint","dependencies":["a443311sqroots25a-h3"],"title":"Answer","text":"The answer is $$\\\\frac{3p \\\\sqrt{102}}{17} q^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a443311sqroots26","title":"Simplifying Dvided Square Roots","body":"Simplify the square root espression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Divide Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a443311sqroots26a","stepAnswer":["$$\\\\frac{\\\\sqrt{2}}{2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\sqrt{98}}{14}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{\\\\sqrt{2}}{2}$$","hints":{"DefaultPathway":[{"id":"a443311sqroots26a-h1","type":"hint","dependencies":[],"title":"Split","text":"Split the radicals into the product of two square roots, one of which can be simplified to a single number such as $$\\\\sqrt{4}$$ or $$\\\\sqrt{16}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots26a-h2","type":"hint","dependencies":["a443311sqroots26a-h1"],"title":"Simplify","text":"Now, simplify the radicals so that there is a product of a constant(or variable) and a square root.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots26a-h3","type":"hint","dependencies":["a443311sqroots26a-h2"],"title":"Remove common factors","text":"Remove the common factors of the numerator and denominator by dividing them out.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots26a-h4","type":"hint","dependencies":["a443311sqroots26a-h3"],"title":"Answer","text":"The answer is $$\\\\frac{\\\\sqrt{2}}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a443311sqroots27","title":"Simplifying Dvided Square Roots","body":"Simplify the square root espression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Divide Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a443311sqroots27a","stepAnswer":["$$\\\\frac{1+\\\\sqrt{5}}{3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{5+\\\\sqrt{125}}{15}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1+\\\\sqrt{5}}{3}$$","hints":{"DefaultPathway":[{"id":"a443311sqroots27a-h1","type":"hint","dependencies":[],"title":"Split","text":"Split the radicals into the product of two square roots, one of which can be simplified to a single number such as $$\\\\sqrt{4}$$ or $$\\\\sqrt{16}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots27a-h2","type":"hint","dependencies":["a443311sqroots27a-h1"],"title":"Simplify","text":"Now, simplify the radicals so that there is a product of a constant(or variable) and a square root.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots27a-h3","type":"hint","dependencies":["a443311sqroots27a-h2"],"title":"Remove common factors","text":"Remove the common factors of the numerator and denominator by dividing them out.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots27a-h4","type":"hint","dependencies":["a443311sqroots27a-h3"],"title":"Answer","text":"The answer is $$\\\\frac{1+\\\\sqrt{5}}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a443311sqroots28","title":"Simplifying Dvided Square Roots","body":"Simplify the square root espression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Divide Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a443311sqroots28a","stepAnswer":["$$\\\\frac{4}{5}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\sqrt{96}}{\\\\sqrt{150}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{4}{5}$$","hints":{"DefaultPathway":[{"id":"a443311sqroots28a-h1","type":"hint","dependencies":[],"title":"Split","text":"Split the radicals into the product of two square roots, one of which can be simplified to a single number such as $$\\\\sqrt{4}$$ or $$\\\\sqrt{16}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots28a-h2","type":"hint","dependencies":["a443311sqroots28a-h1"],"title":"Simplify","text":"Now, simplify the radicals so that there is a product of a constant(or variable) and a square root.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots28a-h3","type":"hint","dependencies":["a443311sqroots28a-h2"],"title":"Remove common factors","text":"Remove the common factors of the numerator and denominator by dividing them out.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots28a-h4","type":"hint","dependencies":["a443311sqroots28a-h3"],"title":"Answer","text":"The answer is $$\\\\frac{4}{5}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a443311sqroots29","title":"Simplifying Dvided Square Roots","body":"Simplify the square root espression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Divide Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a443311sqroots29a","stepAnswer":["$$y^3 \\\\sqrt{13}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\sqrt{26y^7}}{\\\\sqrt{2y}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y^3 \\\\sqrt{13}$$","hints":{"DefaultPathway":[{"id":"a443311sqroots29a-h1","type":"hint","dependencies":[],"title":"Split","text":"Split the radicals into the product of two square roots, one of which can be simplified to a single number such as $$\\\\sqrt{4}$$ or $$\\\\sqrt{16}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots29a-h2","type":"hint","dependencies":["a443311sqroots29a-h1"],"title":"Simplify","text":"Now, simplify the radicals so that there is a product of a constant(or variable) and a square root.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots29a-h3","type":"hint","dependencies":["a443311sqroots29a-h2"],"title":"Remove common factors","text":"Remove the common factors of the numerator and denominator by dividing them out.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots29a-h4","type":"hint","dependencies":["a443311sqroots29a-h3"],"title":"Answer","text":"The answer is $$y^3 \\\\sqrt{13}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a443311sqroots3","title":"Simplifying Dvided Square Roots","body":"Simplify the square root espression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Divide Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a443311sqroots3a","stepAnswer":["$$\\\\frac{3-\\\\sqrt{6}}{6}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{6-\\\\sqrt{24}}{12}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3-\\\\sqrt{6}}{6}$$","hints":{"DefaultPathway":[{"id":"a443311sqroots3a-h1","type":"hint","dependencies":[],"title":"Split","text":"Split the radical into the product of two square roots, one of which can be simplified to a single number such as $$\\\\sqrt{4}$$ or $$\\\\sqrt{16}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots3a-h2","type":"hint","dependencies":["a443311sqroots3a-h1"],"title":"Simplify","text":"Now, simplify the radical so that there is a product of a constant and a square root.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots3a-h3","type":"hint","dependencies":["a443311sqroots3a-h2"],"title":"Remove common factors","text":"Remove the common factors of the numerator and denominator by dividing them out.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots3a-h4","type":"hint","dependencies":["a443311sqroots3a-h3"],"title":"Answer","text":"The answer is $$\\\\frac{3-\\\\sqrt{6}}{6}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a443311sqroots30","title":"Simplifying Dvided Square Roots","body":"Simplify the square root espression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Divide Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a443311sqroots30a","stepAnswer":["$$\\\\frac{8n}{3m^3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\sqrt{320m n^5}}{\\\\sqrt{45m^7 n^3}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{8n}{3m^3}$$","hints":{"DefaultPathway":[{"id":"a443311sqroots30a-h1","type":"hint","dependencies":[],"title":"Split","text":"Split the radicals into the product of two square roots, one of which can be simplified to a single number such as $$\\\\sqrt{4}$$ or $$\\\\sqrt{16}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots30a-h2","type":"hint","dependencies":["a443311sqroots30a-h1"],"title":"Simplify","text":"Now, simplify the radicals so that there is a product of a constant(or variable) and a square root.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots30a-h3","type":"hint","dependencies":["a443311sqroots30a-h2"],"title":"Remove common factors","text":"Remove the common factors of the numerator and denominator by dividing them out.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots30a-h4","type":"hint","dependencies":["a443311sqroots30a-h3"],"title":"Answer","text":"The answer is $$\\\\frac{8n}{3m^3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a443311sqroots4","title":"Simplifying Dvided Square Roots","body":"Simplify the square root espression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Divide Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a443311sqroots4a","stepAnswer":["$$\\\\frac{4-\\\\sqrt{10}}{5}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{8-\\\\sqrt{40}}{10}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{4-\\\\sqrt{10}}{5}$$","hints":{"DefaultPathway":[{"id":"a443311sqroots4a-h1","type":"hint","dependencies":[],"title":"Split","text":"Split the radical into the product of two square roots, one of which can be simplified to a single number such as $$\\\\sqrt{4}$$ or $$\\\\sqrt{16}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots4a-h2","type":"hint","dependencies":["a443311sqroots4a-h1"],"title":"Simplify","text":"Now, simplify the radical so that there is a product of a constant and a square root.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots4a-h3","type":"hint","dependencies":["a443311sqroots4a-h2"],"title":"Remove common factors","text":"Remove the common factors of the numerator and denominator by dividing them out.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots4a-h4","type":"hint","dependencies":["a443311sqroots4a-h3"],"title":"Answer","text":"The answer is $$\\\\frac{4-\\\\sqrt{10}}{5}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a443311sqroots5","title":"Simplifying Dvided Square Roots","body":"Simplify the square root espression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Divide Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a443311sqroots5a","stepAnswer":["$$\\\\frac{3}{5}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\sqrt{27}}{\\\\sqrt{75}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{5}$$","hints":{"DefaultPathway":[{"id":"a443311sqroots5a-h1","type":"hint","dependencies":[],"title":"Combine","text":"Combine the radicals into one radical, since $$\\\\frac{\\\\sqrt{a}}{\\\\sqrt{b}}$$ $$=$$ $$\\\\sqrt{\\\\frac{a}{b}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots5a-h2","type":"hint","dependencies":["a443311sqroots5a-h1"],"title":"Simplify","text":"Now, simplify the quotient under the radical as much as possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots5a-h3","type":"hint","dependencies":["a443311sqroots5a-h2"],"title":"Reduce","text":"If necessary, simplify the radical further by splitting it apart and reducing it to a constant and another radical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots5a-h4","type":"hint","dependencies":["a443311sqroots5a-h3"],"title":"Answer","text":"The answer is $$\\\\frac{3}{5}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a443311sqroots6","title":"Simplifying Dvided Square Roots","body":"Simplify the square root espression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Divide Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a443311sqroots6a","stepAnswer":["$$\\\\frac{2}{3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\sqrt{48}}{\\\\sqrt{108}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2}{3}$$","hints":{"DefaultPathway":[{"id":"a443311sqroots6a-h1","type":"hint","dependencies":[],"title":"Combine","text":"Combine the radicals into one radical, since $$\\\\frac{\\\\sqrt{a}}{\\\\sqrt{b}}$$ $$=$$ $$\\\\sqrt{\\\\frac{a}{b}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots6a-h2","type":"hint","dependencies":["a443311sqroots6a-h1"],"title":"Simplify","text":"Now, simplify the quotient under the radical as much as possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots6a-h3","type":"hint","dependencies":["a443311sqroots6a-h2"],"title":"Reduce","text":"If necessary, simplify the radical further by splitting it apart and reducing it to a constant and another radical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots6a-h4","type":"hint","dependencies":["a443311sqroots6a-h3"],"title":"Answer","text":"The answer is $$\\\\frac{2}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a443311sqroots7","title":"Simplifying Dvided Square Roots","body":"Simplify the square root espression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Divide Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a443311sqroots7a","stepAnswer":["$$y^2 \\\\sqrt{3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\sqrt{6y^5}}{\\\\sqrt{2y}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y^2 \\\\sqrt{3}$$","hints":{"DefaultPathway":[{"id":"a443311sqroots7a-h1","type":"hint","dependencies":[],"title":"Combine","text":"Combine the radicals into one radical, since $$\\\\frac{\\\\sqrt{a}}{\\\\sqrt{b}}$$ $$=$$ $$\\\\sqrt{\\\\frac{a}{b}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots7a-h2","type":"hint","dependencies":["a443311sqroots7a-h1"],"title":"Simplify","text":"Now, simplify the quotient under the radical as much as possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots7a-h3","type":"hint","dependencies":["a443311sqroots7a-h2"],"title":"Reduce","text":"If necessary, simplify the radical further by splitting it apart and reducing it to a constant (or variable) and another radical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots7a-h4","type":"hint","dependencies":["a443311sqroots7a-h3"],"title":"Answer","text":"The answer is $$y^2 \\\\sqrt{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a443311sqroots8","title":"Simplifying Dvided Square Roots","body":"Simplify the square root espression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Divide Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a443311sqroots8a","stepAnswer":["$$\\\\frac{2x}{3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\sqrt{72x^3}}{\\\\sqrt{162x}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2x}{3}$$","hints":{"DefaultPathway":[{"id":"a443311sqroots8a-h1","type":"hint","dependencies":[],"title":"Combine","text":"Combine the radicals into one radical, since $$\\\\frac{\\\\sqrt{a}}{\\\\sqrt{b}}$$ $$=$$ $$\\\\sqrt{\\\\frac{a}{b}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots8a-h2","type":"hint","dependencies":["a443311sqroots8a-h1"],"title":"Simplify","text":"Now, simplify the quotient under the radical as much as possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots8a-h3","type":"hint","dependencies":["a443311sqroots8a-h2"],"title":"Reduce","text":"If necessary, simplify the radical further by splitting it apart and reducing it to a constant (or variable) and another radical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots8a-h4","type":"hint","dependencies":["a443311sqroots8a-h3"],"title":"Answer","text":"The answer is $$\\\\frac{2x}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a443311sqroots9","title":"Simplifying Dvided Square Roots","body":"Simplify the square root espression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Divide Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a443311sqroots9a","stepAnswer":["$$\\\\frac{7b^2}{a}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\sqrt{147a b^8}}{\\\\sqrt{3a^3 b^4}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{7b^2}{a}$$","hints":{"DefaultPathway":[{"id":"a443311sqroots9a-h1","type":"hint","dependencies":[],"title":"Combine","text":"Combine the radicals into one radical, since $$\\\\frac{\\\\sqrt{a}}{\\\\sqrt{b}}$$ $$=$$ $$\\\\sqrt{\\\\frac{a}{b}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots9a-h2","type":"hint","dependencies":["a443311sqroots9a-h1"],"title":"Simplify","text":"Now, simplify the quotient under the radical as much as possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots9a-h3","type":"hint","dependencies":["a443311sqroots9a-h2"],"title":"Reduce","text":"If necessary, simplify the radical further by splitting it apart and reducing it to a constant (or variable) and another radical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a443311sqroots9a-h4","type":"hint","dependencies":["a443311sqroots9a-h3"],"title":"Answer","text":"The answer is $$\\\\frac{7b^2}{a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4488b1Fraction1","title":"How to Simplify a Fraction?","body":"Simplify the following fraction","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Fractions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4488b1Fraction1a","stepAnswer":["$$\\\\frac{-23}{40}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{-69}{120}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-23}{40}$$","hints":{"DefaultPathway":[{"id":"a4488b1Fraction1a-h1","type":"hint","dependencies":[],"title":"Principle","text":"Find the common factor of numerator and denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction1a-h2","type":"hint","dependencies":["a4488b1Fraction1a-h1"],"title":"Factor","text":"Factor the numerator to 3x23","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction1a-h3","type":"hint","dependencies":["a4488b1Fraction1a-h2"],"title":"Factor","text":"Factor the denominator to 2x2x2x3x5","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a4488b1Fraction1a-h3"],"title":"Organizing","text":"What is the common factor?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction1a-h5","type":"hint","dependencies":["a4488b1Fraction1a-h4"],"title":"Division","text":"Dividing both sides by the common factor","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4488b1Fraction10","title":"Evaluate Variable Expressions with Fraction","body":"Evaluate the following expression","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Fractions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4488b1Fraction10a","stepAnswer":["$$\\\\frac{-1}{2}$$"],"problemType":"TextBox","stepTitle":"$$3{ab}^2$$ when $$a=\\\\frac{-2}{3}$$ and $$b=\\\\frac{-1}{2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-1}{2}$$","hints":{"DefaultPathway":[{"id":"a4488b1Fraction10a-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Find $$b^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction10a-h2","type":"hint","dependencies":[],"title":"Substitution","text":"The equation becomes $$3\\\\left(-\\\\frac{2}{3}\\\\right) {\\\\left(-\\\\frac{1}{2}\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":[],"title":"Organizing","text":"What is the common factor of the fraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction10a-h4","type":"hint","dependencies":[],"title":"Division","text":"Dividing both sides by the common factor","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4488b1Fraction11","title":"Evaluate Variable Expressions with Fraction","body":"Evaluate the following expression","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Fractions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4488b1Fraction11a","stepAnswer":["$$\\\\frac{2}{3}$$"],"problemType":"TextBox","stepTitle":"$$4c^3 d$$ when $$c=\\\\frac{-1}{2}$$ and $$d=\\\\frac{-4}{3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2}{3}$$","hints":{"DefaultPathway":[{"id":"a4488b1Fraction11a-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Find $$c^3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction11a-h2","type":"hint","dependencies":[],"title":"Substitution","text":"The equation becomes $$4{\\\\left(-\\\\frac{1}{2}\\\\right)}^3 \\\\left(-\\\\frac{4}{3}\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction11a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":[],"title":"Organizing","text":"What is the common factor of the fraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction11a-h4","type":"hint","dependencies":[],"title":"Division","text":"Dividing both sides by the common factor","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4488b1Fraction12","title":"How to Add or Subtract Fractions?","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Fractions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4488b1Fraction12a","stepAnswer":["$$\\\\frac{103}{60}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{13}{15}+\\\\frac{17}{20}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{103}{60}$$","hints":{"DefaultPathway":[{"id":"a4488b1Fraction12a-h1","type":"hint","dependencies":[],"title":"Principle","text":"They don\'t have a common denominator. Find the least common denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction12a-h2","type":"hint","dependencies":["a4488b1Fraction12a-h1"],"title":"Factor","text":"Factor the first denominator to 3x5","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction12a-h3","type":"hint","dependencies":["a4488b1Fraction12a-h2"],"title":"Factor","text":"Factor the second denominator to 2x2x5","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$60$$"],"dependencies":["a4488b1Fraction12a-h3"],"title":"Calculation","text":"What is the least common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction12a-h5","type":"hint","dependencies":["a4488b1Fraction12a-h4"],"title":"Calculation","text":"Multiply the common factor by the non-common factor","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction12a-h6","type":"hint","dependencies":["a4488b1Fraction12a-h5"],"title":"Multiplication","text":"Multiply the numerator and denominator to reach the least common denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction12a-h7","type":"hint","dependencies":["a4488b1Fraction12a-h6"],"title":"Multiplication","text":"The equation becomes: $$\\\\frac{52}{60}+\\\\frac{51}{60}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction12a-h8","type":"hint","dependencies":["a4488b1Fraction12a-h7"],"title":"Addition","text":"Adding the numerators","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4488b1Fraction13","title":"How to Simplify a Fraction?","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Fractions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4488b1Fraction13a","stepAnswer":["$$\\\\frac{2}{3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\left(-\\\\frac{5}{14}\\\\right)}{\\\\left(-\\\\frac{15}{28}\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2}{3}$$","hints":{"DefaultPathway":[{"id":"a4488b1Fraction13a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The minus signs can be canceled out in multiplication or division","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction13a-h2","type":"hint","dependencies":[],"title":"Principle","text":"Dividing $$=$$ multiplying the reciprocal of the second term","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction13a-h3","type":"hint","dependencies":[],"title":"Principle","text":"Find the common factor of numerator and denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction13a-h4","type":"hint","dependencies":["a4488b1Fraction13a-h3"],"title":"Factor","text":"Factor the numerator to 2x2x5x7","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction13a-h5","type":"hint","dependencies":["a4488b1Fraction13a-h4"],"title":"Factor","text":"Factor the denominator to 2x3x5x7","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction13a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$70$$"],"dependencies":["a4488b1Fraction13a-h5"],"title":"Organizing","text":"What is the common factor?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction13a-h7","type":"hint","dependencies":["a4488b1Fraction13a-h6"],"title":"Multiplication","text":"Multiply the rest of the factors","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4488b1Fraction14","title":"Add or Subtract Fractions?","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Fractions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4488b1Fraction14a","stepAnswer":["$$\\\\frac{27a-32}{36}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{3a}{4}-\\\\frac{8}{9}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{27a-32}{36}$$","hints":{"DefaultPathway":[{"id":"a4488b1Fraction14a-h1","type":"hint","dependencies":[],"title":"Principle","text":"They don\'t have a common denominator. Find the least common denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$36$$"],"dependencies":["a4488b1Fraction14a-h1"],"title":"Calculation","text":"What is the least common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction14a-h3","type":"hint","dependencies":["a4488b1Fraction14a-h2"],"title":"Multiplication","text":"Multiply the numerator and denominator to reach the least common denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction14a-h4","type":"hint","dependencies":["a4488b1Fraction14a-h3"],"title":"Multiplication","text":"The equation becomes: $$\\\\frac{27a}{36}-\\\\frac{32}{36}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction14a-h5","type":"hint","dependencies":["a4488b1Fraction14a-h4"],"title":"Substraction","text":"Substracting the numerators","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a4488b1Fraction14b","stepAnswer":["$$\\\\frac{2a}{3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{3a}{4} \\\\frac{8}{9}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2a}{3}$$","hints":{"DefaultPathway":[{"id":"a4488b1Fraction14b-h1","type":"hint","dependencies":[],"title":"Principle","text":"Find the common factor of numerator and denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction14b-h2","type":"hint","dependencies":["a4488b1Fraction14b-h1"],"title":"Factor","text":"Factor the numerator to 2x2x2x3xa","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction14b-h3","type":"hint","dependencies":["a4488b1Fraction14b-h2"],"title":"Factor","text":"Factor the denominator to 2x2x3x3","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction14b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a4488b1Fraction14b-h3"],"title":"Organizing","text":"What is the common factor?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction14b-h5","type":"hint","dependencies":["a4488b1Fraction14b-h4"],"title":"Division","text":"Dividing both sides by the common factor","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4488b1Fraction15","title":"Simplify Complex Functions","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Fractions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4488b1Fraction15a","stepAnswer":["$$\\\\frac{2}{7}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\frac{2}{3}-\\\\frac{1}{2}}{\\\\frac{1}{4}+\\\\frac{1}{3}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2}{7}$$","hints":{"DefaultPathway":[{"id":"a4488b1Fraction15a-h1","type":"hint","dependencies":[],"title":"Principle","text":"Find the least common denominators in numerator and denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a4488b1Fraction15a-h1"],"title":"Calculation","text":"What is the LCD in numerator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a4488b1Fraction15a-h2"],"title":"Calculation","text":"What is the LCD in denomerator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction15a-h4","type":"hint","dependencies":["a4488b1Fraction15a-h3"],"title":"Multiplication","text":"Multiply the LCD, the fraction becomes $$\\\\frac{\\\\frac{4}{6}-\\\\frac{3}{6}}{\\\\frac{3}{12}+\\\\frac{4}{12}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction15a-h5","type":"hint","dependencies":["a4488b1Fraction15a-h4"],"title":"Calculation","text":"Calculate the value of numerator and denomerator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction15a-h6","type":"hint","dependencies":["a4488b1Fraction15a-h5"],"title":"Calculation","text":"The fraction becomes $$\\\\frac{\\\\frac{1}{6}}{\\\\frac{7}{12}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction15a-h7","type":"hint","dependencies":["a4488b1Fraction15a-h6"],"title":"Principle","text":"Convert the division to multiplying the reciprocal of the second term: $$\\\\frac{1}{6} \\\\frac{12}{7}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction15a-h8","type":"hint","dependencies":["a4488b1Fraction15a-h7"],"title":"Factor","text":"Factor the common factor in numerator and denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4488b1Fraction2","title":"How to Simplify a Fraction?","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Fractions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4488b1Fraction2a","stepAnswer":["-33a"],"problemType":"TextBox","stepTitle":"$$\\\\frac{11}{3}$$ * $$(-9a)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a4488b1Fraction2a-h1","type":"hint","dependencies":[],"title":"Factor","text":"Factor $$(-9a)$$ to -3x3xa","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction2a-h2","type":"hint","dependencies":[],"title":"Principle","text":"Find the common factor of numerator and denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a4488b1Fraction2a-h2"],"title":"Organizing","text":"What is the common factor?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction2a-h4","type":"hint","dependencies":["a4488b1Fraction2a-h3"],"title":"Division","text":"Dividing both sides by the common factor","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4488b1Fraction3","title":"How to Simplify a Fraction?","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Fractions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4488b1Fraction3a","stepAnswer":["$$\\\\frac{4}{15}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\left(-\\\\frac{7}{27}\\\\right)}{\\\\left(-\\\\frac{35}{36}\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{4}{15}$$","hints":{"DefaultPathway":[{"id":"a4488b1Fraction3a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The minus signs can be canceled out in multiplication or division","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction3a-h2","type":"hint","dependencies":[],"title":"Principle","text":"Dividing $$=$$ multiplying the reciprocal of the second term","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction3a-h3","type":"hint","dependencies":[],"title":"Principle","text":"Find the common factor of numerator and denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction3a-h4","type":"hint","dependencies":["a4488b1Fraction3a-h3"],"title":"Factor","text":"Factor the numerator to 3x3x2x2x7","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction3a-h5","type":"hint","dependencies":["a4488b1Fraction3a-h4"],"title":"Factor","text":"Factor the denominator to 3x3x3x5x7","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction3a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$63$$"],"dependencies":["a4488b1Fraction3a-h5"],"title":"Organizing","text":"What is the common factor?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction3a-h7","type":"hint","dependencies":["a4488b1Fraction3a-h6"],"title":"Multiplication","text":"Multiply the rest of the factors","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4488b1Fraction4","title":"How to Simplify a Fraction?","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Fractions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4488b1Fraction4a","stepAnswer":["$$\\\\frac{3}{4b}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\frac{a}{8}}{\\\\frac{ab}{6}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{4b}$$","hints":{"DefaultPathway":[{"id":"a4488b1Fraction4a-h1","type":"hint","dependencies":[],"title":"Principle","text":"Convert the division to multiplying the reciprocal of the second term: $$\\\\frac{a}{8} \\\\frac{6}{ab}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction4a-h2","type":"hint","dependencies":[],"title":"Principle","text":"Find the common factor of numerator and denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction4a-h3","type":"hint","dependencies":["a4488b1Fraction4a-h2"],"title":"Factor","text":"Factor the numerator to 2x3xa","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction4a-h4","type":"hint","dependencies":["a4488b1Fraction4a-h3"],"title":"Factor","text":"Factor the denominator to 2x2x2xaxb","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction4a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["2a"],"dependencies":["a4488b1Fraction4a-h4"],"title":"Organizing","text":"What is the common factor?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction4a-h6","type":"hint","dependencies":["a4488b1Fraction4a-h5"],"title":"Division","text":"Dividing both sides by the common factor","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4488b1Fraction5","title":"How to Add or Subtract Fractions?","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Fractions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4488b1Fraction5a","stepAnswer":["$$\\\\frac{79}{60}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{7}{12}+\\\\frac{11}{15}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{79}{60}$$","hints":{"DefaultPathway":[{"id":"a4488b1Fraction5a-h1","type":"hint","dependencies":[],"title":"Principle","text":"They don\'t have a common denominator. Find the least common denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction5a-h2","type":"hint","dependencies":["a4488b1Fraction5a-h1"],"title":"Factor","text":"Factor the first denominator to 2x2x3","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction5a-h3","type":"hint","dependencies":["a4488b1Fraction5a-h2"],"title":"Factor","text":"Factor the second denominator to 3x5","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$60$$"],"dependencies":["a4488b1Fraction5a-h3"],"title":"Calculation","text":"What is the least common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction5a-h5","type":"hint","dependencies":["a4488b1Fraction5a-h4"],"title":"Calculation","text":"Multiply the common factor by the non-common factor","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction5a-h6","type":"hint","dependencies":["a4488b1Fraction5a-h5"],"title":"Multiplication","text":"Multiply the numerator and denominator to reach the least common denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction5a-h7","type":"hint","dependencies":["a4488b1Fraction5a-h6"],"title":"Multiplication","text":"The equation becomes: $$\\\\frac{35}{60}+\\\\frac{44}{60}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction5a-h8","type":"hint","dependencies":["a4488b1Fraction5a-h7"],"title":"Addition","text":"Adding the numerators","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4488b1Fraction6","title":"Add or Subtract Fractions?","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Fractions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4488b1Fraction6a","stepAnswer":["$$\\\\frac{24k-5}{30}$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(\\\\frac{4k}{5}-\\\\frac{1}{6}\\\\right) \\\\frac{4k}{5} \\\\frac{1}{6}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{24k-5}{30}$$","hints":{"DefaultPathway":[{"id":"a4488b1Fraction6a-h1","type":"hint","dependencies":[],"title":"Principle","text":"They don\'t have a common denominator. Find the least common denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$30$$"],"dependencies":["a4488b1Fraction6a-h1"],"title":"Calculation","text":"What is the least common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction6a-h3","type":"hint","dependencies":["a4488b1Fraction6a-h2"],"title":"Multiplication","text":"Multiply the numerator and denominator to reach the least common denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction6a-h4","type":"hint","dependencies":["a4488b1Fraction6a-h3"],"title":"Multiplication","text":"The equation becomes: $$\\\\frac{24k}{30}-\\\\frac{5}{30}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction6a-h5","type":"hint","dependencies":["a4488b1Fraction6a-h4"],"title":"Substraction","text":"Substracting the numerators","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a4488b1Fraction6b","stepAnswer":["$$\\\\frac{2k}{15}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{4k}{5} \\\\frac{1}{6}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2k}{15}$$","hints":{"DefaultPathway":[{"id":"a4488b1Fraction6b-h1","type":"hint","dependencies":[],"title":"Principle","text":"Find the common factor of numerator and denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction6b-h2","type":"hint","dependencies":["a4488b1Fraction6b-h1"],"title":"Factor","text":"Factor the numerator to 2x2xk","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction6b-h3","type":"hint","dependencies":["a4488b1Fraction6b-h2"],"title":"Factor","text":"Factor the denominator to 2x3x5","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction6b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a4488b1Fraction6b-h3"],"title":"Organizing","text":"What is the common factor?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction6b-h5","type":"hint","dependencies":["a4488b1Fraction6b-h4"],"title":"Division","text":"Dividing both sides by the common factor","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4488b1Fraction7","title":"Placement of Negative Sign in a Fraction","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Fractions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4488b1Fraction7a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"[8*(-2)+4(-3)]/[(-5)*2+3]","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a4488b1Fraction7a-h1","type":"hint","dependencies":[],"title":"Order of Operation","text":"Calculate the numerator and denominator first","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction7a-h2","type":"hint","dependencies":[],"title":"Organizing","text":"The fraction becomes $$\\\\frac{\\\\left(-28\\\\right)}{\\\\left(-7\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction7a-h3","type":"hint","dependencies":[],"title":"Principle","text":"The minus signs can be canceled out in multiplication or division","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction7a-h4","type":"hint","dependencies":[],"title":"Division","text":"Complete the division","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4488b1Fraction8","title":"How to Simplify Complex Fractions?","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Fractions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4488b1Fraction8a","stepAnswer":["$$\\\\frac{1}{90}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{{\\\\left(\\\\frac{1}{3}\\\\right)}^2}{2^3+2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{90}$$","hints":{"DefaultPathway":[{"id":"a4488b1Fraction8a-h1","type":"hint","dependencies":[],"title":"Order of Operation","text":"Calculate the numerator and denominator first","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction8a-h2","type":"hint","dependencies":[],"title":"Organizing","text":"The fraction becomes $$\\\\frac{\\\\frac{1}{9}}{10}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction8a-h3","type":"hint","dependencies":[],"title":"Principle","text":"Convert the division to multiplying the reciprocal of the second term: $$\\\\frac{1}{9} \\\\frac{1}{10}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{90}$$"],"dependencies":["a4488b1Fraction8a-h3"],"title":"Calculation","text":"What is the result of $$\\\\frac{1}{9} \\\\frac{1}{10}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4488b1Fraction9","title":"Simplify Complex Functions","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Fractions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4488b1Fraction9a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\frac{1}{3}+\\\\frac{1}{2}}{\\\\frac{3}{4}-\\\\frac{1}{3}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a4488b1Fraction9a-h1","type":"hint","dependencies":[],"title":"Principle","text":"Find the least common denominators in numerator and denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a4488b1Fraction9a-h1"],"title":"Calculation","text":"What is the LCD in numerator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a4488b1Fraction9a-h1"],"title":"Calculation","text":"What is the LCD in denomerator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction9a-h4","type":"hint","dependencies":[],"title":"Multiplication","text":"Multiply the LCD, the fraction becomes $$\\\\frac{\\\\frac{2}{6}+\\\\frac{3}{6}}{\\\\frac{9}{12}-\\\\frac{4}{12}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction9a-h5","type":"hint","dependencies":[],"title":"Calculation","text":"Calculate the value of numerator and denomerator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction9a-h6","type":"hint","dependencies":[],"title":"Calculation","text":"The fraction becomes $$\\\\frac{\\\\frac{5}{6}}{\\\\frac{5}{12}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction9a-h7","type":"hint","dependencies":[],"title":"Principle","text":"Convert the division to multiplying the reciprocal of the second term: $$\\\\frac{5}{6} \\\\frac{12}{5}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1Fraction9a-h8","type":"hint","dependencies":[],"title":"Factor","text":"Factor the common factor in numerator and denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4488b1fractions1","title":"How to Simplify a Fraction?","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Fractions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4488b1fractions1a","stepAnswer":["$$\\\\frac{-12}{7}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{-108}{63}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-12}{7}$$","hints":{"DefaultPathway":[{"id":"a4488b1fractions1a-h1","type":"hint","dependencies":[],"title":"Rewrite","text":"Rewrite the fraction as a product of primes in both the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1fractions1a-h2","type":"hint","dependencies":["a4488b1fractions1a-h1"],"title":"Eliminate","text":"Eliminate any commonly occuring primes in both the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1fractions1a-h3","type":"hint","dependencies":["a4488b1fractions1a-h2"],"title":"Multiply","text":"Multiply the product back into two terms, one in the numerator, and one in the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4488b1fractions10","title":"Perform the indicated operation","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Fractions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4488b1fractions10a","stepAnswer":["$$\\\\frac{1}{4}$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(-\\\\frac{33}{60}\\\\right) \\\\left(-\\\\frac{40}{88}\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{4}$$","hints":{"DefaultPathway":[{"id":"a4488b1fractions10a-h1","type":"hint","dependencies":[],"title":"Use diagonals","text":"Simplify the fraction diagonally. Split one term from the numerator of one fraction into the product of prime numbers. Do the same for the denominator of the other fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1fractions10a-h2","type":"hint","dependencies":["a4488b1fractions10a-h1"],"title":"Simplify","text":"Simplify diagonally, eliminating prime numbers. Then multiply the product together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1fractions10a-h3","type":"hint","dependencies":["a4488b1fractions10a-h2"],"title":"Multiply","text":"Multiply the two fractions together, multiplying the numerators together and the denominators together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4488b1fractions11","title":"Perform the indicated operation","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Fractions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4488b1fractions11a","stepAnswer":["$$\\\\frac{33}{4x}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\frac{3}{4}}{\\\\frac{x}{11}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{33}{4x}$$","hints":{"DefaultPathway":[{"id":"a4488b1fractions11a-h1","type":"hint","dependencies":[],"title":"Reciprocal","text":"Take the reciprocal of the second fraction. The reciprocal is the opposite of the current fraction. In other words, flip the fraction so the top is at the bottom, and the bottom is at the top.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1fractions11a-h2","type":"hint","dependencies":["a4488b1fractions11a-h1"],"title":"Eliminate diagonally","text":"Eliminate any common terms that may occur in the diagonals of the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1fractions11a-h3","type":"hint","dependencies":["a4488b1fractions11a-h2"],"title":"Multiply","text":"Multiply the numerators together and the deonominators together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4488b1fractions12","title":"Perform the indicated operation","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Fractions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4488b1fractions12a","stepAnswer":["$$\\\\frac{-4}{9}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\frac{5}{18}}{\\\\left(-\\\\frac{15}{24}\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-4}{9}$$","hints":{"DefaultPathway":[{"id":"a4488b1fractions12a-h1","type":"hint","dependencies":[],"title":"Reciprocal","text":"Take the reciprocal of the second fraction. The reciprocal is the opposite of the current fraction. In other words, flip the fraction so the top is at the bottom, and the bottom is at the top.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1fractions12a-h2","type":"hint","dependencies":["a4488b1fractions12a-h1"],"title":"Eliminate diagonally","text":"Eliminate any common terms that may occur in the diagonals of the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1fractions12a-h3","type":"hint","dependencies":["a4488b1fractions12a-h2"],"title":"Multiply","text":"Multiply the numerators together and the deonominators together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4488b1fractions13","title":"Perform the indicated operation","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Fractions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4488b1fractions13a","stepAnswer":["$$\\\\frac{10u}{9v}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\frac{8u}{15}}{\\\\frac{12v}{25}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{10u}{9v}$$","hints":{"DefaultPathway":[{"id":"a4488b1fractions13a-h1","type":"hint","dependencies":[],"title":"Reciprocal","text":"Take the reciprocal of the second fraction. The reciprocal is the opposite of the current fraction. In other words, flip the fraction so the top is at the bottom, and the bottom is at the top.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1fractions13a-h2","type":"hint","dependencies":["a4488b1fractions13a-h1"],"title":"Eliminate diagonally","text":"Eliminate any common terms that may occur in the diagonals of the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1fractions13a-h3","type":"hint","dependencies":["a4488b1fractions13a-h2"],"title":"Multiply","text":"Multiply the numerators together and the deonominators together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4488b1fractions14","title":"Perform the indicated operation","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Fractions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4488b1fractions14a","stepAnswer":["$$\\\\frac{-1}{16}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\frac{3}{4}}{\\\\left(-12\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-1}{16}$$","hints":{"DefaultPathway":[{"id":"a4488b1fractions14a-h1","type":"hint","dependencies":[],"title":"Reciprocal","text":"Take the reciprocal of the second fraction. The reciprocal is the opposite of the current fraction. In other words, flip the fraction so the top is at the bottom, and the bottom is at the top.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1fractions14a-h2","type":"hint","dependencies":["a4488b1fractions14a-h1"],"title":"Eliminate diagonally","text":"Eliminate any common terms that may occur in the diagonals of the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1fractions14a-h3","type":"hint","dependencies":["a4488b1fractions14a-h2"],"title":"Multiply","text":"Multiply the numerators together and the deonominators together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4488b1fractions15","title":"Perform the indicated operation","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Fractions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4488b1fractions15a","stepAnswer":["$$\\\\frac{18}{5y}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\frac{2}{5}}{\\\\frac{y}{9}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{18}{5y}$$","hints":{"DefaultPathway":[{"id":"a4488b1fractions15a-h1","type":"hint","dependencies":[],"title":"Reciprocal","text":"Take the reciprocal of the second fraction. The reciprocal is the opposite of the current fraction. In other words, flip the fraction so the top is at the bottom, and the bottom is at the top.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1fractions15a-h2","type":"hint","dependencies":["a4488b1fractions15a-h1"],"title":"Eliminate diagonally","text":"Eliminate any common terms that may occur in the diagonals of the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1fractions15a-h3","type":"hint","dependencies":["a4488b1fractions15a-h2"],"title":"Multiply","text":"Multiply the numerators together and the deonominators together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4488b1fractions2","title":"How to Simplify a Fraction?","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Fractions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4488b1fractions2a","stepAnswer":["$$\\\\frac{10}{21}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{120}{252}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{10}{21}$$","hints":{"DefaultPathway":[{"id":"a4488b1fractions2a-h1","type":"hint","dependencies":[],"title":"Rewrite","text":"Rewrite the fraction as a product of primes in both the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1fractions2a-h2","type":"hint","dependencies":["a4488b1fractions2a-h1"],"title":"Eliminate","text":"Eliminate any commonly occuring primes in both the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1fractions2a-h3","type":"hint","dependencies":["a4488b1fractions2a-h2"],"title":"Multiply","text":"Multiply the product back into two terms, one in the numerator, and one in the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4488b1fractions3","title":"How to Simplify a Fraction?","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Fractions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4488b1fractions3a","stepAnswer":["$$\\\\frac{2x^2}{3y}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{14x^2}{21} y$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2x^2}{3y}$$","hints":{"DefaultPathway":[{"id":"a4488b1fractions3a-h1","type":"hint","dependencies":[],"title":"Rewrite","text":"Rewrite the fraction as a product of primes in both the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1fractions3a-h2","type":"hint","dependencies":["a4488b1fractions3a-h1"],"title":"Eliminate","text":"Eliminate any commonly occuring primes in both the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1fractions3a-h3","type":"hint","dependencies":["a4488b1fractions3a-h2"],"title":"Multiply","text":"Multiply the product back into two terms, one in the numerator, and one in the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4488b1fractions4","title":"How to Simplify a Fraction?","body":"Simplify the following expression:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Fractions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4488b1fractions4a","stepAnswer":["$$\\\\frac{-21a^2}{11b^2}$$"],"problemType":"TextBox","stepTitle":"-210a**2/110b**","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-21a^2}{11b^2}$$","hints":{"DefaultPathway":[{"id":"a4488b1fractions4a-h1","type":"hint","dependencies":[],"title":"Rewrite","text":"Rewrite the fraction as a product of primes in both the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1fractions4a-h2","type":"hint","dependencies":["a4488b1fractions4a-h1"],"title":"Eliminate","text":"Eliminate any commonly occuring primes in both the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1fractions4a-h3","type":"hint","dependencies":["a4488b1fractions4a-h2"],"title":"Multiply","text":"Multiply the product back into two terms, one in the numerator, and one in the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4488b1fractions5","title":"Perform the operation","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Fractions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4488b1fractions5a","stepAnswer":["$$\\\\frac{1}{3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(-\\\\frac{3}{4}\\\\right) \\\\left(-\\\\frac{4}{9}\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{3}$$","hints":{"DefaultPathway":[{"id":"a4488b1fractions5a-h1","type":"hint","dependencies":[],"title":"Use diagonals","text":"Simplify the fraction diagonally. Split one term from the numerator of one fraction into the product of prime numbers. Do the same for the denominator of the other fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1fractions5a-h2","type":"hint","dependencies":["a4488b1fractions5a-h1"],"title":"Simplify","text":"Simplify diagonally, eliminating prime numbers. Then multiply the product together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1fractions5a-h3","type":"hint","dependencies":["a4488b1fractions5a-h2"],"title":"Multiply","text":"Multiply the two fractions together, multiplying the numerators together and the denominators together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4488b1fractions6","title":"Perform the indicated operation","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Fractions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4488b1fractions6a","stepAnswer":["$$\\\\frac{-1}{10}$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(-\\\\frac{3}{8}\\\\right) \\\\frac{4}{15}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-1}{10}$$","hints":{"DefaultPathway":[{"id":"a4488b1fractions6a-h1","type":"hint","dependencies":[],"title":"Use diagonals","text":"Simplify the fraction diagonally. Split one term from the numerator of one fraction into the product of prime numbers. Do the same for the denominator of the other fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1fractions6a-h2","type":"hint","dependencies":["a4488b1fractions6a-h1"],"title":"Simplify","text":"Simplify diagonally, eliminating prime numbers. Then multiply the product together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1fractions6a-h3","type":"hint","dependencies":["a4488b1fractions6a-h2"],"title":"Multiply","text":"Multiply the two fractions together, multiplying the numerators together and the denominators together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4488b1fractions7","title":"Perform the indicated operation","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Fractions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4488b1fractions7a","stepAnswer":["$$\\\\frac{-21}{50}$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(-\\\\frac{14}{15}\\\\right) \\\\frac{9}{20}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-21}{50}$$","hints":{"DefaultPathway":[{"id":"a4488b1fractions7a-h1","type":"hint","dependencies":[],"title":"Use diagonals","text":"Simplify the fraction diagonally. 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Then multiply the product together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1fractions7a-h3","type":"hint","dependencies":["a4488b1fractions7a-h2"],"title":"Multiply","text":"Multiply the two fractions together, multiplying the numerators together and the denominators together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4488b1fractions8","title":"Perform the indicated operation","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Fractions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4488b1fractions8a","stepAnswer":["$$\\\\frac{-15}{22}$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(-\\\\frac{9}{10}\\\\right) \\\\frac{25}{33}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-15}{22}$$","hints":{"DefaultPathway":[{"id":"a4488b1fractions8a-h1","type":"hint","dependencies":[],"title":"Use diagonals","text":"Simplify the fraction diagonally. 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Then multiply the product together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1fractions8a-h3","type":"hint","dependencies":["a4488b1fractions8a-h2"],"title":"Multiply","text":"Multiply the two fractions together, multiplying the numerators together and the denominators together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4488b1fractions9","title":"Perform the indicated operation","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Fractions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4488b1fractions9a","stepAnswer":["$$\\\\frac{11}{30}$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(-\\\\frac{63}{84}\\\\right) \\\\left(-\\\\frac{44}{90}\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{11}{30}$$","hints":{"DefaultPathway":[{"id":"a4488b1fractions9a-h1","type":"hint","dependencies":[],"title":"Use diagonals","text":"Simplify the fraction diagonally. 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Then multiply the product together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4488b1fractions9a-h3","type":"hint","dependencies":["a4488b1fractions9a-h2"],"title":"Multiply","text":"Multiply the two fractions together, multiplying the numerators together and the denominators together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a453be6realnumbers1","title":"Calculating Square Roots #1","body":"Simplify the following expressions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.8 The Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a453be6realnumbers1a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{25}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"a453be6realnumbers1a-h1","type":"hint","dependencies":[],"title":"Square Root of a Number","text":"if $$n^2=m$$, then $$n$$ is the square root of $$m$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers1a-h2","type":"hint","dependencies":["a453be6realnumbers1a-h1"],"title":"Example Calculation","text":"$${25}^2=625$$, so $$25$$ is the square root of $$625$$. In equation form, $$\\\\sqrt{625}=25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers1a-h3","type":"hint","dependencies":["a453be6realnumbers1a-h2"],"title":"Finding \\"n\\"","text":"$$5^2=25$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a453be6realnumbers1b","stepAnswer":["$$11$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{121}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$11$$","hints":{"DefaultPathway":[{"id":"a453be6realnumbers1b-h1","type":"hint","dependencies":[],"title":"Square Root of a Number","text":"if $$n^2=m$$, then $$n$$ is the square root of $$m$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers1b-h2","type":"hint","dependencies":["a453be6realnumbers1b-h1"],"title":"Example Calculation","text":"$${25}^2=625$$, so $$25$$ is the square root of $$625$$. In equation form, $$\\\\sqrt{625}=25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers1b-h3","type":"hint","dependencies":["a453be6realnumbers1b-h2"],"title":"Finding \\"n\\"","text":"$${11}^2=121$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a453be6realnumbers10","title":"Calculating Square Roots #4","body":"Simplify the following expressions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.8 The Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a453be6realnumbers10a","stepAnswer":["$$8$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{64}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8$$","hints":{"DefaultPathway":[{"id":"a453be6realnumbers10a-h1","type":"hint","dependencies":[],"title":"Square Root of a Number","text":"if $$n^2=m$$, then $$n$$ is the square root of $$m$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers10a-h2","type":"hint","dependencies":["a453be6realnumbers10a-h1"],"title":"Example Calculation","text":"$${25}^2=625$$, so $$25$$ is the square root of $$625$$. In equation form, $$\\\\sqrt{625}=25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers10a-h3","type":"hint","dependencies":["a453be6realnumbers10a-h2"],"title":"Finding \\"n\\"","text":"$$8^2=64$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a453be6realnumbers11","title":"Identifying Rational and Irrational Numbers #1","body":"Given the numbers, identify if they are irrational or rational.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.8 The Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a453be6realnumbers11a","stepAnswer":["Rational"],"problemType":"MultipleChoice","stepTitle":"$$0.583...$$, where the $$3$$ repeats forever.","stepBody":"","answerType":"string","variabilization":{},"choices":["Rational","Irrational"],"hints":{"DefaultPathway":[{"id":"a453be6realnumbers11a-h1","type":"hint","dependencies":[],"title":"Definition of a Rational Number","text":"A rational number is one that either stops or repeats a decimal pattern forever. For example, $$4$$, 4,43, and $$4.3$$, where the $$3$$ repeats forever, are rational numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers11a-h2","type":"hint","dependencies":["a453be6realnumbers11a-h1"],"title":"Definition of an Irrational Number","text":"An irrational number is one that does not stop or repeat a decimal pattern forever. An example of an irrational number is $$\\"2.01001000100001...\\"$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a453be6realnumbers11b","stepAnswer":["Rational"],"problemType":"MultipleChoice","stepTitle":"$$0.47$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Rational","Irrational"],"hints":{"DefaultPathway":[{"id":"a453be6realnumbers11b-h1","type":"hint","dependencies":[],"title":"Definition of a Rational Number","text":"A rational number is one that either stops or repeats a decimal pattern forever. 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An example of an irrational number is $$\\"2.01001000100001...\\"$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a453be6realnumbers11c","stepAnswer":["Irrational"],"problemType":"MultipleChoice","stepTitle":"$$3.605551275..$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["Rational","Irrational"],"hints":{"DefaultPathway":[{"id":"a453be6realnumbers11c-h1","type":"hint","dependencies":[],"title":"Definition of a Rational Number","text":"A rational number is one that either stops or repeats a decimal pattern forever. For example, $$4$$, 4,43, and $$4.3$$, where the $$3$$ repeats forever, are rational numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers11c-h2","type":"hint","dependencies":["a453be6realnumbers11c-h1"],"title":"Definition of an Irrational Number","text":"An irrational number is one that does not stop or repeat a decimal pattern forever. An example of an irrational number is $$\\"2.01001000100001...\\"$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a453be6realnumbers12","title":"Identifying Rational and Irrational Numbers #2","body":"Given the numbers, identify if they are irrational or rational.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.8 The Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a453be6realnumbers12a","stepAnswer":["Rational"],"problemType":"MultipleChoice","stepTitle":"$$0.29$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Rational","Irrational"],"hints":{"DefaultPathway":[{"id":"a453be6realnumbers12a-h1","type":"hint","dependencies":[],"title":"Definition of a Rational Number","text":"A rational number is one that either stops or repeats a decimal pattern forever. For example, $$4$$, 4,43, and $$4.3$$, where the $$3$$ repeats forever, are rational numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers12a-h2","type":"hint","dependencies":["a453be6realnumbers12a-h1"],"title":"Definition of an Irrational Number","text":"An irrational number is one that does not stop or repeat a decimal pattern forever. An example of an irrational number is $$\\"2.01001000100001...\\"$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a453be6realnumbers12b","stepAnswer":["Rational"],"problemType":"MultipleChoice","stepTitle":"$$0.816...$$, where the $$6$$ repeats forever.","stepBody":"","answerType":"string","variabilization":{},"choices":["Rational","Irrational"],"hints":{"DefaultPathway":[{"id":"a453be6realnumbers12b-h1","type":"hint","dependencies":[],"title":"Definition of a Rational Number","text":"A rational number is one that either stops or repeats a decimal pattern forever. For example, $$4$$, 4,43, and $$4.3$$, where the $$3$$ repeats forever, are rational numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers12b-h2","type":"hint","dependencies":["a453be6realnumbers12b-h1"],"title":"Definition of an Irrational Number","text":"An irrational number is one that does not stop or repeat a decimal pattern forever. An example of an irrational number is $$\\"2.01001000100001...\\"$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a453be6realnumbers12c","stepAnswer":["Irrational"],"problemType":"MultipleChoice","stepTitle":"$$2.515115111..$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["Rational","Irrational"],"hints":{"DefaultPathway":[{"id":"a453be6realnumbers12c-h1","type":"hint","dependencies":[],"title":"Definition of a Rational Number","text":"A rational number is one that either stops or repeats a decimal pattern forever. For example, $$4$$, 4,43, and $$4.3$$, where the $$3$$ repeats forever, are rational numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers12c-h2","type":"hint","dependencies":["a453be6realnumbers12c-h1"],"title":"Definition of an Irrational Number","text":"An irrational number is one that does not stop or repeat a decimal pattern forever. An example of an irrational number is $$\\"2.01001000100001...\\"$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a453be6realnumbers13","title":"Identifying Rational and Irrational Numbers #3","body":"Given the numbers, identify if they are irrational or rational.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.8 The Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a453be6realnumbers13a","stepAnswer":["Rational"],"problemType":"MultipleChoice","stepTitle":"$$2.63...$$, where the $$3$$ repeats forever.","stepBody":"","answerType":"string","variabilization":{},"choices":["Rational","Irrational"],"hints":{"DefaultPathway":[{"id":"a453be6realnumbers13a-h1","type":"hint","dependencies":[],"title":"Definition of a Rational Number","text":"A rational number is one that either stops or repeats a decimal pattern forever. For example, $$4$$, 4,43, and $$4.3$$, where the $$3$$ repeats forever, are rational numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers13a-h2","type":"hint","dependencies":["a453be6realnumbers13a-h1"],"title":"Definition of an Irrational Number","text":"An irrational number is one that does not stop or repeat a decimal pattern forever. An example of an irrational number is $$\\"2.01001000100001...\\"$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a453be6realnumbers13b","stepAnswer":["Rational"],"problemType":"MultipleChoice","stepTitle":"$$0.125$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Rational","Irrational"],"hints":{"DefaultPathway":[{"id":"a453be6realnumbers13b-h1","type":"hint","dependencies":[],"title":"Definition of a Rational Number","text":"A rational number is one that either stops or repeats a decimal pattern forever. For example, $$4$$, 4,43, and $$4.3$$, where the $$3$$ repeats forever, are rational numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers13b-h2","type":"hint","dependencies":["a453be6realnumbers13b-h1"],"title":"Definition of an Irrational Number","text":"An irrational number is one that does not stop or repeat a decimal pattern forever. An example of an irrational number is $$\\"2.01001000100001...\\"$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a453be6realnumbers13c","stepAnswer":["Irrational"],"problemType":"MultipleChoice","stepTitle":"$$0.418302..$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["Rational","Irrational"],"hints":{"DefaultPathway":[{"id":"a453be6realnumbers13c-h1","type":"hint","dependencies":[],"title":"Definition of a Rational Number","text":"A rational number is one that either stops or repeats a decimal pattern forever. For example, $$4$$, 4,43, and $$4.3$$, where the $$3$$ repeats forever, are rational numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers13c-h2","type":"hint","dependencies":["a453be6realnumbers13c-h1"],"title":"Definition of an Irrational Number","text":"An irrational number is one that does not stop or repeat a decimal pattern forever. An example of an irrational number is $$\\"2.01001000100001...\\"$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a453be6realnumbers14","title":"Identifying Real Numbers #1","body":"For each number given, identify whether it is a real number or not.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.8 The Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a453be6realnumbers14a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$\\\\sqrt{-169}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a453be6realnumbers14a-h1","type":"hint","dependencies":[],"title":"Definition of Real Numbers","text":"A number is real if it is rational or irrational.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers14a-h2","type":"hint","dependencies":["a453be6realnumbers14a-h1"],"title":"Definition of a Rational Number","text":"A rational number is one that either stops or repeats a decimal pattern forever. For example, $$4$$, 4,43, and $$4.3$$, where the $$3$$ repeats forever, are rational numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers14a-h3","type":"hint","dependencies":["a453be6realnumbers14a-h2"],"title":"Definition of an Irrational Number","text":"An irrational number is one that does not stop or repeat a decimal pattern forever. An example of an irrational number is $$\\"2.01001000100001...\\"$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers14a-h4","type":"hint","dependencies":["a453be6realnumbers14a-h3"],"title":"Square Root of a Negative Number","text":"There is no real number whose square is a negative number. Both negative and positive numbers have positive square roots. Thus, the square root of a negative number is not a real number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a453be6realnumbers14b","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$-\\\\sqrt{64}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a453be6realnumbers14b-h1","type":"hint","dependencies":[],"title":"Definition of Real Numbers","text":"A number is real if it is rational or irrational.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers14b-h2","type":"hint","dependencies":["a453be6realnumbers14b-h1"],"title":"Definition of a Rational Number","text":"A rational number is one that either stops or repeats a decimal pattern forever. For example, $$4$$, 4,43, and $$4.3$$, where the $$3$$ repeats forever, are rational numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers14b-h3","type":"hint","dependencies":["a453be6realnumbers14b-h2"],"title":"Definition of an Irrational Number","text":"An irrational number is one that does not stop or repeat a decimal pattern forever. An example of an irrational number is $$\\"2.01001000100001...\\"$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers14b-h4","type":"hint","dependencies":["a453be6realnumbers14b-h3"],"title":"Value of $$\\\\sqrt{64}$$","text":"$$\\\\sqrt{64}=8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a453be6realnumbers15","title":"Identifying Real Numbers #2","body":"For each number given, identify whether it is a real number or not.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.8 The Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a453be6realnumbers15a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$\\\\sqrt{-196}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a453be6realnumbers15a-h1","type":"hint","dependencies":[],"title":"Definition of Real Numbers","text":"A number is real if it is rational or irrational.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers15a-h2","type":"hint","dependencies":["a453be6realnumbers15a-h1"],"title":"Definition of a Rational Number","text":"A rational number is one that either stops or repeats a decimal pattern forever. For example, $$4$$, 4,43, and $$4.3$$, where the $$3$$ repeats forever, are rational numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers15a-h3","type":"hint","dependencies":["a453be6realnumbers15a-h2"],"title":"Definition of an Irrational Number","text":"An irrational number is one that does not stop or repeat a decimal pattern forever. An example of an irrational number is $$\\"2.01001000100001...\\"$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers15a-h4","type":"hint","dependencies":["a453be6realnumbers15a-h3"],"title":"Square Root of a Negative Number","text":"There is no real number whose square is a negative number. Both negative and positive numbers have positive square roots. Thus, the square root of a negative number is not a real number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a453be6realnumbers15b","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$-\\\\sqrt{81}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a453be6realnumbers15b-h1","type":"hint","dependencies":[],"title":"Definition of Real Numbers","text":"A number is real if it is rational or irrational.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers15b-h2","type":"hint","dependencies":["a453be6realnumbers15b-h1"],"title":"Definition of a Rational Number","text":"A rational number is one that either stops or repeats a decimal pattern forever. For example, $$4$$, 4,43, and $$4.3$$, where the $$3$$ repeats forever, are rational numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers15b-h3","type":"hint","dependencies":["a453be6realnumbers15b-h2"],"title":"Definition of an Irrational Number","text":"An irrational number is one that does not stop or repeat a decimal pattern forever. An example of an irrational number is $$\\"2.01001000100001...\\"$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers15b-h4","type":"hint","dependencies":["a453be6realnumbers15b-h3"],"title":"Value of $$\\\\sqrt{81}$$","text":"$$\\\\sqrt{81}=9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a453be6realnumbers16","title":"Identifying Real Numbers #3","body":"For each number given, identify whether it is a real number or not.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.8 The Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a453be6realnumbers16a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$-\\\\sqrt{49}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a453be6realnumbers16a-h1","type":"hint","dependencies":[],"title":"Definition of Real Numbers","text":"A number is real if it is rational or irrational.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers16a-h2","type":"hint","dependencies":["a453be6realnumbers16a-h1"],"title":"Definition of a Rational Number","text":"A rational number is one that either stops or repeats a decimal pattern forever. For example, $$4$$, 4,43, and $$4.3$$, where the $$3$$ repeats forever, are rational numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers16a-h3","type":"hint","dependencies":["a453be6realnumbers16a-h2"],"title":"Definition of an Irrational Number","text":"An irrational number is one that does not stop or repeat a decimal pattern forever. An example of an irrational number is $$\\"2.01001000100001...\\"$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers16a-h4","type":"hint","dependencies":["a453be6realnumbers16a-h3"],"title":"Value of $$\\\\sqrt{49}$$","text":"$$\\\\sqrt{49}=7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a453be6realnumbers16b","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$\\\\sqrt{-121}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a453be6realnumbers16b-h1","type":"hint","dependencies":[],"title":"Definition of Real Numbers","text":"A number is real if it is rational or irrational.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers16b-h2","type":"hint","dependencies":["a453be6realnumbers16b-h1"],"title":"Definition of a Rational Number","text":"A rational number is one that either stops or repeats a decimal pattern forever. For example, $$4$$, 4,43, and $$4.3$$, where the $$3$$ repeats forever, are rational numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers16b-h3","type":"hint","dependencies":["a453be6realnumbers16b-h2"],"title":"Definition of an Irrational Number","text":"An irrational number is one that does not stop or repeat a decimal pattern forever. An example of an irrational number is $$\\"2.01001000100001...\\"$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers16b-h4","type":"hint","dependencies":["a453be6realnumbers16b-h3"],"title":"Square Root of a Negative Number","text":"There is no real number whose square is a negative number. Both negative and positive numbers have positive square roots. Thus, the square root of a negative number is not a real number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a453be6realnumbers17","title":"Identifying Real Numbers #4","body":"For each number given, identify whether it is a real number or not.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.8 The Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a453be6realnumbers17a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$-\\\\sqrt{64}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a453be6realnumbers17a-h1","type":"hint","dependencies":[],"title":"Definition of Real Numbers","text":"A number is real if it is rational or irrational.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers17a-h2","type":"hint","dependencies":["a453be6realnumbers17a-h1"],"title":"Definition of a Rational Number","text":"A rational number is one that either stops or repeats a decimal pattern forever. For example, $$4$$, 4,43, and $$4.3$$, where the $$3$$ repeats forever, are rational numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers17a-h3","type":"hint","dependencies":["a453be6realnumbers17a-h2"],"title":"Definition of an Irrational Number","text":"An irrational number is one that does not stop or repeat a decimal pattern forever. An example of an irrational number is $$\\"2.01001000100001...\\"$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers17a-h4","type":"hint","dependencies":["a453be6realnumbers17a-h3"],"title":"Value of $$\\\\sqrt{64}$$","text":"$$\\\\sqrt{64}=8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a453be6realnumbers17b","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$\\\\sqrt{-9}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a453be6realnumbers17b-h1","type":"hint","dependencies":[],"title":"Definition of Real Numbers","text":"A number is real if it is rational or irrational.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers17b-h2","type":"hint","dependencies":["a453be6realnumbers17b-h1"],"title":"Definition of a Rational Number","text":"A rational number is one that either stops or repeats a decimal pattern forever. For example, $$4$$, 4,43, and $$4.3$$, where the $$3$$ repeats forever, are rational numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers17b-h3","type":"hint","dependencies":["a453be6realnumbers17b-h2"],"title":"Definition of an Irrational Number","text":"An irrational number is one that does not stop or repeat a decimal pattern forever. An example of an irrational number is $$\\"2.01001000100001...\\"$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers17b-h4","type":"hint","dependencies":["a453be6realnumbers17b-h3"],"title":"Square Root of a Negative Number","text":"There is no real number whose square is a negative number. Both negative and positive numbers have positive square roots. Thus, the square root of a negative number is not a real number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a453be6realnumbers18","title":"Identifying Correct Inequality Signs #1","body":"Fill in the blanks with the correct inequality sign.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.8 The Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a453be6realnumbers18a","stepAnswer":[">"],"problemType":"MultipleChoice","stepTitle":"0.64_0.6","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">"],"hints":{"DefaultPathway":[{"id":"a453be6realnumbers18a-h1","type":"hint","dependencies":[],"title":"When to Use the \\"Less Than\\" (<) Inequality Symbol","text":"$$a<b$$ \u201ca is less than b\u201d when a is to the left of $$b$$ on the number line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers18a-h2","type":"hint","dependencies":["a453be6realnumbers18a-h1"],"title":"When to Use the \\"Greater Than\\" (>) Inequality Symbol","text":"$$a>b$$ \u201ca is greater than b\u201d when a is to the right of $$b$$ on the number line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers18a-h3","type":"hint","dependencies":["a453be6realnumbers18a-h2"],"title":"Example of a Number Line","text":"In the picture attached to this hint is a number line indicating the positions of $$0.04$$ and $$0.4$$. We can see that $$0.04$$ is to the left of $$0.4$$ on the number line, so $$0.04<0.4$$. You can also see that this means $$0.4$$ is the right of $$0.04$$ on the number line, so $$0.4>0.04$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers18a-h4","type":"hint","dependencies":["a453be6realnumbers18a-h3"],"title":"$$0.64$$ Relative to $$0.6$$ on the Number Line","text":"$$0.64$$ is to the right of $$0.6$$ on a number line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a453be6realnumbers19","title":"Identifying Correct Inequality Signs #2","body":"Fill in the blanks with the correct inequality sign.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.8 The Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a453be6realnumbers19a","stepAnswer":[">"],"problemType":"MultipleChoice","stepTitle":"0.42_0.4","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">"],"hints":{"DefaultPathway":[{"id":"a453be6realnumbers19a-h1","type":"hint","dependencies":[],"title":"When to Use the \\"Less Than\\" (<) Inequality Symbol","text":"$$a<b$$ \u201ca is less than b\u201d when a is to the left of $$b$$ on the number line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers19a-h2","type":"hint","dependencies":["a453be6realnumbers19a-h1"],"title":"When to Use the \\"Greater Than\\" (>) Inequality Symbol","text":"$$a>b$$ \u201ca is greater than b\u201d when a is to the right of $$b$$ on the number line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers19a-h3","type":"hint","dependencies":["a453be6realnumbers19a-h2"],"title":"Example of a Number Line","text":"In the picture attached to this hint is a number line indicating the positions of $$0.04$$ and $$0.4$$. We can see that $$0.04$$ is to the left of $$0.4$$ on the number line, so $$0.04<0.4$$. You can also see that this means $$0.4$$ is the right of $$0.04$$ on the number line, so $$0.4>0.04$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers19a-h4","type":"hint","dependencies":["a453be6realnumbers19a-h3"],"title":"$$0.42$$ Relative to $$0.4$$ on the Number Line","text":"$$0.42$$ is to the right of $$0.4$$ on a number line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a453be6realnumbers2","title":"Calculating Square Roots #2","body":"Simplify the following expressions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.8 The Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a453be6realnumbers2a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{36}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"a453be6realnumbers2a-h1","type":"hint","dependencies":[],"title":"Square Root of a Number","text":"if $$n^2=m$$, then $$n$$ is the square root of $$m$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers2a-h2","type":"hint","dependencies":["a453be6realnumbers2a-h1"],"title":"Example Calculation","text":"$${25}^2=625$$, so $$25$$ is the square root of $$625$$. In equation form, $$\\\\sqrt{625}=25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers2a-h3","type":"hint","dependencies":["a453be6realnumbers2a-h2"],"title":"Finding \\"n\\"","text":"$$6^2=36$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a453be6realnumbers2b","stepAnswer":["$$13$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{169}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$13$$","hints":{"DefaultPathway":[{"id":"a453be6realnumbers2b-h1","type":"hint","dependencies":[],"title":"Square Root of a Number","text":"if $$n^2=m$$, then $$n$$ is the square root of $$m$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers2b-h2","type":"hint","dependencies":["a453be6realnumbers2b-h1"],"title":"Example Calculation","text":"$${25}^2=625$$, so $$25$$ is the square root of $$625$$. In equation form, $$\\\\sqrt{625}=25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers2b-h3","type":"hint","dependencies":["a453be6realnumbers2b-h2"],"title":"Finding \\"n\\"","text":"$${13}^2=169$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a453be6realnumbers20","title":"Identifying Correct Inequality Signs #3","body":"Fill in the blanks with the correct inequality sign.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.8 The Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a453be6realnumbers20a","stepAnswer":[">"],"problemType":"MultipleChoice","stepTitle":"0.18_0.1","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">"],"hints":{"DefaultPathway":[{"id":"a453be6realnumbers20a-h1","type":"hint","dependencies":[],"title":"When to Use the \\"Less Than\\" (<) Inequality Symbol","text":"$$a<b$$ \u201ca is less than b\u201d when a is to the left of $$b$$ on the number line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers20a-h2","type":"hint","dependencies":["a453be6realnumbers20a-h1"],"title":"When to Use the \\"Greater Than\\" (>) Inequality Symbol","text":"$$a>b$$ \u201ca is greater than b\u201d when a is to the right of $$b$$ on the number line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers20a-h3","type":"hint","dependencies":["a453be6realnumbers20a-h2"],"title":"Example of a Number Line","text":"In the picture attached to this hint is a number line indicating the positions of $$0.04$$ and $$0.4$$. We can see that $$0.04$$ is to the left of $$0.4$$ on the number line, so $$0.04<0.4$$. You can also see that this means $$0.4$$ is the right of $$0.04$$ on the number line, so $$0.4>0.04$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers20a-h4","type":"hint","dependencies":["a453be6realnumbers20a-h3"],"title":"$$0.18$$ Relative to $$0.1$$ on the Number Line","text":"$$0.18$$ is to the right of $$0.1$$ on a number line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a453be6realnumbers21","title":"Identifying Correct Inequality Signs #4","body":"Fill in the blanks with the correct inequality sign.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.8 The Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a453be6realnumbers21a","stepAnswer":[">"],"problemType":"MultipleChoice","stepTitle":"0.83_0.803","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">"],"hints":{"DefaultPathway":[{"id":"a453be6realnumbers21a-h1","type":"hint","dependencies":[],"title":"When to Use the \\"Less Than\\" (<) Inequality Symbol","text":"$$a<b$$ \u201ca is less than b\u201d when a is to the left of $$b$$ on the number line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers21a-h2","type":"hint","dependencies":["a453be6realnumbers21a-h1"],"title":"When to Use the \\"Greater Than\\" (>) Inequality Symbol","text":"$$a>b$$ \u201ca is greater than b\u201d when a is to the right of $$b$$ on the number line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers21a-h3","type":"hint","dependencies":["a453be6realnumbers21a-h2"],"title":"Example of a Number Line","text":"In the picture attached to this hint is a number line indicating the positions of $$0.04$$ and $$0.4$$. We can see that $$0.04$$ is to the left of $$0.4$$ on the number line, so $$0.04<0.4$$. You can also see that this means $$0.4$$ is the right of $$0.04$$ on the number line, so $$0.4>0.04$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers21a-h4","type":"hint","dependencies":["a453be6realnumbers21a-h3"],"title":"Answer","text":"$$0.803$$ has one more decimal place than $$0.83$$. $$0.83$$ is equal to $$0.830$$, and because $$830>803$$, $$830$$ thousanths is greater than $$803$$ thousandths.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a453be6realnumbers22","title":"Identifying Correct Inequality Signs #5","body":"Fill in the blanks with the correct inequality sign.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.8 The Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a453be6realnumbers22a","stepAnswer":[">"],"problemType":"MultipleChoice","stepTitle":"0.76_0.706","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">"],"hints":{"DefaultPathway":[{"id":"a453be6realnumbers22a-h1","type":"hint","dependencies":[],"title":"When to Use the \\"Less Than\\" (<) Inequality Symbol","text":"$$a<b$$ \u201ca is less than b\u201d when a is to the left of $$b$$ on the number line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers22a-h2","type":"hint","dependencies":["a453be6realnumbers22a-h1"],"title":"When to Use the \\"Greater Than\\" (>) Inequality Symbol","text":"$$a>b$$ \u201ca is greater than b\u201d when a is to the right of $$b$$ on the number line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers22a-h3","type":"hint","dependencies":["a453be6realnumbers22a-h2"],"title":"Example of a Number Line","text":"In the picture attached to this hint is a number line indicating the positions of $$0.04$$ and $$0.4$$. We can see that $$0.04$$ is to the left of $$0.4$$ on the number line, so $$0.04<0.4$$. You can also see that this means $$0.4$$ is the right of $$0.04$$ on the number line, so $$0.4>0.04$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers22a-h4","type":"hint","dependencies":["a453be6realnumbers22a-h3"],"title":"Answer","text":"$$0.706$$ has one more decimal place than $$0.76$$. $$0.76$$ is equal to $$0.760$$, and because $$760>706$$, $$760$$ thousanths is greater than $$706$$ thousandths.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a453be6realnumbers23","title":"Identifying Correct Inequality Signs #6","body":"Fill in the blanks with the correct inequality sign.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.8 The Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a453be6realnumbers23a","stepAnswer":["<"],"problemType":"MultipleChoice","stepTitle":"0.305_0.35","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">"],"hints":{"DefaultPathway":[{"id":"a453be6realnumbers23a-h1","type":"hint","dependencies":[],"title":"When to Use the \\"Less Than\\" (<) Inequality Symbol","text":"$$a<b$$ \u201ca is less than b\u201d when a is to the left of $$b$$ on the number line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers23a-h2","type":"hint","dependencies":["a453be6realnumbers23a-h1"],"title":"When to Use the \\"Greater Than\\" (>) Inequality Symbol","text":"$$a>b$$ \u201ca is greater than b\u201d when a is to the right of $$b$$ on the number line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers23a-h3","type":"hint","dependencies":["a453be6realnumbers23a-h2"],"title":"Example of a Number Line","text":"In the picture attached to this hint is a number line indicating the positions of $$0.04$$ and $$0.4$$. We can see that $$0.04$$ is to the left of $$0.4$$ on the number line, so $$0.04<0.4$$. You can also see that this means $$0.4$$ is the right of $$0.04$$ on the number line, so $$0.4>0.04$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers23a-h4","type":"hint","dependencies":["a453be6realnumbers23a-h3"],"title":"Answer","text":"$$0.305$$ has one more decimal place than $$0.35$$. $$0.35$$ is equal to $$0.350$$, and because $$305<350$$, $$305$$ thousanths is less than $$350$$ thousandths.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a453be6realnumbers24","title":"Identifying Correct Inequality Signs #7","body":"Fill in the blanks with the correct inequality sign.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.8 The Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a453be6realnumbers24a","stepAnswer":["<"],"problemType":"MultipleChoice","stepTitle":"0.37_0.63","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">"],"hints":{"DefaultPathway":[{"id":"a453be6realnumbers24a-h1","type":"hint","dependencies":[],"title":"When to Use the \\"Less Than\\" (<) Inequality Symbol","text":"$$a<b$$ \u201ca is less than b\u201d when a is to the left of $$b$$ on the number line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers24a-h2","type":"hint","dependencies":["a453be6realnumbers24a-h1"],"title":"When to Use the \\"Greater Than\\" (>) Inequality Symbol","text":"$$a>b$$ \u201ca is greater than b\u201d when a is to the right of $$b$$ on the number line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers24a-h3","type":"hint","dependencies":["a453be6realnumbers24a-h2"],"title":"Example of a Number Line","text":"In the picture attached to this hint is a number line indicating the positions of $$0.04$$ and $$0.4$$. We can see that $$0.04$$ is to the left of $$0.4$$ on the number line, so $$0.04<0.4$$. You can also see that this means $$0.4$$ is the right of $$0.04$$ on the number line, so $$0.4>0.04$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers24a-h4","type":"hint","dependencies":["a453be6realnumbers24a-h3"],"title":"Answer","text":"$$63$$ thousandths is greater than $$37$$ thousandths, so $$0.63$$ is to the right of $$0.37$$ on the number line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a453be6realnumbers25","title":"Identifying Correct Inequality Signs #8","body":"Fill in the blanks with the correct inequality sign.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.8 The Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a453be6realnumbers25a","stepAnswer":[">"],"problemType":"MultipleChoice","stepTitle":"0.86_0.69","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">"],"hints":{"DefaultPathway":[{"id":"a453be6realnumbers25a-h1","type":"hint","dependencies":[],"title":"When to Use the \\"Less Than\\" (<) Inequality Symbol","text":"$$a<b$$ \u201ca is less than b\u201d when a is to the left of $$b$$ on the number line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers25a-h2","type":"hint","dependencies":["a453be6realnumbers25a-h1"],"title":"When to Use the \\"Greater Than\\" (>) Inequality Symbol","text":"$$a>b$$ \u201ca is greater than b\u201d when a is to the right of $$b$$ on the number line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers25a-h3","type":"hint","dependencies":["a453be6realnumbers25a-h2"],"title":"Example of a Number Line","text":"In the picture attached to this hint is a number line indicating the positions of $$0.04$$ and $$0.4$$. We can see that $$0.04$$ is to the left of $$0.4$$ on the number line, so $$0.04<0.4$$. You can also see that this means $$0.4$$ is the right of $$0.04$$ on the number line, so $$0.4>0.04$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers25a-h4","type":"hint","dependencies":["a453be6realnumbers25a-h3"],"title":"Answer","text":"$$86$$ thousandths is greater than $$69$$ thousandths, so $$0.86$$ is to the right of $$0.69$$ on the number line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a453be6realnumbers26","title":"Identifying Correct Inequality Signs #9","body":"Fill in the blanks with the correct inequality sign.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.8 The Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a453be6realnumbers26a","stepAnswer":[">"],"problemType":"MultipleChoice","stepTitle":"0.91_0.901","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">"],"hints":{"DefaultPathway":[{"id":"a453be6realnumbers26a-h1","type":"hint","dependencies":[],"title":"When to Use the \\"Less Than\\" (<) Inequality Symbol","text":"$$a<b$$ \u201ca is less than b\u201d when a is to the left of $$b$$ on the number line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers26a-h2","type":"hint","dependencies":["a453be6realnumbers26a-h1"],"title":"When to Use the \\"Greater Than\\" (>) Inequality Symbol","text":"$$a>b$$ \u201ca is greater than b\u201d when a is to the right of $$b$$ on the number line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers26a-h3","type":"hint","dependencies":["a453be6realnumbers26a-h2"],"title":"Example of a Number Line","text":"In the picture attached to this hint is a number line indicating the positions of $$0.04$$ and $$0.4$$. We can see that $$0.04$$ is to the left of $$0.4$$ on the number line, so $$0.04<0.4$$. You can also see that this means $$0.4$$ is the right of $$0.04$$ on the number line, so $$0.4>0.04$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers26a-h4","type":"hint","dependencies":["a453be6realnumbers26a-h3"],"title":"Answer","text":"$$0.901$$ has one more decimal place than $$0.91$$. $$0.91$$ is equal to $$0.910$$, and because $$910>901$$, $$910$$ thousanths is greater than $$901$$ thousandths.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a453be6realnumbers27","title":"Identifying Correct Inequality Signs #10","body":"Fill in the blanks with the correct inequality sign.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.8 The Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a453be6realnumbers27a","stepAnswer":["<"],"problemType":"MultipleChoice","stepTitle":"-0.5_-0.3","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">"],"hints":{"DefaultPathway":[{"id":"a453be6realnumbers27a-h1","type":"hint","dependencies":[],"title":"When to Use the \\"Less Than\\" (<) Inequality Symbol","text":"$$a<b$$ \u201ca is less than b\u201d when a is to the left of $$b$$ on the number line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers27a-h2","type":"hint","dependencies":["a453be6realnumbers27a-h1"],"title":"When to Use the \\"Greater Than\\" (>) Inequality Symbol","text":"$$a>b$$ \u201ca is greater than b\u201d when a is to the right of $$b$$ on the number line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers27a-h3","type":"hint","dependencies":["a453be6realnumbers27a-h2"],"title":"Example of a Number Line","text":"In the picture attached to this hint is a number line indicating the positions of $$0.04$$ and $$0.4$$. We can see that $$0.04$$ is to the left of $$0.4$$ on the number line, so $$0.04<0.4$$. You can also see that this means $$0.4$$ is the right of $$0.04$$ on the number line, so $$0.4>0.04$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers27a-h4","type":"hint","dependencies":["a453be6realnumbers27a-h3"],"title":"$$-0.5$$ Relative to $$-0.3$$ on the Number Line","text":"$$-0.5$$ is to the left of $$-0.3$$ on the number line, showing that $$-0.5$$ is less than $$-0.3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a453be6realnumbers28","title":"Identifying Correct Inequality Signs #11","body":"Fill in the blanks with the correct inequality sign.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.8 The Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a453be6realnumbers28a","stepAnswer":[">"],"problemType":"MultipleChoice","stepTitle":"-0.1_-0.4","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">"],"hints":{"DefaultPathway":[{"id":"a453be6realnumbers28a-h1","type":"hint","dependencies":[],"title":"When to Use the \\"Less Than\\" (<) Inequality Symbol","text":"$$a<b$$ \u201ca is less than b\u201d when a is to the left of $$b$$ on the number line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers28a-h2","type":"hint","dependencies":["a453be6realnumbers28a-h1"],"title":"When to Use the \\"Greater Than\\" (>) Inequality Symbol","text":"$$a>b$$ \u201ca is greater than b\u201d when a is to the right of $$b$$ on the number line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers28a-h3","type":"hint","dependencies":["a453be6realnumbers28a-h2"],"title":"Example of a Number Line","text":"In the picture attached to this hint is a number line indicating the positions of $$0.04$$ and $$0.4$$. We can see that $$0.04$$ is to the left of $$0.4$$ on the number line, so $$0.04<0.4$$. You can also see that this means $$0.4$$ is the right of $$0.04$$ on the number line, so $$0.4>0.04$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers28a-h4","type":"hint","dependencies":["a453be6realnumbers28a-h3"],"title":"$$-0.1$$ Relative to $$-0.4$$ on the Number Line","text":"$$-0.1$$ is to the right of $$-0.4$$ on the number line, showing that $$-0.1$$ is greater than $$-0.4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a453be6realnumbers29","title":"Identifying Correct Inequality Signs #12","body":"Fill in the blanks with the correct inequality sign.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.8 The Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a453be6realnumbers29a","stepAnswer":["<"],"problemType":"MultipleChoice","stepTitle":"-0.62_-0.619","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">"],"hints":{"DefaultPathway":[{"id":"a453be6realnumbers29a-h1","type":"hint","dependencies":[],"title":"When to Use the \\"Less Than\\" (<) Inequality Symbol","text":"$$a<b$$ \u201ca is less than b\u201d when a is to the left of $$b$$ on the number line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers29a-h2","type":"hint","dependencies":["a453be6realnumbers29a-h1"],"title":"When to Use the \\"Greater Than\\" (>) Inequality Symbol","text":"$$a>b$$ \u201ca is greater than b\u201d when a is to the right of $$b$$ on the number line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers29a-h3","type":"hint","dependencies":["a453be6realnumbers29a-h2"],"title":"Example of a Number Line","text":"In the picture attached to this hint is a number line indicating the positions of $$0.04$$ and $$0.4$$. We can see that $$0.04$$ is to the left of $$0.4$$ on the number line, so $$0.04<0.4$$. You can also see that this means $$0.4$$ is the right of $$0.04$$ on the number line, so $$0.4>0.04$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers29a-h4","type":"hint","dependencies":["a453be6realnumbers29a-h3"],"title":"$$-0.62$$ Relative to $$-0.619$$ on the Number Line","text":"$$-0.62$$ is to the left of $$-0.619$$ on the number line, showing that $$-0.62$$ is less than $$-0.619$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a453be6realnumbers3","title":"Calculating Square Roots #3","body":"Simplify the following expressions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.8 The Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a453be6realnumbers3a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{16}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a453be6realnumbers3a-h1","type":"hint","dependencies":[],"title":"Square Root of a Number","text":"if $$n^2=m$$, then $$n$$ is the square root of $$m$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers3a-h2","type":"hint","dependencies":["a453be6realnumbers3a-h1"],"title":"Example Calculation","text":"$${25}^2=625$$, so $$25$$ is the square root of $$625$$. In equation form, $$\\\\sqrt{625}=25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers3a-h3","type":"hint","dependencies":["a453be6realnumbers3a-h2"],"title":"Finding \\"n\\"","text":"$$4^2=16$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a453be6realnumbers3b","stepAnswer":["$$14$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{196}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$14$$","hints":{"DefaultPathway":[{"id":"a453be6realnumbers3b-h1","type":"hint","dependencies":[],"title":"Square Root of a Number","text":"if $$n^2=m$$, then $$n$$ is the square root of $$m$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers3b-h2","type":"hint","dependencies":["a453be6realnumbers3b-h1"],"title":"Example Calculation","text":"$${25}^2=625$$, so $$25$$ is the square root of $$625$$. In equation form, $$\\\\sqrt{625}=25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers3b-h3","type":"hint","dependencies":["a453be6realnumbers3b-h2"],"title":"Finding \\"n\\"","text":"$${14}^2=196$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a453be6realnumbers30","title":"Identifying Correct Inequality Signs #13","body":"Fill in the blanks with the correct inequality sign.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.8 The Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a453be6realnumbers30a","stepAnswer":["<"],"problemType":"MultipleChoice","stepTitle":"-7.31_7.3","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">"],"hints":{"DefaultPathway":[{"id":"a453be6realnumbers30a-h1","type":"hint","dependencies":[],"title":"When to Use the \\"Less Than\\" (<) Inequality Symbol","text":"$$a<b$$ \u201ca is less than b\u201d when a is to the left of $$b$$ on the number line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers30a-h2","type":"hint","dependencies":["a453be6realnumbers30a-h1"],"title":"When to Use the \\"Greater Than\\" (>) Inequality Symbol","text":"$$a>b$$ \u201ca is greater than b\u201d when a is to the right of $$b$$ on the number line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers30a-h3","type":"hint","dependencies":["a453be6realnumbers30a-h2"],"title":"Example of a Number Line","text":"In the picture attached to this hint is a number line indicating the positions of $$0.04$$ and $$0.4$$. We can see that $$0.04$$ is to the left of $$0.4$$ on the number line, so $$0.04<0.4$$. You can also see that this means $$0.4$$ is the right of $$0.04$$ on the number line, so $$0.4>0.04$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers30a-h4","type":"hint","dependencies":["a453be6realnumbers30a-h3"],"title":"$$-7.31$$ Relative to $$-7.3$$ on the Number Line","text":"$$-7.31$$ is to the left of $$-7.3$$ on the number line, showing that $$-7.31$$ is less than $$-7.3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a453be6realnumbers4","title":"Calculating Opposites of Square Roots #1","body":"Simplify the following expressions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.8 The Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a453be6realnumbers4a","stepAnswer":["$$-3$$"],"problemType":"TextBox","stepTitle":"$$-\\\\sqrt{9}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-3$$","hints":{"DefaultPathway":[{"id":"a453be6realnumbers4a-h1","type":"hint","dependencies":[],"title":"Square Root of a Number","text":"if $$n^2=m$$, then $$n$$ is the square root of $$m$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers4a-h2","type":"hint","dependencies":["a453be6realnumbers4a-h1"],"title":"Example Calculation","text":"$${25}^2=625$$, so $$25$$ is the square root of $$625$$. In equation form, $$\\\\sqrt{625}=25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers4a-h3","type":"hint","dependencies":["a453be6realnumbers4a-h2"],"title":"Finding \\"n\\"","text":"$$3^2=9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers4a-h4","type":"hint","dependencies":["a453be6realnumbers4a-h3"],"title":"Opposite of a Square Root","text":"$$-\\\\sqrt{n}$$ means \\"the opposite of the square root of n.\\" For example, $$-\\\\sqrt{625}=-25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a453be6realnumbers4b","stepAnswer":["$$-12$$"],"problemType":"TextBox","stepTitle":"$$-\\\\sqrt{144}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-12$$","hints":{"DefaultPathway":[{"id":"a453be6realnumbers4b-h1","type":"hint","dependencies":[],"title":"Square Root of a Number","text":"if $$n^2=m$$, then $$n$$ is the square root of $$m$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers4b-h2","type":"hint","dependencies":["a453be6realnumbers4b-h1"],"title":"Example Calculation","text":"$${25}^2=625$$, so $$25$$ is the square root of $$625$$. In equation form, $$\\\\sqrt{625}=25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers4b-h3","type":"hint","dependencies":["a453be6realnumbers4b-h2"],"title":"Finding \\"n\\"","text":"$${12}^2=144$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers4b-h4","type":"hint","dependencies":["a453be6realnumbers4b-h3"],"title":"Opposite of a Square Root","text":"$$-\\\\sqrt{n}$$ means \\"the opposite of the square root of n.\\" For example, $$-\\\\sqrt{625}=-25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a453be6realnumbers5","title":"Calculating Opposites of Square Roots #2","body":"Simplify the following expressions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.8 The Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a453be6realnumbers5a","stepAnswer":["$$-2$$"],"problemType":"TextBox","stepTitle":"$$-\\\\sqrt{4}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-2$$","hints":{"DefaultPathway":[{"id":"a453be6realnumbers5a-h1","type":"hint","dependencies":[],"title":"Square Root of a Number","text":"if $$n^2=m$$, then $$n$$ is the square root of $$m$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers5a-h2","type":"hint","dependencies":["a453be6realnumbers5a-h1"],"title":"Example Calculation","text":"$${25}^2=625$$, so $$25$$ is the square root of $$625$$. In equation form, $$\\\\sqrt{625}=25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers5a-h3","type":"hint","dependencies":["a453be6realnumbers5a-h2"],"title":"Finding \\"n\\"","text":"$$2^2=4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers5a-h4","type":"hint","dependencies":["a453be6realnumbers5a-h3"],"title":"Opposite of a Square Root","text":"$$-\\\\sqrt{n}$$ means \\"the opposite of the square root of n.\\" For example, $$-\\\\sqrt{625}=-25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a453be6realnumbers5b","stepAnswer":["$$-15$$"],"problemType":"TextBox","stepTitle":"$$-\\\\sqrt{225}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-15$$","hints":{"DefaultPathway":[{"id":"a453be6realnumbers5b-h1","type":"hint","dependencies":[],"title":"Square Root of a Number","text":"if $$n^2=m$$, then $$n$$ is the square root of $$m$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers5b-h2","type":"hint","dependencies":["a453be6realnumbers5b-h1"],"title":"Example Calculation","text":"$${25}^2=625$$, so $$25$$ is the square root of $$625$$. In equation form, $$\\\\sqrt{625}=25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers5b-h3","type":"hint","dependencies":["a453be6realnumbers5b-h2"],"title":"Finding \\"n\\"","text":"$${15}^2=225$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers5b-h4","type":"hint","dependencies":["a453be6realnumbers5b-h3"],"title":"Opposite of a Square Root","text":"$$-\\\\sqrt{n}$$ means \\"the opposite of the square root of n.\\" For example, $$-\\\\sqrt{625}=-25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a453be6realnumbers6","title":"Calculating Opposites of Square Roots #3","body":"Simplify the following expressions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.8 The Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a453be6realnumbers6a","stepAnswer":["$$-9$$"],"problemType":"TextBox","stepTitle":"$$-\\\\sqrt{81}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-9$$","hints":{"DefaultPathway":[{"id":"a453be6realnumbers6a-h1","type":"hint","dependencies":[],"title":"Square Root of a Number","text":"if $$n^2=m$$, then $$n$$ is the square root of $$m$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers6a-h2","type":"hint","dependencies":["a453be6realnumbers6a-h1"],"title":"Example Calculation","text":"$${25}^2=625$$, so $$25$$ is the square root of $$625$$. In equation form, $$\\\\sqrt{625}=25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers6a-h3","type":"hint","dependencies":["a453be6realnumbers6a-h2"],"title":"Finding \\"n\\"","text":"$$9^2=81$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers6a-h4","type":"hint","dependencies":["a453be6realnumbers6a-h3"],"title":"Opposite of a Square Root","text":"$$-\\\\sqrt{n}$$ means \\"the opposite of the square root of n.\\" For example, $$-\\\\sqrt{625}=-25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a453be6realnumbers6b","stepAnswer":["$$-10$$"],"problemType":"TextBox","stepTitle":"$$-\\\\sqrt{100}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-10$$","hints":{"DefaultPathway":[{"id":"a453be6realnumbers6b-h1","type":"hint","dependencies":[],"title":"Square Root of a Number","text":"if $$n^2=m$$, then $$n$$ is the square root of $$m$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers6b-h2","type":"hint","dependencies":["a453be6realnumbers6b-h1"],"title":"Example Calculation","text":"$${25}^2=625$$, so $$25$$ is the square root of $$625$$. In equation form, $$\\\\sqrt{625}=25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers6b-h3","type":"hint","dependencies":["a453be6realnumbers6b-h2"],"title":"Finding \\"n\\"","text":"$${10}^2=100$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers6b-h4","type":"hint","dependencies":["a453be6realnumbers6b-h3"],"title":"Opposite of a Square Root","text":"$$-\\\\sqrt{n}$$ means \\"the opposite of the square root of n.\\" For example, $$-\\\\sqrt{625}=-25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a453be6realnumbers7","title":"Identifying Different Types of Numbers #1","body":"Write the following numbers as the ratio of two integers.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.8 The Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a453be6realnumbers7a","stepAnswer":["$$\\\\frac{-27}{1}$$"],"problemType":"TextBox","stepTitle":"$$-27$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-27}{1}$$","hints":{"DefaultPathway":[{"id":"a453be6realnumbers7a-h1","type":"hint","dependencies":[],"title":"Writing Numbers as Fractions WIth Denominator $$1$$","text":"Solve this problem by writing the integer as a fraction with denominator $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers7a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a453be6realnumbers7a-h1"],"title":"Writing Numbers as Fractions WIth Denominator $$1$$","text":"Does $$-27=\\\\frac{-27}{1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}},{"id":"a453be6realnumbers7b","stepAnswer":["$$\\\\frac{731}{100}$$"],"problemType":"TextBox","stepTitle":"$$7.31$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{731}{100}$$","hints":{"DefaultPathway":[{"id":"a453be6realnumbers7b-h1","type":"hint","dependencies":[],"title":"Writing the Number As a Mixed Number","text":"The first step is to write the number as a mixed number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers7b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a453be6realnumbers7b-h1"],"title":"Writing the Number As a Mixed Number","text":"Is $$7.31$$ the same as \\"7 and 31/100\\"?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a453be6realnumbers7b-h3","type":"hint","dependencies":["a453be6realnumbers7b-h2"],"title":"Writing Mixed Numbers As Improper Fractions","text":"Then, write the mixed number as an improper fraction for the final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers7b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a453be6realnumbers7b-h3"],"title":"Writing Mixed Numbers As Improper Fractions","text":"Is \\"7 and 31/100\\" the same as \\"731/100\\"?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a453be6realnumbers8","title":"Identifying Different Types of Numbers #1","body":"Write the following numbers as the ratio of two integers.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.8 The Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a453be6realnumbers8a","stepAnswer":["$$\\\\frac{-24}{1}$$"],"problemType":"TextBox","stepTitle":"$$-24$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-24}{1}$$","hints":{"DefaultPathway":[{"id":"a453be6realnumbers8a-h1","type":"hint","dependencies":[],"title":"Writing Numbers as Fractions WIth Denominator $$1$$","text":"Solve this problem by writing the integer as a fraction with denominator $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers8a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a453be6realnumbers8a-h1"],"title":"Writing Numbers as Fractions WIth Denominator $$1$$","text":"Does $$-24=\\\\frac{-24}{1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}},{"id":"a453be6realnumbers8b","stepAnswer":["$$\\\\frac{357}{100}$$"],"problemType":"TextBox","stepTitle":"$$3.57$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{357}{100}$$","hints":{"DefaultPathway":[{"id":"a453be6realnumbers8b-h1","type":"hint","dependencies":[],"title":"Writing the Number As a Mixed Number","text":"The first step is to write the number as a mixed number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers8b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a453be6realnumbers8b-h1"],"title":"Writing the Number As a Mixed Number","text":"Is $$3.57$$ the same as \\"3 and 57/100\\"?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a453be6realnumbers8b-h3","type":"hint","dependencies":["a453be6realnumbers8b-h2"],"title":"Writing Mixed Numbers As Improper Fractions","text":"Then, write the mixed number as an improper fraction for the final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers8b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a453be6realnumbers8b-h3"],"title":"Writing Mixed Numbers As Improper Fractions","text":"Is \\"3 and 57/100\\" the same as \\"357/100\\"?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a453be6realnumbers9","title":"Identifying Different Types of Numbers #1","body":"Write the following numbers as the ratio of two integers.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.8 The Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a453be6realnumbers9a","stepAnswer":["$$\\\\frac{-19}{1}$$"],"problemType":"TextBox","stepTitle":"$$-19$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-19}{1}$$","hints":{"DefaultPathway":[{"id":"a453be6realnumbers9a-h1","type":"hint","dependencies":[],"title":"Writing Numbers as Fractions WIth Denominator $$1$$","text":"Solve this problem by writing the integer as a fraction with denominator $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers9a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a453be6realnumbers9a-h1"],"title":"Writing Numbers as Fractions WIth Denominator $$1$$","text":"Does $$-19=\\\\frac{-19}{1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}},{"id":"a453be6realnumbers9b","stepAnswer":["$$\\\\frac{841}{100}$$"],"problemType":"TextBox","stepTitle":"$$8.41$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{841}{100}$$","hints":{"DefaultPathway":[{"id":"a453be6realnumbers9b-h1","type":"hint","dependencies":[],"title":"Writing the Number As a Mixed Number","text":"The first step is to write the number as a mixed number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers9b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a453be6realnumbers9b-h1"],"title":"Writing the Number As a Mixed Number","text":"Is $$8.41$$ the same as \\"8 and 41/100\\"?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a453be6realnumbers9b-h3","type":"hint","dependencies":["a453be6realnumbers9b-h2"],"title":"Writing Mixed Numbers As Improper Fractions","text":"Then, write the mixed number as an improper fraction for the final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a453be6realnumbers9b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a453be6realnumbers9b-h3"],"title":"Writing Mixed Numbers As Improper Fractions","text":"Is \\"8 and 41/100\\" the same as \\"841/100\\"?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a4593e0functrans1","title":"Transformations of Functions: Part A","body":"These questions test your knowledge of the core concepts. Below is the graph of $$y=f(x)$$ represented in $$\\\\frac{red}{solid}$$, where the domain of f is [0,4]. Choose the graph representing the functions below.\\\\n##figure3.gif","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Transformations of Functions","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a4593e0functrans1a","stepAnswer":["$$\\\\frac{Blue}{Dashed}$$"],"problemType":"MultipleChoice","stepTitle":"$$y=f(x-4)$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{Blue}{Dashed}$$","choices":["$$\\\\frac{Blue}{Dashed}$$","$$\\\\frac{Green}{Dotted}$$","$$\\\\frac{Orange}{Bold}-Dotted$$","$$\\\\frac{Purple}{Bold}-Dashed$$"],"hints":{"DefaultPathway":[{"id":"a4593e0functrans1a-h1","type":"hint","dependencies":[],"title":"Understanding $$f(x-h)$$","text":"For some value $$h$$, $$f(x-h)$$ means that f(x) is translated to the right by $$h$$ units. Similarly, $$f{\\\\left(x+h\\\\right)}$$ means that f(x) is translated to the left by $$h$$ units.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans1a-h2","type":"hint","dependencies":["a4593e0functrans1a-h1"],"title":"Understanding $$f(x-h)$$","text":"Since $$h$$ is $$4$$, $$f(x-4)$$ means that the graph is translated four units to the right.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans1a-h3","type":"hint","dependencies":["a4593e0functrans1a-h2"],"title":"Understanding $$f(x-h)$$","text":"$$f(x-4)$$ means that $$f(4)=2$$, $$f(6)=0$$, and $$f(8)=1$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]}},{"id":"a4593e0functrans1b","stepAnswer":["$$\\\\frac{Blue}{Dashed}$$"],"problemType":"MultipleChoice","stepTitle":"$$y=f(-x)$$","stepBody":"##figure2.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{Blue}{Dashed}$$","choices":["$$\\\\frac{Blue}{Dashed}$$","$$\\\\frac{Green}{Dotted}$$","$$\\\\frac{Orange}{Bold}-Dotted$$","$$\\\\frac{Purple}{Bold}-Dashed$$"],"hints":{"DefaultPathway":[{"id":"a4593e0functrans1b-h1","type":"hint","dependencies":[],"title":"Understanding $$f(-x)$$","text":"$$f(-x)$$ means that f(x) is reflected across the y-axis. Similarly, $$-f(x)$$ means that f(x) is reflected across the x-axis.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans1b-h2","type":"hint","dependencies":["a4593e0functrans1b-h1"],"title":"Understanding $$f(-x)$$","text":"$$f(-x)$$ means that $$f(0)=2$$, $$f(-2)=0$$, and $$f(-4)=1$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a4593e0functrans10","title":"Transformations of Functions: Part A","body":"These questions test your knowledge of the core concepts.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Transformations of Functions","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a4593e0functrans10a","stepAnswer":["$$y=4+\\\\sqrt{-x+3}$$"],"problemType":"MultipleChoice","stepTitle":"Consider the function $$y=\\\\sqrt{x}$$. Find a formula (in terms of x) for the function obtained by translation to the left by $$3$$ units, up by $$4$$ units, then reflection over the y-axis.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=4+\\\\sqrt{-x+3}$$","choices":["$$y=4+\\\\sqrt{-x+3}$$","$$y=4-\\\\sqrt{x+3}$$","$$y=4+\\\\sqrt{-x-3}$$","$$y=4-\\\\sqrt{x-3}$$"],"hints":{"DefaultPathway":[{"id":"a4593e0functrans10a-h1","type":"hint","dependencies":[],"title":"Understanding Left and Right Translations","text":"For some value $$h$$, $$f(x-h)$$ means that f(x) is translated to the right by $$h$$ units. Similarly, $$f{\\\\left(x+h\\\\right)}$$ means that f(x) is translated to the left by $$h$$ units.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans10a-h2","type":"hint","dependencies":["a4593e0functrans10a-h1"],"title":"Understanding Left and Right Translations","text":"A translation to the left by $$3$$ is represented by $$f{\\\\left(x+3\\\\right)}=\\\\sqrt{x+3}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans10a-h3","type":"hint","dependencies":["a4593e0functrans10a-h2"],"title":"Understanding Up and Down Translations","text":"For some value $$h$$, $$f{\\\\left(x\\\\right)}+h$$ means that f(x) is translated up by $$h$$ units. Similarly, $$f(x)-h$$ means that f(x) is translated down by $$h$$ units.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans10a-h4","type":"hint","dependencies":["a4593e0functrans10a-h3"],"title":"Understanding Up and Down Translations","text":"A translation up by $$4$$ is represented by $$f{\\\\left(x+3\\\\right)}+4=4+\\\\sqrt{x+3}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans10a-h5","type":"hint","dependencies":["a4593e0functrans10a-h4"],"title":"Understanding Reflections","text":"$$f(-x)$$ means that f(x) is reflected across the y-axis. Similarly, $$-f(x)$$ means that f(x) is reflected across the x-axis.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans10a-h6","type":"hint","dependencies":["a4593e0functrans10a-h5"],"title":"Understanding Reflections","text":"A reflection over the y-axis is represented by $$f\\\\left(-x+3\\\\right)+4=4+\\\\sqrt{-x+3}$$","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a4593e0functrans100","title":"Transformations of Functions: Part B","body":"These problems are generally harder, often highlighting an important subtlety. On the graph below, the red dotted graph is $$y=h(x)$$. Write a formula for each of the other functions in terms of h(x).\\\\n##figure1.gif","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Transformations of Functions","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a4593e0functrans100a","stepAnswer":["Translate right by $$3$$, then translate up by $$1$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{Blue}{Dash}-Dotted$$: $$y=h{\\\\left(x-3\\\\right)}+1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Translate right by $$3$$, then translate up by $$1$$","choices":["Translate right by $$3$$, then translate up by $$1$$","Translate left by $$3$$, then translate up by $$1$$","Translate right by $$3$$, then translate down by $$1$$","Translate left by $$3$$, then translate down by $$1$$"],"hints":{"DefaultPathway":[{"id":"a4593e0functrans100a-h1","type":"hint","dependencies":[],"title":"Understanding $$f(x-a)$$","text":"For some value a, $$f(x-a)$$ means that f(x) is translated to the right by a units. Similarly, $$f{\\\\left(x+a\\\\right)}$$ means that f(x) is translated to the left by a units.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans100a-h2","type":"hint","dependencies":["a4593e0functrans100a-h1"],"title":"Understanding $$f(x-a)$$","text":"Since a is $$4$$, $$h(x-3)$$ means that the graph is translated right by $$3$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans100a-h3","type":"hint","dependencies":["a4593e0functrans100a-h2"],"title":"Understanding $$f{\\\\left(x\\\\right)}+a$$","text":"For some value a, $$f{\\\\left(x\\\\right)}+a$$ means that f(x) is translated up by a units. Similarly, $$f(x)-a$$ means that f(x) is translated down by a units.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans100a-h4","type":"hint","dependencies":["a4593e0functrans100a-h3"],"title":"Understanding $$f{\\\\left(x\\\\right)}+a$$","text":"Since a is $$1$$, $$h{\\\\left(x-3\\\\right)}+1$$ means that the graph is translated up by $$1$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]}},{"id":"a4593e0functrans100b","stepAnswer":["Translate left by 6,then reflect across $$y-axis$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{Green}{Dashed}$$: $$y=h\\\\left(-x+6\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Translate left by 6,then reflect across $$y-axis$$","Translate right by 6,then reflect across $$y-axis$$","Translate left by 6,then reflect across $$x-axis$$","Translate right by 6,then reflect across $$x-axis$$"],"hints":{"DefaultPathway":[{"id":"a4593e0functrans100b-h1","type":"hint","dependencies":[],"title":"Understanding $$f(x-a)$$","text":"For some value a, $$f(x-a)$$ means that f(x) is translated to the right by a units. Similarly, $$f{\\\\left(x+a\\\\right)}$$ means that f(x) is translated to the left by a units.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans100b-h2","type":"hint","dependencies":["a4593e0functrans100b-h1"],"title":"Understanding $$f(x-a)$$","text":"Since a is $$-6$$, $$h{\\\\left(x+6\\\\right)}$$ means that the graph is translated left by $$6$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans100b-h3","type":"hint","dependencies":["a4593e0functrans100b-h2"],"title":"Understanding $$f(-x)$$","text":"$$f(-x)$$ means that f(x) is reflected across the y-axis. Similarly, $$-f(x)$$ means that f(x) is reflected across the x-axis.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans100b-h4","type":"hint","dependencies":["a4593e0functrans100b-h3"],"title":"Understanding $$f(-x)$$","text":"$$h\\\\left(-x+6\\\\right)$$ means that the graph is reflected across the y-axis.","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a4593e0functrans1000","title":"Transformations of Functions: Part C","body":"These questions are challenging, requiring mastery of each concept and their interrelations. Fill out the sequence of transformations whose composition transforms $$y=3-2\\\\sqrt{1-2x}$$ into $$y=\\\\sqrt{x}$$.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Transformations of Functions","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a4593e0functrans1000a","stepAnswer":["$$\\\\frac{3}{2}-\\\\frac{1}{2} f\\\\left(-\\\\frac{1}{2} x+\\\\frac{1}{2}\\\\right)$$"],"problemType":"MultipleChoice","stepTitle":"Rewrite the function in terms of f(x).","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{3}{2}-\\\\frac{1}{2} f\\\\left(-\\\\frac{1}{2} x+\\\\frac{1}{2}\\\\right)$$","choices":["$$\\\\frac{3}{2}-\\\\frac{1}{2} f\\\\left(-\\\\frac{1}{2} x+\\\\frac{1}{2}\\\\right)$$","$$3-\\\\sqrt{f{\\\\left(\\\\frac{1}{2} x\\\\right)}}$$","$$3-f\\\\left(-\\\\frac{1}{2} x+\\\\frac{1}{2}\\\\right)$$","$$\\\\frac{3}{2}-\\\\frac{1}{2} \\\\sqrt{-\\\\left(\\\\frac{1}{2}\\\\right) f{\\\\left(x\\\\right)}+\\\\frac{1}{2}}$$"],"hints":{"DefaultPathway":[{"id":"a4593e0functrans1000a-h1","type":"hint","dependencies":[],"title":"Understanding the Inverse Property","text":"To transform a function into a similar form, any value can be applied as long as its inverse is also in the function. For example, $$x=3+x-3$$ or $$x=\\\\frac{4}{4} x$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans1000a-h2","type":"hint","dependencies":["a4593e0functrans1000a-h1"],"title":"Transforming $$\\\\sqrt{x}$$","text":"Add and subtract $$1$$ within the square root: $$\\\\sqrt{1+x-1}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans1000a-h3","type":"hint","dependencies":["a4593e0functrans1000a-h2"],"title":"Transforming $$\\\\sqrt{x}$$","text":"Multiply $$\\\\frac{-2}{-2}$$ to $$(x-1)$$: $$\\\\sqrt{1+\\\\left(-\\\\frac{2}{-2}\\\\right) \\\\left(x-1\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans1000a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-\\\\left(\\\\frac{1}{2}\\\\right) x+\\\\frac{1}{2}$$"],"dependencies":["a4593e0functrans1000a-h3"],"title":"Transforming $$\\\\sqrt{x}$$","text":"Distribute $$\\\\frac{-1}{2}$$ to $$(x-1)$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans1000a-h5","type":"hint","dependencies":["a4593e0functrans1000a-h4"],"title":"Transforming $$\\\\sqrt{x}$$","text":"To recap, $$\\\\sqrt{x}=\\\\sqrt{1-2\\\\left(-\\\\frac{1}{2} x+\\\\frac{1}{2}\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans1000a-h6","type":"hint","dependencies":["a4593e0functrans1000a-h5"],"title":"Transforming $$\\\\sqrt{x}$$","text":"Multiply $$\\\\frac{-2}{-2}$$: $$-\\\\left(\\\\frac{1}{2}\\\\right) \\\\left(-2\\\\sqrt{1-2\\\\left(-\\\\frac{1}{2} x+\\\\frac{1}{2}\\\\right)}\\\\right)$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans1000a-h7","type":"hint","dependencies":["a4593e0functrans1000a-h6"],"title":"Transforming $$\\\\sqrt{x}$$","text":"Add and subtract $$3$$ within the parantheses: $$-\\\\left(\\\\frac{1}{2}\\\\right) \\\\left(-3+3-2\\\\sqrt{1-2\\\\left(-\\\\frac{1}{2} x+\\\\frac{1}{2}\\\\right)}\\\\right)$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans1000a-h8","type":"hint","dependencies":["a4593e0functrans1000a-h7"],"title":"Transforming $$\\\\sqrt{x}$$","text":"Pull out the $$-3$$ from the parantheses by multiplying the equation by $$\\\\frac{-1}{2}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans1000a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{2}$$"],"dependencies":["a4593e0functrans1000a-h8"],"title":"Transforming $$\\\\sqrt{x}$$","text":"What is $$-3\\\\left(-\\\\frac{1}{2}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans1000a-h10","type":"hint","dependencies":["a4593e0functrans1000a-h9"],"title":"Transforming $$\\\\sqrt{x}$$","text":"The term $$3-2\\\\sqrt{1-2\\\\left(-\\\\frac{1}{2} x+\\\\frac{1}{2}\\\\right)}$$ can be rewritten in terms of $$f(x)=3-2\\\\sqrt{1-2x}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans1000a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-\\\\left(\\\\frac{1}{2}\\\\right) x+\\\\frac{1}{2}$$"],"dependencies":["a4593e0functrans1000a-h10"],"title":"Transforming $$\\\\sqrt{x}$$","text":"What value of $$x$$ can be substituted to make $$3-2\\\\sqrt{1-2x}$$ equal to $$3-2\\\\sqrt{1-2\\\\left(-\\\\frac{1}{2} x+\\\\frac{1}{2}\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans1000a-h12","type":"hint","dependencies":["a4593e0functrans1000a-h11"],"title":"Transforming $$\\\\sqrt{x}$$","text":"If $$f\\\\left(-\\\\frac{1}{2} x+\\\\frac{1}{2}\\\\right)=3-2\\\\sqrt{1-2\\\\left(-\\\\frac{1}{2} x+\\\\frac{1}{2}\\\\right)}$$, then $$\\\\frac{3}{2}-\\\\frac{1}{2} \\\\left(3-2\\\\sqrt{1-2\\\\left(-\\\\frac{1}{2} x+\\\\frac{1}{2}\\\\right)}\\\\right)=\\\\frac{3}{2}-\\\\frac{1}{2} f\\\\left(-\\\\frac{1}{2} x+\\\\frac{1}{2}\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]}},{"id":"a4593e0functrans1000b","stepAnswer":["Translate left by $$\\\\frac{1}{2}$$"],"problemType":"MultipleChoice","stepTitle":"What step can be taken to transform f(x) to $$f{\\\\left(x+\\\\frac{1}{2}\\\\right)}$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Translate left by $$\\\\frac{1}{2}$$","choices":["Translate left by $$\\\\frac{1}{2}$$","Translate up by $$\\\\frac{1}{2}$$","Translate right by $$\\\\frac{1}{2}$$","Translate down by $$\\\\frac{1}{2}$$"],"hints":{"DefaultPathway":[{"id":"a4593e0functrans1000b-h1","type":"hint","dependencies":[],"title":"Input or Output?","text":"A transformation that affects the input is within the function paranetheses, while a transformation that affects the output is outside the function paranetheses.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans1000b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Input"],"dependencies":["a4593e0functrans1000b-h1"],"title":"Input or Output?","text":"Does the step affect the input or output of the function?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Input","Output"]},{"id":"a4593e0functrans1000b-h3","type":"hint","dependencies":["a4593e0functrans1000b-h2"],"title":"Horizontal or Vertical?","text":"A translation that affects the input is horizontal, while a translation that affects the output is vertical.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans1000b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Horizontal"],"dependencies":["a4593e0functrans1000b-h3"],"title":"Horizontal or Vertical?","text":"Is the translation horizontal or vertical?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Horizontal","Vertical"]},{"id":"a4593e0functrans1000b-h5","type":"hint","dependencies":["a4593e0functrans1000b-h4"],"title":"Understanding Translation","text":"A hoirzontal translation means that the input is added by some value b: $$f{\\\\left(x+b\\\\right)}$$. Translating left adds $$b$$, while translating right subtracts $$b$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans1000b-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Left"],"dependencies":["a4593e0functrans1000b-h5"],"title":"Understanding Translation","text":"Is the horizontal translation left or right?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Left","Right"]}]}},{"id":"a4593e0functrans1000c","stepAnswer":["Stretch horizontally by $$2$$"],"problemType":"MultipleChoice","stepTitle":"What step can be taken to transform $$f{\\\\left(x+\\\\frac{1}{2}\\\\right)}$$ to $$f{\\\\left(\\\\frac{1}{2} x+\\\\frac{1}{2}\\\\right)}$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Stretch horizontally by $$2$$","choices":["Stretch horizontally by $$2$$","Compress horizontally by $$2$$","Stretch vertically by $$2$$","Compress vertically by $$2$$"],"hints":{"DefaultPathway":[{"id":"a4593e0functrans1000c-h1","type":"hint","dependencies":[],"title":"Input or Output?","text":"A transformation that affects the input is within the function paranetheses, while a transformation that affects the output is outside the function paranetheses.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans1000c-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Input"],"dependencies":["a4593e0functrans1000c-h1"],"title":"Input or Output?","text":"Does the step affect the input or output of the function?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Input","Output"]},{"id":"a4593e0functrans1000c-h3","type":"hint","dependencies":["a4593e0functrans1000c-h2"],"title":"Horizontal or Vertical?","text":"A $$\\\\frac{stretch}{compression}$$ that affects the input is horizontal, while a $$\\\\frac{stretch}{compression}$$ that affects the output is vertical.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans1000c-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Horizontal"],"dependencies":["a4593e0functrans1000c-h3"],"title":"Horizontal or Vertical?","text":"Is the $$\\\\frac{stretch}{compression}$$ horizontal or vertical?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Horizontal","Vertical"]},{"id":"a4593e0functrans1000c-h5","type":"hint","dependencies":["a4593e0functrans1000c-h4"],"title":"Understanding $$\\\\frac{Stretch}{Compression}$$","text":"A horizontal $$\\\\frac{stretch}{compression}$$ means that the input is multiplied by some value b: $$f{\\\\left(b x\\\\right)}$$. Compressing multiplies $$b$$, while stretching multiplies $$\\\\frac{1}{b}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans1000c-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Stretch"],"dependencies":["a4593e0functrans1000c-h5"],"title":"Understanding $$\\\\frac{Stretch}{Compression}$$","text":"Is the horizontal transformation a stretch or compression?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Stretch","Compression"]}]}},{"id":"a4593e0functrans1000d","stepAnswer":["Reflect across $$y-axis$$"],"problemType":"MultipleChoice","stepTitle":"What step can be taken to transform $$f{\\\\left(\\\\frac{1}{2} x+\\\\frac{1}{2}\\\\right)}$$ to $$f\\\\left(-\\\\frac{1}{2} x+\\\\frac{1}{2}\\\\right)$$?","stepBody":"","answerType":"string","variabilization":{},"choices":["Reflect across $$y-axis$$","Reflect across $$x-axis$$"],"hints":{"DefaultPathway":[{"id":"a4593e0functrans1000d-h1","type":"hint","dependencies":[],"title":"Input or Output?","text":"A transformation that affects the input is within the function paranetheses, while a transformation that affects the output is outside the function paranetheses.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans1000d-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Input"],"dependencies":["a4593e0functrans1000d-h1"],"title":"Input or Output?","text":"Does the step affect the input or output of the function?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Input","Output"]},{"id":"a4593e0functrans1000d-h3","type":"hint","dependencies":["a4593e0functrans1000d-h2"],"title":"Understanding Reflections","text":"A reflection that affects the input is over the y-axis, while a reflection that affects the output is over the x-axis.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans1000d-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y-axis$$"],"dependencies":["a4593e0functrans1000d-h3"],"title":"Understanding Reflections","text":"Is the reflection over the y-axis or x-axis?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$x-axis$$","$$y-axis$$"]}]}},{"id":"a4593e0functrans1000e","stepAnswer":["Reflect across $$x-axis$$"],"problemType":"MultipleChoice","stepTitle":"What step can be taken to transform $$f\\\\left(-\\\\frac{1}{2} x+\\\\frac{1}{2}\\\\right)$$ to $$-f\\\\left(-\\\\frac{1}{2} x+\\\\frac{1}{2}\\\\right)$$?","stepBody":"","answerType":"string","variabilization":{},"choices":["Reflect across $$y-axis$$","Reflect across $$x-axis$$"],"hints":{"DefaultPathway":[{"id":"a4593e0functrans1000e-h1","type":"hint","dependencies":[],"title":"Input or Output?","text":"A transformation that affects the input is within the function paranetheses, while a transformation that affects the output is outside the function paranetheses.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans1000e-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Output"],"dependencies":["a4593e0functrans1000e-h1"],"title":"Input or Output?","text":"Does the step affect the input or output of the function?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Input","Output"]},{"id":"a4593e0functrans1000e-h3","type":"hint","dependencies":["a4593e0functrans1000e-h2"],"title":"Understanding Reflections","text":"A reflection that affects the input is over the y-axis, while a reflection that affects the output is over the x-axis.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans1000e-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x-axis$$"],"dependencies":["a4593e0functrans1000e-h3"],"title":"Understanding Reflections","text":"Is the reflection over the y-axis or x-axis?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$x-axis$$","$$y-axis$$"]}]}},{"id":"a4593e0functrans1000f","stepAnswer":["Compress vertically by $$2$$"],"problemType":"MultipleChoice","stepTitle":"What step can be taken to transform $$-f\\\\left(-\\\\frac{1}{2} x+\\\\frac{1}{2}\\\\right)$$ to $$-\\\\left(\\\\frac{1}{2}\\\\right) f\\\\left(-\\\\frac{1}{2} x+\\\\frac{1}{2}\\\\right)$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Compress vertically by $$2$$","choices":["Stretch horizontally by $$2$$","Compress horizontally by $$2$$","Stretch vertically by $$2$$","Compress vertically by $$2$$"],"hints":{"DefaultPathway":[{"id":"a4593e0functrans1000f-h1","type":"hint","dependencies":[],"title":"Input or Output?","text":"A transformation that affects the input is within the function paranetheses, while a transformation that affects the output is outside the function paranetheses.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans1000f-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Output"],"dependencies":["a4593e0functrans1000f-h1"],"title":"Input or Output?","text":"Does the step affect the input or output of the function?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Input","Output"]},{"id":"a4593e0functrans1000f-h3","type":"hint","dependencies":["a4593e0functrans1000f-h2"],"title":"Horizontal or Vertical?","text":"A $$\\\\frac{stretch}{compression}$$ that affects the input is horizontal, while a $$\\\\frac{stretch}{compression}$$ that affects the output is vertical.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans1000f-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Vertical"],"dependencies":["a4593e0functrans1000f-h3"],"title":"Horizontal or Vertical?","text":"Is the $$\\\\frac{stretch}{compression}$$ horizontal or vertical?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Horizontal","Vertical"]},{"id":"a4593e0functrans1000f-h5","type":"hint","dependencies":["a4593e0functrans1000f-h4"],"title":"Understanding $$\\\\frac{Stretch}{Compression}$$","text":"A vertical $$\\\\frac{stretch}{compression}$$ means that the output is multiplied by some value b: $$f{\\\\left(b x\\\\right)}$$. Stretching multiplies $$b$$, while compressing multiplies $$\\\\frac{1}{b}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans1000f-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Compression"],"dependencies":["a4593e0functrans1000f-h5"],"title":"Understanding $$\\\\frac{Stretch}{Compression}$$","text":"Is the vertical transformation a stretch or compression?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Stretch","Compression"]}]}},{"id":"a4593e0functrans1000g","stepAnswer":["Translate up by $$\\\\frac{3}{2}$$"],"problemType":"MultipleChoice","stepTitle":"What step can be taken to transform $$-\\\\left(\\\\frac{1}{2}\\\\right) f\\\\left(-\\\\frac{1}{2} x+\\\\frac{1}{2}\\\\right)$$ to $$\\\\frac{3}{2}-\\\\frac{1}{2} f\\\\left(-\\\\frac{1}{2} x+\\\\frac{1}{2}\\\\right)$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Translate up by $$\\\\frac{3}{2}$$","choices":["Translate left by $$\\\\frac{3}{2}$$","Translate up by $$\\\\frac{3}{2}$$","Translate right by $$\\\\frac{3}{2}$$","Translate down by $$\\\\frac{3}{2}$$"],"hints":{"DefaultPathway":[{"id":"a4593e0functrans1000g-h1","type":"hint","dependencies":[],"title":"Input or Output?","text":"A transformation that affects the input is within the function paranetheses, while a transformation that affects the output is outside the function paranetheses.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans1000g-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Output"],"dependencies":["a4593e0functrans1000g-h1"],"title":"Input or Output?","text":"Does the step affect the input or output of the function?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Input","Output"]},{"id":"a4593e0functrans1000g-h3","type":"hint","dependencies":["a4593e0functrans1000g-h2"],"title":"Horizontal or Vertical?","text":"A translation that affects the input is horizontal, while a translation that affects the output is vertical.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans1000g-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Vertical"],"dependencies":["a4593e0functrans1000g-h3"],"title":"Horizontal or Vertical?","text":"Is the translation horizontal or vertical?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Horizontal","Vertical"]},{"id":"a4593e0functrans1000g-h5","type":"hint","dependencies":["a4593e0functrans1000g-h4"],"title":"Understanding Translation","text":"A vertical translation means that the output is added by some value b: $$f{\\\\left(x\\\\right)}+b$$. Translating up adds $$b$$, while translating down subtracts $$b$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans1000g-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Up"],"dependencies":["a4593e0functrans1000g-h5"],"title":"Understanding Translation","text":"Is the vertical translation up or down?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Up","Down"]}]}}]},{"id":"a4593e0functrans101","title":"Transformations of Functions: Part B","body":"These problems are generally harder, often highlighting an important subtlety. With certain functions, transformations can be interpreted as horizontal or vertical. Give details on how to build the function $$y=3x-6$$ as a composition of two transformations as follows.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Transformations of Functions","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a4593e0functrans101a","stepAnswer":["Translate down by $$2$$, then stretch vertically by $$3$$"],"problemType":"MultipleChoice","stepTitle":"A vertical translation, then a vertical $$\\\\frac{stretch}{compression}$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Translate down by $$2$$, then stretch vertically by $$3$$","choices":["Translate down by $$2$$, then stretch vertically by $$3$$","Translate down by $$6$$, then stretch vertically by $$3$$","Translate down by $$2$$, then compress vertically by $$3$$","Translate down by $$6$$, then compress vertically by $$3$$"],"hints":{"DefaultPathway":[{"id":"a4593e0functrans101a-h1","type":"hint","dependencies":[],"title":"Understanding Vertical Translations","text":"A vertical translation means that the output is translated by some amount of units a: $$f{\\\\left(x\\\\right)}+a$$. Translating up adds a while translating down subtracts a.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans101a-h2","type":"hint","dependencies":["a4593e0functrans101a-h1"],"title":"Understanding Vertical $$\\\\frac{Stretch}{Compression}$$","text":"A vertical $$\\\\frac{stretch}{compression}$$ means that the output is multiplied by some value b: $$b f{\\\\left(x\\\\right)}$$. Stretching multiplies $$b$$, while compressing multiplies $$\\\\frac{1}{b}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans101a-h3","type":"hint","dependencies":["a4593e0functrans101a-h2"],"title":"Solving the Equation","text":"A vertical translation by a units, then a vertical $$\\\\frac{stretch}{compression}$$ by $$b$$ units can be combined to create one equation to solve: $$y=b \\\\left(f{\\\\left(x\\\\right)}+a\\\\right)=3x-6$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans101a-h4","type":"hint","dependencies":["a4593e0functrans101a-h3"],"title":"Solving the Equation","text":"Since the transformations are from $$y=x$$, the equation to solve is $$b \\\\left(x+a\\\\right)=3x-6$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans101a-h5","type":"hint","dependencies":["a4593e0functrans101a-h4"],"title":"Solving the Equation","text":"The equation can be solved as a system of linear equations by looking at like terms:\\\\n\\\\n$$b=3$$\\\\n$$b a=-6$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans101a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a4593e0functrans101a-h5"],"title":"Solving the Equation","text":"What is the value of $$b$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans101a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a4593e0functrans101a-h5"],"title":"Solving the Equation","text":"What is the value of a?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans101a-h8","type":"hint","dependencies":["a4593e0functrans101a-h6","a4593e0functrans101a-h7"],"title":"Translating into Words","text":"Since $$a=-2$$ and $$b=3$$, the transformations can now be written out. Translating up adds a while translating down subtracts a. Stretching multiplies $$b$$, while compressing multiplies $$\\\\frac{1}{b}$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]}},{"id":"a4593e0functrans101b","stepAnswer":["Compress horizontally by $$3$$, then translate right by $$2$$"],"problemType":"MultipleChoice","stepTitle":"A horizontal $$\\\\frac{stretch}{compression}$$, then a horizontal translation.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Compress horizontally by $$3$$, then translate right by $$2$$","choices":["Compress horizontally by $$3$$, then translate right by $$2$$","Compress horizontally by $$3$$, then translate right by $$6$$","Stretch horizontally by $$3$$, then translate right by $$2$$","Stretch horizontally by $$3$$, then translate right by $$6$$"],"hints":{"DefaultPathway":[{"id":"a4593e0functrans101b-h1","type":"hint","dependencies":[],"title":"Understanding Horizontal Translations","text":"A horizontal translation means that the input is translated by some amount of units a: $$f{\\\\left(x+a\\\\right)}$$. Translating left adds a while translating right subtracts a.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans101b-h2","type":"hint","dependencies":["a4593e0functrans101b-h1"],"title":"Understanding Horizontal $$\\\\frac{Stretch}{Compression}$$","text":"A horizontal $$\\\\frac{stretch}{compression}$$ means that the input is multiplied by some value b: $$f{\\\\left(b x\\\\right)}$$. Compressing multiplies $$b$$, while stretching multiplies $$\\\\frac{1}{b}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans101b-h3","type":"hint","dependencies":["a4593e0functrans101b-h2"],"title":"Solving the Equation","text":"A horizontal $$\\\\frac{stretch}{compression}$$ by $$b$$ units, then a horizontal translation by a units can be combined to create one equation to solve: $$y=f{\\\\left(b \\\\left(x+a\\\\right)\\\\right)}=3x-6$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans101b-h4","type":"hint","dependencies":["a4593e0functrans101b-h3"],"title":"Solving the Equation","text":"Since the transformations are from $$y=x$$, the equation to solve is $$b \\\\left(x+a\\\\right)=3x-6$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans101b-h5","type":"hint","dependencies":["a4593e0functrans101b-h4"],"title":"Solving the Equation","text":"The equation can be solved as a system of linear equations by looking at like terms:\\\\n\\\\n$$b=3$$\\\\n$$b a=-6$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans101b-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a4593e0functrans101b-h5"],"title":"Solving the Equation","text":"What is the value of $$b$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans101b-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a4593e0functrans101b-h5"],"title":"Solving the Equation","text":"What is the value of a?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans101b-h8","type":"hint","dependencies":["a4593e0functrans101b-h6","a4593e0functrans101b-h7"],"title":"Translating into Words","text":"Since $$a=-2$$ and $$b=3$$, the transformations can now be written out. Translating left adds a while translating right subtracts a. Compressing multiplies $$b$$, while stretching multiplies $$\\\\frac{1}{b}$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a4593e0functrans11","title":"Transformations of Functions: Part A","body":"These questions test your knowledge of the core concepts. In words, list a sequence of two geometric transformations that will transform the graph of $$y=g(x)$$ into the given graph. Check whether the order matters in each example.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Transformations of Functions","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a4593e0functrans11a","stepAnswer":["Translate right by $$2$$, then stretch vertically by 3;order does not matter"],"problemType":"MultipleChoice","stepTitle":"$$y=3g{\\\\left(x-2\\\\right)}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Translate right by $$2$$, then stretch vertically by 3;order does not matter","choices":["Translate right by $$2$$, then stretch vertically by 3;order does not matter","Translate left by $$2$$, then stretch horizontally by 3;order does not matter","Translate right by $$2$$, then stretch vertically by 3;order does matter","Translate left by $$2$$, then stretch horizontally by 3;order does matter"],"hints":{"DefaultPathway":[{"id":"a4593e0functrans11a-h1","type":"hint","dependencies":[],"title":"Understanding $$f(x-h)$$","text":"For some value $$h$$, $$f(x-h)$$ means that f(x) is translated to the right by $$h$$ units. Similarly, $$f{\\\\left(x+h\\\\right)}$$ means that f(x) is translated to the left by $$h$$ units.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans11a-h2","type":"hint","dependencies":["a4593e0functrans11a-h1"],"title":"Understanding $$f(x-h)$$","text":"Since $$h$$ is $$2$$, $$f(x-2)$$ means that the graph is translated right by $$2$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans11a-h3","type":"hint","dependencies":["a4593e0functrans11a-h2"],"title":"Understanding $$h f{\\\\left(x\\\\right)}$$","text":"For some value $$h$$ not equal to $$0$$, $$h f{\\\\left(x\\\\right)}$$ means that f(x) is stretched vertically by $$h$$. Similarly, $$\\\\frac{f{\\\\left(x\\\\right)}}{h}$$ means that f(x) is compressed vertically by $$h$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans11a-h4","type":"hint","dependencies":["a4593e0functrans11a-h3"],"title":"Understanding $$h f{\\\\left(x\\\\right)}$$","text":"Since $$h$$ is $$3$$, $$3f{\\\\left(x-2\\\\right)}$$ means that the graph is stretched vertically by $$3$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans11a-h5","type":"hint","dependencies":["a4593e0functrans11a-h4"],"title":"Does Order Matter?","text":"The order of transformation does not matter if each transformation is performed independently on the input (within the paranethesis of the function) and output (outside the paranethesis of the function).","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans11a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Input"],"dependencies":["a4593e0functrans11a-h5"],"title":"Does Order Matter?","text":"Is the translation right performed on the input or output of the function?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Input","Output"]},{"id":"a4593e0functrans11a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Output"],"dependencies":["a4593e0functrans11a-h5"],"title":"Does Order Matter?","text":"Is the stretch vertically performed on the input or output of the function?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Input","Output"]},{"id":"a4593e0functrans11a-h8","type":"hint","dependencies":["a4593e0functrans11a-h6","a4593e0functrans11a-h7"],"title":"Does Order Matter?","text":"Since the transformations were performed independently of each other, the order does not matter.","variabilization":{},"oer":"https://OATutor.io","license":""}]}},{"id":"a4593e0functrans11b","stepAnswer":["Translate right by $$4$$, then stretch horizontally by 2;order does not matter"],"problemType":"MultipleChoice","stepTitle":"$$y=g{\\\\left(\\\\frac{x}{2}-4\\\\right)}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Translate right by $$4$$, then stretch horizontally by 2;order does not matter","choices":["Translate right by $$4$$, then stretch horizontally by 2;order does not matter","Translate left by $$4$$, then stretch vertically by 2;order does not matter","Translate right by $$4$$, then stretch horizontally by 2;order does matter","Translate left by $$4$$, then stretch vertically by 2;order does matter"],"hints":{"DefaultPathway":[{"id":"a4593e0functrans11b-h1","type":"hint","dependencies":[],"title":"Understanding $$f(x-h)$$","text":"For some value $$h$$, $$f(x-h)$$ means that f(x) is translated to the right by $$h$$ units. Similarly, $$f{\\\\left(x+h\\\\right)}$$ means that f(x) is translated to the left by $$h$$ units.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans11b-h2","type":"hint","dependencies":["a4593e0functrans11b-h1"],"title":"Understanding $$f(x-h)$$","text":"Since $$h$$ is $$4$$, $$f(x-4)$$ means that the graph is translated right by $$4$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans11b-h3","type":"hint","dependencies":["a4593e0functrans11b-h2"],"title":"Understanding $$f{\\\\left(\\\\frac{x}{h}\\\\right)}$$","text":"For some value $$h$$ not equal to $$0$$, $$f{\\\\left(\\\\frac{x}{h}\\\\right)}$$ means that f(x) is stretched horizontally by $$h$$. Similarly, $$f{\\\\left(h x\\\\right)}$$ means that f(x) is compressed horizontally by $$h$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans11b-h4","type":"hint","dependencies":["a4593e0functrans11b-h3"],"title":"Understanding $$f{\\\\left(\\\\frac{x}{h}\\\\right)}$$","text":"Since $$h$$ is $$\\\\frac{1}{2}$$, $$f{\\\\left(\\\\frac{x}{2}-4\\\\right)}$$ means that the graph is stretched horizontally by $$2$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans11b-h5","type":"hint","dependencies":["a4593e0functrans11b-h4"],"title":"Does Order Matter?","text":"The order of transformation does not matter if each transformation is performed independently on the input (within the paranethesis of the function) and output (outside the paranethesis of the function).","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans11b-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Input"],"dependencies":["a4593e0functrans11b-h5"],"title":"Does Order Matter?","text":"Is the translation right performed on the input or output of the function?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Input","Output"]},{"id":"a4593e0functrans11b-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Input"],"dependencies":["a4593e0functrans11b-h5"],"title":"Does Order Matter?","text":"Is the stretch horizontally performed on the input or output of the function?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Input","Output"]},{"id":"a4593e0functrans11b-h8","type":"hint","dependencies":["a4593e0functrans11b-h6","a4593e0functrans11b-h7"],"title":"Does Order Matter?","text":"Since the transformations were both performed on the input, the order does matter.","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a4593e0functrans110","title":"Transformations of Functions: Part B","body":"These problems are generally harder, often highlighting an important subtlety. Use the \\"safe order of transformations\\" for transforming $$y=f(x)$$ $$\\\\rightsquigarrow$$ $$y=A+B f{\\\\left(C x+D\\\\right)}$$ to give a sequence of five elementary transformations whose composition transforms $$y=f(x)$$ into $$y=-8+\\\\frac{1}{4} f{\\\\left(\\\\left(-\\\\frac{1}{2}\\\\right) x-3\\\\right)}$$. The first step is completed for you.\\\\n\\\\nStart with the original: $$y=f(x)$$\\\\n\\\\nFirst translate right $$3$$ units, $$y=f(x-3)$$","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Transformations of Functions","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a4593e0functrans110a","stepAnswer":["Stretch horizontally by $$2$$"],"problemType":"MultipleChoice","stepTitle":"What describes the second step: $$f{\\\\left(\\\\frac{1}{2} x-3\\\\right)}$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Stretch horizontally by $$2$$","choices":["Stretch horizontally by $$2$$","Compress horizontally by $$2$$","Stretch vertically by $$2$$","Compress vertically by $$2$$"],"hints":{"DefaultPathway":[{"id":"a4593e0functrans110a-h1","type":"hint","dependencies":[],"title":"Input or Output?","text":"A transformation that affects the input is within the function paranetheses, while a transformation that affects the output is outside the function paranetheses.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans110a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Input"],"dependencies":["a4593e0functrans110a-h1"],"title":"Input or Output?","text":"Does the step affect the input or output of the function?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Input","Output"]},{"id":"a4593e0functrans110a-h3","type":"hint","dependencies":["a4593e0functrans110a-h2"],"title":"Horizontal or Vertical?","text":"A $$\\\\frac{stretch}{compression}$$ that affects the input is horizontal, while a $$\\\\frac{stretch}{compression}$$ that affects the output is vertical.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans110a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Horizontal"],"dependencies":["a4593e0functrans110a-h3"],"title":"Horizontal or Vertical?","text":"Is the $$\\\\frac{stretch}{compression}$$ horizontal or vertical?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Horizontal","Vertical"]},{"id":"a4593e0functrans110a-h5","type":"hint","dependencies":["a4593e0functrans110a-h4"],"title":"Understanding $$\\\\frac{Stretch}{Compression}$$","text":"A horizontal $$\\\\frac{stretch}{compression}$$ means that the input is multiplied by some value b: $$f{\\\\left(b x\\\\right)}$$. Compressing multiplies $$b$$, while stretching multiplies $$\\\\frac{1}{b}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans110a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Stretch"],"dependencies":["a4593e0functrans110a-h5"],"title":"Understanding $$\\\\frac{Stretch}{Compression}$$","text":"Is the horizontal transformation a stretch or compression?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Stretch","Compression"]}]}},{"id":"a4593e0functrans110b","stepAnswer":["Reflect across the $$y-axis$$"],"problemType":"MultipleChoice","stepTitle":"What describes the third step: $$f\\\\left(-\\\\frac{1}{2} x-3\\\\right)$$?","stepBody":"","answerType":"string","variabilization":{},"choices":["Reflect across the $$y-axis$$","Reflect across the $$x-axis$$"],"hints":{"DefaultPathway":[{"id":"a4593e0functrans110b-h1","type":"hint","dependencies":[],"title":"Input or Output?","text":"A transformation that affects the input is within the function paranetheses, while a transformation that affects the output is outside the function paranetheses.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans110b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Input"],"dependencies":["a4593e0functrans110b-h1"],"title":"Input or Output?","text":"Does the step affect the input or output of the function?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Input","Output"]},{"id":"a4593e0functrans110b-h3","type":"hint","dependencies":["a4593e0functrans110b-h2"],"title":"Understanding Reflections","text":"A reflection that affects the input is over the y-axis, while a reflection that affects the output is over the x-axis.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans110b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y-axis$$"],"dependencies":["a4593e0functrans110b-h3"],"title":"Understanding Reflections","text":"Is the reflection over the y-axis or x-axis?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$x-axis$$","$$y-axis$$"]}]}}]},{"id":"a4593e0functrans111","title":"Transformations of Functions: Part C","body":"These questions are challenging, requiring mastery of each concept and their interrelations. Fill out the sequence of transformations whose composition transforms $$y=3^x$$ into $$y=\\\\frac{14-3^{\\\\left(-5x-2\\\\right)}}{7}$$.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Transformations of Functions","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a4593e0functrans111a","stepAnswer":["$$2-\\\\frac{1}{7} f\\\\left(-5x+2\\\\right)$$"],"problemType":"MultipleChoice","stepTitle":"Rewrite the function in terms of f(x).","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2-\\\\frac{1}{7} f\\\\left(-5x+2\\\\right)$$","choices":["$$2-\\\\frac{1}{7} f\\\\left(-5x+2\\\\right)$$","$$2-\\\\frac{1}{7} 3^{\\\\left(-5f{\\\\left(x\\\\right)}+2\\\\right)}$$","$$\\\\frac{2-f\\\\left(-5x+2\\\\right)}{7}$$","$$2-3^{f\\\\left(-5x+2\\\\right)}$$"],"hints":{"DefaultPathway":[{"id":"a4593e0functrans111a-h1","type":"hint","dependencies":[],"title":"Simplifying the Expression","text":"For some a,b,c not equal to $$0$$, $$\\\\frac{a+b}{c}=\\\\frac{a}{c}+\\\\frac{b}{c}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans111a-h2","type":"hint","dependencies":["a4593e0functrans111a-h1"],"title":"Simplifying the Expression","text":"Split the expression into individual parts to simplify: $$\\\\frac{14}{7}-\\\\frac{1}{7} 3^{\\\\left(-5x+2\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans111a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a4593e0functrans111a-h2"],"title":"Simplifying the Expression","text":"What is $$\\\\frac{14}{7}$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans111a-h4","type":"hint","dependencies":["a4593e0functrans111a-h3"],"title":"Simplifying the Expression","text":"The term $$3^{\\\\left(-5x+2\\\\right)}$$ can be rewritten in terms of $$f(x)=3^x$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans111a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5x+2$$"],"dependencies":["a4593e0functrans111a-h4"],"title":"Simplifying the Expression","text":"What value of $$x$$ can be substituted to make $$3^x$$ equal to $$3^{\\\\left(-5x+2\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans111a-h6","type":"hint","dependencies":["a4593e0functrans111a-h5"],"title":"Simplifying the Expression","text":"If $$f\\\\left(-5x+2\\\\right)=3^{\\\\left(-5x+2\\\\right)}$$, then $$\\\\frac{1}{7} 3^{\\\\left(-5x+2\\\\right)}=\\\\frac{1}{7} f\\\\left(-5x+2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]}},{"id":"a4593e0functrans111b","stepAnswer":["Translate left by $$2$$"],"problemType":"MultipleChoice","stepTitle":"What step can be taken to transform f(x) to $$f{\\\\left(x+2\\\\right)}$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Translate left by $$2$$","choices":["Translate left by $$2$$","Translate right by $$2$$","Translate up by $$2$$","Translate down by $$2$$"],"hints":{"DefaultPathway":[{"id":"a4593e0functrans111b-h1","type":"hint","dependencies":[],"title":"Input or Output?","text":"A transformation that affects the input is within the function paranetheses, while a transformation that affects the output is outside the function paranetheses.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans111b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Input"],"dependencies":["a4593e0functrans111b-h1"],"title":"Input or Output?","text":"Does the step affect the input or output of the function?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Input","Output"]},{"id":"a4593e0functrans111b-h3","type":"hint","dependencies":["a4593e0functrans111b-h2"],"title":"Horizontal or Vertical?","text":"A translation that affects the input is horizontal, while a translation that affects the output is vertical.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans111b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Horizontal"],"dependencies":["a4593e0functrans111b-h3"],"title":"Horizontal or Vertical?","text":"Is the translation horizontal or vertical?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Horizontal","Vertical"]},{"id":"a4593e0functrans111b-h5","type":"hint","dependencies":["a4593e0functrans111b-h4"],"title":"Understanding Translation","text":"A hoirzontal translation means that the input is added by some value b: $$f{\\\\left(x+b\\\\right)}$$. Translating left adds $$b$$, while translating right subtracts $$b$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans111b-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Left"],"dependencies":["a4593e0functrans111b-h5"],"title":"Understanding Translation","text":"Is the horizontal translation left or right?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Left","Right"]}]}}]},{"id":"a4593e0functrans2","title":"Transformations of Functions: Part A","body":"These questions test your knowledge of the core concepts. Below is the graph of $$y=f(x)$$ represented in $$\\\\frac{red}{solid}$$, where the domain of f is [0,4]. Choose the graph representing the functions below.\\\\n##figure3.gif","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Transformations of Functions","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a4593e0functrans2a","stepAnswer":["$$\\\\frac{Purple}{Bold}-Dashed$$"],"problemType":"MultipleChoice","stepTitle":"$$y=f{\\\\left(\\\\frac{x}{2}\\\\right)}$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{Purple}{Bold}-Dashed$$","choices":["$$\\\\frac{Blue}{Dashed}$$","$$\\\\frac{Green}{Dotted}$$","$$\\\\frac{Orange}{Bold}-Dotted$$","$$\\\\frac{Purple}{Bold}-Dashed$$"],"hints":{"DefaultPathway":[{"id":"a4593e0functrans2a-h1","type":"hint","dependencies":[],"title":"Understanding $$f{\\\\left(\\\\frac{x}{h}\\\\right)}$$","text":"For some value $$h$$ not equal to $$0$$, $$f{\\\\left(\\\\frac{x}{h}\\\\right)}$$ means that f(x) is stretched horizontally by $$h$$. Similarly, $$f{\\\\left(h x\\\\right)}$$ means that f(x) is compressed horizontally by $$h$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans2a-h2","type":"hint","dependencies":["a4593e0functrans2a-h1"],"title":"Understanding $$f{\\\\left(\\\\frac{x}{h}\\\\right)}$$","text":"Since $$h$$ is $$\\\\frac{1}{2}$$, $$f{\\\\left(\\\\frac{x}{2}\\\\right)}$$ means that the graph is stretched horizontally by $$2$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans2a-h3","type":"hint","dependencies":["a4593e0functrans2a-h2"],"title":"Understanding $$f{\\\\left(\\\\frac{x}{h}\\\\right)}$$","text":"$$f{\\\\left(\\\\frac{x}{2}\\\\right)}$$ means that $$f(0)=2$$, $$f(4)=0$$, and $$f(8)=1$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]}},{"id":"a4593e0functrans2b","stepAnswer":["$$\\\\frac{Orange}{Bold}-Dotted$$"],"problemType":"MultipleChoice","stepTitle":"$$y=f(x)-2$$","stepBody":"##figure2.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{Orange}{Bold}-Dotted$$","choices":["$$\\\\frac{Blue}{Dashed}$$","$$\\\\frac{Green}{Dotted}$$","$$\\\\frac{Orange}{Bold}-Dotted$$","$$\\\\frac{Purple}{Bold}-Dashed$$"],"hints":{"DefaultPathway":[{"id":"a4593e0functrans2b-h1","type":"hint","dependencies":[],"title":"Understanding $$f{\\\\left(x\\\\right)}+h$$","text":"For some value $$h$$, $$f{\\\\left(x\\\\right)}+h$$ means that f(x) is translated up by $$h$$ units. Similarly, $$f(x)-h$$ means that f(x) is translated down by $$h$$ units.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans2b-h2","type":"hint","dependencies":["a4593e0functrans2b-h1"],"title":"Understanding $$f{\\\\left(x\\\\right)}+h$$","text":"Since $$h$$ is $$-2$$, $$f(x)-2$$ means that the graph is translated down by $$2$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans2b-h3","type":"hint","dependencies":["a4593e0functrans2b-h2"],"title":"Understanding $$f{\\\\left(x\\\\right)}+h$$","text":"$$f(x)-2$$ means that $$f(0)=0$$, $$f(2)=-2$$, and $$f(4)=-1$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a4593e0functrans200","title":"Transformations of Functions: Part B","body":"These problems are generally harder, often highlighting an important subtlety. On the graph below, the red dotted graph is $$y=h(x)$$. Write a formula for each of the other functions in terms of h(x).\\\\n##figure1.gif","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Transformations of Functions","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a4593e0functrans200a","stepAnswer":["Translate left by $$3$$, then stretch vertically by $$2$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{Orange}{Solid}$$: $$y=2h{\\\\left(x+3\\\\right)}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Translate left by $$3$$, then stretch vertically by $$2$$","choices":["Translate left by $$3$$, then stretch vertically by $$2$$","Translate right by $$3$$, then stretch vertically by $$2$$","Translate left by $$3$$, then stretch horizontally by $$2$$","Translate right by $$3$$, then stretch horizontally by $$2$$"],"hints":{"DefaultPathway":[{"id":"a4593e0functrans200a-h1","type":"hint","dependencies":[],"title":"Understanding $$f(x-a)$$","text":"For some value a, $$f(x-a)$$ means that f(x) is translated to the right by a units. Similarly, $$f{\\\\left(x+a\\\\right)}$$ means that f(x) is translated to the left by a units.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans200a-h2","type":"hint","dependencies":["a4593e0functrans200a-h1"],"title":"Understanding $$f(x-a)$$","text":"Since a is $$-3$$, $$h{\\\\left(x+3\\\\right)}$$ means that the graph is translated left by $$3$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans200a-h3","type":"hint","dependencies":["a4593e0functrans200a-h2"],"title":"Understanding $$a f{\\\\left(x\\\\right)}$$","text":"For some value a not equal to $$0$$, $$a f{\\\\left(x\\\\right)}$$ means that f(x) is stretched vertically by a. Similarly, $$\\\\frac{f{\\\\left(x\\\\right)}}{a}$$ means that f(x) is compressed vertically by a.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans200a-h4","type":"hint","dependencies":["a4593e0functrans200a-h3"],"title":"Understanding $$a f{\\\\left(x\\\\right)}$$","text":"Since a is $$2$$, $$2h{\\\\left(x+3\\\\right)}$$ means that the graph is stretched vertically by $$2$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a4593e0functrans3","title":"Transformations of Functions: Part A","body":"These questions test your knowledge of the core concepts. Below is the graph of $$y=f(x)$$ represented in $$\\\\frac{red}{solid}$$, where the domain of f is [0,4]. Choose the graph representing the functions below.\\\\n##figure2.gif","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Transformations of Functions","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a4593e0functrans3a","stepAnswer":["$$\\\\frac{Green}{Dotted}$$"],"problemType":"MultipleChoice","stepTitle":"$$y=2f{\\\\left(x+4\\\\right)}$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{Green}{Dotted}$$","choices":["$$\\\\frac{Blue}{Dashed}$$","$$\\\\frac{Green}{Dotted}$$","$$\\\\frac{Orange}{Bold}-Dotted$$","$$\\\\frac{Purple}{Bold}-Dashed$$"],"hints":{"DefaultPathway":[{"id":"a4593e0functrans3a-h1","type":"hint","dependencies":[],"title":"Understanding $$f(x-h)$$","text":"For some value $$h$$, $$f(x-h)$$ means that f(x) is translated to the right by $$h$$ units. Similarly, $$f{\\\\left(x+h\\\\right)}$$ means that f(x) is translated to the left by $$h$$ units.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans3a-h2","type":"hint","dependencies":["a4593e0functrans3a-h1"],"title":"Understanding $$f(x-h)$$","text":"Since $$h$$ is $$-4$$, $$f{\\\\left(x+4\\\\right)}$$ means that the graph is translated four units to the left.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans3a-h3","type":"hint","dependencies":["a4593e0functrans3a-h2"],"title":"Understanding $$f(x-h)$$","text":"$$f{\\\\left(x+4\\\\right)}$$ means that $$f(-4)=2$$, $$f(-2)=0$$, and $$f(0)=1$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans3a-h4","type":"hint","dependencies":["a4593e0functrans3a-h3"],"title":"Understanding $$h f{\\\\left(x\\\\right)}$$","text":"For some value $$h$$ not equal to $$0$$, $$h f{\\\\left(x\\\\right)}$$ means that f(x) is stretched vertically by $$h$$. Similarly, $$\\\\frac{f{\\\\left(x\\\\right)}}{h}$$ means that f(x) is compressed vertically by $$h$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans3a-h5","type":"hint","dependencies":["a4593e0functrans3a-h4"],"title":"Understanding $$h f{\\\\left(x\\\\right)}$$","text":"Since $$h$$ is $$2$$, $$2f{\\\\left(x\\\\right)}$$ means that the graph is stretched vertically by $$2$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans3a-h6","type":"hint","dependencies":["a4593e0functrans3a-h5"],"title":"Understanding $$h f{\\\\left(x\\\\right)}$$","text":"$$2f{\\\\left(x+4\\\\right)}$$ means that $$f(-4)=4$$, $$f(-2)=0$$, and $$f(0)=2$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a4593e0functrans310","title":"Transformations of Functions: Part B","body":"These problems are generally harder, often highlighting an important subtlety. Use the \\"safe order of transformations\\" for transforming $$y=f(x)$$ $$\\\\rightsquigarrow$$ $$y=A+B f{\\\\left(C x+D\\\\right)}$$ to give a sequence of five elementary transformations whose composition transforms $$y=f(x)$$ into $$y=-8+\\\\frac{1}{4} f{\\\\left(\\\\left(-\\\\frac{1}{2}\\\\right) x-3\\\\right)}$$. 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Stretching multiplies $$b$$, while compressing multiplies $$\\\\frac{1}{b}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans310a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Compression"],"dependencies":["a4593e0functrans310a-h5"],"title":"Understanding $$\\\\frac{Stretch}{Compression}$$","text":"Is the vertical transformation a stretch or compression?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Stretch","Compression"]}]}},{"id":"a4593e0functrans310b","stepAnswer":["Translate down by $$8$$"],"problemType":"MultipleChoice","stepTitle":"What describes the final step: $$-8+\\\\frac{1}{4} f\\\\left(-\\\\frac{1}{2} x-3\\\\right)$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Translate down by $$8$$","choices":["Translate down by $$8$$","Translate up by $$8$$","Translate right by $$8$$","Translate left by $$8$$"],"hints":{"DefaultPathway":[{"id":"a4593e0functrans310b-h1","type":"hint","dependencies":[],"title":"Input or Output?","text":"A transformation that affects the input is within the function paranetheses, while a transformation that affects the output is outside the function paranetheses.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans310b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Output"],"dependencies":["a4593e0functrans310b-h1"],"title":"Input or Output?","text":"Does the step affect the input or output of the function?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Input","Output"]},{"id":"a4593e0functrans310b-h3","type":"hint","dependencies":["a4593e0functrans310b-h2"],"title":"Horizontal or Vertical?","text":"A translation that affects the input is horizontal, while a translation that affects the output is vertical.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans310b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Vertical"],"dependencies":["a4593e0functrans310b-h3"],"title":"Horizontal or Vertical?","text":"Is the translation horizontal or vertical?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Horizontal","Vertical"]},{"id":"a4593e0functrans310b-h5","type":"hint","dependencies":["a4593e0functrans310b-h4"],"title":"Understanding Translation","text":"A vertical translation means that the output is added by some value b: $$f{\\\\left(x\\\\right)}+b$$. 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Fill out the sequence of transformations whose composition transforms $$y=3^x$$ into $$y=\\\\frac{14-3^{\\\\left(-5x-2\\\\right)}}{7}$$.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Transformations of Functions","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a4593e0functrans411a","stepAnswer":["Compress horizontally by $$5$$"],"problemType":"MultipleChoice","stepTitle":"What step can be taken to transform $$f{\\\\left(x+2\\\\right)}$$ to $$f{\\\\left(5x+2\\\\right)}$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Compress horizontally by $$5$$","choices":["Compress horizontally by $$5$$","Stretch horizontally by $$5$$","Compress vertically by $$5$$","Stretch vertically by $$5$$"],"hints":{"DefaultPathway":[{"id":"a4593e0functrans411a-h1","type":"hint","dependencies":[],"title":"Input or Output?","text":"A transformation that affects the input is within the function paranetheses, while a transformation that affects the output is outside the function paranetheses.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans411a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Input"],"dependencies":["a4593e0functrans411a-h1"],"title":"Input or Output?","text":"Does the step affect the input or output of the function?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Input","Output"]},{"id":"a4593e0functrans411a-h3","type":"hint","dependencies":["a4593e0functrans411a-h2"],"title":"Horizontal or Vertical?","text":"A $$\\\\frac{stretch}{compression}$$ that affects the input is horizontal, while a $$\\\\frac{stretch}{compression}$$ that affects the output is vertical.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans411a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Horizontal"],"dependencies":["a4593e0functrans411a-h3"],"title":"Horizontal or Vertical?","text":"Is the $$\\\\frac{stretch}{compression}$$ horizontal or vertical?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Horizontal","Vertical"]},{"id":"a4593e0functrans411a-h5","type":"hint","dependencies":["a4593e0functrans411a-h4"],"title":"Understanding $$\\\\frac{Stretch}{Compression}$$","text":"A horizontal $$\\\\frac{stretch}{compression}$$ means that the input is multiplied by some value b: $$f{\\\\left(b x\\\\right)}$$. Compressing multiplies $$b$$, while stretching multiplies $$\\\\frac{1}{b}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans411a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Compression"],"dependencies":["a4593e0functrans411a-h5"],"title":"Understanding $$\\\\frac{Stretch}{Compression}$$","text":"Is the horizontal transformation a stretch or compression?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Stretch","Compression"]}]}},{"id":"a4593e0functrans411b","stepAnswer":["Reflect across $$y-axis$$"],"problemType":"MultipleChoice","stepTitle":"What step can be taken to transform $$f{\\\\left(5x+2\\\\right)}$$ to $$f\\\\left(-5x+2\\\\right)$$?","stepBody":"","answerType":"string","variabilization":{},"choices":["Reflect across $$y-axis$$","Reflect across $$x-axis$$"],"hints":{"DefaultPathway":[{"id":"a4593e0functrans411b-h1","type":"hint","dependencies":[],"title":"Input or Output?","text":"A transformation that affects the input is within the function paranetheses, while a transformation that affects the output is outside the function paranetheses.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans411b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Input"],"dependencies":["a4593e0functrans411b-h1"],"title":"Input or Output?","text":"Does the step affect the input or output of the function?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Input","Output"]},{"id":"a4593e0functrans411b-h3","type":"hint","dependencies":["a4593e0functrans411b-h2"],"title":"Understanding Reflections","text":"A reflection that affects the input is over the y-axis, while a reflection that affects the output is over the x-axis.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans411b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y-axis$$"],"dependencies":["a4593e0functrans411b-h3"],"title":"Understanding Reflections","text":"Is the reflection over the y-axis or x-axis?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$x-axis$$","$$y-axis$$"]}]}}]},{"id":"a4593e0functrans511","title":"Transformations of Functions: Part C","body":"These questions are challenging, requiring mastery of each concept and their interrelations. Fill out the sequence of transformations whose composition transforms $$y=3^x$$ into $$y=\\\\frac{14-3^{\\\\left(-5x-2\\\\right)}}{7}$$.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Transformations of Functions","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a4593e0functrans511a","stepAnswer":["Reflect across $$x-axis$$"],"problemType":"MultipleChoice","stepTitle":"What step can be taken to transform $$f\\\\left(-5x+2\\\\right)$$ to $$-f\\\\left(-5x+2\\\\right)$$?","stepBody":"","answerType":"string","variabilization":{},"choices":["Reflect across $$y-axis$$","Reflect across $$x-axis$$"],"hints":{"DefaultPathway":[{"id":"a4593e0functrans511a-h1","type":"hint","dependencies":[],"title":"Input or Output?","text":"A transformation that affects the input is within the function paranetheses, while a transformation that affects the output is outside the function paranetheses.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans511a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Output"],"dependencies":["a4593e0functrans511a-h1"],"title":"Input or Output?","text":"Does the step affect the input or output of the function?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Input","Output"]},{"id":"a4593e0functrans511a-h3","type":"hint","dependencies":["a4593e0functrans511a-h2"],"title":"Understanding Reflections","text":"A reflection that affects the input is over the y-axis, while a reflection that affects the output is over the x-axis.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans511a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x-axis$$"],"dependencies":["a4593e0functrans511a-h3"],"title":"Understanding Reflections","text":"Is the reflection over the y-axis or x-axis?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$x-axis$$","$$y-axis$$"]}]}},{"id":"a4593e0functrans511b","stepAnswer":["Compress vertically by $$7$$"],"problemType":"MultipleChoice","stepTitle":"What step can be taken to transform $$-f\\\\left(-5x+2\\\\right)$$ to $$-\\\\left(\\\\frac{1}{7}\\\\right) f\\\\left(-5x+2\\\\right)$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Compress vertically by $$7$$","choices":["Compress horizontally by $$7$$","Stretch horizontally by $$7$$","Compress vertically by $$7$$","Stretch vertically by $$7$$"],"hints":{"DefaultPathway":[{"id":"a4593e0functrans511b-h1","type":"hint","dependencies":[],"title":"Input or Output?","text":"A transformation that affects the input is within the function paranetheses, while a transformation that affects the output is outside the function paranetheses.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans511b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Output"],"dependencies":["a4593e0functrans511b-h1"],"title":"Input or Output?","text":"Does the step affect the input or output of the function?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Input","Output"]},{"id":"a4593e0functrans511b-h3","type":"hint","dependencies":["a4593e0functrans511b-h2"],"title":"Horizontal or Vertical?","text":"A $$\\\\frac{stretch}{compression}$$ that affects the input is horizontal, while a $$\\\\frac{stretch}{compression}$$ that affects the output is vertical.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans511b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Vertical"],"dependencies":["a4593e0functrans511b-h3"],"title":"Horizontal or Vertical?","text":"Is the $$\\\\frac{stretch}{compression}$$ horizontal or vertical?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Horizontal","Vertical"]},{"id":"a4593e0functrans511b-h5","type":"hint","dependencies":["a4593e0functrans511b-h4"],"title":"Understanding $$\\\\frac{Stretch}{Compression}$$","text":"A vertical $$\\\\frac{stretch}{compression}$$ means that the output is multiplied by some value b: $$f{\\\\left(b x\\\\right)}$$. Stretching multiplies $$b$$, while compressing multiplies $$\\\\frac{1}{b}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans511b-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Compression"],"dependencies":["a4593e0functrans511b-h5"],"title":"Understanding $$\\\\frac{Stretch}{Compression}$$","text":"Is the vertical transformation a stretch or compression?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Stretch","Compression"]}]}},{"id":"a4593e0functrans511c","stepAnswer":["Translate up by $$2$$"],"problemType":"MultipleChoice","stepTitle":"What step can be taken to transform $$-\\\\left(\\\\frac{1}{7}\\\\right) f\\\\left(-5x+2\\\\right)$$ to $$2-\\\\frac{1}{7} f\\\\left(-5x+2\\\\right)$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Translate up by $$2$$","choices":["Translate up by $$2$$","Translate down by $$2$$","Translate left by $$2$$","Translate right by $$2$$"],"hints":{"DefaultPathway":[{"id":"a4593e0functrans511c-h1","type":"hint","dependencies":[],"title":"Input or Output?","text":"A transformation that affects the input is within the function paranetheses, while a transformation that affects the output is outside the function paranetheses.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans511c-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Output"],"dependencies":["a4593e0functrans511c-h1"],"title":"Input or Output?","text":"Does the step affect the input or output of the function?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Input","Output"]},{"id":"a4593e0functrans511c-h3","type":"hint","dependencies":["a4593e0functrans511c-h2"],"title":"Horizontal or Vertical?","text":"A translation that affects the input is horizontal, while a translation that affects the output is vertical.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans511c-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Vertical"],"dependencies":["a4593e0functrans511c-h3"],"title":"Horizontal or Vertical?","text":"Is the translation horizontal or vertical?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Horizontal","Vertical"]},{"id":"a4593e0functrans511c-h5","type":"hint","dependencies":["a4593e0functrans511c-h4"],"title":"Understanding Translation","text":"A vertical translation means that the output is added by some value b: $$f{\\\\left(x\\\\right)}+b$$. Translating up adds $$b$$, while translating down subtracts $$b$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4593e0functrans511c-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Up"],"dependencies":["a4593e0functrans511c-h5"],"title":"Understanding Translation","text":"Is the vertical translation up or down?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Up","Down"]}]}}]},{"id":"a47ce72graphquad1","title":"Graphing Quadratic Functions Using Transformations","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Graph Quadratic Functions Using Transformations","courseName":"OpenStax: Intermediate 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vertical transformation down.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a47ce72graphquad10","title":"Graphing Quadratic Functions Using Transformations","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Graph Quadratic Functions Using Transformations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a47ce72graphquad10a","stepAnswer":["D"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=x^2+2$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a47ce72graphquad10a-h1","type":"hint","dependencies":[],"title":"Transformation","text":"Is this a vertical or a horizontal transformation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad10a-h2","type":"hint","dependencies":["a47ce72graphquad10a-h1"],"title":"Vertical Transformation","text":"Since this is a vertical transformation, is it up or down?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad10a-h3","type":"hint","dependencies":["a47ce72graphquad10a-h2"],"title":"Up Transformation","text":"Since it is $$2$$, it\'s a vertical transformation up.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a47ce72graphquad11","title":"Graphing Quadratic Functions Using Transformations","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad11a-h3","type":"hint","dependencies":["a47ce72graphquad11a-h2"],"title":"Left Transformation","text":"Since it is $$2$$, it\'s a horizontal transformation to the left.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a47ce72graphquad12","title":"Graphing Quadratic Functions Using Transformations","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Graph Quadratic Functions Using Transformations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a47ce72graphquad12a","stepAnswer":["B"],"problemType":"MultipleChoice","stepTitle":"$$f(x)={\\\\left(x-2\\\\right)}^2$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a47ce72graphquad12a-h1","type":"hint","dependencies":[],"title":"Transformation","text":"Is this a vertical or a horizontal transformation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad12a-h2","type":"hint","dependencies":["a47ce72graphquad12a-h1"],"title":"Horizontal Transformation","text":"Since this is a horizontal transformation, is it to the left or right?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad12a-h3","type":"hint","dependencies":["a47ce72graphquad12a-h2"],"title":"Right Transformation","text":"Since it is $$-2$$, it\'s a horizontal transformation to the right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a47ce72graphquad13","title":"Graphing Quadratic Functions Using Transformations","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Graph Quadratic Functions Using Transformations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a47ce72graphquad13a","stepAnswer":["B"],"problemType":"MultipleChoice","stepTitle":"$$f(x)={\\\\left(x+2\\\\right)}^2+1$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a47ce72graphquad13a-h1","type":"hint","dependencies":[],"title":"Transformations","text":"Think about the kinds of transformations that are occuring.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad13a-h2","type":"hint","dependencies":["a47ce72graphquad13a-h1"],"title":"Equation","text":"Think about the equation in the form of $${\\\\left(a \\\\left(x+b\\\\right)\\\\right)}^2+k$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad13a-h3","type":"hint","dependencies":["a47ce72graphquad13a-h2"],"title":"Vertical Transformation","text":"Which variable represents the vertical transformation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad13a-h4","type":"hint","dependencies":["a47ce72graphquad13a-h3"],"title":"Vertical Transformation","text":"Since k represents the vertical transformation, what is k in our case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad13a-h5","type":"hint","dependencies":["a47ce72graphquad13a-h4"],"title":"Vertical Transformation","text":"Since our k is $$1$$, is this a vertical shift up or down?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad13a-h6","type":"hint","dependencies":["a47ce72graphquad13a-h5"],"title":"Up Transformation","text":"It is a vertical shift up by $$1$$, since our k is $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad13a-h7","type":"hint","dependencies":["a47ce72graphquad13a-h6"],"title":"Horizontal Transformation","text":"Which variable represents the horizontal transformation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad13a-h8","type":"hint","dependencies":["a47ce72graphquad13a-h7"],"title":"Horizontal Transformation","text":"Since $$b$$ represents the horizontal transformation, what is $$b$$ in our case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad13a-h9","type":"hint","dependencies":["a47ce72graphquad13a-h8"],"title":"Horizontal Transformation","text":"Since our $$b$$ is $$2$$, is this a horizontal shift to the left or right?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad13a-h10","type":"hint","dependencies":["a47ce72graphquad13a-h9"],"title":"Left Transformation","text":"It is a horizontal shift to the left by $$2$$, since $$b$$ is $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad13a-h11","type":"hint","dependencies":["a47ce72graphquad13a-h10"],"title":"Width","text":"Which variable represents the width of our graph?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad13a-h12","type":"hint","dependencies":["a47ce72graphquad13a-h11"],"title":"Width","text":"Since a represents the width of our graph, what is a in this case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad13a-h13","type":"hint","dependencies":["a47ce72graphquad13a-h12"],"title":"Width","text":"Since a is $$1$$, what does that tell us about the width? Will it be skinnier, wider, or normal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad13a-h14","type":"hint","dependencies":["a47ce72graphquad13a-h13"],"title":"Normal Width","text":"Since a is $$1$$, it will have a normal width.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a47ce72graphquad14","title":"Graphing Quadratic Functions Using Transformations","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Graph Quadratic Functions Using Transformations","courseName":"OpenStax: Intermediate 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4.0>"},{"id":"a47ce72graphquad14a-h3","type":"hint","dependencies":["a47ce72graphquad14a-h2"],"title":"Vertical Transformation","text":"Which variable represents the vertical transformation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad14a-h4","type":"hint","dependencies":["a47ce72graphquad14a-h3"],"title":"Vertical Transformation","text":"Since k represents the vertical transformation, what is k in our case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad14a-h5","type":"hint","dependencies":["a47ce72graphquad14a-h4"],"title":"Vertical Transformation","text":"Since our k is $$-1$$, is this a vertical shift up or down?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad14a-h6","type":"hint","dependencies":["a47ce72graphquad14a-h5"],"title":"Down Transformation","text":"It is a vertical shift down by $$1$$, since our k is $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad14a-h7","type":"hint","dependencies":["a47ce72graphquad14a-h6"],"title":"Horizontal Transformation","text":"Which variable represents the horizontal transformation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad14a-h8","type":"hint","dependencies":["a47ce72graphquad14a-h7"],"title":"Horizontal Transformation","text":"Since $$b$$ represents the horizontal transformation, what is $$b$$ in our case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad14a-h9","type":"hint","dependencies":["a47ce72graphquad14a-h8"],"title":"Horizontal Transformation","text":"Since our $$b$$ is $$2$$, is this a horizontal shift to the left or right?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad14a-h10","type":"hint","dependencies":["a47ce72graphquad14a-h9"],"title":"Left Transformation","text":"It is a horizontal shift to the left by $$2$$, since $$b$$ is $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad14a-h11","type":"hint","dependencies":["a47ce72graphquad14a-h10"],"title":"Width","text":"Which variable represents the width of our graph?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad14a-h12","type":"hint","dependencies":["a47ce72graphquad14a-h11"],"title":"Width","text":"Since a represents the width of our graph, what is a in this case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad14a-h13","type":"hint","dependencies":["a47ce72graphquad14a-h12"],"title":"Width","text":"Since a is $$1$$, what does that tell us about the width? Will it be skinnier, wider, or normal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad14a-h14","type":"hint","dependencies":["a47ce72graphquad14a-h13"],"title":"Normal Width","text":"Since a is $$1$$, it will have a normal width.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a47ce72graphquad15","title":"Graphing Quadratic Functions Using Transformations","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Graph Quadratic Functions Using Transformations","courseName":"OpenStax: Intermediate 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4.0>"},{"id":"a47ce72graphquad15a-h3","type":"hint","dependencies":["a47ce72graphquad15a-h2"],"title":"Vertical Transformation","text":"Which variable represents the vertical transformation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad15a-h4","type":"hint","dependencies":["a47ce72graphquad15a-h3"],"title":"Vertical Transformation","text":"Since k represents the vertical transformation, what is k in our case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad15a-h5","type":"hint","dependencies":["a47ce72graphquad15a-h4"],"title":"Vertical Transformation","text":"Since our k is $$-1$$, is this a vertical shift up or down?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad15a-h6","type":"hint","dependencies":["a47ce72graphquad15a-h5"],"title":"Down Transformation","text":"It is a vertical shift down by $$1$$, since our k is $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad15a-h7","type":"hint","dependencies":["a47ce72graphquad15a-h6"],"title":"Horizontal Transformation","text":"Which variable represents the horizontal transformation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad15a-h8","type":"hint","dependencies":["a47ce72graphquad15a-h7"],"title":"Horizontal Transformation","text":"Since $$b$$ represents the horizontal transformation, what is $$b$$ in our case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad15a-h9","type":"hint","dependencies":["a47ce72graphquad15a-h8"],"title":"Horizontal Transformation","text":"Since our $$b$$ is $$-2$$, is this a horizontal shift to the left or right?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad15a-h10","type":"hint","dependencies":["a47ce72graphquad15a-h9"],"title":"Right Transformation","text":"It is a horizontal shift to the right by $$2$$, since $$b$$ is $$-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad15a-h11","type":"hint","dependencies":["a47ce72graphquad15a-h10"],"title":"Width","text":"Which variable represents the width of our graph?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad15a-h12","type":"hint","dependencies":["a47ce72graphquad15a-h11"],"title":"Width","text":"Since a represents the width of our graph, what is a in this case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad15a-h13","type":"hint","dependencies":["a47ce72graphquad15a-h12"],"title":"Width","text":"Since a is $$1$$, what does that tell us about the width? Will it be skinnier, wider, or normal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad15a-h14","type":"hint","dependencies":["a47ce72graphquad15a-h13"],"title":"Normal Width","text":"Since a is $$1$$, it will have a normal width.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a47ce72graphquad16","title":"Graphing Quadratic Functions Using Transformations","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Graph Quadratic Functions Using Transformations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a47ce72graphquad16a","stepAnswer":["C"],"problemType":"MultipleChoice","stepTitle":"$$f(x)={\\\\left(x-2\\\\right)}^2+1$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a47ce72graphquad16a-h1","type":"hint","dependencies":[],"title":"Transformations","text":"Think about the kinds of transformations that are occuring.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad16a-h2","type":"hint","dependencies":["a47ce72graphquad16a-h1"],"title":"Equation","text":"Think about the equation in the form of $${\\\\left(a \\\\left(x+b\\\\right)\\\\right)}^2+k$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad16a-h3","type":"hint","dependencies":["a47ce72graphquad16a-h2"],"title":"Vertical Transformation","text":"Which variable represents the vertical transformation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad16a-h4","type":"hint","dependencies":["a47ce72graphquad16a-h3"],"title":"Vertical Transformation","text":"Since k represents the vertical transformation, what is k in our case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad16a-h5","type":"hint","dependencies":["a47ce72graphquad16a-h4"],"title":"Vertical Transformation","text":"Since our k is $$1$$, is this a vertical shift up or down?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad16a-h6","type":"hint","dependencies":["a47ce72graphquad16a-h5"],"title":"Up Transformation","text":"It is a vertical shift up by $$1$$, since our k is $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad16a-h7","type":"hint","dependencies":["a47ce72graphquad16a-h6"],"title":"Horizontal Transformation","text":"Which variable represents the horizontal transformation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad16a-h8","type":"hint","dependencies":["a47ce72graphquad16a-h7"],"title":"Horizontal Transformation","text":"Since $$b$$ represents the horizontal transformation, what is $$b$$ in our case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad16a-h9","type":"hint","dependencies":["a47ce72graphquad16a-h8"],"title":"Horizontal Transformation","text":"Since our $$b$$ is $$-2$$, is this a horizontal shift to the left or right?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad16a-h10","type":"hint","dependencies":["a47ce72graphquad16a-h9"],"title":"Right Transformation","text":"It is a horizontal shift to the right by $$2$$, since $$b$$ is $$-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad16a-h11","type":"hint","dependencies":["a47ce72graphquad16a-h10"],"title":"Width","text":"Which variable represents the width of our graph?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad16a-h12","type":"hint","dependencies":["a47ce72graphquad16a-h11"],"title":"Width","text":"Since a represents the width of our graph, what is a in this case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad16a-h13","type":"hint","dependencies":["a47ce72graphquad16a-h12"],"title":"Width","text":"Since a is $$1$$, what does that tell us about the width? Will it be skinnier, wider, or normal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad16a-h14","type":"hint","dependencies":["a47ce72graphquad16a-h13"],"title":"Normal Width","text":"Since a is $$1$$, it will have a normal width.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a47ce72graphquad17","title":"Graphing Quadratic Functions Using Transformations","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Graph Quadratic Functions Using Transformations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a47ce72graphquad17a","stepAnswer":["C"],"problemType":"MultipleChoice","stepTitle":"$$f(x)={\\\\left(x+4\\\\right)}^2+2$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a47ce72graphquad17a-h1","type":"hint","dependencies":[],"title":"Transformations","text":"Think about the kinds of transformations that are occuring.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad17a-h2","type":"hint","dependencies":["a47ce72graphquad17a-h1"],"title":"Equation","text":"Think about the equation in the form of $${\\\\left(a \\\\left(x+b\\\\right)\\\\right)}^2+k$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad17a-h3","type":"hint","dependencies":["a47ce72graphquad17a-h2"],"title":"Vertical Transformation","text":"Which variable represents the vertical transformation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad17a-h4","type":"hint","dependencies":["a47ce72graphquad17a-h3"],"title":"Vertical Transformation","text":"Since k represents the vertical transformation, what is k in our case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad17a-h5","type":"hint","dependencies":["a47ce72graphquad17a-h4"],"title":"Vertical Transformation","text":"Since our k is $$2$$, is this a vertical shift up or down?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad17a-h6","type":"hint","dependencies":["a47ce72graphquad17a-h5"],"title":"Up Transformation","text":"It is a vertical shift up by $$2$$, since our k is $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad17a-h7","type":"hint","dependencies":["a47ce72graphquad17a-h6"],"title":"Horizontal Transformation","text":"Which variable represents the horizontal transformation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad17a-h8","type":"hint","dependencies":["a47ce72graphquad17a-h7"],"title":"Horizontal Transformation","text":"Since $$b$$ represents the horizontal transformation, what is $$b$$ in our case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad17a-h9","type":"hint","dependencies":["a47ce72graphquad17a-h8"],"title":"Horizontal Transformation","text":"Since our $$b$$ is $$4$$, is this a horizontal shift to the left or right?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad17a-h10","type":"hint","dependencies":["a47ce72graphquad17a-h9"],"title":"Left Transformation","text":"It is a horizontal shift to the left by $$4$$, since $$b$$ is $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad17a-h11","type":"hint","dependencies":["a47ce72graphquad17a-h10"],"title":"Width","text":"Which variable represents the width of our graph?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad17a-h12","type":"hint","dependencies":["a47ce72graphquad17a-h11"],"title":"Width","text":"Since a represents the width of our graph, what is a in this case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad17a-h13","type":"hint","dependencies":["a47ce72graphquad17a-h12"],"title":"Width","text":"Since a is $$1$$, what does that tell us about the width? Will it be skinnier, wider, or normal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad17a-h14","type":"hint","dependencies":["a47ce72graphquad17a-h13"],"title":"Normal Width","text":"Since a is $$1$$, it will have a normal width.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a47ce72graphquad18","title":"Graphing Quadratic Functions Using Transformations","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Graph Quadratic Functions Using Transformations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a47ce72graphquad18a","stepAnswer":["B"],"problemType":"MultipleChoice","stepTitle":"$$f(x)={\\\\left(x+4\\\\right)}^2-2$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a47ce72graphquad18a-h1","type":"hint","dependencies":[],"title":"Transformations","text":"Think about the kinds of transformations that are occuring.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad18a-h2","type":"hint","dependencies":["a47ce72graphquad18a-h1"],"title":"Equation","text":"Think about the equation in the form of $${\\\\left(a \\\\left(x+b\\\\right)\\\\right)}^2+k$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad18a-h3","type":"hint","dependencies":["a47ce72graphquad18a-h2"],"title":"Vertical Transformation","text":"Which variable represents the vertical transformation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad18a-h4","type":"hint","dependencies":["a47ce72graphquad18a-h3"],"title":"Vertical Transformation","text":"Since k represents the vertical transformation, what is k in our case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad18a-h5","type":"hint","dependencies":["a47ce72graphquad18a-h4"],"title":"Vertical Transformation","text":"Since our k is $$-2$$, is this a vertical shift up or down?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad18a-h6","type":"hint","dependencies":["a47ce72graphquad18a-h5"],"title":"Down Transformation","text":"It is a vertical shift down by $$2$$, since our k is $$-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad18a-h7","type":"hint","dependencies":["a47ce72graphquad18a-h6"],"title":"Horizontal Transformation","text":"Which variable represents the horizontal transformation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad18a-h8","type":"hint","dependencies":["a47ce72graphquad18a-h7"],"title":"Horizontal Transformation","text":"Since $$b$$ represents the horizontal transformation, what is $$b$$ in our case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad18a-h9","type":"hint","dependencies":["a47ce72graphquad18a-h8"],"title":"Horizontal Transformation","text":"Since our $$b$$ is $$4$$, is this a horizontal shift to the left or right?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad18a-h10","type":"hint","dependencies":["a47ce72graphquad18a-h9"],"title":"Left Transformation","text":"It is a horizontal shift to the left by $$4$$, since $$b$$ is $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad18a-h11","type":"hint","dependencies":["a47ce72graphquad18a-h10"],"title":"Width","text":"Which variable represents the width of our graph?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad18a-h12","type":"hint","dependencies":["a47ce72graphquad18a-h11"],"title":"Width","text":"Since a represents the width of our graph, what is a in this case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad18a-h13","type":"hint","dependencies":["a47ce72graphquad18a-h12"],"title":"Width","text":"Since a is $$1$$, what does that tell us about the width? Will it be skinnier, wider, or normal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad18a-h14","type":"hint","dependencies":["a47ce72graphquad18a-h13"],"title":"Normal Width","text":"Since a is $$1$$, it will have a normal width.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a47ce72graphquad19","title":"Graphing Quadratic Functions Using Transformations","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Graph Quadratic Functions Using Transformations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a47ce72graphquad19a","stepAnswer":["A"],"problemType":"MultipleChoice","stepTitle":"$$f(x)={\\\\left(x-4\\\\right)}^2-2$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a47ce72graphquad19a-h1","type":"hint","dependencies":[],"title":"Transformations","text":"Think about the kinds of transformations that are occuring.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad19a-h2","type":"hint","dependencies":["a47ce72graphquad19a-h1"],"title":"Equation","text":"Think about the equation in the form of $${\\\\left(a \\\\left(x+b\\\\right)\\\\right)}^2+k$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad19a-h3","type":"hint","dependencies":["a47ce72graphquad19a-h2"],"title":"Vertical Transformation","text":"Which variable represents the vertical transformation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad19a-h4","type":"hint","dependencies":["a47ce72graphquad19a-h3"],"title":"Vertical Transformation","text":"Since k represents the vertical transformation, what is k in our case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad19a-h5","type":"hint","dependencies":["a47ce72graphquad19a-h4"],"title":"Vertical Transformation","text":"Since our k is $$-2$$, is this a vertical shift up or down?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad19a-h6","type":"hint","dependencies":["a47ce72graphquad19a-h5"],"title":"Down Transformation","text":"It is a vertical shift down by $$2$$, since our k is $$-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad19a-h7","type":"hint","dependencies":["a47ce72graphquad19a-h6"],"title":"Horizontal Transformation","text":"Which variable represents the horizontal transformation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad19a-h8","type":"hint","dependencies":["a47ce72graphquad19a-h7"],"title":"Horizontal Transformation","text":"Since $$b$$ represents the horizontal transformation, what is $$b$$ in our case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad19a-h9","type":"hint","dependencies":["a47ce72graphquad19a-h8"],"title":"Horizontal Transformation","text":"Since our $$b$$ is $$4$$, is this a horizontal shift to the left or right?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad19a-h10","type":"hint","dependencies":["a47ce72graphquad19a-h9"],"title":"Right Transformation","text":"It is a horizontal shift to the right by $$4$$, since $$b$$ is $$-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad19a-h11","type":"hint","dependencies":["a47ce72graphquad19a-h10"],"title":"Width","text":"Which variable represents the width of our graph?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad19a-h12","type":"hint","dependencies":["a47ce72graphquad19a-h11"],"title":"Width","text":"Since a represents the width of our graph, what is a in this case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad19a-h13","type":"hint","dependencies":["a47ce72graphquad19a-h12"],"title":"Width","text":"Since a is $$1$$, what does that tell us about the width? Will it be skinnier, wider, or normal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad19a-h14","type":"hint","dependencies":["a47ce72graphquad19a-h13"],"title":"Normal Width","text":"Since a is $$1$$, it will have a normal width.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a47ce72graphquad2","title":"Graphing Quadratic Functions Using Transformations","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Graph Quadratic Functions Using Transformations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a47ce72graphquad2a","stepAnswer":["A"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=x^2+3$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a47ce72graphquad2a-h1","type":"hint","dependencies":[],"title":"Transformation","text":"Is this a vertical or a horizontal transformation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad2a-h2","type":"hint","dependencies":["a47ce72graphquad2a-h1"],"title":"Vertical Transformation","text":"Since this is a vertical transformation, is it up or down?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad2a-h3","type":"hint","dependencies":["a47ce72graphquad2a-h2"],"title":"Up Transformation","text":"Since it is $$3$$, it\'s a vertical transformation up.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a47ce72graphquad20","title":"Graphing Quadratic Functions Using Transformations","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Graph Quadratic Functions Using Transformations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a47ce72graphquad20a","stepAnswer":["D"],"problemType":"MultipleChoice","stepTitle":"$$f(x)={\\\\left(x-4\\\\right)}^2+2$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a47ce72graphquad20a-h1","type":"hint","dependencies":[],"title":"Transformations","text":"Think about the kinds of transformations that are occuring.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad20a-h2","type":"hint","dependencies":["a47ce72graphquad20a-h1"],"title":"Equation","text":"Think about the equation in the form of $${\\\\left(a \\\\left(x+b\\\\right)\\\\right)}^2+k$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad20a-h3","type":"hint","dependencies":["a47ce72graphquad20a-h2"],"title":"Vertical Transformation","text":"Which variable represents the vertical transformation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad20a-h4","type":"hint","dependencies":["a47ce72graphquad20a-h3"],"title":"Vertical Transformation","text":"Since k represents the vertical transformation, what is k in our case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad20a-h5","type":"hint","dependencies":["a47ce72graphquad20a-h4"],"title":"Vertical Transformation","text":"Since our k is $$2$$, is this a vertical shift up or down?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad20a-h6","type":"hint","dependencies":["a47ce72graphquad20a-h5"],"title":"Up Transformation","text":"It is a vertical shift down by $$2$$, since our k is $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad20a-h7","type":"hint","dependencies":["a47ce72graphquad20a-h6"],"title":"Horizontal Transformation","text":"Which variable represents the horizontal transformation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad20a-h8","type":"hint","dependencies":["a47ce72graphquad20a-h7"],"title":"Horizontal Transformation","text":"Since $$b$$ represents the horizontal transformation, what is $$b$$ in our case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad20a-h9","type":"hint","dependencies":["a47ce72graphquad20a-h8"],"title":"Horizontal Transformation","text":"Since our $$b$$ is $$4$$, is this a horizontal shift to the left or right?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad20a-h10","type":"hint","dependencies":["a47ce72graphquad20a-h9"],"title":"Right Transformation","text":"It is a horizontal shift to the right by $$4$$, since $$b$$ is $$-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad20a-h11","type":"hint","dependencies":["a47ce72graphquad20a-h10"],"title":"Width","text":"Which variable represents the width of our graph?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad20a-h12","type":"hint","dependencies":["a47ce72graphquad20a-h11"],"title":"Width","text":"Since a represents the width of our graph, what is a in this case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad20a-h13","type":"hint","dependencies":["a47ce72graphquad20a-h12"],"title":"Width","text":"Since a is $$1$$, what does that tell us about the width? Will it be skinnier, wider, or normal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad20a-h14","type":"hint","dependencies":["a47ce72graphquad20a-h13"],"title":"Normal Width","text":"Since a is $$1$$, it will have a normal width.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a47ce72graphquad21","title":"Graphing Quadratic Functions Using Transformations","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Graph Quadratic Functions Using Transformations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a47ce72graphquad21a","stepAnswer":["B"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=-2x^2$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a47ce72graphquad21a-h1","type":"hint","dependencies":[],"title":"Transformations","text":"Are there any horizontal or vertical transformations?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad21a-h2","type":"hint","dependencies":["a47ce72graphquad21a-h1"],"title":"Transformations","text":"There are no horizontal or vertical transformations since it is not in the form of $${\\\\left(x+b\\\\right)}^2$$ or $$x^2+k$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad21a-h3","type":"hint","dependencies":["a47ce72graphquad21a-h2"],"title":"Width","text":"Are there any changes to the width?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad21a-h4","type":"hint","dependencies":["a47ce72graphquad21a-h3"],"title":"Width","text":"Since there is a change, it it going to be wide or skinny?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad21a-h5","type":"hint","dependencies":["a47ce72graphquad21a-h4"],"title":"Skinny","text":"Since the absolute value of a is larger than $$1$$, the graph will be skinnier.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad21a-h6","type":"hint","dependencies":["a47ce72graphquad21a-h5"],"title":"Direction","text":"What is the sign of a?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad21a-h7","type":"hint","dependencies":["a47ce72graphquad21a-h6"],"title":"Direction","text":"Since a is negative, does the graph go up or down?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad21a-h8","type":"hint","dependencies":["a47ce72graphquad21a-h7"],"title":"Down","text":"Since a is negative, the graph goes down.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a47ce72graphquad22","title":"Graphing Quadratic Functions Using Transformations","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Graph Quadratic Functions Using Transformations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a47ce72graphquad22a","stepAnswer":["A"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=2x^2$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a47ce72graphquad22a-h1","type":"hint","dependencies":[],"title":"Transformations","text":"Are there any horizontal or vertical transformations?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad22a-h2","type":"hint","dependencies":["a47ce72graphquad22a-h1"],"title":"Transformations","text":"There are no horizontal or vertical transformations since it is not in the form of $${\\\\left(x+b\\\\right)}^2$$ or $$x^2+k$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad22a-h3","type":"hint","dependencies":["a47ce72graphquad22a-h2"],"title":"Width","text":"Are there any changes to the width?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad22a-h4","type":"hint","dependencies":["a47ce72graphquad22a-h3"],"title":"Width","text":"Since there is a change, it it going to be wide or skinny?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad22a-h5","type":"hint","dependencies":["a47ce72graphquad22a-h4"],"title":"Skinny","text":"Since the absolute value of a is larger than $$1$$, the graph will be skinnier.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad22a-h6","type":"hint","dependencies":["a47ce72graphquad22a-h5"],"title":"Direction","text":"What is the sign of a?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad22a-h7","type":"hint","dependencies":["a47ce72graphquad22a-h6"],"title":"Direction","text":"Since a is positive, does the graph go up or down?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad22a-h8","type":"hint","dependencies":["a47ce72graphquad22a-h7"],"title":"Up","text":"Since a is positive, the graph goes up.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a47ce72graphquad23","title":"Graphing Quadratic Functions Using Transformations","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Graph Quadratic Functions Using Transformations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a47ce72graphquad23a","stepAnswer":["D"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{1}{2} x^2$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a47ce72graphquad23a-h1","type":"hint","dependencies":[],"title":"Transformations","text":"Are there any horizontal or vertical transformations?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad23a-h2","type":"hint","dependencies":["a47ce72graphquad23a-h1"],"title":"Transformations","text":"There are no horizontal or vertical transformations since it is not in the form of $${\\\\left(x+b\\\\right)}^2$$ or $$x^2+k$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad23a-h3","type":"hint","dependencies":["a47ce72graphquad23a-h2"],"title":"Width","text":"Are there any changes to the width?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad23a-h4","type":"hint","dependencies":["a47ce72graphquad23a-h3"],"title":"Width","text":"Since there is a change, it it going to be wide or skinny?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad23a-h5","type":"hint","dependencies":["a47ce72graphquad23a-h4"],"title":"Wide","text":"Since the absolute value of a is less than $$1$$, the graph will be wider.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad23a-h6","type":"hint","dependencies":["a47ce72graphquad23a-h5"],"title":"Direction","text":"What is the sign of a?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad23a-h7","type":"hint","dependencies":["a47ce72graphquad23a-h6"],"title":"Direction","text":"Since a is positive, does the graph go up or down?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad23a-h8","type":"hint","dependencies":["a47ce72graphquad23a-h7"],"title":"Up","text":"Since a is positive, the graph goes up.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a47ce72graphquad24","title":"Graphing Quadratic Functions Using Transformations","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Graph Quadratic Functions Using Transformations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a47ce72graphquad24a","stepAnswer":["C"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\left(-\\\\frac{1}{2}\\\\right) x^2$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a47ce72graphquad24a-h1","type":"hint","dependencies":[],"title":"Transformations","text":"Are there any horizontal or vertical transformations?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad24a-h2","type":"hint","dependencies":["a47ce72graphquad24a-h1"],"title":"Transformations","text":"There are no horizontal or vertical transformations since it is not in the form of $${\\\\left(x+b\\\\right)}^2$$ or $$x^2+k$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad24a-h3","type":"hint","dependencies":["a47ce72graphquad24a-h2"],"title":"Width","text":"Are there any changes to the width?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad24a-h4","type":"hint","dependencies":["a47ce72graphquad24a-h3"],"title":"Width","text":"Since there is a change, it it going to be wide or skinny?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad24a-h5","type":"hint","dependencies":["a47ce72graphquad24a-h4"],"title":"Wide","text":"Since the absolute value of a is less than $$1$$, the graph will be wider.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad24a-h6","type":"hint","dependencies":["a47ce72graphquad24a-h5"],"title":"Direction","text":"What is the sign of a?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad24a-h7","type":"hint","dependencies":["a47ce72graphquad24a-h6"],"title":"Direction","text":"Since a is negative, does the graph go up or down?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad24a-h8","type":"hint","dependencies":["a47ce72graphquad24a-h7"],"title":"Down","text":"Since a is negative, the graph goes down.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a47ce72graphquad25","title":"Graphing Quadratic Functions Using Transformations","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Graph Quadratic Functions Using Transformations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a47ce72graphquad25a","stepAnswer":["C"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\left(-3{\\\\left(x+2\\\\right)}^2\\\\right)+7$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a47ce72graphquad25a-h1","type":"hint","dependencies":[],"title":"Transformations","text":"Think about the kinds of transformations that are occuring.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad25a-h2","type":"hint","dependencies":["a47ce72graphquad25a-h1"],"title":"Equation","text":"Think about the equation in the form of $${\\\\left(a \\\\left(x+b\\\\right)\\\\right)}^2+k$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad25a-h3","type":"hint","dependencies":["a47ce72graphquad25a-h2"],"title":"Vertical Transformation","text":"Which variable represents the vertical transformation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad25a-h4","type":"hint","dependencies":["a47ce72graphquad25a-h3"],"title":"Vertical Transformation","text":"Since k represents the vertical transformation, what is k in our case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad25a-h5","type":"hint","dependencies":["a47ce72graphquad25a-h4"],"title":"Vertical Transformation","text":"Since our k is $$7$$, is this a vertical shift up or down?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad25a-h6","type":"hint","dependencies":["a47ce72graphquad25a-h5"],"title":"Up Transformation","text":"It is a vertical shift up by $$7$$, since our k is $$7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad25a-h7","type":"hint","dependencies":["a47ce72graphquad25a-h6"],"title":"Horizontal Transformation","text":"Which variable represents the horizontal transformation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad25a-h8","type":"hint","dependencies":["a47ce72graphquad25a-h7"],"title":"Horizontal Transformation","text":"Since $$b$$ represents the horizontal transformation, what is $$b$$ in our case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad25a-h9","type":"hint","dependencies":["a47ce72graphquad25a-h8"],"title":"Horizontal Transformation","text":"Since our $$b$$ is $$2$$, is this a horizontal shift to the left or right?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad25a-h10","type":"hint","dependencies":["a47ce72graphquad25a-h9"],"title":"Left Transformation","text":"It is a horizontal shift to the right by $$2$$, since $$b$$ is $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad25a-h11","type":"hint","dependencies":["a47ce72graphquad25a-h10"],"title":"Width","text":"Which variable represents the width of our graph?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad25a-h12","type":"hint","dependencies":["a47ce72graphquad25a-h11"],"title":"Width","text":"Since a represents the width of our graph, what is a in this case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad25a-h13","type":"hint","dependencies":["a47ce72graphquad25a-h12"],"title":"Width","text":"Since a is $$-3$$, what does that tell us about the width? Will it be skinnier, wider, or normal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad25a-h14","type":"hint","dependencies":["a47ce72graphquad25a-h13"],"title":"Skinny Width","text":"Since a is $$-3$$, it\'s absolute value is larger than $$1$$, so it will have a skinnier width.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad25a-h15","type":"hint","dependencies":["a47ce72graphquad25a-h14"],"title":"Direction","text":"What is the sign of a and what does this tell us about the direction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad25a-h16","type":"hint","dependencies":["a47ce72graphquad25a-h15"],"title":"Direction","text":"Since a is $$-3$$, what direction does the graph go?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad25a-h17","type":"hint","dependencies":["a47ce72graphquad25a-h16"],"title":"Down","text":"Since a is negative, it will go downwards.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a47ce72graphquad26","title":"Graphing Quadratic Functions Using Transformations","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Graph Quadratic Functions Using Transformations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a47ce72graphquad26a","stepAnswer":["D"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=3{\\\\left(x-2\\\\right)}^2-7$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a47ce72graphquad26a-h1","type":"hint","dependencies":[],"title":"Transformations","text":"Think about the kinds of transformations that are occuring.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad26a-h2","type":"hint","dependencies":["a47ce72graphquad26a-h1"],"title":"Equation","text":"Think about the equation in the form of $${\\\\left(a \\\\left(x+b\\\\right)\\\\right)}^2+k$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad26a-h3","type":"hint","dependencies":["a47ce72graphquad26a-h2"],"title":"Vertical Transformation","text":"Which variable represents the vertical transformation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad26a-h4","type":"hint","dependencies":["a47ce72graphquad26a-h3"],"title":"Vertical Transformation","text":"Since k represents the vertical transformation, what is k in our case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad26a-h5","type":"hint","dependencies":["a47ce72graphquad26a-h4"],"title":"Vertical Transformation","text":"Since our k is $$-7$$, is this a vertical shift up or down?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad26a-h6","type":"hint","dependencies":["a47ce72graphquad26a-h5"],"title":"Down Transformation","text":"It is a vertical shift down by $$7$$, since our k is $$-7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad26a-h7","type":"hint","dependencies":["a47ce72graphquad26a-h6"],"title":"Horizontal Transformation","text":"Which variable represents the horizontal transformation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad26a-h8","type":"hint","dependencies":["a47ce72graphquad26a-h7"],"title":"Horizontal Transformation","text":"Since $$b$$ represents the horizontal transformation, what is $$b$$ in our case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad26a-h9","type":"hint","dependencies":["a47ce72graphquad26a-h8"],"title":"Horizontal Transformation","text":"Since our $$b$$ is $$-2$$, is this a horizontal shift to the left or right?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad26a-h10","type":"hint","dependencies":["a47ce72graphquad26a-h9"],"title":"Right Transformation","text":"It is a horizontal shift to the right by $$2$$, since $$b$$ is $$-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad26a-h11","type":"hint","dependencies":["a47ce72graphquad26a-h10"],"title":"Width","text":"Which variable represents the width of our graph?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad26a-h12","type":"hint","dependencies":["a47ce72graphquad26a-h11"],"title":"Width","text":"Since a represents the width of our graph, what is a in this case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad26a-h13","type":"hint","dependencies":["a47ce72graphquad26a-h12"],"title":"Width","text":"Since a is $$3$$, what does that tell us about the width? Will it be skinnier, wider, or normal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad26a-h14","type":"hint","dependencies":["a47ce72graphquad26a-h13"],"title":"Skinny Width","text":"Since a is $$3$$, it\'s absolute value is larger than $$1$$, so it will have a skinnier width.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad26a-h15","type":"hint","dependencies":["a47ce72graphquad26a-h14"],"title":"Direction","text":"What is the sign of a and what does this tell us about the direction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad26a-h16","type":"hint","dependencies":["a47ce72graphquad26a-h15"],"title":"Direction","text":"Since a is $$3$$, what direction does the graph go?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad26a-h17","type":"hint","dependencies":["a47ce72graphquad26a-h16"],"title":"Up","text":"Since a is positive, it will go upwards.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a47ce72graphquad27","title":"Graphing Quadratic Functions Using Transformations","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Graph Quadratic Functions Using Transformations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a47ce72graphquad27a","stepAnswer":["B"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\left(-3{\\\\left(x-2\\\\right)}^2\\\\right)-7$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a47ce72graphquad27a-h1","type":"hint","dependencies":[],"title":"Transformations","text":"Think about the kinds of transformations that are occuring.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad27a-h2","type":"hint","dependencies":["a47ce72graphquad27a-h1"],"title":"Equation","text":"Think about the equation in the form of $${\\\\left(a \\\\left(x+b\\\\right)\\\\right)}^2+k$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad27a-h3","type":"hint","dependencies":["a47ce72graphquad27a-h2"],"title":"Vertical Transformation","text":"Which variable represents the vertical transformation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad27a-h4","type":"hint","dependencies":["a47ce72graphquad27a-h3"],"title":"Vertical Transformation","text":"Since k represents the vertical transformation, what is k in our case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad27a-h5","type":"hint","dependencies":["a47ce72graphquad27a-h4"],"title":"Vertical Transformation","text":"Since our k is $$-7$$, is this a vertical shift up or down?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad27a-h6","type":"hint","dependencies":["a47ce72graphquad27a-h5"],"title":"Down Transformation","text":"It is a vertical shift down by $$7$$, since our k is $$-7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad27a-h7","type":"hint","dependencies":["a47ce72graphquad27a-h6"],"title":"Horizontal Transformation","text":"Which variable represents the horizontal transformation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad27a-h8","type":"hint","dependencies":["a47ce72graphquad27a-h7"],"title":"Horizontal Transformation","text":"Since $$b$$ represents the horizontal transformation, what is $$b$$ in our case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad27a-h9","type":"hint","dependencies":["a47ce72graphquad27a-h8"],"title":"Horizontal Transformation","text":"Since our $$b$$ is $$-2$$, is this a horizontal shift to the left or right?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad27a-h10","type":"hint","dependencies":["a47ce72graphquad27a-h9"],"title":"Right Transformation","text":"It is a horizontal shift to the right by $$2$$, since $$b$$ is $$-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad27a-h11","type":"hint","dependencies":["a47ce72graphquad27a-h10"],"title":"Width","text":"Which variable represents the width of our graph?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad27a-h12","type":"hint","dependencies":["a47ce72graphquad27a-h11"],"title":"Width","text":"Since a represents the width of our graph, what is a in this case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad27a-h13","type":"hint","dependencies":["a47ce72graphquad27a-h12"],"title":"Width","text":"Since a is $$-3$$, what does that tell us about the width? Will it be skinnier, wider, or normal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad27a-h14","type":"hint","dependencies":["a47ce72graphquad27a-h13"],"title":"Skinny Width","text":"Since a is $$-3$$, it\'s absolute value is larger than $$1$$, so it will have a skinnier width.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad27a-h15","type":"hint","dependencies":["a47ce72graphquad27a-h14"],"title":"Direction","text":"What is the sign of a and what does this tell us about the direction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad27a-h16","type":"hint","dependencies":["a47ce72graphquad27a-h15"],"title":"Direction","text":"Since a is $$-3$$, what direction does the graph go?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad27a-h17","type":"hint","dependencies":["a47ce72graphquad27a-h16"],"title":"Down","text":"Since a is negative, it will go downwards.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a47ce72graphquad28","title":"Graphing Quadratic Functions Using Transformations","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Graph Quadratic Functions Using Transformations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a47ce72graphquad28a","stepAnswer":["A"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{1}{3} {\\\\left(x-2\\\\right)}^2-7$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a47ce72graphquad28a-h1","type":"hint","dependencies":[],"title":"Transformations","text":"Think about the kinds of transformations that are occuring.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad28a-h2","type":"hint","dependencies":["a47ce72graphquad28a-h1"],"title":"Equation","text":"Think about the equation in the form of $${\\\\left(a \\\\left(x+b\\\\right)\\\\right)}^2+k$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad28a-h3","type":"hint","dependencies":["a47ce72graphquad28a-h2"],"title":"Vertical Transformation","text":"Which variable represents the vertical transformation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad28a-h4","type":"hint","dependencies":["a47ce72graphquad28a-h3"],"title":"Vertical Transformation","text":"Since k represents the vertical transformation, what is k in our case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad28a-h5","type":"hint","dependencies":["a47ce72graphquad28a-h4"],"title":"Vertical Transformation","text":"Since our k is $$-7$$, is this a vertical shift up or down?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad28a-h6","type":"hint","dependencies":["a47ce72graphquad28a-h5"],"title":"Down Transformation","text":"It is a vertical shift down by $$7$$, since our k is $$-7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad28a-h7","type":"hint","dependencies":["a47ce72graphquad28a-h6"],"title":"Horizontal Transformation","text":"Which variable represents the horizontal transformation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad28a-h8","type":"hint","dependencies":["a47ce72graphquad28a-h7"],"title":"Horizontal Transformation","text":"Since $$b$$ represents the horizontal transformation, what is $$b$$ in our case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad28a-h9","type":"hint","dependencies":["a47ce72graphquad28a-h8"],"title":"Horizontal Transformation","text":"Since our $$b$$ is $$-2$$, is this a horizontal shift to the left or right?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad28a-h10","type":"hint","dependencies":["a47ce72graphquad28a-h9"],"title":"Right Transformation","text":"It is a horizontal shift to the right by $$2$$, since $$b$$ is $$-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad28a-h11","type":"hint","dependencies":["a47ce72graphquad28a-h10"],"title":"Width","text":"Which variable represents the width of our graph?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad28a-h12","type":"hint","dependencies":["a47ce72graphquad28a-h11"],"title":"Width","text":"Since a represents the width of our graph, what is a in this case?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad28a-h13","type":"hint","dependencies":["a47ce72graphquad28a-h12"],"title":"Width","text":"Since a is $$\\\\frac{1}{3}$$, what does that tell us about the width? Will it be skinnier, wider, or normal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad28a-h14","type":"hint","dependencies":["a47ce72graphquad28a-h13"],"title":"Wide Width","text":"Since a is $$\\\\frac{1}{3}$$, which has an absolute value less than $$1$$, it will have a wider width.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad28a-h15","type":"hint","dependencies":["a47ce72graphquad28a-h14"],"title":"Direction","text":"What is the sign of a and what does this tell us about the direction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad28a-h16","type":"hint","dependencies":["a47ce72graphquad28a-h15"],"title":"Direction","text":"Since a is $$\\\\frac{1}{3}$$, what direction does the graph go?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad28a-h17","type":"hint","dependencies":["a47ce72graphquad28a-h16"],"title":"Up","text":"Since a is positive, it will go upwards.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a47ce72graphquad29","title":"Graphing Quadratic Functions Using Transformations","body":"Rewrite the function in the $$a {\\\\left(x+b\\\\right)}^2+k$$ form.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Graph Quadratic Functions Using Transformations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a47ce72graphquad29a","stepAnswer":["$$3{\\\\left(x+1\\\\right)}^2-4$$"],"problemType":"TextBox","stepTitle":"$$f(x)=3x^2+6x-1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3{\\\\left(x+1\\\\right)}^2-4$$","hints":{"DefaultPathway":[{"id":"a47ce72graphquad29a-h1","type":"hint","dependencies":[],"title":"Finding a","text":"First, we must factor by grouping to find a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad29a-h2","type":"hint","dependencies":["a47ce72graphquad29a-h1"],"title":"Factor by Grouping","text":"What is the GCF between $$3x^2$$ and $$6x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad29a-h3","type":"hint","dependencies":["a47ce72graphquad29a-h2"],"title":"Factor by Grouping","text":"Since the GCF is $$3$$, we factor both terms by $$3$$ and now have (3*((x**2)+(2*x))-1.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad29a-h4","type":"hint","dependencies":["a47ce72graphquad29a-h3"],"title":"Completing the Square","text":"We must complete the square to rewrite the equation in the $$a {\\\\left(x+b\\\\right)}^2+k$$ form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad29a-h5","type":"hint","dependencies":["a47ce72graphquad29a-h4"],"title":"Completing the Square","text":"First, we divide the second term of the polynomial by $$2$$, which in this case is $$1$$, since $$\\\\frac{2}{2}$$ is $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad29a-h6","type":"hint","dependencies":["a47ce72graphquad29a-h5"],"title":"Completing the Square","text":"Next, we square term we got from dividing and add it inside of the polynomial, which in this case is $$1$$, since $$1^2$$ is $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad29a-h7","type":"hint","dependencies":["a47ce72graphquad29a-h6"],"title":"Completing the Square","text":"After this, we notice that we can\'t just add in a $$1$$, so we subtract $$3$$ from our total equation outside of the parentheses, since a is $$3$$ and $$3\\\\times1$$ is $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad29a-h8","type":"hint","dependencies":["a47ce72graphquad29a-h7"],"title":"Completing the Square","text":"Finally, we simplify our polynomial to $$3{\\\\left(x+1\\\\right)}^2-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a47ce72graphquad3","title":"Graphing Quadratic Functions Using Transformations","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Graph Quadratic Functions Using Transformations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a47ce72graphquad3a","stepAnswer":["C"],"problemType":"MultipleChoice","stepTitle":"$$f(x)={\\\\left(x-3\\\\right)}^2$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a47ce72graphquad3a-h1","type":"hint","dependencies":[],"title":"Transformation","text":"Is this a vertical or a horizontal transformation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad3a-h2","type":"hint","dependencies":["a47ce72graphquad3a-h1"],"title":"Horizontal Transformation","text":"Since this is a horizontal transformation, is it to the left or right?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad3a-h3","type":"hint","dependencies":["a47ce72graphquad3a-h2"],"title":"Right Transformation","text":"Since it is $$-3$$, it\'s a horizontal transformation to the right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a47ce72graphquad30","title":"Graphing Quadratic Functions Using Transformations","body":"Rewrite the function in the $$a {\\\\left(x+b\\\\right)}^2+k$$ form.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Graph Quadratic Functions Using Transformations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a47ce72graphquad30a","stepAnswer":["$$2{\\\\left(x-3\\\\right)}^2-11$$"],"problemType":"TextBox","stepTitle":"$$f(x)=2x^2-12x+7$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2{\\\\left(x-3\\\\right)}^2-11$$","hints":{"DefaultPathway":[{"id":"a47ce72graphquad30a-h1","type":"hint","dependencies":[],"title":"Finding a","text":"First, we must factor by grouping to find a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad30a-h2","type":"hint","dependencies":["a47ce72graphquad30a-h1"],"title":"Factor by Grouping","text":"What is the GCF between $$2x^2$$ and $$-12x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad30a-h3","type":"hint","dependencies":["a47ce72graphquad30a-h2"],"title":"Factor by Grouping","text":"Since the GCF is $$2$$, we factor both terms by $$2$$ and now have (2*((x**2)-(6*x))+7.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad30a-h4","type":"hint","dependencies":["a47ce72graphquad30a-h3"],"title":"Completing the Square","text":"We must complete the square to rewrite the equation in the $$a {\\\\left(x+b\\\\right)}^2+k$$ form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad30a-h5","type":"hint","dependencies":["a47ce72graphquad30a-h4"],"title":"Completing the Square","text":"First, we divide the second term of the polynomial by $$2$$, which in this case is $$-3$$, since $$\\\\frac{-6}{2}$$ is $$-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad30a-h6","type":"hint","dependencies":["a47ce72graphquad30a-h5"],"title":"Completing the Square","text":"Next, we square term we got from dividing and add it inside of the polynomial, which in this case is $$9$$, since $${\\\\left(-3\\\\right)}^2$$ is $$9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad30a-h7","type":"hint","dependencies":["a47ce72graphquad30a-h6"],"title":"Completing the Square","text":"After this, we notice that we can\'t just add in a $$9$$, so we subtract $$18$$ from our total equation outside of the parentheses, since a is $$2$$ and $$2\\\\times9$$ is $$18$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a47ce72graphquad30a-h8","type":"hint","dependencies":["a47ce72graphquad30a-h7"],"title":"Completing the Square","text":"Finally, we simplify our polynomial to $$2{\\\\left(x-3\\\\right)}^2-11$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a47ce72graphquad4","title":"Graphing Quadratic Functions Using Transformations","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Graph Quadratic Functions Using Transformations","courseName":"OpenStax: Intermediate 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot13a-h2","type":"hint","dependencies":["a4a0f7dradicalroot13a-h1"],"title":"Order of Operation","text":"Solving the square root","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot13a-h3","type":"hint","dependencies":["a4a0f7dradicalroot13a-h2"],"title":"Calculation","text":"Square root can be removed by squaring both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x-5$$"],"dependencies":["a4a0f7dradicalroot13a-h3"],"title":"Squaring","text":"What is the square of $$\\\\sqrt{3x-5}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot13a-h5","type":"hint","dependencies":["a4a0f7dradicalroot13a-h4"],"title":"Operation","text":"Square root can be canceled by squaring","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot13a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x-5$$"],"dependencies":["a4a0f7dradicalroot13a-h5"],"title":"Operation","text":"What is $$\\\\sqrt{3x-5}$$ without square root?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot13a-h7","type":"hint","dependencies":["a4a0f7dradicalroot13a-h6"],"title":"Principle of Operation","text":"$$5$$ on the right side also has to be squared","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot13a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["a4a0f7dradicalroot13a-h7"],"title":"Squaring","text":"What is $$5^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot13a-h9","type":"hint","dependencies":["a4a0f7dradicalroot13a-h8"],"title":"Organizing","text":"The equation becomes $$3x-5=25$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot13a-h10","type":"hint","dependencies":["a4a0f7dradicalroot13a-h9"],"title":"Operation","text":"Add $$5$$ on both side to isolate $$x$$ term","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot13a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x$$"],"dependencies":["a4a0f7dradicalroot13a-h10"],"title":"Addition","text":"What is $$3x-5+5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a4a0f7dradicalroot13a-h11-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$30$$"],"dependencies":[],"title":"Addition","text":"What is $$25+5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a4a0f7dradicalroot13a-h12","type":"hint","dependencies":["a4a0f7dradicalroot13a-h11"],"title":"Organizing","text":"The equation becomes $$3x=30$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot13a-h13","type":"hint","dependencies":["a4a0f7dradicalroot13a-h12"],"title":"Dividing","text":"Diving $$3x$$ by $$3$$ to produce a single $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot13a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a4a0f7dradicalroot13a-h13"],"title":"Dividing","text":"What is $$30$$ divided by 3?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4a0f7dradicalroot14","title":"Solving Radical Equations","body":"Find the value of the variable.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Solve Equations with Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4a0f7dradicalroot14a","stepAnswer":["$$\\\\frac{1}{3}$$"],"problemType":"TextBox","stepTitle":"Solve: $$\\\\sqrt{3p+3}+3=5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{3}$$","hints":{"DefaultPathway":[{"id":"a4a0f7dradicalroot14a-h1","type":"hint","dependencies":[],"title":"Setup","text":"Simplifying the equation to $$p=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot14a-h2","type":"hint","dependencies":["a4a0f7dradicalroot14a-h1"],"title":"Setup","text":"Isolating $$p$$ function","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot14a-h3","type":"hint","dependencies":["a4a0f7dradicalroot14a-h2"],"title":"Operation","text":"Subtracting $$3$$ on both sides to remove the $$+3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot14a-h4","type":"hint","dependencies":["a4a0f7dradicalroot14a-h3"],"title":"Order of Operation","text":"Solving the square root","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot14a-h5","type":"hint","dependencies":["a4a0f7dradicalroot14a-h4"],"title":"Calculation","text":"Square root can be removed by squaring both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot14a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3p+3$$"],"dependencies":["a4a0f7dradicalroot14a-h5"],"title":"Squaring","text":"What is the square of $$\\\\sqrt{3p+3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot14a-h7","type":"hint","dependencies":["a4a0f7dradicalroot14a-h6"],"title":"Operation","text":"Square root can be canceled by squaring","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot14a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3p+3$$"],"dependencies":["a4a0f7dradicalroot14a-h7"],"title":"Operation","text":"What is $$\\\\sqrt{3p+3}$$ without square root?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot14a-h9","type":"hint","dependencies":["a4a0f7dradicalroot14a-h8"],"title":"Principle of Operation","text":"$$2$$ on the right side also has to be squared","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot14a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a4a0f7dradicalroot14a-h9"],"title":"Squaring","text":"What is $$2^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot14a-h11","type":"hint","dependencies":["a4a0f7dradicalroot14a-h10"],"title":"Organizing","text":"The equation becomes $$3p+3=4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot14a-h12","type":"hint","dependencies":["a4a0f7dradicalroot14a-h11"],"title":"Operation","text":"Subtracting $$3$$ on both side","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot14a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3p$$"],"dependencies":["a4a0f7dradicalroot14a-h12"],"title":"Subtraction","text":"What is $$3p+3-3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot14a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a4a0f7dradicalroot14a-h13"],"title":"Subtraction","text":"What is $$4-3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot14a-h15","type":"hint","dependencies":["a4a0f7dradicalroot14a-h14"],"title":"Organizing","text":"The equation becomes $$3p=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot14a-h16","type":"hint","dependencies":["a4a0f7dradicalroot14a-h15"],"title":"Dividing","text":"Diving $$3p$$ by $$3$$ to produce a single $$p$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot14a-h17","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["a4a0f7dradicalroot14a-h16"],"title":"Dividing","text":"What is $$1$$ divided by 3?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4a0f7dradicalroot15","title":"Solving Radical Equations","body":"Find the value of the variable.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Solve Equations with Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4a0f7dradicalroot15a","stepAnswer":["$$\\\\frac{43+15\\\\sqrt{5}}{2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{n-4}+5=\\\\sqrt{3n+3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{43+15\\\\sqrt{5}}{2}$$","hints":{"DefaultPathway":[{"id":"a4a0f7dradicalroot15a-h1","type":"hint","dependencies":[],"title":"Order of Operation","text":"Solving the square root","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot15a-h2","type":"hint","dependencies":["a4a0f7dradicalroot15a-h1"],"title":"Calculation","text":"Square root can be removed by squaring both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3n+3$$"],"dependencies":["a4a0f7dradicalroot15a-h2"],"title":"Squaring","text":"What is the square of $$\\\\sqrt{3n+3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot15a-h4","type":"hint","dependencies":["a4a0f7dradicalroot15a-h3"],"title":"Operation","text":"Apply binomial formula on $${\\\\left(\\\\sqrt{n-4}+5\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot15a-h5","type":"hint","dependencies":["a4a0f7dradicalroot15a-h4"],"title":"Substitution","text":"$$a=\\\\sqrt{n-4}$$, $$b=5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot15a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$n-4$$"],"dependencies":["a4a0f7dradicalroot15a-h5"],"title":"Substitution","text":"What is $$a^2$$ when $$a=\\\\sqrt{n-4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot15a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10\\\\sqrt{n-4}$$"],"dependencies":["a4a0f7dradicalroot15a-h6"],"title":"Substitution","text":"What is 2ab when $$a=\\\\sqrt{n-4}$$, $$b=5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot15a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["a4a0f7dradicalroot15a-h7"],"title":"Substitution","text":"What is $$b^2$$ when $$b=5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot15a-h9","type":"hint","dependencies":["a4a0f7dradicalroot15a-h8"],"title":"Organizing","text":"The equation becomes $$n-4+10\\\\sqrt{n-4}+25=3n+3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot15a-h10","type":"hint","dependencies":["a4a0f7dradicalroot15a-h9"],"title":"Organizing","text":"Isolating the $$\\\\sqrt{n-4}$$ term to left side","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot15a-h11","type":"hint","dependencies":["a4a0f7dradicalroot15a-h10"],"title":"Organizing","text":"The equation becomes $$10\\\\sqrt{n-4}=2n-18$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot15a-h12","type":"hint","dependencies":["a4a0f7dradicalroot15a-h11"],"title":"Calculation","text":"Square root can be removed by squaring both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot15a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$100n-400$$"],"dependencies":["a4a0f7dradicalroot15a-h12"],"title":"Squaring","text":"What is the square of $$10\\\\sqrt{n-4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot15a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$100n-400$$"],"dependencies":["a4a0f7dradicalroot15a-h13"],"title":"Squaring","text":"What is $${10}^2 {\\\\sqrt{n-4}}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot15a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$100n-400$$"],"dependencies":["a4a0f7dradicalroot15a-h14"],"title":"Multiplication","text":"What is $$100\\\\left(n-4\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot15a-h16","type":"hint","dependencies":["a4a0f7dradicalroot15a-h15"],"title":"Operation","text":"Apply binomial formula on $${\\\\left(2n-18\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot15a-h17","type":"hint","dependencies":["a4a0f7dradicalroot15a-h16"],"title":"Substitution","text":"$$a=2n$$, $$b=-18$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot15a-h18","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4n^2$$"],"dependencies":["a4a0f7dradicalroot15a-h17"],"title":"Substitution","text":"What is $$a^2$$ when $$a=2n$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot15a-h19","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-72n$$"],"dependencies":["a4a0f7dradicalroot15a-h18"],"title":"Substitution","text":"What is 2ab when $$a=2n$$, $$b=-18$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot15a-h20","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$324$$"],"dependencies":["a4a0f7dradicalroot15a-h19"],"title":"Substitution","text":"What is $$b^2$$ when $$b=-18$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot15a-h21","type":"hint","dependencies":["a4a0f7dradicalroot15a-h20"],"title":"Organizing","text":"The equation becomes $$100n-400=4n^2-72n+324$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot15a-h22","type":"hint","dependencies":["a4a0f7dradicalroot15a-h21"],"title":"Organizing","text":"Move every term to right side","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot15a-h23","type":"hint","dependencies":["a4a0f7dradicalroot15a-h22"],"title":"Organizing","text":"The equation becomes $$0=4n^2-172n+724$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot15a-h24","type":"hint","dependencies":["a4a0f7dradicalroot15a-h23"],"title":"Applying Function","text":"Applying the quadratic formula","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot15a-h25","type":"hint","dependencies":["a4a0f7dradicalroot15a-h24"],"title":"Substitution","text":"$$a=4$$, $$b=-172$$, $$c=724$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4a0f7dradicalroot16","title":"Solving Equations with Square Roots","body":"Solve $$\\\\sqrt{5y+1}=4$$ (Please input all answers as $$variable=answer$$, if defined)","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Solve Equations with Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4a0f7dradicalroot16a","stepAnswer":["$$y=3$$"],"problemType":"TextBox","stepTitle":"Solve $$\\\\sqrt{5y+1}=4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=3$$","hints":{"DefaultPathway":[{"id":"a4a0f7dradicalroot16a-h1","type":"hint","dependencies":[],"title":"Isolating the Radical","text":"First you must isolate the radical. If this is already done then think about how we can isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot16a-h2","type":"hint","dependencies":["a4a0f7dradicalroot16a-h1"],"title":"Isolating the Variable","text":"Square both sides of the equation and isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot16a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5y=15$$"],"dependencies":["a4a0f7dradicalroot16a-h2"],"title":"Isolating the Variable","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot16a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y=3$$"],"dependencies":["a4a0f7dradicalroot16a-h3"],"title":"Solving the Equation","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot16a-h5","type":"hint","dependencies":["a4a0f7dradicalroot16a-h4"],"title":"Double Checking","text":"Plug the answer back in to check if the answer is correct. This needs to be done because the radical can lead to some equations having no answers. If it doesn\'t work then there are no solutions to the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4a0f7dradicalroot17","title":"Solving Equations with Square Roots","body":"Solve $$\\\\sqrt{7z+15}=6$$ (Please input all answers as $$variable=answer$$, if defined)","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Solve Equations with Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4a0f7dradicalroot17a","stepAnswer":["$$z=3$$"],"problemType":"TextBox","stepTitle":"Solve $$\\\\sqrt{7z+15}=6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$z=3$$","hints":{"DefaultPathway":[{"id":"a4a0f7dradicalroot17a-h1","type":"hint","dependencies":[],"title":"Isolating the Radical","text":"First you must isolate the radical. If this is already done then think about how we can isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot17a-h2","type":"hint","dependencies":["a4a0f7dradicalroot17a-h1"],"title":"Isolating the Variable","text":"Square both sides of the equation and isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot17a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7z=21$$"],"dependencies":["a4a0f7dradicalroot17a-h2"],"title":"Isolating the Variable","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot17a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$z=3$$"],"dependencies":["a4a0f7dradicalroot17a-h3"],"title":"Solving the Equation","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot17a-h5","type":"hint","dependencies":["a4a0f7dradicalroot17a-h4"],"title":"Double Checking","text":"Plug the answer back in to check if the answer is correct. This needs to be done because the radical can lead to some equations having no answers. If it doesn\'t work then there are no solutions to the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4a0f7dradicalroot18","title":"Solving Equations with Square Roots","body":"Solve $$\\\\sqrt{5x-6}=8$$ (Please input all answers as $$variable=answer$$, if defined)","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Solve Equations with Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4a0f7dradicalroot18a","stepAnswer":["$$x=14$$"],"problemType":"TextBox","stepTitle":"Solve $$\\\\sqrt{5x-6}=8$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x=14$$","hints":{"DefaultPathway":[{"id":"a4a0f7dradicalroot18a-h1","type":"hint","dependencies":[],"title":"Isolating the Radical","text":"First you must isolate the radical. If this is already done then think about how we can isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot18a-h2","type":"hint","dependencies":["a4a0f7dradicalroot18a-h1"],"title":"Isolating the Variable","text":"Square both sides of the equation and isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot18a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5x=70$$"],"dependencies":["a4a0f7dradicalroot18a-h2"],"title":"Isolating the Variable","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot18a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=14$$"],"dependencies":["a4a0f7dradicalroot18a-h3"],"title":"Solving the Equation","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot18a-h5","type":"hint","dependencies":["a4a0f7dradicalroot18a-h4"],"title":"Double Checking","text":"Plug the answer back in to check if the answer is correct. This needs to be done because the radical can lead to some equations having no answers. If it doesn\'t work then there are no solutions to the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4a0f7dradicalroot19","title":"Solving Equations with Square Roots","body":"Solve $$\\\\sqrt{4x-3}=7$$ (Please input all answers as $$variable=answer$$, if defined)","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Solve Equations with Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4a0f7dradicalroot19a","stepAnswer":["$$x=13$$"],"problemType":"TextBox","stepTitle":"Solve $$\\\\sqrt{4x-3}=7$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x=13$$","hints":{"DefaultPathway":[{"id":"a4a0f7dradicalroot19a-h1","type":"hint","dependencies":[],"title":"Isolating the Radical","text":"First you must isolate the radical. If this is already done then think about how we can isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot19a-h2","type":"hint","dependencies":["a4a0f7dradicalroot19a-h1"],"title":"Isolating the Variable","text":"Square both sides of the equation and isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot19a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4x=52$$"],"dependencies":["a4a0f7dradicalroot19a-h2"],"title":"Isolating the Variable","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot19a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=13$$"],"dependencies":["a4a0f7dradicalroot19a-h3"],"title":"Solving the Equation","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot19a-h5","type":"hint","dependencies":["a4a0f7dradicalroot19a-h4"],"title":"Double Checking","text":"Plug the answer back in to check if the answer is correct. This needs to be done because the radical can lead to some equations having no answers. If it doesn\'t work then there are no solutions to the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4a0f7dradicalroot2","title":"Solving Radical Equations","body":"Find the value of $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Solve Equations with Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4a0f7dradicalroot2a","stepAnswer":["$$25$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{2x-1}=7$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$25$$","hints":{"DefaultPathway":[{"id":"a4a0f7dradicalroot2a-h1","type":"hint","dependencies":[],"title":"Setup","text":"Simplifying the equation to $$x=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot2a-h2","type":"hint","dependencies":["a4a0f7dradicalroot2a-h1"],"title":"Order of Operation","text":"Solving the square root","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot2a-h3","type":"hint","dependencies":["a4a0f7dradicalroot2a-h2"],"title":"Calculation","text":"Square root can be removed by squaring both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x-1$$"],"dependencies":["a4a0f7dradicalroot2a-h3"],"title":"Squaring","text":"What is the square of $$\\\\sqrt{2x-1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot2a-h5","type":"hint","dependencies":["a4a0f7dradicalroot2a-h4"],"title":"Operation","text":"Square root can be canceled by squaring","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x-1$$"],"dependencies":["a4a0f7dradicalroot2a-h5"],"title":"Operation","text":"What is $$\\\\sqrt{2x-1}$$ without square root?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot2a-h7","type":"hint","dependencies":["a4a0f7dradicalroot2a-h6"],"title":"Principle of Operation","text":"$$7$$ on the right side also has to be squared","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot2a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$49$$"],"dependencies":["a4a0f7dradicalroot2a-h7"],"title":"Squaring","text":"What is $$7^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot2a-h9","type":"hint","dependencies":["a4a0f7dradicalroot2a-h8"],"title":"Organizing","text":"The equation becomes $$2x-1=49$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot2a-h10","type":"hint","dependencies":["a4a0f7dradicalroot2a-h9"],"title":"Operation","text":"Add $$1$$ on both side","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot2a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x$$"],"dependencies":["a4a0f7dradicalroot2a-h10"],"title":"Addition","text":"What is $$2x-1+1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot2a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$50$$"],"dependencies":["a4a0f7dradicalroot2a-h11"],"title":"Addition","text":"What is $$49+1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot2a-h13","type":"hint","dependencies":["a4a0f7dradicalroot2a-h12"],"title":"Organizing","text":"The equation becomes $$2x=50$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot2a-h14","type":"hint","dependencies":["a4a0f7dradicalroot2a-h13"],"title":"Dividing","text":"Diving $$2x$$ by $$2$$ to produce a single $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot2a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["a4a0f7dradicalroot2a-h14"],"title":"Dividing","text":"What is $$50$$ divided by 2?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4a0f7dradicalroot20","title":"Solving Equations with Square Roots","body":"Solve $$\\\\sqrt{2m-3}-5=0$$ (Please input all answers as $$variable=answer$$, if defined)","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Solve Equations with Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4a0f7dradicalroot20a","stepAnswer":["$$m=14$$"],"problemType":"TextBox","stepTitle":"Solve $$\\\\sqrt{2m-3}-5=0$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$m=14$$","hints":{"DefaultPathway":[{"id":"a4a0f7dradicalroot20a-h1","type":"hint","dependencies":[],"title":"Isolating the Radical","text":"First you must isolate the radical. If this is already done then think about how we can isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{2m-3}=5$$"],"dependencies":["a4a0f7dradicalroot20a-h1"],"title":"Isolating the Radical","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot20a-h3","type":"hint","dependencies":["a4a0f7dradicalroot20a-h2"],"title":"Isolating the Variable","text":"Square both sides of the equation and isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot20a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2m=28$$"],"dependencies":["a4a0f7dradicalroot20a-h3"],"title":"Isolating the Variable","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot20a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$m=14$$"],"dependencies":["a4a0f7dradicalroot20a-h4"],"title":"Solving the Equation","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot20a-h6","type":"hint","dependencies":["a4a0f7dradicalroot20a-h5"],"title":"Double Checking","text":"Plug the answer back in to check if the answer is correct. This needs to be done because the radical can lead to some equations having no answers. If it doesn\'t work then there are no solutions to the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4a0f7dradicalroot21","title":"Solving Equations with Square Roots","body":"Solve $$\\\\sqrt{2n-1}-3=0$$ (Please input all answers as $$variable=answer$$, if defined)","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Solve Equations with Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4a0f7dradicalroot21a","stepAnswer":["$$n=5$$"],"problemType":"TextBox","stepTitle":"Solve $$\\\\sqrt{2n-1}-3=0$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$n=5$$","hints":{"DefaultPathway":[{"id":"a4a0f7dradicalroot21a-h1","type":"hint","dependencies":[],"title":"Isolating the Radical","text":"First you must isolate the radical. If this is already done then think about how we can isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot21a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{2n-1}=3$$"],"dependencies":["a4a0f7dradicalroot21a-h1"],"title":"Isolating the Radical","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot21a-h3","type":"hint","dependencies":["a4a0f7dradicalroot21a-h2"],"title":"Isolating the Variable","text":"Square both sides of the equation and isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot21a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2n=10$$"],"dependencies":["a4a0f7dradicalroot21a-h3"],"title":"Isolating the Variable","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot21a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$n=5$$"],"dependencies":["a4a0f7dradicalroot21a-h4"],"title":"Solving the Equation","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot21a-h6","type":"hint","dependencies":["a4a0f7dradicalroot21a-h5"],"title":"Double Checking","text":"Plug the answer back in to check if the answer is correct. This needs to be done because the radical can lead to some equations having no answers. If it doesn\'t work then there are no solutions to the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4a0f7dradicalroot22","title":"Solving Equations with Square Roots","body":"Solve $$\\\\sqrt{6v-2}=10$$ (Please input all answers as $$variable=answer$$, if defined)","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Solve Equations with Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4a0f7dradicalroot22a","stepAnswer":["$$v=17$$"],"problemType":"TextBox","stepTitle":"Solve $$\\\\sqrt{6v-2}=10$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$v=17$$","hints":{"DefaultPathway":[{"id":"a4a0f7dradicalroot22a-h1","type":"hint","dependencies":[],"title":"Isolating the Radical","text":"First you must isolate the radical. If this is already done then think about how we can isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot22a-h2","type":"hint","dependencies":["a4a0f7dradicalroot22a-h1"],"title":"Isolating the Variable","text":"Square both sides of the equation and isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot22a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6v=102$$"],"dependencies":["a4a0f7dradicalroot22a-h2"],"title":"Isolating the Variable","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot22a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$v=17$$"],"dependencies":["a4a0f7dradicalroot22a-h3"],"title":"Solving the Equation","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot22a-h5","type":"hint","dependencies":["a4a0f7dradicalroot22a-h4"],"title":"Double Checking","text":"Plug the answer back in to check if the answer is correct. This needs to be done because the radical can lead to some equations having no answers. If it doesn\'t work then there are no solutions to the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4a0f7dradicalroot23","title":"Solving Equations with Square Roots","body":"Solve $$\\\\sqrt{4u+2}-6=0$$ (Please input all answers as $$variable=answer$$, if defined, with decimals if necessary)","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Solve Equations with Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4a0f7dradicalroot23a","stepAnswer":["$$u=8.5$$"],"problemType":"TextBox","stepTitle":"Solve $$\\\\sqrt{4u+2}-6=0$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$u=8.5$$","hints":{"DefaultPathway":[{"id":"a4a0f7dradicalroot23a-h1","type":"hint","dependencies":[],"title":"Isolating the Radical","text":"First you must isolate the radical. If this is already done then think about how we can isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot23a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{4u+2}=6$$"],"dependencies":["a4a0f7dradicalroot23a-h1"],"title":"Isolating the Radical","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot23a-h3","type":"hint","dependencies":["a4a0f7dradicalroot23a-h2"],"title":"Isolating the Variable","text":"Square both sides of the equation and isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot23a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4u=34$$"],"dependencies":["a4a0f7dradicalroot23a-h3"],"title":"Isolating the Variable","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot23a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$u=8.5$$"],"dependencies":["a4a0f7dradicalroot23a-h4"],"title":"Solving the Equation","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot23a-h6","type":"hint","dependencies":["a4a0f7dradicalroot23a-h5"],"title":"Double Checking","text":"Plug the answer back in to check if the answer is correct. This needs to be done because the radical can lead to some equations having no answers. If it doesn\'t work then there are no solutions to the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4a0f7dradicalroot24","title":"Solving Equations with Square Roots","body":"Solve $$\\\\sqrt{5q+3}-4=0$$ (Please input all answers as $$variable=answer$$, if defined, with decimals if necessary)","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Solve Equations with Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4a0f7dradicalroot24a","stepAnswer":["$$q=2.6$$"],"problemType":"TextBox","stepTitle":"Solve $$\\\\sqrt{5q+3}-4=0$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$q=2.6$$","hints":{"DefaultPathway":[{"id":"a4a0f7dradicalroot24a-h1","type":"hint","dependencies":[],"title":"Isolating the Radical","text":"First you must isolate the radical. If this is already done then think about how we can isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot24a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{5q+3}=4$$"],"dependencies":["a4a0f7dradicalroot24a-h1"],"title":"Isolating the Radical","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot24a-h3","type":"hint","dependencies":["a4a0f7dradicalroot24a-h2"],"title":"Isolating the Variable","text":"Square both sides of the equation and isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot24a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5q=13$$"],"dependencies":["a4a0f7dradicalroot24a-h3"],"title":"Isolating the Variable","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot24a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$q=2.6$$"],"dependencies":["a4a0f7dradicalroot24a-h4"],"title":"Solving the Equation","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot24a-h6","type":"hint","dependencies":["a4a0f7dradicalroot24a-h5"],"title":"Double Checking","text":"Plug the answer back in to check if the answer is correct. This needs to be done because the radical can lead to some equations having no answers. If it doesn\'t work then there are no solutions to the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4a0f7dradicalroot25","title":"Solving Equations with Square Roots","body":"Solve $$\\\\sqrt{4m+2}+2=6$$ (Please input all answers as $$variable=answer$$, if defined, with decimals if necessary)","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Solve Equations with Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4a0f7dradicalroot25a","stepAnswer":["$$m=3.5$$"],"problemType":"TextBox","stepTitle":"Solve $$\\\\sqrt{4m+2}+2=6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$m=3.5$$","hints":{"DefaultPathway":[{"id":"a4a0f7dradicalroot25a-h1","type":"hint","dependencies":[],"title":"Isolating the Radical","text":"First you must isolate the radical. If this is already done then think about how we can isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot25a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{4m+2}=4$$"],"dependencies":["a4a0f7dradicalroot25a-h1"],"title":"Isolating the Radical","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot25a-h3","type":"hint","dependencies":["a4a0f7dradicalroot25a-h2"],"title":"Isolating the Variable","text":"Square both sides of the equation and isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot25a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4m=14$$"],"dependencies":["a4a0f7dradicalroot25a-h3"],"title":"Isolating the Variable","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot25a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$m=3$$ or $$m=5$$"],"dependencies":["a4a0f7dradicalroot25a-h4"],"title":"Solving the Equation","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$m=3$$ or $$m=4$$","$$m=2$$ or $$m=5$$","$$m=3$$ or $$m=5$$"]},{"id":"a4a0f7dradicalroot25a-h6","type":"hint","dependencies":["a4a0f7dradicalroot25a-h5"],"title":"Double Checking","text":"Plug the answer back in to check if the answer is correct. This needs to be done because the radical can lead to some equations having no answers. If it doesn\'t work then there are no solutions to the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4a0f7dradicalroot26","title":"Solving Equations with Square Roots","body":"Solve $$\\\\sqrt{6n+1}+4=8$$ (Please input all answers as $$variable=answer$$, if defined, with decimals if necessary)","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Solve Equations with Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4a0f7dradicalroot26a","stepAnswer":["$$n=2.5$$"],"problemType":"TextBox","stepTitle":"Solve $$\\\\sqrt{6n+1}+4=8$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$n=2.5$$","hints":{"DefaultPathway":[{"id":"a4a0f7dradicalroot26a-h1","type":"hint","dependencies":[],"title":"Isolating the Radical","text":"First you must isolate the radical. If this is already done then think about how we can isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot26a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{6n+1}=4$$"],"dependencies":["a4a0f7dradicalroot26a-h1"],"title":"Isolating the Radical","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot26a-h3","type":"hint","dependencies":["a4a0f7dradicalroot26a-h2"],"title":"Isolating the Variable","text":"Square both sides of the equation and isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot26a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6n=15$$"],"dependencies":["a4a0f7dradicalroot26a-h3"],"title":"Isolating the Variable","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot26a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$n=2.5$$"],"dependencies":["a4a0f7dradicalroot26a-h4"],"title":"Solving the Equation","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot26a-h6","type":"hint","dependencies":["a4a0f7dradicalroot26a-h5"],"title":"Double Checking","text":"Plug the answer back in to check if the answer is correct. This needs to be done because the radical can lead to some equations having no answers. If it doesn\'t work then there are no solutions to the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4a0f7dradicalroot27","title":"Solving Equations with Square Roots","body":"Solve $$\\\\sqrt{2u-3}+2=0$$ (Please input all answers as $$variable=answer$$, if defined, with decimals if necessary. If there is no answer simply write DNE (Does Not Exist)","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Solve Equations with Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4a0f7dradicalroot27a","stepAnswer":["DNE"],"problemType":"TextBox","stepTitle":"Solve $$\\\\sqrt{2u-3}+2=0$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a4a0f7dradicalroot27a-h1","type":"hint","dependencies":[],"title":"Isolating the Radical","text":"First you must isolate the radical. If this is already done then think about how we can isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot27a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{2u-3}=-2$$"],"dependencies":["a4a0f7dradicalroot27a-h1"],"title":"Isolating the Radical","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot27a-h3","type":"hint","dependencies":["a4a0f7dradicalroot27a-h2"],"title":"Isolating the Variable","text":"Square both sides of the equation and isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot27a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2u=7$$"],"dependencies":["a4a0f7dradicalroot27a-h3"],"title":"Isolating the Variable","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot27a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$u=3.5$$"],"dependencies":["a4a0f7dradicalroot27a-h4"],"title":"Solving the Equation","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot27a-h6","type":"hint","dependencies":["a4a0f7dradicalroot27a-h5"],"title":"Double Checking","text":"Plug the answer back in to check if the answer is correct. This needs to be done because the radical can lead to some equations having no answers. If it doesn\'t work then there are no solutions to the equation. The solution does not exist.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4a0f7dradicalroot28","title":"Solving Equations with Square Roots","body":"Solve $$\\\\sqrt{5v-2}+5=0$$ (Please input all answers as $$variable=answer$$, if defined, with decimals if necessary. If there is no answer simply write DNE (Does Not Exist)","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Solve Equations with Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4a0f7dradicalroot28a","stepAnswer":["DNE"],"problemType":"TextBox","stepTitle":"Solve $$\\\\sqrt{5v-2}+5=0$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a4a0f7dradicalroot28a-h1","type":"hint","dependencies":[],"title":"Isolating the Radical","text":"First you must isolate the radical. If this is already done then think about how we can isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot28a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{5v-2}=-5$$"],"dependencies":["a4a0f7dradicalroot28a-h1"],"title":"Isolating the Radical","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot28a-h3","type":"hint","dependencies":["a4a0f7dradicalroot28a-h2"],"title":"Isolating the Variable","text":"Square both sides of the equation and isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot28a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5v=27$$"],"dependencies":["a4a0f7dradicalroot28a-h3"],"title":"Isolating the Variable","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot28a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$v=5.4$$"],"dependencies":["a4a0f7dradicalroot28a-h4"],"title":"Solving the Equation","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot28a-h6","type":"hint","dependencies":["a4a0f7dradicalroot28a-h5"],"title":"Double Checking","text":"Plug the answer back in to check if the answer is correct. This needs to be done because the radical can lead to some equations having no answers. If it doesn\'t work then there are no solutions to the equation. The solution does not exist.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4a0f7dradicalroot29","title":"Solving Equations with Square Roots","body":"Solve $$\\\\sqrt{3z-5}+2=0$$ (Please input all answers as $$variable=answer$$, if defined, with decimals if necessary. If there is no answer simply write DNE (Does Not Exist)","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Solve Equations with Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4a0f7dradicalroot29a","stepAnswer":["DNE"],"problemType":"TextBox","stepTitle":"Solve $$\\\\sqrt{3z-5}+2=0$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a4a0f7dradicalroot29a-h1","type":"hint","dependencies":[],"title":"Isolating the Radical","text":"First you must isolate the radical. If this is already done then think about how we can isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot29a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{3z-5}=-2$$"],"dependencies":["a4a0f7dradicalroot29a-h1"],"title":"Isolating the Radical","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot29a-h3","type":"hint","dependencies":["a4a0f7dradicalroot29a-h2"],"title":"Isolating the Variable","text":"Square both sides of the equation and isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot29a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3z=9$$"],"dependencies":["a4a0f7dradicalroot29a-h3"],"title":"Isolating the Variable","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot29a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$z=3$$"],"dependencies":["a4a0f7dradicalroot29a-h4"],"title":"Solving the Equation","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot29a-h6","type":"hint","dependencies":["a4a0f7dradicalroot29a-h5"],"title":"Double Checking","text":"Plug the answer back in to check if the answer is correct. This needs to be done because the radical can lead to some equations having no answers. If it doesn\'t work then there are no solutions to the equation. The solution does not exist.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4a0f7dradicalroot3","title":"Solving Radical Equations","body":"Find the value of $$y$$.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Solve Equations with Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4a0f7dradicalroot3a","stepAnswer":["$$\\\\frac{4}{3}$$"],"problemType":"TextBox","stepTitle":"Solve: $$\\\\sqrt{3y+5}+2=5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{4}{3}$$","hints":{"DefaultPathway":[{"id":"a4a0f7dradicalroot3a-h1","type":"hint","dependencies":[],"title":"Setup","text":"Simplifying the equation to $$y=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot3a-h2","type":"hint","dependencies":["a4a0f7dradicalroot3a-h1"],"title":"Setup","text":"Isolating $$y$$ function","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot3a-h3","type":"hint","dependencies":["a4a0f7dradicalroot3a-h2"],"title":"Operation","text":"Subtracting $$2$$ on both sides to remove the $$+2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot3a-h4","type":"hint","dependencies":["a4a0f7dradicalroot3a-h3"],"title":"Order of Operation","text":"Solving the square root","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot3a-h5","type":"hint","dependencies":["a4a0f7dradicalroot3a-h4"],"title":"Calculation","text":"Square root can be removed by squaring both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot3a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3y+5$$"],"dependencies":["a4a0f7dradicalroot3a-h5"],"title":"Squaring","text":"What is the square of $$\\\\sqrt{3y+5}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot3a-h7","type":"hint","dependencies":["a4a0f7dradicalroot3a-h6"],"title":"Operation","text":"Square root can be canceled by squaring","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot3a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3y+5$$"],"dependencies":["a4a0f7dradicalroot3a-h7"],"title":"Operation","text":"What is $$\\\\sqrt{3y+5}$$ without square root?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot3a-h9","type":"hint","dependencies":["a4a0f7dradicalroot3a-h8"],"title":"Principle of Operation","text":"$$-2$$ on the right side also has to be squared","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot3a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a4a0f7dradicalroot3a-h9"],"title":"Squaring","text":"What is $$3^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot3a-h11","type":"hint","dependencies":["a4a0f7dradicalroot3a-h10"],"title":"Organizing","text":"The equation becomes $$3y+5=9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot3a-h12","type":"hint","dependencies":["a4a0f7dradicalroot3a-h11"],"title":"Operation","text":"Subtracting $$5$$ on both side","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot3a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3y$$"],"dependencies":["a4a0f7dradicalroot3a-h12"],"title":"Subtraction","text":"What is $$3y+5-5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot3a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a4a0f7dradicalroot3a-h13"],"title":"Subtraction","text":"What is $$9-5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot3a-h15","type":"hint","dependencies":["a4a0f7dradicalroot3a-h14"],"title":"Organizing","text":"The equation becomes $$2x=50$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot3a-h16","type":"hint","dependencies":["a4a0f7dradicalroot3a-h15"],"title":"Dividing","text":"Diving $$3y$$ by $$3$$ to produce a single $$y$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot3a-h17","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{4}{3}$$"],"dependencies":["a4a0f7dradicalroot3a-h16"],"title":"Dividing","text":"What is $$4$$ divided by 3?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4a0f7dradicalroot30","title":"Solving Equations with Square Roots","body":"Solve $$\\\\sqrt{2m+1}+4=0$$ (Please input all answers as $$variable=answer$$, if defined, with decimals if necessary. If there is no answer simply write DNE, which stands for Does Not Exist.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Solve Equations with Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4a0f7dradicalroot30a","stepAnswer":["DNE"],"problemType":"TextBox","stepTitle":"Solve $$\\\\sqrt{2m+1}+4=0$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a4a0f7dradicalroot30a-h1","type":"hint","dependencies":[],"title":"Isolating the Radical","text":"First you must isolate the radical. If this is already done then think about how we can isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot30a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{2m+1}=-4$$"],"dependencies":["a4a0f7dradicalroot30a-h1"],"title":"Isolating the Radical","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot30a-h3","type":"hint","dependencies":["a4a0f7dradicalroot30a-h2"],"title":"Isolating the Variable","text":"Square both sides of the equation and isolate the variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot30a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2m=15$$"],"dependencies":["a4a0f7dradicalroot30a-h3"],"title":"Isolating the Variable","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot30a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$m=7.5$$"],"dependencies":["a4a0f7dradicalroot30a-h4"],"title":"Solving the Equation","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot30a-h6","type":"hint","dependencies":["a4a0f7dradicalroot30a-h5"],"title":"Double Checking","text":"Plug the answer back in to check if the answer is correct. This needs to be done because the radical can lead to some equations having no answers. If it doesn\'t work then there are no solutions to the equation. The solution does not exist.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4a0f7dradicalroot4","title":"Solving Radical Equations","body":"Find the value of $$m$$.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Solve Equations with Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4a0f7dradicalroot4a","stepAnswer":["$$7$$"],"problemType":"TextBox","stepTitle":"Solve:sqrt(m+9)=m+3","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7$$","hints":{"DefaultPathway":[{"id":"a4a0f7dradicalroot4a-h1","type":"hint","dependencies":[],"title":"Setup","text":"Simplifying the equation to $$m=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot4a-h2","type":"hint","dependencies":["a4a0f7dradicalroot4a-h1"],"title":"Setup","text":"Isolating $$m$$ function by moving $$-m$$ to right side","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot4a-h3","type":"hint","dependencies":["a4a0f7dradicalroot4a-h2"],"title":"Operation","text":"Subtracting $$3$$ on both sides to remove the $$+3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot4a-h4","type":"hint","dependencies":["a4a0f7dradicalroot4a-h3"],"title":"Organizing","text":"The equation becomes $$\\\\sqrt{m+9}=m-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot4a-h5","type":"hint","dependencies":["a4a0f7dradicalroot4a-h4"],"title":"Order of Operation","text":"Solving the square root","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot4a-h6","type":"hint","dependencies":["a4a0f7dradicalroot4a-h5"],"title":"Calculation","text":"Square root can be removed by squaring both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot4a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$m+9$$"],"dependencies":["a4a0f7dradicalroot4a-h6"],"title":"Squaring","text":"What is the square of $$\\\\sqrt{m+9}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot4a-h8","type":"hint","dependencies":["a4a0f7dradicalroot4a-h7"],"title":"Operation","text":"Square root can be canceled by squaring","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot4a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$m+9$$"],"dependencies":["a4a0f7dradicalroot4a-h8"],"title":"Operation","text":"What is $$\\\\sqrt{m+9}$$ without square root?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot4a-h10","type":"hint","dependencies":["a4a0f7dradicalroot4a-h9"],"title":"Principle of Operation","text":"$$m-3$$ on the right side also has to be squared","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot4a-h11","type":"hint","dependencies":["a4a0f7dradicalroot4a-h10"],"title":"Binomial Squares","text":"Recall the binomial square formula: $${\\\\left(a+b\\\\right)}^2=a^2+2ab+b^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot4a-h12","type":"hint","dependencies":["a4a0f7dradicalroot4a-h11"],"title":"Substitution","text":"For $$m-3$$, $$a=m$$ $$b=-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot4a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$m^2-2m+1$$"],"dependencies":["a4a0f7dradicalroot4a-h12"],"title":"Substitution","text":"What is $${\\\\left(m-1\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot4a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$m^2$$"],"dependencies":["a4a0f7dradicalroot4a-h13"],"title":"Substitution","text":"What is $$a^2$$ when $$a=m$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot4a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6m$$"],"dependencies":["a4a0f7dradicalroot4a-h14"],"title":"Substitution","text":"What is 2ab when $$a=m$$, $$b=-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot4a-h16","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a4a0f7dradicalroot4a-h15"],"title":"Substitution","text":"What is $$b^2$$ when $$b=-3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot4a-h17","type":"hint","dependencies":["a4a0f7dradicalroot4a-h16"],"title":"Organizing","text":"The equation becomes $$m+9=m^2-6m+9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot4a-h18","type":"hint","dependencies":["a4a0f7dradicalroot4a-h17"],"title":"Subtraction","text":"Subtracting $$9$$ on both sides to remove the $$+9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot4a-h19","type":"hint","dependencies":["a4a0f7dradicalroot4a-h18"],"title":"Organizing","text":"Moving the $$m$$ term on left side to right side, forming a new equation as $$m^2-7m$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot4a-h20","type":"hint","dependencies":["a4a0f7dradicalroot4a-h19"],"title":"Factor","text":"Factor out $$m$$ from $$m^2-7m$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot4a-h21","type":"hint","dependencies":["a4a0f7dradicalroot4a-h20"],"title":"Organizing","text":"The equation becomes $$m(m-7)=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot4a-h22","type":"hint","dependencies":["a4a0f7dradicalroot4a-h21"],"title":"Principle","text":"To make equation to $$0$$, either $$m=0$$ or $$m-7=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot4a-h23","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a4a0f7dradicalroot4a-h22"],"title":"Calculation","text":"If $$m-7=0$$, $$m=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4a0f7dradicalroot5","title":"Solving Radical Equations","body":"Find the value of the variable.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Solve Equations with Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4a0f7dradicalroot5a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"Solve: $$\\\\sqrt{4z-3}=\\\\sqrt{3z+2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"a4a0f7dradicalroot5a-h1","type":"hint","dependencies":[],"title":"Order of Operation","text":"Solving the square root","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot5a-h2","type":"hint","dependencies":["a4a0f7dradicalroot5a-h1"],"title":"Calculation","text":"Square root can be removed by squaring both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4z-3$$"],"dependencies":["a4a0f7dradicalroot5a-h2"],"title":"Squaring","text":"What is the square of $$\\\\sqrt{4z-3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3z+2$$"],"dependencies":["a4a0f7dradicalroot5a-h3"],"title":"Squaring","text":"What is the square of $$\\\\sqrt{3z+2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot5a-h5","type":"hint","dependencies":["a4a0f7dradicalroot5a-h4"],"title":"Organizing","text":"The equation becomes $$4z-3=3z+2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot5a-h6","type":"hint","dependencies":["a4a0f7dradicalroot5a-h5"],"title":"Organizing","text":"Isolating the $$z$$ term to left side","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot5a-h7","type":"hint","dependencies":["a4a0f7dradicalroot5a-h6"],"title":"Organizing","text":"The equation becomes $$4z-3z=3+2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot5a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$z$$"],"dependencies":["a4a0f7dradicalroot5a-h7"],"title":"Calculation","text":"What is $$4z-3z$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot5a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a4a0f7dradicalroot5a-h8"],"title":"Addition","text":"What is $$3+2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4a0f7dradicalroot6","title":"Solving Radical Equations","body":"Find the value of the variable.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Solve Equations with Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4a0f7dradicalroot6a","stepAnswer":["$$\\\\frac{16}{36}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{x}+3=\\\\sqrt{x+5}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{16}{36}$$","hints":{"DefaultPathway":[{"id":"a4a0f7dradicalroot6a-h1","type":"hint","dependencies":[],"title":"Order of Operation","text":"Solving the square root","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot6a-h2","type":"hint","dependencies":["a4a0f7dradicalroot6a-h1"],"title":"Calculation","text":"Square root can be removed by squaring both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+5$$"],"dependencies":["a4a0f7dradicalroot6a-h2"],"title":"Squaring","text":"What is the square of $$\\\\sqrt{x+5}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot6a-h4","type":"hint","dependencies":["a4a0f7dradicalroot6a-h3"],"title":"Operation","text":"Apply binomial formula on $${\\\\left(\\\\sqrt{x}+3\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot6a-h5","type":"hint","dependencies":["a4a0f7dradicalroot6a-h4"],"title":"Substitution","text":"$$a=\\\\sqrt{x}$$, $$b=3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot6a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x$$"],"dependencies":["a4a0f7dradicalroot6a-h5"],"title":"Substitution","text":"What is $$a^2$$ when $$a=\\\\sqrt{x}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a4a0f7dradicalroot6a-h6-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6\\\\sqrt{x}$$"],"dependencies":[],"title":"Substitution","text":"What is 2ab when $$a=\\\\sqrt{x}$$, $$b=3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot6a-h6-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":[],"title":"Substitution","text":"What is $$b^2$$ when $$b=3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a4a0f7dradicalroot6a-h7","type":"hint","dependencies":["a4a0f7dradicalroot6a-h6"],"title":"Organizing","text":"The equation becomes $$x+6\\\\sqrt{x}+9=x+5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot6a-h8","type":"hint","dependencies":["a4a0f7dradicalroot6a-h7"],"title":"Organizing","text":"Isolating the $$x$$ term to left side","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot6a-h9","type":"hint","dependencies":["a4a0f7dradicalroot6a-h8"],"title":"Organizing","text":"The equation becomes $$x+6\\\\sqrt{x}-x=5-9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot6a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6\\\\sqrt{x}$$"],"dependencies":["a4a0f7dradicalroot6a-h9"],"title":"Calculation","text":"What does $$x+6\\\\sqrt{x}-x=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot6a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["a4a0f7dradicalroot6a-h10"],"title":"Calculation","text":"What does $$5-9=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot6a-h12","type":"hint","dependencies":["a4a0f7dradicalroot6a-h11"],"title":"Organizing","text":"The equation becomes $$6\\\\sqrt{x}=-4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot6a-h13","type":"hint","dependencies":["a4a0f7dradicalroot6a-h12"],"title":"Dividing","text":"Divding both side by $$6$$ to obtain $$\\\\sqrt{x}=\\\\frac{-4}{6}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot6a-h14","type":"hint","dependencies":["a4a0f7dradicalroot6a-h13"],"title":"Calculation","text":"Square root can be removed by squaring both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot6a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{16}{36}$$"],"dependencies":["a4a0f7dradicalroot6a-h14"],"title":"Squaring","text":"What is $${\\\\left(-\\\\frac{4}{6}\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4a0f7dradicalroot7","title":"Solving Radical Equations","body":"Find the value of the variable.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Solve Equations with Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4a0f7dradicalroot7a","stepAnswer":["$$q=6$$ or $$q=2$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\sqrt{q-2}+3=\\\\sqrt{4q+1}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$q=6$$ or $$q=2$$","choices":["$$q=2$$ or $$q=3$$","$$q=5$$ or $$q=2$$","$$q=6$$ or $$q=2$$"],"hints":{"DefaultPathway":[{"id":"a4a0f7dradicalroot7a-h1","type":"hint","dependencies":[],"title":"Order of Operation","text":"Solving the square root","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot7a-h2","type":"hint","dependencies":["a4a0f7dradicalroot7a-h1"],"title":"Calculation","text":"Square root can be removed by squaring both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4q+1$$"],"dependencies":["a4a0f7dradicalroot7a-h2"],"title":"Squaring","text":"What is the square of $$\\\\sqrt{4q+1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot7a-h4","type":"hint","dependencies":["a4a0f7dradicalroot7a-h3"],"title":"Operation","text":"Apply binomial formula on $${\\\\left(\\\\sqrt{q-2}+3\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot7a-h5","type":"hint","dependencies":["a4a0f7dradicalroot7a-h4"],"title":"Substitution","text":"$$a=\\\\sqrt{q-2}$$, $$b=3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot7a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["q-2"],"dependencies":["a4a0f7dradicalroot7a-h5"],"title":"Substitution","text":"What is $$a^2$$ when $$a=\\\\sqrt{q-2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a4a0f7dradicalroot7a-h6-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6\\\\sqrt{q-2}$$"],"dependencies":[],"title":"Substitution","text":"What is 2ab when $$a=\\\\sqrt{q-2}$$, $$b=3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot7a-h6-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":[],"title":"Substitution","text":"What is $$b^2$$ when $$b=3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a4a0f7dradicalroot7a-h7","type":"hint","dependencies":["a4a0f7dradicalroot7a-h6"],"title":"Organizing","text":"The equation becomes $$q-2+6\\\\sqrt{q-2}+9=4q+1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot7a-h8","type":"hint","dependencies":["a4a0f7dradicalroot7a-h7"],"title":"Organizing","text":"Isolating the $$\\\\sqrt{q-2}$$ term to left side","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot7a-h9","type":"hint","dependencies":["a4a0f7dradicalroot7a-h8"],"title":"Organizing","text":"The equation becomes $$6\\\\sqrt{q-2}=3q-6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot7a-h10","type":"hint","dependencies":["a4a0f7dradicalroot7a-h9"],"title":"Calculation","text":"Square root can be removed by squaring both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot7a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["36q-72"],"dependencies":["a4a0f7dradicalroot7a-h10"],"title":"Squaring","text":"What is the square of $$6\\\\sqrt{q-2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a4a0f7dradicalroot7a-h11-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["36q-72"],"dependencies":[],"title":"Squaring","text":"What is $$6^2 {\\\\sqrt{q-2}}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot7a-h11-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["36q-72"],"dependencies":[],"title":"Multiplication","text":"What is $$36\\\\left(q-2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a4a0f7dradicalroot7a-h12","type":"hint","dependencies":["a4a0f7dradicalroot7a-h11"],"title":"Operation","text":"Apply binomial formula on $${\\\\left(3q-6\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot7a-h13","type":"hint","dependencies":["a4a0f7dradicalroot7a-h12"],"title":"Substitution","text":"$$a=3q$$, $$b=-6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot7a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["9q"],"dependencies":["a4a0f7dradicalroot7a-h13"],"title":"Substitution","text":"What is $$a^2$$ when $$a=3q$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot7a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["-36q"],"dependencies":["a4a0f7dradicalroot7a-h14"],"title":"Substitution","text":"What is 2ab when $$a=3q$$, $$b=-6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot7a-h16","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$36$$"],"dependencies":["a4a0f7dradicalroot7a-h15"],"title":"Substitution","text":"What is $$b^2$$ when $$b=6$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot7a-h17","type":"hint","dependencies":["a4a0f7dradicalroot7a-h16"],"title":"Organizing","text":"The equation becomes $$36q-72=9q^2-36q+36$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot7a-h18","type":"hint","dependencies":["a4a0f7dradicalroot7a-h17"],"title":"Organizing","text":"Move every term to right side","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot7a-h19","type":"hint","dependencies":["a4a0f7dradicalroot7a-h18"],"title":"Organizing","text":"The equation becomes $$0=9q^2-72q+108$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot7a-h20","type":"hint","dependencies":["a4a0f7dradicalroot7a-h19"],"title":"Factoring","text":"The equation can be factored out as $$0=9(q-6)(q-2)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot7a-h21","type":"hint","dependencies":["a4a0f7dradicalroot7a-h20"],"title":"Principle","text":"To make the equation valid, either $$(q-6)$$ or $$(q-2)$$ has to be $$0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot7a-h22","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a4a0f7dradicalroot7a-h21"],"title":"Calculation","text":"If $$q-6=0$$, $$q=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot7a-h23","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a4a0f7dradicalroot7a-h22"],"title":"Calculation","text":"If $$q-2=0$$, $$q=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4a0f7dradicalroot8","title":"Use Square Roots in Application","body":"Mike and Lychelle want to make a square patio. They have enough concrete to pave an area of $$200$$ square feet.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Solve Equations with Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4a0f7dradicalroot8a","stepAnswer":["$$14.1$$"],"problemType":"TextBox","stepTitle":"Use the formula $$s=\\\\sqrt{A}$$ to find the length of each side of the patio. Round your answer to the nearest tenth of a foot.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$14.1$$","hints":{"DefaultPathway":[{"id":"a4a0f7dradicalroot8a-h1","type":"hint","dependencies":[],"title":"Identifying","text":"$$A=area$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot8a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Side"],"dependencies":["a4a0f7dradicalroot8a-h1"],"title":"Identifying","text":"What is s?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Perimeter","Diagonal","Side"]},{"id":"a4a0f7dradicalroot8a-h3","type":"hint","dependencies":["a4a0f7dradicalroot8a-h2"],"title":"Formula","text":"For a square, its $$area={side}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{200}$$"],"dependencies":["a4a0f7dradicalroot8a-h3"],"title":"Substitution","text":"What is s when $$A=200$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a4a0f7dradicalroot8a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$14.1$$"],"dependencies":[],"title":"Square Root","text":"What is $$\\\\sqrt{200}$$? (To the nearest tenth)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a4a0f7dradicalroot9","title":"Falling Objects","body":"Christy dropped her sunglasses from a bridge $$400$$ feet above a river.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Solve Equations with Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4a0f7dradicalroot9a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"Use the formula $$t=\\\\frac{\\\\sqrt{h}}{4}$$ to find how many seconds it took for the sunglasses to reach the river.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"a4a0f7dradicalroot9a-h1","type":"hint","dependencies":[],"title":"Setup","text":"$$h=height(in$$ feet), $$t=time(in$$ second)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a4a0f7dradicalroot9a-h1"],"title":"Substitution","text":"What is s when $$h=400$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4a0f7dradicalroot9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a4a0f7dradicalroot9a-h2"],"title":"Square Root","text":"What is $$\\\\sqrt{400}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a4a0f7dradicalroot9a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":[],"title":"Dividing","text":"What is $$20$$ divided by 4?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a4b48f4RealNums1","title":"Combining Like Terms","body":"Simplify this expression by combining like terms.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Properties of Real Numbers","courseName":"OpenStax: Intermediate 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Using this knowledge we can scale down $$12$$ in the numerator and $$6$$ in the denominator to $$\\\\frac{2}{1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4b48f4RealNums26a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10p$$"],"dependencies":["a4b48f4RealNums26a-h2"],"title":"Simplifying the Expression","text":"What is the simplified expression once fully multiplied?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4b48f4RealNums27","title":"Use the Commutative and Associative Properties","body":"Simplify the following exercise.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Properties of Real Numbers","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4b48f4RealNums27a","stepAnswer":["$$12q$$"],"problemType":"TextBox","stepTitle":"$$20\\\\frac{3}{5} q$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12q$$","hints":{"DefaultPathway":[{"id":"a4b48f4RealNums27a-h1","type":"hint","dependencies":[],"title":"Applying Associative Property","text":"Using the Associative Property of Multiplication, we know that the three terms can be multiplied in any order.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4b48f4RealNums27a-h2","type":"hint","dependencies":["a4b48f4RealNums27a-h1"],"title":"Identifying Simplifications","text":"Notice that $$20$$ is a multiple of $$5$$ by a factor of $$4$$. Using this knowledge we can scale down $$20$$ in the numerator and $$5$$ in the denominator to $$\\\\frac{4}{1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4b48f4RealNums27a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12q$$"],"dependencies":["a4b48f4RealNums27a-h2"],"title":"Simplifying the Expression","text":"What is the simplified expression once fully multiplied?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4b48f4RealNums28","title":"Use the Commutative and Associative Properties","body":"Simplify the following exercise.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Properties of Real Numbers","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4b48f4RealNums28a","stepAnswer":["$$a+\\\\frac{6}{5} b$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{5}{6} a+\\\\frac{3}{10} b+\\\\frac{1}{6} a+\\\\frac{9}{10} b$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$a+\\\\frac{6}{5} b$$","hints":{"DefaultPathway":[{"id":"a4b48f4RealNums28a-h1","type":"hint","dependencies":[],"title":"Applying the Commutative Property","text":"Use the Commutative Property of addition to reorder so that like terms are together: $$\\\\frac{5}{6} a+\\\\frac{1}{6} a+\\\\frac{3}{10} b+\\\\frac{9}{10} b$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4b48f4RealNums28a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$a+\\\\frac{6}{5} b$$"],"dependencies":["a4b48f4RealNums28a-h1"],"title":"Simplifying the Expression","text":"What is the simplified expression once like terms are added?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4b48f4RealNums29","title":"Use the Commutative and Associative Properties","body":"Simplify the following exercise.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Properties of Real Numbers","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4b48f4RealNums29a","stepAnswer":["$$\\\\frac{23}{12}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{11}{12}+\\\\frac{4}{9}+\\\\frac{5}{9}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{23}{12}$$","hints":{"DefaultPathway":[{"id":"a4b48f4RealNums29a-h1","type":"hint","dependencies":[],"title":"Applying the Associative Property","text":"The Associative Property of addition shows that the three numbers can be added in any order.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4b48f4RealNums29a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{23}{12}$$"],"dependencies":["a4b48f4RealNums29a-h1"],"title":"Simplifying the Expression","text":"What is the simplified expression once all terms are added?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4b48f4RealNums3","title":"Combining Like Terms","body":"Simplify this expression by combining like terms.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Properties of Real Numbers","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4b48f4RealNums3a","stepAnswer":["$$41m+6n$$"],"problemType":"TextBox","stepTitle":"$$37m+21n+4m-15n$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$41m+6n$$","hints":{"DefaultPathway":[{"id":"a4b48f4RealNums3a-h1","type":"hint","dependencies":[],"title":"Identifying Like Terms","text":"Rearrange the expression so that like terms are next to each other. Write the variable that comes first in the alphabet first for all answers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4b48f4RealNums3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$37m+4m-15n+21n$$"],"dependencies":["a4b48f4RealNums3a-h1"],"title":"Identifying Like Terms","text":"What is the rearranged expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4b48f4RealNums3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$41m+6n$$"],"dependencies":["a4b48f4RealNums3a-h2"],"title":"Combining Like Terms","text":"Combine the terms, what is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4b48f4RealNums30","title":"Use the Commutative and Associative Properties","body":"Simplify the following exercise.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Properties of Real Numbers","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4b48f4RealNums30a","stepAnswer":["$$\\\\frac{11}{6}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{5}{6}+\\\\frac{8}{15}+\\\\frac{7}{15}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{11}{6}$$","hints":{"DefaultPathway":[{"id":"a4b48f4RealNums30a-h1","type":"hint","dependencies":[],"title":"Applying the Associative Property","text":"The Associative Property of addition shows that the three numbers can be added in any order.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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3/4)+1/4","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{18}{13}$$","hints":{"DefaultPathway":[{"id":"a4b48f4RealNums4a-h1","type":"hint","dependencies":[],"title":"Identify Terms with Common Denominators","text":"Notice that the last two terms have the same denominator and they add to $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4b48f4RealNums4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{18}{13}$$"],"dependencies":["a4b48f4RealNums4a-h1"],"title":"Identify Terms with Common Denominators","text":"What is $$1+\\\\frac{5}{13}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4b48f4RealNums5","title":"Adding Fractions","body":"Find the sum of the fractions, written as an improper fraction","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Properties of Real Numbers","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4b48f4RealNums5a","stepAnswer":["$$\\\\frac{22}{15}$$"],"problemType":"TextBox","stepTitle":"(7/15 + 5/8)+3/8","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{22}{15}$$","hints":{"DefaultPathway":[{"id":"a4b48f4RealNums5a-h1","type":"hint","dependencies":[],"title":"Identify Terms with Common Denominators","text":"Notice that the last two terms have the same denominator and they add to $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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terms.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Properties of Real Numbers","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4b48f4RealNums7a","stepAnswer":["$$-73n$$"],"problemType":"TextBox","stepTitle":"$$-84n+\\\\left(-73n\\\\right)+84n$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-73n$$","hints":{"DefaultPathway":[{"id":"a4b48f4RealNums7a-h1","type":"hint","dependencies":[],"title":"Identifying Like Terms","text":"Notice that the first and last terms are opposites of each other meaning they add to equal $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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Terms","body":"Simplify this expression by combining like terms.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Properties of Real Numbers","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4b48f4RealNums9a","stepAnswer":["$$-92x$$"],"problemType":"TextBox","stepTitle":"$$39x+\\\\left(-92x\\\\right)+\\\\left(-39x\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-92x$$","hints":{"DefaultPathway":[{"id":"a4b48f4RealNums9a-h1","type":"hint","dependencies":[],"title":"Identifying Like Terms","text":"Notice that the first and last terms are opposites of each other meaning they add to equal $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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$$ \\\\neq $$ $$0$$, $$m>n$$, $$\\\\frac{a^m}{a^n}$$ $$=$$ $$\\\\frac{1}{a^{n-m}}$$, a $$ \\\\neq $$ $$0$$, $$m>n$$,","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4b9bbfrationalnums15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2$$"],"dependencies":["a4b9bbfrationalnums15a-h2"],"title":"Simplify","text":"Simplify the whole expression","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a4b9bbfrationalnums15b","stepAnswer":["$$y^3$$"],"problemType":"TextBox","stepTitle":"Simplify: (y**(4/3)*y)/(y**(-2/3)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y^3$$","hints":{"DefaultPathway":[{"id":"a4b9bbfrationalnums15b-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{y^{\\\\frac{7}{3}}}{y^{\\\\left(-\\\\frac{2}{3}\\\\right)}}$$"],"dependencies":[],"title":"Product Property","text":"The Power Property: $${\\\\left(a^m\\\\right)}^n$$ $$=$$ $$a^{m n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4b9bbfrationalnums15b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y^{\\\\frac{9}{3}}$$"],"dependencies":["a4b9bbfrationalnums15b-h1"],"title":"Quotient Property","text":"The Quotient Property: $$\\\\frac{a^m}{a^n}$$ $$=$$ $$a^{\\\\frac{m}{n}}$$, a $$ \\\\neq $$ $$0$$, $$m>n$$, $$\\\\frac{a^m}{a^n}$$ $$=$$ $$\\\\frac{1}{a^{n-m}}$$, a $$ \\\\neq $$ $$0$$, $$m>n$$,","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4b9bbfrationalnums15b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y^3$$"],"dependencies":["a4b9bbfrationalnums15b-h2"],"title":"Simplify","text":"Simplify the whole expression","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4b9bbfrationalnums16","title":"Radical Expressions","body":"Write as a radical expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Rational Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4b9bbfrationalnums16a","stepAnswer":["$$\\\\sqrt{r}$$"],"problemType":"TextBox","stepTitle":"Write as a radical expression","stepBody":"$$r^{\\\\frac{1}{2}}$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\sqrt{r}$$","hints":{"DefaultPathway":[{"id":"a4b9bbfrationalnums16a-h1","type":"hint","dependencies":[],"title":"Form","text":"We want to write each expression in the form $$\\\\sqrt[n]{a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4b9bbfrationalnums16a-h2","type":"hint","dependencies":["a4b9bbfrationalnums16a-h1"],"title":"Denominator of exponent","text":"The denominator of the exponent is $$2$$, therefore the index of the radical is also $$2$$. We do not show the index of the radical when it is $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4b9bbfrationalnums17","title":"Radical Expressions","body":"Write as a radical expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Rational Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4b9bbfrationalnums17a","stepAnswer":["$$\\\\sqrt[3]{s}$$"],"problemType":"TextBox","stepTitle":"$$s^{\\\\frac{1}{3}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\sqrt[3]{s}$$","hints":{"DefaultPathway":[{"id":"a4b9bbfrationalnums17a-h1","type":"hint","dependencies":[],"title":"Form","text":"We want to write each expression in the form $$\\\\sqrt[n]{a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4b9bbfrationalnums17a-h2","type":"hint","dependencies":["a4b9bbfrationalnums17a-h1"],"title":"Denominator of exponent","text":"The denominator of the exponent is $$3$$, therefore the index of the radical is also $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4b9bbfrationalnums18","title":"Radical Expressions","body":"Write as a radical expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Rational Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4b9bbfrationalnums18a","stepAnswer":["$$\\\\sqrt[4]{t}$$"],"problemType":"TextBox","stepTitle":"$$t^{\\\\frac{1}{4}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\sqrt[4]{t}$$","hints":{"DefaultPathway":[{"id":"a4b9bbfrationalnums18a-h1","type":"hint","dependencies":[],"title":"Form","text":"We want to write each expression in the form $$\\\\sqrt[n]{a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4b9bbfrationalnums18a-h2","type":"hint","dependencies":["a4b9bbfrationalnums18a-h1"],"title":"Denominator of exponent","text":"The denominator of the exponent is $$4$$, therefore the index of the radical is also $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4b9bbfrationalnums19","title":"Radical Expressions","body":"Write as a radical expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Rational Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4b9bbfrationalnums19a","stepAnswer":["$$\\\\sqrt[7]{g}$$"],"problemType":"TextBox","stepTitle":"$$g^{\\\\frac{1}{7}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\sqrt[7]{g}$$","hints":{"DefaultPathway":[{"id":"a4b9bbfrationalnums19a-h1","type":"hint","dependencies":[],"title":"Form","text":"We want to write each expression in the form $$\\\\sqrt[n]{a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4b9bbfrationalnums19a-h2","type":"hint","dependencies":["a4b9bbfrationalnums19a-h1"],"title":"Denominator of exponent","text":"The denominator of the exponent is $$7$$, therefore the index of the radical is also $$7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4b9bbfrationalnums2","title":"Simplify Expressions with $$a^{\\\\frac{1}{n}}$$","body":"Simplify the following expressions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Rational Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4b9bbfrationalnums2a","stepAnswer":["$$x^{\\\\frac{1}{2}}$$"],"problemType":"TextBox","stepTitle":"Wrtie with a rational exponent: $$\\\\sqrt{x}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^{\\\\frac{1}{2}}$$","hints":{"DefaultPathway":[{"id":"a4b9bbfrationalnums2a-h1","type":"hint","dependencies":[],"title":"Radical Expression","text":"No index is shown, so it is $$2$$. The denominator of the exponent will be $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a4b9bbfrationalnums2b","stepAnswer":["$$y^{\\\\frac{1}{3}}$$"],"problemType":"TextBox","stepTitle":"Wrtie with a rational exponent: $$\\\\sqrt[3]{y}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y^{\\\\frac{1}{3}}$$","hints":{"DefaultPathway":[{"id":"a4b9bbfrationalnums2b-h1","type":"hint","dependencies":[],"title":"Radical Expression","text":"The index is $$3$$, so the denominator of the exponent is $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a4b9bbfrationalnums2c","stepAnswer":["$$z^{\\\\frac{1}{4}}$$"],"problemType":"TextBox","stepTitle":"Wrtie with a rational exponent: $$\\\\sqrt[4]{z}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$z^{\\\\frac{1}{4}}$$","hints":{"DefaultPathway":[{"id":"a4b9bbfrationalnums2c-h1","type":"hint","dependencies":[],"title":"Radical Expression","text":"The index is $$4$$, so the denominator of the exponent is $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4b9bbfrationalnums20","title":"Radical Expressions","body":"Write as a radical expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Rational Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4b9bbfrationalnums20a","stepAnswer":["$$\\\\sqrt[5]{h}$$"],"problemType":"TextBox","stepTitle":"$$h^{\\\\frac{1}{5}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\sqrt[5]{h}$$","hints":{"DefaultPathway":[{"id":"a4b9bbfrationalnums20a-h1","type":"hint","dependencies":[],"title":"Form","text":"We want to write each expression in the form $$\\\\sqrt[n]{a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4b9bbfrationalnums20a-h2","type":"hint","dependencies":["a4b9bbfrationalnums20a-h1"],"title":"Denominator of exponent","text":"The denominator of the exponent is $$5$$, therefore the index of the radical is also $$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4b9bbfrationalnums21","title":"Radical Expressions","body":"Write as a radical expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Rational Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4b9bbfrationalnums21a","stepAnswer":["$$\\\\sqrt[25]{j}$$"],"problemType":"TextBox","stepTitle":"$$j^{\\\\frac{1}{25}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\sqrt[25]{j}$$","hints":{"DefaultPathway":[{"id":"a4b9bbfrationalnums21a-h1","type":"hint","dependencies":[],"title":"Form","text":"We want to write each expression in the form $$\\\\sqrt[n]{a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4b9bbfrationalnums21a-h2","type":"hint","dependencies":["a4b9bbfrationalnums21a-h1"],"title":"Denominator of exponent","text":"The denominator of the exponent is $$25$$, therefore the index of the radical is also $$25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4b9bbfrationalnums22","title":"Wrting Radical Expressions in Exponential Form","body":"Write the radical as an expression with an exponent.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Rational Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4b9bbfrationalnums22a","stepAnswer":["$$-\\\\left(x^{\\\\frac{1}{7}}\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$-\\\\sqrt[7]{x}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-\\\\left(x^{\\\\frac{1}{7}}\\\\right)$$","hints":{"DefaultPathway":[{"id":"a4b9bbfrationalnums22a-h1","type":"hint","dependencies":[],"title":"Form","text":"We want to write each expression in the form $$a^n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4b9bbfrationalnums22a-h2","type":"hint","dependencies":["a4b9bbfrationalnums22a-h1"],"title":"Denominator of exponent","text":"The index of the radical is $$7$$, therefore the denominator of the power should also be $$7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4b9bbfrationalnums23","title":"Wrting Radical Expressions in Exponential Form","body":"Write the radical as an expression with an exponent.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Rational Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4b9bbfrationalnums23a","stepAnswer":["$$s^{\\\\frac{1}{10}}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[10]{s}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$s^{\\\\frac{1}{10}}$$","hints":{"DefaultPathway":[{"id":"a4b9bbfrationalnums23a-h1","type":"hint","dependencies":[],"title":"Form","text":"We want to write each expression in the form $$a^n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4b9bbfrationalnums23a-h2","type":"hint","dependencies":["a4b9bbfrationalnums23a-h1"],"title":"Denominator of exponent","text":"The index of the radical is $$10$$, therefore the denominator of the power should also be $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4b9bbfrationalnums24","title":"Wrting Radical Expressions in Exponential Form","body":"Write the radical as an expression with an exponent.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Rational Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4b9bbfrationalnums24a","stepAnswer":["$$t^{\\\\frac{1}{4}}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[4]{t}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$t^{\\\\frac{1}{4}}$$","hints":{"DefaultPathway":[{"id":"a4b9bbfrationalnums24a-h1","type":"hint","dependencies":[],"title":"Form","text":"We want to write each expression in the form $$a^n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4b9bbfrationalnums24a-h2","type":"hint","dependencies":["a4b9bbfrationalnums24a-h1"],"title":"Denominator of exponent","text":"The index of the radical is $$4$$, therefore the denominator of the power should also be $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4b9bbfrationalnums25","title":"Wrting Radical Expressions in Exponential Form","body":"Write the radical as an expression with an exponent.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Rational Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4b9bbfrationalnums25a","stepAnswer":["$$u^{\\\\frac{1}{5}}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[5]{u}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$u^{\\\\frac{1}{5}}$$","hints":{"DefaultPathway":[{"id":"a4b9bbfrationalnums25a-h1","type":"hint","dependencies":[],"title":"Form","text":"We want to write each expression in the form $$a^n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4b9bbfrationalnums25a-h2","type":"hint","dependencies":["a4b9bbfrationalnums25a-h1"],"title":"Denominator of exponent","text":"The index of the radical is $$5$$, therefore the denominator of the power should also be $$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4b9bbfrationalnums26","title":"Wrting Radical Expressions in Exponential Form","body":"Write the radical as an expression with an exponent.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Rational Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4b9bbfrationalnums26a","stepAnswer":["$$x^{\\\\frac{7}{4}}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[4]{x^7}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^{\\\\frac{7}{4}}$$","hints":{"DefaultPathway":[{"id":"a4b9bbfrationalnums26a-h1","type":"hint","dependencies":[],"title":"Form","text":"We want to write each expression in the form $$a^n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4b9bbfrationalnums26a-h2","type":"hint","dependencies":["a4b9bbfrationalnums26a-h1"],"title":"Denominator of exponent","text":"The index of the radical is $$4$$, and the exponent inside the root is $$7$$, so the base should be to the power of $$\\\\frac{7}{4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4b9bbfrationalnums27","title":"Wrting Radical Expressions in Exponential Form","body":"Write the radical as an expression with an exponent.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Rational Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4b9bbfrationalnums27a","stepAnswer":["$$x^{\\\\frac{3}{5}}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[5]{x^3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^{\\\\frac{3}{5}}$$","hints":{"DefaultPathway":[{"id":"a4b9bbfrationalnums27a-h1","type":"hint","dependencies":[],"title":"Form","text":"We want to write each expression in the form $$a^n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4b9bbfrationalnums27a-h2","type":"hint","dependencies":["a4b9bbfrationalnums27a-h1"],"title":"Denominator of exponent","text":"The index of the radical is $$5$$, and the exponent inside the root is $$3$$, so the base should be to the power of $$\\\\frac{3}{5}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4b9bbfrationalnums28","title":"Wrting Radical Expressions in Exponential Form","body":"Write the radical as an expression with an exponent.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Rational Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4b9bbfrationalnums28a","stepAnswer":["$$x^{\\\\frac{7}{3}}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[3]{x^7}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^{\\\\frac{7}{3}}$$","hints":{"DefaultPathway":[{"id":"a4b9bbfrationalnums28a-h1","type":"hint","dependencies":[],"title":"Form","text":"We want to write each expression in the form $$a^n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4b9bbfrationalnums28a-h2","type":"hint","dependencies":["a4b9bbfrationalnums28a-h1"],"title":"Denominator of exponent","text":"The index of the radical is $$3$$, and the exponent inside the root is $$7$$, so the base should be to the power of $$\\\\frac{7}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4b9bbfrationalnums29","title":"Wrting Radical Expressions in Exponential Form","body":"Write the radical as an expression with an exponent.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Rational Exponents","courseName":"OpenStax: Elementary 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Expressions with $$a^{\\\\frac{1}{n}}$$","body":"Simplify the following expressions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Rational Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4b9bbfrationalnums3a","stepAnswer":["$${\\\\left(5y\\\\right)}^{\\\\frac{1}{2}}$$"],"problemType":"TextBox","stepTitle":"Write with a rational exponent: $$\\\\sqrt{5y}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(5y\\\\right)}^{\\\\frac{1}{2}}$$","hints":{"DefaultPathway":[{"id":"a4b9bbfrationalnums3a-h1","type":"hint","dependencies":[],"title":"Radical Expression","text":"No index is shown, so it is $$2$$. The denominator of the exponent will be $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a4b9bbfrationalnums3b","stepAnswer":["$${\\\\left(4x\\\\right)}^{\\\\frac{1}{3}}$$"],"problemType":"TextBox","stepTitle":"Write with a rational exponent: $$\\\\sqrt[3]{4x}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(4x\\\\right)}^{\\\\frac{1}{3}}$$","hints":{"DefaultPathway":[{"id":"a4b9bbfrationalnums3b-h1","type":"hint","dependencies":[],"title":"Radical Expression","text":"The index is $$3$$, so the denominator of the exponent is $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a4b9bbfrationalnums3c","stepAnswer":["$$3{\\\\left(5z\\\\right)}^{\\\\frac{1}{4}}$$"],"problemType":"TextBox","stepTitle":"Write with a rational exponent: $$3\\\\sqrt[4]{5z}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3{\\\\left(5z\\\\right)}^{\\\\frac{1}{4}}$$","hints":{"DefaultPathway":[{"id":"a4b9bbfrationalnums3c-h1","type":"hint","dependencies":[],"title":"Radical Expression","text":"The index is $$4$$, so the denominator of the exponent is $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4b9bbfrationalnums30","title":"Wrting Radical Expressions in Exponential Form","body":"Write the radical as an expression with an exponent.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Rational Exponents","courseName":"OpenStax: Elementary 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Expressions with $$a^{\\\\frac{m}{n}}$$","body":"Simplify the following expressions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Rational Exponents","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4b9bbfrationalnums7a","stepAnswer":["$$y^{\\\\frac{3}{2}}$$"],"problemType":"TextBox","stepTitle":"Write with a rational exponent in the form $$a^{\\\\frac{m}{n}}$$ $$=$$ $$\\\\sqrt[n]{m}$$: $$\\\\sqrt{y^3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y^{\\\\frac{3}{2}}$$","hints":{"DefaultPathway":[{"id":"a4b9bbfrationalnums7a-h1","type":"hint","dependencies":[],"title":"The numerator of the exponent is the exponent of $$y$$, 3; the denominator of the exponent is the index of the radical, $$2$$.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a4b9bbfrationalnums7b","stepAnswer":["$$x^{\\\\frac{2}{3}}$$"],"problemType":"TextBox","stepTitle":"Write with a rational exponent in the form $$a^{\\\\frac{m}{n}}$$ $$=$$ $$\\\\sqrt[n]{m}$$: $$\\\\sqrt[3]{x^2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^{\\\\frac{2}{3}}$$","hints":{"DefaultPathway":[{"id":"a4b9bbfrationalnums7b-h1","type":"hint","dependencies":[],"title":"The numerator of the exponent is the exponent of $$x$$, 2; the denominator of the exponent is the index of the radical, $$3$$.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a4b9bbfrationalnums7c","stepAnswer":["$$z^{\\\\frac{3}{4}}$$"],"problemType":"TextBox","stepTitle":"Write with a rational exponent in the form $$a^{\\\\frac{m}{n}}$$ $$=$$ $$\\\\sqrt[n]{m}$$: $$\\\\sqrt[4]{z^3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$z^{\\\\frac{3}{4}}$$","hints":{"DefaultPathway":[{"id":"a4b9bbfrationalnums7c-h1","type":"hint","dependencies":[],"title":"The numerator of the exponent is the exponent of $$z$$, 3; the denominator of the exponent is the index of the radical, $$4$$.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4d2b33use1","title":"Evaluating Expressions","body":"Evaluate $$7x-4$$ when:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Use the Language of Algebra","courseName":"OpenStax: Elementary 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Expressions","body":"When $$x=6$$, evaluate:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Use the Language of Algebra","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4d2b33use10a","stepAnswer":["$$216$$"],"problemType":"TextBox","stepTitle":"$$x^3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$216$$","hints":{"DefaultPathway":[{"id":"a4d2b33use10a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$6$$ for $$x$$ and simplify the expression $$6^3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4d2b33use12","title":"Evaluating Expressions","body":"When $$x=2$$, evaluate:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Use the Language of Algebra","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4d2b33use12a","stepAnswer":["$$9$$"],"problemType":"TextBox","stepTitle":"$$6x^2-4x-7$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9$$","hints":{"DefaultPathway":[{"id":"a4d2b33use12a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$2$$ for $$x$$ and simplify the expression $${6\\\\left(2\\\\right)}^2-4\\\\left(2\\\\right)-7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4d2b33use12a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a4d2b33use12a-h4"],"title":"Subtraction","text":"What is $$24-8-7$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4d2b33use13","title":"Identifying Coefficients","body":"Identify the coefficient of the following expressions:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Use the Language of Algebra","courseName":"OpenStax: Elementary 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value that is muliplied to the variable of the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a4d2b33use13c","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"$$z$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a4d2b33use13c-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Identify the Coefficient","text":"What is the value that is muliplied to the variable of the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4d2b33use14","title":"Identifying Coefficients","body":"Identify the coefficient of the following expressions:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary 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4.0>"}]}},{"id":"a4d2b33use14b","stepAnswer":["$$13$$"],"problemType":"TextBox","stepTitle":"$$13a^3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$13$$","hints":{"DefaultPathway":[{"id":"a4d2b33use14b-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$13$$"],"dependencies":[],"title":"Identify the Coefficient","text":"What is the value that is muliplied to the variable of the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a4d2b33use14c","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"$$y^3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a4d2b33use14c-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Identify the Coefficient","text":"What is the value that is muliplied to the variable of the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4d2b33use15","title":"Finding Like Terms","body":"Identify the like terms:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Use the Language of Algebra","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4d2b33use15a","stepAnswer":["$$9$$ and $$15$$, $$2x^3$$ and $$8x^3$$, $$11y^2$$ and $$y^2$$"],"problemType":"MultipleChoice","stepTitle":"$$9$$, $$2x^3$$, $$y^2$$, $$8x^3$$, $$15$$, $$9y$$, $$11y^2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$9$$ and $$15$$, $$2x^3$$ and $$8x^3$$, $$11y^2$$ and $$y^2$$","choices":["$$9$$ and $$15$$, $$2x^3$$ and $$8x^3$$, $$11y^2$$ and $$y^2$$","$$9$$ and $$2x^3$$, $$15$$ and $$8x^3$$, $$11y^2$$ and $$y^2$$","$$2x^3$$ and $$11y^2$$, $$8x^3$$ an $${dy}^2$$, $$9$$ and $$15$$"],"hints":{"DefaultPathway":[{"id":"a4d2b33use15a-h1","type":"hint","dependencies":[],"title":"Like Terms","text":"Terms that are either constants or have the same variables raised to the same powers are called like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4d2b33use16","title":"Finding Like Terms","body":"Identify the like terms:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Use the Language of Algebra","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4d2b33use16a","stepAnswer":["$$19$$ and $$24$$, $$4x^3$$ and $$6x^3$$, $$3x^2$$ and $$8^2$$"],"problemType":"MultipleChoice","stepTitle":"$$4x^3$$, $$8x^2$$, $$19$$, $$3x^2$$, $$24$$, $$6x^3$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$19$$ and $$24$$, $$4x^3$$ and $$6x^3$$, $$3x^2$$ and $$8^2$$","choices":["$$19$$ and $$24$$, $$4x^3$$ and $$6x^3$$, $$3x^2$$ and $$8^2$$","$$19$$ and $$24$$, $$4x^3$$ and $$6x^3$$, $$3x^2$$ and $$8x^2$$","$$19$$ and $$4x^3$$, $$24$$ and $$3x^2$$, $$6x^3$$ and $$8x^2$$","$$24$$ and $$4x^3$$, $$6x^3$$ and $$3x^2$$, $$19$$ and $$8x^2$$"],"hints":{"DefaultPathway":[{"id":"a4d2b33use16a-h1","type":"hint","dependencies":[],"title":"Like Terms","text":"Terms that are either constants or have the same variables raised to the same powers are called like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4d2b33use17","title":"Identifying Terms","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary 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$$3^4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4d2b33use20","title":"Determing Expressions Versus Equations","body":"Determine whether the following are expressions or equations.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Use the Language of Algebra","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4d2b33use20a","stepAnswer":["Equation"],"problemType":"MultipleChoice","stepTitle":"$$2\\\\left(x+3\\\\right)=10$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Expression","Equation"],"hints":{"DefaultPathway":[{"id":"a4d2b33use20a-h1","type":"hint","dependencies":[],"title":"Equations Versus Expressions","text":"In an equation, two expressions are connected with an equal sign, such as \\"5x+8=4.\\" In an expression, there is no equal sign, such as in \\"8x+3.\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a4d2b33use20b","stepAnswer":["Expression"],"problemType":"MultipleChoice","stepTitle":"$$4\\\\left(y-1\\\\right)+1$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Expression","Equation"],"hints":{"DefaultPathway":[{"id":"a4d2b33use20b-h1","type":"hint","dependencies":[],"title":"Equations Versus Expressions","text":"In an equation, two expressions are connected with an equal sign, such as \\"5x+8=4.\\" In an expression, there is no equal sign, such as in \\"8x+3.\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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connected with an equal sign, such as \\"5x+8=4.\\" In an expression, there is no equal sign, such as in \\"8x+3.\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4d2b33use21","title":"Simplifying an Expression Raised to a Power","body":"Simplify the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Use the Language of Algebra","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4d2b33use21a","stepAnswer":["$$81$$"],"problemType":"TextBox","stepTitle":"$$3^4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$81$$","hints":{"DefaultPathway":[{"id":"a4d2b33use21a-h1","type":"hint","dependencies":[],"title":"Expanding the Expression","text":"The first step is to expand the expression. $$3^4=3\\\\times3\\\\times3\\\\times3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4d2b33use21a-h2","type":"hint","dependencies":["a4d2b33use21a-h1"],"title":"Multiplying From Left to Right, Part $$1$$","text":"The second step is to multiply from left to right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4d2b33use21a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a4d2b33use21a-h2"],"title":"First Multiplication","text":"$$3\\\\times3=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4d2b33use21a-h4","type":"hint","dependencies":["a4d2b33use21a-h3"],"title":"Multiplying From Left to Right, Part $$2$$","text":"Multiply from left to right again.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4d2b33use21a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$27$$"],"dependencies":["a4d2b33use21a-h4"],"title":"Second Multiplication","text":"$$9\\\\times3=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4d2b33use21a-h6","type":"hint","dependencies":["a4d2b33use21a-h5"],"title":"Multiplying From Left to Right, Part $$3$$","text":"Finally, multiply the remaining two numbers together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4d2b33use21a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$81$$"],"dependencies":["a4d2b33use21a-h6"],"title":"Thrid 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subtraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a4d2b33use22a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["a4d2b33use22a-h6"],"title":"Second Operation","text":"What is $$4+21$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a4d2b33use22b","stepAnswer":["$$49$$"],"problemType":"TextBox","stepTitle":"$$7\\\\left(4+3\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$49$$","hints":{"DefaultPathway":[{"id":"a4d2b33use22b-h1","type":"hint","dependencies":[],"title":"Using PEMDAS","text":"The Order of Operations, PEMDAS, is Parenthese, Exponents, Multiplication, Division, Addition, and Subtraction.","variabilization":{},"oer":"https://OATutor.io 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always $$1$$, we can tell that $$1^7=1$$. This is because no matter how many times we multiply $$1$$ by itself, it will always be $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4d2b33use3","title":"Evaluating Expressions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Use the Language of Algebra","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4d2b33use3a","stepAnswer":["$$52$$"],"problemType":"TextBox","stepTitle":"Evaluate $$2x^2+3x+8$$ when $$x=4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$52$$","hints":{"DefaultPathway":[{"id":"a4d2b33use3a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$4$$ for $$x$$ and simplify the expression 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$$7+16$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4d2b33use35","title":"Simplifying Expressions","body":"Simplify the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Use the Language of Algebra","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4d2b33use35a","stepAnswer":["$$86$$"],"problemType":"TextBox","stepTitle":"$$9+5^3-4\\\\left(9+3\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$86$$","hints":{"DefaultPathway":[{"id":"a4d2b33use35a-h1","type":"hint","dependencies":[],"title":"Using PEMDAS","text":"The Order of Operations, PEMDAS, is Parenthese, Exponents, Multiplication, Division, Addition, and Subtraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4d2b33use35a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a4d2b33use35a-h1"],"title":"P-Parentheses","text":"Are there any parentheses?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a4d2b33use35a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a4d2b33use35a-h2"],"title":"First Operation","text":"What is $$9+3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4d2b33use35a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$48$$"],"dependencies":["a4d2b33use35a-h3"],"title":"Second Operation","text":"What is $$4$$ times the result of the first operation (the value in the inner parentheses)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4d2b33use35a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a4d2b33use35a-h4"],"title":"E-Exponents","text":"Are there any exponents?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a4d2b33use35a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$125$$"],"dependencies":["a4d2b33use35a-h5"],"title":"Third Operation","text":"What is $$5^3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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$$9+125-48$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4d2b33use36","title":"Simplifying Expressions","body":"Simplify the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Use the Language of Algebra","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4d2b33use36a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"$$7^2-2\\\\left(4\\\\left(5+1\\\\right)\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a4d2b33use36a-h1","type":"hint","dependencies":[],"title":"Using PEMDAS","text":"The Order of Operations, PEMDAS, is Parenthese, Exponents, Multiplication, Division, Addition, and Subtraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4d2b33use36a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a4d2b33use36a-h1"],"title":"P-Parentheses","text":"Are there any parentheses?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a4d2b33use36a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a4d2b33use36a-h2"],"title":"First Operation","text":"What is $$5+1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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The answer to this will be the value in the outer parentheses.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4d2b33use36a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a4d2b33use36a-h4"],"title":"E-Exponents","text":"Are there any exponents?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a4d2b33use36a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$49$$"],"dependencies":["a4d2b33use36a-h5"],"title":"Third Operation","text":"What is $$7^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4d2b33use36a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a4d2b33use36a-h6"],"title":"MD- Multiplication or Division","text":"Is there any multiplication or division?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a4d2b33use36a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$48$$"],"dependencies":["a4d2b33use36a-h7"],"title":"Fourth Operation","text":"What is $$2$$ times the value of the outer parentheses?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4d2b33use36a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a4d2b33use36a-h8"],"title":"AS- Addition or Subtraction","text":"Is there any addition or subtraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a4d2b33use36a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a4d2b33use36a-h9"],"title":"Fifth Operation","text":"What is $$49-48$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4d2b33use4","title":"Finding the Coefficient","body":"Identify the coefficient of each term:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Use the Language of Algebra","courseName":"OpenStax: Elementary 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that is muliplied to the variable of the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a4d2b33use4c","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"a","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a4d2b33use4c-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Identify the Coefficient","text":"What is the value that is muliplied to the variable of the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4d2b33use5","title":"Finding Like Terms","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Use the Language of Algebra","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4d2b33use5a","stepAnswer":["$$y^3$$ and $$4y^3$$, $$7x^2$$ and $$5x^2$$, $$14$$ and $$23$$"],"problemType":"MultipleChoice","stepTitle":"Identify the like terms: $$y^3$$, $$7x^2$$, $$14$$, $$23$$, $$4y^3$$, $$9x$$, $$5x^2$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y^3$$ and $$4y^3$$, $$7x^2$$ and $$5x^2$$, $$14$$ and $$23$$","choices":["$$y^3$$ and $$7x^2$$, $$y^3$$ and $$14$$, $$14$$ and $$7x^2$$","$$y^3$$ and $$4y^3$$, $$7x^2$$ and $$5x^2$$, $$14$$ and $$23$$","$$y^3$$ and $$7x^2$$, $$5x^2$$ and $$14$$, $$23$$ and $$4y^3$$"],"hints":{"DefaultPathway":[{"id":"a4d2b33use5a-h1","type":"hint","dependencies":[],"title":"Like Terms","text":"Terms that are either constants or have the same variables raised to the same powers are called like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4d2b33use6","title":"Identifying Terms","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Use the Language of Algebra","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a4d2b33use6a","stepAnswer":["$$9x^2$$, $$7x$$, $$12$$"],"problemType":"MultipleChoice","stepTitle":"Identify the terms in $$9x^2+7x+12$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$9x^2$$, $$7x$$, $$12$$","choices":["$$9$$, $$x^2$$, $$7$$, $$x$$, $$12$$","$$9$$, $$x$$, $$2$$, $$7$$, $$x$$, $$12$$","$$9x^2$$, $$7x$$, $$12$$"],"hints":{"DefaultPathway":[{"id":"a4d2b33use6a-h1","type":"hint","dependencies":[],"title":"Defining Terms","text":"The terms are the individual expressions that are added or subtracted that make up the whole 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4.0>"}]}}]},{"id":"a4dc570tri1","title":"Trigonometric Functions","body":"These questions test your knowledge of the core concepts.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Trigonometric Functions","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a4dc570tri1a","stepAnswer":["$$120$$"],"problemType":"TextBox","stepTitle":"In the circular sector below, assume the radius of the circle is $$5$$, and the arc length of the sector is $$\\\\frac{10\\\\pi}{3}$$. What is the degree measure of the angle \u03b8?","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$120$$","hints":{"DefaultPathway":[{"id":"a4dc570tri1a-h1","type":"hint","dependencies":[],"title":"Arc Length Formula","text":"Use the formula for arc length in radians: Arc $$Length(s)=r \\\\theta$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4dc570tri1a-h2","type":"hint","dependencies":["a4dc570tri1a-h1"],"title":"Solve the equation","text":"In this case, the arc length (s) is given as $$\\\\frac{10\\\\pi}{3}$$, and the radius (r) is $$5$$. So, you can solve for \u03b8: $$\\\\frac{10\\\\pi}{3}$$ $$=$$ 5\u03b8","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4dc570tri1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{3} \\\\pi$$"],"dependencies":["a4dc570tri1a-h2"],"title":"Solve the equation","text":"What is \u03b8 (in radians)?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4dc570tri1a-h4","type":"hint","dependencies":["a4dc570tri1a-h3"],"title":"Conversion","text":"$$2\\\\pi$$ radians $$=$$ $$360$$ degrees","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4dc570tri1a-h5","type":"hint","dependencies":["a4dc570tri1a-h4"],"title":"Conversion","text":"To convert radians to degrees, we can do $$\\\\frac{2}{3} \\\\pi \\\\frac{360}{2\\\\pi}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4dc570tri1a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$120$$"],"dependencies":["a4dc570tri1a-h5"],"title":"Conversion","text":"What is \u03b8 (in degrees)?","variabilization":{},"oer":"https://OATutor.io","license":"","subHints":[{"id":"a4dc570tri1a-h6-s1","type":"hint","dependencies":[],"title":"Conversion","text":"$$\\\\frac{2}{3} \\\\pi \\\\frac{360}{2\\\\pi}=\\\\frac{360}{3}=120$$","variabilization":{},"oer":"https://OATutor.io","license":""}]}]}},{"id":"a4dc570tri1b","stepAnswer":["no change"],"problemType":"MultipleChoice","stepTitle":"What would change if the circular sector above had radius $$30$$ and arc length $$20\\\\pi$$?","stepBody":"","answerType":"string","variabilization":{},"choices":["no change","\u03b8 increases","\u03b8 decreases"],"hints":{"DefaultPathway":[{"id":"a4dc570tri1b-h1","type":"hint","dependencies":[],"title":"Arc Length Formula","text":"Use the formula for arc length in radians: arc $$length=r \\\\theta$$. So $$\\\\theta=\\\\frac{20\\\\pi}{30}=\\\\frac{2}{3} \\\\pi$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4dc570tri1b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["no change"],"dependencies":["a4dc570tri1b-h1"],"title":"Change or not","text":"Has the degree measure changed compared to the previous question?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["no change","\u03b8 increases","\u03b8 decreases"]}]}}]},{"id":"a4dc570tri2","title":"Trigonometric Functions","body":"These questions test your knowledge of the core concepts.\\\\n##figure2.gif","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Trigonometric Functions","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a4dc570tri2a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"On the half circle below, mark the angles corresponding to $$0$$, $$30$$, $$45$$, $$60$$, $$90$$, $$120$$, $$135$$, $$150$$, and $$180$$ with their radian measures. Is this the correct graph?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a4dc570tri2a-h1","type":"hint","dependencies":[],"title":"Conversion","text":"Remember that $$1\\\\pi$$ radians is equivalent to $$180$$ degrees, so $$1$$ degree is $$\\\\frac{\\\\pi}{180}$$ radians. This relationship will be useful for marking the angles in both degrees and radians.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4dc570tri2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a4dc570tri2a-h1"],"title":"Conversion","text":"What is $$0$$ degree in radians?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4dc570tri2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\pi}{6}$$"],"dependencies":["a4dc570tri2a-h2"],"title":"Conversion","text":"What is $$30$$ degree in radians?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4dc570tri2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\pi}{4}$$"],"dependencies":["a4dc570tri2a-h3"],"title":"Conversion","text":"What is $$45$$ degree in radians?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4dc570tri2a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\pi}{3}$$"],"dependencies":["a4dc570tri2a-h4"],"title":"Conversion","text":"What is $$60$$ degree in radians?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4dc570tri2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\pi}{2}$$"],"dependencies":["a4dc570tri2a-h5"],"title":"Conversion","text":"What is $$90$$ degree in radians?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4dc570tri2a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2\\\\pi}{3}$$"],"dependencies":["a4dc570tri2a-h6"],"title":"Conversion","text":"What is $$120$$ degree in radians?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4dc570tri2a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3\\\\pi}{4}$$"],"dependencies":["a4dc570tri2a-h7"],"title":"Conversion","text":"What is $$135$$ degree in radians?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4dc570tri2a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5\\\\pi}{6}$$"],"dependencies":["a4dc570tri2a-h8"],"title":"Conversion","text":"What is $$150$$ degree in radians?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4dc570tri2a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["pi"],"dependencies":["a4dc570tri2a-h9"],"title":"Conversion","text":"What is $$180$$ degree in radians?","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a4dc570tri3","title":"Trigonometric Functions","body":"These questions test your knowledge of the core concepts.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Trigonometric Functions","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a4dc570tri3a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"The point on the unit circle corresponding to $$\\\\theta=\\\\frac{\\\\pi}{4}$$ is $$P=(\\\\frac{1}{\\\\sqrt{2}},\\\\frac{1}{\\\\sqrt{2}})$$. Find $$cos\\\\left(\\\\frac{\\\\pi}{4}\\\\right)$$, $$sin\\\\left(\\\\frac{\\\\pi}{4}\\\\right)$$, $$tan\\\\left(\\\\frac{\\\\pi}{4}\\\\right)$$, $$\\\\operatorname{sec}\\\\left(\\\\frac{\\\\pi}{4}\\\\right)$$, $$\\\\operatorname{csc}\\\\left(\\\\frac{\\\\pi}{4}\\\\right)$$ and $$\\\\operatorname{cot}\\\\left(\\\\frac{\\\\pi}{4}\\\\right)$$ using the point P. Is it possible to find them?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a4dc570tri3a-h1","type":"hint","dependencies":[],"title":"$$cos\\\\left(\\\\frac{\\\\pi}{4}\\\\right)$$","text":"cos is the x-coordinate of the point on the unit circle. So, $$cos\\\\left(\\\\frac{\\\\pi}{4}\\\\right)$$ $$=$$ $$\\\\frac{1}{\\\\sqrt{2}}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4dc570tri3a-h2","type":"hint","dependencies":["a4dc570tri3a-h1"],"title":"$$sin\\\\left(\\\\frac{\\\\pi}{4}\\\\right)$$","text":"sin is the y-coordinate of the point on the unit circle. So, $$sin\\\\left(\\\\frac{\\\\pi}{4}\\\\right)$$ $$=$$ $$\\\\frac{1}{\\\\sqrt{2}}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4dc570tri3a-h3","type":"hint","dependencies":["a4dc570tri3a-h2"],"title":"$$tan\\\\left(\\\\frac{\\\\pi}{4}\\\\right)$$","text":"Tangent is the ratio of sin to cos.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4dc570tri3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a4dc570tri3a-h3"],"title":"$$tan\\\\left(\\\\frac{\\\\pi}{4}\\\\right)$$","text":"What is $$tan\\\\left(\\\\frac{\\\\pi}{4}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4dc570tri3a-h5","type":"hint","dependencies":["a4dc570tri3a-h4"],"title":"$$\\\\operatorname{sec}\\\\left(\\\\frac{\\\\pi}{4}\\\\right)$$","text":"sec is the reciprocal of cos.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4dc570tri3a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\sqrt{2}$$"],"dependencies":["a4dc570tri3a-h5"],"title":"$$\\\\operatorname{sec}\\\\left(\\\\frac{\\\\pi}{4}\\\\right)$$","text":"What is $$\\\\operatorname{sec}\\\\left(\\\\frac{\\\\pi}{4}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\sqrt{2}$$","$$-\\\\sqrt{2}$$"]},{"id":"a4dc570tri3a-h7","type":"hint","dependencies":["a4dc570tri3a-h6"],"title":"$$\\\\operatorname{csc}\\\\left(\\\\frac{\\\\pi}{4}\\\\right)$$","text":"csc is the reciprocal of sin.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4dc570tri3a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\sqrt{2}$$"],"dependencies":["a4dc570tri3a-h7"],"title":"$$\\\\operatorname{csc}\\\\left(\\\\frac{\\\\pi}{4}\\\\right)$$","text":"What is $$\\\\operatorname{csc}\\\\left(\\\\frac{\\\\pi}{4}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\sqrt{2}$$","$$-\\\\sqrt{2}$$"]},{"id":"a4dc570tri3a-h9","type":"hint","dependencies":["a4dc570tri3a-h8"],"title":"$$\\\\operatorname{cot}\\\\left(\\\\frac{\\\\pi}{4}\\\\right)$$","text":"cot is the reciprocal of tan.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4dc570tri3a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a4dc570tri3a-h9"],"title":"$$\\\\operatorname{cot}\\\\left(\\\\frac{\\\\pi}{4}\\\\right)$$","text":"What is $$\\\\operatorname{cot}\\\\left(\\\\frac{\\\\pi}{4}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":""}]}},{"id":"a4dc570tri3b","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"The point on the unit circle corresponding to $$\\\\theta=\\\\frac{\\\\pi}{4}$$ is $$P=(\\\\frac{1}{\\\\sqrt{2}},\\\\frac{1}{\\\\sqrt{2}})$$. Find two more angles, one positive and one negative, with the same values of cosine and sine. Sketch arcs representing all three angles on the unit circles below. Is the graph correct?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a4dc570tri3b-h1","type":"hint","dependencies":[],"title":"See the graph","text":"The graph shows the correct examples.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a4dc570tri4","title":"Trigonometric Functions","body":"These questions test your knowledge of the core concepts.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Trigonometric Functions","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a4dc570tri4a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"On the unit circle below, sketch an arc that represents the angle $$\\\\theta=\\\\frac{9\\\\pi}{2}$$ and mark the corresponding point P. Using this, determine $$cos\\\\left(\\\\frac{9\\\\pi}{2}\\\\right)$$ and $$sin\\\\left(\\\\frac{9\\\\pi}{2}\\\\right)$$. What about $$tan\\\\left(\\\\frac{9\\\\pi}{2}\\\\right)$$? Is it defined?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a4dc570tri4a-h1","type":"hint","dependencies":[],"title":"$$\\\\frac{9\\\\pi}{2}$$","text":"$$\\\\frac{9\\\\pi}{2}=4\\\\pi+\\\\frac{\\\\pi}{2}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4dc570tri4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a4dc570tri4a-h1"],"title":"$$cos\\\\left(\\\\frac{9\\\\pi}{2}\\\\right)$$","text":"What is $$cos\\\\left(\\\\frac{9\\\\pi}{2}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4dc570tri4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a4dc570tri4a-h2"],"title":"$$sin\\\\left(\\\\frac{9\\\\pi}{2}\\\\right)$$","text":"What is $$sin\\\\left(\\\\frac{9\\\\pi}{2}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4dc570tri4a-h4","type":"hint","dependencies":["a4dc570tri4a-h3"],"title":"$$tan\\\\left(\\\\frac{9\\\\pi}{2}\\\\right)$$","text":"Tangent is the ratio of sin to cos.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4dc570tri4a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a4dc570tri4a-h4"],"title":"$$tan\\\\left(\\\\frac{9\\\\pi}{2}\\\\right)$$","text":"Is $$tan\\\\left(\\\\frac{9\\\\pi}{2}\\\\right)$$ defined?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"],"subHints":[{"id":"a4dc570tri4a-h5-s1","type":"hint","dependencies":[],"title":"$$tan\\\\left(\\\\frac{9\\\\pi}{2}\\\\right)$$","text":"$$tan\\\\left(\\\\frac{9\\\\pi}{2}\\\\right)=\\\\fracsin^c\\\\left(\\\\frac{9\\\\pi}{2}\\\\right)os\\\\left(\\\\frac{9\\\\pi}{2}\\\\right)}$$. $$cos\\\\left(\\\\frac{9\\\\pi}{2}\\\\right)=0$$, the denominator is $$0$$, so $$tan\\\\left(\\\\frac{9\\\\pi}{2}\\\\right)$$ is undefined.","variabilization":{},"oer":"https://OATutor.io","license":""}]}]}}]},{"id":"a4dc570tri5","title":"Trigonometric Functions","body":"These problems are generally harder, often highlighting an important subtlety","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Trigonometric Functions","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a4dc570tri5a","stepAnswer":["$$\\\\frac{-5\\\\pi}{2}$$, $$\\\\frac{-\\\\pi}{2}$$, $$\\\\frac{3\\\\pi}{2}$$"],"problemType":"MultipleChoice","stepTitle":"Determine all angles \u03b8, in the interval $$(-3\\\\pi,-3\\\\pi)$$, such that $$sin(\\\\theta)=-1$$. Hint: Where on the unit circle does this correspond to?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{-5\\\\pi}{2}$$, $$\\\\frac{-\\\\pi}{2}$$, $$\\\\frac{3\\\\pi}{2}$$","choices":["$$\\\\frac{-5\\\\pi}{2}$$, $$\\\\frac{-\\\\pi}{2}$$, $$\\\\frac{3\\\\pi}{2}$$","$$\\\\frac{3\\\\pi}{2}+2n \\\\pi$$"],"hints":{"DefaultPathway":[{"id":"a4dc570tri5a-h1","type":"hint","dependencies":[],"title":"Follow the hint","text":"Identify the point on the unit circle where $$sin(\\\\theta)=-1$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4dc570tri5a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(0,-1)$$"],"dependencies":["a4dc570tri5a-h1"],"title":"Follow the hint","text":"What is the corresponding point?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$(0,-1)$$","$$(0,1)$$","$$(1,0)$$","$$(-1,0)$$"]},{"id":"a4dc570tri5a-h3","type":"hint","dependencies":["a4dc570tri5a-h2"],"title":"Find all angles","text":"\u03b8 can be $$\\\\frac{3\\\\pi}{2}$$, $$\\\\frac{3\\\\pi}{2}+2\\\\pi$$, $$\\\\frac{3\\\\pi}{2}-2\\\\pi$$, $$\\\\frac{3\\\\pi}{2}+4\\\\pi$$, $$\\\\frac{3\\\\pi}{2}-4\\\\pi$$ ......","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4dc570tri5a-h4","type":"hint","dependencies":["a4dc570tri5a-h3"],"title":"Interval","text":"\u03b8 is in $$(-3\\\\pi,3\\\\pi)$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4dc570tri5a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{-5\\\\pi}{2}$$, $$\\\\frac{-\\\\pi}{2}$$, $$\\\\frac{3\\\\pi}{2}$$"],"dependencies":["a4dc570tri5a-h4"],"title":"Interval","text":"What are possible values of \u03b8?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{-5\\\\pi}{2}$$, $$\\\\frac{-\\\\pi}{2}$$, $$\\\\frac{3\\\\pi}{2}$$","$$\\\\frac{3\\\\pi}{2}+2n \\\\pi$$"]}]}},{"id":"a4dc570tri5b","stepAnswer":["$$0$$, $$\\\\pm 2 \\\\pi$$"],"problemType":"MultipleChoice","stepTitle":"Determine all angles \u03b8, in the interval $$(-3\\\\pi,3\\\\pi)$$, such that $$cos(\\\\theta)=1$$.","stepBody":"##figure2.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$0$$, $$\\\\pm 2 \\\\pi$$","choices":["$$0$$, $$\\\\pm 2 \\\\pi$$","$$0$$, $$2\\\\pi$$","$$2n \\\\pi$$"],"hints":{"DefaultPathway":[{"id":"a4dc570tri5b-h1","type":"hint","dependencies":[],"title":"Follow the hint","text":"Identify the point on the unit circle where $$cos(\\\\theta)=1$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4dc570tri5b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(1,0)$$"],"dependencies":["a4dc570tri5b-h1"],"title":"Follow the hint","text":"What is the corresponding point?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$(0,-1)$$","$$(0,1)$$","$$(1,0)$$","$$(-1,0)$$"]},{"id":"a4dc570tri5b-h3","type":"hint","dependencies":["a4dc570tri5b-h2"],"title":"Find all angles","text":"To find all angles \u03b8 in the given interval with $$cos(\\\\theta)=1$$, express the solutions in the general form: $$\\\\theta=0+2n \\\\pi$$, where $$n$$ is an integer.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4dc570tri5b-h4","type":"hint","dependencies":["a4dc570tri5b-h3"],"title":"Interval","text":"\u03b8 is in $$(-3\\\\pi,3\\\\pi)$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a4dc570tri5b-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$0$$, $$\\\\pm 2 \\\\pi$$"],"dependencies":["a4dc570tri5b-h4"],"title":"Interval","text":"What are possible values of \u03b8?\\\\n##figure3.gif##","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$0$$, $$\\\\pm 2 \\\\pi$$","$$0$$, $$2\\\\pi$$","$$2n \\\\pi$$"]}]}},{"id":"a4dc570tri5c","stepAnswer":["$$(-2\\\\pi,\\\\frac{\\\\left(-3\\\\pi\\\\right)}{2})$$ $$\\\\cup$$ $$(0,2\\\\pi)$$"],"problemType":"MultipleChoice","stepTitle":"Describe all angles \u03b8, in the interval $$(-2\\\\pi,2\\\\pi)$$, such that $$cos\\\\left(\\\\theta\\\\right)>0$$ and $$sin\\\\left(\\\\theta\\\\right)>0$$. 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One number is nine less than the other. Find the numbers.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{9}{2}$$","choices":["$$\\\\frac{11}{2}\\\\operatorname{and}\\\\frac{-11}{2}$$","$$\\\\frac{13}{2}\\\\operatorname{and}\\\\frac{-13}{2}$$","$$\\\\frac{9}{2}\\\\operatorname{and}\\\\frac{-9}{2}$$","$$\\\\frac{9}{2}$$"],"hints":{"DefaultPathway":[{"id":"a4edf7dEquation1a-h1","type":"hint","dependencies":[],"title":"Assumption","text":"Assume one number is $$m$$ and the other is $$n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation1a-h2","type":"hint","dependencies":["a4edf7dEquation1a-h1"],"title":"Translation","text":"Converting the first statement to $$m+n=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation1a-h3","type":"hint","dependencies":["a4edf7dEquation1a-h1"],"title":"Translation","text":"Converting the second statement to $$m-n=9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation1a-h4","type":"hint","dependencies":["a4edf7dEquation1a-h2","a4edf7dEquation1a-h3"],"title":"Addition","text":"Combining the two equation together","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation1a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{9}{2}$$"],"dependencies":["a4edf7dEquation1a-h4"],"title":"Calculation","text":"What is the value of $$m$$ when $$2m=9$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation1a-h6","type":"hint","dependencies":["a4edf7dEquation1a-h5"],"title":"Calculation","text":"Find the value of $$n$$ with the known value of $$m$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4edf7dEquation10","title":"Solve uniform motion applications","body":"Solve the following problem:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Solve Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4edf7dEquation10a","stepAnswer":["29;236"],"problemType":"MultipleChoice","stepTitle":"Translate to a system of equations and then solve:\\\\nA small jet can fly 1,325 miles in $$5$$ hours with a tailwind but only 1,035 miles in $$5$$ hours into a headwind. Find the speed of the jet in still air and the speed of the wind.","stepBody":"","answerType":"string","variabilization":{},"choices":["30;230","34;242","29;236"],"hints":{"DefaultPathway":[{"id":"a4edf7dEquation10a-h1","type":"hint","dependencies":[],"title":"Assumption","text":"Assume the jet speed in still air is $$m$$ and the wind speed is $$n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation10a-h2","type":"hint","dependencies":["a4edf7dEquation10a-h1"],"title":"Principle","text":"The tailwind speed for the ship is $$m+n$$ the headwind speed for the ship is $$m-n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation10a-h3","type":"hint","dependencies":["a4edf7dEquation10a-h1"],"title":"Translation","text":"$$Speed Time=Distance$$, $$5\\\\left(m+n\\\\right)=1325;$$ $$5(m-n)=1035$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation10a-h4","type":"hint","dependencies":["a4edf7dEquation10a-h2","a4edf7dEquation10a-h3"],"title":"Addition","text":"Combining the two equation together","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$236$$"],"dependencies":["a4edf7dEquation10a-h4"],"title":"Calculation","text":"What is the value of $$m$$ when $$10m=2360$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation10a-h6","type":"hint","dependencies":[],"title":"Calculation","text":"Find the value of $$n$$ with the known value of $$m$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4edf7dEquation11","title":"Translate to a system of equations and solve.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Solve Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4edf7dEquation11a","stepAnswer":["$$13$$ and $$17$$"],"problemType":"MultipleChoice","stepTitle":"The sum of two number is $$30$$. 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One is $$n+m=30$$, and the other $$n=m-4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation11a-h3","type":"hint","dependencies":["a4edf7dEquation11a-h2"],"title":"Solve","text":"Now, solve for $$n$$ or $$m$$, and plug it into the equation to find the other variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation11a-h4","type":"hint","dependencies":["a4edf7dEquation11a-h3"],"title":"Answer","text":"The answer is $$13$$ and $$17$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4edf7dEquation12","title":"Translate to a system of equations and solve.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Solve Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4edf7dEquation12a","stepAnswer":["$$6$$ and $$9$$"],"problemType":"MultipleChoice","stepTitle":"The sum of two number is $$15$$. One number is $$3$$ less than the other. Find the numbers.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$6$$ and $$9$$","choices":["$$6$$ and $$9$$","$$3$$ and $$12$$","$$4$$ and $$11$$","$$2$$ and $$13$$"],"hints":{"DefaultPathway":[{"id":"a4edf7dEquation12a-h1","type":"hint","dependencies":[],"title":"Name variables","text":"Let $$n$$ $$=$$ the first number and $$m$$ $$=$$ the second number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation12a-h2","type":"hint","dependencies":["a4edf7dEquation12a-h1"],"title":"Equations","text":"We know the sum of the two numbers is $$15$$, and that one number is $$3$$ less than the other. Using this, we can create two equations. One is $$n+m=15$$, and the other $$n=m-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation12a-h3","type":"hint","dependencies":["a4edf7dEquation12a-h2"],"title":"Solve","text":"Now, solve for $$n$$ or $$m$$, and plug it into the equation to find the other variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation12a-h4","type":"hint","dependencies":["a4edf7dEquation12a-h3"],"title":"Answer","text":"The answer is $$6$$ and $$9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4edf7dEquation13","title":"Translate to a system of equations and solve.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Solve Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4edf7dEquation13a","stepAnswer":["$$-18$$ and $$2$$"],"problemType":"MultipleChoice","stepTitle":"The sum of two number is $$-16$$. One number is $$20$$ less than the other. Find the numbers.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-18$$ and $$2$$","choices":["$$6$$ and $$9$$","$$3$$ and $$12$$","$$4$$ and $$11$$","$$2$$ and $$13$$","$$-18$$ and $$2$$"],"hints":{"DefaultPathway":[{"id":"a4edf7dEquation13a-h1","type":"hint","dependencies":[],"title":"Name variables","text":"Let $$n$$ $$=$$ the first number and $$m$$ $$=$$ the second number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation13a-h2","type":"hint","dependencies":["a4edf7dEquation13a-h1"],"title":"Equations","text":"We know the sum of the two numbers is $$-16$$, and that one number is $$20$$ less than the other. Using this, we can create two equations. One is $$n+m=-16$$, and the other $$n=m-20$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation13a-h3","type":"hint","dependencies":["a4edf7dEquation13a-h2"],"title":"Solve","text":"Now, solve for $$n$$ or $$m$$, and plug it into the equation to find the other variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation13a-h4","type":"hint","dependencies":["a4edf7dEquation13a-h3"],"title":"Answer","text":"The answer is $$-18$$ and $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4edf7dEquation14","title":"Translate to a system of equations and solve.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Solve Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4edf7dEquation14a","stepAnswer":["$$-7$$ and $$-19$$"],"problemType":"MultipleChoice","stepTitle":"The sum of two number is $$-26$$. One number is $$12$$ less than the other. Find the numbers.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-7$$ and $$-19$$","choices":["$$-7$$ and $$-19$$","$$3$$ and $$12$$","$$4$$ and $$11$$","$$2$$ and $$13$$","$$-18$$ and $$2$$"],"hints":{"DefaultPathway":[{"id":"a4edf7dEquation14a-h1","type":"hint","dependencies":[],"title":"Name variables","text":"Let $$n$$ $$=$$ the first number and $$m$$ $$=$$ the second number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation14a-h2","type":"hint","dependencies":["a4edf7dEquation14a-h1"],"title":"Equations","text":"We know the sum of the two numbers is $$-26$$, and that one number is $$12$$ less than the other. Using this, we can create two equations. One is $$n+m=-26$$, and the other $$n=m-12$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation14a-h3","type":"hint","dependencies":["a4edf7dEquation14a-h2"],"title":"Solve","text":"Now, solve for $$n$$ or $$m$$, and plug it into the equation to find the other variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation14a-h4","type":"hint","dependencies":["a4edf7dEquation14a-h3"],"title":"Answer","text":"The answer is $$-7$$ and $$-19$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4edf7dEquation15","title":"Translate to a system of equations and solve.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Solve Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4edf7dEquation15a","stepAnswer":["$$45$$ and $$20$$"],"problemType":"MultipleChoice","stepTitle":"The sum of two numbers is $$65$$. Their difference is $$25$$. Find the numbers.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$45$$ and $$20$$","choices":["$$45$$ and $$20$$","$$55$$ and $$10$$","$$35$$ and $$20$$","$$35$$ and $$30$$"],"hints":{"DefaultPathway":[{"id":"a4edf7dEquation15a-h1","type":"hint","dependencies":[],"title":"Name variables","text":"Let $$n$$ $$=$$ the first number and $$m$$ $$=$$ the second number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation15a-h2","type":"hint","dependencies":["a4edf7dEquation15a-h1"],"title":"Equations","text":"We know the sum of the two numbers is $$-65$$, and that the difference is $$25$$. Using this, we can create two equations. One is $$n+m=65$$, and the other $$n-m=25$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation15a-h3","type":"hint","dependencies":["a4edf7dEquation15a-h2"],"title":"Solve","text":"Now, solve for $$n$$ or $$m$$, and plug it into the equation to find the other variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation15a-h4","type":"hint","dependencies":["a4edf7dEquation15a-h3"],"title":"Answer","text":"The answer is $$45$$ and $$20$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4edf7dEquation16","title":"Translate to a system of equations and solve.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Solve Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4edf7dEquation16a","stepAnswer":["$$14$$ and $$23$$"],"problemType":"MultipleChoice","stepTitle":"The sum of two numbers is $$37$$. Their difference is $$9$$. Find the numbers.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$14$$ and $$23$$","choices":["$$14$$ and $$23$$","$$55$$ and $$10$$","$$35$$ and $$20$$","$$35$$ and $$30$$"],"hints":{"DefaultPathway":[{"id":"a4edf7dEquation16a-h1","type":"hint","dependencies":[],"title":"Name variables","text":"Let $$n$$ $$=$$ the first number and $$m$$ $$=$$ the second number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation16a-h2","type":"hint","dependencies":["a4edf7dEquation16a-h1"],"title":"Equations","text":"We know the sum of the two numbers is $$37$$, and that the difference is $$9$$. Using this, we can create two equations. One is $$n+m=37$$, and the other $$n-m=9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation16a-h3","type":"hint","dependencies":["a4edf7dEquation16a-h2"],"title":"Solve","text":"Now, solve for $$n$$ or $$m$$, and plug it into the equation to find the other variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation16a-h4","type":"hint","dependencies":["a4edf7dEquation16a-h3"],"title":"Answer","text":"The answer is $$14$$ and $$23$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4edf7dEquation17","title":"Translate to a system of equations and solve.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Solve Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4edf7dEquation17a","stepAnswer":["$$-38$$ and $$11$$"],"problemType":"MultipleChoice","stepTitle":"The sum of two numbers is $$-27$$. Their difference is $$-59$$. Find the numbers.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-38$$ and $$11$$","choices":["$$-38$$ and $$11$$","$$55$$ and $$10$$","$$35$$ and $$20$$","$$35$$ and $$30$$"],"hints":{"DefaultPathway":[{"id":"a4edf7dEquation17a-h1","type":"hint","dependencies":[],"title":"Name variables","text":"Let $$n$$ $$=$$ the first number and $$m$$ $$=$$ the second number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation17a-h2","type":"hint","dependencies":["a4edf7dEquation17a-h1"],"title":"Equations","text":"We know the sum of the two numbers is $$-27$$, and that the difference is $$-59$$. Using this, we can create two equations. One is $$n+m=-17$$, and the other $$n-m=-59$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation17a-h3","type":"hint","dependencies":["a4edf7dEquation17a-h2"],"title":"Solve","text":"Now, solve for $$n$$ or $$m$$, and plug it into the equation to find the other variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation17a-h4","type":"hint","dependencies":["a4edf7dEquation17a-h3"],"title":"Answer","text":"The answer is $$-38$$ and $$11$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4edf7dEquation18","title":"Translate to a system of equations and solve.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Solve Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4edf7dEquation18a","stepAnswer":["$$80$$"],"problemType":"TextBox","stepTitle":"Jackie has been offered positions by two cable companies. The first company pays a salary of $14,000 plus a commission of $100 for each cable package sold. The second pays a salary of $20,000 plus a commission of $25 for each cable package sold. How many cable packages would need to be sold to make the total pay the same?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$80$$","hints":{"DefaultPathway":[{"id":"a4edf7dEquation18a-h1","type":"hint","dependencies":[],"title":"Name variables","text":"Let $$n$$ $$=$$ the amount of packages sold and $$m$$ $$=$$ the salary.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation18a-h2","type":"hint","dependencies":["a4edf7dEquation18a-h1"],"title":"Equations","text":"We know the first company pays $14000 + $100 for every cable package sold, and the second company pays $20000 + $25 for each package sold. Using this, we can create two equations. One is $$m=14000+100n$$, and the other $$m=20000+25n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation18a-h3","type":"hint","dependencies":["a4edf7dEquation18a-h2"],"title":"Solve","text":"Now, solve for $$n$$ or $$m$$, and plug it into the equation to find the other variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation18a-h4","type":"hint","dependencies":["a4edf7dEquation18a-h3"],"title":"Answer","text":"The answer is $$80$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4edf7dEquation19","title":"Translate to a system of equations and solve.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Solve Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4edf7dEquation19a","stepAnswer":["$$22$$ and $$-67$$"],"problemType":"MultipleChoice","stepTitle":"The sum of two numbers is $$-45$$. Their difference is $$-89$$. Find the numbers.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$22$$ and $$-67$$","choices":["$$22$$ and $$-67$$","$$55$$ and $$10$$","$$35$$ and $$20$$","$$35$$ and $$30$$"],"hints":{"DefaultPathway":[{"id":"a4edf7dEquation19a-h1","type":"hint","dependencies":[],"title":"Name variables","text":"Let $$n$$ $$=$$ the first number and $$m$$ $$=$$ the second number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation19a-h2","type":"hint","dependencies":["a4edf7dEquation19a-h1"],"title":"Equations","text":"We know the sum of the two numbers is $$-45$$, and that the difference is $$-89$$. Using this, we can create two equations. One is $$n+m=-45$$, and the other $$n-m=-89$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation19a-h3","type":"hint","dependencies":["a4edf7dEquation19a-h2"],"title":"Solve","text":"Now, solve for $$n$$ or $$m$$, and plug it into the equation to find the other variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation19a-h4","type":"hint","dependencies":["a4edf7dEquation19a-h3"],"title":"Answer","text":"The answer is $$22$$ and $$-67$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4edf7dEquation2","title":"Solve Direct Translation Applications","body":"Solve the following problem:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Solve Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4edf7dEquation2a","stepAnswer":["$$600$$"],"problemType":"TextBox","stepTitle":"Heather has been offered two options for her salary as a trainer at the gym. Option A would pay her $25,000 plus $15 for each training session. Option B would pay her $10,000+$40 for each training session. How many training sessions would make the salary options equal?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$600$$","hints":{"DefaultPathway":[{"id":"a4edf7dEquation2a-h1","type":"hint","dependencies":[],"title":"Assumption","text":"Assume the number of session is $$n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation2a-h2","type":"hint","dependencies":["a4edf7dEquation2a-h1"],"title":"Translation","text":"Converting the option A to $$25000+15n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation2a-h3","type":"hint","dependencies":["a4edf7dEquation2a-h1"],"title":"Translation","text":"Converting the option B to $$10000+40n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation2a-h4","type":"hint","dependencies":["a4edf7dEquation2a-h2","a4edf7dEquation2a-h3"],"title":"Combination","text":"Set an equation so option A $$=$$ option B, $$25000+15n$$ $$=$$ $$10000+40n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation2a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$600$$"],"dependencies":["a4edf7dEquation2a-h4"],"title":"Calculation","text":"What is the value of $$m$$ when $$25n=15000$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4edf7dEquation20","title":"Translate to a system of equations and solve.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Solve Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4edf7dEquation20a","stepAnswer":["$$120$$"],"problemType":"TextBox","stepTitle":"Mitchell currently sells stoves for company A at a salary of $12,000 plus a $150 commission for each stove he sells. Company B offers him a position with a salary of $24,000 plus a $50 commission for each stove he sells. How many stoves would Mitchell need to sell for the options to be equal?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$120$$","hints":{"DefaultPathway":[{"id":"a4edf7dEquation20a-h1","type":"hint","dependencies":[],"title":"Name variables","text":"Let $$n$$ $$=$$ the amount of stoves sold and $$m$$ $$=$$ the salary.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation20a-h2","type":"hint","dependencies":["a4edf7dEquation20a-h1"],"title":"Equations","text":"We know the company A pays $12000 + $150 for every cable package sold, and the second company pays $24000 + $50 for each package sold. Using this, we can create two equations. One is $$m=12000+150n$$, and the other $$m=24000+50n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation20a-h3","type":"hint","dependencies":["a4edf7dEquation20a-h2"],"title":"Solve","text":"Now, solve for $$n$$ or $$m$$, and plug it into the equation to find the other variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation20a-h4","type":"hint","dependencies":["a4edf7dEquation20a-h3"],"title":"Answer","text":"The answer is $$120$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4edf7dEquation21","title":"Geometry Applications of Systems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Solve Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4edf7dEquation21a","stepAnswer":["$$72.5, 17.5$$"],"problemType":"MultipleChoice","stepTitle":"The difference of two complementary angles is $$55$$ degrees. Find the measures of the angles. Enter your answers in form a,b where a is greater than $$b$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$72.5, 17.5$$","choices":["$$72.5, 17.5$$","$$17.5, 72.5$$","$$67.5, 22.5$$"],"hints":{"DefaultPathway":[{"id":"a4edf7dEquation21a-h1","type":"hint","dependencies":[],"title":"Creating an Equation","text":"The complementary angles add to $$90$$ and have a difference of $$55$$ degrees. So, we have: $$x+y=90, x-y=55$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation21a-h2","type":"hint","dependencies":["a4edf7dEquation21a-h1"],"title":"Elimination","text":"We can eliminate $$y$$ from this equation by adding them together. We get $$2x=145$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation21a-h3","type":"hint","dependencies":["a4edf7dEquation21a-h2"],"title":"Solving for X, Y","text":"$$x=\\\\frac{145}{2}$$, $$x=72.5$$. Plugginig this into an original equation, we get $$y=17.5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4edf7dEquation22","title":"Geometry Applications of Systems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Solve Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4edf7dEquation22a","stepAnswer":["$$53.5, 36.5$$"],"problemType":"MultipleChoice","stepTitle":"The difference of two complementary angles is $$17$$ degrees. Find the measures of the angles. Enter your answers in form a,b where a is greater than $$b$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$53.5, 36.5$$","choices":["$$53.5, 36.5$$","$$36.5, 53.5$$","$$57.5, 32.5$$"],"hints":{"DefaultPathway":[{"id":"a4edf7dEquation22a-h1","type":"hint","dependencies":[],"title":"Creating an Equation","text":"The complementary angles add to $$90$$ and have a difference of $$17$$ degrees. So, we have: $$x+y=90, x-y=17$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation22a-h2","type":"hint","dependencies":["a4edf7dEquation22a-h1"],"title":"Elimination","text":"We can eliminate $$y$$ from this equation by adding them together. We get $$2x=107$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation22a-h3","type":"hint","dependencies":["a4edf7dEquation22a-h2"],"title":"Solving for X, Y","text":"$$x=\\\\frac{107}{2}$$, $$x=53.5$$. Plugging this into an original equation, we get $$y=36.5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4edf7dEquation23","title":"Geometry Applications of Systems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Solve Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4edf7dEquation23a","stepAnswer":["56,34"],"problemType":"MultipleChoice","stepTitle":"Two angles are complementary. The measure of the larger angle is twelve less than twice the measure of the smaller angle. Find the measures of both angles. Enter your answers in form a,b where a is greater than $$b$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["56,34","34,56","38,52"],"hints":{"DefaultPathway":[{"id":"a4edf7dEquation23a-h1","type":"hint","dependencies":[],"title":"Creating an Equation","text":"Complementary angles add to $$90$$, so we have $$x+y=90$$. We also know that $$-x+2y=12$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation23a-h2","type":"hint","dependencies":["a4edf7dEquation23a-h1"],"title":"Elimination","text":"We can eliminate $$x$$ from the equation by adding. We have $$3y=102$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation23a-h3","type":"hint","dependencies":["a4edf7dEquation23a-h2"],"title":"Solving for X, Y","text":"$$y=\\\\frac{102}{3}=34$$. Plugging back $$y$$, we get $$x=90-34=56$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4edf7dEquation24","title":"Geometry Applications of Systems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Solve Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4edf7dEquation24a","stepAnswer":["74,16"],"problemType":"MultipleChoice","stepTitle":"Two angles are complementary. The measure of the larger angle is ten more than four times the measure of the smaller angle. Find the measures of both angles. Enter your answers in form a,b where a is greater than $$b$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["74,16","32,58","16,74"],"hints":{"DefaultPathway":[{"id":"a4edf7dEquation24a-h1","type":"hint","dependencies":[],"title":"Creating an Equation","text":"Complementary angles add to $$90$$, so we have $$x+y=90$$. We also know that $$x-4y=10$$ from the problem.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation24a-h2","type":"hint","dependencies":["a4edf7dEquation24a-h1"],"title":"Elimination","text":"Let us multiply the first equation by $$-1$$ to get $$-x-y=-90$$. We can now add to eliminate $$x$$. $$-5y=-80$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation24a-h3","type":"hint","dependencies":["a4edf7dEquation24a-h2"],"title":"Solving for X, Y","text":"$$y=\\\\frac{\\\\left(-80\\\\right)}{\\\\left(-5\\\\right)}=16$$. Plugging $$y$$ back into an original equationi, we get $$90-16=x=74$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4edf7dEquation25","title":"Geometry Applications of Systems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Solve Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4edf7dEquation25a","stepAnswer":["94,86"],"problemType":"MultipleChoice","stepTitle":"The difference of two supplementary angles is $$8$$ degrees. Find the measures of the angles. Enter your answers in form a,b where a is greater than $$b$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["86,94","94,86","100,80"],"hints":{"DefaultPathway":[{"id":"a4edf7dEquation25a-h1","type":"hint","dependencies":[],"title":"Creating an Equation","text":"The supplementary angles add to $$90$$ and have a difference of $$55$$ degrees. So, we have: $$x+y=180, x-y=8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation25a-h2","type":"hint","dependencies":["a4edf7dEquation25a-h1"],"title":"Elimination","text":"We can eliminate $$y$$ from this equation by adding them together. We get $$2x=188$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation25a-h3","type":"hint","dependencies":["a4edf7dEquation25a-h2"],"title":"Solving for X, Y","text":"$$x=\\\\frac{188}{2}=94$$. Plugginig this into an original equation, we get $$y=86$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4edf7dEquation26","title":"Geometry Applications of Systems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Solve Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4edf7dEquation26a","stepAnswer":["134,46"],"problemType":"MultipleChoice","stepTitle":"The difference of two supplementary angles is $$88$$ degrees. Find the measures of the angles. Enter your answers in form a,b where a is greater than $$b$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["46,134","94,86","134,46"],"hints":{"DefaultPathway":[{"id":"a4edf7dEquation26a-h1","type":"hint","dependencies":[],"title":"Creating an Equation","text":"The supplementary angles add to $$180$$ and have a difference of $$88$$ degrees. So, we have: $$x+y=180, x-y=88$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation26a-h2","type":"hint","dependencies":["a4edf7dEquation26a-h1"],"title":"Elimination","text":"We can eliminate $$y$$ from this equation by adding them together. We get $$2x=268$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation26a-h3","type":"hint","dependencies":["a4edf7dEquation26a-h2"],"title":"Solving for X, Y","text":"$$x=\\\\frac{268}{2}=134$$. Plugginig this into an original equation, we get $$y=46$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4edf7dEquation27","title":"Geometry Applications of Systems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Solve Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4edf7dEquation27a","stepAnswer":["143,37"],"problemType":"MultipleChoice","stepTitle":"Two angles are supplementary. The measure of the larger angle is five less than four times the measure of the smaller angle. Find the measures of both angles. Enter your answers in form a,b where a is greater than $$b$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["146,34","143,37","134,46"],"hints":{"DefaultPathway":[{"id":"a4edf7dEquation27a-h1","type":"hint","dependencies":[],"title":"Creating an Equation","text":"Supplementary angles add to $$180$$, so we have $$x+y=180$$. We also know that $$-x+4y=5$$ from the problem.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation27a-h2","type":"hint","dependencies":["a4edf7dEquation27a-h1"],"title":"Elimination","text":"Let us add the equations to eliminaite $$x$$. $$5y=185$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation27a-h3","type":"hint","dependencies":["a4edf7dEquation27a-h2"],"title":"Solving for X, Y","text":"$$y=\\\\frac{185}{5}=37$$. Plugging $$y$$ back in, we get $$x=180-37=143$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4edf7dEquation28","title":"Geometry Applications of Systems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Solve Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4edf7dEquation28a","stepAnswer":["135,45"],"problemType":"MultipleChoice","stepTitle":"Two angles are supplementary. The measure of the larger angle is four more than three times the measure of the smaller angle. Find the measures of both angles. Enter your answers in form a,b where a is greater than $$b$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["135,45","143,37","134,46"],"hints":{"DefaultPathway":[{"id":"a4edf7dEquation28a-h1","type":"hint","dependencies":[],"title":"Creating an Equation","text":"Supplementary angles add to $$180$$, so we have $$x+y=180$$. We also know that $$x-3y=4$$ from the problem.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation28a-h2","type":"hint","dependencies":["a4edf7dEquation28a-h1"],"title":"Elimination","text":"We can multiply the first equation by $$3$$. $$3x+3y=540$$. Now, we can add the equations to eliminiate $$y$$. $$4x=184$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation28a-h3","type":"hint","dependencies":["a4edf7dEquation28a-h2"],"title":"Solving for X, Y","text":"$$x=\\\\frac{184}{4}=46$$. Plugging $$x$$ back in, we get $$y=135$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4edf7dEquation29","title":"Geometry Applications of Systems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Solve Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4edf7dEquation29a","stepAnswer":["71,19"],"problemType":"MultipleChoice","stepTitle":"The measure of one of the small angles of a right triangle is $$14$$ more than $$3$$ times the measure of the other small angle. Find the measure of both angles. Enter your answers in form a,b where a is greater than $$b$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["135,45","71,19","92,18"],"hints":{"DefaultPathway":[{"id":"a4edf7dEquation29a-h1","type":"hint","dependencies":[],"title":"Creating an Equation","text":"The small angles of a right triangle add to $$90$$. $$x+y=90$$. From the problem, we also know that $$x-3y=14$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation29a-h2","type":"hint","dependencies":["a4edf7dEquation29a-h1"],"title":"Elimination","text":"We can multiply the first equation by $$3$$ to get $$3x+3y=270$$. Now, we add the equations to eliminate $$y$$. $$4x=284$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation29a-h3","type":"hint","dependencies":["a4edf7dEquation29a-h2"],"title":"Solving for X, Y","text":"$$x=\\\\frac{284}{4}=71$$. Plugging $$x$$ back into the equation, we get $$y=90-71=19$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4edf7dEquation3","title":"Solve Geometry Applications","body":"Solve the following problem:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Solve Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4edf7dEquation3a","stepAnswer":["58;32"],"problemType":"MultipleChoice","stepTitle":"Translate to a system of equations and then solve. The difference of two complementary angles is $$26$$ degrees. Find the measures of the angles.","stepBody":"","answerType":"string","variabilization":{},"choices":["56;34","58;32","48;42"],"hints":{"DefaultPathway":[{"id":"a4edf7dEquation3a-h1","type":"hint","dependencies":[],"title":"Assumption","text":"Assume one angle is $$m$$ and the other angle is $$n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation3a-h2","type":"hint","dependencies":["a4edf7dEquation3a-h1"],"title":"Principle","text":"Two complementary angles means the addition of the two angles are $$90$$, so $$m+n=90$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation3a-h3","type":"hint","dependencies":["a4edf7dEquation3a-h1"],"title":"Translation","text":"Converting the other statement to $$m-n=26$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation3a-h4","type":"hint","dependencies":["a4edf7dEquation3a-h2","a4edf7dEquation3a-h3"],"title":"Addition","text":"Combining the two equation together","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$58$$"],"dependencies":["a4edf7dEquation3a-h4"],"title":"Calculation","text":"What is the value of $$m$$ when $$2m=116$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation3a-h6","type":"hint","dependencies":[],"title":"Calculation","text":"Find the value of $$n$$ with the known value of $$m$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4edf7dEquation30","title":"Geometry Applications of Systems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Solve Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4edf7dEquation30a","stepAnswer":["74,16"],"problemType":"MultipleChoice","stepTitle":"The measure of one of the small angles of a right triangle is $$26$$ more than $$3$$ times the measure of the other small angle. Find the measure of both angles. Enter your answers in form a,b where a is greater than $$b$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["72,18","71,19","74,16"],"hints":{"DefaultPathway":[{"id":"a4edf7dEquation30a-h1","type":"hint","dependencies":[],"title":"Creating an Equation","text":"The small angles of a right triangle add to $$90$$. $$x+y=90$$. From the problem, we also know that $$x-3y=26$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation30a-h2","type":"hint","dependencies":["a4edf7dEquation30a-h1"],"title":"Elimination","text":"We can multiply the first equation by $$3$$ to get $$3x+3y=270$$. Now, we add the equations to eliminate $$y$$. $$4x=296$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation30a-h3","type":"hint","dependencies":["a4edf7dEquation30a-h2"],"title":"Solving for X, Y","text":"$$x=\\\\frac{296}{4}=74$$. Plugging $$x$$ back into the equation, we get $$y=90-74=16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4edf7dEquation4","title":"Solve Geometry Applications","body":"Solve the following problem:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Solve Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4edf7dEquation4a","stepAnswer":["148;32"],"problemType":"MultipleChoice","stepTitle":"Translate to a system of equations and then solve. The difference of two complementary angles is $$26$$ degrees. Find the measures of the angles.","stepBody":"","answerType":"string","variabilization":{},"choices":["148;32","162;18","144;36"],"hints":{"DefaultPathway":[{"id":"a4edf7dEquation4a-h1","type":"hint","dependencies":[],"title":"Assumption","text":"Assume one angle is $$m$$ and the other angle is $$n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation4a-h2","type":"hint","dependencies":["a4edf7dEquation4a-h1"],"title":"Principle","text":"Two supplementary angles means the addition of the two angles are $$180$$, so $$m+n=180$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation4a-h3","type":"hint","dependencies":["a4edf7dEquation4a-h1"],"title":"Translation","text":"Converting the other statement to $$m=5n-12$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation4a-h4","type":"hint","dependencies":["a4edf7dEquation4a-h2","a4edf7dEquation4a-h3"],"title":"Addition","text":"Combining the two equation together","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation4a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$32$$"],"dependencies":["a4edf7dEquation4a-h4"],"title":"Calculation","text":"What is the value of $$n$$ when $$6n=192$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation4a-h6","type":"hint","dependencies":[],"title":"Calculation","text":"Find the value of $$m$$ with the known value of $$n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4edf7dEquation5","title":"Solve Geometry Applications","body":"Solve the following problem:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Solve Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4edf7dEquation5a","stepAnswer":["20;70"],"problemType":"MultipleChoice","stepTitle":"The measure of one of the small angles of a right triangle is ten more than three times the measure of the other small angle. Find the measures of both angles.","stepBody":"","answerType":"string","variabilization":{},"choices":["23;67","30;60","20;70"],"hints":{"DefaultPathway":[{"id":"a4edf7dEquation5a-h1","type":"hint","dependencies":[],"title":"Assumption","text":"Assume one angle is $$m$$ and the other angle is $$n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation5a-h2","type":"hint","dependencies":["a4edf7dEquation5a-h1"],"title":"Principle","text":"The addition of two small angles in a right triangle is $$90$$, so $$m+n=90$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation5a-h3","type":"hint","dependencies":["a4edf7dEquation5a-h1"],"title":"Translation","text":"Converting the other statement to $$m=3n+10$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation5a-h4","type":"hint","dependencies":["a4edf7dEquation5a-h2","a4edf7dEquation5a-h3"],"title":"Addition","text":"Combining the two equation together","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a4edf7dEquation5a-h4"],"title":"Calculation","text":"What is the value of $$n$$ when $$4n=80$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation5a-h6","type":"hint","dependencies":[],"title":"Calculation","text":"Find the value of $$m$$ with the known value of $$n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4edf7dEquation6","title":"Solve Geometry Applications","body":"Solve the following problem:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Solve Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4edf7dEquation6a","stepAnswer":["20;85"],"problemType":"MultipleChoice","stepTitle":"Translate to a system of equations and then solve:\\\\nRandall has $$125$$ feet of fencing to enclose the part of his backyard adjacent to his house. He will only need to fence around three sides, because the fourth side will be the wall of the house. He wants the length of the fenced yard (parallel to the house wall) to be $$5$$ feet more than four times as long as the width. Find the length and the width.","stepBody":"","answerType":"string","variabilization":{},"choices":["20;85","23;67","24;65"],"hints":{"DefaultPathway":[{"id":"a4edf7dEquation6a-h1","type":"hint","dependencies":[],"title":"Assumption","text":"Assume the width is $$m$$ and the length is $$n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation6a-h2","type":"hint","dependencies":["a4edf7dEquation6a-h1"],"title":"Translation","text":"The fence only include a length and two widths, so $$2m+n=125$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation6a-h3","type":"hint","dependencies":["a4edf7dEquation6a-h1"],"title":"Translation","text":"Converting the other statement to $$n=4m+5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation6a-h4","type":"hint","dependencies":["a4edf7dEquation6a-h2","a4edf7dEquation6a-h3"],"title":"Addition","text":"Combining the two equation together","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a4edf7dEquation6a-h4"],"title":"Calculation","text":"What is the value of $$m$$ when $$6m=120$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation6a-h6","type":"hint","dependencies":[],"title":"Calculation","text":"Find the value of $$n$$ with the known value of $$m$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4edf7dEquation7","title":"Solve uniform motion applications","body":"Solve the following problem:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Solve Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4edf7dEquation7a","stepAnswer":["$$2.5$$"],"problemType":"TextBox","stepTitle":"Translate to a system of equations and then solve:\\\\nJoni left St. Louis on the interstate, driving west towards Denver at a speed of $$65$$ miles per hour. Half an hour later, Kelly left St. Louis on the same route as Joni, driving $$78$$ miles per hour. How long will it take Kelly to catch up to Joni?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2.5$$","hints":{"DefaultPathway":[{"id":"a4edf7dEquation7a-h1","type":"hint","dependencies":[],"title":"Assumption","text":"Assume the travel time for Joni is $$m$$ and the travel time for Kelly is $$n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation7a-h2","type":"hint","dependencies":["a4edf7dEquation7a-h1"],"title":"Translation","text":"Kelly left half an hour later, so $$n=m-\\\\frac{1}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation7a-h3","type":"hint","dependencies":["a4edf7dEquation7a-h1"],"title":"Translation","text":"$$Speed Time=Distance$$, so $$65m=78n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation7a-h4","type":"hint","dependencies":["a4edf7dEquation7a-h2","a4edf7dEquation7a-h3"],"title":"Addition","text":"Combining the two equation together","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation7a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a4edf7dEquation7a-h4"],"title":"Calculation","text":"What is the value of $$m$$ when $$13m=39$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation7a-h6","type":"hint","dependencies":[],"title":"Calculation","text":"We want to find $$n$$, so find the value of $$n$$ with the known value of $$m$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4edf7dEquation8","title":"Solve uniform motion applications","body":"Solve the following problem:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Solve Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4edf7dEquation8a","stepAnswer":["$$1.5;13.5$$"],"problemType":"MultipleChoice","stepTitle":"Translate to a system of equations and then solve:\\\\nA river cruise ship sailed $$60$$ miles downstream for $$4$$ hours and then took $$5$$ hours sailing upstream to return to the dock. Find the speed of the ship in still water and the speed of the river current.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$1.5;13.5$$","choices":["$$1.5;13.5$$","2;14","$$2.5;12.5$$"],"hints":{"DefaultPathway":[{"id":"a4edf7dEquation8a-h1","type":"hint","dependencies":[],"title":"Assumption","text":"Assume the ship speed in still water is $$m$$ and the water speed is $$n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation8a-h2","type":"hint","dependencies":["a4edf7dEquation8a-h1"],"title":"Principle","text":"The downstream speed for the ship is $$m+n$$ the upstream speed for the ship is $$m-n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation8a-h3","type":"hint","dependencies":["a4edf7dEquation8a-h1"],"title":"Translation","text":"$$Speed Time=Distance$$, distance is constant, so $$4\\\\left(m+n\\\\right)=60;$$ $$5(m-n)=60$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation8a-h4","type":"hint","dependencies":["a4edf7dEquation8a-h2","a4edf7dEquation8a-h3"],"title":"Addition","text":"Combining the two equation together","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation8a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$13.5$$"],"dependencies":["a4edf7dEquation8a-h4"],"title":"Calculation","text":"What is the value of $$m$$ when $$40m=540$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation8a-h6","type":"hint","dependencies":[],"title":"Calculation","text":"Find the value of $$n$$ with the known value of $$m$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a4edf7dEquation9","title":"Solve uniform motion applications","body":"Solve the following problem:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Solve Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a4edf7dEquation9a","stepAnswer":["18;347"],"problemType":"MultipleChoice","stepTitle":"Translate to a system of equations and then solve:\\\\nA private jet can fly 1,095 miles in three hours with a tailwind but only $$987$$ miles in three hours into a headwind. Find the speed of the jet in still air and the speed of the wind.","stepBody":"","answerType":"string","variabilization":{},"choices":["18;345","18;347","16;345"],"hints":{"DefaultPathway":[{"id":"a4edf7dEquation9a-h1","type":"hint","dependencies":[],"title":"Assumption","text":"Assume the jet speed in still air is $$m$$ and the wind speed is $$n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation9a-h2","type":"hint","dependencies":["a4edf7dEquation9a-h1"],"title":"Principle","text":"The tailwind speed for the ship is $$m+n$$ the headwind speed for the ship is $$m-n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation9a-h3","type":"hint","dependencies":["a4edf7dEquation9a-h1"],"title":"Translation","text":"$$Speed Time=Distance$$, $$3\\\\left(m+n\\\\right)=1095;$$ $$3(m-n)=987$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation9a-h4","type":"hint","dependencies":["a4edf7dEquation9a-h2","a4edf7dEquation9a-h3"],"title":"Addition","text":"Combining the two equation together","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation9a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$347$$"],"dependencies":["a4edf7dEquation9a-h4"],"title":"Calculation","text":"What is the value of $$m$$ when $$6m=2082$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a4edf7dEquation9a-h6","type":"hint","dependencies":[],"title":"Calculation","text":"Find the value of $$n$$ with the known value of $$m$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a50bb85hypo1","title":"Television Survey","body":"In a recent survey, it was stated that Americans watch television on average four hours per day. Assume that \u03c3 $$=$$ $$2$$. Using your class as the sample, conduct a hypothesis test to determine if the average for students at your school is lower.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Hypothesis Testing of a Single Mean and Single Proportion","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a50bb85hypo1a","stepAnswer":["$$H_0=4$$"],"problemType":"MultipleChoice","stepTitle":"What is the null hypothesis $$H_0$$ in this context?","stepBody":"Choose the best answer from the following.","answerType":"string","variabilization":{},"answerLatex":"$$H_0=4$$","choices":["$$H_0<4$$","$$H_0=4$$","$$H_0=2$$","$$H_0$$ $$ \\\\neq $$ $$4$$"],"hints":{"DefaultPathway":[{"id":"a50bb85hypo1a-h1","type":"hint","dependencies":[],"title":"Null Hypothesis vs. Alternative Hypothesis","text":"The null hypothesis is generally the complement of the alternative hypothesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a50bb85hypo1a-h2","type":"hint","dependencies":["a50bb85hypo1a-h1"],"title":"Sample Statistics","text":"What are some sample statistics stated in the problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a50bb85hypo1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a50bb85hypo1a-h2"],"title":"Sample Mean \u03bc","text":"What is the sample Mean \u03bc stated in the question?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a50bb85hypo2","title":"Television Survey","body":"In a recent survey, it was stated that Americans watch television on average four hours per day. Assume that \u03c3 $$=$$ $$2$$. Using your class as the sample, conduct a hypothesis test to determine if the average for students at your school is lower.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Hypothesis Testing of a Single Mean and Single Proportion","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a50bb85hypo2a","stepAnswer":["$$H_1<4$$"],"problemType":"MultipleChoice","stepTitle":"What is the alternative hypothesis $$H_1$$ in this context?","stepBody":"Choose the best answer from the following.","answerType":"string","variabilization":{},"answerLatex":"$$H_1<4$$","choices":["$$H_1<4$$","$$H_1=4$$","$$H_1>4$$","$$H_1$$ $$ \\\\neq $$ $$4$$"],"hints":{"DefaultPathway":[{"id":"a50bb85hypo2a-h1","type":"hint","dependencies":[],"title":"Sample Statistics","text":"What are some sample statistics stated in the problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a50bb85hypo2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a50bb85hypo2a-h1"],"title":"Sample Mean \u03bc","text":"What is the sample Mean \u03bc stated in the question?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a50bb85hypo2a-h3","type":"hint","dependencies":["a50bb85hypo2a-h2"],"title":"Null Hypothesis vs. Alternative Hypothesis","text":"The null hypothesis is generally the complement of the alternative hypothesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a50bb85hypo2a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Lower to sample mean"],"dependencies":["a50bb85hypo2a-h3"],"title":"One Sided vs. Two Sided Hypothesis Test","text":"What is the guess on the number of hours spent by Americans watching television?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Greater than than sample mean","Lower or equal to smaple mean","Lower to sample mean","Greater than or equal to sample mean"]}]}}]},{"id":"a50bb85hypo3","title":"Language Survey","body":"About $$42.3\\\\%$$ of Californians and $$19.6\\\\%$$ of all Americans over age five speak a language other than English at home. Using your class as the sample, conduct a hypothesis test to determine if the percent of the students at your school who speak a language other than English at home is different from $$42.3\\\\%$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Hypothesis Testing of a Single Mean and Single Proportion","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a50bb85hypo3a","stepAnswer":["$$H_0=42.3\\\\%$$"],"problemType":"MultipleChoice","stepTitle":"What is the null hypothesis $$H_0$$ in this context?","stepBody":"Choose the best answer from the following.","answerType":"string","variabilization":{},"answerLatex":"$$H_0=42.3\\\\%$$","choices":["$$H_0$$ < $$42.3\\\\%$$","$$H_0=19.6\\\\%$$","$$H_0=42.3\\\\%$$","$$H_0$$ $$ \\\\neq $$ $$42.3\\\\%$$"],"hints":{"DefaultPathway":[{"id":"a50bb85hypo3a-h1","type":"hint","dependencies":[],"title":"Null Hypothesis vs. Alternative Hypothesis","text":"The null hypothesis is generally the complement of the alternative hypothesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a50bb85hypo3a-h2","type":"hint","dependencies":["a50bb85hypo3a-h1"],"title":"Sample Statistics","text":"What are some sample statistics stated in the problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a50bb85hypo3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$42.3$$"],"dependencies":["a50bb85hypo3a-h2"],"title":"Sample Mean \u03bc","text":"What is the sample Mean \u03bc stated in the question?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a50bb85hypo4","title":"Language Survey","body":"About $$42.3\\\\%$$ of Californians and $$19.6\\\\%$$ of all Americans over age five speak a language other than English at home. Using your class as the sample, conduct a hypothesis test to determine if the percent of the students at your school who speak a language other than English at home is different from $$42.3\\\\%$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Hypothesis Testing of a Single Mean and Single Proportion","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a50bb85hypo4a","stepAnswer":["$$H_0$$ $$ \\\\neq $$ $$42.3\\\\%$$"],"problemType":"MultipleChoice","stepTitle":"What is the alternative hypothesis $$H_1$$ in this context?","stepBody":"Choose the best answer from the following.","answerType":"string","variabilization":{},"answerLatex":"$$H_0$$ $$ \\\\neq $$ $$42.3\\\\%$$","choices":["$$H_0$$ < $$42.3\\\\%$$","$$H_0=19.6\\\\%$$","$$H_0=42.3\\\\%$$","$$H_0$$ $$ \\\\neq $$ $$42.3\\\\%$$"],"hints":{"DefaultPathway":[{"id":"a50bb85hypo4a-h1","type":"hint","dependencies":[],"title":"Sample Statistics","text":"What are some sample statistics stated in the problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a50bb85hypo4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$42.3$$"],"dependencies":["a50bb85hypo4a-h1"],"title":"Sample Mean \u03bc","text":"What is the sample Mean \u03bc percentage stated in the question?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a50bb85hypo4a-h3","type":"hint","dependencies":["a50bb85hypo4a-h2"],"title":"Null Hypothesis vs. Alternative Hypothesis","text":"The null hypothesis is generally the complement of the alternative hypothesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a50bb85hypo4a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Not equal to the sample mean"],"dependencies":["a50bb85hypo4a-h3"],"title":"One Sided vs. Two Sided Hypothesis Test","text":"What is the guess on the number of hours spent by Americans watching television?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Not equal to the sample mean","Lower or equal to smaple mean","Lower to sample mean","Greater than or equal to sample mean"]}]}}]},{"id":"a50bb85hypo5","title":"Jeans Survey","body":"Suppose that young adults own an average of three pairs of jeans. Survey eight people from your class to determine if the average is higher than three. Assume the population is normal.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Hypothesis Testing of a Single Mean and Single Proportion","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a50bb85hypo5a","stepAnswer":["$$H_0=3$$"],"problemType":"MultipleChoice","stepTitle":"What is the null hypothesis $$H_0$$ in this context?","stepBody":"Choose the best answer from the following.","answerType":"string","variabilization":{},"answerLatex":"$$H_0=3$$","choices":["$$H_0>3$$","$$H_0=3$$","$$H_0=8$$","$$H_0$$ $$ \\\\neq $$ $$3$$"],"hints":{"DefaultPathway":[{"id":"a50bb85hypo5a-h1","type":"hint","dependencies":[],"title":"Null Hypothesis vs. Alternative Hypothesis","text":"The null hypothesis is generally the complement of the alternative hypothesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a50bb85hypo5a-h2","type":"hint","dependencies":["a50bb85hypo5a-h1"],"title":"Sample Statistics","text":"What are some sample statistics stated in the problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a50bb85hypo5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a50bb85hypo5a-h2"],"title":"Sample Mean \u03bc","text":"What is the sample average \u03bc stated in the question?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a50bb85hypo6","title":"Jeans Survey","body":"Suppose that young adults own an average of three pairs of jeans. Survey eight people from your class to determine if the average is higher than three. Assume the population is normal.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Hypothesis Testing of a Single Mean and Single Proportion","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a50bb85hypo6a","stepAnswer":["$$H_0>3$$"],"problemType":"MultipleChoice","stepTitle":"What is the alternative hypothesis $$H_1$$ in this context?","stepBody":"Choose the best answer from the following.","answerType":"string","variabilization":{},"answerLatex":"$$H_0>3$$","choices":["$$H_0>3$$","$$H_0=3$$","$$H_0=8$$","$$H_0$$ $$ \\\\neq $$ $$3$$"],"hints":{"DefaultPathway":[{"id":"a50bb85hypo6a-h1","type":"hint","dependencies":[],"title":"Sample Statistics","text":"What are some sample statistics stated in the problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a50bb85hypo6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a50bb85hypo6a-h1"],"title":"Sample Mean \u03bc","text":"What is the sample Mean \u03bc percentage stated in the question?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a50bb85hypo6a-h3","type":"hint","dependencies":["a50bb85hypo6a-h2"],"title":"Null Hypothesis vs. Alternative Hypothesis","text":"The null hypothesis is generally the complement of the alternative hypothesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a50bb85hypo6a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Greater than sample mean"],"dependencies":["a50bb85hypo6a-h3"],"title":"One Sided vs. Two Sided Hypothesis Test","text":"What is the guess on the number of hours spent by Americans watching television?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Not equal to the sample mean","Lower or equal to smaple mean","Lower to sample mean","Greater than sample mean"]}]}}]},{"id":"a512f5aexplog1","title":"Solve Exponential Equations","body":"Solve the following equation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Exponential and Logarithmic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a512f5aexplog1a","stepAnswer":["$$\\\\frac{-1}{3}$$"],"problemType":"TextBox","stepTitle":"$$64\\\\times4^{3x}$$ $$=$$ $$16$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-1}{3}$$","hints":{"DefaultPathway":[{"id":"a512f5aexplog1a-h1","type":"hint","dependencies":[],"title":"Rewrite equation so all powers have the same base.","text":"The first step is to identify the common base of all the terms in the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a512f5aexplog1a-h1"],"title":"Finding the common base.","text":"What is $$4\\\\times4$$? How does this relate to exponents?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$64$$"],"dependencies":["a512f5aexplog1a-h2"],"title":"Finding the common base.","text":"What is $$4\\\\times4\\\\times4$$? How does this relate to exponents?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog1a-h4","type":"hint","dependencies":["a512f5aexplog1a-h3"],"title":"Combining terms with the same base.","text":"The next step is to rewrite the terms so that they all have the same base and combine the terms on each side of the equation and then solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog1a-h5","type":"hint","dependencies":["a512f5aexplog1a-h4"],"title":"Properties of exponents","text":"When two terms with the same base are multiplied together, their exponents are added together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog1a-h6","type":"hint","dependencies":["a512f5aexplog1a-h5"],"title":"Taking a power of a power.","text":"To take a power of a power, multiply exponents.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog1a-h7","type":"hint","dependencies":["a512f5aexplog1a-h6"],"title":"One-to-one property of exponents.","text":"The one-to-one property of exponential functions states that for any algebraic expressions S and T, and any positive real number $$b$$ (except when $$b$$ $$=$$ 1), $$b^S$$ $$=$$ $$b^T$$ if and only if S $$=$$ T.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a512f5aexplog10","title":"Solving Exponential Equations using Logarithms","body":"Solve the following equation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Exponential and Logarithmic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a512f5aexplog10a","stepAnswer":["$$\\\\frac{\\\\ln(\\\\frac{3}{5})-3}{8}$$"],"problemType":"TextBox","stepTitle":"$$10e^{8x+3}+2=8$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{\\\\ln(\\\\frac{3}{5})-3}{8}$$","hints":{"DefaultPathway":[{"id":"a512f5aexplog10a-h1","type":"hint","dependencies":[],"title":"Isolating the terms with exponents","text":"The first step is to move all the terms with exponents to one side of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog10a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a512f5aexplog10a-h1"],"title":"Determine if terms have a common base.","text":"Do the terms have a common base?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a512f5aexplog10a-h3","type":"hint","dependencies":["a512f5aexplog10a-h2"],"title":"Solving for $$x$$","text":"The next step is to take ln of both sides. While the choice of which base does not matter, it will be easier to use ln in this case.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog10a-h4","type":"hint","dependencies":["a512f5aexplog10a-h3"],"title":"Properties of logarithms","text":"Recall that $${\\\\ln(b)}^a$$ $$=$$ $$a \\\\ln(b)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog10a-h5","type":"hint","dependencies":["a512f5aexplog10a-h4"],"title":"Distributive Property","text":"Recall that $$a \\\\left(x+y\\\\right)$$ $$=$$ $$a x+a y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a512f5aexplog11","title":"Solving Exponential Equations using Logarithms","body":"How many solutions does the equation $$8e^{\\\\left(-5x-2\\\\right)}-4=-90$$ have?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Exponential and Logarithmic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a512f5aexplog11a","stepAnswer":["No Solutions"],"problemType":"MultipleChoice","stepTitle":"How many solutions does the equation $$8e^{\\\\left(-5x-2\\\\right)}-4=-90$$ have?","stepBody":"","answerType":"string","variabilization":{},"choices":["One Solution","Two Solutions","No Solutions"],"hints":{"DefaultPathway":[{"id":"a512f5aexplog11a-h1","type":"hint","dependencies":[],"title":"Isolating the term with the exponent","text":"The first step is to isolate all the terms with exponents to one side of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog11a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y>-4$$"],"dependencies":["a512f5aexplog11a-h1"],"title":"The range of the exponential function","text":"For the function $$y$$ $$=$$ $$e^{\\\\left(-5x-2\\\\right)}-4$$, what values of $$y$$ are possible?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y>0$$","$$y<0$$","$$y>-4$$","$$y<-4$$","All real numbers"]}]}}]},{"id":"a512f5aexplog12","title":"Solving Exponential Functions in Quadratic Form","body":"Solve the following equation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Exponential and Logarithmic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a512f5aexplog12a","stepAnswer":["ln(3)"],"problemType":"TextBox","stepTitle":"$$e^{2x}-e^x-6=0$$","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a512f5aexplog12a-h1","type":"hint","dependencies":[],"title":"Using substitution","text":"Substitute $$e^x$$ with another variable. What does the equation look like? Substitue that variable into the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog12a-h2","type":"hint","dependencies":["a512f5aexplog12a-h1"],"title":"Factoring","text":"The next step is to factor the equation by the FOIL method and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog12a-h3","type":"hint","dependencies":["a512f5aexplog12a-h2"],"title":"Solving for $$x$$","text":"The last step is to substitute $$e^x$$ back into the equation and solve for $$x$$. Eliminate any extraneous solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog12a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y>0$$"],"dependencies":["a512f5aexplog12a-h3"],"title":"Extraneous solutions","text":"What is the range of the function $$y$$ $$=$$ $$e^x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["All real numbers","$$y<0$$","$$y>0$$"]}]}}]},{"id":"a512f5aexplog13","title":"Using the definition of a Logarithm to solve Logarithmic Equations","body":"Solve the following equation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Exponential and Logarithmic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a512f5aexplog13a","stepAnswer":["$$49$$"],"problemType":"TextBox","stepTitle":"$$5*\\\\log_{7}\\\\left(n\\\\right)$$ $$=$$ $$10$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$49$$","hints":{"DefaultPathway":[{"id":"a512f5aexplog13a-h1","type":"hint","dependencies":[],"title":"Isolate the logarithmic term","text":"The first step is to move all the logarithmic terms to one side of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog13a-h2","type":"hint","dependencies":["a512f5aexplog13a-h1"],"title":"Solve for $$n$$","text":"The next step is to rewrite the equation in exponential form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog13a-h3","type":"hint","dependencies":["a512f5aexplog13a-h2"],"title":"Definition of Logarithms","text":"The definition of a logarithm states that for any algebraic expression S and real numbers $$b$$ and c, where $$b$$ > $$0$$ and $$b$$ does NOT equal $$1$$, $$\\\\log_{b}\\\\left(S\\\\right)=c$$ if and only if $$b^c=S$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a512f5aexplog13a-h3"],"title":"Applying the definition of logarithms to solve the equation","text":"If we were to rewrite the equation $$\\\\log_{7}\\\\left(n\\\\right)$$ $$=$$ $$2$$ in exponential form, what is the value of the exponent?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog13a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a512f5aexplog13a-h4"],"title":"Applying the definition of logarithms to solve the equation","text":"If we were to rewrite the equation $$\\\\log_{7}\\\\left(n\\\\right)$$ $$=$$ $$2$$ in exponential form, what is the value of the base?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a512f5aexplog14","title":"Solving Equations using the One-to-One Property of Logarithms","body":"Solve the following equation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Exponential and Logarithmic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a512f5aexplog14a","stepAnswer":["Both $$\\\\frac{-10}{3}$$ and $$\\\\frac{10}{3}$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\ln(x^2-10)+\\\\ln(9)=ln(10)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Both $$\\\\frac{-10}{3}$$ and $$\\\\frac{10}{3}$$","choices":["$$\\\\frac{-10}{3}$$","$$\\\\frac{10}{3}$$","Both $$\\\\frac{-10}{3}$$ and $$\\\\frac{10}{3}$$","None of the above"],"hints":{"DefaultPathway":[{"id":"a512f5aexplog14a-h1","type":"hint","dependencies":[],"title":"Using properties of logarithms to combine logarithmic terms","text":"If one side of the equation has multiple logarithmic terms, combine them using properties of logarithms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog14a-h2","type":"hint","dependencies":["a512f5aexplog14a-h1"],"title":"Properties of logs","text":"Recall that ln(a) + ln(b) $$=$$ $$\\\\ln(a b)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog14a-h3","type":"hint","dependencies":["a512f5aexplog14a-h2"],"title":"One-to-One property of logarithms","text":"The One-to-One property of logarithms states that for any algebraic expressions S and T and any positive real number $$b$$, where $$b$$ does NOT equal $$1$$, log(base b)S $$=$$ log(base b)T if and only if $$S=T$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog14a-h4","type":"hint","dependencies":["a512f5aexplog14a-h3"],"title":"Extraneous solutions","text":"Use the domain of the logarithmic function to determine whether a solution is extraneous.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog14a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x$$ < $$-\\\\sqrt{10}$$ or $$x$$ > $$\\\\sqrt{10}$$"],"dependencies":["a512f5aexplog14a-h4"],"title":"Finding extraneous solutions","text":"For what values of $$x$$ will $$\\\\ln(x^2-10)$$ be defined?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x$$ < $$-\\\\sqrt{10}$$ or $$x$$ > $$\\\\sqrt{10}$$","$$x$$ < $$-\\\\sqrt{10}$$ and $$x$$ > $$\\\\sqrt{10}$$","All real numbers"]}]}}]},{"id":"a512f5aexplog15","title":"Solving Logarithmic Equations","body":"Solve the following equation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Exponential and Logarithmic Equations","courseName":"OpenStax: College 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4.0>"},{"id":"a512f5aexplog15a-h3","type":"hint","dependencies":["a512f5aexplog15a-h2"],"title":"One-to-One property of logarithms","text":"The One-to-One property of logarithms states that for any algebraic expressions S and T and any positive real number $$b$$, where $$b$$ does NOT equal $$1$$, log(base b)S $$=$$ log(base b)T if and only if $$S=T$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog15a-h4","type":"hint","dependencies":["a512f5aexplog15a-h3"],"title":"Determining Extraneous Solutions","text":"Recall that for ln(a) to be defined, where a represents a number, a must be greater than $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a512f5aexplog16","title":"Solving Logarithmic Equations","body":"Solve the following equation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Exponential and Logarithmic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a512f5aexplog16a","stepAnswer":["$$\\\\frac{3}{4}$$"],"problemType":"TextBox","stepTitle":"$$ln(3)-ln(3-3x)=ln(4)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{4}$$","hints":{"DefaultPathway":[{"id":"a512f5aexplog16a-h1","type":"hint","dependencies":[],"title":"Using properties of logarithms to combine logarithmic terms","text":"If one side of the equation has multiple logarithmic terms, combine them using properties of logarithms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog16a-h2","type":"hint","dependencies":["a512f5aexplog16a-h1"],"title":"Properties of logs","text":"Recall that ln(a) - ln(b) $$=$$ $$\\\\ln(\\\\frac{a}{b})$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog16a-h3","type":"hint","dependencies":["a512f5aexplog16a-h2"],"title":"One-to-One property of logarithms","text":"The One-to-One property of logarithms states that for any algebraic expressions S and T and any positive real number $$b$$, where $$b$$ does NOT equal $$1$$, log(base b)S $$=$$ log(base b)T if and only if $$S=T$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog16a-h4","type":"hint","dependencies":["a512f5aexplog16a-h3"],"title":"Determining Extraneous Solutions","text":"Recall that for ln(a) to be defined, where a represents a number, a must be greater than $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a512f5aexplog17","title":"Solving an Exponential Equation with a Common Base #1","body":"Solve for $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Exponential and Logarithmic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a512f5aexplog17a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"Solve $$2^{x-1}=2^{2x-4}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a512f5aexplog17a-h1","type":"hint","dependencies":[],"title":"One-To-One Property of Exponential Functions","text":"For any algebraic expressions s and $$t$$, and any positive real number $$b$$ that is not equal to $$1$$, $$b^s=b^t$$ if and only if $$s=t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog17a-h2","type":"hint","dependencies":["a512f5aexplog17a-h1"],"title":"Appylying the One-To-Property to the Problem","text":"Both expressions have a base of $$2$$, so using the property, we see that $$x-1=2x-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a512f5aexplog18","title":"Solving an Exponential Equation with a Common Base #2","body":"Solve for $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Exponential and Logarithmic Equations","courseName":"OpenStax: College 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4.0>"}]}}]},{"id":"a512f5aexplog19","title":"Solving Equations by Rewriting Them to Have a Common Base","body":"Solve for $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Exponential and Logarithmic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a512f5aexplog19a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"Solve $$8^{x+2}={16}^{x+1}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a512f5aexplog19a-h1","type":"hint","dependencies":[],"title":"Rewriting the Base","text":"The first step is to rewrite the equation so that both sides have a common base. Since $$8$$ and $$16$$ are both powers of $$2$$, we can use $$2$$ as a common base.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog19a-h2","type":"hint","dependencies":["a512f5aexplog19a-h1"],"title":"Rewriting the Base","text":"Rewriting both sides into expressions with a base of $$2$$, we get $$2^{3\\\\left(x+2\\\\right)}=2^{4\\\\left(x+1\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog19a-h3","type":"hint","dependencies":["a512f5aexplog19a-h2"],"title":"One-To-One Property of Exponential Functions","text":"For any algebraic expressions s and $$t$$, and any positive real number $$b$$ that is not equal to $$1$$, $$b^s=b^t$$ if and only if $$s=t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog19a-h4","type":"hint","dependencies":["a512f5aexplog19a-h3"],"title":"Appylying the One-To-Property to the Problem","text":"Both expressions have a base of $$2$$, so using the property, we see that $$3\\\\left(x+2\\\\right)=4\\\\left(x+1\\\\right)$$. This simplifies to $$3x+6=4x+4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a512f5aexplog2","title":"Solve Exponential Equations","body":"Solve the following equation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Exponential and Logarithmic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a512f5aexplog2a","stepAnswer":["$$-1$$"],"problemType":"TextBox","stepTitle":"$$2^{\\\\left(-3x\\\\right)} \\\\frac{1}{4}$$ $$=$$ $$2^{x+2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1$$","hints":{"DefaultPathway":[{"id":"a512f5aexplog2a-h1","type":"hint","dependencies":[],"title":"Rewrite equation so all powers have the same base.","text":"The first step is to identify the common base of all the terms in the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog2a-h2","type":"hint","dependencies":["a512f5aexplog2a-h1"],"title":"Finding the common base.","text":"Recall that for base a and exponent $$b$$, $$a^{\\\\left(-b\\\\right)}$$ $$=$$ $$\\\\frac{1}{a^b}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog2a-h3","type":"hint","dependencies":["a512f5aexplog2a-h2"],"title":"Combining terms with the same base.","text":"The next step is to rewrite the terms so that they all have the same base and combine the terms on each side of the equation and then solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog2a-h4","type":"hint","dependencies":["a512f5aexplog2a-h3"],"title":"Properties of exponents","text":"When two terms with the same base are multiplied together, their exponents are added together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog2a-h5","type":"hint","dependencies":["a512f5aexplog2a-h4"],"title":"Taking a power of a power.","text":"To take a power of a power, multiply exponents.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog2a-h6","type":"hint","dependencies":["a512f5aexplog2a-h5"],"title":"One-to-one property of exponents.","text":"The one-to-one property of exponential functions states that for any algebraic expressions S and T, and any positive real number $$b$$ (except when $$b$$ $$=$$ 1), $$b^S$$ $$=$$ $$b^T$$ if and only if S $$=$$ T.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a512f5aexplog20","title":"Solving Equations by Rewriting Them to Have a Common Base","body":"Solve for $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Exponential and Logarithmic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a512f5aexplog20a","stepAnswer":["$$-1$$"],"problemType":"TextBox","stepTitle":"Solve $$5^{2x}={25}^{3x+2}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1$$","hints":{"DefaultPathway":[{"id":"a512f5aexplog20a-h1","type":"hint","dependencies":[],"title":"Rewriting the Base","text":"The first step is to rewrite the equation so that both sides have a common base. Since $$5$$ and $$25$$ are both powers of $$5$$, we can use $$5$$ as a common base.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog20a-h2","type":"hint","dependencies":["a512f5aexplog20a-h1"],"title":"Rewriting the Base","text":"The left side of the equation already has a base of $$5$$. We can rewrite the right side of the equation as $$5^{2\\\\left(3x+2\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog20a-h3","type":"hint","dependencies":["a512f5aexplog20a-h2"],"title":"One-To-One Property of Exponential Functions","text":"For any algebraic expressions s and $$t$$, and any positive real number $$b$$ that is not equal to $$1$$, $$b^s=b^t$$ if and only if $$s=t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog20a-h4","type":"hint","dependencies":["a512f5aexplog20a-h3"],"title":"Appylying the One-To-Property to the Problem","text":"Both expressions have a base of $$5$$, so using the property, we see that $$2x=2\\\\left(3x+2\\\\right)$$, This simplifies to $$2x=6x+4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a512f5aexplog21","title":"Solving an Exponential Equation with a Common Base #3","body":"Solve for $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Exponential and Logarithmic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a512f5aexplog21a","stepAnswer":["$$\\\\frac{1}{10}$$"],"problemType":"TextBox","stepTitle":"$$2^{5x}=\\\\sqrt{2}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{10}$$","hints":{"DefaultPathway":[{"id":"a512f5aexplog21a-h1","type":"hint","dependencies":[],"title":"Rewriting the Root into an Exponent","text":"First, rewrite $$\\\\sqrt{2}$$ into $$2^{\\\\frac{1}{2}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog21a-h2","type":"hint","dependencies":["a512f5aexplog21a-h1"],"title":"One-To-One Property of Exponential Functions","text":"For any algebraic expressions s and $$t$$, and any positive real number $$b$$ that is not equal to $$1$$, $$b^s=b^t$$ if and only if $$s=t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog21a-h3","type":"hint","dependencies":["a512f5aexplog21a-h2"],"title":"Appylying the One-To-Property to the Problem","text":"Both expressions have a base of $$2$$, so using the property, we see that $$5x=\\\\frac{1}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a512f5aexplog22","title":"Solving an Equation with Positive and Negative Powers #1","body":"For the following question, think about the range of an exponential function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Exponential and Logarithmic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a512f5aexplog22a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Does $$3^{x+1}=-2$$ have a solution?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a512f5aexplog22a-h1","type":"hint","dependencies":[],"title":"Range of an Exponential Function","text":"Recall that the range of an exponential function is always positive. We can graph the two functions out to see that they never intersect.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a512f5aexplog23","title":"Solving an Equation with Positive and Negative Powers #2","body":"For the following question, think about the range of an exponential function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Exponential and Logarithmic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a512f5aexplog23a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Does $$2^x=-100$$ have a solution?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a512f5aexplog23a-h1","type":"hint","dependencies":[],"title":"Range of an Exponential Function","text":"Recall that the range of an exponential function is always positive. We can graph the two functions out to see that they never intersect.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a512f5aexplog24","title":"Using Algebra to Solve a Logarithmic Equation #1","body":"Solve for $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Exponential and Logarithmic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a512f5aexplog24a","stepAnswer":["$$e^2$$"],"problemType":"TextBox","stepTitle":"$$2lnx+3=7$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$e^2$$","hints":{"DefaultPathway":[{"id":"a512f5aexplog24a-h1","type":"hint","dependencies":[],"title":"Definition of ln","text":"ln is a logarithm with base e.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog24a-h2","type":"hint","dependencies":["a512f5aexplog24a-h1"],"title":"Solving an equation $$lnx=a$$","text":"To solve an equation $$lnx=a$$, make both sides exponents of e. $$e^{lnx}=e^a$$, and from the properties of logarithms, since ln has base e, the left side of the $$equation=x$$. Therefore, $$x=e^a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a512f5aexplog25","title":"Using Algebra to Solve a Logarithmic Equation #2","body":"Solve for $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Exponential and Logarithmic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a512f5aexplog25a","stepAnswer":["$$e^2$$"],"problemType":"TextBox","stepTitle":"$$2lnx+3=7$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$e^2$$","hints":{"DefaultPathway":[{"id":"a512f5aexplog25a-h1","type":"hint","dependencies":[],"title":"Definition of ln","text":"ln is a logarithm with base e.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog25a-h2","type":"hint","dependencies":["a512f5aexplog25a-h1"],"title":"Solving an equation $$lnx=a$$","text":"To solve an equation $$lnx=a$$, make both sides exponents of e. $$e^{lnx}=e^a$$, and from the properties of logarithms, since ln has base e, the left side of the $$equation=x$$. Therefore, $$x=e^a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a512f5aexplog26","title":"Using Algebra to Solve a Logarithmic Equation #2","body":"Solve for $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Exponential and Logarithmic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a512f5aexplog26a","stepAnswer":["$$\\\\frac{1}{6\\\\left(e^{\\\\frac{7}{2}}\\\\right)}$$"],"problemType":"TextBox","stepTitle":"$$2ln(6x)=7$$","stepBody":"Using Like Bases to Solve Exponential Equations","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{6\\\\left(e^{\\\\frac{7}{2}}\\\\right)}$$","hints":{"DefaultPathway":[{"id":"a512f5aexplog26a-h1","type":"hint","dependencies":[],"title":"Definition of ln","text":"ln is a logarithm with base e.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog26a-h2","type":"hint","dependencies":["a512f5aexplog26a-h1"],"title":"Solving an equation $$lnx=a$$","text":"To solve an equation $$lnx=a$$, make both sides exponents of e. $$e^{lnx}=e^a$$, and from the properties of logarithms, since ln has base e, the left side of the $$equation=x$$. Therefore, $$x=e^a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a512f5aexplog3","title":"Solving Exponential Equations","body":"Solve the following equation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Exponential and Logarithmic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a512f5aexplog3a","stepAnswer":["$$\\\\frac{6}{5}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{{36}^{3x}}{{36}^{2x}}$$ $$=$$ $${216}^{2-x}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{6}{5}$$","hints":{"DefaultPathway":[{"id":"a512f5aexplog3a-h1","type":"hint","dependencies":[],"title":"Rewrite equation so all powers have the same base.","text":"The first step is to identify the common base of all the terms in the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$36$$"],"dependencies":["a512f5aexplog3a-h1"],"title":"Finding the common base.","text":"What is $$6\\\\times6$$? How does this relate to exponents?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$216$$"],"dependencies":["a512f5aexplog3a-h2"],"title":"Finding the common base.","text":"What is $$6\\\\times6\\\\times6$$? How does this relate to exponents?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog3a-h4","type":"hint","dependencies":["a512f5aexplog3a-h3"],"title":"Combining terms with the same base.","text":"The next step is to rewrite the terms so that they all have the same base and combine the terms on each side of the equation and then solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog3a-h5","type":"hint","dependencies":["a512f5aexplog3a-h4"],"title":"Properties of exponents","text":"When two terms with the same base are being divided, their exponents are subtracted.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog3a-h6","type":"hint","dependencies":["a512f5aexplog3a-h5"],"title":"Taking a power of a power.","text":"To take a power of a power, multiply exponents.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog3a-h7","type":"hint","dependencies":["a512f5aexplog3a-h6"],"title":"One-to-one property of exponents.","text":"The one-to-one property of exponential functions states that for any algebraic expressions S and T, and any positive real number $$b$$ (except when $$b$$ $$=$$ 1), $$b^S$$ $$=$$ $$b^T$$ if and only if S $$=$$ T.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a512f5aexplog4","title":"Solving Exponential Equations using Logarithms.","body":"Solve the following equation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Exponential and Logarithmic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a512f5aexplog4a","stepAnswer":["$$10$$"],"problemType":"TextBox","stepTitle":"$$9^{x-10}$$ $$=$$ $$1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10$$","hints":{"DefaultPathway":[{"id":"a512f5aexplog4a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":[],"title":"Determine if terms have a common base.","text":"Do the terms have a common base?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a512f5aexplog4a-h2","type":"hint","dependencies":["a512f5aexplog4a-h1"],"title":"Using logarithms to solve for $$x$$.","text":"The next step is to take the log of both sides. The choice of the base for the logs does not matter. For this problem, the easiest choice would be log base $$9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog4a-h3","type":"hint","dependencies":["a512f5aexplog4a-h2"],"title":"Using properties of logs to bring the exponent down.","text":"Recall that $${\\\\ln(a)}^b$$ $$=$$ $$b \\\\ln(a)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog4a-h4","type":"hint","dependencies":["a512f5aexplog4a-h3"],"title":"Isolate $$x$$.","text":"The next step is to isolate $$x$$ on one side of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog4a-h5","type":"hint","dependencies":["a512f5aexplog4a-h4"],"title":"Duality between logarithms and exponents.","text":"Recall that $$\\\\log_{a}\\\\left(b\\\\right)=x$$ can be rewritten as $$a^x=b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog4a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a512f5aexplog4a-h5"],"title":"Solving logarithms.","text":"What is $$\\\\log_{9}\\\\left(9\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog4a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a512f5aexplog4a-h6"],"title":"Solving logarithms.","text":"What is $$\\\\log_{9}\\\\left(1\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog4a-h8","type":"hint","dependencies":["a512f5aexplog4a-h7"],"title":"Solving for $$x$$.","text":"The last step is to solve for $$x$$. The equation should be a simple algebraic equation!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a512f5aexplog5","title":"Solving Exponential Equations using Logarithms","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Exponential and Logarithmic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a512f5aexplog5a","stepAnswer":["$$0$$ (No solution)"],"problemType":"MultipleChoice","stepTitle":"How many solutions does the equation $$e^{r+10}$$ - $$10$$ $$=$$ $$-42$$ have?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$0$$ (No solution)","choices":["infinite","$$2$$","$$1$$","$$0$$ (No solution)"],"hints":{"DefaultPathway":[{"id":"a512f5aexplog5a-h1","type":"hint","dependencies":[],"title":"Isolating the term with the exponent","text":"The first step is to isolate all the terms with exponents to one side of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog5a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y$$ > $$0$$"],"dependencies":["a512f5aexplog5a-h1"],"title":"The range of the exponential function","text":"For the function $$y$$ $$=$$ $$e^{x+10}$$, what values of $$y$$ are possible?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["All real numbers","$$y$$ < $$0$$","$$y$$ > $$0$$"]}]}}]},{"id":"a512f5aexplog6","title":"Solving Exponential Equations using Logarithms","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Exponential and Logarithmic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a512f5aexplog6a","stepAnswer":["$$\\\\ln(\\\\frac{17}{8})$$ - $$7$$"],"problemType":"TextBox","stepTitle":"$$-8{10}^{x+7}-7$$ $$=-24$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\ln(\\\\frac{17}{8})$$ - $$7$$","hints":{"DefaultPathway":[{"id":"a512f5aexplog6a-h1","type":"hint","dependencies":[],"title":"Isolating the terms with exponents","text":"The first step is to move all the terms with exponents to one side of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog6a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a512f5aexplog6a-h1"],"title":"Determine if terms have a common base.","text":"Do the terms have a common base?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a512f5aexplog6a-h3","type":"hint","dependencies":["a512f5aexplog6a-h2"],"title":"Solving for $$x$$","text":"The next step is to take the log of both sides. While the choice of which base does not matter, it will be easier to use log(base 10) in this case.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog6a-h4","type":"hint","dependencies":["a512f5aexplog6a-h3"],"title":"Properties of logarithms","text":"Recall that $${\\\\ln(b)}^a$$ $$=$$ $$a \\\\ln(b)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog6a-h5","type":"hint","dependencies":["a512f5aexplog6a-h4"],"title":"Distributive Property","text":"Recall that $$a \\\\left(x+y\\\\right)$$ $$=$$ $$a x$$ + $$a y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a512f5aexplog7","title":"Solving Exponential Equations using Logarithms","body":"Solve the following equation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Exponential and Logarithmic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a512f5aexplog7a","stepAnswer":["$$\\\\frac{-\\\\left(\\\\ln(\\\\frac{38}{3})-8\\\\right)}{3}$$"],"problemType":"TextBox","stepTitle":"$$e^{\\\\left(-3k\\\\right)}+6=$$ $$44$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-\\\\left(\\\\ln(\\\\frac{38}{3})-8\\\\right)}{3}$$","hints":{"DefaultPathway":[{"id":"a512f5aexplog7a-h1","type":"hint","dependencies":[],"title":"Isolating the terms with exponents","text":"The first step is to move all the terms with exponents to one side of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog7a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a512f5aexplog7a-h1"],"title":"Determine if terms have a common base.","text":"Do the terms have a common base?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a512f5aexplog7a-h3","type":"hint","dependencies":["a512f5aexplog7a-h2"],"title":"Solving for $$x$$","text":"The next step is to take ln of both sides. While the choice of which base does not matter, it will be easier to use ln in this case.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog7a-h4","type":"hint","dependencies":["a512f5aexplog7a-h3"],"title":"Properties of logarithms","text":"Recall that $${\\\\ln(b)}^a$$ $$=$$ $$a \\\\ln(b)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a512f5aexplog8","title":"Solving Exponential Equations with Logarithms","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Exponential and Logarithmic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a512f5aexplog8a","stepAnswer":["(ln(38/3) - 8)/9"],"problemType":"TextBox","stepTitle":"$$-6e^{9x-8}$$ + $$2$$ $$=$$ $$-74$$","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a512f5aexplog8a-h1","type":"hint","dependencies":[],"title":"Isolating the terms with exponents","text":"The first step is to move all the terms with exponents to one side of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog8a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a512f5aexplog8a-h1"],"title":"Determine if terms have a common base.","text":"Do the terms have a common base?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a512f5aexplog8a-h3","type":"hint","dependencies":["a512f5aexplog8a-h2"],"title":"Solving for $$x$$","text":"The next step is to take ln of both sides. While the choice of which base does not matter, it will be easier to use ln in this case.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog8a-h4","type":"hint","dependencies":["a512f5aexplog8a-h3"],"title":"Properties of logarithms","text":"Recall that $${\\\\ln(b)}^a$$ $$=$$ $$a \\\\ln(b)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog8a-h5","type":"hint","dependencies":["a512f5aexplog8a-h4"],"title":"Distributive Property","text":"Recall that $$a \\\\left(x+y\\\\right)$$ $$=$$ $$a x$$ + $$a y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a512f5aexplog9","title":"Solving Exponential Functions in Quadratic Form","body":"Solve the following equation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Exponential and Logarithmic Equations","courseName":"OpenStax: College Algebra","steps":[{"id":"a512f5aexplog9a","stepAnswer":["ln(12)"],"problemType":"TextBox","stepTitle":"$$e^{2x}-e^x-132=0$$","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a512f5aexplog9a-h1","type":"hint","dependencies":[],"title":"Using substitution","text":"Substitute $$e^x$$ with another variable. What does the equation look like? Substitue that variable into the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog9a-h2","type":"hint","dependencies":["a512f5aexplog9a-h1"],"title":"Factoring","text":"The next step is to factor the equation by the FOIL method and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog9a-h3","type":"hint","dependencies":["a512f5aexplog9a-h2"],"title":"Solving for $$x$$","text":"The last step is to substitute $$e^x$$ back into the equation and solve for $$x$$. Eliminate any extraneous solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a512f5aexplog9a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y$$ > $$0$$"],"dependencies":["a512f5aexplog9a-h3"],"title":"Extraneous solutions","text":"What is the range of the function $$y$$ $$=$$ $$e^x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y$$ > $$0$$","$$y$$ < $$0$$","All real numbers"]}]}}]},{"id":"a524aa3PreciseLim1","title":"Write the appropriate \u03b5-\u03b4 defintion for the given statement, $$\\\\lim_{x\\\\toa} f(x)=N$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.5 The Precise Definition of a Limit","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a524aa3PreciseLim1a","stepAnswer":["$$\\\\epsilon>0$$"],"problemType":"MultipleChoice","stepTitle":"For every $$___$$.","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"answerLatex":"$$\\\\epsilon>0$$","choices":["$$\\\\epsilon<0$$","$$\\\\varepsilon=0$$","$$\\\\epsilon>0$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim1a-h1","type":"hint","dependencies":[],"title":"Universal quantifier","text":"Remember that epsilon in a \u03b5-\u03b4 defintion is used to quantify the distance between the function and what the limit is equal to.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim1a-h2","type":"hint","dependencies":["a524aa3PreciseLim1a-h1"],"title":"Distance","text":"The distance between the function and what it is equal to, N,in this case, cannot be negative, thus what must epsilon be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim1b","stepAnswer":["\u03b4 >0"],"problemType":"MultipleChoice","stepTitle":"There exists a $$___$$.","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"choices":["\u03b4 $$=0$$","\u03b4 >0","\u03b4 >0","$$\\\\delta<0$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim1b-h1","type":"hint","dependencies":[],"title":"Existential quantifier","text":"Remember tha delta in a \u03b5-\u03b4 defintion must be greater than the distance between the variable(x in this case) and what the variable is going to (a in this case).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim1b-h2","type":"hint","dependencies":["a524aa3PreciseLim1b-h1"],"title":"Distance","text":"Since delta must be greater than the distance between $$x$$ and a, then delta must be positive","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim1c","stepAnswer":["$$x-a$$"],"problemType":"MultipleChoice","stepTitle":"so that if 0<abs(___)|<\u03b4 .","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"answerLatex":"$$x-a$$","choices":["$$x-a$$","$$N-a$$","$$a-N$$","$$a-x$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim1c-h1","type":"hint","dependencies":[],"title":"Distance between $$x$$ and a","text":"We need to represent the distance between $$x$$ and a, such that $$x$$ is greater than and does not equal a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim1d","stepAnswer":["$$f(x)-N$$"],"problemType":"MultipleChoice","stepTitle":"Then abs(___)<\u03b5.","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"answerLatex":"$$f(x)-N$$","choices":["$$f(x)-N$$","f(x)","$$x-a$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim1d-h1","type":"hint","dependencies":[],"title":"Distance for the function","text":"For the limit, the function must be closer to epislon than N. This step is asking how we can represent this mathematically using f(x) and N.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a524aa3PreciseLim10","title":"The graph of the function satisfies $$\\\\lim_{x\\\\to3} f(x)=2$$. determine a value of $$\\\\delta>0$$ that satisfies the value of epsilon such that the precise definition of limit holds true.","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.5 The Precise Definition of a Limit","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a524aa3PreciseLim10a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"$$epsilon=3$$","stepBody":"(Note: round answer to nearest $$0.25)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim10a-h1","type":"hint","dependencies":[],"title":"Translating the limit","text":"Based on the precise definition of a limit, then if we use the information from the limit we get If $$0<|x-3|<\\\\delta$$, then $$|f{\\\\left(x\\\\right)}-2|<\\\\epsilon$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim10a-h2","type":"hint","dependencies":["a524aa3PreciseLim10a-h1"],"title":"Using the precise definition of a limit","text":"The precise definition of a limit tells us that if $$x$$ is closer than delta to $$3$$ and $$x$$ is not equal to $$3$$, then f(x) is closer to than epsilon to $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim10a-h3","type":"hint","dependencies":["a524aa3PreciseLim10a-h2"],"title":"Visualizing","text":"In essence, we are asked to find how close to $$3$$ must $$x$$ be, if $$y$$ is within epsilon unit of $$2$$ on the graph. We can find this by drawing boundary lines on the graph that fulfill these conditions and making delta the distance from $$x=3$$ to the closer of the $$2$$ $$x-boundary$$ lines.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a524aa3PreciseLim11","title":"Use a graphing calculator to find a number, delta, such that the statement hold true.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.5 The Precise Definition of a Limit","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a524aa3PreciseLim11a","stepAnswer":["$$0.39$$"],"problemType":"TextBox","stepTitle":"$$|\\\\sqrt{x-4}-2|<0.1$$, whenever $$|x-8|<\\\\delta$$. What is delta equal to?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.39$$","hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim11a-h1","type":"hint","dependencies":[],"title":"Using the precise definition of a limit","text":"First, we need to compare the given statement to the conditional statement portion of the precise defintion of a limit form (e.g. if $$|x-a|<\\\\delta$$, then $$|f{\\\\left(x\\\\right)}-L|<\\\\epsilon$$ for the limit $$\\\\lim_{x\\\\toa} f(x)=L)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim11a-h2","type":"hint","dependencies":["a524aa3PreciseLim11a-h1"],"title":"Using information from the statement","text":"After comparing the statement to the conditional statement portion of the precise defintion of a limit, we gain the following key information: f(x) $$=$$ $$\\\\sqrt{x-4}$$, epsilon $$=$$ $$0.1$$, and our limit is $$\\\\lim_{x\\\\to8} f(x)=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim11a-h3","type":"hint","dependencies":["a524aa3PreciseLim11a-h2"],"title":"Graphing the function","text":"Graph the function $$f(x)=\\\\sqrt{x-4}$$ on your graphing calculator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim11a-h4","type":"hint","dependencies":["a524aa3PreciseLim11a-h3"],"title":"Solving for delta","text":"We need to find how close to $$8$$ must $$x$$ be, if $$y$$ is within $$epsilon=0.1$$ units of $$2$$ on the graph. We can find this by drawing boundary lines on the graph that fulfill these conditions and making delta the distance from $$x=8$$ to the closer of the $$2$$ $$x-boundary$$ lines.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a524aa3PreciseLim12","title":"Use a graphing calculator to find a number, delta, such that the statement hold true.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.5 The Precise Definition of a Limit","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a524aa3PreciseLim12a","stepAnswer":["$$0.056$$"],"problemType":"TextBox","stepTitle":"$$|sin\\\\left(2x\\\\right)-\\\\frac{1}{2}|<0.1$$, whenever $$|x-\\\\frac{\\\\pi}{12}|<\\\\delta$$. What is delta equal to?","stepBody":"Input your answer as a decimal and round your answer to the thousandth place.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.056$$","hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim12a-h1","type":"hint","dependencies":[],"title":"Using the precise definition of a limit","text":"First, we need to compare the given statement to the conditional statement portion of the precise defintion of a limit form (e.g. if $$|x-a|<\\\\delta$$, then $$|f{\\\\left(x\\\\right)}-L|<\\\\epsilon$$ for the limit $$\\\\lim_{x\\\\toa} f(x)=L)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim12a-h2","type":"hint","dependencies":["a524aa3PreciseLim12a-h1"],"title":"Using information from the statement","text":"After comparing the statement to the conditional statement portion of the precise defintion of a limit, we gain the following key information: f(x) $$=$$ $$sin\\\\left(2x\\\\right)$$, epsilon $$=$$ $$0.1$$, and our limit is $$/lim{x,(pi/12)$$, $$f(x)}=\\\\frac{1}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim12a-h3","type":"hint","dependencies":["a524aa3PreciseLim12a-h2"],"title":"Graphing the function","text":"Graph the function $$f(x)=\\\\sqrt{x-4}$$ on your graphing calculator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim12a-h4","type":"hint","dependencies":["a524aa3PreciseLim12a-h3"],"title":"Solving for delta","text":"We need to find how close to $$\\\\frac{\\\\pi}{12}$$ must $$x$$ be, if $$y$$ is within epsilon $$=$$ $$0.1$$ units of $$\\\\frac{1}{2}$$ on the graph. We can find this by drawing boundary lines on the graph that fulfill these conditions and making delta the distance from $$x=\\\\frac{\\\\pi}{12}$$ to the closer of the $$2$$ $$x-boundary$$ lines.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a524aa3PreciseLim13","title":"Use the precise defintion of limits to prove the given limit, $$\\\\lim_{x\\\\to3} \\\\frac{x^2-9}{x-3}=6$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.5 The Precise Definition of a Limit","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a524aa3PreciseLim13a","stepAnswer":["j"],"problemType":"TextBox","stepTitle":"What must we let delta equal in terms of epsilon?","stepBody":"To input your answer, let j represent epsilon (e.g. $$5j=5\\\\varepsilon)$$","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim13a-h1","type":"hint","dependencies":[],"title":"Determining epsilon","text":"To solve, we need to look at the limit of our function and see what operation must be done to get $$x$$ to have a power and coefficient of $$1$$ only and apply the same operations to epsilon.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim13b","stepAnswer":["$$x-3$$"],"problemType":"MultipleChoice","stepTitle":"If 0<abs(___)<epsilon","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"answerLatex":"$$x-3$$","choices":["$$x-3$$","$$x+6$$","$$x+3$$","$$x-6$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim13b-h1","type":"hint","dependencies":[],"title":"Precise defintion of limits","text":"Here, we need to represent the distance between $$x$$ and what $$x$$ approaches (in this case 3), such that $$x$$ is greater than and does not equal what it approaches.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim13c","stepAnswer":["$$x+3-6$$"],"problemType":"MultipleChoice","stepTitle":"Then abs(___)=abs(___)<\u03b5.","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"answerLatex":"$$x+3-6$$","choices":["$$x+3-6$$","$$x+6-3$$","$$x+3-6$$","$$x-3+6$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim13c-h1","type":"hint","dependencies":[],"title":"Using the precise definition of a limit","text":"We are trying to prove that for the limit, the function must be closer to epsilon than L for the limit $$\\\\lim_{x\\\\toa} f(x)=L)$$. The general form we are proving is $$|f{\\\\left(x\\\\right)}-L|<\\\\epsilon$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim13c-h2","type":"hint","dependencies":["a524aa3PreciseLim13c-h1"],"title":"Determining the function","text":"First we need to determine the function, we can do this by simplifying the function inside the limit into $$x+3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim13c-h3","type":"hint","dependencies":["a524aa3PreciseLim13c-h2"],"title":"Determining L","text":"From the aforementioned general form, the L we are looking for is on the opposite side of the equal sign of the limit.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim13c-h4","type":"hint","dependencies":["a524aa3PreciseLim13c-h3"],"title":"Defining epsilon","text":"Since we set delta $$=$$ epsilon earlier, then our epsilon must be greater than $$|x-3|$$ as we found in the previous step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a524aa3PreciseLim14","title":"Use the precise defintion of limits to prove the given limit, $$\\\\lim_{x\\\\to2} 5x+8=18$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.5 The Precise Definition of a Limit","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a524aa3PreciseLim14a","stepAnswer":["$$\\\\frac{j}{5}$$"],"problemType":"TextBox","stepTitle":"What must we let delta equal in terms of epsilon?","stepBody":"To input your answer, let j represent epsilon (e.g. 5j $$=$$ 5\u03b5)","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{j}{5}$$","hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim14a-h1","type":"hint","dependencies":[],"title":"Determining epsilon","text":"To solve, we need to look at the limit of our function and see what operation must be done to get $$x$$ to have a coefficient and power of $$1$$ only and apply the same operations to epsilon.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim14b","stepAnswer":["$$x-2$$"],"problemType":"MultipleChoice","stepTitle":"If 0<abs(___)<epsilon/5","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"answerLatex":"$$x-2$$","choices":["$$x-2$$","$$x+2$$","$$x-5$$","$$x+18$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim14b-h1","type":"hint","dependencies":[],"title":"Precise defintion of limits","text":"Here, we need to represent the distance between $$x$$ and what $$x$$ approaches (in this case 2), such that $$x$$ is greater than and does not equal what it approaches.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim14c","stepAnswer":["$$5x+8-18$$"],"problemType":"MultipleChoice","stepTitle":"Then abs(___)=abs(___)<\u03b5.","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"answerLatex":"$$5x+8-18$$","choices":["$$x+8-18$$","$$5x-8-18$$","$$5x+8-18$$","$$5x+8-18$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim14c-h1","type":"hint","dependencies":[],"title":"Using the precise definition of a limit","text":"We are trying to prove that for the limit, the function must be closer to epsilon than L for the limit $$\\\\lim_{x\\\\toa} f(x)=L)$$. The general form we are proving is $$|f{\\\\left(x\\\\right)}-L|<\\\\epsilon$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim14c-h2","type":"hint","dependencies":["a524aa3PreciseLim14c-h1"],"title":"Determining the function","text":"First we need to determine the function, we can do this by looking at the function inside the limit, which is $$5x+18$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim14c-h3","type":"hint","dependencies":["a524aa3PreciseLim14c-h2"],"title":"Determining L","text":"From the aforementioned general form, the L we are looking for is on the opposite side of the equal sign of the limit.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim14c-h4","type":"hint","dependencies":["a524aa3PreciseLim14c-h3"],"title":"Defining epsilon","text":"Since we set delta $$=$$ $$\\\\frac{\\\\epsilon}{5}$$ earlier, then our epsilon must be greater than $$5|x-2|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a524aa3PreciseLim15","title":"Use the precise defintion of limits to prove the given limit, $$\\\\lim_{x\\\\to3} \\\\frac{2x^2-3x-2}{x-2}=5$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.5 The Precise Definition of a Limit","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a524aa3PreciseLim15a","stepAnswer":["$$\\\\frac{j}{2}$$"],"problemType":"TextBox","stepTitle":"What must we let delta equal in terms of epsilon?","stepBody":"To input your answer, let j represent epsilon (e.g. 5j $$=$$ 5\u03b5)","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{j}{2}$$","hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim15a-h1","type":"hint","dependencies":[],"title":"Determining epsilon","text":"To solve, we need to look at the limit of our function and see what operation must be done to get $$x$$ to have a power and coefficient of $$1$$ only and apply the same operations to epsilon.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim15b","stepAnswer":["$$x-2$$"],"problemType":"MultipleChoice","stepTitle":"If 0<abs(___)<epsilon/2","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"answerLatex":"$$x-2$$","choices":["$$x-2$$","$$x+2$$","$$x-5$$","$$x+3$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim15b-h1","type":"hint","dependencies":[],"title":"Precise defintion of limits","text":"Here, we need to represent the distance between $$x$$ and what $$x$$ approaches (in this case 2), such that $$x$$ is greater than and does not equal what it approaches.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim15c","stepAnswer":["$$2x+1-5$$"],"problemType":"MultipleChoice","stepTitle":"Then abs(___)=abs(___)<\u03b5.","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"answerLatex":"$$2x+1-5$$","choices":["$$2x+1-5$$","$$x+1-5$$","$$2x+1-5$$","$$2x-5$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim15c-h1","type":"hint","dependencies":[],"title":"Using the precise definition of a limit","text":"We are trying to prove that for the limit, the function must be closer to epsilon than L for the limit $$\\\\lim_{x\\\\toa} f(x)=L)$$. The general form we are proving is $$|f{\\\\left(x\\\\right)}-L|<\\\\epsilon$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim15c-h2","type":"hint","dependencies":["a524aa3PreciseLim15c-h1"],"title":"Determining the function","text":"First we need to determine the function, we can do this by simplifying the function inside the limit into $$2x+1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim15c-h3","type":"hint","dependencies":["a524aa3PreciseLim15c-h2"],"title":"Determining L","text":"From the aforementioned general form, the L we are looking for is on the opposite side of the equal sign of the limit.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim15c-h4","type":"hint","dependencies":["a524aa3PreciseLim15c-h3"],"title":"Defining epsilon","text":"Since we set $$delta=\\\\frac{\\\\epsilon}{2}$$ earlier, then our epsilon must be greater than $$|2x-4|$$ *2","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a524aa3PreciseLim16","title":"Use the precise defintion of limits to prove the given limit, $$\\\\lim_{x\\\\to0} x^4=0$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.5 The Precise Definition of a Limit","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a524aa3PreciseLim16a","stepAnswer":["$$j^{\\\\frac{1}{4}}$$"],"problemType":"TextBox","stepTitle":"What must we let delta equal in terms of epsilon?","stepBody":"To input your answer, let j represent epsilon (e.g. 5j $$=$$ 5\u03b5)","answerType":"arithmetic","variabilization":{},"answerLatex":"$$j^{\\\\frac{1}{4}}$$","hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim16a-h1","type":"hint","dependencies":[],"title":"Determining epsilon","text":"To solve, we need to look at the limit of our function and see what operation must be done to get $$x$$ to have a power and coefficient of $$1$$ only and apply the same operations to epsilon.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim16b","stepAnswer":["$$x$$"],"problemType":"MultipleChoice","stepTitle":"If 0<abs(___)<epsilon**(1/4)","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"answerLatex":"$$x$$","choices":["$$x$$","$$x+2$$","$$x-\\\\frac{1}{4}$$","$$x+3$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim16b-h1","type":"hint","dependencies":[],"title":"Precise defintion of limits","text":"Here, we need to represent the distance between $$x$$ and what $$x$$ approaches (in this case 0), such that $$x$$ is greater than and does not equal what it approaches.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim16c","stepAnswer":["$$x^4$$"],"problemType":"MultipleChoice","stepTitle":"Then abs(___)=abs(___)<\u03b5.","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"answerLatex":"$$x^4$$","choices":["$$x^4$$","$$x^4$$","$$x^{\\\\frac{1}{4}}$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim16c-h1","type":"hint","dependencies":[],"title":"Using the precise definition of a limit","text":"We are trying to prove that for the limit, the function must be closer to epsilon than L for the limit $$\\\\lim_{x\\\\toa} f(x)=L)$$. The general form we are proving is $$|f{\\\\left(x\\\\right)}-L|<\\\\epsilon$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim16c-h2","type":"hint","dependencies":["a524aa3PreciseLim16c-h1"],"title":"Determining the function","text":"First we need to determine the function, we can do this by looking for the function inside the limit, $$x^4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim16c-h3","type":"hint","dependencies":["a524aa3PreciseLim16c-h2"],"title":"Determining L","text":"From the aforementioned general form, the L we are looking for is on the opposite side of the equal sign of the limit.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim16c-h4","type":"hint","dependencies":["a524aa3PreciseLim16c-h3"],"title":"Defining epsilon","text":"Since we set delta $$=$$ $${\\\\epsilon}^{\\\\frac{1}{4}}$$ earlier, then our epsilon must be greater than $${\\\\left(x-0\\\\right)}^4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a524aa3PreciseLim17","title":"Use the precise defintion of limits to prove the given limit, /lim{x,2,(x**2)+2*x)}=8","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.5 The Precise Definition of a Limit","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a524aa3PreciseLim17a","stepAnswer":["$$j^{\\\\frac{1}{2}}$$"],"problemType":"TextBox","stepTitle":"What must we let delta equal in terms of epsilon?","stepBody":"To input your answer, let j represent epsilon (e.g. 5j $$=$$ 5\u03b5)","answerType":"arithmetic","variabilization":{},"answerLatex":"$$j^{\\\\frac{1}{2}}$$","hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim17a-h1","type":"hint","dependencies":[],"title":"Determining epsilon","text":"To solve, we need to look at the limit of our function and see what operation must be done to get $$x$$ to have a power and coefficient of $$1$$ only and apply the same operations to epsilon.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim17b","stepAnswer":["$$x-2$$"],"problemType":"MultipleChoice","stepTitle":"If 0<abs(___)<epsilon**(1/2)","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"answerLatex":"$$x-2$$","choices":["$$x-2$$","$$x+2$$","$$x-8$$","$$x+3$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim17b-h1","type":"hint","dependencies":[],"title":"Precise defintion of limits","text":"Here, we need to represent the distance between $$x$$ and what $$x$$ approaches (in this case 2), such that $$x$$ is greater than and does not equal what it approaches.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim17c","stepAnswer":["$$x^2+2x-8$$"],"problemType":"MultipleChoice","stepTitle":"Then abs(___)=abs(___)<\u03b5.","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"answerLatex":"$$x^2+2x-8$$","choices":["$$x^2+2x-8$$","$$x^2+x-8$$","$$x^2+2x-8$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim17c-h1","type":"hint","dependencies":[],"title":"Using the precise definition of a limit","text":"We are trying to prove that for the limit, the function must be closer to epsilon than L for the limit $$\\\\lim_{x\\\\toa} f(x)=L)$$. The general form we are proving is $$|f{\\\\left(x\\\\right)}-L|<\\\\epsilon$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim17c-h2","type":"hint","dependencies":["a524aa3PreciseLim17c-h1"],"title":"Determining the function","text":"First we need to determine the function, we can do this by looking for the function inside the limit, which is $$x^2+2x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim17c-h3","type":"hint","dependencies":["a524aa3PreciseLim17c-h2"],"title":"Determining L","text":"From the aforementioned general form, the L we are looking for is on the opposite side of the equal sign of the limit.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim17c-h4","type":"hint","dependencies":["a524aa3PreciseLim17c-h3"],"title":"Defining epsilon","text":"Since we set delta $$=$$ $$\\\\frac{{\\\\epsilon}^1}{2}$$ earlier, then our epsilon must be greater than $$2\\\\left(x-2\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a524aa3PreciseLim18","title":"Use the precise defintion of limits to prove the given one-sided limit, $$\\\\lim_{x\\\\to5} \\\\sqrt{5-x}=0$$ where $$x$$ is approaching from the left.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.5 The Precise Definition of a Limit","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a524aa3PreciseLim18a","stepAnswer":["$$j^2$$"],"problemType":"TextBox","stepTitle":"What must we let delta equal in terms of epsilon?","stepBody":"To input your answer, let j represent epsilon (e.g. 5j $$=$$ 5\u03b5)","answerType":"arithmetic","variabilization":{},"answerLatex":"$$j^2$$","hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim18a-h1","type":"hint","dependencies":[],"title":"Determining epsilon","text":"To solve, we need to look at the limit of our function and see what operation must be done to get $$x$$ to have a power and coefficient of $$1$$ only and apply the same operations to epsilon.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim18b","stepAnswer":["$$5-{\\\\epsilon}^2$$"],"problemType":"MultipleChoice","stepTitle":"If (___)<x<5","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"answerLatex":"$$5-{\\\\epsilon}^2$$","choices":["$$5-{\\\\epsilon}^2$$","$${\\\\epsilon}^2$$","$$5+{\\\\epsilon}^2$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim18b-h1","type":"hint","dependencies":[],"title":"Precise defintion of limits","text":"The general form is $$-\\\\delta<x-a<0$$, which we can manipulate into $$a-\\\\delta<x<a$$ to fit our statement above, where a is what $$x$$ approaches.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim18c","stepAnswer":["$$\\\\sqrt{5-x}$$"],"problemType":"MultipleChoice","stepTitle":"Then abs(___)=(___)<\u03b5.","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"answerLatex":"$$\\\\sqrt{5-x}$$","choices":["$$\\\\sqrt{5-x}$$","$$\\\\sqrt{5-x}$$","$$\\\\sqrt{5-x}$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim18c-h1","type":"hint","dependencies":[],"title":"Using the precise definition of a limit","text":"We are trying to prove that for the limit, the function must be closer to epsilon than L for the limit $$\\\\lim_{x\\\\toa} f(x)=L)$$. The general form we are proving is $$|f{\\\\left(x\\\\right)}-L|<\\\\epsilon$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim18c-h2","type":"hint","dependencies":["a524aa3PreciseLim18c-h1"],"title":"Determining the function","text":"First we need to determine the function, we can do this by looking for the function inside the limit, which is $$\\\\sqrt{5-x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim18c-h3","type":"hint","dependencies":["a524aa3PreciseLim18c-h2"],"title":"Determining L","text":"From the aforementioned general form, the L we are looking for is on the opposite side of the equal sign of the limit.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim18c-h4","type":"hint","dependencies":["a524aa3PreciseLim18c-h3"],"title":"Defining epsilon","text":"Since we set delta $$=$$ $${\\\\epsilon}^2$$ earlier, then we must solve our statement from the previous step such that epsilon has a power and coefficient of positive $$1$$ only. This will make it such that epsilon is greater than $$\\\\sqrt{5-x}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a524aa3PreciseLim19","title":"Use the precise defintion of limits to prove the given one-sided limit, $$\\\\lim_{x\\\\to0} f(x)=-2$$ where $$x$$ is approaching from the right and if $$x<0$$ then $$f(x)=8x-3$$, but if $$x$$ is greater than or equal to $$0$$, then $$f(x)=4x-2$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.5 The Precise Definition of a Limit","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a524aa3PreciseLim19a","stepAnswer":["$$\\\\frac{j}{4}$$"],"problemType":"TextBox","stepTitle":"What must we let delta equal in terms of epsilon?","stepBody":"To input your answer, let j represent epsilon (e.g. 5j $$=$$ 5\u03b5)","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{j}{4}$$","hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim19a-h1","type":"hint","dependencies":[],"title":"Determining epsilon","text":"To solve, we need to look at the limit of our function and see what operation must be done to get $$x$$ to have a power and coefficient of $$1$$ only and apply the same operations to epsilon.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim19b","stepAnswer":["$$\\\\frac{\\\\epsilon}{4}$$"],"problemType":"MultipleChoice","stepTitle":"If 0<x<(___)","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{\\\\epsilon}{4}$$","choices":["$$-\\\\left(\\\\frac{\\\\epsilon}{8}\\\\right)$$","$$\\\\frac{\\\\epsilon}{8}$$","$$-(\\\\varepsilon)$$","$$\\\\frac{\\\\epsilon}{4}$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim19b-h1","type":"hint","dependencies":[],"title":"Precise defintion of limits","text":"The general form is $$0<x-a<\\\\delta$$, which we can manipulate into $$a<x<a+\\\\delta$$ to fit our statement above, where a is what $$x$$ approaches.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim19c","stepAnswer":["$$f{\\\\left(x\\\\right)}+2$$"],"problemType":"MultipleChoice","stepTitle":"Then abs(___)=(___)<\u03b5.","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"answerLatex":"$$f{\\\\left(x\\\\right)}+2$$","choices":["$$f{\\\\left(x\\\\right)}+2$$","$$f(x)-2;8x$$","$$f{\\\\left(x\\\\right)}+2$$","$$f{\\\\left(x\\\\right)}+2$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim19c-h1","type":"hint","dependencies":[],"title":"Using the precise definition of a limit","text":"We are trying to prove that for the limit, the function must be closer to epsilon than L for the limit $$\\\\lim_{x\\\\toa} f(x)=L)$$. The general form we are proving is $$|f{\\\\left(x\\\\right)}-L|<\\\\epsilon$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim19c-h2","type":"hint","dependencies":["a524aa3PreciseLim19c-h1"],"title":"Determining the function","text":"First we need to determine the function, which since the function changes, we can leave as f(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim19c-h3","type":"hint","dependencies":["a524aa3PreciseLim19c-h2"],"title":"Determining L","text":"From the aforementioned general form, the L we are looking for is on the opposite side of the equal sign of the limit.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim19c-h4","type":"hint","dependencies":["a524aa3PreciseLim19c-h3"],"title":"Defining epsilon","text":"Since we set delta $$=$$ $$\\\\frac{\\\\epsilon}{4}$$ earlier, then we must solve our statement from the previous step such that epsilon has a power and coefficient of positive $$1$$ only. This will make it such that epsilon is greater than $$4x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a524aa3PreciseLim2","title":"Write the appropriate \u03b5-\u03b4 defintion for the given statement, $$\\\\lim_{t\\\\tob} g(t)=M$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.5 The Precise Definition of a Limit","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a524aa3PreciseLim2a","stepAnswer":["$$\\\\epsilon>0$$"],"problemType":"MultipleChoice","stepTitle":"For every $$___$$.","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"answerLatex":"$$\\\\epsilon>0$$","choices":["$$\\\\epsilon<0$$","$$\\\\varepsilon=0$$","$$\\\\epsilon>0$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim2a-h1","type":"hint","dependencies":[],"title":"Universal quantifier","text":"Remember that epsilon in a \u03b5-\u03b4 defintion is used to quantify the distance between the function and what the limit is equal to.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim2a-h2","type":"hint","dependencies":["a524aa3PreciseLim2a-h1"],"title":"Distance","text":"The distance between the function and what it is equal to, M in this case, cannot be negative, thus what must epsilon be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim2b","stepAnswer":["\u03b4 >0"],"problemType":"MultipleChoice","stepTitle":"There exists a $$___$$.","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"choices":["\u03b4 $$=0$$","\u03b4 >0","\u03b4 >0","$$\\\\delta<0$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim2b-h1","type":"hint","dependencies":[],"title":"Existential quantifier","text":"Remember tha delta in a \u03b5-\u03b4 defintion must be greater than the distance between the variable(t in this case) and what the variable is going to (b in this case).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim2b-h2","type":"hint","dependencies":["a524aa3PreciseLim2b-h1"],"title":"Distance","text":"Since delta must be greater than the distance between $$t$$ and $$b$$, then delta must be positive","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim2c","stepAnswer":["$$t-b$$"],"problemType":"MultipleChoice","stepTitle":"so that if 0<abs(___)|<\u03b4 .","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"answerLatex":"$$t-b$$","choices":["$$t-b$$","$$b-t$$","$$a-x$$","$$g(t)-t$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim2c-h1","type":"hint","dependencies":[],"title":"Distance between $$x$$ and a","text":"We need to represent the distance between $$t$$ and $$b$$, such that $$t$$ is greater than and does not equal $$b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim2d","stepAnswer":["$$g(t)-M$$"],"problemType":"MultipleChoice","stepTitle":"Then abs(___)<\u03b5.","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"answerLatex":"$$g(t)-M$$","choices":["$$M-g(t)$$","$$g(t)-M$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim2d-h1","type":"hint","dependencies":[],"title":"Distance for the function","text":"For the limit, the function must be closer to epislon than M. This step is asking how we can represent this mathematically using g(t) and M.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a524aa3PreciseLim20","title":"Use the precise defintion of limits to prove the given one-sided limit, $$\\\\lim_{x\\\\to1} f(x)=3$$ where $$x$$ is approaching from the left and if $$x<1$$ then $$f(x)=5x-2$$, but if $$x$$ is greater than or equal to $$1$$, then $$f(x)=7x-1$$.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.5 The Precise Definition of a Limit","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a524aa3PreciseLim20a","stepAnswer":["$$\\\\frac{j}{5}$$"],"problemType":"TextBox","stepTitle":"What must we let delta equal in terms of epsilon?","stepBody":"To input your answer, let j represent epsilon (e.g. 5j $$=$$ 5\u03b5)","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{j}{5}$$","hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim20a-h1","type":"hint","dependencies":[],"title":"Determining epsilon","text":"To solve, we need to look at the limit of our function and see what operation must be done to get $$x$$ to have a power and coefficient of $$1$$ only and apply the same operations to epsilon.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim20b","stepAnswer":["$$1-\\\\frac{\\\\epsilon}{5}$$"],"problemType":"MultipleChoice","stepTitle":"If (___)<x<1","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"answerLatex":"$$1-\\\\frac{\\\\epsilon}{5}$$","choices":["1-(3\u03b5/5)","$$1-\\\\varepsilon$$","$$5-\\\\frac{\\\\epsilon}{5}$$","$$1-\\\\frac{\\\\epsilon}{5}$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim20b-h1","type":"hint","dependencies":[],"title":"Precise defintion of limits","text":"The general form is $$-\\\\delta<x-a<0$$, which we can manipulate into $$a-\\\\delta<x<a$$ to fit our statement above, where a is what $$x$$ approaches.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim20c","stepAnswer":["$$f{\\\\left(x\\\\right)}-3$$"],"problemType":"MultipleChoice","stepTitle":"Then abs(___)=(___)<\u03b5.","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"answerLatex":"$$f{\\\\left(x\\\\right)}-3$$","choices":["$$f(x)-3;x-5$$","$$f{\\\\left(x\\\\right)}-3$$","$$f{\\\\left(x\\\\right)}-2$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim20c-h1","type":"hint","dependencies":[],"title":"Using the precise definition of a limit","text":"We are trying to prove that for the limit, the function must be closer to epsilon than L for the limit $$\\\\lim_{x\\\\toa} f(x)=L)$$. The general form we are proving is $$|f{\\\\left(x\\\\right)}-L|<\\\\epsilon$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim20c-h2","type":"hint","dependencies":["a524aa3PreciseLim20c-h1"],"title":"Determining the function","text":"First we need to determine the function, which since the function changes, we can leave as f(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim20c-h3","type":"hint","dependencies":["a524aa3PreciseLim20c-h2"],"title":"Determining L","text":"From the aforementioned general form, the L we are looking for is on the opposite side of the equal sign of the limit.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim20c-h4","type":"hint","dependencies":["a524aa3PreciseLim20c-h3"],"title":"Defining epsilon","text":"Since we set delta $$=$$ $$\\\\frac{\\\\epsilon}{5}$$ earlier, then we must solve our statement from the previous step such that epsilon has a power and coefficient of positive $$1$$ only. This will make it such that epsilon is greater than $$5\\\\left(x-1\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a524aa3PreciseLim21","title":"Use the precise defintion of limits to prove the given infinite limit, /lim{x,-1,(3/((x+1)**2)}=inf","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.5 The Precise Definition of a Limit","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a524aa3PreciseLim21a","stepAnswer":["$$\\\\sqrt{\\\\frac{3}{M}}$$"],"problemType":"TextBox","stepTitle":"What must we let delta equal in terms of M?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\sqrt{\\\\frac{3}{M}}$$","hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim21a-h1","type":"hint","dependencies":[],"title":"Determining M","text":"To solve, we need to look at the limit of our function and see what operation must be done to get $$x$$ to have a power and coefficient of $$1$$ only and apply the same operations to M.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim21b","stepAnswer":["$$x+1$$"],"problemType":"MultipleChoice","stepTitle":"If 0<abs(___)<sqrt(3/M)","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"answerLatex":"$$x+1$$","choices":["$$x+1$$","$$x-1$$","$$x+\\\\infty$$","$$x-\\\\infty$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim21b-h1","type":"hint","dependencies":[],"title":"Precise defintion of limits","text":"The general form is $$0<|x-a|<\\\\delta$$, where a is what $$x$$ approaches.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim21c","stepAnswer":[">M"],"problemType":"MultipleChoice","stepTitle":"Then $$f(x)=\\\\frac{3}{{\\\\left(x+1\\\\right)}^2}___$$","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"choices":[">M","<M",">(-M)","<(-M)"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim21c-h1","type":"hint","dependencies":[],"title":"Using the precise definition of a limit","text":"For positive infinite limits, we are trying to prove that the function of the limit is greater than M","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a524aa3PreciseLim22","title":"Use the precise defintion of limits to prove the given infinite limit, $$\\\\lim_{x\\\\to0} \\\\frac{1}{x^2}=\\\\infty$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.5 The Precise Definition of a Limit","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a524aa3PreciseLim22a","stepAnswer":["$$\\\\sqrt{\\\\frac{1}{M}}$$"],"problemType":"TextBox","stepTitle":"What must we let delta equal in terms of M?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\sqrt{\\\\frac{1}{M}}$$","hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim22a-h1","type":"hint","dependencies":[],"title":"Determining M","text":"To solve, we need to look at the limit of our function and see what operation must be done to get $$x$$ to have a power and coefficient of $$1$$ only and apply the same operations to M.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim22b","stepAnswer":["$$x$$"],"problemType":"MultipleChoice","stepTitle":"If 0<abs(___)<sqrt(1/M)","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"answerLatex":"$$x$$","choices":["$$x$$","$$x-1$$","$$x+\\\\infty$$","$$x-\\\\infty$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim22b-h1","type":"hint","dependencies":[],"title":"Precise defintion of limits","text":"The general form is $$0<|x-a|<\\\\delta$$, where a is what $$x$$ approaches.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim22c","stepAnswer":[">M"],"problemType":"MultipleChoice","stepTitle":"Then $$f(x)=\\\\frac{1}{x^2}___$$","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"choices":[">M","<M",">(-M)","<(-M)"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim22c-h1","type":"hint","dependencies":[],"title":"Using the precise definition of a limit","text":"For positive infinite limits, we are trying to prove that the function of the limit is greater than M","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a524aa3PreciseLim23","title":"Use the precise defintion of limits to prove the given infinite limit, $$\\\\lim_{x\\\\to2} \\\\frac{\\\\left(-1\\\\right)}{{\\\\left(x-2\\\\right)}^2}=-\\\\infty$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.5 The Precise Definition of a Limit","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a524aa3PreciseLim23a","stepAnswer":["$$-\\\\sqrt{\\\\frac{1}{M}}$$"],"problemType":"TextBox","stepTitle":"What must we let delta equal in terms of M?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-\\\\sqrt{\\\\frac{1}{M}}$$","hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim23a-h1","type":"hint","dependencies":[],"title":"Determining M","text":"To solve, we need to look at the limit of our function and see what operation must be done to get $$x$$ to have a power and coefficient of $$1$$ only and apply the same operations to M.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim23b","stepAnswer":["$$x-2$$"],"problemType":"MultipleChoice","stepTitle":"If 0<abs(___)<-sqrt(1/M)","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"answerLatex":"$$x-2$$","choices":["$$x+1$$","$$x+\\\\infty$$","$$x-1$$","$$x-2$$","$$x-\\\\infty$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim23b-h1","type":"hint","dependencies":[],"title":"Precise defintion of limits","text":"The general form is $$0<|x-a|<\\\\delta$$, where a is what $$x$$ approaches.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim23c","stepAnswer":["<(-M)"],"problemType":"MultipleChoice","stepTitle":"Then f(x)=(-1/((x-2)**2)___","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"choices":[">M","<M",">(-M)","<(-M)"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim23c-h1","type":"hint","dependencies":[],"title":"Using the precise definition of a limit","text":"For negative infinite limits, we are trying to prove that the function of the limit is less than -M","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a524aa3PreciseLim24","title":"An engineer is using a machine to cut a flat square of Aerogel of area $$144$$ $${cm}^2$$. There is a maximum error tolerance in the area of $$8$$ $${cm}^2$$.","body":"Recommended to have a graphing calculator for this problem","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.5 The Precise Definition of a Limit","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a524aa3PreciseLim24a","stepAnswer":["$$0.328$$"],"problemType":"TextBox","stepTitle":"How accurately must the engineer cut on the side, assuming all sides have the same length?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.328$$","hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim24a-h1","type":"hint","dependencies":[],"title":"Precise defintion of limits","text":"This problem can be solved the same way we solved for delta in precise definition of limits problems","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim24a-h2","type":"hint","dependencies":["a524aa3PreciseLim24a-h1"],"title":"Translating the Problem","text":"Since the engineer is trying to cut a square of area $$144{cm}^2$$ equal side lengths which we can denote $$x$$, then we can get the limit $$\\\\lim_{x\\\\to12} x^2=144$$. The error tolerance is the same as the epsilon in a precise definition problem.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim24a-h3","type":"hint","dependencies":["a524aa3PreciseLim24a-h2"],"title":"Translating to a precise defintion of limits statement","text":"Now we need to translate our information into the general form of theconditional statement portion of the precise defintion of a limit. The general form is if $$|x-a|<\\\\delta$$, then $$|f{\\\\left(x\\\\right)}-L|<\\\\epsilon$$ for the limit $$\\\\lim_{x\\\\toa} f(x)=L)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim24a-h4","type":"hint","dependencies":["a524aa3PreciseLim24a-h3"],"title":"Graphing the function","text":"To solve, we need to first graph the function, $$x^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim24a-h5","type":"hint","dependencies":["a524aa3PreciseLim24a-h4"],"title":"Solving for delta","text":"Here, we need to find how close to $$12$$ must $$x$$ be, if $$y$$ is within epsilon $$=$$ $$8$$ units of $$144$$ on the graph. We can find this by drawing boundary lines on the graph that fulfill these conditions and making delta the distance from $$x=12$$ to the closer of the $$2$$ $$x-boundary$$ lines.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim24b","stepAnswer":["$$\\\\varepsilon=8;\u03b4=0.328;a=12;L=144$$"],"problemType":"MultipleChoice","stepTitle":"How do these numbers relate to delta, epsilon, a, and L?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\varepsilon=8;\u03b4=0.328;a=12;L=144$$","choices":["$$\\\\varepsilon=8;\u03b4=0.328;a=12;L=144$$","$$\\\\varepsilon=12;\u03b4=0.328;a=12;L=8$$","$$\\\\varepsilon=8;\u03b4=-0.328;a=12;L=144$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim24b-h1","type":"hint","dependencies":[],"title":"Epsilon","text":"Error tolerance is similar to the idea of how close $$x$$ must to a number if $$y$$ is epsilon away from another number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim24b-h2","type":"hint","dependencies":["a524aa3PreciseLim24b-h1"],"title":"Delta","text":"The corresponding accuracy is similar to how close our estimate of delta for $$x$$ if $$y$$ is epsilon units from another number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim24b-h3","type":"hint","dependencies":["a524aa3PreciseLim24b-h2"],"title":"\\"a\\" in this contex","text":"a is what $$x$$, the side lengths, are approaching.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim24b-h4","type":"hint","dependencies":["a524aa3PreciseLim24b-h3"],"title":"\\"L\\" in this context","text":"L is the maximum value the limit if approaching, which is similar to the maximum area.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a524aa3PreciseLim25","title":"Using precise definition of limits, prove $$\\\\lim_{x\\\\to1} \\\\frac{|x-1|}{x-1}$$ does not exist.","body":"There are many ways to answer this problem, but we will briefly guide you through one possible way.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.5 The Precise Definition of a Limit","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a524aa3PreciseLim25a","stepAnswer":["j"],"problemType":"TextBox","stepTitle":"What must we let delta equal in terms of epsilon?","stepBody":"To input your answer, let j represent epsilon (e.g. 5j $$=$$ 5\u03b5)","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim25a-h1","type":"hint","dependencies":[],"title":"Determining epsilon","text":"To solve, we need to look at the limit of our function and see what operation must be done to get $$x$$ to have a power and coefficient of $$1$$ only and apply the same operations to epsilon.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim25b","stepAnswer":["$$x-1$$"],"problemType":"MultipleChoice","stepTitle":"If 0<abs(___)<epsilon","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"answerLatex":"$$x-1$$","choices":["$$x-1$$","$$x+2$$","$$x-\\\\frac{1}{4}$$","$$x+3$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim25b-h1","type":"hint","dependencies":[],"title":"Precise defintion of limits","text":"Here, we need to represent the distance between $$x$$ and what $$x$$ approaches (in this case 0), such that $$x$$ is greater than and does not equal what it approaches.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim25c","stepAnswer":["The statement does not hold"],"problemType":"MultipleChoice","stepTitle":"Then(abs(x-1))/x-1)-L=abs(x-1)<\u03b5? What is wrong with this?","stepBody":"","answerType":"string","variabilization":{},"choices":["The statement does not hold","epsilon is undefined","L is undefined"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim25c-h1","type":"hint","dependencies":[],"title":"The main issue","text":"The main issue is that no matter what L or $$x$$ is, the statement will not hold true. Try plugging in values for L and $$x$$ to check.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a524aa3PreciseLim3","title":"Write the appropriate \u03b5-\u03b4 defintion for the given statement, $$\\\\lim_{x\\\\toc} h(x)=L$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.5 The Precise Definition of a Limit","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a524aa3PreciseLim3a","stepAnswer":["$$\\\\epsilon>0$$"],"problemType":"MultipleChoice","stepTitle":"For every $$___$$.","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"answerLatex":"$$\\\\epsilon>0$$","choices":["$$\\\\epsilon<0$$","$$\\\\varepsilon=0$$","$$\\\\epsilon>0$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim3a-h1","type":"hint","dependencies":[],"title":"Universal quantifier","text":"Remember that epsilon in a \u03b5-\u03b4 defintion is used to quantify the distance between the function and what the limit is equal to.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim3a-h2","type":"hint","dependencies":["a524aa3PreciseLim3a-h1"],"title":"Distance","text":"The distance between the function and what it is equal to, L in this case, cannot be negative, thus what must epsilon be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim3b","stepAnswer":["\u03b4 >0"],"problemType":"MultipleChoice","stepTitle":"There exists a $$___$$.","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"choices":["\u03b4 $$=0$$","\u03b4 >0","\u03b4 >0","$$\\\\delta<0$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim3b-h1","type":"hint","dependencies":[],"title":"Existential quantifier","text":"Remember tha delta in a \u03b5-\u03b4 defintion must be greater than the distance between the variable(x in this case) and what the variable is going to (c in this case).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim3b-h2","type":"hint","dependencies":["a524aa3PreciseLim3b-h1"],"title":"Distance","text":"Since delta must be greater than the distance between $$t$$ and $$b$$, then delta must be positive","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim3c","stepAnswer":["$$x-c$$"],"problemType":"MultipleChoice","stepTitle":"so that if 0<abs(___)|<\u03b4 .","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"answerLatex":"$$x-c$$","choices":["$$c-x$$","$$x-c$$","$$x-L$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim3c-h1","type":"hint","dependencies":[],"title":"Distance between $$x$$ and a","text":"We need to represent the distance between $$x$$ and c, such that $$x$$ is greater than and does not equal c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim3d","stepAnswer":["$$h(x)-L$$"],"problemType":"MultipleChoice","stepTitle":"Then abs(___)<\u03b5.","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"answerLatex":"$$h(x)-L$$","choices":["$$L-h(x)$$","$$h(x)-L$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim3d-h1","type":"hint","dependencies":[],"title":"Distance for the function","text":"For the limit, the function must be closer to epislon than L. This step is asking how we can represent this mathematically using h(x) and L.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a524aa3PreciseLim4","title":"Write the appropriate \u03b5-\u03b4 defintion for the given statement, $$\\\\lim_{x\\\\toa} w(x)=A$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.5 The Precise Definition of a Limit","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a524aa3PreciseLim4a","stepAnswer":["$$\\\\epsilon>0$$"],"problemType":"MultipleChoice","stepTitle":"For every $$___$$.","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"answerLatex":"$$\\\\epsilon>0$$","choices":["$$\\\\epsilon<0$$","$$\\\\varepsilon=0$$","$$\\\\epsilon>0$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim4a-h1","type":"hint","dependencies":[],"title":"Universal quantifier","text":"Remember that epsilon in a \u03b5-\u03b4 defintion is used to quantify the distance between the function and what the limit is equal to.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim4a-h2","type":"hint","dependencies":["a524aa3PreciseLim4a-h1"],"title":"Distance","text":"The distance between the function and what it is equal to, A in this case, cannot be negative, thus what must epsilon be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim4b","stepAnswer":["\u03b4 >0"],"problemType":"MultipleChoice","stepTitle":"There exists a $$___$$.","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"choices":["\u03b4 $$=0$$","\u03b4 >0","\u03b4 >0","$$\\\\delta<0$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim4b-h1","type":"hint","dependencies":[],"title":"Existential quantifier","text":"Remember tha delta in a \u03b5-\u03b4 defintion must be greater than the distance between the variable(x in this case) and what the variable is going to (a in this case).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim4b-h2","type":"hint","dependencies":["a524aa3PreciseLim4b-h1"],"title":"Distance","text":"Since delta must be greater than the distance between $$t$$ and $$b$$, then delta must be positive","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim4c","stepAnswer":["$$x-a$$"],"problemType":"MultipleChoice","stepTitle":"so that if 0<abs(___)|<\u03b4 .","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"answerLatex":"$$x-a$$","choices":["$$x-a$$","$$A-a$$","$$a-A$$","$$a-x$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim4c-h1","type":"hint","dependencies":[],"title":"Distance between $$x$$ and a","text":"We need to represent the distance between $$x$$ and a, such that $$x$$ is greater than and does not equal a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a524aa3PreciseLim4d","stepAnswer":["$$w(x)-A$$"],"problemType":"MultipleChoice","stepTitle":"Then abs(___)<\u03b5.","stepBody":"Fill in the blank with the correct multiple choice answer.","answerType":"string","variabilization":{},"answerLatex":"$$w(x)-A$$","choices":["$$A-w(x)$$","$$w(x)-A$$"],"hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim4d-h1","type":"hint","dependencies":[],"title":"Distance for the function","text":"For the limit, the function must be closer to epislon than A. This step is asking how we can represent this mathematically using w(x) and A.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a524aa3PreciseLim5","title":"The graph of the function satisfies $$\\\\lim_{x\\\\to2} f(x)=2$$. determine a value of $$\\\\delta>0$$ that satisfies the statement.","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.5 The Precise Definition of a Limit","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a524aa3PreciseLim5a","stepAnswer":["$$0.5$$"],"problemType":"TextBox","stepTitle":"If $$0<|x-2|<\\\\delta$$, then $$|f{\\\\left(x\\\\right)}-2|<1$$. What must delta be less than or equal to?","stepBody":"(Note: round answer to nearest $$0.25)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.5$$","hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim5a-h1","type":"hint","dependencies":[],"title":"Translating the statement","text":"The statement tell us that if $$x$$ is closer than delta to $$2$$ and $$x$$ is not equal to $$2$$, then f(x) is closer to than $$1$$ to $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim5a-h2","type":"hint","dependencies":["a524aa3PreciseLim5a-h1"],"title":"Visualizing","text":"In essence, we are asked to find how close to $$2$$ must $$x$$ be, if $$y$$ is within $$1$$ unit of $$2$$ on the graph. We can find this by drawing boundary lines on the graph that fulfill these conditions and making delta the distance from $$x=2$$ to the closer of the two $$x-boundary$$ lines.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a524aa3PreciseLim6","title":"The graph of the function satisfies $$\\\\lim_{x\\\\to2} f(x)=2$$. determine a value of $$\\\\delta>0$$ that satisfies the statement.","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.5 The Precise Definition of a Limit","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a524aa3PreciseLim6a","stepAnswer":["$$0.25$$"],"problemType":"TextBox","stepTitle":"If $$0<|x-2|<\\\\delta$$, then $$|f{\\\\left(x\\\\right)}-2|<0.5$$. What must delta be less than or equal to?","stepBody":"(Note: round answer to nearest $$0.25)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.25$$","hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim6a-h1","type":"hint","dependencies":[],"title":"Translating the statement","text":"The statement tell us that if $$x$$ is closer than delta to $$2$$ and $$x$$ is not equal to $$2$$, then f(x) is closer to than $$0.5$$ to $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim6a-h2","type":"hint","dependencies":["a524aa3PreciseLim6a-h1"],"title":"Visualizing","text":"In essence, we are asked to find how close to $$2$$ must $$x$$ be, if $$y$$ is within $$0.5$$ of $$2$$ on the graph. We can find this by drawing boundary lines on the graph that fulfill these conditions and making delta the distance from $$x=2$$ to the closer of the $$2$$ $$x-boundary$$ lines.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a524aa3PreciseLim7","title":"The graph of the function satisfies $$\\\\lim_{x\\\\to3} f(x)=-1$$. determine a value of $$\\\\delta>0$$ that satisfies the statement.","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.5 The Precise Definition of a Limit","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a524aa3PreciseLim7a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"If $$0<|x-3|<\\\\delta$$, then $$|f{\\\\left(x\\\\right)}+1|<1$$. What must delta be less than or equal to?","stepBody":"(Note: round answer to nearest $$0.25)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim7a-h1","type":"hint","dependencies":[],"title":"Translating the statement","text":"The statement tell us that if $$x$$ is closer than delta to $$3$$ and $$x$$ is not equal to $$3$$, then f(x) is closer to than $$1$$ to $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim7a-h2","type":"hint","dependencies":["a524aa3PreciseLim7a-h1"],"title":"Visualizing","text":"In essence, we are asked to find how close to $$3$$ must $$x$$ be, if $$y$$ is within $$1$$ unit of $$-1$$ on the graph. We can find this by drawing boundary lines on the graph that fulfill these conditions and making delta the distance from $$x=3$$ to the closer of the $$2$$ $$x-boundary$$ lines.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a524aa3PreciseLim8","title":"The graph of the function satisfies $$\\\\lim_{x\\\\to3} f(x)=-1$$. determine a value of $$\\\\delta>0$$ that satisfies the statement.","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.5 The Precise Definition of a Limit","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a524aa3PreciseLim8a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"If $$0<|x-3|<\\\\delta$$, then $$|f{\\\\left(x\\\\right)}+1|<2$$. What must delta be less than or equal to?","stepBody":"(Note: round answer to nearest $$0.25)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim8a-h1","type":"hint","dependencies":[],"title":"Translating the statement","text":"The statement tell us that if $$x$$ is closer than delta to $$3$$ and $$x$$ is not equal to $$3$$, then f(x) is closer to than $$2$$ than $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim8a-h2","type":"hint","dependencies":["a524aa3PreciseLim8a-h1"],"title":"Visualizing","text":"In essence, we are asked to find how close to $$3$$ must $$x$$ be, if $$y$$ is within $$2$$ of $$-1$$ on the graph. We can find this by drawing boundary lines on the graph that fulfill these conditions and making delta the distance from $$x=3$$ to the closer of the $$2$$ $$x-boundary$$ lines.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a524aa3PreciseLim9","title":"The graph of the function satisfies $$\\\\lim_{x\\\\to3} f(x)=2$$. determine a value of $$\\\\delta>0$$ that satisfies the value of epsilon such that the precise definition of limit holds true.","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.5 The Precise Definition of a Limit","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a524aa3PreciseLim9a","stepAnswer":["$$0.25$$"],"problemType":"TextBox","stepTitle":"$$epsilon=1.5$$","stepBody":"(Note: round answer to nearest $$0.25)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.25$$","hints":{"DefaultPathway":[{"id":"a524aa3PreciseLim9a-h1","type":"hint","dependencies":[],"title":"Translating the limit","text":"Based on the precise definition of a limit, then if we use the information from the limit we get If $$0<|x-3|<\\\\delta$$, then $$|f{\\\\left(x\\\\right)}-2|<\\\\epsilon$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim9a-h2","type":"hint","dependencies":["a524aa3PreciseLim9a-h1"],"title":"Using the precise definition of a limit","text":"The precise definition of a limit tells us that if $$x$$ is closer than delta to $$3$$ and $$x$$ is not equal to $$3$$, then f(x) is closer to than epsilon to $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a524aa3PreciseLim9a-h3","type":"hint","dependencies":["a524aa3PreciseLim9a-h2"],"title":"Visualizing","text":"In essence, we are asked to find how close to $$3$$ must $$x$$ be, if $$y$$ is within epsilon unit of $$2$$ on the graph. We can find this by drawing boundary lines on the graph that fulfill these conditions and making delta the distance from $$x=3$$ to the closer of the $$2$$ $$x-boundary$$ lines.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a526a0cintegration1","title":"Integrating a Function Using the Power Rule","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.4 Integration Formulas and the Net Change Theorem","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a526a0cintegration1a","stepAnswer":["$$\\\\frac{256}{15}$$"],"problemType":"TextBox","stepTitle":"Use the power rule to integrate the function $$\\\\int_{1}^{4} \\\\sqrt{t} \\\\left(1+t\\\\right) \\\\,dt$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{256}{15}$$","hints":{"DefaultPathway":[{"id":"a526a0cintegration1a-h1","type":"hint","dependencies":[],"title":"Rearrange","text":"The first step is to rewrite the function and simplify it so we can apply the power rule: $$\\\\int_{1}^{4} \\\\sqrt{t} \\\\left(1+t\\\\right) \\\\,dt=\\\\int_{1}^{4} t^{\\\\frac{1}{2}} \\\\left(1+t\\\\right) \\\\,dt$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration1a-h2","type":"hint","dependencies":["a526a0cintegration1a-h1"],"title":"Rearrange","text":"Contribute $$t^{\\\\frac{1}{2}}$$ into the term inside parenthesis to obtain $$\\\\int_{1}^{4} t^{\\\\frac{1}{2}}+t^{\\\\frac{3}{2}} \\\\,dt$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration1a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{2}{3} t^{\\\\frac{3}{2}}$$"],"dependencies":["a526a0cintegration1a-h2"],"title":"Integrate each term seperately","text":"In term of $$t$$, what will the first term be after being integrated?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\frac{2}{3} t^{\\\\frac{3}{2}}$$","$$t^{\\\\frac{3}{2}}$$","$$t^{\\\\left(-\\\\frac{1}{2}\\\\right)}$$","$$\\\\left(-\\\\frac{1}{2}\\\\right) t^{\\\\frac{3}{2}}$$"]},{"id":"a526a0cintegration1a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{2}{5} t^{\\\\frac{5}{2}}$$"],"dependencies":["a526a0cintegration1a-h3"],"title":"Integrate each term seperately","text":"In term of $$t$$, what will the second term be after being integrated?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\frac{2}{5} t^{\\\\frac{5}{2}}$$","$$\\\\frac{2}{5} t^{\\\\frac{2}{5}}$$"]},{"id":"a526a0cintegration1a-h5","type":"hint","dependencies":["a526a0cintegration1a-h4"],"title":"Evaluate the integral","text":"As the integral is bounded from $$x=1$$ to $$x=4$$, we have $$\\\\frac{2}{3} 4^{\\\\frac{3}{2}}+\\\\frac{2}{5} 4^{\\\\frac{5}{2}}-\\\\frac{2}{3} 1^{\\\\frac{3}{2}}+\\\\frac{2}{5} 1^{\\\\frac{5}{2}}=\\\\frac{256}{15}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a526a0cintegration10","title":"Use basic integration formulas to compute the following antiderivatives or definite integrals.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.4 Integration Formulas and the Net Change Theorem","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a526a0cintegration10a","stepAnswer":["ln|x|+x**(-1)+C"],"problemType":"TextBox","stepTitle":"$$\\\\int \\\\frac{x-1}{x^2} \\\\,dx$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$ln|x|\\\\left(+x^{\\\\left(-1\\\\right)}\\\\right)+C$$","hints":{"DefaultPathway":[{"id":"a526a0cintegration10a-h1","type":"hint","dependencies":[],"title":"Seperating the terms","text":"We will start by seperating the $$2$$ terms in this integrals and apply the basic inegration rules on each of them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration10a-h2","type":"hint","dependencies":["a526a0cintegration10a-h1"],"title":"Seperating the terms","text":"$$\\\\int \\\\frac{x-1}{x^2} \\\\,dx=\\\\int \\\\frac{x}{x^2} \\\\,dx-\\\\int \\\\frac{1}{x^2} \\\\,dx=\\\\int \\\\frac{1}{x} \\\\,dx-\\\\int \\\\frac{1}{x^2} \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration10a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["ln|x|"],"dependencies":["a526a0cintegration10a-h2"],"title":"Integrating the first term","text":"What is the integration of the first term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration10a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-\\\\left(x^{\\\\left(-1\\\\right)}\\\\right)$$"],"dependencies":["a526a0cintegration10a-h3"],"title":"Integrating the second term","text":"What is the integration of the second term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$-\\\\left(x^{\\\\left(-1\\\\right)}\\\\right)$$","$$x^{\\\\left(-1\\\\right)}$$"],"subHints":[{"id":"a526a0cintegration10a-h4-s1","type":"hint","dependencies":[],"title":"Integrating the second term","text":"$$-\\\\int \\\\frac{1}{x^2} \\\\,dx=-\\\\int x^{\\\\left(-2\\\\right)} \\\\,dx=-\\\\left(x^{\\\\left(-1\\\\right)}\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"a526a0cintegration10a-h5","type":"hint","dependencies":["a526a0cintegration10a-h4"],"title":"Combine two integrations","text":"$$ln|x|-\\\\left(-x^{\\\\left(-1\\\\right)}\\\\right)=ln|x|\\\\left(+x^{\\\\left(-1\\\\right)}\\\\right)+C$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a526a0cintegration11","title":"Use basic integration formulas to compute the following antiderivatives or definite integrals.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.4 Integration Formulas and the Net Change Theorem","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a526a0cintegration11a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"$$\\\\int_{0}^{pi} sinx-cosx \\\\,dx$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a526a0cintegration11a-h1","type":"hint","dependencies":[],"title":"Seperating the terms","text":"We will start by seperating the $$2$$ terms in this integrals and apply the basic inegration rules on each of them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration11a-h2","type":"hint","dependencies":["a526a0cintegration11a-h1"],"title":"Seperating the terms","text":"$$\\\\int_{0}^{pi} sinx-cosx \\\\,dx=\\\\int_{0}^{pi} sinx \\\\,dx-\\\\int_{0}^{pi} cosx \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration11a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["-cosx"],"dependencies":["a526a0cintegration11a-h2"],"title":"Integrating the first term","text":"What is the integration of the first term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration11a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["sinx"],"dependencies":["a526a0cintegration11a-h3"],"title":"Integrating the second term","text":"What is the integration of the second term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration11a-h5","type":"hint","dependencies":["a526a0cintegration11a-h4"],"title":"Combine two integrations","text":"$$(-cos(pi)-cos(0))-(sin(pi)-sin(0))=\\\\left(-\\\\left(-1\\\\right)+1\\\\right)-0-0=2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a526a0cintegration12","title":"Use basic integration formulas to compute the following antiderivatives or definite integrals.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.4 Integration Formulas and the Net Change Theorem","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a526a0cintegration12a","stepAnswer":["$$18s^2$$"],"problemType":"MultipleChoice","stepTitle":"Write an integral that quantifies the change in the area of the surface of a cube when its side length doubles from s unit to 2s units and evaluate the integral.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$18s^2$$","choices":["$$18s^2$$","$$18s$$"],"hints":{"DefaultPathway":[{"id":"a526a0cintegration12a-h1","type":"hint","dependencies":[],"title":"Set up the integral","text":"We will start by converting the word problem into the mathematical expession. According to the Net change theorem, the rate of change of surface of a cube formula will be integrated with the boundaries go from $$x=s$$ to $$x=2s$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration12a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\int_{s}^{2s} 12x \\\\,dx$$"],"dependencies":["a526a0cintegration12a-h1"],"title":"Set up the integral","text":"With the information given in hint $$1$$, how can we express the integral?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\int_{s}^{2s} 12x \\\\,dx$$","$$\\\\int_{s}^{2s} 2x \\\\,dx$$","$$\\\\int_{s}^{2s} 6x \\\\,dx$$","$$\\\\int_{2s}^{4s} x \\\\,dx$$"],"subHints":[{"id":"a526a0cintegration12a-h2-s1","type":"hint","dependencies":[],"title":"Set up the integral","text":"The surface of a cube formula originally is $$6x^2$$ with $$x$$ is the length of the cube.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"a526a0cintegration12a-h3","type":"hint","dependencies":["a526a0cintegration12a-h2"],"title":"Net change theorem","text":"The Net Change Theorem giving the integral for determining change in a function F(x) when $$x$$ changes from a to $$b$$ is stated as $$F(b)-F(a)=\\\\int_{a}^{b} \\\\frac{d}{\\\\operatorname{dx}\\\\left(F\\\\left(x\\\\right)\\\\right)} \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration12a-h4","type":"hint","dependencies":["a526a0cintegration12a-h3"],"title":"Calculate the derivative","text":"$$\\\\frac{d}{dx} F\\\\left(x\\\\right)=\\\\frac{d}{dx} 6x^2=12x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration12a-h5","type":"hint","dependencies":["a526a0cintegration12a-h4"],"title":"Compute the integral","text":"$$\\\\int_{s}^{2s} 12x \\\\,dx=\\\\frac{12x^2}{2}$$ with $$x$$ running from $$x=s$$ to $$x=2s$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration12a-h6","type":"hint","dependencies":["a526a0cintegration12a-h5"],"title":"Evaluate","text":"$$\\\\frac{12{\\\\left(2s\\\\right)}^2}{2}-\\\\frac{12s^2}{2}=18s^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a526a0cintegration13","title":"Use basic integration formulas to compute the following antiderivatives or definite integrals.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.4 Integration Formulas and the Net Change Theorem","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a526a0cintegration13a","stepAnswer":["$$12\\\\pi R$$"],"problemType":"MultipleChoice","stepTitle":"Write an integral that quantifies the increase in the surface area of a sphere as its radius doubles from R unit to $$2R$$ units and evaluate the integral.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$12\\\\pi R$$","choices":["$$12\\\\pi R$$","$$12\\\\pi R^2$$"],"hints":{"DefaultPathway":[{"id":"a526a0cintegration13a-h1","type":"hint","dependencies":[],"title":"Set up the integral","text":"We will start by converting the word problem in to the mathematical expession. The surface of a sphere formula will be integrated with the double side length and the boundaries run from $$x=R$$ to $$x=2R$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration13a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$8*pi*\\\\int_{R}^{2R} x \\\\,dx$$"],"dependencies":["a526a0cintegration13a-h1"],"title":"Set up the integral","text":"With the information given in hint $$1$$, how can we express the integral?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$4*pi*\\\\int_{R}^{2R} x \\\\,dx$$","$$8*pi*\\\\int_{R}^{2R} 2x \\\\,dx$$","$$8*pi*\\\\int_{R}^{2R} x \\\\,dx$$","$$8*pi*\\\\int_{R}^{2R} x \\\\,dx$$"]},{"id":"a526a0cintegration13a-s1","type":"hint","dependencies":[],"title":"Set up the integral","text":"The surface of a cube formula originally is $$8\\\\pi x$$ with $$x$$ is a radius of the sphere. The term $$8\\\\pi$$ can be treated as a constant and moved out of the integral.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration13a-h3","type":"hint","dependencies":["a526a0cintegration13a-h2"],"title":"Net change theorem","text":"The Net Change Theorem giving the integral for determining change in a function F(x) when $$x$$ changes from a to $$b$$ is stated as $$F(b)-F(a)=\\\\int_{a}^{b} \\\\frac{d}{dx} F\\\\left(x\\\\right) \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration13a-h4","type":"hint","dependencies":["a526a0cintegration13a-h3"],"title":"Calculate the derivative","text":"$$\\\\frac{d}{dx} F\\\\left(x\\\\right)=\\\\frac{d}{dx} 4\\\\pi r^2=8\\\\pi x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration13a-h5","type":"hint","dependencies":["a526a0cintegration13a-h4"],"title":"Compute the integral","text":"$$8*pi*\\\\int_{R}^{2R} x \\\\,dx=\\\\frac{8\\\\pi x^2}{2}$$ with $$x$$ running from $$x=R$$ to $$x=2R$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration13a-h6","type":"hint","dependencies":["a526a0cintegration13a-h5"],"title":"Evaluate","text":"$$\\\\frac{8\\\\pi {\\\\left(2R\\\\right)}^2}{2}-\\\\frac{8\\\\pi R^2}{2}=12\\\\pi R$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a526a0cintegration14","title":"Use basic integration formulas to compute the following antiderivatives or definite integrals.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.4 Integration Formulas and the Net Change Theorem","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a526a0cintegration14a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"The area A(t) of a circular shape is growing at a constant rate. If the area increases from $$4\\\\pi$$ units to $$9\\\\pi$$ units between times $$t=2$$ and $$t=3$$, find the net change in the radius during that time.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a526a0cintegration14a-h1","type":"hint","dependencies":[],"title":"Find the radius","text":"Use the area of a circle to find the radius at each time then subtract the radius at the time $$t=3$$ with the time $$t=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a526a0cintegration14a-h1"],"title":"At $$t=2$$","text":"What is the radius of the circle at $$t=2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"a526a0cintegration14a-h2-s1","type":"hint","dependencies":[],"title":"At $$t=2$$","text":"With the $$Area=4\\\\pi$$ given when $$t=2$$, the radius can be calculated as $$Area=4\\\\pi=\\\\pi r^2$$ therefore $$r=2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"a526a0cintegration14a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a526a0cintegration14a-h1"],"title":"At $$t=3$$","text":"What is the radius of the circle at $$t=3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"a526a0cintegration14a-h3-s1","type":"hint","dependencies":[],"title":"At $$t=3$$","text":"With the $$Area=9\\\\pi$$ given when $$t=3$$, the radius can be calculated as $$Area=4\\\\pi=\\\\pi r^2$$ therefore $$r=3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"a526a0cintegration14a-h4","type":"hint","dependencies":["a526a0cintegration14a-h2","a526a0cintegration14a-h3"],"title":"Find the net change","text":"$$\u0394r=3-2=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a526a0cintegration15","title":"Use basic integration formulas to compute the following antiderivatives or definite integrals.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.4 Integration Formulas and the Net Change Theorem","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a526a0cintegration15a","stepAnswer":["$$225$$"],"problemType":"TextBox","stepTitle":"Sandra is a 25-year old woman who weighs $$120$$ lb. She burns $$300-50t$$ $$\\\\frac{cal}{hr}$$ while walking on her treadmill. Her caloric intake from drinking Gatorade is $$100t$$ calories during the tth hour. What is her net decrease in calories after walking for $$3$$ hours?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$225$$","hints":{"DefaultPathway":[{"id":"a526a0cintegration15a-h1","type":"hint","dependencies":[],"title":"Set up the integral","text":"The calories lost after \u2018t\u2019 hours have to be the difference between the calories burnt and calories consumed after these \u2018t\u2019 hours. Therefore the calories intake will carry a negative sign $$-100t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration15a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\int_{0}^{3} 300-50t-100t \\\\,dt$$"],"dependencies":["a526a0cintegration15a-h1"],"title":"Set up the integral","text":"How can we express the integral?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\int_{0}^{3} 300-50t-100t \\\\,dt$$","$$\\\\int_{1}^{3} 300-50t-100t \\\\,dt$$"]},{"id":"a526a0cintegration15a-h3","type":"hint","dependencies":["a526a0cintegration15a-h2"],"title":"Simplify the integral","text":"$$\\\\int_{0}^{3} 300-50t-100t \\\\,dt$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration15a-h4","type":"hint","dependencies":["a526a0cintegration15a-h3"],"title":"Seperating the terms","text":"Seperate and conpute the integral of each term $$\\\\int_{0}^{3} 300 \\\\,dt-\\\\int_{0}^{3} 150t \\\\,dt$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration15a-h5","type":"hint","dependencies":["a526a0cintegration15a-h4"],"title":"Use basic integration formulas to compute the following antiderivatives or definite integrals.","text":"$$300\\\\times3-150\\\\frac{3^2}{2}=225$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a526a0cintegration16","title":"Finding Net Displacement","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.4 Integration Formulas and the Net Change Theorem","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a526a0cintegration16a","stepAnswer":["$$\\\\frac{-3}{2}$$"],"problemType":"TextBox","stepTitle":"Given a velocity function $$v(t)=3t-5$$ (in meters per second) for a partical in motion from time $$t=0$$ to time $$t=3$$, find the net displacement of the particle.","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-3}{2}$$","hints":{"DefaultPathway":[{"id":"a526a0cintegration16a-h1","type":"hint","dependencies":[],"title":"Expression","text":"We can begin by setting the upper and the lower boundaries of the definite integral as the function v(t) starts from $$t=0$$ and $$t=3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration16a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\int_{1}^{3} 3t-5 \\\\,dt$$"],"dependencies":["a526a0cintegration16a-h1"],"title":"Expression","text":"What is the expression of this definite integral?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\int_{1}^{3} 3t-5 \\\\,dt$$","$$\\\\int_{1}^{3} t-5 \\\\,dt$$"]},{"id":"a526a0cintegration16a-h3","type":"hint","dependencies":["a526a0cintegration16a-h2"],"title":"Compute the integral","text":"We obtain $$\\\\frac{3}{2} t^2-5t$$ with $$t=0$$ as the lower boundary and $$t=3$$ as the upper boundary.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration16a-h4","type":"hint","dependencies":["a526a0cintegration16a-h3"],"title":"Evaluate the integral","text":"(3/2)*(3**2)-5*3)-((3/2)*(0**2)-5*0)=-3/2","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a526a0cintegration16b","stepAnswer":["$$1.5+40t-4.9t^2$$"],"problemType":"MultipleChoice","stepTitle":"A ball is thrown upward from a height of $$1.5$$ $$m$$ at an initial speed of $$40$$ $$\\\\frac{m}{sec}$$. Acceleration resulting from gravity is $$-9.8$$ $$\\\\frac{m}{{sec}^2}$$. Neglecting air resistance, solve for the velocity v(t) and the height h(t) of the ball $$t$$ seconds after it is thrown and before it returns to the ground.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$1.5+40t-4.9t^2$$","choices":["$$1.5+40t-4.9t^2$$","$$40t-4.9t^2$$"],"hints":{"DefaultPathway":[{"id":"a526a0cintegration16b-h1","type":"hint","dependencies":[],"title":"Velocity function","text":"$$v(t)=v_0+g t$$ where g is the acceleration due to gravity and $$t$$ is the time of function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration16b-h2","type":"hint","dependencies":["a526a0cintegration16b-h1"],"title":"Substituting the values","text":"$$v(t)=40+\\\\left(-9.8\\\\right) t=40-9.8t$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration16b-h3","type":"hint","dependencies":["a526a0cintegration16b-h2"],"title":"Set up a function for the height","text":"$$h(t)=h-v_0 t+\\\\frac{1}{2} g t^2=1.5+40t+\\\\frac{1}{2} \\\\left(-9.8t^2\\\\right)=1.5+40t-4.9t^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a526a0cintegration22","title":"Finding the Total Distance Traveled","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.4 Integration Formulas and the Net Change Theorem","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a526a0cintegration22a","stepAnswer":["$$\\\\frac{41}{6}$$"],"problemType":"TextBox","stepTitle":"Use Example $$5.24$$ to find the total distance traveled by a particle according to the velocity function $$v(t)=3t-5$$ $$\\\\frac{m}{sec}$$ over a time interval [0,3].","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{41}{6}$$","hints":{"DefaultPathway":[{"id":"a526a0cintegration22a-h1","type":"hint","dependencies":[],"title":"Finding the $$t-intercept$$","text":"The total distance traveled includes both the positive and the negative values. Therefore, we must integrate the absolute value of the velocity function to find the total distance traveled. To continue with the example, use two integrals to find the total distance. First, find the $$t-intercept$$ of the function, since that is where the division of the interval occurs. Set the equation equal to zero and solve for $$t$$. Thus,","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration22a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{3}$$"],"dependencies":[],"title":"Finding the $$t-intercept$$","text":"What is the $$t-intercept$$ of this function?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration22a-h3","type":"hint","dependencies":["a526a0cintegration22a-h2"],"title":"Subintervals","text":"The two subintervals are $$[0,\\\\frac{5}{3}]$$ and $$[\\\\frac{5}{3},3]$$. To find the total distance traveled, integrate the absolute value of the function. Since the function is negative over the interval $$[0,\\\\frac{5}{3}]$$ and positive over $$[\\\\frac{5}{3},3]$$, we obtain /int{-(3*t)-5),0,5/3,t}+/int{(3*t)-5,5/3,3,t}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration22a-h4","type":"hint","dependencies":["a526a0cintegration22a-h3"],"title":"Compute the integral","text":"Power rules for integrals gives us $$\\\\left(-\\\\frac{3t^2}{2}\\\\right)+5t$$ with $$t=$$ $$0$$ as a lower boundary and $$t=\\\\frac{5}{3}$$ as an upper boundary added to $$\\\\left(-\\\\frac{3t^2}{2}\\\\right)+5t$$ with $$t=$$ $$\\\\frac{5}{3}$$ as a lower boundary and $$t=3$$ as an upper boundary.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration22a-h5","type":"hint","dependencies":["a526a0cintegration22a-h4"],"title":"Evaluate the integral","text":"-3*((5/3)**2)/2+5*(5/3)+27/2-15-(-3*((5/3)**2)/2)-25/3)=41/6","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration22a-h6","type":"hint","dependencies":["a526a0cintegration22a-h5"],"title":"Conclusion","text":"So the total distance traveled is $$\\\\frac{41}{6}$$ $$m$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a526a0cintegration5","title":"How Many Gallons of Gasoline Are Consumed?","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.4 Integration Formulas and the Net Change Theorem","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a526a0cintegration5a","stepAnswer":["$$9.6$$"],"problemType":"TextBox","stepTitle":"If the motor on a motorboat is started at $$t=0$$ and the boat consumes gasoline at the rate of $$5-0.1t^3$$ $$\\\\frac{gal}{hr}$$, how much gasoline is used in the first $$2$$ hours?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9.6$$","hints":{"DefaultPathway":[{"id":"a526a0cintegration5a-h1","type":"hint","dependencies":[],"title":"Expression","text":"Express the problem as a definite integral, integrate, and evaluate using the Fundamental Theorem of Calculus. The limits of integration are the endpoints of the interval [0,2].","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration5a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$5t-\\\\frac{0.1t^4}{4}$$"],"dependencies":["a526a0cintegration5a-h1"],"title":"Compute the integral","text":"What is the interal of the function with the lower boundary of $$0$$ and upper boundary of 2?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$5t-\\\\frac{0.1t^4}{4}$$","$$5t-0.1t^4$$"]},{"id":"a526a0cintegration5a-h3","type":"hint","dependencies":["a526a0cintegration5a-h2"],"title":"Evaluate the integral","text":"$$5\\\\times2-\\\\frac{0.1\\\\times2^4}{4}-0=9.6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration5a-h4","type":"hint","dependencies":["a526a0cintegration5a-h3"],"title":"Conclusion","text":"Therefore, the motorboat uses $$9.6$$ gal of gas in $$2$$ hours.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a526a0cintegration6","title":"Integrating an Even Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.4 Integration Formulas and the Net Change Theorem","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a526a0cintegration6a","stepAnswer":["$$\\\\frac{500}{3}$$"],"problemType":"TextBox","stepTitle":"Integrate the even function $$\\\\int_{-2}^{2} 3x-2 \\\\,dx$$ and verify that the integration formula for even functions holds.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{500}{3}$$","hints":{"DefaultPathway":[{"id":"a526a0cintegration6a-h1","type":"hint","dependencies":[],"title":"Symmetry","text":"Graph (a) shows the region below the curve and above the x-axis. We have to zoom in to this graph by a huge amount to see the region. Graph (b) shows the region above the curve and below the x-axis. The signed area of this region is negative. Both views illustrate the symmetry about the y-axis of an even function.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration6a-h2","type":"hint","dependencies":["a526a0cintegration6a-h1"],"title":"Set up the integral","text":"$$\\\\int_{-2}^{2} 3x-2 \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration6a-h3","type":"hint","dependencies":["a526a0cintegration6a-h2"],"title":"Compute the integral","text":"$$\\\\frac{x^9}{3}-2x$$ from $$x=-2$$ to $$x=2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration6a-h4","type":"hint","dependencies":["a526a0cintegration6a-h3"],"title":"Evaluate","text":"$$\\\\frac{2^9}{3}-2\\\\times2-\\\\frac{{\\\\left(-2\\\\right)}^9}{3}-2\\\\left(-2\\\\right)=\\\\frac{1000}{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration6a-h5","type":"hint","dependencies":["a526a0cintegration6a-h4"],"title":"Verifying","text":"To verify the integration formula for even functions, we can calculate the integral from $$0$$ to $$2$$ and double it, then check to make sure we get the same answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration6a-h6","type":"hint","dependencies":["a526a0cintegration6a-h5"],"title":"Set up the integral","text":"$$\\\\int_{0}^{2} 3x-2 \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration6a-h7","type":"hint","dependencies":["a526a0cintegration6a-h6"],"title":"Compute the integral","text":"$$\\\\frac{x^9}{3}-2x$$ from $$x=-2$$ to $$x=2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration6a-h8","type":"hint","dependencies":["a526a0cintegration6a-h7"],"title":"Evaluate","text":"$$\\\\frac{512}{3}-4=\\\\frac{500}{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration6a-h9","type":"hint","dependencies":["a526a0cintegration6a-h8"],"title":"Conclusion","text":"Since $$\\\\frac{2\\\\times500}{3}=\\\\frac{1000}{3}$$, we have verified the formula for even functions in this particular example.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a526a0cintegration7","title":"Integrating an Odd Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.4 Integration Formulas and the Net Change Theorem","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a526a0cintegration7a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"Evaluate the definite integral of the odd function $$-5sinx$$ over the inteval [-pi,pi].","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a526a0cintegration7a-h1","type":"hint","dependencies":[],"title":"Symmetry","text":"We can see the symmetry about the origin by the positive area above the x-axis over [-pi,0] and the negative area below the -axis over [0,pi].","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration7a-h2","type":"hint","dependencies":["a526a0cintegration7a-h1"],"title":"Set up the integral","text":"$$\\\\int_{-pi}^{pi} -5sinx \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration7a-h3","type":"hint","dependencies":["a526a0cintegration7a-h2"],"title":"Compute the integral","text":"$$-5\\\\left(-cosx\\\\right)$$ from pi to -pi","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration7a-h4","type":"hint","dependencies":["a526a0cintegration7a-h3"],"title":"Evaluate","text":"$$5cos\\\\left(\\\\pi\\\\right)-5cos\\\\left(-\\\\pi\\\\right)=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a526a0cintegration8","title":"Integrating a Function Using the Power Rule","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.4 Integration Formulas and the Net Change Theorem","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a526a0cintegration8a","stepAnswer":["$$\\\\frac{-10}{3}$$"],"problemType":"TextBox","stepTitle":"Find the definite integral of $$f(x)=x^2-3x$$ over the interval [1,3].","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-10}{3}$$","hints":{"DefaultPathway":[{"id":"a526a0cintegration8a-h1","type":"hint","dependencies":[],"title":"Power rule","text":"Using power rule for integral to compute the integral with the lower bound $$x=1$$ and upper bound $$x=3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration8a-h2","type":"hint","dependencies":["a526a0cintegration8a-h1"],"title":"Expression","text":"We will start by setting the function insisde the integral going from $$x=1$$ to $$x=3$$. We then obtain $$\\\\int_{1}^{3} x^2-3x \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-10}{3}$$"],"dependencies":["a526a0cintegration8a-h2"],"title":"Compute the integral","text":"What is the answer for this integral?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration8a-h4","type":"hint","dependencies":["a526a0cintegration8a-h3"],"title":"Compute the integral","text":"The antiderivative is $$\\\\frac{x^3}{3}-\\\\frac{3x^2}{2}$$ with the boundaries of $$x$$ going from $$1$$ to $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration8a-h5","type":"hint","dependencies":["a526a0cintegration8a-h4"],"title":"Evaluate the integral","text":"On the next step, we will simply subsitute $$x=1$$ and $$x=3$$ into the antiderivative and substract F(3) from F(1) where F(x) indicates the antiderivative of the given function f(x)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration8a-h6","type":"hint","dependencies":["a526a0cintegration8a-h5"],"title":"Evaluate the integral","text":"(x**3)/3-(3*x**2)/2)-(x**3)/3-(3*x**2)/2=-10/3","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a526a0cintegration8b","stepAnswer":["$$\\\\frac{64}{5}$$"],"problemType":"TextBox","stepTitle":"Integrate the function $$\\\\int_{-2}^{2} x^4 \\\\,dx$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{64}{5}$$","hints":{"DefaultPathway":[{"id":"a526a0cintegration8b-h1","type":"hint","dependencies":[],"title":"Compute the integral","text":"$$\\\\frac{x^5}{5}$$ from $$x=-2$$ to $$x=2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration8b-h2","type":"hint","dependencies":["a526a0cintegration8b-h1"],"title":"Evaluate","text":"$$\\\\frac{2^5}{5}-\\\\frac{{\\\\left(-2\\\\right)}^5}{5}=\\\\frac{64}{5}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a526a0cintegration9","title":"Use basic integration formulas to compute the following antiderivatives or definite integrals.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.4 Integration Formulas and the Net Change Theorem","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a526a0cintegration9a","stepAnswer":["$$\\\\frac{1}{2} log|x|+C$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\int \\\\frac{1}{2x} \\\\,dx$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{1}{2} log|x|+C$$","choices":["$$\\\\frac{1}{2} log$$","$$\\\\frac{1}{2} log|x|+C$$","$$+C$$","$$+C$$","log","$$x$$","$$x$$"],"hints":{"DefaultPathway":[{"id":"a526a0cintegration9a-h1","type":"hint","dependencies":[],"title":"Reciprocal Function","text":"Using one of the basic integration formula $$\\\\int \\\\frac{1}{x} \\\\,dx=log|x|+C$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a526a0cintegration9a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1}{2} log|x|+C$$"],"dependencies":["a526a0cintegration9a-h1"],"title":"Integration","text":"What is the result of integration?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\frac{1}{2} log$$","$$x$$","$$+C$$","log","$$x$$","$$+C$$"],"subHints":[{"id":"a526a0cintegration9a-h2-s1","type":"hint","dependencies":[],"title":"Failing to apply Power Rule","text":"As the function can be rewritten as $$(1/2)*\\\\int x^{-1} \\\\,dx$$, we realize the Power Rule for Integrals will result in the denominator equal $$0$$ which is undefined. We have to use another method to approach the problem.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}]}}]},{"id":"a536a53Uni1","title":"Uniform Distributions and Their Properties","body":"Suppose that we are examining the smiling times of an eight-week-old baby (measured in seconds). The smiling times follow a uniform distribution between zero and $$23$$ seconds, meaning that any smiling time from zero to $$23$$ seconds is equally likely.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.2 The Uniform Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a536a53Uni1a","stepAnswer":["$$11.5$$"],"problemType":"TextBox","stepTitle":"What is the mean amount of time that the baby is smiling?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$11.5$$","hints":{"DefaultPathway":[{"id":"a536a53Uni1a-h1","type":"hint","dependencies":[],"title":"Mean Formula","text":"The formula for calculating the mean of a uniform distribution is $$\\\\frac{a+b}{2}$$, where a and $$b$$ are the $$\\\\frac{start}{end}$$ times of our interval respectively.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a536a53Uni1b","stepAnswer":["$$11.5$$"],"problemType":"TextBox","stepTitle":"Knowing the formula to calculate the mean, what is our final answer?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$11.5$$","hints":{"DefaultPathway":[]}}]},{"id":"a536a53Uni10","title":"Uniform Distributions and Their Properties","body":"A distribution is given as X ~ U $$(0,20)$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.2 The Uniform Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a536a53Uni10a","stepAnswer":["$$\\\\frac{3}{20}$$"],"problemType":"TextBox","stepTitle":"What is $$P\\\\left(10<x<13\\\\right)$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{20}$$","hints":{"DefaultPathway":[{"id":"a536a53Uni10a-h1","type":"hint","dependencies":[],"title":"Probability Function","text":"We know that we are looking at a uniform distribution, and also that we are looking at a continuous probability function. What would this probability function look like? As it turns out, this function will look like a straight line from our start to end points. We can calculate the area under this function to get our desired probability.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a536a53Uni10a-h1"],"title":"Interval","text":"If X goes from $$0$$ to $$20$$, how long is the interval of X?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni10a-h3","type":"hint","dependencies":["a536a53Uni10a-h2"],"title":"Probability $$1$$","text":"$$P\\\\left(0<x<1\\\\right)$$ $$=$$ $$\\\\frac{1}{20}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni10a-h4","type":"hint","dependencies":["a536a53Uni10a-h3"],"title":"Probability $$2$$","text":"$$P\\\\left(0<x<2\\\\right)$$ $$=$$ $$\\\\frac{2}{20}$$. Starting to see a pattern? We can get the length in the interval we want, and divide that by the length of our total interval to get our answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{20}$$"],"dependencies":["a536a53Uni10a-h4"],"title":"Combining","text":"Knowing how to calculate the probability, what will our answer be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a536a53Uni11","title":"Uniform Distributions and Their Properties","body":"The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between zero and $$15$$ minutes, inclusive.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.2 The Uniform Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a536a53Uni11a","stepAnswer":["$$7.5$$"],"problemType":"TextBox","stepTitle":"What is the mean amount of time someone will wait for the bus?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7.5$$","hints":{"DefaultPathway":[{"id":"a536a53Uni11a-h1","type":"hint","dependencies":[],"title":"Mean Formula","text":"The formula for calculating the mean of a uniform distribution is $$\\\\frac{a+b}{2}$$, where a and $$b$$ are the $$\\\\frac{start}{end}$$ times of our interval respectively.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a536a53Uni11b","stepAnswer":["$$7.5$$"],"problemType":"TextBox","stepTitle":"Knowing the formula to calculate the mean, what is our final answer?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7.5$$","hints":{"DefaultPathway":[]}}]},{"id":"a536a53Uni12","title":"Uniform Distributions and Their Properties","body":"The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between zero and $$15$$ minutes, inclusive.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.2 The Uniform Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a536a53Uni12a","stepAnswer":["$$4.33$$"],"problemType":"TextBox","stepTitle":"What is the standard deviation of the amount of time that someone will wait for the bus?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4.33$$","hints":{"DefaultPathway":[{"id":"a536a53Uni12a-h1","type":"hint","dependencies":[],"title":"Standard Deviation Formula","text":"The formula for calculating the standard deviation of a uniform distribution is $$\\\\sqrt{\\\\frac{{\\\\left(b-a\\\\right)}^2}{12}}$$, where a and $$b$$ are the $$\\\\frac{start}{end}$$ times of our interval respectively.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a536a53Uni12b","stepAnswer":["$$4.33$$"],"problemType":"TextBox","stepTitle":"Knowing the formula to calculate the standard deviation, what is our final answer?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4.33$$","hints":{"DefaultPathway":[]}}]},{"id":"a536a53Uni13","title":"Uniform Distributions and Their Properties","body":"The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between zero and $$15$$ minutes, inclusive.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.2 The Uniform Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a536a53Uni13a","stepAnswer":["$$\\\\frac{10}{15}$$"],"problemType":"TextBox","stepTitle":"What is the probability that someone will wait $$10$$ minutes or less for the bus?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{10}{15}$$","hints":{"DefaultPathway":[{"id":"a536a53Uni13a-h1","type":"hint","dependencies":[],"title":"Probability Function","text":"We know that we are looking at a uniform distribution, and also that we are looking at a continuous probability function. What would this probability function look like? As it turns out, this function will look like a straight line from our start to end points. We can calculate the area under this function to get our desired probability.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["a536a53Uni13a-h1"],"title":"Interval","text":"If X goes from $$0$$ to $$15$$, how long is the interval of X?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni13a-h3","type":"hint","dependencies":["a536a53Uni13a-h2"],"title":"Probability $$1$$","text":"The probability that someone waits $$1$$ minute or less for the bus is $$\\\\frac{1}{15}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni13a-h4","type":"hint","dependencies":["a536a53Uni13a-h3"],"title":"Probability $$2$$","text":"The probability that someone waits $$2$$ minutes or less for the bus is $$\\\\frac{2}{15}$$. Starting to see a pattern? We can get the length in the interval we want, and divide that by the length of our total interval to get our answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni13a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{10}{15}$$"],"dependencies":["a536a53Uni13a-h4"],"title":"Combining","text":"Knowing how to calculate the probability, what will our answer be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a536a53Uni14","title":"Uniform Distributions and Their Properties","body":"The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between zero and $$15$$ minutes, inclusive.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.2 The Uniform Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a536a53Uni14a","stepAnswer":["$$\\\\frac{7}{15}$$"],"problemType":"TextBox","stepTitle":"What is the probability that someone will wait $$7$$ minutes or less for the bus?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{7}{15}$$","hints":{"DefaultPathway":[{"id":"a536a53Uni14a-h1","type":"hint","dependencies":[],"title":"Probability Function","text":"We know that we are looking at a uniform distribution, and also that we are looking at a continuous probability function. What would this probability function look like? As it turns out, this function will look like a straight line from our start to end points. 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Starting to see a pattern? We can get the length in the interval we want, and divide that by the length of our total interval to get our answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni14a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{7}{15}$$"],"dependencies":["a536a53Uni14a-h4"],"title":"Combining","text":"Knowing how to calculate the probability, what will our answer be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a536a53Uni15","title":"Uniform Distributions and Their Properties","body":"The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between zero and $$15$$ minutes, inclusive.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.2 The Uniform Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a536a53Uni15a","stepAnswer":["$$\\\\frac{11}{15}$$"],"problemType":"TextBox","stepTitle":"What is the probability that someone will wait between $$2$$ and $$13$$ minutes for the bus?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{11}{15}$$","hints":{"DefaultPathway":[{"id":"a536a53Uni15a-h1","type":"hint","dependencies":[],"title":"Probability Function","text":"We know that we are looking at a uniform distribution, and also that we are looking at a continuous probability function. What would this probability function look like? As it turns out, this function will look like a straight line from our start to end points. We can calculate the area under this function to get our desired probability.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["a536a53Uni15a-h1"],"title":"Interval","text":"If X goes from $$0$$ to $$15$$, how long is the interval of X?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni15a-h3","type":"hint","dependencies":["a536a53Uni15a-h2"],"title":"Probability $$1$$","text":"The probability that someone waits $$1$$ minute or less for the bus is $$\\\\frac{1}{15}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni15a-h4","type":"hint","dependencies":["a536a53Uni15a-h3"],"title":"Probability $$2$$","text":"The probability that someone waits $$2$$ minutes or less for the bus is $$\\\\frac{2}{15}$$. Starting to see a pattern? We can get the length in the interval we want, and divide that by the length of our total interval to get our answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni15a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{11}{15}$$"],"dependencies":["a536a53Uni15a-h4"],"title":"Combining","text":"Knowing how to calculate the probability, what will our answer be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a536a53Uni16","title":"Uniform Distributions and Their Properties","body":"The total duration of baseball games in the major league in the $$2011$$ season is uniformly distributed between $$447$$ hours and $$521$$ hours inclusive.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.2 The Uniform Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a536a53Uni16a","stepAnswer":["$$484$$"],"problemType":"TextBox","stepTitle":"What is the mean amount of time for the duration of MLB Baseball games in 2011?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$484$$","hints":{"DefaultPathway":[{"id":"a536a53Uni16a-h1","type":"hint","dependencies":[],"title":"Mean Formula","text":"The formula for calculating the mean of a uniform distribution is $$\\\\frac{a+b}{2}$$, where a and $$b$$ are the $$\\\\frac{start}{end}$$ times of our interval respectively.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a536a53Uni16b","stepAnswer":["$$484$$"],"problemType":"TextBox","stepTitle":"Knowing the formula to calculate the mean, what is our final answer?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$484$$","hints":{"DefaultPathway":[]}}]},{"id":"a536a53Uni17","title":"Uniform Distributions and Their Properties","body":"The total duration of baseball games in the major league in the $$2011$$ season is uniformly distributed between $$447$$ hours and $$521$$ hours inclusive.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.2 The Uniform Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a536a53Uni17a","stepAnswer":["$$21.36$$"],"problemType":"TextBox","stepTitle":"What is the standard deviation of the duration of MLB Baseball games in 2011?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$21.36$$","hints":{"DefaultPathway":[{"id":"a536a53Uni17a-h1","type":"hint","dependencies":[],"title":"Standard Deviation Formula","text":"The formula for calculating the standard deviation of a uniform distribution is $$\\\\sqrt{\\\\frac{{\\\\left(b-a\\\\right)}^2}{12}}$$, where a and $$b$$ are the $$\\\\frac{start}{end}$$ times of our interval respectively.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a536a53Uni17b","stepAnswer":["$$21.36$$"],"problemType":"TextBox","stepTitle":"Knowing the formula to calculate the standard deviation, what is our final answer?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$21.36$$","hints":{"DefaultPathway":[]}}]},{"id":"a536a53Uni18","title":"Uniform Distributions and Their Properties","body":"The total duration of baseball games in the major league in the $$2011$$ season is uniformly distributed between $$447$$ hours and $$521$$ hours inclusive.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.2 The Uniform Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a536a53Uni18a","stepAnswer":["$$\\\\frac{20}{74}$$"],"problemType":"TextBox","stepTitle":"What is the probability that the duration of games for a team for the $$2011$$ season is between $$480$$ and $$500$$ hours?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{20}{74}$$","hints":{"DefaultPathway":[{"id":"a536a53Uni18a-h1","type":"hint","dependencies":[],"title":"Probability Function","text":"We know that we are looking at a uniform distribution, and also that we are looking at a continuous probability function. What would this probability function look like? As it turns out, this function will look like a straight line from our start to end points. We can calculate the area under this function to get our desired probability.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$74$$"],"dependencies":["a536a53Uni18a-h1"],"title":"Interval","text":"If X goes from $$447$$ to $$521$$, how long is the interval of X?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni18a-h3","type":"hint","dependencies":["a536a53Uni18a-h2"],"title":"Probability $$1$$","text":"The probability that the duration of MLB games in a season is between $$447$$ and $$448$$ is $$\\\\frac{1}{74}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni18a-h4","type":"hint","dependencies":["a536a53Uni18a-h3"],"title":"Probability $$2$$","text":"The probability that the duration of MLB games in a season is between $$447$$ and $$449$$ is $$\\\\frac{2}{74}$$. Starting to see a pattern? We can get the length in the interval we want, and divide that by the length of our total interval to get our answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni18a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{20}{74}$$"],"dependencies":["a536a53Uni18a-h4"],"title":"Combining","text":"Knowing how to calculate the probability, what will our answer be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a536a53Uni19","title":"Uniform Distributions and Their Properties","body":"The total duration of baseball games in the major league in the $$2011$$ season is uniformly distributed between $$447$$ hours and $$521$$ hours inclusive.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.2 The Uniform Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a536a53Uni19a","stepAnswer":["$$\\\\frac{50}{74}$$"],"problemType":"TextBox","stepTitle":"What is the probability that the duration of games for a team for the $$2011$$ season is between $$447$$ and $$497$$ hours?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{50}{74}$$","hints":{"DefaultPathway":[{"id":"a536a53Uni19a-h1","type":"hint","dependencies":[],"title":"Probability Function","text":"We know that we are looking at a uniform distribution, and also that we are looking at a continuous probability function. What would this probability function look like? As it turns out, this function will look like a straight line from our start to end points. We can calculate the area under this function to get our desired probability.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$74$$"],"dependencies":["a536a53Uni19a-h1"],"title":"Interval","text":"If X goes from $$447$$ to $$521$$, how long is the interval of X?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni19a-h3","type":"hint","dependencies":["a536a53Uni19a-h2"],"title":"Probability $$1$$","text":"The probability that the duration of MLB games in a season is between $$447$$ and $$448$$ is $$\\\\frac{1}{74}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni19a-h4","type":"hint","dependencies":["a536a53Uni19a-h3"],"title":"Probability $$2$$","text":"The probability that the duration of MLB games in a season is between $$447$$ and $$449$$ is $$\\\\frac{2}{74}$$. Starting to see a pattern? We can get the length in the interval we want, and divide that by the length of our total interval to get our answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni19a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{50}{74}$$"],"dependencies":["a536a53Uni19a-h4"],"title":"Combining","text":"Knowing how to calculate the probability, what will our answer be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a536a53Uni2","title":"Uniform Distributions and Their Properties","body":"Suppose that we are examining the smiling times of an eight-week-old baby (measured in seconds). The smiling times follow a uniform distribution between zero and $$23$$ seconds, meaning that any smiling time from zero to $$23$$ seconds is equally likely.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.2 The Uniform Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a536a53Uni2a","stepAnswer":["$$6.64$$"],"problemType":"TextBox","stepTitle":"What is the standard deviation of the amount of time that the baby is smiling?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6.64$$","hints":{"DefaultPathway":[{"id":"a536a53Uni2a-h1","type":"hint","dependencies":[],"title":"Standard Deviation Formula","text":"The formula for calculating the standard deviation of a uniform distribution is $$\\\\sqrt{\\\\frac{{\\\\left(b-a\\\\right)}^2}{12}}$$, where a and $$b$$ are the $$\\\\frac{start}{end}$$ times of our interval respectively.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a536a53Uni2b","stepAnswer":["$$6.64$$"],"problemType":"TextBox","stepTitle":"Knowing the formula to calculate the standard deviation, what is our final answer?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6.64$$","hints":{"DefaultPathway":[]}}]},{"id":"a536a53Uni20","title":"Uniform Distributions and Their Properties","body":"The total duration of baseball games in the major league in the $$2011$$ season is uniformly distributed between $$447$$ hours and $$521$$ hours inclusive.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.2 The Uniform Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a536a53Uni20a","stepAnswer":["$$\\\\frac{46}{74}$$"],"problemType":"TextBox","stepTitle":"What is the probability that the duration of games for a team for the $$2011$$ season is between $$465$$ and $$511$$ hours?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{46}{74}$$","hints":{"DefaultPathway":[{"id":"a536a53Uni20a-h1","type":"hint","dependencies":[],"title":"Probability Function","text":"We know that we are looking at a uniform distribution, and also that we are looking at a continuous probability function. What would this probability function look like? As it turns out, this function will look like a straight line from our start to end points. We can calculate the area under this function to get our desired probability.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$74$$"],"dependencies":["a536a53Uni20a-h1"],"title":"Interval","text":"If X goes from $$447$$ to $$521$$, how long is the interval of X?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni20a-h3","type":"hint","dependencies":["a536a53Uni20a-h2"],"title":"Probability $$1$$","text":"The probability that the duration of MLB games in a season is between $$447$$ and $$448$$ is $$\\\\frac{1}{74}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni20a-h4","type":"hint","dependencies":["a536a53Uni20a-h3"],"title":"Probability $$2$$","text":"The probability that the duration of MLB games in a season is between $$447$$ and $$449$$ is $$\\\\frac{2}{74}$$. Starting to see a pattern? We can get the length in the interval we want, and divide that by the length of our total interval to get our answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni20a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{46}{74}$$"],"dependencies":["a536a53Uni20a-h4"],"title":"Combining","text":"Knowing how to calculate the probability, what will our answer be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a536a53Uni3","title":"Uniform Distributions and Their Properties","body":"Suppose that we are examining the smiling times of an eight-week-old baby (measured in seconds). The smiling times follow a uniform distribution between zero and $$23$$ seconds, meaning that any smiling time from zero to $$23$$ seconds is equally likely.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.2 The Uniform Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a536a53Uni3a","stepAnswer":["$$\\\\frac{16}{23}$$"],"problemType":"TextBox","stepTitle":"What is the probability that the eight-year-old baby smiles between two and $$18$$ seconds?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{16}{23}$$","hints":{"DefaultPathway":[{"id":"a536a53Uni3a-h1","type":"hint","dependencies":[],"title":"Probability Function","text":"We know that we are looking at a uniform distribution, and also that we are looking at a continuous probability function. What would this probability function look like? As it turns out, this function will look like a straight line from $$0$$ to $$23$$. We can calculate the area under this function to get our desired probability.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$23$$"],"dependencies":["a536a53Uni3a-h1"],"title":"Interval","text":"If the baby can smile from anywhere between $$0$$ and $$23$$ seconds, how long is the interval (in seconds) in which a baby can smile?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni3a-h3","type":"hint","dependencies":["a536a53Uni3a-h2"],"title":"Chances","text":"There is a $$\\\\frac{1}{23}$$ chance that the baby smiles between $$2$$ and $$3$$ seconds long.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni3a-h4","type":"hint","dependencies":["a536a53Uni3a-h3"],"title":"Chances","text":"There is a $$\\\\frac{2}{23}$$ chance that the baby smiles between $$2$$ and $$4$$ seconds long. Starting to see a pattern? We can count the total number of seconds in the interval we want (2 to $$4$$ seconds has a $$2$$ second interval), and divide that by our total interval (0 to $$23$$ seconds has a $$23$$ second interval) to get our answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{16}{23}$$"],"dependencies":["a536a53Uni3a-h4"],"title":"Combining","text":"Knowing how to calculate the probability, what will our answer be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a536a53Uni4","title":"Uniform Distributions and Their Properties","body":"Suppose that we are examining the smiling times of an eight-week-old baby (measured in seconds). The smiling times follow a uniform distribution between zero and $$23$$ seconds, meaning that any smiling time from zero to $$23$$ seconds is equally likely.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.2 The Uniform Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a536a53Uni4a","stepAnswer":["$$\\\\frac{19}{23}$$"],"problemType":"TextBox","stepTitle":"What is the probability that the eight-year-old baby smiles between $$1$$ and $$20$$ seconds?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{19}{23}$$","hints":{"DefaultPathway":[{"id":"a536a53Uni4a-h1","type":"hint","dependencies":[],"title":"Probability Function","text":"We know that we are looking at a uniform distribution, and also that we are looking at a continuous probability function. What would this probability function look like? As it turns out, this function will look like a straight line from $$0$$ to $$23$$. We can calculate the area under this function to get our desired probability.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$23$$"],"dependencies":["a536a53Uni4a-h1"],"title":"Interval","text":"If the baby can smile from anywhere between $$0$$ and $$23$$ seconds, how long is the interval (in seconds) in which a baby can smile?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni4a-h3","type":"hint","dependencies":["a536a53Uni4a-h2"],"title":"Chances","text":"There is a $$\\\\frac{1}{23}$$ chance that the baby smiles between $$2$$ and $$3$$ seconds long.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni4a-h4","type":"hint","dependencies":["a536a53Uni4a-h3"],"title":"Chances","text":"There is a $$\\\\frac{2}{23}$$ chance that the baby smiles between $$2$$ and $$4$$ seconds long. Starting to see a pattern? We can count the total number of seconds in the interval we want (2 to $$4$$ seconds has a $$2$$ second interval), and divide that by our total interval (0 to $$23$$ seconds has a $$23$$ second interval) to get our answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni4a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{19}{23}$$"],"dependencies":["a536a53Uni4a-h4"],"title":"Combining","text":"Knowing how to calculate the probability, what will our answer be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a536a53Uni5","title":"Uniform Distributions and Their Properties","body":"Suppose that we are examining the smiling times of an eight-week-old baby (measured in seconds). The smiling times follow a uniform distribution between zero and $$23$$ seconds, meaning that any smiling time from zero to $$23$$ seconds is equally likely.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.2 The Uniform Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a536a53Uni5a","stepAnswer":["$$\\\\frac{5}{23}$$"],"problemType":"TextBox","stepTitle":"What is the probability that the eight-year-old baby smiles between $$5.5$$ and $$10.5$$ seconds?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5}{23}$$","hints":{"DefaultPathway":[{"id":"a536a53Uni5a-h1","type":"hint","dependencies":[],"title":"Probability Function","text":"We know that we are looking at a uniform distribution, and also that we are looking at a continuous probability function. What would this probability function look like? As it turns out, this function will look like a straight line from $$0$$ to $$23$$. We can calculate the area under this function to get our desired probability.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$23$$"],"dependencies":["a536a53Uni5a-h1"],"title":"Interval","text":"If the baby can smile from anywhere between $$0$$ and $$23$$ seconds, how long is the interval (in seconds) in which a baby can smile?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni5a-h3","type":"hint","dependencies":["a536a53Uni5a-h2"],"title":"Chances","text":"There is a $$\\\\frac{1}{23}$$ chance that the baby smiles between $$2$$ and $$3$$ seconds long.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni5a-h4","type":"hint","dependencies":["a536a53Uni5a-h3"],"title":"Chances","text":"There is a $$\\\\frac{2}{23}$$ chance that the baby smiles between $$2$$ and $$4$$ seconds long. Starting to see a pattern? We can count the total number of seconds in the interval we want (2 to $$4$$ seconds has a $$2$$ second interval), and divide that by our total interval (0 to $$23$$ seconds has a $$23$$ second interval) to get our answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{23}$$"],"dependencies":["a536a53Uni5a-h4"],"title":"Combining","text":"Knowing how to calculate the probability, what will our answer be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a536a53Uni6","title":"Uniform Distributions and Their Properties","body":"A distribution is given as X ~ U $$(0,20)$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.2 The Uniform Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a536a53Uni6a","stepAnswer":["$$10$$"],"problemType":"TextBox","stepTitle":"What is the mean of X?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10$$","hints":{"DefaultPathway":[{"id":"a536a53Uni6a-h1","type":"hint","dependencies":[],"title":"Mean Formula","text":"The formula for calculating the mean of a uniform distribution is $$\\\\frac{a+b}{2}$$, where a and $$b$$ are the $$\\\\frac{start}{end}$$ times of our interval respectively.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a536a53Uni6b","stepAnswer":["$$10$$"],"problemType":"TextBox","stepTitle":"Knowing the formula to calculate the mean, what is our final answer?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10$$","hints":{"DefaultPathway":[]}}]},{"id":"a536a53Uni7","title":"Uniform Distributions and Their Properties","body":"A distribution is given as X ~ U $$(0,20)$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.2 The Uniform Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a536a53Uni7a","stepAnswer":["$$5.774$$"],"problemType":"TextBox","stepTitle":"What is the standard deviation of X?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5.774$$","hints":{"DefaultPathway":[{"id":"a536a53Uni7a-h1","type":"hint","dependencies":[],"title":"Mean Formula","text":"The formula for calculating the standard deviation of a uniform distribution is $$\\\\sqrt{\\\\frac{{\\\\left(b-a\\\\right)}^2}{12}}$$, where a and $$b$$ are the $$\\\\frac{start}{end}$$ times of our interval respectively.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a536a53Uni7b","stepAnswer":["$$5.774$$"],"problemType":"TextBox","stepTitle":"Knowing the formula to calculate the standard deviation, what is our final answer?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5.774$$","hints":{"DefaultPathway":[]}}]},{"id":"a536a53Uni8","title":"Uniform Distributions and Their Properties","body":"A distribution is given as X ~ U $$(0,20)$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.2 The Uniform Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a536a53Uni8a","stepAnswer":["$$\\\\frac{1}{2}$$"],"problemType":"TextBox","stepTitle":"What is $$P\\\\left(0<x<10\\\\right)$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{2}$$","hints":{"DefaultPathway":[{"id":"a536a53Uni8a-h1","type":"hint","dependencies":[],"title":"Probability Function","text":"We know that we are looking at a uniform distribution, and also that we are looking at a continuous probability function. What would this probability function look like? As it turns out, this function will look like a straight line from our start to end points. We can calculate the area under this function to get our desired probability.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a536a53Uni8a-h1"],"title":"Interval","text":"If X goes from $$0$$ to $$20$$, how long is the interval of X?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni8a-h3","type":"hint","dependencies":["a536a53Uni8a-h2"],"title":"Probability $$1$$","text":"$$P\\\\left(0<x<1\\\\right)$$ $$=$$ $$\\\\frac{1}{20}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni8a-h4","type":"hint","dependencies":["a536a53Uni8a-h3"],"title":"Probability $$2$$","text":"$$P\\\\left(0<x<2\\\\right)$$ $$=$$ $$\\\\frac{2}{20}$$. Starting to see a pattern? We can get the length in the interval we want, and divide that by the length of our total interval to get our answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni8a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a536a53Uni8a-h4"],"title":"Combining","text":"Knowing how to calculate the probability, what will our answer be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a536a53Uni9","title":"Uniform Distributions and Their Properties","body":"A distribution is given as X ~ U $$(0,20)$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.2 The Uniform Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a536a53Uni9a","stepAnswer":["$$\\\\frac{16}{20}$$"],"problemType":"TextBox","stepTitle":"What is $$P\\\\left(1<x<17\\\\right)$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{16}{20}$$","hints":{"DefaultPathway":[{"id":"a536a53Uni9a-h1","type":"hint","dependencies":[],"title":"Probability Function","text":"We know that we are looking at a uniform distribution, and also that we are looking at a continuous probability function. What would this probability function look like? As it turns out, this function will look like a straight line from our start to end points. We can calculate the area under this function to get our desired probability.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a536a53Uni9a-h1"],"title":"Interval","text":"If X goes from $$0$$ to $$20$$, how long is the interval of X?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni9a-h3","type":"hint","dependencies":["a536a53Uni9a-h2"],"title":"Probability $$1$$","text":"$$P\\\\left(0<x<1\\\\right)$$ $$=$$ $$\\\\frac{1}{20}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni9a-h4","type":"hint","dependencies":["a536a53Uni9a-h3"],"title":"Probability $$2$$","text":"$$P\\\\left(0<x<2\\\\right)$$ $$=$$ $$\\\\frac{2}{20}$$. Starting to see a pattern? We can get the length in the interval we want, and divide that by the length of our total interval to get our answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a536a53Uni9a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{16}{20}$$"],"dependencies":["a536a53Uni9a-h4"],"title":"Combining","text":"Knowing how to calculate the probability, what will our answer be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a53b893whole1","title":"How to Round Whole Numbers","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Introduction to Whole Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a53b893whole1a","stepAnswer":["$$23700$$"],"problemType":"TextBox","stepTitle":"Round 23,658 to the nearest hundred.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$23700$$","hints":{"DefaultPathway":[{"id":"a53b893whole1a-h1","type":"hint","dependencies":[],"title":"Locate","text":"Locate the hundreds place in the number","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a53b893whole1a-h1"],"title":"Value","text":"What is the value of the hundreds place","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole1a-h3","type":"hint","dependencies":["a53b893whole1a-h2"],"title":"Next Value","text":"Check the next value to the right of the indicated hundreds number","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole1a-h4","type":"hint","dependencies":["a53b893whole1a-h3"],"title":"Rule","text":"If the number to the right of the hundreds number is greater than or equal to $$5$$ then add one to the hundreds place number and make everything after to 0s. If it is less than $$5$$, then leave the hundreds place number the same and change everything after to 0s.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a53b893whole10","title":"How to Find the Prime Factorization of a Composite Number","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Introduction to Whole Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a53b893whole10a","stepAnswer":["$$2\\\\times2\\\\times2\\\\times2\\\\times3$$"],"problemType":"MultipleChoice","stepTitle":"Factor $$48$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2\\\\times2\\\\times2\\\\times2\\\\times3$$","choices":["$$2\\\\times2\\\\times2\\\\times3$$","$$2\\\\times2\\\\times2\\\\times3\\\\times5$$","$$2\\\\times2\\\\times2\\\\times2\\\\times3$$","$$2\\\\times3\\\\times3\\\\times3\\\\times3$$"],"hints":{"DefaultPathway":[{"id":"a53b893whole10a-h1","type":"hint","dependencies":[],"title":"Finding Two Factors Whose Product is the Given Number","text":"The first step is to find two factors whose product is the given number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole10a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole10a-h1"],"title":"Finding Two Factors Whose Product is the Given Number","text":"Are $$24$$ and $$2$$ factors of $$48$$ that multiply together to make 48?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole10a-h3","type":"hint","dependencies":["a53b893whole10a-h2"],"title":"Prime Number Factors","text":"If a factor is prime, that means it can\'t be divided further. Thus, it is a prime factor of the number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole10a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole10a-h3"],"title":"Verifying if a Factor is Prime","text":"Is $$2$$ prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole10a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a53b893whole10a-h3"],"title":"Verifying if a Factor is Prime","text":"Is $$24$$ prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole10a-h6","type":"hint","dependencies":["a53b893whole10a-h4","a53b893whole10a-h5"],"title":"Proceeding With a Factor that is Not Prime","text":"The next step is to divide factors that are not prime into two more factors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole10a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole10a-h6"],"title":"Finding Two Factors Whose Product is the Given Number","text":"Are $$4$$ and $$6$$ factors of $$24$$ that multiply together to make 24?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole10a-h8","type":"hint","dependencies":["a53b893whole10a-h7"],"title":"Proceeding With a Factor that is Not Prime","text":"Continue to divide factors that are not prime into two more factors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole10a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole10a-h8"],"title":"Finding Two Factors Whose Product is the Given Number","text":"Are $$2$$ and $$2$$ factors of $$4$$ that multiply together to make 4?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole10a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole10a-h9"],"title":"Finding Two Factors Whose Product is the Given Number","text":"Are $$2$$ and $$3$$ factors of $$6$$ that multiply together to make 6?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole10a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole10a-h10"],"title":"Verifying if a Factor is Prime","text":"Are both $$2$$ and $$3$$ prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole10a-h12","type":"hint","dependencies":["a53b893whole10a-h11"],"title":"Final Step","text":"The final step is to write the composite number as the product of all the prime factors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole10a-h13","type":"hint","dependencies":["a53b893whole10a-h12"],"title":"Answer","text":"The found prime factors of $$48$$ were $$2$$, $$2$$, $$2$$, $$2$$, and $$3$$. Thus $$2\\\\times2\\\\times2\\\\times2\\\\times3=48$$, and $$2\\\\times2\\\\times2\\\\times2\\\\times3$$ is the prime factorization of $$48$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a53b893whole11","title":"Finding the Prime Factorization of a Composite Number","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Introduction to Whole Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a53b893whole11a","stepAnswer":["$$2\\\\times2\\\\times3\\\\times3\\\\times7$$"],"problemType":"MultipleChoice","stepTitle":"What is the factorization of 252?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2\\\\times2\\\\times3\\\\times3\\\\times7$$","choices":["$$2\\\\times2\\\\times3\\\\times3\\\\times7$$","$$2\\\\times2\\\\times3\\\\times5\\\\times7$$","$$2\\\\times3\\\\times3\\\\times7\\\\times7$$","$$2\\\\times3\\\\times3\\\\times3\\\\times7$$"],"hints":{"DefaultPathway":[{"id":"a53b893whole11a-h1","type":"hint","dependencies":[],"title":"Finding Two Factors Whose Product is the Given Number","text":"The first step is to find two factors whose product is the given number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole11a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole11a-h1"],"title":"Finding Two Factors Whose Product is the Given Number","text":"Are $$12$$ and $$21$$ factors of $$252$$ that multiply together to make 252?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole11a-h3","type":"hint","dependencies":["a53b893whole11a-h2"],"title":"Prime Number Factors","text":"If a factor is prime, that means it can\'t be divided further. Thus, it is a prime factor of the number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole11a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a53b893whole11a-h3"],"title":"Verifying if a Factor is Prime","text":"Is $$12$$ prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole11a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a53b893whole11a-h4"],"title":"Verifying if a Factor is Prime","text":"Is $$21$$ prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole11a-h6","type":"hint","dependencies":["a53b893whole11a-h5"],"title":"Proceeding With a Factor that is Not Prime","text":"The next step is to divide factors that are not prime into two more factors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole11a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole11a-h6"],"title":"Finding Two Factors Whose Product is the Given Number","text":"Are $$2$$ and $$6$$ factors of $$12$$ that multiply together to make 12?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole11a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole11a-h7"],"title":"Verifying if a Factor is Prime","text":"Is $$2$$ prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole11a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a53b893whole11a-h8"],"title":"Verifying if a Factor is Prime","text":"Is $$6$$ prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole11a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole11a-h9"],"title":"Finding Two Factors Whose Product is the Given Number","text":"Are $$3$$ and $$7$$ factors of $$21$$ that multiply together to make 21?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole11a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole11a-h10"],"title":"Verifying if a Factor is Prime","text":"Is $$3$$ prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole11a-h12","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole11a-h11"],"title":"Verifying if a Factor is Prime","text":"Is $$7$$ prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole11a-h13","type":"hint","dependencies":["a53b893whole11a-h12"],"title":"Proceeding With a Factor that is Not Prime","text":"Continue to divide factors that are not prime into two more factors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole11a-h14","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole11a-h13"],"title":"Finding Two Factors Whose Product is the Given Number","text":"Are $$2$$ and $$3$$ factors of $$6$$ that multiply together to make 6?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole11a-h15","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole11a-h14"],"title":"Verifying if a Factor is Prime","text":"Are both $$2$$ and $$3$$ prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole11a-h16","type":"hint","dependencies":["a53b893whole11a-h15"],"title":"Final Step","text":"The final step is to write the composite number as the product of all the prime factors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole11a-h17","type":"hint","dependencies":["a53b893whole11a-h16"],"title":"Answer","text":"The found prime factors of $$252$$ were 2,2,3,3, and $$7$$. and $$3$$. Thus $$2\\\\times2\\\\times3\\\\times3\\\\times7=252$$, and $$2\\\\times2\\\\times3\\\\times3\\\\times7$$ is the prime factorization of $$252$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a53b893whole12","title":"Finding the Least Common Multiple By Listing Multiples","body":"Find the least common multiple of the two numbers by listing their multiples.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Introduction to Whole Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a53b893whole12a","stepAnswer":["$$60$$"],"problemType":"TextBox","stepTitle":"$$15$$ and $$20$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$60$$","hints":{"DefaultPathway":[{"id":"a53b893whole12a-h1","type":"hint","dependencies":[],"title":"Listing Multiples of $$15$$ and $$20$$","text":"The first step is to list the multiples of $$15$$ and $$20$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole12a-h2","type":"hint","dependencies":["a53b893whole12a-h1"],"title":"Multiples of $$15$$","text":"Multiples of $$15$$ are $$15$$, $$30$$, $$45$$, $$60$$, $$75$$, $$90$$, $$105$$, $$120..$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole12a-h3","type":"hint","dependencies":["a53b893whole12a-h2"],"title":"Multiples of $$20$$","text":"Multiples of $$20$$ are $$20$$, $$40$$, $$60$$, $$80$$, $$100$$, $$120$$, $$140$$, $$160..$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole12a-h4","type":"hint","dependencies":["a53b893whole12a-h3"],"title":"Finding the Least Common Multiple By Comparing the Lists","text":"The smallest number that appears on both lists is the least common multiple","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole12a-h5","type":"hint","dependencies":["a53b893whole12a-h4"],"title":"Smallest Number on Both Lists","text":"$$60$$ is the smallest number that appears on both of the lists.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a53b893whole13","title":"Finding the Least Common Multiple Using the Prime Factors Method","body":"Find the least common multiple of the two numbers using the prime factors method.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Introduction to Whole Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a53b893whole13a","stepAnswer":["$$36$$"],"problemType":"TextBox","stepTitle":"$$12$$ and $$18$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$36$$","hints":{"DefaultPathway":[{"id":"a53b893whole13a-h1","type":"hint","dependencies":[],"title":"Writing Each Number as a Product of Primes","text":"Refer to the image for the first step.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole13a-h2","type":"hint","dependencies":["a53b893whole13a-h1"],"title":"Listing the Primes of Each Number","text":"Refer to the image for the second step.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole13a-h3","type":"hint","dependencies":["a53b893whole13a-h2"],"title":"Bringing Down the Number From Each Column","text":"Refer to the image for the third step.\\\\n##figure3.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole13a-h4","type":"hint","dependencies":["a53b893whole13a-h3"],"title":"Multiplying the Factors","text":"Refer to the image for the fourth step.\\\\n##figure4.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a53b893whole14","title":"Find the Prime Factorization of a Composite Number","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Introduction to Whole Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a53b893whole14a","stepAnswer":["$$2\\\\times2\\\\times2\\\\times2\\\\times5$$"],"problemType":"MultipleChoice","stepTitle":"Factor $$80$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2\\\\times2\\\\times2\\\\times2\\\\times5$$","choices":["$$2\\\\times2\\\\times2\\\\times3\\\\times5$$","$$2\\\\times2\\\\times3\\\\times3\\\\times5$$","$$2\\\\times2\\\\times2\\\\times5\\\\times5$$","$$2\\\\times2\\\\times2\\\\times2\\\\times5$$"],"hints":{"DefaultPathway":[{"id":"a53b893whole14a-h1","type":"hint","dependencies":[],"title":"Finding Two Factors Whose Product is the Given Number","text":"The first step is to find two factors whose product is the given number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole14a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole14a-h1"],"title":"Finding Two Factors Whose Product is the Given Number","text":"Are $$2$$ and $$40$$ factors of $$80$$ that multiply together to make 80?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole14a-h3","type":"hint","dependencies":["a53b893whole14a-h2"],"title":"Prime Number Factors","text":"If a factor is prime, that means it can\'t be divided further. Thus, it is a prime factor of the number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole14a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole14a-h3"],"title":"Verifying if a Factor is Prime","text":"Is $$2$$ prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole14a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a53b893whole14a-h4"],"title":"Verifying if a Factor is Prime","text":"Is $$40$$ prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole14a-h6","type":"hint","dependencies":["a53b893whole14a-h5"],"title":"Proceeding With a Factor that is Not Prime","text":"The next step is to divide factors that are not prime into two more factors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole14a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole14a-h6"],"title":"Finding Two Factors Whose Product is the Given Number","text":"Are $$2$$ and $$20$$ factors of $$40$$ that multiply together to make 40?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole14a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole14a-h7"],"title":"Verifying if a Factor is Prime","text":"Is $$2$$ prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole14a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a53b893whole14a-h8"],"title":"Verifying if a Factor is Prime","text":"Is $$20$$ prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole14a-h10","type":"hint","dependencies":["a53b893whole14a-h9"],"title":"Proceeding With a Factor that is Not Prime","text":"Continue to divide factors that are not prime into two more factors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole14a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole14a-h10"],"title":"Finding Two Factors Whose Product is the Given Number","text":"Are $$2$$ and $$10$$ factors of $$20$$ that multiply together to make 20?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole14a-h12","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole14a-h11"],"title":"Verifying if a Factor is Prime","text":"Is $$2$$ prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole14a-h13","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a53b893whole14a-h12"],"title":"Finding Two Factors Whose Product is the Given Number","text":"Is $$10$$ prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole14a-h14","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole14a-h13"],"title":"Finding Two Factors Whose Product is the Given Number","text":"Are $$2$$ and $$5$$ factors of $$10$$ that multiply together to make 10?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole14a-h15","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole14a-h14"],"title":"Verifying if a Factor is Prime","text":"Is $$2$$ prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole14a-h16","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole14a-h15"],"title":"Verifying if a Factor is Prime","text":"Is $$5$$ prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole14a-h17","type":"hint","dependencies":["a53b893whole14a-h16"],"title":"Final Step","text":"The final step is to write the composite number as the product of all the prime factors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole14a-h18","type":"hint","dependencies":["a53b893whole14a-h17"],"title":"Answer","text":"The found prime factors of $$80$$ were $$2$$, $$2$$, $$2$$, $$2$$, and $$5$$. Thus $$2\\\\times2\\\\times2\\\\times2\\\\times5=80$$, and $$2\\\\times2\\\\times2\\\\times2\\\\times5$$ is the prime factorization of $$80$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a53b893whole15","title":"Find the Prime Factorization of a Composite Number","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Introduction to Whole Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a53b893whole15a","stepAnswer":["$$2\\\\times2\\\\times3\\\\times5$$"],"problemType":"MultipleChoice","stepTitle":"Factor $$60$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2\\\\times2\\\\times3\\\\times5$$","choices":["$$2\\\\times3\\\\times3\\\\times5$$","$$2\\\\times2\\\\times3\\\\times5$$","$$2\\\\times2\\\\times3\\\\times7$$","$$2\\\\times3\\\\times5\\\\times7$$"],"hints":{"DefaultPathway":[{"id":"a53b893whole15a-h1","type":"hint","dependencies":[],"title":"Finding Two Factors Whose Product is the Given Number","text":"The first step is to find two factors whose product is the given number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole15a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole15a-h1"],"title":"Finding Two Factors Whose Product is the Given Number","text":"Are $$2$$ and $$30$$ factors of 60?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole15a-h3","type":"hint","dependencies":["a53b893whole15a-h2"],"title":"Prime Number Factors","text":"If a factor is prime, that means it can\'t be divided further. Thus, it is a prime factor of the number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole15a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole15a-h3"],"title":"Verifying if a Factor is Prime","text":"Is $$2$$ prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole15a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a53b893whole15a-h4"],"title":"Verifying if a Factor is Prime","text":"Is $$30$$ prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole15a-h6","type":"hint","dependencies":["a53b893whole15a-h5"],"title":"Proceeding With a Factor that is Not Prime","text":"The next step is to divide factors that are not prime into two more factors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole15a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole15a-h6"],"title":"Finding Two Factors Whose Product is the Given Number","text":"Are $$2$$ and $$15$$ factors of 30?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole15a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole15a-h7"],"title":"Verifying if a Factor is Prime","text":"Is $$2$$ prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole15a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a53b893whole15a-h8"],"title":"Verifying if a Factor is Prime","text":"Is $$15$$ prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole15a-h10","type":"hint","dependencies":["a53b893whole15a-h9"],"title":"Proceeding With a Factor that is Not Prime","text":"Continue to divide factors that are not prime into two more factors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole15a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole15a-h10"],"title":"Finding Two Factors Whose Product is the Given Number","text":"Are $$3$$ and $$5$$ factors of 15?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole15a-h12","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole15a-h11"],"title":"Verifying if a Factor is Prime","text":"Is $$3$$ prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole15a-h13","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole15a-h12"],"title":"Finding Two Factors Whose Product is the Given Number","text":"Is $$5$$ prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole15a-h14","type":"hint","dependencies":["a53b893whole15a-h13"],"title":"Final Step","text":"The final step is to write the composite number as the product of all the prime factors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole15a-h15","type":"hint","dependencies":["a53b893whole15a-h14"],"title":"Answer","text":"The found prime factors of $$60$$ were $$2$$, $$2$$, $$3$$, and $$5$$. Thus $$2\\\\times2\\\\times3\\\\times5=60$$, and $$2\\\\times2\\\\times3\\\\times5$$ is the factorization of $$60$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a53b893whole16","title":"Find the Prime Factorization of a Composite Number","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Introduction to Whole Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a53b893whole16a","stepAnswer":["$$2\\\\times3\\\\times3\\\\times7$$"],"problemType":"MultipleChoice","stepTitle":"Factor $$126$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2\\\\times3\\\\times3\\\\times7$$","choices":["$$3\\\\times3\\\\times3\\\\times7$$","$$2\\\\times3\\\\times3\\\\times7$$","$$2\\\\times3\\\\times5\\\\times7$$","$$2\\\\times63$$"],"hints":{"DefaultPathway":[{"id":"a53b893whole16a-h1","type":"hint","dependencies":[],"title":"Finding Two Factors Whose Product is the Given Number","text":"The first step is to find two factors whose product is the given number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole16a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole16a-h1"],"title":"Finding Two Factors Whose Product is the Given Number","text":"Are $$2$$ and $$63$$ factors of 60?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole16a-h3","type":"hint","dependencies":["a53b893whole16a-h2"],"title":"Prime Number Factors","text":"If a factor is prime, that means it can\'t be divided further. Thus, it is a prime factor of the number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole16a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole16a-h3"],"title":"Verifying if a Factor is Prime","text":"Is $$2$$ prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole16a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a53b893whole16a-h4"],"title":"Verifying if a Factor is Prime","text":"Is $$63$$ prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole16a-h6","type":"hint","dependencies":["a53b893whole16a-h5"],"title":"Proceeding With a Factor that is Not Prime","text":"The next step is to divide factors that are not prime into two more factors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole16a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole16a-h6"],"title":"Finding Two Factors Whose Product is the Given Number","text":"Are $$7$$ and $$9$$ factors of 63?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole16a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole16a-h7"],"title":"Verifying if a Factor is Prime","text":"Is $$7$$ prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole16a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a53b893whole16a-h8"],"title":"Verifying if a Factor is Prime","text":"Is $$9$$ prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole16a-h10","type":"hint","dependencies":["a53b893whole16a-h9"],"title":"Proceeding With a Factor that is Not Prime","text":"Continue to divide factors that are not prime into two more factors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole16a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole16a-h10"],"title":"Finding Two Factors Whose Product is the Given Number","text":"Are $$3$$ and $$3$$ factors of 9?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole16a-h12","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole16a-h11"],"title":"Verifying if a Factor is Prime","text":"Is $$3$$ prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole16a-h13","type":"hint","dependencies":["a53b893whole16a-h12"],"title":"Final Step","text":"The final step is to write the composite number as the product of all the prime factors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole16a-h14","type":"hint","dependencies":["a53b893whole16a-h13"],"title":"Answer","text":"The found prime factors of $$126$$ were 2,7,3, and $$3$$. Thus $$2\\\\times3\\\\times3\\\\times7=126$$, and $$2\\\\times3\\\\times3\\\\times7$$ is the factorization of $$126$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a53b893whole17","title":"Find the Prime Factorization of a Composite Number","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Introduction to Whole Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a53b893whole17a","stepAnswer":["$$2\\\\times3\\\\times7\\\\times7$$"],"problemType":"MultipleChoice","stepTitle":"Factor $$294$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2\\\\times3\\\\times7\\\\times7$$","choices":["$$3\\\\times3\\\\times7\\\\times7$$","$$3\\\\times3\\\\times3\\\\times7$$","$$2\\\\times3\\\\times7\\\\times7$$","$$3\\\\times3\\\\times5\\\\times7$$"],"hints":{"DefaultPathway":[{"id":"a53b893whole17a-h1","type":"hint","dependencies":[],"title":"Finding Two Factors Whose Product is the Given Number","text":"The first step is to find two factors whose product is the given number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole17a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole17a-h1"],"title":"Finding Two Factors Whose Product is the Given Number","text":"Are $$2$$ and $$147$$ factors of 294?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole17a-h3","type":"hint","dependencies":["a53b893whole17a-h2"],"title":"Prime Number Factors","text":"If a factor is prime, that means it can\'t be divided further. Thus, it is a prime factor of the number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole17a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole17a-h3"],"title":"Verifying if a Factor is Prime","text":"Is $$2$$ prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole17a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a53b893whole17a-h4"],"title":"Verifying if a Factor is Prime","text":"Is $$147$$ prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole17a-h6","type":"hint","dependencies":["a53b893whole17a-h5"],"title":"Proceeding With a Factor that is Not Prime","text":"The next step is to divide factors that are not prime into two more factors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole17a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole17a-h6"],"title":"Finding Two Factors Whose Product is the Given Number","text":"Are $$3$$ and $$49$$ factors of 147?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole17a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole17a-h7"],"title":"Verifying if a Factor is Prime","text":"Is $$3$$ prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole17a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a53b893whole17a-h8"],"title":"Verifying if a Factor is Prime","text":"Is $$49$$ prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole17a-h10","type":"hint","dependencies":["a53b893whole17a-h9"],"title":"Proceeding With a Factor that is Not Prime","text":"Continue to divide factors that are not prime into two more factors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole17a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole17a-h10"],"title":"Finding Two Factors Whose Product is the Given Number","text":"Are $$7$$ and $$7$$ factors of 49?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole17a-h12","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole17a-h11"],"title":"Verifying if a Factor is Prime","text":"Is $$7$$ prime?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole17a-h13","type":"hint","dependencies":["a53b893whole17a-h12"],"title":"Final Step","text":"The final step is to write the composite number as the product of all the prime factors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole17a-h14","type":"hint","dependencies":["a53b893whole17a-h13"],"title":"Answer","text":"The found prime factors of $$294$$ were 2,3,7, and $$7$$. Thus $$2\\\\times3\\\\times7\\\\times7=294$$, and $$2\\\\times3\\\\times7\\\\times7$$ is the factorization of $$294$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a53b893whole18","title":"Finding the Least Common Multiple By Listing Multiples","body":"Find the least common multiple of the two numbers by listing their multiples.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Introduction to Whole Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a53b893whole18a","stepAnswer":["$$36$$"],"problemType":"TextBox","stepTitle":"$$9$$ and $$12$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$36$$","hints":{"DefaultPathway":[{"id":"a53b893whole18a-h1","type":"hint","dependencies":[],"title":"Listing Multiples of $$15$$ and $$20$$","text":"The first step is to list the multiples of $$9$$ and $$12$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole18a-h2","type":"hint","dependencies":["a53b893whole18a-h1"],"title":"Multiples of $$9$$","text":"Multiples of $$9$$ are $$9$$, $$18$$, $$27$$, $$36$$, $$45$$, $$54$$, $$63..$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole18a-h3","type":"hint","dependencies":["a53b893whole18a-h2"],"title":"Multiples of $$12$$","text":"Multiples of $$12$$ are $$12$$, $$24$$, $$36$$, $$48$$, $$60$$, $$72$$, $$84..$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole18a-h4","type":"hint","dependencies":["a53b893whole18a-h3"],"title":"Finding the Least Common Multiple By Comparing the Lists","text":"The smallest number that appears on both lists is the least common multiple","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole18a-h5","type":"hint","dependencies":["a53b893whole18a-h4"],"title":"Smallest Number on Both Lists","text":"$$36$$ is the smallest number that appears on both of the lists.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a53b893whole19","title":"Finding the Least Common Multiple By Listing Multiples","body":"Find the least common multiple of the two numbers by listing their multiples.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Introduction to Whole Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a53b893whole19a","stepAnswer":["$$72$$"],"problemType":"TextBox","stepTitle":"$$18$$ and $$24$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$72$$","hints":{"DefaultPathway":[{"id":"a53b893whole19a-h1","type":"hint","dependencies":[],"title":"Listing Multiples of $$15$$ and $$20$$","text":"The first step is to list the multiples of $$18$$ and $$24$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole19a-h2","type":"hint","dependencies":["a53b893whole19a-h1"],"title":"Multiples of $$18$$","text":"Multiples of $$18$$ are $$18$$, $$36$$, $$54$$, $$72$$, $$90$$, $$108$$, $$126..$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole19a-h3","type":"hint","dependencies":["a53b893whole19a-h2"],"title":"Multiples of $$24$$","text":"Multiples of $$24$$ are $$24$$, $$48$$, $$72$$, $$96$$, $$120$$, $$144$$, $$168..$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole19a-h4","type":"hint","dependencies":["a53b893whole19a-h3"],"title":"Finding the Least Common Multiple By Comparing the Lists","text":"The smallest number that appears on both lists is the least common multiple","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole19a-h5","type":"hint","dependencies":["a53b893whole19a-h4"],"title":"Smallest Number on Both Lists","text":"$$72$$ is the smallest number that appears on both of the lists.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a53b893whole2","title":"How to Round Whole Numbers","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Introduction to Whole Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a53b893whole2a","stepAnswer":["$$104000$$"],"problemType":"TextBox","stepTitle":"Round 103,978 to the nearest hundred","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$104000$$","hints":{"DefaultPathway":[{"id":"a53b893whole2a-h1","type":"hint","dependencies":[],"title":"Locate","text":"Locate the hundreds place in the number","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a53b893whole2a-h1"],"title":"Value","text":"What is the value of the hundreds place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole2a-h3","type":"hint","dependencies":["a53b893whole2a-h2"],"title":"Next Value","text":"Check the next value to the right of thr indicated hundreds number","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole2a-h4","type":"hint","dependencies":["a53b893whole2a-h3"],"title":"Rule","text":"If the number to the right of the hundreds number is greater than or equal to $$5$$ then add one to the hundreds place number and make everything after to 0s. If it is less than $$5$$, then leave the hundreds place number the same and change everything after to 0s.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole2a-h5","type":"hint","dependencies":["a53b893whole2a-h4"],"title":"Carrying","text":"Since $$7$$ is greater than or equal to $$5$$, add $$1$$ to the $$9$$. Add $$1$$ to $$9$$ is $$10$$, so we need to replace the $$9$$ with $$0$$ and carry the $$1$$ to its left.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a53b893whole2b","stepAnswer":["$$104000$$"],"problemType":"TextBox","stepTitle":"Round 103,978 to the nearest thousand","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$104000$$","hints":{"DefaultPathway":[{"id":"a53b893whole2b-h1","type":"hint","dependencies":[],"title":"Locate","text":"Locate the thousands place in the number","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole2b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a53b893whole2b-h1"],"title":"Value","text":"What is the value of the thousands place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole2b-h3","type":"hint","dependencies":["a53b893whole2b-h2"],"title":"Next Value","text":"Check the next value to the right of the indicated thousands number","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole2b-h4","type":"hint","dependencies":["a53b893whole2b-h3"],"title":"Rule","text":"If the number to the right of the thousands number is greater than or equal to $$5$$ then add one to the hundreds place number and make everything after to 0s. If it is less than $$5$$, then leave the thousands place number the same and change everything after to 0s.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a53b893whole2c","stepAnswer":["$$100000$$"],"problemType":"TextBox","stepTitle":"Round 103,978 to the nearest ten thousand","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$100000$$","hints":{"DefaultPathway":[{"id":"a53b893whole2c-h1","type":"hint","dependencies":[],"title":"Locate","text":"Locate the ten thousands place in the number","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole2c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a53b893whole2c-h1"],"title":"Value","text":"What is the value of the ten thousand place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole2c-h3","type":"hint","dependencies":["a53b893whole2c-h2"],"title":"Next Value","text":"Check the next value to the right of the indicated ten-thousands number","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole2c-h4","type":"hint","dependencies":["a53b893whole2c-h3"],"title":"Rule","text":"If the number to the right of the ten thousands number is greater than or equal to $$5$$ then add one to the hundreds place number and make everything after to 0s. If it is less than $$5$$, then leave the ten thousands place number the same and change everything after to 0s.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a53b893whole20","title":"Finding the Least Common Multiple Using the Prime Factors Method","body":"Find the least common multiple of the two numbers using the prime factors method.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Introduction to Whole Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a53b893whole20a","stepAnswer":["$$36$$"],"problemType":"TextBox","stepTitle":"$$9$$ and $$12$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$36$$","hints":{"DefaultPathway":[{"id":"a53b893whole20a-h1","type":"hint","dependencies":[],"title":"Writing Each Number as a Product of Primes","text":"Refer to the image for the first step. Make sure to substitute in the correct numbers, as the problem in the example uses different numbers.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole20a-h2","type":"hint","dependencies":["a53b893whole20a-h1"],"title":"Listing the Primes of Each Number","text":"Refer to the image for the second step.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole20a-h3","type":"hint","dependencies":["a53b893whole20a-h2"],"title":"Bringing Down the Number From Each Column","text":"Refer to the image for the third step.\\\\n##figure3.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole20a-h4","type":"hint","dependencies":["a53b893whole20a-h3"],"title":"Multiplying the Factors","text":"Refer to the image for the fourth step.\\\\n##figure4.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a53b893whole21","title":"Finding the Least Common Multiple Using the Prime Factors Method","body":"Find the least common multiple of the two numbers using the prime factors method.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Introduction to Whole Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a53b893whole21a","stepAnswer":["$$72$$"],"problemType":"TextBox","stepTitle":"$$18$$ and $$24$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$72$$","hints":{"DefaultPathway":[{"id":"a53b893whole21a-h1","type":"hint","dependencies":[],"title":"Writing Each Number as a Product of Primes","text":"Refer to the image for the first step. Make sure to substitute in the correct numbers, as the problem in the example uses different numbers.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole21a-h2","type":"hint","dependencies":["a53b893whole21a-h1"],"title":"Listing the Primes of Each Number","text":"Refer to the image for the second step.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole21a-h3","type":"hint","dependencies":["a53b893whole21a-h2"],"title":"Bringing Down the Number From Each Column","text":"Refer to the image for the third step.\\\\n##figure3.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole21a-h4","type":"hint","dependencies":["a53b893whole21a-h3"],"title":"Multiplying the Factors","text":"Refer to the image for the fourth step.\\\\n##figure4.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a53b893whole22","title":"Finding Place Values","body":"Given the number 63,407,218, find the place value of the following digits.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Introduction to Whole Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a53b893whole22a","stepAnswer":["thousands"],"problemType":"MultipleChoice","stepTitle":"$$7$$","stepBody":"","answerType":"string","variabilization":{},"choices":["ten thousands","thousands","hundrend","tens"],"hints":{"DefaultPathway":[{"id":"a53b893whole22a-h1","type":"hint","dependencies":[],"title":"Focusing on the Digits","text":"Let\'s start by ignoring the place values larger than the one with $$7$$. Focus on 7,218.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole22a-h2","type":"hint","dependencies":["a53b893whole22a-h1"],"title":"Rewriting the Number","text":"7,218 can be rewritten as 7,000+200+10+8.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole22a-h3","type":"hint","dependencies":["a53b893whole22a-h2"],"title":"Rewriting the Number","text":"This is the same thing as $$7$$ $$thousands+2$$ $$hundred+1$$ $$ten+8$$ ones.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole22a-h4","type":"hint","dependencies":["a53b893whole22a-h3"],"title":"Thinking","text":"$$7$$ is in the thousands place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a53b893whole22b","stepAnswer":["ten thousands"],"problemType":"MultipleChoice","stepTitle":"$$0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["tens","hundreds","thousands","ten thousands"],"hints":{"DefaultPathway":[{"id":"a53b893whole22b-h1","type":"hint","dependencies":[],"title":"Focusing on the Digits","text":"Let\'s only look at the relavant digits. Focus on 407,218.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole22b-h2","type":"hint","dependencies":["a53b893whole22b-h1"],"title":"Rewriting the Number","text":"407,218 can be rewritten as 400,000,000+7,000+200+10+8.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole22b-h3","type":"hint","dependencies":["a53b893whole22b-h2"],"title":"Rewriting the Number","text":"This is the same thing as $$4$$ hundred $$thousands+0$$ ten $$thousands+7$$ $$thousands+2$$ $$hundred+1$$ $$ten+8$$ ones.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole22b-h4","type":"hint","dependencies":["a53b893whole22b-h3"],"title":"Thinking","text":"$$0$$ is in the ten thousands place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a53b893whole22c","stepAnswer":["tens"],"problemType":"MultipleChoice","stepTitle":"$$1$$","stepBody":"","answerType":"string","variabilization":{},"choices":["tens","hundreds","thousands","ten thousands"],"hints":{"DefaultPathway":[{"id":"a53b893whole22c-h1","type":"hint","dependencies":[],"title":"Focusing on the Digits","text":"Let\'s start by ignoring the place values larger than the one with $$1$$. Focus on $$18$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole22c-h2","type":"hint","dependencies":["a53b893whole22c-h1"],"title":"Rewriting the Number","text":"$$18$$ can be rewritten as $$10+8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole22c-h3","type":"hint","dependencies":["a53b893whole22c-h2"],"title":"Rewriting the Number","text":"This is the same thing as $$1$$ $$ten+8$$ ones.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole22c-h4","type":"hint","dependencies":["a53b893whole22c-h3"],"title":"Thinking","text":"$$1$$ is in the tens place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a53b893whole22d","stepAnswer":["ten millions"],"problemType":"MultipleChoice","stepTitle":"$$6$$","stepBody":"","answerType":"string","variabilization":{},"choices":["ten thousands","thousands","ten millions","millions"],"hints":{"DefaultPathway":[{"id":"a53b893whole22d-h1","type":"hint","dependencies":[],"title":"Rewriting the Number","text":"63,407,218 can be rewritten as 60,000,000+3,000,000+400,000+7,000+200+10+8.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole22d-h2","type":"hint","dependencies":["a53b893whole22d-h1"],"title":"Rewriting the Number","text":"This is the same thing as $$6$$ ten $$millions+3$$ $$millions+4$$ hundred $$thousands+7$$ $$thousands+2$$ $$hundred+1$$ $$ten+8$$ ones.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole22d-h3","type":"hint","dependencies":["a53b893whole22d-h2"],"title":"Thinking","text":"$$6$$ is in the ten millions place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a53b893whole22e","stepAnswer":["millions"],"problemType":"MultipleChoice","stepTitle":"$$3$$","stepBody":"","answerType":"string","variabilization":{},"choices":["ten thousands","thousands","ten millions","millions"],"hints":{"DefaultPathway":[{"id":"a53b893whole22e-h1","type":"hint","dependencies":[],"title":"Focusing on the Digits","text":"Let\'s start by ignoring the place values larger than the one with $$3$$. Focus on 3,407,218.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole22e-h2","type":"hint","dependencies":["a53b893whole22e-h1"],"title":"Rewriting the Number","text":"3,407,218 can be rewritten as 3,000,000+400,000+7,000+200+10+8.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole22e-h3","type":"hint","dependencies":["a53b893whole22e-h2"],"title":"Rewriting the Number","text":"This is the same thing as $$3$$ $$millions+4$$ hundred $$thousands+7$$ $$thousands+2$$ $$hundred+1$$ $$ten+8$$ ones.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole22e-h4","type":"hint","dependencies":["a53b893whole22e-h3"],"title":"Thinking","text":"$$3$$ is in the millions place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a53b893whole23","title":"Naming Numbers","body":"Name the number using words.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Introduction to Whole Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a53b893whole23a","stepAnswer":["eight trillion, one hundred $$sixty-five$$ billion, four hundred $$thirty-two$$ million, $$ninety-eight$$ thousand, seven hundred ten"],"problemType":"MultipleChoice","stepTitle":"8,165,432,098,710","stepBody":"","answerType":"string","variabilization":{},"choices":["eight trillion, one hundred $$sixty-five$$ billion, four hundred $$thirty-two$$ million, $$ninety-eight$$ thousand, seven hundred ten","eight billion, one hundred $$sixty-five$$ thousand, four hundred $$thirty-two$$ hundred, $$ninety-eight$$","eight quintillion, one hundred $$sixty-five$$ trillion, four hundred $$thirty-two$$ billion, $$ninety-eight$$ million, seven hundred ten","one hundred $$sixty-five$$ billion, four hundred $$thirty-two$$ million, $$ninety-eight$$ thousand, seven hundred ten"],"hints":{"DefaultPathway":[{"id":"a53b893whole23a-h1","type":"hint","dependencies":[],"title":"Rewriting the Number","text":"When we separate the number 8,165,432,098,710 by its commas, it can be rewritten as 8,000,000,000+165,000,000,000+432,000,000,000+498,000+710.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole23a-h2","type":"hint","dependencies":["a53b893whole23a-h1"],"title":"Rewriting the Number","text":"This is the same thing as $$8$$ $$trillion+165$$ $$billion+432$$ $$million+498$$ $$thousand+710$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a53b893whole24","title":"Word Form to Number Form","body":"Write the following as a whole number using digits.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Introduction to Whole Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a53b893whole24a","stepAnswer":["9,246,073,189"],"problemType":"MultipleChoice","stepTitle":"nine billion, two hundred forty-six million, seventy-three thousand, one hundred eighty-nine","stepBody":"","answerType":"string","variabilization":{},"choices":["90,246,073,189","9,246,065,375","9,246,073,189","8,246,074,290"],"hints":{"DefaultPathway":[{"id":"a53b893whole24a-h1","type":"hint","dependencies":[],"title":"Commas","text":"We know that every comma in the the words indicates a comma in the number form. Then, we write the digits between each comma","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole24a-h2","type":"hint","dependencies":["a53b893whole24a-h1"],"title":"Rewriting the Number","text":"There needs to be three digits in every comma, so this becomes $$8$$ billion, $$246$$ million, $$073$$ thousand, $$189$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a53b893whole25","title":"Finding Place Values","body":"Given the number 27,493,615, find the place value of the following digits.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Introduction to Whole Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a53b893whole25a","stepAnswer":["ten millions"],"problemType":"MultipleChoice","stepTitle":"$$2$$","stepBody":"","answerType":"string","variabilization":{},"choices":["ten thousands","thousands","ten millions","millions"],"hints":{"DefaultPathway":[{"id":"a53b893whole25a-h1","type":"hint","dependencies":[],"title":"Rewriting the Number","text":"27,493,615 can be rewritten as 20,000,000+7,000,000+400,000+90,000+3,000+600+10+5.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole25a-h2","type":"hint","dependencies":["a53b893whole25a-h1"],"title":"Rewriting the Number","text":"This is the same thing as $$2$$ ten $$millions+7$$ $$millions+4$$ hundred $$thousands+9$$ ten $$thousands+3$$ $$thousands+6$$ $$hundred+1$$ $$ten+5$$ ones.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole25a-h3","type":"hint","dependencies":["a53b893whole25a-h2"],"title":"Thinking","text":"$$6$$ is in the ten millions place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a53b893whole25b","stepAnswer":["tens"],"problemType":"MultipleChoice","stepTitle":"$$1$$","stepBody":"","answerType":"string","variabilization":{},"choices":["tens","hundreds","thousands","ten thousands"],"hints":{"DefaultPathway":[{"id":"a53b893whole25b-h1","type":"hint","dependencies":[],"title":"Focusing on the Digits","text":"Let\'s start by ignoring the place values larger than the one with $$1$$. Focus on $$15$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole25b-h2","type":"hint","dependencies":["a53b893whole25b-h1"],"title":"Rewriting the Number","text":"$$15$$ can be rewritten as $$10+5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole25b-h3","type":"hint","dependencies":["a53b893whole25b-h2"],"title":"Rewriting the Number","text":"This is the same thing as $$1$$ $$ten+5$$ ones.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole25b-h4","type":"hint","dependencies":["a53b893whole25b-h3"],"title":"Thinking","text":"$$1$$ is in the tens place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a53b893whole25c","stepAnswer":["hundred thousands"],"problemType":"MultipleChoice","stepTitle":"$$4$$","stepBody":"","answerType":"string","variabilization":{},"choices":["hundred thousands","thousands","ten millions","millions"],"hints":{"DefaultPathway":[{"id":"a53b893whole25c-h1","type":"hint","dependencies":[],"title":"Focusing on the Digits","text":"Let\'s start by ignoring the place values larger than the one with $$4$$. Focus on 493,615.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole25c-h2","type":"hint","dependencies":["a53b893whole25c-h1"],"title":"Rewriting the Number","text":"493,615 can be rewritten as 400,000+90,000+3,000+600+10+5.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole25c-h3","type":"hint","dependencies":["a53b893whole25c-h2"],"title":"Rewriting the Number","text":"This is the same thing as $$4$$ hundred $$thousands+9$$ ten $$thousands+3$$ $$thousands+6$$ $$hundreds+1$$ $$ten+5$$ ones.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole25c-h4","type":"hint","dependencies":["a53b893whole25c-h3"],"title":"Thinking","text":"$$3$$ is in the hundred thousands place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a53b893whole25d","stepAnswer":["millions"],"problemType":"MultipleChoice","stepTitle":"$$7$$","stepBody":"","answerType":"string","variabilization":{},"choices":["ten thousands","thousands","ten millions","millions"],"hints":{"DefaultPathway":[{"id":"a53b893whole25d-h1","type":"hint","dependencies":[],"title":"Focusing on the Digits","text":"Let\'s start by ignoring the place values larger than the one with $$7$$. Focus on 7,493,615.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole25d-h2","type":"hint","dependencies":["a53b893whole25d-h1"],"title":"Rewriting the Number","text":"7,493,615 can be rewritten as 7,000,000+400,000+90,000+3,000+600+10+5.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole25d-h3","type":"hint","dependencies":["a53b893whole25d-h2"],"title":"Rewriting the Number","text":"This is the same thing as $$7$$ $$millions+4$$ hundred $$thousands+9$$ $$thousands+6$$ $$hundreds+1$$ $$tens+5$$ ones.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole25d-h4","type":"hint","dependencies":["a53b893whole25d-h3"],"title":"Thinking","text":"$$7$$ is in the millions place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a53b893whole25e","stepAnswer":["ones"],"problemType":"MultipleChoice","stepTitle":"$$5$$","stepBody":"","answerType":"string","variabilization":{},"choices":["ones","tens","hundreds","thousands"],"hints":{"DefaultPathway":[{"id":"a53b893whole25e-h1","type":"hint","dependencies":[],"title":"Focusing on the Digits","text":"Let\'s start by ignoring the place values larger than the one with $$5$$. Focus on $$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole25e-h2","type":"hint","dependencies":["a53b893whole25e-h1"],"title":"Rewriting the Number","text":"This is the same thing as $$5$$ ones.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole25e-h3","type":"hint","dependencies":["a53b893whole25e-h2"],"title":"Thinking","text":"$$5$$ is in the ones place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a53b893whole26","title":"Finding Place Values","body":"For the number 519,711,641,328, find the place value of each digit:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Introduction to Whole Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a53b893whole26a","stepAnswer":["trillions"],"problemType":"MultipleChoice","stepTitle":"$$9$$","stepBody":"","answerType":"string","variabilization":{},"choices":["billions","tens","millions","trillions"],"hints":{"DefaultPathway":[{"id":"a53b893whole26a-h1","type":"hint","dependencies":[],"title":"Focusing on the Digits","text":"Let\'s start by ignoring the place values larger than the one with $$9$$. Focus on 9,711,641,327.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole26a-h2","type":"hint","dependencies":["a53b893whole26a-h1"],"title":"Rewriting the Number","text":"9,711,641,327 can be rewritten as 9,000,000,000+700,000,000+10,000,000+1,000,000+600,000+40,000+1,000+300+20+7.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole26a-h3","type":"hint","dependencies":["a53b893whole26a-h2"],"title":"Rewriting the Number","text":"This is the same thing as $$9$$ $$trillions+7$$ hundred $$millions+1$$ ten $$millions+1$$ $$millions+6$$ hundred $$thousands+4$$ ten $$thousands+1$$ $$thousands+3$$ $$hundreds+2$$ $$tens+7$$ ones.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole26a-h4","type":"hint","dependencies":["a53b893whole26a-h3"],"title":"Thinking","text":"$$9$$ is in the trillions place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a53b893whole26b","stepAnswer":["ten thousands"],"problemType":"MultipleChoice","stepTitle":"$$4$$","stepBody":"","answerType":"string","variabilization":{},"choices":["ten thousands","thousands","ten millions","millions"],"hints":{"DefaultPathway":[{"id":"a53b893whole26b-h1","type":"hint","dependencies":[],"title":"Focusing on the Digits","text":"Let\'s start by ignoring the place values larger than the one with $$4$$. Focus on 41,327.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole26b-h2","type":"hint","dependencies":["a53b893whole26b-h1"],"title":"Rewriting the Number","text":"41,327 can be rewritten as 40,000+1,000+300+20+7.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole26b-h3","type":"hint","dependencies":["a53b893whole26b-h2"],"title":"Rewriting the Number","text":"This is the same thing as $$4$$ ten $$thousands+1$$ $$thousands+3$$ $$hundreds+2$$ $$tens+7$$ ones.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole26b-h4","type":"hint","dependencies":["a53b893whole26b-h3"],"title":"Thinking","text":"$$4$$ is in the ten thousands place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a53b893whole26c","stepAnswer":["tens"],"problemType":"MultipleChoice","stepTitle":"$$2$$","stepBody":"","answerType":"string","variabilization":{},"choices":["ones","tens","hundreds","thousands"],"hints":{"DefaultPathway":[{"id":"a53b893whole26c-h1","type":"hint","dependencies":[],"title":"Focusing on the Digits","text":"Let\'s start by ignoring the place values larger than the one with $$2$$. Focus on $$27$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole26c-h2","type":"hint","dependencies":["a53b893whole26c-h1"],"title":"Rewriting the Number","text":"$$27$$ can be rewritten as $$20+7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole26c-h3","type":"hint","dependencies":["a53b893whole26c-h2"],"title":"Rewriting the Number","text":"This is the same thing as $$2$$ $$tens+7$$ ones.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole26c-h4","type":"hint","dependencies":["a53b893whole26c-h3"],"title":"Thinking","text":"$$2$$ is in the tens place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a53b893whole26d","stepAnswer":["hundred thousands"],"problemType":"MultipleChoice","stepTitle":"$$6$$","stepBody":"","answerType":"string","variabilization":{},"choices":["hundred thousands","thousands","ten millions","millions"],"hints":{"DefaultPathway":[{"id":"a53b893whole26d-h1","type":"hint","dependencies":[],"title":"Focusing on the Digits","text":"Let\'s start by ignoring the place values larger than the one with $$6$$. Focus on 641,327.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole26d-h2","type":"hint","dependencies":["a53b893whole26d-h1"],"title":"Rewriting the Number","text":"641,327 can be rewritten as 600,000+40,000+1,000+300+20+7.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole26d-h3","type":"hint","dependencies":["a53b893whole26d-h2"],"title":"Rewriting the Number","text":"This is the same thing as $$6$$ hundred $$thousands+4$$ ten $$thousands+1$$ $$thousands+3$$ $$hundreds+2$$ $$tens+7$$ ones.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole26d-h4","type":"hint","dependencies":["a53b893whole26d-h3"],"title":"Thinking","text":"$$6$$ is in the hundred thousands place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a53b893whole26e","stepAnswer":["hundred millions"],"problemType":"MultipleChoice","stepTitle":"$$7$$","stepBody":"","answerType":"string","variabilization":{},"choices":["hundred thousands","thousands","hundred millions","millions"],"hints":{"DefaultPathway":[{"id":"a53b893whole26e-h1","type":"hint","dependencies":[],"title":"Focusing on the Digits","text":"Let\'s start by ignoring the place values larger than the one with $$7$$. 711,641,327.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole26e-h2","type":"hint","dependencies":["a53b893whole26e-h1"],"title":"Rewriting the Number","text":"711,641,327 can be rewritten as 700,000,000+10,000,000+1,000,000+600,000+40,000+1,000+300+20+7.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole26e-h3","type":"hint","dependencies":["a53b893whole26e-h2"],"title":"Rewriting the Number","text":"This is the same thing as $$7$$ hundred $$millions+1$$ ten $$millions+1$$ $$millions+6$$ hundred $$thousands+4$$ ten $$thousands+1$$ $$thousands+3$$ $$hundreds+2$$ $$tens+7$$ ones.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole26e-h4","type":"hint","dependencies":["a53b893whole26e-h3"],"title":"Thinking","text":"$$7$$ is in the hundred millions place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a53b893whole27","title":"Naming Numbers","body":"Name the number using words.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Introduction to Whole Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a53b893whole27a","stepAnswer":["nine trillion, two hundred $$fifty-eight$$ billion, one hundred $$thirty-seven$$ million, nine hundred four thousand, $$sixty-one$$"],"problemType":"MultipleChoice","stepTitle":"9,258,137,904,061","stepBody":"","answerType":"string","variabilization":{},"choices":["eight trillion, one hundred $$sixty-seven$$ billion, one hundred $$twenty-nine$$ million, six hundred three thousand, $$sixty-one$$","nine trillion, two hundred $$fifty-eight$$ billion, one hundred $$thirty-seven$$ million, nine hundred four thousand, $$sixty-one$$","nine quintillion, one hundred $$sixty-five$$ trillion, four hundred $$thirty-two$$ billion, $$ninety-eight$$ million, seven hundred ten","three hundred $$sixty-five$$ billion, four hundred $$thirty-two$$ million, $$ninety-eight$$ thousand, seven hundred ten"],"hints":{"DefaultPathway":[{"id":"a53b893whole27a-h1","type":"hint","dependencies":[],"title":"Rewriting the Number","text":"When we separate the number 9,258,137,904,061 by its commas, it can be rewritten as 9,000,000,000+258,000,000,000+137,000,000,000+904,000+61.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole27a-h2","type":"hint","dependencies":["a53b893whole27a-h1"],"title":"Rewriting the Number","text":"This is the same thing as $$9$$ $$trillion+258$$ $$billion+137$$ $$million+904$$ $$thousand+61$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a53b893whole28","title":"Naming Numbers","body":"Name the number using words.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Introduction to Whole Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a53b893whole28a","stepAnswer":["seventeen trillion, eight hundred $$sixty-four$$ billion, three hundred $$twenty-five$$ million, six hundred nineteen thousand, four"],"problemType":"MultipleChoice","stepTitle":"17,864,325,619,004","stepBody":"","answerType":"string","variabilization":{},"choices":["seven trillion, two hundred $$thirty-eight$$ billion, nine hundred $$sixty-nine$$ million, eight hundred five thousand, $$twenty-one$$","six trillion, five hundred $$twenty-eight$$ billion, three hundred $$twenty-three$$ million, five hundred three thousand, $$fifty-six$$","three quintillion, two hundred $$five-nine$$ trillion, four hundred $$thirty-four$$ billion, $$sixty-nine$$ million, two hundred five","seventeen trillion, eight hundred $$sixty-four$$ billion, three hundred $$twenty-five$$ million, six hundred nineteen thousand, four"],"hints":{"DefaultPathway":[{"id":"a53b893whole28a-h1","type":"hint","dependencies":[],"title":"Rewriting the Number","text":"When we separate the number 17,864,325,619,004 by its commas, it can be rewritten as 17,000,000,000+864,000,000,000+325,000,000,000+619,000+4.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole28a-h2","type":"hint","dependencies":["a53b893whole28a-h1"],"title":"Rewriting the Number","text":"This is the same thing as $$17$$ $$trillion+864$$ $$billion+325$$ $$million+619$$ $$thousand+4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a53b893whole29","title":"Word Form to Digits","body":"Write the following number as a whole number using digits.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Introduction to Whole Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a53b893whole29a","stepAnswer":["2,466,714,051"],"problemType":"MultipleChoice","stepTitle":"two billion, four hundred sixty-six million, seven hundred fourteen thousand, fifty-one","stepBody":"","answerType":"string","variabilization":{},"choices":["42,554,765,123","123,563,768,324","2,466,714,051","3,123,546,234"],"hints":{"DefaultPathway":[{"id":"a53b893whole29a-h1","type":"hint","dependencies":[],"title":"Commas","text":"We know that every comma in the the words indicates a comma in the number form. Then, we write the digits between each comma.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole29a-h2","type":"hint","dependencies":["a53b893whole29a-h1"],"title":"Rewriting the Number","text":"There needs to be three digits in every comma, so this becomes $$2$$ billion, $$466$$ million, $$714$$ thousand, $$051$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a53b893whole3","title":"Divisibility Tests","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Introduction to Whole Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a53b893whole3a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Is 5,625 divisible by 2?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a53b893whole3a-h1","type":"hint","dependencies":[],"title":"Rule","text":"If the last digit is $$0$$, $$2$$, $$4$$, $$6$$, or $$8$$, then it is divisible by $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole3a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a53b893whole3a-h1"],"title":"Divisibility","text":"Is 5,625 divisible by $$2$$ based on the information above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}},{"id":"a53b893whole3b","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Is 5,625 divisible by 3?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a53b893whole3b-h1","type":"hint","dependencies":[],"title":"Rule","text":"If the sum of the digits is divisible by $$3$$, then the number is divisible by $$3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole3b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18$$"],"dependencies":["a53b893whole3b-h1"],"title":"Sum","text":"What is the sum of the digits?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole3b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole3b-h2"],"title":"Divisibility","text":"Based on the information above, is $$5625$$ divisible by 3?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}},{"id":"a53b893whole3c","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Is 5,625 divisible by 5?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a53b893whole3c-h1","type":"hint","dependencies":[],"title":"Rule","text":"A number is divisible by $$5$$ if the last digit is $$5$$ or $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole3c-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole3c-h1"],"title":"Divisibility","text":"Based on the information above, is $$5625$$ divisible by 5?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}},{"id":"a53b893whole3d","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Is 5,625 divisible by 6?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a53b893whole3d-h1","type":"hint","dependencies":[],"title":"Rule","text":"A number is divisible by $$6$$ if it is divisible by both $$2$$ and $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole3d-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a53b893whole3d-h1"],"title":"Question","text":"Is the number divisible by $$2$$ and 3?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole3d-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a53b893whole3d-h2"],"title":"Divisibility","text":"Based on the information above, is $$5625$$ divisible by 6?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}},{"id":"a53b893whole3e","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Is 5,625 divisible by 10?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a53b893whole3e-h1","type":"hint","dependencies":[],"title":"Rule","text":"A number is divisible by $$10$$ if it ends with $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole3e-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a53b893whole3e-h1"],"title":"Divisibility","text":"Based on the information above, is $$5625$$ divisible by 10?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a53b893whole30","title":"Word Form to Digits","body":"Write the following number as a whole number using digits.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Introduction to Whole Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a53b893whole30a","stepAnswer":["11,921,830,106"],"problemType":"MultipleChoice","stepTitle":"eleven billion, nine hundred twenty-one million, eight hundred thirty thousand, one hundred six","stepBody":"","answerType":"string","variabilization":{},"choices":["11,921,830,106","110,921,830,106","11,921,837,106"],"hints":{"DefaultPathway":[{"id":"a53b893whole30a-h1","type":"hint","dependencies":[],"title":"Commas","text":"We know that every comma in the the words indicates a comma in the number form. Then, we write the digits between each comma.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole30a-h2","type":"hint","dependencies":["a53b893whole30a-h1"],"title":"Rewriting the Number","text":"There needs to be three digits in every comma, so this becomes $$11$$ billion, $$921$$ million, $$830$$ thousand, $$106$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a53b893whole4","title":"Rounding Whole Numbers","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Introduction to Whole Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a53b893whole4a","stepAnswer":["$$17900$$"],"problemType":"TextBox","stepTitle":"Round to the nearest hundred: 17,852.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$17900$$","hints":{"DefaultPathway":[{"id":"a53b893whole4a-h1","type":"hint","dependencies":[],"title":"Locate","text":"Locate the hundreds place in the number","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a53b893whole4a-h1"],"title":"Value","text":"What is the value of the hundreds place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole4a-h3","type":"hint","dependencies":["a53b893whole4a-h2"],"title":"Next Value","text":"Check the next value to the right of the indicated hundreds number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole4a-h4","type":"hint","dependencies":["a53b893whole4a-h3"],"title":"Rule","text":"If the number to the right of the hundreds number is greater than or equal to $$5$$ then add one to the hundreds place number and make everything after to 0s. If it is less than $$5$$, then leave the hundreds place number the same and change everything after to 0s.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a53b893whole5","title":"Rounding Whole Numbers","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Introduction to Whole Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a53b893whole5a","stepAnswer":["$$468800$$"],"problemType":"TextBox","stepTitle":"Round to the nearest hundred: 468,751.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$468800$$","hints":{"DefaultPathway":[{"id":"a53b893whole5a-h1","type":"hint","dependencies":[],"title":"Locate","text":"Locate the hundreds place in the number","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a53b893whole5a-h1"],"title":"Value","text":"What is the value of the hundreds place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole5a-h3","type":"hint","dependencies":["a53b893whole5a-h2"],"title":"Next Value","text":"Check the next value to the right of thr indicated hundreds number","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole5a-h4","type":"hint","dependencies":["a53b893whole5a-h3"],"title":"Rule","text":"If the number to the right of the hundreds number is greater than or equal to $$5$$ then add one to the hundreds place number and make everything after to 0s. If it is less than $$5$$, then leave the hundreds place number the same and change everything after to 0s.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a53b893whole6","title":"Rounding Whole Numbers","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Introduction to Whole Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a53b893whole6a","stepAnswer":["$$207000$$"],"problemType":"TextBox","stepTitle":"Round 206,981 to the nearest hundred","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$207000$$","hints":{"DefaultPathway":[{"id":"a53b893whole6a-h1","type":"hint","dependencies":[],"title":"Locate","text":"Locate the hundreds place in the number","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a53b893whole6a-h1"],"title":"Value","text":"What is the value of the hundreds place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole6a-h3","type":"hint","dependencies":["a53b893whole6a-h2"],"title":"Next Value","text":"Check the next value to the right of the indicated hundreds number","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole6a-h4","type":"hint","dependencies":["a53b893whole6a-h3"],"title":"Rule","text":"If the number to the right of the hundreds number is greater than or equal to $$5$$ then add one to the hundreds place number and make everything after to 0s. If it is less than $$5$$, then leave the hundreds place number the same and change everything after to 0s.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole6a-h5","type":"hint","dependencies":["a53b893whole6a-h4"],"title":"Carrying","text":"Adding $$1$$ to $$9$$ gives us $$10$$, so we will need to replace the hundreds place with $$0$$ and carry the $$1$$ over to the place on its left, which gives us 207,000.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a53b893whole6b","stepAnswer":["$$207000$$"],"problemType":"TextBox","stepTitle":"Round 206,981 to the nearest thousand","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$207000$$","hints":{"DefaultPathway":[{"id":"a53b893whole6b-h1","type":"hint","dependencies":[],"title":"Locate","text":"Locate the thousands place in the number","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole6b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a53b893whole6b-h1"],"title":"Value","text":"What is the value of the thousands place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole6b-h3","type":"hint","dependencies":["a53b893whole6b-h2"],"title":"Next Value","text":"Check the next value to the right of the indicated thousands number","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole6b-h4","type":"hint","dependencies":["a53b893whole6b-h3"],"title":"Rule","text":"If the number to the right of the thousands number is greater than or equal to $$5$$ then add one to the thousands place number and make everything after to 0s. If it is less than $$5$$, then leave the thousands place number the same and change everything after to 0s.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a53b893whole6c","stepAnswer":["$$210000$$"],"problemType":"TextBox","stepTitle":"Round 206,981 to the nearest ten thousand","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$210000$$","hints":{"DefaultPathway":[{"id":"a53b893whole6c-h1","type":"hint","dependencies":[],"title":"Locate","text":"Locate the ten thousands place in the number","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole6c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a53b893whole6c-h1"],"title":"Value","text":"What is the value of the ten thousands place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole6c-h3","type":"hint","dependencies":["a53b893whole6c-h2"],"title":"Next Value","text":"Check the next value to the right of the indicated ten thousands number","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole6c-h4","type":"hint","dependencies":["a53b893whole6c-h3"],"title":"Rule","text":"If the number to the right of the ten thousands number is greater than or equal to $$5$$ then add one to the ten thousands place number and make everything after to 0s. If it is less than $$5$$, then leave the ten thousands place number the same and change everything after to 0s.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a53b893whole7","title":"Rounding Whole Numbers","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Introduction to Whole Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a53b893whole7a","stepAnswer":["$$785000$$"],"problemType":"TextBox","stepTitle":"Round 784,951 to the nearest hundred","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$785000$$","hints":{"DefaultPathway":[{"id":"a53b893whole7a-h1","type":"hint","dependencies":[],"title":"Locate","text":"Locate the hundreds place in the number","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a53b893whole7a-h1"],"title":"Value","text":"What is the value of the hundreds place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole7a-h3","type":"hint","dependencies":["a53b893whole7a-h2"],"title":"Next Value","text":"Check the next value to the right of the indicated hundreds number","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole7a-h4","type":"hint","dependencies":["a53b893whole7a-h3"],"title":"Rule","text":"If the number to the right of the hundreds number is greater than or equal to $$5$$ then add one to the hundreds place number and make everything after to 0s. If it is less than $$5$$, then leave the hundreds place number the same and change everything after to 0s.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole7a-h5","type":"hint","dependencies":["a53b893whole7a-h4"],"title":"Carrying","text":"Adding $$1$$ to $$9$$ gives us $$10$$, so we will need to replace the hundreds place with $$0$$ and carry the $$1$$ over to the place on its left, which gives us 785,000.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a53b893whole7b","stepAnswer":["$$785000$$"],"problemType":"TextBox","stepTitle":"Round 784,951 to the nearest thousand","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$785000$$","hints":{"DefaultPathway":[{"id":"a53b893whole7b-h1","type":"hint","dependencies":[],"title":"Locate","text":"Locate the thousands place in the number","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole7b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a53b893whole7b-h1"],"title":"Value","text":"What is the value of the thousands place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole7b-h3","type":"hint","dependencies":["a53b893whole7b-h2"],"title":"Next Value","text":"Check the next value to the right of the indicated thousands number","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole7b-h4","type":"hint","dependencies":["a53b893whole7b-h3"],"title":"Rule","text":"If the number to the right of the thousands number is greater than or equal to $$5$$ then add one to the thousands place number and make everything after to 0s. If it is less than $$5$$, then leave the thousands place number the same and change everything after to 0s.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a53b893whole7c","stepAnswer":["$$780000$$"],"problemType":"TextBox","stepTitle":"Round 784,951 to the nearest ten thousand","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$780000$$","hints":{"DefaultPathway":[{"id":"a53b893whole7c-h1","type":"hint","dependencies":[],"title":"Locate","text":"Locate the ten thousands place in the number","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole7c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a53b893whole7c-h1"],"title":"Value","text":"What is the value of the ten thousands place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole7c-h3","type":"hint","dependencies":["a53b893whole7c-h2"],"title":"Next Value","text":"Check the next value to the right of the indicated ten thousands number","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole7c-h4","type":"hint","dependencies":["a53b893whole7c-h3"],"title":"Rule","text":"If the number to the right of the ten thousands number is greater than or equal to $$5$$ then add one to the ten thousands place number and make everything after to 0s. If it is less than $$5$$, then leave the ten thousands place number the same and change everything after to 0s.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a53b893whole8","title":"Divisibility Tests","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Introduction to Whole Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a53b893whole8a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Determine whether 4,962 is divisible by $$2$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a53b893whole8a-h1","type":"hint","dependencies":[],"title":"Rule","text":"If the last digit is $$0$$, $$2$$, $$4$$, $$6$$, or $$8$$, then it is divisible by $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole8a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole8a-h1"],"title":"Divisibility","text":"Is 4,962 divisible by $$2$$ based on the information above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}},{"id":"a53b893whole8b","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Determine whether 4,962 is divisible by $$3$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a53b893whole8b-h1","type":"hint","dependencies":[],"title":"Rule","text":"If the sum of the digits is divisible by $$3$$, then the number is divisible by $$3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole8b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$21$$"],"dependencies":["a53b893whole8b-h1"],"title":"Sum","text":"What is the sum of the digits?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole8b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole8b-h2"],"title":"Divisibility","text":"Based on the information above, is 4,962 divisible by 3?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}},{"id":"a53b893whole8c","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Determine whether 4,962 is divisible by $$5$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a53b893whole8c-h1","type":"hint","dependencies":[],"title":"Rule","text":"A number is divisible by $$5$$ if the last digit is $$5$$ or $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole8c-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a53b893whole8c-h1"],"title":"Divisibility","text":"Based on the information above, is 4,962 divisible by 5?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}},{"id":"a53b893whole8d","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Determine whether 4,962 is divisible by $$6$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a53b893whole8d-h1","type":"hint","dependencies":[],"title":"Rule","text":"A number is divisible by $$6$$ if it is divisible by both $$2$$ and $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole8d-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole8d-h1"],"title":"Question","text":"Is the number divisible by $$2$$ and 3?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole8d-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole8d-h2"],"title":"Divisibility","text":"Based on the information above, is $$4962$$ divisible by 6?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}},{"id":"a53b893whole8e","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Determine whether 4,962 is divisible by $$10$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a53b893whole8e-h1","type":"hint","dependencies":[],"title":"Rule","text":"A number is divisible by $$10$$ if it ends with $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole8e-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a53b893whole8e-h1"],"title":"Divisibility","text":"Based on the information above, is $$4962$$ divisible by 10?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a53b893whole9","title":"Divisibility Tests","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Introduction to Whole Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a53b893whole9a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Determine whether 3,765 is divisible by $$2$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a53b893whole9a-h1","type":"hint","dependencies":[],"title":"Rule","text":"If the last digit is $$0$$, $$2$$, $$4$$, $$6$$, or $$8$$, then it is divisible by $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole9a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a53b893whole9a-h1"],"title":"Divisibility","text":"Is 3,765 divisible by $$2$$ based on the information above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}},{"id":"a53b893whole9b","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Determine whether 3,765 is divisible by $$3$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a53b893whole9b-h1","type":"hint","dependencies":[],"title":"Rule","text":"If the sum of the digits is divisible by $$3$$, then the number is divisible by $$3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole9b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$21$$"],"dependencies":["a53b893whole9b-h1"],"title":"Sum","text":"What is the sum of the digits?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole9b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole9b-h2"],"title":"Divisibility","text":"Based on the information above, is 3,765 divisible by 3?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}},{"id":"a53b893whole9c","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Determine whether 3,765 is divisible by $$5$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a53b893whole9c-h1","type":"hint","dependencies":[],"title":"Rule","text":"A number is divisible by $$5$$ if the last digit is $$5$$ or $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole9c-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a53b893whole9c-h1"],"title":"Divisibility","text":"Based on the information above, is 3,765 divisible by 5?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}},{"id":"a53b893whole9d","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Determine whether 3,765 is divisible by $$6$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a53b893whole9d-h1","type":"hint","dependencies":[],"title":"Rule","text":"A number is divisible by $$6$$ if it is divisible by both $$2$$ and $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole9d-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a53b893whole9d-h1"],"title":"Question","text":"Is the number divisible by $$2$$ and 3?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a53b893whole9d-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a53b893whole9d-h2"],"title":"Divisibility","text":"Based on the information above, is 3,765 divisible by 6?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}},{"id":"a53b893whole9e","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Determine whether 3,765 is divisible by $$10$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a53b893whole9e-h1","type":"hint","dependencies":[],"title":"Rule","text":"A number is divisible by $$10$$ if it ends with $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a53b893whole9e-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a53b893whole9e-h1"],"title":"Divisibility","text":"Based on the information above, is 3,765 divisible by 10?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a54f58cgraphineq1","title":"Determine whether an ordered pair is a solution of a system of linear inequalities","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a54f58cgraphineq1a","stepAnswer":["yes"],"problemType":"MultipleChoice","stepTitle":"Determine () whether $$(-2,4)$$ is a solution to {$$x+4y \\\\geq 10$$, $$3x-2y<12$$}","stepBody":"","answerType":"string","variabilization":{},"choices":["yes","no"],"hints":{"DefaultPathway":[{"id":"a54f58cgraphineq1a-h1","type":"hint","dependencies":[],"title":"Plugging in $$x$$ and $$y$$","text":"$$(-2,4)$$ is a solution because both equations are true when $$x=-2$$ and $$y=4$$ are plugged in.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a54f58cgraphineq1b","stepAnswer":["$$n$$"],"problemType":"MultipleChoice","stepTitle":"Determine () whether $$(3,1)$$ is a solution to {$$x+4y \\\\geq 10$$, $$3x-2y<12$$}","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$n$$","choices":["$$n$$","no","yes"],"hints":{"DefaultPathway":[{"id":"a54f58cgraphineq1b-h1","type":"hint","dependencies":[],"title":"Plugging in $$x$$ and $$y$$","text":"$$(3,1)$$ is not a solution because both equations are not true when $$x=3$$ and $$y=1$$ are plugged in.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a54f58cgraphineq10","title":"Solve a System of Linear Inequalities by Graphing","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a54f58cgraphineq10a","stepAnswer":["$$Blue-red$$"],"problemType":"MultipleChoice","stepTitle":"Solve the system by graphing: {y>=3x-2,y<-1}. Which shaded region is that of the solutions?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Blue","Red","$$Blue-red$$"],"hints":{"DefaultPathway":[{"id":"a54f58cgraphineq10a-h1","type":"hint","dependencies":[],"title":"Graph the First Inequality","text":"We must graph $$y \\\\geq 3x-2$$. We make the line solid and shade above it since we have $$ \\\\geq $$.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a54f58cgraphineq10a-h2","type":"hint","dependencies":["a54f58cgraphineq10a-h1"],"title":"Graph the Second Inequality","text":"We must graph $$y<-1$$ We make the line dashed and shade below it since we have <.\\\\n##figure3.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a54f58cgraphineq10a-h3","type":"hint","dependencies":["a54f58cgraphineq10a-h2"],"title":"The solution is region where shading overlaps: Blue-red","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a54f58cgraphineq11","title":"Solve a System of Linear Inequalities by Graphing","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a54f58cgraphineq11a","stepAnswer":["$$Blue-red$$"],"problemType":"MultipleChoice","stepTitle":"Solve the system by graphing: {x>-4,x-2y>=-4}. Which shaded region is that of the solutions?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Blue","Red","$$Blue-red$$"],"hints":{"DefaultPathway":[{"id":"a54f58cgraphineq11a-h1","type":"hint","dependencies":[],"title":"Graph the First Inequality","text":"We must graph $$x>-4$$, which is a vertical line. We shade to the right.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a54f58cgraphineq11a-h2","type":"hint","dependencies":["a54f58cgraphineq11a-h1"],"title":"Graph the Second Inequality","text":"We must graph $$x-2y \\\\geq -4$$, which rearranges to $$y \\\\leq \\\\frac{-\\\\left(-4-x\\\\right)}{2}$$. We shade below since we have $$ \\\\leq $$.\\\\n##figure3.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a54f58cgraphineq11a-h3","type":"hint","dependencies":["a54f58cgraphineq11a-h2"],"title":"The solution is region where shading overlaps: Blue-red","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a54f58cgraphineq12","title":"Solve a System of Linear Inequalities by Graphing","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a54f58cgraphineq12a","stepAnswer":["No overlap"],"problemType":"MultipleChoice","stepTitle":"Solve the system by graphing: {4x+3y>=12,y<-4/3x+1}. Which shaded region is that of the solutions?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Blue","Red","No overlap"],"hints":{"DefaultPathway":[{"id":"a54f58cgraphineq12a-h1","type":"hint","dependencies":[],"title":"Graph the First Inequality","text":"We must graph $$4x+3y \\\\geq 12$$. We shade above since we have $$ \\\\geq $$\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a54f58cgraphineq12a-h2","type":"hint","dependencies":["a54f58cgraphineq12a-h1"],"title":"Graph the Second Inequality","text":"We must graph $$y<\\\\frac{-4}{3} x+1$$. We shade below since we have <.\\\\n##figure3.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a54f58cgraphineq12a-h3","type":"hint","dependencies":["a54f58cgraphineq12a-h2"],"title":"The solution is region where shading overlaps: No overlap","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a54f58cgraphineq13","title":"Determine if an Ordered Pair is a Solution","body":"Determine whether the ordered pair is a solution to the system of inequalities.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Intermediate 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inequalities.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a54f58cgraphineq16a-h2","type":"hint","dependencies":["a54f58cgraphineq16a-h1"],"title":"Solution to the System","text":"If the ordered pair makes all of the inequalities in the system true, then it is a solution to the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a54f58cgraphineq17","title":"Determine if an Ordered Pair is a Solution","body":"Determine whether the ordered pair is a solution to the system of inequalities.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Intermediate 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a54f58cgraphineq18","title":"Determine if an Ordered Pair is a Solution","body":"Determine whether the ordered pair is a solution to the system of inequalities.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a54f58cgraphineq18a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Is $$(3,0)$$ a solution to the system of inequalities $$y>\\\\frac{2}{3} x-5$$, $$x+\\\\frac{1}{2} y \\\\leq 4$$?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a54f58cgraphineq18a-h1","type":"hint","dependencies":[],"title":"Checking Validity in First Equation","text":"Plug the $$x$$ and $$y$$ values of the ordered pair into the inequalities.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a54f58cgraphineq18a-h2","type":"hint","dependencies":["a54f58cgraphineq18a-h1"],"title":"Solution to the System","text":"If the ordered pair makes all of the inequalities in the system true, then it is a solution to the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a54f58cgraphineq19","title":"Determine if an Ordered Pair is a Solution","body":"Determine whether the ordered pair is a solution to the system of inequalities.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a54f58cgraphineq19a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Is $$(-4,-1)$$ a solution to the system of inequalities $$y<\\\\frac{3x}{2}+3$$, $$\\\\frac{3x}{4}-2y<5$$?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a54f58cgraphineq19a-h1","type":"hint","dependencies":[],"title":"Checking Validity in First Equation","text":"Plug the $$x$$ and $$y$$ values of the ordered pair into the inequalities.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a54f58cgraphineq19a-h2","type":"hint","dependencies":["a54f58cgraphineq19a-h1"],"title":"Solution to the System","text":"If the ordered pair makes all of the inequalities in the system true, then it is a solution to the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a54f58cgraphineq2","title":"Determine whether an ordered pair is a solution of a system of linear inequalities","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a54f58cgraphineq2a","stepAnswer":["no"],"problemType":"MultipleChoice","stepTitle":"Determine () whether $$(3,-1)$$ is a solution to {$$x-5y>10$$, $$2x+3y>-2$$}","stepBody":"","answerType":"string","variabilization":{},"choices":["yes","no"],"hints":{"DefaultPathway":[{"id":"a54f58cgraphineq2a-h1","type":"hint","dependencies":[],"title":"Plugging in $$x$$ and $$y$$","text":"We must plug in $$x=3$$ and $$y=-1$$ into both inequalities and determine whether they are true. 3-5(-1)>10 is false. So, the point is not a solution","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a54f58cgraphineq2b","stepAnswer":["yes"],"problemType":"MultipleChoice","stepTitle":"Determine () whether $$(6,-3)$$ is a solution to {$$x-5y>10$$, $$2x+3y>-2$$}","stepBody":"","answerType":"string","variabilization":{},"choices":["yes","no"],"hints":{"DefaultPathway":[{"id":"a54f58cgraphineq2b-h1","type":"hint","dependencies":[],"title":"Plugging in $$x$$ and $$y$$","text":"We must plug in $$x=6$$ and $$y=-3$$ into both inequalities and determine whether they are true. 6-5(-3)>10 is true. 2(6)+3(-3)>-2 is true. So, the point is a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a54f58cgraphineq20","title":"Determine if an Ordered Pair is a Solution","body":"Determine whether the ordered pair is a solution to the system of inequalities.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a54f58cgraphineq20a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Is $$(8,3)$$ a solution to the system of inequalities $$y<\\\\frac{3x}{2}+3$$, 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inequalities.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a54f58cgraphineq21a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Is $$(2,3)$$ a solution to the system of inequalities $$7x+2y>14$$, $$5x-y \\\\leq 8$$?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a54f58cgraphineq21a-h1","type":"hint","dependencies":[],"title":"Checking Validity in First Equation","text":"Plug the $$x$$ and $$y$$ values of the ordered pair into the inequalities.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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inequalities.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a54f58cgraphineq23a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Is $$(1,-3)$$ a solution to the system of inequalities $$6x-5y<20$$, $$-2x+7y>-8$$?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a54f58cgraphineq23a-h1","type":"hint","dependencies":[],"title":"Checking Validity in First Equation","text":"Plug the $$x$$ and $$y$$ values of the ordered pair into the inequalities.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a54f58cgraphineq23a-h2","type":"hint","dependencies":["a54f58cgraphineq23a-h1"],"title":"Solution to the System","text":"If the ordered pair makes all of the inequalities in the system true, then it is a solution to the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a54f58cgraphineq24","title":"Determine if an Ordered Pair is a Solution","body":"Determine whether the ordered pair is a solution to the system of inequalities.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a54f58cgraphineq24a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Is $$(-4,4)$$ a solution to the system of inequalities 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graphing.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a54f58cgraphineq25a","stepAnswer":["https://openstax.org/apps/archive/20220118.185250/resources/25e4568262e5473528b84fb990e76e3875748569"],"problemType":"MultipleChoice","stepTitle":"Select the graph where the grey region represents the solution of the system of inequalities y<-2x+2,y>=-x-1.","stepBody":"","answerType":"string","variabilization":{},"choices":["https://openstax.org/apps/archive/20220118.185250/resources/25e4568262e5473528b84fb990e76e3875748569","https://openstax.org/apps/archive/20220118.185250/resources/40923204d07f37575d7520b060300667289d1e2a https://openstax.org/apps/archive/20220118.185250/resources/d6bf71c52b8002980d722d182722b2dee0a74b44 https://openstax.org/apps/archive/20220118.185250/resources/eb64294f4a6a5b5c340589016aa3b909deab759f https://openstax.org/apps/archive/20220118.185250/resources/cb8a835d22d99a0ef2900627ca0563992e7a7a7d https://openstax.org/apps/archive/20220118.185250/resources/25e4568262e5473528b84fb990e76e3875748569"],"hints":{"DefaultPathway":[{"id":"a54f58cgraphineq25a-h1","type":"hint","dependencies":[],"title":"Graphing a System of Linear Inequalities","text":"First, graph the first inequality by graphing the the line $$y=3x+2$$. Graph it as a dotted line if the points on the line aren\'t solutions, and as a solid line if the points are solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a54f58cgraphineq25a-h2","type":"hint","dependencies":["a54f58cgraphineq25a-h1"],"title":"Graphing a System of Linear Inequalities","text":"Lightly shade the region, above or below the line, that represents solutions of the first inequality. Do the same thing with the second inequality-- first graph the line that separates the solutions from the non solutions, and then lightly shade the side of solutions. The overlapping area is the area of solutions for both inequalities, which all the solutions to the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a54f58cgraphineq26","title":"Solve a System of Linear Inequalities by Graphing","body":"Solve the system by graphing.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a54f58cgraphineq26a","stepAnswer":["https://openstax.org/apps/archive/20220118.185250/resources/eb64294f4a6a5b5c340589016aa3b909deab759f"],"problemType":"MultipleChoice","stepTitle":"Select the graph where the grey region represents the solution of the system of inequalities y>=-2x/3+2,y>=2x-3.","stepBody":"","answerType":"string","variabilization":{},"choices":["https://openstax.org/apps/archive/20220118.185250/resources/40923204d07f37575d7520b060300667289d1e2a https://openstax.org/apps/archive/20220118.185250/resources/d6bf71c52b8002980d722d182722b2dee0a74b44 https://openstax.org/apps/archive/20220118.185250/resources/eb64294f4a6a5b5c340589016aa3b909deab759f https://openstax.org/apps/archive/20220118.185250/resources/cb8a835d22d99a0ef2900627ca0563992e7a7a7d https://openstax.org/apps/archive/20220118.185250/resources/25e4568262e5473528b84fb990e76e3875748569","https://openstax.org/apps/archive/20220118.185250/resources/eb64294f4a6a5b5c340589016aa3b909deab759f"],"hints":{"DefaultPathway":[{"id":"a54f58cgraphineq26a-h1","type":"hint","dependencies":[],"title":"Graphing a System of Linear Inequalities","text":"First, graph the first inequality by graphing the the line $$y=\\\\frac{-2x}{3}+2$$. Graph it as a dotted line if the points on the line aren\'t solutions, and as a solid line if the points are solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a54f58cgraphineq26a-h2","type":"hint","dependencies":["a54f58cgraphineq26a-h1"],"title":"Graphing a System of Linear Inequalities","text":"Lightly shade the region, above or below the line, that represents solutions of the first inequality. Do the same thing with the second inequality-- first graph the line that separates the solutions from the non solutions, and then lightly shade the side of solutions. The overlapping area is the area of solutions for both inequalities, which all the solutions to the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a54f58cgraphineq27","title":"Solve a System of Linear Inequalities by Graphing","body":"Solve the system by graphing.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a54f58cgraphineq27a","stepAnswer":["https://openstax.org/apps/archive/20220118.185250/resources/cb8a835d22d99a0ef2900627ca0563992e7a7a7d"],"problemType":"MultipleChoice","stepTitle":"Select the graph where the grey region represents the solution of the system of inequalities x+2y<4,y<x-2.","stepBody":"","answerType":"string","variabilization":{},"choices":["https://openstax.org/apps/archive/20220118.185250/resources/40923204d07f37575d7520b060300667289d1e2a https://openstax.org/apps/archive/20220118.185250/resources/d6bf71c52b8002980d722d182722b2dee0a74b44 https://openstax.org/apps/archive/20220118.185250/resources/eb64294f4a6a5b5c340589016aa3b909deab759f https://openstax.org/apps/archive/20220118.185250/resources/cb8a835d22d99a0ef2900627ca0563992e7a7a7d https://openstax.org/apps/archive/20220118.185250/resources/25e4568262e5473528b84fb990e76e3875748569","https://openstax.org/apps/archive/20220118.185250/resources/cb8a835d22d99a0ef2900627ca0563992e7a7a7d"],"hints":{"DefaultPathway":[{"id":"a54f58cgraphineq27a-h1","type":"hint","dependencies":[],"title":"Graphing a System of Linear Inequalities","text":"First, graph the first inequality by graphing the the line $$x+2y<4$$. Graph it as a dotted line if the points on the line aren\'t solutions, and as a solid line if the points are solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a54f58cgraphineq27a-h2","type":"hint","dependencies":["a54f58cgraphineq27a-h1"],"title":"Graphing a System of Linear Inequalities","text":"Lightly shade the region, above or below the line, that represents solutions of the first inequality. Do the same thing with the second inequality-- first graph the line that separates the solutions from the non solutions, and then lightly shade the side of solutions. The overlapping area is the area of solutions for both inequalities, which all the solutions to the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a54f58cgraphineq3","title":"Determine whether an ordered pair is a solution of a system of linear inequalities","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a54f58cgraphineq3a","stepAnswer":["yes"],"problemType":"MultipleChoice","stepTitle":"Determine () whether $$(-2,1)$$ is a solution to {$$y>4x-2$$, $$4x-y<20$$}","stepBody":"","answerType":"string","variabilization":{},"choices":["yes","no"],"hints":{"DefaultPathway":[{"id":"a54f58cgraphineq3a-h1","type":"hint","dependencies":[],"title":"Plugging in $$x$$ and $$y$$","text":"We must plug in $$x=-2$$ and $$y=1$$ into both inequalities and determine whether they are true. 1>4(-2)-2 is true. 4(-2)-1<20 is true. So, the point is a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a54f58cgraphineq3b","stepAnswer":["no"],"problemType":"MultipleChoice","stepTitle":"Determine () whether $$(4,-1)$$ is a solution to {$$y>4x-2$$, $$4x-y<20$$}","stepBody":"","answerType":"string","variabilization":{},"choices":["yes","no"],"hints":{"DefaultPathway":[{"id":"a54f58cgraphineq3b-h1","type":"hint","dependencies":[],"title":"Plugging in $$x$$ and $$y$$","text":"We must plug in $$x=4$$ and $$y=-1$$ into both inequalities and determine whether they are true. $$-1>4\\\\left(4\\\\right)-2$$ is false. So, the point is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a54f58cgraphineq4","title":"Solve a System of Linear Inequalities by Graphing","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a54f58cgraphineq4a","stepAnswer":["Dark blue"],"problemType":"MultipleChoice","stepTitle":"Solve the system by graphing: {y>=2x-1,y<x+1}. Which shaded region is that of the solutions?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Light red","Dark blue","Light blue"],"hints":{"DefaultPathway":[{"id":"a54f58cgraphineq4a-h1","type":"hint","dependencies":[],"title":"Graph the First Inequality","text":"We must graph $$y \\\\geq 2x-1$$. We graph the line as solid, and shade above since we have $$ \\\\geq $$.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a54f58cgraphineq4a-h2","type":"hint","dependencies":["a54f58cgraphineq4a-h1"],"title":"Graph the Second Inequality","text":"We must graph $$y<x+1$$. We graph the line as dashed, and shade below since we have <.\\\\n##figure3.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a54f58cgraphineq4a-h3","type":"hint","dependencies":["a54f58cgraphineq4a-h2"],"title":"The solution is region where shading overlaps: Dark blue.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a54f58cgraphineq5","title":"Solve a System of Linear Inequalities by Graphing","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a54f58cgraphineq5a","stepAnswer":["$$Blue-red$$"],"problemType":"MultipleChoice","stepTitle":"Solve the system by graphing: {y<3x+2,y>-x+1}. Which shaded region is that of the solutions?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Blue","Red","$$Blue-red$$"],"hints":{"DefaultPathway":[{"id":"a54f58cgraphineq5a-h1","type":"hint","dependencies":[],"title":"Graph the First Inequality","text":"We must graph $$y<3x+2$$. We graph the line as dashed, and shade below since we have <.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a54f58cgraphineq5a-h2","type":"hint","dependencies":["a54f58cgraphineq5a-h1"],"title":"Graph the Second Inequality","text":"We must graph $$y>-x+1$$. We graph the line as dashed, and shade above since we have >\\\\n##figure3.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a54f58cgraphineq5a-h3","type":"hint","dependencies":["a54f58cgraphineq5a-h2"],"title":"The solution is region where shading overlaps Blue-red","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a54f58cgraphineq6","title":"Solve a System of Linear Inequalities by Graphing","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a54f58cgraphineq6a","stepAnswer":["$$Blue-red$$"],"problemType":"MultipleChoice","stepTitle":"Solve the system by graphing: {$$y<\\\\frac{-1}{2} x+3$$, $$y<3x-4$$.} Which shaded region is that of the solutions?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Blue","Red","$$Blue-red$$"],"hints":{"DefaultPathway":[{"id":"a54f58cgraphineq6a-h1","type":"hint","dependencies":[],"title":"Graph the First Inequality","text":"We must graph $$y<\\\\frac{-1}{2} x+3$$. We graph the line as dashed, and shade below since we have <.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a54f58cgraphineq6a-h2","type":"hint","dependencies":["a54f58cgraphineq6a-h1"],"title":"Graph the Second Inequality","text":"We must graph $$y<3x-4$$. We graph the line as dashed, and shade below since we have <.\\\\n##figure3.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a54f58cgraphineq6a-h3","type":"hint","dependencies":["a54f58cgraphineq6a-h2"],"title":"The solution is region where shading overlaps: Blue-red","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a54f58cgraphineq7","title":"Solve a System of Linear Inequalities by Graphing","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a54f58cgraphineq7a","stepAnswer":["$$Blue-red$$"],"problemType":"MultipleChoice","stepTitle":"Solve the system by graphing: {$$x-y>3$$, $$y<\\\\frac{-1}{5} x+4$$.} Which shaded region is that of the solutions?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Blue","Red","$$Blue-red$$"],"hints":{"DefaultPathway":[{"id":"a54f58cgraphineq7a-h1","type":"hint","dependencies":[],"title":"Graph the First Inequality","text":"We must graph $$x-y>3$$, which rearranges to $$y<x-3$$. We make the line dashed and shade below it since we have <.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a54f58cgraphineq7a-h2","type":"hint","dependencies":["a54f58cgraphineq7a-h1"],"title":"Graph the Second Inequality","text":"We must graph $$y<\\\\frac{-1}{5} x+4$$. We make the line dashed and shade below it since we have <.\\\\n##figure3.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a54f58cgraphineq7a-h3","type":"hint","dependencies":["a54f58cgraphineq7a-h2"],"title":"The solution is region where shading overlaps: Blue-red","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a54f58cgraphineq8","title":"Solve a System of Linear Inequalities by Graphing","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a54f58cgraphineq8a","stepAnswer":["$$Blue-red$$"],"problemType":"MultipleChoice","stepTitle":"Solve the system by graphing: {$$x+y \\\\leq 2$$, $$y \\\\geq \\\\frac{2}{3} x-1$$.} Which shaded region is that of the solutions?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Blue","Red","$$Blue-red$$"],"hints":{"DefaultPathway":[{"id":"a54f58cgraphineq8a-h1","type":"hint","dependencies":[],"title":"Graph the First Inequality","text":"We must graph $$x+y \\\\leq 2$$, which rearranges to $$y \\\\leq 2-x$$. We make the line solid and shade below it since we have $$ \\\\leq $$.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a54f58cgraphineq8a-h2","type":"hint","dependencies":["a54f58cgraphineq8a-h1"],"title":"Graph the Second Inequality","text":"We must graph $$y \\\\geq \\\\frac{2}{3} x-1$$. We make the line solid and shade above it since we have >.\\\\n##figure3.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a54f58cgraphineq8a-h3","type":"hint","dependencies":["a54f58cgraphineq8a-h2"],"title":"The solution is region where shading overlaps: Blue-red","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a54f58cgraphineq9","title":"Solve a System of Linear Inequalities by Graphing","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a54f58cgraphineq9a","stepAnswer":["$$Blue-red$$"],"problemType":"MultipleChoice","stepTitle":"Solve the system by graphing: {$$3x-2y \\\\leq 6$$, $$y>\\\\frac{-1}{4} x+5$$.} Which shaded region is that of the solutions?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Blue","Red","$$Blue-red$$"],"hints":{"DefaultPathway":[{"id":"a54f58cgraphineq9a-h1","type":"hint","dependencies":[],"title":"Graph the First Inequality","text":"We must graph $$3x-2y \\\\leq 6$$, which rearranges to $$y \\\\geq \\\\frac{-\\\\left(6-3x\\\\right)}{2}$$. We make the line solid and shade above it since we have $$ \\\\geq $$.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a54f58cgraphineq9a-h2","type":"hint","dependencies":["a54f58cgraphineq9a-h1"],"title":"Graph the Second Inequality","text":"We must graph $$y>\\\\frac{-1}{4} x+5$$. We make the line dashed and shade above it since we have >.\\\\n##figure3.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a54f58cgraphineq9a-h3","type":"hint","dependencies":["a54f58cgraphineq9a-h2"],"title":"The solution is region where shading overlaps: Blue-red","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57cb5ccenter1","title":"Calculate Mean and Median","body":"AIDS data indicating the number of months a patient with AIDS lives after taking a new antibody drug are as follows (smallest to largest): 3; 4; 8; 8; 10; 11; 12; 13; 14; 15; 15; 16; 16; 17; 17; 18; 21; 22; 22; 24; 24; 25; 26; 26; 27; 27; 29; 29; 31; 32; 33; 33; 34; 34; 35; 37; 40; 44; 44; $$47$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Measures of the Center of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a57cb5ccenter1a","stepAnswer":["$$23.6$$"],"problemType":"TextBox","stepTitle":"Calculate the mean.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$23.6$$","hints":{"DefaultPathway":[{"id":"a57cb5ccenter1a-h1","type":"hint","dependencies":[],"title":"Definition of Mean","text":"The mean (also called the average) is formed by adding up all the data points and then divide by the total number of data points.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$944$$"],"dependencies":["a57cb5ccenter1a-h1"],"title":"Finding the Total Sum","text":"What is the total sum of all the data points?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$40$$"],"dependencies":["a57cb5ccenter1a-h2"],"title":"Finding the Total Number of Data Points","text":"How many total data points are there? In other words, how many numbers did you sum together in the last part?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$23.6$$"],"dependencies":["a57cb5ccenter1a-h3"],"title":"Calculating Mean from Total Sum and Total Number of Data Points","text":"What is the mean?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a57cb5ccenter1a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$23.6$$"],"dependencies":[],"title":"Calculating Mean from Total Sum and Total Number of Data Points","text":"What is $$\\\\frac{944}{40}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}},{"id":"a57cb5ccenter1b","stepAnswer":["$$24$$"],"problemType":"TextBox","stepTitle":"Calculate the median.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$24$$","hints":{"DefaultPathway":[{"id":"a57cb5ccenter1b-h5","type":"hint","dependencies":["a57cb5ccenter1a-h4"],"title":"Find the Location of the Median","text":"To find the median of a data set, we want to first determine the location of the median. Remember the formula for that data point is $$\\\\frac{n+1}{2}$$ where $$n$$ represents the total number of data points in the set.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter1b-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20.5$$"],"dependencies":["a57cb5ccenter1b-h5"],"title":"Determining Location of the Median","text":"What is the location of the median? What is $$\\\\frac{n+1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a57cb5ccenter1b-h6-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20.5$$"],"dependencies":[],"title":"Determining Location of the Median","text":"What is $$\\\\frac{40+1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a57cb5ccenter1b-h7","type":"hint","dependencies":["a57cb5ccenter1b-h6"],"title":"Finding the Correct Data Points to Calculate Median","text":"Because the location of the median is $$20.5$$, this means we need to find the average between the 20th and 21st data points. Since the data is in order, we need to just find those two data points.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter1b-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$24$$, $$24$$"],"dependencies":["a57cb5ccenter1b-h7"],"title":"Finding the 20th and 21st Data Values","text":"What are the 20th and 21st data values?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$24$$, $$24$$","$$22$$, $$24$$","$$24$$, $$25$$","$$25$$, $$26$$"]},{"id":"a57cb5ccenter1b-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$24$$"],"dependencies":["a57cb5ccenter1b-h8"],"title":"Calculating Median from Data Points","text":"What is the median? In other words, what is the average between the 20th and 21st data values?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a57cb5ccenter1b-h9-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$24$$"],"dependencies":[],"title":"Calculating Median from Data Points","text":"What is $$\\\\frac{24+24}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a57cb5ccenter10","title":"Estimating Mean","body":"Use the following table in the calculations for this problem.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Measures of the Center of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a57cb5ccenter10a","stepAnswer":["$$10.78$$"],"problemType":"TextBox","stepTitle":"Find the best estimate of the mean.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10.78$$","hints":{"DefaultPathway":[{"id":"a57cb5ccenter10a-h1","type":"hint","dependencies":[],"title":"Formula for Calculating Mean from a Frequency Table","text":"To calculate mean from a frequency table, we want to first find all the midpoints of the grade intervals, and then sum up the product of each interval frequency with the midpoint. Lastly, we\'ll need to divide that sum by the total frequency. Let\'s first start by finding all the midpoints.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$54.5$$"],"dependencies":["a57cb5ccenter10a-h1"],"title":"Midpoint for Interval $$49.5-59.5$$","text":"What is the midpoint for the interval $$49.5-59.5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a57cb5ccenter10a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$54.5$$"],"dependencies":[],"title":"Midpoint for Interval $$49.5-59.5$$","text":"What is $$\\\\frac{49.5+59.5}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a57cb5ccenter10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$64.5$$"],"dependencies":["a57cb5ccenter10a-h2"],"title":"Midpoint for Interval $$59.5-69.5$$","text":"What is the midpoint for the interval $$59.5-69.5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$74.5$$"],"dependencies":["a57cb5ccenter10a-h3"],"title":"Midpoint for Interval $$69.5-79.5$$","text":"What is the midpoint for the interval $$69.5-79.5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$84.5$$"],"dependencies":["a57cb5ccenter10a-h4"],"title":"Midpoint for Interval $$79.5-89.5$$","text":"What is the midpoint for the interval $$79.5-89.5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter10a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$94.5$$"],"dependencies":["a57cb5ccenter10a-h5"],"title":"Midpoint for Interval $$89.5-99.5$$","text":"What is the midpoint for the interval $$89.5-99.5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter10a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2385$$"],"dependencies":["a57cb5ccenter10a-h6"],"title":"Determining Total Sum","text":"Now that we know each midpoint for each interval, what is the sum of the product of each interval frequency (right column) and midpoint (just calculated)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a57cb5ccenter10a-h7-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2385$$"],"dependencies":[],"title":"Determining Total Sum","text":"What is $$54.5\\\\times2+64.5\\\\times3+74.5\\\\times8+84.5\\\\times12+94.5\\\\times5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a57cb5ccenter10a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$30$$"],"dependencies":["a57cb5ccenter10a-h7"],"title":"Determining Total Frequency","text":"What is the total frequency? As in, how many data points are in the table?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter10a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$79.5$$"],"dependencies":["a57cb5ccenter10a-h8"],"title":"Estimating the Mean","text":"What is an estimate of the mean? In other words, what is the total sum that we calculated divided by the total frequency?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57cb5ccenter11","title":"Estimating Mean","body":"Use the following table in the calculations for this problem.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Measures of the Center of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a57cb5ccenter11a","stepAnswer":["$$60.94$$"],"problemType":"TextBox","stepTitle":"Find the best estimate of the mean. 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Let\'s first start by finding all the midpoints.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$54.5$$"],"dependencies":["a57cb5ccenter11a-h1"],"title":"Midpoint for Interval $$49.5-59.5$$","text":"What is the midpoint for the interval $$49.5-59.5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a57cb5ccenter11a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$54.5$$"],"dependencies":[],"title":"Midpoint for Interval $$49.5-59.5$$","text":"What is $$\\\\frac{49.5+59.5}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a57cb5ccenter11a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$64.5$$"],"dependencies":["a57cb5ccenter11a-h2"],"title":"Midpoint for Interval $$59.5-69.5$$","text":"What is the midpoint for the interval $$59.5-69.5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$74.5$$"],"dependencies":["a57cb5ccenter11a-h3"],"title":"Midpoint for Interval $$69.5-79.5$$","text":"What is the midpoint for the interval $$69.5-79.5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$84.5$$"],"dependencies":["a57cb5ccenter11a-h4"],"title":"Midpoint for Interval $$79.5-89.5$$","text":"What is the midpoint for the interval $$79.5-89.5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter11a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$94.5$$"],"dependencies":["a57cb5ccenter11a-h5"],"title":"Midpoint for Interval $$89.5-99.5$$","text":"What is the midpoint for the interval $$89.5-99.5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter11a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6154.5$$"],"dependencies":["a57cb5ccenter11a-h6"],"title":"Determining Total Sum","text":"Now that we know each midpoint for each interval, what is the sum of the product of each interval frequency (right column) and midpoint (just calculated)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a57cb5ccenter11a-h7-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6154.5$$"],"dependencies":[],"title":"Determining Total Sum","text":"What is $$54.5\\\\times53+64.5\\\\times32+74.5\\\\times15+84.5\\\\times1+94.5\\\\times0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a57cb5ccenter11a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$101$$"],"dependencies":["a57cb5ccenter11a-h7"],"title":"Determining Total Frequency","text":"What is the total frequency? As in, how many data points are in the table?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter11a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$60.94$$"],"dependencies":["a57cb5ccenter11a-h8"],"title":"Estimating the Mean","text":"What is an estimate of the mean? In other words, what is the total sum that we calculated divided by the total frequency? Round to the nearest hundredths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57cb5ccenter12","title":"Estimating Mean","body":"Use the following table in the calculations for this problem.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Measures of the Center of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a57cb5ccenter12a","stepAnswer":["$$70.66$$"],"problemType":"TextBox","stepTitle":"Find the best estimate of the mean. 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Let\'s first start by finding all the midpoints.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$54.5$$"],"dependencies":["a57cb5ccenter12a-h1"],"title":"Midpoint for Interval $$49.5-59.5$$","text":"What is the midpoint for the interval $$49.5-59.5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a57cb5ccenter12a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$54.5$$"],"dependencies":[],"title":"Midpoint for Interval $$49.5-59.5$$","text":"What is $$\\\\frac{49.5+59.5}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a57cb5ccenter12a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$64.5$$"],"dependencies":["a57cb5ccenter12a-h2"],"title":"Midpoint for Interval $$59.5-69.5$$","text":"What is the midpoint for the interval $$59.5-69.5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$74.5$$"],"dependencies":["a57cb5ccenter12a-h3"],"title":"Midpoint for Interval $$69.5-79.5$$","text":"What is the midpoint for the interval $$69.5-79.5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter12a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$84.5$$"],"dependencies":["a57cb5ccenter12a-h4"],"title":"Midpoint for Interval $$79.5-89.5$$","text":"What is the midpoint for the interval $$79.5-89.5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter12a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$94.5$$"],"dependencies":["a57cb5ccenter12a-h5"],"title":"Midpoint for Interval $$89.5-99.5$$","text":"What is the midpoint for the interval $$89.5-99.5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter12a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6077$$"],"dependencies":["a57cb5ccenter12a-h6"],"title":"Determining Total Sum","text":"Now that we know each midpoint for each interval, what is the sum of the product of each interval frequency (right column) and midpoint (just calculated)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a57cb5ccenter12a-h7-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6077$$"],"dependencies":[],"title":"Determining Total Sum","text":"What is $$54.5\\\\times14+64.5\\\\times32+74.5\\\\times15+84.5\\\\times23+94.5\\\\times2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a57cb5ccenter12a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$86$$"],"dependencies":["a57cb5ccenter12a-h7"],"title":"Determining Total Frequency","text":"What is the total frequency? As in, how many data points are in the table?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter12a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$70.66$$"],"dependencies":["a57cb5ccenter12a-h8"],"title":"Estimating the Mean","text":"What is an estimate of the mean? In other words, what is the total sum that we calculated divided by the total frequency? Round to the nearest hundredths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57cb5ccenter13","title":"Calculating Mean","body":"The following data show the lengths of boats moored in a marina. The data are ordered from smallest to largest: 16; 17; 19; 20; 20; 21; 23; 24; 25; 25; 25; 26; 26; 27; 27; 27; 28; 29; 30; 32; 33; 33; 34; 35; 37; 39; $$40$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Measures of the Center of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a57cb5ccenter13a","stepAnswer":["$$27.33$$"],"problemType":"TextBox","stepTitle":"Calculate the mean. 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In other words, how many numbers did you sum together in the last part?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$27.33$$"],"dependencies":["a57cb5ccenter13a-h3"],"title":"Calculating Mean from Total Sum and Total Number of Data Points","text":"What is the mean? Round your answer to the nearest hundredths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a57cb5ccenter13a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$27.33$$"],"dependencies":[],"title":"Calculating Mean from Total Sum and Total Number of Data Points","text":"What is $$\\\\frac{738}{27}$$ rounded to the nearest hundredths place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a57cb5ccenter14","title":"Calculating Median","body":"The following data show the lengths of boats moored in a marina. The data are ordered from smallest to largest: 16; 17; 19; 20; 20; 21; 23; 24; 25; 25; 25; 26; 26; 27; 27; 27; 28; 29; 30; 32; 33; 33; 34; 35; 37; 39; $$40$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Measures of the Center of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a57cb5ccenter14a","stepAnswer":["$$27$$"],"problemType":"TextBox","stepTitle":"Calculate the Median.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$27$$","hints":{"DefaultPathway":[{"id":"a57cb5ccenter14a-h1","type":"hint","dependencies":[],"title":"Find the Location of the Median","text":"To find the median of a data set, we want to first determine the location of the median. Remember the formula for that data point is $$\\\\frac{n+1}{2}$$ where $$n$$ represents the total number of data points in the set.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$14$$"],"dependencies":["a57cb5ccenter14a-h1"],"title":"Determining Location of the Median","text":"What is the location of the median? What is $$\\\\frac{n+1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a57cb5ccenter14a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$14$$"],"dependencies":[],"title":"Determining Location of the Median","text":"What is $$\\\\frac{27+1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a57cb5ccenter14a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$27$$"],"dependencies":["a57cb5ccenter14a-h2"],"title":"Finding the 14th Data Value","text":"What is the median? In other words, what is the 14th data point?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57cb5ccenter15","title":"Calculating Mode","body":"The following data show the lengths of boats moored in a marina. The data are ordered from smallest to largest: 16; 17; 19; 20; 20; 21; 23; 24; 25; 25; 25; 26; 26; 27; 27; 27; 28; 29; 30; 32; 33; 33; 34; 35; 37; 39; $$40$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Measures of the Center of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a57cb5ccenter15a","stepAnswer":["$$25$$, $$27$$"],"problemType":"MultipleChoice","stepTitle":"Calculate the Mode(s) of the data set.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$25$$, $$27$$","choices":["$$25$$, $$27$$","$$25$$","$$26$$","$$26$$, $$27$$"],"hints":{"DefaultPathway":[{"id":"a57cb5ccenter15a-h1","type":"hint","dependencies":[],"title":"Definition of Mode","text":"The mode of a data set is the most frequent value that occurs in that data set. For instance, for a data set containing the values 1; 1; 2; $$3$$, the mode would be $$1$$ as it appears twice (so has a frequency of 2) while $$2$$ and $$3$$ only appear once in the data set. Let\'s determine the frequencies of each of the different values to find the mode.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a57cb5ccenter15a-h1"],"title":"Determining Frequency of $$16$$","text":"How many times does $$16$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a57cb5ccenter15a-h2"],"title":"Determining Frequency of $$17$$","text":"How many times does $$17$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a57cb5ccenter15a-h3"],"title":"Determining Frequency of $$19$$","text":"How many times does $$19$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter15a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a57cb5ccenter15a-h4"],"title":"Determining Frequency of $$20$$","text":"How many times does $$20$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter15a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a57cb5ccenter15a-h5"],"title":"Determining Frequency of $$21$$","text":"How many times does $$21$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter15a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a57cb5ccenter15a-h6"],"title":"Determining Frequency of $$23$$","text":"How many times does $$23$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter15a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a57cb5ccenter15a-h7"],"title":"Determining Frequency of $$24$$","text":"How many times does $$24$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter15a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a57cb5ccenter15a-h8"],"title":"Determining Frequency of $$25$$","text":"How many times does $$25$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter15a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a57cb5ccenter15a-h9"],"title":"Determining Frequency of $$26$$","text":"How many times does $$26$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter15a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a57cb5ccenter15a-h10"],"title":"Determining Frequency of $$27$$","text":"How many times does $$27$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter15a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a57cb5ccenter15a-h11"],"title":"Determining Frequency of $$28$$","text":"How many times does $$28$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter15a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a57cb5ccenter15a-h12"],"title":"Determining Frequency of $$29$$","text":"How many times does $$29$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter15a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a57cb5ccenter15a-h13"],"title":"Determining Frequency of $$30$$","text":"How many times does $$30$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter15a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a57cb5ccenter15a-h14"],"title":"Determining Frequency of $$32$$","text":"How many times does $$32$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter15a-h16","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a57cb5ccenter15a-h15"],"title":"Determining Frequency of $$33$$","text":"How many times does $$33$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter15a-h17","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a57cb5ccenter15a-h16"],"title":"Determining Frequency of $$34$$","text":"How many times does $$34$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter15a-h18","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a57cb5ccenter15a-h17"],"title":"Determining Frequency of $$35$$","text":"How many times does $$35$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter15a-h19","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a57cb5ccenter15a-h18"],"title":"Determining Frequency of $$37$$","text":"How many times does $$37$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter15a-h20","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a57cb5ccenter15a-h19"],"title":"Determining Frequency of $$39$$","text":"How many times does $$39$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter15a-h21","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a57cb5ccenter15a-h20"],"title":"Determining Frequency of $$40$$","text":"How many times does $$40$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter15a-h22","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$25$$, $$27$$"],"dependencies":["a57cb5ccenter15a-h21"],"title":"Finding the Mode","text":"What are the modes? What data points showed up thrice each in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$25$$, $$27$$","$$25$$","$$26$$","$$26$$, $$27$$"]}]}}]},{"id":"a57cb5ccenter16","title":"Calculating Sample Mean","body":"Sixty-five randomly selected car salespersons were asked the number of cars they generally sell in one week. Fourteen people answered that they generally sell three cars; nineteen generally sell four cars; twelve generally sell five cars; nine generally sell six cars; eleven generally sell seven cars.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Measures of the Center of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a57cb5ccenter16a","stepAnswer":["$$4.75$$"],"problemType":"TextBox","stepTitle":"Calculate the sample mean. Round to the nearest hundredths place.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4.75$$","hints":{"DefaultPathway":[{"id":"a57cb5ccenter16a-h1","type":"hint","dependencies":[],"title":"Definition of Sample Mean","text":"The sample mean is a type of mean that specifically focuses on a randomly selected sample of the broader population. This mean is formed by adding up all the data points and then divide by the total number of data points, like a normal mean is.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$309$$"],"dependencies":["a57cb5ccenter16a-h1"],"title":"Finding the Total Sum","text":"What is the total sum of all the data points? We\'ll note in this specific case that to get the total sum of the data points, we\'ll have to multiply each data value by its frequency. For instance, a data point is $$14$$ and its frequency is $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter16a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$309$$"],"dependencies":["a57cb5ccenter16a-h2"],"title":"Finding the Total Sum","text":"What is $$14\\\\times3+19\\\\times4+12\\\\times5+9\\\\times6+11\\\\times7$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter16a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$65$$"],"dependencies":["a57cb5ccenter16a-h3"],"title":"Finding the Total Number of Data Points","text":"How many total data points are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter16a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4.75$$"],"dependencies":["a57cb5ccenter16a-h4"],"title":"Calculating Mean from Total Sum and Total Number of Data Points","text":"What is the mean? Round your answer to the nearest hundredths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a57cb5ccenter16a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4.75$$"],"dependencies":[],"title":"Calculating Mean from Total Sum and Total Number of Data Points","text":"What is $$\\\\frac{309}{65}$$ rounded to the nearest hundredths place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a57cb5ccenter17","title":"Calculating Median","body":"Sixty-five randomly selected car salespersons were asked the number of cars they generally sell in one week. Fourteen people answered that they generally sell three cars; nineteen generally sell four cars; twelve generally sell five cars; nine generally sell six cars; eleven generally sell seven cars.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Measures of the Center of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a57cb5ccenter17a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"Calculate the median.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a57cb5ccenter17a-h1","type":"hint","dependencies":[],"title":"Find the Location of the Median","text":"To find the median of a data set, we want to first determine the location of the median. Remember the formula for that data point is $$\\\\frac{n+1}{2}$$ where $$n$$ represents the total number of data points in the set.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter17a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$33$$"],"dependencies":["a57cb5ccenter17a-h1"],"title":"Determining Location of the Median","text":"What is the location of the median? What is $$\\\\frac{n+1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a57cb5ccenter17a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$33$$"],"dependencies":[],"title":"Determining Location of the Median","text":"What is $$\\\\frac{65+1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a57cb5ccenter17a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a57cb5ccenter17a-h2"],"title":"Finding the 33rd Data Value","text":"What is the median? In other words, what is the 33rd data point?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter17a-h4","type":"hint","dependencies":["a57cb5ccenter17a-h3"],"title":"Determining Where the 33rd Data Point Is","text":"To determine where the 33rd data point is at, we know that the frequency of selling $$3$$ cars is $$14$$, the frequency of selling $$4$$ cars is $$19$$, and the frequency of selling $$5$$ cars is $$12$$. To find the 33rd data point, add up the frequencies until you get to $$33$$ and that number of cars will be the median value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter17a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$14$$"],"dependencies":["a57cb5ccenter17a-h4"],"title":"Subset to Determining Location of 33rd Data Point","text":"What is the total frequency of people who have sold $$3$$ cars (the minimum)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter17a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$33$$"],"dependencies":["a57cb5ccenter17a-h5"],"title":"Subset to Determining Location of 33rd Data Point","text":"What is the total frequency of people who have sold $$3$$ or $$4$$ cars? Add up the frequencies.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter17a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a57cb5ccenter17a-h6"],"title":"Subset to Determining Location of 33rd Data Point","text":"Is $$33$$, the total frequency of selling $$3$$ or $$4$$ cars, greater than or equal to $$33$$, the desired location of the median?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a57cb5ccenter17a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$4$$ cars"],"dependencies":["a57cb5ccenter17a-h7"],"title":"Final Step to Determining 33rd Data Point","text":"If $$14$$ was the total frequency of selling $$3$$ cars and $$33$$ was the total frequency of selling $$3$$ or $$4$$ cars, is the 33rd data point (the median value) selling $$3$$ cars or $$4$$ cars?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$3$$ cars","$$4$$ cars"]}]}}]},{"id":"a57cb5ccenter18","title":"Calculating Median","body":"Sixty-five randomly selected car salespersons were asked the number of cars they generally sell in one week. Fourteen people answered that they generally sell three cars; nineteen generally sell four cars; twelve generally sell five cars; nine generally sell six cars; eleven generally sell seven cars.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Measures of the Center of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a57cb5ccenter18a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"Calculate the mode.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a57cb5ccenter18a-h1","type":"hint","dependencies":[],"title":"Definition of Mode","text":"The mode of a data set is the most frequent value that occurs in that data set. For instance, for a data set containing the values 1; 1; 2; $$3$$, the mode would be $$1$$ as it appears twice (so has a frequency of 2) while $$2$$ and $$3$$ only appear once in the data set. Let\'s determine the frequencies of each of the different values to find the mode.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$14$$"],"dependencies":["a57cb5ccenter18a-h1"],"title":"Determining Frequency of Selling $$3$$ Cars","text":"How many times does selling $$3$$ cars show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter18a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$14$$"],"dependencies":["a57cb5ccenter18a-h2"],"title":"Determining Frequency of Selling $$3$$ Cars","text":"How many of the $$65$$ randomly selected car salespersons said they generally sell three cars?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter18a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$19$$"],"dependencies":["a57cb5ccenter18a-h3"],"title":"Determining Frequency of Selling $$4$$ Cars","text":"How many times does selling $$4$$ cars show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter18a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a57cb5ccenter18a-h4"],"title":"Determining Frequency of Selling $$5$$ Cars","text":"How many times does selling $$5$$ cars show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter18a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a57cb5ccenter18a-h5"],"title":"Determining Frequency of Selling $$6$$ Cars","text":"How many times does selling $$6$$ cars show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter18a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11$$"],"dependencies":["a57cb5ccenter18a-h6"],"title":"Determining Frequency of Selling $$7$$ Cars","text":"How many times does selling $$7$$ cars show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter18a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a57cb5ccenter18a-h7"],"title":"Finding the Mode","text":"What is the mode? What data point (number of cars sold) showed up the most in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57cb5ccenter19","title":"Estimating the Mean","body":"The most obese countries in the world have obesity rates that range from $$11.4\\\\%$$ to $$74.6\\\\%$$. This data is summarized in the following table.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Measures of the Center of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a57cb5ccenter19a","stepAnswer":["$$23.32$$"],"problemType":"TextBox","stepTitle":"Find the best estimate of the mean. Round to the nearest hundredths place.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$23.32$$","hints":{"DefaultPathway":[{"id":"a57cb5ccenter19a-h1","type":"hint","dependencies":[],"title":"Formula for Calculating Mean from a Frequency Table","text":"To calculate mean from a frequency table, we want to first find all the midpoints of the grade intervals, and then sum up the product of each interval frequency with the midpoint. Lastly, we\'ll need to divide that sum by the total frequency. 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As in, how many data points are in the table?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter19a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$23.32$$"],"dependencies":["a57cb5ccenter19a-h11"],"title":"Estimating the Mean","text":"What is an estimate of the mean? In other words, what is the total sum that we calculated divided by the total frequency? Round your answer to the nearest hundredths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a57cb5ccenter19b","stepAnswer":["Above"],"problemType":"MultipleChoice","stepTitle":"The United States has an average obesity rate of $$33.9\\\\%$$. Is this rate above average or below?","stepBody":"","answerType":"string","variabilization":{},"choices":["Above","Below"],"hints":{"DefaultPathway":[{"id":"a57cb5ccenter19b-h13","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Greater Than"],"dependencies":["a57cb5ccenter19a-h12"],"title":"Numerical Analysis on Percentages","text":"Is $$33.9\\\\%$$ (the United States average) greater than $$23.32\\\\%$$, what we found to be an estimate of the average obesity percentage for the countries in the table?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Greater Than","Less Than"]}]}}]},{"id":"a57cb5ccenter2","title":"Calculate Mean and Median","body":"The following data show the number of months patients typically wait on a transplant list before getting surgery. The data are ordered from smallest to largest. 3; 4; 5; 7; 7; 7; 7; 8; 8; 9; 9; 10; 10; 10; 10; 10; 11; 12; 12; 13; 14; 14; 15; 15; 17; 17; 18; 19; 19; 19; 21; 21; 22; 22; 23; 24; 24; 24; $$24$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Measures of the Center of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a57cb5ccenter2a","stepAnswer":["$$13.95$$"],"problemType":"TextBox","stepTitle":"Calculate the mean. 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In other words, how many numbers did you sum together in the last part?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$13.95$$"],"dependencies":["a57cb5ccenter2a-h3"],"title":"Calculating Mean from Total Sum and Total Number of Data Points","text":"What is the mean? 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Remember the formula for that data point is $$\\\\frac{n+1}{2}$$ where $$n$$ represents the total number of data points in the set.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter2b-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a57cb5ccenter2b-h5"],"title":"Determining Location of the Median","text":"What is the location of the median? What is $$\\\\frac{n+1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a57cb5ccenter2b-h6-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":[],"title":"Determining Location of the Median","text":"What is $$\\\\frac{39+1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a57cb5ccenter2b-h7","type":"hint","dependencies":["a57cb5ccenter2b-h6"],"title":"Finding the Correct Data Points to Calculate Median","text":"Because the location of the median is $$20$$, a whole number, this means that the median is just the data point at that location.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter2b-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$13$$"],"dependencies":["a57cb5ccenter2b-h7"],"title":"Finding the 20th Data Value","text":"What is the median? 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As in, how many data points are in the table?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter20a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$26.57$$"],"dependencies":["a57cb5ccenter20a-h9"],"title":"Estimating the Mean","text":"What is an estimate of the mean? In other words, what is the total sum that we calculated divided by the total frequency? Round your answer to the nearest hundredths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57cb5ccenter3","title":"Determining Best Measure of the \\"Center\\"","body":"Suppose that in a small town of $$50$$ people, one person earns $5,000,000 per year and the other $$49$$ each earn $30,000.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Measures of the Center of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a57cb5ccenter3a","stepAnswer":["$$129400$$"],"problemType":"TextBox","stepTitle":"What is the mean of the data set?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$129400$$","hints":{"DefaultPathway":[{"id":"a57cb5ccenter3a-h1","type":"hint","dependencies":[],"title":"Definition of Mean","text":"The mean (also called the average) is formed by adding up all the data points and then divide by the total number of data points.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6470000$$"],"dependencies":["a57cb5ccenter3a-h1"],"title":"Finding the Total Sum","text":"What is the total sum of the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$50$$"],"dependencies":["a57cb5ccenter3a-h2"],"title":"Finding the Total Number of Data Points","text":"What is the total number of data points in the set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$129400$$"],"dependencies":["a57cb5ccenter3a-h3"],"title":"Determining Mean from Total Sum and Total Number of Data Points","text":"What is the mean? What is the total sum divided by the total number of data points?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a57cb5ccenter3b","stepAnswer":["$$30000$$"],"problemType":"TextBox","stepTitle":"What is the median of the data set?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$30000$$","hints":{"DefaultPathway":[{"id":"a57cb5ccenter3b-h5","type":"hint","dependencies":["a57cb5ccenter3a-h4"],"title":"Find the Location of the Median","text":"To find the median of a data set, we want to first determine the location of the median. Remember the formula for that data point is $$\\\\frac{n+1}{2}$$ where $$n$$ represents the total number of data points in the set.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter3b-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$30000$$"],"dependencies":["a57cb5ccenter3b-h5"],"title":"Finding Median","text":"Note that out of the $$50$$ data points, we have one at $5,000,000 and the other $$49$$ each at $30,000. The location of the median will be at the $$\\\\frac{25+1}{2}=13$$ data point. What is the 13th data point?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a57cb5ccenter3c","stepAnswer":["Median"],"problemType":"MultipleChoice","stepTitle":"Which is the better measure of the \\"center\\": the mean or the median?","stepBody":"","answerType":"string","variabilization":{},"choices":["Median","Mean"],"hints":{"DefaultPathway":[{"id":"a57cb5ccenter3c-h7","type":"hint","dependencies":["a57cb5ccenter3b-h6"],"title":"Finding Outliers","text":"In order to determine which is the better measure of the data, let\'s consider if there are outliers. Note that if there are, the median may be a better measure of the \\"center\\" of the data as the median is less pulled by outliers to skew than the mean is.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter3c-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["5,000,000"],"dependencies":["a57cb5ccenter3c-h7"],"title":"Determining the Outliers","text":"What are the outliers?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["5,000,000","30,000","5,000,000 and 30,000","There are none."]},{"id":"a57cb5ccenter3c-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Median"],"dependencies":["a57cb5ccenter3c-h8"],"title":"Determining the Overall Better Measure","text":"Now, knowing that there is an outlier, based on the previous hints given, is the median a better measure of the \\"center\\" (or middle of the data) or the mean?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Median","Mean"]}]}}]},{"id":"a57cb5ccenter4","title":"Determining Best Measure of the \\"Center\\"","body":"In a sample of $$60$$ households, one house is worth $2,500,000. Twenty-nine houses are worth $280,000, and all the others are worth $315,000.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Measures of the Center of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a57cb5ccenter4a","stepAnswer":["$$334500$$"],"problemType":"TextBox","stepTitle":"What is the mean of the data set?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$334500$$","hints":{"DefaultPathway":[{"id":"a57cb5ccenter4a-h1","type":"hint","dependencies":[],"title":"Definition of Mean","text":"The mean (also called the average) is formed by adding up all the data points and then divide by the total number of data points.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20070000$$"],"dependencies":["a57cb5ccenter4a-h1"],"title":"Finding the Total Sum","text":"What is the total sum of the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$60$$"],"dependencies":["a57cb5ccenter4a-h2"],"title":"Finding the Total Number of Data Points","text":"What is the total number of data points in the set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$334500$$"],"dependencies":["a57cb5ccenter4a-h3"],"title":"Determining Mean from Total Sum and Total Number of Data Points","text":"What is the mean? What is the total sum divided by the total number of data points?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a57cb5ccenter4b","stepAnswer":["$$315000$$"],"problemType":"TextBox","stepTitle":"What is the median of the data set?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$315000$$","hints":{"DefaultPathway":[{"id":"a57cb5ccenter4b-h5","type":"hint","dependencies":["a57cb5ccenter4a-h4"],"title":"Determining the Houses worth $315,000","text":"Note that out of the $$60$$ data points, we have one at $2,500,000, twenty-nine at $280,000, and the others at $315,000. Since we don\'t know how many houses are worth $315,000, we want to determine that before trying to find the location of the median or the median itself.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter4b-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$30$$"],"dependencies":["a57cb5ccenter4b-h5"],"title":"Determining the Houses worth $315,000","text":"How many households are worth $315,000?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a57cb5ccenter4b-h6-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$30$$"],"dependencies":[],"title":"Determining the Houses worth $315,000","text":"What is $$60-1-29$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a57cb5ccenter4b-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$30.5$$"],"dependencies":["a57cb5ccenter4b-h6"],"title":"Determining Location of the Median","text":"What is the location of the median? What is $$\\\\frac{n+1}{2}$$ where $$n$$ represents the total number of households in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter4b-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["315,000 and 315,000"],"dependencies":["a57cb5ccenter4b-h7"],"title":"Determining the 30th and 31st Data Values","text":"Because the location of the median is $$30.5$$, we must take the average of the 30th and 31st data point. What are the 30th and 31st data points respectively?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["315,000 and 315,000","280,000 and 315,000","315,000 and 2,500,000","280,000 and 280,000"]},{"id":"a57cb5ccenter4b-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$315000$$"],"dependencies":["a57cb5ccenter4b-h8"],"title":"Determining the Median Knowing Location","text":"What is the median? What is the average between the 30th and 31st data point?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a57cb5ccenter4c","stepAnswer":["Median"],"problemType":"MultipleChoice","stepTitle":"Which is the better measure of the \\"center\\": the mean or the median?","stepBody":"","answerType":"string","variabilization":{},"choices":["Median","Mean"],"hints":{"DefaultPathway":[{"id":"a57cb5ccenter4c-h10","type":"hint","dependencies":["a57cb5ccenter4b-h9"],"title":"Finding Outliers","text":"In order to determine which is the better measure of the data, let\'s consider if there are outliers. Note that if there are, the median may be a better measure of the \\"center\\" of the data as the median is less pulled by outliers to skew than the mean is.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter4c-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["2,500,000"],"dependencies":["a57cb5ccenter4c-h10"],"title":"Determining the Outliers","text":"What are the outliers?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["2,500,000","280,000 and 2,500,000","280,000 and 315,000","There are none."]},{"id":"a57cb5ccenter4c-h12","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Median"],"dependencies":["a57cb5ccenter4c-h11"],"title":"Determining the Overall Better Measure","text":"Now, knowing that there is an outlier, based on the previous hints given, is the median a better measure of the \\"center\\" (or middle of the data) or the mean?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Median","Mean"]}]}}]},{"id":"a57cb5ccenter5","title":"Determining Mode","body":"Statistics exam scores for $$20$$ students are as follows: 50; 53; 59; 59; 63; 63; 72; 72; 72; 72; 72; 76; 78; 81; 83; 84; 84; 84; 90; $$93$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Measures of the Center of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a57cb5ccenter5a","stepAnswer":["$$72$$"],"problemType":"TextBox","stepTitle":"Find the mode.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$72$$","hints":{"DefaultPathway":[{"id":"a57cb5ccenter5a-h1","type":"hint","dependencies":[],"title":"Definition of Mode","text":"The mode of a data set is the most frequent value that occurs in that data set. For instance, for a data set containing the values 1; 1; 2; $$3$$, the mode would be $$1$$ as it appears twice (so has a frequency of 2) while $$2$$ and $$3$$ only appear once in the data set. Let\'s determine the frequencies of each of the different values to find the mode.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a57cb5ccenter5a-h1"],"title":"Determining Frequency of $$50$$","text":"How many times does $$50$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a57cb5ccenter5a-h2"],"title":"Determining Frequency of $$53$$","text":"How many times does $$53$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a57cb5ccenter5a-h3"],"title":"Determining Frequency of $$59$$","text":"How many times does $$59$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a57cb5ccenter5a-h4"],"title":"Determining Frequency of $$63$$","text":"How many times does $$63$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter5a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a57cb5ccenter5a-h5"],"title":"Determining Frequency of $$72$$","text":"How many times does $$72$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter5a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a57cb5ccenter5a-h6"],"title":"Determining Frequency of $$76$$","text":"How many times does $$76$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter5a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a57cb5ccenter5a-h7"],"title":"Determining Frequency of $$78$$","text":"How many times does $$78$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter5a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a57cb5ccenter5a-h8"],"title":"Determining Frequency of $$81$$","text":"How many times does $$81$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter5a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a57cb5ccenter5a-h9"],"title":"Determining Frequency of $$83$$","text":"How many times does $$83$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter5a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a57cb5ccenter5a-h10"],"title":"Determining Frequency of $$84$$","text":"How many times does $$84$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter5a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a57cb5ccenter5a-h11"],"title":"Determining Frequency of $$90$$","text":"How many times does $$90$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter5a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a57cb5ccenter5a-h12"],"title":"Determining Frequency of $$93$$","text":"How many times does $$93$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter5a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$72$$"],"dependencies":["a57cb5ccenter5a-h13"],"title":"Finding the Mode","text":"What is the mode? What is the most frequent score that occurred for the statistics exam scores?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57cb5ccenter6","title":"Determining Mode","body":"The number of books checked out from the library from $$25$$ students are as follows: 0; 0; 0; 1; 2; 3; 3; 4; 4; 5; 5; 7; 7; 7; 7; 8; 8; 8; 9; 10; 10; 11; 11; 12; $$12$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Measures of the Center of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a57cb5ccenter6a","stepAnswer":["$$7$$"],"problemType":"TextBox","stepTitle":"Find the mode.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7$$","hints":{"DefaultPathway":[{"id":"a57cb5ccenter6a-h1","type":"hint","dependencies":[],"title":"Definition of Mode","text":"The mode of a data set is the most frequent value that occurs in that data set. For instance, for a data set containing the values 1; 1; 2; $$3$$, the mode would be $$1$$ as it appears twice (so has a frequency of 2) while $$2$$ and $$3$$ only appear once in the data set. Let\'s determine the frequencies of each of the different values to find the mode.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a57cb5ccenter6a-h1"],"title":"Determining Frequency of $$0$$","text":"How many times does $$0$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a57cb5ccenter6a-h2"],"title":"Determining Frequency of $$1$$","text":"How many times does $$1$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a57cb5ccenter6a-h3"],"title":"Determining Frequency of $$2$$","text":"How many times does $$2$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a57cb5ccenter6a-h4"],"title":"Determining Frequency of $$3$$","text":"How many times does $$3$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter6a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a57cb5ccenter6a-h5"],"title":"Determining Frequency of $$4$$","text":"How many times does $$4$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter6a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a57cb5ccenter6a-h6"],"title":"Determining Frequency of $$5$$","text":"How many times does $$5$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter6a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a57cb5ccenter6a-h7"],"title":"Determining Frequency of $$7$$","text":"How many times does $$7$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter6a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a57cb5ccenter6a-h8"],"title":"Determining Frequency of $$8$$","text":"How many times does $$8$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter6a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a57cb5ccenter6a-h9"],"title":"Determining Frequency of $$9$$","text":"How many times does $$9$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter6a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a57cb5ccenter6a-h10"],"title":"Determining Frequency of $$10$$","text":"How many times does $$10$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter6a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a57cb5ccenter6a-h11"],"title":"Determining Frequency of $$11$$","text":"How many times does $$11$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter6a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a57cb5ccenter6a-h12"],"title":"Determining Frequency of $$12$$","text":"How many times does $$12$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter6a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a57cb5ccenter6a-h13"],"title":"Finding the Mode","text":"What is the mode? What is the most frequent data point?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57cb5ccenter7","title":"Understanding Unimodal and Bimodal","body":"Five real estate exam scores are $$430$$, $$430$$, $$480$$, $$480$$, $$495$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Measures of the Center of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a57cb5ccenter7a","stepAnswer":["Bimodal"],"problemType":"MultipleChoice","stepTitle":"Determine whether the set is unimodal or bimodal.","stepBody":"","answerType":"string","variabilization":{},"choices":["Unimodal","Bimodal"],"hints":{"DefaultPathway":[{"id":"a57cb5ccenter7a-h1","type":"hint","dependencies":[],"title":"Definitions of Unimodal and Bimodal","text":"The mode of a data set is the most frequent value that occurs in that data set. A set is considered unimodal if there is only one mode. A set is considered bimodal if it has two modes. In order to determine the mode(s) of the data set, let\'s list out the frequencies of each of the points.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a57cb5ccenter7a-h1"],"title":"Determining Frequency of $$430$$","text":"How many times does $$430$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a57cb5ccenter7a-h2"],"title":"Determining Frequency of $$480$$","text":"How many times does $$480$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a57cb5ccenter7a-h3"],"title":"Determining Frequency of $$495$$","text":"How many times does $$495$$ show up in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter7a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$430$$, $$480$$"],"dependencies":["a57cb5ccenter7a-h4"],"title":"Determining the Mode(s)","text":"What is/are the mode(s) of the data? Which value(s) have the highest frequency?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$430$$, $$480$$","$$430$$","$$480$$","$$480$$, $$495$$"]},{"id":"a57cb5ccenter7a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Bimodal"],"dependencies":["a57cb5ccenter7a-h5"],"title":"Determining Unimodal or Bimodal","text":"Based on the number of modes you determined, is this data set unimodal (only has one mode) or bimodal (has two modes)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Unimodal","Bimodal"]}]}}]},{"id":"a57cb5ccenter8","title":"Estimating Mean from a Frequency Table","body":"A frequency table displaying Professor Blount\'s last statistic test is shown.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Measures of the Center of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a57cb5ccenter8a","stepAnswer":["$$76.86$$"],"problemType":"TextBox","stepTitle":"Find the best estimate of the class mean. Round to the nearest hundredths place.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$76.86$$","hints":{"DefaultPathway":[{"id":"a57cb5ccenter8a-h1","type":"hint","dependencies":[],"title":"Formula for Calculating Mean from a Frequency Table","text":"To calculate mean from a frequency table, we want to first find all the midpoints of the grade intervals, and then sum up the product of each interval frequency with the midpoint. Lastly, we\'ll need to divide that sum by the total frequency. Let\'s first start by finding all the midpoints.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$53.25$$"],"dependencies":["a57cb5ccenter8a-h1"],"title":"Midpoint for Grade Interval $$50-56.5$$","text":"What is the midpoint for the grade interval $$50-56.5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a57cb5ccenter8a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$53.25$$"],"dependencies":[],"title":"Midpoint for Grade Interval $$50-56.5$$","text":"What is $$\\\\frac{50+56.5}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a57cb5ccenter8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$59.5$$"],"dependencies":["a57cb5ccenter8a-h2"],"title":"Midpoint for Grade Interval $$56.5-62.5$$","text":"What is the midpoint for the grade interval $$56.5-62.5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$65.5$$"],"dependencies":["a57cb5ccenter8a-h3"],"title":"Midpoint for Grade Interval $$62.5-68.5$$","text":"What is the midpoint for the grade interval $$62.5-68.5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter8a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$71.5$$"],"dependencies":["a57cb5ccenter8a-h4"],"title":"Midpoint for Grade Interval $$68.5-74.5$$","text":"What is the midpoint for the grade interval $$68.5-74.5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter8a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$77.5$$"],"dependencies":["a57cb5ccenter8a-h5"],"title":"Midpoint for Grade Interval $$74.5-80.5$$","text":"What is the midpoint for the grade interval $$74.5-80.5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter8a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$83.5$$"],"dependencies":["a57cb5ccenter8a-h6"],"title":"Midpoint for Grade Interval $$80.5-86.5$$","text":"What is the midpoint for the grade interval $$80.5-86.5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter8a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$89.5$$"],"dependencies":["a57cb5ccenter8a-h7"],"title":"Midpoint for Grade Interval $$86.5-92.5$$","text":"What is the midpoint for the grade interval $$86.5-92.5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter8a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$95.5$$"],"dependencies":["a57cb5ccenter8a-h8"],"title":"Midpoint for Grade Interval $$92.5-98.5$$","text":"What is the midpoint for the grade interval $$92.5-98.5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter8a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1460.25$$"],"dependencies":["a57cb5ccenter8a-h9"],"title":"Determining Total Sum","text":"Now that we know each midpoint for each interval, what is the sum of the product of each interval frequency (right column) and midpoint (just calculated)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a57cb5ccenter8a-h10-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1460.25$$"],"dependencies":[],"title":"Determining Total Sum","text":"What is $$53.25\\\\times1+59.5\\\\times0+65.5\\\\times4+71.5\\\\times4+77.5\\\\times2+83.5\\\\times3+89.5\\\\times4+95.5\\\\times1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a57cb5ccenter8a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$19$$"],"dependencies":["a57cb5ccenter8a-h10"],"title":"Determining Total Frequency","text":"What is the total frequency? As in, how many data points are in the table?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter8a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$76.86$$"],"dependencies":["a57cb5ccenter8a-h11"],"title":"Estimating the Mean","text":"What is an estimate of the mean? In other words, what is the total sum that we calculated divided by the total frequency? Round your answer to the nearest hundredths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57cb5ccenter9","title":"Estimating Mean from a Frequency Table","body":"Maris conducted a study on the effect that playing video games has on memory recall. As part of her study, she compiled the following data provided in the table.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Measures of the Center of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a57cb5ccenter9a","stepAnswer":["$$10.78$$"],"problemType":"TextBox","stepTitle":"Find the best estimate of the mean hours teenagers spend on video games. Round to the nearest hundredths place.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10.78$$","hints":{"DefaultPathway":[{"id":"a57cb5ccenter9a-h1","type":"hint","dependencies":[],"title":"Formula for Calculating Mean from a Frequency Table","text":"To calculate mean from a frequency table, we want to first find all the midpoints of the grade intervals, and then sum up the product of each interval frequency with the midpoint. Lastly, we\'ll need to divide that sum by the total frequency. Let\'s first start by finding all the midpoints.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.75$$"],"dependencies":["a57cb5ccenter9a-h1"],"title":"Midpoint for Hours Teenagers Spend on Video Games $$0-3.5$$","text":"What is the midpoint for the interval $$0-3.5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a57cb5ccenter9a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.75$$"],"dependencies":[],"title":"Midpoint for Hours Teenagers Spend on Video Games $$0-3.5$$","text":"What is $$\\\\frac{0+3.5}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a57cb5ccenter9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5.5$$"],"dependencies":["a57cb5ccenter9a-h2"],"title":"Midpoint for Hours Teenagers Spend on Video Games $$3.5-7.5$$","text":"What is the midpoint for the interval $$3.5-7.5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9.5$$"],"dependencies":["a57cb5ccenter9a-h3"],"title":"Midpoint for Hours Teenagers Spend on Video Games $$7.5-11.5$$","text":"What is the midpoint for the interval $$7.5-11.5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter9a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$13.5$$"],"dependencies":["a57cb5ccenter9a-h4"],"title":"Midpoint for Hours Teenagers Spend on Video Games $$11.5-15.5$$","text":"What is the midpoint for the interval $$11.5-15.5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter9a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$17.5$$"],"dependencies":["a57cb5ccenter9a-h5"],"title":"Midpoint for Hours Teenagers Spend on Video Games $$15.5-19.5$$","text":"What is the midpoint for the interval $$15.5-19.5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter9a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$409.75$$"],"dependencies":["a57cb5ccenter9a-h6"],"title":"Determining Total Sum","text":"Now that we know each midpoint for each interval, what is the sum of the product of each interval frequency (right column) and midpoint (just calculated)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a57cb5ccenter9a-h7-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$409.75$$"],"dependencies":[],"title":"Determining Total Sum","text":"What is $$1.75\\\\times3+5.5\\\\times7+9.5\\\\times12+13.5\\\\times7+17.5\\\\times9$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a57cb5ccenter9a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$38$$"],"dependencies":["a57cb5ccenter9a-h7"],"title":"Determining Total Frequency","text":"What is the total frequency? As in, how many data points are in the table?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57cb5ccenter9a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10.78$$"],"dependencies":["a57cb5ccenter9a-h8"],"title":"Estimating the Mean","text":"What is an estimate of the mean? In other words, what is the total sum that we calculated divided by the total frequency? Round your answer to the nearest hundredths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57f999ser1","title":"Using Formulas for Arithmetic Sequences","body":"Use the formula for the sum of the first $$n$$ terms of each arithmetic sequence to find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Series and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a57f999ser1a","stepAnswer":["$$\\\\frac{5\\\\left(\\\\frac{3}{2}+\\\\frac{7}{2}\\\\right)}{2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{3}{2}+2+\\\\frac{5}{2}+3+\\\\frac{7}{2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5\\\\left(\\\\frac{3}{2}+\\\\frac{7}{2}\\\\right)}{2}$$","hints":{"DefaultPathway":[{"id":"a57f999ser1a-h1","type":"hint","dependencies":[],"title":"Find $$a_1$$ and $$a_n$$","text":"We are given $$a_1=\\\\frac{3}{2}$$ and $$a_n=\\\\frac{7}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser1a-h2","type":"hint","dependencies":["a57f999ser1a-h1"],"title":"Find $$n$$","text":"Count the number of terms in the sequence to find $$n=5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser1a-h3","type":"hint","dependencies":["a57f999ser1a-h2"],"title":"Formula for Sum of the first $$n$$ terms of Arithmetic Sequence","text":"Substitute values for $$a_1$$, $$a_n$$ , and $$n$$ into the formula: $$S_n=\\\\frac{n \\\\left(a_1+a_n\\\\right)}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser1a-h4","type":"hint","dependencies":["a57f999ser1a-h3"],"title":"Formula for Sum of the first $$5$$ terms of Arithmetic Sequence","text":"The formula for the sum of the first $$5$$ terms of the arithmetic sequence is $$S_5=\\\\frac{5\\\\left(\\\\frac{3}{2}+\\\\frac{7}{2}\\\\right)}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57f999ser10","title":"Using Formulas for Arithmetic Sequences","body":"Use the formula for the sum of the first $$n$$ terms of each arithmetic sequence to find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Series and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a57f999ser10a","stepAnswer":["$$\\\\frac{55}{2}$$"],"problemType":"TextBox","stepTitle":"image1","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{55}{2}$$","hints":{"DefaultPathway":[{"id":"a57f999ser10a-h1","type":"hint","dependencies":[],"title":"Find $$a_1$$","text":"To find $$a_1$$, substitute $$k=1$$ into the given explicit formula: $$a_k=\\\\frac{k}{2}-\\\\frac{1}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a57f999ser10a-h1"],"title":"Solve for $$a_1$$","text":"What is $$a_1=\\\\frac{1}{2}-\\\\frac{1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser10a-h3","type":"hint","dependencies":["a57f999ser10a-h2"],"title":"Find $$a_k$$","text":"We are given that $$n=11$$. To find $$a_{11}$$, substitute $$k=11$$ into the given explicit formula: $$a_k=\\\\frac{k}{2}-\\\\frac{1}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a57f999ser10a-h3"],"title":"Solve for $$a_{11}$$","text":"What is $$a_{11}=\\\\frac{11}{2}-\\\\frac{1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser10a-h5","type":"hint","dependencies":["a57f999ser10a-h4"],"title":"Formula for Sum of the first $$n$$ terms of Arithmetic Series","text":"Substitute values for $$a_1$$, $$a_n$$ , and $$n$$ into the formula: $$S_n=\\\\frac{n \\\\left(a_1+a_n\\\\right)}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser10a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{55}{2}$$"],"dependencies":["a57f999ser10a-h5"],"title":"Formula for Sum of the first $$11$$ terms of Arithmetic Series","text":"What is $$S_{11}=\\\\frac{11\\\\left(0+5\\\\right)}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser10a-h7","type":"hint","dependencies":["a57f999ser10a-h6"],"title":"Sum of the first $$11$$ terms of Arithmetic Series","text":"The Sum of the first $$11$$ terms of Arithmetic Series is $$\\\\frac{55}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57f999ser11","title":"Using Formulas for Arithmetic Sequences","body":"Use the formula for the sum of the first $$n$$ terms of each geometric sequence to find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Series and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a57f999ser11a","stepAnswer":["$$5.336$$"],"problemType":"TextBox","stepTitle":"$$S_{11}$$ for the series 8+-4+2+...","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5.336$$","hints":{"DefaultPathway":[{"id":"a57f999ser11a-h1","type":"hint","dependencies":[],"title":"Identify $$a_1$$","text":"The first term is $$a_1=8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{2}$$"],"dependencies":["a57f999ser11a-h1"],"title":"Identify $$r$$","text":"To find $$r$$, divide the 2nd term by the 1st term. What is $$r=\\\\frac{-4}{8}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser11a-h3","type":"hint","dependencies":["a57f999ser11a-h2"],"title":"Identify $$n$$","text":"We are given that $$n=11$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser11a-h4","type":"hint","dependencies":["a57f999ser11a-h3"],"title":"Formula for the Sum of the first $$n$$ terms of a Geometric Series","text":"Substitute values for $$a_1$$, $$r$$, and $$n$$ into the formula: $$S_n=\\\\frac{a_1 \\\\left(1-r^n\\\\right)}{1-r}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5.336$$"],"dependencies":["a57f999ser11a-h4"],"title":"Simplify to find $$S_{11}$$","text":"What is $$S_{11}=\\\\frac{8\\\\left(1-{\\\\left(-\\\\frac{1}{2}\\\\right)}^{11}\\\\right)}{1-\\\\left(-\\\\frac{1}{2}\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser11a-h6","type":"hint","dependencies":["a57f999ser11a-h5"],"title":"Partial Sum","text":"The partial sum is $$5.336$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57f999ser12","title":"Using Formulas for Arithmetic Sequences","body":"Use the formula for the sum of the first $$n$$ terms of each arithmetic sequence to find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Series and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a57f999ser12a","stepAnswer":["$$378$$"],"problemType":"TextBox","stepTitle":"image1","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$378$$","hints":{"DefaultPathway":[{"id":"a57f999ser12a-h1","type":"hint","dependencies":[],"title":"Find $$a_1$$","text":"Find $$a_1$$ by substituting $$k=1$$ into the given explicit formula: $$3\\\\times2^k$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a57f999ser12a-h1"],"title":"Solve for $$a_1$$","text":"What is $$a_1=3\\\\times2^1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser12a-h3","type":"hint","dependencies":["a57f999ser12a-h2"],"title":"Identify $$r$$","text":"We can see from the given explicit formula that $$r=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser12a-h4","type":"hint","dependencies":["a57f999ser12a-h3"],"title":"Identify $$n$$","text":"The upper limit of summation is $$6$$, so $$n=6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser12a-h5","type":"hint","dependencies":["a57f999ser12a-h4"],"title":"Formula for the Sum of the first $$n$$ terms of a Geometric Series","text":"Substitute values for $$a_1$$, $$r$$, and $$n$$ into the formula: $$S_n=\\\\frac{a_1 \\\\left(1-r^n\\\\right)}{1-r}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser12a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$378$$"],"dependencies":["a57f999ser12a-h5"],"title":"Simplify to find $$S_6$$","text":"What is $$S_6=\\\\frac{6\\\\left(1-2^6\\\\right)}{1-2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57f999ser13","title":"Using Formulas for Arithmetic Sequences","body":"Use the formula for the sum of the first $$n$$ terms of each arithmetic sequence to find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Series and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a57f999ser13a","stepAnswer":["$$-7812$$"],"problemType":"TextBox","stepTitle":"$$S_6$$ for the series $$-2-10-50-250..$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-7812$$","hints":{"DefaultPathway":[{"id":"a57f999ser13a-h1","type":"hint","dependencies":[],"title":"Identify $$a_1$$","text":"The first term is $$a_1=-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a57f999ser13a-h1"],"title":"Identify $$r$$","text":"To find $$r$$, divide the 2nd term by the 1st term. What is $$r=\\\\frac{-10}{-2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser13a-h3","type":"hint","dependencies":["a57f999ser13a-h2"],"title":"Identify $$n$$","text":"We are given that $$n=6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser13a-h4","type":"hint","dependencies":["a57f999ser13a-h3"],"title":"Formula for the Sum of the first $$n$$ terms of a Geometric Series","text":"Substitute values for $$a_1$$, $$r$$, and $$n$$ into the formula: $$S_n=\\\\frac{a_1 \\\\left(1-r^n\\\\right)}{1-r}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser13a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-7812$$"],"dependencies":["a57f999ser13a-h4"],"title":"Simplify to find $$S_6$$","text":"What is $$S_6=\\\\frac{-2\\\\left(1-5^6\\\\right)}{1-5}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser13a-h6","type":"hint","dependencies":["a57f999ser13a-h5"],"title":"Partial Sum","text":"The partial sum is $$-7812$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57f999ser14","title":"Using Formulas for Arithmetic Sequences","body":"Use the formula for the sum of the first $$n$$ terms of each arithmetic sequence to find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Series and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a57f999ser14a","stepAnswer":["$$5208.4$$"],"problemType":"TextBox","stepTitle":"$$S_7$$ for the series 0.4-2+10-50...","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5208.4$$","hints":{"DefaultPathway":[{"id":"a57f999ser14a-h1","type":"hint","dependencies":[],"title":"Identify $$a_1$$","text":"The first term is $$a_1=0.4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a57f999ser14a-h1"],"title":"Identify $$r$$","text":"To find $$r$$, divide the 2nd term by the 1st term. What is $$r=\\\\frac{-2}{0.4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser14a-h3","type":"hint","dependencies":["a57f999ser14a-h2"],"title":"Identify $$n$$","text":"We are given that $$n=7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser14a-h4","type":"hint","dependencies":["a57f999ser14a-h3"],"title":"Formula for the Sum of the first $$n$$ terms of a Geometric Series","text":"Substitute values for $$a_1$$, $$r$$, and $$n$$ into the formula: $$S_n=\\\\frac{a_1 \\\\left(1-r^n\\\\right)}{1-r}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser14a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5208.4$$"],"dependencies":["a57f999ser14a-h4"],"title":"Simplify to find $$S_7$$","text":"What is $$S_7=\\\\frac{0.4\\\\left(1-{\\\\left(-5\\\\right)}^7\\\\right)}{1-\\\\left(-5\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser14a-h6","type":"hint","dependencies":["a57f999ser14a-h5"],"title":"Partial Sum","text":"The partial sum is $$5208.4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57f999ser15","title":"Use the formula for the sum of the first $$n$$ terms of a geometric series to find the partial sum.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Series and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a57f999ser15a","stepAnswer":["$$511$$"],"problemType":"TextBox","stepTitle":"Solving Summations","stepBody":"Solve the summation in the attached image.##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$511$$","hints":{"DefaultPathway":[{"id":"a57f999ser15a-h1","type":"hint","dependencies":[],"title":"Find $$a_1$$","text":"Find $$a_1$$ by substituting $$k=1$$ into the given explicit formula: $$2^{k-1}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a57f999ser15a-h1"],"title":"Solve for $$a_1$$","text":"What is $$a_1=2^{1-1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser15a-h3","type":"hint","dependencies":["a57f999ser15a-h2"],"title":"Identify $$r$$","text":"We can see from the given explicit formula that $$r=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser15a-h4","type":"hint","dependencies":["a57f999ser15a-h3"],"title":"Identify $$n$$","text":"The upper limit of summation is $$9$$, so $$n=9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser15a-h5","type":"hint","dependencies":["a57f999ser15a-h4"],"title":"Formula for the Sum of the first $$n$$ terms of a Geometric Series","text":"Substitute values for $$a_1$$, $$r$$, and $$n$$ into the formula: $$S_n=\\\\frac{a_1 \\\\left(1-r^n\\\\right)}{1-r}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser15a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$511$$"],"dependencies":["a57f999ser15a-h5"],"title":"Simplify to find $$S_9$$","text":"What is $$S_9=\\\\frac{1\\\\left(1-2^9\\\\right)}{1-2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57f999ser16","title":"Use the formula for the sum of the first $$n$$ terms of a geometric series to find the partial sum.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Series and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a57f999ser16a","stepAnswer":["$$\\\\frac{-1023}{256}$$"],"problemType":"TextBox","stepTitle":"Solving Summations","stepBody":"Solve the summation in the attached image.##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-1023}{256}$$","hints":{"DefaultPathway":[{"id":"a57f999ser16a-h1","type":"hint","dependencies":[],"title":"Find $$a_1$$","text":"Find $$a_1$$ by substituting $$n=1$$ into the given explicit formula: $$-2{\\\\left(\\\\frac{1}{2}\\\\right)}^{n-1}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a57f999ser16a-h1"],"title":"Solve for $$a_1$$","text":"What is $$a_1=-2{\\\\left(\\\\frac{1}{2}\\\\right)}^{1-1}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser16a-h3","type":"hint","dependencies":["a57f999ser16a-h2"],"title":"Identify $$r$$","text":"We can see from the given explicit formula that $$r=\\\\frac{1}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser16a-h4","type":"hint","dependencies":["a57f999ser16a-h3"],"title":"Identify $$n$$","text":"The upper limit of summation is $$10$$, so $$n=10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser16a-h5","type":"hint","dependencies":["a57f999ser16a-h4"],"title":"Formula for the Sum of the first $$n$$ terms of a Geometric Series","text":"Substitute values for $$a_1$$, $$r$$, and $$n$$ into the formula: $$S_n=\\\\frac{a_1 \\\\left(1-r^n\\\\right)}{1-r}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser16a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1023}{256}$$"],"dependencies":["a57f999ser16a-h5"],"title":"Simplify to find $$S_{10}$$","text":"What is $$S_{10}=\\\\frac{-2\\\\left(1-{\\\\left(\\\\frac{1}{2}\\\\right)}^{10}\\\\right)}{1-\\\\frac{1}{2}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57f999ser17","title":"Interpreting the Sum of Infinite Series","body":"Determine whether the sum of each infinite series is defined.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Series and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a57f999ser17a","stepAnswer":["Sum not defined"],"problemType":"MultipleChoice","stepTitle":"12+8+4+...","stepBody":"","answerType":"string","variabilization":{},"choices":["Sum not defined","Sum defined"],"hints":{"DefaultPathway":[{"id":"a57f999ser17a-h1","type":"hint","dependencies":[],"title":"Find the ratio of $$a_2$$ to $$a_1$$.","text":"The ratio of the second term to the first term is $$\\\\frac{8}{12}=\\\\frac{2}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser17a-h2","type":"hint","dependencies":["a57f999ser17a-h1"],"title":"Find the ratio of $$a_3$$ to $$a_2$$.","text":"The ratio of the third term to the second term is $$\\\\frac{4}{8}=\\\\frac{1}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser17a-h3","type":"hint","dependencies":["a57f999ser17a-h2"],"title":"Determining if there is a Common Ratio","text":"Since $$\\\\frac{2}{3} \\\\neq \\\\frac{1}{2}$$, there is no common ratio, the series is not geometric.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser17a-h4","type":"hint","dependencies":["a57f999ser17a-h3"],"title":"Sum of Infinite Geometric Series","text":"Since the series is not geometric, the sum is not defined.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57f999ser18","title":"Interpreting the Sum of Infinite Series","body":"Determine whether the sum of each infinite series is defined.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Series and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a57f999ser18a","stepAnswer":["Sum defined"],"problemType":"MultipleChoice","stepTitle":"3/4+1/2+1/3+...","stepBody":"","answerType":"string","variabilization":{},"choices":["Sum not defined","Sum defined"],"hints":{"DefaultPathway":[{"id":"a57f999ser18a-h1","type":"hint","dependencies":[],"title":"Find the ratio of $$a_2$$ to $$a_1$$.","text":"The ratio of the second term to the first term is $$\\\\frac{\\\\frac{1}{2}}{\\\\frac{3}{4}}=\\\\frac{2}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser18a-h2","type":"hint","dependencies":["a57f999ser18a-h1"],"title":"Find the ratio of $$a_3$$ to $$a_2$$.","text":"The ratio of the third term to the second term is $$\\\\frac{\\\\frac{1}{3}}{\\\\frac{1}{2}}=\\\\frac{2}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser18a-h3","type":"hint","dependencies":["a57f999ser18a-h2"],"title":"Determining if there is a Constant Ratio","text":"Since $$\\\\frac{2}{3}=\\\\frac{2}{3}$$, there is a common ratio, the series is geometric.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser18a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["T"],"dependencies":["a57f999ser18a-h3"],"title":"Confirm that $$-1<r<1$$","text":"Is $$-1<\\\\frac{2}{3}<1$$ True or False?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"a57f999ser18a-h5","type":"hint","dependencies":["a57f999ser18a-h4"],"title":"Sum of Infinite Geometric Series","text":"Since $$-1<\\\\frac{2}{3}<1$$ is T, then sum is defined.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57f999ser19","title":"Interpreting the Sum of Infinite Series","body":"Determine whether the sum of each infinite series is defined.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Series and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a57f999ser19a","stepAnswer":["Sum defined"],"problemType":"MultipleChoice","stepTitle":"image1","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Sum not defined","Sum defined"],"hints":{"DefaultPathway":[{"id":"a57f999ser19a-h1","type":"hint","dependencies":[],"title":"Explicit Formula","text":"The given formula, $$27{\\\\left(\\\\frac{1}{3}\\\\right)}^k$$, is exponential with a base of $$\\\\frac{1}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser19a-h2","type":"hint","dependencies":["a57f999ser19a-h1"],"title":"Identify $$r$$","text":"The series is geometric with a common ratio of $$\\\\frac{1}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser19a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a57f999ser19a-h2"],"title":"Confirm that $$-1<r<1$$","text":"Is $$-1<\\\\frac{1}{3}<1$$ True or False?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a57f999ser19a-h4","type":"hint","dependencies":["a57f999ser19a-h3"],"title":"Sum of Infinite Geometric Series","text":"Since $$-1<\\\\frac{1}{3}<1$$ is T, then sum is defined.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57f999ser2","title":"Using Formulas for Arithmetic Sequences","body":"Use the formula for the sum of the first $$n$$ terms of each arithmetic sequence to find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Series and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a57f999ser2a","stepAnswer":["$$\\\\frac{10\\\\left(19+73\\\\right)}{2}$$"],"problemType":"TextBox","stepTitle":"$$19+25+31+...+73$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{10\\\\left(19+73\\\\right)}{2}$$","hints":{"DefaultPathway":[{"id":"a57f999ser2a-h1","type":"hint","dependencies":[],"title":"Find $$a_1$$ and $$a_n$$","text":"We are given $$a_1=19$$ and $$a_n=73$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser2a-h2","type":"hint","dependencies":["a57f999ser2a-h1"],"title":"Find $$n$$","text":"To find $$n$$, use the formula for the general term of an arithmetic sequence: $$a_n=a_1+\\\\left(n-1\\\\right) d$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser2a-h3","type":"hint","dependencies":["a57f999ser2a-h2"],"title":"Find $$d$$","text":"The common difference can be found by subtracting the first term from the second term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a57f999ser2a-h3"],"title":"Find $$d$$","text":"What is $$d=25-19$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser2a-h5","type":"hint","dependencies":["a57f999ser2a-h4"],"title":"Find $$n$$","text":"Substitute values for $$a_1$$, $$a_n$$, and $$d$$ into the arithmetic sequence formula to get $$73=19+6\\\\left(n-1\\\\right)$$ and solve for $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a57f999ser2a-h5"],"title":"Find $$n$$","text":"What is $$n$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser2a-h7","type":"hint","dependencies":["a57f999ser2a-h6"],"title":"Formula for Sum of the first $$n$$ terms of Arithmetic Sequence","text":"Substitute values for $$a_1$$, $$a_n$$ , and $$n$$ into the formula: $$S_n=\\\\frac{n \\\\left(a_1+a_n\\\\right)}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser2a-h8","type":"hint","dependencies":["a57f999ser2a-h7"],"title":"Formula for Sum of the first $$10$$ terms of Arithmetic Sequence","text":"The formula for the sum of the first $$10$$ terms of the arithmetic sequence is $$S_{10}=\\\\frac{10\\\\left(19+73\\\\right)}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57f999ser20","title":"Interpreting the Sum of Infinite Series","body":"Determine whether the sum of each infinite series is defined.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Series and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a57f999ser20a","stepAnswer":["Sum not defined"],"problemType":"MultipleChoice","stepTitle":"image1","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Sum not defined","Sum defined"],"hints":{"DefaultPathway":[{"id":"a57f999ser20a-h1","type":"hint","dependencies":[],"title":"Explicit Formula","text":"The given formula, $$5k$$, is not exponential.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser20a-h2","type":"hint","dependencies":["a57f999ser20a-h1"],"title":"Is series geometric?","text":"The series is not geometric because the terms are increasing, and so cannot yield a finite sum.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser20a-h3","type":"hint","dependencies":["a57f999ser20a-h2"],"title":"Sum of Infinite Geometric Series","text":"The sum is not defined.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57f999ser21","title":"Interpreting the Sum of Infinite Series","body":"Determine whether the infinite series has a sum. If so, write the formula for the sum.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Series and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a57f999ser21a","stepAnswer":["Sum not defined"],"problemType":"MultipleChoice","stepTitle":"12+18+24+30+...","stepBody":"","answerType":"string","variabilization":{},"choices":["Sum not defined","$$S=\\\\frac{12}{1-\\\\frac{3}{2}}$$","$$S=\\\\frac{12}{1-\\\\frac{4}{3}}$$"],"hints":{"DefaultPathway":[{"id":"a57f999ser21a-h1","type":"hint","dependencies":[],"title":"Find the ratio of $$a_2$$ to $$a_1$$.","text":"The ratio of the second term to the first term is $$\\\\frac{18}{12}=\\\\frac{3}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser21a-h2","type":"hint","dependencies":["a57f999ser21a-h1"],"title":"Find the ratio of $$a_3$$ to $$a_2$$.","text":"The ratio of the third term to the second term is $$\\\\frac{24}{18}=\\\\frac{4}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser21a-h3","type":"hint","dependencies":["a57f999ser21a-h2"],"title":"Determining if there is a Constant Ratio","text":"Since $$\\\\frac{3}{2} \\\\neq \\\\frac{4}{3}$$, there is no common ratio, the series is not geometric.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser21a-h4","type":"hint","dependencies":["a57f999ser21a-h3"],"title":"Sum of Infinite Geometric Series","text":"Since the series is not geometric, the sum is not defined.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57f999ser22","title":"Interpreting the Sum of Infinite Series","body":"Determine whether the infinite series has a sum. If so, write the formula for the sum.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Series and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a57f999ser22a","stepAnswer":["$$S=\\\\frac{2}{1-0.8}$$"],"problemType":"MultipleChoice","stepTitle":"2+1.6+1.28+1.024+...","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$S=\\\\frac{2}{1-0.8}$$","choices":["Sum not defined","$$S=\\\\frac{2}{1-0.8}$$","$$S=\\\\frac{0.8}{1-2}$$"],"hints":{"DefaultPathway":[{"id":"a57f999ser22a-h1","type":"hint","dependencies":[],"title":"Find the ratio of $$a_2$$ to $$a_1$$.","text":"The ratio of the second term to the first term is $$\\\\frac{1.6}{0.8}=0.8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser22a-h2","type":"hint","dependencies":["a57f999ser22a-h1"],"title":"Find the ratio of $$a_3$$ to $$a_2$$.","text":"The ratio of the third term to the second term is $$\\\\frac{1.28}{1.6}=0.8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser22a-h3","type":"hint","dependencies":["a57f999ser22a-h2"],"title":"Find the ratio of $$a_4$$ to $$a_3$$.","text":"The ratio of the fourth term to the third term is $$\\\\frac{1.024}{1.28}=0.8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser22a-h4","type":"hint","dependencies":["a57f999ser22a-h3"],"title":"Determining if there is a Constant Ratio","text":"Since there is a common ratio of $$0.8$$, the series is geometric.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser22a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["T"],"dependencies":["a57f999ser22a-h4"],"title":"Confirm that $$-1<r<1$$","text":"Is $$-1<0.8<1$$ True or False?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"a57f999ser22a-h6","type":"hint","dependencies":["a57f999ser22a-h5"],"title":"Sum of Infinite Geometric Series","text":"Since $$-1<0.8<1$$ is T, then sum is defined.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser22a-h7","type":"hint","dependencies":["a57f999ser22a-h6"],"title":"Identify $$a_1$$ and $$r$$","text":"The first term is $$a_1=2$$ and $$r=0.8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser22a-h8","type":"hint","dependencies":["a57f999ser22a-h7"],"title":"Formula for the Sum of an Infinite Geometric Series","text":"Substitute values for $$a_1$$ and $$r$$ into the formula: $$S=\\\\frac{a_1}{1-r}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser22a-h9","type":"hint","dependencies":["a57f999ser22a-h8"],"title":"Formula for the Sum of an Infinite Geometric Series","text":"The formula of the sum is $$S=\\\\frac{2}{1-0.8}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57f999ser23","title":"Interpreting the Sum of Infinite Series","body":"Determine whether the infinite series has a sum. If so, write the formula for the sum.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Series and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a57f999ser23a","stepAnswer":["Sum not defined"],"problemType":"MultipleChoice","stepTitle":"image1","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Sum not defined","$$S=\\\\frac{2}{1-4}$$","$$S=\\\\frac{4}{1-2}$$"],"hints":{"DefaultPathway":[{"id":"a57f999ser23a-h1","type":"hint","dependencies":[],"title":"Explicit Formula","text":"The given formula, $$4^{m-1}$$, is exponential with a base of $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser23a-h2","type":"hint","dependencies":["a57f999ser23a-h1"],"title":"Identify $$r$$","text":"The series is geometric with a common ratio of $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser23a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["F"],"dependencies":["a57f999ser23a-h2"],"title":"Confirm that $$-1<r<1$$","text":"Is $$-1<4<1$$ True or False?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"a57f999ser23a-h4","type":"hint","dependencies":["a57f999ser23a-h3"],"title":"Sum of Infinite Geometric Series","text":"Since $$-1<4<1$$ is F; then sum is not defined.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57f999ser24","title":"Interpreting the Sum of Infinite Series","body":"Determine whether the infinite series has a sum. If so, write the formula for the sum.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Series and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a57f999ser24a","stepAnswer":["$$S=\\\\frac{-1}{1-\\\\left(-\\\\frac{1}{2}\\\\right)}$$"],"problemType":"MultipleChoice","stepTitle":"image1","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$S=\\\\frac{-1}{1-\\\\left(-\\\\frac{1}{2}\\\\right)}$$","choices":["Sum not defined","$$S=\\\\frac{-1}{1-\\\\left(-\\\\frac{1}{2}\\\\right)}$$","$$S=\\\\frac{\\\\left(-\\\\frac{1}{2}\\\\right)}{1-\\\\left(-1\\\\right)}$$"],"hints":{"DefaultPathway":[{"id":"a57f999ser24a-h1","type":"hint","dependencies":[],"title":"Explicit Formula","text":"The given formula, $$-\\\\left({\\\\left(-\\\\frac{1}{2}\\\\right)}^{k-1}\\\\right)$$, is exponential with a base of $$\\\\frac{-1}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser24a-h2","type":"hint","dependencies":["a57f999ser24a-h1"],"title":"Identify $$r$$","text":"The series is geometric with a common ratio of $$\\\\frac{-1}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser24a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["T"],"dependencies":["a57f999ser24a-h2"],"title":"Confirm that $$-1<r<1$$","text":"Is $$-1<\\\\frac{-1}{2}<1$$ True or False?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"a57f999ser24a-h4","type":"hint","dependencies":["a57f999ser24a-h3"],"title":"Sum of Infinite Geometric Series","text":"Since $$-1<\\\\frac{-1}{2}<1$$ is F; then sum is defined.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser24a-h5","type":"hint","dependencies":["a57f999ser24a-h4"],"title":"Identify $$a_1$$ and $$r$$","text":"From the given formula, we are given that $$a_1=-1$$ and $$r=\\\\frac{-1}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser24a-h6","type":"hint","dependencies":["a57f999ser24a-h5"],"title":"Formula for the Sum of an Infinite Geometric Series","text":"Substitute values for $$a_1$$ and $$r$$ into the formula: $$S=\\\\frac{a_1}{1-r}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser24a-h7","type":"hint","dependencies":["a57f999ser24a-h6"],"title":"Formula for the Sum of an Infinite Geometric Series","text":"The formula of the sum is $$S=\\\\frac{-1}{1-\\\\left(-\\\\frac{1}{2}\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57f999ser25","title":"Interpreting the Sum of Geometric Series","body":"Find the sum of the infinite geometric series if it exisits.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Series and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a57f999ser25a","stepAnswer":["Sum does not exist"],"problemType":"MultipleChoice","stepTitle":"10+9+8+7+...","stepBody":"","answerType":"string","variabilization":{},"choices":["Sum does not exist","$$-1$$","$$34$$"],"hints":{"DefaultPathway":[{"id":"a57f999ser25a-h1","type":"hint","dependencies":[],"title":"Find $$r$$","text":"The ratio between the $$a_2$$ and $$a_1$$ is $$\\\\frac{9}{10}$$. The ratio between $$a_3$$ and $$a_2$$ is $$\\\\frac{8}{9}$$. Therefore there is no constant ratio.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser25a-h2","type":"hint","dependencies":["a57f999ser25a-h1"],"title":"Is series geometric?","text":"Since there is no common ratio, the series is not geometric.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser25a-h3","type":"hint","dependencies":["a57f999ser25a-h2"],"title":"Sum of Infinite Geometric Series","text":"Since the series is not geometric, the sum does not exist.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57f999ser26","title":"Interpreting the Sum of Geometric Series","body":"Find the sum of the infinite geometric series if it exist. Round to the first decimal place.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Series and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a57f999ser26a","stepAnswer":["$$414.3$$"],"problemType":"TextBox","stepTitle":"248.6+99.44+39.776+...","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$414.3$$","hints":{"DefaultPathway":[{"id":"a57f999ser26a-h1","type":"hint","dependencies":[],"title":"Identify $$a_1$$","text":"The first term is $$a_1=248.6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser26a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.4$$"],"dependencies":["a57f999ser26a-h1"],"title":"Identify $$r$$","text":"To find $$r$$, divide the 2nd term by the 1st term. What is $$r=\\\\frac{99.44}{248.6}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser26a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["T"],"dependencies":["a57f999ser26a-h2"],"title":"Confirm that $$-1<r<1$$","text":"Is $$-1<0.4<1$$ True or False?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"a57f999ser26a-h4","type":"hint","dependencies":["a57f999ser26a-h3"],"title":"Formula for the Sum of an Infinite Geometric Series","text":"Substitute values for $$a_1$$ and $$r$$ into the formula: $$S=\\\\frac{a_1}{1-r}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser26a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$414.3$$"],"dependencies":["a57f999ser26a-h4"],"title":"Solve for S","text":"What is $$S=\\\\frac{248.6}{1-0.4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser26a-h6","type":"hint","dependencies":["a57f999ser26a-h5"],"title":"Sum of Infinite Geometric Series","text":"The sum of the infinite geometric series is $$414.3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57f999ser27","title":"Interpreting the Sum of Geometric Series","body":"Find the sum of the infinite geometric series.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Series and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a57f999ser27a","stepAnswer":["$$3280.5$$"],"problemType":"TextBox","stepTitle":"image1","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3280.5$$","hints":{"DefaultPathway":[{"id":"a57f999ser27a-h1","type":"hint","dependencies":[],"title":"Explicit Formula","text":"The given formula, $$4374{\\\\left(-\\\\frac{1}{3}\\\\right)}^{k-1}$$, is exponential with a base of $$\\\\frac{-1}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser27a-h2","type":"hint","dependencies":["a57f999ser27a-h1"],"title":"Identify $$r$$","text":"The series is geometric with a common ratio of $$\\\\frac{-1}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser27a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["T"],"dependencies":["a57f999ser27a-h2"],"title":"Confirm that $$-1<r<1$$","text":"Is $$-1<\\\\frac{-1}{3}<1$$ True or False?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"a57f999ser27a-h4","type":"hint","dependencies":["a57f999ser27a-h3"],"title":"Sum of Infinite Geometric Series","text":"Since $$-1<\\\\frac{-1}{3}<1$$ is F; then sum is defined.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser27a-h5","type":"hint","dependencies":["a57f999ser27a-h4"],"title":"Identify $$a_1$$ and $$r$$","text":"From the given formula, we are given that $$a_1=4374$$ and $$r=\\\\frac{-1}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser27a-h6","type":"hint","dependencies":["a57f999ser27a-h5"],"title":"Formula for the Sum of an Infinite Geometric Series","text":"Substitute values for $$a_1$$ and $$r$$ into the formula: $$S=\\\\frac{a_1}{1-r}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser27a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3280.5$$"],"dependencies":["a57f999ser27a-h6"],"title":"Formula for the Sum of an Infinite Geometric Series","text":"What is $$S=\\\\frac{4374}{1-\\\\left(-\\\\frac{1}{3}\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser27a-h8","type":"hint","dependencies":["a57f999ser27a-h7"],"title":"Sum of Infinite Geometric Series","text":"The sum of the infinite geometric series is $$3280.5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57f999ser28","title":"Interpreting the Sum of Geometric Series","body":"Find the sum of the infinite geometric series.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Series and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a57f999ser28a","stepAnswer":["Sum does not exist"],"problemType":"MultipleChoice","stepTitle":"image1","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Sum does not exist","$$\\\\frac{-1}{3}$$","$$\\\\frac{3}{2}$$"],"hints":{"DefaultPathway":[{"id":"a57f999ser28a-h1","type":"hint","dependencies":[],"title":"Explicit Formula","text":"The given formula, $$\\\\frac{1}{9} {\\\\left(\\\\frac{4}{3}\\\\right)}^k$$, is exponential with a base of $$\\\\frac{4}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser28a-h2","type":"hint","dependencies":["a57f999ser28a-h1"],"title":"Identify $$r$$","text":"The series is geometric with a common ratio of $$\\\\frac{4}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser28a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["F"],"dependencies":["a57f999ser28a-h2"],"title":"Confirm that $$-1<r<1$$","text":"Is $$-1<\\\\frac{4}{3}<1$$ True or False?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"a57f999ser28a-h4","type":"hint","dependencies":["a57f999ser28a-h3"],"title":"Sum of Infinite Geometric Series","text":"Since $$-1<\\\\frac{4}{3}<1$$ is F; then sum does not exisit.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57f999ser29","title":"Interpreting the Sum of Geometric Series","body":"Find the sum of the infinite geometric series.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Series and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a57f999ser29a","stepAnswer":["$$8$$"],"problemType":"TextBox","stepTitle":"4+2+1+1/2...","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8$$","hints":{"DefaultPathway":[{"id":"a57f999ser29a-h1","type":"hint","dependencies":[],"title":"Identify $$a_1$$","text":"The first term is $$a_1=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser29a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a57f999ser29a-h1"],"title":"Identify $$r$$","text":"To find $$r$$, divide the 2nd term by the 1st term. What is $$r=\\\\frac{2}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser29a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["T"],"dependencies":["a57f999ser29a-h2"],"title":"Confirm that $$-1<r<1$$","text":"Is $$-1<\\\\frac{1}{2}<1$$ True or False?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"a57f999ser29a-h4","type":"hint","dependencies":["a57f999ser29a-h3"],"title":"Formula for the Sum of an Infinite Geometric Series","text":"Substitute values for $$a_1$$ and $$r$$ into the formula: $$S=\\\\frac{a_1}{1-r}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser29a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a57f999ser29a-h4"],"title":"Solve for S","text":"What is $$S=\\\\frac{4}{1-\\\\frac{1}{2}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser29a-h6","type":"hint","dependencies":["a57f999ser29a-h5"],"title":"Sum of Infinite Geometric Series","text":"The sum of the infinite geometric series is $$8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57f999ser3","title":"Using Formulas for Arithmetic Sequences","body":"Use the formula for the sum of the first $$n$$ terms of each arithmetic sequence to find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Series and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a57f999ser3a","stepAnswer":["$$\\\\frac{13\\\\left(3.2+5.6\\\\right)}{2}$$"],"problemType":"TextBox","stepTitle":"$$3.2+3.4+3.6+...+5.6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{13\\\\left(3.2+5.6\\\\right)}{2}$$","hints":{"DefaultPathway":[{"id":"a57f999ser3a-h1","type":"hint","dependencies":[],"title":"Find $$a_1$$ and $$a_n$$","text":"We are given $$a_1=3.2$$ and $$a_n=5.6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser3a-h2","type":"hint","dependencies":["a57f999ser3a-h1"],"title":"Find $$n$$","text":"To find $$n$$, use the formula for the general term of an arithmetic sequence: $$a_n=a_1+\\\\left(n-1\\\\right) d$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser3a-h3","type":"hint","dependencies":["a57f999ser3a-h2"],"title":"Find $$d$$","text":"The common difference can be found by subtracting the first term from the second term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.2$$"],"dependencies":["a57f999ser3a-h3"],"title":"Find $$d$$","text":"What is $$3.4-3.2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser3a-h5","type":"hint","dependencies":["a57f999ser3a-h4"],"title":"Find $$n$$","text":"Substitute values for $$a_1$$, $$a_n$$, and $$d$$ into the arithmetic sequence formula to get $$5.6=3.2+0.2\\\\left(n-1\\\\right)$$ and solve for $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser3a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$13$$"],"dependencies":["a57f999ser3a-h5"],"title":"Find $$n$$","text":"What is $$n$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser3a-h7","type":"hint","dependencies":["a57f999ser3a-h6"],"title":"Formula for Sum of the first $$n$$ terms of Arithmetic Sequence","text":"Substitute values for $$a_1$$, $$a_n$$ , and $$n$$ into the formula: $$S_n=\\\\frac{n \\\\left(a_1+a_n\\\\right)}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser3a-h8","type":"hint","dependencies":["a57f999ser3a-h7"],"title":"Formula for Sum of the first $$13$$ terms of Arithmetic Sequence","text":"The formula for the sum of the first $$13$$ terms of the arithmetic sequence is $$S_{13}=\\\\frac{13\\\\left(3.2+5.6\\\\right)}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57f999ser30","title":"Interpreting the Sum of Geometric Series","body":"Find the sum of the infinite geometric series.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Series and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a57f999ser30a","stepAnswer":["$$\\\\frac{-4}{3}$$"],"problemType":"TextBox","stepTitle":"-1-1/4-1/16-1/64...","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-4}{3}$$","hints":{"DefaultPathway":[{"id":"a57f999ser30a-h1","type":"hint","dependencies":[],"title":"Identify $$a_1$$","text":"The first term is $$a_1=-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser30a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4}$$"],"dependencies":["a57f999ser30a-h1"],"title":"Identify $$r$$","text":"To find $$r$$, divide the 2nd term by the 1st term. What is $$r=\\\\frac{\\\\left(-\\\\frac{1}{4}\\\\right)}{-1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser30a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["T"],"dependencies":["a57f999ser30a-h2"],"title":"Confirm that $$-1<r<1$$","text":"Is $$-1<\\\\frac{1}{4}<1$$ T of F?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"a57f999ser30a-h4","type":"hint","dependencies":["a57f999ser30a-h3"],"title":"Formula for the Sum of an Infinite Geometric Series","text":"Substitute values for $$a_1$$ and $$r$$ into the formula: $$S=\\\\frac{a_1}{1-r}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser30a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-4}{3}$$"],"dependencies":["a57f999ser30a-h4"],"title":"Solve for S","text":"What is $$S=\\\\frac{-1}{1-\\\\frac{1}{4}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser30a-h6","type":"hint","dependencies":["a57f999ser30a-h5"],"title":"Sum of Infinite Geometric Series","text":"The sum of the infinite geometric series is $$\\\\frac{-4}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57f999ser4","title":"Using Formulas for Arithmetic Sequences","body":"Use the formula for the sum of the first $$n$$ terms of each arithmetic sequence to find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Series and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a57f999ser4a","stepAnswer":["$$185$$"],"problemType":"TextBox","stepTitle":"$$5+8+11+14+17+20+23+26+29+32$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$185$$","hints":{"DefaultPathway":[{"id":"a57f999ser4a-h1","type":"hint","dependencies":[],"title":"Find $$a_1$$ and $$a_n$$","text":"We are given $$a_1=5$$ and $$a_n=32$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser4a-h2","type":"hint","dependencies":["a57f999ser4a-h1"],"title":"Find $$n$$","text":"Count the number of terms in the sequence to find $$n=10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser4a-h3","type":"hint","dependencies":["a57f999ser4a-h2"],"title":"Formula for Sum of the first $$n$$ terms of Arithmetic Sequence","text":"Substitute values for $$a_1$$, $$a_n$$ , and $$n$$ into the formula: $$S_n=\\\\frac{n \\\\left(a_1+a_n\\\\right)}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$185$$"],"dependencies":["a57f999ser4a-h3"],"title":"Formula for Sum of the first $$10$$ terms of Arithmetic Series","text":"What is $$S_{10}=\\\\frac{10\\\\left(5+32\\\\right)}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser4a-h5","type":"hint","dependencies":["a57f999ser4a-h4"],"title":"Sum of the first $$10$$ terms of Arithmetic Series","text":"The Sum of the first $$10$$ terms of Arithmetic Series is $$185$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57f999ser5","title":"Using Formulas for Arithmetic Sequences","body":"Use the formula for the sum of the first $$n$$ terms of each arithmetic sequence to find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Series and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a57f999ser5a","stepAnswer":["$$-225$$"],"problemType":"TextBox","stepTitle":"$$20+15+10+...+-50$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-225$$","hints":{"DefaultPathway":[{"id":"a57f999ser5a-h1","type":"hint","dependencies":[],"title":"Find $$a_1$$ and $$a_n$$","text":"We are given $$a_1=20$$ and $$a_n=-50$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser5a-h2","type":"hint","dependencies":["a57f999ser5a-h1"],"title":"Find $$n$$","text":"To find $$n$$, use the formula for the general term of an arithmetic sequence: $$a_n=a_1+\\\\left(n-1\\\\right) d$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser5a-h3","type":"hint","dependencies":["a57f999ser5a-h2"],"title":"Find $$d$$","text":"The common difference can be found by subtracting the first term from the second term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["a57f999ser5a-h3"],"title":"Find $$d$$","text":"What is $$15-20$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser5a-h5","type":"hint","dependencies":["a57f999ser5a-h4"],"title":"Find $$n$$","text":"Substitute values for $$a_1$$, $$a_n$$, and $$d$$ into the arithmetic sequence formula to get $$-50=20+-5\\\\left(n-1\\\\right)$$ and solve for $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser5a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["a57f999ser5a-h5"],"title":"Find $$n$$","text":"What is $$n$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser5a-h7","type":"hint","dependencies":["a57f999ser5a-h6"],"title":"Formula for Sum of the first $$n$$ terms of Arithmetic Series","text":"Substitute values for $$a_1$$, $$a_n$$ , and $$n$$ into the formula: $$S_n=\\\\frac{n \\\\left(a_1+a_n\\\\right)}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser5a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-225$$"],"dependencies":["a57f999ser5a-h7"],"title":"Formula for Sum of the first $$15$$ terms of Arithmetic Series","text":"What is $$S_{15}=\\\\frac{15\\\\left(20-50\\\\right)}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser5a-h9","type":"hint","dependencies":["a57f999ser5a-h8"],"title":"Sum of the first $$15$$ terms of Arithmetic Series","text":"The Sum of the first $$15$$ terms of Arithmetic Series is $$-225$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57f999ser6","title":"Using Formulas for Arithmetic Sequences","body":"Use the formula for the sum of the first $$n$$ terms of each arithmetic sequence to find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Series and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a57f999ser6a","stepAnswer":["$$138$$"],"problemType":"TextBox","stepTitle":"image1","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$138$$","hints":{"DefaultPathway":[{"id":"a57f999ser6a-h1","type":"hint","dependencies":[],"title":"Find $$a_1$$","text":"To find $$a_1$$, substitute $$k=1$$ into the given explicit formula: $$a_k=3k-8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["a57f999ser6a-h1"],"title":"Solve for $$a_1$$","text":"What is $$a_1=3\\\\times1-8$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser6a-h3","type":"hint","dependencies":["a57f999ser6a-h2"],"title":"Find $$a_k$$","text":"We are given that $$n=12$$. To find $$a_{12}$$, substitute $$k=12$$ into the given explicit formula: $$a_k=3k-8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$28$$"],"dependencies":["a57f999ser6a-h3"],"title":"Solve for $$a_{12}$$","text":"What is $$a_{12}=3\\\\times12-8$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser6a-h5","type":"hint","dependencies":["a57f999ser6a-h4"],"title":"Formula for Sum of the first $$n$$ terms of Arithmetic Series","text":"Substitute values for $$a_1$$, $$a_n$$ , and $$n$$ into the formula: $$S_n=\\\\frac{n \\\\left(a_1+a_n\\\\right)}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser6a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$138$$"],"dependencies":["a57f999ser6a-h5"],"title":"Formula for Sum of the first $$12$$ terms of Arithmetic Series","text":"What is $$S_{12}=\\\\frac{12\\\\left(-5+28\\\\right)}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser6a-h7","type":"hint","dependencies":["a57f999ser6a-h6"],"title":"Sum of the first $$12$$ terms of Arithmetic Series","text":"The Sum of the first $$12$$ terms of Arithmetic Series is $$138$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57f999ser7","title":"Using Formulas for Arithmetic Sequences","body":"Use the formula for the sum of the first $$n$$ terms of each arithmetic sequence to find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Series and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a57f999ser7a","stepAnswer":["$$9.3$$"],"problemType":"TextBox","stepTitle":"$$-1.7+-0.4+0.9+2.2+3.5+4.8$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9.3$$","hints":{"DefaultPathway":[{"id":"a57f999ser7a-h1","type":"hint","dependencies":[],"title":"Find $$a_1$$ and $$a_n$$","text":"We are given $$a_1=-1.7$$ and $$a_n=4.8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser7a-h2","type":"hint","dependencies":["a57f999ser7a-h1"],"title":"Find $$n$$","text":"Count the number of terms in the sequence to find $$n=6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser7a-h3","type":"hint","dependencies":["a57f999ser7a-h2"],"title":"Formula for Sum of the first $$n$$ terms of Arithmetic Sequence","text":"Substitute values for $$a_1$$, $$a_n$$ , and $$n$$ into the formula: $$S_n=\\\\frac{n \\\\left(a_1+a_n\\\\right)}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9.3$$"],"dependencies":["a57f999ser7a-h3"],"title":"Solve for $$S_6$$","text":"What is $$S_6=\\\\frac{6\\\\left(-1.7+4.8\\\\right)}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser7a-h5","type":"hint","dependencies":["a57f999ser7a-h4"],"title":"Sum of the first $$6$$ terms of Arithmetic Sequence","text":"The Sum of the first $$6$$ terms of Arithmetic Series is $$9.3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57f999ser8","title":"Using Formulas for Arithmetic Sequences","body":"Use the formula for the sum of the first $$n$$ terms of each arithmetic sequence to find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Series and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a57f999ser8a","stepAnswer":["$$\\\\frac{147}{2}$$"],"problemType":"TextBox","stepTitle":"$$6+\\\\frac{15}{2}+9+\\\\frac{21}{2}+12+\\\\frac{27}{2}+15$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{147}{2}$$","hints":{"DefaultPathway":[{"id":"a57f999ser8a-h1","type":"hint","dependencies":[],"title":"Find $$a_1$$ and $$a_n$$","text":"We are given $$a_1=6$$ and $$a_n=15$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser8a-h2","type":"hint","dependencies":["a57f999ser8a-h1"],"title":"Find $$n$$","text":"Count the number of terms in the sequence to find $$n=7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser8a-h3","type":"hint","dependencies":["a57f999ser8a-h2"],"title":"Formula for Sum of the first $$n$$ terms of Arithmetic Sequence","text":"Substitute values for $$a_1$$, $$a_n$$ , and $$n$$ into the formula: $$S_n=\\\\frac{n \\\\left(a_1+a_n\\\\right)}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{147}{2}$$"],"dependencies":["a57f999ser8a-h3"],"title":"Solve for $$S_7$$","text":"What is $$S_7=\\\\frac{7\\\\left(6+15\\\\right)}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser8a-h5","type":"hint","dependencies":["a57f999ser8a-h4"],"title":"Sum of the first $$7$$ terms of Arithmetic Sequence","text":"The Sum of the first $$6$$ terms of Arithmetic Sequence is $$\\\\frac{147}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a57f999ser9","title":"Using Formulas for Arithmetic Sequences","body":"Use the formula for the sum of the first $$n$$ terms of each arithmetic sequence to find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Series and Their Notations","courseName":"OpenStax: College Algebra","steps":[{"id":"a57f999ser9a","stepAnswer":["$$135$$"],"problemType":"TextBox","stepTitle":"$$-1+3+7+...+31$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$135$$","hints":{"DefaultPathway":[{"id":"a57f999ser9a-h1","type":"hint","dependencies":[],"title":"Find $$a_1$$ and $$a_n$$","text":"We are given $$a_1=-1$$ and $$a_n=31$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser9a-h2","type":"hint","dependencies":["a57f999ser9a-h1"],"title":"Find $$n$$","text":"To find $$n$$, use the formula for the general term of an arithmetic sequence: $$a_n=a_1+\\\\left(n-1\\\\right) d$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser9a-h3","type":"hint","dependencies":["a57f999ser9a-h2"],"title":"Find $$d$$","text":"The common difference can be found by subtracting the first term from the second term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a57f999ser9a-h3"],"title":"Find $$d$$","text":"What is $$d=3-(-1)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser9a-h5","type":"hint","dependencies":["a57f999ser9a-h4"],"title":"Find $$n$$","text":"Substitute values for $$a_1$$, $$a_n$$, and $$d$$ into the arithmetic sequence formula to get $$31=-1+4\\\\left(n-1\\\\right)$$ and solve for $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser9a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a57f999ser9a-h5"],"title":"Find $$n$$","text":"What is $$n$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser9a-h7","type":"hint","dependencies":["a57f999ser9a-h6"],"title":"Formula for Sum of the first $$n$$ terms of Arithmetic Sequence","text":"Substitute values for $$a_1$$, $$a_n$$ , and $$n$$ into the formula: $$S_n=\\\\frac{n \\\\left(a_1+a_n\\\\right)}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser9a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$135$$"],"dependencies":["a57f999ser9a-h7"],"title":"Solve for $$S_9$$","text":"What is $$S_9=\\\\frac{9\\\\left(-1+31\\\\right)}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a57f999ser9a-h9","type":"hint","dependencies":["a57f999ser9a-h8"],"title":"Sum of the first $$9$$ terms of Arithmetic Sequence","text":"The Sum of the first $$9$$ terms of Arithmetic Sequence is $$135$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a58aac7box1","title":"Understanding Box Plots","body":"Given the following box plot, answer the questions.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Box Plots","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a58aac7box1a","stepAnswer":["State University conducted a survey to see how involved its students are in community service. The box plot shows the number of community service hours logged by participants over the past year."],"problemType":"MultipleChoice","stepTitle":"Think of an example (in words) where the data might fit into the above box plot.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["State University conducted a survey to see how involved its students are in community service. The box plot shows the number of community service hours logged by participants over the past year.","State University conducted a survey to see how many hours a student sleeps per day. The box plot shows the number of sleep hours logged by participants over the past year."],"hints":{"DefaultPathway":[{"id":"a58aac7box1a-h1","type":"hint","dependencies":[],"title":"Looking at the min and max values.","text":"First, you should look at the box plot and understand the what the min and max values would indicate in the context of your example.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a58aac7box1a-h2","type":"hint","dependencies":["a58aac7box1a-h1"],"title":"Finding a plausible scenario.","text":"Secondly, you must look for an example that makes sense in the context. i.e. a maximum value of $$150$$ would make it impossible for it to indicate hours of sleep per day.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a58aac7box10","title":"Fewest data points.","body":"A survey was conducted of $$130$$ purchasers of new BMW $$3$$ series cars, $$130$$ purchasers of new BMW $$5$$ series cars, and $$130$$ purchasers of new BMW $$7$$ series cars. In it, people were asked the age they were when they purchased their car. The following box plots display the results.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Box Plots","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a58aac7box10a","stepAnswer":["$$31-35$$"],"problemType":"MultipleChoice","stepTitle":"Look at the BMW $$5$$ series. Which interval has the fewest data in it? How do you know this?\\\\n$$31-35$$\\\\n$$38-41$$\\\\n$$41-64$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$31-35$$","choices":["$$31-35$$","$$31-35$$","$$38-41$$","$$41-64$$"],"hints":{"DefaultPathway":[{"id":"a58aac7box10a-h1","type":"hint","dependencies":[],"title":"Look at how much all the intervals encompass.","text":"First, find what percentage of data values fall in each interval that is clear. $$38-41$$ has 25%, and so does $$41-64$$ and $$31-38$$. This means that less than 25% is between $$31-35$$, making it the smallest interval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a58aac7box11","title":"Describe the box plot.","body":"In a survey of 20-year-olds in China, Germany, and the United States, people were asked the number of foreign countries they had visited in their lifetime. The following box plots display the results.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Box Plots","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a58aac7box11a","stepAnswer":["The box plot for China has no quartiles meaning the data is only a minimum and maximum. For Germany, it is skewed left so it is more variable for the bottom 50%. For US, it is skewed right so it is more variable for the top 50%."],"problemType":"MultipleChoice","stepTitle":"Describe what the shape of each box plot implies about the distribution of the data collected.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["The box plot for China has no quartiles meaning the data is only a minimum and maximum. For Germany, it is skewed left so it is more variable for the bottom 50%. For US, it is skewed right so it is more variable for the top 50%.","The box plot for China has no quartiles meaning the data is only a minimum and maximum. For Germany, it is skewed right so it is more variable for the top 50%. For US, it is skewed left so it is more variable for the bottom 50%."],"hints":{"DefaultPathway":[{"id":"a58aac7box11a-h1","type":"hint","dependencies":[],"title":"Finding skewness","text":"First, you must find the skew of each box plot. Having more data on the left side means it is skewed right. More data on right side means skewed left.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a58aac7box12","title":"Amounts in box plots.","body":"In a survey of 20-year-olds in China, Germany, and the United States, people were asked the number of foreign countries they had visited in their lifetime. The following box plots display the results.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Box Plots","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a58aac7box12a","stepAnswer":["Not enough data"],"problemType":"MultipleChoice","stepTitle":"Have more Americans or more Germans surveyed been to over eight foreign countries?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Americans","Germans","Not enough data"],"hints":{"DefaultPathway":[{"id":"a58aac7box12a-h1","type":"hint","dependencies":[],"title":"Uncertainty in box plots","text":"In box plots, we are only given $$5$$ key numbers, so we do not know the amount of people at each point on the graph. This means that we do not know the specific amount of people that have to more than $$8$$ foreign countries.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a58aac7box13","title":"Comparing box plots","body":"In a survey of 20-year-olds in China, Germany, and the United States, people were asked the number of foreign countries they had visited in their lifetime. The following box plots display the results.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Box Plots","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a58aac7box13a","stepAnswer":["Comparing the medians, Germany tends to travel to more countries than the US. However, this is not a rule, because there is so much variability in each data set."],"problemType":"MultipleChoice","stepTitle":"Compare the three box plots. What do they imply about the foreign travel of 20-year-old residents of the three countries when compared to each other?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Comparing the medians, Germany tends to travel to more countries than the US. However, this is not a rule, because there is so much variability in each data set.","Comparing the medians, the United States tends to travel to more countries than Germany. However, this is not a rule, because there is so much variability in each data set."],"hints":{"DefaultPathway":[{"id":"a58aac7box13a-h1","type":"hint","dependencies":[],"title":"Comparing medians","text":"First, find the medians: China doesn\'t have a median while Germany has one at $$8$$ and the United States at $$2$$. $$8$$ > $$2$$ so it is most likely that Germany travels more, but this dataset is too small for any conclusive claims.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a58aac7box2","title":"Understanding Box Plots","body":"Given the following box plot, answer the questions.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Box Plots","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a58aac7box2a","stepAnswer":["Because the first and second quartiles are close, the data in this quarter is very similar. There is not much variation in the values. The data in the third quarter is much more variable, or spread out. This is clear because the second quartile is so far away from the third quartile."],"problemType":"MultipleChoice","stepTitle":"What does it mean to have the first and second quartiles so close together, while the second to third quartiles are far apart?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Because the first and second quartiles are close, the data in this quarter is very similar. There is not much variation in the values. The data in the third quarter is much more variable, or spread out. This is clear because the second quartile is so far away from the third quartile.","Because the first and second quartiles are close, the data in this quarter is not very similar. There is variation in the values. The data in the third quarter is less variable. This is clear because the second quartile is so far away from the third quartile."],"hints":{"DefaultPathway":[{"id":"a58aac7box2a-h1","type":"hint","dependencies":[],"title":"Identifying the quartiles.","text":"First, you must look at where the first, second, and third quartiles are. The first is at $$0$$, the second at $$20$$, and the third slightly before $$100$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a58aac7box2a-h2","type":"hint","dependencies":["a58aac7box2a-h1"],"title":"Understanding distance in box plots.","text":"Secondly, you must understand that to be close in a box plot means that the data in those quarties is similar as there is not much variation. If they are far apart, there is more variation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a58aac7box3","title":"Describing a Box Plot","body":"A survey was conducted of $$130$$ purchasers of new BMW $$3$$ series cars, $$130$$ purchasers of new BMW $$5$$ series cars, and $$130$$ purchasers of new BMW $$7$$ series cars. In it, people were asked the age they were when they purchased their car. The following box plots display the results.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Box Plots","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a58aac7box3a","stepAnswer":["Each box plot is spread out more in the greater values. Each plot is skewed to the right, so the ages of the top 50% of buyers are more variable than the ages of the lower 50%."],"problemType":"MultipleChoice","stepTitle":"Describe what the shape of each box plot implies about the distribution of the data collected for that car series.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Each box plot is spread out more in the greater values. Each plot is skewed to the left, so the ages of the top 50% of buyers are less variable than the ages of the lower 50%.","Each box plot is spread out more in the greater values. Each plot is skewed to the right, so the ages of the top 50% of buyers are more variable than the ages of the lower 50%."],"hints":{"DefaultPathway":[{"id":"a58aac7box3a-h1","type":"hint","dependencies":[],"title":"Determining the shape of each graph.","text":"First, you need to determine the shape of each box plot. All through tend to have more greater values which means they are all skewed right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a58aac7box3a-h2","type":"hint","dependencies":["a58aac7box3a-h1"],"title":"Understanding what skewness implies.","text":"Secondly, you must understand what a right skew means in context. Here, it shows that the ages of the top 50% of buyers are more variable than the ages of the lower 50%.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a58aac7box4","title":"Understanding Outliers","body":"A survey was conducted of $$130$$ purchasers of new BMW $$3$$ series cars, $$130$$ purchasers of new BMW $$5$$ series cars, and $$130$$ purchasers of new BMW $$7$$ series cars. In it, people were asked the age they were when they purchased their car. The following box plots display the results.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Box Plots","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a58aac7box4a","stepAnswer":["BMW $$3$$"],"problemType":"MultipleChoice","stepTitle":"Which group is most likely to have an outlier?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"BMW $$3$$","choices":["BMW $$3$$","BMW $$5$$","BMW $$7$$"],"hints":{"DefaultPathway":[{"id":"a58aac7box4a-h1","type":"hint","dependencies":[],"title":"Determining outliers.","text":"Think about what properties of a box plot is affected by outliers. The box plot with the longest whisker will be most likely to have an outlier (BMW3).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a58aac7box5","title":"Comparing Box Plots","body":"A survey was conducted of $$130$$ purchasers of new BMW $$3$$ series cars, $$130$$ purchasers of new BMW $$5$$ series cars, and $$130$$ purchasers of new BMW $$7$$ series cars. In it, people were asked the age they were when they purchased their car. The following box plots display the results.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Box Plots","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a58aac7box5a","stepAnswer":["Comparing the median ages, younger people tend to buy the BMW $$3$$ series, while older people tend to buy the BMW $$7$$ series. However, this is not a rule, because there is so much variability in each data set."],"problemType":"MultipleChoice","stepTitle":"Compare the three box plots. What do they imply about the age of purchasing a BMW from the series when compared to each other?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"Comparing the median ages, younger people tend to buy the BMW $$3$$ series, while older people tend to buy the BMW $$7$$ series. However, this is not a rule, because there is so much variability in each data set.","choices":["Comparing the median ages, older people tend to buy the BMW $$3$$ series, while younger people tend to buy the BMW $$7$$ series. However, this is not a rule, because there is so much variability in each data set.","Comparing the median ages, younger people tend to buy the BMW $$3$$ series, while older people tend to buy the BMW $$7$$ series. However, this is not a rule, because there is so much variability in each data set."],"hints":{"DefaultPathway":[{"id":"a58aac7box5a-h1","type":"hint","dependencies":[],"title":"Finding the medians.","text":"First, you should look for the medians of each of the box plots and compare them in relation to each other. From this, we can see that the median of $$7$$ > median of $$5$$ > median of $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a58aac7box6","title":"Finding the smallest quarter and spread.","body":"A survey was conducted of $$130$$ purchasers of new BMW $$3$$ series cars, $$130$$ purchasers of new BMW $$5$$ series cars, and $$130$$ purchasers of new BMW $$7$$ series cars. In it, people were asked the age they were when they purchased their car. The following box plots display the results.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Box Plots","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a58aac7box6a","stepAnswer":["2nd, $$3$$"],"problemType":"MultipleChoice","stepTitle":"Look at the BMW $$5$$ series. Which quarter has the smallest spread of data? What is the spread?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"2nd, $$3$$","choices":["1st, $$3$$","2nd, $$3$$","2nd, $$2$$","3rd, $$3$$"],"hints":{"DefaultPathway":[{"id":"a58aac7box6a-h1","type":"hint","dependencies":[],"title":"Finding the smallest quarter.","text":"First, to pinpoint the smallest quarter, split the box plot into the $$4$$ quarters based on the quartiles and end points and find the smallest one in length (2nd).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a58aac7box6a-h2","type":"hint","dependencies":["a58aac7box6a-h1"],"title":"Finding the spread.","text":"To find the spread of the 2nd quarter, calculate the difference between the first quartile and the median which is $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a58aac7box7","title":"Finding the largest quarter and spread.","body":"A survey was conducted of $$130$$ purchasers of new BMW $$3$$ series cars, $$130$$ purchasers of new BMW $$5$$ series cars, and $$130$$ purchasers of new BMW $$7$$ series cars. In it, people were asked the age they were when they purchased their car. The following box plots display the results.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Box Plots","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a58aac7box7a","stepAnswer":["3rd, $$14$$"],"problemType":"MultipleChoice","stepTitle":"Look at the BMW $$5$$ series. Which quarter has the largest spread of data? What is the spread?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"3rd, $$14$$","choices":["1st, $$14$$","2nd, $$14$$","3rd, $$13$$","3rd, $$14$$"],"hints":{"DefaultPathway":[{"id":"a58aac7box7a-h1","type":"hint","dependencies":[],"title":"Finding the largest quarter.","text":"First, to pinpoint the largest quarter, split the box plot into the $$4$$ quarters based on the quartiles and end points and find the largest one in length (3rd).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a58aac7box7a-h2","type":"hint","dependencies":["a58aac7box7a-h1"],"title":"Finding the spread.","text":"To find the spread of the 3rd quarter, calculate the difference between the median and the third quartile which is $$14$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a58aac7box8","title":"Estimated the IQR.","body":"A survey was conducted of $$130$$ purchasers of new BMW $$3$$ series cars, $$130$$ purchasers of new BMW $$5$$ series cars, and $$130$$ purchasers of new BMW $$7$$ series cars. In it, people were asked the age they were when they purchased their car. The following box plots display the results.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Box Plots","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a58aac7box8a","stepAnswer":["$$17$$"],"problemType":"TextBox","stepTitle":"Look at the BMW $$5$$ series. Estimate the interquartile range (IQR).","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$17$$","hints":{"DefaultPathway":[{"id":"a58aac7box8a-h1","type":"hint","dependencies":[],"title":"Calculating IQR","text":"To calculate IQR, you must take the 3rd quartile and subtract eh 1st quartile. This results in $$55$$ - $$38$$ $$=$$ $$17$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a58aac7box9","title":"Amount of data in each quarter.","body":"A survey was conducted of $$130$$ purchasers of new BMW $$3$$ series cars, $$130$$ purchasers of new BMW $$5$$ series cars, and $$130$$ purchasers of new BMW $$7$$ series cars. In it, people were asked the age they were when they purchased their car. The following box plots display the results.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Box Plots","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a58aac7box9a","stepAnswer":["Not enough information"],"problemType":"MultipleChoice","stepTitle":"Look at the BMW $$5$$ series. Are there more data in the interval $$31$$ to $$38$$ or in the interval $$45$$ to 55?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["$$31-38$$","$$45-55$$","Not enough information"],"hints":{"DefaultPathway":[{"id":"a58aac7box9a-h1","type":"hint","dependencies":[],"title":"Finding out where the intervals lie.","text":"First look at where the intervals lie on the box plot. Here, these intervals are within a quarter so we cannot tell exactly where the data in the quarter is concentrated.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5991b4add1","title":"Identify Polynomials, Monomials, Binomials and Trinomials","body":"Determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Add and Subtract Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5991b4add1a","stepAnswer":["trinomial"],"problemType":"MultipleChoice","stepTitle":"$$81b^5-24b^3+1$$","stepBody":"","answerType":"string","variabilization":{},"choices":["monomial","binomial","trinomial","polynomial"],"hints":{"DefaultPathway":[{"id":"a5991b4add1a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":[],"title":"Number of terms","text":"How many terms are in this expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5991b4add1a-h2","type":"hint","dependencies":["a5991b4add1a-h1"],"title":"Prefix","text":"The prefix for $$3$$ is tri-.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a5991b4add1b","stepAnswer":["polynomial"],"problemType":"MultipleChoice","stepTitle":"$$5c^3+11c^2-c-8$$","stepBody":"","answerType":"string","variabilization":{},"choices":["monomial","binomial","trinomial","polynomial"],"hints":{"DefaultPathway":[{"id":"a5991b4add1b-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":[],"title":"Number of terms","text":"How many terms are in this expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5991b4add1b-h2","type":"hint","dependencies":["a5991b4add1b-h1"],"title":"Prefix","text":"The prefix used for $$4$$ or more is poly-.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a5991b4add1c","stepAnswer":["binomial"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{14}{15} y+\\\\frac{1}{7}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["monomial","binomial","trinomial","polynomial"],"hints":{"DefaultPathway":[{"id":"a5991b4add1c-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Number of terms","text":"How many terms are in this expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5991b4add1c-h2","type":"hint","dependencies":["a5991b4add1c-h1"],"title":"Prefix","text":"The prefix used for $$2$$ or more is bi-.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a5991b4add1d","stepAnswer":["monomial"],"problemType":"MultipleChoice","stepTitle":"$$5$$","stepBody":"","answerType":"string","variabilization":{},"choices":["monomial","binomial","trinomial","polynomial"],"hints":{"DefaultPathway":[{"id":"a5991b4add1d-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Number of terms","text":"How many terms are in this expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5991b4add1d-h2","type":"hint","dependencies":["a5991b4add1d-h1"],"title":"Prefix","text":"The prefix used for $$1$$ or more is mono-.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a5991b4add1e","stepAnswer":["binomial"],"problemType":"MultipleChoice","stepTitle":"$$4y+17$$","stepBody":"","answerType":"string","variabilization":{},"choices":["monomial","binomial","trinomial","polynomial"],"hints":{"DefaultPathway":[{"id":"a5991b4add1e-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Number of terms","text":"How many terms are in this expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5991b4add1e-h2","type":"hint","dependencies":["a5991b4add1e-h1"],"title":"Prefix","text":"The prefix used for $$2$$ or more is bi-.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5991b4add10","title":"Add and Subtract Monomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Add and Subtract Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5991b4add10a","stepAnswer":["-5a"],"problemType":"TextBox","stepTitle":"4a-9a","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a5991b4add10a-h1","type":"hint","dependencies":[],"title":"Combine like terms","text":"If the monomials are like terms, combine them by adding or subtracting the coefficient.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5991b4add10a-h2","type":"hint","dependencies":["a5991b4add10a-h1"],"title":"Coefficients","text":"The coefficients involved in this expression are $$4$$ and $$-9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5991b4add11","title":"Add and Subtract Monomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Add and Subtract Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5991b4add11a","stepAnswer":["40a"],"problemType":"TextBox","stepTitle":"$$28x-(-12x)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a5991b4add11a-h1","type":"hint","dependencies":[],"title":"Combine like terms","text":"If the monomials are like terms, combine them by adding or subtracting the coefficient.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5991b4add11a-h2","type":"hint","dependencies":["a5991b4add11a-h1"],"title":"Coefficients","text":"The coefficients involved in this expression are $$28$$ and $$-12$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5991b4add12","title":"Add and Subtract Monomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Add and Subtract Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5991b4add12a","stepAnswer":["$$-22b$$"],"problemType":"TextBox","stepTitle":"$$-5b-17b$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-22b$$","hints":{"DefaultPathway":[{"id":"a5991b4add12a-h1","type":"hint","dependencies":[],"title":"Combine like terms","text":"If the monomials are like terms, combine them by adding or subtracting the coefficient.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5991b4add12a-h2","type":"hint","dependencies":["a5991b4add12a-h1"],"title":"Coefficients","text":"The coefficients involved in this expression are $$-5$$ and $$-17$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5991b4add13","title":"Add and Subtract Monomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Add and Subtract Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5991b4add13a","stepAnswer":["$$-10a+5b$$"],"problemType":"TextBox","stepTitle":"$$12a+5b-22a$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-10a+5b$$","hints":{"DefaultPathway":[{"id":"a5991b4add13a-h1","type":"hint","dependencies":[],"title":"Combine like terms","text":"If the monomials are like terms, combine them by adding or subtracting the coefficient.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5991b4add13a-h2","type":"hint","dependencies":["a5991b4add13a-h1"],"title":"Coefficients","text":"12a and -22a are like terms so you may combine the coefficients. $$5b$$ is not a like term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5991b4add14","title":"Add and Subtract Monomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Add and Subtract Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5991b4add14a","stepAnswer":["$$-4a^2+b^2$$"],"problemType":"TextBox","stepTitle":"$$2a^2+b^2-6a^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-4a^2+b^2$$","hints":{"DefaultPathway":[{"id":"a5991b4add14a-h1","type":"hint","dependencies":[],"title":"Combine like terms","text":"If the monomials are like terms, combine them by adding or subtracting the coefficient.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5991b4add14a-h2","type":"hint","dependencies":["a5991b4add14a-h1"],"title":"Coefficients","text":"$$2a^2$$ and $$-6a^2$$ are like terms so you may combine the coefficients. $$b^2$$ is not a like term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5991b4add15","title":"Add and Subtract Monomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Add and Subtract Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5991b4add15a","stepAnswer":["$$x\\\\left(y^2\\\\right)-5x-5y^2$$"],"problemType":"TextBox","stepTitle":"$${xy}^2-5x-5y^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x\\\\left(y^2\\\\right)-5x-5y^2$$","hints":{"DefaultPathway":[{"id":"a5991b4add15a-h1","type":"hint","dependencies":[],"title":"Combine like terms","text":"If the monomials are like terms, combine them by adding or subtracting the coefficient.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5991b4add15a-h2","type":"hint","dependencies":["a5991b4add15a-h1"],"title":"Like terms","text":"There are no like terms in this expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5991b4add16","title":"Identifying Polynomial Type","body":"Identify the type of the polynomial:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Add and Subtract Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5991b4add16a","stepAnswer":["Trinomial"],"problemType":"MultipleChoice","stepTitle":"$$4y^2-8y-6$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Monomial","Binomial","Trinomial","Other"],"hints":{"DefaultPathway":[{"id":"a5991b4add16a-h1","type":"hint","dependencies":[],"title":"Number of Terms","text":"Since there are three terms, the expression is a trinomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5991b4add17","title":"Identifying Polynomial Type","body":"Identify the type of the polynomial: -","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Add and Subtract Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5991b4add17a","stepAnswer":["Monomial"],"problemType":"MultipleChoice","stepTitle":"$$5a^4 b^2$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Monomial","Binomial","Trinomial","Other"],"hints":{"DefaultPathway":[{"id":"a5991b4add17a-h1","type":"hint","dependencies":[],"title":"Number of Terms","text":"Since there is one term, the expression is a trinomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5991b4add18","title":"Identifying Polynomial Type","body":"Identify the type of the polynomial:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Add and Subtract Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5991b4add18a","stepAnswer":["Other"],"problemType":"MultipleChoice","stepTitle":"$$2x^5-5x^3-9x^2+3x+4$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Monomial","Binomial","Trinomial","Other"],"hints":{"DefaultPathway":[{"id":"a5991b4add18a-h1","type":"hint","dependencies":[],"title":"Number of Terms","text":"Since there are $$5$$ terms, the expression is neither a monomial, binomial, nor trinomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5991b4add19","title":"Identifying Polynomial Type","body":"Identify the type of the polynomial:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Add and Subtract Polynomials","courseName":"OpenStax: Elementary 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Elementary Algebra","steps":[{"id":"a5991b4add2a","stepAnswer":["binomial"],"problemType":"MultipleChoice","stepTitle":"$$x^2-y^2$$","stepBody":"","answerType":"string","variabilization":{},"choices":["monomial","binomial","trinomial","polynomial"],"hints":{"DefaultPathway":[{"id":"a5991b4add2a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Number of terms","text":"How many terms are in this expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5991b4add2a-h2","type":"hint","dependencies":["a5991b4add2a-h1"],"title":"Prefix","text":"The prefix for $$2$$ is bi-.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a5991b4add2b","stepAnswer":["monomial"],"problemType":"MultipleChoice","stepTitle":"$$-13c^4$$","stepBody":"","answerType":"string","variabilization":{},"choices":["monomial","binomial","trinomial","polynomial"],"hints":{"DefaultPathway":[{"id":"a5991b4add2b-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Number of terms","text":"How many terms are in this expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5991b4add2b-h2","type":"hint","dependencies":["a5991b4add2b-h1"],"title":"Prefix","text":"The prefix used for $$1$$ or more is mono-.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a5991b4add2c","stepAnswer":["trinomial"],"problemType":"MultipleChoice","stepTitle":"$$x^2+5x-7$$","stepBody":"","answerType":"string","variabilization":{},"choices":["monomial","binomial","trinomial","polynomial"],"hints":{"DefaultPathway":[{"id":"a5991b4add2c-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":[],"title":"Number of terms","text":"How many terms are in this expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5991b4add2c-h2","type":"hint","dependencies":["a5991b4add2c-h1"],"title":"Prefix","text":"The prefix used for $$3$$ or more is tri-.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a5991b4add2d","stepAnswer":["trinomial"],"problemType":"MultipleChoice","stepTitle":"$$x^2 y^2-2xy+8$$","stepBody":"","answerType":"string","variabilization":{},"choices":["monomial","binomial","trinomial","polynomial"],"hints":{"DefaultPathway":[{"id":"a5991b4add2d-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":[],"title":"Number of terms","text":"How many terms are in this expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5991b4add2d-h2","type":"hint","dependencies":["a5991b4add2d-h1"],"title":"Prefix","text":"The prefix used for $$3$$ or more is tri-.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a5991b4add2e","stepAnswer":["monomial"],"problemType":"MultipleChoice","stepTitle":"$$19$$","stepBody":"","answerType":"string","variabilization":{},"choices":["monomial","binomial","trinomial","polynomial"],"hints":{"DefaultPathway":[{"id":"a5991b4add2e-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Number of terms","text":"How many terms are in this expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5991b4add2e-h2","type":"hint","dependencies":["a5991b4add2e-h1"],"title":"Prefix","text":"The prefix used for $$1$$ or more is mono-.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5991b4add20","title":"Identifying Polynomial Type","body":"Identify the type of the polynomial:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Add and Subtract Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5991b4add20a","stepAnswer":["Monomial"],"problemType":"MultipleChoice","stepTitle":"q","stepBody":"","answerType":"string","variabilization":{},"choices":["Monomial","Binomial","Trinomial","Other"],"hints":{"DefaultPathway":[{"id":"a5991b4add20a-h1","type":"hint","dependencies":[],"title":"Number of Terms","text":"Since there is $$1$$ term, the expression is a monomial..","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5991b4add21","title":"Adding and Subtracting Polynomials","body":"Simplify the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Add and Subtract Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5991b4add21a","stepAnswer":["$$40y^2$$"],"problemType":"TextBox","stepTitle":"Add $$25y^2+15y^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$40y^2$$","choices":["$$40y^2$$","Binomial","Monomial","Other","Trinomial"],"hints":{"DefaultPathway":[{"id":"a5991b4add21a-h1","type":"hint","dependencies":[],"title":"Combining Like Terms","text":"Since both terms are like, we can add their coefficients to get $$40y^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5991b4add22","title":"Adding and Subtracting Polynomials","body":"Simplify the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Add and Subtract Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5991b4add22a","stepAnswer":["$$21q^2$$"],"problemType":"TextBox","stepTitle":"Add $$12q^2+9q^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$21q^2$$","hints":{"DefaultPathway":[{"id":"a5991b4add22a-h1","type":"hint","dependencies":[],"title":"Combining Like Terms","text":"Since both terms are like, we can add their coefficients to get $$21q^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5991b4add23","title":"Adding and Subtracting Polynomials","body":"Simplify the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Add and Subtract Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5991b4add23a","stepAnswer":["$$-7c^2$$"],"problemType":"TextBox","stepTitle":"Add $$-15c^2+8c^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-7c^2$$","hints":{"DefaultPathway":[{"id":"a5991b4add23a-h1","type":"hint","dependencies":[],"title":"Combining Like Terms","text":"Since both terms are like, we can add their coefficients to get $$-7c^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5991b4add24","title":"Adding and Subtracting Polynomials","body":"Simplify the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Add and Subtract Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5991b4add24a","stepAnswer":["$$23p$$"],"problemType":"TextBox","stepTitle":"Subtract $$16p-(-7p)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$23p$$","hints":{"DefaultPathway":[{"id":"a5991b4add24a-h1","type":"hint","dependencies":[],"title":"Combining Like Terms","text":"Since both terms are like, we can subtract their coefficients to get $$23p$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5991b4add25","title":"Adding and Subtracting Polynomials","body":"Simplify the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Add and Subtract Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5991b4add25a","stepAnswer":["$$-10z^3$$"],"problemType":"TextBox","stepTitle":"Subtract $$-15z^3-\\\\left(-5z^3\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-10z^3$$","hints":{"DefaultPathway":[{"id":"a5991b4add25a-h1","type":"hint","dependencies":[],"title":"Combining Like Terms","text":"Since both terms are like, we can subtract their coefficients to get $$-10z^3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5991b4add26","title":"Adding and Subtracting Polynomials","body":"Simplify the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Add and Subtract Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5991b4add26a","stepAnswer":["$$13m$$"],"problemType":"TextBox","stepTitle":"Subtract $$8m-(-5m)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$13m$$","hints":{"DefaultPathway":[{"id":"a5991b4add26a-h1","type":"hint","dependencies":[],"title":"Combining Like Terms","text":"Since both terms are like, we can subtract their coefficients to get $$13m$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5991b4add27","title":"Adding and Subtracting Polynomials","body":"Simplify the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Add and Subtract Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5991b4add27a","stepAnswer":["$$-5c^2+7d^2$$"],"problemType":"TextBox","stepTitle":"Simplify $$c^2+7d^2-6c^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-5c^2+7d^2$$","hints":{"DefaultPathway":[{"id":"a5991b4add27a-h1","type":"hint","dependencies":[],"title":"Combining Like Terms","text":"We must combine like terms by adding coefficients. We get $$-5c^2+7d^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5991b4add28","title":"Adding and Subtracting Polynomials","body":"Simplify the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Add and Subtract Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5991b4add28a","stepAnswer":["$$5y^2+3z^2$$"],"problemType":"TextBox","stepTitle":"Add $$8y^2+3z^2-3y^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5y^2+3z^2$$","hints":{"DefaultPathway":[{"id":"a5991b4add28a-h1","type":"hint","dependencies":[],"title":"Combining Like Terms","text":"We must combine like terms by adding coefficients. We get $$5y^2+3z^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5991b4add29","title":"Adding and Subtracting Polynomials","body":"Simplify the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Add and Subtract Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5991b4add29a","stepAnswer":["$$-4m^2+n^2$$"],"problemType":"TextBox","stepTitle":"Add $$3m^2+n^2-7m^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-4m^2+n^2$$","hints":{"DefaultPathway":[{"id":"a5991b4add29a-h1","type":"hint","dependencies":[],"title":"Combining Like Terms","text":"We must combine like terms by adding coefficients. We get $$-4m^2+n^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5991b4add3","title":"Identify Polynomials, Monomials, Binomials and Trinomials","body":"Determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Add and Subtract Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5991b4add3a","stepAnswer":["binomial"],"problemType":"MultipleChoice","stepTitle":"$$8-3x$$","stepBody":"","answerType":"string","variabilization":{},"choices":["monomial","binomial","trinomial","polynomial"],"hints":{"DefaultPathway":[{"id":"a5991b4add3a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Number of terms","text":"How many terms are in this expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5991b4add3a-h2","type":"hint","dependencies":["a5991b4add3a-h1"],"title":"Prefix","text":"The prefix for $$2$$ is bi-.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a5991b4add3b","stepAnswer":["polynomial"],"problemType":"MultipleChoice","stepTitle":"$$y^3-8y^2+2y-16$$","stepBody":"","answerType":"string","variabilization":{},"choices":["monomial","binomial","trinomial","polynomial"],"hints":{"DefaultPathway":[{"id":"a5991b4add3b-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":[],"title":"Number of terms","text":"How many terms are in this expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5991b4add3b-h2","type":"hint","dependencies":["a5991b4add3b-h1"],"title":"Prefix","text":"The prefix used for $$4$$ or more is poly-.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a5991b4add3c","stepAnswer":["trinomial"],"problemType":"MultipleChoice","stepTitle":"$$z^2-5z-6$$","stepBody":"","answerType":"string","variabilization":{},"choices":["monomial","binomial","trinomial","polynomial"],"hints":{"DefaultPathway":[{"id":"a5991b4add3c-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":[],"title":"Number of terms","text":"How many terms are in this expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5991b4add3c-h2","type":"hint","dependencies":["a5991b4add3c-h1"],"title":"Prefix","text":"The prefix used for $$3$$ or more is tri-.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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What is the degree?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a5991b4add7b","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"$$z^2-5z-6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a5991b4add7b-h1","type":"hint","dependencies":[],"title":"Degree of a polynomial","text":"The degree of a polynomial is the highest degree of all its terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5991b4add7b-h2","type":"hint","dependencies":["a5991b4add7b-h1"],"title":"Degree of a term","text":"The degree of a term is the sum of the exponents of its variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5991b4add7b-h3","type":"hint","dependencies":["a5991b4add7b-h1"],"title":"Term with the highest degree","text":"$$z^2$$ is the term with the highest degree. What is the degree?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a5991b4add7c","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"$$y^3-8y^2+2y-16$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a5991b4add7c-h1","type":"hint","dependencies":[],"title":"Degree of a polynomial","text":"The degree of a polynomial is the highest degree of all its terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5991b4add7c-h2","type":"hint","dependencies":["a5991b4add7c-h1"],"title":"Degree of a term","text":"The degree of a term is the sum of the exponents of its variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5991b4add7c-h3","type":"hint","dependencies":["a5991b4add7c-h1"],"title":"Term with the highest degree","text":"$$y^3$$ is the term with the highest degree. What is the degree?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a5991b4add7d","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"$$23{ab}^2-14$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a5991b4add7d-h1","type":"hint","dependencies":[],"title":"Degree of a polynomial","text":"The degree of a polynomial is the highest degree of all its terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5991b4add7d-h2","type":"hint","dependencies":["a5991b4add7d-h1"],"title":"Degree of a term","text":"The degree of a term is the sum of the exponents of its variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5991b4add7d-h3","type":"hint","dependencies":["a5991b4add7d-h1"],"title":"Term with the highest degree","text":"$$23{ab}^2$$ is the term with the highest degree. What is the degree?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a5991b4add7e","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"$$-3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a5991b4add7e-h1","type":"hint","dependencies":[],"title":"Degree of a polynomial","text":"The degree of a polynomial is the highest degree of all its terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5991b4add7e-h2","type":"hint","dependencies":["a5991b4add7e-h1"],"title":"Degree of a term","text":"The degree of a term is the sum of the exponents of its variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5991b4add7e-h3","type":"hint","dependencies":["a5991b4add7e-h1"],"title":"Degree of a constant","text":"The degree of a constant is $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5991b4add8","title":"Add and Subtract Monomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Add and Subtract Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5991b4add8a","stepAnswer":["$$12x^2$$"],"problemType":"TextBox","stepTitle":"$$7x^2+5x^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12x^2$$","hints":{"DefaultPathway":[{"id":"a5991b4add8a-h1","type":"hint","dependencies":[],"title":"Combine like terms","text":"If the monomials are like terms, combine them by adding or subtracting the coefficient.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5991b4add8a-h2","type":"hint","dependencies":["a5991b4add8a-h1"],"title":"Coefficients","text":"The coefficients involved in this expression are $$7$$ and $$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5991b4add9","title":"Add and Subtract Monomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Add and Subtract Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5991b4add9a","stepAnswer":["6w"],"problemType":"TextBox","stepTitle":"$$-12w+18w$$","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a5991b4add9a-h1","type":"hint","dependencies":[],"title":"Combine like terms","text":"If the monomials are like terms, combine them by adding or subtracting the coefficient.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5991b4add9a-h2","type":"hint","dependencies":["a5991b4add9a-h1"],"title":"Coefficients","text":"The coefficients involved in this expression are $$-12$$ and $$18$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5a11cdclt1","title":"The Central Limit Theorem for Sums","body":"An unknown distribution has a mean of $$90$$ and a standard deviation of $$15$$. A sample of size $$80$$ is drawn randomly from the population.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 The Central Limit Theorem for Sums","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a5a11cdclt1a","stepAnswer":["$$0.0127$$"],"problemType":"TextBox","stepTitle":"Find the probability that the sum of the $$80$$ values (or the total of the $$80$$ values) is more than 7,500.","stepBody":"Round your answer to four decimal places.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.0127$$","hints":{"DefaultPathway":[{"id":"a5a11cdclt1a-h1","type":"hint","dependencies":[],"title":"Defining X","text":"Let X $$=$$ one value from the original unknown population. The probability question asks you to find a probability for \u2211x, the sum (or total of) $$80$$ values, where \u2211x ~ N(mean of sums, standard deviation of the sums). Begin by solving for the mean of sums and the standard deviation of the sums.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7200$$"],"dependencies":["a5a11cdclt1a-h1"],"title":"Normalcdf Mean of Sums","text":"What is the mean of sums? In other words, what is (n)(\u03bc\u2093) where $$n$$ is the sample size and \u03bc\u2093 is the mean of X?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5a11cdclt1a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$80$$"],"dependencies":[],"title":"Sample Size","text":"What is $$n$$, the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt1a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$90$$"],"dependencies":[],"title":"Mean of X","text":"What is \u03bc\u2093, the mean of X?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt1a-h2-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7200$$"],"dependencies":[],"title":"Calculating the Mean of Sums","text":"What is (80)(90)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a5a11cdclt1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$134.16$$"],"dependencies":["a5a11cdclt1a-h1"],"title":"Standard Deviation of the Sums","text":"What is the standard deviation of the sums? In other words, what is $$\\\\sqrt{n}$$ \u03c3\u2093 where $$n$$ is the sample size and \u03c3\u2093 is the standard deviation of X? Round to the nearest hundredths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5a11cdclt1a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8.944$$"],"dependencies":[],"title":"Solving for Square Root of the Sample Size","text":"What is $$\\\\sqrt{n}$$ where $$n$$ is the sample size? Round to the nearest thousandths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt1a-h3-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":[],"title":"Standard Deviation of X","text":"What is \u03c3\u2093, the standard deviation of X?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt1a-h3-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$134.16$$"],"dependencies":[],"title":"Standard Deviation of the Sums","text":"What is $$15\\\\sqrt{80}$$? Round to the nearest hundredths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a5a11cdclt1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7500$$"],"dependencies":["a5a11cdclt1a-h2","a5a11cdclt1a-h3"],"title":"\u2211x","text":"What is the lower value of \u2211x, the sum (or total of) $$80$$ values? Look back at the original $$\\\\frac{problem}{question}$$ if you are stuck.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt1a-h5","type":"hint","dependencies":["a5a11cdclt1a-h4"],"title":"P(\u2211x > Sum of Values)","text":"Find P(\u2211x > 7500). Using a TI-83, 83+, $$84$$, 84+ calculator, plug in the solved values for the normalcdf(lower value, upper value, mean of sums, stdev of sums) function. To navigate normalcdf, press the 2nd key, vars, and then $$2$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt1a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.0127$$"],"dependencies":["a5a11cdclt1a-h5"],"title":"Plugging Into Normalcdf","text":"What is normalcdf(7500,1E99, (80)(90), sqrt(80)(15))? Round your answer to four decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5a11cdclt10","title":"The Central Limit Theorem for Sums","body":"The mean number of minutes for app engagement by a tablet user is $$8.2$$ minutes. Suppose the standard deviation is one minute. Take a sample of size $$70$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 The Central Limit Theorem for Sums","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a5a11cdclt10a","stepAnswer":["$$0.0009$$"],"problemType":"TextBox","stepTitle":"Find the probability that the sum of the sample is at least ten hours.","stepBody":"Round your answer to four decimal places.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.0009$$","hints":{"DefaultPathway":[{"id":"a5a11cdclt10a-h1","type":"hint","dependencies":[],"title":"Converting Hours to Minutes","text":"Convert ten hours to minutes (60 minutes $$=$$ $$1$$ hour).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$600$$"],"dependencies":["a5a11cdclt10a-h1"],"title":"$$10$$ Hours is ? Minutes","text":"How many minutes are in ten hours?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5a11cdclt10a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$600$$"],"dependencies":[],"title":"Calculating Hours to Minutes","text":"What is $$10\\\\times60$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a5a11cdclt10a-h3","type":"hint","dependencies":["a5a11cdclt10a-h2"],"title":"P(\u2211x $$ \\\\geq $$ Sum of Values)","text":"Find P(\u2211x $$ \\\\geq $$ 600). Using a TI-83, 83+, $$84$$, 84+ calculator, plug in values for the normalcdf(lower value, upper value, mean of sums, stdev of sums) function. To navigate normalcdf, press the 2nd key, vars, and then $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$600$$"],"dependencies":["a5a11cdclt10a-h3"],"title":"Lower Value","text":"What is the lower value?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5a11cdclt10a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$600$$"],"dependencies":[],"title":"At Least Ten Hours","text":"In the original $$\\\\frac{problem}{question}$$, we are looking for the probability that the sum of the sample is at least ten hours. As such, what is the lower value (in minutes) that the sum can be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a5a11cdclt10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["E99"],"dependencies":["a5a11cdclt10a-h3"],"title":"Upper Value","text":"What is the upper value?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt10a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1700$$"],"dependencies":["a5a11cdclt10a-h3"],"title":"Mean of Sums","text":"What is the mean of sums? In other words, what is (n)(\u03bc\u2093) where $$n$$ is the sample size and \u03bc\u2093 is the mean of X?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5a11cdclt10a-h6-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$70$$"],"dependencies":[],"title":"Sample Size","text":"What is $$n$$, the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt10a-h6-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8.2$$"],"dependencies":[],"title":"Mean of Tablet User","text":"What is \u03bc\u2093, the mean of X (tablet user)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt10a-h6-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$574$$"],"dependencies":[],"title":"Calculating the Mean of Sums","text":"What is $$(70)(8.2)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a5a11cdclt10a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8.37$$"],"dependencies":["a5a11cdclt10a-h3"],"title":"Standard Deviation of the Sums","text":"What is the standard deviation of the sums? In other words, what is $$\\\\sqrt{n}$$ \u03c3\u2093 where $$n$$ is the sample size and \u03c3\u2093 is the standard deviation of X (tablet users)? Round to the nearest hundredths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5a11cdclt10a-h7-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8.367$$"],"dependencies":[],"title":"Solving for the Square Root of the Sample Size","text":"What is $$\\\\sqrt{n}$$? Round to the nearest thousandths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt10a-h7-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Standard Deviation of Tablet User","text":"What is \u03c3\u2093, the standard deviation of X (tablet user)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt10a-h7-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8.37$$"],"dependencies":[],"title":"Standard Deviation of the Sums","text":"What is $$1\\\\sqrt{70}$$? Round to the nearest hundredths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a5a11cdclt10a-h8","type":"hint","dependencies":["a5a11cdclt10a-h1","a5a11cdclt10a-h2","a5a11cdclt10a-h3","a5a11cdclt10a-h4","a5a11cdclt10a-h5","a5a11cdclt10a-h6","a5a11cdclt10a-h7"],"title":"Normalcdf Equation","text":"Using a TI-83, 83+, $$84$$, 84+ calculator, plug in the solved values for the normalcdf(lower value, upper value, mean of sums, stdev of sums) function. To navigate normalcdf, press the 2nd key, vars, and then $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt10a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.0009$$"],"dependencies":["a5a11cdclt10a-h8"],"title":"Plugging in Normalcdf on the Calculator","text":"What is normalcdf(600, E99, $$(70)(8.2)$$, sqrt(70)(1))? Round your answer to four decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5a11cdclt2","title":"The Central Limit Theorem for Sums","body":"An unknown distribution has a mean of $$90$$ and a standard deviation of $$15$$. A sample of size $$80$$ is drawn randomly from the population.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 The Central Limit Theorem for Sums","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a5a11cdclt2a","stepAnswer":["$$7401.2$$"],"problemType":"TextBox","stepTitle":"Find the sum that is $$1.5$$ standard deviations above the mean of the sums.","stepBody":"Round to the nearest tenths place.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7401.2$$","hints":{"DefaultPathway":[{"id":"a5a11cdclt2a-h1","type":"hint","dependencies":[],"title":"What to Solve For","text":"Find \u2211x where $$z$$ $$=$$ $$1.5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt2a-h2","type":"hint","dependencies":["a5a11cdclt2a-h1"],"title":"\u2211x Equation","text":"\u2211x $$=$$ (n)(\u03bc\u2093) + $$z \\\\sqrt{n}$$ \u03c3\u2093, where $$x$$ is one value from the original unknown population, $$n$$ is the sample size, \u03bc\u2093 is the mean of X, $$z$$ is is $$z-score$$, and \u03c3\u2093 is the standard deviation of X.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$80$$"],"dependencies":["a5a11cdclt2a-h2"],"title":"Sample Size","text":"What is $$n$$, the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$90$$"],"dependencies":["a5a11cdclt2a-h2"],"title":"Mean of X","text":"What is \u03bc\u2093, the mean of X (one value from the original unknown population)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt2a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.5$$"],"dependencies":["a5a11cdclt2a-h2"],"title":"Z-Score","text":"What is $$z$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8.944$$"],"dependencies":["a5a11cdclt2a-h2"],"title":"$$\\\\sqrt{n}$$","text":"What is $$\\\\sqrt{n}$$? Round to the nearest thousandths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt2a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["a5a11cdclt2a-h2"],"title":"Standard Deviation of X","text":"What is \u03c3\u2093, the standard deviation of X (one value from the original unknown population)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt2a-h8","type":"hint","dependencies":["a5a11cdclt2a-h1","a5a11cdclt2a-h2","a5a11cdclt2a-h3","a5a11cdclt2a-h4","a5a11cdclt2a-h5","a5a11cdclt2a-h6","a5a11cdclt2a-h7"],"title":"Plugging Into the \u2211x Equation","text":"Plug your values into the equation: \u2211x $$=$$ (n)(\u03bc\u2093) + $$z \\\\sqrt{n}$$ \u03c3\u2093. Remember to round your answer to the nearest tenths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt2a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7401.2$$"],"dependencies":["a5a11cdclt2a-h8"],"title":"Plugged-In \u2211x Equation","text":"What is \u2211x $$=$$ (80)(90) + $$(1.5)(\u221a80)(15)$$? Round to the nearest tenths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5a11cdclt3","title":"The Central Limit Theorem for Sums","body":"In a recent study reported Oct. $$29$$, $$2012$$ on the Flurry Blog, the mean age of tablet users is $$34$$ years. Suppose the standard deviation is $$15$$ years. The sample of size is $$50$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 The Central Limit Theorem for Sums","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a5a11cdclt3a","stepAnswer":["$$1700$$"],"problemType":"TextBox","stepTitle":"What is the mean for the sum of the ages of tablet users?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1700$$","hints":{"DefaultPathway":[{"id":"a5a11cdclt3a-h1","type":"hint","dependencies":[],"title":"Mean Equation","text":"To find the mean (\u03bc\u2211\u2093), solve for \u03bc\u2211\u2093 $$=$$ n\u03bc\u2093 where $$n$$ is the sample size and \u03bc\u2093 is the mean of X (tablet users).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1700$$"],"dependencies":["a5a11cdclt3a-h1"],"title":"Solving for n\u03bc\u2093","text":"What is n\u03bc\u2093?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5a11cdclt3a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$50$$"],"dependencies":[],"title":"Sample Size","text":"What is $$n$$, the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt3a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$34$$"],"dependencies":[],"title":"Mean of Tablet Users","text":"What is \u03bc\u2093, the mean of X (tablet users)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt3a-h2-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1700$$"],"dependencies":[],"title":"Plugging Into n\u03bc\u2093","text":"What is 50(34)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a5a11cdclt4","title":"The Central Limit Theorem for Sums","body":"In a recent study reported Oct. $$29$$, $$2012$$ on the Flurry Blog, the mean age of tablet users is $$34$$ years. Suppose the standard deviation is $$15$$ years. The sample of size is $$50$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 The Central Limit Theorem for Sums","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a5a11cdclt4a","stepAnswer":["$$106.07$$"],"problemType":"TextBox","stepTitle":"What is the standard deviation for the sum of the ages of tablet users?","stepBody":"Round to the nearest hundredths place.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$106.07$$","hints":{"DefaultPathway":[{"id":"a5a11cdclt4a-h1","type":"hint","dependencies":[],"title":"Standard Deviation Equation","text":"To find the standard deviation (\u03c3_\u2211x), solve for \u03c3_\u2211x $$=$$ $$\\\\sqrt{n}$$ \u03c3\u2093 where $$n$$ is the sample size and \u03c3\u2093 is the standard deviation of X (tablet users).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$106.07$$"],"dependencies":["a5a11cdclt4a-h1"],"title":"Solving for $$\\\\sqrt{n}$$ \u03c3\u2093","text":"What is $$\\\\sqrt{n}$$ \u03c3\u2093? Round to the nearest hundredths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5a11cdclt4a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$50$$"],"dependencies":[],"title":"Sample Size","text":"What is $$n$$, the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt4a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":[],"title":"Standard Deviation of Tablet Users","text":"What is \u03c3\u2093, the standard deviation of X (tablet users)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt4a-h2-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$106.07$$"],"dependencies":[],"title":"Plugging Into $$\\\\sqrt{n}$$ \u03c3\u2093","text":"What is $$15\\\\sqrt{50}$$? Round to the nearest hundredths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a5a11cdclt5","title":"The Central Limit Theorem for Sums","body":"In a recent study reported Oct. $$29$$, $$2012$$ on the Flurry Blog, the mean age of tablet users is $$34$$ years. Suppose the standard deviation is $$15$$ years. The sample of size is $$50$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 The Central Limit Theorem for Sums","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a5a11cdclt5a","stepAnswer":["$$0.7974$$"],"problemType":"TextBox","stepTitle":"Find the probability that the sum of the ages is between 1,500 and 1,800 years.","stepBody":"Round your answer to four decimal places.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.7974$$","hints":{"DefaultPathway":[{"id":"a5a11cdclt5a-h1","type":"hint","dependencies":[],"title":"\u2211x Equation Boundaries","text":"Solve for P(1500 < \u2211x < 1800).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt5a-h2","type":"hint","dependencies":["a5a11cdclt5a-h1"],"title":"Normalcdf Equivalence","text":"To find P(1500 < \u2211x < 1800), use the normalcdf(lower value, upper value, mean of sums, stdev of sums) function on a calculator. To navigate normalcdf, press the 2nd key, vars, and then $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1500$$"],"dependencies":["a5a11cdclt5a-h2"],"title":"Lower Value","text":"What is the lower value?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1800$$"],"dependencies":["a5a11cdclt5a-h2"],"title":"Upper Value","text":"What is the upper value?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1700$$"],"dependencies":["a5a11cdclt5a-h2"],"title":"Mean of Sums","text":"What is the mean of sums? In other words, what is (n)(\u03bc\u2093) where $$n$$ is the sample size and \u03bc\u2093 is the mean of X?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5a11cdclt5a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$50$$"],"dependencies":[],"title":"Sample Size","text":"What is $$n$$, the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt5a-h5-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$34$$"],"dependencies":[],"title":"Mean of Tablet Users","text":"What is \u03bc\u2093, the mean of X (tablet users)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt5a-h5-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1700$$"],"dependencies":[],"title":"Calculating the Mean of Sums","text":"What is (50)(34)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a5a11cdclt5a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$106.067$$"],"dependencies":["a5a11cdclt5a-h2"],"title":"Standard Deviation of the Sums","text":"What is the standard deviation of the sums? In other words, what is $$\\\\sqrt{n}$$ \u03c3\u2093 where $$n$$ is the sample size and \u03c3\u2093 is the standard deviation of X (tablet users)? Round to the nearest hundredths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5a11cdclt5a-h6-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7.071$$"],"dependencies":[],"title":"Solving for sqrt(Sample Size)","text":"What is $$\\\\sqrt{n}$$? Round to the nearest thousandths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt5a-h6-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":[],"title":"Standard Deviation of Tablet Users","text":"What is \u03c3\u2093, the standard deviation of X (tablet users)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt5a-h6-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$106.067$$"],"dependencies":[],"title":"Standard Deviation of the Sums","text":"What is $$15\\\\sqrt{50}$$? Round to the nearest hundredths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a5a11cdclt5a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.7974$$"],"dependencies":["a5a11cdclt5a-h3","a5a11cdclt5a-h4","a5a11cdclt5a-h5","a5a11cdclt5a-h6"],"title":"Plugging Into Normalcdf","text":"Using the calculator, what is normalcdf(1500, $$1800$$, (50)(34), sqrt(50)(15))? Round your answer to four decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5a11cdclt6","title":"The Central Limit Theorem for Sums","body":"In a recent study reported Oct. $$29$$, $$2012$$ on the Flurry Blog, the mean age of tablet users is $$34$$ years. Suppose the standard deviation is $$15$$ years. The sample of size is $$50$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 The Central Limit Theorem for Sums","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a5a11cdclt6a","stepAnswer":["$$1789.3$$"],"problemType":"TextBox","stepTitle":"Find the 80th percentile for the sum of the $$50$$ ages.","stepBody":"Round to the nearest tenth place.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1789.3$$","hints":{"DefaultPathway":[{"id":"a5a11cdclt6a-h1","type":"hint","dependencies":[],"title":"Using the Calculator","text":"Plug values into the invNorm(percentile, mean of sums, stdev of sums) function on the calculator. To navigate invNorm, press the 2nd key, vars, and then $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.8$$"],"dependencies":["a5a11cdclt6a-h1"],"title":"Percentile","text":"What is the percentile? Enter your answer in decimal form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1700$$"],"dependencies":["a5a11cdclt6a-h2"],"title":"Mean of Sums","text":"What is the mean of sums? In other words, what is (n)(\u03bc\u2093) where $$n$$ is the sample size and \u03bc\u2093 is the mean of X?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5a11cdclt6a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$50$$"],"dependencies":[],"title":"Sample Size","text":"What is $$n$$, the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt6a-h3-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$34$$"],"dependencies":[],"title":"Mean of Tablet Users","text":"What is \u03bc\u2093, the mean of X (tablet users)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt6a-h3-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1700$$"],"dependencies":[],"title":"Calculating the Mean of Sums","text":"What is (50)(34)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a5a11cdclt6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$106.067$$"],"dependencies":["a5a11cdclt6a-h2"],"title":"Standard Deviation of the Sums","text":"What is the standard deviation of the sums? In other words, what is $$\\\\sqrt{n}$$ \u03c3\u2093 where $$n$$ is the sample size and \u03c3\u2093 is the standard deviation of X (tablet users)? Round to the nearest hundredths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5a11cdclt6a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7.071$$"],"dependencies":[],"title":"Solving for Square Root of the Sample Size","text":"What is $$\\\\sqrt{n}$$? Round to the nearest thousandths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt6a-h4-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":[],"title":"Standard Deviation of Tablet Users","text":"What is \u03c3\u2093, the standard deviation of X (tablet users)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt6a-h4-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$106.067$$"],"dependencies":[],"title":"Standard Deviation of the Sums","text":"What is $$15\\\\sqrt{50}$$? Round to the nearest hundredths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a5a11cdclt6a-h5","type":"hint","dependencies":["a5a11cdclt6a-h3","a5a11cdclt6a-h4"],"title":"Plugging Into the \u2211x Equation","text":"Plug your values into the invNorm(percentile, mean of sums, stdev of sums) function on the calculator. Remember to round your answer to the nearest tenths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt6a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1789.3$$"],"dependencies":["a5a11cdclt6a-h5"],"title":"Plugging Into invNorm","text":"Using the calculator, what is $$invNorm(0.80$$, (50)(34), sqrt(50)(15))? Round to the nearest tenth place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5a11cdclt7","title":"The Central Limit Theorem for Sums","body":"The mean number of minutes for app engagement by a tablet user is $$8.2$$ minutes. Suppose the standard deviation is one minute. Take a sample of size $$70$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 The Central Limit Theorem for Sums","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a5a11cdclt7a","stepAnswer":["$$574$$"],"problemType":"TextBox","stepTitle":"What is the mean for the sums (in minutes)?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$574$$","hints":{"DefaultPathway":[{"id":"a5a11cdclt7a-h1","type":"hint","dependencies":[],"title":"Mean Equation","text":"To find the mean (\u03bc\u2211\u2093), solve for \u03bc\u2211\u2093 $$=$$ n\u03bc\u2093 where $$n$$ is the sample size and \u03bc\u2093 is the mean of X (tablet user).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$574$$"],"dependencies":["a5a11cdclt7a-h1"],"title":"Solving for n\u03bc\u2093","text":"What is n\u03bc\u2093?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5a11cdclt7a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$70$$"],"dependencies":[],"title":"Sample Size","text":"What is $$n$$, the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt7a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8.2$$"],"dependencies":[],"title":"Mean of Tablet User","text":"What is \u03bc\u2093, the mean of X (tablet user)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt7a-h2-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$574$$"],"dependencies":[],"title":"Plugging Into n\u03bc\u2093","text":"What is $$(70)(8.2)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a5a11cdclt8","title":"The Central Limit Theorem for Sums","body":"The mean number of minutes for app engagement by a tablet user is $$8.2$$ minutes. Suppose the standard deviation is one minute. Take a sample of size $$70$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 The Central Limit Theorem for Sums","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a5a11cdclt8a","stepAnswer":["$$8.37$$"],"problemType":"TextBox","stepTitle":"What is the standard deviation for the sums (in minutes)?","stepBody":"Round to the nearest hundredths place.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8.37$$","hints":{"DefaultPathway":[{"id":"a5a11cdclt8a-h1","type":"hint","dependencies":[],"title":"Standard Deviation Equation","text":"To find the standard deviation (\u03c3_\u2211x), solve for \u03c3_\u2211x $$=$$ $$\\\\sqrt{n} sigmax$$ where $$n$$ is the sample size and \u03c3\u2093 is the standard deviation of X (tablet user).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8.37$$"],"dependencies":["a5a11cdclt8a-h1"],"title":"Solving for $$\\\\sqrt{n}$$ \u03c3\u2093","text":"What is $$\\\\sqrt{n}$$ \u03c3\u2093? Round to the nearest hundredths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5a11cdclt8a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$70$$"],"dependencies":[],"title":"Sample Size","text":"What is $$n$$, the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt8a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Standard Deviation of Tablet User","text":"What is \u03c3\u2093, the standard deviation of X (tablet user)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt8a-h2-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8.37$$"],"dependencies":[],"title":"Plugging Into $$\\\\sqrt{n} sigmax$$","text":"What is $$1\\\\sqrt{70}$$? Round to the nearest hundredths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a5a11cdclt9","title":"The Central Limit Theorem for Sums","body":"The mean number of minutes for app engagement by a tablet user is $$8.2$$ minutes. Suppose the standard deviation is one minute. Take a sample of size $$70$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 The Central Limit Theorem for Sums","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a5a11cdclt9a","stepAnswer":["$$587.76$$"],"problemType":"TextBox","stepTitle":"Find the 95th percentile for the sum of the sample (in minutes).","stepBody":"Round to the nearest hundredths place.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$587.76$$","hints":{"DefaultPathway":[{"id":"a5a11cdclt9a-h1","type":"hint","dependencies":[],"title":"Using the Calculator","text":"To find the 95th percentile for the sum of the sample, plug values into the invNorm(percentile, mean of sums, stdev of sums) function on the calculator. To navigate invNorm, press the 2nd key, vars, and then $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.95$$"],"dependencies":["a5a11cdclt9a-h1"],"title":"Percentile","text":"What is the percentile? Enter your answer in decimal form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1700$$"],"dependencies":["a5a11cdclt9a-h2"],"title":"Mean of Sums","text":"What is the mean of sums? In other words, what is (n)(\u03bc\u2093) where $$n$$ is the sample size and \u03bc\u2093 is the mean of X?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5a11cdclt9a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$70$$"],"dependencies":[],"title":"Sample Size","text":"What is $$n$$, the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt9a-h3-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8.2$$"],"dependencies":[],"title":"Mean of Tablet User","text":"What is \u03bc\u2093, the mean of X (tablet user)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt9a-h3-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$574$$"],"dependencies":[],"title":"Calculating the Mean of Sums","text":"What is $$(70)(8.2)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a5a11cdclt9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8.37$$"],"dependencies":["a5a11cdclt9a-h2"],"title":"Standard Deviation of the Sums","text":"What is the standard deviation of the sums? In other words, what is $$\\\\sqrt{n}$$ \u03c3\u2093 where $$n$$ is the sample size and \u03c3\u2093 is the standard deviation of X (tablet users)? Round to the nearest hundredths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5a11cdclt9a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8.367$$"],"dependencies":[],"title":"Solving for the Square Root of the Sample Size","text":"What is $$\\\\sqrt{n}$$? Round to the nearest thousandths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt9a-h4-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Standard Deviation of Tablet User","text":"What is \u03c3\u2093, the standard deviation of X (tablet user)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt9a-h4-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8.37$$"],"dependencies":[],"title":"Standard Deviation of the Sums","text":"What is $$1\\\\sqrt{70}$$? Round to the nearest hundredths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a5a11cdclt9a-h5","type":"hint","dependencies":["a5a11cdclt9a-h3","a5a11cdclt9a-h4"],"title":"Plugging Into invNorm","text":"Plug your values into the invNorm(percentile, mean of sums, stdev of sums) function on the calculator. Remember to round your answer to the nearest hundredths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt9a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$587.76$$"],"dependencies":["a5a11cdclt9a-h5"],"title":"Values in invNorm","text":"Using the calculator, what is $$invNorm(0.95$$, $$(70)(8.2)$$, sqrt(70)(1))? Round to the nearest hundredths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5a11cdclt9a-h7","type":"hint","dependencies":["a5a11cdclt9a-h6"],"title":"Conceptual Understanding","text":"Ninety five percent of the sums of app engagement times are at most $$587.76$$ minutes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5b6f42rationals1","title":"Analyzing Rational Functions #1","body":"Find the domain, vertical asymptotes, and horizontal asymptotes of the function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Rational Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5b6f42rationals1a","stepAnswer":["Domain: all real numbers except $$1$$, Vertical Asymptote: $$x=1;$$ Horizontal Asymptote: $$y=0$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{4}{x-1}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Domain: all real numbers except $$1$$, Vertical Asymptote: $$x=1;$$ Horizontal Asymptote: $$y=0$$","choices":["Domain: all real numbers except $$1$$, Vertical Asymptote: $$x=0;$$ Horizontal Asymptote: $$y=1$$","Domain: all real numbers except $$1$$, Vertical Asymptote: $$x=1;$$ Horizontal Asymptote: $$y=0$$"],"hints":{"DefaultPathway":[{"id":"a5b6f42rationals1a-h1","type":"hint","dependencies":[],"title":"Definition of Domain","text":"The domain of a rational function includes all real numbers except those that cause the denominator to equal zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals1a-h2","type":"hint","dependencies":["a5b6f42rationals1a-h1"],"title":"Definition of a Vertical Asymptote","text":"A vertical asymptote of a graph is a vertical line $$x=a$$ where the graph tends toward positive or negative infinity as the inputs approach a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals1a-h3","type":"hint","dependencies":["a5b6f42rationals1a-h2"],"title":"Definition of a Horizontal Asymptote","text":"A horizontal asymptote of a graph is a horizontal line $$y=b$$ where the graph approaches the line as the inputs increase or decrease without bound.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5b6f42rationals10","title":"Analyzing Rational Functions #","body":"Find the domain, vertical asymptotes, and horizontal asymptotes of the function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Rational Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5b6f42rationals10a","stepAnswer":["Domain: $$(-\\\\infty,\\\\frac{1}{3}) \\\\cup (\\\\frac{1}{3},\\\\infty)$$, Vertical Asymptote: $$x=\\\\frac{1}{3}$$, Horizontal Asymptote: $$y=\\\\frac{-2}{3}$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{4-2x}{3x-1}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Domain: $$(-\\\\infty,\\\\frac{1}{3}) \\\\cup (\\\\frac{1}{3},\\\\infty)$$, Vertical Asymptote: $$x=\\\\frac{1}{3}$$, Horizontal Asymptote: $$y=\\\\frac{-2}{3}$$","choices":["Domain: $$(-\\\\infty,\\\\frac{1}{3}) \\\\cup (\\\\frac{1}{3},\\\\infty)$$, Vertical Asymptote: $$x=\\\\frac{1}{3}$$, Horizontal Asymptote: $$y=\\\\frac{-1}{3}$$","Domain: $$(-\\\\infty,\\\\frac{1}{3}) \\\\cup (\\\\frac{1}{3},\\\\infty)$$, Vertical Asymptote: $$x=\\\\frac{1}{3}$$, Horizontal Asymptote: $$y=\\\\frac{1}{3}$$","Domain: $$(-\\\\infty,\\\\frac{1}{3}) \\\\cup (\\\\frac{1}{3},\\\\infty)$$, Vertical Asymptote: $$x=\\\\frac{1}{3}$$, Horizontal Asymptote: $$y=\\\\frac{-2}{3}$$"],"hints":{"DefaultPathway":[{"id":"a5b6f42rationals10a-h1","type":"hint","dependencies":[],"title":"Definition of Domain","text":"The domain of a rational function includes all real numbers except those that cause the denominator to equal zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals10a-h2","type":"hint","dependencies":["a5b6f42rationals10a-h1"],"title":"Definition of a Vertical Asymptote","text":"A vertical asymptote of a graph is a vertical line $$x=a$$ where the graph tends toward positive or negative infinity as the inputs approach a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals10a-h3","type":"hint","dependencies":["a5b6f42rationals10a-h2"],"title":"Definition of a Horizontal Asymptote","text":"A horizontal asymptote of a graph is a horizontal line $$y=b$$ where the graph approaches the line as the inputs increase or decrease without bound.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5b6f42rationals11","title":"Finding the Domain of Rational Functions","body":"For the following exercises, find the domain of the rational functions.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Rational Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5b6f42rationals11a","stepAnswer":["$$(-\\\\infty,-3) \\\\cup (3,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"Find the domain of $$f(x)=x+\\\\frac{3}{x^2-9}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,-3) \\\\cup (3,\\\\infty)$$","choices":["$$(-\\\\infty,-3) \\\\cup (3,\\\\infty)$$","$$(-\\\\infty,3) \\\\cup (-3,\\\\infty)$$","$$(\\\\infty,-3) \\\\cup (3,\\\\infty)$$","$$(-3,3)$$"],"hints":{"DefaultPathway":[{"id":"a5b6f42rationals11a-h1","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["3, -3"],"dependencies":[],"title":"Finding Undefined Values","text":"What values make the function undefined?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals11a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a5b6f42rationals11a-h1"],"title":"Questioning your answer","text":"Do all other values for $$x$$ return a real number?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a5b6f42rationals12","title":"Finding the Domain of Rational Functions","body":"For the following exercises, find the domain of the rational functions.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Rational Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5b6f42rationals12a","stepAnswer":["$$(-\\\\infty,-2) \\\\cup (-2,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=x-\\\\frac{1}{x}+2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,-2) \\\\cup (-2,\\\\infty)$$","choices":["$$(-\\\\infty,-2) \\\\cup (-2,\\\\infty)$$","$$(-\\\\infty,2) \\\\cup (-3,\\\\infty)$$","$$(\\\\infty,-2) \\\\cup (4,\\\\infty)$$","$$(-2,2)$$"],"hints":{"DefaultPathway":[{"id":"a5b6f42rationals12a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":[],"title":"Finding Undefined Values","text":"What values make the function undefined?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals12a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a5b6f42rationals12a-h1"],"title":"Questioning Your Answer","text":"Do all other values for $$x$$ return a real number?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a5b6f42rationals13","title":"Finding the Domain of Rational Functions","body":"For the following exercises, find the domain of the rational functions.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Rational Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5b6f42rationals13a","stepAnswer":["$$(-\\\\infty,1) \\\\cup (1,5) \\\\cup (5,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=$$ $$\\\\frac{4x}{5\\\\left(x-1\\\\right) \\\\left(x-5\\\\right)}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,1) \\\\cup (1,5) \\\\cup (5,\\\\infty)$$","choices":["$$(-\\\\infty,1) \\\\cup (1,5) \\\\cup (5,\\\\infty)$$","$$(\\\\infty,1) \\\\cup (1,5) \\\\cup (5,\\\\infty)$$","$$(-\\\\infty,1) \\\\cup (1,2) \\\\cup (5,\\\\infty)$$","$$(1,5)$$"],"hints":{"DefaultPathway":[{"id":"a5b6f42rationals13a-h1","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["1, 5"],"dependencies":[],"title":"Finding Undefined Values","text":"What values make the function undefined?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals13a-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a5b6f42rationals13a-h1"],"title":"Questioning your answer","text":"Do all other values for $$x$$ return a real number?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a5b6f42rationals14","title":"Identifying Vertical Asymptotes","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Rational Functions","courseName":"OpenStax: College 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is the $$y$$ intercept","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals15a-h3","type":"hint","dependencies":["a5b6f42rationals15a-h2"],"title":"Definition","text":"The $$x$$ intercept is the value of $$x$$ that makes f(x) $$=$$ $$0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["a5b6f42rationals15a-h3"],"title":"Putting it Together","text":"What value of the numerator makes the whole expression 0?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5b6f42rationals16","title":"Finding verical asymptotes and removable 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$$x=7$$"],"hints":{"DefaultPathway":[{"id":"a5b6f42rationals16a-h1","type":"hint","dependencies":[],"title":"Factoring","text":"Factor the denominator and numerator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals16a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["x=5"],"dependencies":["a5b6f42rationals16a-h1"],"title":"What is the common factor in the numerator and denominator? 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This is the vertical asymptote.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5b6f42rationals17","title":"Finding vertical asymptotes and removable discontinuities","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Rational Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5b6f42rationals17a","stepAnswer":["Vertical Asymptote: $$x=-2$$ Removable Discontunity: $$x=2$$"],"problemType":"MultipleChoice","stepTitle":"k(x) $$=$$ $$x-\\\\frac{2}{x^2-4}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Vertical Asymptote: $$x=-2$$ Removable Discontunity: $$x=2$$","choices":["Vertical Asymptote: $$x=-4$$ Removable Discontunity: $$x=2$$","Vertical Asymptote: $$x=2$$ Removable Discontunity: $$x=2$$","Vertical Asymptote: $$x=-2$$ Removable Discontunity: $$x=2$$"],"hints":{"DefaultPathway":[{"id":"a5b6f42rationals17a-h1","type":"hint","dependencies":[],"title":"Factoring Denominators","text":"Factor the denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals17a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["x=2"],"dependencies":["a5b6f42rationals17a-h1"],"title":"What is the common factor in the numerator and denominator? This is the removable discontinuity","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals17a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["x=-2"],"dependencies":["a5b6f42rationals17a-h2"],"title":"What is the other factor? This is the vertical asymptote.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5b6f42rationals18","title":"Finding X and Y intercepts of rational functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Rational Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5b6f42rationals18a","stepAnswer":["$$y$$ intercept: $$(0,\\\\frac{5}{4})$$, $$x$$ intercept: $$(-5,0)$$"],"problemType":"MultipleChoice","stepTitle":"f(x) $$=$$ $$\\\\frac{x-2\\\\left(x-3\\\\right)}{\\\\left(x-1\\\\right) \\\\left(x+2\\\\right) \\\\left(x+5\\\\right)}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y$$ intercept: $$(0,\\\\frac{5}{4})$$, $$x$$ intercept: $$(-5,0)$$","choices":["$$y$$ intercept: $$(0,\\\\frac{4}{5})$$, $$x$$ intercept: $$(-5,0)$$","$$y$$ intercept: $$(0,\\\\frac{5}{4})$$, $$x$$ intercept: $$(-5,0)$$","$$y$$ intercept: $$(\\\\frac{5}{4},0)$$, $$x$$ intercept: $$(-5,0)$$","$$y$$ intercept: $$(0,\\\\frac{5}{4})$$, $$x$$ intercept: $$(0,5)$$"],"hints":{"DefaultPathway":[{"id":"a5b6f42rationals18a-h1","type":"hint","dependencies":[],"title":"Definition","text":"The $$y$$ intercept value is the value that results when all the values of $$x$$ are $$0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-3}{5}$$"],"dependencies":["a5b6f42rationals18a-h1"],"title":"Putting it Together","text":"What is the $$y$$ intercept","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals18a-h3","type":"hint","dependencies":["a5b6f42rationals18a-h2"],"title":"Definition","text":"The $$x$$ intercept is the value of $$x$$ that makes f(x) $$=$$ $$0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals18a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["2, 3"],"dependencies":["a5b6f42rationals18a-h3"],"title":"Putting it Together","text":"What value(s) of the numerator makes the whole expression 0?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5b6f42rationals19","title":"Identifying Horizontal Asymptotes","body":"Find the horizontal asymptote and interpret it in context of the problem.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College 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function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Rational Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5b6f42rationals2a","stepAnswer":["Domain: all real numbers except $$\\\\frac{-2}{5}$$ Vertical Asymptote: $$x=\\\\frac{-2}{5}$$ Horizontal Asymptote: $$y=0$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{2}{5x+2}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Domain: all real numbers except $$\\\\frac{-2}{5}$$ Vertical Asymptote: $$x=\\\\frac{-2}{5}$$ Horizontal Asymptote: $$y=0$$","choices":["Domain: all real numbers, Vertical Asymptote: $$x=2;$$ Horizontal Asymptote: $$y=5$$","Domain: all real numbers except $$\\\\frac{-2}{5}$$ Vertical Asymptote: $$x=\\\\frac{-2}{5}$$ Horizontal Asymptote: $$y=0$$"],"hints":{"DefaultPathway":[{"id":"a5b6f42rationals2a-h1","type":"hint","dependencies":[],"title":"Definition of Domain","text":"The domain of a rational function includes all real numbers except those that cause the denominator to equal zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals2a-h2","type":"hint","dependencies":["a5b6f42rationals2a-h1"],"title":"Definition of a Vertical Asymptote","text":"A vertical asymptote of a graph is a vertical line $$x=a$$ where the graph tends toward positive or negative infinity as the inputs approach a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals2a-h3","type":"hint","dependencies":["a5b6f42rationals2a-h2"],"title":"Definition of a Horizontal Asymptote","text":"A horizontal asymptote of a graph is a horizontal line $$y=b$$ where the graph approaches the line as the inputs increase or decrease without bound.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5b6f42rationals20","title":"Identifying Horizontal and Vertical Asymptotes","body":"Find the vertical and horizontal asymptotes of the function:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Rational Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5b6f42rationals20a","stepAnswer":["Vertical Asymptotes: $$x=1$$, $$x=-2$$, $$x=5$$, Horizontal Asymptotes: $$y=0$$"],"problemType":"MultipleChoice","stepTitle":"f(x) $$=$$ $$\\\\frac{x-2\\\\left(x-3\\\\right)}{\\\\left(x-1\\\\right) \\\\left(x+2\\\\right) \\\\left(x+5\\\\right)}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Vertical Asymptotes: $$x=1$$, $$x=-2$$, $$x=5$$, Horizontal Asymptotes: $$y=0$$","choices":["Vertical Asymptotes: $$x=1$$, $$x=-2$$, $$x=5$$, Horizontal Asymptotes: $$y=0$$","Vertical Asymptotes: $$x=4$$, $$x=-2$$, $$x=5$$, Horizontal Asymptotes: $$y=3$$","Vertical Asymptotes: $$x=1$$, $$x=-2$$, $$x=5$$, Horizontal Asymptotes: $$y=1$$","Vertical Asymptotes: $$x=1$$, $$x=-2$$, $$x=5$$, Horizontal Asymptotes: $$y=5$$"],"hints":{"DefaultPathway":[{"id":"a5b6f42rationals20a-h1","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["x=1,-2,5"],"dependencies":[],"title":"What are the values that cause it to be undefined? These are the vertical asymptotes.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y=0$$"],"dependencies":["a5b6f42rationals20a-h1"],"title":"As $$x$$ goes to $$\\\\infty$$, what value does f(x) approach?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5b6f42rationals21","title":"Finding the Intercepts of a Rational Function","body":"Find the $$x-$$ and y-intercepts of the rational function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Rational Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5b6f42rationals21a","stepAnswer":["(0,5/4),(-5,0)"],"problemType":"TextBox","stepTitle":"$$f(x)=\\\\frac{x+5}{x^2+4}$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a5b6f42rationals21a-h1","type":"hint","dependencies":[],"title":"Finding the Y-Intercept","text":"We must set $$x$$ equal to $$0$$ and then solve for f(x). When we do this, we get $$f(x)=\\\\frac{5}{4}$$. Thus, our y-intercept is $$(0,\\\\frac{5}{4})$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals21a-h2","type":"hint","dependencies":["a5b6f42rationals21a-h1"],"title":"Finding the X-Intercept","text":"Now, we must set f(x) equal to $$0$$ and solve for $$x$$. $$\\\\frac{x+5}{x^2+4}=0$$. This is $$0$$ when the numerator is equal to $$0$$. $$x+5=0$$. This means that the x-intercept is $$(-5,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5b6f42rationals22","title":"Finding the Intercepts of a Rational Function","body":"Find the $$x-$$ and y-intercepts of the rational function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Rational Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5b6f42rationals22a","stepAnswer":["None"],"problemType":"TextBox","stepTitle":"$$f(x)=\\\\frac{x}{x^2-x}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a5b6f42rationals22a-h1","type":"hint","dependencies":[],"title":"Finding the Y-Intercept","text":"We must set $$x$$ equal to $$0$$ and then solve for f(x). $$f(x)=\\\\frac{0}{0}$$. There is no y-intercept.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals22a-h2","type":"hint","dependencies":["a5b6f42rationals22a-h1"],"title":"Finding the X-Intercept","text":"We must now set $$f(x)=0$$ and solve for $$x$$. Since the denominator becomes undefined, there is no x-intercept.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5b6f42rationals23","title":"Finding the Intercepts of a Rational Function","body":"Find the $$x-$$ and y-intercepts of the function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Rational Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5b6f42rationals23a","stepAnswer":["(-7,0),(-1,0),(0,7/30)"],"problemType":"TextBox","stepTitle":"$$f(x)=\\\\frac{x^2+8x+7}{x^2+11x+30}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-7,0),(-1,0),(0,\\\\frac{7}{30})$$","hints":{"DefaultPathway":[{"id":"a5b6f42rationals23a-h1","type":"hint","dependencies":[],"title":"Finding the Y-Intercept","text":"We must now set $$x$$ equal to $$0$$ and then solve for f(x). This leaves us with $$\\\\frac{7}{30}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals23a-h2","type":"hint","dependencies":["a5b6f42rationals23a-h1"],"title":"Finding the X-Intercept","text":"We must now set f(x) equal to $$0$$ and then solve for $$x$$. $$x^2+8x+7=0$$. $$\\\\left(x+7\\\\right) \\\\left(x+1\\\\right)=0$$. This means that $$x=-7$$ and $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5b6f42rationals24","title":"Finding the Intercepts of a Rational Function","body":"Find the $$x-$$ and y-intercepts of the function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Rational Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5b6f42rationals24a","stepAnswer":["(0,1/4)"],"problemType":"TextBox","stepTitle":"$$f(x)=\\\\frac{x^2+x+6}{x^2-10x+24}$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,\\\\frac{1}{4})$$","hints":{"DefaultPathway":[{"id":"a5b6f42rationals24a-h1","type":"hint","dependencies":[],"title":"Finding the Y-Intercept","text":"We must set $$x$$ equal to $$0$$ and solve for f(x) to find the y-intercept. This leavees us with $$\\\\frac{6}{24}$$, or $$\\\\frac{1}{4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals24a-h2","type":"hint","dependencies":["a5b6f42rationals24a-h1"],"title":"Finding the X-Intercept","text":"Now, we must set f(x) equal to $$0$$ and solve $$x$$. There are no x-intercepts.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5b6f42rationals25","title":"Finding the Intercepts of a Rational Function","body":"Find the $$x-$$ and y-intercepts of the function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Rational Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5b6f42rationals25a","stepAnswer":["(0,-94/12),(sqrt(47),0),(-sqrt(47),0)"],"problemType":"TextBox","stepTitle":"$$f(x)=\\\\frac{94-2x^2}{3x^2-12}$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a5b6f42rationals25a-h1","type":"hint","dependencies":[],"title":"Finding the Y-Intercept","text":"We must setet $$x$$ equal to $$0$$ and solve for f(x) in order to find the y-intercept. This leaves us with $$\\\\frac{-94}{12}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals25a-h2","type":"hint","dependencies":["a5b6f42rationals25a-h1"],"title":"Finding the X-Intercept","text":"We must now set f(x) equal to $$0$$ and solve for $$x$$. $$94-2x^2=0$$. $$2x^2=94$$ x=sqrt(47),-sqrt(47)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5b6f42rationals3","title":"Analyzing Rational Functions #3","body":"Find the domain, vertical asymptotes, and horizontal asymptotes of the function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Rational Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5b6f42rationals3a","stepAnswer":["Domain: all real numbers except $$-3$$ and $$3$$, Vertical Asymptotes: $$x=-3$$ and $$x=3$$, Horizontal Asymptotes: $$y=0$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{x}{x^2-9}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Domain: all real numbers except $$-3$$ and $$3$$, Vertical Asymptotes: $$x=-3$$ and $$x=3$$, Horizontal Asymptotes: $$y=0$$","choices":["Domain: all real numbers except $$-3$$ and $$3$$, Vertical Asymptotes: $$x=-3$$ and $$x=3$$, Horizontal Asymptotes: $$y=0$$","Domain: all real numbers except $$-9$$ and $$9$$, Vertical Asymptotes: $$x=-9$$ and $$x=9$$, Horizontal Asymptotes: $$y=0$$"],"hints":{"DefaultPathway":[{"id":"a5b6f42rationals3a-h1","type":"hint","dependencies":[],"title":"Factoring f(x)","text":"The denominator of f(x), $$x^2-9$$, can be factored to $$\\\\left(x+3\\\\right) \\\\left(x-3\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals3a-h2","type":"hint","dependencies":["a5b6f42rationals3a-h1"],"title":"Definition of Domain","text":"The domain of a rational function includes all real numbers except those that cause the denominator to equal zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals3a-h3","type":"hint","dependencies":["a5b6f42rationals3a-h2"],"title":"Definition of a Vertical Asymptote","text":"A vertical asymptote of a graph is a vertical line $$x=a$$ where the graph tends toward positive or negative infinity as the inputs approach a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals3a-h4","type":"hint","dependencies":["a5b6f42rationals3a-h3"],"title":"Definition of a Horizontal Asymptote","text":"A horizontal asymptote of a graph is a horizontal line $$y=b$$ where the graph approaches the line as the inputs increase or decrease without bound.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5b6f42rationals4","title":"Analyzing Rational Functions #4","body":"Find the domain, vertical asymptotes, and horizontal asymptotes of the function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Rational Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5b6f42rationals4a","stepAnswer":["Domain: $$(-\\\\infty,-9) \\\\cup (-9,4) \\\\cup (4,\\\\infty)$$, Vertical Asymptotes: $$x=-9, 4$$, Horizontal Asymptotes: $$y=0$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{x}{x^2+5x-36}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Domain: $$(-\\\\infty,-9) \\\\cup (-9,4) \\\\cup (4,\\\\infty)$$, Vertical Asymptotes: $$x=-9, 4$$, Horizontal Asymptotes: $$y=0$$","choices":["Domain: $$(-\\\\infty,-9) \\\\cup (-9,4) \\\\cup (4,\\\\infty)$$, Vertical Asymptotes: $$x=-9, 4$$, Horizontal Asymptotes: $$y=0$$","Domain: $$(-\\\\infty,-9) \\\\cup (-9,4) \\\\cup (4,\\\\infty)$$, Vertical Asymptotes: $$x=4$$, Horizontal Asymptotes: $$y=0$$","Domain: $$(-\\\\infty,-9) \\\\cup (-9,4) \\\\cup (4,\\\\infty)$$, Vertical Asymptotes: $$x=-9, 9, 4$$, Horizontal Asymptotes: $$y=0$$"],"hints":{"DefaultPathway":[{"id":"a5b6f42rationals4a-h1","type":"hint","dependencies":[],"title":"Factoring f(x)","text":"The denominator of f(x), $$x^2+5x-36$$, can be factored to $$\\\\left(x+9\\\\right) \\\\left(x-4\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals4a-h2","type":"hint","dependencies":["a5b6f42rationals4a-h1"],"title":"Definition of Domain","text":"The domain of a rational function includes all real numbers except those that cause the denominator to equal zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals4a-h3","type":"hint","dependencies":["a5b6f42rationals4a-h2"],"title":"Definition of a Vertical Asymptote","text":"A vertical asymptote of a graph is a vertical line $$x=a$$ where the graph tends toward positive or negative infinity as the inputs approach a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals4a-h4","type":"hint","dependencies":["a5b6f42rationals4a-h3"],"title":"Definition of a Horizontal Asymptote","text":"A horizontal asymptote of a graph is a horizontal line $$y=b$$ where the graph approaches the line as the inputs increase or decrease without bound.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5b6f42rationals5","title":"Analyzing Rational Functions #4","body":"Find the domain, vertical asymptotes, and horizontal asymptotes of the function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Rational Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5b6f42rationals5a","stepAnswer":["Domain: $$(-\\\\infty,3) \\\\cup (3,\\\\infty)$$, Vertical Asymptote: $$x=3$$, Horizontal Asymptote: $$y=0$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{3+x}{x^3-27}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Domain: $$(-\\\\infty,3) \\\\cup (3,\\\\infty)$$, Vertical Asymptote: $$x=3$$, Horizontal Asymptote: $$y=0$$","choices":["Domain: all real numbers, Vertical Asymptote: $$x=3$$, Horizontal Asymptote: $$y=0$$","Domain: all real numbers except $$-3$$, Vertical Asymptote: $$x=3$$, Horizontal Asymptote: $$y=0$$","Domain: $$(-\\\\infty,3) \\\\cup (3,\\\\infty)$$, Vertical Asymptote: $$x=3$$, Horizontal Asymptote: $$y=0$$"],"hints":{"DefaultPathway":[{"id":"a5b6f42rationals5a-h1","type":"hint","dependencies":[],"title":"Factoring f(x)","text":"The denominator of f(x), $$x^2+5x-36$$, can be factored to $$\\\\left(x+9\\\\right) \\\\left(x-4\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals5a-h2","type":"hint","dependencies":["a5b6f42rationals5a-h1"],"title":"Definition of Domain","text":"The domain of a rational function includes all real numbers except those that cause the denominator to equal zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals5a-h3","type":"hint","dependencies":["a5b6f42rationals5a-h2"],"title":"Definition of a Vertical Asymptote","text":"A vertical asymptote of a graph is a vertical line $$x=a$$ where the graph tends toward positive or negative infinity as the inputs approach a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals5a-h4","type":"hint","dependencies":["a5b6f42rationals5a-h3"],"title":"Definition of a Horizontal Asymptote","text":"A horizontal asymptote of a graph is a horizontal line $$y=b$$ where the graph approaches the line as the inputs increase or decrease without bound.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5b6f42rationals6","title":"Analyzing Rational Functions #","body":"Find the domain, vertical asymptotes, and horizontal asymptotes of the function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Rational Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5b6f42rationals6a","stepAnswer":["Domain: $$(-\\\\infty,-4) \\\\cup (-4,0) \\\\cup (0,4) \\\\cup (4,\\\\infty)$$, Vertical Asymptotes: $$x=-4, 0, 4$$, Horizontal Asymptote: $$y=0$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{3x-4}{x^3-16x}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Domain: $$(-\\\\infty,-4) \\\\cup (-4,0) \\\\cup (0,4) \\\\cup (4,\\\\infty)$$, Vertical Asymptotes: $$x=-4, 0, 4$$, Horizontal Asymptote: $$y=0$$","choices":["Domain: $$(-\\\\infty,-4) \\\\cup (-4,0) \\\\cup (0,4)$$, Vertical Asymptotes: $$x=-4, 0, 4$$, Horizontal Asymptote: $$y=0$$","Domain: $$(-\\\\infty,-4) \\\\cup (-4,0) \\\\cup (0,4) \\\\cup (4,\\\\infty)$$, Vertical Asymptotes: $$x=-4, 0, 4$$, Horizontal Asymptote: $$y=0$$","Domain: $$(-\\\\infty,-4) \\\\cup (-4,0) \\\\cup (0,4) \\\\cup (4,\\\\infty)$$, Vertical Asymptotes: $$x=-4, 4$$, Horizontal Asymptote: $$y=0$$","Domain: $$(-\\\\infty,-4) \\\\cup (-4,0) \\\\cup (0,4) \\\\cup (4,\\\\infty)$$, Vertical Asymptotes: $$x=-4, 0, 4$$, Horizontal Asymptote: $$y=0$$"],"hints":{"DefaultPathway":[{"id":"a5b6f42rationals6a-h1","type":"hint","dependencies":[],"title":"Definition of Domain","text":"The domain of a rational function includes all real numbers except those that cause the denominator to equal zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals6a-h2","type":"hint","dependencies":["a5b6f42rationals6a-h1"],"title":"Definition of a Vertical Asymptote","text":"A vertical asymptote of a graph is a vertical line $$x=a$$ where the graph tends toward positive or negative infinity as the inputs approach a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals6a-h3","type":"hint","dependencies":["a5b6f42rationals6a-h2"],"title":"Definition of a Horizontal Asymptote","text":"A horizontal asymptote of a graph is a horizontal line $$y=b$$ where the graph approaches the line as the inputs increase or decrease without bound.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5b6f42rationals7","title":"Analyzing Rational Functions #","body":"Find the domain, vertical asymptotes, and horizontal asymptotes of the function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Rational Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5b6f42rationals7a","stepAnswer":["Domain: $$(-\\\\infty,-7) \\\\cup (-7,-2) \\\\cup (-2,0) \\\\cup (0,\\\\infty)$$, Vertical Asymptotes: $$x=-7, -2, 0$$, Horizontal Asymptote: $$y=0$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{x^2-1}{x^3+9x^2+14x}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Domain: $$(-\\\\infty,-7) \\\\cup (-7,-2) \\\\cup (-2,0) \\\\cup (0,\\\\infty)$$, Vertical Asymptotes: $$x=-7, -2, 0$$, Horizontal Asymptote: $$y=0$$","choices":["Domain: $$(-\\\\infty,-2) \\\\cup (-2,0) \\\\cup (0,\\\\infty)$$, Vertical Asymptotes: $$x=-7, -2, 0$$, Horizontal Asymptote: $$y=0$$","Domain: $$(-\\\\infty,-7) \\\\cup (-7,-2) \\\\cup (-2,0) \\\\cup (0,\\\\infty)$$, Vertical Asymptotes: $$x=-7, -2, 0$$, Horizontal Asymptote: $$y=0$$","Domain: $$(-\\\\infty,-7) \\\\cup (-7,-2) \\\\cup (-2,\\\\infty)$$, Vertical Asymptotes: $$x=-7, -2, 7$$ Horizontal Asymptote: $$y=0$$"],"hints":{"DefaultPathway":[{"id":"a5b6f42rationals7a-h1","type":"hint","dependencies":[],"title":"Definition of Domain","text":"The domain of a rational function includes all real numbers except those that cause the denominator to equal zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals7a-h2","type":"hint","dependencies":["a5b6f42rationals7a-h1"],"title":"Definition of a Vertical Asymptote","text":"A vertical asymptote of a graph is a vertical line $$x=a$$ where the graph tends toward positive or negative infinity as the inputs approach a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals7a-h3","type":"hint","dependencies":["a5b6f42rationals7a-h2"],"title":"Definition of a Horizontal Asymptote","text":"A horizontal asymptote of a graph is a horizontal line $$y=b$$ where the graph approaches the line as the inputs increase or decrease without bound.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5b6f42rationals8","title":"Analyzing Rational Functions #","body":"Find the domain, vertical asymptotes, and horizontal asymptotes of the function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Rational Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5b6f42rationals8a","stepAnswer":["Domain: $$(-\\\\infty,-5) \\\\cup (-5,5) \\\\cup (5,\\\\infty)$$, Vertical Asymptote: $$x=5$$, Horizontal Asymptote: $$y=0$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{x+5}{x^2-25}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Domain: $$(-\\\\infty,-5) \\\\cup (-5,5) \\\\cup (5,\\\\infty)$$, Vertical Asymptote: $$x=5$$, Horizontal Asymptote: $$y=0$$","choices":["Domain: $$(-\\\\infty,5) \\\\cup (5,\\\\infty)$$, Vertical Asymptote: $$x=-5, 5$$, Horizontal Asymptote: $$y=0$$","Domain: $$(-\\\\infty,-5) \\\\cup (-5,5) \\\\cup (5,\\\\infty)$$, Vertical Asymptote: $$x=5$$, Horizontal Asymptote: $$y=0, 5$$","Domain: $$(-\\\\infty,-5) \\\\cup (-5,5) \\\\cup (5,\\\\infty)$$, Vertical Asymptote: $$x=5$$, Horizontal Asymptote: $$y=0$$"],"hints":{"DefaultPathway":[{"id":"a5b6f42rationals8a-h1","type":"hint","dependencies":[],"title":"Definition of Domain","text":"The domain of a rational function includes all real numbers except those that cause the denominator to equal zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals8a-h2","type":"hint","dependencies":["a5b6f42rationals8a-h1"],"title":"Definition of a Vertical Asymptote","text":"A vertical asymptote of a graph is a vertical line $$x=a$$ where the graph tends toward positive or negative infinity as the inputs approach a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals8a-h3","type":"hint","dependencies":["a5b6f42rationals8a-h2"],"title":"Definition of a Horizontal Asymptote","text":"A horizontal asymptote of a graph is a horizontal line $$y=b$$ where the graph approaches the line as the inputs increase or decrease without bound.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5b6f42rationals9","title":"Analyzing Rational Functions #","body":"Find the domain, vertical asymptotes, and horizontal asymptotes of the function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Rational Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5b6f42rationals9a","stepAnswer":["Domain: $$(-\\\\infty,6) \\\\cup (6,\\\\infty)$$, Vertical Asymptote: $$x=6$$, Horizontal Asymptote: $$y=1$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{x-4}{x-6}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Domain: $$(-\\\\infty,6) \\\\cup (6,\\\\infty)$$, Vertical Asymptote: $$x=6$$, Horizontal Asymptote: $$y=1$$","choices":["Domain: $$(-\\\\infty,6) \\\\cup (6,\\\\infty)$$, Vertical Asymptote: $$x=6$$, Horizontal Asymptote: $$y=1$$","Domain: $$(-\\\\infty,6) \\\\cup (6,\\\\infty)$$, Vertical Asymptote: $$x=1$$, Horizontal Asymptote: $$y=6$$","Domain: $$(-\\\\infty,1) \\\\cup (1,\\\\infty)$$, Vertical Asymptote: $$x=6$$, Horizontal Asymptote: $$y=1$$"],"hints":{"DefaultPathway":[{"id":"a5b6f42rationals9a-h1","type":"hint","dependencies":[],"title":"Definition of Domain","text":"The domain of a rational function includes all real numbers except those that cause the denominator to equal zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals9a-h2","type":"hint","dependencies":["a5b6f42rationals9a-h1"],"title":"Definition of a Vertical Asymptote","text":"A vertical asymptote of a graph is a vertical line $$x=a$$ where the graph tends toward positive or negative infinity as the inputs approach a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5b6f42rationals9a-h3","type":"hint","dependencies":["a5b6f42rationals9a-h2"],"title":"Definition of a Horizontal Asymptote","text":"A horizontal asymptote of a graph is a horizontal line $$y=b$$ where the graph approaches the line as the inputs increase or decrease without bound.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c2168rotation1","title":"Identifying a Conic from Its General Form","body":"Identify the graph of each of the following nondegenerate conic sections.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Rotation of Axes","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c2168rotation1a","stepAnswer":["Hyperbola"],"problemType":"MultipleChoice","stepTitle":"$$4x^2-9y^2+36x+36y-125=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Circle","Parabola","Ellipse","Hyperbola"],"hints":{"DefaultPathway":[{"id":"a5c2168rotation1a-h1","type":"hint","dependencies":[],"title":"Identifying a Conic","text":"1) Rewrite the equation in the general form, $$A x^2+B x y+C y^2+D x+E y+F=0$$.\\\\n2) Identify the values of A and C from the general form.\\\\na. If A and C are nonzero, have the same sign, and are not equal to each other, then the graph may be an ellipse.\\\\n$$b$$. If A and C are equal and nonzero and have the same sign, then the graph may be a circle.\\\\nc. If A and C are nonzero and have opposite signs, then the graph may be a hyperbola.\\\\n$$d$$. If either A or C is zero, then the graph may be a parabola.\\\\nIf $$B=0$$, the conic section will have a vertical and/or horizontal axes. If B does not equal $$0$$, as shown below, the conic section is rotated. Notice the phrase \u201cmay be\u201d in the definitions. That is because the equation may not represent a conic section at all, depending on the values of A, B, C, D, E, and F. For example, the degenerate case of a circle or an ellipse is a point:\\\\n$$A x^2+B y^2=0$$, when A and B have the same sign.\\\\nThe degenerate case of a hyperbola is two intersecting straight lines: $$A x^2+B y^2=0$$, when A and B have opposite signs.\\\\nOn the other hand, the equation, $$A x^2+B y^2+1=0$$, when A and B are positive does not represent a graph at all, since there are no real ordered pairs which satisfy it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation1a-h2","type":"hint","dependencies":["a5c2168rotation1a-h1"],"title":"General Form","text":"Referring to the general form $$A x^2+B x y+C y^2+D x+E y+F=0$$, we can identify the type of conic by comparing the coefficients.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a5c2168rotation1a-h2"],"title":"General Form","text":"What is the coefficient of $$x^2$$, A, equals to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":["a5c2168rotation1a-h3"],"title":"General Form","text":"What is the coefficient of $$y^2$$, C, equals to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation1a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Hyperbola"],"dependencies":["a5c2168rotation1a-h4"],"title":"General Form","text":"We notice that A and C have opposite signs. Based on the previous hint, what type of conic is this?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Circle","Parabola","Ellipse","Hyperbola"]}]}},{"id":"a5c2168rotation1b","stepAnswer":["Parabola"],"problemType":"MultipleChoice","stepTitle":"$$9y^2+16x+36y-10=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Circle","Parabola","Ellipse","Hyperbola"],"hints":{"DefaultPathway":[{"id":"a5c2168rotation1b-h1","type":"hint","dependencies":[],"title":"Identifying a Conic","text":"1) Rewrite the equation in the general form, $$A x^2+B x y+C y^2+D x+E y+F=0$$.\\\\n2) Identify the values of A and C from the general form.\\\\na. If A and C are nonzero, have the same sign, and are not equal to each other, then the graph may be an ellipse.\\\\n$$b$$. If A and C are equal and nonzero and have the same sign, then the graph may be a circle.\\\\nc. If A and C are nonzero and have opposite signs, then the graph may be a hyperbola.\\\\n$$d$$. If either A or C is zero, then the graph may be a parabola.\\\\nIf $$B=0$$, the conic section will have a vertical and/or horizontal axes. If B does not equal $$0$$, as shown below, the conic section is rotated. Notice the phrase \u201cmay be\u201d in the definitions. That is because the equation may not represent a conic section at all, depending on the values of A, B, C, D, E, and F. For example, the degenerate case of a circle or an ellipse is a point:\\\\n$$A x^2+B y^2=0$$, when A and B have the same sign.\\\\nThe degenerate case of a hyperbola is two intersecting straight lines: $$A x^2+B y^2=0$$, when A and B have opposite signs.\\\\nOn the other hand, the equation, $$A x^2+B y^2+1=0$$, when A and B are positive does not represent a graph at all, since there are no real ordered pairs which satisfy it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation1b-h2","type":"hint","dependencies":["a5c2168rotation1b-h1"],"title":"General Form","text":"Referring to the general form $$A x^2+B x y+C y^2+D x+E y+F=0$$, we can identify the type of conic by comparing the coefficients.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation1b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a5c2168rotation1b-h2"],"title":"General Form","text":"What is the coefficient of $$x^2$$, A, equals to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation1b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a5c2168rotation1b-h3"],"title":"General Form","text":"What is the coefficient of $$y^2$$, C, equals to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation1b-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Parabola"],"dependencies":["a5c2168rotation1b-h4"],"title":"General Form","text":"We notice that A is zero. Based on the previous hint, what type of conic is this?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Circle","Parabola","Ellipse","Hyperbola"]}]}},{"id":"a5c2168rotation1c","stepAnswer":["Circle"],"problemType":"MultipleChoice","stepTitle":"$$3x^2+3y^2-2x-6y-4=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Circle","Parabola","Ellipse","Hyperbola"],"hints":{"DefaultPathway":[{"id":"a5c2168rotation1c-h1","type":"hint","dependencies":[],"title":"Identifying a Conic","text":"1) Rewrite the equation in the general form, $$A x^2+B x y+C y^2+D x+E y+F=0$$.\\\\n2) Identify the values of A and C from the general form.\\\\na. If A and C are nonzero, have the same sign, and are not equal to each other, then the graph may be an ellipse.\\\\n$$b$$. If A and C are equal and nonzero and have the same sign, then the graph may be a circle.\\\\nc. If A and C are nonzero and have opposite signs, then the graph may be a hyperbola.\\\\n$$d$$. If either A or C is zero, then the graph may be a parabola.\\\\nIf $$B=0$$, the conic section will have a vertical and/or horizontal axes. If B does not equal $$0$$, as shown below, the conic section is rotated. Notice the phrase \u201cmay be\u201d in the definitions. That is because the equation may not represent a conic section at all, depending on the values of A, B, C, D, E, and F. For example, the degenerate case of a circle or an ellipse is a point:\\\\n$$A x^2+B y^2=0$$, when A and B have the same sign.\\\\nThe degenerate case of a hyperbola is two intersecting straight lines: $$A x^2+B y^2=0$$, when A and B have opposite signs.\\\\nOn the other hand, the equation, $$A x^2+B y^2+1=0$$, when A and B are positive does not represent a graph at all, since there are no real ordered pairs which satisfy it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation1c-h2","type":"hint","dependencies":["a5c2168rotation1c-h1"],"title":"General Form","text":"Referring to the general form $$A x^2+B x y+C y^2+D x+E y+F=0$$, we can identify the type of conic by comparing the coefficients.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation1c-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a5c2168rotation1c-h2"],"title":"General Form","text":"What is the coefficient of $$x^2$$, A, equals to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation1c-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a5c2168rotation1c-h3"],"title":"General Form","text":"What is the coefficient of $$y^2$$, C, equals to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation1c-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Circle"],"dependencies":["a5c2168rotation1c-h4"],"title":"General Form","text":"We notice that A and C are equal. Based on the previous hint, what type of conic is this?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Circle","Parabola","Ellipse","Hyperbola"]}]}},{"id":"a5c2168rotation1d","stepAnswer":["Ellipse"],"problemType":"MultipleChoice","stepTitle":"$$-25x^2-4y^2+100x+16y+20=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Circle","Parabola","Ellipse","Hyperbola"],"hints":{"DefaultPathway":[{"id":"a5c2168rotation1d-h1","type":"hint","dependencies":[],"title":"Identifying a Conic","text":"1) Rewrite the equation in the general form, $$A x^2+B x y+C y^2+D x+E y+F=0$$.\\\\n2) Identify the values of A and C from the general form.\\\\na. If A and C are nonzero, have the same sign, and are not equal to each other, then the graph may be an ellipse.\\\\n$$b$$. If A and C are equal and nonzero and have the same sign, then the graph may be a circle.\\\\nc. If A and C are nonzero and have opposite signs, then the graph may be a hyperbola.\\\\n$$d$$. If either A or C is zero, then the graph may be a parabola.\\\\nIf $$B=0$$, the conic section will have a vertical and/or horizontal axes. If B does not equal $$0$$, as shown below, the conic section is rotated. Notice the phrase \u201cmay be\u201d in the definitions. That is because the equation may not represent a conic section at all, depending on the values of A, B, C, D, E, and F. For example, the degenerate case of a circle or an ellipse is a point:\\\\n$$A x^2+B y^2=0$$, when A and B have the same sign.\\\\nThe degenerate case of a hyperbola is two intersecting straight lines: $$A x^2+B y^2=0$$, when A and B have opposite signs.\\\\nOn the other hand, the equation, $$A x^2+B y^2+1=0$$, when A and B are positive does not represent a graph at all, since there are no real ordered pairs which satisfy it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation1d-h2","type":"hint","dependencies":["a5c2168rotation1d-h1"],"title":"General Form","text":"Referring to the general form $$A x^2+B x y+C y^2+D x+E y+F=0$$, we can identify the type of conic by comparing the coefficients.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation1d-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-25$$"],"dependencies":["a5c2168rotation1d-h2"],"title":"General Form","text":"What is the coefficient of $$x^2$$, A, equals to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation1d-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["a5c2168rotation1d-h3"],"title":"General Form","text":"What is the coefficient of $$y^2$$, C, equals to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation1d-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Ellipse"],"dependencies":["a5c2168rotation1d-h4"],"title":"General Form","text":"We notice that A and C are the same sign but are not equal to each other. Based on the previous hint, what type of conic is this?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Circle","Parabola","Ellipse","Hyperbola"]}]}}]},{"id":"a5c2168rotation10","title":"Identify","body":"Determine which conic section is represented based on the given equation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Rotation of Axes","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c2168rotation10a","stepAnswer":["Hyperbola"],"problemType":"MultipleChoice","stepTitle":"$$4x^2-y^2+8x-1=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Ellipse","Circle","Hyperbola","Parabola"],"hints":{"DefaultPathway":[{"id":"a5c2168rotation10a-h1","type":"hint","dependencies":[],"title":"Rewrite","text":"Rewrite the equation in general form $${Ax}^2+Bxy+{Cy}^2+Dx+Ey+F=0$$, if it is not already.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation10a-h2","type":"hint","dependencies":["a5c2168rotation10a-h1"],"title":"Use A and C","text":"Use the definitions of A and C to find the type of equation this conic is. If A and C are nonzero, have the same sign, and are not equal to each other, then the graph is an ellipse. If A and C are equal and nonzero and have the same sign, then the graph is a circle. If A and C are nonzero and have opposite signs, then the graph is a hyperbola. If either A or C is zero, then the graph is a parabola.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation10a-h3","type":"hint","dependencies":["a5c2168rotation10a-h2"],"title":"Answer","text":"As A and B have opposite signs, the equation is a hyperbola.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c2168rotation11","title":"Identify","body":"Determine which conic section is represented based on the given equation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Rotation of Axes","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c2168rotation11a","stepAnswer":["Parabola"],"problemType":"MultipleChoice","stepTitle":"$$4y^2-5x+9y+1=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Ellipse","Circle","Hyperbola","Parabola"],"hints":{"DefaultPathway":[{"id":"a5c2168rotation11a-h1","type":"hint","dependencies":[],"title":"Rewrite","text":"Rewrite the equation in general form $${Ax}^2+Bxy+{Cy}^2+Dx+Ey+F=0$$, if it is not already.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation11a-h2","type":"hint","dependencies":["a5c2168rotation11a-h1"],"title":"Use A and C","text":"Use the definitions of A and C to find the type of equation this conic is. If A and C are nonzero, have the same sign, and are not equal to each other, then the graph is an ellipse. If A and C are equal and nonzero and have the same sign, then the graph is a circle. If A and C are nonzero and have opposite signs, then the graph is a hyperbola. If either A or C is zero, then the graph is a parabola.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation11a-h3","type":"hint","dependencies":["a5c2168rotation11a-h2"],"title":"Answer","text":"As A is zero, the equation is a parabola.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c2168rotation12","title":"Identify","body":"Determine which conic section is represented based on the given equation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Rotation of Axes","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c2168rotation12a","stepAnswer":["Ellipse"],"problemType":"MultipleChoice","stepTitle":"$$2x^2+3y^2-8x-12y+2=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Ellipse","Circle","Hyperbola","Parabola"],"hints":{"DefaultPathway":[{"id":"a5c2168rotation12a-h1","type":"hint","dependencies":[],"title":"Rewrite","text":"Rewrite the equation in general form $${Ax}^2+Bxy+{Cy}^2+Dx+Ey+F=0$$, if it is not already.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation12a-h2","type":"hint","dependencies":["a5c2168rotation12a-h1"],"title":"Use A and C","text":"Use the definitions of A and C to find the type of equation this conic is. If A and C are nonzero, have the same sign, and are not equal to each other, then the graph is an ellipse. If A and C are equal and nonzero and have the same sign, then the graph is a circle. If A and C are nonzero and have opposite signs, then the graph is a hyperbola. If either A or C is zero, then the graph is a parabola.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation12a-h3","type":"hint","dependencies":["a5c2168rotation12a-h2"],"title":"Answer","text":"As A and C are nonzero, have the same sign, and are not equal to each other, the graph is an ellipse.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c2168rotation13","title":"Rotation of Axes","body":"$$4x^2-xy+4y^2-2=0$$, $$\\\\theta=45\\\\degree$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Rotation of Axes","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c2168rotation13a","stepAnswer":["$$7{x\'}^2+9{y\'}^2-4=0$$"],"problemType":"TextBox","stepTitle":"Finding a New Representation of the Given Equation after Rotating through a Given Angle","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7{x\'}^2+9{y\'}^2-4=0$$","hints":{"DefaultPathway":[{"id":"a5c2168rotation13a-h1","type":"hint","dependencies":[],"title":"Equations of Rotation","text":"The equations of rotation are $$x=x\'cos(\\\\theta)-y\'sin(\\\\theta)$$ and $$y=\\\\operatorname{x\'sin}\\\\left(\\\\theta\\\\right)+\\\\operatorname{x\'cos}\\\\left(\\\\theta\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$45$$"],"dependencies":["a5c2168rotation13a-h1"],"title":"Interpreting the Problem","text":"What is \ud835\udf03?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation13a-h3","type":"hint","dependencies":["a5c2168rotation13a-h2"],"title":"Plugging in the Angle","text":"Because $$\\\\theta=45$$, plug in the value into the equations of rotation for $$x$$ and $$y$$. Simplify so you are left with an algebraic function (with no trignometric functions)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation13a-h4","type":"hint","dependencies":["a5c2168rotation13a-h3"],"title":"Substituting Equation of Rotations","text":"Substitute $$x=\\\\frac{x\'-y\'}{\\\\sqrt{2}}$$ and $$y=\\\\frac{x\'+y\'}{\\\\sqrt{2}}$$ into $$4x^2-xy+4y^2-2=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation13a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7{x\'}^2+9{y\'}^2-4=0$$"],"dependencies":["a5c2168rotation13a-h4"],"title":"Algebraic Simplifications","text":"Using FOIL method, combining like terms, and other simplifications, determine the new representation of the equation. Write the answer so that there are no fractions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c2168rotation14","title":"Rotation of Axes","body":"$$3x^2+xy+3y^2-5=0$$, $$\\\\theta=45\\\\degree$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Rotation of Axes","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c2168rotation14a","stepAnswer":["$$7{x\'}^2+5{y\'}^2-5=0$$"],"problemType":"TextBox","stepTitle":"Finding a New Representation of the Given Equation after Rotating through a Given Angle","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7{x\'}^2+5{y\'}^2-5=0$$","hints":{"DefaultPathway":[{"id":"a5c2168rotation14a-h1","type":"hint","dependencies":[],"title":"Equations of Rotation","text":"The equations of rotation are $$x=x\'cos(\\\\theta)-y\'sin(\\\\theta)$$ and $$y=\\\\operatorname{x\'sin}\\\\left(\\\\theta\\\\right)+\\\\operatorname{x\'cos}\\\\left(\\\\theta\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$45$$"],"dependencies":["a5c2168rotation14a-h1"],"title":"Interpreting the Problem","text":"What is \ud835\udf03?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation14a-h3","type":"hint","dependencies":["a5c2168rotation14a-h2"],"title":"Plugging in the Angle","text":"Because $$\\\\theta=45$$, plug in the value into the equations of rotation for $$x$$ and $$y$$. Simplify so you are left with an algebraic function (with no trignometric functions)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation14a-h4","type":"hint","dependencies":["a5c2168rotation14a-h3"],"title":"Substituting Equation of Rotations","text":"Substitute $$x=\\\\frac{x\'-y\'}{\\\\sqrt{2}}$$ and $$y=\\\\frac{x\'+y\'}{\\\\sqrt{2}}$$ into $$3x^2+xy+3y^2-5=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation14a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7{x\'}^2+5{y\'}^2-5=0$$"],"dependencies":["a5c2168rotation14a-h4"],"title":"Algebraic Simplifications","text":"Using FOIL method, combining like terms, and other simplifications, determine the new representation of the equation. Write the answer so that there are no fractions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c2168rotation15","title":"Rotation of Axes","body":"$$\\\\left(-2x^2\\\\right)+8xy+1=0$$, $$\\\\theta=45\\\\degree$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Rotation of Axes","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c2168rotation15a","stepAnswer":["$$3{x\'}^2+2x\'y\'-5{y\'}^2+1=0$$"],"problemType":"TextBox","stepTitle":"Finding a New Representation of the Given Equation after Rotating through a Given Angle","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3{x\'}^2+2x\'y\'-5{y\'}^2+1=0$$","hints":{"DefaultPathway":[{"id":"a5c2168rotation15a-h1","type":"hint","dependencies":[],"title":"Equations of Rotation","text":"The equations of rotation are $$x=x\'cos(\\\\theta)-y\'sin(\\\\theta)$$ and $$y=\\\\operatorname{x\'sin}\\\\left(\\\\theta\\\\right)+\\\\operatorname{x\'cos}\\\\left(\\\\theta\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$45$$"],"dependencies":["a5c2168rotation15a-h1"],"title":"Interpreting the Problem","text":"What is \ud835\udf03?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation15a-h3","type":"hint","dependencies":["a5c2168rotation15a-h2"],"title":"Plugging in the Angle","text":"Because $$\\\\theta=45$$, plug in the value into the equations of rotation for $$x$$ and $$y$$. Simplify so you are left with an algebraic function (with no trignometric functions)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation15a-h4","type":"hint","dependencies":["a5c2168rotation15a-h3"],"title":"Substituting Equation of Rotations","text":"Substitute $$x=\\\\frac{x\'-y\'}{\\\\sqrt{2}}$$ and $$y=\\\\frac{x\'+y\'}{\\\\sqrt{2}}$$ into $$\\\\left(-2x^2\\\\right)+8xy+1=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation15a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3{x\'}^2+2x\'y\'-5{y\'}^2+1=0$$"],"dependencies":["a5c2168rotation15a-h4"],"title":"Algebraic Simplifications","text":"Using FOIL method, combining like terms, and other simplifications, determine the new representation of the equation. Write the answer so that there are no fractions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c2168rotation16","title":"Rotation of Axes","body":"$$2x^2+8xy-1=0$$, $$\\\\theta=30\\\\degree$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Rotation of Axes","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c2168rotation16a","stepAnswer":["$$\\\\left(3+2\\\\sqrt{3}\\\\right) {x\'}^2+\\\\left(1-2\\\\sqrt{3}\\\\right) {y\'}^2+\\\\left(4-2\\\\sqrt{3}\\\\right) x\'y\'-1=0$$"],"problemType":"TextBox","stepTitle":"Finding a New Representation of the Given Equation after Rotating through a Given Angle","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(3+2\\\\sqrt{3}\\\\right) {x\'}^2+\\\\left(1-2\\\\sqrt{3}\\\\right) {y\'}^2+\\\\left(4-2\\\\sqrt{3}\\\\right) x\'y\'-1=0$$","hints":{"DefaultPathway":[{"id":"a5c2168rotation16a-h1","type":"hint","dependencies":[],"title":"Equations of Rotation","text":"The equations of rotation are $$x=x\'cos(\\\\theta)-y\'sin(\\\\theta)$$ and $$y=\\\\operatorname{x\'sin}\\\\left(\\\\theta\\\\right)+\\\\operatorname{x\'cos}\\\\left(\\\\theta\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$30$$"],"dependencies":["a5c2168rotation16a-h1"],"title":"Interpreting the Problem","text":"What is \ud835\udf03?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation16a-h3","type":"hint","dependencies":["a5c2168rotation16a-h2"],"title":"Plugging in the Angle","text":"Because $$\\\\theta=30$$, plug in the value into the equations of rotation for $$x$$ and $$y$$. Simplify so you are left with an algebraic function (with no trignometric functions)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation16a-h4","type":"hint","dependencies":["a5c2168rotation16a-h3"],"title":"Substituting Equation of Rotations","text":"Substitute $$x=\\\\operatorname{x\'}\\\\left(\\\\frac{\\\\sqrt{3}}{2}\\\\right)-\\\\operatorname{y\'}\\\\left(\\\\frac{1}{2}\\\\right)$$ and $$y=\\\\operatorname{x\'}\\\\left(\\\\frac{1}{2}\\\\right)+\\\\operatorname{y\'}\\\\left(\\\\frac{\\\\sqrt{3}}{2}\\\\right)$$ into $$2x^2+8xy-1=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation16a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(3+2\\\\sqrt{3}\\\\right) {x\'}^2+\\\\left(1-2\\\\sqrt{3}\\\\right) {y\'}^2+\\\\left(4-2\\\\sqrt{3}\\\\right) x\'y\'-1=0$$"],"dependencies":["a5c2168rotation16a-h4"],"title":"Algebraic Simplifications","text":"Using FOIL method, combining like terms, and other simplifications, determine the new representation of the equation. Write the answer so that there are no fractions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c2168rotation17","title":"Rotation of Axes","body":"(x**2)+(3sqrt(3)xy)+4y**2)+y-2=0","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Rotation of Axes","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c2168rotation17a","stepAnswer":["$$60$$"],"problemType":"TextBox","stepTitle":"Finding the Angle","stepBody":"Determine the angle \ud835\udf03 that will eliminate the xy term (in degrees)","answerType":"arithmetic","variabilization":{},"answerLatex":"$$60$$","hints":{"DefaultPathway":[{"id":"a5c2168rotation17a-h1","type":"hint","dependencies":[],"title":"Standard Form of Conic","text":"The standard form of a conic is $${Ax}^2+{BxCy}^2+Dx+Ey+F=0$$. Using the standard form, match it with the given equation to solve for $$cot(2\\\\theta)=\\\\frac{A-C}{B}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation17a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{\\\\sqrt{3}}$$"],"dependencies":["a5c2168rotation17a-h1"],"title":"Find cot(2\ud835\udf03)","text":"What is cot(2\ud835\udf03)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation17a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$60$$"],"dependencies":["a5c2168rotation17a-h2"],"title":"Solve for \ud835\udf03","text":"Using trignometric rules, solve for \ud835\udf03 (in degrees)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a5c2168rotation17b","stepAnswer":["$$11{x\'}^2-{y\'}^2+\\\\sqrt{3} x\'+y\'-4=0$$"],"problemType":"TextBox","stepTitle":"Finding a New Representation of the Given Equation after Rotating through a Given Angle","stepBody":"Write the corresponding equation without the xy term when rotated about the angle \ud835\udf03","answerType":"arithmetic","variabilization":{},"answerLatex":"$$11{x\'}^2-{y\'}^2+\\\\sqrt{3} x\'+y\'-4=0$$","hints":{"DefaultPathway":[{"id":"a5c2168rotation17b-h1","type":"hint","dependencies":[],"title":"Equations of Rotation","text":"The equations of rotation are $$x=x\'cos(\\\\theta)-y\'sin(\\\\theta)$$ and $$y=\\\\operatorname{x\'sin}\\\\left(\\\\theta\\\\right)+\\\\operatorname{x\'cos}\\\\left(\\\\theta\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation17b-h2","type":"hint","dependencies":["a5c2168rotation17b-h1"],"title":"Plugging in the Angle","text":"Because $$\\\\theta=60$$, plug in the value into the equations of rotation for $$x$$ and $$y$$. Simplify so you are left with an algebraic function (with no trignometric functions)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation17b-h3","type":"hint","dependencies":["a5c2168rotation17b-h2"],"title":"Substituting Equation of Rotations","text":"Substitute $$x=\\\\operatorname{x\'}\\\\left(\\\\frac{1}{2}\\\\right)-\\\\operatorname{y\'}\\\\left(\\\\frac{\\\\sqrt{3}}{2}\\\\right)$$ and $$y=\\\\operatorname{x\'}\\\\left(\\\\frac{\\\\sqrt{3}}{2}\\\\right)+\\\\operatorname{y\'}\\\\left(\\\\frac{1}{2}\\\\right)$$ into (x**2)+(3sqrt(3)xy)+4y**2)+y-2=0","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation17b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11{x\'}^2-{y\'}^2+\\\\sqrt{3} x\'+y\'-4=0$$"],"dependencies":["a5c2168rotation17b-h3"],"title":"Algebraic Simplifications","text":"Using FOIL method, combining like terms, and other simplifications, determine the new representation of the equation. Write the answer so that there are no fractions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c2168rotation18","title":"Rotation of Axes","body":"$$9x^2-3\\\\sqrt{3} xy+6y^2+4y-3=0$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Rotation of Axes","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c2168rotation18a","stepAnswer":["$$150$$"],"problemType":"TextBox","stepTitle":"Finding the Angle","stepBody":"Determine the angle \ud835\udf03 that will eliminate the xy term (in degrees)","answerType":"arithmetic","variabilization":{},"answerLatex":"$$150$$","hints":{"DefaultPathway":[{"id":"a5c2168rotation18a-h1","type":"hint","dependencies":[],"title":"Standard Form of Conic","text":"The standard form of a conic is $${Ax}^2+{BxCy}^2+Dx+Ey+F=0$$. Using the standard form, match it with the given equation to solve for $$cot(2\\\\theta)=\\\\frac{A-C}{B}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{\\\\sqrt{3}}$$"],"dependencies":["a5c2168rotation18a-h1"],"title":"Find cot(2\ud835\udf03)","text":"What is cot(2\ud835\udf03)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation18a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$150$$"],"dependencies":["a5c2168rotation18a-h2"],"title":"Solve for \ud835\udf03","text":"Using trignometric rules, solve for \ud835\udf03 (in degrees)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a5c2168rotation18b","stepAnswer":["$$21{x\'}^2+9{y\'}^2+4x\'-4\\\\sqrt{3} y\'-6=0$$"],"problemType":"TextBox","stepTitle":"Finding a New Representation of the Given Equation after Rotating through a Given Angle","stepBody":"Write the corresponding equation without the xy term when rotated about the angle \ud835\udf03","answerType":"arithmetic","variabilization":{},"answerLatex":"$$21{x\'}^2+9{y\'}^2+4x\'-4\\\\sqrt{3} y\'-6=0$$","hints":{"DefaultPathway":[{"id":"a5c2168rotation18b-h1","type":"hint","dependencies":[],"title":"Equations of Rotation","text":"The equations of rotation are $$x=x\'cos(\\\\theta)-y\'sin(\\\\theta)$$ and $$y=\\\\operatorname{x\'sin}\\\\left(\\\\theta\\\\right)+\\\\operatorname{x\'cos}\\\\left(\\\\theta\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation18b-h2","type":"hint","dependencies":["a5c2168rotation18b-h1"],"title":"Plugging in the Angle","text":"Because $$\\\\theta=150$$, plug in the value into the equations of rotation for $$x$$ and $$y$$. Simplify so you are left with an algebraic function (with no trignometric functions)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation18b-h3","type":"hint","dependencies":["a5c2168rotation18b-h2"],"title":"Substituting Equation of Rotations","text":"Substitute $$x=\\\\operatorname{x\'negneg}\\\\left(\\\\frac{\\\\sqrt{3}}{2}\\\\right)-\\\\operatorname{y\'}\\\\left(\\\\frac{1}{2}\\\\right)$$ and $$y=\\\\operatorname{x\'}\\\\left(\\\\frac{1}{2}\\\\right)+\\\\operatorname{y\'negneg}\\\\left(\\\\frac{\\\\sqrt{3}}{2}\\\\right)$$ into $$9x^2-3\\\\sqrt{3} xy+6y^2+4y-3=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation18b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$21{x\'}^2+9{y\'}^2+4x\'-4\\\\sqrt{3} y\'-6=0$$"],"dependencies":["a5c2168rotation18b-h3"],"title":"Algebraic Simplifications","text":"Using FOIL method, combining like terms, and other simplifications, determine the new representation of the equation. Write the answer so that there are no fractions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c2168rotation19","title":"Rotation of Axes","body":"$$x^2+4xy+y^2-2x+1=0$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Rotation of Axes","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c2168rotation19a","stepAnswer":["$$45$$"],"problemType":"TextBox","stepTitle":"Finding the Angle","stepBody":"Determine the angle \ud835\udf03 that will eliminate the xy term (in degrees)","answerType":"arithmetic","variabilization":{},"answerLatex":"$$45$$","hints":{"DefaultPathway":[{"id":"a5c2168rotation19a-h1","type":"hint","dependencies":[],"title":"Standard Form of Conic","text":"The standard form of a conic is $${Ax}^2+{BxCy}^2+Dx+Ey+F=0$$. Using the standard form, match it with the given equation to solve for $$cot(2\\\\theta)=\\\\frac{A-C}{B}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a5c2168rotation19a-h1"],"title":"Find cot(2\ud835\udf03)","text":"What is cot(2\ud835\udf03)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation19a-h3","type":"hint","dependencies":["a5c2168rotation19a-h2"],"title":"Interpreting Standard Form","text":"If $$A=C$$, then $$\\\\theta=45$$. In this case $$A=C=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a5c2168rotation19b","stepAnswer":["$$21{x\'}^2+9{y\'}^2+4x\'-4\\\\sqrt{3} y\'-6=0$$"],"problemType":"TextBox","stepTitle":"Finding a New Representation of the Given Equation after Rotating through a Given Angle","stepBody":"Write the corresponding equation without the xy term when rotated about the angle \ud835\udf03","answerType":"arithmetic","variabilization":{},"answerLatex":"$$21{x\'}^2+9{y\'}^2+4x\'-4\\\\sqrt{3} y\'-6=0$$","hints":{"DefaultPathway":[{"id":"a5c2168rotation19b-h1","type":"hint","dependencies":[],"title":"Equations of Rotation","text":"The equations of rotation are $$x=x\'cos(\\\\theta)-y\'sin(\\\\theta)$$ and $$y=\\\\operatorname{x\'sin}\\\\left(\\\\theta\\\\right)+\\\\operatorname{x\'cos}\\\\left(\\\\theta\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation19b-h2","type":"hint","dependencies":["a5c2168rotation19b-h1"],"title":"Plugging in the Angle","text":"Because $$\\\\theta=45$$, plug in the value into the equations of rotation for $$x$$ and $$y$$. Simplify so you are left with an algebraic function (with no trignometric functions)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation19b-h3","type":"hint","dependencies":["a5c2168rotation19b-h2"],"title":"Substituting Equation of Rotations","text":"Substitute $$x=\\\\frac{x\'-y\'}{\\\\sqrt{2}}$$ and $$y=\\\\frac{x\'+y\'}{\\\\sqrt{2}}$$ into $$x^2+4xy+y^2-2x+1=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation19b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$21{x\'}^2+9{y\'}^2+4x\'-4\\\\sqrt{3} y\'-6=0$$"],"dependencies":["a5c2168rotation19b-h3"],"title":"Algebraic Simplifications","text":"Using FOIL method, combining like terms, and other simplifications, determine the new representation of the equation. Write the answer so that there are no fractions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c2168rotation2","title":"Finding a New Representation of an Equation after Rotating through a Given Angle","body":"Find a new representation of the following equation after rotating through an angle of /theta=45**o.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Rotation of Axes","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c2168rotation2a","stepAnswer":["$$\\\\frac{{x\'}^2}{20}+\\\\frac{{y\'}^2}{12}=1$$"],"problemType":"MultipleChoice","stepTitle":"$$2x^2-x y+2y^2-30=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{{x\'}^2}{20}+\\\\frac{{y\'}^2}{12}=1$$","choices":["$$\\\\frac{{x\'}^2}{20}+\\\\frac{{y\'}^2}{12}=1$$","$$\\\\frac{{x\'}^2}{12}+\\\\frac{{y\'}^2}{20}=1$$","$$\\\\frac{{x\'}^2}{20}-\\\\frac{{y\'}^2}{12}=1$$"],"hints":{"DefaultPathway":[{"id":"a5c2168rotation2a-h1","type":"hint","dependencies":[],"title":"Equations of Rotation","text":"If a point (x,y) on the Cartesian plane is represented on a new coordinate plane where the axes of rotation are formed by rotating an angle $$\\\\theta$$ from the positive x-axis, then the coordinates of the point with respect to the new axes are (x\u2032,y\u2032). We can use the following equations of rotation to define the relationship between (x,y) and (x\u2032,y\u2032):\\\\nx=x\u2032*cos(\\\\theta)-y\u2032*sin(\\\\theta)\\\\nand\\\\ny=x\u2032*sin(\\\\theta)+y\u2032*cos(\\\\theta)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation2a-h2","type":"hint","dependencies":["a5c2168rotation2a-h1"],"title":"Substitution","text":"We want to substitute x=x\u2032*cos(\\\\theta)-y\u2032*sin(\\\\theta) and y=x\u2032*sin(\\\\theta)+y\u2032*cos(\\\\theta) into the equation so that we can manipulate the equation into the new representation. Before we do so, we can also substitute $$\\\\theta={45}^o$$ into the $$sin\\\\left(\\\\theta\\\\right)$$ and $$cos\\\\left(\\\\theta\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{\\\\sqrt{2}}$$"],"dependencies":["a5c2168rotation2a-h2"],"title":"Substitution","text":"$$sin\\\\left({45}^o\\\\right)=cos\\\\left({45}^o\\\\right)$$. What are they equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{x\'-y\'}{\\\\sqrt{2}}$$"],"dependencies":["a5c2168rotation2a-h3"],"title":"Substitution","text":"Simplifying the equation of rotation for the $$x$$ term where x=x\u2032*cos(\\\\theta)-y\u2032*sin(\\\\theta), what is $$x$$ equals to after substituting in $$\\\\theta={45}^o$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation2a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{x\'+y\'}{\\\\sqrt{2}}$$"],"dependencies":["a5c2168rotation2a-h4"],"title":"Substitution","text":"Simplifying the equation of rotation for the $$y$$ term where y=x\u2032*sin(\\\\theta)+y\u2032*cos(\\\\theta), what is $$y$$ equals to after substituting in $$\\\\theta={45}^o$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation2a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2{\\\\left(\\\\frac{x\'-y\'}{\\\\sqrt{2}}\\\\right)}^2-\\\\frac{x\'-y\'}{\\\\sqrt{2}} \\\\frac{x\'+y\'}{\\\\sqrt{2}}+2{\\\\left(\\\\frac{x\'+y\'}{\\\\sqrt{2}}\\\\right)}^2-30=0$$"],"dependencies":["a5c2168rotation2a-h5"],"title":"Substitution","text":"Substituting $$x=\\\\frac{x\'-y\'}{\\\\sqrt{2}}$$ and $$y=\\\\frac{x\'+y\'}{\\\\sqrt{2}}$$ into $$2x^2-x y+2y^2-30=0$$, what is the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2{\\\\left(\\\\frac{x\'-y\'}{\\\\sqrt{2}}\\\\right)}^2-\\\\frac{x\'-y\'}{\\\\sqrt{2}} \\\\frac{x\'+y\'}{\\\\sqrt{2}}+2{\\\\left(\\\\frac{x\'+y\'}{\\\\sqrt{2}}\\\\right)}^2-30=0$$","$$2{\\\\left(\\\\frac{x\'-y\'}{\\\\sqrt{2}}\\\\right)}^2-\\\\frac{x\'+y\'}{\\\\sqrt{2}} \\\\frac{x\'+y\'}{\\\\sqrt{2}}+2{\\\\left(\\\\frac{x\'-y\'}{\\\\sqrt{2}}\\\\right)}^2-30=0$$"]},{"id":"a5c2168rotation2a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\left(x\'-y\'\\\\right) \\\\left(x\'-y\'\\\\right)-\\\\frac{\\\\left(x\'-y\'\\\\right) \\\\left(x\'+y\'\\\\right)}{2}+\\\\left(x\'+y\'\\\\right) \\\\left(x\'+y\'\\\\right)-30=0$$"],"dependencies":["a5c2168rotation2a-h6"],"title":"Simplification","text":"We can start our simplification by first squaring the denominators and cancelling out with the coefficients of the squared terms. What is the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\left(x\'-y\'\\\\right) \\\\left(x\'-y\'\\\\right)-\\\\frac{\\\\left(x\'-y\'\\\\right) \\\\left(x\'+y\'\\\\right)}{2}+\\\\left(x\'+y\'\\\\right) \\\\left(x\'+y\'\\\\right)-30=0$$","$$\\\\left(x\'+y\'\\\\right) \\\\left(x\'-y\'\\\\right)-\\\\frac{\\\\left(x\'-y\'\\\\right) \\\\left(x\'+y\'\\\\right)}{2}+\\\\frac{\\\\left(x\'+y\'\\\\right) \\\\left(x\'+y\'\\\\right)}{2}-30=0$$"]},{"id":"a5c2168rotation2a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${x\'}^2-2x\' y\'+{y\'}^2-\\\\frac{{x\'}^2-{y\'}^2}{2}+{x\'}^2+2x\' y\'+{y\'}^2-30=0$$"],"dependencies":["a5c2168rotation2a-h7"],"title":"Simplification","text":"With the equation that we obtained, we will first start by expanding out the binomials using the FOIL method. What is the equation after doing so?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$${x\'}^2-2x\' y\'+{y\'}^2-\\\\frac{{x\'}^2-{y\'}^2}{2}+{x\'}^2+2x\' y\'+{y\'}^2-30=0$$","$${x\'}^2+2x\' y\'+{y\'}^2-\\\\frac{{x\'}^2-{y\'}^2}{2}+{x\'}^2+2x\' y\'+{y\'}^2-30=0$$"]},{"id":"a5c2168rotation2a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2{x\'}^2+2{y\'}^2-\\\\frac{{x\'}^2-{y\'}^2}{2}-30=0$$"],"dependencies":["a5c2168rotation2a-h8"],"title":"Simplification","text":"Next we will combine the like terms. What is the current equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2{x\'}^2+2{y\'}^2-\\\\frac{{x\'}^2-{y\'}^2}{2}-30=0$$","$$2{x\'}^2+2{y\'}^2+\\\\frac{{x\'}^2-{y\'}^2}{2}-30=0$$"]},{"id":"a5c2168rotation2a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3{x\'}^2+5{y\'}^2=60$$"],"dependencies":["a5c2168rotation2a-h9"],"title":"Simplification","text":"Multiply all the terms by $$2$$ and shift the constant term to the RHS. What is the equation now?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$3{x\'}^2+5{y\'}^2=60$$","$$3{x\'}^2+3{y\'}^2=60$$"]},{"id":"a5c2168rotation2a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{{x\'}^2}{20}+\\\\frac{{y\'}^2}{12}=1$$"],"dependencies":["a5c2168rotation2a-h10"],"title":"Simplification","text":"Set the RHS to $$1$$ by dividing by $$60$$. We can write the equation with x\' and y\' in the standard form by dividing each term by the constant in the denominator. What is the equation now? The equation is an ellipse.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{{x\'}^2}{20}+\\\\frac{{y\'}^2}{12}=1$$","$$\\\\frac{{x\'}^2}{12}+\\\\frac{{y\'}^2}{20}=1$$","$$\\\\frac{{x\'}^2}{20}-\\\\frac{{y\'}^2}{12}=1$$"]}]}}]},{"id":"a5c2168rotation3","title":"Rewriting an Equation with respect to the x\' and y\' axes without the x\'y\' term","body":"Rewrite the following equation in the x\'y\' system without an x\'y\' term.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Rotation of Axes","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c2168rotation3a","stepAnswer":["$$\\\\frac{{x\'}^2}{4}+\\\\frac{{y\'}^2}{1}=1$$"],"problemType":"MultipleChoice","stepTitle":"$$8x^2-12x y+17y^2=20$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{{x\'}^2}{4}+\\\\frac{{y\'}^2}{1}=1$$","choices":["$$\\\\frac{{x\'}^2}{4}+\\\\frac{{y\'}^2}{1}=1$$","$$\\\\frac{{x\'}^2}{1}+\\\\frac{{y\'}^2}{4}=1$$"],"hints":{"DefaultPathway":[{"id":"a5c2168rotation3a-h1","type":"hint","dependencies":[],"title":"$$\\\\operatorname{Cot}\\\\left(2\\\\theta\\\\right)$$","text":"To transform the equation of a conic given in the form $$A x^2+B x y+C y^2+D x+E y+F=0$$ into standard form by rotating the axes, we will rewrite the general form as an equation in the x\' and y\' coordinate system without the x\'y\' term, by rotating the axes by a measure of $$\\\\theta$$ that satisfies $$\\\\operatorname{cot}\\\\left(2\\\\theta\\\\right)=\\\\frac{A-C}{B}$$.\\\\nIf $$\\\\operatorname{cot}\\\\left(2\\\\theta\\\\right)>0$$, then $$2\\\\theta$$ is in the first quadrant, and $$\\\\theta$$ is between $$(0^o,{45}^o)$$.\\\\nIf $$\\\\operatorname{cot}\\\\left(2\\\\theta\\\\right)<0$$, then $$2\\\\theta$$ is in the second quadrant, and $$\\\\theta$$ is between $$({45}^o,{90}^o)$$.\\\\nIf $$A=C$$, then $$\\\\theta={45}^o$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation3a-h2","type":"hint","dependencies":["a5c2168rotation3a-h1"],"title":"Equations of Rotation","text":"If a point (x,y) on the Cartesian plane is represented on a new coordinate plane where the axes of rotation are formed by rotating an angle $$\\\\theta$$ from the positive x-axis, then the coordinates of the point with respect to the new axes are (x\u2032,y\u2032). We can use the following equations of rotation to define the relationship between (x,y) and (x\u2032,y\u2032):\\\\nx=x\u2032*cos(\\\\theta)-y\u2032*sin(\\\\theta)\\\\nand\\\\ny=x\u2032*sin(\\\\theta)+y\u2032*cos(\\\\theta)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation3a-h3","type":"hint","dependencies":["a5c2168rotation3a-h2"],"title":"Finding $$\\\\operatorname{Cot}\\\\left(2\\\\theta\\\\right)$$","text":"We want to find $$sin\\\\left(\\\\theta\\\\right)$$ and $$cos\\\\left(\\\\theta\\\\right)$$ so that we can use them for substitution later on. To do so, we would start by finding $$\\\\operatorname{cot}\\\\left(2\\\\theta\\\\right)=\\\\frac{A-C}{B}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a5c2168rotation3a-h3"],"title":"Finding $$\\\\operatorname{Cot}\\\\left(2\\\\theta\\\\right)$$","text":"What is A?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-12$$"],"dependencies":["a5c2168rotation3a-h4"],"title":"Finding $$\\\\operatorname{Cot}\\\\left(2\\\\theta\\\\right)$$","text":"What is B?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation3a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$17$$"],"dependencies":["a5c2168rotation3a-h5"],"title":"Finding $$\\\\operatorname{Cot}\\\\left(2\\\\theta\\\\right)$$","text":"What is C?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation3a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{4}$$"],"dependencies":["a5c2168rotation3a-h6"],"title":"Finding $$\\\\operatorname{Cot}\\\\left(2\\\\theta\\\\right)$$","text":"What is $$\\\\operatorname{cot}\\\\left(2\\\\theta\\\\right)=\\\\frac{A-C}{B}$$?\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation3a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a5c2168rotation3a-h7"],"title":"Hypotenuse","text":"For the angle $$2\\\\theta$$, the adjacent is of length $$3$$ unit and the opposite side is of length $$4$$ unit. What is the length of the hypotenuse? You can use the Pythagorean Theorem to calculate.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation3a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{\\\\sqrt{5}}$$"],"dependencies":["a5c2168rotation3a-h8"],"title":"Finding $$\\\\operatorname{Sin}\\\\left(\\\\theta\\\\right)$$","text":"Since we have the angle $$2\\\\theta$$, we will use the trigonometry identity, sin(\\\\theta)=sqrt((1-cos(2*\\\\theta)/2). We can calculate $$cos\\\\left(2\\\\theta\\\\right)$$ from the diagram as $$\\\\frac{adjacent}{hypotenuse}$$ for an angle $$2\\\\theta$$. What is $$sin\\\\left(\\\\theta\\\\right)$$?\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation3a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{\\\\sqrt{5}}$$"],"dependencies":["a5c2168rotation3a-h9"],"title":"Finding $$\\\\operatorname{Cos}\\\\left(\\\\theta\\\\right)$$","text":"Since we have the angle $$2\\\\theta$$, we will use the trigonometry identity, cos(\\\\theta)=sqrt((1+cos(2*\\\\theta)/2). We can calculate $$cos\\\\left(2\\\\theta\\\\right)$$ from the diagram as $$\\\\frac{adjacent}{hypotenuse}$$ for an angle $$2\\\\theta$$. What is $$cos\\\\left(\\\\theta\\\\right)$$?\\\\n##figure3.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation3a-h11","type":"hint","dependencies":["a5c2168rotation3a-h10"],"title":"Substitution","text":"We want to substitute x=x\u2032*cos(\\\\theta)-y\u2032*sin(\\\\theta) and y=x\u2032*sin(\\\\theta)+y\u2032*cos(\\\\theta) into the equation so that we can manipulate the equation into the new representation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation3a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2x\'-y\'}{\\\\sqrt{5}}$$"],"dependencies":["a5c2168rotation3a-h11"],"title":"Substitution","text":"Simplifying the equation of rotation for the $$x$$ term where x=x\u2032*cos(\\\\theta)-y\u2032*sin(\\\\theta), what is $$x$$ equals to after substituting in $$sin\\\\left(\\\\theta\\\\right)$$ and $$cos\\\\left(\\\\theta\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation3a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{x\'+2y\'}{\\\\sqrt{5}}$$"],"dependencies":["a5c2168rotation3a-h12"],"title":"Substitution","text":"Simplifying the equation of rotation for the $$y$$ term where y=x\u2032*sin(\\\\theta)+y\u2032*cos(\\\\theta), what is $$y$$ equals to after substituting in $$sin\\\\left(\\\\theta\\\\right)$$ and $$cos\\\\left(\\\\theta\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation3a-h14","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$8{\\\\left(\\\\frac{2x\'-y\'}{\\\\sqrt{5}}\\\\right)}^2-12\\\\frac{2x\'-y\'}{\\\\sqrt{5}} \\\\frac{x\'+2y\'}{\\\\sqrt{5}}+17{\\\\left(\\\\frac{x\'+2y\'}{\\\\sqrt{5}}\\\\right)}^2=20$$"],"dependencies":["a5c2168rotation3a-h13"],"title":"Substitution","text":"Substituting $$x=\\\\frac{2x\'-y\'}{\\\\sqrt{5}}$$ and $$y=\\\\frac{x\'+2y\'}{\\\\sqrt{5}}$$ into $$8x^2-12x y+17y^2=20$$, what is the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$8{\\\\left(\\\\frac{2x\'-y\'}{\\\\sqrt{5}}\\\\right)}^2-12\\\\frac{2x\'-y\'}{\\\\sqrt{5}} \\\\frac{x\'+2y\'}{\\\\sqrt{5}}+17{\\\\left(\\\\frac{x\'+2y\'}{\\\\sqrt{5}}\\\\right)}^2=20$$","$$8{\\\\left(\\\\frac{x\'+2y\'}{\\\\sqrt{5}}\\\\right)}^2-12\\\\frac{2x\'-y\'}{\\\\sqrt{5}} \\\\frac{x\'+2y\'}{\\\\sqrt{5}}+17{\\\\left(\\\\frac{2x\'-y\'}{\\\\sqrt{5}}\\\\right)}^2=20$$"]},{"id":"a5c2168rotation3a-h15","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$8\\\\frac{\\\\left(2x\'-y\'\\\\right) \\\\left(2x\'-y\'\\\\right)}{5}-12\\\\frac{\\\\left(2x\'-y\'\\\\right) \\\\left(x\'+2y\'\\\\right)}{5}+17\\\\frac{\\\\left(x\'+2y\'\\\\right) \\\\left(x\'+2y\'\\\\right)}{5}=20$$"],"dependencies":["a5c2168rotation3a-h14"],"title":"Simplification","text":"We can start our simplification by first squaring the denominators,. What is the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$8\\\\frac{\\\\left(2x\'-y\'\\\\right) \\\\left(2x\'-y\'\\\\right)}{5}-12\\\\frac{\\\\left(2x\'-y\'\\\\right) \\\\left(x\'+2y\'\\\\right)}{5}+17\\\\frac{\\\\left(x\'+2y\'\\\\right) \\\\left(x\'+2y\'\\\\right)}{5}=20$$","$$8\\\\frac{\\\\left(x\'+2y\'\\\\right) \\\\left(x\'+2y\'\\\\right)}{5}-12\\\\frac{\\\\left(2x\'-y\'\\\\right) \\\\left(x\'+2y\'\\\\right)}{5}+17\\\\frac{\\\\left(2x\'-y\'\\\\right) \\\\left(2x\'-y\'\\\\right)}{5}=20$$"]},{"id":"a5c2168rotation3a-h16","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$8\\\\left(4{x\'}^2-4x\' y\'+{y\'}^2\\\\right)-12\\\\left(2{x\'}^2+3x\' y\'-2{y\'}^2\\\\right)+17\\\\left({x\'}^2+4x\' y\'+4{y\'}^2\\\\right)=100$$"],"dependencies":["a5c2168rotation3a-h15"],"title":"Simplification","text":"Next, we will multiply by $$5$$ on both sides so that we can remove the denominators and expand out the binomials using the FOIL method. What is the equation now?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$8\\\\left(4{x\'}^2-4x\' y\'+{y\'}^2\\\\right)-12\\\\left(2{x\'}^2+3x\' y\'-2{y\'}^2\\\\right)+17\\\\left({x\'}^2+4x\' y\'+4{y\'}^2\\\\right)=100$$","$$8\\\\left(4{x\'}^2+4x\' y\'+{y\'}^2\\\\right)-12\\\\left(2{x\'}^2-3x\' y\'-2{y\'}^2\\\\right)+17\\\\left({x\'}^2-4x\' y\'+4{y\'}^2\\\\right)=100$$"]},{"id":"a5c2168rotation3a-h17","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$25{x\'}^2+100{y\'}^2=100$$"],"dependencies":["a5c2168rotation3a-h16"],"title":"Simplification","text":"Next we will distribute the scalar multiples and combine like terms. What is the current equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$25{x\'}^2+100{y\'}^2=100$$","$$100{x\'}^2+25{y\'}^2=100$$"]},{"id":"a5c2168rotation3a-h18","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{{x\'}^2}{4}+\\\\frac{{y\'}^2}{1}=1$$"],"dependencies":["a5c2168rotation3a-h17"],"title":"Simplification","text":"Set the RHS to $$1$$ by dividing by $$100$$. We can write the equation with x\' and y\' in the standard form by dividing each term by the constant in the denominator. What is the equation now? The equation is an ellipse.\\\\n##figure4.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{{x\'}^2}{4}+\\\\frac{{y\'}^2}{1}=1$$","$$\\\\frac{{x\'}^2}{1}+\\\\frac{{y\'}^2}{4}=1$$"]}]}}]},{"id":"a5c2168rotation5","title":"Identifying the Conic without Rotating Axes","body":"Identify the conic for each of the following without rotating axes.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Rotation of Axes","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c2168rotation5a","stepAnswer":["Ellipse"],"problemType":"MultipleChoice","stepTitle":"$$5x^2+2\\\\sqrt{3} x y+2y^2-5=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Circle","Parabola","Ellipse","Hyperbola"],"hints":{"DefaultPathway":[{"id":"a5c2168rotation5a-h1","type":"hint","dependencies":[],"title":"Identifying Conics with Discriminant","text":"If the equation $$A x^2+B x y+C y^2+D x+E y+F=0$$ is transformed by rotating axes into the equation $$A\' {x\'}^2+B\' x\' y\'+C\' {y\'}^2+D\' x\'+E\' y\'+F\'=0$$, then B**2-4*A*C=B\u2032**2-4*A\u2032*C\u2032.\\\\n\\\\nThe equation $$A x^2+B x y+C y^2+D x+E y+F=0$$ is an ellipse, a parabola, or a hyperbola, or a degenerate case of one of these.\\\\nIf the discriminant, $$B^2-4A C$$, is\\\\n<0, the conic section is an ellipse\\\\n$$=0$$, the conic section is a parabola\\\\n>0, the conic section is a hyperbola","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation5a-h2","type":"hint","dependencies":["a5c2168rotation5a-h1"],"title":"Finding the Discriminant","text":"Referring to the general form $$A x^2+B x y+C y^2+D x+E y+F=0$$, we can identify the type of conic by comparing finding the discriminant.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a5c2168rotation5a-h2"],"title":"Finding the Discriminant","text":"What is the coefficient of $$x^2$$, A, equals to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2\\\\sqrt{3}$$"],"dependencies":["a5c2168rotation5a-h3"],"title":"Finding the Discriminant","text":"What is the coefficient of $$x y$$, B, equals to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a5c2168rotation5a-h4"],"title":"Finding the Discriminant","text":"What is the coefficient of $$y^2$$, C, equals to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation5a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-28$$"],"dependencies":["a5c2168rotation5a-h5"],"title":"Finding the Discriminant","text":"What is the discriminant, $$B^2-4A C$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation5a-h7","type":"hint","dependencies":["a5c2168rotation5a-h6"],"title":"Discriminant","text":"Since the discriminant is less than zero, what does this tell us about the conics based on the previous hint.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a5c2168rotation5b","stepAnswer":["Ellipse"],"problemType":"MultipleChoice","stepTitle":"$$5x^2+2\\\\sqrt{3} x y+12y^2-5=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Circle","Parabola","Ellipse","Hyperbola"],"hints":{"DefaultPathway":[{"id":"a5c2168rotation5b-h1","type":"hint","dependencies":[],"title":"Identifying Conics with Discriminant","text":"If the equation $$A x^2+B x y+C y^2+D x+E y+F=0$$ is transformed by rotating axes into the equation $$A\' {x\'}^2+B\' x\' y\'+C\' {y\'}^2+D\' x\'+E\' y\'+F\'=0$$, then B**2-4*A*C=B\u2032**2-4*A\u2032*C\u2032.\\\\n\\\\nThe equation $$A x^2+B x y+C y^2+D x+E y+F=0$$ is an ellipse, a parabola, or a hyperbola, or a degenerate case of one of these.\\\\nIf the discriminant, $$B^2-4A C$$, is\\\\n<0, the conic section is an ellipse\\\\n$$=0$$, the conic section is a parabola\\\\n>0, the conic section is a hyperbola","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation5b-h2","type":"hint","dependencies":["a5c2168rotation5b-h1"],"title":"Finding the Discriminant","text":"Referring to the general form $$A x^2+B x y+C y^2+D x+E y+F=0$$, we can identify the type of conic by comparing finding the discriminant.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation5b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a5c2168rotation5b-h2"],"title":"Finding the Discriminant","text":"What is the coefficient of $$x^2$$, A, equals to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation5b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2\\\\sqrt{3}$$"],"dependencies":["a5c2168rotation5b-h3"],"title":"Finding the Discriminant","text":"What is the coefficient of $$x y$$, B, equals to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation5b-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a5c2168rotation5b-h4"],"title":"Finding the Discriminant","text":"What is the coefficient of $$y^2$$, C, equals to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation5b-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-228$$"],"dependencies":["a5c2168rotation5b-h5"],"title":"Finding the Discriminant","text":"What is the discriminant, $$B^2-4A C$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation5b-h7","type":"hint","dependencies":["a5c2168rotation5b-h6"],"title":"Discriminant","text":"Since the discriminant is less than zero, what does this tell us about the conics based on the previous hint.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c2168rotation6","title":"Identifying the Conic without Rotating Axes","body":"Identify the conic for each of the following without rotating axes.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Rotation of Axes","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c2168rotation6a","stepAnswer":["Hyperbola"],"problemType":"MultipleChoice","stepTitle":"$$x^2-9x y+3y^2-12=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Circle","Parabola","Ellipse","Hyperbola"],"hints":{"DefaultPathway":[{"id":"a5c2168rotation6a-h1","type":"hint","dependencies":[],"title":"Identifying Conics with Discriminant","text":"If the equation $$A x^2+B x y+C y^2+D x+E y+F=0$$ is transformed by rotating axes into the equation $$A\' {x\'}^2+B\' x\' y\'+C\' {y\'}^2+D\' x\'+E\' y\'+F\'=0$$, then B**2-4*A*C=B\u2032**2-4*A\u2032*C\u2032.\\\\n\\\\nThe equation $$A x^2+B x y+C y^2+D x+E y+F=0$$ is an ellipse, a parabola, or a hyperbola, or a degenerate case of one of these.\\\\nIf the discriminant, $$B^2-4A C$$, is\\\\n<0, the conic section is an ellipse\\\\n$$=0$$, the conic section is a 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4.0>"},{"id":"a5c2168rotation6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":["a5c2168rotation6a-h3"],"title":"Finding the Discriminant","text":"What is the coefficient of $$x y$$, B, equals to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a5c2168rotation6a-h4"],"title":"Finding the Discriminant","text":"What is the coefficient of $$y^2$$, C, equals to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation6a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$69$$"],"dependencies":["a5c2168rotation6a-h5"],"title":"Finding the Discriminant","text":"What is the discriminant, $$B^2-4A C$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation6a-h7","type":"hint","dependencies":["a5c2168rotation6a-h6"],"title":"Discriminant","text":"Since the discriminant is greater than zero, what does this tell us about the conics based on the previous hint.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a5c2168rotation6b","stepAnswer":["Ellipse"],"problemType":"MultipleChoice","stepTitle":"$$10x^2-9x y+4y^2-4=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Circle","Parabola","Ellipse","Hyperbola"],"hints":{"DefaultPathway":[{"id":"a5c2168rotation6b-h1","type":"hint","dependencies":[],"title":"Identifying Conics with Discriminant","text":"If the equation $$A x^2+B x y+C y^2+D x+E y+F=0$$ is transformed by rotating axes into the equation $$A\' {x\'}^2+B\' x\' y\'+C\' {y\'}^2+D\' x\'+E\' y\'+F\'=0$$, then B**2-4*A*C=B\u2032**2-4*A\u2032*C\u2032.\\\\n\\\\nThe equation $$A x^2+B x y+C y^2+D x+E y+F=0$$ is an ellipse, a parabola, or a hyperbola, or a degenerate case of one of these.\\\\nIf the discriminant, $$B^2-4A C$$, is\\\\n<0, the conic section is an ellipse\\\\n$$=0$$, the conic section is a parabola\\\\n>0, the conic section is a hyperbola","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation6b-h2","type":"hint","dependencies":["a5c2168rotation6b-h1"],"title":"Finding the Discriminant","text":"Referring to the general form $$A x^2+B x y+C y^2+D x+E y+F=0$$, we can identify the type of conic by comparing finding the discriminant.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation6b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a5c2168rotation6b-h2"],"title":"Finding the Discriminant","text":"What is the coefficient of $$x^2$$, A, equals to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation6b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":["a5c2168rotation6b-h3"],"title":"Finding the Discriminant","text":"What is the coefficient of $$x y$$, B, equals to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation6b-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a5c2168rotation6b-h4"],"title":"Finding the Discriminant","text":"What is the coefficient of $$y^2$$, C, equals to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation6b-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-79$$"],"dependencies":["a5c2168rotation6b-h5"],"title":"Finding the Discriminant","text":"What is the discriminant, $$B^2-4A C$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation6b-h7","type":"hint","dependencies":["a5c2168rotation6b-h6"],"title":"Discriminant","text":"Since the discriminant is less than zero, what does this tell us about the conics based on the previous hint.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c2168rotation7","title":"Identify","body":"Determine which conic section is represented based on the given equation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Rotation of Axes","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c2168rotation7a","stepAnswer":["Ellipse"],"problemType":"MultipleChoice","stepTitle":"$$9x^2+4y^2+72x+36y-500=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Ellipse","Circle","Hyperbola","Parabola"],"hints":{"DefaultPathway":[{"id":"a5c2168rotation7a-h1","type":"hint","dependencies":[],"title":"Rewrite","text":"Rewrite the equation in general form $${Ax}^2+Bxy+{Cy}^2+Dx+Ey+F=0$$, if it is not already.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation7a-h2","type":"hint","dependencies":["a5c2168rotation7a-h1"],"title":"Use A and C","text":"Use the definitions of A and C to find the type of equation this conic is. If A and C are nonzero, have the same sign, and are not equal to each other, then the graph is an ellipse. If A and C are equal and nonzero and have the same sign, then the graph is a circle. If A and C are nonzero and have opposite signs, then the graph is a hyperbola. If either A or C is zero, then the graph is a parabola.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation7a-h3","type":"hint","dependencies":["a5c2168rotation7a-h2"],"title":"Answer","text":"As A and C are nonzero, have the same sign, and are not equal to each other, the graph is an ellipse.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c2168rotation8","title":"Identify","body":"Determine which conic section is represented based on the given equation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Rotation of Axes","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c2168rotation8a","stepAnswer":["Parabola"],"problemType":"MultipleChoice","stepTitle":"$$x^2-10x+4y-10=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Ellipse","Circle","Hyperbola","Parabola"],"hints":{"DefaultPathway":[{"id":"a5c2168rotation8a-h1","type":"hint","dependencies":[],"title":"Rewrite","text":"Rewrite the equation in general form $${Ax}^2+Bxy+{Cy}^2+Dx+Ey+F=0$$, if it is not already.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation8a-h2","type":"hint","dependencies":["a5c2168rotation8a-h1"],"title":"Use A and C","text":"Use the definitions of A and C to find the type of equation this conic is. If A and C are nonzero, have the same sign, and are not equal to each other, then the graph is an ellipse. If A and C are equal and nonzero and have the same sign, then the graph is a circle. If A and C are nonzero and have opposite signs, then the graph is a hyperbola. If either A or C is zero, then the graph is a parabola.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation8a-h3","type":"hint","dependencies":["a5c2168rotation8a-h2"],"title":"Answer","text":"As B is zero, the equation is a parabola.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c2168rotation9","title":"Identify","body":"Determine which conic section is represented based on the given equation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Rotation of Axes","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c2168rotation9a","stepAnswer":["Hyperbola"],"problemType":"MultipleChoice","stepTitle":"$$2x^2-2y^2+4x-6y-2=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Ellipse","Circle","Hyperbola","Parabola"],"hints":{"DefaultPathway":[{"id":"a5c2168rotation9a-h1","type":"hint","dependencies":[],"title":"Rewrite","text":"Rewrite the equation in general form $${Ax}^2+Bxy+{Cy}^2+Dx+Ey+F=0$$, if it is not already.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation9a-h2","type":"hint","dependencies":["a5c2168rotation9a-h1"],"title":"Use A and C","text":"Use the definitions of A and C to find the type of equation this conic is. If A and C are nonzero, have the same sign, and are not equal to each other, then the graph is an ellipse. If A and C are equal and nonzero and have the same sign, then the graph is a circle. If A and C are nonzero and have opposite signs, then the graph is a hyperbola. If either A or C is zero, then the graph is a parabola.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c2168rotation9a-h3","type":"hint","dependencies":["a5c2168rotation9a-h2"],"title":"Answer","text":"As A and B have opposite signs, the equation is a hyperbola.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c70bfsquareroots1","title":"Add and Subtract Square Roots","body":"Simplify the following exercise:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Add and Subtract Square Roots","courseName":"OpenStax: Elementary 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4.0>"}]}}]},{"id":"a5c70bfsquareroots10","title":"Add and Subtract Square Roots","body":"Simplify the following exercise:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Add and Subtract Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5c70bfsquareroots10a","stepAnswer":["$$-5\\\\sqrt{b}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{b}-6\\\\sqrt{b}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-5\\\\sqrt{b}$$","hints":{"DefaultPathway":[{"id":"a5c70bfsquareroots10a-h1","type":"hint","dependencies":[],"title":"$$\\\\frac{Adding}{Subtracting}$$ Coefficients","text":"Since the radicals are like, we subtract the coefficients.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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We leave the expression as is.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c70bfsquareroots13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8\\\\sqrt{a}-2\\\\sqrt{b}$$"],"dependencies":["a5c70bfsquareroots13a-h1"],"title":"Simpligying Expression","text":"What is the simplified form of the problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c70bfsquareroots14","title":"Add and Subtract Square Roots","body":"Simplify the following exercise:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Add and Subtract Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5c70bfsquareroots14a","stepAnswer":["$$5\\\\sqrt{c}-3\\\\sqrt{d}$$"],"problemType":"TextBox","stepTitle":"$$5\\\\sqrt{c}-3\\\\sqrt{d}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5\\\\sqrt{c}-3\\\\sqrt{d}$$","hints":{"DefaultPathway":[{"id":"a5c70bfsquareroots14a-h1","type":"hint","dependencies":[],"title":"$$\\\\frac{Adding}{Subtracting}$$ Coefficients","text":"Since the radicals are not like, we cannot subtract them. We leave the expression as is.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c70bfsquareroots14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5\\\\sqrt{c}-3\\\\sqrt{d}$$"],"dependencies":["a5c70bfsquareroots14a-h1"],"title":"Simpligying Expression","text":"What is the simplified form of the problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c70bfsquareroots15","title":"Add and Subtract Square Roots","body":"Simplify the following exercise:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Add and Subtract Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5c70bfsquareroots15a","stepAnswer":["$$5\\\\sqrt{m}+\\\\sqrt{n}$$"],"problemType":"TextBox","stepTitle":"$$5\\\\sqrt{m}+\\\\sqrt{n}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5\\\\sqrt{m}+\\\\sqrt{n}$$","hints":{"DefaultPathway":[{"id":"a5c70bfsquareroots15a-h1","type":"hint","dependencies":[],"title":"$$\\\\frac{Adding}{Subtracting}$$ Coefficients","text":"Since the radicals are not like, we cannot subtract them. We leave the expression as is.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c70bfsquareroots15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5\\\\sqrt{m}+\\\\sqrt{n}$$"],"dependencies":["a5c70bfsquareroots15a-h1"],"title":"Simpligying Expression","text":"What is the simplified form of the problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c70bfsquareroots16","title":"Adding and Subtracting Like Square Roots","body":"Simplify the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Add and Subtract Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5c70bfsquareroots16a","stepAnswer":["$$-5\\\\sqrt{2}$$"],"problemType":"TextBox","stepTitle":"$$2\\\\sqrt{2}-7\\\\sqrt{2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-5\\\\sqrt{2}$$","hints":{"DefaultPathway":[{"id":"a5c70bfsquareroots16a-h1","type":"hint","dependencies":[],"title":"Operations with Like Radicals","text":"Since the radicals are like, we subtract the coefficients. For example, $$4\\\\sqrt{3}-5\\\\sqrt{3}$$ $$=$$ $$\\\\left(4-5\\\\right) \\\\sqrt{3}$$ $$=$$ $$-\\\\sqrt{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c70bfsquareroots17","title":"Adding and Subtracting Like Square Roots","body":"Simplify the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Add and Subtract Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5c70bfsquareroots17a","stepAnswer":["$$-\\\\sqrt{2}$$"],"problemType":"TextBox","stepTitle":"$$8\\\\sqrt{2}-9\\\\sqrt{2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-\\\\sqrt{2}$$","hints":{"DefaultPathway":[{"id":"a5c70bfsquareroots17a-h1","type":"hint","dependencies":[],"title":"Operations with Like Radicals","text":"Since the radicals are like, we subtract the coefficients. For example, $$4\\\\sqrt{3}-5\\\\sqrt{3}$$ $$=$$ $$\\\\left(4-5\\\\right) \\\\sqrt{3}$$ $$=$$ $$-\\\\sqrt{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c70bfsquareroots18","title":"Adding and Subtracting Like Square Roots","body":"Simplify the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Add and Subtract Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5c70bfsquareroots18a","stepAnswer":["$$-4\\\\sqrt{3}$$"],"problemType":"TextBox","stepTitle":"$$5\\\\sqrt{3}-9\\\\sqrt{3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-4\\\\sqrt{3}$$","hints":{"DefaultPathway":[{"id":"a5c70bfsquareroots18a-h1","type":"hint","dependencies":[],"title":"Operations with Like Radicals","text":"Since the radicals are like, we subtract the coefficients. For example, $$4\\\\sqrt{3}-5\\\\sqrt{3}$$ $$=$$ $$\\\\left(4-5\\\\right) \\\\sqrt{3}$$ $$=$$ $$-\\\\sqrt{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c70bfsquareroots19","title":"Adding and Subtracting Like Square Roots","body":"Simplify the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Add and Subtract Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5c70bfsquareroots19a","stepAnswer":["$$7\\\\sqrt{y}$$"],"problemType":"TextBox","stepTitle":"$$3\\\\sqrt{y}+4\\\\sqrt{y}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7\\\\sqrt{y}$$","hints":{"DefaultPathway":[{"id":"a5c70bfsquareroots19a-h1","type":"hint","dependencies":[],"title":"Operations with Like Radicals","text":"Since the radicals are like, we add the coefficients. For example, $$4\\\\sqrt{3}+5\\\\sqrt{3}$$ $$=$$ $$\\\\left(4+5\\\\right) \\\\sqrt{3}$$ $$=$$ $$9\\\\sqrt{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c70bfsquareroots2","title":"Add and Subtract Square Roots","body":"Simplify the following exercise:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Add and Subtract Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5c70bfsquareroots2a","stepAnswer":["$$4\\\\sqrt{2}$$"],"problemType":"TextBox","stepTitle":"$$7\\\\sqrt{2}-3\\\\sqrt{2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4\\\\sqrt{2}$$","hints":{"DefaultPathway":[{"id":"a5c70bfsquareroots2a-h1","type":"hint","dependencies":[],"title":"$$\\\\frac{Adding}{Subtracting}$$ Coefficients","text":"Since the radicals are like, we subtract the coefficients.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c70bfsquareroots2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4\\\\sqrt{2}$$"],"dependencies":["a5c70bfsquareroots2a-h1"],"title":"Simpligying Expression","text":"What is the simplified form of the problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c70bfsquareroots20","title":"Adding and Subtracting Like Square Roots","body":"Simplify the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Add and Subtract Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5c70bfsquareroots20a","stepAnswer":["$$9\\\\sqrt{x}$$"],"problemType":"TextBox","stepTitle":"$$2\\\\sqrt{x}+7\\\\sqrt{x}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9\\\\sqrt{x}$$","hints":{"DefaultPathway":[{"id":"a5c70bfsquareroots20a-h1","type":"hint","dependencies":[],"title":"Operations with Like Radicals","text":"Since the radicals are like, we add the coefficients. For example, $$4\\\\sqrt{3}+5\\\\sqrt{3}$$ $$=$$ $$\\\\left(4+5\\\\right) \\\\sqrt{3}$$ $$=$$ $$9\\\\sqrt{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c70bfsquareroots21","title":"Adding and Subtracting Like Square Roots","body":"Simplify the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Add and Subtract Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5c70bfsquareroots21a","stepAnswer":["$$8\\\\sqrt{u}$$"],"problemType":"TextBox","stepTitle":"$$5\\\\sqrt{u}+3\\\\sqrt{u}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8\\\\sqrt{u}$$","hints":{"DefaultPathway":[{"id":"a5c70bfsquareroots21a-h1","type":"hint","dependencies":[],"title":"Operations with Like Radicals","text":"Since the radicals are like, we add the coefficients. For example, $$4\\\\sqrt{3}+5\\\\sqrt{3}$$ $$=$$ $$\\\\left(4+5\\\\right) \\\\sqrt{3}$$ $$=$$ $$9\\\\sqrt{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c70bfsquareroots22","title":"Adding and Subtracting Like Square Roots","body":"Simplify the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Add and Subtract Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5c70bfsquareroots22a","stepAnswer":["$$4\\\\sqrt{x}-2\\\\sqrt{y}$$"],"problemType":"TextBox","stepTitle":"$$4\\\\sqrt{x}-2\\\\sqrt{y}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4\\\\sqrt{x}-2\\\\sqrt{y}$$","hints":{"DefaultPathway":[{"id":"a5c70bfsquareroots22a-h1","type":"hint","dependencies":[],"title":"Operations without Like Radicals","text":"Since the radicals are not like, we cannot subtract them. We leave the expression as is.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c70bfsquareroots23","title":"Adding and Subtracting Like Square Roots","body":"Simplify the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Add and Subtract Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5c70bfsquareroots23a","stepAnswer":["$$7\\\\sqrt{p}-6\\\\sqrt{q}$$"],"problemType":"TextBox","stepTitle":"$$7\\\\sqrt{p}-6\\\\sqrt{q}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7\\\\sqrt{p}-6\\\\sqrt{q}$$","hints":{"DefaultPathway":[{"id":"a5c70bfsquareroots23a-h1","type":"hint","dependencies":[],"title":"Operations without Like Radicals","text":"Since the radicals are not like, we cannot subtract them. We leave the expression as is.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c70bfsquareroots24","title":"Adding and Subtracting Like Square Roots","body":"Simplify the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Add and Subtract Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5c70bfsquareroots24a","stepAnswer":["$$6\\\\sqrt{a}-3\\\\sqrt{b}$$"],"problemType":"TextBox","stepTitle":"$$6\\\\sqrt{a}-3\\\\sqrt{b}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6\\\\sqrt{a}-3\\\\sqrt{b}$$","hints":{"DefaultPathway":[{"id":"a5c70bfsquareroots24a-h1","type":"hint","dependencies":[],"title":"Operations without Like Radicals","text":"Since the radicals are not like, we cannot subtract them. We leave the expression as is.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c70bfsquareroots25","title":"Adding and Subtracting Like Square Roots","body":"Simplify the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Add and Subtract Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5c70bfsquareroots25a","stepAnswer":["$$11\\\\sqrt{13}$$"],"problemType":"TextBox","stepTitle":"$$5\\\\sqrt{13}+4\\\\sqrt{13}+2\\\\sqrt{13}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$11\\\\sqrt{13}$$","hints":{"DefaultPathway":[{"id":"a5c70bfsquareroots25a-h1","type":"hint","dependencies":[],"title":"Operations with Like Radicals","text":"Since the radicals are like, we add the coefficients. For example, $$4\\\\sqrt{3}+5\\\\sqrt{3}$$ $$=$$ $$\\\\left(4+5\\\\right) \\\\sqrt{3}$$ $$=$$ $$9\\\\sqrt{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c70bfsquareroots26","title":"Adding and Subtracting Like Square Roots","body":"Simplify the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Add and Subtract Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5c70bfsquareroots26a","stepAnswer":["$$9\\\\sqrt{11}$$"],"problemType":"TextBox","stepTitle":"$$4\\\\sqrt{11}+2\\\\sqrt{11}+3\\\\sqrt{11}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9\\\\sqrt{11}$$","hints":{"DefaultPathway":[{"id":"a5c70bfsquareroots26a-h1","type":"hint","dependencies":[],"title":"Operations with Like Radicals","text":"Since the radicals are like, we add the coefficients. For example, $$4\\\\sqrt{3}+5\\\\sqrt{3}$$ $$=$$ $$\\\\left(4+5\\\\right) \\\\sqrt{3}$$ $$=$$ $$9\\\\sqrt{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c70bfsquareroots27","title":"Adding and Subtracting Like Square Roots","body":"Simplify the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Add and Subtract Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5c70bfsquareroots27a","stepAnswer":["$$11\\\\sqrt{10}$$"],"problemType":"TextBox","stepTitle":"$$6\\\\sqrt{10}+2\\\\sqrt{10}+3\\\\sqrt{10}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$11\\\\sqrt{10}$$","hints":{"DefaultPathway":[{"id":"a5c70bfsquareroots27a-h1","type":"hint","dependencies":[],"title":"Operations with Like Radicals","text":"Since the radicals are like, we add the coefficients. For example, $$4\\\\sqrt{3}+5\\\\sqrt{3}$$ $$=$$ $$\\\\left(4+5\\\\right) \\\\sqrt{3}$$ $$=$$ $$9\\\\sqrt{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c70bfsquareroots28","title":"Adding and Subtracting Like Square Roots","body":"Simplify the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Add and Subtract Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5c70bfsquareroots28a","stepAnswer":["$$-4\\\\sqrt{6}+3\\\\sqrt{3}$$"],"problemType":"TextBox","stepTitle":"$$2\\\\sqrt{6}-6\\\\sqrt{6}+3\\\\sqrt{3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-4\\\\sqrt{6}+3\\\\sqrt{3}$$","hints":{"DefaultPathway":[{"id":"a5c70bfsquareroots28a-h1","type":"hint","dependencies":[],"title":"Operations with Like Radicals","text":"When radicals are like, we subtract or add the coefficients depending on the operation. For example, $$4\\\\sqrt{3}+5\\\\sqrt{3}$$ $$=$$ $$\\\\left(4+5\\\\right) \\\\sqrt{3}$$ $$=$$ $$9\\\\sqrt{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c70bfsquareroots28a-h2","type":"hint","dependencies":["a5c70bfsquareroots28a-h1"],"title":"Operations without Like Radicals","text":"When radicals are not like, we cannot add or subtract them. We leave the expression as is.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c70bfsquareroots29","title":"Adding and Subtracting Like Square Roots","body":"Simplify the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Add and Subtract Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5c70bfsquareroots29a","stepAnswer":["$$\\\\sqrt{5}+2\\\\sqrt{6}$$"],"problemType":"TextBox","stepTitle":"$$5\\\\sqrt{5}-4\\\\sqrt{5}+2\\\\sqrt{6}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\sqrt{5}+2\\\\sqrt{6}$$","hints":{"DefaultPathway":[{"id":"a5c70bfsquareroots29a-h1","type":"hint","dependencies":[],"title":"Operations with Like Radicals","text":"When radicals are like, we subtract or add the coefficients depending on the operation. For example, $$4\\\\sqrt{3}+5\\\\sqrt{3}$$ $$=$$ $$\\\\left(4+5\\\\right) \\\\sqrt{3}$$ $$=$$ $$9\\\\sqrt{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c70bfsquareroots29a-h2","type":"hint","dependencies":["a5c70bfsquareroots29a-h1"],"title":"Operations without Like Radicals","text":"When radicals are not like, we cannot add or subtract them. We leave the expression as is.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c70bfsquareroots3","title":"Add and Subtract Square Roots","body":"Simplify the following exercise:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Add and Subtract Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5c70bfsquareroots3a","stepAnswer":["$$9\\\\sqrt{5}$$"],"problemType":"TextBox","stepTitle":"$$3\\\\sqrt{5}+6\\\\sqrt{5}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9\\\\sqrt{5}$$","hints":{"DefaultPathway":[{"id":"a5c70bfsquareroots3a-h1","type":"hint","dependencies":[],"title":"$$\\\\frac{Adding}{Subtracting}$$ Coefficients","text":"Since the radicals are like, we add the coefficients.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c70bfsquareroots3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9\\\\sqrt{5}$$"],"dependencies":["a5c70bfsquareroots3a-h1"],"title":"Simpligying Expression","text":"What is the simplified form of the problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c70bfsquareroots30","title":"Adding and Subtracting Like Square Roots","body":"Simplify the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Add and Subtract Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5c70bfsquareroots30a","stepAnswer":["$$-5\\\\sqrt{7}+2\\\\sqrt{5}$$"],"problemType":"TextBox","stepTitle":"$$3\\\\sqrt{7}-8\\\\sqrt{7}+2\\\\sqrt{5}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-5\\\\sqrt{7}+2\\\\sqrt{5}$$","hints":{"DefaultPathway":[{"id":"a5c70bfsquareroots30a-h1","type":"hint","dependencies":[],"title":"Operations with Like Radicals","text":"When radicals are like, we subtract or add the coefficients depending on the operation. For example, $$4\\\\sqrt{3}+5\\\\sqrt{3}$$ $$=$$ $$\\\\left(4+5\\\\right) \\\\sqrt{3}$$ $$=$$ $$9\\\\sqrt{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c70bfsquareroots30a-h2","type":"hint","dependencies":["a5c70bfsquareroots30a-h1"],"title":"Operations without Like Radicals","text":"When radicals are not like, we cannot add or subtract them. We leave the expression as is.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c70bfsquareroots4","title":"Add and Subtract Square Roots","body":"Simplify the following exercise:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Add and Subtract Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5c70bfsquareroots4a","stepAnswer":["$$12\\\\sqrt{5}$$"],"problemType":"TextBox","stepTitle":"$$4\\\\sqrt{5}+8\\\\sqrt{5}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12\\\\sqrt{5}$$","hints":{"DefaultPathway":[{"id":"a5c70bfsquareroots4a-h1","type":"hint","dependencies":[],"title":"$$\\\\frac{Adding}{Subtracting}$$ Coefficients","text":"Since the radicals are like, we add the coefficients.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c70bfsquareroots4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12\\\\sqrt{5}$$"],"dependencies":["a5c70bfsquareroots4a-h1"],"title":"Simpligying Expression","text":"What is the simplified form of the problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c70bfsquareroots5","title":"Add and Subtract Square Roots","body":"Simplify the following exercise:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Add and Subtract Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5c70bfsquareroots5a","stepAnswer":["$$-\\\\sqrt{7}$$"],"problemType":"TextBox","stepTitle":"$$9\\\\sqrt{7}-10\\\\sqrt{7}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-\\\\sqrt{7}$$","hints":{"DefaultPathway":[{"id":"a5c70bfsquareroots5a-h1","type":"hint","dependencies":[],"title":"$$\\\\frac{Adding}{Subtracting}$$ Coefficients","text":"Since the radicals are like, we subtract the coefficients.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c70bfsquareroots5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-\\\\sqrt{7}$$"],"dependencies":["a5c70bfsquareroots5a-h1"],"title":"Simpligying Expression","text":"What is the simplified form of the problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c70bfsquareroots6","title":"Add and Subtract Square Roots","body":"Simplify the following exercise:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Add and Subtract Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5c70bfsquareroots6a","stepAnswer":["$$-\\\\sqrt{7}$$"],"problemType":"TextBox","stepTitle":"$$11\\\\sqrt{7}-12\\\\sqrt{7}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-\\\\sqrt{7}$$","hints":{"DefaultPathway":[{"id":"a5c70bfsquareroots6a-h1","type":"hint","dependencies":[],"title":"$$\\\\frac{Adding}{Subtracting}$$ Coefficients","text":"Since the radicals are like, we subtract the coefficients.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c70bfsquareroots6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-\\\\sqrt{7}$$"],"dependencies":["a5c70bfsquareroots6a-h1"],"title":"Simpligying Expression","text":"What is the simplified form of the problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c70bfsquareroots7","title":"Add and Subtract Square Roots","body":"Simplify the following exercise:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Add and Subtract Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5c70bfsquareroots7a","stepAnswer":["$$9\\\\sqrt{y}$$"],"problemType":"TextBox","stepTitle":"$$7\\\\sqrt{y}+2\\\\sqrt{y}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9\\\\sqrt{y}$$","hints":{"DefaultPathway":[{"id":"a5c70bfsquareroots7a-h1","type":"hint","dependencies":[],"title":"$$\\\\frac{Adding}{Subtracting}$$ Coefficients","text":"Since the radicals are like, we add the coefficients.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c70bfsquareroots7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9\\\\sqrt{y}$$"],"dependencies":["a5c70bfsquareroots7a-h1"],"title":"Simpligying Expression","text":"What is the simplified form of the problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c70bfsquareroots8","title":"Add and Subtract Square Roots","body":"Simplify the following exercise:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Add and Subtract Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5c70bfsquareroots8a","stepAnswer":["$$12\\\\sqrt{n}$$"],"problemType":"TextBox","stepTitle":"$$9\\\\sqrt{n}+3\\\\sqrt{n}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12\\\\sqrt{n}$$","hints":{"DefaultPathway":[{"id":"a5c70bfsquareroots8a-h1","type":"hint","dependencies":[],"title":"$$\\\\frac{Adding}{Subtracting}$$ Coefficients","text":"Since the radicals are like, we add the coefficients.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c70bfsquareroots8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12\\\\sqrt{n}$$"],"dependencies":["a5c70bfsquareroots8a-h1"],"title":"Simpligying Expression","text":"What is the simplified form of the problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c70bfsquareroots9","title":"Add and Subtract Square Roots","body":"Simplify the following exercise:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Add and Subtract Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a5c70bfsquareroots9a","stepAnswer":["$$-3\\\\sqrt{a}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{a}-4\\\\sqrt{a}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-3\\\\sqrt{a}$$","hints":{"DefaultPathway":[{"id":"a5c70bfsquareroots9a-h1","type":"hint","dependencies":[],"title":"$$\\\\frac{Adding}{Subtracting}$$ Coefficients","text":"Since the radicals are like, we subtract the coefficients.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c70bfsquareroots9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3\\\\sqrt{a}$$"],"dependencies":["a5c70bfsquareroots9a-h1"],"title":"Simpligying Expression","text":"What is the simplified form of the problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c95e8polyzero10","title":"Using the Remainder Theorem to Find the Remainder","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Zeros of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c95e8polyzero10a","stepAnswer":["$$-44791$$"],"problemType":"TextBox","stepTitle":"Find the remainder of: $$\\\\frac{5x^5-4x^4+3x^3-2x^2+x-1}{x+6}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-44791$$","hints":{"DefaultPathway":[{"id":"a5c95e8polyzero10a-h1","type":"hint","dependencies":[],"title":"Remainder Theorem Definition","text":"The remainder theorem is as follows: if f(x) is being divided by $$(x-a)$$, then the remainder is equal to f(a). This means that we must plug in $$x=-6$$ into our dividend.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero10a-h2","type":"hint","dependencies":["a5c95e8polyzero10a-h1"],"title":"Plugging In","text":"$$5x^5-4x^4+3x^3-2x^2+x-1$$ becomes 5(-6)**5-4(-6)**4+3(-6)**3-2(-6)**2+(-6)-1. This simplifies to $$-44791$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c95e8polyzero11","title":"Using the Remainder Theorem to Find the Remainder","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Zeros of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c95e8polyzero11a","stepAnswer":["$$255$$"],"problemType":"TextBox","stepTitle":"Find the remainder of: $$\\\\frac{x^4-1}{x-4}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$255$$","hints":{"DefaultPathway":[{"id":"a5c95e8polyzero11a-h1","type":"hint","dependencies":[],"title":"Remainder Theorem Definition","text":"The remainder theorem is as follows: if f(x) is being divided by $$(x-a)$$, then the remainder is equal to f(a). This means that we must plug in $$x=4$$ into our dividend.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero11a-h2","type":"hint","dependencies":["a5c95e8polyzero11a-h1"],"title":"Plugging In","text":"$$x^4-1$$ becomes $$4^4-1$$. Our remainder is $$255$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c95e8polyzero12","title":"Using the Remainder Theorem to Find the Remainder","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Zeros of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c95e8polyzero12a","stepAnswer":["$$95$$"],"problemType":"TextBox","stepTitle":"Find the remainder of: $$\\\\frac{3x^3+4x^2-8x+2}{x-3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$95$$","hints":{"DefaultPathway":[{"id":"a5c95e8polyzero12a-h1","type":"hint","dependencies":[],"title":"Remainder Theorem Definition","text":"The remainder theorem is as follows: if f(x) is being divided by $$(x-a)$$, then the remainder is equal to f(a). This means that we must plug in $$x=3$$ into our dividend.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero12a-h2","type":"hint","dependencies":["a5c95e8polyzero12a-h1"],"title":"Plugging In","text":"$$3x^3+4x^2-8x+2$$ becomes $${3\\\\left(3\\\\right)}^3+{4\\\\left(3\\\\right)}^2-8\\\\left(3\\\\right)+2$$. This simplifies to $$95$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c95e8polyzero13","title":"Using the Remainder Theorem to Find the Remainder","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Zeros of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c95e8polyzero13a","stepAnswer":["$$-1$$"],"problemType":"TextBox","stepTitle":"Find the remainder of: $$\\\\frac{4x^3+5x^2-2x+7}{x+2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1$$","hints":{"DefaultPathway":[{"id":"a5c95e8polyzero13a-h1","type":"hint","dependencies":[],"title":"Remainder Theorem Definition","text":"The remainder theorem is as follows: if f(x) is being divided by $$(x-a)$$, then the remainder is equal to f(a). This means that we must plug in $$x=-2$$ into our dividend.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero13a-h2","type":"hint","dependencies":["a5c95e8polyzero13a-h1"],"title":"Plugging In","text":"$$4x^3+5x^2-2x+7$$ becomes 4(-2)**3+5(-2)**2-2(-2)+7. This means our remainder is $$-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c95e8polyzero16","title":"Using the Rational Zero Theorem #1","body":"Use the Rational Zero Theorem to find the real solution(s) to the equation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Zeros of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c95e8polyzero16a","stepAnswer":["$$x=2, 4, -3$$"],"problemType":"MultipleChoice","stepTitle":"$$x^3-3x^2-10x+24=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=2, 4, -3$$","choices":["$$x=1, 4, -3$$","$$x=5, 4, 3$$","$$x=2, 4, -3$$"],"hints":{"DefaultPathway":[{"id":"a5c95e8polyzero16a-h1","type":"hint","dependencies":[],"title":"Definition of the Rational Zero Theorem","text":"The Rational Zero Theorem tells us that if $$\\\\frac{p}{q}$$ is a zero of f(x), then $$p$$ is a factor of $$1$$ and q is a factor of $$2$$. $$p$$ is a factor of the constant term and q is a factor of the leading coefficient.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero16a-h2","type":"hint","dependencies":["a5c95e8polyzero16a-h1"],"title":"Determining Factors","text":"The first step is to determine all factors of the constant term and all factors of the leading coefficient.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero16a-h3","type":"hint","dependencies":["a5c95e8polyzero16a-h2"],"title":"Determing Values of $$\\\\frac{p}{q}$$","text":"Next, determine all possible values of $$\\\\frac{p}{q}$$. Be sure to include both positive and negative candidates.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero16a-h4","type":"hint","dependencies":["a5c95e8polyzero16a-h3"],"title":"Using Subsitution","text":"Finally, determine which possible zeros are actual zeros by evaluating each case of $$f{\\\\left(\\\\frac{p}{q}\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c95e8polyzero17","title":"Using the Rational Zero Theorem #2","body":"Use the Rational Zero Theorem to find the real solution(s) to the equation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Zeros of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c95e8polyzero17a","stepAnswer":["$$x=2-\\\\frac{3}{2}-4$$"],"problemType":"MultipleChoice","stepTitle":"$$2x^3+7x^2-10x-24=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=2-\\\\frac{3}{2}-4$$","choices":["$$x=\\\\frac{2}{3}-\\\\frac{3}{2}-4$$","$$x=2-\\\\frac{3}{2}-4$$","$$x=2, -3, -4$$"],"hints":{"DefaultPathway":[{"id":"a5c95e8polyzero17a-h1","type":"hint","dependencies":[],"title":"Definition of the Rational Zero Theorem","text":"The Rational Zero Theorem tells us that if $$\\\\frac{p}{q}$$ is a zero of f(x), then $$p$$ is a factor of $$1$$ and q is a factor of $$2$$. $$p$$ is a factor of the constant term and q is a factor of the leading coefficient.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero17a-h2","type":"hint","dependencies":["a5c95e8polyzero17a-h1"],"title":"Determining Factors","text":"The first step is to determine all factors of the constant term and all factors of the leading coefficient.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero17a-h3","type":"hint","dependencies":["a5c95e8polyzero17a-h2"],"title":"Determing Values of $$\\\\frac{p}{q}$$","text":"Next, determine all possible values of $$\\\\frac{p}{q}$$. 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Be sure to include both positive and negative candidates.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero18a-h4","type":"hint","dependencies":["a5c95e8polyzero18a-h3"],"title":"Using Subsitution","text":"Finally, determine which possible zeros are actual zeros by evaluating each case of $$f{\\\\left(\\\\frac{p}{q}\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c95e8polyzero19","title":"Using the Rational Zero Theorem #4","body":"Use the Rational Zero Theorem to find the real solution(s) to the equation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Zeros of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c95e8polyzero19a","stepAnswer":["$$x=-5, -4, 4$$"],"problemType":"MultipleChoice","stepTitle":"$$x^3+5x^2-16x-80=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=-5, -4, 4$$","choices":["$$x=-5, -4, 4$$","$$x=-2, -4, 4$$","$$x=-5, -4, 3$$"],"hints":{"DefaultPathway":[{"id":"a5c95e8polyzero19a-h1","type":"hint","dependencies":[],"title":"Definition of the Rational Zero Theorem","text":"The Rational Zero Theorem tells us that if $$\\\\frac{p}{q}$$ is a zero of f(x), then $$p$$ is a factor of $$1$$ and q is a factor of $$2$$. $$p$$ is a factor of the constant term and q is a factor of the leading coefficient.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero19a-h2","type":"hint","dependencies":["a5c95e8polyzero19a-h1"],"title":"Determining Factors","text":"The first step is to determine all factors of the constant term and all factors of the leading coefficient.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero19a-h3","type":"hint","dependencies":["a5c95e8polyzero19a-h2"],"title":"Determing Values of $$\\\\frac{p}{q}$$","text":"Next, determine all possible values of $$\\\\frac{p}{q}$$. Be sure to include both positive and negative candidates.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero19a-h4","type":"hint","dependencies":["a5c95e8polyzero19a-h3"],"title":"Using Subsitution","text":"Finally, determine which possible zeros are actual zeros by evaluating each case of $$f{\\\\left(\\\\frac{p}{q}\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c95e8polyzero20","title":"Using the Rational Zero Theorem #5","body":"Use the Rational Zero Theorem to find the real solution(s) to the equation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Zeros of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c95e8polyzero20a","stepAnswer":["$$x=3, -5, 5$$"],"problemType":"MultipleChoice","stepTitle":"$$x^3-3x^2-25x+75=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=3, -5, 5$$","choices":["$$x=3, -2, 5$$","$$x=3, -5, 5$$","$$x=1, -5, 5$$"],"hints":{"DefaultPathway":[{"id":"a5c95e8polyzero20a-h1","type":"hint","dependencies":[],"title":"Definition of the Rational Zero Theorem","text":"The Rational Zero Theorem tells us that if $$\\\\frac{p}{q}$$ is a zero of f(x), then $$p$$ is a factor of $$1$$ and q is a factor of $$2$$. $$p$$ is a factor of the constant term and q is a factor of the leading coefficient.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero20a-h2","type":"hint","dependencies":["a5c95e8polyzero20a-h1"],"title":"Determining Factors","text":"The first step is to determine all factors of the constant term and all factors of the leading coefficient.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero20a-h3","type":"hint","dependencies":["a5c95e8polyzero20a-h2"],"title":"Determing Values of $$\\\\frac{p}{q}$$","text":"Next, determine all possible values of $$\\\\frac{p}{q}$$. Be sure to include both positive and negative candidates.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero20a-h4","type":"hint","dependencies":["a5c95e8polyzero20a-h3"],"title":"Using Subsitution","text":"Finally, determine which possible zeros are actual zeros by evaluating each case of $$f{\\\\left(\\\\frac{p}{q}\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c95e8polyzero21","title":"Using the Rational Zero Theorem #6","body":"Use the Rational Zero Theorem to find the real solution(s) to the equation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Zeros of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c95e8polyzero21a","stepAnswer":["x=-1/2,5,-3"],"problemType":"MultipleChoice","stepTitle":"$$2x^3-3x^2-32x-15=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["x=-1,5/2,-3","$$x=-2, 5, -3$$","x=-1/2,5,-3"],"hints":{"DefaultPathway":[{"id":"a5c95e8polyzero21a-h1","type":"hint","dependencies":[],"title":"Definition of the Rational Zero Theorem","text":"The Rational Zero Theorem tells us that if $$\\\\frac{p}{q}$$ is a zero of f(x), then $$p$$ is a factor of $$1$$ and q is a factor of $$2$$. $$p$$ is a factor of the constant term and q is a factor of the leading coefficient.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero21a-h2","type":"hint","dependencies":["a5c95e8polyzero21a-h1"],"title":"Determining Factors","text":"The first step is to determine all factors of the constant term and all factors of the leading coefficient.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero21a-h3","type":"hint","dependencies":["a5c95e8polyzero21a-h2"],"title":"Determing Values of $$\\\\frac{p}{q}$$","text":"Next, determine all possible values of $$\\\\frac{p}{q}$$. Be sure to include both positive and negative candidates.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero21a-h4","type":"hint","dependencies":["a5c95e8polyzero21a-h3"],"title":"Using Subsitution","text":"Finally, determine which possible zeros are actual zeros by evaluating each case of $$f{\\\\left(\\\\frac{p}{q}\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c95e8polyzero22","title":"Using the Rational Zero Theorem #7","body":"Use the Rational Zero Theorem to find the real solution(s) to the equation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Zeros of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c95e8polyzero22a","stepAnswer":["x=-1,-3/2,2"],"problemType":"MultipleChoice","stepTitle":"$$2x^3+x^2-7x-6=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["x=-1,-3/2,2","$$x=-1, -3, 2$$","x=-1,2,2/5"],"hints":{"DefaultPathway":[{"id":"a5c95e8polyzero22a-h1","type":"hint","dependencies":[],"title":"Definition of the Rational Zero Theorem","text":"The Rational Zero Theorem tells us that if $$\\\\frac{p}{q}$$ is a zero of f(x), then $$p$$ is a factor of $$1$$ and q is a factor of $$2$$. $$p$$ is a factor of the constant term and q is a factor of the leading coefficient.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero22a-h2","type":"hint","dependencies":["a5c95e8polyzero22a-h1"],"title":"Determining Factors","text":"The first step is to determine all factors of the constant term and all factors of the leading coefficient.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero22a-h3","type":"hint","dependencies":["a5c95e8polyzero22a-h2"],"title":"Determing Values of $$\\\\frac{p}{q}$$","text":"Next, determine all possible values of $$\\\\frac{p}{q}$$. Be sure to include both positive and negative candidates.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero22a-h4","type":"hint","dependencies":["a5c95e8polyzero22a-h3"],"title":"Using Subsitution","text":"Finally, determine which possible zeros are actual zeros by evaluating each case of $$f{\\\\left(\\\\frac{p}{q}\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c95e8polyzero23","title":"Using the Rational Zero Theorem #8","body":"Use the Rational Zero Theorem to find the real solution(s) to the equation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Zeros of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c95e8polyzero23a","stepAnswer":["x=1/2,(1+sqrt(5))/2,(1-sqrt(5))/2"],"problemType":"MultipleChoice","stepTitle":"$$2x^3-3x^2-x+1=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["x=3/4,(1+sqrt(5))/2,(1-sqrt(5))/2","x=3/2,(1+sqrt(5))/2,(1-sqrt(5))/2","x=1/2,(1+sqrt(5))/2,(1-sqrt(5))/2"],"hints":{"DefaultPathway":[{"id":"a5c95e8polyzero23a-h1","type":"hint","dependencies":[],"title":"Definition of the Rational Zero Theorem","text":"The Rational Zero Theorem tells us that if $$\\\\frac{p}{q}$$ is a zero of f(x), then $$p$$ is a factor of $$1$$ and q is a factor of $$2$$. $$p$$ is a factor of the constant term and q is a factor of the leading coefficient.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero23a-h2","type":"hint","dependencies":["a5c95e8polyzero23a-h1"],"title":"Determining Factors","text":"The first step is to determine all factors of the constant term and all factors of the leading coefficient.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero23a-h3","type":"hint","dependencies":["a5c95e8polyzero23a-h2"],"title":"Determing Values of $$\\\\frac{p}{q}$$","text":"Next, determine all possible values of $$\\\\frac{p}{q}$$. Be sure to include both positive and negative candidates.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero23a-h4","type":"hint","dependencies":["a5c95e8polyzero23a-h3"],"title":"Using Subsitution","text":"Finally, determine which possible zeros are actual zeros by evaluating each case of $$f{\\\\left(\\\\frac{p}{q}\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c95e8polyzero24","title":"Using the Rational Zero Theorem #9","body":"Use the Rational Zero Theorem to find the real solution(s) to the equation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Zeros of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c95e8polyzero24a","stepAnswer":["x=-2/3,(1+sqrt(13))/2,(1-sqrt(13))/2"],"problemType":"MultipleChoice","stepTitle":"$$3x^3-x^2-11x-6=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["x=-1/2,(1+sqrt(13))/2,(1-sqrt(13))/2","x=-2/5,(1+sqrt(13))/2,(1-sqrt(13))/2","x=-2/3,(1+sqrt(13))/2,(1-sqrt(13))/2"],"hints":{"DefaultPathway":[{"id":"a5c95e8polyzero24a-h1","type":"hint","dependencies":[],"title":"Definition of the Rational Zero Theorem","text":"The Rational Zero Theorem tells us that if $$\\\\frac{p}{q}$$ is a zero of f(x), then $$p$$ is a factor of $$1$$ and q is a factor of $$2$$. $$p$$ is a factor of the constant term and q is a factor of the leading coefficient.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero24a-h2","type":"hint","dependencies":["a5c95e8polyzero24a-h1"],"title":"Determining Factors","text":"The first step is to determine all factors of the constant term and all factors of the leading coefficient.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero24a-h3","type":"hint","dependencies":["a5c95e8polyzero24a-h2"],"title":"Determing Values of $$\\\\frac{p}{q}$$","text":"Next, determine all possible values of $$\\\\frac{p}{q}$$. Be sure to include both positive and negative candidates.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero24a-h4","type":"hint","dependencies":["a5c95e8polyzero24a-h3"],"title":"Using Subsitution","text":"Finally, determine which possible zeros are actual zeros by evaluating each case of $$f{\\\\left(\\\\frac{p}{q}\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c95e8polyzero25","title":"Using the Rational Zero Theorem #10","body":"Use the Rational Zero Theorem to find the real solution(s) to the equation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Zeros of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c95e8polyzero25a","stepAnswer":["$$x=\\\\frac{3}{2}$$"],"problemType":"MultipleChoice","stepTitle":"$$2x^3-5x^2+9x-9=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=\\\\frac{3}{2}$$","choices":["$$x=\\\\frac{3}{2}$$","x=3/2,-1,4","x=3/2,-5,3"],"hints":{"DefaultPathway":[{"id":"a5c95e8polyzero25a-h1","type":"hint","dependencies":[],"title":"Definition of the Rational Zero Theorem","text":"The Rational Zero Theorem tells us that if $$\\\\frac{p}{q}$$ is a zero of f(x), then $$p$$ is a factor of $$1$$ and q is a factor of $$2$$. $$p$$ is a factor of the constant term and q is a factor of the leading coefficient.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero25a-h2","type":"hint","dependencies":["a5c95e8polyzero25a-h1"],"title":"Determining Factors","text":"The first step is to determine all factors of the constant term and all factors of the leading coefficient.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero25a-h3","type":"hint","dependencies":["a5c95e8polyzero25a-h2"],"title":"Determing Values of $$\\\\frac{p}{q}$$","text":"Next, determine all possible values of $$\\\\frac{p}{q}$$. Be sure to include both positive and negative candidates.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero25a-h4","type":"hint","dependencies":["a5c95e8polyzero25a-h3"],"title":"Using Subsitution","text":"Finally, determine which possible zeros are actual zeros by evaluating each case of $$f{\\\\left(\\\\frac{p}{q}\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c95e8polyzero26","title":"Using the Remainder Theorem to Evaluate a Polynomial","body":"According to the Remainder Theorem, if a polynomial f(x) is divided by (x - k), then the remainder is the value f(k).","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Zeros of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c95e8polyzero26a","stepAnswer":["$$25$$"],"problemType":"TextBox","stepTitle":"Use the Remainder Theorem to evaluate $$f(x)=6x^4-x^3-15x^2+2x-7$$ at $$x=2$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$25$$","hints":{"DefaultPathway":[{"id":"a5c95e8polyzero26a-h1","type":"hint","dependencies":[],"title":"Using the Remainder Theorem","text":"To find the remainder using the Remainder Theorem, use synthetic division to divide the polynomial by $$(x-2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero26a-h2","type":"hint","dependencies":["a5c95e8polyzero26a-h1"],"title":"Using Synthetic Division","text":"Use synthetic division to divide the polynomial by $$(x-2)$$. The quotient after dividing by $$(x-2)$$ is $$6x^3$$ + $$11x^2$$ + $$7x$$ + $$16$$ and the remainder is $$25$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c95e8polyzero27","title":"Listing All Possible Rational Zeros","body":"The Rational Zero Theorem states that, if the polynomial $$f(x)=a_n x^n+a_n-1x^{n-1}+...+a_1 x+a_0$$ has integer coefficients, then every rational zero of f(x) has the form $$\\\\frac{p}{q}$$ where $$p$$ is a factor of the constant term $$a_0$$ and q is a factor of the leading coefficient $$a_n$$. When the leading coefficient is $$1$$, the possible rational zeros are the factors of the constant term.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Zeros of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c95e8polyzero27a","stepAnswer":["-4, -2, -1, -1/2, 1/2, 1, 2, 4"],"problemType":"TextBox","stepTitle":"List all possible rational zeros of $$f(x)=2x^4-5x^3+x^2-4$$. (List them in ascending order as such: $$-2$$, $$-1$$, $$\\\\frac{-1}{2}$$, $$\\\\frac{1}{2}$$, $$1$$, ...)","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-4$$, $$-2$$, $$-1$$, $$\\\\frac{-1}{2}$$, $$\\\\frac{1}{2}$$, $$1$$, $$2$$, $$4$$","hints":{"DefaultPathway":[{"id":"a5c95e8polyzero27a-h1","type":"hint","dependencies":[],"title":"What terms contribute to the possible rational zeros?","text":"The only possible rational zeros of f(x) are the quotients of the factors of the last term, $$-4$$, and the factors of the leading coefficient, $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero27a-h2","type":"hint","dependencies":["a5c95e8polyzero27a-h1"],"title":"Factors of the Constant Term","text":"What are the factors of the constant term, -4? From here on, these factors will be denoted as $$p$$, factors of the constant term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero27a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["-4, -2, -1, 1, 2, 4"],"dependencies":["a5c95e8polyzero27a-h2"],"title":"What are the factors of the constant term, -4? (List them in ascending order as such: $$-2$$, $$-1$$, $$1$$, ...)","text":"There are $$6$$ in total. Consider that there are negative factors as well.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero27a-h4","type":"hint","dependencies":["a5c95e8polyzero27a-h1"],"title":"What are the factors of the leading coefficient, 2? From here on, these factors will be denoted as q, factors of the leading coefficient.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero27a-h5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["-2, -1, 1, 2"],"dependencies":["a5c95e8polyzero27a-h4"],"title":"What are the factors of the leading coefficient, 2? (List them in ascending order as such: $$-2$$, $$-1$$, $$1$$, ...)","text":"There are $$4$$ in total. Consider that there are negative factors as well.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero27a-h4","type":"hint","dependencies":["a5c95e8polyzero27a-h1"],"title":"What are the different combination of rational zeros?","text":"Consider that a rational zero is of the form $$\\\\frac{p}{q}$$ where $$p$$ are factors of the constant term and q are factors of the leading coefficient. It would help to list out all your $$p$$ and q to write out the different combination.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero27a-h5","type":"hint","dependencies":["a5c95e8polyzero27a-h4"],"title":"What are the possible rational zeros with $$p$$ $$=$$ $$+-1$$?","text":"Recall what are the factors of $$2$$, q, that you\'ve previously found.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero27a-h6","type":"hint","dependencies":["a5c95e8polyzero27a-h5"],"title":"The combination of all q with $$p$$ $$=$$ +- $$1$$ are $$+-\\\\left(\\\\frac{1}{1}\\\\right)$$, $$+-\\\\left(\\\\frac{1}{2}\\\\right)$$","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero27a-h7","type":"hint","dependencies":["a5c95e8polyzero27a-h4"],"title":"What are the possible rational zeros with $$p=+-2$$?","text":"Recall what are the factors of $$2$$, q, that you\'ve previously found.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero27a-h8","type":"hint","dependencies":["a5c95e8polyzero27a-h7"],"title":"The combination of all q with p=+- $$2$$ are $$+-\\\\left(\\\\frac{2}{1}\\\\right)$$, $$+-\\\\left(\\\\frac{2}{2}\\\\right)$$","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero27a-h9","type":"hint","dependencies":["a5c95e8polyzero27a-h4"],"title":"What are the possible rational zeros with $$p=+-4$$?","text":"Recall what are the factors of $$2$$, q, that you\'ve previously found.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero27a-h10","type":"hint","dependencies":["a5c95e8polyzero27a-h9"],"title":"The combination of all q with p=+- $$4$$ are $$+-\\\\left(\\\\frac{4}{1}\\\\right)$$, $$+-\\\\left(\\\\frac{4}{2}\\\\right)$$","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero27a-h11","type":"hint","dependencies":["a5c95e8polyzero27a-h4"],"title":"Simplify all the rational zeros that you have found so that there are no duplicates.","text":"For example, $$\\\\frac{2}{2}=\\\\frac{1}{1}=1$$ and $$\\\\frac{-4}{2}=\\\\frac{-2}{1}=-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c95e8polyzero28","title":"Using the Rational Zero Theorem to Find Rational Zeros","body":"The Rational Zero Theorem states that, if the polynomial $$f(x)=a_n x^n+a_n$$ - 1*x**(n - 1)+...+a_1*x+a_0 has integer coefficients, then every rational zero of f(x) has the form $$\\\\frac{p}{q}$$ where $$p$$ is a factor of the constant term $$a_0$$ and q is a factor of the leading coefficient $$a_n$$. When the leading coefficient is $$1$$, the possible rational zeros are the factors of the constant term.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Zeros of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c95e8polyzero28a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"Use the Rational Zero Theorem to find the rational zeros of f(x) $$=$$ $$2x^3$$ + $$x^2$$ - $$4x$$ + $$1$$. (List them in ascending order as such: $$-2$$, $$-1$$, $$\\\\frac{-1}{2}$$, $$\\\\frac{1}{2}$$, $$1$$, ...)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a5c95e8polyzero28a-h1","type":"hint","dependencies":[],"title":"What terms contribute to the possible rational zeros?","text":"The only possible rational zeros of f(x) are the quotients of the factors of the constant term, $$1$$, and the factors of the leading coefficient, $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero28a-h2","type":"hint","dependencies":["a5c95e8polyzero28a-h1"],"title":"What are the factors of the constant term, 1? From here on, these factors will be denoted as $$p$$, factors of the constant term.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero28a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["-1, 1"],"dependencies":["a5c95e8polyzero28a-h2"],"title":"What are the factors of the constant term, 1? (List them in ascending order as such: $$-2$$, $$-1$$, $$1$$, ...)","text":"There are $$2$$ in total. Consider that there are negative factors as well.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero28a-h4","type":"hint","dependencies":["a5c95e8polyzero28a-h3"],"title":"What are the factors of the leading coefficient, 2? From here on, these factors will be denoted as q, factors of the leading coefficient.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero28a-h5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["-2, -1, 1, 2"],"dependencies":["a5c95e8polyzero28a-h4"],"title":"What are the factors of the leading coefficient, 2? (List them in ascending order as such: $$-2$$, $$-1$$, $$1$$, ...)","text":"There are $$4$$ in total. Consider that there are negative factors as well.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero28a-h6","type":"hint","dependencies":["a5c95e8polyzero28a-h5"],"title":"What are the different combination of rational zeros?","text":"Consider that a rational zero is of the form $$\\\\frac{p}{q}$$ where $$p$$ are factors of the constant term and q are factors of the leading coefficient. It would help to list out all your $$p$$ and q to write out the different combination.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero28a-h7","type":"hint","dependencies":["a5c95e8polyzero28a-h6"],"title":"What are the possible rational zeros with $$p=+-1$$?","text":"Recall what are the factors of $$2$$, q, that you\'ve previously found.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero28a-h8","type":"hint","dependencies":["a5c95e8polyzero28a-h7"],"title":"The combination of all q with p=+- $$1$$ are $$+-\\\\left(\\\\frac{1}{1}\\\\right)$$, $$+-\\\\left(\\\\frac{1}{2}\\\\right)$$","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero28a-h9","type":"hint","dependencies":["a5c95e8polyzero28a-h8"],"title":"Determine if the possible zeros are actual zeros by substituting these values for $$x$$ in f(x).","text":"If the result of f(k) is non-zero, then there is a remainder and k is not a zero of f(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero28a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a5c95e8polyzero28a-h9"],"title":"What is the remainder of $$f(-1)$$?","text":"Substitute $$x=-1$$ into $$f(x)=2x^3+x^2-4x+1$$. The remainder is the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero28a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a5c95e8polyzero28a-h10"],"title":"What is the remainder of f(1)?","text":"Substitute $$x=1$$ into $$f(x)=2x^3+x^2-4x+1$$. The remainder is the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero28a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a5c95e8polyzero28a-h11"],"title":"What is the remainder of $$f\\\\left(-\\\\frac{1}{2}\\\\right)$$?","text":"Substitute $$x=\\\\frac{-1}{2}$$ into $$f(x)=2x^3+x^2-4x+1$$. The remainder is the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero28a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{2}$$"],"dependencies":["a5c95e8polyzero28a-h12"],"title":"What is the remainder of $$f{\\\\left(\\\\frac{1}{2}\\\\right)}$$?","text":"Substitute $$x=\\\\frac{1}{2}$$ into $$f(x)=2x^3+x^2-4x+1$$. The remainder is the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c95e8polyzero29","title":"Finding the Zeros of a Polynomial Function with Repeated Real Zeros","body":"Find the possible rational zeros for the function using the Rational Zero Theorem.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Zeros of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c95e8polyzero29a","stepAnswer":["-1/2, 1"],"problemType":"TextBox","stepTitle":"Find the zeros of f(x) $$=$$ $$4x^3-3x-1$$. (List them in ascending order as such: $$-2$$, $$-1$$, $$\\\\frac{-1}{2}$$, $$\\\\frac{1}{2}$$, $$1$$, ...)","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{-1}{2}$$, $$1$$","hints":{"DefaultPathway":[{"id":"a5c95e8polyzero29a-h1","type":"hint","dependencies":[],"title":"Definition of the Rational Zero Theorem","text":"The Rational Zero Theorem states that, if the polynomial f(x) $$=$$ $$a_n x^n$$ + $$a_n$$ - 1*x**(n - 1) + ... + $$a_1 x$$ + $$a_0$$ has integer coefficients, then every rational zero of f(x) has the form $$\\\\frac{p}{q}$$ where $$p$$ is a factor of the constant term $$a_0$$ and q is a factor of the leading coefficient $$a_n$$. When the leading coefficient is $$1$$, the possible rational zeros are the factors of the constant term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero29a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["-1, 1"],"dependencies":["a5c95e8polyzero29a-h1"],"title":"What are the factors of the constant term, -1? From here on, these factors will be denoted as $$p$$, factors of the constant term.(List them in ascending order as such: $$-2$$, $$-1$$, $$1$$, ...)","text":"There are $$2$$ in total. Consider that there are negative factors as well.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero29a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["-4, -2, -1, 1, 2, 4"],"dependencies":["a5c95e8polyzero29a-h2"],"title":"What are the factors of the leading coefficient, 4? From here on, these factors will be denoted as $$p$$, factors of the constant term.(List them in ascending order as such: $$-2$$, $$-1$$, $$1$$, ...)","text":"There are $$6$$ in total. Consider that there are negative factors as well.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero29a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["-1, -1/2, -1/4, 1/4, 1/2, 1"],"dependencies":["a5c95e8polyzero29a-h3"],"title":"What are the different combination of rational zeros?","text":"Consider that a rational zero is of the form $$\\\\frac{p}{q}$$ where $$p$$ are factors of the constant term and q are factors of the leading coefficient. It would help to list out all your $$p$$ and q to write out the different combination. There are $$6$$ in total. Consider that there are negative factors as well. (List them in ascending order as such: $$-2$$, $$-1$$, $$\\\\frac{-1}{2}$$, $$\\\\frac{1}{2}$$, $$1$$, ...)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero29a-h5","type":"hint","dependencies":["a5c95e8polyzero29a-h4"],"title":"Use synthetic division to divide the polynomial by (x + 2).","text":"Now that we have found a list of possible rational zeros for the function. We will use the synthetic division to evaluate each possible zero until we find one that gives a remainder of $$0$$. We can begin with $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero29a-h6","type":"hint","dependencies":["a5c95e8polyzero29a-h5"],"title":"Use synthetic division to divide the function by (x - 1).","text":"The quotient after dividing by $$(x-1)$$ is $$4x^2+4x+1$$ and the remainder is $$0$$. Therefore, $$1$$ is a zero of the function and the polynomial can be rewritten as $$\\\\left(x-1\\\\right) \\\\left(4x^2+4x+1\\\\right)$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero29a-h7","type":"hint","dependencies":["a5c95e8polyzero29a-h6"],"title":"Factorize the remaining quadratic equation, $$4x^2$$ + $$4x$$ + $$1$$, to find the remaining zeros.","text":"We can do so by either factoring, using the quadratic formula $$x$$ $$=$$ $$(-b$$ +- sqrt(b**2 - 4*a*c)) / $$2a$$, or by expressing the polynomial in terms of the standard form f(x) $$=$$ a*(x - h)**2 + k and solving for $$x$$ when f(x) $$=$$ $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c95e8polyzero30","title":"Finding the Zeros of a Polynomial Function with Complex Zeros","body":"Find the zeroes of the following function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Zeros of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c95e8polyzero30a","stepAnswer":["-3, i*sqrt(3)/3, -i*sqrt(3)/3"],"problemType":"TextBox","stepTitle":"f(x) $$=$$ $$3x^3+9x^2+x+3$$. (List the real roots first, then the complex roots starting with the positive complex roots. Rationalize any surds. Example: $$1$$, $$\\\\frac{i \\\\sqrt{2}}{2}$$, -i*sqrt(2)/2)","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-3$$, $$\\\\frac{i \\\\sqrt{3}}{3}$$, $$\\\\frac{-i \\\\sqrt{3}}{3}$$","hints":{"DefaultPathway":[{"id":"a5c95e8polyzero30a-h1","type":"hint","dependencies":[],"title":"Find the possible rational zeros for the function using the Rational Zero Theorem.","text":"The Rational Zero Theorem states that, if the polynomial f(x) $$=$$ a_n*x**n+a_n-1*x**(n - 1)+...+a_1*x+a_0 has integer coefficients, then every rational zero of f(x) has the form $$\\\\frac{p}{q}$$ where $$p$$ is a factor of the constant term $$a_0$$ and q is a factor of the leading coefficient $$a_n$$. When the leading coefficient is $$1$$, the possible rational zeros are the factors of the constant term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero30a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["-3,-1, 1, 3"],"dependencies":["a5c95e8polyzero30a-h1"],"title":"What are the factors of the constant term, 3? From here on, these factors will be denoted as $$p$$, factors of the constant term.(List them in ascending order as such: $$-2$$, $$-1$$, $$1$$, ...)","text":"There are $$4$$ in total. Consider that there are negative factors as well.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero30a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["-3,-1, 1, 3"],"dependencies":["a5c95e8polyzero30a-h1"],"title":"What are the factors of the leading coefficient, 3? From here on, these factors will be denoted as $$p$$, factors of the constant term.(List them in ascending order as such: $$-2$$, $$-1$$, $$1$$, ...)","text":"There are $$4$$ in total. Consider that there are negative factors as well.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero30a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["-3, -1, -1/3, 1/3, 1, 3"],"dependencies":["a5c95e8polyzero30a-h1"],"title":"What are the different combination of rational zeros?","text":"Consider that a rational zero is of the form $$\\\\frac{p}{q}$$ where $$p$$ are factors of the constant term and q are factors of the leading coefficient. It would help to list out all your $$p$$ and q to write out the different combination. There are $$6$$ in total. Consider that there are negative factors as well. (List them in ascending order as such: $$-2$$, $$-1$$, $$\\\\frac{-1}{2}$$, $$\\\\frac{1}{2}$$, $$1$$, ...)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero30a-h5","type":"hint","dependencies":["a5c95e8polyzero30a-h4"],"title":"Factoring out a zero to simplify the equation.","text":"Now that we have found a list of possible rational zeros for the function. We will use the synthetic division to evaluate each possible zero until we find one that gives a remainder of $$0$$. We can begin with $$-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero30a-h6","type":"hint","dependencies":["a5c95e8polyzero30a-h5"],"title":"Use synthetic division to divide the function by (x + 3).","text":"The quotient after dividing by $$x+3$$ is $$3x^2+1$$ and the remainder is $$0$$. Therefore, $$-3$$ is a zero of the function and the polynomial can be rewritten as $$\\\\left(x+3\\\\right) \\\\left(3x^2+1\\\\right)$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero30a-h7","type":"hint","dependencies":["a5c95e8polyzero30a-h6"],"title":"Solving for the other zeros.","text":"Note that because there is no $$x$$ term, we can set directly set quadratic polynomial to $$0$$ to solve for $$x$$, the remaining roots.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero30a-h8","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["i*sqrt(3)/3, -i*sqrt(3)/3"],"dependencies":["a5c95e8polyzero30a-h7"],"title":"Solving for the other zeros. (List the complex roots starting with the positive complex roots. Rationalize any surds. Example: $$1$$, $$\\\\frac{i \\\\sqrt{2}}{2}$$, -i*sqrt(2)/2)","text":"Set $$3x^2+1=0$$ and make $$x$$ the subject. The roots are complex","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c95e8polyzero31","title":"Using the Linear Factorization Theorem to Find a Polynomial with Given Zeros","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Zeros of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c95e8polyzero31a","stepAnswer":["$$-5x^4-5x^3+25x^2-5x+30$$"],"problemType":"TextBox","stepTitle":"Find a fourth degree polynomial with real coefficients that has zeros of $$-3$$, $$2$$, i, such that $$f(-2)=100$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-5x^4-5x^3+25x^2-5x+30$$","hints":{"DefaultPathway":[{"id":"a5c95e8polyzero31a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["i"],"dependencies":[],"title":"Finding the fourth root.","text":"By the Complex Conjugate Theorem, since i is a zero, what is the last zero","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero31a-h2","type":"hint","dependencies":["a5c95e8polyzero31a-h1"],"title":"Expressing f(x) in terms of the roots, then expand it.","text":"We can express f(x) as the multiplication of the four factors scaled by a constant a. In this case, we know that the factors are (x + 3), (x - 2), (x - i), (x + i). As such, we can rewrite f(x) $$=$$ a*(x + 3)*(x - 2)*(x - i)*(x + i).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero31a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2$$ + $$x$$ - $$6$$"],"dependencies":["a5c95e8polyzero31a-h2"],"title":"Expand the real factors of f(x).","text":"What is $$\\\\left(x+3\\\\right) \\\\left(x-2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero31a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2+1$$"],"dependencies":["a5c95e8polyzero31a-h2"],"title":"Expand the complex factors of f(x).","text":"What is $$\\\\left(x-i\\\\right) \\\\left(x+i\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero31a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["a*(x**4 + $$x^3$$ - $$5x^2$$ + $$x$$ - 6)"],"dependencies":["a5c95e8polyzero31a-h3","a5c95e8polyzero31a-h4"],"title":"Expanding the rest of the polynomial. Leave the constant scaling factor a outside. (Example: a*(x**2 + 1)","text":"What is (x**2+x - 6)*(x**2+1)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero31a-h6","type":"hint","dependencies":["a5c95e8polyzero31a-h5"],"title":"Solving for the scaling factor, a.","text":"Use the fact that $$f(-2)=100$$ to find a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero31a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["a5c95e8polyzero31a-h6"],"title":"Solving for the scaling factor, a.","text":"Substitute $$x=-2$$ and $$f(-2)=100$$ into the f(x) $$=$$ $$a \\\\left(x^4+x^3-5x^2+x-6\\\\right)$$ that was previously found. Then, solve for a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero31a-h8","type":"hint","dependencies":["a5c95e8polyzero31a-h7"],"title":"Multiply a that was found into the rest of the polynomial.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c95e8polyzero32","title":"Using Descartes\u2019 Rule of Signs","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Zeros of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c95e8polyzero32a","stepAnswer":["(2, 2, 0), (2, 0, 2), (0, 2, 2), (0, 0, 4)"],"problemType":"TextBox","stepTitle":"Use Descartes\u2019 Rule of Signs to determine the possible numbers of positive and negative real zeros for f(x) $$=$$ $$-\\\\left(x^4\\\\right)$$ - $$3x^3$$ + $$6x^2$$ - $$4x$$ - $$12$$. (Consider that there are complex roots as well, provide the answer as a list of coordinates of the form: (number of positive real zeros, number of negative real zeros, number of complex zeros). Sort the list in descending order, i.e. compare the first coordinate and place whichever has a higher value earlier in the list. If there is a tie, look at the second, then third coordinate. Example: (4, $$0$$, 0), (2, $$2$$, 0), (0, 4,0), (0, $$2$$, 2) )","stepBody":"According to Descartes\u2019 Rule of Signs, if we let f(x) $$=$$ $$a_n x^n$$ + $$a_n-1x^{n-1}$$ + ... + $$a_1 x$$ + $$a_0$$ be a polynomial function with real coefficients:\\\\nThe number of positive real zeros is either equal to the number of sign changes of f(x) or is less than the number of sign changes by an even integer.\\\\nThe number of negative real zeros is either equal to the number of sign changes of $$f(-x)$$ or is less than the number of sign changes by an even integer.","answerType":"string","variabilization":{},"answerLatex":"(2, $$2$$, 0), (2, $$0$$, 2), (0, $$2$$, 2), (0, $$0$$, 4)","hints":{"DefaultPathway":[{"id":"a5c95e8polyzero32a-h1","type":"hint","dependencies":[],"title":"Determine the number of positive real roots.","text":"Count the number of sign changes in f(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero32a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["2, 0"],"dependencies":["a5c95e8polyzero32a-h1"],"title":"Determine the number of positive real roots.","text":"The number of positive real zeros is either equal to the number of sign changes of f(x) or is less than the number of sign changes by an even integer. (List the number of possibilities in descending order like so: $$5$$, $$3$$, 1)\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero32a-h3","type":"hint","dependencies":[],"title":"Determine the number of negative real roots.","text":"Count the number of sign changes in $$f(-x)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero32a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["2, 0"],"dependencies":["a5c95e8polyzero32a-h3"],"title":"Determine the number of negative real roots.","text":"The number of negative real zeros is either equal to the number of sign changes of $$f(-x)$$ or is less than the number of sign changes by an even integer. (List the number of possibilities in descending order like so: $$5$$, $$3$$, 1)\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero32a-h5","type":"hint","dependencies":["a5c95e8polyzero32a-h1","a5c95e8polyzero32a-h3"],"title":"Different possibilities of roots. Consider that roots can be complex as well","text":"Recall that since the coefficients are real, by the Complex Conjugate Theorem, complex roots comes in pair as well. (There are $$4$$ different possibilities)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c95e8polyzero33","title":"Solving Polynomial Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Zeros of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c95e8polyzero33a","stepAnswer":["13, 9, 3"],"problemType":"TextBox","stepTitle":"A new bakery offers decorated sheet cakes for children\u2019s birthday parties and other special occasions. The bakery wants the volume of a small cake to be $$351$$ cubic inches. The cake is in the shape of a rectangular solid. They want the length of the cake to be four inches longer than the width of the cake and the height of the cake to be one-third of the width. What should the dimensions of the cake pan be?","stepBody":"List the answer by length, width then height. Example: $$10$$, $$8$$, $$12$$","answerType":"string","variabilization":{},"answerLatex":"$$13$$, $$9$$, $$3$$","hints":{"DefaultPathway":[{"id":"a5c95e8polyzero33a-h1","type":"hint","dependencies":[],"title":"Formulating the problem.","text":"Let l, w, $$h$$ denote length, width, height respectively. What is the volume, V, in terms of l, w, $$h$$? What is the relationship between length and width? What is the relationship between height and width?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero33a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["w + $$4$$"],"dependencies":["a5c95e8polyzero33a-h1"],"title":"Relationship between length and width.","text":"The length of the cake is four inches longer than the width. Express l in terms of w.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero33a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{w}{3}$$"],"dependencies":["a5c95e8polyzero33a-h1"],"title":"Relationship between height and width.","text":"The height of the cake is one-third of the width. Express $$h$$ in terms of w.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero33a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$l w h$$"],"dependencies":["a5c95e8polyzero33a-h1"],"title":"Relationship between volume and the parameters.","text":"Volume is given by the product of length, width and height. Express V in terms of l, w, $$h$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero33a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{w^3}{3}+\\\\frac{4w^2}{3}$$"],"dependencies":["a5c95e8polyzero33a-h4"],"title":"Expressing V in terms of w.","text":"We want a polynomial that is in terms of a single variable so that we are able to apply what we have learnt to solve it. Substitute the previous expressions that were found.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero33a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$w^3+7w^2-1053$$"],"dependencies":["a5c95e8polyzero33a-h5"],"title":"Substitute the given volume into the equation so that we can solve for the roots.","text":"Substitute V $$=$$ $$351$$ into V $$=$$ $$\\\\frac{w^3}{3}+\\\\frac{4w^2}{3}$$, then shift everything to a single side so that the equation is equal to $$0$$. Multiply by $$3$$ to remove the fractions so that calculation is easier. What is the polynomial function of w, f(w)? (Recall that the if k is a root of the function f(x), then f(k) $$=$$ $$0$$. In this case we want to make one side of the equation zero so that the polynomial function of w on the other side can be used to solve for the root.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero33a-h7","type":"hint","dependencies":["a5c95e8polyzero33a-h6"],"title":"How many positive real solutions are there?","text":"Since the width has to be a positive real number, we can use the Descartes\' Rule of Signs.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero33a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a5c95e8polyzero33a-h7"],"title":"How many positive real solutions are there?","text":"How many sign changes are there? The number of positive real zeros is either equal to the number of sign changes of f(x) or is less than the number of sign changes by an even integer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero33a-h9","type":"hint","dependencies":["a5c95e8polyzero33a-h8"],"title":"Find the possible rational zeros for the function using the Rational Zero Theorem.","text":"We can use the Rational Zero Theorem to tell us how many rational zeros that could potentially be roots. The Rational Zero Theorem states that, if the polynomial f(x) $$=$$ $$a_n x^n+a_n$$ - 1*x**(n - 1)+...+a_1*x+a_0 has integer coefficients, then every rational zero of f(x) has the form $$\\\\frac{p}{q}$$ where $$p$$ is a factor of the constant term $$a_0$$ and q is a factor of the leading coefficient $$a_n$$. When the leading coefficient is $$1$$, the possible rational zeros are the factors of the constant term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero33a-h10","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["1, 3, 9, 13, 27, 39, 81, 117, 351, 1053"],"dependencies":["a5c95e8polyzero33a-h9"],"title":"What are the factors of the constant term, 1053? From here on, these factors will be denoted as $$p$$, factors of the constant term. (List only the positive factors for this question. List them in ascending order as such: $$1$$, $$2$$, ...)","text":"There are $$20$$ factors in total, including the negative factors. We will only be listing the positive factors. There are $$10$$ of them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero33a-h11","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["1"],"dependencies":["a5c95e8polyzero33a-h9"],"title":"What are the factors of the leading coefficient, 1? From here on, these factors will be denoted as $$p$$, factors of the constant term. (List only the positive factors for this question. List them in ascending order as such: $$1$$, $$2$$, ...)","text":"There are $$2$$ factors in total, including the negative factors. We will only be listing the positive factors. There is $$1$$ of them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero33a-h12","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["1, 3, 9, 13, 27, 39, 81, 117, 351, 1053"],"dependencies":["a5c95e8polyzero33a-h9"],"title":"What are the different combination of rational zeros?","text":"Consider that a rational zero is of the form $$\\\\frac{p}{q}$$ where $$p$$ are factors of the constant term and q are factors of the leading coefficient. It would help to list out all your $$p$$ and q to write out the different combination. There are $$20$$ in total, including the negative rational zeros. We will only be listing the positive rational zeros. There are $$10$$ of them. (List them in ascending order as such: $$\\\\frac{1}{2}$$, $$1$$, ...)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero33a-h13","type":"hint","dependencies":["a5c95e8polyzero33a-h9"],"title":"Testing for roots.","text":"Use synthetic division to check if the potential zero is actually a root. We would want to start testing from the most logical values. Thus, we would ignore negative potential zeros and start from $$1$$. By Descartes\' Rule of Signs, how many roots are we looking for?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero33a-h14","type":"hint","dependencies":["a5c95e8polyzero33a-h13"],"title":"Testing for roots.","text":"Is w $$=$$ $$1$$ a root? A root would result in a remainder of zero.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero33a-h15","type":"hint","dependencies":["a5c95e8polyzero33a-h13"],"title":"Testing for roots.","text":"Is w $$=$$ $$3$$ a root? A root would result in a remainder of zero.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero33a-h16","type":"hint","dependencies":["a5c95e8polyzero33a-h13"],"title":"Testing for roots.","text":"Is w $$=$$ $$9$$ a root? A root would result in a remainder of zero.\\\\n##figure3.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero33a-h17","type":"hint","dependencies":["a5c95e8polyzero33a-h13"],"title":"Finding the other parameter, length and height.","text":"Substitute the w that we found into the expression for l and $$h$$ from before.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c95e8polyzero34","title":"Solving Polynomial Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Zeros of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c95e8polyzero34a","stepAnswer":["4,3,7"],"problemType":"TextBox","stepTitle":"A shipping container in the shape of a rectangular solid must have a volume of $$84$$ cubic meters. The client tells the manufacturer that, because of the contents, the length of the container must be one meter longer than the width, and the height must be one meter greater than twice the width. What should the dimensions of the container be?","stepBody":"List the answer by length, width then height. Example: 10,8,12","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a5c95e8polyzero34a-h1","type":"hint","dependencies":[],"title":"Formulating the problem.","text":"Let l, w, $$h$$ denote length, width, height respectively. What is the volume, V, in terms of l, w, $$h$$? What is the relationship between length and width? What is the relationship between height and width?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero34a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["w + $$1$$"],"dependencies":["a5c95e8polyzero34a-h1"],"title":"Relationship between length and width.","text":"The length of the container is one meter longer than the width. Express l in terms of w.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero34a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2w+1$$"],"dependencies":["a5c95e8polyzero34a-h1"],"title":"Relationship between height and width.","text":"The height of the container is one meter greater than twice the width. Express $$h$$ in terms of w.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero34a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$l w h$$"],"dependencies":["a5c95e8polyzero34a-h1"],"title":"Relationship between volumn and the parameters.","text":"Volume is given by the product of length, width and height. Express V in terms of l, w, $$h$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero34a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2w^3+3w^2+w$$"],"dependencies":["a5c95e8polyzero34a-h4"],"title":"Expressing V in terms of w.","text":"We want a polynomial that is in terms of a single variable so that we are able to apply what we have learnt to solve it. Substitute the previous expressions that were found.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero34a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2w^3+3w^2+w-84$$"],"dependencies":["a5c95e8polyzero34a-h5"],"title":"Substitute the given volume into the equation so that we can solve for the roots.","text":"Substitute V $$=$$ $$84$$ into V $$=$$ $$2w^3+3w^2+w$$, then shift everything to a single side so that the equation is equal to $$0$$. What is the polynomial function of w, f(w)? (Recall that the if k is a root of the function f(x), then f(k) $$=$$ $$0$$. In this case we want to make one side of the equation zero so that the polynomial function of w on the other side can be used to solve for the root.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero34a-h7","type":"hint","dependencies":[],"title":"How many positive real solutions are there?","text":"Relationship between volume and the parameters.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero34a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a5c95e8polyzero34a-h7"],"title":"How many positive real solutions are there?","text":"How many sign changes are there? The number of positive real zeros is either equal to the number of sign changes of f(x) or is less than the number of sign changes by an even integer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero34a-h9","type":"hint","dependencies":[],"title":"Find the possible rational zeros for the function using the Rational Zero Theorem.","text":"We can use the Rational Zero Theorem to tell us how many rational zeros that could potentially be roots. The Rational Zero Theorem states that, if the polynomial f(x) $$=$$ $$a_n x^n+a_n-1x^{n-1}+...+a_1 x+a_0$$ has integer coefficients, then every rational zero of f(x) has the form $$\\\\frac{p}{q}$$ where $$p$$ is a factor of the constant term $$a_0$$ and q is a factor of the leading coefficient $$a_n$$. When the leading coefficient is $$1$$, the possible rational zeros are the factors of the constant term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero34a-h10","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84"],"dependencies":["a5c95e8polyzero34a-h9"],"title":"What are the factors of the constant term, -84? From here on, these factors will be denoted as $$p$$, factors of the constant term. (List only the positive factors for this question. List them in ascending order as such: $$1$$, $$2$$, ...)","text":"There are $$24$$ factors in total, including the negative factors. We will only be listing the positive factors. There are $$12$$ of them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero34a-h11","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["1, 2"],"dependencies":["a5c95e8polyzero34a-h9"],"title":"What are the factors of the leading coefficient, 2? From here on, these factors will be denoted as $$p$$, factors of the constant term. (List only the positive factors for this question. List them in ascending order as such: $$1$$, $$2$$, ...)","text":"There are $$4$$ factors in total, including the negative factors. We will only be listing the positive factors. There are $$2$$ of them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero34a-h12","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["1, 3/2, 2, 3, 7/2, 4, 6, 7, 21/2, 12, 14, 21, 28, 42, 84"],"dependencies":["a5c95e8polyzero34a-h9"],"title":"What are the different combination of rational zeros?","text":"Consider that a rational zero is of the form $$\\\\frac{p}{q}$$ where $$p$$ are factors of the constant term and q are factors of the leading coefficient. It would help to list out all your $$p$$ and q to write out the different combination. There are $$30$$ in total, including the negative rational zeros. We will only be listing the positive rational zeros. There are $$15$$ of them. (List them in ascending order as such: $$\\\\frac{1}{2}$$, $$1$$, ...)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero34a-h13","type":"hint","dependencies":["a5c95e8polyzero34a-h9"],"title":"Testing for roots.","text":"Use synthetic division to check if the potential zero is actually a root. We would want to start testing from the most logical values. Thus, we would ignore negative potential zeros and start from $$1$$. By Descartes\' Rule of Signs, how many roots are we looking for?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero34a-h14","type":"hint","dependencies":["a5c95e8polyzero34a-h13"],"title":"Finding the other parameter, length and height.","text":"Substitute the w that we found into the expression for l and $$h$$ from before.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c95e8polyzero35","title":"Using the Factor Theorem to Find the Zeros of a Polynomial Expression","body":"According to the Factor Theorem, k is a zero of f(x) if and only if $$(x-k)$$ is a factor of f(x).","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Zeros of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c95e8polyzero35a","stepAnswer":["-2, 3, 5"],"problemType":"TextBox","stepTitle":"Show that $$x+2$$ is a factor of $$x^3-6x^2-x+30$$. Find the remaining factors. Use the factors to determine the zeros of the polynomial. (List the zeros in ascending order like so: $$1$$, $$2$$, 3)","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-2$$, $$3$$, $$5$$","hints":{"DefaultPathway":[{"id":"a5c95e8polyzero35a-h1","type":"hint","dependencies":[],"title":"Deciding on a Method of Division","text":"We can use synthetic division to show that $$x+2$$ is a factor of the polynomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero35a-h2","type":"hint","dependencies":["a5c95e8polyzero35a-h1"],"title":"Using Synthetic Division","text":"Observe that the remainder is zero, so $$x+2$$ is a factor of the polynomial. The quotient is $$x^2-8x$$ + $$15$$. Thus, we can use the Division Algorithm to write the polynomial as the product of the divisor and the quotient $$\\\\left(x+2\\\\right) \\\\left(x^2-8x+15\\\\right)$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero35a-h3","type":"hint","dependencies":["a5c95e8polyzero35a-h2"],"title":"We can factor the quadratic polynomial $$x^2-8x+15$$","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero35a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["3, 5"],"dependencies":["a5c95e8polyzero35a-h3"],"title":"Finding the zeroes of $$x^2-8x+15$$. (List the zeros in ascending order like so: $$1$$, $$2$$, 3)","text":"We can do so by either factoring, using the quadratic formula $$x=\\\\frac{\\\\left(-b+-\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$, or by expressing the polynomial in terms of the standard form $$f(x)=a {\\\\left(x-h\\\\right)}^2+k$$ and solving for $$x$$ when $$f(x)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero35a-h5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["-2, 3, 5"],"dependencies":["a5c95e8polyzero35a-h4"],"title":"What are all the zeros that we have found? (List the zeros in ascending order like so: $$1$$, $$2$$, 3)","text":"Rewriting the polynomial after finding all the zeroes, we get $$\\\\left(x+2\\\\right) \\\\left(x-3\\\\right) \\\\left(x-5\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c95e8polyzero6","title":"Using the Remainder Theorem to Find the Remainder","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Zeros of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c95e8polyzero6a","stepAnswer":["$$-6$$"],"problemType":"TextBox","stepTitle":"Find the remainder of: $$\\\\frac{x^4-9x^2+14}{x-2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-6$$","hints":{"DefaultPathway":[{"id":"a5c95e8polyzero6a-h1","type":"hint","dependencies":[],"title":"Remainder Theorem Definition","text":"The remainder theorem is as follows: if f(x) is being divided by $$(x-a)$$, then the remainder is equal to f(a). This means that we must plug in $$x=2$$ into our dividend.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero6a-h2","type":"hint","dependencies":["a5c95e8polyzero6a-h1"],"title":"Plugging In","text":"$$x^4-9x^2+14$$ becomes $$2^4-{9\\\\left(2\\\\right)}^2+14$$ after plugging in. This simplifies to $$06$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c95e8polyzero7","title":"Using the Remainder Theorem to Find the Remainder","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Zeros of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c95e8polyzero7a","stepAnswer":["$$-106$$"],"problemType":"TextBox","stepTitle":"Find the remainder of: $$\\\\frac{3x^3-2x^2+x-4}{x+3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-106$$","hints":{"DefaultPathway":[{"id":"a5c95e8polyzero7a-h1","type":"hint","dependencies":[],"title":"Remainder Theorem Definition","text":"The remainder theorem is as follows: if f(x) is being divided by $$(x-a)$$, then the remainder is equal to f(a). This means that we must plug in $$x=-3$$ into our dividend.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero7a-h2","type":"hint","dependencies":["a5c95e8polyzero7a-h1"],"title":"Plugging In","text":"$$3x^3-2x^2+x-4$$ becomes 3(-3)**3-2(-3)**2+(-3)-4 after plugging in. This simplifies to $$-106$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c95e8polyzero8","title":"Using the Remainder Theorem to Find the Remainder","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Zeros of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c95e8polyzero8a","stepAnswer":["$$-17$$"],"problemType":"TextBox","stepTitle":"Find the remainder of: $$\\\\frac{x^4+5x^3-4x-17}{x+1}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-17$$","hints":{"DefaultPathway":[{"id":"a5c95e8polyzero8a-h1","type":"hint","dependencies":[],"title":"Remainder Theorem Definition","text":"The remainder theorem is as follows: if f(x) is being divided by $$(x-a)$$, then the remainder is equal to f(a). This means that we must plug in $$x=-1$$ into our dividend.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero8a-h2","type":"hint","dependencies":["a5c95e8polyzero8a-h1"],"title":"Plugging In","text":"$$x^4+5x^3-4x-17$$ becomes (-1)**4+5(-1)**3-4(-1)-17. This means the remaindere is $$-17$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c95e8polyzero9","title":"Using the Remainder Theorem to Find the Remainder","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.5 Zeros of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a5c95e8polyzero9a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"Find the remainder of: $$\\\\frac{\\\\left(-3x^2+6x+24\\\\right)}{x-4}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a5c95e8polyzero9a-h1","type":"hint","dependencies":[],"title":"Remainder Theorem Definition","text":"The remainder theorem is as follows: if f(x) is being divided by $$(x-a)$$, then the remainder is equal to f(a). This means that we must plug in $$x=4$$ into our dividend.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c95e8polyzero9a-h2","type":"hint","dependencies":["a5c95e8polyzero9a-h1"],"title":"Plugging In","text":"$$-3x^2+6x+24$$ becomes $$-\\\\left({3\\\\left(4\\\\right)}^2\\\\right)+6\\\\left(4\\\\right)+24$$. This simplifies to $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c984cfdist1","title":"F Distribution","body":"As part of an experiment to see how different types of soil cover would affect slicing tomato production, Marist College students grew tomato plants under different soil cover conditions. Groups of three plants each had one of the following treatments: bare soil, a commercial ground cover, black plastic, straw, or compost. All plants grew under the same conditions and were the same variety. Students recorded the weight (in grams) of tomatoes produced by each of the $$n$$ $$=$$ $$15$$ plants in Table $$13.4$$. The means of the tomato yields under the five mulching conditions are represented by \u03bc1, \u03bc2, \u03bc3, \u03bc4, \u03bc5.\\\\n##figure3.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"13.3 Facts About the F Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a5c984cfdist1a","stepAnswer":["Different"],"problemType":"MultipleChoice","stepTitle":"Are all the means the same or is at least one different? Use a significance level of 5% to test the null hypothesis that there is no difference in mean yields among the five groups against the alternative hypothesis that at least one mean is different from the rest.","stepBody":"","answerType":"string","variabilization":{},"choices":["Same","Different"],"hints":{"DefaultPathway":[{"id":"a5c984cfdist1a-h1","type":"hint","dependencies":[],"title":"Null and Alternative Hypotheses","text":"The null and alternative hypotheses are the following, respectively: $$H_0$$: \u03bc1 $$=$$ \u03bc2 $$=$$ \u03bc3 $$=$$ \u03bc4 $$=$$ \u03bc5 and $$H_a$$: \u03bci $$ \\\\neq $$ \u03bcj some i $$ \\\\neq $$ j","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c984cfdist1a-h2","type":"hint","dependencies":["a5c984cfdist1a-h1"],"title":"Distribution","text":"Distribution for the test: df(num) $$=$$ $$5$$ - $$1$$ $$=$$ $$4$$ and df(denom) $$=$$ $$15$$ - $$5$$ $$=$$ $$10$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c984cfdist1a-h3","type":"hint","dependencies":["a5c984cfdist1a-h2"],"title":"Statistic F","text":"The F statistic is a ratio (a fraction). There are two sets of degrees of freedom: one for the numerator and one for the denominator. To solve for F, find the numerator mean square divided by the denominator mean square. If you would like, you can use a calculator by putting the data in the table into lists L1, L2, L3, and L4. Press STAT and arrow over to TESTS. Arrow down to F:ANOVA. Press ENTER and Enter (L1,L2,L3,L4). The calculator displays the F statistic, the $$p-value$$ and the values for the one-way ANOVA table.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c984cfdist1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4.481$$"],"dependencies":["a5c984cfdist1a-h3"],"title":"Solving for F","text":"What is F? Round to three decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5c984cfdist1a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4.481$$"],"dependencies":[],"title":"Plugging into F","text":"What is 9,162,140/2,044,672.6? Round to three decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a5c984cfdist1a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.0248$$"],"dependencies":["a5c984cfdist1a-h4"],"title":"Probability Statement","text":"What is $$p-value$$ $$=$$ P(F > $$4.481)$$? Round to four decimal places.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c984cfdist1a-h6","type":"hint","dependencies":["a5c984cfdist1a-h5"],"title":"Compare \ud835\udefc and the $$p-value$$","text":"Since we are testing at a significance level of 5%, \ud835\udefc $$=$$ $$0.05$$. The $$p-value$$ $$=$$ $$0.0248$$. If \ud835\udefc > $$p-value$$, reject $$H_0$$ if \ud835\udefc < $$p-value$$, accept $$H_0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c984cfdist1a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a5c984cfdist1a-h6"],"title":"Rejecting or Accepting the Null Hypothesis","text":"Is \ud835\udefc > $$p-value$$ such that you reject the null hypothesis?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a5c984cfdist1a-h8","type":"hint","dependencies":["a5c984cfdist1a-h7"],"title":"Conclusion","text":"At the 5% significance level, we have reasonably strong evidence that differences in mean yields for slicing tomato plants grown under different mulching conditions are unlikely to be due to chance alone. We may conclude that at least some of mulches led to different mean yields.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c984cfdist2","title":"F Distribution","body":"Four sororities took a random sample of sisters regarding their grade means for the past term. The results are shown in Table 13.7.\\\\n##figure3.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"13.3 Facts About the F Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a5c984cfdist2a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Using a significance level of 1%, is there a difference in mean grades among the sororities?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a5c984cfdist2a-h1","type":"hint","dependencies":[],"title":"Understanding the Problem","text":"Let \u03bc1, \u03bc2, \u03bc3, \u03bc4 be the population means of the sororities. Remember that the null hypothesis claims that the sorority groups are from the same normal distribution. The alternate hypothesis says that at least two of the sorority groups come from populations with different normal distributions. Notice that the four sample sizes are each five.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c984cfdist2a-h2","type":"hint","dependencies":["a5c984cfdist2a-h1"],"title":"Null and Alternative Hypotheses","text":"The null and alternative hypotheses are the following, respectively: $$H_0$$: \u03bc1 $$=$$ \u03bc2 $$=$$ \u03bc3 $$=$$ \u03bc4 and $$H_a$$: Not all of the means \u03bc1, \u03bc2, \u03bc3, \u03bc4 are equal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c984cfdist2a-h3","type":"hint","dependencies":["a5c984cfdist2a-h2"],"title":"Test Distribution","text":"The distribution for the test: k $$=$$ $$4$$ groups and $$n$$ $$=$$ $$20$$ samples in total where df(num) $$=$$ k - $$1$$ $$=$$ $$4$$ - $$1$$ $$=$$ $$3$$ and df(denom) $$=$$ $$n$$ - k $$=$$ $$20$$ - $$4$$ $$=$$ $$16$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c984cfdist2a-h4","type":"hint","dependencies":["a5c984cfdist2a-h3"],"title":"Statistic F","text":"The F test statistic is a ratio (a fraction). There are two sets of degrees of freedom: one for the numerator and one for the denominator. To solve for F, find the numerator mean square divided by the denominator mean square, where SS is the sum of squares. Using a calculator, put the data in the table into lists L1, L2, L3, and L4. Press STAT and arrow over to TESTS. Arrow down to F:ANOVA. Press ENTER and Enter (L1,L2,L3,L4). The calculator displays the F statistic, the $$p-value$$ and the values for the one-way ANOVA table.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c984cfdist2a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.23$$"],"dependencies":["a5c984cfdist2a-h4"],"title":"Solving for F","text":"What is F? Round to the nearest hundredths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c984cfdist2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1241$$"],"dependencies":["a5c984cfdist2a-h5"],"title":"Probability Statement","text":"What is $$p-value$$ $$=$$ P(F > $$2.23)$$? Round to four decimal places.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c984cfdist2a-h7","type":"hint","dependencies":["a5c984cfdist2a-h6"],"title":"Compare \ud835\udefc and the $$p-value$$","text":"Since we are testing at a significance level of 1%, \ud835\udefc $$=$$ $$0.01$$. The $$p-value$$ $$=$$ $$0.1241$$. If \ud835\udefc > $$p-value$$, reject $$H_0$$ if \ud835\udefc < $$p-value$$, accept $$H_0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c984cfdist2a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a5c984cfdist2a-h7"],"title":"Rejecting or Accepting the Null Hypothesis","text":"Is \ud835\udefc > $$p-value$$ such that you reject the null hypothesis?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a5c984cfdist2a-h9","type":"hint","dependencies":["a5c984cfdist2a-h8"],"title":"Conclusion","text":"There is not sufficient evidence to conclude that there is a difference among the mean grades for the sororities.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5c984cfdist3","title":"F Distribution","body":"A fourth grade class is studying the environment. One of the assignments is to grow bean plants in different soils. Tommy chose to grow his bean plants in soil found outside his classroom mixed with dryer lint. Tara chose to grow her bean plants in potting soil bought at the local nursery. Nick chose to grow his bean plants in soil from his mother\'s garden. No chemicals were used on the plants, only water. They were grown inside the classroom next to a large window. Each child grew five plants. At the end of the growing period, each plant was measured, producing the data (in inches) in Table 13.9.\\\\n##figure4.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"13.3 Facts About the F Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a5c984cfdist3a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Does it appear that the three media in which the bean plants were grown produce the same mean height? Test at a 3% level of significance.","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a5c984cfdist3a-h1","type":"hint","dependencies":[],"title":"Understanding the Problem","text":"We will perform the calculations that lead to the F\' statistic. Notice that each group has the same number of plants, so we will use the following formula:\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c984cfdist3a-h2","type":"hint","dependencies":["a5c984cfdist3a-h1"],"title":"Calculations","text":"First, calculate the sample mean and sample variance of each group.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c984cfdist3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$24.2$$"],"dependencies":["a5c984cfdist3a-h2"],"title":"Tommy\'s Sample Mean","text":"What is Tommy\'s sample mean?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5c984cfdist3a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$24.2$$"],"dependencies":[],"title":"Calculating Tommy\'s Sample Mean","text":"Add up the Tommy\'s Plants column and divide by the total number of Tommy\'s plants: what is (24 + $$21$$ + $$23$$ + $$30$$ + 23)/5?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a5c984cfdist3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11.7$$"],"dependencies":["a5c984cfdist3a-h2"],"title":"Tommy\'s Sample Variance","text":"What is Tommy\'s sample variance?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c984cfdist3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25.4$$"],"dependencies":["a5c984cfdist3a-h2"],"title":"Tara\'s Sample Mean","text":"What is Tara\'s sample mean?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5c984cfdist3a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25.4$$"],"dependencies":[],"title":"Calculating Tara\'s Sample Mean","text":"Add up the Tara\'s Plants column and divide by the total number of Tara\'s plants: what is (25 + $$31$$ + $$23$$ + $$20$$ + 28)/5?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a5c984cfdist3a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18.3$$"],"dependencies":["a5c984cfdist3a-h2"],"title":"Tara\'s Sample Variance","text":"What is Tara\'s sample variance?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c984cfdist3a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$24.4$$"],"dependencies":["a5c984cfdist3a-h2"],"title":"Nick\'s Sample Mean","text":"What is Nick\'s sample mean?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5c984cfdist3a-h7-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$24.4$$"],"dependencies":[],"title":"Calculating Nick\'s Sample Mean","text":"Add up the Nick\'s Plants column and divide by the total number of Nick\'s plants: what is (23 + $$27$$ + $$22$$ + $$30$$ + 20)/5?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a5c984cfdist3a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16.3$$"],"dependencies":["a5c984cfdist3a-h2"],"title":"Nick\'s Sample Variance","text":"What is Nick\'s sample variance?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c984cfdist3a-h9","type":"hint","dependencies":["a5c984cfdist3a-h3","a5c984cfdist3a-h4","a5c984cfdist3a-h5","a5c984cfdist3a-h6","a5c984cfdist3a-h7","a5c984cfdist3a-h8"],"title":"Sample Mean and Variance","text":"The sample means and variances are as follows:\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c984cfdist3a-h10","type":"hint","dependencies":["a5c984cfdist3a-h9"],"title":"Variance of Group Means","text":"Next, calculate the variance of the three group means (calculate the variance of $$24.2$$, $$25.4$$, and $$24.4)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c984cfdist3a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.413$$"],"dependencies":["a5c984cfdist3a-h10"],"title":"Calculating the Variance of Group Means","text":"What is the variance of the three group means? Round to the thousandths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c984cfdist3a-h12","type":"hint","dependencies":["a5c984cfdist3a-h11"],"title":"Mean of Sample Variances","text":"$$MS_{between}$$ $$=$$ $${s_{x\u0304}}^2$$ $$=$$ $$(5)(0.413)$$ where $$n$$ $$=$$ $$5$$ is the sample size (number of plants each child grew). Calculate the mean of the three sample variances (calculate the mean of $$11.7$$, $$18.3$$, and $$16.3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c984cfdist3a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15.433$$"],"dependencies":["a5c984cfdist3a-h12"],"title":"Calculating the Mean of Sample Variances","text":"What is the mean of the sample variances? In other words, what is $$s^2$$ pooled? Round to the thousandths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c984cfdist3a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.134$$"],"dependencies":["a5c984cfdist3a-h13"],"title":"F Statistic","text":"The dfs for the numerator $$=$$ the number of groups - $$1$$ $$=$$ $$3$$ - $$1$$ $$=$$ $$2$$, and the dfs for the denominator $$=$$ the total number of samples - the number of groups $$=$$ $$15$$ - $$3$$ $$=$$ $$12$$. The F statistic (or F ratio) is the following. Plugging in your solved values, what is F? Round to the thousandths place.\\\\n##figure3.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5c984cfdist3a-h14-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.134$$"],"dependencies":[],"title":"Calculating F","text":"What is $$\\\\frac{{s_{x\u0304}}^2}{s^2}$$ pooled $$=$$ $$\\\\frac{5\\\\times0.413}{15.433}$$? Round to the thousandths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a5c984cfdist3a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.8759$$"],"dependencies":["a5c984cfdist3a-h14"],"title":"Probability Statement","text":"What is $$p-value$$ $$=$$ P(F > $$0.134)$$? Round to four decimal places. To calculate the $$p-value$$ on a calculator, press 2nd DISTR, arrow down to Fcdf(and press ENTER), enter $$(0.134$$, E99, $$2$$, 12), and press ENTER.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c984cfdist3a-h16","type":"hint","dependencies":["a5c984cfdist3a-h15"],"title":"Compare \ud835\udefc and the $$p-value$$","text":"Since we are testing at a significance level of 3%, \ud835\udefc $$=$$ $$0.03$$. The $$p-value$$ $$=$$ $$0.8759$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5c984cfdist3a-h17","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Not reject"],"dependencies":["a5c984cfdist3a-h16"],"title":"Rejecting or Accepting the Null Hypothesis","text":"Do you reject or not reject the null hypothesis?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Reject","Not reject"]},{"id":"a5c984cfdist3a-h18","type":"hint","dependencies":["a5c984cfdist3a-h17"],"title":"Conclusion","text":"With a 3% level of significance, from the sample data, the evidence is not sufficient to conclude that the mean heights of the bean plants are different.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5ccaf6logarithmsandlogistic1","title":"Solving Logarithmic Equations #1","body":"A logarithmic model is given by the equation $$h(p)=67.682-5.792ln(p)$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.8 Fitting Exponential Models to Data","courseName":"OpenStax: College Algebra","steps":[{"id":"a5ccaf6logarithmsandlogistic1a","stepAnswer":["$$2.67$$"],"problemType":"TextBox","stepTitle":"To the nearest hundredth, for what value of $$p$$ does $$h(p)=62$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2.67$$","hints":{"DefaultPathway":[{"id":"a5ccaf6logarithmsandlogistic1a-h1","type":"hint","dependencies":[],"title":"Isolating $$p$$","text":"To solve this problem, isolate $$p$$ to the left side and plug in the value $$h(p)=62$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5ccaf6logarithmsandlogistic1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$p=e^{\\\\frac{5.682}{5.792}}$$"],"dependencies":["a5ccaf6logarithmsandlogistic1a-h1"],"title":"Identifying the Rewritten Equation","text":"What do you get after rewriting the equation to isolate $$p$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5ccaf6logarithmsandlogistic10","title":"Solving Logistic Equations #9","body":"The population P of an endangered species habitat for wolves is modeled by the function $$P(x)=\\\\frac{558}{1+54.8e^{-0.462 x}}$$, where $$x$$ is given in years.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.8 Fitting Exponential Models to Data","courseName":"OpenStax: College Algebra","steps":[{"id":"a5ccaf6logarithmsandlogistic10a","stepAnswer":["$$38$$"],"problemType":"TextBox","stepTitle":"How many wolves will the habitat have after three years? Round to the nearest whole number.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$38$$","hints":{"DefaultPathway":[{"id":"a5ccaf6logarithmsandlogistic10a-h1","type":"hint","dependencies":[],"title":"Identifying What Each Term Represents","text":"$$x$$ represents the number of months that have passed, and P(x) represents the number of wolves in the habitat at $$x$$ months.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5ccaf6logarithmsandlogistic10a-h2","type":"hint","dependencies":["a5ccaf6logarithmsandlogistic10a-h1"],"title":"Converting From Yeras to Months","text":"A year has $$12$$ months, so one and a half years has $$12+6=18$$ months.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5ccaf6logarithmsandlogistic11","title":"Solving Logistic Equations #10","body":"The population P of an endangered species habitat for wolves is modeled by the function $$P(x)=\\\\frac{558}{1+54.8e^{-0.462 x}}$$, where $$x$$ is given in years.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.8 Fitting Exponential Models to Data","courseName":"OpenStax: College Algebra","steps":[{"id":"a5ccaf6logarithmsandlogistic11a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"How many years will it take before there are $$100$$ wolves in the pond? Round to the nearest whole number.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"a5ccaf6logarithmsandlogistic11a-h1","type":"hint","dependencies":[],"title":"Identifying What Represents the Answer","text":"In this problem, we are trying to find what $$x$$ is when $$P(x)=100$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5ccaf6logarithmsandlogistic11a-h2","type":"hint","dependencies":["a5ccaf6logarithmsandlogistic11a-h1"],"title":"Isolating $$x$$","text":"After plugging in $$P(x)=100$$, we get $$100=\\\\frac{558}{1+54.8e^{-0.462 x}}$$. After isolating $$x$$, we will get the answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5ccaf6logarithmsandlogistic12","title":"Matching Graphs With Equations #1","body":"Match the given scatterplot with the function of best fit.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.8 Fitting Exponential Models to Data","courseName":"OpenStax: College Algebra","steps":[{"id":"a5ccaf6logarithmsandlogistic12a","stepAnswer":["$$y={\\\\operatorname{2.104}\\\\left(1.479\\\\right)}^x$$"],"problemType":"MultipleChoice","stepTitle":"Which function matches this graph?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$y={\\\\operatorname{2.104}\\\\left(1.479\\\\right)}^x$$","choices":["$$y=10.209e^{-0.294 x}$$","$$y=5.598-1.912ln(x)$$","$$y={\\\\operatorname{2.104}\\\\left(1.479\\\\right)}^x$$","$$y=4.607+2.733\\\\ln(x)$$","$$y=\\\\frac{14.005}{1+2.79e^{-0.812 x}}$$"],"hints":{"DefaultPathway":[{"id":"a5ccaf6logarithmsandlogistic12a-h1","type":"hint","dependencies":[],"title":"Plugging in Values","text":"You can plug in values of an x-coordinate to each equation to see if the output is the same as in the graph.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5ccaf6logarithmsandlogistic12a-h2","type":"hint","dependencies":["a5ccaf6logarithmsandlogistic12a-h1"],"title":"Inferring the Shape of the Graph","text":"Think about what conditions have to be true for an exponential, logarithmic, or logistic graph to increase or decrease.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5ccaf6logarithmsandlogistic13","title":"Matching Graphs With Equations #2","body":"Match the given scatterplot with the function of best fit.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.8 Fitting Exponential Models to Data","courseName":"OpenStax: College Algebra","steps":[{"id":"a5ccaf6logarithmsandlogistic13a","stepAnswer":["$$y=4.607+2.733\\\\ln(x)$$"],"problemType":"MultipleChoice","stepTitle":"Which function matches this graph?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$y=4.607+2.733\\\\ln(x)$$","choices":["$$y=10.209e^{-0.294 x}$$","$$y=5.598-1.912ln(x)$$","$$y={\\\\operatorname{2.104}\\\\left(1.479\\\\right)}^x$$","$$y=4.607+2.733\\\\ln(x)$$","$$y=\\\\frac{14.005}{1+2.79e^{-0.812 x}}$$"],"hints":{"DefaultPathway":[{"id":"a5ccaf6logarithmsandlogistic13a-h1","type":"hint","dependencies":[],"title":"Plugging in Values","text":"You can plug in values of an x-coordinate to each equation to see if the output is the same as in the graph.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5ccaf6logarithmsandlogistic13a-h2","type":"hint","dependencies":["a5ccaf6logarithmsandlogistic13a-h1"],"title":"Inferring the Shape of the Graph","text":"Think about what conditions have to be true for an exponential, logarithmic, or logistic graph to increase or decrease.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5ccaf6logarithmsandlogistic14","title":"Matching Graphs With Equations #3","body":"Match the given scatterplot with the function of best fit.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.8 Fitting Exponential Models to Data","courseName":"OpenStax: College Algebra","steps":[{"id":"a5ccaf6logarithmsandlogistic14a","stepAnswer":["$$y=5.598-1.912ln(x)$$"],"problemType":"MultipleChoice","stepTitle":"Which function matches this graph?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$y=5.598-1.912ln(x)$$","choices":["$$y=10.209e^{-0.294 x}$$","$$y=5.598-1.912ln(x)$$","$$y={\\\\operatorname{2.104}\\\\left(1.479\\\\right)}^x$$","$$y=4.607+2.733\\\\ln(x)$$","y=14.005/(1+2.79e**(-.812x)"],"hints":{"DefaultPathway":[{"id":"a5ccaf6logarithmsandlogistic14a-h1","type":"hint","dependencies":[],"title":"Plugging in Values","text":"You can plug in values of an x-coordinate to each equation to see if the output is the same as in the graph.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5ccaf6logarithmsandlogistic14a-h2","type":"hint","dependencies":["a5ccaf6logarithmsandlogistic14a-h1"],"title":"Inferring the Shape of the Graph","text":"Think about what conditions have to be true for an exponential, logarithmic, or logistic graph to increase or decrease.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5ccaf6logarithmsandlogistic15","title":"Matching Graphs With Equations #4","body":"Match the given scatterplot with the function of best fit.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.8 Fitting Exponential Models to Data","courseName":"OpenStax: College Algebra","steps":[{"id":"a5ccaf6logarithmsandlogistic15a","stepAnswer":["$$y=\\\\frac{14.005}{1+2.79e^{-0.812 x}}$$"],"problemType":"MultipleChoice","stepTitle":"Which function matches this graph?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$y=\\\\frac{14.005}{1+2.79e^{-0.812 x}}$$","choices":["$$y=10.209e^{-0.294 x}$$","$$y=5.598-1.912ln(x)$$","$$y={\\\\operatorname{2.104}\\\\left(1.479\\\\right)}^x$$","$$y=4.607+2.733\\\\ln(x)$$","$$y=\\\\frac{14.005}{1+2.79e^{-0.812 x}}$$"],"hints":{"DefaultPathway":[{"id":"a5ccaf6logarithmsandlogistic15a-h1","type":"hint","dependencies":[],"title":"Plugging in Values","text":"You can plug in values of an x-coordinate to each equation to see if the output is the same as in the graph.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5ccaf6logarithmsandlogistic15a-h2","type":"hint","dependencies":["a5ccaf6logarithmsandlogistic15a-h1"],"title":"Inferring the Shape of the Graph","text":"Think about what conditions have to be true for an exponential, logarithmic, or logistic graph to increase or decrease.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5ccaf6logarithmsandlogistic16","title":"Matching Graphs With Equations #5","body":"Match the given scatterplot with the function of best fit.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.8 Fitting Exponential Models to Data","courseName":"OpenStax: College Algebra","steps":[{"id":"a5ccaf6logarithmsandlogistic16a","stepAnswer":["$$y=10.209e^{-0.294 x}$$"],"problemType":"MultipleChoice","stepTitle":"Which function matches this graph?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$y=10.209e^{-0.294 x}$$","choices":["$$y=10.209e^{-0.294 x}$$","$$y=5.598-1.912ln(x)$$","$$y={\\\\operatorname{2.104}\\\\left(1.479\\\\right)}^x$$","$$y=4.607+2.733\\\\ln(x)$$","y=14.005/(1+2.79e**(-.812x)"],"hints":{"DefaultPathway":[{"id":"a5ccaf6logarithmsandlogistic16a-h1","type":"hint","dependencies":[],"title":"Plugging in Values","text":"You can plug in values of an x-coordinate to each equation to see if the output is the same as in the graph.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5ccaf6logarithmsandlogistic16a-h2","type":"hint","dependencies":["a5ccaf6logarithmsandlogistic16a-h1"],"title":"Inferring the Shape of the Graph","text":"Think about what conditions have to be true for an exponential, logarithmic, or logistic graph to increase or decrease.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5ccaf6logarithmsandlogistic17","title":"Solving Logistic Equations #11","body":"A population is modeled by the logistic equation $$P(t)=\\\\frac{175}{1+6.995e^{-0.68 t}}$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.8 Fitting Exponential Models to Data","courseName":"OpenStax: College Algebra","steps":[{"id":"a5ccaf6logarithmsandlogistic17a","stepAnswer":["$$22$$"],"problemType":"TextBox","stepTitle":"To the nearest whole number, what is the initial value?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$22$$","hints":{"DefaultPathway":[{"id":"a5ccaf6logarithmsandlogistic17a-h1","type":"hint","dependencies":[],"title":"How to Find the Initial Value","text":"The initial value is P(t) when $$t=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5ccaf6logarithmsandlogistic18","title":"Solving Logistic Equations #12","body":"A population is modeled by the logistic equation $$P(t)=\\\\frac{175}{1+6.995e^{-0.68 t}}$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.8 Fitting Exponential Models to Data","courseName":"OpenStax: College Algebra","steps":[{"id":"a5ccaf6logarithmsandlogistic18a","stepAnswer":["$$175$$"],"problemType":"TextBox","stepTitle":"What is the carrying capacity?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$175$$","hints":{"DefaultPathway":[{"id":"a5ccaf6logarithmsandlogistic18a-h1","type":"hint","dependencies":[],"title":"Carrying Capacity of a Logistic Equation","text":"For a logistic equation $$y=\\\\frac{c}{1+{ae}^{\\\\left(-bx\\\\right)}}$$, the carrying capacity is c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5ccaf6logarithmsandlogistic19","title":"Using Exponential Regression to Fit a Model to Data","body":"In $$2007$$, a university study was published investigating the crash risk of alcohol impaired driving. Data from 2,871 crashes were used to measure the association of a person\u2019s blood alcohol level (BAC) with the risk of being in an accident. The attached table shows results from the study. The relative risk is a measure of how many times more likely a person is to crash. So, for example, a person with a BAC of $$0.09$$ is $$3.54$$ times as likely to crash as a person who has not been drinking alcohol.\\\\n##figure2.gif","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.8 Fitting Exponential Models to Data","courseName":"OpenStax: College Algebra","steps":[{"id":"a5ccaf6logarithmsandlogistic19a","stepAnswer":["$$y={\\\\operatorname{0.58304829}\\\\left(22072021300\\\\right)}^x$$"],"problemType":"MultipleChoice","stepTitle":"Let $$x$$ represent the BAC level, and let $$y$$ represent the corresponding relative risk. Use exponential regression to fit a model to these data.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y={\\\\operatorname{0.58304829}\\\\left(22072021300\\\\right)}^x$$","choices":["$$y={\\\\operatorname{0.58276929}\\\\left(25085021300\\\\right)}^x$$","$$y={\\\\operatorname{0.58304829}\\\\left(22072021300\\\\right)}^x$$","$$y={\\\\operatorname{0.5865229}\\\\left(21073621760\\\\right)}^x$$"],"hints":{"DefaultPathway":[{"id":"a5ccaf6logarithmsandlogistic19a-h1","type":"hint","dependencies":[],"title":"Verifying the Scatterplot\'s Pattern","text":"Using the STAT then EDIT menu on a graphing utility, list the BAC values in L1 and the relative risk values in L2. Then use the STATPLOT feature to verify that the scatterplot follows the exponential pattern shown in the attached figure.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5ccaf6logarithmsandlogistic19a-h2","type":"hint","dependencies":["a5ccaf6logarithmsandlogistic19a-h1"],"title":"Using ExpReg on the Calculator","text":"Use the \u201cExpReg\u201d command from the STAT then CALC menu to obtain the exponential model, y=0.58304829(22,072,021,300)**x.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5ccaf6logarithmsandlogistic2","title":"Solving Logistic Equations #1","body":"A logistic model is given by the equation $$P(t)=\\\\frac{90}{1+5e^{-0.42 t}}$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.8 Fitting Exponential Models to Data","courseName":"OpenStax: College Algebra","steps":[{"id":"a5ccaf6logarithmsandlogistic2a","stepAnswer":["$$3.83$$"],"problemType":"TextBox","stepTitle":"To the nearest hundredth, for what value of $$t$$ does $$P(t)=45$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3.83$$","hints":{"DefaultPathway":[{"id":"a5ccaf6logarithmsandlogistic2a-h1","type":"hint","dependencies":[],"title":"Subsituting in P(t)","text":"Subsitute in $$P(t)=45$$ and solve for $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5ccaf6logarithmsandlogistic20","title":"Using Logarithmic Regression to Fit a Model to Data","body":"Due to advances in medicine and higher standards of living, life expectancy has been increasing in most developed countries since the beginning of the 20th century. The attached table shows the average life expectancies, in years, of Americans from 1900-2010.\\\\n##figure2.gif","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.8 Fitting Exponential Models to Data","courseName":"OpenStax: College Algebra","steps":[{"id":"a5ccaf6logarithmsandlogistic20a","stepAnswer":["$$y=42.52722583+13.85752327\\\\ln(x)$$"],"problemType":"MultipleChoice","stepTitle":"Let $$x$$ represent time in decades starting with $$x=1$$ for the year $$1900$$, $$x=2$$ for the year $$1910$$, and so on. Let $$y$$ represent the corresponding life expectancy. Use logarithmic regression to fit a model to these data.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=42.52722583+13.85752327\\\\ln(x)$$","choices":["$$y=42.52722583+13.85752327\\\\ln(x)$$","$$y=45.57892583+17.85752327\\\\ln(x)$$","$$y=40.52723583+14.857812327\\\\ln(x)$$"],"hints":{"DefaultPathway":[{"id":"a5ccaf6logarithmsandlogistic20a-h1","type":"hint","dependencies":[],"title":"Verifying the Scatterplot\'s Pattern","text":"Using the STAT then EDIT menu on a graphing utility, list the years using values $$1-12$$ in L1 and the corresponding life expectancy in L2. Then use the STATPLOT feature to verify that the scatterplot follows a logarithmic pattern as shown in the attached figure.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5ccaf6logarithmsandlogistic20a-h2","type":"hint","dependencies":["a5ccaf6logarithmsandlogistic20a-h1"],"title":"Using LnReg on the Calculator","text":"Use the \u201cLnReg\u201d command from the STAT then CALC menu to obtain the logarithmic model, $$y=42.52722583+13.85752327\\\\ln(x)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5ccaf6logarithmsandlogistic3","title":"Solving Logistic Equations #2","body":"A logistic model is given by the equation $$P(t)=\\\\frac{90}{1+5e^{-0.42 t}}$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.8 Fitting Exponential Models to Data","courseName":"OpenStax: College Algebra","steps":[{"id":"a5ccaf6logarithmsandlogistic3a","stepAnswer":["$$15$$"],"problemType":"TextBox","stepTitle":"To the nearest hundredth, what is the $$y$$ intercept?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$15$$","hints":{"DefaultPathway":[{"id":"a5ccaf6logarithmsandlogistic3a-h1","type":"hint","dependencies":[],"title":"How to Find the $$y$$ intercept","text":"The $$y$$ intercept is P(t) when $$t=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5ccaf6logarithmsandlogistic4","title":"Solving Logistic Equations #3","body":"The population P of a koi pond over $$x$$ months is modeled by the function $$P(x)=1+16e^{-0.28 x}$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.8 Fitting Exponential Models to Data","courseName":"OpenStax: College Algebra","steps":[{"id":"a5ccaf6logarithmsandlogistic4a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"What was the initial population of koi?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a5ccaf6logarithmsandlogistic4a-h1","type":"hint","dependencies":[],"title":"Identifying What Each Term Represents","text":"$$x$$ represents the number of months that have passed, and P(x) represents the number of koi in the pond at $$x$$ months.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5ccaf6logarithmsandlogistic4a-h2","type":"hint","dependencies":["a5ccaf6logarithmsandlogistic4a-h1"],"title":"How to Find the Initial Population of Koi","text":"The initial population of koi is P(x) when $$x=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5ccaf6logarithmsandlogistic5","title":"Solving Logistic Equations #4","body":"The population P of a koi pond over $$x$$ months is modeled by the function $$P(x)=\\\\frac{68}{1+16e^{-0.28 x}}$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.8 Fitting Exponential Models to Data","courseName":"OpenStax: College Algebra","steps":[{"id":"a5ccaf6logarithmsandlogistic5a","stepAnswer":["$$48$$"],"problemType":"TextBox","stepTitle":"How many koi will the pond have after one and a half years? Round to the nearest whole number.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$48$$","hints":{"DefaultPathway":[{"id":"a5ccaf6logarithmsandlogistic5a-h1","type":"hint","dependencies":[],"title":"Identifying What Each Term Represents","text":"$$x$$ represents the number of months that have passed, and P(x) represents the number of koi in the pond at $$x$$ months.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5ccaf6logarithmsandlogistic5a-h2","type":"hint","dependencies":["a5ccaf6logarithmsandlogistic5a-h1"],"title":"Converting From Yeras to Months","text":"A year has $$12$$ months, so one and a half years has $$12+6=18$$ months.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5ccaf6logarithmsandlogistic6","title":"Solving Logistic Equations #5","body":"The population P of a koi pond over $$x$$ months is modeled by the function $$P(x)=\\\\frac{68}{1+16e^{-0.28 x}}$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.8 Fitting Exponential Models to Data","courseName":"OpenStax: College Algebra","steps":[{"id":"a5ccaf6logarithmsandlogistic6a","stepAnswer":["$$7$$"],"problemType":"TextBox","stepTitle":"How many months will it take before there are $$20$$ koi in the pond? Round to the nearest whole number.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7$$","hints":{"DefaultPathway":[{"id":"a5ccaf6logarithmsandlogistic6a-h1","type":"hint","dependencies":[],"title":"Identifying What Represents the Answer","text":"In this problem, we are trying to find what $$x$$ is when $$P(x)=20$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5ccaf6logarithmsandlogistic6a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["x=-(ln(2.4)/0.28"],"dependencies":["a5ccaf6logarithmsandlogistic6a-h1"],"title":"Isolating $$x$$","text":"We solve for $$x$$ by isolating it. After plugging in $$P(x)=20$$, we get $$20=\\\\frac{68}{1+16e^{-0.28 x}}$$. What equation can we rewrite this into?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["x=-(ln(2.4)/68","x=-(ln(0.28)/2.4","x=-(ln(2.4)/0.28"]}]}}]},{"id":"a5ccaf6logarithmsandlogistic7","title":"Solving Logistic Equations #6","body":"The population P of a koi pond over $$x$$ months is modeled by the function $$P(x)=\\\\frac{68}{1+16e^{-0.28 x}}$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.8 Fitting Exponential Models to Data","courseName":"OpenStax: College Algebra","steps":[{"id":"a5ccaf6logarithmsandlogistic7a","stepAnswer":["$$10$$"],"problemType":"TextBox","stepTitle":"Use the intersect feature of your calculator to approximate the number of months it will take before the population of the pond reaches half its carrying capacity. Round to the nearest whole number.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10$$","hints":{"DefaultPathway":[{"id":"a5ccaf6logarithmsandlogistic7a-h1","type":"hint","dependencies":[],"title":"Carring Capacity of the Problem","text":"In this problem, the carrying capacity is $$68$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5ccaf6logarithmsandlogistic7a-h2","type":"hint","dependencies":["a5ccaf6logarithmsandlogistic7a-h1"],"title":"Using the Intersect Feature On a Calculator","text":"Graph $$y=34$$ and $$y=\\\\frac{68}{1+16e^{-0.028 x}}$$. Then, find the $$x$$ coordinate of their intersection by going to the calculate menu, selecting the intersect option, and identifying the two different $$\\\\frac{curves}{lines}$$ on the graph.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5ccaf6logarithmsandlogistic8","title":"Solving Logistic Equations #7","body":"The population P of an endangered species habitat for wolves is modeled by the function $$P(x)=\\\\frac{558}{1+54.8e^{-0.462 x}}$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.8 Fitting Exponential Models to Data","courseName":"OpenStax: College Algebra","steps":[{"id":"a5ccaf6logarithmsandlogistic8a","stepAnswer":["$$9$$"],"problemType":"TextBox","stepTitle":"Use the intersect feature to approximate the number of months it will take before the population of the pond reaches half its carrying capacity. Round to the nearest whole number.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9$$","hints":{"DefaultPathway":[{"id":"a5ccaf6logarithmsandlogistic8a-h1","type":"hint","dependencies":[],"title":"Carring Capacity of the Problem","text":"In this problem, the carrying capacity is $$558$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5ccaf6logarithmsandlogistic8a-h2","type":"hint","dependencies":["a5ccaf6logarithmsandlogistic8a-h1"],"title":"Using the Intersect Feature On a Calculator","text":"Graph $$y=279$$ and $$\\\\frac{558}{1+54.8e^{-0.462 x}}$$. Then, find the $$x$$ coordinate of their intersection by going to the calculate menu, selecting the intersect option, and identifying the two different $$\\\\frac{curves}{lines}$$ on the graph.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5ccaf6logarithmsandlogistic9","title":"Solving Logistic Equations #8","body":"The population P of an endangered species habitat for wolves is modeled by the function $$P(x)=\\\\frac{558}{1+54.8e^{-0.462 x}}$$, where $$x$$ is given in years.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.8 Fitting Exponential Models to Data","courseName":"OpenStax: College Algebra","steps":[{"id":"a5ccaf6logarithmsandlogistic9a","stepAnswer":["$$10$$"],"problemType":"TextBox","stepTitle":"What was the initial population of wolves?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10$$","hints":{"DefaultPathway":[{"id":"a5ccaf6logarithmsandlogistic9a-h1","type":"hint","dependencies":[],"title":"Identifying What Each Term Represents","text":"$$x$$ represents the number of months that have passed, and P(x) represents the number of wolves in the habitat at $$x$$ months.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5ccaf6logarithmsandlogistic9a-h2","type":"hint","dependencies":["a5ccaf6logarithmsandlogistic9a-h1"],"title":"How to Find the Initial Population of Wolves","text":"The initial population of wolves is P(x) when $$x=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5d54dagcf1","title":"Finding the Greatest Common Factor of Two or More Expressions","body":"Find the greatest common factor of the three expressions.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5d54dagcf1a","stepAnswer":["$$3x$$"],"problemType":"TextBox","stepTitle":"$$21x^3$$, $$9x^2$$, $$15x$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3x$$","hints":{"DefaultPathway":[{"id":"a5d54dagcf1a-h1","type":"hint","dependencies":[],"title":"Factoring","text":"The first step is to factor out the primes of all the coefficients and write the variables with exponents in expanded form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf1a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["3,7"],"dependencies":["a5d54dagcf1a-h1"],"title":"Factoring","text":"What are the prime factors of 21? (Write with commas in between numbers and no spaces)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf1a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["3,3"],"dependencies":["a5d54dagcf1a-h1"],"title":"Factoring","text":"What are the prime factors of 9? (Write with commas in between numbers and no spaces)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf1a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["3,5"],"dependencies":["a5d54dagcf1a-h1"],"title":"Factoring","text":"What are the prime factors of 15? (Write with commas in between numbers and no spaces)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf1a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a5d54dagcf1a-h2","a5d54dagcf1a-h3","a5d54dagcf1a-h4"],"title":"Greatest Common Factor","text":"What is the greatest common factor of only the coefficients?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf1a-h6","type":"hint","dependencies":["a5d54dagcf1a-h5"],"title":"Greatest Common Factor","text":"The final step is to find the greatest common factor between the variables with exponents, namely $$x$$, $$x^2$$, and $$x^3$$, which is $$x$$. That combined with the coefficient greatest common factor gives a solutions of $$3x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5d54dagcf10","title":"Factor the Greatest Common Factor from a Polynomial","body":"Find the greatest common factor of the polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5d54dagcf10a","stepAnswer":["$$3p\\\\left(p^2-2pq+3q^3\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor: $$3p^3-6p^2 q+9{pq}^3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3p\\\\left(p^2-2pq+3q^3\\\\right)$$","hints":{"DefaultPathway":[{"id":"a5d54dagcf10a-h1","type":"hint","dependencies":[],"title":"Find the Greatest Common Factor of all the terms of the polynomial.","text":"The first step is to find the GCF of all the terms of the polynomial (3p**3,6p**2q,9pq**3).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3p$$"],"dependencies":["a5d54dagcf10a-h1"],"title":"Find the Greatest Common Factor of all the terms of the polynomial.","text":"What is the greatest common factor of the terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf10a-h3","type":"hint","dependencies":["a5d54dagcf10a-h2"],"title":"Rewriting each term as a product using the GCF","text":"The next step is to rewrite the equation with each term as a product between the GCF and another term. EX for first term: $$3p^3$$ -> $$3p p^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf10a-h4","type":"hint","dependencies":["a5d54dagcf10a-h3"],"title":"Using the Distributive Property","text":"The final step is to use the reverse distributive property to take the GCF out of the equation. This results in the equation $$3p\\\\left(p^2-2pq+3q^3\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5d54dagcf11","title":"Finding the Greatest Common Factor","body":"Find the greatest common factor of the two expressions.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5d54dagcf11a","stepAnswer":["$$2p q$$"],"problemType":"TextBox","stepTitle":"$$10p^3 q$$, $$12p q^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2p q$$","hints":{"DefaultPathway":[{"id":"a5d54dagcf11a-h1","type":"hint","dependencies":[],"title":"Break Into Smaller Questions","text":"We can divide up this question by first determining the greatest common factor of the constant (numerical) terms and then looking at the greatest common factor of the variable terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a5d54dagcf11a-h1"],"title":"GCF for Constants","text":"What is the greatest common factor between $$10$$ and 12?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5d54dagcf11a-h2-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2\\\\times5$$"],"dependencies":[],"title":"Prime Factorization of $$10$$","text":"What is the prime factorization of 10?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2\\\\times5$$","$$2\\\\times6$$","$$3\\\\times5$$","$$3\\\\times4$$"]},{"id":"a5d54dagcf11a-h2-s2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2\\\\times2\\\\times3$$"],"dependencies":[],"title":"Prime Factorization of $$12$$","text":"What is the prime factorization of 12?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2\\\\times2\\\\times3$$","$$6\\\\times3$$","$$2\\\\times2\\\\times4$$","$$2\\\\times3\\\\times3$$"]},{"id":"a5d54dagcf11a-h2-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Common Factor","text":"What is the common factor between $$2\\\\times5$$ and $$2\\\\times2\\\\times3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a5d54dagcf11a-h3","type":"hint","dependencies":[],"title":"GCF for Variables","text":"Knowing the greatest common numerical factor, we can multiply that with the greatest common variable factor. Let\'s determine the greatest common numerical factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf11a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$p q$$"],"dependencies":["a5d54dagcf11a-h3"],"title":"GCF for Variables","text":"What is the greatest common factor between $$p^3 q$$ and $$p q^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$p q$$","$$p q^2$$","$$p^3 q$$","$$p^2 q^2$$"],"subHints":[{"id":"a5d54dagcf11a-h4-s4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$p p p q$$"],"dependencies":[],"title":"Expand $$p^3 q$$","text":"How can we rewrite (\\"factorize\\") $$p^3 q$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$p p p q$$","$$3p q$$","$$p q q q$$","$$p q$$"]},{"id":"a5d54dagcf11a-h4-s5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$p q q$$"],"dependencies":[],"title":"Expand $$p q^2$$","text":"How can we rewrite (\\"factorize\\") $$p q^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$p q q$$","$$2p q$$","$$p p q$$","$$p p q q$$"]},{"id":"a5d54dagcf11a-h4-s6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$p q$$"],"dependencies":[],"title":"Common Factor","text":"What is the common factor between $$p p p q$$ and $$p q q$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$p q$$","$$p p p q$$","$$p q q$$","$$p p q q$$"]}]},{"id":"a5d54dagcf11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2p q$$"],"dependencies":["a5d54dagcf11a-h4"],"title":"Final Answer","text":"We can multiply the constant GCF and the variable GCF to get the final answer. What is the product?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5d54dagcf12","title":"Finding the Greatest Common Factor","body":"Find the greatest common factor of the two expressions.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5d54dagcf12a","stepAnswer":["$$6m^2 n^3$$"],"problemType":"TextBox","stepTitle":"$$12m^2 n^3$$, $$30m^5 n^3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6m^2 n^3$$","hints":{"DefaultPathway":[{"id":"a5d54dagcf12a-h1","type":"hint","dependencies":[],"title":"Break Into Smaller Questions","text":"We can divide up this question by first determining the greatest common factor of the constant (numerical) terms and then looking at the greatest common factor of the variable terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a5d54dagcf12a-h1"],"title":"GCF for Constants","text":"What is the greatest common factor between $$12$$ and 30?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5d54dagcf12a-h2-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2\\\\times2\\\\times3$$"],"dependencies":[],"title":"Prime Factorization of $$12$$","text":"What is the prime factorization of 12?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2\\\\times2\\\\times3$$","$$6\\\\times3$$","$$2\\\\times2\\\\times4$$","$$2\\\\times3\\\\times3$$"]},{"id":"a5d54dagcf12a-h2-s2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2\\\\times3\\\\times5$$"],"dependencies":[],"title":"Prime Factorization of $$30$$","text":"What is the prime factorization of 30?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2\\\\times3\\\\times5$$","$$6\\\\times6$$","$$2\\\\times5$$","$$2\\\\times4\\\\times5$$"]},{"id":"a5d54dagcf12a-h2-s3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2$$, $$3$$"],"dependencies":[],"title":"Common Factor","text":"What are the common factors between $$2\\\\times2\\\\times3$$ and $$2\\\\times3\\\\times5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2$$, $$3$$","$$2$$, $$5$$","$$2$$, $$2$$","$$3$$, $$5$$"]}]},{"id":"a5d54dagcf12a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a5d54dagcf12a-h2"],"title":"Determine Constant GCF","text":"$$2\\\\times3$$ is the greatest common factor for the numerical (constant) expressions. What is $$2\\\\times3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf12a-h4","type":"hint","dependencies":[],"title":"GCF for Variables","text":"Knowing the greatest common numerical factor, we can multiply that with the greatest common variable factor. Let\'s determine the greatest common numerical factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf12a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$m^2 n^3$$"],"dependencies":["a5d54dagcf12a-h4"],"title":"GCF for Variables","text":"What is the greatest common factor between $$m^2 n^3$$ and $$m^5 n^3$$ written in exponential form with bases $$m$$ and $$n$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5d54dagcf12a-h5-s4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$m m n n n$$"],"dependencies":[],"title":"Expand $$m^2 n^3$$","text":"How can we rewrite (\\"factorize\\") $$m^2 n^3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$m m n n n$$","$$m m m n n$$","$$m m n n$$","$$m m m n n n$$"]},{"id":"a5d54dagcf12a-h5-s5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$m m m m m n n n$$"],"dependencies":[],"title":"Expand $$m^5 n^3$$","text":"How can we rewrite (\\"factorize\\") $$m^5 n^3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$m m m m m n n n$$","$$m m m n n n n n$$","$$m m m n n n$$","$$m m m m n n n n$$"]},{"id":"a5d54dagcf12a-h5-s6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$m m n n n$$"],"dependencies":[],"title":"Common Factor","text":"What is the common factor between $$m m n n n$$ and $$m m m m m n n n$$ using no exponents?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a5d54dagcf12a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$m^2 n^3$$"],"dependencies":["a5d54dagcf12a-h5"],"title":"Exponentiate","text":"Rewrite the expression just found $$m m n n n$$ as exponent multiplication with bases $$m$$ and $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf12a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6m^2 n^3$$"],"dependencies":["a5d54dagcf12a-h6"],"title":"Final Answer","text":"We can multiply the constant GCF and the variable GCF to get the final answer. What is the product?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5d54dagcf13","title":"Finding the Greatest Common Factor","body":"Find the Greatest Common Factor of the three expressions.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5d54dagcf13a","stepAnswer":["$$2a$$"],"problemType":"TextBox","stepTitle":"$$10a^3$$, $$12a^2$$, $$14a$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2a$$","hints":{"DefaultPathway":[{"id":"a5d54dagcf13a-h1","type":"hint","dependencies":[],"title":"Break Into Smaller Questions","text":"We can divide up this question by first determining the greatest common factor of the constant (numerical) terms and then looking at the greatest common factor of the variable terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a5d54dagcf13a-h1"],"title":"GCF for Constants","text":"What is the greatest common factor between $$10$$, $$12$$, and 14?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5d54dagcf13a-h2-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2\\\\times5$$"],"dependencies":[],"title":"Prime Factorization of $$12$$","text":"What is the prime factorization of 10?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2\\\\times5$$","$$2\\\\times2\\\\times3$$","$$2\\\\times7$$","$$2\\\\times2\\\\times2$$"]},{"id":"a5d54dagcf13a-h2-s2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2\\\\times2\\\\times3$$"],"dependencies":[],"title":"Prime Factorization of $$30$$","text":"What is the prime factorization of 12?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2\\\\times5$$","$$2\\\\times2\\\\times3$$","$$2\\\\times7$$","$$2\\\\times2\\\\times2$$"]},{"id":"a5d54dagcf13a-h2-s3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2\\\\times7$$"],"dependencies":[],"title":"Prime Factorization of $$30$$","text":"What is the prime factorization of 14?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2\\\\times5$$","$$2\\\\times2\\\\times3$$","$$2\\\\times7$$","$$2\\\\times2\\\\times2$$"]},{"id":"a5d54dagcf13a-h2-s4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Common Factor","text":"What is the common factor between $$2\\\\times5$$, $$2\\\\times2\\\\times3$$, and $$2\\\\times7$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a5d54dagcf13a-h3","type":"hint","dependencies":[],"title":"GCF for Variables","text":"Knowing the greatest common numerical factor, we can multiply that with the greatest common variable factor. Let\'s determine the greatest common numerical factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["a"],"dependencies":["a5d54dagcf13a-h3"],"title":"GCF for Variables","text":"What is the greatest common factor between $$a^3$$, $$a^2$$ and a?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5d54dagcf13a-h4-s5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$a a a$$"],"dependencies":[],"title":"Expand $$a^3$$","text":"How can we rewrite (\\"factorize\\") $$a^3$$ with no exponents?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$a a a$$","a","$$a a$$","$$a a a a$$"]},{"id":"a5d54dagcf13a-h4-s6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$a a a$$"],"dependencies":[],"title":"Expand $$a^2$$","text":"How can we rewrite (\\"factorize\\") $$a^2$$ with no exponents?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$a a a$$","a","$$a a$$","$$a a a a$$"]},{"id":"a5d54dagcf13a-h4-s7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["a"],"dependencies":[],"title":"Common Factor","text":"What is the common factor between $$a a a$$, $$a a$$, and a using no exponents?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a5d54dagcf13a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2a$$"],"dependencies":["a5d54dagcf13a-h4"],"title":"Final Answer","text":"We can multiply the constant GCF and the variable GCF to get the final answer. What is the product?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5d54dagcf14","title":"Finding the Greatest Common Factor","body":"Find the Greatest Common Factor of the three expressions.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5d54dagcf14a","stepAnswer":["$$5x^3 y$$"],"problemType":"TextBox","stepTitle":"$$35x^3 y^2$$, $$10x^4 y$$, $$5x^5 y^3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5x^3 y$$","hints":{"DefaultPathway":[{"id":"a5d54dagcf14a-h1","type":"hint","dependencies":[],"title":"Break Into Smaller Questions","text":"We can divide up this question by first determining the greatest common factor of the constant (numerical) terms and then looking at the greatest common factor of the variable terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a5d54dagcf14a-h1"],"title":"GCF for Constants","text":"What is the greatest common factor between $$35$$, $$10$$, and 5?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5d54dagcf14a-h2-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$5\\\\times7$$"],"dependencies":[],"title":"Prime Factorization of $$12$$","text":"What is the prime factorization of 35?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$5\\\\times7$$","$$2\\\\times5$$","$$3\\\\times5$$","$$6\\\\times7$$"]},{"id":"a5d54dagcf14a-h2-s2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2\\\\times5$$"],"dependencies":[],"title":"Prime Factorization of $$30$$","text":"What is the prime factorization of 10?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$5\\\\times7$$","$$2\\\\times5$$","$$3\\\\times5$$","$$6\\\\times7$$"]},{"id":"a5d54dagcf14a-h2-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":[],"title":"Common Factor","text":"What is the common factor between $$5\\\\times7$$, $$2\\\\times5$$, and 5?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a5d54dagcf14a-h3","type":"hint","dependencies":["a5d54dagcf14a-h2"],"title":"GCF for Variables","text":"Knowing the greatest common numerical factor, we can multiply that with the greatest common variable factor. Let\'s determine the greatest common numerical factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf14a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^3 y$$"],"dependencies":["a5d54dagcf14a-h3"],"title":"GCF for Variables","text":"What is the greatest common factor between $$x^3 y^2$$, $$x^4 y$$, $$x^5 y^3$$ in exponential form with bases $$x$$ and $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5d54dagcf14a-h4-s4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x x x y$$"],"dependencies":[],"title":"Expand $$x^3 y^2$$","text":"How can we rewrite (\\"factorize\\") $$x^3 y^2$$ with no exponents?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x x x y y$$","$$x x x x y$$","$$x x x x x y y y$$","$$x x x y y y$$"]},{"id":"a5d54dagcf14a-h4-s5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x x x x y$$"],"dependencies":[],"title":"Expand $$x^4 y$$","text":"How can we rewrite (\\"factorize\\") $$x^4 y$$ with no exponents?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x x x y y$$","$$x x x x y$$","$$x x x x x y y y$$","$$x x x y y y$$"]},{"id":"a5d54dagcf14a-h4-s6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x x x x x y y y$$"],"dependencies":[],"title":"Expand $$x^5 y^3$$","text":"How can we rewrite (\\"factorize\\") $$x^5 y^3$$ with no exponents?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x x x y y$$","$$x x x x y$$","$$x x x x x y y y$$","$$x x x y y y$$"]}]},{"id":"a5d54dagcf14a-h5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["x*x*x*y"],"dependencies":["a5d54dagcf14a-h4"],"title":"Common Factor","text":"What is the common factor between $$x x x y y$$, $$x x x x y$$, $$x x x x x y y y$$ using no exponents?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf14a-h6","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["x**3*y"],"dependencies":["a5d54dagcf14a-h5"],"title":"Exponentiate","text":"Rewrite the expression $$x x x y$$ as exponents with base $$x$$ and $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf14a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5x^3 y$$"],"dependencies":["a5d54dagcf14a-h6"],"title":"Final Answer","text":"We can multiply the constant GCF and the variable GCF to get the final answer. What is the product?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5d54dagcf15","title":"Finding the Greatest Common Factor","body":"Find the Greatest Common Factor of the polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5d54dagcf15a","stepAnswer":["$$9\\\\left(n-7\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$9n-63$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9\\\\left(n-7\\\\right)$$","hints":{"DefaultPathway":[{"id":"a5d54dagcf15a-h1","type":"hint","dependencies":[],"title":"Distributive Property","text":"We can start by finding the greatest common factor of the constants.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a5d54dagcf15a-h1"],"title":"GCF for Constants","text":"What is the greatest common factor between $$9$$ and 63?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5d54dagcf15a-h2-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["yes"],"dependencies":[],"title":"Divisibility","text":"Is $$63$$ divisible by 9?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["yes","no"]},{"id":"a5d54dagcf15a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":[],"title":"Divisibility","text":"What is $$\\\\frac{63}{9}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a5d54dagcf15a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["n-7"],"dependencies":["a5d54dagcf15a-h2"],"title":"Distributive Property Conclusion","text":"Factor $$9$$ out of $$9n-63$$. What is the resulting expression, as in, what is $$\\\\frac{9n-63}{9}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5d54dagcf15a-h3-s3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["n"],"dependencies":[],"title":"Factor $$9n$$","text":"What is the result of factoring $$9$$ out of $$9n$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf15a-h3-s4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-7$$"],"dependencies":[],"title":"Factor $$-63$$","text":"What is the result of factoring $$9$$ out of -63?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf15a-h3-s5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["n-7"],"dependencies":[],"title":"Sum the factors","text":"Using the \\"reverse\\" Distributive Property, what is the result of factoring out $$9$$ from $$9n-63$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a5d54dagcf15a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$9(n-7)$$"],"dependencies":["a5d54dagcf15a-h3"],"title":"Final Answer","text":"Multiply the numerical greatest common factor with what you get when you got in the previous question. What answer do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$9(n-7)$$","$$9\\\\left(n+7\\\\right)$$","9(-n+7)","$$9(-n-7)$$"]}]}}]},{"id":"a5d54dagcf17","title":"Finding the Greatest Common Factor","body":"Find the Greatest Common Factor of the polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5d54dagcf17a","stepAnswer":["3*(x**2+2x-3)"],"problemType":"TextBox","stepTitle":"$$3x^2+6x-9$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$3\\\\left(x^2+2x-3\\\\right)$$","hints":{"DefaultPathway":[{"id":"a5d54dagcf17a-h1","type":"hint","dependencies":[],"title":"Greatest Common Factor","text":"The first step is determining the greatest common factor between the three terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf17a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a5d54dagcf17a-h1"],"title":"GCF","text":"What is the GCF for the three terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5d54dagcf17a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":[],"title":"GCF for Constants","text":"What is the GCF for $$3$$, $$6$$, and 9? As in, what is the greatest number that evenly divides into $$3$$, $$6$$, and 9?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf17a-h2-s2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$1$$"],"dependencies":[],"title":"GCF for Variables","text":"What is the GCF for $$x^2$$, $$x$$, and 1?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$1$$","$$x$$","$$x^2$$","$$x^3$$"]},{"id":"a5d54dagcf17a-h2-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":[],"title":"Overall GCF","text":"Multiply the two previous partial GCFS you found to get the overall GCF. What is it?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a5d54dagcf17a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["x**2+2x-3"],"dependencies":["a5d54dagcf17a-h2"],"title":"Factorization","text":"Now, factor each of the terms separately. What is the sum of the terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5d54dagcf17a-h3-s4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["x**2"],"dependencies":[],"title":"Factor $$3x^2$$","text":"Factor out a $$3$$ from $$3x^2$$. What remains?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf17a-h3-s5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["2x"],"dependencies":[],"title":"Factor $$6x$$","text":"Factor out a $$3$$ from $$6x$$. What remains?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf17a-h3-s6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":[],"title":"Factor $$-9$$","text":"Factor out a $$3$$ from $$-9$$. What remains?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf17a-h3-s7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x^2+2x-3$$"],"dependencies":[],"title":"Summation of factors","text":"Using the \\"reverse\\" Distributive Property, add together the three factors previously found. What is the sum?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x^2+2x-3$$","$$3x^2+2x-1$$","$$2x^2+3x-1$$","$$x^2+3x-2$$"]}]},{"id":"a5d54dagcf17a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["3*(x**2+2x-3)"],"dependencies":["a5d54dagcf17a-h3"],"title":"Final Answer","text":"Using knowledge of the distributive property, multiply the GCF with the sum of the factored values to get the final answer for the factorization of $$3x^2+6x-9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5d54dagcf18","title":"Finding the Greatest Common Factor","body":"Find the Greatest Common Factor of the polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5d54dagcf18a","stepAnswer":["$$5x \\\\left(x^2-3x+4\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$5x^3-15x^2+20x$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5x \\\\left(x^2-3x+4\\\\right)$$","hints":{"DefaultPathway":[{"id":"a5d54dagcf18a-h1","type":"hint","dependencies":[],"title":"Greatest Common Factor","text":"The first step is determining the greatest common factor between the three terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf18a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["5*x"],"dependencies":["a5d54dagcf18a-h1"],"title":"GCF","text":"What is the GCF for the three terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5d54dagcf18a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":[],"title":"GCF for Constants","text":"What is the GCF for $$5$$, $$15$$, and 20?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf18a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["x"],"dependencies":[],"title":"GCF for Variables","text":"What is the GCF for $$x^3$$, $$x^2$$, and $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf18a-h2-s3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["5*x"],"dependencies":[],"title":"Overall GCF","text":"Multiply the two previous partial GCFS you found to get the overall GCF. What is it?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a5d54dagcf18a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["x**2-3x+4"],"dependencies":["a5d54dagcf18a-h2"],"title":"Factorization","text":"Now, factor each of the terms separately. What is the sum of the terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5d54dagcf18a-h3-s4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["x**2"],"dependencies":[],"title":"Factor $$3x^2$$","text":"Factor out $$5x$$ from $$5x^3$$. What remains?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf18a-h3-s5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["-3*x"],"dependencies":[],"title":"Factor $$6x$$","text":"Factor out $$5x$$ from $$-15x^2$$. What remains?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf18a-h3-s6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":[],"title":"Factor $$-9$$","text":"Factor out $$5x$$ from $$20x$$. What remains?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf18a-h3-s7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x^2-3x+4$$"],"dependencies":[],"title":"Summation of factors","text":"Using the \\"reverse\\" Distributive Property, add together the three factors previously found. What is the sum?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x^2-3x+4$$","$$3x^2-4x+1$$","$$4x^2-3x+1$$","$$x^2-4x+3$$"]}]},{"id":"a5d54dagcf18a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["5*x*(x**2-3x+4)"],"dependencies":["a5d54dagcf18a-h3"],"title":"Final Answer","text":"Using knowledge of the distributive property, multiply the GCF with the sum of the factored values to get the final answer for the factorization of $$5x^3-15x^2+20x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5d54dagcf19","title":"Finding the Greatest Common Factor","body":"Find the Greatest Common Factor of the polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5d54dagcf19a","stepAnswer":["$$3x \\\\left(8x^2-4x+5\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$24x^3-12x^2+15x$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3x \\\\left(8x^2-4x+5\\\\right)$$","hints":{"DefaultPathway":[{"id":"a5d54dagcf19a-h1","type":"hint","dependencies":[],"title":"Greatest Common Factor","text":"The first step is determining the greatest common factor between the three terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf19a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["3*x"],"dependencies":["a5d54dagcf19a-h1"],"title":"GCF","text":"What is the GCF for the three terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5d54dagcf19a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":[],"title":"GCF for Constants","text":"What is the GCF for $$24$$, $$-12$$, and 15?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf19a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["x"],"dependencies":[],"title":"GCF for Variables","text":"What is the GCF for $$x^3$$, $$x^2$$, and $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf19a-h2-s3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["3*x"],"dependencies":[],"title":"Overall GCF","text":"Multiply the two previous partial GCFS you found to get the overall GCF. What is it?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a5d54dagcf19a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["8*x**2-4*x+5"],"dependencies":["a5d54dagcf19a-h2"],"title":"Factorization","text":"Now, factor each of the terms separately. What is the sum of the terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5d54dagcf19a-h3-s4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["8*x**2"],"dependencies":[],"title":"Factor $$3x^2$$","text":"Factor out $$3x$$ from $$24x^3$$. What remains?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf19a-h3-s5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["-4x"],"dependencies":[],"title":"Factor $$6x$$","text":"Factor out $$3x$$ from $$-12x^2$$. What remains?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf19a-h3-s6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":[],"title":"Factor $$-9$$","text":"Factor out $$3x$$ from $$15x$$. What remains?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf19a-h3-s7","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["8*x**2-4x+5"],"dependencies":[],"title":"Summation of factors","text":"Using the \\"reverse\\" Distributive Property, add together the three factors previously found. What is the sum?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a5d54dagcf19a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["3*x*(8*x**2-4x+5)"],"dependencies":["a5d54dagcf19a-h3"],"title":"Final Answer","text":"Using knowledge of the distributive property, multiply the GCF with the sum of the factored values to get the final answer for the factorization of $$24x^3-12x^2+15x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5d54dagcf2","title":"Factor the Greatest Common Factor from a Polynomial","body":"Find the greatest common factor of the polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5d54dagcf2a","stepAnswer":["$$4m\\\\left(2m^2-3mn+5n^2\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$8m^3-12m^2 n+20{mn}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4m\\\\left(2m^2-3mn+5n^2\\\\right)$$","hints":{"DefaultPathway":[{"id":"a5d54dagcf2a-h1","type":"hint","dependencies":[],"title":"Find the Greatest Common Factor of all the terms of the polynomial.","text":"The first step is to find the Greatest Common Factor of all the terms of the polynomial (8m**3,12m**2n,20mn**2).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4m$$"],"dependencies":["a5d54dagcf2a-h1"],"title":"Find the Greatest Common Factor of all the terms of the polynomial.","text":"What is the greatest common factor of the terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf2a-h3","type":"hint","dependencies":["a5d54dagcf2a-h2"],"title":"Rewriting each term as a product using the GCF","text":"The next step is to rewrite the equation with each term as a product between the GCF and another term. EX for first term: $$8m^3$$ -> $$4m\\\\times2 m^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf2a-h4","type":"hint","dependencies":["a5d54dagcf2a-h3"],"title":"Using the distributive property","text":"The final step is to use the reverse distributive property to take the GCF out of the equation. This results in the equation $$4m\\\\left(2m^2-3mn+5n^2\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5d54dagcf20","title":"Finding the Greatest Common Factor","body":"Find the Greatest Common Factor of the polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5d54dagcf20a","stepAnswer":["$$6y^2 \\\\left(2x+3x^2-5y\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$12x y^2+18x^2 y^2-30y^3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6y^2 \\\\left(2x+3x^2-5y\\\\right)$$","hints":{"DefaultPathway":[{"id":"a5d54dagcf20a-h1","type":"hint","dependencies":[],"title":"Greatest Common Factor","text":"The first step is determining the greatest common factor between the three terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf20a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["6*y**2"],"dependencies":["a5d54dagcf20a-h1"],"title":"GCF","text":"What is the GCF for the three terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5d54dagcf20a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":[],"title":"GCF for Constants","text":"What is the GCF for $$12$$, $$18$$, and -30?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf20a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["y**2"],"dependencies":[],"title":"GCF for Variables","text":"What is the GCF for $$x y^2$$, $$x^2 y^2$$, and $$y^3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf20a-h2-s3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["6*y**2"],"dependencies":[],"title":"Overall GCF","text":"Multiply the two previous partial GCFS you found to get the overall GCF. What is it?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a5d54dagcf20a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["2x+3*x**2-5y"],"dependencies":["a5d54dagcf20a-h2"],"title":"Factorization","text":"Now, factor each of the terms separately. What is the sum of the terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5d54dagcf20a-h3-s4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["2x"],"dependencies":[],"title":"Factor $$3x^2$$","text":"Factor out $$6y^2$$ from $$12x y^2$$. What remains?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf20a-h3-s5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["3*x**2"],"dependencies":[],"title":"Factor $$6x$$","text":"Factor out $$6y^2$$ from $$18x^2 y^2$$. What remains?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf20a-h3-s6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5y$$"],"dependencies":[],"title":"Factor $$-9$$","text":"Factor out $$6y$$ from $$-30y^3$$. What remains?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf20a-h3-s7","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["2x+3*x**2-5y"],"dependencies":[],"title":"Summation of factors","text":"Using the \\"reverse\\" Distributive Property, add together the three factors previously found. What is the sum?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a5d54dagcf20a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["6*y**2*(2x+3*x**2-5y)"],"dependencies":["a5d54dagcf20a-h3"],"title":"Final Answer","text":"Using knowledge of the distributive property, multiply the GCF with the sum of the factored values to get the final answer for the factorization of $$24x^3-12x^2+15x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5d54dagcf21","title":"Factor the Greatest Common Factor from a Polynomial","body":"In the following exercise, factor the greatest common factor from each polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5d54dagcf21a","stepAnswer":["$$-2\\\\left(x+2\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$-2x-4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-2\\\\left(x+2\\\\right)$$","hints":{"DefaultPathway":[{"id":"a5d54dagcf21a-h1","type":"hint","dependencies":[],"title":"Greatest Common Factor","text":"The first step is to choose what number (or variable) we can factor out of the equation. This is done by finding the greatest common factor of each item in the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf21a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a5d54dagcf21a-h1"],"title":"Greatest Common Factor","text":"What number are we able to factor out from the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf21a-h3","type":"hint","dependencies":["a5d54dagcf21a-h2"],"title":"Greatest Common Factor","text":"The next step is to actually factor out the item from the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf21a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2\\\\left(x+2\\\\right)$$"],"dependencies":["a5d54dagcf21a-h3"],"title":"Greatest Common Factor","text":"What does our final equation look like once we factor out the item?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5d54dagcf22","title":"Factor the Greatest Common Factor from a Polynomial","body":"In the following exercise, factor the greatest common factor from each polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5d54dagcf22a","stepAnswer":["$$-2x \\\\left(x^2-9x+4\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$-2x^3+18x^2-8x$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-2x \\\\left(x^2-9x+4\\\\right)$$","hints":{"DefaultPathway":[{"id":"a5d54dagcf22a-h1","type":"hint","dependencies":[],"title":"Greatest Common Factor","text":"The first step is to choose what number (or variable) we can factor out of the equation. This is done by finding the greatest common factor of each item in the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf22a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2x$$"],"dependencies":["a5d54dagcf22a-h1"],"title":"Greatest Common Factor","text":"What number are we able to factor out from the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf22a-h3","type":"hint","dependencies":["a5d54dagcf22a-h2"],"title":"Greatest Common Factor","text":"The next step is to actually factor out the item from the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf22a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2x \\\\left(x^2-9x+4\\\\right)$$"],"dependencies":["a5d54dagcf22a-h3"],"title":"Greatest Common Factor","text":"What does our final equation look like once we factor out the item?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5d54dagcf23","title":"Factor the Greatest Common Factor from a Polynomial","body":"In the following exercise, factor the greatest common factor from each polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5d54dagcf23a","stepAnswer":["$$-5y \\\\left(y^2-7y+3\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$-5y^3+35y^2-15y$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-5y \\\\left(y^2-7y+3\\\\right)$$","hints":{"DefaultPathway":[{"id":"a5d54dagcf23a-h1","type":"hint","dependencies":[],"title":"Greatest Common Factor","text":"The first step is to choose what number (or variable) we can factor out of the equation. This is done by finding the greatest common factor of each item in the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf23a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5y$$"],"dependencies":["a5d54dagcf23a-h1"],"title":"Greatest Common Factor","text":"What number are we able to factor out from the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf23a-h3","type":"hint","dependencies":["a5d54dagcf23a-h2"],"title":"Greatest Common Factor","text":"The next step is to actually factor out the item from the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf23a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5y \\\\left(y^2-7y+3\\\\right)$$"],"dependencies":["a5d54dagcf23a-h3"],"title":"Greatest Common Factor","text":"What does our final equation look like once we factor out the item?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5d54dagcf24","title":"Factor the Greatest Common Factor from a Polynomial","body":"In the following exercise, factor the greatest common factor from each polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5d54dagcf24a","stepAnswer":["$$\\\\left(x+1\\\\right) \\\\left(5x+3\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$5x \\\\left(x+1\\\\right)+3\\\\left(x+1\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(x+1\\\\right) \\\\left(5x+3\\\\right)$$","hints":{"DefaultPathway":[{"id":"a5d54dagcf24a-h1","type":"hint","dependencies":[],"title":"Greatest Common Factor","text":"The first step is to choose what number (or variable) we can factor out of the equation. This is done by finding the greatest common factor of each item in the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf24a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+1$$"],"dependencies":["a5d54dagcf24a-h1"],"title":"Greatest Common Factor","text":"What number are we able to factor out from the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf24a-h3","type":"hint","dependencies":["a5d54dagcf24a-h2"],"title":"Greatest Common Factor","text":"The next step is to actually factor out the item from the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf24a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x+1\\\\right) \\\\left(5x+3\\\\right)$$"],"dependencies":["a5d54dagcf24a-h3"],"title":"Greatest Common Factor","text":"What does our final equation look like once we factor out the item?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5d54dagcf25","title":"Factor by Grouping","body":"In the following exercise, factor by grouping.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5d54dagcf25a","stepAnswer":["$$\\\\left(b+5\\\\right) \\\\left(a+3\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$ab+5a+3b+15$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(b+5\\\\right) \\\\left(a+3\\\\right)$$","hints":{"DefaultPathway":[{"id":"a5d54dagcf25a-h1","type":"hint","dependencies":[],"title":"Greatest Common Factor","text":"The first step is to factor out the GCF from the first two groups of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf25a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$a \\\\left(b+5\\\\right)+3\\\\left(b+5\\\\right)$$"],"dependencies":["a5d54dagcf25a-h1"],"title":"Greatest Common Factor","text":"What will our equation look like if we factor out the first two groups?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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This results in the equation $$-4a\\\\left(a^2-9a+2\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5d54dagcf5","title":"Factor the Greatest Common Factor from a Polynomial","body":"Find the greatest common factor of the polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5d54dagcf5a","stepAnswer":["$$-4b\\\\left(b^2-4b+2\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor: $$-4b^3+16b^2-8b$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-4b\\\\left(b^2-4b+2\\\\right)$$","hints":{"DefaultPathway":[{"id":"a5d54dagcf5a-h1","type":"hint","dependencies":[],"title":"Finding GCF","text":"Find the GCF first","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4b$$"],"dependencies":["a5d54dagcf5a-h1"],"title":"Finding GCF","text":"What is the GCF?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf5a-h3","type":"hint","dependencies":["a5d54dagcf5a-h2"],"title":"Rewriting each term","text":"The next step is to rewrite each term as a product of the GCF and another term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf5a-h4","type":"hint","dependencies":["a5d54dagcf5a-h3"],"title":"Using the distributive property","text":"The final step is to use the reverse distributive property to take the GCF out of the equation. This results in the equation $$-4b\\\\left(b^2-4b+2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5d54dagcf6","title":"Factor the Greatest Common Factor from a Polynomial","body":"Find the greatest common factor of the polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5d54dagcf6a","stepAnswer":["$$-7a\\\\left(a^2-3a+2\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor: $$-7a^3+21a^2-14a$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-7a\\\\left(a^2-3a+2\\\\right)$$","hints":{"DefaultPathway":[{"id":"a5d54dagcf6a-h1","type":"hint","dependencies":[],"title":"Finding GCF","text":"Find the GCF first","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["-7a"],"dependencies":["a5d54dagcf6a-h1"],"title":"Finding GCF","text":"What is the GCF?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf6a-h3","type":"hint","dependencies":["a5d54dagcf6a-h2"],"title":"Rewriting each term","text":"The next step is to rewrite each term as a product of the GCF and another term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf6a-h4","type":"hint","dependencies":["a5d54dagcf6a-h3"],"title":"Using the distributive property","text":"The final step is to use the reverse distributive property to take the GCF out of the equation. This results in the equation $$-7a\\\\left(a^2-3a+2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5d54dagcf7","title":"How to Factor a Polynomial by Grouping","body":"Factor by grouping:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5d54dagcf7a","stepAnswer":["$$\\\\left(x+3\\\\right) \\\\left(y+2\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$xy+3y+2x+6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(x+3\\\\right) \\\\left(y+2\\\\right)$$","hints":{"DefaultPathway":[{"id":"a5d54dagcf7a-h1","type":"hint","dependencies":[],"title":"Group terms with common factors","text":"The first step is to check if there is a GCF among all terms. If there isn\'t then separate the equation into the terms with like variables. In this case, we would separate the first two terms from the second two.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf7a-h2","type":"hint","dependencies":["a5d54dagcf7a-h1"],"title":"Factoring","text":"The next step is to factor the GCF from the first two terms and second two terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y$$"],"dependencies":["a5d54dagcf7a-h2"],"title":"Factoring","text":"What is the GCF of the first two terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a5d54dagcf7a-h2"],"title":"Factoring","text":"What is the GCF of the second two terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf7a-h5","type":"hint","dependencies":["a5d54dagcf7a-h3","a5d54dagcf7a-h4"],"title":"Factoring","text":"Notice that each term has a common factor of $$x+3$$. Factor out the common factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf7a-h6","type":"hint","dependencies":["a5d54dagcf7a-h5"],"title":"Factoring","text":"This leaves you with $$\\\\left(x+3\\\\right) \\\\left(y+2\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5d54dagcf8","title":"How to Factor a Polynomial by Grouping","body":"Factor by grouping:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5d54dagcf8a","stepAnswer":["$$\\\\left(x+8\\\\right) \\\\left(y+3\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor by grouping: $$xy+8y+3x+24$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(x+8\\\\right) \\\\left(y+3\\\\right)$$","hints":{"DefaultPathway":[{"id":"a5d54dagcf8a-h1","type":"hint","dependencies":[],"title":"Group terms with common factors","text":"The first step is to check if there is a GCF among all terms. If there isn\'t then separate the equation into the terms with like variables. In this case, we would separate the first two terms from the second two.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf8a-h2","type":"hint","dependencies":["a5d54dagcf8a-h1"],"title":"Factoring","text":"The next step is to factor the GCF from the first two terms and second two terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y$$"],"dependencies":["a5d54dagcf8a-h2"],"title":"Factoring","text":"What is the GCF of the first two terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a5d54dagcf8a-h2"],"title":"Factoring","text":"What is the GCF of the second two terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf8a-h5","type":"hint","dependencies":["a5d54dagcf8a-h3","a5d54dagcf8a-h4"],"title":"Factoring","text":"Notice that each term has a common factor of $$x+8$$. Factor out the common factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf8a-h6","type":"hint","dependencies":["a5d54dagcf8a-h5"],"title":"Factoring","text":"This leaves you with $$\\\\left(x+8\\\\right) \\\\left(y+3\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5d54dagcf9","title":"How to Factor a Polynomial by Grouping","body":"Factor by grouping:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Greatest Common Factor and Factor by Grouping","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5d54dagcf9a","stepAnswer":["$$\\\\left(a+7\\\\right) \\\\left(b+8\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor by grouping: $$ab+7b+8a+56$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(a+7\\\\right) \\\\left(b+8\\\\right)$$","hints":{"DefaultPathway":[{"id":"a5d54dagcf9a-h1","type":"hint","dependencies":[],"title":"Group terms with common factors","text":"The first step is to check if there is a GCF among all terms. If there isn\'t then separate the equation into the terms with like variables. In this case, we would separate the first two terms from the second two.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf9a-h2","type":"hint","dependencies":["a5d54dagcf9a-h1"],"title":"Factoring","text":"The next step is to factor the GCF from the first two terms and second two terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$b$$"],"dependencies":["a5d54dagcf9a-h2"],"title":"Factoring","text":"What is the GCF of the first two terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a5d54dagcf9a-h2"],"title":"Factoring","text":"What is the GCF of the second two terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf9a-h5","type":"hint","dependencies":["a5d54dagcf9a-h3","a5d54dagcf9a-h4"],"title":"Factoring","text":"Notice that each term has a common factor of $$x+8$$. Factor out the common factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5d54dagcf9a-h6","type":"hint","dependencies":["a5d54dagcf9a-h5"],"title":"Factoring","text":"This leaves you with $$\\\\left(a+7\\\\right) \\\\left(b+8\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5e282bSolvquad1","title":"Solve Quadratic Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5e282bSolvquad1a","stepAnswer":["(-5*sqrt(2), 5*sqrt(2))"],"problemType":"TextBox","stepTitle":"$$x^2-50$$ $$=$$ $$0$$","stepBody":"If there is more than one solution, please enter you answer as (a,b) where $$a<b$$ and a, $$b$$ are in the simplest form. If you can\'t further simplify the answer into a whole number, please eneter it in a form of $$b \\\\sqrt{c}$$ where $$b$$ is whole numbers and c is natural number.","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a5e282bSolvquad1a-h1","type":"hint","dependencies":[],"title":"Isolate the Quadratic Term and Make Its Coefficient One","text":"We add $$50$$ to both sides to get $$x^2$$ by itself which gives $$x^2=50$$. Since the coefficient of $$x^2$$ is $$1$$ already, we do not need to take extra step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad1a-h2","type":"hint","dependencies":["a5e282bSolvquad1a-h1"],"title":"Use Square root Property","text":"The square root property: if $$x^2$$ $$=$$ k, then $$x$$ $$=$$ $$\\\\sqrt{k}$$, or $$x$$ $$=$$ $$-\\\\sqrt{k}$$. $$x^2=50$$, we can take square root both sides and get $$x=\\\\sqrt{50}$$ or $$x=-\\\\sqrt{50}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5\\\\sqrt{2}$$"],"dependencies":["a5e282bSolvquad1a-h2"],"title":"Simplify the Radical","text":"Simplify $$-\\\\sqrt{50}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5\\\\sqrt{2}$$"],"dependencies":["a5e282bSolvquad1a-h3"],"title":"Simplify the Radical","text":"Simplify $$\\\\sqrt{50}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad1a-h5","type":"hint","dependencies":["a5e282bSolvquad1a-h4"],"title":"Final Answer","text":"Therefore, $$x^2-50$$ $$=$$ $$0$$ has two solutions which are $$x=5\\\\sqrt{2}$$ and $$x=-5\\\\sqrt{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5e282bSolvquad10","title":"Solve Quadratic Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5e282bSolvquad10a","stepAnswer":["(-7,7)"],"problemType":"TextBox","stepTitle":"$$3z^2-147$$ $$=$$ $$0$$","stepBody":"If there is more than one solution, please enter you answer as (a,b) where $$a<b$$ and a, $$b$$ are in the simplest form. If you can\'t further simplify the answer into a whole number, please eneter it in a form of $$b \\\\sqrt{c}$$ where $$b$$ is whole numbers and c is natural number.","answerType":"string","variabilization":{},"answerLatex":"$$(-7,7)$$","hints":{"DefaultPathway":[{"id":"a5e282bSolvquad10a-h1","type":"hint","dependencies":[],"title":"Make the Leading Coefficient \\"1\\"","text":"We can divide both side by $$3$$ and get $$z^2-49=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad10a-h2","type":"hint","dependencies":["a5e282bSolvquad10a-h1"],"title":"Isolate the Quadratic Term","text":"We add $$49$$ to both sides to get $$z^2$$ by itself which gives $$z^2=49$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad10a-h3","type":"hint","dependencies":["a5e282bSolvquad10a-h2"],"title":"Use Square root Property","text":"The square root property: if $$x^2$$ $$=$$ k, then $$x$$ $$=$$ $$\\\\sqrt{k}$$, or $$x$$ $$=$$ $$-\\\\sqrt{k}$$. $$z^2=49$$, we can take square root both sides and get $$z=\\\\sqrt{49}$$ or $$z=-\\\\sqrt{49}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-7$$"],"dependencies":["a5e282bSolvquad10a-h3"],"title":"Simplify the Radical","text":"Simplify $$-\\\\sqrt{49}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a5e282bSolvquad10a-h4"],"title":"Simplify the Radical","text":"Simplify $$\\\\sqrt{49}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad10a-h6","type":"hint","dependencies":["a5e282bSolvquad10a-h5"],"title":"Final Answer","text":"Therefore, $$3z^2-147$$ $$=$$ $$0$$ has two solutions which are $$z=-7$$ and $$z=7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5e282bSolvquad11","title":"Solve Quadratic Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5e282bSolvquad11a","stepAnswer":["(-3*(sqrt(2)),3*(sqrt(2)))"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2}{3} u^2$$ $$+5=17$$","stepBody":"If there is more than one solution, please enter you answer as (a,b) where $$a<b$$ and a, $$b$$ are in the simplest form. If you can\'t further simplify the answer into a whole number, please eneter it in a form of $$b \\\\sqrt{c}$$ where $$b$$ is whole numbers and c is natural number.","answerType":"string","variabilization":{},"answerLatex":"$$(-3\\\\sqrt{2},3\\\\sqrt{2})$$","hints":{"DefaultPathway":[{"id":"a5e282bSolvquad11a-h1","type":"hint","dependencies":[],"title":"Isolate the Quadratic Term","text":"We minus $$5$$ on both sides to get $$u^2$$ by itself which gives $$\\\\frac{2}{3} u^2=12$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{3}$$"],"dependencies":["a5e282bSolvquad11a-h1"],"title":"Find Leading Coefficient","text":"What\'s the coefficient of $$u^2$$ now?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad11a-h3","type":"hint","dependencies":["a5e282bSolvquad11a-h2"],"title":"Make the Leading Coefficient \\"1\\"","text":"Divide by the leading coefficient on both sides which gives $$u^2$$ $$=18$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad11a-h4","type":"hint","dependencies":["a5e282bSolvquad11a-h3"],"title":"Use Square root Property","text":"The square root property: if $$x^2$$ $$=$$ k, then $$x$$ $$=$$ $$\\\\sqrt{k}$$, or $$x$$ $$=$$ $$-\\\\sqrt{k}$$, or $$x$$ $$=\\\\sqrt{k}$$. $$u^2=18$$, we can take square root both sides and get $$u=\\\\sqrt{18}$$ or $$u=-\\\\sqrt{18}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3\\\\sqrt{2}$$"],"dependencies":["a5e282bSolvquad11a-h4"],"title":"Simplify the Radical","text":"Simplify $$-\\\\sqrt{18}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad11a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3\\\\sqrt{2}$$"],"dependencies":["a5e282bSolvquad11a-h5"],"title":"Simplify the Radical","text":"Simplify $$\\\\sqrt{18}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad11a-h7","type":"hint","dependencies":["a5e282bSolvquad11a-h6"],"title":"Final Answer","text":"Therefore, $$\\\\frac{2}{3} u^2$$ $$+5=17$$ has two solutions which are $$u=-3\\\\sqrt{2}$$ $$andu=3\\\\sqrt{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5e282bSolvquad12","title":"Solve Quadratic Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5e282bSolvquad12a","stepAnswer":["(-2*(sqrt(10)),2*(sqrt(10)))"],"problemType":"TextBox","stepTitle":"$$\\\\frac{1}{2} u^2+4=24$$","stepBody":"If there is more than one solution, please enter you answer as (a,b) where $$a<b$$ and a, $$b$$ are in the simplest form. If you can\'t further simplify the answer into a whole number, please eneter it in a form of $$b \\\\sqrt{c}$$ where $$b$$ is whole numbers and c is natural number.","answerType":"string","variabilization":{},"answerLatex":"$$(-2\\\\sqrt{10},2\\\\sqrt{10})$$","hints":{"DefaultPathway":[{"id":"a5e282bSolvquad12a-h1","type":"hint","dependencies":[],"title":"Isolate the Quadratic Term","text":"We minus $$4$$ on both sides to get $$u^2$$ by itself which gives $$\\\\frac{1}{2} u^2=20$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a5e282bSolvquad12a-h1"],"title":"Find Leading Coefficient","text":"What\'s the coefficient of $$u^2$$ now?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad12a-h3","type":"hint","dependencies":["a5e282bSolvquad12a-h2"],"title":"Make the Leading Coefficient \\"1\\"","text":"Divide by the leading coefficient on both sides which gives $$u^2$$ $$=40$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad12a-h4","type":"hint","dependencies":["a5e282bSolvquad12a-h3"],"title":"Use Square root Property","text":"The square root property: if $$x^2$$ $$=$$ k, then $$x$$ $$=$$ $$\\\\sqrt{k}$$, or $$x$$ $$=$$ $$-\\\\sqrt{k}$$, or $$x$$ $$=\\\\sqrt{k}$$. $$u^2=40$$, we can take square root both sides and get $$u=\\\\sqrt{40}$$ or $$u=-\\\\sqrt{40}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad12a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2\\\\sqrt{10}$$"],"dependencies":["a5e282bSolvquad12a-h4"],"title":"Simplify the Radical","text":"Simplify $$-\\\\sqrt{40}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad12a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2\\\\sqrt{10}$$"],"dependencies":["a5e282bSolvquad12a-h5"],"title":"Simplify the Radical","text":"Simplify $$\\\\sqrt{40}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad12a-h7","type":"hint","dependencies":["a5e282bSolvquad12a-h6"],"title":"Final Answer","text":"Therefore, $$\\\\frac{1}{2} u^2+4=24$$ has two solutions which are $$u=-2\\\\sqrt{10}$$, and $$u=2\\\\sqrt{10}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5e282bSolvquad13","title":"Solve Quadratic Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5e282bSolvquad13a","stepAnswer":["(-2*(sqrt(7)),2*(sqrt(7)))"],"problemType":"TextBox","stepTitle":"$$\\\\frac{3}{4} y^2-3=18$$","stepBody":"If there is more than one solution, please enter you answer as (a,b) where $$a<b$$ and a, $$b$$ are in the simplest form. If you can\'t further simplify the answer into a whole number, please eneter it in a form of $$b \\\\sqrt{c}$$ where $$b$$ is whole numbers and c is natural number.","answerType":"string","variabilization":{},"answerLatex":"$$(-2\\\\sqrt{7},2\\\\sqrt{7})$$","hints":{"DefaultPathway":[{"id":"a5e282bSolvquad13a-h1","type":"hint","dependencies":[],"title":"Isolate the Quadratic Term","text":"We add $$3$$ on both sides to get $$y^2$$ by itself which gives $$\\\\frac{3}{4} y^2=21$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{4}$$"],"dependencies":["a5e282bSolvquad13a-h1"],"title":"Find Leading Coefficient","text":"What\'s the coefficient of $$u^2$$ now?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad13a-h3","type":"hint","dependencies":["a5e282bSolvquad13a-h2"],"title":"Make the Leading Coefficient \\"1\\"","text":"Divide by the leading coefficient on both sides which gives $$y^2=28$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad13a-h4","type":"hint","dependencies":["a5e282bSolvquad13a-h3"],"title":"Use Square root Property","text":"The square root property: if $$x^2$$ $$=$$ k, then $$x$$ $$=$$ $$\\\\sqrt{k}$$, or $$x$$ $$=$$ $$-\\\\sqrt{k}$$, or $$x$$ $$=\\\\sqrt{k}$$. $$y^2=28$$, we can take square root both sides and get $$y=\\\\sqrt{28}$$ or $$y=-\\\\sqrt{28}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad13a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2\\\\sqrt{7}$$"],"dependencies":["a5e282bSolvquad13a-h4"],"title":"Simplify the Radical","text":"Simplify $$-\\\\sqrt{28}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad13a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2\\\\sqrt{7}$$"],"dependencies":["a5e282bSolvquad13a-h5"],"title":"Simplify the Radical","text":"Simplify $$\\\\sqrt{28}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad13a-h7","type":"hint","dependencies":["a5e282bSolvquad13a-h6"],"title":"Final Answer","text":"Therefore, $$\\\\frac{3}{4} y^2-3=18$$ has two solutions which are $$y=-2\\\\sqrt{7}$$, and $$y=2\\\\sqrt{7}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5e282bSolvquad14","title":"Solve Quadratic Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5e282bSolvquad14a","stepAnswer":["(-7*(sqrt(1/2)),7*(sqrt(1/2)))"],"problemType":"TextBox","stepTitle":"$$2x^2-8=41$$","stepBody":"If there is more than one solution, please enter you answer as (a,b) where $$a<b$$ and a, $$b$$ are in the simplest form. If you can\'t further simplify the answer into a whole number, please eneter it in a form of $$b \\\\sqrt{c}$$ where $$b$$ is whole numbers and c is natural number or fraction.","answerType":"string","variabilization":{},"answerLatex":"$$(-7\\\\sqrt{\\\\frac{1}{2}},7\\\\sqrt{\\\\frac{1}{2}})$$","hints":{"DefaultPathway":[{"id":"a5e282bSolvquad14a-h1","type":"hint","dependencies":[],"title":"Isolate the Quadratic Term","text":"We add $$8$$ on both sides to get $$x^2$$ by itself which gives $$2x^2=49$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a5e282bSolvquad14a-h1"],"title":"Find Leading Coefficient","text":"What\'s the coefficient of $$u^2$$ now?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad14a-h3","type":"hint","dependencies":["a5e282bSolvquad14a-h2"],"title":"Make the Leading Coefficient \\"1\\"","text":"Divide by the leading coefficient on both sides which gives $$x^2=\\\\frac{49}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad14a-h4","type":"hint","dependencies":["a5e282bSolvquad14a-h3"],"title":"Use Square root Property","text":"The square root property: if $$x^2$$ $$=$$ k, then $$x$$ $$=$$ $$\\\\sqrt{k}$$, or $$x$$ $$=$$ $$-\\\\sqrt{k}$$, or $$x$$ $$=\\\\sqrt{k}$$. $$x^2=\\\\frac{49}{2}$$, we can take square root both sides and get $$x=\\\\sqrt{\\\\frac{49}{2}}$$ or $$x=-\\\\sqrt{\\\\frac{49}{2}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad14a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-7\\\\sqrt{\\\\frac{1}{2}}$$"],"dependencies":["a5e282bSolvquad14a-h4"],"title":"Simplify the Radical","text":"Simplify $$-\\\\sqrt{\\\\frac{49}{2}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad14a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7\\\\sqrt{\\\\frac{1}{2}}$$"],"dependencies":["a5e282bSolvquad14a-h5"],"title":"Simplify the Radical","text":"Simplify $$\\\\sqrt{\\\\frac{49}{2}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad14a-h7","type":"hint","dependencies":["a5e282bSolvquad14a-h6"],"title":"Final Answer","text":"Therefore, $$2x^2-8=41$$ has two solutions which are $$x=-7\\\\sqrt{\\\\frac{1}{2}}$$, and $$x=7\\\\sqrt{\\\\frac{1}{2}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5e282bSolvquad15","title":"Solve Quadratic Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5e282bSolvquad15a","stepAnswer":["(-6*(sqrt(1/5)),6*(sqrt(1/5)))"],"problemType":"TextBox","stepTitle":"$$5r^2-2$$ $$=$$ $$34$$","stepBody":"If there is more than one solution, please enter you answer as (a,b) where $$a<b$$ and a, $$b$$ are in the simplest form. If you can\'t further simplify the answer into a whole number, please eneter it in a form of $$b \\\\sqrt{c}$$ where $$b$$ is whole numbers and c is natural number or fraction.","answerType":"string","variabilization":{},"answerLatex":"$$(-6\\\\sqrt{\\\\frac{1}{5}},6\\\\sqrt{\\\\frac{1}{5}})$$","hints":{"DefaultPathway":[{"id":"a5e282bSolvquad15a-h1","type":"hint","dependencies":[],"title":"Isolate the Quadratic Term","text":"We add $$2$$ on both sides to get $$r^2$$ by itself which gives $$5r^2=36$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a5e282bSolvquad15a-h1"],"title":"Find Leading Coefficient","text":"What\'s the coefficient of $$u^2$$ now?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad15a-h3","type":"hint","dependencies":["a5e282bSolvquad15a-h2"],"title":"Make the Leading Coefficient \\"1\\"","text":"Divide by the leading coefficient on both sides which gives $$r^2=\\\\frac{36}{5}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad15a-h4","type":"hint","dependencies":["a5e282bSolvquad15a-h3"],"title":"Use Square root Property","text":"The square root property: if $$x^2$$ $$=$$ k, then $$x$$ $$=$$ $$\\\\sqrt{k}$$, or $$x$$ $$=$$ $$-\\\\sqrt{k}$$, or $$x$$ $$=\\\\sqrt{k}$$. $$r^2=\\\\frac{36}{5}$$, we can take square root both sides and get $$r=\\\\sqrt{\\\\frac{36}{5}}$$ or $$r=-\\\\sqrt{\\\\frac{36}{5}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad15a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6\\\\sqrt{\\\\frac{1}{5}}$$"],"dependencies":["a5e282bSolvquad15a-h4"],"title":"Simplify the Radical","text":"Simplify $$-\\\\sqrt{\\\\frac{36}{5}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad15a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6\\\\sqrt{\\\\frac{1}{5}}$$"],"dependencies":["a5e282bSolvquad15a-h5"],"title":"Simplify the Radical","text":"Simplify $$\\\\sqrt{\\\\frac{36}{5}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad15a-h7","type":"hint","dependencies":["a5e282bSolvquad15a-h6"],"title":"Final Answer","text":"Therefore, $$5r^2-2$$ $$=$$ $$34$$ has two solutions which are $$r=-6\\\\sqrt{\\\\frac{1}{5}}$$, and $$r=6\\\\sqrt{\\\\frac{1}{5}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5e282bSolvquad16","title":"Solve Quadratic Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5e282bSolvquad16a","stepAnswer":["(-8*(sqrt(1/3)),8*(sqrt(1/3)))"],"problemType":"TextBox","stepTitle":"$$3r^2+6=70$$","stepBody":"If there is more than one solution, please enter you answer as (a,b) where $$a<b$$ and a, $$b$$ are in the simplest form. If you can\'t further simplify the answer into a whole number, please eneter it in a form of $$b \\\\sqrt{c}$$ where $$b$$ is whole numbers and c is natural number or fraction.","answerType":"string","variabilization":{},"answerLatex":"$$(-8\\\\sqrt{\\\\frac{1}{3}},8\\\\sqrt{\\\\frac{1}{3}})$$","hints":{"DefaultPathway":[{"id":"a5e282bSolvquad16a-h1","type":"hint","dependencies":[],"title":"Isolate the Quadratic Term","text":"We subtract $$6$$ on both sides to get $$r^2$$ by itself which gives $$3r^2=64$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a5e282bSolvquad16a-h1"],"title":"Find Leading Coefficient","text":"What\'s the coefficient of $$u^2$$ now?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad16a-h3","type":"hint","dependencies":["a5e282bSolvquad16a-h2"],"title":"Make the Leading Coefficient \\"1\\"","text":"Divide by the leading coefficient on both sides which gives $$r^2=\\\\frac{64}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad16a-h4","type":"hint","dependencies":["a5e282bSolvquad16a-h3"],"title":"Use Square Root Property","text":"The square root property: if $$x^2$$ $$=$$ k, then $$x$$ $$=$$ $$\\\\sqrt{k}$$, or $$x$$ $$=$$ $$-\\\\sqrt{k}$$, or $$x$$ $$=\\\\sqrt{k}$$. $$r^2=\\\\frac{64}{3}$$, we can take square root both sides and get $$r=\\\\sqrt{\\\\frac{64}{3}}$$ or $$r=-\\\\sqrt{\\\\frac{64}{3}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad16a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8\\\\sqrt{\\\\frac{1}{3}}$$"],"dependencies":["a5e282bSolvquad16a-h4"],"title":"Simplify the Radical","text":"Simplify $$\\\\sqrt{\\\\frac{64}{3}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad16a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-8\\\\sqrt{\\\\frac{1}{3}}$$"],"dependencies":["a5e282bSolvquad16a-h5"],"title":"Simplify the Radical","text":"Simplify $$-\\\\sqrt{\\\\frac{64}{3}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad16a-h7","type":"hint","dependencies":["a5e282bSolvquad16a-h6"],"title":"Final Answer","text":"Therefore, $$3r^2+6=70$$ has two solutions which are $$r=-8\\\\sqrt{\\\\frac{1}{3}}$$, and $$r=8\\\\sqrt{\\\\frac{1}{3}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5e282bSolvquad17","title":"Solve Quadratic Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5e282bSolvquad17a","stepAnswer":["(7-2*sqrt(3),7+2*sqrt(3))"],"problemType":"TextBox","stepTitle":"$$4{\\\\left(y-7\\\\right)}^2=48$$","stepBody":"If there is more than one solution, please enter you answer as (a,b) where $$a<b$$ and a, $$b$$ are in the simplest form. If you can\'t further simplify the answer into a whole number, please eneter it in a form of $$a+b \\\\sqrt{c}$$ where a, $$b$$ are whole numbers and c is natural number or fraction.","answerType":"string","variabilization":{},"answerLatex":"$$(7-2\\\\sqrt{3},7+2\\\\sqrt{3})$$","hints":{"DefaultPathway":[{"id":"a5e282bSolvquad17a-h1","type":"hint","dependencies":[],"title":"Make the Leading Coefficient \\"1\\"","text":"Divide by the leading coefficient on both sides which gives $${\\\\left(y-7\\\\right)}^2=12$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad17a-h2","type":"hint","dependencies":["a5e282bSolvquad17a-h1"],"title":"Use Square Root Property","text":"The square root property: if $$x^2$$ $$=$$ k, then $$x$$ $$=$$ $$\\\\sqrt{k}$$, or $$x$$ $$=$$ $$-\\\\sqrt{k}$$, or $$x$$ $$=\\\\sqrt{k}$$. $${\\\\left(y-7\\\\right)}^2=12$$, we can take square root both sides and get $$y-7=\\\\sqrt{12}$$ or $$y-7=-\\\\sqrt{12}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad17a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{12}+7$$"],"dependencies":["a5e282bSolvquad17a-h2"],"title":"Find $$y$$ in $$y-7=\\\\sqrt{12}$$","text":"What is $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad17a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-\\\\sqrt{12}+7$$"],"dependencies":["a5e282bSolvquad17a-h3"],"title":"Find $$y$$ in $$y-7=-\\\\sqrt{12}$$","text":"What is $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad17a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7+2\\\\sqrt{3}$$"],"dependencies":["a5e282bSolvquad17a-h4"],"title":"Simplify the Radical","text":"Simplify $$\\\\sqrt{12}+7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad17a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7-2\\\\sqrt{3}$$"],"dependencies":["a5e282bSolvquad17a-h5"],"title":"Simplify the Radical","text":"Simplify $$-\\\\sqrt{12}+7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad17a-h7","type":"hint","dependencies":["a5e282bSolvquad17a-h6"],"title":"Final Answer","text":"Therefore, $$4{\\\\left(y-7\\\\right)}^2=48$$ has two solutions which are $$y=-7+2\\\\sqrt{3}$$, and $$y=7+2\\\\sqrt{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5e282bSolvquad18","title":"Solve Quadratic Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5e282bSolvquad18a","stepAnswer":["(3-3*sqrt(2),3 + 3*sqrt(2))"],"problemType":"TextBox","stepTitle":"$$3{\\\\left(a-3\\\\right)}^2$$ $$=54$$","stepBody":"If there is more than one solution, please enter you answer as (a,b) where $$a<b$$ and a, $$b$$ are in the simplest form. If you can\'t further simplify the answer into a whole number, please eneter it in a form of $$a+b \\\\sqrt{c}$$ where a, $$b$$ are whole numbers and c is natural number or fraction.","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a5e282bSolvquad18a-h1","type":"hint","dependencies":[],"title":"Make the Leading Coefficient \\"1\\"","text":"Divide by the leading coefficient on both sides which gives $${\\\\left(a-3\\\\right)}^2$$ $$=18$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad18a-h2","type":"hint","dependencies":["a5e282bSolvquad18a-h1"],"title":"Use Square Root Property","text":"The square root property: if $$x^2$$ $$=$$ k, then $$x$$ $$=$$ $$\\\\sqrt{k}$$, or $$x$$ $$=$$ $$-\\\\sqrt{k}$$, or $$x$$ $$=\\\\sqrt{k}$$. $${\\\\left(a-3\\\\right)}^2$$ $$=18$$, we can take square root both sides and get $$a-3=\\\\sqrt{18}$$ or $$a-3=-\\\\sqrt{18}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad18a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3-\\\\sqrt{18}$$"],"dependencies":["a5e282bSolvquad18a-h2"],"title":"Find a in $$a-3=\\\\sqrt{18}$$","text":"What is a?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad18a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3+\\\\sqrt{18}$$"],"dependencies":["a5e282bSolvquad18a-h3"],"title":"Find a in $$a-3=-\\\\sqrt{18}$$","text":"What is a?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad18a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3-3\\\\sqrt{2}$$"],"dependencies":["a5e282bSolvquad18a-h4"],"title":"Simplify the Radical","text":"Simplify $$3-\\\\sqrt{18}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad18a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3+3\\\\sqrt{2}$$"],"dependencies":["a5e282bSolvquad18a-h5"],"title":"Simplify the Radical","text":"Simplify $$3+\\\\sqrt{18}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad18a-h7","type":"hint","dependencies":["a5e282bSolvquad18a-h6"],"title":"Final Answer","text":"Therefore, $$3{\\\\left(a-3\\\\right)}^2$$ $$=54$$ has two solutions which are $$a=3+3\\\\sqrt{2}$$, and $$a=3-3\\\\sqrt{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5e282bSolvquad19","title":"Solve Quadratic Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5e282bSolvquad19a","stepAnswer":["(-2-2*sqrt(10),-2+2*sqrt(10))"],"problemType":"TextBox","stepTitle":"$$2{\\\\left(a+2\\\\right)}^2=80$$","stepBody":"If there is more than one solution, please enter you answer as (a,b) where $$a<b$$ and a, $$b$$ are in the simplest form. If you can\'t further simplify the answer into a whole number, please eneter it in a form of $$a+b \\\\sqrt{c}$$ where a, $$b$$ are whole numbers and c is natural number or fraction.","answerType":"string","variabilization":{},"answerLatex":"$$(-2-2\\\\sqrt{10},-2+2\\\\sqrt{10})$$","hints":{"DefaultPathway":[{"id":"a5e282bSolvquad19a-h1","type":"hint","dependencies":[],"title":"Make the Leading Coefficient \\"1\\"","text":"Divide by the leading coefficient on both sides which gives $${\\\\left(a+2\\\\right)}^2=40$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad19a-h2","type":"hint","dependencies":["a5e282bSolvquad19a-h1"],"title":"Use Square Root Property","text":"The square root property: if $$x^2$$ $$=$$ k, then $$x$$ $$=$$ $$\\\\sqrt{k}$$, or $$x$$ $$=$$ $$-\\\\sqrt{k}$$, or $$x$$ $$=\\\\sqrt{k}$$. $${\\\\left(a+2\\\\right)}^2=40$$, we can take square root both sides and get $$a+2=\\\\sqrt{40}$$ or $$a+2=-\\\\sqrt{40}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad19a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2-\\\\sqrt{40}$$"],"dependencies":["a5e282bSolvquad19a-h2"],"title":"Find a in $$a+2=\\\\sqrt{40}$$","text":"What is a?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad19a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2+\\\\sqrt{40}$$"],"dependencies":["a5e282bSolvquad19a-h3"],"title":"Find a in $$a+2=-\\\\sqrt{40}$$","text":"What is a?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad19a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2+2\\\\sqrt{10}$$"],"dependencies":["a5e282bSolvquad19a-h4"],"title":"Simplify the Radical","text":"Simplify $$-2+\\\\sqrt{40}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad19a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2-2\\\\sqrt{10}$$"],"dependencies":["a5e282bSolvquad19a-h5"],"title":"Simplify the Radical","text":"Simplify $$-2-\\\\sqrt{40}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad19a-h7","type":"hint","dependencies":["a5e282bSolvquad19a-h6"],"title":"Final Answer","text":"Therefore, $$2{\\\\left(a+2\\\\right)}^2=80$$ has two solutions which are $$a=-2-2\\\\sqrt{10}$$, and $$a=-2+2\\\\sqrt{10}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5e282bSolvquad2","title":"Solve Quadratic Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5e282bSolvquad2a","stepAnswer":["(-4*(sqrt(3)),4*(sqrt(3)))"],"problemType":"TextBox","stepTitle":"$$x^2-48$$ $$=$$ $$0$$","stepBody":"If there is more than one solution, please enter you answer as (a,b) where $$a<b$$ and a, $$b$$ are in the simplest form. If you can\'t further simplify the answer into a whole number, please eneter it in a form of $$b \\\\sqrt{c}$$ where $$b$$ is whole numbers and c is natural number.","answerType":"string","variabilization":{},"answerLatex":"$$(-4\\\\sqrt{3},4\\\\sqrt{3})$$","hints":{"DefaultPathway":[{"id":"a5e282bSolvquad2a-h1","type":"hint","dependencies":[],"title":"Isolate the Quadratic Term and Make Its Coefficient One","text":"We add $$48$$ to both sides to get $$x^2$$ by itself which gives $$x^2=48$$. Since the coefficient of $$x^2$$ is $$1$$ already, we do not need to take extra step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad2a-h2","type":"hint","dependencies":["a5e282bSolvquad2a-h1"],"title":"Use Square root Property","text":"The square root property: if $$x^2$$ $$=$$ k, then $$x$$ $$=$$ $$\\\\sqrt{k}$$, or $$x$$ $$=$$ $$-\\\\sqrt{k}$$. $$x^2=48$$, we can take square root both sides and get $$x=\\\\sqrt{48}$$ or $$x=-\\\\sqrt{48}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4\\\\sqrt{3}$$"],"dependencies":["a5e282bSolvquad2a-h2"],"title":"Simplify the Radical","text":"Simplify $$-\\\\sqrt{48}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4\\\\sqrt{3}$$"],"dependencies":["a5e282bSolvquad2a-h3"],"title":"Simplify the Radical","text":"Simplify $$\\\\sqrt{48}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad2a-h5","type":"hint","dependencies":["a5e282bSolvquad2a-h4"],"title":"Final Answer","text":"Therefore, $$x^2-48$$ $$=$$ $$0$$ has two solutions which are $$x=4\\\\sqrt{3}$$ and $$x=-4\\\\sqrt{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5e282bSolvquad20","title":"Solve Quadratic Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5e282bSolvquad20a","stepAnswer":["(2-3*sqrt(3),2+3*sqrt(3))"],"problemType":"TextBox","stepTitle":"$$2{\\\\left(x-2\\\\right)}^2+3=57$$","stepBody":"If there is more than one solution, please enter you answer as (a,b) where $$a<b$$ and a, $$b$$ are in the simplest form. If you can\'t further simplify the answer into a whole number, please eneter it in a form of $$a+b \\\\sqrt{c}$$ where a, $$b$$ are whole numbers and c is natural number or fraction.","answerType":"string","variabilization":{},"answerLatex":"$$(2-3\\\\sqrt{3},2+3\\\\sqrt{3})$$","hints":{"DefaultPathway":[{"id":"a5e282bSolvquad20a-h1","type":"hint","dependencies":[],"title":"Isolate the Quadratic term","text":"We minus $$3$$ on both sides to get $$2{\\\\left(x-2\\\\right)}^2$$ by itself which gives $$2{\\\\left(x-2\\\\right)}^2=54$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a5e282bSolvquad20a-h1"],"title":"Find Leading Coefficient","text":"What\'s the leading coefficient now?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad20a-h3","type":"hint","dependencies":["a5e282bSolvquad20a-h2"],"title":"Make the Leading Coefficient \\"1\\"","text":"Divide by the leading coefficient on both sides which gives $${\\\\left(x-2\\\\right)}^2=27$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad20a-h4","type":"hint","dependencies":["a5e282bSolvquad20a-h3"],"title":"Use Square Root Property","text":"The square root property: if $$x^2$$ $$=$$ k, then $$x$$ $$=$$ $$\\\\sqrt{k}$$, or $$x$$ $$=$$ $$-\\\\sqrt{k}$$, or $$x$$ $$=\\\\sqrt{k}$$. $${\\\\left(x-2\\\\right)}^2=27$$, we can take square root both sides and get $$x-2=\\\\sqrt{27}$$ or $$x-2=-\\\\sqrt{27}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad20a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-\\\\sqrt{27}+2$$"],"dependencies":["a5e282bSolvquad20a-h4"],"title":"Find $$x$$ in $$x-2=-\\\\sqrt{27}$$","text":"What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad20a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{27}+2$$"],"dependencies":["a5e282bSolvquad20a-h5"],"title":"Find $$x$$ in $$x-2=\\\\sqrt{27}$$","text":"What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad20a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2-3\\\\sqrt{3}$$"],"dependencies":["a5e282bSolvquad20a-h6"],"title":"Simplify the Radical","text":"Simplify $$-\\\\sqrt{27}+2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad20a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2+3\\\\sqrt{3}$$"],"dependencies":["a5e282bSolvquad20a-h7"],"title":"Simplify the Radical","text":"Simplify $$\\\\sqrt{27}+2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad20a-h9","type":"hint","dependencies":["a5e282bSolvquad20a-h8"],"title":"Final Answer","text":"Therefore, $$2{\\\\left(x-2\\\\right)}^2+3=57$$ has two solutions which are $$x=2-3\\\\sqrt{3}$$, and $$x=2+3\\\\sqrt{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5e282bSolvquad21","title":"Solve Quadratic Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5e282bSolvquad21a","stepAnswer":["(5-2*sqrt(5),5+2*sqrt(5))"],"problemType":"TextBox","stepTitle":"$$5{\\\\left(x-5\\\\right)}^2+4=104$$","stepBody":"If there is more than one solution, please enter you answer as (a,b) where $$a<b$$ and a, $$b$$ are in the simplest form. If you can\'t further simplify the answer into a whole number, please eneter it in a form of $$a+b \\\\sqrt{c}$$ where a, $$b$$ are whole numbers and c is natural number or fraction.","answerType":"string","variabilization":{},"answerLatex":"$$(5-2\\\\sqrt{5},5+2\\\\sqrt{5})$$","hints":{"DefaultPathway":[{"id":"a5e282bSolvquad21a-h1","type":"hint","dependencies":[],"title":"Isolate the Quadratic term","text":"We minus $$4$$ on both sides to get $$5{\\\\left(x-5\\\\right)}^2$$ isolated by itself which gives $$5{\\\\left(x-5\\\\right)}^2=100$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad21a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a5e282bSolvquad21a-h1"],"title":"Find Leading Coefficient","text":"What\'s the leading coefficient now?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad21a-h3","type":"hint","dependencies":["a5e282bSolvquad21a-h2"],"title":"Make the Leading Coefficient \\"1\\"","text":"Divide by the leading coefficient on both sides which gives $${\\\\left(x-5\\\\right)}^2=20$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad21a-h4","type":"hint","dependencies":["a5e282bSolvquad21a-h3"],"title":"Use Square Root Property","text":"The square root property: if $$x^2$$ $$=$$ k, then $$x$$ $$=$$ $$\\\\sqrt{k}$$, or $$x$$ $$=$$ $$-\\\\sqrt{k}$$, or $$x$$ $$=\\\\sqrt{k}$$. $${\\\\left(x-5\\\\right)}^2=20$$, we can take square root both sides and get $$x-5=\\\\sqrt{20}$$ or $$x-5=-\\\\sqrt{20}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad21a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5+\\\\sqrt{20}$$"],"dependencies":["a5e282bSolvquad21a-h4"],"title":"Find $$x$$ in $$x-5=-\\\\sqrt{20}$$","text":"What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad21a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5-\\\\sqrt{20}$$"],"dependencies":["a5e282bSolvquad21a-h5"],"title":"Find $$x$$ in $$x-5=\\\\sqrt{20}$$","text":"What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad21a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5-2\\\\sqrt{5}$$"],"dependencies":["a5e282bSolvquad21a-h6"],"title":"Simplify the Radical","text":"Simplify $$5-\\\\sqrt{20}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad21a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5+2\\\\sqrt{5}$$"],"dependencies":["a5e282bSolvquad21a-h7"],"title":"Simplify the Radical","text":"Simplify $$5+\\\\sqrt{20}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad21a-h9","type":"hint","dependencies":["a5e282bSolvquad21a-h8"],"title":"Final Answer","text":"Therefore, $$5{\\\\left(x-5\\\\right)}^2+4=104$$ has two solutions which are $$x=5-2\\\\sqrt{5}$$, and $$x=5+2\\\\sqrt{5}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5e282bSolvquad22","title":"Solve Quadratic Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5e282bSolvquad22a","stepAnswer":["(-3-4*sqrt(2),-3+4*sqrt(2))"],"problemType":"TextBox","stepTitle":"$$3{\\\\left(b+3\\\\right)}^2-8$$ $$=88$$","stepBody":"If there is more than one solution, please enter you answer as (a,b) where $$a<b$$ and a, $$b$$ are in the simplest form. If you can\'t further simplify the answer into a whole number, please eneter it in a form of $$a+b \\\\sqrt{c}$$ where a, $$b$$ are whole numbers and c is natural number or fraction.","answerType":"string","variabilization":{},"answerLatex":"$$(-3-4\\\\sqrt{2},-3+4\\\\sqrt{2})$$","hints":{"DefaultPathway":[{"id":"a5e282bSolvquad22a-h1","type":"hint","dependencies":[],"title":"Isolate the Quadratic term","text":"We add $$8$$ on both sides to get $$3{\\\\left(b+3\\\\right)}^2$$ isolated by itself which gives $$3{\\\\left(b+3\\\\right)}^2=96$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad22a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a5e282bSolvquad22a-h1"],"title":"Find Leading Coefficient","text":"What\'s the leading coefficient now?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad22a-h3","type":"hint","dependencies":["a5e282bSolvquad22a-h2"],"title":"Make the Leading Coefficient \\"1\\"","text":"Divide by the leading coefficient on both sides which gives $${\\\\left(b+3\\\\right)}^2=32$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad22a-h4","type":"hint","dependencies":["a5e282bSolvquad22a-h3"],"title":"Use Square Root Property","text":"The square root property: if $$x^2$$ $$=$$ k, then $$x$$ $$=$$ $$\\\\sqrt{k}$$, or $$x$$ $$=$$ $$-\\\\sqrt{k}$$, or $$x$$ $$=\\\\sqrt{k}$$. $${\\\\left(b+3\\\\right)}^2=32$$, we can take square root both sides and get $$b+3=\\\\sqrt{32}$$ or $$b+3=-\\\\sqrt{32}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad22a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-\\\\sqrt{32}-3$$"],"dependencies":["a5e282bSolvquad22a-h4"],"title":"Find $$b$$ in $$b+3=-\\\\sqrt{32}$$","text":"What is $$b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad22a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{32}-3$$"],"dependencies":["a5e282bSolvquad22a-h5"],"title":"Find $$b$$ in $$b+3=\\\\sqrt{32}$$","text":"What is $$b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad22a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4\\\\sqrt{2}-3$$"],"dependencies":["a5e282bSolvquad22a-h6"],"title":"Simplify the Radical","text":"Simplify $$-\\\\sqrt{32}-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad22a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4\\\\sqrt{2}-3$$"],"dependencies":["a5e282bSolvquad22a-h7"],"title":"Simplify the Radical","text":"Simplify $$\\\\sqrt{32}-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad22a-h9","type":"hint","dependencies":["a5e282bSolvquad22a-h8"],"title":"Final Answer","text":"Therefore, $$3{\\\\left(b+3\\\\right)}^2-8$$ $$=88$$ has two solutions which are $$b=-4\\\\sqrt{2}-3$$, and $$b=4\\\\sqrt{2}-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5e282bSolvquad23","title":"Solve Quadratic Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5e282bSolvquad23a","stepAnswer":["(-5/2,3/2)"],"problemType":"TextBox","stepTitle":"$$4n^2+4n+1=16$$","stepBody":"If there is more than one solution, please enter you answer as (a,b) where $$a<b$$ and a, $$b$$ are in the simplest form. If you can\'t further simplify the answer into a whole number, please eneter it in a form of $$a+b \\\\sqrt{c}$$ where a, $$b$$ are whole numbers and c is natural number or fraction.","answerType":"string","variabilization":{},"answerLatex":"$$(\\\\frac{-5}{2},\\\\frac{3}{2})$$","hints":{"DefaultPathway":[{"id":"a5e282bSolvquad23a-h1","type":"hint","dependencies":[],"title":"Factor the Perfect Square","text":"Does the expression match any of these formats: $${\\\\left(a x\\\\right)}^2+2a b x$$ + $$b^2={\\\\left(a x+b\\\\right)}^2$$, $${\\\\left(a x\\\\right)}^2-2a b x+b^2={\\\\left(a x-b\\\\right)}^2$$ ?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad23a-h2","type":"hint","dependencies":["a5e282bSolvquad23a-h1"],"title":"Factor the Perfect Square","text":"In this case, we can see that 4n**2+4n+1as $${\\\\left(a x\\\\right)}^2+2a b x$$ + $$b^2={\\\\left(a x+b\\\\right)}^2$$ where $$ax=2n$$ and $$b=1$$. We can factor $$4n^2+4n+1$$ as $${\\\\left(2n+1\\\\right)}^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad23a-h3","type":"hint","dependencies":["a5e282bSolvquad23a-h2"],"title":"Rewrite the Equation","text":"We can rewrite $$4n^2+4n+1=16$$ as $${\\\\left(2n+1\\\\right)}^2=16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad23a-h4","type":"hint","dependencies":["a5e282bSolvquad23a-h3"],"title":"Solve for Unknown Variable","text":"$${\\\\left(2n+1\\\\right)}^2=16$$. What is $$n$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad23a-h5","type":"hint","dependencies":["a5e282bSolvquad23a-h4"],"title":"Use Square Root Property","text":"The square root property: if $$x^2$$ $$=$$ k, then $$x$$ $$=$$ $$\\\\sqrt{k}$$, or $$x$$ $$=$$ $$-\\\\sqrt{k}$$, or $$x$$ $$=\\\\sqrt{k}$$. $${\\\\left(2n+1\\\\right)}^2=16$$, we can take square root both sides and get $$2n+1=4$$ or $$2n+1=-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad23a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-5}{2}$$"],"dependencies":["a5e282bSolvquad23a-h5"],"title":"Solve for $$n$$ in $$2n+1=-4$$","text":"What is $$n$$ when $$2n+1=-4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad23a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{2}$$"],"dependencies":["a5e282bSolvquad23a-h6"],"title":"Solve for $$n$$ in $$2n+1=4$$","text":"What is $$n$$ when $$2n+1=4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad23a-h8","type":"hint","dependencies":["a5e282bSolvquad23a-h7"],"title":"Final Answer","text":"Therefore, $$4n^2+4n+1=16$$ has two solutions which are $$n=\\\\frac{-5}{2}$$ and $$n=\\\\frac{3}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5e282bSolvquad24","title":"Solve Quadratic Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5e282bSolvquad24a","stepAnswer":["(-1,7/3)"],"problemType":"TextBox","stepTitle":"$$9n^2-12n+4=25$$","stepBody":"If there is more than one solution, please enter you answer as (a,b) where $$a<b$$ and a, $$b$$ are in the simplest form. If you can\'t further simplify the answer into a whole number or fraction, please eneter it in a form of $$a+b \\\\sqrt{c}$$ where a, $$b$$ are whole numbers and c is natural number or fraction.","answerType":"string","variabilization":{},"answerLatex":"$$(-1,\\\\frac{7}{3})$$","hints":{"DefaultPathway":[{"id":"a5e282bSolvquad24a-h1","type":"hint","dependencies":[],"title":"Factor the Perfect Square","text":"Does the expression match any of these formats: $${\\\\left(a x\\\\right)}^2+2a b x$$ + $$b^2$$ $$=$$ $${\\\\left(a x+b\\\\right)}^2$$, $${\\\\left(a x\\\\right)}^2-2a b x+b^2$$ $$=$$ $${\\\\left(a x-b\\\\right)}^2$$ ? Yes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad24a-h2","type":"hint","dependencies":["a5e282bSolvquad24a-h1"],"title":"Factor the Perfect Square","text":"In this case, we can see that $$9n^2-12n+4$$ as $${\\\\left(a x\\\\right)}^2-2a b x+b^2={\\\\left(ax-b\\\\right)}^2$$ where $$ax=3n$$ and $$b=2$$. We can factor $$9n^2-12n+4=25$$ as $${\\\\left(3n-2\\\\right)}^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad24a-h3","type":"hint","dependencies":["a5e282bSolvquad24a-h2"],"title":"Rewrite the Equation","text":"We can rewrite $$9n^2-12n+4=25$$ as $${\\\\left(3n-2\\\\right)}^2=25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad24a-h4","type":"hint","dependencies":["a5e282bSolvquad24a-h3"],"title":"Solve for Unknown Variable","text":"$${\\\\left(3n-2\\\\right)}^2=25$$. What is $$n$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad24a-h5","type":"hint","dependencies":["a5e282bSolvquad24a-h4"],"title":"Use Square Root Property","text":"The square root property: if $$x^2$$ $$=$$ k, then $$x$$ $$=$$ $$\\\\sqrt{k}$$, or $$x$$ $$=$$ $$-\\\\sqrt{k}$$, or $$x$$ $$=\\\\sqrt{k}$$. $${\\\\left(3n-2\\\\right)}^2=25$$, we can take square root both sides and get $$3n-2=5$$ or $$3n-2=-5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad24a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{7}{3}$$"],"dependencies":["a5e282bSolvquad24a-h5"],"title":"Solve for $$n$$ in $$3n-2=5$$","text":"What is $$n$$ when $$3n-2=5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad24a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a5e282bSolvquad24a-h6"],"title":"Solve for $$n$$ in $$3n-2=-5$$","text":"What is $$n$$ $$when3n-2=-5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad24a-h8","type":"hint","dependencies":["a5e282bSolvquad24a-h7"],"title":"Final Answer","text":"Therefore, $$9n^2-12n+4=25$$ has two solutions which are $$n=-1$$ and $$n=\\\\frac{7}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5e282bSolvquad25","title":"Solve Quadratic Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5e282bSolvquad25a","stepAnswer":["(-7/4,-3/4)"],"problemType":"TextBox","stepTitle":"$$16n^2+40n+25=4$$","stepBody":"If there is more than one solution, please enter you answer as (a,b) where $$a<b$$ and a, $$b$$ are in the simplest form. If you can\'t further simplify the answer into a whole number or fraction, please eneter it in a form of $$a+b \\\\sqrt{c}$$ where a, $$b$$ are whole numbers and c is natural number or fraction.","answerType":"string","variabilization":{},"answerLatex":"$$(\\\\frac{-7}{4},\\\\frac{-3}{4})$$","hints":{"DefaultPathway":[{"id":"a5e282bSolvquad25a-h1","type":"hint","dependencies":[],"title":"Factor the Perfect Square","text":"Does the expression match any of these formats: $${\\\\left(a x\\\\right)}^2+2a b x$$ + $$b^2$$ $$=$$ $${\\\\left(a x+b\\\\right)}^2$$, $${\\\\left(a x\\\\right)}^2-2a b x+b^2$$ $$=$$ $${\\\\left(a x-b\\\\right)}^2$$ ? Yes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad25a-h2","type":"hint","dependencies":["a5e282bSolvquad25a-h1"],"title":"Factor the Perfect Square","text":"In this case, we can see that $$16n^2+40n+25$$ as $${\\\\left(a x\\\\right)}^2+2a b x+b^2={\\\\left(ax+b\\\\right)}^2$$ where $$ax=4n$$ and $$b=5$$. We can factor $$16n^2+40n+25$$ as $${\\\\left(4n+5\\\\right)}^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad25a-h3","type":"hint","dependencies":["a5e282bSolvquad25a-h2"],"title":"Rewrite the Equation","text":"We can rewrite $$16n^2+40n+25=4$$ as $${\\\\left(4n+5\\\\right)}^2=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad25a-h4","type":"hint","dependencies":["a5e282bSolvquad25a-h3"],"title":"Solve for Unknown Variable","text":"$${\\\\left(4n+5\\\\right)}^2=4$$. What is $$n$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad25a-h5","type":"hint","dependencies":["a5e282bSolvquad25a-h4"],"title":"Use Square Root Property","text":"The square root property: if $$x^2$$ $$=$$ k, then $$x$$ $$=$$ $$\\\\sqrt{k}$$, or $$x$$ $$=$$ $$-\\\\sqrt{k}$$, or $$x$$ $$=\\\\sqrt{k}$$. $${\\\\left(4n+5\\\\right)}^2=4$$, we can take square root both sides and get $$4n+5=-2$$ or $$4n+5=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad25a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-7}{4}$$"],"dependencies":["a5e282bSolvquad25a-h5"],"title":"Solve for $$n$$ in $$4n+5=-2$$","text":"What is $$n$$ when $$4n+5=-2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad25a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-3}{4}$$"],"dependencies":["a5e282bSolvquad25a-h6"],"title":"Solve for $$n$$ in $$4n+5=2$$","text":"What is $$n$$ when $$4n+5=2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad25a-h8","type":"hint","dependencies":["a5e282bSolvquad25a-h7"],"title":"Final Answer","text":"Therefore, $$16n^2+40n+25=4$$ has two solutions which are $$n=\\\\frac{-7}{4}$$ and $$n=\\\\frac{-3}{4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5e282bSolvquad26","title":"Solve Quadratic Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5e282bSolvquad26a","stepAnswer":["(1,9)"],"problemType":"TextBox","stepTitle":"$$n^2-10n+25=16$$","stepBody":"If there is more than one solution, please enter you answer as (a,b) where $$a<b$$ and a, $$b$$ are in the simplest form. If you can\'t further simplify the answer into a whole number or fraction, please eneter it in a form of $$a+b \\\\sqrt{c}$$ where a, $$b$$ are whole numbers and c is natural number or fraction.","answerType":"string","variabilization":{},"answerLatex":"$$(1,9)$$","hints":{"DefaultPathway":[{"id":"a5e282bSolvquad26a-h1","type":"hint","dependencies":[],"title":"Factor the Perfect Square","text":"Does the expression match any of these formats: $${\\\\left(a x\\\\right)}^2+2a b x$$ + $$b^2$$ $$=$$ $${\\\\left(a x+b\\\\right)}^2$$, $${\\\\left(a x\\\\right)}^2-2a b x+b^2$$ $$=$$ $${\\\\left(a x-b\\\\right)}^2$$ ? Yes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad26a-h2","type":"hint","dependencies":["a5e282bSolvquad26a-h1"],"title":"Factor the Perfect Square","text":"In this case, we can see that $$n^2-10n+25$$ as $${\\\\left(a x\\\\right)}^2-2a b x+b^2={\\\\left(ax-b\\\\right)}^2$$ where $$ax=n$$ and $$b=5$$. We can factor $$n^2-10n+25$$ as $${\\\\left(n+5\\\\right)}^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad26a-h3","type":"hint","dependencies":["a5e282bSolvquad26a-h2"],"title":"Rewrite the Equation","text":"We can rewrite $$n^2-10n+25=16$$ as $${\\\\left(n-5\\\\right)}^2=16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad26a-h4","type":"hint","dependencies":["a5e282bSolvquad26a-h3"],"title":"Solve for Unknown Variable","text":"$${\\\\left(n-5\\\\right)}^2=16$$. What is $$n$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad26a-h5","type":"hint","dependencies":["a5e282bSolvquad26a-h4"],"title":"Use Square Root Property","text":"The square root property: if $$x^2$$ $$=$$ k, then $$x$$ $$=$$ $$\\\\sqrt{k}$$, or $$x$$ $$=$$ $$-\\\\sqrt{k}$$, or $$x$$ $$=\\\\sqrt{k}$$. $${\\\\left(n-5\\\\right)}^2=16$$, we can take square root both sides and get $$n-5=-4$$ or $$n-5=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad26a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a5e282bSolvquad26a-h5"],"title":"Solve for $$n$$ in $$n-5=-4$$","text":"What is $$n$$ when $$n-5=-4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad26a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a5e282bSolvquad26a-h6"],"title":"Solve for $$n$$ in $$n-5=4$$","text":"What is $$n$$ when $$n-5=4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad26a-h8","type":"hint","dependencies":["a5e282bSolvquad26a-h7"],"title":"Final Answer","text":"Therefore, $$n^2-10n+25=16$$ has two solutions which are $$n=1$$ and $$n=9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5e282bSolvquad27","title":"Solve Quadratic Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5e282bSolvquad27a","stepAnswer":["(-9/7,1)"],"problemType":"TextBox","stepTitle":"$$49n^2+14n+1=64$$","stepBody":"If there is more than one solution, please enter you answer as (a,b) where $$a<b$$ and a, $$b$$ are in the simplest form. If you can\'t further simplify the answer into a whole number or fraction, please eneter it in a form of $$a+b \\\\sqrt{c}$$ where a, $$b$$ are whole numbers and c is natural number or fraction.","answerType":"string","variabilization":{},"answerLatex":"$$(\\\\frac{-9}{7},1)$$","hints":{"DefaultPathway":[{"id":"a5e282bSolvquad27a-h1","type":"hint","dependencies":[],"title":"Factor the Perfect Square","text":"Does the expression match any of these formats: $${\\\\left(a x\\\\right)}^2+2a b x$$ + $$b^2$$ $$=$$ $${\\\\left(a x+b\\\\right)}^2$$, $${\\\\left(a x\\\\right)}^2-2a b x+b^2$$ $$=$$ $${\\\\left(a x-b\\\\right)}^2$$ ? Yes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad27a-h2","type":"hint","dependencies":["a5e282bSolvquad27a-h1"],"title":"Factor the Perfect Square","text":"In this case, we can see that $$49n^2+14n+1$$ as $${\\\\left(a x\\\\right)}^2+2a b x+b^2={\\\\left(ax+b\\\\right)}^2$$ where $$ax=7n$$ and $$b=1$$. We can factor $$49n^2+14n+1$$ as $${\\\\left(7n+1\\\\right)}^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad27a-h3","type":"hint","dependencies":["a5e282bSolvquad27a-h2"],"title":"Rewrite the Equation","text":"We can rewrite $$49n^2+14n+1=64$$ as $${\\\\left(7n+1\\\\right)}^2=64$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad27a-h4","type":"hint","dependencies":["a5e282bSolvquad27a-h3"],"title":"Solve for Unknown Variable","text":"$${\\\\left(7n+1\\\\right)}^2=64$$. What is $$n$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad27a-h5","type":"hint","dependencies":["a5e282bSolvquad27a-h4"],"title":"Use Square Root Property","text":"The square root property: if $$x^2$$ $$=$$ k, then $$x$$ $$=$$ $$\\\\sqrt{k}$$, or $$x$$ $$=$$ $$-\\\\sqrt{k}$$, or $$x$$ $$=\\\\sqrt{k}$$. $${\\\\left(7n+1\\\\right)}^2=64$$, we can take square root both sides and get $$7n+1=8$$ or $$7n+1=-8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad27a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a5e282bSolvquad27a-h5"],"title":"Solve for $$n$$ in $$7n+1=8$$","text":"What is $$n$$ when $$7n+1=8$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad27a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-9}{7}$$"],"dependencies":["a5e282bSolvquad27a-h6"],"title":"Solve for $$n$$ in $$7n+1=-8$$","text":"What is $$n$$ when $$7n+1=-8$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad27a-h8","type":"hint","dependencies":["a5e282bSolvquad27a-h7"],"title":"Final Answer","text":"Therefore, $$49n^2+14n+1=64$$ has two solutions which are $$n=1$$ and $$n=\\\\frac{-9}{7}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5e282bSolvquad28","title":"Solve Quadratic Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5e282bSolvquad28a","stepAnswer":["((-2/5)+(3/5)*sqrt(2), (-2/5)-(3/5)*sqrt(2))"],"problemType":"TextBox","stepTitle":"$$25n^2+20n+4=18$$","stepBody":"If there is more than one solution, please enter you answer as (j,k) where $$j<k$$ and j, k are in the simplest form. If you can\'t further simplify the answer into a whole number or fraction, please eneter it in a form of $$a+b \\\\sqrt{c}$$ where a, $$b$$ and c are either natural numbers or fractions.","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a5e282bSolvquad28a-h1","type":"hint","dependencies":[],"title":"Factor the Perfect Square","text":"Does the expression match any of these formats: $${\\\\left(a x\\\\right)}^2+2a b x$$ + $$b^2$$ $$=$$ $${\\\\left(a x+b\\\\right)}^2$$, $${\\\\left(a x\\\\right)}^2-2a b x+b^2$$ $$=$$ $${\\\\left(a x-b\\\\right)}^2$$ ? Yes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad28a-h2","type":"hint","dependencies":["a5e282bSolvquad28a-h1"],"title":"Factor the Perfect Square","text":"In this case, we can see that $$25n^2+20n+4$$ as $${\\\\left(a x\\\\right)}^2+2a b x+b^2={\\\\left(ax+b\\\\right)}^2$$ where $$ax=5n$$ and $$b=2$$. We can factor $$25n^2+20n+4$$ as $${\\\\left(5n+2\\\\right)}^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad28a-h3","type":"hint","dependencies":["a5e282bSolvquad28a-h2"],"title":"Rewrite the Equation","text":"We can rewrite $$25n^2+20n+4=18$$ as $${\\\\left(5n+2\\\\right)}^2=18$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad28a-h4","type":"hint","dependencies":["a5e282bSolvquad28a-h3"],"title":"Solve for Unknown Variable","text":"$${\\\\left(5n+2\\\\right)}^2=18$$. What is $$n$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad28a-h5","type":"hint","dependencies":["a5e282bSolvquad28a-h4"],"title":"Use Square Root Property","text":"The square root property: if $$x^2$$ $$=$$ k, then $$x$$ $$=$$ $$\\\\sqrt{k}$$, or $$x$$ $$=$$ $$-\\\\sqrt{k}$$, or $$x$$ $$=\\\\sqrt{k}$$. $${\\\\left(5n+2\\\\right)}^2=18$$, we can take square root both sides and get $$5n+2=\\\\sqrt{18}$$ and $$5n+2=-\\\\sqrt{18}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad28a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{5}+\\\\frac{3}{5} \\\\sqrt{2}$$"],"dependencies":["a5e282bSolvquad28a-h5"],"title":"Solve for $$n$$ in $$5n+2=\\\\sqrt{18}$$","text":"What is $$n$$ when $$5n+2=-\\\\sqrt{18}$$? Simplify the square root as much as possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad28a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{5}-\\\\frac{3}{5} \\\\sqrt{2}$$"],"dependencies":["a5e282bSolvquad28a-h6"],"title":"Solve for $$n$$ in $$5n+2=-\\\\sqrt{18}$$","text":"What is $$n$$ when $$5n+2=\\\\sqrt{18}$$? Simplify the square root as much as possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad28a-h8","type":"hint","dependencies":["a5e282bSolvquad28a-h7"],"title":"Final Answer","text":"Therefore, $$25n^2+20n+4=18$$ has two solutions which are $$n=\\\\frac{2}{5}+\\\\frac{3}{5} \\\\sqrt{2}$$ and $$n=\\\\frac{2}{5}-\\\\frac{3}{5} \\\\sqrt{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5e282bSolvquad29","title":"Solve Quadratic Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5e282bSolvquad29a","stepAnswer":["((-5/3)-(2/3)*sqrt(2),(-5/3)+(2/3)*sqrt(2))"],"problemType":"TextBox","stepTitle":"$$9n^2+30n+25=8$$","stepBody":"If there is more than one solution, please enter you answer as (j,k) where $$j<k$$ and j, k are in the simplest form. If you can\'t further simplify the answer into a whole number or fraction, please eneter it in a form of $$a+b \\\\sqrt{c}$$ where a, $$b$$ and c are either natural numbers or fractions.","answerType":"string","variabilization":{},"answerLatex":"$$(\\\\left(-\\\\frac{5}{3}\\\\right)-\\\\frac{2}{3} \\\\sqrt{2},\\\\left(-\\\\frac{5}{3}\\\\right)+\\\\frac{2}{3} \\\\sqrt{2})$$","hints":{"DefaultPathway":[{"id":"a5e282bSolvquad29a-h1","type":"hint","dependencies":[],"title":"Factor the Perfect Square","text":"Does the expression match any of these formats: $${\\\\left(a x\\\\right)}^2+2a b x$$ + $$b^2$$ $$=$$ $${\\\\left(a x+b\\\\right)}^2$$, $${\\\\left(a x\\\\right)}^2-2a b x+b^2$$ $$=$$ $${\\\\left(a x-b\\\\right)}^2$$ ? Yes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad29a-h2","type":"hint","dependencies":["a5e282bSolvquad29a-h1"],"title":"Factor the Perfect Square","text":"In this case, we can see that $$9n^2+30n+25$$ as $${\\\\left(a x\\\\right)}^2+2a b x+b^2={\\\\left(ax+b\\\\right)}^2$$ where $$ax=3n$$ and $$b=5$$. We can factor $$9n^2+30n+25$$ as $${\\\\left(3n+5\\\\right)}^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad29a-h3","type":"hint","dependencies":["a5e282bSolvquad29a-h2"],"title":"Rewrite the Equation","text":"We can rewrite $$9n^2+30n+25=8$$ as $${\\\\left(3n+5\\\\right)}^2=8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad29a-h4","type":"hint","dependencies":["a5e282bSolvquad29a-h3"],"title":"Solve for Unknown Variable","text":"$${\\\\left(3n+5\\\\right)}^2=8$$. What is $$n$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad29a-h5","type":"hint","dependencies":["a5e282bSolvquad29a-h4"],"title":"Use Square Root Property","text":"The square root property: if $$x^2$$ $$=$$ k, then $$x$$ $$=$$ $$\\\\sqrt{k}$$, or $$x$$ $$=$$ $$-\\\\sqrt{k}$$, or $$x$$ $$=\\\\sqrt{k}$$. $${\\\\left(3n+5\\\\right)}^2=8$$, we can take square root both sides and get $$3n+5=\\\\sqrt{8}$$ and $$3n+5=-\\\\sqrt{8}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad29a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(-\\\\frac{5}{3}\\\\right)+\\\\frac{2}{3} \\\\sqrt{2}$$"],"dependencies":["a5e282bSolvquad29a-h5"],"title":"Solve for $$n$$ in $$3n+5=\\\\sqrt{8}$$","text":"What is $$n$$ when $$3n+5=\\\\sqrt{8}$$? Simplify the square root as much as possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad29a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(-\\\\frac{5}{3}\\\\right)-\\\\frac{2}{3} \\\\sqrt{2}$$"],"dependencies":["a5e282bSolvquad29a-h6"],"title":"Solve for $$n$$ in $$3n+5=-\\\\sqrt{8}$$","text":"What is $$n$$ when $$3n+5=-\\\\sqrt{8}$$? Simplify the square root as much as possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad29a-h8","type":"hint","dependencies":["a5e282bSolvquad29a-h7"],"title":"Final Answer","text":"Therefore, $$9n^2+30n+25=8$$ has two solutions which are $$n=\\\\left(-\\\\frac{5}{3}\\\\right)+\\\\frac{2}{3} \\\\sqrt{2}$$ and $$n=\\\\left(-\\\\frac{5}{3}\\\\right)-\\\\frac{2}{3} \\\\sqrt{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5e282bSolvquad3","title":"Solve Quadratic Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5e282bSolvquad3a","stepAnswer":["(-3*(sqrt(3)),3*(sqrt(3)))"],"problemType":"TextBox","stepTitle":"$$x^2-27$$ $$=$$ $$0$$","stepBody":"If there is more than one solution, please enter you answer as (a,b) where $$a<b$$ and a, $$b$$ are in the simplest form. If you can\'t further simplify the answer into a whole number, please eneter it in a form of $$b \\\\sqrt{c}$$ where $$b$$ is whole numbers and c is natural number.","answerType":"string","variabilization":{},"answerLatex":"$$(-3\\\\sqrt{3},3\\\\sqrt{3})$$","hints":{"DefaultPathway":[{"id":"a5e282bSolvquad3a-h1","type":"hint","dependencies":[],"title":"Isolate the Quadratic Term and Make Its Coefficient One","text":"We add $$27$$ to both sides to get $$x^2$$ by itself which gives $$x^2=27$$. Since the coefficient of $$x^2$$ is $$1$$ already, we do not need to take extra step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad3a-h2","type":"hint","dependencies":["a5e282bSolvquad3a-h1"],"title":"Use Square root Property","text":"The square root property: if $$x^2$$ $$=$$ k, then $$x$$ $$=$$ $$\\\\sqrt{k}$$, or $$x$$ $$=$$ $$-\\\\sqrt{k}$$. $$x^2=27$$, we can take square root both sides and get $$x=\\\\sqrt{27}$$ or $$x=-\\\\sqrt{27}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3\\\\sqrt{3}$$"],"dependencies":["a5e282bSolvquad3a-h2"],"title":"Simplify the Radical","text":"Simplify $$-\\\\sqrt{27}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3\\\\sqrt{3}$$"],"dependencies":["a5e282bSolvquad3a-h3"],"title":"Simplify the Radical","text":"Simplify $$\\\\sqrt{27}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad3a-h5","type":"hint","dependencies":["a5e282bSolvquad3a-h4"],"title":"Final Answer","text":"Therefore, $$x^2-27$$ $$=$$ $$0$$ has two solutions which are $$x=3\\\\sqrt{3}$$ and $$x=-3\\\\sqrt{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5e282bSolvquad30","title":"Solve Quadratic Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5e282bSolvquad30a","stepAnswer":["(-2-2*sqrt(6),-2+2*sqrt(6))"],"problemType":"TextBox","stepTitle":"$$5{\\\\left(b+2\\\\right)}^2-2=118$$","stepBody":"If there is more than one solution, please enter you answer as (j,k) where $$j<k$$ and j, k are in the simplest form. If you can\'t further simplify the answer into a whole number or fraction, please eneter it in a form of $$a+b \\\\sqrt{c}$$ where a, $$b$$ and c are either natural numbers or fractions.","answerType":"string","variabilization":{},"answerLatex":"$$(-2-2\\\\sqrt{6},-2+2\\\\sqrt{6})$$","hints":{"DefaultPathway":[{"id":"a5e282bSolvquad30a-h1","type":"hint","dependencies":[],"title":"Isolate the Quadratic term","text":"We add $$2$$ on both sides to get $$5{\\\\left(b+2\\\\right)}^2$$ by itself which gives $$5{\\\\left(b+2\\\\right)}^2=120$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad30a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a5e282bSolvquad30a-h1"],"title":"Find Leading Coefficient","text":"What\'s the leading coefficient now?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad30a-h3","type":"hint","dependencies":["a5e282bSolvquad30a-h2"],"title":"Make the Leading Coefficient \\"1\\"","text":"Divide by the leading coefficient on both sides which gives $${\\\\left(b+2\\\\right)}^2=24$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad30a-h4","type":"hint","dependencies":["a5e282bSolvquad30a-h3"],"title":"Use Square Root Property","text":"The square root property: if $$x^2$$ $$=$$ k, then $$x$$ $$=$$ $$\\\\sqrt{k}$$, or $$x$$ $$=$$ $$-\\\\sqrt{k}$$, or $$x$$ $$=\\\\sqrt{k}$$. $${\\\\left(b+2\\\\right)}^2=24$$, we can take square root both sides and get $$b+2=\\\\sqrt{24}$$ or $$b+2=-\\\\sqrt{24}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad30a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2\\\\sqrt{6}-2$$"],"dependencies":["a5e282bSolvquad30a-h4"],"title":"Find $$b$$ in $$b+2=-\\\\sqrt{24}$$","text":"What is $$b$$ when $$b+2=-\\\\sqrt{24}$$? Simplify the square root as much as possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad30a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2\\\\sqrt{6}-2$$"],"dependencies":["a5e282bSolvquad30a-h5"],"title":"Find $$b$$ in $$b+2=\\\\sqrt{24}$$","text":"$$b+2=\\\\sqrt{24}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad30a-h7","type":"hint","dependencies":["a5e282bSolvquad30a-h6"],"title":"Final Answer","text":"Therefore, $$5{\\\\left(b+2\\\\right)}^2-2=118$$ has two solutions which are $$x=-2\\\\sqrt{6}-2$$, and $$x=2\\\\sqrt{6}-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5e282bSolvquad4","title":"Solve Quadratic Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5e282bSolvquad4a","stepAnswer":["(-6*(sqrt(2)), 6*(sqrt(2)))"],"problemType":"TextBox","stepTitle":"$$x^2-72$$ $$=$$ $$0$$","stepBody":"If there is more than one solution, please enter you answer as (a,b) where $$a<b$$ and a, $$b$$ are in the simplest form. If you can\'t further simplify the answer into a whole number, please eneter it in a form of $$b \\\\sqrt{c}$$ where $$b$$ is whole numbers and c is natural number.","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a5e282bSolvquad4a-h1","type":"hint","dependencies":[],"title":"Isolate the Quadratic Term and Make Its Coefficient One","text":"We add $$72$$ to both sides to get $$x^2$$ by itself which gives $$x^2=72$$. Since the coefficient of $$x^2$$ is $$1$$ already, we do not need to take extra step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad4a-h2","type":"hint","dependencies":["a5e282bSolvquad4a-h1"],"title":"Use Square root Property","text":"The square root property: if $$x^2$$ $$=$$ k, then $$x$$ $$=$$ $$\\\\sqrt{k}$$, or $$x$$ $$=$$ $$-\\\\sqrt{k}$$. $$x^2=72$$, we can take square root both sides and get $$x=\\\\sqrt{72}$$ or $$x=-\\\\sqrt{72}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6\\\\sqrt{2}$$"],"dependencies":["a5e282bSolvquad4a-h2"],"title":"Simplify the Radical","text":"Simplify $$-\\\\sqrt{72}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6\\\\sqrt{2}$$"],"dependencies":["a5e282bSolvquad4a-h3"],"title":"Simplify the Radical","text":"Simplify $$\\\\sqrt{72}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad4a-h5","type":"hint","dependencies":["a5e282bSolvquad4a-h4"],"title":"Final Answer","text":"Therefore, $$x^2-72$$ $$=$$ $$0$$ has two solutions which are $$x=-6\\\\sqrt{2}$$ and $$x=6\\\\sqrt{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5e282bSolvquad5","title":"Solve Quadratic Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5e282bSolvquad5a","stepAnswer":["(-3*(sqrt(6)),3*(sqrt(6)))"],"problemType":"TextBox","stepTitle":"$$x^2-54$$ $$=$$ $$0$$","stepBody":"If there is more than one solution, please enter you answer as (a,b) where $$a<b$$ and a, $$b$$ are in the simplest form. If you can\'t further simplify the answer into a whole number, please eneter it in a form of $$b \\\\sqrt{c}$$ where $$b$$ is whole numbers and c is natural number.","answerType":"string","variabilization":{},"answerLatex":"$$(-3\\\\sqrt{6},3\\\\sqrt{6})$$","hints":{"DefaultPathway":[{"id":"a5e282bSolvquad5a-h1","type":"hint","dependencies":[],"title":"Isolate the Quadratic Term and Make Its Coefficient One","text":"We add $$54$$ to both sides to get $$x^2$$ by itself which gives $$x^2=54$$. Since the coefficient of $$x^2$$ is $$1$$ already, we do not need to take extra step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad5a-h2","type":"hint","dependencies":["a5e282bSolvquad5a-h1"],"title":"Use Square root Property","text":"The square root property: if $$x^2$$ $$=$$ k, then $$x$$ $$=$$ $$\\\\sqrt{k}$$, or $$x$$ $$=$$ $$-\\\\sqrt{k}$$. $$x^2=54$$, we can take square root both sides and get $$x=\\\\sqrt{54}$$ or $$x=-\\\\sqrt{54}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3\\\\sqrt{6}$$"],"dependencies":["a5e282bSolvquad5a-h2"],"title":"Simplify the Radical","text":"Simplify $$-\\\\sqrt{54}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3\\\\sqrt{6}$$"],"dependencies":["a5e282bSolvquad5a-h3"],"title":"Simplify the Radical","text":"Simplify $$\\\\sqrt{54}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad5a-h5","type":"hint","dependencies":["a5e282bSolvquad5a-h4"],"title":"Final Answer","text":"Therefore, $$x^2-54$$ $$=$$ $$0$$ has two solutions which are $$x=-3\\\\sqrt{6}$$ and $$x=3\\\\sqrt{6}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5e282bSolvquad6","title":"Solve Quadratic Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5e282bSolvquad6a","stepAnswer":["(-6,6)"],"problemType":"TextBox","stepTitle":"$$3x^2-108$$ $$=$$ $$0$$","stepBody":"If there is more than one solution, please enter you answer as (a,b) where $$a<b$$ and a, $$b$$ are in the simplest form. If you can\'t further simplify the answer into a whole number, please eneter it in a form of $$b \\\\sqrt{c}$$ where $$b$$ is whole numbers and c is natural number.","answerType":"string","variabilization":{},"answerLatex":"$$(-6,6)$$","hints":{"DefaultPathway":[{"id":"a5e282bSolvquad6a-h1","type":"hint","dependencies":[],"title":"Make the Leading Coefficient \\"1\\"","text":"We can divide both side by $$3$$ and get $$x^2-36=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad6a-h2","type":"hint","dependencies":["a5e282bSolvquad6a-h1"],"title":"Isolate the Quadratic Term","text":"We add $$36$$ to both sides to get $$x^2$$ by itself which gives $$x^2=36$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad6a-h3","type":"hint","dependencies":["a5e282bSolvquad6a-h2"],"title":"Use Square root Property","text":"The square root property: if $$x^2$$ $$=$$ k, then $$x$$ $$=$$ $$\\\\sqrt{k}$$, or $$x$$ $$=$$ $$-\\\\sqrt{k}$$. $$x^2=36$$, we can take square root both sides and get $$x=\\\\sqrt{36}$$ or $$x=-\\\\sqrt{36}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6$$"],"dependencies":["a5e282bSolvquad6a-h3"],"title":"Simplify the Radical","text":"Simplify $$-\\\\sqrt{36}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a5e282bSolvquad6a-h4"],"title":"Simplify the Radical","text":"Simplify $$\\\\sqrt{36}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad6a-h6","type":"hint","dependencies":["a5e282bSolvquad6a-h5"],"title":"Final Answer","text":"Therefore, $$3x^2-108=0$$ has two solutions which are $$x=-6$$ and $$x=6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5e282bSolvquad7","title":"Solve Quadratic Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5e282bSolvquad7a","stepAnswer":["(-7,7)"],"problemType":"TextBox","stepTitle":"$$2x^2-98=0$$","stepBody":"If there is more than one solution, please enter you answer as (a,b) where $$a<b$$ and a, $$b$$ are in the simplest form. If you can\'t further simplify the answer into a whole number, please eneter it in a form of $$b \\\\sqrt{c}$$ where $$b$$ is whole numbers and c is natural number.","answerType":"string","variabilization":{},"answerLatex":"$$(-7,7)$$","hints":{"DefaultPathway":[{"id":"a5e282bSolvquad7a-h1","type":"hint","dependencies":[],"title":"Make the Leading Coefficient \\"1\\"","text":"We can divide both side by $$2$$ and get $$x^2-49=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad7a-h2","type":"hint","dependencies":["a5e282bSolvquad7a-h1"],"title":"Isolate the Quadratic Term","text":"We add $$49$$ to both sides to get $$x^2$$ by itself which gives $$x^2=49$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad7a-h3","type":"hint","dependencies":["a5e282bSolvquad7a-h2"],"title":"Use Square root Property","text":"The square root property: if $$x^2$$ $$=$$ k, then $$x$$ $$=$$ $$\\\\sqrt{k}$$, or $$x$$ $$=$$ $$-\\\\sqrt{k}$$. $$x^2=49$$, we can take square root both sides and get $$x=\\\\sqrt{49}$$ or $$x=-\\\\sqrt{49}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-7$$"],"dependencies":["a5e282bSolvquad7a-h3"],"title":"Simplify the Radical","text":"Simplify $$-\\\\sqrt{49}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad7a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a5e282bSolvquad7a-h4"],"title":"Simplify the Radical","text":"Simplify $$\\\\sqrt{49}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad7a-h6","type":"hint","dependencies":["a5e282bSolvquad7a-h5"],"title":"Final Answer","text":"Therefore, $$2x^2-98=0$$ has two solutions which are $$x=-7$$ and $$x=7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5e282bSolvquad8","title":"Solve Quadratic Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5e282bSolvquad8a","stepAnswer":["(-4,4)"],"problemType":"TextBox","stepTitle":"$$5x^2-80=0$$","stepBody":"If there is more than one solution, please enter you answer as (a,b) where $$a<b$$ and a, $$b$$ are in the simplest form. If you can\'t further simplify the answer into a whole number, please eneter it in a form of $$b \\\\sqrt{c}$$ where $$b$$ is whole numbers and c is natural number.","answerType":"string","variabilization":{},"answerLatex":"$$(-4,4)$$","hints":{"DefaultPathway":[{"id":"a5e282bSolvquad8a-h1","type":"hint","dependencies":[],"title":"Make the Leading Coefficient \\"1\\"","text":"We can divide both side by $$5$$ and get $$x^2-16=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad8a-h2","type":"hint","dependencies":["a5e282bSolvquad8a-h1"],"title":"Isolate the Quadratic Term","text":"We add $$16$$ to both sides to get $$x^2$$ by itself which gives $$x^2=16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad8a-h3","type":"hint","dependencies":["a5e282bSolvquad8a-h2"],"title":"Use Square root Property","text":"The square root property: if $$x^2$$ $$=$$ k, then $$x$$ $$=$$ $$\\\\sqrt{k}$$, or $$x$$ $$=$$ $$-\\\\sqrt{k}$$. $$x^2=16$$, we can take square root both sides and get $$x=\\\\sqrt{16}$$ or $$x=-\\\\sqrt{16}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["a5e282bSolvquad8a-h3"],"title":"Simplify the Radical","text":"Simplify $$-\\\\sqrt{16}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad8a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a5e282bSolvquad8a-h4"],"title":"Simplify the Radical","text":"Simplify $$\\\\sqrt{16}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad8a-h6","type":"hint","dependencies":["a5e282bSolvquad8a-h5"],"title":"Final Answer","text":"Therefore, $$5x^2-80=0$$ has two solutions which are $$x=-4$$ and $$x=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5e282bSolvquad9","title":"Solve Quadratic Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5e282bSolvquad9a","stepAnswer":["(-5,5)"],"problemType":"TextBox","stepTitle":"$$5y^2-125$$ $$=$$ $$0$$","stepBody":"If there is more than one solution, please enter you answer as (a,b) where $$a<b$$ and a, $$b$$ are in the simplest form. If you can\'t further simplify the answer into a whole number, please eneter it in a form of $$b \\\\sqrt{c}$$ where $$b$$ is whole numbers and c is natural number.","answerType":"string","variabilization":{},"answerLatex":"$$(-5,5)$$","hints":{"DefaultPathway":[{"id":"a5e282bSolvquad9a-h1","type":"hint","dependencies":[],"title":"Make the Leading Coefficient \\"1\\"","text":"We can divide both side by $$5$$ and get $$y^2-25=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad9a-h2","type":"hint","dependencies":["a5e282bSolvquad9a-h1"],"title":"Isolate the Quadratic Term","text":"We add $$25$$ to both sides to get $$y^2$$ by itself which gives $$y^2=25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad9a-h3","type":"hint","dependencies":["a5e282bSolvquad9a-h2"],"title":"Use Square root Property","text":"The square root property: if $$x^2$$ $$=$$ k, then $$x$$ $$=$$ $$\\\\sqrt{k}$$, or $$x$$ $$=$$ $$-\\\\sqrt{k}$$. $$y^2=25$$, we can take square root both sides and get $$y=\\\\sqrt{25}$$ or $$y=-\\\\sqrt{25}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["a5e282bSolvquad9a-h3"],"title":"Simplify the Radical","text":"Simplify $$-\\\\sqrt{25}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad9a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a5e282bSolvquad9a-h4"],"title":"Simplify the Radical","text":"Simplify $$\\\\sqrt{25}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5e282bSolvquad9a-h6","type":"hint","dependencies":["a5e282bSolvquad9a-h5"],"title":"Final Answer","text":"Therefore, $$5y^2-125=0$$ has two solutions which are $$y=-5$$ and $$y=5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5f7784homogen1","title":"Test for Homogeneity","body":"Suppose that $$250$$ randomly selected male college students and $$300$$ randomly selected female college students were asked about their living arrangements: dormitory, apartment, with parents, other. The results are shown in Table 11.19.\\\\n##figure2.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.4 Test for Homogeneity","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a5f7784homogen1a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Do male and female college students have the same distribution of living arrangements? Use a level of significance of $$0.05$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a5f7784homogen1a-h1","type":"hint","dependencies":[],"title":"Null and Alternative Hypotheses","text":"$$H_0$$: the distribution of living arrangements for male college students is the same as the distribution of living arrangements for female college students. $$H_a$$: the distribution of living arrangements for male college students is not the same as the distribution of living arrangements for female college students.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5f7784homogen1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a5f7784homogen1a-h1"],"title":"Degrees of Freedom (df)","text":"What is df, the degrees of freedom?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5f7784homogen1a-h2-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":[],"title":"df Equation","text":"If df $$=$$ number of columns - $$1$$, what is df?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5f7784homogen1a-h2-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":[],"title":"Solving for df","text":"df $$=$$ number of columns - $$1$$ where the number of columns is $$4$$. What is df?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a5f7784homogen1a-h5","type":"hint","dependencies":["a5f7784homogen1a-h2"],"title":"Test Distribution","text":"The distribution for the test is as follows:\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5f7784homogen1a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10.1287$$"],"dependencies":["a5f7784homogen1a-h5"],"title":"Test Statistic","text":"Using a calculator or computer, what is the test statistic $$X^2$$? Round to four decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5f7784homogen1a-h7","type":"hint","dependencies":["a5f7784homogen1a-h6"],"title":"Probability Statement","text":"$$p-value$$ $$=$$ P((X**2) > $$10.1287)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5f7784homogen1a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.0175$$"],"dependencies":["a5f7784homogen1a-h7"],"title":"P-Value","text":"What is the $$p-value$$? Round your answer to four decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5f7784homogen1a-h9","type":"hint","dependencies":["a5f7784homogen1a-h8"],"title":"Comparing \ud835\udefc and the P-Value","text":"Since the problem asks us to use a level of significance of $$0.05$$, \ud835\udefc $$=$$ $$0.05$$. The $$p-value$$ $$=$$ $$0.0175$$. If \ud835\udefc > $$p-value$$, reject $$H_0$$. If \ud835\udefc < $$p-value$$, accept $$H_0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5f7784homogen1a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a5f7784homogen1a-h9"],"title":"Greater Than or Less Than the P-Value","text":"Is \ud835\udefc > $$p-value$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a5f7784homogen1a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Reject"],"dependencies":["a5f7784homogen1a-h10"],"title":"Reject or Accept $$H_0$$ Based on the P-Value","text":"Based on your previous answer, would you reject or not reject $$H_0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Do not reject","Reject"]},{"id":"a5f7784homogen1a-h12","type":"hint","dependencies":["a5f7784homogen1a-h11"],"title":"Conclusion","text":"At a 5% level of significance, from the data, there is sufficient evidence to conclude that the distributions of living arrangements for male and female college students are not the same. Notice that the conclusion is only that the distributions are not the same. We cannot use the test for homogeneity to draw any conclusions about how they differ.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5f7784homogen2","title":"Test for Homogeneity","body":"Both before and after a recent earthquake, surveys were conducted asking voters which of the three candidates they planned on voting for in the upcoming city council election. Table $$11.21$$ shows the results of the survey.\\\\n##figure2.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.4 Test for Homogeneity","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a5f7784homogen2a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Has there been a change in the distribution of voter preferences since the earthquake? Use a level of significance of $$0.05$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a5f7784homogen2a-h1","type":"hint","dependencies":[],"title":"Null and Alternative Hypotheses","text":"$$H_0$$: the distribution of voter preferences was the same before and after the earthquake. $$H_a$$: the distribution of voter preferences was not the same before and after the earthquake.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5f7784homogen2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a5f7784homogen2a-h1"],"title":"Degrees of Freedom (df)","text":"What is df, the degrees of freedom?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a5f7784homogen2a-h2-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"df Equation","text":"If df $$=$$ number of columns - $$1$$, what is df?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5f7784homogen2a-h2-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Solving for df","text":"df $$=$$ number of columns - $$1$$ where the number of columns is $$3$$. What is df?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a5f7784homogen2a-h5","type":"hint","dependencies":["a5f7784homogen2a-h2"],"title":"Test Distribution","text":"The distribution for the test is as follows:\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5f7784homogen2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3.2603$$"],"dependencies":["a5f7784homogen2a-h5"],"title":"Test Statistic","text":"Using a calculator or computer, what is the test statistic $$X^2$$? Round to four decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5f7784homogen2a-h7","type":"hint","dependencies":["a5f7784homogen2a-h6"],"title":"Probability Statement","text":"$$p-value$$ $$=$$ P((X**2) > $$3.2603)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5f7784homogen2a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1959$$"],"dependencies":["a5f7784homogen2a-h7"],"title":"P-Value","text":"What is the $$p-value$$? Round your answer to four decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5f7784homogen2a-h9","type":"hint","dependencies":["a5f7784homogen2a-h8"],"title":"Comparing \ud835\udefc and the P-Value","text":"Since the problem asks us to use a level of significance of $$0.05$$, \ud835\udefc $$=$$ $$0.05$$. The $$p-value$$ $$=$$ $$0.1959$$. If \ud835\udefc > $$p-value$$, reject $$H_0$$ if \ud835\udefc < $$p-value$$, accept $$H_0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5f7784homogen2a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a5f7784homogen2a-h9"],"title":"Greater Than or Less Than the P-Value","text":"Is \ud835\udefc > $$p-value$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a5f7784homogen2a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Do not reject"],"dependencies":["a5f7784homogen2a-h10"],"title":"Reject or Accept $$H_0$$ Based on the P-Value","text":"Based on your previous answer, would you reject or not reject $$H_0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Do not reject","Reject"]},{"id":"a5f7784homogen2a-h12","type":"hint","dependencies":["a5f7784homogen2a-h11"],"title":"Conclusion","text":"At a 5% level of significance, from the data, there is insufficient evidence to conclude that the distribution of voter preferences was not the same before and after the earthquake.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5fd810distance1","title":"Find the distance between the points","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.1 Distance and Midpoint Formulas; Circles","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5fd810distance1a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"$$(2,0)$$ and $$(5,4)$$","stepBody":"Write your answer in exact form. Simplify your answer as much as possible.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"a5fd810distance1a-h1","type":"hint","dependencies":[],"title":"Distance formula","text":"The distance formula is $$\\\\sqrt{{\\\\left(x_2-x_1\\\\right)}^2+{\\\\left(y_2-y_1\\\\right)}^2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance1a-h2","type":"hint","dependencies":["a5fd810distance1a-h1"],"title":"Plug in","text":"Plug in the values for the equation. Choose a point to be the $$x_2$$ and $$y_2$$ pair, and use the other as the $$x_1$$ and $$y_1$$ pair. We get $$\\\\sqrt{{\\\\left(5-2\\\\right)}^2+{\\\\left(4-0\\\\right)}^2}=\\\\sqrt{25}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance1a-h3","type":"hint","dependencies":["a5fd810distance1a-h2"],"title":"Simplify","text":"Simplify the answer into one term. $$\\\\sqrt{25}=5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5fd810distance10","title":"Find the distance between the points","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.1 Distance and Midpoint Formulas; Circles","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5fd810distance10a","stepAnswer":["$$2\\\\sqrt{10}$$"],"problemType":"TextBox","stepTitle":"$$(-1,-2)$$ and $$(-3,4)$$","stepBody":"Write your answer in exact form. Simplify your answer as much as possible.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2\\\\sqrt{10}$$","hints":{"DefaultPathway":[{"id":"a5fd810distance10a-h1","type":"hint","dependencies":[],"title":"Distance formula","text":"The distance formula is $$\\\\sqrt{{\\\\left(x_2-x_1\\\\right)}^2+{\\\\left(y_2-y_1\\\\right)}^2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance10a-h2","type":"hint","dependencies":["a5fd810distance10a-h1"],"title":"Plug in","text":"Plug in the values for the equation. Choose a point to be the $$x_2$$ and $$y_2$$ pair, and use the other as the $$x_1$$ and $$y_1$$ pair. We get $$\\\\sqrt{{\\\\left(-3-\\\\left(--1\\\\right)\\\\right)}^2+{\\\\left(4-\\\\left(-2\\\\right)\\\\right)}^2}=\\\\sqrt{40}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance10a-h3","type":"hint","dependencies":["a5fd810distance10a-h2"],"title":"Simplify","text":"Simplify the answer into one term. $$\\\\sqrt{40}=\\\\sqrt{4\\\\times10}=\\\\sqrt{4} \\\\sqrt{10}=2\\\\sqrt{10}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5fd810distance11","title":"Find the distance between the points","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.1 Distance and Midpoint Formulas; Circles","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5fd810distance11a","stepAnswer":["$$2\\\\sqrt{17}$$"],"problemType":"TextBox","stepTitle":"$$(3,-1)$$ and $$(1,7)$$","stepBody":"Write your answer in exact form. Simplify your answer as much as possible.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2\\\\sqrt{17}$$","hints":{"DefaultPathway":[{"id":"a5fd810distance11a-h1","type":"hint","dependencies":[],"title":"Distance formula","text":"The distance formula is $$\\\\sqrt{{\\\\left(x_2-x_1\\\\right)}^2+{\\\\left(y_2-y_1\\\\right)}^2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance11a-h2","type":"hint","dependencies":["a5fd810distance11a-h1"],"title":"Plug in","text":"Plug in the values for the equation. Choose a point to be the $$x_2$$ and $$y_2$$ pair, and use the other as the $$x_1$$ and $$y_1$$ pair. We get $$\\\\sqrt{{\\\\left(1-3\\\\right)}^2+{\\\\left(7-\\\\left(-1\\\\right)\\\\right)}^2}=\\\\sqrt{68}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance11a-h3","type":"hint","dependencies":["a5fd810distance11a-h2"],"title":"Simplify","text":"Simplify the answer into one term. $$\\\\sqrt{68}=\\\\sqrt{4\\\\times17}=\\\\sqrt{4} \\\\sqrt{17}=2\\\\sqrt{17}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5fd810distance12","title":"Find the distance between the points","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.1 Distance and Midpoint Formulas; Circles","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5fd810distance12a","stepAnswer":["$$\\\\sqrt{202}$$"],"problemType":"TextBox","stepTitle":"$$(-4,-5)$$ and $$(7,4)$$","stepBody":"Write your answer in exact form. Simplify your answer as much as possible.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\sqrt{202}$$","hints":{"DefaultPathway":[{"id":"a5fd810distance12a-h1","type":"hint","dependencies":[],"title":"Distance formula","text":"The distance formula is $$\\\\sqrt{{\\\\left(x_2-x_1\\\\right)}^2+{\\\\left(y_2-y_1\\\\right)}^2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance12a-h2","type":"hint","dependencies":["a5fd810distance12a-h1"],"title":"Plug in","text":"Plug in the values for the equation. Choose a point to be the $$x_2$$ and $$y_2$$ pair, and use the other as the $$x_1$$ and $$y_1$$ pair. We get $$\\\\sqrt{{\\\\left(7-\\\\left(-4\\\\right)\\\\right)}^2+{\\\\left(4-\\\\left(-5\\\\right)\\\\right)}^2}=\\\\sqrt{202}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance12a-h3","type":"hint","dependencies":["a5fd810distance12a-h2"],"title":"Simplify","text":"Simplify the answer into one term. There is no way to further simplify $$\\\\sqrt{202}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5fd810distance13","title":"Find the midpoint of the line segments whose endpoints are given","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.1 Distance and Midpoint Formulas; Circles","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5fd810distance13a","stepAnswer":["$$(2,-4)$$"],"problemType":"MultipleChoice","stepTitle":"$$(0,-5)$$ and $$(4,-3)$$","stepBody":"Please select the correct answer.","answerType":"string","variabilization":{},"answerLatex":"$$(2,-4)$$","choices":["$$(2,-4)$$","$$(-3,3)$$","$$(1,-3)$$","$$(7,1)$$"],"hints":{"DefaultPathway":[{"id":"a5fd810distance13a-h1","type":"hint","dependencies":[],"title":"Midpoint formula","text":"The midpoint of the line segment whose endpoints are the two points $$(x_1,y_1)$$ and $$(x_2,y_2)$$ is $$(\\\\frac{x_1+x_2}{2},\\\\frac{y_1+y_2}{2})$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance13a-h2","type":"hint","dependencies":["a5fd810distance13a-h1"],"title":"Plug in","text":"Plug in the values for the equation. Choose a point to be the $$x_2$$ and $$y_2$$ pair, and use the other as the $$x_1$$ and $$y_1$$ pair. We get $$(\\\\frac{x_1+x_2}{2},\\\\frac{y_1+y_2}{2})=(\\\\frac{4+0}{2},\\\\left(-3+\\\\frac{\\\\left(-5\\\\right)}{2}\\\\right))$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance13a-h3","type":"hint","dependencies":["a5fd810distance13a-h2"],"title":"Simplify","text":"Simplify the answer. $$(\\\\frac{4+0}{2},\\\\frac{\\\\left(-3+\\\\left(-5\\\\right)\\\\right)}{2})=(2,-4)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5fd810distance14","title":"Find the midpoint of the line segments whose endpoints are given","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.1 Distance and Midpoint Formulas; Circles","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5fd810distance14a","stepAnswer":["$$(2,-4)$$"],"problemType":"MultipleChoice","stepTitle":"$$(-2,-6)$$ and $$(6,-2)$$","stepBody":"Please select the correct answer.","answerType":"string","variabilization":{},"answerLatex":"$$(2,-4)$$","choices":["$$(2,-4)$$","$$(-3,3)$$","$$(1,-3)$$","$$(7,1)$$"],"hints":{"DefaultPathway":[{"id":"a5fd810distance14a-h1","type":"hint","dependencies":[],"title":"Midpoint formula","text":"The midpoint of the line segment whose endpoints are the two points $$(x_1,y_1)$$ and $$(x_2,y_2)$$ is $$(\\\\frac{x_1+x_2}{2},\\\\frac{y_1+y_2}{2})$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance14a-h2","type":"hint","dependencies":["a5fd810distance14a-h1"],"title":"Plug in","text":"Plug in the values for the equation. Choose a point to be the $$x_2$$ and $$y_2$$ pair, and use the other as the $$x_1$$ and $$y_1$$ pair. We get $$(\\\\frac{x_1+x_2}{2},\\\\frac{y_1+y_2}{2})=(\\\\frac{6+\\\\left(-2\\\\right)}{2},\\\\frac{\\\\left(-6+\\\\left(-2\\\\right)\\\\right)}{2})$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance14a-h3","type":"hint","dependencies":["a5fd810distance14a-h2"],"title":"Simplify","text":"Simplify the answer. $$(\\\\frac{6+\\\\left(-2\\\\right)}{2},\\\\frac{\\\\left(-6+\\\\left(-2\\\\right)\\\\right)}{2})=(2,-4)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5fd810distance15","title":"Find the midpoint of the line segments whose endpoints are given","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.1 Distance and Midpoint Formulas; Circles","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5fd810distance15a","stepAnswer":["$$(3.5, -1.5)$$"],"problemType":"MultipleChoice","stepTitle":"$$(3,-1)$$ and $$(4,-2)$$","stepBody":"Please select the correct answer.","answerType":"string","variabilization":{},"answerLatex":"$$(3.5, -1.5)$$","choices":["$$(2,-4)$$","$$(-3,3)$$","$$(1,-3)$$","$$(3.5, -1.5)$$"],"hints":{"DefaultPathway":[{"id":"a5fd810distance15a-h1","type":"hint","dependencies":[],"title":"Midpoint formula","text":"The midpoint of the line segment whose endpoints are the two points $$(x_1,y_1)$$ and $$(x_2,y_2)$$ is $$(\\\\frac{x_1+x_2}{2},\\\\frac{y_1+y_2}{2})$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance15a-h2","type":"hint","dependencies":["a5fd810distance15a-h1"],"title":"Plug in","text":"Plug in the values for the equation. Choose a point to be the $$x_2$$ and $$y_2$$ pair, and use the other as the $$x_1$$ and $$y_1$$ pair. We get ((x_1+x_2)/2,(y_1+y_2)/2)=((4+3)/2),((-1+(-2))/2)).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance15a-h3","type":"hint","dependencies":["a5fd810distance15a-h2"],"title":"Simplify","text":"Simplify the answer. ((4+3)/2),((-1+(-2))/2))=(3.5,-1.5)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5fd810distance16","title":"Find the midpoint of the line segments whose endpoints are given","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.1 Distance and Midpoint Formulas; Circles","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5fd810distance16a","stepAnswer":["$$(\\\\frac{1}{5},-2)$$"],"problemType":"MultipleChoice","stepTitle":"$$(-3,-3)$$ and $$(6,-1)$$","stepBody":"Please select the correct answer.","answerType":"string","variabilization":{},"answerLatex":"$$(\\\\frac{1}{5},-2)$$","choices":["$$(2,-4)$$","$$(-3,3)$$","$$(1,-3)$$","$$(\\\\frac{1}{5},-2)$$"],"hints":{"DefaultPathway":[{"id":"a5fd810distance16a-h1","type":"hint","dependencies":[],"title":"Midpoint formula","text":"The midpoint of the line segment whose endpoints are the two points $$(x_1,y_1)$$ and $$(x_2,y_2)$$ is $$(\\\\frac{x_1+x_2}{2},\\\\frac{y_1+y_2}{2})$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance16a-h2","type":"hint","dependencies":["a5fd810distance16a-h1"],"title":"Plug in","text":"Plug in the values for the equation. Choose a point to be the $$x_2$$ and $$y_2$$ pair, and use the other as the $$x_1$$ and $$y_1$$ pair. We get $$(\\\\frac{x_1+x_2}{2},\\\\frac{y_1+y_2}{2})=(\\\\frac{6+\\\\left(-3\\\\right)}{2},\\\\frac{\\\\left(-1+\\\\left(-3\\\\right)\\\\right)}{2})$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance16a-h3","type":"hint","dependencies":["a5fd810distance16a-h2"],"title":"Simplify","text":"Simplify the answer. $$(\\\\frac{6+\\\\left(-3\\\\right)}{2},\\\\frac{\\\\left(-1+\\\\left(-3\\\\right)\\\\right)}{2})=(\\\\frac{1}{5},-2)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5fd810distance17","title":"Find the center and the radius of the circle","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.1 Distance and Midpoint Formulas; Circles","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5fd810distance17a","stepAnswer":["Center: $$(-5,-3)$$, Radius: $$1$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(x+5\\\\right)}^2+{\\\\left(y+3\\\\right)}^2=1$$","stepBody":"Please select the correct answer.","answerType":"string","variabilization":{},"answerLatex":"Center: $$(-5,-3)$$, Radius: $$1$$","choices":["Center: $$(-5,-3)$$, Radius: $$1$$","Center: $$(2,4)$$, Radius: $$1$$","Center: $$(-5,7)$$, Radius: $$2$$","Center: $$(3,2)$$, Radius: $$2$$"],"hints":{"DefaultPathway":[{"id":"a5fd810distance17a-h1","type":"hint","dependencies":[],"title":"Standard form of a circle","text":"The standard form of a circle is $${\\\\left(x-h\\\\right)}^2+{\\\\left(y-k\\\\right)}^2=r^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance17a-h2","type":"hint","dependencies":["a5fd810distance17a-h1"],"title":"Center","text":"The center of an equation of a circle in standard form is (h,k). We can rewrite the equation as $${\\\\left(x-\\\\left(-5\\\\right)\\\\right)}^2+{\\\\left(y-\\\\left(-3\\\\right)\\\\right)}^2=1^2$$. From here we get the center is $$(-5,-3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance17a-h3","type":"hint","dependencies":["a5fd810distance17a-h2"],"title":"Radius","text":"The radius of an equation of a circle in standard form is simply the $$r$$ value. Note that in the standard form of the circle, $$r$$ is squared. We can rewrite the equation as $${\\\\left(x-\\\\left(-5\\\\right)\\\\right)}^2+{\\\\left(y-\\\\left(-3\\\\right)\\\\right)}^2=1^2$$. We get the radius is $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5fd810distance18","title":"Find the center and the radius of the circle","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.1 Distance and Midpoint Formulas; Circles","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5fd810distance18a","stepAnswer":["Center: $$(2,3)$$, Radius: $$3$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(x-2\\\\right)}^2+{\\\\left(y-3\\\\right)}^2=9$$","stepBody":"Please select the correct answer.","answerType":"string","variabilization":{},"answerLatex":"Center: $$(2,3)$$, Radius: $$3$$","choices":["Center: $$(2,3)$$, Radius: $$3$$","Center: $$(2,4)$$, Radius: $$1$$","Center: $$(-5,7)$$, Radius: $$2$$","Center: $$(3,2)$$, Radius: $$2$$"],"hints":{"DefaultPathway":[{"id":"a5fd810distance18a-h1","type":"hint","dependencies":[],"title":"Standard form of a circle","text":"The standard form of a circle is $${\\\\left(x-h\\\\right)}^2+{\\\\left(y-k\\\\right)}^2=r^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance18a-h2","type":"hint","dependencies":["a5fd810distance18a-h1"],"title":"Center","text":"The center of an equation of a circle in standard form is (h,k). We can rewrite the equation as $${\\\\left(x-2\\\\right)}^2+{\\\\left(y-3\\\\right)}^2=3^2$$. From here we get the center is $$(2,3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance18a-h3","type":"hint","dependencies":["a5fd810distance18a-h2"],"title":"Radius","text":"The radius of an equation of a circle in standard form is simply the $$r$$ value. Note that in the standard form of the circle, $$r$$ is squared. We can rewrite the equation as $${\\\\left(x-2\\\\right)}^2+{\\\\left(y-3\\\\right)}^2=3^2$$. From here we get the radius is $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5fd810distance19","title":"Find the center and the radius of the circle","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.1 Distance and Midpoint Formulas; Circles","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5fd810distance19a","stepAnswer":["Center: $$(4,-2)$$, Radius: $$4$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(x-4\\\\right)}^2+{\\\\left(y+2\\\\right)}^2=16$$","stepBody":"Please select the correct answer.","answerType":"string","variabilization":{},"answerLatex":"Center: $$(4,-2)$$, Radius: $$4$$","choices":["Center: $$(2,3)$$, Radius: $$3$$","Center: $$(4,-2)$$, Radius: $$4$$","Center: $$(-5,7)$$, Radius: $$2$$","Center: $$(3,2)$$, Radius: $$2$$"],"hints":{"DefaultPathway":[{"id":"a5fd810distance19a-h1","type":"hint","dependencies":[],"title":"Standard form of a circle","text":"The standard form of a circle is $${\\\\left(x-h\\\\right)}^2+{\\\\left(y-k\\\\right)}^2=r^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance19a-h2","type":"hint","dependencies":["a5fd810distance19a-h1"],"title":"Center","text":"The center of an equation of a circle in standard form is (h,k). We can rewrite the equation as $${\\\\left(x-4\\\\right)}^2+{\\\\left(y-\\\\left(-2\\\\right)\\\\right)}^2=4^2$$. From here we get the center is $$(4,-2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance19a-h3","type":"hint","dependencies":["a5fd810distance19a-h2"],"title":"Radius","text":"The radius of an equation of a circle in standard form is simply the $$r$$ value. Note that in the standard form of the circle, $$r$$ is squared. We can rewrite the equation as $${\\\\left(x-4\\\\right)}^2+{\\\\left(y-\\\\left(-2\\\\right)\\\\right)}^2=4^2$$. From here we get the radius is $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5fd810distance2","title":"Find the distance between the points","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.1 Distance and Midpoint Formulas; Circles","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5fd810distance2a","stepAnswer":["$$10$$"],"problemType":"TextBox","stepTitle":"$$(-4,-3)$$ and $$(2,5)$$","stepBody":"Write your answer in exact form. Simplify your answer as much as possible.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10$$","hints":{"DefaultPathway":[{"id":"a5fd810distance2a-h1","type":"hint","dependencies":[],"title":"Distance formula","text":"The distance formula is $$\\\\sqrt{{\\\\left(x_2-x_1\\\\right)}^2+{\\\\left(y_2-y_1\\\\right)}^2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance2a-h2","type":"hint","dependencies":["a5fd810distance2a-h1"],"title":"Plug in","text":"Plug in the values for the equation. Choose a point to be the $$x_2$$ and $$y_2$$ pair, and use the other as the $$x_1$$ and $$y_1$$ pair. We get $$\\\\sqrt{{\\\\left(2-\\\\left(-4\\\\right)\\\\right)}^2+{\\\\left(5-\\\\left(-3\\\\right)\\\\right)}^2}=\\\\sqrt{100}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance2a-h3","type":"hint","dependencies":["a5fd810distance2a-h2"],"title":"Simplify","text":"Simplify the answer into one term. $$\\\\sqrt{100}=10$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5fd810distance20","title":"Find the center and the radius of the circle","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.1 Distance and Midpoint Formulas; Circles","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5fd810distance20a","stepAnswer":["Center: $$(-2,5)$$, Radius: $$2$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(x+2\\\\right)}^2+{\\\\left(y-5\\\\right)}^2=4$$","stepBody":"Please select the correct answer.","answerType":"string","variabilization":{},"answerLatex":"Center: $$(-2,5)$$, Radius: $$2$$","choices":["Center: $$(2,3)$$, Radius: $$3$$","Center: $$(-2,5)$$, Radius: $$2$$","Center: $$(-5,7)$$, Radius: $$2$$","Center: $$(3,2)$$, Radius: $$2$$"],"hints":{"DefaultPathway":[{"id":"a5fd810distance20a-h1","type":"hint","dependencies":[],"title":"Standard form of a circle","text":"The standard form of a circle is $${\\\\left(x-h\\\\right)}^2+{\\\\left(y-k\\\\right)}^2=r^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance20a-h2","type":"hint","dependencies":["a5fd810distance20a-h1"],"title":"Center","text":"The center of an equation of a circle in standard form is (h,k). We can rewrite the equation as $${\\\\left(x-\\\\left(-2\\\\right)\\\\right)}^2+{\\\\left(y-5\\\\right)}^2=2^2$$. From here we get the center is $$(-2,5)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance20a-h3","type":"hint","dependencies":["a5fd810distance20a-h2"],"title":"Radius","text":"The radius of an equation of a circle in standard form is simply the $$r$$ value. Note that in the standard form of the circle, $$r$$ is squared. We can rewrite the equation as $${\\\\left(x-\\\\left(-2\\\\right)\\\\right)}^2+{\\\\left(y-5\\\\right)}^2=2^2$$. From here we get the radius is $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5fd810distance21","title":"Find the center and the radius of the circle","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.1 Distance and Midpoint Formulas; Circles","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5fd810distance21a","stepAnswer":["Center: $$(0,-2)$$, Radius: $$5$$"],"problemType":"MultipleChoice","stepTitle":"$$x^2+{\\\\left(y+2\\\\right)}^2=25$$","stepBody":"Please select the correct answer.","answerType":"string","variabilization":{},"answerLatex":"Center: $$(0,-2)$$, Radius: $$5$$","choices":["Center: $$(2,3)$$, Radius: $$3$$","Center: $$(0,-2)$$, Radius: $$5$$","Center: $$(-5,7)$$, Radius: $$2$$","Center: $$(3,2)$$, Radius: $$2$$"],"hints":{"DefaultPathway":[{"id":"a5fd810distance21a-h1","type":"hint","dependencies":[],"title":"Standard form of a circle","text":"The standard form of a circle is $${\\\\left(x-h\\\\right)}^2+{\\\\left(y-k\\\\right)}^2=r^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance21a-h2","type":"hint","dependencies":["a5fd810distance21a-h1"],"title":"Center","text":"The center of an equation of a circle in standard form is (h,k). We can rewrite the equation as $${\\\\left(x-0\\\\right)}^2+{\\\\left(y-\\\\left(-2\\\\right)\\\\right)}^2=5^2$$. From here we get the center is $$(0,-2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance21a-h3","type":"hint","dependencies":["a5fd810distance21a-h2"],"title":"Radius","text":"The radius of an equation of a circle in standard form is simply the $$r$$ value. Note that in the standard form of the circle, $$r$$ is squared. We can rewrite the equation as $${\\\\left(x-0\\\\right)}^2+{\\\\left(y-\\\\left(-2\\\\right)\\\\right)}^2=5^2$$. From here we get the radius is $$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5fd810distance22","title":"Find the center and the radius of the circle","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.1 Distance and Midpoint Formulas; Circles","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5fd810distance22a","stepAnswer":["Center: $$(1,0)$$, Radius: $$6$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(x-1\\\\right)}^2+y^2=36$$","stepBody":"Please select the correct answer.","answerType":"string","variabilization":{},"answerLatex":"Center: $$(1,0)$$, Radius: $$6$$","choices":["Center: $$(2,3)$$, Radius: $$3$$","Center: $$(0,-2)$$, Radius: $$5$$","Center: $$(-5,7)$$, Radius: $$2$$","Center: $$(1,0)$$, Radius: $$6$$"],"hints":{"DefaultPathway":[{"id":"a5fd810distance22a-h1","type":"hint","dependencies":[],"title":"Standard form of a circle","text":"The standard form of a circle is $${\\\\left(x-h\\\\right)}^2+{\\\\left(y-k\\\\right)}^2=r^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance22a-h2","type":"hint","dependencies":["a5fd810distance22a-h1"],"title":"Center","text":"The center of an equation of a circle in standard form is (h,k). We can rewrite the equation as $${\\\\left(x-1\\\\right)}^2+{\\\\left(y-0\\\\right)}^2=6^2$$. From here we get the center is $$(1,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance22a-h3","type":"hint","dependencies":["a5fd810distance22a-h2"],"title":"Radius","text":"The radius of an equation of a circle in standard form is simply the $$r$$ value. Note that in the standard form of the circle, $$r$$ is squared. We can rewrite the equation as $${\\\\left(x-1\\\\right)}^2+{\\\\left(y-0\\\\right)}^2=6^2$$. From here we get the radius is $$6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5fd810distance23","title":"Find the center and the radius of the circle","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.1 Distance and Midpoint Formulas; Circles","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5fd810distance23a","stepAnswer":["Center: $$(1.5, -2.5)$$, Radius: $$0.5$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(x-1.5\\\\right)}^2+{\\\\left(y+2.5\\\\right)}^2=0.25$$","stepBody":"Please select the correct answer.","answerType":"string","variabilization":{},"answerLatex":"Center: $$(1.5, -2.5)$$, Radius: $$0.5$$","choices":["Center: $$(2,3)$$, Radius: $$3$$","Center: $$(0,-2)$$, Radius: $$5$$","Center: $$(1.5, -2.5)$$, Radius: $$0.5$$","Center: $$(1,0)$$, Radius: $$6$$"],"hints":{"DefaultPathway":[{"id":"a5fd810distance23a-h1","type":"hint","dependencies":[],"title":"Standard form of a circle","text":"The standard form of a circle is $${\\\\left(x-h\\\\right)}^2+{\\\\left(y-k\\\\right)}^2=r^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance23a-h2","type":"hint","dependencies":["a5fd810distance23a-h1"],"title":"Center","text":"The center of an equation of a circle in standard form is (h,k). We can rewrite the equation as $${\\\\left(x-1.5\\\\right)}^2+{\\\\left(y-\\\\left(-2.5\\\\right)\\\\right)}^2={0.5}^2$$. From here we get the center is $$(1.5, -2.5)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance23a-h3","type":"hint","dependencies":["a5fd810distance23a-h2"],"title":"Radius","text":"The radius of an equation of a circle in standard form is simply the $$r$$ value. Note that in the standard form of the circle, $$r$$ is squared. We can rewrite the equation as $${\\\\left(x-1.5\\\\right)}^2+{\\\\left(y-\\\\left(-2.5\\\\right)\\\\right)}^2={0.5}^2$$. From here we get the radius is $$0.5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5fd810distance24","title":"Find the center and the radius of the circle","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.1 Distance and Midpoint Formulas; Circles","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5fd810distance24a","stepAnswer":["Center: $$(1,3)$$, Radius: $$\\\\frac{3}{2}$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(x-1\\\\right)}^2+{\\\\left(y-3\\\\right)}^2=\\\\frac{9}{4}$$","stepBody":"Please select the correct answer.","answerType":"string","variabilization":{},"answerLatex":"Center: $$(1,3)$$, Radius: $$\\\\frac{3}{2}$$","choices":["Center: $$(2,3)$$, Radius: $$3$$","Center: $$(1,3)$$, Radius: $$\\\\frac{3}{2}$$","Center: $$(1.5, -2.5)$$, Radius: $$0.5$$","Center: $$(1,0)$$, Radius: $$6$$"],"hints":{"DefaultPathway":[{"id":"a5fd810distance24a-h1","type":"hint","dependencies":[],"title":"Standard form of a circle","text":"The standard form of a circle is $${\\\\left(x-h\\\\right)}^2+{\\\\left(y-k\\\\right)}^2=r^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance24a-h2","type":"hint","dependencies":["a5fd810distance24a-h1"],"title":"Center","text":"The center of an equation of a circle in standard form is (h,k). We can rewrite the equation as $${\\\\left(x-1\\\\right)}^2+{\\\\left(y-3\\\\right)}^2={\\\\left(\\\\frac{3}{2}\\\\right)}^2$$. From here we get the center is $$(1,3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance24a-h3","type":"hint","dependencies":["a5fd810distance24a-h2"],"title":"Radius","text":"The radius of an equation of a circle in standard form is simply the $$r$$ value. Note that in the standard form of the circle, $$r$$ is squared. We can rewrite the equation as $${\\\\left(x-1\\\\right)}^2+{\\\\left(y-3\\\\right)}^2={\\\\left(\\\\frac{3}{2}\\\\right)}^2$$. From here we get the radius is $$\\\\frac{3}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5fd810distance25","title":"Find the center and the radius of the circle","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.1 Distance and Midpoint Formulas; Circles","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5fd810distance25a","stepAnswer":["Center: $$(0,0)$$, Radius: $$8$$"],"problemType":"MultipleChoice","stepTitle":"$$x^2+y^2=64$$","stepBody":"Please select the correct answer.","answerType":"string","variabilization":{},"answerLatex":"Center: $$(0,0)$$, Radius: $$8$$","choices":["Center: $$(0,0)$$, Radius: $$8$$","Center: $$(1,3)$$, Radius: $$\\\\frac{3}{2}$$","Center: $$(1.5, -2.5)$$, Radius: $$0.5$$","Center: $$(1,0)$$, Radius: $$6$$"],"hints":{"DefaultPathway":[{"id":"a5fd810distance25a-h1","type":"hint","dependencies":[],"title":"Standard form of a circle","text":"The standard form of a circle is $${\\\\left(x-h\\\\right)}^2+{\\\\left(y-k\\\\right)}^2=r^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance25a-h2","type":"hint","dependencies":["a5fd810distance25a-h1"],"title":"Center","text":"The center of an equation of a circle in standard form is (h,k). We can rewrite the equation as $${\\\\left(x-0\\\\right)}^2+{\\\\left(y-0\\\\right)}^2=8^2$$. From here we get the center is $$(0,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance25a-h3","type":"hint","dependencies":["a5fd810distance25a-h2"],"title":"Radius","text":"The radius of an equation of a circle in standard form is simply the $$r$$ value. Note that in the standard form of the circle, $$r$$ is squared. We can rewrite the equation as $${\\\\left(x-0\\\\right)}^2+{\\\\left(y-0\\\\right)}^2=8^2$$. From here we get the radius is $$8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5fd810distance26","title":"Find the center and the radius of the circle","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.1 Distance and Midpoint Formulas; Circles","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5fd810distance26a","stepAnswer":["Center: $$(0,0)$$, Radius: $$7$$"],"problemType":"MultipleChoice","stepTitle":"$$x^2+y^2=49$$","stepBody":"Please select the correct answer.","answerType":"string","variabilization":{},"answerLatex":"Center: $$(0,0)$$, Radius: $$7$$","choices":["Center: $$(0,0)$$, Radius: $$8$$","Center: $$(0,0)$$, Radius: $$7$$","Center: $$(1.5, -2.5)$$, Radius: $$0.5$$","Center: $$(1,0)$$, Radius: $$6$$"],"hints":{"DefaultPathway":[{"id":"a5fd810distance26a-h1","type":"hint","dependencies":[],"title":"Standard form of a circle","text":"The standard form of a circle is $${\\\\left(x-h\\\\right)}^2+{\\\\left(y-k\\\\right)}^2=r^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance26a-h2","type":"hint","dependencies":["a5fd810distance26a-h1"],"title":"Center","text":"The center of an equation of a circle in standard form is (h,k). We can rewrite the equation as $${\\\\left(x-0\\\\right)}^2+{\\\\left(y-0\\\\right)}^2=7^2$$. From here we get the center is $$(0,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance26a-h3","type":"hint","dependencies":["a5fd810distance26a-h2"],"title":"Radius","text":"The radius of an equation of a circle in standard form is simply the $$r$$ value. Note that in the standard form of the circle, $$r$$ is squared. We can rewrite the equation as $${\\\\left(x-0\\\\right)}^2+{\\\\left(y-0\\\\right)}^2=7^2$$. From here we get the radius is $$7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5fd810distance27","title":"Find the center and the radius of the circle","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.1 Distance and Midpoint Formulas; Circles","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5fd810distance27a","stepAnswer":["Center: $$(0,0)$$, Radius: $$2$$"],"problemType":"MultipleChoice","stepTitle":"$$2x^2+2y^2=8$$","stepBody":"Please select the correct answer.","answerType":"string","variabilization":{},"answerLatex":"Center: $$(0,0)$$, Radius: $$2$$","choices":["Center: $$(0,0)$$, Radius: $$8$$","Center: $$(0,0)$$, Radius: $$7$$","Center: $$(0,0)$$, Radius: $$2$$","Center: $$(1,0)$$, Radius: $$6$$"],"hints":{"DefaultPathway":[{"id":"a5fd810distance27a-h1","type":"hint","dependencies":[],"title":"Standard form of a circle","text":"The standard form of a circle is $${\\\\left(x-h\\\\right)}^2+{\\\\left(y-k\\\\right)}^2=r^2$$. Manipulate the equation into standard form by dividing or multiplying on both sides if necessary.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance27a-h2","type":"hint","dependencies":["a5fd810distance27a-h1"],"title":"Center","text":"The center of an equation of a circle in standard form is (h,k). We can rewrite the equation as $${\\\\left(x-0\\\\right)}^2+{\\\\left(y-0\\\\right)}^2=4=2^2$$. From here we get the center is $$(0,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance27a-h3","type":"hint","dependencies":["a5fd810distance27a-h2"],"title":"Radius","text":"The radius of an equation of a circle in standard form is simply the $$r$$ value. Note that in the standard form of the circle, $$r$$ is squared. We can rewrite the equation as $${\\\\left(x-0\\\\right)}^2+{\\\\left(y-0\\\\right)}^2=2^2$$. From here we get the radius is $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5fd810distance28","title":"Find the center and the radius of the circle","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.1 Distance and Midpoint Formulas; Circles","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5fd810distance28a","stepAnswer":["Center: $$(0,0)$$, Radius: $$6$$"],"problemType":"MultipleChoice","stepTitle":"$$6x^2+6y^2=216$$","stepBody":"Please select the correct answer.","answerType":"string","variabilization":{},"answerLatex":"Center: $$(0,0)$$, Radius: $$6$$","choices":["Center: $$(0,0)$$, Radius: $$8$$","Center: $$(0,0)$$, Radius: $$7$$","Center: $$(0,0)$$, Radius: $$2$$","Center: $$(0,0)$$, Radius: $$6$$"],"hints":{"DefaultPathway":[{"id":"a5fd810distance28a-h1","type":"hint","dependencies":[],"title":"Standard form of a circle","text":"The standard form of a circle is $${\\\\left(x-h\\\\right)}^2+{\\\\left(y-k\\\\right)}^2=r^2$$. Manipulate the equation into standard form by dividing or multiplying on both sides if necessary.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance28a-h2","type":"hint","dependencies":["a5fd810distance28a-h1"],"title":"Center","text":"The center of an equation of a circle in standard form is (h,k). We can rewrite the equation as $${\\\\left(x-0\\\\right)}^2+{\\\\left(y-0\\\\right)}^2=36=6^2$$. From here we get the center is $$(0,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance28a-h3","type":"hint","dependencies":["a5fd810distance28a-h2"],"title":"Radius","text":"The radius of an equation of a circle in standard form is simply the $$r$$ value. Note that in the standard form of the circle, $$r$$ is squared. We can rewrite the equation as $${\\\\left(x-0\\\\right)}^2+{\\\\left(y-0\\\\right)}^2=36=6^2$$. From here we get the radius is $$6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5fd810distance29","title":"Find the center and the radius of the circle","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.1 Distance and Midpoint Formulas; Circles","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5fd810distance29a","stepAnswer":["Center: $$(3,7)$$, Radius: $$2$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(x-3\\\\right)}^2+{\\\\left(y-7\\\\right)}^2=4$$","stepBody":"Please select the correct answer.","answerType":"string","variabilization":{},"answerLatex":"Center: $$(3,7)$$, Radius: $$2$$","choices":["Center: $$(3,7)$$, Radius: $$2$$","Center: $$(0,0)$$, Radius: $$7$$","Center: $$(0,0)$$, Radius: $$2$$","Center: $$(0,0)$$, Radius: $$6$$"],"hints":{"DefaultPathway":[{"id":"a5fd810distance29a-h1","type":"hint","dependencies":[],"title":"Standard form of a circle","text":"The standard form of a circle is $${\\\\left(x-h\\\\right)}^2+{\\\\left(y-k\\\\right)}^2=r^2$$. Manipulate the equation into standard form by dividing or multiplying on both sides if necessary.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance29a-h2","type":"hint","dependencies":["a5fd810distance29a-h1"],"title":"Center","text":"The center of an equation of a circle in standard form is (h,k). We can rewrite the equation as $${\\\\left(x-3\\\\right)}^2+{\\\\left(y-7\\\\right)}^2=2^2$$. From here we get the center is $$(3,7)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance29a-h3","type":"hint","dependencies":["a5fd810distance29a-h2"],"title":"Radius","text":"The radius of an equation of a circle in standard form is simply the $$r$$ value. Note that in the standard form of the circle, $$r$$ is squared. We can rewrite the equation as $${\\\\left(x-3\\\\right)}^2+{\\\\left(y-7\\\\right)}^2=2^2$$. From here we get the radius is $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5fd810distance3","title":"Find the distance between the points","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.1 Distance and Midpoint Formulas; Circles","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5fd810distance3a","stepAnswer":["$$13$$"],"problemType":"TextBox","stepTitle":"$$(-4,-3)$$ and $$(8,2)$$","stepBody":"Write your answer in exact form. Simplify your answer as much as possible.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$13$$","hints":{"DefaultPathway":[{"id":"a5fd810distance3a-h1","type":"hint","dependencies":[],"title":"Distance formula","text":"The distance formula is $$\\\\sqrt{{\\\\left(x_2-x_1\\\\right)}^2+{\\\\left(y_2-y_1\\\\right)}^2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance3a-h2","type":"hint","dependencies":["a5fd810distance3a-h1"],"title":"Plug in","text":"Plug in the values for the equation. Choose a point to be the $$x_2$$ and $$y_2$$ pair, and use the other as the $$x_1$$ and $$y_1$$ pair. We get $$\\\\sqrt{{\\\\left(8-\\\\left(-4\\\\right)\\\\right)}^2+{\\\\left(2-\\\\left(-3\\\\right)\\\\right)}^2}=\\\\sqrt{169}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance3a-h3","type":"hint","dependencies":["a5fd810distance3a-h2"],"title":"Simplify","text":"Simplify the answer into one term. $$\\\\sqrt{169}=13$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5fd810distance30","title":"Find the center and the radius of the circle","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.1 Distance and Midpoint Formulas; Circles","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5fd810distance30a","stepAnswer":["Center: $$(4,-3)$$, Radius: $$2$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(x+4\\\\right)}^2+{\\\\left(y-3\\\\right)}^2=4$$","stepBody":"Please select the correct answer.","answerType":"string","variabilization":{},"answerLatex":"Center: $$(4,-3)$$, Radius: $$2$$","choices":["Center: $$(3,7)$$, Radius: $$2$$","Center: $$(0,0)$$, Radius: $$7$$","Center: $$(4,-3)$$, Radius: $$2$$","Center: $$(0,0)$$, Radius: $$6$$"],"hints":{"DefaultPathway":[{"id":"a5fd810distance30a-h1","type":"hint","dependencies":[],"title":"Standard form of a circle","text":"The standard form of a circle is $${\\\\left(x-h\\\\right)}^2+{\\\\left(y-k\\\\right)}^2=r^2$$. Manipulate the equation into standard form by dividing or multiplying on both sides if necessary.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance30a-h2","type":"hint","dependencies":["a5fd810distance30a-h1"],"title":"Center","text":"The center of an equation of a circle in standard form is (h,k). We can rewrite the equation as $${\\\\left(x-\\\\left(-4\\\\right)\\\\right)}^2+{\\\\left(y-3\\\\right)}^2=2^2$$. From here we get the center is $$(-4,3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance30a-h3","type":"hint","dependencies":["a5fd810distance30a-h2"],"title":"Radius","text":"The radius of an equation of a circle in standard form is simply the $$r$$ value. Note that in the standard form of the circle, $$r$$ is squared. We can rewrite the equation as $${\\\\left(x-\\\\left(-4\\\\right)\\\\right)}^2+{\\\\left(y-3\\\\right)}^2=2^2$$. From here we get the radius is $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5fd810distance4","title":"Find the distance between the points","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.1 Distance and Midpoint Formulas; Circles","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5fd810distance4a","stepAnswer":["$$17$$"],"problemType":"TextBox","stepTitle":"$$(-7,-3)$$ and $$(8,5)$$","stepBody":"Write your answer in exact form. Simplify your answer as much as possible.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$17$$","hints":{"DefaultPathway":[{"id":"a5fd810distance4a-h1","type":"hint","dependencies":[],"title":"Distance formula","text":"The distance formula is $$\\\\sqrt{{\\\\left(x_2-x_1\\\\right)}^2+{\\\\left(y_2-y_1\\\\right)}^2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance4a-h2","type":"hint","dependencies":["a5fd810distance4a-h1"],"title":"Plug in","text":"Plug in the values for the equation. Choose a point to be the $$x_2$$ and $$y_2$$ pair, and use the other as the $$x_1$$ and $$y_1$$ pair. We get $$\\\\sqrt{{\\\\left(8-\\\\left(-7\\\\right)\\\\right)}^2+{\\\\left(5-\\\\left(-3\\\\right)\\\\right)}^2}=\\\\sqrt{289}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance4a-h3","type":"hint","dependencies":["a5fd810distance4a-h2"],"title":"Simplify","text":"Simplify the answer into one term. $$\\\\sqrt{289}=17$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5fd810distance5","title":"Find the distance between the points","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.1 Distance and Midpoint Formulas; Circles","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5fd810distance5a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"$$(-1,4)$$ and $$(2,0)$$","stepBody":"Write your answer in exact form. Simplify your answer as much as possible.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"a5fd810distance5a-h1","type":"hint","dependencies":[],"title":"Distance formula","text":"The distance formula is $$\\\\sqrt{{\\\\left(x_2-x_1\\\\right)}^2+{\\\\left(y_2-y_1\\\\right)}^2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance5a-h2","type":"hint","dependencies":["a5fd810distance5a-h1"],"title":"Plug in","text":"Plug in the values for the equation. Choose a point to be the $$x_2$$ and $$y_2$$ pair, and use the other as the $$x_1$$ and $$y_1$$ pair. We get $$\\\\sqrt{{\\\\left(2-\\\\left(-1\\\\right)\\\\right)}^2+{\\\\left(0-4\\\\right)}^2}=\\\\sqrt{25}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance5a-h3","type":"hint","dependencies":["a5fd810distance5a-h2"],"title":"Simplify","text":"Simplify the answer into one term. $$\\\\sqrt{25}=5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5fd810distance6","title":"Find the distance between the points","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.1 Distance and Midpoint Formulas; Circles","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5fd810distance6a","stepAnswer":["$$10$$"],"problemType":"TextBox","stepTitle":"$$(-1,3)$$ and $$(5,-5)$$","stepBody":"Write your answer in exact form. Simplify your answer as much as possible.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10$$","hints":{"DefaultPathway":[{"id":"a5fd810distance6a-h1","type":"hint","dependencies":[],"title":"Distance formula","text":"The distance formula is $$\\\\sqrt{{\\\\left(x_2-x_1\\\\right)}^2+{\\\\left(y_2-y_1\\\\right)}^2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance6a-h2","type":"hint","dependencies":["a5fd810distance6a-h1"],"title":"Plug in","text":"Plug in the values for the equation. Choose a point to be the $$x_2$$ and $$y_2$$ pair, and use the other as the $$x_1$$ and $$y_1$$ pair. We get $$\\\\sqrt{{\\\\left(5-\\\\left(-1\\\\right)\\\\right)}^2+{\\\\left(-5-3\\\\right)}^2}=\\\\sqrt{100}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance6a-h3","type":"hint","dependencies":["a5fd810distance6a-h2"],"title":"Simplify","text":"Simplify the answer into one term. $$\\\\sqrt{100}=10$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5fd810distance7","title":"Find the distance between the points","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.1 Distance and Midpoint Formulas; Circles","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5fd810distance7a","stepAnswer":["$$13$$"],"problemType":"TextBox","stepTitle":"$$(1,-4)$$ and $$(6,8)$$","stepBody":"Write your answer in exact form. Simplify your answer as much as possible.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$13$$","hints":{"DefaultPathway":[{"id":"a5fd810distance7a-h1","type":"hint","dependencies":[],"title":"Distance formula","text":"The distance formula is $$\\\\sqrt{{\\\\left(x_2-x_1\\\\right)}^2+{\\\\left(y_2-y_1\\\\right)}^2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance7a-h2","type":"hint","dependencies":["a5fd810distance7a-h1"],"title":"Plug in","text":"Plug in the values for the equation. Choose a point to be the $$x_2$$ and $$y_2$$ pair, and use the other as the $$x_1$$ and $$y_1$$ pair. We get $$\\\\sqrt{{\\\\left(6-1\\\\right)}^2+{\\\\left(8-\\\\left(-4\\\\right)\\\\right)}^2}=\\\\sqrt{169}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance7a-h3","type":"hint","dependencies":["a5fd810distance7a-h2"],"title":"Simplify","text":"Simplify the answer into one term. $$\\\\sqrt{169}=13$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5fd810distance8","title":"Find the distance between the points","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.1 Distance and Midpoint Formulas; Circles","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5fd810distance8a","stepAnswer":["$$17$$"],"problemType":"TextBox","stepTitle":"$$(-8,-2)$$ and $$(7,6)$$","stepBody":"Write your answer in exact form. Simplify your answer as much as possible.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$17$$","hints":{"DefaultPathway":[{"id":"a5fd810distance8a-h1","type":"hint","dependencies":[],"title":"Distance formula","text":"The distance formula is $$\\\\sqrt{{\\\\left(x_2-x_1\\\\right)}^2+{\\\\left(y_2-y_1\\\\right)}^2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance8a-h2","type":"hint","dependencies":["a5fd810distance8a-h1"],"title":"Plug in","text":"Plug in the values for the equation. Choose a point to be the $$x_2$$ and $$y_2$$ pair, and use the other as the $$x_1$$ and $$y_1$$ pair. We get $$\\\\sqrt{{\\\\left(7-\\\\left(-8\\\\right)\\\\right)}^2+{\\\\left(6-\\\\left(-2\\\\right)\\\\right)}^2}=\\\\sqrt{289}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance8a-h3","type":"hint","dependencies":["a5fd810distance8a-h2"],"title":"Simplify","text":"Simplify the answer into one term. $$\\\\sqrt{289}=17$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a5fd810distance9","title":"Find the distance between the points","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.1 Distance and Midpoint Formulas; Circles","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a5fd810distance9a","stepAnswer":["$$3\\\\sqrt{5}$$"],"problemType":"TextBox","stepTitle":"$$(-3,-5)$$ and $$(0,1)$$","stepBody":"Write your answer in exact form. Simplify your answer as much as possible.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3\\\\sqrt{5}$$","hints":{"DefaultPathway":[{"id":"a5fd810distance9a-h1","type":"hint","dependencies":[],"title":"Distance formula","text":"The distance formula is $$\\\\sqrt{{\\\\left(x_2-x_1\\\\right)}^2+{\\\\left(y_2-y_1\\\\right)}^2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance9a-h2","type":"hint","dependencies":["a5fd810distance9a-h1"],"title":"Plug in","text":"Plug in the values for the equation. Choose a point to be the $$x_2$$ and $$y_2$$ pair, and use the other as the $$x_1$$ and $$y_1$$ pair. We get $$\\\\sqrt{{\\\\left(0-\\\\left(-3\\\\right)\\\\right)}^2+{\\\\left(1-\\\\left(-5\\\\right)\\\\right)}^2}=\\\\sqrt{45}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a5fd810distance9a-h3","type":"hint","dependencies":["a5fd810distance9a-h2"],"title":"Simplify","text":"Simplify the answer into one term. $$\\\\sqrt{45}=\\\\sqrt{9} \\\\sqrt{5}=3\\\\sqrt{5}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6014eaSubAdd1","title":"Verify a Solution of an Equation","body":"Determine whether $$x=\\\\frac{3}{2}$$ is a solution of the equation","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Solve Equations Using the Subtraction and Addition Properties of Equality","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6014eaSubAdd1a","stepAnswer":["TRUE"],"problemType":"MultipleChoice","stepTitle":"$$4x-2=2x+1$$","stepBody":"","answerType":"string","variabilization":{},"choices":["TRUE","FALSE"],"hints":{"DefaultPathway":[{"id":"a6014eaSubAdd1a-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Substitute the number in for the variable in the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6014eaSubAdd1a-h2","type":"hint","dependencies":["a6014eaSubAdd1a-h1"],"title":"Substitution","text":"After substituting $$x=\\\\frac{3}{2}$$ into the equation, we get $$4\\\\frac{3}{2}-2=2\\\\frac{3}{2}+1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6014eaSubAdd1a-h3","type":"hint","dependencies":["a6014eaSubAdd1a-h2"],"title":"Simplification","text":"Simplify the expressions on both sides of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6014eaSubAdd1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6-2$$"],"dependencies":["a6014eaSubAdd1a-h3"],"title":"Simplification","text":"Simplify the left side of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6014eaSubAdd1a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3+1$$"],"dependencies":["a6014eaSubAdd1a-h4"],"title":"Simplification","text":"Simplify the right side of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6014eaSubAdd1a-h6","type":"hint","dependencies":["a6014eaSubAdd1a-h5"],"title":"Comparison","text":"Determine whether the resulting equation is true.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6014eaSubAdd1a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a6014eaSubAdd1a-h6"],"title":"Comparison","text":"Determine whether $$6-2$$ equals $$3+1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]}]}}]},{"id":"a6014eaSubAdd10","title":"Solve Equations Using the Addition Property of Equality","body":"Solve the equation","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Solve Equations Using the Subtraction and Addition Properties of Equality","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6014eaSubAdd10a","stepAnswer":["$$-32$$"],"problemType":"TextBox","stepTitle":"$$p-41=-73$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-32$$","hints":{"DefaultPathway":[{"id":"a6014eaSubAdd10a-h1","type":"hint","dependencies":[],"title":"Addition property of equality","text":"When you add the same quantity to both sides of an equation, you still have equality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6014eaSubAdd10a-h2","type":"hint","dependencies":["a6014eaSubAdd10a-h1"],"title":"Addition","text":"After adding $$41$$ to each side of the equation, we get $$p-41+41=-73+41$$.","variabilization":{},"oer":"https://OATutor.io 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$$-73$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]}]}}]},{"id":"a6014eaSubAdd11","title":"","body":"Solve the equation","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Solve Equations Using the Subtraction and Addition Properties of Equality","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6014eaSubAdd11a","stepAnswer":["$$x=\\\\frac{11}{8}$$"],"problemType":"TextBox","stepTitle":"$$x-\\\\frac{5}{8}=\\\\frac{3}{4}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x=\\\\frac{11}{8}$$","hints":{"DefaultPathway":[{"id":"a6014eaSubAdd11a-h1","type":"hint","dependencies":[],"title":"Addition property of equality","text":"When you add the same quantity to both sides of an equation, you 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Translate to an equals sign $$(=)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6014eaSubAdd33a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["7a-6a"],"dependencies":["a6014eaSubAdd33a-h1"],"title":"Left Side","text":"Translate the words to the left of the \\"equals\\" word(s) into an algebraic expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6014eaSubAdd33a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-8$$"],"dependencies":["a6014eaSubAdd33a-h2"],"title":"Right Side","text":"Translate the words to the right of the \\"equals\\" word(s) into an algebraic expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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$$6\\\\frac{5}{3}+10=12\\\\frac{5}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6014eaSubAdd4a-h3","type":"hint","dependencies":["a6014eaSubAdd4a-h2"],"title":"Simplification","text":"Simplify the expressions on both sides of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6014eaSubAdd4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10+10$$"],"dependencies":["a6014eaSubAdd4a-h3"],"title":"Simplification","text":"Simplify the left side of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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<CC BY 4.0>","choices":["TRUE","FALSE"]}]}},{"id":"a6014eaSubAdd4b","stepAnswer":["FALSE"],"problemType":"MultipleChoice","stepTitle":"Is $$u=\\\\frac{-1}{2}$$ a solution of $$8u-1=6u$$?","stepBody":"","answerType":"string","variabilization":{},"choices":["TRUE","FALSE"],"hints":{"DefaultPathway":[{"id":"a6014eaSubAdd4b-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Substitute the number in for the variable in the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6014eaSubAdd4b-h2","type":"hint","dependencies":["a6014eaSubAdd4b-h1"],"title":"Substitution","text":"After substituting $$u=\\\\frac{-1}{2}$$ into the equation, we get $$8\\\\left(-\\\\frac{1}{2}\\\\right)-1=6\\\\left(-\\\\frac{1}{2}\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6014eaSubAdd4b-h3","type":"hint","dependencies":["a6014eaSubAdd4b-h2"],"title":"Simplification","text":"Simplify the expressions on both sides of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6014eaSubAdd4b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4-1$$"],"dependencies":["a6014eaSubAdd4b-h3"],"title":"Simplification","text":"Simplify the left side of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6014eaSubAdd4b-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a6014eaSubAdd4b-h4"],"title":"Simplification","text":"Simplify the right side of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6014eaSubAdd4b-h6","type":"hint","dependencies":["a6014eaSubAdd4b-h5"],"title":"Comparison","text":"Determine whether the resulting equation is true.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6014eaSubAdd4b-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["FALSE"],"dependencies":["a6014eaSubAdd4b-h6"],"title":"Comparison","text":"Determine whether $$-4-1$$ equals $$-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]}]}}]},{"id":"a6014eaSubAdd5","title":"Solve Equations Using the Subtraction Property of Equality","body":"Solve the equation","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6014eaSubAdd5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-50$$"],"dependencies":["a6014eaSubAdd5a-h2"],"title":"Simplification","text":"What do we get for $$y$$ after simplifying the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6014eaSubAdd5a-h4","type":"hint","dependencies":["a6014eaSubAdd5a-h3"],"title":"Verification","text":"Check whether the result is a solution of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6014eaSubAdd5a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a6014eaSubAdd5a-h4"],"title":"Verification","text":"Check whether $$-50+37$$ equals $$-13$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]}]}}]},{"id":"a6014eaSubAdd6","title":"Solve Equations Using the Subtraction Property of Equality","body":"Solve the equation","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Solve Equations Using the Subtraction and Addition Properties of Equality","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6014eaSubAdd6a","stepAnswer":["$$-46$$"],"problemType":"TextBox","stepTitle":"$$x+19=-27$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-46$$","hints":{"DefaultPathway":[{"id":"a6014eaSubAdd6a-h1","type":"hint","dependencies":[],"title":"Subtraction property of equality","text":"When you subtract the same quantity from both sides of an 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the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6014eaSubAdd6a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a6014eaSubAdd6a-h4"],"title":"Verification","text":"Check whether $$-46+19$$ equals $$-27$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]}]}}]},{"id":"a6014eaSubAdd7","title":"Solve Equations Using the Subtraction Property of Equality","body":"Solve the equation","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Solve Equations Using the Subtraction and Addition Properties of Equality","courseName":"OpenStax: Elementary 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4.0>"},{"id":"a6014eaSubAdd7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-50$$"],"dependencies":["a6014eaSubAdd7a-h2"],"title":"Simplification","text":"What do we get for $$x$$ after simplifying the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6014eaSubAdd7a-h4","type":"hint","dependencies":["a6014eaSubAdd7a-h3"],"title":"Verification","text":"Check whether the result is a solution of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6014eaSubAdd7a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a6014eaSubAdd7a-h4"],"title":"Verification","text":"Check whether $$-50+16$$ equals $$-34$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]}]}}]},{"id":"a6014eaSubAdd8","title":"Solve Equations Using the Addition Property of Equality","body":"Solve the equation","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Solve Equations Using the Subtraction and Addition Properties of Equality","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6014eaSubAdd8a","stepAnswer":["$$-9$$"],"problemType":"TextBox","stepTitle":"$$a-28=-37$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-9$$","hints":{"DefaultPathway":[{"id":"a6014eaSubAdd8a-h1","type":"hint","dependencies":[],"title":"Addition property of equality","text":"When you add the same quantity to both sides of an equation, you still have equality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6014eaSubAdd8a-h2","type":"hint","dependencies":["a6014eaSubAdd8a-h1"],"title":"Addition","text":"After adding $$28$$ to each side of the equation, we get $$a-28+28=-37+28$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6014eaSubAdd8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":["a6014eaSubAdd8a-h2"],"title":"Simplification","text":"What do we get for a after simplifying the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6014eaSubAdd8a-h4","type":"hint","dependencies":["a6014eaSubAdd8a-h3"],"title":"Verification","text":"Check whether the result is a solution of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6014eaSubAdd8a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a6014eaSubAdd8a-h4"],"title":"Verification","text":"Check whether $$-9-28$$ equals $$-37$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]}]}}]},{"id":"a6014eaSubAdd9","title":"Solve Equations Using the Addition Property of Equality","body":"Solve the equation","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Solve Equations Using the Subtraction and Addition Properties of Equality","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6014eaSubAdd9a","stepAnswer":["$$-14$$"],"problemType":"TextBox","stepTitle":"$$n-61=-75$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-14$$","hints":{"DefaultPathway":[{"id":"a6014eaSubAdd9a-h1","type":"hint","dependencies":[],"title":"Addition property of equality","text":"When you add the same quantity to both sides of an equation, you still have equality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6014eaSubAdd9a-h2","type":"hint","dependencies":["a6014eaSubAdd9a-h1"],"title":"Addition","text":"After adding $$61$$ to each side of the equation, we get $$n-61+61=-75+61$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6014eaSubAdd9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-14$$"],"dependencies":["a6014eaSubAdd9a-h2"],"title":"Simplification","text":"What do we get for $$n$$ after simplifying the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6014eaSubAdd9a-h4","type":"hint","dependencies":["a6014eaSubAdd9a-h3"],"title":"Verification","text":"Check whether the result is a solution of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6014eaSubAdd9a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a6014eaSubAdd9a-h4"],"title":"Verification","text":"Check whether $$-14-61$$ equals $$-75$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]}]}}]},{"id":"a6050f0Rare1","title":"Fill in the blanks.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Rare Events, the Sample, Decision and Conclusion","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a6050f0Rare1a","stepAnswer":["the $$p-value$$ is less than the established alpha value, and the results of the sample data support the alternative hypothesis."],"problemType":"MultipleChoice","stepTitle":"Reject the null hypothesis when","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"the $$p-value$$ is less than the established alpha value, and the results of the sample data support the alternative hypothesis.","choices":["the $$p-value$$ is less than the established alpha value, and the results of the sample data support the alternative hypothesis.","the $$p-value$$ is less than the established alpha value, and the results of the sample data do not support the alternative hypothesis","the $$p-value$$ is less than the established beta value","the $$p-value$$ is more than the established beta value"],"hints":{"DefaultPathway":[{"id":"a6050f0Rare1a-h1","type":"hint","dependencies":[],"title":"$$p-value$$","text":"Think about $$p-value$$ and null hypothesis","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6050f0Rare2","title":"Fill in the blanks.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.4 Rare Events, the Sample, Decision and Conclusion","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a6050f0Rare2a","stepAnswer":["the $$p-value$$ is less than the established alpha value"],"problemType":"MultipleChoice","stepTitle":"Do not reject the null when hypothesis when","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"the $$p-value$$ is less than the established alpha value","choices":["the $$p-value$$ is greater than the established alpha value. The results of the sample data do not support the alternative hypothesis.","the $$p-value$$ is greater than the established alpha value. The results of the sample data do support the alternative hypothesis.","the $$p-value$$ is less than the established alpha value","the $$p-value$$ is less than the established alpha value. The results of the sample data do not support the alternative hypothesis.","the $$p-value$$ is less than the established alpha value. The results of the sample data do support the alternative hypothesis."],"hints":{"DefaultPathway":[{"id":"a6050f0Rare2a-h1","type":"hint","dependencies":[],"title":"$$p-value$$","text":"Think about $$p-value$$ and null hypothesis","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a60a373solverat1","title":"Solving a Rational Equation","body":"Solve for a. Please input the answer as $$variable=answer$$. If there are multiple answers, separate them with a comma and a space.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Rational Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a60a373solverat1a","stepAnswer":["a=5, a=-3"],"problemType":"TextBox","stepTitle":"$$1-\\\\frac{2}{a}=\\\\frac{-15}{a^2}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$a=5$$, $$a=-3$$","hints":{"DefaultPathway":[{"id":"a60a373solverat1a-h1","type":"hint","dependencies":[],"title":"Lowest Common Denominator","text":"The first step is to find the lowest common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$a^2$$"],"dependencies":["a60a373solverat1a-h1"],"title":"Lowest Common Denominator","text":"What is the lowest common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat1a-h3","type":"hint","dependencies":["a60a373solverat1a-h2"],"title":"Distribute and multiply","text":"Distribute and multiply the lowest common denominator to all the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat1a-h4","type":"hint","dependencies":["a60a373solverat1a-h3"],"title":"Quadratic Form","text":"Rearrange the equation into the quadratic form and factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat1a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(a-5\\\\right) \\\\left(a+3\\\\right)=0$$"],"dependencies":["a60a373solverat1a-h4"],"title":"Quadratic Form","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat1a-h6","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["a=5, a=-3"],"dependencies":["a60a373solverat1a-h5"],"title":"Zero Product Property","text":"Solve for a. Please input the answer as $$variable=answer$$. If there are multiple answers, separate them with a comma and a space.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat1a-h7","type":"hint","dependencies":["a60a373solverat1a-h6"],"title":"Double Check","text":"Plug answers for a back into the original equation to make the sure answer is correct and not undefined at any point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a60a373solverat10","title":"Solving a Rational Equation","body":"Solve for $$x$$. Please input the answer as $$variable=answer$$. If there are multiple answers, separate them with a comma and a space.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Rational Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a60a373solverat10a","stepAnswer":["$$x=-2$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{x}{18}+\\\\frac{x+6}{9} x=\\\\frac{2}{3x}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x=-2$$","hints":{"DefaultPathway":[{"id":"a60a373solverat10a-h1","type":"hint","dependencies":[],"title":"Lowest Common Denominator","text":"The first step is to find the lowest common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18x$$"],"dependencies":["a60a373solverat10a-h1"],"title":"Lowest Common Denominator","text":"What is the lowest common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat10a-h3","type":"hint","dependencies":["a60a373solverat10a-h2"],"title":"Distribute and multiply","text":"Distribute and multiply the lowest common denominator to all the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2+2x+12=12$$"],"dependencies":["a60a373solverat10a-h3"],"title":"Distribute and multiply","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat10a-h5","type":"hint","dependencies":["a60a373solverat10a-h4"],"title":"Quadratic Form","text":"Rearrange the equation into the quadratic form and factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat10a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x\\\\left(x+2\\\\right)=0$$"],"dependencies":["a60a373solverat10a-h5"],"title":"Quadratic Form","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat10a-h7","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["x=0,x=-2"],"dependencies":["a60a373solverat10a-h6"],"title":"Zero Product Property","text":"Solve for $$x$$. Please input the answer as $$variable=answer$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat10a-h8","type":"hint","dependencies":["a60a373solverat10a-h7"],"title":"Double Check","text":"Plug answers for a back into the original equation to make the sure answer is correct and not undefined at any point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat10a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=0$$"],"dependencies":["a60a373solverat10a-h8"],"title":"Double Check","text":"Which of the solutions for $$x$$ is invalid?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a60a373solverat11","title":"Solving a Rational Equation","body":"Solve for $$y$$. Please input the answer as $$variable=answer$$. If there are multiple answers, separate them with a comma and a space.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Rational Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a60a373solverat11a","stepAnswer":["$$y=-3$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{y+5}{5} y+\\\\frac{y}{15}=\\\\frac{1}{y}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=-3$$","hints":{"DefaultPathway":[{"id":"a60a373solverat11a-h1","type":"hint","dependencies":[],"title":"Lowest Common Denominator","text":"The first step is to find the lowest common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15y$$"],"dependencies":["a60a373solverat11a-h1"],"title":"Lowest Common Denominator","text":"What is the lowest common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat11a-h3","type":"hint","dependencies":["a60a373solverat11a-h2"],"title":"Distribute and multiply","text":"Distribute and multiply the lowest common denominator to all the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3y+15+y^2=15$$"],"dependencies":["a60a373solverat11a-h3"],"title":"Distribute and multiply","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat11a-h5","type":"hint","dependencies":["a60a373solverat11a-h4"],"title":"Quadratic Form","text":"Rearrange the equation into the quadratic form and factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat11a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y^2+3y=0$$"],"dependencies":["a60a373solverat11a-h5"],"title":"Quadratic Form","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat11a-h7","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["y=0,y=-3"],"dependencies":["a60a373solverat11a-h6"],"title":"Zero Product Property","text":"Solve for $$y$$. Please input the answer as $$variable=answer$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat11a-h8","type":"hint","dependencies":["a60a373solverat11a-h7"],"title":"Double Check","text":"Plug answers for a back into the original equation to make the sure answer is correct and not undefined at any point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat11a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y=-3$$"],"dependencies":["a60a373solverat11a-h8"],"title":"Double Check","text":"Which of the solutions for $$x$$ is invalid?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a60a373solverat12","title":"Solving a Rational Equation","body":"Solve for $$x$$. Please input the answer as $$variable=answer$$. If there are multiple answers, separate them with a comma and a space.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Rational Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a60a373solverat12a","stepAnswer":["$$x=3$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{x}{x+4}=\\\\frac{32}{x^2-16}+5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x=3$$","hints":{"DefaultPathway":[{"id":"a60a373solverat12a-h1","type":"hint","dependencies":[],"title":"Lowest Common Denominator","text":"The first step is to find the lowest common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2-16$$"],"dependencies":["a60a373solverat12a-h1"],"title":"Lowest Common Denominator","text":"What is the lowest common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat12a-h3","type":"hint","dependencies":["a60a373solverat12a-h2"],"title":"Distribute and multiply","text":"Distribute and multiply the lowest common denominator to all the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2-4x=32+5x^2-80$$"],"dependencies":["a60a373solverat12a-h3"],"title":"Distribute and multiply","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat12a-h5","type":"hint","dependencies":["a60a373solverat12a-h4"],"title":"Quadratic Form","text":"Rearrange the equation into the quadratic form and factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat12a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4x^2+4x-48=0$$"],"dependencies":["a60a373solverat12a-h5"],"title":"Quadratic Form","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat12a-h7","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["x=-4,x=3"],"dependencies":["a60a373solverat12a-h6"],"title":"Zero Product Property","text":"Solve for $$x$$. Please input the answer as $$variable=answer$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat12a-h8","type":"hint","dependencies":["a60a373solverat12a-h7"],"title":"Double Check","text":"Plug answers for a back into the original equation to make the sure answer is correct and not undefined at any point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat12a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=-4$$"],"dependencies":["a60a373solverat12a-h8"],"title":"Double Check","text":"Which of the solutions for $$x$$ is invalid?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a60a373solverat13","title":"Solving a Rational Equation","body":"Solve for L in terms of A and W.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Rational Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a60a373solverat13a","stepAnswer":["$$L=\\\\frac{A}{W}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{A}{L}=W$$","stepBody":"Solve for L in terms of A and W","answerType":"arithmetic","variabilization":{},"answerLatex":"$$L=\\\\frac{A}{W}$$","hints":{"DefaultPathway":[{"id":"a60a373solverat13a-h1","type":"hint","dependencies":[],"title":"Clear the fractions","text":"The first step is to clear the fractions by multiplying the entire equation by the lowest common denominator: L","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat13a-h2","type":"hint","dependencies":["a60a373solverat13a-h1"],"title":"Isolate L","text":"Isolate L by dividing the equation by W","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat13a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$L=\\\\frac{A}{W}$$"],"dependencies":["a60a373solverat13a-h2"],"title":"Isolate L","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a60a373solverat14","title":"Solving a Rational Equation","body":"Solve for A in terms of F and M.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Rational Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a60a373solverat14a","stepAnswer":["$$A=\\\\frac{F}{M}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{F}{A}=M$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$A=\\\\frac{F}{M}$$","hints":{"DefaultPathway":[{"id":"a60a373solverat14a-h1","type":"hint","dependencies":[],"title":"Clear the fractions","text":"The first step is to clear the fractions by multiplying the entire equation by the lowest common denominator: A","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat14a-h2","type":"hint","dependencies":["a60a373solverat14a-h1"],"title":"Isolate A","text":"Isolate A by dividing the equation by M","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat14a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$A=\\\\frac{F}{M}$$"],"dependencies":["a60a373solverat14a-h2"],"title":"Isolate A","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a60a373solverat15","title":"Solving a Rational Equation","body":"Solve for a in terms of $$b$$ and c","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Rational Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a60a373solverat15a","stepAnswer":["$$a=\\\\frac{b}{1-bc}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{1}{a}+\\\\frac{1}{b}=c$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$a=\\\\frac{b}{1-bc}$$","hints":{"DefaultPathway":[{"id":"a60a373solverat15a-h1","type":"hint","dependencies":[],"title":"Clear the fractions","text":"The first step is to clear the fractions by multiplying the entire equation by the lowest common denominator: ab","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat15a-h2","type":"hint","dependencies":["a60a373solverat15a-h1"],"title":"Isolate a","text":"To isolate a, we must first bring all a values to one side and then factoring out the a values","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$a(1-bc)=b$$"],"dependencies":["a60a373solverat15a-h2"],"title":"Isolate a","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat15a-h4","type":"hint","dependencies":["a60a373solverat15a-h3"],"title":"Isolate a","text":"Divide the entire equation by $$(1-bc)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat15a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$a=\\\\frac{b}{1-bc}$$"],"dependencies":["a60a373solverat15a-h4"],"title":"Isolate a","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a60a373solverat16","title":"Solving a Rational Equation","body":"Solve for a. Please input the answer as $$variable=answer$$.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Rational Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a60a373solverat16a","stepAnswer":["$$a=10$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{1}{a}+\\\\frac{2}{5}=\\\\frac{1}{2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$a=10$$","hints":{"DefaultPathway":[{"id":"a60a373solverat16a-h1","type":"hint","dependencies":[],"title":"Lowest Common Denominator","text":"The first step is to find the lowest common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["10a"],"dependencies":["a60a373solverat16a-h1"],"title":"Lowest Common Denominator","text":"What is the lowest common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat16a-h3","type":"hint","dependencies":["a60a373solverat16a-h2"],"title":"Distribute and multiply","text":"Distribute and multiply the lowest common denominator to all the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat16a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10+4a=5a$$"],"dependencies":["a60a373solverat16a-h3"],"title":"Distribute and multiply","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat16a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$a=10$$"],"dependencies":["a60a373solverat16a-h4"],"title":"Simple Algebra","text":"Solve for a. Please input the answer as $$variable=answer$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat16a-h6","type":"hint","dependencies":["a60a373solverat16a-h5"],"title":"Double Check","text":"Plug answers for a back into the original equation to make the sure answer is correct and not undefined at any point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a60a373solverat17","title":"Solving a Rational Equation","body":"Solve for $$b$$. Please input the answer as $$variable=answer$$.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Rational Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a60a373solverat17a","stepAnswer":["$$b=-6$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{5}{6}+\\\\frac{3}{b}=\\\\frac{1}{3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$b=-6$$","hints":{"DefaultPathway":[{"id":"a60a373solverat17a-h1","type":"hint","dependencies":[],"title":"Lowest Common Denominator","text":"The first step is to find the lowest common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat17a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6b$$"],"dependencies":["a60a373solverat17a-h1"],"title":"Lowest Common Denominator","text":"What is the lowest common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat17a-h3","type":"hint","dependencies":["a60a373solverat17a-h2"],"title":"Distribute and multiply","text":"Distribute and multiply the lowest common denominator to all the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat17a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5b+18=2b$$"],"dependencies":["a60a373solverat17a-h3"],"title":"Distribute and multiply","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat17a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$b=-6$$"],"dependencies":["a60a373solverat17a-h4"],"title":"Simple Algebra","text":"Solve for $$b$$. Please input the answer as $$variable=answer$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat17a-h6","type":"hint","dependencies":["a60a373solverat17a-h5"],"title":"Double Check","text":"Plug answers for a back into the original equation to make the sure answer is correct and not undefined at any point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a60a373solverat18","title":"Solving a Rational Equation","body":"Solve for c. Please input the answer as $$variable=answer$$.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Rational Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a60a373solverat18a","stepAnswer":["$$c=\\\\frac{4}{7}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{5}{2}-\\\\frac{1}{c}=\\\\frac{3}{4}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$c=\\\\frac{4}{7}$$","hints":{"DefaultPathway":[{"id":"a60a373solverat18a-h1","type":"hint","dependencies":[],"title":"Lowest Common Denominator","text":"The first step is to find the lowest common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["4c"],"dependencies":["a60a373solverat18a-h1"],"title":"Lowest Common Denominator","text":"What is the lowest common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat18a-h3","type":"hint","dependencies":["a60a373solverat18a-h2"],"title":"Distribute and multiply","text":"Distribute and multiply the lowest common denominator to all the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat18a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10c-4=3c$$"],"dependencies":["a60a373solverat18a-h3"],"title":"Distribute and multiply","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat18a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$c=\\\\frac{4}{7}$$"],"dependencies":["a60a373solverat18a-h4"],"title":"Simple Algebra","text":"Solve for c. Please input the answer as $$variable=answer$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat18a-h6","type":"hint","dependencies":["a60a373solverat18a-h5"],"title":"Double Check","text":"Plug answers for a back into the original equation to make the sure answer is correct and not undefined at any point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a60a373solverat19","title":"Solving a Rational Equation","body":"Solve for $$d$$. Please input the answer as $$variable=answer$$.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Rational Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a60a373solverat19a","stepAnswer":["$$d=\\\\frac{9}{7}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{6}{3}-\\\\frac{2}{d}=\\\\frac{4}{9}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$d=\\\\frac{9}{7}$$","hints":{"DefaultPathway":[{"id":"a60a373solverat19a-h1","type":"hint","dependencies":[],"title":"Lowest Common Denominator","text":"The first step is to find the lowest common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9d$$"],"dependencies":["a60a373solverat19a-h1"],"title":"Lowest Common Denominator","text":"What is the lowest common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat19a-h3","type":"hint","dependencies":["a60a373solverat19a-h2"],"title":"Distribute and multiply","text":"Distribute and multiply the lowest common denominator to all the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat19a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18d-18=4d$$"],"dependencies":["a60a373solverat19a-h3"],"title":"Distribute and multiply","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat19a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$d=\\\\frac{9}{7}$$"],"dependencies":["a60a373solverat19a-h4"],"title":"Simple Algebra","text":"Solve for $$d$$. Please input the answer as $$variable=answer$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat19a-h6","type":"hint","dependencies":["a60a373solverat19a-h5"],"title":"Double Check","text":"Plug answers for a back into the original equation to make the sure answer is correct and not undefined at any point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a60a373solverat2","title":"Solving a Rational Equation","body":"Solve for $$b$$. Please input the answer as $$variable=answer$$. If there are multiple answers, separate them with a comma and a space.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Rational Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a60a373solverat2a","stepAnswer":["b=6, b=-2"],"problemType":"TextBox","stepTitle":"$$1-\\\\frac{4}{b}=\\\\frac{12}{b^2}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$b=6$$, $$b=-2$$","hints":{"DefaultPathway":[{"id":"a60a373solverat2a-h1","type":"hint","dependencies":[],"title":"Lowest Common Denominator","text":"The first step is to find the lowest common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$b^2$$"],"dependencies":["a60a373solverat2a-h1"],"title":"Lowest Common Denominator","text":"What is the lowest common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat2a-h3","type":"hint","dependencies":["a60a373solverat2a-h2"],"title":"Distribute and multiply","text":"Distribute and multiply the lowest common denominator to all the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat2a-h4","type":"hint","dependencies":["a60a373solverat2a-h3"],"title":"Quadratic Form","text":"Rearrange the equation into the quadratic form and factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat2a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(b-6\\\\right) \\\\left(b+2\\\\right)=0$$"],"dependencies":["a60a373solverat2a-h4"],"title":"Quadratic Form","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat2a-h6","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["b=6, b=-2"],"dependencies":["a60a373solverat2a-h5"],"title":"Zero Product Property","text":"Solve for $$b$$. Please input the answer as $$variable=answer$$. If there are multiple answers, separate them with a comma and a space.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat2a-h7","type":"hint","dependencies":["a60a373solverat2a-h6"],"title":"Double Check","text":"Plug answers for a back into the original equation to make the sure answer is correct and not undefined at any point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a60a373solverat20","title":"Solving a Rational Equation","body":"Solve for v. Please input the answer as $$variable=answer$$.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Rational Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a60a373solverat20a","stepAnswer":["$$v=\\\\frac{40}{21}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{4}{5}+\\\\frac{1}{4}=\\\\frac{2}{v}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$v=\\\\frac{40}{21}$$","hints":{"DefaultPathway":[{"id":"a60a373solverat20a-h1","type":"hint","dependencies":[],"title":"Lowest Common Denominator","text":"The first step is to find the lowest common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["20v"],"dependencies":["a60a373solverat20a-h1"],"title":"Lowest Common Denominator","text":"What is the lowest common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat20a-h3","type":"hint","dependencies":["a60a373solverat20a-h2"],"title":"Distribute and multiply","text":"Distribute and multiply the lowest common denominator to all the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat20a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16v+5v=40$$"],"dependencies":["a60a373solverat20a-h3"],"title":"Distribute and multiply","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat20a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$v=\\\\frac{40}{21}$$"],"dependencies":["a60a373solverat20a-h4"],"title":"Simple Algebra","text":"Solve for v. Please input the answer as $$variable=answer$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat20a-h6","type":"hint","dependencies":["a60a373solverat20a-h5"],"title":"Double Check","text":"Plug answers for a back into the original equation to make the sure answer is correct and not undefined at any point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a60a373solverat21","title":"Solving a Rational Equation","body":"Solve for w. Please input the answer as $$variable=answer$$.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Rational Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a60a373solverat21a","stepAnswer":["$$w=\\\\frac{21}{23}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{3}{7}+\\\\frac{2}{3}=\\\\frac{1}{w}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$w=\\\\frac{21}{23}$$","hints":{"DefaultPathway":[{"id":"a60a373solverat21a-h1","type":"hint","dependencies":[],"title":"Lowest Common Denominator","text":"The first step is to find the lowest common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat21a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["21w"],"dependencies":["a60a373solverat21a-h1"],"title":"Lowest Common Denominator","text":"What is the lowest common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat21a-h3","type":"hint","dependencies":["a60a373solverat21a-h2"],"title":"Distribute and multiply","text":"Distribute and multiply the lowest common denominator to all the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat21a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9w+14w=21$$"],"dependencies":["a60a373solverat21a-h3"],"title":"Distribute and multiply","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat21a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$w=\\\\frac{21}{23}$$"],"dependencies":["a60a373solverat21a-h4"],"title":"Simple Algebra","text":"Solve for w. Please input the answer as $$variable=answer$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat21a-h6","type":"hint","dependencies":["a60a373solverat21a-h5"],"title":"Double Check","text":"Plug answers for a back into the original equation to make the sure answer is correct and not undefined at any point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a60a373solverat22","title":"Solving a Rational Equation","body":"Solve for $$x$$. Please input the answer as $$variable=answer$$.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Rational Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a60a373solverat22a","stepAnswer":["$$x=-9$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{7}{9}+\\\\frac{1}{x}=\\\\frac{2}{3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x=-9$$","hints":{"DefaultPathway":[{"id":"a60a373solverat22a-h1","type":"hint","dependencies":[],"title":"Lowest Common Denominator","text":"The first step is to find the lowest common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat22a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9x$$"],"dependencies":["a60a373solverat22a-h1"],"title":"Lowest Common Denominator","text":"What is the lowest common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat22a-h3","type":"hint","dependencies":["a60a373solverat22a-h2"],"title":"Distribute and multiply","text":"Distribute and multiply the lowest common denominator to all the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat22a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7x+9=6x$$"],"dependencies":["a60a373solverat22a-h3"],"title":"Distribute and multiply","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat22a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=-9$$"],"dependencies":["a60a373solverat22a-h4"],"title":"Simple Algebra","text":"Solve for $$x$$. Please input the answer as $$variable=answer$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat22a-h6","type":"hint","dependencies":["a60a373solverat22a-h5"],"title":"Double Check","text":"Plug answers for a back into the original equation to make the sure answer is correct and not undefined at any point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a60a373solverat23","title":"Solving a Rational Equation","body":"Solve for $$y$$. Please input the answer as $$variable=answer$$.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Rational Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a60a373solverat23a","stepAnswer":["$$y=-16$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{3}{8}+\\\\frac{2}{y}=\\\\frac{1}{4}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=-16$$","hints":{"DefaultPathway":[{"id":"a60a373solverat23a-h1","type":"hint","dependencies":[],"title":"Lowest Common Denominator","text":"The first step is to find the lowest common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat23a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8y$$"],"dependencies":["a60a373solverat23a-h1"],"title":"Lowest Common Denominator","text":"What is the lowest common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat23a-h3","type":"hint","dependencies":["a60a373solverat23a-h2"],"title":"Distribute and multiply","text":"Distribute and multiply the lowest common denominator to all the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat23a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3y+16=2y$$"],"dependencies":["a60a373solverat23a-h3"],"title":"Distribute and multiply","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat23a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y=-16$$"],"dependencies":["a60a373solverat23a-h4"],"title":"Simple Algebra","text":"Solve for $$y$$. Please input the answer as $$variable=answer$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat23a-h6","type":"hint","dependencies":["a60a373solverat23a-h5"],"title":"Double Check","text":"Plug answers for a back into the original equation to make the sure answer is correct and not undefined at any point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a60a373solverat24","title":"Solving a Rational Equation","body":"Solve for $$m$$. Please input the answer as $$variable=answer$$.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Rational Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a60a373solverat24a","stepAnswer":["m=4, m=-2"],"problemType":"TextBox","stepTitle":"$$1-\\\\frac{2}{m}=\\\\frac{8}{m^2}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$m=4$$, $$m=-2$$","hints":{"DefaultPathway":[{"id":"a60a373solverat24a-h1","type":"hint","dependencies":[],"title":"Lowest Common Denominator","text":"The first step is to find the lowest common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat24a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$m^2$$"],"dependencies":["a60a373solverat24a-h1"],"title":"Lowest Common Denominator","text":"What is the lowest common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat24a-h3","type":"hint","dependencies":["a60a373solverat24a-h2"],"title":"Distribute and multiply","text":"Distribute and multiply the lowest common denominator to all the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat24a-h4","type":"hint","dependencies":["a60a373solverat24a-h3"],"title":"Quadratic Form","text":"Rearrange the equation into the quadratic form and factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat24a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(m-4\\\\right) \\\\left(m+2\\\\right)=0$$"],"dependencies":["a60a373solverat24a-h4"],"title":"Quadratic Form","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat24a-h6","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["m=4, m=-2"],"dependencies":["a60a373solverat24a-h5"],"title":"Zero Product Property","text":"Solve for $$m$$. Please input the answer as $$variable=answer$$. If there are two answers, please input the answer as $$variable=answer1$$, $$variable=answer2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat24a-h7","type":"hint","dependencies":["a60a373solverat24a-h6"],"title":"Double Check","text":"Plug answers for a back into the original equation to make the sure answer is correct and not undefined at any point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a60a373solverat25","title":"Solving a Rational Equation","body":"Solve for $$n$$. Please input the answer as $$variable=answer$$. If there are two answers, please input the answer as $$variable=answer1$$, $$variable=answer2$$.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Rational Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a60a373solverat25a","stepAnswer":["n=-7, n=3"],"problemType":"TextBox","stepTitle":"$$1+\\\\frac{4}{n}=\\\\frac{21}{n^2}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$n=-7$$, $$n=3$$","hints":{"DefaultPathway":[{"id":"a60a373solverat25a-h1","type":"hint","dependencies":[],"title":"Lowest Common Denominator","text":"The first step is to find the lowest common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat25a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$n^2$$"],"dependencies":["a60a373solverat25a-h1"],"title":"Lowest Common Denominator","text":"What is the lowest common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat25a-h3","type":"hint","dependencies":["a60a373solverat25a-h2"],"title":"Distribute and multiply","text":"Distribute and multiply the lowest common denominator to all the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat25a-h4","type":"hint","dependencies":["a60a373solverat25a-h3"],"title":"Quadratic Form","text":"Rearrange the equation into the quadratic form and factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat25a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(n+7\\\\right) \\\\left(n-3\\\\right)=0$$"],"dependencies":["a60a373solverat25a-h4"],"title":"Quadratic Form","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat25a-h6","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["n=-7, n=3"],"dependencies":["a60a373solverat25a-h5"],"title":"Zero Product Property","text":"Solve for $$n$$. Please input the answer as $$variable=answer$$. If there are two answers, please input the answer as $$variable=answer1$$, $$variable=answer2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat25a-h7","type":"hint","dependencies":["a60a373solverat25a-h6"],"title":"Double Check","text":"Plug answers for a back into the original equation to make the sure answer is correct and not undefined at any point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a60a373solverat26","title":"Solving a Rational Equation","body":"Solve for $$p$$. Please input the answer as $$variable=answer$$. If there are two answers, please input the answer as $$variable=answer1$$, $$variable=answer2$$.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Rational Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a60a373solverat26a","stepAnswer":["p=-4, p=-5"],"problemType":"TextBox","stepTitle":"$$1+\\\\frac{9}{p}=\\\\left(-\\\\frac{20}{p^2}\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$p=-4$$, $$p=-5$$","hints":{"DefaultPathway":[{"id":"a60a373solverat26a-h1","type":"hint","dependencies":[],"title":"Lowest Common Denominator","text":"The first step is to find the lowest common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat26a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$p^2$$"],"dependencies":["a60a373solverat26a-h1"],"title":"Lowest Common Denominator","text":"What is the lowest common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat26a-h3","type":"hint","dependencies":["a60a373solverat26a-h2"],"title":"Distribute and multiply","text":"Distribute and multiply the lowest common denominator to all the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat26a-h4","type":"hint","dependencies":["a60a373solverat26a-h3"],"title":"Quadratic Form","text":"Rearrange the equation into the quadratic form and factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat26a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(p+5\\\\right) \\\\left(p+4\\\\right)=0$$"],"dependencies":["a60a373solverat26a-h4"],"title":"Quadratic Form","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat26a-h6","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["p=-4, p=-5"],"dependencies":["a60a373solverat26a-h5"],"title":"Zero Product Property","text":"Solve for $$p$$. Please input the answer as $$variable=answer$$. If there are two answers, please input the answer as $$variable=answer1$$, $$variable=answer2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat26a-h7","type":"hint","dependencies":["a60a373solverat26a-h6"],"title":"Double Check","text":"Plug answers for a back into the original equation to make the sure answer is correct and not undefined at any point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a60a373solverat27","title":"Solving a Rational Equation","body":"Solve for q. Please input the answer as $$variable=answer$$. If there are two answers, please input the answer as $$variable=answer1$$, $$variable=answer2$$.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Rational Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a60a373solverat27a","stepAnswer":["q=6, q=1"],"problemType":"TextBox","stepTitle":"$$1-\\\\frac{7}{q}=\\\\left(-\\\\frac{6}{q^2}\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$q=6$$, $$q=1$$","hints":{"DefaultPathway":[{"id":"a60a373solverat27a-h1","type":"hint","dependencies":[],"title":"Lowest Common Denominator","text":"The first step is to find the lowest common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat27a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$q^2$$"],"dependencies":["a60a373solverat27a-h1"],"title":"Lowest Common Denominator","text":"What is the lowest common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat27a-h3","type":"hint","dependencies":["a60a373solverat27a-h2"],"title":"Distribute and multiply","text":"Distribute and multiply the lowest common denominator to all the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat27a-h4","type":"hint","dependencies":["a60a373solverat27a-h3"],"title":"Quadratic Form","text":"Rearrange the equation into the quadratic form and factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat27a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$(q-6)(q-1)=0$$"],"dependencies":["a60a373solverat27a-h4"],"title":"Quadratic Form","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat27a-h6","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["q=6, q=1"],"dependencies":["a60a373solverat27a-h5"],"title":"Zero Product Property","text":"Solve for q. Please input the answer as $$variable=answer$$. If there are two answers, please input the answer as $$variable=answer1$$, $$variable=answer2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat27a-h7","type":"hint","dependencies":["a60a373solverat27a-h6"],"title":"Double Check","text":"Plug answers for a back into the original equation to make the sure answer is correct and not undefined at any point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a60a373solverat28","title":"Solving a Rational Equation","body":"Solve for $$r$$. Please input the answer as $$variable=answer$$.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Rational Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a60a373solverat28a","stepAnswer":["$$r=-6$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{1}{r+3}=\\\\frac{4}{2} r$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$r=-6$$","hints":{"DefaultPathway":[{"id":"a60a373solverat28a-h1","type":"hint","dependencies":[],"title":"Lowest Common Denominator","text":"The first step is to find the lowest common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat28a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2r \\\\left(r+3\\\\right)$$"],"dependencies":["a60a373solverat28a-h1"],"title":"Lowest Common Denominator","text":"What is the lowest common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat28a-h3","type":"hint","dependencies":["a60a373solverat28a-h2"],"title":"Distribute and multiply","text":"Distribute and multiply the lowest common denominator to all the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat28a-h4","type":"hint","dependencies":["a60a373solverat28a-h3"],"title":"Quadratic Form","text":"Rearrange the equation into the quadratic form and factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat28a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2r=4r+12$$"],"dependencies":["a60a373solverat28a-h4"],"title":"Quadratic Form","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat28a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$r=-6$$"],"dependencies":["a60a373solverat28a-h5"],"title":"Zero Product Property","text":"Solve for $$r$$. Please input the answer as $$variable=answer$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat28a-h7","type":"hint","dependencies":["a60a373solverat28a-h6"],"title":"Double Check","text":"Plug answers for a back into the original equation to make the sure answer is correct and not undefined at any point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a60a373solverat3","title":"Solving a Rational Equation","body":"Solve for $$x$$. Please input the answer as $$variable=answer$$. If there are multiple answers, separate them with a comma and a space.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Rational Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a60a373solverat3a","stepAnswer":["$$x=-2$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{1}{x-1}=\\\\frac{2}{3x}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x=-2$$","hints":{"DefaultPathway":[{"id":"a60a373solverat3a-h1","type":"hint","dependencies":[],"title":"Lowest Common Denominator","text":"The first step is to find the lowest common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$(3x)(x-1)$$"],"dependencies":["a60a373solverat3a-h1"],"title":"Lowest Common Denominator","text":"What is the lowest common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat3a-h3","type":"hint","dependencies":["a60a373solverat3a-h2"],"title":"Distribute and multiply","text":"Distribute and multiply the lowest common denominator to all the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x=2x-2$$"],"dependencies":["a60a373solverat3a-h3"],"title":"Distribute and multiply","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=-2$$"],"dependencies":["a60a373solverat3a-h4"],"title":"Simple Algebra","text":"Solve for $$x$$. Please input the answer as $$variable=answer$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat3a-h6","type":"hint","dependencies":["a60a373solverat3a-h5"],"title":"Double Check","text":"Plug answers for a back into the original equation to make the sure answer is correct and not undefined at any point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a60a373solverat4","title":"Solving a Rational Equation","body":"Solve for $$n$$. Please input the answer as $$variable=answer$$. If there are multiple answers, separate them with a comma and a space.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Rational Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a60a373solverat4a","stepAnswer":["$$n=-2$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{3}{5n+1}=\\\\frac{2}{3n}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$n=-2$$","hints":{"DefaultPathway":[{"id":"a60a373solverat4a-h1","type":"hint","dependencies":[],"title":"Lowest Common Denominator","text":"The first step is to find the lowest common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3n \\\\left(5n+1\\\\right)$$"],"dependencies":["a60a373solverat4a-h1"],"title":"Lowest Common Denominator","text":"What is the lowest common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat4a-h3","type":"hint","dependencies":["a60a373solverat4a-h2"],"title":"Distribute and multiply","text":"Distribute and multiply the lowest common denominator to all the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9n=10n+2$$"],"dependencies":["a60a373solverat4a-h3"],"title":"Distribute and multiply","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat4a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$n=-2$$"],"dependencies":["a60a373solverat4a-h4"],"title":"Simple Algebra","text":"Solve for $$n$$. Please input the answer as $$variable=answer$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat4a-h6","type":"hint","dependencies":["a60a373solverat4a-h5"],"title":"Double Check","text":"Plug answers for a back into the original equation to make the sure answer is correct and not undefined at any point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a60a373solverat5","title":"Solving a Rational Equation","body":"Solve for $$x$$. Please input the answer as $$variable=answer$$. If there are multiple answers, separate them with a comma and a space.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Rational Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a60a373solverat5a","stepAnswer":["$$x=\\\\frac{2}{3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2}{x+1}+\\\\frac{1}{x-1}=\\\\frac{1}{\\\\left(x+1\\\\right) \\\\left(x-1\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x=\\\\frac{2}{3}$$","hints":{"DefaultPathway":[{"id":"a60a373solverat5a-h1","type":"hint","dependencies":[],"title":"Lowest Common Denominator","text":"The first step is to find the lowest common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x+1\\\\right) \\\\left(x-1\\\\right)$$"],"dependencies":["a60a373solverat5a-h1"],"title":"Lowest Common Denominator","text":"What is the lowest common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat5a-h3","type":"hint","dependencies":["a60a373solverat5a-h2"],"title":"Distribute and multiply","text":"Distribute and multiply the lowest common denominator to all the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x-2+x+1=1$$"],"dependencies":["a60a373solverat5a-h3"],"title":"Distribute and multiply","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=\\\\frac{2}{3}$$"],"dependencies":["a60a373solverat5a-h4"],"title":"Simple Algebra","text":"Solve for $$x$$. Please input the answer as $$variable=answer$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat5a-h6","type":"hint","dependencies":["a60a373solverat5a-h5"],"title":"Double Check","text":"Plug answers for a back into the original equation to make the sure answer is correct and not undefined at any point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a60a373solverat6","title":"Solving a Rational Equation","body":"Solve for $$y$$. Please input the answer as $$variable=answer$$. If there are multiple answers, separate them with a comma and a space.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Rational Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a60a373solverat6a","stepAnswer":["$$y=2$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{5}{y+3}+\\\\frac{2}{y-3}=\\\\frac{5}{y^2-9}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=2$$","hints":{"DefaultPathway":[{"id":"a60a373solverat6a-h1","type":"hint","dependencies":[],"title":"Lowest Common Denominator","text":"The first step is to find the lowest common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y^2-9$$"],"dependencies":["a60a373solverat6a-h1"],"title":"Lowest Common Denominator","text":"What is the lowest common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat6a-h3","type":"hint","dependencies":["a60a373solverat6a-h2"],"title":"Distribute and multiply","text":"Distribute and multiply the lowest common denominator to all the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5y-15+2y+6=5$$"],"dependencies":["a60a373solverat6a-h3"],"title":"Distribute and multiply","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y=2$$"],"dependencies":["a60a373solverat6a-h4"],"title":"Simple Algebra","text":"Solve for $$y$$. Please input the answer as $$variable=answer$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat6a-h6","type":"hint","dependencies":["a60a373solverat6a-h5"],"title":"Double Check","text":"Plug answers for a back into the original equation to make the sure answer is correct and not undefined at any point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a60a373solverat7","title":"Solving a Rational Equation","body":"Solve for $$x$$. Please input the answer as $$variable=answer$$. If there are multiple answers, separate them with a comma and a space.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Rational Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a60a373solverat7a","stepAnswer":["x=-2, x=-1"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2}{x+5}-\\\\frac{1}{x-1}=1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=-2$$, $$x=-1$$","hints":{"DefaultPathway":[{"id":"a60a373solverat7a-h1","type":"hint","dependencies":[],"title":"Lowest Common Denominator","text":"The first step is to find the lowest common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x+5\\\\right) \\\\left(x-1\\\\right)$$"],"dependencies":["a60a373solverat7a-h1"],"title":"Lowest Common Denominator","text":"What is the lowest common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat7a-h3","type":"hint","dependencies":["a60a373solverat7a-h2"],"title":"Distribute and multiply","text":"Distribute and multiply the lowest common denominator to all the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x-2-x-5=x^2+4x-5$$"],"dependencies":["a60a373solverat7a-h3"],"title":"Distribute and multiply","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat7a-h5","type":"hint","dependencies":["a60a373solverat7a-h4"],"title":"Quadratic Form","text":"Rearrange the equation into the quadratic form and factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat7a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x+2\\\\right) \\\\left(x+1\\\\right)=0$$"],"dependencies":["a60a373solverat7a-h5"],"title":"Quadratic Form","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat7a-h7","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["x=-2, x=-1"],"dependencies":["a60a373solverat7a-h6"],"title":"Zero Product Property","text":"Solve for $$x$$. Please input the answer as $$variable=answer$$. If there are two answers, input your answer in the format $$x=a$$, $$x=b$$ where a and $$b$$ are the values of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat7a-h8","type":"hint","dependencies":["a60a373solverat7a-h7"],"title":"Double Check","text":"Plug answers for a back into the original equation to make the sure answer is correct and not undefined at any point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a60a373solverat8","title":"Solving a Rational Equation","body":"Solve for $$x$$. Please input the answer as $$variable=answer$$. If there are multiple answers, separate them with a comma and a space.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Rational Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a60a373solverat8a","stepAnswer":["x=-3, x=2"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2}{x+8}-\\\\frac{2}{x-2}=1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=-3$$, $$x=2$$","hints":{"DefaultPathway":[{"id":"a60a373solverat8a-h1","type":"hint","dependencies":[],"title":"Lowest Common Denominator","text":"The first step is to find the lowest common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x+8\\\\right) \\\\left(x-2\\\\right)$$"],"dependencies":["a60a373solverat8a-h1"],"title":"Lowest Common Denominator","text":"What is the lowest common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat8a-h3","type":"hint","dependencies":["a60a373solverat8a-h2"],"title":"Distribute and multiply","text":"Distribute and multiply the lowest common denominator to all the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x-6-2x-16=x^2+6x+16$$"],"dependencies":["a60a373solverat8a-h3"],"title":"Distribute and multiply","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat8a-h5","type":"hint","dependencies":["a60a373solverat8a-h4"],"title":"Quadratic Form","text":"Rearrange the equation into the quadratic form and factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat8a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x+3\\\\right) \\\\left(x-2\\\\right)=0$$"],"dependencies":["a60a373solverat8a-h5"],"title":"Quadratic Form","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat8a-h7","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["x=-3, x=2"],"dependencies":["a60a373solverat8a-h6"],"title":"Zero Product Property","text":"Solve for $$x$$. If there are two answers, input your answer in the format $$x=a$$, $$x=b$$ where a and $$b$$ are the values of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat8a-h8","type":"hint","dependencies":["a60a373solverat8a-h7"],"title":"Double Check","text":"Plug answers for a back into the original equation to make the sure answer is correct and not undefined at any point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a60a373solverat9","title":"Solving a Rational Equation","body":"Solve for $$y$$. Please input the answer as $$variable=answer$$. If there are multiple answers, separate them with a comma and a space.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Rational Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a60a373solverat9a","stepAnswer":["$$y=-5$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{y-14}{y^2+3y-4}=\\\\frac{2}{y+4}+\\\\frac{7}{y-1}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=-5$$","hints":{"DefaultPathway":[{"id":"a60a373solverat9a-h1","type":"hint","dependencies":[],"title":"Lowest Common Denominator","text":"The first step is to find the lowest common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y^2+3y-4$$"],"dependencies":["a60a373solverat9a-h1"],"title":"Lowest Common Denominator","text":"What is the lowest common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat9a-h3","type":"hint","dependencies":["a60a373solverat9a-h2"],"title":"Distribute and multiply","text":"Distribute and multiply the lowest common denominator to all the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y-14=2y-2+7y+28$$"],"dependencies":["a60a373solverat9a-h3"],"title":"Distribute and multiply","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat9a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y=-5$$"],"dependencies":["a60a373solverat9a-h4"],"title":"Simple Algebra","text":"Solve for $$y$$. Please input the answer as $$variable=answer$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a60a373solverat9a-h6","type":"hint","dependencies":["a60a373solverat9a-h5"],"title":"Double Check","text":"Plug answers for a back into the original equation to make the sure answer is correct and not undefined at any point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a612a9eprop1","title":"Hive Medication","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Comparing Two Independent Population Proportions","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a612a9eprop1a","stepAnswer":["Insufficient Evidence"],"problemType":"MultipleChoice","stepTitle":"Two types of medication for hives are being tested to determine if there is a difference in the proportions of adult patient reactions. Twenty out of a random sample of $$200$$ adults given medication A still had hives $$30$$ minutes after taking the medication. Twelve out of another random sample of $$200$$ adults given medication B still had hives $$30$$ minutes after taking the medication. Test at a 1% level of significance.","stepBody":"","answerType":"string","variabilization":{},"choices":["Medication A is more effective","Medication B is more effective","Insufficient Evidence"],"hints":{"DefaultPathway":[{"id":"a612a9eprop1a-h1","type":"hint","dependencies":[],"title":"Type of Test","text":"Test of Two Proportions","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop1a-h2","type":"hint","dependencies":["a612a9eprop1a-h1"],"title":"One-tailed or Two-tailed?","text":"This is a two-tailed test as the question asks for the difference in proportions","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1404$$"],"dependencies":["a612a9eprop1a-h2"],"title":"P-value","text":"What is the $$p-value$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop1a-s1","type":"hint","dependencies":["a612a9eprop1a-h3"],"title":"Calculating P-value","text":"In order to calculate the $$p-value$$, you can use the 2PropZTest function on your calculator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a612a9eprop10","title":"Shopping for Electronics","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Comparing Two Independent Population Proportions","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a612a9eprop10a","stepAnswer":["Insufficient Evidence"],"problemType":"MultipleChoice","stepTitle":"While her husband spent 2\xbd hours picking out new speakers, a statistician decided to determine whether the percent of men who enjoy shopping for electronic equipment is higher than the percent of women who enjoy shopping for electronic equipment. The population was Saturday afternoon shoppers. Out of $$67$$ men, $$24$$ said they enjoyed the activity. Eight of the $$24$$ women surveyed claimed to enjoy the activity. Interpret the results of the survey.","stepBody":"","answerType":"string","variabilization":{},"choices":["A greater proportion of men enjoy shopping for electronic equipment than women","Insufficient Evidence"],"hints":{"DefaultPathway":[{"id":"a612a9eprop10a-h1","type":"hint","dependencies":[],"title":"Type of Test","text":"Test of Two Proportions","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop10a-h2","type":"hint","dependencies":["a612a9eprop10a-h1"],"title":"One-tailed or Two-tailed?","text":"More than implies that the test is right-tailed","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.4133$$"],"dependencies":["a612a9eprop10a-h2"],"title":"P-value","text":"What is the $$p-value$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop10a-s1","type":"hint","dependencies":["a612a9eprop10a-h3"],"title":"Calculating P-value","text":"In order to calculate the $$p-value$$, you can use the 2PropZTest function on your calculator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a612a9eprop11","title":"Ear Piercings","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Comparing Two Independent Population Proportions","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a612a9eprop11a","stepAnswer":["There is a difference in the proportion of men and women that have at least one piercing"],"problemType":"MultipleChoice","stepTitle":"Joan Nguyen recently claimed that the proportion of college-age males with at least one pierced ear is as high as the proportion of college-age females. She conducted a survey in her classes. Out of $$107$$ males, $$20$$ had at least one pierced ear. Out of $$92$$ females, $$47$$ had at least one pierced ear. Do you believe that the proportion of males has reached the proportion of females?","stepBody":"","answerType":"string","variabilization":{},"choices":["There is a difference in the proportion of men and women that have at least one piercing","Insufficient Evidence"],"hints":{"DefaultPathway":[{"id":"a612a9eprop11a-h1","type":"hint","dependencies":[],"title":"Type of Test","text":"Test of Two Proportions","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop11a-h2","type":"hint","dependencies":["a612a9eprop11a-h1"],"title":"One-tailed or Two-tailed?","text":"As high as implies that the test is two-tailed","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop11a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a612a9eprop11a-h2"],"title":"P-value","text":"What is the $$p-value$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop11a-s1","type":"hint","dependencies":["a612a9eprop11a-h3"],"title":"Calculating P-value","text":"In order to calculate the $$p-value$$, you can use the 2PropZTest function on your calculator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a612a9eprop12","title":"State Demographics","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Comparing Two Independent Population Proportions","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a612a9eprop12a","stepAnswer":["Nevada has a greater proportion of people of two or more races than North Dakota"],"problemType":"MultipleChoice","stepTitle":"In the recent Census, three percent of the U.S. population reported being of two or more races. However, the percent varies tremendously from state to state. Suppose that two random surveys are conducted. In the first random survey, out of 1,000 North Dakotans, only nine people reported being of two or more races. In the second random survey, out of $$500$$ Nevadans, $$17$$ people reported being of two or more races. Conduct a hypothesis test to determine if the population percents are the same for the two states or if the percent for Nevada is statistically higher than for North Dakota.","stepBody":"","answerType":"string","variabilization":{},"choices":["Nevada has a greater proportion of people of two or more races than North Dakota","Insufficient Evidence"],"hints":{"DefaultPathway":[{"id":"a612a9eprop12a-h1","type":"hint","dependencies":[],"title":"Type of Test","text":"Test of Two Proportions","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop12a-h2","type":"hint","dependencies":["a612a9eprop12a-h1"],"title":"One-tailed or Two-tailed?","text":"Higher than implies that the test is right-tailed","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop12a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.00023$$"],"dependencies":["a612a9eprop12a-h2"],"title":"P-value","text":"What is the $$p-value$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop12a-s1","type":"hint","dependencies":["a612a9eprop12a-h3"],"title":"Calculating P-value","text":"In order to calculate the $$p-value$$, you can use the 2PropZTest function on your calculator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a612a9eprop13","title":"Operating System Crashes","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Comparing Two Independent Population Proportions","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a612a9eprop13a","stepAnswer":["Insufficient Evidence"],"problemType":"MultipleChoice","stepTitle":"Two types of phone operating system are being tested to determine if there is a difference in the proportions of system failures (crashes). Fifteen out of a random sample of $$150$$ phones with OS1 had system failures within the first eight hours of operation. Nine out of another random sample of $$150$$ phones with OS2 had system failures within the first eight hours of operation. OS2 is believed to be more stable (have fewer crashes) than OS1.","stepBody":"","answerType":"string","variabilization":{},"choices":["OS1 has a greater proportion of system crashes than OS2","Insufficient Evidence"],"hints":{"DefaultPathway":[{"id":"a612a9eprop13a-h1","type":"hint","dependencies":[],"title":"Type of Test","text":"Test of Two Proportions","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop13a-h2","type":"hint","dependencies":["a612a9eprop13a-h1"],"title":"One-tailed or Two-tailed?","text":"The test is right-tailed","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop13a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1018$$"],"dependencies":["a612a9eprop13a-h2"],"title":"P-value","text":"What is the $$p-value$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop13a-s1","type":"hint","dependencies":["a612a9eprop13a-h3"],"title":"Calculating P-value","text":"In order to calculate the $$p-value$$, you can use the 2PropZTest function on your calculator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a612a9eprop14","title":"Neuroinvasive West Nile Virus","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Comparing Two Independent Population Proportions","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a612a9eprop14a","stepAnswer":["There is sufficient evidence to conclude that the proportion of people in the United States in $$2011$$ who contracted neuroinvasive West Nile disease is more than the proportion of people in the United States in $$2010$$ who contracted neuroinvasive West Nile disease."],"problemType":"MultipleChoice","stepTitle":"In the United States in $$2010$$ there were $$629$$ reported cases of neuroinvasive West Nile virus out of a total of 1,021 reported cases and there were $$486$$ neuroinvasive reported cases out of a total of $$712$$ cases reported in $$2011$$. Is the $$2011$$ proportion of neuroinvasive West Nile virus cases more than the $$2010$$ proportion of neuroinvasive West Nile virus cases? Using a 1% level of significance, conduct an appropriate hypothesis test.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"There is sufficient evidence to conclude that the proportion of people in the United States in $$2011$$ who contracted neuroinvasive West Nile disease is more than the proportion of people in the United States in $$2010$$ who contracted neuroinvasive West Nile disease.","choices":["Insufficient Evidence","There is sufficient evidence to conclude that the proportion of people in the United States in $$2011$$ who contracted neuroinvasive West Nile disease is more than the proportion of people in the United States in $$2010$$ who contracted neuroinvasive West Nile disease","There is sufficient evidence to conclude that the proportion of people in the United States in $$2011$$ who contracted neuroinvasive West Nile disease is more than the proportion of people in the United States in $$2010$$ who contracted neuroinvasive West Nile disease."],"hints":{"DefaultPathway":[{"id":"a612a9eprop14a-h1","type":"hint","dependencies":[],"title":"Type of Test","text":"Test of Two Proportions","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop14a-h2","type":"hint","dependencies":["a612a9eprop14a-h1"],"title":"One-tailed or Two-tailed?","text":"More than implies that the test is right-tailed","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop14a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.0022$$"],"dependencies":["a612a9eprop14a-h2"],"title":"P-value","text":"What is the $$p-value$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop14a-s1","type":"hint","dependencies":["a612a9eprop14a-h3"],"title":"Calculating P-value","text":"In order to calculate the $$p-value$$, you can use the 2PropZTest function on your calculator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a612a9eprop15","title":"Chocolate Bar","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Comparing Two Independent Population Proportions","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a612a9eprop15a","stepAnswer":["A greater proportion of children like the chocolate bar than adults"],"problemType":"MultipleChoice","stepTitle":"A new chocolate bar is taste-tested on consumers. Of interest is whether the proportion of children who like the new chocolate bar is greater than the proportion of adults who like it. Of the $$95$$ children sampled, $$79$$ liked the chocolate bar and of the $$143$$ adults, $$92$$ liked the chocolate bar. Conduct a hypothesis test at a 5% significance level","stepBody":"","answerType":"string","variabilization":{},"choices":["A greater proportion of children like the chocolate bar than adults","Insufficient Evidence"],"hints":{"DefaultPathway":[{"id":"a612a9eprop15a-h1","type":"hint","dependencies":[],"title":"Type of Test","text":"Test of Two Proportions","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop15a-h2","type":"hint","dependencies":["a612a9eprop15a-h1"],"title":"One-tailed or Two-tailed?","text":"Greater than implies that the test is right-tailed","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.00079$$"],"dependencies":["a612a9eprop15a-h2"],"title":"P-value","text":"What is the $$p-value$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop15a-s1","type":"hint","dependencies":["a612a9eprop15a-h3"],"title":"Calculating P-value","text":"In order to calculate the $$p-value$$, you can use the 2PropZTest function on your calculator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a612a9eprop3","title":"Adult Smartphone Usage (Race)","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Comparing Two Independent Population Proportions","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a612a9eprop3a","stepAnswer":["White cellphone users use iPhones more"],"problemType":"MultipleChoice","stepTitle":"Researchers conducted a study of smartphone use among adults. A cell phone company claimed that iPhone smartphones are more popular with White people $$(non-Hispanic)$$ than with African Americans. The results of the survey indicate that of the $$232$$ African American cell phone owners randomly sampled, 5% have an iPhone. Of the 1,343 White cell phone owners randomly sampled, 10% own an iPhone. Test at the 5% level of significance. Is the proportion of White iPhone owners greater than the proportion of African American iPhone owners?","stepBody":"","answerType":"string","variabilization":{},"choices":["White cellphone users use iPhones more","Insufficient Evidence"],"hints":{"DefaultPathway":[{"id":"a612a9eprop3a-h1","type":"hint","dependencies":[],"title":"Type of Test","text":"Test of Two Proportions","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop3a-h2","type":"hint","dependencies":["a612a9eprop3a-h1"],"title":"One-tailed or Two-tailed","text":"The words \\"more popular\\" indicate that the test is right-tailed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.0077$$"],"dependencies":["a612a9eprop3a-h2"],"title":"P-value","text":"What is the $$p-value$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop3a-s1","type":"hint","dependencies":["a612a9eprop3a-h3"],"title":"Calculating P-value","text":"In order to calculate the $$p-value$$, you can use the 2PropZTest function on your calculator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a612a9eprop4","title":"High School Drug Usage","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Comparing Two Independent Population Proportions","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a612a9eprop4a","stepAnswer":["Insufficient Evidence"],"problemType":"MultipleChoice","stepTitle":"A recent drug survey showed an increase in the use of drugs and alcohol among local high school seniors as compared to the national percent. Suppose that a survey of $$100$$ local seniors and $$100$$ national seniors is conducted to see if the proportion of drug and alcohol use is higher locally than nationally. Locally, $$65$$ seniors reported using drugs or alcohol within the past month, while $$60$$ national seniors reported using them. Conduct a hypothesis.","stepBody":"","answerType":"string","variabilization":{},"choices":["Local high schoolers use drugs at a higher proportion","Insufficient Evidence"],"hints":{"DefaultPathway":[{"id":"a612a9eprop4a-h1","type":"hint","dependencies":[],"title":"Type of Test","text":"Test of Two Proportions","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop4a-h2","type":"hint","dependencies":["a612a9eprop4a-h1"],"title":"One-tailed or Two-tailed","text":"The word increase implies that the test is left-tailed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.2326$$"],"dependencies":["a612a9eprop4a-h2"],"title":"P-value","text":"What is the $$p-value$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop4a-s1","type":"hint","dependencies":["a612a9eprop4a-h3"],"title":"Calculating P-value","text":"In order to calculate the $$p-value$$, you can use the 2PropZTest function on your calculator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a612a9eprop5","title":"Suicide proportions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Comparing Two Independent Population Proportions","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a612a9eprop5a","stepAnswer":["Insufficient Evidence"],"problemType":"MultipleChoice","stepTitle":"We are interested in whether the proportions of female suicide victims for ages $$15$$ to $$24$$ are the same for White and Black in the United States. We randomly pick one year, $$1992$$, to compare the races. The number of suicides estimated in the United States in $$1992$$ for White females is 4,930. Five hundred eighty were aged $$15$$ to $$24$$. The estimate for Black females is $$330$$. Forty were aged $$15$$ to $$24$$. We will let female suicide victims be our population.","stepBody":"","answerType":"string","variabilization":{},"choices":["Suicide is more prevelant in White females","Suicide is more prevelant in Black females","Insufficient Evidence"],"hints":{"DefaultPathway":[{"id":"a612a9eprop5a-h1","type":"hint","dependencies":[],"title":"Type of Test","text":"Test of Two Proportions","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop5a-h2","type":"hint","dependencies":["a612a9eprop5a-h1"],"title":"One-tailed or Two-tailed?","text":"This is a two-tailed test as the question asks for the difference in proportions","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.8458$$"],"dependencies":["a612a9eprop5a-h2"],"title":"P-value","text":"What is the $$p-value$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop5a-s1","type":"hint","dependencies":["a612a9eprop5a-h3"],"title":"Calculating P-value","text":"In order to calculate the $$p-value$$, you can use the 2PropZTest function on your calculator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a612a9eprop6","title":"College Demographics","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Comparing Two Independent Population Proportions","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a612a9eprop6a","stepAnswer":["Different proportions of Hispanic students"],"problemType":"MultipleChoice","stepTitle":"A recent year was randomly picked from $$1985$$ to the present. In that year, there were 2,051 Hispanic students at Cabrillo College out of a total of 12,328 students. At Lake Tahoe College, there were $$321$$ Hispanic students out of a total of 2,441 students. In general, do you think that the percent of Hispanic students at the two colleges is basically the same or different?","stepBody":"","answerType":"string","variabilization":{},"choices":["Different proportions of Hispanic students","Insufficient Evidence"],"hints":{"DefaultPathway":[{"id":"a612a9eprop6a-h1","type":"hint","dependencies":[],"title":"Type of Test","text":"Test of Two Proportions","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop6a-h2","type":"hint","dependencies":["a612a9eprop6a-h1"],"title":"One-tailed or Two-tailed?","text":"This is a two-tailed test as the question asks for the difference in proportions","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.00002$$"],"dependencies":["a612a9eprop6a-h2"],"title":"P-value","text":"What is the $$p-value$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop6a-s1","type":"hint","dependencies":["a612a9eprop6a-h3"],"title":"Calculating P-value","text":"In order to calculate the $$p-value$$, you can use the 2PropZTest function on your calculator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a612a9eprop7","title":"eReader Usage","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Comparing Two Independent Population Proportions","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a612a9eprop7a","stepAnswer":["eReader usage is different between $$16$$ to $$29$$ year olds and 30+ year olds"],"problemType":"MultipleChoice","stepTitle":"Researchers conducted a study to find out if there is a difference in the use of eReaders by different age groups. Randomly selected participants were divided into two age groups. In the 16- to 29-year-old group, 7% of the $$628$$ surveyed use eReaders, while 11% of the 2,309 participants $$30$$ years old and older use eReaders. Conduct a hypothesis test at significance level 1%.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"eReader usage is different between $$16$$ to $$29$$ year olds and 30+ year olds","choices":["eReader usage is different between $$16$$ to $$29$$ year olds and 30+ year olds","Insufficient Evidence"],"hints":{"DefaultPathway":[{"id":"a612a9eprop7a-h1","type":"hint","dependencies":[],"title":"Type of Test","text":"Test of Two Proportions","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop7a-h2","type":"hint","dependencies":["a612a9eprop7a-h1"],"title":"One-tailed or Two-tailed?","text":"This is a two-tailed test as the question asks for the difference in proportions","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.0033$$"],"dependencies":["a612a9eprop7a-h2"],"title":"P-value","text":"What is the $$p-value$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop7a-s1","type":"hint","dependencies":["a612a9eprop7a-h3"],"title":"Calculating P-value","text":"In order to calculate the $$p-value$$, you can use the 2PropZTest function on your calculator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a612a9eprop8","title":"Obesity vs Gender","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Comparing Two Independent Population Proportions","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a612a9eprop8a","stepAnswer":["Insufficient Evidence"],"problemType":"MultipleChoice","stepTitle":"Adults aged $$18$$ years old and older were randomly selected for a survey on obesity. Adults are considered obese if their body mass index (BMI) is at least $$30$$. The researchers wanted to determine if the proportion of women who are obese in the south is less than the proportion of southern men who are obese. The results are shown in the table","stepBody":"","answerType":"string","variabilization":{},"choices":["A lower proportion of southern women are obese compared to southern men","Insufficient Evidence"],"hints":{"DefaultPathway":[{"id":"a612a9eprop8a-h1","type":"hint","dependencies":[],"title":"Type of Test","text":"Test of Two Proportions","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop8a-h2","type":"hint","dependencies":["a612a9eprop8a-h1"],"title":"One-tailed or Two-tailed?","text":"Less than implies that the test is left-tailed","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.9997$$"],"dependencies":["a612a9eprop8a-h2"],"title":"P-value","text":"What is the $$p-value$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop8a-s1","type":"hint","dependencies":["a612a9eprop8a-h3"],"title":"Calculating P-value","text":"In order to calculate the $$p-value$$, you can use the 2PropZTest function on your calculator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a612a9eprop9","title":"Adult Smartphone Usage","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Comparing Two Independent Population Proportions","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a612a9eprop9a","stepAnswer":["A greater proportion of men use smartphones than women"],"problemType":"MultipleChoice","stepTitle":"A group of friends debated whether more men use smartphones than women. They consulted a research study of smartphone use among adults. The results of the survey indicate that of the $$973$$ men randomly sampled, $$379$$ use smartphones. For women, $$404$$ of the 1,304 who were randomly sampled use smartphones. Test at the 5% level of significance.","stepBody":"","answerType":"string","variabilization":{},"choices":["A greater proportion of men use smartphones than women","Insufficient Evidence"],"hints":{"DefaultPathway":[{"id":"a612a9eprop9a-h1","type":"hint","dependencies":[],"title":"Type of Test","text":"Test of Two Proportions","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop9a-h2","type":"hint","dependencies":["a612a9eprop9a-h1"],"title":"One-tailed or Two-tailed?","text":"More than implies that the test is right-tailed","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.00004$$"],"dependencies":["a612a9eprop9a-h2"],"title":"P-value","text":"What is the $$p-value$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a612a9eprop9a-s1","type":"hint","dependencies":["a612a9eprop9a-h3"],"title":"Calculating P-value","text":"In order to calculate the $$p-value$$, you can use the 2PropZTest function on your calculator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a61c721rational1","title":"Multiplying Rational Expressions","body":"Multiply the rational expressions and show the product in simplest form:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.6 Rational Expressions","courseName":"OpenStax: College Algebra","steps":[{"id":"a61c721rational1a","stepAnswer":["$$\\\\frac{\\\\left(x-1\\\\right) \\\\left(2x-1\\\\right)}{3\\\\left(x+6\\\\right)}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{x^2+4x-5}{3x+18} \\\\frac{2x-1}{x+5}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{\\\\left(x-1\\\\right) \\\\left(2x-1\\\\right)}{3\\\\left(x+6\\\\right)}$$","hints":{"DefaultPathway":[{"id":"a61c721rational1a-h1","type":"hint","dependencies":[],"title":"Factor the expression","text":"Factoring is the process to split a product into a group of factors (a multiplication of simpler expressions).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational1a-h2","type":"hint","dependencies":["a61c721rational1a-h1"],"title":"Factor the expression","text":"Not all terms can be factored. Factor the terms that can be split further.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational1a-h3","type":"hint","dependencies":["a61c721rational1a-h2"],"title":"Factor the expression","text":"In this case, we can factor $$x^2+4x-5$$ and $$3x+18$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x+5\\\\right) \\\\left(x-1\\\\right)$$"],"dependencies":["a61c721rational1a-h3"],"title":"Factor the expression","text":"What does $$x^2+4x-5$$ factor into?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational1a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3\\\\left(x+6\\\\right)$$"],"dependencies":["a61c721rational1a-h4"],"title":"Factor the expression","text":"What does $$3x+18$$ factor into?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational1a-h6","type":"hint","dependencies":["a61c721rational1a-h5"],"title":"Cancel terms","text":"Now that you have factored all possible terms, if you see the same term in the numerator and the denominator of the product, you can cancel it out.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational1a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+5$$"],"dependencies":["a61c721rational1a-h6"],"title":"Cancel terms","text":"What term can we cancel out?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational1a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\left(x-1\\\\right) \\\\left(2x-1\\\\right)}{3\\\\left(x+6\\\\right)}$$"],"dependencies":["a61c721rational1a-h7"],"title":"Final Answer","text":"After canceling out terms, your final answer will remain in product form. What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a61c721rational10","title":"Rational Expressions","body":"Simplify the rational expressions.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.6 Rational Expressions","courseName":"OpenStax: College Algebra","steps":[{"id":"a61c721rational10a","stepAnswer":["$$3b+3$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{9b^2+18b+9}{3b+3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3b+3$$","hints":{"DefaultPathway":[{"id":"a61c721rational10a-h1","type":"hint","dependencies":[],"title":"Factoring Trinomials","text":"Factor the trinomial in the numerator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9\\\\left(b+1\\\\right) \\\\left(b+1\\\\right)$$"],"dependencies":["a61c721rational10a-h1"],"title":"Factoring Trinomials","text":"What are the factors of numerator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3\\\\left(b+1\\\\right)$$"],"dependencies":["a61c721rational10a-h2"],"title":"Factoring Trinomials","text":"What are the factors of denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational15a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a61c721rational15a-h10"],"title":"Common Factor","text":"What is the simplified expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a61c721rational16","title":"Rational Expressions","body":"Divide the rational expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.6 Rational Expressions","courseName":"OpenStax: College 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4.0>"},{"id":"a61c721rational16a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(11y+2\\\\right) \\\\left(2y+5\\\\right)$$"],"dependencies":["a61c721rational16a-h2"],"title":"Factoring Trinomials","text":"What are the factors of the first numerator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational16a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(6y-1\\\\right) \\\\left(2y+5\\\\right)$$"],"dependencies":["a61c721rational16a-h3"],"title":"Factoring Trinomials","text":"What are the factors of the first denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational16a-h5","type":"hint","dependencies":["a61c721rational16a-h4"],"title":"Factoring Expressions","text":"Factor the numerator and denominator of the second expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational16a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(11y+2\\\\right) \\\\left(y+4\\\\right)$$"],"dependencies":["a61c721rational16a-h5"],"title":"Factoring Trinomials","text":"What are the factors of the second numerator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational16a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$(6y-1)(4y-1)$$"],"dependencies":["a61c721rational16a-h6"],"title":"Factoring Trinomials","text":"What are the factors of the second denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational16a-h8","type":"hint","dependencies":["a61c721rational16a-h7"],"title":"Multiplying Expressions","text":"Multilply numerators and denominators.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational16a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(11y+2\\\\right) \\\\left(2y+5\\\\right) \\\\left(11y+2\\\\right) \\\\left(y+4\\\\right)$$"],"dependencies":["a61c721rational16a-h8"],"title":"Simplifying Expressions","text":"After multiplying the factors, what is the numerator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational16a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(6y-1\\\\right) \\\\left(2y+5\\\\right) \\\\left(6y-1\\\\right) \\\\left(4y-1\\\\right)$$"],"dependencies":["a61c721rational16a-h9"],"title":"Simplifying Expressions","text":"After multiplying the factors, what is the denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational16a-h11","type":"hint","dependencies":["a61c721rational16a-h10"],"title":"Common Factor","text":"Cancel the common factors from the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational16a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{4y-1}{y+4}$$"],"dependencies":["a61c721rational16a-h11"],"title":"Common Factor","text":"What is the simplified expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a61c721rational17","title":"Rational Expressions","body":"Add and subtract the rational expression, and then simplify.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.6 Rational Expressions","courseName":"OpenStax: College Algebra","steps":[{"id":"a61c721rational17a","stepAnswer":["$$\\\\frac{4y+10x}{xy}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{4}{x}+\\\\frac{10}{y}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{4y+10x}{xy}$$","hints":{"DefaultPathway":[{"id":"a61c721rational17a-h1","type":"hint","dependencies":[],"title":"The Least Common Denominator","text":"Find the LCD.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational17a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["xy"],"dependencies":["a61c721rational17a-h1"],"title":"The Least Common Denominator","text":"What is the LCD?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational17a-h3","type":"hint","dependencies":["a61c721rational17a-h2"],"title":"Common Denominator","text":"Multiply each expression by the appropriate form of $$1$$ to obtain xy as the denominator for each fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational17a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{4y}{xy}$$"],"dependencies":["a61c721rational17a-h3"],"title":"Common Denominator","text":"What is the first fraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational17a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{10x}{xy}$$"],"dependencies":["a61c721rational17a-h4"],"title":"Common Denominator","text":"What is the second fraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational17a-h6","type":"hint","dependencies":["a61c721rational17a-h5"],"title":"Adding Fractions","text":"Now that the expressions have the same denominator, we simply add the numerators to find the sum.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational17a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{4y+10x}{xy}$$"],"dependencies":["a61c721rational17a-h6"],"title":"Adding Fractions","text":"What is the final fraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a61c721rational18","title":"Adding Rational Expressions","body":"Add the rational expressions and simplify.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.6 Rational Expressions","courseName":"OpenStax: College Algebra","steps":[{"id":"a61c721rational18a","stepAnswer":["$$\\\\frac{9a-7}{\\\\left(a+1\\\\right) \\\\left(a-3\\\\right)}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{4}{a+1}+\\\\frac{5}{a-3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{9a-7}{\\\\left(a+1\\\\right) \\\\left(a-3\\\\right)}$$","hints":{"DefaultPathway":[{"id":"a61c721rational18a-h1","type":"hint","dependencies":[],"title":"Common Denominator","text":"The first step is to find the least common denominator of two rational expressions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(a+1\\\\right) \\\\left(a-3\\\\right)$$"],"dependencies":["a61c721rational18a-h1"],"title":"Common Denominator","text":"What is the least common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational18a-h3","type":"hint","dependencies":["a61c721rational18a-h2"],"title":"Common Denominator","text":"Since we do not know the value of a, the least common multiple of the denominators is their product. So, the LCD is $$\\\\left(a+1\\\\right) \\\\left(a-3\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational18a-h4","type":"hint","dependencies":["a61c721rational18a-h3"],"title":"Multiplying the First Expression","text":"Now we need to multiply both expressions by a factor equal to $$1$$ that makes the denominators equal to the LCD. Starting with the first expression, we can multiply $$\\\\frac{4}{a+1}$$ by a factor to make the denominator $$\\\\left(a+1\\\\right) \\\\left(a-3\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational18a-h5","type":"hint","dependencies":["a61c721rational18a-h4"],"title":"Multiplying the First Expression","text":"The denominator of $$\\\\frac{4}{a+1}$$ multiplied by $$a+3$$ equals $$\\\\left(a+1\\\\right) \\\\left(a+3\\\\right)$$. So, we need to multiply $$\\\\frac{4}{a+1}$$ by $$\\\\frac{a+3}{a+3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational18a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{4a-12}{\\\\left(a+1\\\\right) \\\\left(a-3\\\\right)}$$"],"dependencies":["a61c721rational18a-h5"],"title":"Multiplying the First Expression","text":"What is $$\\\\frac{4}{a+1} \\\\frac{a-3}{a-3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational18a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["4a-12"],"dependencies":["a61c721rational18a-h6"],"title":"Multiplying the First Expression","text":"What is $$4\\\\left(a-3\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational18a-h8","type":"hint","dependencies":["a61c721rational18a-h7"],"title":"Multiplying the First Expression","text":"When multiplying fractions, we multiply the numerators and multiply the denominators. For the numerators, we can distribute the $$4$$ into the $$a+3$$ by adding $$4a$$ andadding it to $$4\\\\left(-3\\\\right)$$. The numerator then becomes 4a-12. After multiplying the denominators, we can leave the denominator as $$\\\\left(a+1\\\\right) \\\\left(a-3\\\\right)$$ since it is in its factored form. So, the new expression becomes $$\\\\frac{4a-12}{\\\\left(a+1\\\\right) \\\\left(a-3\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational18a-h9","type":"hint","dependencies":["a61c721rational18a-h8"],"title":"Multiplying the Second Expression","text":"Next, we need to multiply the second expression by a fraction to make the denominator equal to the LCD. Because the denominator of $$\\\\frac{5}{a-3}$$ needs to by multiplied by $$a+1$$ to become $$\\\\left(a+1\\\\right) \\\\left(a-3\\\\right)$$, we need to $$\\\\frac{5}{a-3}$$ by $$\\\\frac{a+1}{a+1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational18a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5a+5}{\\\\left(a+1\\\\right) \\\\left(a-3\\\\right)}$$"],"dependencies":["a61c721rational18a-h9"],"title":"Multiplying the Second Expression","text":"What is $$\\\\frac{5}{a-3} \\\\frac{a+1}{a+1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational18a-h11","type":"hint","dependencies":["a61c721rational18a-h10"],"title":"Multiplying the Second Expression","text":"We distribute the $$5$$ into $$a+1$$ to get $$5a+4$$. Then, we leave the denominator in its factored form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational18a-h12","type":"hint","dependencies":["a61c721rational18a-h11"],"title":"Adding the Expressions","text":"Because the expressions now have the same denominator, we can add the numerators to get the sum of the two expressions. We\'ll leave the denominator as it is, but combine the numerator\'s like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational18a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["9a"],"dependencies":["a61c721rational18a-h12"],"title":"Adding the Expressions","text":"What is $$5a+4a$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational18a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-7$$"],"dependencies":["a61c721rational18a-h13"],"title":"Adding the Expressions","text":"What is $$-12+5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational18a-h15","type":"hint","dependencies":["a61c721rational18a-h14"],"title":"Final Expression","text":"Now we have our final expression: $$\\\\frac{9a-7}{\\\\left(a+1\\\\right) \\\\left(a-3\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a61c721rational19","title":"Combining Rational Expressions","body":"Subtract the rational expressions, and then simplify.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.6 Rational Expressions","courseName":"OpenStax: College Algebra","steps":[{"id":"a61c721rational19a","stepAnswer":["$$\\\\frac{9y-3}{\\\\left(y-2\\\\right) \\\\left(y+1\\\\right)}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{y+3}{y-2}-\\\\frac{y-3}{y+1}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{9y-3}{\\\\left(y-2\\\\right) \\\\left(y+1\\\\right)}$$","hints":{"DefaultPathway":[{"id":"a61c721rational19a-h1","type":"hint","dependencies":[],"title":"Common Denominator","text":"The first step is to find the least common denominator of two rational expressions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(y-2\\\\right) \\\\left(y+1\\\\right)$$"],"dependencies":["a61c721rational19a-h1"],"title":"Common Denominator","text":"What is the least common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational19a-h3","type":"hint","dependencies":["a61c721rational19a-h2"],"title":"Common Denominator","text":"Since we do not know the value of $$y$$, the least common multiple of the denominators is their product. So, the LCD is $$\\\\left(y-2\\\\right) \\\\left(y+1\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational19a-h4","type":"hint","dependencies":["a61c721rational19a-h3"],"title":"Multiplying the First Expression","text":"Now we need to multiply both expressions by a factor equal to $$1$$ that makes the denominators equal to the LCD. Starting with the first expression, we can multiply $$\\\\frac{y+3}{y+1}$$ by a factor to make the denominator $$\\\\left(y-2\\\\right) \\\\left(y+1\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational19a-h5","type":"hint","dependencies":["a61c721rational19a-h4"],"title":"Multiplying the First Expression","text":"The denominator of $$\\\\frac{y+3}{y-2}$$ multiplied by $$y+1$$ equals $$\\\\left(y-2\\\\right) \\\\left(y+1\\\\right)$$. So, we need to multiply $$\\\\frac{y+3}{y+1}$$ by $$\\\\left(y+1\\\\right) \\\\left(y+1\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational19a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{y^2+4y+3}{\\\\left(y-2\\\\right) \\\\left(y+1\\\\right)}$$"],"dependencies":["a61c721rational19a-h5"],"title":"Multiplying the First Expression","text":"What is $$\\\\frac{y+3}{y-2} \\\\left(y+1\\\\right) \\\\left(y+1\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational19a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y^2+4y+3$$"],"dependencies":["a61c721rational19a-h6"],"title":"Multiplying the First Expression","text":"$$\\\\left(y+3\\\\right) \\\\left(y+1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational19a-h8","type":"hint","dependencies":["a61c721rational19a-h7"],"title":"Multiplying the First Expression","text":"When multiplying polynomials, we need to use FOIL.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational19a-h9","type":"hint","dependencies":["a61c721rational19a-h8"],"title":"Multiplying the First Expression","text":"First, multiply the first value in each polynomial: $$y y=y^2$$. Next, multiply the outside values: $$1y=y$$. Then, multiply the inside values: $$3y=3y$$. Lastly, multiply the last values: $$3\\\\times1=3$$. Finally, combine like terms: $$y+3y=4y$$. So, the expression reads $$y^2+4y+3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational19a-h10","type":"hint","dependencies":["a61c721rational19a-h9"],"title":"Multiplying the Second Expression","text":"Next, we need to multiply the second expression by a fraction to make the denominator equal to the LCD. Because the denominator of $$\\\\frac{y-3}{y+1}$$ needs to by multiplied by $$y-2$$ to become $$\\\\left(y-2\\\\right) \\\\left(y+1\\\\right)$$, we need to $$\\\\frac{y-3}{y+1}$$ by $$(y-2)(y-2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational19a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y^2-5y+\\\\frac{6}{\\\\left(y-2\\\\right) \\\\left(y+1\\\\right)}$$"],"dependencies":["a61c721rational19a-h10"],"title":"Multiplying the Second Expression","text":"What is $$\\\\frac{y+3}{y-2} \\\\frac{y+1}{y+1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational19a-h12","type":"hint","dependencies":["a61c721rational19a-h11"],"title":"Multiplying the Second Expression","text":"When multiplying polynomials, we need to use FOIL.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational19a-h13","type":"hint","dependencies":["a61c721rational19a-h12"],"title":"Multiplying the Second Expression","text":"First, multiply the first value in each polynomial: $$y y=y^2$$. Next, multiply the outside values: $$1y=y$$. Then, multiply the inside values: $$3y=3y$$. Lastly, multiply the last values: $$3\\\\times1=3$$. Finally, combine like terms: $$y+3y=4y$$. So, the expression reads $$y^2+4y+3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational19a-h14","type":"hint","dependencies":["a61c721rational19a-h13"],"title":"Adding the Expressions","text":"Because the expressions now have the same denominator, we can add the numerators to get the sum of the two expressions. We\'ll leave the denominator as it is, but combine the numerator\'s like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational19a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["9a"],"dependencies":["a61c721rational19a-h14"],"title":"Adding the Expressions","text":"What is $$y^2+4y+3-y^2-5y+6$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational19a-h16","type":"hint","dependencies":["a61c721rational19a-h15"],"title":"Adding the Expressions","text":"Combine the like terms: $$y^2-y^2=0;$$ $$4y+5y=9y;$$ $$3-6=-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational19a-h17","type":"hint","dependencies":["a61c721rational19a-h16"],"title":"Final Expression","text":"Now we have our final expression: $$\\\\frac{2y^2-y+9}{\\\\left(y-2\\\\right) \\\\left(y+1\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a61c721rational2","title":"Dividing Rational Expressions","body":"Divide the rational expressions and express the quotient in simplest form:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.6 Rational Expressions","courseName":"OpenStax: College Algebra","steps":[{"id":"a61c721rational2a","stepAnswer":["$$\\\\frac{\\\\left(2x-3\\\\right) \\\\left(x+1\\\\right)}{\\\\left(x-1\\\\right) \\\\left(x-2\\\\right)}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\frac{2x^2+x-6}{x^2-1}}{\\\\frac{x^2-4}{x^2+2x+1}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{\\\\left(2x-3\\\\right) \\\\left(x+1\\\\right)}{\\\\left(x-1\\\\right) \\\\left(x-2\\\\right)}$$","hints":{"DefaultPathway":[{"id":"a61c721rational2a-h1","type":"hint","dependencies":[],"title":"Factor the expression","text":"Factoring is the process to split a product into a group of factors (a multiplication of simpler expressions).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational2a-h2","type":"hint","dependencies":["a61c721rational2a-h1"],"title":"Factor the expression","text":"Not all terms can be factored. Factor the terms that can be split further.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational2a-h3","type":"hint","dependencies":["a61c721rational2a-h2"],"title":"Factor the expression","text":"In this case, we can factor all the factors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(2x-3\\\\right) \\\\left(x+2\\\\right)$$"],"dependencies":["a61c721rational2a-h3"],"title":"Factor the expression","text":"What does $$2x^2+x-6$$ factor into?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational2a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x+1\\\\right) \\\\left(x-1\\\\right)$$"],"dependencies":["a61c721rational2a-h4"],"title":"Factor the expression","text":"What does $$x^2-1$$ factor into?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x+2\\\\right) \\\\left(x-2\\\\right)$$"],"dependencies":["a61c721rational2a-h5"],"title":"Factor the expression","text":"What does $$x^2-4$$ factor into?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational2a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(x+1\\\\right)}^2$$"],"dependencies":["a61c721rational2a-h6"],"title":"Factor the expression","text":"What does $$x^2+2x+1$$ factor into?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational2a-h8","type":"hint","dependencies":["a61c721rational2a-h7"],"title":"Cancel terms","text":"Now that you have factored all possible terms, if you see the same term in the numerator and the denominator of the product, you can cancel it out.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational2a-h9","type":"hint","dependencies":["a61c721rational2a-h8"],"title":"Cancel terms","text":"In this case, we can cancel out $$x+1$$ and $$x+2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational2a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\left(2x-3\\\\right) \\\\left(x+1\\\\right)}{\\\\left(x-1\\\\right) \\\\left(x-2\\\\right)}$$"],"dependencies":["a61c721rational2a-h9"],"title":"Final Answer","text":"After canceling out terms, your final answer will remain in product form. What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a61c721rational20","title":"Combining Rational Expressions","body":"Subtract the rational expressions, and then simplify.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.6 Rational Expressions","courseName":"OpenStax: College Algebra","steps":[{"id":"a61c721rational20a","stepAnswer":["$$z^2-13z-6$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{3z}{z+1}-\\\\frac{2z+5}{z-2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$z^2-13z-6$$","hints":{"DefaultPathway":[{"id":"a61c721rational20a-h1","type":"hint","dependencies":[],"title":"Common Denominator","text":"The first step is to find the least common denominator of two rational expressions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(z-2\\\\right) \\\\left(z+1\\\\right)$$"],"dependencies":["a61c721rational20a-h1"],"title":"Common Denominator","text":"What is the least common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational20a-h3","type":"hint","dependencies":["a61c721rational20a-h2"],"title":"Common Denominator","text":"Since we do not know the value of $$z$$, the least common multiple of the denominators is their product. So, the LCD is $$\\\\left(z-2\\\\right) \\\\left(z+1\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational20a-h4","type":"hint","dependencies":["a61c721rational20a-h3"],"title":"Multiplying the First Expression","text":"Now we need to multiply both expressions by a factor equal to $$1$$ that makes the denominators equal to the LCD. Starting with the first expression, we can multiply $$\\\\frac{3z}{z+1}$$ by a factor to make the denominator $$\\\\left(z-2\\\\right) \\\\left(z+1\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational20a-h5","type":"hint","dependencies":["a61c721rational20a-h4"],"title":"Multiplying the First Expression","text":"The denominator of $$\\\\frac{3z}{z-2}$$ multiplied by $$z+1$$ equals $$\\\\left(z-2\\\\right) \\\\left(z+1\\\\right)$$. So, we need to multiply $$\\\\frac{3z}{y+1}$$ by $$\\\\frac{z-2}{z-2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational20a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3z^2-6z}{\\\\left(z+1\\\\right) \\\\left(z-2\\\\right)}$$"],"dependencies":["a61c721rational20a-h5"],"title":"Multiplying the First Expression","text":"What is $$\\\\frac{3z}{z+1} \\\\frac{z-2}{z-2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational20a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3z^2-6z$$"],"dependencies":["a61c721rational20a-h6"],"title":"Multiplying the First Expression","text":"What is $$3z \\\\left(z-2\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational20a-h8","type":"hint","dependencies":["a61c721rational20a-h7"],"title":"Multiplying the First Expression","text":"We need to distribute the $$3z$$ into the $$(z-2)$$: $$3z z=3z^2$$ and $$3z \\\\left(-2\\\\right)=-6z$$. So, the numerator is $$\\\\frac{3z^2}{\\\\left(z-2\\\\right) \\\\left(z+1\\\\right)}$$. We can leave the denominator in its factored form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational20a-h9","type":"hint","dependencies":["a61c721rational20a-h8"],"title":"Multiplying the Second Expression","text":"Next, we need to multiply the second expression by a fraction to make the denominator equal to the LCD. Because the denominator of $$\\\\frac{2z+5}{z+1}$$ needs to by multiplied by $$(z-2)$$ to become $$\\\\left(z-2\\\\right) \\\\left(z+1\\\\right)$$, we need to $$\\\\frac{2z+5}{z+1}$$ by $$\\\\frac{z-2}{z-2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational20a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2z^2+7z+5}{\\\\left(z-2\\\\right) \\\\left(z+1\\\\right)}$$"],"dependencies":["a61c721rational20a-h9"],"title":"Multiplying the Second Expression","text":"What is $$\\\\frac{2z+5}{z-2} \\\\frac{z+1}{z+1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational20a-h11","type":"hint","dependencies":["a61c721rational20a-h10"],"title":"Multiplying the Second Expression","text":"When multiplying polynomials, we need to use FOIL.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational20a-h12","type":"hint","dependencies":["a61c721rational20a-h11"],"title":"Multiplying the Second Expression","text":"First, multiply the first value in each polynomial: $$2z z=2z^2$$. Next, multiply the outside values: $$2z\\\\times1=2z$$. Then, multiply the inside values: $$5z=5z$$. Lastly, multiply the last values: $$5\\\\times1=5$$. Finally, combine like terms: $$2z+5z=7z$$. So, the expression reads $$2z^2+7z+5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational20a-h13","type":"hint","dependencies":["a61c721rational20a-h12"],"title":"Subtracting the Expressions","text":"Because the expressions now have the same denominator, we can add the numerators to get the difference of the two expressions. We\'ll leave the denominator as it is, but combine the numerator\'s like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational20a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$z^2-13z-6$$"],"dependencies":["a61c721rational20a-h13"],"title":"Subtracting the Expressions","text":"What is $$3z^2-6z-2z^2+7z+5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational20a-h15","type":"hint","dependencies":["a61c721rational20a-h14"],"title":"Subtracting the Expressions","text":"Combine the like terms: $$3z^2-2z^2=z^2$$ $$-6z-7z=-13z;$$ $$-6$$ stays the same.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational20a-h16","type":"hint","dependencies":["a61c721rational20a-h15"],"title":"Final Expression","text":"Now we have our final expression: $$\\\\frac{z^2-13z-6}{\\\\left(z-2\\\\right) \\\\left(z+1\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a61c721rational21","title":"Combining Rational Expressions","body":"Add the rational expressions, and then simplify.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.6 Rational Expressions","courseName":"OpenStax: College Algebra","steps":[{"id":"a61c721rational21a","stepAnswer":["$$\\\\frac{2xy+x+y}{\\\\left(x+1\\\\right) \\\\left(y+1\\\\right)}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2xy+x+y}{\\\\left(x+1\\\\right) \\\\left(y+1\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2xy+x+y}{\\\\left(x+1\\\\right) \\\\left(y+1\\\\right)}$$","hints":{"DefaultPathway":[{"id":"a61c721rational21a-h1","type":"hint","dependencies":[],"title":"Common Denominator","text":"The first step is to find the least common denominator of two rational expressions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational21a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x+1\\\\right) \\\\left(y+1\\\\right)$$"],"dependencies":["a61c721rational21a-h1"],"title":"Common Denominator","text":"What is the least common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational21a-h3","type":"hint","dependencies":["a61c721rational21a-h2"],"title":"Common Denominator","text":"Since we do not know the value of $$x$$ or $$y$$, the least common multiple of the denominators is their product. So, the LCD is $$\\\\left(x+1\\\\right) \\\\left(y+1\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational21a-h4","type":"hint","dependencies":["a61c721rational21a-h3"],"title":"Multiplying the First Expression","text":"Now we need to multiply both expressions by a factor equal to $$1$$ that makes the denominators equal to the LCD. Starting with the first expression, we can multiply $$\\\\frac{x}{x+1}$$ by a factor to make the denominator $$\\\\left(x+1\\\\right) \\\\left(y+1\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational21a-h5","type":"hint","dependencies":["a61c721rational21a-h4"],"title":"Multiplying the First Expression","text":"The denominator of $$\\\\frac{x}{x+1}$$ multiplied by $$y+1$$ equals $$\\\\left(x+1\\\\right) \\\\left(y+1\\\\right)$$. So, we need to multiply $$\\\\frac{x}{x+1}$$ by $$\\\\frac{y+1}{y+1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational21a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{xy+x}{\\\\left(x+1\\\\right) \\\\left(y+1\\\\right)}$$"],"dependencies":["a61c721rational21a-h5"],"title":"Multiplying the First Expression","text":"What is $$\\\\frac{x}{x+1} \\\\frac{y+1}{y+1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational21a-h7","type":"hint","dependencies":["a61c721rational21a-h6"],"title":"Multiplying the First Expression","text":"We can keep the denominator as it is, but distribute the $$x$$ into $$y+1$$: $$xy+x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational21a-h8","type":"hint","dependencies":["a61c721rational21a-h7"],"title":"Multiplying the Second Expression","text":"Next, we need to multiply the second expression by a fraction to make the denominator equal to the LCD. Because the denominator of $$\\\\frac{y}{y+1}$$ needs to by multiplied by $$x+1$$ to become $$\\\\left(z+1\\\\right) \\\\left(y+1\\\\right)$$, we need to $$\\\\frac{y}{y+1}$$ by $$\\\\frac{y+1}{y+1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational21a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{xy+y}{\\\\left(x+1\\\\right) \\\\left(y+1\\\\right)}$$"],"dependencies":["a61c721rational21a-h8"],"title":"Multiplying the Second Expression","text":"What is $$\\\\frac{y}{y+1} \\\\frac{x+1}{x+1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational21a-h10","type":"hint","dependencies":["a61c721rational21a-h9"],"title":"Multiplying the Second Expression","text":"Distribute the $$y$$ into $$x+1$$: $$x y=xy$$ and $$1y=y$$, so the expression reads $$xy+y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational21a-h11","type":"hint","dependencies":["a61c721rational21a-h10"],"title":"Subtracting the Expressions","text":"Because the expressions now have the same denominator, we can add the numerators to get the sum of the two expressions. We\'ll leave the denominator as it is, but combine the numerator\'s like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational21a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2xy+x+y$$"],"dependencies":["a61c721rational21a-h11"],"title":"Subtracting the Expressions","text":"What is $$xy+x+xy+y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational21a-h13","type":"hint","dependencies":["a61c721rational21a-h12"],"title":"Subtracting the Expressions","text":"Combine the like terms: $$xy+xy=2xy;$$ the $$x$$ and $$y$$ stay the same.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational21a-h14","type":"hint","dependencies":["a61c721rational21a-h13"],"title":"Final Expression","text":"Now we have our final expression: $$\\\\frac{2xy+x+y}{\\\\left(x+1\\\\right) \\\\left(y+1\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a61c721rational22","title":"Rational Expressions","body":"For the following exercise, multiply the rational expressions and express the product in simplest form.","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"1.6 Rational Expressions","courseName":"OpenStax: College Algebra","steps":[{"id":"a61c721rational22a","stepAnswer":["$$\\\\frac{\\\\left(d+5\\\\right) \\\\left(d-5\\\\right)}{5d-1} \\\\left(5d+1\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2d^2+15d+25}{4d^2-25}$$ * $$\\\\frac{2d^2-15d+25}{25d-1}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{\\\\left(d+5\\\\right) \\\\left(d-5\\\\right)}{5d-1} \\\\left(5d+1\\\\right)$$","hints":{"DefaultPathway":[{"id":"a61c721rational22a-h1","type":"hint","dependencies":[],"title":"Factoring","text":"First, we need to factor $$2d^2$$ + $$15d$$ + $$25$$.","variabilization":{},"oer":"","license":""},{"id":"a61c721rational22a-h2","type":"hint","dependencies":["a61c721rational22a-h1"],"title":"Split","text":"Split $$15d$$ to $$10d$$ and $$5d$$.","variabilization":{},"oer":"","license":""},{"id":"a61c721rational22a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2d\\\\left(d+5\\\\right)$$"],"dependencies":["a61c721rational22a-h2"],"title":"Factoring","text":"What is the factoring of $$2d^2$$ + $$10d$$?","variabilization":{},"oer":"","license":""},{"id":"a61c721rational22a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5\\\\left(d+5\\\\right)$$"],"dependencies":["a61c721rational22a-h3"],"title":"Factoring","text":"What is the factoring of $$5d+25$$?","variabilization":{},"oer":"","license":""},{"id":"a61c721rational22a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(2d+5\\\\right) \\\\left(d+5\\\\right)$$"],"dependencies":["a61c721rational22a-h4"],"title":"Factoring","text":"Factor out the common factor $$d+5$$?","variabilization":{},"oer":"","license":""},{"id":"a61c721rational22a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(2d+5\\\\right) \\\\left(2d-5\\\\right)$$"],"dependencies":["a61c721rational22a-h5"],"title":"Factoring","text":"Factor $$4d^2$$ - $$25$$ using difference of squares.","variabilization":{},"oer":"","license":""},{"id":"a61c721rational22a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(5d-1\\\\right) \\\\left(5d+1\\\\right)$$"],"dependencies":["a61c721rational22a-h6"],"title":"Factoring","text":"Factor $$25d^2$$ - $$1$$ using difference of squares.","variabilization":{},"oer":"","license":""},{"id":"a61c721rational22a-h8","type":"hint","dependencies":["a61c721rational22a-h7"],"title":"Factoring","text":"Next, we need to factor $$2d^2$$ - $$15d$$ + $$25$$.","variabilization":{},"oer":"","license":""},{"id":"a61c721rational22a-h9","type":"hint","dependencies":["a61c721rational22a-h8"],"title":"Split","text":"Split $$15d$$ to $$-10d$$ and $$-5d$$.","variabilization":{},"oer":"","license":""},{"id":"a61c721rational22a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2d(d-5)$$"],"dependencies":["a61c721rational22a-h9"],"title":"Factoring","text":"What is the factoring of $$2d^2$$ - $$10d$$?","variabilization":{},"oer":"","license":""},{"id":"a61c721rational22a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5(d-5)$$"],"dependencies":["a61c721rational22a-h10"],"title":"Factoring","text":"What is the factoring of $$-5d+25$$.","variabilization":{},"oer":"","license":""},{"id":"a61c721rational22a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$(2d-5)(d-5)$$"],"dependencies":["a61c721rational22a-h11"],"title":"Factoring","text":"Factor out the common factor $$(d-5)$$.","variabilization":{},"oer":"","license":""},{"id":"a61c721rational22a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\left(d+5\\\\right) \\\\left(d-5\\\\right)}{5d-1} \\\\left(5d+1\\\\right)$$"],"dependencies":["a61c721rational22a-h12"],"title":"Simplify","text":"Cross out the same factors on top and bottom.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a61c721rational3","title":"Adding Rational Expressions","body":"Add the rational expressions:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.6 Rational Expressions","courseName":"OpenStax: College Algebra","steps":[{"id":"a61c721rational3a","stepAnswer":["$$\\\\frac{6x+5y}{xy}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{5}{x}+\\\\frac{6}{y}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{6x+5y}{xy}$$","hints":{"DefaultPathway":[{"id":"a61c721rational3a-h1","type":"hint","dependencies":[],"title":"The least common multiple","text":"The LCM or least common multiple of a set of numbers is the smallest positive integer that is divisible by all those numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["xy"],"dependencies":["a61c721rational3a-h1"],"title":"The least common multiple","text":"To combine the fractions, we want to find the LCM of the denominators. What is it?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational3a-h3","type":"hint","dependencies":["a61c721rational3a-h2"],"title":"Multiplication by $$1$$","text":"We must multiple each expression by the appropriate form of $$1$$ to obtain xy as the denominator for each fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{y}{y}$$"],"dependencies":["a61c721rational3a-h3"],"title":"Multiplication by $$1$$","text":"What should we multiply to $$\\\\frac{5}{x}$$ so that its denominator becomes xy?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{x}{x}$$"],"dependencies":["a61c721rational3a-h4"],"title":"Multiplication by $$1$$","text":"What should we multiply to $$\\\\frac{6}{y}$$ so that its denominator becomes xy?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational3a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{6x+5y}{xy}$$"],"dependencies":["a61c721rational3a-h5"],"title":"Adding Terms","text":"Now that both terms have the same denominator, we can add their numerators. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a61c721rational4","title":"Subtracting Rational Expressions","body":"Subtract the rational expressions:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.6 Rational Expressions","courseName":"OpenStax: College Algebra","steps":[{"id":"a61c721rational4a","stepAnswer":["$$\\\\frac{4\\\\left(x-4\\\\right)}{{\\\\left(x+2\\\\right)}^{2\\\\left(x-2\\\\right)}}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{6}{x^2+4x+4}-\\\\frac{2}{x^2-4}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{4\\\\left(x-4\\\\right)}{{\\\\left(x+2\\\\right)}^{2\\\\left(x-2\\\\right)}}$$","hints":{"DefaultPathway":[{"id":"a61c721rational4a-h1","type":"hint","dependencies":[],"title":"Factor the expression","text":"Factoring is the process to split a product into a group of factors (a multiplication of simpler expressions).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational4a-h2","type":"hint","dependencies":["a61c721rational4a-h1"],"title":"Factor the expression","text":"Not all terms can be factored. Factor the terms that can be split further.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational4a-h3","type":"hint","dependencies":["a61c721rational4a-h2"],"title":"Factor the expression","text":"In this case, we should factor the denominators since they can be broken into individual factors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(x+2\\\\right)}^2$$"],"dependencies":["a61c721rational4a-h3"],"title":"Factor the expression","text":"What does $$x^2+4x+4$$ factor to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational4a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x+2\\\\right) \\\\left(x-2\\\\right)$$"],"dependencies":["a61c721rational4a-h4"],"title":"Factor the expression","text":"What does $$x^2-4$$ factor to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational4a-h6","type":"hint","dependencies":["a61c721rational4a-h5"],"title":"The least common multiple","text":"The LCM or least common multiple of a set of numbers is the smallest positive integer that is divisible by all those numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational4a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(x+2\\\\right)}^{2\\\\left(x-2\\\\right)}$$"],"dependencies":["a61c721rational4a-h6"],"title":"The least common multiple","text":"To combine the fractions, we want to find the LCM of the denominators. What is it?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational4a-h8","type":"hint","dependencies":["a61c721rational4a-h7"],"title":"Multiplication by $$1$$","text":"We must multiple each expression by the appropriate form of $$1$$ to obtain xy as the denominator for each fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational4a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x-2$$"],"dependencies":["a61c721rational4a-h8"],"title":"Multiplication by $$1$$","text":"What should we multiply to $$\\\\frac{6}{x^2+4x+4}$$ so that its denominator becomes $${\\\\left(x+2\\\\right)}^{2\\\\left(x-2\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational4a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+2$$"],"dependencies":["a61c721rational4a-h9"],"title":"Multiplication by $$1$$","text":"What should we multiply to $$\\\\frac{2}{x^2-4}$$ so that its denominator becomes $${\\\\left(x+2\\\\right)}^{2\\\\left(x-2\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational4a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{4\\\\left(x-4\\\\right)}{{\\\\left(x+2\\\\right)}^{2\\\\left(x-2\\\\right)}}$$"],"dependencies":["a61c721rational4a-h10"],"title":"Adding Terms","text":"Now that both terms have the same denominator, we can add their numerators. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a61c721rational5","title":"Multiplying Rational Expressions","body":"Multiply the rational expressions and show the product in simplest form:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.6 Rational Expressions","courseName":"OpenStax: College Algebra","steps":[{"id":"a61c721rational5a","stepAnswer":["$$\\\\frac{\\\\left(x+5\\\\right) \\\\left(x+6\\\\right)}{x+2} \\\\left(x+4\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\left(x^2+11x+30\\\\right) \\\\left(x^2+7x+12\\\\right)}{x^2+5x+6} \\\\left(x^2+8x+16\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{\\\\left(x+5\\\\right) \\\\left(x+6\\\\right)}{x+2} \\\\left(x+4\\\\right)$$","hints":{"DefaultPathway":[{"id":"a61c721rational5a-h1","type":"hint","dependencies":[],"title":"Factor the expression","text":"Factoring is the process to split a product into a group of factors (a multiplication of simpler expressions).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational5a-h2","type":"hint","dependencies":["a61c721rational5a-h1"],"title":"Factor the expression","text":"Not all terms can be factored. Factor the terms that can be split further.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational5a-h3","type":"hint","dependencies":["a61c721rational5a-h2"],"title":"Factor the expression","text":"In this case, we can factor all $$4$$ terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x+5\\\\right) \\\\left(x+6\\\\right)$$"],"dependencies":["a61c721rational5a-h3"],"title":"Factor the expression","text":"What does $$x^2+11x-\\\\left(+30\\\\right)$$ factor into?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x+2\\\\right) \\\\left(x+3\\\\right)$$"],"dependencies":["a61c721rational5a-h4"],"title":"Factor the expression","text":"What does $$x^2+5x+6$$ factor into?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational5a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x+3\\\\right) \\\\left(x+4\\\\right)$$"],"dependencies":["a61c721rational5a-h5"],"title":"Factor the expression","text":"What does $$x^2+7x+12$$ factor into?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational5a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(x+4\\\\right)}^2$$"],"dependencies":["a61c721rational5a-h6"],"title":"Factor the expression","text":"What does $$x^2+8x+16$$ factor into?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational5a-h8","type":"hint","dependencies":["a61c721rational5a-h7"],"title":"Cancel terms","text":"Now that you have factored all possible terms, if you see the same term in the numerator and the denominator of the product, you can cancel it out.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational5a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\left(x+5\\\\right) \\\\left(x+6\\\\right)}{x+2} \\\\left(x+4\\\\right)$$"],"dependencies":["a61c721rational5a-h8"],"title":"Cancel terms","text":"In this case, we can cancel $$x+3$$ and $$x+4$$. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a61c721rational6","title":"Dividing Rational Expressions","body":"Divide the rational expressions and express the quotient in simplest form:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.6 Rational Expressions","courseName":"OpenStax: College Algebra","steps":[{"id":"a61c721rational6a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\frac{9x^2-6}{3x^2+17x-28}}{\\\\frac{3x^2-2x-8}{x^2+5x-14}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a61c721rational6a-h1","type":"hint","dependencies":[],"title":"Factor the expression","text":"Factoring is the process to split a product into a group of factors (a multiplication of simpler expressions).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational6a-h2","type":"hint","dependencies":["a61c721rational6a-h1"],"title":"Factor the expression","text":"Not all terms can be factored. Factor the terms that can be split further.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational6a-h3","type":"hint","dependencies":["a61c721rational6a-h2"],"title":"Factor the expression","text":"In this case, we should factor all the expressions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(3x-4\\\\right) \\\\left(3x+4\\\\right)$$"],"dependencies":["a61c721rational6a-h3"],"title":"Factor the expression","text":"What does $$9x^2-16$$ factor to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(3x-4\\\\right) \\\\left(x+7\\\\right)$$"],"dependencies":["a61c721rational6a-h4"],"title":"Factor the expression","text":"What does $$3x^2+17x-28$$ factor to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational6a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x+7\\\\right) \\\\left(x-2\\\\right)$$"],"dependencies":["a61c721rational6a-h5"],"title":"Factor the expression","text":"What does $$x^2+5x-14$$ factor to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational6a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x-2\\\\right) \\\\left(3x+4\\\\right)$$"],"dependencies":["a61c721rational6a-h6"],"title":"Factor the expression","text":"What does $$3x^2-2x-8$$ factor to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational6a-h8","type":"hint","dependencies":["a61c721rational6a-h7"],"title":"Cancel terms","text":"Now that you have factored all possible terms, if you see the same term in the numerator and the denominator of the product, you can cancel it out.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational6a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a61c721rational6a-h8"],"title":"Cancel terms","text":"In this case, we can cancel every single term out. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a61c721rational7","title":"Subtracting Rational Expressions","body":"Subtract the rational expressions:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.6 Rational Expressions","courseName":"OpenStax: College Algebra","steps":[{"id":"a61c721rational7a","stepAnswer":["$$\\\\frac{2\\\\left(x-7\\\\right)}{x+5} \\\\left(x-3\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{3}{x+5}-\\\\frac{1}{x-3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2\\\\left(x-7\\\\right)}{x+5} \\\\left(x-3\\\\right)$$","hints":{"DefaultPathway":[{"id":"a61c721rational7a-h1","type":"hint","dependencies":[],"title":"The least common multiple","text":"The LCM or least common multiple of a set of numbers is the smallest positive integer that is divisible by all those numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x+5\\\\right) \\\\left(x-3\\\\right)$$"],"dependencies":["a61c721rational7a-h1"],"title":"The least common multiple","text":"To combine the fractions, we want to find the LCM of the denominators. What is it?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational7a-h3","type":"hint","dependencies":["a61c721rational7a-h2"],"title":"Multiplication by $$1$$","text":"We must multiple each expression by the appropriate form of $$1$$ to obtain xy as the denominator for each fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{x-3}{x-3}$$"],"dependencies":["a61c721rational7a-h3"],"title":"Multiplication by $$1$$","text":"What should we multiply to $$\\\\frac{3}{x+5}$$ so that its denominator becomes $$\\\\left(x+5\\\\right) \\\\left(x-3\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational7a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{x+5}{x+5}$$"],"dependencies":["a61c721rational7a-h4"],"title":"Multiplication by $$1$$","text":"What should we multiply to $$\\\\frac{1}{x-3}$$ so that its denominator becomes $$\\\\left(x+5\\\\right) \\\\left(x-3\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational7a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2\\\\left(x-7\\\\right)}{x+5} \\\\left(x-3\\\\right)$$"],"dependencies":["a61c721rational7a-h5"],"title":"Adding Terms","text":"Now that both terms have the same denominator, we can add their numerators. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a61c721rational8","title":"Simplifying Expressions","body":"Simplifying Complex Rational Expressions","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.6 Rational Expressions","courseName":"OpenStax: College Algebra","steps":[{"id":"a61c721rational8a","stepAnswer":["$$\\\\frac{\\\\left(x-y\\\\right) \\\\left(x+y\\\\right)}{x}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\frac{x}{y}-\\\\frac{y}{x}}{y}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{\\\\left(x-y\\\\right) \\\\left(x+y\\\\right)}{x}$$","hints":{"DefaultPathway":[{"id":"a61c721rational8a-h1","type":"hint","dependencies":[],"title":"The least common multiple","text":"The LCM or least common multiple of a set of numbers is the smallest positive integer that is divisible by all those numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["xy"],"dependencies":["a61c721rational8a-h1"],"title":"The least common multiple","text":"To combine the fractions in the numerator, we want to find the LCM of the denominators. What is it?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational8a-h3","type":"hint","dependencies":["a61c721rational8a-h2"],"title":"Multiplication by $$1$$","text":"We must multiple each expression by the appropriate form of $$1$$ to obtain xy as the denominator for each fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{y}{y}$$"],"dependencies":["a61c721rational8a-h3"],"title":"Multiplication by $$1$$","text":"What should we multiply to $$\\\\frac{x}{y}$$ so that its denominator becomes xy?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational8a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{x}{x}$$"],"dependencies":["a61c721rational8a-h4"],"title":"Multiplication by $$1$$","text":"What should we multiply to $$\\\\frac{y}{x}$$ so that its denominator becomes xy?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational8a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\left(x-y\\\\right) \\\\left(x+y\\\\right)}{xy}$$"],"dependencies":["a61c721rational8a-h5"],"title":"Adding Terms","text":"Now that both terms have the same denominator, we can add their numerators. What does the numerator simplify to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational8a-h7","type":"hint","dependencies":["a61c721rational8a-h6"],"title":"Cancel terms","text":"Now that you have factored all possible terms, if you see the same term in the numerator and the denominator of the product, you can cancel it out.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational8a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y$$"],"dependencies":["a61c721rational8a-h7"],"title":"Cancel terms","text":"What term can we cancel out?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational8a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\left(x-y\\\\right) \\\\left(x+y\\\\right)}{x}$$"],"dependencies":["a61c721rational8a-h8"],"title":"Final Answer","text":"After canceling out terms, your final answer will remain in product form. What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a61c721rational9","title":"Rational Expressions","body":"Simplify the rational expressions.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.6 Rational Expressions","courseName":"OpenStax: College Algebra","steps":[{"id":"a61c721rational9a","stepAnswer":["$$\\\\frac{y+5}{y+6}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{y^2+10y+25}{y^2+11y+30}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{y+5}{y+6}$$","hints":{"DefaultPathway":[{"id":"a61c721rational9a-h1","type":"hint","dependencies":[],"title":"Factoring Trinomials","text":"Factor the trinomial in the numerator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(y+5\\\\right) \\\\left(y+5\\\\right)$$"],"dependencies":["a61c721rational9a-h1"],"title":"Factoring Trinomials","text":"What are the two binomial factors of numerator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(y+5\\\\right) \\\\left(y+6\\\\right)$$"],"dependencies":["a61c721rational9a-h2"],"title":"Factoring Trinomials","text":"What are the two binomial factors of denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y+5$$"],"dependencies":["a61c721rational9a-h3"],"title":"Common Factor","text":"What is the common factor?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational9a-h5","type":"hint","dependencies":["a61c721rational9a-h4"],"title":"Common Factor","text":"Cancel the common factor from the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational9a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y+5$$"],"dependencies":["a61c721rational9a-h5"],"title":"Simplifying Expressions","text":"What is remaining in the numerator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational9a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y+6$$"],"dependencies":["a61c721rational9a-h6"],"title":"Simplifying Expressions","text":"What is remaining in the denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a61c721rational9a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{y+5}{y+6}$$"],"dependencies":["a61c721rational9a-h7"],"title":"Simplifying Expressions","text":"What is the simplified expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a62d4a8factoring1","title":"Factoring Polynomials","body":"Factor completely.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Factor Trinomials with Leading Coefficient Oth","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a62d4a8factoring1a","stepAnswer":["$$2\\\\left(n+3\\\\right) \\\\left(n-7\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$2n^2-8n-42$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2\\\\left(n+3\\\\right) \\\\left(n-7\\\\right)$$","hints":{"DefaultPathway":[{"id":"a62d4a8factoring1a-h1","type":"hint","dependencies":[],"title":"Use a preliminary strategy","text":"As there is a greatest common factor(2), factor it out. The expression should look like this: $$2\\\\left(n^2-4n-21\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring1a-h2","type":"hint","dependencies":["a62d4a8factoring1a-h1"],"title":"Identify","text":"Inside the parentheses, is it a binomial, trinomial, or are there more than three terms? Since it is a trinomial with a coefficient of $$1$$, undo FOIL.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring1a-h3","type":"hint","dependencies":["a62d4a8factoring1a-h2"],"title":"Use factors","text":"Use the factors of $$-21$$ to finish FOIL.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring1a-h4","type":"hint","dependencies":["a62d4a8factoring1a-h3"],"title":"Answer","text":"The answer is $$2\\\\left(n+3\\\\right) \\\\left(n-7\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a62d4a8factoring10","title":"Factoring Polynomials","body":"Factor completely.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Factor Trinomials with Leading Coefficient Oth","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a62d4a8factoring10a","stepAnswer":["$$\\\\left(8a+5\\\\right) \\\\left(a-1\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$8a^2-3a-5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(8a+5\\\\right) \\\\left(a-1\\\\right)$$","hints":{"DefaultPathway":[{"id":"a62d4a8factoring10a-h1","type":"hint","dependencies":[],"title":"Find factor pairs of the first term","text":"The factors of $$8a^2$$ are 2a and 4a, and a and 8a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring10a-h2","type":"hint","dependencies":["a62d4a8factoring10a-h1"],"title":"Find factor pairs of the third term","text":"The only factors of $$-5$$ are $$-1$$ and $$5$$ or $$1$$ and $$-5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring10a-h3","type":"hint","dependencies":["a62d4a8factoring10a-h2"],"title":"Test","text":"Test all possibly combinations of the factors until the correct product is found.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring10a-h4","type":"hint","dependencies":["a62d4a8factoring10a-h3"],"title":"Answer","text":"The answer is $$\\\\left(8a+5\\\\right) \\\\left(a-1\\\\right)$$,","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a62d4a8factoring11","title":"Factor Trinomials of the form $${ax}^2+bx+c$$ with a GCF","body":"Factor the expression below completely","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Factor Trinomials with Leading Coefficient Oth","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a62d4a8factoring11a","stepAnswer":["$$3m(m-5)(m-2)$$"],"problemType":"TextBox","stepTitle":"$$3m^3-21m^2+30m$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3m(m-5)(m-2)$$","hints":{"DefaultPathway":[{"id":"a62d4a8factoring11a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3m$$"],"dependencies":[],"title":"Greatest common factor","text":"What is the greatest common factor?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring11a-h2","type":"hint","dependencies":["a62d4a8factoring11a-h1"],"title":"Factor","text":"Factor out the $$3m$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring11a-h3","type":"hint","dependencies":["a62d4a8factoring11a-h2"],"title":"Result","text":"You are left with $$3m \\\\left(m^2-7m+10\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring11a-h4","type":"hint","dependencies":["a62d4a8factoring11a-h3"],"title":"Polynomial type","text":"$$m^2-7m+10$$ is a trinomial with leading coefficient $$1$$, so undo FOIL.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a62d4a8factoring12","title":"Factor Trinomials of the form $${ax}^2+bx+c$$ with a GCF","body":"Factor the expression below completely","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Factor Trinomials with Leading Coefficient Oth","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a62d4a8factoring12a","stepAnswer":["$$5x^{2\\\\left(x-3\\\\right)} \\\\left(x+5\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$5x^4+10x^3-75x^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5x^{2\\\\left(x-3\\\\right)} \\\\left(x+5\\\\right)$$","hints":{"DefaultPathway":[{"id":"a62d4a8factoring12a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5x^2$$"],"dependencies":[],"title":"Greatest common factor","text":"What is the greatest common factor?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring12a-h2","type":"hint","dependencies":["a62d4a8factoring12a-h1"],"title":"Factor","text":"Factor out the $$5x^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring12a-h3","type":"hint","dependencies":["a62d4a8factoring12a-h2"],"title":"Result","text":"You are left with $$5x^2 \\\\left(x^2+2x-15\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring12a-h4","type":"hint","dependencies":["a62d4a8factoring12a-h3"],"title":"Polynomial type","text":"$$x^2+2x-15$$ is a trinomial with leading coefficient $$1$$, so undo FOIL.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a62d4a8factoring13","title":"Factor Trinomials Using Trial and Error","body":"Factor the expression below","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Factor Trinomials with Leading Coefficient Oth","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a62d4a8factoring13a","stepAnswer":["$$\\\\left(2t+5\\\\right) \\\\left(t+1\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$2t^2+7t+5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(2t+5\\\\right) \\\\left(t+1\\\\right)$$","hints":{"DefaultPathway":[{"id":"a62d4a8factoring13a-h1","type":"hint","dependencies":[],"title":"Factor pairs of the first term","text":"The only factors for $$2t^2$$ are $$2t$$, $$1t$$ or $$-2t$$, $$-1t$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring13a-h2","type":"hint","dependencies":["a62d4a8factoring13a-h1"],"title":"Factor pairs for the third term","text":"The only factors of $$5$$ are $$1$$, $$5$$ or $$-1$$, $$-5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring13a-h3","type":"hint","dependencies":["a62d4a8factoring13a-h2"],"title":"Trial and Error","text":"Test all the combinations of the factors until the correct product is found","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a62d4a8factoring14","title":"Factor Trinomials Using Trial and Error","body":"Factor the expression below","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Factor Trinomials with Leading Coefficient Oth","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a62d4a8factoring14a","stepAnswer":["$$\\\\left(11x+1\\\\right) \\\\left(x+3\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$11x^2+34x+3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(11x+1\\\\right) \\\\left(x+3\\\\right)$$","hints":{"DefaultPathway":[{"id":"a62d4a8factoring14a-h1","type":"hint","dependencies":[],"title":"Factor pairs of the first term","text":"The only factors for $$11x^2$$ are $$11x$$, $$x$$ or $$-11x$$, $$-x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring14a-h2","type":"hint","dependencies":["a62d4a8factoring14a-h1"],"title":"Factor pairs for the third term","text":"The only factors of $$3$$ are $$1$$, $$3$$ or $$-1$$, $$-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring14a-h3","type":"hint","dependencies":["a62d4a8factoring14a-h2"],"title":"Trial and Error","text":"Test all the combinations of the factors until the correct product is found","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a62d4a8factoring15","title":"Factor Trinomials Using Trial and Error","body":"Factor the expression below","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Factor Trinomials with Leading Coefficient Oth","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a62d4a8factoring15a","stepAnswer":["$$(4w-1)(w-1)$$"],"problemType":"TextBox","stepTitle":"$$4w^2-5w+1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$(4w-1)(w-1)$$","hints":{"DefaultPathway":[{"id":"a62d4a8factoring15a-h1","type":"hint","dependencies":[],"title":"Factor pairs of the first term","text":"The factors for $$4w^2$$ are 4w, w or 2w, 2w","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring15a-h2","type":"hint","dependencies":["a62d4a8factoring15a-h1"],"title":"Factor pairs for the third term","text":"Find the factors if the last rem. Consider the signs. The coefficient of the middle term is negative, so we use the negative factors. The only factors of $$1$$ are $$-1$$, $$-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring15a-h3","type":"hint","dependencies":["a62d4a8factoring15a-h2"],"title":"Trial and Error","text":"Test all the combinations of the factors until the correct product is found","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a62d4a8factoring16","title":"Factor Trinomials Using Trial and Error","body":"Factor the expression below","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Factor Trinomials with Leading Coefficient Oth","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a62d4a8factoring16a","stepAnswer":["$$(3p-2)(2p-5)$$"],"problemType":"TextBox","stepTitle":"$$6p^2-19p+10$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$(3p-2)(2p-5)$$","hints":{"DefaultPathway":[{"id":"a62d4a8factoring16a-h1","type":"hint","dependencies":[],"title":"Factor pairs of the first term","text":"The factors for $$6p^2$$ are 3p,2p, or $$6p$$, $$1p$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring16a-h2","type":"hint","dependencies":["a62d4a8factoring16a-h1"],"title":"Factor pairs for the third term","text":"Find the factors if the last rem. Consider the signs. The coefficient of the middle term is negative, so we use the negative factors. The factors of $$10$$ are $$-10$$, $$-1$$ or $$-2$$, $$-5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring16a-h3","type":"hint","dependencies":["a62d4a8factoring16a-h2"],"title":"Trial and Error","text":"Test all the combinations of the factors until the correct product is found","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a62d4a8factoring17","title":"Factor Trinomials Using Trial and Error","body":"Factor the expression below","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Factor Trinomials with Leading Coefficient Oth","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a62d4a8factoring17a","stepAnswer":["$$\\\\left(4q+1\\\\right) \\\\left(q-2\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$4q^2-7q-2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(4q+1\\\\right) \\\\left(q-2\\\\right)$$","hints":{"DefaultPathway":[{"id":"a62d4a8factoring17a-h1","type":"hint","dependencies":[],"title":"Factor pairs of the first term","text":"The factors for $$4q^2$$ are 4q, 1q or 2q, 2q","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring17a-h2","type":"hint","dependencies":["a62d4a8factoring17a-h1"],"title":"Factor pairs for the third term","text":"The factors of $$-2$$ are $$-2$$, $$1$$ or $$-1$$, $$2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring17a-h3","type":"hint","dependencies":["a62d4a8factoring17a-h2"],"title":"Trial and Error","text":"Test all the combinations of the factors until the correct product is found","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a62d4a8factoring18","title":"Factor Trinomials Using Trial and Error","body":"Factor the expression below","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Factor Trinomials with Leading Coefficient Oth","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a62d4a8factoring18a","stepAnswer":["$$\\\\left(4p-3\\\\right) \\\\left(p+5\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$4p^2+17p-15$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(4p-3\\\\right) \\\\left(p+5\\\\right)$$","hints":{"DefaultPathway":[{"id":"a62d4a8factoring18a-h1","type":"hint","dependencies":[],"title":"Factor pairs of the first term","text":"The factors for $$4p^2$$ are $$4p$$, $$1p$$ or $$2p$$, $$2p$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring18a-h2","type":"hint","dependencies":["a62d4a8factoring18a-h1"],"title":"Factor pairs for the third term","text":"The factors of $$-15$$ are $$-15$$, $$1$$ or $$-1$$, $$15$$ or $$--3$$, $$5$$ or 3,-5","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring18a-h3","type":"hint","dependencies":["a62d4a8factoring18a-h2"],"title":"Trial and Error","text":"Test all the combinations of the factors until the correct product is found","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a62d4a8factoring19","title":"Factor Trinomials Using Trial and Error","body":"Factor the expression below","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Factor Trinomials with Leading Coefficient Oth","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a62d4a8factoring19a","stepAnswer":["$$16(x-1)(x-1)$$"],"problemType":"TextBox","stepTitle":"$$16x^2-32x+16$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16(x-1)(x-1)$$","hints":{"DefaultPathway":[{"id":"a62d4a8factoring19a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":[],"title":"Greatest common factor","text":"What is the greatest common factor?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring19a-h2","type":"hint","dependencies":["a62d4a8factoring19a-h1"],"title":"Factor","text":"Factor out the $$16$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring19a-h3","type":"hint","dependencies":["a62d4a8factoring19a-h2"],"title":"Polynomial type","text":"$$x^2-2x+1$$ is a trinomial with leading coefficient $$1$$, so undo FOIL.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a62d4a8factoring2","title":"Factoring Polynomials","body":"Factor completely.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Factor Trinomials with Leading Coefficient Oth","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a62d4a8factoring2a","stepAnswer":["$$4\\\\left(m+1\\\\right) \\\\left(m-2\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$4m^2-4m-8$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4\\\\left(m+1\\\\right) \\\\left(m-2\\\\right)$$","hints":{"DefaultPathway":[{"id":"a62d4a8factoring2a-h1","type":"hint","dependencies":[],"title":"Use a preliminary strategy","text":"As there is a greatest common factor(4), factor it out. The expression should look like this: $$4\\\\left(m^2-m-2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring2a-h2","type":"hint","dependencies":["a62d4a8factoring2a-h1"],"title":"Identify","text":"Inside the parentheses, is it a binomial, trinomial, or are there more than three terms? Since it is a trinomial with a coefficient of $$1$$, undo FOIL.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring2a-h3","type":"hint","dependencies":["a62d4a8factoring2a-h2"],"title":"Use factors","text":"Use the factors of $$-2$$ to finish FOIL.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring2a-h4","type":"hint","dependencies":["a62d4a8factoring2a-h3"],"title":"Answer","text":"The answer is $$4\\\\left(m+1\\\\right) \\\\left(m-2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a62d4a8factoring20","title":"Factor Trinomials Using Trial and Error","body":"Factor the expression below","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Factor Trinomials with Leading Coefficient Oth","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a62d4a8factoring20a","stepAnswer":["$$10q\\\\left(3q+2\\\\right) \\\\left(q+4\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$30q^3+140q^2+80q$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10q\\\\left(3q+2\\\\right) \\\\left(q+4\\\\right)$$","hints":{"DefaultPathway":[{"id":"a62d4a8factoring20a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["10q"],"dependencies":[],"title":"Greatest common factor","text":"What is the greatest common factor?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring20a-h2","type":"hint","dependencies":["a62d4a8factoring20a-h1"],"title":"Factor","text":"Factor out the 10q","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring20a-h3","type":"hint","dependencies":["a62d4a8factoring20a-h2"],"title":"Factor pairs of the first term","text":"The factors for $$3q^2$$ are 3q, 1q","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring20a-h4","type":"hint","dependencies":["a62d4a8factoring20a-h3"],"title":"Factor pairs for the third term","text":"The factors of $$-8$$ are $$1$$, $$8$$ or $$2$$, $$4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring20a-h5","type":"hint","dependencies":["a62d4a8factoring20a-h4"],"title":"Trial and Error","text":"Test all the combinations of the factors until the correct product is found","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a62d4a8factoring21","title":"Factor Trinomials Using the \'ac\' Method","body":"Factor the expression below","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Factor Trinomials with Leading Coefficient Oth","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a62d4a8factoring21a","stepAnswer":["$$\\\\left(5n+1\\\\right) \\\\left(n+4\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$5n^2+21n+4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(5n+1\\\\right) \\\\left(n+4\\\\right)$$","hints":{"DefaultPathway":[{"id":"a62d4a8factoring21a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":[],"title":"Product of ac","text":"What is the product of $$a c$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring21a-h2","type":"hint","dependencies":["a62d4a8factoring21a-h1"],"title":"Two numbers $$m$$ and $$n$$","text":"Find $$2$$ numbers that multiply to ac and add to $$b$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring21a-h3","type":"hint","dependencies":["a62d4a8factoring21a-h2"],"title":"Split middle term","text":"Split the middle term using $$m$$ and n: $${ax}^2+bx+c={ax}^2+mx+nx+c$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring21a-h4","type":"hint","dependencies":["a62d4a8factoring21a-h3"],"title":"Factor by grouping","text":"The expression you are left with is an expression with $$4$$ terms. Factor by grouping.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a62d4a8factoring22","title":"Factor Trinomials Using the \'ac\' Method","body":"Factor the expression below","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Factor Trinomials with Leading Coefficient Oth","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a62d4a8factoring22a","stepAnswer":["$$\\\\left(3z+1\\\\right) \\\\left(3z+4\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$9z^2+15z+4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(3z+1\\\\right) \\\\left(3z+4\\\\right)$$","hints":{"DefaultPathway":[{"id":"a62d4a8factoring22a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$36$$"],"dependencies":[],"title":"Product of ac","text":"What is the product of $$a c$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring22a-h2","type":"hint","dependencies":["a62d4a8factoring22a-h1"],"title":"Two numbers $$m$$ and $$n$$","text":"Find $$2$$ numbers that multiply to ac and add to $$b$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring22a-h3","type":"hint","dependencies":["a62d4a8factoring22a-h2"],"title":"Split middle term","text":"Split the middle term using $$m$$ and n: $${ax}^2+bx+c={ax}^2+mx+nx+c$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring22a-h4","type":"hint","dependencies":["a62d4a8factoring22a-h3"],"title":"Factor by grouping","text":"The expression you are left with is an expression with $$4$$ terms. Factor by grouping.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a62d4a8factoring23","title":"Factor Trinomials Using the \'ac\' Method","body":"Factor the expression below","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Factor Trinomials with Leading Coefficient Oth","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a62d4a8factoring23a","stepAnswer":["$$(2k-3)(2k-5)$$"],"problemType":"TextBox","stepTitle":"$$4k^2-16k+15$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$(2k-3)(2k-5)$$","hints":{"DefaultPathway":[{"id":"a62d4a8factoring23a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$60$$"],"dependencies":[],"title":"Product of ac","text":"What is the product of $$a c$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring23a-h2","type":"hint","dependencies":["a62d4a8factoring23a-h1"],"title":"Two numbers $$m$$ and $$n$$","text":"Find $$2$$ numbers that multiply to ac and add to $$b$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring23a-h3","type":"hint","dependencies":["a62d4a8factoring23a-h2"],"title":"Split middle term","text":"Split the middle term using $$m$$ and n: $${ax}^2+bx+c={ax}^2+mx+nx+c$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring23a-h4","type":"hint","dependencies":["a62d4a8factoring23a-h3"],"title":"Factor by grouping","text":"The expression you are left with is an expression with $$4$$ terms. Factor by grouping.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a62d4a8factoring24","title":"Factor Trinomials Using the \'ac\' Method","body":"Factor the expression below","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Factor Trinomials with Leading Coefficient Oth","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a62d4a8factoring24a","stepAnswer":["$$(5s-4)(s-1)$$"],"problemType":"TextBox","stepTitle":"$$5s^2-9s+4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$(5s-4)(s-1)$$","hints":{"DefaultPathway":[{"id":"a62d4a8factoring24a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":[],"title":"Product of ac","text":"What is the product of $$a c$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring24a-h2","type":"hint","dependencies":["a62d4a8factoring24a-h1"],"title":"Two numbers $$m$$ and $$n$$","text":"Find $$2$$ numbers that multiply to ac and add to $$b$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring24a-h3","type":"hint","dependencies":["a62d4a8factoring24a-h2"],"title":"Split middle term","text":"Split the middle term using $$m$$ and n: $${ax}^2+bx+c={ax}^2+mx+nx+c$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring24a-h4","type":"hint","dependencies":["a62d4a8factoring24a-h3"],"title":"Factor by grouping","text":"The expression you are left with is an expression with $$4$$ terms. Factor by grouping.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a62d4a8factoring25","title":"Factor Trinomials Using the \'ac\' Method","body":"Factor the expression below","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Factor Trinomials with Leading Coefficient Oth","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a62d4a8factoring25a","stepAnswer":["$$\\\\left(3y+5\\\\right) \\\\left(2y-3\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$6y^2+y-15$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(3y+5\\\\right) \\\\left(2y-3\\\\right)$$","hints":{"DefaultPathway":[{"id":"a62d4a8factoring25a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-90$$"],"dependencies":[],"title":"Product of ac","text":"What is the product of $$a c$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring25a-h2","type":"hint","dependencies":["a62d4a8factoring25a-h1"],"title":"Two numbers $$m$$ and $$n$$","text":"Find $$2$$ numbers that multiply to ac and add to $$b$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring25a-h3","type":"hint","dependencies":["a62d4a8factoring25a-h2"],"title":"Split middle term","text":"Split the middle term using $$m$$ and n: $${ax}^2+bx+c={ax}^2+mx+nx+c$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring25a-h4","type":"hint","dependencies":["a62d4a8factoring25a-h3"],"title":"Factor by grouping","text":"The expression you are left with is an expression with $$4$$ terms. Factor by grouping.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a62d4a8factoring3","title":"Factoring Polynomials","body":"Factor completely.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Factor Trinomials with Leading Coefficient Oth","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a62d4a8factoring3a","stepAnswer":["$$5\\\\left(k+2\\\\right) \\\\left(k-5\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$5k^2-15k-50$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5\\\\left(k+2\\\\right) \\\\left(k-5\\\\right)$$","hints":{"DefaultPathway":[{"id":"a62d4a8factoring3a-h1","type":"hint","dependencies":[],"title":"Use a preliminary strategy","text":"As there is a greatest common factor(5), factor it out. The expression should look like this: $$5\\\\left(k^2-3k-10\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring3a-h2","type":"hint","dependencies":["a62d4a8factoring3a-h1"],"title":"Identify","text":"Inside the parentheses, is it a binomial, trinomial, or are there more than three terms? Since it is a trinomial with a coefficient of $$1$$, undo FOIL.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring3a-h3","type":"hint","dependencies":["a62d4a8factoring3a-h2"],"title":"Use factors","text":"Use the factors of $$-10$$ to finish FOIL.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring3a-h4","type":"hint","dependencies":["a62d4a8factoring3a-h3"],"title":"Answer","text":"The answer is $$5\\\\left(k+2\\\\right) \\\\left(k-5\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a62d4a8factoring4","title":"Factoring Polynomials","body":"Factor completely.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Factor Trinomials with Leading Coefficient Oth","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a62d4a8factoring4a","stepAnswer":["$$4(y-2)(y-7)$$"],"problemType":"TextBox","stepTitle":"$$4y^2-36y+56$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4(y-2)(y-7)$$","hints":{"DefaultPathway":[{"id":"a62d4a8factoring4a-h1","type":"hint","dependencies":[],"title":"Use a preliminary strategy","text":"As there is a greatest common factor(4), factor it out. The expression should look like this: $$4\\\\left(y^2-9y+12\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring4a-h2","type":"hint","dependencies":["a62d4a8factoring4a-h1"],"title":"Identify","text":"Inside the parentheses, is it a binomial, trinomial, or are there more than three terms? Since it is a trinomial with a coefficient of $$1$$, undo FOIL.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring4a-h3","type":"hint","dependencies":["a62d4a8factoring4a-h2"],"title":"Use factors","text":"Use the factors of $$12$$ to finish FOIL.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring4a-h4","type":"hint","dependencies":["a62d4a8factoring4a-h3"],"title":"Answer","text":"The answer is $$4(y-2)(y-7)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a62d4a8factoring5","title":"Factoring Polynomials","body":"Factor completely.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Factor Trinomials with Leading Coefficient Oth","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a62d4a8factoring5a","stepAnswer":["$$3(r-1)(r-2)$$"],"problemType":"TextBox","stepTitle":"$$3r^2-9r+6$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3(r-1)(r-2)$$","hints":{"DefaultPathway":[{"id":"a62d4a8factoring5a-h1","type":"hint","dependencies":[],"title":"Use a preliminary strategy","text":"As there is a greatest common factor(3), factor it out. The expression should look like this: $$3\\\\left(r^2-3r+2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring5a-h2","type":"hint","dependencies":["a62d4a8factoring5a-h1"],"title":"Identify","text":"Inside the parentheses, is it a binomial, trinomial, or are there more than three terms? Since it is a trinomial with a coefficient of $$1$$, undo FOIL.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring5a-h3","type":"hint","dependencies":["a62d4a8factoring5a-h2"],"title":"Use factors","text":"Use the factors of $$2$$ to finish FOIL.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring5a-h4","type":"hint","dependencies":["a62d4a8factoring5a-h3"],"title":"Answer","text":"The answer is $$3(r-1)(r-2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a62d4a8factoring6","title":"Factoring Polynomials","body":"Factor completely.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Factor Trinomials with Leading Coefficient Oth","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a62d4a8factoring6a","stepAnswer":["$$4u\\\\left(u-1\\\\right) \\\\left(u+5\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$4u^3+16u^2-20u$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4u\\\\left(u-1\\\\right) \\\\left(u+5\\\\right)$$","hints":{"DefaultPathway":[{"id":"a62d4a8factoring6a-h1","type":"hint","dependencies":[],"title":"Use a preliminary strategy","text":"As there is a greatest common factor, which in this case includes a variable(4u), factor it out. The expression should look like this: $$4u\\\\left(u^2+4u-5\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring6a-h2","type":"hint","dependencies":["a62d4a8factoring6a-h1"],"title":"Identify","text":"Inside the parentheses, is it a binomial, trinomial, or are there more than three terms? Since it is a trinomial with a coefficient of $$1$$, undo FOIL.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring6a-h3","type":"hint","dependencies":["a62d4a8factoring6a-h2"],"title":"Use factors","text":"Use the factors of $$5$$ to finish FOIL.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring6a-h4","type":"hint","dependencies":["a62d4a8factoring6a-h3"],"title":"Answer","text":"The answer is $$4u\\\\left(u-1\\\\right) \\\\left(u+5\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a62d4a8factoring7","title":"Factoring Polynomials","body":"Factor completely.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Factor Trinomials with Leading Coefficient Oth","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a62d4a8factoring7a","stepAnswer":["$$\\\\left(2a+3\\\\right) \\\\left(a+1\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$2a^2+5a+3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(2a+3\\\\right) \\\\left(a+1\\\\right)$$","hints":{"DefaultPathway":[{"id":"a62d4a8factoring7a-h1","type":"hint","dependencies":[],"title":"Find factor pairs of the first term","text":"The only factors of $$2a^2$$ are a and 2a. Since its only that pair, lets put them in the parentheses.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring7a-h2","type":"hint","dependencies":["a62d4a8factoring7a-h1"],"title":"Find factor pairs of the third term","text":"The only factors of $$3$$ are $$1$$ and $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring7a-h3","type":"hint","dependencies":["a62d4a8factoring7a-h2"],"title":"Test","text":"Test all possibly combinations of the factors until the correct product is found.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring7a-h4","type":"hint","dependencies":["a62d4a8factoring7a-h3"],"title":"Answer","text":"The answer is $$\\\\left(2a+3\\\\right) \\\\left(a+1\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a62d4a8factoring8","title":"Factoring Polynomials","body":"Factor completely.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Factor Trinomials with Leading Coefficient Oth","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a62d4a8factoring8a","stepAnswer":["$$\\\\left(4b+1\\\\right) \\\\left(b+1\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$4b^2+5b+1$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(4b+1\\\\right) \\\\left(b+1\\\\right)$$","hints":{"DefaultPathway":[{"id":"a62d4a8factoring8a-h1","type":"hint","dependencies":[],"title":"Find factor pairs of the first term","text":"The factors of $$4b^2$$ are $$b$$ and $$4b$$, and $$2b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring8a-h2","type":"hint","dependencies":["a62d4a8factoring8a-h1"],"title":"Find factor pairs of the third term","text":"The only factor of $$1$$ is $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring8a-h3","type":"hint","dependencies":["a62d4a8factoring8a-h2"],"title":"Test","text":"Test all possibly combinations of the factors until the correct product is found.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring8a-h4","type":"hint","dependencies":["a62d4a8factoring8a-h3"],"title":"Answer","text":"The answer is $$\\\\left(4b+1\\\\right) \\\\left(b+1\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a62d4a8factoring9","title":"Factoring Polynomials","body":"Factor completely.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Factor Trinomials with Leading Coefficient Oth","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a62d4a8factoring9a","stepAnswer":["$$(3b-5)(2b-1)$$"],"problemType":"TextBox","stepTitle":"$$6b^2-13b+5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$(3b-5)(2b-1)$$","hints":{"DefaultPathway":[{"id":"a62d4a8factoring9a-h1","type":"hint","dependencies":[],"title":"Find factor pairs of the first term","text":"The factors of $$6b^2$$ are $$2b$$ and $$3b$$, and $$b$$ and $$6b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring9a-h2","type":"hint","dependencies":["a62d4a8factoring9a-h1"],"title":"Find factor pairs of the third term","text":"The only factors of $$5$$ are1 and $$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring9a-h3","type":"hint","dependencies":["a62d4a8factoring9a-h2"],"title":"Test","text":"Test all possibly combinations of the factors until the correct product is found.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a62d4a8factoring9a-h4","type":"hint","dependencies":["a62d4a8factoring9a-h3"],"title":"Answer","text":"The answer is $$(3b-5)(2b-1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a63563bnewton1","title":"Finding a Root of a Polynomial","body":"Use Newton\u2019s method to approximate a root of $$f(x)=x^3-3x+1$$ in the interval [1,2]. Let $$x_0=2$$ and find $$x_1$$, $$x_2$$, $$x_3$$, $$x_4$$, and $$x_5$$. This problem is intended to be done using a calculator.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.9 Newton\u2019s Method","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a63563bnewton1a","stepAnswer":["1.666666667"],"problemType":"TextBox","stepTitle":"Find $$x_1$$ accurate to $$9$$ decimal places.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$1.666666667$$","hints":{"DefaultPathway":[{"id":"a63563bnewton1a-h1","type":"hint","dependencies":[],"title":"Derivative of f(x)","text":"The derivative of f(x) would be needed approximate a root of f(x) using Newton\'s method.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a63563bnewton1a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$f\'(x)=3x^2-3$$"],"dependencies":["a63563bnewton1a-h1"],"title":"Find the Derivative of f(x)","text":"What is the derivative of f(x)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$f\'(x)=3x^2-3$$","$$f\'(x)=3x-3$$","$$f\'(x)=x^3-3$$","$$f\'(x)=3x-3x+1$$"]},{"id":"a63563bnewton1a-h3","type":"hint","dependencies":["a63563bnewton1a-h2"],"title":"Newton\'s Method With $$n=1$$","text":"To find $$x_1$$, use the equation $$x_n=x_n-1-\\\\frac{f{\\\\left(x_n-1\\\\right)}}{\\\\operatorname{f\'}\\\\left(x_n-1\\\\right)}$$ with $$n=1$$ and $$x_0=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a63563bnewton1b","stepAnswer":["1.548611111"],"problemType":"TextBox","stepTitle":"Find $$x_2$$ accurate to $$9$$ decimal places.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$1.548611111$$","hints":{"DefaultPathway":[{"id":"a63563bnewton1b-h1","type":"hint","dependencies":[],"title":"Newton\'s Method With $$n=2$$","text":"To find $$x_2$$, use the equation $$x_n=x_n-1-\\\\frac{f{\\\\left(x_n-1\\\\right)}}{\\\\operatorname{f\'}\\\\left(x_n-1\\\\right)}$$ with $$n=2$$ and the value of $$x_1$$ stored on the calculator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a63563bnewton1c","stepAnswer":["1.532390162"],"problemType":"TextBox","stepTitle":"Find $$x_3$$ accurate to $$9$$ decimal places.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$1.532390162$$","hints":{"DefaultPathway":[{"id":"a63563bnewton1c-h1","type":"hint","dependencies":[],"title":"Newton\'s Method With $$n=3$$","text":"To find $$x_3$$, use the equation $$x_n=x_n-1-\\\\frac{f{\\\\left(x_n-1\\\\right)}}{\\\\operatorname{f\'}\\\\left(x_n-1\\\\right)}$$ with $$n=3$$ and the value of $$x_2$$ stored on the calculator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a63563bnewton1d","stepAnswer":["1.532088989"],"problemType":"TextBox","stepTitle":"Find $$x_4$$ accurate to $$9$$ decimal places.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$1.532088989$$","hints":{"DefaultPathway":[{"id":"a63563bnewton1d-h1","type":"hint","dependencies":[],"title":"Newton\'s Method With $$n=4$$","text":"To find $$x_4$$, use the equation $$x_n=x_n-1-\\\\frac{f{\\\\left(x_n-1\\\\right)}}{\\\\operatorname{f\'}\\\\left(x_n-1\\\\right)}$$ with $$n=4$$ and the value of $$x_3$$ stored on the calculator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a63563bnewton1e","stepAnswer":["1.532088886"],"problemType":"TextBox","stepTitle":"Find $$x_5$$ accurate to $$9$$ decimal places.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$1.532088886$$","hints":{"DefaultPathway":[{"id":"a63563bnewton1e-h1","type":"hint","dependencies":[],"title":"Newton\'s Method With $$n=5$$","text":"To find $$x_5$$, use the equation $$x_n=x_n-1-\\\\frac{f{\\\\left(x_n-1\\\\right)}}{\\\\operatorname{f\'}\\\\left(x_n-1\\\\right)}$$ with $$n=5$$ and the value of $$x_4$$ stored on the calculator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a63563bnewton1e-h2","type":"hint","dependencies":["a63563bnewton1e-h1"],"title":"Same Value With Subsequent Application of Newton\'s Method","text":"$$x_6$$ also approximates to $$1.532088886$$. We note that we obtained the same value for $$x_5$$ and $$x_6$$. Therefore, any subsequent application of Newton\u2019s method will most likely give the same value for $$x_n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a63563bnewton10","title":"Computing $$x_1$$ and $$x_2$$","body":"Compute $$x_1$$ and $$x_2$$ using the specified iterative method: $$x_n+1=\\\\frac{1}{\\\\sqrt{x_n}}$$. This problem is intended to be done using a calculator.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.9 Newton\u2019s Method","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a63563bnewton10a","stepAnswer":["$$1.291$$"],"problemType":"TextBox","stepTitle":"$$x_1$$ with $$x_0=0.6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1.291$$","hints":{"DefaultPathway":[{"id":"a63563bnewton10a-h1","type":"hint","dependencies":[],"title":"Formula to Find $$x_1$$ with $$x_0=0.6$$","text":"To compute $$x_1$$, substitute $$x_0$$ with $$0.6$$ into $$x_1=\\\\frac{1}{\\\\sqrt{x_0}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a63563bnewton10b","stepAnswer":["$$0.8801$$"],"problemType":"TextBox","stepTitle":"$$x_2$$ with $$x_0=0.6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.8801$$","hints":{"DefaultPathway":[{"id":"a63563bnewton10b-h1","type":"hint","dependencies":[],"title":"Formula to Find $$x_2$$ with $$x_0=0.6$$","text":"To compute $$x_2$$, substitute the value of $$x_1$$ into $$x_2=\\\\frac{1}{\\\\sqrt{x_1}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a63563bnewton10c","stepAnswer":["$$0.7071$$"],"problemType":"TextBox","stepTitle":"$$x_1$$ with $$x_0=2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.7071$$","hints":{"DefaultPathway":[{"id":"a63563bnewton10c-h1","type":"hint","dependencies":[],"title":"Formula to Find $$x_1$$ with $$x_0=2$$","text":"To compute $$x_1$$, substitute $$x_0$$ with $$2$$ into $$x_1=\\\\frac{1}{\\\\sqrt{x_0}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a63563bnewton10d","stepAnswer":["$$1.189$$"],"problemType":"TextBox","stepTitle":"$$x_2$$ with $$x_0=2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1.189$$","hints":{"DefaultPathway":[{"id":"a63563bnewton10d-h1","type":"hint","dependencies":[],"title":"Formula to Find $$x_2$$ with $$x_0=2$$","text":"To compute $$x_2$$, substitute the value of $$x_1$$ into $$x_2=\\\\frac{1}{\\\\sqrt{x_1}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a63563bnewton11","title":"Computing $$x_1$$ and $$x_2$$","body":"Compute $$x_1$$ and $$x_2$$ using the specified iterative method: $$x_n+1={x_n}^2+x_n-2$$. This problem is intended to be done using a calculator.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.9 Newton\u2019s Method","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a63563bnewton11a","stepAnswer":["$$\\\\frac{-26}{25}$$"],"problemType":"TextBox","stepTitle":"$$x_1$$ with $$x_0=0.6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-26}{25}$$","hints":{"DefaultPathway":[{"id":"a63563bnewton11a-h1","type":"hint","dependencies":[],"title":"Formula to Find $$x_1$$ with $$x_0=0.6$$","text":"To compute $$x_1$$, substitute $$x_0$$ with $$0.6$$ into $$x_1={x_0}^2+x_0-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a63563bnewton11b","stepAnswer":["$$\\\\frac{-1224}{625}$$"],"problemType":"TextBox","stepTitle":"$$x_2$$ with $$x_0=0.6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-1224}{625}$$","hints":{"DefaultPathway":[{"id":"a63563bnewton11b-h1","type":"hint","dependencies":[],"title":"Formula to Find $$x_2$$ with $$x_0=0.6$$","text":"To compute $$x_2$$, substitute the value of $$x_1$$ into $$x_2={x_1}^2+x_1-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a63563bnewton11c","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"$$x_1$$ with $$x_0=2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a63563bnewton11c-h1","type":"hint","dependencies":[],"title":"Formula to Find $$x_1$$ with $$x_0=2$$","text":"To compute $$x_1$$, substitute $$x_0$$ with $$2$$ into $$x_1={x_0}^2+x_0-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a63563bnewton11d","stepAnswer":["$$18$$"],"problemType":"TextBox","stepTitle":"$$x_2$$ with $$x_0=2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$18$$","hints":{"DefaultPathway":[{"id":"a63563bnewton11d-h1","type":"hint","dependencies":[],"title":"Formula to Find $$x_2$$ with $$x_0=2$$","text":"To compute $$x_2$$, substitute the value of $$x_1$$ into $$x_2={x_1}^2+x_1-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a63563bnewton12","title":"Computing $$x_1$$ and $$x_2$$","body":"Compute $$x_1$$ and $$x_2$$ using the specified iterative method: $$x_n+1=|x_n|$$. This problem is intended to be done using a calculator.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.9 Newton\u2019s Method","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a63563bnewton12a","stepAnswer":["$$\\\\frac{6}{10}$$"],"problemType":"TextBox","stepTitle":"$$x_1$$ with $$x_0=0.6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{6}{10}$$","hints":{"DefaultPathway":[{"id":"a63563bnewton12a-h1","type":"hint","dependencies":[],"title":"Formula to Find $$x_1$$ with $$x_0=0.6$$","text":"To compute $$x_1$$, substitute $$x_0$$ with $$0.6$$ into $$x_1=|x_0|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a63563bnewton12b","stepAnswer":["$$\\\\frac{6}{10}$$"],"problemType":"TextBox","stepTitle":"$$x_2$$ with $$x_0=0.6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{6}{10}$$","hints":{"DefaultPathway":[{"id":"a63563bnewton12b-h1","type":"hint","dependencies":[],"title":"Formula to Find $$x_2$$ with $$x_0=0.6$$","text":"To compute $$x_2$$, insert the value of $$x_1$$ into $$x_2=|x_1|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a63563bnewton12c","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"$$x_1$$ with $$x_0=2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a63563bnewton12c-h1","type":"hint","dependencies":[],"title":"Formula to Find $$x_1$$ with $$x_0=2$$","text":"To compute $$x_1$$, substitute $$x_0$$ with $$2$$ into $$x_1=|x_0|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a63563bnewton12d","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"$$x_2$$ with $$x_0=2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a63563bnewton12d-h1","type":"hint","dependencies":[],"title":"Formula to Find $$x_2$$ with $$x_0=2$$","text":"To compute $$x_2$$, insert the value of $$x_1$$ into $$x_2=|x_1|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a63563bnewton13","title":"Initial Guess","body":"Solve to four decimal places using Newton\u2019s method and a computer or calculator. Choose any initial guess $$x_0$$ that is not the exact root. This problem is intended to be done using a calculator.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.9 Newton\u2019s Method","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a63563bnewton13a","stepAnswer":["$$3.1623$$ or $$-3.1623$$"],"problemType":"MultipleChoice","stepTitle":"$$x^4-100=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$3.1623$$ or $$-3.1623$$","choices":["$$3.1623$$ or $$-3.1623$$","$$3.3510$$ or $$-3.3510$$","$$3.3$$ or $$-3.3$$","$$3.1545$$ or $$-3.1545$$"],"hints":{"DefaultPathway":[{"id":"a63563bnewton13a-h1","type":"hint","dependencies":[],"title":"Newton\'s Method","text":"The formula for Newton\'s Method is given by $$x_n+1=x_n-\\\\frac{f{\\\\left(x_n\\\\right)}}{\\\\operatorname{f\'}\\\\left(x_n\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a63563bnewton13a-h2","type":"hint","dependencies":["a63563bnewton13a-h1"],"title":"Find the Derivative of f(x)","text":"The derivative of f(x) would be needed to use Newton\'s method.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a63563bnewton13a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$f\'(x)=4x^3$$"],"dependencies":["a63563bnewton13a-h2"],"title":"Find the Derivative of f(x)","text":"What is the derivative of $$f(x)=x^4-100$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$f\'(x)=4x$$","$$f\'(x)=4x^3-100$$","$$f\'(x)=4x-100$$","$$f\'(x)=4x^3$$"]}]}}]},{"id":"a63563bnewton14","title":"Initial Guess","body":"Solve to four decimal places using Newton\u2019s method and a computer or calculator. Choose any initial guess $$x_0$$ that is not the exact root. This problem is intended to be done using a calculator.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.9 Newton\u2019s Method","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a63563bnewton14a","stepAnswer":["$$0$$, $$-1$$ or $$1$$"],"problemType":"MultipleChoice","stepTitle":"$$x^3-x=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$0$$, $$-1$$ or $$1$$","choices":["$$-0.5$$, $$0$$ , or $$0.5$$","$$-2$$, $$0$$, or $$2$$","$$0$$, $$-1$$ or $$1$$","$$0$$, $$-1$$, or $$1$$","There is no solution to the equation."],"hints":{"DefaultPathway":[{"id":"a63563bnewton14a-h1","type":"hint","dependencies":[],"title":"Newton\'s Method","text":"The formula for Newton\'s Method is given by $$x_n+1=x_n-\\\\frac{f{\\\\left(x_n\\\\right)}}{\\\\operatorname{f\'}\\\\left(x_n\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a63563bnewton14a-h2","type":"hint","dependencies":["a63563bnewton14a-h1"],"title":"Find the Derivative of f(x)","text":"The derivative of f(x) would be needed to use Newton\'s method.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a63563bnewton14a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$f\'(x)=3x^2-1$$"],"dependencies":["a63563bnewton14a-h2"],"title":"Find the Derivative of f(x)","text":"What is the derivative of $$f(x)=x^3-x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$f\'(x)=3x^2-1$$","$$f\'(x)=3x-1$$","$$f\'(x)=3x^2$$","$$f\'(x)=3x-1$$"]}]}}]},{"id":"a63563bnewton15","title":"Fixed Points","body":"Use Newton\u2019s method to find the fixed points of the function where $$f(x)=x;$$ round to three decimals. This problem is intended to be done using a calculator.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.9 Newton\u2019s Method","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a63563bnewton15a","stepAnswer":["4.493"],"problemType":"TextBox","stepTitle":"tan(x) on $$x=(\\\\frac{\\\\pi}{2},\\\\frac{3\\\\pi}{2})$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$4.493$$","hints":{"DefaultPathway":[{"id":"a63563bnewton15a-h1","type":"hint","dependencies":[],"title":"Newton\'s Method","text":"The formula for Newton\'s Method is given by $$x_n+1=x_n-\\\\frac{f{\\\\left(x_n\\\\right)}}{\\\\operatorname{f\'}\\\\left(x_n\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a63563bnewton16","title":"Fixed Points","body":"Use Newton\u2019s method to find the fixed points of the function where $$f(x)=x;$$ round to three decimals. This problem is intended to be done using a calculator.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.9 Newton\u2019s Method","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a63563bnewton16a","stepAnswer":["$$0.159$$, $$3.146$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\ln(x)+2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$0.159$$, $$3.146$$","choices":["$$0.159$$, $$3.146$$","$$0.159$$, $$3.147$$","$$0.158$$, $$3.146$$","$$0.159$$, $$3.145$$"],"hints":{"DefaultPathway":[{"id":"a63563bnewton16a-h1","type":"hint","dependencies":[],"title":"Newton\'s Method","text":"The formula for Newton\'s Method is given by $$x_n+1=x_n-\\\\frac{f{\\\\left(x_n\\\\right)}}{\\\\operatorname{f\'}\\\\left(x_n\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a63563bnewton17","title":"Formulation of Newton\'s Method","body":"Newton\u2019s method can be used to find maxima and minima of functions in addition to the roots. In this case apply Newton\u2019s method to the derivative function f\u2032(x) to find its roots, instead of the original function. Consider the formulation of the method.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.9 Newton\u2019s Method","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a63563bnewton17a","stepAnswer":["True"],"problemType":"MultipleChoice","stepTitle":"State whether the statement is true or false: We need f to be twice continuously differentiable as a necessary additional restriction on the function f.","stepBody":"","answerType":"string","variabilization":{},"choices":["True","False"],"hints":{"DefaultPathway":[]}}]},{"id":"a63563bnewton18","title":"Local Minima and Maxima","body":"Use Newton\u2019s method to find the location of the local minima; round to three decimals.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.9 Newton\u2019s Method","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a63563bnewton18a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"Minimum of $$f(x)=3x^3+2x^2-16$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a63563bnewton18a-h1","type":"hint","dependencies":[],"title":"Newton\'s Method","text":"The formula for Newton\'s Method is given by $$x_n+1=x_n-\\\\frac{f{\\\\left(x_n\\\\right)}}{\\\\operatorname{f\'}\\\\left(x_n\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a63563bnewton19","title":"Local Minima and Maxima","body":"Use Newton\u2019s method to find the location of the local maxima; round to three decimals.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.9 Newton\u2019s Method","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a63563bnewton19a","stepAnswer":["$$-1$$"],"problemType":"TextBox","stepTitle":"Maximum of $$f(x)=x+\\\\frac{1}{x}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1$$","hints":{"DefaultPathway":[{"id":"a63563bnewton19a-h1","type":"hint","dependencies":[],"title":"Newton\'s Method","text":"The formula for Newton\'s Method is given by $$x_n+1=x_n-\\\\frac{f{\\\\left(x_n\\\\right)}}{\\\\operatorname{f\'}\\\\left(x_n\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a63563bnewton2","title":"Finding a Square Root","body":"Use Newton\u2019s method to approximate $$\\\\sqrt{2}$$. Let $$f(x)=x^2-2$$, let $$x_0=2$$, and calculate $$x_1$$, $$x_2$$, $$x_3$$, $$x_4$$, $$x_5$$. We note that since $$f(x)=x^2-2$$ has a zero at $$\\\\sqrt{2}$$, the initial value $$x_0=2$$ is a reasonable choice to approximate $$\\\\sqrt{2}$$. This problem is intended to be done using a calculator.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.9 Newton\u2019s Method","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a63563bnewton2a","stepAnswer":["$$1.5$$"],"problemType":"TextBox","stepTitle":"Calculate $$x_1$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1.5$$","hints":{"DefaultPathway":[{"id":"a63563bnewton2a-h1","type":"hint","dependencies":[],"title":"Find the Derivative of f(x)","text":"The derivative of f(x) would be needed approximate $$\\\\sqrt{2}$$ using Newton\'s method.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a63563bnewton2a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$f\'(x)=2x$$"],"dependencies":["a63563bnewton2a-h1"],"title":"Find the Derivative of f(x)","text":"What is the derivative of $$f(x)=x^2-2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$f\'(x)=2x$$","$$f\'(x)=x^2$$","$$f\'(x)=2x^2-2$$","$$f\'(x)=2x^2$$"]},{"id":"a63563bnewton2a-h3","type":"hint","dependencies":["a63563bnewton2a-h2"],"title":"Modified Newton\'s Method","text":"After finding the derivative of f(x), we can rewrite $$x_n=x_n-1-\\\\frac{f{\\\\left(x_n-1\\\\right)}}{\\\\operatorname{f\'}\\\\left(x_n-1\\\\right)}$$ as $$x_n=\\\\frac{1}{2} \\\\left(x_n-1+\\\\frac{2}{x_n}-1\\\\right)$$. x_n=x(n-1)-(f(x_n-1)/f\'(x_n-1) $$=x_n-1-\\\\frac{{x_n}^2-2}{2} x_n-1$$ $$=\\\\frac{1}{2\\\\left(x_n-1\\\\right)}+\\\\frac{1}{x_n}-1$$ $$=\\\\frac{1}{2\\\\left(x_n-1+\\\\frac{2}{x_n}-1\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a63563bnewton2a-h4","type":"hint","dependencies":["a63563bnewton2a-h3"],"title":"Newton\'s Method With $$n=1$$","text":"To find $$x_1$$, use the equation $$x_n=\\\\frac{1}{2} \\\\left(x_n-1+\\\\frac{2}{x_n}-1\\\\right)$$ with $$n=1$$ and $$x_0=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a63563bnewton2b","stepAnswer":["1.416666667"],"problemType":"TextBox","stepTitle":"Calculate $$x_2$$ accurate to $$9$$ decimal places.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$1.416666667$$","hints":{"DefaultPathway":[{"id":"a63563bnewton2b-h1","type":"hint","dependencies":[],"title":"Newton\'s Method With $$n=2$$","text":"To find $$x_2$$, use the equation $$x_n=\\\\frac{1}{2} \\\\left(x_n-1+\\\\frac{2}{x_n}-1\\\\right)$$ with $$n=2$$ and the value of $$x_1$$ stored on the calculator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a63563bnewton2c","stepAnswer":["1.414215686"],"problemType":"TextBox","stepTitle":"Calculate $$x_3$$ accurate to $$9$$ decimal places.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$1.414215686$$","hints":{"DefaultPathway":[{"id":"a63563bnewton2c-h1","type":"hint","dependencies":[],"title":"Newton\'s Method With $$n=3$$","text":"To find $$x_3$$, use the equation $$x_n=\\\\frac{1}{2} \\\\left(x_n-1+\\\\frac{2}{x_n}-1\\\\right)$$ with $$n=3$$ and the value of $$x_2$$ stored on the calculator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a63563bnewton2d","stepAnswer":["1.414213562"],"problemType":"TextBox","stepTitle":"Calculate $$x_4$$ accurate to $$9$$ decimal places.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$1.414213562$$","hints":{"DefaultPathway":[{"id":"a63563bnewton2d-h1","type":"hint","dependencies":[],"title":"Newton\'s Method With $$n=4$$","text":"To find $$x_4$$, use the equation $$x_n=\\\\frac{1}{2} \\\\left(x_n-1+\\\\frac{2}{x_n}-1\\\\right)$$ with $$n=4$$ and the value of $$x_3$$ stored on the calculator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a63563bnewton2e","stepAnswer":["1.414213562"],"problemType":"TextBox","stepTitle":"Calculate $$x_5$$ accurate to $$9$$ decimal places.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$1.414213562$$","hints":{"DefaultPathway":[{"id":"a63563bnewton2e-h1","type":"hint","dependencies":[],"title":"Newton\'s Method With $$n=5$$","text":"To find $$x_5$$, use the equation $$x_n=\\\\frac{1}{2} \\\\left(x_n-1+\\\\frac{2}{x_n}-1\\\\right)$$ with $$n=5$$ and the value of $$x_4$$ stored on the calculator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a63563bnewton2e-h2","type":"hint","dependencies":["a63563bnewton2e-h1"],"title":"Subsequent Application of Newton\'s Method","text":"Since we obtained the same value for $$x_4$$ and $$x_5$$, it is unlikely that the value $$x_n$$ will change on any subsequent application of Newton\u2019s method.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a63563bnewton2f","stepAnswer":["sqrt(2)\u22481.414213562"],"problemType":"MultipleChoice","stepTitle":"What can we conclude about $$\\\\sqrt{2}$$?","stepBody":"","answerType":"string","variabilization":{},"choices":["sqrt(2)\u22481.414213562","sqrt(2)\u22481.414215686","sqrt(2)\u22481.416666667","sqrt(2)\u22481.42"],"hints":{"DefaultPathway":[]}}]},{"id":"a63563bnewton20","title":"Local Minima and Maxima","body":"Use Newton\u2019s method to find the location of the local maxima; round to three decimals. This problem is intended to be done using a calculator.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.9 Newton\u2019s Method","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a63563bnewton20a","stepAnswer":["5.619"],"problemType":"TextBox","stepTitle":"Maximum of f(x)=(sqrt(x)-sqrt(3,x))/x. (Three decimal places)","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$5.619$$","hints":{"DefaultPathway":[{"id":"a63563bnewton20a-h1","type":"hint","dependencies":[],"title":"Newton\'s Method","text":"The formula for Newton\'s Method is given by $$x_n+1=x_n-\\\\frac{f{\\\\left(x_n\\\\right)}}{\\\\operatorname{f\'}\\\\left(x_n\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a63563bnewton21","title":"Local Minima and Maxima","body":"Use Newton\u2019s method to find the location of the local minima and/or maxima of the following functions; round to three decimals. This problem is intended to be done using a calculator.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.9 Newton\u2019s Method","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a63563bnewton21a","stepAnswer":["-1.326"],"problemType":"TextBox","stepTitle":"Minimum of $$f(x)=x^4+x^3+3x^2+12x+6$$. (Three decimal places)","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-1.326$$","hints":{"DefaultPathway":[{"id":"a63563bnewton21a-h1","type":"hint","dependencies":[],"title":"Newton\'s Method","text":"The formula for Newton\'s Method is given by $$x_n+1=x_n-\\\\frac{f{\\\\left(x_n\\\\right)}}{\\\\operatorname{f\'}\\\\left(x_n\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a63563bnewton22","title":"Solve the Equation","body":"Use the specified method to solve the equation. If it does not work, explain why it does not work.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.9 Newton\u2019s Method","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a63563bnewton22a","stepAnswer":["There is no solution to the equation."],"problemType":"MultipleChoice","stepTitle":"Newton\u2019s method, $$0=e^x$$","stepBody":"","answerType":"string","variabilization":{},"choices":["There is no solution to the equation.","It enters a cycle."],"hints":{"DefaultPathway":[{"id":"a63563bnewton22a-h1","type":"hint","dependencies":[],"title":"Derivative of f(x)","text":"The derivative of f(x) would be needed to use Newton\'s method.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a63563bnewton22a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$f\'(x)=e^x$$"],"dependencies":["a63563bnewton22a-h1"],"title":"Derivative of f(x)","text":"What is the derivative of $$f(x)=e^x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$f\'(x)={xe}^x$$","$$f\'(x)=e$$","$$f\'(x)=e^x$$","$$f\'(x)=xe$$"]},{"id":"a63563bnewton22a-h3","type":"hint","dependencies":["a63563bnewton22a-h2"],"title":"Failure of Newton\'s Method","text":"Consider what the consequence is when f\'(x) is the same as f(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a63563bnewton23","title":"Solve the Equation","body":"Use the specified method to solve the equation. If it does not work, explain why it does not work.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.9 Newton\u2019s Method","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a63563bnewton23a","stepAnswer":["It enters a cycle."],"problemType":"MultipleChoice","stepTitle":"Solving $$x_n+1=-\\\\left({x_n}^3\\\\right)$$ starting at $$x_0=-1$$","stepBody":"","answerType":"string","variabilization":{},"choices":["It enters a cycle.","There is no solution to the equation."],"hints":{"DefaultPathway":[{"id":"a63563bnewton23a-h1","type":"hint","dependencies":[],"title":"Find $$x_1$$","text":"First find $$x_1$$. To find $$x_1$$, use the formula $$x_n=F\\\\left(x_n-1\\\\right)$$ with $$n=1$$ and $$x_0=-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a63563bnewton23a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a63563bnewton23a-h1"],"title":"Find $$x_1$$","text":"What is the value of $$x_1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a63563bnewton23a-h3","type":"hint","dependencies":["a63563bnewton23a-h2"],"title":"Find $$x_2$$","text":"Second, find $$x_2$$. To find $$x_2$$, use the formula $$x_n=F\\\\left(x_n-1\\\\right)$$ with $$n=2$$ and the value of $$x_1$$ stored on the calculator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a63563bnewton23a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a63563bnewton23a-h3"],"title":"Find $$x_2$$","text":"What is the value of $$x_2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a63563bnewton23a-h5","type":"hint","dependencies":["a63563bnewton23a-h4"],"title":"Failure of Newton\'s Method","text":"Notice that the approximations alternate back and forth between two values. Consequently, the approximations will never approach a root. Continuing to calculate values $$x_3$$, $$x_4$$, $$x_5$$, $$...x_n$$ will result in a cycle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a63563bnewton24","title":"The Secant Method","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.9 Newton\u2019s Method","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a63563bnewton24a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"The secant method is an alternative iterative method to Newton\u2019s method. Find a root to $$0=sinx+3x$$ accurate to four decimal places. This problem is intended to be done using a calculator.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a63563bnewton24a-h1","type":"hint","dependencies":[],"title":"Secant Method","text":"The formula for the secant method is given by x_n=x_n-1-f(x_n-1)(((x_n-1-x_n-2)/(f(x_n-1)-f(x_n-2)).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a63563bnewton25","title":"The Secant Method","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.9 Newton\u2019s Method","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a63563bnewton25a","stepAnswer":["-0.3513"],"problemType":"TextBox","stepTitle":"The secant method is an alternative iterative method to Newton\u2019s method. Find a root to $$\\\\ln(x+2)=\\\\frac{1}{2}$$ accurate to four decimal places. This problem is intended to be done using a calculator.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-0.3513$$","hints":{"DefaultPathway":[{"id":"a63563bnewton25a-h1","type":"hint","dependencies":[],"title":"Secant Method","text":"The formula for the secant method is given by x_n=x_n-1-f(x_n-1)(((x_n-1-x_n-2)/(f(x_n-1)-f(x_n-2)).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a63563bnewton26","title":"Newton\'s Method and Secant Method","body":"Use both Newton\u2019s method and the secant method to calculate a root. Use a calculator or computer to calculate how many iterations of each are needed to reach within three decimal places of the exact answer. For the secant method, use the first guess from Newton\u2019s method.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.9 Newton\u2019s Method","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a63563bnewton26a","stepAnswer":["Newton: $$11$$ iterations, secant: $$16$$ iterations"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=x^2+2x+1$$, $$x_0=1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Newton: $$11$$ iterations, secant: $$16$$ iterations","choices":["Newton: $$10$$ iterations, secant: $$16$$ iterations","Newton: $$11$$ iterations, secant: $$16$$ iterations","Newton: $$11$$ iterations, secant: $$17$$ iterations","Newton: $$10$$ iterations, secant: $$17$$ iterations"],"hints":{"DefaultPathway":[{"id":"a63563bnewton26a-h1","type":"hint","dependencies":[],"title":"Newton\'s Method","text":"The formula for the Newton\'s method is given by $$x_n=x_n-1-\\\\frac{f{\\\\left(x_n-1\\\\right)}}{\\\\operatorname{f\'}\\\\left(x_n-1\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a63563bnewton26a-h2","type":"hint","dependencies":["a63563bnewton26a-h1"],"title":"Secant Method","text":"The formula for the secant method is given by x_n=x_n-1-f(x_n-1)(((x_n-1-x_n-2)/(f(x_n-1)-f(x_n-2)).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a63563bnewton27","title":"Newton\'s Method and Secant Method","body":"Use both Newton\u2019s method and the secant method to calculate a root. Use a calculator or computer to calculate how many iterations of each are needed to reach within three decimal places of the exact answer. For the secant method, use the first guess from Newton\u2019s method.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.9 Newton\u2019s Method","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a63563bnewton27a","stepAnswer":["Newton: three iterations, secant: six iterations"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=sinx$$, $$x_0=1$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Newton: three iterations, secant: six iterations","Newton: three iterations, secant: five iterations","Newton: four iterations, secant: six iterations","Newton: four iterations, secant: five iterations"],"hints":{"DefaultPathway":[{"id":"a63563bnewton27a-h1","type":"hint","dependencies":[],"title":"Newton\'s Method","text":"The formula for the Newton\'s method is given by $$x_n=x_n-1-\\\\frac{f{\\\\left(x_n-1\\\\right)}}{\\\\operatorname{f\'}\\\\left(x_n-1\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a63563bnewton27a-h2","type":"hint","dependencies":["a63563bnewton27a-h1"],"title":"Secant Method","text":"The formula for the secant method is given by x_n=x_n-1-f(x_n-1)(((x_n-1-x_n-2)/(f(x_n-1)-f(x_n-2)).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a63563bnewton28","title":"Newton\'s Method and Secant Method","body":"Use both Newton\u2019s method and the secant method to calculate a root. Use a calculator or computer to calculate how many iterations of each are needed to reach within three decimal places of the exact answer. For the secant method, use the first guess from Newton\u2019s method.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.9 Newton\u2019s Method","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a63563bnewton28a","stepAnswer":["Newton: five iterations, secant: eight iterations"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=x^3+2x+4$$, $$x_0=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Newton: three iterations, secant: nine iterations","Newton: three iterations, secant: eight iterations","Newton: five iterations, secant: eight iterations","Newton: five iterations, secant: nine iterations"],"hints":{"DefaultPathway":[{"id":"a63563bnewton28a-h1","type":"hint","dependencies":[],"title":"Newton\'s Method","text":"The formula for the Newton\'s method is given by $$x_n=x_n-1-\\\\frac{f{\\\\left(x_n-1\\\\right)}}{\\\\operatorname{f\'}\\\\left(x_n-1\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a63563bnewton28a-h2","type":"hint","dependencies":["a63563bnewton28a-h1"],"title":"Secant Method","text":"The formula for the secant method is given by x_n=x_n-1-f(x_n-1)(((x_n-1-x_n-2)/(f(x_n-1)-f(x_n-2)).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a63563bnewton29","title":"Kepler\'s Equation","body":"Consider Kepler\u2019s equation regarding planetary orbits, $$M=E-esin(E)$$, where M is the mean anomaly, E is eccentric anomaly, and e measures eccentricity. For this problem, e will not represent Euler\'s number and instead will represent eccentricity.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.9 Newton\u2019s Method","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a63563bnewton29a","stepAnswer":["4.071"],"problemType":"TextBox","stepTitle":"Use Newton\u2019s method to solve for the eccentric anomaly E when the mean anomaly $$M=\\\\frac{3\\\\pi}{2}$$ and the eccentricity of the orbit $$\\\\varepsilon=0.8;$$ round to three decimals. This problem is intended to be done using a calculator.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$4.071$$","hints":{"DefaultPathway":[{"id":"a63563bnewton29a-h1","type":"hint","dependencies":[],"title":"Kepler\'s Equation","text":"The formula for Kepler\'s equation is $$M=E-esin(E)$$. Remember that e will not represent Euler\'s number and instead will represent eccentricity.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a63563bnewton29a-h2","type":"hint","dependencies":["a63563bnewton29a-h1"],"title":"Set Up the Equation","text":"Plug in given information into Kepler\'s equation to solve for E.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a63563bnewton3","title":"When Newton\'s Method Fails","body":"Consider the function $$f(x)=x^3-2x+2$$. Let $$x_0=0$$. Show that the sequence x1, x_2,\u2026 fails to approach a root of f.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.9 Newton\u2019s Method","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a63563bnewton3a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"Find $$x_1$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a63563bnewton3a-h1","type":"hint","dependencies":[],"title":"Find the Derivative of f(x)","text":"The derivative of f(x) would be needed to find $$x_1$$ using Newton\'s method.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a63563bnewton3a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$f\'(x)=3x^2-2$$"],"dependencies":["a63563bnewton3a-h1"],"title":"Find the Derivative of f(x)","text":"What is the derivative of f(x)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$f\'(x)=3x-2$$","$$f\'(x)=x^3-2x$$","$$f\'(x)=3x^2+2$$","$$f\'(x)=3x^2-2$$"]},{"id":"a63563bnewton3a-h3","type":"hint","dependencies":["a63563bnewton3a-h2"],"title":"Newton\'s Method With $$n=1$$","text":"To find $$x_1$$, use the equation $$x_n=x_n-1-\\\\frac{f{\\\\left(x_n-1\\\\right)}}{\\\\operatorname{f\'}\\\\left(x_n-1\\\\right)}$$ with $$n=1$$ and $$x_0=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a63563bnewton3b","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"Find $$x_2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a63563bnewton3b-h1","type":"hint","dependencies":[],"title":"Newton\'s Method With $$n=2$$","text":"To find $$x_2$$, use the equation $$x_n=x_n-1-\\\\frac{f{\\\\left(x_n-1\\\\right)}}{\\\\operatorname{f\'}\\\\left(x_n-1\\\\right)}$$ with $$n=2$$ and the value of $$x_1$$ stored on the calculator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a63563bnewton3b-h2","type":"hint","dependencies":["a63563bnewton3b-h1"],"title":"Recall the Derivative of f(x)","text":"The derivative of f(x) would be needed to find $$x_2$$ using Newton\'s method.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a63563bnewton3c","stepAnswer":["The numbers continue to bounce back and forth between $$0$$ and $$1$$, therefore failing to approach the root of f."],"problemType":"MultipleChoice","stepTitle":"What can we infer based on the numbers $$x_0$$, $$x_1$$, $$x_2$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"The numbers continue to bounce back and forth between $$0$$ and $$1$$, therefore failing to approach the root of f.","choices":["The numbers continue to bounce back and forth between $$0$$ and $$1$$, therefore failing to approach the root of f.","The numbers quickly approach the root of f with no issue.","The numbers stay at $$1$$ and never gets closer to the root of f."],"hints":{"DefaultPathway":[]}}]},{"id":"a63563bnewton30","title":"Bank Investment","body":"Consider a bank investment. The initial investment is $10,000. After $$25$$ years, the investment has tripled to $30,000.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.9 Newton\u2019s Method","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a63563bnewton30a","stepAnswer":["$$4.39\\\\%$$"],"problemType":"MultipleChoice","stepTitle":"Use Newton\u2019s method to determine the interest rate if the interest was compounded continuously. This problem is intended to be done using a calculator.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$4.39\\\\%$$","choices":["$$4.39\\\\%$$","$$4.28\\\\%$$","$$4.47\\\\%$$","$$4.32\\\\%$$"],"hints":{"DefaultPathway":[]}}]},{"id":"a63563bnewton4","title":"Finding a Limit for an Iterative Process","body":"Let $$F(x)=\\\\frac{1}{2} x+4$$ and let $$x_0=0$$. For all $$n \\\\geq 1$$, let $$x_n=F\\\\left(x_n-1\\\\right)$$.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.9 Newton\u2019s Method","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a63563bnewton4a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"Find $$x_1$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a63563bnewton4a-h1","type":"hint","dependencies":[],"title":"Formula for $$x_n$$","text":"Use the formula $$x_n=F\\\\left(x_n-1\\\\right)$$ to define subsequent numbers $$x_n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a63563bnewton4a-h2","type":"hint","dependencies":["a63563bnewton4a-h1"],"title":"Formula With $$n=1$$","text":"To find $$x_1$$, use the formula $$x_n=F\\\\left(x_n-1\\\\right)$$ with $$n=1$$ and $$x_0=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a63563bnewton4b","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"Find $$x_2$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"a63563bnewton4b-h1","type":"hint","dependencies":[],"title":"Formula for $$x_n$$","text":"Use the formula $$x_n=F\\\\left(x_n-1\\\\right)$$ to define subsequent numbers $$x_n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a63563bnewton4b-h2","type":"hint","dependencies":["a63563bnewton4b-h1"],"title":"Formula With $$n=2$$","text":"To find $$x_2$$, use the formula $$x_n=F\\\\left(x_n-1\\\\right)$$ with $$n=2$$ and the value of $$x_1$$ stored on the calculator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a63563bnewton4c","stepAnswer":["$$7$$"],"problemType":"TextBox","stepTitle":"Find $$x_3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7$$","hints":{"DefaultPathway":[{"id":"a63563bnewton4c-h1","type":"hint","dependencies":[],"title":"Formula for $$x_n$$","text":"Use the formula $$x_n=F\\\\left(x_n-1\\\\right)$$ to define subsequent numbers $$x_n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a63563bnewton4c-h2","type":"hint","dependencies":["a63563bnewton4c-h1"],"title":"Formula With $$n=3$$","text":"To find $$x_3$$, use the formula $$x_n=F\\\\left(x_n-1\\\\right)$$ with $$n=3$$ and the value of $$x_2$$ stored on the calculator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a63563bnewton4d","stepAnswer":["$$7.5$$"],"problemType":"TextBox","stepTitle":"Find $$x_4$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7.5$$","hints":{"DefaultPathway":[{"id":"a63563bnewton4d-h1","type":"hint","dependencies":[],"title":"Formula for $$x_n$$","text":"Use the formula $$x_n=F\\\\left(x_n-1\\\\right)$$ to define subsequent numbers $$x_n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a63563bnewton4d-h2","type":"hint","dependencies":["a63563bnewton4d-h1"],"title":"Formula With $$n=4$$","text":"To find $$x_4$$, use the formula $$x_n=F\\\\left(x_n-1\\\\right)$$ with $$n=4$$ and the value of $$x_3$$ stored on the calculator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a63563bnewton4e","stepAnswer":["$$7.75$$"],"problemType":"TextBox","stepTitle":"Find $$x_5$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7.75$$","hints":{"DefaultPathway":[{"id":"a63563bnewton4e-h1","type":"hint","dependencies":[],"title":"Formula for $$x_n$$","text":"Use the formula $$x_n=F\\\\left(x_n-1\\\\right)$$ to define subsequent numbers $$x_n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a63563bnewton4e-h2","type":"hint","dependencies":["a63563bnewton4e-h1"],"title":"Formula With $$n=5$$","text":"To find $$x_5$$, use the formula $$x_n=F\\\\left(x_n-1\\\\right)$$ with $$n=5$$ and the value of $$x_4$$ stored on the calculator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a63563bnewton4f","stepAnswer":["The values $$x_n$$ approach $$8$$."],"problemType":"MultipleChoice","stepTitle":"Make a Conjecture","stepBody":"Make a conjecture about what happens to the list of numbers $$x_1$$, $$x_2$$, x_3\u2026, x_n,\u2026 as $$n$$ approaches $$\\\\infty$$.","answerType":"string","variabilization":{},"answerLatex":"The values $$x_n$$ approach $$8$$.","choices":["The values $$x_n$$ approach $$6$$.","The values $$x_n$$ approach $$7$$.","The values $$x_n$$ approach $$8$$.","The values $$x_n$$ approach $$9$$."],"hints":{"DefaultPathway":[]}}]},{"id":"a63563bnewton5","title":"Newton\'s Method","body":"Write Newton\u2019s method as $$x_n+1=F\\\\left(x_n\\\\right)$$ for solving $$f(x)=0$$.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.9 Newton\u2019s Method","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a63563bnewton5a","stepAnswer":["$$F\\\\left(x_n\\\\right)=x_n-\\\\frac{{x_n}^3+2x_n+1}{{\\\\left(3x_n\\\\right)}^2+2}$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=x^3+2x+1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$F\\\\left(x_n\\\\right)=x_n-\\\\frac{{x_n}^3+2x_n+1}{{\\\\left(3x_n\\\\right)}^2+2}$$","choices":["$$F\\\\left(x_n\\\\right)=x_n-\\\\frac{{x_n}^3+2x_n+1}{{\\\\left(3x_n\\\\right)}^2+2}$$","$$F\\\\left(x_n\\\\right)=\\\\frac{{x_n}^3+2x_n+1}{{\\\\left(3x_n\\\\right)}^2+2}$$"],"hints":{"DefaultPathway":[{"id":"a63563bnewton5a-h1","type":"hint","dependencies":[],"title":"Find the Derivative of f(x)","text":"The derivative of f(x) would be needed to work with Newton\'s method.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a63563bnewton5a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$f\'(x)=3x^2+2$$"],"dependencies":["a63563bnewton5a-h1"],"title":"Find the Derivative of f(x)","text":"What is the derivative of f(x)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$f\'(x)=3x+2$$","$$f\'(x)=3x^2+2$$","$$f\'(x)=3x^2$$","$$f\'(x)=3x^2+2+1$$"]},{"id":"a63563bnewton5a-h3","type":"hint","dependencies":["a63563bnewton5a-h2"],"title":"Newton\'s Method","text":"The formula for Newton\'s Method is given by $$x_n+1=x_n-\\\\frac{f{\\\\left(x_n\\\\right)}}{\\\\operatorname{f\'}\\\\left(x_n\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a63563bnewton5a-h4","type":"hint","dependencies":["a63563bnewton5a-h3"],"title":"Newton\'s Method","text":"Define $$f{\\\\left(x_n\\\\right)}$$ and $$\\\\operatorname{f\'}\\\\left(x_n\\\\right)$$ then plug into formula.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a63563bnewton6","title":"Newton\'s Method","body":"Write Newton\u2019s method as $$x_n+1=F\\\\left(x_n\\\\right)$$ for solving $$f(x)=0$$.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.9 Newton\u2019s Method","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a63563bnewton6a","stepAnswer":["$$F\\\\left(x_n\\\\right)=x_n-\\\\frac{e^{x_n}}{e^{x_n}}$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=e^x$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$F\\\\left(x_n\\\\right)=x_n-\\\\frac{e^{x_n}}{e^{x_n}}$$","choices":["$$F\\\\left(x_n\\\\right)=x_n-\\\\frac{e^{x_n}}{e^{x_n}}$$","$$F\\\\left(x_n\\\\right)=\\\\frac{e^{x_n}}{e^{x_n}}$$"],"hints":{"DefaultPathway":[{"id":"a63563bnewton6a-h1","type":"hint","dependencies":[],"title":"Find the Derivative of f(x)","text":"The derivative of f(x) would be needed for this problem.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a63563bnewton6a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$f\'(x)=e^x$$"],"dependencies":["a63563bnewton6a-h1"],"title":"Find the Derivative of f(x)","text":"What is the derivative of f(x)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$f\'(x)={xe}^x$$","$$f\'(x)=e$$","$$f\'(x)=e^x$$","$$f\'(x)=xe$$"]},{"id":"a63563bnewton6a-h3","type":"hint","dependencies":["a63563bnewton6a-h2"],"title":"Newton\'s Method","text":"The formula for Newton\'s Method is given by $$x_n+1=x_n-\\\\frac{f{\\\\left(x_n\\\\right)}}{\\\\operatorname{f\'}\\\\left(x_n\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a63563bnewton6a-h4","type":"hint","dependencies":["a63563bnewton6a-h3"],"title":"Newton\'s Method","text":"Define $$f{\\\\left(x_n\\\\right)}$$ and $$\\\\operatorname{f\'}\\\\left(x_n\\\\right)$$ then plug into formula.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a63563bnewton7","title":"Using Iteration","body":"Solve $$f(x)=0$$ using the iteration $$x_n+1=x_n-\\\\operatorname{cf}\\\\left(x_n\\\\right)$$, which differs slightly from Newton\u2019s method. Find a c that works and a c that fails to converge, with the exception of $$c=0$$.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.9 Newton\u2019s Method","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a63563bnewton7a","stepAnswer":["$$|c|>0.5$$ fails, $$|c| \\\\leq 0.5$$ works"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=x^2-4$$, with $$x_0=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$|c|>0.5$$ fails, $$|c| \\\\leq 0.5$$ works","choices":["$$|c|>0.5$$ works, $$|c| \\\\leq 0.5$$ fails","$$|c|>0.5$$ fails, $$|c| \\\\leq 0.5$$ works"],"hints":{"DefaultPathway":[{"id":"a63563bnewton7a-h1","type":"hint","dependencies":[],"title":"Iteration of Newton\'s Method","text":"Recall that the iterative method is given by $$x_n+1=x_n-\\\\operatorname{cf}\\\\left(x_n\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a63563bnewton8","title":"Using Iteration","body":"The iteration $$x_n+1=x_n-\\\\operatorname{cf}\\\\left(x_n\\\\right)$$ differs slightly from Newton\'s method.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.9 Newton\u2019s Method","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a63563bnewton8a","stepAnswer":["$$c=\\\\frac{1}{\\\\operatorname{f\'}\\\\left(x_n\\\\right)}$$"],"problemType":"MultipleChoice","stepTitle":"What is the value of \\"c\\" for Newton\'s Method?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$c=\\\\frac{1}{\\\\operatorname{f\'}\\\\left(x_n\\\\right)}$$","choices":["$$c=\\\\frac{1}{\\\\operatorname{f\'}\\\\left(x_n\\\\right)}$$","$$c=\\\\frac{1}{f{\\\\left(x_n\\\\right)}}$$","$$c=\\\\operatorname{f\'}\\\\left(x_n\\\\right)$$","$$c=f{\\\\left(x_n\\\\right)}$$"],"hints":{"DefaultPathway":[{"id":"a63563bnewton8a-h1","type":"hint","dependencies":[],"title":"Newton\'s Method","text":"The formula for Newton\'s Method is given by $$x_n+1=x_n-\\\\frac{f{\\\\left(x_n\\\\right)}}{\\\\operatorname{f\'}\\\\left(x_n\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a63563bnewton8a-h2","type":"hint","dependencies":["a63563bnewton8a-h1"],"title":"Newton\'s Method","text":"Consider what c would need to be to turn the iteration into the formula.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a63563bnewton9","title":"Computing $$x_1$$ and $$x_2$$","body":"Compute $$x_1$$ and $$x_2$$ using the specified iterative method: $$x_n+1=2\\\\operatorname{x_n}\\\\left(1-x_n\\\\right)$$. This problem is intended to be done using a calculator.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.9 Newton\u2019s Method","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a63563bnewton9a","stepAnswer":["$$\\\\frac{12}{25}$$"],"problemType":"TextBox","stepTitle":"$$x_1$$ with $$x_0=0.6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{12}{25}$$","hints":{"DefaultPathway":[{"id":"a63563bnewton9a-h1","type":"hint","dependencies":[],"title":"Formula to Find $$x_1$$ with $$x_0=0.6$$","text":"To compute $$x_1$$, substitute $$x_0$$ with $$0.6$$ into $$x_1=2\\\\operatorname{x_0}\\\\left(1-x_0\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a63563bnewton9b","stepAnswer":["$$\\\\frac{312}{625}$$"],"problemType":"TextBox","stepTitle":"$$x_2$$ with $$x_0=0.6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{312}{625}$$","hints":{"DefaultPathway":[{"id":"a63563bnewton9b-h1","type":"hint","dependencies":[],"title":"Formula to Find $$x_2$$ with $$x_0=0.6$$","text":"To compute $$x_2$$, substitute the value of $$x_1$$ into $$x_2=2\\\\operatorname{x_1}\\\\left(1-x_1\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a63563bnewton9c","stepAnswer":["$$-4$$"],"problemType":"TextBox","stepTitle":"$$x_1$$ with $$x_0=2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-4$$","hints":{"DefaultPathway":[{"id":"a63563bnewton9c-h1","type":"hint","dependencies":[],"title":"Formula to Find $$x_1$$ with $$x_0=2$$","text":"To compute $$x_1$$, substitute $$x_0$$ with $$2$$ into $$x_1=2\\\\operatorname{x_0}\\\\left(1-x_0\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a63563bnewton9d","stepAnswer":["$$-40$$"],"problemType":"TextBox","stepTitle":"$$x_2$$ with $$x_0=2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-40$$","hints":{"DefaultPathway":[{"id":"a63563bnewton9d-h1","type":"hint","dependencies":[],"title":"Formula to Find $$x_2$$ with $$x_0=2$$","text":"To compute $$x_2$$, substitute the value of $$x_1$$ into $$x_2=2\\\\operatorname{x_1}\\\\left(1-x_1\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a65ae04inequalities1","title":"Solve Absolute Value Equations","body":"Solve for $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Absolute Value Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a65ae04inequalities1a","stepAnswer":["$$x=\\\\pm 4$$"],"problemType":"MultipleChoice","stepTitle":"$$|x|=4$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=\\\\pm 4$$","choices":["$$x=\\\\pm 4$$","$$x=-4$$","$$x=4$$","No Solution"],"hints":{"DefaultPathway":[{"id":"a65ae04inequalities1a-h1","type":"hint","dependencies":[],"title":"Write the equivalent equations using the formula. If $$|x|=a$$, then $$u=-a$$ and $$u=a$$.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities1a-h2","type":"hint","dependencies":["a65ae04inequalities1a-h1"],"title":"Answer","text":"The answer is $$x=\\\\pm 4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a65ae04inequalities10","title":"Solve Absolute Value Equations","body":"Solve for $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Absolute Value Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a65ae04inequalities10a","stepAnswer":["No Solution"],"problemType":"MultipleChoice","stepTitle":"$$|\\\\frac{x}{4}+3|+3=1$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$x=7, 1$$","$$x=2, 3$$","$$x=4, -5$$","No Solution"],"hints":{"DefaultPathway":[{"id":"a65ae04inequalities10a-h1","type":"hint","dependencies":[],"title":"Isolate","text":"Isolate the absolute value on one side of the equation, with constants on the other by using operations on both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities10a-h2","type":"hint","dependencies":["a65ae04inequalities10a-h1"],"title":"Answer","text":"The answer is no solution as there can be no negative of an absolute value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a65ae04inequalities11","title":"Solve Absolute Value Equations","body":"Solve for $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Absolute Value Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a65ae04inequalities11a","stepAnswer":["x=2,1/4"],"problemType":"MultipleChoice","stepTitle":"$$|6x-5|=|2x+3|$$","stepBody":"","answerType":"string","variabilization":{},"choices":["x=2,1/4","x=-3,1/5","$$x=\\\\frac{1}{2}-\\\\frac{2}{3}$$","$$x=8, 3$$"],"hints":{"DefaultPathway":[{"id":"a65ae04inequalities11a-h1","type":"hint","dependencies":[],"title":"Isolate","text":"Isolate the absolute values on both sides of the equation, which in this case is already done.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities11a-h2","type":"hint","dependencies":["a65ae04inequalities11a-h1"],"title":"Split","text":"Split the equation into two, one where $$6x-5=2x+3$$, and one where $$6x-5=-\\\\left(2x+3\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities11a-h3","type":"hint","dependencies":["a65ae04inequalities11a-h2"],"title":"Solve","text":"Now, solve for $$x$$ in both of the above equations to find the two solutions to the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities11a-h4","type":"hint","dependencies":["a65ae04inequalities11a-h3"],"title":"Answer","text":"The answer is x=2,1/4.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a65ae04inequalities12","title":"Solve Absolute Value Equations","body":"Solve for $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Absolute Value Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a65ae04inequalities12a","stepAnswer":["$$x=3, 2$$"],"problemType":"MultipleChoice","stepTitle":"$$|2x-5|+2=3$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=3, 2$$","choices":["$$x=3, 2$$","$$x=2, -4$$","$$x=-6, 6$$","$$x=1, 5$$"],"hints":{"DefaultPathway":[{"id":"a65ae04inequalities12a-h1","type":"hint","dependencies":[],"title":"Isolate","text":"Isolate the absolute value on one side of the equation, with constants on the other by using operations on both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities12a-h2","type":"hint","dependencies":["a65ae04inequalities12a-h1"],"title":"Split","text":"Split the equation into two, one where $$2x-5=1$$, and one where $$2x-5=-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities12a-h3","type":"hint","dependencies":["a65ae04inequalities12a-h2"],"title":"Solve","text":"Now, solve for $$x$$ in both of the above equations to find the two solutions to the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities12a-h4","type":"hint","dependencies":["a65ae04inequalities12a-h3"],"title":"Answer","text":"The answer is $$x=3, 2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a65ae04inequalities13","title":"Solve Absolute Value Equations","body":"Solve for $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Absolute Value Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a65ae04inequalities13a","stepAnswer":["$$x=3-\\\\frac{11}{3}$$"],"problemType":"MultipleChoice","stepTitle":"$$|3x+1|-3=7$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=3-\\\\frac{11}{3}$$","choices":["$$x=3-\\\\frac{11}{3}$$","$$x=2-\\\\frac{4}{3}$$","$$x=-6, -3$$","$$x=7, 8$$"],"hints":{"DefaultPathway":[{"id":"a65ae04inequalities13a-h1","type":"hint","dependencies":[],"title":"Isolate","text":"Isolate the absolute value on one side of the equation, with constants on the other by using operations on both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities13a-h2","type":"hint","dependencies":["a65ae04inequalities13a-h1"],"title":"Split","text":"Split the equation into two, one where $$3x+1=10$$, and one where $$3x+1=-10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities13a-h3","type":"hint","dependencies":["a65ae04inequalities13a-h2"],"title":"Solve","text":"Now, solve for $$x$$ in both of the above equations to find the two solutions to the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities13a-h4","type":"hint","dependencies":["a65ae04inequalities13a-h3"],"title":"Answer","text":"The answer is $$x=3-\\\\frac{11}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a65ae04inequalities14","title":"Solve Absolute Value Equations","body":"Solve for $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Absolute Value Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a65ae04inequalities14a","stepAnswer":["$$x=\\\\frac{3}{2}-\\\\frac{1}{2}$$"],"problemType":"MultipleChoice","stepTitle":"$$5|2x-1|-3=7$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=\\\\frac{3}{2}-\\\\frac{1}{2}$$","choices":["$$x=\\\\frac{3}{2}-\\\\frac{1}{2}$$","$$x=3, -1$$","$$x=11, -5$$","$$x=7, -4$$"],"hints":{"DefaultPathway":[{"id":"a65ae04inequalities14a-h1","type":"hint","dependencies":[],"title":"Isolate","text":"Isolate the absolute value on one side of the equation, with constants on the other by using operations on both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities14a-h2","type":"hint","dependencies":["a65ae04inequalities14a-h1"],"title":"Split","text":"Split the equation into two, one where $$2x-1=2$$, and one where $$2x-1=-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities14a-h3","type":"hint","dependencies":["a65ae04inequalities14a-h2"],"title":"Solve","text":"Now, solve for $$x$$ in both of the above equations to find the two solutions to the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities14a-h4","type":"hint","dependencies":["a65ae04inequalities14a-h3"],"title":"Answer","text":"The answer is $$x=\\\\frac{3}{2}-\\\\frac{1}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a65ae04inequalities15","title":"Solve Absolute Value Equations","body":"Solve for $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Absolute Value Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a65ae04inequalities15a","stepAnswer":["No Solution"],"problemType":"MultipleChoice","stepTitle":"$$|x-7|=-3$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$x=7, 1$$","$$x=2, 3$$","$$x=4, -5$$","No Solution"],"hints":{"DefaultPathway":[{"id":"a65ae04inequalities15a-h1","type":"hint","dependencies":[],"title":"Isolate","text":"Isolate the absolute value on one side of the equation, with constants on the other by using operations on both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities15a-h2","type":"hint","dependencies":["a65ae04inequalities15a-h1"],"title":"Answer","text":"The answer is no solution as there can be no negative of an absolute value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a65ae04inequalities16","title":"Solving Absolute Value Equations","body":"Find the value of $$x$$. If there are two values of $$x$$, enter your answer in the form $$\\"x=a$$ or $$x=b\\"$$ without the quotes, and with a and $$b$$ representing the values of $$x$$. Let a be the smaller value of $$x$$ and $$b$$ be the larger value of $$x$$. Example: $$x=\\\\frac{-1}{5}$$ or $$x=3$$","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Absolute Value Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a65ae04inequalities16a","stepAnswer":["x=-7/5 or x=3"],"problemType":"TextBox","stepTitle":"$$|5x-4|-3=8$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=\\\\frac{-7}{5}$$ or $$x=3$$","hints":{"DefaultPathway":[{"id":"a65ae04inequalities16a-h1","type":"hint","dependencies":[],"title":"Isolating the Expression","text":"First, isolate the absolute value expression. For example, if the problem were $$3|x|-2=6$$, first add two to both sides for the equation, and then divide both sides by $$3$$ to turn it into into $$|x|=\\\\frac{8}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities16a-h2","type":"hint","dependencies":["a65ae04inequalities16a-h1"],"title":"Writing Equivalent Expressions","text":"Write the equivalent expressions since the expression inside the absolute value could be either negative or positive. For example, if the equation were $$|x|=4$$, the two equations would be $$x=4$$ or $$x=-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities16a-h3","type":"hint","dependencies":["a65ae04inequalities16a-h2"],"title":"Solving Each Equation","text":"Solve each equation for the value of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities16a-h4","type":"hint","dependencies":["a65ae04inequalities16a-h3"],"title":"Checking Each Solution","text":"Check each solution by subsituting the values of $$x$$ into the orginal equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a65ae04inequalities17","title":"Solving Absolute Value Equations","body":"Find the value of $$x$$. If there are two values of $$x$$, enter your answer in the form $$\\"x=a$$ or $$x=b\\"$$ without the quotes, and with a and $$b$$ representing the values of $$x$$. Let a be the smaller value of $$x$$ and $$b$$ be the larger value of $$x$$. Example: $$x=\\\\frac{-1}{5}$$ or $$x=3$$","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Absolute Value Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a65ae04inequalities17a","stepAnswer":["x=-2/3 or x=4"],"problemType":"TextBox","stepTitle":"$$|3x-5|-1=6$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=\\\\frac{-2}{3}$$ or $$x=4$$","hints":{"DefaultPathway":[{"id":"a65ae04inequalities17a-h1","type":"hint","dependencies":[],"title":"Isolating the Expression","text":"First, isolate the absolute value expression. For example, if the problem were $$3|x|-2=6$$, first add two to both sides for the equation, and then divide both sides by $$3$$ to turn it into into $$|x|=\\\\frac{8}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities17a-h2","type":"hint","dependencies":["a65ae04inequalities17a-h1"],"title":"Writing Equivalent Expressions","text":"Write the equivalent expressions since the expression inside the absolute value could be either negative or positive. For example, if the equation were $$|x|=4$$, the two equations would be $$x=4$$ or $$x=-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities17a-h3","type":"hint","dependencies":["a65ae04inequalities17a-h2"],"title":"Solving Each Equation","text":"Solve each equation for the value of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities17a-h4","type":"hint","dependencies":["a65ae04inequalities17a-h3"],"title":"Checking Each Solution","text":"Check each solution by subsituting the values of $$x$$ into the orginal equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a65ae04inequalities18","title":"Solving Absolute Value Equations","body":"Find the value of $$x$$. If there are two values of $$x$$, enter your answer in the form $$\\"x=a$$ or $$x=b\\"$$ without the quotes, and with a and $$b$$ representing the values of $$x$$. Let a be the smaller value of $$x$$ and $$b$$ be the larger value of $$x$$. Example: $$x=\\\\frac{-1}{5}$$ or $$x=3$$","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Absolute Value Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a65ae04inequalities18a","stepAnswer":["x=-1 or x=5/2"],"problemType":"TextBox","stepTitle":"$$|4x-3|-5=2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=-1$$ or $$x=\\\\frac{5}{2}$$","hints":{"DefaultPathway":[{"id":"a65ae04inequalities18a-h1","type":"hint","dependencies":[],"title":"Isolating the Expression","text":"First, isolate the absolute value expression. For example, if the problem were $$3|x|-2=6$$, first add two to both sides for the equation, and then divide both sides by $$3$$ to turn it into into $$|x|=\\\\frac{8}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities18a-h2","type":"hint","dependencies":["a65ae04inequalities18a-h1"],"title":"Writing Equivalent Expressions","text":"Write the equivalent expressions since the expression inside the absolute value could be either negative or positive. For example, if the equation were $$|x|=4$$, the two equations would be $$x=4$$ or $$x=-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities18a-h3","type":"hint","dependencies":["a65ae04inequalities18a-h2"],"title":"Solving Each Equation","text":"Solve each equation for the value of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities18a-h4","type":"hint","dependencies":["a65ae04inequalities18a-h3"],"title":"Checking Each Solution","text":"Check each solution by subsituting the values of $$x$$ into the orginal equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a65ae04inequalities19","title":"Solving Absolute Value Equations","body":"Find the value of $$x$$. If there are two values of $$x$$, enter your answer in the form $$\\"x=a$$ or $$x=b\\"$$ without the quotes, and with a and $$b$$ representing the values of $$x$$. Let a be the smaller value of $$x$$ and $$b$$ be the larger value of $$x$$. Example: $$x=\\\\frac{-1}{5}$$ or $$x=3$$","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Absolute Value Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a65ae04inequalities19a","stepAnswer":["x=5 or x=9"],"problemType":"TextBox","stepTitle":"$$2|x-7|+5=9$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=5$$ or $$x=9$$","hints":{"DefaultPathway":[{"id":"a65ae04inequalities19a-h1","type":"hint","dependencies":[],"title":"Isolating the Expression","text":"First, isolate the absolute value expression. For example, if the problem were $$3|x|-2=6$$, first add two to both sides for the equation, and then divide both sides by $$3$$ to turn it into into $$|x|=\\\\frac{8}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities19a-h2","type":"hint","dependencies":["a65ae04inequalities19a-h1"],"title":"Writing Equivalent Expressions","text":"Write the equivalent expressions since the expression inside the absolute value could be either negative or positive. For example, if the equation were $$|x|=4$$, the two equations would be $$x=4$$ or $$x=-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities19a-h3","type":"hint","dependencies":["a65ae04inequalities19a-h2"],"title":"Solving Each Equation","text":"Solve each equation for the value of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities19a-h4","type":"hint","dependencies":["a65ae04inequalities19a-h3"],"title":"Checking Each Solution","text":"Check each solution by subsituting the values of $$x$$ into the orginal equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a65ae04inequalities2","title":"Solve Absolute Value Equations","body":"Solve for $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Absolute Value Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a65ae04inequalities2a","stepAnswer":["No Solution"],"problemType":"MultipleChoice","stepTitle":"$$|x|=-3$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$x=\\\\pm 3$$","$$x=-3$$","$$x=3$$","No Solution"],"hints":{"DefaultPathway":[{"id":"a65ae04inequalities2a-h1","type":"hint","dependencies":[],"title":"Write the equivalent equations using the formula. If $$|x|=a$$, then $$u=-a$$ and $$u=a$$. If $$|x|=-a$$, then there is no solution.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities2a-h2","type":"hint","dependencies":["a65ae04inequalities2a-h1"],"title":"Answer","text":"The answer is no solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a65ae04inequalities20","title":"Solving Absolute Value Equations","body":"Find the value of $$x$$. If there are two values of $$x$$, enter your answer in the form $$\\"x=a$$ or $$x=b\\"$$ without the quotes, and with a and $$b$$ representing the values of $$x$$. Let a be the smaller value of $$x$$ and $$b$$ be the larger value of $$x$$. Example: $$x=\\\\frac{-1}{5}$$ or $$x=3$$","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Absolute Value Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a65ae04inequalities20a","stepAnswer":["x=0 or x=8"],"problemType":"TextBox","stepTitle":"$$3|x-4|-4=8$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=0$$ or $$x=8$$","hints":{"DefaultPathway":[{"id":"a65ae04inequalities20a-h1","type":"hint","dependencies":[],"title":"Isolating the Expression","text":"First, isolate the absolute value expression. For example, if the problem were $$3|x|-2=6$$, first add two to both sides for the equation, and then divide both sides by $$3$$ to turn it into into $$|x|=\\\\frac{8}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities20a-h2","type":"hint","dependencies":["a65ae04inequalities20a-h1"],"title":"Writing Equivalent Expressions","text":"Write the equivalent expressions since the expression inside the absolute value could be either negative or positive. For example, if the equation were $$|x|=4$$, the two equations would be $$x=4$$ or $$x=-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities20a-h3","type":"hint","dependencies":["a65ae04inequalities20a-h2"],"title":"Solving Each Equation","text":"Solve each equation for the value of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities20a-h4","type":"hint","dependencies":["a65ae04inequalities20a-h3"],"title":"Checking Each Solution","text":"Check each solution by subsituting the values of $$x$$ into the orginal equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a65ae04inequalities21","title":"Solving Absolute Value Equations","body":"Find the value of $$x$$. If there are two values of $$x$$, enter your answer in the form $$\\"x=a$$ or $$x=b\\"$$ without the quotes, and with a and $$b$$ representing the values of $$x$$. Let a be the smaller value of $$x$$ and $$b$$ be the larger value of $$x$$. Example: $$x=\\\\frac{-1}{5}$$ or $$x=3$$","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Absolute Value Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a65ae04inequalities21a","stepAnswer":["x=2 or x=8"],"problemType":"TextBox","stepTitle":"$$2|x-5|+3=9$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=2$$ or $$x=8$$","hints":{"DefaultPathway":[{"id":"a65ae04inequalities21a-h1","type":"hint","dependencies":[],"title":"Isolating the Expression","text":"First, isolate the absolute value expression. For example, if the problem were $$3|x|-2=6$$, first add two to both sides for the equation, and then divide both sides by $$3$$ to turn it into into $$|x|=\\\\frac{8}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities21a-h2","type":"hint","dependencies":["a65ae04inequalities21a-h1"],"title":"Writing Equivalent Expressions","text":"Write the equivalent expressions since the expression inside the absolute value could be either negative or positive. For example, if the equation were $$|x|=4$$, the two equations would be $$x=4$$ or $$x=-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities21a-h3","type":"hint","dependencies":["a65ae04inequalities21a-h2"],"title":"Solving Each Equation","text":"Solve each equation for the value of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities21a-h4","type":"hint","dependencies":["a65ae04inequalities21a-h3"],"title":"Checking Each Solution","text":"Check each solution by subsituting the values of $$x$$ into the orginal equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a65ae04inequalities22","title":"Solving Absolute Value Equations","body":"Find the value of $$x$$. If there are two values of $$x$$, enter your answer in the form $$\\"x=a$$ or $$x=b\\"$$ without the quotes, and with a and $$b$$ representing the values of $$x$$. Let a be the smaller value of $$x$$ and $$b$$ be the larger value of $$x$$. Example: $$x=\\\\frac{-1}{5}$$ or $$x=3$$. If there is no solution, write \\"No Solution\\" without the quotes.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Absolute Value Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a65ae04inequalities22a","stepAnswer":["No Solution"],"problemType":"TextBox","stepTitle":"$$|\\\\frac{2}{3} x-4|+11=3$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a65ae04inequalities22a-h1","type":"hint","dependencies":[],"title":"Isolating the Expression","text":"First, isolate the absolute value expression. For example, if the problem were $$3|x|-2=6$$, first add two to both sides for the equation, and then divide both sides by $$3$$ to turn it into into $$|x|=\\\\frac{8}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities22a-h2","type":"hint","dependencies":["a65ae04inequalities22a-h1"],"title":"Writing Equivalent Expressions","text":"Write the equivalent expressions since the expression inside the absolute value could be either negative or positive. For example, if the equation were $$|x|=4$$, the two equations would be $$x=4$$ or $$x=-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities22a-h3","type":"hint","dependencies":["a65ae04inequalities22a-h2"],"title":"Solving Each Equation","text":"Solve each equation for the value of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities22a-h4","type":"hint","dependencies":["a65ae04inequalities22a-h3"],"title":"Checking Each Solution","text":"Check each solution by subsituting the values of $$x$$ into the orginal equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a65ae04inequalities23","title":"Solving Absolute Value Equations","body":"Find the value of $$x$$. If there are two values of $$x$$, enter your answer in the form $$\\"x=a$$ or $$x=b\\"$$ without the quotes, and with a and $$b$$ representing the values of $$x$$. Let a be the smaller value of $$x$$ and $$b$$ be the larger value of $$x$$. Example: $$x=\\\\frac{-1}{5}$$ or $$x=3$$. If there is no solution, write \\"No Solution\\" without the quotes.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Absolute Value Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a65ae04inequalities23a","stepAnswer":["No Solution"],"problemType":"TextBox","stepTitle":"$$|\\\\frac{3}{4} x-5|+9=4$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a65ae04inequalities23a-h1","type":"hint","dependencies":[],"title":"Isolating the Expression","text":"First, isolate the absolute value expression. For example, if the problem were $$3|x|-2=6$$, first add two to both sides for the equation, and then divide both sides by $$3$$ to turn it into into $$|x|=\\\\frac{8}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities23a-h2","type":"hint","dependencies":["a65ae04inequalities23a-h1"],"title":"Writing Equivalent Expressions","text":"Write the equivalent expressions since the expression inside the absolute value could be either negative or positive. For example, if the equation were $$|x|=4$$, the two equations would be $$x=4$$ or $$x=-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities23a-h3","type":"hint","dependencies":["a65ae04inequalities23a-h2"],"title":"Solving Each Equation","text":"Solve each equation for the value of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities23a-h4","type":"hint","dependencies":["a65ae04inequalities23a-h3"],"title":"Checking Each Solution","text":"Check each solution by subsituting the values of $$x$$ into the orginal equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a65ae04inequalities24","title":"Solving Absolute Value Equations","body":"Find the value of $$x$$. If there are two values of $$x$$, enter your answer in the form $$\\"x=a$$ or $$x=b\\"$$ without the quotes, and with a and $$b$$ representing the values of $$x$$. Let a be the smaller value of $$x$$ and $$b$$ be the larger value of $$x$$. Example: $$x=\\\\frac{-1}{5}$$ or $$x=3$$. If there is no solution, write \\"No Solution\\" without the quotes.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Absolute Value Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a65ae04inequalities24a","stepAnswer":["No Solution"],"problemType":"TextBox","stepTitle":"$$|\\\\frac{5}{6} x+3|+8=6$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a65ae04inequalities24a-h1","type":"hint","dependencies":[],"title":"Isolating the Expression","text":"First, isolate the absolute value expression. For example, if the problem were $$3|x|-2=6$$, first add two to both sides for the equation, and then divide both sides by $$3$$ to turn it into into $$|x|=\\\\frac{8}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities24a-h2","type":"hint","dependencies":["a65ae04inequalities24a-h1"],"title":"Writing Equivalent Expressions","text":"Write the equivalent expressions since the expression inside the absolute value could be either negative or positive. For example, if the equation were $$|x|=4$$, the two equations would be $$x=4$$ or $$x=-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities24a-h3","type":"hint","dependencies":["a65ae04inequalities24a-h2"],"title":"Solving Each Equation","text":"Solve each equation for the value of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities24a-h4","type":"hint","dependencies":["a65ae04inequalities24a-h3"],"title":"Checking Each Solution","text":"Check each solution by subsituting the values of $$x$$ into the orginal equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a65ae04inequalities25","title":"Solving Absolute Value Inequalities","body":"Write the solution using interval notation. For example, if the solution is $$-5<x<3$$, input $$\\"(-5,3)\\"$$ without the quotes.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Absolute Value Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a65ae04inequalities25a","stepAnswer":["(-7,7)"],"problemType":"TextBox","stepTitle":"$$|x|<7$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-7,7)$$","hints":{"DefaultPathway":[{"id":"a65ae04inequalities25a-h1","type":"hint","dependencies":[],"title":"Writing the Inequality","text":"Write the equivalent inequality. For example, if the expression is originally $$|x|<3$$, the equivalent inequality is $$-3<x<3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities25a-h2","type":"hint","dependencies":["a65ae04inequalities25a-h1"],"title":"Graphing the Solution","text":"Graph the solution (the possible values of $$x$$ based on the inequality.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities25a-h3","type":"hint","dependencies":["a65ae04inequalities25a-h2"],"title":"Writing the Solution","text":"Lastly, write the solution using interval notation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a65ae04inequalities26","title":"Solving Absolute Value Inequalities","body":"Write the solution using interval notation. For example, if the solution is $$-5<x<3$$, input $$\\"(-5,3)\\"$$ without the quotes.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Absolute Value Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a65ae04inequalities26a","stepAnswer":["(-9,9)"],"problemType":"TextBox","stepTitle":"$$|x|<9$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-9,9)$$","hints":{"DefaultPathway":[{"id":"a65ae04inequalities26a-h1","type":"hint","dependencies":[],"title":"Writing the Inequality","text":"Write the equivalent inequality. For example, if the expression is originally $$|x|<3$$, the equivalent inequality is $$-3<x<3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities26a-h2","type":"hint","dependencies":["a65ae04inequalities26a-h1"],"title":"Graphing the Solution","text":"Graph the solution (the possible values of $$x$$ based on the inequality.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities26a-h3","type":"hint","dependencies":["a65ae04inequalities26a-h2"],"title":"Writing the Solution","text":"Lastly, write the solution using interval notation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a65ae04inequalities27","title":"Solving Absolute Value Inequalities","body":"Write the solution using interval notation. For example, if the solution is $$-5<x<3$$, input $$\\"(-5,3)\\"$$ without the quotes.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Absolute Value Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a65ae04inequalities27a","stepAnswer":["(-1,1)"],"problemType":"TextBox","stepTitle":"$$|x|<1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-1,1)$$","hints":{"DefaultPathway":[{"id":"a65ae04inequalities27a-h1","type":"hint","dependencies":[],"title":"Writing the Inequality","text":"Write the equivalent inequality. For example, if the expression is originally $$|x|<3$$, the equivalent inequality is $$-3<x<3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities27a-h2","type":"hint","dependencies":["a65ae04inequalities27a-h1"],"title":"Graphing the Solution","text":"Graph the solution (the possible values of $$x$$ based on the inequality.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities27a-h3","type":"hint","dependencies":["a65ae04inequalities27a-h2"],"title":"Writing the Solution","text":"Lastly, write the solution using interval notation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a65ae04inequalities28","title":"Solving Absolute Value Inequalities","body":"Write the solution using interval notation. For example, if the solution is $$-5<x<3$$, input $$\\"(-5,3)\\"$$ without the quotes. Pay attention to whether the notation should be with parentheses or brackets based on whether the interval is open or closed.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Absolute Value Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a65ae04inequalities28a","stepAnswer":["[2/5,2]"],"problemType":"TextBox","stepTitle":"$$|5x-6| \\\\leq 4$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[\\\\frac{2}{5},2]$$","hints":{"DefaultPathway":[{"id":"a65ae04inequalities28a-h1","type":"hint","dependencies":[],"title":"Writing the Inequality","text":"Write the equivalent inequality. For example, if the expression is originally $$|x|<3$$, the equivalent inequality is $$-3<x<3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities28a-h2","type":"hint","dependencies":["a65ae04inequalities28a-h1"],"title":"Graphing the Solution","text":"Graph the solution (the possible values of $$x$$ based on the inequality.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities28a-h3","type":"hint","dependencies":["a65ae04inequalities28a-h2"],"title":"Writing the Solution","text":"Lastly, write the solution using interval notation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a65ae04inequalities29","title":"Solving Absolute Value Inequalities","body":"Write the solution using interval notation. For example, if the solution is $$-5<x<3$$, input $$\\"(-5,3)\\"$$ without the quotes. Pay attention to whether the notation should be with parentheses or brackets based on whether the interval is open or closed.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Absolute Value Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a65ae04inequalities29a","stepAnswer":["[-2,3]"],"problemType":"TextBox","stepTitle":"$$|2x-1| \\\\leq 5$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a65ae04inequalities29a-h1","type":"hint","dependencies":[],"title":"Writing the Inequality","text":"Write the equivalent inequality. For example, if the expression is originally $$|x|<3$$, the equivalent inequality is $$-3<x<3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities29a-h2","type":"hint","dependencies":["a65ae04inequalities29a-h1"],"title":"Graphing the Solution","text":"Graph the solution (the possible values of $$x$$ based on the inequality.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities29a-h3","type":"hint","dependencies":["a65ae04inequalities29a-h2"],"title":"Writing the Solution","text":"Lastly, write the solution using interval notation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a65ae04inequalities3","title":"Solve Absolute Value Equations","body":"Solve for $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Absolute Value Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a65ae04inequalities3a","stepAnswer":["$$x=0$$"],"problemType":"MultipleChoice","stepTitle":"$$|x|=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=0$$","choices":["$$x=\\\\pm 0$$","$$x=-0$$","$$x=0$$","No Solution"],"hints":{"DefaultPathway":[{"id":"a65ae04inequalities3a-h1","type":"hint","dependencies":[],"title":"Write the equivalent equations using the formula. If $$|x|=a$$, then $$u=-a$$ and $$u=a$$. If $$|x|=-a$$, then there is no solution.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities3a-h2","type":"hint","dependencies":["a65ae04inequalities3a-h1"],"title":"Answer","text":"The answer is $$x=0$$ as $$0$$ is neither positive or negative, and therefore has no sign.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a65ae04inequalities30","title":"Solving Absolute Value Inequalities","body":"Write the solution using interval notation. For example, if the solution is $$-5<x<3$$, input $$\\"(-5,3)\\"$$ without the quotes. Pay attention to whether the notation should be with parentheses or brackets based on whether the interval is open or closed.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Absolute Value Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a65ae04inequalities30a","stepAnswer":["[1/2,2]"],"problemType":"TextBox","stepTitle":"$$|4x-5| \\\\leq 3$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[\\\\frac{1}{2},2]$$","hints":{"DefaultPathway":[{"id":"a65ae04inequalities30a-h1","type":"hint","dependencies":[],"title":"Writing the Inequality","text":"Write the equivalent inequality. For example, if the expression is originally $$|x|<3$$, the equivalent inequality is $$-3<x<3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities30a-h2","type":"hint","dependencies":["a65ae04inequalities30a-h1"],"title":"Graphing the Solution","text":"Graph the solution (the possible values of $$x$$ based on the inequality.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities30a-h3","type":"hint","dependencies":["a65ae04inequalities30a-h2"],"title":"Writing the Solution","text":"Lastly, write the solution using interval notation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a65ae04inequalities4","title":"Solve Absolute Value Equations","body":"Solve for $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Absolute Value Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a65ae04inequalities4a","stepAnswer":["$$x=1-\\\\frac{1}{2}$$"],"problemType":"MultipleChoice","stepTitle":"$$|4x-1|-3=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=1-\\\\frac{1}{2}$$","choices":["$$x=1-\\\\frac{1}{2}$$","x=2,3/2","$$x=2, 4$$","$$x=\\\\frac{-4}{5}-\\\\frac{1}{2}$$"],"hints":{"DefaultPathway":[{"id":"a65ae04inequalities4a-h1","type":"hint","dependencies":[],"title":"Isolate","text":"Isolate the absolute value on one side of the equation, with constants on the other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities4a-h2","type":"hint","dependencies":["a65ae04inequalities4a-h1"],"title":"Split","text":"Split the equation into two, one where $$4x-1=3$$, and one where $$4x-1=-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities4a-h3","type":"hint","dependencies":["a65ae04inequalities4a-h2"],"title":"Solve","text":"Now, solve for $$x$$ in both of the above equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities4a-h4","type":"hint","dependencies":["a65ae04inequalities4a-h3"],"title":"Answer","text":"The answer is $$x=1-\\\\frac{1}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a65ae04inequalities5","title":"Solve Absolute Value Equations","body":"Solve for $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Absolute Value Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a65ae04inequalities5a","stepAnswer":["$$x=-1-\\\\frac{5}{2}$$"],"problemType":"MultipleChoice","stepTitle":"$$|4x+7|+2=5$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=-1-\\\\frac{5}{2}$$","choices":["$$x=-1-\\\\frac{5}{2}$$","x=2,3/2","$$x=2, 4$$","$$x=\\\\frac{-4}{5}-\\\\frac{1}{2}$$"],"hints":{"DefaultPathway":[{"id":"a65ae04inequalities5a-h1","type":"hint","dependencies":[],"title":"Isolate","text":"Isolate the absolute value on one side of the equation, with constants on the other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities5a-h2","type":"hint","dependencies":["a65ae04inequalities5a-h1"],"title":"Split","text":"Split the equation into two, one where $$4x+7=3$$, and one where $$4x+7=-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities5a-h3","type":"hint","dependencies":["a65ae04inequalities5a-h2"],"title":"Solve","text":"Now, solve for $$x$$ in both of the above equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities5a-h4","type":"hint","dependencies":["a65ae04inequalities5a-h3"],"title":"Answer","text":"The answer is $$x=-1-\\\\frac{5}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a65ae04inequalities6","title":"Solve Absolute Value Equations","body":"Solve for $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Absolute Value Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a65ae04inequalities6a","stepAnswer":["$$x=7, 1$$"],"problemType":"MultipleChoice","stepTitle":"$$3|x-4|+2=11$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=7, 1$$","choices":["$$x=7, 1$$","$$x=-2, 3$$","$$x=8, -4$$","$$x=2, 5$$"],"hints":{"DefaultPathway":[{"id":"a65ae04inequalities6a-h1","type":"hint","dependencies":[],"title":"Isolate","text":"Isolate the absolute value on one side of the equation, with constants on the other by using operations on both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities6a-h2","type":"hint","dependencies":["a65ae04inequalities6a-h1"],"title":"Split","text":"Split the equation into two, one where $$x+4=3$$, and one where $$x+4=-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities6a-h3","type":"hint","dependencies":["a65ae04inequalities6a-h2"],"title":"Solve","text":"Now, solve for $$x$$ in both of the above equations to find the two solutions to the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities6a-h4","type":"hint","dependencies":["a65ae04inequalities6a-h3"],"title":"Answer","text":"The answer is $$x=7, 1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a65ae04inequalities7","title":"Solve Absolute Value Equations","body":"Solve for $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Absolute Value Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a65ae04inequalities7a","stepAnswer":["$$x=1, -5$$"],"problemType":"MultipleChoice","stepTitle":"$$3|x+2|-5=4$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=1, -5$$","choices":["$$x=1, -5$$","$$x=3, -7$$","$$x=-6, 9$$","$$x=1, -2$$"],"hints":{"DefaultPathway":[{"id":"a65ae04inequalities7a-h1","type":"hint","dependencies":[],"title":"Isolate","text":"Isolate the absolute value on one side of the equation, with constants on the other by using operations on both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities7a-h2","type":"hint","dependencies":["a65ae04inequalities7a-h1"],"title":"Split","text":"Split the equation into two, one where $$x+2=3$$, and one where $$x+2=-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities7a-h3","type":"hint","dependencies":["a65ae04inequalities7a-h2"],"title":"Solve","text":"Now, solve for $$x$$ in both of the above equations to find the two solutions to the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities7a-h4","type":"hint","dependencies":["a65ae04inequalities7a-h3"],"title":"Answer","text":"The answer is $$x=1, -5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a65ae04inequalities8","title":"Solve Absolute Value Equations","body":"Solve for $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Absolute Value Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a65ae04inequalities8a","stepAnswer":["$$x=7, 1$$"],"problemType":"MultipleChoice","stepTitle":"$$-3|x-4|+4=-5$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=7, 1$$","choices":["$$x=7, 1$$","$$x=2, 3$$","$$x=4, -5$$","$$x=-6, -5$$"],"hints":{"DefaultPathway":[{"id":"a65ae04inequalities8a-h1","type":"hint","dependencies":[],"title":"Isolate","text":"Isolate the absolute value on one side of the equation, with constants on the other by using operations on both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities8a-h2","type":"hint","dependencies":["a65ae04inequalities8a-h1"],"title":"Split","text":"Split the equation into two, one where $$x-4=3$$, and one where $$x-4=-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities8a-h3","type":"hint","dependencies":["a65ae04inequalities8a-h2"],"title":"Solve","text":"Now, solve for $$x$$ in both of the above equations to find the two solutions to the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities8a-h4","type":"hint","dependencies":["a65ae04inequalities8a-h3"],"title":"Answer","text":"The answer is $$x=7, 1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a65ae04inequalities9","title":"Solve Absolute Value Equations","body":"Solve for $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Absolute Value Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a65ae04inequalities9a","stepAnswer":["No Solution"],"problemType":"MultipleChoice","stepTitle":"$$|\\\\frac{3x}{5}-2|+5=2$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$x=7, 1$$","$$x=2, 3$$","$$x=4, -5$$","No Solution"],"hints":{"DefaultPathway":[{"id":"a65ae04inequalities9a-h1","type":"hint","dependencies":[],"title":"Isolate","text":"Isolate the absolute value on one side of the equation, with constants on the other by using operations on both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a65ae04inequalities9a-h2","type":"hint","dependencies":["a65ae04inequalities9a-h1"],"title":"Answer","text":"The answer is no solution as there can be no negative of an absolute value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6614c6sol1","title":"System of Equations by Elimination","body":"Solve the systems of equations by elimination. Write solution as an ordered pair.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Solve Systems of Equations by Elimination","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6614c6sol1a","stepAnswer":["$$(-2,6)$$"],"problemType":"MultipleChoice","stepTitle":"","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$(-2,6)$$","choices":["$$(-2,6)$$","$$(6,-2)$$","$$(2,-6)$$"],"hints":{"DefaultPathway":[{"id":"a6614c6sol1a-h1","type":"hint","dependencies":[],"title":"Eliminate a Variable","text":"Elimintate the y\'s by multiplying the second equation by $$2$$.\\\\n$$5x+2y=2$$\\\\n$$2\\\\left(-3x-y\\\\right)=2\\\\times0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol1a-h2","type":"hint","dependencies":["a6614c6sol1a-h1"],"title":"Simplify","text":"$$5x+2y=2$$\\\\n$$-6x-2y=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol1a-h3","type":"hint","dependencies":["a6614c6sol1a-h2"],"title":"Add the Equations to Eliminate One Variable","text":"Add the x\'s, y\'s, and constants. $$y$$ will be eliminated and then solve for $$x$$.\\\\n$$-x=2$$\\\\n$$x=-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol1a-h4","type":"hint","dependencies":["a6614c6sol1a-h3"],"title":"Substitute Solution into Original Equations","text":"Substitute $$x=-2$$ into the first equation, $$5x+2y=2$$. Then solve for $$y$$.\\\\n$$5x+2y=2$$\\\\n$$5\\\\left(-2\\\\right)+2y=2$$\\\\n$$-10+2y=2$$\\\\n$$2y=12$$\\\\n$$y=6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol1a-h5","type":"hint","dependencies":["a6614c6sol1a-h3","a6614c6sol1a-h4"],"title":"Solution as Ordered Pair","text":"Solution to the system of equations is $$(-2,6)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6614c6sol10","title":"System of Equations by Elimination","body":"Using elimination, solve the system of equations.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Solve Systems of Equations by Elimination","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6614c6sol10a","stepAnswer":["$$(6,9)$$"],"problemType":"MultipleChoice","stepTitle":"Refer to the image for the system of equations.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$(6,9)$$","choices":["$$(6,9)$$","$$(9,6)$$","$$(-6,9)$$","$$(6,-9)$$","$$(-9,6)$$"],"hints":{"DefaultPathway":[{"id":"a6614c6sol10a-h1","type":"hint","dependencies":[],"title":"Eliminate a Variable","text":"The equations are already in standard form. To get opposite coefficients of $$y$$, multiply the first equation by $$2$$.\\\\n(-3*x+y=-9)*2\\\\n$$x-2y=-12$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol10a-h2","type":"hint","dependencies":["a6614c6sol10a-h1"],"title":"Simplify and Add","text":"Add the two equations.\\\\n$$-6x+2y=-18$$\\\\n$$x-2y=-12$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol10a-h3","type":"hint","dependencies":["a6614c6sol10a-h2"],"title":"Solve for $$x$$","text":"$$y$$ will be eliminated and then solve for $$x$$.\\\\n$$-5x=-30$$\\\\n$$x=6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol10a-h4","type":"hint","dependencies":["a6614c6sol10a-h3"],"title":"Substitute and Solve for $$y$$","text":"Substitute $$x=6$$ into one of the original equations and solve for $$y$$.\\\\n$$-3\\\\times6+y=-9$$\\\\n$$-18+y=-9$$\\\\n$$y=9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol10a-h5","type":"hint","dependencies":["a6614c6sol10a-h4"],"title":"Solution as Ordered Pair","text":"Solution to the system of equations is $$(6,9)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6614c6sol11","title":"System of Equations by Elimination","body":"Using elimination, solve the system of equations.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Solve Systems of Equations by Elimination","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6614c6sol11a","stepAnswer":["$$(-2,1)$$"],"problemType":"MultipleChoice","stepTitle":"Refer to the image for the system of equations.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$(-2,1)$$","choices":["$$(2,1)$$","$$(-2,1)$$","$$(-2,-1)$$","$$(1,-2)$$","$$(-1,2)$$"],"hints":{"DefaultPathway":[{"id":"a6614c6sol11a-h1","type":"hint","dependencies":[],"title":"Eliminate a Variable","text":"The equations are already in standard form. To get opposite coefficients of $$y$$, multiply the first equation by $$2$$.\\\\n(3*x-y=-7)*2\\\\n$$4x+2y=-6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol11a-h2","type":"hint","dependencies":["a6614c6sol11a-h1"],"title":"Simplify and Add","text":"Add the two equations.\\\\n$$6x-2y=-14$$\\\\n$$4x+2y=-6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol11a-h3","type":"hint","dependencies":["a6614c6sol11a-h2"],"title":"Solve for $$x$$","text":"$$y$$ will be eliminated and then solve for $$x$$.\\\\n$$10x=-20$$\\\\n$$x=-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol11a-h4","type":"hint","dependencies":["a6614c6sol11a-h3"],"title":"Substitute and Solve for $$y$$","text":"Substitute $$x=-2$$ into one of the original equations and solve for $$y$$.\\\\n$$3\\\\left(-2\\\\right)-y=-7$$\\\\n$$-6-y=-7$$\\\\n$$y=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol11a-h5","type":"hint","dependencies":["a6614c6sol11a-h4"],"title":"Solution as Ordered Pair","text":"Solution to the system of equations is $$(-2,1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6614c6sol12","title":"System of Equations by Elimination","body":"Using elimination, solve the system of equations.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Solve Systems of Equations by Elimination","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6614c6sol12a","stepAnswer":["$$(-7,-1)$$"],"problemType":"MultipleChoice","stepTitle":"Refer to the image for the system of equations.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$(-7,-1)$$","choices":["$$(-7,-1)$$","$$(7,-1)$$","$$(-7,1)$$","$$(1,-7)$$","$$(-1,7)$$"],"hints":{"DefaultPathway":[{"id":"a6614c6sol12a-h1","type":"hint","dependencies":[],"title":"Eliminate a Variable","text":"The equations are already in standard form. The coefficients of $$y$$ are already opposite.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol12a-h2","type":"hint","dependencies":["a6614c6sol12a-h1"],"title":"Simplify and Add","text":"Add the two equations.\\\\n$$x+y=-8$$\\\\n$$x-y=-6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol12a-h3","type":"hint","dependencies":["a6614c6sol12a-h2"],"title":"Solve for $$x$$","text":"$$y$$ will be eliminated and then solve for $$x$$.\\\\n$$2x=-14$$\\\\n$$x=-7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol12a-h4","type":"hint","dependencies":["a6614c6sol12a-h3"],"title":"Substitute and Solve for $$y$$","text":"Substitute $$x=-7$$ into one of the original equations and solve for $$y$$.\\\\n$$x+y=-8$$\\\\n$$\\\\left(-7\\\\right)+y=-8$$\\\\n$$y=-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol12a-h5","type":"hint","dependencies":["a6614c6sol12a-h4"],"title":"Solution as Ordered Pair","text":"Solution to the system of equations is $$(-7,-1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6614c6sol13","title":"System of Equations by Elimination","body":"Using elimination, solve the system of equations.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Solve Systems of Equations by Elimination","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6614c6sol13a","stepAnswer":["$$(-2,-4)$$"],"problemType":"MultipleChoice","stepTitle":"Refer to the image for the system of equations.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$(-2,-4)$$","choices":["$$(-2,-4)$$","$$(2,-4)$$","$$(-2,4)$$","$$(2,4)$$"],"hints":{"DefaultPathway":[{"id":"a6614c6sol13a-h1","type":"hint","dependencies":[],"title":"Eliminate a Variable","text":"The equations are already in standard form. The coefficients of $$y$$ are already opposite.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol13a-h2","type":"hint","dependencies":["a6614c6sol13a-h1"],"title":"Simplify and Add","text":"Add the two equations.\\\\n$$-7x+6y=-10$$\\\\n$$x-6y=22$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol13a-h3","type":"hint","dependencies":["a6614c6sol13a-h2"],"title":"Solve for $$x$$","text":"$$y$$ will be eliminated and then solve for $$x$$.\\\\n$$-6x=12$$\\\\n$$x=-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol13a-h4","type":"hint","dependencies":["a6614c6sol13a-h3"],"title":"Substitute and Solve for $$y$$","text":"Substitute $$x=-2$$ into one of the original equations and solve for $$y$$.\\\\n$$-7x+6y=-10$$\\\\n$$-7\\\\left(-2\\\\right)+6y=-10$$\\\\n$$14+6y=-10$$\\\\n$$y=-4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol13a-h5","type":"hint","dependencies":["a6614c6sol13a-h4"],"title":"Solution as Ordered Pair","text":"Solution to the system of equations is $$(-2,-4)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6614c6sol14","title":"System of Equations by Elimination","body":"Using elimination, solve the system of equations.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Solve Systems of Equations by Elimination","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6614c6sol14a","stepAnswer":["$$(-1,3)$$"],"problemType":"MultipleChoice","stepTitle":"Refer to the image for the system of equations.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$(-1,3)$$","choices":["$$(-1,3)$$","$$(1,3)$$","$$(1,-3)$$","$$(-1,-3)$$"],"hints":{"DefaultPathway":[{"id":"a6614c6sol14a-h1","type":"hint","dependencies":[],"title":"Eliminate a Variable","text":"The equations are already in standard form. The coefficients of $$x$$ are already opposite.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol14a-h2","type":"hint","dependencies":["a6614c6sol14a-h1"],"title":"Simplify and Add","text":"Add the two equations.\\\\n$$5x+2y=1$$\\\\n$$-5x-4y=-7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol14a-h3","type":"hint","dependencies":["a6614c6sol14a-h2"],"title":"Solve for $$y$$","text":"$$x$$ will be eliminated and then solve for $$y$$.\\\\n$$-2y=-6$$\\\\n$$y=3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol14a-h4","type":"hint","dependencies":["a6614c6sol14a-h3"],"title":"Substitute and Solve for $$x$$","text":"Substitute $$y=3$$ into one of the original equations and solve for $$x$$.\\\\n$$5x+2y=1$$\\\\n$$5x+2\\\\times3=1$$\\\\n$$5x+6=1$$\\\\n$$x=-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol14a-h5","type":"hint","dependencies":["a6614c6sol14a-h4"],"title":"Solution as Ordered Pair","text":"Solution to the system of equations is $$(-1,3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6614c6sol15","title":"System of Equations by Elimination","body":"Using elimination, solve the system of equations.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Solve Systems of Equations by Elimination","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6614c6sol15a","stepAnswer":["$$(-1,2)$$"],"problemType":"MultipleChoice","stepTitle":"Refer to the image for the system of equations.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$(-1,2)$$","choices":["$$(-1,2)$$","$$(1,2)$$","$$(-1,-2)$$","$$(1,-2)$$"],"hints":{"DefaultPathway":[{"id":"a6614c6sol15a-h1","type":"hint","dependencies":[],"title":"Eliminate a Variable","text":"The equations are already in standard form. To get opposite coefficients of $$y$$, multiply the second equation by $$-2$$.\\\\n$$3x-4y=-11$$\\\\n(x-2*y=-5)*-2","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol15a-h2","type":"hint","dependencies":["a6614c6sol15a-h1"],"title":"Simplify and Add","text":"Add the two equations.\\\\n$$3x-4y=-11$$\\\\n$$-2x+4y=10$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol15a-h3","type":"hint","dependencies":["a6614c6sol15a-h2"],"title":"Solve for $$x$$","text":"$$y$$ will be eliminated and then solve for $$x$$.\\\\n$$x=-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol15a-h4","type":"hint","dependencies":["a6614c6sol15a-h3"],"title":"Substitute and Solve for $$y$$","text":"Substitute $$x=-1$$ into one of the original equations and solve for $$y$$.\\\\n$$3x-4y=-11$$\\\\n$$3\\\\left(-1\\\\right)-4y=-11$$\\\\n$$-3-4y=-11$$\\\\n$$y=2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol15a-h5","type":"hint","dependencies":["a6614c6sol15a-h4"],"title":"Solution as Ordered Pair","text":"Solution to the system of equations is $$(-1,2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6614c6sol16","title":"System of Equations by Elimination","body":"Using elimination, solve the system of equations.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Solve Systems of Equations by Elimination","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6614c6sol16a","stepAnswer":["$$(-5,9)$$"],"problemType":"MultipleChoice","stepTitle":"Refer to the image for the system of equations.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$(-5,9)$$","choices":["$$(-5,9)$$","$$(5,9)$$","$$(5,-9)$$","$$(-5,-9)$$"],"hints":{"DefaultPathway":[{"id":"a6614c6sol16a-h1","type":"hint","dependencies":[],"title":"Eliminate a Variable","text":"The equations are already in standard form. To get opposite coefficients of $$x$$, multiply the second equation by $$6$$.\\\\n$$6x-5y=-75$$\\\\n(-x-2*y=-13)*6","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol16a-h2","type":"hint","dependencies":["a6614c6sol16a-h1"],"title":"Simplify and Add","text":"Add the two equations.\\\\n$$6x-5y=-75$$\\\\n$$-6x-12y=-78$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol16a-h3","type":"hint","dependencies":["a6614c6sol16a-h2"],"title":"Solve for $$y$$","text":"$$x$$ will be eliminated and then solve for $$y$$.\\\\n$$-17y=-153$$\\\\n$$y=9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol16a-h4","type":"hint","dependencies":["a6614c6sol16a-h3"],"title":"Substitute and Solve for $$x$$","text":"Substitute $$y=9$$ into one of the original equations and solve for $$x$$.\\\\n$$6x-5y-75$$\\\\n$$6x-5\\\\times9=-75$$\\\\n$$6x-45=-75$$\\\\n$$x=-5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol16a-h5","type":"hint","dependencies":["a6614c6sol16a-h4"],"title":"Solution as Ordered Pair","text":"Solution to the system of equations is $$(-5,9)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6614c6sol17","title":"System of Equations","body":"Translate to a system of equations and solve.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Solve Systems of Equations by Elimination","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6614c6sol17a","stepAnswer":["$$20$$ and $$45$$"],"problemType":"MultipleChoice","stepTitle":"The sum of two numbers is $$65$$. Their difference is $$25$$. Find the numbers.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$20$$ and $$45$$","choices":["$$20$$ and $$45$$","$$30$$ and $$35$$","$$40$$ and $$25$$"],"hints":{"DefaultPathway":[{"id":"a6614c6sol17a-h1","type":"hint","dependencies":[],"title":"Identify What We Are Looking For","text":"We are looking for two numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol17a-h2","type":"hint","dependencies":["a6614c6sol17a-h1"],"title":"Name What We Are Looking For","text":"Choose a variable to represent the quantity. Let $$n=the$$ first number. Let $$m=the$$ second number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol17a-h3","type":"hint","dependencies":["a6614c6sol17a-h2"],"title":"Translate into a System of Equations","text":"The sum of the two numbers is 65: $$n+m=65$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol17a-h4","type":"hint","dependencies":["a6614c6sol17a-h3"],"title":"Translate into a System of Equations","text":"Their difference is 25: $$n-m=25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol17a-h5","type":"hint","dependencies":["a6614c6sol17a-h4"],"title":"Solve the System of Equations","text":"To solve the system of equations, use elimination. The equations are in standard form and the coefficients of $$m$$ are opposites.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol17a-h6","type":"hint","dependencies":["a6614c6sol17a-h5"],"title":"Eliminate a Variable","text":"Add the two equations.\\\\n$$n+m=65$$\\\\n$$n-m=25$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol17a-h7","type":"hint","dependencies":["a6614c6sol17a-h6"],"title":"Solve for $$n$$","text":"$$m$$ will be eliminated and then solve for $$n$$.\\\\n$$2n=90$$\\\\n$$n=45$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol17a-h8","type":"hint","dependencies":["a6614c6sol17a-h7"],"title":"Substitute and Solve for $$m$$","text":"Substitute $$n=45$$ into one of the original equations and solve for $$m$$.\\\\n$$n+m=65$$\\\\n$$45+m=65$$\\\\n$$m=20$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol17a-h9","type":"hint","dependencies":["a6614c6sol17a-h8"],"title":"Two Numbers","text":"The two numbers are $$20$$ and $$45$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6614c6sol18","title":"System of Equations","body":"Translate to a system of equations and solve.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Solve Systems of Equations by Elimination","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6614c6sol18a","stepAnswer":["$$23$$ and $$14$$"],"problemType":"MultipleChoice","stepTitle":"The sum of two numbers is $$37$$. Their difference is $$9$$. Find the numbers.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$23$$ and $$14$$","choices":["$$23$$ and $$14$$","$$20$$ and $$17$$","$$20$$ and $$29$$"],"hints":{"DefaultPathway":[{"id":"a6614c6sol18a-h1","type":"hint","dependencies":[],"title":"Identify What We Are Looking For","text":"We are looking for two numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol18a-h2","type":"hint","dependencies":["a6614c6sol18a-h1"],"title":"Name What We Are Looking For","text":"Choose a variable to represent that quantity. Let $$n=the$$ first number. Let $$m=the$$ second number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol18a-h3","type":"hint","dependencies":["a6614c6sol18a-h2"],"title":"Translate into a System of Equations","text":"The sum of the two numbers is 37: $$n+m=37$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol18a-h4","type":"hint","dependencies":["a6614c6sol18a-h3"],"title":"Translate into a System of Equations","text":"Their difference is 9: $$n-m=9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol18a-h5","type":"hint","dependencies":["a6614c6sol18a-h4"],"title":"Solve the System of Equations","text":"To solve the system of equations, use elimination. The equations are in standard form and the coefficients of $$m$$ are opposites.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol18a-h6","type":"hint","dependencies":["a6614c6sol18a-h5"],"title":"Eliminate a Variable","text":"Add the two equations.\\\\n$$n+m=37$$\\\\n$$n-m=9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol18a-h7","type":"hint","dependencies":["a6614c6sol18a-h6"],"title":"Solve for $$n$$","text":"$$m$$ will be eliminated and then solve for $$n$$.\\\\n$$2n=46$$\\\\n$$n=23$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol18a-h8","type":"hint","dependencies":["a6614c6sol18a-h7"],"title":"Substitute and Solve for $$m$$","text":"Substitute $$n=23$$ into one of the original equations and solve for $$m$$.\\\\n$$n+m=37$$\\\\n$$23+m=37$$\\\\n$$m=14$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol18a-h9","type":"hint","dependencies":["a6614c6sol18a-h8"],"title":"Two Numbers","text":"The two numbers are $$23$$ and $$14$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6614c6sol19","title":"System of Equations","body":"Translate to a system of equations and solve.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Solve Systems of Equations by Elimination","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6614c6sol19a","stepAnswer":["$$16$$ and $$-43$$"],"problemType":"MultipleChoice","stepTitle":"The sum of two numbers is $$-27$$. Their difference is $$-59$$. Find the numbers.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$16$$ and $$-43$$","choices":["$$16$$ and $$43$$","$$-16$$ and $$-43$$","$$16$$ and $$-43$$","$$-16$$ and $$43$$"],"hints":{"DefaultPathway":[{"id":"a6614c6sol19a-h1","type":"hint","dependencies":[],"title":"Identify What We Are Looking For","text":"We are looking for two numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol19a-h2","type":"hint","dependencies":["a6614c6sol19a-h1"],"title":"Name What We Are Looking For","text":"Choose a variable to represent that quantity. Let $$n=the$$ first number. Let $$m=the$$ second number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol19a-h3","type":"hint","dependencies":["a6614c6sol19a-h2"],"title":"Translate into a System of Equations","text":"The sum of the two numbers is -27: $$n+m=-27$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol19a-h4","type":"hint","dependencies":["a6614c6sol19a-h3"],"title":"Translate into a System of Equations","text":"Their difference is -59: $$n-m=-59$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol19a-h5","type":"hint","dependencies":["a6614c6sol19a-h4"],"title":"Solve the System of Equations","text":"To solve the system of equations, use elimination. The equations are in standard form and the coefficients of $$m$$ are opposites.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol19a-h6","type":"hint","dependencies":["a6614c6sol19a-h5"],"title":"Eliminate a Variable","text":"Add the two equations.\\\\n$$n+m=-27$$\\\\n$$n-m=-59$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol19a-h7","type":"hint","dependencies":["a6614c6sol19a-h6"],"title":"Solve for $$n$$","text":"$$m$$ will be eliminated and then solve for $$n$$.\\\\n$$2n=-86$$\\\\n$$n=-43$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol19a-h8","type":"hint","dependencies":["a6614c6sol19a-h7"],"title":"Substitute and Solve for $$m$$","text":"Substitute $$n=-43$$ into one of the original equations and solve for $$m$$.\\\\n$$n+m=-27$$\\\\n$$-43+m=-27$$\\\\n$$m=16$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol19a-h9","type":"hint","dependencies":["a6614c6sol19a-h8"],"title":"Two Numbers","text":"The two numbers are $$16$$ and $$-43$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6614c6sol2","title":"System of Equations by Elimination","body":"Solve the systems of equations by elimination. Write solution as an ordered pair.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Solve Systems of Equations by Elimination","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6614c6sol2a","stepAnswer":["$$(6,9)$$"],"problemType":"MultipleChoice","stepTitle":"","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$(6,9)$$","choices":["$$(-6,-9)$$","$$(6,9)$$","$$(9,6)$$"],"hints":{"DefaultPathway":[{"id":"a6614c6sol2a-h1","type":"hint","dependencies":[],"title":"Eliminate a Variable","text":"Elimintate the x\'s by multiplying the second equation by $$3$$.\\\\n$$-3x+y=-9$$\\\\n$$3\\\\left(x-2y\\\\right)=3\\\\left(-12\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol2a-h2","type":"hint","dependencies":["a6614c6sol2a-h1"],"title":"Simplify","text":"$$-3x+y=-9$$\\\\n$$3x-6y=-36$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol2a-h3","type":"hint","dependencies":["a6614c6sol2a-h2"],"title":"Add the Equations to Eliminate One Variable","text":"Add the x\'s, y\'s, and constants. $$x$$ will be eliminated and then solve for $$y$$.\\\\n$$-5y=-45$$\\\\n$$y=9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol2a-h4","type":"hint","dependencies":["a6614c6sol2a-h3"],"title":"Substitute Solution into Original Equations","text":"Substitute $$y=9$$ into the first equation, $$-3x+y=-9$$. Then solve for $$x$$.\\\\n$$-3x+y=-9$$\\\\n$$-3x+9=-9$$\\\\n$$-3x=-18$$\\\\n$$x=6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol2a-h5","type":"hint","dependencies":["a6614c6sol2a-h3","a6614c6sol2a-h4"],"title":"Solution as Ordered Pair","text":"Solution to the system of equations is $$(6,9)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6614c6sol20","title":"System of Equations","body":"Translate to a system of equations and solve.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Solve Systems of Equations by Elimination","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6614c6sol20a","stepAnswer":["$$-67$$ and $$22$$"],"problemType":"MultipleChoice","stepTitle":"The sum of two numbers is $$-45$$. Their difference is $$-89$$. Find the numbers.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-67$$ and $$22$$","choices":["$$-67$$ and $$22$$","$$67$$ and $$22$$","$$-67$$ and $$-22$$","$$67$$ and $$-22$$"],"hints":{"DefaultPathway":[{"id":"a6614c6sol20a-h1","type":"hint","dependencies":[],"title":"Identify What We Are Looking For","text":"We are looking for two numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol20a-h2","type":"hint","dependencies":["a6614c6sol20a-h1"],"title":"Name What We Are Looking For","text":"Choose a variable to represent that quantity. Let $$n=the$$ first number. Let $$m=the$$ second number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol20a-h3","type":"hint","dependencies":["a6614c6sol20a-h2"],"title":"Translate into a System of Equations","text":"The sum of the two numbers is -45: $$n+m=-45$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol20a-h4","type":"hint","dependencies":["a6614c6sol20a-h3"],"title":"Translate into a System of Equations","text":"Their difference is -89: $$n-m=-89$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol20a-h5","type":"hint","dependencies":["a6614c6sol20a-h4"],"title":"Solve the System of Equations","text":"To solve the system of equations, use elimination. The equations are in standard form and the coefficients of $$m$$ are opposites.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol20a-h6","type":"hint","dependencies":["a6614c6sol20a-h5"],"title":"Eliminate a Variable","text":"Add the two equations.\\\\n$$n+m=-45$$\\\\n$$n-m=-89$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol20a-h7","type":"hint","dependencies":["a6614c6sol20a-h6"],"title":"Solve for $$n$$","text":"$$m$$ will be eliminated and then solve for $$n$$.\\\\n$$2n=-134$$\\\\n$$n=-67$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol20a-h8","type":"hint","dependencies":["a6614c6sol20a-h7"],"title":"Substitute and Solve for $$m$$","text":"Substitute $$n=-67$$ into one of the original equations and solve for $$m$$.\\\\n$$n+m=-45$$\\\\n$$-67+m=-45$$\\\\n$$m=22$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol20a-h9","type":"hint","dependencies":["a6614c6sol20a-h8"],"title":"Two Numbers","text":"The two numbers are $$-67$$ and $$22$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6614c6sol21","title":"Solving Systems of Equations","body":"Andrea is buying some new shirts and sweaters. She is able to buy $$3$$ shirts and $$2$$ sweaters for $114 or she is able to buy $$2$$ shirts and $$4$$ sweaters for $164.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Solve Systems of Equations by Elimination","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6614c6sol21a","stepAnswer":["$$16$$"],"problemType":"TextBox","stepTitle":"How much does a shirt cost?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16$$","hints":{"DefaultPathway":[{"id":"a6614c6sol21a-h1","type":"hint","dependencies":[],"title":"Identify What We Are Looking For","text":"We are looking for the cost of the shirt. We also do not know the cost of the sweater.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol21a-h2","type":"hint","dependencies":["a6614c6sol21a-h1"],"title":"Name What We Are Looking For","text":"Choose a variable to represent that quantity. Let $$n=the$$ cost of the shirt. Let $$m=the$$ cost of the sweater.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol21a-h3","type":"hint","dependencies":["a6614c6sol21a-h2"],"title":"Translate into a System of Equations","text":"$$3$$ shirts and $$2$$ sweaters cost $114: $$3n+2m=114$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol21a-h4","type":"hint","dependencies":["a6614c6sol21a-h3"],"title":"Translate into a System of Equations","text":"$$2$$ shirts and $$4$$ sweaters cost $164: $$2n+4m=164$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol21a-h5","type":"hint","dependencies":["a6614c6sol21a-h4"],"title":"Solve the System of Equations","text":"To solve the system of equations, use elimination. The equations are in standard form. To get opposite coefficients of $$m$$, divide the second equation by $$-2$$.\\\\n$$3n+2m=114$$\\\\n(2*n+4*m=164)/-2","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol21a-h6","type":"hint","dependencies":["a6614c6sol21a-h5"],"title":"Eliminate a Variable","text":"Add the two equations.\\\\n$$3n+2m=114$$\\\\n$$-n-2m=-82$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol21a-h7","type":"hint","dependencies":["a6614c6sol21a-h6"],"title":"Solve for $$n$$","text":"$$m$$ will be eliminated and then solve for $$n$$.\\\\n$$2n=32$$\\\\n$$n=16$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol21a-h8","type":"hint","dependencies":["a6614c6sol21a-h7"],"title":"Cost of Shirt","text":"The shirt cost $16.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6614c6sol21b","stepAnswer":["$$33$$"],"problemType":"TextBox","stepTitle":"How much does a sweater cost?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$33$$","hints":{"DefaultPathway":[{"id":"a6614c6sol21b-h1","type":"hint","dependencies":[],"title":"Substitute and Solve for $$m$$","text":"Substitute $$n=16$$ into one of the original equations and solve for $$m$$.\\\\n$$3n+2m=114$$\\\\n$$3\\\\times16+2m=114$$\\\\n$$48+2m=114$$\\\\n$$m=33$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol21b-h2","type":"hint","dependencies":["a6614c6sol21b-h1"],"title":"Cost of Sweater","text":"The sweater cost $33.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6614c6sol22","title":"Solving Systems of Equations","body":"Peter is buying office supplies. He is able to buy $$3$$ packages of paper and $$4$$ staplers for $40 or he is able to buy $$5$$ packages of paper and $$6$$ staplers for $62.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Solve Systems of Equations by Elimination","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6614c6sol22a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"How much does a package of paper cost?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a6614c6sol22a-h1","type":"hint","dependencies":[],"title":"Identify What We Are Looking For","text":"We are looking for the cost of a package of paper. We also do not know the cost of a stapler.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol22a-h2","type":"hint","dependencies":["a6614c6sol22a-h1"],"title":"Name What We Are Looking For","text":"Choose a variable to represent that quantity. Let $$p=the$$ cost of a package of paper. Let $$s=the$$ cost of a stapler.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol22a-h3","type":"hint","dependencies":["a6614c6sol22a-h2"],"title":"Translate into a System of Equations","text":"$$3$$ packages of paper and $$4$$ staplers cost $40: $$3p+4s=40$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol22a-h4","type":"hint","dependencies":["a6614c6sol22a-h3"],"title":"Translate into a System of Equations","text":"$$5$$ packages of paper and $$6$$ staplers cost $62: $$5n+6m=62$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol22a-h5","type":"hint","dependencies":["a6614c6sol22a-h4"],"title":"Solve the System of Equations","text":"To solve the system of equations, use elimination. The equations are in standard form. To get opposite coefficients of s, multiply the first equation by $$-3$$ and the second equation by $$2$$.\\\\n(3*p+4*s=40)*-3\\\\n(5*n+6*m=62)*2","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol22a-h6","type":"hint","dependencies":["a6614c6sol22a-h5"],"title":"Eliminate a Variable","text":"Add the two equations.\\\\n$$-9p-12s=-120$$\\\\n$$10p+12s=124$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol22a-h7","type":"hint","dependencies":["a6614c6sol22a-h6"],"title":"Solve for $$p$$","text":"s will be eliminated and then solve for $$p$$.\\\\n$$p=4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol22a-h8","type":"hint","dependencies":["a6614c6sol22a-h7"],"title":"Cost of Paper","text":"The package of paper cost $4.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6614c6sol22b","stepAnswer":["$$7$$"],"problemType":"TextBox","stepTitle":"How much does a stapler cost?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7$$","hints":{"DefaultPathway":[{"id":"a6614c6sol22b-h1","type":"hint","dependencies":[],"title":"Substitute and Solve for s","text":"Substitute $$p=4$$ into one of the original equations and solve for s.\\\\n$$3p+4s=40$$\\\\n$$3\\\\times4+4s=40$$\\\\n$$12+4s=40$$\\\\n$$s=7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol22b-h2","type":"hint","dependencies":["a6614c6sol22b-h1"],"title":"Cost of Staplers","text":"The staplers cost $7.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6614c6sol23","title":"Solving Systems of Equations","body":"The total amount of sodium in $$2$$ hot dogs and $$3$$ cups of cottage cheese is $$4720$$ mg. The total amount of sodium in $$5$$ hot dogs and $$2$$ cups of cottage cheese is $$6300$$ mg.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Solve Systems of Equations by Elimination","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6614c6sol23a","stepAnswer":["$$860$$"],"problemType":"TextBox","stepTitle":"How much sodium is in a hot dog?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$860$$","hints":{"DefaultPathway":[{"id":"a6614c6sol23a-h1","type":"hint","dependencies":[],"title":"Identify What We Are Looking For","text":"We are looking for the amount of sodium in a hot dog. We also do not know how much sodium is in a cup of cottage cheese.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol23a-h2","type":"hint","dependencies":["a6614c6sol23a-h1"],"title":"Name What We Are Looking For","text":"Choose a variable to represent that quantity. Let $$h=the$$ amount of sodium in hot dog. Let $$c=the$$ amount of sodium in cottage cheese.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol23a-h3","type":"hint","dependencies":["a6614c6sol23a-h2"],"title":"Translate into a System of Equations","text":"The total amount of sodium in $$2$$ hot dogs and $$3$$ cups of cottage cheese is $$4720$$ mg: $$2h+3c=4720$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol23a-h4","type":"hint","dependencies":["a6614c6sol23a-h3"],"title":"Translate into a System of Equations","text":"The total amount of sodium in $$5$$ hot dogs and $$2$$ cups of cottage cheese is $$6300$$ mg: $$5h+2c=6300$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol23a-h5","type":"hint","dependencies":["a6614c6sol23a-h4"],"title":"Solve the System of Equations","text":"To solve the system of equations, use elimination. The equations are in standard form. To get opposite coefficients of c, multiply the first equation by $$-2$$ and the second equation by $$3$$.\\\\n(2*h+3*c=4720)*-2\\\\n(5*h+2*c=6300)*3","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol23a-h6","type":"hint","dependencies":["a6614c6sol23a-h5"],"title":"Eliminate a Variable","text":"Add the two equations.\\\\n$$-4h-6c=-9440$$\\\\n$$15h+6c=18900$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol23a-h7","type":"hint","dependencies":["a6614c6sol23a-h6"],"title":"Solve for $$h$$","text":"c will be eliminated and then solve for $$h$$.\\\\n$$11h=9460$$\\\\n$$h=860$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol23a-h8","type":"hint","dependencies":["a6614c6sol23a-h7"],"title":"Amount of Sodium","text":"There is $$860$$ mg of sodium in the hotdog.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6614c6sol23b","stepAnswer":["$$1000$$"],"problemType":"TextBox","stepTitle":"How much sodium is in a cup of cottage cheese?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1000$$","hints":{"DefaultPathway":[{"id":"a6614c6sol23b-h1","type":"hint","dependencies":[],"title":"Substitute and Solve for c","text":"Substitute $$h=860$$ into one of the original equations and solve for c.\\\\n$$2h+3c=4720$$\\\\n$$2\\\\times860+3c=4720$$\\\\n$$1720+3c=4720$$\\\\n$$c=1000$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol23b-h2","type":"hint","dependencies":["a6614c6sol23b-h1"],"title":"Amount of Sodium","text":"There is $$1000$$ mg of sodium in a cup of cotage cheese.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6614c6sol24","title":"Solving Systems of Equations","body":"The total number of calories in $$2$$ hot dogs and $$3$$ cups of cottage cheese is $$960$$ calories. The total number of calories in $$5$$ hot dogs and $$2$$ cups of cottage cheese is $$1190$$ calories.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Solve Systems of Equations by Elimination","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6614c6sol24a","stepAnswer":["$$150$$"],"problemType":"TextBox","stepTitle":"How many calories are in a hot dog?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$150$$","hints":{"DefaultPathway":[{"id":"a6614c6sol24a-h1","type":"hint","dependencies":[],"title":"Identify What We Are Looking For","text":"We are looking for the amount of calories in a hot dog. We also do not know how much calories is in a cup of cottage cheese.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol24a-h2","type":"hint","dependencies":["a6614c6sol24a-h1"],"title":"Name What We Are Looking For","text":"Choose a variable to represent that quantity. Let $$h=the$$ amount of calories in hot dog. Let $$c=the$$ amount of calories in cottage cheese.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol24a-h3","type":"hint","dependencies":["a6614c6sol24a-h2"],"title":"Translate into a System of Equations","text":"The total number of calories in $$2$$ hot dogs and $$3$$ cups of cottage cheese is $$960$$ calories: $$2h+3c=960$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol24a-h4","type":"hint","dependencies":["a6614c6sol24a-h3"],"title":"Translate into a System of Equations","text":"The total number of calories in $$5$$ hot dogs and $$2$$ cups of cottage cheese is $$1190$$ calories: $$5h+2c=1190$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol24a-h5","type":"hint","dependencies":["a6614c6sol24a-h4"],"title":"Solve the System of Equations","text":"To solve the system of equations, use elimination. The equations are in standard form. To get opposite coefficients of c, multiply the first equation by $$-2$$ and the second equation by $$3$$.\\\\n(2*h+3*c=960)*-2\\\\n(5*h+2*c=1190)*3","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol24a-h6","type":"hint","dependencies":["a6614c6sol24a-h5"],"title":"Eliminate a Variable","text":"Add the two equations.\\\\n$$-4h-6c=-1920$$\\\\n$$15h+6c=3570$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol24a-h7","type":"hint","dependencies":["a6614c6sol24a-h6"],"title":"Solve for $$h$$","text":"c will be eliminated and then solve for $$h$$.\\\\n$$11h=1650$$\\\\n$$h=150$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol24a-h8","type":"hint","dependencies":["a6614c6sol24a-h7"],"title":"Substitute and Solve for c","text":"There are $$150$$ calories in a hotdog.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6614c6sol24b","stepAnswer":["$$220$$"],"problemType":"TextBox","stepTitle":"How many calories are in a cup of cottage cheese?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$220$$","hints":{"DefaultPathway":[{"id":"a6614c6sol24b-h1","type":"hint","dependencies":[],"title":"Substitute and Solve for c","text":"Substitute $$h=150$$ into one of the original equations and solve for c.\\\\n$$2h+3c=960$$\\\\n$$2\\\\times150+3c=960$$\\\\n$$300+3c=960$$\\\\n$$c=220$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol24b-h2","type":"hint","dependencies":["a6614c6sol24b-h1"],"title":"Substitute and Solve for c","text":"There are $$220$$ calories in a cup of cottage cheese.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6614c6sol3","title":"System of Equations by Elimination","body":"Solve the systems of equations by elimination. Write solution as an ordered pair.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Solve Systems of Equations by Elimination","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6614c6sol3a","stepAnswer":["$$(4,5)$$"],"problemType":"MultipleChoice","stepTitle":"","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$(4,5)$$","choices":["$$(-5,-4)$$","$$(5,4)$$","$$(4,5)$$"],"hints":{"DefaultPathway":[{"id":"a6614c6sol3a-h1","type":"hint","dependencies":[],"title":"Eliminate a Variable","text":"Elimintate the y\'s by multiplying the second equation by $$5$$.\\\\n$$6x-5y=-1$$\\\\n$$5\\\\left(2x+y\\\\right)=5\\\\times13$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol3a-h2","type":"hint","dependencies":["a6614c6sol3a-h1"],"title":"Simplify","text":"$$6x-5y=-1$$\\\\n$$10x+5y=65$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol3a-h3","type":"hint","dependencies":["a6614c6sol3a-h2"],"title":"Add the Equations to Eliminate One Variable","text":"Add the x\'s, y\'s, and constants. $$y$$ will be eliminated and then solve for $$x$$.\\\\n$$16x=64$$\\\\n$$x=4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol3a-h4","type":"hint","dependencies":["a6614c6sol3a-h3"],"title":"Substitute Solution into Original Equations","text":"Substitute $$x=4$$ into the first equation, $$6x-5y=-1$$. Then solve for $$y$$.\\\\n$$6x-5y=-1$$\\\\n$$6\\\\times4-5y=-1$$\\\\n$$24-5y=-1$$\\\\n$$-5y=-25$$\\\\n$$y=5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol3a-h5","type":"hint","dependencies":["a6614c6sol3a-h3","a6614c6sol3a-h4"],"title":"Solution as Ordered Pair","text":"Solution to the system of equations is $$(4,5)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6614c6sol4","title":"System of Equations by Elimination","body":"Solve the systems of equations by elimination. Write solution as an ordered pair.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Solve Systems of Equations by Elimination","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6614c6sol4a","stepAnswer":["$$(-2,1)$$"],"problemType":"MultipleChoice","stepTitle":"","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$(-2,1)$$","choices":["$$(-2,1)$$","$$(1,-2)$$","$$(2,1)$$"],"hints":{"DefaultPathway":[{"id":"a6614c6sol4a-h1","type":"hint","dependencies":[],"title":"Eliminate a Variable","text":"Elimintate the y\'s by multiplying the first equation by $$2$$.\\\\n$$2\\\\left(3x-y\\\\right)=2\\\\left(-7\\\\right)$$\\\\n$$4x+2y=-6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol4a-h2","type":"hint","dependencies":["a6614c6sol4a-h1"],"title":"Simplify","text":"$$6x-2y=-14$$\\\\n$$4x+2y=-6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol4a-h3","type":"hint","dependencies":["a6614c6sol4a-h2"],"title":"Add the Equations to Eliminate One Variable","text":"Add the x\'s, y\'s, and constants. $$y$$ will be eliminated and then solve for $$x$$.\\\\n$$10x=-20$$\\\\n$$x=-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol4a-h4","type":"hint","dependencies":["a6614c6sol4a-h3"],"title":"Substitute Solution into Original Equations","text":"Substitute $$x=-2$$ into the first equation, $$3x-y=-7$$. Then solve for $$y$$.\\\\n$$3x-y=-7$$\\\\n$$3\\\\left(-2\\\\right)-y=-7$$\\\\n$$-6-y=-7$$\\\\n$$-y=-1$$\\\\n$$y=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol4a-h5","type":"hint","dependencies":["a6614c6sol4a-h3","a6614c6sol4a-h4"],"title":"Solution as Ordered Pair","text":"Solution to the system of equations is $$(-2,1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6614c6sol5","title":"System of Equations by Elimination","body":"Solve the systems of equations by elimination. Write solution as an ordered pair.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Solve Systems of Equations by Elimination","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6614c6sol5a","stepAnswer":["$$(-3,2)$$"],"problemType":"MultipleChoice","stepTitle":"","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$(-3,2)$$","choices":["$$(-3,-2)$$","$$(3,-2)$$","$$(-3,2)$$"],"hints":{"DefaultPathway":[{"id":"a6614c6sol5a-h1","type":"hint","dependencies":[],"title":"Eliminate a Variable","text":"The y\'s are already opposites so they will be eliminated once we add up the two equations.\\\\n$$x+y=-1$$\\\\n$$x-y=-5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol5a-h2","type":"hint","dependencies":["a6614c6sol5a-h1"],"title":"Add the Equations to Eliminate One Variable","text":"Add the x\'s, y\'s, and constants. $$y$$ will be eliminated and then solve for $$x$$.\\\\n$$2x=-6$$\\\\n$$x=-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol5a-h3","type":"hint","dependencies":["a6614c6sol5a-h2"],"title":"Substitute Solution into Original Equations","text":"Substitute $$x=-3$$ into the first equation, $$x+y=-1$$. Then solve for $$y$$.\\\\n$$x+y=-1$$\\\\n$$\\\\left(-3\\\\right)+y=-1$$\\\\n$$y=2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol5a-h4","type":"hint","dependencies":["a6614c6sol5a-h2","a6614c6sol5a-h3"],"title":"Solution as Ordered Pair","text":"Solution to the system of equations is $$(-3,2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6614c6sol6","title":"System of Equations by Elimination","body":"Solve the systems of equations by elimination. Write solution as an ordered pair.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Solve Systems of Equations by Elimination","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6614c6sol6a","stepAnswer":["$$(-7,-1)$$"],"problemType":"MultipleChoice","stepTitle":"","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$(-7,-1)$$","choices":["$$(-7,-1)$$","$$(7,-1)$$","$$(-7,1)$$"],"hints":{"DefaultPathway":[{"id":"a6614c6sol6a-h1","type":"hint","dependencies":[],"title":"Eliminate a Variable","text":"The y\'s are already opposites so they will be eliminated once we add up the two equations.\\\\n$$x+y=-8$$\\\\n$$x-y=-6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol6a-h2","type":"hint","dependencies":["a6614c6sol6a-h1"],"title":"Add the Equations to Eliminate One Variable","text":"Add the x\'s, y\'s, and constants. $$y$$ will be eliminated and then solve for $$x$$.\\\\n$$2x=-14$$\\\\n$$x=-7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol6a-h3","type":"hint","dependencies":["a6614c6sol6a-h2"],"title":"Substitute Solution into Original Equations","text":"Substitute $$x=-7$$ into the first equation, $$x+y=-7$$. Then solve for $$y$$.\\\\n$$x+y=-8$$\\\\n$$\\\\left(-7\\\\right)+y=-8$$\\\\n$$y=-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol6a-h4","type":"hint","dependencies":["a6614c6sol6a-h2","a6614c6sol6a-h3"],"title":"Solution as Ordered Pair","text":"Solution to the system of equations is $$(-7,-1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6614c6sol7","title":"System of Equations by Elimination","body":"Solve the systems of equations by elimination. Write solution as an ordered pair.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Solve Systems of Equations by Elimination","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6614c6sol7a","stepAnswer":["$$(5,7)$$"],"problemType":"MultipleChoice","stepTitle":"","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$(5,7)$$","choices":["$$(-7,-5)$$","$$(5,7)$$","$$(7,5)$$"],"hints":{"DefaultPathway":[{"id":"a6614c6sol7a-h1","type":"hint","dependencies":[],"title":"Eliminate a Variable","text":"The y\'s are already opposites so they will be eliminated once we add up the two equations.\\\\n$$3x-2y=1$$\\\\n$$-x+2y=9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol7a-h2","type":"hint","dependencies":["a6614c6sol7a-h1"],"title":"Add the Equations to Eliminate One Variable","text":"Add the x\'s, y\'s, and constants. $$y$$ will be eliminated and then solve for $$x$$.\\\\n$$2x=10$$\\\\n$$x=5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol7a-h3","type":"hint","dependencies":["a6614c6sol7a-h2"],"title":"Substitute Solution into Original Equations","text":"Substitute $$x=5$$ into the first equation, $$3x-2y=1$$. Then solve for $$y$$.\\\\n$$3x-2y=1$$\\\\n$$3\\\\times5-2y=1$$\\\\n$$15-2y=1$$\\\\n$$-2y=-14$$\\\\n$$y=7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol7a-h4","type":"hint","dependencies":["a6614c6sol7a-h2","a6614c6sol7a-h3"],"title":"Solution as Ordered Pair","text":"Solution to the system of equations is $$(5,7)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6614c6sol8","title":"System of Equations by Elimination","body":"Solve the systems of equations by elimination. Write solution as an ordered pair.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Solve Systems of Equations by Elimination","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6614c6sol8a","stepAnswer":["$$(-2,-4)$$"],"problemType":"MultipleChoice","stepTitle":"","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$(-2,-4)$$","choices":["$$(2,-4)$$","$$(2,4)$$","$$(-2,-4)$$"],"hints":{"DefaultPathway":[{"id":"a6614c6sol8a-h1","type":"hint","dependencies":[],"title":"Eliminate a Variable","text":"The y\'s are already opposites so they will be eliminated once we add up the two equations.\\\\n$$-7x+6y=-10$$\\\\n$$x-6y=22$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol8a-h2","type":"hint","dependencies":["a6614c6sol8a-h1"],"title":"Add the Equations to Eliminate One Variable","text":"Add the x\'s, y\'s, and constants. $$y$$ will be eliminated and then solve for $$x$$.\\\\n$$-6x=12$$\\\\n$$x=-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol8a-h3","type":"hint","dependencies":["a6614c6sol8a-h2"],"title":"Substitute Solution into Original Equations","text":"Substitute $$x=-2$$ into the first equation, $$-7x+6y=-10$$. Then solve for $$y$$.\\\\n$$-7x+6y=-10$$\\\\n$$-7\\\\left(-2\\\\right)+6y=-10$$\\\\n$$14+6y=-10$$\\\\n$$6y=-24$$\\\\n$$y=-4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol8a-h4","type":"hint","dependencies":["a6614c6sol8a-h2","a6614c6sol8a-h3"],"title":"Solution as Ordered Pair","text":"Solution to the system of equations is $$(-2,-4)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6614c6sol9","title":"System of Equations by Elimination","body":"Solve the systems of equations by elimination. Write solution as an ordered pair.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Solve Systems of Equations by Elimination","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6614c6sol9a","stepAnswer":["$$(-11,15)$$"],"problemType":"MultipleChoice","stepTitle":"","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$(-11,15)$$","choices":["$$(-11,15)$$","$$(11,15)$$","$$(11,-15)$$"],"hints":{"DefaultPathway":[{"id":"a6614c6sol9a-h1","type":"hint","dependencies":[],"title":"Eliminate a Variable","text":"The y\'s are already opposites so they will be eliminated once we add up the two equations.\\\\n$$3x+2y=-3$$\\\\n$$-x-2y=-19$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol9a-h2","type":"hint","dependencies":["a6614c6sol9a-h1"],"title":"Add the Equations to Eliminate One Variable","text":"Add the x\'s, y\'s, and constants. $$y$$ will be eliminated and then solve for $$x$$.\\\\n$$2x=-22$$\\\\n$$x=-11$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol9a-h3","type":"hint","dependencies":["a6614c6sol9a-h2"],"title":"Substitute Solution into Original Equations","text":"Substitute $$x=-11$$ into the first equation, $$3x+2y=-3$$. Then solve for $$y$$.\\\\n$$3x+2y=-3$$\\\\n$$3\\\\left(-11\\\\right)+2y=-3$$\\\\n$$-33+2y=-3$$\\\\n$$2y=30$$\\\\n$$y=15$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6614c6sol9a-h4","type":"hint","dependencies":["a6614c6sol9a-h2","a6614c6sol9a-h3"],"title":"Solution as Ordered Pair","text":"Solution to the system of equations is $$(-11,15)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a664eb9bin1","title":"Using Pascal\'s Triangle to Expand a polynomial.","body":"Use Pascal\u2019s Triangle to expand the following:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Binomial Theorem","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a664eb9bin1a","stepAnswer":["$$x^6+6x^5 y^1+15x^4 y^2+20x^3 y^3+15x^2 y^4+6x^1 y^5+y^6$$"],"problemType":"TextBox","stepTitle":"Using Pascal\'s Triangle to Expand a polynomial.","stepBody":"$${\\\\left(x+y\\\\right)}^6$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^6+6x^5 y^1+15x^4 y^2+20x^3 y^3+15x^2 y^4+6x^1 y^5+y^6$$","hints":{"DefaultPathway":[{"id":"a664eb9bin1a-h1","type":"hint","dependencies":[],"title":"Finding out the pattern with Pascal\'s Triangle.","text":"We know that the variables in this pattern will follow the pattern in Pascal\'s triangle. You should use the row of Pascal\'s triangle where the second number is $$6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin1a-h2","type":"hint","dependencies":["a664eb9bin1a-h1"],"title":"Finding out the coefficients.","text":"The coefficients according to Pascal\'s triangle are $$1$$, $$6$$, $$15$$, $$20$$, $$15$$, $$6$$, $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin1a-h3","type":"hint","dependencies":["a664eb9bin1a-h2"],"title":"Expanding the polynomial.","text":"Use the formula: $${\\\\left(a+b\\\\right)}^n$$ $$=$$ $$a^n$$ + $$___$$ $$a^n-1b^1$$ + $$___$$ $$a^n-2b^2$$ +... + $$___$$ $$a^1 b^n-1$$ + $$b^n$$ and fill in the blanks with the values from Pascal\'s triangle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin1a-h4","type":"hint","dependencies":["a664eb9bin1a-h3"],"title":"Plugging in the coefficients.","text":"By plugging in the coefficients found earlier, we get $$x^6+6x^5 y^1+15x^4 y^2+20x^3 y^3+15x^2 y^4+6x^1 y^5+y^6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a664eb9bin10","title":"Using the Binomial Theorem to Expand a Polynomial.","body":"Use the Binomial Theorem to expand the following:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Binomial Theorem","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a664eb9bin10a","stepAnswer":["$$243x^5-810x^4 y^1+1080x^3 y^2-720x^2 y^3+240x^1 y^4-32y^5$$"],"problemType":"TextBox","stepTitle":"Using the Binomial Theorem to Expand a Polynomial.","stepBody":"$${\\\\left(3x-2y\\\\right)}^5$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$243x^5-810x^4 y^1+1080x^3 y^2-720x^2 y^3+240x^1 y^4-32y^5$$","hints":{"DefaultPathway":[{"id":"a664eb9bin10a-h1","type":"hint","dependencies":[],"title":"Identify the a and $$b$$.","text":"Identify the a, $$b$$, and $$n$$ of $${\\\\left(a+b\\\\right)}^n$$. Here, $$a=3x$$, $$b=-2y$$, $$n=5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin10a-h2","type":"hint","dependencies":["a664eb9bin10a-h1"],"title":"Use the binomial theorm.","text":"Follow the binomial theorem from this image.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin10a-h3","type":"hint","dependencies":["a664eb9bin10a-h2"],"title":"substitute into the binomial theorem.","text":"Substitute the a, $$b$$, and $$n$$ that you found into the binomial equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin10a-h4","type":"hint","dependencies":["a664eb9bin10a-h3"],"title":"Simplify.","text":"Simplify, resulting in $$243x^5-810x^4 y^1+1080x^3 y^2-720x^2 y^3+240x^1 y^4-32y^5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a664eb9bin11","title":"Using the Binomial Theorem to Expand a Polynomial.","body":"Use the Binomial Theorem to expand the following:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Binomial Theorem","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a664eb9bin11a","stepAnswer":["$$256x^4-768x^3 y^1+864x^2 y^2-432x^1 y^3+81y^4$$"],"problemType":"TextBox","stepTitle":"Using the Binomial Theorem to Expand a Polynomial.","stepBody":"$${\\\\left(4x-3y\\\\right)}^4$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$256x^4-768x^3 y^1+864x^2 y^2-432x^1 y^3+81y^4$$","hints":{"DefaultPathway":[{"id":"a664eb9bin11a-h1","type":"hint","dependencies":[],"title":"Identify the a and $$b$$.","text":"Identify the a, $$b$$, and $$n$$ of $${\\\\left(a+b\\\\right)}^n$$. Here, $$a=4x$$, $$b=-3y$$, $$n=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin11a-h2","type":"hint","dependencies":["a664eb9bin11a-h1"],"title":"Use the binomial theorm.","text":"Follow the binomial theorem from this image.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin11a-h3","type":"hint","dependencies":["a664eb9bin11a-h2"],"title":"substitute into the binomial theorem.","text":"Substitute the a, $$b$$, and $$n$$ that you found into the binomial equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin11a-h4","type":"hint","dependencies":["a664eb9bin11a-h3"],"title":"Simplify.","text":"Simplify, resulting in $$256x^4-768x^3 y^1+864x^2 y^2-432x^1 y^3+81y^4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a664eb9bin12","title":"Using Pascal\'s Triangle to Expand a polynomial.","body":"Use Pascal\u2019s Triangle to expand the following:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Binomial Theorem","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a664eb9bin12a","stepAnswer":["$$x^4+4x^3 y^1+6x^2 y^2+4x^1 y^3+y^4$$"],"problemType":"TextBox","stepTitle":"Using Pascal\'s Triangle to Expand a polynomial.","stepBody":"$${\\\\left(x+y\\\\right)}^4$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^4+4x^3 y^1+6x^2 y^2+4x^1 y^3+y^4$$","hints":{"DefaultPathway":[{"id":"a664eb9bin12a-h1","type":"hint","dependencies":[],"title":"Finding out the pattern with Pascal\'s Triangle.","text":"We know that the variables in this pattern will follow the pattern in Pascal\'s triangle. You should use the row of Pascal\'s triangle where the second number is $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin12a-h2","type":"hint","dependencies":["a664eb9bin12a-h1"],"title":"Finding out the coefficients.","text":"The coefficients according to Pascal\'s triangle are $$1$$, $$4$$, $$6$$, $$4$$, $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin12a-h3","type":"hint","dependencies":["a664eb9bin12a-h2"],"title":"Expanding the polynomial.","text":"Use the formula: $${\\\\left(a+b\\\\right)}^n$$ $$=$$ $$a^n$$ + $$___$$ $$a^n-1b^1$$ + $$___$$ $$a^n-2b^2$$ +... + $$___$$ $$a^1 b^n-1$$ + $$b^n$$ and fill in the blanks with the values from Pascal\'s triangle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin12a-h4","type":"hint","dependencies":["a664eb9bin12a-h3"],"title":"Plugging in the coefficients.","text":"By plugging in the coefficients found earlier, we get $$x^4+4x^3 y^1+6x^2 y^2+4x^1 y^3+y^4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a664eb9bin13","title":"Using Pascal\'s Triangle to Expand a polynomial.","body":"Use Pascal\u2019s Triangle to expand the following:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Binomial Theorem","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a664eb9bin13a","stepAnswer":["$$a^8+8a^7 b^1+28a^6 b^2+56a^5 b^3+70a^4 b^4+56a^3 b^5+28a^2 b^6+8a^1 b^7+b^8$$"],"problemType":"TextBox","stepTitle":"Using Pascal\'s Triangle to Expand a polynomial.","stepBody":"$${\\\\left(a+b\\\\right)}^8$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$a^8+8a^7 b^1+28a^6 b^2+56a^5 b^3+70a^4 b^4+56a^3 b^5+28a^2 b^6+8a^1 b^7+b^8$$","hints":{"DefaultPathway":[{"id":"a664eb9bin13a-h1","type":"hint","dependencies":[],"title":"Finding out the pattern with Pascal\'s Triangle.","text":"We know that the variables in this pattern will follow the pattern in Pascal\'s triangle. You should use the row of Pascal\'s triangle where the second number is $$8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin13a-h2","type":"hint","dependencies":["a664eb9bin13a-h1"],"title":"Finding out the coefficients.","text":"The coefficients according to Pascal\'s triangle are $$1$$, $$8$$, $$28$$, $$56$$, $$70$$, $$56$$, $$28$$, $$8$$, $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin13a-h3","type":"hint","dependencies":["a664eb9bin13a-h2"],"title":"Expanding the polynomial.","text":"Use the formula: $${\\\\left(a+b\\\\right)}^n$$ $$=$$ $$a^n$$ + $$___$$ $$a^n-1b^1$$ + $$___$$ $$a^n-2b^2$$ +... + $$___$$ $$a^1 b^n-1$$ + $$b^n$$ and fill in the blanks with the values from Pascal\'s triangle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin13a-h4","type":"hint","dependencies":["a664eb9bin13a-h3"],"title":"Plugging in the coefficients.","text":"By plugging in the coefficients found earlier, we get $$a^8+8a^7 b^1+28a^6 b^2+56a^5 b^3+70a^4 b^4+56a^3 b^5+28a^2 b^6+8a^1 b^7+b^8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a664eb9bin14","title":"Using Pascal\'s Triangle to Expand a polynomial.","body":"Use Pascal\u2019s Triangle to expand the following:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Binomial Theorem","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a664eb9bin14a","stepAnswer":["$$x^5-5x^4 y^1+10x^3 y^2-10x^2 y^3+5x^1 y^4-y^5$$"],"problemType":"TextBox","stepTitle":"Using Pascal\'s Triangle to Expand a polynomial.","stepBody":"$${\\\\left(x-y\\\\right)}^5$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^5-5x^4 y^1+10x^3 y^2-10x^2 y^3+5x^1 y^4-y^5$$","hints":{"DefaultPathway":[{"id":"a664eb9bin14a-h1","type":"hint","dependencies":[],"title":"Finding out the pattern with Pascal\'s Triangle.","text":"We know that the variables in this pattern will follow the pattern in Pascal\'s triangle. You should use the row of Pascal\'s triangle where the second number is $$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin14a-h2","type":"hint","dependencies":["a664eb9bin14a-h1"],"title":"Finding out the coefficients.","text":"The coefficients according to Pascal\'s triangle are $$1$$, $$5$$, $$10$$, $$10$$, $$5$$, $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin14a-h3","type":"hint","dependencies":["a664eb9bin14a-h2"],"title":"Expanding the polynomial.","text":"Use the formula: $${\\\\left(a+b\\\\right)}^n$$ $$=$$ $$a^n$$ + $$___$$ $$a^n-1b^1$$ + $$___$$ $$a^n-2b^2$$ +... + $$___$$ $$a^1 b^n-1$$ + $$b^n$$ and fill in the blanks with the values from Pascal\'s triangle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin14a-h4","type":"hint","dependencies":["a664eb9bin14a-h3"],"title":"Plugging in the coefficients.","text":"By plugging in the coefficients found earlier, we get x**5-5*x**4*y**1+10*x**3*y**2-10*x**2*y**3+5*x**1*y**4-*y**5.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a664eb9bin15","title":"Using Pascal\'s Triangle to Expand a Polynomial and Simplify.","body":"Use Pascal\u2019s Triangle to expand the following:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Binomial Theorem","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a664eb9bin15a","stepAnswer":["$$x^4+16x^3+96x^2+256x^1+256$$"],"problemType":"TextBox","stepTitle":"Using Pascal\'s Triangle to Expand a polynomial.","stepBody":"$${\\\\left(x+4\\\\right)}^4$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^4+16x^3+96x^2+256x^1+256$$","hints":{"DefaultPathway":[{"id":"a664eb9bin15a-h1","type":"hint","dependencies":[],"title":"Finding out the pattern with Pascal\'s Triangle.","text":"We know that the variables in this pattern will follow the pattern in Pascal\'s triangle. You should use the row of Pascal\'s triangle where the second number is $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin15a-h2","type":"hint","dependencies":["a664eb9bin15a-h1"],"title":"Finding out the coefficients.","text":"The coefficients according to Pascal\'s triangle are $$1$$, $$4$$, $$6$$, $$4$$, $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin15a-h3","type":"hint","dependencies":["a664eb9bin15a-h2"],"title":"Expanding the polynomial.","text":"Use the formula: $${\\\\left(a+b\\\\right)}^n$$ $$=$$ $$a^n$$ + $$___$$ $$a^n-1b^1$$ + $$___$$ $$a^n-2b^2$$ +... + $$___$$ $$a^1 b^n-1$$ + $$b^n$$ and fill in the blanks with the values from Pascal\'s triangle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin15a-h4","type":"hint","dependencies":["a664eb9bin15a-h3"],"title":"Plugging in the coefficients.","text":"Plug in the $$x$$, $$y$$, and coefficents.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin15a-h5","type":"hint","dependencies":["a664eb9bin15a-h4"],"title":"Simplify the equation.","text":"Simplifying the equation results in the solution: $$x^4+16x^3+96x^2+256x^1+256$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a664eb9bin16","title":"Using Pascal\'s Triangle to Expand a Polynomial and Simplify.","body":"Use Pascal\u2019s Triangle to expand the following:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Binomial Theorem","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a664eb9bin16a","stepAnswer":["$$x^3+15x^2+75x^1+125$$"],"problemType":"TextBox","stepTitle":"Using Pascal\'s Triangle to Expand a polynomial.","stepBody":"$${\\\\left(x+5\\\\right)}^3$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^3+15x^2+75x^1+125$$","hints":{"DefaultPathway":[{"id":"a664eb9bin16a-h1","type":"hint","dependencies":[],"title":"Finding out the pattern with Pascal\'s Triangle.","text":"We know that the variables in this pattern will follow the pattern in Pascal\'s triangle. You should use the row of Pascal\'s triangle where the second number is $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin16a-h2","type":"hint","dependencies":["a664eb9bin16a-h1"],"title":"Finding out the coefficients.","text":"The coefficients according to Pascal\'s triangle are $$1$$, $$3$$, $$3$$, $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin16a-h3","type":"hint","dependencies":["a664eb9bin16a-h2"],"title":"Expanding the polynomial.","text":"Use the formula: $${\\\\left(a+b\\\\right)}^n$$ $$=$$ $$a^n$$ + $$___$$ $$a^n-1b^1$$ + $$___$$ $$a^n-2b^2$$ +... + $$___$$ $$a^1 b^n-1$$ + $$b^n$$ and fill in the blanks with the values from Pascal\'s triangle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin16a-h4","type":"hint","dependencies":["a664eb9bin16a-h3"],"title":"Plugging in the coefficients.","text":"Plug in the $$x$$, $$y$$, and coefficents.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin16a-h5","type":"hint","dependencies":["a664eb9bin16a-h4"],"title":"Simplify the equation.","text":"Simplifying the equation results in the solution: $$x^3+15x^2+75x^1+125$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a664eb9bin17","title":"Using Pascal\'s Triangle to Expand a Polynomial and Simplify.","body":"Use Pascal\u2019s Triangle to expand the following:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Binomial Theorem","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a664eb9bin17a","stepAnswer":["$$x^5+10x^4+40x^3+80x^2+80x^1+32$$"],"problemType":"TextBox","stepTitle":"Using Pascal\'s Triangle to Expand a polynomial.","stepBody":"$${\\\\left(x+2\\\\right)}^5$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^5+10x^4+40x^3+80x^2+80x^1+32$$","hints":{"DefaultPathway":[{"id":"a664eb9bin17a-h1","type":"hint","dependencies":[],"title":"Finding out the pattern with Pascal\'s Triangle.","text":"We know that the variables in this pattern will follow the pattern in Pascal\'s triangle. You should use the row of Pascal\'s triangle where the second number is $$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin17a-h2","type":"hint","dependencies":["a664eb9bin17a-h1"],"title":"Finding out the coefficients.","text":"The coefficients according to Pascal\'s triangle are $$1$$, $$5$$, $$10$$, $$10$$, $$5$$, $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin17a-h3","type":"hint","dependencies":["a664eb9bin17a-h2"],"title":"Expanding the polynomial.","text":"Use the formula: $${\\\\left(a+b\\\\right)}^n$$ $$=$$ $$a^n$$ + $$___$$ $$a^n-1b^1$$ + $$___$$ $$a^n-2b^2$$ +... + $$___$$ $$a^1 b^n-1$$ + $$b^n$$ and fill in the blanks with the values from Pascal\'s triangle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin17a-h4","type":"hint","dependencies":["a664eb9bin17a-h3"],"title":"Plugging in the coefficients.","text":"Plug in the $$x$$, $$y$$, and coefficents.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin17a-h5","type":"hint","dependencies":["a664eb9bin17a-h4"],"title":"Simplify the equation.","text":"Simplifying the equation results in the solution: $$x^5+10x^4+40x^3+80x^2+80x^1+32$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a664eb9bin18","title":"Using Pascal\'s Triangle to Expand a Polynomial and Simplify.","body":"Use Pascal\u2019s Triangle to expand the following:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Binomial Theorem","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a664eb9bin18a","stepAnswer":["$$x^7+7x^6+21x^5+35x^4+35x^3+21x^2+7x^1+1$$"],"problemType":"TextBox","stepTitle":"Using Pascal\'s Triangle to Expand a polynomial.","stepBody":"$${\\\\left(x+1\\\\right)}^7$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^7+7x^6+21x^5+35x^4+35x^3+21x^2+7x^1+1$$","hints":{"DefaultPathway":[{"id":"a664eb9bin18a-h1","type":"hint","dependencies":[],"title":"Finding out the pattern with Pascal\'s Triangle.","text":"We know that the variables in this pattern will follow the pattern in Pascal\'s triangle. You should use the row of Pascal\'s triangle where the second number is $$7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin18a-h2","type":"hint","dependencies":["a664eb9bin18a-h1"],"title":"Finding out the coefficients.","text":"The coefficients according to Pascal\'s triangle are $$1$$, $$7$$, $$21$$, $$35$$, $$35$$, $$21$$, $$7$$, $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin18a-h3","type":"hint","dependencies":["a664eb9bin18a-h2"],"title":"Expanding the polynomial.","text":"Use the formula: $${\\\\left(a+b\\\\right)}^n$$ $$=$$ $$a^n$$ + $$___$$ $$a^n-1b^1$$ + $$___$$ $$a^n-2b^2$$ +... + $$___$$ $$a^1 b^n-1$$ + $$b^n$$ and fill in the blanks with the values from Pascal\'s triangle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin18a-h4","type":"hint","dependencies":["a664eb9bin18a-h3"],"title":"Plugging in the coefficients.","text":"Plug in the $$x$$, $$y$$, and coefficents.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin18a-h5","type":"hint","dependencies":["a664eb9bin18a-h4"],"title":"Simplify the equation.","text":"Simplifying the equation results in the solution: $$x^7+7x^6+21x^5+35x^4+35x^3+21x^2+7x^1+1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a664eb9bin19","title":"Using Pascal\'s Triangle to Expand a Polynomial and Simplify.","body":"Use Pascal\u2019s Triangle to expand the following:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Binomial Theorem","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a664eb9bin19a","stepAnswer":["$$x^5-15x^4+90x^3-270x^2+405x^1-243$$"],"problemType":"TextBox","stepTitle":"Using Pascal\'s Triangle to Expand a polynomial.","stepBody":"$${\\\\left(x-3\\\\right)}^5$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^5-15x^4+90x^3-270x^2+405x^1-243$$","hints":{"DefaultPathway":[{"id":"a664eb9bin19a-h1","type":"hint","dependencies":[],"title":"Finding out the pattern with Pascal\'s Triangle.","text":"We know that the variables in this pattern will follow the pattern in Pascal\'s triangle. You should use the row of Pascal\'s triangle where the second number is $$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin19a-h2","type":"hint","dependencies":["a664eb9bin19a-h1"],"title":"Finding out the coefficients.","text":"The coefficients according to Pascal\'s triangle are $$1$$, $$5$$, $$10$$, $$10$$, $$5$$, $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin19a-h3","type":"hint","dependencies":["a664eb9bin19a-h2"],"title":"Expanding the polynomial.","text":"Use the formula: $${\\\\left(a+b\\\\right)}^n$$ $$=$$ $$a^n$$ + $$___$$ $$a^n-1b^1$$ + $$___$$ $$a^n-2b^2$$ +... + $$___$$ $$a^1 b^n-1$$ + $$b^n$$ and fill in the blanks with the values from Pascal\'s triangle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin19a-h4","type":"hint","dependencies":["a664eb9bin19a-h3"],"title":"Plugging in the coefficients.","text":"Plug in the $$x$$, $$y$$, and coefficents.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin19a-h5","type":"hint","dependencies":["a664eb9bin19a-h4"],"title":"Simplify the equation.","text":"Simplifying the equation results in the solution: $$x^5-15x^4+90x^3-270x^2+405x^1-243$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a664eb9bin2","title":"Using Pascal\'s Triangle to Expand a polynomial.","body":"Use Pascal\u2019s Triangle to expand the following:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Binomial Theorem","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a664eb9bin2a","stepAnswer":["$$x^5+5x^4 y^1+10x^3 y^2+10x^2 y^3+5x^1 y^4+y^5$$"],"problemType":"TextBox","stepTitle":"Using Pascal\'s Triangle to Expand a polynomial.","stepBody":"$${\\\\left(x+y\\\\right)}^5$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^5+5x^4 y^1+10x^3 y^2+10x^2 y^3+5x^1 y^4+y^5$$","hints":{"DefaultPathway":[{"id":"a664eb9bin2a-h1","type":"hint","dependencies":[],"title":"Finding out the pattern with Pascal\'s Triangle.","text":"We know that the variables in this pattern will follow the pattern in Pascal\'s triangle. You should use the row of Pascal\'s triangle where the second number is $$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin2a-h2","type":"hint","dependencies":["a664eb9bin2a-h1"],"title":"Finding out the coefficients.","text":"The coefficients according to Pascal\'s triangle are $$1$$, $$5$$, $$10$$, $$10$$, $$5$$, $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin2a-h3","type":"hint","dependencies":["a664eb9bin2a-h2"],"title":"Expanding the polynomial.","text":"Use the formula: $${\\\\left(a+b\\\\right)}^n$$ $$=$$ $$a^n$$ + $$___$$ $$a^n-1b^1$$ + $$___$$ $$a^n-2b^2$$ +... + $$___$$ $$a^1 b^n-1$$ + $$b^n$$ and fill in the blanks with the values from Pascal\'s triangle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin2a-h4","type":"hint","dependencies":["a664eb9bin2a-h3"],"title":"Plugging in the coefficients.","text":"By plugging in the coefficients found earlier, we get $$x^5+5x^4 y^1+10x^3 y^2+10x^2 y^3+5x^1 y^4+y^5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a664eb9bin20","title":"Using Pascal\'s Triangle to Expand a Polynomial and Simplify.","body":"Use Pascal\u2019s Triangle to expand the following:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Binomial Theorem","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a664eb9bin20a","stepAnswer":["$$x^6-12x^5+60x^4-160x^3+240x^2-192x^1+64$$"],"problemType":"TextBox","stepTitle":"Using Pascal\'s Triangle to Expand a polynomial.","stepBody":"$${\\\\left(x-2\\\\right)}^6$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^6-12x^5+60x^4-160x^3+240x^2-192x^1+64$$","hints":{"DefaultPathway":[{"id":"a664eb9bin20a-h1","type":"hint","dependencies":[],"title":"Finding out the pattern with Pascal\'s Triangle.","text":"We know that the variables in this pattern will follow the pattern in Pascal\'s triangle. You should use the row of Pascal\'s triangle where the second number is $$6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin20a-h2","type":"hint","dependencies":["a664eb9bin20a-h1"],"title":"Finding out the coefficients.","text":"The coefficients according to Pascal\'s triangle are $$1$$, $$6$$, $$15$$, $$20$$, $$15$$, $$6$$, $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin20a-h3","type":"hint","dependencies":["a664eb9bin20a-h2"],"title":"Expanding the polynomial.","text":"Use the formula: $${\\\\left(a+b\\\\right)}^n$$ $$=$$ $$a^n$$ + $$___$$ $$a^n-1b^1$$ + $$___$$ $$a^n-2b^2$$ +... + $$___$$ $$a^1 b^n-1$$ + $$b^n$$ and fill in the blanks with the values from Pascal\'s triangle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin20a-h4","type":"hint","dependencies":["a664eb9bin20a-h3"],"title":"Plugging in the coefficients.","text":"Plug in the $$x$$, $$y$$, and coefficents.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin20a-h5","type":"hint","dependencies":["a664eb9bin20a-h4"],"title":"Simplify the equation.","text":"Simplifying the equation results in the solution: $$x^6-12x^5+60x^4-160x^3+240x^2-192x^1+64$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a664eb9bin21","title":"Using the Binomial Theorem to Expand a Polynomial.","body":"Use the Binomial Theorem to expand the following:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Binomial Theorem","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a664eb9bin21a","stepAnswer":["$$x^4-8x^3+24x^2-32x^1+16$$"],"problemType":"TextBox","stepTitle":"Using the Binomial Theorem to Expand a Polynomial.","stepBody":"$${\\\\left(x-2\\\\right)}^4$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^4-8x^3+24x^2-32x^1+16$$","hints":{"DefaultPathway":[{"id":"a664eb9bin21a-h1","type":"hint","dependencies":[],"title":"Identify the a and $$b$$.","text":"Identify the a, $$b$$, and $$n$$ of $${\\\\left(a+b\\\\right)}^n$$. Here, $$a=x$$, $$b=-2$$, $$n=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin21a-h2","type":"hint","dependencies":["a664eb9bin21a-h1"],"title":"Use the binomial theorm.","text":"Follow the binomial theorem from this image.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin21a-h3","type":"hint","dependencies":["a664eb9bin21a-h2"],"title":"substitute into the binomial theorem.","text":"Substitute the a, $$b$$, and $$n$$ that you found into the binomial equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin21a-h4","type":"hint","dependencies":["a664eb9bin21a-h3"],"title":"Simplify.","text":"Simplify, resulting in $$x^4-8x^3+24x^2-32x^1+16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a664eb9bin22","title":"Using the Binomial Theorem to Expand a Polynomial.","body":"Use the Binomial Theorem to expand the following:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Binomial Theorem","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a664eb9bin22a","stepAnswer":["$$x^4-12x^3+54x^2-108x^1+81$$"],"problemType":"TextBox","stepTitle":"Using the Binomial Theorem to Expand a Polynomial.","stepBody":"$${\\\\left(x-3\\\\right)}^4$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^4-12x^3+54x^2-108x^1+81$$","hints":{"DefaultPathway":[{"id":"a664eb9bin22a-h1","type":"hint","dependencies":[],"title":"Identify the a and $$b$$.","text":"Identify the a, $$b$$, and $$n$$ of $${\\\\left(a+b\\\\right)}^n$$. Here, $$a=x$$, $$b=-3$$, $$n=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin22a-h2","type":"hint","dependencies":["a664eb9bin22a-h1"],"title":"Use the binomial theorm.","text":"Follow the binomial theorem from this image.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin22a-h3","type":"hint","dependencies":["a664eb9bin22a-h2"],"title":"substitute into the binomial theorem.","text":"Substitute the a, $$b$$, and $$n$$ that you found into the binomial equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin22a-h4","type":"hint","dependencies":["a664eb9bin22a-h3"],"title":"Simplify.","text":"Simplify, resulting in $$x^4-12x^3+54x^2-108x^1+81$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a664eb9bin23","title":"Using the Binomial Theorem to Expand a Polynomial.","body":"Use the Binomial Theorem to expand the following:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Binomial Theorem","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a664eb9bin23a","stepAnswer":["$$x^5-5x^4+10x^3-10x^2+5x^1-1$$"],"problemType":"TextBox","stepTitle":"Using the Binomial Theorem to Expand a Polynomial.","stepBody":"$${\\\\left(x-1\\\\right)}^5$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^5-5x^4+10x^3-10x^2+5x^1-1$$","hints":{"DefaultPathway":[{"id":"a664eb9bin23a-h1","type":"hint","dependencies":[],"title":"Identify the a and $$b$$.","text":"Identify the a, $$b$$, and $$n$$ of $${\\\\left(a+b\\\\right)}^n$$. Here, $$a=x$$, $$b=-1$$, $$n=5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin23a-h2","type":"hint","dependencies":["a664eb9bin23a-h1"],"title":"Use the binomial theorm.","text":"Follow the binomial theorem from this image.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin23a-h3","type":"hint","dependencies":["a664eb9bin23a-h2"],"title":"substitute into the binomial theorem.","text":"Substitute the a, $$b$$, and $$n$$ that you found into the binomial equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin23a-h4","type":"hint","dependencies":["a664eb9bin23a-h3"],"title":"Simplify.","text":"Simplify, resulting in $$x^5-5x^4+10x^3-10x^2+5x^1-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a664eb9bin24","title":"Using the Binomial Theorem to Expand a Polynomial.","body":"Use the Binomial Theorem to expand the following:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Binomial Theorem","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a664eb9bin24a","stepAnswer":["$$x^3-12x^2+48x^1-64$$"],"problemType":"TextBox","stepTitle":"Using the Binomial Theorem to Expand a Polynomial.","stepBody":"$${\\\\left(x-4\\\\right)}^3$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^3-12x^2+48x^1-64$$","hints":{"DefaultPathway":[{"id":"a664eb9bin24a-h1","type":"hint","dependencies":[],"title":"Identify the a and $$b$$.","text":"Identify the a, $$b$$, and $$n$$ of $${\\\\left(a+b\\\\right)}^n$$. Here, $$a=x$$, $$b=-4$$, $$n=3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin24a-h2","type":"hint","dependencies":["a664eb9bin24a-h1"],"title":"Use the binomial theorm.","text":"Follow the binomial theorem from this image.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin24a-h3","type":"hint","dependencies":["a664eb9bin24a-h2"],"title":"substitute into the binomial theorem.","text":"Substitute the a, $$b$$, and $$n$$ that you found into the binomial equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin24a-h4","type":"hint","dependencies":["a664eb9bin24a-h3"],"title":"Simplify.","text":"Simplify, resulting in $$x^3-12x^2+48x^1-64$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a664eb9bin25","title":"Using the Binomial Theorem to Expand a Polynomial.","body":"Use the Binomial Theorem to expand the following:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Binomial Theorem","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a664eb9bin25a","stepAnswer":["$$x^3+3x^2 y^1+3x^1 y^2+y^3$$"],"problemType":"TextBox","stepTitle":"Using the Binomial Theorem to Expand a Polynomial.","stepBody":"$${\\\\left(x+y\\\\right)}^3$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^3+3x^2 y^1+3x^1 y^2+y^3$$","hints":{"DefaultPathway":[{"id":"a664eb9bin25a-h1","type":"hint","dependencies":[],"title":"Identify the a and $$b$$.","text":"Identify the a, $$b$$, and $$n$$ of $${\\\\left(a+b\\\\right)}^n$$. Here, $$a=x$$, $$b=y$$, $$n=3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin25a-h2","type":"hint","dependencies":["a664eb9bin25a-h1"],"title":"Use the binomial theorm.","text":"Follow the binomial theorem from this image.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin25a-h3","type":"hint","dependencies":["a664eb9bin25a-h2"],"title":"substitute into the binomial theorem.","text":"Substitute the a, $$b$$, and $$n$$ that you found into the binomial equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin25a-h4","type":"hint","dependencies":["a664eb9bin25a-h3"],"title":"Simplify.","text":"Simplify, resulting in $$x^3+3x^2 y^1+3x^1 y^2+y^3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a664eb9bin26","title":"Using the Binomial Theorem to Expand a Polynomial.","body":"Use the Binomial Theorem to expand the following:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Binomial Theorem","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a664eb9bin26a","stepAnswer":["$$m^5+5m^4 n^1+10m^3 n^2+10m^2 n^3+5m^1 n^4+n^5$$"],"problemType":"TextBox","stepTitle":"Using the Binomial Theorem to Expand a Polynomial.","stepBody":"$${\\\\left(m+n\\\\right)}^5$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$m^5+5m^4 n^1+10m^3 n^2+10m^2 n^3+5m^1 n^4+n^5$$","hints":{"DefaultPathway":[{"id":"a664eb9bin26a-h1","type":"hint","dependencies":[],"title":"Identify the a and $$b$$.","text":"Identify the a, $$b$$, and $$n$$ of $${\\\\left(a+b\\\\right)}^n$$. Here, $$a=m$$, $$b=n$$, $$n=5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin26a-h2","type":"hint","dependencies":["a664eb9bin26a-h1"],"title":"Use the binomial theorm.","text":"Follow the binomial theorem from this image.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin26a-h3","type":"hint","dependencies":["a664eb9bin26a-h2"],"title":"substitute into the binomial theorem.","text":"Substitute the a, $$b$$, and $$n$$ that you found into the binomial equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin26a-h4","type":"hint","dependencies":["a664eb9bin26a-h3"],"title":"Simplify.","text":"Simplify, resulting in $$m^5+5m^4 n^1+10m^3 n^2+10m^2 n^3+5m^1 n^4+n^5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a664eb9bin27","title":"Using the Binomial Theorem to Expand a Polynomial.","body":"Use the Binomial Theorem to expand the following:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Binomial Theorem","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a664eb9bin27a","stepAnswer":["$$243x^5-405x^4 y^1+270x^3 y^2-90x^2 y^3+15x^1 y^4-y^5$$"],"problemType":"TextBox","stepTitle":"Using the Binomial Theorem to Expand a Polynomial.","stepBody":"$${\\\\left(3x-y\\\\right)}^5$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$243x^5-405x^4 y^1+270x^3 y^2-90x^2 y^3+15x^1 y^4-y^5$$","hints":{"DefaultPathway":[{"id":"a664eb9bin27a-h1","type":"hint","dependencies":[],"title":"Identify the a and $$b$$.","text":"Identify the a, $$b$$, and $$n$$ of $${\\\\left(a+b\\\\right)}^n$$. Here, $$a=3x$$, $$b=-y$$, $$n=5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin27a-h2","type":"hint","dependencies":["a664eb9bin27a-h1"],"title":"Use the binomial theorm.","text":"Follow the binomial theorem from this image.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin27a-h3","type":"hint","dependencies":["a664eb9bin27a-h2"],"title":"substitute into the binomial theorem.","text":"Substitute the a, $$b$$, and $$n$$ that you found into the binomial equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin27a-h4","type":"hint","dependencies":["a664eb9bin27a-h3"],"title":"Simplify.","text":"Simplify, resulting in $$243x^5-405x^4 y^1+270x^3 y^2-90x^2 y^3+15x^1 y^4-y^5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a664eb9bin28","title":"Using the Binomial Theorem to Expand a Polynomial.","body":"Use the Binomial Theorem to expand the following:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Binomial Theorem","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a664eb9bin28a","stepAnswer":["$$625x^4-1000x^3 y^1+600x^2 y^2-160x^1 y^3+16y^4$$"],"problemType":"TextBox","stepTitle":"Using the Binomial Theorem to Expand a Polynomial.","stepBody":"$${\\\\left(5x-2y\\\\right)}^4$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$625x^4-1000x^3 y^1+600x^2 y^2-160x^1 y^3+16y^4$$","hints":{"DefaultPathway":[{"id":"a664eb9bin28a-h1","type":"hint","dependencies":[],"title":"Identify the a and $$b$$.","text":"Identify the a, $$b$$, and $$n$$ of $${\\\\left(a+b\\\\right)}^n$$. Here, $$a=5x$$, $$b=-2y$$, $$n=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin28a-h2","type":"hint","dependencies":["a664eb9bin28a-h1"],"title":"Use the binomial theorm.","text":"Follow the binomial theorem from this image.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin28a-h3","type":"hint","dependencies":["a664eb9bin28a-h2"],"title":"substitute into the binomial theorem.","text":"Substitute the a, $$b$$, and $$n$$ that you found into the binomial equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin28a-h4","type":"hint","dependencies":["a664eb9bin28a-h3"],"title":"Simplify.","text":"Simplify, resulting in $$625x^4-1000x^3 y^1+600x^2 y^2-160x^1 y^3+16y^4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a664eb9bin29","title":"Using the Binomial Theorem to Expand a Polynomial.","body":"Use the Binomial Theorem to expand the following:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Binomial Theorem","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a664eb9bin29a","stepAnswer":["$$16x^4-160x^3 y^1+600x^2 y^2-1000x^1 y^3+625y^4$$"],"problemType":"TextBox","stepTitle":"Using the Binomial Theorem to Expand a Polynomial.","stepBody":"$${\\\\left(2x-5y\\\\right)}^4$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16x^4-160x^3 y^1+600x^2 y^2-1000x^1 y^3+625y^4$$","hints":{"DefaultPathway":[{"id":"a664eb9bin29a-h1","type":"hint","dependencies":[],"title":"Identify the a and $$b$$.","text":"Identify the a, $$b$$, and $$n$$ of $${\\\\left(a+b\\\\right)}^n$$. Here, $$a=2x$$, $$b=-5y$$, $$n=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin29a-h2","type":"hint","dependencies":["a664eb9bin29a-h1"],"title":"Use the binomial theorm.","text":"Follow the binomial theorem from this image.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin29a-h3","type":"hint","dependencies":["a664eb9bin29a-h2"],"title":"substitute into the binomial theorem.","text":"Substitute the a, $$b$$, and $$n$$ that you found into the binomial equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin29a-h4","type":"hint","dependencies":["a664eb9bin29a-h3"],"title":"Simplify.","text":"Simplify, resulting in $$16x^4-160x^3 y^1+600x^2 y^2-1000x^1 y^3+625y^4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a664eb9bin3","title":"Using Pascal\'s Triangle to Expand a polynomial.","body":"Use Pascal\u2019s Triangle to expand the following:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Binomial Theorem","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a664eb9bin3a","stepAnswer":["$$p^7+7p^6 q^1+21p^5 q^2+35p^4 q^3+35p^3 q^4+21p^2 q^5+7p^1 q^6+q^7$$"],"problemType":"TextBox","stepTitle":"Using Pascal\'s Triangle to Expand a polynomial.","stepBody":"$${\\\\left(p+q\\\\right)}^7$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$p^7+7p^6 q^1+21p^5 q^2+35p^4 q^3+35p^3 q^4+21p^2 q^5+7p^1 q^6+q^7$$","hints":{"DefaultPathway":[{"id":"a664eb9bin3a-h1","type":"hint","dependencies":[],"title":"Finding out the pattern with Pascal\'s Triangle.","text":"We know that the variables in this pattern will follow the pattern in Pascal\'s triangle. You should use the row of Pascal\'s triangle where the second number is $$7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin3a-h2","type":"hint","dependencies":["a664eb9bin3a-h1"],"title":"Finding out the coefficients.","text":"The coefficients according to Pascal\'s triangle are $$1$$, $$7$$, $$21$$, $$35$$, $$35$$, $$21$$, $$7$$, $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin3a-h3","type":"hint","dependencies":["a664eb9bin3a-h2"],"title":"Expanding the polynomial.","text":"Use the formula: $${\\\\left(a+b\\\\right)}^n$$ $$=$$ $$a^n$$ + $$___$$ $$a^n-1b^1$$ + $$___$$ $$a^n-2b^2$$ +... + $$___$$ $$a^1 b^n-1$$ + $$b^n$$ and fill in the blanks with the values from Pascal\'s triangle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin3a-h4","type":"hint","dependencies":["a664eb9bin3a-h3"],"title":"Plugging in the coefficients.","text":"By plugging in the coefficients found earlier, we get $$p^7+7p^6 q^1+21p^5 q^2+35p^4 q^3+35p^3 q^4+21p^2 q^5+7p^1 q^6+q^7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a664eb9bin30","title":"Using the Binomial Theorem to Expand a Polynomial.","body":"Use the Binomial Theorem to expand the following:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Binomial Theorem","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a664eb9bin30a","stepAnswer":["$$243x^5+1620x^4 y^1+4320x^3 y^2+5760x^2 y^3+3840x^1 y^4+1024y^5$$"],"problemType":"TextBox","stepTitle":"Using the Binomial Theorem to Expand a Polynomial.","stepBody":"$${\\\\left(3x+4y\\\\right)}^5$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$243x^5+1620x^4 y^1+4320x^3 y^2+5760x^2 y^3+3840x^1 y^4+1024y^5$$","hints":{"DefaultPathway":[{"id":"a664eb9bin30a-h1","type":"hint","dependencies":[],"title":"Identify the a and $$b$$.","text":"Identify the a, $$b$$, and $$n$$ of $${\\\\left(a+b\\\\right)}^n$$. Here, $$a=3x$$, $$b=4y$$, $$n=5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin30a-h2","type":"hint","dependencies":["a664eb9bin30a-h1"],"title":"Use the binomial theorm.","text":"Follow the binomial theorem from this image.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin30a-h3","type":"hint","dependencies":["a664eb9bin30a-h2"],"title":"substitute into the binomial theorem.","text":"Substitute the a, $$b$$, and $$n$$ that you found into the binomial equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin30a-h4","type":"hint","dependencies":["a664eb9bin30a-h3"],"title":"Simplify.","text":"Simplify, resulting in $$243x^5+1620x^4 y^1+4320x^3 y^2+5760x^2 y^3+3840x^1 y^4+1024y^5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a664eb9bin4","title":"Using Pascal\'s Triangle to Expand a Polynomial and Simplify.","body":"Use Pascal\u2019s Triangle to expand the following:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Binomial Theorem","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a664eb9bin4a","stepAnswer":["$$x^5+15x^4+90x^3+270x^2+405x^1+243$$"],"problemType":"TextBox","stepTitle":"Using Pascal\'s Triangle to Expand a polynomial.","stepBody":"$${\\\\left(x+3\\\\right)}^5$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^5+15x^4+90x^3+270x^2+405x^1+243$$","hints":{"DefaultPathway":[{"id":"a664eb9bin4a-h1","type":"hint","dependencies":[],"title":"Finding out the pattern with Pascal\'s Triangle.","text":"We know that the variables in this pattern will follow the pattern in Pascal\'s triangle. You should use the row of Pascal\'s triangle where the second number is $$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin4a-h2","type":"hint","dependencies":["a664eb9bin4a-h1"],"title":"Finding out the coefficients.","text":"The coefficients according to Pascal\'s triangle are $$1$$, $$5$$, $$10$$, $$10$$, $$5$$, $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin4a-h3","type":"hint","dependencies":["a664eb9bin4a-h2"],"title":"Expanding the polynomial.","text":"Use the formula: $${\\\\left(a+b\\\\right)}^n$$ $$=$$ $$a^n$$ + $$___$$ $$a^n-1b^1$$ + $$___$$ $$a^n-2b^2$$ +... + $$___$$ $$a^1 b^n-1$$ + $$b^n$$ and fill in the blanks with the values from Pascal\'s triangle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin4a-h4","type":"hint","dependencies":["a664eb9bin4a-h3"],"title":"Plugging in the coefficients.","text":"By plugging in the coefficients found earlier, we get $$x^5+5x^4\\\\times3+10x^3\\\\times9+10x^2\\\\times27+5x^1\\\\times81+243$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin4a-h5","type":"hint","dependencies":["a664eb9bin4a-h4"],"title":"Simplify the equation.","text":"Simplifying the equation results in the solution: $$x^5+15x^4+90x^3+270x^2+405x^1+243$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a664eb9bin5","title":"Using Pascal\'s Triangle to Expand a polynomial.","body":"Use Pascal\u2019s Triangle to expand the following:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Binomial Theorem","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a664eb9bin5a","stepAnswer":["$$x^6+6x^5+15x^4 y+20x^3+15x^2+6x^1+1$$"],"problemType":"TextBox","stepTitle":"Using Pascal\'s Triangle to Expand a polynomial.","stepBody":"$${\\\\left(x+1\\\\right)}^6$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^6+6x^5+15x^4 y+20x^3+15x^2+6x^1+1$$","hints":{"DefaultPathway":[{"id":"a664eb9bin5a-h1","type":"hint","dependencies":[],"title":"Finding out the pattern with Pascal\'s Triangle.","text":"We know that the variables in this pattern will follow the pattern in Pascal\'s triangle. You should use the row of Pascal\'s triangle where the second number is $$6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin5a-h2","type":"hint","dependencies":["a664eb9bin5a-h1"],"title":"Finding out the coefficients.","text":"The coefficients according to Pascal\'s triangle are $$1$$, $$6$$, $$15$$, $$20$$, $$15$$, $$6$$, $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin5a-h3","type":"hint","dependencies":["a664eb9bin5a-h2"],"title":"Expanding the polynomial.","text":"Use the formula: $${\\\\left(a+b\\\\right)}^n$$ $$=$$ $$a^n$$ + $$___$$ $$a^n-1b^1$$ + $$___$$ $$a^n-2b^2$$ +... + $$___$$ $$a^1 b^n-1$$ + $$b^n$$ and fill in the blanks with the values from Pascal\'s triangle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin5a-h4","type":"hint","dependencies":["a664eb9bin5a-h3"],"title":"Plugging in the coefficients.","text":"By plugging in the coefficients found earlier, we get $$x^6+6x^5\\\\times1+15x^4\\\\times1+20x^3\\\\times1+15x^2\\\\times1+6x^1\\\\times1+1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin5a-h5","type":"hint","dependencies":["a664eb9bin5a-h4"],"title":"Simplify the equation.","text":"Simplifying the equation results in the solution: $$x^6+6x^5+15x^4 y+20x^3+15x^2+6x^1+1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a664eb9bin6","title":"Using the Binomial Theorem to Expand a Polynomial.","body":"Use the Binomial Theorem to expand the following:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Binomial Theorem","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a664eb9bin6a","stepAnswer":["$$x^5-10x^4+40x^3-80x^2+80x^1-32$$"],"problemType":"TextBox","stepTitle":"Using the Binomial Theorem to Expand a Polynomial.","stepBody":"$${\\\\left(x-2\\\\right)}^5$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^5-10x^4+40x^3-80x^2+80x^1-32$$","hints":{"DefaultPathway":[{"id":"a664eb9bin6a-h1","type":"hint","dependencies":[],"title":"Identify the a and $$b$$.","text":"Identify the a, $$b$$, and $$n$$ of $${\\\\left(a+b\\\\right)}^n$$. Here, $$a=x$$, $$b=-2$$, $$n=5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin6a-h2","type":"hint","dependencies":["a664eb9bin6a-h1"],"title":"Use the binomial theorm.","text":"Follow the binomial theorem from this image.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin6a-h3","type":"hint","dependencies":["a664eb9bin6a-h2"],"title":"substitute into the binomial theorem.","text":"Substitute the a, $$b$$, and $$n$$ that you found into the binomial equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin6a-h4","type":"hint","dependencies":["a664eb9bin6a-h3"],"title":"Simplify.","text":"Simplify, resulting in $$x^5-10x^4+40x^3-80x^2+80x^1-32$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a664eb9bin7","title":"Using the Binomial Theorem to Expand a Polynomial.","body":"Use the Binomial Theorem to expand the following:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Binomial Theorem","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a664eb9bin7a","stepAnswer":["$$x^5-15x^4+90x^3-270x^2+405x^1-243$$"],"problemType":"TextBox","stepTitle":"Using the Binomial Theorem to Expand a Polynomial.","stepBody":"$${\\\\left(x-3\\\\right)}^5$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^5-15x^4+90x^3-270x^2+405x^1-243$$","hints":{"DefaultPathway":[{"id":"a664eb9bin7a-h1","type":"hint","dependencies":[],"title":"Identify the a and $$b$$.","text":"Identify the a, $$b$$, and $$n$$ of $${\\\\left(a+b\\\\right)}^n$$. Here, $$a=x$$, $$b=-3$$, $$n=5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin7a-h2","type":"hint","dependencies":["a664eb9bin7a-h1"],"title":"Use the binomial theorm.","text":"Follow the binomial theorem from this image.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin7a-h3","type":"hint","dependencies":["a664eb9bin7a-h2"],"title":"substitute into the binomial theorem.","text":"Substitute the a, $$b$$, and $$n$$ that you found into the binomial equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin7a-h4","type":"hint","dependencies":["a664eb9bin7a-h3"],"title":"Simplify.","text":"Simplify, resulting in $$x^5-15x^4+90x^3-270x^2+405x^1-243$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a664eb9bin8","title":"Using the Binomial Theorem to Expand a Polynomial.","body":"Use the Binomial Theorem to expand the following:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Binomial Theorem","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a664eb9bin8a","stepAnswer":["$$x^6-6x^5+15x^4-20x^3+15x^2-6x^1+1$$"],"problemType":"TextBox","stepTitle":"Using the Binomial Theorem to Expand a Polynomial.","stepBody":"$${\\\\left(x-1\\\\right)}^6$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^6-6x^5+15x^4-20x^3+15x^2-6x^1+1$$","hints":{"DefaultPathway":[{"id":"a664eb9bin8a-h1","type":"hint","dependencies":[],"title":"Identify the a and $$b$$.","text":"Identify the a, $$b$$, and $$n$$ of $${\\\\left(a+b\\\\right)}^n$$. Here, $$a=x$$, $$b=-1$$, $$n=6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin8a-h2","type":"hint","dependencies":["a664eb9bin8a-h1"],"title":"Use the binomial theorm.","text":"Follow the binomial theorem from this image.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin8a-h3","type":"hint","dependencies":["a664eb9bin8a-h2"],"title":"substitute into the binomial theorem.","text":"Substitute the a, $$b$$, and $$n$$ that you found into the binomial equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin8a-h4","type":"hint","dependencies":["a664eb9bin8a-h3"],"title":"Simplify.","text":"Simplify, resulting in $$x^6-6x^5+15x^4-20x^3+15x^2-6x^1+1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a664eb9bin9","title":"Using the Binomial Theorem to Expand a Polynomial.","body":"Use the Binomial Theorem to expand the following:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.4 Binomial Theorem","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a664eb9bin9a","stepAnswer":["$$16x^4-96x^3 y^1+16x^2 y^2-216x^1 y^3+81y^4$$"],"problemType":"TextBox","stepTitle":"Using the Binomial Theorem to Expand a Polynomial.","stepBody":"$${\\\\left(2x-3y\\\\right)}^4$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16x^4-96x^3 y^1+16x^2 y^2-216x^1 y^3+81y^4$$","hints":{"DefaultPathway":[{"id":"a664eb9bin9a-h1","type":"hint","dependencies":[],"title":"Identify the a and $$b$$.","text":"Identify the a, $$b$$, and $$n$$ of $${\\\\left(a+b\\\\right)}^n$$. Here, $$a=2x$$, $$b=-3y$$, $$n=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin9a-h2","type":"hint","dependencies":["a664eb9bin9a-h1"],"title":"Use the binomial theorm.","text":"Follow the binomial theorem from this image.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin9a-h3","type":"hint","dependencies":["a664eb9bin9a-h2"],"title":"substitute into the binomial theorem.","text":"Substitute the a, $$b$$, and $$n$$ that you found into the binomial equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a664eb9bin9a-h4","type":"hint","dependencies":["a664eb9bin9a-h3"],"title":"Simplify.","text":"Simplify, resulting in $$16x^4-96x^3 y^1+16x^2 y^2-216x^1 y^3+81y^4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a675767Ellipses1","title":"Graph an Ellipse with Center at the Origin","body":"Choose the graph that represents the given equation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.3 Ellipses","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a675767Ellipses1a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{x^2}{4}+\\\\frac{y^2}{25}=1$$","stepBody":"Please enter the number of graph that best represents the given equation.##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a675767Ellipses1a-h1","type":"hint","dependencies":[],"title":"Write the Equation in Standard Form","text":"It is in standard form $$\\\\frac{x^2}{4}+\\\\frac{y^2}{25}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses1a-h2","type":"hint","dependencies":["a675767Ellipses1a-h1"],"title":"Determine Whether the Major Axis is Horizontal or Vertical","text":"Since $$4<25$$ and $$25$$ is in the $$y^2$$ term, the major axis is vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses1a-h3","type":"hint","dependencies":["a675767Ellipses1a-h2"],"title":"Find the endpoints of the major axis.","text":"The endpoints will be the y-intercept. Since $$b^2=25$$ then $$b=-5$$ or $$b=5$$. The endpoint of the major axis are $$(0,5)$$, $$(0,-5)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses1a-h4","type":"hint","dependencies":["a675767Ellipses1a-h3"],"title":"Elimination of Choices","text":"SInce graph $$1$$ is the only graph with y-intercepts as $$(0,5)$$ and $$(0,-5)$$, graph $$1$$ is the final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a675767Ellipses10","title":"Graph an Ellipse with Center at the Origin","body":"Choose the graph that represents the given equation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.3 Ellipses","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a675767Ellipses10a","stepAnswer":["https://ibb.co/6JyL7m2"],"problemType":"MultipleChoice","stepTitle":"$$16x^2+9y^2=144$$","stepBody":"Choose the graph that represents the given equation.","answerType":"string","variabilization":{},"choices":["https://ibb.co/6JyL7m2","https://ibb.co/MSPzGr8","https://ibb.co/3zDFFXs","https://ibb.co/XJwrbPF"],"hints":{"DefaultPathway":[{"id":"a675767Ellipses10a-h1","type":"hint","dependencies":[],"title":"Write the Equation in Standard Form","text":"The equation is not in standard form. It can be written as standard form by divding $$144$$ both sides and get $$\\\\frac{x^2}{9}+\\\\frac{y^2}{16}=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses10a-h2","type":"hint","dependencies":["a675767Ellipses10a-h1"],"title":"Determine Whether the Major Axis is Horizontal or Vertical","text":"Since $$9<16$$ and $$25$$ is in the $$y^2$$ term, the major axis is verticle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses10a-h3","type":"hint","dependencies":["a675767Ellipses10a-h2"],"title":"Find the endpoints of the major axis.","text":"The endpoints will be the y-intercept. Since $$b^2=16$$ then $$b=-4$$ or $$b=4$$. The endpoint of the major axis are $$(0,4)$$, $$(0,-4)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses10a-h4","type":"hint","dependencies":["a675767Ellipses10a-h3"],"title":"Find the endpoints of the minor axis.","text":"The endpoints will be the x-intercepts. Since $$a^2=9$$, then $$a=-3$$ or $$a=3$$. The endpoints of the minor axis are $$(3,0)$$, $$(-3,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses10a-h5","type":"hint","dependencies":["a675767Ellipses10a-h4"],"title":"Elimination of Choices","text":"The graph with y-intercepts $$(0,4)$$, $$(0,-4)$$ and x-intercepts $$(3,0)$$, $$(-3,0)$$ is the answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a675767Ellipses11","title":"Graph an Ellipse with Center at the Origin","body":"Choose the graph that represents the given equation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.3 Ellipses","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a675767Ellipses11a","stepAnswer":["https://ibb.co/8YV0D7c"],"problemType":"MultipleChoice","stepTitle":"$$16x^2+36y^2=576$$","stepBody":"Choose the graph that represents the given equation.","answerType":"string","variabilization":{},"choices":["https://ibb.co/19XxnzZ","https://ibb.co/MSPzGr8","https://ibb.co/3zDFFXs","https://ibb.co/8YV0D7c"],"hints":{"DefaultPathway":[{"id":"a675767Ellipses11a-h1","type":"hint","dependencies":[],"title":"Write the Equation in Standard Form","text":"The equation is not in standard form. It can be written as standard form by dividing $$576$$ both sides and get $$\\\\frac{x^2}{36}+\\\\frac{y^2}{16}=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses11a-h2","type":"hint","dependencies":["a675767Ellipses11a-h1"],"title":"Determine Whether the Major Axis is Horizontal or Vertical","text":"Since $$16<36$$ and $$25$$ is in the $$x^2$$ term, the major axis is horizontal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses11a-h3","type":"hint","dependencies":["a675767Ellipses11a-h2"],"title":"Find the endpoints of the major axis.","text":"The endpoints will be the x-intercept. Since $$b^2=36$$ then $$b=-6$$ or $$b=6$$. The endpoint of the major axis are $$(-6,0)$$, $$(6,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses11a-h4","type":"hint","dependencies":["a675767Ellipses11a-h3"],"title":"Find the endpoints of the minor axis.","text":"The endpoints will be the y-intercepts. Since $$a^2=16$$, then $$a=-4$$ or $$a=4$$. The endpoints of the minor axis are $$(0,4)$$, $$(0,-4)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses11a-h5","type":"hint","dependencies":["a675767Ellipses11a-h4"],"title":"Elimination of Choices","text":"The graph with y-intercepts $$(0,4)$$, $$(0,-4)$$ and x-intercepts $$(-6,0)$$, $$(6,0)$$ is the answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a675767Ellipses12","title":"Graph an Ellipse with Center at the Origin","body":"Choose the graph that represents the given equation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.3 Ellipses","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a675767Ellipses12a","stepAnswer":["https://ibb.co/rpx6Pfd"],"problemType":"MultipleChoice","stepTitle":"$$9x^2+25y^2=225$$","stepBody":"Choose the graph that represents the given equation.","answerType":"string","variabilization":{},"choices":["https://ibb.co/rpx6Pfd","https://ibb.co/MSPzGr8","https://ibb.co/3zDFFXs","https://ibb.co/8YV0D7c"],"hints":{"DefaultPathway":[{"id":"a675767Ellipses12a-h1","type":"hint","dependencies":[],"title":"Write the Equation in Standard Form","text":"The equation is not in standard form. It can be written as standard form by divding $$225$$ both sides and get $$\\\\frac{x^2}{25}+\\\\frac{y^2}{9}=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses12a-h2","type":"hint","dependencies":["a675767Ellipses12a-h1"],"title":"Determine Whether the Major Axis is Horizontal or Vertical","text":"Since $$9<25$$ and $$25$$ is in the $$x^2$$ term, the major axis is horizontal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses12a-h3","type":"hint","dependencies":["a675767Ellipses12a-h2"],"title":"Find the endpoints of the major axis.","text":"The endpoints will be the x-intercept. Since $$b^2=25$$ then $$b=-5$$ or $$b=5$$. The endpoint of the major axis are $$(-5,0)$$, $$(5,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses12a-h4","type":"hint","dependencies":["a675767Ellipses12a-h3"],"title":"Find the endpoints of the minor axis.","text":"The endpoints will be the y-intercepts. Since $$a^2=9$$, then $$a=-3$$ or $$a=3$$. The endpoints of the minor axis are $$(0,3)$$, $$(0,-3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses12a-h5","type":"hint","dependencies":["a675767Ellipses12a-h4"],"title":"Elimination of Choices","text":"The graph with y-intercepts $$(0,3)$$, $$(0,-3)$$ and x-intercepts $$(5,0)$$, $$(-5,0)$$ is the answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a675767Ellipses13","title":"Find the Equation of an Ellipse with Center at the Origin","body":"In the following exercise, find the equation of the ellipse shown in the graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.3 Ellipses","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a675767Ellipses13a","stepAnswer":["((x**2)/9)+((y**2)/25)=1"],"problemType":"TextBox","stepTitle":"Find the Equation for the Ellipse in the Graph.","stepBody":"Please enter your answer as $$\\\\frac{x^2}{A}+\\\\frac{y^2}{B}=C$$.##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{x^2}{9}+\\\\frac{y^2}{25}=1$$","hints":{"DefaultPathway":[{"id":"a675767Ellipses13a-h1","type":"hint","dependencies":[],"title":"Recognize the Standard Equation of Ellipses","text":"We recognize this as an ellipse that is centered at the origin. So the equation should be in the form of $$\\\\frac{x^2}{a^2}+\\\\frac{y^2}{b^2}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a675767Ellipses13a-h1"],"title":"Find the Length of Minor Axis","text":"The major axis is verticle because the value of the distance from the origin to the horizontal endpoints is less than the distance from the origin to the verticle endpoints. What is the distance from the center of the ellipse (the origin) to the vertex?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses13a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a675767Ellipses13a-h2"],"title":"Find the Length of Major Axis","text":"What is the distance from the center of horizontal endpoints of the ellipse?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses13a-h4","type":"hint","dependencies":["a675767Ellipses13a-h3"],"title":"Plug in Values of a and $$b$$ into Standard Equation","text":"Since the verticle distance is $$5$$ which is associated with the coefficient of $$y$$ terms $$b=5$$ and $$b^2=25$$. Since the horizontal distance is $$3$$ which is associated with the coefficient of $$x$$ terms, $$a=3$$ and $$a^2=9$$. Given $$\\\\frac{x^2}{a^2}+\\\\frac{y^2}{b^2}=1$$, through pluging in $$a^2=9$$ and $$b^2=25$$, you should get $$\\\\frac{x^2}{9}+\\\\frac{y^2}{25}=1$$ as your final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a675767Ellipses14","title":"Find the Equation of an Ellipse with Center at the Origin","body":"In the following exercise, find the equation of the ellipse shown in the graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.3 Ellipses","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a675767Ellipses14a","stepAnswer":["((x**2)/16)+((y**2)/36)=1"],"problemType":"TextBox","stepTitle":"Find the Equation for the Ellipse in the Graph.","stepBody":"Please enter your answer as $$\\\\frac{x^2}{A}+\\\\frac{y^2}{B}=C$$.##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{x^2}{16}+\\\\frac{y^2}{36}=1$$","hints":{"DefaultPathway":[{"id":"a675767Ellipses14a-h1","type":"hint","dependencies":[],"title":"Recognize the Standard Equation of Ellipses","text":"We recognize this as an ellipse that is centered at the origin. So the equation should be in the form of $$\\\\frac{x^2}{a^2}+\\\\frac{y^2}{b^2}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a675767Ellipses14a-h1"],"title":"Find the Length of Minor Axis","text":"The major axis is verticle because the value of the distance from the origin to the horizontal endpoints is less than the distance from the origin to the verticle endpoints. What is the distance from the center of the ellipse (the origin) to the vertex?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses14a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a675767Ellipses14a-h2"],"title":"Find the Length of Major Axis","text":"What is the distance from the center of horizontal endpoints of the ellipse?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses14a-h4","type":"hint","dependencies":["a675767Ellipses14a-h3"],"title":"Plug in Values of a and $$b$$ into Standard Equation","text":"Since the verticle distance is $$6$$ which is associated with the coefficient of $$y$$ terms $$b=6$$ and $$b^2=36$$. Since the horizontal distance is $$4$$ which is associated with the coefficient of $$x$$ terms, $$a=4$$ and $$a^2=16$$. Given $$\\\\frac{x^2}{a^2}+\\\\frac{y^2}{b^2}=1$$, through pluging in $$a^2=16$$ and $$b^2=36$$, you should get $$\\\\frac{x^2}{16}+\\\\frac{y^2}{36}=1$$ as your final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a675767Ellipses15","title":"Find the Equation of an Ellipse with Center at the Origin","body":"In the following exercise, find the equation of the ellipse shown in the graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.3 Ellipses","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a675767Ellipses15a","stepAnswer":["(x**2)/25+(y**2)/4=1"],"problemType":"TextBox","stepTitle":"Find the Equation for the Ellipse in the Graph.","stepBody":"Please enter your answer as $$\\\\frac{x^2}{A}+\\\\frac{y^2}{B}=C$$.##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{x^2}{25}+\\\\frac{y^2}{4}=1$$","hints":{"DefaultPathway":[{"id":"a675767Ellipses15a-h1","type":"hint","dependencies":[],"title":"Recognize the Standard Equation of Ellipses","text":"We recognize this as an ellipse that is centered at the origin. So the equation should be in the form of $$\\\\frac{x^2}{a^2}+\\\\frac{y^2}{b^2}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a675767Ellipses15a-h1"],"title":"Find the Length of Minor Axis","text":"The major axis is horizontal because the value of the distance from the origin to the horizontal endpoints is greater than the distance from the origin to the verticle endpoints. What is the distance from the center of the ellipse (the origin) to the vertex?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a675767Ellipses15a-h2"],"title":"Find the Length of Major Axis","text":"What is the distance from the center of verticle endpoints of the ellipse?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses15a-h4","type":"hint","dependencies":["a675767Ellipses15a-h3"],"title":"Plug in Values of a and $$b$$ into Standard Equation","text":"Since the verticle distance is $$2$$ which is associated with the coefficient of $$y$$ terms $$b=2$$ and $$b^2=4$$. Since the horizontal distance is $$5$$ which is associated with the coefficient of $$x$$ terms, $$a=5$$ and $$a^2=25$$. Given $$\\\\frac{x^2}{a^2}+\\\\frac{y^2}{b^2}=1$$, through pluging in $$a^2=25$$ and $$b^2=4$$, you should get $$\\\\frac{x^2}{25}+\\\\frac{y^2}{4}=1$$ as your final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a675767Ellipses16","title":"Find the Equation of an Ellipse with Center at the Origin","body":"In the following exercise, find the equation of the ellipse shown in the graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.3 Ellipses","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a675767Ellipses16a","stepAnswer":["((x**2)/9)+((y**2)/16)=1"],"problemType":"TextBox","stepTitle":"Find the Equation for the Ellipse in the Graph.","stepBody":"Please enter your answer as $$\\\\frac{x^2}{A}+\\\\frac{y^2}{B}=C$$.##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{x^2}{9}+\\\\frac{y^2}{16}=1$$","hints":{"DefaultPathway":[{"id":"a675767Ellipses16a-h1","type":"hint","dependencies":[],"title":"Recognize the Standard Equation of Ellipses","text":"We recognize this as an ellipse that is centered at the origin. So the equation should be in the form of $$\\\\frac{x^2}{a^2}+\\\\frac{y^2}{b^2}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a675767Ellipses16a-h1"],"title":"Find the Length of Minor Axis","text":"The major axis is verticle because the value of the distance from the origin to the horizontal endpoints is less than the distance from the origin to the verticle endpoints. What is the distance from the center of the ellipse (the origin) to the vertex?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses16a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a675767Ellipses16a-h2"],"title":"Find the Length of Major Axis","text":"What is the distance from the center of horizontal endpoints of the ellipse?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses16a-h4","type":"hint","dependencies":["a675767Ellipses16a-h3"],"title":"Plug in Values of a and $$b$$ into Standard Equation","text":"Since the verticle distance is $$4$$ which is associated with the coefficient of $$y$$ terms $$b=4$$ and $$b^2=16$$. Since the horizontal distance is $$3$$ which is associated with the coefficient of $$x$$ terms, $$a=3$$ and $$a^2=9$$. Given $$\\\\frac{x^2}{a^2}+\\\\frac{y^2}{b^2}=1$$, through pluging in $$a^2=9$$ and $$b^2=16$$, you should get $$\\\\frac{x^2}{9}+\\\\frac{y^2}{16}=1$$ as your final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a675767Ellipses17","title":"Find the Equation of an Ellipse with Center Not at the Origin","body":"In the following exercise, find the graph of the given ellipse equations.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.3 Ellipses","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a675767Ellipses17a","stepAnswer":["https://ibb.co/bdcFCDg"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{{\\\\left(x+1\\\\right)}^2}{4}+\\\\frac{{\\\\left(y+6\\\\right)}^2}{25}=1$$","stepBody":"Choose the graph that represents the given equation.","answerType":"string","variabilization":{},"choices":["https://ibb.co/wc9Vf8M","https://ibb.co/WyVNDsH","https://ibb.co/bdcFCDg","https://ibb.co/Cn9y40H"],"hints":{"DefaultPathway":[{"id":"a675767Ellipses17a-h1","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-1,-6)"],"dependencies":[],"title":"Recognize the Standard Equation of Ellipses","text":"The equation is in standard form, $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$. In the standard form, (h,k) is the center of ellipse. What is the center (h,k)? Enter your answer in the form (h,k).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses17a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a675767Ellipses17a-h1"],"title":"Find a and $$b$$ in $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$.","text":"$$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$ is the standard form of ellipse that is not centered at the origin. Given equation $$\\\\frac{{\\\\left(x+1\\\\right)}^2}{4}+\\\\frac{{\\\\left(y+6\\\\right)}^2}{25}=1$$, what is a?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses17a-h3","type":"hint","dependencies":["a675767Ellipses17a-h2"],"title":"Find a and $$b$$ in $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$.","text":"We know that $$a=2$$ and $$b=5$$ in the given equation. $$2<5$$, so the major axis is verticle. The distance from the center to the vertices is $$5$$. The distance from the center to the endpoints of the minor axis is $$2$$. So this graph is the correct answer.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a675767Ellipses18","title":"Find the Equation of an Ellipse with Center Not at the Origin","body":"In the following exercise, find the graph of the given ellipse equations.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.3 Ellipses","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a675767Ellipses18a","stepAnswer":["https://ibb.co/j4W3kRr"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{{\\\\left(x-3\\\\right)}^2}{25}+\\\\frac{{\\\\left(y+2\\\\right)}^2}{9}=1$$","stepBody":"Choose the graph that represents the given equation.","answerType":"string","variabilization":{},"choices":["https://ibb.co/j4W3kRr","https://ibb.co/WyVNDsH","https://ibb.co/bdcFCDg","https://ibb.co/Cn9y40H"],"hints":{"DefaultPathway":[{"id":"a675767Ellipses18a-h1","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(3,-2)"],"dependencies":[],"title":"Recognize the Standard Equation of Ellipses","text":"The equation is in standard form, $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$. In the standard form, (h,k) is the center of ellipse. What is the center (h,k)? Enter your answer in the form (h,k).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a675767Ellipses18a-h1"],"title":"Find a and $$b$$ in $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$.","text":"$$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$ is the standard form of ellipse that is not centered at the origin. Given equation $$\\\\frac{{\\\\left(x-3\\\\right)}^2}{25}+\\\\frac{{\\\\left(y+2\\\\right)}^2}{9}=1$$, what is a?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses18a-h3","type":"hint","dependencies":["a675767Ellipses18a-h2"],"title":"Find a and $$b$$ in $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$.","text":"We know that $$a=5$$ and $$b=3$$ in the given equation. $$3<5$$, so the major axis is horizontal. The distance from the center to the vertices is $$5$$. The distance from the center to the endpoints of the minor axis is $$3$$. So this graph is the correct answer.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a675767Ellipses19","title":"Find the Equation of an Ellipse with Center Not at the Origin","body":"In the following exercise, find the graph of the given ellipse equations.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.3 Ellipses","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a675767Ellipses19a","stepAnswer":["https://ibb.co/FgtXC2P"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{{\\\\left(x+4\\\\right)}^2}{4}+\\\\frac{{\\\\left(y-2\\\\right)}^2}{9}=1$$","stepBody":"Choose the graph that represents the given equation.","answerType":"string","variabilization":{},"choices":["https://ibb.co/j4W3kRr","https://ibb.co/WyVNDsH","https://ibb.co/bdcFCDg","https://ibb.co/FgtXC2P"],"hints":{"DefaultPathway":[{"id":"a675767Ellipses19a-h1","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-4,2)"],"dependencies":[],"title":"Recognize the Standard Equation of Ellipses","text":"The equation is in standard form, $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$. In the standard form, (h,k) is the center of ellipse. What is the center (h,k)? Enter your answer in the form (h,k).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a675767Ellipses19a-h1"],"title":"Find a and $$b$$ in $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$.","text":"$$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$ is the standard form of ellipse that is not centered at the origin. Given equation $$\\\\frac{{\\\\left(x+4\\\\right)}^2}{4}+\\\\frac{{\\\\left(y-2\\\\right)}^2}{9}=1$$, what is a?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses19a-h3","type":"hint","dependencies":["a675767Ellipses19a-h2"],"title":"Find a and $$b$$ in $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$.","text":"We know that $$a=2$$ and $$b=3$$ in the given equation. $$2<3$$, so the major axis is verticle. The distance from the center to the vertices is $$3$$. The distance from the center to the endpoints of the minor axis is $$2$$. So this graph is the correct answer.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a675767Ellipses2","title":"Graph an Ellipse with Center at the Origin","body":"Choose the graph that represents the given equation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.3 Ellipses","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a675767Ellipses2a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{x^2}{9}+\\\\frac{y^2}{25}=1$$","stepBody":"Choose the graph that represents the given equation.##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a675767Ellipses2a-h1","type":"hint","dependencies":[],"title":"Write the Equation in Standard Form","text":"It is in standard form $$\\\\frac{x^2}{9}+\\\\frac{y^2}{25}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses2a-h2","type":"hint","dependencies":["a675767Ellipses2a-h1"],"title":"Determine Whether the Major Axis is Horizontal or Vertical","text":"Since $$9<25$$ and $$25$$ is in the $$y^2$$ term, the major axis is vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses2a-h3","type":"hint","dependencies":["a675767Ellipses2a-h2"],"title":"Find the endpoints of the major axis.","text":"The endpoints will be the y-intercept. Since $$b^2=25$$ then $$b=-5$$ or $$b=5$$. The endpoint of the major axis are $$(0,5)$$, $$(0,-5)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses2a-h4","type":"hint","dependencies":["a675767Ellipses2a-h3"],"title":"Find the endpoints of the minor axis.","text":"The endpoints will be the x-intercepts. Since $$a^2=9$$, then $$a=-3$$ or $$a=3$$. The endpoints of the minor axis are $$(3,0)$$, $$(-3,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses2a-h5","type":"hint","dependencies":["a675767Ellipses2a-h4"],"title":"Elimination of Choices","text":"SInce graph $$2$$ is the only graph with y-intercepts as $$(0,5)$$ and $$(0,-5)$$ and x-intercept as $$(-3,0)$$ and $$(3,0)$$, graph $$2$$ is the final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a675767Ellipses20","title":"Find the Equation of an Ellipse with Center Not at the Origin","body":"In the following exercise, find the graph of the given ellipse equations.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.3 Ellipses","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a675767Ellipses20a","stepAnswer":["https://ibb.co/k5bTQ3g"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{{\\\\left(x-4\\\\right)}^2}{9}+\\\\frac{{\\\\left(y-1\\\\right)}^2}{16}=1$$","stepBody":"Choose the graph that represents the given equation.","answerType":"string","variabilization":{},"choices":["https://ibb.co/k5bTQ3g","https://ibb.co/WyVNDsH","https://ibb.co/bdcFCDg","https://ibb.co/FgtXC2P"],"hints":{"DefaultPathway":[{"id":"a675767Ellipses20a-h1","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(4,1)"],"dependencies":[],"title":"Recognize the Standard Equation of Ellipses","text":"The equation is in standard form, $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$. In the standard form, (h,k) is the center of ellipse. What is the center (h,k)? Enter your answer in the form (h,k).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a675767Ellipses20a-h1"],"title":"Find a and $$b$$ in $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$.","text":"$$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$ is the standard form of ellipse that is not centered at the origin. Given equation $$\\\\frac{{\\\\left(x-4\\\\right)}^2}{9}+\\\\frac{{\\\\left(y-1\\\\right)}^2}{16}=1$$, what is a?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses20a-h3","type":"hint","dependencies":["a675767Ellipses20a-h2"],"title":"Find a and $$b$$ in $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$.","text":"We know that $$a=3$$ and $$b=4$$ in the given equation. $$3<4$$, so the major axis is verticle. The distance from the center to the vertices is $$4$$. The distance from the center to the endpoints of the minor axis is $$3$$. So this graph is the correct answer.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a675767Ellipses21","title":"Find the Equation of an Ellipse with Center Not at the Origin","body":"In the following exercise, find the graph of the given ellipse equations.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.3 Ellipses","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a675767Ellipses21a","stepAnswer":["https://ibb.co/sJ4NSmd"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{{\\\\left(x-3\\\\right)}^2}{4}+\\\\frac{{\\\\left(y-7\\\\right)}^2}{25}=1$$","stepBody":"Choose the graph that represents the given equation.","answerType":"string","variabilization":{},"choices":["https://ibb.co/B2zw0MV","https://ibb.co/NpPzFTb","https://ibb.co/DMqddCx","https://ibb.co/sJ4NSmd"],"hints":{"DefaultPathway":[{"id":"a675767Ellipses21a-h1","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(3,7)"],"dependencies":[],"title":"Recognize the Standard Equation of Ellipses","text":"The equation is in standard form, $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$. In the standard form, (h,k) is the center of ellipse. What is the center (h,k)? Enter your answer in the form (h,k).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses21a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a675767Ellipses21a-h1"],"title":"Find a and $$b$$ in $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$.","text":"$$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$ is the standard form of ellipse that is not centered at the origin. Given equation $$\\\\frac{{\\\\left(x-3\\\\right)}^2}{4}+\\\\frac{{\\\\left(y-7\\\\right)}^2}{25}=1$$, what is a?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses21a-h3","type":"hint","dependencies":["a675767Ellipses21a-h2"],"title":"Find a and $$b$$ in $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$.","text":"We know that $$a=2$$ and $$b=5$$ in the given equation. $$2<5$$, so the major axis is verticle. The distance from the center to the vertices is $$5$$. The distance from the center to the endpoints of the minor axis is $$2$$. So this graph is the correct answer.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a675767Ellipses22","title":"Find the Equation of an Ellipse with Center Not at the Origin","body":"In the following exercise, find the graph of the given ellipse equations.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.3 Ellipses","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a675767Ellipses22a","stepAnswer":["https://ibb.co/6v1czrq"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{{\\\\left(x+6\\\\right)}^2}{16}+\\\\frac{{\\\\left(y+5\\\\right)}^2}{4}=1$$","stepBody":"Choose the graph that represents the given equation.","answerType":"string","variabilization":{},"choices":["https://ibb.co/6v1czrq","https://ibb.co/NpPzFTb","https://ibb.co/DMqddCx","https://ibb.co/sJ4NSmd"],"hints":{"DefaultPathway":[{"id":"a675767Ellipses22a-h1","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-6,-5)"],"dependencies":[],"title":"Recognize the Standard Equation of Ellipses","text":"The equation is in standard form, $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$. In the standard form, (h,k) is the center of ellipse. What is the center (h,k)? Enter your answer in the form (h,k).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses22a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a675767Ellipses22a-h1"],"title":"Find a and $$b$$ in $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$.","text":"$$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$ is the standard form of ellipse that is not centered at the origin. Given equation $$\\\\frac{{\\\\left(x+6\\\\right)}^2}{16}+\\\\frac{{\\\\left(y+5\\\\right)}^2}{4}=1$$, what is a?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses22a-h3","type":"hint","dependencies":["a675767Ellipses22a-h2"],"title":"Find a and $$b$$ in $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$.","text":"We know that $$a=4$$ and $$b=2$$ in the given equation. $$2<4$$, so the major axis is horizontal. The distance from the center to the vertices is $$4$$. The distance from the center to the endpoints of the minor axis is $$2$$. So this graph is the correct answer.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a675767Ellipses23","title":"Find the Equation of an Ellipse with Center Not at the Origin","body":"In the following exercise, find the graph of the given ellipse equations.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.3 Ellipses","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a675767Ellipses23a","stepAnswer":["https://ibb.co/0hkVZB2"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{{\\\\left(x-5\\\\right)}^2}{9}+\\\\frac{{\\\\left(y+4\\\\right)}^2}{25}=1$$","stepBody":"Choose the graph that represents the given equation.","answerType":"string","variabilization":{},"choices":["https://ibb.co/B2zw0MV","https://ibb.co/NpPzFTb","https://ibb.co/DMqddCx","https://ibb.co/0hkVZB2"],"hints":{"DefaultPathway":[{"id":"a675767Ellipses23a-h1","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(5,-4)"],"dependencies":[],"title":"Recognize the Standard Equation of Ellipses","text":"The equation is in standard form, $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$. In the standard form, (h,k) is the center of ellipse. What is the center (h,k)? Enter your answer in the form (h,k).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses23a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a675767Ellipses23a-h1"],"title":"Find a and $$b$$ in $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$.","text":"$$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$ is the standard form of ellipse that is not centered at the origin. Given equation $$\\\\frac{{\\\\left(x-5\\\\right)}^2}{9}+\\\\frac{{\\\\left(y+4\\\\right)}^2}{25}=1$$, what is a?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses23a-h3","type":"hint","dependencies":["a675767Ellipses23a-h2"],"title":"Find a and $$b$$ in $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$.","text":"We know that $$a=3$$ and $$b=5$$ in the given equation. $$3<5$$, so the major axis is verticle. The distance from the center to the vertices is $$5$$. The distance from the center to the endpoints of the minor axis is $$3$$. So this graph is the correct answer.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a675767Ellipses24","title":"Find the Equation of an Ellipse with Center Not at the Origin","body":"In the following exercise, find the graph of the given ellipse equations.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.3 Ellipses","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a675767Ellipses24a","stepAnswer":["https://ibb.co/D4zvc1y"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{{\\\\left(x+5\\\\right)}^2}{36}+\\\\frac{{\\\\left(y-3\\\\right)}^2}{16}=1$$","stepBody":"Choose the graph that represents the given equation.","answerType":"string","variabilization":{},"choices":["https://ibb.co/D4zvc1y","https://ibb.co/NpPzFTb","https://ibb.co/DMqddCx","https://ibb.co/sJ4NSmd"],"hints":{"DefaultPathway":[{"id":"a675767Ellipses24a-h1","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-5,3)"],"dependencies":[],"title":"Recognize the Standard Equation of Ellipses","text":"The equation is in standard form, $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$. In the standard form, (h,k) is the center of ellipse. What is the center (h,k)? Enter your answer in the form (h,k).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses24a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a675767Ellipses24a-h1"],"title":"Find a and $$b$$ in $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$.","text":"$$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$ is the standard form of ellipse that is not centered at the origin. Given equation $$\\\\frac{{\\\\left(x+5\\\\right)}^2}{36}+\\\\frac{{\\\\left(y-3\\\\right)}^2}{16}=1$$, what is a?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses24a-h3","type":"hint","dependencies":["a675767Ellipses24a-h2"],"title":"Find a and $$b$$ in $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$.","text":"We know that $$a=6$$ and $$b=4$$ in the given equation. $$4<6$$, so the major axis is horizontal. The distance from the center to the vertices is $$6$$. The distance from the center to the endpoints of the minor axis is $$4$$. So this graph is the correct answer.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a675767Ellipses25","title":"Find the Equation of an Ellipse with Center Not at the Origin","body":"In the following exercise, find the graph of the given ellipse equations.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.3 Ellipses","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a675767Ellipses25a","stepAnswer":["https://ibb.co/p0BFP4Q"],"problemType":"MultipleChoice","stepTitle":"$$x^2+y^2=49$$","stepBody":"Choose the graph that represents the given equation.","answerType":"string","variabilization":{},"choices":["https://ibb.co/D4zvc1y","https://ibb.co/NpPzFTb","https://ibb.co/DMqddCx","https://ibb.co/p0BFP4Q"],"hints":{"DefaultPathway":[{"id":"a675767Ellipses25a-h1","type":"hint","dependencies":[],"title":"Recognize the Standard Equation","text":"The equation is in the form of $$x^2+y^2=r^2$$ which is the standard form of a circle with center at origin. In this case $$r=7$$. Therefore the equation represents the circle with radius $$7$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a675767Ellipses26","title":"Find the Equation of an Ellipse with Center Not at the Origin","body":"In the following exercise, find the graph of the given ellipse equations.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.3 Ellipses","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a675767Ellipses26a","stepAnswer":["https://ibb.co/dJmp7MT"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(x+5\\\\right)}^2+{\\\\left(y+2\\\\right)}^2=4$$","stepBody":"Choose the graph that represents the given equation.","answerType":"string","variabilization":{},"choices":["https://ibb.co/dJmp7MT","https://ibb.co/NpPzFTb","https://ibb.co/DMqddCx","https://ibb.co/p0BFP4Q"],"hints":{"DefaultPathway":[{"id":"a675767Ellipses26a-h1","type":"hint","dependencies":[],"title":"Recognize the Standard Equation","text":"The equation is in the form of $${\\\\left(x-h\\\\right)}^2+{\\\\left(y-k\\\\right)}^2=r^2$$ which is the standard form of a circle with center at (h,k). In this case $$r=2$$. Therefore the equation represents the circle centered at $$(-5,-2)$$ with radius $$2$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a675767Ellipses27","title":"Find the Equation of an Ellipse with Center Not at the Origin","body":"In the following exercise, find the graph of the given ellipse equations.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.3 Ellipses","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a675767Ellipses27a","stepAnswer":["https://ibb.co/RBh2PKT"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{{\\\\left(x+3\\\\right)}^2}{16}+\\\\frac{{\\\\left(y+1\\\\right)}^2}{4}=1$$","stepBody":"Choose the graph that represents the given equation.","answerType":"string","variabilization":{},"choices":["https://ibb.co/D4zvc1y","https://ibb.co/RBh2PKT","https://ibb.co/DMqddCx","https://ibb.co/sJ4NSmd"],"hints":{"DefaultPathway":[{"id":"a675767Ellipses27a-h1","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-3,-1)"],"dependencies":[],"title":"Recognize the Standard Equation of Ellipses","text":"The equation is in standard form, $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$. In the standard form, (h,k) is the center of ellipse. What is the center (h,k)? Enter your answer in the form (h,k).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses27a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a675767Ellipses27a-h1"],"title":"Find a and $$b$$ in $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$.","text":"$$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$ is the standard form of ellipse that is not centered at the origin. Given $$\\\\operatorname{equation}\\\\left(\\\\frac{{\\\\left(x+3\\\\right)}^2}{16}\\\\right)+\\\\frac{{\\\\left(y+1\\\\right)}^2}{4}=1$$, what is a?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses27a-h3","type":"hint","dependencies":["a675767Ellipses27a-h2"],"title":"Find a and $$b$$ in $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}+\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$.","text":"We know that $$a=4$$ and $$b=2$$ in the given equation. $$2<4$$, so the major axis is horizontal. The distance from the center to the vertices is $$4$$. The distance from the center to the endpoints of the minor axis is $$2$$. So this graph is the correct answer.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a675767Ellipses28","title":"Find the Equation of an Ellipse with Center Not at the Origin","body":"In the following exercise, find the graph of the given ellipse equations.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.3 Ellipses","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a675767Ellipses28a","stepAnswer":["https://ibb.co/Z2zN1LV"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(x-2\\\\right)}^2+{\\\\left(y-3\\\\right)}^2=9$$","stepBody":"Choose the graph that represents the given equation.","answerType":"string","variabilization":{},"choices":["https://ibb.co/dJmp7MT","https://ibb.co/Z2zN1LV","https://ibb.co/DMqddCx","https://ibb.co/p0BFP4Q"],"hints":{"DefaultPathway":[{"id":"a675767Ellipses28a-h1","type":"hint","dependencies":[],"title":"Recognize the Standard Equation of Circle","text":"The equation is in the form of $${\\\\left(x-h\\\\right)}^2+{\\\\left(y-k\\\\right)}^2=r^2$$ which is the standard form of a circle with center at (h,k). In this case $$r=3$$. Therefore the equation represents the circle centered at $$(2,3)$$ with radius $$3$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a675767Ellipses29","title":"Find the Equation of an Ellipse with Center Not at the Origin","body":"In the following exercise, find the graph of the given ellipse equations.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.3 Ellipses","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a675767Ellipses29a","stepAnswer":["https://ibb.co/vdQVthZ"],"problemType":"MultipleChoice","stepTitle":"$$x^2+y^2=64$$","stepBody":"Choose the graph that represents the given equation.","answerType":"string","variabilization":{},"choices":["https://ibb.co/D4zvc1y","https://ibb.co/NpPzFTb","https://ibb.co/vdQVthZ","https://ibb.co/p0BFP4Q"],"hints":{"DefaultPathway":[{"id":"a675767Ellipses29a-h1","type":"hint","dependencies":[],"title":"Recognize the Standard Equation","text":"The equation is in the form of $$x^2+y^2=r^2$$ which is the standard form of a circle with center at origin. In this case $$r=8$$. Therefore the equation represents the circle with radius $$8$$ centered at the origin.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a675767Ellipses3","title":"Graph an Ellipse with Center at the Origin","body":"Choose the graph that represents the given equation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.3 Ellipses","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a675767Ellipses3a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{x^2}{25}+\\\\frac{y^2}{36}=1$$","stepBody":"Choose the graph that represents the given equation.##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a675767Ellipses3a-h1","type":"hint","dependencies":[],"title":"Write the Equation in Standard Form","text":"It is in standard form $$\\\\frac{x^2}{25}+\\\\frac{y^2}{36}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses3a-h2","type":"hint","dependencies":["a675767Ellipses3a-h1"],"title":"Determine Whether the Major Axis is Horizontal or Vertical","text":"Since $$25<36$$ and $$36$$ is in the $$y^2$$ term, the major axis is vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses3a-h3","type":"hint","dependencies":["a675767Ellipses3a-h2"],"title":"Find the endpoints of the major axis.","text":"The endpoints will be the y-intercept. Since $$b^2=36$$ then $$b=-6$$ or $$b=6$$. The endpoint of the major axis are $$(0,6)$$, $$(0,-6)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses3a-h4","type":"hint","dependencies":["a675767Ellipses3a-h3"],"title":"Find the endpoints of the minor axis.","text":"The endpoints will be the x-intercepts. Since $$a^2=25$$, then $$a=-5$$ or $$a=5$$. The endpoints of the minor axis are $$(5,0)$$, $$(-5,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses3a-h5","type":"hint","dependencies":["a675767Ellipses3a-h4"],"title":"Elimination of Choices","text":"SInce graph $$3$$ is the only graph with y-intercepts as $$(0,6)$$ and $$(0,-6)$$ and x-intercept as $$(5,0)$$, $$(-5,0)$$, graph $$3$$ is the final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a675767Ellipses30","title":"Graph an Ellipse with Center at the Origin","body":"Choose the graph that represents the given equation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.3 Ellipses","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a675767Ellipses30a","stepAnswer":["https://ibb.co/KxQk2D2"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{x^2}{25}+\\\\frac{y^2}{36}=1$$","stepBody":"Choose the graph that represents the given equation.","answerType":"string","variabilization":{},"choices":["https://ibb.co/KxQk2D2","https://ibb.co/MSPzGr8","https://ibb.co/3zDFFXs","https://ibb.co/bXtPNZR"],"hints":{"DefaultPathway":[{"id":"a675767Ellipses30a-h1","type":"hint","dependencies":[],"title":"Write the Equation in Standard Form","text":"It is in standard form as $$\\\\frac{x^2}{25}+\\\\frac{y^2}{36}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses30a-h2","type":"hint","dependencies":["a675767Ellipses30a-h1"],"title":"Determine Whether the Major Axis is Horizontal or Vertical","text":"Since $$25<36$$ and $$9$$ is in the $$y^2$$ term, the major axis is verticle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses30a-h3","type":"hint","dependencies":["a675767Ellipses30a-h2"],"title":"Find the endpoints of the major axis.","text":"The vertices will be the y-intercept. Since $$b^2=36$$ then $$b=-6$$ or $$b=6$$. The endpoint of the major axis are $$(0,6)$$, $$(0,-6)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses30a-h4","type":"hint","dependencies":["a675767Ellipses30a-h3"],"title":"Find the endpoints of the minor axis.","text":"The endpoints will be the x-intercepts. Since $$a^2=25$$, then $$a=-5$$ or $$a=5$$. The endpoints of the minor axis are $$(5,0)$$, $$(-5,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses30a-h5","type":"hint","dependencies":["a675767Ellipses30a-h4"],"title":"Elimination of Choices","text":"The graph with y-intercepts $$(0,6)$$, $$(0,-6)$$ and x-intercepts $$(-5,0)$$, $$(5,0)$$ is the answer.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a675767Ellipses4","title":"Graph an Ellipse with Center at the Origin","body":"Choose the graph that represents the given equation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.3 Ellipses","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a675767Ellipses4a","stepAnswer":["https://ibb.co/bXtPNZR\\\\n"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{x^2}{16}+\\\\frac{y^2}{36}=1$$","stepBody":"Choose the graph that represents the given equation.","answerType":"string","variabilization":{},"choices":["https://i.ibb.co/1RfXZYr/Screen-Shot-2022-05-13-at-10-59-06-PM.png","https://i.ibb.co/M7dsP45/4.png","https://i.ibb.co/cQXyNhZ/Screen-Shot-2022-05-13-at-10-58-55-PM.png","https://i.ibb.co/vY7dd5B/Screen-Shot-2022-05-13-at-10-59-16-PM.png","https://ibb.co/bXtPNZR\\\\n"],"hints":{"DefaultPathway":[{"id":"a675767Ellipses4a-h1","type":"hint","dependencies":[],"title":"Write the Equation in Standard Form","text":"It is in standard form $$\\\\frac{x^2}{16}+\\\\frac{y^2}{36}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses4a-h2","type":"hint","dependencies":["a675767Ellipses4a-h1"],"title":"Determine Whether the Major Axis is Horizontal or Vertical","text":"Since $$16<36$$ and $$36$$ is in the $$y^2$$ term, the major axis is vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses4a-h3","type":"hint","dependencies":["a675767Ellipses4a-h2"],"title":"Find the endpoints of the major axis.","text":"The endpoints will be the y-intercept. Since $$b^2=36$$ then $$b=-6or$$ $$b=6$$. The endpoint of the major axis are $$(0,6)$$, $$(0,-6)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses4a-h4","type":"hint","dependencies":["a675767Ellipses4a-h3"],"title":"Find the endpoints of the minor axis.","text":"The endpoints will be the x-intercepts. Since $$a^2=16$$, then $$a=-4$$ or $$a=4$$. The endpoints of the minor axis are $$(4,0)$$, $$(-4,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses4a-h5","type":"hint","dependencies":["a675767Ellipses4a-h4"],"title":"Elimination of Choices","text":"The graph with y-intercepts $$(0,6)$$, $$(0,-6)$$ and x-intercepts $$(4,0)$$, $$(-4,0)$$ is the answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a675767Ellipses5","title":"Graph an Ellipse with Center at the Origin","body":"Choose the graph that represents the given equation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.3 Ellipses","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a675767Ellipses5a","stepAnswer":["https://ibb.co/pQGw204"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{x^2}{25}+\\\\frac{y^2}{9}=1$$","stepBody":"Choose the graph that represents the given equation.","answerType":"string","variabilization":{},"choices":["https://ibb.co/pQGw204","https://ibb.co/Zc6gdRB","https://ibb.co/3zDFFXs","https://ibb.co/bXtPNZR"],"hints":{"DefaultPathway":[{"id":"a675767Ellipses5a-h1","type":"hint","dependencies":[],"title":"Write the Equation in Standard Form","text":"It is in standard form $$\\\\frac{x^2}{25}+\\\\frac{y^2}{9}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses5a-h2","type":"hint","dependencies":["a675767Ellipses5a-h1"],"title":"Determine Whether the Major Axis is Horizontal or Vertical","text":"Since $$9<25$$ and $$25$$ is in the $$x^2$$ term, the major axis is horizontal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses5a-h3","type":"hint","dependencies":["a675767Ellipses5a-h2"],"title":"Find the endpoints of the major axis.","text":"The endpoints will be the x-intercept. Since $$b^2=25$$ then $$b=-5or$$ $$b=5$$. The endpoint of the major axis are $$(-5,0)$$, $$(5,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses5a-h4","type":"hint","dependencies":["a675767Ellipses5a-h3"],"title":"Find the endpoints of the minor axis.","text":"The endpoints will be the y-intercepts. Since $$a^2=9$$, then $$a=-3$$ or $$a=3$$. The endpoints of the minor axis are $$(0,3)$$, $$(0,-3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses5a-h5","type":"hint","dependencies":["a675767Ellipses5a-h4"],"title":"Elimination of Choices","text":"The graph with x-intercepts $$(-5,0)$$, $$(5,0)$$ and y-intercepts $$(0,3)$$, $$(0,-3)$$ is the answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a675767Ellipses6","title":"Graph an Ellipse with Center at the Origin","body":"Choose the graph that represents the given equation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.3 Ellipses","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a675767Ellipses6a","stepAnswer":["https://ibb.co/Ht2Hj00"],"problemType":"MultipleChoice","stepTitle":"$$x^2+\\\\frac{y^2}{4}=1$$","stepBody":"Choose the graph that represents the given equation.","answerType":"string","variabilization":{},"choices":["https://ibb.co/Ht2Hj00","https://ibb.co/Zc6gdRB","https://ibb.co/3zDFFXs","https://ibb.co/bXtPNZR"],"hints":{"DefaultPathway":[{"id":"a675767Ellipses6a-h1","type":"hint","dependencies":[],"title":"Write the Equation in Standard Form","text":"It can be written in the standard form $$\\\\frac{x^2}{1}+\\\\frac{y^2}{4}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses6a-h2","type":"hint","dependencies":["a675767Ellipses6a-h1"],"title":"Determine Whether the Major Axis is Horizontal or Vertical","text":"Since $$1<4$$ and $$4$$ is in the $$y^2$$ term, the major axis is horizontal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses6a-h3","type":"hint","dependencies":["a675767Ellipses6a-h2"],"title":"Find the endpoints of the major axis.","text":"The endpoints will be the y-intercept. Since $$b^2=4$$ then $$b=-2$$ or $$b=2$$. The endpoint of the major axis are $$(0,2)$$, $$(0,-2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses6a-h4","type":"hint","dependencies":["a675767Ellipses6a-h3"],"title":"Find the endpoints of the minor axis.","text":"The endpoints will be the x-intercepts. Since $$a^2=1$$, then $$a=-1$$ or $$a=1$$. The endpoints of the minor axis are $$(1,0)$$, $$(-1,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses6a-h5","type":"hint","dependencies":["a675767Ellipses6a-h4"],"title":"Elimination of Choices","text":"The graph with y-intercepts $$(0,2)$$, $$(0,-2)$$ and x-intercepts $$(1,0)$$, $$(-1,0)$$ is the answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a675767Ellipses7","title":"Graph an Ellipse with Center at the Origin","body":"Choose the graph that represents the given equation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.3 Ellipses","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a675767Ellipses7a","stepAnswer":["https://ibb.co/19XxnzZ"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{x^2}{36}+\\\\frac{y^2}{16}=1$$","stepBody":"Choose the graph that represents the given equation.","answerType":"string","variabilization":{},"choices":["https://ibb.co/19XxnzZ","https://ibb.co/Zc6gdRB","https://ibb.co/3zDFFXs","https://ibb.co/bXtPNZR"],"hints":{"DefaultPathway":[{"id":"a675767Ellipses7a-h1","type":"hint","dependencies":[],"title":"Write the Equation in Standard Form","text":"It is in standard form $$\\\\frac{x^2}{36}+\\\\frac{y^2}{16}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses7a-h2","type":"hint","dependencies":["a675767Ellipses7a-h1"],"title":"Determine Whether the Major Axis is Horizontal or Vertical","text":"Since $$16<36$$ and $$36$$ is in the $$x^2$$ term, the major axis is horizontal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses7a-h3","type":"hint","dependencies":["a675767Ellipses7a-h2"],"title":"Find the endpoints of the major axis.","text":"The endpoints will be the x-intercept. Since $$b^2=36$$ then $$b=-6$$ or $$b=6$$. The endpoint of the major axis are $$(-6,0)$$, $$(6,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses7a-h4","type":"hint","dependencies":["a675767Ellipses7a-h3"],"title":"Find the endpoints of the minor axis.","text":"The endpoints will be the y-intercepts. Since $$a^2=16$$, then $$a=-4$$ or $$a=4$$. The endpoints of the minor axis are $$(0,4)$$, $$(0,-4)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses7a-h5","type":"hint","dependencies":["a675767Ellipses7a-h4"],"title":"Elimination of Choices","text":"The graph with y-intercepts $$(0,4)$$, $$(0,-4)$$ and x-intercepts $$(-6,0)$$, $$(6,0)$$ is the answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a675767Ellipses8","title":"Graph an Ellipse with Center at the Origin","body":"Choose the graph that represents the given equation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.3 Ellipses","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a675767Ellipses8a","stepAnswer":["https://ibb.co/MSPzGr8"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{x^2}{9}+y^2=1$$","stepBody":"Choose the graph that represents the given equation.","answerType":"string","variabilization":{},"choices":["https://ibb.co/19XxnzZ","https://ibb.co/MSPzGr8","https://ibb.co/3zDFFXs","https://ibb.co/bXtPNZR"],"hints":{"DefaultPathway":[{"id":"a675767Ellipses8a-h1","type":"hint","dependencies":[],"title":"Write the Equation in Standard Form","text":"It can be written in standard form as $$\\\\frac{x^2}{9}+\\\\frac{y^2}{1}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses8a-h2","type":"hint","dependencies":["a675767Ellipses8a-h1"],"title":"Determine Whether the Major Axis is Horizontal or Vertical","text":"Since $$1<9$$ and $$9$$ is in the $$x^2$$ term, the major axis is horizontal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses8a-h3","type":"hint","dependencies":["a675767Ellipses8a-h2"],"title":"Find the endpoints of the major axis.","text":"The endpoints will be the x-intercept. Since $$a^2=9$$ then $$a=-3$$ or $$a=3$$. The endpoint of the major axis are $$(-3,0)$$, $$(3,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses8a-h4","type":"hint","dependencies":["a675767Ellipses8a-h3"],"title":"Find the endpoints of the minor axis.","text":"The endpoints will be the y-intercepts. Since $$b^2=1$$, then $$b=-1$$ or $$b=1$$. The endpoints of the minor axis are $$(0,1)$$, $$(0,-1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses8a-h5","type":"hint","dependencies":["a675767Ellipses8a-h4"],"title":"Elimination of Choices","text":"The graph with y-intercepts $$(0,1)$$, $$(0,-1)$$ and x-intercepts $$(-3,0)$$, $$(3,0)$$ is the answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a675767Ellipses9","title":"Graph an Ellipse with Center at the Origin","body":"Choose the graph that represents the given equation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.3 Ellipses","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a675767Ellipses9a","stepAnswer":["https://ibb.co/XJwrbPF"],"problemType":"MultipleChoice","stepTitle":"$$4x^2+25y^2=100$$","stepBody":"Choose the graph that represents the given equation.","answerType":"string","variabilization":{},"choices":["https://ibb.co/19XxnzZ","https://ibb.co/MSPzGr8","https://ibb.co/3zDFFXs","https://ibb.co/XJwrbPF"],"hints":{"DefaultPathway":[{"id":"a675767Ellipses9a-h1","type":"hint","dependencies":[],"title":"Write the Equation in Standard Form","text":"The equation is not in standard form. It can be written as standard form by divding $$100$$ both sides and get $$\\\\frac{x^2}{25}+\\\\frac{y^2}{4}=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses9a-h2","type":"hint","dependencies":["a675767Ellipses9a-h1"],"title":"Determine Whether the Major Axis is Horizontal or Vertical","text":"Since $$4<25$$ and $$25$$ is in the $$x^2$$ term, the major axis is horizontal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses9a-h3","type":"hint","dependencies":["a675767Ellipses9a-h2"],"title":"Find the endpoints of the major axis.","text":"The endpoints will be the x-intercept. Since $$b^2=25$$ then $$b=-5$$ or $$b=5$$. The endpoint of the major axis are $$(-5,0)$$, $$(5,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses9a-h4","type":"hint","dependencies":["a675767Ellipses9a-h3"],"title":"Find the endpoints of the minor axis.","text":"The endpoints will be the y-intercepts. Since $$a^2=4$$, then $$a=-2$$ or $$a=2$$. The endpoints of the minor axis are $$(0,2)$$, $$(0,-2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a675767Ellipses9a-h5","type":"hint","dependencies":["a675767Ellipses9a-h4"],"title":"Elimination of Choices","text":"The graph with y-intercepts $$(0,2)$$, $$(0,-2)$$ and x-intercepts $$(-5,0)$$, $$(5,0)$$ is the answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6b9f29radfunc1","title":"Evaluating Radical Function","body":"Evaluate the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Use Radicals in Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a6b9f29radfunc1a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"$$f(x)=\\\\sqrt{2x-1}$$, find f(5).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a6b9f29radfunc1a-h1","type":"hint","dependencies":[],"title":"Substitution in Function","text":"How can we substitute $$5$$ into the equation? We can replace $$x$$ with $$5$$ in the given function expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc1a-h2","type":"hint","dependencies":["a6b9f29radfunc1a-h1"],"title":"Evaluating the Function","text":"How can we simplify $$\\\\sqrt{2\\\\times5-1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc1a-h3","type":"hint","dependencies":["a6b9f29radfunc1a-h2"],"title":"Taking the Square Root.","text":"What is $$\\\\sqrt{9}$$, what can be multiplied by itself to get 9? $$3\\\\times3=9$$ so $$\\\\sqrt{9}=3$$. Therefore $$f(5)=3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6b9f29radfunc10","title":"Finding the Domain of a Radical Function","body":"Find the domain of the function and enter your answer in the form of [x,y], where [] is including the endpoints and () is excluding the endpoints. Also, write infinity as $$\\\\infty$$ or $$-\\\\infty$$.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Use Radicals in Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a6b9f29radfunc10a","stepAnswer":["[5/6,inf)"],"problemType":"TextBox","stepTitle":"$$f(x)=\\\\sqrt{6x-5}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[\\\\frac{5}{6},\\\\infty)$$","hints":{"DefaultPathway":[{"id":"a6b9f29radfunc10a-h1","type":"hint","dependencies":[],"title":"Radical Index","text":"What is the index of the radical?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc10a-h2","type":"hint","dependencies":["a6b9f29radfunc10a-h1"],"title":"Even radical index","text":"Since the radical index is $$2$$, it is even, so, the radicand must be greater than or equal to $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc10a-h3","type":"hint","dependencies":["a6b9f29radfunc10a-h2"],"title":"Solving the Radicand","text":"Solve $$6x-5 \\\\geq 0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc10a-h4","type":"hint","dependencies":["a6b9f29radfunc10a-h3"],"title":"Solve the Inequality","text":"Add $$5$$ to both sides of the inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc10a-h5","type":"hint","dependencies":["a6b9f29radfunc10a-h4"],"title":"Solve the Inequality","text":"Divide both sides of the inequality by $$6$$. Then we get $$x$$ is greater than equal to $$\\\\frac{5}{6}$$. We can write it in interval notation $$[\\\\frac{5}{6},\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6b9f29radfunc11","title":"Finding the Domain of a Radical Function","body":"Find the domain of the function and enter your answer in the form of [x,y], where [] is including the endpoints and () is excluding the endpoints. Also, write infinity as $$\\\\infty$$ or $$-\\\\infty$$.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Use Radicals in Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a6b9f29radfunc11a","stepAnswer":["(1,inf)"],"problemType":"TextBox","stepTitle":"$$f(x)=\\\\sqrt{\\\\frac{6}{x-1}}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(1,\\\\infty)$$","hints":{"DefaultPathway":[{"id":"a6b9f29radfunc11a-h1","type":"hint","dependencies":[],"title":"Radical Index","text":"What is the index of the radical?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc11a-h2","type":"hint","dependencies":["a6b9f29radfunc11a-h1"],"title":"Even radical index","text":"The radical index is $$2$$, it is even. However, the denominator can not equal $$0$$. We need the value of $$\\\\frac{6}{x-1}$$ be greater than $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc11a-h3","type":"hint","dependencies":["a6b9f29radfunc11a-h2"],"title":"Solving the Radicand","text":"Solve $$\\\\frac{6}{x-1}>0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc11a-h4","type":"hint","dependencies":["a6b9f29radfunc11a-h3"],"title":"Solve the Inequality","text":"Since the numerator is positive, to get $$\\\\frac{6}{x-1}>0$$, we only need to ensure the denominator is positive. Solve $$x-1>0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc11a-h5","type":"hint","dependencies":["a6b9f29radfunc11a-h4"],"title":"Find the domain","text":"To solve $$x-1>0$$, we can add $$1$$ both side which gives $$x>1$$ as the final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6b9f29radfunc12","title":"Finding the Domain of a Radical Function","body":"Find the domain of the function and enter your answer in the form of [x,y], where [] is including the endpoints and () is excluding the endpoints. Also, write infinity as $$\\\\infty$$ or $$-\\\\infty$$.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Use Radicals in Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a6b9f29radfunc12a","stepAnswer":["(-3,inf)"],"problemType":"TextBox","stepTitle":"$$f(x)=\\\\sqrt{\\\\frac{4}{x+3}}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-3,\\\\infty)$$","hints":{"DefaultPathway":[{"id":"a6b9f29radfunc12a-h1","type":"hint","dependencies":[],"title":"Radical Index","text":"What is the index of the radical?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc12a-h2","type":"hint","dependencies":["a6b9f29radfunc12a-h1"],"title":"Even radical index","text":"Since the radical index is $$2$$, it is even, so, the radicand must be greater than or equal to $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc12a-h3","type":"hint","dependencies":["a6b9f29radfunc12a-h2"],"title":"Solving the Radicand","text":"Since the numerator is positive, to get $$\\\\frac{4}{x+3}>0$$, we only need to ensure the denominator is positive. Solve $$x+3>0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc12a-h4","type":"hint","dependencies":["a6b9f29radfunc12a-h3"],"title":"Solve the Inequality","text":"Solve $$x+3>0$$. Subtract $$3$$ to both sides of the inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc12a-h5","type":"hint","dependencies":["a6b9f29radfunc12a-h4"],"title":"Find the domain","text":"We get $$x>-3$$ as the final answer. We can write $$x>3$$ in interval notation as $$(-3,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6b9f29radfunc13","title":"Finding the Domain of a Radical Function","body":"Find the domain of the function and enter your answer in the form of [x,y], where [] is including the endpoints and () is excluding the endpoints. Also, write infinity as $$\\\\infty$$ or $$-\\\\infty$$.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Use Radicals in Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a6b9f29radfunc13a","stepAnswer":["(5,inf)"],"problemType":"TextBox","stepTitle":"$$f(x)=\\\\sqrt{\\\\frac{9}{x-5}}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(5,\\\\infty)$$","hints":{"DefaultPathway":[{"id":"a6b9f29radfunc13a-h1","type":"hint","dependencies":[],"title":"Radical Index","text":"What is the index of the radical?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc13a-h2","type":"hint","dependencies":["a6b9f29radfunc13a-h1"],"title":"Even radical index","text":"Since the radical index is $$2$$, it is even, so, the radicand must be greater than or equal to $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc13a-h3","type":"hint","dependencies":["a6b9f29radfunc13a-h2"],"title":"Solving the Radicand","text":"Solve $$x-5 \\\\geq 0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc13a-h4","type":"hint","dependencies":["a6b9f29radfunc13a-h3"],"title":"Solve the Inequality","text":"Add $$5$$ to both sides of the inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc13a-h5","type":"hint","dependencies":["a6b9f29radfunc13a-h4"],"title":"Find the domain","text":"Since $$x$$ can\'t be $$5$$, or else we are dividing by $$0$$, we exclude $$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6b9f29radfunc14","title":"Evaluating Radical Function","body":"Evaluate the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Use Radicals in Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a6b9f29radfunc14a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"$$f(x)=\\\\sqrt{4x-4}$$, find f(5).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a6b9f29radfunc14a-h1","type":"hint","dependencies":[],"title":"Substitution in Function","text":"How can we substitute $$5$$ into the equation? We can replace $$x$$ with $$5$$ in the given function expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc14a-h2","type":"hint","dependencies":["a6b9f29radfunc14a-h1"],"title":"Evaluating the Function","text":"How can we simplify $$\\\\sqrt{4\\\\times5-4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc14a-h3","type":"hint","dependencies":["a6b9f29radfunc14a-h2"],"title":"Taking the Square Root.","text":"What is $$\\\\sqrt{16}$$, what can be multiplied by itself to get 16? $$4\\\\times4=16$$. Therefore $$\\\\sqrt{16}=4=f(5)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6b9f29radfunc15","title":"Evaluating Radical Function","body":"Evaluate the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Use Radicals in Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a6b9f29radfunc15a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"$$f(x)=\\\\sqrt{6x-5}$$, find f(5).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"a6b9f29radfunc15a-h1","type":"hint","dependencies":[],"title":"Substitution in Function","text":"How can we substitute $$5$$ into the equation? We can replace $$x$$ with $$5$$ in the given function expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc15a-h2","type":"hint","dependencies":["a6b9f29radfunc15a-h1"],"title":"Evaluating the Function","text":"How can we simplify $$\\\\sqrt{6\\\\times5-5}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc15a-h3","type":"hint","dependencies":["a6b9f29radfunc15a-h2"],"title":"Taking the Square Root.","text":"What is $$\\\\sqrt{25}$$, what can be multiplied by itself to get 25? $$5\\\\times5=25$$. Therefore, $$\\\\sqrt{25}=f(5)=5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6b9f29radfunc16","title":"Evaluating Radical Function","body":"Evaluate the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Use Radicals in Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a6b9f29radfunc16a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"$$f(x)=\\\\sqrt{6x+1}$$, find f(4).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"a6b9f29radfunc16a-h1","type":"hint","dependencies":[],"title":"Substitution in Function","text":"How can we substitute $$4$$ into the equation? We can replace $$x$$ with $$4$$ in the given function expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc16a-h2","type":"hint","dependencies":["a6b9f29radfunc16a-h1"],"title":"Evaluating the Function","text":"How can we simplify $$\\\\sqrt{6\\\\times4+1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc16a-h3","type":"hint","dependencies":["a6b9f29radfunc16a-h2"],"title":"Taking the Square Root.","text":"What is $$\\\\sqrt{25}$$, what can be multiplied by itself to get 25? $$5\\\\times5=25$$. Therefore, $$\\\\sqrt{25}=f(4)=5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6b9f29radfunc17","title":"Evaluating Radical Function","body":"Evaluate the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Use Radicals in Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a6b9f29radfunc17a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"$$f(x)=\\\\sqrt{3x+1}$$, find f(8).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"a6b9f29radfunc17a-h1","type":"hint","dependencies":[],"title":"Substitution in Function","text":"How can we substitute $$8$$ into the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc17a-h2","type":"hint","dependencies":["a6b9f29radfunc17a-h1"],"title":"Substitution in Function","text":"How can we substitute $$8$$ into the equation? We can replace $$x$$ with $$8$$ in the given function expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc17a-h3","type":"hint","dependencies":["a6b9f29radfunc17a-h2"],"title":"Taking the Square Root.","text":"What is $$\\\\sqrt{25}$$, what can be multiplied by itself to get 25? $$5\\\\times5=25$$. Therefore, $$\\\\sqrt{25}=f(8)=5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6b9f29radfunc18","title":"Evaluating Radical Function","body":"Evaluate the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Use Radicals in Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a6b9f29radfunc18a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"$$f(x)=\\\\sqrt{3-2x}$$, find f(1).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a6b9f29radfunc18a-h1","type":"hint","dependencies":[],"title":"Substitution in Function","text":"How can we substitute $$1$$ into the equation? We can replace $$x$$ with $$1$$ in the given function expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc18a-h2","type":"hint","dependencies":["a6b9f29radfunc18a-h1"],"title":"Evaluating the Function","text":"How can we simplify $$\\\\sqrt{3-2\\\\times1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc18a-h3","type":"hint","dependencies":["a6b9f29radfunc18a-h2"],"title":"Taking the Square Root.","text":"What is $$\\\\sqrt{1}$$, what can be multiplied by itself to get 1? $$1\\\\times1=1$$. Therefore, $$\\\\sqrt{1}=f(1)=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6b9f29radfunc19","title":"Evaluating Radical Function","body":"Evaluate the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Use Radicals in Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a6b9f29radfunc19a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"$$f(x)=\\\\sqrt{8-4x}$$, find f(1).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a6b9f29radfunc19a-h1","type":"hint","dependencies":[],"title":"Substitution in Function","text":"How can we substitute $$1$$ into the equation? We can replace $$x$$ with $$1$$ in the given function expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc19a-h2","type":"hint","dependencies":["a6b9f29radfunc19a-h1"],"title":"Evaluating the Function","text":"How can we simplify $$\\\\sqrt{8-4\\\\times1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc19a-h3","type":"hint","dependencies":["a6b9f29radfunc19a-h2"],"title":"Taking the Square Root.","text":"What is $$\\\\sqrt{4}$$, what can be multiplied by itself to get 4? $$2\\\\times2=4$$. Therefore, $$\\\\sqrt{4}=2=f(1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6b9f29radfunc2","title":"Evaluating Radical Function","body":"Evaluate the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Use Radicals in Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a6b9f29radfunc2a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"$$f(x)=\\\\sqrt{3x-2}$$, find f(6).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a6b9f29radfunc2a-h1","type":"hint","dependencies":[],"title":"Substitution in Function","text":"How can we substitute $$6$$ into the equation? We can replace $$x$$ with $$6$$ in the given function expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc2a-h2","type":"hint","dependencies":["a6b9f29radfunc2a-h1"],"title":"Evaluating the Function","text":"How can we simplify $$\\\\sqrt{3\\\\times6-2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc2a-h3","type":"hint","dependencies":["a6b9f29radfunc2a-h2"],"title":"Taking the Square Root.","text":"What is $$\\\\sqrt{16}$$, what can be multiplied by itself to get 16? $$4\\\\times4=16$$, so $$\\\\sqrt{16}=4$$. Therefore $$f(6)=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6b9f29radfunc20","title":"Evaluating Radical Function","body":"Evaluate the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Use Radicals in Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a6b9f29radfunc20a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"$$f(x)=\\\\sqrt{5x-1}$$, find f(2).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a6b9f29radfunc20a-h1","type":"hint","dependencies":[],"title":"Substitution in Function","text":"How can we substitute $$2$$ into the equation? We can replace $$x$$ with $$2$$ in the given function expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc20a-h2","type":"hint","dependencies":["a6b9f29radfunc20a-h1"],"title":"Evaluating the Function","text":"How can we simplify sqrt((5*2)-1))?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc20a-h3","type":"hint","dependencies":["a6b9f29radfunc20a-h2"],"title":"Taking the Square Root.","text":"What is $$\\\\sqrt{9}$$, what can be multiplied by itself to get 9? $$3\\\\times3=9$$. Therefore, $$\\\\sqrt{9}=3=f(2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6b9f29radfunc21","title":"Evaluating Radical Function","body":"Evaluate the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Use Radicals in Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a6b9f29radfunc21a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"$$f(x)=\\\\sqrt{4x+1}$$, find f(2).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a6b9f29radfunc21a-h1","type":"hint","dependencies":[],"title":"Substitution in Function","text":"How can we substitute $$2$$ into the equation? We can replace $$x$$ with $$2$$ in the given function expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc21a-h2","type":"hint","dependencies":["a6b9f29radfunc21a-h1"],"title":"Evaluating the Function","text":"How can we simplify sqrt((4*2)+1))?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc21a-h3","type":"hint","dependencies":["a6b9f29radfunc21a-h2"],"title":"Taking the Square Root.","text":"What is $$\\\\sqrt{9}$$, what can be multiplied by itself to get 9? $$3\\\\times3=9$$. Therefore, $$\\\\sqrt{9}=3=f(2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6b9f29radfunc22","title":"Finding the Domain of a Radical Function","body":"Find the domain of the function and enter your answer in the form of [x,y], where [] is including the endpoints and () is excluding the endpoints. Also, write infinity as $$\\\\infty$$ or $$-\\\\infty$$.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Use Radicals in Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a6b9f29radfunc22a","stepAnswer":["[1/3,inf)"],"problemType":"TextBox","stepTitle":"$$f(x)=\\\\sqrt{3x-1}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[\\\\frac{1}{3},\\\\infty)$$","hints":{"DefaultPathway":[{"id":"a6b9f29radfunc22a-h1","type":"hint","dependencies":[],"title":"Radical Index","text":"What is the index of the radical?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc22a-h2","type":"hint","dependencies":["a6b9f29radfunc22a-h1"],"title":"Even radical index","text":"Since the radical index is $$2$$, it is even, so, the radicand must be greater than or equal to $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc22a-h3","type":"hint","dependencies":["a6b9f29radfunc22a-h2"],"title":"Solving the Radicand","text":"Solve $$3x-1 \\\\geq 0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc22a-h4","type":"hint","dependencies":["a6b9f29radfunc22a-h3"],"title":"Solve the Inequality","text":"Add $$1$$ to both sides of the inequality and get $$3x$$ is greater than and equal to $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc22a-h5","type":"hint","dependencies":["a6b9f29radfunc22a-h4"],"title":"Solve the Inequality","text":"Divide both sides of the inequality by $$3$$ and get $$x$$ is greater than and equal to $$\\\\frac{1}{3}$$. We can turn it in to interval notation and get $$[\\\\frac{1}{3},\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6b9f29radfunc23","title":"Finding the Domain of a Radical Function","body":"Find the domain of the function and enter your answer in the form of [x,y], where [] is including the endpoints and () is excluding the endpoints. Also, write infinity as $$\\\\infty$$ or $$-\\\\infty$$.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Use Radicals in Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a6b9f29radfunc23a","stepAnswer":["[1/2,inf)"],"problemType":"TextBox","stepTitle":"$$f(x)=\\\\sqrt{4x-2}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[\\\\frac{1}{2},\\\\infty)$$","hints":{"DefaultPathway":[{"id":"a6b9f29radfunc23a-h1","type":"hint","dependencies":[],"title":"Radical Index","text":"What is the index of the radical?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc23a-h2","type":"hint","dependencies":["a6b9f29radfunc23a-h1"],"title":"Even radical index","text":"Since the radical index is $$2$$, it is even, so, the radicand must be greater than or equal to $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc23a-h3","type":"hint","dependencies":["a6b9f29radfunc23a-h2"],"title":"Solving the Radicand","text":"Solve $$4x-2 \\\\geq 0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc23a-h4","type":"hint","dependencies":["a6b9f29radfunc23a-h3"],"title":"Solve the Inequality","text":"Add $$2$$ to both sides of the inequality and get $$4x$$ is greater than and equal to $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc23a-h5","type":"hint","dependencies":["a6b9f29radfunc23a-h4"],"title":"Solve the Inequality","text":"Divide both sides of the inequality by $$4$$ which gives $$x$$ is greater than and equal to $$\\\\frac{1}{2}$$. We can turn it into interval notation as $$[\\\\frac{1}{2},\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6b9f29radfunc24","title":"Finding the Domain of a Radical Function","body":"Find the domain of the function and enter your answer in the form of [x,y], where [] is including the endpoints and () is excluding the endpoints. Also, write infinity as $$\\\\infty$$ or $$-\\\\infty$$.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Use Radicals in Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a6b9f29radfunc24a","stepAnswer":["(-inf,2/3]"],"problemType":"TextBox","stepTitle":"$$f(x)=\\\\sqrt{2-3x}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\frac{2}{3}]$$","hints":{"DefaultPathway":[{"id":"a6b9f29radfunc24a-h1","type":"hint","dependencies":[],"title":"Radical Index","text":"What is the index of the radical?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc24a-h2","type":"hint","dependencies":["a6b9f29radfunc24a-h1"],"title":"Even radical index","text":"Since the radical index is $$2$$, it is even, so, the radicand must be greater than or equal to $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc24a-h3","type":"hint","dependencies":["a6b9f29radfunc24a-h2"],"title":"Solving the Radicand","text":"Solve $$2-3x \\\\geq 0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc24a-h4","type":"hint","dependencies":["a6b9f29radfunc24a-h3"],"title":"Solve the Inequality","text":"Subtract $$2$$ from both sides of the inequality which gives $$-3x$$ is greater than and equal to $$-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc24a-h5","type":"hint","dependencies":["a6b9f29radfunc24a-h4"],"title":"Solve the Inequality","text":"Divide both sides of the inequality by $$-3$$ which gives $$x$$ is less than and equal to $$\\\\frac{2}{3}$$. We can turn it into interval notation as $$(-\\\\infty,\\\\frac{2}{3}]$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6b9f29radfunc25","title":"Finding the Domain of a Radical Function","body":"Find the domain of the function and enter your answer in the form of [x,y], where [] is including the endpoints and () is excluding the endpoints. Also, write infinity as $$\\\\infty$$ or $$-\\\\infty$$.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Use Radicals in Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a6b9f29radfunc25a","stepAnswer":["(-inf,8]"],"problemType":"TextBox","stepTitle":"$$f(x)=\\\\sqrt{8-x}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,8]$$","hints":{"DefaultPathway":[{"id":"a6b9f29radfunc25a-h1","type":"hint","dependencies":[],"title":"Radical Index","text":"What is the index of the radical?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc25a-h2","type":"hint","dependencies":["a6b9f29radfunc25a-h1"],"title":"Even radical index","text":"Since the radical index is $$2$$, it is even, so, the radicand must be greater than or equal to $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc25a-h3","type":"hint","dependencies":["a6b9f29radfunc25a-h2"],"title":"Solving the Radicand","text":"Solve $$8-x \\\\geq 0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc25a-h4","type":"hint","dependencies":["a6b9f29radfunc25a-h3"],"title":"Solve the Inequality","text":"Subtract $$8$$ from both sides of the inequality which gives $$-x$$ is less than and equal to $$-8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc25a-h5","type":"hint","dependencies":["a6b9f29radfunc25a-h4"],"title":"Solve the Inequality","text":"Divide both sides of the inequality by -1and get $$x$$ is less than equal to $$8$$. We can turn it into interval notation as $$(-\\\\infty,8]$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6b9f29radfunc26","title":"Finding the Domain of a Radical Function","body":"Find the domain of the function and enter your answer in the form of [x,y], where [] is including the endpoints and () is excluding the endpoints. Also, write infinity as $$\\\\infty$$ or $$-\\\\infty$$.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Use Radicals in Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a6b9f29radfunc26a","stepAnswer":["(2,inf)"],"problemType":"TextBox","stepTitle":"$$f(x)=\\\\sqrt{\\\\frac{5}{x-2}}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(2,\\\\infty)$$","hints":{"DefaultPathway":[{"id":"a6b9f29radfunc26a-h1","type":"hint","dependencies":[],"title":"Radical Index","text":"What is the index of the radical?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc26a-h2","type":"hint","dependencies":["a6b9f29radfunc26a-h1"],"title":"Even radical index","text":"Since the radical index is $$2$$, it is even, so, the radicand must be greater than or equal to $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc26a-h3","type":"hint","dependencies":["a6b9f29radfunc26a-h2"],"title":"Solving the Radicand","text":"We need $$\\\\frac{5}{x-2} \\\\geq 0$$. The denominator can not be zero. Since the numerator is positive and we only need to ensure the denominator is positive. Solve $$x-2>0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc26a-h4","type":"hint","dependencies":["a6b9f29radfunc26a-h3"],"title":"Solve the Inequality","text":"Add $$2$$ to both sides of the inequality and get $$x>2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc26a-h5","type":"hint","dependencies":["a6b9f29radfunc26a-h4"],"title":"Find the domain","text":"We can turn $$x>2$$ into interval notation as $$(2,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6b9f29radfunc27","title":"Finding the Domain of a Radical Function","body":"Find the domain of the function and enter your answer in the form of [x,y], where [] is including the endpoints and () is excluding the endpoints. Also, write infinity as $$\\\\infty$$ or $$-\\\\infty$$.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Use Radicals in Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a6b9f29radfunc27a","stepAnswer":["(-3,inf)"],"problemType":"TextBox","stepTitle":"$$f(x)=\\\\sqrt{\\\\frac{6}{x+3}}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-3,\\\\infty)$$","hints":{"DefaultPathway":[{"id":"a6b9f29radfunc27a-h1","type":"hint","dependencies":[],"title":"Radical Index","text":"What is the index of the radical?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc27a-h2","type":"hint","dependencies":["a6b9f29radfunc27a-h1"],"title":"Even radical index","text":"Since the radical index is $$2$$, it is even, so, the radicand must be greater than or equal to $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc27a-h3","type":"hint","dependencies":["a6b9f29radfunc27a-h2"],"title":"Solving the Radicand","text":"We need $$\\\\frac{6}{x+3} \\\\geq 0$$. The denominator can not be zero. Since the numerator is positive and we only need to ensure the denominator is positive. Solve $$x+3>0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc27a-h4","type":"hint","dependencies":["a6b9f29radfunc27a-h3"],"title":"Solve the Inequality","text":"Subtract $$3$$ to both sides of the inequality and get $$x>-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc27a-h5","type":"hint","dependencies":["a6b9f29radfunc27a-h4"],"title":"Find the domain","text":"We can turn $$x>-3$$ into interval notation as $$(-3,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6b9f29radfunc28","title":"Finding the Domain of a Radical Function","body":"Find the domain of the function and enter your answer in the form of [x,y], where [] is including the endpoints and () is excluding the endpoints. Also, write infinity as $$\\\\infty$$ or $$-\\\\infty$$.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Use Radicals in Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a6b9f29radfunc28a","stepAnswer":["[-1,inf)"],"problemType":"TextBox","stepTitle":"$$f(x)=\\\\sqrt{x+1}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[-1,\\\\infty)$$","hints":{"DefaultPathway":[{"id":"a6b9f29radfunc28a-h1","type":"hint","dependencies":[],"title":"Radical Index","text":"What is the index of the radical?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc28a-h2","type":"hint","dependencies":["a6b9f29radfunc28a-h1"],"title":"Even radical index","text":"Since the radical index is $$2$$, it is even, so, the radicand must be greater than or equal to $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc28a-h3","type":"hint","dependencies":["a6b9f29radfunc28a-h2"],"title":"Solving the Radicand","text":"Solve $$x+1 \\\\geq 0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc28a-h4","type":"hint","dependencies":["a6b9f29radfunc28a-h3"],"title":"Solve the Inequality","text":"Subtract $$1$$ from both sides of the inequality which gives $$x$$ is greater than and equal to $$-1$$. We can turn it into interval notation as $$[-1,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6b9f29radfunc29","title":"Finding the Domain of a Radical Function","body":"Find the domain of the function and enter your answer in the form of [x,y], where [] is including the endpoints and () is excluding the endpoints. Also, write infinity as $$\\\\infty$$ or $$-\\\\infty$$.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Use Radicals in Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a6b9f29radfunc29a","stepAnswer":["[1,inf)"],"problemType":"TextBox","stepTitle":"$$f(x)=\\\\sqrt{x-1}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[1,\\\\infty)$$","hints":{"DefaultPathway":[{"id":"a6b9f29radfunc29a-h1","type":"hint","dependencies":[],"title":"Radical Index","text":"What is the index of the radical?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc29a-h2","type":"hint","dependencies":["a6b9f29radfunc29a-h1"],"title":"Even radical index","text":"Since the radical index is $$2$$, it is even, so, the radicand must be greater than or equal to $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc29a-h3","type":"hint","dependencies":["a6b9f29radfunc29a-h2"],"title":"Solving the Radicand","text":"Solve $$x-1 \\\\geq 0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc29a-h4","type":"hint","dependencies":["a6b9f29radfunc29a-h3"],"title":"Solve the Inequality","text":"Add $$1$$ to both sides of the inequality which gives $$x$$ is greater than and equal to $$1$$. We can turn it into interval notation as $$[1,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6b9f29radfunc3","title":"Evaluating Radical Function","body":"Evaluate the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Use Radicals in Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a6b9f29radfunc3a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"$$f(x)=\\\\sqrt{5x+5}$$, find f(4).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"a6b9f29radfunc3a-h1","type":"hint","dependencies":[],"title":"Substitution in Function","text":"How can we substitute $$4$$ into the equation? We can replace $$x$$ with $$4$$ in the given function expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc3a-h2","type":"hint","dependencies":["a6b9f29radfunc3a-h1"],"title":"Evaluating the Function","text":"How can we simplify $$\\\\sqrt{5\\\\times4+5}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc3a-h3","type":"hint","dependencies":["a6b9f29radfunc3a-h2"],"title":"Taking the Square Root.","text":"What is $$\\\\sqrt{25}$$? What can be multiplied by itself to get 25? $$5\\\\times5=25$$, so $$\\\\sqrt{25}=5$$. Therefore $$f(4)=5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6b9f29radfunc4","title":"Evaluating Radical Function","body":"Evaluate the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Use Radicals in Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a6b9f29radfunc4a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"f(x)=sqrt(3, $$x-6)$$, find f(14).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a6b9f29radfunc4a-h1","type":"hint","dependencies":[],"title":"Substitution in Function","text":"How can we substitute $$14$$ into the equation? We can replace $$x$$ with $$14$$ in the given function expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc4a-h2","type":"hint","dependencies":["a6b9f29radfunc4a-h1"],"title":"Evaluating the Function","text":"How can we simplify $$\\\\sqrt[3]{14-6}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc4a-h3","type":"hint","dependencies":["a6b9f29radfunc4a-h2"],"title":"Taking the Cube Root.","text":"What is $$\\\\sqrt[3]{8}$$, what can be multiplied by itself $$3$$ times to get 8? $$2\\\\times2\\\\times2=8$$, so $$\\\\sqrt[3]{8}=2$$. Therefore $$f(14)=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6b9f29radfunc5","title":"Evaluating Radical Function","body":"Evaluate the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Use Radicals in Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a6b9f29radfunc5a","stepAnswer":["$$-1$$"],"problemType":"TextBox","stepTitle":"f(x)=sqrt(3, (3*x)-4), find f(1).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1$$","hints":{"DefaultPathway":[{"id":"a6b9f29radfunc5a-h1","type":"hint","dependencies":[],"title":"Substitution in Function","text":"How can we substitute $$1$$ into the equation? We can replace $$x$$ with $$1$$ in the given function expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc5a-h2","type":"hint","dependencies":["a6b9f29radfunc5a-h1"],"title":"Evaluating the Function","text":"How can we simplify $$\\\\sqrt[3]{-1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc5a-h3","type":"hint","dependencies":["a6b9f29radfunc5a-h2"],"title":"Taking the Cube Root.","text":"The cube root of a negative number is negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc5a-h4","type":"hint","dependencies":["a6b9f29radfunc5a-h3"],"title":"Taking the Cube Root.","text":"What is $$\\\\sqrt[3]{-1}$$, what can be multiplied by itself $$3$$ times to get -1? $$\\\\left(-1\\\\right) \\\\left(-1\\\\right) \\\\left(-1\\\\right)=-1$$, so $$\\\\sqrt[3]{-1}=-1$$. Therefore $$f(1)=-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6b9f29radfunc6","title":"Evaluating Radical Function","body":"Evaluate the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Use Radicals in Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a6b9f29radfunc6a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"f(x)=sqrt(4, (5*x)-4), find f(4).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a6b9f29radfunc6a-h1","type":"hint","dependencies":[],"title":"Substitution in Function","text":"How can we substitute $$4$$ into the equation? We can replace $$x$$ with $$4$$ in the given function expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc6a-h2","type":"hint","dependencies":["a6b9f29radfunc6a-h1"],"title":"Evaluating the Function","text":"How can we simplify sqrt(4, (5*4)-4)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc6a-h3","type":"hint","dependencies":["a6b9f29radfunc6a-h2"],"title":"Taking the Fourth Root.","text":"What is $$\\\\sqrt[4]{16}$$, what can be multiplied by itself $$4$$ times to get 16? $$2^4=16$$. Therefore $$\\\\sqrt[4]{16}=2=f(4)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6b9f29radfunc7","title":"Evaluating Radical Function","body":"Evaluate the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Use Radicals in Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a6b9f29radfunc7a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"f(x)=sqrt(4, (3*x)+4), find f(4).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a6b9f29radfunc7a-h1","type":"hint","dependencies":[],"title":"Substitution in Function","text":"How can we substitute $$4$$ into the equation? We can replace $$x$$ with $$4$$ in the given function expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc7a-h2","type":"hint","dependencies":["a6b9f29radfunc7a-h1"],"title":"Evaluating the Function","text":"How can we simplify sqrt(4, (3*4)+4)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc7a-h3","type":"hint","dependencies":["a6b9f29radfunc7a-h2"],"title":"Taking the Fourth Root.","text":"What is $$\\\\sqrt[4]{16}$$, what can be multiplied by itself $$4$$ times to get 16? $$2^4=16$$. Therefore, $$\\\\sqrt[4]{16}=2=f(4)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6b9f29radfunc8","title":"Evaluating Radical Function","body":"Evaluate the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Use Radicals in Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a6b9f29radfunc8a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"f(x)=sqrt(4, (5*x)+1), find f(16).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a6b9f29radfunc8a-h1","type":"hint","dependencies":[],"title":"Substitution in Function","text":"How can we substitute $$16$$ into the equation? We can replace $$x$$ with $$16$$ in the given function expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc8a-h2","type":"hint","dependencies":["a6b9f29radfunc8a-h1"],"title":"Evaluating the Function","text":"How can we simplify sqrt(4, (5*16)+1)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc8a-h3","type":"hint","dependencies":["a6b9f29radfunc8a-h2"],"title":"Taking the Fourth Root.","text":"What is $$\\\\sqrt[4]{81}$$, what can be multiplied by itself $$4$$ times to get 81? $$3^4=81$$, so $$\\\\sqrt[4]{81}=3=f(16)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6b9f29radfunc9","title":"Finding the Domain of a Radical Function","body":"Find the domain of the function and enter your answer in the form of [x,y], where [] is including the endpoints and () is excluding the endpoints. Also, write infinity as $$\\\\infty$$ or $$-\\\\infty$$.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.7 Use Radicals in Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a6b9f29radfunc9a","stepAnswer":["[4/3,inf)"],"problemType":"TextBox","stepTitle":"$$f(x)=\\\\sqrt{3x-4}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[\\\\frac{4}{3},\\\\infty)$$","hints":{"DefaultPathway":[{"id":"a6b9f29radfunc9a-h1","type":"hint","dependencies":[],"title":"Radical Index","text":"What is the index of the radical?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc9a-h2","type":"hint","dependencies":["a6b9f29radfunc9a-h1"],"title":"Even radical index","text":"Since the radical index is $$2$$, it is even, so the radicand must be greater than or equal to $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc9a-h3","type":"hint","dependencies":["a6b9f29radfunc9a-h2"],"title":"Solving the Radicand","text":"Solve $$3x-4 \\\\geq 0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc9a-h4","type":"hint","dependencies":["a6b9f29radfunc9a-h3"],"title":"Solve the Inequality","text":"Add $$4$$ to both sides of the inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6b9f29radfunc9a-h5","type":"hint","dependencies":["a6b9f29radfunc9a-h4"],"title":"Solve the Inequality","text":"Divide both sides of the inequality by $$3$$. Then we get $$x$$ is greater than equal to $$\\\\frac{4}{3}$$. We can write it in interval notation $$[\\\\frac{4}{3},\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d0007exp1","title":"Using the Product Rule","body":"Write each of the following products with a single base.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Exponents and Scientific Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"a6d0007exp1a","stepAnswer":["$$t^8$$"],"problemType":"MultipleChoice","stepTitle":"$$t^5 t^3$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$t^8$$","choices":["$$t^2$$","$$t^8$$","$$t^{15}$$","$$t^{10}$$"],"hints":{"DefaultPathway":[{"id":"a6d0007exp1a-h1","type":"hint","dependencies":[],"title":"Product Rule of Exponents","text":"For any real number a and natural numbers $$m$$ and $$n$$, the product rule of exponents states that $$a^m a^n=a \\\\left(m+n\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp1a-h2","type":"hint","dependencies":["a6d0007exp1a-h1"],"title":"Product Rule of Exponents","text":"Use the product rule to simplify the expression: $$t^5 t^3=t^{5+3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6d0007exp1b","stepAnswer":["$${\\\\left(-3\\\\right)}^6$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(-3\\\\right)}^5 \\\\left(-3\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$${\\\\left(-3\\\\right)}^6$$","choices":["$${\\\\left(-3\\\\right)}^5$$","$${\\\\left(-3\\\\right)}^6$$","$${\\\\left(-3\\\\right)}^4$$","$${\\\\left(-3\\\\right)}^{-5}$$"],"hints":{"DefaultPathway":[{"id":"a6d0007exp1b-h1","type":"hint","dependencies":[],"title":"Product Rule of Exponents","text":"For any real number a and natural numbers $$m$$ and $$n$$, the product rule of exponents states that $$a^m a^n=a \\\\left(m+n\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp1b-h2","type":"hint","dependencies":["a6d0007exp1b-h1"],"title":"Product Rule of Exponents","text":"Use the product rule to simplify the expression: $${\\\\left(-3\\\\right)}^5 \\\\left(-3\\\\right)={\\\\left(-3\\\\right)}^5 {\\\\left(-3\\\\right)}^1={\\\\left(-3\\\\right)}^{5+1}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6d0007exp1c","stepAnswer":["$$x^{10}$$"],"problemType":"MultipleChoice","stepTitle":"$$x^2 x^5 x^3$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x^{10}$$","choices":["$$x^{10}$$","$$x^{30}$$","$$x^{13}$$","$$x^{17}$$"],"hints":{"DefaultPathway":[{"id":"a6d0007exp1c-h1","type":"hint","dependencies":[],"title":"Product Rule of Exponents","text":"For any real number a and natural numbers $$m$$ and $$n$$, the product rule of exponents states that $$a^m a^n=a \\\\left(m+n\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp1c-h2","type":"hint","dependencies":["a6d0007exp1c-h1"],"title":"Product Rule of Exponents","text":"Use the product rule to simplify the expression: $$x^2 x^5 x^3=x^{2+5+3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d0007exp10","title":"The Quotient Rule of Exponents","body":"Simplify each expression and write the answer with positive exponents only.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Exponents and Scientific Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"a6d0007exp10a","stepAnswer":["$$\\\\frac{1}{{\\\\left(-3t\\\\right)}^6}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{{\\\\left(-3t\\\\right)}^2}{{\\\\left(-3t\\\\right)}^8}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{{\\\\left(-3t\\\\right)}^6}$$","hints":{"DefaultPathway":[{"id":"a6d0007exp10a-h1","type":"hint","dependencies":[],"title":"Quotient Rule of Exponents","text":"For any real number a and natural numbers $$m$$ and $$n$$, such that $$m>n$$, the quotient rule of exponents states that $$\\\\frac{a^m}{a^n}=a^{m-n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp10a-h2","type":"hint","dependencies":["a6d0007exp10a-h1"],"title":"Quotient Rule of Exponents","text":"Use the quotient rule to simplify the expression: $$\\\\frac{{\\\\left(-3t\\\\right)}^2}{{\\\\left(-3t\\\\right)}^8}={\\\\left(-3t\\\\right)}^{2-8}={\\\\left(-3t\\\\right)}^{\\\\left(-6\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp10a-h3","type":"hint","dependencies":["a6d0007exp10a-h2"],"title":"Negative Exponent Rule","text":"For any nonzero real number a and natural number $$n$$, the negative rule of exponents states that $$a -n=\\\\frac{1}{a^n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6d0007exp10b","stepAnswer":["$$\\\\frac{1}{f^3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{f^{47}}{f^{49} f}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{f^3}$$","hints":{"DefaultPathway":[{"id":"a6d0007exp10b-h1","type":"hint","dependencies":[],"title":"Product Rule of Exponents","text":"For any real number a and natural numbers $$m$$ and $$n$$, the product rule of exponents states that $$a^m a^n=a \\\\left(m+n\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp10b-h2","type":"hint","dependencies":["a6d0007exp10b-h1"],"title":"Product Rule of Exponents","text":"Use the product rule to simplify the expression: $$\\\\frac{f^{47}}{f^{49} f}=\\\\frac{f^{47}}{f^{49} f^1}=\\\\frac{f^{47}}{f^{50}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp10b-h3","type":"hint","dependencies":["a6d0007exp10b-h2"],"title":"Quotient Rule of Exponents","text":"For any real number a and natural numbers $$m$$ and $$n$$, such that $$m>n$$, the quotient rule of exponents states that $$\\\\frac{a^m}{a^n}=a^{m-n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp10b-h4","type":"hint","dependencies":["a6d0007exp10b-h3"],"title":"Quotient Rule of Exponents","text":"Use the quotient rule to simplify the expression: $$\\\\frac{f^{47}}{f^{50}}=f^{47-50}=f^{\\\\left(-3\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp10b-h5","type":"hint","dependencies":["a6d0007exp10b-h4"],"title":"Negative Exponent Rule","text":"For any nonzero real number a and natural number $$n$$, the negative rule of exponents states that $$a -n=\\\\frac{1}{a^n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6d0007exp10c","stepAnswer":["$$\\\\frac{2}{5k^3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2k^4}{5k^7}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2}{5k^3}$$","hints":{"DefaultPathway":[{"id":"a6d0007exp10c-h1","type":"hint","dependencies":[],"title":"Quotient Rule of Exponents","text":"For any real number a and natural numbers $$m$$ and $$n$$, such that $$m>n$$, the quotient rule of exponents states that $$\\\\frac{a^m}{a^n}=a^{m-n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp10c-h2","type":"hint","dependencies":["a6d0007exp10c-h1"],"title":"Quotient Rule of Exponents","text":"Use the quotient rule to simplify the expression: $$\\\\frac{2k^4}{5} k^7=$$ $$\\\\frac{2}{5} k^{4-7}=\\\\frac{2}{5} k^{\\\\left(-3\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp10c-h3","type":"hint","dependencies":["a6d0007exp10c-h2"],"title":"Negative Exponent Rule","text":"For any nonzero real number a and natural number $$n$$, the negative rule of exponents states that $$a -n=\\\\frac{1}{a^n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d0007exp11","title":"Using the Product and Quotient Rules","body":"Write each of the following products with a single base. Do not simplify further. Write answers with positive exponents.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Exponents and Scientific Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"a6d0007exp11a","stepAnswer":["$$\\\\frac{1}{b^6}$$"],"problemType":"TextBox","stepTitle":"$$b^2 b^{-8}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{b^6}$$","hints":{"DefaultPathway":[{"id":"a6d0007exp11a-h1","type":"hint","dependencies":[],"title":"Product Rule of Exponents","text":"For any real number a and natural numbers $$m$$ and $$n$$, the product rule of exponents states that $$a^m a^n=a^{m+n}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp11a-h2","type":"hint","dependencies":["a6d0007exp11a-h1"],"title":"Product Rule of Exponents","text":"Use the product rule to simplify the expression: $$b^2 b^{-8}=b^{2-8}=b^{\\\\left(-6\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp11a-h3","type":"hint","dependencies":["a6d0007exp11a-h2"],"title":"Negative Exponent Rule","text":"For any nonzero real number a and natural number $$n$$, the negative rule of exponents states that $$a -n=\\\\frac{1}{a^n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6d0007exp11b","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(-x\\\\right)}^5 {\\\\left(-x\\\\right)}^{-5}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a6d0007exp11b-h1","type":"hint","dependencies":[],"title":"Product Rule of Exponents","text":"For any real number a and natural numbers $$m$$ and product rule of exponents states that $$a^m a^n=a \\\\left(m+n\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp11b-h2","type":"hint","dependencies":["a6d0007exp11b-h1"],"title":"Product Rule of Exponents","text":"Use the product rule to simplify the expression: $${\\\\left(-x\\\\right)}^5 {\\\\left(-x\\\\right)}^{-5}={\\\\left(-x\\\\right)}^{5-5}={\\\\left(-x\\\\right)}^0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp11b-h3","type":"hint","dependencies":["a6d0007exp11b-h2"],"title":"Zero Exponent Rule","text":"For any nonzero real number a, the zero exponent rule of exponents states that $$a^0=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6d0007exp11c","stepAnswer":["$$\\\\frac{1}{{\\\\left(-7z\\\\right)}^4}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{-7z}{{\\\\left(-7z\\\\right)}^5}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{{\\\\left(-7z\\\\right)}^4}$$","hints":{"DefaultPathway":[{"id":"a6d0007exp11c-h1","type":"hint","dependencies":[],"title":"Quotient Rule of Exponents","text":"For any real number a and natural numbers $$m$$ and $$n$$, such that $$m>n$$, the quotient rule of exponents states that $$\\\\frac{a^m}{a^n}=a^{m-n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp11c-h2","type":"hint","dependencies":["a6d0007exp11c-h1"],"title":"Quotient Rule of Exponents","text":"Use the quotient rule to simplify the expression: $$\\\\frac{-7z}{{\\\\left(-7z\\\\right)}^5}=\\\\frac{-7z^1}{{\\\\left(-7z\\\\right)}^5}=-7z^{1-5}=-7z^{\\\\left(-4\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp11c-h3","type":"hint","dependencies":["a6d0007exp11c-h2"],"title":"Negative Exponent Rule","text":"For any nonzero real number a and natural number $$n$$, the negative rule of exponents states that $$a -n=\\\\frac{1}{a^n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d0007exp12","title":"The Product and Quotient Rules of Exponents","body":"Write each of the following products with a single base. Do not simplify further. Write answers with positive exponents.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Exponents and Scientific Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"a6d0007exp12a","stepAnswer":["$$\\\\frac{1}{t^5}$$"],"problemType":"TextBox","stepTitle":"$$t^{-11} t^6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{t^5}$$","hints":{"DefaultPathway":[{"id":"a6d0007exp12a-h1","type":"hint","dependencies":[],"title":"Product Rule of Exponents","text":"For any real number a and natural numbers $$m$$ and $$n$$, the product rule of exponents states that $$a^m a^n=a \\\\left(m+n\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp12a-h2","type":"hint","dependencies":["a6d0007exp12a-h1"],"title":"Product Rule of Exponents","text":"Use the product rule to simplify the expression: $$b^2 b^{-8}=b^{2-8}=b^{\\\\left(-6\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp12a-h3","type":"hint","dependencies":["a6d0007exp12a-h2"],"title":"Negative Exponent Rule","text":"For any nonzero real number a and natural number $$n$$, the negative rule of exponents states that $$a -n=\\\\frac{1}{a^n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6d0007exp12b","stepAnswer":["$$\\\\frac{1}{25}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{{25}^{12}}{{25}^{13}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{25}$$","hints":{"DefaultPathway":[{"id":"a6d0007exp12b-h1","type":"hint","dependencies":[],"title":"Quotient Rule of Exponents","text":"For any real number a and natural numbers $$m$$ and $$n$$, such that $$m>n$$, the quotient rule of exponents states that $$\\\\frac{a^m}{a^n}=a^{m-n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp12b-h2","type":"hint","dependencies":["a6d0007exp12b-h1"],"title":"Quotient Rule of Exponents","text":"Use the quotient rule to simplify the expression: $$\\\\frac{{25}^{12}}{{25}^{13}}={25}^{12-13}={25}^{\\\\left(-1\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp12b-h3","type":"hint","dependencies":["a6d0007exp12b-h2"],"title":"Negative Exponent Rule","text":"For any nonzero real number a and natural number $$n$$, the negative rule of exponents states that $$a -n=\\\\frac{1}{a^n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d0007exp13","title":"Using the Power of a Product Rule","body":"Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Exponents and Scientific Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"a6d0007exp13a","stepAnswer":["$$a^3 b^6$$"],"problemType":"TextBox","stepTitle":"$${\\\\left({ab}^2\\\\right)}^3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$a^3 b^6$$","hints":{"DefaultPathway":[{"id":"a6d0007exp13a-h1","type":"hint","dependencies":[],"title":"Power of a Product Rule","text":"For any real numbers a and $$b$$ and any integer $$n$$, the power of a product rule of exponents states that $${ab}^n=a^n b^n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp13a-h2","type":"hint","dependencies":["a6d0007exp13a-h1"],"title":"Power of a Product Rule","text":"Use the power of a product rule to simplify the expression: $${\\\\left({ab}^2\\\\right)}^3=$$ $$a^3 {\\\\left(b^2\\\\right)}^3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp13a-h3","type":"hint","dependencies":["a6d0007exp13a-h2"],"title":"Power Rule of Exponents","text":"For any real number a and positive integers $$m$$ and $$n$$, the power rule of exponents states that $${\\\\left(a^m\\\\right)}^n=a^{m n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp13a-h4","type":"hint","dependencies":["a6d0007exp13a-h3"],"title":"Power Rule of Exponents","text":"Use the power rule to simplify the expression: $$a^3 {\\\\left(b^2\\\\right)}^3=a^{1\\\\times3} b^{2\\\\times3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6d0007exp13b","stepAnswer":["$$32768t^{15}$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(2t\\\\right)}^{15}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$32768t^{15}$$","hints":{"DefaultPathway":[{"id":"a6d0007exp13b-h1","type":"hint","dependencies":[],"title":"Power of a Product Rule","text":"For any real numbers a and $$b$$ and any integer $$n$$, the power of a product rule of exponents states that $${ab}^n=a^n b^n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp13b-h2","type":"hint","dependencies":["a6d0007exp13b-h1"],"title":"Power of a Product Rule","text":"Use the power of a product rule to simplify the expression: $${\\\\left(2t\\\\right)}^{15}=2^{15} t^{15}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6d0007exp13c","stepAnswer":["$$-8w^9$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(-2w^3\\\\right)}^3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-8w^9$$","hints":{"DefaultPathway":[{"id":"a6d0007exp13c-h1","type":"hint","dependencies":[],"title":"Power of a Product Rule","text":"For any real numbers a and $$b$$ and any integer $$n$$, the power of a product rule of exponents states that $${\\\\left(a b\\\\right)}^n=a^n b^n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp13c-h2","type":"hint","dependencies":["a6d0007exp13c-h1"],"title":"Power of a Product Rule","text":"Use the power of a product rule to simplify the expression: $${\\\\left(-2w^3\\\\right)}^3={\\\\left(-2\\\\right)}^3 {\\\\left(w^3\\\\right)}^3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp13c-h3","type":"hint","dependencies":["a6d0007exp13c-h2"],"title":"Power Rule of Exponents","text":"For any real number a and positive integers $$m$$ and $$n$$, the power rule of exponents states that $${\\\\left(a^m\\\\right)}^n=a^{m n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp13c-h4","type":"hint","dependencies":["a6d0007exp13c-h3"],"title":"Power Rule of Exponents","text":"Use the power rule to simplify the expression: $$a^3 {\\\\left(b^2\\\\right)}^3={\\\\left(-2\\\\right)}^3 {\\\\left(w^3\\\\right)}^3=-8w^{3\\\\times3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d0007exp14","title":"The Power of a Product Rule of Exponents","body":"Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Exponents and Scientific Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"a6d0007exp14a","stepAnswer":["$$g^{10} h^{15}$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(g^2 h^3\\\\right)}^5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$g^{10} h^{15}$$","hints":{"DefaultPathway":[{"id":"a6d0007exp14a-h1","type":"hint","dependencies":[],"title":"Power of a Product Rule","text":"For any real numbers a and $$b$$ and any integer $$n$$, the power of a product rule of exponents states that $${ab}^n=a^n b^n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp14a-h2","type":"hint","dependencies":["a6d0007exp14a-h1"],"title":"Power of a Product Rule","text":"Use the power of a product rule to simplify the expression: $${\\\\left(g^2 h^3\\\\right)}^5={\\\\left(g^2\\\\right)}^5 {\\\\left(h^3\\\\right)}^5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp14a-h3","type":"hint","dependencies":["a6d0007exp14a-h2"],"title":"Power Rule of Exponents","text":"For any real number a and positive integers $$m$$ and $$n$$, the power rule of exponents states that $${\\\\left(a^m\\\\right)}^n=a^{m n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp14a-h4","type":"hint","dependencies":["a6d0007exp14a-h3"],"title":"Power Rule of Exponents","text":"Use the power rule to simplify the expression: $${\\\\left(g^2\\\\right)}^5 {\\\\left(h^3\\\\right)}^5=g^{2\\\\times5} h^{3\\\\times5}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6d0007exp14b","stepAnswer":["$$125t^3$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(5t\\\\right)}^3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$125t^3$$","hints":{"DefaultPathway":[{"id":"a6d0007exp14b-h1","type":"hint","dependencies":[],"title":"Power of a Product Rule","text":"For any real numbers a and $$b$$ and any integer $$n$$, the power of a product rule of exponents states that $${ab}^n=a^n b^n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp14b-h2","type":"hint","dependencies":["a6d0007exp14b-h1"],"title":"Power of a Product Rule","text":"Use the power of a product rule to simplify the expression: $${\\\\left(5t\\\\right)}^3=5^3 t^3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6d0007exp14c","stepAnswer":["$$-27y^{15}$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(-3y^5\\\\right)}^3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-27y^{15}$$","hints":{"DefaultPathway":[{"id":"a6d0007exp14c-h1","type":"hint","dependencies":[],"title":"Power of a Product Rule","text":"For any real numbers a and $$b$$ and any integer $$n$$, the power of a product rule of exponents states that $${ab}^n=a^n b^n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp14c-h2","type":"hint","dependencies":["a6d0007exp14c-h1"],"title":"Power of a Product Rule","text":"Use the power of a product rule to simplify the expression: $${\\\\left(-3y^5\\\\right)}^3={\\\\left(-3\\\\right)}^3 {\\\\left(y^5\\\\right)}^3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp14c-h3","type":"hint","dependencies":["a6d0007exp14c-h2"],"title":"Power Rule of Exponents","text":"For any real number a and positive integers $$m$$ and $$n$$, the power rule of exponents states that $${\\\\left(a^m\\\\right)}^n=a^{m n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp14c-h4","type":"hint","dependencies":["a6d0007exp14c-h3"],"title":"Power Rule of Exponents","text":"Use the power rule to simplify the expression: $${\\\\left(g^2\\\\right)}^5 {\\\\left(h^3\\\\right)}^5=g^{2\\\\times5} h^{3\\\\times5}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d0007exp15","title":"Using the Power of a Quotient Rule of Exponents","body":"Simplify each of the following quotients as much as possible using the power of a quotient rule. Write answers with positive exponents.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Exponents and Scientific Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"a6d0007exp15a","stepAnswer":["$$\\\\frac{64}{z^{33}}$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(\\\\frac{4}{z^{11}}\\\\right)}^3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{64}{z^{33}}$$","hints":{"DefaultPathway":[{"id":"a6d0007exp15a-h1","type":"hint","dependencies":[],"title":"Power of a Quotient Rule","text":"For any real numbers a and $$b$$ and any integer $$n$$, the power of a quotient rule of exponents states that $${\\\\left(\\\\frac{a}{b}\\\\right)}^n=\\\\frac{a^n}{b^n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp15a-h2","type":"hint","dependencies":["a6d0007exp15a-h1"],"title":"Power of a Quotient Rule","text":"Use the power of a quotient rule to simplify the expression: $${\\\\left(\\\\frac{4}{z^{11}}\\\\right)}^3=\\\\frac{4^3}{{\\\\left(z^{11}\\\\right)}^3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp15a-h3","type":"hint","dependencies":["a6d0007exp15a-h2"],"title":"Power Rule of Exponents","text":"For any real number a and positive integers $$m$$ and $$n$$, the power rule of exponents states that $${\\\\left(a^m\\\\right)}^n=a^{m n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp15a-h4","type":"hint","dependencies":["a6d0007exp15a-h3"],"title":"Power Rule of Exponents","text":"Use the power rule to simplify the expression: $$\\\\frac{4^3}{{\\\\left(z^{11}\\\\right)}^3}=\\\\frac{64}{z^{11\\\\times3}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6d0007exp15b","stepAnswer":["$$\\\\frac{p^6}{q^{18}}$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(\\\\frac{p}{q^3}\\\\right)}^6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{p^6}{q^{18}}$$","hints":{"DefaultPathway":[{"id":"a6d0007exp15b-h1","type":"hint","dependencies":[],"title":"Power of a Quotient Rule","text":"For any real numbers a and $$b$$ and any integer $$n$$, the power of a quotient rule of exponents states that $${\\\\left(\\\\frac{a}{b}\\\\right)}^n=\\\\frac{a^n}{b^n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp15b-h2","type":"hint","dependencies":["a6d0007exp15b-h1"],"title":"Power of a Quotient Rule","text":"Use the power of a quotient rule to simplify the expression: $${\\\\left(\\\\frac{p}{q^3}\\\\right)}^6=\\\\frac{p^6}{{\\\\left(q^3\\\\right)}^6}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp15b-h3","type":"hint","dependencies":["a6d0007exp15b-h2"],"title":"Power Rule of Exponents","text":"For any real number a and positive integers $$m$$ and $$n$$, the power rule of exponents states that $${\\\\left(a^m\\\\right)}^n=a^{m n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp15b-h4","type":"hint","dependencies":["a6d0007exp15b-h3"],"title":"Power Rule of Exponents","text":"Use the power rule to simplify the expression: $$\\\\frac{p^6}{{\\\\left(q^3\\\\right)}^6}=\\\\frac{p^6}{q^{3\\\\times6}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6d0007exp15c","stepAnswer":["$$\\\\frac{j^{12}}{k^8}$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(j^3 k^{-2}\\\\right)}^4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{j^{12}}{k^8}$$","hints":{"DefaultPathway":[{"id":"a6d0007exp15c-h1","type":"hint","dependencies":[],"title":"Negative Exponent Rule","text":"For any nonzero real number a and natural number $$n$$, the negative rule of exponents states that $$a^{-n}=\\\\frac{1}{a^n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp15c-h2","type":"hint","dependencies":["a6d0007exp15c-h1"],"title":"Negative Exponent Rule","text":"Use the negative exponent rule to simplify the expression: $${\\\\left(j^3 k^{-2}\\\\right)}^4={\\\\left(\\\\frac{j^3}{k^2}\\\\right)}^4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp15c-h3","type":"hint","dependencies":["a6d0007exp15c-h2"],"title":"Power of a Quotient Rule","text":"For any real numbers a and $$b$$ and any integer $$n$$, the power of a quotient rule of exponents states that $${\\\\left(\\\\frac{a}{b}\\\\right)}^n=\\\\frac{a^n}{b^n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp15c-h4","type":"hint","dependencies":["a6d0007exp15c-h3"],"title":"Power of a Quotient Rule","text":"Use the power of a quotient rule to simplify the expression: $${\\\\left(\\\\frac{j^3}{k^2}\\\\right)}^4=\\\\frac{{\\\\left(j^3\\\\right)}^4}{{\\\\left(k^2\\\\right)}^4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp15c-h5","type":"hint","dependencies":["a6d0007exp15c-h4"],"title":"Power Rule of Exponents","text":"For any real number a and positive integers $$m$$ and $$n$$, the power rule of exponents states that $${\\\\left(a^m\\\\right)}^n=a^{m n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp15c-h6","type":"hint","dependencies":["a6d0007exp15c-h5"],"title":"Power Rule of Exponents","text":"Use the power rule to simplify the expression: $$\\\\frac{{\\\\left(j^3\\\\right)}^4}{{\\\\left(k^2\\\\right)}^4}=\\\\frac{j^{3\\\\times4}}{k^{2\\\\times4}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d0007exp16","title":"The Power of a Quotient Rule of Exponents","body":"Simplify each of the following quotients as much as possible using the power of a quotient rule. Write answers with positive exponents.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Exponents and Scientific Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"a6d0007exp16a","stepAnswer":["$$\\\\frac{b^{15}}{c^3}$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(\\\\frac{b^5}{c}\\\\right)}^3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{b^{15}}{c^3}$$","hints":{"DefaultPathway":[{"id":"a6d0007exp16a-h1","type":"hint","dependencies":[],"title":"Power of a Quotient Rule","text":"For any real numbers a and $$b$$ and any integer $$n$$, the power of a quotient rule of exponents states that $${\\\\left(\\\\frac{a}{b}\\\\right)}^n=\\\\frac{a^n}{b^n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp16a-h2","type":"hint","dependencies":["a6d0007exp16a-h1"],"title":"Power of a Quotient Rule","text":"Use the power of a quotient rule to simplify the expression: $${\\\\left(\\\\frac{b^5}{c}\\\\right)}^3=\\\\frac{{\\\\left(b^5\\\\right)}^3}{c^3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp16a-h3","type":"hint","dependencies":["a6d0007exp16a-h2"],"title":"Power Rule of Exponents","text":"For any real number a and positive integers $$m$$ and $$n$$, the power rule of exponents states that $${\\\\left(a^m\\\\right)}^n=a^{m n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp16a-h4","type":"hint","dependencies":["a6d0007exp16a-h3"],"title":"Power Rule of Exponents","text":"Use the power rule to simplify the expression: $$\\\\frac{{\\\\left(b^5\\\\right)}^3}{c^3}=\\\\frac{b^{5\\\\times3}}{c^3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6d0007exp16b","stepAnswer":["$$\\\\frac{625}{u^{32}}$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(\\\\frac{5}{u^8}\\\\right)}^4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{625}{u^{32}}$$","hints":{"DefaultPathway":[{"id":"a6d0007exp16b-h1","type":"hint","dependencies":[],"title":"Power of a Quotient Rule","text":"For any real numbers a and $$b$$ and any integer $$n$$, the power of a quotient rule of exponents states that $${\\\\left(\\\\frac{a}{b}\\\\right)}^n=\\\\frac{a^n}{b^n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp16b-h2","type":"hint","dependencies":["a6d0007exp16b-h1"],"title":"Power of a Quotient Rule","text":"Use the power of a quotient rule to simplify the expression: $${\\\\left(\\\\frac{5}{u^8}\\\\right)}^4=\\\\frac{5^4}{{\\\\left(u^8\\\\right)}^4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp16b-h3","type":"hint","dependencies":["a6d0007exp16b-h2"],"title":"Power Rule of Exponents","text":"For any real number a and positive integers $$m$$ and $$n$$, the power rule of exponents states that $${\\\\left(a^m\\\\right)}^n=a^{m n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp16b-h4","type":"hint","dependencies":["a6d0007exp16b-h3"],"title":"Power Rule of Exponents","text":"Use the power rule to simplify the expression: $${\\\\left(\\\\frac{5}{u^8}\\\\right)}^4=\\\\frac{5^4}{{\\\\left(u^8\\\\right)}^4}=\\\\frac{625}{u^{8\\\\times4}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6d0007exp16c","stepAnswer":["$$\\\\frac{q^{24}}{p^{32}}$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(p^{-4} q^3\\\\right)}^8$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{q^{24}}{p^{32}}$$","hints":{"DefaultPathway":[{"id":"a6d0007exp16c-h1","type":"hint","dependencies":[],"title":"Negative Exponent Rule","text":"For any nonzero real number a and natural number $$n$$, the negative rule of exponents states that $$a^{-n}=\\\\frac{1}{a^n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp16c-h2","type":"hint","dependencies":["a6d0007exp16c-h1"],"title":"Negative Exponent Rule","text":"Use the negative exponent rule to simplify the expression: $${\\\\left(p^{-4} q^3\\\\right)}^8={\\\\left(\\\\frac{q^3}{p^4}\\\\right)}^8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp16c-h3","type":"hint","dependencies":["a6d0007exp16c-h2"],"title":"Power of a Quotient Rule","text":"For any real numbers a and $$b$$ and any integer $$n$$, the power of a quotient rule of exponents states that $${\\\\left(\\\\frac{a}{b}\\\\right)}^n=\\\\frac{a^n}{b^n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp16c-h4","type":"hint","dependencies":["a6d0007exp16c-h3"],"title":"Power of a Quotient Rule","text":"Use the power of a quotient rule to simplify the expression: $${\\\\left(\\\\frac{q^3}{p^4}\\\\right)}^8=\\\\frac{{\\\\left(q^3\\\\right)}^8}{{\\\\left(p^4\\\\right)}^8}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp16c-h5","type":"hint","dependencies":["a6d0007exp16c-h4"],"title":"Power Rule of Exponents","text":"For any real number a and positive integers $$m$$ and $$n$$, the power rule of exponents states that $${\\\\left(a^m\\\\right)}^n=a^{m n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp16c-h6","type":"hint","dependencies":["a6d0007exp16c-h5"],"title":"Power Rule of Exponents","text":"Use the power rule to simplify the expression: $$\\\\frac{{\\\\left(q^3\\\\right)}^8}{{\\\\left(p^4\\\\right)}^8}=\\\\frac{q^{3\\\\times8}}{k^{4\\\\times8}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d0007exp17","title":"Converting Standard Notation to Scientific Notation","body":"Write each number in scientific notation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Exponents and Scientific Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"a6d0007exp17a","stepAnswer":["$$2.4{10}^{22}$$"],"problemType":"TextBox","stepTitle":"Distance to Andromeda Galaxy from Earth: 24,000,000,000,000,000,000,000 $$m$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2.4{10}^{22}$$","hints":{"DefaultPathway":[{"id":"a6d0007exp17a-h1","type":"hint","dependencies":[],"title":"Scientific Notation","text":"A number is written in scientific notation if it is written in the form $$a {10}^n$$, where 1<=|a|<10 and $$n$$ is an integer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp17a-h2","type":"hint","dependencies":["a6d0007exp17a-h1"],"title":"Turning a Number Into Scientific Notation","text":"To write a number in scientific notation, move the decimal point to the right of the first digit in the number. Write the digits as a decimal number between $$1$$ and $$10$$. Count the number of places $$n$$ that you moved the decimal point. Multiply the decimal number by $$10$$ raised to a power of $$n$$. If you moved the decimal left as in a very large number, $$n$$ is positive. If you moved the decimal right as in a small large number, $$n$$ is negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp17a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Left"],"dependencies":["a6d0007exp17a-h2"],"title":"Moving the Decimal","text":"In which direction do you have to move the decimal for the number 24,000,000,000,000,000,000,000?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Left","Right"]},{"id":"a6d0007exp17a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$22$$"],"dependencies":["a6d0007exp17a-h3"],"title":"Moving the Decimal","text":"How many places to the left do you need to move the decimal for the number 24,000,000,000,000,000,000,000?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6d0007exp17b","stepAnswer":["$$1{10}^{12}$$"],"problemType":"TextBox","stepTitle":"Number of stars in Andromeda Galaxy: 1,000,000,000,000","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1{10}^{12}$$","hints":{"DefaultPathway":[{"id":"a6d0007exp17b-h1","type":"hint","dependencies":[],"title":"Scientific Notation","text":"A number is written in scientific notation if it is written in the form $$a {10}^n$$, where 1<=|a|<10 and $$n$$ is an integer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp17b-h2","type":"hint","dependencies":["a6d0007exp17b-h1"],"title":"Turning a Number Into Scientific Notation","text":"To write a number in scientific notation, move the decimal point to the right of the first digit in the number. Write the digits as a decimal number between $$1$$ and $$10$$. Count the number of places $$n$$ that you moved the decimal point. Multiply the decimal number by $$10$$ raised to a power of $$n$$. If you moved the decimal left as in a very large number, $$n$$ is positive. If you moved the decimal right as in a small large number, $$n$$ is negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp17b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Left"],"dependencies":["a6d0007exp17b-h2"],"title":"Moving the Decimal","text":"In which direction do you have to move the decimal for the number 1,000,000,000,000?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Left","Right"]},{"id":"a6d0007exp17b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a6d0007exp17b-h3"],"title":"Moving the Decimal","text":"How many places to the left do you need to move the decimal for the number 1,000,000,000,000?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6d0007exp17c","stepAnswer":["$$9.4{10}^{-13}$$"],"problemType":"TextBox","stepTitle":"Diameter of electron: $$0.00000000000094$$ $$m$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9.4{10}^{-13}$$","hints":{"DefaultPathway":[{"id":"a6d0007exp17c-h1","type":"hint","dependencies":[],"title":"Scientific Notation","text":"A number is written in scientific notation if it is written in the form $$a {10}^n$$, where 1<=|a|<10 and $$n$$ is an integer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp17c-h2","type":"hint","dependencies":["a6d0007exp17c-h1"],"title":"Turning a Number Into Scientific Notation","text":"To write a number in scientific notation, move the decimal point to the right of the first digit in the number. Write the digits as a decimal number between $$1$$ and $$10$$. Count the number of places $$n$$ that you moved the decimal point. Multiply the decimal number by $$10$$ raised to a power of $$n$$. If you moved the decimal left as in a very large number, $$n$$ is positive. If you moved the decimal right as in a small large number, $$n$$ is negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp17c-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Right"],"dependencies":["a6d0007exp17c-h2"],"title":"Moving the Decimal","text":"In which direction do you have to move the decimal for the number $$0.00000000000094$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Left","Right"]},{"id":"a6d0007exp17c-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$13$$"],"dependencies":["a6d0007exp17c-h3"],"title":"Moving the Decimal","text":"How many places to the right do you need to move the decimal for the number $$0.00000000000094$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d0007exp18","title":"Converting Numbers into Scientific Notation","body":"Write each number in scientific notation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Exponents and Scientific Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"a6d0007exp18a","stepAnswer":["$$7.158\\\\times {10}^9$$"],"problemType":"TextBox","stepTitle":"World population (April 2014): 7,158,000,000","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7.158\\\\times {10}^9$$","hints":{"DefaultPathway":[{"id":"a6d0007exp18a-h1","type":"hint","dependencies":[],"title":"Scientific Notation","text":"A number is written in scientific notation if it is written in the form $$a {10}^n$$, where 1<=|a|<10 and $$n$$ is an integer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp18a-h2","type":"hint","dependencies":["a6d0007exp18a-h1"],"title":"Turning a Number Into Scientific Notation","text":"To write a number in scientific notation, move the decimal point to the right of the first digit in the number. Write the digits as a decimal number between $$1$$ and $$10$$. Count the number of places $$n$$ that you moved the decimal point. Multiply the decimal number by $$10$$ raised to a power of $$n$$. If you moved the decimal left as in a very large number, $$n$$ is positive. If you moved the decimal right as in a small large number, $$n$$ is negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp18a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Left"],"dependencies":["a6d0007exp18a-h2"],"title":"Moving the Decimal","text":"In which direction do you have to move the decimal for the number 7,158,000,000?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Left","Right"]},{"id":"a6d0007exp18a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a6d0007exp18a-h3"],"title":"Moving the Decimal","text":"How many places to the left do you need to move the decimal for the number 7,158,000,000?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6d0007exp18b","stepAnswer":["$$3.34\\\\times {10}^{-9}$$"],"problemType":"TextBox","stepTitle":"Time for light to travel $$1$$ m: $$0.00000000334$$ s","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3.34\\\\times {10}^{-9}$$","hints":{"DefaultPathway":[{"id":"a6d0007exp18b-h1","type":"hint","dependencies":[],"title":"Scientific Notation","text":"A number is written in scientific notation if it is written in the form $$a {10}^n$$, where 1<=|a|<10 and $$n$$ is an integer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp18b-h2","type":"hint","dependencies":["a6d0007exp18b-h1"],"title":"Turning a Number Into Scientific Notation","text":"To write a number in scientific notation, move the decimal point to the right of the first digit in the number. Write the digits as a decimal number between $$1$$ and $$10$$. Count the number of places $$n$$ that you moved the decimal point. Multiply the decimal number by $$10$$ raised to a power of $$n$$. If you moved the decimal left as in a very large number, $$n$$ is positive. 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Remember, if $$n$$ is positive, the value of the number is greater than $$1$$, and if $$n$$ is negative, the value of the number is less than one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp20d-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Left"],"dependencies":["a6d0007exp20d-h1"],"title":"Moving the Decimal","text":"In which direction do you have to move the decimal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Left","Right"]},{"id":"a6d0007exp20d-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a6d0007exp20d-h2"],"title":"Moving the Decimal","text":"How many places to the left do you need to move the decimal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d0007exp21","title":"Using Scientific Notation in Applications","body":"Perform the operations and write the answer in scientific notation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Exponents and Scientific Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"a6d0007exp21a","stepAnswer":["$$5.291\\\\times {10}^4$$"],"problemType":"TextBox","stepTitle":"Simplify $$8.14\\\\times {10}^{-7} 6.5{10}^{10}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5.291\\\\times {10}^4$$","hints":{"DefaultPathway":[{"id":"a6d0007exp21a-h1","type":"hint","dependencies":[],"title":"Commutative and Associative Properties of Multiplication","text":"Use the commutative and associative properties of multiplication to simplify the expression: $$8.14\\\\times {10}^{-7} 6.5{10}^{10}=8.14\\\\times6.5 {10}^{-7} {10}^{10}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp21a-h2","type":"hint","dependencies":["a6d0007exp21a-h1"],"title":"Product Rule of Exponents","text":"Use the product rule of exponents to simplify the expression: $$8.14\\\\times6.5 {10}^{-7} {10}^{10}=52.91{10}^3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp21a-h3","type":"hint","dependencies":["a6d0007exp21a-h2"],"title":"Scientific Notation","text":"Rewrite the expression in scientific notation: $$52.91{10}^3=5.291\\\\times {10}^4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6d0007exp21b","stepAnswer":["$$1.25\\\\times {10}^2$$"],"problemType":"TextBox","stepTitle":"What is $$\\\\frac{1.2{10}^8}{9.6{10}^5}$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1.25\\\\times {10}^2$$","hints":{"DefaultPathway":[{"id":"a6d0007exp21b-h1","type":"hint","dependencies":[],"title":"Commutative and Associative Properties of Multiplication","text":"Use the commutative and associative properties of multiplication to simplify the expression: $$\\\\frac{1.2{10}^8}{9.6{10}^5}=\\\\frac{1.2}{9.6} \\\\frac{{10}^8}{{10}^5}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp21b-h2","type":"hint","dependencies":["a6d0007exp21b-h1"],"title":"Quotient Rule of Exponents","text":"Use the quotient rule of exponents to simplify the expression: $$\\\\frac{1.2}{9.6} \\\\frac{{10}^8}{{10}^5}=0.125{10}^3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp21b-h3","type":"hint","dependencies":["a6d0007exp21b-h2"],"title":"Scientific Notation","text":"Rewrite the expression in scientific notation: $$0.125{10}^3=1.25\\\\times {10}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d0007exp22","title":"Thickness of a Dime","body":"A dime is the thinnest coin in U.S. currency. A dime\u2019s thickness measures $$1.35\\\\times {10}^{-3}$$ $$m$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Exponents and Scientific Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"a6d0007exp22a","stepAnswer":["$$0.00135$$"],"problemType":"TextBox","stepTitle":"Rewrite the number in standard notation.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.00135$$","hints":{"DefaultPathway":[{"id":"a6d0007exp22a-h1","type":"hint","dependencies":[],"title":"Converting to Standard Notation","text":"To convert a number in scientific notation to standard notation, simply reverse the process. Move the decimal $$n$$ places to the right if $$n$$ is positive or $$n$$ places to the left if $$n$$ is negative and add zeros as needed. Remember, if $$n$$ is positive, the value of the number is greater than $$1$$, and if $$n$$ is negative, the value of the number is less than one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp22a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Left"],"dependencies":["a6d0007exp22a-h1"],"title":"Moving the Decimal","text":"In which direction do you have to move the decimal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Left","Right"]},{"id":"a6d0007exp22a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a6d0007exp22a-h2"],"title":"Moving the Decimal","text":"How many places to the left do you need to move the decimal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d0007exp23","title":"Terabyte Size","body":"A terabyte is made of approximately 1,099,500,000,000 bytes.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Exponents and Scientific Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"a6d0007exp23a","stepAnswer":["$$1.0995\\\\times {10}^{12}$$"],"problemType":"TextBox","stepTitle":"Rewrite in scientific notation.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1.0995\\\\times {10}^{12}$$","hints":{"DefaultPathway":[{"id":"a6d0007exp23a-h1","type":"hint","dependencies":[],"title":"Scientific Notation","text":"A number is written in scientific notation if it is written in the form $$a {10}^n$$, where 1<=|a|<10 and $$n$$ is an integer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp23a-h2","type":"hint","dependencies":["a6d0007exp23a-h1"],"title":"Turning a Number Into Scientific Notation","text":"To write a number in scientific notation, move the decimal point to the right of the first digit in the number. Write the digits as a decimal number between $$1$$ and $$10$$. Count the number of places $$n$$ that you moved the decimal point. Multiply the decimal number by $$10$$ raised to a power of $$n$$. If you moved the decimal left as in a very large number, $$n$$ is positive. If you moved the decimal right as in a small large number, $$n$$ is negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp23a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Left"],"dependencies":["a6d0007exp23a-h2"],"title":"Moving the Decimal","text":"In which direction do you have to move the decimal for the number 1,099,500,000,000?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Left","Right"]},{"id":"a6d0007exp23a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a6d0007exp23a-h3"],"title":"Moving the Decimal","text":"How many places to the left do you need to move the decimal for the number 1,099,500,000,000?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d0007exp24","title":"Length of a Picometer","body":"One picometer is approximately $$3.397\\\\times {10}^{-11}$$ in.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Exponents and Scientific Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"a6d0007exp24a","stepAnswer":["$$0.00000000003397$$"],"problemType":"TextBox","stepTitle":"Rewrite this length using standard notation.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.00000000003397$$","hints":{"DefaultPathway":[{"id":"a6d0007exp24a-h1","type":"hint","dependencies":[],"title":"Converting to Standard Notation","text":"To convert a number in scientific notation to standard notation, simply reverse the process. Move the decimal $$n$$ places to the right if $$n$$ is positive or $$n$$ places to the left if $$n$$ is negative and add zeros as needed. Remember, if $$n$$ is positive, the value of the number is greater than $$1$$, and if $$n$$ is negative, the value of the number is less than one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp24a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Left"],"dependencies":["a6d0007exp24a-h1"],"title":"Moving the Decimal","text":"In which direction do you have to move the decimal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Left","Right"]},{"id":"a6d0007exp24a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11$$"],"dependencies":["a6d0007exp24a-h2"],"title":"Moving the Decimal","text":"How many places to the left do you need to move the decimal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d0007exp25","title":"Distance Between Earth and Sun","body":"The average distance between Earth and the Sun is 92,960,000 mi.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Exponents and Scientific Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"a6d0007exp25a","stepAnswer":["$$9.296\\\\times {10}^7$$"],"problemType":"TextBox","stepTitle":"Rewrite the distance using scientific notation.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9.296\\\\times {10}^7$$","hints":{"DefaultPathway":[{"id":"a6d0007exp25a-h1","type":"hint","dependencies":[],"title":"Scientific Notation","text":"A number is written in scientific notation if it is written in the form $$a {10}^n$$, where 1<=|a|<10 and $$n$$ is an integer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp25a-h2","type":"hint","dependencies":["a6d0007exp25a-h1"],"title":"Turning a Number Into Scientific Notation","text":"To write a number in scientific notation, move the decimal point to the right of the first digit in the number. Write the digits as a decimal number between $$1$$ and $$10$$. Count the number of places $$n$$ that you moved the decimal point. Multiply the decimal number by $$10$$ raised to a power of $$n$$. If you moved the decimal left as in a very large number, $$n$$ is positive. If you moved the decimal right as in a small large number, $$n$$ is negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp25a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Left"],"dependencies":["a6d0007exp25a-h2"],"title":"Moving the Decimal","text":"In which direction do you have to move the decimal for the number 92,960,000?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Left","Right"]},{"id":"a6d0007exp25a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a6d0007exp25a-h3"],"title":"Moving the Decimal","text":"How many places to the left do you need to move the decimal for the number 92,960,000?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d0007exp3","title":"Using the Quotient Rule","body":"Write each of the following products with a single base. Do not simplify further.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Exponents and Scientific Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"a6d0007exp3a","stepAnswer":["$${\\\\left(-2\\\\right)}^5$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{{\\\\left(-2\\\\right)}^{14}}{{\\\\left(-2\\\\right)}^9}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(-2\\\\right)}^5$$","hints":{"DefaultPathway":[{"id":"a6d0007exp3a-h1","type":"hint","dependencies":[],"title":"Quotient Rule of Exponents","text":"For any real number a and natural numbers $$m$$ and $$n$$, such that $$m>n$$, the quotient rule of exponents states that $$\\\\frac{a^m}{a^n}=a^{m-n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp3a-h2","type":"hint","dependencies":["a6d0007exp3a-h1"],"title":"Quotient Rule of Exponents","text":"Use the quotient rule to simplify the expression: $$\\\\frac{{\\\\left(-2\\\\right)}^{14}}{{\\\\left(-2\\\\right)}^9}={\\\\left(-2\\\\right)}^{14-9}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6d0007exp3b","stepAnswer":["$$t^8$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{t^{23}}{t^{15}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$t^8$$","hints":{"DefaultPathway":[{"id":"a6d0007exp3b-h1","type":"hint","dependencies":[],"title":"Quotient Rule of Exponents","text":"For any real number a and natural numbers $$m$$ and $$n$$, such that $$m>n$$, the quotient rule of exponents states that $$\\\\frac{a^m}{a^n}=a^{m-n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp3b-h2","type":"hint","dependencies":["a6d0007exp3b-h1"],"title":"Quotient Rule of Exponents","text":"Use the quotient rule to simplify the expression: $$\\\\frac{t^{23}}{t^{15}}=t^{23-15}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6d0007exp3c","stepAnswer":["$${\\\\left(z \\\\sqrt{2}\\\\right)}^4$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{{\\\\left(z \\\\sqrt{2}\\\\right)}^5}{z \\\\sqrt{2}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(z \\\\sqrt{2}\\\\right)}^4$$","hints":{"DefaultPathway":[{"id":"a6d0007exp3c-h1","type":"hint","dependencies":[],"title":"Quotient Rule of Exponents","text":"For any real number a and natural numbers $$m$$ and $$n$$, such that $$m>n$$, the quotient rule of exponents states that $$\\\\frac{a^m}{a^n}=a^{m-n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp3c-h2","type":"hint","dependencies":["a6d0007exp3c-h1"],"title":"Quotient Rule of Exponents","text":"Use the quotient rule to simplify the expression: $$\\\\frac{{\\\\left(z \\\\sqrt{2}\\\\right)}^5}{z \\\\sqrt{2}}=$$ (z*sqrt(2))**5/ $${\\\\left(z \\\\sqrt{2}\\\\right)}^1={\\\\left(z \\\\sqrt{2}\\\\right)}^{5-1}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d0007exp4","title":"The Quotient Rule of Exponents","body":"Write each of the following products with a single base. Do not simplify further.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Exponents and Scientific Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"a6d0007exp4a","stepAnswer":["$$s^7$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{s^{75}}{s^{68}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$s^7$$","hints":{"DefaultPathway":[{"id":"a6d0007exp4a-h1","type":"hint","dependencies":[],"title":"Quotient Rule of Exponents","text":"For any real number a and natural numbers $$m$$ and $$n$$, such that $$m>n$$, the quotient rule of exponents states that $$\\\\frac{a^m}{a^n}=a^{m-n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp4a-h2","type":"hint","dependencies":["a6d0007exp4a-h1"],"title":"Quotient Rule of Exponents","text":"Use the quotient rule to simplify the expression: $$\\\\frac{s^{75}}{s^{68}}=$$ $$s^{75-68}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6d0007exp4b","stepAnswer":["$${\\\\left(-3\\\\right)}^5$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{{\\\\left(-3\\\\right)}^6}{-3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(-3\\\\right)}^5$$","hints":{"DefaultPathway":[{"id":"a6d0007exp4b-h1","type":"hint","dependencies":[],"title":"Quotient Rule of Exponents","text":"For any real number a and natural numbers $$m$$ and $$n$$, such that $$m>n$$, the quotient rule of exponents states that $$\\\\frac{a^m}{a^n}=a^{m-n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp4b-h2","type":"hint","dependencies":["a6d0007exp4b-h1"],"title":"Quotient Rule of Exponents","text":"Use the quotient rule to simplify the expression: $$\\\\frac{{\\\\left(-3\\\\right)}^6}{-3}=\\\\frac{{\\\\left(-3\\\\right)}^6}{{\\\\left(-3\\\\right)}^1}={\\\\left(-3\\\\right)}^{6-1}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6d0007exp4c","stepAnswer":["$${\\\\left({ef}^2\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{{\\\\left({ef}^2\\\\right)}^5}{{\\\\left({ef}^2\\\\right)}^3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left({ef}^2\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"a6d0007exp4c-h1","type":"hint","dependencies":[],"title":"Quotient Rule of Exponents","text":"For any real number a and natural numbers $$m$$ and $$n$$, such that $$m>n$$, the quotient rule of exponents states that $$\\\\frac{a^m}{a^n}=a^{m-n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp4c-h2","type":"hint","dependencies":["a6d0007exp4c-h1"],"title":"Quotient Rule of Exponents","text":"Use the quotient rule to simplify the expression: $$\\\\frac{{\\\\left({ef}^2\\\\right)}^5}{{\\\\left({ef}^2\\\\right)}^3}={\\\\left({ef}^2\\\\right)}^{5-3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d0007exp5","title":"Using the Power Rule","body":"Write each of the following products with a single base. Do not simplify further.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Exponents and Scientific Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"a6d0007exp5a","stepAnswer":["$$x^{14}$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(x^2\\\\right)}^7$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^{14}$$","hints":{"DefaultPathway":[{"id":"a6d0007exp5a-h1","type":"hint","dependencies":[],"title":"Power Rule of Exponents","text":"For any real number a and positive integers $$m$$ and $$n$$, the power rule of exponents states that $${\\\\left(a^m\\\\right)}^n=a^{m n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp5a-h2","type":"hint","dependencies":["a6d0007exp5a-h1"],"title":"Power Rule of Exponents","text":"Use the power rule to simplify the expression: $${\\\\left(x^2\\\\right)}^7=x^{2\\\\times7}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6d0007exp5b","stepAnswer":["$${\\\\left(2t\\\\right)}^{15}$$"],"problemType":"TextBox","stepTitle":"$${\\\\left({\\\\left(2t\\\\right)}^5\\\\right)}^3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(2t\\\\right)}^{15}$$","hints":{"DefaultPathway":[{"id":"a6d0007exp5b-h1","type":"hint","dependencies":[],"title":"Power Rule of Exponents","text":"For any real number a and positive integers $$m$$ and $$n$$, the power rule of exponents states that $${\\\\left(a^m\\\\right)}^n=a^{m n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp5b-h2","type":"hint","dependencies":["a6d0007exp5b-h1"],"title":"Power Rule of Exponents","text":"Use the power rule to simplify the expression: $${\\\\left({\\\\left(2t\\\\right)}^5\\\\right)}^3={\\\\left(2t\\\\right)}^{5\\\\times3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6d0007exp5c","stepAnswer":["$${\\\\left(-3\\\\right)}^{55}$$"],"problemType":"TextBox","stepTitle":"$${\\\\left({\\\\left(-3\\\\right)}^5\\\\right)}^{11}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(-3\\\\right)}^{55}$$","hints":{"DefaultPathway":[{"id":"a6d0007exp5c-h1","type":"hint","dependencies":[],"title":"Power Rule of Exponents","text":"For any real number a and positive integers $$m$$ and $$n$$, the power rule of exponents states that $${\\\\left(a^m\\\\right)}^n=a^{m n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp5c-h2","type":"hint","dependencies":["a6d0007exp5c-h1"],"title":"Power Rule of Exponents","text":"Use the power rule to simplify the expression: $${\\\\left({\\\\left(-3\\\\right)}^5\\\\right)}^{11}={\\\\left(-3\\\\right)}^{5\\\\times11}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d0007exp6","title":"The Power Rule of Exponents","body":"Write each of the following products with a single base. Do not simplify further.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Exponents and Scientific Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"a6d0007exp6a","stepAnswer":["$${\\\\left(3y\\\\right)}^{24}$$"],"problemType":"TextBox","stepTitle":"$${\\\\left({\\\\left(3y\\\\right)}^8\\\\right)}^3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(3y\\\\right)}^{24}$$","hints":{"DefaultPathway":[{"id":"a6d0007exp6a-h1","type":"hint","dependencies":[],"title":"Power Rule of Exponents","text":"For any real number a and positive integers $$m$$ and $$n$$, the power rule of exponents states that $${\\\\left(a^m\\\\right)}^n=a^{m n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp6a-h2","type":"hint","dependencies":["a6d0007exp6a-h1"],"title":"Power Rule of Exponents","text":"Use the power rule to simplify the expression: $${\\\\left({\\\\left(3y\\\\right)}^8\\\\right)}^3={\\\\left(3y\\\\right)}^{8\\\\times3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6d0007exp6b","stepAnswer":["$$t^{35}$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(t^5\\\\right)}^7$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$t^{35}$$","hints":{"DefaultPathway":[{"id":"a6d0007exp6b-h1","type":"hint","dependencies":[],"title":"Power Rule of Exponents","text":"For any real number a and positive integers $$m$$ and $$n$$, the power rule of exponents states that $${\\\\left(a^m\\\\right)}^n=a^{m n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp6b-h2","type":"hint","dependencies":["a6d0007exp6b-h1"],"title":"Power Rule of Exponents","text":"Use the power rule to simplify the expression: $${\\\\left(t^5\\\\right)}^7=t^{5\\\\times7}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6d0007exp6c","stepAnswer":["$${\\\\left(-g\\\\right)}^{16}$$"],"problemType":"TextBox","stepTitle":"$${\\\\left({\\\\left(-g\\\\right)}^4\\\\right)}^4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(-g\\\\right)}^{16}$$","hints":{"DefaultPathway":[{"id":"a6d0007exp6c-h1","type":"hint","dependencies":[],"title":"Power Rule of Exponents","text":"For any real number a and positive integers $$m$$ and $$n$$, the power rule of exponents states that $${\\\\left(a^m\\\\right)}^n=a^{m n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp6c-h2","type":"hint","dependencies":["a6d0007exp6c-h1"],"title":"Power Rule of Exponents","text":"Use the power rule to simplify the expression: $${\\\\left({\\\\left(-g\\\\right)}^4\\\\right)}^4={\\\\left(-g\\\\right)}^{4\\\\times4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d0007exp7","title":"Using the Zero Exponent Rule","body":"Simplify each expression using the zero exponent rule of exponents.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Exponents and Scientific Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"a6d0007exp7a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{c^3}{c^3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a6d0007exp7a-h1","type":"hint","dependencies":[],"title":"Quotient Rule of Exponents","text":"For any real number a and natural numbers $$m$$ and $$n$$, such that $$m>n$$, the quotient rule of exponents states that $$\\\\frac{a^m}{a^n}=a^{m-n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp7a-h2","type":"hint","dependencies":["a6d0007exp7a-h1"],"title":"Quotient Rule of Exponents","text":"Use the quotient rule to simplify the expression: $$\\\\frac{c^3}{c^3}=c^{3-3}=c^0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp7a-h3","type":"hint","dependencies":["a6d0007exp7a-h2"],"title":"Zero Exponent Rule","text":"For any nonzero real number a, the zero exponent rule of exponents states that $$a^0=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6d0007exp7b","stepAnswer":["$$-3$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{-3x^5}{x^5}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-3$$","hints":{"DefaultPathway":[{"id":"a6d0007exp7b-h1","type":"hint","dependencies":[],"title":"Quotient Rule of Exponents","text":"For any real number a and natural numbers $$m$$ and $$n$$, such that $$m>n$$, the quotient rule of exponents states that $$\\\\frac{a^m}{a^n}=a^{m-n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp7b-h2","type":"hint","dependencies":["a6d0007exp7b-h1"],"title":"Quotient Rule of Exponents","text":"Use the quotient rule to simplify the expression: $$\\\\frac{-3x^5}{x^5}=-3x^{5-5}=-3x^0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp7b-h3","type":"hint","dependencies":["a6d0007exp7b-h2"],"title":"Zero Exponent Rule","text":"For any nonzero real number a, the zero exponent rule of exponents states that $$a^0=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp7b-h4","type":"hint","dependencies":["a6d0007exp7b-h3"],"title":"Simplify","text":"Use the Zero Exponent Rule to simplify the expression: $$-3x^0=-3(1)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6d0007exp7c","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{{\\\\left(j^2 k\\\\right)}^4}{j^2 k {\\\\left(j^2 k\\\\right)}^3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a6d0007exp7c-h1","type":"hint","dependencies":[],"title":"Product Rule of Exponents","text":"For any real number a and natural numbers $$m$$ and $$n$$, the product rule of exponents states that $$a^m a^n=a \\\\left(m+n\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp7c-h2","type":"hint","dependencies":["a6d0007exp7c-h1"],"title":"Product Rule of Exponents","text":"Use the product rule to simplify the expression: $$\\\\frac{{\\\\left(j^2 k\\\\right)}^4}{j^2 k {\\\\left(j^2 k\\\\right)}^3}=\\\\frac{{\\\\left(j^2 k\\\\right)}^4}{{\\\\left(j^2 k\\\\right)}^1 {\\\\left(j^2 k\\\\right)}^3}=\\\\frac{{\\\\left(j^2 k\\\\right)}^4}{{\\\\left(j^2 k\\\\right)}^{1+3}}=\\\\frac{{\\\\left(j^2 k\\\\right)}^4}{{\\\\left(j^2 k\\\\right)}^4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp7c-h3","type":"hint","dependencies":["a6d0007exp7c-h2"],"title":"Quotient Rule of Exponents","text":"For any real number a and natural numbers $$m$$ and $$n$$, such that $$m>n$$, the quotient rule of exponents states that $$\\\\frac{a^m}{a^n}=a^{m-n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp7c-h4","type":"hint","dependencies":["a6d0007exp7c-h3"],"title":"Quotient Rule of Exponents","text":"Use the quotient rule to simplify the expression: $$\\\\frac{{\\\\left(j^2 k\\\\right)}^4}{{\\\\left(j^2 k\\\\right)}^4}={\\\\left(j^2 k\\\\right)}^{4-4}={\\\\left(j^2 k\\\\right)}^0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp7c-h5","type":"hint","dependencies":["a6d0007exp7c-h4"],"title":"Zero Exponent Rule","text":"For any nonzero real number a, the zero exponent rule of exponents states that $$a^0=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d0007exp8","title":"The Zero Exponent Rule of Exponents","body":"Simplify each expression using the zero exponent rule of exponents.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Exponents and Scientific Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"a6d0007exp8a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{t^7}{t^7}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a6d0007exp8a-h1","type":"hint","dependencies":[],"title":"Quotient Rule of Exponents","text":"For any real number a and natural numbers $$m$$ and $$n$$, such that $$m>n$$, the quotient rule of exponents states that $$\\\\frac{a^m}{a^n}=a^{m-n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp8a-h2","type":"hint","dependencies":["a6d0007exp8a-h1"],"title":"Quotient Rule of Exponents","text":"Use the quotient rule to simplify the expression: $$\\\\frac{t^7}{t^7}=t^{7-7}=t^0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp8a-h3","type":"hint","dependencies":["a6d0007exp8a-h2"],"title":"Zero Exponent Rule","text":"For any nonzero real number a, the zero exponent rule of exponents states that $$a^0=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6d0007exp8b","stepAnswer":["$$\\\\frac{1}{2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{{\\\\left({de}^2\\\\right)}^{11}}{{2\\\\left({de}^2\\\\right)}^{11}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{2}$$","hints":{"DefaultPathway":[{"id":"a6d0007exp8b-h1","type":"hint","dependencies":[],"title":"Quotient Rule of Exponents","text":"For any real number a and natural numbers $$m$$ and $$n$$, such that $$m>n$$, the quotient rule of exponents states that $$\\\\frac{a^m}{a^n}=a^{m-n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp8b-h2","type":"hint","dependencies":["a6d0007exp8b-h1"],"title":"Quotient Rule of Exponents","text":"Use the quotient rule to simplify the expression: $$\\\\frac{{\\\\left({de}^2\\\\right)}^{11}}{{2\\\\left({de}^2\\\\right)}^{11}}=\\\\frac{1}{2} {\\\\left({de}^2\\\\right)}^{11-11}=\\\\frac{1}{2} {\\\\left({de}^2\\\\right)}^0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp8b-h3","type":"hint","dependencies":["a6d0007exp8b-h2"],"title":"Zero Exponent Rule","text":"For any nonzero real number a, the zero exponent rule of exponents states that $$a^0=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp8b-h4","type":"hint","dependencies":["a6d0007exp8b-h3"],"title":"Simplify","text":"Use the Zero Exponent Rule to simplify the expression: $$\\\\frac{1}{2} {\\\\left({de}^2\\\\right)}^0=1\\\\frac{1}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6d0007exp8c","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{w^4 w^2}{w^6}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a6d0007exp8c-h1","type":"hint","dependencies":[],"title":"Product Rule of Exponents","text":"For any real number a and natural numbers $$m$$ and $$n$$, the product rule of exponents states that $$a^m a^n=a \\\\left(m+n\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp8c-h2","type":"hint","dependencies":["a6d0007exp8c-h1"],"title":"Product Rule of Exponents","text":"Use the product rule to simplify the expression: $$\\\\frac{w^4 w^2}{w^6}=\\\\frac{w^{4+2}}{w^6}=\\\\frac{w^6}{w^6}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp8c-h3","type":"hint","dependencies":["a6d0007exp8c-h2"],"title":"Quotient Rule of Exponents","text":"For any real number a and natural numbers $$m$$ and $$n$$, such that $$m>n$$, the quotient rule of exponents states that $$\\\\frac{a^m}{a^n}=a^{m-n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp8c-h4","type":"hint","dependencies":["a6d0007exp8c-h3"],"title":"Quotient Rule of Exponents","text":"Use the quotient rule to simplify the expression: $$\\\\frac{w^6}{w^6}=w^{6-6}=w^0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp8c-h5","type":"hint","dependencies":["a6d0007exp8c-h4"],"title":"Zero Exponent Rule","text":"For any nonzero real number a, the zero exponent rule of exponents states that $$a^0=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d0007exp9","title":"Using the Negative Exponent Rule","body":"Write each of the following quotients with a single base. Do not simplify further. Write answers with positive exponents. (Note: Type theta to get \u03b8 in answer.)","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Exponents and Scientific Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"a6d0007exp9a","stepAnswer":["$$\\\\frac{1}{{\\\\theta}^7}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{{\\\\theta}^3}{{\\\\theta}^{10}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{{\\\\theta}^7}$$","hints":{"DefaultPathway":[{"id":"a6d0007exp9a-h1","type":"hint","dependencies":[],"title":"Quotient Rule of Exponents","text":"For any real number a and natural numbers $$m$$ and $$n$$, such that $$m>n$$, the quotient rule of exponents states that $$\\\\frac{a^m}{a^n}=a^{m-n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp9a-h2","type":"hint","dependencies":["a6d0007exp9a-h1"],"title":"Quotient Rule of Exponents","text":"Use the quotient rule to simplify the expression: $$\\\\frac{{\\\\theta}^3}{{\\\\theta}^{10}}={\\\\theta}^{3-10}={\\\\theta}^{\\\\left(-7\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp9a-h3","type":"hint","dependencies":["a6d0007exp9a-h2"],"title":"Negative Exponent Rule","text":"For any nonzero real number a and natural number $$n$$, the negative rule of exponents states that $$a -n=\\\\frac{1}{a^n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6d0007exp9b","stepAnswer":["$$\\\\frac{1}{z}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{z^2 z}{z^4}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{z}$$","hints":{"DefaultPathway":[{"id":"a6d0007exp9b-h1","type":"hint","dependencies":[],"title":"Product Rule of Exponents","text":"For any real number a and natural numbers $$m$$ and $$n$$, the product rule of exponents states that $$a^m a^n=a \\\\left(m+n\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp9b-h2","type":"hint","dependencies":["a6d0007exp9b-h1"],"title":"Product Rule of Exponents","text":"Use the product rule to simplify the expression: $$\\\\frac{z^2 z}{z^4}=\\\\frac{z^2 z^1}{z^4}=\\\\frac{z^{2+1}}{z^4}=\\\\frac{z^3}{z^4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp9b-h3","type":"hint","dependencies":["a6d0007exp9b-h2"],"title":"Quotient Rule of Exponents","text":"For any real number a and natural numbers $$m$$ and $$n$$, such that $$m>n$$, the quotient rule of exponents states that $$\\\\frac{a^m}{a^n}=a^{m-n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp9b-h4","type":"hint","dependencies":["a6d0007exp9b-h3"],"title":"Quotient Rule of Exponents","text":"Use the quotient rule to simplify the expression: $$\\\\frac{z^3}{z^4}=z^{3-4}=z^{\\\\left(-1\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp9b-h5","type":"hint","dependencies":["a6d0007exp9b-h4"],"title":"Negative Exponent Rule","text":"For any nonzero real number a and natural number $$n$$, the negative rule of exponents states that $$a -n=\\\\frac{1}{a^n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6d0007exp9c","stepAnswer":["$$\\\\frac{1}{{\\\\left(-5t^3\\\\right)}^4}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{{\\\\left(-5t^3\\\\right)}^4}{{\\\\left(-5t^3\\\\right)}^8}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{{\\\\left(-5t^3\\\\right)}^4}$$","hints":{"DefaultPathway":[{"id":"a6d0007exp9c-h1","type":"hint","dependencies":[],"title":"Quotient Rule of Exponents","text":"For any real number a and natural numbers $$m$$ and $$n$$, such that $$m>n$$, the quotient rule of exponents states that $$\\\\frac{a^m}{a^n}=a^{m-n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp9c-h2","type":"hint","dependencies":["a6d0007exp9c-h1"],"title":"Quotient Rule of Exponents","text":"Use the quotient rule to simplify the expression: $$\\\\frac{{\\\\left(-5t^3\\\\right)}^4}{{\\\\left(-5t^3\\\\right)}^8}={\\\\left(-5t^3\\\\right)}^{4-8}={\\\\left(-5t^3\\\\right)}^{\\\\left(-4\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d0007exp9c-h3","type":"hint","dependencies":["a6d0007exp9c-h2"],"title":"Negative Exponent Rule","text":"For any nonzero real number a and natural number $$n$$, the negative rule of exponents states that $$a -n=\\\\frac{1}{a^n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d440d7.3q1","title":"Computing the Mean","body":"A study involving stress is conducted among the students on a college campus. The stress scores follow a uniform distribution with the lowest stress score equal to one and the highest equal to five. The individual stress scores follow a uniform distribution, X ~ $$ \\\\cup (1,5)$$ where a $$=$$ $$1$$ and $$b$$ $$=$$ $$5$$, Using a sample of $$75$$ students, to solve the following questions","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Using the Central Limit Theorem","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a6d440d7.3q1a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"Find the mean stress score for the $$75$$ students.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a6d440d7.3q1a-h1","type":"hint","dependencies":[],"title":"Equation for the mean value","text":"Remember \u03bc $$=$$ (a + b)/2","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d440d7.3q10","title":"Computing the Probability","body":"A study was done about violence against prostitutes and the symptoms of the posttraumatic stress that they developed. The age range of the prostitutes was $$14$$ to $$61$$. The mean age was $$30.9$$ years with a standard deviation of nine years.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Using the Central Limit Theorem","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a6d440d7.3q10a","stepAnswer":["$$0.9886$$"],"problemType":"TextBox","stepTitle":"In a sample of $$25$$ prostitutes, what is the probability that the mean age of the prostitutes is less than 35?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.9886$$","hints":{"DefaultPathway":[{"id":"a6d440d7.3q10a-h1","type":"hint","dependencies":[],"title":"Find the k Value","text":"P(x\u0304 < 35) $$=$$ $$normalcdf(-E99, 35, 30.9, 1.8)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d440d7.3q11","title":"Computing the Probability","body":"A study was done about violence against prostitutes and the symptoms of the posttraumatic stress that they developed. The age range of the prostitutes was $$14$$ to $$61$$. The mean age was $$30.9$$ years with a standard deviation of nine years.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Using the Central Limit Theorem","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a6d440d7.3q11a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Is it likely that the mean age of the sample group could be more than $$50$$ years? Interpret the results.","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No","Not Sure"],"hints":{"DefaultPathway":[{"id":"a6d440d7.3q11a-h1","type":"hint","dependencies":[],"title":"Find the k Value","text":"P(x\u0304 < 35) $$=$$ normalcdf(50, $$E99, 30.9, 1.8)$$ \u2248 $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d440d7.3q11a-h2","type":"hint","dependencies":["a6d440d7.3q11a-h1"],"title":"Explanation","text":"For this sample group, it is almost impossible for the group\u2019s average age to be more than $$50$$. However, it is still possible for an individual in this group to have an age greater than $$50$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d440d7.3q12","title":"Computing the Probability","body":"A study was done about violence against prostitutes and the symptoms of the posttraumatic stress that they developed. The age range of the prostitutes was $$14$$ to $$61$$. The mean age was $$30.9$$ years with a standard deviation of nine years.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Using the Central Limit Theorem","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a6d440d7.3q12a","stepAnswer":["$$0.0864$$"],"problemType":"TextBox","stepTitle":"In a sample of $$49$$ prostitutes, what is the probability that the sum of the ages is no less than 1,600?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.0864$$","hints":{"DefaultPathway":[{"id":"a6d440d7.3q12a-h1","type":"hint","dependencies":[],"title":"Normal CDF","text":"P(\u03a3x $$ \\\\geq $$ 1,600) $$=$$ $$normalcdf(1600, E99, 1514.10, 63)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d440d7.3q13","title":"Computing the Probability","body":"A study was done about violence against prostitutes and the symptoms of the posttraumatic stress that they developed. The age range of the prostitutes was $$14$$ to $$61$$. The mean age was $$30.9$$ years with a standard deviation of nine years.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Using the Central Limit Theorem","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a6d440d7.3q13a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Is it likely that the sum of the ages of the $$49$$ prostitutes is at most 1,595?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No","Not Sure"],"hints":{"DefaultPathway":[{"id":"a6d440d7.3q13a-h1","type":"hint","dependencies":[],"title":"Find the Probability","text":"P(\u03a3x $$ \\\\leq $$ 1,595) $$=$$ $$normalcdf(-E99, 1595, 1514.10, 63)$$ $$=$$ $$0.9005$$. This","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d440d7.3q13a-h2","type":"hint","dependencies":["a6d440d7.3q13a-h1"],"title":"Explanation","text":"This means that there is a 90% chance that the sum of the ages for the sample group $$n$$ $$=$$ $$49$$ is at most $$1595$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d440d7.3q14","title":"Computing the Probability","body":"A study was done about violence against prostitutes and the symptoms of the posttraumatic stress that they developed. The age range of the prostitutes was $$14$$ to $$61$$. The mean age was $$30.9$$ years with a standard deviation of nine years.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Using the Central Limit Theorem","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a6d440d7.3q14a","stepAnswer":["$$32.7$$"],"problemType":"TextBox","stepTitle":"Find the 95th percentile for the sample mean age of $$65$$ prostitutes.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$32.7$$","hints":{"DefaultPathway":[{"id":"a6d440d7.3q14a-h1","type":"hint","dependencies":[],"title":"Find the Probability","text":"The 95th percentile $$=$$ $$invNorm(0.95, 30.9, 1.1)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d440d7.3q14a-h2","type":"hint","dependencies":["a6d440d7.3q14a-h1"],"title":"Explanation","text":"This indicates that 95% of the prostitutes in the sample of $$65$$ are younger than $$32.7$$ years, on average.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d440d7.3q15","title":"Computing the Probability","body":"A study was done about violence against prostitutes and the symptoms of the posttraumatic stress that they developed. The age range of the prostitutes was $$14$$ to $$61$$. The mean age was $$30.9$$ years with a standard deviation of nine years.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Using the Central Limit Theorem","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a6d440d7.3q15a","stepAnswer":["$$2101.5$$"],"problemType":"TextBox","stepTitle":"Find the 90th percentile for the sum of the ages of $$65$$ prostitutes.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2101.5$$","hints":{"DefaultPathway":[{"id":"a6d440d7.3q15a-h1","type":"hint","dependencies":[],"title":"Find the Probability","text":"The 90th percentile $$=$$ $$invNorm(0.90, 2008.5, 72.56)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d440d7.3q15a-h2","type":"hint","dependencies":["a6d440d7.3q15a-h1"],"title":"Explanation","text":"This indicates that 90% of the prostitutes in the sample of $$65$$ have a sum of ages less than $$2, 101.5$$ years.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d440d7.3q2","title":"Computing the Mean","body":"A study involving stress is conducted among the students on a college campus. The stress scores follow a uniform distribution with the lowest stress score equal to one and the highest equal to five. The individual stress scores follow a uniform distribution, X ~ $$ \\\\cup (1,5)$$ where a $$=$$ $$1$$ and $$b$$ $$=$$ $$5$$, Using a sample of $$75$$ students, to solve the following questions","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Using the Central Limit Theorem","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a6d440d7.3q2a","stepAnswer":["$$\\\\frac{1.15}{\\\\sqrt{75}}$$"],"problemType":"TextBox","stepTitle":"Find the standard deviation stress score for the $$75$$ students.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1.15}{\\\\sqrt{75}}$$","hints":{"DefaultPathway":[{"id":"a6d440d7.3q2a-h1","type":"hint","dependencies":[],"title":"Equation for the standard deviation value","text":"Remember \u03c3x $$=$$ $$\\\\sqrt{\\\\frac{{\\\\left(a+b\\\\right)}^2}{12\\\\times75}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d440d7.3q3","title":"Computing the Mean","body":"A study involving stress is conducted among the students on a college campus. The stress scores follow a uniform distribution with the lowest stress score equal to one and the highest equal to five. The individual stress scores follow a uniform distribution, X ~ $$ \\\\cup (1,5)$$ where a $$=$$ $$1$$ and $$b$$ $$=$$ $$5$$, Using a sample of $$75$$ students, to solve the following questions","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Using the Central Limit Theorem","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a6d440d7.3q3a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"Find $$P(x \\\\leq $$ 2)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a6d440d7.3q3a-h1","type":"hint","dependencies":[],"title":"Normal Distribution","text":"The normal distribution is N~(3, 1.15/sqrt(75))","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d440d7.3q3a-h2","type":"hint","dependencies":["a6d440d7.3q3a-h1"],"title":"Reminder","text":"The smallest stress score is one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d440d7.3q3a-h3","type":"hint","dependencies":["a6d440d7.3q3a-h2"],"title":"Calculator Function","text":"Use the calculator function: normalcdf(lowest, highest, mean, sd)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d440d7.3q4","title":"Computing the Mean","body":"A study involving stress is conducted among the students on a college campus. The stress scores follow a uniform distribution with the lowest stress score equal to one and the highest equal to five. The individual stress scores follow a uniform distribution, X ~ $$ \\\\cup (1,5)$$ where a $$=$$ $$1$$ and $$b$$ $$=$$ $$5$$, Using a sample of $$75$$ students, to solve the following questions","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Using the Central Limit Theorem","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a6d440d7.3q4a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"Find the 90th percentile for the mean of $$75$$ stress scores.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a6d440d7.3q4a-h1","type":"hint","dependencies":[],"title":"Define variables","text":"Let k $$=$$ the 90th precentile. Find k, where $$P(x \\\\leq $$ k) $$=$$ $$0.90$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d440d7.3q4a-h2","type":"hint","dependencies":["a6d440d7.3q4a-h1"],"title":"Calculator Function","text":"Use the calculator function: InvNorm(percentage, mean, sd)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d440d7.3q4a-h3","type":"hint","dependencies":["a6d440d7.3q4a-h2"],"title":"Explanation","text":"The 90th percentile for the mean of $$75$$ scores is about $$3.2$$. This tells us that 90% of all the means of $$75$$ stress scores are at most $$3.2$$, and that 10% are at least $$3.2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d440d7.3q5","title":"Computing the Probability","body":"A study involving stress is conducted among the students on a college campus. The stress scores follow a uniform distribution with the lowest stress score equal to one and the highest equal to five. The individual stress scores follow a uniform distribution, X ~ $$ \\\\cup (1,5)$$ where a $$=$$ $$1$$ and $$b$$ $$=$$ $$5$$, Using a sample of $$75$$ students, to solve the following questions","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Using the Central Limit Theorem","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a6d440d7.3q5a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"Find P(\u03a3x < 200). Draw the graph to help you, and give an approximated answer.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a6d440d7.3q5a-h1","type":"hint","dependencies":[],"title":"Mean","text":"The mean of the sum of $$75$$ stress scores is (75)(3) $$=$$ $$225$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d440d7.3q5a-h2","type":"hint","dependencies":["a6d440d7.3q5a-h1"],"title":"Standard Deviation","text":"The standard deviation of the sum of $$75$$ stress scores is $$1.15\\\\sqrt{75}$$ $$=$$ $$9.96$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d440d7.3q5a-h3","type":"hint","dependencies":["a6d440d7.3q5a-h2"],"title":"Reminder","text":"The smallest total of $$75$$ stress scores is $$75$$, because the smallest single score is one","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d440d7.3q5a-h4","type":"hint","dependencies":["a6d440d7.3q5a-h3"],"title":"Normal CDF","text":"normalcdf (start value $$=$$ $$75$$ ,end value,mean,SD). The probability that the total of $$75$$ scores is less than $$200$$ is about zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d440d7.3q6","title":"Computing the Probability","body":"A study involving stress is conducted among the students on a college campus. The stress scores follow a uniform distribution with the lowest stress score equal to one and the highest equal to five. The individual stress scores follow a uniform distribution, X ~ $$ \\\\cup (1,5)$$ where a $$=$$ $$1$$ and $$b$$ $$=$$ $$5$$, Using a sample of $$75$$ students, to solve the following questions","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Using the Central Limit Theorem","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a6d440d7.3q6a","stepAnswer":["$$237.8$$"],"problemType":"TextBox","stepTitle":"Find the 90th percentile for the total of $$75$$ stress scores.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$237.8$$","hints":{"DefaultPathway":[{"id":"a6d440d7.3q6a-h1","type":"hint","dependencies":[],"title":"Find the k Value","text":"Let k $$=$$ the 90th percentile. Find k where P(\u03a3x < k) $$=$$ $$0.90$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d440d7.3q6a-h2","type":"hint","dependencies":["a6d440d7.3q6a-h1"],"title":"Find the k Value","text":"k $$=$$ $$237.8$$, draw a graph to help you comprehend","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d440d7.3q6a-h3","type":"hint","dependencies":["a6d440d7.3q6a-h2"],"title":"Explanation","text":"The 90th percentile for the sum of $$75$$ scores is about $$237.8$$. This tells us that 90% of all the sums of $$75$$ scores are no more than $$237.8$$ and 10% are no less than $$237.8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d440d7.3q6a-h4","type":"hint","dependencies":["a6d440d7.3q6a-h3"],"title":"Normal CDF","text":"$$invNorm(0.90$$, $$75\\\\times3$$, sqrt(75)*(1.15)) $$=$$ $$237.8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d440d7.3q7","title":"Using the clt to find probability","body":"Suppose that a market research analyst for a cell phone company conducts a study of their customers who exceed the time allowance included on their basic cell phone contract; the analyst finds that for those people who exceed the time included in their basic contract, the excess time used follows an exponential distribution with a mean of $$22$$ minutes.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Using the Central Limit Theorem","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a6d440d7.3q7a","stepAnswer":["$$0.792$$"],"problemType":"TextBox","stepTitle":"Find the probability that the mean excess time used by the $$80$$ customers in the sample is longer than $$20$$ minutes. This is asking us to find P(x\u0304 > 20). Write your answer in $$3$$ significant figures","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.792$$","hints":{"DefaultPathway":[{"id":"a6d440d7.3q7a-h1","type":"hint","dependencies":[],"title":"Compute P( x\u0304 >(20))","text":"normalcdf(20,1E99,22,22/sqrt(80)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d440d7.3q7a-h2","type":"hint","dependencies":["a6d440d7.3q7a-h1"],"title":"Find the k Value","text":"1E99 $$=$$ $${10}^{99}$$ and -1E99 $$=$$ $$-\\\\left({10}^{99}\\\\right)$$. Press the EE key for E. Or just use $${10}^{99}$$ instead of 1E99.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d440d7.3q7a-h3","type":"hint","dependencies":["a6d440d7.3q7a-h2"],"title":"Explanation","text":"The probability is $$0.7919$$ that the mean excess time used is more than $$20$$ minutes, for a sample of $$80$$ customers who exceed their contracted time allowance.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d440d7.3q8","title":"Using the clt to find probability","body":"Suppose that a market research analyst for a cell phone company conducts a study of their customers who exceed the time allowance included on their basic cell phone contract; the analyst finds that for those people who exceed the time included in their basic contract, the excess time used follows an exponential distribution with a mean of $$22$$ minutes.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Using the Central Limit Theorem","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a6d440d7.3q8a","stepAnswer":["$$0.403$$"],"problemType":"TextBox","stepTitle":"Suppose that one customer who exceeds the time limit for his cell phone contract is randomly selected. Find the probability that this individual customer\'s excess time is longer than $$20$$ minutes. This is asking us to find P(x > 20). Write your answer in $$3$$ significant figures.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.403$$","hints":{"DefaultPathway":[{"id":"a6d440d7.3q8a-h1","type":"hint","dependencies":[],"title":"Exponential Distribution for an Individual","text":"Find P(x > 20). Remember to use the exponential distribution for an individual: X~ $$\\\\operatorname{EXP}\\\\left(\\\\frac{1}{22}\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d440d7.3q8a-h2","type":"hint","dependencies":["a6d440d7.3q8a-h1"],"title":"Exponential Distribution for an Individual","text":"$$P\\\\left(x>20\\\\right)$$ $$=$$ $$e^{\\\\left(-20\\\\frac{1}{22}\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6d440d7.3q9","title":"Using the clt to find percentiles","body":"Suppose that a market research analyst for a cell phone company conducts a study of their customers who exceed the time allowance included on their basic cell phone contract; the analyst finds that for those people who exceed the time included in their basic contract, the excess time used follows an exponential distribution with a mean of $$22$$ minutes.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Using the Central Limit Theorem","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a6d440d7.3q9a","stepAnswer":["$$26$$"],"problemType":"TextBox","stepTitle":"Find the 95th percentile for the sample mean excess time for samples of $$80$$ customers who exceed their basic contract time allowances. Draw a graph.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$26$$","hints":{"DefaultPathway":[{"id":"a6d440d7.3q9a-h1","type":"hint","dependencies":[],"title":"Find k","text":"Let k $$=$$ the 95th percentile. Find k where P( x\u0304 < k) $$=$$ $$0.95$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d440d7.3q9a-h2","type":"hint","dependencies":["a6d440d7.3q9a-h1"],"title":"invNorm","text":"$$invNorm(0.95, 22$$, 22/sqrt(80)) $$=$$ $$26.0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6d440d7.3q9a-h3","type":"hint","dependencies":["a6d440d7.3q9a-h2"],"title":"Explanation","text":"The 95th percentile for the sample mean excess time used is about $$26.0$$ minutes for random samples of $$80$$ customers who exceed their contractual allowed time.\\\\nNinety five percent of such samples would have means under $$26$$ minutes; only five percent of such samples would have means above $$26$$ minutes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6e12dcextrema1","title":"For the following exercise, find the critical point(s) in the domain of the following function.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.3 Maxima and Minima","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a6e12dcextrema1a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"$$y=4\\\\sqrt{x}-x^2$$","stepBody":"Input \\"12345\\" if there are no critical points.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a6e12dcextrema1a-h1","type":"hint","dependencies":[],"title":"Finding the derivative","text":"To find the critical points, we need to find the first derivative of the function. For reference, we can use the power rule to differentiate this function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema1a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y\'=\\\\frac{2}{x^{\\\\frac{1}{2}}}+2x$$"],"dependencies":["a6e12dcextrema1a-h1"],"title":"Finding the first derivative","text":"Find the first derivative of $$y=4x^{\\\\frac{1}{2}}-x^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$y\'=\\\\frac{2}{x^{\\\\frac{1}{2}}}+2x$$","$$y\'=\\\\frac{2}{x^{\\\\left(-\\\\frac{1}{2}\\\\right)}}+2x$$","$$y\'=\\\\frac{2}{x^{\\\\frac{1}{2}}}-2x$$"]},{"id":"a6e12dcextrema1a-h3","type":"hint","dependencies":["a6e12dcextrema1a-h2"],"title":"Finding the critical point(s)","text":"The critical point(s) are the values of $$x$$ which cause the first derivative of the function to be equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a6e12dcextrema1a-h3"],"title":"Solve for the value of $$x$$ that sets the first derivative of the function to zero.","text":"Solve for $$x$$ in the following: $$0=\\\\frac{2}{x^{\\\\frac{1}{2}}}+2x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a6e12dcextrema10","title":"For the following exercise, find the critical point(s) in the domain of the following function.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.3 Maxima and Minima","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a6e12dcextrema10a","stepAnswer":["$$-1;1$$"],"problemType":"MultipleChoice","stepTitle":"$$y=x+\\\\frac{1}{x}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$-1;1$$","$$\\\\frac{-1}{5}$$","$$-1.2;1.2$$","None"],"hints":{"DefaultPathway":[{"id":"a6e12dcextrema10a-h1","type":"hint","dependencies":[],"title":"Finding the derivative","text":"To find the critical points, we need to find the first derivative of the function. For reference, the quotient rule is helpful for this differentiating this function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema10a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y\'=1-\\\\frac{1}{x^2}$$"],"dependencies":["a6e12dcextrema10a-h1"],"title":"Finding the first derivative","text":"Find the first derivative of $$y=x+\\\\frac{1}{x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$y\'=1-\\\\frac{1}{x^2}$$","$$y\'=1+\\\\frac{1}{x^2}$$","$$y\'=3-\\\\frac{1}{x^2}$$"]},{"id":"a6e12dcextrema10a-h3","type":"hint","dependencies":["a6e12dcextrema10a-h2"],"title":"Finding the critical point(s)","text":"The critical point(s) are the values of $$x$$ which cause the first derivative of the function to be equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema10a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-1;1$$"],"dependencies":["a6e12dcextrema10a-h3"],"title":"Solve for the value(s) of $$x$$ that set the first derivative of the function to zero.","text":"Solve for value(s) of $$x$$ in the following: $$0=1-\\\\frac{1}{x^2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$-1;1$$","$$\\\\frac{-1}{5}$$","$$-1.2;1.2$$"]}]}}]},{"id":"a6e12dcextrema11","title":"For the following exercise, find the local and/or maxima in the specified domain of the following function.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.3 Maxima and Minima","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a6e12dcextrema11a","stepAnswer":["Absolute maximum: $$x=4, y=\\\\frac{33}{2}$$ Absolute $$minimum:x=1, y=3$$"],"problemType":"MultipleChoice","stepTitle":"$$y=x^2+\\\\frac{2}{x}$$ over [1,4]","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Absolute maximum: $$x=4, y=\\\\frac{33}{2}$$ Absolute $$minimum:x=1, y=3$$","choices":["Absolute maximum: $$x=4, y=\\\\frac{33}{2}$$ Absolute $$minimum:x=1, y=3$$","Absolute maximum: $$x=4, y=\\\\frac{32}{2}$$ Absolute $$minimum:x=3, y=3$$","Absolute maximum: $$x=2, y=\\\\frac{33}{2}$$ Absolute $$minimum:x=1.5, y=3$$","None"],"hints":{"DefaultPathway":[{"id":"a6e12dcextrema11a-h1","type":"hint","dependencies":[],"title":"Applying the given endpoints to the function","text":"The first step is to evaluate $$y=x^2+\\\\frac{2}{x}$$ at $$x=1$$ and $$x=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema11a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$f(1)=3;f(4)=16.5$$"],"dependencies":["a6e12dcextrema11a-h1"],"title":"Evaluate the function at the given endpoints","text":"$$y=x^2+\\\\frac{2}{x}$$ at $$x=1$$ and $$x=4$$. Remember $$y=f(x)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$f(1)=3;f(4)=16.5$$","$$f(1)=3;f(4)=16$$","$$f(1)=2;f(4)=16.5$$"]},{"id":"a6e12dcextrema11a-h3","type":"hint","dependencies":["a6e12dcextrema11a-h2"],"title":"Finding the derivative","text":"To find the critical points, we need to find the first derivative of the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema11a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y\'=2x-2x^{\\\\left(-2\\\\right)}$$"],"dependencies":["a6e12dcextrema11a-h3"],"title":"Finding the first derivative","text":"Find the first derivative of $$y=x^2+\\\\frac{2}{x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$y\'=2x-2x^{\\\\left(-2\\\\right)}$$","$$y\'=2x-2x^2$$","$$y\'=2x+2x^{\\\\left(-2\\\\right)}$$"]},{"id":"a6e12dcextrema11a-h5","type":"hint","dependencies":["a6e12dcextrema11a-h4"],"title":"Finding the critical point(s)","text":"The critical point(s) are the values of $$x$$ which cause the first derivative of the function to be equal to zero within the given domain","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema11a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$1$$"],"dependencies":["a6e12dcextrema11a-h5"],"title":"Solve for the value(s) of $$x$$ that set the first derivative of the function to zero.","text":"Solve for value(s) of $$x$$ in the following: $$0=2x-2x^{\\\\left(-2\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$1$$","$$0$$","0;1","None"]},{"id":"a6e12dcextrema11a-h7","type":"hint","dependencies":["a6e12dcextrema11a-h6"],"title":"Finding the local and absolute maxima","text":"The final step is to compare the resulting $$y$$ values from the endpoints and critical values. The $$x$$ value with the highest corresponding $$y$$ value is the absolute maximum. The $$x$$ value with the lowest corresponding $$y$$ value is the absolute minimum. Any other continuous critical points can be local maxima with the same conditions holding true as for absolute maxima.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a6e12dcextrema12","title":"For the following exercise, find the local and/or maxima in the specified domain of the following function.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.3 Maxima and Minima","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a6e12dcextrema12a","stepAnswer":["Absolute maximum: $$x=4, y=16;$$ Absolute $$minimum:x=0, y=3$$"],"problemType":"MultipleChoice","stepTitle":"$$y=x^2+3$$ over [-1,4]","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Absolute maximum: $$x=4, y=16;$$ Absolute $$minimum:x=0, y=3$$","choices":["Absolute maximum: $$x=4, y=16;$$ Absolute $$minimum:x=0, y=3$$","Absolute maximum: $$x=4, y=\\\\frac{32}{2}$$ Absolute $$minimum:x=3, y=3$$","Absolute maximum: $$x=2, y=\\\\frac{33}{2}$$ Absolute $$minimum:x=1.5, y=3$$","None"],"hints":{"DefaultPathway":[{"id":"a6e12dcextrema12a-h1","type":"hint","dependencies":[],"title":"Applying the given endpoints to the function","text":"The first step is to evaluate $$y=x^2+3$$ at $$x=-1$$ and $$x=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema12a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$f(-1)=4;f(4)=19$$"],"dependencies":["a6e12dcextrema12a-h1"],"title":"Evaluate the function at the given endpoints","text":"$$y=x^2+3$$ at $$x=-1$$ and $$x=4$$. Remember $$y=f(x)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$f(-1)=2;f(4)=16$$","$$f(-1)=3;f(4)=16$$","$$f(-1)=4;f(4)=19$$"]},{"id":"a6e12dcextrema12a-h3","type":"hint","dependencies":["a6e12dcextrema12a-h2"],"title":"Finding the derivative","text":"To find the critical points, we need to find the first derivative of the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema12a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y\'=2x$$"],"dependencies":["a6e12dcextrema12a-h3"],"title":"Finding the first derivative","text":"Find the first derivative of $$y=x^2+3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$y\'=2x$$","$$y\'=2x-2x$$","$$y\'=2x^{\\\\left(-2\\\\right)}$$"]},{"id":"a6e12dcextrema12a-h5","type":"hint","dependencies":["a6e12dcextrema12a-h4"],"title":"Finding the critical point(s)","text":"The critical point(s) are the values of $$x$$ which cause the first derivative of the function to be equal to zero within the given domain","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema12a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$0$$"],"dependencies":["a6e12dcextrema12a-h5"],"title":"Solve for the value(s) of $$x$$ that set the first derivative of the function to zero.","text":"Solve for value(s) of $$x$$ in the following: $$0=2x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$0$$","$$2$$","$$-2$$","None"]},{"id":"a6e12dcextrema12a-h7","type":"hint","dependencies":["a6e12dcextrema12a-h6"],"title":"Finding the local and absolute maxima","text":"The final step is to compare the resulting $$y$$ values from the endpoints and critical values. The $$x$$ value with the highest corresponding $$y$$ value is the absolute maximum. The $$x$$ value with the lowest corresponding $$y$$ value is the absolute minimum. Any other continuous critical points can be local maxima with the same conditions holding true as for absolute maxima.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a6e12dcextrema13","title":"For the following exercise, find the local and/or maxima in the specified domain of the following function.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.3 Maxima and Minima","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a6e12dcextrema13a","stepAnswer":["Absolute maximum: $$x=-1, y=4;$$ Absolute $$minimum:x=0, 1, y=0;$$ Local maximum: $$x=0.5, y=0.063$$"],"problemType":"MultipleChoice","stepTitle":"$$y={\\\\left(x-x^2\\\\right)}^2$$ over [-1,1]","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Absolute maximum: $$x=-1, y=4;$$ Absolute $$minimum:x=0, 1, y=0;$$ Local maximum: $$x=0.5, y=0.063$$","choices":["Absolute maximum: $$x=-1, y=4;$$ Absolute $$minimum:x=0, 1, y=0;$$ Local maximum: $$x=0.5, y=0.063$$","Absolute maximum: $$x=-1, y=4;$$ Absolute $$minimum:x=0, 1, y=0;$$","Absolute maximum: $$x=-1, y=4;$$ Absolute $$minimum:x=1, y=0;$$","None"],"hints":{"DefaultPathway":[{"id":"a6e12dcextrema13a-h1","type":"hint","dependencies":[],"title":"Applying the given endpoints to the function","text":"The first step is to evaluate $$y={\\\\left(x-x^2\\\\right)}^2$$ at $$x=-1$$ and $$x=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema13a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$f(-1)=4;f(1)=0$$"],"dependencies":["a6e12dcextrema13a-h1"],"title":"Evaluate the function at the given endpoints","text":"$$y={\\\\left(x-x^2\\\\right)}^2$$ at $$x=-1$$ and $$x=1$$. Remember $$y=f(x)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$f(-1)=4;f(1)=0$$","$$f(-1)=4;f(1)=4$$","$$f(-1)=0;f(1)=0$$"]},{"id":"a6e12dcextrema13a-h3","type":"hint","dependencies":["a6e12dcextrema13a-h2"],"title":"Finding the derivative","text":"To find the critical points, we need to find the first derivative of the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema13a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y\'=2\\\\left(x-x^2\\\\right) \\\\left(1-2x\\\\right)$$"],"dependencies":["a6e12dcextrema13a-h3"],"title":"Finding the first derivative","text":"Find the first derivative of $$y={\\\\left(x-x^2\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$y\'=2\\\\left(x-x^2\\\\right) \\\\left(1-2x\\\\right)$$","$$y\'=2\\\\left(3x-x^2\\\\right) \\\\left(1-2x\\\\right)$$","$$y\'=2\\\\left(x-x^2\\\\right) \\\\left(1-x\\\\right)$$"]},{"id":"a6e12dcextrema13a-h5","type":"hint","dependencies":["a6e12dcextrema13a-h4"],"title":"Finding the critical point(s)","text":"The critical point(s) are the values of $$x$$ which cause the first derivative of the function to be equal to zero within the given domain","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema13a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$0$$"],"dependencies":["a6e12dcextrema13a-h5"],"title":"Solve for the value(s) of $$x$$ that set the first derivative of the function to zero.","text":"Solve for value(s) of $$x$$ in the following: $$0=2\\\\left(x-x^2\\\\right) \\\\left(1-2x\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$0$$","$$0$$","0;2;1","None"]},{"id":"a6e12dcextrema13a-h7","type":"hint","dependencies":["a6e12dcextrema13a-h6"],"title":"Finding the local and absolute maxima","text":"The final step is to compare the resulting $$y$$ values from the endpoints and critical values. The $$x$$ value with the highest corresponding $$y$$ value is the absolute maximum. The $$x$$ value with the lowest corresponding $$y$$ value is the absolute minimum. Any other continuous critical points can be local maxima with the same conditions holding true as for absolute maxima.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a6e12dcextrema14","title":"For the following exercise, find the local and/or maxima in the specified domain of the following function.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.3 Maxima and Minima","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a6e12dcextrema14a","stepAnswer":["Absolute minimum:x=(1/2),y=4"],"problemType":"MultipleChoice","stepTitle":"$$y=\\\\frac{1}{x-x^2}$$ over $$(0,1)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Absolute minimum:x=(1/2),y=4","Absolute minimum:x=(1/2),y=3","Absolute minimum:x=(1/3),y=4","None"],"hints":{"DefaultPathway":[{"id":"a6e12dcextrema14a-h1","type":"hint","dependencies":[],"title":"Applying the given endpoints to the function","text":"The first step would normally be to evaluate the function at the given endpoints, but these endpoints are non-inclusive so can\'t test them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema14a-h2","type":"hint","dependencies":["a6e12dcextrema14a-h1"],"title":"Finding the derivative","text":"To find the critical points, we need to find the first derivative of the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema14a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y\'=\\\\frac{-\\\\left(1-2x\\\\right)}{{\\\\left(x-x^2\\\\right)}^2}$$"],"dependencies":["a6e12dcextrema14a-h2"],"title":"Finding the first derivative","text":"Find the first derivative of $$y=\\\\frac{1}{x-x^2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$y\'=\\\\frac{-\\\\left(1-2x\\\\right)}{x-x^2}$$","$$y\'=\\\\frac{-\\\\left(1-4x\\\\right)}{{\\\\left(x-x^2\\\\right)}^2}$$","$$y\'=\\\\frac{-\\\\left(1-2x\\\\right)}{{\\\\left(x-x^2\\\\right)}^2}$$"]},{"id":"a6e12dcextrema14a-h4","type":"hint","dependencies":["a6e12dcextrema14a-h3"],"title":"Finding the critical point(s)","text":"The critical point(s) are the values of $$x$$ which cause the first derivative of the function to be equal to zero within the given domain","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema14a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$0.5$$"],"dependencies":["a6e12dcextrema14a-h4"],"title":"Solve for the value(s) of $$x$$ that set the first derivative of the function to zero.","text":"Solve for value(s) of $$x$$ in the following: $$0=\\\\frac{-\\\\left(1-2x\\\\right)}{{\\\\left(x-x^2\\\\right)}^2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$0.5$$","$$0$$","$$1$$","None"]},{"id":"a6e12dcextrema14a-h6","type":"hint","dependencies":["a6e12dcextrema14a-h5"],"title":"Finding the local and absolute maxima","text":"The final step is to compare the resulting $$y$$ values from the critical values. The $$x$$ value with the highest corresponding $$y$$ value is the absolute maximum. The $$x$$ value with the lowest corresponding $$y$$ value is the absolute minimum. Any other continuous critical points can be local maxima with the same conditions holding true as for absolute maxima.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a6e12dcextrema15","title":"For the following exercise, find the local and/or maxima in the specified domain of the following function.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.3 Maxima and Minima","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a6e12dcextrema15a","stepAnswer":["Absolute maximum: $$x=1, y=\\\\sqrt{8}$$ Absolute $$minimum:x=9, y=0$$"],"problemType":"MultipleChoice","stepTitle":"$$y={\\\\left(9-x\\\\right)}^{\\\\frac{1}{2}}$$ over [1,9]","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Absolute maximum: $$x=1, y=\\\\sqrt{8}$$ Absolute $$minimum:x=9, y=0$$","choices":["Absolute maximum: $$x=1, y=\\\\sqrt{8}$$ Absolute $$minimum:x=9, y=0$$","Absolute maximum: $$x=1, y=\\\\sqrt{8}$$ Absolute $$minimum:x=4, y=0$$","None"],"hints":{"DefaultPathway":[{"id":"a6e12dcextrema15a-h1","type":"hint","dependencies":[],"title":"Applying the given endpoints to the function","text":"The first step is to evaluate $$y={\\\\left(9-x\\\\right)}^{\\\\frac{1}{2}}$$ at $$x=1$$ and $$x=9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema15a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$f(1)=\\\\sqrt{8}=0$$"],"dependencies":["a6e12dcextrema15a-h1"],"title":"Evaluate the function at the given endpoints","text":"$$y={\\\\left(9-x\\\\right)}^{\\\\frac{1}{2}}$$ at $$x=1$$ and $$x=9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$f(1)=\\\\sqrt{8}=2$$","$$f(1)=\\\\sqrt{2}=0$$","$$f(1)=\\\\sqrt{8}=0$$"]},{"id":"a6e12dcextrema15a-h3","type":"hint","dependencies":["a6e12dcextrema15a-h2"],"title":"Finding the derivative","text":"To find the critical points, we need to find the first derivative of the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema15a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y\'=\\\\frac{-1}{2\\\\sqrt{9-x}}$$"],"dependencies":["a6e12dcextrema15a-h3"],"title":"Finding the first derivative","text":"Find the first derivative of $$y={\\\\left(9-x\\\\right)}^{\\\\frac{1}{2}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$y\'=\\\\frac{-2}{2\\\\sqrt{9-x}}$$","$$y\'=\\\\frac{-1}{2\\\\sqrt{9-x}}$$","$$y\'=\\\\frac{-1}{2\\\\sqrt{9-2x}}$$"]},{"id":"a6e12dcextrema15a-h5","type":"hint","dependencies":["a6e12dcextrema15a-h4"],"title":"Finding the critical point(s)","text":"The critical point(s) are the values of $$x$$ which cause the first derivative of the function to be equal to zero within the given domain","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema15a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["None"],"dependencies":["a6e12dcextrema15a-h5"],"title":"Solve for the value(s) of $$x$$ that set the first derivative of the function to zero.","text":"Solve for value(s) of $$x$$ in the following: $$0=\\\\frac{-1}{2\\\\sqrt{9-x}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["None","$$5$$","$$1$$","$$0$$"]},{"id":"a6e12dcextrema15a-h7","type":"hint","dependencies":["a6e12dcextrema15a-h6"],"title":"Finding the local and absolute maxima","text":"The final step is to compare the resulting $$y$$ values from the endpoints and critical values. The $$x$$ value with the highest corresponding $$y$$ value is the absolute maximum. The $$x$$ value with the lowest corresponding $$y$$ value is the absolute minimum. Any other continuous critical points can be local maxima with the same conditions holding true as for absolute maxima.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a6e12dcextrema16","title":"For the following exercise, find the local and/or maxima in the specified domain of the following function.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.3 Maxima and Minima","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a6e12dcextrema16a","stepAnswer":["Absolute maximum: x=2pi,y=2pi; Absolute $$minimum:x=0, y=0$$"],"problemType":"MultipleChoice","stepTitle":"$$y=x+sin\\\\left(x\\\\right)$$ over $$[0,2\\\\pi]$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Absolute maximum: x=2pi,y=2pi; Absolute $$minimum:x=0, y=0$$","choices":["Absolute maximum: x=2pi,y=2pi; Absolute $$minimum:x=0, y=0$$","Absolute maximum: x=2pi,y=pi; Absolute $$minimum:x=0, y=0$$","Absolute maximum: x=2pi,y=2pi; Absolute $$minimum:x=0, y=1$$","None"],"hints":{"DefaultPathway":[{"id":"a6e12dcextrema16a-h1","type":"hint","dependencies":[],"title":"Applying the given endpoints to the function","text":"The first step is to evaluate $$y=x+sin\\\\left(x\\\\right)$$ at $$x=0$$ and $$x=2\\\\pi$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema16a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$f(0)=0=2\\\\pi$$"],"dependencies":["a6e12dcextrema16a-h1"],"title":"Evaluate the function at the given endpoints","text":"$$y=x+sin\\\\left(x\\\\right)$$ at $$x=0and$$ $$x=2\\\\pi$$. Remember $$y=f(x)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$f(0)=0=2\\\\pi$$","$$f(0)=0=2$$","$$f(0)=0=pi$$"]},{"id":"a6e12dcextrema16a-h3","type":"hint","dependencies":["a6e12dcextrema16a-h2"],"title":"Finding the derivative","text":"To find the critical points, we need to find the first derivative of the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema16a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y\'=1+cos\\\\left(x\\\\right)$$"],"dependencies":["a6e12dcextrema16a-h3"],"title":"Finding the first derivative","text":"Find the first derivative of $$y=x+sin\\\\left(x\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$y\'=1+cos\\\\left(x\\\\right)$$","$$y\'=1-cos(x)$$","$$y\'=1+sin\\\\left(x\\\\right)$$"]},{"id":"a6e12dcextrema16a-h5","type":"hint","dependencies":["a6e12dcextrema16a-h4"],"title":"Finding the critical point(s)","text":"The critical point(s) are the values of $$x$$ which cause the first derivative of the function to be equal to zero within the given domain","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema16a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["0;pi"],"dependencies":["a6e12dcextrema16a-h5"],"title":"Solve for the value(s) of $$x$$ that set the first derivative of the function to zero.","text":"Solve for value(s) of $$x$$ in the following: $$0=1+cos\\\\left(x\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["0;pi","$$0$$","pi","None"]},{"id":"a6e12dcextrema16a-h7","type":"hint","dependencies":["a6e12dcextrema16a-h6"],"title":"Finding the local and absolute maxima","text":"The final step is to compare the resulting $$y$$ values from the endpoints and critical values. The $$x$$ value with the highest corresponding $$y$$ value is the absolute maximum. The $$x$$ value with the lowest corresponding $$y$$ value is the absolute minimum. Any other continuous critical points can be local maxima with the same conditions holding true as for absolute maxima.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a6e12dcextrema17","title":"For the following exercise, find the local and/or maxima in the specified domain of the following function.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.3 Maxima and Minima","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a6e12dcextrema17a","stepAnswer":["Absolute maximum: $$x=100, y=\\\\frac{101}{100}$$ Absolute $$minimum:x=0, y=0$$"],"problemType":"MultipleChoice","stepTitle":"$$y=\\\\frac{x}{1+x}$$ over [0,100]","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Absolute maximum: $$x=100, y=\\\\frac{101}{100}$$ Absolute $$minimum:x=0, y=0$$","choices":["Absolute maximum: $$x=100, y=\\\\frac{101}{100}$$ Absolute $$minimum:x=0, y=0$$","Absolute maximum: $$x=100, y=\\\\frac{101}{100}$$ Absolute $$minimum:x=0, y=1$$","Absolute maximum: $$x=100, y=\\\\frac{101}{100}$$ Absolute $$minimum:x=12.5, y=0$$","Absolute maximum: $$x=100, y=\\\\frac{102}{100}$$ Absolute $$minimum:x=0, y=0$$","None"],"hints":{"DefaultPathway":[{"id":"a6e12dcextrema17a-h1","type":"hint","dependencies":[],"title":"Applying the given endpoints to the function","text":"The first step is to evaluate $$y=\\\\frac{x}{1+x}$$ at $$x=0$$ and $$x=100$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema17a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$f(0)=0;f(100)=\\\\frac{100}{101}$$"],"dependencies":["a6e12dcextrema17a-h1"],"title":"Evaluate the function at the given endpoints","text":"$$y=\\\\frac{x}{1+x}$$ at $$x=0$$ and $$x=100$$. Remember $$y=f(x)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$f(0)=0;f(100)=\\\\frac{100}{101}$$","$$f(0)=0;f(100)=\\\\frac{100}{99}$$","$$f(0)=1;f(100)=\\\\frac{100}{101}$$"]},{"id":"a6e12dcextrema17a-h3","type":"hint","dependencies":["a6e12dcextrema17a-h2"],"title":"Finding the derivative","text":"To find the critical points, we need to find the first derivative of the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema17a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y\'={\\\\left(x+1\\\\right)}^{\\\\left(-2\\\\right)}$$"],"dependencies":["a6e12dcextrema17a-h3"],"title":"Finding the first derivative","text":"Find the first derivative of $$y=\\\\frac{x}{x+1}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$y\'={\\\\left(x+1\\\\right)}^{\\\\left(-2\\\\right)}$$","$$y\'=\\\\frac{2}{{\\\\left(x+1\\\\right)}^{\\\\left(-2\\\\right)}}$$","$$y\'={\\\\left(x+1\\\\right)}^2$$"]},{"id":"a6e12dcextrema17a-h5","type":"hint","dependencies":["a6e12dcextrema17a-h4"],"title":"Finding the critical point(s)","text":"The critical point(s) are the values of $$x$$ which cause the first derivative of the function to be equal to zero within the given domain","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema17a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["None"],"dependencies":["a6e12dcextrema17a-h5"],"title":"Solve for the value(s) of $$x$$ that set the first derivative of the function to zero.","text":"Solve for value(s) of $$x$$ in the following: $$0={\\\\left(x+1\\\\right)}^{\\\\left(-2\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$0$$","$$2$$","$$-2$$","None"]},{"id":"a6e12dcextrema17a-h7","type":"hint","dependencies":["a6e12dcextrema17a-h6"],"title":"Finding the local and absolute maxima","text":"The final step is to compare the resulting $$y$$ values from the endpoints and critical values. The $$x$$ value with the highest corresponding $$y$$ value is the absolute maximum. The $$x$$ value with the lowest corresponding $$y$$ value is the absolute minimum. Any other continuous critical points can be local maxima with the same conditions holding true as for absolute maxima.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a6e12dcextrema18","title":"For the following exercise, find the local and/or maxima in the specified domain of the following function.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.3 Maxima and Minima","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a6e12dcextrema18a","stepAnswer":["Absolute maximum: $$x=-3, y=6;$$ Absolute minimum:x=-1<x<1,y=2"],"problemType":"MultipleChoice","stepTitle":"$$y=|x+1|+|x-1|$$ over [-3,2]","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Absolute maximum: $$x=-3, y=6;$$ Absolute minimum:x=-1<x<1,y=2","choices":["Absolute maximum: $$x=-3, y=6;$$ Absolute minimum:x=-1<x<1,y=2","Absolute maximum: $$x=-3;$$ Absolute minimum:x=x<1,y=2","Absolute maximum: $$x=-1;$$ Absolute minimum:x=-1<x<1,y=2","None"],"hints":{"DefaultPathway":[{"id":"a6e12dcextrema18a-h1","type":"hint","dependencies":[],"title":"Applying the given endpoints to the function","text":"The first step is to evaluate $$y=|x+1|+|x-1|$$ at $$x=-3$$ and $$x=2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema18a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$f(-3)=6;f(2)=4$$"],"dependencies":["a6e12dcextrema18a-h1"],"title":"Evaluate the function at the given endpoints","text":"$$y=|x+1|+|x-1|$$ at $$x=-3$$ and $$x=2$$. Remember $$y=f(x)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$f(-3)=6;f(2)=4$$","$$f(-3)=6;f(2)=3$$","$$f(-3)=6;f(2)=1$$","$$f(-3)=3;f(2)=4$$"]},{"id":"a6e12dcextrema18a-h3","type":"hint","dependencies":["a6e12dcextrema18a-h2"],"title":"Finding the derivative","text":"To find the critical points, we need to find the first derivative of the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema18a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y\'=\\\\frac{x+1}{|x+1|}+\\\\frac{x-1}{|x-1|}$$"],"dependencies":["a6e12dcextrema18a-h3"],"title":"Finding the first derivative","text":"Find the first derivative of $$y=|x+1|+|x-1|$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$y\'=\\\\frac{x+1}{|x+1|}+\\\\frac{x-1}{|x-1|}$$","$$y\'=\\\\frac{x+1}{|x-1|}+\\\\frac{x-1}{|x-1|}$$","$$y\'=\\\\frac{x+1}{|x+1|}+\\\\frac{x-1}{|x+1|}$$"]},{"id":"a6e12dcextrema18a-h5","type":"hint","dependencies":["a6e12dcextrema18a-h4"],"title":"Finding the critical point(s)","text":"The critical point(s) are the values of $$x$$ which cause the first derivative of the function to be equal to zero within the given domain","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema18a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["None"],"dependencies":["a6e12dcextrema18a-h5"],"title":"Solve for the value(s) of $$x$$ that set the first derivative of the function to zero.","text":"Solve for value(s) of $$x$$ in the following: $$0=\\\\frac{x+1}{|x+1|}+\\\\frac{x-1}{|x-1|}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$0$$","$$2$$","$$-2$$","None"]},{"id":"a6e12dcextrema18a-h7","type":"hint","dependencies":["a6e12dcextrema18a-h6"],"title":"Finding the local and absolute maxima","text":"The final step is to compare the resulting $$y$$ values from the endpoints and critical values. The $$x$$ value with the highest corresponding $$y$$ value is the absolute maximum. The $$x$$ value with the lowest corresponding $$y$$ value is the absolute minimum. Any other continuous critical points can be local maxima with the same conditions holding true as for absolute maxima.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a6e12dcextrema19","title":"For the following exercise, find the local and/or maxima in the specified domain of the following function.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.3 Maxima and Minima","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a6e12dcextrema19a","stepAnswer":["Absolute maximum: x=1/3,y=0.385; Absolute $$minimum:x=4, y=-6$$"],"problemType":"MultipleChoice","stepTitle":"$$y=\\\\sqrt{x}-\\\\sqrt{x^3}$$ over [0,4]","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Absolute maximum: x=1/3,y=0.385; Absolute $$minimum:x=4, y=-6$$","choices":["Absolute maximum: x=1/3,y=0.385; Absolute $$minimum:x=4, y=-6$$","None","Absolute maximum: x=1/3,y=0.385; Absolute $$minimum:x=4, y=-3$$","Absolute maximum: x=1/3,y=0.385; Absolute $$minimum:x=0, y=0$$"],"hints":{"DefaultPathway":[{"id":"a6e12dcextrema19a-h1","type":"hint","dependencies":[],"title":"Applying the given endpoints to the function","text":"The first step is to evaluate $$y=\\\\sqrt{x}-\\\\sqrt{x^3}$$ at $$x=0$$ and $$x=4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema19a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$f(0)=0;f(4)=-6$$"],"dependencies":["a6e12dcextrema19a-h1"],"title":"Evaluate the function at the given endpoints","text":"$$y=\\\\sqrt{x}-\\\\sqrt{x^3}$$ at $$x=0$$ and $$x=4$$. Remember $$y=f(x)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$f(0)=0;f(4)=-6$$","$$f(0)=0;f(4)=-3$$","$$f(0)=0;f(4)=-8$$"]},{"id":"a6e12dcextrema19a-h3","type":"hint","dependencies":["a6e12dcextrema19a-h2"],"title":"Finding the derivative","text":"To find the critical points, we need to find the first derivative of the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema19a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y\'=\\\\frac{-\\\\left(3x-1\\\\right)}{2\\\\sqrt{x}}$$"],"dependencies":["a6e12dcextrema19a-h3"],"title":"Finding the first derivative","text":"Find the first derivative of $$y=\\\\sqrt{x}-\\\\sqrt{x^3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$y\'=\\\\frac{-\\\\left(3x-1\\\\right)}{2\\\\sqrt{x}}$$","$$y\'=\\\\frac{-\\\\left(3x-2\\\\right)}{2\\\\sqrt{x}}$$","$$y\'=\\\\frac{-\\\\left(3x-1\\\\right)}{\\\\sqrt{x}}$$"]},{"id":"a6e12dcextrema19a-h5","type":"hint","dependencies":["a6e12dcextrema19a-h4"],"title":"Finding the critical point(s)","text":"The critical point(s) are the values of $$x$$ which cause the first derivative of the function to be equal to zero within the given domain","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema19a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["a6e12dcextrema19a-h5"],"title":"Solve for the value(s) of $$x$$ that set the first derivative of the function to zero.","text":"Solve for value(s) of $$x$$ in the following: $$0=\\\\frac{-\\\\left(3x-1\\\\right)}{2\\\\sqrt{x}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$0$$","$$\\\\frac{1}{3}$$","$$\\\\frac{1}{2}$$","None"]},{"id":"a6e12dcextrema19a-h7","type":"hint","dependencies":["a6e12dcextrema19a-h6"],"title":"Finding the local and absolute maxima","text":"The final step is to compare the resulting $$y$$ values from the endpoints and critical values. The $$x$$ value with the highest corresponding $$y$$ value is the absolute maximum. The $$x$$ value with the lowest corresponding $$y$$ value is the absolute minimum. Any other continuous critical points can be local maxima with the same conditions holding true as for absolute maxima.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a6e12dcextrema2","title":"For the following exercise, find the critical point(s) in the domain of the following function.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.3 Maxima and Minima","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a6e12dcextrema2a","stepAnswer":["(1/2),(-1/2)"],"problemType":"MultipleChoice","stepTitle":"$$y=4x^3-3x$$","stepBody":"Input \\"12345\\" if there are no critical points.","answerType":"string","variabilization":{},"choices":["(1/2),(-1/2)","$$\\\\frac{1}{2}$$","(1/2),(-1/4)"],"hints":{"DefaultPathway":[{"id":"a6e12dcextrema2a-h1","type":"hint","dependencies":[],"title":"Finding the derivative","text":"To find the critical points, we need to find the first derivative of the function. For reference, we can use the power rule to differentiate this function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema2a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y\'=12x^2-3$$"],"dependencies":["a6e12dcextrema2a-h1"],"title":"Finding the first derivative","text":"Find the first derivative of $$y=4x^3-3x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$y\'=12x^2-3$$","$$y\'=12x^2+3$$","$$y\'=12x^2-x$$"]},{"id":"a6e12dcextrema2a-h3","type":"hint","dependencies":["a6e12dcextrema2a-h2"],"title":"Finding the critical point(s)","text":"The critical point(s) are the values of $$x$$ which cause the first derivative of the function to be equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema2a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["(1/2),(-1/2)"],"dependencies":["a6e12dcextrema2a-h3"],"title":"Solve for the value(s) of $$x$$ that set the first derivative of the function to zero.","text":"Solve for value(s) of $$x$$ in the following: $$0=12x^2-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["(1/2),(-1/2)","$$\\\\frac{1}{2}$$","(1/2),(-1/4)"]}]}}]},{"id":"a6e12dcextrema20","title":"For the following exercise, find the local and/or maxima in the specified domain of the following function.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.3 Maxima and Minima","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a6e12dcextrema20a","stepAnswer":["Absolute maximum: x=pi/4,y=sqrt(2); Absolute minimum:x=5pi/4,y=-sqrt(2)"],"problemType":"MultipleChoice","stepTitle":"$$y=sin\\\\left(x\\\\right)+cos\\\\left(x\\\\right)$$ over $$[0,2\\\\pi]$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Absolute maximum: x=pi/4,y=sqrt(2); Absolute minimum:x=5pi/4,y=-sqrt(2)","Absolute maximum: x=pi/4,y=sqrt(2); Absolute minimum:x=pi/4,y=-sqrt(2)","Absolute maximum: x=pi/4,y=sqrt(2); Absolute minimum:x=5pi/4,y=-sqrt(3)","None"],"hints":{"DefaultPathway":[{"id":"a6e12dcextrema20a-h1","type":"hint","dependencies":[],"title":"Applying the given endpoints to the function","text":"The first step is to evaluate $$y=sin\\\\left(x\\\\right)+cos\\\\left(x\\\\right)$$ at $$x=0$$ and $$x=2\\\\pi$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema20a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$f(0)=1=1$$"],"dependencies":["a6e12dcextrema20a-h1"],"title":"Evaluate the function at the given endpoints","text":"$$y=sin\\\\left(x\\\\right)+cos\\\\left(x\\\\right)$$ at $$x=0$$ and $$x=2\\\\pi$$. Remember $$y=f(x)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$f(0)=1=1$$","$$f(0)=1=\\\\frac{\\\\pi}{2}$$","$$f(0)=\\\\pi=1$$"]},{"id":"a6e12dcextrema20a-h3","type":"hint","dependencies":["a6e12dcextrema20a-h2"],"title":"Finding the derivative","text":"To find the critical points, we need to find the first derivative of the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema20a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y\'=cos(x)-sin(x)$$"],"dependencies":["a6e12dcextrema20a-h3"],"title":"Finding the first derivative","text":"Find the first derivative of $$y=sin\\\\left(x\\\\right)+cos\\\\left(x\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$y\'=cos(x)-sin(x)$$","$$y\'=cos\\\\left(x\\\\right)+sin\\\\left(x\\\\right)$$","$$y\'=-cos(x)-sin(x)$$"]},{"id":"a6e12dcextrema20a-h5","type":"hint","dependencies":["a6e12dcextrema20a-h4"],"title":"Finding the critical point(s)","text":"The critical point(s) are the values of $$x$$ which cause the first derivative of the function to be equal to zero within the given domain","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema20a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{\\\\pi}{4}$$"],"dependencies":["a6e12dcextrema20a-h5"],"title":"Solve for the value(s) of $$x$$ that set the first derivative of the function to zero.","text":"Solve for value(s) of $$x$$ in the following: $$0=cos(x)-sin(x)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\frac{\\\\pi}{4}$$","$$0$$","$$\\\\frac{5\\\\pi}{4}$$","None"]},{"id":"a6e12dcextrema20a-h7","type":"hint","dependencies":["a6e12dcextrema20a-h6"],"title":"Finding the local and absolute maxima","text":"The final step is to compare the resulting $$y$$ values from the endpoints and critical values. The $$x$$ value with the highest corresponding $$y$$ value is the absolute maximum. The $$x$$ value with the lowest corresponding $$y$$ value is the absolute minimum. Any other continuous critical points can be local maxima with the same conditions holding true as for absolute maxima.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a6e12dcextrema21","title":"For the following exercise, find the local and/or maxima in the specified domain of the following function.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.3 Maxima and Minima","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a6e12dcextrema21a","stepAnswer":["Absolute maximum: $$x=2.214, y=5;$$ Absolute $$minimum:x=5.356, y=-5$$"],"problemType":"MultipleChoice","stepTitle":"$$y=4sin\\\\left(x\\\\right)-3cos\\\\left(x\\\\right)$$ over $$[0,2\\\\pi]$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Absolute maximum: $$x=2.214, y=5;$$ Absolute $$minimum:x=5.356, y=-5$$","choices":["Absolute maximum: $$x=2.214, y=5;$$ Absolute $$minimum:x=5.356, y=-5$$","Absolute maximum: $$x=2.214, y=5;$$ Absolute minimum:x=4pi/3,y=-5","Absolute maximum: x=2pi,y=5; Absolute minimum:x=5pi/2,y=-4","None"],"hints":{"DefaultPathway":[{"id":"a6e12dcextrema21a-h1","type":"hint","dependencies":[],"title":"Applying the given endpoints to the function","text":"The first step is to evaluate $$y=4sin\\\\left(x\\\\right)-3cos\\\\left(x\\\\right)$$ at $$x=0$$ and $$x=2\\\\pi$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema21a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$f(0)=-3=-3$$"],"dependencies":["a6e12dcextrema21a-h1"],"title":"Evaluate the function at the given endpoints","text":"$$y=4sin\\\\left(x\\\\right)-3cos\\\\left(x\\\\right)$$ at $$x=0$$ and $$x=2\\\\pi$$. Remember $$y=f(x)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$f(0)=-3=-3$$","$$f(0)=-2=-3$$","$$f(0)=-2=-2$$"]},{"id":"a6e12dcextrema21a-h3","type":"hint","dependencies":["a6e12dcextrema21a-h2"],"title":"Finding the derivative","text":"To find the critical points, we need to find the first derivative of the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema21a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y\'=4cos\\\\left(x\\\\right)+3sin\\\\left(x\\\\right)$$"],"dependencies":["a6e12dcextrema21a-h3"],"title":"Finding the first derivative","text":"Find the first derivative of $$y=4sin\\\\left(x\\\\right)-3cos\\\\left(x\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$y\'=4cos\\\\left(x\\\\right)+3sin\\\\left(x\\\\right)$$","$$y\'=-4cos\\\\left(x\\\\right)+3sin\\\\left(x\\\\right)$$","$$y\'=4cos\\\\left(x\\\\right)-3sin\\\\left(x\\\\right)$$"]},{"id":"a6e12dcextrema21a-h5","type":"hint","dependencies":["a6e12dcextrema21a-h4"],"title":"Finding the critical point(s)","text":"The critical point(s) are the values of $$x$$ which cause the first derivative of the function to be equal to zero within the given domain","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema21a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2.214;5.356$$"],"dependencies":["a6e12dcextrema21a-h5"],"title":"Solve for the value(s) of $$x$$ that set the first derivative of the function to zero.","text":"Solve for value(s) of $$x$$ in the following: $$0=4cos\\\\left(x\\\\right)+3sin\\\\left(x\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$0$$","$$2.214;5.356$$","$$0;5.356$$","None"]},{"id":"a6e12dcextrema21a-h7","type":"hint","dependencies":["a6e12dcextrema21a-h6"],"title":"Finding the local and absolute maxima","text":"The final step is to compare the resulting $$y$$ values from the endpoints and critical values. The $$x$$ value with the highest corresponding $$y$$ value is the absolute maximum. The $$x$$ value with the lowest corresponding $$y$$ value is the absolute minimum. Any other continuous critical points can be local maxima with the same conditions holding true as for absolute maxima.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a6e12dcextrema22","title":"For the following exercise, find the local and/or maxima for the functions over $$(-\\\\infty,\\\\infty)$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.3 Maxima and Minima","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a6e12dcextrema22a","stepAnswer":["Absolute $$minimum:x=-2, y=1$$"],"problemType":"MultipleChoice","stepTitle":"$$y=x^2+4x+5$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Absolute $$minimum:x=-2, y=1$$","choices":["Absolute $$minimum:x=-2, y=1$$","Absolute $$minimum:x=-3, y=1$$","Absolute $$minimum:x=-2, y=2$$","None"],"hints":{"DefaultPathway":[{"id":"a6e12dcextrema22a-h1","type":"hint","dependencies":[],"title":"Finding the derivative","text":"To find the critical points, we need to find the first derivative of the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema22a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y\'=2x+4$$"],"dependencies":["a6e12dcextrema22a-h1"],"title":"Finding the first derivative","text":"Find the first derivative of $$y=x^2+4x+5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$y\'=2x+4$$","$$y\'=3x+4$$","$$y\'=2x+3$$"]},{"id":"a6e12dcextrema22a-h3","type":"hint","dependencies":["a6e12dcextrema22a-h2"],"title":"Finding the critical point(s)","text":"The critical point(s) are the values of $$x$$ which cause the first derivative of the function to be equal to zero within the given domain","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema22a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-2$$"],"dependencies":["a6e12dcextrema22a-h3"],"title":"Solve for the value(s) of $$x$$ that set the first derivative of the function to zero.","text":"Solve for value(s) of $$x$$ in the following: $$0=2x+4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$-2$$","$$0$$","$$3$$","None"]},{"id":"a6e12dcextrema22a-h5","type":"hint","dependencies":["a6e12dcextrema22a-h4"],"title":"Finding the local and absolute maxima","text":"The final step is to compare the resulting $$y$$ values from the critical values. The $$x$$ value with the highest corresponding $$y$$ value is the absolute maximum. The $$x$$ value with the lowest corresponding $$y$$ value is the absolute minimum. Any other continuous critical points can be local maxima with the same conditions holding true as for absolute maxima.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a6e12dcextrema23","title":"For the following exercise, find the local and/or maxima for the functions over $$(-\\\\infty,\\\\infty)$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.3 Maxima and Minima","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a6e12dcextrema23a","stepAnswer":["Local maximum: $$x=-2, y=16;$$ Local minimum: $$x=2, y=-16$$"],"problemType":"MultipleChoice","stepTitle":"$$y=x^3-12x$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Local maximum: $$x=-2, y=16;$$ Local minimum: $$x=2, y=-16$$","choices":["Local maximum: $$x=-2, y=16;$$ Local minimum: $$x=2, y=-16$$","Local maximum: $$x=-1, y=16;$$ Local minimum: $$x=2, y=-16$$","Local maximum: $$x=-2, y=16;$$ Local minimum: $$x=1, y=-6$$","None"],"hints":{"DefaultPathway":[{"id":"a6e12dcextrema23a-h1","type":"hint","dependencies":[],"title":"Finding the derivative","text":"To find the critical points, we need to find the first derivative of the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema23a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y\'=3x^2-12$$"],"dependencies":["a6e12dcextrema23a-h1"],"title":"Finding the first derivative","text":"Find the first derivative of $$y=x^3-12x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$y\'=3x^2-12$$","$$y\'=x^2-12$$","$$y\'=3x^2-8$$"]},{"id":"a6e12dcextrema23a-h3","type":"hint","dependencies":["a6e12dcextrema23a-h2"],"title":"Finding the critical point(s)","text":"The critical point(s) are the values of $$x$$ which cause the first derivative of the function to be equal to zero within the given domain","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema23a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-2;2$$"],"dependencies":["a6e12dcextrema23a-h3"],"title":"Solve for the value(s) of $$x$$ that set the first derivative of the function to zero.","text":"Solve for value(s) of $$x$$ in the following: $$0=3x^2-12$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$-2;2$$","$$-1;2$$","$$-2;1$$","None"]},{"id":"a6e12dcextrema23a-h5","type":"hint","dependencies":["a6e12dcextrema23a-h4"],"title":"Finding the local and absolute maxima","text":"The final step is to compare the resulting $$y$$ values from the critical values. Because this function is a third degree polynomial, we know that there cannot be any absolute extrema. Instead,the $$x$$ value with the highest corresponding $$y$$ value is the local maximum. The $$x$$ value with the lowest corresponding $$y$$ value is the local minimum.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a6e12dcextrema24","title":"For the following exercise, find the local and/or maxima for the functions over $$(-\\\\infty,\\\\infty)$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.3 Maxima and Minima","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a6e12dcextrema24a","stepAnswer":["Absolute minimum: $$x=-3, y=-135;$$ Local Maximum: $$x=0, y=0;$$ Local minimum: $$x=1, y=-7$$"],"problemType":"MultipleChoice","stepTitle":"$$y=3x^4+8x^3-18x^2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Absolute minimum: $$x=-3, y=-135;$$ Local Maximum: $$x=0, y=0;$$ Local minimum: $$x=1, y=-7$$","choices":["Absolute minimum: $$x=-4, y=-135;$$ Local Maximum: $$x=0, y=0;$$ Local minimum: $$x=1, y=-7$$","Absolute minimum: $$x=-3, y=-135;$$ Local Maximum: $$x=0, y=0;$$ Local minimum: $$x=1, y=4$$","Absolute minimum: $$x=-3, y=-135;$$ Local Maximum: $$x=0, y=0;$$ Local minimum: $$x=1, y=-7$$","None"],"hints":{"DefaultPathway":[{"id":"a6e12dcextrema24a-h1","type":"hint","dependencies":[],"title":"Finding the derivative","text":"To find the critical points, we need to find the first derivative of the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema24a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y\'=12x^3+24x^2-36x$$"],"dependencies":["a6e12dcextrema24a-h1"],"title":"Finding the first derivative","text":"Find the first derivative of $$y=3x^4+8x^3-18x^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$y\'=12x^3+24x^2-36x$$","$$y\'=6x^3+24x^2-36x$$","$$y\'=12x^3+24x^2-24x$$"]},{"id":"a6e12dcextrema24a-h3","type":"hint","dependencies":["a6e12dcextrema24a-h2"],"title":"Finding the critical point(s)","text":"The critical point(s) are the values of $$x$$ which cause the first derivative of the function to be equal to zero within the given domain","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema24a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-3;0;1$$"],"dependencies":["a6e12dcextrema24a-h3"],"title":"Solve for the value(s) of $$x$$ that set the first derivative of the function to zero.","text":"Solve for value(s) of $$x$$ in the following: $$0=12x^3+24x^2-36x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$-3;0;1$$","$$3;0;-1$$","0;1;3","None"]},{"id":"a6e12dcextrema24a-h5","type":"hint","dependencies":["a6e12dcextrema24a-h4"],"title":"Finding the local and absolute maxima","text":"The final step is to compare the resulting $$y$$ values from the critical values. Because this function is a positive, even degree polynomial, we know that there cannot be an absolute maximum. Instead,the $$x$$ value with the highest corresponding $$y$$ value is the local maximum. The $$x$$ value with the lowest corresponding $$y$$ value is the absolute minimum and any other continuous critical points are local minimums.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a6e12dcextrema25","title":"For the following exercise, find the local and/or maxima for the functions over $$(-\\\\infty,\\\\infty)$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.3 Maxima and Minima","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a6e12dcextrema25a","stepAnswer":["None"],"problemType":"MultipleChoice","stepTitle":"$$y=\\\\frac{x^2-1}{x-1}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Local maximum: $$x=3, y=16;$$ Local minimum: $$x=4, y=-6$$","Local maximum: $$x=0, y=0;$$ Local minimum: $$x=2, y=-4$$","Local maximum: $$x=4, y=16;$$ Local minimum: $$x=1, y=-6$$","None"],"hints":{"DefaultPathway":[{"id":"a6e12dcextrema25a-h1","type":"hint","dependencies":[],"title":"Finding the derivative","text":"To find the critical points, we need to find the first derivative of the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema25a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y\'=\\\\frac{2x}{x-1}-\\\\frac{x^2-1}{{\\\\left(x-1\\\\right)}^2}$$"],"dependencies":["a6e12dcextrema25a-h1"],"title":"Finding the first derivative","text":"Find the first derivative of $$y=\\\\frac{x^2-1}{x-1}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$y\'=\\\\frac{2x}{x-1}-\\\\frac{x^2-1}{{\\\\left(x-1\\\\right)}^2}$$","$$y\'=\\\\frac{2}{x-1}-\\\\frac{x^2-1}{{\\\\left(x-1\\\\right)}^2}$$","$$y\'=\\\\frac{2x}{x-1}-\\\\frac{x^2-2}{{\\\\left(x-1\\\\right)}^2}$$"]},{"id":"a6e12dcextrema25a-h3","type":"hint","dependencies":["a6e12dcextrema25a-h2"],"title":"Finding the critical point(s)","text":"The critical point(s) are the values of $$x$$ which cause the first derivative of the function to be equal to zero within the given domain","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema25a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["None"],"dependencies":["a6e12dcextrema25a-h3"],"title":"Solve for the value(s) of $$x$$ that set the first derivative of the function to zero.","text":"Solve for value(s) of $$x$$ in the following: $$0=3x^2-12$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$-2$$","$$0$$","$$4$$","None"]},{"id":"a6e12dcextrema25a-h5","type":"hint","dependencies":["a6e12dcextrema25a-h4"],"title":"Finding the local and absolute maxima","text":"The final step is to compare the resulting $$y$$ values from the critical values. Because we have no critical points, there are no extrema.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a6e12dcextrema3","title":"For the following exercise, find the critical point(s) in the domain of the following function.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.3 Maxima and Minima","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a6e12dcextrema3a","stepAnswer":["$$12345$$"],"problemType":"TextBox","stepTitle":"$$y=\\\\frac{1}{x-1}$$","stepBody":"Input \\"12345\\" if there are no critical points.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12345$$","hints":{"DefaultPathway":[{"id":"a6e12dcextrema3a-h1","type":"hint","dependencies":[],"title":"Finding the derivative","text":"To find the critical points, we need to find the first derivative of the function. For reference, we can use the power rule to differentiate this function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema3a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y\'=-\\\\left({\\\\left(x-1\\\\right)}^{\\\\left(-2\\\\right)}\\\\right)$$"],"dependencies":["a6e12dcextrema3a-h1"],"title":"Finding the first derivative","text":"Find the first derivative of $$y=\\\\frac{1}{x-1}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$y\'=-\\\\left({\\\\left(x-1\\\\right)}^{\\\\left(-2\\\\right)}\\\\right)$$","$$y\'=-\\\\left({\\\\left(x-1\\\\right)}^2\\\\right)$$","$$y\'={\\\\left(x-1\\\\right)}^{\\\\left(-2\\\\right)}$$"]},{"id":"a6e12dcextrema3a-h3","type":"hint","dependencies":["a6e12dcextrema3a-h2"],"title":"Finding the critical point(s)","text":"The critical point(s) are the values of $$x$$ which cause the first derivative of the function to be equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12345$$"],"dependencies":["a6e12dcextrema3a-h3"],"title":"Solve for the value(s) of $$x$$ that set the first derivative of the function to zero.","text":"Solve for value(s) of $$x$$ in the following: $$0=-\\\\left({\\\\left(x-1\\\\right)}^{\\\\left(-2\\\\right)}\\\\right)$$. Input \\"12345\\" if there are no values of $$x$$ which satisfy this condition.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a6e12dcextrema4","title":"For the following exercise, find the critical point(s) in the domain of the following function.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.3 Maxima and Minima","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a6e12dcextrema4a","stepAnswer":["$$12345$$"],"problemType":"TextBox","stepTitle":"$$y=ln(x-2)$$","stepBody":"Input \\"12345\\" if there are no critical points.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12345$$","hints":{"DefaultPathway":[{"id":"a6e12dcextrema4a-h1","type":"hint","dependencies":[],"title":"Finding the derivative","text":"To find the critical points, we need to find the first derivative of the function. For reference, we can use the chain rule to differentiate this function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema4a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y\'={\\\\left(x-2\\\\right)}^{\\\\left(-1\\\\right)}$$"],"dependencies":["a6e12dcextrema4a-h1"],"title":"Finding the first derivative","text":"Find the first derivative of $$y=ln(x-2)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$y\'={\\\\left(x-2\\\\right)}^{\\\\left(-1\\\\right)}$$","$$y\'={\\\\left(x-2\\\\right)}^{\\\\left(-2\\\\right)}$$","$$y\'={\\\\left(2x-2\\\\right)}^{\\\\left(-1\\\\right)}$$"]},{"id":"a6e12dcextrema4a-h3","type":"hint","dependencies":["a6e12dcextrema4a-h2"],"title":"Finding the critical point(s)","text":"The critical point(s) are the values of $$x$$ which cause the first derivative of the function to be equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12345$$"],"dependencies":["a6e12dcextrema4a-h3"],"title":"Solve for the value(s) of $$x$$ that set the first derivative of the function to zero.","text":"Solve for value(s) of $$x$$ in the following: $$0={\\\\left(x-2\\\\right)}^{\\\\left(-1\\\\right)}$$. Input \\"12345\\" if there are no values of $$x$$ which satisfy this condition.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a6e12dcextrema5","title":"For the following exercise, find the critical point(s) in the domain of the following function.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.3 Maxima and Minima","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a6e12dcextrema5a","stepAnswer":["$$12345$$"],"problemType":"TextBox","stepTitle":"$$y=tan(x)$$","stepBody":"Input \\"12345\\" if there are no critical points.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12345$$","hints":{"DefaultPathway":[{"id":"a6e12dcextrema5a-h1","type":"hint","dependencies":[],"title":"Finding the derivative","text":"To find the critical points, we need to find the first derivative of the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema5a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y\'=\\\\operatorname{sec}\\\\left(x\\\\right) \\\\operatorname{sec}\\\\left(x\\\\right)$$"],"dependencies":["a6e12dcextrema5a-h1"],"title":"Finding the first derivative","text":"Find the first derivative of $$y=tan(x)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$y\'=\\\\operatorname{sec}\\\\left(x\\\\right) \\\\operatorname{sec}\\\\left(x\\\\right)$$","$$y\'=tan\\\\left(x\\\\right) tan\\\\left(x\\\\right)$$","$$y\'=sec(x)$$","$$y\'=tan(x)$$"]},{"id":"a6e12dcextrema5a-h3","type":"hint","dependencies":["a6e12dcextrema5a-h2"],"title":"Finding the critical point(s)","text":"The critical point(s) are the values of $$x$$ which cause the first derivative of the function to be equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12345$$"],"dependencies":["a6e12dcextrema5a-h3"],"title":"Solve for the value(s) of $$x$$ that set the first derivative of the function to zero.","text":"Solve for value(s) of $$x$$ in the following: $$0=\\\\operatorname{sec}\\\\left(x\\\\right) \\\\operatorname{sec}\\\\left(x\\\\right)$$. Input \\"12345\\" if there are no values of $$x$$ which satisfy this condition.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a6e12dcextrema6","title":"For the following exercise, find the critical point(s) in the domain of the following function.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.3 Maxima and Minima","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a6e12dcextrema6a","stepAnswer":["$$0;-2;2$$"],"problemType":"MultipleChoice","stepTitle":"$$y=\\\\sqrt{4-x^2}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$0;-2;2$$","$$0;-2$$","$$-2;2$$","$$0$$","None"],"hints":{"DefaultPathway":[{"id":"a6e12dcextrema6a-h1","type":"hint","dependencies":[],"title":"Finding the derivative","text":"To find the critical points, we need to find the first derivative of the function. For reference, we can use the power rule to differentiate this function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema6a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y\'=\\\\left(-\\\\frac{x}{{\\\\left(4-x^2\\\\right)}^{\\\\frac{1}{2}}}\\\\right)$$"],"dependencies":["a6e12dcextrema6a-h1"],"title":"Finding the first derivative","text":"Find the first derivative of $$y={\\\\left(4-x^2\\\\right)}^{\\\\frac{1}{2}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$y\'=\\\\left(-\\\\frac{x}{{\\\\left(4-x^2\\\\right)}^{\\\\frac{1}{2}}}\\\\right)$$","$$y\'=\\\\left(-\\\\frac{x}{{\\\\left(4-x^2\\\\right)}^{\\\\left(-\\\\frac{1}{2}\\\\right)}}\\\\right)$$","$$y\'=\\\\left(-\\\\frac{x}{{\\\\left(2x\\\\right)}^{\\\\frac{1}{2}}}\\\\right)$$"]},{"id":"a6e12dcextrema6a-h3","type":"hint","dependencies":["a6e12dcextrema6a-h2"],"title":"Finding the critical point(s)","text":"The critical point(s) are the values of $$x$$ which cause the first derivative of the function to be equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema6a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$0;-2;2$$"],"dependencies":["a6e12dcextrema6a-h3"],"title":"Solve for the value(s) of $$x$$ that set the first derivative of the function to zero.","text":"Solve for value(s) of $$x$$ in the following: $$0=\\\\left(-\\\\frac{x}{{\\\\left(4-x^2\\\\right)}^{\\\\frac{1}{2}}}\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$0;-2;2$$","$$0;-2$$","$$-2;2$$","$$0$$","None"]}]}}]},{"id":"a6e12dcextrema7","title":"For the following exercise, find the critical point(s) in the domain of the following function.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.3 Maxima and Minima","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a6e12dcextrema7a","stepAnswer":["$$0$$"],"problemType":"MultipleChoice","stepTitle":"$$y=x^{\\\\frac{3}{2}}-3x^{\\\\frac{5}{2}}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$0$$","choices":["$$0$$","0;5","$$0$$","None"],"hints":{"DefaultPathway":[{"id":"a6e12dcextrema7a-h1","type":"hint","dependencies":[],"title":"Finding the derivative","text":"To find the critical points, we need to find the first derivative of the function. For reference, we can use the power rule to differentiate this function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema7a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y\'=\\\\frac{3}{2} x^{\\\\frac{1}{2}}-\\\\frac{15}{2} x^{\\\\frac{3}{2}}$$"],"dependencies":["a6e12dcextrema7a-h1"],"title":"Finding the first derivative","text":"Find the first derivative of $$y=x^{\\\\frac{3}{2}}-3x^{\\\\frac{5}{2}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$y\'=\\\\frac{3}{2} x^{\\\\frac{1}{2}}-\\\\frac{15}{2} x^{\\\\frac{3}{2}}$$","$$y\'=\\\\frac{3}{2} x^{\\\\frac{1}{2}}-\\\\frac{5}{2} x^{\\\\frac{3}{2}}$$","$$y\'=\\\\frac{3}{2} x^{\\\\frac{1}{2}}+\\\\frac{15}{2} x^{\\\\frac{3}{2}}$$"]},{"id":"a6e12dcextrema7a-h3","type":"hint","dependencies":["a6e12dcextrema7a-h2"],"title":"Finding the critical point(s)","text":"The critical point(s) are the values of $$x$$ which cause the first derivative of the function to be equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema7a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$0$$"],"dependencies":["a6e12dcextrema7a-h3"],"title":"Solve for the value(s) of $$x$$ that set the first derivative of the function to zero.","text":"Solve for value(s) of $$x$$ in the following: $$0=\\\\frac{3}{2} x^{\\\\frac{1}{2}}-\\\\frac{15}{2} x^{\\\\frac{3}{2}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$0$$","0;5","$$0$$","None"]}]}}]},{"id":"a6e12dcextrema8","title":"For the following exercise, find the critical point(s) in the domain of the following function.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.3 Maxima and Minima","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a6e12dcextrema8a","stepAnswer":["$$12345$$"],"problemType":"TextBox","stepTitle":"$$y=\\\\frac{x^2-1}{x^2+2x-3}$$","stepBody":"Input \\"12345\\" if there are no critical points.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12345$$","hints":{"DefaultPathway":[{"id":"a6e12dcextrema8a-h1","type":"hint","dependencies":[],"title":"Finding the derivative","text":"To find the critical points, we need to find the first derivative of the function. For reference, the quotient rule is helpful for this differentiating this function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema8a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y\'=\\\\frac{2}{{\\\\left(x+3\\\\right)}^2}$$"],"dependencies":["a6e12dcextrema8a-h1"],"title":"Finding the first derivative","text":"Find the first derivative of $$y=\\\\frac{x^2-1}{x^2+2x-3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$y\'=\\\\frac{2}{{\\\\left(x+3\\\\right)}^2}$$","$$y\'=\\\\frac{2}{{\\\\left(x-3\\\\right)}^2}$$","$$y\'=\\\\frac{2x}{{\\\\left(x+3\\\\right)}^2}$$"]},{"id":"a6e12dcextrema8a-h3","type":"hint","dependencies":["a6e12dcextrema8a-h2"],"title":"Finding the critical point(s)","text":"The critical point(s) are the values of $$x$$ which cause the first derivative of the function to be equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e12dcextrema8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12345$$"],"dependencies":["a6e12dcextrema8a-h3"],"title":"Solve for the value(s) of $$x$$ that set the first derivative of the function to zero.","text":"Solve for value(s) of $$x$$ in the following: $$0=\\\\frac{2}{{\\\\left(x+3\\\\right)}^2}$$. 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Estimate the number of points that statisfy the Mean Value Theorem.","variabilization":{},"oer":"https://openstax.org/books/calculus-volume-1/pages/4-4-the-mean-value-theorem <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.4 The Mean Value Theorem","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a6e13b4MVT1a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"$$3x^3+2x+1$$ from [-1,1]","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a6e13b4MVT1a-h1","type":"hint","dependencies":[],"title":"Mean Value Theorem","text":"Remember what the definition of the MVT is","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e13b4MVT1a-h2","type":"hint","dependencies":["a6e13b4MVT1a-h1"],"title":"Secant Line","text":"Draw the secant line between the $$2$$ end points then estimate the number of tangent lines parallel to the secant 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the following functions, determine if the MVT is applicable from [0,3]","variabilization":{},"oer":"https://openstax.org/books/calculus-volume-1/pages/4-4-the-mean-value-theorem <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.4 The Mean Value Theorem","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a6e13b4MVT11a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{1}{|x+1|}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a6e13b4MVT11a-h1","type":"hint","dependencies":[],"title":"MVT","text":"Remember for the MVT we must be able to take a derivative and the function must be continous","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a6e13b4MVT12","title":"Mean Value Theorem","body":"For the following functions, determine if the MVT is applicable from [-1,1]","variabilization":{},"oer":"https://openstax.org/books/calculus-volume-1/pages/4-4-the-mean-value-theorem <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.4 The Mean Value Theorem","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a6e13b4MVT12a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{x^2+3x+2}{x}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a6e13b4MVT12a-h1","type":"hint","dependencies":[],"title":"MVT","text":"Remember for the MVT we must be able to take a derivative and the function must be continous","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a6e13b4MVT13","title":"Mean Value Theorem","body":"For the following functions, determine if the MVT is applicable from [0, e-1]","variabilization":{},"oer":"https://openstax.org/books/calculus-volume-1/pages/4-4-the-mean-value-theorem <OpenStax: Calculus Volume 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1","steps":[{"id":"a6e13b4MVT14a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$5+|x|$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a6e13b4MVT14a-h1","type":"hint","dependencies":[],"title":"MVT","text":"Remember for the MVT we must be able to take a derivative and the function must be continous","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a6e13b4MVT15","title":"Mean Value Theorem","body":"For the following function, determin if there exists an interval where the MVT is applicable","variabilization":{},"oer":"https://openstax.org/books/calculus-volume-1/pages/4-4-the-mean-value-theorem <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.4 The Mean Value Theorem","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a6e13b4MVT15a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{1}{x^3}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a6e13b4MVT15a-h1","type":"hint","dependencies":[],"title":"MVT","text":"Remember for the MVT we must be able to take a derivative and the function must be continous","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a6e13b4MVT2","title":"Mean Value Theorem","body":"Graph the following function on a calculator and draw the secant line that connects the endpoints. Estimate the number of points that statisfy the Mean Value Theorem.","variabilization":{},"oer":"https://openstax.org/books/calculus-volume-1/pages/4-4-the-mean-value-theorem <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.4 The Mean Value Theorem","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a6e13b4MVT2a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"$$x^2 cos\\\\left(\\\\pi x\\\\right)$$ from [-2,2]","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"a6e13b4MVT2a-h1","type":"hint","dependencies":[],"title":"Mean Value Theorem","text":"Remember what the definition of the MVT is","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e13b4MVT2a-h2","type":"hint","dependencies":["a6e13b4MVT2a-h1"],"title":"Secant Line","text":"Draw the secant line between the $$2$$ end points then estimate the number of tangent lines parallel to the secant line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a6e13b4MVT3","title":"Mean Value Theorem","body":"For the following function, find all points c that satisfy the Mean Value Theorem from $$0$$ < c < $$2$$","variabilization":{},"oer":"https://openstax.org/books/calculus-volume-1/pages/4-4-the-mean-value-theorem <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.4 The Mean Value Theorem","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a6e13b4MVT3a","stepAnswer":["$$\\\\frac{2\\\\times3^{\\\\frac{1}{2}}}{3}$$"],"problemType":"TextBox","stepTitle":"$$x^3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2\\\\times3^{\\\\frac{1}{2}}}{3}$$","hints":{"DefaultPathway":[{"id":"a6e13b4MVT3a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":[],"title":"Mean Value Theorem","text":"First, solve for the slope of the tangent line using the MVT","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e13b4MVT3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2\\\\times3^{\\\\frac{1}{2}}}{3}$$"],"dependencies":["a6e13b4MVT3a-h1"],"title":"Solve for c","text":"Now equal the slope to the derivative to find c","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a6e13b4MVT4","title":"Mean Value Theorem","body":"For the following function, find all points c that satisfy the Mean Value Theorem from $$0$$ < c < $$2$$","variabilization":{},"oer":"https://openstax.org/books/calculus-volume-1/pages/4-4-the-mean-value-theorem <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.4 The Mean Value Theorem","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a6e13b4MVT4a","stepAnswer":["$$\\\\frac{1}{2}$$, $$1$$, $$\\\\frac{2}{3}$$"],"problemType":"MultipleChoice","stepTitle":"$$cos\\\\left(2\\\\pi x\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{1}{2}$$, $$1$$, $$\\\\frac{2}{3}$$","choices":["$$\\\\frac{3}{4}$$, $$2$$, $$\\\\frac{4}{3}$$","$$2$$, $$\\\\frac{1}{2}$$, $$\\\\frac{2}{3}$$","$$3$$, $$\\\\frac{1}{4}$$, $$\\\\frac{5}{6}$$","$$\\\\frac{1}{2}$$, $$1$$, $$\\\\frac{2}{3}$$","$$\\\\frac{1}{2}$$, $$1$$, $$\\\\frac{2}{3}$$"],"hints":{"DefaultPathway":[{"id":"a6e13b4MVT4a-h1","type":"hint","dependencies":[],"title":"Mean Value Theorem","text":"First, solve for the slope of the tangent line using the MVT","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e13b4MVT4a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1}{2}$$, $$1$$, $$\\\\frac{2}{3}$$"],"dependencies":["a6e13b4MVT4a-h1"],"title":"Solve for c","text":"Now equal the slope to the derivative to find c","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\frac{1}{2}$$, $$1$$, $$\\\\frac{2}{3}$$","$$\\\\frac{3}{4}$$, $$2$$, $$\\\\frac{4}{3}$$","$$3$$, $$\\\\frac{1}{4}$$, $$\\\\frac{5}{6}$$","$$2$$, $$\\\\frac{1}{2}$$, $$\\\\frac{2}{3}$$"]}]}}]},{"id":"a6e13b4MVT5","title":"Mean Value Theorem","body":"For the following function, find all points c that satisfy the Mean Value Theorem from $$0$$ < c < $$2$$","variabilization":{},"oer":"https://openstax.org/books/calculus-volume-1/pages/4-4-the-mean-value-theorem <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.4 The Mean Value Theorem","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a6e13b4MVT5a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(x-1\\\\right)}^{10}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a6e13b4MVT5a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":[],"title":"Mean Value Theorem","text":"First, solve for the slope of the tangent line using the MVT","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a6e13b4MVT5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a6e13b4MVT5a-h1"],"title":"Solve for c","text":"Now equal the slope to the derivative to find c","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a6e13b4MVT6","title":"Mean Value Theorem","body":"For the following functions, determine if the MVT is applicable from [-1,1]","variabilization":{},"oer":"https://openstax.org/books/calculus-volume-1/pages/4-4-the-mean-value-theorem <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.4 The Mean Value Theorem","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a6e13b4MVT6a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$|x-\\\\frac{1}{2}|$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a6e13b4MVT6a-h1","type":"hint","dependencies":[],"title":"MVT","text":"Remember for the MVT we must be able to take a derivative","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a6e13b4MVT7","title":"Mean Value Theorem","body":"For the following functions, determine if the MVT is applicable from [-1,1]","variabilization":{},"oer":"https://openstax.org/books/calculus-volume-1/pages/4-4-the-mean-value-theorem <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.4 The Mean Value Theorem","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a6e13b4MVT7a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$${|x|}^{\\\\frac{1}{2}}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a6e13b4MVT7a-h1","type":"hint","dependencies":[],"title":"MVT","text":"Remember for the MVT we must be able to take a derivative","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a6e13b4MVT8","title":"Mean Value Theorem","body":"For the following functions, determine if the MVT is applicable from [0,1]","variabilization":{},"oer":"https://openstax.org/books/calculus-volume-1/pages/4-4-the-mean-value-theorem <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.4 The Mean Value Theorem","courseName":"OpenStax: Calculus Volume 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When possible, simplify your answer.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 Multiply and Divide Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6ead19multiplingyrational16a","stepAnswer":["$$\\\\frac{3}{10}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{12}{16}+\\\\frac{4}{10}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{10}$$","hints":{"DefaultPathway":[{"id":"a6ead19multiplingyrational16a-h1","type":"hint","dependencies":[],"title":"Multiply","text":"Multiply the numerators and the denominators of each fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational16a-h2","type":"hint","dependencies":["a6ead19multiplingyrational16a-h1"],"title":"Common Factors","text":"Find common factors, and remove them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational16a-h3","type":"hint","dependencies":["a6ead19multiplingyrational16a-h2"],"title":"Simplify","text":"Simplify the fractions, such that there is one number in the numerator, and one in the denomiator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6ead19multiplingyrational17","title":"Adding Rational Expressions","body":"Find the value of the expression. 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When possible, simplify your answer.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 Multiply and Divide Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6ead19multiplingyrational19a","stepAnswer":["$$\\\\frac{8a^4}{3} b^2$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{12a^3 b b^2 2{ab}^2}{9b^3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{8a^4}{3} b^2$$","hints":{"DefaultPathway":[{"id":"a6ead19multiplingyrational19a-h1","type":"hint","dependencies":[],"title":"Multiply","text":"Multiply the numerators and the denominators of each fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational19a-h2","type":"hint","dependencies":["a6ead19multiplingyrational19a-h1"],"title":"Common Factors","text":"Find common factors between the numerator and denomiator, and group them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational19a-h3","type":"hint","dependencies":["a6ead19multiplingyrational19a-h2"],"title":"Power Rules","text":"Use the power rules to simplify variables, subtracting when powers are divided, and adding powers when they are multiplied.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational19a-h4","type":"hint","dependencies":["a6ead19multiplingyrational19a-h3"],"title":"Simplify","text":"Simplify the fractions, such that there is only one instance of a variable in the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6ead19multiplingyrational20","title":"Multiplying Rational Expressions","body":"Find the value of the expression. 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When possible, simplify your answer.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 Multiply and Divide Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6ead19multiplingyrational21a","stepAnswer":["p(p\u22124)/2(p\u22129)"],"problemType":"TextBox","stepTitle":"$$\\\\left(\\\\frac{5p^2}{p^2}-5p-36\\\\right) \\\\left(p^2-\\\\frac{16}{10} p\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{p\\\\left(p-4\\\\right)}{2\\\\left(p-9\\\\right)}$$","hints":{"DefaultPathway":[{"id":"a6ead19multiplingyrational21a-h1","type":"hint","dependencies":[],"title":"Factor","text":"Factor the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational21a-h2","type":"hint","dependencies":["a6ead19multiplingyrational21a-h1"],"title":"Multiply","text":"Multiply the numerators and the denominators of each fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational21a-h3","type":"hint","dependencies":["a6ead19multiplingyrational21a-h2"],"title":"Common Factors","text":"Find common factors between the numerator and denomiator, and group them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational21a-h4","type":"hint","dependencies":["a6ead19multiplingyrational21a-h3"],"title":"Power Rules","text":"Use the power rules to simplify variables, subtracting when powers are divided, and adding powers when they are multiplied.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational21a-h5","type":"hint","dependencies":["a6ead19multiplingyrational21a-h4"],"title":"Simplify","text":"Simplify the fractions, such that there is only one instance of a variable in the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6ead19multiplingyrational22","title":"Multiplying Rational Expressions","body":"Find the value of the expression. When possible, simplify your answer.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 Multiply and Divide Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6ead19multiplingyrational22a","stepAnswer":["r+5/2r(r+2)"],"problemType":"TextBox","stepTitle":"$$\\\\left(\\\\frac{4r}{r^2}-3r-10\\\\right) \\\\left(r^2-\\\\frac{25}{8} r^2\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$r+\\\\frac{5}{2} r\\\\left(r+2\\\\right)$$","hints":{"DefaultPathway":[{"id":"a6ead19multiplingyrational22a-h1","type":"hint","dependencies":[],"title":"Factor","text":"Factor the denominator and numerator where possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational22a-h2","type":"hint","dependencies":["a6ead19multiplingyrational22a-h1"],"title":"Multiply","text":"Multiply the numerators and the denominators of each fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational22a-h3","type":"hint","dependencies":["a6ead19multiplingyrational22a-h2"],"title":"Common Factors","text":"Find common factors between the numerator and denomiator, and group them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational22a-h4","type":"hint","dependencies":["a6ead19multiplingyrational22a-h3"],"title":"Power Rules","text":"Use the power rules to simplify variables, subtracting when powers are divided, and adding powers when they are multiplied.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational22a-h5","type":"hint","dependencies":["a6ead19multiplingyrational22a-h4"],"title":"Simplify","text":"Simplify the fractions, such that there is only one instance of a variable in the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6ead19multiplingyrational23","title":"Multiplying Rational Expressions","body":"Find the value of the expression. When possible, simplify your answer.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 Multiply and Divide Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6ead19multiplingyrational23a","stepAnswer":["x\u22127/4(x+3)"],"problemType":"TextBox","stepTitle":"$$\\\\left(x^2-\\\\frac{7x}{x^2}+6x+9\\\\right) \\\\left(x+\\\\frac{3}{4} x\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x-\\\\frac{7}{4\\\\left(x+3\\\\right)}$$","hints":{"DefaultPathway":[{"id":"a6ead19multiplingyrational23a-h1","type":"hint","dependencies":[],"title":"Factor","text":"Factor the denominator and numerator where possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational23a-h2","type":"hint","dependencies":["a6ead19multiplingyrational23a-h1"],"title":"Multiply","text":"Multiply the numerators and the denominators of each fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational23a-h3","type":"hint","dependencies":["a6ead19multiplingyrational23a-h2"],"title":"Common Factors","text":"Find common factors between the numerator and denomiator, and group them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational23a-h4","type":"hint","dependencies":["a6ead19multiplingyrational23a-h3"],"title":"Power Rules","text":"Use the power rules to simplify variables, subtracting when powers are divided, and adding powers when they are multiplied.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational23a-h5","type":"hint","dependencies":["a6ead19multiplingyrational23a-h4"],"title":"Simplify","text":"Simplify the fractions, such that there is only one instance of a variable in the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6ead19multiplingyrational24","title":"Multiplying Rational Expressions","body":"Find the value of the expression. When possible, simplify your answer.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 Multiply and Divide Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6ead19multiplingyrational24a","stepAnswer":["z+3/z(z+1)"],"problemType":"TextBox","stepTitle":"$$\\\\left(z^2+\\\\frac{3z}{z^2}-3z-4\\\\right) \\\\left(z-\\\\frac{4}{z^2}\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$z+\\\\frac{3}{z\\\\left(z+1\\\\right)}$$","hints":{"DefaultPathway":[{"id":"a6ead19multiplingyrational24a-h1","type":"hint","dependencies":[],"title":"Factor","text":"Factor the denominator and numerator where possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational24a-h2","type":"hint","dependencies":["a6ead19multiplingyrational24a-h1"],"title":"Multiply","text":"Multiply the numerators and the denominators of each fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational24a-h3","type":"hint","dependencies":["a6ead19multiplingyrational24a-h2"],"title":"Common Factors","text":"Find common factors between the numerator and denomiator, and group them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational24a-h4","type":"hint","dependencies":["a6ead19multiplingyrational24a-h3"],"title":"Power Rules","text":"Use the power rules to simplify variables, subtracting when powers are divided, and adding powers when they are multiplied.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational24a-h5","type":"hint","dependencies":["a6ead19multiplingyrational24a-h4"],"title":"Simplify","text":"Simplify the fractions, such that there is only one instance of a variable in the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6ead19multiplingyrational25","title":"Multiplying Rational Expressions","body":"Find the value of the expression. When possible, simplify your answer.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 Multiply and Divide Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6ead19multiplingyrational25a","stepAnswer":["-4(b+9)/3(b+7)"],"problemType":"TextBox","stepTitle":"$$\\\\left(28-\\\\frac{4b}{3} b-3\\\\right) \\\\left(b^2+8b-\\\\frac{9}{b^2}-49\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{-4\\\\left(b+9\\\\right)}{3\\\\left(b+7\\\\right)}$$","hints":{"DefaultPathway":[{"id":"a6ead19multiplingyrational25a-h1","type":"hint","dependencies":[],"title":"Factor","text":"Factor the denominator and numerator where possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational25a-h2","type":"hint","dependencies":["a6ead19multiplingyrational25a-h1"],"title":"Multiply","text":"Multiply the numerators and the denominators of each fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational25a-h3","type":"hint","dependencies":["a6ead19multiplingyrational25a-h2"],"title":"Common Factors","text":"Find common factors between the numerator and denomiator, and group them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational25a-h4","type":"hint","dependencies":["a6ead19multiplingyrational25a-h3"],"title":"Power Rules","text":"Use the power rules to simplify variables, subtracting when powers are divided, and adding powers when they are multiplied.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational25a-h5","type":"hint","dependencies":["a6ead19multiplingyrational25a-h4"],"title":"Simplify","text":"Simplify the fractions, such that there is only one instance of a variable in the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6ead19multiplingyrational26","title":"Multiply Rational Expressions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 Multiply and Divide Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6ead19multiplingyrational26a","stepAnswer":["$$-7$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\frac{35d-7d^2}{d^2+7d} \\\\left(d^2+12d+35\\\\right)}{d^2-25}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-7$$","hints":{"DefaultPathway":[{"id":"a6ead19multiplingyrational26a-h1","type":"hint","dependencies":[],"title":"Factor","text":"Factor the denominator and numerator where possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational26a-h2","type":"hint","dependencies":["a6ead19multiplingyrational26a-h1"],"title":"Multiply","text":"Multiply the numerators and the denominators of each fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational26a-h3","type":"hint","dependencies":["a6ead19multiplingyrational26a-h2"],"title":"Common Factors","text":"Find common factors between the numerator and denomiator, and group them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational26a-h4","type":"hint","dependencies":["a6ead19multiplingyrational26a-h3"],"title":"Power Rules","text":"Use the power rules to simplify variables, subtracting when powers are divided, and adding powers when they are multiplied.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational26a-h5","type":"hint","dependencies":["a6ead19multiplingyrational26a-h4"],"title":"Simplify","text":"Simplify the fractions, such that there is only one instance of a variable in the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6ead19multiplingyrational27","title":"Multiply Rational Expressions","body":"Find the value of the expression. When possible, simplify your answer.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 Multiply and Divide Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6ead19multiplingyrational27a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{4n+20}{n^2+n-20} \\\\left(n^2-\\\\frac{16}{4} n+16\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a6ead19multiplingyrational27a-h1","type":"hint","dependencies":[],"title":"Factor","text":"Factor the denominator and numerator where possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational27a-h2","type":"hint","dependencies":["a6ead19multiplingyrational27a-h1"],"title":"Multiply","text":"Multiply the numerators and the denominators of each fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational27a-h3","type":"hint","dependencies":["a6ead19multiplingyrational27a-h2"],"title":"Common Factors","text":"Find common factors between the numerator and denomiator, and group them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational27a-h4","type":"hint","dependencies":["a6ead19multiplingyrational27a-h3"],"title":"Power Rules","text":"Use the power rules to simplify variables, subtracting when powers are divided, and adding powers when they are multiplied.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational27a-h5","type":"hint","dependencies":["a6ead19multiplingyrational27a-h4"],"title":"Simplify","text":"Simplify the fractions, such that there is only one instance of a variable in the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6ead19multiplingyrational28","title":"Multiply Rational Expressions","body":"Find the value of the expression. When possible, simplify your answer.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 Multiply and Divide Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6ead19multiplingyrational28a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(q^2-\\\\frac{2q}{q^2}+6q-16\\\\right) \\\\left(q^2-\\\\frac{64}{q^2}-8q\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a6ead19multiplingyrational28a-h1","type":"hint","dependencies":[],"title":"Factor","text":"Factor the denominator and numerator where possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational28a-h2","type":"hint","dependencies":["a6ead19multiplingyrational28a-h1"],"title":"Multiply","text":"Multiply the numerators and the denominators of each fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational28a-h3","type":"hint","dependencies":["a6ead19multiplingyrational28a-h2"],"title":"Common Factors","text":"Find common factors between the numerator and denomiator, and group them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational28a-h4","type":"hint","dependencies":["a6ead19multiplingyrational28a-h3"],"title":"Power Rules","text":"Use the power rules to simplify variables, subtracting when powers are divided, and adding powers when they are multiplied.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational28a-h5","type":"hint","dependencies":["a6ead19multiplingyrational28a-h4"],"title":"Simplify","text":"Simplify the fractions, such that there is only one instance of a variable in the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6ead19multiplingyrational29","title":"Dividing Rational Expressions","body":"Find the value of the expression. When possible, simplify your answer.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 Multiply and Divide Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6ead19multiplingyrational29a","stepAnswer":["$$\\\\frac{\\\\left(6-t\\\\right) \\\\left(t+3\\\\right)}{t-5}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{t-\\\\frac{6}{3}-t}{t-\\\\frac{5}{t^2}-9}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{\\\\left(6-t\\\\right) \\\\left(t+3\\\\right)}{t-5}$$","hints":{"DefaultPathway":[{"id":"a6ead19multiplingyrational29a-h1","type":"hint","dependencies":[],"title":"Factor","text":"Factor the denominator and numerator where possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational29a-h2","type":"hint","dependencies":["a6ead19multiplingyrational29a-h1"],"title":"Reciprocal","text":"Take the reciprocal of the second fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational29a-h3","type":"hint","dependencies":["a6ead19multiplingyrational29a-h2"],"title":"Multiply","text":"Multiply the numerators and the denominators of each fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational29a-h4","type":"hint","dependencies":["a6ead19multiplingyrational29a-h3"],"title":"Common Factors","text":"Find common factors between the numerator and denomiator, and group them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational29a-h5","type":"hint","dependencies":["a6ead19multiplingyrational29a-h4"],"title":"Power Rules","text":"Use the power rules to simplify variables, subtracting when powers are divided, and adding powers when they are multiplied.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational29a-h6","type":"hint","dependencies":["a6ead19multiplingyrational29a-h5"],"title":"Simplify","text":"Simplify the fractions, such that there is only one instance of a variable in the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6ead19multiplingyrational30","title":"Dividing Rational Expressions","body":"Find the value of the expression. When possible, simplify your answer.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 Multiply and Divide Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6ead19multiplingyrational30a","stepAnswer":["$$\\\\frac{-1}{10-w}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{10+\\\\frac{w}{w}-8}{100-\\\\frac{w^2}{8}-w}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-1}{10-w}$$","hints":{"DefaultPathway":[{"id":"a6ead19multiplingyrational30a-h1","type":"hint","dependencies":[],"title":"Factor","text":"Factor the denominator and numerator where possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational30a-h2","type":"hint","dependencies":["a6ead19multiplingyrational30a-h1"],"title":"Reciprocal","text":"Take the reciprocal of the second fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational30a-h3","type":"hint","dependencies":["a6ead19multiplingyrational30a-h2"],"title":"Multiply","text":"Multiply the numerators and the denominators of each fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational30a-h4","type":"hint","dependencies":["a6ead19multiplingyrational30a-h3"],"title":"Common Factors","text":"Find common factors between the numerator and denomiator, and group them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational30a-h5","type":"hint","dependencies":["a6ead19multiplingyrational30a-h4"],"title":"Power Rules","text":"Use the power rules to simplify variables, subtracting when powers are divided, and adding powers when they are multiplied.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplingyrational30a-h6","type":"hint","dependencies":["a6ead19multiplingyrational30a-h5"],"title":"Simplify","text":"Simplify the fractions, such that there is only one instance of a variable in the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6ead19multiplyingrationals1","title":"Multiplying Rational Expressions","body":"Find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 Multiply and Divide Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6ead19multiplyingrationals1a","stepAnswer":["$$\\\\frac{4}{21}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{10}{28}$$ * $$\\\\frac{8}{15}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{4}{21}$$","hints":{"DefaultPathway":[{"id":"a6ead19multiplyingrationals1a-h1","type":"hint","dependencies":[],"title":"Multiplying the Numerator and Denominator","text":"We can multiply the numerators to get $$80$$ and the denominators to get $$420$$. Thus, we have $$\\\\frac{80}{420}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplyingrationals1a-h2","type":"hint","dependencies":["a6ead19multiplyingrationals1a-h1"],"title":"Simplifying if Necessary","text":"We must simply $$\\\\frac{80}{420}$$ to put it into a fraction of lowest terms. We can divide the numerator and denominator by $$20$$ to get $$\\\\frac{4}{21}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6ead19multiplyingrationals10","title":"Multiplying Rational Expressions","body":"Find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 Multiply and Divide Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6ead19multiplyingrationals10a","stepAnswer":["(x-4)/(x+3)"],"problemType":"TextBox","stepTitle":"Multiply $$\\\\frac{x^2-4x}{x^2+5x+6}$$ * $$\\\\frac{x+2}{x}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{x-4}{x+3}$$","hints":{"DefaultPathway":[{"id":"a6ead19multiplyingrationals10a-h1","type":"hint","dependencies":[],"title":"Multiplying the Numerator and Denominator","text":"We can multiply the numerator and denominators to get $$\\\\frac{\\\\left(x^2-4x\\\\right) \\\\left(x+2\\\\right)}{x\\\\left(x^2+5x+6\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplyingrationals10a-h2","type":"hint","dependencies":["a6ead19multiplyingrationals10a-h1"],"title":"Simplifying if Necessary","text":"This fraction can be simplified through factoring. $$\\\\frac{\\\\left(x^2-4x\\\\right) \\\\left(x+2\\\\right)}{x\\\\left(x^2+5x+6\\\\right)}=\\\\frac{x\\\\left(x-4\\\\right) \\\\left(x+2\\\\right)}{x\\\\left(x+3\\\\right) \\\\left(x+2\\\\right)}=\\\\frac{x-4}{x+3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6ead19multiplyingrationals11","title":"Multiplying Rational Expressions","body":"Find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 Multiply and Divide Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6ead19multiplyingrationals11a","stepAnswer":["-2(x+1)/(x+4)"],"problemType":"TextBox","stepTitle":"Multiply $$\\\\frac{16-4x}{2x-12}$$ * $$\\\\frac{x^2-5x-5}{x^2-16}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{-2\\\\left(x+1\\\\right)}{x+4}$$","hints":{"DefaultPathway":[{"id":"a6ead19multiplyingrationals11a-h1","type":"hint","dependencies":[],"title":"Multiplying the Numerator and Denominator","text":"We can multiply the numerator and denominators to get $$\\\\frac{\\\\left(16-4x\\\\right) \\\\left(x^2-4x-6\\\\right)}{\\\\left(2x-12\\\\right) \\\\left(x^2-16\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplyingrationals11a-h2","type":"hint","dependencies":["a6ead19multiplyingrationals11a-h1"],"title":"Simplifying if Necessary","text":"This fraction can be simplified through factoring. $$\\\\frac{\\\\left(16-4x\\\\right) \\\\left(x^2-5x-6\\\\right)}{\\\\left(2x-12\\\\right) \\\\left(x^2-16\\\\right)}=\\\\frac{\\\\left(16-4x\\\\right) \\\\left(x-6\\\\right) \\\\left(x+1\\\\right)}{\\\\left(2x-12\\\\right) \\\\left(x+4\\\\right) \\\\left(x-4\\\\right)}=\\\\frac{-2\\\\left(x+1\\\\right)}{x+4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6ead19multiplyingrationals12","title":"multiplying rational Expressions","body":"Find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 Multiply and Divide Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6ead19multiplyingrationals12a","stepAnswer":["-6(x+3)/(x+2)"],"problemType":"TextBox","stepTitle":"Multiply $$\\\\frac{12x-6x^2}{x^2+8x}$$ * $$\\\\left(x^2+11x+24\\\\right) \\\\left(x^2-4\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{-6\\\\left(x+3\\\\right)}{x+2}$$","hints":{"DefaultPathway":[{"id":"a6ead19multiplyingrationals12a-h1","type":"hint","dependencies":[],"title":"Multiplying the Numerator and Denominator","text":"We can multiply the numerator and denominators to get $$\\\\frac{\\\\left(12x-6x^2\\\\right) \\\\left(x^2+11x+24\\\\right)}{\\\\left(x^2+8x\\\\right) \\\\left(x^2-4\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplyingrationals12a-h2","type":"hint","dependencies":["a6ead19multiplyingrationals12a-h1"],"title":"Simplifying if Necessary","text":"This fraction can be simplified through factoring. $$\\\\frac{\\\\left(12x-6x^2\\\\right) \\\\left(x^2+11x+24\\\\right)}{\\\\left(x^2+8x\\\\right) \\\\left(x^2-4\\\\right)}=\\\\frac{-6x\\\\left(x-2\\\\right) \\\\left(x+8\\\\right) \\\\left(x+3\\\\right)}{x\\\\left(x+8\\\\right) \\\\left(x+2\\\\right) \\\\left(x-2\\\\right)}=\\\\frac{-6\\\\left(x+3\\\\right)}{x+2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6ead19multiplyingrationals13","title":"Multiplying Rational Expressions","body":"Find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 Multiply and Divide Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6ead19multiplyingrationals13a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"Multiply $$\\\\frac{2x-6}{x^2-8x+15}$$ * $$\\\\frac{x^2-25}{2x+10}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a6ead19multiplyingrationals13a-h1","type":"hint","dependencies":[],"title":"Multiplying the Numerator and Denominator","text":"We can multiply the numerator and denominators to get $$\\\\frac{\\\\left(2x-6\\\\right) \\\\left(x^2-25\\\\right)}{\\\\left(x^2-8x+15\\\\right) \\\\left(2x+10\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplyingrationals13a-h2","type":"hint","dependencies":["a6ead19multiplyingrationals13a-h1"],"title":"Simplifying if Necessary","text":"This fraction can be simplified through factoring. (2x-6)(x**2-25)/((x**2-8x+15)(2x+10))=2(x-3)(x+5)(x-5)/(2(x-5)(x-3)(x+5)=1/1=1","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6ead19multiplyingrationals2","title":"Multiplying Rational Expressions","body":"Find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 Multiply and Divide Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6ead19multiplyingrationals2a","stepAnswer":["$$\\\\frac{3}{4}$$"],"problemType":"TextBox","stepTitle":"Multiply $$\\\\frac{15\\\\frac{6}{10}}{12}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{4}$$","hints":{"DefaultPathway":[{"id":"a6ead19multiplyingrationals2a-h1","type":"hint","dependencies":[],"title":"Multiplying the Numerator and Denominator","text":"We can multiply the numerators to get $$90$$ and the denominators to get $$120$$. Thus, we have $$\\\\frac{90}{120}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplyingrationals2a-h2","type":"hint","dependencies":["a6ead19multiplyingrationals2a-h1"],"title":"Simplifying if Necessary","text":"$$\\\\frac{90}{120}$$ can be simplified to $$\\\\frac{9}{12}$$, which can in turn be simplified to $$\\\\frac{3}{4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6ead19multiplyingrationals2b","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"Multiply $$\\\\frac{6\\\\frac{20}{15}}{8}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a6ead19multiplyingrationals2b-h1","type":"hint","dependencies":[],"title":"Multiplying the Numerator and Denominator","text":"We can multiply the numerators and denominators to get $$\\\\frac{120}{120}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplyingrationals2b-h2","type":"hint","dependencies":["a6ead19multiplyingrationals2b-h1"],"title":"Simplifying if Necessary","text":"$$\\\\frac{120}{120}$$ simplifies to $$\\\\frac{1}{1}$$, or $$1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6ead19multiplyingrationals3","title":"Multiplying Rational Expressions","body":"Find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 Multiply and Divide Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6ead19multiplyingrationals3a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"Multiply $$\\\\frac{6\\\\frac{3x}{3y^2} {xy}^3}{x^2 y}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a6ead19multiplyingrationals3a-h1","type":"hint","dependencies":[],"title":"Multiplying the Numerator and Denominator","text":"We can multiply the numerators and denominators to get $$\\\\frac{12x^2 y^3}{3y^3 x^2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplyingrationals3a-h2","type":"hint","dependencies":["a6ead19multiplyingrationals3a-h1"],"title":"Simplifying if Necessary","text":"We can simplify this fraction by dividing the common terms from the numerator and denominator. If we do this, we will get $$\\\\frac{12}{3}$$, which becomes $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6ead19multiplyingrationals4","title":"Multiplying Rational Expressions","body":"Find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 Multiply and Divide Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6ead19multiplyingrationals4a","stepAnswer":["$$\\\\frac{5p^2}{2} q$$"],"problemType":"TextBox","stepTitle":"Multiply $$\\\\frac{3pq}{q^2}$$ * $$\\\\frac{5p^2 q}{6pq}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5p^2}{2} q$$","hints":{"DefaultPathway":[{"id":"a6ead19multiplyingrationals4a-h1","type":"hint","dependencies":[],"title":"Multiplying the Numerator and Denominator","text":"We can multiply the numerators and denominators to get $$\\\\frac{15p^3 q^2}{6{pq}^3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplyingrationals4a-h2","type":"hint","dependencies":["a6ead19multiplyingrationals4a-h1"],"title":"Simplifying if Necessary","text":"We can simplify this rational expression by dividing the common terms from the numerator and denominator. Diong this, we get $$\\\\frac{15p^2}{6q}$$, which is equal to $$\\\\frac{5p^2}{2} q$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6ead19multiplyingrationals4b","stepAnswer":["12y**3/7"],"problemType":"TextBox","stepTitle":"Multiply $$\\\\frac{6x^3 y}{7x^2}$$ * $$\\\\frac{2{xy}^3}{x^2 y}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{12y^3}{7}$$","hints":{"DefaultPathway":[{"id":"a6ead19multiplyingrationals4b-h1","type":"hint","dependencies":[],"title":"Multiplying the Numerator and Denominator","text":"We can multiply the numerator and denominators to get $$\\\\frac{12x^4 y^4}{7x^4 y}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplyingrationals4b-h2","type":"hint","dependencies":["a6ead19multiplyingrationals4b-h1"],"title":"Simplifying if Necessary","text":"To simplify this fraction, we must cancel out common terms from the numerator and denominator. Doing this, we get $$\\\\frac{12y^3}{7}$$, which is fully simplified.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6ead19multiplyingrationals5","title":"Multiplying Rational Expressions","body":"Find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 Multiply and Divide Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6ead19multiplyingrationals5a","stepAnswer":["x(x+3)/(x-4)(3x**2)"],"problemType":"TextBox","stepTitle":"Multiply $$\\\\frac{2x}{x^2-7x+12}$$ * $$\\\\frac{x^2-9}{6x^2}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{x\\\\left(x+3\\\\right)}{x-4} 3x^2$$","hints":{"DefaultPathway":[{"id":"a6ead19multiplyingrationals5a-h1","type":"hint","dependencies":[],"title":"Multiplying the Numerator and Denominator","text":"We can multiply the numerator and denominators to get $$\\\\frac{2x\\\\left(x^2-9\\\\right)}{\\\\left(x^2-7x+12\\\\right) 6x^2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplyingrationals5a-h2","type":"hint","dependencies":["a6ead19multiplyingrationals5a-h1"],"title":"Simplifying if Necessary","text":"This fraction can be simplified through factoring. $$\\\\frac{2x\\\\left(x+3\\\\right) \\\\left(x-3\\\\right)}{\\\\left(x-4\\\\right) \\\\left(x-3\\\\right) 6x^2}=\\\\frac{x\\\\left(x+3\\\\right)}{x-4} 3x^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6ead19multiplyingrationals6","title":"Multiplying Rational Expressions","body":"Find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 Multiply and Divide Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6ead19multiplyingrationals6a","stepAnswer":["(x-4)/(2(x+3))"],"problemType":"TextBox","stepTitle":"Multiply $$\\\\frac{5x}{x^2+5x+6}$$ * $$\\\\frac{x^2-4}{10x}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{x-4}{2\\\\left(x+3\\\\right)}$$","hints":{"DefaultPathway":[{"id":"a6ead19multiplyingrationals6a-h1","type":"hint","dependencies":[],"title":"Multiplying the Numerator and Denominator","text":"We can multiply the numerator and denominators to get $$\\\\frac{5x\\\\left(x^2-4\\\\right)}{\\\\left(x^2+5x+6\\\\right) 10x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplyingrationals6a-h2","type":"hint","dependencies":["a6ead19multiplyingrationals6a-h1"],"title":"Simplifying if Necessary","text":"This fraction can be simplified through factoring. $$\\\\frac{5x\\\\left(x^2-4\\\\right)}{\\\\left(x^2+5x+6\\\\right) 10x}=\\\\frac{5x\\\\left(x+2\\\\right) \\\\left(x-4\\\\right)}{\\\\left(x+3\\\\right) \\\\left(x+2\\\\right) 10x}=\\\\frac{x-4}{2\\\\left(x+3\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6ead19multiplyingrationals7","title":"Multiplying Rational Expressions","body":"Find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 Multiply and Divide Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6ead19multiplyingrationals7a","stepAnswer":["3(x-6)/(x+5)"],"problemType":"TextBox","stepTitle":"Multiply $$\\\\frac{\\\\frac{9x^2}{x^2+11x+30} \\\\left(x^2-36\\\\right)}{3x^2}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{3\\\\left(x-6\\\\right)}{x+5}$$","hints":{"DefaultPathway":[{"id":"a6ead19multiplyingrationals7a-h1","type":"hint","dependencies":[],"title":"Multiplying the Numerator and Denominator","text":"We can multiply the numerator and denominators to get $$\\\\frac{9x^{2\\\\left(x^2-36\\\\right)}}{\\\\left(x^2+11x+30\\\\right) 3x^2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplyingrationals7a-h2","type":"hint","dependencies":["a6ead19multiplyingrationals7a-h1"],"title":"Simplifying if Necessary","text":"This fraction can be simplified through factoring. $$\\\\frac{9x^{2\\\\left(x^2-36\\\\right)}}{\\\\left(x^2+11x+30\\\\right) 3x^2}=\\\\frac{9x^{2\\\\left(x+6\\\\right)} \\\\left(x-6\\\\right)}{\\\\left(x+6\\\\right) \\\\left(x+5\\\\right) 3x^2}=\\\\frac{3\\\\left(x-6\\\\right)}{x+5}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6ead19multiplyingrationals8","title":"Multiplying Rational Expressions","body":"Find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 Multiply and Divide Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6ead19multiplyingrationals8a","stepAnswer":["(n-7)/(2(n+1))"],"problemType":"TextBox","stepTitle":"Multiply $$\\\\frac{n^2-7n}{n^2+2n+1}$$ * $$\\\\frac{n+1}{2n}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{n-7}{2\\\\left(n+1\\\\right)}$$","hints":{"DefaultPathway":[{"id":"a6ead19multiplyingrationals8a-h1","type":"hint","dependencies":[],"title":"Multiplying the Numerator and Denominator","text":"We can multiply the numerator and denominators to get $$\\\\frac{\\\\left(n^2-7n\\\\right) \\\\left(n+1\\\\right)}{\\\\left(n^2+2n+1\\\\right) 2n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplyingrationals8a-h2","type":"hint","dependencies":["a6ead19multiplyingrationals8a-h1"],"title":"Simplifying if Necessary","text":"This fraction can be simplified through factoring. $$\\\\frac{\\\\left(n^2-7n\\\\right) \\\\left(n+1\\\\right)}{\\\\left(n^2+2n+1\\\\right) 2n}=\\\\frac{n\\\\left(n-7\\\\right) \\\\left(n+1\\\\right)}{\\\\left(n+1\\\\right) \\\\left(n+1\\\\right) 2n}=\\\\frac{n-7}{2\\\\left(n+1\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6ead19multiplyingrationals9","title":"Multiplying Rational Expressions","body":"Find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 Multiply and Divide Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a6ead19multiplyingrationals9a","stepAnswer":["(x+5)/x"],"problemType":"TextBox","stepTitle":"Multiply $$\\\\frac{x^2-25}{x^2-3x-10}$$ * $$\\\\frac{x+2}{x}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{x+5}{x}$$","hints":{"DefaultPathway":[{"id":"a6ead19multiplyingrationals9a-h1","type":"hint","dependencies":[],"title":"Multiplying the Numerator and Denominator","text":"We can multiply the numerator and denominators to get $$\\\\frac{\\\\left(x+2\\\\right) \\\\left(x^2-25\\\\right)}{\\\\left(x^2-3x-10\\\\right) x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6ead19multiplyingrationals9a-h2","type":"hint","dependencies":["a6ead19multiplyingrationals9a-h1"],"title":"Simplifying if Necessary","text":"This fraction can be simplified through factoring. $$\\\\frac{\\\\left(x+2\\\\right) \\\\left(x^2-25\\\\right)}{\\\\left(x^2-3x-10\\\\right) x}=\\\\frac{\\\\left(x+2\\\\right) \\\\left(x-5\\\\right) \\\\left(x+5\\\\right)}{\\\\left(x-5\\\\right) \\\\left(x+2\\\\right) x}=\\\\frac{x+5}{x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6f9727real1","title":"Identifying Rational Numbers","body":"Write each of the following rational numbers as either a terminating or repeating decimal.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Real Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a6f9727real1a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"What is $$\\\\frac{15}{5}$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a6f9727real1a-h1","type":"hint","dependencies":[],"title":"Fraction Explanation","text":"$$\\\\frac{15}{5}$$ implies having $$15$$ pieces needed to be split amongst $$5$$ people.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a6f9727real1a-h1"],"title":"Dividing","text":"We can express the above idea as $$15$$ divided by $$5$$. What is that equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6f9727real1b","stepAnswer":["$$0.52$$"],"problemType":"TextBox","stepTitle":"What is $$\\\\frac{13}{25}$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.52$$","hints":{"DefaultPathway":[{"id":"a6f9727real1b-h1","type":"hint","dependencies":[],"title":"Fraction Explanation","text":"$$\\\\frac{13}{25}$$ implies having $$13$$ pieces needed to be split amongst $$25$$ people.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real1b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.52$$"],"dependencies":["a6f9727real1b-h1"],"title":"Dividing","text":"We can express the above idea as $$13$$ divided by $$25$$. What is that equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6f9727real1c","stepAnswer":["$$-0.71$$"],"problemType":"MultipleChoice","stepTitle":"What is $$\\\\frac{-5}{7}$$? (To the nearest hundredth)","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-0.71$$","choices":["$$-0.25$$","$$-0.71$$","$$0.57$$","$$-0.63$$"],"hints":{"DefaultPathway":[{"id":"a6f9727real1c-h1","type":"hint","dependencies":[],"title":"Dividing","text":"We can solve this problem by dividing $$-5$$ into $$7$$ pieces.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6f9727real10","title":"Identifying Rational Numbers","body":"Decide whether each of the following numbers is rational or irrational.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Real Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a6f9727real10a","stepAnswer":["Rational"],"problemType":"MultipleChoice","stepTitle":"$$\\\\sqrt{4}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Rational","Irrational"],"hints":{"DefaultPathway":[{"id":"a6f9727real10a-h1","type":"hint","dependencies":[],"title":"Set of Rational Numbers","text":"The set of rational numbers includes fractions written as $$\\\\frac{m}{n}$$ where $$m$$ and $$n$$ are integers, and $$n$$ does not equal $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real10a-h2","type":"hint","dependencies":["a6f9727real10a-h1"],"title":"Set of Irrational Numbers","text":"The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real10a-h3","type":"hint","dependencies":["a6f9727real10a-h2"],"title":"Square Root","text":"$$\\\\sqrt{4}$$ $$=$$ $$2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real10a-h4","type":"hint","dependencies":["a6f9727real10a-h3"],"title":"Rationality of Integers","text":"$$2$$ is rational.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6f9727real10b","stepAnswer":["Irrational"],"problemType":"MultipleChoice","stepTitle":"$$\\\\sqrt{2}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Rational","Irrational"],"hints":{"DefaultPathway":[{"id":"a6f9727real10b-h1","type":"hint","dependencies":[],"title":"Set of Rational Numbers","text":"The set of rational numbers includes fractions written as $$\\\\frac{m}{n}$$ where $$m$$ and $$n$$ are integers, and $$n$$ does not equal $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real10b-h2","type":"hint","dependencies":["a6f9727real10b-h1"],"title":"Set of Irrational Numbers","text":"The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real10b-h3","type":"hint","dependencies":["a6f9727real10b-h2"],"title":"Square Roots of Prime Numbers","text":"The square root of a prime number is always irrational.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real10b-h4","type":"hint","dependencies":["a6f9727real10b-h3"],"title":"Prime Number","text":"$$2$$ is a prime number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6f9727real10c","stepAnswer":["Rational"],"problemType":"MultipleChoice","stepTitle":"$$1.414213$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Rational","Irrational"],"hints":{"DefaultPathway":[{"id":"a6f9727real10c-h1","type":"hint","dependencies":[],"title":"Set of Rational Numbers","text":"The set of rational numbers includes fractions written as $$\\\\frac{m}{n}$$ where $$m$$ and $$n$$ are integers, and $$n$$ does not equal $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real10c-h2","type":"hint","dependencies":["a6f9727real10c-h1"],"title":"Set of Irrational Numbers","text":"The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real10c-h3","type":"hint","dependencies":["a6f9727real10c-h2"],"title":"Fraction","text":"$$1.414213$$ is a terminating decimal and thus can be written as a fraction of the form $$\\\\frac{1414213}{100000}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6f9727real11","title":"Using a Formula","body":"A right circular cylinder with radius $$r$$ and height $$h$$ has the surface area S (in square units) given by the formula $$S=2\\\\pi r\\\\left(r+h\\\\right)$$. \\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Real Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a6f9727real11a","stepAnswer":["$$180\\\\pi$$"],"problemType":"TextBox","stepTitle":"Find the surface area of a cylinder with radius $$6$$ in. and height $$9$$ in. Leave the answer in terms of pi.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$180\\\\pi$$","hints":{"DefaultPathway":[{"id":"a6f9727real11a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitiute $$r=6$$ and $$h=9$$ into the equation to obtain $$2\\\\operatorname{\\\\pi}\\\\left(6\\\\right) \\\\left(6+9\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real11a-h2","type":"hint","dependencies":["a6f9727real11a-h1"],"title":"Parentheses","text":"Simplify the parentheses.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real11a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["a6f9727real11a-h2"],"title":"Parentheses","text":"What is $$6+9$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real11a-h4","type":"hint","dependencies":["a6f9727real11a-h3"],"title":"Multiplication","text":"The next step is to simplify multiplication and division.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$180$$"],"dependencies":["a6f9727real11a-h4"],"title":"Multiplication","text":"What is $$2\\\\times6\\\\times15$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real11a-h6","type":"hint","dependencies":["a6f9727real11a-h5"],"title":"Multiply by pi","text":"Multiply by pi to obtain $$180\\\\pi$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6f9727real12","title":"Simplifying Algebraic Expressions","body":"Simplify each algebraic expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Real Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a6f9727real12a","stepAnswer":["$$4x-5y-7$$"],"problemType":"MultipleChoice","stepTitle":"$$3x-2y+x-3y-7$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$4x-5y-7$$","choices":["$$4x-5y-7$$","$$2x-5y+7$$","$$4x+y-7$$","$$2x+y+7$$"],"hints":{"DefaultPathway":[{"id":"a6f9727real12a-h1","type":"hint","dependencies":[],"title":"Commutative Property of Addition","text":"The equation can be re-arranged in the form $$3x+x-2y-3y-7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real12a-h2","type":"hint","dependencies":["a6f9727real12a-h1"],"title":"Like Terms","text":"You can add like terms (terms with the same variable) together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real12a-h3","type":"hint","dependencies":["a6f9727real12a-h2"],"title":"Like Terms","text":"$$3x+x=4x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real12a-h4","type":"hint","dependencies":["a6f9727real12a-h2"],"title":"Like Terms","text":"$$-2y-3y=-5y$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6f9727real12b","stepAnswer":["$$7r-11$$"],"problemType":"MultipleChoice","stepTitle":"$$2r-5\\\\left(3-r\\\\right)+4$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$7r-11$$","choices":["$$7r-11$$","$$-3r+19$$","$$7r+19$$","$$-3r-11$$"],"hints":{"DefaultPathway":[{"id":"a6f9727real12b-h1","type":"hint","dependencies":[],"title":"Distributive Property","text":"The distributive property states that the product of a factor times a sum is the sum of the factor times each term in the sum. $$a \\\\left(b+c\\\\right)=a b+a c$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real12b-h2","type":"hint","dependencies":["a6f9727real12b-h1"],"title":"Distributive Property","text":"Using the distributive property the equation can be re-arranged in the form $$2r-15+5r+4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real12b-h3","type":"hint","dependencies":["a6f9727real12b-h2"],"title":"Commutative Property of Addition","text":"The equation can be re-arranged in the form $$2r+5r-15+4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real12b-h4","type":"hint","dependencies":["a6f9727real12b-h3"],"title":"Like Terms","text":"$$2r+5r=7r$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real12b-h5","type":"hint","dependencies":["a6f9727real12b-h3"],"title":"Like Terms","text":"$$-15+4=-11$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6f9727real12c","stepAnswer":["$$5mn-5m+n$$"],"problemType":"MultipleChoice","stepTitle":"$$2mn-5m+3mn+n$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$5mn-5m+n$$","choices":["$$5mn-5m+n$$","$$-mn-5m+n$$","$$-3m+4n$$","$$2mn-2m+n$$"],"hints":{"DefaultPathway":[{"id":"a6f9727real12c-h1","type":"hint","dependencies":[],"title":"Commutative Property of Addition","text":"The equation can be re-arranged in the form $$2mn+3mn-5m+n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real12c-h2","type":"hint","dependencies":["a6f9727real12c-h1"],"title":"Like Terms","text":"You can add like terms (terms with the same variable) together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real12c-h3","type":"hint","dependencies":["a6f9727real12c-h2"],"title":"Like Terms","text":"$$2mn+3mn=5mn$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6f9727real13","title":"Simplifying a Formula","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Real Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a6f9727real13a","stepAnswer":["$$2\\\\left(L+W\\\\right)$$"],"problemType":"TextBox","stepTitle":"A rectangle with length L and width W has a perimeter P given by $$P=L+W+L+W$$. Simplify this expression.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2\\\\left(L+W\\\\right)$$","hints":{"DefaultPathway":[{"id":"a6f9727real13a-h1","type":"hint","dependencies":[],"title":"Commutative Property of Addition","text":"The equation can be re-arranged in the form $$L+L+W+W$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real13a-h2","type":"hint","dependencies":["a6f9727real13a-h1"],"title":"Simplify","text":"$$L+L+W+W=2L+2W$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real13a-h3","type":"hint","dependencies":["a6f9727real13a-h2"],"title":"Common Multiple","text":"Both 2L and 2W have $$2$$ as a common multiple.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real13a-h4","type":"hint","dependencies":["a6f9727real13a-h3"],"title":"Common Multiple","text":"You can take a common factor of $$2$$ outside the equation to get $$2\\\\left(L+W\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6f9727real14","title":"Firefighter Allocation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Real Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a6f9727real14a","stepAnswer":["$$10$$"],"problemType":"TextBox","stepTitle":"A town\'s total allocation for firefighter\'s wages and bene\ufb01ts in a new budget is $600,000. If wages are calculated at $40,000 per firefighter and bene\ufb01ts at $20,000 per firefighter, how many firefighters can the town employ if they spend their whole budget?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10$$","hints":{"DefaultPathway":[{"id":"a6f9727real14a-h1","type":"hint","dependencies":[],"title":"Equation","text":"If $$x$$ represents the maximum number of firemen that could be employed, then we can represent this problem with the equation 600,000=40,000x+20,000x.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real14a-h2","type":"hint","dependencies":["a6f9727real14a-h1"],"title":"Combining Like Terms","text":"40,000x+20,000x=60,000x","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real14a-h3","type":"hint","dependencies":["a6f9727real14a-h2"],"title":"Isolate the Variable","text":"To isolate $$x$$, both sides can be divided by 60,000.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real14a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a6f9727real14a-h3"],"title":"Solve for the Variable","text":"What is 600,000/60,000","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6f9727real15","title":"Baking Cookies","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Real Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a6f9727real15a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"Alice, Raul, and Maria are baking cookies together. They need $$\\\\frac{3}{4}$$ cup of flour and $$\\\\frac{1}{3}$$ cup of butter to make a dozen cookies. They each brought the ingredients they had at home. Alice brought $$2$$ cups of flour and $$\\\\frac{1}{4}$$ cup of butter, Raul brought $$1$$ cup of flour and $$\\\\frac{1}{2}$$ cup of butter, and Maria brought $$\\\\frac{5}{4}$$ cups of flour and $$\\\\frac{3}{4}$$ cup of butter. If the students have plenty of the other ingredients they need (sugar, salt, baking soda, etc.), how many whole batches of a dozen cookies can they make?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a6f9727real15a-h1","type":"hint","dependencies":[],"title":"Total Cups of Butter and Flour","text":"The first step is to calculate how many cups of each ingredient the students brought.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{17}{4}$$"],"dependencies":["a6f9727real15a-h1"],"title":"Total Cups of Flour","text":"How many cups of Flour did the students bring?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a6f9727real15a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{17}{4}$$"],"dependencies":[],"title":"Total Cups of Flour","text":"What is $$2+1+\\\\frac{5}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a6f9727real15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{2}$$"],"dependencies":["a6f9727real15a-h1"],"title":"Total Cups of Butter","text":"How many cups of Butter did the students bring?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a6f9727real15a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{2}$$"],"dependencies":[],"title":"Total Cups of Butter","text":"What is $$\\\\frac{1}{4}+\\\\frac{1}{2}+\\\\frac{3}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a6f9727real15a-h4","type":"hint","dependencies":["a6f9727real15a-h2","a6f9727real15a-h3"],"title":"Batches of Cookies","text":"Next, calculate how many batches of cookies each ingredient can contribute to.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real15a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{17}{3}$$"],"dependencies":["a6f9727real15a-h4"],"title":"Flour for Cookies","text":"What is $$\\\\frac{\\\\frac{17}{4}}{\\\\frac{3}{4}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real15a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{9}{2}$$"],"dependencies":["a6f9727real15a-h4"],"title":"Butter for Cookies","text":"What is $$\\\\frac{\\\\frac{3}{2}}{\\\\frac{1}{3}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real15a-h7","type":"hint","dependencies":["a6f9727real15a-h5","a6f9727real15a-h6"],"title":"Limting Factor","text":"The butter serves as a limiting factor since it can only make $$4$$ batches while the flour can make $$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6f9727real16","title":"Anna in D.C.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Real Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a6f9727real16a","stepAnswer":["$22"],"problemType":"MultipleChoice","stepTitle":"Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is 10%. She leaves a 15% tip on the price of her meal before the sales tax is added, and the tax is calculated on the pre-tip amount. She spends a total of $$\\\\$27.50$$ for dinner. What is the cost of her dinner without tax or tip?","stepBody":"","answerType":"string","variabilization":{},"choices":["$20","$21","$22","$24"],"hints":{"DefaultPathway":[{"id":"a6f9727real16a-h1","type":"hint","dependencies":[],"title":"Increased Cost of Meal","text":"The tax is 10% of the price of the meal and the tip is 15% of the price of the meal. Combining the meal, the tax, and the tip we get 125% of the cost of the meal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real16a-h2","type":"hint","dependencies":["a6f9727real16a-h1"],"title":"Cost of Meal","text":"Since Anna paid $$\\\\$27.50$$ total this means that $$1.25$$ times the cost of the meal is $$\\\\$27.50$$. So dividing $$\\\\$27.50$$ by $$1.25$$ gives the cost of the meal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real16a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$22$$"],"dependencies":["a6f9727real16a-h2"],"title":"Cost of Meal","text":"What is $$\\\\frac{27.5}{1.25}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6f9727real17","title":"Numeric Expressions","body":"Simplify the given expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Real Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a6f9727real17a","stepAnswer":["$$-2$$"],"problemType":"TextBox","stepTitle":"What is $$3-12\\\\times2+19$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-2$$","hints":{"DefaultPathway":[{"id":"a6f9727real17a-h1","type":"hint","dependencies":[],"title":"Multiplication","text":"The first step is to simplify multiplication and division.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real17a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-24$$"],"dependencies":["a6f9727real17a-h1"],"title":"Multiplication","text":"What is $$-12\\\\times2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real17a-h3","type":"hint","dependencies":["a6f9727real17a-h2"],"title":"Addition","text":"The next step is to simplify addition and subtraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real17a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a6f9727real17a-h3"],"title":"Addition","text":"What is $$3-24+19$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6f9727real17b","stepAnswer":["$$-6$$"],"problemType":"TextBox","stepTitle":"What is $$\\\\frac{25}{5^2}-7$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-6$$","hints":{"DefaultPathway":[{"id":"a6f9727real17b-h1","type":"hint","dependencies":[],"title":"Exponent","text":"The first step is to simplify any exponents.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real17b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["a6f9727real17b-h1"],"title":"Exponent","text":"What is $$5^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real17b-h3","type":"hint","dependencies":["a6f9727real17b-h2"],"title":"Division","text":"The next step is to simplify multiplication and division.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real17b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a6f9727real17b-h3"],"title":"Division","text":"What is $$\\\\frac{25}{25}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real17b-h5","type":"hint","dependencies":["a6f9727real17b-h4"],"title":"Subtraction","text":"The final step is to simplify addition and subtraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real17b-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6$$"],"dependencies":["a6f9727real17b-h5"],"title":"Subtraction","text":"What is $$1-7$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6f9727real18","title":"Algebraic Equations","body":"For the following exercises, solve for the variable.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Real Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a6f9727real18a","stepAnswer":["$$14$$"],"problemType":"MultipleChoice","stepTitle":"$$4y+8-2y$$ for $$y=3$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$14$$","choices":["$$14$$","$$26$$","$$17$$","$$23$$"],"hints":{"DefaultPathway":[{"id":"a6f9727real18a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$y=3$$ into the equation to obtain $$4\\\\left(3\\\\right)+8-2\\\\left(3\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a6f9727real18a-h1"],"title":"Multiplication","text":"What is 4(3)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real18a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6$$"],"dependencies":["a6f9727real18a-h1"],"title":"Multiplication","text":"What is $$-2(3)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real18a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$14$$"],"dependencies":["a6f9727real18a-h2","a6f9727real18a-h3"],"title":"Addition","text":"What is $$12+8-6$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6f9727real18b","stepAnswer":["$$-66$$"],"problemType":"MultipleChoice","stepTitle":"$$4z-2z\\\\left(1+4\\\\right)-36$$ for $$z=5$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-66$$","choices":["$$-66$$","$$-86$$","$$-30$$","$$16$$"],"hints":{"DefaultPathway":[{"id":"a6f9727real18b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$z=5$$ into the equation to obtain $$4\\\\left(5\\\\right)-2\\\\left(5\\\\right) \\\\left(1+4\\\\right)-36$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real18b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a6f9727real18b-h1"],"title":"Parentheses","text":"What is $$1+4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real18b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a6f9727real18b-h2"],"title":"Multiplication","text":"What is 4(5)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real18b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-50$$"],"dependencies":["a6f9727real18b-h3"],"title":"Multiplication","text":"What is $$-2(5)(5)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real18b-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-66$$"],"dependencies":["a6f9727real18b-h3","a6f9727real18b-h4"],"title":"Addition","text":"What is $$20-50-36$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6f9727real19","title":"Simplifying Algebraic Equations","body":"For the following exercises, simplify the expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Real Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a6f9727real19a","stepAnswer":["$$-14y-11$$"],"problemType":"MultipleChoice","stepTitle":"$$2y-4^2 y-11$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-14y-11$$","choices":["$$-14y-11$$","$$18y-11$$","$$-14y+11$$","$$18y+11$$"],"hints":{"DefaultPathway":[{"id":"a6f9727real19a-h1","type":"hint","dependencies":[],"title":"Parentheses","text":"Start by evaluating everything inside the parentheses.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["a6f9727real19a-h1"],"title":"Parentheses","text":"What is $$4^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real19a-h3","type":"hint","dependencies":["a6f9727real19a-h2"],"title":"Combining Like Terms","text":"$$2y-16y=-14y$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6f9727real19b","stepAnswer":["$$-4b+1$$"],"problemType":"MultipleChoice","stepTitle":"$$8b-4b\\\\left(3\\\\right)+1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-4b+1$$","choices":["$$-4b+1$$","$$4b+3$$","$$20b+1$$","$$12b+3$$"],"hints":{"DefaultPathway":[{"id":"a6f9727real19b-h1","type":"hint","dependencies":[],"title":"Individual Terms","text":"Start by simplifying each individual term first.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real19b-h2","type":"hint","dependencies":["a6f9727real19b-h1"],"title":"Multiplication","text":"$$4b(3)=3\\\\times4 b=12b$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real19b-h3","type":"hint","dependencies":["a6f9727real19b-h2"],"title":"Combining Like Terms","text":"$$8b-12b=-4b$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6f9727real2","title":"Converting Fractions of a Unit into a Smaller Unit","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Real Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a6f9727real2a","stepAnswer":["$$\\\\frac{24}{5}$$"],"problemType":"TextBox","stepTitle":"Five brothers are going to take turns watching their family\'s new puppy. How much time will each brother spend watching the puppy in a single day if they all watch him for an equal length of time?","stepBody":"Write your answer using only hours.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{24}{5}$$","hints":{"DefaultPathway":[{"id":"a6f9727real2a-h1","type":"hint","dependencies":[],"title":"Hours in a Day","text":"The day has $$24$$ hours.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real2a-h2","type":"hint","dependencies":["a6f9727real2a-h1"],"title":"Number Line","text":"The problem can be solved by drawing a number line of length $$24$$ and separating it into $$5$$ equal parts.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real2a-h3","type":"hint","dependencies":["a6f9727real2a-h2"],"title":"Fraction","text":"The separation of the number line can be represented as a fraction or a decimal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6f9727real2b","stepAnswer":["$$\\\\frac{25}{6}$$"],"problemType":"TextBox","stepTitle":"Mrs. Hinojosa had $$75$$ feet of ribbon. If each of the $$18$$ students in her class gets an equal length of ribbon, how long will each piece be?","stepBody":"Write your answer using only feet.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{25}{6}$$","hints":{"DefaultPathway":[{"id":"a6f9727real2b-h1","type":"hint","dependencies":[],"title":"Number Line","text":"The problem can be solved by drawing a number line of length $$75$$ and separating it into $$18$$ equal parts.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real2b-h2","type":"hint","dependencies":["a6f9727real2b-h1"],"title":"Fraction","text":"The separation of the number line can be represented as a fraction or a decimal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real2b-h3","type":"hint","dependencies":["a6f9727real2b-h2"],"title":"Simplify","text":"The process can be made easier by noticing both the length of the ribbon and the number of students are divisible by $$6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6f9727real20","title":"Fruit Salad","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Real Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a6f9727real20a","stepAnswer":["$$64$$"],"problemType":"MultipleChoice","stepTitle":"A fruit salad consists of blueberries, raspberries, grapes, and cherries. The fruit salad has a total of $$280$$ pieces of fruit. There are twice as many raspberries as blueberries, three times as many grapes as cherries, and four times as many cherries as raspberries. How many cherries are there in the fruit salad?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$64$$","choices":["$$8$$","$$16$$","$$25$$","$$64$$","$$96$$"],"hints":{"DefaultPathway":[{"id":"a6f9727real20a-h1","type":"hint","dependencies":[],"title":"Writing an Equation","text":"We are looking for the number of cherries in the fruit salad and will introduce a variable to relate the number of pieces of each fruit. If we let our variable denote the number of cherries then some work is needed to set up our relationship because the first sentence in the problem deals with blueberries and raspberries. There are twice as many raspberries as blueberries so it is natural to let $$x$$ denote the number of blueberries.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real20a-h2","type":"hint","dependencies":["a6f9727real20a-h1"],"title":"Writing an Equation","text":"We find that Blueberries can be represented by $$x$$, Rasberries by $$2x$$, Cherries by $$8x$$, and Grapes by $$24x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real20a-h3","type":"hint","dependencies":["a6f9727real20a-h2"],"title":"Writing an Equation","text":"We can express all the information we have gathered with the equation $$35x=280$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real20a-h4","type":"hint","dependencies":["a6f9727real20a-h3"],"title":"Solving the Equation","text":"To solve the equation, divide both side by $$35$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real20a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a6f9727real20a-h4"],"title":"Solving the Equation","text":"What is $$\\\\frac{280}{35}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real20a-h6","type":"hint","dependencies":["a6f9727real20a-h5"],"title":"Interpeting the Result","text":"This result tells us that there are $$8$$ times as many cherries as blueberries so there are $$64$$ cherries.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6f9727real21","title":"Discounted Books","body":"Katie and Margarita have $$\\\\$20.00$$ each to spend at Students\' Choice book store, where all students receive a 20% discount. They both want to purchase a copy of the same book which normally sells for $$\\\\$22.50$$ plus 10% sales tax.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Real Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a6f9727real21a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"To check if she has enough to purchase the book, Katie takes 20% of $$\\\\$22.50$$ and subtracts that amount from the normal price. She takes 10% of the discounted selling price and adds it back to find the purchase amount.","stepBody":"Is Katie\'s Method correct?","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a6f9727real21a-h1","type":"hint","dependencies":[],"title":"Distributive Property","text":"Using the distributive property, we see that subtracting 20% is the same as multiplying by $$(1-0.20)$$: $$22.50-(0.20(22.50))=(1-0.20)(22.50)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real21a-h2","type":"hint","dependencies":["a6f9727real21a-h1"],"title":"Multiplication","text":"Multiplying by $$1-0.20=0.80$$ is the same thing as finding $$80$$ percent. Adding 10% is the same as multiplying by $$1+0.1$$: $$18+\\\\operatorname{0.1}\\\\left(18\\\\right)=18\\\\left(1+0.1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real21a-h3","type":"hint","dependencies":["a6f9727real21a-h2"],"title":"Multiplication","text":"Multiplying by $$1+0.1=1.10$$ is the same thing as finding $$110$$ percent. Thus, Katie\'s method is correct.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6f9727real21b","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Margarita takes 80% of the normal purchase price and then computes 110% of the reduced price.","stepBody":"Does Margarita have enough money to purchase the book?","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a6f9727real21b-h1","type":"hint","dependencies":[],"title":"Percentage","text":"Margarita first computes 80% of the original price: $$(0.80)22.50=18.00$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real21b-h2","type":"hint","dependencies":["a6f9727real21b-h1"],"title":"Percentage","text":"Next, she computes 110% of the new amount: $$(1.10)18.00=19.80$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real21b-h3","type":"hint","dependencies":["a6f9727real21b-h2"],"title":"Purchase Cost","text":"Margarita has $$\\\\$20.00$$, thus she can buy the book.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6f9727real22","title":"Writing Integers as Equivalent Rational Numbers","body":"Which of the following is equivalent to the given numbers?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Real Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a6f9727real22a","stepAnswer":["$$\\\\frac{11}{1}$$"],"problemType":"MultipleChoice","stepTitle":"$$11$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{11}{1}$$","choices":["$$\\\\frac{11}{1}$$","$$\\\\frac{21}{2}$$","$$\\\\frac{11}{0}$$","$$\\\\frac{21}{1}$$","$$\\\\frac{21}{3}$$"],"hints":{"DefaultPathway":[{"id":"a6f9727real22a-h1","type":"hint","dependencies":[],"title":"Fraction Explanation","text":"Any whole number can be wrriten as a fraction as itself over $$1$$. This is equivalent to dividing the whole number by $$1$$, thus returning its original value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real22a-h2","type":"hint","dependencies":["a6f9727real22a-h1"],"title":"Fraction Explanation","text":"$$11$$ can be written as a fraction in the form $$\\\\frac{11}{1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6f9727real22b","stepAnswer":["$$\\\\frac{9}{3}$$"],"problemType":"MultipleChoice","stepTitle":"$$3$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{9}{3}$$","choices":["$$\\\\frac{3}{0}$$","$$\\\\frac{6}{3}$$","$$\\\\frac{9}{3}$$","$$\\\\frac{12}{3}$$","$$\\\\frac{15}{4}$$"],"hints":{"DefaultPathway":[{"id":"a6f9727real22b-h1","type":"hint","dependencies":[],"title":"Fraction Explanation","text":"Any whole number can be wrriten as a fraction as itself over $$1$$. This is equivalent to dividing the whole number by $$1$$, thus returning its original value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real22b-h2","type":"hint","dependencies":["a6f9727real22b-h1"],"title":"Fraction Explanation","text":"$$3$$ can be written as a fraction in the form $$\\\\frac{3}{1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real22b-h3","type":"hint","dependencies":["a6f9727real22b-h2"],"title":"Multiplication","text":"If $$\\\\frac{3}{1}$$ is multiplied by $$1$$ it would still hold its value. $$\\\\frac{3}{3}$$ can also be used to represent a value of $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real22b-h4","type":"hint","dependencies":["a6f9727real22b-h3"],"title":"Multiplication","text":"$$\\\\frac{3\\\\frac{3}{1}}{3}=\\\\frac{9}{3}=\\\\frac{3}{1}=3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6f9727real23","title":"Identifying Rationality of Numbers","body":"Decide whether each of the following numbers is rational or irrational.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Real Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a6f9727real23a","stepAnswer":["Rational"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{7}{77}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Rational","Irrational"],"hints":{"DefaultPathway":[{"id":"a6f9727real23a-h1","type":"hint","dependencies":[],"title":"Set of Rational Numbers","text":"The set of rational numbers includes fractions written as $$\\\\frac{m}{n}$$ where $$m$$ and $$n$$ are integers, and $$n$$ does not equal $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real23a-h2","type":"hint","dependencies":["a6f9727real23a-h1"],"title":"Set of Irrational Numbers","text":"The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real23a-h3","type":"hint","dependencies":["a6f9727real23a-h2"],"title":"Integers","text":"Both $$7$$ and $$77$$ are integers, and $$77$$ does not equal $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6f9727real23b","stepAnswer":["Rational"],"problemType":"MultipleChoice","stepTitle":"$$\\\\sqrt{81}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Rational","Irrational"],"hints":{"DefaultPathway":[{"id":"a6f9727real23b-h1","type":"hint","dependencies":[],"title":"Set of Rational Numbers","text":"The set of rational numbers includes fractions written as $$\\\\frac{m}{n}$$ where $$m$$ and $$n$$ are integers, and $$n$$ does not equal $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real23b-h2","type":"hint","dependencies":["a6f9727real23b-h1"],"title":"Set of Irrational Numbers","text":"The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real23b-h3","type":"hint","dependencies":["a6f9727real23b-h2"],"title":"Square Root","text":"$$\\\\sqrt{81}$$ $$=$$ $$9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real23b-h4","type":"hint","dependencies":["a6f9727real23b-h3"],"title":"Rationality of Integers","text":"$$9$$ is rational.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6f9727real23c","stepAnswer":["Irrational"],"problemType":"MultipleChoice","stepTitle":"$$4.27027002700027\u2026$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Rational","Irrational"],"hints":{"DefaultPathway":[{"id":"a6f9727real23c-h1","type":"hint","dependencies":[],"title":"Set of Rational Numbers","text":"The set of rational numbers includes fractions written as $$\\\\frac{m}{n}$$ where $$m$$ and $$n$$ are integers, and $$n$$ does not equal $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real23c-h2","type":"hint","dependencies":["a6f9727real23c-h1"],"title":"Set of Irrational Numbers","text":"The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real23c-h3","type":"hint","dependencies":["a6f9727real23c-h2"],"title":"Nonrepeating and Nonterminating Numbers","text":"The given number is not rational, are nonrepeating, and are nonterminating, thus it must be irrational.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6f9727real24","title":"Determining Rationality of Numbers","body":"Decide whether each of the following numbers is rational or irrational.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Real Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a6f9727real24a","stepAnswer":["Rational"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{-10}{3}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Rational","Irrational"],"hints":{"DefaultPathway":[{"id":"a6f9727real24a-h1","type":"hint","dependencies":[],"title":"Set of Rational Numbers","text":"The set of rational numbers includes fractions written as $$\\\\frac{m}{n}$$ where $$m$$ and $$n$$ are integers, and $$n$$ does not equal $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real24a-h2","type":"hint","dependencies":["a6f9727real24a-h1"],"title":"Set of Irrational Numbers","text":"The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real24a-h3","type":"hint","dependencies":["a6f9727real24a-h2"],"title":"Integers","text":"Both $$-10$$ and $$3$$ are integers, and $$3$$ does not equal $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6f9727real24b","stepAnswer":["Irrational"],"problemType":"MultipleChoice","stepTitle":"$$\\\\sqrt{5}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Rational","Irrational"],"hints":{"DefaultPathway":[{"id":"a6f9727real24b-h1","type":"hint","dependencies":[],"title":"Set of Rational Numbers","text":"The set of rational numbers includes fractions written as $$\\\\frac{m}{n}$$ where $$m$$ and $$n$$ are integers, and $$n$$ does not equal $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real24b-h2","type":"hint","dependencies":["a6f9727real24b-h1"],"title":"Set of Irrational Numbers","text":"The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real24b-h3","type":"hint","dependencies":["a6f9727real24b-h2"],"title":"Square Roots of Prime Numbers","text":"The square root of a prime number is always irrational.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real24b-h4","type":"hint","dependencies":["a6f9727real24b-h3"],"title":"Prime Number","text":"$$5$$ is a prime number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6f9727real24c","stepAnswer":["Rational"],"problemType":"MultipleChoice","stepTitle":"$$0.615384615384\u2026$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Rational","Irrational"],"hints":{"DefaultPathway":[{"id":"a6f9727real24c-h1","type":"hint","dependencies":[],"title":"Set of Rational Numbers","text":"The set of rational numbers includes fractions written as $$\\\\frac{m}{n}$$ where $$m$$ and $$n$$ are integers, and $$n$$ does not equal $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real24c-h2","type":"hint","dependencies":["a6f9727real24c-h1"],"title":"Set of Irrational Numbers","text":"The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real24c-h3","type":"hint","dependencies":["a6f9727real24c-h2"],"title":"Repeating Decimal","text":"The given number is a repeating decimal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6f9727real25","title":"Classifying Rational Numbers","body":"Classify each number as either rational or irrational.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Real Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a6f9727real25a","stepAnswer":["Irrational"],"problemType":"MultipleChoice","stepTitle":"$$-6\\\\pi$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Rational","Irrational"],"hints":{"DefaultPathway":[{"id":"a6f9727real25a-h1","type":"hint","dependencies":[],"title":"Set of Rational Numbers","text":"The set of rational numbers includes fractions written as $$\\\\frac{m}{n}$$ where $$m$$ and $$n$$ are integers, and $$n$$ does not equal $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real25a-h2","type":"hint","dependencies":["a6f9727real25a-h1"],"title":"Set of Irrational Numbers","text":"The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real25a-h3","type":"hint","dependencies":["a6f9727real25a-h2"],"title":"Properties of pi","text":"pi is not rational, nonrepeating, and nonterminating","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6f9727real25b","stepAnswer":["Rational"],"problemType":"MultipleChoice","stepTitle":"$$-\\\\sqrt{289}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Rational","Irrational"],"hints":{"DefaultPathway":[{"id":"a6f9727real25b-h1","type":"hint","dependencies":[],"title":"Simplify","text":"$$-\\\\sqrt{289}=-\\\\sqrt{{17}^2}=-17$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real25b-h2","type":"hint","dependencies":["a6f9727real25b-h1"],"title":"Set of Rational Numbers","text":"The set of rational numbers includes fractions written as $$\\\\frac{m}{n}$$ where $$m$$ and $$n$$ are integers, and $$n$$ does not equal $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real25b-h3","type":"hint","dependencies":["a6f9727real25b-h2"],"title":"Set of Irrational Numbers","text":"The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real25b-h4","type":"hint","dependencies":["a6f9727real25b-h3"],"title":"Rationality of Integers","text":"$$-17$$ is rational.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6f9727real25c","stepAnswer":["Rational"],"problemType":"MultipleChoice","stepTitle":"$$-11.411411411\u2026$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Rational","Irrational"],"hints":{"DefaultPathway":[{"id":"a6f9727real25c-h1","type":"hint","dependencies":[],"title":"Set of Rational Numbers","text":"The set of rational numbers includes fractions written as $$\\\\frac{m}{n}$$ where $$m$$ and $$n$$ are integers, and $$n$$ does not equal $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real25c-h2","type":"hint","dependencies":["a6f9727real25c-h1"],"title":"Set of Irrational Numbers","text":"The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real25c-h3","type":"hint","dependencies":["a6f9727real25c-h2"],"title":"Repeating Decimal","text":"The given number is a repeating decimal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6f9727real26","title":"Order of Operations","body":"Evaluate the following numerical expressions.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Real Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a6f9727real26a","stepAnswer":["$$20$$"],"problemType":"TextBox","stepTitle":"$$2\\\\times5+3\\\\times2+4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$20$$","hints":{"DefaultPathway":[{"id":"a6f9727real26a-h1","type":"hint","dependencies":[],"title":"Multiplication","text":"The first step is to simplify multiplication and division.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real26a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a6f9727real26a-h1"],"title":"Multiplication","text":"What is $$2\\\\times5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real26a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a6f9727real26a-h1"],"title":"Multiplication","text":"What is $$3\\\\times2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real26a-h4","type":"hint","dependencies":["a6f9727real26a-h2","a6f9727real26a-h3"],"title":"Addition","text":"The next step is to simplify addition and subtraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real26a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a6f9727real26a-h4"],"title":"Addition","text":"What is $$10+6+4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6f9727real26b","stepAnswer":["$$30$$"],"problemType":"TextBox","stepTitle":"$$2\\\\left(5+3\\\\times2+4\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$30$$","hints":{"DefaultPathway":[{"id":"a6f9727real26b-h1","type":"hint","dependencies":[],"title":"Parentheses","text":"The first step is to simplify the parentheses.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real26b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a6f9727real26b-h1"],"title":"Parentheses","text":"What is $$3\\\\times2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real26b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["a6f9727real26b-h2"],"title":"Parentheses","text":"What is $$5+6+4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real26b-h4","type":"hint","dependencies":["a6f9727real26b-h3"],"title":"Multiplication","text":"The next step is to simplify multiplication and division.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real26b-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$30$$"],"dependencies":["a6f9727real26b-h4"],"title":"Multiplication","text":"What is $$2\\\\times15$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6f9727real26c","stepAnswer":["$$96$$"],"problemType":"TextBox","stepTitle":"$$2\\\\left(5+3\\\\right) \\\\left(2+4\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$96$$","hints":{"DefaultPathway":[{"id":"a6f9727real26c-h1","type":"hint","dependencies":[],"title":"Parentheses","text":"The first step is to simplify the parentheses.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real26c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a6f9727real26c-h1"],"title":"Parentheses","text":"What is $$5+3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real26c-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a6f9727real26c-h1"],"title":"Parentheses","text":"What is $$2+4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real26c-h4","type":"hint","dependencies":["a6f9727real26c-h2","a6f9727real26c-h3"],"title":"Multiplication","text":"The next step is to simplify multiplication and division.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real26c-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$96$$"],"dependencies":["a6f9727real26c-h4"],"title":"Multiplication","text":"What is $$2\\\\times8\\\\times6$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6f9727real27","title":"Order of Operations in Mathematical Expressions","body":"Evaluate the following numerical expressions.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Real Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a6f9727real27a","stepAnswer":["$$45$$"],"problemType":"TextBox","stepTitle":"$$5{\\\\left(5-2\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$45$$","hints":{"DefaultPathway":[{"id":"a6f9727real27a-h1","type":"hint","dependencies":[],"title":"Parentheses","text":"The first step is to simplify the parentheses.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real27a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a6f9727real27a-h1"],"title":"Parentheses","text":"What is $$5-2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real27a-h3","type":"hint","dependencies":["a6f9727real27a-h2"],"title":"Exponent","text":"The next step is to simplify any exponents.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real27a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a6f9727real27a-h3"],"title":"Exponent","text":"What is $$3^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real27a-h5","type":"hint","dependencies":["a6f9727real27a-h4"],"title":"Multiplication","text":"The next step is to simplify multiplication and division.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real27a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$45$$"],"dependencies":["a6f9727real27a-h5"],"title":"Multiplication","text":"What is $$5\\\\times9$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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4.0>"},{"id":"a6f9727real28a-h3","type":"hint","dependencies":["a6f9727real28a-h2"],"title":"Addition","text":"The next step is to simplify addition and subtraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real28a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$44$$"],"dependencies":["a6f9727real28a-h3"],"title":"Addition","text":"What is $$9+5+30$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6f9727real28b","stepAnswer":["$$40$$"],"problemType":"MultipleChoice","stepTitle":"$$3+8\\\\times5-3$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$40$$","choices":["$$40$$","$$52$$","$$22$$","$$46$$"],"hints":{"DefaultPathway":[{"id":"a6f9727real28b-h1","type":"hint","dependencies":[],"title":"Multiplication","text":"The first 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$$3+40-3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6f9727real3","title":"Sets of Numbers","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Real Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a6f9727real3a","stepAnswer":["$$0$$"],"problemType":"MultipleChoice","stepTitle":"Which of the following numbers does not belong in the set of natural number?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$0$$","choices":["$$0$$","$$1$$","$$2$$","$$3$$"],"hints":{"DefaultPathway":[{"id":"a6f9727real3a-h1","type":"hint","dependencies":[],"title":"Counting","text":"The natural numbers consist of the numbers used for counting.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real3a-h2","type":"hint","dependencies":["a6f9727real3a-h1"],"title":"Counting","text":"Counting generally starts form the number $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6f9727real3b","stepAnswer":["$$\\\\frac{0}{4}$$"],"problemType":"MultipleChoice","stepTitle":"Which of the following numbers belongs to the set of whole numbers?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{0}{4}$$","choices":["$$-2$$","$$\\\\frac{0}{4}$$","$$4.5$$","$$\\\\frac{3}{7}$$"],"hints":{"DefaultPathway":[{"id":"a6f9727real3b-h1","type":"hint","dependencies":[],"title":"Whole Numbers","text":"The set of whole numbers is the set of natural numbers plus zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC 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(where $$x$$ is nonzero)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6f9727real3c","stepAnswer":["Set of Irrational Numbers"],"problemType":"MultipleChoice","stepTitle":"In which of the following sets can you find pi?","stepBody":"","answerType":"string","variabilization":{},"choices":["Set of Rational Numbers","Set of Integers","Set of Natural Numbers","Set of Irrational Numbers"],"hints":{"DefaultPathway":[{"id":"a6f9727real3c-h1","type":"hint","dependencies":[],"title":"Set of Rational Numbers","text":"The set of rational numbers includes fractions written as $$\\\\frac{m}{n}$$ where $$m$$ and $$n$$ are integers, and $$n$$ does not equal $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real3c-h2","type":"hint","dependencies":["a6f9727real3c-h1"],"title":"Set of Irrational Numbers","text":"The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real3c-h3","type":"hint","dependencies":["a6f9727real3c-h2"],"title":"Properties of pi","text":"pi is not rational, nonrepeating, and nonterminating","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6f9727real4","title":"Properties of Real Numbers","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Real Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a6f9727real4a","stepAnswer":["Real Number"],"problemType":"MultipleChoice","stepTitle":"The Closure 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real5b-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-13$$"],"dependencies":["a6f9727real5b-h4"],"title":"Parentheses","text":"What is $$3-16$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real5b-h6","type":"hint","dependencies":["a6f9727real5b-h5"],"title":"Multiplication","text":"The next step is to simplify multiplication and division.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real5b-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$105$$"],"dependencies":["a6f9727real5b-h6"],"title":"Multiplication","text":"What is $$7\\\\times15$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real5b-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$26$$"],"dependencies":["a6f9727real5b-h7"],"title":"Multiplication","text":"What is $$\\\\left(-2\\\\right) \\\\left(-13\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real5b-h9","type":"hint","dependencies":["a6f9727real5b-h8"],"title":"Addition","text":"The final step is to simplify addition and subtraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real5b-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$132$$"],"dependencies":["a6f9727real5b-h9"],"title":"Addition","text":"What is $$105+26+1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6f9727real6","title":"Distributive Property","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Real Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a6f9727real6a","stepAnswer":["$$20$$"],"problemType":"TextBox","stepTitle":"What is $$4\\\\left(12+\\\\left(-7\\\\right)\\\\right)$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$20$$","hints":{"DefaultPathway":[{"id":"a6f9727real6a-h1","type":"hint","dependencies":[],"title":"Distributive Property","text":"The distributive property states that the product of a factor times a sum is the sum of the factor times each term in the sum. $$a \\\\left(b+c\\\\right)=a b+a c$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$48$$"],"dependencies":["a6f9727real6a-h1"],"title":"Multiplication","text":"What is $$4\\\\times12$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-28$$"],"dependencies":["a6f9727real6a-h1"],"title":"Multiplication","text":"What is $$4\\\\left(-7\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a6f9727real6a-h2","a6f9727real6a-h3"],"title":"Addition","text":"What is $$48+\\\\left(-28\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6f9727real6b","stepAnswer":["$$-163$$"],"problemType":"MultipleChoice","stepTitle":"What is $$\\\\operatorname{100}\\\\left(0.75+\\\\left(-2.38\\\\right)\\\\right)$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-163$$","choices":["$$-149$$","$$-163$$","$$-313$$","$$-326$$"],"hints":{"DefaultPathway":[{"id":"a6f9727real6b-h1","type":"hint","dependencies":[],"title":"Distributive Property","text":"The distributive property states that the product of a factor times a sum is the sum of the factor times each term in the sum. $$a \\\\left(b+c\\\\right)=a b+a c$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real6b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$75$$"],"dependencies":["a6f9727real6b-h1"],"title":"Multiplication","text":"What is $$100\\\\times0.75$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real6b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-238$$"],"dependencies":["a6f9727real6b-h1"],"title":"Multiplication","text":"What is $$100\\\\left(-2.38\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real6b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-163$$"],"dependencies":["a6f9727real6b-h2","a6f9727real6b-h3"],"title":"Addition","text":"What is $$75+\\\\left(-238\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6f9727real7","title":"Combining Properties of Real Numbers","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Real Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a6f9727real7a","stepAnswer":["$$\\\\frac{2}{3}$$"],"problemType":"MultipleChoice","stepTitle":"What is $$\\\\frac{4}{7} \\\\frac{2}{3} \\\\frac{7}{4}$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{2}{3}$$","choices":["$$\\\\frac{2}{3}$$","$$\\\\frac{7}{4}$$","$$\\\\frac{4}{7}$$","$$1$$"],"hints":{"DefaultPathway":[{"id":"a6f9727real7a-h1","type":"hint","dependencies":[],"title":"Commutative Property of Multiplication","text":"The commutative property of addition states that if a and $$b$$ are real numbers, then $$a b=b a$$","variabilization":{},"oer":"","license":""},{"id":"a6f9727real7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a6f9727real7a-h1"],"title":"Multiplication","text":"What is $$\\\\frac{4}{7} \\\\frac{7}{4}$$?","variabilization":{},"oer":"","license":""},{"id":"a6f9727real7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{3}$$"],"dependencies":["a6f9727real7a-h2"],"title":"Multiplication","text":"what is $$1\\\\frac{2}{3}$$?","variabilization":{},"oer":"","license":""}]}},{"id":"a6f9727real7b","stepAnswer":["$$5$$"],"problemType":"MultipleChoice","stepTitle":"What is $$5+8+\\\\left(-8\\\\right)$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$5$$","choices":["$$8$$","$$-5$$","$$0$$","$$5$$"],"hints":{"DefaultPathway":[{"id":"a6f9727real7b-h1","type":"hint","dependencies":[],"title":"Commutative Property of Addition","text":"The closure proeprty states that if a and $$b$$ are real numbers, then $$a+b$$ is a unique real number, and $$a b$$ is a unique real number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real7b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a6f9727real7b-h1"],"title":"Addition","text":"What is $$8+\\\\left(-8\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real7b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a6f9727real7b-h1"],"title":"Addition","text":"What is $$0+5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6f9727real8","title":"Evaluating an Algebraic Expression at Different Values","body":"Evaluate the expression $$2x-7$$ for each value for $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Real Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a6f9727real8a","stepAnswer":["$$-7$$"],"problemType":"TextBox","stepTitle":"If $$x=0$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-7$$","hints":{"DefaultPathway":[{"id":"a6f9727real8a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Plug in $$0$$ in place of $$x$$. You then get the equation $$2(0)-7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a6f9727real8a-h1"],"title":"Multiplication","text":"What is 2(0)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-7$$"],"dependencies":["a6f9727real8a-h2"],"title":"Subtraction","text":"What is $$0-7$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6f9727real8b","stepAnswer":["$$-6$$"],"problemType":"TextBox","stepTitle":"If $$x=\\\\frac{1}{2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-6$$","hints":{"DefaultPathway":[{"id":"a6f9727real8b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Plug in $$\\\\frac{1}{2}$$ in place of $$x$$. You then get the equation $$2\\\\left(\\\\frac{1}{2}\\\\right)-7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real8b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a6f9727real8b-h1"],"title":"Multiplication","text":"What is $$2\\\\left(\\\\frac{1}{2}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real8b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6$$"],"dependencies":["a6f9727real8b-h2"],"title":"Subtraction","text":"What is $$1-7$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6f9727real8c","stepAnswer":["$$-15$$"],"problemType":"TextBox","stepTitle":"If $$x=-4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-15$$","hints":{"DefaultPathway":[{"id":"a6f9727real8c-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Plug in $$0$$ in place of $$x$$. You then get the equation $$2(-4)-7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real8c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-8$$"],"dependencies":["a6f9727real8c-h1"],"title":"Multiplication","text":"What is $$2(-4)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real8c-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-15$$"],"dependencies":["a6f9727real8c-h2"],"title":"Subtraction","text":"What is $$(-8)-7$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a6f9727real9","title":"Evaluating Algebraic Expressions At Specific Values","body":"Evaluate each expression for the given values.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Real Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a6f9727real9a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"Solve $$x+5$$ for $$x=-5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a6f9727real9a-h1","type":"hint","dependencies":[],"title":"Subsititute","text":"Plug in $$-5$$ in place of $$x$$. You then get the equation $$-5+5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a6f9727real9a-h1"],"title":"Addition","text":"What is $$-5+5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a6f9727real9b","stepAnswer":["$$\\\\frac{10}{19}$$"],"problemType":"MultipleChoice","stepTitle":"Solve $$\\\\frac{t}{2t-1}$$ for $$t=10$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{10}{19}$$","choices":["$$\\\\frac{10}{19}$$","$$\\\\frac{10}{21}$$","$$\\\\frac{10}{20}$$","$$\\\\frac{10}{22}$$"],"hints":{"DefaultPathway":[{"id":"a6f9727real9b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Plug in $$10$$ for $$t$$. You get the equation $$\\\\frac{10}{2\\\\left(10\\\\right)-1}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real9b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a6f9727real9b-h1"],"title":"Multiplication","text":"What is $$2\\\\times10$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real9b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$19$$"],"dependencies":["a6f9727real9b-h2"],"title":"Subtraction","text":"What is $$20-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a6f9727real9b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{10}{19}$$"],"dependencies":["a6f9727real9b-h3"],"title":"Substitute","text":"What is $$\\\\frac{10}{2\\\\left(10\\\\right)-1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7079e8inverse1","title":"For the following function, find (f**(-1))\'(a)","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.7 Derivatives of Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a7079e8inverse1a","stepAnswer":["$$\\\\frac{1}{5}$$"],"problemType":"TextBox","stepTitle":"$$f(x)=x^3+2x+3$$, $$a=0$$","stepBody":"Input your answer as a fraction excluding the $$\\"f(x)=\\"$$ part.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{5}$$","hints":{"DefaultPathway":[{"id":"a7079e8inverse1a-h1","type":"hint","dependencies":[],"title":"The inverse function theorem","text":"The inverse function theorem is that (f**(-1))\'*(x)=1/(f\'*(f**-1)*(x)).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse1a-h2","type":"hint","dependencies":["a7079e8inverse1a-h1"],"title":"Finding the Inverse of f(x) at $$a=0$$.","text":"To find the inverse of f(x) where $$a=0$$, then we need to first find what value of $$x$$ when we plug it into f(x) will give us a value of $$0$$. In this case, it is when $$x=-1$$, so we know that $$0f^{\\\\left(-1\\\\right)}=-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse1a-h3","type":"hint","dependencies":["a7079e8inverse1a-h2"],"title":"Finding the derivative of f(x)","text":"Next, to find the derivative of f(x), we can use the power rule. To do this, we can multiply the coefficient of each $$x$$ by its current exponent, and then subtract $$1$$ from the exponent. Remember, constants become $$0$$ when we take the derivative of a function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse1a-h4","type":"hint","dependencies":["a7079e8inverse1a-h3"],"title":"Applying the Inverse Function Theorem","text":"Since we already found $$0f^{\\\\left(-1\\\\right)}=-1$$ and the derivative of f(x), then we can plug what we know into the inverse function theorem.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a7079e8inverse10","title":"For the given function, $$f(x)={\\\\left(-x\\\\right)}^3-x+2$$, $$P(-8,2)$$, find following.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.7 Derivatives of Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a7079e8inverse10a","stepAnswer":["$$\\\\frac{-1}{13}$$"],"problemType":"TextBox","stepTitle":"Find the slope of the tangent line to its inverse function $$f^{\\\\left(-1\\\\right)}$$ at the indicated point P","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-1}{13}$$","hints":{"DefaultPathway":[{"id":"a7079e8inverse10a-h1","type":"hint","dependencies":[],"title":"The inverse function theorem will tell us the slope of the tangent line to the inverse function.","text":"The inverse function theorem is that (f**(-1))\'*(x)=1/(f\'*(f**-1)*(x)).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse10a-h2","type":"hint","dependencies":["a7079e8inverse10a-h1"],"title":"Finding the Inverse of f(x) at $$a=0$$.","text":"To find the inverse of f(x) where $$a=-8$$ (remember that a is equal to the x-value of a point), then we need to first find what value of $$x$$ when we plug it into f(x) will give us a value of $$-8$$. In this case, it is when $$x=2$$, so we know that $$f^{\\\\left(-1\\\\right)} \\\\left(-8\\\\right)=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse10a-h3","type":"hint","dependencies":["a7079e8inverse10a-h2"],"title":"Finding the derivative of f(x)","text":"Next, to find the derivative of f(x), we can use the power rule. To do this, we can multiply the coefficient of each $$x$$ by its current exponent, and then subtract $$1$$ from the exponent. Remember, constants become $$0$$ when we take the derivative of a function and the derivative of tan(x) is $${sec}^2 x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse10a-h4","type":"hint","dependencies":["a7079e8inverse10a-h3"],"title":"Applying the Inverse Function Theorem","text":"Since we already found $$f^{\\\\left(-1\\\\right)} \\\\left(-8\\\\right)=2$$ and the derivative of f(x), then we can plug what we know into the inverse function theorem. Our resulting slope is $$\\\\frac{-1}{13}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a7079e8inverse10b","stepAnswer":["$$y=-\\\\left(\\\\frac{x}{13}\\\\right)+\\\\frac{18}{13}$$"],"problemType":"MultipleChoice","stepTitle":"Find the equation of the tangent line to the graph of $$f^{\\\\left(-1\\\\right)}$$ at the indicated point P.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=-\\\\left(\\\\frac{x}{13}\\\\right)+\\\\frac{18}{13}$$","choices":["$$y=-\\\\left(\\\\frac{x}{13}\\\\right)+\\\\frac{18}{13}$$","$$y=\\\\frac{x}{13}+\\\\frac{18}{13}$$","$$y=-\\\\left(\\\\frac{x}{13}\\\\right)-\\\\frac{18}{13}$$","$$y=-\\\\left(\\\\frac{x}{13}\\\\right)+\\\\frac{34}{13}$$"],"hints":{"DefaultPathway":[{"id":"a7079e8inverse10b-h1","type":"hint","dependencies":[],"title":"Use the point-slope form for the equation of a line.","text":"The point-slope form for the equation of a line is $$y-y1=m \\\\left(x-x1\\\\right)$$, We can plug in what we know, which is $$m=\\\\frac{-1}{13}$$, $$x1=-8$$, $$y1=2$$, to solve for the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a7079e8inverse11","title":"For the given function, $$f(x)=x^5+3x^3-4x-8$$, $$P(-8,1)$$, find following.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.7 Derivatives of Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a7079e8inverse11a","stepAnswer":["$$\\\\frac{1}{10}$$"],"problemType":"TextBox","stepTitle":"Find the slope of the tangent line to its inverse function $$f^{\\\\left(-1\\\\right)}$$ at the indicated point P","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{10}$$","hints":{"DefaultPathway":[{"id":"a7079e8inverse11a-h1","type":"hint","dependencies":[],"title":"The inverse function theorem will tell us the slope of the tangent line to the inverse function.","text":"The inverse function theorem is that (f**(-1))\'*(x)=1/(f\'*(f**-1)*(x)).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse11a-h2","type":"hint","dependencies":["a7079e8inverse11a-h1"],"title":"Finding the Inverse of f(x) at $$a=0$$.","text":"To find the inverse of f(x) where $$a=-8$$ (remember that a is equal to the x-value of a point), then we need to first find what value of $$x$$ when we plug it into f(x) will give us a value of $$-8$$. In this case, it is when $$x=1$$, so we know that $$f^{\\\\left(-1\\\\right)} \\\\left(-8\\\\right)=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse11a-h3","type":"hint","dependencies":["a7079e8inverse11a-h2"],"title":"Finding the derivative of f(x)","text":"Next, to find the derivative of f(x), we can use the power rule. To do this, we can multiply the coefficient of each $$x$$ by its current exponent, and then subtract $$1$$ from the exponent. Remember, constants become $$0$$ when we take the derivative of a function and the derivative of tan(x) is $${sec}^2 x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse11a-h4","type":"hint","dependencies":["a7079e8inverse11a-h3"],"title":"Applying the Inverse Function Theorem","text":"Since we already found $$f^{\\\\left(-1\\\\right)} \\\\left(-8\\\\right)=1$$ and the derivative of f(x), then we can plug what we know into the inverse function theorem. Our resulting slope is $$\\\\frac{1}{10}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a7079e8inverse11b","stepAnswer":["$$y=\\\\frac{x}{10}+\\\\frac{18}{10}$$"],"problemType":"MultipleChoice","stepTitle":"Find the equation of the tangent line to the graph of $$f^{\\\\left(-1\\\\right)}$$ at the indicated point P.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=\\\\frac{x}{10}+\\\\frac{18}{10}$$","choices":["$$y=\\\\frac{x}{10}+\\\\frac{18}{10}$$","$$y=\\\\frac{x}{10}+\\\\frac{8}{10}$$","$$y=\\\\frac{x}{10}-\\\\frac{18}{10}$$"],"hints":{"DefaultPathway":[{"id":"a7079e8inverse11b-h1","type":"hint","dependencies":[],"title":"Use the point-slope form for the equation of a line.","text":"The point-slope form for the equation of a line is $$y-y1=m \\\\left(x-x1\\\\right)$$, We can plug in what we know, which is $$m=\\\\frac{1}{10}$$, $$x1=-8$$, $$y1=1$$, to solve for the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a7079e8inverse12","title":"For the following exercise, find the $$\\\\frac{dy}{dx}$$ for the given function.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.7 Derivatives of Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a7079e8inverse12a","stepAnswer":["$$\\\\frac{2x}{\\\\sqrt{1-x^4}}$$"],"problemType":"TextBox","stepTitle":"$$y={sin}^{-1} x^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2x}{\\\\sqrt{1-x^4}}$$","hints":{"DefaultPathway":[{"id":"a7079e8inverse12a-h1","type":"hint","dependencies":[],"title":"Derivative of inverse trigonometric functions:sin(x)","text":"The formula for the derivative of Inverse sin(x) is $$\\\\frac{d}{dx} {sin}^{-1} x=\\\\frac{1}{\\\\sqrt{1-x^2}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse12a-h2","type":"hint","dependencies":["a7079e8inverse12a-h1"],"title":"Applying the chain rule","text":"Because the inverse of sin has a function inside its parentheses, \\"x**2\\" in this case, we need to apply the chain rule before using the formula from the previous hint. For reference, the chain rule is $$(f(g(x)))\'=\\\\operatorname{f\'}\\\\left(g{\\\\left(x\\\\right)}\\\\right) \\\\operatorname{g\'}\\\\left(x\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a7079e8inverse13","title":"For the following exercise, find the $$\\\\frac{dy}{dx}$$ for the given function.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.7 Derivatives of Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a7079e8inverse13a","stepAnswer":["$$\\\\frac{-1}{2\\\\sqrt{-\\\\left(x^2\\\\right)+x}}$$"],"problemType":"MultipleChoice","stepTitle":"$$y={cos}^{-1} \\\\sqrt{x}$$","stepBody":"Make sure to simplify your square roots when multiplying at the end for your answer.","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{-1}{2\\\\sqrt{-\\\\left(x^2\\\\right)+x}}$$","choices":["$$\\\\frac{-1}{2\\\\sqrt{-\\\\left(x^2\\\\right)+x}}$$","$$\\\\frac{1}{2\\\\sqrt{-\\\\left(x^2\\\\right)+x}}$$"],"hints":{"DefaultPathway":[{"id":"a7079e8inverse13a-h1","type":"hint","dependencies":[],"title":"Derivative of inverse trigonometric functions: cos(x)","text":"The formula for the derivative of inverse cos(x) is $$\\\\frac{d}{dx} {cos}^{-1} x=\\\\frac{-1}{\\\\sqrt{1-x^2}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse13a-h2","type":"hint","dependencies":["a7079e8inverse13a-h1"],"title":"Applying the chain rule","text":"Because the inverse of cos has a function inside its parentheses, $$\\\\sqrt{x}$$, in this case, we need to apply the chain rule before using the formula from the previous hint. For reference, the chain rule is $$(f(g(x)))\'=\\\\operatorname{f\'}\\\\left(g{\\\\left(x\\\\right)}\\\\right) \\\\operatorname{g\'}\\\\left(x\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a7079e8inverse14","title":"For the following exercise, find the $$\\\\frac{dy}{dx}$$ for the given function.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.7 Derivatives of Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a7079e8inverse14a","stepAnswer":["$$\\\\frac{-1}{\\\\sqrt{1-x^2}}$$"],"problemType":"TextBox","stepTitle":"$$y={sec}^{-1} \\\\frac{1}{x}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-1}{\\\\sqrt{1-x^2}}$$","hints":{"DefaultPathway":[{"id":"a7079e8inverse14a-h1","type":"hint","dependencies":[],"title":"Derivative of inverse trigonometric functions:sec(x)","text":"The formula for the derivative of inverse sec(x) is $$\\\\frac{d}{dx} {sec}^{-1} x=\\\\frac{1}{|x| \\\\sqrt{x^2-1}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse14a-h2","type":"hint","dependencies":["a7079e8inverse14a-h1"],"title":"Applying the chain rule","text":"Because the inverse of sec has a function inside its parentheses, \\"1/x\\" in this case, we need to apply the chain rule before using the formula from the previous hint. For reference, the chain rule is $$(f(g(x)))\'=\\\\operatorname{f\'}\\\\left(g{\\\\left(x\\\\right)}\\\\right) \\\\operatorname{g\'}\\\\left(x\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a7079e8inverse15","title":"For the following exercise, find the $$\\\\frac{dy}{dx}$$ for the given function.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.7 Derivatives of Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a7079e8inverse15a","stepAnswer":["$$\\\\frac{-1}{2\\\\sqrt{1-\\\\frac{1}{x^2}} x^2 \\\\sqrt{{cos}^{\\\\left(-1\\\\right)} x}}$$"],"problemType":"MultipleChoice","stepTitle":"$$y=\\\\sqrt{{csc}^{\\\\left(-1\\\\right)} x}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{-1}{2\\\\sqrt{1-\\\\frac{1}{x^2}} x^2 \\\\sqrt{{cos}^{\\\\left(-1\\\\right)} x}}$$","choices":["$$\\\\frac{-1}{2\\\\sqrt{1-\\\\frac{1}{x^2}} x^2 \\\\sqrt{{cos}^{\\\\left(-1\\\\right)} x}}$$","$$\\\\frac{1}{2\\\\sqrt{1-\\\\frac{1}{x^2}} x^2 \\\\sqrt{{cos}^{\\\\left(-1\\\\right)} x}}$$"],"hints":{"DefaultPathway":[{"id":"a7079e8inverse15a-h1","type":"hint","dependencies":[],"title":"Derivative of inverse trigonometric functions:csc(x)","text":"The formula for the derivative of inverse csc(x) is $$\\\\frac{d}{dx} {csc}^{-1} x=\\\\frac{-1}{|x| \\\\sqrt{x^2-1}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse15a-h2","type":"hint","dependencies":["a7079e8inverse15a-h1"],"title":"Applying the chain rule","text":"Because the function $$\\\\sqrt{x}$$ or $$x^{\\\\frac{1}{2}}$$ has a function inside its parentheses,cos**(-1)(x) in this case, we need to apply the chain rule before using the formula from the previous hint. For reference, the chain rule is $$(f(g(x)))\'=\\\\operatorname{f\'}\\\\left(g{\\\\left(x\\\\right)}\\\\right) \\\\operatorname{g\'}\\\\left(x\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a7079e8inverse16","title":"For the following exercise, find the $$\\\\frac{dy}{dx}$$ for the given function.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.7 Derivatives of Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a7079e8inverse16a","stepAnswer":["$$\\\\frac{{\\\\left(3\\\\left(1+{tan}^{\\\\left(-1\\\\right)} x\\\\right)\\\\right)}^2}{1+x^2}$$"],"problemType":"TextBox","stepTitle":"$$y={\\\\left(1+{tan}^{\\\\left(-1\\\\right)} x\\\\right)}^3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{{\\\\left(3\\\\left(1+{tan}^{\\\\left(-1\\\\right)} x\\\\right)\\\\right)}^2}{1+x^2}$$","hints":{"DefaultPathway":[{"id":"a7079e8inverse16a-h1","type":"hint","dependencies":[],"title":"Derivative of inverse trigonometric functions:tan(x)","text":"The formula for the derivative of inverse tan(x) is $$\\\\frac{d}{dx} {tan}^{\\\\left(-1\\\\right)} x=\\\\frac{1}{1+x^2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse16a-h2","type":"hint","dependencies":["a7079e8inverse16a-h1"],"title":"Applying the chain rule","text":"Because the function $$x^3$$ has a function inside its parentheses, \\"1+tan**(-1)(x)\\" in this case, we need to apply the chain rule before using the formula from the previous hint. For reference, the chain rule is $$(f(g(x)))\'=\\\\operatorname{f\'}\\\\left(g{\\\\left(x\\\\right)}\\\\right) \\\\operatorname{g\'}\\\\left(x\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a7079e8inverse17","title":"For the following exercise, find the $$\\\\frac{dy}{dx}$$ for the given function.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.7 Derivatives of Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a7079e8inverse17a","stepAnswer":["(-2*sin**(-1)(2*x))/sqrt(1-4*x**2))+(2*cos**(-1)(2*x))/sqrt(1-4*x**2))"],"problemType":"TextBox","stepTitle":"$$y={cos}^{\\\\left(-1\\\\right)} 2x {sin}^{\\\\left(-1\\\\right)} 2x$$","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a7079e8inverse17a-h1","type":"hint","dependencies":[],"title":"Applying the product rule.","text":"Because we are multiplying $$2$$ functions, we need to use the product rule in order to find the derivative. For reference the product rule is $$\\\\frac{d}{dy} f{\\\\left(x\\\\right)} g{\\\\left(x\\\\right)}=\\\\operatorname{f\'}\\\\left(x\\\\right) g{\\\\left(x\\\\right)}+f{\\\\left(x\\\\right)} \\\\operatorname{g\'}\\\\left(x\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse17a-h2","type":"hint","dependencies":["a7079e8inverse17a-h1"],"title":"Derivative of inverse trigonometric functions:cos(x) and sin(x)","text":"The formula for the derivative of Inverse sin(x) is $$\\\\frac{d}{dx} {sin}^{-1} x=\\\\frac{1}{\\\\sqrt{1-x^2}}$$. The formula for the derivative of inverse cos(x) is $$\\\\frac{d}{dx} {cos}^{-1} x=\\\\frac{-1}{\\\\sqrt{1-x^2}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse17a-h3","type":"hint","dependencies":["a7079e8inverse17a-h2"],"title":"Applying the chain rule","text":"Because the inverse of cos and sin have a function inside theirs parentheses, \\"2x\\" in this case, we need to apply the chain rule before using the formula from the previous hint. For reference, the chain rule is $$(f(g(x)))\'=\\\\operatorname{f\'}\\\\left(g{\\\\left(x\\\\right)}\\\\right) \\\\operatorname{g\'}\\\\left(x\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a7079e8inverse18","title":"For the following exercise, find the $$\\\\frac{dy}{dx}$$ for the given function.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.7 Derivatives of Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a7079e8inverse18a","stepAnswer":["$$\\\\frac{-1}{\\\\left(1+x^2\\\\right) {\\\\left({tan}^{\\\\left(-1\\\\right)} x\\\\right)}^2}$$"],"problemType":"TextBox","stepTitle":"$$y=\\\\frac{1}{{tan}^{-x}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-1}{\\\\left(1+x^2\\\\right) {\\\\left({tan}^{\\\\left(-1\\\\right)} x\\\\right)}^2}$$","hints":{"DefaultPathway":[{"id":"a7079e8inverse18a-h1","type":"hint","dependencies":[],"title":"Applying the reciprocal rule of derivation","text":"Because the given equation in the form $$\\\\frac{1}{f{\\\\left(x\\\\right)}}$$, we can use the reciprocal rule which states that $$\\\\frac{d}{dx} \\\\frac{1}{f{\\\\left(x\\\\right)}}=\\\\frac{\\\\left(-\\\\operatorname{f\'}\\\\left(x\\\\right)\\\\right)}{{f{\\\\left(x\\\\right)}}^2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse18a-h2","type":"hint","dependencies":["a7079e8inverse18a-h1"],"title":"Derivative of inverse trigonometric functions:tan(x)","text":"The formula for the derivative of inverse tan(x) is $$\\\\frac{d}{dx} {tan}^{\\\\left(-1\\\\right)} x=\\\\frac{1}{1+x^2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a7079e8inverse19","title":"For the following exercise, find the $$\\\\frac{dy}{dx}$$ for the given function.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.7 Derivatives of Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a7079e8inverse19a","stepAnswer":["$$\\\\frac{-1}{x \\\\sqrt{x^2-1}}$$"],"problemType":"TextBox","stepTitle":"$$y={sec}^{\\\\left(-1\\\\right)} \\\\left(-x\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-1}{x \\\\sqrt{x^2-1}}$$","hints":{"DefaultPathway":[{"id":"a7079e8inverse19a-h1","type":"hint","dependencies":[],"title":"Derivative of inverse trigonometric functions:sec(x)","text":"The formula for the derivative of inverse sec(x) is $$\\\\frac{d}{dx} {sec}^{-1} x=\\\\frac{1}{|x| \\\\sqrt{x^2-1}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse19a-h2","type":"hint","dependencies":["a7079e8inverse19a-h1"],"title":"Applying the chain rule","text":"Because the inverse of sec has a function inside its parentheses, $$-x$$ in this case, we need to apply the chain rule before using the formula from the previous hint. For reference, the chain rule is $$(f(g(x)))\'=\\\\operatorname{f\'}\\\\left(g{\\\\left(x\\\\right)}\\\\right) \\\\operatorname{g\'}\\\\left(x\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a7079e8inverse2","title":"For the following function, find (f**(-1))\'(a)","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.7 Derivatives of Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a7079e8inverse2a","stepAnswer":["$$\\\\frac{1}{2}$$"],"problemType":"TextBox","stepTitle":"$$f(x)=x^2+3x+2$$, $$x$$ greater than or equal to $$\\\\frac{-3}{2}$$, $$a=2$$","stepBody":"Input your answer as a fraction excluding the $$\\"f(x)=\\"$$ part.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{2}$$","hints":{"DefaultPathway":[{"id":"a7079e8inverse2a-h1","type":"hint","dependencies":[],"title":"The inverse function theorem","text":"The inverse function theorem is that (f**(-1))\'*(x)=1/(f\'*(f**-1)*(x)).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse2a-h2","type":"hint","dependencies":["a7079e8inverse2a-h1"],"title":"Finding the Inverse of f(x) at $$a=0$$.","text":"To find the inverse of f(x) where $$a=2$$, then we need to first find what value of $$x$$ when we plug it into f(x) will give us a value of $$2$$. In this case, it is when $$x=-1$$, so we know that $$2f^{\\\\left(-1\\\\right)}=-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse2a-h3","type":"hint","dependencies":["a7079e8inverse2a-h2"],"title":"Finding the derivative of f(x)","text":"Next, to find the derivative of f(x), we can use the power rule. To do this, we can multiply the coefficient of each $$x$$ by its current exponent, and then subtract $$1$$ from the exponent. Remember, constants become $$0$$ when we take the derivative of a function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse2a-h4","type":"hint","dependencies":["a7079e8inverse2a-h3"],"title":"Applying the Inverse Function Theorem","text":"Since we already found $$2f^{\\\\left(-1\\\\right)}=-1$$ and the derivative of f(x), then we can plug what we know into the inverse function theorem.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a7079e8inverse20","title":"For the following exercise, find the $$\\\\frac{dy}{dx}$$ for the given function.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.7 Derivatives of Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a7079e8inverse20a","stepAnswer":["x/((5-(x**2))*(sqrt(4-(x**2)))"],"problemType":"TextBox","stepTitle":"$$y={cot}^{\\\\left(-1\\\\right)} \\\\sqrt{4-x^2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a7079e8inverse20a-h1","type":"hint","dependencies":[],"title":"Derivative of inverse trigonometric functions: cot(x)","text":"The formula for the derivative of inverse cot(x) is $$\\\\frac{d}{dx} {cot}^{\\\\left(-1\\\\right)} x=\\\\left(-\\\\frac{1}{1+x^2}\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse20a-h2","type":"hint","dependencies":["a7079e8inverse20a-h1"],"title":"Applying the chain rule","text":"Because the inverse of cot has a function inside its parentheses, \\"sqrt(4-(x**2))\\" in this case, we need to apply the chain rule before using the formula from the previous hint. For reference, the chain rule is $$(f(g(x)))\'=\\\\operatorname{f\'}\\\\left(g{\\\\left(x\\\\right)}\\\\right) \\\\operatorname{g\'}\\\\left(x\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a7079e8inverse21","title":"For the following exercise, find the $$\\\\frac{dy}{dx}$$ for the given function.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.7 Derivatives of Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a7079e8inverse21a","stepAnswer":["(csc**(-1)(x))-(1/(x*(sqrt(1-(1/(x**2)))))"],"problemType":"TextBox","stepTitle":"$$y=x {csc}^{\\\\left(-1\\\\right)} x$$","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a7079e8inverse21a-h1","type":"hint","dependencies":[],"title":"Applying the product rule.","text":"Because we are multiplying $$2$$ functions, we need to use the product rule in order to find the derivative. For reference the product rule is $$\\\\frac{d}{dy} f{\\\\left(x\\\\right)} g{\\\\left(x\\\\right)}=\\\\operatorname{f\'}\\\\left(x\\\\right) g{\\\\left(x\\\\right)}+f{\\\\left(x\\\\right)} \\\\operatorname{g\'}\\\\left(x\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse21a-h2","type":"hint","dependencies":["a7079e8inverse21a-h1"],"title":"Derivative of inverse trigonometric functions:csc(x)","text":"The formula for the derivative of inverse csc(x) is $$\\\\frac{d}{dx} {csc}^{-1} x=\\\\frac{-1}{|x| \\\\sqrt{x^2-1}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a7079e8inverse22","title":"For the following function, find (f**(-1))\'*(a), using the given values.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.7 Derivatives of Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a7079e8inverse22a","stepAnswer":["$$-1$$"],"problemType":"TextBox","stepTitle":"$$f(pi)=0$$, $$f\'(pi)=-1, a=0$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1$$","hints":{"DefaultPathway":[{"id":"a7079e8inverse22a-h1","type":"hint","dependencies":[],"title":"The inverse function theorem","text":"The inverse function theorem is that (f**(-1))\'*(x)=1/(f\'*(f**-1)*(x)).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse22a-h2","type":"hint","dependencies":["a7079e8inverse22a-h1"],"title":"Finding the Inverse of f(x) at $$a=0$$.","text":"To find the inverse of f(x) where $$a=0$$, then we need to first find what value of $$x$$ when we plug it into f(x) will give us a value of $$0$$. In this case, it is when $$x=pi$$, so we know that $$0f^{\\\\left(-1\\\\right)}=pi$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse22a-h3","type":"hint","dependencies":["a7079e8inverse22a-h2"],"title":"Finding the derivative of f(f**(-1)(x)))","text":"We found that the inverse of f(x) $$=pi$$ in the previous hint, therefore we need next know what f\'(pi) equals, which is given to us, it is $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse22a-h4","type":"hint","dependencies":["a7079e8inverse22a-h3"],"title":"Applying the Inverse Function Theorem","text":"Since we know that $$\\\\operatorname{f\'}\\\\left(f^{\\\\left(-1\\\\right)} \\\\pi\\\\right)$$ equals $$-1$$, we can can apply the inverse function theorem by doing 1/(our answer). In this case this is $$\\\\frac{1}{\\\\left(-1\\\\right)}=-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a7079e8inverse23","title":"For the following function, find (f**(-1))\'(a), using the given values.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.7 Derivatives of Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a7079e8inverse23a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"$$f(6)=2$$, $$f\'(6)=\\\\frac{1}{3}$$, $$a=2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a7079e8inverse23a-h1","type":"hint","dependencies":[],"title":"The inverse function theorem","text":"The inverse function theorem is that (f**(-1))\'(x)=(1/f\'((f**-1)(x))).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse23a-h2","type":"hint","dependencies":["a7079e8inverse23a-h1"],"title":"Finding the Inverse of f(x) at $$a=0$$.","text":"To find the inverse of f(x) where $$a=2$$, then we need to first find what value of $$x$$ when we plug it into f(x) will give us a value of $$2$$. In this case, it is when $$x=6$$, so we know that $$2f^{\\\\left(-1\\\\right)}=6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse23a-h3","type":"hint","dependencies":["a7079e8inverse23a-h2"],"title":"Finding the derivative of f(f**(-1)(x)))","text":"We found that the inverse of f(x) $$=6$$ in the previous hint, therefore we need next know what f\'(6) equals, which is given to us, it is $$\\\\frac{1}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse23a-h4","type":"hint","dependencies":["a7079e8inverse23a-h3"],"title":"Applying the Inverse Function Theorem","text":"Since we know that $$\\\\operatorname{f\'}\\\\left(6f^{\\\\left(-1\\\\right)}\\\\right)$$ equals $$\\\\frac{1}{3}$$, we can can apply the inverse function theorem by doing 1/(our answer). In this case this is $$\\\\frac{1}{\\\\frac{1}{3}}=3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a7079e8inverse24","title":"For the following function, find (f**(-1))\'(a), using the given values.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.7 Derivatives of Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a7079e8inverse24a","stepAnswer":["$$\\\\frac{1}{2}$$"],"problemType":"TextBox","stepTitle":"$$f{\\\\left(\\\\frac{1}{3}\\\\right)}=-8$$, $$\\\\operatorname{f\'}\\\\left(\\\\frac{1}{3}\\\\right)=2$$, $$a=-8$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{2}$$","hints":{"DefaultPathway":[{"id":"a7079e8inverse24a-h1","type":"hint","dependencies":[],"title":"The inverse function theorem","text":"The inverse function theorem is that (f**(-1))\'(x)=(1/f\'((f**-1)(x))).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse24a-h2","type":"hint","dependencies":["a7079e8inverse24a-h1"],"title":"Finding the Inverse of f(x) at $$a=0$$.","text":"To find the inverse of f(x) where $$a=-8$$, then we need to first find what value of $$x$$ when we plug it into f(x) will give us a value of $$-8$$. In this case, it is when $$x=\\\\frac{1}{3}$$, so we know that $$f^{\\\\left(-1\\\\right)} \\\\left(-8\\\\right)=\\\\frac{1}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse24a-h3","type":"hint","dependencies":["a7079e8inverse24a-h2"],"title":"Finding the derivative of f(f**(-1)(x)))","text":"We found that the inverse of f(x) $$=\\\\frac{1}{3}$$ in the previous hint, therefore we need next know what $$\\\\operatorname{f\'}\\\\left(\\\\frac{1}{3}\\\\right)$$ equals, which is given to us, it is $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse24a-h4","type":"hint","dependencies":["a7079e8inverse24a-h3"],"title":"Applying the Inverse Function Theorem","text":"Since we know that $$\\\\operatorname{f\'}\\\\left(f^{\\\\left(-1\\\\right)} \\\\frac{1}{3}\\\\right)$$ equals $$2$$, we can can apply the inverse function theorem by doing 1/(our answer). In this case this is $$\\\\frac{1}{2}=\\\\frac{1}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a7079e8inverse25","title":"For the following function, find (f**(-1))\'(a), using the given values.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.7 Derivatives of Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a7079e8inverse25a","stepAnswer":["$$\\\\frac{1}{10}$$"],"problemType":"TextBox","stepTitle":"$$f(1)=-3$$, $$f\'(1)=10$$, $$a=-3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{10}$$","hints":{"DefaultPathway":[{"id":"a7079e8inverse25a-h1","type":"hint","dependencies":[],"title":"The inverse function theorem","text":"The inverse function theorem is that (f**(-1))\'(x)=(1/f\'((f**-1)(x))).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse25a-h2","type":"hint","dependencies":["a7079e8inverse25a-h1"],"title":"Finding the Inverse of f(x) at $$a=0$$.","text":"To find the inverse of f(x) where $$a=-3$$, then we need to first find what value of $$x$$ when we plug it into f(x) will give us a value of $$-3$$. In this case, it is when $$x=1$$, so we know that f**-1(-3)=1.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse25a-h3","type":"hint","dependencies":["a7079e8inverse25a-h2"],"title":"Finding the derivative of f(f**(-1)(x)))","text":"We found that the inverse of f(x) $$=1$$ in the previous hint, therefore we need next know what f\'(1) equals, which is given to us, it is $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse25a-h4","type":"hint","dependencies":["a7079e8inverse25a-h3"],"title":"Applying the Inverse Function Theorem","text":"Since we know that $$\\\\operatorname{f\'}\\\\left(1f^{\\\\left(-1\\\\right)}\\\\right)$$ equals $$10$$, we can can apply the inverse function theorem by doing 1/(our answer). In this case this is $$\\\\frac{1}{10}=\\\\frac{1}{10}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a7079e8inverse3","title":"For the following function, find (f**(-1))\'(a)","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.7 Derivatives of Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a7079e8inverse3a","stepAnswer":["$$\\\\frac{2}{3}$$"],"problemType":"TextBox","stepTitle":"$$f(x)=x+\\\\sqrt{x}$$, $$a=2$$","stepBody":"Input your answer as a fraction excluding the $$\\"f(x)=\\"$$ part.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2}{3}$$","hints":{"DefaultPathway":[{"id":"a7079e8inverse3a-h1","type":"hint","dependencies":[],"title":"The inverse function theorem","text":"The inverse function theorem is that (f**(-1))\'*(x)=1/(f\'*(f**-1)*(x)).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse3a-h2","type":"hint","dependencies":["a7079e8inverse3a-h1"],"title":"Finding the Inverse of f(x) at $$a=0$$.","text":"To find the inverse of f(x) where $$a=2$$, then we need to first find what value of $$x$$ when we plug it into f(x) will give us a value of $$2$$. In this case, it is when $$x=1$$, so we know that $$2f^{\\\\left(-1\\\\right)}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse3a-h3","type":"hint","dependencies":["a7079e8inverse3a-h2"],"title":"Finding the derivative of f(x)","text":"Next, to find the derivative of f(x), we can use the power rule. To do this, we can multiply the coefficient of each $$x$$ by its current exponent, and then subtract $$1$$ from the exponent. Remember, constants become $$0$$ when we take the derivative of a function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse3a-h4","type":"hint","dependencies":["a7079e8inverse3a-h3"],"title":"Applying the Inverse Function Theorem","text":"Since we already found $$2f^{\\\\left(-1\\\\right)}=1$$ and the derivative of f(x), then we can plug what we know into the inverse function theorem.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a7079e8inverse4","title":"For the following function, find (f**(-1))\'(a)","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.7 Derivatives of Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a7079e8inverse4a","stepAnswer":["$$\\\\frac{1}{3}$$"],"problemType":"TextBox","stepTitle":"$$f(x)=x-\\\\frac{2}{x}$$, $$x<0$$, $$a=1$$","stepBody":"Input your answer as a fraction excluding the $$\\"f(x)=\\"$$ part.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{3}$$","hints":{"DefaultPathway":[{"id":"a7079e8inverse4a-h1","type":"hint","dependencies":[],"title":"The inverse function theorem","text":"The inverse function theorem is that (f**(-1))\'*(x)=1/(f\'*(f**-1)*(x)).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse4a-h2","type":"hint","dependencies":["a7079e8inverse4a-h1"],"title":"Finding the Inverse of f(x) at $$a=0$$.","text":"To find the inverse of f(x) where $$a=1$$, then we need to first find what value of $$x$$ when we plug it into f(x) will give us a value of $$1$$. In this case, it is when $$x=-1$$, so we know that $$1f^{\\\\left(-1\\\\right)}=-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse4a-h3","type":"hint","dependencies":["a7079e8inverse4a-h2"],"title":"Finding the derivative of f(x)","text":"Next, to find the derivative of f(x), we can use the power rule. To do this, we can multiply the coefficient of each $$x$$ by its current exponent, and then subtract $$1$$ from the exponent. Remember, constants become $$0$$ when we take the derivative of a function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse4a-h4","type":"hint","dependencies":["a7079e8inverse4a-h3"],"title":"Applying the Inverse Function Theorem","text":"Since we already found $$1f^{\\\\left(-1\\\\right)}=-1$$ and the derivative of f(x), then we can plug what we know into the inverse function theorem.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a7079e8inverse5","title":"For the following function, find (f**(-1))\'(a)","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.7 Derivatives of Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a7079e8inverse5a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"$$f(x)=x+sin\\\\left(x\\\\right)$$, $$a=0$$","stepBody":"Input your answer as a fraction excluding the $$\\"f(x)=\\"$$ part.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a7079e8inverse5a-h1","type":"hint","dependencies":[],"title":"The inverse function theorem","text":"The inverse function theorem is that (f**(-1))\'*(x)=1/(f\'*(f**-1)*(x)).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse5a-h2","type":"hint","dependencies":["a7079e8inverse5a-h1"],"title":"Finding the Inverse of f(x) at $$a=0$$.","text":"To find the inverse of f(x) where $$a=0$$, then we need to first find what value of $$x$$ when we plug it into f(x) will give us a value of $$0$$. In this case, it is when $$x=0$$, so we know that $$0f^{\\\\left(-1\\\\right)}=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse5a-h3","type":"hint","dependencies":["a7079e8inverse5a-h2"],"title":"Finding the derivative of f(x)","text":"Next, to find the derivative of f(x), we can use the power rule. To do this, we can multiply the coefficient of each $$x$$ by its current exponent, and then subtract $$1$$ from the exponent. Remember, constants become $$0$$ when we take the derivative of a function and the derivative of sin(x) is cos(x)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse5a-h4","type":"hint","dependencies":["a7079e8inverse5a-h3"],"title":"Applying the Inverse Function Theorem","text":"Since we already found $$0f^{\\\\left(-1\\\\right)}=0$$ and the derivative of f(x), then we can plug what we know into the inverse function theorem.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a7079e8inverse6","title":"For the following function, find (f**(-1))\'(a)","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.7 Derivatives of Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a7079e8inverse6a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"$$f(x)=tan\\\\left(x\\\\right)+3x^2$$, $$a=0$$","stepBody":"Input your answer as a fraction excluding the $$\\"f(x)=\\"$$ part.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a7079e8inverse6a-h1","type":"hint","dependencies":[],"title":"The inverse function theorem","text":"The inverse function theorem is that (f**(-1))\'*(x)=1/(f\'*(f**-1)*(x)).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse6a-h2","type":"hint","dependencies":["a7079e8inverse6a-h1"],"title":"Finding the Inverse of f(x) at $$a=0$$.","text":"To find the inverse of f(x) where $$a=0$$, then we need to first find what value of $$x$$ when we plug it into f(x) will give us a value of $$0$$. In this case, it is when $$x=0$$, so we know that $$0f^{\\\\left(-1\\\\right)}=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse6a-h3","type":"hint","dependencies":["a7079e8inverse6a-h2"],"title":"Finding the derivative of f(x)","text":"Next, to find the derivative of f(x), we can use the power rule. To do this, we can multiply the coefficient of each $$x$$ by its current exponent, and then subtract $$1$$ from the exponent. Remember, constants become $$0$$ when we take the derivative of a function and the derivative of tan(x) is $${sec}^2 x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse6a-h4","type":"hint","dependencies":["a7079e8inverse6a-h3"],"title":"Applying the Inverse Function Theorem","text":"Since we already found $$0f^{\\\\left(-1\\\\right)}=0$$ and the derivative of f(x), then we can plug what we know into the inverse function theorem.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a7079e8inverse7","title":"For the given function, $$f(x)=\\\\sqrt{x-4}$$, $$P(2,8)$$, find following.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.7 Derivatives of Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a7079e8inverse7a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"Find the slope of the tangent line to its inverse function $$f^{\\\\left(-1\\\\right)}$$ at the indicated point P","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a7079e8inverse7a-h1","type":"hint","dependencies":[],"title":"The inverse function theorem will tell us the slope of the tangent line to the inverse function.","text":"The inverse function theorem is that (f**(-1))\'*(x)=1/(f\'*(f**-1)*(x)).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse7a-h2","type":"hint","dependencies":["a7079e8inverse7a-h1"],"title":"Finding the Inverse of f(x) at $$a=0$$.","text":"To find the inverse of f(x) where $$a=2$$ (remember that a is equal to the x-value of a point), then we need to first find what value of $$x$$ when we plug it into f(x) will give us a value of $$2$$. In this case, it is when $$x=8$$, so we know that $$2f^{\\\\left(-1\\\\right)}=8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse7a-h3","type":"hint","dependencies":["a7079e8inverse7a-h2"],"title":"Finding the derivative of f(x)","text":"Next, to find the derivative of f(x), we can use the power rule. To do this, we can multiply the coefficient of each $$x$$ by its current exponent, and then subtract $$1$$ from the exponent. Remember, constants become $$0$$ when we take the derivative of a function and the derivative of tan(x) is $${sec}^2 x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse7a-h4","type":"hint","dependencies":["a7079e8inverse7a-h3"],"title":"Applying the Inverse Function Theorem","text":"Since we already found $$2f^{\\\\left(-1\\\\right)}=8$$ and the derivative of f(x), then we can plug what we know into the inverse function theorem. Our resulting slope is $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a7079e8inverse7b","stepAnswer":["$$y=4x$$"],"problemType":"MultipleChoice","stepTitle":"Find the equation of the tangent line to the graph of $$f^{\\\\left(-1\\\\right)}$$ at the indicated point P.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=4x$$","choices":["$$y=4x$$","$$y=3x$$","$$y=4x+1$$"],"hints":{"DefaultPathway":[{"id":"a7079e8inverse7b-h1","type":"hint","dependencies":[],"title":"Use the point-slope form for the equation of a line.","text":"The point-slope form for the equation of a line is $$y-y1=m \\\\left(x-x1\\\\right)$$, We can plug in what we know, which is $$m=4$$, $$x1=2$$, $$y1=8$$, to solve for the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a7079e8inverse8","title":"For the given function, $$f(x)=\\\\frac{4}{1+x^2}$$, $$P(2,1)$$, find following.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.7 Derivatives of Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a7079e8inverse8a","stepAnswer":["$$\\\\frac{-1}{2}$$"],"problemType":"TextBox","stepTitle":"Find the slope of the tangent line to its inverse function $$f^{\\\\left(-1\\\\right)}$$ at the indicated point P","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-1}{2}$$","hints":{"DefaultPathway":[{"id":"a7079e8inverse8a-h1","type":"hint","dependencies":[],"title":"The inverse function theorem will tell us the slope of the tangent line to the inverse function.","text":"The inverse function theorem is that (f**(-1))\'*(x)=1/(f\'*(f**-1)*(x)).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse8a-h2","type":"hint","dependencies":["a7079e8inverse8a-h1"],"title":"Finding the Inverse of f(x) at $$a=0$$.","text":"To find the inverse of f(x) where $$a=2$$ (remember that a is equal to the x-value of a point), then we need to first find what value of $$x$$ when we plug it into f(x) will give us a value of $$2$$. In this case, it is when $$x=1$$, so we know that $$2f^{\\\\left(-1\\\\right)}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse8a-h3","type":"hint","dependencies":["a7079e8inverse8a-h2"],"title":"Finding the derivative of f(x)","text":"Next, to find the derivative of f(x), we can use the power rule. To do this, we can multiply the coefficient of each $$x$$ by its current exponent, and then subtract $$1$$ from the exponent. Remember, constants become $$0$$ when we take the derivative of a function and the derivative of tan(x) is $${sec}^2 x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse8a-h4","type":"hint","dependencies":["a7079e8inverse8a-h3"],"title":"Applying the Inverse Function Theorem","text":"Since we already found $$2f^{\\\\left(-1\\\\right)}=1$$ and the derivative of f(x), then we can plug what we know into the inverse function theorem. Our resulting slope is $$\\\\frac{-1}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a7079e8inverse8b","stepAnswer":["$$y=-\\\\left(\\\\frac{x}{2}\\\\right)+2$$"],"problemType":"MultipleChoice","stepTitle":"Find the equation of the tangent line to the graph of $$f^{\\\\left(-1\\\\right)}$$ at the indicated point P.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=-\\\\left(\\\\frac{x}{2}\\\\right)+2$$","choices":["$$y=-\\\\left(\\\\frac{x}{2}\\\\right)+2$$","$$y=\\\\frac{x}{2}$$","$$y=-\\\\left(\\\\frac{x}{3}\\\\right)$$","$$y=(2x)$$"],"hints":{"DefaultPathway":[{"id":"a7079e8inverse8b-h1","type":"hint","dependencies":[],"title":"Use the point-slope form for the equation of a line.","text":"The point-slope form for the equation of a line is $$y-y1=m \\\\left(x-x1\\\\right)$$, We can plug in what we know, which is $$m=\\\\frac{-1}{2}$$, $$x1=2$$, $$y1=1$$, to solve for the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a7079e8inverse9","title":"For the given function, $$f(x)={\\\\left(x^3+1\\\\right)}^4$$, $$P(16,1)$$, find following.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.7 Derivatives of Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a7079e8inverse9a","stepAnswer":["$$\\\\frac{1}{96}$$"],"problemType":"TextBox","stepTitle":"Find the slope of the tangent line to its inverse function $$f^{\\\\left(-1\\\\right)}$$ at the indicated point P","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{96}$$","hints":{"DefaultPathway":[{"id":"a7079e8inverse9a-h1","type":"hint","dependencies":[],"title":"The inverse function theorem will tell us the slope of the tangent line to the inverse function.","text":"The inverse function theorem is that (f**(-1))\'*(x)=1/(f\'*(f**-1)*(x)).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse9a-h2","type":"hint","dependencies":["a7079e8inverse9a-h1"],"title":"Finding the Inverse of f(x) at $$a=0$$.","text":"To find the inverse of f(x) where $$a=16$$ (remember that a is equal to the x-value of a point), then we need to first find what value of $$x$$ when we plug it into f(x) will give us a value of $$16$$. In this case, it is when $$x=1$$, so we know that $$16f^{\\\\left(-1\\\\right)}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse9a-h3","type":"hint","dependencies":["a7079e8inverse9a-h2"],"title":"Finding the derivative of f(x)","text":"Next, to find the derivative of f(x), we can use the power rule. To do this, we can multiply the coefficient of each $$x$$ by its current exponent, and then subtract $$1$$ from the exponent. Remember, constants become $$0$$ when we take the derivative of a function and the derivative of tan(x) is $${sec}^2 x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a7079e8inverse9a-h4","type":"hint","dependencies":["a7079e8inverse9a-h3"],"title":"Applying the Inverse Function Theorem","text":"Since we already found $$16f^{\\\\left(-1\\\\right)}=1$$ and the derivative of f(x), then we can plug what we know into the inverse function theorem. Our resulting slope is $$\\\\frac{1}{96}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a7079e8inverse9b","stepAnswer":["$$y=\\\\frac{x}{96}-5$$"],"problemType":"MultipleChoice","stepTitle":"Find the equation of the tangent line to the graph of $$f^{\\\\left(-1\\\\right)}$$ at the indicated point P.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=\\\\frac{x}{96}-5$$","choices":["$$y=\\\\frac{x}{96}-5$$","$$y=\\\\frac{x}{96}$$","$$y=\\\\frac{x}{96}-7$$"],"hints":{"DefaultPathway":[{"id":"a7079e8inverse9b-h1","type":"hint","dependencies":[],"title":"Use the point-slope form for the equation of a line.","text":"The point-slope form for the equation of a line is $$y-y1=m \\\\left(x-x1\\\\right)$$, We can plug in what we know, which is $$m=\\\\frac{1}{96}$$, $$x1=16$$, $$y1=1$$, to solve for the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a73ccaalimits1","title":"For the following exercise, examine the graph. Identify where the vertical asymptote(s) are located.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.6 Limits at Infinity and Asymptotes","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a73ccaalimits1a","stepAnswer":["$$x=1$$"],"problemType":"MultipleChoice","stepTitle":"Identify where the vertical asymptote(s) are located.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$x=1$$","choices":["$$x=1$$","$$x=2$$","$$x=0$$"],"hints":{"DefaultPathway":[{"id":"a73ccaalimits1a-h1","type":"hint","dependencies":[],"title":"Identifying vertical asymptotes visually","text":"To identify vertical asymptotes visually, look for the value of $$x$$ at which the function approaches y-values of infinity or negative $$\\\\infty$$. Remember that a function cannot cross a vertical asymptote.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a73ccaalimits10","title":"For the following function, determine whether there is an asymptote at $$x=a$$ without using a graphing calculator.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.6 Limits at Infinity and Asymptotes","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a73ccaalimits10a","stepAnswer":["No, there is not a vertical asymptote at $$a=1$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=1+x^{\\\\left(-\\\\frac{2}{5}\\\\right)}$$ where $$a=1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"No, there is not a vertical asymptote at $$a=1$$","choices":["Yes, there is a vertical asymptote at $$a=1$$","No, there is not a vertical asymptote at $$a=1$$"],"hints":{"DefaultPathway":[{"id":"a73ccaalimits10a-h1","type":"hint","dependencies":[],"title":"Simplify the function","text":"The first step to determine whether there is an asymptote at the given $$x=a$$ is to simplify the function if possible. Remember if we can cancel out something after factoring the numerator and denominator there is a hole at this cancelled out value, which is different from a vertical asymptote.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a73ccaalimits10a-h2","type":"hint","dependencies":["a73ccaalimits10a-h1"],"title":"Determining if there is an asymptote at the given value of $$x=a$$","text":"To determine if there is an asymptote at the given value of $$a=1$$, we need to plug in our value of $$x=1$$ into the simplifed function. If we get a real number, then there is no vertical asymptote. If we do not get a real number (e.g. 5/0), then there is a vertical asymptote at the given value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a73ccaalimits11","title":"For the following exercise, evaluate the limit.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.6 Limits at Infinity and Asymptotes","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a73ccaalimits11a","stepAnswer":["$$0$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{1}{3x+6}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$0$$","choices":["$$\\\\infty$$","$$-\\\\infty$$","$$1$$","$$0$$"],"hints":{"DefaultPathway":[{"id":"a73ccaalimits11a-h1","type":"hint","dependencies":[],"title":"Evaluating the limit","text":"The first step to evaluate the limit is plug in infinity for $$x$$ in the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a73ccaalimits11a-h2","type":"hint","dependencies":["a73ccaalimits11a-h1"],"title":"Property of Infinity","text":"Remember that infinity is theoretically larger than any other real number, so it will have the greatest effect on the function. Thus we can ignore any numbers or coefficent that do not have the \\"largest infinity\\". (e.g. $$2$$ times infinity squared is larger than 1,000 times $$\\\\infty$$, so we focus on the effect on $$2$$ times infinity squared)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a73ccaalimits11a-h3","type":"hint","dependencies":["a73ccaalimits11a-h2"],"title":"What we are really looking at","text":"What we are really looking at in this problem is $$\\\\frac{1}{\\\\infty}$$ since the $$6$$ in the denominator and the coefficent of $$3$$ will not have as large an effect on the limit compared to $$\\\\infty$$. Thus, since $$\\\\frac{1}{\\\\infty}$$ is infinitely small, the limit as $$x$$ approaches infinity approaches closer to $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a73ccaalimits12","title":"For the following exercise, evaluate the limit.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.6 Limits at Infinity and Asymptotes","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a73ccaalimits12a","stepAnswer":["$$\\\\frac{1}{2}$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{2x-5}{4x}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{1}{2}$$","choices":["$$\\\\infty$$","$$-\\\\infty$$","$$0$$","$$\\\\frac{1}{2}$$"],"hints":{"DefaultPathway":[{"id":"a73ccaalimits12a-h1","type":"hint","dependencies":[],"title":"Evaluating the limit","text":"The first step to evaluate the limit is plug in infinity for $$x$$ in the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a73ccaalimits12a-h2","type":"hint","dependencies":["a73ccaalimits12a-h1"],"title":"Property of Infinity","text":"Remember that infinity is theoretically larger than any other real number, so it will have the greatest effect on the function. Thus we can ignore any numbers or coefficent that do not have the \\"largest infinity\\". (e.g. $$2$$ times infinity squared is larger than 1,000 times $$\\\\infty$$, so we focus on the effect on $$2$$ times infinity squared)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a73ccaalimits12a-h3","type":"hint","dependencies":["a73ccaalimits12a-h2"],"title":"What we are really looking at","text":"What we are really looking at in this problem is $$\\\\frac{2\\\\infty}{4\\\\infty}$$ since the negative $$5$$ in the numerator not have as large an effect on the limit compared to $$\\\\infty$$. We can cancel out the infinities since they are of the same power, which leaves us with $$\\\\frac{2}{4}$$ or $$\\\\frac{1}{2}$$ as our answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a73ccaalimits13","title":"For the following exercise, evaluate the limit.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.6 Limits at Infinity and Asymptotes","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a73ccaalimits13a","stepAnswer":["$$\\\\infty$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{x^2-2x+5}{x+2}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\infty$$","choices":["$$\\\\infty$$","$$-\\\\infty$$","$$1$$","$$0$$"],"hints":{"DefaultPathway":[{"id":"a73ccaalimits13a-h1","type":"hint","dependencies":[],"title":"Evaluating the limit","text":"The first step to evaluate the limit is plug in infinity for $$x$$ in the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a73ccaalimits13a-h2","type":"hint","dependencies":["a73ccaalimits13a-h1"],"title":"Property of Infinity","text":"Remember that infinity is theoretically larger than any other real number, so it will have the greatest effect on the function. Thus we can ignore any numbers or coefficent that do not have the \\"largest infinity\\". (e.g. $$2$$ times infinity squared is larger than 1,000 times $$\\\\infty$$, so we focus on the effect on $$2$$ times infinity squared)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a73ccaalimits13a-h3","type":"hint","dependencies":["a73ccaalimits13a-h2"],"title":"What we are really looking at","text":"What we are really looking at in this problem is $$\\\\frac{{\\\\infty}^2}{\\\\infty}$$ since the $$-2x+5$$ and $$+2$$ will not have as large an effect on the limit compared to $$\\\\infty$$. We can cancel out an infinity from the numerator and denominator, which leaves us with an answer of positive $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a73ccaalimits14","title":"For the following exercise, evaluate the limit.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.6 Limits at Infinity and Asymptotes","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a73ccaalimits14a","stepAnswer":["$$-\\\\infty$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\lim_{x\\\\to-\\\\infty} \\\\frac{3x^3-2x}{x^2+2x+8}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-\\\\infty$$","choices":["$$\\\\infty$$","$$-\\\\infty$$","$$1$$","$$0$$"],"hints":{"DefaultPathway":[{"id":"a73ccaalimits14a-h1","type":"hint","dependencies":[],"title":"Evaluating the limit","text":"The first step to evaluate the limit is plug in negative infinity for $$x$$ in the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a73ccaalimits14a-h2","type":"hint","dependencies":["a73ccaalimits14a-h1"],"title":"Property of Infinity","text":"Remember that absolute value of infinity is theoretically larger than any other real number, so it will have the greatest effect on the function. Thus we can ignore any numbers or coefficent that do not have the \\"largest infinity\\". (e.g. absolute valie of $$2$$ times infinity squared is larger than the absolute value of 1,000 times $$\\\\infty$$, so we focus on the effect on $$2$$ times infinity squared)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a73ccaalimits14a-h3","type":"hint","dependencies":["a73ccaalimits14a-h2"],"title":"What we are really looking at","text":"What we are really looking at in this problem is $$\\\\frac{3\\\\left(-{\\\\infty}^3\\\\right)}{\\\\left(-{\\\\infty}^2\\\\right)}$$ since the $$-2x$$ and $$\\\\left(+2\\\\right) x+2$$ will not have as large an effect on the limit compared to $$\\\\infty$$. We can cancel out a $$\\\\left(-{\\\\infty}^2\\\\right)$$ and are left with $$3\\\\left(-\\\\infty\\\\right)$$ which is practically the same as just negative $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a73ccaalimits15","title":"For the following exercise, evaluate the limit.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.6 Limits at Infinity and Asymptotes","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a73ccaalimits15a","stepAnswer":["$$\\\\frac{-1}{7}$$"],"problemType":"TextBox","stepTitle":"$$\\\\lim_{x\\\\to-\\\\infty} \\\\frac{x^4-4x^3+1}{\\\\left(-7x^4-2x^2+2\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-1}{7}$$","hints":{"DefaultPathway":[{"id":"a73ccaalimits15a-h1","type":"hint","dependencies":[],"title":"Evaluating the limit","text":"The first step to evaluate the limit is plug in negative infinity for $$x$$ in the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a73ccaalimits15a-h2","type":"hint","dependencies":["a73ccaalimits15a-h1"],"title":"Property of Infinity","text":"Remember that absolute value of infinity is theoretically larger than any other real number, so it will have the greatest effect on the function. Thus we can ignore any numbers or coefficent that do not have the \\"largest infinity\\". (e.g. absolute valie of $$2$$ times infinity squared is larger than the absolute value of 1,000 times $$\\\\infty$$, so we focus on the effect on $$2$$ times infinity squared)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a73ccaalimits15a-h3","type":"hint","dependencies":["a73ccaalimits15a-h2"],"title":"What we are really looking at","text":"What we are really looking at in this problem is $$\\\\frac{\\\\left(-{\\\\infty}^4\\\\right)}{\\\\left(-7\\\\left(-{\\\\infty}^4\\\\right)\\\\right)}$$ since the $$\\\\left(-4x^3\\\\right)+1$$ and $$\\\\left(-2x^2\\\\right)+2$$ will not have as large an effect on the limit compared to $$\\\\infty$$. We can cancel out $$\\\\left(-{\\\\infty}^4\\\\right)$$ and are left with $$\\\\frac{1}{\\\\left(-7\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a73ccaalimits16","title":"For the following exercise, evaluate the limit.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.6 Limits at Infinity and Asymptotes","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a73ccaalimits16a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{3x}{\\\\sqrt{x^2+1}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a73ccaalimits16a-h1","type":"hint","dependencies":[],"title":"Evaluating the limit","text":"The first step to evaluate the limit is plug in infinity for $$x$$ in the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a73ccaalimits16a-h2","type":"hint","dependencies":["a73ccaalimits16a-h1"],"title":"Property of Infinity","text":"Remember that absolute value of infinity is theoretically larger than any other real number, so it will have the greatest effect on the function. Thus we can ignore any numbers or coefficent that do not have the \\"largest infinity\\". (e.g. absolute valie of $$2$$ times infinity squared is larger than the absolute value of 1,000 times $$\\\\infty$$, so we focus on the effect on $$2$$ times infinity squared)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a73ccaalimits16a-h3","type":"hint","dependencies":["a73ccaalimits16a-h2"],"title":"What we are really looking at","text":"What we are really looking at in this problem is $$\\\\frac{3\\\\infty}{\\\\infty}$$. We can cancel out the infinity and are left with $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a73ccaalimits17","title":"For the following exercise, evaluate the limit.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.6 Limits at Infinity and Asymptotes","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a73ccaalimits17a","stepAnswer":["$$-2$$"],"problemType":"TextBox","stepTitle":"$$\\\\lim_{x\\\\to-\\\\infty} \\\\frac{\\\\sqrt{4x^2-1}}{x+2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-2$$","hints":{"DefaultPathway":[{"id":"a73ccaalimits17a-h1","type":"hint","dependencies":[],"title":"Evaluating the limit","text":"The first step to evaluate the limit is plug in negative infinity for $$x$$ in the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a73ccaalimits17a-h2","type":"hint","dependencies":["a73ccaalimits17a-h1"],"title":"Property of Infinity","text":"Remember that absolute value of infinity is theoretically larger than any other real number, so it will have the greatest effect on the function. Thus we can ignore any numbers or coefficent that do not have the \\"largest infinity\\". (e.g. absolute valie of $$2$$ times infinity squared is larger than the absolute value of 1,000 times $$\\\\infty$$, so we focus on the effect on $$2$$ times infinity squared)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a73ccaalimits17a-h3","type":"hint","dependencies":["a73ccaalimits17a-h2"],"title":"What we are really looking at","text":"What we are really looking at in this problem is $$\\\\frac{2\\\\infty}{\\\\left(-\\\\infty\\\\right)}$$. We can cancel out an infinity and are left with $$\\\\frac{2}{-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a73ccaalimits18","title":"For the following exercise, evaluate the limit.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.6 Limits at Infinity and Asymptotes","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a73ccaalimits18a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{4x}{\\\\sqrt{x^2-1}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a73ccaalimits18a-h1","type":"hint","dependencies":[],"title":"Evaluating the limit","text":"The first step to evaluate the limit is plug in infinity for $$x$$ in the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a73ccaalimits18a-h2","type":"hint","dependencies":["a73ccaalimits18a-h1"],"title":"Property of Infinity","text":"Remember that absolute value of infinity is theoretically larger than any other real number, so it will have the greatest effect on the function. Thus we can ignore any numbers or coefficent that do not have the \\"largest infinity\\". (e.g. absolute valie of $$2$$ times infinity squared is larger than the absolute value of 1,000 times $$\\\\infty$$, so we focus on the effect on $$2$$ times infinity squared)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a73ccaalimits18a-h3","type":"hint","dependencies":["a73ccaalimits18a-h2"],"title":"What we are really looking at","text":"What we are really looking at in this problem is $$\\\\frac{4\\\\infty}{\\\\infty}$$. We can cancel out infinity and are left with $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a73ccaalimits19","title":"For the following exercise, evaluate the limit.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.6 Limits at Infinity and Asymptotes","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a73ccaalimits19a","stepAnswer":["$$-4$$"],"problemType":"TextBox","stepTitle":"$$\\\\lim_{x\\\\to-\\\\infty} \\\\frac{4x}{\\\\sqrt{x^2-1}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-4$$","hints":{"DefaultPathway":[{"id":"a73ccaalimits19a-h1","type":"hint","dependencies":[],"title":"Evaluating the limit","text":"The first step to evaluate the limit is plug in negative infinity for $$x$$ in the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a73ccaalimits19a-h2","type":"hint","dependencies":["a73ccaalimits19a-h1"],"title":"Property of Infinity","text":"Remember that absolute value of infinity is theoretically larger than any other real number, so it will have the greatest effect on the function. Thus we can ignore any numbers or coefficent that do not have the \\"largest infinity\\". (e.g. absolute valie of $$2$$ times infinity squared is larger than the absolute value of 1,000 times $$\\\\infty$$, so we focus on the effect on $$2$$ times infinity squared)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a73ccaalimits19a-h3","type":"hint","dependencies":["a73ccaalimits19a-h2"],"title":"What we are really looking at","text":"What we are really looking at in this problem is $$\\\\frac{4-\\\\infty}{\\\\infty}$$. We can cancel out infinity and are left with $$-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a73ccaalimits2","title":"For the following exercise, examine the graph. Identify where the vertical asymptote(s) are located.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.6 Limits at Infinity and Asymptotes","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a73ccaalimits2a","stepAnswer":["$$x=-3;x=2$$"],"problemType":"MultipleChoice","stepTitle":"Identify where the vertical asymptote(s) are located.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$x=-3;x=2$$","choices":["$$x=-3;x=2$$","$$x=-3;x=-2$$","$$x=3;x=2$$"],"hints":{"DefaultPathway":[{"id":"a73ccaalimits2a-h1","type":"hint","dependencies":[],"title":"Identifying vertical asymptotes visually","text":"To identify vertical asymptotes visually, look for the value of $$x$$ at which the function approaches y-values of infinity or negative $$\\\\infty$$. Remember that a function cannot cross a vertical asymptote.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a73ccaalimits20","title":"For the following exercise, evaluate the limit.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.6 Limits at Infinity and Asymptotes","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a73ccaalimits20a","stepAnswer":["$$0$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\lim_{x\\\\to\\\\infty} \\\\frac{2\\\\sqrt{x}}{x-\\\\sqrt{x}+1}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$0$$","choices":["$$\\\\infty$$","$$-\\\\infty$$","$$1$$","$$0$$"],"hints":{"DefaultPathway":[{"id":"a73ccaalimits20a-h1","type":"hint","dependencies":[],"title":"Evaluating the limit","text":"The first step to evaluate the limit is plug in infinity for $$x$$ in the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a73ccaalimits20a-h2","type":"hint","dependencies":["a73ccaalimits20a-h1"],"title":"Property of Infinity","text":"Remember that infinity is theoretically larger than any other real number, so it will have the greatest effect on the function. Thus we can ignore any numbers or coefficent that do not have the \\"largest infinity\\". (e.g. $$2$$ times infinity squared is larger than 1,000 times $$\\\\infty$$, so we focus on the effect on $$2$$ times infinity squared)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a73ccaalimits20a-h3","type":"hint","dependencies":["a73ccaalimits20a-h2"],"title":"What we are really looking at","text":"What we are really looking at in this problem is $$\\\\frac{1}{\\\\infty}$$ since the $$2\\\\sqrt{x}$$ and $$-\\\\sqrt{x}+1$$ will not have as large an effect on the limit compared to $$\\\\infty$$. Thus, since $$\\\\frac{1}{\\\\infty}$$ is infinitely small, the limit as $$x$$ approaches infinity approaches closer to $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a73ccaalimits21","title":"For the following exercise, find the horizontal and vertical asymptotes.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.6 Limits at Infinity and Asymptotes","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a73ccaalimits21a","stepAnswer":["Horizontal Asymptotes: None, Vertical Asymptotes: $$x=0$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=x-\\\\frac{9}{x}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Horizontal Asymptotes: None, Vertical Asymptotes: $$x=0$$","choices":["Horizontal Asymptotes: None, Vertical Asymptotes: $$x=0$$","Horizontal Asymptotes: None, Vertical Asymptotes: $$x=2$$","Horizontal Asymptotes: None, Vertical Asymptotes: $$x=1$$","Horizontal Asymptotes: $$y=0$$, Vertical Asymptotes: None"],"hints":{"DefaultPathway":[{"id":"a73ccaalimits21a-h1","type":"hint","dependencies":[],"title":"Determining if there is a Vertical Asymptote","text":"To determine if there is a vertical asymptote, we need to look for values of $$x$$ which would make the function undefined. In this case, if we plugged in $$x=0$$, then the $$\\\\frac{9}{x}$$ would become $$\\\\frac{9}{0}$$ which is undefined.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a73ccaalimits21a-h2","type":"hint","dependencies":["a73ccaalimits21a-h1"],"title":"Determining if there is a Horizontal Asymptote","text":"To determine if there is a horizontal asymptote, we need to find the limit of the function as $$x$$ approaches infinity and negative $$\\\\infty$$. If we get a real number, then there is a horizontal asymptote there. 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If not, then there is no horizontal asymptote for the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a73ccaalimits23","title":"For the following exercise, find the horizontal and vertical asymptotes.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.6 Limits at Infinity and Asymptotes","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a73ccaalimits23a","stepAnswer":["Horizontal Asymptotes:None,Vertical $$Asymptotes:x=2;x=-2$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{x^3}{4-x^2}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Horizontal Asymptotes:None,Vertical $$Asymptotes:x=2;x=-2$$","choices":["Horizontal $$Asymptotes:y=1:y=-1, Vertical$$ $$Asymptotes:x=0$$","Horizontal Asymptotes:None,Vertical $$Asymptotes:x=2;x=-2$$","Horizontal Asymptotes:None,Vertical $$Asymptotes:x=1$$","Horizontal $$Asymptotes:y=0, Vertical$$ Asymptotes:None"],"hints":{"DefaultPathway":[{"id":"a73ccaalimits23a-h1","type":"hint","dependencies":[],"title":"Determining if there is a Vertical Asymptote","text":"To determine if there is a vertical asymptote, we need to look for values of $$x$$ which would make the function undefined. In this case, if we plugged in $$x=2$$ or $$x=-2$$, then the function would become $$\\\\frac{8}{0}$$ or $$\\\\frac{-8}{0}$$ which are undefined.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a73ccaalimits23a-h2","type":"hint","dependencies":["a73ccaalimits23a-h1"],"title":"Determining if there is a Horizontal Asymptote","text":"To determine if there is a horizontal asymptote, we need to find the limit of the function as $$x$$ approaches infinity and negative $$\\\\infty$$. If we get a real number, then there is a horizontal asymptote there. If not, then there is no horizontal asymptote for the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a73ccaalimits24","title":"For the following exercise, find the horizontal and vertical asymptotes.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.6 Limits at Infinity and Asymptotes","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a73ccaalimits24a","stepAnswer":["Horizontal $$Asymptotes:y=1, Vertical$$ Asymptotes:None"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{x^2+3}{x^2+1}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Horizontal $$Asymptotes:y=1, Vertical$$ Asymptotes:None","choices":["Horizontal $$Asymptotes:y=1, Vertical$$ Asymptotes:None","Horizontal Asymptotes:None,Vertical $$Asymptotes:x=2;x=-2$$","Horizontal Asymptotes:None,Vertical $$Asymptotes:x=1$$","Horizontal $$Asymptotes:y=0, Vertical$$ Asymptotes:None"],"hints":{"DefaultPathway":[{"id":"a73ccaalimits24a-h1","type":"hint","dependencies":[],"title":"Determining if there is a Vertical Asymptote","text":"To determine if there is a vertical asymptote, we need to look for values of $$x$$ which would make the function undefined.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a73ccaalimits24a-h2","type":"hint","dependencies":["a73ccaalimits24a-h1"],"title":"Determining if there is a Horizontal Asymptote","text":"To determine if there is a horizontal asymptote, we need to find the limit of the function as $$x$$ approaches infinity and negative $$\\\\infty$$. If we get a real number, then there is a horizontal asymptote there. If not, then there is no horizontal asymptote for the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a73ccaalimits25","title":"For the following exercise, find the horizontal and vertical asymptotes.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.6 Limits at Infinity and Asymptotes","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a73ccaalimits25a","stepAnswer":["Horizontal $$Asymptotes:y=1, Vertical$$ $$Asymptotes:x=1$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{x^3+1}{x^3-1}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Horizontal $$Asymptotes:y=1, Vertical$$ $$Asymptotes:x=1$$","choices":["Horizontal $$Asymptotes:y=1, Vertical$$ $$Asymptotes:x=1$$","Horizontal Asymptotes:None,Vertical $$Asymptotes:x=2;x=-2$$","Horizontal Asymptotes:None,Vertical $$Asymptotes:x=1$$","Horizontal $$Asymptotes:y=0, Vertical$$ Asymptotes:None"],"hints":{"DefaultPathway":[{"id":"a73ccaalimits25a-h1","type":"hint","dependencies":[],"title":"Determining if there is a Vertical Asymptote","text":"To determine if there is a vertical asymptote, we need to look for values of $$x$$ which would make the function undefined. In this case, if we plugged in $$x=1$$, then the function would become $$\\\\frac{2}{0}$$ which is undefined.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a73ccaalimits25a-h2","type":"hint","dependencies":["a73ccaalimits25a-h1"],"title":"Determining if there is a Horizontal Asymptote","text":"To determine if there is a horizontal asymptote, we need to find the limit of the function as $$x$$ approaches infinity and negative $$\\\\infty$$. If we get a real number, then there is a horizontal asymptote there. If not, then there is no horizontal asymptote for the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a73ccaalimits3","title":"For the following exercise, examine the graph. 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Remember that a function cannot cross a vertical asymptote.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a73ccaalimits4","title":"For the following exercise, examine the graph. Identify where the vertical asymptote(s) are located.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.6 Limits at Infinity and Asymptotes","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a73ccaalimits4a","stepAnswer":["$$x=0;x=1;x=2$$"],"problemType":"MultipleChoice","stepTitle":"Identify where the vertical asymptote(s) are located.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$x=0;x=1;x=2$$","choices":["$$x=0;x=1;x=2$$","$$x=0;x=-1;x=2$$","$$x=-2;x=1;x=2$$"],"hints":{"DefaultPathway":[{"id":"a73ccaalimits4a-h1","type":"hint","dependencies":[],"title":"Identifying vertical asymptotes visually","text":"To identify vertical asymptotes visually, look for the value of $$x$$ at which the function approaches y-values of infinity or negative $$\\\\infty$$. Remember that a function cannot cross a vertical asymptote.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a73ccaalimits5","title":"For the following exercise, examine the graph. Identify where the vertical asymptote(s) are located.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.6 Limits at Infinity and Asymptotes","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a73ccaalimits5a","stepAnswer":["$$x=0$$"],"problemType":"MultipleChoice","stepTitle":"Identify where the vertical asymptote(s) are located.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$x=0$$","choices":["$$x=1$$","$$x=2$$","$$x=0$$"],"hints":{"DefaultPathway":[{"id":"a73ccaalimits5a-h1","type":"hint","dependencies":[],"title":"Identifying vertical asymptotes visually","text":"To identify vertical asymptotes visually, look for the value of $$x$$ at which the function approaches y-values of infinity or negative $$\\\\infty$$. Remember that a function cannot cross a vertical asymptote.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a73ccaalimits6","title":"For the following function, determine whether there is an asymptote at $$x=a$$ without using a graphing calculator.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.6 Limits at Infinity and Asymptotes","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a73ccaalimits6a","stepAnswer":["No, there is not a vertical asymptote at $$a=-1$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{x+1}{x^2+5x+4}$$, $$a=-1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"No, there is not a vertical asymptote at $$a=-1$$","choices":["Yes, there is a vertical asymptote at $$a=-1$$","No, there is not a vertical asymptote at $$a=-1$$"],"hints":{"DefaultPathway":[{"id":"a73ccaalimits6a-h1","type":"hint","dependencies":[],"title":"Simplify the function","text":"The first step to determine whether there is an asymptote at the given $$x=a$$ is to simplify the function if possible. In this case, we can do this by factoring the numerator and denominator and cancelling out $$x+1$$ which meaning $$x=-1$$ is undefined and there is a hole at this value, which is different from a vertical asymptote.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a73ccaalimits6a-h2","type":"hint","dependencies":["a73ccaalimits6a-h1"],"title":"Determining if there is an asymptote at the given value of $$x=a$$","text":"To determine if there is an asymptote at the given value of $$a=-1$$, we need to plug in our value of $$x=-1$$ into the simplifed function. If we get a real number, then there is no vertical asymptote. If we do not get a real number, then there is a vertical asymptote at the given value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a73ccaalimits7","title":"For the following function, determine whether there is an asymptote at $$x=a$$ without using a graphing calculator.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.6 Limits at Infinity and Asymptotes","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a73ccaalimits7a","stepAnswer":["Yes, there is a vertical asymptote at $$a=2$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{x}{x-2}$$ where $$a=2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Yes, there is a vertical asymptote at $$a=2$$","choices":["Yes, there is a vertical asymptote at $$a=2$$","No, there is not a vertical asymptote at $$a=2$$"],"hints":{"DefaultPathway":[{"id":"a73ccaalimits7a-h1","type":"hint","dependencies":[],"title":"Simplify the function","text":"The first step to determine whether there is an asymptote at the given $$x=a$$ is to simplify the function if possible. 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If we do not get a real number (e.g. 5/0), then there is a vertical asymptote at the given value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a73ccaalimits8","title":"For the following function, determine whether there is an asymptote at $$x=a$$ without using a graphing calculator.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.6 Limits at Infinity and Asymptotes","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a73ccaalimits8a","stepAnswer":["No, there is not a vertical asymptote at $$a=-2$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)={\\\\left(x+2\\\\right)}^{\\\\frac{3}{2}}$$ where $$a=-2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"No, there is not a vertical asymptote at $$a=-2$$","choices":["Yes, there is a vertical asymptote at $$a=-2$$","No, there is not a vertical asymptote at $$a=-2$$"],"hints":{"DefaultPathway":[{"id":"a73ccaalimits8a-h1","type":"hint","dependencies":[],"title":"Simplify the function","text":"The first step to determine whether there is an asymptote at the given $$x=a$$ is to simplify the function if possible. 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Remember if we can cancel out something after factoring the numerator and denominator there is a hole at this cancelled out value, which is different from a vertical asymptote.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a73ccaalimits9a-h2","type":"hint","dependencies":["a73ccaalimits9a-h1"],"title":"Determining if there is an asymptote at the given value of $$x=a$$","text":"To determine if there is an asymptote at the given value of $$a=1$$, we need to plug in our value of $$x=1$$ into the simplifed function. If we get a real number, then there is no vertical asymptote. If we do not get a real number (e.g. 5/0), then there is a vertical asymptote at the given value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a73cd4bprobbasic1","title":"Probability Topics","body":"Suppose we have $$40$$ M&M\'s with the following amounts of each color: $$10$$ blue, $$10$$ red, $$10$$ yellow, $$5$$ orange, $$5$$ brown.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Probability Topics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a73cd4bprobbasic1a","stepAnswer":["$$\\\\frac{1}{4}$$"],"problemType":"TextBox","stepTitle":"What is the probability that we choose a blue M&M if we select one M&M at random?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{4}$$","hints":{"DefaultPathway":[{"id":"a73cd4bprobbasic1a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of our situation, we first need to find out how many M&M\'s we can choose from.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$40$$"],"dependencies":["a73cd4bprobbasic1a-h1"],"title":"Total M&M\'s","text":"How many M&M\'s are there in total?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic1a-h3","type":"hint","dependencies":["a73cd4bprobbasic1a-h2"],"title":"Probability Rules","text":"Next, we need to find the amount of M&M\'s that correspond to our initial condition.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a73cd4bprobbasic1a-h3"],"title":"Total Blue M&M\'s","text":"How many blue M&M\'s are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic1a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4}$$"],"dependencies":["a73cd4bprobbasic1a-h4"],"title":"Answer","text":"To find our answer, we can take our first proportion and divide it by our second to get the probability. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a73cd4bprobbasic10","title":"Probability Topics","body":"Suppose we have $$40$$ M&M\'s with the following amounts of each color: $$10$$ blue, $$10$$ red, $$10$$ yellow, $$5$$ orange, $$5$$ brown.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Probability Topics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a73cd4bprobbasic10a","stepAnswer":["$$\\\\frac{1}{128}$$"],"problemType":"TextBox","stepTitle":"What is the probability that we select a red, then yellow, then orange M&M in $$3$$ trials with replacement?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{128}$$","hints":{"DefaultPathway":[{"id":"a73cd4bprobbasic10a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of our situation, we will use the multiplication rule to combine our two probabilities.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4}$$"],"dependencies":["a73cd4bprobbasic10a-h1"],"title":"M&M Probability","text":"What is the probability we get a red M&M?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4}$$"],"dependencies":["a73cd4bprobbasic10a-h2"],"title":"M&M Probability","text":"What is the probability we get a yellow M&M?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{8}$$"],"dependencies":["a73cd4bprobbasic10a-h3"],"title":"M&M Probability","text":"What is the probability we get an orange M&M?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{128}$$"],"dependencies":["a73cd4bprobbasic10a-h4"],"title":"Answer","text":"To find our answer, we will calculate our probability as we do normally, then use the multiplication rule to find our new probability. Essentially, we will be multiplying two of our probabilities together. Keep in mind that we don\'t care whether or not we get [Orange, Yellow] or [Yellow, Orange], meaning we will multiply our final probability by $$2$$. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a73cd4bprobbasic11","title":"Probability Topics","body":"Suppose that we are doing a study on High School students and their activities outside of school. Out of $$1200$$ students, $$250$$ play some kind of sport, $$200$$ do some other activity outside of school, $$150$$ students play sports and do an activity, $$400$$ students have jobs, $$50$$ students have a job and play sports, and $$150$$ students don\'t do anything after school.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Probability Topics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a73cd4bprobbasic11a","stepAnswer":["$$\\\\frac{5}{24}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a randomly selected student only plays a sport?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5}{24}$$","hints":{"DefaultPathway":[{"id":"a73cd4bprobbasic11a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of our situation, we first need to find out how many students we can choose from.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1200$$"],"dependencies":["a73cd4bprobbasic11a-h1"],"title":"Total Students","text":"How many students are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic11a-h3","type":"hint","dependencies":["a73cd4bprobbasic11a-h2"],"title":"Probability Rules","text":"Next, we need to find the amount of students that correspond to our initial condition.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$250$$"],"dependencies":["a73cd4bprobbasic11a-h3"],"title":"Total Athletes","text":"How many students exclusively do sports?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{24}$$"],"dependencies":["a73cd4bprobbasic11a-h4"],"title":"Answer","text":"To find our answer, we can take our first proportion and divide it by our second to get the probability. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a73cd4bprobbasic12","title":"Probability Topics","body":"Suppose that we are doing a study on High School students and their activities outside of school. Out of $$1200$$ students, $$250$$ play some kind of sport, $$200$$ do some other activity outside of school, $$150$$ students play sports and do an activity, $$400$$ students have jobs, $$50$$ students have a job and play sports, and $$150$$ students don\'t do anything after school.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Probability Topics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a73cd4bprobbasic12a","stepAnswer":["$$\\\\frac{3}{24}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a randomly selected student does nothing after school?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{24}$$","hints":{"DefaultPathway":[{"id":"a73cd4bprobbasic12a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of our situation, we first need to find out how many students we can choose from.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1200$$"],"dependencies":["a73cd4bprobbasic12a-h1"],"title":"Total Students","text":"How many students are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic12a-h3","type":"hint","dependencies":["a73cd4bprobbasic12a-h2"],"title":"Probability Rules","text":"Next, we need to find the amount of students that correspond to our initial condition.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$150$$"],"dependencies":["a73cd4bprobbasic12a-h3"],"title":"Total Regular Students","text":"How many students don\'t do anything after school?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic12a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{24}$$"],"dependencies":["a73cd4bprobbasic12a-h4"],"title":"Answer","text":"To find our answer, we can take our first proportion and divide it by our second to get the probability. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a73cd4bprobbasic13","title":"Probability Topics","body":"Suppose that we are doing a study on High School students and their activities outside of school. Out of $$1200$$ students, $$250$$ play some kind of sport, $$200$$ do some other activity outside of school, $$150$$ students play sports and do an activity, $$400$$ students have jobs, $$50$$ students have a job and play sports, and $$150$$ students don\'t do anything after school.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Probability Topics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a73cd4bprobbasic13a","stepAnswer":["$$\\\\frac{9}{24}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a randomly selected student plays a sport?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{9}{24}$$","hints":{"DefaultPathway":[{"id":"a73cd4bprobbasic13a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of our situation, we first need to find out how many students we can choose from.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1200$$"],"dependencies":["a73cd4bprobbasic13a-h1"],"title":"Total Students","text":"How many students are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic13a-h3","type":"hint","dependencies":["a73cd4bprobbasic13a-h2"],"title":"Probability Rules","text":"Next, we need to find the amount of students that correspond to our initial condition.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$450$$"],"dependencies":["a73cd4bprobbasic13a-h3"],"title":"Total Athletes","text":"How many students exclusively do sports?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic13a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{9}{24}$$"],"dependencies":["a73cd4bprobbasic13a-h4"],"title":"Answer","text":"To find our answer, we can take our first proportion and divide it by our second to get the probability. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a73cd4bprobbasic14","title":"Probability Topics","body":"Suppose that we are doing a study on High School students and their activities outside of school. Out of $$1200$$ students, $$250$$ play some kind of sport, $$200$$ do some other activity outside of school, $$150$$ students play sports and do an activity, $$400$$ students have jobs, $$50$$ students have a job and play sports, and $$150$$ students don\'t do anything after school.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Probability Topics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a73cd4bprobbasic14a","stepAnswer":["$$\\\\frac{17}{24}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a randomly selected student has a job or plays a sport?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{17}{24}$$","hints":{"DefaultPathway":[{"id":"a73cd4bprobbasic14a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of our situation, we first need to find out how many students we can choose from.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1200$$"],"dependencies":["a73cd4bprobbasic14a-h1"],"title":"Total Students","text":"How many students are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic14a-h3","type":"hint","dependencies":["a73cd4bprobbasic14a-h2"],"title":"Probability Rules","text":"Next, we need to find the amount of students that correspond to our initial condition.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic14a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$850$$"],"dependencies":["a73cd4bprobbasic14a-h3"],"title":"Total Athletes and Employees","text":"How many students have a job or play a sport?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic14a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{17}{24}$$"],"dependencies":["a73cd4bprobbasic14a-h4"],"title":"Answer","text":"To find our answer, we can take our first proportion and divide it by our second to get the probability. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a73cd4bprobbasic15","title":"Probability Topics","body":"Suppose that we are doing a study on High School students and their activities outside of school. Out of $$1200$$ students, $$250$$ play some kind of sport, $$200$$ do some other activity outside of school, $$150$$ students play sports and do an activity, $$400$$ students have jobs, $$50$$ students have a job and play sports, and $$150$$ students don\'t do anything after school.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Probability Topics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a73cd4bprobbasic15a","stepAnswer":["$$\\\\frac{3}{24}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a randomly selected student plays a sport and does another activity outside of school?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{24}$$","hints":{"DefaultPathway":[{"id":"a73cd4bprobbasic15a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of our situation, we first need to find out how many students we can choose from.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1200$$"],"dependencies":["a73cd4bprobbasic15a-h1"],"title":"Total Students","text":"How many students are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic15a-h3","type":"hint","dependencies":["a73cd4bprobbasic15a-h2"],"title":"Probability Rules","text":"Next, we need to find the amount of students that correspond to our initial condition.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$150$$"],"dependencies":["a73cd4bprobbasic15a-h3"],"title":"Total Athletes and Activities","text":"How many students play sports and do another activity outside of school?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic15a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{24}$$"],"dependencies":["a73cd4bprobbasic15a-h4"],"title":"Answer","text":"To find our answer, we can take our first proportion and divide it by our second to get the probability. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a73cd4bprobbasic16","title":"Probability Topics","body":"Suppose that we are doing a study on High School students and their activities outside of school. Out of $$1200$$ students, $$250$$ play some kind of sport, $$200$$ do some other activity outside of school, $$150$$ students play sports and do an activity, $$400$$ students have jobs, $$50$$ students have a job and play sports, and $$150$$ students don\'t do anything after school.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Probability Topics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a73cd4bprobbasic16a","stepAnswer":["$$\\\\frac{15}{24}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a randomly selected student does not play a sport?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{15}{24}$$","hints":{"DefaultPathway":[{"id":"a73cd4bprobbasic16a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of our situation, we first need to find out how many students we can choose from.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1200$$"],"dependencies":["a73cd4bprobbasic16a-h1"],"title":"Total Students","text":"How many students are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic16a-h3","type":"hint","dependencies":["a73cd4bprobbasic16a-h2"],"title":"Probability Rules","text":"Next, we need to find the amount of students that correspond to our initial condition.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic16a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$750$$"],"dependencies":["a73cd4bprobbasic16a-h3"],"title":"Total Non-Athletes","text":"How many students do not play sports?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic16a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{15}{24}$$"],"dependencies":["a73cd4bprobbasic16a-h4"],"title":"Answer","text":"To find our answer, we can take our first proportion and divide it by our second to get the probability. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a73cd4bprobbasic17","title":"Probability Topics","body":"Suppose that we are doing a study on High School students and their activities outside of school. Out of $$1200$$ students, $$250$$ play some kind of sport, $$200$$ do some other activity outside of school, $$150$$ students play sports and do an activity, $$400$$ students have jobs, $$50$$ students have a job and play sports, and $$150$$ students don\'t do anything after school.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Probability Topics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a73cd4bprobbasic17a","stepAnswer":["$$\\\\frac{1}{9}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a randomly selected student plays a sport given that they also have a job?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{9}$$","hints":{"DefaultPathway":[{"id":"a73cd4bprobbasic17a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of our situation, we first need to find out how many students we can choose from.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic17a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$450$$"],"dependencies":["a73cd4bprobbasic17a-h1"],"title":"Total Employees","text":"How many students have a job?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic17a-h3","type":"hint","dependencies":["a73cd4bprobbasic17a-h2"],"title":"Probability Rules","text":"Next, we need to find the amount of students that correspond to our initial condition.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic17a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$50$$"],"dependencies":["a73cd4bprobbasic17a-h3"],"title":"Total Working Athletes","text":"How many students play sports and have a job?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic17a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{9}$$"],"dependencies":["a73cd4bprobbasic17a-h4"],"title":"Answer","text":"To find our answer, we can take our first proportion and divide it by our second to get the probability. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a73cd4bprobbasic18","title":"Probability Topics","body":"Suppose that we are doing a study on High School students and their activities outside of school. Out of $$1200$$ students, $$250$$ play some kind of sport, $$200$$ do some other activity outside of school, $$150$$ students play sports and do an activity, $$400$$ students have jobs, $$50$$ students have a job and play sports, and $$150$$ students don\'t do anything after school.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Probability Topics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a73cd4bprobbasic18a","stepAnswer":["$$\\\\frac{1}{3}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a randomly selected student does an activity outside of school given that they have a job?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{3}$$","hints":{"DefaultPathway":[{"id":"a73cd4bprobbasic18a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of our situation, we first need to find out how many students we can choose from.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$450$$"],"dependencies":["a73cd4bprobbasic18a-h1"],"title":"Total Employees","text":"How many students have a job?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic18a-h3","type":"hint","dependencies":["a73cd4bprobbasic18a-h2"],"title":"Probability Rules","text":"Next, we need to find the amount of students that correspond to our initial condition.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic18a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$150$$"],"dependencies":["a73cd4bprobbasic18a-h3"],"title":"Total Activity Students","text":"How many students do an activity and play a sport outside of school?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic18a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["a73cd4bprobbasic18a-h4"],"title":"Answer","text":"To find our answer, we can take our first proportion and divide it by our second to get the probability. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a73cd4bprobbasic19","title":"Probability Topics","body":"Suppose that we are doing a study on High School students and their activities outside of school. Out of $$1200$$ students, $$250$$ play some kind of sport, $$200$$ do some other activity outside of school, $$150$$ students play sports and do an activity, $$400$$ students have jobs, $$50$$ students have a job and play sports, and $$150$$ students don\'t do anything after school.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Probability Topics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a73cd4bprobbasic19a","stepAnswer":["$$\\\\frac{7}{8}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a randomly selected student does at least one thing (Sports, Job, Activity) outside of school?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{7}{8}$$","hints":{"DefaultPathway":[{"id":"a73cd4bprobbasic19a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of our situation, we first need to find out how many students we can choose from.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1200$$"],"dependencies":["a73cd4bprobbasic19a-h1"],"title":"Total Students","text":"How many students are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic19a-h3","type":"hint","dependencies":["a73cd4bprobbasic19a-h2"],"title":"Probability Rules","text":"Next, we need to find the amount of students that correspond to our initial condition.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic19a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1050$$"],"dependencies":["a73cd4bprobbasic19a-h3"],"title":"Total Students Not Doing Nothing","text":"How many students do something after school?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic19a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{7}{8}$$"],"dependencies":["a73cd4bprobbasic19a-h4"],"title":"Answer","text":"To find our answer, we can take our first proportion and divide it by our second to get the probability. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a73cd4bprobbasic2","title":"Probability Topics","body":"Suppose we have $$40$$ M&M\'s with the following amounts of each color: $$10$$ blue, $$10$$ red, $$10$$ yellow, $$5$$ orange, $$5$$ brown.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Probability Topics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a73cd4bprobbasic2a","stepAnswer":["$$\\\\frac{1}{4}$$"],"problemType":"TextBox","stepTitle":"What is the probability that we choose a brown or orange M&M if we select one M&M at random?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{4}$$","hints":{"DefaultPathway":[{"id":"a73cd4bprobbasic2a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of our situation, we first need to find out how many M&M\'s we can choose from.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$40$$"],"dependencies":["a73cd4bprobbasic2a-h1"],"title":"Total M&M\'s","text":"How many M&M\'s are there in total?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic2a-h3","type":"hint","dependencies":["a73cd4bprobbasic2a-h2"],"title":"Probability Rules","text":"Next, we need to find the amount of M&M\'s that correspond to our initial condition.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a73cd4bprobbasic2a-h3"],"title":"Total Orange and Brown M&M\'s","text":"How many brown and orange M&M\'s are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic2a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4}$$"],"dependencies":["a73cd4bprobbasic2a-h4"],"title":"Answer","text":"To find our answer, we can take our first proportion and divide it by our second to get the probability. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a73cd4bprobbasic20","title":"Probability Topics","body":"Suppose that we are doing a study on High School students and their activities outside of school. Out of $$1200$$ students, $$250$$ play some kind of sport, $$200$$ do some other activity outside of school, $$150$$ students play sports and do an activity, $$400$$ students have jobs, $$50$$ students have a job and play sports, and $$150$$ students don\'t do anything after school.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Probability Topics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a73cd4bprobbasic20a","stepAnswer":["With Jobs"],"problemType":"MultipleChoice","stepTitle":"If we added $$500$$ more students with jobs to this study, would there be more students with or without jobs outside of school?","stepBody":"","answerType":"string","variabilization":{},"choices":["With Jobs","Without Jobs"],"hints":{"DefaultPathway":[{"id":"a73cd4bprobbasic20a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of our situation, we first need to find out how many students we can choose from.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1700$$"],"dependencies":["a73cd4bprobbasic20a-h1"],"title":"Total Students","text":"How many students are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$950$$ / $$1700$$"]},{"id":"a73cd4bprobbasic20a-h3","type":"hint","dependencies":["a73cd4bprobbasic20a-h2"],"title":"Probability Rules","text":"Next, we need to find the amount of students that correspond to our initial condition.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic20a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$950$$"],"dependencies":["a73cd4bprobbasic20a-h3"],"title":"Total Employees","text":"How many students have a job?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic20a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{95}{170}$$"],"dependencies":["a73cd4bprobbasic20a-h4"],"title":"Students with Jobs Proportion","text":"What is the proportion of students with jobs? Is this greater than or less than $$.5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic20a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["With Jobs"],"dependencies":["a73cd4bprobbasic20a-h5"],"title":"Answer","text":"Now that we have our proportion, are there more students with or without jobs in this study?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["With Jobs","Without Jobs"]}]}}]},{"id":"a73cd4bprobbasic21","title":"Probability Topics","body":"Suppose we survey $$150$$ office workers about their eating habits during lunch time. According to the study, $$100$$ employees eat at an establishment outside of their work, $$20$$ employees eat in their office, $$15$$ employees eat in their break room, and $$15$$ employees choose to skip lunch or just have a snack.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Probability Topics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a73cd4bprobbasic21a","stepAnswer":["$$\\\\frac{2}{3}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a randomly selected employee eats at an establishment during lunch?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2}{3}$$","hints":{"DefaultPathway":[{"id":"a73cd4bprobbasic21a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of our situation, we first need to find out how many office workers we can choose from.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic21a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$150$$"],"dependencies":["a73cd4bprobbasic21a-h1"],"title":"Total Employees","text":"How many employees are there in total?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic21a-h3","type":"hint","dependencies":["a73cd4bprobbasic21a-h2"],"title":"Probability Rules","text":"Next, we need to find the amount of employees that correspond to our initial condition.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic21a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$100$$"],"dependencies":["a73cd4bprobbasic21a-h3"],"title":"Total Establishment Eaters","text":"How many employees eat at an establishment during their lunch break?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic21a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{3}$$"],"dependencies":["a73cd4bprobbasic21a-h4"],"title":"Answer","text":"To find our answer, we can take our first proportion and divide it by our second to get the probability. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a73cd4bprobbasic22","title":"Probability Topics","body":"Suppose we survey $$150$$ office workers about their eating habits during lunch time. According to the study, $$100$$ employees eat at an establishment outside of their work, $$20$$ employees eat in their office, $$15$$ employees eat in their break room, and $$15$$ employees choose to skip lunch or just have a snack.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Probability Topics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a73cd4bprobbasic22a","stepAnswer":["$$\\\\frac{1}{10}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a randomly selected office worker does not eat lunch?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{10}$$","hints":{"DefaultPathway":[{"id":"a73cd4bprobbasic22a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of our situation, we first need to find out how many office workers we can choose from.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic22a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$150$$"],"dependencies":["a73cd4bprobbasic22a-h1"],"title":"Total Employees","text":"How many employees are there in total?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic22a-h3","type":"hint","dependencies":["a73cd4bprobbasic22a-h2"],"title":"Probability Rules","text":"Next, we need to find the amount of employees that correspond to our initial condition.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic22a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["a73cd4bprobbasic22a-h3"],"title":"Total Non-Eaters","text":"How many employees don\'t eat lunch?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic22a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{10}$$"],"dependencies":["a73cd4bprobbasic22a-h4"],"title":"Answer","text":"To find our answer, we can take our first proportion and divide it by our second to get the probability. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a73cd4bprobbasic23","title":"Probability Topics","body":"Suppose we survey $$150$$ office workers about their eating habits during lunch time. According to the study, $$100$$ employees eat at an establishment outside of their work, $$20$$ employees eat in their office, $$15$$ employees eat in their break room, and $$15$$ employees choose to skip lunch or just have a snack.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Probability Topics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a73cd4bprobbasic23a","stepAnswer":["$$\\\\frac{1}{3}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a randomly selected worker does not eat out at an establishment during lunch?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{3}$$","hints":{"DefaultPathway":[{"id":"a73cd4bprobbasic23a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of our situation, we first need to find out how many office workers we can choose from.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic23a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$150$$"],"dependencies":["a73cd4bprobbasic23a-h1"],"title":"Total Employees","text":"How many employees are there in total?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic23a-h3","type":"hint","dependencies":["a73cd4bprobbasic23a-h2"],"title":"Probability Rules","text":"Next, we need to find the amount of employees that correspond to our initial condition.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic23a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$50$$"],"dependencies":["a73cd4bprobbasic23a-h3"],"title":"Total Non-Establishment Eaters","text":"How many employees don\'t eat at an establishment during their lunch break?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic23a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["a73cd4bprobbasic23a-h4"],"title":"Answer","text":"To find our answer, we can take our first proportion and divide it by our second to get the probability. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a73cd4bprobbasic24","title":"Probability Topics","body":"Suppose we survey $$150$$ office workers about their eating habits during lunch time. According to the study, $$100$$ employees eat at an establishment outside of their work, $$20$$ employees eat in their office, $$15$$ employees eat in their break room, and $$15$$ employees choose to skip lunch or just have a snack.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Probability Topics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a73cd4bprobbasic24a","stepAnswer":["$$\\\\frac{7}{30}$$"],"problemType":"TextBox","stepTitle":"What is the probability that a randomly selected employee eats in their office or in their break room?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{7}{30}$$","hints":{"DefaultPathway":[{"id":"a73cd4bprobbasic24a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of our situation, we first need to find out how many office workers we can choose from.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic24a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$150$$"],"dependencies":["a73cd4bprobbasic24a-h1"],"title":"Total Employees","text":"How many employees are there in total?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic24a-h3","type":"hint","dependencies":["a73cd4bprobbasic24a-h2"],"title":"Probability Rules","text":"Next, we need to find the amount of employees that correspond to our initial condition.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic24a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$35$$"],"dependencies":["a73cd4bprobbasic24a-h3"],"title":"Total $$\\\\frac{Office}{Break}$$ Room Eaters","text":"How many employees eat in their office or in their break room?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic24a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{7}{30}$$"],"dependencies":["a73cd4bprobbasic24a-h4"],"title":"Answer","text":"To find our answer, we can take our first proportion and divide it by our second to get the probability. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a73cd4bprobbasic25","title":"Probability Topics","body":"Suppose we survey $$150$$ office workers about their eating habits during lunch time. According to the study, $$100$$ employees eat at an establishment outside of their work, $$20$$ employees eat in their office, $$15$$ employees eat in their break room, and $$15$$ employees choose to skip lunch or just have a snack.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Probability Topics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a73cd4bprobbasic25a","stepAnswer":["$$\\\\frac{3}{10}$$"],"problemType":"TextBox","stepTitle":"What is the probability that an employee does not eat lunch given that they do not go out to an establishment during lunch time?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{10}$$","hints":{"DefaultPathway":[{"id":"a73cd4bprobbasic25a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of our situation, we first need to find out how many office workers we can choose from.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic25a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$50$$"],"dependencies":["a73cd4bprobbasic25a-h1"],"title":"Total Employees Not Going Out","text":"How many employees do not go to an establishment during lunch time?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic25a-h3","type":"hint","dependencies":["a73cd4bprobbasic25a-h2"],"title":"Probability Rules","text":"Next, we need to find the amount of employees that correspond to our initial condition.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic25a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["a73cd4bprobbasic25a-h3"],"title":"Total Non-Eaters","text":"How many employees do not eat lunch?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic25a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{10}$$"],"dependencies":["a73cd4bprobbasic25a-h4"],"title":"Answer","text":"To find our answer, we can take our first proportion and divide it by our second to get the probability. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a73cd4bprobbasic3","title":"Probability Topics","body":"Suppose we have $$40$$ M&M\'s with the following amounts of each color: $$10$$ blue, $$10$$ red, $$10$$ yellow, $$5$$ orange, $$5$$ brown.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Probability Topics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a73cd4bprobbasic3a","stepAnswer":["$$\\\\frac{9}{39}$$"],"problemType":"TextBox","stepTitle":"What is the probability that we choose a blue M&M given we just chose a blue M&M before? Assume we are choosing M&M\'s without replacement.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{9}{39}$$","hints":{"DefaultPathway":[{"id":"a73cd4bprobbasic3a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of our situation, we first need to find out how many M&M\'s we can choose from.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$39$$"],"dependencies":["a73cd4bprobbasic3a-h1"],"title":"Total M&M\'s","text":"How many M&M\'s are there in total given that we just selected one without replacement earlier?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic3a-h3","type":"hint","dependencies":["a73cd4bprobbasic3a-h2"],"title":"Probability Rules","text":"Next, we need to find the amount of M&M\'s that correspond to our initial condition.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a73cd4bprobbasic3a-h3"],"title":"Total Blue M&M\'s","text":"How many blue M&M\'s are there, knowing that we just selected a blue M&M previously. Keep in mind we are selecting M&M\'s without replacement.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{9}{39}$$"],"dependencies":["a73cd4bprobbasic3a-h4"],"title":"Answer","text":"To find our answer, we can take our first proportion and divide it by our second to get the probability. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a73cd4bprobbasic4","title":"Probability Topics","body":"Suppose we have $$40$$ M&M\'s with the following amounts of each color: $$10$$ blue, $$10$$ red, $$10$$ yellow, $$5$$ orange, $$5$$ brown.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Probability Topics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a73cd4bprobbasic4a","stepAnswer":["$$\\\\frac{1}{16}$$"],"problemType":"TextBox","stepTitle":"What is the probability that we choose a blue, then a red M&M from $$2$$ selections with replacement?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{16}$$","hints":{"DefaultPathway":[{"id":"a73cd4bprobbasic4a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of our situation, we will use the multiplication rule to combine our two probabilities.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4}$$"],"dependencies":["a73cd4bprobbasic4a-h1"],"title":"M&M Probability","text":"What is the probability we get a blue M&M?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4}$$"],"dependencies":["a73cd4bprobbasic4a-h2"],"title":"M&M Probability","text":"What is the probability we get a red M&M?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{16}$$"],"dependencies":["a73cd4bprobbasic4a-h3"],"title":"Answer","text":"To find our answer, we will calculate our probability as we do normally, then use the multiplication rule to find our new probability. Essentially, we will be multiplying two of our probabilities together. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a73cd4bprobbasic5","title":"Probability Topics","body":"Suppose we have $$40$$ M&M\'s with the following amounts of each color: $$10$$ blue, $$10$$ red, $$10$$ yellow, $$5$$ orange, $$5$$ brown.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Probability Topics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a73cd4bprobbasic5a","stepAnswer":["$$\\\\frac{5}{78}$$"],"problemType":"TextBox","stepTitle":"What is the probability that we choose a blue M&M, then a red one, assuming that we choose M&M\'s without replacement?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5}{78}$$","hints":{"DefaultPathway":[{"id":"a73cd4bprobbasic5a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of our situation, we will use the multiplication rule to combine our two probabilities.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4}$$"],"dependencies":["a73cd4bprobbasic5a-h1"],"title":"M&M Probability","text":"What is the probability we get a blue M&M?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{10}{39}$$"],"dependencies":["a73cd4bprobbasic5a-h2"],"title":"M&M Probability","text":"What is the probability we get a red M&M given that we just chose a blue one without replacement?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{78}$$"],"dependencies":["a73cd4bprobbasic5a-h3"],"title":"Answer","text":"To find our answer, we will calculate our probability as we do normally, then use the multiplication rule to find our new probability. Essentially, we will be multiplying two of our probabilities together. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a73cd4bprobbasic6","title":"Probability Topics","body":"Suppose we have $$40$$ M&M\'s with the following amounts of each color: $$10$$ blue, $$10$$ red, $$10$$ yellow, $$5$$ orange, $$5$$ brown.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Probability Topics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a73cd4bprobbasic6a","stepAnswer":["$$\\\\frac{1}{3}$$"],"problemType":"TextBox","stepTitle":"What is the probability that we select a yellow M&M given that the next M&M we choose will be either blue, yellow, or red?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{3}$$","hints":{"DefaultPathway":[{"id":"a73cd4bprobbasic6a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of our situation, we first need to find out how many M&M\'s we can choose from.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$30$$"],"dependencies":["a73cd4bprobbasic6a-h1"],"title":"Total M&M\'s","text":"How many blue, yellow and red M&M\'s are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic6a-h3","type":"hint","dependencies":["a73cd4bprobbasic6a-h2"],"title":"Probability Rules","text":"Next, we need to find the amount of M&M\'s that correspond to our initial condition.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a73cd4bprobbasic6a-h3"],"title":"Total Yellow M&M\'s","text":"How many yellow M&M\'s are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["a73cd4bprobbasic6a-h4"],"title":"Answer","text":"To find our answer, we can take our first proportion and divide it by our second to get the probability. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a73cd4bprobbasic7","title":"Probability Topics","body":"Suppose we have $$40$$ M&M\'s with the following amounts of each color: $$10$$ blue, $$10$$ red, $$10$$ yellow, $$5$$ orange, $$5$$ brown.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Probability Topics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a73cd4bprobbasic7a","stepAnswer":["$$\\\\frac{1}{16}$$"],"problemType":"TextBox","stepTitle":"What is the probability that we select an orange M&M and a yellow M&M if we select two random M&Ms?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{16}$$","hints":{"DefaultPathway":[{"id":"a73cd4bprobbasic7a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of our situation, we will use the multiplication rule to combine our two probabilities.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{8}$$"],"dependencies":["a73cd4bprobbasic7a-h1"],"title":"M&M Probability","text":"What is the probability we get an orange M&M?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4}$$"],"dependencies":["a73cd4bprobbasic7a-h2"],"title":"M&M Probability","text":"What is the probability we get a yellow M&M?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{16}$$"],"dependencies":["a73cd4bprobbasic7a-h3"],"title":"Answer","text":"To find our answer, we will calculate our probability as we do normally, then use the multiplication rule to find our new probability. Essentially, we will be multiplying two of our probabilities together. Keep in mind that we don\'t care whether or not we get [Orange, Yellow] or [Yellow, Orange], meaning we will multiply our final probability by $$2$$. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a73cd4bprobbasic8","title":"Probability Topics","body":"Suppose we have $$40$$ M&M\'s with the following amounts of each color: $$10$$ blue, $$10$$ red, $$10$$ yellow, $$5$$ orange, $$5$$ brown.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Probability Topics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a73cd4bprobbasic8a","stepAnswer":["$$\\\\frac{1}{2}$$"],"problemType":"TextBox","stepTitle":"What is the probability that we select a brown M&M given that our next M&M is going to be brown or orange?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{2}$$","hints":{"DefaultPathway":[{"id":"a73cd4bprobbasic8a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of our situation, we first need to find out how many M&M\'s we can choose from.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a73cd4bprobbasic8a-h1"],"title":"Total M&M\'s","text":"How many M&M\'s are brown or orange?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic8a-h3","type":"hint","dependencies":["a73cd4bprobbasic8a-h2"],"title":"Probability Rules","text":"Next, we need to find the amount of M&M\'s that correspond to our initial condition.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a73cd4bprobbasic8a-h3"],"title":"Total Brown M&M\'s","text":"How many M&M\'s are brown?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic8a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a73cd4bprobbasic8a-h4"],"title":"Answer","text":"To find our answer, we can take our first proportion and divide it by our second to get the probability. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a73cd4bprobbasic9","title":"Probability Topics","body":"Suppose we have $$40$$ M&M\'s with the following amounts of each color: $$10$$ blue, $$10$$ red, $$10$$ yellow, $$5$$ orange, $$5$$ brown.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Probability Topics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a73cd4bprobbasic9a","stepAnswer":["$$\\\\frac{3}{4}$$"],"problemType":"TextBox","stepTitle":"What is the probability that we don\'t select a blue M&M if we choose one at random?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{4}$$","hints":{"DefaultPathway":[{"id":"a73cd4bprobbasic9a-h1","type":"hint","dependencies":[],"title":"Probability Rules","text":"To find the probability of our situation, we first need to find out how many M&M\'s we can choose from.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$40$$"],"dependencies":["a73cd4bprobbasic9a-h1"],"title":"Total M&M\'s","text":"How many M&M\'s are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic9a-h3","type":"hint","dependencies":["a73cd4bprobbasic9a-h2"],"title":"Probability Rules","text":"Next, we need to find the amount of M&M\'s that correspond to our initial condition.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$30$$"],"dependencies":["a73cd4bprobbasic9a-h3"],"title":"Total Brown M&M\'s","text":"How many M&M\'s are not blue?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a73cd4bprobbasic9a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{4}$$"],"dependencies":["a73cd4bprobbasic9a-h4"],"title":"Answer","text":"To find our answer, we can take our first proportion and divide it by our second to get the probability. What is our final answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a746c1bFactoring1","title":"Factoring Completely","body":"Please factor the following polynomial","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 General Strategy for Factoring Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a746c1bFactoring1a","stepAnswer":["$$7x \\\\left(x+2\\\\right) \\\\left(x-5\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$7x^3-21x^2-70x$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7x \\\\left(x+2\\\\right) \\\\left(x-5\\\\right)$$","hints":{"DefaultPathway":[{"id":"a746c1bFactoring1a-h1","type":"hint","dependencies":[],"title":"Find the Greatest Common Factor","text":"Find the Greatest Common Factor of all three terms in the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7x$$"],"dependencies":["a746c1bFactoring1a-h1"],"title":"Find the Greatest Common Factor","text":"What is the greatest factor of all three terms $$7x^3$$, $$21x^2$$, and $$70x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7x \\\\left(x^2-3x-10\\\\right)$$"],"dependencies":["a746c1bFactoring1a-h2"],"title":"Factor out the GCF","text":"Rewrite the expression by factoring out the greatest common factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a746c1bFactoring1a-h3-s1","type":"hint","dependencies":[],"title":"Divide each term","text":"We can divide each term by $$7x$$, then add those terms together. That will give you the answer of $$7x^3-21x^2-10x$$ divide by $$7x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring1a-h3-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2$$"],"dependencies":["a746c1bFactoring1a-h3-s1"],"title":"Divide","text":"What is $$7x^3$$ divide by $$7x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring1a-h3-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3x$$"],"dependencies":["a746c1bFactoring1a-h3-s1"],"title":"Divide","text":"What is $$-21x^2$$ divide by $$7x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring1a-h3-s4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-10$$"],"dependencies":["a746c1bFactoring1a-h3-s1"],"title":"Divide","text":"What is $$-70x$$ divide by $$7x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a746c1bFactoring1a-h4","type":"hint","dependencies":["a746c1bFactoring1a-h3"],"title":"Factorable Trinomial","text":"Given that $$x^2-3x-10$$ is a trinomial, we can try to factor out linear term from the trinomial. Note that not all trinomial is factorable, but in this case, it is factorable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring1a-h5","type":"hint","dependencies":["a746c1bFactoring1a-h4"],"title":"Methods","text":"$$x^2-3x-10$$ is also a quadratic, there are several approaches we can take to factor this quadratic---completing square, Reversing foil, and using quadratic formula.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring1a-h6","type":"hint","dependencies":["a746c1bFactoring1a-h5"],"title":"Method of Choice","text":"In this case, we will use reverse foil method and leave the other approaches as exercise on your own.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring1a-h7","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-5,2)"],"dependencies":["a746c1bFactoring1a-h6"],"title":"a and $$b$$","text":"We know that $$\\\\left(x+b\\\\right) \\\\left(x+a\\\\right)=x^2+a b x+ab$$. In this case, our $$a b$$ is $$-10$$, assume that both a, $$b$$ is integer. Then we get some possible combination for a, $$b$$ --- (1 and -10) OR (-1 and 10) OR (-2 and 5) OR (2 and -5). Which combination will both satisfy $$a b=-10$$ and $$a+b=-3$$. Please enter your answer in the form (x,y) by having $$x$$ less than $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring1a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x-5\\\\right) \\\\left(x+2\\\\right)$$"],"dependencies":["a746c1bFactoring1a-h7"],"title":"Use formula","text":"Using the answer you get from last step, write out the linear factor form for $$x^2-3x-10$$ in the form $$\\\\left(x-b\\\\right) \\\\left(x+a\\\\right)$$ where a, $$b$$ are positive integers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring1a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7x \\\\left(x+2\\\\right) \\\\left(x-5\\\\right)$$"],"dependencies":["a746c1bFactoring1a-h8"],"title":"Putting it all together","text":"As the final step, put together all the factors. $$7x^3-21x^2-70x=$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a746c1bFactoring10","title":"Factoring Completely","body":"Please factor the following polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 General Strategy for Factoring Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a746c1bFactoring10a","stepAnswer":["$$3x y \\\\left(9y^2+16\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$27x y^3+48x y$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3x y \\\\left(9y^2+16\\\\right)$$","hints":{"DefaultPathway":[{"id":"a746c1bFactoring10a-h1","type":"hint","dependencies":[],"title":"Find the common factor","text":"Is there a greatest common factor of all the terms in the expression $$27x y^3+48x y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x y$$"],"dependencies":["a746c1bFactoring10a-h1"],"title":"Find the common factor","text":"What is the greatest common factor of $$27x y^3$$ and $$48x y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9y^2+16$$"],"dependencies":["a746c1bFactoring10a-h2"],"title":"Factor out the greatest common term","text":"(27*x*(y**3))+(48*x*y)=3*x*y*(?). What is ?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring10a-h4","type":"hint","dependencies":["a746c1bFactoring10a-h3"],"title":"Factoring Completely","text":"There is no way to further factor $$3x y$$. Are there any ways to factor $$9y^2+16$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring10a-h5","type":"hint","dependencies":["a746c1bFactoring10a-h4"],"title":"Finished Factoring","text":"We can rewrite $$9y^2+16$$ as $${\\\\left(3y\\\\right)}^2+4^2$$. Note that sum of squares is prime so it is not factorable. Therefore, we can not further factor $$9y^2+16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring10a-h6","type":"hint","dependencies":["a746c1bFactoring10a-h5"],"title":"Put together all factors","text":"Since we can not further factor $$9y^2+16$$ and $$3x y$$, $$27x y^3+48x y$$ can factor completely into $$3x y \\\\left(9y^2+16\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a746c1bFactoring11","title":"Factoring Completely","body":"Please factor the following polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 General Strategy for Factoring Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a746c1bFactoring11a","stepAnswer":["$$3\\\\left(2x+3y\\\\right) \\\\left(4x^2-6x y+9y^2\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$24x^3+81y^3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3\\\\left(2x+3y\\\\right) \\\\left(4x^2-6x y+9y^2\\\\right)$$","hints":{"DefaultPathway":[{"id":"a746c1bFactoring11a-h1","type":"hint","dependencies":[],"title":"Factor out greatest common term","text":"What is the greatest common factor of all the terms in the expression $$24x^3+81y^3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a746c1bFactoring11a-h1"],"title":"Factor out greatest common term","text":"What is the greatest common factor of $$24x^3$$ and $$81y^3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring11a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8x^3+27y^3$$"],"dependencies":["a746c1bFactoring11a-h2"],"title":"Find Values","text":"(24*x**3)+(81*y**3)=3*(?) What is ?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x$$"],"dependencies":["a746c1bFactoring11a-h3"],"title":"Find a","text":"Are there any ways to write $$8x^3$$ into $$a^3$$? If so, what is a?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a746c1bFactoring11a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x$$"],"dependencies":[],"title":"Find Values","text":"Given that $${\\\\left(a b\\\\right)}^c$$ where a, $$b$$ and c are real numbers, $${\\\\left(a b\\\\right)}^c=a^c b^c$$. $$8x^3=8x^3$$ and $$8=2^3$$, so 8*x**3=(2**3)*(x**3)=(?)**3. What is (?)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a746c1bFactoring11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3y$$"],"dependencies":["a746c1bFactoring11a-h3"],"title":"Find $$b$$","text":"Are there any ways to rewrite $$27y^3$$ as $$b^3$$? If so, what is $$b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a746c1bFactoring11a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3y$$"],"dependencies":[],"title":"Find Values","text":"Given that $${\\\\left(a b\\\\right)}^c$$ where a, $$b$$ and c are real numbers, $${\\\\left(a b\\\\right)}^c=a^c b^c$$. $$27y^3=27y^3$$ and $$27=3^3$$, so 27*y**3=(3**3)*(y**3)=(?)**3. What is (?)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a746c1bFactoring11a-h6","type":"hint","dependencies":["a746c1bFactoring11a-h4","a746c1bFactoring11a-h5"],"title":"Use sum of cube rule","text":"We observe that $$8x^3+27y^3={\\\\left(2x\\\\right)}^3+{\\\\left(3y\\\\right)}^3$$ which is the sum of cubes. We can use the sum of cubes rule to factor the expression. Sum of cube rule $$\\\\operatorname{statesv}\\\\left(a^3\\\\right)+b^3$$ can be factored completely into $$\\\\left(a+b\\\\right) \\\\left(a^2-a b+b^2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring11a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(2x+3y\\\\right) \\\\left({\\\\left(2x\\\\right)}^2-2x 3y+{\\\\left(3y\\\\right)}^2\\\\right)$$"],"dependencies":["a746c1bFactoring11a-h6"],"title":"Factor using sum of cubes rule","text":"By using the sum of cube rule, we can factor $$8x^3+27y^3$$ as?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring11a-h8","type":"hint","dependencies":["a746c1bFactoring11a-h7"],"title":"Put together all factors","text":"Since there is no way to further factor $$2x+3y$$ and $${\\\\left(2x\\\\right)}^2-2x 3y+{\\\\left(3y\\\\right)}^2$$, $$24x^3+81y^3$$ can be factored completely into $$3\\\\left(2x+3y\\\\right) \\\\left({\\\\left(2x\\\\right)}^2-2x 3y+{\\\\left(3y\\\\right)}^2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a746c1bFactoring12","title":"Factoring Completely","body":"Please factor the following polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 General Strategy for Factoring Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a746c1bFactoring12a","stepAnswer":["$$2\\\\left(p+3q\\\\right) \\\\left(p^2-p 3q+{\\\\left(3q\\\\right)}^2\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$2p^3+54q^3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2\\\\left(p+3q\\\\right) \\\\left(p^2-p 3q+{\\\\left(3q\\\\right)}^2\\\\right)$$","hints":{"DefaultPathway":[{"id":"a746c1bFactoring12a-h1","type":"hint","dependencies":[],"title":"Factor out the greatest common factor","text":"What is the greatest common factor of all the terms in the expression $$2p^3+54q^3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a746c1bFactoring12a-h1"],"title":"Factor out the greatest common factor","text":"What is the greatest common factor of $$2p^3$$ and $$54q^3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring12a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$p^3+27q^3$$"],"dependencies":["a746c1bFactoring12a-h2"],"title":"Factor out the greatest common factor","text":"(2*p**3)+(54*q**3)=2*(?) What is (?)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3q$$"],"dependencies":["a746c1bFactoring12a-h3"],"title":"Rewrite the individual terms","text":"Are there any ways to write $$27q^3$$ into $$a^3$$? If so, what is a?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a746c1bFactoring12a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3q$$"],"dependencies":[],"title":"Rewrite the individual terms","text":"Given that $${\\\\left(a b\\\\right)}^c$$ where a, $$b$$ and c are real numbers, $${\\\\left(a b\\\\right)}^c=a^c b^c$$, $$27q^3=27q^3$$ and $$27=3^3$$, we can rewrite the expression as 27*q**3=(3**3)*(q**3)=(?)**3. What is (?)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a746c1bFactoring12a-h5","type":"hint","dependencies":["a746c1bFactoring12a-h4"],"title":"Use sum of cube rule","text":"We observe that $$p^3+27q^3=p^3+{\\\\left(3q\\\\right)}^3$$ which is the sum of cubes. We can use the sum of cubes rule to factor. Sum of cube rule: $$a^3+b^3$$ can be factored completely into $$\\\\left(a+b\\\\right) \\\\left(a^2-a b+b^2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring12a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(p+3q\\\\right) \\\\left(p^2-p 3q+{\\\\left(3q\\\\right)}^2\\\\right)$$"],"dependencies":["a746c1bFactoring12a-h5"],"title":"Use sum of cube rule","text":"By using the sum of cube rule, we can factor $$p^3+27q^3$$ as?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring12a-h7","type":"hint","dependencies":["a746c1bFactoring12a-h6"],"title":"Put together all factors","text":"Since there is no way to further factor $$p+3q$$ and $$p^2-p 3q+{\\\\left(3q\\\\right)}^2$$, $$2p^3+54q^3$$ can be factored completely into $$2\\\\left(p+3q\\\\right) \\\\left(p^2-p 3q+{\\\\left(3q\\\\right)}^2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a746c1bFactoring13","title":"Factoring Completely","body":"Please factor the following polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 General Strategy for Factoring Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a746c1bFactoring13a","stepAnswer":["$$4\\\\left(x+2b\\\\right) \\\\left(x-a\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$4x^2+8b x-4a x-8a b$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4\\\\left(x+2b\\\\right) \\\\left(x-a\\\\right)$$","hints":{"DefaultPathway":[{"id":"a746c1bFactoring13a-h1","type":"hint","dependencies":[],"title":"Factor out the greatest common term","text":"What is the greatest common factor of all terms in the expression $$4x^2+8b x-4a x-8a b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a746c1bFactoring13a-h1"],"title":"Factor out the greatest common term","text":"What is the greatest common factor of $$4x^2$$, $$8b x$$, $$4a x$$ and $$8a b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring13a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2+2b x-a x-2a b$$"],"dependencies":["a746c1bFactoring13a-h2"],"title":"Factor out the greatest common term","text":"$$4x^2+8b x-4a x-8a b=$$ 4*(?)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring13a-h4","type":"hint","dependencies":["a746c1bFactoring13a-h3"],"title":"Factor by grouping","text":"Next, we will group the terms with similar factor together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring13a-h5","type":"hint","dependencies":["a746c1bFactoring13a-h4"],"title":"Factor by grouping","text":"In the expression, $$x^2+2b x-a x-2a b$$, we will group $$-\\\\left(a x\\\\right)$$ and $$-\\\\left(2a b\\\\right)$$ together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring13a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-a \\\\left(x+2b\\\\right)$$"],"dependencies":["a746c1bFactoring13a-h5"],"title":"Factor by grouping","text":"Factor $$-\\\\left(a x\\\\right)-2a b$$:","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring13a-h7","type":"hint","dependencies":["a746c1bFactoring13a-h4"],"title":"Factor by grouping","text":"We will group $$x^2$$ and $$2b x$$ together so that we can factor out the common factor $$x$$. Then we get $$x^2+2b x=x \\\\left(x+2b\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring13a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+2b$$"],"dependencies":["a746c1bFactoring13a-h7"],"title":"Factor by grouping","text":"By putting together the result of factoring by grouping , we yield $$x^2+2b x-a x-2a b=x \\\\left(x+2b\\\\right)-a \\\\left(x+2b\\\\right)$$. What is the greatest common factor in $$x \\\\left(x+2b\\\\right)$$ and $$-a \\\\left(x+2b\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring13a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x-a$$"],"dependencies":["a746c1bFactoring13a-h8"],"title":"Factor by grouping","text":"$$x \\\\left(x+2b\\\\right)-a \\\\left(x+2b\\\\right)$$ has the common factor $$x+2b$$, so we can further factor $$x \\\\left(x+2b\\\\right)-a \\\\left(x+2b\\\\right)$$ into (x+2b)*(?). what is (?)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring13a-h10","type":"hint","dependencies":["a746c1bFactoring13a-h9"],"title":"Put together all factors","text":"Based on the above steps, we get $$4x^2+8b x-4a x-8a b=4\\\\left(x \\\\left(x+2b\\\\right)-a \\\\left(x+2b\\\\right)\\\\right)=4\\\\left(x+2b\\\\right) \\\\left(x-a\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring13a-h11","type":"hint","dependencies":["a746c1bFactoring13a-h10"],"title":"Final remark","text":"Alternatively, you can also group (-a*x with x**2), (2*b*x with -2*a*b) then do factoring by group. You should yield the same final answer. You can try it as exercise.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a746c1bFactoring14","title":"Factoring Completely","body":"Please factor the following polynomial","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 General Strategy for Factoring Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a746c1bFactoring14a","stepAnswer":["$$6\\\\left(x+b\\\\right) \\\\left(x-2c\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$6x^2-12x c+6b x-12b c$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6\\\\left(x+b\\\\right) \\\\left(x-2c\\\\right)$$","hints":{"DefaultPathway":[{"id":"a746c1bFactoring14a-h1","type":"hint","dependencies":[],"title":"Factor out the greatest common term","text":"What is the greatest common factor of all the terms in the expression $$6x^2-12x c+6b x-12b c$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a746c1bFactoring14a-h1"],"title":"Finding the GCF","text":"What is the greatest common factor of $$6x^2$$, $$-\\\\left(12x c\\\\right)$$, $$6b x$$ and $$-\\\\left(12b c\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring14a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2-2x c+b x-2b c$$"],"dependencies":["a746c1bFactoring14a-h2"],"title":"Finding Values","text":"6*x**2-(12*x*c)+(6*b*x)-(12*b*c)=6*(?), what is ?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring14a-h4","type":"hint","dependencies":["a746c1bFactoring14a-h3"],"title":"Group terms","text":"Next, we will group the terms with similar factor together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring14a-h5","type":"hint","dependencies":["a746c1bFactoring14a-h4"],"title":"Group terms","text":"In the expression, $$x^2-2x c+b x-2b c$$, we will group $$x^2$$ and -(2*x*c)) together","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring14a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x \\\\left(x-2c\\\\right)$$"],"dependencies":["a746c1bFactoring14a-h5"],"title":"Factor","text":"Factor $$x^2-2x c$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring14a-h7","type":"hint","dependencies":["a746c1bFactoring14a-h6"],"title":"Factor by grouping","text":"We will group $$b x$$ and $$-\\\\left(2b c\\\\right)$$ together, so we can factor out the common factor of $$b$$. $$b x-2b c=b \\\\left(x-2c\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring14a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x-2c$$"],"dependencies":["a746c1bFactoring14a-h7"],"title":"Finding the GCF","text":"Based on the above steps, we yield $$x^2-2x c+b x-2b c=x \\\\left(x-2c\\\\right)+b \\\\left(x-2c\\\\right)$$. What is the greatest common factor of $$x \\\\left(x-2c\\\\right)$$ and $$b \\\\left(x-2c\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring14a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+b$$"],"dependencies":["a746c1bFactoring14a-h8"],"title":"Finding Values","text":"$$x \\\\left(x-2c\\\\right)$$ and $$b \\\\left(x-2c\\\\right)$$ has the greatest common factor $$(x-2c)$$, so we can further factor $$x \\\\left(x-2c\\\\right)+b \\\\left(x-2c\\\\right)$$ into (x-2*c)*(?). what is (?)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring14a-h10","type":"hint","dependencies":["a746c1bFactoring14a-h9"],"title":"Put together all factors","text":"Based on the above steps, we get $$6x^2-12x c+6b x-12b c=6\\\\left(x^2-2x c+b x-2b c\\\\right)=6\\\\left(x+b\\\\right) \\\\left(x-2c\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring14a-h11","type":"hint","dependencies":["a746c1bFactoring14a-h10"],"title":"Final remark","text":"Alternatively, you can also group (x**2 with b*x), (-2*x*c with -2*b*c) then do factoring by grouping. You should yield the same final answer. You can try it as exercise.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a746c1bFactoring15","title":"Factoring Completely","body":"Please factor the following polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 General Strategy for Factoring Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a746c1bFactoring15a","stepAnswer":["$$2\\\\left(2x+3y\\\\right) \\\\left(4x-1\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$16x^2+24x y-4x-6y$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2\\\\left(2x+3y\\\\right) \\\\left(4x-1\\\\right)$$","hints":{"DefaultPathway":[{"id":"a746c1bFactoring15a-h1","type":"hint","dependencies":[],"title":"Factor out the greatest common term","text":"What is the greatest common factor of all the terms in the expression $$16x^2+24x y-4x-6y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a746c1bFactoring15a-h1"],"title":"Finding the GCF","text":"What is the greatest common factor of $$16x^2$$, $$24x y$$, $$-4x$$ and $$-6y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8x^2+12x y-2x-3y$$"],"dependencies":["a746c1bFactoring15a-h2"],"title":"Finding Values","text":"(16*x**2)+(24*x*y)-4*x-6*y=2*(?) what is (?)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring15a-h4","type":"hint","dependencies":["a746c1bFactoring15a-h3"],"title":"Group terms","text":"Next, we will group the terms with similar factor together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring15a-h5","type":"hint","dependencies":["a746c1bFactoring15a-h4"],"title":"Group terms","text":"In the expression $$8x^2+12x y-2x-3y$$, we will group $$8x^2$$ and $$12x y$$ together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring15a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4x \\\\left(2x+3y\\\\right)$$"],"dependencies":["a746c1bFactoring15a-h5"],"title":"Factor","text":"Factor $$8x^2+12x y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring15a-h7","type":"hint","dependencies":["a746c1bFactoring15a-h4"],"title":"Factor by grouping","text":"We will group $$-2x$$ with $$-3y$$ together and factor out $$-1$$. Then we get $$\\\\left(-2x-3y\\\\right)=-1\\\\left(2x+3y\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring15a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x+3y$$"],"dependencies":["a746c1bFactoring15a-h7"],"title":"Finding the GCF","text":"Put together the result of factoring by grouping , we yield $$8x^2+12x y-2x-3y=4x \\\\left(2x+3y\\\\right)-1\\\\left(2x+3y\\\\right)$$. What is the greatest common factor of $$4x \\\\left(2x+3y\\\\right)$$ and $$-1\\\\left(2x+3y\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring15a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4x-1$$"],"dependencies":["a746c1bFactoring15a-h8"],"title":"Finding Values","text":"$$4x \\\\left(2x+3y\\\\right)$$ and $$-1\\\\left(2x+3y\\\\right)$$ has the greatest common factor of $$2x+3y$$, so we can further factor $$4x \\\\left(2x+3y\\\\right)-1\\\\left(2x+3y\\\\right)$$ into (2*x+3*y)*(?). what is ?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring15a-h10","type":"hint","dependencies":["a746c1bFactoring15a-h9"],"title":"Put together all factors","text":"Based on the above steps, we get $$16x^2+24x y-4x-6y=2\\\\left(8x^2+12x y-2x-3y\\\\right)=2\\\\left(2x+3y\\\\right) \\\\left(4x-1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring15a-h11","type":"hint","dependencies":["a746c1bFactoring15a-h10"],"title":"Final remark","text":"Alternatively, you can also group (8*x**2 with -2*x), (12*x*y with -3*y) then do factoring by grouping. You should yield the same final answer. You can try it as exercise.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a746c1bFactoring2","title":"Factoring Completely","body":"Please factor the following polynomial","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 General Strategy for Factoring Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a746c1bFactoring2a","stepAnswer":["$$6\\\\left(2y-5\\\\right) \\\\left(2y+5\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$24y^2-150$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6\\\\left(2y-5\\\\right) \\\\left(2y+5\\\\right)$$","hints":{"DefaultPathway":[{"id":"a746c1bFactoring2a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":[],"title":"Find the Greatest Common Factor","text":"Find the Greatest Common Factor for the two terms in the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4y^2-25$$"],"dependencies":["a746c1bFactoring2a-h1"],"title":"Factor out common factor","text":"(24*y**2)-150=6*(?), what is ?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2y$$"],"dependencies":["a746c1bFactoring2a-h2"],"title":"Difference of square","text":"We observe that $$25$$ can be rewritten as square of $$5$$. $$4y^2$$ can be rewritten as square of?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(2y+5\\\\right) \\\\left(2y-5\\\\right)$$"],"dependencies":["a746c1bFactoring2a-h3"],"title":"Formula","text":"Recall that $$\\\\left(a+b\\\\right) \\\\left(a-b\\\\right)=a^2-b^2$$. How do we factor $$4y^2-25$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a746c1bFactoring2a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(2y+5\\\\right) \\\\left(2y-5\\\\right)$$"],"dependencies":[],"title":"Rewrite","text":"We can rewrite $$4y^2-25$$ as $${\\\\left(2y\\\\right)}^2-5^2$$, use the formula $$a^2-b^2=\\\\left(a-b\\\\right) \\\\left(a+b\\\\right)$$. we can factor $$4y^2-25$$ as?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a746c1bFactoring2a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6\\\\left(2y-5\\\\right) \\\\left(2y+5\\\\right)$$"],"dependencies":["a746c1bFactoring2a-h4"],"title":"Put together all factors","text":"$$24y^2-150=6\\\\left(4y^2-25\\\\right)$$, and $$4y^2-25=\\\\left(2y+5\\\\right) \\\\left(2y-5\\\\right)$$. What is the linear factor form of $$24y^2-150$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a746c1bFactoring3","title":"Factoring Completely","body":"Please factor the following polynomial","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 General Strategy for Factoring Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a746c1bFactoring3a","stepAnswer":["$$4x \\\\left(2x-3\\\\right) \\\\left(2x+3\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$16x^3-36x$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4x \\\\left(2x-3\\\\right) \\\\left(2x+3\\\\right)$$","hints":{"DefaultPathway":[{"id":"a746c1bFactoring3a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4x$$"],"dependencies":[],"title":"Find the Greatest Common Factor","text":"Find the greatest common factor of all terms in the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4x^2-9$$"],"dependencies":["a746c1bFactoring3a-h1"],"title":"Factor out common factor","text":"(16*x**3)-36*x=4*x*(?)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(2x\\\\right)}^2$$"],"dependencies":["a746c1bFactoring3a-h2"],"title":"Difference of square","text":"We observe that $$9$$ can be rewrite $$3^2$$. $$4x^2$$ can be rewritten as square of ?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(2x+3\\\\right) \\\\left(2x-3\\\\right)$$"],"dependencies":["a746c1bFactoring3a-h3"],"title":"Formula","text":"Recall that $$\\\\left(a+b\\\\right) \\\\left(a-b\\\\right)=a^2-b^2$$. How do we factor $$4x^2-9^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a746c1bFactoring3a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(2x+3\\\\right) \\\\left(2x-3\\\\right)$$"],"dependencies":[],"title":"Rewrite","text":"We can rewrite $$4x^2-9^2$$ as $${\\\\left(2x\\\\right)}^2-3^2$$. Use the formula $$a^2-b^2=\\\\left(a-b\\\\right) \\\\left(a+b\\\\right)$$. How do you factor $$4x^2-9$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a746c1bFactoring3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4x \\\\left(2x-3\\\\right) \\\\left(2x+3\\\\right)$$"],"dependencies":["a746c1bFactoring3a-h4"],"title":"Put together all factors","text":"$$16x^3-36x=4x \\\\left(4x^2-9\\\\right)$$ and $$4x^2-9=\\\\left(2x+3\\\\right) \\\\left(2x-3\\\\right)$$. What is the linear factor form of $$16x^3-36x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a746c1bFactoring4","title":"Factoring Completely","body":"Please factor the following polynomial","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 General Strategy for Factoring Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a746c1bFactoring4a","stepAnswer":["$$3\\\\left(3y-4\\\\right) \\\\left(3y+4\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$27y^2-48$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3\\\\left(3y-4\\\\right) \\\\left(3y+4\\\\right)$$","hints":{"DefaultPathway":[{"id":"a746c1bFactoring4a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":[],"title":"Find the Greatest Common Factor","text":"Find the greatest common factor of all terms in the expression","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9y^2-16$$"],"dependencies":["a746c1bFactoring4a-h1"],"title":"Factor out common factor","text":"(27*y**2)-48=3*(?)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(3y\\\\right)}^2$$"],"dependencies":["a746c1bFactoring4a-h2"],"title":"Difference of square","text":"We observe that $$16$$ can be rewritten $$4^2$$. $$9y^2$$ can be rewritten as ?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(3y-4\\\\right) \\\\left(3y+4\\\\right)$$"],"dependencies":["a746c1bFactoring4a-h3"],"title":"Formula","text":"Recall that $$\\\\left(a+b\\\\right) \\\\left(a-b\\\\right)=a^2-b^2$$, so how can we factor the difference of square for $$9y^2-16$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a746c1bFactoring4a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(3y-4\\\\right) \\\\left(3y+4\\\\right)$$"],"dependencies":[],"title":"Rewrite","text":"We can rewrite $$9y^2-16$$ as $${\\\\left(3y\\\\right)}^2-4^2$$. Use the formula $$a^2-b^2=\\\\left(a-b\\\\right) \\\\left(a+b\\\\right)$$. We can factor $$9y^2-16$$ as?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a746c1bFactoring4a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3\\\\left(3y-4\\\\right) \\\\left(3y+4\\\\right)$$"],"dependencies":["a746c1bFactoring4a-h4"],"title":"Put together all factors","text":"$$27y^2-48=3\\\\left(9y^2-16\\\\right)$$ and $$9y^2-16=\\\\left(3y-4\\\\right) \\\\left(3y+4\\\\right)$$. What is the linear factor form of $$27y^2-48$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a746c1bFactoring5","title":"Factoring Completely","body":"Please factor the following polynomial","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 General Strategy for Factoring Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a746c1bFactoring5a","stepAnswer":["$${\\\\left(2a-3b\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$4a^2-12a b+9b^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(2a-3b\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"a746c1bFactoring5a-h1","type":"hint","dependencies":[],"title":"Greatest Common Factor","text":"The greatest common factor amongst the three terms in the expression is $$1$$. So we need to use other approaches.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(2a\\\\right)}^2$$"],"dependencies":["a746c1bFactoring5a-h1"],"title":"Perfect square","text":"Is the first term in the expression a perfect square? How do you rewrite $$4a^2$$ into a perfect square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(3b\\\\right)}^2$$"],"dependencies":["a746c1bFactoring5a-h1"],"title":"Perfect square","text":"Is the last term in the expression a perfect square? How do you rewrite $$9b^2$$ into perfect square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring5a-h4","type":"hint","dependencies":["a746c1bFactoring5a-h3","a746c1bFactoring5a-h2"],"title":"Formula","text":"We can rewrite the expression $$4a^2-12a b+9b^2={\\\\left(2a\\\\right)}^2-12a b+{\\\\left(3b\\\\right)}^2$$. Does it fit the pattern $$x^2-2x y+y^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2a$$"],"dependencies":["a746c1bFactoring5a-h4"],"title":"Finding $$x$$","text":"If $$4a^2-12a b+9b^2={\\\\left(2a\\\\right)}^2-12a b+{\\\\left(3b\\\\right)}^2$$ fits the pattern $$x^2-2x y+y^2$$, then what is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring5a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3b$$"],"dependencies":["a746c1bFactoring5a-h4"],"title":"Finding $$y$$","text":"If $$4a^2-12a b+9b^2={\\\\left(2a\\\\right)}^2-12a b+{\\\\left(3b\\\\right)}^2$$ fits the pattern $$x^2-2x y+y^2$$, then what is $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring5a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12a b$$"],"dependencies":["a746c1bFactoring5a-h5","a746c1bFactoring5a-h6"],"title":"Finding 2xy","text":"If $$x=2a$$ and $$y=3b$$, then what is $$2x y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring5a-h8","type":"hint","dependencies":["a746c1bFactoring5a-h7"],"title":"Find the pattern","text":"If we let $$x=2a$$, $$y=3b$$, then $$2x y=12a b$$, and $$x^2-2x y+y^2=4a^2-12a b+9b^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring5a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(2a-3b\\\\right)}^2$$"],"dependencies":["a746c1bFactoring5a-h8"],"title":"Factor first term","text":"Given that $$x^2-2x y+y^2=2\\\\left(x-y\\\\right)$$, how do we factor $$4a^2-12a b+9b^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a746c1bFactoring5a-h9-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(2a-3b\\\\right)}^2$$"],"dependencies":[],"title":"Rewrite","text":"Based on the above steps, we observe that when $$x=2a$$, $$y=3b$$, $$4a^2-12a b+9b^2$$ fits the pattern $$x^2-2x y+y^2$$. Given that $$x^2-2x y+y^2=2\\\\left(x-y\\\\right)$$. How do we rewrite $$4a^2-12a b+9b^2$$ as a perfect square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a746c1bFactoring6","title":"Factoring Completely","body":"Please factor the following polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 General Strategy for Factoring Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a746c1bFactoring6a","stepAnswer":["$${\\\\left(2x+5y\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$4x^2+20x y+25y^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(2x+5y\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"a746c1bFactoring6a-h1","type":"hint","dependencies":[],"title":"Greatest Common Factor","text":"The greatest common factor amongst the three terms in the expression is $$1$$. So we need to use other approaches.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(2x\\\\right)}^2$$"],"dependencies":["a746c1bFactoring6a-h1"],"title":"Perfect square","text":"Is the first term of the expression a perfect square? How do you rewrite $$4x^2$$ into perfect square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(5y\\\\right)}^2$$"],"dependencies":["a746c1bFactoring6a-h1"],"title":"Perfect square","text":"Is the last term of the expression a perfect square? How do you rewrite $$25y^2$$ into perfect square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring6a-h4","type":"hint","dependencies":["a746c1bFactoring6a-h3","a746c1bFactoring6a-h2"],"title":"Formula","text":"We can rewrite the expression $$4x^2+20x y+25y^2={\\\\left(2x\\\\right)}^2+20x y+{\\\\left(5y\\\\right)}^2$$. Does it fit the pattern $$a^2+2a b+b^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x$$"],"dependencies":["a746c1bFactoring6a-h4"],"title":"Finding $$x$$","text":"If $$4x^2+20x y+25y^2={\\\\left(2x\\\\right)}^2+20x y+{\\\\left(5y\\\\right)}^2$$ fits the pattern $$a^2+2a b+b^2$$, then what is a?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring6a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5y$$"],"dependencies":["a746c1bFactoring6a-h4"],"title":"Finding $$y$$","text":"If $$4x^2+20x y+25y^2={\\\\left(2x\\\\right)}^2+20x y+{\\\\left(5y\\\\right)}^2$$ fits the pattern $$a^2+2a b+b^2$$, then what is $$b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring6a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20x y$$"],"dependencies":["a746c1bFactoring6a-h5","a746c1bFactoring6a-h6"],"title":"Finding 2xy","text":"If $$a=2x$$ and $$b=5y$$, then what is $$2a b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring6a-h8","type":"hint","dependencies":["a746c1bFactoring6a-h7"],"title":"Find the pattern","text":"If we have $$a=2x$$, $$b=5y$$, and $$2a b=20x y$$, then $$a^2+2a b+b^2=4x^2+20x y+25y^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring6a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(2x+5b\\\\right)}^2$$"],"dependencies":["a746c1bFactoring6a-h8"],"title":"Factor first term","text":"Given that $$a^2+2a b+b^2=2\\\\left(a+b\\\\right)$$, how do we factor $$4x^2+20x y+25y^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a746c1bFactoring6a-h9-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(2x+5b\\\\right)}^2$$"],"dependencies":[],"title":"Rewrite","text":"Based on the above steps, we observe that if $$a=2x$$ and $$b=5y$$, then $$4x^2+20x y+25y^2$$ fits the pattern $$a^2+2a b+b^2$$. Given that $$a^2+2a b+b^2=2\\\\left(a+b\\\\right)$$, how do we rewrite $$4x^2+20x y+25y^2$$ as a perfect square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a746c1bFactoring7","title":"Factoring Completely","body":"Please factor the following polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 General Strategy for Factoring Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a746c1bFactoring7a","stepAnswer":["$${\\\\left(3x-4y\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$9x^2-24x y+16y^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(3x-4y\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"a746c1bFactoring7a-h1","type":"hint","dependencies":[],"title":"Greatest Common Factor","text":"The greatest common factor amongst the three terms in the expression is $$1$$. So we need to use other approaches.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(3x\\\\right)}^2$$"],"dependencies":["a746c1bFactoring7a-h1"],"title":"Perfect square","text":"Is the first term of the expression a perfect square? How do you rewrite $$9x^2$$ as a perfect square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(4y\\\\right)}^2$$"],"dependencies":["a746c1bFactoring7a-h1"],"title":"Perfect square","text":"Is the last term in the expression a perfect square? How do you rewrite $$16y^2$$ as a perfect square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring7a-h4","type":"hint","dependencies":["a746c1bFactoring7a-h3","a746c1bFactoring7a-h2"],"title":"Formula","text":"We can rewrite the expression $$9x^2-24x y+16y^2={\\\\left(3x\\\\right)}^2-24x y+{\\\\left(4y\\\\right)}^2$$. Does it fit the pattern $$a^2-2a b+b^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring7a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x$$"],"dependencies":["a746c1bFactoring7a-h4"],"title":"Finding $$x$$","text":"If $$9x^2-24x y+16y^2={\\\\left(3x\\\\right)}^2-24x y+{\\\\left(4y\\\\right)}^2$$ fits the pattern $$a^2-2a b+b^2$$, then what is a?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring7a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4y$$"],"dependencies":["a746c1bFactoring7a-h4"],"title":"Finding $$y$$","text":"If $$9x^2-24x y+16y^2={\\\\left(3x\\\\right)}^2-24x y+{\\\\left(4y\\\\right)}^2$$ fits the pattern $$a^2-2a b+b^2$$, then what is $$b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring7a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$24x y$$"],"dependencies":["a746c1bFactoring7a-h5","a746c1bFactoring7a-h6"],"title":"Finding 2xy","text":"If $$a=3x$$ and $$b=4y$$, then what is $$2a b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring7a-h8","type":"hint","dependencies":["a746c1bFactoring7a-h7"],"title":"Find the pattern","text":"If we let $$a=3x$$ and $$b=4y$$, and $$2a b=24x y$$, then $$a^2-2a b+b^2=9x^2-24x y+16y^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring7a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(3x-4y\\\\right)}^2$$"],"dependencies":["a746c1bFactoring7a-h8"],"title":"Factor first term","text":"Given that $$a^2-2a b+b^2=2\\\\left(a-b\\\\right)$$, how do we factor $$9x^2-24x y+16y^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a746c1bFactoring7a-h9-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(3x-4y\\\\right)}^2$$"],"dependencies":[],"title":"Rewrite","text":"Based on the above steps, we observe that if $$a=3x$$ and $$b=4y$$, then $$3x^2-24x y+16y^2$$ fits the pattern $$a^2-2a b+b^2$$. Given that $$a^2-2a b+b^2=2\\\\left(a-b\\\\right)$$, how do we rewrite $$9x^2-24x y+16y^2$$ as a perfect square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a746c1bFactoring8","title":"Factoring Completely","body":"Please factor the following polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 General Strategy for Factoring Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a746c1bFactoring8a","stepAnswer":["$$3x y^2 \\\\left(4x^2+25\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$12x^3 y^2+75x y^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3x y^2 \\\\left(4x^2+25\\\\right)$$","hints":{"DefaultPathway":[{"id":"a746c1bFactoring8a-h1","type":"hint","dependencies":[],"title":"Find the common factor","text":"What is the greatest common factor of all terms in the expression $$12x^3 y^2+75x y^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x y^2$$"],"dependencies":["a746c1bFactoring8a-h1"],"title":"Find the common factor","text":"What is the greatest common factor of $$12x^3 y^2$$ and $$75x y^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4x^2+25$$"],"dependencies":["a746c1bFactoring8a-h2"],"title":"Factor out the greatest common term","text":"(12*(x**3)*(y**2))+(75*x*y**2)=3*x*(y**2)*(?). What is ?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring8a-h4","type":"hint","dependencies":["a746c1bFactoring8a-h3"],"title":"Factoring Completely","text":"There is no way to further factor $$3x y^2$$. Are there any ways to factor $$4x^2+25$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring8a-h5","type":"hint","dependencies":["a746c1bFactoring8a-h4"],"title":"Finished Factoring","text":"We can rewrite $$4x^2+25$$ as $${\\\\left(2x\\\\right)}^2+5^2$$. Note that sum of squares is prime so it is not factorable. Therefore, we can not further than the factor $$4x^2+25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring8a-h6","type":"hint","dependencies":["a746c1bFactoring8a-h5"],"title":"Put together all factors","text":"Since we cannot further factor $$3x y^2 \\\\left(4x^2+25\\\\right)$$, $$12x^3 y^2+75x y^2$$ can factor completely into $$3x y^2 \\\\left(4x^2+25\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a746c1bFactoring9","title":"Factoring Completely","body":"Please factor the following polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 General Strategy for Factoring Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a746c1bFactoring9a","stepAnswer":["$$2x y \\\\left(25x^2+36\\\\right)$$"],"problemType":"TextBox","stepTitle":"(50*(x**3)*y))+72*x*y","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2x y \\\\left(25x^2+36\\\\right)$$","hints":{"DefaultPathway":[{"id":"a746c1bFactoring9a-h1","type":"hint","dependencies":[],"title":"Find the common factor","text":"What is the greatest common factor of all the terms in the expression (50*(x**3)*y))+72*x*y?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x y$$"],"dependencies":["a746c1bFactoring9a-h1"],"title":"Find the common factor","text":"What is the greatest common factor of (50*(x**3)*y)) and $$72x y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25x^2+36$$"],"dependencies":["a746c1bFactoring9a-h2"],"title":"Factor out the greatest common term","text":"(50*(x**3)*y))+72*x*y=2*x*y*(?).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring9a-h4","type":"hint","dependencies":["a746c1bFactoring9a-h3"],"title":"Factoring Completely","text":"There is no way to further factor $$2x y$$. Are there any ways to factor $$25x^2+36$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring9a-h5","type":"hint","dependencies":["a746c1bFactoring9a-h4"],"title":"Finished Factoring","text":"We can rewrite $$25x^2+36$$ as $${\\\\left(5x\\\\right)}^2+6^2$$. Note that sum of squares is prime so it is not factorable. Therefore, we can not further factor $$25x^2+36$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bFactoring9a-h6","type":"hint","dependencies":["a746c1bFactoring9a-h5"],"title":"Put together all factors","text":"Since there is no way to further factor $$25x^2+36$$ and $$2x y$$, $$50x^3 y+72x y$$ can factor completely into $$2x y \\\\left(25x^2+36\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a746c1bfactoringpoly1","title":"Factoring Polynomials","body":"Factor the Polynomial completely.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 General Strategy for Factoring Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a746c1bfactoringpoly1a","stepAnswer":["$$\\\\left(2x-1\\\\right) \\\\left(x+7\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$2x^2+13x-7$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(2x-1\\\\right) \\\\left(x+7\\\\right)$$","hints":{"DefaultPathway":[{"id":"a746c1bfactoringpoly1a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":[],"title":"Looking for a GCF","text":"Is there a greatest common factor? Enter the greatest common factor or $$0$$ if there isn\'t one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a746c1bfactoringpoly1a-h1"],"title":"Identifying terms","text":"Is this a binomial, trinomial, or are there more than $$3$$ terms? How many different terms are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a746c1bfactoringpoly1a-h2"],"title":"Leading Coefficient","text":"Since this is a trinomial, what is the leading coefficient?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly1a-h4","type":"hint","dependencies":["a746c1bfactoringpoly1a-h3"],"title":"FOIL","text":"Undo the FOIL for the polynomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly1a-h5","type":"hint","dependencies":["a746c1bfactoringpoly1a-h4"],"title":"Factoring out","text":"Trying factoring out the polynomial with the leading coefficient in a way to get $$2x^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly1a-h6","type":"hint","dependencies":["a746c1bfactoringpoly1a-h5"],"title":"Constants","text":"What are the constants that can be multiplied by the leading constant?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a746c1bfactoringpoly10","title":"Factoring Polynomials","body":"Factor the polynomial completely.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 General Strategy for Factoring Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a746c1bfactoringpoly10a","stepAnswer":["$${\\\\left(7x-8\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$49x^2-112x+64$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(7x-8\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"a746c1bfactoringpoly10a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":[],"title":"Looking for a GCF","text":"Is there a greatest common factor? Enter the greatest common factor or $$0$$ if there isn\'t one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a746c1bfactoringpoly10a-h1"],"title":"Factoring Polynomials","text":"Is this a binomial, trinomial, or are there more than $$3$$ terms? How many different terms are there in the parentheses?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$49$$"],"dependencies":["a746c1bfactoringpoly10a-h2"],"title":"Leading Coefficient","text":"Since this is a trinomial, what is the leading coefficient?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly10a-h4","type":"hint","dependencies":["a746c1bfactoringpoly10a-h3"],"title":"FOIL","text":"Undo the FOIL for the polynomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly10a-h5","type":"hint","dependencies":["a746c1bfactoringpoly10a-h4"],"title":"Factors for the leading coefficient","text":"Think about some factors that multiply to get $$49x^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly10a-h6","type":"hint","dependencies":["a746c1bfactoringpoly10a-h5"],"title":"Suggestions","text":"How about $$7$$ and $$7$$, and $$1$$ and 49?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly10a-h7","type":"hint","dependencies":["a746c1bfactoringpoly10a-h6"],"title":"Factors for C","text":"What are some factors that can be multiplied to get 64?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly10a-h8","type":"hint","dependencies":["a746c1bfactoringpoly10a-h7"],"title":"Factors for C","text":"How about $$8$$ and $$8$$, $$16$$ and $$4$$, and $$32$$ and $$2$$, $$64$$ and 1?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a746c1bfactoringpoly11","title":"Factoring Polynomials","body":"Factor the polynomial completely.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 General Strategy for Factoring Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a746c1bfactoringpoly11a","stepAnswer":["$${\\\\left(x+7y\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"(x**2)+((14*x)*y))+(49*(y**2))","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(x+7y\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"a746c1bfactoringpoly11a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":[],"title":"Looking for a GCF","text":"Is there a greatest common factor? Enter the greatest common factor or $$0$$ if there isn\'t one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a746c1bfactoringpoly11a-h1"],"title":"Factoring Polynomials","text":"Is this a binomial, trinomial, or are there more than $$3$$ terms? How many different terms are there in the parentheses?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly11a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a746c1bfactoringpoly11a-h2"],"title":"Leading Coefficient","text":"Since this is a trinomial, what is the leading coefficient?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly11a-h4","type":"hint","dependencies":["a746c1bfactoringpoly11a-h3"],"title":"FOIL","text":"Undo the FOIL for the polynomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly11a-h5","type":"hint","dependencies":["a746c1bfactoringpoly11a-h4"],"title":"Factors for the leading coefficient","text":"Think about some factors that multiply to get $$x^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly11a-h6","type":"hint","dependencies":["a746c1bfactoringpoly11a-h5"],"title":"Suggestions","text":"How about $$1$$ and 1?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly11a-h7","type":"hint","dependencies":["a746c1bfactoringpoly11a-h6"],"title":"Factors for C","text":"What are some factors that can be multiplied to get $$49y^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly11a-h8","type":"hint","dependencies":["a746c1bfactoringpoly11a-h7"],"title":"Factors for C","text":"How about $$7$$ and $$7$$, $$49$$ and 1?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a746c1bfactoringpoly12","title":"Factoring Polynomials","body":"Factor the polynomial completely.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 General Strategy for Factoring Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a746c1bfactoringpoly12a","stepAnswer":["$${\\\\left(8x+y\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"(64*(x**2))+((16*x)*y))+(y**2)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(8x+y\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"a746c1bfactoringpoly12a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":[],"title":"Looking for a GCF","text":"Is there a greatest common factor? Enter the greatest common factor or $$0$$ if there isn\'t one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a746c1bfactoringpoly12a-h1"],"title":"Factoring Polynomials","text":"Is this a binomial, trinomial, or are there more than $$3$$ terms? How many different terms are there in the parentheses?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly12a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$64$$"],"dependencies":["a746c1bfactoringpoly12a-h2"],"title":"Leading Coefficient","text":"Since this is a trinomial, what is the leading coefficient?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly12a-h4","type":"hint","dependencies":["a746c1bfactoringpoly12a-h3"],"title":"FOIL","text":"Undo the FOIL for the polynomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly12a-h5","type":"hint","dependencies":["a746c1bfactoringpoly12a-h4"],"title":"Factors for the leading coefficient","text":"Think about some factors that multiply to get $$64x^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly12a-h6","type":"hint","dependencies":["a746c1bfactoringpoly12a-h5"],"title":"Suggestions","text":"How about $$8$$ and $$8$$, $$4$$ and $$16$$, $$2$$ and $$32$$, $$1$$ and 64?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly12a-h7","type":"hint","dependencies":["a746c1bfactoringpoly12a-h6"],"title":"Factors for C","text":"What are some factors that can be multiplied to get $$y^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly12a-h8","type":"hint","dependencies":["a746c1bfactoringpoly12a-h7"],"title":"Factors for C","text":"How about $$1$$ and 1?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a746c1bfactoringpoly13","title":"Factoring Polynomials","body":"Factor the polynomial completely.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 General Strategy for Factoring Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a746c1bfactoringpoly13a","stepAnswer":["$$7\\\\left(x+3\\\\right) \\\\left(x-2\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$7x^2+7x-42$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7\\\\left(x+3\\\\right) \\\\left(x-2\\\\right)$$","hints":{"DefaultPathway":[{"id":"a746c1bfactoringpoly13a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":[],"title":"Looking for a GCF","text":"Is there a greatest common factor? Enter the greatest common factor or $$0$$ if there isn\'t one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a746c1bfactoringpoly13a-h1"],"title":"Factoring Polynomials","text":"Is this a binomial, trinomial, or are there more than $$3$$ terms? How many different terms are there in the parentheses?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly13a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a746c1bfactoringpoly13a-h2"],"title":"Leading Coefficient","text":"Since this is a trinomial, what is the leading coefficient?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly13a-h4","type":"hint","dependencies":["a746c1bfactoringpoly13a-h3"],"title":"FOIL","text":"Undo the FOIL for the polynomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly13a-h5","type":"hint","dependencies":["a746c1bfactoringpoly13a-h4"],"title":"Factors for the leading coefficient","text":"Think about some factors that multiply to get $$x^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly13a-h6","type":"hint","dependencies":["a746c1bfactoringpoly13a-h5"],"title":"Suggestions","text":"How about $$1$$ and 1?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly13a-h7","type":"hint","dependencies":["a746c1bfactoringpoly13a-h6"],"title":"Factors for C","text":"What are some factors that can be multiplied to get -6?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly13a-h8","type":"hint","dependencies":["a746c1bfactoringpoly13a-h7"],"title":"Suggestions","text":"How about $$3$$ and $$2$$, and $$1$$ and 2?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly13a-h9","type":"hint","dependencies":["a746c1bfactoringpoly13a-h8"],"title":"Factors for C","text":"What should the signs be for the factors of C?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a746c1bfactoringpoly14","title":"Factoring Polynomials","body":"Factor the polynomial completely.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 General Strategy for Factoring Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a746c1bfactoringpoly14a","stepAnswer":["$$6\\\\left(5x^2+5x+12\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$30x^2+30x+72$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6\\\\left(5x^2+5x+12\\\\right)$$","hints":{"DefaultPathway":[{"id":"a746c1bfactoringpoly14a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":[],"title":"Looking for a GCF","text":"Is there a greatest common factor? Enter the greatest common factor or $$0$$ if there isn\'t one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a746c1bfactoringpoly14a-h1"],"title":"Factoring Polynomials","text":"Is this a binomial, trinomial, or are there more than $$3$$ terms? How many different terms are there in the parentheses?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly14a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$30$$"],"dependencies":["a746c1bfactoringpoly14a-h2"],"title":"Leading Coeffecient","text":"Since this is a trinomial, what is the leading coefficient?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly14a-h4","type":"hint","dependencies":["a746c1bfactoringpoly14a-h3"],"title":"FOIL","text":"Undo the FOIL for the polynomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly14a-h5","type":"hint","dependencies":["a746c1bfactoringpoly14a-h4"],"title":"Finished?","text":"Can this be factored anymore?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a746c1bfactoringpoly15","title":"Factoring Polynomials","body":"Factor the polynomial completely.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 General Strategy for Factoring Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a746c1bfactoringpoly15a","stepAnswer":["$$3x y \\\\left(x-3\\\\right) \\\\left(x^2+3x+9\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$3x^4 y-81x y$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3x y \\\\left(x-3\\\\right) \\\\left(x^2+3x+9\\\\right)$$","hints":{"DefaultPathway":[{"id":"a746c1bfactoringpoly15a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x y$$"],"dependencies":[],"title":"Looking for a GCF","text":"Is there a greatest common factor? Enter the greatest common factor or $$0$$ if there isn\'t one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a746c1bfactoringpoly15a-h1"],"title":"Factoring Polynomials","text":"Is this a binomial, trinomial, or are there more than $$3$$ terms? How many different terms are there in the parentheses?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly15a-h3","type":"hint","dependencies":["a746c1bfactoringpoly15a-h2"],"title":"Cubes","text":"Since this is a binomial, is it a sum or difference of squares or cubes?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly15a-h4","type":"hint","dependencies":["a746c1bfactoringpoly15a-h3"],"title":"Factoring More","text":"Difference of cubes can be factored down further.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly15a-h5","type":"hint","dependencies":["a746c1bfactoringpoly15a-h4"],"title":"Formula","text":"Remember: the formula for difference of cubes is, $$x^3-y^3=\\\\left(x-y\\\\right) \\\\left(x^2+x y+y^2\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a746c1bfactoringpoly2","title":"Factoring Polynomials","body":"Factor the polynomial completely.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 General Strategy for Factoring Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a746c1bfactoringpoly2a","stepAnswer":["$$8x^2-9x-3$$"],"problemType":"TextBox","stepTitle":"$$8x^2-9x-3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8x^2-9x-3$$","hints":{"DefaultPathway":[{"id":"a746c1bfactoringpoly2a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":[],"title":"Factoring Polynomials","text":"Is there a greatest common factor? Enter the greatest common factor or $$0$$ if there isn\'t one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a746c1bfactoringpoly2a-h1"],"title":"Terms","text":"Is this a binomial, trinomial, or are there more than $$3$$ terms? How many different terms are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a746c1bfactoringpoly2a-h2"],"title":"Leading Coefficient","text":"Since this is a trinomial, what is the leading coefficient?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly2a-h4","type":"hint","dependencies":["a746c1bfactoringpoly2a-h3"],"title":"FOIL","text":"Undo the FOIL for the polynomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly2a-h5","type":"hint","dependencies":["a746c1bfactoringpoly2a-h4"],"title":"Finished?","text":"Is this polynomial factorable or completely simplified?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a746c1bfactoringpoly3","title":"Factoring Polynomials","body":"Factor the polynomial completely.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 General Strategy for Factoring Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a746c1bfactoringpoly3a","stepAnswer":["$$x^3 \\\\left(x^2+9\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$x^5+9x^3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^3 \\\\left(x^2+9\\\\right)$$","hints":{"DefaultPathway":[{"id":"a746c1bfactoringpoly3a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^3$$"],"dependencies":[],"title":"Looking for a GCF","text":"Is there a greatest common factor? Enter the greatest common factor or $$0$$ if there isn\'t one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a746c1bfactoringpoly3a-h1"],"title":"Identifying terms","text":"Is this a binomial, trinomial, or are there more than $$3$$ terms? How many different terms are there in the parentheses?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly3a-h3","type":"hint","dependencies":["a746c1bfactoringpoly3a-h2"],"title":"Squares","text":"Since this is a binomial, is it a sum of squares or difference of squares?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly3a-h4","type":"hint","dependencies":["a746c1bfactoringpoly3a-h3"],"title":"Finished?","text":"Sum of squares cannot be factored anymore.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a746c1bfactoringpoly4","title":"Factoring Polynomials","body":"Factor the polynomial completely.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 General Strategy for Factoring Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a746c1bfactoringpoly4a","stepAnswer":["$$x^3 \\\\left(x^2+9\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$75x^3+12x$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^3 \\\\left(x^2+9\\\\right)$$","hints":{"DefaultPathway":[{"id":"a746c1bfactoringpoly4a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x$$"],"dependencies":[],"title":"Factoring Polynomials","text":"Is there a greatest common factor? Enter the greatest common factor or $$0$$ if there isn\'t one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a746c1bfactoringpoly4a-h1"],"title":"Terms","text":"Is this a binomial, trinomial, or are there more than $$3$$ terms? How many different terms are there in the parentheses?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly4a-h3","type":"hint","dependencies":["a746c1bfactoringpoly4a-h2"],"title":"Squares","text":"Since this is a binomial, is it a sum of squares or difference of squares?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly4a-h4","type":"hint","dependencies":["a746c1bfactoringpoly4a-h3"],"title":"Finished?","text":"Sum of squares cannot be factored anymore.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a746c1bfactoringpoly5","title":"Factoring Polynomials","body":"Factor the polynomial completely.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 General Strategy for Factoring Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a746c1bfactoringpoly5a","stepAnswer":["$$\\\\left(11x+y\\\\right) \\\\left(11x-y\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$121x^2-y^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(11x+y\\\\right) \\\\left(11x-y\\\\right)$$","hints":{"DefaultPathway":[{"id":"a746c1bfactoringpoly5a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":[],"title":"Factoring Polynomials","text":"Is there a greatest common factor? Enter the greatest common factor or $$0$$ if there isn\'t one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a746c1bfactoringpoly5a-h1"],"title":"Terms","text":"Is this a binomial, trinomial, or are there more than $$3$$ terms? How many different terms are there in the parentheses?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly5a-h3","type":"hint","dependencies":["a746c1bfactoringpoly5a-h2"],"title":"Squares","text":"Since this is a binomial, is it a sum of squares or difference of squares?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly5a-h4","type":"hint","dependencies":["a746c1bfactoringpoly5a-h3"],"title":"Finished?","text":"Difference of squares can be factored further.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a746c1bfactoringpoly6","title":"Factoring Polynomials","body":"Factor the polynomial completely.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 General Strategy for Factoring Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a746c1bfactoringpoly6a","stepAnswer":["$$\\\\left(7x-6y\\\\right) \\\\left(7x+6y\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$49x^2-36y^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(7x-6y\\\\right) \\\\left(7x+6y\\\\right)$$","hints":{"DefaultPathway":[{"id":"a746c1bfactoringpoly6a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":[],"title":"Looking for a GCF","text":"Is there a greatest common factor? Enter the greatest common factor or $$0$$ if there isn\'t one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a746c1bfactoringpoly6a-h1"],"title":"Factoring Polynomials","text":"Is this a binomial, trinomial, or are there more than $$3$$ terms? How many different terms are there in the parentheses?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly6a-h3","type":"hint","dependencies":["a746c1bfactoringpoly6a-h2"],"title":"Squares","text":"Since this is a binomial, is it a sum of squares or difference of squares?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly6a-h4","type":"hint","dependencies":["a746c1bfactoringpoly6a-h3"],"title":"Finished?","text":"Difference of squares can be factored further.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a746c1bfactoringpoly7","title":"Factoring Polynomials","body":"Factor the polynomial completely.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 General Strategy for Factoring Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a746c1bfactoringpoly7a","stepAnswer":["$$8\\\\left(x-2\\\\right) \\\\left(x+2\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$8x^2-32$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8\\\\left(x-2\\\\right) \\\\left(x+2\\\\right)$$","hints":{"DefaultPathway":[{"id":"a746c1bfactoringpoly7a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":[],"title":"Looking for a GCF","text":"Is there a greatest common factor? Enter the greatest common factor or $$0$$ if there isn\'t one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a746c1bfactoringpoly7a-h1"],"title":"Factoring Polynomials","text":"Is this a binomial, trinomial, or are there more than $$3$$ terms? How many different terms are there in the parentheses?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly7a-h3","type":"hint","dependencies":["a746c1bfactoringpoly7a-h2"],"title":"Squares","text":"Since this is a binomial, is it a sum of squares or difference of squares?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly7a-h4","type":"hint","dependencies":["a746c1bfactoringpoly7a-h3"],"title":"Finished?","text":"Difference of squares can be factored further.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a746c1bfactoringpoly8","title":"Factoring Polynomials","body":"Factor the polynomial completely.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 General Strategy for Factoring Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a746c1bfactoringpoly8a","stepAnswer":["$$4\\\\left(3x-5\\\\right) \\\\left(3x+5\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$36x^2-100$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4\\\\left(3x-5\\\\right) \\\\left(3x+5\\\\right)$$","hints":{"DefaultPathway":[{"id":"a746c1bfactoringpoly8a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":[],"title":"Looking for a GCF","text":"Is there a greatest common factor? Enter the greatest common factor or $$0$$ if there isn\'t one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a746c1bfactoringpoly8a-h1"],"title":"Factoring Polynomials","text":"Is this a binomial, trinomial, or are there more than $$3$$ terms? How many different terms are there in the parentheses?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly8a-h3","type":"hint","dependencies":["a746c1bfactoringpoly8a-h2"],"title":"Squares","text":"Since this is a binomial, is it a sum of squares or difference of squares?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly8a-h4","type":"hint","dependencies":["a746c1bfactoringpoly8a-h3"],"title":"Finished?","text":"Difference of squares can be factored further.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly8a-h5","type":"hint","dependencies":["a746c1bfactoringpoly8a-h4"],"title":"Formula","text":"$$x^2-y^2=\\\\left(x+y\\\\right) \\\\left(x-y\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a746c1bfactoringpoly9","title":"Factoring Polynomials","body":"Factor the polynomial completely.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 General Strategy for Factoring Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a746c1bfactoringpoly9a","stepAnswer":["$${\\\\left(5x-6\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$25x^2-60x+36$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(5x-6\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"a746c1bfactoringpoly9a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":[],"title":"Looking for a GCF","text":"Is there a greatest common factor? Enter the greatest common factor or $$0$$ if there isn\'t one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a746c1bfactoringpoly9a-h1"],"title":"Factoring Polynomials","text":"Is this a binomial, trinomial, or are there more than $$3$$ terms? How many different terms are there in the parentheses?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["a746c1bfactoringpoly9a-h2"],"title":"Leading Coefficient","text":"Since this is a trinomial, what is the leading coefficient?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly9a-h4","type":"hint","dependencies":["a746c1bfactoringpoly9a-h3"],"title":"FOIL","text":"Undo the FOIL for the polynomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly9a-h5","type":"hint","dependencies":["a746c1bfactoringpoly9a-h4"],"title":"Factors for the leading coefficient","text":"Think about some factors that multiply to get $$25x^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly9a-h6","type":"hint","dependencies":["a746c1bfactoringpoly9a-h5"],"title":"Suggestions","text":"How about $$5$$ and $$5$$, and $$1$$ and 25?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly9a-h7","type":"hint","dependencies":["a746c1bfactoringpoly9a-h6"],"title":"Factors for C","text":"What are some factors that can be multiplied to get 36?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a746c1bfactoringpoly9a-h8","type":"hint","dependencies":["a746c1bfactoringpoly9a-h7"],"title":"Factors for C","text":"How about $$6$$ and $$6$$, $$9$$ and $$4$$, and $$12$$ and 3?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a750210trig1","title":"Converting from Degrees to Radians","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.3 Trigonometric Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a750210trig1a","stepAnswer":["$$\\\\frac{4\\\\pi}{3}$$ radians"],"problemType":"MultipleChoice","stepTitle":"Convert the angle, $$240$$ degrees, from degrees to radians.","stepBody":"Write the answer as a multiple of pi.","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{4\\\\pi}{3}$$ radians","choices":["$$\\\\frac{4\\\\pi}{3}$$ radians","$$\\\\frac{4}{3}$$ radians","$$\\\\frac{8\\\\pi}{3}$$ radians","$$\\\\frac{5\\\\pi}{3}$$ radians"],"hints":{"DefaultPathway":[{"id":"a750210trig1a-h1","type":"hint","dependencies":[],"title":"Conversion between degrees and radians","text":"Remember that $$180$$ degrees is equal to pi radians.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a750210trig1a-h2","type":"hint","dependencies":["a750210trig1a-h1"],"title":"Conversion Factor","text":"$$1=(pi$$ rad)/180 degrees $$=$$ $$180$$ degrees/(pi rad)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a750210trig1a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{4\\\\pi}{3}$$ radians"],"dependencies":["a750210trig1a-h2"],"title":"Calculate","text":"$$240$$ degrees*((pi rad)/180 degrees)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\frac{4\\\\pi}{3}$$ radians","$$\\\\frac{4}{3}$$ radians","$$\\\\frac{8\\\\pi}{3}$$ radians","$$\\\\frac{5\\\\pi}{3}$$ radians"]}]}}]},{"id":"a750210trig10","title":"Converting from Radians to Degrees","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.3 Trigonometric Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a750210trig10a","stepAnswer":["$$75$$ degrees"],"problemType":"MultipleChoice","stepTitle":"Convert the $$\\\\operatorname{angle}\\\\left(\\\\frac{5\\\\pi}{12}\\\\right)$$ radians, from radians to degrees.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$75$$ degrees","choices":["$$120$$ degrees","$$75$$ degrees","$$-180$$ degrees","$$90$$ degrees"],"hints":{"DefaultPathway":[{"id":"a750210trig10a-h1","type":"hint","dependencies":[],"title":"Conversion between degrees and radians","text":"Remember that pi radians is equal to $$180$$ degrees.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a750210trig10a-h2","type":"hint","dependencies":["a750210trig10a-h1"],"title":"Conversion Factor","text":"$$1=((pi$$ rad)/180 $$degrees)=(180$$ degrees/(pi rad))","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a750210trig10a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$75$$ degrees"],"dependencies":["a750210trig10a-h2"],"title":"Calculate","text":"$$\\\\frac{5\\\\pi}{12}$$ radians * (180 degrees/(pi radians))","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$120$$ degrees","$$75$$ degrees","$$-180$$ degrees","$$90$$ degrees"]}]}}]},{"id":"a750210trig11","title":"Evaluating Function Values","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.3 Trigonometric Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a750210trig11a","stepAnswer":["$$-0.5$$"],"problemType":"TextBox","stepTitle":"Evaluate the expression, $$cos\\\\left(\\\\frac{4\\\\pi}{3}\\\\right)$$ .","stepBody":"It may be useful to pull up a picture of the unit circle for reference.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-0.5$$","hints":{"DefaultPathway":[{"id":"a750210trig11a-h1","type":"hint","dependencies":[],"title":"Unit Circle $$1$$","text":"The value inside the parenthesis of the cos refers to an angle which also corresponds to a point on the unit circle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a750210trig11a-h2","type":"hint","dependencies":["a750210trig11a-h1"],"title":"Unit Circle $$2$$","text":"The angle $$\\\\frac{4\\\\pi}{3}$$ corresponds to the point $$(\\\\left(-\\\\frac{1}{2}\\\\right),\\\\left(-\\\\frac{\\\\sqrt{3}}{2}\\\\right))$$ on the unit circle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a750210trig11a-h3","type":"hint","dependencies":["a750210trig11a-h2"],"title":"Definition of trigonometric functions","text":"Letting $$x$$ be an angle, sinx $$=$$ $$y$$, cosx $$=$$ $$x$$, and tanx $$=$$ $$\\\\frac{y}{x}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a750210trig11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-0.5$$"],"dependencies":["a750210trig11a-h3"],"title":"Combining","text":"Based on the above information, what is the value of cos $$x$$ at the angle $$\\\\frac{4\\\\pi}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a750210trig12","title":"Evaluating Function Values","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.3 Trigonometric Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a750210trig12a","stepAnswer":["$$-1$$"],"problemType":"TextBox","stepTitle":"Evaluate the expression, $$tan\\\\left(\\\\frac{19\\\\pi}{4}\\\\right)$$ .","stepBody":"It may be useful to pull up a picture of the unit circle for reference.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1$$","hints":{"DefaultPathway":[{"id":"a750210trig12a-h1","type":"hint","dependencies":[],"title":"Unit Circle $$1$$","text":"The value inside the parenthesis of the tan refers to an angle which also corresponds to a point on the unit circle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a750210trig12a-h2","type":"hint","dependencies":["a750210trig12a-h1"],"title":"Unit Circle $$2$$","text":"The angle $$\\\\frac{19\\\\pi}{4}$$ corresponds to more than $$2$$ revolutions around the unit circle and is equal to $$2\\\\pi+2\\\\pi+\\\\frac{3\\\\pi}{4}$$, thus we can continue to subtract by $$2\\\\pi$$ or $$1$$ rotation until we get an angle between $$0$$ and $$2\\\\pi$$ to find the corresponding value on the unit circle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a750210trig12a-h3","type":"hint","dependencies":["a750210trig12a-h2"],"title":"Unit Circle $$3$$","text":"We know that the angle $$\\\\frac{3\\\\pi}{4}$$ corresponds to the point $$(\\\\left(-\\\\frac{\\\\sqrt{2}}{2}\\\\right),\\\\frac{\\\\sqrt{2}}{2})$$ on the unit circle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a750210trig12a-h4","type":"hint","dependencies":["a750210trig12a-h3"],"title":"Definition of trigonometric functions","text":"Letting $$x$$ be an angle, sinx $$=$$ $$y$$, cosx $$=$$ $$x$$, and tanx $$=$$ $$\\\\frac{y}{x}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a750210trig12a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a750210trig12a-h4"],"title":"Combining","text":"Based on the above information, what is the value of tanx at the angle $$\\\\frac{19\\\\pi}{4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a750210trig13","title":"Evaluating Function Values","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.3 Trigonometric Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a750210trig13a","stepAnswer":["$$\\\\frac{-\\\\sqrt{2}}{2}$$"],"problemType":"MultipleChoice","stepTitle":"Evaluate the expression, $$sin\\\\left(\\\\frac{\\\\left(-3\\\\pi\\\\right)}{4}\\\\right)$$ .","stepBody":"It may be useful to pull up a picture of the unit circle for reference.","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{-\\\\sqrt{2}}{2}$$","choices":["$$\\\\frac{-\\\\sqrt{2}}{2}$$","$$\\\\frac{\\\\sqrt{2}}{2}$$"],"hints":{"DefaultPathway":[{"id":"a750210trig13a-h1","type":"hint","dependencies":[],"title":"Unit Circle $$1$$","text":"The value inside the parenthesis of the sin refers to an angle which also corresponds to a point on the unit circle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a750210trig13a-h2","type":"hint","dependencies":["a750210trig13a-h1"],"title":"Unit Circle $$2$$","text":"The angle $$\\\\frac{\\\\left(-3\\\\pi\\\\right)}{4}$$ corresponds to a revolution in the negative direction, thus we need to add $$2\\\\pi$$ to get the angle $$\\\\frac{5\\\\pi}{4}$$ which corresponds to the point $$(\\\\left(-\\\\frac{\\\\sqrt{2}}{2}\\\\right),\\\\left(-\\\\frac{\\\\sqrt{2}}{2}\\\\right))$$ on the unit circle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a750210trig13a-h3","type":"hint","dependencies":["a750210trig13a-h2"],"title":"Definition of trigonometric functions","text":"Letting $$x$$ be an angle, sinx $$=$$ $$y$$, cosx $$=$$ $$x$$, and tanx $$=$$ $$\\\\frac{y}{x}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a750210trig13a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{-\\\\sqrt{2}}{2}$$"],"dependencies":["a750210trig13a-h3"],"title":"Combining","text":"Based on the above information, what is the value of sinx at the angle $$\\\\frac{\\\\left(-3\\\\pi\\\\right)}{4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\frac{-\\\\sqrt{2}}{2}$$","$$\\\\frac{\\\\sqrt{2}}{2}$$"]}]}}]},{"id":"a750210trig14","title":"Evaluating Function Values","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.3 Trigonometric Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a750210trig14a","stepAnswer":["$$\\\\frac{2}{\\\\sqrt{3}}$$"],"problemType":"MultipleChoice","stepTitle":"Evaluate the expression, $$\\\\operatorname{sec}\\\\left(\\\\frac{\\\\pi}{6}\\\\right)$$ .","stepBody":"It may be useful to pull up a picture of the unit circle for reference.","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{2}{\\\\sqrt{3}}$$","choices":["$$\\\\frac{2}{\\\\sqrt{3}}$$","$$\\\\frac{-2}{\\\\sqrt{3}}$$"],"hints":{"DefaultPathway":[{"id":"a750210trig14a-h1","type":"hint","dependencies":[],"title":"Unit Circle $$1$$","text":"The value inside the parenthesis of the sin refers to an angle which also corresponds to a point on the unit circle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a750210trig14a-h2","type":"hint","dependencies":["a750210trig14a-h1"],"title":"Unit Circle $$2$$","text":"The angle $$\\\\frac{\\\\pi}{6}$$ corresponds to the point $$(\\\\frac{\\\\sqrt{3}}{2},\\\\frac{1}{2})$$ on the unit circle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a750210trig14a-h3","type":"hint","dependencies":["a750210trig14a-h2"],"title":"Definition of trigonometric 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and $$\\\\frac{\\\\pi}{4}$$. (I.e. $$\\\\frac{\\\\pi}{12}$$ $$=$$ pi/3-pi/4)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a750210trig15a-h3","type":"hint","dependencies":["a750210trig15a-h2"],"title":"Subtraction formula of sinx","text":"The subtraction formula for sinx is sin $$(A-B)=sinA CosB-cosA sinB$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a750210trig15a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{\\\\sqrt{3}-1}{2-\\\\sqrt{2}}$$"],"dependencies":["a750210trig15a-h3"],"title":"Calculate","text":"Calculate $$sin\\\\left(\\\\frac{\\\\pi}{3}\\\\right) cos\\\\left(\\\\frac{\\\\pi}{4}\\\\right)-cos\\\\left(\\\\frac{\\\\pi}{3}\\\\right) sin\\\\left(\\\\frac{\\\\pi}{4}\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["(sqrt(3)-1)/(2-sqrt(2)","(sqrt(2)-1)/(3-sqrt(2)","(sqrt(1)-1)/(3-sqrt(2)","(sqrt(3)-1)/(1-sqrt(2)"]},{"id":"a750210trig15a-h5","type":"hint","dependencies":["a750210trig15a-h3"],"title":"Unit Circle","text":"The replacement angles $$\\\\frac{\\\\pi}{3}$$ and $$\\\\frac{\\\\pi}{4}$$ are on the common unit circle and sinx $$=$$ the $$y$$ value while cosx $$=$$ the $$x$$ value of the corresponding point on the unit circle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a750210trig16","title":"Evaluating Function Values","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.3 Trigonometric Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a750210trig16a","stepAnswer":["(sqrt(2-sqrt(3))/2"],"problemType":"MultipleChoice","stepTitle":"Evaluate the expression, $$cos\\\\left(\\\\frac{5\\\\pi}{12}\\\\right)$$ .","stepBody":"It may be useful to pull up a picture of the unit circle for reference.","answerType":"string","variabilization":{},"choices":["(sqrt(2-sqrt(2))/2","(sqrt(1-sqrt(3))/2","(sqrt(4-sqrt(3))/2","(sqrt(2-sqrt(3))/2"],"hints":{"DefaultPathway":[{"id":"a750210trig16a-h1","type":"hint","dependencies":[],"title":"Angles","text":"The value inside the parenthesis of the cos refers to an angle which can also correspond to a point on the unit circle, but in this case does not.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a750210trig16a-h2","type":"hint","dependencies":["a750210trig16a-h1"],"title":"Half angle formula","text":"Since $$\\\\frac{5\\\\pi}{12}$$ is not a common angle on the unit circle, we need to use the half angle formula. The half angle formula is $$cos\\\\left(\\\\frac{x}{2}\\\\right)$$ $$=$$ $$\\\\pm \\\\sqrt{\\\\frac{1+cosx}{2}}$$. 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Find the missing side of the triangle.","stepBody":"For this step round to $$1$$ decimal place","answerType":"string","variabilization":{},"answerLatex":"$$b=20$$","choices":["$$b=20$$","$$b=21$$","$$b=22$$","$$b=23$$"],"hints":{"DefaultPathway":[{"id":"a750210trig18a-h1","type":"hint","dependencies":[],"title":"Pythagorean theorem","text":"$$a^2+b^2=c^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a750210trig18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a750210trig18a-h1"],"title":"Solve for c","text":"$${21}^2+b^2={29}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a750210trig18b","stepAnswer":["$$\\\\frac{21}{29}$$"],"problemType":"TextBox","stepTitle":"Find the trigonometric function value of sin for the angle at 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<OATutor>","license":"","choices":["$$990$$ degrees","$$330$$ degrees","$$15$$ degrees","$$-330$$ degrees"]}]}}]},{"id":"a750210trig9","title":"Converting from Radians to Degrees","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.3 Trigonometric Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a750210trig9a","stepAnswer":["$$-540$$ degrees"],"problemType":"MultipleChoice","stepTitle":"Convert the angle,(-3pi) radians, from radians to degrees.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-540$$ degrees","choices":["$$240$$ degrees","$$-540$$ degrees","$$-180$$ degrees","$$-330$$ degrees"],"hints":{"DefaultPathway":[{"id":"a750210trig9a-h1","type":"hint","dependencies":[],"title":"Conversion between degrees and radians","text":"Remember that pi radians is equal to $$180$$ degrees.","variabilization":{},"oer":"https://OATutor.io 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4.0>"},{"id":"a75d03cRecCord1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=\\\\frac{-4}{3}$$"],"dependencies":["a75d03cRecCord1a-h2"],"title":"Solving the equation","text":"Solve for x: $$0=-3x-4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a75d03cRecCord1b","stepAnswer":["(0,-4)"],"problemType":"TextBox","stepTitle":"What is the y-intercept?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,-4)$$","hints":{"DefaultPathway":[{"id":"a75d03cRecCord1b-h1","type":"hint","dependencies":[],"title":"Definition of y-intercepts","text":"The y-intercept is the point at which the graph crosses the y-axis","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord1b-h2","type":"hint","dependencies":["a75d03cRecCord1b-h1"],"title":"How to determine the x-intercept?","text":"Set $$x$$ equal to zero and solve for $$y$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord1b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y=-4$$"],"dependencies":["a75d03cRecCord1b-h2"],"title":"Solving the equation","text":"Solve for y: $$y=-3(0)-4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a75d03cRecCord10","title":"Finding the Distance between Two Points","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 The Rectangular Coordinate Systems and Graph","courseName":"OpenStax: College Algebra","steps":[{"id":"a75d03cRecCord10a","stepAnswer":["$$5\\\\sqrt{5}$$"],"problemType":"TextBox","stepTitle":"Find the distance between the points $$(1,4)$$ and $$(11,9)$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5\\\\sqrt{5}$$","hints":{"DefaultPathway":[{"id":"a75d03cRecCord10a-h1","type":"hint","dependencies":[],"title":"Distance Formula","text":"Given endpoints $$(x_1,y_1)$$, and $$(x_2,y_2)$$, the distance between two points is given by $$\\\\sqrt{{\\\\left(x_2-x_1\\\\right)}^2+{\\\\left(y_2-y_1\\\\right)}^2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord10a-h2","type":"hint","dependencies":["a75d03cRecCord10a-h1"],"title":"Plug into the formula","text":"We should first calculate $$11-1$$ and $$9-4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a75d03cRecCord10a-h2"],"title":"Subtraction","text":"What is $$11-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a75d03cRecCord10a-h2"],"title":"Subtraction","text":"What is $$9-4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord10a-h5","type":"hint","dependencies":["a75d03cRecCord10a-h3","a75d03cRecCord10a-h4"],"title":"Simplification","text":"We need to simplify the expression $${10}^2+5^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord10a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$100$$"],"dependencies":["a75d03cRecCord10a-h5"],"title":"Square","text":"What is $${10}^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord10a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["a75d03cRecCord10a-h5"],"title":"Square","text":"What is $$5^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord10a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$125$$"],"dependencies":["a75d03cRecCord10a-h6","a75d03cRecCord10a-h7"],"title":"Addition","text":"What is $$100+25$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord10a-h9","type":"hint","dependencies":["a75d03cRecCord10a-h8"],"title":"Extraction of square root","text":"The thid step is computing the principal square root.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord10a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5\\\\sqrt{5}$$"],"dependencies":["a75d03cRecCord10a-h9"],"title":"Square root","text":"What is the principal square root of 125?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a75d03cRecCord11","title":"Finding the Distance between Two Locations","body":"Tracie set out from Elmhurst, IL, $$(0,0)$$ to go to Franklin Park. On the way, she made a few stops to do errands. Each stop is indicated by a red dot in figure.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 The Rectangular Coordinate Systems and Graph","courseName":"OpenStax: College Algebra","steps":[{"id":"a75d03cRecCord11a","stepAnswer":["$$15000$$"],"problemType":"TextBox","stepTitle":"Find the total distance that Tracie traveled in feet(Each grid unit represent 1,000 feet, only enter the final number for the answer).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$15000$$","hints":{"DefaultPathway":[{"id":"a75d03cRecCord11a-h1","type":"hint","dependencies":[],"title":"Identify location","text":"The first thing we should do is to identify ordered pairs to describe each position.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord11a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(1,1)"],"dependencies":["a75d03cRecCord11a-h1"],"title":"Ordered pairs","text":"Where is her first stop?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord11a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(5,1)"],"dependencies":["a75d03cRecCord11a-h1"],"title":"Ordered pairs","text":"Where is her second stop?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord11a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(8,3)"],"dependencies":["a75d03cRecCord11a-h1"],"title":"Ordered pairs","text":"Where is her third stop?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord11a-h5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(8,7)"],"dependencies":["a75d03cRecCord11a-h1"],"title":"Ordered pairs","text":"Where is her fourth stop?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord11a-h6","type":"hint","dependencies":["a75d03cRecCord11a-h2","a75d03cRecCord11a-h3","a75d03cRecCord11a-h4","a75d03cRecCord11a-h5"],"title":"Calculate the distance","text":"The number of grid units that Tracie traveled from $$(x_1,y_1)$$ to $$(x_2,y_2)$$ is $$|x_2-x_1|+|y_2-y_1|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord11a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a75d03cRecCord11a-h6"],"title":"Calculate the distance","text":"What is |1-0|+|1-0|?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord11a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a75d03cRecCord11a-h6"],"title":"Calculate the distance","text":"What is |5-1|+|1-1|?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord11a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a75d03cRecCord11a-h6"],"title":"Calculate the distance","text":"What is |8-5|+|3-1|?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord11a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a75d03cRecCord11a-h6"],"title":"Calculate the distance","text":"What is |8-8|+|7-3|?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord11a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["a75d03cRecCord11a-h7","a75d03cRecCord11a-h8","a75d03cRecCord11a-h9","a75d03cRecCord11a-h10"],"title":"Total distance","text":"What is $$2+4+5+4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord11a-h12","type":"hint","dependencies":["a75d03cRecCord11a-h11"],"title":"Conversion of Units","text":"Since each grid unit represents $$1000$$ feet, so the total distance that Tracie traveled is $$15000$$ feet.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a75d03cRecCord11b","stepAnswer":["$$\\\\sqrt{113}$$"],"problemType":"TextBox","stepTitle":"Find the distance between her starting and final positions.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\sqrt{113}$$","hints":{"DefaultPathway":[{"id":"a75d03cRecCord11b-h1","type":"hint","dependencies":[],"title":"Distance Formula","text":"Given endpoints $$(x_1,y_1)$$, and $$(x_2,y_2)$$, the distance between two points is given by $$\\\\sqrt{{\\\\left(x_2-x_1\\\\right)}^2+{\\\\left(y_2-y_1\\\\right)}^2}$$ .","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord11b-h2","type":"hint","dependencies":["a75d03cRecCord11b-h1"],"title":"Plug into the formula","text":"We should first calculate $$8-0$$ and $$7-0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord11b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a75d03cRecCord11b-h2"],"title":"Subtraction","text":"What is $$8-0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord11b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a75d03cRecCord11b-h2"],"title":"Subtraction","text":"What is $$7-0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord11b-h5","type":"hint","dependencies":["a75d03cRecCord11b-h3","a75d03cRecCord11b-h4"],"title":"Simplification","text":"We need to simplify the expression $$8^2+7^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord11b-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$64$$"],"dependencies":["a75d03cRecCord11b-h5"],"title":"Square","text":"What is $$8^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord11b-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$49$$"],"dependencies":["a75d03cRecCord11b-h5"],"title":"Square","text":"What is $$7^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord11b-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$113$$"],"dependencies":["a75d03cRecCord11b-h6","a75d03cRecCord11b-h7"],"title":"Addition","text":"What is $$64+49$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord11b-h9","type":"hint","dependencies":["a75d03cRecCord11b-h8"],"title":"Extraction of square root","text":"The thid step is computing the principal square root.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord11b-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{113}$$"],"dependencies":["a75d03cRecCord11b-h9"],"title":"Square root","text":"What is the principal square root of 113?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a75d03cRecCord12","title":"Finding the Distance between Two Points","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 The Rectangular Coordinate Systems and Graph","courseName":"OpenStax: College Algebra","steps":[{"id":"a75d03cRecCord12a","stepAnswer":["$$\\\\sqrt{74}$$"],"problemType":"TextBox","stepTitle":"Find the distance between the points $$(-4,1)$$ and $$(3,-4)$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\sqrt{74}$$","hints":{"DefaultPathway":[{"id":"a75d03cRecCord12a-h1","type":"hint","dependencies":[],"title":"Distance Formula","text":"Given endpoints $$(x_1,y_1)$$, and $$(x_2,y_2)$$, the distance between two points is given by $$\\\\sqrt{{\\\\left(x_2-x_1\\\\right)}^2+{\\\\left(y_2-y_1\\\\right)}^2}$$ .","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord12a-h2","type":"hint","dependencies":["a75d03cRecCord12a-h1"],"title":"Plug into the formula","text":"We should first calculate $$3-(-4)$$ and $$(-4)-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord12a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a75d03cRecCord12a-h2"],"title":"Subtraction","text":"What is $$3-(-4)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["a75d03cRecCord12a-h2"],"title":"Subtraction","text":"What is $$(-4)-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord12a-h5","type":"hint","dependencies":["a75d03cRecCord12a-h3","a75d03cRecCord12a-h4"],"title":"Simplification","text":"We need to simplify the expression $$7^2+{\\\\left(-5\\\\right)}^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord12a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$49$$"],"dependencies":["a75d03cRecCord12a-h5"],"title":"Square","text":"What is $$7^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord12a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["a75d03cRecCord12a-h5"],"title":"Square","text":"What is $${\\\\left(-5\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord12a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$74$$"],"dependencies":["a75d03cRecCord12a-h6","a75d03cRecCord12a-h7"],"title":"Addition","text":"What is $$49+25$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord12a-h9","type":"hint","dependencies":["a75d03cRecCord12a-h8"],"title":"Extraction of square root","text":"The thid step is computing the principal square root.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord12a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{74}$$"],"dependencies":["a75d03cRecCord12a-h9"],"title":"Square root","text":"What is the principal square root of 74?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a75d03cRecCord13","title":"Finding the Distance between Two Points","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 The Rectangular Coordinate Systems and Graph","courseName":"OpenStax: College Algebra","steps":[{"id":"a75d03cRecCord13a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"Find the distance between the points $$(5,0)$$ and $$(5,6)$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"a75d03cRecCord13a-h1","type":"hint","dependencies":[],"title":"Distance Formula","text":"Given endpoints $$(x_1,y_1)$$, and $$(x_2,y_2)$$, the distance between two points is given by $$\\\\sqrt{{\\\\left(x_2-x_1\\\\right)}^2+{\\\\left(y_2-y_1\\\\right)}^2}$$ .","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord13a-h2","type":"hint","dependencies":["a75d03cRecCord13a-h1"],"title":"Plug into the formula","text":"We should first calculate $$5-5$$ and $$6-0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord13a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a75d03cRecCord13a-h2"],"title":"Subtraction","text":"What is $$5-5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a75d03cRecCord13a-h2"],"title":"Subtraction","text":"What is $$6-0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord13a-h5","type":"hint","dependencies":["a75d03cRecCord13a-h3","a75d03cRecCord13a-h4"],"title":"Simplification","text":"We need to simplify the expression $$0^2+6^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord13a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a75d03cRecCord13a-h5"],"title":"Square","text":"What is $$0^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord13a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$36$$"],"dependencies":["a75d03cRecCord13a-h5"],"title":"Square","text":"What is $$6^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord13a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$36$$"],"dependencies":["a75d03cRecCord13a-h6","a75d03cRecCord13a-h7"],"title":"Addition","text":"What is $$0+36$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord13a-h9","type":"hint","dependencies":["a75d03cRecCord13a-h8"],"title":"Extraction of square root","text":"The thid step is computing the principal square root.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord13a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a75d03cRecCord13a-h9"],"title":"Square root","text":"What is the principal square root of 36?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a75d03cRecCord14","title":"Finding the Distance between Two Points","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College 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should first calculate $$41-19$$ and $$71-12$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord14a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$22$$"],"dependencies":["a75d03cRecCord14a-h2"],"title":"Subtraction","text":"What is $$41-19$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord14a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$59$$"],"dependencies":["a75d03cRecCord14a-h2"],"title":"Subtraction","text":"What is $$71-12$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord14a-h5","type":"hint","dependencies":["a75d03cRecCord14a-h3","a75d03cRecCord14a-h4"],"title":"Simplification","text":"We need to simplify the expression $${22}^2+{59}^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord14a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$484$$"],"dependencies":["a75d03cRecCord14a-h5"],"title":"Square","text":"What is $${22}^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord14a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3481$$"],"dependencies":["a75d03cRecCord14a-h5"],"title":"Square","text":"What is $${59}^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord14a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3965$$"],"dependencies":["a75d03cRecCord14a-h6","a75d03cRecCord14a-h7"],"title":"Addition","text":"Whta is $$484+3481$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord14a-h9","type":"hint","dependencies":["a75d03cRecCord14a-h8"],"title":"Extraction of square root","text":"The thid step is computing the principal square root.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord14a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{3965}$$"],"dependencies":["a75d03cRecCord14a-h9"],"title":"Square root","text":"What is the principal square root of 3965?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a75d03cRecCord15","title":"Finding the Midpoint of the Line Segment","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 The Rectangular Coordinate Systems and Graph","courseName":"OpenStax: College Algebra","steps":[{"id":"a75d03cRecCord15a","stepAnswer":["(8,3/2)"],"problemType":"TextBox","stepTitle":"Find the midpoint of the line segment with the endpoints $$(7,-2)$$ and $$(9,5)$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(8,\\\\frac{3}{2})$$","hints":{"DefaultPathway":[{"id":"a75d03cRecCord15a-h1","type":"hint","dependencies":[],"title":"Midpoint Formula","text":"Given the endpoints of a line segment, $$(x_1,y_1)$$ and $$(x_2,y_2)$$, the midpoint formula states how to find the coordinates of the midpoint M. $$M=($$ $$\\\\frac{x_1+x_2}{2}$$, $$\\\\frac{y_1+y_2}{2}$$ ).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["a75d03cRecCord15a-h1"],"title":"Addition","text":"What is $$7+9$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a75d03cRecCord15a-h2"],"title":"Division","text":"What is $$\\\\frac{16}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a75d03cRecCord15a-h1"],"title":"Addition","text":"What is $$-2+5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord15a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{2}$$"],"dependencies":["a75d03cRecCord15a-h4"],"title":"Division","text":"What is $$\\\\frac{3}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a75d03cRecCord16","title":"Finding the Midpoint of the Line Segment","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 The Rectangular 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord16a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{2}$$"],"dependencies":["a75d03cRecCord16a-h4"],"title":"Division","text":"What is $$\\\\frac{5}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a75d03cRecCord17","title":"Finding the Center of a Circle","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 The Rectangular Coordinate Systems and Graph","courseName":"OpenStax: College Algebra","steps":[{"id":"a75d03cRecCord17a","stepAnswer":["(2,-4)"],"problemType":"TextBox","stepTitle":"The diameter of a circle has endpoints $$(-1,-4)$$ and $$(5,-4)$$. 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord17a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["a75d03cRecCord17a-h5"],"title":"Division","text":"What is $$\\\\frac{-8}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a75d03cRecCord18","title":"Finding the Midpoint of the Line Segment","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 The Rectangular Coordinate Systems and Graph","courseName":"OpenStax: College Algebra","steps":[{"id":"a75d03cRecCord18a","stepAnswer":["(3,-3/2)"],"problemType":"TextBox","stepTitle":"Find the midpoint of the line segment with the endpoints $$(-1,1)$$ and 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a75d03cRecCord19","title":"Finding the Midpoint of the Line Segment","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 The Rectangular Coordinate Systems and Graph","courseName":"OpenStax: College Algebra","steps":[{"id":"a75d03cRecCord19a","stepAnswer":["(2,-1)"],"problemType":"TextBox","stepTitle":"Find the midpoint of the line segment with the endpoints $$(0,7)$$ and $$(4,-9)$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(2,-1)$$","hints":{"DefaultPathway":[{"id":"a75d03cRecCord19a-h1","type":"hint","dependencies":[],"title":"Midpoint Formula","text":"Given the endpoints of a line segment, $$(x_1,y_1)$$ and $$(x_2,y_2)$$, the midpoint formula states how to find the coordinates of the midpoint M. $$M=($$ $$\\\\frac{x_1+x_2}{2}$$, $$\\\\frac{y_1+y_2}{2}$$ ).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a75d03cRecCord19a-h1"],"title":"Addition","text":"What is $$0+4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord19a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a75d03cRecCord19a-h2"],"title":"Division","text":"What is $$\\\\frac{4}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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<CC BY 4.0>"},{"id":"a75d03cRecCord2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=4$$"],"dependencies":["a75d03cRecCord2a-h2"],"title":"Solving the equation","text":"Solve for x: $$0=\\\\left(-\\\\frac{3}{4}\\\\right) x+3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a75d03cRecCord2b","stepAnswer":["(0,3)"],"problemType":"TextBox","stepTitle":"What is the y-intercept?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,3)$$","hints":{"DefaultPathway":[{"id":"a75d03cRecCord2b-h1","type":"hint","dependencies":[],"title":"Definition of y-intercepts","text":"The y-intercept is the point at which the graph crosses y-axis","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord2b-h2","type":"hint","dependencies":[],"title":"How to determine the y-intercept?","text":"Set $$x$$ equal to zero and solve for $$y$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord2b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y=3$$"],"dependencies":["a75d03cRecCord2b-h2"],"title":"Solving the equation","text":"Solve for y: $$y=\\\\frac{-3}{4\\\\left(0\\\\right)}+3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a75d03cRecCord20","title":"Finding Intercepts: Numeric","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 The Rectangular Coordinate Systems and Graph","courseName":"OpenStax: College Algebra","steps":[{"id":"a75d03cRecCord20a","stepAnswer":["$$(0,-4)$$ $$(3,0)$$"],"problemType":"MultipleChoice","stepTitle":"Find the $$x$$ and $$y$$ intercepts of the function $$4x-3y=12$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,-4)$$ $$(3,0)$$","choices":["$$(0,-4)$$ $$(3,0)$$","$$(0,4)$$ $$(3,0)$$","$$(0,-4)$$ $$(-3,0)$$","$$(0,4)$$ $$(-3,0)$$"],"hints":{"DefaultPathway":[{"id":"a75d03cRecCord20a-h1","type":"hint","dependencies":[],"title":"Intercepts","text":"An intercept is where the function intersects either the $$x$$ or $$y$$ axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord20a-h2","type":"hint","dependencies":["a75d03cRecCord20a-h1"],"title":"Intercepts","text":"To find an intercept, set either $$x$$ or $$y$$ to $$0$$ and solve for the other variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord20a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["a75d03cRecCord20a-h2"],"title":"Intercepts","text":"In this case, let\'s set $$x$$ to $$0$$. What is the $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord20a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a75d03cRecCord20a-h3"],"title":"Intercepts","text":"Let\'s also set $$y$$ to $$0$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord20a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(0,-4)$$ $$(3,0)$$"],"dependencies":["a75d03cRecCord20a-h4"],"title":"Final Answer","text":"Write our two intercepts as points (x,y).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(0,-4)$$ $$(3,0)$$","$$(0,4)$$ $$(3,0)$$","$$(0,-4)$$ $$(-3,0)$$","$$(0,4)$$ $$(-3,0)$$"]}]}}]},{"id":"a75d03cRecCord21","title":"Finding Intercepts: Numeric","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 The Rectangular Coordinate Systems and Graph","courseName":"OpenStax: College Algebra","steps":[{"id":"a75d03cRecCord21a","stepAnswer":["$$(0,-4)$$ $$(8,0)$$"],"problemType":"MultipleChoice","stepTitle":"Find the $$x$$ and $$y$$ intercepts of the function $$x-2y=8$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,-4)$$ $$(8,0)$$","choices":["$$(0,4)$$ $$(8,0)$$","$$(0,-4)$$ $$(8,0)$$","$$(0,4)$$ $$(-8,0)$$","$$(0,-4)$$ $$(-8,0)$$"],"hints":{"DefaultPathway":[{"id":"a75d03cRecCord21a-h1","type":"hint","dependencies":[],"title":"Intercepts","text":"An intercept is where the function intersects either the $$x$$ or $$y$$ axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord21a-h2","type":"hint","dependencies":["a75d03cRecCord21a-h1"],"title":"Intercepts","text":"To find an intercept, set either $$x$$ or $$y$$ to $$0$$ and solve for the other variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord21a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["a75d03cRecCord21a-h2"],"title":"Intercepts","text":"In this case, let\'s set $$x$$ to $$0$$. What is the $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord21a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a75d03cRecCord21a-h3"],"title":"Intercepts","text":"Let\'s also set $$y$$ to $$0$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord21a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(0,-4)$$ $$(8,0)$$"],"dependencies":["a75d03cRecCord21a-h4"],"title":"Final Answer","text":"Write our two intercepts as points (x,y).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(0,4)$$ $$(8,0)$$","$$(0,-4)$$ $$(8,0)$$","$$(0,4)$$ $$(-8,0)$$","$$(0,-4)$$ $$(-8,0)$$"]}]}}]},{"id":"a75d03cRecCord22","title":"Finding Intercepts: Numeric","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 The Rectangular Coordinate Systems and Graph","courseName":"OpenStax: College Algebra","steps":[{"id":"a75d03cRecCord22a","stepAnswer":["$$(0,5)$$ $$(-1,0)$$"],"problemType":"MultipleChoice","stepTitle":"Find the $$x$$ and $$y$$ intercepts of the function $$y-5=5x$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,5)$$ $$(-1,0)$$","choices":["$$(0,-5)$$ $$(-1,0)$$","$$(0,5)$$ $$(-1,0)$$","$$(0,-5)$$ $$(1,0)$$","$$(0,5)$$ $$(1,0)$$"],"hints":{"DefaultPathway":[{"id":"a75d03cRecCord22a-h1","type":"hint","dependencies":[],"title":"Intercepts","text":"An intercept is where the function intersects either the $$x$$ or $$y$$ axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord22a-h2","type":"hint","dependencies":["a75d03cRecCord22a-h1"],"title":"Intercepts","text":"To find an intercept, set either $$x$$ or $$y$$ to $$0$$ and solve for the other variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord22a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a75d03cRecCord22a-h2"],"title":"Intercepts","text":"In this case, let\'s set $$x$$ to $$0$$. What is the $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord22a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a75d03cRecCord22a-h3"],"title":"Intercepts","text":"Let\'s also set $$y$$ to $$0$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord22a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(0,5)$$ $$(-1,0)$$"],"dependencies":["a75d03cRecCord22a-h4"],"title":"Final Answer","text":"Write our two intercepts as points (x,y).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(0,-5)$$ $$(-1,0)$$","$$(0,5)$$ $$(-1,0)$$","$$(0,-5)$$ $$(1,0)$$","$$(0,5)$$ $$(1,0)$$"]}]}}]},{"id":"a75d03cRecCord23","title":"Finding Intercepts: Numeric","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 The Rectangular Coordinate Systems and Graph","courseName":"OpenStax: College Algebra","steps":[{"id":"a75d03cRecCord23a","stepAnswer":["$$(0,2)$$ $$(3,0)$$"],"problemType":"MultipleChoice","stepTitle":"Find the $$x$$ and $$y$$ intercepts of the function $$3y=-2x+6$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,2)$$ $$(3,0)$$","choices":["$$(0,2)$$ $$(3,0)$$","$$(0,3)$$ $$(2,0)$$","$$(0,-2)$$ $$(-3,0)$$","$$(0,-3)$$ $$(-2,0)$$"],"hints":{"DefaultPathway":[{"id":"a75d03cRecCord23a-h1","type":"hint","dependencies":[],"title":"Intercepts","text":"An intercept is where the function intersects either the $$x$$ or $$y$$ axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord23a-h2","type":"hint","dependencies":["a75d03cRecCord23a-h1"],"title":"Intercepts","text":"To find an intercept, set either $$x$$ or $$y$$ to $$0$$ and solve for the other variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord23a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a75d03cRecCord23a-h2"],"title":"Intercepts","text":"In this case, let\'s set $$x$$ to $$0$$. What is the $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord23a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a75d03cRecCord23a-h3"],"title":"Intercepts","text":"Let\'s also set $$y$$ to $$0$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord23a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(0,2)$$ $$(3,0)$$"],"dependencies":["a75d03cRecCord23a-h4"],"title":"Final Answer","text":"Write our two intercepts as points (x,y).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(0,2)$$ $$(3,0)$$","$$(0,3)$$ $$(2,0)$$","$$(0,-2)$$ $$(-3,0)$$","$$(0,-3)$$ $$(-2,0)$$"]}]}}]},{"id":"a75d03cRecCord24","title":"Finding Intercepts: Numeric","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 The Rectangular Coordinate Systems and Graph","courseName":"OpenStax: College Algebra","steps":[{"id":"a75d03cRecCord24a","stepAnswer":["$$(0, -1.5)$$ $$(3,0)$$"],"problemType":"MultipleChoice","stepTitle":"Find the $$x$$ and $$y$$ intercepts of the function $$y=\\\\frac{x-3}{2}$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0, -1.5)$$ $$(3,0)$$","choices":["$$(0,-3)$$ $$(1.5, 0)$$","$$(0, 1.5)$$ $$(-3,0)$$","$$(0,3)$$ $$(-1.5, 0)$$","$$(0, -1.5)$$ $$(3,0)$$"],"hints":{"DefaultPathway":[{"id":"a75d03cRecCord24a-h1","type":"hint","dependencies":[],"title":"Intercepts","text":"An intercept is where the function intersects either the $$x$$ or $$y$$ axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord24a-h2","type":"hint","dependencies":["a75d03cRecCord24a-h1"],"title":"Intercepts","text":"To find an intercept, set either $$x$$ or $$y$$ to $$0$$ and solve for the other variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord24a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1.5$$"],"dependencies":["a75d03cRecCord24a-h2"],"title":"Intercepts","text":"In this case, let\'s set $$x$$ to $$0$$. What is the $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord24a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a75d03cRecCord24a-h3"],"title":"Intercepts","text":"Let\'s also set $$y$$ to $$0$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord24a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(0, -1.5)$$ $$(3,0)$$"],"dependencies":["a75d03cRecCord24a-h4"],"title":"Final Answer","text":"Write our two intercepts as points (x,y).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(0,-3)$$ $$(1.5, 0)$$","$$(0, 1.5)$$ $$(-3,0)$$","$$(0,3)$$ $$(-1.5, 0)$$","$$(0, -1.5)$$ $$(3,0)$$"]}]}}]},{"id":"a75d03cRecCord25","title":"Real-World Applications","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 The Rectangular Coordinate Systems and Graph","courseName":"OpenStax: College Algebra","steps":[{"id":"a75d03cRecCord25a","stepAnswer":["$$93$$"],"problemType":"TextBox","stepTitle":"The coordinates on a map for San Francisco are $$(53,17)$$ and those for Sacramento are $$(123,78)$$. Note that coordinates represent miles. Find the distance between the cities to the nearest mile.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$93$$","hints":{"DefaultPathway":[{"id":"a75d03cRecCord25a-h1","type":"hint","dependencies":[],"title":"Interpreting the Problem","text":"To find the distance between the cities, we need to find the distance between the points.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord25a-h2","type":"hint","dependencies":["a75d03cRecCord25a-h1"],"title":"Pythagorean Theorem","text":"We will use the Pythagorean Theorem to find the distance. The Pythagorean theorem allows us to relate two distances to get a third distance. The theorem states that $$c^2=a^2+b^2$$, where c is the hypotenuse and a, $$b$$ are the non-hypotenuse sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord25a-h3","type":"hint","dependencies":["a75d03cRecCord25a-h2"],"title":"Pythagorean Theorem","text":"To use the Pythagorean theorem, we need to find the horizontal and vertical distances between the points. This can be done by finding the difference between their $$x$$ and $$y$$ coordinates, respectively.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord25a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$70$$"],"dependencies":["a75d03cRecCord25a-h3"],"title":"Hotizontal Distance","text":"To find the horizontal distance, find the positive difference between the $$x$$ coordinates of both points. What is it?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord25a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$61$$"],"dependencies":["a75d03cRecCord25a-h3"],"title":"Veritical Distance","text":"To find the vertical distance, find the positive difference between the $$y$$ coordinates of both points. What is it?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord25a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$93$$"],"dependencies":["a75d03cRecCord25a-h4","a75d03cRecCord25a-h5"],"title":"Final Answer","text":"With these distances, we can add their squares and take the square root of their sum. What is the final distance?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a75d03cRecCord26","title":"Real-World Applications","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 The Rectangular Coordinate Systems and Graph","courseName":"OpenStax: College Algebra","steps":[{"id":"a75d03cRecCord26a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"A small craft in Lake Ontario sends out a distress signal. The coordinates of the boat in trouble were $$(49,64)$$. One rescue boat is at the coordinates $$(60,82)$$ and a second Coast Guard craft is at coordinates $$(58,47)$$. Assuming both rescue craft travel at the same rate, which one would get to the distressed boat the fastest?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a75d03cRecCord26a-h1","type":"hint","dependencies":[],"title":"Interpreting the Problem","text":"To find which boat got their the fastest, we must find the distances between the $$2$$ boats and the small craft since both boats travel at the same rate.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord26a-h2","type":"hint","dependencies":["a75d03cRecCord26a-h1"],"title":"Pythagorean Theorem","text":"We will use the Pythagorean Theorem to find the distance. The Pythagorean theorem allows us to relate two distances to get a third distance. The theorem states that $$c^2=a^2+b^2$$, where c is the hypotenuse and a, $$b$$ are the non-hypotenuse sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord26a-h3","type":"hint","dependencies":["a75d03cRecCord26a-h2"],"title":"Pythagorean Theorem","text":"To use the Pythagorean theorem, we need to find the horizontal and vertical distances between the points. This can be done by finding the difference between their $$x$$ and $$y$$ coordinates, respectively.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord26a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11$$"],"dependencies":["a75d03cRecCord26a-h3"],"title":"1st Horizontal Distance","text":"To find the horizontal distance of the first boat, find the positive difference between the $$x$$ coordinates of both points. What is it?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord26a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a75d03cRecCord26a-h3"],"title":"2nd Horizontal Distance","text":"To find the horizontal distance of thesecond boat, find the positive difference between the $$x$$ coordinates of both points. What is it?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord26a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18$$"],"dependencies":["a75d03cRecCord26a-h3"],"title":"1st Vertical Distance","text":"To find the vertical distance for the first boat, find the positive difference between the $$y$$ coordinates of both points. What is it?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord26a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$17$$"],"dependencies":["a75d03cRecCord26a-h3"],"title":"2nd Vertical Distance","text":"To find the vertical distance for the second boat, find the positive difference between the $$y$$ coordinates of both points. What is it?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord26a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$21$$"],"dependencies":["a75d03cRecCord26a-h4","a75d03cRecCord26a-h6"],"title":"1st Distance","text":"With these distances, we can add their squares and take the square root of their sum. What is the final distance for the first boat (rounded to the nearest whole number)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord26a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$19$$"],"dependencies":["a75d03cRecCord26a-h5","a75d03cRecCord26a-h7"],"title":"2nd Distance","text":"With these distances, we can add their squares and take the square root of their sum. What is the final distance for the second boat (rounded to the nearest whole number)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a75d03cRecCord26a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a75d03cRecCord26a-h8","a75d03cRecCord26a-h9"],"title":"Final Answer","text":"By comparing the $$2$$ distances, which boat reaches the signal the fastest (give boat number)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a75d03cRecCord27","title":"Real-World Applications","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 The Rectangular Coordinate Systems and Graph","courseName":"OpenStax: College Algebra","steps":[{"id":"a75d03cRecCord27a","stepAnswer":["$$54$$"],"problemType":"TextBox","stepTitle":"A person on the top of a building wants to have a guy wire extend to a point on the ground $$20$$ ft from the building. 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Or consider $$P\\\\left(X>x\\\\right)$$ as $$1-P\\\\left(X<x\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7811e5normal10","title":"Smart Users","body":"In the United States the ages $$13$$ to 55+ of smartphone users approximately follow a normal distribution with approximate mean and standard deviation of $$36.9$$ years and $$13.9$$ years, respectively.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Using the Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a7811e5normal10a","stepAnswer":["$$48.6$$"],"problemType":"TextBox","stepTitle":"Find the 80th percentile of this distribution.","stepBody":"Round answers to one decimal place.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$48.6$$","hints":{"DefaultPathway":[{"id":"a7811e5normal10a-h1","type":"hint","dependencies":[],"title":"Using a Calculator (TI-83, 83+, $$84$$, 84+ Calculator)","text":"Using invNorm","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$48.6$$"],"dependencies":["a7811e5normal10a-h1"],"title":"$$invNorm(0.80$$, $$36.9$$, $$13.9)$$","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7811e5normal11","title":"Smart Users","body":"In the United States the ages $$13$$ to 55+ of smartphone users approximately follow a normal distribution with approximate mean and standard deviation of $$36.9$$ years and $$13.9$$ years respectively.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Using the Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a7811e5normal11a","stepAnswer":["$$18.8$$"],"problemType":"TextBox","stepTitle":"Calculate the interquartile range(IQR)","stepBody":"Round answers to one decimal place.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$18.8$$","hints":{"DefaultPathway":[{"id":"a7811e5normal11a-h1","type":"hint","dependencies":[],"title":"$$IQR=Q_3-Q_1$$","text":"Calculate $$Q_3=75th$$ percentile and $$Q_1=25th$$ percentile.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$46.2754$$"],"dependencies":["a7811e5normal11a-h1"],"title":"Calculate $$Q_3$$","text":"$$invNorm(0.75, 36.9, 13.9)$$. Round answers to four decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal11a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$27.5246$$"],"dependencies":["a7811e5normal11a-h1"],"title":"Calculate $$Q_1$$","text":"$$invNorm(0.25, 36.9, 13.9)$$. Round answers to four decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18.8$$"],"dependencies":["a7811e5normal11a-h1"],"title":"Calculate IQR","text":"$$IQR=Q_3-Q_1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7811e5normal12","title":"Smart Users","body":"In the United States the ages $$13$$ to 55+ of smartphone users approximately follow a normal distribution with approximate mean and standard deviation of $$36.9$$ years and $$13.9$$ years respectively.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Using the Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a7811e5normal12a","stepAnswer":["$$40.4$$"],"problemType":"TextBox","stepTitle":"Forty percent of the smartphone users from $$13$$ to 55+ are at least what age?","stepBody":"Round answers to one decimal place.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$40.4$$","hints":{"DefaultPathway":[{"id":"a7811e5normal12a-h1","type":"hint","dependencies":[],"title":"Find k where $$P(x \\\\geq k)=0.40$$","text":"At least translates to greater than or equal to","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal12a-h2","type":"hint","dependencies":[],"title":"Find the area to the left and right?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal12a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.4$$"],"dependencies":["a7811e5normal12a-h2"],"title":"Find the area to the right?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.6$$"],"dependencies":["a7811e5normal12a-h2"],"title":"Find the area to the left?","text":"$$1-0.4=0.6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal12a-h5","type":"hint","dependencies":["a7811e5normal12a-h2"],"title":"Using a Calculator (TI-83, 83+, $$84$$, 84+ Calculator)","text":"Using invNorm","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal12a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.4$$"],"dependencies":["a7811e5normal12a-h5"],"title":"$$invNorm(0.60, 36.9, 13.9)$$","text":"Round answers to one decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7811e5normal13","title":"A citrus farmer who grows mandarin oranges find that the diameters of mandarin oranges harvested on his farm follow a normal distribution with a mean diameter of $$5.85$$ cm and a standard deviation of $$0.24$$ cm.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Using the Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a7811e5normal13a","stepAnswer":["$$0.266$$"],"problemType":"TextBox","stepTitle":"Find the probability that a randomly selected mandarin orange from this farm has a diameter larger than $$6.0$$ cm.","stepBody":"Round answers to four decimal place.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.266$$","hints":{"DefaultPathway":[{"id":"a7811e5normal13a-h1","type":"hint","dependencies":[],"title":"Identify \u03bc and \u03c3","text":"Let $$X=diameter$$ of randomly selected mandarin oranges. What is \u03bc and \u03c3 in $$X~N(\u03bc,\u03c3)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5.85$$"],"dependencies":["a7811e5normal13a-h1"],"title":"Identify \u03bc","text":"What is the mean \u03bc in here?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal13a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.24$$"],"dependencies":["a7811e5normal13a-h1"],"title":"Identify \u03c3","text":"What is the standard deviation \u03c3?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal13a-h4","type":"hint","dependencies":["a7811e5normal13a-h1"],"title":"Using a Calculator (TI-83, 83+, $$84$$, 84+ Calculator)","text":"We know X~N(5.85,0.24). Compute $$P\\\\left(x>6\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal13a-h5","type":"hint","dependencies":["a7811e5normal13a-h4"],"title":"Compute the area probability between $$x>6$$. $$P\\\\left(x>6\\\\right)$$.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal13a-h6","type":"hint","dependencies":["a7811e5normal13a-h5"],"title":"Upper value","text":"Use 1E99 (pressing $$1$$, the EE key (a 2nd key) and then 99) or enter $${10}^{99}$$ instead.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal13a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.266$$"],"dependencies":["a7811e5normal13a-h6"],"title":"normalcdf(6,10**99,5.85,0.24)","text":"Round to 4th digits after decimal.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7811e5normal14","title":"A citrus farmer who grows mandarin oranges find that the diameters of mandarin oranges harvested on his farm follow a normal distribution with a mean diameter of $$5.85$$ cm and a standard deviation of $$0.24$$ cm.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Using the Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a7811e5normal14a","stepAnswer":["$$5.79$$ $$5.91$$"],"problemType":"TextBox","stepTitle":"The middle 20% of mandarin oranges from this farm have diameters between $$___$$ and $$___$$ .","stepBody":"Separated by space.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5.79$$ $$5.91$$","hints":{"DefaultPathway":[{"id":"a7811e5normal14a-h1","type":"hint","dependencies":[],"title":"What correspond to the middle 20%?","text":"Find k1, the 40th percentile, and k2, the 60th percentile.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal14a-h2","type":"hint","dependencies":[],"title":"Find k1.","text":"Calculate the 40th percentile.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7811e5normal15","title":"A citrus farmer who grows mandarin oranges find that the diameters of mandarin oranges harvested on his farm follow a normal distribution with a mean diameter of $$5.85$$ cm and a standard deviation of $$0.24$$ cm.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Using the Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a7811e5normal15a","stepAnswer":["$$6.16$$"],"problemType":"TextBox","stepTitle":"Find the 90th percentile for the diameters of mandarin oranges, and interpret it in a complete sentence.","stepBody":"Round answers to two decimal 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Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a7811e5normal2a","stepAnswer":["$$0.3446$$"],"problemType":"TextBox","stepTitle":"Find the probability that a randomly selected student scored more than $$65$$ on the exam.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.3446$$","hints":{"DefaultPathway":[{"id":"a7811e5normal2a-h1","type":"hint","dependencies":[],"title":"Identify \u03bc and \u03c3","text":"We know the final exam score is normally distributed, so we let $$X=a$$ score on the final exam. X follows $$X~N(\u03bc,\u03c3)$$. What is \u03bc and \u03c3 in this context?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$63$$"],"dependencies":["a7811e5normal2a-h1"],"title":"Identify \u03bc","text":"What is \u03bc mean in here?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a7811e5normal2a-h1"],"title":"Identify \u03c3","text":"What is \u03c3 standard deviation in here?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal2a-h4","type":"hint","dependencies":["a7811e5normal2a-h1"],"title":"Calculating the Z-score","text":"What is the $$z-score$$? Utilize the formula (x-mean)/(standard deviation)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal2a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.4$$"],"dependencies":["a7811e5normal2a-h4"],"title":"Z-score","text":"$$\\\\frac{65-63}{5}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal2a-h6","type":"hint","dependencies":["a7811e5normal2a-h4"],"title":"Using a Calculator to compute the area to the left (TI-83, 83+, $$84$$, 84+ Calculator)","text":"Go into 2nd DISTR. --\x3e 2:normalcdf. Follow the syntax normalcdf(lower value, upper value, mean, standard deviation)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal2a-h7","type":"hint","dependencies":["a7811e5normal2a-h4"],"title":"Upper value","text":"Use 1E99 (pressing $$1$$, the EE key (a 2nd key) and then 99) or enter $${10}^{99}$$ instead.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal2a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.3446$$"],"dependencies":["a7811e5normal2a-h7"],"title":"Area to the right","text":"Known the area to the left is $$0.6554$$, what is the area to the right?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7811e5normal3","title":"Final Exam Scores","body":"The final exam scores in statistics class were normally distributed with a mean of $$63$$ and a standard deviation of five.","variabilization":{},"oer":"https://openstax.org","license":0,"lesson":"6.2 Using the Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a7811e5normal3a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"Find the probability that a randomly selected student scored less than $$85$$. Round to the nearest integer.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a7811e5normal3a-h1","type":"hint","dependencies":[],"title":"Area to the left","text":"What is $$P\\\\left(x<85\\\\right)$$?","variabilization":{},"oer":"","license":""},{"id":"a7811e5normal3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a7811e5normal3a-h1"],"title":"Using a Calculator to compute the area to the left (TI-83, 83+, $$84$$, 84+ Calculator)","text":"Compute the normalcdf(0,85,63,5). Round Up.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a7811e5normal4","title":"Final Exam Scores","body":"The final exam scores in statistics class were normally distributed with a mean of $$63$$ and a standard deviation of five.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Using the Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a7811e5normal4a","stepAnswer":["$$69.4$$"],"problemType":"TextBox","stepTitle":"Find the 90th percentile.","stepBody":"Round answers to one decimal place.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$69.4$$","hints":{"DefaultPathway":[{"id":"a7811e5normal4a-h1","type":"hint","dependencies":[],"title":"Draw a graph and shade the area that corresponds to the 90th percentile.","text":"The 90th percentile corresponds to score k that has 90% of the scores below k and 10% of the scores above k.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal4a-h2","type":"hint","dependencies":[],"title":"Using a Calculator to compute k.","text":"Using invNorm","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$69.4$$"],"dependencies":["a7811e5normal4a-h2"],"title":"Follow the syntax invNorm(area to the left, mean, standard deviation)","text":"click invNorm in 2nd DISTR -- $$invNorm(0.90, 63, 5)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7811e5normal5","title":"Final Exam Scores","body":"The final exam scores in statistics class were normally distributed with a mean of $$63$$ and a standard deviation of five.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Using the Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a7811e5normal5a","stepAnswer":["$$65.5$$"],"problemType":"TextBox","stepTitle":"Find the 70th percentile.","stepBody":"Round answers to one decimal place.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$65.5$$","hints":{"DefaultPathway":[{"id":"a7811e5normal5a-h1","type":"hint","dependencies":[],"title":"What is 70th percentile?","text":"Find the 70th percentile means to find the score k such that 70% of scores are below k and 30% of the scores are above k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal5a-h2","type":"hint","dependencies":[],"title":"Probability area under the curve","text":"Let $$k=the$$ 70th percentile. What is $$P\\\\left(x<k\\\\right)$$ equal to? Consider the Total area under the normal distributed curve is $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.7$$"],"dependencies":["a7811e5normal5a-h2"],"title":"Consider the Total area under the normal distributed curve is $$1$$.","text":"k is the split point at x-axis","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal5a-h4","type":"hint","dependencies":["a7811e5normal5a-h2"],"title":"Using a Calculator (TI-83, 83+, $$84$$, 84+ Calculator)","text":"Click invNorm in 2nd DISTR.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$65.6$$"],"dependencies":["a7811e5normal5a-h4"],"title":"Follow the syntax invNorm(area to the left, mean, standard deviation)","text":"$$invNorm(0.70, 63, 5)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7811e5normal6","title":"Computer For Entertainment","body":"A personal computer is used for office work at home, research, communication, personal finances, education, entertainment, social networking, and a myrid of other things. Suppose that the average number of hours a household personal computer is used for entertainment is two hours per day. Assume the times for entertainment are normally distributed and the standard deviation for the times is half an hour.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Using the Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a7811e5normal6a","stepAnswer":["$$0.5886$$"],"problemType":"TextBox","stepTitle":"Find the probability that a household personal computer is used for entertainment between $$1.8$$ and $$2.75$$ hours per day.","stepBody":"Round answers to four decimal place.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.5886$$","hints":{"DefaultPathway":[{"id":"a7811e5normal6a-h1","type":"hint","dependencies":[],"title":"Identify \u03bc and \u03c3","text":"Let $$X=amount$$ of time (in hours) a household personal computer is used for entertainment. What is \u03bc and \u03c3 in $$X~N(\u03bc,\u03c3)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a7811e5normal6a-h1"],"title":"Identify \u03bc","text":"What is the mean \u03bc?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.5$$"],"dependencies":["a7811e5normal6a-h1"],"title":"Identify \u03c3","text":"What is the standard deviation \u03c3 in decimal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal6a-h4","type":"hint","dependencies":["a7811e5normal6a-h1"],"title":"Using a Calculator (TI-83, 83+, $$84$$, 84+ Calculator)","text":"We know X~N(2,0.5). Compute P(1.8<x <2.75).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal6a-h5","type":"hint","dependencies":["a7811e5normal6a-h4"],"title":"Compute the area probability between $$x=1.8$$ and $$x=2.75$$. $$P\\\\left(1.8<x<2.75\\\\right)$$.","text":"\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal6a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.5886$$"],"dependencies":["a7811e5normal6a-h5"],"title":"$$normalcdf(1.8, 2.75, 2, 0.5)$$","text":"Round answers to four decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7811e5normal7","title":"Computer For Entertainment","body":"A personal computer is used for office work at home, research, communication, personal finances, education, entertainment, social networking, and a myrid of other things. Suppose that the average number of hours a household personal computer is used for entertainment is two hours per day. Assume the times for entertainment are normally distributed and the standard deviation for the times is half an hour.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Using the Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a7811e5normal7a","stepAnswer":["$$1.66$$"],"problemType":"TextBox","stepTitle":"Find the maximum number of hours per day that the buttom quartile of households uses a personal computer for entertainment.","stepBody":"Round answers to two decimal place.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1.66$$","hints":{"DefaultPathway":[{"id":"a7811e5normal7a-h1","type":"hint","dependencies":[],"title":"What does it mean to find the bottom quartile?","text":"Find the 25th quartile, k, where $$P\\\\left(x<k\\\\right)=0.25$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal7a-h2","type":"hint","dependencies":[],"title":"Using a Calculator (TI-83, 83+, $$84$$, 84+ Calculator)","text":"Using invNorm","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.66$$"],"dependencies":["a7811e5normal7a-h2"],"title":"$$invNorm(0.25, 2, 0.5)$$","text":"\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7811e5normal8","title":"Smart Users","body":"In the United States the ages $$13$$ to 55+ of smartphone users approximately follow a normal distribution with approximate mean and standard deviation of $$36.9$$ years and $$13.9$$ years, respectively.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Using the Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a7811e5normal8a","stepAnswer":["$$0.8186$$"],"problemType":"TextBox","stepTitle":"Determine the probability that a random smartphone user in the age range $$13$$ to 55+ is between $$23$$ and $$64.7$$ years old.","stepBody":"Round answers to four decimal place.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.8186$$","hints":{"DefaultPathway":[{"id":"a7811e5normal8a-h1","type":"hint","dependencies":[],"title":"Identify \u03bc and \u03c3","text":"Let $$X=age$$ of smart phone users. What is \u03bc and \u03c3 in $$X~N(\u03bc,\u03c3)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$36.9$$"],"dependencies":["a7811e5normal8a-h1"],"title":"Identify \u03bc","text":"What is the mean \u03bc in here?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$13.9$$"],"dependencies":["a7811e5normal8a-h1"],"title":"Identify \u03c3","text":"What is the standard deviation \u03c3?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal8a-h4","type":"hint","dependencies":["a7811e5normal8a-h1"],"title":"Using a Calculator (TI-83, 83+, $$84$$, 84+ Calculator)","text":"We know X~N(36.9,13.9). Compute $$P\\\\left(23<x<64.7\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal8a-h5","type":"hint","dependencies":["a7811e5normal8a-h4"],"title":"Compute the area probability between $$x=23$$ and $$x=64.7$$. $$P\\\\left(23<x<64.7\\\\right)$$.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal8a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.8186$$"],"dependencies":["a7811e5normal8a-h5"],"title":"$$normalcdf(23, 64, 36.9, 13.9)$$","text":"Round to 4th digits after decimal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7811e5normal9","title":"Smart Users","body":"In the United States the ages $$13$$ to 55+ of smartphone users approximately follow a normal distribution with approximate mean and standard deviation of $$36.9$$ years and $$13.9$$ years, respectively.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Using the Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a7811e5normal9a","stepAnswer":["$$0.8413$$"],"problemType":"TextBox","stepTitle":"Determine the probability that a randomly selected smartphone user in the age range $$13$$ to 55+ is at most $$50.8$$ years old.","stepBody":"Round answers to four decimal place.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.8413$$","hints":{"DefaultPathway":[{"id":"a7811e5normal9a-h1","type":"hint","dependencies":[],"title":"Identify \u03bc and \u03c3","text":"Let $$X=age$$ of smart phone users. What is \u03bc and \u03c3 in $$X~N(\u03bc,\u03c3)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$36.9$$"],"dependencies":["a7811e5normal9a-h1"],"title":"Identify \u03bc","text":"What is the mean \u03bc in here?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$13.9$$"],"dependencies":["a7811e5normal9a-h1"],"title":"Identify \u03c3","text":"What is the standard deviation \u03c3?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal9a-h4","type":"hint","dependencies":["a7811e5normal9a-h1"],"title":"Using a Calculator (TI-83, 83+, $$84$$, 84+ Calculator)","text":"We know X~N(36.9,13.9). Compute $$P(x \\\\leq 50.8)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal9a-h5","type":"hint","dependencies":["a7811e5normal9a-h4"],"title":"Compute the area probability between $$x=50.8$$. $$P\\\\left(x<50.8\\\\right)$$.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal9a-h6","type":"hint","dependencies":["a7811e5normal9a-h5"],"title":"Lower Bound","text":"To enter negative $$infinitity$$, use -1E99 (pressing $$-1$$, the EE key (a 2nd key) and then 99) or enter $$-\\\\left({10}^{99}\\\\right)$$ instead.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7811e5normal9a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.8413$$"],"dependencies":["a7811e5normal9a-h6"],"title":"normalcdf(-10**99,50.8,36.9,13.9)","text":"Round answers to four decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a78ba6esubsitution1","title":"Solving Systems of Equations Using Subsitution","body":"Use subsitution to solve for $$x$$ and $$y$$ in the given system of equations. Choose the correct answer in the form of an ordered pair.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.2 Solving Systems of Equations by Substitution","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a78ba6esubsitution1a","stepAnswer":["$$(4,-1)$$"],"problemType":"MultipleChoice","stepTitle":"$$2x+y=7$$, $$x-2y=6$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(4,-1)$$","choices":["$$(4,-1)$$","$$(3,-2)$$","$$(4,0)$$"],"hints":{"DefaultPathway":[{"id":"a78ba6esubsitution1a-h1","type":"hint","dependencies":[],"title":"Solving for Y in the First Equation","text":"First, solve the first equation for $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a78ba6esubsitution1a-h2","type":"hint","dependencies":["a78ba6esubsitution1a-h1"],"title":"Subsituting in Y","text":"Replace $$y$$ in the second equation with the value solved from the first equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a78ba6esubsitution1a-h3","type":"hint","dependencies":["a78ba6esubsitution1a-h2"],"title":"Solving a One-Variable Equation","text":"After subsituting in y\'s value from the first equation, the second equation now has only one variable, $$x$$. Solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a78ba6esubsitution1a-h4","type":"hint","dependencies":["a78ba6esubsitution1a-h3"],"title":"Using One Variable to Solve for Another","text":"Using the numerical value of $$x$$, plug $$x$$ back into the original first equation to solve for the numerical value of $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a78ba6esubsitution1a-h5","type":"hint","dependencies":["a78ba6esubsitution1a-h4"],"title":"Checking Values","text":"Subsitute the values back into the original equations to make sure they are true.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a78ba6esubsitution10","title":"Solving Systems of Equations Using Subsitution","body":"Use subsitution to solve for $$x$$ and $$y$$ in the given system of equations. 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Choose the correct answer in the form of an ordered pair.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.2 Solving Systems of Equations by Substitution","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a78ba6esubsitution4a","stepAnswer":["$$(4,2)$$"],"problemType":"MultipleChoice","stepTitle":"$$x+3y=10$$, $$4x+y=18$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(4,2)$$","choices":["$$(4,2)$$","$$(3,1)$$","$$(-4,5)$$"],"hints":{"DefaultPathway":[{"id":"a78ba6esubsitution4a-h1","type":"hint","dependencies":[],"title":"Step 1: Solving for One Variable","text":"Solve one of the equations for either variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a78ba6esubsitution4a-h2","type":"hint","dependencies":["a78ba6esubsitution4a-h1"],"title":"Step 2: Using Subsitution to Create a One-Variable Equation","text":"Subsitute the expression from Step $$1$$ into the other equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a78ba6esubsitution4a-h3","type":"hint","dependencies":["a78ba6esubsitution4a-h2"],"title":"Step 3: Solving a One-Variable Equation","text":"Solve the resulting one-variable equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a78ba6esubsitution4a-h4","type":"hint","dependencies":["a78ba6esubsitution4a-h3"],"title":"Step 4: Solving for the Numerical Value of the Other Variable","text":"Subsitute the solution from Step $$3$$ into the other equation to find the value of the other variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a78ba6esubsitution4a-h5","type":"hint","dependencies":["a78ba6esubsitution4a-h4"],"title":"Step 5: Checking Values","text":"Check that the ordered pair is a solution to both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a78ba6esubsitution5","title":"Solving Systems of Equations Using Subsitution","body":"Use subsitution to solve for $$x$$ and $$y$$ in the given system of equations. Choose the correct answer in the form of an ordered pair.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.2 Solving Systems of Equations by Substitution","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a78ba6esubsitution5a","stepAnswer":["$$(-1,-3)$$"],"problemType":"MultipleChoice","stepTitle":"$$2x-y=1$$, $$y=-3x-6$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-1,-3)$$","choices":["$$(5,-3)$$","$$(2,3)$$","$$(-1,-3)$$"],"hints":{"DefaultPathway":[{"id":"a78ba6esubsitution5a-h1","type":"hint","dependencies":[],"title":"Step 1: Solving for One Variable","text":"Solve one of the equations for either variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a78ba6esubsitution5a-h2","type":"hint","dependencies":["a78ba6esubsitution5a-h1"],"title":"Step 2: Using Subsitution to Create a One-Variable Equation","text":"Subsitute the expression from Step $$1$$ into the other equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a78ba6esubsitution5a-h3","type":"hint","dependencies":["a78ba6esubsitution5a-h2"],"title":"Step 3: Solving a One-Variable Equation","text":"Solve the resulting one-variable equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a78ba6esubsitution5a-h4","type":"hint","dependencies":["a78ba6esubsitution5a-h3"],"title":"Step 4: Solving for the Numerical Value of the Other Variable","text":"Subsitute the solution from Step $$3$$ into the other equation to find the value of the other variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a78ba6esubsitution5a-h5","type":"hint","dependencies":["a78ba6esubsitution5a-h4"],"title":"Step 5: Checking Values","text":"Check that the ordered pair is a solution to both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a78ba6esubsitution6","title":"Solving Systems of Equations Using Subsitution","body":"Use subsitution to solve for $$x$$ and $$y$$ in the given system of equations. Choose the correct answer in the form of an ordered pair.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.2 Solving Systems of Equations by Substitution","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a78ba6esubsitution6a","stepAnswer":["$$(3,-4)$$"],"problemType":"MultipleChoice","stepTitle":"$$3x+y=5$$, $$2x+4y=-10$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(3,-4)$$","choices":["$$(1,4)$$","$$(2,-2)$$","$$(3,-4)$$"],"hints":{"DefaultPathway":[{"id":"a78ba6esubsitution6a-h1","type":"hint","dependencies":[],"title":"Step 1: Solving for One Variable","text":"Solve one of the equations for either variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a78ba6esubsitution6a-h2","type":"hint","dependencies":["a78ba6esubsitution6a-h1"],"title":"Step 2: Using Subsitution to Create a One-Variable Equation","text":"Subsitute the expression from Step $$1$$ into the other equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a78ba6esubsitution6a-h3","type":"hint","dependencies":["a78ba6esubsitution6a-h2"],"title":"Step 3: Solving a One-Variable Equation","text":"Solve the resulting one-variable equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a78ba6esubsitution6a-h4","type":"hint","dependencies":["a78ba6esubsitution6a-h3"],"title":"Step 4: Solving for the Numerical Value of the Other Variable","text":"Subsitute the solution from Step $$3$$ into the other equation to find the value of the other variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a78ba6esubsitution6a-h5","type":"hint","dependencies":["a78ba6esubsitution6a-h4"],"title":"Step 5: Checking Values","text":"Check that the ordered pair is a solution to both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a78ba6esubsitution6b","stepAnswer":["$$(1,-2)$$"],"problemType":"MultipleChoice","stepTitle":"$$4x+y=2$$, $$3x+2y=-1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(1,-2)$$","choices":["$$(1,-3)$$","$$(1,-1)$$","$$(1,-2)$$","$$(1,0)$$"],"hints":{"DefaultPathway":[{"id":"a78ba6esubsitution6b-h1","type":"hint","dependencies":[],"title":"Step 1: Solving for One Variable","text":"Solve one of the equations for either variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a78ba6esubsitution6b-h2","type":"hint","dependencies":["a78ba6esubsitution6b-h1"],"title":"Step 2: Using Subsitution to Create a One-Variable Equation","text":"Subsitute the expression from Step $$1$$ into the other equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a78ba6esubsitution6b-h3","type":"hint","dependencies":["a78ba6esubsitution6b-h2"],"title":"Step 3: Solving a One-Variable Equation","text":"Solve the resulting one-variable equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a78ba6esubsitution6b-h4","type":"hint","dependencies":["a78ba6esubsitution6b-h3"],"title":"Step 4: Solving for the Numerical Value of the Other Variable","text":"Subsitute the solution from Step $$3$$ into the other equation to find the value of the other variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a78ba6esubsitution6b-h5","type":"hint","dependencies":["a78ba6esubsitution6b-h4"],"title":"Step 5: Checking Values","text":"Check that the ordered pair is a solution to both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a78ba6esubsitution7","title":"Solving Systems of Equations Using Subsitution","body":"Use subsitution to solve for $$x$$ and $$y$$ in the given system of equations. Choose the correct answer in the form of an ordered pair.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.2 Solving Systems of Equations by Substitution","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a78ba6esubsitution7a","stepAnswer":["$$(2,6)$$"],"problemType":"MultipleChoice","stepTitle":"$$-x+y=4$$, $$4x-y=2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(2,6)$$","choices":["$$(1,5)$$","$$(2,6)$$","$$(3,7)$$"],"hints":{"DefaultPathway":[{"id":"a78ba6esubsitution7a-h1","type":"hint","dependencies":[],"title":"Step 1: Solving for One Variable","text":"Solve one of the equations for either variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a78ba6esubsitution7a-h2","type":"hint","dependencies":["a78ba6esubsitution7a-h1"],"title":"Step 2: Using Subsitution to Create a One-Variable Equation","text":"Subsitute the expression from Step $$1$$ into the other equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a78ba6esubsitution7a-h3","type":"hint","dependencies":["a78ba6esubsitution7a-h2"],"title":"Step 3: Solving a One-Variable Equation","text":"Solve the resulting one-variable equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a78ba6esubsitution7a-h4","type":"hint","dependencies":["a78ba6esubsitution7a-h3"],"title":"Step 4: Solving for the Numerical Value of the Other Variable","text":"Subsitute the solution from Step $$3$$ into the other equation to find the value of the other variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a78ba6esubsitution7a-h5","type":"hint","dependencies":["a78ba6esubsitution7a-h4"],"title":"Step 5: Checking Values","text":"Check that the ordered pair is a solution to both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a78ba6esubsitution8","title":"Solving Systems of Equations Using Subsitution","body":"Use subsitution to solve for $$x$$ and $$y$$ in the given system of equations. Choose the correct answer in the form of an ordered pair.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.2 Solving Systems of Equations by Substitution","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a78ba6esubsitution8a","stepAnswer":["$$(8,5)$$"],"problemType":"MultipleChoice","stepTitle":"$$x-2y=-2$$, $$3x+2y=34$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(8,5)$$","choices":["$$(8,5)$$","$$(4,2)$$","$$(9,3)$$"],"hints":{"DefaultPathway":[{"id":"a78ba6esubsitution8a-h1","type":"hint","dependencies":[],"title":"Step 1: Solving for One Variable","text":"Solve one of the equations for either variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a78ba6esubsitution8a-h2","type":"hint","dependencies":["a78ba6esubsitution8a-h1"],"title":"Step 2: Using Subsitution to Create a One-Variable Equation","text":"Subsitute the expression from Step $$1$$ into the other equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a78ba6esubsitution8a-h3","type":"hint","dependencies":["a78ba6esubsitution8a-h2"],"title":"Step 3: Solving a One-Variable Equation","text":"Solve the resulting one-variable equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a78ba6esubsitution8a-h4","type":"hint","dependencies":["a78ba6esubsitution8a-h3"],"title":"Step 4: Solving for the Numerical Value of the Other Variable","text":"Subsitute the solution from Step $$3$$ into the other equation to find the value of the other variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a78ba6esubsitution8a-h5","type":"hint","dependencies":["a78ba6esubsitution8a-h4"],"title":"Step 5: Checking Values","text":"Check that the ordered pair is a solution to both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a78ba6esubsitution9","title":"Solving Systems of Equations Using Subsitution","body":"Use subsitution to solve for $$x$$ and $$y$$ in the given system of equations. Choose the correct answer in the form of an ordered pair.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.2 Solving Systems of Equations by Substitution","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a78ba6esubsitution9a","stepAnswer":["$$(-2,-3)$$"],"problemType":"MultipleChoice","stepTitle":"$$x-5y=13$$, $$4x-3y=1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-2,-3)$$","choices":["$$(-2,3)$$","$$(1,-3)$$","$$(-2,-3)$$"],"hints":{"DefaultPathway":[{"id":"a78ba6esubsitution9a-h1","type":"hint","dependencies":[],"title":"Step 1: Solving for One Variable","text":"Solve one of the equations for either variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a78ba6esubsitution9a-h2","type":"hint","dependencies":["a78ba6esubsitution9a-h1"],"title":"Step 2: Using Subsitution to Create a One-Variable Equation","text":"Subsitute the expression from Step $$1$$ into the other equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a78ba6esubsitution9a-h3","type":"hint","dependencies":["a78ba6esubsitution9a-h2"],"title":"Step 3: Solving a One-Variable Equation","text":"Solve the resulting one-variable equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a78ba6esubsitution9a-h4","type":"hint","dependencies":["a78ba6esubsitution9a-h3"],"title":"Step 4: Solving for the Numerical Value of the Other Variable","text":"Subsitute the solution from Step $$3$$ into the other equation to find the value of the other variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a78ba6esubsitution9a-h5","type":"hint","dependencies":["a78ba6esubsitution9a-h4"],"title":"Step 5: Checking Values","text":"Check that the ordered pair is a solution to both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a79a53dgood1","title":"Absenteeism","body":"Absenteeism of college students from math classes is a major concern to math instructors because missing class appears to increase the drop rate. Suppose that a study was done to determine if the actual student absenteeism rate follows faculty perception. The faculty expected that a group of $$100$$ students would miss class according to Table $$1$$. A random survey across all mathematics courses was then done to determine the actual number (observed) of absences in a course. The chart in Table $$2$$ displays the results of that survey.\\\\n##figure3.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Goodness-of-Fit Test","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a79a53dgood1a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"A random survey across all mathematics courses was then done to determine the actual number (observed) of absences in a course. The chart in Table $$2$$ displays the results of that survey. Determine the null and alternative hypotheses needed to conduct a goodness-of-fit test.\\\\nH0: Student absenteeism fits faculty perception.\\\\n\\\\nThe alternative hypothesis is the opposite of the null hypothesis.\\\\nHa: Student absenteeism does not fit faculty perception.","stepBody":"Can you use the information as it appears in the charts to conduct the goodness-of-fit test?##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a79a53dgood1a-h1","type":"hint","dependencies":[],"title":"Look at expected number of absences for the \\"12+\\" entry on both tables.","text":"Notice that the expected number of absences for the \\"12+\\" entry is less than five (it is two and four). The expected value for each cell needs to be at least five in order for you to use this test.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood1a-h2","type":"hint","dependencies":["a79a53dgood1a-h1"],"title":"Combine that group with the $$\\"9-11\\"$$ group to create new tables where the number of students for each entry are at least five.","text":"The new results are in Table $$11.3$$ and Table $$11.4$$.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a79a53dgood10","title":"College Majors of Graduating Females","body":"UCLA conducted a survey of more than 263,000 college freshmen from $$385$$ colleges in fall $$2005$$. The results of students\' expected majors by gender were reported in The Chronicle of Higher Education $$\\\\frac{\\\\frac{2}{2}}{2006}$$. Suppose a survey of 5,000 graduating females and 5,000 graduating males was done as a follow-up last year to determine what their actual majors were. The results are shown in the tables for Exercise $$11.77$$ and Exercise $$11.78$$. The second column in each table does not add to 100% because of rounding.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Goodness-of-Fit Test","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a79a53dgood10a","stepAnswer":["There is insufficient evidence to conclude that the distribution of actual college majors of graduating females do not fit the distribution of their expected majors."],"problemType":"MultipleChoice","stepTitle":"Conduct a goodness-of-fit test to determine if the actual college majors of graduating females fit the distribution of their expected majors.","stepBody":"Choose the answer from the following.##figure1.gif## ","answerType":"string","variabilization":{},"choices":["There is insufficient evidence to conclude that the distribution of actual college majors of graduating females do not fit the distribution of their expected majors.","There is sufficient evidence to conclude that the distribution of actual college majors of graduating females do not fit the distribution of their expected majors.","There is insufficient evidence to conclude that the distribution of actual college majors of graduating females do fit the distribution of their expected majors.","There is sufficient evidence to conclude that the distribution of actual college majors of graduating females do fit the distribution of their expected majors."],"hints":{"DefaultPathway":[{"id":"a79a53dgood10a-h1","type":"hint","dependencies":[],"title":"Hypotheses","text":"What are the Null Hypothesis $$H_0$$ and Alternative Hypothesis in this context?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood10a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$H_0$$: The actual college majors of graduating females fit the distribution of their expected majors"],"dependencies":["a79a53dgood10a-h1"],"title":"Null Hypothesis","text":"Choose the answer from the following.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$H_0$$: The actual college majors of graduating females fit the distribution of their expected majors","$$H_0$$: The actual college majors of graduating females do not fit the distribution of their expected majors"]},{"id":"a79a53dgood10a-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$H_a$$: The actual college majors of graduating females do not fit the distribution of their expected majors"],"dependencies":[],"title":"Alternative Hypothesis","text":"Choose the answer from the following.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$H_a$$: The actual college majors of graduating females fit the distribution of their expected majors","$$H_a$$: The actual college majors of graduating females do not fit the distribution of their expected majors"]},{"id":"a79a53dgood10a-h3","type":"hint","dependencies":["a79a53dgood10a-h2"],"title":"Consider what distribution is most appropriate to use in this problem.","text":"Is this a Goodness-of-Fit problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a79a53dgood10a-h3"],"title":"What is the degree of freedom?","text":"$$df=11-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood10a-h5","type":"hint","dependencies":["a79a53dgood10a-h4"],"title":"Compute the chi-square test statistics","text":"$$X^2$$ $$=$$ $$11.48$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood10a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.3211$$"],"dependencies":["a79a53dgood10a-h5"],"title":"Compute the $$p-value$$.","text":"Round the answer to four decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood10a-h7","type":"hint","dependencies":["a79a53dgood10a-h6"],"title":"Compare with the significant level $$0.05$$.","text":"At 5% significance level, \\\\alpha $$=$$ $$0.05$$. For this data, P < \\\\alpha. Reject the null hypothesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood10a-h8","type":"hint","dependencies":["a79a53dgood10a-h7"],"title":"Interpret the Result","text":"At the 5% level of significance, from the data, there is insufficient evidence to conclude that the distribution of actual college majors of graduating females do not fit the distribution of their expected majors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a79a53dgood11","title":"Marital Status","body":"The marital status distribution of the U.S. male population, ages $$15$$ and older, is as shown in table.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Goodness-of-Fit Test","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a79a53dgood11a","stepAnswer":["Data does not fit the distribution."],"problemType":"MultipleChoice","stepTitle":"Conduct a goodness-of-fit test","stepBody":"Choose the answer from the following.##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Data does not fit the distribution.","Data does fit the distribution."],"hints":{"DefaultPathway":[{"id":"a79a53dgood11a-h1","type":"hint","dependencies":[],"title":"Hypotheses","text":"What are the Null Hypothesis $$H_0$$ and Alternative Hypothesis in this context?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood11a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$H_0$$: Data does fit the distribution."],"dependencies":["a79a53dgood11a-h1"],"title":"Null Hypothesis","text":"Choose the answer from the following.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$H_0$$: Data does not fit the distribution.","$$H_0$$: Data does fit the distribution."]},{"id":"a79a53dgood11a-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$H_a$$: Data does not fit the distribution."],"dependencies":[],"title":"Alternative Hypothesis","text":"Choose the answer from the following.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$H_a$$: Data does not fit the distribution.","$$H_a$$: Data does fit the distribution."]},{"id":"a79a53dgood11a-h3","type":"hint","dependencies":["a79a53dgood11a-h2"],"title":"Consider what distribution is most appropriate to use in this problem.","text":"Is this a Goodness-of-Fit problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a79a53dgood11a-h3"],"title":"What is the degree of freedom?","text":"$$df=4-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood11a-h5","type":"hint","dependencies":["a79a53dgood11a-h4"],"title":"Compute the chi-square test statistics","text":"$$X^2$$ $$=$$ $$19.27$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood11a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.0002$$"],"dependencies":["a79a53dgood11a-h5"],"title":"Compute the $$p-value$$.","text":"Round the answer to four decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood11a-h7","type":"hint","dependencies":["a79a53dgood11a-h6"],"title":"Compare with the significant level $$0.05$$.","text":"At 5% significance level, \\\\alpha $$=$$ $$0.05$$. For this data, P < \\\\alpha. Reject the null hypothesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood11a-h8","type":"hint","dependencies":["a79a53dgood11a-h7"],"title":"Interpret the Result","text":"At the 5% level of significance, from the data, it does not fit the distribution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a79a53dgood12","title":"Coin Flip","body":"Suppose you flip two coins $$100$$ times. The results are $$20$$ HH, $$27$$ HT, $$30$$ TH, and $$23$$ TT.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Goodness-of-Fit Test","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a79a53dgood12a","stepAnswer":["There is insufficient evidence to conclude that the coins are not fair."],"problemType":"MultipleChoice","stepTitle":"Are the coins fair?","stepBody":"Test at a 5% significance level.","answerType":"string","variabilization":{},"choices":["There is insufficient evidence to conclude that the coins are not fair.","There is sufficient evidence to conclude that the coins are not fair.","There is insufficient evidence to conclude that the coins are fair.","There is sufficient evidence to conclude that the coins are fair."],"hints":{"DefaultPathway":[{"id":"a79a53dgood12a-h1","type":"hint","dependencies":[],"title":"Hypotheses","text":"What are the Null Hypothesis $$H_0$$ and Alternative Hypothesis in this context?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood12a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$H_0$$: The coins are fair."],"dependencies":["a79a53dgood12a-h1"],"title":"Null Hypothesis","text":"Choose the answer from the following.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$H_0$$: The coins are fair.","$$H_0$$: The data does not fit the distribution."]},{"id":"a79a53dgood12a-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$H_a$$: The data does not fit the distribution."],"dependencies":[],"title":"Alternative Hypothesis","text":"Choose the answer from the following.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$H_a$$: The coins are fair.","$$H_a$$: The data does not fit the distribution."]},{"id":"a79a53dgood12a-h3","type":"hint","dependencies":["a79a53dgood12a-h2"],"title":"Consider what distribution is most appropriate to use in this problem.","text":"Is this a Goodness-of-Fit problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a79a53dgood12a-h3"],"title":"What is the degree of freedom?","text":"$$df=3-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood12a-h5","type":"hint","dependencies":["a79a53dgood12a-h4"],"title":"Compute the chi-square test statistics","text":"$$X^2$$ $$=$$ $$2.14$$\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood12a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.343$$"],"dependencies":["a79a53dgood12a-h5"],"title":"Compute the $$p-value$$, P(X**2 > $$2.14)$$","text":"Round the answer to three decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood12a-h7","type":"hint","dependencies":["a79a53dgood12a-h6"],"title":"Compare with the significant level $$0.05$$.","text":"At 5% significance level, \\\\alpha $$=$$ $$0.05$$. For this data, \\\\alpha < P. Do not reject the null hypothesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood12a-h8","type":"hint","dependencies":["a79a53dgood12a-h7"],"title":"Interpret the Result","text":"At the 5% level of significance, from the data, there is insufficient evidence to conclude that the coins are not fair.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a79a53dgood13","title":"Absences","body":"Employers want to know which days of the week employees are absent in a five-day work week. Most employers would like to believe that employees are absent equally during the week. Suppose a random sample of $$60$$ managers were asked on which day of the week they had the highest number of employee absences.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Goodness-of-Fit Test","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a79a53dgood13a","stepAnswer":["There is not sufficient evidence to conclude that the absent days do not occur with equal frequencies."],"problemType":"MultipleChoice","stepTitle":"The results were distributed as in Table $$11.7$$. For the population of employees, do the days for the highest number of absences occur with equal frequencies during a five-day work week?","stepBody":"Test at a 5% significance level.##figure1.gif## ","answerType":"string","variabilization":{},"choices":["There is not sufficient evidence to conclude that the absent days do not occur with equal frequencies.","There is sufficient evidence to conclude that the absent days do not occur with equal frequencies.","there is not sufficient evidence to conclude that the absent days do occur with equal frequencies.","There is sufficient evidence to conclude that the absent days do occur with equal frequencies."],"hints":{"DefaultPathway":[{"id":"a79a53dgood13a-h1","type":"hint","dependencies":[],"title":"Hypotheses","text":"What are the Null Hypothesis $$H_0$$ and Alternative Hypothesis in this context?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood13a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$H_0$$: The absent days occur with equal frequencies, that is, they fit a uniform distribution."],"dependencies":["a79a53dgood13a-h1"],"title":"Null Hypothesis","text":"Choose the answer from the following.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$H_0$$: The absent days occur with equal frequencies, that is, they fit a uniform distribution.","$$H_0$$: The absent days occur with unequal frequencies, that is, they do not fit a uniform distribution."]},{"id":"a79a53dgood13a-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$H_a$$: The absent days occur with unequal frequencies, that is, they do not fit a uniform distribution."],"dependencies":[],"title":"Alternative Hypothesis","text":"Choose the answer from the following.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$H_a$$: The absent days occur with equal frequencies, that is, they fit a uniform distribution.","$$H_a$$: The absent days occur with unequal frequencies, that is, they do not fit a uniform distribution."]},{"id":"a79a53dgood13a-h3","type":"hint","dependencies":["a79a53dgood13a-h2"],"title":"Consider what distribution is most appropriate to use in this problem.","text":"Is this a Goodness-of-Fit problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a79a53dgood13a-h3"],"title":"What is the degree of freedom?","text":"$$df=5-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood13a-h5","type":"hint","dependencies":["a79a53dgood13a-h4"],"title":"Compute the chi-square test statistics","text":"$$X^2$$ $$=$$ $$3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood13a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.5578$$"],"dependencies":["a79a53dgood13a-h5"],"title":"Compute the $$p-value$$, P(X**2 > 3)","text":"Round the answer to three decimal places.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood13a-h7","type":"hint","dependencies":["a79a53dgood13a-h6"],"title":"Compare with the significant level $$0.05$$.","text":"At 5% significance level, \\\\alpha $$=$$ $$0.05$$. For this data, \\\\alpha < P. Do not reject the null hypothesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood13a-h8","type":"hint","dependencies":["a79a53dgood13a-h7"],"title":"Interpret the Result","text":"At a 5% level of significance, from the sample data, there is not sufficient evidence to conclude that the absent days do not occur with equal frequencies.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a79a53dgood14","title":"Televisions","body":"One study indicates that the number of televisions that American families have is distributed (this is the given distribution for the American population) as in Table 11.9.\\\\n##figure2.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Goodness-of-Fit Test","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a79a53dgood14a","stepAnswer":["At the 1% significance level, from the data, there is sufficient evidence to conclude that the \\"number of televisions\\" distribution for the far western United States is different from the \\"number of televisions\\" distribution for the American population as a whole."],"problemType":"MultipleChoice","stepTitle":"The table contains observed (O) frequency values.","stepBody":"At the 1% significance level, does it appear that the distribution \\"number of televisions\\" of far western United States families is different from the distribution for the American population as a whole?","answerType":"string","variabilization":{},"choices":["At the 1% significance level, from the data, there is sufficient evidence to conclude that the \\"number of televisions\\" distribution for the far western United States is different from the \\"number of televisions\\" distribution for the American population as a whole.","At the 1% significance level, from the data, there is insufficient evidence to conclude that the \\"number of televisions\\" distribution for the far western United States is different from the \\"number of televisions\\" distribution for the American population as a whole.\\\\n"],"hints":{"DefaultPathway":[{"id":"a79a53dgood14a-h1","type":"hint","dependencies":[],"title":"Hypotheses","text":"What are the Null Hypothesis $$H_0$$ and Alternative Hypothesis in this context?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood14a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$H_0$$: The \\"number of televisions\\" distribution of far western United States families is the same as the \\"number of televisions\\" distribution of the American population."],"dependencies":["a79a53dgood14a-h1"],"title":"Null Hypothesis","text":"Choose the answer from the following.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$H_0$$: The \\"number of televisions\\" distribution of far western United States families is the same as the \\"number of televisions\\" distribution of the American population.","$$H_0$$: The \\"number of televisions\\" distribution of far western United States families is different from the \\"number of televisions\\" distribution of the American population."]},{"id":"a79a53dgood14a-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Ha: The \\"number of televisions\\" distribution of far western United States families is different from the \\"number of televisions\\" distribution of the American population.\\\\n\\\\n"],"dependencies":[],"title":"Alternative Hypothesis","text":"Choose the answer from the following.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$H_a$$: The \\"number of televisions\\" distribution of far western United States families is the same as the \\"number of televisions\\" distribution of the American population.","$$H_a$$: The \\"number of televisions\\" distribution of far western United States families is different from the \\"number of televisions\\" distribution of the American population."]},{"id":"a79a53dgood14a-h3","type":"hint","dependencies":["a79a53dgood14a-h2"],"title":"Consider what distribution is most appropriate to use in this problem.","text":"Is this a Goodness-of-Fit problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood14a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a79a53dgood14a-h3"],"title":"What is the degree of freedom?","text":"$$df=5-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood14a-h5","type":"hint","dependencies":["a79a53dgood14a-h4"],"title":"Compute the chi-square test statistics","text":"$$X^2$$ $$=$$ $$29.65$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood14a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.000006$$"],"dependencies":["a79a53dgood14a-h5"],"title":"Compute the $$p-value$$, P(X**2 > $$29.65)$$","text":"Round the answer to six decimal places.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood14a-h7","type":"hint","dependencies":["a79a53dgood14a-h6"],"title":"Compare with the significant level $$0.05$$.","text":"At 5% significance level, \\\\alpha $$=$$ $$0.05$$. For this data, \\\\alpha < P. Do not reject the null hypothesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood14a-h8","type":"hint","dependencies":["a79a53dgood14a-h7"],"title":"Interpret the Result","text":"At the 1% significance level, from the data, there is sufficient evidence to conclude that the \\"number of televisions\\" distribution for the far western United States is different from the \\"number of televisions\\" distribution for the American population as a whole.\\\\n\\\\n","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a79a53dgood15","title":"Final Exam","body":"A teacher predicts that the distribution of grades on the final exam will be and they are recorded in Table $$11.27$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Goodness-of-Fit Test","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a79a53dgood15a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"DF","stepBody":"What is the number of degrees of freedom (df)?##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a79a53dgood15a-h1","type":"hint","dependencies":[],"title":"Tables","text":"Take a closer look at the two tables given in the problem.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a79a53dgood15a-h1"],"title":"Rows of the Tables","text":"How many \\"cells\\" or categories in each of the new tables?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood15a-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":[],"title":"Degree of Freedom","text":"df $$=$$ number of cells - $$1$$ $$=$$ $$4$$ - $$1$$ $$=$$ $$3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a79a53dgood2","title":"Absenteeism","body":"Absenteeism of college students from math classes is a major concern to math instructors because missing class appears to increase the drop rate. Suppose that a study was done to determine if the actual student absenteeism rate follows faculty perception. The faculty expected that a group of $$100$$ students would miss class according to Table $$1$$. A random survey across all mathematics courses was then done to determine the actual number (observed) of absences in a course. The chart in Table $$2$$ displays the results of that survey.\\\\n##figure2.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Goodness-of-Fit Test","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a79a53dgood2a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"A random survey across all mathematics courses was then done to determine the actual number (observed) of absences in a course. The chart in Table $$2$$ displays the results of that survey. Determine the null and alternative hypotheses needed to conduct a goodness-of-fit test.\\\\nH0: Student absenteeism fits faculty perception.\\\\n\\\\nThe alternative hypothesis is the opposite of the null hypothesis.\\\\nHa: Student absenteeism does not fit faculty perception.","stepBody":"What is the number of degrees of freedom (df)?##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a79a53dgood2a-h1","type":"hint","dependencies":[],"title":"Tables","text":"Take a closer look at the two tables given in the problem.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a79a53dgood2a-h1"],"title":"Rows of the Tables","text":"How many \\"cells\\" or categories in each of the new tables?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood2a-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":[],"title":"Degree of Freedom","text":"df $$=$$ number of cells - $$1$$ $$=$$ $$4$$ - $$1$$ $$=$$ $$3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a79a53dgood3","title":"Defective Products","body":"A factory manager needs to understand how many products are defective versus how many are produced. The number of expected defects is listed in Table $$11.5$$. A random sample was taken to determine the actual number of defects. Table $$11.6$$ shows the results of the survey.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org","license":0,"lesson":"11.2 Goodness-of-Fit Test","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a79a53dgood3a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"State the degrees of freedom.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a79a53dgood3a-h1","type":"hint","dependencies":[],"title":"Table","text":"Take a closer look at the table given in the problem.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a79a53dgood3a-h1"],"title":"Rows of the Table","text":"How many \\"cells\\" or categories in the table?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood3a-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a79a53dgood3a-h2"],"title":"Degree of Freedom","text":"df $$=$$ number of cells - $$1$$ $$=$$ $$4$$ - $$1$$ $$=$$ $$3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a79a53dgood4","title":"Adult Literacy Rate","body":"Students in a social studies class hypothesize that the literacy rates across the world for every region are 82%.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Goodness-of-Fit Test","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a79a53dgood4a","stepAnswer":["$$10$$"],"problemType":"TextBox","stepTitle":"Table $$11.14$$ shows the actual literacy rates across the world broken down by region.","stepBody":"What are the test statistic and the degrees of freedom?##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10$$","hints":{"DefaultPathway":[{"id":"a79a53dgood4a-h1","type":"hint","dependencies":[],"title":"Table","text":"Take a closer look at the table given in the problem.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a79a53dgood4a-h1"],"title":"Rows of the Table","text":"How many \\"cells\\" or categories in the table?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood4a-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a79a53dgood4a-h2"],"title":"Degree of Freedom","text":"df $$=$$ number of cells - $$1$$ $$=$$ $$4$$ - $$1$$ $$=$$ $$3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a79a53dgood5","title":"Goodness-of-Fit Test","body":"Read the statement and decide whether it is true or false.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Goodness-of-Fit Test","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a79a53dgood5a","stepAnswer":["TRUE"],"problemType":"MultipleChoice","stepTitle":"In a goodness-of-fit test, the expected values are the values we would expect if the null hypothesis were true.","stepBody":"Choose the answer from the following.","answerType":"string","variabilization":{},"choices":["TRUE","FALSE"],"hints":{"DefaultPathway":[{"id":"a79a53dgood5a-h1","type":"hint","dependencies":[],"title":"Null Hypothesis and Expected Value","text":"The expected value is equal to the population value assume that the null hypothesis is true.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a79a53dgood6","title":"Goodness-of-Fit Test","body":"Read the statement and decide whether it is true or false.\\\\n","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Goodness-of-Fit Test","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a79a53dgood6a","stepAnswer":["TRUE"],"problemType":"MultipleChoice","stepTitle":"Use a goodness-of-fit test to determine if high school principals believe that students are absent equally during the week or not.","stepBody":"Choose the answer from the following.","answerType":"string","variabilization":{},"choices":["TRUE","FALSE"],"hints":{"DefaultPathway":[{"id":"a79a53dgood6a-h1","type":"hint","dependencies":[],"title":"Null Hypothesis","text":"$$H_0$$: The absent days occur with equal frequencies, that is, they fit a uniform distribution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood6a-h2","type":"hint","dependencies":[],"title":"Alternative Hypothesis","text":"$$H_a$$: The absent days occur with unequal frequencies, that is, they do not fit a uniform distribution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood6a-h3","type":"hint","dependencies":["a79a53dgood6a-h2"],"title":"Sampling","text":"We can sample from the population to ask on which day of the week they had the highest number of student. absences.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a79a53dgood7","title":"Goodness-of-Fit Test","body":"Read the statement and decide whether it is true or false.\\\\n","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Goodness-of-Fit Test","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a79a53dgood7a","stepAnswer":["FALSE"],"problemType":"MultipleChoice","stepTitle":"In a goodness-of fit test, if the $$p-value$$ is $$0.0113$$, in general, do not reject the null hypothesis.","stepBody":"Choose the answer from the following.","answerType":"string","variabilization":{},"choices":["TRUE","FALSE"],"hints":{"DefaultPathway":[{"id":"a79a53dgood7a-h1","type":"hint","dependencies":[],"title":"P-Value","text":"Because the $$p-value$$ is less than our significance level of $$0.05$$, we will to reject the null hypothesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a79a53dgood8","title":"USA obese population","body":"Table $$11.42$$ contains information from a survey among $$499$$ participants classified according to their age groups. The second column shows the percentage of obese people per age class among the study participants. The last column comes from a different study at the national level that shows the corresponding percentages of obese people in the same age classes in the USA.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Goodness-of-Fit Test","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a79a53dgood8a","stepAnswer":["The surveyed obese do not fit the distribution of expected obese."],"problemType":"MultipleChoice","stepTitle":"Perform a hypothesis test at the 5% significance level to determine whether the survey participants are a representative sample of the USA obese population.","stepBody":"Choose the answer from the following.##figure1.gif## ","answerType":"string","variabilization":{},"choices":["The surveyed obese do not fit the distribution of expected obese.","The surveyed obese do fit the distribution of expected obese."],"hints":{"DefaultPathway":[{"id":"a79a53dgood8a-h1","type":"hint","dependencies":[],"title":"Hypotheses","text":"What are the Null Hypothesis $$H_0$$ and Alternative Hypothesis in this context?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood8a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$H_0$$: Surveyed obese fit the distribution of expected obese"],"dependencies":["a79a53dgood8a-h1"],"title":"Null Hypothesis","text":"Choose the answer from the following.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$H_0$$: Surveyed obese fit the distribution of expected obese","$$H_0$$: Surveyed obese do not fit the distribution of expected obese"]},{"id":"a79a53dgood8a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$H_a$$: Surveyed obese do not fit the distribution of expected obese"],"dependencies":["a79a53dgood8a-h2"],"title":"Alternative Hypothesis","text":"Choose the answer from the following.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$H_a$$: Surveyed obese fit the distribution of expected obese","$$H_a$$: Surveyed obese do not fit the distribution of expected obese"]},{"id":"a79a53dgood8a-h4","type":"hint","dependencies":["a79a53dgood8a-h3"],"title":"Consider what distribution is most appropriate to use in this problem.","text":"Is this a Goodness-of-Fit problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood8a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a79a53dgood8a-h4"],"title":"What is the degree of freedom?","text":"$$df=5-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood8a-h6","type":"hint","dependencies":["a79a53dgood8a-h5"],"title":"Compute the chi-square test statistics","text":"$$X^2$$ $$=$$ $$9.85$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood8a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.043$$"],"dependencies":["a79a53dgood8a-h6"],"title":"Compute the $$p-value$$.","text":"Round the answer to three decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood8a-h8","type":"hint","dependencies":["a79a53dgood8a-h7"],"title":"Compare with the significant level $$0.05$$.","text":"At 5% significance level, \\\\alpha $$=$$ $$0.05$$. For this data, P < \\\\alpha. Reject the null hypothesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood8a-h9","type":"hint","dependencies":["a79a53dgood8a-h8"],"title":"Interpret the Result","text":"At the 5% level of significance, from the data, there is sufficient evidence to conclude that the surveyed obese do not fit the distribution of expected obese.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a79a53dgood9","title":"AP Exam","body":"The columns in Table $$11.37$$ contain the $$\\\\frac{Race}{Ethnicity}$$ of U.S. Public Schools for a recent year, the percentages for the Advanced Placement Examinee Population for that class, and the Overall Student Population. Suppose the right column contains the result of a survey of 1,000 local students from that year who took an AP Exam.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Goodness-of-Fit Test","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a79a53dgood9a","stepAnswer":["There is insufficient evidence to conclude that local data do not follow the distribution of the U.S. AP examinee distribution."],"problemType":"MultipleChoice","stepTitle":"Perform a goodness-of-fit test to determine whether the local results follow the distribution of U.S. AP examinee population, based on ethnicity.","stepBody":"Choose the answer from the following.##figure1.gif## ","answerType":"string","variabilization":{},"choices":["There is insufficient evidence to conclude that local data do not follow the distribution of the U.S. AP examinee distribution.","There is sufficient evidence to conclude that local data do not follow the distribution of the U.S. AP examinee distribution.","There is insufficient evidence to conclude that local data do follow the distribution of the U.S. AP examinee distribution.","There is sufficient evidence to conclude that local data do follow the distribution of the U.S. AP examinee distribution."],"hints":{"DefaultPathway":[{"id":"a79a53dgood9a-h1","type":"hint","dependencies":[],"title":"Hypotheses","text":"What are the Null Hypothesis $$H_0$$ and Alternative Hypothesis in this context?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood9a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$H_0$$: The local results follow the distribution of the U.S. AP examinee population"],"dependencies":["a79a53dgood9a-h1"],"title":"Null Hypothesis","text":"Choose the answer from the following.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$H_0$$: The local results follow the distribution of the U.S. AP examinee population","$$H_0$$: The local results do not follow the distribution of the U.S. AP examinee population"]},{"id":"a79a53dgood9a-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$H_a$$: The local results do not follow the distribution of the U.S. AP examinee population"],"dependencies":[],"title":"Alternative Hypothesis","text":"Choose the answer from the following.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$H_a$$: The local results follow the distribution of the U.S. AP examinee population","$$H_a$$: The local results do not follow the distribution of the U.S. AP examinee population"]},{"id":"a79a53dgood9a-h3","type":"hint","dependencies":["a79a53dgood9a-h2"],"title":"Consider what distribution is most appropriate to use in this problem.","text":"Is this a Goodness-of-Fit problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a79a53dgood9a-h3"],"title":"What is the degree of freedom?","text":"$$df=6-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood9a-h5","type":"hint","dependencies":["a79a53dgood9a-h4"],"title":"Compute the chi-square test statistics","text":"$$X^2$$ $$=$$ $$13.4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood9a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.0199$$"],"dependencies":["a79a53dgood9a-h5"],"title":"Compute the $$p-value$$.","text":"Round the answer to four decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood9a-h7","type":"hint","dependencies":["a79a53dgood9a-h6"],"title":"Compare with the significant level $$0.05$$.","text":"At 5% significance level, \\\\alpha $$=$$ $$0.05$$. For this data, P < \\\\alpha. Reject the null hypothesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a79a53dgood9a-h8","type":"hint","dependencies":["a79a53dgood9a-h7"],"title":"Interpret the Result","text":"At the 5% level of significance, from the data, there is insufficient evidence to conclude that local data do not follow the distribution of the U.S. AP examinee distribution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a79dde1funccomp1","title":"Composition of Functions: Part A","body":"These questions test your knowledge of the core concepts. Suppose $$f(x)=3x^2+x+1$$. Find formulas for each of the following values.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Composition of Functions","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a79dde1funccomp1a","stepAnswer":["$$3w^2+w+1$$"],"problemType":"TextBox","stepTitle":"f(w)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3w^2+w+1$$","hints":{"DefaultPathway":[{"id":"a79dde1funccomp1a-h1","type":"hint","dependencies":[],"title":"Substituting $$x$$","text":"For all instances of $$x$$, replace with w.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3w^2$$"],"dependencies":["a79dde1funccomp1a-h1"],"title":"Substituting $$x$$ in $$3x^2$$","text":"What is $$3x^2$$ replaced with w?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["w"],"dependencies":["a79dde1funccomp1a-h1"],"title":"Substituting $$x$$ in $$x$$","text":"What is $$x$$ replaced with w?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a79dde1funccomp1a-h1"],"title":"Substituting $$x$$ in $$1$$","text":"What is $$1$$ replaced with w?","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a79dde1funccomp10","title":"Composition of Functions: Part A","body":"These questions test your knowledge of the core concepts. Suppose $$f(x)=\\\\sqrt{2x+1}$$ and $$g(x)=3x-1$$. Find the domains of the following functions.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Composition of Functions","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a79dde1funccomp10a","stepAnswer":["$$[\\\\frac{-1}{2},\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"The sum $$f+g$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[\\\\frac{-1}{2},\\\\infty)$$","choices":["$$[\\\\frac{-1}{2},\\\\infty)$$","$$(-\\\\infty,\\\\frac{1}{2}]$$","$$[\\\\frac{-1}{2},\\\\frac{1}{2}]$$","$$(-\\\\infty,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"a79dde1funccomp10a-h1","type":"hint","dependencies":[],"title":"Understanding $$f+g$$","text":"$$f+g$$ is the same as saying $$f{\\\\left(x\\\\right)}+g{\\\\left(x\\\\right)}=\\\\sqrt{2x+1}+3x-1$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp10a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Cannot take the square root of negatives"],"dependencies":["a79dde1funccomp10a-h1"],"title":"Restrictions of f","text":"Are there any restrictions to the domain for the function $$f(x)=\\\\sqrt{2x+1}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Cannot take the square root of negatives","No restrictions","Can only take the square root of positives","Cannot be any number"],"subHints":[{"id":"a79dde1funccomp10a-h2-s1","type":"hint","dependencies":[],"title":"Bounds of the Square Root","text":"For some a, $$\\\\sqrt{a}$$ if and only if a $$\\\\geq$$ $$0$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a79dde1funccomp10a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2x+1$$ $$\\\\geq$$ $$0$$"],"dependencies":["a79dde1funccomp10a-h2"],"title":"Restrictions of f","text":"Which inequality represents the restriction placed on $$f(x)=\\\\sqrt{2x+1}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$2x+1$$ $$\\\\geq$$ $$0$$","$$2x+1$$ $$\\\\leq$$ $$0$$","$$2x+1$$ $$>$$ $$0$$","$$2x+1$$ $$<$$ $$0$$"]},{"id":"a79dde1funccomp10a-h4","type":"hint","dependencies":["a79dde1funccomp10a-h3"],"title":"Restrictions of f","text":"Rearrange the inequality such that all the \'x\'s are on one side and the constants are on the other.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp10a-h5","type":"hint","dependencies":["a79dde1funccomp10a-h4"],"title":"Restrictions of f","text":"Subtract $$1$$ from the left to get $$2x$$ $$\\\\geq$$ $$0-1$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp10a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a79dde1funccomp10a-h5"],"title":"Restrictions of f","text":"What is $$0-1$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp10a-h7","type":"hint","dependencies":["a79dde1funccomp10a-h6"],"title":"Restrictions of f","text":"Divide $$2$$ from both sides of $$2x$$ $$\\\\geq$$ $$-1$$ to isolate \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp10a-h8","type":"hint","dependencies":["a79dde1funccomp10a-h7"],"title":"Restrictions of f","text":"The inequality $$x$$ $$\\\\geq$$ $$\\\\frac{-1}{2}$$ can be written in interval notation by specifying the lower bound first, followed by the upper bound.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp10a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{-1}{2}$$"],"dependencies":["a79dde1funccomp10a-h8"],"title":"Restrictions of f","text":"What is the lower bound of the inequality $$x$$ $$\\\\geq$$ $$\\\\frac{-1}{2}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{-1}{2}$$","$$-\\\\infty$$"]},{"id":"a79dde1funccomp10a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\infty$$"],"dependencies":["a79dde1funccomp10a-h8"],"title":"Restrictions of f","text":"What is the upper bound of the inequality $$x$$ $$\\\\geq$$ $$\\\\frac{-1}{2}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{-1}{2}$$","$$\\\\infty$$"]},{"id":"a79dde1funccomp10a-h11","type":"hint","dependencies":["a79dde1funccomp10a-h9","a79dde1funccomp10a-h10"],"title":"Restrictions of f","text":"If the bound itself should be included as a valid answer of \'x\', meaning it can be equal to that value, then a bracket (\\"[\\" or \\"]\\") should be used. Otherwise, use a paranthesis (\\"(\\" or \\")\\").","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp10a-h12","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a79dde1funccomp10a-h11"],"title":"Restrictions of f","text":"Is the lower bound $$\\\\frac{-1}{2}$$ included as a valid value of \'x\' in $$x$$ $$\\\\geq$$ $$\\\\frac{-1}{2}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"a79dde1funccomp10a-h13","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a79dde1funccomp10a-h11"],"title":"Restrictions of f","text":"Is the upper bound $$\\\\infty$$ included as a valid value of \'x\' in $$x$$ $$\\\\geq$$ $$\\\\frac{-1}{2}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"],"subHints":[{"id":"a79dde1funccomp10a-h13-s1","type":"hint","dependencies":[],"title":"Infinity in Interval Notation","text":"As $$\\\\infty$$ is not a number, it cannot be included as part of a valid value or bound for \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a79dde1funccomp10a-h14","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No restrictions"],"dependencies":["a79dde1funccomp10a-h12","a79dde1funccomp10a-h13"],"title":"Restrictions of g","text":"Are there any restrictions to the domain for the function $$g(x)=3x-1$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Cannot be a negative number","No restrictions","Cannot be $$0$$","Cannot be any number"]},{"id":"a79dde1funccomp10a-h15","type":"hint","dependencies":["a79dde1funccomp10a-h14"],"title":"Restrictions of $$f+g$$","text":"Since g does not have any restrictions, the domain will be represented by the restrictions of f.","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a79dde1funccomp100","title":"Composition of Functions: Part B","body":"These problems are generally harder, often highlighting an important subtlety. It\'s perfectly fine to have compositions of more than two functions. Let $$f(x)=x^2$$, $$g(x)=\\\\frac{1}{x-2}$$, and $$h(x)=3-x$$. Compute the following values or explain why they are undefined.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Composition of Functions","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a79dde1funccomp100a","stepAnswer":["$$\\\\frac{11}{4}$$"],"problemType":"MultipleChoice","stepTitle":"(h $$\\\\circ$$ f $$\\\\circ$$ g)(4)","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{11}{4}$$","choices":["$$\\\\frac{11}{4}$$","$$-1$$","$$2$$","Undefined"],"hints":{"DefaultPathway":[{"id":"a79dde1funccomp100a-h1","type":"hint","dependencies":[],"title":"Understanding $$h$$ $$\\\\circ$$ f $$\\\\circ$$ g","text":"$$h$$ $$\\\\circ$$ f $$\\\\circ$$ $$g=h(f(g(x)))$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp100a-h2","type":"hint","dependencies":["a79dde1funccomp100a-h1"],"title":"Solving g(4)","text":"$$g(4)=\\\\frac{1}{4-2}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp100a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a79dde1funccomp100a-h2"],"title":"Solving g(4)","text":"What is $$\\\\frac{1}{4-2}$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp100a-h4","type":"hint","dependencies":["a79dde1funccomp100a-h3"],"title":"Solving f(g(4))","text":"$$f(g(4))={\\\\left(\\\\frac{1}{2}\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp100a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4}$$"],"dependencies":["a79dde1funccomp100a-h4"],"title":"Solving f(g(4))","text":"What is $${\\\\left(\\\\frac{1}{2}\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp100a-h6","type":"hint","dependencies":["a79dde1funccomp100a-h5"],"title":"Solving h(f(g(4)))","text":"$$h(f(g(4)))=3-\\\\frac{1}{4}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp100a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{11}{4}$$"],"dependencies":["a79dde1funccomp100a-h6"],"title":"Solving h(f(g(4)))","text":"What is $$3-\\\\frac{1}{4}$$?","variabilization":{},"oer":"https://OATutor.io","license":""}]}},{"id":"a79dde1funccomp100b","stepAnswer":["Undefined"],"problemType":"MultipleChoice","stepTitle":"(g $$\\\\circ$$ $$h$$ $$\\\\circ$$ f)(1)","stepBody":"","answerType":"string","variabilization":{},"choices":["Undefined","$$0$$","$$16$$","$$2$$"],"hints":{"DefaultPathway":[{"id":"a79dde1funccomp100b-h1","type":"hint","dependencies":[],"title":"Understanding g $$\\\\circ$$ $$h$$ $$\\\\circ$$ f","text":"g $$\\\\circ$$ $$h$$ $$\\\\circ$$ $$f=g(h(f(x)))$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp100b-h2","type":"hint","dependencies":["a79dde1funccomp100b-h1"],"title":"Solving f(1)","text":"$$f(1)=1^2$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp100b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a79dde1funccomp100b-h2"],"title":"Solving f(1)","text":"What is $$1^2$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp100b-h4","type":"hint","dependencies":["a79dde1funccomp100b-h3"],"title":"Solving h(f(1))","text":"$$h(f(1))=3-1$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp100b-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a79dde1funccomp100b-h4"],"title":"Solving h(f(1))","text":"What is $$3-1$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp100b-h6","type":"hint","dependencies":["a79dde1funccomp100b-h5"],"title":"Solving g(h(f(1)))","text":"$$g(h(f(1)))=\\\\frac{1}{2-2}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp100b-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a79dde1funccomp100b-h6"],"title":"Solving g(h(f(1)))","text":"What is $$2-2$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp100b-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a79dde1funccomp100b-h7"],"title":"Solving g(h(f(1)))","text":"Is $$\\\\frac{1}{0}$$ a defined value?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]}]}},{"id":"a79dde1funccomp100c","stepAnswer":["$$\\\\frac{1}{{\\\\left(6-\\\\sqrt{5}\\\\right)}^2-2}$$"],"problemType":"MultipleChoice","stepTitle":"(g $$\\\\circ$$ f $$\\\\circ$$ h)( $$\\\\sqrt{5}$$ -3)","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{1}{{\\\\left(6-\\\\sqrt{5}\\\\right)}^2-2}$$","choices":["$$\\\\frac{1}{{\\\\left(6-\\\\sqrt{5}\\\\right)}^2-2}$$","$$\\\\frac{1}{{\\\\left(4-\\\\sqrt{5}\\\\right)}^2}$$","$$3-\\\\frac{1}{{\\\\left(\\\\sqrt{5}-3\\\\right)}^2}$$","Undefined"],"hints":{"DefaultPathway":[{"id":"a79dde1funccomp100c-h1","type":"hint","dependencies":[],"title":"Understanding g $$\\\\circ$$ f $$\\\\circ$$ $$h$$","text":"g $$\\\\circ$$ f $$\\\\circ$$ $$h=g(f(h(x)))$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp100c-h2","type":"hint","dependencies":["a79dde1funccomp100c-h1"],"title":"Solving $$h{\\\\left(\\\\sqrt{5}-3\\\\right)}$$","text":"$$h{\\\\left(\\\\sqrt{5}-3\\\\right)}=3-\\\\sqrt{5}-3$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp100c-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6-\\\\sqrt{5}$$"],"dependencies":["a79dde1funccomp100c-h2"],"title":"Solving $$h{\\\\left(\\\\sqrt{5}-3\\\\right)}$$","text":"What is $$3-\\\\sqrt{5}-3$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp100c-h4","type":"hint","dependencies":["a79dde1funccomp100c-h3"],"title":"Solving $$f{\\\\left(h{\\\\left(\\\\sqrt{5}-3\\\\right)}\\\\right)}$$","text":"$$f{\\\\left(h{\\\\left(\\\\sqrt{5}-3\\\\right)}\\\\right)}={\\\\left(6-\\\\sqrt{5}\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp100c-h5","type":"hint","dependencies":["a79dde1funccomp100c-h4"],"title":"Solving $$g{\\\\left(f{\\\\left(h{\\\\left(\\\\sqrt{5}-3\\\\right)}\\\\right)}\\\\right)}$$","text":"$$g{\\\\left(f{\\\\left(h{\\\\left(\\\\sqrt{5}-3\\\\right)}\\\\right)}\\\\right)}=\\\\frac{1}{{\\\\left(6-\\\\sqrt{5}\\\\right)}^2-2}$$","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a79dde1funccomp1000","title":"Composition of Functions: Part C","body":"These questions are challenging, requiring mastery of each concept and their interrelations. Let f and g be decreasing functions. Show, directly from the definition, that $$f$$ $$\\\\circ$$ $$g$$ is increasing by completing the following proof:\\\\n\\\\nSay $$x<y$$.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Composition of Functions","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a79dde1funccomp1000a","stepAnswer":["$$g{\\\\left(y\\\\right)}<g{\\\\left(x\\\\right)}$$"],"problemType":"MultipleChoice","stepTitle":"Since g is decreasing...","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$g{\\\\left(y\\\\right)}<g{\\\\left(x\\\\right)}$$","choices":["$$g{\\\\left(x\\\\right)}<g{\\\\left(y\\\\right)}$$","$$g{\\\\left(y\\\\right)}<g{\\\\left(x\\\\right)}$$"],"hints":{"DefaultPathway":[{"id":"a79dde1funccomp1000a-h1","type":"hint","dependencies":[],"title":"Decreasing Functions","text":"For some decreasing function a, a(x) grows smaller as $$x$$ grows larger.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp1000a-h2","type":"hint","dependencies":["a79dde1funccomp1000a-h1"],"title":"Decreasing Functions","text":"For two values i,j>i;a(i)>a(j) when a is a decreasing function.","variabilization":{},"oer":"https://OATutor.io","license":""}]}},{"id":"a79dde1funccomp1000b","stepAnswer":["$$f{\\\\left(g{\\\\left(x\\\\right)}\\\\right)}<f{\\\\left(g{\\\\left(y\\\\right)}\\\\right)}$$"],"problemType":"MultipleChoice","stepTitle":"Since f is decreasing...","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$f{\\\\left(g{\\\\left(x\\\\right)}\\\\right)}<f{\\\\left(g{\\\\left(y\\\\right)}\\\\right)}$$","choices":["$$g{\\\\left(x\\\\right)}<g{\\\\left(y\\\\right)}$$","$$g{\\\\left(y\\\\right)}<g{\\\\left(x\\\\right)}$$","$$g{\\\\left(x\\\\right)}<g{\\\\left(y\\\\right)}$$","$$f{\\\\left(g{\\\\left(y\\\\right)}\\\\right)}<f{\\\\left(g{\\\\left(x\\\\right)}\\\\right)}$$","$$f{\\\\left(g{\\\\left(x\\\\right)}\\\\right)}<f{\\\\left(g{\\\\left(y\\\\right)}\\\\right)}$$"],"hints":{"DefaultPathway":[{"id":"a79dde1funccomp1000b-h1","type":"hint","dependencies":[],"title":"Decreasing Functions","text":"For some decreasing function a, a(x) grows smaller as $$x$$ grows larger.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp1000b-h2","type":"hint","dependencies":["a79dde1funccomp1000b-h1"],"title":"Decreasing Functions","text":"For two values i,j>i;a(i)>a(j) when a is a decreasing function.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp1000b-h3","type":"hint","dependencies":["a79dde1funccomp1000b-h2"],"title":"Decreasing Functions","text":"For another decreasing function $$b$$, since $$a\\\\left(i\\\\right)>a\\\\left(j\\\\right)$$, then $$b\\\\left(a\\\\left(j\\\\right)\\\\right)<b\\\\left(a\\\\left(i\\\\right)\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]}},{"id":"a79dde1funccomp1000c","stepAnswer":["(f $$\\\\circ$$ g)(x)<(f $$\\\\circ$$ g)(y)"],"problemType":"MultipleChoice","stepTitle":"Since (f $$\\\\circ$$ $$g)(z)=f(g(z))$$ by the definition of function composition...","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"(f $$\\\\circ$$ g)(x)<(f $$\\\\circ$$ g)(y)","choices":["$$f{\\\\left(g{\\\\left(y\\\\right)}\\\\right)}<f{\\\\left(g{\\\\left(x\\\\right)}\\\\right)}$$","$$f{\\\\left(g{\\\\left(x\\\\right)}\\\\right)}<f{\\\\left(g{\\\\left(y\\\\right)}\\\\right)}$$","(f $$\\\\circ$$ g)(y)<(f $$\\\\circ$$ g)(x)","(f $$\\\\circ$$ g)(x)<(f $$\\\\circ$$ g)(y)"],"hints":{"DefaultPathway":[{"id":"a79dde1funccomp1000c-h1","type":"hint","dependencies":[],"title":"Function Composition","text":"As (a $$\\\\circ$$ $$b)(i)=a(b(i))$$, any inequalities would remain the same: $$a\\\\left(b\\\\left(i\\\\right)\\\\right)<a\\\\left(b\\\\left(j\\\\right)\\\\right)$$ is the same as (a $$\\\\circ$$ b)(i)<(a $$\\\\circ$$ b)(j).","variabilization":{},"oer":"https://OATutor.io","license":""}]}},{"id":"a79dde1funccomp1000d","stepAnswer":["Because $$x<y$$ is the same as (f $$\\\\circ$$ g)(x)<(f $$\\\\circ$$ g)(y), then (f $$\\\\circ$$ g) is increasing."],"problemType":"MultipleChoice","stepTitle":"What is the final line of the proof?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Because $$x<y$$ is the same as (f $$\\\\circ$$ g)(x)<(f $$\\\\circ$$ g)(y), then (f $$\\\\circ$$ g) is increasing.","choices":["Because $$x<y$$ is the same as (f $$\\\\circ$$ g)(x)<(f $$\\\\circ$$ g)(y), then (f $$\\\\circ$$ g) is increasing.","Because $$x<y$$ is the same as (g $$\\\\circ$$ f)(x)<(g $$\\\\circ$$ f)(y), then (f $$\\\\circ$$ g) is increasing.","Because $$x<y$$ is the same as (f $$\\\\circ$$ g)(y)<(f $$\\\\circ$$ g)(x), then (f $$\\\\circ$$ g) is increasing.","Because $$x<y$$ is the same as (f $$\\\\circ$$ g)(x)<(g $$\\\\circ$$ f)(y), then (f $$\\\\circ$$ g) is increasing."],"hints":{"DefaultPathway":[{"id":"a79dde1funccomp1000d-h1","type":"hint","dependencies":[],"title":"Combining the Previous Steps","text":"Remember that $$x<y$$ from the first assumption of the proof.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp1000d-h2","type":"hint","dependencies":["a79dde1funccomp1000d-h1"],"title":"Combining the Previous Steps","text":"Remember that after manipulating the functions using definitions, (f $$\\\\circ$$ g)(x)<(f $$\\\\circ$$ g)(y).","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a79dde1funccomp101","title":"Composition of Functions: Part B","body":"These problems are generally harder, often highlighting an important subtlety. If we wanted to justify the claim that the composition of two odd functions is another odd function, i.e. if f and g are odd functions, then f $$\\\\circ$$ g is an odd function, we could show the following steps:\\\\n\\\\n(f $$\\\\circ$$ $$g)(-x)=f(g(-x))=f(-g(x))=-f(g(x))=-(f$$ $$\\\\circ$$ g(x))\\\\n\\\\nGive justification for each step. The first step is done for you.\\\\n\\\\n(f $$\\\\circ$$ $$g)(-x)=f(g(-x))$$ is true by the definition of function composition.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Composition of Functions","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a79dde1funccomp101a","stepAnswer":["because g is an odd function"],"problemType":"MultipleChoice","stepTitle":"$$f(g(-x))=f(-g(x))$$ is true...","stepBody":"","answerType":"string","variabilization":{},"choices":["by the definition of function composition","because g is an odd function","because f is an odd function","by the definition of a function"],"hints":{"DefaultPathway":[{"id":"a79dde1funccomp101a-h1","type":"hint","dependencies":[],"title":"Properties of Functions","text":"A function a is odd if $$a(-x)=-a(x)$$, for all $$x$$ in the domain of a.","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a79dde1funccomp101b","title":"Composition of Functions: Part B","body":"These problems are generally harder, often highlighting an important subtlety. If we wanted to justify the claim that the composition of two odd functions is another odd function, i.e. if f and g are odd functions, then f $$\\\\circ$$ g is an odd function, we could show the following steps:\\\\n\\\\n(f $$\\\\circ$$ $$g)(-x)=f(g(-x))=f(-g(x))=-f(g(x))=-(f$$ $$\\\\circ$$ g(x))\\\\n\\\\nGive justification for each step. 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The first step is done for you.\\\\n\\\\n(f $$\\\\circ$$ $$g)(-x)=f(g(-x))$$ is true by the definition of function composition.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Composition of Functions","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a79dde1funccomp101ca","stepAnswer":["by the definition of function composition"],"problemType":"MultipleChoice","stepTitle":"$$-f(g(x))=-(f$$ $$\\\\circ$$ g(x)) is true...","stepBody":"","answerType":"string","variabilization":{},"choices":["by the definition of function composition","because g is an odd function","because f is an odd function","by the definition of a function"],"hints":{"DefaultPathway":[{"id":"a79dde1funccomp101ca-h1","type":"hint","dependencies":[],"title":"Properties of Functions","text":"For any two functions a,b;the function composition a $$\\\\circ$$ $$b=a(b(x))$$","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a79dde1funccomp10b","title":"Composition of Functions: Part A","body":"These questions test your knowledge of the core concepts. Suppose $$f(x)=\\\\sqrt{2x+1}$$ and $$g(x)=3x-1$$. Find the domains of the following functions.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Composition of Functions","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a79dde1funccomp10ba","stepAnswer":["$$[\\\\frac{-1}{2},\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"The product $$f g$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[\\\\frac{-1}{2},\\\\infty)$$","choices":["$$[\\\\frac{-1}{2},\\\\infty)$$","$$(-\\\\infty,\\\\frac{1}{2}]$$","$$[\\\\frac{-1}{2},\\\\frac{1}{2}]$$","$$(-\\\\infty,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"a79dde1funccomp10ba-h1","type":"hint","dependencies":[],"title":"Understanding $$f g$$","text":"$$f g$$ is the same as saying $$f{\\\\left(x\\\\right)} g{\\\\left(x\\\\right)}=\\\\sqrt{2x+1} \\\\left(3x-1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp10ba-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Cannot take the square root of negatives"],"dependencies":["a79dde1funccomp10ba-h1"],"title":"Restrictions of f","text":"Are there any restrictions to the domain for the function $$f(x)=\\\\sqrt{2x+1}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Cannot take the square root of negatives","No restrictions","Can only take the square root of positives","Cannot be any number"],"subHints":[{"id":"a79dde1funccomp10ba-h2-s1","type":"hint","dependencies":[],"title":"Bounds of the Square Root","text":"For some a, $$\\\\sqrt{a}$$ if and only if a $$\\\\geq$$ $$0$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a79dde1funccomp10ba-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2x+1$$ $$\\\\geq$$ $$0$$"],"dependencies":["a79dde1funccomp10ba-h2"],"title":"Restrictions of f","text":"Which inequality represents the restriction placed on $$f(x)=\\\\sqrt{2x+1}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$2x+1$$ $$\\\\geq$$ $$0$$","$$2x+1$$ $$\\\\leq$$ $$0$$","$$2x+1$$ $$>$$ $$0$$","$$2x+1$$ $$<$$ $$0$$"]},{"id":"a79dde1funccomp10ba-h4","type":"hint","dependencies":["a79dde1funccomp10ba-h3"],"title":"Restrictions of f","text":"Rearrange the inequality such that all the \'x\'s are on one side and the constants are on the other.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp10ba-h5","type":"hint","dependencies":["a79dde1funccomp10ba-h4"],"title":"Restrictions of f","text":"Subtract $$1$$ from the left to get $$2x$$ $$\\\\geq$$ $$0-1$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp10ba-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a79dde1funccomp10ba-h5"],"title":"Restrictions of f","text":"What is $$0-1$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp10ba-h7","type":"hint","dependencies":["a79dde1funccomp10ba-h6"],"title":"Restrictions of f","text":"Divide $$2$$ from both sides of $$2x$$ $$\\\\geq$$ $$-1$$ to isolate \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp10ba-h8","type":"hint","dependencies":["a79dde1funccomp10ba-h7"],"title":"Restrictions of f","text":"The inequality $$x$$ $$\\\\geq$$ $$\\\\frac{-1}{2}$$ can be written in interval notation by specifying the lower bound first, followed by the upper bound.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp10ba-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{-1}{2}$$"],"dependencies":["a79dde1funccomp10ba-h8"],"title":"Restrictions of f","text":"What is the lower bound of the inequality $$x$$ $$\\\\geq$$ $$\\\\frac{-1}{2}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{-1}{2}$$","$$-\\\\infty$$"]},{"id":"a79dde1funccomp10ba-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\infty$$"],"dependencies":["a79dde1funccomp10ba-h8"],"title":"Restrictions of f","text":"What is the upper bound of the inequality $$x$$ $$\\\\geq$$ $$\\\\frac{-1}{2}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{-1}{2}$$","$$\\\\infty$$"]},{"id":"a79dde1funccomp10ba-h11","type":"hint","dependencies":["a79dde1funccomp10ba-h9","a79dde1funccomp10ba-h10"],"title":"Restrictions of f","text":"If the bound itself should be included as a valid answer of \'x\', meaning it can be equal to that value, then a bracket (\\"[\\" or \\"]\\") should be used. Otherwise, use a paranthesis (\\"(\\" or \\")\\").","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp10ba-h12","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a79dde1funccomp10ba-h11"],"title":"Restrictions of f","text":"Is the lower bound $$\\\\frac{-1}{2}$$ included as a valid value of \'x\' in $$x$$ $$\\\\geq$$ $$\\\\frac{-1}{2}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"a79dde1funccomp10ba-h13","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a79dde1funccomp10ba-h11"],"title":"Restrictions of f","text":"Is the upper bound $$\\\\infty$$ included as a valid value of \'x\' in $$x$$ $$\\\\geq$$ $$\\\\frac{-1}{2}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"],"subHints":[{"id":"a79dde1funccomp10ba-h13-s1","type":"hint","dependencies":[],"title":"Infinity in Interval Notation","text":"As $$\\\\infty$$ is not a number, it cannot be included as part of a valid value or bound for \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a79dde1funccomp10ba-h14","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No restrictions"],"dependencies":["a79dde1funccomp10ba-h12","a79dde1funccomp10ba-h13"],"title":"Restrictions of g","text":"Are there any restrictions to the domain for the function $$g(x)=3x-1$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Cannot be a negative number","No restrictions","Cannot be $$0$$","Cannot be any number"]},{"id":"a79dde1funccomp10ba-h15","type":"hint","dependencies":["a79dde1funccomp10ba-h14"],"title":"Restrictions of $$f g$$","text":"Since g does not have any restrictions, the domain will be represented by the restrictions of f.","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a79dde1funccomp10c","title":"Composition of Functions: Part A","body":"These questions test your knowledge of the core concepts. Suppose $$f(x)=\\\\sqrt{2x+1}$$ and $$g(x)=3x-1$$. Find the domains of the following functions.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Composition of Functions","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a79dde1funccomp10ca","stepAnswer":["$$[\\\\frac{-1}{2},\\\\frac{1}{3})$$ $$\\\\cup$$ $$(\\\\frac{1}{3},\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"The quotient $$\\\\frac{f}{g}$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[\\\\frac{-1}{2},\\\\frac{1}{3})$$ $$\\\\cup$$ $$(\\\\frac{1}{3},\\\\infty)$$","choices":["$$[\\\\frac{-1}{2},\\\\frac{1}{3})$$ $$\\\\cup$$ $$(\\\\frac{1}{3},\\\\infty)$$","$$[\\\\frac{-1}{2},\\\\infty)$$","$$[\\\\frac{1}{3},\\\\infty)$$","$$(-\\\\infty,\\\\frac{1}{3})$$ $$\\\\cup$$ $$(\\\\frac{1}{3},\\\\frac{1}{2}]$$"],"hints":{"DefaultPathway":[{"id":"a79dde1funccomp10ca-h1","type":"hint","dependencies":[],"title":"Understanding $$\\\\frac{f}{g}$$","text":"$$\\\\frac{f}{g}$$ is the same as saying $$\\\\frac{f{\\\\left(x\\\\right)}}{g{\\\\left(x\\\\right)}}=\\\\frac{\\\\sqrt{2x+1}}{3x-1}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp10ca-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Cannot take the square root of negatives"],"dependencies":["a79dde1funccomp10ca-h1"],"title":"Restrictions of f","text":"Are there any restrictions to the domain for the function $$f(x)=\\\\sqrt{2x+1}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Cannot take the square root of negatives","No restrictions","Can only take the square root of positives","Cannot be any number"],"subHints":[{"id":"a79dde1funccomp10ca-h2-s1","type":"hint","dependencies":[],"title":"Bounds of the Square Root","text":"For some a, $$\\\\sqrt{a}$$ if and only if a $$\\\\geq$$ $$0$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a79dde1funccomp10ca-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2x+1$$ $$\\\\geq$$ $$0$$"],"dependencies":["a79dde1funccomp10ca-h2"],"title":"Restrictions of f","text":"Which inequality represents the restriction placed on $$f(x)=\\\\sqrt{2x+1}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$2x+1$$ $$\\\\geq$$ $$0$$","$$2x+1$$ $$\\\\leq$$ $$0$$","$$2x+1$$ $$>$$ $$0$$","$$2x+1$$ $$<$$ $$0$$"]},{"id":"a79dde1funccomp10ca-h4","type":"hint","dependencies":["a79dde1funccomp10ca-h3"],"title":"Restrictions of f","text":"Rearrange the inequality such that all the \'x\'s are on one side and the constants are on the other.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp10ca-h5","type":"hint","dependencies":["a79dde1funccomp10ca-h4"],"title":"Restrictions of f","text":"Subtract $$1$$ from the left to get $$2x$$ $$\\\\geq$$ $$0-1$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp10ca-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a79dde1funccomp10ca-h5"],"title":"Restrictions of f","text":"What is $$0-1$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp10ca-h7","type":"hint","dependencies":["a79dde1funccomp10ca-h6"],"title":"Restrictions of f","text":"Divide $$2$$ from both sides of $$2x$$ $$\\\\geq$$ $$-1$$ to isolate \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp10ca-h8","type":"hint","dependencies":["a79dde1funccomp10ca-h7"],"title":"Restrictions of f","text":"The inequality $$x$$ $$\\\\geq$$ $$\\\\frac{-1}{2}$$ can be written in interval notation by specifying the lower bound first, followed by the upper bound.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp10ca-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{-1}{2}$$"],"dependencies":["a79dde1funccomp10ca-h8"],"title":"Restrictions of f","text":"What is the lower bound of the inequality $$x$$ $$\\\\geq$$ $$\\\\frac{-1}{2}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{-1}{2}$$","$$-\\\\infty$$"]},{"id":"a79dde1funccomp10ca-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\infty$$"],"dependencies":["a79dde1funccomp10ca-h8"],"title":"Restrictions of f","text":"These questions are challenging, requiring mastery of each concept and their interrelations. Let f and g be decreasing functions. Show, directly from the definition, that f $$\\\\circ$$ g is increasing.","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{-1}{2}$$","$$\\\\infty$$"]},{"id":"a79dde1funccomp10ca-h11","type":"hint","dependencies":["a79dde1funccomp10ca-h9","a79dde1funccomp10ca-h10"],"title":"Restrictions of f","text":"If the bound itself should be included as a valid answer of \'x\', meaning it can be equal to that value, then a bracket (\\"[\\" or \\"]\\") should be used. Otherwise, use a paranthesis (\\"(\\" or \\")\\").","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp10ca-h12","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a79dde1funccomp10ca-h11"],"title":"Restrictions of f","text":"Is the lower bound $$\\\\frac{-1}{2}$$ included as a valid value of \'x\' in $$x$$ $$\\\\geq$$ $$\\\\frac{-1}{2}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"a79dde1funccomp10ca-h13","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a79dde1funccomp10ca-h11"],"title":"Restrictions of f","text":"Is the upper bound $$\\\\infty$$ included as a valid value of \'x\' in $$x$$ $$\\\\geq$$ $$\\\\frac{-1}{2}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"],"subHints":[{"id":"a79dde1funccomp10ca-h13-s1","type":"hint","dependencies":[],"title":"Infinity in Interval Notation","text":"As $$\\\\infty$$ is not a number, it cannot be included as part of a valid value or bound for \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a79dde1funccomp10ca-h14","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Cannot be $$0$$"],"dependencies":["a79dde1funccomp10ca-h12","a79dde1funccomp10ca-h13"],"title":"Restrictions of $$\\\\frac{1}{g}$$","text":"Are there any restrictions to the domain for the function $$\\\\frac{1}{g{\\\\left(x\\\\right)}}=\\\\frac{1}{3x-1}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Cannot be a negative number","No restrictions","Cannot be $$0$$","Cannot be any number"],"subHints":[{"id":"a79dde1funccomp10ca-h14-s1","type":"hint","dependencies":[],"title":"Bounds of a Reciprocal","text":"For some a, $$\\\\frac{1}{a}$$ if and only if a $$\\\\neq$$ $$0$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a79dde1funccomp10ca-h15","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3x-1 \\\\neq 0$$"],"dependencies":["a79dde1funccomp10ca-h14"],"title":"Restrictions of $$\\\\frac{1}{g}$$","text":"Which inequality represents the restriction placed on $$\\\\frac{1}{g{\\\\left(x\\\\right)}}=\\\\frac{1}{3x-1}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$3x-1 \\\\neq 0$$","$$3x-1=0$$","$$3x-1$$ $$\\\\geq$$ $$0$$","$$3x-1$$ $$\\\\leq$$ $$0$$"]},{"id":"a79dde1funccomp10ca-h16","type":"hint","dependencies":["a79dde1funccomp10ca-h15"],"title":"Restrictions of $$\\\\frac{1}{g}$$","text":"Rearrange the inequality such that all the \'x\'s are on one side and the constants are on the other.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp10ca-h17","type":"hint","dependencies":["a79dde1funccomp10ca-h16"],"title":"Restrictions of $$\\\\frac{1}{g}$$","text":"Add $$1$$ from the left to get $$3x$$ not equal to $$0+1$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp10ca-h18","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a79dde1funccomp10ca-h17"],"title":"Restrictions of $$\\\\frac{1}{g}$$","text":"What is $$0+1$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp10ca-h19","type":"hint","dependencies":["a79dde1funccomp10ca-h18"],"title":"Restrictions of $$\\\\frac{1}{g}$$","text":"Divide $$3$$ from both sides of $$3x \\\\neq 1$$ to isolate \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp10ca-h20","type":"hint","dependencies":["a79dde1funccomp10ca-h19"],"title":"Restrictions of $$\\\\frac{f}{g}$$","text":"Since f\'s domain is $$[\\\\frac{-1}{2},\\\\infty)$$ and g\'s domain is $$x \\\\neq \\\\frac{1}{3}$$, the final domain will be where both statements are true.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp10ca-h21","type":"hint","dependencies":["a79dde1funccomp10ca-h20"],"title":"Restrictions of $$\\\\frac{f}{g}$$","text":"For some interval [a,b] where c is between a and $$b$$, if $$x \\\\neq c$$, then the interval would be [a,c) $$\\\\cup$$ (c,b].","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a79dde1funccomp10d","title":"Composition of Functions: Part A","body":"These questions test your knowledge of the core concepts. Suppose $$f(x)=\\\\sqrt{2x+1}$$ and $$g(x)=3x-1$$. Find the domains of the following functions.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Composition of Functions","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a79dde1funccomp10da","stepAnswer":["$$[\\\\frac{1}{6},\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"The composition f $$\\\\circ$$ g.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[\\\\frac{1}{6},\\\\infty)$$","choices":["$$[\\\\frac{1}{6},\\\\infty)$$","$$(-\\\\infty,\\\\frac{1}{2}]$$","$$[\\\\frac{-1}{2},\\\\frac{1}{6}]$$","$$[\\\\frac{-1}{2},\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"a79dde1funccomp10da-h1","type":"hint","dependencies":[],"title":"Understanding f $$\\\\circ$$ g","text":"f $$\\\\circ$$ g is the same as saying $$f(g(x))=\\\\sqrt{2\\\\left(3x-1\\\\right)+1}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp10da-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6x-1$$"],"dependencies":["a79dde1funccomp10da-h1"],"title":"Simplifying f $$\\\\circ$$ g","text":"What is $$2\\\\left(3x-1\\\\right)+1$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp10da-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Cannot take the square root of negatives"],"dependencies":["a79dde1funccomp10da-h2"],"title":"Restrictions of f $$\\\\circ$$ g","text":"Are there any restrictions to the domain for the function f $$\\\\circ$$ $$g=\\\\sqrt{6x-1}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Cannot take the square root of negatives","No restrictions","Can only take the square root of positives","Cannot be any 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\'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp10da-h9","type":"hint","dependencies":["a79dde1funccomp10da-h8"],"title":"Restrictions of f $$\\\\circ$$ g","text":"The inequality $$x$$ $$\\\\geq$$ $$\\\\frac{1}{6}$$ can be written in interval notation by specifying the lower bound first, followed by the upper bound.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp10da-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1}{6}$$"],"dependencies":["a79dde1funccomp10da-h9"],"title":"Restrictions of f $$\\\\circ$$ g","text":"What is the lower bound of the inequality $$x$$ $$\\\\geq$$ $$\\\\frac{1}{6}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{1}{6}$$","$$-\\\\infty$$"]},{"id":"a79dde1funccomp10da-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\infty$$"],"dependencies":["a79dde1funccomp10da-h9"],"title":"Restrictions of f $$\\\\circ$$ g","text":"What is the upper bound of the inequality $$x$$ $$\\\\geq$$ $$\\\\frac{1}{6}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{1}{6}$$","$$\\\\infty$$"]},{"id":"a79dde1funccomp10da-h12","type":"hint","dependencies":["a79dde1funccomp10da-h10","a79dde1funccomp10da-h11"],"title":"Restrictions of f $$\\\\circ$$ g","text":"If the bound itself should be included as a valid answer of \'x\', meaning it can be equal to that value, then a bracket (\\"[\\" or \\"]\\") should be used. 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Let $$f(x)=3x-4$$ and $$g(x)=\\\\frac{1}{3} \\\\left(x-4\\\\right)$$. Answer the following questions to show that in this special case f $$\\\\circ$$ $$g=g$$ $$\\\\circ$$ f.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Composition of Functions","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a79dde1funccomp11a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Do f and g have the same domain?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a79dde1funccomp11a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No restrictions"],"dependencies":[],"title":"Restrictions of f","text":"Are there any restrictions to the domain for the function $$f(x)=3x-4$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Cannot be a negative number","No restrictions","Cannot be $$0$$","Cannot be any number"]},{"id":"a79dde1funccomp11a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No restrictions"],"dependencies":[],"title":"Restrictions of g","text":"Are there any restrictions to the domain for the function $$g(x)=\\\\frac{1}{3} \\\\left(x-4\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Cannot be a negative number","No restrictions","Cannot be $$0$$","Cannot be any number"]}]}},{"id":"a79dde1funccomp11b","stepAnswer":["$$x$$"],"problemType":"TextBox","stepTitle":"What function represents f $$\\\\circ$$ g?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x$$","hints":{"DefaultPathway":[{"id":"a79dde1funccomp11b-h1","type":"hint","dependencies":[],"title":"Understanding f $$\\\\circ$$ g","text":"f $$\\\\circ$$ $$g=f(g(x))=3\\\\frac{1}{3} \\\\left(x+4\\\\right)-4$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp11b-h2","type":"hint","dependencies":["a79dde1funccomp11b-h1"],"title":"Simplifying f $$\\\\circ$$ g","text":"$$3\\\\frac{1}{3} \\\\left(x+4\\\\right)-4$$ can be simplified by first multiplying $$\\\\frac{3\\\\times1}{3}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp11b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a79dde1funccomp11b-h2"],"title":"Simplifying f $$\\\\circ$$ g","text":"What is $$3\\\\frac{1}{3}$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp11b-h4","type":"hint","dependencies":["a79dde1funccomp11b-h3"],"title":"Simplifying f $$\\\\circ$$ g","text":"$$1\\\\left(x+4\\\\right)-4=x+4-4$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp11b-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x$$"],"dependencies":["a79dde1funccomp11b-h4"],"title":"Simplifying f $$\\\\circ$$ g","text":"What is $$x+4-4$$?","variabilization":{},"oer":"https://OATutor.io","license":""}]}},{"id":"a79dde1funccomp11c","stepAnswer":["$$x$$"],"problemType":"TextBox","stepTitle":"What function represents g $$\\\\circ$$ f?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x$$","hints":{"DefaultPathway":[{"id":"a79dde1funccomp11c-h1","type":"hint","dependencies":[],"title":"Understanding g $$\\\\circ$$ f","text":"g $$\\\\circ$$ $$f=g(f(x))=\\\\frac{1}{3} \\\\left(3x-4+4\\\\right)$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp11c-h2","type":"hint","dependencies":["a79dde1funccomp11c-h1"],"title":"Simplifying g $$\\\\circ$$ f","text":"$$\\\\frac{1}{3} \\\\left(3x-4+4\\\\right)$$ can be simplified by first adding $$3x-4+4$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp11c-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x$$"],"dependencies":["a79dde1funccomp11c-h2"],"title":"Simplifying g $$\\\\circ$$ f","text":"What is $$3x-4+4$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp11c-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x$$"],"dependencies":["a79dde1funccomp11c-h3"],"title":"Simplifying g $$\\\\circ$$ f","text":"What is $$\\\\frac{1}{3} 3x$$?","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a79dde1funccomp110","title":"Composition of Functions: Part C","body":"These questions are challenging, requiring mastery of each concept and their interrelations. Note that since $$0<1$$ $$\\\\leq$$ $$x^2+1$$ for all values of $$x$$, the algebraic properties of inequalities imply that $$0<\\\\frac{1}{x^2+1}$$ $$\\\\leq$$ $$\\\\frac{1}{1}=1$$. Thus, the function $$g(x)=\\\\frac{1}{x^2+1}$$ is bounded.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Composition of Functions","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a79dde1funccomp110a","stepAnswer":["$$h(x)=x$$"],"problemType":"MultipleChoice","stepTitle":"Determine which function $$h$$ such that $$h$$ $$\\\\circ$$ g is bounded.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$h(x)=x$$","choices":["$$h(x)=x$$","$$h(x)=\\\\frac{1}{x}$$"],"hints":{"DefaultPathway":[{"id":"a79dde1funccomp110a-h1","type":"hint","dependencies":[],"title":"Bounds of $$h$$","text":"Since g is already bounded between $$0$$ and $$1$$, $$h$$ can be an identity function to maintain the bounds of g.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp110a-h2","type":"hint","dependencies":["a79dde1funccomp110a-h1"],"title":"Bounds of $$h$$","text":"An identity function is one where the input of the function is the output.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp110a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a79dde1funccomp110a-h2"],"title":"Bounds of $$h$$","text":"Is $$h(x)=x$$ an identity function?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"a79dde1funccomp110a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a79dde1funccomp110a-h2"],"title":"Bounds of $$h$$","text":"Is $$h(x)=\\\\frac{1}{x}$$ an identity function?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]}]}}]},{"id":"a79dde1funccomp110b","title":"Composition of Functions: Part C","body":"These questions are challenging, requiring mastery of each concept and their interrelations. Note that since $$0<1$$ $$\\\\leq$$ $$x^2+1$$ for all values of $$x$$, the algebraic properties of inequalities imply that $$0<\\\\frac{1}{x^2+1}$$ $$\\\\leq$$ $$\\\\frac{1}{1}=1$$. Thus, the function $$g(x)=\\\\frac{1}{x^2+1}$$ is bounded.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Composition of Functions","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a79dde1funccomp110ba","stepAnswer":["$$f(x)=\\\\frac{1}{x}$$"],"problemType":"MultipleChoice","stepTitle":"Determine which function f such that f $$\\\\circ$$ g is not bounded.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$f(x)=\\\\frac{1}{x}$$","choices":["$$f(x)=x$$","$$f(x)=\\\\frac{1}{x}$$"],"hints":{"DefaultPathway":[{"id":"a79dde1funccomp110ba-h1","type":"hint","dependencies":[],"title":"Unbounded Functions","text":"A function is not bounded when its range, or output, can reach $$\\\\infty$$ or $$-\\\\infty$$ as $$x$$ grows larger or smaller.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp110ba-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1}{x^2+1}$$"],"dependencies":["a79dde1funccomp110ba-h1"],"title":"Bounds of f","text":"What is (f $$\\\\circ$$ g)(x)?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{1}{x^2+1}$$","$$x^2+1$$"]},{"id":"a79dde1funccomp110ba-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a79dde1funccomp110ba-h2"],"title":"Bounds of f","text":"Is (f $$\\\\circ$$ g)(x) bounded?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"],"subHints":[{"id":"a79dde1funccomp110ba-h3-s1","type":"hint","dependencies":[],"title":"Bounds of f $$\\\\circ$$ g","text":"Since $$f(g(x))=g(x)$$ for all $$x$$, then f $$\\\\circ$$ $$g=g$$. The bounds of the composition are the same as g.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a79dde1funccomp110ba-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x^2+1$$"],"dependencies":["a79dde1funccomp110ba-h3"],"title":"Bounds of f","text":"What is (f $$\\\\circ$$ g)(1/x)?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{1}{x^2+1}$$","$$x^2+1$$"]},{"id":"a79dde1funccomp110ba-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a79dde1funccomp110ba-h4"],"title":"Bounds of f","text":"Is (f $$\\\\circ$$ g)(1/x) bounded?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"],"subHints":[{"id":"a79dde1funccomp110ba-h5-s1","type":"hint","dependencies":[],"title":"Bounds of f $$\\\\circ$$ g","text":"As $$x$$ approaches $$\\\\infty$$, $$x^2+1$$ approaches $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]}]}}]},{"id":"a79dde1funccomp111","title":"Composition of Functions: Part C","body":"These questions are challenging, requiring mastery of each concept and their interrelations.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Composition of Functions","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a79dde1funccomp111a","stepAnswer":["$$(-4,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"Let $$f(x)=\\\\log_{2}\\\\left(x\\\\right)$$, $$g(x)=x^3-1$$, and $$h(x)=x+5$$. Determine the domain of the function f $$\\\\circ$$ g $$\\\\circ$$ $$h$$. Express your answer in interval notation.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-4,\\\\infty)$$","choices":["$$(0,\\\\infty)$$","$$(1,\\\\infty)$$","$$(-4,\\\\infty)$$","$$(-\\\\infty,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"a79dde1funccomp111a-h1","type":"hint","dependencies":[],"title":"Bounds of f","text":"The logarithm of some value \'b\' cannot be negative or zero, so $$b>0$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp111a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x>0$$"],"dependencies":["a79dde1funccomp111a-h1"],"title":"Bounds of f","text":"What inequality represents the bounds of f?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$x>0$$","$$x$$ $$\\\\geq$$ $$0$$","$$x$$ $$\\\\leq$$ $$0$$","$$x<0$$"]},{"id":"a79dde1funccomp111a-h3","type":"hint","dependencies":["a79dde1funccomp111a-h2"],"title":"Bounds of f $$\\\\circ$$ g","text":"The function g can be plugged into the previous inequality for $$x>0$$ to get the bounds of f $$\\\\circ$$ g.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp111a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x^3-1>0$$"],"dependencies":["a79dde1funccomp111a-h3"],"title":"Bounds of f $$\\\\circ$$ g","text":"What inequality represents the bounds of f $$\\\\circ$$ g?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$x^3-1>0$$","$$x>0$$","$$x>x^3-1$$","$$x^3-1>x^3-1$$"]},{"id":"a79dde1funccomp111a-h5","type":"hint","dependencies":["a79dde1funccomp111a-h4"],"title":"Bounds of f $$\\\\circ$$ g","text":"Add $$1$$ from the left side to get $$x^3>0+1$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp111a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a79dde1funccomp111a-h5"],"title":"Bounds of f $$\\\\circ$$ g","text":"What is $$0+1$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp111a-h7","type":"hint","dependencies":["a79dde1funccomp111a-h6"],"title":"Bounds of f $$\\\\circ$$ g","text":"Take the cubed root of both sides of $$x^3>1$$ to isolate \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp111a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a79dde1funccomp111a-h7"],"title":"Bounds of f $$\\\\circ$$ g","text":"What is $$\\\\sqrt[3]{1}$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp111a-h9","type":"hint","dependencies":["a79dde1funccomp111a-h8"],"title":"Bounds of f $$\\\\circ$$ g $$\\\\circ$$ $$h$$","text":"The function g can be plugged into the previous inequality $$x>1$$ to get the bounds of f $$\\\\circ$$ g $$\\\\circ$$ $$h$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp111a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x+5>1$$"],"dependencies":["a79dde1funccomp111a-h9"],"title":"Bounds of f $$\\\\circ$$ g $$\\\\circ$$ $$h$$","text":"What inequality represents the bounds of f $$\\\\circ$$ g $$\\\\circ$$ $$h$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$x+5>1$$","$$x^3-1>1$$","$$x>1$$","$$x+5>0$$"]},{"id":"a79dde1funccomp111a-h11","type":"hint","dependencies":["a79dde1funccomp111a-h10"],"title":"Bounds of f $$\\\\circ$$ g $$\\\\circ$$ $$h$$","text":"Subtract $$5$$ from the left side to get $$x>1-5$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp111a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["a79dde1funccomp111a-h11"],"title":"Bounds of f $$\\\\circ$$ g $$\\\\circ$$ $$h$$","text":"What is $$1-5$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp111a-h13","type":"hint","dependencies":["a79dde1funccomp111a-h12"],"title":"Interval Notation","text":"The inequality $$x>-4$$ can be written in interval notation by specifying the lower bound first, followed by the upper bound.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp111a-h14","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-4$$"],"dependencies":["a79dde1funccomp111a-h13"],"title":"Interval Notation","text":"What is the lower bound of the inequality $$x>-4$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$-4$$","$$-\\\\infty$$"]},{"id":"a79dde1funccomp111a-h15","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\infty$$"],"dependencies":["a79dde1funccomp111a-h13"],"title":"Interval Notation","text":"What is the upper bound of the inequality $$x>-4$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$-4$$","$$\\\\infty$$"]},{"id":"a79dde1funccomp111a-h16","type":"hint","dependencies":["a79dde1funccomp111a-h14","a79dde1funccomp111a-h15"],"title":"Interval Notation","text":"If the bound itself should be included as a valid answer of \'x\', meaning it can be equal to that value, then a bracket (\\"[\\" or \\"]\\") should be used. Otherwise, use a paranthesis (\\"(\\" or \\")\\").","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp111a-h17","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a79dde1funccomp111a-h16"],"title":"Interval Notation","text":"Is the lower bound $$-4$$ included as a valid value of \'x\' in $$x>-4$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"a79dde1funccomp111a-h18","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a79dde1funccomp111a-h16"],"title":"Interval Notation","text":"Is the upper bound $$\\\\infty$$ included as a valid value of \'x\' in $$x>-4$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"],"subHints":[{"id":"a79dde1funccomp111a-h18-s1","type":"hint","dependencies":[],"title":"Infinity in Interval Notation","text":"As $$\\\\infty$$ is not a number, it cannot be included as part of a valid value or bound for \'x\'.","variabilization":{},"oer":"https://OATutor.io","license":""}]}]}}]},{"id":"a79dde1funccomp2","title":"Composition of Functions: Part A","body":"These questions test your knowledge of the core concepts. Suppose $$f(x)=3x^2+x+1$$. Find formulas for each of the following values.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Composition of Functions","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a79dde1funccomp2a","stepAnswer":["$$48x^8+4x^4+1$$"],"problemType":"TextBox","stepTitle":"$$f{\\\\left(4x^4\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$48x^8+4x^4+1$$","hints":{"DefaultPathway":[{"id":"a79dde1funccomp2a-h1","type":"hint","dependencies":[],"title":"Substituting $$x$$","text":"For all instances of $$x$$, replace with $$4x^4$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp2a-h2","type":"hint","dependencies":["a79dde1funccomp2a-h1"],"title":"Substituting $$x$$ in $$3x^2$$","text":"Replacing $$x$$ within $$3x^2$$ results in $$3{\\\\left(4x^4\\\\right)}^2$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp2a-h3","type":"hint","dependencies":["a79dde1funccomp2a-h2"],"title":"Substituting $$x$$ in $$3x^2$$","text":"For a,b,c;(a*b)**c=(a**c)*(b**c)","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp2a-h4","type":"hint","dependencies":["a79dde1funccomp2a-h3"],"title":"Substituting $$x$$ in $$3x^2$$","text":"Distribute the exponent to each component: $$3\\\\times4^2 {\\\\left(x^4\\\\right)}^2$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp2a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$48$$"],"dependencies":["a79dde1funccomp2a-h4"],"title":"Substituting $$x$$ in $$3x^2$$","text":"What is $$3\\\\times4^2$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^8$$"],"dependencies":["a79dde1funccomp2a-h4"],"title":"Substituting $$x$$ in $$3x^2$$","text":"What is $${\\\\left(x^4\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io","license":"","subHints":[{"id":"a79dde1funccomp2a-h6-s1","type":"hint","dependencies":[],"title":"Power of a Power","text":"For a,b,c;(a**b)**c=a**(b*c).","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a79dde1funccomp2a-h7","type":"hint","dependencies":["a79dde1funccomp2a-h5","a79dde1funccomp2a-h6"],"title":"Substituting $$x$$ in $$3x^2$$","text":"Multiply the two components together to get the final term: $$48x^8$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp2a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4x^4$$"],"dependencies":["a79dde1funccomp2a-h7"],"title":"Substituting $$x$$ in $$x$$","text":"What is $$x$$ replaced with $$4x^4$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp2a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a79dde1funccomp2a-h8"],"title":"Substituting $$x$$ in $$1$$","text":"What is $$1$$ replaced with $$4x^4$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp2a-h10","type":"hint","dependencies":["a79dde1funccomp2a-h9"],"title":"Adding the Terms","text":"Add up $$48x^8$$, $$4x^4$$, and $$1$$ to get the final result.","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a79dde1funccomp3","title":"Composition of Functions: Part A","body":"These questions test your knowledge of the core concepts. Suppose $$f(x)=3x^2+x+1$$. Find formulas for each of the following values.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Composition of Functions","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a79dde1funccomp3a","stepAnswer":["$$12x^2-10x+3$$"],"problemType":"TextBox","stepTitle":"$$f{\\\\left(2x-1\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12x^2-10x+3$$","hints":{"DefaultPathway":[{"id":"a79dde1funccomp3a-h1","type":"hint","dependencies":[],"title":"Substituting $$x$$","text":"For all instances of $$x$$, replace with $$2x-1$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp3a-h2","type":"hint","dependencies":["a79dde1funccomp3a-h1"],"title":"Substituting $$x$$ in $$3x^2$$","text":"Replacing $$x$$ within $$3x^2$$ results in $$3{\\\\left(2x-1\\\\right)}^2$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4x^2-4x+1$$"],"dependencies":["a79dde1funccomp3a-h2"],"title":"Substituting $$x$$ in $$3x^2$$","text":"What is $${\\\\left(2x-1\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io","license":"","subHints":[{"id":"a79dde1funccomp3a-h3-s1","type":"hint","dependencies":[],"title":"Multiplying Binomials","text":"For a,b, $${\\\\left(a x+b\\\\right)}^2=a^2 x^2+2a b x+b^2$$","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a79dde1funccomp3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12x^2-12x+3$$"],"dependencies":["a79dde1funccomp3a-h3"],"title":"Substituting $$x$$ in $$3x^2$$","text":"What is $$3\\\\left(4x^2-4x+1\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":"","subHints":[{"id":"a79dde1funccomp3a-h4-s1","type":"hint","dependencies":[],"title":"Distributing Terms","text":"For a,b,c, $$a \\\\left(b+c\\\\right)=a b+a c$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a79dde1funccomp3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x-1$$"],"dependencies":["a79dde1funccomp3a-h4"],"title":"Substituting $$x$$ in $$x$$","text":"What is $$x$$ replaced with $$2x-1$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp3a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a79dde1funccomp3a-h5"],"title":"Substituting $$x$$ in $$1$$","text":"What is $$1$$ replaced with $$2x-1$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp3a-h7","type":"hint","dependencies":["a79dde1funccomp3a-h6"],"title":"Adding the Terms","text":"Add up $$12x^2-12x+3$$, $$2x-1$$, and $$1$$ by combining like terms to get the final result.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp3a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-10x$$"],"dependencies":["a79dde1funccomp3a-h7"],"title":"Adding the Terms","text":"What is $$-12x+2x$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp3a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a79dde1funccomp3a-h7"],"title":"Adding the Terms","text":"What is $$3-1+1$$?","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a79dde1funccomp4","title":"Composition of Functions: Part A","body":"These questions test your knowledge of the core concepts. Suppose $$f(x)=3x^2+x+1$$. Find formulas for each of the following values.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Composition of Functions","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"a79dde1funccomp4a","stepAnswer":["$$\\\\frac{5x^2+x+5}{{\\\\left(x-2\\\\right)}^2}$$"],"problemType":"TextBox","stepTitle":"$$f{\\\\left(\\\\frac{x+1}{x-2}\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5x^2+x+5}{{\\\\left(x-2\\\\right)}^2}$$","hints":{"DefaultPathway":[{"id":"a79dde1funccomp4a-h1","type":"hint","dependencies":[],"title":"Substituting $$x$$","text":"For all instances of $$x$$, replace with $$\\\\frac{x+1}{x-2}$$: $$3{\\\\left(\\\\frac{x+1}{x-2}\\\\right)}^2+\\\\frac{x+1}{x-2}+1$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp4a-h2","type":"hint","dependencies":["a79dde1funccomp4a-h1"],"title":"Finding the Least Common Denominator","text":"To simplify the equation further, the terms can be added together by finding the least common denominator and then multiplying each term such that each term is over the same denominator.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp4a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${\\\\left(x-2\\\\right)}^2$$"],"dependencies":["a79dde1funccomp4a-h2"],"title":"Finding the Least Common Denominator","text":"What is the Least Common Denominator?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$1$$","$$x-2$$","$${\\\\left(x-2\\\\right)}^2$$","$$\\\\frac{x+1}{x-2}$$"]},{"id":"a79dde1funccomp4a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$1$$"],"dependencies":["a79dde1funccomp4a-h3"],"title":"Simplifying $$3{\\\\left(\\\\frac{x+1}{x-2}\\\\right)}^2$$","text":"What should $$3{\\\\left(\\\\frac{x+1}{x-2}\\\\right)}^2$$ be multiplied by to have its denominator equal $${\\\\left(x-2\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$1$$","$$\\\\frac{x-2}{x-2}$$","$$\\\\frac{{\\\\left(x-2\\\\right)}^2}{{\\\\left(x-2\\\\right)}^2}$$"]},{"id":"a79dde1funccomp4a-h5","type":"hint","dependencies":["a79dde1funccomp4a-h4"],"title":"Simplifying $$3{\\\\left(\\\\frac{x+1}{x-2}\\\\right)}^2$$","text":"For a,b>0,c, $${\\\\left(\\\\frac{a}{b}\\\\right)}^c=\\\\frac{a^c}{b^c}$$, $$\\\\frac{3{\\\\left(x+1\\\\right)}^2}{{\\\\left(x-2\\\\right)}^2}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp4a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2+2x+1$$"],"dependencies":["a79dde1funccomp4a-h5"],"title":"Simplifying $$3{\\\\left(\\\\frac{x+1}{x-2}\\\\right)}^2$$","text":"What is $${\\\\left(x+1\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io","license":"","subHints":[{"id":"a79dde1funccomp4a-h6-s1","type":"hint","dependencies":[],"title":"Multiplying Binomials","text":"For a,b, $${\\\\left(a x+b\\\\right)}^2=a^2 x^2+2a b x+b^2$$","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a79dde1funccomp4a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x^2+6x+3$$"],"dependencies":["a79dde1funccomp4a-h6"],"title":"Simplifying $$3{\\\\left(\\\\frac{x+1}{x-2}\\\\right)}^2$$","text":"What is $$3\\\\left(x^2+2x+1\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":"","subHints":[{"id":"a79dde1funccomp4a-h7-s1","type":"hint","dependencies":[],"title":"Distributing Terms","text":"For a,b,c, $$a \\\\left(b+c\\\\right)=a b+a c$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a79dde1funccomp4a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{x-2}{x-2}$$"],"dependencies":["a79dde1funccomp4a-h7"],"title":"Simplifying $$\\\\frac{x+1}{x-2}$$","text":"What should $$\\\\frac{x+1}{x-2}$$ be multiplied by to have its denominator equal $${\\\\left(x-2\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$1$$","$$\\\\frac{x-2}{x-2}$$","$$\\\\frac{{\\\\left(x-2\\\\right)}^2}{{\\\\left(x-2\\\\right)}^2}$$"]},{"id":"a79dde1funccomp4a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2-x-2$$"],"dependencies":["a79dde1funccomp4a-h8"],"title":"Simplifying $$\\\\frac{x+1}{x-2}$$","text":"What is $$\\\\left(x+1\\\\right) \\\\left(x-2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":"","subHints":[{"id":"a79dde1funccomp4a-h9-s1","type":"hint","dependencies":[],"title":"Multiplying Binomials","text":"For a,b, $$\\\\left(x+a\\\\right) \\\\left(x+b\\\\right)=x^2+\\\\left(a+b\\\\right) x+a b$$","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a79dde1funccomp4a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{{\\\\left(x-2\\\\right)}^2}{{\\\\left(x-2\\\\right)}^2}$$"],"dependencies":["a79dde1funccomp4a-h9"],"title":"Simplifying $$1$$","text":"What should $$1$$ be multiplied by to have its denominator equal $${\\\\left(x-2\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$1$$","$$\\\\frac{x-2}{x-2}$$","$$\\\\frac{{\\\\left(x-2\\\\right)}^2}{{\\\\left(x-2\\\\right)}^2}$$"]},{"id":"a79dde1funccomp4a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2-4x+4$$"],"dependencies":["a79dde1funccomp4a-h10"],"title":"Simplifying $$1$$","text":"What is $${\\\\left(x-2\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io","license":"","subHints":[{"id":"a79dde1funccomp4a-h11-s1","type":"hint","dependencies":[],"title":"Multiplying Binomials","text":"For a,b, $${\\\\left(a x+b\\\\right)}^2=a^2 x^2+2a b x+b^2$$","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"a79dde1funccomp4a-h12","type":"hint","dependencies":["a79dde1funccomp4a-h11"],"title":"Adding the Terms","text":"Add up $$3x^2+6x+3$$, $$x^2-x-2$$, and $$x^2-4x+4$$ by combining like terms to get the numerator of the final result. Remember that the denominator is $${\\\\left(x-2\\\\right)}^2$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp4a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5x^2$$"],"dependencies":["a79dde1funccomp4a-h12"],"title":"Adding the Terms","text":"What is $$3x^2+x^2+x^2$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp4a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x$$"],"dependencies":["a79dde1funccomp4a-h12"],"title":"Adding the Terms","text":"What is $$6x-x-4x$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"a79dde1funccomp4a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a79dde1funccomp4a-h12"],"title":"Adding the Terms","text":"What is $$3-2+4$$?","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"a7b98f9comp1","title":"Performing Algebraic Operations on Functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Composition of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7b98f9comp1a","stepAnswer":["$$x \\\\left(x-1\\\\right)$$"],"problemType":"MultipleChoice","stepTitle":"Find and simplify the function $$(g-f)(x)$$, given $$f(x)=x-1$$ and $$g(x)=x^2-1$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x \\\\left(x-1\\\\right)$$","choices":["$$x \\\\left(x-1\\\\right)$$","$$x^2 \\\\left(x-1\\\\right)$$","$$x \\\\left(x-2\\\\right)$$"],"hints":{"DefaultPathway":[{"id":"a7b98f9comp1a-h1","type":"hint","dependencies":[],"title":"General Form","text":"We should begin by writing the general form, and then substitute the given functions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9comp1a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$g(x)-f(x)$$"],"dependencies":["a7b98f9comp1a-h1"],"title":"General Form","text":"What is the general form of $$(g-f)(x)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$g(x)-f(x)$$","$$g{\\\\left(x\\\\right)}+f{\\\\left(x\\\\right)}$$","$$g{\\\\left(x\\\\right)} f{\\\\left(x\\\\right)}$$"]},{"id":"a7b98f9comp1a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x^2-1-x-1$$"],"dependencies":["a7b98f9comp1a-h2"],"title":"Substitution","text":"What do we get after substituting the given functions?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x^2-1+x-1$$","$$x^2-1-x-1$$","$$\\\\left(x^2-1\\\\right) \\\\left(x-1\\\\right)$$"]},{"id":"a7b98f9comp1a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x \\\\left(x-1\\\\right)$$"],"dependencies":["a7b98f9comp1a-h3"],"title":"Rearrangement","text":"What do we get after rearranging the above equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x \\\\left(x-1\\\\right)$$","$$x^2 \\\\left(x-1\\\\right)$$","$$x \\\\left(x-2\\\\right)$$"]}]}},{"id":"a7b98f9comp1b","stepAnswer":["$$x+1$$"],"problemType":"MultipleChoice","stepTitle":"Find and simplify the function $$\\\\frac{g}{f} x$$, given $$f(x)=x-1$$ and $$g(x)=x^2-1$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x+1$$","choices":["$$x-1$$","$$x^2+1$$","$$x+1$$"],"hints":{"DefaultPathway":[{"id":"a7b98f9comp1b-h1","type":"hint","dependencies":[],"title":"General Form","text":"We should begin by writing the general form, and then substitute the given functions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9comp1b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{g{\\\\left(x\\\\right)}}{f{\\\\left(x\\\\right)}}$$"],"dependencies":["a7b98f9comp1b-h1"],"title":"General Form","text":"What is the general form of $$\\\\frac{g}{f} x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{g{\\\\left(x\\\\right)}}{f{\\\\left(x\\\\right)}}$$","$$g{\\\\left(x\\\\right)}+f{\\\\left(x\\\\right)}$$","$$g{\\\\left(x\\\\right)} f{\\\\left(x\\\\right)}$$"]},{"id":"a7b98f9comp1b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{x^2-1}{x-1}$$"],"dependencies":["a7b98f9comp1b-h2"],"title":"Substitution","text":"What do we get after substituting the given functions?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{x^2-1}{x-1}$$","$$x^2-1+x-1$$","$$\\\\left(x^2-1\\\\right) \\\\left(x-1\\\\right)$$"]},{"id":"a7b98f9comp1b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\left(x+1\\\\right) \\\\left(x-1\\\\right)$$"],"dependencies":["a7b98f9comp1b-h3"],"title":"Rearrangement","text":"How can we rewrite $$x^2-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\left(x+1\\\\right) \\\\left(x-1\\\\right)$$","$$x^2$$","$$\\\\left(x+2\\\\right) \\\\left(x-\\\\frac{1}{2}\\\\right)$$"]},{"id":"a7b98f9comp1b-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x+1$$"],"dependencies":["a7b98f9comp1b-h4"],"title":"Rearrangement","text":"What do we get after rearranging the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x-1$$","$$x^2+1$$","$$x+1$$"]}]}}]},{"id":"a7b98f9comp2","title":"Determining whether Composition of Functions is Commutative","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Composition of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7b98f9comp2a","stepAnswer":["$$7-2x$$"],"problemType":"MultipleChoice","stepTitle":"Using the functions provided, find f(g(x)). $$f(x)=2x+1$$, $$g(x)=3-x$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$7-2x$$","choices":["$$6+3x$$","$$7+3x$$","$$7-2x$$"],"hints":{"DefaultPathway":[{"id":"a7b98f9comp2a-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Let\u2019s begin by substituting g(x) into f(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9comp2a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2\\\\left(3-x\\\\right)+1$$"],"dependencies":["a7b98f9comp2a-h1"],"title":"Substitution","text":"What do we get after substituting the given functions?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$3-2x+1$$","$$2\\\\left(3+x\\\\right)-1$$","$$2\\\\left(3-x\\\\right)+1$$"]},{"id":"a7b98f9comp2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7-2x$$"],"dependencies":["a7b98f9comp2a-h2"],"title":"Simplify","text":"What do we get after rearranging and simplifying the above equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a7b98f9comp2b","stepAnswer":["$$-2x+2$$"],"problemType":"TextBox","stepTitle":"Using the functions provided, find g(f(x)). $$f(x)=2x+1$$, $$g(x)=3-x$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-2x+2$$","hints":{"DefaultPathway":[{"id":"a7b98f9comp2b-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Let\u2019s begin by substituting f(x) into g(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9comp2b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3-2x+1$$"],"dependencies":["a7b98f9comp2b-h1"],"title":"Substitution","text":"What do we get after substituting the given functions?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$3-2x+1$$","$$2\\\\left(3+x\\\\right)-1$$","$$2\\\\left(3-x\\\\right)+1$$"]},{"id":"a7b98f9comp2b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2x+2$$"],"dependencies":["a7b98f9comp2b-h2"],"title":"Simplify","text":"What do we get after rearranging and simplifying the above equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7b98f9comp3","title":"Interpreting Composite Functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Composition of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7b98f9comp3a","stepAnswer":["the number of calories burned in $$3$$ minutes (by doing $$sit-ups)$$"],"problemType":"MultipleChoice","stepTitle":"The function c(s) gives the number of calories burned completing s sit-ups, and s(t) gives the number of sit-ups a person can complete in $$t$$ minutes. Interpret c(s(3)).","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"the number of calories burned in $$3$$ minutes (by doing sit-ups)","choices":["the number of $$sit-ups$$ in $$3$$ minutes","the number of calories burned in $$3$$ minutes (by doing $$sit-ups)$$","the number of $$sit-ups$$ the person has to do in order to burn $$3$$ kcal of calories"],"hints":{"DefaultPathway":[{"id":"a7b98f9comp3a-h1","type":"hint","dependencies":[],"title":"Interpreting s(3)","text":"Because the input to the s-function is time, $$t=3$$ represents $$3$$ minutes, and s(3) is the number of sit-ups completed in $$3$$ minutes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9comp3a-h2","type":"hint","dependencies":["a7b98f9comp3a-h1"],"title":"Putting it together","text":"Using s(3) as the input to the function c(s) gives us the number of calories burned during the number of sit-ups that can be completed in $$3$$ minutes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7b98f9comp4","title":"Investigating the Order of Function Composition","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Composition of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7b98f9comp4a","stepAnswer":["g(f(x))"],"problemType":"MultipleChoice","stepTitle":"Suppose f(x) gives miles that can be driven in $$x$$ hours and g(y) gives the gallons of gas used in driving $$y$$ miles. Which of these expressions is meaningful: f(g(y)) or g(f(x))?","stepBody":"","answerType":"string","variabilization":{},"choices":["f(g(y))","g(f(x))"],"hints":{"DefaultPathway":[{"id":"a7b98f9comp4a-h1","type":"hint","dependencies":[],"title":"Interpreting f(x)","text":"The function $$y=f(x)$$ is a function whose output is the number of miles driven corresponding to the number of hours driven.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9comp4a-h2","type":"hint","dependencies":["a7b98f9comp4a-h1"],"title":"Interpreting g(y)","text":"The function g(y) is a function whose output is the number of gallons used corresponding to the number of miles driven.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9comp4a-h3","type":"hint","dependencies":["a7b98f9comp4a-h2"],"title":"Interpreting f(g(y))","text":"The function f(x) requires a number of hours as the input. Trying to input a number of gallons does not make sense. The expression f(g(y)) is meaningless.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9comp4a-h4","type":"hint","dependencies":["a7b98f9comp4a-h3"],"title":"Interpreting g(f(x))","text":"The function g(y) requires a number of miles as the input. Using f(x) (miles driven) as an input value for g(y), where gallons of gas depends on miles driven, does make sense. The expression g(f(x)) makes sense, and will yield the number of gallons of gas used, g, driving a certain number of miles, f(x), in $$x$$ hours.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7b98f9composition1","title":"Decomposing a Function","body":"For the following exercise, find functions f(x) and g(x) so the given function can be expressed as $$h(x)=f(g(x))$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Composition of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7b98f9composition1a","stepAnswer":["$$f(x)=x^{\\\\frac{1}{3}}$$, $$g(x)=x-1$$"],"problemType":"MultipleChoice","stepTitle":"$$h(x)={\\\\left(x-1\\\\right)}^{\\\\frac{1}{3}}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$f(x)=x^{\\\\frac{1}{3}}$$, $$g(x)=x-1$$","choices":["$$f(x)={\\\\left(-1\\\\right)}^{\\\\frac{1}{3}}$$, $$g(x)=x^{\\\\frac{1}{3}}$$","$$f(x)=x^{\\\\frac{1}{3}}$$, $$g(x)={\\\\left(-1\\\\right)}^{\\\\frac{1}{3}}$$","$$f(x)=x^{\\\\frac{1}{3}}$$, $$g(x)=x-1$$","$$f(x)=x-1$$, $$g(x)=x^{\\\\frac{1}{3}}$$"],"hints":{"DefaultPathway":[{"id":"a7b98f9composition1a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$f(x)=x^{\\\\frac{1}{3}}$$"],"dependencies":[],"title":"First Function","text":"What is the function in which $$x-1$$ is inserted to get its cube root?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$f(x)=x-1$$","$$f(x)=x^{\\\\frac{1}{3}}$$"]},{"id":"a7b98f9composition1a-h2","type":"hint","dependencies":["a7b98f9composition1a-h1"],"title":"Second Function","text":"The other function is what you put into the previous function to get the given function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition1a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$g(x)=x-1$$"],"dependencies":["a7b98f9composition1a-h2"],"title":"Second Function","text":"What is the second function?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$g(x)=x-1$$","$$g(x)=x^{\\\\frac{1}{3}}$$"]}]}}]},{"id":"a7b98f9composition10","title":"Decomposing a Function","body":"Find functions f(x) and g(x) so the given function can be expressed as $$h(x)=f(g(x))$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Composition of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7b98f9composition10a","stepAnswer":["$$g(x)=\\\\sqrt[3]{x}$$, $$f(x)=4+x$$"],"problemType":"MultipleChoice","stepTitle":"$$h(x)=4+\\\\sqrt[3]{x}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$g(x)=\\\\sqrt[3]{x}$$, $$f(x)=4+x$$","choices":["$$g(x)=\\\\sqrt[3]{x}$$, $$f(x)=4$$","$$g(x)=x$$, $$f(x)=x^3+4$$","$$g(x)=\\\\sqrt[3]{x}$$, $$f(x)=4+x$$"],"hints":{"DefaultPathway":[{"id":"a7b98f9composition10a-h1","type":"hint","dependencies":[],"title":"Looking for an Inner Function","text":"The first step is to look for a function inside a function in the formula for f(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition10a-h2","type":"hint","dependencies":["a7b98f9composition10a-h1"],"title":"Inner Function","text":"The function contains the expression $$\\\\sqrt[3]{x}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition10a-h3","type":"hint","dependencies":["a7b98f9composition10a-h2"],"title":"Outer Function","text":"$$4$$ is added to $$\\\\sqrt[3]{x}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition10a-h4","type":"hint","dependencies":["a7b98f9composition10a-h3"],"title":"Answer","text":"$$g(x)=\\\\sqrt[3]{x}$$, $$f(x)=4+x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7b98f9composition11","title":"Decomposing a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Composition of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7b98f9composition11a","stepAnswer":["$$g(x)=\\\\frac{1}{2x-3}$$, $$f(x)=\\\\sqrt[3]{x}$$"],"problemType":"MultipleChoice","stepTitle":"$$h(x)=\\\\sqrt[3]{\\\\frac{1}{2x-3}}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$g(x)=\\\\frac{1}{2x-3}$$, $$f(x)=\\\\sqrt[3]{x}$$","choices":["$$g(x)=\\\\frac{1}{2x-3}$$, $$f(x)=\\\\sqrt[3]{x}$$","$$g(x)=2x-3$$, $$f(x)=\\\\frac{1}{x}$$","$$g(x)=\\\\sqrt[3]{x}$$, $$f(x)=2x-3$$"],"hints":{"DefaultPathway":[{"id":"a7b98f9composition11a-h1","type":"hint","dependencies":[],"title":"Looking for an Inner Function","text":"The first step is to look for a function inside a function in the formula for f(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition11a-h2","type":"hint","dependencies":["a7b98f9composition11a-h1"],"title":"Inner Function","text":"Inside the cube root is the expression $$\\\\frac{1}{2x-3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition11a-h3","type":"hint","dependencies":["a7b98f9composition11a-h2"],"title":"Outer Function","text":"Then, the cube root of $$\\\\frac{1}{2x-3}$$ is taken.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition11a-h4","type":"hint","dependencies":["a7b98f9composition11a-h3"],"title":"Answer","text":"$$g(x)=\\\\frac{1}{2x-3}$$, $$f(x)=\\\\sqrt[3]{x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7b98f9composition12","title":"Decomposing a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Composition of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7b98f9composition12a","stepAnswer":["$$g(x)={\\\\left(3x^2-4\\\\right)}^{\\\\left(-3\\\\right)}$$, $$f(x)=\\\\frac{1}{x}$$"],"problemType":"MultipleChoice","stepTitle":"$$h(x)=\\\\frac{1}{{\\\\left(3x^2-4\\\\right)}^{\\\\left(-3\\\\right)}}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$g(x)={\\\\left(3x^2-4\\\\right)}^{\\\\left(-3\\\\right)}$$, $$f(x)=\\\\frac{1}{x}$$","choices":["$$g(x)={\\\\left(3x^2-4\\\\right)}^{\\\\left(-3\\\\right)}$$, $$f(x)=\\\\frac{1}{x}$$","$$g(x)=x-4$$, $$f(x)=\\\\frac{1}{3} x^2$$","$$g(x)=3x^2-4$$, $$f(x)=\\\\frac{1}{x}$$"],"hints":{"DefaultPathway":[{"id":"a7b98f9composition12a-h1","type":"hint","dependencies":[],"title":"Looking for an Inner Function","text":"The first step is to look for a function inside a function in the formula for f(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition12a-h2","type":"hint","dependencies":["a7b98f9composition12a-h1"],"title":"Inner Function","text":"The denominator of the function is $${\\\\left(3x^2-4\\\\right)}^{\\\\left(-3\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition12a-h3","type":"hint","dependencies":["a7b98f9composition12a-h2"],"title":"Outer Function","text":"Then, $$1$$ is divided by $${\\\\left(3x^2-4\\\\right)}^{\\\\left(-3\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition12a-h4","type":"hint","dependencies":["a7b98f9composition12a-h3"],"title":"Answer","text":"$$g(x)={\\\\left(3x^2-4\\\\right)}^{\\\\left(-3\\\\right)}$$, $$f(x)=\\\\frac{1}{x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7b98f9composition13","title":"Decomposing a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Composition of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7b98f9composition13a","stepAnswer":["$$g(x)=\\\\frac{3x-2}{x+5}$$, $$f(x)=\\\\sqrt[4]{x}$$"],"problemType":"MultipleChoice","stepTitle":"$$h(x)=\\\\sqrt[4]{\\\\frac{3x-2}{x+5}}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$g(x)=\\\\frac{3x-2}{x+5}$$, $$f(x)=\\\\sqrt[4]{x}$$","choices":["$$g(x)=\\\\frac{3x-2}{x+5}$$, $$f(x)=\\\\sqrt[4]{x}$$","$$g(x)=\\\\frac{3x-2}{x+5}$$, $$f(x)=\\\\sqrt{x}$$","$$g(x)=3x-2)$$, $$f(x)=\\\\frac{\\\\sqrt[4]{x}}{x+5}$$"],"hints":{"DefaultPathway":[{"id":"a7b98f9composition13a-h1","type":"hint","dependencies":[],"title":"Looking for an Inner Function","text":"The first step is to look for a function inside a function in the formula for f(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition13a-h2","type":"hint","dependencies":["a7b98f9composition13a-h1"],"title":"Inner Function","text":"Inside the fourth root is the expression $$\\\\frac{3x-2}{x+5}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition13a-h3","type":"hint","dependencies":["a7b98f9composition13a-h2"],"title":"Outer Function","text":"Then, the fourth root of $$\\\\frac{3x-2}{x+5}$$ is taken.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition13a-h4","type":"hint","dependencies":["a7b98f9composition13a-h3"],"title":"Answer","text":"$$g(x)=\\\\frac{3x-2}{x+5}$$, $$f(x)=\\\\sqrt[4]{x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7b98f9composition14","title":"Decomposing a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Composition of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7b98f9composition14a","stepAnswer":["$$g(x)=\\\\frac{8+x^3}{8-x^3}$$, $$f(x)=x^4$$"],"problemType":"MultipleChoice","stepTitle":"$$h(x)={\\\\left(\\\\frac{8+x^3}{8-x^3}\\\\right)}^4$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$g(x)=\\\\frac{8+x^3}{8-x^3}$$, $$f(x)=x^4$$","choices":["$$g(x)=\\\\frac{8+x^3}{8-x^3}$$, $$f(x)=x^4$$","$$g(x)=8+x^3$$, $$f(x)={\\\\left(8-x^3\\\\right)}^4$$","g(x)=x**4,f(x)=8-x"],"hints":{"DefaultPathway":[{"id":"a7b98f9composition14a-h1","type":"hint","dependencies":[],"title":"Looking for an Inner Function","text":"The first step is to look for a function inside a function in the formula for f(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition14a-h2","type":"hint","dependencies":["a7b98f9composition14a-h1"],"title":"Inner Function","text":"The function contains the expression $$\\\\frac{8+x^3}{8-x^3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition14a-h3","type":"hint","dependencies":["a7b98f9composition14a-h2"],"title":"Outer Function","text":"The expression $$\\\\frac{8+x^3}{8-x^3}$$ is then raised to the fourth power.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition14a-h4","type":"hint","dependencies":["a7b98f9composition14a-h3"],"title":"Answer","text":"$$g(x)=\\\\frac{8+x^3}{8-x^3}$$, $$f(x)=x^4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7b98f9composition15","title":"Decomposing a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Composition of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7b98f9composition15a","stepAnswer":["g(x)=2x+6,f(x)=sqrt(x)"],"problemType":"MultipleChoice","stepTitle":"$$h(x)=\\\\sqrt{2x+6}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["g(x)=2x+6,f(x)=sqrt(x)","$$g(x)=2x, f(x)=\\\\sqrt{x}+6$$","g(x)=x+6,f(x)=sqrt(2x)"],"hints":{"DefaultPathway":[{"id":"a7b98f9composition15a-h1","type":"hint","dependencies":[],"title":"Looking for an Inner Function","text":"The first step is to look for a function inside a function in the formula for f(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition15a-h2","type":"hint","dependencies":["a7b98f9composition15a-h1"],"title":"Inner Function","text":"Inside the square root is the expression $$2x+6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition15a-h3","type":"hint","dependencies":["a7b98f9composition15a-h2"],"title":"Outer Function","text":"Then, the square root of $$2x+6$$ is taken.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition15a-h4","type":"hint","dependencies":["a7b98f9composition15a-h3"],"title":"Answer","text":"g(x)=2x+6,f(x)=sqrt(x)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7b98f9composition16","title":"Composition of Funtions With Exponents","body":"For $$f(x)=x^2+2x$$ and $$g(x)=6-x^2$$, solve the following.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Composition of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7b98f9composition16a","stepAnswer":["$$2x+6$$"],"problemType":"TextBox","stepTitle":"$$f+g$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2x+6$$","hints":{"DefaultPathway":[{"id":"a7b98f9composition16a-h1","type":"hint","dependencies":[],"title":"Addition","text":"Remember that $$f+g$$ is the same as $$\\\\left(f+g\\\\right) x$$. All we have to do is add the two functions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x+6$$"],"dependencies":["a7b98f9composition16a-h1"],"title":"Addition","text":"What is $$x^2+2x+6x-x^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a7b98f9composition16a-h2-s1","type":"hint","dependencies":[],"title":"Combining Like Terms","text":"$$x^2-x^2=0$$, so this term drops out. We are left with $$2x$$ and $$6$$, so the final answer is $$2x+6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}},{"id":"a7b98f9composition16b","stepAnswer":["$$2x^2+2x-6$$"],"problemType":"TextBox","stepTitle":"f-g","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2x^2+2x-6$$","hints":{"DefaultPathway":[{"id":"a7b98f9composition16b-h1","type":"hint","dependencies":[],"title":"Subtraction","text":"Remember that f-g is the same as $$(f-g)(x)$$. All we have to do is subtract the two functions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition16b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x+6$$"],"dependencies":["a7b98f9composition16b-h1"],"title":"Subtraction","text":"What is $$x^2+2x-6x-x^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a7b98f9composition16b-h2-s1","type":"hint","dependencies":[],"title":"Combining Like Terms","text":"$$x^2+x^2=2x^2$$, and we are left with $$2x$$ and $$6$$, so the final answer is $$2x^2+2x-6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}},{"id":"a7b98f9composition16c","stepAnswer":["$$-\\\\left(x^4\\\\right)-2x^3+6x^2+12x$$"],"problemType":"TextBox","stepTitle":"fg","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-\\\\left(x^4\\\\right)-2x^3+6x^2+12x$$","hints":{"DefaultPathway":[{"id":"a7b98f9composition16c-h1","type":"hint","dependencies":[],"title":"Multiplication","text":"Remember that fg is the same as (fg)(x). All we have to do is use foil to multiply the two functions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition16c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-\\\\left(x^4\\\\right)-2x^3+6x^2+12x$$"],"dependencies":["a7b98f9composition16c-h1"],"title":"Multiplication","text":"What is $$\\\\left(x^2+2x\\\\right) \\\\left(6x-x^2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition16c-h3","type":"hint","dependencies":["a7b98f9composition16c-h2"],"title":"Combining Like Terms","text":"After multiplying the products, there are no like terms to combine, so we will be left with $$-\\\\left(x^4\\\\right)-2x^3+6x+12x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a7b98f9composition16d","stepAnswer":["$$\\\\frac{x^2+2x}{6-x^2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{f}{g}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{x^2+2x}{6-x^2}$$","hints":{"DefaultPathway":[{"id":"a7b98f9composition16d-h1","type":"hint","dependencies":[],"title":"Division","text":"Remember that $$\\\\frac{f}{g}$$ is the same as $$\\\\frac{f}{g} x$$. All we have to do divide the two functions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition16d-h2","type":"hint","dependencies":["a7b98f9composition16d-h1"],"title":"Division","text":"After dividing the two expressions, there is nothing to simplify so we will be left with $$\\\\frac{x^2+2x}{6-x^2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7b98f9composition17","title":"Composition of Funtions With Constants","body":"Given $$f(x)=-3x^2+x$$ and $$g(x)=5$$, find the following.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Composition of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7b98f9composition17a","stepAnswer":["$$-3x^2+x+5$$"],"problemType":"TextBox","stepTitle":"$$f+g$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-3x^2+x+5$$","hints":{"DefaultPathway":[{"id":"a7b98f9composition17a-h1","type":"hint","dependencies":[],"title":"Addition","text":"Remember that $$f+g$$ is the same as $$\\\\left(f+g\\\\right) x$$. All we have to do is add the two functions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition17a-h2","type":"hint","dependencies":["a7b98f9composition17a-h1"],"title":"Combining Like Terms","text":"There are no like terms to combine, so the final answer $$is3 x^2+x+5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a7b98f9composition17b","stepAnswer":["$$-3x^2+x-5$$"],"problemType":"TextBox","stepTitle":"f-g","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-3x^2+x-5$$","hints":{"DefaultPathway":[{"id":"a7b98f9composition17b-h1","type":"hint","dependencies":[],"title":"Subtraction","text":"Remember that f-g is the same as $$(f-g)(x)$$. All we have to do is subtract the two functions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition17b-h2","type":"hint","dependencies":["a7b98f9composition17b-h1"],"title":"Combining Like Terms","text":"There are no like terms to combine, so the final answer is $$-3x^2+x-5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a7b98f9composition17c","stepAnswer":["$$-15x^2+5x$$"],"problemType":"TextBox","stepTitle":"fg","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-15x^2+5x$$","hints":{"DefaultPathway":[{"id":"a7b98f9composition17c-h1","type":"hint","dependencies":[],"title":"Multiplication","text":"Remember that fg is the same as (fg)(x). All we have to do is use foil to multiply the two functions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition17c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-15x^2+5x$$"],"dependencies":["a7b98f9composition17c-h1"],"title":"Multiplication","text":"What is $$5\\\\left(-3x^2+x\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a7b98f9composition17c-h2-s1","type":"hint","dependencies":[],"title":"Combining Like Terms","text":"$$5\\\\left(-3x^2+x\\\\right)$$ is the same as 5(-3x**2+x). After distributing the $$5$$, we are left with $$-15x^2+5x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}},{"id":"a7b98f9composition17d","stepAnswer":["$$\\\\frac{\\\\left(-3x^2+x\\\\right)}{5}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{f}{g}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{\\\\left(-3x^2+x\\\\right)}{5}$$","hints":{"DefaultPathway":[{"id":"a7b98f9composition17d-h1","type":"hint","dependencies":[],"title":"Division","text":"Remember that $$\\\\frac{f}{g}$$ is the same as $$\\\\frac{f}{g} x$$. All we have to do is divide the two functions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition17d-h2","type":"hint","dependencies":["a7b98f9composition17d-h1"],"title":"Division","text":"After dividing the terms, there is nothing to simplify so we will be left with $$\\\\frac{\\\\left(-3x^2+x\\\\right)}{5}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7b98f9composition18","title":"Composition of Funtions With Exponents","body":"Given $$f(x)=2x^2+4x$$ and $$g(x)=12x$$, find the following.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Composition of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7b98f9composition18a","stepAnswer":["$$2x^2+16x$$"],"problemType":"TextBox","stepTitle":"$$f+g$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2x^2+16x$$","hints":{"DefaultPathway":[{"id":"a7b98f9composition18a-h1","type":"hint","dependencies":[],"title":"Addition","text":"Remember that $$f+g$$ is the same as $$\\\\left(f+g\\\\right) x$$. All we have to do is add the two functions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x^2+16x$$"],"dependencies":["a7b98f9composition18a-h1"],"title":"Addition","text":"What is $$2x^2+2x+12x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a7b98f9composition18a-h2-s1","type":"hint","dependencies":[],"title":"Combining Like Terms","text":"$$4x+12x=16x$$, but there is no other $$x^2$$ term, so the final answer is $$2x^2+16x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}},{"id":"a7b98f9composition18b","stepAnswer":["$$2x^2-8x$$"],"problemType":"TextBox","stepTitle":"f-g","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2x^2-8x$$","hints":{"DefaultPathway":[{"id":"a7b98f9composition18b-h1","type":"hint","dependencies":[],"title":"Subtraction","text":"Remember that f-g is the same as $$(f-g)(x)$$. All we have to do is subtract the two functions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition18b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x^2-8x$$"],"dependencies":["a7b98f9composition18b-h1"],"title":"Subtraction","text":"What is $$2x^2+4x-12x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a7b98f9composition18b-h2-s1","type":"hint","dependencies":[],"title":"Combining Like Terms","text":"$$4x-12x=-8x$$, but there is no other $$x^2$$ term, so the final answer is $$2x^2-8x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}},{"id":"a7b98f9composition18c","stepAnswer":["$$24x^3+48x^2$$"],"problemType":"TextBox","stepTitle":"fg","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$24x^3+48x^2$$","hints":{"DefaultPathway":[{"id":"a7b98f9composition18c-h1","type":"hint","dependencies":[],"title":"Multiplication","text":"Remember that fg is the same as $$f g x$$. All we have to do is use foil to multiply the two functions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition18c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$24x^3+48x^2$$"],"dependencies":["a7b98f9composition18c-h1"],"title":"Multiplication","text":"What is $$\\\\left(2x^2+4x\\\\right) 12x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition18c-h3","type":"hint","dependencies":["a7b98f9composition18c-h2"],"title":"Combining Like Terms","text":"After multiplying the products, there are no like terms to combine, so we will be left with $$24x^3+48x^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a7b98f9composition18d","stepAnswer":["$$\\\\frac{x+2}{6}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{f}{g}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{x+2}{6}$$","hints":{"DefaultPathway":[{"id":"a7b98f9composition18d-h1","type":"hint","dependencies":[],"title":"Division","text":"Remember that $$\\\\frac{f}{g}$$ is the same as $$\\\\frac{f}{g} x$$. All we have to do is divide the two functions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition18d-h2","type":"hint","dependencies":["a7b98f9composition18d-h1"],"title":"Division","text":"Let\'s start by putting the equation in fractional form, $$\\\\frac{2x^2+4x}{12x}$$,","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition18d-h3","type":"hint","dependencies":["a7b98f9composition18d-h2"],"title":"Division","text":"Now, we can factor out $$2x$$ from the numerator and denominator: $$\\\\frac{2x\\\\left(x+2\\\\right)}{2x\\\\times6}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition18d-h4","type":"hint","dependencies":["a7b98f9composition18d-h3"],"title":"Division","text":"The $$2x$$ in the numerator and denominatory cancel each other out, so we are left with $$\\\\frac{x+2}{6}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7b98f9composition19","title":"Composition of Rational Functions","body":"Given $$f(x)=\\\\frac{1}{x-4}$$ and $$g(x)=\\\\frac{1}{6-x}$$, find the following.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Composition of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7b98f9composition19a","stepAnswer":["$$\\\\frac{2}{\\\\left(x-4\\\\right) \\\\left(6-x\\\\right)}$$"],"problemType":"TextBox","stepTitle":"$$f+g$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2}{\\\\left(x-4\\\\right) \\\\left(6-x\\\\right)}$$","hints":{"DefaultPathway":[{"id":"a7b98f9composition19a-h1","type":"hint","dependencies":[],"title":"Addition","text":"Remember that $$f+g$$ is the same as $$\\\\left(f+g\\\\right) x$$. All we have to do is add the two functions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition19a-h2","type":"hint","dependencies":["a7b98f9composition19a-h1"],"title":"Addition","text":"First, because these functions are fractions, we need to find a common denominator. Our common denominator is the product of the two existing denominators: $$(x-4)(6-x)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition19a-h3","type":"hint","dependencies":["a7b98f9composition19a-h2"],"title":"Addition","text":"Now we need to multiply the fractions by a factor of $$1$$ to get them to have the common denominator. (1/(x-4))*((6-x)(6-x))=(6-x)/((x-4)(6-x), and (1/(6-x))*((x-4)(x-4))=(x-4)/((x-4)(6-x). Now, we need to add these new fractions together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition19a-h4","type":"hint","dependencies":["a7b98f9composition19a-h3"],"title":"Addition","text":"Our last step is to add the numerators. By adding the $$x$$ terms and the constants, $$\\\\frac{x-4+6-x}{\\\\left(x-4\\\\right) \\\\left(6-x\\\\right)}$$ becomes our final solution: $$\\\\frac{2}{\\\\left(x-4\\\\right) \\\\left(6-x\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a7b98f9composition19b","stepAnswer":["$$\\\\frac{10-2x}{\\\\left(x-4\\\\right) \\\\left(6-x\\\\right)}$$"],"problemType":"TextBox","stepTitle":"f-g","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{10-2x}{\\\\left(x-4\\\\right) \\\\left(6-x\\\\right)}$$","hints":{"DefaultPathway":[{"id":"a7b98f9composition19b-h1","type":"hint","dependencies":[],"title":"Subtraction","text":"Remember that f-g is the same as $$(f-g)(x)$$. All we have to do is subtract the two functions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition19b-h2","type":"hint","dependencies":["a7b98f9composition19b-h1"],"title":"Subtraction","text":"Again, because these functions are fractions, we need to find a common denominator. Our common denominator is still $$(x-4)(6-x)$$. Just like in the addition step, we need to multiply the fractions by a factor of $$1$$ to get them to have the common denominator. We are left with(6-x)/((x-4)(6-x) and (x-4)/((x-4)(6-x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition19b-h3","type":"hint","dependencies":["a7b98f9composition19b-h2"],"title":"Subtraction","text":"Our last step is to add the numerators. By subtracting the $$x$$ terms and the constants, $$\\\\frac{6-x-x-4}{\\\\left(x-4\\\\right) \\\\left(6-x\\\\right)}$$ becomes our final solution: $$\\\\frac{10-2x}{\\\\left(x-4\\\\right) \\\\left(6-x\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a7b98f9composition19c","stepAnswer":["$$\\\\frac{1}{\\\\left(x-4\\\\right) \\\\left(6-x\\\\right)}$$"],"problemType":"TextBox","stepTitle":"fg","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{\\\\left(x-4\\\\right) \\\\left(6-x\\\\right)}$$","hints":{"DefaultPathway":[{"id":"a7b98f9composition19c-h1","type":"hint","dependencies":[],"title":"Multiplication","text":"Remember that fg is the same as $$f g x$$. All we have to do is use foil to multiply the two functions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition19c-h2","type":"hint","dependencies":["a7b98f9composition19c-h1"],"title":"Multiplication","text":"Since we are multiplying fractions, we muliply the numerators together and the denominators together. $$1\\\\times1=1$$, so the numerator stays the same. The denominator becomes $$(x-4)(6-x)$$ which we can leave in this factored form. Our answer is $$\\\\frac{1}{\\\\left(x-4\\\\right) \\\\left(6-x\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a7b98f9composition19d","stepAnswer":["$$\\\\frac{6-x}{x-4}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{f}{g}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{6-x}{x-4}$$","hints":{"DefaultPathway":[{"id":"a7b98f9composition19d-h1","type":"hint","dependencies":[],"title":"Division","text":"Remember that $$\\\\frac{f}{g}$$ is the same as $$\\\\frac{f}{g} x$$. All we have to do is divide the two functions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition19d-h2","type":"hint","dependencies":["a7b98f9composition19d-h1"],"title":"Division","text":"Since these functions are fractions, we need to keep, change, and switch to divide them. We keep $$\\\\frac{1}{x-4}$$ the same, change the division to muliplication, and switch $$\\\\frac{1}{6-x}$$ to $$\\\\frac{6-x}{1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition19d-h3","type":"hint","dependencies":["a7b98f9composition19d-h2"],"title":"Division","text":"We now multiply the numerators and denominators together. $$1\\\\left(6-x\\\\right)=6-x$$, and $$1\\\\left(x-4\\\\right)=x-4$$. So, the final answer is $$\\\\frac{6-x}{x-4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7b98f9composition2","title":"Decomposing a Function","body":"For the following exercise, find functions f(x) and g(x) so the given function can be expressed as $$h(x)=f(g(x))$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Composition of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7b98f9composition2a","stepAnswer":["$$f(x)=|x|$$, $$g(x)=x^2+7$$"],"problemType":"MultipleChoice","stepTitle":"$$h(x)=|x^2+7|$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$f(x)=|x|$$, $$g(x)=x^2+7$$","choices":["$$f(x)=|x|$$, $$g(x)=x^2+7$$","$$f(x)=x^2+7$$, $$g(x)=|x|$$","$$f(x)=x^2 g{\\\\left(x\\\\right)}=|x+7|$$","$$f(x)=|x^2|$$, $$g(x)=x+7$$"],"hints":{"DefaultPathway":[{"id":"a7b98f9composition2a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$f(x)=|x|$$"],"dependencies":[],"title":"First Function","text":"What is the function in which $$x^2+7$$ is inserted to get its absolute value?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$f(x)=|x|$$","$$f(x)=x^2+7$$","$$f(x)=x^2$$"]},{"id":"a7b98f9composition2a-h2","type":"hint","dependencies":["a7b98f9composition2a-h1"],"title":"Second Function","text":"The other function is what you put into the previous function to get the given function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition2a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$g(x)=x^2+7$$"],"dependencies":["a7b98f9composition2a-h2"],"title":"Second Function","text":"What is the second function?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$g(x)=|x|$$","$$g(x)=x^2+7$$","$$g(x)=x^2$$"]}]}}]},{"id":"a7b98f9composition20","title":"Composition of Functions with Square Roots","body":"Given $$f(x)=3x^2$$ and $$g(x)=\\\\sqrt{x-5}$$, find the following.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Composition of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7b98f9composition20a","stepAnswer":["$$3x^2+\\\\sqrt{x-5}$$"],"problemType":"TextBox","stepTitle":"$$f+g$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3x^2+\\\\sqrt{x-5}$$","hints":{"DefaultPathway":[{"id":"a7b98f9composition20a-h1","type":"hint","dependencies":[],"title":"Addition","text":"Remember that $$f+g$$ is the same as $$\\\\left(f+g\\\\right) x$$. All we have to do is add the two functions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition20a-h2","type":"hint","dependencies":["a7b98f9composition20a-h1"],"title":"Addition","text":"There are no like terms to combine, so the answer is just the sum of the two functions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a7b98f9composition20b","stepAnswer":["$$3x^2-\\\\sqrt{x-5}$$"],"problemType":"TextBox","stepTitle":"f-g","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3x^2-\\\\sqrt{x-5}$$","hints":{"DefaultPathway":[{"id":"a7b98f9composition20b-h1","type":"hint","dependencies":[],"title":"Subtraction","text":"Remember that f-g is the same as $$(f-g)(x)$$. All we have to do is add the two functions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition20b-h2","type":"hint","dependencies":["a7b98f9composition20b-h1"],"title":"Subtraction","text":"There are no like terms to combine, so the answer is just the difference of the two functions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a7b98f9composition20c","stepAnswer":["$$3x^2 \\\\sqrt{x-5}$$"],"problemType":"TextBox","stepTitle":"fg","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3x^2 \\\\sqrt{x-5}$$","hints":{"DefaultPathway":[{"id":"a7b98f9composition20c-h1","type":"hint","dependencies":[],"title":"Multiplication","text":"Remember that fg is the same as $$f g x$$. All we have to do is use foil to multiply the two functions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition20c-h2","type":"hint","dependencies":["a7b98f9composition20c-h1"],"title":"Multiplication","text":"There is no way to simplify the product of these functions, so we leave the answer as one function multiplied by the other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a7b98f9composition20d","stepAnswer":["$$\\\\frac{3x^2 \\\\sqrt{x-5}}{x-5}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{f}{g}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3x^2 \\\\sqrt{x-5}}{x-5}$$","hints":{"DefaultPathway":[{"id":"a7b98f9composition20d-h1","type":"hint","dependencies":[],"title":"Division","text":"Remember that $$\\\\frac{f}{g}$$ is the same as $$\\\\frac{f}{g} x$$. All we have to do is divide the two functions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition20d-h2","type":"hint","dependencies":["a7b98f9composition20d-h1"],"title":"Division","text":"Let\'s write the quotient of these functions as a fraction: (3x**2)/(sqrt(x-5). Because the bottom of the function is a square root, we need to rationalize the function by getting rid of the square root in the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition20d-h3","type":"hint","dependencies":["a7b98f9composition20d-h2"],"title":"Rationalization","text":"To do this, we need to square $$\\\\sqrt{x-5}$$ so that the denominator doesn\'t have a square root. So, we have to multiply (3x**2)/(sqrt(x-5) by $$\\\\frac{\\\\sqrt{x-5}}{\\\\sqrt{x-5}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition20d-h4","type":"hint","dependencies":["a7b98f9composition20d-h3"],"title":"Multiplication","text":"Since there is no way to simplify the product of $$3x^2 \\\\sqrt{x-5}$$, this leaves us with ((3x**2)(sqrt(x-5))/(x-5)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7b98f9composition21","title":"Composition of Functions with Square Roots and Absolute Value","body":"For $$f(x)=\\\\sqrt{x}$$ and $$g(x)=|x-3|$$, find the following.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Composition of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7b98f9composition21a","stepAnswer":["$$\\\\frac{\\\\sqrt{x} |x-3|}{x}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{g}{f}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{\\\\sqrt{x} |x-3|}{x}$$","hints":{"DefaultPathway":[{"id":"a7b98f9composition21a-h1","type":"hint","dependencies":[],"title":"Division","text":"Remember that $$\\\\frac{g}{f}$$ is the same as $$\\\\frac{g}{f} x$$. All we have to do is divide the two functions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition21a-h2","type":"hint","dependencies":["a7b98f9composition21a-h1"],"title":"Division","text":"Let\'s write the quotient of these functions as a fraction: $$\\\\frac{|x-3|}{\\\\sqrt{x}}$$. Because the bottom of the function is a square root, we need to rationalize the function by getting rid of the square root in the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition21a-h3","type":"hint","dependencies":["a7b98f9composition21a-h2"],"title":"Rationalization","text":"To do this, we need to square $$\\\\sqrt{x}$$ so that the denominator doesn\'t have a square root. So, we have to multiply $$\\\\frac{|x-3|}{\\\\sqrt{x}}$$ by $$\\\\frac{\\\\sqrt{x}}{\\\\sqrt{x}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition21a-h4","type":"hint","dependencies":["a7b98f9composition21a-h3"],"title":"Multiplication","text":"Since there is no way to simplify the product of $$\\\\sqrt{x} |x-3|$$, this leaves us with $$\\\\frac{\\\\sqrt{x} |x-3|}{x}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7b98f9composition22","title":"Computing Compositions of Functions","body":"Find the indicated function given $$f(x)=2x^2+1$$ and $$g(x)=3x-5$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Composition of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7b98f9composition22a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"f(g(2))","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a7b98f9composition22a-h1","type":"hint","dependencies":[],"title":"Composition","text":"Remember that to solve f(g(2)), we need to first find the value of g(2), then plug that value in to f(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition22a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a7b98f9composition22a-h1"],"title":"Finding g(2)","text":"To find g(2), we need to plug in $$2$$ for the $$x$$ in the function g: $$3(2)-5$$. What is g(2)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a7b98f9composition22a-h2-s1","type":"hint","dependencies":[],"title":"Multiplication","text":"Based off the Order of Operations, the first step is the multiplication: $$3\\\\times2=6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition22a-h2-s2","type":"hint","dependencies":["a7b98f9composition22a-h2-s1"],"title":"Subtraction","text":"Now, we need to subtract $$5$$ from $$6$$. $$6-5=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a7b98f9composition22a-h3","type":"hint","dependencies":["a7b98f9composition22a-h2"],"title":"Finding f(g(2))","text":"Now, we can plug in the value of g(2) as the $$x$$ into f(x). In other words, we are evaluating f(1).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition22a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a7b98f9composition22a-h3"],"title":"Finding f(1)","text":"To find f(1), we plug in $$1$$ for all $$x$$ in f(x), and we get the expression $$2\\\\times1^2+1$$. What does this evaluate to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a7b98f9composition22b","stepAnswer":["$$6x^2-2$$"],"problemType":"TextBox","stepTitle":"g(f(x))","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6x^2-2$$","hints":{"DefaultPathway":[{"id":"a7b98f9composition22b-h1","type":"hint","dependencies":[],"title":"Composition","text":"Remember that to solve g(f(x)), we need to plug in the value of f(x) into the value of $$x$$ in g(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition22b-h2","type":"hint","dependencies":["a7b98f9composition22b-h1"],"title":"Composition","text":"We need to plug in f(x), or $$2x^2+1$$, for the $$x$$ in the function g: $$3\\\\left(2x^2+1\\\\right)+5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition22b-h3","type":"hint","dependencies":["a7b98f9composition22b-h2"],"title":"Multiplication","text":"Based off the Order of Operations, the first step is the multiply and distribute the 3: $$6x^2+3-5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition22b-h4","type":"hint","dependencies":["a7b98f9composition22b-h3"],"title":"Subtraction","text":"Now, our composition function reads $$6x^2+3-5$$. The next step is to subtract: $$6x^2+3-5=6x^2-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a7b98f9composition22c","stepAnswer":["$$9$$"],"problemType":"TextBox","stepTitle":"$$f(f(-2))$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9$$","hints":{"DefaultPathway":[{"id":"a7b98f9composition22c-h1","type":"hint","dependencies":[],"title":"Composition","text":"Remember that to solve $$f(f(-2))$$, we need to first find the value of $$f(-2)$$, then plug that value back in to f(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition22c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a7b98f9composition22c-h1"],"title":"Finding $$f(-2)$$","text":"To find $$f(-2)$$, we need to plug in $$-2$$ for the $$x$$ in the function f: (2(-2)**2)+1. What does this evaluate to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a7b98f9composition22c-h2-s1","type":"hint","dependencies":[],"title":"Exponents","text":"Based off the Order of Operations, the first step is the exponent: $${\\\\left(-2\\\\right)}^2=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition22c-h2-s2","type":"hint","dependencies":["a7b98f9composition22c-h2-s1"],"title":"Multiplication","text":"Now, our expression reads $$2\\\\times4+1$$. The next step is the multiplication: $$2\\\\times4=8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition22c-h2-s3","type":"hint","dependencies":["a7b98f9composition22c-h2-s2"],"title":"Addition","text":"Lastly, our expression is $$8+1$$, which is equal to $$9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a7b98f9composition22c-h3","type":"hint","dependencies":["a7b98f9composition22c-h2"],"title":"Finding $$f(f(-2))$$","text":"Our next step is to plug the value of $$f(-2)$$ into f. In other words, we are evaluating f(9).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition22c-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a7b98f9composition22c-h3"],"title":"Finding $$f(f(-2))$$","text":"To find $$f(-2)$$, we need to plug in $$-2$$ for the $$x$$ in the function f: (2(-2)**2)+1. What does this evaluate to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7b98f9composition23","title":"Composition of Functions","body":"For $$f(x)=x^2+1$$ and $$g(x)=\\\\sqrt{x+2}$$, find the following.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Composition of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7b98f9composition23a","stepAnswer":["$$x+3$$"],"problemType":"TextBox","stepTitle":"f(g(x))","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x+3$$","hints":{"DefaultPathway":[{"id":"a7b98f9composition23a-h1","type":"hint","dependencies":[],"title":"Composition","text":"Remember that to solve f(g(x)), we need to plug in the value of g(x) into the value of $$x$$ in f(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition23a-h2","type":"hint","dependencies":["a7b98f9composition23a-h1"],"title":"Composition","text":"We need to plug in g(x), or $$\\\\sqrt{x-2}$$, for the $$x$$ in the function f: $${\\\\sqrt{x+2}}^2+1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition23a-h3","type":"hint","dependencies":["a7b98f9composition23a-h2"],"title":"Exponents","text":"Based off the Order of Operations, the first step is the exponents: $${\\\\sqrt{x+2}}^2=x+2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition23a-h4","type":"hint","dependencies":["a7b98f9composition23a-h3"],"title":"Multiplication","text":"Now, our composition function reads $$x+2+1$$. The next step is to add the constants: $$x+2+1=x+3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a7b98f9composition23b","stepAnswer":["$$\\\\sqrt{x^2+3}$$"],"problemType":"TextBox","stepTitle":"g(f(x))","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\sqrt{x^2+3}$$","hints":{"DefaultPathway":[{"id":"a7b98f9composition23b-h1","type":"hint","dependencies":[],"title":"Composition","text":"Remember that to solve g(f(x)), we need to plug in the value of f(x) into the value of $$x$$ in g(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition23b-h2","type":"hint","dependencies":["a7b98f9composition23b-h1"],"title":"Composition","text":"We need to plug in f(x), or $$x^2+1$$, for the $$x$$ in the function f: $$\\\\sqrt{x^2+1+2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition23b-h3","type":"hint","dependencies":["a7b98f9composition23b-h2"],"title":"Addition","text":"We need to add the constants: $$\\\\sqrt{x^2+2+1}=\\\\sqrt{x^2+3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7b98f9composition24","title":"Composition of Functions","body":"For $$f(x)=\\\\sqrt{x}+2$$ and $$g(x)=x^2+3$$, find the following.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Composition of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7b98f9composition24a","stepAnswer":["$$\\\\sqrt{x^2+3}+2$$"],"problemType":"TextBox","stepTitle":"f(g(x))","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\sqrt{x^2+3}+2$$","hints":{"DefaultPathway":[{"id":"a7b98f9composition24a-h1","type":"hint","dependencies":[],"title":"Composition","text":"Remember that to solve f(g(x)), we need to plug in the value of g(x) into the value of $$x$$ in f(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition24a-h2","type":"hint","dependencies":["a7b98f9composition24a-h1"],"title":"Composition","text":"We need to plug in g(x), or $$x^2+3$$, for the $$x$$ in the function f: $$\\\\sqrt{x^2+3}+2$$. There is no way to simplify this expression, so we are left with our final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a7b98f9composition24b","stepAnswer":["$$x+4\\\\left(\\\\sqrt{x}\\\\right)+7$$"],"problemType":"TextBox","stepTitle":"g(f(x))","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x+4\\\\left(\\\\sqrt{x}\\\\right)+7$$","hints":{"DefaultPathway":[{"id":"a7b98f9composition24b-h1","type":"hint","dependencies":[],"title":"Composition","text":"Remember that to solve g(f(x)), we need to plug in the value of f(x) into the value of $$x$$ in g(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition24b-h2","type":"hint","dependencies":["a7b98f9composition24b-h1"],"title":"Composition","text":"We need to plug in f(x), or $$\\\\sqrt{x}+2$$, for the $$x$$ in the function f: $${\\\\left(\\\\sqrt{x}+2\\\\right)}^2+3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition24b-h3","type":"hint","dependencies":["a7b98f9composition24b-h2"],"title":"Exponents","text":"Based off the Order of Operations, the first step is the exponent: $${\\\\left(\\\\sqrt{x}+2\\\\right)}^2=x+4\\\\left(\\\\sqrt{x}\\\\right)+4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition24b-h4","type":"hint","dependencies":["a7b98f9composition24b-h3"],"title":"Addition","text":"The expression now reads $$x+4\\\\left(\\\\sqrt{x}\\\\right)+4+3$$. The last step is to add the constants: $$x+4\\\\left(\\\\sqrt{x}\\\\right)+4+3=$$ $$x+4\\\\left(\\\\sqrt{x}\\\\right)+7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7b98f9composition3","title":"Decomposing a Function","body":"For the following exercise, find functions f(x) and g(x) so the given function can be expressed as $$h(x)=f(g(x))$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Composition of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7b98f9composition3a","stepAnswer":["$$f(x)=\\\\frac{1}{x^2}$$, $$g(x)=x-2$$"],"problemType":"MultipleChoice","stepTitle":"$$h(x)=$$ (1/(x-2)**2","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$f(x)=\\\\frac{1}{x^2}$$, $$g(x)=x-2$$","choices":["$$f(x)=x^2$$, $$g(x)=x-2$$","$$f(x)=x-2$$, $$g(x)=x^2$$","$$f(x)=\\\\frac{1}{x^2}$$, $$g(x)=x-2$$","$$f(x)=x-2$$, $$g(x)=\\\\frac{1}{x^2}$$"],"hints":{"DefaultPathway":[{"id":"a7b98f9composition3a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$f(x)=\\\\frac{1}{x^2}$$"],"dependencies":[],"title":"First Function","text":"What is the function in which $$x-2$$ is inserted to get the given function?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$f(x)=x^2$$","$$f(x)=x-2$$","$$f(x)=\\\\frac{1}{x^2}$$","$$f(x)=x-2$$"]},{"id":"a7b98f9composition3a-h2","type":"hint","dependencies":["a7b98f9composition3a-h1"],"title":"Second Function","text":"The other function is what you put into the previous function to get the given function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition3a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$g(x)=x-2$$"],"dependencies":["a7b98f9composition3a-h2"],"title":"Second Function","text":"What is the second function?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$g(x)=x-2$$","$$g(x)=x^2$$","$$g(x)=\\\\frac{1}{x^2}$$"]}]}}]},{"id":"a7b98f9composition4","title":"Decomposing a Function","body":"For the following exercise, find functions f(x) and g(x) so the given function can be expressed as $$h(x)=f(g(x))$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Composition of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7b98f9composition4a","stepAnswer":["$$f(x)={\\\\left(\\\\frac{1}{x}\\\\right)}^2$$, $$g(x)=2x-3$$"],"problemType":"MultipleChoice","stepTitle":"$$h(x)={\\\\left(\\\\frac{1}{2x-3}\\\\right)}^2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$f(x)={\\\\left(\\\\frac{1}{x}\\\\right)}^2$$, $$g(x)=2x-3$$","choices":["$$f(x)=x^2$$, $$g(x)=2x-3$$","$$f(x)=2x-3$$, $$g(x)=x^2$$","$$f(x)={\\\\left(\\\\frac{1}{x}\\\\right)}^2$$, $$g(x)=2x-3$$","$$f(x)=2x$$, $$g(x)=\\\\frac{1}{x}$$"],"hints":{"DefaultPathway":[{"id":"a7b98f9composition4a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$f(x)={\\\\left(\\\\frac{1}{x}\\\\right)}^2$$"],"dependencies":[],"title":"First Function","text":"What is the function in which $$2x-3$$ is inserted to get the given function?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$f(x)=x^2$$","$$f(x)=2x-3$$","$$f(x)={\\\\left(\\\\frac{1}{x}\\\\right)}^2$$","$$f(x)=2x$$"]},{"id":"a7b98f9composition4a-h2","type":"hint","dependencies":["a7b98f9composition4a-h1"],"title":"Second Function","text":"The other function is what you put into the previous function to get the given function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition4a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$g(x)=2x-3$$"],"dependencies":["a7b98f9composition4a-h2"],"title":"Second Function","text":"What is the second function?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$g(x)=2x-3$$","$$g(x)=x^2$$","$$g(x)=\\\\frac{1}{x}$$"]}]}}]},{"id":"a7b98f9composition5","title":"Decomposing a Function","body":"For the following exercise, find functions f(x) and g(x) so the given function can be expressed as $$h(x)=f(g(x))$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Composition of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7b98f9composition5a","stepAnswer":["$$f(x)=\\\\sqrt{x}$$, $$g(x)=\\\\frac{2x-1}{3x+4}$$"],"problemType":"MultipleChoice","stepTitle":"$$h(x)=\\\\sqrt{\\\\frac{2x-1}{3x+4}}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$f(x)=\\\\sqrt{x}$$, $$g(x)=\\\\frac{2x-1}{3x+4}$$","choices":["$$f(x)=\\\\frac{2x-1}{3x+4}$$, $$g(x)=\\\\sqrt{x}$$","$$f(x)=\\\\sqrt{x}$$, $$g(x)=\\\\frac{2x-1}{3x+4}$$","$$f(x)=\\\\frac{2}{3} x$$, $$g(x)=\\\\sqrt{x+3}$$","$$f(x)=\\\\sqrt{x+3}$$, $$g(x)=\\\\frac{2}{3} x$$"],"hints":{"DefaultPathway":[{"id":"a7b98f9composition5a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$f(x)=\\\\sqrt{x}$$"],"dependencies":[],"title":"First Function","text":"What is the function in which $$\\\\frac{2x+1}{3x+4}$$ is inserted to get the given function?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$f(x)=\\\\frac{2x-1}{3x+4}$$","$$f(x)=\\\\sqrt{x}$$","$$f(x)=\\\\frac{2}{3} x$$","$$f(x)=\\\\sqrt{x+3}$$"]},{"id":"a7b98f9composition5a-h2","type":"hint","dependencies":["a7b98f9composition5a-h1"],"title":"Second Function","text":"The other function is what you put into the previous function to get the given function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition5a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$g(x)=\\\\frac{2x-1}{3x+4}$$"],"dependencies":["a7b98f9composition5a-h2"],"title":"Second Function","text":"What is the second function?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$g(x)=\\\\sqrt{x}$$","$$g(x)=\\\\frac{2x-1}{3x+4}$$","$$g(x)=\\\\sqrt{x+3}$$","$$g(x)=\\\\frac{2}{3} x$$"]}]}}]},{"id":"a7b98f9composition6","title":"Decomposing a Function","body":"Find functions f(x) and g(x) so the given function can be expressed as $$h(x)=f(g(x))$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Composition of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7b98f9composition6a","stepAnswer":["$$g(x)=x+2$$, $$f(x)=x^2$$"],"problemType":"MultipleChoice","stepTitle":"$$h(x)={\\\\left(x+2\\\\right)}^2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$g(x)=x+2$$, $$f(x)=x^2$$","choices":["$$g(x)=x$$, $$f(x)=x+2$$","$$g(x)=x+2$$, $$f(x)=x^2$$","$$g(x)=x^2$$, $$f(x)=x+2$$"],"hints":{"DefaultPathway":[{"id":"a7b98f9composition6a-h1","type":"hint","dependencies":[],"title":"Looking for an Inner Function","text":"The first step is to look for a function inside a function in the formula for f(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition6a-h2","type":"hint","dependencies":["a7b98f9composition6a-h1"],"title":"Inner Function","text":"The expression $$x+2$$ is on the inside of the square.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition6a-h3","type":"hint","dependencies":["a7b98f9composition6a-h2"],"title":"Outer Function","text":"The quantity $$x+2$$ is then squared.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition6a-h4","type":"hint","dependencies":["a7b98f9composition6a-h3"],"title":"Answer","text":"$$g(x)=x+2$$, $$f(x)=x^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7b98f9composition7","title":"Decomposing a Function","body":"Find functions f(x) and g(x) so the given function can be expressed as $$h(x)=f(g(x))$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Composition of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7b98f9composition7a","stepAnswer":["$$g(x)=x-5$$, $$f(x)=x^3$$"],"problemType":"MultipleChoice","stepTitle":"$$h(x)={\\\\left(x-5\\\\right)}^3$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$g(x)=x-5$$, $$f(x)=x^3$$","choices":["$$g(x)={\\\\left(x-5\\\\right)}^3$$, $$f(x)=x-5$$","$$g(x)=x^3$$, $$f(x)=x-5$$","$$g(x)=x-5$$, $$f(x)=x^3$$"],"hints":{"DefaultPathway":[{"id":"a7b98f9composition7a-h1","type":"hint","dependencies":[],"title":"Looking for an Inner Function","text":"The first step is to look for a function inside a function in the formula for f(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition7a-h2","type":"hint","dependencies":["a7b98f9composition7a-h1"],"title":"Inner Function","text":"The expression $$x-5$$ is on the inside of the cubed expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition7a-h3","type":"hint","dependencies":["a7b98f9composition7a-h2"],"title":"Outer Function","text":"The quantity $$x-5$$ is cubed (raised to the power of three.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition7a-h4","type":"hint","dependencies":["a7b98f9composition7a-h3"],"title":"Answer","text":"$$g(x)=x-5$$, $$f(x)=x^3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7b98f9composition8","title":"Decomposing a Function","body":"Find functions f(x) and g(x) so the given function can be expressed as $$h(x)=f(g(x))$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Composition of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7b98f9composition8a","stepAnswer":["$$g(x)=x-5$$, $$f(x)=\\\\frac{3}{x}$$"],"problemType":"MultipleChoice","stepTitle":"$$h(x)=\\\\frac{3}{x-5}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$g(x)=x-5$$, $$f(x)=\\\\frac{3}{x}$$","choices":["$$g(x)=x-5$$, $$f(x)=\\\\frac{3}{x-5}$$","$$g(x)=x-5$$, $$f(x)=\\\\frac{3}{x}$$","$$g(x)=\\\\frac{3}{x}$$, $$f(x)=x-5$$"],"hints":{"DefaultPathway":[{"id":"a7b98f9composition8a-h1","type":"hint","dependencies":[],"title":"Looking for an Inner Function","text":"The first step is to look for a function inside a function in the formula for f(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition8a-h2","type":"hint","dependencies":["a7b98f9composition8a-h1"],"title":"Inner Function","text":"The expression $$x-5$$ is the denominator of the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition8a-h3","type":"hint","dependencies":["a7b98f9composition8a-h2"],"title":"Outer Function","text":"$$3$$ is divided by the denominator of the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition8a-h4","type":"hint","dependencies":["a7b98f9composition8a-h3"],"title":"Answer","text":"$$g(x)=x-5$$, $$f(x)=\\\\frac{3}{x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7b98f9composition9","title":"Decomposing a Function","body":"Find functions f(x) and g(x) so the given function can be expressed as $$h(x)=f(g(x))$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Composition of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7b98f9composition9a","stepAnswer":["$$g(x)={\\\\left(x+2\\\\right)}^2$$, $$f(x)=\\\\frac{4}{x}$$"],"problemType":"MultipleChoice","stepTitle":"$$h(x)=\\\\frac{4}{{\\\\left(x+2\\\\right)}^2}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$g(x)={\\\\left(x+2\\\\right)}^2$$, $$f(x)=\\\\frac{4}{x}$$","choices":["$$g(x)={\\\\left(x+2\\\\right)}^2$$, $$f(x)=\\\\frac{4}{x}$$","$$g(x)=x+2$$ $$f(x)=x^2$$","$$g(x)=\\\\frac{4}{x}$$, $$f(x)=x+2$$"],"hints":{"DefaultPathway":[{"id":"a7b98f9composition9a-h1","type":"hint","dependencies":[],"title":"Looking for an Inner Function","text":"The first step is to look for a function inside a function in the formula for f(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition9a-h2","type":"hint","dependencies":["a7b98f9composition9a-h1"],"title":"Inner Function","text":"The expression $${\\\\left(x+2\\\\right)}^2$$ is the denominator of the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition9a-h3","type":"hint","dependencies":["a7b98f9composition9a-h2"],"title":"Outer Function","text":"$$4$$ is divided by the denominator of the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7b98f9composition9a-h4","type":"hint","dependencies":["a7b98f9composition9a-h3"],"title":"Answer","text":"$$g(x)={\\\\left(x+2\\\\right)}^2$$, 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Suppose the ball was instead thrown from the top of a $$10m$$ building. Relate this new height function b(t) to h(t) and then find a formula for b(t).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$b(t)=-4.9t^2+30t+10$$","hints":{"DefaultPathway":[{"id":"a7dc5fftransformation1a-h1","type":"hint","dependencies":[],"title":"Moving higher from the ground by $$10m$$ means that you are throwing the ball $$10m$$ higher every time because you are on a building.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$b(t)=-4.9t^2+30t+10$$"],"dependencies":["a7dc5fftransformation1a-h1"],"title":"What is the final expression if we increase everything by 10?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7dc5fftransformation10","title":"Finding Transformed Functions","body":"Write a formula for the function g that results when the graph of a given function is transformed as described.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Transformation of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7dc5fftransformation10a","stepAnswer":["$$-\\\\sqrt{\\\\frac{1}{2} x}$$"],"problemType":"MultipleChoice","stepTitle":"The graph of $$f(x)=\\\\sqrt{x}$$ is reflected over the x-axis and horizontally stretched by a factor of $$2$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-\\\\sqrt{\\\\frac{1}{2} x}$$","choices":["$$\\\\sqrt{\\\\frac{1}{2} x}$$","$$-\\\\sqrt{\\\\frac{1}{2} x}$$","$$-\\\\sqrt{2x}$$","$$\\\\sqrt{2x}$$"],"hints":{"DefaultPathway":[{"id":"a7dc5fftransformation10a-h1","type":"hint","dependencies":[],"title":"Order of Transformations","text":"The first step is to recognize the order of transformations. 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The coefficient needed for a horizontal stretch or compression is the reciprocal of the stretch or compression. So to compress the graph horizontally by a scale factor of $$2$$, we need a coefficient of $$\\\\frac{1}{2}$$ in our function. Therefore, replace every $$x$$ in the function with $$\\\\frac{1}{2} x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation10a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-\\\\sqrt{\\\\frac{1}{2} x}$$"],"dependencies":["a7dc5fftransformation10a-h4"],"title":"Horizontal Compression","text":"What is the function after it has been horizontally compressed by a factor of $$\\\\frac{1}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\sqrt{\\\\frac{1}{2} x}$$","$$-\\\\sqrt{\\\\frac{1}{2} x}$$","$$-\\\\sqrt{2x}$$","$$\\\\sqrt{2x}$$"]}]}}]},{"id":"a7dc5fftransformation11","title":"Finding Transformed Functions","body":"Write a formula for the function g that results when the graph of a given function is transformed as described.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Transformation of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7dc5fftransformation11a","stepAnswer":["$$\\\\frac{1}{3{\\\\left(x+2\\\\right)}^2}-3$$"],"problemType":"MultipleChoice","stepTitle":"The graph of $$f(x)=\\\\frac{1}{x^2}$$ is vertically compressed by a factor of $$\\\\frac{1}{3}$$, then shifted to the left $$2$$ units and down $$3$$ units.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{1}{3{\\\\left(x+2\\\\right)}^2}-3$$","choices":["$$\\\\frac{1}{\\\\frac{1}{3} {\\\\left(x+2\\\\right)}^2}-3$$","$$\\\\frac{1}{\\\\frac{1}{3} {\\\\left(x-2\\\\right)}^2}+3$$","$$\\\\frac{1}{3{\\\\left(x-2\\\\right)}^2}+3$$","$$\\\\frac{1}{3{\\\\left(x+2\\\\right)}^2}-3$$"],"hints":{"DefaultPathway":[{"id":"a7dc5fftransformation11a-h1","type":"hint","dependencies":[],"title":"Order of Transformations","text":"The first step is to recognize the order of transformations. First, $$\\\\frac{1}{x^2}$$ is vertically compressed by a factor of $$\\\\frac{1}{3}$$, then, shifted to the left $$2$$ units, and lastly, shifted down $$3$$ units.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation11a-h2","type":"hint","dependencies":["a7dc5fftransformation11a-h1"],"title":"Vertically Compressing by a Factor of $$3$$","text":"To vertically compress the function by a factor of $$\\\\frac{1}{3}$$, multiply the entire function by $$\\\\frac{1}{3}$$ to get $$\\\\frac{1}{3} f{\\\\left(x\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation11a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1}{3x^2}$$"],"dependencies":["a7dc5fftransformation11a-h2"],"title":"Vertically Compressing by a Factor of $$3$$","text":"What is $$\\\\frac{1}{3} f{\\\\left(x\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{1}{\\\\frac{1}{3} x^2}$$","$$\\\\frac{1}{3x^2}$$"]},{"id":"a7dc5fftransformation11a-h4","type":"hint","dependencies":["a7dc5fftransformation11a-h3"],"title":"Shifting Left Two Units","text":"To shift the function left two units, replace $$x$$ with $$x+2$$ in the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation11a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1}{3{\\\\left(x+2\\\\right)}^2}$$"],"dependencies":["a7dc5fftransformation11a-h4"],"title":"Shifting Left Two Units","text":"What is the function after it has been shifted left $$2$$ units?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{1}{3{\\\\left(x+2\\\\right)}^2}$$","$$\\\\frac{1}{3{\\\\left(x-2\\\\right)}^2}$$"]},{"id":"a7dc5fftransformation11a-h6","type":"hint","dependencies":["a7dc5fftransformation11a-h5"],"title":"Shifting Down Three Units","text":"To shift the function down three units, subtract $$3$$ from the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation11a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1}{3{\\\\left(x+2\\\\right)}^2}-3$$"],"dependencies":["a7dc5fftransformation11a-h6"],"title":"Shifting Down Three Units","text":"What is the function after it has been shifted down $$3$$ units?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{1}{\\\\frac{1}{3} {\\\\left(x+2\\\\right)}^2}-3$$","$$\\\\frac{1}{\\\\frac{1}{3} {\\\\left(x-2\\\\right)}^2}+3$$","$$\\\\frac{1}{3{\\\\left(x-2\\\\right)}^2}+3$$","$$\\\\frac{1}{3{\\\\left(x+2\\\\right)}^2}-3$$"]}]}}]},{"id":"a7dc5fftransformation12","title":"Finding Transformed Functions","body":"Write a formula for the function g that results when the graph of a given function is transformed as described.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Transformation of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7dc5fftransformation12a","stepAnswer":["$$8\\\\left(x-4\\\\right)-2$$"],"problemType":"MultipleChoice","stepTitle":"The graph of $$f(x)=\\\\frac{1}{x}$$ is vertically stretched by a factor of $$8$$, then shifted to the right $$4$$ units and up $$2$$ units.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$8\\\\left(x-4\\\\right)-2$$","choices":["$$8\\\\left(x+4\\\\right)+2$$","$$8\\\\left(x-4\\\\right)-2$$","$$\\\\frac{1}{8} \\\\left(x-4\\\\right)-2$$","$$\\\\frac{1}{8} \\\\left(x+4\\\\right)+2$$"],"hints":{"DefaultPathway":[{"id":"a7dc5fftransformation12a-h1","type":"hint","dependencies":[],"title":"Order of Transformations","text":"The first step is to recognize the order of transformations. First, $$\\\\frac{1}{x}$$ is vertically stretched by a factor of $$8$$, then, shifted to the right $$4$$ units, and lastly, shifted down $$2$$ units.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation12a-h2","type":"hint","dependencies":["a7dc5fftransformation12a-h1"],"title":"Vertically Stretching by a Factor of $$8$$","text":"To vertically stretch the function by a factor of $$8$$, multiply the entire function by $$8$$ to get 8f(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation12a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{8}{x}$$"],"dependencies":["a7dc5fftransformation12a-h2"],"title":"Vertically Stretching by a Factor of $$8$$","text":"What is $$8f{\\\\left(x\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{8}{x}$$","$$\\\\frac{1}{8x}$$"]},{"id":"a7dc5fftransformation12a-h4","type":"hint","dependencies":["a7dc5fftransformation12a-h3"],"title":"Shifting Right Four Units","text":"To shift the function right four units, replace $$x$$ with $$x-4$$ in the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation12a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{8}{x-4}$$"],"dependencies":["a7dc5fftransformation12a-h4"],"title":"Shifting Right Four Units","text":"What is the function after it has been shifted left four units?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{8}{x+4}$$","$$\\\\frac{8}{x-4}$$"]},{"id":"a7dc5fftransformation12a-h6","type":"hint","dependencies":["a7dc5fftransformation12a-h5"],"title":"Shifting Down Two Units","text":"To shift the function down two units, subtract $$2$$ from the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation12a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$8\\\\left(x-4\\\\right)-2$$"],"dependencies":["a7dc5fftransformation12a-h6"],"title":"Shifting Down Two Units","text":"What is the function after it has been shifted down $$2$$ units?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$8\\\\left(x+4\\\\right)+2$$","$$8\\\\left(x-4\\\\right)-2$$","$$\\\\frac{1}{8} \\\\left(x-4\\\\right)-2$$","$$\\\\frac{1}{8} \\\\left(x+4\\\\right)+2$$"]}]}}]},{"id":"a7dc5fftransformation13","title":"Finding Transformed Functions","body":"Write a formula for the function g that results when the graph of a given function is transformed as described.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Transformation of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7dc5fftransformation13a","stepAnswer":["$$\\\\frac{1}{2} {\\\\left(x-5\\\\right)}^2+1$$"],"problemType":"MultipleChoice","stepTitle":"The graph of $$f(x)=x^2$$ is vertically compressed by a factor of $$\\\\frac{1}{2}$$, then shifted to the right $$5$$ units and up $$1$$ unit.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{1}{2} {\\\\left(x-5\\\\right)}^2+1$$","choices":["$$\\\\frac{1}{2} {\\\\left(x+5\\\\right)}^2+1$$","$$\\\\frac{1}{2} {\\\\left(x-5\\\\right)}^2+1$$","$$\\\\frac{1}{2} {\\\\left(x-5\\\\right)}^2-1$$","$$2{\\\\left(x+5\\\\right)}^2-1$$"],"hints":{"DefaultPathway":[{"id":"a7dc5fftransformation13a-h1","type":"hint","dependencies":[],"title":"Order of Transformations","text":"The first step is to recognize the order of transformations. First, $$x^2$$ is vertically compressed by a factor of $$\\\\frac{1}{2}$$, then, shifted to the right $$5$$ units, and lastly, shifted up $$1$$ unit.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation13a-h2","type":"hint","dependencies":["a7dc5fftransformation13a-h1"],"title":"Vertically Compressing by a Factor of $$3$$","text":"To vertically compress the function by a factor of $$\\\\frac{1}{2}$$, multiply the entire function by $$\\\\frac{1}{2}$$ to get $$\\\\frac{1}{2} f{\\\\left(x\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation13a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1}{2} x^2$$"],"dependencies":["a7dc5fftransformation13a-h2"],"title":"Vertically Compressing by a Factor of $$3$$","text":"What is $$\\\\frac{1}{2} f{\\\\left(x\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2x^2$$","$$\\\\frac{1}{2} x^2$$"]},{"id":"a7dc5fftransformation13a-h4","type":"hint","dependencies":["a7dc5fftransformation13a-h3"],"title":"Shifting Right Five Units","text":"To shift the function right five units, replace $$x$$ with $$x-5$$ in the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation13a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1}{2} {\\\\left(x-5\\\\right)}^2$$"],"dependencies":["a7dc5fftransformation13a-h4"],"title":"Shifting Right Five Units","text":"What is the function after it has been shifted right $$5$$ units?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{1}{2} {\\\\left(x-5\\\\right)}^2$$","$$\\\\frac{1}{2} {\\\\left(x+5\\\\right)}^2$$"]},{"id":"a7dc5fftransformation13a-h6","type":"hint","dependencies":["a7dc5fftransformation13a-h5"],"title":"Shifting Up One Unit","text":"To shift the function up one unit, add $$1$$ to the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation13a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1}{2} {\\\\left(x-5\\\\right)}^2+1$$"],"dependencies":["a7dc5fftransformation13a-h6"],"title":"Shifting Up One Unit","text":"What is the function after it has been shifted up $$1$$ unit?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{1}{2} {\\\\left(x+5\\\\right)}^2+1$$","$$\\\\frac{1}{2} {\\\\left(x-5\\\\right)}^2+1$$","$$\\\\frac{1}{2} {\\\\left(x-5\\\\right)}^2-1$$","$$2{\\\\left(x+5\\\\right)}^2-1$$"]}]}}]},{"id":"a7dc5fftransformation14","title":"Finding Transformed Functions","body":"Write a formula for the function g that results when the graph of a given function is transformed as described.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Transformation of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7dc5fftransformation14a","stepAnswer":["$$\\\\frac{1}{9} {\\\\left(x-4\\\\right)}^2-3$$"],"problemType":"MultipleChoice","stepTitle":"The graph of $$f(x)=x^2$$ is horizontally stretched by a factor of $$3$$, then shifted to the left $$4$$ units and down $$3$$ units.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{1}{9} {\\\\left(x-4\\\\right)}^2-3$$","choices":["$$\\\\frac{1}{3} {\\\\left(x-4\\\\right)}^2+3$$","$$\\\\frac{1}{9} {\\\\left(x-4\\\\right)}^2-3$$","$$9{\\\\left(x-4\\\\right)}^2-3$$","$$3{\\\\left(x-4\\\\right)}^2+3$$"],"hints":{"DefaultPathway":[{"id":"a7dc5fftransformation14a-h1","type":"hint","dependencies":[],"title":"Order of Transformations","text":"The first step is to recognize the order of transformations. First, $$x^2$$ is horizontally stretched by a factor of $$3$$, then, shifted to the right $$4$$ units, and lastly, shifted down $$3$$ units.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation14a-h2","type":"hint","dependencies":["a7dc5fftransformation14a-h1"],"title":"How to Horizontally Compress a Function","text":"The second step is to stretch the graph horizontally by a scale factor of $$3$$. The coefficient needed for a horizontal stretch or compression is the reciprocal of the stretch or compression. So to compress the graph horizontally by a scale factor of $$3$$, we need a coefficient of $$\\\\frac{1}{3}$$ in our function. Therefore, replace every $$x$$ in the function with $$\\\\frac{1}{3} x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation14a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1}{9} x^2$$"],"dependencies":["a7dc5fftransformation14a-h2"],"title":"Horizontal Compression","text":"What is the function after it has been horizontally compressed by a factor of 3?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{1}{9} x^2$$","$$\\\\frac{1}{3} x^2$$","$$9x^2$$","$$3x^2$$"]},{"id":"a7dc5fftransformation14a-h4","type":"hint","dependencies":["a7dc5fftransformation14a-h3"],"title":"Shifting Right Four Units","text":"To shift the function right four units, replace $$x$$ with $$x-4$$ in the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation14a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1}{9} {\\\\left(x-4\\\\right)}^2$$"],"dependencies":["a7dc5fftransformation14a-h4"],"title":"Shifting Right Four Units","text":"What is the function after it has been shifted left four units?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{1}{9} {\\\\left(x+4\\\\right)}^2$$","$$\\\\frac{1}{9} {\\\\left(x-4\\\\right)}^2$$"]},{"id":"a7dc5fftransformation14a-h6","type":"hint","dependencies":["a7dc5fftransformation14a-h5"],"title":"Shifting Down Three Units","text":"To shift the function down three units, subtract $$3$$ from the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation14a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1}{9} {\\\\left(x-4\\\\right)}^2-3$$"],"dependencies":["a7dc5fftransformation14a-h6"],"title":"Shifting Down Three Units","text":"What is the function after it has been shifted down $$3$$ units?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{1}{9} {\\\\left(x-4\\\\right)}^2-3$$","$$\\\\frac{1}{9} {\\\\left(x-4\\\\right)}^2+3$$"]}]}}]},{"id":"a7dc5fftransformation15","title":"Reflections","body":"Find the equation of the function $$s(t)=\\\\sqrt{t}$$ after the following transformations.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Transformation of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7dc5fftransformation15a","stepAnswer":["$$-\\\\sqrt{t}$$"],"problemType":"MultipleChoice","stepTitle":"Vertical Reflection","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-\\\\sqrt{t}$$","choices":["$$\\\\sqrt{t}$$","$$-\\\\sqrt{t}$$","$$-\\\\sqrt{-t}$$","$$\\\\sqrt{-t}$$"],"hints":{"DefaultPathway":[{"id":"a7dc5fftransformation15a-h1","type":"hint","dependencies":[],"title":"Where to Reflect","text":"Because this is a vertical reflection, we have to reflect the function over the x-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation15a-h2","type":"hint","dependencies":["a7dc5fftransformation15a-h1"],"title":"X-Axis","text":"Reflections over the x-axis mean the y-values for each x-value are the opposite of what they would normally be. These are denoted as $$g(x)=-f(x)$$, where g(x) is the reflected function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation15a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-\\\\sqrt{t}$$"],"dependencies":["a7dc5fftransformation15a-h2"],"title":"X-Axis","text":"What is the expression of s(t) after its reflection?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\sqrt{t}$$","$$-\\\\sqrt{t}$$","$$-\\\\sqrt{-t}$$","$$\\\\sqrt{-t}$$"]},{"id":"a7dc5fftransformation15a-h4","type":"hint","dependencies":["a7dc5fftransformation15a-h3"],"title":"X-Axis","text":"In this case, $$-f(x)$$ is the same as $$-s(t)$$, or $$-\\\\sqrt{t}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a7dc5fftransformation15b","stepAnswer":["$$\\\\sqrt{-t}$$"],"problemType":"MultipleChoice","stepTitle":"Horizontal Reflection","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\sqrt{-t}$$","choices":["$$\\\\sqrt{t}$$","$$-\\\\sqrt{t}$$","$$-\\\\sqrt{-t}$$","$$\\\\sqrt{-t}$$"],"hints":{"DefaultPathway":[{"id":"a7dc5fftransformation15b-h1","type":"hint","dependencies":[],"title":"Where to Reflect","text":"Because this is a horizontal reflection, we have to reflect the function over the y-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation15b-h2","type":"hint","dependencies":["a7dc5fftransformation15b-h1"],"title":"Y-Axis","text":"Reflections over the y-axis mean the x-values for each y-value are the opposite of what they would normally be. These are denoted as $$g(x)=f(-x)$$, where g(x) is the reflected function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation15b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\sqrt{-t}$$"],"dependencies":["a7dc5fftransformation15b-h2"],"title":"Y-Axis","text":"What is the expression of s(t) after its reflection?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\sqrt{t}$$","$$-\\\\sqrt{t}$$","$$-\\\\sqrt{-t}$$","$$\\\\sqrt{-t}$$"]},{"id":"a7dc5fftransformation15b-h4","type":"hint","dependencies":["a7dc5fftransformation15b-h3"],"title":"Y-Axis","text":"In this case, $$f(-x)$$ is the same as $$s-(t)$$, or $$\\\\sqrt{-t}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7dc5fftransformation16","title":"Odd, Even, or Neither?","body":"Is the following function even, odd, or neither?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Transformation of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7dc5fftransformation16a","stepAnswer":["odd"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=x^3+2x$$","stepBody":"","answerType":"string","variabilization":{},"choices":["even","odd","neither"],"hints":{"DefaultPathway":[{"id":"a7dc5fftransformation16a-h1","type":"hint","dependencies":[],"title":"Testing Even Functions","text":"Let\'s start by checking if the function is even. Even functions are defined as functions where $$f(x)=f(-x)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation16a-h2","type":"hint","dependencies":["a7dc5fftransformation16a-h1"],"title":"Testing Even Functions","text":"Evaluate $$f(-x)$$. $$f(-x)=-\\\\left(x^3\\\\right)-2x$$. Since this is not equivalent to $$x^3+2x$$, the function is not even.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation16a-h3","type":"hint","dependencies":["a7dc5fftransformation16a-h2"],"title":"Testing Odd Functions","text":"Now, let\'s check if the function is odd. Odd functions are defined as functions where $$-f(x)=f(-x)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation16a-h4","type":"hint","dependencies":["a7dc5fftransformation16a-h3"],"title":"Testing Odd Functions","text":"We already found $$f(-x)$$ when we tested if the function was even, so we just have to evaluate $$-f(x)$$. $$-f(x)=-\\\\left(x^3+2x\\\\right)=-x$$. Since this is equivalent to $$-\\\\left(x^3\\\\right)-2x$$, the function is odd.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7dc5fftransformation17","title":"Algebreic Transformations","body":"Write a formula for the function obtained when it is shifted as described.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Transformation of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7dc5fftransformation17a","stepAnswer":["$$f(x)=\\\\sqrt{x+2}+1$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\sqrt{x}$$ is shifted up $$1$$ unit and to the left $$2$$ units.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$f(x)=\\\\sqrt{x+2}+1$$","choices":["$$f(x)=\\\\sqrt{x-2}-1$$","$$f(x)=\\\\sqrt{x-2}+1$$","$$f(x)=\\\\sqrt{x+2}+1$$","$$f(x)=\\\\sqrt{x+2}-1$$"],"hints":{"DefaultPathway":[{"id":"a7dc5fftransformation17a-h1","type":"hint","dependencies":[],"title":"Horizontal Shift","text":"Let\'s start with the horizontal shift. Horizontal shifts are denoted as addition (shift to the left) or subtraction (shift to the right) inside of the parentheses. To shift two units to the left, we add two inside the parentheses.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation17a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$f(x)=\\\\sqrt{x+2}$$"],"dependencies":["a7dc5fftransformation17a-h1"],"title":"Horizontal Shift","text":"What does the function look like now?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$f(x)=\\\\sqrt{x+2}$$","$$f(x)=\\\\sqrt{x-2}$$"]},{"id":"a7dc5fftransformation17a-h3","type":"hint","dependencies":["a7dc5fftransformation17a-h2"],"title":"Vertical Shift","text":"Vertical shifts denoted as addition (shift upwards) or subtraction (shift downwards) outside of the parentheses. To shift up one unit, we add one units outside the parentheses.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation17a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$f(x)=\\\\sqrt{x+2}+1$$"],"dependencies":["a7dc5fftransformation17a-h3"],"title":"Vertical Shift","text":"What does the function look like now?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$f(x)=\\\\sqrt{x+2}-1$$","$$f(x)=\\\\sqrt{x+2}+1$$"]}]}}]},{"id":"a7dc5fftransformation18","title":"Algebreic Transformations","body":"Write a formula for the function obtained when it is shifted as described.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Transformation of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7dc5fftransformation18a","stepAnswer":["$$|x-1|-3$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=|x|$$ is shifted down $$3$$ units and to the right $$1$$ unit.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$|x-1|-3$$","choices":["$$|x-1|-3$$","$$|x+1|-3$$","$$|x-1|+3$$","$$|x+1|+3$$"],"hints":{"DefaultPathway":[{"id":"a7dc5fftransformation18a-h1","type":"hint","dependencies":[],"title":"Horizontal Shift","text":"Let\'s start with the horizontal shift. Horizontal shifts are denoted as addition (shift to the left) or subtraction (shift to the right) inside of the parentheses, or in this case, the absolute value signs. To shift one unit to the right, we subtract one inside the the absolute value backets.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation18a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$|x-1|$$"],"dependencies":["a7dc5fftransformation18a-h1"],"title":"Horizontal Shift","text":"What does the function look like now?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$|x-1|$$","$$|x+1|$$"]},{"id":"a7dc5fftransformation18a-h3","type":"hint","dependencies":["a7dc5fftransformation18a-h2"],"title":"Vertical Shift","text":"Vertical shifts denoted as addition (shift upwards) or subtraction (shift downwards) outside of the absolute value sign. To shift down one unit, we subtract three outside the brackets.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation18a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$|x-1|-3$$"],"dependencies":["a7dc5fftransformation18a-h3"],"title":"Vertical Shift","text":"What does the function look?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$|x-1|-3$$","$$|x+1|-3$$","$$|x-1|+3$$","$$|x+1|+3$$"]}]}}]},{"id":"a7dc5fftransformation19","title":"Algebreic Transformations","body":"Write a formula for the function obtained when it is shifted as described.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Transformation of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7dc5fftransformation19a","stepAnswer":["$$\\\\frac{1}{x-3}-4$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{1}{x}$$ is shifted down $$4$$ units and to the right $$3$$ units.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{1}{x-3}-4$$","choices":["$$\\\\frac{1}{x-3}-4$$","$$\\\\frac{1}{x-3}+4$$","$$\\\\frac{1}{x+3}+4$$","$$\\\\frac{1}{x+3}-4$$"],"hints":{"DefaultPathway":[{"id":"a7dc5fftransformation19a-h1","type":"hint","dependencies":[],"title":"Horizontal Shift","text":"Let\'s start with the horizontal shift. Horizontal shifts are denoted as addition (shift to the left) or subtraction (shift to the right) inside of the parentheses, or in this case, next to the x-value in the denominator. To shift three units to the right, we subtract three from $$x$$ in the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation19a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1}{x-3}$$"],"dependencies":["a7dc5fftransformation19a-h1"],"title":"Horizontal Shift","text":"What does the function look like now?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{1}{x+3}$$","$$\\\\frac{1}{x-3}$$"]},{"id":"a7dc5fftransformation19a-h3","type":"hint","dependencies":["a7dc5fftransformation19a-h2"],"title":"Vertical Shift","text":"Vertical shifts denoted as addition (shift upwards) or subtraction (shift downwards) outside of the fraction. To shift down four units, we subtract four outside of the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation19a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1}{x-3}-4$$"],"dependencies":["a7dc5fftransformation19a-h3"],"title":"Vertical Shift","text":"What does the function look?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{1}{x-3}+4$$","$$\\\\frac{1}{x-3}-4$$"]}]}}]},{"id":"a7dc5fftransformation2","title":"Odd, Even Functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Transformation of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7dc5fftransformation2a","stepAnswer":["Even"],"problemType":"MultipleChoice","stepTitle":"Is the function f(s) $$=$$ $$s^4$$ + $$3s^2$$ $$+7$$ even, odd, or neither","stepBody":"","answerType":"string","variabilization":{},"choices":["Odd","Even","Neither"],"hints":{"DefaultPathway":[{"id":"a7dc5fftransformation2a-h1","type":"hint","dependencies":[],"title":"If f(s) is even then $$f(-s)$$ should be equal to f(s)","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$s^4$$ + $$3s^2$$ + $$7$$"],"dependencies":["a7dc5fftransformation2a-h1"],"title":"What is $$f(-s)$$","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation2a-h3","type":"hint","dependencies":["a7dc5fftransformation2a-h2"],"title":"If f(s) is odd then $$-f(-s)$$ should be equal to f(s)","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-\\\\left(s^4\\\\right)$$ - $$3s^2$$ - $$7$$"],"dependencies":["a7dc5fftransformation2a-h3"],"title":"What is $$-f(-s)$$","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation2a-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Even"],"dependencies":["a7dc5fftransformation2a-h4"],"title":"Is the function even, odd, or neither?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Odd","Even"]}]}}]},{"id":"a7dc5fftransformation20","title":"Algebreic Transformations","body":"Write a formula for the function obtained when it is shifted as described.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Transformation of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7dc5fftransformation20a","stepAnswer":["$$\\\\frac{1}{{\\\\left(x-3\\\\right)}^2}+2$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{1}{x^2}$$ is shifted up $$2$$ units and to the left $$4$$ units.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{1}{{\\\\left(x-3\\\\right)}^2}+2$$","choices":["$$\\\\frac{1}{{\\\\left(x-3\\\\right)}^2}-2$$","$$\\\\frac{1}{{\\\\left(x-3\\\\right)}^2}+2$$","$$\\\\frac{1}{{\\\\left(x+3\\\\right)}^2}+2$$","$$\\\\frac{1}{{\\\\left(x+3\\\\right)}^2}-2$$"],"hints":{"DefaultPathway":[{"id":"a7dc5fftransformation20a-h1","type":"hint","dependencies":[],"title":"Horizontal Shift","text":"Let\'s start with the horizontal shift. Horizontal shifts are denoted as addition (shift to the left) or subtraction (shift to the right) inside of the parentheses. Because $$x$$ is being squared in this function, we need to add the shift to $$x$$ before it is squared. To shift four units to the left, we add four to $$x$$ in the denominator, then square that value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation20a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1}{{\\\\left(x-3\\\\right)}^2}$$"],"dependencies":["a7dc5fftransformation20a-h1"],"title":"Horizontal Shift","text":"What does the function look like now?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{1}{{\\\\left(x+3\\\\right)}^2}$$","$$\\\\frac{1}{{\\\\left(x-3\\\\right)}^2}$$"]},{"id":"a7dc5fftransformation20a-h3","type":"hint","dependencies":["a7dc5fftransformation20a-h2"],"title":"Vertical Shift","text":"Vertical shifts denoted as addition (shift upwards) or subtraction (shift downwards) outside of the fraction. To shift up two units, we add two outside of the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation20a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1}{{\\\\left(x-3\\\\right)}^2}+2$$"],"dependencies":["a7dc5fftransformation20a-h3"],"title":"Vertical Shift","text":"What does the function look?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{1}{{\\\\left(x-3\\\\right)}^2}+2$$","$$\\\\frac{1}{{\\\\left(x-3\\\\right)}^2}-2$$"]}]}}]},{"id":"a7dc5fftransformation21","title":"Increasing and Decreasing Intervals","body":"Determine the interval(s) on which the function is increasing and decreasing.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Transformation of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7dc5fftransformation21a","stepAnswer":["increasing on $$(-1,\\\\infty);$$ decreasing on $$(-\\\\infty,-1)$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)={4\\\\left(x+1\\\\right)}^2-5$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"increasing on $$(-1,\\\\infty);$$ decreasing on $$(-\\\\infty,-1)$$","choices":["increasing on $$(-1,\\\\infty);$$ decreasing on $$(-\\\\infty,-1)$$","increasing on $$(-\\\\infty,-1);$$ decreasing on $$(-1,\\\\infty)$$","increasing on $$(1,\\\\infty);$$ decreasing on $$(-\\\\infty,1)$$","increasing on $$(-\\\\infty,1);$$ decreasing on $$(\\\\infty,1)$$"],"hints":{"DefaultPathway":[{"id":"a7dc5fftransformation21a-h1","type":"hint","dependencies":[],"title":"Base Function","text":"Since intervals are based on x-values, we only have to pay attention to horizontal shifts and reflections. There are no reflections, so we only have to look at the horizontal shifts. The base function of f(x) is $$x^2$$. In $$x^2$$ functions, the graph decreases for every x-value before the vertex, and increases for every x-value after the vertex.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation21a-h2","type":"hint","dependencies":["a7dc5fftransformation21a-h1"],"title":"Base Function","text":"This means that if we find the the x-value of the vertex, we can find where the function increases and decreases.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation21a-h3","type":"hint","dependencies":["a7dc5fftransformation21a-h2"],"title":"Vertex","text":"We have to find the x-value of the vertex of the transformations. The function is horizontally shifted one x-value to the left.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation21a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a7dc5fftransformation21a-h3"],"title":"Vertex","text":"What is the x-value of the vertex?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation21a-h5","type":"hint","dependencies":["a7dc5fftransformation21a-h4"],"title":"Intervals","text":"So, since the vertex is at $$x=-1$$, and $$x^2$$ increases after the vertex and decreases after the vertex, we know what intervals the function increases and decreases on.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7dc5fftransformation22","title":"Increasing and Decreasing Intervals","body":"Determine the interval(s) on which the function is increasing and decreasing.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Transformation of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7dc5fftransformation22a","stepAnswer":["increasing on $$(-3,\\\\infty);$$ decreasing on $$(-\\\\infty,-3)$$"],"problemType":"MultipleChoice","stepTitle":"$$g(x)={5\\\\left(x+3\\\\right)}^2-2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"increasing on $$(-3,\\\\infty);$$ decreasing on $$(-\\\\infty,-3)$$","choices":["increasing on $$(-\\\\infty,-3);$$ decreasing on $$(-3,-\\\\infty)$$","increasing on $$(-3,\\\\infty);$$ decreasing on $$(-\\\\infty,-3)$$","increasing on $$(3,\\\\infty);$$ decreasing on $$(-\\\\infty,3)$$","increasing on $$(-\\\\infty,3);$$ decreasing on $$(\\\\infty,3)$$"],"hints":{"DefaultPathway":[{"id":"a7dc5fftransformation22a-h1","type":"hint","dependencies":[],"title":"Base Function","text":"Since intervals are based on x-values, we only have to pay attention to horizontal shifts and reflections. There are no reflections, so we only have to look at the horizontal shifts. The base function of f(x) is $$x^2$$. In $$x^2$$ functions, the graph decreases for every x-value before the vertex, and increases for every x-value after the vertex.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation22a-h2","type":"hint","dependencies":["a7dc5fftransformation22a-h1"],"title":"Base Function","text":"This means that if we find the the x-value of the vertex, we can find where the function increases and decreases.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation22a-h3","type":"hint","dependencies":["a7dc5fftransformation22a-h2"],"title":"Vertex","text":"We have to find the x-value of the vertex of the transformations. The function is horizontally shifted three x-value to the left.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation22a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a7dc5fftransformation22a-h3"],"title":"Vertex","text":"What is the x-value of the vertex?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation22a-h5","type":"hint","dependencies":["a7dc5fftransformation22a-h4"],"title":"Intervals","text":"So, since the vertex is at $$x=-3$$, and $$x^2$$ increases after the vertex and decreases after the vertex, we know what intervals the function increases and decreases on.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7dc5fftransformation23","title":"Increasing and Decreasing Intervals","body":"Determine the interval(s) on which the function is increasing and decreasing.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Transformation of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7dc5fftransformation23a","stepAnswer":["never increases; decreasing on $$(-\\\\infty,4)$$"],"problemType":"MultipleChoice","stepTitle":"$$a(x)=\\\\sqrt{-x+4}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"never increases; decreasing on $$(-\\\\infty,4)$$","choices":["increasing on $$(4,\\\\infty);$$ never decreases","never increases; decreasing on $$(4,\\\\infty)$$","never increases; decreasing on $$(-\\\\infty,4)$$","increasing on $$(-\\\\infty,-4);$$ never decreases"],"hints":{"DefaultPathway":[{"id":"a7dc5fftransformation23a-h1","type":"hint","dependencies":[],"title":"Base Function","text":"Since intervals are based on x-values, we only have to pay attention to horizontal shifts and reflections. There is a reflection over the y-axis (denoted by the -x), so we know that the increasing and decreasing intervals will be reversed. The base function of f(x) is $$\\\\sqrt{x}$$. In $$\\\\sqrt{x}$$ functions, the graph increases for every x-value before the vertex starting with $$0$$ and never decreases.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation23a-h2","type":"hint","dependencies":["a7dc5fftransformation23a-h1"],"title":"Base Function","text":"This means that if we find the the x-value that makes $$y=0$$, we can find where the function decreases.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation23a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["a7dc5fftransformation23a-h2"],"title":"$$y=0$$","text":"Based on the fact that the function is shifted four units to the left $$+4$$, what is the x-value for $$y=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation23a-h4","type":"hint","dependencies":["a7dc5fftransformation23a-h3"],"title":"Intervals","text":"So, since $$y=0$$ at $$x=-4$$, and $$\\\\sqrt{-x}$$ decreases before $$y=0$$, we know what intervals the function decreases on.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7dc5fftransformation24","title":"Increasing and Decreasing Intervals","body":"Determine the interval(s) on which the function is increasing and decreasing.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Transformation of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7dc5fftransformation24a","stepAnswer":["increasing on $$(-\\\\infty,0);$$ never decreases"],"problemType":"MultipleChoice","stepTitle":"$$k(x)=-3\\\\sqrt{x}-1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"increasing on $$(-\\\\infty,0);$$ never decreases","choices":["increasing on $$(3,\\\\infty);$$ never decreases","never increases; decreasing on $$(3,\\\\infty)$$","never increases; decreasing on $$(-\\\\infty,-3)$$","increasing on $$(-\\\\infty,0);$$ never decreases"],"hints":{"DefaultPathway":[{"id":"a7dc5fftransformation24a-h1","type":"hint","dependencies":[],"title":"Base Function","text":"Since intervals are based on x-values, we only have to pay attention to horizontal shifts and reflections. There is no horizontal shift, so we only look at the reflections. There is a reflection over the x-axis (denoted by the -3), so we know that the increasing and decreasing intervals will be reversed. The base function of f(x) is $$\\\\sqrt{x}$$. In $$\\\\sqrt{x}$$ functions, the graph increases for every x-value before the vertex starting with $$0$$ and never decreases.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation24a-h2","type":"hint","dependencies":["a7dc5fftransformation24a-h1"],"title":"Base Function","text":"This means that if we just need to reverse the intervals where $$\\\\sqrt{x}$$ increases.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation24a-h3","type":"hint","dependencies":["a7dc5fftransformation24a-h2"],"title":"Intervals","text":"So, since $$\\\\sqrt{x}$$ increases on the interval (0, -inf), and k(x) is the opposite of that, we know the interval on which k(x) decreases.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7dc5fftransformation3","title":"Determining Even and Odd Functions","body":"Determine whether the function is odd, even, or neither.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Transformation of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7dc5fftransformation3a","stepAnswer":["Even"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=3x^4$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Even","Odd","Neither"],"hints":{"DefaultPathway":[{"id":"a7dc5fftransformation3a-h1","type":"hint","dependencies":[],"title":"Definition of an Even Function","text":"A function is called an even function if for every input $$x$$, $$f(x)=f(-x)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation3a-h2","type":"hint","dependencies":["a7dc5fftransformation3a-h1"],"title":"Definition of an Odd Function","text":"A function is called an odd function if for every input $$x$$, $$f(x)=-f(-x)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation3a-h3","type":"hint","dependencies":["a7dc5fftransformation3a-h2"],"title":"How to Find $$f(-x)$$","text":"Find $$f(-x)$$ by subsituting $$-x$$ in wherever $$x$$ appears in the original equation. $${\\\\left(-x\\\\right)}^n=x^n$$ when $$n$$ is an even, and $${\\\\left(-x\\\\right)}^n=-\\\\left(x^n\\\\right)$$ when $$n$$ is odd.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation3a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3x^4$$"],"dependencies":["a7dc5fftransformation3a-h3"],"title":"Determining $$f(-x)$$","text":"What is $$f(-x)$$ equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$-3x^4$$","$$3x^4$$"]},{"id":"a7dc5fftransformation3a-h5","type":"hint","dependencies":["a7dc5fftransformation3a-h4"],"title":"How to Find $$-f(-x)$$","text":"Find $$-f(-x)$$ by multiplying each term of $$f(-x)$$ by $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation3a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-3x^4$$"],"dependencies":["a7dc5fftransformation3a-h5"],"title":"Determining $$-f(-x)$$","text":"What is $$-f(-x)$$ equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$-3x^4$$","$$3x^4$$"]},{"id":"a7dc5fftransformation3a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a7dc5fftransformation3a-h6"],"title":"Checking if f(x) is Even","text":"Does $$f(x)=f(-x)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a7dc5fftransformation3a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a7dc5fftransformation3a-h7"],"title":"Checking if f(x) is Odd","text":"Does $$f(x)=-f(-x)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a7dc5fftransformation4","title":"Determining Even and Odd Functions","body":"Determine whether the function is odd, even, or neither.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Transformation of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7dc5fftransformation4a","stepAnswer":["Neither"],"problemType":"MultipleChoice","stepTitle":"$$g(x)=\\\\sqrt{x}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Even","Odd","Neither"],"hints":{"DefaultPathway":[{"id":"a7dc5fftransformation4a-h1","type":"hint","dependencies":[],"title":"Definition of an Even Function","text":"A function is called an even function if for every input $$x$$, $$g(x)=g(-x)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation4a-h2","type":"hint","dependencies":["a7dc5fftransformation4a-h1"],"title":"Definition of an Odd Function","text":"A function is called an odd function if for every input $$x$$, $$g(x)=-g(-x)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation4a-h3","type":"hint","dependencies":["a7dc5fftransformation4a-h2"],"title":"How to Find $$g(-x)$$","text":"Find $$g(-x)$$ by subsituting $$-x$$ in wherever $$x$$ appears in the original equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation4a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\sqrt{-x}$$"],"dependencies":["a7dc5fftransformation4a-h3"],"title":"Determining $$g(-x)$$","text":"What is $$g(-x)$$ equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\sqrt{-x}$$","$$-\\\\sqrt{-x}$$"]},{"id":"a7dc5fftransformation4a-h5","type":"hint","dependencies":["a7dc5fftransformation4a-h4"],"title":"How to Find $$-g(-x)$$","text":"Find $$-g(-x)$$ by multiplying each term of $$g(-x)$$ by $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation4a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-\\\\sqrt{-x}$$"],"dependencies":["a7dc5fftransformation4a-h5"],"title":"Determining $$-g(-x)$$","text":"What is $$-g(-x)$$ equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\sqrt{-x}$$","$$-\\\\sqrt{-x}$$"]},{"id":"a7dc5fftransformation4a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a7dc5fftransformation4a-h6"],"title":"Checking if g(x) is Even","text":"Does $$g(x)=g(-x)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a7dc5fftransformation4a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a7dc5fftransformation4a-h7"],"title":"Checking if g(x) is Odd","text":"Does $$g(x)=-g(-x)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a7dc5fftransformation5","title":"Determining Even and Odd Functions","body":"Determine whether the function is odd, even, or neither.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Transformation of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7dc5fftransformation5a","stepAnswer":["Odd"],"problemType":"MultipleChoice","stepTitle":"$$h(x)=\\\\frac{1}{x}+3x$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Even","Odd","Neither"],"hints":{"DefaultPathway":[{"id":"a7dc5fftransformation5a-h1","type":"hint","dependencies":[],"title":"Definition of an Even Function","text":"A function is called an even function if for every input $$x$$, $$h(x)=h(-x)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation5a-h2","type":"hint","dependencies":["a7dc5fftransformation5a-h1"],"title":"Definition of an Odd Function","text":"A function is called an odd function if for every input $$x$$, $$h(x)=-h(-x)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation5a-h3","type":"hint","dependencies":["a7dc5fftransformation5a-h2"],"title":"How to Find $$h(-x)$$","text":"Find $$h(-x)$$ by subsituting $$-x$$ in wherever $$x$$ appears in the original equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation5a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{-1}{x}-3x$$"],"dependencies":["a7dc5fftransformation5a-h3"],"title":"Determining $$h(-x)$$","text":"What is $$h(-x)$$ equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{-1}{x}-3x$$","$$\\\\frac{1}{x}+3x$$","$$\\\\frac{1}{x}-3x$$","$$\\\\frac{-1}{x}+3x$$"]},{"id":"a7dc5fftransformation5a-h5","type":"hint","dependencies":["a7dc5fftransformation5a-h4"],"title":"How to Find $$-h(-x)$$","text":"Find $$-h(-x)$$ by multiplying each term of $$g(-x)$$ by $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation5a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1}{x}+3x$$"],"dependencies":["a7dc5fftransformation5a-h5"],"title":"Determining $$-h(-x)$$","text":"What is $$-h(-x)$$ equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{-1}{x}-3x$$","$$\\\\frac{1}{x}+3x$$","$$\\\\frac{1}{x}-3x$$","$$\\\\frac{-1}{x}+3x$$"]},{"id":"a7dc5fftransformation5a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a7dc5fftransformation5a-h6"],"title":"Checking if h(x) is Even","text":"Does $$h(x)=h(-x)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a7dc5fftransformation5a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a7dc5fftransformation5a-h7"],"title":"Checking if h(x) is Odd","text":"Does $$h(x)=-h(-x)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a7dc5fftransformation6","title":"Determining Even and Odd Functions","body":"Determine whether the function is odd, even, or neither.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Transformation of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7dc5fftransformation6a","stepAnswer":["Even"],"problemType":"MultipleChoice","stepTitle":"$$f(x)={\\\\left(x-2\\\\right)}^2$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Even","Odd","Neither"],"hints":{"DefaultPathway":[{"id":"a7dc5fftransformation6a-h1","type":"hint","dependencies":[],"title":"Definition of an Even Function","text":"A function is called an even function if for every input $$x$$, $$f(x)=f(-x)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation6a-h2","type":"hint","dependencies":["a7dc5fftransformation6a-h1"],"title":"Definition of an Odd Function","text":"A function is called an odd function if for every input $$x$$, $$f(x)=-f(-x)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation6a-h3","type":"hint","dependencies":["a7dc5fftransformation6a-h2"],"title":"How to Find $$f(-x)$$","text":"Find $$f(-x)$$ by subsituting $$-x$$ in wherever $$x$$ appears in the original equation. $${\\\\left(-x\\\\right)}^n=x^n$$ when $$n$$ is an even, and $${\\\\left(-x\\\\right)}^n=-\\\\left(x^n\\\\right)$$ when $$n$$ is odd-- for example, $${\\\\left(x+5\\\\right)}^2$$ $$=$$ $${\\\\left(-\\\\left(-x-5\\\\right)\\\\right)}^2={\\\\left(x+5\\\\right)}^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation6a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${\\\\left(x+2\\\\right)}^2$$"],"dependencies":["a7dc5fftransformation6a-h3"],"title":"Determining $$f(-x)$$","text":"What is $$f(-x)$$ equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$-\\\\left({\\\\left(x+2\\\\right)}^2\\\\right)$$","$${\\\\left(x+2\\\\right)}^2$$","$${\\\\left(x-2\\\\right)}^2$$"]},{"id":"a7dc5fftransformation6a-h5","type":"hint","dependencies":["a7dc5fftransformation6a-h4"],"title":"How to Find $$-f(-x)$$","text":"Find $$-f(-x)$$ by multiplying each term of $$f(-x)$$ by $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation6a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-\\\\left({\\\\left(x+2\\\\right)}^2\\\\right)$$"],"dependencies":["a7dc5fftransformation6a-h5"],"title":"Determining $$-f(-x)$$","text":"What is $$-f(-x)$$ equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$-\\\\left({\\\\left(x+2\\\\right)}^2\\\\right)$$","$${\\\\left(x+2\\\\right)}^2$$","$${\\\\left(x-2\\\\right)}^2$$"]},{"id":"a7dc5fftransformation6a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a7dc5fftransformation6a-h6"],"title":"Checking if f(x) is Even","text":"Does $$f(x)=f(-x)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a7dc5fftransformation6a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a7dc5fftransformation6a-h7"],"title":"Checking if f(x) is Odd","text":"Does $$f(x)=-f(-x)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a7dc5fftransformation7","title":"Determining Even and Odd Functions","body":"Determine whether the function is odd, even, or neither.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Transformation of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7dc5fftransformation7a","stepAnswer":["Even"],"problemType":"MultipleChoice","stepTitle":"$$g(x)=2x^4$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Even","Odd","Neither"],"hints":{"DefaultPathway":[{"id":"a7dc5fftransformation7a-h1","type":"hint","dependencies":[],"title":"Definition of an Even Function","text":"A function is called an even function if for every input $$x$$, $$g(x)=g(-x)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation7a-h2","type":"hint","dependencies":["a7dc5fftransformation7a-h1"],"title":"Definition of an Odd Function","text":"A function is called an odd function if for every input $$x$$, $$g(x)=-g(-x)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation7a-h3","type":"hint","dependencies":["a7dc5fftransformation7a-h2"],"title":"How to Find $$g(-x)$$","text":"Find $$g(-x)$$ by subsituting $$-x$$ in wherever $$x$$ appears in the original equation. $${\\\\left(-x\\\\right)}^n=x^n$$ when $$n$$ is an even, and $${\\\\left(-x\\\\right)}^n=-\\\\left(x^n\\\\right)$$ when $$n$$ is odd.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation7a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2x^4$$"],"dependencies":["a7dc5fftransformation7a-h3"],"title":"Determining $$g(-x)$$","text":"What is $$g(-x)$$ equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$-2x^4$$","$$2x^4$$"]},{"id":"a7dc5fftransformation7a-h5","type":"hint","dependencies":["a7dc5fftransformation7a-h4"],"title":"How to Find $$-g(-x)$$","text":"Find $$-g(-x)$$ by multiplying each term of $$g(-x)$$ by $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation7a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-2x^4$$"],"dependencies":["a7dc5fftransformation7a-h5"],"title":"Determining $$-g(-x)$$","text":"What is $$-g(-x)$$ equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$-2x^4$$","$$2x^4$$"]},{"id":"a7dc5fftransformation7a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a7dc5fftransformation7a-h6"],"title":"Checking if g(x) is Even","text":"Does $$g(x)=g(-x)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a7dc5fftransformation7a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a7dc5fftransformation7a-h7"],"title":"Checking if g(x) is Odd","text":"Does $$g(x)=-g(-x)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a7dc5fftransformation8","title":"Determining Even and Odd Functions","body":"Determine whether the function is odd, even, or neither.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Transformation of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7dc5fftransformation8a","stepAnswer":["Odd"],"problemType":"MultipleChoice","stepTitle":"$$h(x)=2x-x^3$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Even","Odd","Neither"],"hints":{"DefaultPathway":[{"id":"a7dc5fftransformation8a-h1","type":"hint","dependencies":[],"title":"Definition of an Even Function","text":"A function is called an even function if for every input $$x$$, $$h(x)=h(-x)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation8a-h2","type":"hint","dependencies":["a7dc5fftransformation8a-h1"],"title":"Definition of an Odd Function","text":"A function is called an odd function if for every input $$x$$, $$h(x)=-h(-x)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation8a-h3","type":"hint","dependencies":["a7dc5fftransformation8a-h2"],"title":"How to Find $$h(-x)$$","text":"Find $$h(-x)$$ by subsituting $$-x$$ in wherever $$x$$ appears in the original equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation8a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-2x+x^3$$"],"dependencies":["a7dc5fftransformation8a-h3"],"title":"Determining $$h(-x)$$","text":"What is $$h(-x)$$ equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$-2x+x^3$$","$$-2x-x^3$$","$$2x+x^3$$","$$2x-x^3$$"]},{"id":"a7dc5fftransformation8a-h5","type":"hint","dependencies":["a7dc5fftransformation8a-h4"],"title":"How to Find $$-h(-x)$$","text":"Find $$-h(-x)$$ by multiplying each term of $$g(-x)$$ by $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation8a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2x-x^3$$"],"dependencies":["a7dc5fftransformation8a-h5"],"title":"Determining $$-h(-x)$$","text":"What is $$-h(-x)$$ equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$-2x+x^3$$","$$-2x-x^3$$","$$2x+x^3$$","$$2x-x^3$$"]},{"id":"a7dc5fftransformation8a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a7dc5fftransformation8a-h6"],"title":"Checking if h(x) is Even","text":"Does $$h(x)=h(-x)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a7dc5fftransformation8a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a7dc5fftransformation8a-h7"],"title":"Checking if h(x) is Odd","text":"Does $$h(x)=-h(-x)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a7dc5fftransformation9","title":"Finding Transformed Functions","body":"Write a formula for the function g that results when the graph of a given function is transformed as described.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Transformation of Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7dc5fftransformation9a","stepAnswer":["$$|4x|$$"],"problemType":"MultipleChoice","stepTitle":"The graph of $$f(x)=|x|$$ is reflected over the y-axis and horizontally compressed by a factor of $$\\\\frac{1}{4}$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$|4x|$$","choices":["$$|4x|$$","$$-|4x|$$","$$|\\\\frac{1}{4} x|$$","$$-|\\\\frac{1}{4} x|$$"],"hints":{"DefaultPathway":[{"id":"a7dc5fftransformation9a-h1","type":"hint","dependencies":[],"title":"Order of Transformations","text":"The first step is to recognize the order of transformations. First, $$|x|$$ is reflected over the $$y$$ axis, and then it is horizontally compressed by a factor of $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation9a-h2","type":"hint","dependencies":["a7dc5fftransformation9a-h1"],"title":"Reflecting over the $$y$$ axis","text":"The second step is to reflect the function over the $$y$$ axis by replacing $$x$$ with $$-x$$ in the function. So, find $$f(-x)$$. Remember that absolute value is always positive, so $$|x|=|-x|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation9a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$|x|$$"],"dependencies":["a7dc5fftransformation9a-h2"],"title":"Reflecting over the $$y$$ axis","text":"What is $$f(-x)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$|-x|$$","$$|x|$$","$$-|x|$$","$$-|-x|$$"]},{"id":"a7dc5fftransformation9a-h4","type":"hint","dependencies":["a7dc5fftransformation9a-h3"],"title":"How to Horizontally Compress a Function","text":"The last step is to compress the graph horizontally by a scale factor of $$\\\\frac{1}{4}$$. The coefficient needed for a horizontal stretch or compression is the reciprocal of the stretch or compression. So to compress the graph horizontally by a scale factor of $$\\\\frac{1}{4}$$, we need a coefficient of $$4$$ in our function. Therefore, replace every $$x$$ in the function with $$4x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7dc5fftransformation9a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$|4x|$$"],"dependencies":["a7dc5fftransformation9a-h4"],"title":"Horizontal Compression","text":"What is the function after it has been horizontally compressed by a factor of $$\\\\frac{1}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$|4x|$$","$$-|4x|$$","$$|\\\\frac{1}{4} x|$$","$$-|\\\\frac{1}{4} x|$$"]}]}}]},{"id":"a7ea646graph1","title":"Finding the $$x-Intercepts$$ of a Polynomial Function by Factoring #1","body":"Find the $$x$$ intercepts of the following function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Graphs of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7ea646graph1a","stepAnswer":["$$(0,0)$$, $$(1,0)$$, $$(-1,0)$$, $$(\\\\sqrt{2},0)$$, $$(-\\\\sqrt{2},0)$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=x^6-3x^4+2x^2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,0)$$, $$(1,0)$$, $$(-1,0)$$, $$(\\\\sqrt{2},0)$$, $$(-\\\\sqrt{2},0)$$","choices":["$$(0,0)$$, $$(1,0)$$, $$(-1,0)$$, $$(-\\\\sqrt{2},0)$$","$$(0,0)$$, $$(1,0)$$, $$(-1,0)$$, $$(\\\\sqrt{2},0)$$, $$(-\\\\sqrt{2},0)$$","$$(0,0)$$, $$(1,0)$$, $$(-1,0)$$, $$(\\\\sqrt{2},0)$$, $$(-\\\\sqrt{2},0)$$, $$(\\\\sqrt{3},0)$$, $$(-\\\\sqrt{3},0)$$"],"hints":{"DefaultPathway":[{"id":"a7ea646graph1a-h1","type":"hint","dependencies":[],"title":"How to Factor","text":"First, factor out the greatest common factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2$$"],"dependencies":["a7ea646graph1a-h1"],"title":"Identifying the GCF","text":"What is the greatest common factor?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph1a-h3","type":"hint","dependencies":["a7ea646graph1a-h2"],"title":"Continued Factoring","text":"Next, factor the remaining trinomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph1a-h4","type":"hint","dependencies":["a7ea646graph1a-h3"],"title":"Solving for Zeros","text":"Finally, set each factor equal to $$0$$ and find $$x$$ at those values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ea646graph10","title":"Finding the X-Intercept of a Polynomial Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Graphs of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7ea646graph10a","stepAnswer":["(0, 0),(3,0),(1,0)"],"problemType":"TextBox","stepTitle":"Find the $$t-intercepts$$ of $$C(t)=2t^4-8t^3+6t^2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,0),(3,0),(1,0)$$","hints":{"DefaultPathway":[{"id":"a7ea646graph10a-h1","type":"hint","dependencies":[],"title":"Taking Out the Common Term","text":"First, we must take out $$2t^2$$ using the distributive property. We now have $$C(t)=2t^2 \\\\left(t^2-4t+3\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph10a-h2","type":"hint","dependencies":["a7ea646graph10a-h1"],"title":"Factoring the Interior Quadratic","text":"We can now factor the expression inside the parenthesis since it is a quadratic. $$t^2-4t+3=(t-3)(t-1)$$. So, our equation is $$C(t)=2t^{2\\\\left(t-3\\\\right)} \\\\left(t-1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph10a-h3","type":"hint","dependencies":["a7ea646graph10a-h2"],"title":"Solving for the Intercepts","text":"We must set the whole equation to $$0$$ and solve for $$t$$. $$0=2t^{2\\\\left(t-3\\\\right)} \\\\left(t-1\\\\right)$$. This means that $$t=0$$, $$3$$, and $$1$$. So, our $$t-intercepts$$ are $$(0,0)$$, $$(3,0)$$, and $$(1,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ea646graph11","title":"Finding the X-Intercept of a Polynomial Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Graphs of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7ea646graph11a","stepAnswer":["(0,0),(-5,0),(2,0)"],"problemType":"TextBox","stepTitle":"Find the $$t-intercepts$$ of $$C(t)=4t^4+12t^3-40t^2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,0),(-5,0),(2,0)$$","hints":{"DefaultPathway":[{"id":"a7ea646graph11a-h1","type":"hint","dependencies":[],"title":"Taking Out the Common Term","text":"We can now factor the expression inside the parenthesis since it is a quadratic. $$t^2-4t+3=(t-3)(t-1)$$. So, our equation is $$C(t)=2t^2 \\\\left(t-3\\\\right) \\\\left(t-1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph11a-h2","type":"hint","dependencies":["a7ea646graph11a-h1"],"title":"Factoring the Interior Quadratic","text":"We can now factor the expression inside the parentheses. $$t^2+3t-10=\\\\left(t+5\\\\right) \\\\left(t-2\\\\right)$$. We now have $$C(t)=4t^2 \\\\left(t+5\\\\right) \\\\left(t-2\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph11a-h3","type":"hint","dependencies":["a7ea646graph11a-h2"],"title":"Solving for the Intercepts","text":"We must now set the whole equation to $$0$$ and solve for $$t$$. $$0=4t^{2\\\\left(t+5\\\\right)} \\\\left(t-2\\\\right)$$. So, $$t=0$$, $$-5$$, and $$2$$. This means our $$t-intercepts$$ are $$(0,0)$$, $$(-5,0)$$, and $$(2,0)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ea646graph12","title":"Finding the X-Intercept of a Polynomial Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Graphs of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7ea646graph12a","stepAnswer":["(0,0),(1,0),(-1,0)"],"problemType":"TextBox","stepTitle":"We must now set the whole equation to $$0$$ and solve for $$t$$. $$0=4t^2 \\\\left(t+5\\\\right) \\\\left(t-2\\\\right)$$. So, $$t=0$$, $$-5$$, and $$2$$. This means our $$t-intercepts$$ are $$(0,0)$$, $$(-5,0)$$, and $$(2,0)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,0),(1,0),(-1,0)$$","hints":{"DefaultPathway":[{"id":"a7ea646graph12a-h1","type":"hint","dependencies":[],"title":"Taking Out the Common Term","text":"First, we must take out $$x^2$$ using the distributive property. We now have $$f(x)=x^2 \\\\left(x^2-1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph12a-h2","type":"hint","dependencies":["a7ea646graph12a-h1"],"title":"Factoring the Interior Quadratic","text":"We can now factor the expression inside the parantheses. $$x^2-1=\\\\left(x+1\\\\right) \\\\left(x-1\\\\right)$$. We now have $$f(x)=x^{2\\\\left(x+1\\\\right)} \\\\left(x-1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph12a-h3","type":"hint","dependencies":["a7ea646graph12a-h2"],"title":"Solving for the Intercepts","text":"We must now set the whole equation $$0$$ and solve for $$x$$. $$0=x^{2\\\\left(x+1\\\\right)} \\\\left(x-1\\\\right)$$. So, $$x=0, 1$$, and $$-1$$. This means our intercepts are $$(0,0),(1,0)$$, and $$(-1,0)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ea646graph13","title":"Finding the X-Intercept of a Polynomial Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Graphs of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7ea646graph13a","stepAnswer":["(0, 0),(5,0),(-4,0)"],"problemType":"TextBox","stepTitle":"Find the x-intercepts of $$f(x)=x^3+x^2-20x$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,0),(5,0),(-4,0)$$","hints":{"DefaultPathway":[{"id":"a7ea646graph13a-h1","type":"hint","dependencies":[],"title":"Taking Out the Common Term","text":"First, we must take out $$x$$ using the distributive property. We now have $$f(x)=x\\\\left(x^2+x-20\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph13a-h2","type":"hint","dependencies":["a7ea646graph13a-h1"],"title":"Factoring the Interior Quadratic","text":"We must now factor the interior quadratic. $$x^2+x-20=\\\\left(x-5\\\\right) \\\\left(x+4\\\\right)$$. We now have $$f(x)=x\\\\left(x-5\\\\right) \\\\left(x+4\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph13a-h3","type":"hint","dependencies":["a7ea646graph13a-h2"],"title":"Solving for the Intercepts","text":"We can now set the whole equation to $$0$$ and solve for $$x$$. $$0=x\\\\left(x-5\\\\right) \\\\left(x+4\\\\right)$$. This means that $$x=0, 5$$, and $$-4$$. Thus, our x-intercepts are: $$(0,0),(5,0),(-4,0)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ea646graph14","title":"Finding the X-Intercept of a Polynomial Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Graphs of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7ea646graph14a","stepAnswer":["(0,0),(-7,0),(1,0)"],"problemType":"TextBox","stepTitle":"Find the x-intercepts of $$f(x)=x^3+6x^2-7x$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,0),(-7,0),(1,0)$$","hints":{"DefaultPathway":[{"id":"a7ea646graph14a-h1","type":"hint","dependencies":[],"title":"Taking Out the Common Term","text":"First, we must take out $$x^2$$ using the distributive property. We now have $$f(x)=x\\\\left(x^2+6x-7\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph14a-h2","type":"hint","dependencies":["a7ea646graph14a-h1"],"title":"Factoring the Interior Quadratic","text":"Now, we must factor the expression inside the parantheses. $$x^2+6x-7=\\\\left(x+7\\\\right) \\\\left(x-1\\\\right)$$. So, we have $$f(x)=x\\\\left(x+7\\\\right) \\\\left(x-1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph14a-h3","type":"hint","dependencies":["a7ea646graph14a-h2"],"title":"Solving for the Intercepts","text":"We can now solve the whole equation by setting it equal to $$0$$. $$0=x\\\\left(x+7\\\\right) \\\\left(x-1\\\\right)$$. This means that our x-intercepts are: $$(0,0),(-7,0),(1,0)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ea646graph15","title":"Finding the X-Intercept of a Polynomial Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Graphs of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7ea646graph15a","stepAnswer":["(-2,0),(2,0),(-1,0)"],"problemType":"TextBox","stepTitle":"Find the x-intercepts of $$f(x)=x^3+x^2-4x-4$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-2,0),(2,0),(-1,0)$$","hints":{"DefaultPathway":[{"id":"a7ea646graph15a-h1","type":"hint","dependencies":[],"title":"Pairing Together Terms","text":"We can pair together the first and second terms and the third and fourth terms as shown: $$f(x)=x^3+x^2-4x+4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph15a-h2","type":"hint","dependencies":["a7ea646graph15a-h1"],"title":"Taking Out the Common Term","text":"Now, we must take out the common term in both pairs of terms. $$f(x)=x^{2\\\\left(x+1\\\\right)}-4\\\\left(x+1\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph15a-h3","type":"hint","dependencies":["a7ea646graph15a-h2"],"title":"Joining Together Both Terms","text":"We can now put together both terms into a single quadratic. $$f(x)=\\\\left(x^2-4\\\\right) \\\\left(x+1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph15a-h4","type":"hint","dependencies":["a7ea646graph15a-h3"],"title":"Factoring the Quadratic","text":"We can now factor the quadratic. $$x^2-4=\\\\left(x+2\\\\right) \\\\left(x-2\\\\right)$$. We now have $$f(x)=\\\\left(x+2\\\\right) \\\\left(x-2\\\\right) \\\\left(x+1\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph15a-h5","type":"hint","dependencies":["a7ea646graph15a-h4"],"title":"Solving for the Intercepts","text":"We must now set the whole equation to $$0$$ to solve for the x-intercepts. $$0=\\\\left(x+2\\\\right) \\\\left(x-2\\\\right) \\\\left(x+1\\\\right)$$. This means that $$x=-2, 2, -1$$. Our x-intercepts are therefore: $$(-2,0),(2,0),(-1,0)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ea646graph16","title":"Finding the X-Intercept of a Polynomial Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Graphs of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7ea646graph16a","stepAnswer":["(-3,0),(2,0),(-2,0)"],"problemType":"TextBox","stepTitle":"Find the x-intercept of the function $$f(x)=x^3+2x^2-9x-18$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-3,0),(2,0),(-2,0)$$","hints":{"DefaultPathway":[{"id":"a7ea646graph16a-h1","type":"hint","dependencies":[],"title":"Pairing Together Terms","text":"We can pair together the first and second terms and the third and fourth terms so that we can later factor. This can be done as shown. $$f(x)=x^3+2x^2+\\\\left(-9x-18\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph16a-h2","type":"hint","dependencies":["a7ea646graph16a-h1"],"title":"Taking Out the Common Term","text":"We must now take out the common term(s) in each pair. $$f(x)=x^{2\\\\left(x+2\\\\right)}-9\\\\left(x+2\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph16a-h3","type":"hint","dependencies":["a7ea646graph16a-h2"],"title":"Joining Together Both Terms","text":"Now we must join together both terms by using the common term that was pulled out. $$f(x)=\\\\left(x^2-9\\\\right) \\\\left(x+2\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph16a-h4","type":"hint","dependencies":["a7ea646graph16a-h3"],"title":"Factoring the Quadratic","text":"Finally, we can factor each expression in the parantheses. $$x^2-9=\\\\left(x+3\\\\right) \\\\left(x-3\\\\right)$$. This means that $$f(x)=\\\\left(x+3\\\\right) \\\\left(x-2\\\\right) \\\\left(x+2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph16a-h5","type":"hint","dependencies":["a7ea646graph16a-h4"],"title":"Solving for the Intercepts","text":"We can set the factored equation from the previous step equal to $$0$$ in order to solve for $$x$$. $$0=\\\\left(x+3\\\\right) \\\\left(x-2\\\\right) \\\\left(x+2\\\\right)$$. This means that $$x=-3$$, $$2$$, and $$-2$$. This, then, means that our x-intercepts are $$(-3,0),(2,0),(-2,0)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ea646graph17","title":"Finding the X-Intercept of a Polynomial Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Graphs of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7ea646graph17a","stepAnswer":["(-2,0),(2,0),(1/2,0)"],"problemType":"TextBox","stepTitle":"Find the x-intercept of the function $$f(x)=2x^3-x^2-8x+4$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a7ea646graph17a-h1","type":"hint","dependencies":[],"title":"Pairing Together Terms","text":"We can pair together the first and second terms and the third and fourth terms as shown: $$f(x)=2x^3-x^2+\\\\left(-8x+4\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph17a-h2","type":"hint","dependencies":["a7ea646graph17a-h1"],"title":"Taking Out the Common Term","text":"We can now take out a common term from each of the groupings as shown. $$f(x)=x^{2\\\\left(2x-1\\\\right)}-4\\\\left(2x-1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph17a-h3","type":"hint","dependencies":["a7ea646graph17a-h2"],"title":"Joining Together Both Terms","text":"Now, we must merge both terms into a single expression. $$f(x)=\\\\left(x^2-4\\\\right) \\\\left(2x-1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph17a-h4","type":"hint","dependencies":["a7ea646graph17a-h3"],"title":"Factoring the Quadratic","text":"We can actually factor $$x^2-4$$ into $$\\\\left(x+2\\\\right) \\\\left(x-2\\\\right)$$. Now, $$f(x)=\\\\left(x+2\\\\right) \\\\left(x-2\\\\right) \\\\left(2x-1\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph17a-h5","type":"hint","dependencies":["a7ea646graph17a-h4"],"title":"Solving for the Intercepts","text":"We can set the equation equal to $$0$$. $$0=\\\\left(x+2\\\\right) \\\\left(x-2\\\\right) \\\\left(2x-1\\\\right)$$. This means that $$x=-2$$, $$2$$, and $$\\\\frac{1}{2}$$. So, our x-intercepts are (-2,0),(2,0),(1/2,0)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ea646graph18","title":"Finding the X-Intercept of a Polynomial Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Graphs of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7ea646graph18a","stepAnswer":["(2,0),(-1,0)"],"problemType":"TextBox","stepTitle":"Find the x-intercepts of the function $$f(x)=x^6-7x^3-8$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(2,0),(-1,0)$$","hints":{"DefaultPathway":[{"id":"a7ea646graph18a-h1","type":"hint","dependencies":[],"title":"Substituting $$t$$","text":"We can let $$t=x^3$$ and substitute into the equation. We now have $$g(t)=t^2-7t-8$$. This is a quadratic that we can factor normally.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph18a-h2","type":"hint","dependencies":["a7ea646graph18a-h1"],"title":"Factoring the Quadratic","text":"Using factoring, we can convert $$t^2-7t-8$$ to $$\\\\left(t-8\\\\right) \\\\left(t+1\\\\right)$$. So, our equation is now $$g(t)=\\\\left(t-8\\\\right) \\\\left(t+1\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph18a-h3","type":"hint","dependencies":["a7ea646graph18a-h2"],"title":"Solving for $$t$$","text":"We can now solve for $$t$$ by setting the equation equal to $$0$$. $$0=\\\\left(t-8\\\\right) \\\\left(t+1\\\\right)$$. $$t=8, -1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph18a-h4","type":"hint","dependencies":["a7ea646graph18a-h3"],"title":"Solving for the Intercepts","text":"Since we know the values of $$t$$, we can take their cube root to find the values of $$x$$. $$x=2, -1$$. This means our intercepts are $$(2,0)$$ and $$(-1,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ea646graph19","title":"Finding the X-Intercept of a Polynomial Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Graphs of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7ea646graph19a","stepAnswer":["(1,0),(-1,0)"],"problemType":"TextBox","stepTitle":"Find the x-intercept of the function $$f(x)=2x^4+6x^2-8$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(1,0),(-1,0)$$","hints":{"DefaultPathway":[{"id":"a7ea646graph19a-h1","type":"hint","dependencies":[],"title":"Substituting $$t$$","text":"We can let $$t=x^2$$ and substitute into the equation. We now have $$g(t)=2t^2+6t-8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph19a-h2","type":"hint","dependencies":["a7ea646graph19a-h1"],"title":"Factoring the Quadratic","text":"We now factor the quadratic from the previous step. $$g(t)=2\\\\left(t+4\\\\right) \\\\left(t-1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph19a-h3","type":"hint","dependencies":["a7ea646graph19a-h2"],"title":"Solving for $$t$$","text":"We can set the equation equal to $$0$$. $$0=2\\\\left(t+4\\\\right) \\\\left(t-1\\\\right)$$. This means that $$t=1$$ and $$-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph19a-h4","type":"hint","dependencies":["a7ea646graph19a-h3"],"title":"Solving for the Intercepts","text":"We can now take the square root of the values of $$t$$ to get the values of $$x$$. This means that $$x=1$$ and $$-1$$, since there is no real square root of $$-4$$. So, our intercept is $$(1,0),(-1,0)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ea646graph2","title":"Finding the $$x-Intercepts$$ of a Polynomial Function by Factoring #2","body":"Finding the $$x-Intercepts$$ of a Polynomial Function by Factoring #2","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Graphs of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7ea646graph2a","stepAnswer":["$$(-1,0),(1,0),(5,0)$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=x^3-5x^2-x+5$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-1,0),(1,0),(5,0)$$","choices":["$$(-1,0),(1,0),(5,0)$$","$$(1,0)$$, $$(-5,0)$$, $$(5,0)$$, $$(6,0)$$","$$(1,0)$$, $$(-5,0)$$, $$(5,0)$$, $$(-6,0)$$, $$(6,0)$$"],"hints":{"DefaultPathway":[{"id":"a7ea646graph2a-h1","type":"hint","dependencies":[],"title":"How to Factor","text":"First, factor by grouping.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x^2-1\\\\right) \\\\left(x-5\\\\right)$$"],"dependencies":["a7ea646graph2a-h1"],"title":"Result of Factoring","text":"What is the result after factoring by grouping? $$f(x)=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph2a-h3","type":"hint","dependencies":["a7ea646graph2a-h2"],"title":"Continued Factoring","text":"Next, factor out the difference of squares.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph2a-h4","type":"hint","dependencies":["a7ea646graph2a-h3"],"title":"Solving for Zeros","text":"Finally, set each factor to $$0$$ and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ea646graph20","title":"Finding the X-Intercept of a Polynomial Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Graphs of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7ea646graph20a","stepAnswer":["(-1,0),(1,0),(3,0)"],"problemType":"TextBox","stepTitle":"Find the x-intercept of the function $$f(x)=x^3-3x^2-x+3$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-1,0),(1,0),(3,0)$$","hints":{"DefaultPathway":[{"id":"a7ea646graph20a-h1","type":"hint","dependencies":[],"title":"Pairing Together Terms","text":"We can pair together the first and second terms and the second and third terms as shown. $$f(x)=x^3-3x^2+\\\\left(-x+3\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph20a-h2","type":"hint","dependencies":["a7ea646graph20a-h1"],"title":"Taking Out the Common Term","text":"We can now take out the common terms as shown: $$f(x)=x^{2\\\\left(x-3\\\\right)}-x-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph20a-h3","type":"hint","dependencies":["a7ea646graph20a-h2"],"title":"Joining Together Both Terms","text":"We can now join the whole equation into a single expression. $$f(x)=\\\\left(x^2-1\\\\right) \\\\left(x-3\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph20a-h4","type":"hint","dependencies":["a7ea646graph20a-h3"],"title":"Factoring the Quadratic","text":"Since $$x^2-1=\\\\left(x+1\\\\right) \\\\left(x-1\\\\right)$$, we can simplify the function to $$f(x)=\\\\left(x+1\\\\right) \\\\left(x-1\\\\right) \\\\left(x-3\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph20a-h5","type":"hint","dependencies":["a7ea646graph20a-h4"],"title":"Solving for the Intercepts","text":"We must now solve for $$x$$ by setting the whole equation to $$0$$. $$0=\\\\left(x+1\\\\right) \\\\left(x-1\\\\right) \\\\left(x-3\\\\right)$$. This means that $$x=-1, 1$$, and $$3$$. Our intercepts are thus: $$(-1,0),(1,0),(3,0)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ea646graph21","title":"Finding Zeroes and Multiplicity","body":"For the following exercises, find the zeros and give the multiplicity of each.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Graphs of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7ea646graph21a","stepAnswer":["$$-2$$ with multiplicity of $$3$$, $$3$$ with multiplicity of $$-2$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)={\\\\left(x+2\\\\right)}^3 {\\\\left(x-3\\\\right)}^2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-2$$ with multiplicity of $$3$$, $$3$$ with multiplicity of $$-2$$","choices":["$$-3$$ with multiplicity of $$3$$, $$3$$ with multiplicity of $$-2$$","$$-4$$ with multiplicity of $$3$$, $$-5$$ with multiplicity of $$-2$$","$$2$$ with multiplicity of $$3$$, $$-3$$ with multiplicity of $$-2$$","$$-2$$ with multiplicity of $$3$$, $$3$$ with multiplicity of $$-2$$","$$-2$$ with multiplicity of $$3$$, $$3$$ with multiplicity of $$-2$$"],"hints":{"DefaultPathway":[{"id":"a7ea646graph21a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-2$$, $$3$$"],"dependencies":[],"title":"Finding Zeroes","text":"What are the values of $$x$$ that make the expression 0?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$-3$$, $$2$$","$$-2$$, $$3$$","$$2$$, $$3$$","$$3$$, $$2$$"]},{"id":"a7ea646graph21a-h2","type":"hint","dependencies":["a7ea646graph21a-h1"],"title":"Definition of Multiplicity","text":"The multiplicity is the power to which each part of the 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph30a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["0, 2"],"dependencies":["a7ea646graph30a-h1"],"title":"Finding Zeroes","text":"What are the values of $$x$$ that make the expression 0?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph30a-h3","type":"hint","dependencies":["a7ea646graph30a-h2"],"title":"Definition of Multiplicity","text":"The multiplicity is the power to which each part of the expression is raised","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ea646graph31","title":"Find the $$t-intercepts$$ of the Following Function #2","body":"Find the $$t-intercepts$$ of the following function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Graphs of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7ea646graph31a","stepAnswer":["$$(-2,0),(3,0),(-5,0)$$"],"problemType":"MultipleChoice","stepTitle":"$$C(t)=3\\\\left(t+2\\\\right) \\\\left(t-3\\\\right) \\\\left(t+5\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-2,0),(3,0),(-5,0)$$","choices":["$$(-2,0),(-3,0),(-5,0)$$","$$(2,0),(3,0),(-5,0)$$","$$(-2,0),(3,0),(-5,0)$$"],"hints":{"DefaultPathway":[{"id":"a7ea646graph31a-h1","type":"hint","dependencies":[],"title":"Find the $$t$$ values when $$C(t)=0$$.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ea646graph32","title":"Find the $$t-intercepts$$ of the Following Function #1","body":"Find the $$t-intercepts$$ of the following function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Graphs of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7ea646graph32a","stepAnswer":["$$(4,0),(-1,0),(6,0)$$"],"problemType":"MultipleChoice","stepTitle":"$$C(t)=2\\\\left(t-4\\\\right) \\\\left(t+1\\\\right) \\\\left(t-6\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(4,0),(-1,0),(6,0)$$","choices":["$$(4,0),(-1,0),(-6,0)$$","$$(4,0),(-1,0),(6,0)$$","$$(-4,0),(-1,0),(6,0)$$"],"hints":{"DefaultPathway":[{"id":"a7ea646graph32a-h1","type":"hint","dependencies":[],"title":"Find the $$t$$ values when $$C(t)=0$$.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ea646graph4","title":"Finding the $$x-Intercepts$$ of a Polynomial Function Using a Graph","body":"Using the graph, find the x-intercepts of the following function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Graphs of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7ea646graph4a","stepAnswer":["$$(-3,0)$$, $$(-2,0)$$, $$(1,0)$$"],"problemType":"MultipleChoice","stepTitle":"$$h(x)=x^3+4x^2+x-6$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$(-3,0)$$, $$(-2,0)$$, $$(1,0)$$","choices":["$$(3,0)$$, $$(-2,0)$$, $$(1,0)$$","$$(-4,0)$$, $$(-3,0)$$, $$(2,0)$$","$$(-3,0)$$, $$(-2,0)$$, $$(1,0)$$"],"hints":{"DefaultPathway":[{"id":"a7ea646graph4a-h1","type":"hint","dependencies":[],"title":"Interpreting the Graph","text":"Looking at the graph of the function, it appears that there are $$x$$ intercepts at $$x=-3$$, $$x=-2$$, and $$x=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ea646graph5","title":"Finding the Maximum Number of Turning Points Using the Degree of a Polynomial Function","body":"Find the maximum number of turning points of each polynomial function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Graphs of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7ea646graph5a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"$$f(x)=-\\\\left(x^3\\\\right)+4x^5-3x^2+1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a7ea646graph5a-h1","type":"hint","dependencies":[],"title":"Definition of a Turning Point","text":"A turning point is a point of the graph where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph5a-h2","type":"hint","dependencies":["a7ea646graph5a-h1"],"title":"Turning Points of a Polynomial of Degree $$n$$","text":"A polynomial of degree $$n$$ has at most $$n-1$$ turning points.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a7ea646graph5b","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"$$f(x)=-\\\\left({\\\\left(x-1\\\\right)}^2\\\\right) \\\\left(1+2x^2\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a7ea646graph5b-h1","type":"hint","dependencies":[],"title":"Definition of a Turning Point","text":"A turning point is a point of the graph where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph5b-h2","type":"hint","dependencies":["a7ea646graph5b-h1"],"title":"Expanding the Polynomial","text":"Expand the polynomial to identify the degree of its leading term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph5b-h3","type":"hint","dependencies":["a7ea646graph5b-h2"],"title":"Turning Points of a Polynomial of Degree $$n$$","text":"A polynomial of degree $$n$$ has at most $$n-1$$ turning points.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ea646graph6","title":"Writing a Formula for a Polynomial Function from the Graph","body":"Write a formula for the polynomial function in the graph.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Graphs of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7ea646graph6a","stepAnswer":["$$\\\\frac{1}{30} \\\\left(x+3\\\\right) {\\\\left(x-2\\\\right)}^2 \\\\left(x-5\\\\right)$$"],"problemType":"TextBox","stepTitle":"What is the formula for the displayed polynomial? $$f(x)=$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{30} \\\\left(x+3\\\\right) {\\\\left(x-2\\\\right)}^2 \\\\left(x-5\\\\right)$$","hints":{"DefaultPathway":[{"id":"a7ea646graph6a-h1","type":"hint","dependencies":[],"title":"Identifying Intercepts","text":"This graph has three x-intercepts: $$x=-3, 2$$, and $$5$$. The y-intercept is located at $$(0,2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph6a-h2","type":"hint","dependencies":["a7ea646graph6a-h1"],"title":"Identifying Traits of the Factors","text":"At $$x=-3$$ and $$x=5$$, the graph passes through the axis linearly, suggesting the corresponding factors of the polynomial will be linear.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph6a-h3","type":"hint","dependencies":["a7ea646graph6a-h2"],"title":"Identifying the Degree of the Polynomial","text":"At $$x=2$$, the graph bounces at the intercept, suggesting the corresponding factor of the polynomial will be second degree (quadratic).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph6a-h4","type":"hint","dependencies":["a7ea646graph6a-h3"],"title":"Putting Identified Traits Together","text":"Together, this gives us $$f(x)=a\\\\left(x+3\\\\right) {\\\\left(x-2\\\\right)}^2 \\\\left(x-5\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph6a-h5","type":"hint","dependencies":["a7ea646graph6a-h4"],"title":"Determing the Stretch Factor","text":"To determine the stretch factor, we utilize another point on the graph. We will use the y-intercept $$(0,-2)$$, to solve for a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ea646graph6a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{30}$$"],"dependencies":["a7ea646graph6a-h5"],"title":"Identifying the Stretch Factor","text":"When $$f(x)=-2$$ and $$x=0$$, $$a=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ea646graph7","title":"Find the $$t-intercepts$$ of the Following Function #3","body":"Find the $$t-intercepts$$ of the following function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Graphs of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7ea646graph7a","stepAnswer":["$$(0,0),(2,0),(-1,0)$$"],"problemType":"MultipleChoice","stepTitle":"$$C(t)=4{t\\\\left(t-2\\\\right)}^2 \\\\left(t+1\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,0),(2,0),(-1,0)$$","choices":["$$(0,0),(2,0),(-1,0)$$","$$(0,0),(-2,0),(1,0)$$","$$(0,0),(2,0),(1,0)$$"],"hints":{"DefaultPathway":[{"id":"a7ea646graph7a-h1","type":"hint","dependencies":[],"title":"Find the $$t$$ values when $$C(t)=0$$.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ea646graph8","title":"Find the $$t-intercepts$$ of the Following Function #4","body":"Find the $$t-intercepts$$ of the following function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Graphs of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7ea646graph8a","stepAnswer":["$$(0,0),(2,0),(-1,0)$$"],"problemType":"MultipleChoice","stepTitle":"$$C(t)=4{t\\\\left(t-2\\\\right)}^2 \\\\left(t+1\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,0),(2,0),(-1,0)$$","choices":["$$(0,0),(2,0),(-1,0)$$","$$(0,0),(-2,0),(1,0)$$","$$(0,0),(2,0),(1,0)$$"],"hints":{"DefaultPathway":[{"id":"a7ea646graph8a-h1","type":"hint","dependencies":[],"title":"Find the $$t$$ values when $$C(t)=0$$.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ea646graph9","title":"Find the $$t-intercepts$$ of the Following Function #5","body":"Find the $$t-intercepts$$ of the following function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.3 Graphs of Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a7ea646graph9a","stepAnswer":["$$(0,0),(3,0),(-1,0)$$"],"problemType":"MultipleChoice","stepTitle":"$$C(t)=2t\\\\left(t-3\\\\right) {\\\\left(t+1\\\\right)}^2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,0),(3,0),(-1,0)$$","choices":["$$(0,0),(-3,0),(-1,0)$$","$$(0,0),(3,0),(-1,0)$$","$$(0,0),(3,0),(1,0)$$"],"hints":{"DefaultPathway":[{"id":"a7ea646graph9a-h1","type":"hint","dependencies":[],"title":"Find the $$t$$ values when $$C(t)=0$$.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ecb8ffactoring1","title":"Factoring Trinomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Factor Trinomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a7ecb8ffactoring1a","stepAnswer":["$$\\\\left(x+6\\\\right) \\\\left(x+5\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor $$x^2+11x+30$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(x+6\\\\right) \\\\left(x+5\\\\right)$$","hints":{"DefaultPathway":[{"id":"a7ecb8ffactoring1a-h1","type":"hint","dependencies":[],"title":"General Factoring Procedure","text":"When given a trinomial in the form $$x^2+bx+c$$, we must find two numbers (m and n) that multiply to c and add to $$b$$. The factored solution is $$\\\\left(x+m\\\\right) \\\\left(x+n\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ecb8ffactoring1a-h2","type":"hint","dependencies":["a7ecb8ffactoring1a-h1"],"title":"Finding $$m$$ and $$n$$","text":"$$c=30$$, and $$b=11$$. $$m=6$$ and $$m=5$$ multiply to $$30$$ and add to $$11$$. So, our solution is $$\\\\left(x+6\\\\right) \\\\left(x+5\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ecb8ffactoring10","title":"Factoring Trinomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Factor Trinomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a7ecb8ffactoring10a","stepAnswer":["$$(x-10)(x-3)$$"],"problemType":"TextBox","stepTitle":"Factor $$x^2-13x+30$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$(x-10)(x-3)$$","hints":{"DefaultPathway":[{"id":"a7ecb8ffactoring10a-h1","type":"hint","dependencies":[],"title":"General Factoring Procedure","text":"When given a trinomial in the form $$x^2+bx+c$$, we must find two numbers (m and n) that multiply to c and add to $$b$$. The factored solution is $$\\\\left(x+m\\\\right) \\\\left(x+n\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ecb8ffactoring10a-h2","type":"hint","dependencies":["a7ecb8ffactoring10a-h1"],"title":"Finding $$m$$ and $$n$$","text":"$$c=30$$, and $$b=-13$$. $$m=-10$$ and $$m=-3$$ multiply to c and add to $$b$$. So, our solution is $$(x-10)(x-3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ecb8ffactoring11","title":"Factoring Trinomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Factor Trinomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a7ecb8ffactoring11a","stepAnswer":["$$(x-7)(x-1)$$"],"problemType":"TextBox","stepTitle":"Factor $$x^2-8x+7$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$(x-7)(x-1)$$","hints":{"DefaultPathway":[{"id":"a7ecb8ffactoring11a-h1","type":"hint","dependencies":[],"title":"General Factoring Procedure","text":"When given a trinomial in the form $$x^2+bx+c$$, we must find two numbers (m and n) that multiply to c and add to $$b$$. The factored solution is $$\\\\left(x+m\\\\right) \\\\left(x+n\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ecb8ffactoring11a-h2","type":"hint","dependencies":["a7ecb8ffactoring11a-h1"],"title":"Finding $$m$$ and $$n$$","text":"$$c=7$$, and $$b=-8$$. $$m=-7$$ and $$m=-1$$ multiply to c and add to $$b$$. So, our solution is $$(x-7)(x-1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ecb8ffactoring12","title":"Factoring Trinomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Factor Trinomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a7ecb8ffactoring12a","stepAnswer":["$$(x-3)(x-2)$$"],"problemType":"TextBox","stepTitle":"Factor $$x^2-5x+6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$(x-3)(x-2)$$","hints":{"DefaultPathway":[{"id":"a7ecb8ffactoring12a-h1","type":"hint","dependencies":[],"title":"General Factoring Procedure","text":"When given a trinomial in the form $$x^2+bx+c$$, we must find two numbers (m and n) that multiply to c and add to $$b$$. The factored solution is $$\\\\left(x+m\\\\right) \\\\left(x+n\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ecb8ffactoring12a-h2","type":"hint","dependencies":["a7ecb8ffactoring12a-h1"],"title":"Finding $$m$$ and $$n$$","text":"$$c=6$$, and $$b=-5$$. $$m=-3$$ and $$m=-2$$ multiply to c and add to $$b$$. So, our solution is $$(x-3)(x-2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ecb8ffactoring13","title":"Factoring Trinomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Factor Trinomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a7ecb8ffactoring13a","stepAnswer":["$$\\\\left(x+6\\\\right) \\\\left(x-1\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor $$5x-6+x^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(x+6\\\\right) \\\\left(x-1\\\\right)$$","hints":{"DefaultPathway":[{"id":"a7ecb8ffactoring13a-h1","type":"hint","dependencies":[],"title":"General Factoring Procedure","text":"When given a trinomial in the form $$x^2+bx+c$$, we must find two numbers (m and n) that multiply to c and add to $$b$$. The factored solution is $$\\\\left(x+m\\\\right) \\\\left(x+n\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ecb8ffactoring13a-h2","type":"hint","dependencies":["a7ecb8ffactoring13a-h1"],"title":"Rearranging our Trinomial","text":"We must rearrange our trinomial to fix the form $$x^2+bx+c$$. We get $$x^2+5x-6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ecb8ffactoring13a-h3","type":"hint","dependencies":["a7ecb8ffactoring13a-h2"],"title":"Finding $$m$$ and $$n$$","text":"$$c=-6$$, and $$b=5$$. $$m=6$$ and $$m=-1$$ multiply to c and add to $$b$$. So, our solution is $$\\\\left(x+6\\\\right) \\\\left(x-1\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ecb8ffactoring14","title":"Factoring Trinomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Factor Trinomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a7ecb8ffactoring14a","stepAnswer":["$$\\\\left(x+7\\\\right) \\\\left(x-1\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor $$6x-7+x^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(x+7\\\\right) \\\\left(x-1\\\\right)$$","hints":{"DefaultPathway":[{"id":"a7ecb8ffactoring14a-h1","type":"hint","dependencies":[],"title":"General Factoring Procedure","text":"When given a trinomial in the form $$x^2+bx+c$$, we must find two numbers (m and n) that multiply to c and add to $$b$$. The factored solution is $$\\\\left(x+m\\\\right) \\\\left(x+n\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ecb8ffactoring14a-h2","type":"hint","dependencies":["a7ecb8ffactoring14a-h1"],"title":"Rearranging our Trinomial","text":"We must rearrange our trinomial to fix the form $$x^2+bx+c$$. We get $$x^2+6x-7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ecb8ffactoring14a-h3","type":"hint","dependencies":["a7ecb8ffactoring14a-h2"],"title":"Finding $$m$$ and $$n$$","text":"$$c=-7$$, and $$b=6$$. $$m=7$$ and $$m=-1$$ multiply to c and add to $$b$$. So, our solution is $$\\\\left(x+7\\\\right) \\\\left(x-1\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ecb8ffactoring15","title":"Factoring Trinomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Factor Trinomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a7ecb8ffactoring15a","stepAnswer":["$$(x-4)(x-2)$$"],"problemType":"TextBox","stepTitle":"Factor $$8-6x+x^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$(x-4)(x-2)$$","hints":{"DefaultPathway":[{"id":"a7ecb8ffactoring15a-h1","type":"hint","dependencies":[],"title":"General Factoring Procedure","text":"When given a trinomial in the form $$x^2+bx+c$$, we must find two numbers (m and n) that multiply to c and add to $$b$$. The factored solution is $$\\\\left(x+m\\\\right) \\\\left(x+n\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ecb8ffactoring15a-h2","type":"hint","dependencies":["a7ecb8ffactoring15a-h1"],"title":"Rearranging our Trinomial","text":"We must rearrange our trinomial to fix the form $$x^2+bx+c$$. We get $$x^2-6x+8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ecb8ffactoring15a-h3","type":"hint","dependencies":["a7ecb8ffactoring15a-h2"],"title":"Finding $$m$$ and $$n$$","text":"$$c=8$$, and $$b=-6$$. $$m=-4$$ and $$m=-2$$ multiply to c and add to $$b$$. So, our solution is $$(x-4)(x-2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ecb8ffactoring2","title":"Factoring Trinomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Factor Trinomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a7ecb8ffactoring2a","stepAnswer":["$$\\\\left(x+7\\\\right) \\\\left(x+3\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor $$x^2+10x+21$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(x+7\\\\right) \\\\left(x+3\\\\right)$$","hints":{"DefaultPathway":[{"id":"a7ecb8ffactoring2a-h1","type":"hint","dependencies":[],"title":"General Factoring Procedure","text":"When given a trinomial in the form $$x^2+bx+c$$, we must find two numbers (m and n) that multiply to c and add to $$b$$. The factored solution is $$\\\\left(x+m\\\\right) \\\\left(x+n\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ecb8ffactoring2a-h2","type":"hint","dependencies":["a7ecb8ffactoring2a-h1"],"title":"Finding $$m$$ and $$n$$","text":"$$c=21$$, and $$b=10$$. $$m=7$$ and $$m=3$$ multiply to c and add to $$b$$. So, our solution is $$\\\\left(x+7\\\\right) \\\\left(x+3\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ecb8ffactoring3","title":"Factoring Trinomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Factor Trinomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a7ecb8ffactoring3a","stepAnswer":["$$\\\\left(x+16\\\\right) \\\\left(x+3\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor $$x^2+19x+48$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(x+16\\\\right) \\\\left(x+3\\\\right)$$","hints":{"DefaultPathway":[{"id":"a7ecb8ffactoring3a-h1","type":"hint","dependencies":[],"title":"General Factoring Procedure","text":"When given a trinomial in the form $$x^2+bx+c$$, we must find two numbers (m and n) that multiply to c and add to $$b$$. The factored solution is $$\\\\left(x+m\\\\right) \\\\left(x+n\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ecb8ffactoring3a-h2","type":"hint","dependencies":["a7ecb8ffactoring3a-h1"],"title":"Finding $$m$$ and $$n$$","text":"$$c=48$$, and $$b=19$$. $$m=16$$ and $$m=3$$ multiply to c and add to $$b$$. So, our solution is $$\\\\left(x+16\\\\right) \\\\left(x+3\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ecb8ffactoring4","title":"Factoring Trinomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Factor Trinomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a7ecb8ffactoring4a","stepAnswer":["$$\\\\left(x+8\\\\right) \\\\left(x+6\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor $$x^2+14x+48$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(x+8\\\\right) \\\\left(x+6\\\\right)$$","hints":{"DefaultPathway":[{"id":"a7ecb8ffactoring4a-h1","type":"hint","dependencies":[],"title":"General Factoring Procedure","text":"When given a trinomial in the form $$x^2+bx+c$$, we must find two numbers (m and n) that multiply to c and add to $$b$$. The factored solution is $$\\\\left(x+m\\\\right) \\\\left(x+n\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ecb8ffactoring4a-h2","type":"hint","dependencies":["a7ecb8ffactoring4a-h1"],"title":"Finding $$m$$ and $$n$$","text":"$$c=48$$, and $$b=14$$. $$m=8$$ and $$m=6$$ multiply to c and add to $$b$$. So, our solution is $$\\\\left(x+8\\\\right) \\\\left(x+6\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ecb8ffactoring5","title":"Factoring Trinomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Factor Trinomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a7ecb8ffactoring5a","stepAnswer":["$$\\\\left(x+20\\\\right) \\\\left(x+5\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor $$x^2+25x+100$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(x+20\\\\right) \\\\left(x+5\\\\right)$$","hints":{"DefaultPathway":[{"id":"a7ecb8ffactoring5a-h1","type":"hint","dependencies":[],"title":"General Factoring Procedure","text":"When given a trinomial in the form $$x^2+bx+c$$, we must find two numbers (m and n) that multiply to c and add to $$b$$. The factored solution is $$\\\\left(x+m\\\\right) \\\\left(x+n\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ecb8ffactoring5a-h2","type":"hint","dependencies":["a7ecb8ffactoring5a-h1"],"title":"Finding $$m$$ and $$n$$","text":"$$c=100$$, and $$b=25$$. $$m=20$$ and $$m=5$$ multiply to c and add to $$b$$. So, our solution is $$\\\\left(x+20\\\\right) \\\\left(x+5\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ecb8ffactoring6","title":"Factoring Trinomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Factor Trinomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a7ecb8ffactoring6a","stepAnswer":["$$\\\\left(x+100\\\\right) \\\\left(x+1\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor $$x^2+101x+100$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(x+100\\\\right) \\\\left(x+1\\\\right)$$","hints":{"DefaultPathway":[{"id":"a7ecb8ffactoring6a-h1","type":"hint","dependencies":[],"title":"General Factoring Procedure","text":"When given a trinomial in the form $$x^2+bx+c$$, we must find two numbers (m and n) that multiply to c and add to $$b$$. The factored solution is $$\\\\left(x+m\\\\right) \\\\left(x+n\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ecb8ffactoring6a-h2","type":"hint","dependencies":["a7ecb8ffactoring6a-h1"],"title":"Finding $$m$$ and $$n$$","text":"$$c=101$$, and $$b=100$$. $$m=100$$ and $$m=1$$ multiply to c and add to $$b$$. So, our solution is $$\\\\left(x+100\\\\right) \\\\left(x+1\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ecb8ffactoring7","title":"Factoring Trinomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Factor Trinomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a7ecb8ffactoring7a","stepAnswer":["$$(x-6)(x-2)$$"],"problemType":"TextBox","stepTitle":"Factor $$x^2-8x+12$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$(x-6)(x-2)$$","hints":{"DefaultPathway":[{"id":"a7ecb8ffactoring7a-h1","type":"hint","dependencies":[],"title":"General Factoring Procedure","text":"When given a trinomial in the form $$x^2+bx+c$$, we must find two numbers (m and n) that multiply to c and add to $$b$$. The factored solution is $$\\\\left(x+m\\\\right) \\\\left(x+n\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ecb8ffactoring7a-h2","type":"hint","dependencies":["a7ecb8ffactoring7a-h1"],"title":"Finding $$m$$ and $$n$$","text":"$$c=12$$, and $$b=-8$$. $$m=-6$$ and $$m=-2$$ multiply to c and add to $$b$$. So, our solution is $$(x-6)(x-2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ecb8ffactoring8","title":"Factoring Trinomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Factor Trinomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a7ecb8ffactoring8a","stepAnswer":["$$(x-9)(x-4)$$"],"problemType":"TextBox","stepTitle":"Factor $$x^2-13x+36$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$(x-9)(x-4)$$","hints":{"DefaultPathway":[{"id":"a7ecb8ffactoring8a-h1","type":"hint","dependencies":[],"title":"General Factoring Procedure","text":"When given a trinomial in the form $$x^2+bx+c$$, we must find two numbers (m and n) that multiply to c and add to $$b$$. The factored solution is $$\\\\left(x+m\\\\right) \\\\left(x+n\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ecb8ffactoring8a-h2","type":"hint","dependencies":["a7ecb8ffactoring8a-h1"],"title":"Finding $$m$$ and $$n$$","text":"$$c=36$$, and $$b=-12$$. $$m=-9$$ and $$m=-4$$ multiply to c and add to $$b$$. So, our solution is $$(x-9)(x-4)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7ecb8ffactoring9","title":"Factoring Trinomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.2 Factor Trinomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a7ecb8ffactoring9a","stepAnswer":["$$(x-15)(x-3)$$"],"problemType":"TextBox","stepTitle":"Factor $$x^2+18x+45$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$(x-15)(x-3)$$","hints":{"DefaultPathway":[{"id":"a7ecb8ffactoring9a-h1","type":"hint","dependencies":[],"title":"General Factoring Procedure","text":"When given a trinomial in the form $$x^2+bx+c$$, we must find two numbers (m and n) that multiply to c and add to $$b$$. The factored solution is $$\\\\left(x+m\\\\right) \\\\left(x+n\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7ecb8ffactoring9a-h2","type":"hint","dependencies":["a7ecb8ffactoring9a-h1"],"title":"Finding $$m$$ and $$n$$","text":"$$c=45$$, and $$b=-18$$. $$m=-15$$ and $$m=-3$$ multiply to c and add to $$b$$. So, our solution is $$(x-15)(x-3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7fd07f5.1continuous1","title":"Distributions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Continuous Probability Functions","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a7fd07f5.1continuous1a","stepAnswer":["$$0$$"],"problemType":"MultipleChoice","stepTitle":"For a continuous probability distribution, $$0$$ $$ \\\\leq X$$ $$ \\\\leq $$ $$15$$. What is P(x > 15)?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$0$$","choices":["$$0$$","$$1$$","$$0.5$$","$$0.15$$"],"hints":{"DefaultPathway":[{"id":"a7fd07f5.1continuous1a-h1","type":"hint","dependencies":[],"title":"Interpretation","text":"There is very little information given hinting that if we know some fundamental characteristics of continuous probability distribution, we can find the probability.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7fd07f5.1continuous1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a7fd07f5.1continuous1a-h1"],"title":"Distributions","text":"In a continuous probability distribution, the total area in the distribution is $$_{}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7fd07f5.1continuous1a-h3","type":"hint","dependencies":["a7fd07f5.1continuous1a-h2"],"title":"Interpretation","text":"The area under the curve of a continuous probability distribution is $$1$$, so the area that is outside of the curve is $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7fd07f5.1continuous2","title":"Distributions","body":"Distributions","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Continuous Probability Functions","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a7fd07f5.1continuous2a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"What is the area under f(x) if the function is a continuous probability density function?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a7fd07f5.1continuous2a-h1","type":"hint","dependencies":[],"title":"Distributions","text":"The area under probability density functions must be $$1$$ in total to accurately model the experiment.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7fd07f5.1continuous3","title":"Distributions","body":"Distributions","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Continuous Probability Functions","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a7fd07f5.1continuous3a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"For a continuous probability distribution, $$0 \\\\leq x \\\\leq 10$$. What is $$P(x=7)$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a7fd07f5.1continuous3a-h1","type":"hint","dependencies":[],"title":"Distributions","text":"The total area under the curve of a continuous probability distribution function is $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7fd07f5.1continuous3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a7fd07f5.1continuous3a-h1"],"title":"Area","text":"We are trying to find the area of a line $$x=7$$ under a curve whose total area equals $$1$$. The line has $$width=0$$. What is the area of the line?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7fd07f5.1intro","title":"Continuous Probability Distributions","body":"Consider the function f(x) $$=$$ $$\\\\frac{1}{20}$$ for $$0 \\\\leq $$ $$x \\\\leq 20$$. $$x=a$$ real number. The graph of $$f(x)=120$$ is a horizontal line. However, since $$0 \\\\leq x \\\\leq 20$$, f(x) is restricted to the portion between $$x=0$$ and $$x=20$$, inclusive. $$f(x)=120$$ for $$0 \\\\leq x \\\\leq 20$$. The graph of f(x) $$=$$ $$120$$ is a horizontal line segment when $$0$$ $$ \\\\leq $$ $$x$$ $$ \\\\leq $$ 20.The area between f(x) $$=$$ $$120$$ where $$0 \\\\leq x \\\\leq 20$$ and the x-axis is the area of a rectangle with $$base=20$$ and height=120.\\\\n##figure2.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Continuous Probability Functions","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a7fd07f5.1introa","stepAnswer":["$$20$$"],"problemType":"TextBox","stepTitle":"What is the area of the rectangle formed by f(x)?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$20$$","hints":{"DefaultPathway":[{"id":"a7fd07f5.1introa-h1","type":"hint","dependencies":[],"title":"Interpretation","text":"To find the area of the rectangle, multiply base and height.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7fd07f5.1introa-h2","type":"hint","dependencies":["a7fd07f5.1introa-h1"],"title":"Area","text":"The height given is $$h=\\\\frac{1}{20}$$, and the base is $$b=20$$. $$Area=b h$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a7fd07f5.1introb","stepAnswer":["$$0.1$$"],"problemType":"TextBox","stepTitle":"Suppose we want to find the area between $$f(x)=20$$ and the x-axis where $$0<X<2$$. What is the area?","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.1$$","hints":{"DefaultPathway":[{"id":"a7fd07f5.1introb-h1","type":"hint","dependencies":[],"title":"Finding Area","text":"Find the base and height of the newly formed rectangle indicated by the blue shaded region.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7fd07f5.1introb-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a7fd07f5.1introb-h1"],"title":"Finding Area","text":"What is the base? Assume the base to be the length on the x-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7fd07f5.1introb-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{20}$$"],"dependencies":["a7fd07f5.1introb-h2"],"title":"Finding Area","text":"What is the height? Assume the height to be the length on the y-axis","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7fd07f5.1introb-h4","type":"hint","dependencies":["a7fd07f5.1introb-h3"],"title":"Calculating Area","text":"$$Area=base height$$, we have $$base=2$$, $$height=\\\\frac{1}{20}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a7fd07f5.1introc","stepAnswer":["$$0.55$$"],"problemType":"TextBox","stepTitle":"The area shaded corresponds to a probability. The probability that $$x$$ is between zero and two is $$0.1$$, which can be written mathematically as P(0 < $$x$$ < 2) $$=$$ P(x < 2) $$=$$ $$0.1$$. Suppose we want to find the area between $$f(x)=20$$ and the x-axis where $$4<X<15$$. What is the area?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.55$$","hints":{"DefaultPathway":[{"id":"a7fd07f5.1introc-h1","type":"hint","dependencies":[],"title":"Base and Height","text":"Find the base and height of the newly formed rectangle indicated by the blue shaded region.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7fd07f5.1introc-h2","type":"hint","dependencies":["a7fd07f5.1introc-h1"],"title":"Finding Base","text":"Find the base. Assume the base to be the length on the x-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7fd07f5.1introc-h3","type":"hint","dependencies":["a7fd07f5.1introc-h2"],"title":"Finding Height","text":"Find the height. Assume the height to be the length on the y-axis","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7fd07f5.1introc-h4","type":"hint","dependencies":["a7fd07f5.1introc-h3"],"title":"Finding Area","text":"$$Area=base height$$, we have $$base=15-4=11$$, $$height=\\\\frac{1}{20}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7fd07f5.1normal","title":"Distributions","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Continuous Probability Functions","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a7fd07f5.1normala","stepAnswer":["normal"],"problemType":"MultipleChoice","stepTitle":"Which type of distribution does the graph illustrate?","stepBody":"","answerType":"string","variabilization":{},"choices":["skewed right","skewed left","normal","uniform"],"hints":{"DefaultPathway":[{"id":"a7fd07f5.1normala-h1","type":"hint","dependencies":[],"title":"Distributions","text":"The x-axis indicates symmetry about the center at the line $$x=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7fd07f5.1normala-h2","type":"hint","dependencies":["a7fd07f5.1normala-h1"],"title":"Distributions","text":"The symmetry and bell shape of the curve signals that this is a $$_{}$$ distribution","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7fd07f5.1rectangle","title":"Distributions","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Continuous Probability Functions","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a7fd07f5.1rectanglea","stepAnswer":["Uniform Distribution"],"problemType":"MultipleChoice","stepTitle":"Which type of distribution does the graph illustrate?","stepBody":"","answerType":"string","variabilization":{},"choices":["Uniform Distribution","Normal Distribution","Skewed Distribution","Binomial Distribution"],"hints":{"DefaultPathway":[{"id":"a7fd07f5.1rectanglea-h1","type":"hint","dependencies":[],"title":"Distributions","text":"Notice that every event is equally likely to occur.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7fd07f5.1rectanglea-h2","type":"hint","dependencies":["a7fd07f5.1rectanglea-h1"],"title":"Distributions","text":"Since all events are likely and the distribution forms a rectangle shape, this is considered to be a uniform distribution. Remember that uniform distributions can either be continous or discrete.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7fd07f5.1rectangle2","title":"Distributions","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Continuous Probability Functions","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a7fd07f5.1rectangle2a","stepAnswer":["2,5"],"problemType":"MultipleChoice","stepTitle":"What does the shaded area represent? $$P\\\\left(_{}<x<_{}\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["2,5","3,4","1,2","5,8"],"hints":{"DefaultPathway":[{"id":"a7fd07f5.1rectangle2a-h1","type":"hint","dependencies":[],"title":"Notation","text":"The shaded area should be covered by whatever the X limits are in the inquality statement.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7fd07f5.1rectangle2a-h2","type":"hint","dependencies":["a7fd07f5.1rectangle2a-h1"],"title":"Interpretation","text":"The lower bound should be the lowest value to the left that that the shaded area exists, and the higher bound should be the highest value to the right that the shaded area exists. Therefore, the lower bound is $$2$$, and the upper bound is $$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7fd07f5.1wedge","title":"Distributions","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Continuous Probability Functions","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a7fd07f5.1wedgea","stepAnswer":["skewed right"],"problemType":"MultipleChoice","stepTitle":"Which type of distribution does the graph illustrate?","stepBody":"","answerType":"string","variabilization":{},"choices":["skewed right","skewed left","normal","uniform"],"hints":{"DefaultPathway":[{"id":"a7fd07f5.1wedgea-h1","type":"hint","dependencies":[],"title":"Distributions","text":"Notice that the graph shows a moot point curve towards the right. This is an indication of a skewed graph.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7fd07f5.1wedgea-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["right"],"dependencies":["a7fd07f5.1wedgea-h1"],"title":"Distributions","text":"To name skewed graphs, the tail end with lower probability is the direction of the skew. What direction does the tail point?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["left","right"]},{"id":"a7fd07f5.1wedgea-h3","type":"hint","dependencies":["a7fd07f5.1wedgea-h2"],"title":"Distributions","text":"Since the tail is to the right, we call this distribution right skewed","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a7fd07f5.1wedge2","title":"Distributions","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Continuous Probability Functions","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a7fd07f5.1wedge2a","stepAnswer":["6,7"],"problemType":"MultipleChoice","stepTitle":"What does the shaded area represent? $$P\\\\left(_{}<x<_{}\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["6,7","0,10","0,6","6,8"],"hints":{"DefaultPathway":[{"id":"a7fd07f5.1wedge2a-h1","type":"hint","dependencies":[],"title":"Notation","text":"The shaded area should be covered by whatever the X limits are in the inquality statement.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a7fd07f5.1wedge2a-h2","type":"hint","dependencies":["a7fd07f5.1wedge2a-h1"],"title":"Interpretation","text":"The lower bound should be the lowest value to the left that that the shaded area exists, and the higher bound should be the highest value to the right that the shaded area exists. Therefore, the lower bound is $$6$$, and the upper bound is $$7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a805812corr1","title":"Testing the Significance of the Correlation Coefficient","body":"For a given line of best fit, you computed that $$r$$ $$=$$ $$0.6501$$ using $$n$$ $$=$$ $$12$$ data points.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.3 The Regression Equation","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a805812corr1a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Can the line be used for prediction?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a805812corr1a-h1","type":"hint","dependencies":[],"title":"Understanding the Question","text":"The 95% Critical Values of the Sample Correlation Coefficient Table can be used to determine whether the computed value of $$r$$, the sample correlation coefficient, is significant or not. Compare $$r$$ to the appropriate critical value in the table. If $$r$$ is not between the positive and negative critical values, then the correlation coefficient is significant. If $$r$$ is significant, then you may want to use the line for prediction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a805812corr1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a805812corr1a-h1"],"title":"Degrees of Freedom","text":"What are the degrees of freedom (df): $$n$$ - $$2$$, where $$n$$ is the number of data points?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a805812corr1a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":[],"title":"Finding $$n$$","text":"What is $$n$$, the number of data points?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a805812corr1a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":[],"title":"Solving for Degrees of Freedom","text":"Plugging $$n$$ $$=$$ $$12$$ into the df equation, what is $$n$$ - 2?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a805812corr1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.576$$"],"dependencies":["a805812corr1a-h2"],"title":"Critical Values","text":"What positive and negative critical value is associated with df $$=$$ 10? Reference the 95% Critical Values of the Sample Correlation Coefficient Table.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a805812corr1a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a805812corr1a-h3"],"title":"$$r$$ Significance","text":"The critical values associated with df $$=$$ $$10$$ are $$-0.576$$ and $$+0.576$$. Keep in mind that $$r$$ $$=$$ $$0.6501$$. If $$r$$ < negative critical value or $$r$$ > positive critical value, then $$r$$ is significant. Is $$r$$ significant?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a805812corr1a-h5","type":"hint","dependencies":["a805812corr1a-h4"],"title":"Prediction","text":"Since $$r$$ $$=$$ $$0.6501$$ and $$0.6501$$ > $$0.576$$, $$r$$ is significant and the line may be used for prediction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a805812corr2","title":"Testing the Significance of the Correlation Coefficient","body":"For a given line of best fit, you compute that $$r$$ $$=$$ $$0.5204$$ using $$n$$ $$=$$ $$9$$ data points.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.3 The Regression Equation","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a805812corr2a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Can the line be used for prediction?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a805812corr2a-h1","type":"hint","dependencies":[],"title":"Understanding the Question","text":"The 95% Critical Values of the Sample Correlation Coefficient Table can be used to determine whether the computed value of $$r$$, the sample correlation coefficient, is significant or not. Compare $$r$$ to the appropriate critical value in the table. If $$r$$ is not between the positive and negative critical values, then the correlation coefficient is significant. If $$r$$ is significant, then you may want to use the line for prediction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a805812corr2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a805812corr2a-h1"],"title":"Degrees of Freedom","text":"What are the degrees of freedom (df): $$n$$ - $$2$$, where $$n$$ is the number of data points?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a805812corr2a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":[],"title":"Finding $$n$$","text":"What is $$n$$, the number of data points?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a805812corr2a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":[],"title":"Solving for Degrees of Freedom","text":"Plugging $$n$$ $$=$$ $$9$$ into the df equation, what is $$n$$ - 2?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a805812corr2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.666$$"],"dependencies":["a805812corr2a-h2"],"title":"Critical Values","text":"What positive and negative critical value is associated with df $$=$$ 7? Reference the 95% Critical Values of the Sample Correlation Coefficient Table.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a805812corr2a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a805812corr2a-h3"],"title":"$$r$$ Significance","text":"The critical values associated with df $$=$$ $$7$$ are $$-0.666$$ and $$+0.666$$. Keep in mind that $$r$$ $$=$$ $$0.5204$$. If $$r$$ < negative critical value or $$r$$ > positive critical value, then $$r$$ is significant. Is $$r$$ significant?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a805812corr2a-h5","type":"hint","dependencies":["a805812corr2a-h4"],"title":"Prediction","text":"Since $$r$$ $$=$$ $$0.5204$$ and $$-0.666$$ < $$0.5204$$ < $$0.666$$, $$r$$ is not significant and the line should not be used for prediction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a805812corr3","title":"Testing the Significance of the Correlation Coefficient","body":"For a given line of best fit, you compute that $$r$$ $$=$$ $$-0.7204$$ using $$n$$ $$=$$ $$8$$ data points.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.3 The Regression Equation","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a805812corr3a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Can the line be used for prediction?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a805812corr3a-h1","type":"hint","dependencies":[],"title":"Understanding the Question","text":"The 95% Critical Values of the Sample Correlation Coefficient Table can be used to determine whether the computed value of $$r$$, the sample correlation coefficient, is significant or not. Compare $$r$$ to the appropriate critical value in the table. If $$r$$ is not between the positive and negative critical values, then the correlation coefficient is significant. If $$r$$ is significant, then you may want to use the line for prediction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a805812corr3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a805812corr3a-h1"],"title":"Degrees of Freedom","text":"What are the degrees of freedom (df): $$n$$ - $$2$$, where $$n$$ is the number of data points?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a805812corr3a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":[],"title":"Finding $$n$$","text":"What is $$n$$, the number of data points?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a805812corr3a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":[],"title":"Solving for Degrees of Freedom","text":"Plugging $$n$$ $$=$$ $$8$$ into the df equation, what is $$n$$ - 2?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a805812corr3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.707$$"],"dependencies":["a805812corr3a-h2"],"title":"Critical Values","text":"What positive and negative critical value is associated with df $$=$$ 6? Reference the 95% Critical Values of the Sample Correlation Coefficient Table.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a805812corr3a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a805812corr3a-h3"],"title":"$$r$$ Significance","text":"The critical values associated with df $$=$$ $$6$$ are $$-0.707$$ and $$+0.707$$. Keep in mind that $$r$$ $$=$$ $$-0.7204$$. If $$r$$ < negative critical value or $$r$$ > positive critical value, then $$r$$ is significant. Is $$r$$ significant?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a805812corr3a-h5","type":"hint","dependencies":["a805812corr3a-h4"],"title":"Prediction","text":"Since $$r$$ $$=$$ $$-0.7204$$ and $$-0.7204$$ < $$-0.707$$, $$r$$ is significant and the line may be used for prediction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a805812corr4","title":"Testing the Significance of the Correlation Coefficient","body":"Suppose you computed the following correlation coefficient $$r$$ $$=$$ $$-0.567$$ where the sample size, $$n$$, is $$19$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.3 The Regression Equation","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a805812corr4a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Using the table, is $$r$$ is significant such that the line of best fit associated with each $$r$$ can be used to predict a $$y$$ value?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a805812corr4a-h1","type":"hint","dependencies":[],"title":"Understanding the Question","text":"The 95% Critical Values of the Sample Correlation Coefficient Table (simplified version) can be used to determine whether the computed value of $$r$$, the sample correlation coefficient, is significant or not. Compare $$r$$ to the appropriate critical value in the table. If $$r$$ is not between the positive and negative critical values, then the correlation coefficient is significant. If $$r$$ is significant, then you may want to use the line for prediction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a805812corr4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$17$$"],"dependencies":["a805812corr4a-h1"],"title":"Degrees of Freedom","text":"What are the degrees of freedom (df): $$n$$ - $$2$$, where $$n$$ is the number of data points?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a805812corr4a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$19$$"],"dependencies":[],"title":"Finding $$n$$","text":"What is $$n$$, the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a805812corr4a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$17$$"],"dependencies":[],"title":"Solving for Degrees of Freedom","text":"Plugging $$n$$ $$=$$ $$19$$ into the df equation, what is $$n$$ - 2?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a805812corr4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.456$$"],"dependencies":["a805812corr4a-h2"],"title":"Critical Values","text":"What positive and negative critical value is associated with df $$=$$ 17? Reference the 95% Critical Values of the Sample Correlation Coefficient Table.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a805812corr4a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a805812corr4a-h3"],"title":"$$r$$ Significance","text":"The critical values associated with df $$=$$ $$17$$ are $$-0.456$$ and $$+0.456$$. Keep in mind that $$r$$ $$=$$ $$-0.567$$. If $$r$$ < negative critical value or $$r$$ > positive critical value, then $$r$$ is significant. Is $$r$$ significant?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a805812corr4a-h5","type":"hint","dependencies":["a805812corr4a-h4"],"title":"Prediction","text":"Since $$r$$ $$=$$ $$-0.567$$ and $$-0.567$$ < $$-0.456$$, $$r$$ is significant and the line may be used for prediction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a805812corr5","title":"Testing the Significance of the Correlation Coefficient","body":"Suppose you computed the following correlation coefficient $$r$$ $$=$$ $$0.708$$, where the sample size, $$n$$, is nine.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.3 The Regression Equation","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a805812corr5a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Using the table, is $$r$$ is significant such that the line of best fit associated with each $$r$$ can be used to predict a $$y$$ value?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a805812corr5a-h1","type":"hint","dependencies":[],"title":"Understanding the Question","text":"The 95% Critical Values of the Sample Correlation Coefficient Table (simplified version) can be used to determine whether the computed value of $$r$$, the sample correlation coefficient, is significant or not. Compare $$r$$ to the appropriate critical value in the table. If $$r$$ is not between the positive and negative critical values, then the correlation coefficient is significant. If $$r$$ is significant, then you may want to use the line for prediction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a805812corr5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a805812corr5a-h1"],"title":"Degrees of Freedom","text":"What are the degrees of freedom (df): $$n$$ - $$2$$, where $$n$$ is the number of data points?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a805812corr5a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":[],"title":"Finding $$n$$","text":"What is $$n$$, the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a805812corr5a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":[],"title":"Solving for Degrees of Freedom","text":"Plugging $$n$$ $$=$$ $$9$$ into the df equation, what is $$n$$ - 2?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a805812corr5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.666$$"],"dependencies":["a805812corr5a-h2"],"title":"Critical Values","text":"What positive and negative critical value is associated with df $$=$$ 7? Reference the 95% Critical Values of the Sample Correlation Coefficient Table.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a805812corr5a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a805812corr5a-h3"],"title":"$$r$$ Significance","text":"The critical values associated with df $$=$$ $$7$$ are $$-0.666$$ and $$+0.666$$. Keep in mind that $$r$$ $$=$$ $$0.708$$. If $$r$$ < negative critical value or $$r$$ > positive critical value, then $$r$$ is significant. Is $$r$$ significant?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a805812corr5a-h5","type":"hint","dependencies":["a805812corr5a-h4"],"title":"Prediction","text":"Since $$r$$ $$=$$ $$0.708$$ and $$0.708$$ > $$0.666$$, $$r$$ is significant and the line may be used for prediction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a805812corr6","title":"Testing the Significance of the Correlation Coefficient","body":"Suppose you computed the following correlation coefficient $$r$$ $$=$$ $$0.134$$, where the sample size, $$n$$, is $$14$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.3 The Regression Equation","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a805812corr6a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Using the table, is $$r$$ is significant such that the line of best fit associated with each $$r$$ can be used to predict a $$y$$ value?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a805812corr6a-h1","type":"hint","dependencies":[],"title":"Understanding the Question","text":"The 95% Critical Values of the Sample Correlation Coefficient Table (simplified version) can be used to determine whether the computed value of $$r$$, the sample correlation coefficient, is significant or not. Compare $$r$$ to the appropriate critical value in the table. If $$r$$ is not between the positive and negative critical values, then the correlation coefficient is significant. If $$r$$ is significant, then you may want to use the line for prediction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a805812corr6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a805812corr6a-h1"],"title":"Degrees of Freedom","text":"What are the degrees of freedom (df): $$n$$ - $$2$$, where $$n$$ is the number of data points?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a805812corr6a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$14$$"],"dependencies":[],"title":"Finding $$n$$","text":"What is $$n$$, the sample size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a805812corr6a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":[],"title":"Solving for Degrees of Freedom","text":"Plugging $$n$$ $$=$$ $$14$$ into the df equation, what is $$n$$ - 2?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a805812corr6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.532$$"],"dependencies":["a805812corr6a-h2"],"title":"Critical Values","text":"What positive and negative critical value is associated with df $$=$$ 12? Reference the 95% Critical Values of the Sample Correlation Coefficient Table.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a805812corr6a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a805812corr6a-h3"],"title":"$$r$$ Significance","text":"The critical values associated with df $$=$$ $$12$$ are $$-0.532$$ and $$+0.532$$. Keep in mind that $$r$$ $$=$$ $$0.134$$. If $$r$$ < negative critical value or $$r$$ > positive critical value, then $$r$$ is significant. Is $$r$$ significant?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a805812corr6a-h5","type":"hint","dependencies":["a805812corr6a-h4"],"title":"Prediction","text":"Since $$r$$ $$=$$ $$0.134$$ and $$-0.532$$ < $$0.134$$ < $$0.532$$, $$r$$ is not significant and the line should not be used for prediction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a805812corr7","title":"Testing the Significance of the Correlation Coefficient","body":"Suppose you computed the following correlation coefficient $$r$$ $$=$$ $$0$$, where the sample size, $$n$$, is five.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.3 The Regression Equation","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a805812corr7a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Using the table, is $$r$$ is significant such that the line of best fit associated with each $$r$$ can be used to predict a $$y$$ value?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a805812corr7a-h1","type":"hint","dependencies":[],"title":"Understanding the Question","text":"The 95% Critical Values of the Sample Correlation Coefficient Table (simplified version) can be used to determine whether the computed value of $$r$$, the sample correlation coefficient, is significant or not. If $$r$$ is not between the positive and negative critical values, then the correlation coefficient is significant. If $$r$$ is significant, then you may want to use the line for prediction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a805812corr7a-h2","type":"hint","dependencies":["a805812corr7a-h1"],"title":"$$r$$ $$=$$ $$0$$","text":"No matter what the dfs are, $$r$$ $$=$$ $$0$$ is between the two critical values, so $$r$$ is not significant.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a81d715SolRadical1","title":"Solve Radical Equations","body":"In the following exercises, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Radical Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a81d715SolRadical1a","stepAnswer":["$$14$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{5x-6}=8$$","stepBody":"Please enter your solution. If there is no solution for this equation, please enter \\"None\\"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$14$$","hints":{"DefaultPathway":[{"id":"a81d715SolRadical1a-h1","type":"hint","dependencies":[],"title":"Isolate the Radical Terms","text":"Isolate the radical on one side of the equation. Since the given equation already have radical isolated on one side, we can proceed to the next step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical1a-h2","type":"hint","dependencies":["a81d715SolRadical1a-h1"],"title":"Raise Both Sides Of the Equation To The Power Of The Index","text":"$${\\\\sqrt{5x-6}}^2=8^2$$\\\\nAfter squaring both sides, we get $$5x-6=64$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical1a-h3","type":"hint","dependencies":["a81d715SolRadical1a-h2"],"title":"Check For Remaining Radical Terms","text":"Are there any more radicals? No. We can proceed to the next step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical1a-h4","type":"hint","dependencies":["a81d715SolRadical1a-h3"],"title":"Solve For Equation","text":"Solve for the equation $$5x-6=64$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical1a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$14$$"],"dependencies":["a81d715SolRadical1a-h4"],"title":"Solve For Equation","text":"What is the value of $$x$$ for $$5x-6=64$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a81d715SolRadical10","title":"Solve Radical Equations","body":"In the following exercises, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Radical Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a81d715SolRadical10a","stepAnswer":["$$10$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{12u+1}-11=0$$","stepBody":"Please enter your solution. If there is no solution for this equation, please enter \\"None\\"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10$$","hints":{"DefaultPathway":[{"id":"a81d715SolRadical10a-h1","type":"hint","dependencies":[],"title":"Isolate the Radical","text":"Isolate the radical on one side of the equation. We can add $$11$$ on both sides get $$\\\\sqrt{12u+1}=11$$ which has radicals isolated on one side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical10a-h2","type":"hint","dependencies":["a81d715SolRadical10a-h1"],"title":"Raise Both Sides Of the Equation To The Power Of The Index","text":"$${\\\\sqrt{12u+1}}^2={11}^2$$\\\\nAfter squaring both sides, we get $$12u+1=121$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical10a-h3","type":"hint","dependencies":["a81d715SolRadical10a-h2"],"title":"Check For Remaining Radicals","text":"Are there any more radicals? No. We can proceed to the next step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical10a-h4","type":"hint","dependencies":["a81d715SolRadical10a-h3"],"title":"Solve For Equation","text":"Solve for the equation $$12u+1=121$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a81d715SolRadical10a-h4"],"title":"Solve For Equation","text":"What is the value of u for $$12u+1=121$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a81d715SolRadical11","title":"Solve Radical Equations","body":"In the following exercises, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Radical Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a81d715SolRadical11a","stepAnswer":["$$\\\\frac{7}{2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{4m+2}+2=6$$","stepBody":"Please enter your solution. If there is no solution for this equation, please enter \\"None\\"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{7}{2}$$","hints":{"DefaultPathway":[{"id":"a81d715SolRadical11a-h1","type":"hint","dependencies":[],"title":"Isolate the Radical","text":"Isolate the radical on one side of the equation. We can subtract $$2$$ on both sides get $$\\\\sqrt{4m+2}=4$$ which has radicals isolated on one side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical11a-h2","type":"hint","dependencies":["a81d715SolRadical11a-h1"],"title":"Raise Both Sides Of the Equation To The Power Of The Index","text":"$${\\\\sqrt{4m+2}}^2=4^2$$\\\\nAfter squaring both sides, we get $$4m+2=16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical11a-h3","type":"hint","dependencies":["a81d715SolRadical11a-h2"],"title":"Check For Remaining Radicals","text":"Are there any more radicals? No. We can proceed to the next step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical11a-h4","type":"hint","dependencies":["a81d715SolRadical11a-h3"],"title":"Solve For Equation","text":"Solve for the equation $$4m+2=16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{7}{2}$$"],"dependencies":["a81d715SolRadical11a-h4"],"title":"Solve For Equation","text":"What is the value of $$m$$ for $$4m+2=16$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a81d715SolRadical12","title":"Solve Radical Equations","body":"In the following exercises, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Radical Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a81d715SolRadical12a","stepAnswer":["$$\\\\frac{5}{2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{6n+1}+4=8$$","stepBody":"Please enter your solution. If there is no solution for this equation, please enter \\"None\\"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5}{2}$$","hints":{"DefaultPathway":[{"id":"a81d715SolRadical12a-h1","type":"hint","dependencies":[],"title":"Isolate the Radical","text":"Isolate the radical on one side of the equation. We can subtract $$4$$ on both sides get $$\\\\sqrt{6n+1}=4$$ which has radicals isolated on one side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical12a-h2","type":"hint","dependencies":["a81d715SolRadical12a-h1"],"title":"Raise Both Sides Of the Equation To The Power Of The Index","text":"$${\\\\sqrt{6n+1}}^2=4^2$$\\\\nAfter squaring both sides, we get $$6n+1=16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical12a-h3","type":"hint","dependencies":["a81d715SolRadical12a-h2"],"title":"Check For Remaining Radicals","text":"Are there any more radicals? No. We can proceed to the next step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical12a-h4","type":"hint","dependencies":["a81d715SolRadical12a-h3"],"title":"Solve For Equation","text":"Solve for the equation $$6n+1=16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical12a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{2}$$"],"dependencies":["a81d715SolRadical12a-h4"],"title":"Solve For Equation","text":"What is the value of $$n$$ for $$6n+1=16$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a81d715SolRadical13","title":"Solve Radical Equations","body":"In the following exercises, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Radical Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a81d715SolRadical13a","stepAnswer":["None"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{2u-3}+2=0$$","stepBody":"Please enter your solution. If there is no solution for this equation, please enter \\"None\\"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a81d715SolRadical13a-h1","type":"hint","dependencies":[],"title":"Isolate the Radical","text":"Isolate the radical on one side of the equation. We can subtract $$2$$ on both sides get $$\\\\sqrt{2u-3}=-2$$ which has radicals isolated on one side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical13a-h2","type":"hint","dependencies":["a81d715SolRadical13a-h1"],"title":"Check for Value of Square Root Function","text":"The square root functions only take positive $$y$$ value. In other word, $$\\\\sqrt{x}=y$$ is defined when $$y$$ is non negative real number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical13a-h3","type":"hint","dependencies":["a81d715SolRadical13a-h2"],"title":"Check for Value of Square Root Function","text":"$$\\\\sqrt{2u-3}=-2$$ does not have solution since $$-2$$ is negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a81d715SolRadical14","title":"Solve Radical Equations","body":"In the following exercises, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Radical Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a81d715SolRadical14a","stepAnswer":["None"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{5v-2}+5=0$$","stepBody":"Please enter your solution. If there is no solution for this equation, please enter \\"None\\"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a81d715SolRadical14a-h1","type":"hint","dependencies":[],"title":"Isolate the Radical","text":"Isolate the radical on one side of the equation. We can subtract $$2$$ on both sides get $$\\\\sqrt{5v-2}=-5$$ which has radicals isolated on one side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical14a-h2","type":"hint","dependencies":["a81d715SolRadical14a-h1"],"title":"Check for Value of Square Root Function","text":"The square root functions only take positive $$y$$ value. In other word, $$\\\\sqrt{x}=y$$ is defined when $$y$$ is non negative real number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical14a-h3","type":"hint","dependencies":["a81d715SolRadical14a-h2"],"title":"Check for Value of Square Root Function","text":"$$\\\\sqrt{5v-2}=-5$$ does not have solution since $$-5$$ is negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a81d715SolRadical15","title":"Solve Radical Equations","body":"In the following exercises, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Radical Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a81d715SolRadical15a","stepAnswer":["$$10$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[3]{6x+4}=4$$","stepBody":"Please enter your solution. If there is no solution for this equation, please enter \\"None\\"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10$$","hints":{"DefaultPathway":[{"id":"a81d715SolRadical15a-h1","type":"hint","dependencies":[],"title":"Isolate the Radical","text":"Isolate the radical on one side of the equation. Since the given equation already have radical isolated on one side, we can proceed to the next step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical15a-h2","type":"hint","dependencies":["a81d715SolRadical15a-h1"],"title":"Raise Both Sides Of the Equation To The Power Of The Index","text":"$${\\\\sqrt[3]{6x+4}}^3=4^3$$\\\\nAfter cubing both sides, we get $$6x+4=64$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical15a-h3","type":"hint","dependencies":["a81d715SolRadical15a-h2"],"title":"Check For Remaining Radicals","text":"Are there any more radicals? No. We can proceed to the next step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical15a-h4","type":"hint","dependencies":["a81d715SolRadical15a-h3"],"title":"Solve For Equation","text":"Solve for the equation $$6x+4=64$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical15a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a81d715SolRadical15a-h4"],"title":"Solve For Equation","text":"What is the value of $$x$$ for $$6x+4=64$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a81d715SolRadical16","title":"Solve Radical Equations","body":"In the following exercises, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Radical Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a81d715SolRadical16a","stepAnswer":["$$11$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[3]{11x+4}=5$$","stepBody":"Please enter your solution. If there is no solution for this equation, please enter \\"None\\"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$11$$","hints":{"DefaultPathway":[{"id":"a81d715SolRadical16a-h1","type":"hint","dependencies":[],"title":"Isolate the Radical","text":"Isolate the radical on one side of the equation. Since the given equation already have radical isolated on one side, we can proceed to the next step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical16a-h2","type":"hint","dependencies":["a81d715SolRadical16a-h1"],"title":"Raise Both Sides Of the Equation To The Power Of The Index","text":"$${\\\\sqrt[3]{11x+4}}^3=5^3$$\\\\nAfter cubing both sides, we get $$11x+4=125$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical16a-h3","type":"hint","dependencies":["a81d715SolRadical16a-h2"],"title":"Check For Remaining Radicals","text":"Are there any more radicals? No. We can proceed to the next step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical16a-h4","type":"hint","dependencies":["a81d715SolRadical16a-h3"],"title":"Solve For Equation","text":"Solve for the equation $$11x+4=125$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical16a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11$$"],"dependencies":["a81d715SolRadical16a-h4"],"title":"Solve For Equation","text":"What is the value of $$x$$ for $$11x+4=125$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a81d715SolRadical17","title":"Solve Radical Equations","body":"In the following exercises, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Radical Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a81d715SolRadical17a","stepAnswer":["$$-8$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[3]{4x+5}-2=-5$$","stepBody":"Please enter your solution. If there is no solution for this equation, please enter \\"None\\"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-8$$","hints":{"DefaultPathway":[{"id":"a81d715SolRadical17a-h1","type":"hint","dependencies":[],"title":"Isolate the Radical","text":"Isolate the radical on one side of the equation. We can add $$2$$ on both sides get $$\\\\sqrt[3]{4x+5}=-3$$ which has radicals isolated on one side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical17a-h2","type":"hint","dependencies":["a81d715SolRadical17a-h1"],"title":"Raise Both Sides Of the Equation To The Power Of The Index","text":"$${\\\\sqrt[3]{4x+5}}^3={\\\\left(-3\\\\right)}^3$$\\\\nAfter cubing both sides, we get $$4x+5=-27$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical17a-h3","type":"hint","dependencies":["a81d715SolRadical17a-h2"],"title":"Check For Remaining Radicals","text":"Are there any more radicals? No. We can proceed to the next step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical17a-h4","type":"hint","dependencies":["a81d715SolRadical17a-h3"],"title":"Solve For Equation","text":"Solve for the equation $$4x+5=-27$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical17a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-8$$"],"dependencies":["a81d715SolRadical17a-h4"],"title":"Solve For Equation","text":"What is the value of $$x$$ for $$4x+5=-27$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a81d715SolRadical18","title":"Solve Radical Equations","body":"In the following exercises, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Radical Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a81d715SolRadical18a","stepAnswer":["$$-7$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[3]{9x-1}-1=-5$$","stepBody":"Please enter your solution. If there is no solution for this equation, please enter \\"None\\"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-7$$","hints":{"DefaultPathway":[{"id":"a81d715SolRadical18a-h1","type":"hint","dependencies":[],"title":"Isolate the Radical","text":"Isolate the radical on one side of the equation. We can add $$1$$ on both sides get $$\\\\sqrt[3]{9x-1}=-4$$ which has radicals isolated on one side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical18a-h2","type":"hint","dependencies":["a81d715SolRadical18a-h1"],"title":"Raise Both Sides Of the Equation To The Power Of The Index","text":"$${\\\\sqrt[3]{9x-1}}^3={\\\\left(-4\\\\right)}^3$$\\\\nAfter cubing both sides, we get $$9x-1=-64$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical18a-h3","type":"hint","dependencies":["a81d715SolRadical18a-h2"],"title":"Check For Remaining Radicals","text":"Are there any more radicals? No. We can proceed to the next step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical18a-h4","type":"hint","dependencies":["a81d715SolRadical18a-h3"],"title":"Solve For Equation","text":"Solve for the equation $$9x-1=-64$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical18a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-7$$"],"dependencies":["a81d715SolRadical18a-h4"],"title":"Solve For Equation","text":"What is the value of $$x$$ for $$9x-1=-64$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a81d715SolRadical19","title":"Solve Radical Equations","body":"In the following exercises, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Radical Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a81d715SolRadical19a","stepAnswer":["$$8$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(6x+1\\\\right)}^{\\\\frac{1}{2}}-3=4$$","stepBody":"Please enter your solution. If there is no solution for this equation, please enter \\"None\\"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8$$","hints":{"DefaultPathway":[{"id":"a81d715SolRadical19a-h1","type":"hint","dependencies":[],"title":"Isolate the Radical Exponent Terms","text":"Isolate the radical exponent terms on one side of the equation. We can add $$3$$ on both sides get $${\\\\left(6x+1\\\\right)}^{\\\\frac{1}{2}}=7$$ which has radical terms isolated on one side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical19a-h2","type":"hint","dependencies":["a81d715SolRadical19a-h1"],"title":"Raise Both Sides Of the Equation To Second Power","text":"$${\\\\left(6x+1\\\\right)}^{\\\\frac{1}{2}}=7^2$$\\\\nAfter squaring both sides, we get $$6x+1=49$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical19a-h3","type":"hint","dependencies":["a81d715SolRadical19a-h2"],"title":"Check For Remaining Radicals Exponent Terms","text":"Are there any more radicals? No. We can proceed to the next step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical19a-h4","type":"hint","dependencies":["a81d715SolRadical19a-h3"],"title":"Solve For Equation","text":"Solve for the equation $$6x+1=49$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical19a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a81d715SolRadical19a-h4"],"title":"Solve For Equation","text":"What is the value of $$x$$ for $$6x+1=49$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a81d715SolRadical2","title":"Solve Radical Equations","body":"In the following exercises, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Radical Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a81d715SolRadical2a","stepAnswer":["$$13$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{4x-3}=7$$","stepBody":"Please enter your solution. If there is no solution for this equation, please enter \\"None\\"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$13$$","hints":{"DefaultPathway":[{"id":"a81d715SolRadical2a-h1","type":"hint","dependencies":[],"title":"Isolate the Radical","text":"Isolate the radical on one side of the equation. Since the given equation already have radical isolated on one side, we can proceed to the next step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical2a-h2","type":"hint","dependencies":["a81d715SolRadical2a-h1"],"title":"Raise Both Sides Of the Equation To The Power Of The Index","text":"$${\\\\sqrt{4x-3}}^2=7^2$$\\\\nAfter squaring both sides, we get $$4x-3=49$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical2a-h3","type":"hint","dependencies":["a81d715SolRadical2a-h2"],"title":"Check For Remaining Radicals","text":"Are there any more radicals? No. We can proceed to the next step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical2a-h4","type":"hint","dependencies":["a81d715SolRadical2a-h3"],"title":"Solve For Equation","text":"Solve for the equation $$4x-3=49$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical2a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$13$$"],"dependencies":["a81d715SolRadical2a-h4"],"title":"Solve For Equation","text":"What is the value of $$x$$ for $$4x-3=49$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a81d715SolRadical20","title":"Solve Radical Equations","body":"In the following exercises, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Radical Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a81d715SolRadical20a","stepAnswer":["$$9$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(3x-2\\\\right)}^{\\\\frac{1}{2}}+1=6$$","stepBody":"Please enter your solution. If there is no solution for this equation, please enter \\"None\\"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9$$","hints":{"DefaultPathway":[{"id":"a81d715SolRadical20a-h1","type":"hint","dependencies":[],"title":"Isolate the Radical Exponent Terms","text":"Isolate the radicals exponent terms on one side of the equation. We can subtract $$1$$ on both sides get $${\\\\left(3x-2\\\\right)}^{\\\\frac{1}{2}}=5$$ which has radicals exponent terms isolated on one side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical20a-h2","type":"hint","dependencies":["a81d715SolRadical20a-h1"],"title":"Raise Both Sides Of the Equation To The Second Power","text":"$${\\\\left({\\\\left(3x-2\\\\right)}^{\\\\frac{1}{2}}\\\\right)}^2=5^2$$\\\\nAfter cubing both sides, we get $$3x-2=25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical20a-h3","type":"hint","dependencies":["a81d715SolRadical20a-h2"],"title":"Check For Remaining Radicals Exponent Terms","text":"Are there any more radicals exponent terms? No. We can proceed to the next step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical20a-h4","type":"hint","dependencies":["a81d715SolRadical20a-h3"],"title":"Solve For Equation","text":"Solve for the equation $$3x-2=25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical20a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a81d715SolRadical20a-h4"],"title":"Solve For Equation","text":"What is the value of $$x$$ for $$3x-2=25$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a81d715SolRadical21","title":"Solve Radical Equations","body":"In the following exercises, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Radical Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a81d715SolRadical21a","stepAnswer":["$$-4$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(8x+5\\\\right)}^{\\\\frac{1}{3}}+2=-1$$","stepBody":"Please enter your solution. If there is no solution for this equation, please enter \\"None\\"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-4$$","hints":{"DefaultPathway":[{"id":"a81d715SolRadical21a-h1","type":"hint","dependencies":[],"title":"Isolate the Radicals Exponent Terms","text":"Isolate the radicals exponent terms on one side of the equation. We can subtract $$2$$ on both sides get $${\\\\left(8x+5\\\\right)}^{\\\\frac{1}{3}}=-3$$ which has radicals exponent terms isolated on one side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical21a-h2","type":"hint","dependencies":["a81d715SolRadical21a-h1"],"title":"Raise Both Sides Of the Equation To The Third Power","text":"$${\\\\left({\\\\left(8x+5\\\\right)}^{\\\\frac{1}{3}}\\\\right)}^3={\\\\left(-3\\\\right)}^3$$\\\\nAfter cubing both sides, we get $$8x+5=-27$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical21a-h3","type":"hint","dependencies":["a81d715SolRadical21a-h2"],"title":"Check For Remaining Radicals Exponent Terms","text":"Are there any more radicals exponent terms? No. We can proceed to the next step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical21a-h4","type":"hint","dependencies":["a81d715SolRadical21a-h3"],"title":"Solve For Equation","text":"Solve for the equation $$8x+5=-27$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical21a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["a81d715SolRadical21a-h4"],"title":"Solve For Equation","text":"What is the value of $$x$$ for $$8x+5=-27$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a81d715SolRadical22","title":"Solve Radical Equations","body":"In the following exercises, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Radical Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a81d715SolRadical22a","stepAnswer":["$$-10$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(12x-5\\\\right)}^{\\\\frac{1}{3}}+8=3$$","stepBody":"Please enter your solution. If there is no solution for this equation, please enter \\"None\\"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-10$$","hints":{"DefaultPathway":[{"id":"a81d715SolRadical22a-h1","type":"hint","dependencies":[],"title":"Isolate the Radicals Exponent Terms","text":"Isolate the radicals exponent terms on one side of the equation. We can subtract $$8$$ on both sides get $${\\\\left(12x-5\\\\right)}^{\\\\frac{1}{3}}=-5$$ which has radicals exponent terms isolated on one side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical22a-h2","type":"hint","dependencies":["a81d715SolRadical22a-h1"],"title":"Raise Both Sides Of the Equation To The Third Power","text":"$${\\\\left({\\\\left(12x-5\\\\right)}^{\\\\frac{1}{3}}\\\\right)}^3={\\\\left(-5\\\\right)}^3$$\\\\nAfter cubing both sides, we get $$12x-5=-125$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical22a-h3","type":"hint","dependencies":["a81d715SolRadical22a-h2"],"title":"Check For Remaining Radicals Exponent Terms","text":"Are there any more radicals exponent terms? No. We can proceed to the next step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical22a-h4","type":"hint","dependencies":["a81d715SolRadical22a-h3"],"title":"Solve For Equation","text":"Solve for the equation $$12x-5=-125$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical22a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-10$$"],"dependencies":["a81d715SolRadical22a-h4"],"title":"Solve For Equation","text":"What is the value of $$x$$ for $$12x-5=-125$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a81d715SolRadical23","title":"Solve Radical Equations","body":"In the following exercises, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Radical Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a81d715SolRadical23a","stepAnswer":["$$7$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(12x-3\\\\right)}^{\\\\frac{1}{4}}-5=-2$$","stepBody":"Please enter your solution. If there is no solution for this equation, please enter \\"None\\"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7$$","hints":{"DefaultPathway":[{"id":"a81d715SolRadical23a-h1","type":"hint","dependencies":[],"title":"Isolate the Radicals Exponent Terms","text":"Isolate the radicals exponent terms on one side of the equation. We can add $$5$$ on both sides get $${\\\\left(12x-3\\\\right)}^{\\\\frac{1}{4}}=3$$ which has radicals exponent terms isolated on one side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical23a-h2","type":"hint","dependencies":["a81d715SolRadical23a-h1"],"title":"Raise Both Sides Of the Equation To The Fourth Power","text":"$${\\\\left({\\\\left(12x-3\\\\right)}^{\\\\frac{1}{4}}\\\\right)}^4=3^4$$\\\\nAfter rasing both sides to the fourth power, we get $$12x-3=81$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical23a-h3","type":"hint","dependencies":["a81d715SolRadical23a-h2"],"title":"Check For Remaining Radicals Exponent Terms","text":"Are there any more radicals exponent terms? No. We can proceed to the next step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical23a-h4","type":"hint","dependencies":["a81d715SolRadical23a-h3"],"title":"Solve For Equation","text":"Solve for the equation $$12x-3=81$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical23a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a81d715SolRadical23a-h4"],"title":"Solve For Equation","text":"What is the value of $$x$$ for $$12x-3=81$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a81d715SolRadical24","title":"Solve Radical Equations","body":"In the following exercises, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Radical Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a81d715SolRadical24a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(5x-4\\\\right)}^{\\\\frac{1}{4}}+7=9$$","stepBody":"Please enter your solution. If there is no solution for this equation, please enter \\"None\\"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a81d715SolRadical24a-h1","type":"hint","dependencies":[],"title":"Isolate the Radicals Exponent Terms","text":"Isolate the radicals exponent terms on one side of the equation. We can subtract $$7$$ on both sides get $${\\\\left(5x-4\\\\right)}^{\\\\frac{1}{4}}=2$$ which has radicals exponent terms isolated on one side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical24a-h2","type":"hint","dependencies":["a81d715SolRadical24a-h1"],"title":"Raise Both Sides Of the Equation To The Fourth Power","text":"$${\\\\left(5x-4\\\\right)}^{\\\\frac{1}{4}}=2$$\\\\nAfter rasing both sides to the fourth power, we get $$5x-4=16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical24a-h3","type":"hint","dependencies":["a81d715SolRadical24a-h2"],"title":"Check For Remaining Radicals Exponent Terms","text":"Are there any more radicals exponent terms? No. We can proceed to the next step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical24a-h4","type":"hint","dependencies":["a81d715SolRadical24a-h3"],"title":"Solve For Equation","text":"Solve for the equation $$5x-4=16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical24a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a81d715SolRadical24a-h4"],"title":"Solve For Equation","text":"What is the value of $$x$$ for $$5x-4=16$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a81d715SolRadical25","title":"Solve Radical Equations","body":"In the following exercises, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Radical Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a81d715SolRadical25a","stepAnswer":["$$42$$"],"problemType":"TextBox","stepTitle":"$$3\\\\sqrt{2x-3}-20=7$$","stepBody":"Please enter your solution. If there is no solution for this equation, please enter \\"None\\"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$42$$","hints":{"DefaultPathway":[{"id":"a81d715SolRadical25a-h1","type":"hint","dependencies":[],"title":"Isolate Radical Term","text":"Isolate the radical term on one side of the equation. We can add $$20$$ on both sides get $$3\\\\sqrt{2x-3}-20=7$$ which has radicals isolated on one side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical25a-h2","type":"hint","dependencies":["a81d715SolRadical25a-h1"],"title":"Isolate The Radical","text":"We can isolate the radical by dividing by $$3$$ both sides and get $$\\\\sqrt{2x-3}=9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical25a-h3","type":"hint","dependencies":["a81d715SolRadical25a-h2"],"title":"Raise Both Sides Of the Equation To The Power Of The Index","text":"$${\\\\sqrt{2x-3}}^2=9^2$$\\\\nAfter squaring both sides, we get $$2x-3=81$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical25a-h4","type":"hint","dependencies":["a81d715SolRadical25a-h3"],"title":"Check For Remaining Radicals","text":"Are there any more radicals? No. We can proceed to the next step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical25a-h5","type":"hint","dependencies":["a81d715SolRadical25a-h4"],"title":"Solve For Equation","text":"Solve for the equation $$2x-3=81$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical25a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$42$$"],"dependencies":["a81d715SolRadical25a-h5"],"title":"Solve For Equation","text":"What is the value of xfor $$2x-3=81$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a81d715SolRadical26","title":"Solve Radical Equations","body":"In the following exercises, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Radical Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a81d715SolRadical26a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"$$2\\\\sqrt{5x+1}-8=0$$","stepBody":"Please enter your solution. If there is no solution for this equation, please enter \\"None\\"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a81d715SolRadical26a-h1","type":"hint","dependencies":[],"title":"Isolate Radical Term","text":"Isolate the radical term on one side of the equation. We can add $$8$$ on both sides get $$2\\\\sqrt{5x+1}=8$$ which has radical term isolated on one side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical26a-h2","type":"hint","dependencies":["a81d715SolRadical26a-h1"],"title":"Isolate The Radical","text":"We can isolate the radical by dividing by $$2$$ both sides and get $$\\\\sqrt{5x+1}=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical26a-h3","type":"hint","dependencies":["a81d715SolRadical26a-h2"],"title":"Raise Both Sides Of the Equation To The Power Of The Index","text":"$${\\\\sqrt{5x+1}}^2=4^2$$\\\\nAfter squaring both sides, we get $$5x+1=16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical26a-h4","type":"hint","dependencies":["a81d715SolRadical26a-h3"],"title":"Check For Remaining Radicals","text":"Are there any more radicals? No. We can proceed to the next step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical26a-h5","type":"hint","dependencies":["a81d715SolRadical26a-h4"],"title":"Solve For Equation","text":"Solve for the equation $$5x+1=16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical26a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a81d715SolRadical26a-h5"],"title":"Solve For Equation","text":"What is the value of xfor $$5x+1=16$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a81d715SolRadical27","title":"Solve Radical Equations","body":"In the following exercises, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Radical Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a81d715SolRadical27a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"$$2\\\\sqrt{8r+1}-8=2$$","stepBody":"Please enter your solution. If there is no solution for this equation, please enter \\"None\\"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a81d715SolRadical27a-h1","type":"hint","dependencies":[],"title":"Isolate Radical Term","text":"Isolate the radical term on one side of the equation. We can add $$8$$ on both sides get $$2\\\\sqrt{8r+1}=10$$ which has radical term isolated on one side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical27a-h2","type":"hint","dependencies":["a81d715SolRadical27a-h1"],"title":"Isolate The Radical","text":"We can isolate the radical by dividing by $$2$$ both sides and get $$\\\\sqrt{8r+1}=5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical27a-h3","type":"hint","dependencies":["a81d715SolRadical27a-h2"],"title":"Raise Both Sides Of the Equation To The Power Of The Index","text":"$${\\\\sqrt{8r+1}}^2=5^2$$\\\\nAfter squaring both sides, we get $$8r+1=25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical27a-h4","type":"hint","dependencies":["a81d715SolRadical27a-h3"],"title":"Check For Remaining Radicals","text":"Are there any more radicals? No. We can proceed to the next step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical27a-h5","type":"hint","dependencies":["a81d715SolRadical27a-h4"],"title":"Solve For Equation","text":"Solve for the equation $$8r+1=25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical27a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a81d715SolRadical27a-h5"],"title":"Solve For Equation","text":"What is the value of $$r$$ for $$8r+1=25$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a81d715SolRadical28","title":"Solve Radical Equations","body":"In the following exercises, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Radical Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a81d715SolRadical28a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"$$3\\\\sqrt{7y+1}-10=8$$","stepBody":"Please enter your solution. If there is no solution for this equation, please enter \\"None\\"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"a81d715SolRadical28a-h1","type":"hint","dependencies":[],"title":"Isolate Radical Term","text":"Isolate the radical term on one side of the equation. We can add $$8$$ on both sides get $$3\\\\sqrt{7y+1}=18$$ which has radical term isolated on one side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical28a-h2","type":"hint","dependencies":["a81d715SolRadical28a-h1"],"title":"Isolate The Radical","text":"We can isolate the radical by dividing by $$3$$ both sides and get $$\\\\sqrt{7y+1}=6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical28a-h3","type":"hint","dependencies":["a81d715SolRadical28a-h2"],"title":"Raise Both Sides Of the Equation To The Power Of The Index","text":"$${\\\\sqrt{7y+1}}^2=6^2$$\\\\nAfter squaring both sides, we get $$7y+1=36$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical28a-h4","type":"hint","dependencies":["a81d715SolRadical28a-h3"],"title":"Check For Remaining Radicals","text":"Are there any more radicals? No. We can proceed to the next step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical28a-h5","type":"hint","dependencies":["a81d715SolRadical28a-h4"],"title":"Solve For Equation","text":"Solve for the equation $$7y+1=36$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical28a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a81d715SolRadical28a-h5"],"title":"Solve For Equation","text":"What is the value of $$y$$ for $$7y+1=36$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a81d715SolRadical29","title":"Solve Radical Equations","body":"In the following exercises, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Radical Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a81d715SolRadical29a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{3u+7}=\\\\sqrt{5u+1}$$","stepBody":"Please enter your solution. If there is no solution for this equation, please enter \\"None\\"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a81d715SolRadical29a-h1","type":"hint","dependencies":[],"title":"Isolate the Radical terms","text":"The radical terms are isolated. We can proceed to the next step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical29a-h2","type":"hint","dependencies":["a81d715SolRadical29a-h1"],"title":"Raise Both Sides Of the Equation To The Power Of The Index","text":"$${\\\\sqrt{3u+7}}^2={\\\\sqrt{5u+1}}^2$$\\\\nAfter squaring both sides, we get $$3u+7=5u+1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical29a-h3","type":"hint","dependencies":["a81d715SolRadical29a-h2"],"title":"Check For Remaining Radicals","text":"Are there any more radicals? No. We can proceed to the next step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical29a-h4","type":"hint","dependencies":["a81d715SolRadical29a-h3"],"title":"Solve For Equation","text":"Solve for the equation $$3u+7=5u+1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical29a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a81d715SolRadical29a-h4"],"title":"Solve For Equation","text":"What is the value of u for $$3u+7=5u+1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical29a-h6","type":"hint","dependencies":["a81d715SolRadical29a-h5"],"title":"Solve For Equation","text":"To solve $$3u+7=5u+1$$, we need to isolate the u on one side by itself. We can subtract 3u both side to remove 3u from the left handside which gives $$7=2u+1$$. Then we can subtract $$1$$ both sides to get 2u isolated on the right side. It gives $$6=2u$$. We can divide both sides by $$2$$ and get $$u=3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a81d715SolRadical3","title":"Solve Radical Equations","body":"In the following exercises, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Radical Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a81d715SolRadical3a","stepAnswer":["None"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{5x+1}=-3$$","stepBody":"Please enter your solution as fraction. If there is no solution for this equation, please enter \\"None\\"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a81d715SolRadical3a-h1","type":"hint","dependencies":[],"title":"Check for Value of Square Root Function","text":"The square root functions only take positive $$y$$ value. In other word, $$\\\\sqrt{x}=y$$ is defined when $$y$$ is non negative real number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical3a-h2","type":"hint","dependencies":["a81d715SolRadical3a-h1"],"title":"Check for Value of Square Root Function","text":"$$\\\\sqrt{5x+1}=-3$$ does not have solution since $$-3$$ is negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a81d715SolRadical30","title":"Solve Radical Equations","body":"In the following exercises, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Radical Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a81d715SolRadical30a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{4v+1}=\\\\sqrt{3v+3}$$","stepBody":"Please enter your solution. If there is no solution for this equation, please enter \\"None\\"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a81d715SolRadical30a-h1","type":"hint","dependencies":[],"title":"Isolate the Radical terms","text":"The radical terms are isolated. We can proceed to the next step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical30a-h2","type":"hint","dependencies":["a81d715SolRadical30a-h1"],"title":"Raise Both Sides Of the Equation To The Power Of The Index","text":"$${\\\\sqrt{4v+1}}^2={\\\\sqrt{3v+3}}^2$$\\\\nAfter squaring both sides, we get $$4v+1=3v+3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical30a-h3","type":"hint","dependencies":["a81d715SolRadical30a-h2"],"title":"Check For Remaining Radicals","text":"Are there any more radicals? No. We can proceed to the next step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical30a-h4","type":"hint","dependencies":["a81d715SolRadical30a-h3"],"title":"Solve For Equation","text":"Solve for the equation $$4v+1=3v+3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical30a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a81d715SolRadical30a-h4"],"title":"Solve For Equation","text":"What is the value of v for $$4v+1=3v+3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical30a-h6","type":"hint","dependencies":["a81d715SolRadical30a-h5"],"title":"Solve For Equation","text":"To solve $$4v+1=3v+3$$, we need to isolate the v on one side by itself. We can subtract 3v both side to remove 3v from the right handside which gives $$v+1=3$$. Then we can subtract $$1$$ both sides to get v isolated on the left side. It gives $$v=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a81d715SolRadical4","title":"Solve Radical Equations","body":"In the following exercises, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Radical Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a81d715SolRadical4a","stepAnswer":["None"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{3y-4}=-2$$","stepBody":"Please enter your solution as fraction. If there is no solution for this equation, please enter \\"None\\".","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a81d715SolRadical4a-h1","type":"hint","dependencies":[],"title":"Check for Value of Square Root Function","text":"The square root functions only take positive $$y$$ value. In other word, $$\\\\sqrt{x}=y$$ is defined when $$y$$ is non negative real number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical4a-h2","type":"hint","dependencies":["a81d715SolRadical4a-h1"],"title":"Check for Value of Square Root Function","text":"$$\\\\sqrt{3y-4}=-2$$ does not have solution since $$-2$$ is negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a81d715SolRadical5","title":"Solve Radical Equations","body":"In the following exercises, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Radical Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a81d715SolRadical5a","stepAnswer":["$$-4$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[3]{2x}=(-2)$$","stepBody":"Please enter your solution. If there is no solution for this equation, please enter \\"None\\"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-4$$","hints":{"DefaultPathway":[{"id":"a81d715SolRadical5a-h1","type":"hint","dependencies":[],"title":"Isolate the Radical","text":"Isolate the radical on one side of the equation. Since the given equation already has a radical isolated on one side, we can proceed to the next step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical5a-h2","type":"hint","dependencies":["a81d715SolRadical5a-h1"],"title":"Raise Both Sides Of the Equation To The Power Of The Index","text":"$${\\\\sqrt[3]{2x}}^3={\\\\left(-2\\\\right)}^3$$\\\\nAfter cubing both sides, we get $$2x=-8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical5a-h3","type":"hint","dependencies":["a81d715SolRadical5a-h2"],"title":"Check For Remaining Radicals","text":"Are there any more radicals? No. We can proceed to the next step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical5a-h4","type":"hint","dependencies":["a81d715SolRadical5a-h3"],"title":"Solve For Equation","text":"Solve for the equation $$2x=-8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["a81d715SolRadical5a-h4"],"title":"Solve For Equation","text":"What is the value of $$x$$ for $$2x=-8$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a81d715SolRadical6","title":"Solve Radical Equations","body":"In the following exercises, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Radical Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a81d715SolRadical6a","stepAnswer":["$$7$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[3]{4x-1}=3$$","stepBody":"Please enter your solution. If there is no solution for this equation, please enter \\"None\\"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7$$","hints":{"DefaultPathway":[{"id":"a81d715SolRadical6a-h1","type":"hint","dependencies":[],"title":"Isolate the Radical","text":"Isolate the radical on one side of the equation. Since the given equation already have radical isolated on one side, we can proceed to the next step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical6a-h2","type":"hint","dependencies":["a81d715SolRadical6a-h1"],"title":"Raise Both Sides Of the Equation To The Power Of The Index","text":"$${\\\\sqrt[3]{4x-1}}^3=3^3$$\\\\nAfter cubing both sides, we get $$4x-1=27$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical6a-h3","type":"hint","dependencies":["a81d715SolRadical6a-h2"],"title":"Check For Remaining Radicals","text":"Are there any more radicals? No. We can proceed to the next step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical6a-h4","type":"hint","dependencies":["a81d715SolRadical6a-h3"],"title":"Solve For Equation","text":"Solve for the equation $$4x-1=27$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a81d715SolRadical6a-h4"],"title":"Solve For Equation","text":"What is the value of $$x$$ for $$4x-1=27$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a81d715SolRadical7","title":"Solve Radical Equations","body":"In the following exercises, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Radical Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a81d715SolRadical7a","stepAnswer":["$$14$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{2m-3}-5=0$$","stepBody":"Please enter your solution. If there is no solution for this equation, please enter \\"None\\"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$14$$","hints":{"DefaultPathway":[{"id":"a81d715SolRadical7a-h1","type":"hint","dependencies":[],"title":"Isolate the Radical","text":"Isolate the radical on one side of the equation. We can add $$5$$ on both sides get $$\\\\sqrt{2m-3}=5$$ which has radicals isolated on one side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical7a-h2","type":"hint","dependencies":["a81d715SolRadical7a-h1"],"title":"Raise Both Sides Of the Equation To The Power Of The Index","text":"$${\\\\sqrt{2m-3}}^2=5^2$$\\\\nAfter squaring both sides, we get $$2m-3=25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical7a-h3","type":"hint","dependencies":["a81d715SolRadical7a-h2"],"title":"Check For Remaining Radicals","text":"Are there any more radicals? No. We can proceed to the next step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical7a-h4","type":"hint","dependencies":["a81d715SolRadical7a-h3"],"title":"Solve For Equation","text":"Solve for the equation $$2m-3=25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical7a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$14$$"],"dependencies":["a81d715SolRadical7a-h4"],"title":"Solve For Equation","text":"What is the value of $$m$$ for $$2m-3=25$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a81d715SolRadical8","title":"Solve Radical Equations","body":"In the following exercises, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Radical Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a81d715SolRadical8a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{2n-1}-3=0$$","stepBody":"Please enter your solution. If there is no solution for this equation, please enter \\"None\\"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"a81d715SolRadical8a-h1","type":"hint","dependencies":[],"title":"Isolate the Radical","text":"Isolate the radical on one side of the equation. We can add $$3$$ on both sides get $$\\\\sqrt{2n-1}=3$$ which has radicals isolated on one side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical8a-h2","type":"hint","dependencies":["a81d715SolRadical8a-h1"],"title":"Raise Both Sides Of the Equation To The Power Of The Index","text":"$${\\\\sqrt{2n-1}}^2=3^2$$\\\\nAfter squaring both sides, we get $$2n-1=9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical8a-h3","type":"hint","dependencies":["a81d715SolRadical8a-h2"],"title":"Check For Remaining Radicals","text":"Are there any more radicals? No. We can proceed to the next step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical8a-h4","type":"hint","dependencies":["a81d715SolRadical8a-h3"],"title":"Solve For Equation","text":"Solve for the equation $$2n-1=9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical8a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a81d715SolRadical8a-h4"],"title":"Solve For Equation","text":"What is the value of $$n$$ for $$2n-1=9$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a81d715SolRadical9","title":"Solve Radical Equations","body":"In the following exercises, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.6 Solve Radical Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a81d715SolRadical9a","stepAnswer":["$$17$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{6v-2}-10=0$$","stepBody":"Please enter your solution. If there is no solution for this equation, please enter \\"None\\"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$17$$","hints":{"DefaultPathway":[{"id":"a81d715SolRadical9a-h1","type":"hint","dependencies":[],"title":"Isolate the Radical","text":"Isolate the radical on one side of the equation. We can add $$10$$ on both sides get $$\\\\sqrt{6v-2}=10$$ which has radicals isolated on one side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical9a-h2","type":"hint","dependencies":["a81d715SolRadical9a-h1"],"title":"Raise Both Sides Of the Equation To The Power Of The Index","text":"$${\\\\sqrt{6v-2}}^2={10}^2$$\\\\nAfter squaring both sides, we get $$6v-2=100$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical9a-h3","type":"hint","dependencies":["a81d715SolRadical9a-h2"],"title":"Check For Remaining Radicals","text":"Are there any more radicals? No. We can proceed to the next step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical9a-h4","type":"hint","dependencies":["a81d715SolRadical9a-h3"],"title":"Solve For Equation","text":"Solve for the equation $$6v-2=100$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a81d715SolRadical9a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$17$$"],"dependencies":["a81d715SolRadical9a-h4"],"title":"Solve For Equation","text":"What is the value of v for $$6v-2=100$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a82ea5csubstitution1","title":"Using Substitution to Find an Antiderivative","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.5 Substitution","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a82ea5csubstitution1a","stepAnswer":["$$\\\\frac{{\\\\left(3x^2+4\\\\right)}^5}{5}+C$$"],"problemType":"MultipleChoice","stepTitle":"Use substitution to find the antiderivative $$\\\\int 6{x\\\\left(3x^2+4\\\\right)}^4 \\\\,dx$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{{\\\\left(3x^2+4\\\\right)}^5}{5}+C$$","choices":["$$\\\\frac{{\\\\left(3x^2+4\\\\right)}^5}{5}+C$$","$$\\\\frac{{\\\\left(3x^2+4\\\\right)}^5}{5}$$"],"hints":{"DefaultPathway":[{"id":"a82ea5csubstitution1a-h1","type":"hint","dependencies":[],"title":"Choose an Expression for u","text":"First choose an expression for u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a82ea5csubstitution1a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$u=3x^2+4$$"],"dependencies":["a82ea5csubstitution1a-h1"],"title":"Choose an Expression for u","text":"Which expression for u would be most appropriate for substitution?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$u=6x^4$$","$$u=3x^2+4$$","$$u=6x$$","$$u={\\\\left(3x^2+4\\\\right)}^4$$"]},{"id":"a82ea5csubstitution1a-h3","type":"hint","dependencies":["a82ea5csubstitution1a-h2"],"title":"Choose an Expression for u","text":"We choose $$u=3x^2+4$$ because then $$du=6xdx$$, and we already have du in the integrand.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a82ea5csubstitution1a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\int u^4 \\\\,du$$"],"dependencies":["a82ea5csubstitution1a-h3"],"title":"Write the Integral in Terms of u","text":"Write the integral in terms of u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\int u^4 \\\\,du$$","$$\\\\int u \\\\,du$$"]},{"id":"a82ea5csubstitution1a-h5","type":"hint","dependencies":["a82ea5csubstitution1a-h4"],"title":"Evaluate the Integral","text":"Remember that du is the derivative of the expresion chosen for u, regardless of what is inside the integrand. Evaluate the integral with respect to u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a82ea5csubstitution1a-h6","type":"hint","dependencies":["a82ea5csubstitution1a-h5"],"title":"Evaluate the Integral","text":"Replace u with $$3x^2+4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a82ea5csubstitution10","title":"Finding the Antiderivative using Substitution","body":"Find the antiderivative using the indicated substitution.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.5 Substitution","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a82ea5csubstitution10a","stepAnswer":["$$\\\\frac{1}{5} {\\\\left(x+1\\\\right)}^5+C$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\int {\\\\left(x+1\\\\right)}^4 \\\\,dx;$$ $$u=x+1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{1}{5} {\\\\left(x+1\\\\right)}^5+C$$","choices":["$$\\\\frac{1}{5} {\\\\left(x+1\\\\right)}^5+C$$","$${\\\\left(x+1\\\\right)}^5+C$$"],"hints":{"DefaultPathway":[{"id":"a82ea5csubstitution10a-h1","type":"hint","dependencies":[],"title":"Substitute $$x+1$$ Into the Integral","text":"For this problem, we would want to use the expression $$u=x+1$$ to find the antiderivative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a82ea5csubstitution11","title":"Finding the Antiderivative using Substitution","body":"Find the antiderivative using the indicated substitution.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.5 Substitution","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a82ea5csubstitution11a","stepAnswer":["$$\\\\sqrt{x^2+1}+C$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\int \\\\frac{x}{\\\\sqrt{x^2+1}} \\\\,dx;$$ $$u=x^2+1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\sqrt{x^2+1}+C$$","choices":["$$\\\\sqrt{x^2+1}+C$$","$$\\\\sqrt{x^2+1}$$"],"hints":{"DefaultPathway":[{"id":"a82ea5csubstitution11a-h1","type":"hint","dependencies":[],"title":"Substitute $$x^2+1$$ Into the Integral","text":"For this problem, we would want to use the expression $$u=x^2+1$$ to find the antiderivative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a82ea5csubstitution12","title":"Finding the Antiderivative using Substitution","body":"Find the antiderivative using the indicated substitution.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.5 Substitution","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a82ea5csubstitution12a","stepAnswer":["$$\\\\frac{1}{8} {\\\\left(x^2-2x\\\\right)}^4+C$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\int \\\\left(x-1\\\\right) {\\\\left(x^2 x\\\\right)}^3 \\\\,dx;$$ $$u=x^2-2x$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{1}{8} {\\\\left(x^2-2x\\\\right)}^4+C$$","choices":["$$\\\\frac{1}{8} {\\\\left(x^2-2x\\\\right)}^4+C$$","$${\\\\left(x^2-2x\\\\right)}^4+C$$"],"hints":{"DefaultPathway":[{"id":"a82ea5csubstitution12a-h1","type":"hint","dependencies":[],"title":"Substitute $$x^2-2x$$ Into the Integral","text":"For this problem, we would want to use the expression $$u=x^2-2x$$ to find the antiderivative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a82ea5csubstitution13","title":"Finding the Antiderivative using Substitution","body":"Find the antiderivative using the indicated substitution.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.5 Substitution","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a82ea5csubstitution13a","stepAnswer":["$$sintheta-\\\\frac{{sin}^3 \\\\theta}{3}+C$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\int {cos}^3 \\\\theta \\\\,dtheta;$$ $$u=sintheta$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$sintheta-\\\\frac{{sin}^3 \\\\theta}{3}+C$$","choices":["$$sintheta-\\\\frac{{sin}^3 \\\\theta}{3}+C$$","$$sintheta-\\\\frac{{sin}^3 \\\\theta}{3}$$"],"hints":{"DefaultPathway":[{"id":"a82ea5csubstitution13a-h1","type":"hint","dependencies":[],"title":"Trigonometric Identity","text":"Use the trigonometric identity $${cos}^2 \\\\theta=1-{sin}^2 \\\\theta$$ to find the antiderivative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a82ea5csubstitution13a-h2","type":"hint","dependencies":["a82ea5csubstitution13a-h1"],"title":"Constant of Integration","text":"Since we are determining the indefinite integral, we will be including the constant of integration C at the end of our answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a82ea5csubstitution14","title":"Change of Variables: Indefinite Integral","body":"Use a suitable change of variables to determine the indefinite integral.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.5 Substitution","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a82ea5csubstitution14a","stepAnswer":["$$\\\\frac{-1}{22{\\\\left(11x-7\\\\right)}^2}+C$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\int {\\\\left(11x-7\\\\right)}^{\\\\left(-3\\\\right)} \\\\,dx$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{-1}{22{\\\\left(11x-7\\\\right)}^2}+C$$","choices":["$$\\\\frac{-1}{22{\\\\left(11x-7\\\\right)}^2}+C$$","$$\\\\frac{-1}{22{\\\\left(11x-7\\\\right)}^2}$$","$$\\\\frac{1}{22{\\\\left(11x-7\\\\right)}^2}+C$$"],"hints":{"DefaultPathway":[{"id":"a82ea5csubstitution14a-h1","type":"hint","dependencies":[],"title":"Constant of Integration","text":"Since we are determining the indefinite integral, we will be including the constant of integration C at the end of our answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a82ea5csubstitution15","title":"Change of Variables: Indefinite Integral","body":"Use a suitable change of variables to determine the indefinite integral.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.5 Substitution","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a82ea5csubstitution15a","stepAnswer":["$$-\\\\left(\\\\frac{cos^3\\\\left(\\\\pi t\\\\right)}{3} \\\\pi\\\\right)+C$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\int cos^2\\\\left(\\\\pi t\\\\right) sin\\\\left(\\\\pi t\\\\right) \\\\,dt$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-\\\\left(\\\\frac{cos^3\\\\left(\\\\pi t\\\\right)}{3} \\\\pi\\\\right)+C$$","choices":["$$-\\\\left(\\\\frac{cos^3\\\\left(\\\\pi t\\\\right)}{3} \\\\pi\\\\right)+C$$","$$-\\\\left(\\\\frac{cos^3\\\\left(\\\\pi t\\\\right)}{3} \\\\pi\\\\right)$$"],"hints":{"DefaultPathway":[{"id":"a82ea5csubstitution15a-h1","type":"hint","dependencies":[],"title":"Constant of Integration","text":"Since we are determining the indefinite integral, we will be including the constant of integration C at the end of our answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a82ea5csubstitution16","title":"Change of Variables: Indefinite Integral","body":"Use a suitable change of variables to determine the indefinite integral.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.5 Substitution","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a82ea5csubstitution16a","stepAnswer":["1/33(1-cos**3theta)**11+C"],"problemType":"TextBox","stepTitle":"$$\\\\int {\\\\left(1-{cos}^3 \\\\theta\\\\right)}^{10} {cos}^2 sintheta \\\\,dtheta$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{1}{{\\\\operatorname{33}\\\\left(1-{cos}^3 \\\\theta\\\\right)}^{11}}+C$$","hints":{"DefaultPathway":[{"id":"a82ea5csubstitution16a-h1","type":"hint","dependencies":[],"title":"Constant of Integration","text":"Since we are determining the indefinite integral, we will be including the constant of integration C at the end of our answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a82ea5csubstitution17","title":"Estimating the Area Under Curves","body":"Use a calculator to estimate the area under the curve using left Riemann sums with $$50$$ terms, then use substitution to solve for the exact answer.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.5 Substitution","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a82ea5csubstitution17a","stepAnswer":["$$L_{50}=-8.5779$$. The exact area is $$\\\\frac{-81}{8}$$."],"problemType":"MultipleChoice","stepTitle":"$$[T]y={x\\\\left(1-x^2\\\\right)}^3$$ over [-1, 2]","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$L_{50}=-8.5779$$. The exact area is $$\\\\frac{-81}{8}$$.","choices":["$$L_{50}=-8.5779$$. The exact area is $$\\\\frac{-81}{8}$$.","$$L_{50}=-7.5779$$. The exact area is $$\\\\frac{-79}{8}$$.","$$L_{50}=-8.5886$$. The exact area is $$\\\\frac{-80}{7}$$.","$$L_{50}=-8.4886$$. The exact area is $$\\\\frac{-80}{7}$$."],"hints":{"DefaultPathway":[]}}]},{"id":"a82ea5csubstitution18","title":"Change of Variables: Definite Integral","body":"Use a change of variables to evaluate the definite integral.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.5 Substitution","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a82ea5csubstitution18a","stepAnswer":["(2/3)(sqrt(2)-1)"],"problemType":"TextBox","stepTitle":"$$\\\\int_{0}^{1} \\\\frac{t^2}{\\\\sqrt{1+t^3}} \\\\,dt$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{2}{3} \\\\left(\\\\sqrt{2}-1\\\\right)$$","hints":{"DefaultPathway":[{"id":"a82ea5csubstitution18a-h1","type":"hint","dependencies":[],"title":"Choose an Expression","text":"Let $$u=1+t^3$$, so $$du=2t^2 dt$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a82ea5csubstitution18a-h2","type":"hint","dependencies":["a82ea5csubstitution18a-h1"],"title":"Adjust the Limits of Integration","text":"To adjust the limits of integration, we note that when $$t=0$$, $$u=1+0^3$$, and when $$t=1$$, $$u=1+1^3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a82ea5csubstitution18a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$1/3*\\\\int 12u^{\\\\left(-\\\\frac{1}{2}\\\\right)} \\\\,du$$"],"dependencies":["a82ea5csubstitution18a-h2"],"title":"Adjust the Limits of Integration","text":"What is the result of adjusting the limits of integration?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$1/3*\\\\int 12u^{\\\\left(-\\\\frac{1}{2}\\\\right)} \\\\,du$$","$$1/3*\\\\int 21u^{\\\\left(-\\\\frac{1}{2}\\\\right)} \\\\,du$$","$$\\\\int 12u^{\\\\left(-\\\\frac{1}{2}\\\\right)} \\\\,du$$","$$\\\\int 21u^{\\\\left(-\\\\frac{1}{2}\\\\right)} \\\\,du$$"]},{"id":"a82ea5csubstitution18a-h4","type":"hint","dependencies":["a82ea5csubstitution18a-h3"],"title":"Evaluate the Definite Integral","text":"Evaluate the definite integral.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a82ea5csubstitution19","title":"Change of Variables: Definite Integral","body":"Use a change of variables to evaluate the definite integral.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.5 Substitution","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a82ea5csubstitution19a","stepAnswer":["sqrt(2)-1"],"problemType":"TextBox","stepTitle":"$$\\\\int_{0}^{1} \\\\frac{x}{\\\\sqrt{1+x^2}} \\\\,dx$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\sqrt{2}-1$$","hints":{"DefaultPathway":[{"id":"a82ea5csubstitution19a-h1","type":"hint","dependencies":[],"title":"Choose an Expression","text":"Let $$u=1+x^2$$, so $$du=2xdx$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a82ea5csubstitution19a-h2","type":"hint","dependencies":["a82ea5csubstitution19a-h1"],"title":"Adjust the Limits of Integration","text":"To adjust the limits of integration, we note that when $$x=0$$, $$u=1+0^2$$, and when $$x=1$$, $$u=1+1^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a82ea5csubstitution19a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$1/2*\\\\int_{1}^{2} u^{\\\\left(-\\\\frac{1}{2}\\\\right)} \\\\,du$$"],"dependencies":["a82ea5csubstitution19a-h2"],"title":"Adjust the Limits of Integration","text":"What is the result of adjusting the limits of integration?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$1/2*\\\\int_{1}^{2} u^{\\\\left(-\\\\frac{1}{2}\\\\right)} \\\\,du$$","$$1/2*\\\\int_{2}^{1} u^{\\\\left(-\\\\frac{1}{2}\\\\right)} \\\\,du$$","$$\\\\int_{1}^{2} u^{\\\\left(-\\\\frac{1}{2}\\\\right)} \\\\,du$$","$$\\\\int_{2}^{1} u^{\\\\left(-\\\\frac{1}{2}\\\\right)} \\\\,du$$"]},{"id":"a82ea5csubstitution19a-h4","type":"hint","dependencies":["a82ea5csubstitution19a-h3"],"title":"Evaluate the Definite Integral","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a82ea5csubstitution2","title":"Using Substitution with Alteration","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.5 Substitution","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a82ea5csubstitution2a","stepAnswer":["$$\\\\frac{1}{3} {\\\\left(z^2-5\\\\right)}^{\\\\frac{3}{2}}+C$$"],"problemType":"MultipleChoice","stepTitle":"Use substitution to find $$\\\\int z \\\\sqrt{z^2-5} \\\\,dz$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{1}{3} {\\\\left(z^2-5\\\\right)}^{\\\\frac{3}{2}}+C$$","choices":["$$\\\\frac{1}{3} {\\\\left(z^2-5\\\\right)}^{\\\\frac{3}{2}}+C$$","$$\\\\frac{1}{3} {\\\\left(z^2-5\\\\right)}^{\\\\frac{3}{2}}$$"],"hints":{"DefaultPathway":[{"id":"a82ea5csubstitution2a-h1","type":"hint","dependencies":[],"title":"Rewrite the Integral","text":"Rewrite the integral as $$\\\\int z\\\\left({\\\\left(z^2-5\\\\right)}^{\\\\frac{1}{2}}\\\\right) \\\\,dz$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a82ea5csubstitution2a-h2","type":"hint","dependencies":["a82ea5csubstitution2a-h1"],"title":"Choose an Expression","text":"Let $$u=z^2-5$$ and $$du=2zdz$$. Our new concern is that $$du=2zdz$$ and the original expression has only xdz.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a82ea5csubstitution2a-h3","type":"hint","dependencies":["a82ea5csubstitution2a-h2"],"title":"Alter the Expression by Multiplying by $$\\\\frac{1}{2}$$","text":"Alter the expression for du or the integral in u will be twice as large as it should be. We can do this by multiplying both sides of the du equation by $$\\\\frac{1}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a82ea5csubstitution2a-h4","type":"hint","dependencies":["a82ea5csubstitution2a-h3"],"title":"Write the Integral in Terms of u","text":"Write the integral in terms of u, but pull the $$\\\\frac{1}{2}$$ outside the integration symbol.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a82ea5csubstitution2a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$1/2*\\\\int u \\\\frac{1}{2} \\\\,du$$"],"dependencies":["a82ea5csubstitution2a-h4"],"title":"Write the Integral in Terms of u","text":"What would the integral look like?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$1/2*\\\\int u \\\\frac{1}{2} \\\\,du$$","$$1/2*\\\\int u \\\\,du$$","$$\\\\int u \\\\frac{1}{2} \\\\,du$$","$$\\\\int u \\\\,du$$"]},{"id":"a82ea5csubstitution2a-h6","type":"hint","dependencies":["a82ea5csubstitution2a-h5"],"title":"Integrate the Expression in Terms of u","text":"Integrate the expression in u.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a82ea5csubstitution20","title":"Estimating Coefficients","body":"[T] The following graph is of a function of the form f(x)=acos(nt)+bcos(mt).\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.5 Substitution","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a82ea5csubstitution20a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"Estimate the coefficients a and $$b$$ and the frequency $$n$$ and $$m$$. Use these estimates to approximate $$\\\\int_{0}^{pi} f(t) \\\\,dt$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a82ea5csubstitution20a-h1","type":"hint","dependencies":[],"title":"Estimate the Coefficients","text":"The graph is a function of the form $$f(x)=\\\\arccos(nt)+\\\\operatorname{bcos}\\\\left(mt\\\\right)$$. Take a look at the graph and note the frequencies. Note the frequency parameters are $$n$$ and $$m$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a82ea5csubstitution20a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$f(t)=2cos(3t)-cos(2t)$$"],"dependencies":["a82ea5csubstitution20a-h1"],"title":"Estimate the Coefficients","text":"2cos","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$f(t)=2cos(3t)-cos(2t)$$","$$f(t)=3cos(2t)-cos(3t)$$","$$f(t)=cos(3t)-cos(2t)$$","$$f(t)=3cos(3t)-cos(2t)$$"]},{"id":"a82ea5csubstitution20a-h3","type":"hint","dependencies":["a82ea5csubstitution20a-h2"],"title":"Set Up the Integral","text":"Use the correct function to approximate $$\\\\int_{0}^{pi} f(t) \\\\,dt$$. Evaluate the integral.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a82ea5csubstitution20a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a82ea5csubstitution20a-h3"],"title":"Approximate the Integral","text":"The integral would be set up as $$\\\\int_{0}^{pi} 2cos(3t)-cos(2t) \\\\,dt$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"a82ea5csubstitution20a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":[],"title":"Approximate the Integral","text":"What is $$0-0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}]}}]},{"id":"a82ea5csubstitution3","title":"Using Substitution with Integrals of Trigonometric Functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.5 Substitution","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a82ea5csubstitution3a","stepAnswer":["1/(2cos**2t))+C"],"problemType":"TextBox","stepTitle":"Use substitution to evaluate the integral $$\\\\int \\\\frac{sin t}{cos t^3} \\\\,dt$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a82ea5csubstitution3a-h1","type":"hint","dependencies":[],"title":"Choose an Expression","text":"We know the derivative of cost is -sint, so we set $$u=cost$$. Then $$du=-sindt$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a82ea5csubstitution3a-h2","type":"hint","dependencies":["a82ea5csubstitution3a-h1"],"title":"Substitute cost Into the Integral","text":"Substitute u with cost into the integral.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a82ea5csubstitution3a-h3","type":"hint","dependencies":["a82ea5csubstitution3a-h2"],"title":"Evaluate the Integral","text":"Evaluate the integral.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a82ea5csubstitution3a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-\\\\left(-\\\\frac{1}{2}\\\\right) u^{-2}+C$$"],"dependencies":["a82ea5csubstitution3a-h3"],"title":"Evaluate the Integral","text":"What is the result of evaluating the integral?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$-\\\\left(-\\\\frac{1}{2}\\\\right) u^{-2}+C$$","$$\\\\frac{-1}{2} u^{-2}+C$$","$$-\\\\left(-\\\\frac{1}{2}\\\\right) u^{+C}$$","$$u^{-2}+C$$"]},{"id":"a82ea5csubstitution3a-h5","type":"hint","dependencies":["a82ea5csubstitution3a-h4"],"title":"Put Answer Back","text":"Put the answer back in terms of $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a82ea5csubstitution4","title":"Finding an Antiderivative Using u-Substitution","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.5 Substitution","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a82ea5csubstitution4a","stepAnswer":["$$\\\\frac{2}{3} {\\\\left(x-1\\\\right)}^{\\\\frac{1}{2}} \\\\left(x+2\\\\right)+C$$"],"problemType":"MultipleChoice","stepTitle":"Use substitution to find the antiderivative $$\\\\int \\\\frac{x}{\\\\sqrt{x-1}} \\\\,dx$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{2}{3} {\\\\left(x-1\\\\right)}^{\\\\frac{1}{2}} \\\\left(x+2\\\\right)+C$$","choices":["$$\\\\frac{2}{3} {\\\\left(x-1\\\\right)}^{\\\\frac{1}{2}} \\\\left(x+2\\\\right)+C$$","$$\\\\frac{2}{3} {\\\\left(x-1\\\\right)}^{\\\\frac{1}{2}} \\\\left(x+2\\\\right)$$"],"hints":{"DefaultPathway":[{"id":"a82ea5csubstitution4a-h1","type":"hint","dependencies":[],"title":"Choose an Expression","text":"Express $$x$$ in terms of u. If $$u=x-1$$, then $$x=u+1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a82ea5csubstitution4a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\int u^{\\\\frac{1}{2}}+u^{\\\\left(-\\\\frac{1}{2}\\\\right)} \\\\,du$$"],"dependencies":["a82ea5csubstitution4a-h1"],"title":"Rewrite the Integral in Terms of u","text":"What is the integral in terms of u?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\int u^{\\\\frac{1}{2}}+u^{\\\\left(-\\\\frac{1}{2}\\\\right)} \\\\,du$$","$$\\\\int u^{\\\\frac{1}{2}} \\\\,du$$"]},{"id":"a82ea5csubstitution4a-h3","type":"hint","dependencies":["a82ea5csubstitution4a-h2"],"title":"Integrate the Expression","text":"Integrate by replacing u with the original expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a82ea5csubstitution4a-h4","type":"hint","dependencies":["a82ea5csubstitution4a-h3"],"title":"Factor and Simplify","text":"Factor and simplify the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a82ea5csubstitution5","title":"Using Substitution to Evaluate a Definite Integral","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.5 Substitution","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a82ea5csubstitution5a","stepAnswer":["$$\\\\frac{182}{9}$$"],"problemType":"TextBox","stepTitle":"Use substitution to evaluate $$\\\\int x^{{2\\\\left(1+2x^3\\\\right)}^5} \\\\,dx$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{182}{9}$$","hints":{"DefaultPathway":[{"id":"a82ea5csubstitution5a-h1","type":"hint","dependencies":[],"title":"Choose an Expression","text":"Let $$u=1+2x^3$$, so $$du=6x^2 dx$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a82ea5csubstitution5a-h2","type":"hint","dependencies":["a82ea5csubstitution5a-h1"],"title":"Choose an Expression","text":"Since the original function includes one factor of $$x^2$$ and $$du=6x^2 dx$$, multiply both sides of the du equation by $$\\\\frac{1}{6}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a82ea5csubstitution5a-h3","type":"hint","dependencies":["a82ea5csubstitution5a-h2"],"title":"Adjust the Limits of Integration","text":"To adjust the limits of integration, note that when $$x=0$$, $$u=1+2\\\\left(0\\\\right)=1$$, and when $$x=1$$, $$u=1+2\\\\left(1\\\\right)=3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a82ea5csubstitution5a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$1/6*\\\\int_{1}^{3} u^5 \\\\,du$$"],"dependencies":["a82ea5csubstitution5a-h3"],"title":"Adjust the Limits of Integration","text":"What is the result of adjusting the limits of integration?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$1/6*\\\\int_{1}^{3} u^5 \\\\,du$$","$$\\\\int_{1}^{3} u^5 \\\\,du$$","$$1/6*\\\\int_{1}^{3} u^4 \\\\,du$$","$$1/6*\\\\int_{1}^{3} u^4 \\\\,du$$"]},{"id":"a82ea5csubstitution5a-h5","type":"hint","dependencies":["a82ea5csubstitution5a-h4"],"title":"Evaluate the Expression","text":"Evaluate the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a82ea5csubstitution6","title":"Using Substitution with an Exponential Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.5 Substitution","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a82ea5csubstitution6a","stepAnswer":["134.568"],"problemType":"TextBox","stepTitle":"Use substitution to evaluate $$\\\\int_{0}^{1} x e^{4x^2+3} \\\\,dx$$. Round to three decimals.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$134.568$$","hints":{"DefaultPathway":[{"id":"a82ea5csubstitution6a-h1","type":"hint","dependencies":[],"title":"Choose an Expression","text":"Let $$u=4x^2+3$$. Then, $$du=8xdx$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a82ea5csubstitution6a-h2","type":"hint","dependencies":["a82ea5csubstitution6a-h1"],"title":"Adjust the Limits of Integration","text":"To adjust the limits of integration, we note that when $$x=0$$, $$u=3$$, and when $$x=1$$, $$u=7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a82ea5csubstitution6a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$1/8*\\\\int_{3}^{7} e^u \\\\,du$$"],"dependencies":["a82ea5csubstitution6a-h2"],"title":"Adjust the Limits of Integration","text":"What is the result of adjusting the limits of integration?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$1/8*\\\\int_{3}^{7} e^u \\\\,du$$","$$\\\\int_{3}^{7} e^u \\\\,du$$","$$1/8*\\\\int_{7}^{3} e^u \\\\,du$$","$$\\\\int_{7}^{3} e^u \\\\,du$$"]},{"id":"a82ea5csubstitution6a-h4","type":"hint","dependencies":["a82ea5csubstitution6a-h3"],"title":"Integrate","text":"Integrate the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a82ea5csubstitution7","title":"Using Substitution to Evaluate a Trigonometric Integral","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.5 Substitution","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a82ea5csubstitution7a","stepAnswer":["pi/4"],"problemType":"TextBox","stepTitle":"Use substitution to evaluate $$\\\\int_{0}^{\\\\frac{\\\\pi}{2}} {cos}^2 \\\\theta \\\\,dtheta$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{\\\\pi}{4}$$","hints":{"DefaultPathway":[{"id":"a82ea5csubstitution7a-h1","type":"hint","dependencies":[],"title":"Rewrite the Integral","text":"First use the trigonometric identity $${cos}^2 \\\\theta=\\\\frac{1+cos2 \\\\theta}{2}$$ to rewrite the integral.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a82ea5csubstitution7a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\int_{0}^{\\\\frac{\\\\pi}{2}} \\\\frac{1+cos2 \\\\theta}{2} \\\\,dtheta$$"],"dependencies":["a82ea5csubstitution7a-h1"],"title":"Rewrite the Integral","text":"What is the new integral?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\int_{0}^{\\\\frac{\\\\pi}{2}} \\\\frac{1+cos2 \\\\theta}{2} \\\\,dtheta$$","$$\\\\int_{0}^{\\\\frac{\\\\pi}{2}} \\\\frac{cos2 \\\\theta}{2} \\\\,dtheta$$"]},{"id":"a82ea5csubstitution7a-h3","type":"hint","dependencies":["a82ea5csubstitution7a-h2"],"title":"Evaluate the Second Integral","text":"Make a substitution to evaluate the second integral by letting $$u=2theta$$. Then, $$du=2dtheta$$, or $$\\\\frac{1}{2} du=dtheta$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a82ea5csubstitution7a-h4","type":"hint","dependencies":["a82ea5csubstitution7a-h3"],"title":"Express the Second Integral in Terms of u","text":"When $$theta=0$$, $$u=0$$, and when $$theta=\\\\frac{\\\\pi}{2}$$, $$u=pi$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a82ea5csubstitution8","title":"Verifying Identities Using Differentiation","body":"Verify each identity using differentiation. Then, using the indicated u-substitution, identify f such that the integral takes the form $$\\\\int f(u) \\\\,du$$.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.5 Substitution","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a82ea5csubstitution8a","stepAnswer":["$$f(u)=\\\\frac{{\\\\left(u+1\\\\right)}^2}{\\\\sqrt{u}}$$"],"problemType":"MultipleChoice","stepTitle":"For $$x>1$$: $$\\\\int \\\\frac{x^2}{\\\\sqrt{x-1}} \\\\,dx=\\\\frac{2}{\\\\operatorname{15}\\\\left(\\\\sqrt{x-1}\\\\right)} \\\\left(3x^2+4x+8\\\\right)+C$$ $$u=x-1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$f(u)=\\\\frac{{\\\\left(u+1\\\\right)}^2}{\\\\sqrt{u}}$$","choices":["$$f(u)=\\\\frac{{\\\\left(u+1\\\\right)}^2}{\\\\sqrt{u}}$$","$$f(u)=\\\\frac{u+1}{\\\\sqrt{u}}$$","$$f(u)=\\\\frac{{\\\\left(u+1\\\\right)}^2}{u}$$","$$f(u)=\\\\frac{{\\\\left(u-1\\\\right)}^2}{\\\\sqrt{u}}$$"],"hints":{"DefaultPathway":[]}}]},{"id":"a82ea5csubstitution9","title":"Verifying Identities Using Differentiation","body":"Verify each identity using differentiation. Then, using the indicated u-substitution, identify f such that the integral takes the form $$\\\\int f(u) \\\\,du$$.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.5 Substitution","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a82ea5csubstitution9a","stepAnswer":["$$du=8xdx;$$ $$f(u)=\\\\frac{1}{8\\\\sqrt{u}}$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\int \\\\frac{x}{{\\\\left(4x^2+9\\\\right)}^2} \\\\,dx=\\\\frac{-1}{8\\\\left(4x^2+9\\\\right)}$$ $$u=4x^2+9$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$du=8xdx;$$ $$f(u)=\\\\frac{1}{8\\\\sqrt{u}}$$","choices":["$$du=8xdx;$$ $$f(u)=\\\\frac{1}{8\\\\sqrt{u}}$$","$$du=8xdx;$$ $$f(u)=\\\\frac{1}{8}$$","$$du=xdx;$$ $$f(u)=1$$","$$du=\\\\frac{1}{8} xdx$$ $$f(u)=\\\\frac{1}{8\\\\sqrt{u}}$$"],"hints":{"DefaultPathway":[]}}]},{"id":"a833c22linear1","title":"Linear Functions","body":"For the following exercises, determine whether the equation of the curve can be written as a linear function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a833c22linear1a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$y=3x-5$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a833c22linear1a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":[],"title":"Slope","text":"What is the slope of the function?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear1a-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a833c22linear1a-h1"],"title":"Constant","text":"Is this slope constant for every increase in $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a833c22linear1a-s2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a833c22linear1a-s1"],"title":"Meaning","text":"Does this mean it can be written as a linear function?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a833c22linear10","title":"Linear Functions","body":"Find the slope of the line that passes through the two given points.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a833c22linear10a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"$$(1,5)$$ and $$(4,11)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a833c22linear10a-h1","type":"hint","dependencies":[],"title":"Subtract","text":"Subtract y2 - y1.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a833c22linear10a-h1"],"title":"Value","text":"What is this value?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear10a-h3","type":"hint","dependencies":["a833c22linear10a-h2"],"title":"Subtract","text":"Subtract x2 - x1.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a833c22linear10a-h3"],"title":"Value","text":"What is this value?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear10a-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a833c22linear10a-h4"],"title":"Divide","text":"What is y2-y1 over x2-x1?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a833c22linear11","title":"Finding the Slope of a Line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a833c22linear11a","stepAnswer":["$$\\\\frac{-1}{2}$$"],"problemType":"TextBox","stepTitle":"If f(x) is a linear function, and $$(2,3)$$ and $$(0,4)$$ are points on the line, find the slope.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-1}{2}$$","hints":{"DefaultPathway":[{"id":"a833c22linear11a-h1","type":"hint","dependencies":[],"title":"The Slope Formula","text":"The slope formula can be defined as follows: $$m=\\\\frac{y1-y2}{x1-x2}$$, where (x1,y1) and (x2,y2) are the points on the line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear11a-h2","type":"hint","dependencies":["a833c22linear11a-h1"],"title":"Using the Slope Formula","text":"We can plug in the points $$(2,3)$$ and $$(0,4)$$ into the slope formula as shown. $$m$$ $$=$$ $$\\\\frac{4-3}{0-2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear11a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{2}$$"],"dependencies":["a833c22linear11a-h2"],"title":"Final Slope","text":"What is the numerical result of the previous step?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a833c22linear12","title":"Town Population","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a833c22linear12a","stepAnswer":["$$142$$"],"problemType":"TextBox","stepTitle":"The population of a small town increased from 1,442 to 1,868 between $$2009$$ and $$2012$$. Find the change of population per year if we assume the change was constant from $$2009$$ to $$2012$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$142$$","hints":{"DefaultPathway":[{"id":"a833c22linear12a-h1","type":"hint","dependencies":[],"title":"Creating Two Points","text":"We can create two points with the information given. We know that the town\'s population in $$2009$$ was $$1442$$, so we can create the point $$(2009,1442)$$. We are also given that the town\'s population in $$2012$$ was $$1868$$, so we can create another point $$(2012,1868)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear12a-h2","type":"hint","dependencies":["a833c22linear12a-h1"],"title":"The Slope Formula","text":"The slope formula can be defined as follows: $$m=\\\\frac{y1-y2}{x1-x2}$$, where (x1,y1) and (x2,y2) are the points on the line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear12a-h3","type":"hint","dependencies":["a833c22linear12a-h2"],"title":"Using the Slope Formula","text":"We can plug in the points $$(2009,1442)$$ and $$(2012,1868)$$ into the slope formula as shown. $$m$$ $$=$$ $$\\\\frac{1868-1442}{2012-2009}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$142$$"],"dependencies":["a833c22linear12a-h3"],"title":"Final Slope","text":"What is the numerical result of the previous step?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a833c22linear13","title":"Finding the Slope of a Line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a833c22linear13a","stepAnswer":["$$y=-7x+3$$"],"problemType":"TextBox","stepTitle":"If f(x) is a linear function, with $$f(2)=-11$$, and $$f(4)=-25$$, find an equation for the function in slope-intercept form.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=-7x+3$$","hints":{"DefaultPathway":[{"id":"a833c22linear13a-h1","type":"hint","dependencies":[],"title":"Creating Two Points","text":"We are given that $$f(2)=-11and$$ $$f(4)=-25$$, so we can create the points $$(2,-11)$$ and $$(4,-25)$$, respectively.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear13a-h2","type":"hint","dependencies":["a833c22linear13a-h1"],"title":"Finding the Slope of the Line","text":"We can now use the points to calculate the slope using the formula $$m$$ $$=$$ $$\\\\frac{y1-y2}{x1-x2}$$. $$m=\\\\frac{\\\\left(-25\\\\right)-\\\\left(-11\\\\right)}{4-2}$$. $$m=-7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear13a-h3","type":"hint","dependencies":["a833c22linear13a-h2"],"title":"Plugging into Point-Slope Form","text":"Using the calculated slope and one of the previous points, we can create an equation in point-slope form as shown. $$y+11=-7(x-2)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear13a-h4","type":"hint","dependencies":["a833c22linear13a-h3"],"title":"Transforming into Slope-Intercept Form","text":"We now rewrite the equation such that it is in point-slope form. $$y=-7x+3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a833c22linear14","title":"Finding the x-intercept of a Line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a833c22linear14a","stepAnswer":["$$16$$"],"problemType":"TextBox","stepTitle":"Find the $$x$$ value of the x-intercept of the function $$f(x)=\\\\frac{1}{4} x-4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16$$","hints":{"DefaultPathway":[{"id":"a833c22linear14a-h1","type":"hint","dependencies":[],"title":"Setting the Equation Equal to Zero","text":"We must now set the equation equal to $$0$$ as shown: $$0=\\\\frac{1}{4} x-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear14a-h2","type":"hint","dependencies":["a833c22linear14a-h1"],"title":"Solving for $$x$$","text":"We now solve the equation by isolating the $$x$$ term. $$4=\\\\frac{1}{4} x$$ --\x3e $$x=16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear14a-h3","type":"hint","dependencies":["a833c22linear14a-h2"],"title":"Extracting the x-intercept","text":"Since we solve for $$x$$ as $$16$$, we know the x-intercept is $$(16,0)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a833c22linear15","title":"Writing the Equation of a Horizontal Line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a833c22linear15a","stepAnswer":["$$y=-4$$"],"problemType":"TextBox","stepTitle":"Write the equation of the line graphed.","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=-4$$","hints":{"DefaultPathway":[{"id":"a833c22linear15a-h1","type":"hint","dependencies":[],"title":"Finding the Equation","text":"For any x-value, the y-value is $$-4$$. So the equation of the line must be $$y=-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a833c22linear16","title":"Writing the Equation of a Vertical Line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a833c22linear16a","stepAnswer":["$$x=7$$"],"problemType":"TextBox","stepTitle":"Write the equation of the line graphed.","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x=7$$","hints":{"DefaultPathway":[{"id":"a833c22linear16a-h1","type":"hint","dependencies":[],"title":"Finding the Equation","text":"The x-value is constantly $$7$$, so the equation of the line must be $$x=7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a833c22linear17","title":"Writing the Equation of a Line Parallel or Perpendicular to a Given Line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a833c22linear17a","stepAnswer":["$$y=3x-9$$"],"problemType":"TextBox","stepTitle":"Find a line parallel to the graph of $$f(x)=3x+6$$ given that the line passes through the point $$(3,0)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=3x-9$$","hints":{"DefaultPathway":[{"id":"a833c22linear17a-h1","type":"hint","dependencies":[],"title":"Finding the Slope of the Parallel Line","text":"Because we know that parallel lines have the same slope, we need to find the slope of the original line. The equation is in slope-intercept form, so we know the slope is 3; this means that the slope of the parallel line is also $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear17a-h2","type":"hint","dependencies":["a833c22linear17a-h1"],"title":"Substituting into Slope-Intercept Form","text":"If we substitute $$m=3$$ into $$y=mx+b$$, we have $$y=3x+b$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear17a-h3","type":"hint","dependencies":["a833c22linear17a-h2"],"title":"Solving for $$b$$","text":"We now solve for $$b$$ in the equation $$y=3x+b$$. Plugging in the point $$(3,0)$$, we get $$0=9+b$$ --\x3e $$b=-0$$. So, the final equation is $$y=3x-9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a833c22linear18","title":"Finding the Equation of a Perpendicular Line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a833c22linear18a","stepAnswer":["$$y=\\\\left(-\\\\frac{1}{3}\\\\right) x+1$$"],"problemType":"TextBox","stepTitle":"Find the equation of a line that is perpendicular to the graph $$y=3x+3$$ and passes through $$(0,0)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=\\\\left(-\\\\frac{1}{3}\\\\right) x+1$$","hints":{"DefaultPathway":[{"id":"a833c22linear18a-h1","type":"hint","dependencies":[],"title":"Finding the Slope of the Perpendicular Line","text":"We know that a perpendicular line must have a slope that is the negative reciprocal of the original line\'s slope. The negative reciprocal of the original slope, $$3$$, is $$\\\\frac{-1}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear18a-h2","type":"hint","dependencies":["a833c22linear18a-h1"],"title":"Substituting into Slope-Intercept Form","text":"If we substitute $$m=\\\\frac{-1}{3}$$ into the form $$y=mx+b$$, we get $$y=\\\\left(-\\\\frac{1}{3}\\\\right) x+b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear18a-h3","type":"hint","dependencies":["a833c22linear18a-h2"],"title":"Solving for $$b$$","text":"Since we know that the line passes through the point $$(3,0)$$, we can create the equation $$0=-1+b$$. From this, we get $$b=1$$. So, $$y=\\\\left(-\\\\frac{1}{3}\\\\right) x+1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a833c22linear19","title":"Finding the Equation of a Line Perpendicular to a Given Line Passing through a Point\\\\n","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a833c22linear19a","stepAnswer":["$$y=6x-19$$"],"problemType":"TextBox","stepTitle":"A line passes through the points $$(-2,6)$$ and $$(4,5)$$. Find the equation of a perpendicular line that passes through the point $$(4,5)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=6x-19$$","hints":{"DefaultPathway":[{"id":"a833c22linear19a-h1","type":"hint","dependencies":[],"title":"Finding the Original Line\'s Slope","text":"Using the slope formula, $$m=\\\\frac{y1-y2}{x1-x2}$$, we can calculate the slope. $$m=\\\\frac{6-5}{\\\\left(-2-4\\\\right)}$$. $$m=\\\\frac{-1}{6}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear19a-h2","type":"hint","dependencies":["a833c22linear19a-h1"],"title":"Finding the Perpendicular Line\'s Slope","text":"Since we know that a perpendicular line has a slope that is the negative reciprocal of the original line\'s, we can calculate the slope. The negative reciprocal of $$\\\\frac{-1}{6}$$ is $$6$$, so $$6$$ is the slope of the perpendicular line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear19a-h3","type":"hint","dependencies":["a833c22linear19a-h2"],"title":"Substituting into Slope-Intercept Form","text":"We can now plug in $$m=6$$ into the equation $$y=mx+b$$. We now have $$y=6x+b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear19a-h4","type":"hint","dependencies":["a833c22linear19a-h3"],"title":"Solving for $$b$$","text":"We can plug in the point $$(4,5)$$ since we know the line passes through this point. We now have $$5=24+b$$. Solving for $$b$$, we get $$b=-19$$. Now we must plug this into the equation derived in the previous step: $$y=6x-19$$ is the equation of the perpendicular line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a833c22linear2","title":"Linear Functions","body":"For the following exercises, determine whether the equation of the curve can be written as a linear function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a833c22linear2a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$3x+5y=15$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a833c22linear2a-h1","type":"hint","dependencies":[],"title":"Subtract","text":"Subtract $$3x$$ from both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear2a-h2","type":"hint","dependencies":["a833c22linear2a-h1"],"title":"Divide","text":"Divide both sides by $$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-3}{5}$$"],"dependencies":["a833c22linear2a-h2"],"title":"Slope","text":"What is the slope of the function?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear2a-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a833c22linear2a-h3"],"title":"Constant","text":"Is this slope constant for every increase in $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a833c22linear2a-s2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a833c22linear2a-s1"],"title":"Meaning","text":"Does this mean it can be written as a linear function?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a833c22linear20","title":"Deciding Whether a Function Is Increasing, Decreasing, or Constant.","body":"Some recent studies suggest that a teenager sends an average of $$60$$ texts per day. For each of the following scenarios, find the linear function that describes the relationship between the input value and the output value. Then, determine whether the graph of the function is increasing, decreasing, or constant.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a833c22linear20a","stepAnswer":["increasing"],"problemType":"MultipleChoice","stepTitle":"The total number of texts a teen sends is considered a function of time in days. The input is the number of days, and output is the total number of texts sent.","stepBody":"","answerType":"string","variabilization":{},"choices":["increasing","constant","decreasing"],"hints":{"DefaultPathway":[{"id":"a833c22linear20a-h1","type":"hint","dependencies":[],"title":"Rewrite","text":"Let\'s try to rewrite this in equation form. We can make our variable $$x$$ where $$x$$ $$=$$ the number of days","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear20a-h2","type":"hint","dependencies":["a833c22linear20a-h1"],"title":"Rewrite","text":"We can rewrite it to be f(x) $$=$$ $$60x$$ because that is how many texts are sent each day","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear20a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["positive"],"dependencies":["a833c22linear20a-h2"],"title":"Checking for if function is increasing, decreasing, or constant.","text":"To see if our function is increasing, decreasing, or constant, we have to check the slope. Is the slope positive, constant, or negative?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["positive","constant","negative"]},{"id":"a833c22linear20a-h4","type":"hint","dependencies":["a833c22linear20a-h3"],"title":"Checking for if function is increasing, decreasing, or constant.","text":"We see that $$60$$ is positive which means our function is increasing","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a833c22linear20b","stepAnswer":["decreasing"],"problemType":"MultipleChoice","stepTitle":"A teen has a limit of $$500$$ texts per month in his or her data plan. The input is the number of days, and output is the total number of texts remaining for the month.","stepBody":"","answerType":"string","variabilization":{},"choices":["increasing","constant","decreasing"],"hints":{"DefaultPathway":[{"id":"a833c22linear20b-h1","type":"hint","dependencies":[],"title":"Rewrite","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear20b-h2","type":"hint","dependencies":["a833c22linear20b-h1"],"title":"Rewrite","text":"Let\'s try to rewrite this in equation form. We can make our variable $$x$$ where $$x$$ $$=$$ the number of days","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear20b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["negative"],"dependencies":["a833c22linear20b-h2"],"title":"Checking for if function is increasing, decreasing, or constant.","text":"To see if our function is increasing, decreasing, or constant, we have to check the slope. Is the slope positive, constant, or negative?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["positive","constant","negative"]},{"id":"a833c22linear20b-h4","type":"hint","dependencies":["a833c22linear20b-h3"],"title":"Checking for if function is increasing, decreasing, or constant.","text":"We see that $$-60$$ is negative which means our function is decreasing","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a833c22linear20c","stepAnswer":["constant"],"problemType":"MultipleChoice","stepTitle":"A teen has an unlimited number of texts in his or her data plan for a cost of $50 per month. The input is the number of days, and output is the total cost of texting each month.","stepBody":"","answerType":"string","variabilization":{},"choices":["increasing","constant","decreasing"],"hints":{"DefaultPathway":[{"id":"a833c22linear20c-h1","type":"hint","dependencies":[],"title":"Rewrite","text":"Let\'s try to rewrite this in equation form. We can make our variable $$x$$ where $$x$$ $$=$$ the number of days","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear20c-h2","type":"hint","dependencies":["a833c22linear20c-h1"],"title":"Rewrite","text":"We can rewrite it to be f(x) $$=$$ $$50$$ because the number of days does not affect the price","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear20c-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["constant"],"dependencies":["a833c22linear20c-h2"],"title":"Checking for if function is increasing, decreasing, or constant.","text":"To see if our function is increasing, decreasing, or constant, we have to check the slope. Is the slope positive, constant, or negative?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["positive","constant","negative"]},{"id":"a833c22linear20c-h4","type":"hint","dependencies":["a833c22linear20c-h3"],"title":"Checking for if function is increasing, decreasing, or constant.","text":"We see that our slope is $$0$$, which means our function is constant.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a833c22linear21","title":"Finding the Slope of a Linear Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a833c22linear21a","stepAnswer":["Increasing"],"problemType":"MultipleChoice","stepTitle":"If f(x) is a linear function, and $$(3,-2)$$ and $$(8,1)$$ are points on the line, find the slope. Is this function increasing or decreasing?","stepBody":"","answerType":"string","variabilization":{},"choices":["Increasing","Decreasing"],"hints":{"DefaultPathway":[{"id":"a833c22linear21a-h1","type":"hint","dependencies":[],"title":"Formula","text":"We know that to find if the function is increasing, constant, or decreasing we must find the slope. To do so, we can use our slope formula.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear21a-h2","type":"hint","dependencies":["a833c22linear21a-h1"],"title":"Formula","text":"The formula is (change in output)/(change in input).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear21a-h3","type":"hint","dependencies":["a833c22linear21a-h2"],"title":"Plug in Values","text":"$$m=\\\\frac{1-\\\\left(-2\\\\right)}{8-3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear21a-h4","type":"hint","dependencies":["a833c22linear21a-h3"],"title":"Solve","text":"When we solve, we get $$\\\\frac{3}{5}$$ as our slope","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear21a-h5","type":"hint","dependencies":["a833c22linear21a-h4"],"title":"Checking for if function is increasing, decreasing, or constant.","text":"We see that our slope is $$\\\\frac{3}{5}$$, a positive number, which means our function is increasing.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a833c22linear22","title":"Finding the Population Change from a Linear Function","body":"The population of a city increased from 23,400 to 27,800 between $$2008$$ and $$2012$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a833c22linear22a","stepAnswer":["$$1100$$"],"problemType":"TextBox","stepTitle":"What is the change of population per year if we assume the change was constant from $$2008$$ to 2012?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1100$$","hints":{"DefaultPathway":[{"id":"a833c22linear22a-h1","type":"hint","dependencies":[],"title":"Change in people","text":"First, let\'s calculate the change in people","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear22a-h2","type":"hint","dependencies":["a833c22linear22a-h1"],"title":"Change in people","text":"The change in people is $$27800-23400=4400$$ people","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear22a-h3","type":"hint","dependencies":["a833c22linear22a-h2"],"title":"Number of years","text":"Next, let\'s calculate the number of years","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear22a-h4","type":"hint","dependencies":["a833c22linear22a-h3"],"title":"Number of years","text":"The number of years is $$2012-2008=4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear22a-h5","type":"hint","dependencies":["a833c22linear22a-h4"],"title":"Rate of Change","text":"To find the rate of change divide the change in people over the change in years","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear22a-h6","type":"hint","dependencies":["a833c22linear22a-h5"],"title":"Rate of Change","text":"$$\\\\frac{4400}{4}$$ $$=$$ $$1100$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a833c22linear23","title":"Writing an Equation for a Linear Function","body":"Write an equation for a linear function given its graph.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a833c22linear23a","stepAnswer":["$$y=3x+2$$"],"problemType":"MultipleChoice","stepTitle":"Based on the attached graph, what is the equation of the linear function?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$y=3x+2$$","choices":["$$y=2x+2$$","$$y=6x+2$$","$$y=3x+3$$","$$y=3x+2$$"],"hints":{"DefaultPathway":[{"id":"a833c22linear23a-h1","type":"hint","dependencies":[],"title":"Identify points to calculate slope","text":"Identify two points on the line, such as $$(0,2)$$ and $$(-2,-4)$$. Use the points to calculate the slope.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear23a-h2","type":"hint","dependencies":["a833c22linear23a-h1"],"title":"Use Formula","text":"$$m=\\\\frac{y2-y1}{x2-x1}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear23a-h3","type":"hint","dependencies":["a833c22linear23a-h2"],"title":"Plug in Values","text":"$$\\\\frac{\\\\left(-4-2\\\\right)}{\\\\left(-2-0\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear23a-h4","type":"hint","dependencies":["a833c22linear23a-h3"],"title":"Solve","text":"$$3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear23a-h5","type":"hint","dependencies":["a833c22linear23a-h4"],"title":"Substitute into point-slope form","text":"Substitute the slope and the coordinates of one of the points into the point-slope form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear23a-h6","type":"hint","dependencies":["a833c22linear23a-h5"],"title":"Use Formula","text":"$$y-y1=m(x-x1)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear23a-h7","type":"hint","dependencies":["a833c22linear23a-h6"],"title":"Plug in Values","text":"$$y-(-4)$$ $$=$$ $$3(x-(-2))$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear23a-h8","type":"hint","dependencies":["a833c22linear23a-h7"],"title":"Simplify","text":"$$y+4=3\\\\left(x+2\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear23a-h9","type":"hint","dependencies":["a833c22linear23a-h8"],"title":"Rewrite to slope-intercept form","text":"Rewrite in form $$y=mx+b$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear23a-h10","type":"hint","dependencies":["a833c22linear23a-h9"],"title":"Rewrite to slope-intercept form","text":"$$y=3x+2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a833c22linear24","title":"Writing an Equation for a Linear Cost Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a833c22linear24a","stepAnswer":["$$5000$$"],"problemType":"TextBox","stepTitle":"Suppose Ben starts a company in which he incurs a fixed cost of $1,250 per month for the overhead, which includes his office rent. His production costs are $$\\\\$37.50$$ per item. Write a linear function C where C(x) is the cost for $$x$$ items produced in a given month.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5000$$","hints":{"DefaultPathway":[{"id":"a833c22linear24a-h1","type":"hint","dependencies":[],"title":"Rewrite","text":"We can rewrite this to C(x) $$=$$ $$1270+37.5x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear24a-h2","type":"hint","dependencies":["a833c22linear24a-h1"],"title":"Plug in Value","text":"We can plug in $$100$$ for $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear24a-h3","type":"hint","dependencies":["a833c22linear24a-h2"],"title":"Solve","text":"We can solve $$1250+\\\\operatorname{37.5}\\\\left(100\\\\right)$$ $$=$$ $$5000$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a833c22linear25","title":"Writing an Equation for a Linear Function Given Two Points","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a833c22linear25a","stepAnswer":["$$y=\\\\frac{3}{5} x-\\\\frac{19}{5}$$"],"problemType":"MultipleChoice","stepTitle":"If f is a linear function, with $$f(3)=-2$$, and $$f(8)=1$$, find an equation for the function in slope-intercept form.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=\\\\frac{3}{5} x-\\\\frac{19}{5}$$","choices":["$$y=\\\\frac{3}{5} x-\\\\frac{19}{5}$$","$$y=\\\\frac{3}{4} x-\\\\frac{19}{5}$$","$$y=\\\\frac{5}{3} x-\\\\frac{13}{5}$$","$$y=\\\\frac{3}{5} x-\\\\frac{13}{5}$$"],"hints":{"DefaultPathway":[{"id":"a833c22linear25a-h1","type":"hint","dependencies":[],"title":"Rewrite in coordinate form","text":"We can rewrite the given to coordinate form to get $$(3,-2)$$ and $$(8,1)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear25a-h2","type":"hint","dependencies":["a833c22linear25a-h1"],"title":"Use Formula","text":"$$m=\\\\frac{y2-y1}{x2-x1}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear25a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{5}$$"],"dependencies":["a833c22linear25a-h2"],"title":"Plug in Values","text":"What is (1-(-2)/(8-3)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear25a-h4","type":"hint","dependencies":["a833c22linear25a-h3"],"title":"Substitute into point-slope form","text":"Substitute the slope and the coordinates of one of the points into the point-slope form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear25a-h5","type":"hint","dependencies":["a833c22linear25a-h4"],"title":"Use Formula","text":"$$y-y1=m(x-x1)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear25a-h6","type":"hint","dependencies":["a833c22linear25a-h5"],"title":"Plug in Values","text":"$$y-(-2)$$ $$=$$ (3/5)(x-3))","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear25a-h7","type":"hint","dependencies":["a833c22linear25a-h6"],"title":"Simplify","text":"$$y+2=\\\\frac{3}{5} \\\\left(x-3\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear25a-h8","type":"hint","dependencies":["a833c22linear25a-h7"],"title":"Rewrite to slope-intercept form","text":"rewrite in form $$y=mx+b$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear25a-h9","type":"hint","dependencies":["a833c22linear25a-h8"],"title":"Rewrite to slope-intercept form","text":"$$y=\\\\frac{3}{5} x-\\\\frac{19}{5}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a833c22linear26","title":"Using a Linear Function to Determine the Number of Songs in a Music Collection","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a833c22linear26a","stepAnswer":["$$380$$"],"problemType":"TextBox","stepTitle":"Marcus currently has $$200$$ songs in his music collection. Every month, he adds $$15$$ new songs. Write a formula for the number of songs, N, in his collection as a function of time, $$t$$, the number of months. How many songs will he own at the end of one year?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$380$$","hints":{"DefaultPathway":[{"id":"a833c22linear26a-h1","type":"hint","dependencies":[],"title":"Find the \\"b\\" value","text":"In our slope-intercept form, our $$b$$ value is the initial amount which is $$200$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear26a-h2","type":"hint","dependencies":["a833c22linear26a-h1"],"title":"Find the \\"m\\" value","text":"Our $$m$$ value is how much it is increasing per month which would be $$15$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear26a-h3","type":"hint","dependencies":["a833c22linear26a-h2"],"title":"Equation","text":"Our equation is now N(t) $$=$$ $$15t$$ + $$200$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear26a-h4","type":"hint","dependencies":["a833c22linear26a-h3"],"title":"Plug in values","text":"If we plug in $$12$$ for $$t$$ which is the number of months in a year, we get N(12) $$=$$ $$15\\\\times12$$ + $$200$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear26a-h5","type":"hint","dependencies":["a833c22linear26a-h4"],"title":"Solving","text":"This equals $$380$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a833c22linear27","title":"Using a Linear Function to Calculate Salary Based on Commission","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a833c22linear27a","stepAnswer":["I(n) $$=$$ $$80n$$ + $$520$$"],"problemType":"MultipleChoice","stepTitle":"Working as an insurance salesperson, Ilya earns a base salary plus a commission on each new policy. Therefore, Ilya\u2019s weekly income \ud835\udc3c, depends on the number of new policies, \ud835\udc5b, he sells during the week. Last week he sold $$3$$ new policies, and earned $760 for the week. The week before, he sold $$5$$ new policies and earned $920. Find an equation for \ud835\udc3c(\ud835\udc5b), and interpret the meaning of the components of the equation.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"I(n) $$=$$ $$80n$$ + $$520$$","choices":["I(n) $$=$$ $$70n$$ + $$510$$","I(n) $$=$$ $$80n$$ + $$520$$","I(n) $$=$$ $$90n$$ + $$530$$","I(n) $$=$$ $$100n$$ + $$540$$"],"hints":{"DefaultPathway":[{"id":"a833c22linear27a-h1","type":"hint","dependencies":[],"title":"Find \\"m\\"","text":"We are given two point pairs: $$(3,760)$$ and $$(5,920)$$ and we can plug it into our slope formula to find the slope","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear27a-h2","type":"hint","dependencies":["a833c22linear27a-h1"],"title":"Find \\"m\\"","text":"$$m=\\\\frac{920-760}{5-3}$$ $$=$$ $80 per policy","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear27a-h3","type":"hint","dependencies":["a833c22linear27a-h2"],"title":"Find $$b$$","text":"We can put it into slope-intercept form to get I(n) $$=$$ $$80n$$ + $$b$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear27a-h4","type":"hint","dependencies":["a833c22linear27a-h3"],"title":"Find $$b$$","text":"We know I(3) $$=$$ $$760$$ so $$760-80(3)$$ $$=$$ $$520$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear27a-h5","type":"hint","dependencies":["a833c22linear27a-h4"],"title":"Rewrite","text":"Rewrite the final equation as I(n) $$=$$ $$80n$$ + $$520$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a833c22linear28","title":"Using Tabular Form to Write an Equation for a Linear Function","body":"The attached table relates the number of rats in a population to time, in weeks.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a833c22linear28a","stepAnswer":["$$P(w)=40w+1000$$"],"problemType":"MultipleChoice","stepTitle":"Use the table to write a linear equation.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$P(w)=40w+1000$$","choices":["$$P(w)=-40w+1000$$","$$P(w)=40w-1000$$","$$P(w)=40w+1000$$","$$P(w)=-40w-1000$$"],"hints":{"DefaultPathway":[{"id":"a833c22linear28a-h1","type":"hint","dependencies":[],"title":"Find $$b$$","text":"We know that the intital value is $$1000$$ so $$b=1000$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear28a-h2","type":"hint","dependencies":["a833c22linear28a-h1"],"title":"Find $$m$$","text":"We see that the number increases by $$80$$ every $$2$$ weeks, so $$m=40$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear28a-h3","type":"hint","dependencies":["a833c22linear28a-h2"],"title":"Rewrite in Slope-Intercept Form","text":"When we rewrite, we get P(w) $$=$$ $$40w+1000$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a833c22linear29","title":"Matching Linear Functions to Their Graphs","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a833c22linear29a","stepAnswer":["$$I=a;$$ $$II=d;$$ $$III=b;$$ $$IV=c$$"],"problemType":"MultipleChoice","stepTitle":"Match each equation of the linear functions with one of the lines in the attached graph.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$I=a;$$ $$II=d;$$ $$III=b;$$ $$IV=c$$","choices":["$$I=d;$$ $$II=c;$$ $$III=a;$$ $$IV=b$$","$$I=a;$$ $$II=d;$$ $$III=b;$$ $$IV=c$$","$$I=a;$$ $$II=b;$$ $$III=c;$$ $$IV=d$$","$$I=b;$$ $$II=d;$$ $$III=c;$$ $$IV=a$$"],"hints":{"DefaultPathway":[{"id":"a833c22linear29a-h1","type":"hint","dependencies":[],"title":"Equation a","text":"This function has a slope of $$2$$ and a y-intercept of $$3$$. It must pass through the point $$(0,3)$$ and slant upward from left to right. We can use two points to find the slope, or we can compare it with the other functions listed. Function \ud835\udc54 has the same slope, but a different y-intercept. Lines I and III have the same slant because they have the same slope. Line III does not pass through $$(0,3)$$ so \ud835\udc53 must be represented by line I.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear29a-h2","type":"hint","dependencies":["a833c22linear29a-h1"],"title":"Equation $$b$$","text":"This function also has a slope of $$2$$, but a y-intercept of -3.-3. It must pass through the point $$(0,-3)(0,-3)$$ and slant upward from left to right. It must be represented by line III.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear29a-h3","type":"hint","dependencies":["a833c22linear29a-h2"],"title":"Equation c","text":"This function has a slope of $$-2$$ and a y-intercept of $$3$$. This is the only function listed with a negative slope, so it must be represented by line IV because it slants downward from left to right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear29a-h4","type":"hint","dependencies":["a833c22linear29a-h3"],"title":"Equation $$d$$","text":"This function has a slope of $$1212$$ and a y-intercept of $$3$$. It must pass through the point $$(0,3)$$ and slant upward from left to right. Lines I and II pass through $$(0,3),(0,3)$$, but the slope of \ud835\udc57j is less than the slope of \ud835\udc53f so the line for \ud835\udc57j must be flatter. This function is represented by Line II.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a833c22linear3","title":"Linear Functions","body":"For the following exercises, determine whether the equation of the curve can be written as a linear function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a833c22linear3a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$3x+5y^2=15$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a833c22linear3a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Power","text":"What is the power of the $$y$$ variable?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear3a-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a833c22linear3a-h1"],"title":"Meaning","text":"Does this mean it can be written as a linear function?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a833c22linear4","title":"Linear Functions","body":"For the following exercises, determine whether the equation of the curve can be written as a linear function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a833c22linear4a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{\\\\left(-x-35\\\\right)}{2}$$ $$=$$ $$y$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a833c22linear4a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Power","text":"What is the power of the $$y$$ variable?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear4a-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a833c22linear4a-h1"],"title":"Meaning","text":"If both the powers of $$y$$ and $$x$$ are $$1$$, does this mean you can write this as a linear function?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a833c22linear5","title":"Linear Functions","body":"For the following exercises, determine whether each function is increasing or decreasing.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a833c22linear5a","stepAnswer":["Decreasing"],"problemType":"MultipleChoice","stepTitle":"$$g(x)=5x-6$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Increasing","Decreasing"],"hints":{"DefaultPathway":[{"id":"a833c22linear5a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":[],"title":"Slope","text":"What is the slope of the function?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear5a-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a833c22linear5a-h1"],"title":"Sign","text":"Is this slope positive?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a833c22linear5a-s2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a833c22linear5a-s1"],"title":"Meaning","text":"Does this mean it is increasing?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a833c22linear6","title":"Linear Functions","body":"For the following exercises, determine whether each function is increasing or decreasing.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a833c22linear6a","stepAnswer":["Decreasing"],"problemType":"MultipleChoice","stepTitle":"$$g(x)=8-3x$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Increasing","Decreasing"],"hints":{"DefaultPathway":[{"id":"a833c22linear6a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":[],"title":"Slope","text":"What is the slope of the function?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear6a-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a833c22linear6a-h1"],"title":"Sign","text":"Is this slope positive?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a833c22linear6a-s2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a833c22linear6a-s1"],"title":"Meaning","text":"Does this mean it is increasing?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a833c22linear7","title":"Linear Functions","body":"For the following exercises, determine whether each function is increasing or decreasing.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a833c22linear7a","stepAnswer":["Decreasing"],"problemType":"MultipleChoice","stepTitle":"$$g(x)=8-3x$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Increasing","Decreasing"],"hints":{"DefaultPathway":[{"id":"a833c22linear7a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":[],"title":"Slope","text":"What is the slope of the function?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear7a-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a833c22linear7a-h1"],"title":"Sign","text":"Is this slope positive?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a833c22linear7a-s2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a833c22linear7a-s1"],"title":"Meaning","text":"Does this mean it is increasing?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a833c22linear8","title":"Linear Functions","body":"For the following exercises, determine whether each function is increasing or decreasing.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a833c22linear8a","stepAnswer":["Increasing"],"problemType":"MultipleChoice","stepTitle":"$$g(x)=\\\\frac{1}{4} x-5$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Increasing","Decreasing"],"hints":{"DefaultPathway":[{"id":"a833c22linear8a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4}$$"],"dependencies":[],"title":"Slope","text":"What is the slope of the function?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear8a-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a833c22linear8a-h1"],"title":"Sign","text":"Is this slope positive?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a833c22linear8a-s2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a833c22linear8a-s1"],"title":"Meaning","text":"Does this mean it is increasing?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a833c22linear9","title":"Linear Functions","body":"For the following exercises, determine whether each function is increasing or decreasing.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a833c22linear9a","stepAnswer":["Decreasing"],"problemType":"MultipleChoice","stepTitle":"$$m(x)=\\\\frac{-3}{8} x+3$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Increasing","Decreasing"],"hints":{"DefaultPathway":[{"id":"a833c22linear9a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":[],"title":"Slope","text":"What is the slope of the function?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a833c22linear9a-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a833c22linear9a-h1"],"title":"Sign","text":"Is this slope positive?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a833c22linear9a-s2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a833c22linear9a-s1"],"title":"Meaning","text":"Does this mean it is increasing?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a8583b4FracDec1","title":"How to Solve Equations with Fraction Coefficients","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Equations with Fractions or Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8583b4FracDec1a","stepAnswer":["$$7$$"],"problemType":"TextBox","stepTitle":"Solve: $$\\\\frac{1}{6} y-\\\\frac{1}{3}=\\\\frac{5}{6}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7$$","hints":{"DefaultPathway":[{"id":"a8583b4FracDec1a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":[],"title":"Getting Rid of Fraction Denominators","text":"What is the LCD (largest common denominator) of $$\\\\frac{1}{6}$$, $$\\\\frac{1}{3}$$, and $$\\\\frac{5}{6}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec1a-h2","type":"hint","dependencies":["a8583b4FracDec1a-h1"],"title":"Getting Rid of Fraction Denominators","text":"We must now multiply both sides of the equation by $$6$$ and use the distributive property, so we have $$6\\\\frac{1}{6} y-6\\\\frac{1}{3}=6\\\\frac{5}{6}$$. This simplifies to $$y-2=5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec1a-h3","type":"hint","dependencies":["a8583b4FracDec1a-h2"],"title":"Solving the New Equation","text":"We must now isolate the \\"y\\" term in the equation to solve.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a8583b4FracDec1a-h3"],"title":"Solving the New Equation","text":"To solve for the \\"y\\" term, we must add $$2$$ to both sides of the equation. What is the value of $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8583b4FracDec10","title":"How to Solve Equations with Fraction Coefficients","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Equations with Fractions or Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8583b4FracDec10a","stepAnswer":["$$-3$$"],"problemType":"TextBox","stepTitle":"Solve: $$-5=\\\\frac{1}{4\\\\left(8x+4\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-3$$","hints":{"DefaultPathway":[{"id":"a8583b4FracDec10a-h1","type":"hint","dependencies":[],"title":"Getting Rid of Fraction Denominators","text":"Notice that we can just use the distributive property here to get rid of the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec10a-h2","type":"hint","dependencies":["a8583b4FracDec10a-h1"],"title":"Getting Rid of Fraction Denominators","text":"We can just multiply $$\\\\frac{1}{4}$$ by the values inside the parantheses. We now have $$-5=2x+1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec10a-h3","type":"hint","dependencies":["a8583b4FracDec10a-h2"],"title":"Solving the New Equation","text":"Now, we must isolate the \\"x\\" term so that we can solve.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec10a-h4","type":"hint","dependencies":["a8583b4FracDec10a-h3"],"title":"Solving the New Equation","text":"We can isolate $$x$$ by subtracting $$1$$ from each side of the equation. We now have $$2x=-6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a8583b4FracDec10a-h4"],"title":"Solving the New Equation","text":"Finally, we can divide both sides of the equation by $$2$$. What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8583b4FracDec11","title":"How to Solve Equations with Fraction Coefficients","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Equations with Fractions or Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8583b4FracDec11a","stepAnswer":["$$-4$$"],"problemType":"TextBox","stepTitle":"Solve: $$-11=\\\\frac{1}{2\\\\left(6p+2\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-4$$","hints":{"DefaultPathway":[{"id":"a8583b4FracDec11a-h1","type":"hint","dependencies":[],"title":"Getting Rid of Fraction Denominators","text":"Notice that we can just use the distributive property here to get rid of the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec11a-h2","type":"hint","dependencies":["a8583b4FracDec11a-h1"],"title":"Getting Rid of Fraction Denominators","text":"We can just multiply $$\\\\frac{1}{2}$$ by the values inside the parantheses. We now have $$-11=3p+1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec11a-h3","type":"hint","dependencies":["a8583b4FracDec11a-h2"],"title":"Solving the New Equation","text":"We can now isolate the \\"p\\" term to solve the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec11a-h4","type":"hint","dependencies":["a8583b4FracDec11a-h3"],"title":"Solving the New Equation","text":"To isolate the \\"s\\" term, we can subtract $$1$$ from each side of the equation We now have $$3p=-12$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["a8583b4FracDec11a-h4"],"title":"Solving the New Equation","text":"Finally, we can divide both sides of the equation by $$3$$. What is the value of $$p$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8583b4FracDec12","title":"How to Solve Equations with Fraction Coefficients","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Equations with Fractions or Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8583b4FracDec12a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"Solve: $$8=\\\\frac{1}{3\\\\left(9q+6\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a8583b4FracDec12a-h1","type":"hint","dependencies":[],"title":"Getting Rid of Fraction Denominators","text":"Notice that we can just use the distributive property here to get rid of the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec12a-h2","type":"hint","dependencies":["a8583b4FracDec12a-h1"],"title":"Getting Rid of Fraction Denominators","text":"All we need to do is multiply $$\\\\frac{1}{3}$$ by the values inside the parantheses. We now have $$8=3q+2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec12a-h3","type":"hint","dependencies":["a8583b4FracDec12a-h2"],"title":"Solving the New Equation","text":"To solve the equation now, we must isolate the \\"q\\" terms on one side of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec12a-h4","type":"hint","dependencies":["a8583b4FracDec12a-h3"],"title":"Solving the New Equation","text":"We can subtract $$2$$ from both sides of the equation to isolate the q term. We now have $$6=3q$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec12a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a8583b4FracDec12a-h4"],"title":"Solving the New Equation","text":"Finally, we can divide both sides of the equation by $$3$$ to solve for q. What is the value of q?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8583b4FracDec13","title":"How to Solve Equations with Fraction Coefficients","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Equations with Fractions or Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8583b4FracDec13a","stepAnswer":["$$9$$"],"problemType":"TextBox","stepTitle":"Solve: $$\\\\frac{1}{2\\\\left(y-5\\\\right)}=\\\\frac{1}{4\\\\left(y-1\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9$$","hints":{"DefaultPathway":[{"id":"a8583b4FracDec13a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":[],"title":"Getting Rid of Fraction Denominators","text":"What is the LCD (largest common denominator) of $$\\\\frac{1}{2}$$ and $$\\\\frac{1}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec13a-h2","type":"hint","dependencies":["a8583b4FracDec13a-h1"],"title":"Getting Rid of Fraction Denominators","text":"Now, we must multiply by the LCD on both sides of the equation. We now have $$\\\\frac{4\\\\times1}{2\\\\left(y-5\\\\right)}=\\\\frac{4\\\\times1}{4\\\\left(y-1\\\\right)}$$. This simplifies to $$2(y-5)=(y-1)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec13a-h3","type":"hint","dependencies":["a8583b4FracDec13a-h2"],"title":"Distributive Property","text":"We can now use the distributive property to simplify the left-hand side of the equation. We now have $$2y-10=y-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec13a-h4","type":"hint","dependencies":["a8583b4FracDec13a-h3"],"title":"Solving the New Equation","text":"Now, we must isolate the \\"y\\" term to solve the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec13a-h5","type":"hint","dependencies":["a8583b4FracDec13a-h4"],"title":"Solving the New Equation","text":"We can isolate the \\"y\\" term by adding $$10$$ to both sides of the equation. We now have $$2y=y+9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec13a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a8583b4FracDec13a-h5"],"title":"Solving the New Equation","text":"Now, we can subtract $$y$$ from both sides of the equation. What is the value of $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8583b4FracDec14","title":"How to Solve Equations with Fraction Coefficients","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Equations with Fractions or Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8583b4FracDec14a","stepAnswer":["$$-1$$"],"problemType":"TextBox","stepTitle":"Solve: $$\\\\frac{1}{2\\\\left(m-3\\\\right)}=\\\\frac{1}{4\\\\left(m-7\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1$$","hints":{"DefaultPathway":[{"id":"a8583b4FracDec14a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":[],"title":"Getting Rid of Fraction Denominators","text":"What is the LCD (largest common denominator) of $$\\\\frac{1}{2}$$ and $$\\\\frac{1}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec14a-h2","type":"hint","dependencies":["a8583b4FracDec14a-h1"],"title":"Getting Rid of Fraction Denominators","text":"Now we must multiply both sides by the LCD. We now have $$\\\\frac{4\\\\times1}{2\\\\left(m-3\\\\right)}=\\\\frac{4\\\\times1}{4\\\\left(m-7\\\\right)}$$. This simplifies to $$2(m-3)=(m-7)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec14a-h3","type":"hint","dependencies":["a8583b4FracDec14a-h2"],"title":"Distributive Property","text":"We can now use the distributive property to simplify the left-hand side of the equation. We now have $$2m-6=m-7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec14a-h4","type":"hint","dependencies":["a8583b4FracDec14a-h3"],"title":"Solving the New Equation","text":"We must now isolate the \\"m\\" term so that we can solve the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec14a-h5","type":"hint","dependencies":["a8583b4FracDec14a-h4"],"title":"Solving the New Equation","text":"We can do this by first adding $$6$$ to both sides of the equation. We now have $$2m=m-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec14a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a8583b4FracDec14a-h5"],"title":"Solving the New Equation","text":"Now, we can subtract $$m$$ from both sides of the equation. What is the solution to this equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8583b4FracDec15","title":"How to Solve Equations with Fraction Coefficients","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Equations with Fractions or Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8583b4FracDec15a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"Solve: $$\\\\frac{5x-3}{4}=\\\\frac{x}{2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a8583b4FracDec15a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":[],"title":"Getting Rid of Fraction Denominators","text":"What is the LCD (largest common denominator) of this equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec15a-h2","type":"hint","dependencies":["a8583b4FracDec15a-h1"],"title":"Getting Rid of Fraction Denominators","text":"Now, we must multiply by the LCD on both sides of the equation. We now have $$(5x-3)=2x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec15a-h3","type":"hint","dependencies":["a8583b4FracDec15a-h2"],"title":"Solving the New Equation","text":"To solve this equation, we must first isolate the \\"x\\" term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec15a-h4","type":"hint","dependencies":["a8583b4FracDec15a-h3"],"title":"Solving the New Equation","text":"We can isolate the \\"x\\" term by subtracting $$2x$$ from both sides of the equatoin. We now have $$3x-3=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec15a-h5","type":"hint","dependencies":["a8583b4FracDec15a-h4"],"title":"Solving the New Equation","text":"Now, we must add $$3$$ to both sides of the equation. We now have $$3x=3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec15a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a8583b4FracDec15a-h5"],"title":"Solving the New Equation","text":"Finally, we can divide both sides of the equation by $$3$$. What is the solution of the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8583b4FracDec16","title":"Solve Equations with Fraction Coefficients","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Equations with Fractions or Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8583b4FracDec16a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"Solve: $$\\\\frac{3}{4} x-\\\\frac{1}{2}=\\\\frac{1}{4}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a8583b4FracDec16a-h1","type":"hint","dependencies":[],"title":"Isolate the term with $$x$$","text":"The first step is to move all the constant terms to one side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec16a-h2","type":"hint","dependencies":["a8583b4FracDec16a-h1"],"title":"Adding and Subtracting Fractions","text":"In order to add or subtract fractions, the fractions must have the same denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec16a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a8583b4FracDec16a-h2"],"title":"Finding an equivalent fraction for $$\\\\frac{1}{2}$$","text":"What number should I multiply to the numerator and denominator of $$\\\\frac{1}{2}$$ so that the equivalent fraction will have a denominator of 4?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec16a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{4}$$"],"dependencies":["a8583b4FracDec16a-h3"],"title":"Adding and Subtracting Fractions","text":"What is $$\\\\frac{1}{4}+\\\\frac{2}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a8583b4FracDec16a-h4-s1","type":"hint","dependencies":[],"title":"Adding and Subtracting Fractions","text":"When $$\\\\frac{adding}{subtracting}$$ fractions, the denominator stays the same.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a8583b4FracDec16a-h5","type":"hint","dependencies":["a8583b4FracDec16a-h4"],"title":"Solving for $$x$$","text":"The next step is to multiply both sides by the reciprocal of the coefficient of $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec16a-h6","type":"hint","dependencies":["a8583b4FracDec16a-h5"],"title":"Definition of reciprocal","text":"The reciprocal of the fraction $$\\\\frac{a}{b}$$ is the fraction $$\\\\frac{b}{a}$$ such that $$\\\\frac{a}{b} \\\\frac{b}{a}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec16a-h7","type":"hint","dependencies":["a8583b4FracDec16a-h6"],"title":"Multiplying fractions","text":"When multiplying fractions, multiply the numerators together and the denominators together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8583b4FracDec17","title":"Solve Equations with Fraction Coefficients","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Equations with Fractions or Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8583b4FracDec17a","stepAnswer":["$$-144$$"],"problemType":"TextBox","stepTitle":"Solve: $$\\\\frac{5}{6} n-\\\\frac{1}{4} n-\\\\frac{1}{2} n=-12$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-144$$","hints":{"DefaultPathway":[{"id":"a8583b4FracDec17a-h1","type":"hint","dependencies":[],"title":"Changing the coefficients to whole numbers","text":"The first step is to multiply both sides of the equation by some number to change the fraction coefficients to whole numbers. This makes the equation easier to solve.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec17a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a8583b4FracDec17a-h1"],"title":"Finding the number to multiply","text":"What is the smallest multiple of $$2$$, $$4$$, and 6?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec17a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a8583b4FracDec17a-h2"],"title":"Multiplying by a Whole Number","text":"What is $$12\\\\frac{5}{6}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec17a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a8583b4FracDec17a-h2"],"title":"Multiplying by a Whole Number","text":"What is $$12-\\\\left(\\\\frac{1}{4}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec17a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6$$"],"dependencies":["a8583b4FracDec17a-h2"],"title":"Multiplying by a Whole Number","text":"What is $$12-\\\\left(\\\\frac{1}{2}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec17a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-144$$"],"dependencies":["a8583b4FracDec17a-h2"],"title":"Multiplying by a Whole Number","text":"Remember that to make the equality hold true, the right hand side also has to be multiplied by the same number. What is $$-12\\\\times12$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec17a-h7","type":"hint","dependencies":["a8583b4FracDec17a-h3","a8583b4FracDec17a-h4","a8583b4FracDec17a-h5","a8583b4FracDec17a-h6"],"title":"Combine Left Hand Side","text":"The left hand side becomes $$10n-3n-6n=n$$. Therefore, we get the answer $$n=-144$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8583b4FracDec18","title":"Solve Equations with Fraction Coefficients","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Equations with Fractions or Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8583b4FracDec18a","stepAnswer":["$$-2$$"],"problemType":"TextBox","stepTitle":"Solve: $$\\\\frac{3}{2} z+\\\\frac{1}{3}=z-\\\\frac{2}{3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-2$$","hints":{"DefaultPathway":[{"id":"a8583b4FracDec18a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":[],"title":"Getting Rid of Fraction Denominators","text":"What is the LCD (largest common denominator) of $$\\\\frac{3}{2}$$, $$\\\\frac{1}{3}$$, and $$\\\\frac{2}{3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec18a-h2","type":"hint","dependencies":["a8583b4FracDec18a-h1"],"title":"Getting Rid of Fraction Denominators","text":"We must now multiply both sides of the equation by $$6$$ and use the distributive property, so we have $$6\\\\frac{3}{2} z+6\\\\frac{1}{3}=6z-6\\\\frac{2}{3}$$. This simplifies to $$9z+2=6z-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec18a-h3","type":"hint","dependencies":["a8583b4FracDec18a-h2"],"title":"Solving the New Equation","text":"To solve this equation, we must isolate the \\"z\\" term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec18a-h4","type":"hint","dependencies":["a8583b4FracDec18a-h3"],"title":"Solving the New Equation","text":"To isolate the \\"z\\" term, we can begin by subtracting $$6z$$ from both sides of the equation. We now have $$3z+2=-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec18a-h5","type":"hint","dependencies":["a8583b4FracDec18a-h4"],"title":"Solving the New Equation","text":"Now, we must subtract $$2$$ from both sides so that only the \\"z\\" term is on the left-hand side. We now get $$3z=-6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec18a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a8583b4FracDec18a-h5"],"title":"Solving the New Equation","text":"Finally, we can divide both sides by $$3$$ to get the \\"z\\" term by itself on the left-hand side. What is the value of $$z$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8583b4FracDec19","title":"Solve Equations with Fraction Coefficients","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Equations with Fractions or Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8583b4FracDec19a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"Solve: $$1=\\\\frac{1}{5} \\\\left(15x-10\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a8583b4FracDec19a-h1","type":"hint","dependencies":[],"title":"Using Distributive Property","text":"The first step is to use distributive property to clear the fraction","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec19a-h2","type":"hint","dependencies":["a8583b4FracDec19a-h1"],"title":"Definition of Distributive Property","text":"Distributive Property is the property such that $$a \\\\left(b+c\\\\right)=a b+a c$$, where a,b, and c are constants","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec19a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a8583b4FracDec19a-h2"],"title":"Multiplying fractions and numbers","text":"What is $$15\\\\frac{1}{5}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a8583b4FracDec19a-h3-s1","type":"hint","dependencies":[],"title":"Multiplying fractions and numbers","text":"$$\\\\frac{1}{a} b$$ is the same thing as \\"b divided by a\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a8583b4FracDec19a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a8583b4FracDec19a-h2"],"title":"Multiplying fractions and numbers","text":"What is $$-10\\\\frac{1}{5}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a8583b4FracDec19a-h4-s1","type":"hint","dependencies":[],"title":"Multiplying fractions and numbers","text":"$$\\\\frac{1}{a} b$$ is the same thing as \\"b divided by a\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a8583b4FracDec19a-h5","type":"hint","dependencies":["a8583b4FracDec19a-h3","a8583b4FracDec19a-h4"],"title":"Isolate terms with variables","text":"The next step is to move all the terms with variables to one side of the equation and all the constant terms to the other. By adding $$2$$ to both sides, we get $$3x=3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec19a-h6","type":"hint","dependencies":["a8583b4FracDec19a-h5"],"title":"Solving for $$x$$","text":"The next step is to multiply both sides of the equation by the reciprocal of the coefficient of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec19a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["a8583b4FracDec19a-h6"],"title":"Reciprocal of $$3$$","text":"What is the reciprocal of 3?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec19a-h8","type":"hint","dependencies":["a8583b4FracDec19a-h7"],"title":"Solving for $$x$$","text":"Multiplying both sides by $$\\\\frac{1}{3}$$, we get $$x=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8583b4FracDec2","title":"How to Solve Equations with Fraction Coefficients","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Equations with Fractions or Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8583b4FracDec2a","stepAnswer":["$$\\\\frac{1}{2}$$"],"problemType":"TextBox","stepTitle":"Solve: $$\\\\frac{1}{4} x+\\\\frac{1}{2}=\\\\frac{5}{8}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{2}$$","hints":{"DefaultPathway":[{"id":"a8583b4FracDec2a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":[],"title":"Getting Rid of Fraction Denominators","text":"What is the LCD (largest common denominator) of $$\\\\frac{1}{4}$$, $$\\\\frac{1}{2}$$, and $$\\\\frac{5}{8}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec2a-h2","type":"hint","dependencies":["a8583b4FracDec2a-h1"],"title":"Getting Rid of Fraction Denominators","text":"We must now multiply both sides of the equation by $$8$$ and use the distributive property. We now have $$8\\\\frac{1}{4} x+8\\\\frac{1}{2}=\\\\frac{8\\\\times5}{8}$$. This simplifies to $$2x+4=5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec2a-h3","type":"hint","dependencies":["a8583b4FracDec2a-h2"],"title":"Solving the New Equation","text":"We must now isolate the \\"x\\" term in the equation to solve.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec2a-h4","type":"hint","dependencies":["a8583b4FracDec2a-h3"],"title":"Solving the New Equation","text":"To solve for the \\"x\\" term, we must add subtract $$4$$ from both sides of the equation. We now have $$2x=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec2a-h5","type":"hint","dependencies":["a8583b4FracDec2a-h4"],"title":"Solving the New Equation","text":"Now, we can divide both sides by $$2$$. We now have $$x=\\\\frac{1}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8583b4FracDec20","title":"Solve Equations with Fraction Coefficients","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Equations with Fractions or Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8583b4FracDec20a","stepAnswer":["$$7$$"],"problemType":"TextBox","stepTitle":"Solve: $$\\\\frac{1}{5} \\\\left(q+3\\\\right)=\\\\frac{1}{2} \\\\left(q-3\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7$$","hints":{"DefaultPathway":[{"id":"a8583b4FracDec20a-h1","type":"hint","dependencies":[],"title":"Clearing out the fractions","text":"The first step is to multiply both sides of the equation by a number that will clear out the fractions. In other words, we will multiply both sides of the equation by the least common multiple of the two denominators.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a8583b4FracDec20a-h1"],"title":"Finding the least common multiple","text":"What is the smallest number that is a multiple of $$2$$ and 5?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec20a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a8583b4FracDec20a-h2"],"title":"Multiplying numbers","text":"What is $$10\\\\frac{1}{5}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec20a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a8583b4FracDec20a-h2"],"title":"Multiplying numbers","text":"What is $$10\\\\frac{1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec20a-h5","type":"hint","dependencies":["a8583b4FracDec20a-h3","a8583b4FracDec20a-h4"],"title":"Using Distributive Property","text":"Now we get the cleared-up equation $$2\\\\left(q+3\\\\right)=5(q-3)$$. The next step is to use distributive property to get rid of the parantheses.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec20a-h6","type":"hint","dependencies":["a8583b4FracDec20a-h5"],"title":"Definition of Distributive Property","text":"Distributive Property is the property such that $$a \\\\left(b+c\\\\right)=a b+a c$$, where a,b, and c are constants","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec20a-h7","type":"hint","dependencies":["a8583b4FracDec20a-h6"],"title":"Using Distributive Property","text":"After distributing, we get the equation $$2q+6=5q-15$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec20a-h8","type":"hint","dependencies":["a8583b4FracDec20a-h7"],"title":"Solving for q","text":"The next step is to move all the terms with q to one side of the expression and all the constants to another. In other words, we want to isolate q by algebraically manipulate the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec20a-h9","type":"hint","dependencies":["a8583b4FracDec20a-h8"],"title":"Solving for q","text":"We can start by subtracting 2q from both sides, which gives us the equation $$3=3q-15$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec20a-h10","type":"hint","dependencies":["a8583b4FracDec20a-h9"],"title":"Solving for q","text":"The next step is to add $$15$$ to both sides of the equation, which gives us $$3q=21$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec20a-h11","type":"hint","dependencies":["a8583b4FracDec20a-h10"],"title":"Solving for q","text":"The last step is to divide both sides by the coefficient of q, which gives us the final answer $$q=7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8583b4FracDec21","title":"Solve Equations with Fraction Coefficients","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Equations with Fractions or Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8583b4FracDec21a","stepAnswer":["$$-1$$"],"problemType":"TextBox","stepTitle":"Solve: $$\\\\frac{4m+2}{6}=\\\\frac{m}{3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1$$","hints":{"DefaultPathway":[{"id":"a8583b4FracDec21a-h1","type":"hint","dependencies":[],"title":"Clearing out the fractions","text":"The first step is to multiply both sides of the equation by a number that will clear out the fractions. In other words, we will multiply both sides of the equation by the least common multiple of the two denominators.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec21a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a8583b4FracDec21a-h1"],"title":"Finding the least common multiple","text":"What is the smallest number that is a multiple of $$6$$ and 3?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec21a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a8583b4FracDec21a-h2"],"title":"Multiplying and dividing numbers","text":"What is $$6\\\\frac{1}{6}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec21a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a8583b4FracDec21a-h2"],"title":"Multiplying and dividing numbers","text":"What is $$6\\\\frac{1}{3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec21a-h5","type":"hint","dependencies":["a8583b4FracDec21a-h3","a8583b4FracDec21a-h4"],"title":"New Equation","text":"Now, we get the new equation $$4m+2=2m$$, and we need to solve for $$m$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec21a-h6","type":"hint","dependencies":["a8583b4FracDec21a-h5"],"title":"Solving for $$m$$","text":"The next step is to move all the terms with $$m$$ to one side of the expression and all the constants to another. In other words, we want to isolate $$m$$ by algebraically manipulate the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec21a-h7","type":"hint","dependencies":["a8583b4FracDec21a-h6"],"title":"Solving for $$m$$","text":"We can start by subtracting $$2m$$ from both sides, which gives us the equation $$2m+2=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec21a-h8","type":"hint","dependencies":["a8583b4FracDec21a-h7"],"title":"Solving for $$m$$","text":"The next step is to subtract $$2$$ from both sides of the equation, which gives us $$2m=-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec21a-h9","type":"hint","dependencies":["a8583b4FracDec21a-h8"],"title":"Solving for $$m$$","text":"The last step is to divide both sides of the equation by the coefficient of $$m$$, which gives us $$m=-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8583b4FracDec22","title":"Solve Equations with Fraction Coefficients","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Equations with Fractions or Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8583b4FracDec22a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"Solve: $$\\\\frac{3v-6}{2}+5=\\\\frac{11v-4}{5}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a8583b4FracDec22a-h1","type":"hint","dependencies":[],"title":"Clearing out the fractions","text":"The first step is to multiply both sides of the equation by a number that will clear out the fractions. In other words, we will multiply both sides of the equation by the least common multiple of the denominators.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec22a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a8583b4FracDec22a-h1"],"title":"Finding the least common multiple","text":"What is the smallest number that is a multiple of $$2$$ and 5?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec22a-h3","type":"hint","dependencies":["a8583b4FracDec22a-h2"],"title":"Using Distributive Property","text":"The next step is to use distributive property to clear out the fractions","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec22a-h4","type":"hint","dependencies":["a8583b4FracDec22a-h3"],"title":"Definition of Distributive Property","text":"Distributive Property is the property such that $$a \\\\left(b+c\\\\right)=a b+a c$$, where a,b, and c are constants","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec22a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a8583b4FracDec22a-h4"],"title":"Multiplying numbers","text":"What is $$10\\\\frac{1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec22a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$50$$"],"dependencies":["a8583b4FracDec22a-h4"],"title":"Multiplying numbers","text":"What is $$10\\\\times5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec22a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a8583b4FracDec22a-h4"],"title":"Multiplying numbers","text":"What is $$10\\\\frac{1}{5}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec22a-h8","type":"hint","dependencies":["a8583b4FracDec22a-h5","a8583b4FracDec22a-h6","a8583b4FracDec22a-h7"],"title":"New Equation","text":"Now, we get the new equation $$5\\\\left(3v-6\\\\right)+50=2(11v-4)$$, and we need to solve for v.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec22a-h9","type":"hint","dependencies":["a8583b4FracDec22a-h8"],"title":"Solving for v","text":"The first step is to distribute the parentheses. Multiplying, we get $$15v-30+50=22v-8$$, which is equivalent to $$15v+20=22v-8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec22a-h10","type":"hint","dependencies":["a8583b4FracDec22a-h9"],"title":"Solving for v","text":"The next step is to move all the terms with v to one side of the expression and all the constants to another. In other words, we want to isolate v by algebraically manipulate the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec22a-h11","type":"hint","dependencies":["a8583b4FracDec22a-h10"],"title":"Solving for v","text":"We can start by subtracting 15v from both sides, which gives us the equation $$20=7v-8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec22a-h12","type":"hint","dependencies":["a8583b4FracDec22a-h11"],"title":"Solving for v","text":"The next step is to add $$8$$ to both sides of the equation, which gives us $$7v=28$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec22a-h13","type":"hint","dependencies":["a8583b4FracDec22a-h12"],"title":"Solving for v","text":"The last step is to divide both sides of the equation by the coefficient of v, which gives us $$v=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8583b4FracDec23","title":"Solve Equations with Decimal Coefficients","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Equations with Fractions or Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8583b4FracDec23a","stepAnswer":["$$15$$"],"problemType":"TextBox","stepTitle":"Solve: $$0.4y-4=2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$15$$","hints":{"DefaultPathway":[{"id":"a8583b4FracDec23a-h1","type":"hint","dependencies":[],"title":"Isolating the variable term","text":"The first step is to move all the terms with a variable to one side of the equation and all the constant terms to the other. To do this, we add $$4$$ to both sides of the equation, which gives us $$0.4y=6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec23a-h2","type":"hint","dependencies":["a8583b4FracDec23a-h1"],"title":"Clearing the decimal","text":"To clear the decimal, multiply both sides of the equation by some factor of $$10$$. In this case, we can mulitply the equation by $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec23a-h3","type":"hint","dependencies":["a8583b4FracDec23a-h2"],"title":"Equation Without Decimals","text":"Multiplying both sides of the equation by $$10$$, we get $$10\\\\times0.4y=10\\\\times6$$, which simplifies to $$4y=60$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec23a-h4","type":"hint","dependencies":["a8583b4FracDec23a-h3"],"title":"Solving for $$y$$","text":"To solve for $$y$$, we simply need to divide both sides of the equation by $$4$$, which gives us $$y=15$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8583b4FracDec24","title":"Solve Equations with Decimal Coefficients","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Equations with Fractions or Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8583b4FracDec24a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"Solve: $$2.1k+3=7.2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a8583b4FracDec24a-h1","type":"hint","dependencies":[],"title":"Isolating the variable","text":"The first step is to move all the terms with a variable to one side and all the constants to the other. To do this, we subtract $$3$$ from both sides of the equation, which gives us $$2.1k=4.2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec24a-h2","type":"hint","dependencies":["a8583b4FracDec24a-h1"],"title":"Clearing the decimal","text":"To clear the decimal, multiply both sides of the equation by some factor of $$10$$. In this case, we can mulitply the equation by $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec24a-h3","type":"hint","dependencies":["a8583b4FracDec24a-h2"],"title":"Equation Without Decimals","text":"Multiplying both sides of the equation by $$10$$, we get $$10\\\\times2.1k=10\\\\times4.2$$, which simplifies to $$21k=42$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec24a-h4","type":"hint","dependencies":["a8583b4FracDec24a-h3"],"title":"Solving for k","text":"To solve for k, we simply need to divide both sides of the equation by $$21$$, which gives us $$k=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8583b4FracDec25","title":"Solve Equations with Decimal Coefficients","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Equations with Fractions or Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8583b4FracDec25a","stepAnswer":["$$20$$"],"problemType":"TextBox","stepTitle":"Solve: $$0.7x+0.4=0.6x+2.4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$20$$","hints":{"DefaultPathway":[{"id":"a8583b4FracDec25a-h1","type":"hint","dependencies":[],"title":"Isolating the variable","text":"The first step is to move all the terms with a variable to one side and all the constants to the other. To do this, we can subtract $$0.6x$$ and $$0.4$$ from both sides, which gives us the equation $$0.1x=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec25a-h2","type":"hint","dependencies":["a8583b4FracDec25a-h1"],"title":"Clearing the decimal","text":"To clear the decimal, multiply both sides of the equation by some factor of $$10$$. In this case, we can mulitply the equation by $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec25a-h3","type":"hint","dependencies":["a8583b4FracDec25a-h2"],"title":"Equation Without Decimals","text":"Multiplying both sides of the equation by $$10$$, we get $$10\\\\times0.1x=10\\\\times2$$, which simplifies to $$x=20$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8583b4FracDec26","title":"Solve Equations with Decimal Coefficients","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Equations with Fractions or Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8583b4FracDec26a","stepAnswer":["$$22$$"],"problemType":"TextBox","stepTitle":"$$0.48x+1.56=0.58x-0.64$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$22$$","hints":{"DefaultPathway":[{"id":"a8583b4FracDec26a-h1","type":"hint","dependencies":[],"title":"Isolating the variable","text":"The first step is to move all the terms with a variable to one side and all the constants to the other. To do this, we can subtract $$0.48x$$ and add $$0.64$$ to both sides, which gives us the equation $$0.1x=2.2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec26a-h2","type":"hint","dependencies":[],"title":"Clearing the decimal","text":"To clear the decimal, multiply both sides of the equation by some factor of $$10$$. In this case, we can mulitply the equation by $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec26a-h3","type":"hint","dependencies":[],"title":"Equation Without Decimals","text":"Multiplying both sides of the equation by $$10$$, we get $$10\\\\times0.1x=10\\\\times2.2$$, which simplifies to $$x=22$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8583b4FracDec27","title":"Solve Equations with Decimal Coefficients","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Equations with Fractions or Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8583b4FracDec27a","stepAnswer":["$$8$$"],"problemType":"TextBox","stepTitle":"$$1.2x-0.91=0.8x+2.29$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8$$","hints":{"DefaultPathway":[{"id":"a8583b4FracDec27a-h1","type":"hint","dependencies":[],"title":"Isolating the variable","text":"The first step is to move all the terms with a variable to one side and all the constants to the other. To do this, we can subtract $$0.8x$$ and add $$0.91$$ to both sides, which gives us the equation $$0.4x=3.2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec27a-h2","type":"hint","dependencies":["a8583b4FracDec27a-h1"],"title":"Clearing the decimal","text":"To clear the decimal, multiply both sides of the equation by some factor of $$10$$. In this case, we can mulitply the equation by $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec27a-h3","type":"hint","dependencies":["a8583b4FracDec27a-h2"],"title":"Equation Without Decimals","text":"Multiplying both sides of the equation by $$10$$, we get $$10\\\\times0.4x=10\\\\times3.2$$, which simplifies to $$4x=32$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec27a-h4","type":"hint","dependencies":["a8583b4FracDec27a-h3"],"title":"Solving for $$x$$","text":"To solve for $$x$$, we simply need to divide both sides of the equation by $$4$$, which gives us $$x=8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8583b4FracDec28","title":"Solve Equations with Decimal Coefficients","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Equations with Fractions or Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8583b4FracDec28a","stepAnswer":["$$19$$"],"problemType":"TextBox","stepTitle":"Solve: $$0.05n+\\\\operatorname{0.1}\\\\left(n+7\\\\right)=3.55$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$19$$","hints":{"DefaultPathway":[{"id":"a8583b4FracDec28a-h1","type":"hint","dependencies":[],"title":"Distributive Property","text":"The first step is to use distributive property to clear the parantheses.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec28a-h2","type":"hint","dependencies":["a8583b4FracDec28a-h1"],"title":"Definition of Distributive Property","text":"Distributive Property is the property such that $$a \\\\left(b+c\\\\right)=a b+a c$$, where a,b, and c are constants","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec28a-h3","type":"hint","dependencies":["a8583b4FracDec28a-h2"],"title":"Applying the Distributive Property","text":"Applying the distributive property, we get the equation $$0.05n+0.1n+0.7=3.55$$, which simplifies to $$0.15n+0.7=3.55$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec28a-h4","type":"hint","dependencies":["a8583b4FracDec28a-h3"],"title":"Isolating the variable","text":"The first step is to move all the terms with a variable to one side and all the constants to the other. To do this, we can subtract $$0.7$$ from both sides, which gives us the equation $$0.15n=2.85$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec28a-h5","type":"hint","dependencies":["a8583b4FracDec28a-h4"],"title":"Clearing the decimal","text":"To clear the decimal, multiply both sides of the equation by some factor of $$10$$. In this case, we can mulitply the equation by $$100$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec28a-h6","type":"hint","dependencies":["a8583b4FracDec28a-h5"],"title":"Equation Without Decimals","text":"Multiplying both sides of the equation by $$100$$, we get $$100\\\\times0.15n=100\\\\times2.85$$, which simplifies to $$15n=285$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec28a-h7","type":"hint","dependencies":["a8583b4FracDec28a-h6"],"title":"Solving for $$n$$","text":"To solve for $$n$$, we simply need to divide both sides of the equation by $$15$$, which gives us $$n=19$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8583b4FracDec29","title":"Solve Equations with Decimal Coefficients","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Equations with Fractions or Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8583b4FracDec29a","stepAnswer":["$$10$$"],"problemType":"TextBox","stepTitle":"Solve: $$0.1d+\\\\operatorname{0.25}\\\\left(d+7\\\\right)=5.25$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10$$","hints":{"DefaultPathway":[{"id":"a8583b4FracDec29a-h1","type":"hint","dependencies":[],"title":"Distributive Property","text":"The first step is to use distributive property to clear the parantheses.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec29a-h2","type":"hint","dependencies":["a8583b4FracDec29a-h1"],"title":"Definition of Distributive Property","text":"Distributive Property is the property such that $$a \\\\left(b+c\\\\right)=a b+a c$$, where a,b, and c are constants","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec29a-h3","type":"hint","dependencies":["a8583b4FracDec29a-h2"],"title":"Applying the Distributive Property","text":"Applying the distributive property, we get the equation $$0.1d+0.25d+0.25\\\\times7=5.25$$, which simplifies to $$0.35d+1.75=5.25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec29a-h4","type":"hint","dependencies":["a8583b4FracDec29a-h3"],"title":"Isolating the variable","text":"The first step is to move all the terms with a variable to one side and all the constants to the other. To do this, we can subtract $$1.75$$ from both sides, which gives us the equation $$0.35d=3.5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec29a-h5","type":"hint","dependencies":["a8583b4FracDec29a-h4"],"title":"Clearing the decimal","text":"To clear the decimal, multiply both sides of the equation by some factor of $$10$$. In this case, we can mulitply the equation by $$100$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec29a-h6","type":"hint","dependencies":["a8583b4FracDec29a-h5"],"title":"Equation Without Decimals","text":"Multiplying both sides of the equation by $$100$$, we get $$100\\\\times0.35d=100\\\\times3.5$$, which simplifies to $$35d=350$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec29a-h7","type":"hint","dependencies":["a8583b4FracDec29a-h6"],"title":"Solving for $$d$$","text":"To solve for $$d$$, we simply need to divide both sides of the equation by $$35$$, which gives us $$d=10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8583b4FracDec3","title":"How to Solve Equations with Fraction Coefficients","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Equations with Fractions or Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8583b4FracDec3a","stepAnswer":["$$-2$$"],"problemType":"TextBox","stepTitle":"Solve: $$\\\\frac{1}{8} x+\\\\frac{1}{2}=\\\\frac{1}{4}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-2$$","hints":{"DefaultPathway":[{"id":"a8583b4FracDec3a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":[],"title":"Getting Rid of Fraction Denominators","text":"What is the LCD (largest common denominator) of $$\\\\frac{1}{2}$$, $$\\\\frac{1}{8}$$, and $$\\\\frac{1}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec3a-h2","type":"hint","dependencies":["a8583b4FracDec3a-h1"],"title":"Getting Rid of Fraction Denominators","text":"We must now multiply both sides of the equation by the LCD and use the distributive property. We now have $$8\\\\frac{1}{8} x+\\\\frac{8\\\\times1}{2}=\\\\frac{8\\\\times1}{4}$$. This simplifies to $$x+4=2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec3a-h3","type":"hint","dependencies":["a8583b4FracDec3a-h2"],"title":"Solving the New Equation","text":"We must now isolate the \\"x\\" term in the equation to solve.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a8583b4FracDec3a-h3"],"title":"Solving the New Equation","text":"We can isolate $$x$$ by subtracting $$4$$ from each side of the equation. What is the final value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8583b4FracDec30","title":"Solve Equations with Decimal Coefficients","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Equations with Fractions or Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8583b4FracDec30a","stepAnswer":["$$15$$"],"problemType":"TextBox","stepTitle":"Solve: $$\\\\operatorname{0.05}\\\\left(q-8\\\\right)+0.25q=4.10$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$15$$","hints":{"DefaultPathway":[{"id":"a8583b4FracDec30a-h1","type":"hint","dependencies":[],"title":"Distributive Property","text":"The first step is to use distributive property to clear the parantheses.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec30a-h2","type":"hint","dependencies":["a8583b4FracDec30a-h1"],"title":"Definition of Distributive Property","text":"Distributive Property is the property such that $$a \\\\left(b+c\\\\right)=a b+a c$$, where a,b, and c are constants","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec30a-h3","type":"hint","dependencies":["a8583b4FracDec30a-h2"],"title":"Applying the Distributive Property","text":"Applying the distributive property, we get the equation $$0.05q-0.05\\\\times8+0.25q=4.10$$, which simplifies to $$0.3q-0.4=4.1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec30a-h4","type":"hint","dependencies":["a8583b4FracDec30a-h3"],"title":"Isolating the variable","text":"The first step is to move all the terms with a variable to one side and all the constants to the other. To do this, we can add $$0.4$$ to both sides, which gives us the equation $$0.3q=4.5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec30a-h5","type":"hint","dependencies":["a8583b4FracDec30a-h4"],"title":"Clearing the decimal","text":"To clear the decimal, multiply both sides of the equation by some factor of $$10$$. In this case, we can mulitply the equation by $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec30a-h6","type":"hint","dependencies":["a8583b4FracDec30a-h5"],"title":"Equation Without Decimals","text":"Multiplying both sides of the equation by $$10$$, we get $$10\\\\times0.3q=10\\\\times4.5$$, which simplifies to $$3q=45$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec30a-h7","type":"hint","dependencies":["a8583b4FracDec30a-h6"],"title":"Solving for q","text":"To solve for q, we simply need to divide both sides of the equation by $$3$$, which gives us $$q=15$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8583b4FracDec4","title":"How to Solve Equations with Fraction Coefficients","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Equations with Fractions or Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8583b4FracDec4a","stepAnswer":["$$40$$"],"problemType":"TextBox","stepTitle":"Solve: $$6=\\\\frac{1}{2} v+\\\\frac{2}{5} v-\\\\frac{3}{4} v$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$40$$","hints":{"DefaultPathway":[{"id":"a8583b4FracDec4a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":[],"title":"Getting Rid of Fraction Denominators","text":"What is the LCD (largest common denominator) of $$\\\\frac{1}{2}$$, $$\\\\frac{2}{5}$$, and $$\\\\frac{-3}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec4a-h2","type":"hint","dependencies":["a8583b4FracDec4a-h1"],"title":"Getting Rid of Fraction Denominators","text":"We must now multiply both sides of the equation by the LCD to get rid of the fractions. We now have $$6\\\\times20=\\\\operatorname{20}\\\\left(\\\\frac{1}{2} v\\\\right)+\\\\operatorname{20}\\\\left(\\\\frac{2}{5} v\\\\right)-\\\\operatorname{20}\\\\left(\\\\frac{3}{4} v\\\\right)$$. This simplifies to $$120=3v$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec4a-h3","type":"hint","dependencies":["a8583b4FracDec4a-h2"],"title":"Solving the New Equation","text":"To solve this equation for v, we must first isolate the \\"v\\" term in the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$40$$"],"dependencies":["a8583b4FracDec4a-h3"],"title":"Solving the New Equation","text":"We can isolate v by dividing both sides of the equation by $$3$$. What is the value of v?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8583b4FracDec5","title":"How to Solve Equations with Fraction Coefficients","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Equations with Fractions or Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8583b4FracDec5a","stepAnswer":["$$12$$"],"problemType":"TextBox","stepTitle":"Solve: $$7=\\\\frac{1}{2} x+\\\\frac{3}{4} x-\\\\frac{2}{3} x$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12$$","hints":{"DefaultPathway":[{"id":"a8583b4FracDec5a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":[],"title":"Getting Rid of Fraction Denominators","text":"What is the LCD (largest common denominator) of $$\\\\frac{1}{2}$$, $$\\\\frac{3}{4}$$, and $$\\\\frac{-2}{3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec5a-h2","type":"hint","dependencies":["a8583b4FracDec5a-h1"],"title":"Getting Rid of Fraction Denominators","text":"We must now multiply both sides of the equation by the LCD. We get: $$12\\\\times7=\\\\frac{12\\\\times1}{2} x+\\\\frac{12\\\\times3}{4} x-\\\\frac{12\\\\times2}{3} x$$. This simplifies to $$7x=84$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec5a-h3","type":"hint","dependencies":["a8583b4FracDec5a-h2"],"title":"Solving the New Equation","text":"To solve this equation, we must now isolate the \\"x\\" term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a8583b4FracDec5a-h3"],"title":"Solving the New Equation","text":"To isolate $$x$$, we must divide both sides of the equation by $$7$$. What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8583b4FracDec6","title":"How to Solve Equations with Fraction Coefficients","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Equations with Fractions or Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8583b4FracDec6a","stepAnswer":["$$-12$$"],"problemType":"TextBox","stepTitle":"Solve $$-1=\\\\frac{1}{2} u+\\\\frac{1}{4} u-\\\\frac{2}{3} u$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-12$$","hints":{"DefaultPathway":[{"id":"a8583b4FracDec6a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":[],"title":"Getting Rid of Fraction Denominators","text":"What is the LCD (largest common denominator) of $$\\\\frac{1}{2}$$, $$\\\\frac{1}{4}$$, and $$\\\\frac{-2}{3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec6a-h2","type":"hint","dependencies":["a8583b4FracDec6a-h1"],"title":"Getting Rid of Fraction Denominators","text":"We must now multiply both sides of the equation by the LCD. We get $$-1\\\\times12=\\\\frac{12\\\\times1}{2} u+\\\\frac{12\\\\times1}{4} u-12\\\\frac{2}{3} u$$. This simplifies to $$6u+3u-8u=-12$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-12$$"],"dependencies":["a8583b4FracDec6a-h2"],"title":"Solving the New Equation","text":"To solve this equation, all we have to do is simplify the left-hand side. We now have $$u=-12$$. What is the value of u?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8583b4FracDec7","title":"How to Solve Equations with Fraction Coefficients","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Equations with Fractions or Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8583b4FracDec7a","stepAnswer":["$$-2$$"],"problemType":"TextBox","stepTitle":"Solve: $$a+\\\\frac{3}{4}=\\\\frac{3}{8} a-\\\\frac{1}{2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-2$$","hints":{"DefaultPathway":[{"id":"a8583b4FracDec7a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":[],"title":"Getting Rid of Fraction Denominators","text":"What is the LCD (largest common denominator) of $$\\\\frac{3}{4}$$, $$\\\\frac{3}{8}$$, and $$\\\\frac{-1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec7a-h2","type":"hint","dependencies":["a8583b4FracDec7a-h1"],"title":"Getting Rid of Fraction Denominators","text":"We must now multiply both sides of the equation by the LCD. We now have $$8a+\\\\frac{8\\\\times3}{4}=\\\\frac{8\\\\times3}{8} a-\\\\frac{8\\\\times1}{2}$$. This simplifies to $$8a+6=3a-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec7a-h3","type":"hint","dependencies":["a8583b4FracDec7a-h2"],"title":"Solving the New Equation","text":"To solve this equation, we must isolate the \\"a\\" term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec7a-h4","type":"hint","dependencies":["a8583b4FracDec7a-h3"],"title":"Solving the New Equation","text":"To isolate the \\"a\\" term, we can begin by subtracting 3a from both sides of the equation. We now have $$5a+6=-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec7a-h5","type":"hint","dependencies":["a8583b4FracDec7a-h4"],"title":"Solving the New Equation","text":"Now, we must subtract $$6$$ from both sides so that only the \\"a\\" term is on the left-hand side. We now get $$5a=-10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec7a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a8583b4FracDec7a-h5"],"title":"Solving the New Equation","text":"Finally, we can divide both sides by $$5$$ to get the \\"a\\" term by itself on the left-hand side. What is the value of a?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8583b4FracDec8","title":"How to Solve Equations with Fraction Coefficients","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Equations with Fractions or Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8583b4FracDec8a","stepAnswer":["$$-1$$"],"problemType":"TextBox","stepTitle":"Solve: $$x+\\\\frac{1}{3}=\\\\frac{1}{6} x-\\\\frac{1}{2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1$$","hints":{"DefaultPathway":[{"id":"a8583b4FracDec8a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":[],"title":"Getting Rid of Fraction Denominators","text":"What is the LCD (largest common denominator) of $$\\\\frac{1}{3}$$, $$\\\\frac{1}{6}$$, and $$\\\\frac{-1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec8a-h2","type":"hint","dependencies":["a8583b4FracDec8a-h1"],"title":"Getting Rid of Fraction Denominators","text":"We must now multiply both sides of the equation by the LCD. We now have $$6x+2=x-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec8a-h3","type":"hint","dependencies":["a8583b4FracDec8a-h2"],"title":"Solving the New Equation","text":"To solve this equation for $$x$$, we must isolate all \\"x\\" terms on one side of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec8a-h4","type":"hint","dependencies":["a8583b4FracDec8a-h3"],"title":"Solving the New Equation","text":"We can subtract $$x$$ from both sides of the equation. We now have $$5x+2=-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec8a-h5","type":"hint","dependencies":["a8583b4FracDec8a-h4"],"title":"Solving the New Equation","text":"Now, we must subtract $$2$$ from both sides of the equation so that $$x$$ term is by itself. We now have $$5x=-5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec8a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a8583b4FracDec8a-h5"],"title":"Solving the New Equation","text":"Finally, we must divide by $$5$$ on both sides of the equation so we can completely isolate $$x$$. What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8583b4FracDec9","title":"How to Solve Equations with Fraction Coefficients","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Equations with Fractions or Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8583b4FracDec9a","stepAnswer":["$$-2$$"],"problemType":"TextBox","stepTitle":"Solve: $$c+\\\\frac{3}{4}=\\\\frac{1}{2} c-\\\\frac{1}{4}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-2$$","hints":{"DefaultPathway":[{"id":"a8583b4FracDec9a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":[],"title":"Getting Rid of Fraction Denominators","text":"What is the LCD (largest common denominator) of $$\\\\frac{3}{4}$$, $$\\\\frac{1}{2}$$, and $$\\\\frac{-1}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec9a-h2","type":"hint","dependencies":["a8583b4FracDec9a-h1"],"title":"Getting Rid of Fraction Denominators","text":"We must now multiply both sides of the equation by the LCD. We now have $$4c+\\\\frac{4\\\\times3}{4}=\\\\frac{4\\\\times1}{2} c-\\\\frac{4\\\\times1}{4}$$. This simplifies to $$4c+3=2c-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec9a-h3","type":"hint","dependencies":["a8583b4FracDec9a-h2"],"title":"Solving the New Equation","text":"To solve this equation, we must first isolate the \\"c\\" term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec9a-h4","type":"hint","dependencies":["a8583b4FracDec9a-h3"],"title":"Solving the New Equation","text":"We can subtract 2c from both sides of the equation. We now have $$2c+3=-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec9a-h5","type":"hint","dependencies":["a8583b4FracDec9a-h4"],"title":"Solving the New Equation","text":"Now, we can subtract $$3$$ from both sides of the equation. We have $$2c=-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8583b4FracDec9a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a8583b4FracDec9a-h5"],"title":"Solving the New Equation","text":"Finally, we can divide both sides of the equation by $$2$$. What is the value of c?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a870b02DivMul1","title":"Solve Equations Using the Division Property of Equality","body":"Solve the equation for the variable.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Solve Equations using the Division and Multiplication Properties of Equality","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a870b02DivMul1a","stepAnswer":["$$\\\\frac{-27}{5}$$"],"problemType":"TextBox","stepTitle":"$$5x=-27$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-27}{5}$$","hints":{"DefaultPathway":[{"id":"a870b02DivMul1a-h1","type":"hint","dependencies":[],"title":"Division property of equality","text":"When you divide both sides of an equation by any non-zero number, you still have equality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{5} x=\\\\frac{-27}{5}$$"],"dependencies":["a870b02DivMul1a-h1"],"title":"Division","text":"Divide $$5$$ from each side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-27}{5}$$"],"dependencies":["a870b02DivMul1a-h2"],"title":"Simplification","text":"What do we get for $$x$$ after simplifying the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul1a-h4","type":"hint","dependencies":["a870b02DivMul1a-h3"],"title":"Verification","text":"Check whether the result is a solution of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul1a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a870b02DivMul1a-h4"],"title":"Verification","text":"Check whether $$5\\\\left(-\\\\frac{27}{5}\\\\right)$$ equals $$-27$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]}]}}]},{"id":"a870b02DivMul10","title":"Solve the equation for the variable.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["a870b02DivMul10a-h2"],"title":"Simplification","text":"What do we get for $$x$$ after simplifying the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul10a-h4","type":"hint","dependencies":["a870b02DivMul10a-h3"],"title":"Verification","text":"Check whether the result is a solution of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul16f-h4","type":"hint","dependencies":["a870b02DivMul16f-h3"],"title":"Verification","text":"Check whether the result is a solution of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul16f-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a870b02DivMul16f-h4"],"title":"Verification","text":"Check whether $$-13\\\\times0$$ equals $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]}]}},{"id":"a870b02DivMul16h","stepAnswer":["$$100$$"],"problemType":"TextBox","stepTitle":"$$-20=\\\\frac{q}{\\\\left(-5\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$100$$","hints":{"DefaultPathway":[{"id":"a870b02DivMul16h-h1","type":"hint","dependencies":[],"title":"Multiplication property of equality","text":"If you multiply both sides of an equation by the same number, you still have equality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul16h-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(-20\\\\right) \\\\left(-5\\\\right)=\\\\frac{q}{\\\\left(-5\\\\right)} \\\\left(-5\\\\right)$$"],"dependencies":["a870b02DivMul16h-h1"],"title":"Multiplication","text":"Multiple $$-5$$ from each side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul16h-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$100$$"],"dependencies":["a870b02DivMul16h-h2"],"title":"Simplification","text":"What do we get for q after simplifying the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul16h-h4","type":"hint","dependencies":["a870b02DivMul16h-h3"],"title":"Verification","text":"Check whether the result is a solution of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul16h-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["TRUE"],"dependencies":["a870b02DivMul16h-h4"],"title":"Verification","text":"Check whether $$-20$$ equals $$\\\\frac{100}{\\\\left(-5\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]}]}},{"id":"a870b02DivMul16i","stepAnswer":["$$-144$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{y}{9}=-16$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-144$$","hints":{"DefaultPathway":[{"id":"a870b02DivMul16i-h1","type":"hint","dependencies":[],"title":"Multiplication property of equality","text":"If you multiply both sides of an equation by the same number, you still have equality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul16i-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a870b02DivMul16i-h4"],"title":"Verification","text":"Check whether $$\\\\frac{-144}{9}$$ equals $$-16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]}]}},{"id":"a870b02DivMul16j","stepAnswer":["$$-540$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{m}{\\\\left(-12\\\\right)}=45$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-540$$","hints":{"DefaultPathway":[{"id":"a870b02DivMul16j-h1","type":"hint","dependencies":[],"title":"Multiplication property of equality","text":"If you multiply both sides of an equation by the same number, you still have equality.","variabilization":{},"oer":"https://OATutor.io 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4.0>"},{"id":"a870b02DivMul16j-h4","type":"hint","dependencies":["a870b02DivMul16j-h3"],"title":"Verification","text":"Check whether the result is a solution of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul16j-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a870b02DivMul16j-h4"],"title":"Verification","text":"Check whether $$\\\\frac{-540}{\\\\left(-12\\\\right)}$$ equals $$45$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]}]}},{"id":"a870b02DivMul16l","stepAnswer":["$$72$$"],"problemType":"TextBox","stepTitle":"$$-v=-72$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$72$$","hints":{"DefaultPathway":[{"id":"a870b02DivMul16l-h1","type":"hint","dependencies":[],"title":"Multiplication property of equality","text":"If you multiply both sides of an equation by the same number, you still have equality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul16l-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(-v\\\\right) \\\\left(-1\\\\right)=\\\\left(-72\\\\right) \\\\left(-1\\\\right)$$"],"dependencies":["a870b02DivMul16l-h1"],"title":"Multiplication","text":"Multiple $$-1$$ from each side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul16l-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$72$$"],"dependencies":["a870b02DivMul16l-h2"],"title":"Simplification","text":"What do we get for v after simplifying the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul16l-h4","type":"hint","dependencies":["a870b02DivMul16l-h3"],"title":"Verification","text":"Check whether the result is a solution of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul16m-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a870b02DivMul16m-h4"],"title":"Verification","text":"Check whether $$72\\\\frac{2}{3}$$ equals $$48$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]}]}},{"id":"a870b02DivMul16n","stepAnswer":["$$-64$$"],"problemType":"TextBox","stepTitle":"$$-\\\\left(\\\\frac{5}{8}\\\\right) w=40$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-64$$","hints":{"DefaultPathway":[{"id":"a870b02DivMul16n-h1","type":"hint","dependencies":[],"title":"Multiplication property of equality","text":"If you multiply both sides of an equation by the same number, you still have 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For example, let $$c=the$$ cost of one pound.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul38a-h5","type":"hint","dependencies":["a870b02DivMul38a-h4"],"title":"Translation","text":"Translate into an equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul38a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6c=10.74$$"],"dependencies":["a870b02DivMul38a-h5"],"title":"Translation","text":"Write the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul38a-h7","type":"hint","dependencies":["a870b02DivMul38a-h6"],"title":"Division property of equality","text":"When you divide both sides of an equation by any non-zero number, you still have equality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul38a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$c=1.79$$"],"dependencies":["a870b02DivMul38a-h7"],"title":"Division","text":"Divide both sides by $$6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul38a-h9","type":"hint","dependencies":["a870b02DivMul38a-h8"],"title":"Verification","text":"Check whether the result is a solution of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul38a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a870b02DivMul38a-h9"],"title":"Verification","text":"Check whether $$6\\\\times1.79$$ equals $$10.74$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a870b02DivMul38a-h11","type":"hint","dependencies":["a870b02DivMul38a-h10"],"title":"Explanation","text":"Since $$c=1.79$$, the cost of $$1$$ pound of grapes is $$\\\\$1.79$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a870b02DivMul39","title":"Translate and Solve Applications","body":"Arianna bought a 24-pack of water bottles for $$\\\\$9.36$$.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Solve Equations using the Division and Multiplication Properties of Equality","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a870b02DivMul39a","stepAnswer":["$0.39"],"problemType":"TextBox","stepTitle":"What was the cost of one water bottle?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\$0.39$$","hints":{"DefaultPathway":[{"id":"a870b02DivMul39a-h1","type":"hint","dependencies":[],"title":"Translation","text":"We need to find what we are asked to find and which sentence gives the information to find it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul39a-h2","type":"hint","dependencies":["a870b02DivMul39a-h1"],"title":"Objective","text":"We are asked to find the cost of one water bottle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul39a-h3","type":"hint","dependencies":["a870b02DivMul39a-h2"],"title":"Information","text":"The cost of $$24$$ water bottles is $$\\\\$9.36$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul39a-h4","type":"hint","dependencies":["a870b02DivMul39a-h3"],"title":"Assignment","text":"Assign a variable to what we want to find. 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$$24\\\\times0.39$$ equals $$9.36$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a870b02DivMul39a-h11","type":"hint","dependencies":["a870b02DivMul39a-h10"],"title":"Explanation","text":"Since $$c=0.39$$, the cost of one water bottle is $$\\\\$0.39$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a870b02DivMul4","title":"Solve Equations Using the Multiplication Property of Equality","body":"Solve the equation for the variable.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Solve Equations using the Division and Multiplication Properties of Equality","courseName":"OpenStax: Elementary 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For example, let $$c=the$$ cost for each person.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul40a-h5","type":"hint","dependencies":["a870b02DivMul40a-h4"],"title":"Translation","text":"Translate into an equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul40a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6c=34.98$$"],"dependencies":["a870b02DivMul40a-h5"],"title":"Translation","text":"Write the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul40a-h7","type":"hint","dependencies":["a870b02DivMul40a-h6"],"title":"Division property of equality","text":"When you divide both sides of an equation by any non-zero number, you still have 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$$6\\\\times5.83$$ equals $$34.98$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"a870b02DivMul40a-h11","type":"hint","dependencies":["a870b02DivMul40a-h10"],"title":"Explanation","text":"Since $$c=5.83$$, the cost for each person is $$\\\\$5.83$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a870b02DivMul5","title":"Solve Equations Using the Multiplication Property of Equality","body":"Solve the equation","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Solve Equations using the Division and Multiplication Properties of Equality","courseName":"OpenStax: Elementary 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$294$$"],"dependencies":["a870b02DivMul5a-h2"],"title":"Simplification","text":"What do we get for a after simplifying the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul5a-h4","type":"hint","dependencies":["a870b02DivMul5a-h3"],"title":"Verification","text":"Check whether the result is a solution of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul5a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a870b02DivMul5a-h4"],"title":"Verification","text":"Check whether $$\\\\frac{294}{\\\\left(-7\\\\right)}$$ equals $$42$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]}]}}]},{"id":"a870b02DivMul6","title":"Solve Equations Using the Multiplication Property of Equality","body":"Solve the equation for the variable.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Solve Equations using the Division and Multiplication Properties of Equality","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a870b02DivMul6a","stepAnswer":["$$144$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{b}{\\\\left(-6\\\\right)}=-24$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$144$$","hints":{"DefaultPathway":[{"id":"a870b02DivMul6a-h1","type":"hint","dependencies":[],"title":"Multiplication property of equality","text":"If 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul6a-h4","type":"hint","dependencies":["a870b02DivMul6a-h3"],"title":"Verification","text":"Check whether the result is a solution of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul6a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a870b02DivMul6a-h4"],"title":"Verification","text":"Check whether $$\\\\frac{144}{\\\\left(-6\\\\right)}$$ equals $$-24$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]}]}}]},{"id":"a870b02DivMul7","title":"Solve Equations Using the Division Property of Equality","body":"Solve the equation for the 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any non-zero number, you still have equality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-n}{\\\\left(-1\\\\right)}=\\\\frac{\\\\left(-9\\\\right)}{\\\\left(-1\\\\right)}$$"],"dependencies":["a870b02DivMul7a-h2"],"title":"Division","text":"Divide $$-1$$ from each side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":["a870b02DivMul7a-h3"],"title":"Simplification","text":"What do we get for $$n$$ after simplifying the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul7a-h5","type":"hint","dependencies":["a870b02DivMul7a-h4"],"title":"Verification","text":"Check whether the result is a solution of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul7a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a870b02DivMul7a-h5"],"title":"Verification","text":"Check whether $$-(-9)$$ equals $$9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]}]}}]},{"id":"a870b02DivMul8","title":"Solve Equations Using the Division Property of Equality","body":"Solve the equation for the variable.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Solve Equations using the Division and Multiplication Properties of Equality","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a870b02DivMul8a","stepAnswer":["$$-8$$"],"problemType":"TextBox","stepTitle":"$$-k=8$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-8$$","hints":{"DefaultPathway":[{"id":"a870b02DivMul8a-h1","type":"hint","dependencies":[],"title":"Minus sign","text":"Remember -k is equivalent to $$\\\\left(-1\\\\right) k$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul8a-h2","type":"hint","dependencies":["a870b02DivMul8a-h1"],"title":"Division property of equality","text":"When you divide both sides of an equation by any non-zero number, you still have equality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-k}{\\\\left(-1\\\\right)}=\\\\frac{8}{\\\\left(-1\\\\right)}$$"],"dependencies":["a870b02DivMul8a-h2"],"title":"Division","text":"Divide $$-1$$ from each side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-8$$"],"dependencies":["a870b02DivMul8a-h3"],"title":"Simplification","text":"What do we get for k after simplifying the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul8a-h5","type":"hint","dependencies":["a870b02DivMul8a-h4"],"title":"Verification","text":"Check whether the result is a solution of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul8a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a870b02DivMul8a-h5"],"title":"Verification","text":"Check whether $$-(-8)$$ equals $$8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]}]}}]},{"id":"a870b02DivMul9","title":"Solve Equations Using the Division Property of Equality","body":"Solve the equation for the variable.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Solve Equations using the Division and Multiplication Properties of Equality","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a870b02DivMul9a","stepAnswer":["$$-3$$"],"problemType":"TextBox","stepTitle":"$$-g=3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-3$$","hints":{"DefaultPathway":[{"id":"a870b02DivMul9a-h1","type":"hint","dependencies":[],"title":"Minus sign","text":"Remember -g is equivalent to $$\\\\left(-1\\\\right) g$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul9a-h2","type":"hint","dependencies":["a870b02DivMul9a-h1"],"title":"Division property of equality","text":"When you divide both sides of an equation by any non-zero number, you still have equality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-g}{\\\\left(-1\\\\right)}=\\\\frac{3}{\\\\left(-1\\\\right)}$$"],"dependencies":["a870b02DivMul9a-h2"],"title":"Division","text":"Divide $$-1$$ from each side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a870b02DivMul9a-h3"],"title":"Simplification","text":"What do we get for g after simplifying the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul9a-h5","type":"hint","dependencies":["a870b02DivMul9a-h4"],"title":"Verification","text":"Check whether the result is a solution of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a870b02DivMul9a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a870b02DivMul9a-h5"],"title":"Verification","text":"Check whether $$-(-3)$$ equals $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]}]}}]},{"id":"a8802f1std1","title":"Two Population Means with Unknown Standard Deviations","body":"The average amount of time boys and girls aged seven to $$11$$ spend playing sports each day is believed to be the same. A study is done and data are collected, resulting in the data in the table. Each population has a normal distribution.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Two Population Means with Unknown Standard Deviations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a8802f1std1a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Is there a difference in the mean amount of time boys and girls aged seven to $$11$$ play sports each day? Test at the 5% level of significance.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a8802f1std1a-h1","type":"hint","dependencies":[],"title":"Representations","text":"The population standard deviations are not known. Let g be the subscript for girls and $$b$$ be the subscript for boys such that $$\u03bc_g$$ is the population mean for girls and $$\u03bc_b$$ is the population mean for boys. This is a test of two independent groups, two population means.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std1a-h2","type":"hint","dependencies":["a8802f1std1a-h1"],"title":"Random Variable","text":"Two Population Means with Unknown Standard Deviations, Degrees of Freedom, T-Score, Standard Error","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std1a-h3","type":"hint","dependencies":["a8802f1std1a-h2"],"title":"Null and Alternative Hypotheses","text":"$$H_0$$ is the null hypothesis and $$H_a$$ is the alternative hypothesis. $$H_0$$: $$\u03bc_g$$ $$=$$ $$\u03bc_b$$ where $$\u03bc_g$$ - $$\u03bc_b$$ $$=$$ $$0$$ and $$H_a$$: $$\u03bc_g$$ $$ \\\\neq $$ $$\u03bc_b$$ where $$H_a$$: $$\u03bc_g$$ $$=$$ $$\u03bc_b$$ $$ \\\\neq $$ $$0$$. When the problems says that the means are the same, this tells you that $$H_0$$ has an equals sign. Since there are no other words to indicate $$H_a$$, assume it says that the means are different. This is a two-tailed test.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std1a-h4","type":"hint","dependencies":["a8802f1std1a-h3"],"title":"Distribution for the Test","text":"Solve for $$t_{df}$$ (the number of degrees of freedom) where df is calculated using the df formula for independent groups, two population means. Use a calculator to solve for df. Do not pool the variances.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std1a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18.8462$$"],"dependencies":["a8802f1std1a-h4"],"title":"Solving for df","text":"Using a calculator, what is df? Round to the four decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std1a-h6","type":"hint","dependencies":["a8802f1std1a-h5"],"title":"P-Value","text":"Calculate the $$p-value$$ using a student\'s $$t-distribution$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std1a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.0054$$"],"dependencies":["a8802f1std1a-h6"],"title":"Solving for the $$p-value$$","text":"What is the $$p-value$$?\\\\n##figure3.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std1a-h8","type":"hint","dependencies":["a8802f1std1a-h7"],"title":"$$s_g$$ and $$s_b$$","text":"Solve for $$s_g$$ and $$s_b$$, the sample standard deviations and estimates of $$\\\\sigma_g$$ and $$\\\\sigma_b$$ (the unknown population standard deviations), respectively. When both sample sizes n1 and n2 are five or larger, the Student\'s $$t$$ approximation is very good. Notice that the sample variances $${s1}^2$$ and $${s2}^2$$ are not pooled.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std1a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.866$$"],"dependencies":["a8802f1std1a-h8"],"title":"$$s_g$$","text":"What is $$s_g$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std1a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a8802f1std1a-h8"],"title":"$$s_b$$","text":"What is $$s_b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std1a-h11","type":"hint","dependencies":["a8802f1std1a-h9","a8802f1std1a-h10"],"title":"$$x\u0304_g$$ - $$x\u0304_b$$","text":"Solve for $$x\u0304_g$$ - $$x\u0304_b$$, the difference of the sample means.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std1a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a8802f1std1a-h11"],"title":"Solving $$x\u0304_g$$","text":"What is $$x\u0304_g$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std1a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3.2$$"],"dependencies":["a8802f1std1a-h12"],"title":"Solving $$x\u0304_b$$","text":"What is $$x\u0304_b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std1a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1.2$$"],"dependencies":["a8802f1std1a-h13"],"title":"Plugging Into $$x\u0304_g$$ - $$x\u0304_b$$","text":"Plugging your values into the equation, what is $$x\u0304_g$$ - $$x\u0304_b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std1a-h15","type":"hint","dependencies":["a8802f1std1a-h14"],"title":"Rejecting or Accepting $$H_0$$","text":"If \ud835\udefc > $$p-value$$, reject $$H_0$$. This means that you would be rejecting $$\u03bc_g$$ $$=$$ $$\u03bc_b$$ such that the means are different. If \ud835\udefc $$ \\\\leq $$ $$p-value$$, accept $$H_0$$ such that $$\u03bc_g$$ $$=$$ $$\u03bc_b$$, and the means are the same.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std1a-h16","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a8802f1std1a-h15"],"title":"Null Hypothesis","text":"Is the null hypothesis $$H_0$$ rejected?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a8802f1std1a-h17","type":"hint","dependencies":["a8802f1std1a-h16"],"title":"Understanding Null Hypothesis","text":"Since \ud835\udefc > $$p-value$$, reject $$H_0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8802f1std10","title":"Two Population Means with Unknown Standard Deviations","body":"A professor at a large community college wanted to determine whether there is a difference in the means of final exam scores between students who took his statistics course online and the students who took his face-to-face statistics class. He believed that the mean of the final exam scores for the online class would be lower than that of the face-to-face class. The randomly selected $$30$$ final exam scores from each group are listed in Table $$10.3$$ and Table $$10.4$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Two Population Means with Unknown Standard Deviations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a8802f1std10a","stepAnswer":["Means"],"problemType":"MultipleChoice","stepTitle":"Is this a test of two means or two proportions?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Means","Proportions"],"hints":{"DefaultPathway":[{"id":"a8802f1std10a-h1","type":"hint","dependencies":[],"title":"Mean or Proportion","text":"Mean is the average value in a sample. Proportion is the number of observations of a characteristic in comparison to the entire sample.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std10a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Means"],"dependencies":["a8802f1std10a-h1"],"title":"Understanding the Problem","text":"When reading the problem, is the study comparing the difference in means (averages) of the math classes or difference in proportions of final exam scores?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Means","Proportions"]}]}}]},{"id":"a8802f1std11","title":"Two Population Means with Unknown Standard Deviations","body":"A professor at a large community college wanted to determine whether there is a difference in the means of final exam scores between students who took his statistics course online and the students who took his face-to-face statistics class. He believed that the mean of the final exam scores for the online class would be lower than that of the face-to-face class. The randomly selected $$30$$ final exam scores from each group are listed in Table $$10.3$$ and Table $$10.4$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Two Population Means with Unknown Standard Deviations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a8802f1std11a","stepAnswer":["Unknown"],"problemType":"MultipleChoice","stepTitle":"Are the populations\' standard deviations known or unknown?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Known","Unknown"],"hints":{"DefaultPathway":[{"id":"a8802f1std11a-h1","type":"hint","dependencies":[],"title":"Unknown or Known","text":"Examine the problem to determine whether or not the standard deviations of the populations were stated.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std11a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a8802f1std11a-h1"],"title":"Sample or Population","text":"Does the problem state the populations\' standard deviations anywhere?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a8802f1std11a-h3","type":"hint","dependencies":["a8802f1std11a-h2"],"title":"Standard Deviations","text":"If the standard deviations were never stated, the standard deviations of the populations are unknown.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8802f1std12","title":"Two Population Means with Unknown Standard Deviations","body":"A professor at a large community college wanted to determine whether there is a difference in the means of final exam scores between students who took his statistics course online and the students who took his face-to-face statistics class. He believed that the mean of the final exam scores for the online class would be lower than that of the face-to-face class. The randomly selected $$30$$ final exam scores from each group are listed in Table $$10.3$$ and Table $$10.4$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Two Population Means with Unknown Standard Deviations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a8802f1std12a","stepAnswer":["Student\'s $$t$$"],"problemType":"MultipleChoice","stepTitle":"Which distribution do you use to perform the test?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"Student\'s $$t$$","choices":["Student\'s $$t$$","$$P-Value$$","$$x\u0304_A$$ $$-$$ $$x\u0304_B$$"],"hints":{"DefaultPathway":[{"id":"a8802f1std12a-h1","type":"hint","dependencies":[],"title":"Walking Through Each Choice","text":"$$x\u0304_A$$ - $$x\u0304_B$$ $$=$$ difference in the sample mean amount of time girls and boys play (also known as the random variable), $$p-value$$ $$=$$ how likely the data occurred under the null hypothesis, and $$t_{df}$$ $$=$$ distribution for the test.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std12a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Student\'s $$t$$"],"dependencies":["a8802f1std12a-h1"],"title":"Distribution","text":"Which of the following solves for the distribution of the test: Student\'s $$t$$, $$p-value$$, or $$x\u0304_A$$ - $$x\u0304_B$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Student\'s $$t$$","$$P-Value$$","$$x\u0304_A$$ $$-$$ $$x\u0304_B$$"]}]}}]},{"id":"a8802f1std13","title":"Two Population Means with Unknown Standard Deviations","body":"A professor at a large community college wanted to determine whether there is a difference in the means of final exam scores between students who took his statistics course online and the students who took his face-to-face statistics class. He believed that the mean of the final exam scores for the online class would be lower than that of the face-to-face class. The randomly selected $$30$$ final exam scores from each group are listed in Table $$10.3$$ and Table $$10.4$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Two Population Means with Unknown Standard Deviations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a8802f1std13a","stepAnswer":["$$x\u0304_A$$ $$-$$ $$x\u0304_B$$"],"problemType":"MultipleChoice","stepTitle":"What is the random variable?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$x\u0304_A$$ - $$x\u0304_B$$","choices":["Student\'s $$t$$","$$P-Value$$","$$x\u0304_A$$ $$-$$ $$x\u0304_B$$"],"hints":{"DefaultPathway":[{"id":"a8802f1std13a-h1","type":"hint","dependencies":[],"title":"Walking Through Each Choice","text":"$$x\u0304_A$$ - $$x\u0304_B$$ $$=$$ difference in the sample mean amount of time girls and boys play (also known as the random variable), $$p-value$$ $$=$$ how likely the data occurred under the null hypothesis, and $$t_{df}$$ $$=$$ distribution for the test.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std13a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x\u0304_A$$ $$-$$ $$x\u0304_B$$"],"dependencies":["a8802f1std13a-h1"],"title":"Random Variable","text":"Which of the following solves for the random variable: Student\'s $$t$$, $$p-value$$, or $$x\u0304_A$$ - $$x\u0304_B$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Student\'s $$t$$","$$P-Value$$","$$x\u0304_A$$ $$-$$ $$x\u0304_B$$"]}]}}]},{"id":"a8802f1std14","title":"Two Population Means with Unknown Standard Deviations","body":"A professor at a large community college wanted to determine whether there is a difference in the means of final exam scores between students who took his statistics course online and the students who took his face-to-face statistics class. He believed that the mean of the final exam scores for the online class would be lower than that of the face-to-face class. The randomly selected $$30$$ final exam scores from each group are listed in Table $$10.3$$ and Table $$10.4$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Two Population Means with Unknown Standard Deviations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a8802f1std14a","stepAnswer":["$$\u03bc_1$$ $$=$$ $$\u03bc_2$$"],"problemType":"MultipleChoice","stepTitle":"Which is the null hypothesis $$H_0$$, where $$\u03bc_1$$ and $$\u03bc_2$$ are the population means?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$\u03bc_1$$ $$=$$ $$\u03bc_2$$","choices":["$$\u03bc_1$$ $$=$$ $$\u03bc_2$$","$$\u03bc_1$$ < $$\u03bc_2$$"],"hints":{"DefaultPathway":[{"id":"a8802f1std14a-h1","type":"hint","dependencies":[],"title":"Defining the Null Hypothesis","text":"Null hypothesis: the means of the final exam scores are equal for the online and face-to-face statistics classes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std14a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\u03bc_1$$ $$=$$ $$\u03bc_2$$"],"dependencies":["a8802f1std14a-h1"],"title":"Selecting Null","text":"In which of the following choices do the means, $$\u03bc_1$$ and $$\u03bc_2$$, equal each other?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\u03bc_1$$ $$=$$ $$\u03bc_2$$","$$\u03bc_1$$ < $$\u03bc_2$$"]}]}}]},{"id":"a8802f1std15","title":"Two Population Means with Unknown Standard Deviations","body":"A professor at a large community college wanted to determine whether there is a difference in the means of final exam scores between students who took his statistics course online and the students who took his face-to-face statistics class. He believed that the mean of the final exam scores for the online class would be lower than that of the face-to-face class. The randomly selected $$30$$ final exam scores from each group are listed in Table $$10.3$$ and Table $$10.4$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Two Population Means with Unknown Standard Deviations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a8802f1std15a","stepAnswer":["$$\u03bc_1$$ < $$\u03bc_2$$"],"problemType":"MultipleChoice","stepTitle":"Which is the alternative hypothesis $$H_0$$, where $$\u03bc_1$$ and $$\u03bc_2$$ are the population means?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$\u03bc_1$$ < $$\u03bc_2$$","choices":["$$\u03bc_1$$ $$=$$ $$\u03bc_2$$","$$\u03bc_1$$ < $$\u03bc_2$$"],"hints":{"DefaultPathway":[{"id":"a8802f1std15a-h1","type":"hint","dependencies":[],"title":"Defining the Alternative Hypothesis","text":"Alternative hypothesis: the mean of the final exam scores of the online class is less than the mean of the final exam scores of the face-to-face class.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std15a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\u03bc_1$$ < $$\u03bc_2$$"],"dependencies":["a8802f1std15a-h1"],"title":"Selecting Alternative","text":"Which of the following choices is the mean of the final exam scores of the online class, $$\u03bc_1$$, less than the mean of the final exam scores of the face-to-face class, $$\u03bc_2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\u03bc_1$$ $$=$$ $$\u03bc_2$$","$$\u03bc_1$$ < $$\u03bc_2$$"]}]}}]},{"id":"a8802f1std16","title":"Two Population Means with Unknown Standard Deviations","body":"A professor at a large community college wanted to determine whether there is a difference in the means of final exam scores between students who took his statistics course online and the students who took his face-to-face statistics class. He believed that the mean of the final exam scores for the online class would be lower than that of the face-to-face class. The randomly selected $$30$$ final exam scores from each group are listed in Table $$10.3$$ and Table $$10.4$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Two Population Means with Unknown Standard Deviations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a8802f1std16a","stepAnswer":["Left"],"problemType":"MultipleChoice","stepTitle":"Is this test right-, left-, or two-tailed?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Left","Right","Two"],"hints":{"DefaultPathway":[{"id":"a8802f1std16a-h1","type":"hint","dependencies":[],"title":"Using the Calculator","text":"Graph the test.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std16a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Left"],"dependencies":["a8802f1std16a-h1"],"title":"Visual Graph","text":"Looking at the graph, on what side is the colored region on?\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Left","Right","Two"]}]}}]},{"id":"a8802f1std17","title":"Two Population Means with Unknown Standard Deviations","body":"A professor at a large community college wanted to determine whether there is a difference in the means of final exam scores between students who took his statistics course online and the students who took his face-to-face statistics class. He believed that the mean of the final exam scores for the online class would be lower than that of the face-to-face class. The randomly selected $$30$$ final exam scores from each group are listed in Table $$10.3$$ and Table $$10.4$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Two Population Means with Unknown Standard Deviations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a8802f1std17a","stepAnswer":["$$0.0011$$"],"problemType":"TextBox","stepTitle":"Using a calculator, what is the $$p-value$$?","stepBody":"Round your answer to four decimal digits.##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.0011$$","hints":{"DefaultPathway":[]}}]},{"id":"a8802f1std18","title":"Two Population Means with Unknown Standard Deviations","body":"A professor at a large community college wanted to determine whether there is a difference in the means of final exam scores between students who took his statistics course online and the students who took his face-to-face statistics class. He believed that the mean of the final exam scores for the online class would be lower than that of the face-to-face class. The randomly selected $$30$$ final exam scores from each group are listed in Table $$10.3$$ and Table $$10.4$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Two Population Means with Unknown Standard Deviations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a8802f1std18a","stepAnswer":["Reject"],"problemType":"MultipleChoice","stepTitle":"Do you reject or not reject the null hypothesis?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Reject","Do not reject"],"hints":{"DefaultPathway":[{"id":"a8802f1std18a-h1","type":"hint","dependencies":[],"title":"P-Value & Null Hypothesis","text":"The problem says to test at a 1% significance level, meaning that the $$p-value$$ threshold is $$0.01$$. If $$p-value$$ < $$0.01$$, reject $$H_0$$. This means that you would be rejecting $$\u03bc_1$$ $$=$$ $$\u03bc_2$$, and the means are different. If $$p-value$$ $$ \\\\geq $$ $$0.01$$, accept $$H_0$$ such that $$\u03bc_1$$ $$=$$ $$\u03bc_2$$, and the means are the same.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.0011$$"],"dependencies":["a8802f1std18a-h1"],"title":"Finding the P-Value","text":"Using a calculator, what is the $$p-value$$? Round to four decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std18a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Less than"],"dependencies":["a8802f1std18a-h2"],"title":"Testing the Null Hypothesis","text":"Is the $$p-value$$ greater than or less than $$0.01$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Greater than","Less than"]},{"id":"a8802f1std18a-h4","type":"hint","dependencies":["a8802f1std18a-h3"],"title":"Rejecting or Accepting the Null Hypothesis","text":"The professor was correct. The evidence shows that the mean of the final exam scores for the online class is lower than that of the face-to-face class. At the 5% level of significance, from the sample data, there is sufficient evidence to conclude that the mean of the final exam scores for the online class is less than the mean of final exam scores of the face-to-face class.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8802f1std19","title":"Two Population Means with Unknown Standard Deviations","body":"A professor at a large community college wanted to determine whether there is a difference in the means of final exam scores between students who took his statistics course online and the students who took his face-to-face statistics class. He believed that the mean of the final exam scores for the online class would be lower than that of the face-to-face class. The randomly selected $$30$$ final exam scores from each group are listed in Table $$10.3$$ and Table $$10.4$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Two Population Means with Unknown Standard Deviations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a8802f1std19a","stepAnswer":["Large"],"problemType":"MultipleChoice","stepTitle":"Calculate Cohen\u2019s $$d$$. Is the size of the effect small, medium or large?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Small","Medium","Large"],"hints":{"DefaultPathway":[{"id":"a8802f1std19a-h1","type":"hint","dependencies":[],"title":"Calculating Cohen\'s $$d$$","text":"Cohen\'s $$d$$ is a measure of effect size based on the differences between two means. Specifically, it is the measure of the difference between two means divided by the pooled standard deviation. The calculated value of effect size is then compared to Cohen\u2019s standards of small, medium, and large effect sizes: if the size of effect is small, $$d$$ $$=$$ $$0.2;$$ for medium, $$d$$ $$=$$ $$0.5;$$ for large, $$d$$ $$=$$ $$0.8$$.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.834$$"],"dependencies":["a8802f1std19a-h1"],"title":"Solving for $$d$$","text":"What is $$d$$ (the difference between two means divided by the pooled standard deviation)? Round to the nearest thousandths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std19a-h3","type":"hint","dependencies":["a8802f1std19a-h2"],"title":"Sample Sizes and Means","text":"The means of the final exam scores for online class and face-to-face class are as follows respectively: $$x\u0304_A$$ $$=$$ $$\\\\frac{2185.5}{30}$$ $$=$$ $$72.85$$, $$x\u0304_B$$ $$=$$ $$\\\\frac{2549.4}{30}$$ $$=$$ $$84.98$$, and $$n_1$$ and $$n_2$$ $$=$$ $$30$$. Use a calculator to find $$d$$ if needed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std19a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Large"],"dependencies":["a8802f1std19a-h3"],"title":"Small, Medium, or Large Size of Effect","text":"Compare the calculated value of effect size (d) to Cohen\u2019s standards of small, medium, and large effect sizes where if the size of effect is small, $$d$$ $$=$$ $$0.2;$$ for medium, $$d$$ $$=$$ $$0.5;$$ for large, $$d$$ $$=$$ $$0.8$$. What is the effect size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Small","Medium","Large"]},{"id":"a8802f1std19a-h5","type":"hint","dependencies":["a8802f1std19a-h4"],"title":"Explanation","text":"$$d$$ $$=$$ $$0.834;$$ Large because $$0.834$$ is greater than Cohen\u2019s $$0.8$$ for a large effect size. The size of the differences between the means of the final exam scores of online students and students in a face-to-face class is large indicating a significant difference.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8802f1std2","title":"Two Population Means with Unknown Standard Deviations","body":"A study is done by a community group in two neighboring colleges to determine which one graduates students with more math classes. College A samples $$11$$ graduates. Their average is four math classes with a standard deviation of $$1.5$$ math classes. College B samples nine graduates. Their average is $$3.5$$ math classes with a standard deviation of one math class. The community group believes that a student who graduates from college A has taken more math classes, on the average. Both populations have a normal distribution. Test at a 1% significance level.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Two Population Means with Unknown Standard Deviations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a8802f1std2a","stepAnswer":["Means"],"problemType":"MultipleChoice","stepTitle":"Is this a test of two means or two proportions?","stepBody":"","answerType":"string","variabilization":{},"choices":["Means","Proportions"],"hints":{"DefaultPathway":[{"id":"a8802f1std2a-h1","type":"hint","dependencies":[],"title":"Mean or Proportion","text":"Mean is the average value in a sample. Proportion is the number of observations of a characteristic in comparison to the entire sample.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std2a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Means"],"dependencies":["a8802f1std2a-h1"],"title":"Understanding the Problem","text":"When reading the problem, is the study comparing the means (averages) of the math classes or the proportions of the math classes?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Means","Proportions"]}]}}]},{"id":"a8802f1std20","title":"Two Population Means with Unknown Standard Deviations","body":"A study is done to determine if Company A retains its workers longer than Company B. Company A samples $$15$$ workers, and their average time with the company is five years with a standard deviation of $$1.2$$. Company B samples $$20$$ workers, and their average time with the company is $$4.5$$ years with a standard deviation of $$0.8$$. The populations are normally distributed.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Two Population Means with Unknown Standard Deviations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a8802f1std20a","stepAnswer":["Small"],"problemType":"MultipleChoice","stepTitle":"Calculate Cohen\u2019s $$d$$ for Example $$10.2$$. Is the size of the effect small, medium, or large?","stepBody":"","answerType":"string","variabilization":{},"choices":["Small","Medium","Large"],"hints":{"DefaultPathway":[{"id":"a8802f1std20a-h1","type":"hint","dependencies":[],"title":"Calculating Cohen\'s $$d$$","text":"Cohen\'s $$d$$ is a measure of effect size based on the differences between two means. Specifically, it is the measure of the difference between two means divided by the pooled standard deviation. The calculated value of effect size is then compared to Cohen\u2019s standards of small, medium, and large effect sizes: if the size of effect is small, $$d$$ $$=$$ $$0.2;$$ for medium, $$d$$ $$=$$ $$0.5;$$ for large, $$d$$ $$=$$ $$0.8$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a8802f1std20a-h1"],"title":"$$\u03bc_1$$","text":"What is $$\u03bc_1$$ (the population mean of Company A)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std20a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3.5$$"],"dependencies":["a8802f1std20a-h1"],"title":"$$\u03bc_2$$","text":"What is $$\u03bc_2$$ (the population mean of Company B)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std20a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.5$$"],"dependencies":["a8802f1std20a-h1"],"title":"$$s_1$$","text":"What is $$s_1$$, the sample standard deviation, an estimate of \u03c31 (the unknown population standard deviation), of Company A?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std20a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a8802f1std20a-h1"],"title":"$$s_2$$","text":"What is $$s_2$$, the sample standard deviation, an estimate of \u03c32 (the unknown population standard deviation), of Company B?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std20a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11$$"],"dependencies":["a8802f1std20a-h1"],"title":"$$n_1$$","text":"What is $$n_1$$ (the sample size of Company A)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std20a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a8802f1std20a-h1"],"title":"$$n_2$$","text":"What is $$n_2$$ (the sample size of Company B)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std20a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.384$$"],"dependencies":["a8802f1std20a-h1","a8802f1std20a-h2","a8802f1std20a-h3","a8802f1std20a-h4","a8802f1std20a-h5","a8802f1std20a-h6","a8802f1std20a-h7"],"title":"Solving $$d$$","text":"What is $$d$$? Round to the nearest thousandths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std20a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Small"],"dependencies":["a8802f1std20a-h8"],"title":"Small, Medium, or Large Size of Effect","text":"Compare the calculated value of effect size (d) to Cohen\u2019s standards of small, medium, and large effect sizes where if the size of effect is small, $$d$$ $$=$$ $$0.2;$$ for medium, $$d$$ $$=$$ $$0.5;$$ for large, $$d$$ $$=$$ $$0.8$$. What is the effect size?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Small","Medium","Large"]},{"id":"a8802f1std20a-h10","type":"hint","dependencies":["a8802f1std20a-h9"],"title":"Explanation","text":"The effect is small because $$0.384$$ is between Cohen\u2019s value of $$0.2$$ for small effect size and $$0.5$$ for medium effect size. The size of the differences of the means for the two colleges is small indicating that there is not a significant difference between them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8802f1std3","title":"Two Population Means with Unknown Standard Deviations","body":"A study is done by a community group in two neighboring colleges to determine which one graduates students with more math classes. College A samples $$11$$ graduates. Their average is four math classes with a standard deviation of $$1.5$$ math classes. College B samples nine graduates. Their average is $$3.5$$ math classes with a standard deviation of one math class. The community group believes that a student who graduates from college A has taken more math classes, on the average. Both populations have a normal distribution. Test at a 1% significance level.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Two Population Means with Unknown Standard Deviations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a8802f1std3a","stepAnswer":["Unknown"],"problemType":"MultipleChoice","stepTitle":"Are the populations\' standard deviations known or unknown?","stepBody":"","answerType":"string","variabilization":{},"choices":["Known","Unknown"],"hints":{"DefaultPathway":[{"id":"a8802f1std3a-h1","type":"hint","dependencies":[],"title":"Unknown or Known","text":"Examine the problem to determine whether or not the standard deviations of the populations were stated. Keep in mind that samples are only representative of populations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std3a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Samples"],"dependencies":["a8802f1std3a-h1"],"title":"Sample or Population","text":"Does the study use populations or samples?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Samples","Populations"]},{"id":"a8802f1std3a-h3","type":"hint","dependencies":["a8802f1std3a-h2"],"title":"Standard Deviations","text":"Because the study was done on two samples, the standard deviations of the samples are known, but the standard deviations of the populations are unknown.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8802f1std4","title":"Two Population Means with Unknown Standard Deviations","body":"A study is done by a community group in two neighboring colleges to determine which one graduates students with more math classes. College A samples $$11$$ graduates. Their average is four math classes with a standard deviation of $$1.5$$ math classes. College B samples nine graduates. Their average is $$3.5$$ math classes with a standard deviation of one math class. The community group believes that a student who graduates from college A has taken more math classes, on the average. Both populations have a normal distribution. Test at a 1% significance level.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Two Population Means with Unknown Standard Deviations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a8802f1std4a","stepAnswer":["Student\'s $$t$$"],"problemType":"MultipleChoice","stepTitle":"Which distribution do you use to perform the test?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Student\'s $$t$$","choices":["Student\'s $$t$$","$$P-Value$$","$$x\u0304_A$$ $$-$$ $$x\u0304_B$$"],"hints":{"DefaultPathway":[{"id":"a8802f1std4a-h1","type":"hint","dependencies":[],"title":"Walking Through Each Choice","text":"$$x\u0304_A$$ - $$x\u0304_B$$ $$=$$ difference in the sample mean amount of time girls and boys play (also known as the random variable), $$p-value$$ $$=$$ how likely the data occurred under the null hypothesis, and $$t_{df}$$ $$=$$ distribution for the test.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std4a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Student\'s $$t$$"],"dependencies":["a8802f1std4a-h1"],"title":"Distribution","text":"Which of the following solves for the distribution of the test: Student\'s $$t$$, $$p-value$$, or $$x\u0304_A$$ - $$x\u0304_B$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Student\'s $$t$$","$$P-Value$$","$$x\u0304_A$$ $$-$$ $$x\u0304_B$$"]}]}}]},{"id":"a8802f1std5","title":"Two Population Means with Unknown Standard Deviations","body":"A study is done by a community group in two neighboring colleges to determine which one graduates students with more math classes. College A samples $$11$$ graduates. Their average is four math classes with a standard deviation of $$1.5$$ math classes. College B samples nine graduates. Their average is $$3.5$$ math classes with a standard deviation of one math class. The community group believes that a student who graduates from college A has taken more math classes, on the average. Both populations have a normal distribution. Test at a 1% significance level.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Two Population Means with Unknown Standard Deviations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a8802f1std5a","stepAnswer":["$$x\u0304_A$$ $$-$$ $$x\u0304_B$$"],"problemType":"MultipleChoice","stepTitle":"What is the random variable?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x\u0304_A$$ - $$x\u0304_B$$","choices":["Student\'s $$t$$","$$P-Value$$","$$x\u0304_A$$ $$-$$ $$x\u0304_B$$"],"hints":{"DefaultPathway":[{"id":"a8802f1std5a-h1","type":"hint","dependencies":[],"title":"Walking Through Each Choice","text":"$$x\u0304_A$$ - $$x\u0304_B$$ $$=$$ difference in the sample mean amount of time girls and boys play (also known as the random variable), $$p-value$$ $$=$$ how likely the data occurred under the null hypothesis, and $$t_{df}$$ $$=$$ distribution for the test.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std5a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x\u0304_A$$ $$-$$ $$x\u0304_B$$"],"dependencies":["a8802f1std5a-h1"],"title":"Random Variable","text":"Which of the following solves for the random variable: Student\'s $$t$$, $$p-value$$, or $$x\u0304_A$$ - $$x\u0304_B$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Student\'s $$t$$","$$P-Value$$","$$x\u0304_A$$ $$-$$ $$x\u0304_B$$"]}]}}]},{"id":"a8802f1std6","title":"Two Population Means with Unknown Standard Deviations","body":"A study is done by a community group in two neighboring colleges to determine which one graduates students with more math classes. College A samples $$11$$ graduates. Their average is four math classes with a standard deviation of $$1.5$$ math classes. College B samples nine graduates. Their average is $$3.5$$ math classes with a standard deviation of one math class. The community group believes that a student who graduates from college A has taken more math classes, on the average. Both populations have a normal distribution. Test at a 1% significance level.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Two Population Means with Unknown Standard Deviations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a8802f1std6a","stepAnswer":["$$\u03bc_A$$ $$ \\\\leq $$ $$\u03bc_B$$"],"problemType":"MultipleChoice","stepTitle":"Which is the null hypothesis $$H_0$$, where $$\u03bc_A$$ and $$\u03bc_B$$ are the population means?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\u03bc_A$$ $$ \\\\leq $$ $$\u03bc_B$$","choices":["$$\u03bc_A$$ $$ \\\\leq $$ $$\u03bc_B$$","$$\u03bc_A$$ > $$\u03bc_B$$"],"hints":{"DefaultPathway":[{"id":"a8802f1std6a-h1","type":"hint","dependencies":[],"title":"Null Vs. Alternative Hypothesis","text":"The problem states that \\"the community group believes that a student who graduates from college A has taken more math classes, on the average.\\" This statement is the alternative hypothesis (what the community group is trying to prove), where the population mean of college A, $$\u03bc_A$$, is greater than the population mean of college B, $$\u03bc_B$$. The null hypothesis should be the opposite of the alternative hypothesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8802f1std7","title":"Two Population Means with Unknown Standard Deviations","body":"A study is done by a community group in two neighboring colleges to determine which one graduates students with more math classes. College A samples $$11$$ graduates. Their average is four math classes with a standard deviation of $$1.5$$ math classes. College B samples nine graduates. Their average is $$3.5$$ math classes with a standard deviation of one math class. The community group believes that a student who graduates from college A has taken more math classes, on the average. Both populations have a normal distribution. Test at a 1% significance level.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Two Population Means with Unknown Standard Deviations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a8802f1std7a","stepAnswer":["Right"],"problemType":"MultipleChoice","stepTitle":"Is this test right-, left-, or two-tailed?","stepBody":"","answerType":"string","variabilization":{},"choices":["Left","Right","Two"],"hints":{"DefaultPathway":[{"id":"a8802f1std7a-h1","type":"hint","dependencies":[],"title":"Using the Calculator","text":"Graph the test.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std7a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Right"],"dependencies":["a8802f1std7a-h1"],"title":"Visual Graph","text":"Looking at the graph, on what side is the colored region on?\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Left","Right","Two"]}]}}]},{"id":"a8802f1std8","title":"Two Population Means with Unknown Standard Deviations","body":"A study is done by a community group in two neighboring colleges to determine which one graduates students with more math classes. College A samples $$11$$ graduates. Their average is four math classes with a standard deviation of $$1.5$$ math classes. College B samples nine graduates. Their average is $$3.5$$ math classes with a standard deviation of one math class. The community group believes that a student who graduates from college A has taken more math classes, on the average. Both populations have a normal distribution. Test at a 1% significance level.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Two Population Means with Unknown Standard Deviations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a8802f1std8a","stepAnswer":["$$0.1928$$"],"problemType":"TextBox","stepTitle":"Using a calculator, what is the $$p-value$$? Round to four decimal places.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.1928$$","hints":{"DefaultPathway":[]}}]},{"id":"a8802f1std9","title":"Two Population Means with Unknown Standard Deviations","body":"A study is done by a community group in two neighboring colleges to determine which one graduates students with more math classes. College A samples $$11$$ graduates. Their average is four math classes with a standard deviation of $$1.5$$ math classes. College B samples nine graduates. Their average is $$3.5$$ math classes with a standard deviation of one math class. The community group believes that a student who graduates from college A has taken more math classes, on the average. Both populations have a normal distribution. Test at a 1% significance level.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Two Population Means with Unknown Standard Deviations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a8802f1std9a","stepAnswer":["Do not reject"],"problemType":"MultipleChoice","stepTitle":"Do you reject or not reject the null hypothesis?","stepBody":"","answerType":"string","variabilization":{},"choices":["Reject","Do not reject"],"hints":{"DefaultPathway":[{"id":"a8802f1std9a-h1","type":"hint","dependencies":[],"title":"P-Value & Null Hypothesis","text":"The problem says to test at a 1% significance level, meaning that the $$p-value$$ threshold is $$0.01$$. If $$p-value$$ < $$0.01$$, reject $$H_0$$. This means that you would be rejecting $$\u03bc_A$$ $$=$$ $$\u03bc_B$$, and the means are different. If $$p-value$$ $$ \\\\geq $$ $$0.01$$, accept $$H_0$$ such that $$\u03bc_A$$ $$=$$ $$\u03bc_B$$, and the means are the same.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1928$$"],"dependencies":["a8802f1std9a-h1"],"title":"Finding the P-Value","text":"Using a calculator, what is the $$p-value$$? Round to four decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8802f1std9a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Greater than"],"dependencies":["a8802f1std9a-h2"],"title":"Testing the Null Hypothesis","text":"Is the $$p-value$$ greater or less than $$0.01$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Greater than","Less than"]},{"id":"a8802f1std9a-h4","type":"hint","dependencies":["a8802f1std9a-h3"],"title":"Rejecting or Accepting the Null Hypothesis","text":"The $$p-value$$ of $$0.1928$$ is greater than $$0.01$$, meaning that it accepts the null hypothesis. At the 1% level of significance, from the sample data, there is not sufficient evidence to conclude that a student who graduates from college A has taken more math classes, on the average, than a student who graduates from college B.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a88aae9QuadEq1","title":"Solve The Quadratic Equation","body":"Solve:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Solve Quadratic Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a88aae9QuadEq1a","stepAnswer":["2.5,-3"],"problemType":"TextBox","stepTitle":"$$2y^2+y=15$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2.5, -3$$","hints":{"DefaultPathway":[{"id":"a88aae9QuadEq1a-h1","type":"hint","dependencies":[],"title":"Quadratic Formula","text":"Remember the Quadratic Formula, $$\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4ac}\\\\right)}{2a}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a88aae9QuadEq1a-h1"],"title":"Quadratic Formula","text":"What is \\"a\\" in our equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a88aae9QuadEq1a-h2"],"title":"Quadratic Formula","text":"What is \\"b\\" in our equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-15$$"],"dependencies":["a88aae9QuadEq1a-h3"],"title":"Quadratic Formula","text":"What is \\"c\\" in our equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq1a-h5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["2.5,-3"],"dependencies":["a88aae9QuadEq1a-h4"],"title":"Solution","text":"With our \\"a\\", \\"b\\", and \\"c\\", what number will we get once we plug these into the quadratic formula?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a88aae9QuadEq10","title":"Solve The Quadratic Inequality","body":"Solve the following inequality analytically and write where the inequality is true using interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Solve Quadratic Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a88aae9QuadEq10a","stepAnswer":["[1-sqrt(2),1+sqrt(2)]"],"problemType":"TextBox","stepTitle":"$$-\\\\left(x^2\\\\right)+2x+1 \\\\geq 0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[1-\\\\sqrt{2},1+\\\\sqrt{2}]$$","hints":{"DefaultPathway":[{"id":"a88aae9QuadEq10a-h1","type":"hint","dependencies":[],"title":"Procedure","text":"To solve the quadratic inequality, the procedure is as follows. Solve for the roots of the quadratic formula, divide the number line into intervals based on the roots of the quadratic formula, test the values of each interval to see if they are positive or negative, and determine the intervals where the inequality is correct.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq10a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["1~sqrt(2)"],"dependencies":["a88aae9QuadEq10a-h1"],"title":"Quadratic Roots","text":"What are the roots of the quadratic equation? Use ~ for $$\\\\frac{plus}{minus}$$ and i for imaginary numbers in the answer. Write the answer in the form of x1,x2 if applicable as well.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq10a-h3","type":"hint","dependencies":["a88aae9QuadEq10a-h2"],"title":"Interval Checking","text":"Now let\'s check our intervals. We will have $$3$$ distinct intervals to check values for.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a88aae9QuadEq10a-h3"],"title":"Interval Checking","text":"If we plug in $$x=-1$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a88aae9QuadEq10a-h4"],"title":"Interval Checking","text":"If we plug in $$x=1$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq10a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a88aae9QuadEq10a-h5"],"title":"Interval Checking","text":"If we plug in $$x=3$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq10a-h7","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["[1-sqrt(2),1+sqrt(2)]"],"dependencies":["a88aae9QuadEq10a-h6"],"title":"Solution","text":"Since we have checked each interval, which ones have positive values in them?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a88aae9QuadEq11","title":"Solve The Quadratic Inequality","body":"Solve the following inequality analytically and write where the inequality is true using interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Solve Quadratic Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a88aae9QuadEq11a","stepAnswer":["(-inf,4-sqrt(2))&(4+sqrt(2),inf)"],"problemType":"TextBox","stepTitle":"$$-\\\\left(x^2\\\\right)+8x-14<0$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a88aae9QuadEq11a-h1","type":"hint","dependencies":[],"title":"Procedure","text":"To solve the quadratic inequality, the procedure is as follows. Solve for the roots of the quadratic formula, divide the number line into intervals based on the roots of the quadratic formula, test the values of each interval to see if they are positive or negative, and determine the intervals where the inequality is correct.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq11a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["4~sqrt(2)"],"dependencies":["a88aae9QuadEq11a-h1"],"title":"Quadratic Roots","text":"What are the roots of the quadratic equation? Use ~ for $$\\\\frac{plus}{minus}$$ and i for imaginary numbers in the answer. Write the answer in the form of x1,x2 if applicable as well.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq11a-h3","type":"hint","dependencies":["a88aae9QuadEq11a-h2"],"title":"Interval Checking","text":"Now let\'s check our intervals. We will have $$3$$ distinct intervals to check values for.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a88aae9QuadEq11a-h3"],"title":"Interval Checking","text":"If we plug in $$x=2$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a88aae9QuadEq11a-h4"],"title":"Interval Checking","text":"If we plug in $$x=4$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq11a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a88aae9QuadEq11a-h5"],"title":"Interval Checking","text":"If we plug in $$x=6$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq11a-h7","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-inf,4-sqrt(2))&(4+sqrt(2),inf)"],"dependencies":["a88aae9QuadEq11a-h6"],"title":"Solution","text":"Since we have checked each interval, which ones have positive values in them?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a88aae9QuadEq12","title":"Solve The Quadratic Inequality","body":"Solve the following inequality analytically and write where the inequality is true using interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Solve Quadratic Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a88aae9QuadEq12a","stepAnswer":["(-inf,inf)"],"problemType":"TextBox","stepTitle":"$$x^2-3x+4>0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\infty)$$","hints":{"DefaultPathway":[{"id":"a88aae9QuadEq12a-h1","type":"hint","dependencies":[],"title":"Procedure","text":"To solve the quadratic inequality, the procedure is as follows. Solve for the roots of the quadratic formula, divide the number line into intervals based on the roots of the quadratic formula, test the values of each interval to see if they are positive or negative, and determine the intervals where the inequality is correct.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq12a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(3~sqrt(7)i)/2"],"dependencies":["a88aae9QuadEq12a-h1"],"title":"Quadratic Roots","text":"What are the roots of the quadratic equation? Use ~ for $$\\\\frac{plus}{minus}$$ and i for imaginary numbers in the answer. Write the answer in the form of x1,x2 if applicable as well.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq12a-h3","type":"hint","dependencies":["a88aae9QuadEq12a-h2"],"title":"Interval Checking","text":"If our roots are comprised of imaginary numbers, then that root does not seperate two intervals. In other words, we can ignore the imaginary solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a88aae9QuadEq12a-h3"],"title":"Interval Checking","text":"If we plug in $$x=0$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq12a-h5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-inf,inf)"],"dependencies":["a88aae9QuadEq12a-h4"],"title":"Solution","text":"Since we only have $$1$$ interval and it is positive, where will our inequality be true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a88aae9QuadEq13","title":"Solve The Quadratic Inequality","body":"Solve the following inequality analytically and write where the inequality is true using interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Solve Quadratic Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a88aae9QuadEq13a","stepAnswer":["None"],"problemType":"TextBox","stepTitle":"$$x^2-3x+4 \\\\leq 0$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a88aae9QuadEq13a-h1","type":"hint","dependencies":[],"title":"Procedure","text":"To solve the quadratic inequality, the procedure is as follows. Solve for the roots of the quadratic formula, divide the number line into intervals based on the roots of the quadratic formula, test the values of each interval to see if they are positive or negative, and determine the intervals where the inequality is correct.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq13a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(3~sqrt(7)i)/2"],"dependencies":["a88aae9QuadEq13a-h1"],"title":"Quadratic Roots","text":"What are the roots of the quadratic equation? Use ~ for $$\\\\frac{plus}{minus}$$ and i for imaginary numbers in the answer. Write the answer in the form of x1,x2 if applicable as well.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq13a-h3","type":"hint","dependencies":["a88aae9QuadEq13a-h2"],"title":"Interval Checking","text":"If our roots are comprised of imaginary numbers, then that root does not seperate two intervals. In other words, we can ignore the imaginary solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a88aae9QuadEq13a-h3"],"title":"Interval Checking","text":"If we plug in $$x=0$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq13a-h5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["None"],"dependencies":["a88aae9QuadEq13a-h4"],"title":"Solution","text":"Since we only have $$1$$ interval and it is positive, where will our inequality be true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a88aae9QuadEq14","title":"Solve The Quadratic Inequality","body":"Solve the following inequality analytically and write where the inequality is true using interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Solve Quadratic Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a88aae9QuadEq14a","stepAnswer":["(-inf,inf)"],"problemType":"TextBox","stepTitle":"$$-\\\\left(x^2\\\\right)+2x-4 \\\\leq 0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\infty)$$","hints":{"DefaultPathway":[{"id":"a88aae9QuadEq14a-h1","type":"hint","dependencies":[],"title":"Procedure","text":"To solve the quadratic inequality, the procedure is as follows. Solve for the roots of the quadratic formula, divide the number line into intervals based on the roots of the quadratic formula, test the values of each interval to see if they are positive or negative, and determine the intervals where the inequality is correct.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq14a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["1~i*sqrt(3)"],"dependencies":["a88aae9QuadEq14a-h1"],"title":"Quadratic Roots","text":"What are the roots of the quadratic equation? Use ~ for $$\\\\frac{plus}{minus}$$ and i for imaginary numbers in the answer. Write the answer in the form of x1,x2 if applicable as well.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq14a-h3","type":"hint","dependencies":["a88aae9QuadEq14a-h2"],"title":"Interval Checking","text":"If our roots are comprised of imaginary numbers, then that root does not seperate two intervals. In other words, we can ignore the imaginary solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq14a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["a88aae9QuadEq14a-h3"],"title":"Interval Checking","text":"If we plug in $$x=0$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq14a-h5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-inf,inf)"],"dependencies":["a88aae9QuadEq14a-h4"],"title":"Solution","text":"Since we only have $$1$$ interval and it is positive, where will our inequality be true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a88aae9QuadEq15","title":"Solve The Quadratic Inequality","body":"Solve the following inequality analytically and write where the inequality is true using interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Solve Quadratic Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a88aae9QuadEq15a","stepAnswer":["(-inf,inf)"],"problemType":"TextBox","stepTitle":"$$x^2+3x+3>0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\infty)$$","hints":{"DefaultPathway":[{"id":"a88aae9QuadEq15a-h1","type":"hint","dependencies":[],"title":"Procedure","text":"To solve the quadratic inequality, the procedure is as follows. Solve for the roots of the quadratic formula, divide the number line into intervals based on the roots of the quadratic formula, test the values of each interval to see if they are positive or negative, and determine the intervals where the inequality is correct.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq15a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-3~i*sqrt(3))/2"],"dependencies":["a88aae9QuadEq15a-h1"],"title":"Quadratic Roots","text":"What are the roots of the quadratic equation? Use ~ for $$\\\\frac{plus}{minus}$$ and i for imaginary numbers in the answer. Write the answer in the form of x1,x2 if applicable as well.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq15a-h3","type":"hint","dependencies":["a88aae9QuadEq15a-h2"],"title":"Interval Checking","text":"If our roots are comprised of imaginary numbers, then that root does not seperate two intervals. In other words, we can ignore the imaginary solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a88aae9QuadEq15a-h3"],"title":"Interval Checking","text":"If we plug in $$x=0$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq15a-h5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-inf,inf)"],"dependencies":["a88aae9QuadEq15a-h4"],"title":"Solution","text":"Since we only have $$1$$ interval and it is positive, where will our inequality be true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a88aae9QuadEq16","title":"Solve The Quadratic Inequality","body":"Solve the following inequality graphically: That is, graph the inequality and write where the inequality is true using interval notation. Write the answer in the form of (x1, x2) where () exclude a number and [] include a number. In the case of $$infinities$$, write it as $$\\\\infty$$ and if there are multiple ranges, use & to include it as well..","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Solve Quadratic Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a88aae9QuadEq16a","stepAnswer":["(-inf,-5)&(-1,inf)"],"problemType":"TextBox","stepTitle":"$$x^2+6x+5>0$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a88aae9QuadEq16a-h1","type":"hint","dependencies":[],"title":"Positive or Negative?","text":"Since our formula is going to be greater than zero, we want to look at the interval on the graph where our line is above zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq16a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-inf,-5)&(-1,inf)"],"dependencies":["a88aae9QuadEq16a-h1"],"title":"Solution","text":"When we graph our formula, what part of the graph is going to be greater than zero?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a88aae9QuadEq17","title":"Solve The Quadratic Inequality","body":"Solve the following inequality graphically: That is, graph the inequality and write where the inequality is true using interval notation. Write the answer in the form of (x1, x2) where () exclude a number and [] include a number. In the case of $$infinities$$, write it as $$\\\\infty$$ and if there are multiple ranges, use & to include it as well..","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Solve Quadratic Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a88aae9QuadEq17a","stepAnswer":["(-3,-1)"],"problemType":"TextBox","stepTitle":"$$x^2+4x+3 \\\\leq 0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-3,-1)$$","hints":{"DefaultPathway":[{"id":"a88aae9QuadEq17a-h1","type":"hint","dependencies":[],"title":"Positive or Negative?","text":"Since our formula is going to be less than or equal to zero, we want to look at the interval on the graph where our line is underneath or at zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq17a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-3,-1)"],"dependencies":["a88aae9QuadEq17a-h1"],"title":"Solution","text":"When we graph our formula, what part of the graph is going to be less than or equal to zero?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a88aae9QuadEq18","title":"Solve The Quadratic Inequality","body":"Solve the following inequality graphically: That is, graph the inequality and write where the inequality is true using interval notation. Write the answer in the form of (x1, x2) where () exclude a number and [] include a number. In the case of $$infinities$$, write it as $$\\\\infty$$ and if there are multiple ranges, use & to include it as well..","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Solve Quadratic Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a88aae9QuadEq18a","stepAnswer":["(-inf,-6]&[3,inf)"],"problemType":"TextBox","stepTitle":"$$-\\\\left(x^2\\\\right)-3x+18 \\\\leq 0$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a88aae9QuadEq18a-h1","type":"hint","dependencies":[],"title":"Positive or Negative?","text":"Since our formula is going to be less than or equal to zero, we want to look at the interval on the graph where our line is underneath or at zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq18a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-inf,-6]&[3,inf)"],"dependencies":["a88aae9QuadEq18a-h1"],"title":"Solution","text":"When we graph our formula, what part of the graph is going to be less than or equal to zero?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a88aae9QuadEq19","title":"Solve The Quadratic Inequality","body":"Solve the following inequality graphically: That is, graph the inequality and write where the inequality is true using interval notation. Write the answer in the form of (x1, x2) where () exclude a number and [] include a number. In the case of $$infinities$$, write it as $$\\\\infty$$ and if there are multiple ranges, use & to include it as well..","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Solve Quadratic Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a88aae9QuadEq19a","stepAnswer":["[-3,4]"],"problemType":"TextBox","stepTitle":"$$-\\\\left(x^2\\\\right)+x+12 \\\\geq 0$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a88aae9QuadEq19a-h1","type":"hint","dependencies":[],"title":"Positive or Negative?","text":"Since our formula is going to be less than or equal to zero, we want to look at the interval on the graph where our line is at or underneath zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq19a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["[-3,4]"],"dependencies":["a88aae9QuadEq19a-h1"],"title":"Solution","text":"When we graph our formula, what part of the graph is going to be less than zero?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a88aae9QuadEq2","title":"Solve The Quadratic Inequality","body":"Solve the following inequality graphically: That is, graph the inequality and write where the inequality is true using interval notation. Write the answer in the form of (x1, x2) where () exclude a number and [] include a number. In the case of $$infinities$$, write it as $$\\\\infty$$ and if there are multiple ranges, use & to include it as well..","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Solve Quadratic Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a88aae9QuadEq2a","stepAnswer":["(2,4)"],"problemType":"TextBox","stepTitle":"$$x^2-6x+8<0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(2,4)$$","hints":{"DefaultPathway":[{"id":"a88aae9QuadEq2a-h1","type":"hint","dependencies":[],"title":"Positive or Negative?","text":"Since our formula is going to be less than zero, we want to look at the interval on the graph where our line is underneath zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq2a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(2,4)"],"dependencies":["a88aae9QuadEq2a-h1"],"title":"Solution","text":"When we graph our formula, what part of the graph is going to be less than zero?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a88aae9QuadEq20","title":"Solve The Quadratic Inequality","body":"Solve the following inequality analytically and write where the inequality is true using interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Solve Quadratic Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a88aae9QuadEq20a","stepAnswer":["(-inf,-4]&[1,inf)"],"problemType":"TextBox","stepTitle":"$$x^2+3x-4 \\\\geq 0$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a88aae9QuadEq20a-h1","type":"hint","dependencies":[],"title":"Procedure","text":"To solve the quadratic inequality, the procedure is as follows. Solve for the roots of the quadratic formula, divide the number line into intervals based on the roots of the quadratic formula, test the values of each interval to see if they are positive or negative, and determine the intervals where the inequality is correct.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq20a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["-4,1"],"dependencies":["a88aae9QuadEq20a-h1"],"title":"Quadratic Roots","text":"What are the roots of the quadratic equation? Use ~ for $$\\\\frac{plus}{minus}$$ and i for imaginary numbers in the answer. Write the answer in the form of x1,x2 if applicable as well.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq20a-h3","type":"hint","dependencies":["a88aae9QuadEq20a-h2"],"title":"Interval Checking","text":"Now let\'s check our intervals. We will have $$3$$ distinct intervals to check values for.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq20a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a88aae9QuadEq20a-h3"],"title":"Interval Checking","text":"If we plug in $$x=-5$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq20a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["a88aae9QuadEq20a-h4"],"title":"Interval Checking","text":"If we plug in $$x=0$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq20a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a88aae9QuadEq20a-h5"],"title":"Interval Checking","text":"If we plug in $$x=2$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq20a-h7","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-inf,-4]&[1,inf)"],"dependencies":["a88aae9QuadEq20a-h6"],"title":"Solution","text":"Since we have checked each interval, which ones have positive values in them?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a88aae9QuadEq21","title":"Solve The Quadratic Inequality","body":"Solve the following inequality analytically and write where the inequality is true using interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Solve Quadratic Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a88aae9QuadEq21a","stepAnswer":["(2,5)"],"problemType":"TextBox","stepTitle":"$$x^2-7x+10<0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(2,5)$$","hints":{"DefaultPathway":[{"id":"a88aae9QuadEq21a-h1","type":"hint","dependencies":[],"title":"Procedure","text":"To solve the quadratic inequality, the procedure is as follows. Solve for the roots of the quadratic formula, divide the number line into intervals based on the roots of the quadratic formula, test the values of each interval to see if they are positive or negative, and determine the intervals where the inequality is correct.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq21a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["2,5"],"dependencies":["a88aae9QuadEq21a-h1"],"title":"Quadratic Roots","text":"What are the roots of the quadratic equation? Use ~ for $$\\\\frac{plus}{minus}$$ and i for imaginary numbers in the answer. Write the answer in the form of x1,x2 if applicable as well.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq21a-h3","type":"hint","dependencies":["a88aae9QuadEq21a-h2"],"title":"Interval Checking","text":"Now let\'s check our intervals. We will have $$3$$ distinct intervals to check values for.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq21a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a88aae9QuadEq21a-h3"],"title":"Interval Checking","text":"If we plug in $$x=1$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq21a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a88aae9QuadEq21a-h4"],"title":"Interval Checking","text":"If we plug in $$x=4$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq21a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a88aae9QuadEq21a-h5"],"title":"Interval Checking","text":"If we plug in $$x=6$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq21a-h7","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(2,5)"],"dependencies":["a88aae9QuadEq21a-h6"],"title":"Solution","text":"Since we have checked each interval, which ones have negative values in them?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a88aae9QuadEq22","title":"Solve The Quadratic Inequality","body":"Solve the following inequality analytically and write where the inequality is true using interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Solve Quadratic Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a88aae9QuadEq22a","stepAnswer":["(-inf,-5)&(-3,inf)"],"problemType":"TextBox","stepTitle":"$$x^2+8x+15>0$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a88aae9QuadEq22a-h1","type":"hint","dependencies":[],"title":"Procedure","text":"To solve the quadratic inequality, the procedure is as follows. Solve for the roots of the quadratic formula, divide the number line into intervals based on the roots of the quadratic formula, test the values of each interval to see if they are positive or negative, and determine the intervals where the inequality is correct.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq22a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["-5,-3"],"dependencies":["a88aae9QuadEq22a-h1"],"title":"Quadratic Roots","text":"What are the roots of the quadratic equation? Use ~ for $$\\\\frac{plus}{minus}$$ and i for imaginary numbers in the answer. Write the answer in the form of x1,x2 if applicable as well.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq22a-h3","type":"hint","dependencies":["a88aae9QuadEq22a-h2"],"title":"Interval Checking","text":"Now let\'s check our intervals. We will have $$3$$ distinct intervals to check values for.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq22a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a88aae9QuadEq22a-h3"],"title":"Interval Checking","text":"If we plug in $$x=-6$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq22a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a88aae9QuadEq22a-h4"],"title":"Interval Checking","text":"If we plug in $$x=-4$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq22a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a88aae9QuadEq22a-h5"],"title":"Interval Checking","text":"If we plug in $$x=2$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq22a-h7","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-inf,-5)&(-3,inf)"],"dependencies":["a88aae9QuadEq22a-h6"],"title":"Solution","text":"Since we have checked each interval, which ones have positive values in them?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a88aae9QuadEq23","title":"Solve The Quadratic Inequality","body":"Solve the following inequality analytically and write where the inequality is true using interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Solve Quadratic Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a88aae9QuadEq23a","stepAnswer":["[2-sqrt(6),2+sqrt(6)]"],"problemType":"TextBox","stepTitle":"$$x^2-4x+2 \\\\leq 0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[2-\\\\sqrt{6},2+\\\\sqrt{6}]$$","hints":{"DefaultPathway":[{"id":"a88aae9QuadEq23a-h1","type":"hint","dependencies":[],"title":"Procedure","text":"To solve the quadratic inequality, the procedure is as follows. Solve for the roots of the quadratic formula, divide the number line into intervals based on the roots of the quadratic formula, test the values of each interval to see if they are positive or negative, and determine the intervals where the inequality is correct.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq23a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["2~sqrt(6)"],"dependencies":["a88aae9QuadEq23a-h1"],"title":"Quadratic Roots","text":"What are the roots of the quadratic equation? Use ~ for $$\\\\frac{plus}{minus}$$ and i for imaginary numbers in the answer. Write the answer in the form of x1,x2 if applicable as well.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq23a-h3","type":"hint","dependencies":["a88aae9QuadEq23a-h2"],"title":"Interval Checking","text":"Now let\'s check our intervals. We will have $$3$$ distinct intervals to check values for.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq23a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a88aae9QuadEq23a-h3"],"title":"Interval Checking","text":"If we plug in $$x=0$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq23a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a88aae9QuadEq23a-h4"],"title":"Interval Checking","text":"If we plug in $$x=-2$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq23a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a88aae9QuadEq23a-h5"],"title":"Interval Checking","text":"If we plug in $$x=4$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq23a-h7","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["[2-sqrt(6),2+sqrt(6)]"],"dependencies":["a88aae9QuadEq23a-h6"],"title":"Solution","text":"Since we have checked each interval, which ones have negative values in them?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a88aae9QuadEq24","title":"Solve The Quadratic Inequality","body":"Solve the following inequality analytically and write where the inequality is true using interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Solve Quadratic Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a88aae9QuadEq24a","stepAnswer":["(-6,-2)"],"problemType":"TextBox","stepTitle":"$$x^2+8x+12<0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-6,-2)$$","hints":{"DefaultPathway":[{"id":"a88aae9QuadEq24a-h1","type":"hint","dependencies":[],"title":"Procedure","text":"To solve the quadratic inequality, the procedure is as follows. Solve for the roots of the quadratic formula, divide the number line into intervals based on the roots of the quadratic formula, test the values of each interval to see if they are positive or negative, and determine the intervals where the inequality is correct.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq24a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["-6,-2"],"dependencies":["a88aae9QuadEq24a-h1"],"title":"Quadratic Roots","text":"What are the roots of the quadratic equation? Use ~ for $$\\\\frac{plus}{minus}$$ and i for imaginary numbers in the answer. Write the answer in the form of x1,x2 if applicable as well.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq24a-h3","type":"hint","dependencies":["a88aae9QuadEq24a-h2"],"title":"Interval Checking","text":"Now let\'s check our intervals. We will have $$3$$ distinct intervals to check values for.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq24a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a88aae9QuadEq24a-h3"],"title":"Interval Checking","text":"If we plug in $$x=-7$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq24a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["a88aae9QuadEq24a-h4"],"title":"Interval Checking","text":"If we plug in $$x=-4$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq24a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a88aae9QuadEq24a-h5"],"title":"Interval Checking","text":"If we plug in $$x=-1$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq24a-h7","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-6,-2)"],"dependencies":["a88aae9QuadEq24a-h6"],"title":"Solution","text":"Since we have checked each interval, which ones have negative values in them?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a88aae9QuadEq25","title":"Solve The Quadratic Inequality","body":"Solve the following inequality analytically and write where the inequality is true using interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Solve Quadratic Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a88aae9QuadEq25a","stepAnswer":["(-3-sqrt(6),-3+sqrt(6))"],"problemType":"TextBox","stepTitle":"$$x^2+6x+3<0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-3-\\\\sqrt{6},-3+\\\\sqrt{6})$$","hints":{"DefaultPathway":[{"id":"a88aae9QuadEq25a-h1","type":"hint","dependencies":[],"title":"Procedure","text":"To solve the quadratic inequality, the procedure is as follows. Solve for the roots of the quadratic formula, divide the number line into intervals based on the roots of the quadratic formula, test the values of each interval to see if they are positive or negative, and determine the intervals where the inequality is correct.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq25a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["-3~sqrt(6)"],"dependencies":["a88aae9QuadEq25a-h1"],"title":"Quadratic Roots","text":"What are the roots of the quadratic equation? Use ~ for $$\\\\frac{plus}{minus}$$ and i for imaginary numbers in the answer. Write the answer in the form of x1,x2 if applicable as well.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq25a-h3","type":"hint","dependencies":["a88aae9QuadEq25a-h2"],"title":"Interval Checking","text":"Now let\'s check our intervals. We will have $$3$$ distinct intervals to check values for.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq25a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a88aae9QuadEq25a-h3"],"title":"Interval Checking","text":"If we plug in $$x=-6$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq25a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6$$"],"dependencies":["a88aae9QuadEq25a-h4"],"title":"Interval Checking","text":"If we plug in $$x=-3$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq25a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a88aae9QuadEq25a-h5"],"title":"Interval Checking","text":"If we plug in $$x=0$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq25a-h7","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-3-sqrt(6),-3+sqrt(6))"],"dependencies":["a88aae9QuadEq25a-h6"],"title":"Solution","text":"Since we have checked each interval, which ones have negative values in them?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a88aae9QuadEq26","title":"Solve The Quadratic Inequality","body":"Solve the following inequality analytically and write where the inequality is true using interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Solve Quadratic Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a88aae9QuadEq26a","stepAnswer":["(-inf,inf)"],"problemType":"TextBox","stepTitle":"$$-\\\\left(x^2\\\\right)+2x-7<0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\infty)$$","hints":{"DefaultPathway":[{"id":"a88aae9QuadEq26a-h1","type":"hint","dependencies":[],"title":"Procedure","text":"To solve the quadratic inequality, the procedure is as follows. Solve for the roots of the quadratic formula, divide the number line into intervals based on the roots of the quadratic formula, test the values of each interval to see if they are positive or negative, and determine the intervals where the inequality is correct.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq26a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["1~i*sqrt(6)"],"dependencies":["a88aae9QuadEq26a-h1"],"title":"Quadratic Roots","text":"What are the roots of the quadratic equation? Use ~ for $$\\\\frac{plus}{minus}$$ and i for imaginary numbers in the answer. Write the answer in the form of x1,x2 if applicable as well.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq26a-h3","type":"hint","dependencies":["a88aae9QuadEq26a-h2"],"title":"Interval Checking","text":"If our roots are comprised of imaginary numbers, then that root does not seperate two intervals. In other words, we can ignore the imaginary solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq26a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-7$$"],"dependencies":["a88aae9QuadEq26a-h3"],"title":"Interval Checking","text":"If we plug in $$x=0$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq26a-h5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-inf,inf)"],"dependencies":["a88aae9QuadEq26a-h4"],"title":"Solution","text":"Since we only have $$1$$ interval and it is negative, where will our inequality be true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a88aae9QuadEq27","title":"Solve The Quadratic Inequality","body":"Solve the following inequality analytically and write where the inequality is true using interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Solve Quadratic Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a88aae9QuadEq27a","stepAnswer":["(-inf,inf)"],"problemType":"TextBox","stepTitle":"$$-2\\\\left(x^2\\\\right)+8x-10<0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\infty)$$","hints":{"DefaultPathway":[{"id":"a88aae9QuadEq27a-h1","type":"hint","dependencies":[],"title":"Procedure","text":"To solve the quadratic inequality, the procedure is as follows. Solve for the roots of the quadratic formula, divide the number line into intervals based on the roots of the quadratic formula, test the values of each interval to see if they are positive or negative, and determine the intervals where the inequality is correct.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq27a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["2~i"],"dependencies":["a88aae9QuadEq27a-h1"],"title":"Quadratic Roots","text":"What are the roots of the quadratic equation? Use ~ for $$\\\\frac{plus}{minus}$$ and i for imaginary numbers in the answer. Write the answer in the form of x1,x2 if applicable as well.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq27a-h3","type":"hint","dependencies":["a88aae9QuadEq27a-h2"],"title":"Interval Checking","text":"If our roots are comprised of imaginary numbers, then that root does not seperate two intervals. In other words, we can ignore the imaginary solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq27a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-10$$"],"dependencies":["a88aae9QuadEq27a-h3"],"title":"Interval Checking","text":"If we plug in $$x=0$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq27a-h5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-inf,inf)"],"dependencies":["a88aae9QuadEq27a-h4"],"title":"Solution","text":"Since we only have $$1$$ interval and it is negative, where will our inequality be true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a88aae9QuadEq28","title":"Solve The Quadratic Inequality","body":"Solve the following inequality analytically and write where the inequality is true using interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Solve Quadratic Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a88aae9QuadEq28a","stepAnswer":["(-inf,inf)"],"problemType":"TextBox","stepTitle":"$$-\\\\left(x^2\\\\right)-4x-5<0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\infty)$$","hints":{"DefaultPathway":[{"id":"a88aae9QuadEq28a-h1","type":"hint","dependencies":[],"title":"Procedure","text":"To solve the quadratic inequality, the procedure is as follows. Solve for the roots of the quadratic formula, divide the number line into intervals based on the roots of the quadratic formula, test the values of each interval to see if they are positive or negative, and determine the intervals where the inequality is correct.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq28a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["-(2~i)"],"dependencies":["a88aae9QuadEq28a-h1"],"title":"Quadratic Roots","text":"What are the roots of the quadratic equation? Use ~ for $$\\\\frac{plus}{minus}$$ and i for imaginary numbers in the answer. Write the answer in the form of x1,x2 if applicable as well.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq28a-h3","type":"hint","dependencies":["a88aae9QuadEq28a-h2"],"title":"Interval Checking","text":"If our roots are comprised of imaginary numbers, then that root does not seperate two intervals. In other words, we can ignore the imaginary solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq28a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["a88aae9QuadEq28a-h3"],"title":"Interval Checking","text":"If we plug in $$x=0$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq28a-h5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-inf,inf)"],"dependencies":["a88aae9QuadEq28a-h4"],"title":"Solution","text":"Since we only have $$1$$ interval and it is negative, where will our inequality be true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a88aae9QuadEq29","title":"Solve The Quadratic Inequality","body":"Solve the following inequality analytically and write where the inequality is true using interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Solve Quadratic Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a88aae9QuadEq29a","stepAnswer":["None"],"problemType":"TextBox","stepTitle":"$$-\\\\left(x^2\\\\right)+x-7>0$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a88aae9QuadEq29a-h1","type":"hint","dependencies":[],"title":"Procedure","text":"To solve the quadratic inequality, the procedure is as follows. Solve for the roots of the quadratic formula, divide the number line into intervals based on the roots of the quadratic formula, test the values of each interval to see if they are positive or negative, and determine the intervals where the inequality is correct.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq29a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(1~3*i*sqrt(3))/2"],"dependencies":["a88aae9QuadEq29a-h1"],"title":"Quadratic Roots","text":"What are the roots of the quadratic equation? Use ~ for $$\\\\frac{plus}{minus}$$ and i for imaginary numbers in the answer. Write the answer in the form of x1,x2 if applicable as well.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq29a-h3","type":"hint","dependencies":["a88aae9QuadEq29a-h2"],"title":"Interval Checking","text":"If our roots are comprised of imaginary numbers, then that root does not seperate two intervals. In other words, we can ignore the imaginary solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq29a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-7$$"],"dependencies":["a88aae9QuadEq29a-h3"],"title":"Interval Checking","text":"If we plug in $$x=0$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq29a-h5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["None"],"dependencies":["a88aae9QuadEq29a-h4"],"title":"Solution","text":"Since we only have $$1$$ interval and it is negative, where will our inequality be true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a88aae9QuadEq3","title":"Solve The Quadratic Inequality","body":"Solve the following inequality graphically: That is, graph the inequality and write where the inequality is true using interval notation. Write the answer in the form of (x1, x2) where () exclude a number and [] include a number. In the case of $$infinities$$, write it as $$\\\\infty$$ and if there are multiple ranges, use & to include it as well..","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Solve Quadratic Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a88aae9QuadEq3a","stepAnswer":["(-4,2)"],"problemType":"TextBox","stepTitle":"$$x^2+2x-8<0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-4,2)$$","hints":{"DefaultPathway":[{"id":"a88aae9QuadEq3a-h1","type":"hint","dependencies":[],"title":"Positive or Negative?","text":"Since our formula is going to be less than zero, we want to look at the interval on the graph where our line is underneath zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq3a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(2,4)"],"dependencies":["a88aae9QuadEq3a-h1"],"title":"Solution","text":"When we graph our formula, what part of the graph is going to be less than zero?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a88aae9QuadEq30","title":"Solve The Quadratic Inequality","body":"Solve the following inequality analytically and write where the inequality is true using interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Solve Quadratic Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a88aae9QuadEq30a","stepAnswer":["(-inf,inf)"],"problemType":"TextBox","stepTitle":"$$x^2+3x+5>0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\infty)$$","hints":{"DefaultPathway":[{"id":"a88aae9QuadEq30a-h1","type":"hint","dependencies":[],"title":"Procedure","text":"To solve the quadratic inequality, the procedure is as follows. Solve for the roots of the quadratic formula, divide the number line into intervals based on the roots of the quadratic formula, test the values of each interval to see if they are positive or negative, and determine the intervals where the inequality is correct.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq30a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-3~i*sqrt(11))/2"],"dependencies":["a88aae9QuadEq30a-h1"],"title":"Quadratic Roots","text":"What are the roots of the quadratic equation? Use ~ for $$\\\\frac{plus}{minus}$$ and i for imaginary numbers in the answer. Write the answer in the form of x1,x2 if applicable as well.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq30a-h3","type":"hint","dependencies":["a88aae9QuadEq30a-h2"],"title":"Interval Checking","text":"If our roots are comprised of imaginary numbers, then that root does not seperate two intervals. In other words, we can ignore the imaginary solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq30a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a88aae9QuadEq30a-h3"],"title":"Interval Checking","text":"If we plug in $$x=0$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq30a-h5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-inf,inf)"],"dependencies":["a88aae9QuadEq30a-h4"],"title":"Solution","text":"Since we only have $$1$$ interval and it is negative, where will our inequality be true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a88aae9QuadEq4","title":"Solve The Quadratic Inequality","body":"Solve the following inequality graphically: That is, graph the inequality and write where the inequality is true using interval notation. Write the answer in the form of (x1, x2) where () exclude a number and [] include a number. In the case of $$infinities$$, write it as $$\\\\infty$$ and if there are multiple ranges, use & to include it as well..","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Solve Quadratic Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a88aae9QuadEq4a","stepAnswer":["(-inf,2]&[6,inf)"],"problemType":"TextBox","stepTitle":"$$x^2-8x+12 \\\\geq 0$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a88aae9QuadEq4a-h1","type":"hint","dependencies":[],"title":"Positive or Negative?","text":"Since our formula is going to be greater than or equal to zero, we want to look at the interval on the graph where our line is zero or above.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq4a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-inf,2]&[6,inf)"],"dependencies":["a88aae9QuadEq4a-h1"],"title":"Solution","text":"When we graph our formula, what part of the graph is going to be less than zero?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a88aae9QuadEq5","title":"Solve The Quadratic Inequality","body":"Solve the following inequality graphically: That is, graph the inequality and write where the inequality is true using interval notation. Write the answer in the form of (x1, x2) where () exclude a number and [] include a number. In the case of $$infinities$$, write it as $$\\\\infty$$ and if there are multiple ranges, use & to include it as well..","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Solve Quadratic Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a88aae9QuadEq5a","stepAnswer":["(-inf,-6]&[-2,inf)"],"problemType":"TextBox","stepTitle":"$$-\\\\left(x^2\\\\right)-8x-12 \\\\leq 0$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a88aae9QuadEq5a-h1","type":"hint","dependencies":[],"title":"Positive or Negative?","text":"Since our formula is going to be less than or equal to zero, we want to look at the interval on the graph where our line is at or underneath zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq5a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-inf,-6]&[-2,inf)"],"dependencies":["a88aae9QuadEq5a-h1"],"title":"Solution","text":"When we graph our formula, what part of the graph is going to be less than zero?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a88aae9QuadEq6","title":"Solve The Quadratic Inequality","body":"Solve the following inequality analytically and write where the inequality is true using interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Solve Quadratic Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a88aae9QuadEq6a","stepAnswer":["(-inf,-3]&[4,inf)"],"problemType":"TextBox","stepTitle":"$$x^2-x-12 \\\\geq 0$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a88aae9QuadEq6a-h1","type":"hint","dependencies":[],"title":"Procedure","text":"To solve the quadratic inequality, the procedure is as follows. Solve for the roots of the quadratic formula, divide the number line into intervals based on the roots of the quadratic formula, test the values of each interval to see if they are positive or negative, and determine the intervals where the inequality is correct.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq6a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["-3,4"],"dependencies":["a88aae9QuadEq6a-h1"],"title":"Quadratic Roots","text":"What are the roots of the quadratic equation? Use ~ for $$\\\\frac{plus}{minus}$$ and i for imaginary numbers in the answer. Write the answer in the form of x1,x2 if applicable as well.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq6a-h3","type":"hint","dependencies":["a88aae9QuadEq6a-h2"],"title":"Interval Checking","text":"Now let\'s check our intervals. We will have $$3$$ distinct intervals to check values for.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a88aae9QuadEq6a-h3"],"title":"Interval Checking","text":"If we plug in $$x=-4$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-12$$"],"dependencies":["a88aae9QuadEq6a-h4"],"title":"Interval Checking","text":"If we plug in $$x=0$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq6a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a88aae9QuadEq6a-h5"],"title":"Interval Checking","text":"If we plug in $$x=5$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq6a-h7","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-inf,-3]&[4,inf)"],"dependencies":["a88aae9QuadEq6a-h6"],"title":"Solution","text":"Since we have checked each interval, which ones have positive values in them?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a88aae9QuadEq7","title":"Solve The Quadratic Inequality","body":"Solve the following inequality analytically and write where the inequality is true using interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Solve Quadratic Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a88aae9QuadEq7a","stepAnswer":["(-inf,-4]&[2,inf)"],"problemType":"TextBox","stepTitle":"$$x^2+2x-8 \\\\geq 0$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a88aae9QuadEq7a-h1","type":"hint","dependencies":[],"title":"Procedure","text":"To solve the quadratic inequality, the procedure is as follows. Solve for the roots of the quadratic formula, divide the number line into intervals based on the roots of the quadratic formula, test the values of each interval to see if they are positive or negative, and determine the intervals where the inequality is correct.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq7a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["-4,2"],"dependencies":["a88aae9QuadEq7a-h1"],"title":"Quadratic Roots","text":"What are the roots of the quadratic equation? Use ~ for $$\\\\frac{plus}{minus}$$ and i for imaginary numbers in the answer. Write the answer in the form of x1,x2 if applicable as well.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq7a-h3","type":"hint","dependencies":["a88aae9QuadEq7a-h2"],"title":"Interval Checking","text":"Now let\'s check our intervals. We will have $$3$$ distinct intervals to check values for.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a88aae9QuadEq7a-h3"],"title":"Interval Checking","text":"If we plug in $$x=-5$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq7a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-8$$"],"dependencies":["a88aae9QuadEq7a-h4"],"title":"Interval Checking","text":"If we plug in $$x=0$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq7a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a88aae9QuadEq7a-h5"],"title":"Interval Checking","text":"If we plug in $$x=3$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq7a-h7","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-inf,-4]&[2,inf)"],"dependencies":["a88aae9QuadEq7a-h6"],"title":"Solution","text":"Since we have checked each interval, which ones have positive values in them?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a88aae9QuadEq8","title":"Solve The Quadratic Inequality","body":"Solve the following inequality analytically and write where the inequality is true using interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Solve Quadratic Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a88aae9QuadEq8a","stepAnswer":["[-3,5]"],"problemType":"TextBox","stepTitle":"$$x^2-2x-15 \\\\leq 0$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a88aae9QuadEq8a-h1","type":"hint","dependencies":[],"title":"Procedure","text":"To solve the quadratic inequality, the procedure is as follows. Solve for the roots of the quadratic formula, divide the number line into intervals based on the roots of the quadratic formula, test the values of each interval to see if they are positive or negative, and determine the intervals where the inequality is correct.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq8a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["-3,5"],"dependencies":["a88aae9QuadEq8a-h1"],"title":"Quadratic Roots","text":"What are the roots of the quadratic equation? Use ~ for $$\\\\frac{plus}{minus}$$ and i for imaginary numbers in the answer. Write the answer in the form of x1,x2 if applicable as well.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq8a-h3","type":"hint","dependencies":["a88aae9QuadEq8a-h2"],"title":"Interval Checking","text":"Now let\'s check our intervals. We will have $$3$$ distinct intervals to check values for.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a88aae9QuadEq8a-h3"],"title":"Interval Checking","text":"If we plug in $$x=-4$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq8a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-15$$"],"dependencies":["a88aae9QuadEq8a-h4"],"title":"Interval Checking","text":"If we plug in $$x=0$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq8a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a88aae9QuadEq8a-h5"],"title":"Interval Checking","text":"If we plug in $$x=6$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq8a-h7","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["[-3,5]"],"dependencies":["a88aae9QuadEq8a-h6"],"title":"Solution","text":"Since we have checked each interval, which ones have negative values in them?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a88aae9QuadEq9","title":"Solve The Quadratic Inequality","body":"Solve the following inequality analytically and write where the inequality is true using interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.8 Solve Quadratic Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a88aae9QuadEq9a","stepAnswer":["[3-sqrt(2),3+sqrt(2)]"],"problemType":"TextBox","stepTitle":"$$-\\\\left(x^2\\\\right)+6x-7 \\\\geq 0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[3-\\\\sqrt{2},3+\\\\sqrt{2}]$$","hints":{"DefaultPathway":[{"id":"a88aae9QuadEq9a-h1","type":"hint","dependencies":[],"title":"Procedure","text":"To solve the quadratic inequality, the procedure is as follows. Solve for the roots of the quadratic formula, divide the number line into intervals based on the roots of the quadratic formula, test the values of each interval to see if they are positive or negative, and determine the intervals where the inequality is correct.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq9a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["3~sqrt(2)"],"dependencies":["a88aae9QuadEq9a-h1"],"title":"Quadratic Roots","text":"What are the roots of the quadratic equation? Use ~ for $$\\\\frac{plus}{minus}$$ and i for imaginary numbers in the answer. Write the answer in the form of x1,x2 if applicable as well.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq9a-h3","type":"hint","dependencies":["a88aae9QuadEq9a-h2"],"title":"Interval Checking","text":"Now let\'s check our intervals. We will have $$3$$ distinct intervals to check values for.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a88aae9QuadEq9a-h3"],"title":"Interval Checking","text":"If we plug in $$x=1$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq9a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a88aae9QuadEq9a-h4"],"title":"Interval Checking","text":"If we plug in $$x=2$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq9a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a88aae9QuadEq9a-h5"],"title":"Interval Checking","text":"If we plug in $$x=5$$, what will our value be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a88aae9QuadEq9a-h7","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["[3-sqrt(2),3+sqrt(2)]"],"dependencies":["a88aae9QuadEq9a-h6"],"title":"Solution","text":"Since we have checked each interval, which ones have positive values in them?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ad46flog1","title":"Solve Logarithmic Equations Using the Properties of Logarithms","body":"Solve for $$x$$ in the following equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Solve Exponential and Logarithmic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8ad46flog1a","stepAnswer":["$$9$$"],"problemType":"TextBox","stepTitle":"$$2*\\\\log_{5}\\\\left(x\\\\right)=\\\\log_{5}\\\\left(81\\\\right)$$.","stepBody":"Solve for $$x$$.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9$$","hints":{"DefaultPathway":[{"id":"a8ad46flog1a-h1","type":"hint","dependencies":[],"title":"Use Log Rules","text":"We know that $$n*\\\\log_{a}\\\\left(b\\\\right)=\\\\log_{a}\\\\left(b^n\\\\right)$$.\\\\n$$2*\\\\log_{5}\\\\left(x\\\\right)=\\\\log_{5}\\\\left(x^2\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog1a-h2","type":"hint","dependencies":["a8ad46flog1a-h1"],"title":"Matching bases","text":"$$\\\\log_{5}\\\\left(x^2\\\\right)$$ and $$\\\\log_{5}\\\\left(81\\\\right)$$ have the same log base. $$\\\\log_{5}\\\\left(x^2\\\\right)$$ and $$\\\\log_{5}\\\\left(81\\\\right)$$, so we can see it as $$x^2=81$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog1a-h3","type":"hint","dependencies":["a8ad46flog1a-h2"],"title":"Value of $$b$$ in $$\\\\log_{a}\\\\left(b\\\\right)$$","text":"$$x^2=81$$ has two solution $$x=9$$ and $$x=-9$$. Given $$\\\\log_{a}\\\\left(b\\\\right)$$, $$b$$ can not be negative. Therefore we can omit the negative answer. $$x=9$$ is the only solution for the given equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ad46flog10","title":"Solve Exponential Equations Using Logarithms","body":"Solve for $$x$$ in the following equation. Find the exact answer and then approximate it to three decimal places.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Solve Exponential and Logarithmic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8ad46flog10a","stepAnswer":["$$1.49$$"],"problemType":"TextBox","stepTitle":"$$5^x=11$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1.49$$","hints":{"DefaultPathway":[{"id":"a8ad46flog10a-h1","type":"hint","dependencies":[],"title":"Bring Down the Exponential","text":"Take the logarithm of both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog10a-h2","type":"hint","dependencies":["a8ad46flog10a-h1"],"title":"Apply Log Rule","text":"We know that $$n*\\\\log_{a}\\\\left(b\\\\right)=\\\\log_{a}\\\\left(b^n\\\\right)$$. After taking log both sides, the left expression will be $$\\\\log_{10}\\\\left(5^x\\\\right)=x*\\\\log_{10}\\\\left(5\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog10a-h3","type":"hint","dependencies":["a8ad46flog10a-h2"],"title":"Solve for $$x$$ in New Equation","text":"$$x*\\\\log_{10}\\\\left(5\\\\right)=\\\\log_{10}\\\\left(11\\\\right)$$ We can solve for $$x$$ and get $$x=(\\\\log_{10}\\\\left(11\\\\right))/(\\\\log_{10}\\\\left(5\\\\right))$$. $$(\\\\log_{10}\\\\left(11\\\\right))/(\\\\log_{10}\\\\left(5\\\\right))$$ approximately equals $$1.49$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ad46flog11","title":"Solve Exponential Equations Using Logarithms","body":"Solve for $$x$$ in the following equation. Find the exact answer and then approximate it to three decimal places.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Solve Exponential and Logarithmic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8ad46flog11a","stepAnswer":["$$1.933$$"],"problemType":"TextBox","stepTitle":"$$7^x=43$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1.933$$","hints":{"DefaultPathway":[{"id":"a8ad46flog11a-h1","type":"hint","dependencies":[],"title":"Bring Down the Exponential","text":"Take the logarithm of both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog11a-h2","type":"hint","dependencies":["a8ad46flog11a-h1"],"title":"Apply Log Rule","text":"We know that $$n*\\\\log_{a}\\\\left(b\\\\right)=\\\\log_{a}\\\\left(b^n\\\\right)$$. After taking log both sides, the left expression will be $$\\\\log_{10}\\\\left(7^x\\\\right)=x*\\\\log_{10}\\\\left(7\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog11a-h3","type":"hint","dependencies":["a8ad46flog11a-h2"],"title":"Solve for $$x$$ in New Equation","text":"$$x*\\\\log_{10}\\\\left(7\\\\right)=\\\\log_{10}\\\\left(43\\\\right)$$ We can solve for $$x$$ and get $$x=(\\\\log_{10}\\\\left(43\\\\right))/(\\\\log_{10}\\\\left(7\\\\right))$$ which approximatly equal to $$1.933$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ad46flog12","title":"Solve Exponential Equations Using Logarithms","body":"Solve for $$x$$ in the following equation. Find the exact answer and then approximate it to three decimal places.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Solve Exponential and Logarithmic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8ad46flog12a","stepAnswer":["$$2.205$$"],"problemType":"TextBox","stepTitle":"$$8^x=98$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2.205$$","hints":{"DefaultPathway":[{"id":"a8ad46flog12a-h1","type":"hint","dependencies":[],"title":"Bring Down the Exponential","text":"Take the logarithm of both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog12a-h2","type":"hint","dependencies":["a8ad46flog12a-h1"],"title":"Apply Log Rule","text":"We know that $$n*\\\\log_{a}\\\\left(b\\\\right)=\\\\log_{a}\\\\left(b^n\\\\right)$$. After taking log both sides, the left expression will be $$\\\\log_{10}\\\\left(8^x\\\\right)=x*\\\\log_{10}\\\\left(8\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog12a-h3","type":"hint","dependencies":["a8ad46flog12a-h2"],"title":"Solve for $$x$$ in New Equation","text":"$$x*\\\\log_{10}\\\\left(8\\\\right)=\\\\log_{10}\\\\left(98\\\\right)$$ We can solve for $$x$$ and get $$x=(\\\\log_{10}\\\\left(98\\\\right))/(\\\\log_{10}\\\\left(8\\\\right))$$ which approximatly equal to $$2.205$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ad46flog13","title":"Solve Exponential Equations Using Logarithms","body":"Solve for $$x$$ in the following equation. Find the exact answer and then approximate it to three decimal places.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Solve Exponential and Logarithmic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8ad46flog13a","stepAnswer":["$$0.079$$"],"problemType":"TextBox","stepTitle":"$$3e^{x+2}=24$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.079$$","hints":{"DefaultPathway":[{"id":"a8ad46flog13a-h1","type":"hint","dependencies":[],"title":"Isolate the Exponential Term","text":"Divide $$3$$ both sides get $$e^{x+2}=8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog13a-h2","type":"hint","dependencies":["a8ad46flog13a-h1"],"title":"Bring Down the Exponential","text":"Take the natural logarithm of both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog13a-h3","type":"hint","dependencies":["a8ad46flog13a-h2"],"title":"Take Natural log Both Sides","text":"Definition of Natural log: $$ln(a)=\\\\log_{e}\\\\left(a\\\\right)$$\\\\n$$\\\\ln(a^b)=b \\\\ln(a)$$\\\\n$$ln(e)=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog13a-h4","type":"hint","dependencies":["a8ad46flog13a-h3"],"title":"Use the log Rule","text":"$$\\\\ln(e^{x+2})=\\\\left(x+2\\\\right) \\\\ln(e)=x+2=ln$$ $$8$$. $$x=-2+\\\\ln(8)$$ which approximately equals $$0.079$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ad46flog14","title":"Solve Exponential Equations Using Logarithms","body":"Solve for $$x$$ in the following equation. Find the exact answer and then approximate it to three decimal places.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Solve Exponential and Logarithmic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8ad46flog14a","stepAnswer":["$$4.197$$"],"problemType":"TextBox","stepTitle":"$$2e^{x-2}=18$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4.197$$","hints":{"DefaultPathway":[{"id":"a8ad46flog14a-h1","type":"hint","dependencies":[],"title":"Isolate the Exponential Term","text":"Divide by $$2$$ both sides and get $$e^{x-2}=9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog14a-h2","type":"hint","dependencies":["a8ad46flog14a-h1"],"title":"Bring Down the Exponential","text":"Take the logarithm of both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog14a-h3","type":"hint","dependencies":["a8ad46flog14a-h2"],"title":"Take Natural log Both Sides","text":"Definition of Natural log: $$ln(a)=\\\\log_{e}\\\\left(a\\\\right)$$\\\\n$$\\\\ln(a^b)=b \\\\ln(a)$$\\\\n$$ln(e)=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog14a-h4","type":"hint","dependencies":["a8ad46flog14a-h3"],"title":"Use the log Rule","text":"$$\\\\ln(e^{x-2})=\\\\left(x-2\\\\right) \\\\ln(e)=ln(9)$$\\\\n$$x-2=ln(9)$$, $$x=2+\\\\ln(9)$$ $$x$$ approximately equals $$4.197$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ad46flog15","title":"Solve Exponential Equations Using Logarithms","body":"Solve for $$x$$ in the following equation. Find the exact answer and then approximate it to three decimal places.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Solve Exponential and Logarithmic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8ad46flog15a","stepAnswer":["$$0.805$$"],"problemType":"TextBox","stepTitle":"$$5e^{2x}=25$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.805$$","hints":{"DefaultPathway":[{"id":"a8ad46flog15a-h1","type":"hint","dependencies":[],"title":"Isolate the Exponential Term","text":"Divide by $$5$$ both sides and get $$e^{2x}=5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog15a-h2","type":"hint","dependencies":["a8ad46flog15a-h1"],"title":"Bring Down the Exponential","text":"Take the logarithm of both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog15a-h3","type":"hint","dependencies":["a8ad46flog15a-h2"],"title":"Take Natural log Both Sides","text":"Definition of Natural log: $$ln(a)=\\\\log_{e}\\\\left(a\\\\right)$$\\\\n$$\\\\ln(a^b)=b \\\\ln(a)$$\\\\n$$ln(e)=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog15a-h4","type":"hint","dependencies":["a8ad46flog15a-h3"],"title":"Use the log Rule","text":"$$\\\\ln(e^{2x})=2x \\\\ln(e)=ln(5)$$\\\\n$$2x=ln(5)$$ $$x=\\\\frac{1}{2} \\\\ln(5)$$ which approximately equals $$0.805$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ad46flog16","title":"Use Exponential Models in Applications","body":"Solve for $$x$$ in the following equation. Find the exact answer and then approximate it to three decimal places.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Solve Exponential and Logarithmic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8ad46flog16a","stepAnswer":["$$0.095$$"],"problemType":"TextBox","stepTitle":"Jermael\u2019s parents put $10,000 in investments for his college expenses on his first birthday. They hope the investments will be worth $50,000 when he turns $$18$$. If the interest compounds continuously, approximately what rate of growth will they need to achieve their goal?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.095$$","hints":{"DefaultPathway":[{"id":"a8ad46flog16a-h1","type":"hint","dependencies":[],"title":"Find the Equation for New Balance","text":"For a principal, P, invested at an interest rate, $$r$$, for $$t$$ years, the new balance, A is:\\\\n$$A={Pe}^{r t}$$ when compounded continuously.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog16a-h2","type":"hint","dependencies":["a8ad46flog16a-h1"],"title":"Subsitute the Values into Equation","text":"$$A=50000$$, $$P=10000$$, $$t=17$$. $$50000=10000e^{17r}$$. Solve for $$r$$. We get $$r=\\\\frac{\\\\ln(5)}{17}$$ which approximately equals $$0.095$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ad46flog17","title":"Use Exponential Models in Applications","body":"Solve for $$x$$ in the following equation. Find the exact answer and then approximate it to three decimal places.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Solve Exponential and Logarithmic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8ad46flog17a","stepAnswer":["$$656100$$"],"problemType":"TextBox","stepTitle":"Researchers recorded that a certain bacteria population grew from $$100$$ to $$300$$ in $$3$$ hours. At this rate of growth, how many bacteria will there be $$24$$ hours from the start of the experiment?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$656100$$","hints":{"DefaultPathway":[{"id":"a8ad46flog17a-h1","type":"hint","dependencies":[],"title":"Find the Exponential Growth and Decay Equation","text":"For an original amount, $$A_0$$ , that grows or decays at a rate, k, for a certain time, $$t$$, the final amount, A, is:\\\\n$$A=A_0 e^{k t}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog17a-h2","type":"hint","dependencies":["a8ad46flog17a-h1"],"title":"Subsitute the Values into Equation to find k","text":"$$A=300$$, $$A_0=100$$, $$t=3$$ $$300=100e^{3k}$$ Solve for k and get $$k=(ln$$ (3))/3 which approximately equal $$0.366$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog17a-h3","type":"hint","dependencies":["a8ad46flog17a-h2"],"title":"Subsitute the Values into Equation to find new A","text":"We use this rate of growth to predict the number of bacteria there will be in $$24$$ hours. Plug in $$A_0=100$$, $$k=0.366$$, $$t=24$$ get A=100*e**(24*(ln (3))/3) which approximaely equal $$656100$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ad46flog18","title":"Solve Logarithmic Equations Using the Properties of Logarithms","body":"Principle","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Solve Exponential and Logarithmic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8ad46flog18a","stepAnswer":["$$8$$"],"problemType":"TextBox","stepTitle":"Solve for $$x$$ in the following equation","stepBody":"$$2*\\\\log_{4}\\\\left(x\\\\right)=\\\\log_{4}\\\\left(64\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8$$","hints":{"DefaultPathway":[{"id":"a8ad46flog18a-h1","type":"hint","dependencies":[],"title":"Use Log Rules","text":"We know that $$n*\\\\log_{a}\\\\left(b\\\\right)=\\\\log_{a}\\\\left(b^n\\\\right)$$. $$2*\\\\log_{4}\\\\left(x\\\\right)=\\\\log_{4}\\\\left(x^2\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog18a-h2","type":"hint","dependencies":["a8ad46flog18a-h1"],"title":"Match Bases and Solve for $$x$$","text":"Since both sides has the same base, we can take out the logarithm and get $$x^2=64$$. Solve for $$x^2=64$$ which gives $$x=-8$$ and $$x=8$$. Given $$\\\\log_{a}\\\\left(b\\\\right)$$, $$b$$ can not be negative. We can omit the negative answer. Therefore $$x=8$$ is the only solution for this equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ad46flog19","title":"Solve Logarithmic Equations Using the Properties of Logarithms","body":"Principle","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Solve Exponential and Logarithmic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8ad46flog19a","stepAnswer":["$$7$$"],"problemType":"TextBox","stepTitle":"Solve for $$x$$ in the following equation","stepBody":"$$2*\\\\log_{10}\\\\left(x\\\\right)=\\\\log_{10}\\\\left(49\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7$$","hints":{"DefaultPathway":[{"id":"a8ad46flog19a-h1","type":"hint","dependencies":[],"title":"Use Log Rules","text":"We know that $$n*\\\\log_{a}\\\\left(b\\\\right)=\\\\log_{a}\\\\left(b^n\\\\right)$$. $$2*\\\\log_{10}\\\\left(x\\\\right)=\\\\log_{10}\\\\left(x^2\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog19a-h2","type":"hint","dependencies":["a8ad46flog19a-h1"],"title":"Match Bases and Solve for $$x$$","text":"Since $$\\\\log_{10}\\\\left(x^2\\\\right)=\\\\log_{10}\\\\left(49\\\\right)$$ have the same bases both sides, we can take out the logarithm and get $$x^2=49$$. Solve for $$x^2=49$$ which gives $$x=-7$$ and $$x=7$$. Given $$\\\\log_{a}\\\\left(b\\\\right)$$, $$b$$ can not be negative. We can omit the negative answer. Therefore $$x=7$$ is the only solution for this equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ad46flog2","title":"Solve Logarithmic Equations Using the Properties of Logarithms","body":"Solve for $$x$$ in the following equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Solve Exponential and Logarithmic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8ad46flog2a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"Use Log Rules","stepBody":"$$2*\\\\log_{3}\\\\left(x\\\\right)=\\\\log_{3}\\\\left(36\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"a8ad46flog2a-h1","type":"hint","dependencies":[],"title":"Rewrite expression","text":"We know that $$n*\\\\log_{a}\\\\left(b\\\\right)=\\\\log_{a}\\\\left(b^n\\\\right)$$.\\\\n$$2*\\\\log_{3}\\\\left(x\\\\right)=\\\\log_{3}\\\\left(x^2\\\\right)=\\\\log_{3}\\\\left(36\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog2a-h2","type":"hint","dependencies":["a8ad46flog2a-h1"],"title":"Matching bases","text":"$$\\\\log_{3}\\\\left(x^2\\\\right)$$ and $$\\\\log_{3}\\\\left(x^2\\\\right)$$ have the same log base, so $$x^2=36$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog2a-h3","type":"hint","dependencies":["a8ad46flog2a-h2"],"title":"Value of $$b$$ in $$\\\\log_{a}\\\\left(b\\\\right)$$","text":"$$x^2=36$$ $$x=6$$ or $$x=-6$$. Given $$\\\\log_{a}\\\\left(b\\\\right)$$, $$b$$ can not be negative. Therefore we can omit the negative answer. $$x=6$$ is the only solution for the given equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ad46flog20","title":"Solve Logarithmic Equations Using the Properties of Logarithms","body":"Principle","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Solve Exponential and Logarithmic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8ad46flog20a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"Solve for $$x$$ in the following equation","stepBody":"$$3*\\\\log_{3}\\\\left(x\\\\right)=\\\\log_{3}\\\\left(27\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a8ad46flog20a-h1","type":"hint","dependencies":[],"title":"Use Log Rules","text":"We know that $$n*\\\\log_{a}\\\\left(b\\\\right)=\\\\log_{a}\\\\left(b^n\\\\right)$$. $$3*\\\\log_{3}\\\\left(x\\\\right)=\\\\log_{3}\\\\left(x^3\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog20a-h2","type":"hint","dependencies":["a8ad46flog20a-h1"],"title":"Match Bases and Solve for $$x$$","text":"Since $$3*\\\\log_{3}\\\\left(x\\\\right)=\\\\log_{3}\\\\left(x^3\\\\right)=\\\\log_{3}\\\\left(27\\\\right)$$ have the same bases both sides, we can take out the logarithm and get $$x^3=27$$. Solve for $$x^3=27$$ which gives $$x=3$$. Given $$\\\\log_{a}\\\\left(b\\\\right)$$, $$b$$ can not be negative. $$3$$ is non negative. Therefore $$x=3$$ is the only solution for this equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ad46flog21","title":"Solve Logarithmic Equations Using the Properties of Logarithms","body":"Principle","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Solve Exponential and Logarithmic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8ad46flog21a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"Solve for $$x$$ in the following equation","stepBody":"$$3*\\\\log_{6}\\\\left(x\\\\right)=\\\\log_{6}\\\\left(64\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a8ad46flog21a-h1","type":"hint","dependencies":[],"title":"Use Log Rules","text":"We know that $$n*\\\\log_{a}\\\\left(b\\\\right)=\\\\log_{a}\\\\left(b^n\\\\right)$$. $$3*\\\\log_{6}\\\\left(x\\\\right)=\\\\log_{6}\\\\left(x^3\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog21a-h2","type":"hint","dependencies":["a8ad46flog21a-h1"],"title":"Match Bases and Solve for $$x$$","text":"Since $$3*\\\\log_{6}\\\\left(x\\\\right)=\\\\log_{6}\\\\left(x^3\\\\right)=\\\\log_{6}\\\\left(64\\\\right)$$ have the same bases both sides, we can take out the logarithm and get $$x^3=64$$. Solve for $$x^3=64$$ which gives $$x=4$$. Given $$\\\\log_{a}\\\\left(b\\\\right)$$, $$b$$ can not be negative. $$4$$ is non negative. Therefore $$x=4$$ is the only solution for this equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog21a-h1","type":"hint","dependencies":[],"title":"Calculation","text":"$$3*\\\\log_{6}\\\\left(x\\\\right)=\\\\log_{6}\\\\left(x^3\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ad46flog22","title":"Solve Logarithmic Equations Using the Properties of Logarithms","body":"Solve for $$x$$ in the following equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Solve Exponential and Logarithmic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8ad46flog22a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{5}\\\\left(4x-2\\\\right)=\\\\log_{5}\\\\left(10\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a8ad46flog22a-h1","type":"hint","dependencies":[],"title":"Matching bases","text":"With all terms containing same log base, the log can be removed, so it becomes $$4x-2=10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog22a-h2","type":"hint","dependencies":["a8ad46flog22a-h1"],"title":"Solve for $$x$$","text":"$$4x-2=10$$ Solve for $$x$$ get $$x=3$$ which is positive. Given $$\\\\log_{a}\\\\left(b\\\\right)$$, $$b$$ can not be negative. So $$x=3$$ is the solution for the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ad46flog23","title":"Solve Logarithmic Equations Using the Properties of Logarithms","body":"Solve for $$x$$ in the following equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Solve Exponential and Logarithmic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8ad46flog23a","stepAnswer":["1,3"],"problemType":"MultipleChoice","stepTitle":"$$\\\\log_{3}\\\\left(x^2+3\\\\right)=\\\\log_{3}\\\\left(4x\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["1,3","2,4","1,4"],"hints":{"DefaultPathway":[{"id":"a8ad46flog23a-h1","type":"hint","dependencies":[],"title":"Match Bases","text":"With all terms containing same log base, the log can be removed, so it becomes $$x^2+3=4x$$. Solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog23a-h2","type":"hint","dependencies":["a8ad46flog23a-h1"],"title":"Solve for $$x$$","text":"We can simplify into $$x^2-4x+3=0$$ and solve for $$x$$. We can factor the quadratic get $$\\\\left(x-1\\\\right) \\\\left(x-3\\\\right)=0$$ which have $$x=1$$ or $$x=3$$ as solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog23a-h3","type":"hint","dependencies":["a8ad46flog23a-h2"],"title":"Check for Constrains","text":"Given $$\\\\log_{a}\\\\left(b\\\\right)$$, $$b$$ can not be negative. Both $$1$$ and $$3$$ are non negative. Therefore $$x=1$$ and $$x=3$$ are solutions for this equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ad46flog24","title":"Solve Logarithmic Equations Using the Properties of Logarithms","body":"Solve for $$x$$ in the following equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Solve Exponential and Logarithmic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8ad46flog24a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{3}\\\\left(x\\\\right)+\\\\log_{3}\\\\left(x\\\\right)=2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a8ad46flog24a-h1","type":"hint","dependencies":[],"title":"Use Log Rules","text":"Logarithm multiplication rule: $$\\\\log_{a}\\\\left(b\\\\right)+\\\\log_{a}\\\\left(c\\\\right)=\\\\log_{a}\\\\left(b c\\\\right)$$. $$\\\\log_{3}\\\\left(x\\\\right)+\\\\log_{3}\\\\left(x\\\\right)=\\\\log_{3}\\\\left(x^2\\\\right)=2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog24a-h2","type":"hint","dependencies":["a8ad46flog24a-h1"],"title":"Turn the Constant into log","text":"Use the definition of logarithm, we can get $$2=\\\\log_{3}\\\\left(9\\\\right)$$ since $$3^2=9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog24a-h3","type":"hint","dependencies":["a8ad46flog24a-h2"],"title":"Match Bases","text":"With all term containing same log base, the log can be removed, so it becomes $$x^2=9$$. Solve for $$x$$ and get $$x=-3$$ or $$x=3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog24a-h4","type":"hint","dependencies":["a8ad46flog24a-h3"],"title":"Check for Constrains","text":"Given $$\\\\log_{a}\\\\left(b\\\\right)$$, $$b$$ can not be negative. We can omit the negative answer and get $$x=3$$ as the only solution for the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ad46flog25","title":"Solve Logarithmic Equations Using the Properties of Logarithms","body":"Solve for $$x$$ in the following equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Solve Exponential and Logarithmic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8ad46flog25a","stepAnswer":["$$8$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{4}\\\\left(x\\\\right)+\\\\log_{4}\\\\left(x\\\\right)=3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8$$","hints":{"DefaultPathway":[{"id":"a8ad46flog25a-h1","type":"hint","dependencies":[],"title":"Use Log Rules","text":"Logarithm multiplication rule: $$\\\\log_{a}\\\\left(b\\\\right)+\\\\log_{a}\\\\left(c\\\\right)=\\\\log_{a}\\\\left(b c\\\\right)$$. $$\\\\log_{4}\\\\left(x\\\\right)+\\\\log_{4}\\\\left(x\\\\right)=\\\\log_{4}\\\\left(x^2\\\\right)=3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog25a-h2","type":"hint","dependencies":["a8ad46flog25a-h1"],"title":"Turn the Constant into log","text":"Use the definition of logarithm, we can get $$3=\\\\log_{4}\\\\left(64\\\\right)$$ since $$4^3=64$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog25a-h3","type":"hint","dependencies":["a8ad46flog25a-h2"],"title":"Match Bases","text":"With all terms containing same log base, the log can be removed, so it becomes $$x^2=64$$. Solve for $$x$$ and get $$x=-8$$ or $$x=8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog25a-h4","type":"hint","dependencies":["a8ad46flog25a-h3"],"title":"Check for Constrains","text":"Given $$\\\\log_{a}\\\\left(b\\\\right)$$, $$b$$ cannot be negative. We can omit the negative answer and get $$x=8$$ as the only solution for the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ad46flog26","title":"Solve Logarithmic Equations Using the Properties of Logarithms","body":"Solve for $$x$$ in the following equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Solve Exponential and Logarithmic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8ad46flog26a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{3}\\\\left(x\\\\right)+\\\\log_{3}\\\\left(x+6\\\\right)=3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a8ad46flog26a-h1","type":"hint","dependencies":[],"title":"Use Log Rules","text":"Logarithm multiplication rule: $$\\\\log_{a}\\\\left(b\\\\right)+\\\\log_{a}\\\\left(c\\\\right)=\\\\log_{a}\\\\left(b c\\\\right)$$. $$\\\\log_{3}\\\\left(x\\\\right)+\\\\log_{3}\\\\left(x+6\\\\right)=\\\\log_{3}\\\\left(x \\\\left(x+6\\\\right)\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog26a-h2","type":"hint","dependencies":["a8ad46flog26a-h1"],"title":"Turn the Constant into log","text":"Use the definition of logarithm, we can get $$3=\\\\log_{3}\\\\left(27\\\\right)$$ since $$3^3=27$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog26a-h3","type":"hint","dependencies":["a8ad46flog26a-h2"],"title":"Match Bases","text":"With all term containing same log base, the log can be removed, so it becomes $$x \\\\left(x+6\\\\right)=27$$. Solve for $$x$$.\\\\n$$x^2+6-27=0$$ We can factor the quadratic and get $$\\\\left(x+9\\\\right) \\\\left(x-3\\\\right)=0$$. $$x=-9$$ or $$x=3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog26a-h4","type":"hint","dependencies":["a8ad46flog26a-h3"],"title":"Check for Constrains","text":"Given $$\\\\log_{a}\\\\left(b\\\\right)$$, $$b$$ can not be negative. We can omit the negative answer and get $$x=3$$ as the only solution for the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ad46flog27","title":"Solve Logarithmic Equations Using the Properties of Logarithms","body":"Solve for $$x$$ in the following equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Solve Exponential and Logarithmic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8ad46flog27a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{2}\\\\left(x\\\\right)+\\\\log_{2}\\\\left(x-3\\\\right)=2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a8ad46flog27a-h1","type":"hint","dependencies":[],"title":"Use Log Rules","text":"Logarithm multiplication rule: $$\\\\log_{a}\\\\left(b\\\\right)+\\\\log_{a}\\\\left(c\\\\right)=\\\\log_{a}\\\\left(b c\\\\right).\\\\log_{2}\\\\left(x\\\\right)+\\\\log_{2}\\\\left(x-3\\\\right)=\\\\log_{2}\\\\left(x \\\\left(x-3\\\\right)\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog27a-h2","type":"hint","dependencies":["a8ad46flog27a-h1"],"title":"Turn the Constant into log","text":"Use the definition of logarithm, we can get $$2=\\\\log_{2}\\\\left(4\\\\right)$$ since $$2^2=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog27a-h3","type":"hint","dependencies":["a8ad46flog27a-h2"],"title":"Match Bases","text":"With all term containing same log base, the log can be removed, so it becomes $$x \\\\left(x-3\\\\right)=4$$. We can simplify into $$x^2-3x-4=0$$. We can factor the quadratic and get $$\\\\left(x-4\\\\right) \\\\left(x+1\\\\right)=0$$ which has $$x=4$$ and $$x=-1$$ as solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog27a-h4","type":"hint","dependencies":["a8ad46flog27a-h3"],"title":"Check for Constrains","text":"Given $$\\\\log_{a}\\\\left(b\\\\right)$$, $$b$$ can not be negative. We can omit the negative answer and get $$x=4$$ as the only solution for the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ad46flog28","title":"Solve Logarithmic Equations Using the Properties of Logarithms","body":"Solve for $$x$$ in the following equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Solve Exponential and Logarithmic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8ad46flog28a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{10}\\\\left(x\\\\right)+\\\\log_{10}\\\\left(x+3\\\\right)=1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a8ad46flog28a-h1","type":"hint","dependencies":[],"title":"Use Log Rules","text":"Logarithm multiplication rule: $$\\\\log_{a}\\\\left(b\\\\right)+\\\\log_{a}\\\\left(c\\\\right)=\\\\log_{a}\\\\left(b c\\\\right)$$. $$\\\\log_{10}\\\\left(x\\\\right)+\\\\log_{10}\\\\left(x+3\\\\right)=\\\\log_{10}\\\\left(x \\\\left(x+3\\\\right)\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog28a-h2","type":"hint","dependencies":["a8ad46flog28a-h1"],"title":"Turn the Constant into log","text":"$$1=\\\\log_{10}\\\\left(10\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog28a-h3","type":"hint","dependencies":["a8ad46flog28a-h2"],"title":"Match Bases","text":"With all term containing same log base, the log can be removed, so it becomes $$x \\\\left(x+3\\\\right)=10$$. We can simplify it into $$x^2+3x-10=0$$. We can factor the quadratic and get $$\\\\left(x+5\\\\right) \\\\left(x-2\\\\right)=0$$ which has $$x=-5$$ and $$x=2$$ as solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog28a-h4","type":"hint","dependencies":["a8ad46flog28a-h3"],"title":"Check for Constrains","text":"Given $$\\\\log_{a}\\\\left(b\\\\right)$$, $$b$$ can not be negative. We can omit the negative answer and get $$x=2$$ as the only solution for the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ad46flog29","title":"Solve Logarithmic Equations Using the Properties of Logarithms","body":"Solve for $$x$$ in the following equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Solve Exponential and Logarithmic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8ad46flog29a","stepAnswer":["$$20$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{10}\\\\left(x\\\\right)+\\\\log_{10}\\\\left(x-15\\\\right)=2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$20$$","hints":{"DefaultPathway":[{"id":"a8ad46flog29a-h1","type":"hint","dependencies":[],"title":"Use Log Rules","text":"Logarithm multiplication rule: $$\\\\log_{a}\\\\left(b\\\\right)+\\\\log_{a}\\\\left(c\\\\right)=\\\\log_{a}\\\\left(b c\\\\right)$$. log{10}{x}+log{10}{x-15}=og{10}{x*(x-15)}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog29a-h2","type":"hint","dependencies":["a8ad46flog29a-h1"],"title":"Turn the Constant into log","text":"Use the definition of logarithm, we get $$2=\\\\log_{10}\\\\left(100\\\\right)$$ since $${10}^2=100$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog29a-h3","type":"hint","dependencies":["a8ad46flog29a-h2"],"title":"Match Bases","text":"With all term containing same log base, the log can be removed, so it becomes $$x \\\\left(x-15\\\\right)=100$$. $$x^2-15x-100=0$$. We can factor the quadratic as $$\\\\left(x-20\\\\right) \\\\left(x+5\\\\right)=0$$ which has $$x=20$$ and $$x=-5$$ as solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog29a-h4","type":"hint","dependencies":["a8ad46flog29a-h3"],"title":"Check for Constrains","text":"Given $$\\\\log_{a}\\\\left(b\\\\right)$$, $$b$$ can not be negative. We can omit the negative answer and get $$x=20$$ as the only solution for the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ad46flog3","title":"Solve Logarithmic Equations Using the Properties of Logarithms.","body":"Solve for $$x$$ in the following equation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Solve Exponential and Logarithmic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8ad46flog3a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"Use Log Rules","stepBody":"We know that $$n*\\\\log_{a}\\\\left(b\\\\right)=\\\\log_{a}\\\\left(b^n\\\\right)$$. $$3*\\\\log_{10}\\\\left(x\\\\right)=\\\\log_{10}\\\\left(x^3\\\\right)=\\\\log_{10}\\\\left(64\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a8ad46flog3a-h1","type":"hint","dependencies":[],"title":"Matching bases","text":"$$\\\\log_{10}\\\\left(3x\\\\right)$$ and $$\\\\log_{10}\\\\left(x^3\\\\right)$$ have the same log base, so we can see it as $$x^3=64$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog3a-h2","type":"hint","dependencies":["a8ad46flog3a-h1"],"title":"Final Answer","text":"Solve for $$x^3=64$$. Given $$\\\\log_{a}\\\\left(b\\\\right)$$, $$b$$ can not be negative.The only answer is $$x=4$$ which is positive. Therefore, $$x=4$$ is the only solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ad46flog30","title":"Use Exponential Models in Applications","body":"Solve for $$x$$ in the following equation. Find the exact answer and then approximate it to three decimal places.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Solve Exponential and Logarithmic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8ad46flog30a","stepAnswer":["$$0.386$$"],"problemType":"TextBox","stepTitle":"$$4e^{x+1}=16$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.386$$","hints":{"DefaultPathway":[{"id":"a8ad46flog30a-h1","type":"hint","dependencies":[],"title":"Isolate the Exponential Term","text":"Divide by $$4$$ both sides and get $$e^{x+1}=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog30a-h2","type":"hint","dependencies":["a8ad46flog30a-h1"],"title":"Bring Down the Exponential","text":"Take the logarithm of both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog30a-h3","type":"hint","dependencies":["a8ad46flog30a-h2"],"title":"Take Natural log Both Sides","text":"Definition of Natural log: $$ln(a)=\\\\log_{e}\\\\left(a\\\\right)$$\\\\n$$\\\\ln(a^b)=b \\\\ln(a)$$\\\\n$$ln(e)=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog30a-h4","type":"hint","dependencies":["a8ad46flog30a-h3"],"title":"Use the log Rule","text":"$$\\\\ln(e^{x+1})=\\\\left(x+1\\\\right) \\\\ln(e)=x+1=ln(4)$$\\\\n$$x=-1+\\\\ln(4)$$ which approximately equals $$0.386$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ad46flog4","title":"Solve Logarithmic Equations Using the Properties of Logarithms","body":"Solve for $$x$$ in the following equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Solve Exponential and Logarithmic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8ad46flog4a","stepAnswer":["$$9$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{3}\\\\left(x\\\\right)+\\\\log_{3}\\\\left(x-8\\\\right)=2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9$$","hints":{"DefaultPathway":[{"id":"a8ad46flog4a-h1","type":"hint","dependencies":[],"title":"Use Log Rules","text":"Logarithm multiplication rule: $$\\\\log_{a}\\\\left(b\\\\right)+\\\\log_{a}\\\\left(c\\\\right)=\\\\log_{a}\\\\left(b c\\\\right)$$, so $$\\\\log_{3}\\\\left(x\\\\right)+\\\\log_{3}\\\\left(x-8\\\\right)=\\\\log_{3}\\\\left(x \\\\left(x-8\\\\right)\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog4a-h2","type":"hint","dependencies":["a8ad46flog4a-h1"],"title":"Rewrite the Expression","text":"$$2=\\\\log_{3}\\\\left(9\\\\right)$$, so we can rewrite it as $$\\\\log_{3}\\\\left(9\\\\right)=\\\\log_{3}\\\\left(x \\\\left(x-8\\\\right)\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog4a-h3","type":"hint","dependencies":["a8ad46flog4a-h2"],"title":"Matching bases","text":"With all term containing same log base, the log can be removed, so it becomes $$x \\\\left(x-8\\\\right)=9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog4a-h4","type":"hint","dependencies":["a8ad46flog4a-h3"],"title":"Solve for $$x$$","text":"Given $$x \\\\left(x-8\\\\right)=9$$ we can solve for $$x$$. $$x \\\\left(x-8\\\\right)=9$$ can be simplified into $$x^2-8x-9=0$$. We can factor $$x^2-8x-9$$ as $$\\\\left(x-9\\\\right) \\\\left(x+1\\\\right)$$. $$\\\\left(x-9\\\\right) \\\\left(x+1\\\\right)=0$$ has two solutions which are $$x=-1$$ and $$x=9$$. Given $$\\\\log_{a}\\\\left(b\\\\right)$$, $$b$$ can not be negative. We can omit the negative answer. Therefore $$x=9$$ is the only solution for this equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ad46flog5","title":"Solve Logarithmic Equations Using the Properties of Logarithms","body":"Solve for $$x$$ in the following equation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Solve Exponential and Logarithmic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8ad46flog5a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{2}\\\\left(x\\\\right)+\\\\log_{2}\\\\left(x-2\\\\right)=3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a8ad46flog5a-h1","type":"hint","dependencies":[],"title":"Use Log Rules","text":"Logarithm multiplication rule: $$\\\\log_{a}\\\\left(b\\\\right)+\\\\log_{a}\\\\left(c\\\\right)=\\\\log_{a}\\\\left(b c\\\\right)$$, so $$\\\\log_{2}\\\\left(x\\\\right)+\\\\log_{2}\\\\left(x-2\\\\right)=\\\\log_{2}\\\\left(x \\\\left(x-2\\\\right)\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog5a-h2","type":"hint","dependencies":["a8ad46flog5a-h1"],"title":"Rewrite the Expression","text":"$$3=\\\\log_{2}\\\\left(8\\\\right)=\\\\log_{2}\\\\left(x \\\\left(x-2\\\\right)\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog5a-h3","type":"hint","dependencies":["a8ad46flog5a-h2"],"title":"Matching bases","text":"$$\\\\log_{2}\\\\left(8\\\\right)$$ and $$\\\\log_{2}\\\\left(x \\\\left(x-2\\\\right)\\\\right)$$ containing same log base, the log can be removed, so it becomes $$x \\\\left(x-2\\\\right)=8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog5a-h4","type":"hint","dependencies":["a8ad46flog5a-h3"],"title":"Solve for $$x$$","text":"Given $$x \\\\left(x-2\\\\right)=8$$ solve for $$x$$. We can simplify it as $$x^2-2x-8=0$$. Then we can factor the quadratic and get $$\\\\left(x-4\\\\right) \\\\left(x+2\\\\right)=0$$ which gives $$x=4$$ or $$x=-2$$. Given $$\\\\log_{a}\\\\left(b\\\\right)$$, $$b$$ can not be negative. We can omit the negative answer. Therefore $$x=4$$ is the only solution for this equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ad46flog6","title":"Solve Logarithmic Equations Using the Properties of Logarithms","body":"Solve for $$x$$ in the following equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Solve Exponential and Logarithmic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8ad46flog6a","stepAnswer":["$$8$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{2}\\\\left(x\\\\right)+\\\\log_{2}\\\\left(x-6\\\\right)=4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8$$","hints":{"DefaultPathway":[{"id":"a8ad46flog6a-h1","type":"hint","dependencies":[],"title":"Use Log Rules","text":"Logarithm multiplication rule: $$\\\\log_{a}\\\\left(b\\\\right)+\\\\log_{a}\\\\left(c\\\\right)=\\\\log_{a}\\\\left(b c\\\\right)$$, so $$\\\\log_{2}\\\\left(x\\\\right)+\\\\log_{2}\\\\left(x-6\\\\right)=\\\\log_{2}\\\\left(x \\\\left(x-6\\\\right)\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog6a-h2","type":"hint","dependencies":["a8ad46flog6a-h1"],"title":"Rewrite the Expression","text":"$$4=\\\\log_{2}\\\\left(16\\\\right)=\\\\log_{2}\\\\left(x \\\\left(x-6\\\\right)\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog6a-h3","type":"hint","dependencies":["a8ad46flog6a-h2"],"title":"Matching bases","text":"Both $$\\\\log_{2}\\\\left(16\\\\right)$$ and $$\\\\log_{2}\\\\left(x \\\\left(x-6\\\\right)\\\\right)$$ containing same log base, the log can be removed, so it becomes $$x \\\\left(x-6\\\\right)=16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog6a-h4","type":"hint","dependencies":["a8ad46flog6a-h3"],"title":"Solve for $$x$$","text":"Given $$x \\\\left(x-6\\\\right)=16$$ solve for $$x$$. We can simplify it as $$x^2-6x-16=0$$. Then we can factor the quadratic and get $$\\\\left(x-8\\\\right) \\\\left(x+2\\\\right)=0$$ which gives $$x=8$$ or $$x=-2$$. Given $$\\\\log_{a}\\\\left(b\\\\right)$$, $$b$$ can not be negative. We can omit the negative answer. Therefore $$x=8$$ is the only solution for this equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ad46flog7","title":"Solve Logarithmic Equations Using the Properties of Logarithms","body":"Solve for $$x$$ in the following equation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Solve Exponential and Logarithmic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8ad46flog7a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{4}\\\\left(x+6\\\\right)-\\\\log_{4}\\\\left(2x+5\\\\right)=-\\\\log_{4}\\\\left(x\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a8ad46flog7a-h1","type":"hint","dependencies":[],"title":"Use Log Rules","text":"Logarithm Division rule: $$\\\\log_{a}\\\\left(b\\\\right)-\\\\log_{a}\\\\left(c\\\\right)=\\\\log_{a}\\\\left(\\\\frac{b}{c}\\\\right)$$ $$\\\\log_{4}\\\\left(x+6\\\\right)-\\\\log_{4}\\\\left(2x+5\\\\right)=\\\\log_{4}\\\\left(\\\\frac{x+6}{2x+5}\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog7a-h2","type":"hint","dependencies":["a8ad46flog7a-h1"],"title":"Use Log Rules","text":"Logarithm rule: $$k*\\\\log_{a}\\\\left(b\\\\right)=\\\\log_{a}\\\\left(b^k\\\\right)$$ where a and $$b$$ are positive and a does not equal to $$1$$. $$-\\\\log_{4}\\\\left(x\\\\right)=\\\\log_{4}\\\\left(x^{\\\\left(-1\\\\right)}\\\\right)=\\\\log_{4}\\\\left(\\\\frac{1}{x}\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog7a-h3","type":"hint","dependencies":["a8ad46flog7a-h2"],"title":"Match Bases and Solve for $$x$$","text":"$$\\\\log_{4}\\\\left(\\\\frac{x+6}{2x+5}\\\\right)=\\\\log_{4}\\\\left(\\\\frac{1}{x}\\\\right)$$ Since both sides has the same base, we can take out the logarithm and get $$\\\\frac{x+6}{2x+5}=\\\\frac{1}{x}$$. We can cross multiply and get $$x^2+6x=2x+5$$. We can simplify it into $$x^2+4x-5=0$$. We can factor the quadratic and get $$\\\\left(x+5\\\\right) \\\\left(x-1\\\\right)=0$$ which gives $$x=1$$ and $$x=-5$$. Given $$\\\\log_{a}\\\\left(b\\\\right)$$, $$b$$ can not be negative. We can omit the negative answer. Therefore $$x=1$$ is the only solution for this equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ad46flog8","title":"Solve Logarithmic Equations Using the Properties of Logarithms","body":"Solve for $$x$$ in the following equation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Solve Exponential and Logarithmic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8ad46flog8a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{10}\\\\left(x+2\\\\right)-\\\\log_{10}\\\\left(4x+3\\\\right)=-\\\\log_{10}\\\\left(x\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a8ad46flog8a-h1","type":"hint","dependencies":[],"title":"Use Log Rules","text":"Logarithm Division rule: $$\\\\log_{a}\\\\left(b\\\\right)-\\\\log_{a}\\\\left(c\\\\right)=\\\\log_{a}\\\\left(\\\\frac{b}{c}\\\\right)$$ $$\\\\log_{10}\\\\left(x+2\\\\right)-\\\\log_{10}\\\\left(4x+3\\\\right)=\\\\log_{10}\\\\left(\\\\frac{x+2}{4x+3}\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog8a-h2","type":"hint","dependencies":["a8ad46flog8a-h1"],"title":"Use Log Rules","text":"Logarithm rule: $$k*\\\\log_{a}\\\\left(b\\\\right)=\\\\log_{a}\\\\left(b^k\\\\right)$$ where a and $$b$$ are positive and a does not equal to $$1$$. $$-\\\\log_{10}\\\\left(x\\\\right)=\\\\log_{10}\\\\left(x^{\\\\left(-1\\\\right)}\\\\right)=\\\\log_{10}\\\\left(\\\\frac{1}{x}\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog8a-h3","type":"hint","dependencies":["a8ad46flog8a-h2"],"title":"Match Bases and Solve for $$x$$","text":"$$\\\\log_{10}\\\\left(\\\\frac{x+2}{4x+3}\\\\right)=\\\\log_{10}\\\\left(\\\\frac{1}{x}\\\\right)$$ Since both sides has the same base, we can take out the logarithm and get $$\\\\frac{x+2}{4x+3}=\\\\frac{1}{x}$$. We can cross multiply and get $$x^2+2x=4x+3$$. We can simplify it into $$x^2-2x-3=0$$. We can factor the quadratic and get $$\\\\left(x-3\\\\right) \\\\left(x+1\\\\right)=0$$ which gives $$x=-1$$ and $$x=3$$. Given $$\\\\log_{a}\\\\left(b\\\\right)$$, $$b$$ cannot be negative. We can omit the negative answer. Therefore $$x=3$$ is the only solution for this equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ad46flog9","title":"Solve Logarithmic Equations Using the Properties of Logarithms","body":"Solve for $$x$$ in the following equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.5 Solve Exponential and Logarithmic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8ad46flog9a","stepAnswer":["$$8$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{10}\\\\left(x-2\\\\right)-\\\\log_{10}\\\\left(4x+16\\\\right)=\\\\log_{10}\\\\left(\\\\frac{1}{x}\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8$$","hints":{"DefaultPathway":[{"id":"a8ad46flog9a-h1","type":"hint","dependencies":[],"title":"Use Log Rules","text":"Logarithm Division rule: $$\\\\log_{a}\\\\left(b\\\\right)-\\\\log_{a}\\\\left(c\\\\right)=\\\\log_{a}\\\\left(\\\\frac{b}{c}\\\\right)$$ $$\\\\log_{10}\\\\left(x-2\\\\right)-\\\\log_{10}\\\\left(4x+16\\\\right)=\\\\log_{10}\\\\left(\\\\frac{x-2}{4x+16}\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ad46flog9a-h2","type":"hint","dependencies":["a8ad46flog9a-h1"],"title":"Match Bases and Solve for $$x$$","text":"$$\\\\log_{10}\\\\left(\\\\frac{x-2}{4x+16}\\\\right)=\\\\log_{10}\\\\left(\\\\frac{1}{x}\\\\right)$$ Since both sides has the same base, we can take out the logarithm and get $$\\\\frac{x-2}{4x+16}=\\\\frac{1}{x}$$. We can cross multiply and get $$x^2-2x=4x+16$$. We can simplify it into $$x^2-6x-16=0$$. We can factor the quadratic and get $$\\\\left(x-8\\\\right) \\\\left(x+2\\\\right)=0$$ which gives $$x=-2$$ and $$x=8$$. Given $$\\\\log_{a}\\\\left(b\\\\right)$$, $$b$$ can not be negative. We can omit the negative answer. Therefore $$x=8$$ is the only solution for this equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b5fd4mixapp13","title":"Solving Mixture Applications","body":"Marissa wants to blend candy selling for $$\\\\$1.80$$ per pound with candy costing $$\\\\$1.20$$ per pound to get a mixture that costs her $$\\\\$1.40$$ per pound to make. She wants to make $$90$$ pounds of the candy blend. How","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Solve Mixture Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b5fd4mixapp13a","stepAnswer":["30 pounds of $1.80 candy, 60 pounds of $1.20 candy"],"problemType":"TextBox","stepTitle":"How many pounds of each type of candy should she use? Write the answer in the form \\"x pounds of $$\\\\$1.80$$ candy, $$y$$ pounds of $$\\\\$1.20$$ candy\\" without the quotes and with the appropriate values in the places of $$x$$ and $$y$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$30$$ pounds of $$\\\\$1.80$$ candy, $$60$$ pounds of $$\\\\$1.20$$ candy","hints":{"DefaultPathway":[{"id":"a8b5fd4mixapp13a-h1","type":"hint","dependencies":[],"title":"Setting up a System of Equations","text":"Set up a system of equations to represent the situation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b5fd4mixapp13a-h2","type":"hint","dependencies":["a8b5fd4mixapp13a-h1"],"title":"First Equation","text":"The first equation is based on the number of pounds of each type. Using a for $1.80/pound candy and $$b$$ for $1.20/pound candy, we can write this as $$a+b=90$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b5fd4mixapp13a-h3","type":"hint","dependencies":["a8b5fd4mixapp13a-h2"],"title":"Second Equation","text":"The second equation is based on the total value of the candy she wants to make. We can represent this with $$1.8a+1.2b=1.4\\\\left(a+b\\\\right)$$, where $$a+b$$ is the total pounds of candy. We know from the first equation that $$a+b$$ is $$90$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b5fd4mixapp13a-h4","type":"hint","dependencies":["a8b5fd4mixapp13a-h3"],"title":"Solving the System Using the Elimination Method","text":"Solve the system using elimination. Multiply one of the equations by a constant such that one of the variables in the second equation is eliminated when the first equation is added to it. Then you will have the value of the other variable. Solve for eliminated variable by plugging in the value of the known variable to one of the equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b5fd4mixapp14","title":"Solving Interest Applications","body":"Hattie had $3000 to invest and wants to earn $$10.6\\\\%$$ interest per year. She will put some of the money into an account that earns 12% per year and the rest into an account that earns 10% per year.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Solve Mixture Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b5fd4mixapp14a","stepAnswer":["900 and 2100"],"problemType":"TextBox","stepTitle":"How much money should she put into each account? Write your answer in the format \\"x and y\\" with $$x$$ representing the amount of money in dollars she wants to put in the 12% interest per year account, and $$y$$ the amount of dollars she wants to put in the 10% interest per year account.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$900$$ and $$2100$$","hints":{"DefaultPathway":[{"id":"a8b5fd4mixapp14a-h1","type":"hint","dependencies":[],"title":"Setting up a System of Equations","text":"Set up a system of equations to represent the situation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b5fd4mixapp14a-h2","type":"hint","dependencies":["a8b5fd4mixapp14a-h1"],"title":"First Equation","text":"First, set up an equation to represent how the initial money is distributed. Let a be the amount of money she wants to put into the account that earns 12% per year, and $$b$$, the amount of money she wants to put into the account that earns 10% per year. The equation is $$a+b=3000$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b5fd4mixapp14a-h3","type":"hint","dependencies":["a8b5fd4mixapp14a-h2"],"title":"Second Equation","text":"Next, write an equation to represent the interest she wants to earn. In this problem, it is $$0.12a+0.1b=0.106(3000)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b5fd4mixapp14a-h4","type":"hint","dependencies":["a8b5fd4mixapp14a-h3"],"title":"Solving the System Using the Elimination Method","text":"Solve the system using elimination. Multiply one of the equations by a constant such that one of the variables in the second equation is eliminated when the first equation is added to it. Then you will have the value of the other variable. Solve for eliminated variable by plugging in the value of the known variable to one of the equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a8b5fd4mixapp14b","stepAnswer":["1600 and 960"],"problemType":"TextBox","stepTitle":"How much money did she put in each account? Write your answer in the format \\"x and y\\" with $$x$$ representing the amount of money in dollars she put in the 8% interest per year account, and $$y$$ the amount of dollars she put in the 6% interest per year account.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$1600$$ and $$960$$","hints":{"DefaultPathway":[{"id":"a8b5fd4mixapp14b-h1","type":"hint","dependencies":[],"title":"Setting up a System of Equations","text":"Set up a system of equations to represent the situation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b5fd4mixapp14b-h2","type":"hint","dependencies":["a8b5fd4mixapp14b-h1"],"title":"First Equation","text":"First, set up an equation to represent how the initial money is distributed. Let a be the amount of money she put into the account that earns 8% per year, and $$b$$, the amount of money she put into the account that earns 6% per year. The equation is $$a+b=2560$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b5fd4mixapp14b-h3","type":"hint","dependencies":["a8b5fd4mixapp14b-h2"],"title":"Second Equation","text":"Next, write an equation to represent the interest she wants to earn. In this problem, it is $$0.08a+0.06b=0.0725(2560)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b5fd4mixapp14b-h4","type":"hint","dependencies":["a8b5fd4mixapp14b-h3"],"title":"Solving the System Using the Elimination Method","text":"Solve the system using elimination. Multiply one of the equations by a constant such that one of the variables in the second equation is eliminated when the first equation is added to it. Then you will have the value of the other variable. Solve for eliminated variable by plugging in the value of the known variable to one of the equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b5fd4mixapp2","title":"Solve Mixture Applications","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Solve Mixture Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b5fd4mixapp2a","stepAnswer":["$$206$$"],"problemType":"TextBox","stepTitle":"The ticket office at the zoo sold $$553$$ tickets one day. The receipts totaled $3,936. How many $9 adult tickets were sold if child tickets cost $6?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$206$$","hints":{"DefaultPathway":[{"id":"a8b5fd4mixapp2a-h1","type":"hint","dependencies":[],"title":"Translate Into System","text":"Letting $$x=number$$ of adult tickets sold and $$y=number$$ of child tickets sold: $$x+y=553$$, $$9x+6y=3936$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b5fd4mixapp2a-h2","type":"hint","dependencies":["a8b5fd4mixapp2a-h1"],"title":"Eliminating One Variable","text":"Let\'s eliminate $$y$$ so that we can solve for $$x$$. Multiply the first equation by $$-6$$ and then add. $$3x=618$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b5fd4mixapp2a-h3","type":"hint","dependencies":["a8b5fd4mixapp2a-h2"],"title":"Solving the Equation","text":"$$x=\\\\frac{618}{3}=206$$. $$206$$ adult tickets were sold.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a8b5fd4mixapp2b","stepAnswer":["190 child tickets, 110 adult tickets"],"problemType":"TextBox","stepTitle":"How many adult and how many child tickets were sold? Write the answer in the form \\"x child tickets, $$y$$ adult tickets\\" without the quotes and with the appropriate values in the places of $$x$$ and $$y$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$190$$ child tickets, $$110$$ adult tickets","hints":{"DefaultPathway":[{"id":"a8b5fd4mixapp2b-h1","type":"hint","dependencies":[],"title":"Setting up a System of Equations","text":"Set up a system of equations to represent the situation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b5fd4mixapp2b-h2","type":"hint","dependencies":["a8b5fd4mixapp2b-h1"],"title":"First Equation","text":"The first equation is based on the number of tickets sold. Using a for adult tickets and c for child tickets, we can write this as $$a+c=300$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b5fd4mixapp2b-h3","type":"hint","dependencies":["a8b5fd4mixapp2b-h2"],"title":"Second Equation","text":"The second equation is based on the total cost of tickets sold. $$70a+50c=17200$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b5fd4mixapp2b-h4","type":"hint","dependencies":["a8b5fd4mixapp2b-h3"],"title":"Solving the System Using the Elimination Method","text":"Solve the system using elimination. Multiply one of the equations by a constant such that one of the variables in the second equation is eliminated when the first equation is added to it. Then you will have the value of the other variable. Solve for eliminated variable by plugging in the value of the known variable to one of the equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b5fd4mixapp3","title":"Solve Mixture Applications","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Solve Mixture Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b5fd4mixapp3a","stepAnswer":["$$42$$"],"problemType":"TextBox","stepTitle":"The box office at a movie theater sold $$147$$ tickets for the evening show, and receipts totaled $1,302. How many $11 adult were sold if child tickets cost $8?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$42$$","hints":{"DefaultPathway":[{"id":"a8b5fd4mixapp3a-h1","type":"hint","dependencies":[],"title":"Translate Into System","text":"Let\'s set $$x$$ to the number of adult tickets sold and $$y$$ to the number of child tickets sold. Then, we have: $$x+y=147$$, $$11x+8y=1302$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b5fd4mixapp3a-h2","type":"hint","dependencies":["a8b5fd4mixapp3a-h1"],"title":"Eliminating One Variable","text":"Let\'s eliminate $$y$$ so that we can solve for $$x$$. Multiply the first equation by $$-8$$ and then add. $$3x=126$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b5fd4mixapp3a-h3","type":"hint","dependencies":["a8b5fd4mixapp3a-h2"],"title":"Solving the Equation","text":"$$x=\\\\frac{126}{3}=42$$. $$42$$ adult tickets were sold.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a8b5fd4mixapp3b","stepAnswer":["32 child tickets, 40 adult tickets"],"problemType":"TextBox","stepTitle":"How many adult and how many child tickets did Josie buy? Write the answer in the form \\"x child tickets, $$y$$ adult tickets\\" without the quotes and with the appropriate values in the places of $$x$$ and $$y$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$32$$ child tickets, $$40$$ adult tickets","hints":{"DefaultPathway":[{"id":"a8b5fd4mixapp3b-h1","type":"hint","dependencies":[],"title":"Setting up a System of Equations","text":"Set up a system of equations to represent the situation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b5fd4mixapp3b-h2","type":"hint","dependencies":["a8b5fd4mixapp3b-h1"],"title":"First Equation","text":"The first equation is based on the number of tickets sold. Using a for adult tickets and c for child tickets, we can write this as $$a+c=72$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b5fd4mixapp3b-h3","type":"hint","dependencies":["a8b5fd4mixapp3b-h2"],"title":"Second Equation","text":"The second equation is based on the total cost of tickets sold. $$22a+10c=1200$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b5fd4mixapp3b-h4","type":"hint","dependencies":["a8b5fd4mixapp3b-h3"],"title":"Solving the System Using the Elimination Method","text":"Solve the system using elimination. Multiply one of the equations by a constant such that one of the variables in the second equation is eliminated when the first equation is added to it. Then you will have the value of the other variable. Solve for eliminated variable by plugging in the value of the known variable to one of the equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b5fd4mixapp4","title":"Solve Mixture Applications","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Solve Mixture Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b5fd4mixapp4a","stepAnswer":["$$36$$"],"problemType":"TextBox","stepTitle":"Juan has a pocketful of nickels and dimes. The total value of the coins is $$\\\\$8.10$$. The number of dimes is $$9$$ less than twice the number of nickels. How many nickels does Juan have?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$36$$","hints":{"DefaultPathway":[{"id":"a8b5fd4mixapp4a-h1","type":"hint","dependencies":[],"title":"Translate Into System","text":"Let\'s set $$x$$ to the number of dimes and $$y$$ to the number of nickles. Then: x-2y=-9,0.1x+0.05y=8.10","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b5fd4mixapp4a-h2","type":"hint","dependencies":["a8b5fd4mixapp4a-h1"],"title":"Substituting the Variables","text":"Let\'s substitute $$x=-9+2y$$ into the equation. We get 0.1(-9+2y)+0.05y=8.10","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b5fd4mixapp4a-h3","type":"hint","dependencies":["a8b5fd4mixapp4a-h2"],"title":"Solving the Equation","text":"$$0.25y=9.00$$, $$y=36$$. There are $$36$$ nickles","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a8b5fd4mixapp4b","stepAnswer":["6 Main tickets, 10 Terrace tickets"],"problemType":"TextBox","stepTitle":"How many of Main Level and how many Terrace Level tickets did they buy? Write the answer in the form \\"x Main tickets, $$y$$ Terrace tickets\\" without the quotes and with the appropriate values in the places of $$x$$ and $$y$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$6$$ Main tickets, $$10$$ Terrace tickets","hints":{"DefaultPathway":[{"id":"a8b5fd4mixapp4b-h1","type":"hint","dependencies":[],"title":"Setting up a System of Equations","text":"Set up a system of equations to represent the situation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b5fd4mixapp4b-h2","type":"hint","dependencies":["a8b5fd4mixapp4b-h1"],"title":"First Equation","text":"The first equation is based on the number of tickets sold. Using $$m$$ for Main tickets and $$t$$ for Terrace tickets, we can write this as $$m+t=16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b5fd4mixapp4b-h3","type":"hint","dependencies":["a8b5fd4mixapp4b-h2"],"title":"Second Equation","text":"The second equation is based on the total cost of tickets sold. $$69m+39t=804$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b5fd4mixapp4b-h4","type":"hint","dependencies":["a8b5fd4mixapp4b-h3"],"title":"Solving the System Using the Elimination Method","text":"Solve the system using elimination. Multiply one of the equations by a constant such that one of the variables in the second equation is eliminated when the first equation is added to it. Then you will have the value of the other variable. Solve for eliminated variable by plugging in the value of the known variable to one of the equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b5fd4mixapp5","title":"Solve Mixture Applications","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Solve Mixture Applications with Systems of Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b5fd4mixapp5a","stepAnswer":["$$13$$"],"problemType":"TextBox","stepTitle":"Matilda has a handful of quarters and dimes, with a total value of $$\\\\$8.55$$. The number of quarters is $$3$$ more than twice the number of dimes. How many dimes does she have?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$13$$","hints":{"DefaultPathway":[{"id":"a8b5fd4mixapp5a-h1","type":"hint","dependencies":[],"title":"Translate Into System","text":"Let $$x$$ equal the number of quarters and $$y$$ equal the number of dimes. $$0.25x+0.1y=8.55$$, $$x-2y=3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b5fd4mixapp5a-h2","type":"hint","dependencies":["a8b5fd4mixapp5a-h1"],"title":"Substituting the Variables","text":"Let\'s solve for $$x$$ so that we can isolate $$y$$ in the equation. $$x=2y+3$$. Substituting this into the other equation, we get: $$\\\\operatorname{0.25}\\\\left(2y+3\\\\right)+0.1y=8.55$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b5fd4mixapp5a-h3","type":"hint","dependencies":["a8b5fd4mixapp5a-h2"],"title":"Solving the Equation","text":"We can now solve this equation for $$y$$ to find the number of dimes. $$0.6y=7.8$$, $$y=\\\\frac{7.8}{0.6}=13$$. She has $$13$$ dimes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a8b5fd4mixapp5b","stepAnswer":["128 child tickets, 125 adult tickets"],"problemType":"TextBox","stepTitle":"How many adult and how many child tickets did Josie buy? Write the answer in the form \\"x child tickets, $$y$$ adult tickets\\" without the quotes and with the appropriate values in the places of $$x$$ and $$y$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$128$$ child tickets, $$125$$ adult tickets","hints":{"DefaultPathway":[{"id":"a8b5fd4mixapp5b-h1","type":"hint","dependencies":[],"title":"Setting up a System of Equations","text":"Set up a system of equations to represent the situation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b5fd4mixapp5b-h2","type":"hint","dependencies":["a8b5fd4mixapp5b-h1"],"title":"First Equation","text":"The first equation is based on the number of tickets sold. Using a for adult tickets and c for child tickets, we can write this as $$a+c=253$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b5fd4mixapp5b-h3","type":"hint","dependencies":["a8b5fd4mixapp5b-h2"],"title":"Second Equation","text":"The second equation is based on the total cost of tickets sold. $$15a+7c=2771$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b5fd4mixapp5b-h4","type":"hint","dependencies":["a8b5fd4mixapp5b-h3"],"title":"Solving the System Using the Elimination Method","text":"Solve the system using elimination. Multiply one of the equations by a constant such that one of the variables in the second equation is eliminated when the first equation is added to it. Then you will have the value of the other variable. Solve for eliminated variable by plugging in the value of the known variable to one of the equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b6b9fLinear1","title":"Solve Linear Equations Using a General Strategy","body":"Determine whether the values are solutions to the equation: $$5y+3=10y-4$$","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Use a General Strategy to Solve Linear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b6b9fLinear1a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$y=\\\\frac{3}{5}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a8b6b9fLinear1a-h1","type":"hint","dependencies":[],"title":"Substituting the Given Value","text":"We can substitute the given value for $$y$$ into the equation. This gives us $$5\\\\left(\\\\frac{3}{5}\\\\right)+3=\\\\operatorname{10}\\\\left(\\\\frac{3}{5}\\\\right)-4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a8b6b9fLinear1a-h1"],"title":"Calculation","text":"What is $$5\\\\left(\\\\frac{3}{5}\\\\right)+3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a8b6b9fLinear1a-h1"],"title":"Calculation","text":"What is $$\\\\operatorname{10}\\\\left(\\\\frac{3}{5}\\\\right)-4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear1a-h4","type":"hint","dependencies":[],"title":"Equality","text":"$$6$$ does not equal to $$2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a8b6b9fLinear1b","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$y=\\\\frac{7}{5}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a8b6b9fLinear1b-h1","type":"hint","dependencies":[],"title":"Substituting the Given Value","text":"We can substitute the given value for $$y$$ into the equation. This gives us $$5\\\\left(\\\\frac{7}{5}\\\\right)+3=\\\\operatorname{10}\\\\left(\\\\frac{7}{5}\\\\right)-4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear1b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a8b6b9fLinear1b-h1"],"title":"Calculation","text":"What is $$5\\\\left(\\\\frac{7}{5}\\\\right)+3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear1b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a8b6b9fLinear1b-h1"],"title":"Calculation","text":"What is $$\\\\operatorname{10}\\\\left(\\\\frac{7}{5}\\\\right)-4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear1b-h4","type":"hint","dependencies":[],"title":"Equality","text":"Left hand side equals to the right hand side","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b6b9fLinear10","title":"How to Solve a Linear Equation Using a General Strategy","body":"Solve the following equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Use a General Strategy to Solve Linear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b6b9fLinear10a","stepAnswer":["$$-2$$"],"problemType":"TextBox","stepTitle":"$$6(p-3)-7=5\\\\left(4p+3\\\\right)-12$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-2$$","hints":{"DefaultPathway":[{"id":"a8b6b9fLinear10a-h1","type":"hint","dependencies":[],"title":"Separate Constants and Variables on Either Side of the Equation","text":"We must first simplify each side of the equation to get $$6p-18-7=20p+15-12$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear10a-h2","type":"hint","dependencies":[],"title":"Simplify","text":"Organize the constant to the same side to obtain $$14p=-28$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear10a-h3","type":"hint","dependencies":[],"title":"Division","text":"We must make the coefficient of the variable equal to $$1$$, so we divide by $$14$$ on both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b6b9fLinear11","title":"How to Solve a Linear Equation Using a General Strategy","body":"Solve the following equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Use a General Strategy to Solve Linear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b6b9fLinear11a","stepAnswer":["$$-8$$"],"problemType":"TextBox","stepTitle":"$$8\\\\left(q+1\\\\right)-5=3(2q-4)-1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-8$$","hints":{"DefaultPathway":[{"id":"a8b6b9fLinear11a-h1","type":"hint","dependencies":[],"title":"Separate Constants and Variables on Either Side of the Equation","text":"We must first simplify each side of the equation to get $$8q+8-5=6q-12-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear11a-h2","type":"hint","dependencies":[],"title":"Simplify","text":"Organize the constant to the same side to obtain $$2q=-16$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear11a-h3","type":"hint","dependencies":[],"title":"Division","text":"We must make the coefficient of the variable equal to $$1$$, so we divide by $$2$$ on both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b6b9fLinear12","title":"How to Solve a Linear Equation Using a General Strategy","body":"Solve the following equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Use a General Strategy to Solve Linear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b6b9fLinear12a","stepAnswer":["$$-2$$"],"problemType":"TextBox","stepTitle":"$$10(3-8(2s-5))=15(40-5s)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-2$$","hints":{"DefaultPathway":[{"id":"a8b6b9fLinear12a-h1","type":"hint","dependencies":[],"title":"Separate Constants and Variables on Either Side of the Equation","text":"We must first simplify each side of the equation to get $$\\\\operatorname{10}\\\\left(3-16s+40\\\\right)=600-75s$$. $$30-160s+400=600-75s$$. Now, we collect constants and variables on either side. $$-170=85s$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear12a-h2","type":"hint","dependencies":["a8b6b9fLinear12a-h1"],"title":"Simplify","text":"Simplify the equation to get $$30-160s+400=600-75s$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear12a-h3","type":"hint","dependencies":[],"title":"Simplify","text":"Organize the constant to the same side to obtain $$85s=-170$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear12a-h4","type":"hint","dependencies":[],"title":"Division","text":"We must make the coefficient of the variable equal to $$1$$, so we divide by $$85$$ on both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b6b9fLinear13","title":"How to Solve a Linear Equation Using a General Strategy","body":"Solve the following equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Use a General Strategy to Solve Linear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b6b9fLinear13a","stepAnswer":["$$\\\\frac{-17}{5}$$"],"problemType":"TextBox","stepTitle":"$$6(4-2(7y-1))=8(13-8y)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-17}{5}$$","hints":{"DefaultPathway":[{"id":"a8b6b9fLinear13a-h1","type":"hint","dependencies":[],"title":"Separate Constants and Variables on Either Side of the Equation","text":"We must first simplify each side of the equation to get $$6\\\\left(4-14y+2\\\\right)=104-64y$$. $$24-84y+12=104-64y$$. Now, we collect constants and variables on either side. $$-68=20y$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear13a-h2","type":"hint","dependencies":["a8b6b9fLinear13a-h1"],"title":"Simplify","text":"Simplify the equation to get $$24-84y+12=104-64y$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear13a-h3","type":"hint","dependencies":[],"title":"Simplify","text":"Organize the constant to the same side to obtain $$20y=-68$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear13a-h4","type":"hint","dependencies":[],"title":"Division","text":"We must make the coefficient of the variable equal to $$1$$, so we divide by $$20$$ on both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b6b9fLinear14","title":"How to Solve a Linear Equation with Fraction or Decimal Coefficient","body":"Solve the following equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Use a General Strategy to Solve Linear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b6b9fLinear14a","stepAnswer":["$$-1$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{1}{12} x+\\\\frac{5}{6}=\\\\frac{3}{4}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1$$","hints":{"DefaultPathway":[{"id":"a8b6b9fLinear14a-h1","type":"hint","dependencies":[],"title":"Common Denominator","text":"Identify the common denominator as $$12$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear14a-h2","type":"hint","dependencies":["a8b6b9fLinear14a-h1"],"title":"Rewrite","text":"Convert all fraction with denominator as 12: $$\\\\frac{1}{12} x+\\\\frac{10}{12}=\\\\frac{9}{12}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear14a-h3","type":"hint","dependencies":["a8b6b9fLinear14a-h2"],"title":"Simplify","text":"Factor out $$\\\\frac{1}{12}$$: $$x+10=9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b6b9fLinear15","title":"How to Solve a Linear Equation with Fraction or Decimal Coefficient","body":"Solve the following equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Use a General Strategy to Solve Linear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b6b9fLinear15a","stepAnswer":["$$\\\\frac{1}{2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{1}{4} x+\\\\frac{1}{2}=\\\\frac{5}{8}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{2}$$","hints":{"DefaultPathway":[{"id":"a8b6b9fLinear15a-h1","type":"hint","dependencies":[],"title":"Common Denominator","text":"Identify the common denominator as $$8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear15a-h2","type":"hint","dependencies":["a8b6b9fLinear15a-h1"],"title":"Rewrite","text":"Convert all fraction with denominator as 8: $$\\\\frac{2}{8} x+\\\\frac{4}{8}=\\\\frac{5}{8}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear15a-h3","type":"hint","dependencies":["a8b6b9fLinear15a-h2"],"title":"Simplify","text":"Factor out $$\\\\frac{1}{8}$$: $$2x+4=5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b6b9fLinear16","title":"Solve Equations Using the General Strategy","body":"Determine whether the given values are solutions to the equation: $$6y+10=12y$$","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Use a General Strategy to Solve Linear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b6b9fLinear16a","stepAnswer":["TRUE"],"problemType":"MultipleChoice","stepTitle":"$$y=\\\\frac{5}{3}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["TRUE","FALSE"],"hints":{"DefaultPathway":[{"id":"a8b6b9fLinear16a-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Substitute $$y$$ for its numeric value $$\\\\frac{5}{3}$$ on either side of the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a8b6b9fLinear16a-h1"],"title":"Multiplication","text":"What is $$6\\\\frac{5}{3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear16a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a8b6b9fLinear16a-h1"],"title":"Simplifying","text":"What is $$10+10$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear16a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a8b6b9fLinear16a-h1"],"title":"Multiplication","text":"What is $$12\\\\frac{5}{3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear16a-h5","type":"hint","dependencies":[],"title":"Equality","text":"Check if the numbers on both sides are equal","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a8b6b9fLinear16b","stepAnswer":["FALSE"],"problemType":"MultipleChoice","stepTitle":"$$y=\\\\frac{-1}{2}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["TRUE","FALSE"],"hints":{"DefaultPathway":[{"id":"a8b6b9fLinear16b-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Substitute $$y$$ for its numeric value $$\\\\left(-\\\\frac{1}{2}\\\\right)$$ on either side of the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear16b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a8b6b9fLinear16b-h1"],"title":"Substitution","text":"What is $$6\\\\left(-\\\\frac{1}{2}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear16b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-7$$"],"dependencies":["a8b6b9fLinear16b-h1"],"title":"Simplifying","text":"What is $$-3+10$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear16b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6$$"],"dependencies":["a8b6b9fLinear16b-h1"],"title":"Substitution","text":"What is $$12\\\\left(-\\\\frac{1}{2}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear16b-h5","type":"hint","dependencies":[],"title":"Equality","text":"Check if the numbers on both sides are equal","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b6b9fLinear17","title":"Solve Equations Using the General Strategy","body":"Determine whether the given values are solutions to the equation: $$8u-1=6u$$","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Use a General Strategy to Solve Linear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b6b9fLinear17a","stepAnswer":["FALSE"],"problemType":"MultipleChoice","stepTitle":"$$u=\\\\frac{-1}{2}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["TRUE","FALSE"],"hints":{"DefaultPathway":[{"id":"a8b6b9fLinear17a-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Substitute u for its numeric value $$\\\\left(-\\\\frac{1}{2}\\\\right)$$ on either side of the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear17a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["a8b6b9fLinear17a-h1"],"title":"Substitution","text":"What is $$8\\\\left(-\\\\frac{1}{2}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear17a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["a8b6b9fLinear17a-h1"],"title":"Simplifying","text":"What is $$-4-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear17a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a8b6b9fLinear17a-h1"],"title":"Substitution","text":"What is $$6\\\\left(-\\\\frac{1}{2}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear17a-h5","type":"hint","dependencies":[],"title":"Equality","text":"Check if the numbers on both sides are equal","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a8b6b9fLinear17b","stepAnswer":["TRUE"],"problemType":"MultipleChoice","stepTitle":"$$u=\\\\frac{1}{3}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["TRUE","FALSE"],"hints":{"DefaultPathway":[{"id":"a8b6b9fLinear17b-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Substitute u for its numeric value $$\\\\frac{1}{2}$$ on either side of the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear17b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a8b6b9fLinear17b-h1"],"title":"Substitution","text":"What is $$8\\\\frac{1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear17b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a8b6b9fLinear17b-h1"],"title":"Simplifying","text":"What is $$4-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear17b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a8b6b9fLinear17b-h1"],"title":"Substitution","text":"What is $$6\\\\frac{1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear17b-h5","type":"hint","dependencies":[],"title":"Equality","text":"Check if the numbers on both sides are equal","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b6b9fLinear18","title":"Solve Equations Using the General Strategy","body":"Solve the Linear Equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Use a General Strategy to Solve Linear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b6b9fLinear18a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"$$15(y-9)=-60$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"a8b6b9fLinear18a-h1","type":"hint","dependencies":[],"title":"Distributive Property","text":"Use the distributive property to multiply $$15$$ with $$(y-9)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15y-135$$"],"dependencies":["a8b6b9fLinear18a-h1"],"title":"Multiplication","text":"What is $$15\\\\left(y-9\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear18a-h3","type":"hint","dependencies":["a8b6b9fLinear18a-h2"],"title":"Organizing","text":"Once both sides have an equation without parantheses, use algebra to manipulate the equation into the form $$y=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear18a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$75$$"],"dependencies":[],"title":"Addition","text":"What is $$-60+135$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear18a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a8b6b9fLinear18a-h4"],"title":"Division","text":"What is $$\\\\frac{75}{15}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b6b9fLinear19","title":"Solve Equations Using the General Strategy","body":"Solve the Linear Equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Use a General Strategy to Solve Linear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b6b9fLinear19a","stepAnswer":["$$1.5$$"],"problemType":"TextBox","stepTitle":"$$-6+6\\\\left(5-k\\\\right)=15$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1.5$$","hints":{"DefaultPathway":[{"id":"a8b6b9fLinear19a-h1","type":"hint","dependencies":[],"title":"Distributive Property","text":"Use the distributive property to multiply $$6\\\\left(5-k\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$30-6k$$"],"dependencies":["a8b6b9fLinear19a-h1"],"title":"Multiplication","text":"What is $$6\\\\left(5-k\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear19a-h3","type":"hint","dependencies":["a8b6b9fLinear19a-h2"],"title":"Organizing","text":"Once both sides have an equation without parantheses, use algebra to manipulate the equation into the form $$6k=30-6-15$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear19a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$21$$"],"dependencies":["a8b6b9fLinear19a-h3"],"title":"Addition","text":"What is $$15+6$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear19a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a8b6b9fLinear19a-h4"],"title":"Subtraction","text":"What is $$30-21$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear19a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.5$$"],"dependencies":["a8b6b9fLinear19a-h5"],"title":"Division","text":"What is $$\\\\frac{9}{6}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b6b9fLinear2","title":"Solve Linear Equations Using a General Strategy","body":"Determine whether the values are solutions to the equation: $$9y+2=6y+3$$","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Use a General Strategy to Solve Linear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b6b9fLinear2a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$y=\\\\frac{4}{3}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a8b6b9fLinear2a-h1","type":"hint","dependencies":[],"title":"Substituting the Given Value","text":"We can substitute the given value for $$y$$ into the equation. This gives us $$9\\\\left(\\\\frac{4}{3}\\\\right)+2=6\\\\left(\\\\frac{4}{3}\\\\right)-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$14$$"],"dependencies":["a8b6b9fLinear2a-h1"],"title":"Calculation","text":"What is $$9\\\\left(\\\\frac{4}{3}\\\\right)+2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11$$"],"dependencies":["a8b6b9fLinear2a-h1"],"title":"Calculation","text":"What is $$6\\\\left(\\\\frac{4}{3}\\\\right)+3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear2a-h4","type":"hint","dependencies":[],"title":"Equality","text":"$$14$$ does not equal to $$11$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a8b6b9fLinear2b","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"Determine whether $$y=\\\\frac{1}{3}$$ is a solution to $$9y+2=6y-3$$ (0 for non-solution, $$1$$ for solution)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a8b6b9fLinear2b-h1","type":"hint","dependencies":[],"title":"Substituting the Given Value","text":"We can substitute the given value for $$y$$ into the equation. This gives us $$9\\\\left(\\\\frac{1}{3}\\\\right)+2=6\\\\left(\\\\frac{1}{3}\\\\right)-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear2b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a8b6b9fLinear2b-h1"],"title":"Calculation","text":"What is $$9\\\\left(\\\\frac{1}{3}\\\\right)+2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear2b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a8b6b9fLinear2b-h1"],"title":"Calculation","text":"What is $$6\\\\left(\\\\frac{1}{3}\\\\right)+3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear2b-h4","type":"hint","dependencies":[],"title":"Equality","text":"Left hand side equals to the right hand side","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b6b9fLinear20","title":"Solve Equations Using the General Strategy","body":"Solve the Linear Equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Use a General Strategy to Solve Linear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b6b9fLinear20a","stepAnswer":["$$2.5$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{1}{5} \\\\left(15c+10\\\\right)=c+7$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2.5$$","hints":{"DefaultPathway":[{"id":"a8b6b9fLinear20a-h1","type":"hint","dependencies":[],"title":"Distributive Property","text":"Use the distributive property to multiply $$\\\\frac{1}{5} \\\\left(15c+10\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3c+2$$"],"dependencies":["a8b6b9fLinear20a-h1"],"title":"Multiplication","text":"What is $$\\\\frac{1}{5} \\\\left(15c+10\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear20a-h3","type":"hint","dependencies":["a8b6b9fLinear20a-h2"],"title":"Organizing","text":"Once both sides have an equation without parantheses, use algebra to manipulate the equation into the form $$c=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear20a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a8b6b9fLinear20a-h3"],"title":"Subtraction","text":"What is $$7-2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear20a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["2c"],"dependencies":["a8b6b9fLinear20a-h4"],"title":"Subtraction","text":"What is 3c-c?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear20a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.5$$"],"dependencies":["a8b6b9fLinear20a-h5"],"title":"Division","text":"What is $$\\\\frac{5}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b6b9fLinear21","title":"Solve Equations Using the General Strategy","body":"Solve the Linear Equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Use a General Strategy to Solve Linear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b6b9fLinear21a","stepAnswer":["$$-3$$"],"problemType":"TextBox","stepTitle":"$$-15+4\\\\left(2-5y\\\\right)=-7\\\\left(y-4\\\\right)+4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-3$$","hints":{"DefaultPathway":[{"id":"a8b6b9fLinear21a-h1","type":"hint","dependencies":[],"title":"Distributive Property","text":"Use the distributive property to multiply $$4\\\\left(2-5y\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear21a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8-20y$$"],"dependencies":["a8b6b9fLinear21a-h1"],"title":"Multiplication","text":"What is $$4\\\\left(2-5y\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear21a-h3","type":"hint","dependencies":["a8b6b9fLinear21a-h2"],"title":"Distributive Property","text":"Use the distributive property to multiply $$-7\\\\left(y-4\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear21a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-7y+28$$"],"dependencies":["a8b6b9fLinear21a-h3"],"title":"Multiplication","text":"What is $$-7\\\\left(y-4\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear21a-h5","type":"hint","dependencies":["a8b6b9fLinear21a-h4"],"title":"Organizing","text":"Once you\'ve used the distributive property, you get $$-15+8-20y=-7y+28+4$$. Combining the numbers without a $$y$$ coefficient (through addition and subtraction) will make handling the equation easier","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear21a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-7$$"],"dependencies":["a8b6b9fLinear21a-h5"],"title":"Addition","text":"What is $$-15+8$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear21a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$32$$"],"dependencies":["a8b6b9fLinear21a-h6"],"title":"Addition","text":"What is $$28+4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear21a-h8","type":"hint","dependencies":["a8b6b9fLinear21a-h7"],"title":"Organizing","text":"Once both sides have an equation without parantheses, use algebra to manipulate the equation into the form $$y=?$$, given that the current equation is $$-7-20y=-7y+32$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear21a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$39$$"],"dependencies":["a8b6b9fLinear21a-h8"],"title":"Addition","text":"What is $$32+7$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear21a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-13y$$"],"dependencies":["a8b6b9fLinear21a-h9"],"title":"Addition","text":"What is $$-20y+7y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear21a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a8b6b9fLinear21a-h10"],"title":"Division","text":"What is $$\\\\frac{39}{-13}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b6b9fLinear22","title":"Solve Equations Using the General Strategy","body":"Solve the Linear Equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Use a General Strategy to Solve Linear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b6b9fLinear22a","stepAnswer":["$$-4$$"],"problemType":"TextBox","stepTitle":"$$4[5-8(4c-3)]=12(1-13c)-8$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-4$$","hints":{"DefaultPathway":[{"id":"a8b6b9fLinear22a-h1","type":"hint","dependencies":[],"title":"Parantheses Priority","text":"When dealing with parantheses within parantheses, simplify the inner parantheses first, then the outer","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear22a-h2","type":"hint","dependencies":["a8b6b9fLinear22a-h1"],"title":"Distributive Property I","text":"Use the distributive property to multiply $$8(4c-3)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear22a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["32c-24"],"dependencies":["a8b6b9fLinear22a-h2"],"title":"Distributive Property I","text":"What is $$8(4c-3)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear22a-h4","type":"hint","dependencies":["a8b6b9fLinear22a-h3"],"title":"Simplify","text":"After applying the distributive property to the inner parantheses, the outer parantheses can now be simplified","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear22a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$29-32c$$"],"dependencies":["a8b6b9fLinear22a-h4"],"title":"Simplify Parantheses","text":"What is $$5-(32c-24)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear22a-h6","type":"hint","dependencies":["a8b6b9fLinear22a-h5"],"title":"Distributive Property II","text":"Use the distributive property to multiply $$4(29-32c)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear22a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$116-128c$$"],"dependencies":["a8b6b9fLinear22a-h6"],"title":"Distributive Property II","text":"What is $$4(29-32c)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear22a-h8","type":"hint","dependencies":["a8b6b9fLinear22a-h7"],"title":"Distributive Property III","text":"Use the distributive property to multiply $$12(1-13c)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear22a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12-156c$$"],"dependencies":["a8b6b9fLinear22a-h8"],"title":"Distributive Property III","text":"What is $$12(1-13c)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear22a-h10","type":"hint","dependencies":["a8b6b9fLinear22a-h9"],"title":"Organizing","text":"Once you\'ve used the distributive property, you get $$116-128c=12-156c-8$$, reorganize it to $$-128c-(-156c)=12-8-116$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear22a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a8b6b9fLinear22a-h10"],"title":"Subtraction","text":"What is $$12-8$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear22a-h12","type":"hint","dependencies":["a8b6b9fLinear22a-h11"],"title":"Organizing","text":"Once both sides have an equation without parantheses, use algebra to manipulate the equation into the form $$c=?$$,","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear22a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["28c"],"dependencies":["a8b6b9fLinear22a-h12"],"title":"Subtraction","text":"What is $$-128c-(-156c)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear22a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-112$$"],"dependencies":["a8b6b9fLinear22a-h13"],"title":"Subtraction","text":"What is $$4-116$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear22a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["a8b6b9fLinear22a-h14"],"title":"Division","text":"What is $$\\\\frac{-112}{28}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b6b9fLinear23","title":"Classify Equations","body":"Classify the equation as a conditional equation, an identity, or a contradiction, and then state the solution","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Use a General Strategy to Solve Linear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b6b9fLinear23a","stepAnswer":["Identity: all real numbers"],"problemType":"MultipleChoice","stepTitle":"$$23z+19=3\\\\left(5z-9\\\\right)+8z+46$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Conditional equation: $$z=2$$","Identity: all real numbers","Contradiction: no solution"],"hints":{"DefaultPathway":[{"id":"a8b6b9fLinear23a-h1","type":"hint","dependencies":[],"title":"Organizing","text":"Modify the equation into the form $$z=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear23a-h2","type":"hint","dependencies":["a8b6b9fLinear23a-h1"],"title":"Distributive Property I","text":"Use the distributive property to multiply $$3(5z-9)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear23a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15z-27$$"],"dependencies":["a8b6b9fLinear23a-h2"],"title":"Multiplication","text":"What is $$3(5z-9)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear23a-h4","type":"hint","dependencies":["a8b6b9fLinear23a-h3"],"title":"Simplify","text":"Once you get the equation $$23z+19=15z-27+8z+46$$, simplify each side as much as possible","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear23a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$23z+19$$"],"dependencies":["a8b6b9fLinear23a-h4"],"title":"Simplify","text":"Simplify $$15z-27+8z+46$$ as much as possible","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear23a-h6","type":"hint","dependencies":["a8b6b9fLinear23a-h5"],"title":"Classifying the Equation","text":"Determine if the equation contradicts itself, or only valid under certain condition","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b6b9fLinear24","title":"Classify Equations","body":"Classify the equation as a conditional equation, an identity, or a contradiction, and then state the solution","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Use a General Strategy to Solve Linear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b6b9fLinear24a","stepAnswer":["Conditional equation: $$m=\\\\frac{16}{5}$$"],"problemType":"MultipleChoice","stepTitle":"$$22(3m-4)=8\\\\left(2m+9\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Conditional equation: $$m=\\\\frac{16}{5}$$","choices":["Conditional equation: $$m=\\\\frac{16}{5}$$","Identity: all real numbers","Contradiction: no solution"],"hints":{"DefaultPathway":[{"id":"a8b6b9fLinear24a-h1","type":"hint","dependencies":[],"title":"Organizing","text":"Modify the equation into the form $$m=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear24a-h2","type":"hint","dependencies":["a8b6b9fLinear24a-h1"],"title":"Distributive Property I","text":"Use the distributive property to multiply $$22(3m-4)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear24a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$66m-88$$"],"dependencies":["a8b6b9fLinear24a-h2"],"title":"Multiplication","text":"What is $$22(3m-4)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear24a-h4","type":"hint","dependencies":["a8b6b9fLinear24a-h3"],"title":"Distributive Property II","text":"Use the distributive property to multiply $$8\\\\left(2m+9\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear24a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16m+72$$"],"dependencies":["a8b6b9fLinear24a-h4"],"title":"Multiplication","text":"What is $$8\\\\left(2m+9\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear24a-h6","type":"hint","dependencies":["a8b6b9fLinear24a-h5"],"title":"Organizing","text":"Use algebra to manipulate the equation into the form $$m=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear24a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$50m$$"],"dependencies":["a8b6b9fLinear24a-h6"],"title":"Subtraction","text":"What is $$66m-16m$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear24a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$160$$"],"dependencies":["a8b6b9fLinear24a-h7"],"title":"Subtraction","text":"What is $$72-(-88)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear24a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{16}{5}$$"],"dependencies":["a8b6b9fLinear24a-h8"],"title":"Division","text":"What is $$\\\\frac{160}{50}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear24a-h10","type":"hint","dependencies":["a8b6b9fLinear24a-h9"],"title":"Classifying the Equation","text":"Determine if the equation contradicts itself, or only valid under certain condition","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b6b9fLinear25","title":"Classify Equations","body":"Classify the equation as a conditional equation, an identity, or a contradiction, and then state the solution","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Use a General Strategy to Solve Linear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b6b9fLinear25a","stepAnswer":["Identity: all real numbers"],"problemType":"MultipleChoice","stepTitle":"$$9\\\\left(14d+9\\\\right)+4d=\\\\operatorname{13}\\\\left(10d+6\\\\right)+3$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Conditional equation: $$z=2$$","Identity: all real numbers","Contradiction: no solution"],"hints":{"DefaultPathway":[{"id":"a8b6b9fLinear25a-h1","type":"hint","dependencies":[],"title":"Organizing","text":"Rewrite the equation into the form $$d=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear25a-h2","type":"hint","dependencies":[],"title":"Distributive Property I","text":"Use the distributive property to multiply $$9\\\\left(14d+9\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear25a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$126d+81$$"],"dependencies":["a8b6b9fLinear25a-h2"],"title":"Multiplication","text":"What is $$9\\\\left(14d+9\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear25a-h4","type":"hint","dependencies":[],"title":"Distributive Property II","text":"Use the distributive property to multiple $$\\\\operatorname{13}\\\\left(10d+6\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear25a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$130d+78$$"],"dependencies":["a8b6b9fLinear25a-h4"],"title":"Multiplication","text":"What is $$\\\\operatorname{13}\\\\left(10d+6\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear25a-h6","type":"hint","dependencies":[],"title":"Simplify","text":"Once you get the equation $$126d+81+4d=130d+78+3$$, simplify each side as much possible","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear25a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$130d+81$$"],"dependencies":["a8b6b9fLinear25a-h3"],"title":"Addition","text":"Simplify $$126d+81+4d$$ as much as possible","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear25a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$130d+81$$"],"dependencies":["a8b6b9fLinear25a-h5"],"title":"Addition","text":"Simplify $$130d+78+3$$ as much as possible","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear25a-h9","type":"hint","dependencies":[],"title":"Classifying the Equation","text":"Determine if the equation contradicts itself, or only valid under certain condition","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b6b9fLinear26","title":"Solve Equations with Fraction or Decimal Coefficients","body":"Solve the following equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Use a General Strategy to Solve Linear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b6b9fLinear26a","stepAnswer":["$$-1$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{5}{6} y-\\\\frac{2}{3}=\\\\frac{-3}{2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1$$","hints":{"DefaultPathway":[{"id":"a8b6b9fLinear26a-h1","type":"hint","dependencies":[],"title":"Multiply by LCD","text":"Multiply each term by the LCD of all the denominators","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear26a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a8b6b9fLinear26a-h1"],"title":"Identify LCD","text":"What is the LCD of the denominators?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear26a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a8b6b9fLinear26a-h2"],"title":"Multiplication","text":"What is $$6\\\\frac{5}{6}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear26a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a8b6b9fLinear26a-h2"],"title":"Multiplication","text":"What is $$6\\\\frac{2}{3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear26a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":["a8b6b9fLinear26a-h2"],"title":"Multiplication","text":"What is $$6\\\\left(-\\\\frac{3}{2}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear26a-h6","type":"hint","dependencies":[],"title":"Simplify","text":"Use algebra to manipulate the equation into the form $$y=?$$, now that you have an equation $$5y-4=-9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear26a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":[],"title":"Addition","text":"What is $$-9+4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear26a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a8b6b9fLinear26a-h7"],"title":"Division","text":"What is $$\\\\frac{-5}{5}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b6b9fLinear27","title":"Solve Equations with Fraction or Decimal Coefficients","body":"Solve the following equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Use a General Strategy to Solve Linear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b6b9fLinear27a","stepAnswer":["$$\\\\frac{9}{4}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{1}{3} w+\\\\frac{5}{4}=w-\\\\frac{1}{4}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{9}{4}$$","hints":{"DefaultPathway":[{"id":"a8b6b9fLinear27a-h1","type":"hint","dependencies":[],"title":"Multiply by LCD","text":"Multiply each term by the LCD of all the denominators","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear27a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a8b6b9fLinear27a-h1"],"title":"Identify LCD","text":"What is the LCD of the denominators?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear27a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["4w"],"dependencies":["a8b6b9fLinear27a-h1"],"title":"Multiplication","text":"What is $$12\\\\frac{1}{3} w$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear27a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["a8b6b9fLinear27a-h1"],"title":"Multiplication","text":"What is $$12\\\\frac{5}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear27a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["12w"],"dependencies":["a8b6b9fLinear27a-h1"],"title":"Multiplication","text":"What is $$12w$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear27a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a8b6b9fLinear27a-h1"],"title":"Multiplication","text":"What is $$12\\\\frac{1}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear27a-h7","type":"hint","dependencies":[],"title":"Simplify","text":"Use algebra to manipulate the equation into the form $$y=?$$, now that you have an equation $$4w+15=12w-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear27a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-8w$$"],"dependencies":[],"title":"Finding w","text":"What is $$4w-12w$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear27a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-18$$"],"dependencies":["a8b6b9fLinear27a-h4","a8b6b9fLinear27a-h6"],"title":"Finding w","text":"What is $$-3-15$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear27a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{9}{4}$$"],"dependencies":["a8b6b9fLinear27a-h9"],"title":"Finding w","text":"What is $$\\\\frac{-18}{-8}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b6b9fLinear28","title":"Solve Equations with Fraction or Decimal Coefficients","body":"Solve the following equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Use a General Strategy to Solve Linear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b6b9fLinear28a","stepAnswer":["$$-41$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{1}{4} \\\\left(p-7\\\\right)=\\\\frac{1}{3} \\\\left(p+5\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-41$$","hints":{"DefaultPathway":[{"id":"a8b6b9fLinear28a-h1","type":"hint","dependencies":[],"title":"Distributive Property I","text":"Use the distributive property to multiply $$\\\\frac{1}{4} \\\\left(p-7\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear28a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4} p-\\\\frac{7}{4}$$"],"dependencies":["a8b6b9fLinear28a-h1"],"title":"Multiplication","text":"What is $$\\\\frac{1}{4} \\\\left(p-7\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear28a-h3","type":"hint","dependencies":[],"title":"Distributive Property II","text":"Use the distributive property to multiply $$\\\\frac{1}{3} \\\\left(p+5\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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denominators?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear28a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3p$$"],"dependencies":["a8b6b9fLinear28a-h6"],"title":"Multiplication","text":"What is $$12\\\\frac{1}{4} p$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear28a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$21$$"],"dependencies":["a8b6b9fLinear28a-h6"],"title":"Multiplication","text":"What is $$12\\\\frac{7}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear28a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4p$$"],"dependencies":["a8b6b9fLinear28a-h6"],"title":"Multiplication","text":"What is $$12\\\\frac{1}{3} p$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear28a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a8b6b9fLinear28a-h6"],"title":"Multiplication","text":"What is $$12\\\\frac{5}{3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear28a-h11","type":"hint","dependencies":[],"title":"Simply","text":"Use algebra to manipulate the equation into the form $$p=?$$, now that you have an equation $$3p-21=4p+20$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear28a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1p$$"],"dependencies":["a8b6b9fLinear28a-h11"],"title":"Subtraction","text":"What is $$3p-4p$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear28a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$41$$"],"dependencies":[],"title":"Addition","text":"What is $$20+21$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear28a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-41$$"],"dependencies":["a8b6b9fLinear28a-h13"],"title":"Division","text":"What is $$\\\\frac{41}{-1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b6b9fLinear29","title":"Solve Equations with Fraction or Decimal Coefficients","body":"Solve the following equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Use a General Strategy to Solve Linear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b6b9fLinear29a","stepAnswer":["$$18$$"],"problemType":"TextBox","stepTitle":"$$0.4x+0.6=0.5x-1.2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$18$$","hints":{"DefaultPathway":[{"id":"a8b6b9fLinear29a-h1","type":"hint","dependencies":[],"title":"Rewrite the Equation","text":"Write the original equation into the form $$a x=b$$, where a and $$b$$ are some numbers","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear29a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-0.1x$$"],"dependencies":[],"title":"Subtraction","text":"What is $$0.4x-0.5x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear29a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1.8$$"],"dependencies":[],"title":"Subtraction","text":"What is $$-1.2-0.6$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear29a-h4","type":"hint","dependencies":["a8b6b9fLinear29a-h3"],"title":"Removing the Decimals","text":"To get rid of the decimals, multiply every term in the equation (-0.1*x=-1.8) by $$100$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear29a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-10$$"],"dependencies":["a8b6b9fLinear29a-h4"],"title":"Multiplication","text":"What is $$-0.1\\\\times100$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear29a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-180$$"],"dependencies":["a8b6b9fLinear29a-h4"],"title":"Multiplication","text":"What is $$-1.8\\\\times100$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear29a-h7","type":"hint","dependencies":["a8b6b9fLinear29a-h6"],"title":"Simplify","text":"Use algebra to maniputlate the equation into the form $$x=?$$, now that you have the equation $$-10x=-180$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear29a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18$$"],"dependencies":["a8b6b9fLinear29a-h5","a8b6b9fLinear29a-h6"],"title":"Division","text":"What is $$\\\\frac{-180}{-10}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b6b9fLinear3","title":"How to Solve a Linear Equation Using a General Strategy","body":"Solve the following equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Use a General Strategy to Solve Linear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b6b9fLinear3a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"$$7(n-3)-8=-15$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a8b6b9fLinear3a-h1","type":"hint","dependencies":[],"title":"Separate Constants and Variables on Either Side of the Equation","text":"We must first simplify each side of the equation to get $$7n-21-8=-15$$. Now, we collect all constants on the right side. $$7n=14$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear3a-h2","type":"hint","dependencies":[],"title":"Simplify","text":"Simplify the left hand side to $$7n-29=-15$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear3a-h3","type":"hint","dependencies":[],"title":"Simplify","text":"Organize the constant to the same side to obtain $$7n=14$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear3a-h4","type":"hint","dependencies":[],"title":"Division","text":"We must make the coefficient of the variable equal to $$1$$, so we divide by $$7$$ on both sides to get $$n=2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b6b9fLinear30","title":"Solve Equations with Fraction or Decimal Coefficients","body":"Solve the following equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Use a General Strategy to Solve Linear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b6b9fLinear30a","stepAnswer":["$$8$$"],"problemType":"TextBox","stepTitle":"$$0.1d+0.25\\\\left(d+5\\\\right)=4.05$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8$$","hints":{"DefaultPathway":[{"id":"a8b6b9fLinear30a-h1","type":"hint","dependencies":[],"title":"Distributive Property I","text":"Use the distributive property to multiply $$0.25\\\\left(d+5\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear30a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.25d+1.25$$"],"dependencies":["a8b6b9fLinear30a-h1"],"title":"Multiplication","text":"What is $$0.25\\\\left(d+5\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear30a-h3","type":"hint","dependencies":[],"title":"Combine Terms","text":"Simplify each side of the current equation (0.10*d+0.25*d+1.25=4.05) by combining like terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear30a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.35d$$"],"dependencies":["a8b6b9fLinear30a-h3"],"title":"Addition","text":"What is $$0.1d+0.25d$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear30a-h5","type":"hint","dependencies":["a8b6b9fLinear30a-h4"],"title":"Removing the Decimal","text":"To get rid of the decimals, multiply every term in the current equation (0.35*d+1.25=4.05) by $$100$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear30a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$35d$$"],"dependencies":["a8b6b9fLinear30a-h5"],"title":"Multiplication","text":"What is $$100\\\\times0.35 d$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear30a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$125$$"],"dependencies":["a8b6b9fLinear30a-h5"],"title":"Multiplication","text":"What is $$100\\\\times1.25$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear30a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$405$$"],"dependencies":["a8b6b9fLinear30a-h5"],"title":"Multiplication","text":"What is $$100\\\\times4.05$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear30a-h9","type":"hint","dependencies":[],"title":"Simplify","text":"Use algebra to manipulate the equation into the form $$d=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear30a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$280$$"],"dependencies":["a8b6b9fLinear30a-h9"],"title":"Subtraction","text":"What is $$405-125$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear30a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a8b6b9fLinear30a-h10"],"title":"Division","text":"What is $$\\\\frac{280}{35}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b6b9fLinear4","title":"How to Solve a Linear Equation Using a General Strategy","body":"Solve the following equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Use a General Strategy to Solve Linear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b6b9fLinear4a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"$$2\\\\left(m-4\\\\right)+3=-1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a8b6b9fLinear4a-h1","type":"hint","dependencies":[],"title":"Separate Constants and Variables on Either Side of the Equation","text":"We must first simplify each side of the equation to get $$2m-8+3=-$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear4a-h2","type":"hint","dependencies":[],"title":"Simplify","text":"Simplify the left hand side to $$2m-5=-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear4a-h3","type":"hint","dependencies":[],"title":"Simplify","text":"Organize the constant to the same side to obtain $$2m=4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear4a-h4","type":"hint","dependencies":[],"title":"Division","text":"We must make the coefficient of the variable equal to $$1$$, so we divide by $$2$$ on both sides to get $$m=2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b6b9fLinear5","title":"How to Solve a Linear Equation Using a General Strategy","body":"Solve the following equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Use a General Strategy to Solve Linear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b6b9fLinear5a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"$$5\\\\left(a-3\\\\right)+5=-10$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a8b6b9fLinear5a-h1","type":"hint","dependencies":[],"title":"Separate Constants and Variables on Either Side of the Equation","text":"We must first simplify each side of the equation to get $$5a-15+5=-10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear5a-h2","type":"hint","dependencies":[],"title":"Simplify","text":"Simply the left hand side to $$5a-10=-10$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear5a-h3","type":"hint","dependencies":[],"title":"Simplify","text":"Organize the constant to the same side to obtain $$7n=14$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear5a-h4","type":"hint","dependencies":[],"title":"Division","text":"We must make the coefficient of the variable equal to $$1$$, so we divide by $$5$$ on both sides to get $$a=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b6b9fLinear6","title":"How to Solve a Linear Equation Using a General Strategy","body":"Solve the following equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Use a General Strategy to Solve Linear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b6b9fLinear6a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2}{3} \\\\left(3m-6\\\\right)=5-m$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a8b6b9fLinear6a-h1","type":"hint","dependencies":[],"title":"Separate Constants and Variables on Either Side of the Equation","text":"We must first simplify each side of the equation to get $$2m-4=5-m$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear6a-h2","type":"hint","dependencies":[],"title":"Simplify","text":"Organize the constant to the same side to obtain $$3m=9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear6a-h3","type":"hint","dependencies":[],"title":"Division","text":"We must make the coefficient of the variable equal to $$1$$, so we divide by $$3$$ on both sides to get $$m=3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b6b9fLinear7","title":"How to Solve a Linear Equation Using a General Strategy","body":"Solve the following equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Use a General Strategy to Solve Linear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b6b9fLinear7a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{1}{3} \\\\left(6u+3\\\\right)=7-u$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a8b6b9fLinear7a-h1","type":"hint","dependencies":[],"title":"Separate Constants and Variables on Either Side of the Equation","text":"We must first simplify each side of the equation to get $$2u+1=7-u$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear7a-h2","type":"hint","dependencies":[],"title":"Simplify","text":"Organize the constant to the same side to obtain $$3u=6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear7a-h3","type":"hint","dependencies":[],"title":"Division","text":"We must make the coefficient of the variable equal to $$1$$, so we divide by 23on both sides to get $$u=2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b6b9fLinear8","title":"How to Solve a Linear Equation Using a General Strategy","body":"Solve the following equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Use a General Strategy to Solve Linear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b6b9fLinear8a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2}{3} \\\\left(9x-12\\\\right)=8+2x$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a8b6b9fLinear8a-h1","type":"hint","dependencies":[],"title":"Separate Constants and Variables on Either Side of the Equation","text":"We must first simplify each side of the equation to get $$6x-8=8+2x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear8a-h2","type":"hint","dependencies":[],"title":"Simplify","text":"Organize the constant to the same side to obtain $$4x=16$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear8a-h3","type":"hint","dependencies":[],"title":"Division","text":"We must make the coefficient of the variable equal to $$1$$, so we divide by $$4$$ on both sides to get $$x=4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8b6b9fLinear9","title":"How to Solve a Linear Equation Using a General Strategy","body":"Solve the following equation","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Use a General Strategy to Solve Linear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8b6b9fLinear9a","stepAnswer":["$$\\\\frac{-9}{2}$$"],"problemType":"TextBox","stepTitle":"$$4(x-1)-2=5\\\\left(2x+3\\\\right)+6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-9}{2}$$","hints":{"DefaultPathway":[{"id":"a8b6b9fLinear9a-h1","type":"hint","dependencies":[],"title":"Separate Constants and Variables on Either Side of the Equation","text":"We must first simplify each side of the equation to get $$4x-4-2=10x+15+6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear9a-h2","type":"hint","dependencies":[],"title":"Simplify","text":"Organize the constant to the same side to obtain $$6x=-27$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8b6b9fLinear9a-h3","type":"hint","dependencies":[],"title":"Division","text":"We must make the coefficient of the variable equal to $$1$$, so we divide by $$6$$ on both sides to get $$x=\\\\frac{-9}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8d511eDeterminant1","title":"Evaluate the Determinant of a 2\xd72 Matrix","body":"Evaluate the following determinants","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Solve Systems of Equations Using Determinants","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8d511eDeterminant1a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"$$\\\\begin{bmatrix} 4 & -2 \\\\\\\\ 3 & -1 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a8d511eDeterminant1a-h1","type":"hint","dependencies":[],"title":"Principle","text":"Write the matrix in determinant form","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant1a-h2","type":"hint","dependencies":["a8d511eDeterminant1a-h1"],"title":"Calculation","text":"Subtract the products of the diagonals","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a8d511eDeterminant1a-h2"],"title":"Calculation","text":"What is $$4\\\\left(-1\\\\right)-3\\\\left(-2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a8d511eDeterminant1b","stepAnswer":["$$-8$$"],"problemType":"TextBox","stepTitle":"$$\\\\begin{bmatrix} -3 & -4 \\\\\\\\ -2 & 0 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-8$$","hints":{"DefaultPathway":[{"id":"a8d511eDeterminant1b-h1","type":"hint","dependencies":[],"title":"Principle","text":"Write the matrix in determinant form","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant1b-h2","type":"hint","dependencies":["a8d511eDeterminant1b-h1"],"title":"Calculation","text":"Subtract the products of the diagonals","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant1b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-8$$"],"dependencies":["a8d511eDeterminant1b-h2"],"title":"Calculation","text":"What is $$0\\\\left(-3\\\\right)-\\\\left(-4\\\\right) \\\\left(-2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8d511eDeterminant10","title":"Dependent and Inconsistent Systems of Equations","body":"Solve the system of equations using Cramer\u2019s rule","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Solve Systems of Equations Using Determinants","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8d511eDeterminant10a","stepAnswer":["inconsistent"],"problemType":"MultipleChoice","stepTitle":"$$x+3y=4;$$ $$-2x-6y=3$$","stepBody":"","answerType":"string","variabilization":{},"choices":["inconsistent","dependent"],"hints":{"DefaultPathway":[{"id":"a8d511eDeterminant10a-h1","type":"hint","dependencies":[],"title":"Principle","text":"Write a 2x2 determinant with the coefficients of $$x$$ and $$y$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant10a-h2","type":"hint","dependencies":["a8d511eDeterminant10a-h1"],"title":"Operation","text":"Evaluate the 2x2 determinant","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a8d511eDeterminant10a-h2"],"title":"Calculation","text":"What is $$-6-(-6)$$?","variabilization":{},"oer":"https://OATutor.io 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant20a-h3","type":"hint","dependencies":["a8d511eDeterminant20a-h2"],"title":"Simplify","text":"Simplify the determinant down to a single value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8d511eDeterminant21","title":"Evaluate the specified determinant","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Solve Systems of Equations Using Determinants","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8d511eDeterminant21a","stepAnswer":["$$31$$"],"problemType":"TextBox","stepTitle":"$$\\\\begin{bmatrix} 3 & -1 & 4 \\\\\\\\ -1 & 0 & -2 \\\\\\\\ -4 & 1 & 5 \\\\end{bmatrix}$$","stepBody":"Find minor $$b_2$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$31$$","hints":{"DefaultPathway":[{"id":"a8d511eDeterminant21a-h1","type":"hint","dependencies":[],"title":"Write","text":"Eliminate the row and column that contains the given spot in the matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant21a-h2","type":"hint","dependencies":["a8d511eDeterminant21a-h1"],"title":"Find determinant","text":"Now, in the 2x2 matrix that remains, evaluate the determinant.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant21a-h3","type":"hint","dependencies":["a8d511eDeterminant21a-h2"],"title":"Simplify","text":"Simplify the determinant down to a single value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8d511eDeterminant22","title":"Evaluate the specified determinant","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Solve Systems of Equations Using Determinants","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8d511eDeterminant22a","stepAnswer":["$$-1$$"],"problemType":"TextBox","stepTitle":"$$\\\\begin{bmatrix} 3 & -1 & 4 \\\\\\\\ -1 & 0 & -2 \\\\\\\\ -4 & 1 & 5 \\\\end{bmatrix}$$","stepBody":"Find minor $$c_3$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1$$","hints":{"DefaultPathway":[{"id":"a8d511eDeterminant22a-h1","type":"hint","dependencies":[],"title":"Write","text":"Eliminate the row and column that contains the given spot in the matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC 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4.0>"},{"id":"a8d511eDeterminant23a-h3","type":"hint","dependencies":["a8d511eDeterminant23a-h2"],"title":"Simplify","text":"Simplify the determinant down to a single value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8d511eDeterminant24","title":"Evaluate the specified determinant","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Solve Systems of Equations Using Determinants","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8d511eDeterminant24a","stepAnswer":["$$-14$$"],"problemType":"TextBox","stepTitle":"$$\\\\begin{bmatrix} -1 & -3 & 2 \\\\\\\\ 4 & -2 & -1 \\\\\\\\ -2 & 0 & -3 \\\\end{bmatrix}$$","stepBody":"Find minor $$b_1$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-14$$","hints":{"DefaultPathway":[{"id":"a8d511eDeterminant24a-h1","type":"hint","dependencies":[],"title":"Write","text":"Eliminate the row and column that contains the given spot in the matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant24a-h2","type":"hint","dependencies":["a8d511eDeterminant24a-h1"],"title":"Find determinant","text":"Now, in the 2x2 matrix that remains, evaluate the determinant.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant24a-h3","type":"hint","dependencies":["a8d511eDeterminant24a-h2"],"title":"Simplify","text":"Simplify the determinant down to a single value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8d511eDeterminant25","title":"Evaluate the specified determinant","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Solve Systems of Equations Using Determinants","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8d511eDeterminant25a","stepAnswer":["$$-6$$"],"problemType":"TextBox","stepTitle":"$$\\\\begin{bmatrix} -1 & -3 & 2 \\\\\\\\ 4 & -2 & -1 \\\\\\\\ -2 & 0 & -3 \\\\end{bmatrix}$$","stepBody":"Find minor $$c_2$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-6$$","hints":{"DefaultPathway":[{"id":"a8d511eDeterminant25a-h1","type":"hint","dependencies":[],"title":"Write","text":"Eliminate the row and column that contains the given spot in the matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant25a-h2","type":"hint","dependencies":["a8d511eDeterminant25a-h1"],"title":"Find determinant","text":"Now, in the 2x2 matrix that remains, evaluate the determinant.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant25a-h3","type":"hint","dependencies":["a8d511eDeterminant25a-h2"],"title":"Simplify","text":"Simplify the determinant down to a single value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8d511eDeterminant26","title":"Evaluate the specified determinant","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Solve Systems of Equations Using Determinants","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8d511eDeterminant26a","stepAnswer":["$$9$$"],"problemType":"TextBox","stepTitle":"$$\\\\begin{bmatrix} -2 & -2 & 3 \\\\\\\\ 1 & -3 & 0 \\\\\\\\ -2 & 3 & -2 \\\\end{bmatrix}$$","stepBody":"Find minor $$a_3$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9$$","hints":{"DefaultPathway":[{"id":"a8d511eDeterminant26a-h1","type":"hint","dependencies":[],"title":"Write","text":"Eliminate the row and column that contains the given spot in the matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant26a-h2","type":"hint","dependencies":["a8d511eDeterminant26a-h1"],"title":"Find determinant","text":"Now, in the 2x2 matrix that remains, evaluate the determinant.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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$$b_3$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-3$$","hints":{"DefaultPathway":[{"id":"a8d511eDeterminant27a-h1","type":"hint","dependencies":[],"title":"Write","text":"Eliminate the row and column that contains the given spot in the matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant27a-h2","type":"hint","dependencies":["a8d511eDeterminant27a-h1"],"title":"Find determinant","text":"Now, in the 2x2 matrix that remains, evaluate the determinant.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant27a-h3","type":"hint","dependencies":["a8d511eDeterminant27a-h2"],"title":"Simplify","text":"Simplify the determinant down to a single value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8d511eDeterminant28","title":"Evaluate the specified determinant","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Solve Systems of Equations Using Determinants","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8d511eDeterminant28a","stepAnswer":["$$8$$"],"problemType":"TextBox","stepTitle":"$$\\\\begin{bmatrix} -2 & -2 & 3 \\\\\\\\ 1 & -3 & 0 \\\\\\\\ -2 & 3 & -2 \\\\end{bmatrix}$$","stepBody":"Find minor $$c_3$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8$$","hints":{"DefaultPathway":[{"id":"a8d511eDeterminant28a-h1","type":"hint","dependencies":[],"title":"Write","text":"Eliminate the row and column that contains the given spot in the matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant28a-h2","type":"hint","dependencies":["a8d511eDeterminant28a-h1"],"title":"Find determinant","text":"Now, in the 2x2 matrix that remains, evaluate the determinant.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant28a-h3","type":"hint","dependencies":["a8d511eDeterminant28a-h2"],"title":"Simplify","text":"Simplify the determinant down to a single value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8d511eDeterminant29","title":"Evaluate the specified determinant","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Solve Systems of Equations Using Determinants","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8d511eDeterminant29a","stepAnswer":["$$-7$$"],"problemType":"TextBox","stepTitle":"$$\\\\begin{bmatrix} 2 & -3 & -4 \\\\\\\\ -1 & 2 & -3 \\\\\\\\ 0 & -1 & -2 \\\\end{bmatrix}$$","stepBody":"Find minor $$a_1$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-7$$","hints":{"DefaultPathway":[{"id":"a8d511eDeterminant29a-h1","type":"hint","dependencies":[],"title":"Write","text":"Eliminate the row and column that contains the given spot in the matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant29a-h2","type":"hint","dependencies":["a8d511eDeterminant29a-h1"],"title":"Find determinant","text":"Now, in the 2x2 matrix that remains, evaluate the determinant.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant29a-h3","type":"hint","dependencies":["a8d511eDeterminant29a-h2"],"title":"Simplify","text":"Simplify the determinant down to a single value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8d511eDeterminant3","title":"For the determinant $$\\\\begin{bmatrix} 4 & -2 & 3 \\\\\\\\ 1 & 0 & -3 \\\\\\\\ -2 & -4 & 2 \\\\end{bmatrix}$$, find and then evaluate the following minors","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Solve Systems of Equations Using Determinants","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8d511eDeterminant3a","stepAnswer":["$$-12$$"],"problemType":"TextBox","stepTitle":"a1","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-12$$","hints":{"DefaultPathway":[{"id":"a8d511eDeterminant3a-h1","type":"hint","dependencies":[],"title":"Principle","text":"Eliminant the row and column that contains a1","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant3a-h2","type":"hint","dependencies":["a8d511eDeterminant3a-h1"],"title":"Principle","text":"Calculate the 2x2 matrix","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant3a-h3","type":"hint","dependencies":["a8d511eDeterminant3a-h2"],"title":"Calculation","text":"Subtract the products of the diagonals","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-12$$"],"dependencies":["a8d511eDeterminant3a-h3"],"title":"Calculation","text":"What is $$0\\\\times2-\\\\left(-3\\\\right) \\\\left(-4\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a8d511eDeterminant3b","stepAnswer":["$$-15$$"],"problemType":"TextBox","stepTitle":"b3","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-15$$","hints":{"DefaultPathway":[{"id":"a8d511eDeterminant3b-h1","type":"hint","dependencies":[],"title":"Principle","text":"Eliminant the row and column that contains b3","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant3b-h2","type":"hint","dependencies":["a8d511eDeterminant3b-h1"],"title":"Principle","text":"Calculate the 2x2 matrix","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant3b-h3","type":"hint","dependencies":["a8d511eDeterminant3b-h2"],"title":"Calculation","text":"Subtract the products of the diagonals","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant3b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-15$$"],"dependencies":["a8d511eDeterminant3b-h3"],"title":"Calculation","text":"What is $$4\\\\left(-3\\\\right)-1\\\\times3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a8d511eDeterminant3c","stepAnswer":["$$-20$$"],"problemType":"TextBox","stepTitle":"c2","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-20$$","hints":{"DefaultPathway":[{"id":"a8d511eDeterminant3c-h1","type":"hint","dependencies":[],"title":"Principle","text":"Eliminant the row and column that contains c2","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant3c-h2","type":"hint","dependencies":["a8d511eDeterminant3c-h1"],"title":"Principle","text":"Calculate the 2x2 matrix","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant3c-h3","type":"hint","dependencies":["a8d511eDeterminant3c-h2"],"title":"Calculation","text":"Subtract the products of the diagonals","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant3c-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-20$$"],"dependencies":["a8d511eDeterminant3c-h3"],"title":"Calculation","text":"What is $$4\\\\left(-4\\\\right)-\\\\left(-2\\\\right) \\\\left(-2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8d511eDeterminant30","title":"Evaluate the specified determinant","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Solve Systems of Equations Using Determinants","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8d511eDeterminant30a","stepAnswer":["$$-4$$"],"problemType":"TextBox","stepTitle":"$$\\\\begin{bmatrix} 2 & -3 & -4 \\\\\\\\ -1 & 2 & -3 \\\\\\\\ 0 & -1 & -2 \\\\end{bmatrix}$$","stepBody":"Find minor $$b_2$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-4$$","hints":{"DefaultPathway":[{"id":"a8d511eDeterminant30a-h1","type":"hint","dependencies":[],"title":"Write","text":"Eliminate the row and column that contains the given spot in the matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant30a-h2","type":"hint","dependencies":["a8d511eDeterminant30a-h1"],"title":"Find determinant","text":"Now, in the 2x2 matrix that remains, evaluate the determinant.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant30a-h3","type":"hint","dependencies":["a8d511eDeterminant30a-h2"],"title":"Simplify","text":"Simplify the determinant down to a single value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8d511eDeterminant4","title":"Evaluate the Determinant of a 3\xd73 Matrix","body":"For the determinant $$\\\\begin{bmatrix} 4 & -2 & 3 \\\\\\\\ 1 & 0 & -3 \\\\\\\\ -2 & -4 & 2 \\\\end{bmatrix}$$, find and then evaluate the following minors","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Solve Systems of Equations Using Determinants","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8d511eDeterminant4a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"a1","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a8d511eDeterminant4a-h1","type":"hint","dependencies":[],"title":"Principle","text":"Eliminant the row and column that contains a1","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant4a-h2","type":"hint","dependencies":["a8d511eDeterminant4a-h1"],"title":"Principle","text":"Calculate the 2x2 matrix","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant4a-h3","type":"hint","dependencies":["a8d511eDeterminant4a-h2"],"title":"Calculation","text":"Subtract the products of the diagonals","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a8d511eDeterminant4a-h3"],"title":"Calculation","text":"What is $$2\\\\times3-\\\\left(-1\\\\right) \\\\left(-3\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a8d511eDeterminant4b","stepAnswer":["$$11$$"],"problemType":"TextBox","stepTitle":"b2","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$11$$","hints":{"DefaultPathway":[{"id":"a8d511eDeterminant4b-h1","type":"hint","dependencies":[],"title":"Principle","text":"Eliminant the row and column that contains b2","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant4b-h2","type":"hint","dependencies":["a8d511eDeterminant4b-h1"],"title":"Principle","text":"Calculate the 2x2 matrix","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant4b-h3","type":"hint","dependencies":["a8d511eDeterminant4b-h2"],"title":"Calculation","text":"Subtract the products of the diagonals","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant4b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11$$"],"dependencies":["a8d511eDeterminant4b-h3"],"title":"Calculation","text":"What is $$1\\\\times3-4\\\\left(-2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a8d511eDeterminant4c","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"c3","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a8d511eDeterminant4c-h1","type":"hint","dependencies":[],"title":"Principle","text":"Eliminant the row and column that contains c3","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant4c-h2","type":"hint","dependencies":["a8d511eDeterminant4c-h1"],"title":"Principle","text":"Calculate the 2x2 matrix","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant4c-h3","type":"hint","dependencies":["a8d511eDeterminant4c-h2"],"title":"Calculation","text":"Subtract the products of the diagonals","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant4c-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a8d511eDeterminant4c-h3"],"title":"Calculation","text":"What is $$1\\\\times2-0\\\\left(-1\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8d511eDeterminant5","title":"Expanding by Minors Along the First Row to Evaluate a 3x3 determinant","body":"Evaluate the following determinant by expanding by minors along the first row","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Solve Systems of Equations Using Determinants","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8d511eDeterminant5a","stepAnswer":["$$-25$$"],"problemType":"TextBox","stepTitle":"$$\\\\begin{bmatrix} 2 & -3 & -1 \\\\\\\\ 3 & 2 & 0 \\\\\\\\ -1 & -1 & -2 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-25$$","hints":{"DefaultPathway":[{"id":"a8d511eDeterminant5a-h1","type":"hint","dependencies":[],"title":"Principle","text":"Expand the minors along the first row to $$2*\\\\begin{bmatrix} 2 & 0 \\\\\\\\ -1 & -2 \\\\end{bmatrix}-(-3)*\\\\begin{bmatrix} 3 & 0 \\\\\\\\ -1 & -2 \\\\end{bmatrix}+(-1)*\\\\begin{bmatrix} 3 & 2 \\\\\\\\ -1 & -1 \\\\end{bmatrix}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant5a-h2","type":"hint","dependencies":["a8d511eDeterminant5a-h1"],"title":"Operation","text":"Evaluate the 2x2 determinants","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-25$$"],"dependencies":["a8d511eDeterminant5a-h2"],"title":"Calculation","text":"What is $$2\\\\left(-4\\\\right)+3\\\\left(-6\\\\right)-1\\\\left(-1\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8d511eDeterminant6","title":"Expanding by Minors Along the First Row to Evaluate a 3x3 determinant","body":"Evaluate the following determinant by expanding by minors along the first row","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Solve Systems of Equations Using Determinants","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8d511eDeterminant6a","stepAnswer":["$$21$$"],"problemType":"TextBox","stepTitle":"$$\\\\begin{bmatrix} 3 & -2 & 4 \\\\\\\\ 0 & -1 & -2 \\\\\\\\ 2 & 3 & -1 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$21$$","hints":{"DefaultPathway":[{"id":"a8d511eDeterminant6a-h1","type":"hint","dependencies":[],"title":"Principle","text":"Expand the minors along the first row to $$3*\\\\begin{bmatrix} -1 & -2 \\\\\\\\ 3 & -1 \\\\end{bmatrix}-(-2)*\\\\begin{bmatrix} 0 & -1 \\\\\\\\ -2 & 2 \\\\end{bmatrix}+4*\\\\begin{bmatrix} 0 & 3 \\\\\\\\ -1 & 2 \\\\end{bmatrix}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant6a-h2","type":"hint","dependencies":["a8d511eDeterminant6a-h1"],"title":"Operation","text":"Evaluate the 2x2 determinants","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$21$$"],"dependencies":["a8d511eDeterminant6a-h2"],"title":"Calculation","text":"What is $$3\\\\times7+2\\\\times4+4\\\\left(-2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8d511eDeterminant7","title":"Expanding by Minors Along the First Row to Evaluate a 3x3 determinant","body":"Evaluate the following determinant by expanding by minors","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Solve Systems of Equations Using Determinants","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8d511eDeterminant7a","stepAnswer":["$$49$$"],"problemType":"TextBox","stepTitle":"$$\\\\begin{bmatrix} 4 & -1 & -3 \\\\\\\\ 3 & 0 & 2 \\\\\\\\ 5 & -4 & -3 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$49$$","hints":{"DefaultPathway":[{"id":"a8d511eDeterminant7a-h1","type":"hint","dependencies":[],"title":"Principle","text":"Expanding the rows containing $$0$$ will be easy for calculation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant7a-h2","type":"hint","dependencies":["a8d511eDeterminant7a-h1"],"title":"Order of Operation","text":"Expand the minors along the second row","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant7a-h3","type":"hint","dependencies":["a8d511eDeterminant7a-h2"],"title":"Principle","text":"Remember the sign of the second row is (-,+,-)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant7a-h4","type":"hint","dependencies":["a8d511eDeterminant7a-h3"],"title":"Operation","text":"Evaluate the 2x2 determinants","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant7a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$49$$"],"dependencies":["a8d511eDeterminant7a-h4"],"title":"Calculation","text":"What is $$\\\\left(-3\\\\right) \\\\left(-9\\\\right)+0-2\\\\left(-11\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8d511eDeterminant8","title":"Expanding by Minors Along the First Row to Evaluate a 3x3 determinant","body":"Evaluate the following determinant by expanding by minors","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Solve Systems of Equations Using Determinants","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8d511eDeterminant8a","stepAnswer":["$$-11$$"],"problemType":"TextBox","stepTitle":"$$\\\\begin{bmatrix} 2 & -1 & -3 \\\\\\\\ 0 & 3 & -4 \\\\\\\\ 3 & -4 & -3 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-11$$","hints":{"DefaultPathway":[{"id":"a8d511eDeterminant8a-h1","type":"hint","dependencies":[],"title":"Principle","text":"Expanding the rows containing $$0$$ will be easy for calculation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant8a-h2","type":"hint","dependencies":["a8d511eDeterminant8a-h1"],"title":"Order of Operation","text":"Expand the minors along the second row","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant8a-h3","type":"hint","dependencies":["a8d511eDeterminant8a-h2"],"title":"Principle","text":"Remember the sign of the second row is (-,+,-)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant8a-h4","type":"hint","dependencies":["a8d511eDeterminant8a-h3"],"title":"Operation","text":"Evaluate the 2x2 determinants","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant8a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-11$$"],"dependencies":["a8d511eDeterminant8a-h4"],"title":"Calculation","text":"What is $$0+3\\\\times3-\\\\left(-4\\\\right) \\\\left(-5\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8d511eDeterminant9","title":"Cramer\'s Rule for Solving a System of Three Equations","body":"Solving the system of equations using Cramer\'s Rule","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Solve Systems of Equations Using Determinants","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a8d511eDeterminant9a","stepAnswer":["$$(-5, -8, -17)$$"],"problemType":"MultipleChoice","stepTitle":"$$3x-5y+4z=5;$$ $$5x+2y+z=0;$$ $$2x+3y-2z=3$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-5, -8, -17)$$","choices":["$$(-5, -8, -17)$$","(3,2,7)","$$(-1, -4, -6)$$","(1,4,18)"],"hints":{"DefaultPathway":[{"id":"a8d511eDeterminant9a-h1","type":"hint","dependencies":[],"title":"Principle","text":"Write the determinant using the coefficients of $$x$$, $$y$$, and $$z$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant9a-h2","type":"hint","dependencies":["a8d511eDeterminant9a-h1"],"title":"Order of Operation","text":"Expand the minor along the first row","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant9a-h3","type":"hint","dependencies":["a8d511eDeterminant9a-h2"],"title":"Operation","text":"Evaluate the 2x2 determinants","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-37$$"],"dependencies":["a8d511eDeterminant9a-h3"],"title":"Calculation","text":"What is $$3\\\\left(-7\\\\right)-\\\\left(-5\\\\right) \\\\left(-12\\\\right)+4\\\\times11$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant9a-h5","type":"hint","dependencies":["a8d511eDeterminant9a-h4"],"title":"Principle","text":"Replace the $$x$$ coefficient with the constant to find Dx","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant9a-h6","type":"hint","dependencies":["a8d511eDeterminant9a-h5"],"title":"Order of Operation","text":"Expand the minor along the first column","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant9a-h7","type":"hint","dependencies":["a8d511eDeterminant9a-h6"],"title":"Operation","text":"Evaluate the 2x2 determinants","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant9a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-74$$"],"dependencies":["a8d511eDeterminant9a-h7"],"title":"Calculation","text":"What is $$5\\\\left(-7\\\\right)-0+3\\\\left(-13\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant9a-h9","type":"hint","dependencies":["a8d511eDeterminant9a-h8"],"title":"Principle","text":"Replace the $$y$$ coefficient with the constant to find Dy","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant9a-h10","type":"hint","dependencies":["a8d511eDeterminant9a-h9"],"title":"Order of Operation","text":"Expand the minor along the second column","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant9a-h11","type":"hint","dependencies":["a8d511eDeterminant9a-h10"],"title":"Operation","text":"Evaluate the 2x2 determinants","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant9a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$111$$"],"dependencies":["a8d511eDeterminant9a-h11"],"title":"Calculation","text":"What is $$-5\\\\left(-12\\\\right)+0-3\\\\left(-17\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant9a-h13","type":"hint","dependencies":["a8d511eDeterminant9a-h12"],"title":"Principle","text":"Replace the $$z$$ coefficient with the constant to find Dz","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant9a-h14","type":"hint","dependencies":["a8d511eDeterminant9a-h13"],"title":"Order of Operation","text":"Expand the minor along the third column","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant9a-h15","type":"hint","dependencies":["a8d511eDeterminant9a-h14"],"title":"Operation","text":"Evaluate the 2x2 determinants","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant9a-h16","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$148$$"],"dependencies":["a8d511eDeterminant9a-h15"],"title":"Calculation","text":"What is $$5\\\\times11-0+3\\\\times31$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8d511eDeterminant9a-h17","type":"hint","dependencies":["a8d511eDeterminant9a-h16"],"title":"Calculation","text":"Determine the values of the variables with the equation: $$x=\\\\frac{Dx}{D}$$, $$y=\\\\frac{Dy}{D}$$, $$z=\\\\frac{Dz}{D}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8e6481uni1","title":"Speed of trains","body":"An express train and a local train leave Pittsburgh to travel to Washington, D.C. The express train can make the trip in $$4$$ hours and the local train takes $$5$$ hours for the trip. The speed of the express train is $$12$$ miles per hour faster than the speed of the local train.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Solve Uniform Motion Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8e6481uni1a","stepAnswer":["$$60$$"],"problemType":"TextBox","stepTitle":"Find the speed of the express train.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$60$$","hints":{"DefaultPathway":[{"id":"a8e6481uni1a-h1","type":"hint","dependencies":[],"title":"Visualize","text":"Draw a diagram to illustrate what is happening.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni1a-h2","type":"hint","dependencies":["a8e6481uni1a-h1"],"title":"Identify","text":"Identify what you are solving for.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni1a-h3","type":"hint","dependencies":["a8e6481uni1a-h2"],"title":"What Are You Solving For","text":"You are solving for the speed of the express train.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni1a-h4","type":"hint","dependencies":["a8e6481uni1a-h3"],"title":"Make a Variable","text":"Create a variable to represent the speed of the express train. Let\'s call the speed of the express train $$x$$. Since the speed of the local train is $$12$$ mph slower than the speed of the express train, the speed of the local train would be $$x-12$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni1a-h5","type":"hint","dependencies":["a8e6481uni1a-h4"],"title":"Translate to Equation","text":"Translate the situation into an equation using variable you named. Remeber that the distances two trains traveled are the same, and the distance equals $$speed time$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni1a-h6","type":"hint","dependencies":["a8e6481uni1a-h5"],"title":"The Equation","text":"Multiply the speed of the trains by the time to get $$4x=5(x-12)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni1a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$60$$"],"dependencies":["a8e6481uni1a-h6"],"title":"Solving the Equation","text":"After solving the equation, what do you get for $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a8e6481uni1a-h7-s1","type":"hint","dependencies":[],"title":"Solving the Equation","text":"To solve the equation $$4x=5(x-12)$$, we follow these steps: 4x=5x-60--\x3e60=5x-4x--\x3ex=60.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a8e6481uni10","title":"Walking vs. Biking","body":"When Katie Mae walks to school, it takes her $$30$$ minutes. If she rides her bike, it takes her $$15$$ minutes. Her speed is three miles per hour faster when she rides her bike than when she walks.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Solve Uniform Motion Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8e6481uni10a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"What is her walking speed in mph?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a8e6481uni10a-h1","type":"hint","dependencies":[],"title":"Visualize","text":"Draw a diagram representing the situation.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni10a-h2","type":"hint","dependencies":["a8e6481uni10a-h1"],"title":"Identify","text":"Identify what you are solving for.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni10a-h3","type":"hint","dependencies":["a8e6481uni10a-h2"],"title":"What Are You Solving For","text":"You are solving for the Katie Mae\'s walking speed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni10a-h4","type":"hint","dependencies":["a8e6481uni10a-h3"],"title":"Make a Variable","text":"Create a variable to represent the walking speed. Let\'s call the this speed $$x$$. Since her biking speed is $$3$$ mph faster than her walking speed, her biking speed is $$x+3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni10a-h5","type":"hint","dependencies":["a8e6481uni10a-h4"],"title":"Translate to Equation","text":"Translate the situation into an equation using the variable. Remember that the distance from Katie Mae\u2019s home to her school is the same whether she is walking or riding her bike, and distance equals the product of speed and time. Also, you will need to first convert the minutes into hours.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni10a-h6","type":"hint","dependencies":["a8e6481uni10a-h4"],"title":"The Equation","text":"The translated equation is $$0.5x=\\\\operatorname{0.25}\\\\left(x+3\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni10a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a8e6481uni10a-h6"],"title":"Solving the Equation","text":"After solving the equation, what do you get for $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a8e6481uni10a-h7-s1","type":"hint","dependencies":[],"title":"Solving the Equation","text":"To solve the equation $$0.5x=\\\\operatorname{0.25}\\\\left(x+3\\\\right)$$, we follow these steps: 0.5x=0.25x+0.75--\x3e0.25x=0.75--\x3ex=3.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}},{"id":"a8e6481uni10b","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"Find her biking speed.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"a8e6481uni10b-h1","type":"hint","dependencies":[],"title":"Find the Difference","text":"Katie Mae\'s walking speed is $$3$$ mph less than her biking speed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni10b-h2","type":"hint","dependencies":["a8e6481uni10b-h1"],"title":"The Answer","text":"Her biking speed is $$3+3=6$$ mph.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8e6481uni11","title":"Uphill vs. Downhill","body":"Suzy takes $$50$$ minutes to hike uphill from the parking lot to the lookout tower. It takes her $$30$$ minutes to hike back down to the parking lot. Her speed going downhill is $$1.2$$ miles per hour faster than her speed going uphill.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Solve Uniform Motion Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8e6481uni11a","stepAnswer":["$$1.8$$"],"problemType":"TextBox","stepTitle":"Find Suzy\u2019s uphill speed.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1.8$$","hints":{"DefaultPathway":[{"id":"a8e6481uni11a-h1","type":"hint","dependencies":[],"title":"Identify","text":"Identify what you are solving for.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni11a-h2","type":"hint","dependencies":["a8e6481uni11a-h1"],"title":"What Are You Solving For","text":"You are solving for the uphill hiking speed of Suzy.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni11a-h3","type":"hint","dependencies":["a8e6481uni11a-h2"],"title":"Make a Variable","text":"Create a variable to represent the uphill hiking speed. Let\'s call the this speed $$x$$. Since her downhill hiking speed is $$1.2$$ mph faster than her walking speed uphill, her downhill hiking speed is $$x+1.2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni11a-h4","type":"hint","dependencies":["a8e6481uni11a-h3"],"title":"Translate to Equation","text":"Translate the situation into an equation using the variable. Remember that the distance is the same whether she is walking uphill or downhill, and distance equals the product of speed and time. Also, you will need to first convert the minutes into hours.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni11a-h5","type":"hint","dependencies":["a8e6481uni11a-h4"],"title":"The Equation","text":"The translated equation is $$\\\\frac{5}{6} x=\\\\frac{1}{2\\\\left(x+1.2\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni11a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.8$$"],"dependencies":["a8e6481uni11a-h5"],"title":"Solving the Equation","text":"After solving the equation, what do you get for $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a8e6481uni11a-h6-s1","type":"hint","dependencies":[],"title":"Solving the Equation","text":"To solve the equation $$\\\\frac{5}{6} x=\\\\frac{1}{2\\\\left(x+1.2\\\\right)}$$, we follow these steps: (5/6)x=(1/2)x+0.6--\x3e(1/3)x=0.6--\x3ex=1.8.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}},{"id":"a8e6481uni11b","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"Find Suzy\'s downhill speed.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a8e6481uni11b-h1","type":"hint","dependencies":[],"title":"Find the Difference","text":"Suzie\'s downhill speed is $$1.2$$ mph more than her uphill speed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni11b-h2","type":"hint","dependencies":["a8e6481uni11b-h1"],"title":"The Answer","text":"Her downhill speed is $$1.8+1.2=3$$ mph.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8e6481uni12","title":"Upstream vs. Downstream","body":"Llewyn takes $$45$$ minutes to drive his boat upstream from the dock to his favorite fishing spot. It takes him $$30$$ minutes to drive the boat back downstream to the dock. The boat\u2019s speed going downstream is four miles per hour faster than its speed going upstream.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Solve Uniform Motion Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8e6481uni12a","stepAnswer":["$$8$$"],"problemType":"TextBox","stepTitle":"Find the boat\u2019s upstream speed.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8$$","hints":{"DefaultPathway":[{"id":"a8e6481uni12a-h1","type":"hint","dependencies":[],"title":"Identify","text":"Identify what you are solving for.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni12a-h2","type":"hint","dependencies":["a8e6481uni12a-h1"],"title":"What Are You Solving For","text":"You are solving for the boat\'s upstream speed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni12a-h3","type":"hint","dependencies":["a8e6481uni12a-h2"],"title":"Make a Variable","text":"Create a variable to represent the speed going upstream. Let\'s call the this speed $$x$$. Since the speed going downstream is $$4$$ mph faster than the speed going upstream, the downstream speed $$x+4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni12a-h4","type":"hint","dependencies":["a8e6481uni12a-h3"],"title":"Translate to Equation","text":"Translate the situation into an equation using the variable. Remember that the distance is the same whether Llewyn is going upstream or downstream, and distance equals the product of speed and time. Also, you will need to first convert the minutes into hours.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni12a-h5","type":"hint","dependencies":["a8e6481uni12a-h4"],"title":"The Equation","text":"The translated equation is $$0.75x=\\\\operatorname{0.5}\\\\left(x+4\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni12a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a8e6481uni12a-h5"],"title":"Solving the Equation","text":"After solving the equation, what do you get for $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a8e6481uni12a-h6-s1","type":"hint","dependencies":[],"title":"Solving the Equation","text":"To solve the equation $$0.75x=\\\\operatorname{0.5}\\\\left(x+4\\\\right)$$, we follow these steps: 0.75x=0.5x+2--\x3e0.25x=2--\x3ex=8.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}},{"id":"a8e6481uni12b","stepAnswer":["$$12$$"],"problemType":"TextBox","stepTitle":"Find the boat\'s downstream speed.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12$$","hints":{"DefaultPathway":[{"id":"a8e6481uni12b-h1","type":"hint","dependencies":[],"title":"Find the Difference","text":"The downstream speed is $$4$$ mph more than the upstream speed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni12b-h2","type":"hint","dependencies":["a8e6481uni12b-h1"],"title":"The Answer","text":"The downstream speed is $$8+4=12$$ mph.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8e6481uni13","title":"Driving Speeds","body":"Hamilton loves to travel to Las Vegas, $$255$$ miles from his home in Orange County. On his last trip, he left his house at 2:00 pm. The first part of his trip was on congested city freeways. At 4:00 pm, the traffic cleared and he was able to drive through the desert at a speed $$1.75$$ times as fast as when he drove in the congested area. He arrived in Las Vegas at 6:30 pm.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Solve Uniform Motion Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8e6481uni13a","stepAnswer":["$$40$$"],"problemType":"TextBox","stepTitle":"How fast was he driving in the congested freeways?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$40$$","hints":{"DefaultPathway":[{"id":"a8e6481uni13a-h1","type":"hint","dependencies":[],"title":"Visualize","text":"Draw a diagram representing the situation.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni13a-h2","type":"hint","dependencies":["a8e6481uni13a-h1"],"title":"Identify","text":"Identify what you are solving for.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni13a-h3","type":"hint","dependencies":["a8e6481uni13a-h2"],"title":"What Are You Solving For","text":"You are solving for the Hamilton\'s speed in the congested freeways.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni13a-h4","type":"hint","dependencies":["a8e6481uni13a-h3"],"title":"Make a Variable","text":"Create a variable to represent the speed in the congested freeways. Let the variable be $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni13a-h5","type":"hint","dependencies":["a8e6481uni13a-h4"],"title":"Time Elapsed","text":"Determine how much time each part of the journey took using subtraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni13a-h6","type":"hint","dependencies":["a8e6481uni13a-h5"],"title":"Time Elapsed Answer","text":"2:00 PM to 4:00 PM is two hours. 4:00 PM to 6:30 PM is $$2.5$$ hours long. So, he spent two hours in the congested freeways and $$2.5$$ hours in the desert.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni13a-h7","type":"hint","dependencies":["a8e6481uni13a-h6"],"title":"Translate to Equation","text":"Translate the situation into an equation using the variable. Remember that $$distance=speed time$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni13a-h8","type":"hint","dependencies":["a8e6481uni13a-h7"],"title":"The Equation","text":"Multiply the speeds by the time it takes for each respective part of the journey to get the equation $$2x+\\\\operatorname{1.75}\\\\left(2.5x\\\\right)=255$$ (this is the combined distance).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni13a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$40$$"],"dependencies":["a8e6481uni13a-h8"],"title":"Solving the Equation","text":"After solving the equation, what do you get for $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a8e6481uni13a-h9-s1","type":"hint","dependencies":[],"title":"Solving the Equation","text":"To solve the equation $$2x+\\\\operatorname{1.75}\\\\left(2.5x\\\\right)=255$$, we follow these steps: 2x+4.375x=255--\x3e6.375x=255--\x3ex=40.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}},{"id":"a8e6481uni13b","stepAnswer":["$$70$$"],"problemType":"TextBox","stepTitle":"Find Hamilton\'s speed in the desert.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$70$$","hints":{"DefaultPathway":[{"id":"a8e6481uni13b-h1","type":"hint","dependencies":[],"title":"Find the Difference","text":"The desert speed is $$1.75$$ times the speed in the congested freeways.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni13b-h2","type":"hint","dependencies":["a8e6481uni13b-h1"],"title":"The Answer","text":"The desert speed is $$40\\\\times1.75=70$$ mph.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8e6481uni14","title":"Running and Biking Speeds","body":"Cruz is training to compete in a triathlon. He left his house at 6:00 and ran until 7:30. Then he rode his bike until 9:45. He covered a total distance of $$51$$ miles. His speed when biking was $$1.6$$ times his speed when running.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Solve Uniform Motion Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8e6481uni14a","stepAnswer":["$$10$$"],"problemType":"TextBox","stepTitle":"Find Cruz\u2019s running speed.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10$$","hints":{"DefaultPathway":[{"id":"a8e6481uni14a-h1","type":"hint","dependencies":[],"title":"Identify","text":"Identify what you are solving for.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni14a-h2","type":"hint","dependencies":["a8e6481uni14a-h1"],"title":"What Are You Solving For","text":"You are solving for the Cruz\'s running speed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni14a-h3","type":"hint","dependencies":["a8e6481uni14a-h2"],"title":"Make a Variable","text":"Create a variable to represent his running speed. Let the variable be $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni14a-h4","type":"hint","dependencies":["a8e6481uni14a-h3"],"title":"Time Elapsed","text":"Determine how much time each part of the journey took using subtraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni14a-h5","type":"hint","dependencies":["a8e6481uni14a-h4"],"title":"Time Elapsed Answer","text":"6:00 PM to 7:30 PM is $$1.5$$ hours. 7:30 PM to 9:45 PM is $$2.25$$ hours long. So, he spent $$1.5$$ hours running and $$2.25$$ hours biking.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni14a-h6","type":"hint","dependencies":["a8e6481uni14a-h5"],"title":"Translate to Equation","text":"Translate the situation into an equation using the variable. Remember that $$distance=speed time$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni14a-h7","type":"hint","dependencies":["a8e6481uni14a-h6"],"title":"The Equation","text":"Multiply the speeds by the time it takes for each respective part of the training to get the equation $$1.5x+\\\\operatorname{2.25}\\\\left(1.6x\\\\right)=51$$ (this is the overall distance).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni14a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a8e6481uni14a-h7"],"title":"Solving the Equation","text":"After solving the equation, what do you get for $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a8e6481uni14a-h8-s1","type":"hint","dependencies":[],"title":"Solving the Equation","text":"To solve the equation $$1.5x+\\\\operatorname{2.25}\\\\left(1.6x\\\\right)=51$$, we follow these steps: 1.5x+ 3.6x=51--\x3e5.1x=51--\x3ex=10.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}},{"id":"a8e6481uni14b","stepAnswer":["$$16$$"],"problemType":"TextBox","stepTitle":"Find Cruz\'s biking speed.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16$$","hints":{"DefaultPathway":[{"id":"a8e6481uni14b-h1","type":"hint","dependencies":[],"title":"Find the Difference","text":"The biking speed is $$1.6$$ times the running speed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni14b-h2","type":"hint","dependencies":["a8e6481uni14b-h1"],"title":"The Answer","text":"The biking speed of Cruz is $$10\\\\times1.6=16$$ mph.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8e6481uni15","title":"Biking Speeds","body":"Phuong left home on his bicycle at 10:00. He rode on the flat street until 11:15, then rode uphill until 11:45. He rode a total of $$31$$ miles. His speed riding uphill was $$0.6$$ times his speed on the flat street.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Solve Uniform Motion Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8e6481uni15a","stepAnswer":["$$20$$"],"problemType":"TextBox","stepTitle":"Find his speed biking on the flat street.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$20$$","hints":{"DefaultPathway":[{"id":"a8e6481uni15a-h1","type":"hint","dependencies":[],"title":"Identify","text":"Identify what you are solving for.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni15a-h2","type":"hint","dependencies":["a8e6481uni15a-h1"],"title":"What Are You Solving For","text":"You are solving for the Phuong\'s speed on the flat street.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni15a-h3","type":"hint","dependencies":["a8e6481uni15a-h2"],"title":"Make a Variable","text":"Create a variable to represent his speed on a flat street. Let the variable be $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni15a-h4","type":"hint","dependencies":["a8e6481uni15a-h3"],"title":"Time Elapsed","text":"Determine how much time each part of the journey took using subtraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni15a-h5","type":"hint","dependencies":["a8e6481uni15a-h4"],"title":"Time Elapsed Answer","text":"10:00 PM to 11:15 PM is $$1.25$$ hours. 11:15 PM to 11:45 PM is $$0.5$$ hours long. So, he spent $$1.25$$ hours biking on a flat street and $$0.5$$ hours biking uphill.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni15a-h6","type":"hint","dependencies":["a8e6481uni15a-h5"],"title":"Translate to Equation","text":"Translate the situation into an equation using the variable. Remember that $$distance=speed time$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni15a-h7","type":"hint","dependencies":["a8e6481uni15a-h6"],"title":"The Equation","text":"Multiply the speeds by the time it takes for each respective part of the journey to get the equation $$1.25x+\\\\operatorname{0.5}\\\\left(0.6x\\\\right)=31$$ (this is the overall distance).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni15a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a8e6481uni15a-h7"],"title":"Solving the Equation","text":"After solving the equation, what do you get for $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a8e6481uni15a-h8-s1","type":"hint","dependencies":[],"title":"Solving the Equation","text":"To solve the equation $$1.25x+\\\\operatorname{0.5}\\\\left(0.6x\\\\right)=31$$, we follow these steps: 1.25x+0.3x=31--\x3e1.55x=31--\x3ex=20.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}},{"id":"a8e6481uni15b","stepAnswer":["$$12$$"],"problemType":"TextBox","stepTitle":"Find Phuong\'s uphill biking speed.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12$$","hints":{"DefaultPathway":[{"id":"a8e6481uni15b-h1","type":"hint","dependencies":[],"title":"Find the Difference","text":"The uphillbiking speed is $$0.6$$ times the biking speed on a flat street.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni15b-h2","type":"hint","dependencies":["a8e6481uni15b-h1"],"title":"The Answer","text":"The uphill biking speed of Phuong is $$20\\\\times0.6=12$$ mph.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8e6481uni16","title":"Solve Uniform Motion Applications","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Solve Uniform Motion Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8e6481uni16a","stepAnswer":["Mason $$75$$ mph, train $$60$$ mph"],"problemType":"MultipleChoice","stepTitle":"Lilah is moving from Portland to Seattle. It takes her three hours to go by train. Mason loves the train station in Portland and drives to the train station in Seattle with all Lilah\'s boxes in his car. It takes him $$2.4$$ hours to get to Seattle, driving at $$15$$ miles per hour faster than the speed of the train. Find Mason\'s speed and the speed of the train.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Mason $$75$$ mph, train $$60$$ mph","choices":["Mason $$75$$ mph, train $$60$$ mph","Mason $$90$$ mph, train $$50$$ mph","Mason $$60$$ mph, train $$75$$ mph"],"hints":{"DefaultPathway":[{"id":"a8e6481uni16a-h1","type":"hint","dependencies":[],"title":"Read the problem","text":"Make sure all the words and ideas are understood. Draw a diagram to illustrate what is happening.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni16a-h2","type":"hint","dependencies":["a8e6481uni16a-h1"],"title":"Identify and name what we\'re looking for","text":"We are asked to find the speed of Mason and the train. Take $$r$$ as the speed of the train and $$r+15$$ to be the speed of Mason.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni16a-h3","type":"hint","dependencies":["a8e6481uni16a-h2"],"title":"Translate into an equation","text":"Restate what we know about the rates, time, and distance into an equation. $$2.4\\\\left(r+15\\\\right)=3r$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni16a-h4","type":"hint","dependencies":["a8e6481uni16a-h3"],"title":"Solve the equation.","text":"$$r=60=train\'s$$ speed. $$r+15=Mason\'s$$ $$speed=75$$ mph.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8e6481uni17","title":"Solve Uniform Motion Applications","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Solve Uniform Motion Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8e6481uni17a","stepAnswer":["Cheryl $$3$$ mph, Kathy $$5$$ mph"],"problemType":"MultipleChoice","stepTitle":"Kathy and Cheryl are walking in a fundraiser. Kathy completes the course in $$4.8$$ hours and Cheryl completes the course in $$8$$ hours. Kathy walks two miles per hour faster than Cheryl. Find Kathy\'s speed and Cheryl\'s speed.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Cheryl $$3$$ mph, Kathy $$5$$ mph","choices":["Cheryl $$3$$ mph, Kathy $$5$$ mph","Cheryl $$5$$ mph, Kathy $$3$$ mph","Cheryl $$7$$ mph, Kathy $$9$$ mph"],"hints":{"DefaultPathway":[{"id":"a8e6481uni17a-h1","type":"hint","dependencies":[],"title":"Read the problem.","text":"Make sure all the words and ideas are understood. Draw a diagram to illustrate what is happening.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni17a-h2","type":"hint","dependencies":["a8e6481uni17a-h1"],"title":"Identify and name what we\'re looking for","text":"We are asked to find the speed of Kathy and Cheryl. Take the speed of Cheryl to be $$r$$ and $$r+2$$ to be the speed of Kathy","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni17a-h3","type":"hint","dependencies":["a8e6481uni17a-h2"],"title":"Translate into an equation","text":"Restate what we know about the rates, time, and distance into an equation. $$4.8\\\\left(r+2\\\\right)=8r$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni17a-h4","type":"hint","dependencies":["a8e6481uni17a-h3"],"title":"Solve the equation.","text":"$$r=3=speed$$ of Cheryl. $$r+2=5=speed$$ of Kathy","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8e6481uni18","title":"Solve Uniform Motion Applications","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Solve Uniform Motion Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8e6481uni18a","stepAnswer":["Express bus $$75$$ mph, Local bus $$50$$ mph"],"problemType":"MultipleChoice","stepTitle":"Two buses go from Sacramento for San Diego. The express bus makes the trip in $$6.8$$ hour and the local bus takes $$10.2$$ hrou for the trip. The speed of the express bus is $$25$$ mph faster than the speed of the local bus. Find the speed of both busses.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Express bus $$75$$ mph, Local bus $$50$$ mph","choices":["Express bus $$60$$ mph, Local bus $$40$$ mph","Express bus $$75$$ mph, Local bus $$50$$ mph","Express bus $$40$$ mph, Local bus $$50$$ mph"],"hints":{"DefaultPathway":[{"id":"a8e6481uni18a-h1","type":"hint","dependencies":[],"title":"Read the problem.","text":"Make sure all the words and ideas are understood. Draw a diagram to illustrate what is happening.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni18a-h2","type":"hint","dependencies":["a8e6481uni18a-h1"],"title":"Identify and name what we\'re looking for","text":"We are asked to find the speed of both buses. Take the speed of the local bus to be $$r$$ and $$r+25$$ to be the speed of the express bus","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni18a-h3","type":"hint","dependencies":["a8e6481uni18a-h2"],"title":"Translate into an equation","text":"Restate what we know about the rates, time, and distance into an equation. $$6.8\\\\left(r+25\\\\right)=10.2r$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni18a-h4","type":"hint","dependencies":["a8e6481uni18a-h3"],"title":"Solve the equation.","text":"$$r=50;$$ $$r+25=75$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8e6481uni19","title":"Solve Uniform Motion Applications","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Solve Uniform Motion Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8e6481uni19a","stepAnswer":["Private jet $$330$$ mph, Commercial jet $$540$$ mph"],"problemType":"MultipleChoice","stepTitle":"A commercial jet and a private airplane fly from Denver to Phoenix. It takes the commercial jet $$1.1$$ hours for hte flight and it takes the private airplane $$1.8$$ hours. The speed of the commercial jet is $$210$$ miles per hour faster than the speed of the private airplane. Find the speed of both airplanes.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Private jet $$330$$ mph, Commercial jet $$540$$ mph","choices":["Private jet $$300$$ mph, Commercial jet $$510$$ mph","Private jet $$330$$ mph, Commercial jet $$540$$ mph","Private jet $$400$$ mph, Commercial jet $$610$$ mph"],"hints":{"DefaultPathway":[{"id":"a8e6481uni19a-h1","type":"hint","dependencies":[],"title":"Read the problem.","text":"Make sure all the words and ideas are understood. Draw a diagram to illustrate what is happening.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni19a-h2","type":"hint","dependencies":["a8e6481uni19a-h1"],"title":"Identify and name what we\'re looking for","text":"We are asked to find the speed of both airplanes. Take the speed of the private jet to be $$r$$ and $$r+210$$ to be the speed of the commercial jet","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni19a-h3","type":"hint","dependencies":["a8e6481uni19a-h2"],"title":"Translate into an equation","text":"$$1.1\\\\left(210+r\\\\right)=1.8r$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni19a-h4","type":"hint","dependencies":["a8e6481uni19a-h3"],"title":"Solve the equation.","text":"$$r=330;$$ $$r+210=540$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8e6481uni2","title":"Speed of bikers","body":"Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis\u2019s speed is seven miles per hour faster than Wayne\u2019s speed, so it takes Wayne $$2$$ hours to ride to the beach while it takes Dennis $$1.5$$ hours for the ride.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Solve Uniform Motion Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8e6481uni2a","stepAnswer":["$$21$$"],"problemType":"TextBox","stepTitle":"Find the speed of Wayne.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$21$$","hints":{"DefaultPathway":[{"id":"a8e6481uni2a-h1","type":"hint","dependencies":[],"title":"Identify","text":"Identify what you are solving for.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni2a-h2","type":"hint","dependencies":["a8e6481uni2a-h1"],"title":"What Are You Solving For","text":"You are solving for the speed of Wayne.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni2a-h3","type":"hint","dependencies":["a8e6481uni2a-h2"],"title":"Make a Variable","text":"Create a variable to represent the Wayne\'s speed. Let\'s call the Wayne\'s speed $$x$$. Since Dennis\'s speed is $$7$$ mph faster than the Wayne\'s speed, Dennis\'s speed would be $$x+7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni2a-h4","type":"hint","dependencies":["a8e6481uni2a-h3"],"title":"Translate to Equation","text":"Translate the situation into an equation using variable you named. Remeber that the distances two bikers traveled are the same, and the distance equals $$speed time$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni2a-h5","type":"hint","dependencies":["a8e6481uni2a-h4"],"title":"The Equation","text":"Multiply the speed of the bikers by the time to get $$\\\\operatorname{1.5}\\\\left(x+7\\\\right)=2x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$21$$"],"dependencies":["a8e6481uni2a-h5"],"title":"Solving the Equation","text":"After solving the equation, what do you get for $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a8e6481uni2a-h6-s1","type":"hint","dependencies":[],"title":"Solving the Equation","text":"To solve the equation $$\\\\operatorname{1.5}\\\\left(x+7\\\\right)=2x.$$, we follow these steps: 1.5x+10.5=2x--\x3e0.5x=10.5--\x3ex=21.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}},{"id":"a8e6481uni2b","stepAnswer":["$$28$$"],"problemType":"TextBox","stepTitle":"Find the speed of Dennis.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$28$$","hints":{"DefaultPathway":[{"id":"a8e6481uni2b-h1","type":"hint","dependencies":[],"title":"Find the Difference","text":"Dennis\'s speed is $$7$$ mph more than Wayne\'s.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni2b-h2","type":"hint","dependencies":["a8e6481uni2b-h1"],"title":"The Answer","text":"Dennis\'s speed is $$28$$ mph.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8e6481uni20","title":"Solve Uniform Motion Applications","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Solve Uniform Motion Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8e6481uni20a","stepAnswer":["Saul $$82$$ mph, Erwin $$74$$ mph"],"problemType":"MultipleChoice","stepTitle":"Saul drove his truck $$3$$ hours from Dallas towards Kansas City and stopped at a truck stop to get dinner. At the truck stop he met Erwin, who had driven $$4$$ hours from Kansas City towards Dallas. The distance between Dallas and Kansas City is $$542$$ miles, and Erwins speed was eight miles per hour slower than Sauls speed. Find the speed of the two truckers.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Saul $$82$$ mph, Erwin $$74$$ mph","choices":["Saul $$40$$ mph, Erwin $$60$$ mph","Saul $$56$$ mph, Erwin $$74$$ mph","Saul $$82$$ mph, Erwin $$74$$ mph"],"hints":{"DefaultPathway":[{"id":"a8e6481uni20a-h1","type":"hint","dependencies":[],"title":"Read the problem.","text":"Make sure all the words and ideas are understood. Draw a diagram to illustrate what is happening.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni20a-h2","type":"hint","dependencies":["a8e6481uni20a-h1"],"title":"Identify and name what we\'re looking for","text":"We are asked to find the speed of Saul and Erwin. Take the speed of the Erwin to be $$r-8$$ and $$r$$ to be the speed of the Saul.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni20a-h3","type":"hint","dependencies":["a8e6481uni20a-h2"],"title":"Translate into an equation","text":"$$3r=4\\\\left(r-8\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni20a-h4","type":"hint","dependencies":["a8e6481uni20a-h3"],"title":"Solve the equation.","text":"$$3r=4r-32$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8e6481uni21","title":"Solve Uniform Motion Applications","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Solve Uniform Motion Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8e6481uni21a","stepAnswer":["Charlie $$7.1$$ mph, Violet $$17.1$$ mph"],"problemType":"MultipleChoice","stepTitle":"Charlie and Violet met for lunch at a restaurant between Memphis and New Orleans. Charlie ahd left Memphis and drove $$4.8$$ hours towards New Orleans. Violet ahs left New Orleans and drove $$2$$ hours towards Memphis, at a speed $$10$$ miles per hour faster than Charlie\'s\' speed. The distance between Memphis and New Orleans is $$394$$ miles. Find the speed of the two drivers.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Charlie $$7.1$$ mph, Violet $$17.1$$ mph","choices":["Charlie $$7.1$$ mph, Violet $$17.1$$ mph","Charlie $$6$$ mph, Violet $$16$$ mph","Charlie $$11$$ mph, Violet $$21$$ mph"],"hints":{"DefaultPathway":[{"id":"a8e6481uni21a-h1","type":"hint","dependencies":[],"title":"Read the problem.","text":"Make sure all the words and ideas are understood. Draw a diagram to illustrate what is happening.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni21a-h2","type":"hint","dependencies":["a8e6481uni21a-h1"],"title":"Identify and name what we\'re looking for","text":"We are asked to find the speed of Charlie and Violet. Take the speed of the Charlie to be $$r$$ and $$r+10$$ to be the speed of the Violet.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni21a-h3","type":"hint","dependencies":["a8e6481uni21a-h2"],"title":"Translate into an equation","text":"$$4.8r=2\\\\left(r+10\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni21a-h4","type":"hint","dependencies":["a8e6481uni21a-h3"],"title":"Solve the equation.","text":"$$4.8r=2r+20$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8e6481uni22","title":"Solve Uniform Motion Applications","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Solve Uniform Motion Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8e6481uni22a","stepAnswer":["Helen $$57.2$$ mph, Anne $$53.2$$ mph"],"problemType":"MultipleChoice","stepTitle":"Sisters Helen and Anne live $$332$$ miles apart. For Thanksgiving, they met at their other sisters house part way between their homes. Helen drove $$3.2$$ hour and Anne drove $$2.8$$ hours. Helens average speed was four miles per hour faster than Anne\'s. Find Helen\'s average speed and Anne\'s average speed","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Helen $$57.2$$ mph, Anne $$53.2$$ mph","choices":["Helen $$57.2$$ mph, Anne $$53.2$$ mph","Helen $$55$$ mph, Anne $$59$$ mph","Helen $$34$$ mph, Anne $$37.4$$ mph"],"hints":{"DefaultPathway":[{"id":"a8e6481uni22a-h1","type":"hint","dependencies":[],"title":"Read the problem.","text":"Make sure all the words and ideas are understood. Draw a diagram to illustrate what is happening.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni22a-h2","type":"hint","dependencies":["a8e6481uni22a-h1"],"title":"Identify and name what we\'re looking for","text":"We are asked to find the speed of Helen and Anne. Take the speed of the Anne to be $$r$$ and $$r+4$$ to be the speed of the Helen.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni22a-h3","type":"hint","dependencies":["a8e6481uni22a-h2"],"title":"Translate into an equation","text":"$$r+4\\\\times3.2=2.8r$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8e6481uni23","title":"Solve Uniform Motion Applications","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Solve Uniform Motion Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8e6481uni23a","stepAnswer":["Leo $$18$$ mph, Ethan $$24$$ mph"],"problemType":"MultipleChoice","stepTitle":"Ethan and Leo start riding their bikes at the opposite ends of a 65-mile bike path. After Ethan ahs ridden $$1.5$$ hours and Leo has ridden $$2$$ horus, they meet on the path. Ethan\'s speed is six miles per hour faster than Leo\'s speed. Find the speed of the two bikers","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Leo $$18$$ mph, Ethan $$24$$ mph","choices":["Leo $$18$$ mph, Ethan $$24$$ mph","Leo $$16$$ mph, Ethan $$22$$ mph","Leo $$17$$ mph, Ethan $$23$$ mph"],"hints":{"DefaultPathway":[{"id":"a8e6481uni23a-h1","type":"hint","dependencies":[],"title":"Read the problem.","text":"Make sure all the words and ideas are understood. Draw a diagram to illustrate what is happening.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni23a-h2","type":"hint","dependencies":["a8e6481uni23a-h1"],"title":"Identify and name what we\'re looking for","text":"We are asked to find the speed of Leo and Ethan. Take the speed of the Leo to be $$r$$ and $$r+6$$ to be the speed of Ethan.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni23a-h3","type":"hint","dependencies":["a8e6481uni23a-h2"],"title":"Translate into an equation","text":"$$1.5\\\\left(r+6\\\\right)=2r$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni23a-h4","type":"hint","dependencies":["a8e6481uni23a-h3"],"title":"Solve the equation.","text":"$$1.5r+9=2r$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8e6481uni24","title":"Solve Uniform Motion Applications","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Solve Uniform Motion Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8e6481uni24a","stepAnswer":["Aletheia $$2.4$$ mph, Elvira $$3$$ mph"],"problemType":"MultipleChoice","stepTitle":"Elvira and Aletheia live $$3.1$$ miles apart on the same street. They are in a study group that meets at a coffee shop between their houses. It took Elvira half an hour and Aletheia two-thirds of an hour to walk to the coffee shop. Aletheia\'s speed is $$0.6$$ miles per hour slower than Elvira\'s speed. Find both women\'s walking speeds.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Aletheia $$2.4$$ mph, Elvira $$3$$ mph","choices":["Aletheia $$3.2$$ mph, Elvira $$4$$ mph","Aletheia $$2.4$$ mph, Elvira $$3$$ mph","Aletheia $$4$$ mph, Elvira $$5$$ mph"],"hints":{"DefaultPathway":[{"id":"a8e6481uni24a-h1","type":"hint","dependencies":[],"title":"Read the problem.","text":"Make sure all the words and ideas are understood. Draw a diagram to illustrate what is happening.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni24a-h2","type":"hint","dependencies":["a8e6481uni24a-h1"],"title":"Identify and name what we\'re looking for","text":"We are asked to find the speed of Elvira and Aletheia. Take the speed of the Elvira to be $$r$$ and $$r-0.6$$ to be the speed of Aletheia.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni24a-h3","type":"hint","dependencies":["a8e6481uni24a-h2"],"title":"Translate into an equation","text":"$$\\\\frac{2}{3} \\\\left(r-0.6\\\\right)=0.5r$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8e6481uni25","title":"Solve Uniform Motion Applications","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Solve Uniform Motion Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8e6481uni25a","stepAnswer":["Fabian $$18$$ mph, DaMarcus $$12$$ mph"],"problemType":"MultipleChoice","stepTitle":"DaMarcus and Fabian live $$23$$ miles apart and play soccer at a park between their homes. DaMarcus rode his bike for three-quarters of an hour and Fabian rode his bike for half an hour to get ot the park. Fabian\'s speed was six miles per hour faster than DaMarcus\' speed. Find the speed of both soccer players.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Fabian $$18$$ mph, DaMarcus $$12$$ mph","choices":["Fabian $$16$$ mph, DaMarcus $$10$$ mph","Fabian $$18$$ mph, DaMarcus $$12$$ mph","Fabian $$20$$ mph, DaMarcus $$14$$ mph"],"hints":{"DefaultPathway":[{"id":"a8e6481uni25a-h1","type":"hint","dependencies":[],"title":"Read the problem.","text":"Make sure all the words and ideas are understood. Draw a diagram to illustrate what is happening.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni25a-h2","type":"hint","dependencies":["a8e6481uni25a-h1"],"title":"Identify and name what we\'re looking for","text":"We are asked to find the DaMarcus and Fabian. Take the speed of the DaMarcus to be $$r$$ and $$r+6$$ to be the speed of Fabian.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni25a-h3","type":"hint","dependencies":["a8e6481uni25a-h2"],"title":"Translate into an equation","text":"$$\\\\frac{3}{4} r=\\\\frac{1}{2} \\\\left(6+r\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni25a-h4","type":"hint","dependencies":["a8e6481uni25a-h3"],"title":"Solve the equation.","text":"$$0.75r=3+0.5r$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8e6481uni26","title":"Solve Uniform Motion Applications","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Solve Uniform Motion Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8e6481uni26a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"Cindy and Richard leave their dorm in Charleston at the same time. Cindy rides her bicycle north at a speed of $$18$$ miles per hour. Richard rides his bicycle south at a speed of $$14$$ miles per hour. How long will it take them to be $$96$$ miles apart?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a8e6481uni26a-h1","type":"hint","dependencies":[],"title":"Read the problem.","text":"Make sure all the words and ideas are understood. Draw a diagram to illustrate what is happening.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni26a-h2","type":"hint","dependencies":["a8e6481uni26a-h1"],"title":"Identify and name what we\'re looking for","text":"We are looking for the time travelled. Both people will travel the same amount of distance.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni26a-h3","type":"hint","dependencies":["a8e6481uni26a-h2"],"title":"Translate into an equation","text":"$$18t+14t=96$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni26a-h4","type":"hint","dependencies":["a8e6481uni26a-h3"],"title":"Solve the equation.","text":"$$32t=96$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8e6481uni27","title":"Solve Uniform Motion Applications","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Solve Uniform Motion Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8e6481uni27a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"Matt and Chris leave their uncle\'s house in Phoenix at the same time. Matt drives west on I-60 at a speed of $$76$$ miles per hour. Chris drives east on I-60 at a speed of $$82$$ miles per hour. How many hours will it take them to be $$632$$ miles apart?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a8e6481uni27a-h1","type":"hint","dependencies":[],"title":"Read the problem.","text":"Make sure all the words and ideas are understood. Draw a diagram to illustrate what is happening.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni27a-h2","type":"hint","dependencies":["a8e6481uni27a-h1"],"title":"Identify and name what we\'re looking for","text":"We are looking for the time travelled. Both people will travel the same amount of distance.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni27a-h3","type":"hint","dependencies":["a8e6481uni27a-h2"],"title":"Translate into an equation","text":"$$76t+82t=632$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni27a-h4","type":"hint","dependencies":["a8e6481uni27a-h3"],"title":"Solve the equation.","text":"$$158t=632$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8e6481uni28","title":"Solve Uniform Motion Applications","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Solve Uniform Motion Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8e6481uni28a","stepAnswer":["$$3.5$$"],"problemType":"TextBox","stepTitle":"Two busses leave Billings at the same time. The Seattle bus heads west on I-90 at a speed of $$73$$ miles per hour while the Chicago bus heads east at a speed of $$79$$ miles an hour. How many hours will it take them to be $$532$$ miles apart?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3.5$$","hints":{"DefaultPathway":[{"id":"a8e6481uni28a-h1","type":"hint","dependencies":[],"title":"Read the problem.","text":"Make sure all the words and ideas are understood. Draw a diagram to illustrate what is happening.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni28a-h2","type":"hint","dependencies":["a8e6481uni28a-h1"],"title":"Identify and name what we\'re looking for","text":"We are looking for the time travelled.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni28a-h3","type":"hint","dependencies":["a8e6481uni28a-h2"],"title":"Translate into an equation","text":"$$73t+79t=532$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni28a-h4","type":"hint","dependencies":["a8e6481uni28a-h3"],"title":"Solve the equation.","text":"$$152t=532$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8e6481uni29","title":"Solve Uniform Motion Applications","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Solve Uniform Motion Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8e6481uni29a","stepAnswer":["$$4.5$$"],"problemType":"TextBox","stepTitle":"Two boats leave the same dock in Cairo at the same time. One heads north on the Mississippi River while the other heads south. The northbound boat travels four miles per hour. The southbound boat goes eight miles per hour. How long will it take them to be $$54$$ miles apart?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4.5$$","hints":{"DefaultPathway":[{"id":"a8e6481uni29a-h1","type":"hint","dependencies":[],"title":"Read the problem.","text":"Make sure all the words and ideas are understood. Draw a diagram to illustrate what is happening.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni29a-h2","type":"hint","dependencies":["a8e6481uni29a-h1"],"title":"Identify and name what we\'re looking for","text":"We are looking for the time travelled.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni29a-h3","type":"hint","dependencies":["a8e6481uni29a-h2"],"title":"Translate into an equation","text":"$$4t+8t=54$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni29a-h4","type":"hint","dependencies":["a8e6481uni29a-h3"],"title":"Solve the equation.","text":"$$12t=54$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8e6481uni3","title":"Driving Speed","body":"Jeromy can drive from his house in Cleveland to his college in Chicago in $$4.5$$ hours. It takes his mother $$6$$ hours to make the same drive. Jeromy drives $$20$$ miles per hour faster than his mother.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Solve Uniform Motion Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8e6481uni3a","stepAnswer":["$$60$$"],"problemType":"TextBox","stepTitle":"Find his mother\u2019s speed.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$60$$","hints":{"DefaultPathway":[{"id":"a8e6481uni3a-h1","type":"hint","dependencies":[],"title":"Identify","text":"Identify what you are solving for.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni3a-h2","type":"hint","dependencies":["a8e6481uni3a-h1"],"title":"What Are You Solving For","text":"You are solving fo Jeromy\'s mother\'s speed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni3a-h3","type":"hint","dependencies":["a8e6481uni3a-h2"],"title":"Make a Variable","text":"Create a variable to represent the Jeromy\'s mother\'s speed. Let\'s call Jeromy\'s mother\'s speed $$x$$. Since the Jeromy\'s mother\'s speed is $$20$$ mph slower than the Jeromy\'s speed, Jeromy\'s speed would be $$x-20$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni3a-h4","type":"hint","dependencies":["a8e6481uni3a-h3"],"title":"Translate to Equation","text":"Translate the situation into an equation using variable you named. Remeber that the distances Jeromy and his mother traveled are the same, and the distance equals $$speed time$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni3a-h5","type":"hint","dependencies":["a8e6481uni3a-h4"],"title":"The Equation","text":"Multiply the speed of the drivers by the time to get $$6x=\\\\operatorname{4.5}\\\\left(x+20\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni3a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$60$$"],"dependencies":["a8e6481uni3a-h5"],"title":"Solving the Equation","text":"After solving the equation, what do you get for $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a8e6481uni3a-h6-s1","type":"hint","dependencies":[],"title":"Solving the Equation","text":"To solve the equation $$6x=\\\\operatorname{4.5}\\\\left(x+20\\\\right)$$, we follow these steps: 6x=4.5x+90--\x3e1.5x=90--\x3ex=60.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}},{"id":"a8e6481uni3b","stepAnswer":["$$80$$"],"problemType":"TextBox","stepTitle":"Find the Jeromy\'s speed.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$80$$","hints":{"DefaultPathway":[{"id":"a8e6481uni3b-h1","type":"hint","dependencies":[],"title":"Find the Difference","text":"Jeromy\'s speed is $$20$$ mph more than his mother\'s.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni3b-h2","type":"hint","dependencies":["a8e6481uni3b-h1"],"title":"The Answer","text":"The Jeromy\'s speed is $$80$$ mph.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8e6481uni30","title":"Solve Uniform Motion Applications","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Solve Uniform Motion Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8e6481uni30a","stepAnswer":["walking $$3$$ mph, jogging $$4.5$$ mph"],"problemType":"MultipleChoice","stepTitle":"Lorena walks the path around the park in $$30$$ minutes. If she jogs, it takes her $$20$$ minutes. Her jogging speed is $$1.5$$ miles per hour faster than ehr walking speed. Find Lorena\'s walking speed and jogging speed.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"walking $$3$$ mph, jogging $$4.5$$ mph","choices":["walking $$3$$ mph, jogging $$4.5$$ mph","walking $$4$$ mph, jogging $$5.5$$ mph","walking $$5$$ mph, jogging $$6.5$$ mph"],"hints":{"DefaultPathway":[{"id":"a8e6481uni30a-h1","type":"hint","dependencies":[],"title":"Read the problem.","text":"Make sure all the words and ideas are understood. Draw a diagram to illustrate what is happening.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni30a-h2","type":"hint","dependencies":["a8e6481uni30a-h1"],"title":"Translate into an equation","text":"$$30r=20\\\\left(1.5+r\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni30a-h3","type":"hint","dependencies":["a8e6481uni30a-h2"],"title":"Solve the equation.","text":"$$30r=30+20r$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8e6481uni4","title":"Driving Speed","body":"Christopher and his parents live $$115$$ miles apart. They met at a restaurant between their homes to celebrate his mother\u2019s birthday. Christopher drove $$1.5$$ hours while his parents drove $$1$$ hour to get to the restaurant. Christopher\u2019s average speed was $$10$$ miles per hour faster than his parents\u2019 average speed.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Solve Uniform Motion Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8e6481uni4a","stepAnswer":["$$40$$"],"problemType":"TextBox","stepTitle":"Christopher and his parents live $$115$$ miles apart. They met at a restaurant between their homes to celebrate his mother\u2019s birthday. Christopher drove $$1.5$$ hours while his parents drove $$1$$ hour to get to the restaurant. Christopher\u2019s average speed was $$10$$ miles per hour faster than his parents\u2019 average speed. What was the average speed of his parents as they drove to the restaurant?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$40$$","hints":{"DefaultPathway":[{"id":"a8e6481uni4a-h1","type":"hint","dependencies":[],"title":"Visualize","text":"Draw a diagram representing the situation.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni4a-h2","type":"hint","dependencies":["a8e6481uni4a-h1"],"title":"Identify","text":"Identify what you are solving for.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni4a-h3","type":"hint","dependencies":["a8e6481uni4a-h2"],"title":"What Are You Solving For","text":"You are solving for the parents\' driving speed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni4a-h4","type":"hint","dependencies":["a8e6481uni4a-h3"],"title":"Make a Variable","text":"Create a variable to represent the speed of Christopher\'s parents. Let\'s call the this speed $$x$$. Since Christopher\'s average speed is $$10$$ mph faster than his parents\' average speed, Christopher\'s average speed is $$x+10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni4a-h5","type":"hint","dependencies":["a8e6481uni4a-h4"],"title":"Translate to Equation","text":"Translate the situation into an equation using the variable. Remember the distance Christopher travelled plus the distance his parents travel must add up to $$115$$ miles, and distance equals the product of speed and time.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni4a-h6","type":"hint","dependencies":["a8e6481uni4a-h5"],"title":"The Equation","text":"The translated equation is $$\\\\operatorname{1.5}\\\\left(x+10\\\\right)+1x=115$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni4a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$40$$"],"dependencies":["a8e6481uni4a-h6"],"title":"Solving the Equation","text":"After solving the equation, what do you get for $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni4a-h8","type":"hint","dependencies":["a8e6481uni4a-h7"],"title":"Solving the Equation","text":"To solve the equation $$\\\\operatorname{1.5}\\\\left(x+10\\\\right)+1x=115$$, we follow these steps: 4x=5x-60--\x3e60=5x-4x--\x3ex=60.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a8e6481uni4b","stepAnswer":["$$50$$"],"problemType":"TextBox","stepTitle":"Find the speed of Christopher.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$50$$","hints":{"DefaultPathway":[{"id":"a8e6481uni4b-h1","type":"hint","dependencies":[],"title":"Find the Difference","text":"Christopher\'s speed is $$10$$ mph more than that of his parents.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni4b-h2","type":"hint","dependencies":["a8e6481uni4b-h1"],"title":"The Answer","text":"The speed of Christopher is $$40+10=50$$ mph.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8e6481uni5","title":"Driving Speed","body":"Carina is driving from her home in Anaheim to Berkeley on the same day her brother is driving from Berkeley to Anaheim, so they decide to meet for lunch along the way in Buttonwillow. The distance from Anaheim to Berkeley is $$410$$ miles. It takes Carina $$3$$ hours to get to Buttonwillow, while her brother drives $$4$$ hours to get there. The average speed Carina\u2019s brother drove was $$15$$ miles per hour faster than Carina\u2019s average speed.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Solve Uniform Motion Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8e6481uni5a","stepAnswer":["$$50$$"],"problemType":"TextBox","stepTitle":"Find Carina\u2019s average speed.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$50$$","hints":{"DefaultPathway":[{"id":"a8e6481uni5a-h1","type":"hint","dependencies":[],"title":"Identify","text":"Identify what you are solving for.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni5a-h2","type":"hint","dependencies":["a8e6481uni5a-h1"],"title":"What Are You Solving For","text":"You are solving for the speeds of the two drivers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni5a-h3","type":"hint","dependencies":["a8e6481uni5a-h2"],"title":"Make a Variable","text":"Create a variable to represent the speed of Carina. Let\'s call the this speed $$x$$. Since Carina\'s brother\'s average speed is $$15$$ mph faster than Carina\'s average speed, her brother\'s average speed is $$x+15$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni5a-h4","type":"hint","dependencies":["a8e6481uni5a-h3"],"title":"Translate to Equation","text":"Translate the situation into an equation using the variable. Remember the total distance Carina and her brother travelled is $$410$$ miles, and distance equals the product of speed and time.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni5a-h5","type":"hint","dependencies":["a8e6481uni5a-h4"],"title":"The Equation","text":"The translated equation is $$3x+4\\\\left(x+15\\\\right)=410$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni5a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$50$$"],"dependencies":["a8e6481uni5a-h5"],"title":"Solving the Equation","text":"After solving the equation, what do you get for $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a8e6481uni5a-h6-s1","type":"hint","dependencies":[],"title":"Solving the Equation","text":"To solve the equation $$\\\\operatorname{1.5}\\\\left(x+10\\\\right)+1x=115$$, we follow these steps: 4x=5x-60--\x3e60=5x-4x--\x3ex=60.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}},{"id":"a8e6481uni5b","stepAnswer":["$$65$$"],"problemType":"TextBox","stepTitle":"Find the speed of Carina\'s brother.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$65$$","hints":{"DefaultPathway":[{"id":"a8e6481uni5b-h1","type":"hint","dependencies":[],"title":"Find the Difference","text":"Carina\'s brother\'s speed is $$15$$ mph more than that of his sister.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni5b-h2","type":"hint","dependencies":["a8e6481uni5b-h1"],"title":"The Answer","text":"The speed of Carina\'s brother is $$50+15=65$$ mph.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8e6481uni6","title":"Driving Speed","body":"Ashley goes to college in Minneapolis, $$234$$ miles from her home in Sioux Falls. She wants her parents to bring her more winter clothes, so they decide to meet at a restaurant on the road between Minneapolis and Sioux Falls. Ashley and her parents both drove $$2$$ hours to the restaurant. Ashley\u2019s average speed was seven miles per hour faster than her parents\u2019 average speed.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Solve Uniform Motion Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8e6481uni6a","stepAnswer":["$$55$$"],"problemType":"TextBox","stepTitle":"Find her parents\u2019 average speed.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$55$$","hints":{"DefaultPathway":[{"id":"a8e6481uni6a-h1","type":"hint","dependencies":[],"title":"Identify","text":"Identify what you are solving for.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni6a-h2","type":"hint","dependencies":["a8e6481uni6a-h1"],"title":"What Are You Solving For","text":"You are solving for the speed of Ashley\'s parents.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni6a-h3","type":"hint","dependencies":["a8e6481uni6a-h2"],"title":"Make a Variable","text":"Create a variable to represent the speed of Ashley\'s parents. Let\'s call the this speed $$x$$. Since Ashley\'s average speed is $$7$$ mph faster than her parents\' average speed, Ashley\'s average speed is $$x+7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni6a-h4","type":"hint","dependencies":["a8e6481uni6a-h3"],"title":"Translate to Equation","text":"Translate the situation into an equation using the variable. Remember the total distance Ashley and her parents travelled is $$234$$ miles, and distance equals the product of speed and time.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni6a-h5","type":"hint","dependencies":["a8e6481uni6a-h4"],"title":"The Equation","text":"The translated equation is $$2x+2\\\\left(x+7\\\\right)=234$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni6a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$55$$"],"dependencies":["a8e6481uni6a-h5"],"title":"Solving the Equation","text":"After solving the equation, what do you get for $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a8e6481uni6a-h6-s1","type":"hint","dependencies":[],"title":"Solving the Equation","text":"To solve the equation $$2x+2\\\\left(x+7\\\\right)=234$$, we follow these steps: 2x+2x+14=234--\x3e4x=220--\x3ex=55.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}},{"id":"a8e6481uni6b","stepAnswer":["$$62$$"],"problemType":"TextBox","stepTitle":"Find the speed of Ashley.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$62$$","hints":{"DefaultPathway":[{"id":"a8e6481uni6b-h1","type":"hint","dependencies":[],"title":"Find the Difference","text":"Ashley\'s speed is $$7$$ mph more than that of her parents.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni6b-h2","type":"hint","dependencies":["a8e6481uni6b-h1"],"title":"The Answer","text":"The speed of Ashley is $$55+7=62$$ mph.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8e6481uni7","title":"Driving time","body":"Two truck drivers leave a rest area on the interstate at the same time. One truck travels east and the other one travels west. The truck traveling west travels at $$70$$ mph and the truck traveling east has an average speed of $$60$$ mph.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Solve Uniform Motion Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8e6481uni7a","stepAnswer":["$$2.5$$"],"problemType":"TextBox","stepTitle":"How long will they travel before they are $$325$$ miles apart?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2.5$$","hints":{"DefaultPathway":[{"id":"a8e6481uni7a-h1","type":"hint","dependencies":[],"title":"Visualize","text":"Draw a diagram representing the situation.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni7a-h2","type":"hint","dependencies":["a8e6481uni7a-h1"],"title":"Identify","text":"Identify what you are solving for.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni7a-h3","type":"hint","dependencies":["a8e6481uni7a-h2"],"title":"What Are You Solving For","text":"You are solving for the time it takes the two drivers to become $$325$$ miles apart.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni7a-h4","type":"hint","dependencies":["a8e6481uni7a-h3"],"title":"Make a Variable","text":"Create a variable to represent the time that it takes the drivers to become $$325$$ miles apart. Let the variable be $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni7a-h5","type":"hint","dependencies":["a8e6481uni7a-h4"],"title":"Translate to Equation","text":"Translate the situation into an equation using variable you named. Remeber that the time two trucks traveled are the same, but the directions are different, and the distance equals $$speed time$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni7a-h6","type":"hint","dependencies":["a8e6481uni7a-h5"],"title":"The Equation","text":"Multiply the speed of the drivers by the time to get that $$60x+70x=325$$ (since they are moving away from each other).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni7a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.5$$"],"dependencies":["a8e6481uni7a-h6"],"title":"Solving the Equation","text":"After solving the equation, what do you get for $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a8e6481uni7a-h7-s1","type":"hint","dependencies":[],"title":"Solving the Equation","text":"To solve the equation $$60x+70x=325$$, we follow these steps: 60x+70x=325--\x3e130x=325--\x3ex=2.5.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a8e6481uni8","title":"Driving time","body":"Pierre and Monique leave their home in Portland at the same time. Pierre drives north on the turnpike at a speed of $$75$$ miles per hour while Monique drives south at a speed of $$68$$ miles per hour.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Solve Uniform Motion Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8e6481uni8a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"How long will it take them to be $$429$$ miles apart?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a8e6481uni8a-h1","type":"hint","dependencies":[],"title":"Identify","text":"Identify what you are solving for.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni8a-h2","type":"hint","dependencies":["a8e6481uni8a-h1"],"title":"What Are You Solving For","text":"You are solving for the time it takes the two drivers to become $$429$$ miles apart.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni8a-h3","type":"hint","dependencies":["a8e6481uni8a-h2"],"title":"Make a Variable","text":"Create a variable to represent the time that it takes the drivers to become $$429$$ miles apart. Let the variable be $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni8a-h4","type":"hint","dependencies":["a8e6481uni8a-h3"],"title":"Translate to Equation","text":"Translate the situation into an equation using variable you named. Remeber that the time two drivers drived are the same, but the directions are different, and the distance equals $$speed time$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni8a-h5","type":"hint","dependencies":["a8e6481uni8a-h4"],"title":"The Equation","text":"Multiply the speed of the drivers by the time to get that $$75x+68x=429$$ (since they are moving away from each other).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni8a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a8e6481uni8a-h5"],"title":"Solving the Equation","text":"After solving the equation, what do you get for $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a8e6481uni8a-h6-s1","type":"hint","dependencies":[],"title":"Solving the Equation","text":"To solve the equation $$75x+68x=429$$, we follow these steps: 75x+68x=429--\x3e143x=429--\x3ex=3.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a8e6481uni9","title":"Driving time","body":"Thanh and Nhat leave their office in Sacramento at the same time. Thanh drives north on I-5 at a speed of $$72$$ miles per hour. Nhat drives south on I-5 at a speed of $$76$$ miles per hour.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Solve Uniform Motion Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a8e6481uni9a","stepAnswer":["$$2.23$$"],"problemType":"TextBox","stepTitle":"How long will it take them to be $$330$$ miles apart (round to the nearest hundredth)?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2.23$$","hints":{"DefaultPathway":[{"id":"a8e6481uni9a-h1","type":"hint","dependencies":[],"title":"Identify","text":"Identify what you are solving for.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni9a-h2","type":"hint","dependencies":["a8e6481uni9a-h1"],"title":"What Are You Solving For","text":"You are solving for the time it takes the two drivers to become $$330$$ miles apart.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni9a-h3","type":"hint","dependencies":["a8e6481uni9a-h2"],"title":"Make a Variable","text":"Create a variable to represent the time that it takes the drivers to become $$330$$ miles apart. Let the variable be $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni9a-h4","type":"hint","dependencies":["a8e6481uni9a-h3"],"title":"Translate to Equation","text":"Translate the situation into an equation using variable you named. Remeber that the time two drivers drived are the same, but the directions are different, and the distance equals $$speed time$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni9a-h5","type":"hint","dependencies":["a8e6481uni9a-h4"],"title":"The Equation","text":"Multiply the speed of the drivers by the time to get that $$72x+76x=330$$ (since they are moving away from each other).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8e6481uni9a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.23$$"],"dependencies":["a8e6481uni9a-h5"],"title":"Solving the Equation","text":"After solving the equation, what do you get for $$x$$? 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In the context of f\'(x), if $$x$$ would be an inflection point if it had a slope of $$0$$ or undefined.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a8f47f3deriv12","title":"Intervals and Extrema in f\'(x)","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"4.5 Derivatives and the Shape of a Graph","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a8f47f3deriv12a","stepAnswer":["$$x>0$$, $$x<1$$\\\\n"],"problemType":"MultipleChoice","stepTitle":"Analyze the graph of f\u2032, then list all the intervals where f(x) is concave up.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x>0$$, $$x<1$$\\\\n","choices":["$$x<-1$$, $$x>1$$","$$x>-0.5$$","$$x>0$$","$$x>0$$, $$x<1$$","$$x>0$$, $$x<1$$\\\\n"],"hints":{"DefaultPathway":[{"id":"a8f47f3deriv12a-h1","type":"hint","dependencies":[],"title":"The Second Derivative and Concavity","text":"If $$\\\\operatorname{f\'\'}\\\\left(x\\\\right)>0$$ over interval I, then f is concave up over I.\\\\nIf $$\\\\operatorname{f\'\'}\\\\left(x\\\\right)<0$$ over interval I, then f is concave down over I.","variabilization":{},"oer":"","license":""},{"id":"a8f47f3deriv12a-h2","type":"hint","dependencies":["a8f47f3deriv12a-h1"],"title":"Finding f\'\'(x) from f\'(x)","text":"The positive or negative value of f\'\'(x) will correspond with whether the slope of f\'(x) is positive or negative.","variabilization":{},"oer":"","license":""}]}},{"id":"a8f47f3deriv12b","stepAnswer":["$$x=0$$, $$x=1$$"],"problemType":"MultipleChoice","stepTitle":"List all the inflection points of f(x).","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=0$$, $$x=1$$","choices":["$$x=1$$","$$x=0$$, $$x=1$$","$$x=-1$$, $$x=0$$, $$x=1$$","$$x=0$$"],"hints":{"DefaultPathway":[{"id":"a8f47f3deriv12b-h3","type":"hint","dependencies":[],"title":"Inflection Points","text":"If f is continuous at a and f changes concavity at a, the point (a,f(a)) is an inflection point of f. In the context of f\'(x), if $$x$$ would be an inflection point if it had a slope of $$0$$ or undefined.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a8f47f3deriv16","title":"A Third Degree Polynomial","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"4.5 Derivatives and the Shape of a Graph","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a8f47f3deriv16a","stepAnswer":["True"],"problemType":"MultipleChoice","stepTitle":"Consider a third-degree polynomial f(x), which has the properties $$f\u2032(1)=0$$, $$f\u2032(3)=0$$. True or false: $$f\'\'(x)=0$$ for some $$1 \\\\leq x \\\\leq 3$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["False","True","True"],"hints":{"DefaultPathway":[{"id":"a8f47f3deriv16a-h1","type":"hint","dependencies":[],"title":"The Mean Value Theorem","text":"Let f be continuous over the closed interval [a,b] and differentiable over the open interval (a,b). Then, there exists at least one point c in (a,b) such that $$\\\\frac{f{\\\\left(b\\\\right)}-f{\\\\left(a\\\\right)}}{b-a}$$.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a8f47f3deriv17","title":"A Third Degree Polynomial","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"4.5 Derivatives and the Shape of a Graph","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a8f47f3deriv17a","stepAnswer":["True"],"problemType":"MultipleChoice","stepTitle":"Consider a third-degree polynomial f(x), which has the properties $$f\u2032(1)=0$$, $$f\u2032(3)=0$$. True or false: if f(x) has three roots, it has at least one inflection point.","stepBody":"","answerType":"string","variabilization":{},"choices":["False","True","True"],"hints":{"DefaultPathway":[{"id":"a8f47f3deriv17a-h1","type":"hint","dependencies":[],"title":"Second Derivative and Concavity","text":"If f is continuous at a and f changes concavity at a, the point (a,f(a)) is an inflection point of f.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a8f47f3deriv19","title":"Evaluating a Function","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"4.5 Derivatives and the Shape of a Graph","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a8f47f3deriv19a","stepAnswer":["all $$x$$"],"problemType":"MultipleChoice","stepTitle":"Consider $$f(x)=sin\\\\left(x\\\\right)+tan\\\\left(x\\\\right)$$ over x=[-pi/2, pi/2]. Find the intervals where f is increasing.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"all $$x$$","choices":["$$x>\\\\frac{\\\\pi}{2}$$","$$x<\\\\frac{\\\\pi}{2}$$","none of $$x$$","all $$x$$"],"hints":{"DefaultPathway":[{"id":"a8f47f3deriv19a-h1","type":"hint","dependencies":[],"title":"The First Derivative","text":"If f\u2032 is greater than $$0$$, f is increasing. If f\' is less than $$0$$, f is decreasing. If f is $$0$$ or undefined, it may be a local minimum or maximum.","variabilization":{},"oer":"","license":""},{"id":"a8f47f3deriv19a-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$cos\\\\left(x\\\\right)+{sec}^{2\\\\left(x\\\\right)}$$"],"dependencies":["a8f47f3deriv19a-h1"],"title":"Finding f\'(x)","text":"Find the derivative of f, and then solve to find when f\'(x) is greater than $$0$$. What is the derivative of f(x)?","variabilization":{},"oer":"","license":"","choices":["$$tan^2\\\\left(x\\\\right)+sin\\\\left(x\\\\right)$$","$$cos\\\\left(x\\\\right)+{sec}^{2\\\\left(x\\\\right)}$$","$$tan(x)-cos(x)$$","$${sin}^{2\\\\left(x\\\\right)}+{sec}^{2\\\\left(x\\\\right)}$$"]}]}},{"id":"a8f47f3deriv19b","stepAnswer":["none"],"problemType":"MultipleChoice","stepTitle":"Find the local or maxima.","stepBody":"","answerType":"string","variabilization":{},"choices":["none","maximum: $$x=\\\\frac{\\\\pi}{2}$$","maximum: $$x=-2\\\\pi$$","maximum: $$x=-1$$"],"hints":{"DefaultPathway":[{"id":"a8f47f3deriv19b-h2","type":"hint","dependencies":["a8f47f3deriv19a-h1"],"title":"The First Derivative Rule","text":"If f\u2032 changes sign from positive when $$x<c$$ to negative when $$x>c$$, then f(c) is a local maximum of f.\\\\nIf f\u2032 changes sign from negative when $$x<c$$ to positive when $$x>c$$, then f(c) is a local minimum of f.\\\\nIf f\' has the same sign for $$x<c$$ and $$x>c$$, then f(c) is neither a local maximum nor a local minimum of f.","variabilization":{},"oer":"","license":""}]}},{"id":"a8f47f3deriv19c","stepAnswer":["$$x>0$$"],"problemType":"MultipleChoice","stepTitle":"Find where f(x) is concave up.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x>0$$","choices":["$$x>\\\\pi$$","$$x<\\\\pi$$","$$x>0$$","$$x<0$$"],"hints":{"DefaultPathway":[{"id":"a8f47f3deriv19c-h3","type":"hint","dependencies":["a8f47f3deriv19b-h2"],"title":"Second Derivative and Concavity","text":"If $$\\\\operatorname{f\'\'}\\\\left(x\\\\right)>0$$ over interval I, then f is concave up over I.\\\\nIf $$\\\\operatorname{f\'\'}\\\\left(x\\\\right)<0$$ over interval I, then f is concave down over I.","variabilization":{},"oer":"","license":""},{"id":"a8f47f3deriv19c-s2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2{sec}^{2\\\\left(x\\\\right)} tan\\\\left(x\\\\right)-sin\\\\left(x\\\\right)$$"],"dependencies":["a8f47f3deriv19c-h3"],"title":"Finding f\'\'(x)","text":"Find the second derivative of f(x), and then solve to find when f\'\'(x) is greater than $$0$$. 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Is f\'\'(x) positive or negative?","variabilization":{},"oer":"","license":"","choices":["positive","negative"]}]}}]},{"id":"a8f47f3deriv20","title":"Evaluating a Function","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"4.5 Derivatives and the Shape of a Graph","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a8f47f3deriv20a","stepAnswer":["all $$x$$"],"problemType":"MultipleChoice","stepTitle":"Consider $$f(x)=\\\\frac{1}{x-1}$$ where $$x$$ is not $$1$$. Find the intervals where f is increasing.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"all $$x$$","choices":["$$x>1$$","$$x<1$$","none of $$x$$","all $$x$$"],"hints":{"DefaultPathway":[{"id":"a8f47f3deriv20a-h1","type":"hint","dependencies":[],"title":"The First Derivative","text":"If f\u2032 is greater than $$0$$, f is increasing. If f\' is less than $$0$$, f is decreasing. If f is $$0$$ or undefined, it may be a local minimum or maximum.","variabilization":{},"oer":"","license":""},{"id":"a8f47f3deriv20a-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{-1}{{\\\\left(x-1\\\\right)}^2}$$"],"dependencies":["a8f47f3deriv20a-h1"],"title":"Finding f\'(x)","text":"Find the derivative of f, and then solve to find when f\'(x) is greater than $$0$$. What is the derivative of f(x)?","variabilization":{},"oer":"","license":"","choices":["$$\\\\frac{-1}{{\\\\left(x-1\\\\right)}^2}$$","$$\\\\frac{{\\\\left(x+1\\\\right)}^2}{2}$$","$$x^2+x$$"]}]}},{"id":"a8f47f3deriv20b","stepAnswer":["none"],"problemType":"MultipleChoice","stepTitle":"Find the local or maxima.","stepBody":"","answerType":"string","variabilization":{},"choices":["none","maximum: $$x=1$$","maximum: $$x=-1$$, minimum: $$x=0$$","minimum: $$x=-1$$"],"hints":{"DefaultPathway":[{"id":"a8f47f3deriv20b-h2","type":"hint","dependencies":["a8f47f3deriv20a-h1"],"title":"The First Derivative Rule","text":"If f\u2032 changes sign from positive when $$x<c$$ to negative when $$x>c$$, then f(c) is a local maximum of f.\\\\nIf f\u2032 changes sign from negative when $$x<c$$ to positive when $$x>c$$, then f(c) is a local minimum of f.\\\\nIf f\' has the same sign for $$x<c$$ and $$x>c$$, then f(c) is neither a local maximum nor a local minimum of f.","variabilization":{},"oer":"","license":""}]}},{"id":"a8f47f3deriv20c","stepAnswer":["$$x<1$$"],"problemType":"MultipleChoice","stepTitle":"Find where f(x) is concave up.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x<1$$","choices":["$$x>-1$$","$$x<1$$","$$x>0$$","$$x>1$$"],"hints":{"DefaultPathway":[{"id":"a8f47f3deriv20c-h3","type":"hint","dependencies":["a8f47f3deriv20b-h2"],"title":"Second Derivative and Concavity","text":"If $$\\\\operatorname{f\'\'}\\\\left(x\\\\right)>0$$ over interval I, then f is concave up over I.\\\\nIf $$\\\\operatorname{f\'\'}\\\\left(x\\\\right)<0$$ over interval I, then f is concave down over I.","variabilization":{},"oer":"","license":""},{"id":"a8f47f3deriv20c-s2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{-2}{{\\\\left(x-1\\\\right)}^3}$$"],"dependencies":["a8f47f3deriv20c-h3"],"title":"Finding f\'\'(x)","text":"Find the second derivative of f(x), and then solve to find when f\'\'(x) is greater than $$0$$. What is the second derivative of f(x)?","variabilization":{},"oer":"","license":"","choices":["$$\\\\frac{-2}{{\\\\left(x-1\\\\right)}^3}$$","$$\\\\frac{2}{{\\\\left(x-1\\\\right)}^3}$$"]}]}},{"id":"a8f47f3deriv20d","stepAnswer":["none"],"problemType":"MultipleChoice","stepTitle":"Find the inflection points of f(x).","stepBody":"","answerType":"string","variabilization":{},"choices":["$$x=-1$$, $$x=0$$","$$x=1$$, $$x=-1$$","none","$$x=1$$"],"hints":{"DefaultPathway":[{"id":"a8f47f3deriv20d-h4","type":"hint","dependencies":["a8f47f3deriv20c-h3"],"title":"Second Derivative and Concavity","text":"If f is continuous at a and f changes concavity at a, the point (a,f(a)) is an inflection point of f.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a8f47f3deriv3","title":"Intervals in f\'(x)","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"4.5 Derivatives and the Shape of a Graph","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a8f47f3deriv3a","stepAnswer":["Increasing: $$-2<x<-1$$, $$x>2$$ decreasing: $$x<-2$$ and $$-1<x<2$$"],"problemType":"MultipleChoice","stepTitle":"Analyze the graph of f\u2032, then list all intervals where f is increasing or decreasing.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Increasing: $$-2<x<-1$$, $$x>2$$ decreasing: $$x<-2$$ and $$-1<x<2$$","choices":["Increasing: $$-2<x<-1$$, $$x>2$$ decreasing: $$x<-2$$ and $$-1<x<2$$","Increasing: $$x>1$$, $$x<-1.5$$ decreasing: $$-1.5<x<1$$","Increasing: $$x>0$$, $$-2<x<-1$$ decreasing: $$x<-2$$, $$-1<x<0$$"],"hints":{"DefaultPathway":[{"id":"a8f47f3deriv3a-h1","type":"hint","dependencies":[],"title":"The First Derivative","text":"If f\u2032 is greater than $$0$$, f is increasing. 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If f is $$0$$ or undefined, it may be a local minimum or maximum.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a8f47f3deriv4","title":"Intervals in f\'(x)","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"4.5 Derivatives and the Shape of a Graph","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a8f47f3deriv4a","stepAnswer":["Decreasing: $$x<1$$ increasing: $$x>1$$"],"problemType":"MultipleChoice","stepTitle":"Analyze the graph of f\u2032, then list all intervals where f is increasing or decreasing.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Decreasing: $$x<1$$ increasing: $$x>1$$","choices":["Decreasing: $$0<x<1$$ Increasing: $$x<0$$, $$x>1$$","Decreasing: $$0<x<0.5$$ increasing: $$x>0$$, $$x>1$$","Decreasing: $$x<1$$ increasing: $$x>1$$"],"hints":{"DefaultPathway":[{"id":"a8f47f3deriv4a-h1","type":"hint","dependencies":[],"title":"The First Derivative","text":"If f\u2032 is greater than $$0$$, f is increasing. 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If f\' is less than $$0$$, f is decreasing. If f is $$0$$ or undefined, it is a critical point.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a8f47f3deriv7","title":"Intervals and Extrema in f\'(x)","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"4.5 Derivatives and the Shape of a Graph","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a8f47f3deriv7a","stepAnswer":["$$-2<x<-1$$, $$0<x<1$$, $$x>2$$"],"problemType":"MultipleChoice","stepTitle":"Analyze the graph of f\u2032, then list all intervals where f is increasing.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-2<x<-1$$, $$0<x<1$$, $$x>2$$","choices":["$$-1<x<0$$, $$1<x<2$$","$$-2<x<-1.5$$, $$-0.75<x<0.75$$, $$x>1.5$$","$$-2<x<-1$$, $$0<x<1$$, $$x>2$$","$$-2<x<-1$$, $$0<x<1$$, $$x>2$$\\\\n"],"hints":{"DefaultPathway":[{"id":"a8f47f3deriv7a-h1","type":"hint","dependencies":[],"title":"The First Derivative","text":"If f\u2032 is greater than $$0$$, f is increasing. If f\' is less than $$0$$, f is decreasing. If f is $$0$$ or undefined, it is a critical point.","variabilization":{},"oer":"","license":""}]}},{"id":"a8f47f3deriv7b","stepAnswer":["maxima: $$x=-1$$, 1; minima: $$x=-2$$, $$0$$, $$2$$"],"problemType":"MultipleChoice","stepTitle":"List all the intervals where the minima and maxima are located.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"maxima: $$x=-1$$, 1; minima: $$x=-2$$, $$0$$, $$2$$","choices":["maxima: $$-2$$, 0; minima: $$x=-1, 1$$","maxima: $$x=-1.75$$, $$0.75;$$ minima: $$x=-0.75$$, $$1.5$$","maxima: $$x=-1$$, 1; minima: $$x=-2$$, $$0$$, $$2$$","maxima: $$x=-1, 1;$$ minima: $$x=-2$$, $$0$$, $$2$$","minima: $$x=-0.75$$, $$1.5;$$ maxima: $$x=-1.75$$, $$0.75$$"],"hints":{"DefaultPathway":[{"id":"a8f47f3deriv7b-h2","type":"hint","dependencies":[],"title":"The First Derivative Rule","text":"If f\u2032 changes sign from positive when $$x<c$$ to negative when $$x>c$$, then f(c) is a local maximum of f.\\\\nIf f\u2032 changes sign from negative when $$x<c$$ to positive when $$x>c$$, then f(c) is a local minimum of f.\\\\nIf f\' has the same sign for $$x<c$$ and $$x>c$$, then f(c) is neither a local maximum nor a local minimum of f.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a8f47f3deriv8","title":"Intervals and Extrema in f\'(x)","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"4.5 Derivatives and the Shape of a Graph","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a8f47f3deriv8a","stepAnswer":["$$x>0$$"],"problemType":"MultipleChoice","stepTitle":"Analyze the graph of f\u2032, then list all intervals where f is increasing.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x>0$$","choices":["decreasing over all $$x$$","$$x<0$$","$$x>0$$","all $$x$$"],"hints":{"DefaultPathway":[{"id":"a8f47f3deriv8a-h1","type":"hint","dependencies":[],"title":"The First Derivative","text":"If f\u2032 is greater than $$0$$, f is increasing. If f\' is less than $$0$$, f is decreasing. If f is $$0$$ or undefined, it is a critical point.","variabilization":{},"oer":"","license":""}]}},{"id":"a8f47f3deriv8b","stepAnswer":["minimum: $$x=0$$"],"problemType":"MultipleChoice","stepTitle":"List all the intervals where the minima and maxima are located.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"minimum: $$x=0$$","choices":["maximum: $$x=0$$","maxima: $$x=-1;$$ minima: $$x=-1$$","minimum: $$x=0$$","maxima: $$x=1;$$ minima: $$x=-1$$"],"hints":{"DefaultPathway":[{"id":"a8f47f3deriv8b-h2","type":"hint","dependencies":[],"title":"The First Derivative Rule","text":"If f\u2032 changes sign from positive when $$x<c$$ to negative when $$x>c$$, then f(c) is a local maximum of f.\\\\nIf f\u2032 changes sign from negative when $$x<c$$ to positive when $$x>c$$, then f(c) is a local minimum of f.\\\\nIf f\' has the same sign for $$x<c$$ and $$x>c$$, then f(c) is neither a local maximum nor a local minimum of f.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a8f47f3deriv9","title":"Intervals and Extrema in f\'(x)","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"4.5 Derivatives and the Shape of a Graph","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a8f47f3deriv9a","stepAnswer":["all $$x$$"],"problemType":"MultipleChoice","stepTitle":"Analyze the graph of f\u2032, then list all the intervals where f(x) is concave up.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"all $$x$$","choices":["none of $$x$$","all $$x$$","$$x<0$$","$$x>0$$"],"hints":{"DefaultPathway":[{"id":"a8f47f3deriv9a-h1","type":"hint","dependencies":[],"title":"The Second Derivative and Concavity","text":"If $$\\\\operatorname{f\'\'}\\\\left(x\\\\right)>0$$ over interval I, then f is concave up over I.\\\\nIf $$\\\\operatorname{f\'\'}\\\\left(x\\\\right)<0$$ over interval I, then f is concave down over I.","variabilization":{},"oer":"","license":""},{"id":"a8f47f3deriv9a-h2","type":"hint","dependencies":["a8f47f3deriv9a-h1"],"title":"Finding f\'\'(x) from f\'(x)","text":"The positive or negative value of f\'\'(x) will correspond with whether the slope of f\'(x) is positive or negative.","variabilization":{},"oer":"","license":""}]}},{"id":"a8f47f3deriv9b","stepAnswer":["none"],"problemType":"MultipleChoice","stepTitle":"List all the inflection points of f(x).","stepBody":"","answerType":"string","variabilization":{},"choices":["$$x=0$$","none","$$x=0.5$$","$$x=-0.5$$"],"hints":{"DefaultPathway":[{"id":"a8f47f3deriv9b-h3","type":"hint","dependencies":[],"title":"Inflection Points","text":"If f is continuous at a and f changes concavity at a, the point (a,f(a)) is an inflection point of f. In the context of f\'(x), if $$x$$ would be an inflection point if it had a slope of $$0$$ or undefined.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a8ffef9inverses1","title":"Verifying Inverse Functions","body":"We want to show that $$f(x)=\\\\frac{1}{x+1}$$ and $$f^{-1\\\\left(x\\\\right)}=\\\\frac{1}{x}-1$$ are inverses, for $$x \\\\neq 0, -1$$. Evaluate the following function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.7 Inverses and Radical Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a8ffef9inverses1a","stepAnswer":["$$x$$"],"problemType":"TextBox","stepTitle":"$$f^{-1\\\\left(f{\\\\left(x\\\\right)}\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x$$","hints":{"DefaultPathway":[{"id":"a8ffef9inverses1a-h1","type":"hint","dependencies":[],"title":"Verifying Inverses","text":"Recall that $$f^{-1\\\\left(f{\\\\left(x\\\\right)}\\\\right)}=x$$ and $$f{\\\\left(f^{-1\\\\left(x\\\\right)}\\\\right)}=x$$ when f(x) and $$f^{-1\\\\left(x\\\\right)}$$ are inverses.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{\\\\frac{1}{x+1}}-1$$"],"dependencies":["a8ffef9inverses1a-h1"],"title":"Verifying Inverses","text":"$$f^{-1\\\\left(f{\\\\left(x\\\\right)}\\\\right)}$$ can equivalently be written as $$f^{-1\\\\left(\\\\frac{1}{x+1}\\\\right)}$$. What is the function when we substitute $$x$$ with $$f(x)=\\\\frac{1}{x+1}$$ in $$f^{-1\\\\left(x\\\\right)}=\\\\frac{1}{x}-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+1$$"],"dependencies":["a8ffef9inverses1a-h2"],"title":"Simplifying","text":"What is $$\\\\frac{1}{\\\\frac{1}{x+1}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x$$"],"dependencies":["a8ffef9inverses1a-h3"],"title":"Simplifying","text":"What is $$x+1-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a8ffef9inverses1b","stepAnswer":["$$x$$"],"problemType":"TextBox","stepTitle":"$$f{\\\\left(f^{-1\\\\left(x\\\\right)}\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x$$","hints":{"DefaultPathway":[{"id":"a8ffef9inverses1b-h1","type":"hint","dependencies":[],"title":"Verifying Inverses","text":"Recall that $$f^{-1\\\\left(f{\\\\left(x\\\\right)}\\\\right)}=x$$ and $$f{\\\\left(f^{-1\\\\left(x\\\\right)}\\\\right)}=x$$ when f(x) and $$f^{-1\\\\left(x\\\\right)}$$ are inverses.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses1b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{\\\\frac{1}{x}-1+1}$$"],"dependencies":["a8ffef9inverses1b-h1"],"title":"Verifying Inverses","text":"$$f{\\\\left(f^{-1\\\\left(x\\\\right)}\\\\right)}$$ can equivalently be written as $$f{\\\\left(\\\\frac{1}{x}-1\\\\right)}$$. What is the function when we substitute $$x$$ with $$f^{-1\\\\left(x\\\\right)}=\\\\frac{1}{x}-1$$ in $$f(x)=\\\\frac{1}{x+1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses1b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{x}$$"],"dependencies":["a8ffef9inverses1b-h2"],"title":"Simplifying","text":"We can simplify the denominator. Simplify $$\\\\frac{1}{x}-1+1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses1b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x$$"],"dependencies":["a8ffef9inverses1b-h3"],"title":"Simplifying","text":"What is $$\\\\frac{1}{\\\\frac{1}{x}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ffef9inverses10","title":"Finding the Inverse of a Rational Function","body":"Find the inverse of the following function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.7 Inverses and Radical Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a8ffef9inverses10a","stepAnswer":["$$\\\\frac{3+2x}{x-1}$$"],"problemType":"TextBox","stepTitle":"$$f(x)=\\\\frac{x+3}{x-2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3+2x}{x-1}$$","hints":{"DefaultPathway":[{"id":"a8ffef9inverses10a-h1","type":"hint","dependencies":[],"title":"Finding the Inverse","text":"The general procedure for finding an inverse is to:\\\\n1) Replace f(x) with $$y$$.\\\\n2) Interchange $$x$$ and $$y$$.\\\\n3) Solve for $$y$$, and rename the function $$f^{-1\\\\left(x\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses10a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y=\\\\frac{x+3}{x-2}$$"],"dependencies":["a8ffef9inverses10a-h1"],"title":"Replace f(x) with $$y$$","text":"What does the equation look like after replacing f(x) with $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y=\\\\frac{x+3}{x-2}$$","$$f(x)=\\\\frac{x+3}{x-2}$$","$$x=\\\\frac{y+3}{y-2}$$"]},{"id":"a8ffef9inverses10a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x=\\\\frac{y+3}{y-2}$$"],"dependencies":["a8ffef9inverses10a-h2"],"title":"Interchange $$x$$ and $$y$$","text":"What does the equation look like after interchanging $$x$$ and $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y=\\\\frac{x+3}{x-2}$$","$$f(x)=\\\\frac{x+3}{x-2}$$","$$x=\\\\frac{y+3}{y-2}$$"]},{"id":"a8ffef9inverses10a-h4","type":"hint","dependencies":["a8ffef9inverses10a-h3"],"title":"Solve for $$y$$","text":"Make $$y$$ the subject.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses10a-h5","type":"hint","dependencies":["a8ffef9inverses10a-h4"],"title":"Solve for $$y$$","text":"Multiply the denominator of the fraction to both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses10a-h6","type":"hint","dependencies":["a8ffef9inverses10a-h5"],"title":"Solve for $$y$$","text":"Expand out $$x \\\\left(y-2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses10a-h7","type":"hint","dependencies":["a8ffef9inverses10a-h6"],"title":"Solve for $$y$$","text":"Move the $$y$$ terms to the same side of the equation so that we can isolate $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses10a-h8","type":"hint","dependencies":["a8ffef9inverses10a-h7"],"title":"Solve for $$y$$","text":"Factorize out $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses10a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3+2x}{x-1}$$"],"dependencies":["a8ffef9inverses10a-h8"],"title":"Solve for $$y$$","text":"Divide the coefficient of $$y$$ from both sides to make $$y$$ the subject. What is $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses10a-h10","type":"hint","dependencies":["a8ffef9inverses10a-h9"],"title":"Rename the function $$f^{-1\\\\left(x\\\\right)}$$","text":"Replace the $$y$$ with $$f^{-1\\\\left(x\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ffef9inverses11","title":"Factoring a Difference of Squares","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.7 Inverses and Radical Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a8ffef9inverses11a","stepAnswer":["$$\\\\left(3x-5\\\\right) \\\\left(3x+5\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor $$9x^2-25$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(3x-5\\\\right) \\\\left(3x+5\\\\right)$$","hints":{"DefaultPathway":[{"id":"a8ffef9inverses11a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x$$"],"dependencies":[],"title":"Square Root","text":"What is the square root of $$9x^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses11a-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":[],"title":"Square Root","text":"What is the square root of 25?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses11a-h2","type":"hint","dependencies":[],"title":"Difference of Squares","text":"Remember: The Difference of Squares formula is $$\\\\left(a-b\\\\right) \\\\left(a+b\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses11a-h3","type":"hint","dependencies":["a8ffef9inverses11a-h2"],"title":"Variable Values","text":"The value of a is $$3x$$ and the value of $$b$$ is $$5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ffef9inverses12","title":"Factoring a Difference of Squares","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.7 Inverses and Radical Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a8ffef9inverses12a","stepAnswer":["$$\\\\left(9x-10\\\\right) \\\\left(9x+10\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor $$81y^2-100$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(9x-10\\\\right) \\\\left(9x+10\\\\right)$$","hints":{"DefaultPathway":[{"id":"a8ffef9inverses12a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9x$$"],"dependencies":[],"title":"Square Root","text":"What is the square root $$81x^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses12a-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":[],"title":"Square Root","text":"What is the square root of 100?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses12a-h2","type":"hint","dependencies":[],"title":"Difference of Squares","text":"Remember: The Difference of Squares formula is $$\\\\left(a-b\\\\right) \\\\left(a+b\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses12a-h3","type":"hint","dependencies":["a8ffef9inverses12a-h2"],"title":"Variable Values","text":"The value of a is $$9x$$ and the value of $$b$$ is $$10$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ffef9inverses13","title":"Factoring a Sum of Cubes","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.7 Inverses and Radical Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a8ffef9inverses13a","stepAnswer":["$$\\\\left(x+8\\\\right) \\\\left(x^2-8x+64\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$x^3+512$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(x+8\\\\right) \\\\left(x^2-8x+64\\\\right)$$","hints":{"DefaultPathway":[{"id":"a8ffef9inverses13a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x$$"],"dependencies":[],"title":"Cube Root","text":"What is the cube root of $$x^3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses13a-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":[],"title":"Cube Root","text":"What is the cube root of 512?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses13a-h2","type":"hint","dependencies":[],"title":"Sum of Cubes Formula","text":"$$\\\\left(a+b\\\\right) \\\\left(a^2-ab+b^2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses13a-h3","type":"hint","dependencies":["a8ffef9inverses13a-h2"],"title":"Variable Values","text":"The value of a in the equation is $$x$$ and the value of $$b$$ is $$8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ffef9inverses14","title":"Factoring a Sum of Cubes","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.7 Inverses and Radical Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a8ffef9inverses14a","stepAnswer":["$$\\\\left(6a+b\\\\right) \\\\left(36a^2+6ab+b^2\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$216a^3+b^3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(6a+b\\\\right) \\\\left(36a^2+6ab+b^2\\\\right)$$","hints":{"DefaultPathway":[{"id":"a8ffef9inverses14a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["6a"],"dependencies":[],"title":"Cube Root","text":"What is the cube root of $$216a^3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses14a-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$b$$"],"dependencies":[],"title":"Cube Root","text":"What is the cube root of $$b^3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses14a-h2","type":"hint","dependencies":[],"title":"Sum of Cubes Formula","text":"$$\\\\left(a+b\\\\right) \\\\left(a^2-ab+b^2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses14a-h3","type":"hint","dependencies":["a8ffef9inverses14a-h2"],"title":"Variable Values","text":"The value of a in the equation is 6a and the value of $$b$$ is $$b$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ffef9inverses15","title":"Factoring a Difference of Cubes","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.7 Inverses and Radical Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a8ffef9inverses15a","stepAnswer":["$$\\\\left(2x-5\\\\right) \\\\left(4x^2+10x+25\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor $$8x^3-125$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(2x-5\\\\right) \\\\left(4x^2+10x+25\\\\right)$$","hints":{"DefaultPathway":[{"id":"a8ffef9inverses15a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x$$"],"dependencies":[],"title":"Cube Root","text":"What is the cube root of $$8x^3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses15a-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":[],"title":"Cube Root","text":"What is the cube root of 125?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses15a-h2","type":"hint","dependencies":[],"title":"Difference of Cubes Formula","text":"$$\\\\left(a-b\\\\right) \\\\left(a^2+ab+b^2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses15a-h3","type":"hint","dependencies":["a8ffef9inverses15a-h2"],"title":"Variable Values","text":"The value of a is $$2x$$ and the value of $$b$$ is $$5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ffef9inverses16","title":"Factoring a Difference of Cubes","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.7 Inverses and Radical Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a8ffef9inverses16a","stepAnswer":["$$\\\\left(10x-1\\\\right) \\\\left(100x^2+10x+1\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor the difference of cubes: $$1000x^3-1$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(10x-1\\\\right) \\\\left(100x^2+10x+1\\\\right)$$","hints":{"DefaultPathway":[{"id":"a8ffef9inverses16a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":[],"title":"Cube Root","text":"What is the cube root of $${1000}^3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses16a-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Cube Root","text":"What is the cube root of 1?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses16a-h2","type":"hint","dependencies":[],"title":"Difference of Cubes Formula","text":"$$\\\\left(a-b\\\\right) \\\\left(a^2+ab+b^2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses16a-h3","type":"hint","dependencies":["a8ffef9inverses16a-h2"],"title":"Variable Values","text":"The value of a is $$10x$$ and the value of $$b$$ is $$1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ffef9inverses17","title":"Factoring an Expression with Fractional or Negative Exponents","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.7 Inverses and Radical Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a8ffef9inverses17a","stepAnswer":["$${\\\\left(x+2\\\\right)}^{\\\\left(-\\\\frac{1}{3}\\\\right)} \\\\left(7x+8\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$3{x\\\\left(x+2\\\\right)}^{\\\\left(-\\\\frac{1}{3}\\\\right)}+{4\\\\left(x+2\\\\right)}^{\\\\frac{2}{3}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(x+2\\\\right)}^{\\\\left(-\\\\frac{1}{3}\\\\right)} \\\\left(7x+8\\\\right)$$","hints":{"DefaultPathway":[{"id":"a8ffef9inverses17a-h1","type":"hint","dependencies":[],"title":"GCF","text":"Factor out the GCF $${\\\\left(x+2\\\\right)}^{\\\\left(-\\\\frac{1}{3}\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses17a-h2","type":"hint","dependencies":["a8ffef9inverses17a-h1"],"title":"Simplify","text":"Simplify the term other than the GCF","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ffef9inverses18","title":"Factoring an Expression with Fractional or Negative Exponents","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.7 Inverses and Radical Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a8ffef9inverses18a","stepAnswer":["$${\\\\left(5a-1\\\\right)}^{\\\\left(-\\\\frac{1}{4}\\\\right)} \\\\left(17a-2\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{32\\\\left(5a-1\\\\right)}{4}+7{a\\\\left(5a-1\\\\right)}^{\\\\left(-\\\\frac{1}{4}\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(5a-1\\\\right)}^{\\\\left(-\\\\frac{1}{4}\\\\right)} \\\\left(17a-2\\\\right)$$","hints":{"DefaultPathway":[{"id":"a8ffef9inverses18a-h1","type":"hint","dependencies":[],"title":"GCF","text":"Factor out the GCF $${\\\\left(5a-4\\\\right)}^{\\\\left(-\\\\frac{1}{4}\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses18a-h2","type":"hint","dependencies":["a8ffef9inverses18a-h1"],"title":"Simplify","text":"Simplify the term other than the GCF","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ffef9inverses19","title":"Finding the Inverse of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.7 Inverses and Radical Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a8ffef9inverses19a","stepAnswer":["$$y=\\\\sqrt[3]{\\\\frac{x-4}{\\\\left(-2\\\\right)}}$$"],"problemType":"TextBox","stepTitle":"Find the inverse of $$f(x)=4-2x^3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=\\\\sqrt[3]{\\\\frac{x-4}{\\\\left(-2\\\\right)}}$$","hints":{"DefaultPathway":[{"id":"a8ffef9inverses19a-h1","type":"hint","dependencies":[],"title":"Switching $$x$$ and $$y$$","text":"We must now substitute all instances for $$x$$ by $$y$$ and all instance of $$y$$ by $$x$$. $$x=4-2y^3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses19a-h2","type":"hint","dependencies":["a8ffef9inverses19a-h1"],"title":"Solving for $$y$$","text":"Now, we must manipulate the equatiton and solve for $$y$$. $$y=\\\\sqrt[3]{\\\\frac{x-4}{\\\\left(-2\\\\right)}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ffef9inverses20","title":"Finding the Inverse of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.7 Inverses and Radical Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a8ffef9inverses20a","stepAnswer":["$$y=\\\\frac{x^2-1}{2}$$"],"problemType":"TextBox","stepTitle":"Find the inverse of $$f(x)=\\\\sqrt{2x+1}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=\\\\frac{x^2-1}{2}$$","hints":{"DefaultPathway":[{"id":"a8ffef9inverses20a-h1","type":"hint","dependencies":[],"title":"Switching $$x$$ and $$y$$","text":"We must now substitute all instances for $$x$$ by $$y$$ and instances of $$y$$ by $$x$$. $$x=\\\\sqrt{2y+1}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses20a-h2","type":"hint","dependencies":["a8ffef9inverses20a-h1"],"title":"Solving for $$y$$","text":"Now, we must manipulate the equation and solve for $$y$$. $$y=\\\\frac{x^2-1}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ffef9inverses21","title":"Finding the Inverse of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.7 Inverses and Radical Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a8ffef9inverses21a","stepAnswer":["$$\\\\frac{x^2-3}{\\\\left(-4\\\\right)}$$"],"problemType":"TextBox","stepTitle":"Find the inverse of $$f(x)=\\\\sqrt{2x+1}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{x^2-3}{\\\\left(-4\\\\right)}$$","hints":{"DefaultPathway":[{"id":"a8ffef9inverses21a-h1","type":"hint","dependencies":[],"title":"Switching $$x$$ and $$y$$","text":"We must now substitute all instances for $$x$$ by $$y$$ and all instance of $$y$$ by $$x$$. $$x=\\\\sqrt{3-4y}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses21a-h2","type":"hint","dependencies":["a8ffef9inverses21a-h1"],"title":"Solving for $$y$$","text":"Now, we must manipulate the equation and solve for $$y$$. $$y=\\\\frac{x^2-3}{\\\\left(-4\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ffef9inverses22","title":"Finding the Inverse of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.7 Inverses and Radical Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a8ffef9inverses22a","stepAnswer":["$$y=\\\\frac{{\\\\left(x-9\\\\right)}^2+4}{4}$$"],"problemType":"TextBox","stepTitle":"Find the inverse of $$f(x)=9+\\\\sqrt{4x-4}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=\\\\frac{{\\\\left(x-9\\\\right)}^2+4}{4}$$","hints":{"DefaultPathway":[{"id":"a8ffef9inverses22a-h1","type":"hint","dependencies":[],"title":"Switching $$x$$ and $$y$$","text":"We must now substitute all instances for $$x$$ by $$y$$ and instances of $$y$$ by $$x$$. $$x=9+\\\\sqrt{4y-4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses22a-h2","type":"hint","dependencies":["a8ffef9inverses22a-h1"],"title":"Solving for $$y$$","text":"Now, we must manipulate the equation and solve for $$y$$. $$y=\\\\frac{{\\\\left(x-9\\\\right)}^2+4}{4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ffef9inverses23","title":"Finding the Inverse of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.7 Inverses and Radical Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a8ffef9inverses23a","stepAnswer":["$$y=\\\\frac{{\\\\left(x-5\\\\right)}^2+8}{6}$$"],"problemType":"TextBox","stepTitle":"Find the inverse of $$f(x)=\\\\sqrt{6x-8}+5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=\\\\frac{{\\\\left(x-5\\\\right)}^2+8}{6}$$","hints":{"DefaultPathway":[{"id":"a8ffef9inverses23a-h1","type":"hint","dependencies":[],"title":"Switching $$x$$ and $$y$$","text":"We must now substitute all instances for $$x$$ by $$y$$ and instances of $$y$$ by $$x$$. $$x=\\\\sqrt{9y-8}+5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses23a-h2","type":"hint","dependencies":["a8ffef9inverses23a-h1"],"title":"Solving for $$y$$","text":"Now, we must manipulate the equation and solve for $$y$$. $$y=\\\\frac{{\\\\left(x-5\\\\right)}^2+8}{6}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ffef9inverses24","title":"Finding the Inverse of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.7 Inverses and Radical Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a8ffef9inverses24a","stepAnswer":["$$y={\\\\left(\\\\frac{x-9}{2}\\\\right)}^3$$"],"problemType":"TextBox","stepTitle":"Find the inverse of $$f(x)=9+2\\\\sqrt[3]{x}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y={\\\\left(\\\\frac{x-9}{2}\\\\right)}^3$$","hints":{"DefaultPathway":[{"id":"a8ffef9inverses24a-h1","type":"hint","dependencies":[],"title":"Switching $$x$$ and $$y$$","text":"We must now substitute all instances for $$x$$ by $$y$$ and instances of $$y$$ by $$x$$. $$x=9+2\\\\sqrt[3]{y}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses24a-h2","type":"hint","dependencies":["a8ffef9inverses24a-h1"],"title":"Solving for $$y$$","text":"Now, we must manipulate the equation and solve for $$y$$. $$y={\\\\left(\\\\frac{x-9}{2}\\\\right)}^3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ffef9inverses25","title":"Finding the Inverse of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.7 Inverses and Radical Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a8ffef9inverses25a","stepAnswer":["$$y={\\\\left(-x-3\\\\right)}^3$$"],"problemType":"TextBox","stepTitle":"Find the inverse of $$f(x)=3-\\\\sqrt[3]{x}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y={\\\\left(-x-3\\\\right)}^3$$","hints":{"DefaultPathway":[{"id":"a8ffef9inverses25a-h1","type":"hint","dependencies":[],"title":"Switching $$x$$ and $$y$$","text":"We must now substitute all instances for $$x$$ by $$y$$ and instances of $$y$$ by $$x$$. $$x=3-\\\\sqrt[3]{y}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses25a-h2","type":"hint","dependencies":["a8ffef9inverses25a-h1"],"title":"Solving for $$y$$","text":"Now, we must manipulate the equation and solve for $$y$$. $$y={\\\\left(-x-3\\\\right)}^3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ffef9inverses26","title":"Finding the Inverse of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.7 Inverses and Radical Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a8ffef9inverses26a","stepAnswer":["$$y=\\\\frac{2}{x}-8$$"],"problemType":"TextBox","stepTitle":"Find the inverse of $$f(x)=\\\\frac{2}{x+8}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=\\\\frac{2}{x}-8$$","hints":{"DefaultPathway":[{"id":"a8ffef9inverses26a-h1","type":"hint","dependencies":[],"title":"Switching $$x$$ and $$y$$","text":"We must now substitute all instances for $$x$$ by $$y$$ and instances of $$y$$ by $$x$$. $$x=\\\\frac{2}{y+8}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses26a-h2","type":"hint","dependencies":["a8ffef9inverses26a-h1"],"title":"Solving for $$y$$","text":"Now, we must manipulate the equation and solve for $$y$$. $$y=\\\\frac{2}{x}-8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ffef9inverses27","title":"Finding the Inverse of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.7 Inverses and Radical Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a8ffef9inverses27a","stepAnswer":["$$y=\\\\frac{3}{x}+4$$"],"problemType":"TextBox","stepTitle":"Find the inverse of $$f(x)=\\\\frac{3}{x-4}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=\\\\frac{3}{x}+4$$","hints":{"DefaultPathway":[{"id":"a8ffef9inverses27a-h1","type":"hint","dependencies":[],"title":"Switching $$x$$ and $$y$$","text":"We must now substitute all instances for $$x$$ by $$y$$ and instances of $$y$$ by $$x$$. $$x=\\\\frac{3}{y-4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses27a-h2","type":"hint","dependencies":["a8ffef9inverses27a-h1"],"title":"Solving for $$y$$","text":"Now, we must manipulate the equation and solve for $$y$$. $$y=\\\\frac{3}{x}+4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ffef9inverses28","title":"Finding the Inverse of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.7 Inverses and Radical Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a8ffef9inverses28a","stepAnswer":["$$y=\\\\frac{3-7x}{x-1}$$"],"problemType":"TextBox","stepTitle":"Find the inverse of $$f(x)=\\\\frac{x+3}{x+7}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=\\\\frac{3-7x}{x-1}$$","hints":{"DefaultPathway":[{"id":"a8ffef9inverses28a-h1","type":"hint","dependencies":[],"title":"Switching $$x$$ and $$y$$","text":"We must now substitute all instances for $$x$$ by $$y$$ and instances of $$y$$ by $$x$$. $$x=\\\\frac{y+3}{y+7}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses28a-h2","type":"hint","dependencies":["a8ffef9inverses28a-h1"],"title":"Solving for $$y$$","text":"Now, we must manipulate the equation and solve for $$y$$. $$x\\\\left(y+7\\\\right)=y+3$$. $$xy+7x=y+3$$. $$xy=y+3-7x$$. $$xy-y=3-7x$$. $$y(x-1)=3-7x$$. $$y=\\\\frac{3-7x}{x-1}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ffef9inverses3","title":"Restricting the Domain to Find the Inverse of a Polynomial Function","body":"Find the inverse of the following function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.7 Inverses and Radical Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a8ffef9inverses3a","stepAnswer":["$$4+\\\\sqrt{x}$$"],"problemType":"TextBox","stepTitle":"$$f(x)={\\\\left(x-4\\\\right)}^2$$, $$x \\\\geq 4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4+\\\\sqrt{x}$$","hints":{"DefaultPathway":[{"id":"a8ffef9inverses3a-h1","type":"hint","dependencies":[],"title":"Finding the Domain","text":"The original function $$f(x)={\\\\left(x-4\\\\right)}^2$$ is not one-to-one, but the function is restricted to a domain of $$x \\\\geq 4$$ on which it is one-to-one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses3a-h2","type":"hint","dependencies":["a8ffef9inverses3a-h1"],"title":"Finding the Inverse","text":"The general procedure for finding an inverse is to:\\\\n1) Replace f(x) with $$y$$.\\\\n2) Interchange $$x$$ and $$y$$.\\\\n3) Solve for $$y$$, and rename the function $$f^{-1\\\\left(x\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses3a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y={\\\\left(x-4\\\\right)}^2$$"],"dependencies":["a8ffef9inverses3a-h2"],"title":"Replace f(x) with $$y$$","text":"What does the equation look like after replacing f(x) with $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y={\\\\left(x-4\\\\right)}^2$$","$$x={\\\\left(y-4\\\\right)}^2$$","$$f(x)={\\\\left(x-4\\\\right)}^2$$"]},{"id":"a8ffef9inverses3a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x={\\\\left(y-4\\\\right)}^2$$"],"dependencies":["a8ffef9inverses3a-h3"],"title":"Interchange $$x$$ and $$y$$","text":"What does the equation look like after interchanging $$x$$ and $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y={\\\\left(x-4\\\\right)}^2$$","$$x={\\\\left(y-4\\\\right)}^2$$","$$f(x)={\\\\left(x-4\\\\right)}^2$$"]},{"id":"a8ffef9inverses3a-h5","type":"hint","dependencies":["a8ffef9inverses3a-h4"],"title":"Solve for $$y$$","text":"Make $$y$$ the subject.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses3a-h6","type":"hint","dependencies":["a8ffef9inverses3a-h5"],"title":"Solve for $$y$$","text":"Take the square root.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses3a-h7","type":"hint","dependencies":["a8ffef9inverses3a-h6"],"title":"Solve for $$y$$","text":"Add $$4$$ to both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses3a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["f**-1(x)=4+sqrt(x),f**-1(x)=4-sqrt(x)"],"dependencies":["a8ffef9inverses3a-h7"],"title":"Rename the function $$f^{-1\\\\left(x\\\\right)}$$","text":"Replace the $$y$$ with $$f^{-1\\\\left(x\\\\right)}$$. What does the equation look like now?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["f**-1(x)=4+sqrt(x),f**-1(x)=4-sqrt(x)","y=4+sqrt(x),y=4-sqrt(x)","$$f^{-1\\\\left(x\\\\right)}=4+\\\\sqrt{x}$$","$$f^{-1\\\\left(x\\\\right)}=4-\\\\sqrt{x}$$"]},{"id":"a8ffef9inverses3a-h9","type":"hint","dependencies":["a8ffef9inverses3a-h8"],"title":"Domain","text":"The domain of the original function was restricted to $$x \\\\geq 4$$, so the outputs of the inverse need to be the same, i.e. $$f^{-1\\\\left(x\\\\right)} \\\\geq 4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses3a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$f^{-1\\\\left(x\\\\right)}=4+\\\\sqrt{x}$$"],"dependencies":["a8ffef9inverses3a-h9"],"title":"Domain","text":"Which of the inverse would let us satisfy the initial domain of $$x \\\\geq 4$$, i.e. for all $$x$$ in the domain of $$f^{-1\\\\left(x\\\\right)}$$, $$f^{-1\\\\left(x\\\\right)} \\\\geq 4$$? Note that $$\\\\sqrt{x}$$ is always greater than or equal to $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$f^{-1\\\\left(x\\\\right)}=4+\\\\sqrt{x}$$","$$f^{-1\\\\left(x\\\\right)}=4-\\\\sqrt{x}$$"]}]}},{"id":"a8ffef9inverses3b","stepAnswer":["$$4-\\\\sqrt{x}$$"],"problemType":"TextBox","stepTitle":"f(x)=(x-4)**2,x<=4","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4-\\\\sqrt{x}$$","hints":{"DefaultPathway":[{"id":"a8ffef9inverses3b-h1","type":"hint","dependencies":[],"title":"Finding the Domain","text":"The original function $$f(x)={\\\\left(x-4\\\\right)}^2$$ is not one-to-one, but the function is restricted to a domain of $$x \\\\leq 4$$ on which it is one-to-one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses3b-h2","type":"hint","dependencies":["a8ffef9inverses3b-h1"],"title":"Finding the Inverse","text":"The general procedure for finding an inverse is to:\\\\n1) Replace f(x) with $$y$$.\\\\n2) Interchange $$x$$ and $$y$$.\\\\n3) Solve for $$y$$, and rename the function $$f^{-1\\\\left(x\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses3b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y={\\\\left(x-4\\\\right)}^2$$"],"dependencies":["a8ffef9inverses3b-h2"],"title":"Replace f(x) with $$y$$","text":"What does the equation look like after replacing f(x) with $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y={\\\\left(x-4\\\\right)}^2$$","$$x={\\\\left(y-4\\\\right)}^2$$","$$f(x)={\\\\left(x-4\\\\right)}^2$$"]},{"id":"a8ffef9inverses3b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x={\\\\left(y-4\\\\right)}^2$$"],"dependencies":["a8ffef9inverses3b-h3"],"title":"Interchange $$x$$ and $$y$$","text":"What does the equation look like after interchanging $$x$$ and $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y={\\\\left(x-4\\\\right)}^2$$","$$x={\\\\left(y-4\\\\right)}^2$$","$$f(x)={\\\\left(x-4\\\\right)}^2$$"]},{"id":"a8ffef9inverses3b-h5","type":"hint","dependencies":["a8ffef9inverses3b-h4"],"title":"Solve for $$y$$","text":"Make $$y$$ the subject.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses3b-h6","type":"hint","dependencies":["a8ffef9inverses3b-h5"],"title":"Solve for $$y$$","text":"Take the square root.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses3b-h7","type":"hint","dependencies":["a8ffef9inverses3b-h6"],"title":"Solve for $$y$$","text":"Add $$4$$ to both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses3b-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["f**-1(x)=4+sqrt(x),f**-1(x)=4-sqrt(x)"],"dependencies":["a8ffef9inverses3b-h7"],"title":"Rename the function $$f^{-1\\\\left(x\\\\right)}$$","text":"Replace the $$y$$ with $$f^{-1\\\\left(x\\\\right)}$$. What does the equation look like now?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["f**-1(x)=4+sqrt(x),f**-1(x)=4-sqrt(x)","y=4+sqrt(x),y=4-sqrt(x)","$$f^{-1\\\\left(x\\\\right)}=4+\\\\sqrt{x}$$","$$f^{-1\\\\left(x\\\\right)}=4-\\\\sqrt{x}$$"]},{"id":"a8ffef9inverses3b-h9","type":"hint","dependencies":["a8ffef9inverses3b-h8"],"title":"Domain","text":"The domain of the original function was restricted to $$x \\\\leq 4$$, so the outputs of the inverse need to be the same, i.e. $$f^{-1\\\\left(x\\\\right)} \\\\leq 4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses3b-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$f^{-1\\\\left(x\\\\right)}=4-\\\\sqrt{x}$$"],"dependencies":["a8ffef9inverses3b-h9"],"title":"Domain","text":"Which of the inverse would let us satisfy the initial domain of $$x \\\\leq 4$$, i.e. for all $$x$$ in the domain of $$f^{-1\\\\left(x\\\\right)}$$, $$f^{-1\\\\left(x\\\\right)} \\\\leq 4$$? Note that $$\\\\sqrt{x}$$ is always greater than or equal to $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$f^{-1\\\\left(x\\\\right)}=4+\\\\sqrt{x}$$","$$f^{-1\\\\left(x\\\\right)}=4-\\\\sqrt{x}$$"]}]}}]},{"id":"a8ffef9inverses4","title":"Finding the Inverse of a Quadratic Function When the Restriction Is Not Specified","body":"Restrict the domain and then find the inverse of the following function. We will utilize the + case for this question.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.7 Inverses and Radical Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a8ffef9inverses4a","stepAnswer":["$$2+\\\\sqrt{x+3}$$"],"problemType":"TextBox","stepTitle":"$$f(x)={\\\\left(x-2\\\\right)}^2-3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2+\\\\sqrt{x+3}$$","hints":{"DefaultPathway":[{"id":"a8ffef9inverses4a-h1","type":"hint","dependencies":[],"title":"Finding the Inverse","text":"Given a polynomial function, restrict the domain of a function that is not one-to-one and then find the inverse.\\\\n1) Restrict the domain by determining a domain on which the original function is one-to-one.\\\\n2) Replace f(x) with $$y$$.\\\\n3) Interchange $$x$$ and $$y$$.\\\\n4) Solve for $$y$$, and rename the function or pair of function $$f^{-1\\\\left(x\\\\right)}$$.\\\\n5) Revise the formula for $$f^{-1\\\\left(x\\\\right)}$$ by ensuring that the outputs of the inverse function correspond to the restricted domain of the original function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses4a-h2","type":"hint","dependencies":["a8ffef9inverses4a-h1"],"title":"Restricting the Domain","text":"We can restrict this function to a domain on which it will be one-to-one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses4a-h3","type":"hint","dependencies":["a8ffef9inverses4a-h2"],"title":"Restricting the Domain","text":"We can see this is a parabola with vertex at $$(2,-3)$$ that opens upward. Because the graph will be decreasing on one side of the vertex and increasing on the other side, we can restrict this function to a domain on which it will be one-to-one by limiting the domain to $$x \\\\geq 2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses4a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y={\\\\left(x-2\\\\right)}^2-3$$"],"dependencies":["a8ffef9inverses4a-h3"],"title":"Replace f(x) with $$y$$","text":"What does the equation look like after replacing f(x) with $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y={\\\\left(x-2\\\\right)}^2-3$$","$$f(x)={\\\\left(x-2\\\\right)}^2-3$$","$$x={\\\\left(y-2\\\\right)}^2-3$$"]},{"id":"a8ffef9inverses4a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x={\\\\left(y-2\\\\right)}^2-3$$"],"dependencies":["a8ffef9inverses4a-h4"],"title":"Interchange $$x$$ and $$y$$","text":"What does the equation look like after interchanging $$x$$ and $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y={\\\\left(x-2\\\\right)}^2-3$$","$$f(x)={\\\\left(x-2\\\\right)}^2-3$$","$$x={\\\\left(y-2\\\\right)}^2-3$$"]},{"id":"a8ffef9inverses4a-h6","type":"hint","dependencies":["a8ffef9inverses4a-h5"],"title":"Solve for $$y$$","text":"Make $$y$$ the subject.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses4a-h7","type":"hint","dependencies":["a8ffef9inverses4a-h6"],"title":"Solve for $$y$$","text":"Add $$3$$ to both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses4a-h8","type":"hint","dependencies":["a8ffef9inverses4a-h7"],"title":"Take the square root","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses4a-h9","type":"hint","dependencies":["a8ffef9inverses4a-h8"],"title":"Add $$2$$ to both sides","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses4a-h10","type":"hint","dependencies":["a8ffef9inverses4a-h9"],"title":"Rename the function $$f^{-1\\\\left(x\\\\right)}$$","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses4a-h11","type":"hint","dependencies":["a8ffef9inverses4a-h10"],"title":"Domain","text":"Now we need to determine which case to use. Because we restricted our original function to a domain of $$x \\\\geq 2$$, the outputs of the inverse should be the same, i.e. $$f^{-1\\\\left(x\\\\right)} \\\\geq 2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses4a-h12","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$f^{-1\\\\left(x\\\\right)}=2+\\\\sqrt{x+3}$$"],"dependencies":["a8ffef9inverses4a-h11"],"title":"Domain","text":"Which of the inverse would let us satisfy the initial domain of $$x \\\\geq 2$$, i.e. for all $$x$$ in the domain of $$f^{-1\\\\left(x\\\\right)}$$, $$f^{-1\\\\left(x\\\\right)} \\\\geq 2$$? Note that $$\\\\sqrt{x}$$ is always greater than or equal to $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$f^{-1\\\\left(x\\\\right)}=2+\\\\sqrt{x+3}$$","$$f^{-1\\\\left(x\\\\right)}=2-\\\\sqrt{x+3}$$"]}]}}]},{"id":"a8ffef9inverses5","title":"Finding the Inverse of a Radical Function","body":"Restrict the domain of the following function and then find the inverse.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.7 Inverses and Radical Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a8ffef9inverses5a","stepAnswer":["$$x^2+4$$, $$x \\\\geq 0$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\sqrt{x-4}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x^2+4$$, $$x \\\\geq 0$$","choices":["$$x^2+4$$, $$x \\\\geq 0$$","$$x^2+4$$, $$x$$ is real","$$x^2-4$$, $$x \\\\geq 0$$"],"hints":{"DefaultPathway":[{"id":"a8ffef9inverses5a-h1","type":"hint","dependencies":[],"title":"Finding the Inverse","text":"Given a polynomial function, restrict the domain of a function that is not one-to-one and then find the inverse.\\\\n1) Restrict the domain by determining a domain on which the original function is one-to-one.\\\\n2) Replace f(x) with $$y$$.\\\\n3) Interchange $$x$$ and $$y$$.\\\\n4) Solve for $$y$$, and rename the function or pair of function $$f^{-1\\\\left(x\\\\right)}$$.\\\\n5) Revise the formula for $$f^{-1\\\\left(x\\\\right)}$$ by ensuring that the outputs of the inverse function correspond to the restricted domain of the original function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses5a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x \\\\geq 4$$"],"dependencies":["a8ffef9inverses5a-h1"],"title":"Domain","text":"What values of $$x$$ for which f(x) is one-to-one?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x \\\\geq 4$$","$$x \\\\geq 3$$","$$x<4$$"]},{"id":"a8ffef9inverses5a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$f(x) \\\\geq 0$$"],"dependencies":["a8ffef9inverses5a-h2"],"title":"Range","text":"Now that we know what is the domain for f(x), what is the range of f(x)? This would affect the domain of our inverse function. Note that f(x) is a radical function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$f(x) \\\\geq 0$$","f(x) is real"]},{"id":"a8ffef9inverses5a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y=\\\\sqrt{x-4}$$"],"dependencies":["a8ffef9inverses5a-h3"],"title":"Replace f(x) with $$y$$","text":"What does the equation look like after replacing f(x) with $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y=\\\\sqrt{x-4}$$","$$f(x)=\\\\sqrt{x-4}$$","$$x=\\\\sqrt{y-4}$$"]},{"id":"a8ffef9inverses5a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x=\\\\sqrt{y-4}$$"],"dependencies":["a8ffef9inverses5a-h4"],"title":"Interchange $$x$$ and $$y$$","text":"What does the equation look like after interchanging $$x$$ and $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y=\\\\sqrt{x-4}$$","$$f(x)=\\\\sqrt{x-4}$$","$$x=\\\\sqrt{y-4}$$"]},{"id":"a8ffef9inverses5a-h6","type":"hint","dependencies":["a8ffef9inverses5a-h5"],"title":"Solve for $$y$$","text":"Make $$y$$ the subject.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses5a-h7","type":"hint","dependencies":["a8ffef9inverses5a-h6"],"title":"Solve for $$y$$","text":"Square each side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses5a-h8","type":"hint","dependencies":["a8ffef9inverses5a-h7"],"title":"Solve for $$y$$","text":"Add $$4$$ to both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses5a-h9","type":"hint","dependencies":["a8ffef9inverses5a-h8"],"title":"Rename the function $$f^{-1\\\\left(x\\\\right)}$$","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses5a-h10","type":"hint","dependencies":["a8ffef9inverses5a-h9"],"title":"Domain of Inverse Function","text":"Recall that the domain of the function must be limited to the range of the original function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ffef9inverses6","title":"Finding the Inverse of a Radical Function","body":"Restrict the domain of the following function and then find the inverse.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.7 Inverses and Radical Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a8ffef9inverses6a","stepAnswer":["$$\\\\frac{x^2-3}{2}$$"],"problemType":"TextBox","stepTitle":"$$f(x)=\\\\sqrt{2x+3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{x^2-3}{2}$$","hints":{"DefaultPathway":[{"id":"a8ffef9inverses6a-h1","type":"hint","dependencies":[],"title":"Finding the Inverse","text":"Given a polynomial function, restrict the domain of a function that is not one-to-one and then find the inverse.\\\\n1) Restrict the domain by determining a domain on which the original function is one-to-one.\\\\n2) Replace f(x) with $$y$$.\\\\n3) Interchange $$x$$ and $$y$$.\\\\n4) Solve for $$y$$, and rename the function or pair of function $$f^{-1\\\\left(x\\\\right)}$$.\\\\n5) Revise the formula for $$f^{-1\\\\left(x\\\\right)}$$ by ensuring that the outputs of the inverse function correspond to the restricted domain of the original function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses6a-h2","type":"hint","dependencies":["a8ffef9inverses6a-h1"],"title":"Domain","text":"The domain are the values of $$x$$ for which f(x) is one-to-one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-3}{2}$$"],"dependencies":["a8ffef9inverses6a-h2"],"title":"Domain","text":"The domain is the values of $$x$$ for which f(x) is one-to-one. This occurs at the range of f(x) where the output is real. This occurs when the expression within the radical is greater than or equal to zero. What is the minimum value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses6a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y=\\\\sqrt{2x+3}$$"],"dependencies":["a8ffef9inverses6a-h3"],"title":"Replace f(x) with $$y$$","text":"What does the equation look like after replacing f(x) with $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y=\\\\sqrt{2x+3}$$","$$f(x)=\\\\sqrt{2x+3}$$","$$x=\\\\sqrt{2y+3}$$"]},{"id":"a8ffef9inverses6a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x=\\\\sqrt{2y+3}$$"],"dependencies":["a8ffef9inverses6a-h4"],"title":"Interchange $$x$$ and $$y$$","text":"What does the equation look like after interchanging $$x$$ and $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y=\\\\sqrt{2x+3}$$","$$f(x)=\\\\sqrt{2x+3}$$","$$x=\\\\sqrt{2y+3}$$"]},{"id":"a8ffef9inverses6a-h6","type":"hint","dependencies":["a8ffef9inverses6a-h5"],"title":"Solve for $$y$$","text":"Make $$y$$ the subject.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses6a-h7","type":"hint","dependencies":["a8ffef9inverses6a-h6"],"title":"Solve for $$y$$","text":"Square each side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses6a-h8","type":"hint","dependencies":["a8ffef9inverses6a-h7"],"title":"Solve for $$y$$","text":"Subtract $$3$$ from both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses6a-h9","type":"hint","dependencies":["a8ffef9inverses6a-h8"],"title":"Solve for $$y$$","text":"Divide by $$2$$ to make $$y$$ the subject.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses6a-h10","type":"hint","dependencies":["a8ffef9inverses6a-h9"],"title":"Rename the function $$f^{-1\\\\left(x\\\\right)}$$","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ffef9inverses8","title":"Finding the Domain of a Radical Function Composed with a Rational Function","body":"Find the domain of the following function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.7 Inverses and Radical Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a8ffef9inverses8a","stepAnswer":["$$[-2,1),[3,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\sqrt{\\\\frac{\\\\left(x+2\\\\right) \\\\left(x-3\\\\right)}{x-1}}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[-2,1),[3,\\\\infty)$$","choices":["$$[-2,1),[3,\\\\infty)$$","$$[-2,\\\\infty)$$","$$(-\\\\infty,\\\\infty)$$","(-inf,2),[1,3)"],"hints":{"DefaultPathway":[{"id":"a8ffef9inverses8a-h1","type":"hint","dependencies":[],"title":"Change of Sign","text":"Because a square root is only defined when the quantity under the radical is non-negative, we need to determine where $$\\\\frac{\\\\left(x+2\\\\right) \\\\left(x-3\\\\right)}{x-1} \\\\geq 0$$. The output of a rational function can change signs (change from positive to negative or vice versa) at x-intercepts and at vertical asymptotes. For this equation, where can the graph change signs?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses8a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-2, 1, 3$$"],"dependencies":["a8ffef9inverses8a-h1"],"title":"Change of Sign","text":"Where does the equation change sign? They usually occur around zeros and asymptotes, for e.g. consider $$(x-1)$$, $$x-1>0$$ at $$x=1.01$$, $$x-1<0$$ at $$x=0.99$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$-2, 1, 3$$","$$-2, 3$$","$$1$$"]},{"id":"a8ffef9inverses8a-h3","type":"hint","dependencies":["a8ffef9inverses8a-h2"],"title":"Testing Values","text":"For each expression, $$x+2$$, $$(x-3)$$, $$(x-1)$$, we know that there is a point beyond which the expression becomes either positive or negative. By finding the product of the sign of each expressions, we can evaluate the sign of the overall equation. For instance, at $$x=-\\\\infty$$, $$x+2$$, $$(x-3)$$, $$(x-1)$$ < $$0$$, hence their product would yield an overall equation that is negative at $$x=-\\\\infty$$. By finding where $$f(x) \\\\geq 0$$, we can find the domain of the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a8ffef9inverses8a-h3"],"title":"Testing Values","text":"How many different range of values should we test? E.g. $$(-\\\\infty,0]$$, $$(0,\\\\infty)$$ would be considered $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a8ffef9inverses8a-h4-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["(-inf,-2),[-2,1),(1,3),[3,inf)"],"dependencies":[],"title":"Testing Values","text":"How many different range of values should we test? Recall that there are $$3$$ change-of-sign values. Thus we would want to consider the intervals around them. Also note that the change-of-sign value itself could be in the domain if it yields a $$0$$ because f(x) outputs a real value at $$\\\\sqrt{0}$$ and f(x) is one-to-one at this point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["(-inf,-2),[-2,1),(1,3),[3,inf)","(-inf,-2),[-2,1),[1,3),[3,inf)","(-inf,-2),[-2,3),[3,inf)"]}]},{"id":"a8ffef9inverses8a-h5","type":"hint","dependencies":["a8ffef9inverses8a-h4"],"title":"Testing $$(-\\\\infty,-2)$$","text":"Check if the equation, $$\\\\frac{\\\\left(x+2\\\\right) \\\\left(x-3\\\\right)}{x-1}$$, is positive or negative for values in this range. If it is positive then f(x) will have a real output, and thus this range is part of the domain.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses8a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["negative"],"dependencies":["a8ffef9inverses8a-h5"],"title":"Testing $$(-\\\\infty,-2)$$","text":"Check if the equation, $$\\\\frac{\\\\left(x+2\\\\right) \\\\left(x-3\\\\right)}{x-1}$$, is positive or negative for values in this range. Substitute $$x=-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["positive","negative"]},{"id":"a8ffef9inverses8a-h7","type":"hint","dependencies":["a8ffef9inverses8a-h4"],"title":"Testing [-2,1)","text":"Check if the equation, $$\\\\frac{\\\\left(x+2\\\\right) \\\\left(x-3\\\\right)}{x-1}$$, is positive or negative for values in this range. If it is positive then f(x) will have a real output, and thus this range is part of the domain.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses8a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["positive"],"dependencies":["a8ffef9inverses8a-h7"],"title":"Testing [-2,1)","text":"Check if the equation, $$\\\\frac{\\\\left(x+2\\\\right) \\\\left(x-3\\\\right)}{x-1}$$, is positive or negative for values in this range. Substitute $$x=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["positive","negative"]},{"id":"a8ffef9inverses8a-h9","type":"hint","dependencies":["a8ffef9inverses8a-h4"],"title":"Testing $$(1,3)$$","text":"Check if the equation, $$\\\\frac{\\\\left(x+2\\\\right) \\\\left(x-3\\\\right)}{x-1}$$, is positive or negative for values in this range. If it is positive then f(x) will have a real output, and thus this range is part of the domain.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses8a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["negative"],"dependencies":["a8ffef9inverses8a-h9"],"title":"Testing $$(1,3)$$","text":"Check if the equation, $$\\\\frac{\\\\left(x+2\\\\right) \\\\left(x-3\\\\right)}{x-1}$$, is positive or negative for values in this range. Substitute $$x=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["positive","negative"]},{"id":"a8ffef9inverses8a-h11","type":"hint","dependencies":["a8ffef9inverses8a-h4"],"title":"Testing $$[3,\\\\infty)$$","text":"Check if the equation, $$\\\\frac{\\\\left(x+2\\\\right) \\\\left(x-3\\\\right)}{x-1}$$, is positive or negative for values in this range. If it is positive then f(x) will have a real output, and thus this range is part of the domain.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses8a-h12","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["positive"],"dependencies":["a8ffef9inverses8a-h11"],"title":"Testing $$[3,\\\\infty)$$","text":"Check if the equation, $$\\\\frac{\\\\left(x+2\\\\right) \\\\left(x-3\\\\right)}{x-1}$$, is positive or negative for values in this range. Substitute $$x=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["positive","negative"]},{"id":"a8ffef9inverses8a-h13","type":"hint","dependencies":["a8ffef9inverses8a-h6","a8ffef9inverses8a-h8","a8ffef9inverses8a-h10","a8ffef9inverses8a-h12"],"title":"Union","text":"Now that you\'ve found the range for which f(x) is valid, the domain is the intersection of these union of these ranges.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a8ffef9inverses9","title":"Finding the Inverse of a Rational Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.7 Inverses and Radical Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a8ffef9inverses9a","stepAnswer":["$$300$$"],"problemType":"TextBox","stepTitle":"The function $$C=\\\\frac{20+0.4n}{100+n}$$ represents the concentration C of an acid solution after $$n$$ mL of 40% solution has been added to $$100$$ mL of a 20% solution. First, find the inverse of the function; that is, find an expression for $$n$$ in terms of C. Then use your result to determine how much of the 40% solution should be added so that the final mixture is a 35% solution.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$300$$","hints":{"DefaultPathway":[{"id":"a8ffef9inverses9a-h1","type":"hint","dependencies":[],"title":"Finding the Inverse","text":"The general process for finding an inverse is as follow:\\\\nGiven a polynomial function, restrict the domain of a function that is not one-to-one and then find the inverse.\\\\n1) Restrict the domain by determining a domain on which the original function is one-to-one.\\\\n2) Replace f(x) with $$y$$.\\\\n3) Interchange $$x$$ and $$y$$.\\\\n4) Solve for $$y$$, and rename the function or pair of function $$f^{-1\\\\left(x\\\\right)}$$.\\\\n5) Revise the formula for $$f^{-1\\\\left(x\\\\right)}$$ by ensuring that the outputs of the inverse function correspond to the restricted domain of the original function.\\\\nHowever, note that in real-world applications, we do not swap the variables when finding inverses. Instead, we change which variable is considered to be the independent variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses9a-h2","type":"hint","dependencies":["a8ffef9inverses9a-h1"],"title":"Domain","text":"Notice that in this question, $$n$$ and C would not be meaningful when they are less than $$0$$. Thus, the domain and range for them are from zero to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses9a-h3","type":"hint","dependencies":["a8ffef9inverses9a-h2"],"title":"Solve for $$n$$","text":"Make $$n$$ the subject of the equation, $$C=\\\\frac{20+0.4n}{100+n}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses9a-h4","type":"hint","dependencies":["a8ffef9inverses9a-h3"],"title":"Solve for $$n$$","text":"Multiply the denominator in the fraction to both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses9a-h5","type":"hint","dependencies":["a8ffef9inverses9a-h4"],"title":"Solve for $$n$$","text":"Expand out $$C \\\\left(100+n\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses9a-h6","type":"hint","dependencies":["a8ffef9inverses9a-h5"],"title":"Solve for $$n$$","text":"Move the $$n$$ terms to the same side of the equation so that we can isolate $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses9a-h7","type":"hint","dependencies":["a8ffef9inverses9a-h6"],"title":"Solve for $$n$$","text":"Factorize out $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses9a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{100C-20}{0.4-C}$$"],"dependencies":["a8ffef9inverses9a-h7"],"title":"Solve for $$n$$","text":"Divide the coefficient of $$n$$ from both sides to make $$n$$ the subject. What is $$n$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a8ffef9inverses9a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$300$$"],"dependencies":["a8ffef9inverses9a-h8"],"title":"Evaluate $$n$$","text":"Substitute $$C=0.35$$ and solve for $$n$$. Leave your answer in $$3$$ significant figures.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a912db6arithmetic1","title":"Finding Common Differences","body":"Is the sequence arithmetic? If so, find the common difference. Answer N if sequence is not arithmetic. If sequence is arithmetic, answer what the common difference is?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":" 9.2 Arithmetic Sequences","courseName":"OpenStax: College Algebra","steps":[{"id":"a912db6arithmetic1a","stepAnswer":["N"],"problemType":"TextBox","stepTitle":"(1,2,4,8,16,...)","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a912db6arithmetic1a-h1","type":"hint","dependencies":[],"title":"Arithmetic Sequence\'s Characteristics","text":"Subtract each term from the subsequent term to determine whether a common difference exists.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a912db6arithmetic1a-h1"],"title":"Subtracting Subsequent Terms","text":"What is $$2-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a912db6arithmetic1a-h2"],"title":"Subtracting Subsequent Terms","text":"What is $$4-2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a912db6arithmetic1a-h3"],"title":"Subtracting Subsequent Terms","text":"What is $$8-4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic1a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a912db6arithmetic1a-h4"],"title":"Subtracting Subsequent Terms","text":"What is $$16-8$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic1a-h6","type":"hint","dependencies":["a912db6arithmetic1a-h5"],"title":"Analyzing the Common Difference","text":"If the difference between subsequent terms is the same, that is the common difference and the sequence is arithmetic.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic1a-h7","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["N"],"dependencies":["a912db6arithmetic1a-h6"],"title":"Analyzing the Common Difference","text":"Is there a common difference? Answer Y for yes and N for no.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a912db6arithmetic10","title":"Using Explicit Formulas for Arithmetic Sequences","body":"Find the specified term for the arithmetic sequence given the first term and common difference.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":" 9.2 Arithmetic Sequences","courseName":"OpenStax: College Algebra","steps":[{"id":"a912db6arithmetic10a","stepAnswer":["$$41$$"],"problemType":"TextBox","stepTitle":"First term is $$6$$, common difference is $$7$$, find the 6th term.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$41$$","hints":{"DefaultPathway":[{"id":"a912db6arithmetic10a-h1","type":"hint","dependencies":[],"title":"Formula for Writing Arithmetic Sequences","text":"Substitute given values into $$a_n=a_1+\\\\left(n-1\\\\right) d$$ to find terms of sequence.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic10a-h2","type":"hint","dependencies":["a912db6arithmetic10a-h1"],"title":"Substitute","text":"Substitute given values into $$a_n=a_1+\\\\left(n-1\\\\right) d$$ to find terms of sequence.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic10a-h3","type":"hint","dependencies":["a912db6arithmetic10a-h2"],"title":"Substitute","text":"The equation is $$a_n=6+7\\\\left(n-1\\\\right)$$ after substituting given values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic10a-h4","type":"hint","dependencies":["a912db6arithmetic10a-h3"],"title":"Substitute","text":"Substitute $$6$$ for $$n$$ in equation to find the sixth term of the sequence.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$41$$"],"dependencies":["a912db6arithmetic10a-h4"],"title":"Substitute","text":"What is $$a_6$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a912db6arithmetic11","title":"Using Recursive Formulas for Arithmetic Sequences","body":"What is the value of $$d$$ in $$a_n=a_n-1+d$$?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":" 9.2 Arithmetic Sequences","courseName":"OpenStax: College Algebra","steps":[{"id":"a912db6arithmetic11a","stepAnswer":["$$11$$"],"problemType":"TextBox","stepTitle":"$$(-18, -7, 4, 15, 26, ...)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$11$$","hints":{"DefaultPathway":[{"id":"a912db6arithmetic11a-h1","type":"hint","dependencies":[],"title":"Finding Common Differences","text":"Find the common difference by subtracting the subsequent terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11$$"],"dependencies":["a912db6arithmetic11a-h1"],"title":"Finding Common Differences","text":"What is $$-7-(-18)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic11a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11$$"],"dependencies":["a912db6arithmetic11a-h2"],"title":"Finding Common Differences","text":"What is $$4-(-7)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic11a-h4","type":"hint","dependencies":["a912db6arithmetic11a-h3"],"title":"Finding Common Differences","text":"Substitute the common difference for $$d$$ in the recursive formula.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a912db6arithmetic12","title":"Using Recursive Formulas for Arithmetic Sequences","body":"What is the value of $$d$$ in $$a_n=a_n-1+d$$?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic12a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.4$$"],"dependencies":["a912db6arithmetic12a-h2"],"title":"Finding Common Differences","text":"What is $$11.7-10.3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic12a-h4","type":"hint","dependencies":["a912db6arithmetic12a-h3"],"title":"Finding Common Differences","text":"Substitute the common difference for $$d$$ in the recursive formula.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a912db6arithmetic13","title":"Using Recursive Formulas for Arithmetic Sequences","body":"What is the value of $$d$$ in $$a_n=a_n-1+d$$?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":" 9.2 Arithmetic Sequences","courseName":"OpenStax: College Algebra","steps":[{"id":"a912db6arithmetic13a","stepAnswer":["$$9$$"],"problemType":"TextBox","stepTitle":"(17,26,35,...)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9$$","hints":{"DefaultPathway":[{"id":"a912db6arithmetic13a-h1","type":"hint","dependencies":[],"title":"Finding Common Differences","text":"Find the common difference by subtracting the subsequent terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a912db6arithmetic13a-h1"],"title":"Finding Common Differences","text":"What is $$26-17$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic13a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a912db6arithmetic13a-h2"],"title":"Finding Common Differences","text":"What is $$35-26$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic13a-h4","type":"hint","dependencies":["a912db6arithmetic13a-h3"],"title":"Finding Common Differences","text":"Substitute the common difference for $$d$$ in the recursive formula.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a912db6arithmetic14","title":"Using Recursive Formulas for Arithmetic Sequences","body":"What is the value of $$d$$ in $$a_n=a_n-1+d$$?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":" 9.2 Arithmetic Sequences","courseName":"OpenStax: College Algebra","steps":[{"id":"a912db6arithmetic14a","stepAnswer":["$$7$$"],"problemType":"TextBox","stepTitle":"(4,11,18,...)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7$$","hints":{"DefaultPathway":[{"id":"a912db6arithmetic14a-h1","type":"hint","dependencies":[],"title":"Finding Common Differences","text":"Find the common difference by subtracting the subsequent terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a912db6arithmetic14a-h1"],"title":"Finding Common Differences","text":"What is $$11-4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic14a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a912db6arithmetic14a-h2"],"title":"Finding Common Differences","text":"What is $$18-11$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic14a-h4","type":"hint","dependencies":["a912db6arithmetic14a-h3"],"title":"Finding Common Differences","text":"Substitute the common difference for $$d$$ in the recursive formula.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a912db6arithmetic15","title":"Writing Terms of Arithmetic Sequences","body":"Use the explicit formula to write the first $$5$$ terms of the sequence. (Write numbers in order with comma in between each, no spaces, and no brackets.)","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":" 9.2 Arithmetic Sequences","courseName":"OpenStax: College Algebra","steps":[{"id":"a912db6arithmetic15a","stepAnswer":["20,16,12,8,4"],"problemType":"TextBox","stepTitle":"$$a_n=24-4n$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a912db6arithmetic15a-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Substitute $$1$$ for $$n$$ in the formula to get the first term of the sequence. Substitute $$2$$ for the second term, $$3$$ for third, etc...","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a912db6arithmetic15a-h1"],"title":"Substitution","text":"What is $$a_n$$ when $$n=1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["a912db6arithmetic15a-h2"],"title":"Substitution","text":"What is $$a_n$$ when $$n=2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a912db6arithmetic15a-h3"],"title":"Substitution","text":"What is $$a_n$$ when $$n=3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic15a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a912db6arithmetic15a-h4"],"title":"Substitution","text":"What is $$a_n$$ when $$n=4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic15a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a912db6arithmetic15a-h5"],"title":"Substitution","text":"What is $$a_n$$ when $$n=5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a912db6arithmetic3","title":"Finding Common Differences","body":"Is the sequence arithmetic? If so, find the common difference. Answer N if sequence is not arithmetic. If sequence is arithmetic, answer what the common difference is?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":" 9.2 Arithmetic Sequences","courseName":"OpenStax: College Algebra","steps":[{"id":"a912db6arithmetic3a","stepAnswer":["$$-2$$"],"problemType":"TextBox","stepTitle":"(18,16,14,12,10,...)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-2$$","hints":{"DefaultPathway":[{"id":"a912db6arithmetic3a-h1","type":"hint","dependencies":[],"title":"Arithmetic Sequence\'s Characteristics","text":"Subtract each term from the subsequent term to determine whether a common difference exists.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a912db6arithmetic3a-h1"],"title":"Subtracting Subsequent Terms","text":"What is $$16-18$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a912db6arithmetic3a-h2"],"title":"Subtracting Subsequent Terms","text":"What is $$14-16$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a912db6arithmetic3a-h3"],"title":"Subtracting Subsequent Terms","text":"What is $$12-14$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a912db6arithmetic3a-h4"],"title":"Subtracting Subsequent Terms","text":"What is $$12-10$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic3a-h6","type":"hint","dependencies":["a912db6arithmetic3a-h5"],"title":"Analyzing the Common Difference","text":"If the difference between subsequent terms is the same, that is the common difference and the sequence is arithmetic.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic3a-h7","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["Y"],"dependencies":["a912db6arithmetic3a-h6"],"title":"Analyzing the Common Difference","text":"Is there a common difference? Answer Y for yes and N for no.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic3a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a912db6arithmetic3a-h7"],"title":"Analyzing the Common Difference","text":"What is the common difference?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a912db6arithmetic4","title":"Finding Common Differences","body":"Is the sequence arithmetic? If so, find the common difference. Answer N if sequence is not arithmetic. If sequence is arithmetic, answer what the common difference is?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":" 9.2 Arithmetic Sequences","courseName":"OpenStax: College Algebra","steps":[{"id":"a912db6arithmetic4a","stepAnswer":["N"],"problemType":"TextBox","stepTitle":"(1,3,6,10,15,...)","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a912db6arithmetic4a-h1","type":"hint","dependencies":[],"title":"Arithmetic Sequence\'s Characteristics","text":"Subtract each term from the subsequent term to determine whether a common difference exists.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a912db6arithmetic4a-h1"],"title":"Subtracting Subsequent Terms","text":"What is $$3-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a912db6arithmetic4a-h2"],"title":"Subtracting Subsequent Terms","text":"What is $$6-3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a912db6arithmetic4a-h3"],"title":"Subtracting Subsequent Terms","text":"What is $$10-6$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic4a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a912db6arithmetic4a-h4"],"title":"Subtracting Subsequent Terms","text":"What is $$15-10$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic4a-h6","type":"hint","dependencies":["a912db6arithmetic4a-h5"],"title":"Analyzing the Common Difference","text":"If the difference between subsequent terms is the same, that is the common difference and the sequence is arithmetic.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic4a-h7","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["N"],"dependencies":["a912db6arithmetic4a-h6"],"title":"Analyzing the Common Difference","text":"Is there a common difference? Answer Y for yes and N for no.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a912db6arithmetic5","title":"Writing Terms of Arithmetic Sequences","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":" 9.2 Arithmetic Sequences","courseName":"OpenStax: College Algebra","steps":[{"id":"a912db6arithmetic5a","stepAnswer":["17,14,11,8,5 "],"problemType":"TextBox","stepTitle":"Write the first five terms of the arithmetic sequence with $$a\u2081=17$$ and $$\ud835\udc51=-3$$. (Write numbers in order with comma in between each, no spaces, and no brackets.)","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a912db6arithmetic5a-h1","type":"hint","dependencies":[],"title":"Formula for Writing Arithmetic Sequences","text":"Substitute given values into $$a_n=a_1+\\\\left(n-1\\\\right) d$$ to find terms of sequence.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic5a-h2","type":"hint","dependencies":["a912db6arithmetic5a-h1"],"title":"Adding Subsequent Terms","text":"Add $$d$$ to a\u2081 to get second term of sequence. Add $$d$$ to that number to get the third term. Repeat until you get $$5$$ terms of the sequence.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$14$$"],"dependencies":["a912db6arithmetic5a-h2"],"title":"Adding Subsequent Terms","text":"What is $$17+\\\\left(-3\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11$$"],"dependencies":["a912db6arithmetic5a-h3"],"title":"Adding Subsequent Terms","text":"What is $$14+\\\\left(-3\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a912db6arithmetic5a-h4"],"title":"Adding Subsequent Terms","text":"What is $$11+\\\\left(-3\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic5a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a912db6arithmetic5a-h5"],"title":"Adding Subsequent Terms","text":"What is $$8+\\\\left(-3\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic5a-h7","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["17,14,11,8,5 "],"dependencies":["a912db6arithmetic5a-h6"],"title":"Arithmetic Sequences","text":"What are the first five terms of this arithmetic sequence?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a912db6arithmetic6","title":"Writing Terms of Arithmetic Sequences","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":" 9.2 Arithmetic Sequences","courseName":"OpenStax: College Algebra","steps":[{"id":"a912db6arithmetic6a","stepAnswer":["0,2/3,4/3,2,8/3"],"problemType":"TextBox","stepTitle":"Write the first five terms of the arithmetic sequence with $$a\u2081=0$$ and $$\ud835\udc51=\\\\frac{2}{3}$$. (Write numbers in order with comma in between each, no spaces, and no brackets.)","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a912db6arithmetic6a-h1","type":"hint","dependencies":[],"title":"Formula for Writing Arithmetic Sequences","text":"Substitute given values into $$a_n=a_1+\\\\left(n-1\\\\right) d$$ to find terms of sequence.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic6a-h2","type":"hint","dependencies":["a912db6arithmetic6a-h1"],"title":"Adding Subsequent Terms","text":"Add $$d$$ to a\u2081 to get second term of sequence. Add $$d$$ to that number to get the third term. Repeat until you get $$5$$ terms of the sequence.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{3}$$"],"dependencies":["a912db6arithmetic6a-h2"],"title":"Adding Subsequent Terms","text":"What is $$0+\\\\frac{2}{3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{4}{3}$$"],"dependencies":["a912db6arithmetic6a-h3"],"title":"Adding Subsequent Terms","text":"What is $$\\\\frac{2}{3}+\\\\frac{2}{3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a912db6arithmetic6a-h4"],"title":"Adding Subsequent Terms","text":"What is $$\\\\frac{4}{3}+\\\\frac{2}{3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic6a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{8}{3}$$"],"dependencies":["a912db6arithmetic6a-h5"],"title":"Adding Subsequent Terms","text":"What is $$2+\\\\frac{2}{3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic6a-h7","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["17,14,11,8,5 "],"dependencies":["a912db6arithmetic6a-h6"],"title":"Arithmetic Sequences","text":"What are the first five terms of this arithmetic sequence?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a912db6arithmetic7","title":"Finding Common Differences","body":"Is the sequence arithmetic? If so, find the common difference. Answer N if sequence is not arithmetic. If sequence is arithmetic, answer what the common difference is?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":" 9.2 Arithmetic Sequences","courseName":"OpenStax: College Algebra","steps":[{"id":"a912db6arithmetic7a","stepAnswer":["N"],"problemType":"TextBox","stepTitle":"(4,16,64,256,1024,...)","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a912db6arithmetic7a-h1","type":"hint","dependencies":[],"title":"Arithmetic Sequence\'s Characteristics","text":"Subtract each term from the subsequent term to determine whether a common difference exists.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a912db6arithmetic7a-h1"],"title":"Subtracting Subsequent Terms","text":"What is $$16-4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$48$$"],"dependencies":["a912db6arithmetic7a-h2"],"title":"Subtracting Subsequent Terms","text":"What is $$64-16$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$192$$"],"dependencies":["a912db6arithmetic7a-h3"],"title":"Subtracting Subsequent Terms","text":"What is $$256-64$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic7a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$768$$"],"dependencies":["a912db6arithmetic7a-h4"],"title":"Subtracting Subsequent Terms","text":"What is $$1024-256$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic7a-h6","type":"hint","dependencies":["a912db6arithmetic7a-h5"],"title":"Analyzing the Common Difference","text":"If the difference between subsequent terms is the same, that is the common difference and the sequence is arithmetic.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic7a-h7","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["N"],"dependencies":["a912db6arithmetic7a-h6"],"title":"Analyzing the Common Difference","text":"Is there a common difference? Answer Y for yes and N for no.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a912db6arithmetic9","title":"Using Explicit Formulas for Arithmetic Sequences","body":"Find the specified term for the arithmetic sequence given the first term and common difference.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":" 9.2 Arithmetic Sequences","courseName":"OpenStax: College Algebra","steps":[{"id":"a912db6arithmetic9a","stepAnswer":["$$19$$"],"problemType":"TextBox","stepTitle":"First term is $$4$$, common difference is $$5$$, find the 4th term.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$19$$","hints":{"DefaultPathway":[{"id":"a912db6arithmetic9a-h1","type":"hint","dependencies":[],"title":"Formula for Writing Arithmetic Sequences","text":"Substitute given values into $$a_n=a_1+\\\\left(n-1\\\\right) d$$ to find terms of sequence.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic9a-h2","type":"hint","dependencies":["a912db6arithmetic9a-h1"],"title":"Substitute","text":"Substitute given values into $$a_n=a_1+\\\\left(n-1\\\\right) d$$ to find terms of sequence.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic9a-h3","type":"hint","dependencies":["a912db6arithmetic9a-h2"],"title":"Substitute","text":"The equation is $$a_n=4+5\\\\left(n-1\\\\right)$$ after substituting given values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic9a-h4","type":"hint","dependencies":["a912db6arithmetic9a-h3"],"title":"Substitute","text":"Substitute $$4$$ for $$n$$ in equation to find the fourth term of the sequence.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a912db6arithmetic9a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$19$$"],"dependencies":["a912db6arithmetic9a-h4"],"title":"Substitute","text":"What is $$a_4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a91e303graphhyper1","title":"Graphing Hyperbolas with Center $$(0,0)$$","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.4 Hyperbolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a91e303graphhyper1a","stepAnswer":["D"],"problemType":"MultipleChoice","stepTitle":"Graph $$\\\\frac{x^2}{25}-\\\\frac{y^2}{4}=1$$.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a91e303graphhyper1a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"Write the equation in standard form. $$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper1a-h2","type":"hint","dependencies":["a91e303graphhyper1a-h1"],"title":"Horizontal or Vertical","text":"Determine whether the transverse axis is horizontal or vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper1a-h3","type":"hint","dependencies":["a91e303graphhyper1a-h2"],"title":"Horizontal or Vertical","text":"If the $$x^2$$ is positive, then the transverse axis is horizontal, and if the $$y^2$$ term is positive, then the transverse axis is vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper1a-h4","type":"hint","dependencies":["a91e303graphhyper1a-h3"],"title":"Horizontal or Vertical","text":"In our case, the transverse axis is horizontal since $$x^2$$ is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper1a-h5","type":"hint","dependencies":["a91e303graphhyper1a-h4"],"title":"Vertices","text":"Since the transverse axis is horizontal, we want to find the vertices on the $$x$$ axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper1a-h6","type":"hint","dependencies":["a91e303graphhyper1a-h5"],"title":"Vertices","text":"To find the vertices, we find the square root of $$a^2$$ and the square root of $$b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper1a-h7","type":"hint","dependencies":["a91e303graphhyper1a-h6"],"title":"Vertices","text":"In our case, a is $$\\\\pm 5$$, and $$b$$ is $$\\\\pm 2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper1a-h8","type":"hint","dependencies":["a91e303graphhyper1a-h7"],"title":"Asymptotes","text":"Now, we must find the equations of our asymptotes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper1a-h9","type":"hint","dependencies":["a91e303graphhyper1a-h8"],"title":"Asymptotes","text":"Since our transverse axis is horizontal, we will use the equations, $$y=\\\\frac{b}{a} x$$ and $$y$$ $$=$$ $$-\\\\left(\\\\frac{b}{a}\\\\right) x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper1a-h10","type":"hint","dependencies":["a91e303graphhyper1a-h9"],"title":"Asymptotes","text":"We then sketch our equations and draw the the different branches of the hyperbola with the asymptotes as guides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a91e303graphhyper10","title":"Graphing Hyperbolas with Center $$(0,0)$$","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.4 Hyperbolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a91e303graphhyper10a","stepAnswer":["A"],"problemType":"MultipleChoice","stepTitle":"Graph $$4y^2-25x^2=100$$.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a91e303graphhyper10a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"Write the equation in standard form. $$\\\\frac{y^2}{a^2}-\\\\frac{x^2}{b^2}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper10a-h2","type":"hint","dependencies":["a91e303graphhyper10a-h1"],"title":"Standard Form","text":"First, we try to get the right side of the equation to equal to $$1$$. To do this, we divide both sides by $$100$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper10a-h3","type":"hint","dependencies":["a91e303graphhyper10a-h2"],"title":"Standard Form","text":"Next, we simplify the equation and get $$\\\\frac{y^2}{25}-\\\\frac{x^2}{4}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper10a-h4","type":"hint","dependencies":["a91e303graphhyper10a-h3"],"title":"Horizontal or Vertical","text":"Determine whether the transverse axis is horizontal or vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper10a-h5","type":"hint","dependencies":["a91e303graphhyper10a-h4"],"title":"Horizontal or Vertical","text":"If the $$x^2$$ is positive, then the transverse axis is horizontal, and if the $$y^2$$ term is positive, then the transverse axis is vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper10a-h6","type":"hint","dependencies":["a91e303graphhyper10a-h5"],"title":"Horizontal or Vertical","text":"In our case, the transverse axis is vertical since $$y^2$$ is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper10a-h7","type":"hint","dependencies":["a91e303graphhyper10a-h6"],"title":"Vertices","text":"Since the transverse axis is vertical, we want to find the vertices on the $$y$$ axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper10a-h8","type":"hint","dependencies":["a91e303graphhyper10a-h7"],"title":"Vertices","text":"To find the vertices, we find the square root of $$a^2$$ and the square root of $$b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper10a-h9","type":"hint","dependencies":["a91e303graphhyper10a-h8"],"title":"Vertices","text":"In our case, a is $$\\\\pm 5$$, and $$b$$ is $$\\\\pm 2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper10a-h10","type":"hint","dependencies":["a91e303graphhyper10a-h9"],"title":"Asymptotes","text":"Now, we must find the equations of our asymptotes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper10a-h11","type":"hint","dependencies":["a91e303graphhyper10a-h10"],"title":"Asymptotes","text":"Since our transverse axis is vertical, we will use the equations, $$y=\\\\frac{a}{b} x$$ and $$y$$ $$=$$ $$-\\\\left(\\\\frac{a}{b}\\\\right) x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper10a-h12","type":"hint","dependencies":["a91e303graphhyper10a-h11"],"title":"Asymptotes","text":"We then sketch our equations and draw the the different branches of the hyperbola with the asymptotes as guides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a91e303graphhyper11","title":"Graphing Hyperbolas with Center $$(0,0)$$","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.4 Hyperbolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a91e303graphhyper11a","stepAnswer":["D"],"problemType":"MultipleChoice","stepTitle":"Graph $$25y^2-9x^2=225$$.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a91e303graphhyper11a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"Write the equation in standard form. $$\\\\frac{y^2}{a^2}-\\\\frac{x^2}{b^2}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper11a-h2","type":"hint","dependencies":["a91e303graphhyper11a-h1"],"title":"Standard Form","text":"First, we try to get the right side of the equation to equal to $$1$$. To do this, we divide both sides by $$225$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper11a-h3","type":"hint","dependencies":["a91e303graphhyper11a-h2"],"title":"Standard Form","text":"Next, we simplify the equation and get $$\\\\frac{y^2}{9}-\\\\frac{x^2}{25}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper11a-h4","type":"hint","dependencies":["a91e303graphhyper11a-h3"],"title":"Horizontal or Vertical","text":"Determine whether the transverse axis is horizontal or vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper11a-h5","type":"hint","dependencies":["a91e303graphhyper11a-h4"],"title":"Horizontal or Vertical","text":"If the $$x^2$$ is positive, then the transverse axis is horizontal, and if the $$y^2$$ term is positive, then the transverse axis is vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper11a-h6","type":"hint","dependencies":["a91e303graphhyper11a-h5"],"title":"Horizontal or Vertical","text":"In our case, the transverse axis is vertical since $$y^2$$ is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper11a-h7","type":"hint","dependencies":["a91e303graphhyper11a-h6"],"title":"Vertices","text":"Since the transverse axis is vertical, we want to find the vertices on the $$y$$ axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper11a-h8","type":"hint","dependencies":["a91e303graphhyper11a-h7"],"title":"Vertices","text":"To find the vertices, we find the square root of $$a^2$$ and the square root of $$b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper11a-h9","type":"hint","dependencies":["a91e303graphhyper11a-h8"],"title":"Vertices","text":"In our case, a is $$\\\\pm 3$$, and $$b$$ is $$\\\\pm 5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper11a-h10","type":"hint","dependencies":["a91e303graphhyper11a-h9"],"title":"Asymptotes","text":"Now, we must find the equations of our asymptotes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper11a-h11","type":"hint","dependencies":["a91e303graphhyper11a-h10"],"title":"Asymptotes","text":"Since our transverse axis is vertical, we will use the equations, $$y=\\\\frac{a}{b} x$$ and $$y$$ $$=$$ $$-\\\\left(\\\\frac{a}{b}\\\\right) x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper11a-h12","type":"hint","dependencies":["a91e303graphhyper11a-h11"],"title":"Asymptotes","text":"We then sketch our equations and draw the the different branches of the hyperbola with the asymptotes as guides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a91e303graphhyper12","title":"Graphing Hyperbolas with Center $$(0,0)$$","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.4 Hyperbolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a91e303graphhyper12a","stepAnswer":["C"],"problemType":"MultipleChoice","stepTitle":"Graph $$16y^2-9x^2=144$$.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a91e303graphhyper12a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"Write the equation in standard form. $$\\\\frac{y^2}{a^2}-\\\\frac{x^2}{b^2}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper12a-h2","type":"hint","dependencies":["a91e303graphhyper12a-h1"],"title":"Standard Form","text":"First, we try to get the right side of the equation to equal to $$1$$. To do this, we divide both sides by $$144$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper12a-h3","type":"hint","dependencies":["a91e303graphhyper12a-h2"],"title":"Standard Form","text":"Next, we simplify the equation and get $$\\\\frac{y^2}{9}-\\\\frac{x^2}{16}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper12a-h4","type":"hint","dependencies":["a91e303graphhyper12a-h3"],"title":"Horizontal or Vertical","text":"Determine whether the transverse axis is horizontal or vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper12a-h5","type":"hint","dependencies":["a91e303graphhyper12a-h4"],"title":"Horizontal or Vertical","text":"If the $$x^2$$ is positive, then the transverse axis is horizontal, and if the $$y^2$$ term is positive, then the transverse axis is vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper12a-h6","type":"hint","dependencies":["a91e303graphhyper12a-h5"],"title":"Horizontal or Vertical","text":"In our case, the transverse axis is vertical since $$y^2$$ is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper12a-h7","type":"hint","dependencies":["a91e303graphhyper12a-h6"],"title":"Vertices","text":"Since the transverse axis is vertical, we want to find the vertices on the $$y$$ axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper12a-h8","type":"hint","dependencies":["a91e303graphhyper12a-h7"],"title":"Vertices","text":"To find the vertices, we find the square root of $$a^2$$ and the square root of $$b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper12a-h9","type":"hint","dependencies":["a91e303graphhyper12a-h8"],"title":"Vertices","text":"In our case, a is $$\\\\pm 3$$, and $$b$$ is $$\\\\pm 4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper12a-h10","type":"hint","dependencies":["a91e303graphhyper12a-h9"],"title":"Asymptotes","text":"Now, we must find the equations of our asymptotes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper12a-h11","type":"hint","dependencies":["a91e303graphhyper12a-h10"],"title":"Asymptotes","text":"Since our transverse axis is vertical, we will use the equations, $$y=\\\\frac{a}{b} x$$ and $$y$$ $$=$$ $$-\\\\left(\\\\frac{a}{b}\\\\right) x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper12a-h12","type":"hint","dependencies":["a91e303graphhyper12a-h11"],"title":"Asymptotes","text":"We then sketch our equations and draw the the different branches of the hyperbola with the asymptotes as guides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a91e303graphhyper13","title":"Graphing Hyperbolas with Center $$(0,0)$$","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.4 Hyperbolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a91e303graphhyper13a","stepAnswer":["D"],"problemType":"MultipleChoice","stepTitle":"Graph $$4y^2-9x^2=36$$.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a91e303graphhyper13a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"Write the equation in standard form. $$\\\\frac{y^2}{a^2}-\\\\frac{x^2}{b^2}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper13a-h2","type":"hint","dependencies":["a91e303graphhyper13a-h1"],"title":"Standard Form","text":"First, we try to get the right side of the equation to equal to $$1$$. To do this, we divide both sides by $$36$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper13a-h3","type":"hint","dependencies":["a91e303graphhyper13a-h2"],"title":"Standard Form","text":"Next, we simplify the equation and get $$\\\\frac{y^2}{9}-\\\\frac{x^2}{4}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper13a-h4","type":"hint","dependencies":["a91e303graphhyper13a-h3"],"title":"Horizontal or Vertical","text":"Determine whether the transverse axis is horizontal or vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper13a-h5","type":"hint","dependencies":["a91e303graphhyper13a-h4"],"title":"Horizontal or Vertical","text":"If the $$x^2$$ is positive, then the transverse axis is horizontal, and if the $$y^2$$ term is positive, then the transverse axis is vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper13a-h6","type":"hint","dependencies":["a91e303graphhyper13a-h5"],"title":"Horizontal or Vertical","text":"In our case, the transverse axis is vertical since $$y^2$$ is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper13a-h7","type":"hint","dependencies":["a91e303graphhyper13a-h6"],"title":"Vertices","text":"Since the transverse axis is vertical, we want to find the vertices on the $$y$$ axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper13a-h8","type":"hint","dependencies":["a91e303graphhyper13a-h7"],"title":"Vertices","text":"To find the vertices, we find the square root of $$a^2$$ and the square root of $$b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper13a-h9","type":"hint","dependencies":["a91e303graphhyper13a-h8"],"title":"Vertices","text":"In our case, a is $$\\\\pm 3$$, and $$b$$ is $$\\\\pm 2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper13a-h10","type":"hint","dependencies":["a91e303graphhyper13a-h9"],"title":"Asymptotes","text":"Now, we must find the equations of our asymptotes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper13a-h11","type":"hint","dependencies":["a91e303graphhyper13a-h10"],"title":"Asymptotes","text":"Since our transverse axis is vertical, we will use the equations, $$y=\\\\frac{a}{b} x$$ and $$y$$ $$=$$ $$-\\\\left(\\\\frac{a}{b}\\\\right) x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper13a-h12","type":"hint","dependencies":["a91e303graphhyper13a-h11"],"title":"Asymptotes","text":"We then sketch our equations and draw the the different branches of the hyperbola with the asymptotes as guides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a91e303graphhyper14","title":"Graphing Hyperbolas with Center $$(0,0)$$","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.4 Hyperbolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a91e303graphhyper14a","stepAnswer":["A"],"problemType":"MultipleChoice","stepTitle":"Graph $$16y^2-25x^2=400$$.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a91e303graphhyper14a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"Write the equation in standard form. $$\\\\frac{y^2}{a^2}-\\\\frac{x^2}{b^2}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper14a-h2","type":"hint","dependencies":["a91e303graphhyper14a-h1"],"title":"Standard Form","text":"First, we try to get the right side of the equation to equal to $$1$$. To do this, we divide both sides by $$400$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper14a-h3","type":"hint","dependencies":["a91e303graphhyper14a-h2"],"title":"Standard Form","text":"Next, we simplify the equation and get $$\\\\frac{y^2}{25}-\\\\frac{x^2}{16}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper14a-h4","type":"hint","dependencies":["a91e303graphhyper14a-h3"],"title":"Horizontal or Vertical","text":"Determine whether the transverse axis is horizontal or vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper14a-h5","type":"hint","dependencies":["a91e303graphhyper14a-h4"],"title":"Horizontal or Vertical","text":"If the $$x^2$$ is positive, then the transverse axis is horizontal, and if the $$y^2$$ term is positive, then the transverse axis is vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper14a-h6","type":"hint","dependencies":["a91e303graphhyper14a-h5"],"title":"Horizontal or Vertical","text":"In our case, the transverse axis is vertical since $$y^2$$ is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper14a-h7","type":"hint","dependencies":["a91e303graphhyper14a-h6"],"title":"Vertices","text":"Since the transverse axis is vertical, we want to find the vertices on the $$y$$ axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper14a-h8","type":"hint","dependencies":["a91e303graphhyper14a-h7"],"title":"Vertices","text":"To find the vertices, we find the square root of $$a^2$$ and the square root of $$b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper14a-h9","type":"hint","dependencies":["a91e303graphhyper14a-h8"],"title":"Vertices","text":"In our case, a is $$\\\\pm 5$$, and $$b$$ is $$\\\\pm 4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper14a-h10","type":"hint","dependencies":["a91e303graphhyper14a-h9"],"title":"Asymptotes","text":"Now, we must find the equations of our asymptotes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper14a-h11","type":"hint","dependencies":["a91e303graphhyper14a-h10"],"title":"Asymptotes","text":"Since our transverse axis is vertical, we will use the equations, $$y=\\\\frac{a}{b} x$$ and $$y$$ $$=$$ $$-\\\\left(\\\\frac{a}{b}\\\\right) x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper14a-h12","type":"hint","dependencies":["a91e303graphhyper14a-h11"],"title":"Asymptotes","text":"We then sketch our equations and draw the the different branches of the hyperbola with the asymptotes as guides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a91e303graphhyper15","title":"Graphing Hyperbolas with Center $$(0,0)$$","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.4 Hyperbolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a91e303graphhyper15a","stepAnswer":["B"],"problemType":"MultipleChoice","stepTitle":"Graph $$4x^2-16y^2=64$$.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a91e303graphhyper15a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"Write the equation in standard form. $$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper15a-h2","type":"hint","dependencies":["a91e303graphhyper15a-h1"],"title":"Standard Form","text":"First, we try to get the right side of the equation to equal to $$1$$. To do this, we divide both sides by $$64$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper15a-h3","type":"hint","dependencies":["a91e303graphhyper15a-h2"],"title":"Standard Form","text":"Next, we simplify the equation and get $$\\\\frac{x^2}{16}-\\\\frac{y^2}{4}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper15a-h4","type":"hint","dependencies":["a91e303graphhyper15a-h3"],"title":"Horizontal or Vertical","text":"Determine whether the transverse axis is horizontal or vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper15a-h5","type":"hint","dependencies":["a91e303graphhyper15a-h4"],"title":"Horizontal or Vertical","text":"If the $$x^2$$ is positive, then the transverse axis is horizontal, and if the $$y^2$$ term is positive, then the transverse axis is vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper15a-h6","type":"hint","dependencies":["a91e303graphhyper15a-h5"],"title":"Horizontal or Vertical","text":"In our case, the transverse axis is horizontal since $$x^2$$ is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper15a-h7","type":"hint","dependencies":["a91e303graphhyper15a-h6"],"title":"Vertices","text":"Since the transverse axis is horizontal, we want to find the vertices on the $$y$$ axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper15a-h8","type":"hint","dependencies":["a91e303graphhyper15a-h7"],"title":"Vertices","text":"To find the vertices, we find the square root of $$a^2$$ and the square root of $$b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper15a-h9","type":"hint","dependencies":["a91e303graphhyper15a-h8"],"title":"Vertices","text":"In our case, a is $$\\\\pm 4$$, and $$b$$ is $$\\\\pm 2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper15a-h10","type":"hint","dependencies":["a91e303graphhyper15a-h9"],"title":"Asymptotes","text":"Now, we must find the equations of our asymptotes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper15a-h11","type":"hint","dependencies":["a91e303graphhyper15a-h10"],"title":"Asymptotes","text":"Since our transverse axis is horizontal, we will use the equations, $$y=\\\\frac{b}{a} x$$ and $$y$$ $$=$$ $$-\\\\left(\\\\frac{b}{a}\\\\right) x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper15a-h12","type":"hint","dependencies":["a91e303graphhyper15a-h11"],"title":"Asymptotes","text":"We then sketch our equations and draw the the different branches of the hyperbola with the asymptotes as guides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a91e303graphhyper16","title":"Graphing Hyperbolas with Center $$(0,0)$$","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.4 Hyperbolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a91e303graphhyper16a","stepAnswer":["C"],"problemType":"MultipleChoice","stepTitle":"Graph $$9x^2-4y^2=36$$.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a91e303graphhyper16a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"Write the equation in standard form. $$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper16a-h2","type":"hint","dependencies":["a91e303graphhyper16a-h1"],"title":"Standard Form","text":"First, we try to get the right side of the equation to equal to $$1$$. To do this, we divide both sides by $$36$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper16a-h3","type":"hint","dependencies":["a91e303graphhyper16a-h2"],"title":"Standard Form","text":"Next, we simplify the equation and get $$\\\\frac{x^2}{4}-\\\\frac{y^2}{9}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper16a-h4","type":"hint","dependencies":["a91e303graphhyper16a-h3"],"title":"Horizontal or Vertical","text":"Determine whether the transverse axis is horizontal or vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper16a-h5","type":"hint","dependencies":["a91e303graphhyper16a-h4"],"title":"Horizontal or Vertical","text":"If the $$x^2$$ is positive, then the transverse axis is horizontal, and if the $$y^2$$ term is positive, then the transverse axis is vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper16a-h6","type":"hint","dependencies":["a91e303graphhyper16a-h5"],"title":"Horizontal or Vertical","text":"In our case, the transverse axis is horizontal since $$x^2$$ is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper16a-h7","type":"hint","dependencies":["a91e303graphhyper16a-h6"],"title":"Vertices","text":"Since the transverse axis is horizontal, we want to find the vertices on the $$y$$ axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper16a-h8","type":"hint","dependencies":["a91e303graphhyper16a-h7"],"title":"Vertices","text":"To find the vertices, we find the square root of $$a^2$$ and the square root of $$b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper16a-h9","type":"hint","dependencies":["a91e303graphhyper16a-h8"],"title":"Vertices","text":"In our case, a is $$\\\\pm 2$$, and $$b$$ is $$\\\\pm 3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper16a-h10","type":"hint","dependencies":["a91e303graphhyper16a-h9"],"title":"Asymptotes","text":"Now, we must find the equations of our asymptotes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper16a-h11","type":"hint","dependencies":["a91e303graphhyper16a-h10"],"title":"Asymptotes","text":"Since our transverse axis is horizontal, we will use the equations, $$y=\\\\frac{b}{a} x$$ and $$y$$ $$=$$ $$-\\\\left(\\\\frac{b}{a}\\\\right) x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper16a-h12","type":"hint","dependencies":["a91e303graphhyper16a-h11"],"title":"Asymptotes","text":"We then sketch our equations and draw the the different branches of the hyperbola with the asymptotes as guides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a91e303graphhyper17","title":"Graphing Hyperbolas with Center (h,k)","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.4 Hyperbolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a91e303graphhyper17a","stepAnswer":["D"],"problemType":"MultipleChoice","stepTitle":"Graph $$\\\\frac{{\\\\left(x-1\\\\right)}^2}{9}-\\\\frac{{\\\\left(y-2\\\\right)}^2}{16}=1$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a91e303graphhyper17a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"Write the equation in standard form. $$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper17a-h2","type":"hint","dependencies":["a91e303graphhyper17a-h1"],"title":"Horizontal or Vertical","text":"Determine whether the transverse axis is horizontal or vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper17a-h3","type":"hint","dependencies":["a91e303graphhyper17a-h2"],"title":"Horizontal or Vertical","text":"If the $$x^2$$ is positive, then the transverse axis is horizontal, and if the $$y^2$$ term is positive, then the transverse axis is vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper17a-h4","type":"hint","dependencies":["a91e303graphhyper17a-h3"],"title":"Horizontal or Vertical","text":"In our case, the transverse axis is horizontal since $$x^2$$ is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper17a-h5","type":"hint","dependencies":["a91e303graphhyper17a-h4"],"title":"Finding the Center","text":"Since the equation is in the form of $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}-\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$, we can find our center, which is shown from $$h$$ and k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper17a-h6","type":"hint","dependencies":["a91e303graphhyper17a-h5"],"title":"Finding the Center","text":"To find the center, we look for $$h$$ and k, which in this case is $$h=1$$ and $$k=2$$, so our center is located at $$(1,2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper17a-h7","type":"hint","dependencies":["a91e303graphhyper17a-h6"],"title":"Vertices","text":"To find the vertices, we find the square root of $$a^2$$ and the square root of $$b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper17a-h8","type":"hint","dependencies":["a91e303graphhyper17a-h7"],"title":"Vertices","text":"In our case, a is $$\\\\pm 3$$, and $$b$$ is $$\\\\pm 4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper17a-h9","type":"hint","dependencies":["a91e303graphhyper17a-h8"],"title":"Asymptotes","text":"Now, we sketch a rectangle around our center using our found a and $$b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper17a-h10","type":"hint","dependencies":["a91e303graphhyper17a-h9"],"title":"Asymptotes","text":"We then sketch two diagonal lines from the corners of the rectanges and through the center, which will outline our asymptotes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper17a-h11","type":"hint","dependencies":["a91e303graphhyper17a-h10"],"title":"Asymptotes","text":"We can then draw the hyperbola with the rectangle and lines as our guides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a91e303graphhyper18","title":"Graphing Hyperbolas with Center (h,k)","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.4 Hyperbolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a91e303graphhyper18a","stepAnswer":["C"],"problemType":"MultipleChoice","stepTitle":"Graph $$\\\\frac{{\\\\left(x-3\\\\right)}^2}{25}-\\\\frac{{\\\\left(y-1\\\\right)}^2}{9}=1$$.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a91e303graphhyper18a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"Write the equation in standard form. $$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper18a-h2","type":"hint","dependencies":["a91e303graphhyper18a-h1"],"title":"Horizontal or Vertical","text":"Determine whether the transverse axis is horizontal or vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper18a-h3","type":"hint","dependencies":["a91e303graphhyper18a-h2"],"title":"Horizontal or Vertical","text":"If the $$x^2$$ is positive, then the transverse axis is horizontal, and if the $$y^2$$ term is positive, then the transverse axis is vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper18a-h4","type":"hint","dependencies":["a91e303graphhyper18a-h3"],"title":"Horizontal or Vertical","text":"In our case, the transverse axis is horizontal since $$x^2$$ is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper18a-h5","type":"hint","dependencies":["a91e303graphhyper18a-h4"],"title":"Finding the Center","text":"Since the equation is in the form of $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}-\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$, we can find our center, which is shown from $$h$$ and k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper18a-h6","type":"hint","dependencies":["a91e303graphhyper18a-h5"],"title":"Finding the Center","text":"To find the center, we look for $$h$$ and k, which in this case is $$h=3$$ and $$k=1$$, so our center is located at $$(3,1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper18a-h7","type":"hint","dependencies":["a91e303graphhyper18a-h6"],"title":"Vertices","text":"To find the vertices, we find the square root of $$a^2$$ and the square root of $$b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper18a-h8","type":"hint","dependencies":["a91e303graphhyper18a-h7"],"title":"Vertices","text":"In our case, a is $$\\\\pm 5$$, and $$b$$ is $$\\\\pm 3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper18a-h9","type":"hint","dependencies":["a91e303graphhyper18a-h8"],"title":"Asymptotes","text":"Now, we sketch a rectangle around our center using our found a and $$b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper18a-h10","type":"hint","dependencies":["a91e303graphhyper18a-h9"],"title":"Asymptotes","text":"We then sketch two diagonal lines from the corners of the rectanges and through the center, which will outline our asymptotes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper18a-h11","type":"hint","dependencies":["a91e303graphhyper18a-h10"],"title":"Asymptotes","text":"We can then draw the hyperbola with the rectangle and lines as our guides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a91e303graphhyper19","title":"Graphing Hyperbolas with Center (h,k)","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.4 Hyperbolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a91e303graphhyper19a","stepAnswer":["A"],"problemType":"MultipleChoice","stepTitle":"Graph $$\\\\frac{{\\\\left(x-2\\\\right)}^2}{4}-\\\\frac{{\\\\left(y-2\\\\right)}^2}{9}=1$$.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a91e303graphhyper19a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"Write the equation in standard form. $$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper19a-h2","type":"hint","dependencies":["a91e303graphhyper19a-h1"],"title":"Horizontal or Vertical","text":"Determine whether the transverse axis is horizontal or vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper19a-h3","type":"hint","dependencies":["a91e303graphhyper19a-h2"],"title":"Horizontal or Vertical","text":"If the $$x^2$$ is positive, then the transverse axis is horizontal, and if the $$y^2$$ term is positive, then the transverse axis is vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper19a-h4","type":"hint","dependencies":["a91e303graphhyper19a-h3"],"title":"Horizontal or Vertical","text":"In our case, the transverse axis is horizontal since $$x^2$$ is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper19a-h5","type":"hint","dependencies":["a91e303graphhyper19a-h4"],"title":"Finding the Center","text":"Since the equation is in the form of $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}-\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$, we can find our center, which is shown from $$h$$ and k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper19a-h6","type":"hint","dependencies":["a91e303graphhyper19a-h5"],"title":"Finding the Center","text":"To find the center, we look for $$h$$ and k, which in this case is $$h=2$$ and $$k=2$$, so our center is located at $$(2,2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper19a-h7","type":"hint","dependencies":["a91e303graphhyper19a-h6"],"title":"Vertices","text":"To find the vertices, we find the square root of $$a^2$$ and the square root of $$b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper19a-h8","type":"hint","dependencies":["a91e303graphhyper19a-h7"],"title":"Vertices","text":"In our case, a is $$\\\\pm 2$$, and $$b$$ is $$\\\\pm 3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper19a-h9","type":"hint","dependencies":["a91e303graphhyper19a-h8"],"title":"Asymptotes","text":"Now, we sketch a rectangle around our center using our found a and $$b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper19a-h10","type":"hint","dependencies":["a91e303graphhyper19a-h9"],"title":"Asymptotes","text":"We then sketch two diagonal lines from the corners of the rectanges and through the center, which will outline our asymptotes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper19a-h11","type":"hint","dependencies":["a91e303graphhyper19a-h10"],"title":"Asymptotes","text":"We can then draw the hyperbola with the rectangle and lines as our guides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a91e303graphhyper2","title":"Graphing Hyperbolas with Center $$(0,0)$$","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.4 Hyperbolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a91e303graphhyper2a","stepAnswer":["C"],"problemType":"MultipleChoice","stepTitle":"Graph $$\\\\frac{x^2}{16}-\\\\frac{y^2}{4}=1$$.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a91e303graphhyper2a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"Write the equation in standard form. $$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper2a-h2","type":"hint","dependencies":["a91e303graphhyper2a-h1"],"title":"Horizontal or Vertical","text":"Determine whether the transverse axis is horizontal or vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper2a-h3","type":"hint","dependencies":["a91e303graphhyper2a-h2"],"title":"Horizontal or Vertical","text":"If the $$x^2$$ is positive, then the transverse axis is horizontal, and if the $$y^2$$ term is positive, then the transverse axis is vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper2a-h4","type":"hint","dependencies":["a91e303graphhyper2a-h3"],"title":"Horizontal or Vertical","text":"In our case, the transverse axis is horizontal since $$x^2$$ is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper2a-h5","type":"hint","dependencies":["a91e303graphhyper2a-h4"],"title":"Vertices","text":"Since the transverse axis is horizontal, we want to find the vertices on the $$x$$ axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper2a-h6","type":"hint","dependencies":["a91e303graphhyper2a-h5"],"title":"Vertices","text":"To find the vertices, we find the square root of $$a^2$$ and the square root of $$b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper2a-h7","type":"hint","dependencies":["a91e303graphhyper2a-h6"],"title":"Vertices","text":"In our case, a is $$\\\\pm 4$$, and $$b$$ is $$\\\\pm 2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper2a-h8","type":"hint","dependencies":["a91e303graphhyper2a-h7"],"title":"Asymptotes","text":"Now, we must find the equations of our asymptotes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper2a-h9","type":"hint","dependencies":["a91e303graphhyper2a-h8"],"title":"Asymptotes","text":"Since our transverse axis is horizontal, we will use the equations, $$y=\\\\frac{b}{a} x$$ and $$y$$ $$=$$ $$-\\\\left(\\\\frac{b}{a}\\\\right) x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper2a-h10","type":"hint","dependencies":["a91e303graphhyper2a-h9"],"title":"Asymptotes","text":"We then sketch our equations and draw the the different branches of the hyperbola with the asymptotes as guides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a91e303graphhyper20","title":"Graphing Hyperbolas with Center $$(0,0)$$","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.4 Hyperbolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a91e303graphhyper20a","stepAnswer":["B"],"problemType":"MultipleChoice","stepTitle":"Graphing Hyperbolas with Center (h,k)","stepBody":"$$-\\\\left(\\\\frac{{\\\\left(y+2\\\\right)}^2}{4}\\\\right)+\\\\frac{{\\\\left(x+1\\\\right)}^2}{9}=1$$##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a91e303graphhyper20a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"Write the equation in standard form. $$-\\\\left(\\\\frac{y^2}{a^2}\\\\right)+\\\\frac{x^2}{b^2}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper20a-h2","type":"hint","dependencies":["a91e303graphhyper20a-h1"],"title":"Horizontal or Vertical","text":"Determine whether the transverse axis is horizontal or vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper20a-h3","type":"hint","dependencies":["a91e303graphhyper20a-h2"],"title":"Horizontal or Vertical","text":"If the $$x^2$$ is positive, then the transverse axis is horizontal, and if the $$y^2$$ term is positive, then the transverse axis is vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper20a-h4","type":"hint","dependencies":["a91e303graphhyper20a-h3"],"title":"Horizontal or Vertical","text":"In our case, the transverse axis is horizontal since $$x^2$$ is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper20a-h5","type":"hint","dependencies":["a91e303graphhyper20a-h4"],"title":"Finding the Center","text":"Since the equation is in the form of $$-\\\\left(\\\\frac{{\\\\left(y-k\\\\right)}^2}{a^2}\\\\right)+\\\\frac{{\\\\left(x-h\\\\right)}^2}{b^2}=1$$, we can find our center, which is shown from $$h$$ and k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper20a-h6","type":"hint","dependencies":["a91e303graphhyper20a-h5"],"title":"Finding the Center","text":"To find the center, we look for $$h$$ and k, which in this case is $$h=-1$$ and $$k=-2$$, so our center is located at $$(-1,-2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper20a-h7","type":"hint","dependencies":["a91e303graphhyper20a-h6"],"title":"Vertices","text":"To find the vertices, we find the square root of $$a^2$$ and the square root of $$b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper20a-h8","type":"hint","dependencies":["a91e303graphhyper20a-h7"],"title":"Vertices","text":"In our case, a is $$\\\\pm 3$$, and $$b$$ is $$\\\\pm 2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper20a-h9","type":"hint","dependencies":["a91e303graphhyper20a-h8"],"title":"Asymptotes","text":"Now, we sketch a rectangle around our center using our found a and $$b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper20a-h10","type":"hint","dependencies":["a91e303graphhyper20a-h9"],"title":"Asymptotes","text":"We then sketch two diagonal lines from the corners of the rectanges and through the center, which will outline our asymptotes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper20a-h11","type":"hint","dependencies":["a91e303graphhyper20a-h10"],"title":"Asymptotes","text":"We can then draw the hyperbola with the rectangle and lines as our guides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a91e303graphhyper21","title":"Graphing Hyperbolas with Center (h,k)","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.4 Hyperbolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a91e303graphhyper21a","stepAnswer":["B"],"problemType":"MultipleChoice","stepTitle":"Graph $$\\\\frac{{\\\\left(y+3\\\\right)}^2}{16}-\\\\frac{{\\\\left(x+2\\\\right)}^2}{9}=1$$.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a91e303graphhyper21a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"Write the equation in standard form. $$\\\\frac{y^2}{a^2}-\\\\frac{x^2}{b^2}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper21a-h2","type":"hint","dependencies":["a91e303graphhyper21a-h1"],"title":"Horizontal or Vertical","text":"Determine whether the transverse axis is horizontal or vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper21a-h3","type":"hint","dependencies":["a91e303graphhyper21a-h2"],"title":"Horizontal or Vertical","text":"If the $$x^2$$ is positive, then the transverse axis is horizontal, and if the $$y^2$$ term is positive, then the transverse axis is vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper21a-h4","type":"hint","dependencies":["a91e303graphhyper21a-h3"],"title":"Horizontal or Vertical","text":"In our case, the transverse axis is vertical since $$y^2$$ is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper21a-h5","type":"hint","dependencies":["a91e303graphhyper21a-h4"],"title":"Finding the Center","text":"Since the equation is in the form of $$\\\\frac{{\\\\left(y-k\\\\right)}^2}{a^2}-\\\\frac{{\\\\left(x-h\\\\right)}^2}{b^2}=1$$, we can find our center, which is shown from $$h$$ and k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper21a-h6","type":"hint","dependencies":["a91e303graphhyper21a-h5"],"title":"Finding the Center","text":"To find the center, we look for $$h$$ and k, which in this case is $$h=-2$$ and $$k=-3$$, so our center is located at $$(-2,-3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper21a-h7","type":"hint","dependencies":["a91e303graphhyper21a-h6"],"title":"Vertices","text":"To find the vertices, we find the square root of $$a^2$$ and the square root of $$b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper21a-h8","type":"hint","dependencies":["a91e303graphhyper21a-h7"],"title":"Vertices","text":"In our case, a is $$\\\\pm 4$$, and $$b$$ is $$\\\\pm 3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper21a-h9","type":"hint","dependencies":["a91e303graphhyper21a-h8"],"title":"Asymptotes","text":"Now, we sketch a rectangle around our center using our found a and $$b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper21a-h10","type":"hint","dependencies":["a91e303graphhyper21a-h9"],"title":"Asymptotes","text":"We then sketch two diagonal lines from the corners of the rectanges and through the center, which will outline our asymptotes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper21a-h11","type":"hint","dependencies":["a91e303graphhyper21a-h10"],"title":"Asymptotes","text":"We can then draw the hyperbola with the rectangle and lines as our guides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a91e303graphhyper22","title":"Graphing Hyperbolas with Center (h,k)","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.4 Hyperbolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a91e303graphhyper22a","stepAnswer":["A"],"problemType":"MultipleChoice","stepTitle":"Graph $$\\\\frac{{\\\\left(y+2\\\\right)}^2}{9}-\\\\frac{{\\\\left(x+2\\\\right)}^2}{9}=1$$.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a91e303graphhyper22a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"Write the equation in standard form. $$\\\\frac{y^2}{a^2}-\\\\frac{x^2}{b^2}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper22a-h2","type":"hint","dependencies":["a91e303graphhyper22a-h1"],"title":"Horizontal or Vertical","text":"Determine whether the transverse axis is horizontal or vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper22a-h3","type":"hint","dependencies":["a91e303graphhyper22a-h2"],"title":"Horizontal or Vertical","text":"If the $$x^2$$ is positive, then the transverse axis is horizontal, and if the $$y^2$$ term is positive, then the transverse axis is vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper22a-h4","type":"hint","dependencies":["a91e303graphhyper22a-h3"],"title":"Horizontal or Vertical","text":"In our case, the transverse axis is vertical since $$y^2$$ is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper22a-h5","type":"hint","dependencies":["a91e303graphhyper22a-h4"],"title":"Finding the Center","text":"Since the equation is in the form of $$\\\\frac{{\\\\left(y-k\\\\right)}^2}{a^2}-\\\\frac{{\\\\left(x-h\\\\right)}^2}{b^2}=1$$, we can find our center, which is shown from $$h$$ and k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper22a-h6","type":"hint","dependencies":["a91e303graphhyper22a-h5"],"title":"Finding the Center","text":"To find the center, we look for $$h$$ and k, which in this case is $$h=-2$$ and $$k=-2$$, so our center is located at $$(-2,-2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper22a-h7","type":"hint","dependencies":["a91e303graphhyper22a-h6"],"title":"Vertices","text":"To find the vertices, we find the square root of $$a^2$$ and the square root of $$b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper22a-h8","type":"hint","dependencies":["a91e303graphhyper22a-h7"],"title":"Vertices","text":"In our case, a is $$\\\\pm 3$$, and $$b$$ is $$\\\\pm 3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper22a-h9","type":"hint","dependencies":["a91e303graphhyper22a-h8"],"title":"Asymptotes","text":"Now, we sketch a rectangle around our center using our found a and $$b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper22a-h10","type":"hint","dependencies":["a91e303graphhyper22a-h9"],"title":"Asymptotes","text":"We then sketch two diagonal lines from the corners of the rectanges and through the center, which will outline our asymptotes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper22a-h11","type":"hint","dependencies":["a91e303graphhyper22a-h10"],"title":"Asymptotes","text":"We can then draw the hyperbola with the rectangle and lines as our guides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a91e303graphhyper23","title":"Graphing Hyperbolas with Center (h,k)","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.4 Hyperbolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a91e303graphhyper23a","stepAnswer":["C"],"problemType":"MultipleChoice","stepTitle":"Graph $$\\\\frac{{\\\\left(x-1\\\\right)}^2}{16}-\\\\frac{{\\\\left(y-3\\\\right)}^2}{4}=1$$.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a91e303graphhyper23a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"Write the equation in standard form. $$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper23a-h2","type":"hint","dependencies":["a91e303graphhyper23a-h1"],"title":"Horizontal or Vertical","text":"Determine whether the transverse axis is horizontal or vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper23a-h3","type":"hint","dependencies":["a91e303graphhyper23a-h2"],"title":"Horizontal or Vertical","text":"If the $$x^2$$ is positive, then the transverse axis is horizontal, and if the $$y^2$$ term is positive, then the transverse axis is vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper23a-h4","type":"hint","dependencies":["a91e303graphhyper23a-h3"],"title":"Horizontal or Vertical","text":"In our case, the transverse axis is horizontal since $$x^2$$ is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper23a-h5","type":"hint","dependencies":["a91e303graphhyper23a-h4"],"title":"Finding the Center","text":"Since the equation is in the form of $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}-\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$, we can find our center, which is shown from $$h$$ and k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper23a-h6","type":"hint","dependencies":["a91e303graphhyper23a-h5"],"title":"Finding the Center","text":"To find the center, we look for $$h$$ and k, which in this case is $$h=1$$ and $$k=3$$, so our center is located at $$(1,3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper23a-h7","type":"hint","dependencies":["a91e303graphhyper23a-h6"],"title":"Vertices","text":"To find the vertices, we find the square root of $$a^2$$ and the square root of $$b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper23a-h8","type":"hint","dependencies":["a91e303graphhyper23a-h7"],"title":"Vertices","text":"In our case, a is $$\\\\pm 4$$, and $$b$$ is $$\\\\pm 2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper23a-h9","type":"hint","dependencies":["a91e303graphhyper23a-h8"],"title":"Asymptotes","text":"Now, we sketch a rectangle around our center using our found a and $$b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper23a-h10","type":"hint","dependencies":["a91e303graphhyper23a-h9"],"title":"Asymptotes","text":"We then sketch two diagonal lines from the corners of the rectanges and through the center, which will outline our asymptotes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper23a-h11","type":"hint","dependencies":["a91e303graphhyper23a-h10"],"title":"Asymptotes","text":"We can then draw the hyperbola with the rectangle and lines as our guides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a91e303graphhyper24","title":"Graphing Hyperbolas with Center (h,k)","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.4 Hyperbolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a91e303graphhyper24a","stepAnswer":["D"],"problemType":"MultipleChoice","stepTitle":"Graph $$\\\\frac{{\\\\left(x-2\\\\right)}^2}{4}-\\\\frac{{\\\\left(y-3\\\\right)}^2}{16}=1$$.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a91e303graphhyper24a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"Write the equation in standard form. $$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper24a-h2","type":"hint","dependencies":["a91e303graphhyper24a-h1"],"title":"Horizontal or Vertical","text":"Determine whether the transverse axis is horizontal or vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper24a-h3","type":"hint","dependencies":["a91e303graphhyper24a-h2"],"title":"Horizontal or Vertical","text":"If the $$x^2$$ is positive, then the transverse axis is horizontal, and if the $$y^2$$ term is positive, then the transverse axis is vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper24a-h4","type":"hint","dependencies":["a91e303graphhyper24a-h3"],"title":"Horizontal or Vertical","text":"In our case, the transverse axis is horizontal since $$x^2$$ is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper24a-h5","type":"hint","dependencies":["a91e303graphhyper24a-h4"],"title":"Finding the Center","text":"Since the equation is in the form of $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}-\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$, we can find our center, which is shown from $$h$$ and k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper24a-h6","type":"hint","dependencies":["a91e303graphhyper24a-h5"],"title":"Finding the Center","text":"To find the center, we look for $$h$$ and k, which in this case is $$h=2$$ and $$k=3$$, so our center is located at $$(2,3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper24a-h7","type":"hint","dependencies":["a91e303graphhyper24a-h6"],"title":"Vertices","text":"To find the vertices, we find the square root of $$a^2$$ and the square root of $$b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper24a-h8","type":"hint","dependencies":["a91e303graphhyper24a-h7"],"title":"Vertices","text":"In our case, a is $$\\\\pm 2$$, and $$b$$ is $$\\\\pm 4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper24a-h9","type":"hint","dependencies":["a91e303graphhyper24a-h8"],"title":"Asymptotes","text":"Now, we sketch a rectangle around our center using our found a and $$b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper24a-h10","type":"hint","dependencies":["a91e303graphhyper24a-h9"],"title":"Asymptotes","text":"We then sketch two diagonal lines from the corners of the rectanges and through the center, which will outline our asymptotes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper24a-h11","type":"hint","dependencies":["a91e303graphhyper24a-h10"],"title":"Asymptotes","text":"We can then draw the hyperbola with the rectangle and lines as our guides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a91e303graphhyper25","title":"Graphing Hyperbolas with Center (h,k)","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.4 Hyperbolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a91e303graphhyper25a","stepAnswer":["A"],"problemType":"MultipleChoice","stepTitle":"Graph $$\\\\frac{{\\\\left(y-4\\\\right)}^2}{9}-\\\\frac{{\\\\left(x-2\\\\right)}^2}{25}=1$$.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a91e303graphhyper25a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"Write the equation in standard form. $$\\\\frac{y^2}{a^2}-\\\\frac{x^2}{b^2}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper25a-h2","type":"hint","dependencies":["a91e303graphhyper25a-h1"],"title":"Horizontal or Vertical","text":"Determine whether the transverse axis is horizontal or vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper25a-h3","type":"hint","dependencies":["a91e303graphhyper25a-h2"],"title":"Horizontal or Vertical","text":"If the $$x^2$$ is positive, then the transverse axis is horizontal, and if the $$y^2$$ term is positive, then the transverse axis is vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper25a-h4","type":"hint","dependencies":["a91e303graphhyper25a-h3"],"title":"Horizontal or Vertical","text":"In our case, the transverse axis is vertical since $$y^2$$ is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper25a-h5","type":"hint","dependencies":["a91e303graphhyper25a-h4"],"title":"Finding the Center","text":"Since the equation is in the form of $$\\\\frac{{\\\\left(y-k\\\\right)}^2}{a^2}-\\\\frac{{\\\\left(x-h\\\\right)}^2}{b^2}=1$$, we can find our center, which is shown from $$h$$ and k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper25a-h6","type":"hint","dependencies":["a91e303graphhyper25a-h5"],"title":"Finding the Center","text":"To find the center, we look for $$h$$ and k, which in this case is $$h=2$$ and $$k=4$$, so our center is located at $$(2,4)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper25a-h7","type":"hint","dependencies":["a91e303graphhyper25a-h6"],"title":"Vertices","text":"To find the vertices, we find the square root of $$a^2$$ and the square root of $$b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper25a-h8","type":"hint","dependencies":["a91e303graphhyper25a-h7"],"title":"Vertices","text":"In our case, a is $$\\\\pm 3$$, and $$b$$ is $$\\\\pm 5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper25a-h9","type":"hint","dependencies":["a91e303graphhyper25a-h8"],"title":"Asymptotes","text":"Now, we sketch a rectangle around our center using our found a and $$b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper25a-h10","type":"hint","dependencies":["a91e303graphhyper25a-h9"],"title":"Asymptotes","text":"We then sketch two diagonal lines from the corners of the rectanges and through the center, which will outline our asymptotes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper25a-h11","type":"hint","dependencies":["a91e303graphhyper25a-h10"],"title":"Asymptotes","text":"We can then draw the hyperbola with the rectangle and lines as our guides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a91e303graphhyper26","title":"Graphing Hyperbolas with Center (h,k)","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.4 Hyperbolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a91e303graphhyper26a","stepAnswer":["D"],"problemType":"MultipleChoice","stepTitle":"Graph $$\\\\frac{{\\\\left(y-1\\\\right)}^2}{25}-\\\\frac{{\\\\left(x-4\\\\right)}^2}{16}=1$$.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a91e303graphhyper26a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"Write the equation in standard form. $$\\\\frac{y^2}{a^2}-\\\\frac{x^2}{b^2}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper26a-h2","type":"hint","dependencies":["a91e303graphhyper26a-h1"],"title":"Horizontal or Vertical","text":"Determine whether the transverse axis is horizontal or vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper26a-h3","type":"hint","dependencies":["a91e303graphhyper26a-h2"],"title":"Horizontal or Vertical","text":"If the $$x^2$$ is positive, then the transverse axis is horizontal, and if the $$y^2$$ term is positive, then the transverse axis is vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper26a-h4","type":"hint","dependencies":["a91e303graphhyper26a-h3"],"title":"Horizontal or Vertical","text":"In our case, the transverse axis is vertical since $$y^2$$ is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper26a-h5","type":"hint","dependencies":["a91e303graphhyper26a-h4"],"title":"Finding the Center","text":"Since the equation is in the form of $$\\\\frac{{\\\\left(y-k\\\\right)}^2}{a^2}-\\\\frac{{\\\\left(x-h\\\\right)}^2}{b^2}=1$$, we can find our center, which is shown from $$h$$ and k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper26a-h6","type":"hint","dependencies":["a91e303graphhyper26a-h5"],"title":"Finding the Center","text":"To find the center, we look for $$h$$ and k, which in this case is $$h=4$$ and $$k=1$$, so our center is located at $$(4,1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper26a-h7","type":"hint","dependencies":["a91e303graphhyper26a-h6"],"title":"Vertices","text":"To find the vertices, we find the square root of $$a^2$$ and the square root of $$b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper26a-h8","type":"hint","dependencies":["a91e303graphhyper26a-h7"],"title":"Vertices","text":"In our case, a is $$\\\\pm 5$$, and $$b$$ is $$\\\\pm 4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper26a-h9","type":"hint","dependencies":["a91e303graphhyper26a-h8"],"title":"Asymptotes","text":"Now, we sketch a rectangle around our center using our found a and $$b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper26a-h10","type":"hint","dependencies":["a91e303graphhyper26a-h9"],"title":"Asymptotes","text":"We then sketch two diagonal lines from the corners of the rectanges and through the center, which will outline our asymptotes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper26a-h11","type":"hint","dependencies":["a91e303graphhyper26a-h10"],"title":"Asymptotes","text":"We can then draw the hyperbola with the rectangle and lines as our guides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a91e303graphhyper27","title":"Graphing Hyperbolas with Center (h,k)","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.4 Hyperbolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a91e303graphhyper27a","stepAnswer":["B"],"problemType":"MultipleChoice","stepTitle":"Graph $$\\\\frac{{\\\\left(y+4\\\\right)}^2}{25}-\\\\frac{{\\\\left(x+1\\\\right)}^2}{36}=1$$.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a91e303graphhyper27a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"Write the equation in standard form. $$\\\\frac{y^2}{a^2}-\\\\frac{x^2}{b^2}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper27a-h2","type":"hint","dependencies":["a91e303graphhyper27a-h1"],"title":"Horizontal or Vertical","text":"Determine whether the transverse axis is horizontal or vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper27a-h3","type":"hint","dependencies":["a91e303graphhyper27a-h2"],"title":"Horizontal or Vertical","text":"If the $$x^2$$ is positive, then the transverse axis is horizontal, and if the $$y^2$$ term is positive, then the transverse axis is vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper27a-h4","type":"hint","dependencies":["a91e303graphhyper27a-h3"],"title":"Horizontal or Vertical","text":"In our case, the transverse axis is vertical since $$y^2$$ is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper27a-h5","type":"hint","dependencies":["a91e303graphhyper27a-h4"],"title":"Finding the Center","text":"Since the equation is in the form of $$\\\\frac{{\\\\left(y-k\\\\right)}^2}{a^2}-\\\\frac{{\\\\left(x-h\\\\right)}^2}{b^2}=1$$, we can find our center, which is shown from $$h$$ and k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper27a-h6","type":"hint","dependencies":["a91e303graphhyper27a-h5"],"title":"Finding the Center","text":"To find the center, we look for $$h$$ and k, which in this case is $$h=-1$$ and $$k=-4$$, so our center is located at $$(-1,-4)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper27a-h7","type":"hint","dependencies":["a91e303graphhyper27a-h6"],"title":"Vertices","text":"To find the vertices, we find the square root of $$a^2$$ and the square root of $$b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper27a-h8","type":"hint","dependencies":["a91e303graphhyper27a-h7"],"title":"Vertices","text":"In our case, a is $$\\\\pm 5$$, and $$b$$ is $$\\\\pm 6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper27a-h9","type":"hint","dependencies":["a91e303graphhyper27a-h8"],"title":"Asymptotes","text":"Now, we sketch a rectangle around our center using our found a and $$b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper27a-h10","type":"hint","dependencies":["a91e303graphhyper27a-h9"],"title":"Asymptotes","text":"We then sketch two diagonal lines from the corners of the rectanges and through the center, which will outline our asymptotes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper27a-h11","type":"hint","dependencies":["a91e303graphhyper27a-h10"],"title":"Asymptotes","text":"We can then draw the hyperbola with the rectangle and lines as our guides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a91e303graphhyper28","title":"Graphing Hyperbolas with Center (h,k)","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.4 Hyperbolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a91e303graphhyper28a","stepAnswer":["C"],"problemType":"MultipleChoice","stepTitle":"Graph $$\\\\frac{{\\\\left(y+1\\\\right)}^2}{16}-\\\\frac{{\\\\left(x+1\\\\right)}^2}{4}=1$$.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a91e303graphhyper28a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"Write the equation in standard form. $$\\\\frac{y^2}{a^2}-\\\\frac{x^2}{b^2}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper28a-h2","type":"hint","dependencies":["a91e303graphhyper28a-h1"],"title":"Horizontal or Vertical","text":"Determine whether the transverse axis is horizontal or vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper28a-h3","type":"hint","dependencies":["a91e303graphhyper28a-h2"],"title":"Horizontal or Vertical","text":"If the $$x^2$$ is positive, then the transverse axis is horizontal, and if the $$y^2$$ term is positive, then the transverse axis is vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper28a-h4","type":"hint","dependencies":["a91e303graphhyper28a-h3"],"title":"Horizontal or Vertical","text":"In our case, the transverse axis is vertical since $$y^2$$ is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper28a-h5","type":"hint","dependencies":["a91e303graphhyper28a-h4"],"title":"Finding the Center","text":"Since the equation is in the form of $$\\\\frac{{\\\\left(y-k\\\\right)}^2}{a^2}-\\\\frac{{\\\\left(x-h\\\\right)}^2}{b^2}=1$$, we can find our center, which is shown from $$h$$ and k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper28a-h6","type":"hint","dependencies":["a91e303graphhyper28a-h5"],"title":"Finding the Center","text":"To find the center, we look for $$h$$ and k, which in this case is $$h=-1$$ and $$k=-1$$, so our center is located at $$(-1,-1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper28a-h7","type":"hint","dependencies":["a91e303graphhyper28a-h6"],"title":"Vertices","text":"To find the vertices, we find the square root of $$a^2$$ and the square root of $$b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper28a-h8","type":"hint","dependencies":["a91e303graphhyper28a-h7"],"title":"Vertices","text":"In our case, a is $$\\\\pm 4$$, and $$b$$ is $$\\\\pm 2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper28a-h9","type":"hint","dependencies":["a91e303graphhyper28a-h8"],"title":"Asymptotes","text":"Now, we sketch a rectangle around our center using our found a and $$b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper28a-h10","type":"hint","dependencies":["a91e303graphhyper28a-h9"],"title":"Asymptotes","text":"We then sketch two diagonal lines from the corners of the rectanges and through the center, which will outline our asymptotes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper28a-h11","type":"hint","dependencies":["a91e303graphhyper28a-h10"],"title":"Asymptotes","text":"We can then draw the hyperbola with the rectangle and lines as our guides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a91e303graphhyper29","title":"Writing the Hyperbola equation in standard form.","body":"Standardize the given equation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.4 Hyperbolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a91e303graphhyper29a","stepAnswer":["$$\\\\frac{{\\\\left(x-3\\\\right)}^2}{9}-\\\\frac{{\\\\left(y+2\\\\right)}^2}{4}=1$$"],"problemType":"TextBox","stepTitle":"Standardize $$4x^2-9y^2-24x-36y-36=0$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{{\\\\left(x-3\\\\right)}^2}{9}-\\\\frac{{\\\\left(y+2\\\\right)}^2}{4}=1$$","hints":{"DefaultPathway":[{"id":"a91e303graphhyper29a-h1","type":"hint","dependencies":[],"title":"Simplifying","text":"First, add the constant on both sides, to resemble the standard form. In this case, we add $$36$$ to both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper29a-h2","type":"hint","dependencies":["a91e303graphhyper29a-h1"],"title":"Factor by Grouping","text":"If possible, we then group the different variables, which in this case are $$x$$ and $$y$$ and factor their constants. In this case, we factor $$4$$ from the x\'s and $$-9$$ from the y\'s.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper29a-h3","type":"hint","dependencies":["a91e303graphhyper29a-h2"],"title":"Completing the Square","text":"We then complete the square for both the variables, $$x$$ and $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper29a-h4","type":"hint","dependencies":["a91e303graphhyper29a-h3"],"title":"Completing the Square","text":"After completing the square, we have the equation, $$4{\\\\left(x-3\\\\right)}^2-9{\\\\left(y+2\\\\right)}^2=36$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper29a-h5","type":"hint","dependencies":["a91e303graphhyper29a-h4"],"title":"Simplification","text":"To match it with the standard form, we then divide both sides to get $$1$$, so in this case, we divide both sides by $$36$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a91e303graphhyper3","title":"Graphing Hyperbolas with Center $$(0,0)$$","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.4 Hyperbolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a91e303graphhyper3a","stepAnswer":["A"],"problemType":"MultipleChoice","stepTitle":"Graph $$\\\\frac{x^2}{9}-\\\\frac{y^2}{16}=1$$.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a91e303graphhyper3a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"Write the equation in standard form. $$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper3a-h2","type":"hint","dependencies":["a91e303graphhyper3a-h1"],"title":"Horizontal or Vertical","text":"Determine whether the transverse axis is horizontal or vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper3a-h3","type":"hint","dependencies":["a91e303graphhyper3a-h2"],"title":"Horizontal or Vertical","text":"If the $$x^2$$ is positive, then the transverse axis is horizontal, and if the $$y^2$$ term is positive, then the transverse axis is vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper3a-h4","type":"hint","dependencies":["a91e303graphhyper3a-h3"],"title":"Horizontal or Vertical","text":"In our case, the transverse axis is horizontal since $$x^2$$ is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper3a-h5","type":"hint","dependencies":["a91e303graphhyper3a-h4"],"title":"Vertices","text":"Since the transverse axis is horizontal, we want to find the vertices on the $$x$$ axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper3a-h6","type":"hint","dependencies":["a91e303graphhyper3a-h5"],"title":"Vertices","text":"To find the vertices, we find the square root of $$a^2$$ and the square root of $$b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper3a-h7","type":"hint","dependencies":["a91e303graphhyper3a-h6"],"title":"Vertices","text":"In our case, a is $$\\\\pm 3$$, and $$b$$ is $$\\\\pm 4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper3a-h8","type":"hint","dependencies":["a91e303graphhyper3a-h7"],"title":"Asymptotes","text":"Now, we must find the equations of our asymptotes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper3a-h9","type":"hint","dependencies":["a91e303graphhyper3a-h8"],"title":"Asymptotes","text":"Since our transverse axis is horizontal, we will use the equations, $$y=\\\\frac{b}{a} x$$ and $$y$$ $$=$$ $$-\\\\left(\\\\frac{b}{a}\\\\right) x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper3a-h10","type":"hint","dependencies":["a91e303graphhyper3a-h9"],"title":"Asymptotes","text":"We then sketch our equations and draw the the different branches of the hyperbola with the asymptotes as guides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a91e303graphhyper30","title":"Writing the Hyperbola equation in standard form.","body":"Standardize the given equation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.4 Hyperbolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a91e303graphhyper30a","stepAnswer":["$$\\\\frac{{\\\\left(x+1\\\\right)}^2}{16}-\\\\frac{{\\\\left(y-2\\\\right)}^2}{9}=1$$"],"problemType":"TextBox","stepTitle":"Standardize $$9x^2-16y^2+18x+64y-199=0$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{{\\\\left(x+1\\\\right)}^2}{16}-\\\\frac{{\\\\left(y-2\\\\right)}^2}{9}=1$$","hints":{"DefaultPathway":[{"id":"a91e303graphhyper30a-h1","type":"hint","dependencies":[],"title":"Simplifying","text":"First, add the constant on both sides, to resemble the standard form. In this case, we add $$199$$ to both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper30a-h2","type":"hint","dependencies":["a91e303graphhyper30a-h1"],"title":"Factor by Grouping","text":"If possible, we then group the different variables, which in this case are $$x$$ and $$y$$ and factor their constants. In this case, we factor $$9$$ from the x\'s and $$-16$$ from the y\'s.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper30a-h3","type":"hint","dependencies":["a91e303graphhyper30a-h2"],"title":"Completing the Square","text":"We then complete the square for both the variables, $$x$$ and $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper30a-h4","type":"hint","dependencies":["a91e303graphhyper30a-h3"],"title":"Completing the Square","text":"After completing the square, we have the equation, $$9{\\\\left(x+1\\\\right)}^2-16{\\\\left(y-2\\\\right)}^2=144$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper30a-h5","type":"hint","dependencies":["a91e303graphhyper30a-h4"],"title":"Simplification","text":"To match it with the standard form, we then divide both sides to get $$1$$, so in this case, we divide both sides by $$36$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a91e303graphhyper4","title":"Graphing Hyperbolas with Center $$(0,0)$$","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.4 Hyperbolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a91e303graphhyper4a","stepAnswer":["B"],"problemType":"MultipleChoice","stepTitle":"Graph $$\\\\frac{x^2}{9}-\\\\frac{y^2}{4}=1$$.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a91e303graphhyper4a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"Write the equation in standard form. $$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper4a-h2","type":"hint","dependencies":["a91e303graphhyper4a-h1"],"title":"Horizontal or Vertical","text":"Determine whether the transverse axis is horizontal or vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper4a-h3","type":"hint","dependencies":["a91e303graphhyper4a-h2"],"title":"Horizontal or Vertical","text":"If the $$x^2$$ is positive, then the transverse axis is horizontal, and if the $$y^2$$ term is positive, then the transverse axis is vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper4a-h4","type":"hint","dependencies":["a91e303graphhyper4a-h3"],"title":"Horizontal or Vertical","text":"In our case, the transverse axis is horizontal since $$x^2$$ is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper4a-h5","type":"hint","dependencies":["a91e303graphhyper4a-h4"],"title":"Vertices","text":"Since the transverse axis is horizontal, we want to find the vertices on the $$x$$ axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper4a-h6","type":"hint","dependencies":["a91e303graphhyper4a-h5"],"title":"Vertices","text":"To find the vertices, we find the square root of $$a^2$$ and the square root of $$b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper4a-h7","type":"hint","dependencies":["a91e303graphhyper4a-h6"],"title":"Vertices","text":"In our case, a is $$\\\\pm 3$$, and $$b$$ is $$\\\\pm 2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper4a-h8","type":"hint","dependencies":["a91e303graphhyper4a-h7"],"title":"Asymptotes","text":"Now, we must find the equations of our asymptotes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper4a-h9","type":"hint","dependencies":["a91e303graphhyper4a-h8"],"title":"Asymptotes","text":"Since our transverse axis is horizontal, we will use the equations, $$y=\\\\frac{b}{a} x$$ and $$y$$ $$=$$ $$-\\\\left(\\\\frac{b}{a}\\\\right) x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper4a-h10","type":"hint","dependencies":["a91e303graphhyper4a-h9"],"title":"Asymptotes","text":"We then sketch our equations and draw the the different branches of the hyperbola with the asymptotes as guides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a91e303graphhyper5","title":"Graphing Hyperbolas with Center $$(0,0)$$","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.4 Hyperbolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a91e303graphhyper5a","stepAnswer":["B"],"problemType":"MultipleChoice","stepTitle":"Graph $$\\\\frac{x^2}{25}-\\\\frac{y^2}{9}=1$$.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a91e303graphhyper5a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"Write the equation in standard form. $$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper5a-h2","type":"hint","dependencies":["a91e303graphhyper5a-h1"],"title":"Horizontal or Vertical","text":"Determine whether the transverse axis is horizontal or vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper5a-h3","type":"hint","dependencies":["a91e303graphhyper5a-h2"],"title":"Horizontal or Vertical","text":"If the $$x^2$$ is positive, then the transverse axis is horizontal, and if the $$y^2$$ term is positive, then the transverse axis is vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper5a-h4","type":"hint","dependencies":["a91e303graphhyper5a-h3"],"title":"Horizontal or Vertical","text":"In our case, the transverse axis is horizontal since $$x^2$$ is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper5a-h5","type":"hint","dependencies":["a91e303graphhyper5a-h4"],"title":"Vertices","text":"Since the transverse axis is horizontal, we want to find the vertices on the $$x$$ axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper5a-h6","type":"hint","dependencies":["a91e303graphhyper5a-h5"],"title":"Vertices","text":"To find the vertices, we find the square root of $$a^2$$ and the square root of $$b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper5a-h7","type":"hint","dependencies":["a91e303graphhyper5a-h6"],"title":"Vertices","text":"In our case, a is $$\\\\pm 5$$, and $$b$$ is $$\\\\pm 3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper5a-h8","type":"hint","dependencies":["a91e303graphhyper5a-h7"],"title":"Asymptotes","text":"Now, we must find the equations of our asymptotes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper5a-h9","type":"hint","dependencies":["a91e303graphhyper5a-h8"],"title":"Asymptotes","text":"Since our transverse axis is horizontal, we will use the equations, $$y=\\\\frac{b}{a} x$$ and $$y$$ $$=$$ $$-\\\\left(\\\\frac{b}{a}\\\\right) x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper5a-h10","type":"hint","dependencies":["a91e303graphhyper5a-h9"],"title":"Asymptotes","text":"We then sketch our equations and draw the the different branches of the hyperbola with the asymptotes as guides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a91e303graphhyper6","title":"Graphing Hyperbolas with Center $$(0,0)$$","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.4 Hyperbolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a91e303graphhyper6a","stepAnswer":["A"],"problemType":"MultipleChoice","stepTitle":"Graph $$\\\\frac{x^2}{16}-\\\\frac{y^2}{25}=1$$.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a91e303graphhyper6a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"Write the equation in standard form. $$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper6a-h2","type":"hint","dependencies":["a91e303graphhyper6a-h1"],"title":"Horizontal or Vertical","text":"Determine whether the transverse axis is horizontal or vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper6a-h3","type":"hint","dependencies":["a91e303graphhyper6a-h2"],"title":"Horizontal or Vertical","text":"If the $$x^2$$ is positive, then the transverse axis is horizontal, and if the $$y^2$$ term is positive, then the transverse axis is vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper6a-h4","type":"hint","dependencies":["a91e303graphhyper6a-h3"],"title":"Horizontal or Vertical","text":"In our case, the transverse axis is horizontal since $$x^2$$ is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper6a-h5","type":"hint","dependencies":["a91e303graphhyper6a-h4"],"title":"Vertices","text":"Since the transverse axis is horizontal, we want to find the vertices on the $$x$$ axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper6a-h6","type":"hint","dependencies":["a91e303graphhyper6a-h5"],"title":"Vertices","text":"To find the vertices, we find the square root of $$a^2$$ and the square root of $$b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper6a-h7","type":"hint","dependencies":["a91e303graphhyper6a-h6"],"title":"Vertices","text":"In our case, a is $$\\\\pm 4$$, and $$b$$ is $$\\\\pm 5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper6a-h8","type":"hint","dependencies":["a91e303graphhyper6a-h7"],"title":"Asymptotes","text":"Now, we must find the equations of our asymptotes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper6a-h9","type":"hint","dependencies":["a91e303graphhyper6a-h8"],"title":"Asymptotes","text":"Since our transverse axis is horizontal, we will use the equations, $$y=\\\\frac{b}{a} x$$ and $$y$$ $$=$$ $$-\\\\left(\\\\frac{b}{a}\\\\right) x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper6a-h10","type":"hint","dependencies":["a91e303graphhyper6a-h9"],"title":"Asymptotes","text":"We then sketch our equations and draw the the different branches of the hyperbola with the asymptotes as guides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a91e303graphhyper7","title":"Graphing Hyperbolas with Center $$(0,0)$$","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.4 Hyperbolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a91e303graphhyper7a","stepAnswer":["C"],"problemType":"MultipleChoice","stepTitle":"Graph $$\\\\frac{y^2}{25}-\\\\frac{x^2}{4}=1$$.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a91e303graphhyper7a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"Write the equation in standard form. $$\\\\frac{y^2}{a^2}-\\\\frac{x^2}{b^2}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper7a-h2","type":"hint","dependencies":["a91e303graphhyper7a-h1"],"title":"Horizontal or Vertical","text":"Determine whether the transverse axis is horizontal or vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper7a-h3","type":"hint","dependencies":["a91e303graphhyper7a-h2"],"title":"Horizontal or Vertical","text":"If the $$x^2$$ is positive, then the transverse axis is horizontal, and if the $$y^2$$ term is positive, then the transverse axis is vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper7a-h4","type":"hint","dependencies":["a91e303graphhyper7a-h3"],"title":"Horizontal or Vertical","text":"In our case, the transverse axis is vertical since $$y^2$$ is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper7a-h5","type":"hint","dependencies":["a91e303graphhyper7a-h4"],"title":"Vertices","text":"Since the transverse axis is vertical, we want to find the vertices on the $$y$$ axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper7a-h6","type":"hint","dependencies":["a91e303graphhyper7a-h5"],"title":"Vertices","text":"To find the vertices, we find the square root of $$a^2$$ and the square root of $$b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper7a-h7","type":"hint","dependencies":["a91e303graphhyper7a-h6"],"title":"Vertices","text":"In our case, a is $$\\\\pm 5$$, and $$b$$ is $$\\\\pm 2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper7a-h8","type":"hint","dependencies":["a91e303graphhyper7a-h7"],"title":"Asymptotes","text":"Now, we must find the equations of our asymptotes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper7a-h9","type":"hint","dependencies":["a91e303graphhyper7a-h8"],"title":"Asymptotes","text":"Since our transverse axis is vertical, we will use the equations, $$y=\\\\frac{a}{b} x$$ and $$y$$ $$=$$ $$-\\\\left(\\\\frac{a}{b}\\\\right) x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper7a-h10","type":"hint","dependencies":["a91e303graphhyper7a-h9"],"title":"Asymptotes","text":"We then sketch our equations and draw the the different branches of the hyperbola with the asymptotes as guides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a91e303graphhyper8","title":"Graphing Hyperbolas with Center $$(0,0)$$","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.4 Hyperbolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a91e303graphhyper8a","stepAnswer":["D"],"problemType":"MultipleChoice","stepTitle":"Graph $$\\\\frac{y^2}{36}-\\\\frac{x^2}{16}=1$$.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a91e303graphhyper8a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"Write the equation in standard form. $$\\\\frac{y^2}{a^2}-\\\\frac{x^2}{b^2}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper8a-h2","type":"hint","dependencies":["a91e303graphhyper8a-h1"],"title":"Horizontal or Vertical","text":"Determine whether the transverse axis is horizontal or vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper8a-h3","type":"hint","dependencies":["a91e303graphhyper8a-h2"],"title":"Horizontal or Vertical","text":"If the $$x^2$$ is positive, then the transverse axis is horizontal, and if the $$y^2$$ term is positive, then the transverse axis is vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper8a-h4","type":"hint","dependencies":["a91e303graphhyper8a-h3"],"title":"Horizontal or Vertical","text":"In our case, the transverse axis is vertical since $$y^2$$ is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper8a-h5","type":"hint","dependencies":["a91e303graphhyper8a-h4"],"title":"Vertices","text":"Since the transverse axis is vertical, we want to find the vertices on the $$y$$ axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper8a-h6","type":"hint","dependencies":["a91e303graphhyper8a-h5"],"title":"Vertices","text":"To find the vertices, we find the square root of $$a^2$$ and the square root of $$b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper8a-h7","type":"hint","dependencies":["a91e303graphhyper8a-h6"],"title":"Vertices","text":"In our case, a is $$\\\\pm 6$$, and $$b$$ is $$\\\\pm 4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper8a-h8","type":"hint","dependencies":["a91e303graphhyper8a-h7"],"title":"Asymptotes","text":"Now, we must find the equations of our asymptotes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper8a-h9","type":"hint","dependencies":["a91e303graphhyper8a-h8"],"title":"Asymptotes","text":"Since our transverse axis is vertical, we will use the equations, $$y=\\\\frac{a}{b} x$$ and $$y$$ $$=$$ $$-\\\\left(\\\\frac{a}{b}\\\\right) x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper8a-h10","type":"hint","dependencies":["a91e303graphhyper8a-h9"],"title":"Asymptotes","text":"We then sketch our equations and draw the the different branches of the hyperbola with the asymptotes as guides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a91e303graphhyper9","title":"Graphing Hyperbolas with Center $$(0,0)$$","body":"Select the correct graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.4 Hyperbolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a91e303graphhyper9a","stepAnswer":["B"],"problemType":"MultipleChoice","stepTitle":"Graph $$4y^2-16x^2=64$$.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"a91e303graphhyper9a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"Write the equation in standard form. $$\\\\frac{y^2}{a^2}-\\\\frac{x^2}{b^2}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper9a-h2","type":"hint","dependencies":["a91e303graphhyper9a-h1"],"title":"Standard Form","text":"First, we try to get the right side of the equation to equal to $$1$$. To do this, we divide both sides by $$64$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper9a-h3","type":"hint","dependencies":["a91e303graphhyper9a-h2"],"title":"Standard Form","text":"Next, we simplify the equation and get $$\\\\frac{y^2}{16}-\\\\frac{x^2}{4}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper9a-h4","type":"hint","dependencies":["a91e303graphhyper9a-h3"],"title":"Horizontal or Vertical","text":"Determine whether the transverse axis is horizontal or vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper9a-h5","type":"hint","dependencies":["a91e303graphhyper9a-h4"],"title":"Horizontal or Vertical","text":"If the $$x^2$$ is positive, then the transverse axis is horizontal, and if the $$y^2$$ term is positive, then the transverse axis is vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper9a-h6","type":"hint","dependencies":["a91e303graphhyper9a-h5"],"title":"Horizontal or Vertical","text":"In our case, the transverse axis is vertical since $$y^2$$ is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper9a-h7","type":"hint","dependencies":["a91e303graphhyper9a-h6"],"title":"Vertices","text":"Since the transverse axis is vertical, we want to find the vertices on the $$y$$ axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper9a-h8","type":"hint","dependencies":["a91e303graphhyper9a-h7"],"title":"Vertices","text":"To find the vertices, we find the square root of $$a^2$$ and the square root of $$b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper9a-h9","type":"hint","dependencies":["a91e303graphhyper9a-h8"],"title":"Vertices","text":"In our case, a is $$\\\\pm 4$$, and $$b$$ is $$\\\\pm 2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper9a-h10","type":"hint","dependencies":["a91e303graphhyper9a-h9"],"title":"Asymptotes","text":"Now, we must find the equations of our asymptotes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper9a-h11","type":"hint","dependencies":["a91e303graphhyper9a-h10"],"title":"Asymptotes","text":"Since our transverse axis is vertical, we will use the equations, $$y=\\\\frac{a}{b} x$$ and $$y$$ $$=$$ $$-\\\\left(\\\\frac{a}{b}\\\\right) x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a91e303graphhyper9a-h12","type":"hint","dependencies":["a91e303graphhyper9a-h11"],"title":"Asymptotes","text":"We then sketch our equations and draw the the different branches of the hyperbola with the asymptotes as guides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a94b3d5means1","title":"Two Population Means","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Two Population Means with Known Standard Deviations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a94b3d5means1a","stepAnswer":["Insufficient Evidence"],"problemType":"MultipleChoice","stepTitle":"The mean lasting time of two competing floor waxes is to be compared. Twenty floors are randomly assigned to test each wax. Both populations have a normal distributions. The data are recorded in the table. Does the data indicate that wax $$1$$ is more effective than wax 2? Test at a 5% level of significance.","stepBody":"","answerType":"string","variabilization":{},"choices":["True $$-$$ Wax $$1$$ is more effective","False $$-$$ Wax $$2$$ is more effective","Insufficient Evidence"],"hints":{"DefaultPathway":[{"id":"a94b3d5means1a-h1","type":"hint","dependencies":[],"title":"Breaking down the question","text":"The words \\"is more effective\\" says that wax $$1$$ lasts longer than wax $$2$$, on average. \\"Longer\\" is a \u201c>\u201d symbol and it means Alternate Hypothesis is considered. Therefore it\'s a right tailed test.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a94b3d5means2","title":"Two Population Means","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Two Population Means with Known Standard Deviations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a94b3d5means2a","stepAnswer":["Insufficient Evidence"],"problemType":"MultipleChoice","stepTitle":"An interested citizen wanted to know if Democratic U. S. senators are older than Republican U.S. senators, on average. On May $$26$$ $$2013$$, the mean age of $$30$$ randomly selected Republican Senators was $$61$$ years $$247$$ days old $$(61.675$$ years) with a standard deviation of $$10.17$$ years. The mean age of $$30$$ randomly selected Democratic senators was $$61$$ years $$257$$ days old $$(61.704$$ years) with a standard deviation of $$9.55$$ years. Do the data indicate that Democratic senators are older than Republican senators, on average? Test at a 5% level of significance.","stepBody":"","answerType":"string","variabilization":{},"choices":["True $$-$$ Democratic Senators are older","False $$-$$ Republican Senators are older","Insufficient Evidence"],"hints":{"DefaultPathway":[{"id":"a94b3d5means2a-h1","type":"hint","dependencies":[],"title":"Breaking down the question","text":"The words \\"older than\\" translates as a \u201c>\u201d symbol and it means alternate hypothesis is considered. Therefore, this is a right-tailed test.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a94b3d5means3","title":"Two Population Means","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Two Population Means with Known Standard Deviations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a94b3d5means3a","stepAnswer":["Difference in mean speeds of the pitches of the two pitchers"],"problemType":"MultipleChoice","stepTitle":"The mean speeds of fastball pitches from two different baseball pitchers are to be compared. A sample of $$14$$ fastball pitches is measured from each pitcher. The populations have normal distributions. Table shows the result. Scouters believe that Rodriguez pitches a speedier fastball. What is the random variable?","stepBody":"","answerType":"string","variabilization":{},"choices":["Difference in mean speeds of the pitches of the two pitchers","Mean speeds of each of the two pitchers","Highest mean speed of both the pitchers"],"hints":{"DefaultPathway":[{"id":"a94b3d5means3a-h1","type":"hint","dependencies":[],"title":"Random Variable Definition","text":"Random Variable: Type of variable whose possible values depend on the outcomes of a certain random phenomenon","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a94b3d5means4","title":"Two Population Means","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Two Population Means with Known Standard Deviations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a94b3d5means4a","stepAnswer":["Reject the Null Hypothesis"],"problemType":"MultipleChoice","stepTitle":"The mean speeds of fastball pitches from two different baseball pitchers are to be compared. A sample of $$14$$ fastball pitches is measured from each pitcher. The populations have normal distributions. Table shows the result. Scouters believe that Rodriguez pitches a speedier fastball. At the 1% significance level, what is your conclusion?","stepBody":"","answerType":"string","variabilization":{},"choices":["Reject the Alternate Hypothesis","Reject the Null Hypothesis","Insufficient Evidence"],"hints":{"DefaultPathway":[{"id":"a94b3d5means4a-h1","type":"hint","dependencies":[],"title":"Definition of Null Hypothesis","text":"It is a statement of no difference between the variables\u2014they are not related.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a94b3d5means4a-h2","type":"hint","dependencies":["a94b3d5means4a-h1"],"title":"Breaking down the question","text":"There is sufficient data to conclude that the mean speed of Rodriguez\u2019s fastball is faster than Wesley\u2019s.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a94b3d5means5","title":"Two Population Means","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Two Population Means with Known Standard Deviations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a94b3d5means5a","stepAnswer":["null: \u03bc1 $$ \\\\leq $$ \u03bc2 alternate: \u03bc1 > \u03bc2"],"problemType":"MultipleChoice","stepTitle":"A researcher is testing the effects of plant food on plant growth. Nine plants have been given the plant food. Another nine plants have not been given the plant food. The heights of the plants are recorded after eight weeks. The populations have normal distributions. The following table is the result. The researcher thinks the food makes the plants grow taller. State the null and alternate hypotheses.","stepBody":"Let subscripts $$1$$ $$=$$ Food, $$2$$ $$=$$ No Food","answerType":"string","variabilization":{},"answerLatex":"null: \u03bc1 $$ \\\\leq $$ \u03bc2 alternate: \u03bc1 > \u03bc2","choices":["null: \u03bc1 $$ \\\\leq $$ \u03bc2 alternate: \u03bc1 > \u03bc2","null: \u03bc1 $$ \\\\leq $$ \u03bc2 alternate: \u03bc1 < \u03bc2","null: \u03bc1 > \u03bc2 alternate: \u03bc1 > \u03bc2","null: \u03bc1 < \u03bc2 alternate: \u03bc1 $$ \\\\leq $$ \u03bc2"],"hints":{"DefaultPathway":[{"id":"a94b3d5means5a-h1","type":"hint","dependencies":[],"title":"Definition of Null and Alternate Hypothesis","text":"Null: It is a statement of no difference between the variables\u2014they are not related; Alternate: It is a claim about the population that is contradictory to H0 and what we conclude when we reject H0.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a94b3d5means6","title":"Two Population Means","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Two Population Means with Known Standard Deviations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a94b3d5means6a","stepAnswer":["~ $$0.020$$"],"problemType":"MultipleChoice","stepTitle":"A researcher is testing the effects of plant food on plant growth. Nine plants have been given the plant food. Another nine plants have not been given the plant food. The heights of the plants are recorded after eight weeks. The populations have normal distributions. The following table is the result. The researcher thinks the food makes the plants grow taller. What is the $$p-value$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"~ $$0.020$$","choices":["~ $$0.020$$","~ $$0.025$$","~ $$0.050$$","~ $$0.030$$"],"hints":{"DefaultPathway":[{"id":"a94b3d5means6a-h1","type":"hint","dependencies":[],"title":"$$p-value$$ formula","text":"(p1-p0)/(\u221ap0(1-p0)/n) where p1 is sample proportion and p0 is assumed population proportion in null hypothesis","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a94b3d5means7","title":"Two Population Means","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Two Population Means with Known Standard Deviations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a94b3d5means7a","stepAnswer":["null: \u03bc1 $$=$$ \u03bc2 alternate: \u03bc1 $$ \\\\neq $$ \u03bc2"],"problemType":"MultipleChoice","stepTitle":"Two metal alloys are being considered as material for ball bearings. The mean melting point of the two alloys is to be compared. $$15$$ pieces of each metal are being tested. Both populations have normal distributions. The following table is the result. It is believed that Alloy Zeta has a different melting point. What is the null and alternate hypotheses?","stepBody":"Let subscripts $$1$$ $$=$$ Gamma, $$2$$ $$=$$ Zeta","answerType":"string","variabilization":{},"answerLatex":"null: \u03bc1 $$=$$ \u03bc2 alternate: \u03bc1 $$ \\\\neq $$ \u03bc2","choices":["null: \u03bc1 $$=$$ \u03bc2 alternate: \u03bc1 $$ \\\\neq $$ \u03bc2","null: \u03bc1 $$=$$ \u03bc2 alternate: \u03bc1 $$=$$ \u03bc2","null: \u03bc1 $$ \\\\neq $$ \u03bc2 alternate: \u03bc1 $$=$$ \u03bc2","null: \u03bc1 $$ \\\\neq $$ \u03bc2 alternate: \u03bc1 $$ \\\\neq $$ \u03bc2"],"hints":{"DefaultPathway":[{"id":"a94b3d5means7a-h1","type":"hint","dependencies":[],"title":"Definition of Null and Alternate Hypothesis","text":"Null: It is a statement of no difference between the variables\u2014they are not related; Alternate: It is a claim about the population that is contradictory to H0 and what we conclude when we reject H0.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a94b3d5means8","title":"Two Population Means","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Two Population Means with Known Standard Deviations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a94b3d5means8a","stepAnswer":["~ $$0.006$$"],"problemType":"MultipleChoice","stepTitle":"Two metal alloys are being considered as material for ball bearings. The mean melting point of the two alloys is to be compared. $$15$$ pieces of each metal are being tested. Both populations have normal distributions. The following table is the result. It is believed that Alloy Zeta has a different melting point. What is the $$p-value$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"~ $$0.006$$","choices":["~ $$0.006$$","~ $$0.004$$","~ $$0.009$$","~ $$0.01$$"],"hints":{"DefaultPathway":[{"id":"a94b3d5means8a-h1","type":"hint","dependencies":[],"title":"$$p-value$$ formula","text":"(p1-p0)/(\u221ap0(1-p0)/n) where p1 is sample proportion and p0 is assumed population proportion in null hypothesis","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a94b3d5means9","title":"Two Population Means","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Two Population Means with Known Standard Deviations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a94b3d5means9a","stepAnswer":["Reject the Null Hypothesis"],"problemType":"MultipleChoice","stepTitle":"Two metal alloys are being considered as material for ball bearings. The mean melting point of the two alloys is to be compared. $$15$$ pieces of each metal are being tested. Both populations have normal distributions. The following table is the result. It is believed that Alloy Zeta has a different melting point. At the 1% significance level, what is your conclusion?","stepBody":"","answerType":"string","variabilization":{},"choices":["Reject the Alternate Hypothesis","Reject the Null Hypothesis","Insufficient Evidence"],"hints":{"DefaultPathway":[{"id":"a94b3d5means9a-h1","type":"hint","dependencies":[],"title":"Breaking down the question","text":"The data support that the melting point for Alloy Zeta is different from the melting point of Alloy Gamma.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a961302definite1","title":"Evaluating an Integral Using the Definition","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.2 The Definite Integral","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a961302definite1a","stepAnswer":["$$\\\\frac{8}{3}$$"],"problemType":"TextBox","stepTitle":"Use the definition of the definite integral to evaluate $$\\\\int_{0}^{2} x^2 \\\\,dx$$. Use a right-endpoint approximation to generate the Riemann sum.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{8}{3}$$","hints":{"DefaultPathway":[{"id":"a961302definite1a-h1","type":"hint","dependencies":[],"title":"Set Up a Riemann Sum","text":"Based on the limits of integration, we have $$a=0$$ and $$b=2$$. For $$i=0$$, $$1$$, $$2, ...$$, $$n$$, let P={x_i} be a regular partition of [0, 2] then find dx, or $$\\\\frac{b-a}{n}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite1a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{2}{n}$$"],"dependencies":["a961302definite1a-h1"],"title":"Set Up a Riemann Sum","text":"What is dx or $$\\\\frac{b-a}{n}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\frac{2}{n}$$","$$\\\\frac{-2}{n}$$","$$0$$","$$\\\\frac{1}{2} n$$"]},{"id":"a961302definite1a-h3","type":"hint","dependencies":["a961302definite1a-h2"],"title":"Calculate the Function Value at the Right Endpoint","text":"Since we are using a right-endpoint approximation to generate Riemann sums, for each i, we need to calculate the function value at the right endpoint of the interval [x_i-1, x_i]. The right endpoint of the interval is $$x_i$$, and since P is a regular partition, $$x_i=\\\\left(x_0+1\\\\right) dx=\\\\frac{2i}{n}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite1a-h4","type":"hint","dependencies":["a961302definite1a-h3"],"title":"Calculate the Function Value at the Right Endpoint","text":"Find the function value $$f{\\\\left(x_i\\\\right)}$$ at the right endpoint.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite1a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{4i^2}{n^2}$$"],"dependencies":["a961302definite1a-h4"],"title":"Calculate the Function Value at the Right Endpoint","text":"What is $$f{\\\\left(x_i\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\frac{4i^2}{n^2}$$","$$\\\\frac{4i^2}{n}$$"],"subHints":[{"id":"a961302definite1a-h5-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{4i^2}{n^2}$$"],"dependencies":[],"title":"Calculate the Function Value at the Right Endpoint","text":"$$f{\\\\left(x_i\\\\right)}={x_i}^2={\\\\left(\\\\frac{2i}{n}\\\\right)}^2=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\frac{4i^2}{n^2}$$","$$\\\\frac{4i^2}{n}$$"]}]},{"id":"a961302definite1a-h6","type":"hint","dependencies":["a961302definite1a-h5"],"title":"Riemann Sum","text":"Now find the Riemann sum.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite1a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["(8/n**3)sum{i\\\\=1}{n}{i**2}"],"dependencies":["a961302definite1a-h6"],"title":"Riemann Sum","text":"Which of the following is the correct Riemann sum?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["(8/n**3)sum{i\\\\=1}{n}{i**2}","(n**2)sum{i\\\\=1}{n}{i**2}","(4/n**2)sum{i\\\\=1}{n}{i}"]},{"id":"a961302definite1a-h8","type":"hint","dependencies":["a961302definite1a-h7"],"title":"Riemann Sum","text":"Evaluate the Riemann sum.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite1a-h9","type":"hint","dependencies":["a961302definite1a-h8"],"title":"Take the Limit as $$n$$ Approaches Infinity","text":"/int{x**2,0,2,x}=/lim{sum{i\\\\=1}{n}{f(x_i)dx},n,inf}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite1a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{8}{3}$$"],"dependencies":["a961302definite1a-h9"],"title":"Take the Limit as $$n$$ Approaches Infinity","text":"What is /lim{(8/3)+(4/n)+(8/(6n**2))}?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"a961302definite1a-h10-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{8}{3}$$"],"dependencies":[],"title":"Take the Limit as $$n$$ Approaches Infinity","text":"What is $$\\\\frac{8}{3}+0+0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}]}}]},{"id":"a961302definite10","title":"Expressing Limits as Definite Integrals","body":"Express the right limit as $$n$$ approaches infinity as definite integrals, identifying the correct intervals.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.2 The Definite Integral","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a961302definite10a","stepAnswer":["$$\\\\int_{0}^{1} x \\\\,dx$$"],"problemType":"MultipleChoice","stepTitle":"R_n=(1/n)*sum{i\\\\=1}{n}{i/n}","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\int_{0}^{1} x \\\\,dx$$","choices":["$$\\\\int_{0}^{1} x \\\\,dx$$","$$\\\\int_{0}^{0} x \\\\,dx$$"],"hints":{"DefaultPathway":[{"id":"a961302definite10a-h1","type":"hint","dependencies":[],"title":"Expressing Limits as Definite Integrals","text":"Consider the limit of $$R_n$$ as a Riemann sum representing the integral of $$f(x)=x$$ over the interval from $$0$$ to $$1$$.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a961302definite11","title":"Expressing Limits as Definite Integrals","body":"Express the right limit as $$n$$ approaches infinity as definite integrals, identifying the correct intervals.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.2 The Definite Integral","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a961302definite11a","stepAnswer":["$$\\\\int_{1}^{2} x \\\\ln(x^2) \\\\,dx$$"],"problemType":"MultipleChoice","stepTitle":"R_n=(1/n)*sum{i\\\\=1}{n}{i*(1+(i/n)*log((1+i/n)**2))}","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\int_{1}^{2} x \\\\ln(x^2) \\\\,dx$$","choices":["$$\\\\int_{1}^{2} x \\\\ln(x^2) \\\\,dx$$","$$\\\\int_{2}^{2} x \\\\ln(x^2) \\\\,dx$$"],"hints":{"DefaultPathway":[{"id":"a961302definite11a-h1","type":"hint","dependencies":[],"title":"Expressing Limits as Definite Integrals","text":"Consider the limit of $$R_n$$ as a Riemann sum representing the integral of $$f(x)=x$$ over the interval from $$0$$ to $$1$$.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a961302definite12","title":"Areas of Triangles","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.2 The Definite 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See provided figure.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$14$$","hints":{"DefaultPathway":[{"id":"a961302definite12a-h1","type":"hint","dependencies":[],"title":"Set Up Integrals","text":"Set up three definite integrals. Since all triangles are above the x-axis, we are adding them up together. The integrands are already provided for you on the graph. Be sure to use the correct intervals.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$14$$"],"dependencies":[],"title":"Evaluate the Integrals","text":"Evaluate the definite integrals you set up.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"a961302definite12a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$14$$"],"dependencies":[],"title":"Evaluate the Integrals","text":"What is $$1+2\\\\times2+3\\\\times3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}]}}]},{"id":"a961302definite13","title":"Areas of Triangles and Circles","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.2 The Definite Integral","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a961302definite13a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"Evaluate the integrals of the functions graphed using the formulas for areas of triangles and circles, and subtracting the areas below the x-axis if applicable. See provided figure.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"a961302definite13a-h1","type":"hint","dependencies":[],"title":"Set Up Integrals","text":"Set up three definite integrals. Since the middle triangle is below the x-axis, we are subtracting its area from the other two triangles that are aboce the x-axis. The integrands are already provided for you on the graph. Be sure to use the correct intervals.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":[],"title":"Evaluate the Integrals","text":"Evaluate the definite integrals you set up.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"a961302definite13a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":[],"title":"Evaluate the Integrals","text":"What is $$1-4+9$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}]}}]},{"id":"a961302definite14","title":"Evaluating Integrals Using Area Formulas","body":"Evaluate the integral using area formulas.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.2 The Definite Integral","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a961302definite14a","stepAnswer":["$$9$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\int_{0}^{6} 3-|x-3| \\\\,dx$$","stepBody":"The integral is the area of the triangle, which is:","answerType":"string","variabilization":{},"answerLatex":"$$9$$","choices":["$$8$$","$$9$$","$$10$$","$$11$$"],"hints":{"DefaultPathway":[{"id":"a961302definite14a-h1","type":"hint","dependencies":[],"title":"Graph the Function","text":"If we graph function $$3-|x-3|$$, we notice that the graph represents a symmetrical triangle from [0, 6]. We will use the area of a triangle formula.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite14a-h2","type":"hint","dependencies":["a961302definite14a-h1"],"title":"Find the Area of Two Triangles $$A_1$$ and $$A_2$$","text":"We can split the triangle vertically into two triangles $$A_1$$ and $$A_2$$ so they both resemble right triangles. We will find the area for these two triangles then add them up together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite14a-h3","type":"hint","dependencies":["a961302definite14a-h2"],"title":"Formula for Area of a Triangle","text":"The formula for the area of a triangle is $$A=\\\\frac{1}{2} b h$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite14a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$9$$"],"dependencies":["a961302definite14a-h3"],"title":"Add $$A_1$$ and $$A_2$$","text":"What is $$A_1$$ + $$A_2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$8$$","$$9$$","$$10$$","$$11$$"],"subHints":[{"id":"a961302definite14a-h4-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$9$$"],"dependencies":[],"title":"Add $$A_1$$ and $$A_2$$","text":"What is $$4.5+4.5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$8$$","$$9$$","$$10$$","$$11$$"]}]}]}}]},{"id":"a961302definite15","title":"Areas of Triangles and Circles","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.2 The Definite Integral","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a961302definite15a","stepAnswer":["$$10-2\\\\pi$$"],"problemType":"MultipleChoice","stepTitle":"Evaluate the integrals of the functions graphed using the formulas for areas of triangles and circles, and subtracting the areas below the x-axis if applicable.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$10-2\\\\pi$$","choices":["$$10-2\\\\pi$$","$$10+\\\\pi$$"],"hints":{"DefaultPathway":[{"id":"a961302definite15a-h1","type":"hint","dependencies":[],"title":"Set Up Integrals","text":"Set up three definite integrals. Since the semicircle is below the x-axis, we will subtract the area of the semicircle from the total area of the other two triangles. The integrands are already provided for you on the graph. Be sure to use the correct intervals.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite15a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$10-2\\\\pi$$"],"dependencies":["a961302definite15a-h1"],"title":"Evaluate the Integrals","text":"Evaluate the definite integrals you set up.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$10-2\\\\pi$$","$$10+\\\\pi$$"],"subHints":[{"id":"a961302definite15a-h2-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$10-2\\\\pi$$"],"dependencies":[],"title":"Evaluate the Integrals","text":"What is $$1-2\\\\pi+9$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$10-2\\\\pi$$","$$10+\\\\pi$$"]}]}]}}]},{"id":"a961302definite16","title":"Computing Integrals","body":"Evaluate the Integrals and Add Them Up Together","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.2 The Definite Integral","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a961302definite16a","stepAnswer":["$$41$$"],"problemType":"TextBox","stepTitle":"$$\\\\int_{2}^{4} 4f{\\\\left(x\\\\right)}-3g{\\\\left(x\\\\right)} \\\\,dx$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$41$$","hints":{"DefaultPathway":[{"id":"a961302definite16a-h1","type":"hint","dependencies":[],"title":"Apply the Integral of a Sum Property","text":"$$\\\\int_{a}^{b} f{\\\\left(x\\\\right)}+g{\\\\left(x\\\\right)} \\\\,dx=\\\\int_{a}^{b} f(x) \\\\,dx+\\\\int_{a}^{b} g(x) \\\\,dx$$. In other words, the integral of a sum is the sum of the integrals. This means separating the terms into two different integrals.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite16a-h2","type":"hint","dependencies":["a961302definite16a-h1"],"title":"Apply the Integral of the Product Property","text":"$$\\\\int_{a}^{b} c f{\\\\left(x\\\\right)} \\\\,dx=c*\\\\int_{a}^{b} f(x) \\\\,dx$$ for constant c. In other words, the integral of the produce of a constant and a function is equal to the constant multiplied by the integral of the function. This means taking the constant outside and putting it in front of the integral.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite16a-h3","type":"hint","dependencies":["a961302definite16a-h2"],"title":"Evaluate the Integrals and Add Them Up Together","text":"Now that we applied the properties of definite integrals to the problem to manipulate the expressions, we can evaluate them to find the answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite16a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$41$$"],"dependencies":["a961302definite16a-h3"],"title":"Evaluate the Integrals and Add Them Up Together","text":"What is $$4*\\\\int_{2}^{4} f(x) \\\\,dx-3*\\\\int_{2}^{4} g(x) \\\\,dx$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite16a-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$41$$"],"dependencies":[],"title":"Evaluate the Integrals and Add Them Up Together","text":"What is $$32+9$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a961302definite17","title":"Finding the Net Signed Area","body":"Find the net signed area between f(x) and the x-axis. $$\\\\int_{2}^{4} {\\\\left(x-3\\\\right)}^3 \\\\,dx$$","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.2 The Definite Integral","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a961302definite17a","stepAnswer":["Antisymmetric"],"problemType":"MultipleChoice","stepTitle":"Is the integrand symmetric or antisymmetric with respect to $$x=3$$?","stepBody":"","answerType":"string","variabilization":{},"choices":["Antisymmetric","Symmetric"],"hints":{"DefaultPathway":[{"id":"a961302definite17a-h1","type":"hint","dependencies":[],"title":"Graph of f","text":"Take a look at the graph of f.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a961302definite17b","stepAnswer":["The integral is zero."],"problemType":"MultipleChoice","stepTitle":"Select the correct statement.","stepBody":"","answerType":"string","variabilization":{},"choices":["The integral is zero.","The integral is $$1$$.","The integral is $$2$$.","The integral is $$-1$$."],"hints":{"DefaultPathway":[{"id":"a961302definite17b-h1","type":"hint","dependencies":[],"title":"Graph of f","text":"Take a look at the graph of f.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a961302definite18","title":"Computing Integrals","body":"Given that $$\\\\int_{0}^{1} x \\\\,dx=\\\\frac{1}{2}$$, $$\\\\int_{0}^{1} x^2 \\\\,dx=\\\\frac{1}{3}$$, and $$\\\\int_{0}^{1} x^3 \\\\,dx=\\\\frac{1}{4}$$, compute the integrals.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.2 The Definite Integral","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a961302definite18a","stepAnswer":["$$\\\\frac{7}{12}$$"],"problemType":"TextBox","stepTitle":"$$\\\\int_{0}^{1} 1-x+x^2-x^3 \\\\,dx$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{7}{12}$$","hints":{"DefaultPathway":[{"id":"a961302definite18a-h1","type":"hint","dependencies":[],"title":"Apply the Integral of a Sum and Difference Property","text":"For addition, $$\\\\int_{a}^{b} f{\\\\left(x\\\\right)}+g{\\\\left(x\\\\right)} \\\\,dx=\\\\int_{a}^{b} f(x) \\\\,dx+\\\\int_{a}^{b} g(x) \\\\,dx$$. In other words, the integral of a sum is the sum of the integrals. For subtraction, $$\\\\int_{a}^{b} f(x)-g(x) \\\\,dx=\\\\int_{a}^{b} f(x) \\\\,dx-int{g(x),a,b,x}$$. This means separating the terms into three different integrals with the appropriate sign.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite18a-h2","type":"hint","dependencies":[],"title":"Given Information","text":"Consider the given information and substitute in the appropriate values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite18a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{7}{12}$$"],"dependencies":[],"title":"Evaluate the Integrals","text":"After substituting the values e can evaluate the integrals.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"a961302definite18a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{7}{12}$$"],"dependencies":[],"title":"Evaluate the Integrals","text":"What is $$1-\\\\frac{1}{2}+\\\\frac{1}{3}-\\\\frac{1}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}]}}]},{"id":"a961302definite19","title":"Computing Integrals","body":"Given that $$\\\\int_{0}^{1} x \\\\,dx=\\\\frac{1}{2}$$, $$\\\\int_{0}^{1} x^2 \\\\,dx=\\\\frac{1}{3}$$, and $$\\\\int_{0}^{1} x^3 \\\\,dx=\\\\frac{1}{4}$$, compute the integrals.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.2 The Definite Integral","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a961302definite19a","stepAnswer":["$$\\\\frac{23}{4}$$"],"problemType":"TextBox","stepTitle":"$$\\\\int_{0}^{1} 7-5x^3 \\\\,dx$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{23}{4}$$","hints":{"DefaultPathway":[{"id":"a961302definite19a-h1","type":"hint","dependencies":[],"title":"Apply the Integral of a Difference Property","text":"For subtraction, $$\\\\int_{a}^{b} f(x)-g(x) \\\\,dx=\\\\int_{a}^{b} f(x) \\\\,dx-int{g(x),a,b,x}$$. This means separating the terms into three different integrals with the appropriate sign.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite19a-h2","type":"hint","dependencies":[],"title":"Given Information","text":"Consider the given information and substitute in the appropriate values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite19a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{23}{4}$$"],"dependencies":[],"title":"Evaluate the Integrals","text":"After substituting the values e can evaluate the integrals. What is the result of evaluating the integrals?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"a961302definite19a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{23}{4}$$"],"dependencies":[],"title":"Evaluate the Integrals","text":"What is $$7-\\\\frac{5}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}]}}]},{"id":"a961302definite2","title":"Using Geometric Formulas to Calculate Definite Integrals","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.2 The Definite Integral","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a961302definite2a","stepAnswer":["$$7.069$$"],"problemType":"TextBox","stepTitle":"Use the formula for the area of a circle to evaluate $$\\\\int_{3}^{6} \\\\sqrt{9-{\\\\left(x-3\\\\right)}^2} \\\\,dx$$. (Three decimal places).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7.069$$","hints":{"DefaultPathway":[{"id":"a961302definite2a-h1","type":"hint","dependencies":[],"title":"Area of a Circle","text":"The formula for the area of a circle is $$A=\\\\pi r^2$$. Look at the figure provided and made a slight modification to the formula for the area of a circle to find the area of the shaded region.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite2a-h2","type":"hint","dependencies":["a961302definite2a-h1"],"title":"Area of a Semicircle","text":"The function describes a semicircle with a radius $$3$$. The area of a semicircle is just one-half the area of a circle, or $$A=\\\\frac{1}{2} \\\\pi r^2$$. Now that we know that the function describes a semicircle, we need to make another slight modification to the formula of a semicircle to find the area of the shaded region, half a semicircle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite2a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$A=\\\\frac{1}{4} \\\\pi r^2$$"],"dependencies":["a961302definite2a-h2"],"title":"Area of Half a Semicircle","text":"What is the formula for the area of half a semicircle?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$A=\\\\pi r^2$$","$$A=\\\\frac{1}{2} \\\\pi r^2$$","$$A=\\\\frac{1}{4} \\\\pi r^2$$"],"subHints":[{"id":"a961302definite2a-h3-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$A=\\\\frac{1}{4} \\\\pi r^2$$"],"dependencies":[],"title":"Area of Half a Semicircle","text":"Recall the formula for the area of a circle is $$A=\\\\pi r^2$$. Half a semicircle is one quarter, or $$\\\\frac{1}{4}$$, of a circle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$A=\\\\pi r^2$$","$$A=\\\\frac{1}{2} \\\\pi r^2$$","$$A=\\\\frac{1}{4} \\\\pi r^2$$"]}]},{"id":"a961302definite2a-h4","type":"hint","dependencies":["a961302definite2a-h3"],"title":"Evaluate the Limit","text":"Evaluate $$\\\\int_{3}^{6} \\\\sqrt{9-{\\\\left(x-3\\\\right)}^2} \\\\,dx$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite2a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7.069$$"],"dependencies":["a961302definite2a-h4"],"title":"Evaluate the Limit","text":"What is $$\\\\frac{1}{4} {\\\\operatorname{\\\\pi}\\\\left(3\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a961302definite20","title":"Computing Integrals","body":"Suppose that $$\\\\int_{0}^{4} f(x) \\\\,dx=5$$ and $$\\\\int_{0}^{2} f(x) \\\\,dx=-3$$ and $$\\\\int_{0}^{4} g(x) \\\\,dx=1$$ and $$\\\\int_{0}^{2} g(x) \\\\,dx=2$$. Compute the integrals.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.2 The Definite Integral","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a961302definite20a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"$$\\\\int_{2}^{4} f{\\\\left(x\\\\right)}+g{\\\\left(x\\\\right)} \\\\,dx$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"a961302definite20a-h1","type":"hint","dependencies":[],"title":"Apply the Integral of a Sum Property","text":"$$\\\\int_{a}^{b} f{\\\\left(x\\\\right)}+g{\\\\left(x\\\\right)} \\\\,dx=\\\\int_{a}^{b} f(x) \\\\,dx+\\\\int_{a}^{b} g(x) \\\\,dx$$. In other words, the integral of a sum is the sum of the integrals. This means separating the terms into two different integrals.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite20a-h2","type":"hint","dependencies":[],"title":"Evaluate the Integrals and Add Them Up Together","text":"Now that we applied the property of definite integrals to the problem to manipulate the expressions, we can evaluate them to find the answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite20a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":[],"title":"Evaluate the Integrals and Add Them Up Together","text":"What is $$\\\\int_{2}^{4} f(x) \\\\,dx+\\\\int_{2}^{4} g(x) \\\\,dx$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"a961302definite20a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":[],"title":"Evaluate the Integrals and Add Them Up Together","text":"What is $$8-3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}]}}]},{"id":"a961302definite21","title":"Graph of $$sin\\\\left(2pix\\\\right)$$","body":"Answer parts a and $$b$$. From the graph of $$sin\\\\left(2\\\\pi x\\\\right)$$ shown: a. Explain why $$\\\\int_{0}^{1} sin\\\\left(2\\\\pi t\\\\right) \\\\,dt=0$$ and $$b$$. Explain why, in general, $$\\\\int_{a}^{a+1} sin\\\\left(2\\\\pi t\\\\right) \\\\,dt=0$$ for any value..\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.2 The Definite Integral","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a961302definite21a","stepAnswer":["The graph is antisymmetric with respect to $$t=\\\\frac{1}{2}$$ over [0, 1], so the average value is zero."],"problemType":"MultipleChoice","stepTitle":"a. Explain why $$\\\\int_{0}^{1} sin\\\\left(2\\\\pi t\\\\right) \\\\,dt=0$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"The graph is antisymmetric with respect to $$t=\\\\frac{1}{2}$$ over [0, 1], so the average value is zero.","choices":["The graph is antisymmetric with respect to $$t=\\\\frac{1}{2}$$ over [0, 1], so the average value is zero.","The graph is symmetric with respect to $$t=\\\\frac{1}{2}$$ over [0, 1], so the average value is zero."],"hints":{"DefaultPathway":[{"id":"a961302definite21a-h1","type":"hint","dependencies":[],"title":"Look at the Graph","text":"Consider the symmetry of the graph respect to $$t=\\\\frac{1}{2}$$ over [0, 1].","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a961302definite21b","stepAnswer":["For any value of a, the graph between [a, a+1] is a shift of the graph over [0, 1], so the net areas above and below the axis do not change and the average remains zero."],"problemType":"MultipleChoice","stepTitle":"$$b$$. Explain why, in general, $$\\\\int_{a}^{a+1} sin\\\\left(2\\\\pi t\\\\right) \\\\,dt=0$$ for any value.","stepBody":"","answerType":"string","variabilization":{},"choices":["For any value of a, the graph between [a, a+1] is a shift of the graph over [0, 1], so the net areas above and below the axis change and the average remains zero.","For any value of a, the graph between [a, a+1] is a shift of the graph over [0, 1], so the net areas above and below the axis do not change and the average remains zero.","For any value of a, the graph between [a, a+1] is a shift of the graph over [0, 1], so the net areas above and below the axis do not change and the average remains zero."],"hints":{"DefaultPathway":[{"id":"a961302definite21b-h1","type":"hint","dependencies":[],"title":"Look at the Graph","text":"Consider the values of a in the graph in the interval [0, 1] and the net areas both above and below the x-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a961302definite3","title":"Finding the Net Signed Area","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.2 The Definite Integral","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a961302definite3a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"Find the net signed area between the curve of the function $$f(x)=2x$$ and the x-axis over the interval [-3, 3].","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a961302definite3a-h1","type":"hint","dependencies":[],"title":"Find the Area of Triangle $$A_1$$","text":"According to the figure, the function produces a straight line that forms two triangles $$A_1$$ and $$A_2$$. Use the geometric formula for the area of the triangle, $$A=\\\\frac{1}{2} b h$$, to find the area of triangle $$A_1$$ above the axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a961302definite3a-h1"],"title":"Find the Area of Triangle $$A_1$$","text":"What is the area of triangle $$A_1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"a961302definite3a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":[],"title":"Find the Area of Triangle $$A_1$$","text":"What is $$A=\\\\frac{1}{2} b h$$ if $$3$$ is the base and $$6$$ is the height?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"a961302definite3a-h3","type":"hint","dependencies":["a961302definite3a-h2"],"title":"Find the Area of Triangle $$A_2$$","text":"Use the geometric formula for the area of the triangle, $$A=\\\\frac{1}{2} b h$$, to find the area of triangle $$A_2$$, below the axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a961302definite3a-h3"],"title":"Find the Area of Triangle $$A_2$$","text":"What is the area of triangle $$A_2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"a961302definite3a-h4-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":[],"title":"Find the Area of Triangle $$A_2$$","text":"What is $$A=\\\\frac{1}{2} b h$$ if $$3$$ is the base and $$6$$ is the height?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"a961302definite3a-h5","type":"hint","dependencies":["a961302definite3a-h4"],"title":"Subtract $$A_2$$ from $$A_1$$ to Find the Net Area","text":"The net area is $$\\\\int_{-3}^{3} 2x \\\\,dx=A_1-A_2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite3a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a961302definite3a-h5"],"title":"Subtract $$A_2$$ from $$A_1$$ to Find the Net Area","text":"What is $$A_1-A_2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"a961302definite3a-h6-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":[],"title":"Subtract $$A_2$$ from $$A_1$$ to Find the Net Area","text":"$$A_1$$ and $$A_2$$ both equal to $$9$$. What is $$9-9$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"a961302definite3a-h7","type":"hint","dependencies":["a961302definite3a-h6"],"title":"Analysis","text":"If $$A_1$$ is the area above the x-axis and $$A_2$$ is the area below the x-axis, then the net area is $$A_1-A_2$$. Since the areas of the two triangles are equal, the net area is zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a961302definite4","title":"Find the Total Area","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.2 The Definite Integral","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a961302definite4a","stepAnswer":["$$10$$"],"problemType":"TextBox","stepTitle":"Find the total area between $$f(x)=x-2$$ and the x-axis over the interval [0, 6].","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10$$","hints":{"DefaultPathway":[{"id":"a961302definite4a-h1","type":"hint","dependencies":[],"title":"Calculate the $$x-Intercept$$","text":"Calculate the x-intercept as $$(2,0)$$ by setting $$y=0$$ then solving for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite4a-h2","type":"hint","dependencies":["a961302definite4a-h1"],"title":"What is Needed to Find the Total Area","text":"To find the total area, take the area below the x-axis over the subinterval [0, 2] and add it to the area above the x-axis on the subinterval [2, 6]. From this we have $$\\\\int_{0}^{6} |x-2| \\\\,dx=A_2+A_1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite4a-h3","type":"hint","dependencies":["a961302definite4a-h2"],"title":"Find the Area of Two Triangles $$A_2$$ and $$A_1$$","text":"Use the formula for the area of a triangle, $$A=\\\\frac{1}{2} b h$$ to obtain $$A_2$$ and $$A_1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite4a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$A_2=2$$, $$A_1=8$$"],"dependencies":["a961302definite4a-h3"],"title":"Find the Area of Two Triangles $$A_2$$ and $$A_1$$","text":"What is the area of $$A_2$$ and $$A_1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$A_2=2$$, $$A_1=4$$","$$A_2=2$$, $$A_1=4$$","$$A_2=1$$, $$A_1=4$$","$$A_2=2$$, $$A_1=6$$","$$A_2=2$$, $$A_1=8$$"]},{"id":"a961302definite4a-h5","type":"hint","dependencies":["a961302definite4a-h4"],"title":"Add $$A_1$$ and $$A_2$$ to Find the Total Area","text":"Now that we know $$A_1$$ and $$A_2$$, we can add the two areas together to find the total area.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite4a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a961302definite4a-h5"],"title":"Add $$A_1$$ and $$A_2$$ to Find the Total Area","text":"What is $$A_1+A_2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"a961302definite4a-h6-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":[],"title":"Add $$A_1$$ and $$A_2$$ to Find the Total Area","text":"What is $$8+2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}]}}]},{"id":"a961302definite5","title":"Using the Properties of the Definite Integral","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.2 The Definite Integral","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a961302definite5a","stepAnswer":["$$-3\\\\int_{-2}^{1} x^3 \\\\,dx+2\\\\int_{-2}^{1} x \\\\,dx+\\\\int_{-2}^{1} 2 \\\\,dx$$"],"problemType":"MultipleChoice","stepTitle":"Use the properties of the definite integral to express the definite integral of $$f(x)=-3x^3+2x+2$$ over the interval [-2, 1] as the sum of three definite integrals.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-3\\\\int_{-2}^{1} x^3 \\\\,dx+2\\\\int_{-2}^{1} x \\\\,dx+\\\\int_{-2}^{1} 2 \\\\,dx$$","choices":["$$\\\\int_{-2}^{1} x^3 \\\\,dx+\\\\int_{-2}^{1} x \\\\,dx+\\\\int_{-2}^{1} 2 \\\\,dx$$","$$-3\\\\int_{-2}^{1} x^3 \\\\,dx+2\\\\int_{-2}^{1} x \\\\,dx+\\\\int_{-2}^{1} 2 \\\\,dx$$","$$2\\\\int_{-2}^{1} x^3 \\\\,dx-3\\\\int_{-2}^{1} x \\\\,dx+\\\\int_{-2}^{1} 2 \\\\,dx$$"],"hints":{"DefaultPathway":[{"id":"a961302definite5a-h1","type":"hint","dependencies":[],"title":"Convert Function to Integral Notation","text":"Convert the function into integral notation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite5a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\int_{-2}^{1} -3x^3+2x+2 \\\\,dx$$"],"dependencies":["a961302definite5a-h1"],"title":"Convert Function to Integral Notation","text":"Which of the following is the correct integral notation for $$f(x)=-3x^3+2x+2$$ over the interval [-2, 1]?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\int_{1}^{-2} -3x^3+2x+2 \\\\,dx$$","$$\\\\int_{-2}^{1} -3x^3+2x+2 \\\\,dx$$","$$\\\\int_{0}^{-2} -3x^3+2x+2 \\\\,dx$$","$$\\\\int_{0}^{1} -3x^3+2x+2 \\\\,dx$$"]},{"id":"a961302definite5a-h3","type":"hint","dependencies":["a961302definite5a-h2"],"title":"Apply the Integral of a Sum Property","text":"$$\\\\int_{a}^{b} f{\\\\left(x\\\\right)}+g{\\\\left(x\\\\right)} \\\\,dx=\\\\int_{a}^{b} f(x) \\\\,dx+\\\\int_{a}^{b} g(x) \\\\,dx$$. In other words, the integral of a sum is the sum of the integrals. This means separating the terms into three different integrals.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite5a-h4","type":"hint","dependencies":["a961302definite5a-h3"],"title":"Apply the Integral of the Product Property","text":"$$\\\\int_{a}^{b} c f{\\\\left(x\\\\right)} \\\\,dx=c*\\\\int_{a}^{b} f(x) \\\\,dx$$ for constant c. In other words, the integral of the produce of a constant and a function is equal to the constant multiplied by the integral of the function. This means taking the constant outside and putting it in front of the integral.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a961302definite6","title":"Using the Properties of the Definite Integral","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.2 The Definite Integral","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a961302definite6a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"If it is known that $$\\\\int_{0}^{8} f(x) \\\\,dx=10$$ and $$\\\\int_{0}^{5} f(x) \\\\,dx=5$$, find the value of $$\\\\int_{5}^{8} f(x) \\\\,dx$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"a961302definite6a-h1","type":"hint","dependencies":[],"title":"Apply the Integral Property","text":"Refer to the property of the integral $$\\\\int_{a}^{b} f(x) \\\\,dx=\\\\int_{a}^{c} f(x) \\\\,dx+\\\\int_{c}^{b} f(x) \\\\,dx$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite6a-h2","type":"hint","dependencies":["a961302definite6a-h1"],"title":"Substitution","text":"Substitute $$\\\\int_{0}^{8} f(x) \\\\,dx$$ with $$10$$ and $$\\\\int_{0}^{5} f(x) \\\\,dx$$ with $$5$$ then solve for $$\\\\int_{5}^{8} f(x) \\\\,dx$$. Solve like you normally would with algebra.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a961302definite7","title":"Comparing Two Functions over a Given Interval","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.2 The Definite Integral","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a961302definite7a","stepAnswer":["$$\\\\int_{0}^{1} g(x) \\\\,dx \\\\geq \\\\int_{0}^{1} f(x) \\\\,dx$$"],"problemType":"MultipleChoice","stepTitle":"Compare $$f(x)=\\\\sqrt{1+x^2}$$ and $$g(x)=\\\\sqrt{1+x}$$ over the interval [0, 1].","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\int_{0}^{1} g(x) \\\\,dx \\\\geq \\\\int_{0}^{1} f(x) \\\\,dx$$","choices":["$$\\\\int_{0}^{1} g(x) \\\\,dx \\\\geq \\\\int_{0}^{1} f(x) \\\\,dx$$","$$\\\\int_{0}^{1} g(x) \\\\,dx \\\\leq \\\\int_{0}^{1} f(x) \\\\,dx$$","$$\\\\int_{0}^{1} g(x) \\\\,dx>\\\\int_{0}^{1} f(x) \\\\,dx$$","$$\\\\int_{0}^{1} g(x) \\\\,dx<\\\\int_{0}^{1} f(x) \\\\,dx$$"],"hints":{"DefaultPathway":[{"id":"a961302definite7a-h1","type":"hint","dependencies":[],"title":"Graph the Functions","text":"Use a graphing calculator to graph the functions as it is necessary to understand how they compare over the interval [0, 1].","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite7a-h2","type":"hint","dependencies":["a961302definite7a-h1"],"title":"Zoom into Graph","text":"On the interval [0, 1], g(x) is above f(x). Think abour the relationship between f(x) and g(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a961302definite8","title":"Finding the Average Value of a Linear Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.2 The Definite Integral","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a961302definite8a","stepAnswer":["$$\\\\frac{7}{2}$$"],"problemType":"TextBox","stepTitle":"Find the average value of $$f(x)=x+1$$ over the interval [0, 5].","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{7}{2}$$","hints":{"DefaultPathway":[{"id":"a961302definite8a-h1","type":"hint","dependencies":[],"title":"Graph the Function","text":"First, graph the function on the stated interval, as shown in the figure.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite8a-h2","type":"hint","dependencies":["a961302definite8a-h1"],"title":"Area of a Trapezoid","text":"The region is a trapezoid lying on its side, so we can use the area formula for a trapezoid $$A=\\\\frac{1}{2} h{\\\\left(a+b\\\\right)}$$, where $$h$$ represents height, and a and $$b$$ represent the two parallel sides. Then, $$\\\\int_{0}^{5} x+1 \\\\,dx=\\\\frac{1}{2} h{\\\\left(a+b\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{35}{2}$$"],"dependencies":["a961302definite8a-h2"],"title":"Area of a Trapezoid","text":"What is $$\\\\frac{1}{2} h{\\\\left(a+b\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite8a-h4","type":"hint","dependencies":["a961302definite8a-h3"],"title":"Set up the Integral","text":"We can find the average value of the function by setting up the integral as $$1/(5-0)*\\\\int_{0}^{5} x+1 \\\\,dx$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a961302definite8a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{7}{2}$$"],"dependencies":["a961302definite8a-h4"],"title":"Evaluate the Integral","text":"Evaluate the integral $$1/(5-0)*\\\\int_{0}^{5} x+1 \\\\,dx$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"a961302definite8a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{7}{2}$$"],"dependencies":[],"title":"Evaluate the Integral","text":"What is $$\\\\frac{1}{5} \\\\frac{35}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}]}}]},{"id":"a961302definite9","title":"Expressing Limits as Definite Integrals","body":"Express the right limit as $$n$$ approaches infinity as definite integrals, identifying the correct intervals.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.2 The Definite Integral","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a961302definite9a","stepAnswer":["$$\\\\int_{3}^{6} x \\\\,dx$$"],"problemType":"MultipleChoice","stepTitle":"R_n=(3/n)*sum{i\\\\=1}{n}{3+3*(i/n)}","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\int_{3}^{6} x \\\\,dx$$","choices":["$$\\\\int_{3}^{6} x \\\\,dx$$","$$\\\\int_{3}^{6} x^2 \\\\,dx$$"],"hints":{"DefaultPathway":[{"id":"a961302definite9a-h1","type":"hint","dependencies":[],"title":"Expressing Limits as Definite Integrals","text":"Consider the limit of $$R_n$$ as a Riemann sum representing the integral of $$f(x)=x$$ over the interval from $$0$$ to $$1$$.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a969466InverseFunc1","title":"Horizontal Line Test","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.4 Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a969466InverseFunc1a","stepAnswer":["One to one"],"problemType":"MultipleChoice","stepTitle":"Use the horizontal line test to determine whether this graph is one-to-one.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Not one to one","One to one"],"hints":{"DefaultPathway":[{"id":"a969466InverseFunc1a-h1","type":"hint","dependencies":[],"title":"One to one","text":"A function is one-to-one if and only if every horizontal line intersects the graph of the function no more than once.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a969466InverseFunc10","title":"Pairs of Inverse Functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.4 Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a969466InverseFunc10a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=8x$$, $$g(x)=\\\\frac{x}{8}$$. Are these two functions inverses?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a969466InverseFunc10a-h1","type":"hint","dependencies":[],"title":"Composition","text":"If $$f(g(x))=x$$ for all $$x$$ in the domain g, and if $$g(f(x))=x$$ for all $$x$$ in the domain, then f and g are inverses.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a969466InverseFunc10a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x$$"],"dependencies":["a969466InverseFunc10a-h1"],"title":"Functions in functions","text":"What is f(g(x))?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$x$$","f(x)","g(x)"]},{"id":"a969466InverseFunc10a-s2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x$$"],"dependencies":["a969466InverseFunc10a-h1"],"title":"Functions in functions","text":"What is g(f(x))?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$x$$","f(x)","g(x)"]}]}}]},{"id":"a969466InverseFunc11","title":"Pairs of Inverse Functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.4 Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a969466InverseFunc11a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=5x-7$$, $$g(x)=\\\\frac{x+5}{7}$$. Are these two functions inverses?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a969466InverseFunc11a-h1","type":"hint","dependencies":[],"title":"Composition","text":"If $$f(g(x))=x$$ for all $$x$$ in the domain g, and if $$g(f(x))=x$$ for all $$x$$ in the domain, then f and g are inverses.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a969466InverseFunc11a-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a969466InverseFunc11a-h1"],"title":"Functions in functions","text":"Is f(g(x)) equal to $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["Yes","No"]},{"id":"a969466InverseFunc11a-s2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a969466InverseFunc11a-h1"],"title":"Functions in functions","text":"Is f(g(x)) equal to $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["Yes","No"]}]}}]},{"id":"a969466InverseFunc12","title":"Pairs of Inverse Functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.4 Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a969466InverseFunc12a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{1}{x-1}$$ when $$x$$ is not $$1$$. $$g(x)=\\\\frac{1}{x}+1$$, when $$x$$ is not $$0$$. Are these two functions inverses?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a969466InverseFunc12a-h1","type":"hint","dependencies":[],"title":"Composition","text":"If $$f(g(x))=x$$ for all $$x$$ in the domain g, and if $$g(f(x))=x$$ for all $$x$$ in the domain, then f and g are inverses.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a969466InverseFunc12a-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x$$"],"dependencies":["a969466InverseFunc12a-h1"],"title":"Functions in functions","text":"What is f(g(x))?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$x$$","f(x)","g(x)"]},{"id":"a969466InverseFunc12a-s2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x$$"],"dependencies":["a969466InverseFunc12a-h1"],"title":"Functions in functions","text":"What is g(f(x))?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$x$$","f(x)","g(x)"]}]}}]},{"id":"a969466InverseFunc13","title":"Inverse Trig Functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.4 Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a969466InverseFunc13a","stepAnswer":["$$\\\\frac{\u03c0}{6}$$"],"problemType":"TextBox","stepTitle":"Evaluate $${tan}^{\\\\left(-1\\\\right)} \\\\frac{\\\\sqrt{3}}{3}$$ for $$0 \\\\leq \\\\theta \\\\leq \\\\frac{\\\\pi}{2}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{\u03c0}{6}$$","hints":{"DefaultPathway":[{"id":"a969466InverseFunc13a-h1","type":"hint","dependencies":[],"title":"Angle","text":"Find the angle \u03b8 such that $$tan\\\\theta=\\\\frac{\\\\sqrt{3}}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a969466InverseFunc14","title":"Inverse Trig Functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.4 Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a969466InverseFunc14a","stepAnswer":["$$\\\\frac{\u03c0}{4}$$"],"problemType":"TextBox","stepTitle":"Evaluate $$1{cot}^{\\\\left(-1\\\\right)}$$ for $$0 \\\\leq \\\\theta \\\\leq \\\\frac{\\\\pi}{2}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{\u03c0}{4}$$","hints":{"DefaultPathway":[{"id":"a969466InverseFunc14a-h1","type":"hint","dependencies":[],"title":"Angle","text":"Find the angle \u03b8 such that $$cot(\\\\theta)=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a969466InverseFunc14a-h2","type":"hint","dependencies":["a969466InverseFunc14a-h1"],"title":"Cotangent","text":"Cotangent is equal to $$\\\\frac{1}{tan}$$, or $$\\\\frac{adjacent}{opposite}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a969466InverseFunc15","title":"Inverse Trig Functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.4 Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a969466InverseFunc15a","stepAnswer":["$$\\\\frac{\u03c0}{6}$$"],"problemType":"TextBox","stepTitle":"Evaluate $${cos}^{\\\\left(-1\\\\right)} \\\\frac{\\\\sqrt{3}}{2}$$ for $$0 \\\\leq \\\\theta \\\\leq \\\\frac{\\\\pi}{2}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{\u03c0}{6}$$","hints":{"DefaultPathway":[{"id":"a969466InverseFunc15a-h1","type":"hint","dependencies":[],"title":"Angle","text":"Find the angle \u03b8 such that $$cos(\\\\theta)=\\\\frac{\\\\sqrt{3}}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a969466InverseFunc16","title":"Inverse Trig Functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.4 Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a969466InverseFunc16a","stepAnswer":["$$\\\\frac{\u03c0}{6}$$"],"problemType":"TextBox","stepTitle":"Evaluate $$sin\\\\left({cos}^{\\\\left(-1\\\\right)} \\\\frac{\\\\sqrt{2}}{2}\\\\right)$$ for $$0 \\\\leq \\\\theta \\\\leq \\\\frac{\\\\pi}{2}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{\u03c0}{6}$$","hints":{"DefaultPathway":[{"id":"a969466InverseFunc16a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\u03c0}{4}$$"],"dependencies":[],"title":"Inverse Trig Functions","text":"What is the angle \u03b8 such that $$cos(\\\\theta)=\\\\frac{\\\\sqrt{3}}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a969466InverseFunc17","title":"Inverse Trig Functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a969466InverseFunc17a","stepAnswer":["$$\\\\frac{\u03c0}{6}$$"],"problemType":"TextBox","stepTitle":"Evaluate $$tan\\\\left({tan}^{\\\\left(-1\\\\right)} \\\\left(-\\\\frac{\\\\pi}{6}\\\\right)\\\\right)$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{\u03c0}{6}$$","hints":{"DefaultPathway":[{"id":"a969466InverseFunc17a-h1","type":"hint","dependencies":[],"title":"Inverse Trig Functions","text":"If f and g are inverses, $$f(g(x))=x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a969466InverseFunc18","title":"Blood in an Artery","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a969466InverseFunc18a","stepAnswer":["$$\\\\sqrt{0.04-\\\\frac{V}{500}}$$"],"problemType":"MultipleChoice","stepTitle":"The velocity V (in centimeters per second) of blood in an artery at a distance $$x$$ cm from the center of the artery can be modeled by the function $$V=f(x)=\\\\operatorname{500}\\\\left(0.04-x^2\\\\right)$$ for $$0 \\\\leq x \\\\leq 0.2$$. What is the inverse of this function?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\sqrt{0.04-\\\\frac{V}{500}}$$","choices":["500V","$$\\\\sqrt{0.04-\\\\frac{V}{500}}$$","$$\\\\frac{0.04V}{\\\\sqrt{500}}$$","$$0.04\\\\sqrt{V}$$"],"hints":{"DefaultPathway":[{"id":"a969466InverseFunc18a-h1","type":"hint","dependencies":[],"title":"Solving the function","text":"Solve the equation $$y=f(x)$$ for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a969466InverseFunc18a-h2","type":"hint","dependencies":["a969466InverseFunc18a-h1"],"title":"Inverses","text":"Interchange the variables $$x$$ and $$y$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"a969466InverseFunc18b","stepAnswer":["$$0.1$$"],"problemType":"TextBox","stepTitle":"What is the distance from the center of an artery with a velocity of $$15$$ $$\\\\frac{cm}{sec}$$? Round to the nearest decimal place.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.1$$","hints":{"DefaultPathway":[{"id":"a969466InverseFunc18b-h3","type":"hint","dependencies":[],"title":"Inverse Function","text":"This inverse function determines the distance from the center of the artery at which blood is flowing with velocity V. To find distance, plug the correct velocity into V in the inverse function.","variabilization":{},"oer":"","license":""}]}}]},{"id":"a969466InverseFunc2","title":"Horizontal Line Test","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.4 Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a969466InverseFunc2a","stepAnswer":["Not one to one"],"problemType":"MultipleChoice","stepTitle":"Use the horizontal line test to determine whether this graph is one-to-one.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Not one to one","One to one"],"hints":{"DefaultPathway":[{"id":"a969466InverseFunc2a-h1","type":"hint","dependencies":[],"title":"One to one","text":"A function is one-to-one if and only if every horizontal line intersects the graph of the function no more than once.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a969466InverseFunc20","title":"Toxins in Lakes","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.4 Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a969466InverseFunc20a","stepAnswer":["$$31250$$"],"problemType":"TextBox","stepTitle":"The cost to remove a toxin from a lake is modeled by the function $$C(p)=\\\\frac{75p}{85-p}$$, where C is the cost (in thousands of dollars) and $$p$$ is the amount of toxin in a small lake (measured in parts per billion [ppb]). This model is valid only when the amount of toxin is less than $$85$$ ppb. Find the cost to remove $$25$$ ppb of the toxin from the lake.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$31250$$","hints":{"DefaultPathway":[{"id":"a969466InverseFunc20a-h1","type":"hint","dependencies":[],"title":"Solving for Cost","text":"Value $$p$$ is the amount of toxin in the small lake. Plug the correct value into the given function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a969466InverseFunc20a-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{85C}{75+C}$$"],"dependencies":["a969466InverseFunc20a-h1"],"title":"Inverse Function","text":"What is the inverse function for C(p)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\frac{75C}{85+C}$$","$$\\\\frac{85C}{75+C}$$","$$C-\\\\frac{75}{85}$$","$$\\\\frac{C+85}{75}$$"]}]}},{"id":"a969466InverseFunc20b","stepAnswer":["$$92$$"],"problemType":"TextBox","stepTitle":"An airplane\u2019s Mach number M is the ratio of its speed to the speed of sound. When a plane is flying at a constant altitude, then its Mach angle is given by $$\u03bc=2sin-1\\\\left(\\\\frac{1}{M}\\\\right)$$. Find the Mach angle (to the nearest degree) for $$M=1.4$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$92$$","hints":{"DefaultPathway":[{"id":"a969466InverseFunc20b-h2","type":"hint","dependencies":[],"title":"Plugging in","text":"Plug in $$M=1.4$$ into the given equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a969466InverseFunc21","title":"Mach Angle","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.4 Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a969466InverseFunc21a","stepAnswer":["$$3.86$$"],"problemType":"TextBox","stepTitle":"An airplane\u2019s Mach number M is the ratio of its speed to the speed of sound. When a plane is flying at a constant altitude, then its Mach angle is given by $$\u03bc=2sin-1\\\\left(\\\\frac{1}{M}\\\\right)$$.\\\\nFind the Mach number (to $$2$$ decimal places) for $$\u03bc=\\\\frac{\\\\pi}{6}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3.86$$","hints":{"DefaultPathway":[{"id":"a969466InverseFunc21a-h1","type":"hint","dependencies":[],"title":"Inverse Function","text":"The inverse function determines the mach number for angle \u03bc. To find the inverse, solve the equation $$y=f(x)$$ for $$x$$ and then interchange the variables $$x$$ and $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a969466InverseFunc21a-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1}sin^$\\\\left(\\\\frac{\u03bc}{2}\\\\righ$$"],"dependencies":["a969466InverseFunc21a-h1"],"title":"Inverses","text":"What is the inverse function for $$\u03bc=2sin-1\\\\left(\\\\frac{1}{M}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\frac{1}{\\\\sin\\\\left(\\\\frac{\\\\mu}{2}\\\\right)}$$","$$\\\\frac{\\\\sin\\\\left(\\\\frac{\\\\mu}{2}\\\\right)}{2}$$","$${cos}^{\\\\left(-1\\\\right)} \\\\left(\u03bc+1\\\\right)$$","$${cos}^{\\\\left(-1\\\\right)} \\\\frac{\u03bc}{2}$$"]}]}}]},{"id":"a969466InverseFunc22","title":"Mach Angle","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.4 Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a969466InverseFunc22a","stepAnswer":["$$2.3$$"],"problemType":"TextBox","stepTitle":"An airplane\u2019s Mach number M is the ratio of its speed to the speed of sound. When a plane is flying at a constant altitude, then its Mach angle is given by $$\u03bc=2sin-1\\\\left(\\\\frac{1}{M}\\\\right)$$.\\\\nFind the Mach number (to $$2$$ decimal places) for $$\u03bc=\\\\frac{2\\\\pi}{7}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2.3$$","hints":{"DefaultPathway":[{"id":"a969466InverseFunc22a-h1","type":"hint","dependencies":[],"title":"Inverse Function","text":"The inverse function determines the mach number for angle \u03bc. To find the inverse, solve the equation $$y=f(x)$$ for $$x$$ and then interchange the variables $$x$$ and $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a969466InverseFunc22a-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1}sin^$\\\\left(\\\\frac{\u03bc}{2}\\\\righ$$"],"dependencies":["a969466InverseFunc22a-h1"],"title":"Inverses","text":"What is the inverse function for $$\u03bc=2sin-1\\\\left(\\\\frac{1}{M}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$${sin}^{\\\\left(-1\\\\right)} \\\\frac{\\\\frac{\u03bc}{2}}{2}$$","$$\\\\frac{1}{\\\\sin\\\\left(\\\\frac{\\\\mu}{2}\\\\right)}$$","$$cos\\\\left(\u03bc+1\\\\right)$$","$${cos}^{\\\\left(-1\\\\right)} \\\\frac{\u03bc}{2}$$"]}]}}]},{"id":"a969466InverseFunc23","title":"Temperature of a City","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.4 Inverse Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a969466InverseFunc23a","stepAnswer":["$$8.51$$, $$6.69$$"],"problemType":"MultipleChoice","stepTitle":"The average temperature (in degrees Celsius) of a city in the northern United States can be modeled by the function $$T(x)=5+18sin\\\\left(6\\\\pi \\\\left(x-4.6\\\\right)\\\\right)$$, where $$x$$ is time in months and $$x=1.00$$ corresponds to January $$1$$. 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What is this $$p$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a969466InverseFunc9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a969466InverseFunc9a-h3"],"title":"Undefined Point","text":"For the function $$f(x)=\\\\sqrt{x-1}$$, what would $$y$$ be greater than or equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a975ae1MultDivInts1","title":"Multiplying Integers with Signs","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Decimals","courseName":"OpenStax: Intermediate 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That same day, the high temperature in Embarrass, Minnesota was -12\xb0. 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He deposits $225 to the account. What is the new balance?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$187$$","hints":{"DefaultPathway":[{"id":"a975ae1MultDivInts30a-h1","type":"hint","dependencies":[],"title":"Translate","text":"First, translate the problem to an algebraic expression","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a975ae1MultDivInts30a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-38+225$$"],"dependencies":["a975ae1MultDivInts30a-h1"],"title":"Translate","text":"What is the algebraic representation of the problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a975ae1MultDivInts30a-h3","type":"hint","dependencies":["a975ae1MultDivInts30a-h2"],"title":"Simplify","text":"Next, simplify the algebraic expression that you 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Algebra","steps":[{"id":"a975ae1MultDivInts4a","stepAnswer":["$$-8$$"],"problemType":"TextBox","stepTitle":"$$-1\\\\times8$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-8$$","hints":{"DefaultPathway":[{"id":"a975ae1MultDivInts4a-h1","type":"hint","dependencies":[],"title":"Multiplication by $$-1$$","text":"Multiplying by $$-1$$ gives you the same number, with the opposite sign.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a975ae1MultDivInts4a-h2","type":"hint","dependencies":["a975ae1MultDivInts4a-h1"],"title":"Answer","text":"The answer is $$-8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a975ae1MultDivInts4b","stepAnswer":["$$16$$"],"problemType":"TextBox","stepTitle":"$$-1\\\\times-16$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16$$","hints":{"DefaultPathway":[{"id":"a975ae1MultDivInts4b-h1","type":"hint","dependencies":[],"title":"Multiplication by $$-1$$","text":"Multiplying by $$-1$$ gives you the same number, with the opposite sign.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a975ae1MultDivInts5","title":"Dividing Integers with Signs","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Decimals","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a975ae1MultDivInts5a","stepAnswer":["$$-9$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{-27}{3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-9$$","hints":{"DefaultPathway":[{"id":"a975ae1MultDivInts5a-h1","type":"hint","dependencies":[],"title":"Divide Numbers","text":"First, divide the numbers as if they were positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a975ae1MultDivInts5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a975ae1MultDivInts5a-h1"],"title":"Divide Numbers","text":"What is $$\\\\frac{27}{3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a975ae1MultDivInts5a-h3","type":"hint","dependencies":["a975ae1MultDivInts5a-h2"],"title":"Final Sign","text":"If the signs of the integers are the same, then the quotient is positive; otherwise, the quotient is negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a975ae1MultDivInts5b","stepAnswer":["$$25$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{-100}{-4}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$25$$","hints":{"DefaultPathway":[{"id":"a975ae1MultDivInts5b-h1","type":"hint","dependencies":[],"title":"Divide Numbers","text":"First, divide the numbers as if they were positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a975ae1MultDivInts5b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["a975ae1MultDivInts5b-h1"],"title":"Divide Numbers","text":"What is $$\\\\frac{100}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a975ae1MultDivInts5b-h3","type":"hint","dependencies":["a975ae1MultDivInts5b-h2"],"title":"Final Sign","text":"If the signs of the integers are the same, then the quotient is positive; otherwise, the quotient is negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a975ae1MultDivInts6","title":"Dividing Integers with Signs","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Decimals","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a975ae1MultDivInts6a","stepAnswer":["$$-9$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{-63}{7}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-9$$","hints":{"DefaultPathway":[{"id":"a975ae1MultDivInts6a-h1","type":"hint","dependencies":[],"title":"Divide Numbers","text":"First, divide the numbers as if they were positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a975ae1MultDivInts6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a975ae1MultDivInts6a-h1"],"title":"Divide Numbers","text":"What is $$\\\\frac{63}{7}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a975ae1MultDivInts6a-h3","type":"hint","dependencies":["a975ae1MultDivInts6a-h2"],"title":"Final Sign","text":"If the signs of the integers are the same, then the quotient is positive; otherwise, the quotient is negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a975ae1MultDivInts6b","stepAnswer":["$$-23$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{-115}{5}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-23$$","hints":{"DefaultPathway":[{"id":"a975ae1MultDivInts6b-h1","type":"hint","dependencies":[],"title":"Divide Numbers","text":"First, divide the numbers as if they were positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a975ae1MultDivInts6b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$23$$"],"dependencies":["a975ae1MultDivInts6b-h1"],"title":"Divide Numbers","text":"What is $$\\\\frac{115}{5}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a975ae1MultDivInts6b-h3","type":"hint","dependencies":["a975ae1MultDivInts6b-h2"],"title":"Final Sign","text":"If the signs of the integers are the same, then the quotient is positive; otherwise, the quotient is negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a975ae1MultDivInts7","title":"Simplifying Expressions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Decimals","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a975ae1MultDivInts7a","stepAnswer":["$$-48$$"],"problemType":"TextBox","stepTitle":"Simplify the expression to a single number","stepBody":"7(-2)+4(-7)-6","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-48$$","hints":{"DefaultPathway":[{"id":"a975ae1MultDivInts7a-h1","type":"hint","dependencies":[],"title":"Order of Operations","text":"First, simplify what\'s in the parentheses, then simplify any exponents, then $$\\\\frac{multiply}{divide}$$, then $$\\\\frac{add}{subtract}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a975ae1MultDivInts7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-14$$"],"dependencies":["a975ae1MultDivInts7a-h1"],"title":"What is $$7(-2)$$?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a975ae1MultDivInts7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-28$$"],"dependencies":["a975ae1MultDivInts7a-h2"],"title":"What is $$4(-7)$$?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a975ae1MultDivInts7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-48$$"],"dependencies":["a975ae1MultDivInts7a-h3"],"title":"What is $$-14-28-6$$?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a975ae1MultDivInts8","title":"Simplifying Exponent Expressions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Decimals","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a975ae1MultDivInts8a","stepAnswer":["$$16$$"],"problemType":"TextBox","stepTitle":"Simplify the expression to a single number.","stepBody":"$${\\\\left(-2\\\\right)}^4$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16$$","hints":{"DefaultPathway":[{"id":"a975ae1MultDivInts8a-h1","type":"hint","dependencies":[],"title":"Expanded Form","text":"This is the same as $$\\\\left(-2\\\\right) \\\\left(-2\\\\right) \\\\left(-2\\\\right) \\\\left(-2\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a975ae1MultDivInts8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["a975ae1MultDivInts8a-h1"],"title":"Expanded Form","text":"What is $$\\\\left(-2\\\\right) \\\\left(-2\\\\right) \\\\left(-2\\\\right) \\\\left(-2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a975ae1MultDivInts8b","stepAnswer":["$$-16$$"],"problemType":"TextBox","stepTitle":"Simplify the expression to a single number.","stepBody":"$$-\\\\left(2^4\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-16$$","hints":{"DefaultPathway":[{"id":"a975ae1MultDivInts8b-h1","type":"hint","dependencies":[],"title":"Expanded Form","text":"This is the same as $$-\\\\left(2\\\\times2\\\\times2\\\\times2\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a975ae1MultDivInts8b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-16$$"],"dependencies":["a975ae1MultDivInts8b-h1"],"title":"Expanded Form","text":"What is $$-\\\\left(2\\\\times2\\\\times2\\\\times2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a975ae1MultDivInts9","title":"Simplifying Exponent Expressions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Decimals","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a975ae1MultDivInts9a","stepAnswer":["$$49$$"],"problemType":"TextBox","stepTitle":"Simplify the expression to a single number.","stepBody":"$${\\\\left(-7\\\\right)}^2$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$49$$","hints":{"DefaultPathway":[{"id":"a975ae1MultDivInts9a-h1","type":"hint","dependencies":[],"title":"Expanded Form","text":"This is the same as $$\\\\left(-7\\\\right) \\\\left(-7\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a975ae1MultDivInts9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$49$$"],"dependencies":["a975ae1MultDivInts9a-h1"],"title":"Expanded Form","text":"What is $$\\\\left(-7\\\\right) \\\\left(-7\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a975ae1MultDivInts9b","stepAnswer":["$$-49$$"],"problemType":"TextBox","stepTitle":"Simplify the expression to a single number.","stepBody":"$$-\\\\left(7^2\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-49$$","hints":{"DefaultPathway":[{"id":"a975ae1MultDivInts9b-h1","type":"hint","dependencies":[],"title":"Expanded Form","text":"This is the same as $$-\\\\left(7\\\\times7\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a975ae1MultDivInts9b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-49$$"],"dependencies":["a975ae1MultDivInts9b-h1"],"title":"Expanded Form","text":"What is $$-\\\\left(7\\\\times7\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9790f0System1","title":"System of Equations","body":"Solve the system by graphing:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Solve Systems of Nonlinear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9790f0System1a","stepAnswer":["(3,2)"],"problemType":"TextBox","stepTitle":"$$x-3y=-3$$, $$x+y=5$$","stepBody":"Please enter your answer as (a,b).","answerType":"string","variabilization":{},"answerLatex":"$$(3,2)$$","hints":{"DefaultPathway":[{"id":"a9790f0System1a-h1","type":"hint","dependencies":[],"title":"Point of Intersection","text":"To solve the system of equations by graphing, we will want to graph both functions and see where they intersect. This point will be our answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System1a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(3,2)"],"dependencies":["a9790f0System1a-h1"],"title":"Solution","text":"When we graph our two functions, where do they intersect? Please enter your answer as (a,b).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9790f0System10","title":"System of Equations","body":"Solve the system by substitution:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Solve Systems of Nonlinear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9790f0System10a","stepAnswer":["(0,-3),(1,0)"],"problemType":"TextBox","stepTitle":"$$9x^2+y^2=9$$, $$y=3x-3$$","stepBody":"Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","answerType":"string","variabilization":{},"answerLatex":"$$(0,-3),(1,0)$$","hints":{"DefaultPathway":[{"id":"a9790f0System10a-h1","type":"hint","dependencies":[],"title":"Isolating Variable","text":"To solve the system of equations using substitution, we will want to isolate one of our variables. Then we will replace the variable in our other equation with our isolated formula. We can then solve for one of our variables, after which we can solve for the other one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9x^2+{\\\\left(3x-3\\\\right)}^2=9$$"],"dependencies":["a9790f0System10a-h1"],"title":"Plugging it in","text":"What will our other equation look like once we replace $$y$$ with the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System10a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["x=0,1"],"dependencies":["a9790f0System10a-h2"],"title":"Simplifying","text":"What will our expression look like when simplified?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System10a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["y=-3,0"],"dependencies":["a9790f0System10a-h3"],"title":"Plugging it in","text":"When we plug in our values, what will $$y$$ evaluate to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System10a-h5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(0,-3),(1,0)"],"dependencies":["a9790f0System10a-h4"],"title":"Solution","text":"What are our $$x$$ and $$y$$ values equal to? Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9790f0System11","title":"System of Equations","body":"Solve the system by substitution:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Solve Systems of Nonlinear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9790f0System11a","stepAnswer":["(-3,0),(0,1)"],"problemType":"TextBox","stepTitle":"$$x^2+9y^2=9$$, $$y=\\\\frac{1}{3} x+1$$","stepBody":"Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","answerType":"string","variabilization":{},"answerLatex":"$$(-3,0),(0,1)$$","hints":{"DefaultPathway":[{"id":"a9790f0System11a-h1","type":"hint","dependencies":[],"title":"Isolating Variable","text":"To solve the system of equations using substitution, we will want to isolate one of our variables. Then we will replace the variable in our other equation with our isolated formula. We can then solve for one of our variables, after which we can solve for the other one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2+9{\\\\left(\\\\frac{1}{3} x+1\\\\right)}^2=9$$"],"dependencies":["a9790f0System11a-h1"],"title":"Plugging it in","text":"What will our other equation look like once we replace $$y$$ with the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System11a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["x=-3,0"],"dependencies":["a9790f0System11a-h2"],"title":"Simplifying","text":"What will our expression look like when simplified?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System11a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["y=0,1"],"dependencies":["a9790f0System11a-h3"],"title":"Plugging it in","text":"When we plug in our values, what will $$y$$ evaluate to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System11a-h5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-3,0),(0,1)"],"dependencies":["a9790f0System11a-h4"],"title":"Solution","text":"What are our $$x$$ and $$y$$ values equal to? Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9790f0System12","title":"System of Equations","body":"Solve the system by substitution:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Solve Systems of Nonlinear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9790f0System12a","stepAnswer":["(-4/5,6/5),(0,2)"],"problemType":"TextBox","stepTitle":"$$4x^2+y^2=4$$, $$y=x+2$$","stepBody":"Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a9790f0System12a-h1","type":"hint","dependencies":[],"title":"Isolating Variable","text":"To solve the system of equations using substitution, we will want to isolate one of our variables. Then we will replace the variable in our other equation with our isolated formula. We can then solve for one of our variables, after which we can solve for the other one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4x^2+{\\\\left(x+2\\\\right)}^2=4$$"],"dependencies":["a9790f0System12a-h1"],"title":"Plugging it in","text":"What will our other equation look like once we replace $$y$$ with the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System12a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["x=-.8,0"],"dependencies":["a9790f0System12a-h2"],"title":"Simplifying","text":"What will our expression look like when simplified?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System12a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["y=1.2,2"],"dependencies":["a9790f0System12a-h3"],"title":"Plugging it in","text":"When we plug in our values, what will $$y$$ evaluate to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System12a-h5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-.8,1.2),(0,2)"],"dependencies":["a9790f0System12a-h4"],"title":"Solution","text":"What are our $$x$$ and $$y$$ values equal to? Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9790f0System13","title":"System of Equations","body":"Solve the system by substitution:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Solve Systems of Nonlinear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9790f0System13a","stepAnswer":["(-1,1),(2,4)"],"problemType":"TextBox","stepTitle":"$$x^2-y=0$$, $$y=x+2$$","stepBody":"Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","answerType":"string","variabilization":{},"answerLatex":"$$(-1,1),(2,4)$$","hints":{"DefaultPathway":[{"id":"a9790f0System13a-h1","type":"hint","dependencies":[],"title":"Isolating Variable","text":"To solve the system of equations using substitution, we will want to isolate one of our variables. Then we will replace the variable in our other equation with our isolated formula. We can then solve for one of our variables, after which we can solve for the other one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2-x-2=0$$"],"dependencies":["a9790f0System13a-h1"],"title":"Plugging it in","text":"What will our other equation look like once we replace $$y$$ with the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System13a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["x=-1,2"],"dependencies":["a9790f0System13a-h2"],"title":"Simplifying","text":"What will our expression look like when simplified?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System13a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["y=1,4"],"dependencies":["a9790f0System13a-h3"],"title":"Plugging it in","text":"When we plug in our values, what will $$y$$ evaluate to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System13a-h5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-1,1),(2,4)"],"dependencies":["a9790f0System13a-h4"],"title":"Solution","text":"What are our $$x$$ and $$y$$ values equal to? Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9790f0System14","title":"System of Equations","body":"Solve the system by substitution:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Solve Systems of Nonlinear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9790f0System14a","stepAnswer":["(0,0),(2,4)"],"problemType":"TextBox","stepTitle":"$$x^2-y=0$$, $$y=2x$$","stepBody":"Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","answerType":"string","variabilization":{},"answerLatex":"$$(0,0),(2,4)$$","hints":{"DefaultPathway":[{"id":"a9790f0System14a-h1","type":"hint","dependencies":[],"title":"Isolating Variable","text":"To solve the system of equations using substitution, we will want to isolate one of our variables. Then we will replace the variable in our other equation with our isolated formula. We can then solve for one of our variables, after which we can solve for the other one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2-2x=0$$"],"dependencies":["a9790f0System14a-h1"],"title":"Plugging it in","text":"What will our other equation look like once we replace $$y$$ with the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System14a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["x=0,2"],"dependencies":["a9790f0System14a-h2"],"title":"Simplifying","text":"What will our expression look like when simplified?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System14a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["y=0,4"],"dependencies":["a9790f0System14a-h3"],"title":"Plugging it in","text":"When we plug in our values, what will $$y$$ evaluate to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System14a-h5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(0,0),(2,4)"],"dependencies":["a9790f0System14a-h4"],"title":"Solution","text":"What are our $$x$$ and $$y$$ values equal to? Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9790f0System15","title":"System of Equations","body":"Solve the system by substitution:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Solve Systems of Nonlinear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9790f0System15a","stepAnswer":["(0,0),(1,1)"],"problemType":"TextBox","stepTitle":"$$y^2-x=0$$, $$y=x$$","stepBody":"Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","answerType":"string","variabilization":{},"answerLatex":"$$(0,0),(1,1)$$","hints":{"DefaultPathway":[{"id":"a9790f0System15a-h1","type":"hint","dependencies":[],"title":"Isolating Variable","text":"To solve the system of equations using substitution, we will want to isolate one of our variables. Then we will replace the variable in our other equation with our isolated formula. We can then solve for one of our variables, after which we can solve for the other one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2-x=0$$"],"dependencies":["a9790f0System15a-h1"],"title":"Plugging it in","text":"What will our other equation look like once we replace $$y$$ with the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System15a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["x=0,1"],"dependencies":["a9790f0System15a-h2"],"title":"Simplifying","text":"What will our expression look like when simplified?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System15a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["y=0,1"],"dependencies":["a9790f0System15a-h3"],"title":"Plugging it in","text":"When we plug in our values, what will $$y$$ evaluate to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System15a-h5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(0,0),(1,1)"],"dependencies":["a9790f0System15a-h4"],"title":"Solution","text":"What are our $$x$$ and $$y$$ values equal to? Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9790f0System16","title":"System of Equations","body":"Solve the system by elimination:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Solve Systems of Nonlinear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9790f0System16a","stepAnswer":["(-2,0),(-sqrt(3),-1),(sqrt(3),-1),(2,0)"],"problemType":"TextBox","stepTitle":"$$x^2+y^2=4$$, $$x^2-y=4$$","stepBody":"Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a9790f0System16a-h1","type":"hint","dependencies":[],"title":"Elimination","text":"To solve the system of equations using elimination, we need to first manipulate our equations so that we are able to cancel out one of the terms. Then we will add the two equations together term by term and use our output to find the value of $$x$$ or $$y$$. We can then use our value to find the other missing one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2$$"],"dependencies":["a9790f0System16a-h1"],"title":"Opposite Terms","text":"What is one term that we can cancel out?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System16a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-\\\\left(x^2\\\\right)+4=-4$$"],"dependencies":["a9790f0System16a-h2"],"title":"Manipulating Equations","text":"How can we change one of our equations to cancel out our term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System16a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y^2+y=0$$"],"dependencies":["a9790f0System16a-h3"],"title":"Term by Term Addition","text":"What will our equation look like once we add our equations together term by term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System16a-h5","type":"hint","dependencies":["a9790f0System16a-h4"],"title":"Solving Terms","text":"What will our value(s) of $$y$$ be? $$y=0, -1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System16a-h6","type":"hint","dependencies":["a9790f0System16a-h5"],"title":"Solving Terms","text":"What will our value(s) of $$x$$ be when we plug in $$y$$ values into an equation? $$x=\\\\pm 2\\\\pm \\\\sqrt{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System16a-h7","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-2,0),(-sqrt(3),-1),(sqrt(3),-1),(2,0)"],"dependencies":["a9790f0System16a-h6"],"title":"Solution","text":"What are our $$x$$ and $$y$$ values equal to? Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9790f0System17","title":"System of Equations","body":"Solve the system by elimination:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Solve Systems of Nonlinear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9790f0System17a","stepAnswer":["(-3,0),(-sqrt(8),-1),(sqrt(8),-1),(3,0)"],"problemType":"TextBox","stepTitle":"x**2+y**2=9,x**2-y=9","stepBody":"Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a9790f0System17a-h1","type":"hint","dependencies":[],"title":"Elimination","text":"To solve the system of equations using elimination, we need to first manipulate our equations so that we are able to cancel out one of the terms. Then we will add the two equations together term by term and use our output to find the value of $$x$$ or $$y$$. We can then use our value to find the other missing one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System17a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2$$"],"dependencies":["a9790f0System17a-h1"],"title":"Opposite Terms","text":"What is one term that we can cancel out?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System17a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-\\\\left(x^2\\\\right)+y=-9$$"],"dependencies":["a9790f0System17a-h2"],"title":"Manipulating Equations","text":"How can we change one of our equations to cancel out our term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System17a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y^2+y=0$$"],"dependencies":["a9790f0System17a-h3"],"title":"Term by Term Addition","text":"What will our equation look like once we add our equations together term by term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System17a-h5","type":"hint","dependencies":["a9790f0System17a-h4"],"title":"Solving Terms","text":"What will our value(s) of $$y$$ be? $$y=0, -1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System17a-h6","type":"hint","dependencies":["a9790f0System17a-h5"],"title":"Solving Terms","text":"What will our value(s) of $$x$$ be when we plug in $$y$$ values into an equation? $$x=\\\\pm 3\\\\pm \\\\sqrt{8}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System17a-h7","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-3,0),(-sqrt(8),-1),(sqrt(8),-1),(3,0)"],"dependencies":["a9790f0System17a-h6"],"title":"Solution","text":"What are our $$x$$ and $$y$$ values equal to? Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9790f0System18","title":"System of Equations","body":"Solve the system by elimination:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Solve Systems of Nonlinear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9790f0System18a","stepAnswer":["(-1,0),(0,-1),(0,1)"],"problemType":"TextBox","stepTitle":"$$x^2+y^2=1$$, $$-x+y^2=1$$","stepBody":"Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$. $$b<d$$ if $$a=c$$.","answerType":"string","variabilization":{},"answerLatex":"$$(-1,0),(0,-1),(0,1)$$","hints":{"DefaultPathway":[{"id":"a9790f0System18a-h1","type":"hint","dependencies":[],"title":"Elimination","text":"To solve the system of equations using elimination, we need to first manipulate our equations so that we are able to cancel out one of the terms. Then we will add the two equations together term by term and use our output to find the value of $$x$$ or $$y$$. We can then use our value to find the other missing one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y^2$$"],"dependencies":["a9790f0System18a-h1"],"title":"Opposite Terms","text":"What is one term that we can cancel out?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System18a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x-y^2=-1$$"],"dependencies":["a9790f0System18a-h2"],"title":"Manipulating Equations","text":"How can we change one of our equations to cancel out our term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System18a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2+x=0$$"],"dependencies":["a9790f0System18a-h3"],"title":"Term by Term Addition","text":"What will our equation look like once we add our equations together term by term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System18a-h5","type":"hint","dependencies":["a9790f0System18a-h4"],"title":"Solving Terms","text":"What will our value(s) of $$x$$ be? $$x=-1, 0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System18a-h6","type":"hint","dependencies":["a9790f0System18a-h5"],"title":"Solving Terms","text":"What will our value(s) of $$y$$ be when we plug in $$x$$ values into an equation? $$y=0\\\\pm 1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System18a-h7","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-1,0),(0,-1),(0,1)"],"dependencies":["a9790f0System18a-h6"],"title":"Solution","text":"What are our $$x$$ and $$y$$ values equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9790f0System19","title":"System of Equations","body":"Solve the system by graphing:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Solve Systems of Nonlinear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9790f0System19a","stepAnswer":["(1,2),(2,8)"],"problemType":"TextBox","stepTitle":"$$y=6x-4$$, $$y=2\\\\left(x^2\\\\right)$$","stepBody":"Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","answerType":"string","variabilization":{},"answerLatex":"$$(1,2),(2,8)$$","hints":{"DefaultPathway":[{"id":"a9790f0System19a-h1","type":"hint","dependencies":[],"title":"Point of Intersection","text":"To solve the system of equations by graphing, we will want to graph both functions and see where they intersect. The point(s) will be our answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System19a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(1,2),(2,8)"],"dependencies":["a9790f0System19a-h1"],"title":"Solution","text":"When we graph our two functions, where do they intersect? Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9790f0System2","title":"System of Equations","body":"Solve the system by graphing:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Solve Systems of Nonlinear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9790f0System2a","stepAnswer":["(7,5)"],"problemType":"TextBox","stepTitle":"$$x-2y=-3$$, $$x+y=12$$","stepBody":"Please enter your answer as (a,b).","answerType":"string","variabilization":{},"answerLatex":"$$(7,5)$$","hints":{"DefaultPathway":[{"id":"a9790f0System2a-h1","type":"hint","dependencies":[],"title":"Point of Intersection","text":"To solve the system of equations by graphing, we will want to graph both functions and see where they intersect. This point will be our answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System2a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(7,5)"],"dependencies":["a9790f0System2a-h1"],"title":"Solution","text":"When we graph our two functions, where do they intersect? Please enter your answer as (a,b).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9790f0System20","title":"System of Equations","body":"Solve the system by graphing:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Solve Systems of Nonlinear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9790f0System20a","stepAnswer":["(-2,0),(-1,1)"],"problemType":"TextBox","stepTitle":"$$x-y=-2$$, $$x=y^2-2$$","stepBody":"Please enter your answer as (a,b). 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If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9790f0System21","title":"System of Equations","body":"Solve the system by graphing:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Solve Systems of Nonlinear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9790f0System21a","stepAnswer":["(0,-1),(1,0)"],"problemType":"TextBox","stepTitle":"$$y=x-1$$, $$y=x^2-1$$","stepBody":"Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","answerType":"string","variabilization":{},"answerLatex":"$$(0,-1),(1,0)$$","hints":{"DefaultPathway":[{"id":"a9790f0System21a-h1","type":"hint","dependencies":[],"title":"Point of Intersection","text":"To solve the system of equations by graphing, we will want to graph both functions and see where they intersect. The point(s) will be our answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System21a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(0,-1),(1,0)"],"dependencies":["a9790f0System21a-h1"],"title":"Solution","text":"When we graph our two functions, where do they intersect? Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9790f0System22","title":"System of Equations","body":"Solve the system by graphing:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Solve Systems of Nonlinear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9790f0System22a","stepAnswer":["(-2,0)"],"problemType":"TextBox","stepTitle":"$$x=-2$$, $$x^2+y^2=4$$","stepBody":"Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","answerType":"string","variabilization":{},"answerLatex":"$$(-2,0)$$","hints":{"DefaultPathway":[{"id":"a9790f0System22a-h1","type":"hint","dependencies":[],"title":"Point of Intersection","text":"To solve the system of equations by graphing, we will want to graph both functions and see where they intersect. The point(s) will be our answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System22a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-2,0)"],"dependencies":["a9790f0System22a-h1"],"title":"Solution","text":"When we graph our two functions, where do they intersect? Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9790f0System23","title":"System of Equations","body":"Solve the system by graphing:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Solve Systems of Nonlinear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9790f0System23a","stepAnswer":["(0,-4)"],"problemType":"TextBox","stepTitle":"$$y=-4$$, $$x^2+y^2=16$$","stepBody":"Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","answerType":"string","variabilization":{},"answerLatex":"$$(0,-4)$$","hints":{"DefaultPathway":[{"id":"a9790f0System23a-h1","type":"hint","dependencies":[],"title":"Point of Intersection","text":"To solve the system of equations by graphing, we will want to graph both functions and see where they intersect. The point(s) will be our answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System23a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(0,-4)"],"dependencies":["a9790f0System23a-h1"],"title":"Solution","text":"When we graph our two functions, where do they intersect? Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9790f0System24","title":"System of Equations","body":"Solve the system by graphing:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Solve Systems of Nonlinear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9790f0System24a","stepAnswer":["(1,1),(4,-2)"],"problemType":"TextBox","stepTitle":"$$x+y=2$$, $$x=y^2$$","stepBody":"Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","answerType":"string","variabilization":{},"answerLatex":"$$(1,1),(4,-2)$$","hints":{"DefaultPathway":[{"id":"a9790f0System24a-h1","type":"hint","dependencies":[],"title":"Point of Intersection","text":"To solve the system of equations by graphing, we will want to graph both functions and see where they intersect. The point(s) will be our answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System24a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(1,1),(4,-2)"],"dependencies":["a9790f0System24a-h1"],"title":"Solution","text":"When we graph our two functions, where do they intersect? Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9790f0System25","title":"System of Equations","body":"Solve the system by substitution:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Solve Systems of Nonlinear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9790f0System25a","stepAnswer":["(0,-1),(2,0)"],"problemType":"TextBox","stepTitle":"$$x^2+4\\\\left(y^2\\\\right)=4$$, $$y=\\\\frac{1}{2} x-1$$","stepBody":"Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","answerType":"string","variabilization":{},"answerLatex":"$$(0,-1),(2,0)$$","hints":{"DefaultPathway":[{"id":"a9790f0System25a-h1","type":"hint","dependencies":[],"title":"Isolating Variable","text":"To solve the system of equations using substitution, we will want to isolate one of our variables. Then we will replace the variable in our other equation with our isolated formula. We can then solve for one of our variables, after which we can solve for the other one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System25a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2+16{\\\\left(\\\\frac{1}{2} x-1\\\\right)}^2=4$$"],"dependencies":["a9790f0System25a-h1"],"title":"Plugging it in","text":"What will our other equation look like once we replace $$y$$ with the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System25a-h3","type":"hint","dependencies":["a9790f0System25a-h2"],"title":"Simplifying","text":"What will our expression look like when simplified? $$x=0, 2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System25a-h4","type":"hint","dependencies":["a9790f0System25a-h3"],"title":"Plugging it in","text":"When we plug in our values, what will $$y$$ evaluate to? $$y=-1, 0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System25a-h5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(0,-1),(2,0)"],"dependencies":["a9790f0System25a-h4"],"title":"Solution","text":"What are our $$x$$ and $$y$$ values equal to? Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9790f0System26","title":"System of Equations","body":"Solve the system by substitution:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Solve Systems of Nonlinear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9790f0System26a","stepAnswer":["(-1,0),(3,0)"],"problemType":"TextBox","stepTitle":"$$9x^2+y^2=9$$, $$y=3x+3$$","stepBody":"Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","answerType":"string","variabilization":{},"answerLatex":"$$(-1,0),(3,0)$$","hints":{"DefaultPathway":[{"id":"a9790f0System26a-h1","type":"hint","dependencies":[],"title":"Isolating Variable","text":"To solve the system of equations using substitution, we will want to isolate one of our variables. Then we will replace the variable in our other equation with our isolated formula. We can then solve for one of our variables, after which we can solve for the other one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System26a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9x^2+3x+3=9$$"],"dependencies":["a9790f0System26a-h1"],"title":"Plugging it in","text":"What will our other equation look like once we replace $$y$$ with the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System26a-h3","type":"hint","dependencies":["a9790f0System26a-h2"],"title":"Simplifying","text":"What will our expression look like when simplified? $$x=-1, 3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System26a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y=0$$"],"dependencies":["a9790f0System26a-h3"],"title":"Plugging it in","text":"When we plug in our values, what will $$y$$ evaluate to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System26a-h5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-1,0),(3,0)"],"dependencies":["a9790f0System26a-h4"],"title":"Solution","text":"What are our $$x$$ and $$y$$ values equal to? Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9790f0System27","title":"System of Equations","body":"Solve the system by substitution:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Solve Systems of Nonlinear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9790f0System27a","stepAnswer":["(-.6,2.4),(0,3)"],"problemType":"TextBox","stepTitle":"$$9x^2+y^2=9$$, $$y=3x+3$$","stepBody":"Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","answerType":"string","variabilization":{},"answerLatex":"$$(-.6,2.4),(0,3)$$","hints":{"DefaultPathway":[{"id":"a9790f0System27a-h1","type":"hint","dependencies":[],"title":"Isolating Variable","text":"To solve the system of equations using substitution, we will want to isolate one of our variables. Then we will replace the variable in our other equation with our isolated formula. We can then solve for one of our variables, after which we can solve for the other one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System27a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9x^2+{\\\\left(3x+3\\\\right)}^2=9$$"],"dependencies":["a9790f0System27a-h1"],"title":"Plugging it in","text":"What will our other equation look like once we replace $$y$$ with the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System27a-h3","type":"hint","dependencies":["a9790f0System27a-h2"],"title":"Simplifying","text":"What will our expression look like when simplified? $$x=-.6, 0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System27a-h4","type":"hint","dependencies":["a9790f0System27a-h3"],"title":"Plugging it in","text":"When we plug in our values, what will $$y$$ evaluate to? $$y=2.4, 3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System27a-h5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-1,0),(0,3)"],"dependencies":["a9790f0System27a-h4"],"title":"Solution","text":"What are our $$x$$ and $$y$$ values equal to? Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9790f0System28","title":"System of Equations","body":"Solve the system by substitution:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Solve Systems of Nonlinear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9790f0System28a","stepAnswer":["(12,-5),(12,5)"],"problemType":"TextBox","stepTitle":"$$x^2+y^2=169$$, $$x=12$$","stepBody":"Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$. $$b<d$$ if $$a=c$$.","answerType":"string","variabilization":{},"answerLatex":"$$(12,-5),(12,5)$$","hints":{"DefaultPathway":[{"id":"a9790f0System28a-h1","type":"hint","dependencies":[],"title":"Isolating Variable","text":"To solve the system of equations using substitution, we will want to isolate one of our variables. Then we will replace the variable in our other equation with our isolated formula. We can then solve for one of our variables, after which we can solve for the other one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System28a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$144+y^2=169$$"],"dependencies":["a9790f0System28a-h1"],"title":"Plugging it in","text":"What will our other equation look like once we replace $$x$$ with the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System28a-h3","type":"hint","dependencies":["a9790f0System28a-h2"],"title":"Simplifying","text":"What will our expression look like when simplified? $$y=\\\\pm 5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System28a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=12$$"],"dependencies":["a9790f0System28a-h3"],"title":"Plugging it in","text":"When we plug in our values, what will $$x$$ evaluate to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System28a-h5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(12,-5),(12,5)"],"dependencies":["a9790f0System28a-h4"],"title":"Solution","text":"What are our $$x$$ and $$y$$ values equal to? Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9790f0System29","title":"System of Equations","body":"Solve the system by substitution:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Solve Systems of Nonlinear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9790f0System29a","stepAnswer":["(3,4),(5,0)"],"problemType":"TextBox","stepTitle":"$$x^2+y^2=25$$, $$y=10-2x$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(3,4),(5,0)$$","hints":{"DefaultPathway":[{"id":"a9790f0System29a-h1","type":"hint","dependencies":[],"title":"Isolating Variable","text":"To solve the system of equations using substitution, we will want to isolate one of our variables. Then we will replace the variable in our other equation with our isolated formula. We can then solve for one of our variables, after which we can solve for the other one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System29a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2+{\\\\left(10-2x\\\\right)}^2=25$$"],"dependencies":["a9790f0System29a-h1"],"title":"Plugging it in","text":"What will our other equation look like once we replace $$y$$ with the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System29a-h3","type":"hint","dependencies":["a9790f0System29a-h2"],"title":"Simplifying","text":"What will our expression look like when simplified? $$x=3, 5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System29a-h4","type":"hint","dependencies":["a9790f0System29a-h3"],"title":"Plugging it in","text":"When we plug in our values, what will $$y$$ evaluate to? $$y=4, 0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System29a-h5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(3,4),(5,0)"],"dependencies":["a9790f0System29a-h4"],"title":"Solution","text":"What are our $$x$$ and $$y$$ values equal to? Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9790f0System3","title":"System of Equations","body":"Solve the system by substitution:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Solve Systems of Nonlinear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9790f0System3a","stepAnswer":["(2,1.5)"],"problemType":"TextBox","stepTitle":"$$x-4y=-4$$, $$-3x+4y=0$$","stepBody":"Please enter your answer as (a,b).","answerType":"string","variabilization":{},"answerLatex":"$$(2, 1.5)$$","hints":{"DefaultPathway":[{"id":"a9790f0System3a-h1","type":"hint","dependencies":[],"title":"Isolating Variable","text":"To solve the system of equations using substitution, we will want to isolate one of our variables. Then we will replace the variable in our other equation with our isolated formula. We can then solve for one of our variables, after which we can solve for the other one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=4y-4$$"],"dependencies":["a9790f0System3a-h1"],"title":"Isolating Variable","text":"If we have the equation $$x-4y=-4$$ and we isolate $$x$$, what will our equation look like?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3\\\\left(4y-4\\\\right)+4y=0$$"],"dependencies":["a9790f0System3a-h2"],"title":"Plugging it in","text":"With our $$x$$ variable isolated, what will our other equation look like once we replace $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-8y+12=0$$"],"dependencies":["a9790f0System3a-h3"],"title":"Simplify","text":"What will our equation look like simplified?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.5$$"],"dependencies":["a9790f0System3a-h4"],"title":"Solve for $$y$$","text":"What is $$y$$ equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System3a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a9790f0System3a-h5"],"title":"Solve for $$x$$","text":"What does $$x=4(1.5)-4$$ evaluate to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9790f0System3a-h7","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(2,1.5)"],"dependencies":["a9790f0System3a-h6"],"title":"Solution","text":"What are our $$x$$ and $$y$$ equal to? Please enter your answer as (a,b).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9790f0System30","title":"System of Equations","body":"Solve the system by substitution:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Solve Systems of Nonlinear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9790f0System30a","stepAnswer":["(0,3),(1,4)"],"problemType":"TextBox","stepTitle":"$$y=x^2+3$$, $$x=y-3$$","stepBody":"Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","answerType":"string","variabilization":{},"answerLatex":"$$(0,3),(1,4)$$","hints":{"DefaultPathway":[{"id":"a9790f0System30a-h1","type":"hint","dependencies":[],"title":"Isolating Variable","text":"To solve the system of equations using substitution, we will want to isolate one of our variables. Then we will replace the variable in our other equation with our isolated formula. 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Please enter your answer as (a,b). If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9790f0System4","title":"System of Equations","body":"Solve the system by graphing:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Solve Systems of Nonlinear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9790f0System4a","stepAnswer":["(-1,1),(2,4)"],"problemType":"TextBox","stepTitle":"$$x-y=-2$$, $$y=x^2$$","stepBody":"Please enter your answer as (a,b). 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If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9790f0System6","title":"System of Equations","body":"Solve the system by graphing:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Solve Systems of Nonlinear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9790f0System6a","stepAnswer":["(-2,-1), (1,2)"],"problemType":"TextBox","stepTitle":"$$x-y=-1$$, $$y=-\\\\left(x^2\\\\right)+3$$","stepBody":"Please enter your answer as (a,b). 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If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9790f0System7","title":"System of Equations","body":"Solve the system by graphing:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Solve Systems of Nonlinear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9790f0System7a","stepAnswer":["(2,-1)"],"problemType":"TextBox","stepTitle":"$$y=-1$$, $${\\\\left(x-2\\\\right)}^2+{\\\\left(y+3\\\\right)}^2=4$$","stepBody":"Please enter your answer as (a,b). 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If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9790f0System8","title":"System of Equations","body":"Solve the system by graphing:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Solve Systems of Nonlinear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9790f0System8a","stepAnswer":["(-6,1)"],"problemType":"TextBox","stepTitle":"$$x=-6$$, $${\\\\left(x+3\\\\right)}^2+{\\\\left(y-1\\\\right)}^2=9$$","stepBody":"Please enter your answer as (a,b). 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If there is more than one answer, please enter them as (a,b), (c,d) where $$a<c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9790f0System9","title":"System of Equations","body":"Solve the system by graphing:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Solve Systems of Nonlinear Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9790f0System9a","stepAnswer":["(2,-1)"],"problemType":"TextBox","stepTitle":"$$y=-1$$, $${\\\\left(x-2\\\\right)}^2+{\\\\left(y+3\\\\right)}^2=4$$","stepBody":"Please enter your answer as (a,b). 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What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square1a-h3","type":"hint","dependencies":["a985d52Square1a-h2"],"title":"Creating a Binomial Squared","text":"Rewrite as a binomial squared.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(y+6\\\\right)}^2$$"],"dependencies":["a985d52Square1a-h3"],"title":"Creating a Binomial Squared","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a985d52Square10","title":"Completing the Square","body":"Solve the following equation by completing the square.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a985d52Square10a","stepAnswer":["DNE"],"problemType":"MultipleChoice","stepTitle":"$$z^2+8z=-19$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$-4+\\\\sqrt{3}$$","$$-4-\\\\sqrt{3}$$","$$-4\\\\pm \\\\sqrt{3}$$","DNE"],"hints":{"DefaultPathway":[{"id":"a985d52Square10a-h1","type":"hint","dependencies":[],"title":"Creating a Perfect Square Trinomial","text":"First identify the coefficient of the term without the square.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["a985d52Square10a-h1"],"title":"Creating a Perfect Square Trinomial","text":"Take the coefficient, divide it by $$2$$, then take the answer and square it. 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What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square10a-h3","type":"hint","dependencies":["a985d52Square10a-h2"],"title":"Creating a Binomial Squared","text":"Adding the square on both side","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(z+4\\\\right)}^2=-3$$"],"dependencies":["a985d52Square10a-h3"],"title":"Creating a Binomial Squared","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square10a-h5","type":"hint","dependencies":[],"title":"Calculation","text":"Take the square root of both side","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square10a-h6","type":"hint","dependencies":["a985d52Square10a-h4","a985d52Square10a-h5"],"title":"Principle","text":"$$\\\\sqrt{-3}$$ does not exist","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a985d52Square11","title":"Completing the Square","body":"Solve the following equation by completing the square.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a985d52Square11a","stepAnswer":["$$9\\\\pm 5\\\\sqrt{3}$$"],"problemType":"MultipleChoice","stepTitle":"$$p^2-18p=-6$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$9\\\\pm 5\\\\sqrt{3}$$","choices":["$$9+5\\\\sqrt{3}$$","$$9-5\\\\sqrt{3}$$","$$9\\\\pm 5\\\\sqrt{3}$$"],"hints":{"DefaultPathway":[{"id":"a985d52Square11a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(p-9\\\\right)}^2=75$$"],"dependencies":[],"title":"Completing the Square","text":"Create a binomial squared. 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What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square12a-h2","type":"hint","dependencies":["a985d52Square12a-h1"],"title":"Solving the equation","text":"Take the square root of both sides and solve for $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square12a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4\\\\pm 3\\\\sqrt{3}$$"],"dependencies":["a985d52Square12a-h2"],"title":"Solving the equation","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square12a-h4","type":"hint","dependencies":["a985d52Square12a-h3"],"title":"Double Checking","text":"Double check that the answers work by plugging them back into the original equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a985d52Square13","title":"Completing the Square","body":"Solve the following equation by completing the square.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a985d52Square13a","stepAnswer":["$$1$$ or $$-11$$"],"problemType":"MultipleChoice","stepTitle":"$$x^2+10x+4=15$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$1$$ or $$-11$$","choices":["$$1$$ or $$-11$$","$$-1$$ or $$-11$$","$$1$$ or $$11$$"],"hints":{"DefaultPathway":[{"id":"a985d52Square13a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(x+5\\\\right)}^2=36$$"],"dependencies":[],"title":"Completing the Square","text":"Create a binomial squared. What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square13a-h2","type":"hint","dependencies":["a985d52Square13a-h1"],"title":"Solving the equation","text":"Take the square root of both sides and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square13a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$1$$ or $$-11$$"],"dependencies":["a985d52Square13a-h2"],"title":"Solving the equation","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$1$$ or $$-11$$","$$-1$$ or $$-11$$","$$1$$ or $$11$$"]},{"id":"a985d52Square13a-h4","type":"hint","dependencies":["a985d52Square13a-h3"],"title":"Double Checking","text":"Double check that the answers work by plugging them back into the original equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a985d52Square14","title":"Completing the Square","body":"Solve the following equation by completing the square.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a985d52Square14a","stepAnswer":["$$3$$ or $$-7$$"],"problemType":"MultipleChoice","stepTitle":"$$a^2+4a+9=30$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$3$$ or $$-7$$","choices":["$$3$$ or $$-7$$","$$-3$$ or $$-7$$","$$3$$ or $$7$$"],"hints":{"DefaultPathway":[{"id":"a985d52Square14a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(a+2\\\\right)}^2=25$$"],"dependencies":[],"title":"Completing the Square","text":"Create a binomial squared. What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square14a-h2","type":"hint","dependencies":["a985d52Square14a-h1"],"title":"Solving the equation","text":"Take the square root of both sides and solve for a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square14a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3$$ or $$-7$$"],"dependencies":["a985d52Square14a-h2"],"title":"Solving the equation","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$3$$ or $$-7$$","$$-3$$ or $$-7$$","$$3$$ or $$7$$"]},{"id":"a985d52Square14a-h4","type":"hint","dependencies":["a985d52Square14a-h3"],"title":"Double Checking","text":"Double check that the answers work by plugging them back into the original equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a985d52Square15","title":"Completing the Square","body":"Solve the following equation by completing the square.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a985d52Square15a","stepAnswer":["$$\\\\frac{5\\\\pm \\\\sqrt{61}}{2}$$"],"problemType":"MultipleChoice","stepTitle":"$$p^2=5p+9$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{5\\\\pm \\\\sqrt{61}}{2}$$","choices":["$$\\\\frac{5\\\\pm \\\\sqrt{61}}{2}$$","$$\\\\frac{5+\\\\sqrt{61}}{2}$$","$$\\\\frac{5-\\\\sqrt{61}}{2}$$"],"hints":{"DefaultPathway":[{"id":"a985d52Square15a-h1","type":"hint","dependencies":[],"title":"Organizing","text":"Place all variables on one side: $$p^2-5p=9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(p-\\\\frac{5}{2}\\\\right)}^2=\\\\frac{61}{4}$$"],"dependencies":[],"title":"Completing the Square","text":"Create a binomial squared. What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square15a-h3","type":"hint","dependencies":["a985d52Square15a-h2"],"title":"Solving the equation","text":"Take the square root of both sides and solve for a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5\\\\pm \\\\sqrt{61}}{2}$$"],"dependencies":["a985d52Square15a-h3"],"title":"Solving the equation","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square15a-h5","type":"hint","dependencies":["a985d52Square15a-h4"],"title":"Double Checking","text":"Double check that the answers work by plugging them back into the original equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a985d52Square16","title":"Solve Quadratic Equations Using the Quadratic Formula","body":"Solve the following exercise by using the Quadratic Formula:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a985d52Square16a","stepAnswer":["$$0.75$$ and $$-1$$"],"problemType":"MultipleChoice","stepTitle":"$$4m^2+m-3=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$0.75$$ and $$-1$$","choices":["$$0.75$$ and $$-1$$","$$-0.25$$ and $$2$$","$$-4$$ and $$3$$","$$-2$$ and $$1$$"],"hints":{"DefaultPathway":[{"id":"a985d52Square16a-h1","type":"hint","dependencies":[],"title":"Equation in Standard Form","text":"Since the equation is already in standard form, we can identify the a, $$b$$, and c values based on $$a x^2+b x+c=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square16a-h2","type":"hint","dependencies":["a985d52Square16a-h1"],"title":"Plugging Into Quadratic Formula","text":"Write the quadratic formula. Then substitute in the value of a, $$b$$, and c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square16a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$0.75$$ and $$-1$$"],"dependencies":["a985d52Square16a-h2"],"title":"Isolating the Variable","text":"Simplify the fraction and solve for the variable. 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Then substitute in the value of a, $$b$$, and c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square18a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3$$ and $$0.5$$"],"dependencies":["a985d52Square18a-h2"],"title":"Isolating the Variable","text":"Simplify the fraction and solve for the variable. 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What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square4a-h3","type":"hint","dependencies":["a985d52Square4a-h2"],"title":"Creating a Binomial Squared","text":"Rewrite as a binomial squared.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(u-\\\\frac{9}{2}\\\\right)}^2$$"],"dependencies":["a985d52Square4a-h3"],"title":"Creating a Binomial Squared","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a985d52Square5","title":"Completing the Square","body":"First complete the following square by creating a perfect square trinomial then convert the result into a binomial squared.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a985d52Square5a","stepAnswer":["$${\\\\left(p+\\\\frac{1}{8}\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$p^2+\\\\frac{1}{4} p$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(p+\\\\frac{1}{8}\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"a985d52Square5a-h1","type":"hint","dependencies":[],"title":"Creating a Perfect Square Trinomial","text":"First identify the coefficient of the term without the square.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{64}$$"],"dependencies":["a985d52Square5a-h1"],"title":"Creating a Perfect Square Trinomial","text":"Take the coefficient, divide it by $$2$$, then take the answer and square it. This will complete the square. What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square5a-h3","type":"hint","dependencies":["a985d52Square5a-h2"],"title":"Creating a Binomial Squared","text":"Rewrite as a binomial squared.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(p+\\\\frac{1}{8}\\\\right)}^2$$"],"dependencies":["a985d52Square5a-h3"],"title":"Creating a Binomial Squared","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a985d52Square6","title":"Completing the Square","body":"Solve the following equation by completing the square.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a985d52Square6a","stepAnswer":["$${\\\\left(c+2\\\\right)}^2=9$$"],"problemType":"TextBox","stepTitle":"$$c^2+4c=5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(c+2\\\\right)}^2=9$$","hints":{"DefaultPathway":[{"id":"a985d52Square6a-h1","type":"hint","dependencies":[],"title":"Creating a Perfect Square Trinomial","text":"First identify the coefficient of the term without the square.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a985d52Square6a-h1"],"title":"Creating a Perfect Square Trinomial","text":"Take the coefficient, divide it by $$2$$, then take the answer and square it. This will complete the square. What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square6a-h3","type":"hint","dependencies":["a985d52Square6a-h2"],"title":"Creating a Binomial Squared","text":"Adding the square on both side","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(c+2\\\\right)}^2=9$$"],"dependencies":["a985d52Square6a-h3"],"title":"Creating a Binomial Squared","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a985d52Square7","title":"Completing the Square","body":"Solve the following equation by completing the square.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a985d52Square7a","stepAnswer":["$${\\\\left(d+5\\\\right)}^2=16$$"],"problemType":"TextBox","stepTitle":"$$d^2+10d=-9$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(d+5\\\\right)}^2=16$$","hints":{"DefaultPathway":[{"id":"a985d52Square7a-h1","type":"hint","dependencies":[],"title":"Creating a Perfect Square Trinomial","text":"First identify the coefficient of the term without the square.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["a985d52Square7a-h1"],"title":"Creating a Perfect Square Trinomial","text":"Take the coefficient, divide it by $$2$$, then take the answer and square it. This will complete the square. What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square7a-h3","type":"hint","dependencies":["a985d52Square7a-h2"],"title":"Creating a Binomial Squared","text":"Adding the square on both side","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(d+5\\\\right)}^2=16$$"],"dependencies":["a985d52Square7a-h3"],"title":"Creating a Binomial Squared","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a985d52Square8","title":"Completing the Square","body":"Solve the following equation by completing the square.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a985d52Square8a","stepAnswer":["$${\\\\left(y-3\\\\right)}^2=25$$"],"problemType":"TextBox","stepTitle":"$$y^2-6y=16$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(y-3\\\\right)}^2=25$$","hints":{"DefaultPathway":[{"id":"a985d52Square8a-h1","type":"hint","dependencies":[],"title":"Creating a Perfect Square Trinomial","text":"First identify the coefficient of the term without the square.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a985d52Square8a-h1"],"title":"Creating a Perfect Square Trinomial","text":"Take the coefficient, divide it by $$2$$, then take the answer and square it. This will complete the square. What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square8a-h3","type":"hint","dependencies":["a985d52Square8a-h2"],"title":"Creating a Binomial Squared","text":"Adding the square on both side","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(y-3\\\\right)}^2=25$$"],"dependencies":["a985d52Square8a-h3"],"title":"Creating a Binomial Squared","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a985d52Square9","title":"Completing the Square","body":"Solve the following equation by completing the square.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Solve Quadratic Equations Using the Quadratic Formula","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a985d52Square9a","stepAnswer":["DNE"],"problemType":"MultipleChoice","stepTitle":"$$y^2-10y=-35$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$5+\\\\sqrt{10}$$","$$5-\\\\sqrt{10}$$","$$5\\\\pm \\\\sqrt{10}$$","DNE"],"hints":{"DefaultPathway":[{"id":"a985d52Square9a-h1","type":"hint","dependencies":[],"title":"Creating a Perfect Square Trinomial","text":"First identify the coefficient of the term without the square.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["a985d52Square9a-h1"],"title":"Creating a Perfect Square Trinomial","text":"Take the coefficient, divide it by $$2$$, then take the answer and square it. This will complete the square. What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square9a-h3","type":"hint","dependencies":["a985d52Square9a-h2"],"title":"Creating a Binomial Squared","text":"Adding the square on both side","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(y-5\\\\right)}^2=-10$$"],"dependencies":["a985d52Square9a-h3"],"title":"Creating a Binomial Squared","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square9a-h5","type":"hint","dependencies":[],"title":"Calculation","text":"Take the square root of both side","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a985d52Square9a-h6","type":"hint","dependencies":["a985d52Square9a-h4","a985d52Square9a-h5"],"title":"Principle","text":"$$\\\\sqrt{-10}$$ does not exist","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a98b1afgraphlog1","title":"Identifying the Domain of a Logarithmic Shift","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Graphs of Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a98b1afgraphlog1a","stepAnswer":["$$(-3,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"What is the domain of $$f(x)=\\\\log_{2}\\\\left(x+3\\\\right)$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-3,\\\\infty)$$","choices":["$$(3,\\\\infty)$$","$$(-3,\\\\infty)$$","$$(0,\\\\infty)$$","none of the above"],"hints":{"DefaultPathway":[{"id":"a98b1afgraphlog1a-h1","type":"hint","dependencies":[],"title":"Domain of log functions","text":"The domain of the log function $$f(x)=\\\\log_{a}\\\\left(x\\\\right)$$ is $$(0,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a98b1afgraphlog1a-h2","type":"hint","dependencies":[],"title":"Transformations","text":"$$f(x)=\\\\log_{2}\\\\left(x+3\\\\right)$$ translates the parent function $$f(x)=\\\\log_{2}\\\\left(x\\\\right)$$ $$3$$ units to the left. How does this affect the domain?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a98b1afgraphlog10","title":"Vertical asymptotes of log functions","body":"What is the vertical asymptote of $$f(x)=\\\\ln(3x+1)$$? Write as $$x=[number]$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Graphs of Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a98b1afgraphlog10a","stepAnswer":["$$x=\\\\frac{-1}{3}$$"],"problemType":"TextBox","stepTitle":"","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x=\\\\frac{-1}{3}$$","hints":{"DefaultPathway":[{"id":"a98b1afgraphlog10a-h1","type":"hint","dependencies":[],"title":"Characteristics of the graph of the parent function","text":"The function $$f(x)=\\\\log_{b}\\\\left(x\\\\right)$$, $$b>0$$ and $$b$$ not equal to $$1$$, has the following properties: - one-to-one function\\\\n- vertical asymptote: $$x=0$$\\\\n- domain: $$(0,\\\\infty)$$\\\\n- range: $$(-\\\\infty,\\\\infty)$$\\\\n- x-intercept: $$(1,0)$$ and key point (b,1)\\\\n- y-intercept: none\\\\n- increasing if $$b>1$$\\\\n- decreasing if $$0<b<1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a98b1afgraphlog10a-h2","type":"hint","dependencies":[],"title":"Transformations of log functions","text":"To achieve the function above, the parent function is translated $$1$$ unit to the left and horizontally compressed by a factor of $$\\\\frac{1}{3}$$. How does this affect the vertical asymptote?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a98b1afgraphlog11","title":"Vertical asymptotes of log functions","body":"What is the vertical asymptote of $$f(x)=3\\\\ln(-x)+2$$? Write as an equation of $$x$$ (e.g. $$x=[number])$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Graphs of Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a98b1afgraphlog11a","stepAnswer":["$$x=0$$"],"problemType":"TextBox","stepTitle":"","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x=0$$","hints":{"DefaultPathway":[{"id":"a98b1afgraphlog11a-h1","type":"hint","dependencies":[],"title":"Characteristics of the graph of the parent function","text":"The function $$f(x)=\\\\log_{b}\\\\left(x\\\\right)$$, $$b>0$$ and $$b$$ not equal to $$1$$, has the following properties: - one-to-one function\\\\n- vertical asymptote: $$x=0$$\\\\n- domain: $$(0,\\\\infty)$$\\\\n- range: $$(-\\\\infty,\\\\infty)$$\\\\n- x-intercept: $$(1,0)$$ and key point (b,1)\\\\n- y-intercept: none\\\\n- increasing if $$b>1$$\\\\n- decreasing if $$0<b<1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a98b1afgraphlog11a-h2","type":"hint","dependencies":[],"title":"Transformations of log functions","text":"To achieve the function above, the parent function is reflected across the $$y$$ axis, vertically stretched by a factor of $$3$$, and translated up by $$2$$ units. How does this affect the vertical asymptote?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a98b1afgraphlog12","title":"Vertical Asymptotes of log functions","body":"What is the vertical asymptote of the function $$g(x)=-\\\\ln(3x+9)-7$$? Write as an equation of $$x$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Graphs of Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a98b1afgraphlog12a","stepAnswer":["$$x=-3$$"],"problemType":"TextBox","stepTitle":"","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x=-3$$","hints":{"DefaultPathway":[{"id":"a98b1afgraphlog12a-h1","type":"hint","dependencies":[],"title":"Characteristics of the graph of the parent function","text":"The function $$f(x)=\\\\log_{b}\\\\left(x\\\\right)$$, $$b>0$$ and $$b$$ not equal to $$1$$, has the following properties: - one-to-one function\\\\n- vertical asymptote: $$x=0$$\\\\n- domain: $$(0,\\\\infty)$$\\\\n- range: $$(-\\\\infty,\\\\infty)$$\\\\n- x-intercept: $$(1,0)$$ and key point (b,1)\\\\n- y-intercept: none\\\\n- increasing if $$b>1$$\\\\n- decreasing if $$0<b<1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a98b1afgraphlog12a-h2","type":"hint","dependencies":[],"title":"Transformations of log functions","text":"To achieve the function above, the parent function is translated left $$9$$ units, horizontally compressed by a factor of $$\\\\frac{1}{3}$$, reflected across the x-axis, and translated down $$7$$ units. How does this affect the vertical asymptote?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a98b1afgraphlog13","title":"Domain and vertical asymptotes of log functions","body":"What is the domain and vertical asymptote of $$f(x)=ln(2-x)$$?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Graphs of Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a98b1afgraphlog13a","stepAnswer":["domain: (-inf, 2), vertical asymptote: $$x=2$$"],"problemType":"MultipleChoice","stepTitle":"","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"domain: (-inf, 2), vertical asymptote: $$x=2$$","choices":["domain: (-inf, 2), vertical asymptote: $$x=2$$","domain: (2, inf), vertical asymptote: $$x=2$$","none of the above"],"hints":{"DefaultPathway":[{"id":"a98b1afgraphlog13a-h1","type":"hint","dependencies":[],"title":"Characteristics of the graph of the parent function","text":"The function $$f(x)=\\\\log_{b}\\\\left(x\\\\right)$$, $$b>0$$ and $$b$$ not equal to $$1$$, has the following properties: - one-to-one function\\\\n- vertical asymptote: $$x=0$$\\\\n- domain: $$(0,\\\\infty)$$\\\\n- range: $$(-\\\\infty,\\\\infty)$$\\\\n- x-intercept: $$(1,0)$$ and key point (b,1)\\\\n- y-intercept: none\\\\n- increasing if $$b>1$$\\\\n- decreasing if $$0<b<1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a98b1afgraphlog13a-h2","type":"hint","dependencies":[],"title":"Transformations of log functions","text":"To achieve the function above, the parent function is translated left $$2$$ units and then reflected across the y-axis. How does this affect the domain and vertical asymptote?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a98b1afgraphlog14","title":"Domain and vertical asymptotes of log functions","body":"What is the domain and vertical asymptote of the function $$f(x)=\\\\ln(x-\\\\frac{3}{7})$$?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Graphs of Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a98b1afgraphlog14a","stepAnswer":["domain: (3/7, inf), vertical asymptote: $$x=\\\\frac{3}{7}$$"],"problemType":"MultipleChoice","stepTitle":"","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"domain: (3/7, inf), vertical asymptote: $$x=\\\\frac{3}{7}$$","choices":["domain: (3/7, inf), vertical asymptote: $$x=\\\\frac{3}{7}$$","none of the above","domain: (-inf, inf), vertical asymptote: $$x=\\\\frac{3}{7}$$","domain: (3/7, inf), vertical asymptote: $$x=\\\\frac{3}{7}$$"],"hints":{"DefaultPathway":[{"id":"a98b1afgraphlog14a-h1","type":"hint","dependencies":[],"title":"Characteristics of the graph of the parent function","text":"The function $$f(x)=\\\\log_{b}\\\\left(x\\\\right)$$, $$b>0$$ and $$b$$ not equal to $$1$$, has the following properties: - one-to-one function\\\\n- vertical asymptote: $$x=0$$\\\\n- domain: $$(0,\\\\infty)$$\\\\n- range: $$(-\\\\infty,\\\\infty)$$\\\\n- x-intercept: $$(1,0)$$ and key point (b,1)\\\\n- y-intercept: none\\\\n- increasing if $$b>1$$\\\\n- decreasing if $$0<b<1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a98b1afgraphlog14a-h2","type":"hint","dependencies":[],"title":"Transformations of log functions","text":"To achieve the function above, the parent function is translated $$\\\\frac{3}{7}$$ units to the right. How does this affect the domain and vertical asymptote?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a98b1afgraphlog15","title":"Domain and vertical asymptotes of log functions","body":"What is the domain and vertical asymptote of $$f(x)=-\\\\ln(3x-4)+3$$?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Graphs of Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a98b1afgraphlog15a","stepAnswer":["domain: (4/3, inf), vertical asymptote: $$x=\\\\frac{4}{3}$$"],"problemType":"MultipleChoice","stepTitle":"","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"domain: (4/3, inf), vertical asymptote: $$x=\\\\frac{4}{3}$$","choices":["domain: (4/3, inf), vertical asymptote: $$x=\\\\frac{4}{3}$$","domain: (-inf, inf), vertical asymptote: $$x=\\\\frac{4}{3}$$","none of the above"],"hints":{"DefaultPathway":[{"id":"a98b1afgraphlog15a-h1","type":"hint","dependencies":[],"title":"Characteristics of the graph of the parent function","text":"The function $$f(x)=\\\\log_{b}\\\\left(x\\\\right)$$, $$b>0$$ and $$b$$ not equal to $$1$$, has the following properties: - one-to-one function\\\\n- vertical asymptote: $$x=0$$\\\\n- domain: $$(0,\\\\infty)$$\\\\n- range: $$(-\\\\infty,\\\\infty)$$\\\\n- x-intercept: $$(1,0)$$ and key point (b,1)\\\\n- y-intercept: none\\\\n- increasing if $$b>1$$\\\\n- decreasing if $$0<b<1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a98b1afgraphlog15a-h2","type":"hint","dependencies":[],"title":"Transformations of log functions","text":"To achieve the function above, the parent function is translated $$4$$ unites to the right, horizontally compressed by a factor of $$\\\\frac{1}{3}$$, reflected across the x-axis, and translated up $$3$$ units. How does this affect the domain and vertical asymptote?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a98b1afgraphlog16","title":"Domain and vertical asymptotes of log functions","body":"What is the domain and vertical asymptote of $$g(x)=\\\\ln(2x+6)-5$$?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Graphs of Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a98b1afgraphlog16a","stepAnswer":["domain: $$(-3$$, inf), vertical asymptote: $$x=-3$$"],"problemType":"MultipleChoice","stepTitle":"","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"domain: (-3, inf), vertical asymptote: $$x=-3$$","choices":["domain: (-inf, inf), vertical asymptote: $$x=-3$$","none of the above","domain: $$(-3$$, inf), vertical asymptote: $$x=-3$$","domain: $$(-3$$, inf), vertical asymptote: $$x=-3$$"],"hints":{"DefaultPathway":[{"id":"a98b1afgraphlog16a-h1","type":"hint","dependencies":[],"title":"Characteristics of the graph of the parent function","text":"The function $$f(x)=\\\\log_{b}\\\\left(x\\\\right)$$, $$b>0$$ and $$b$$ not equal to $$1$$, has the following properties: - one-to-one function\\\\n- vertical asymptote: $$x=0$$\\\\n- domain: $$(0,\\\\infty)$$\\\\n- range: $$(-\\\\infty,\\\\infty)$$\\\\n- x-intercept: $$(1,0)$$ and key point (b,1)\\\\n- y-intercept: none\\\\n- increasing if $$b>1$$\\\\n- decreasing if $$0<b<1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a98b1afgraphlog16a-h2","type":"hint","dependencies":[],"title":"Transformations of log functions","text":"To achieve the function above, the parent function is translated $$6$$ units to the left, horizontally compressed by a factor of $$\\\\frac{1}{2}$$, and translated $$5$$ units down. How does this affect the domain and vertical asymptote?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a98b1afgraphlog17","title":"Domain and vertical asymptotes of log functions","body":"What is the domain and vertical asymptote of log{3}(15-5x)+6?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Graphs of Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a98b1afgraphlog17a","stepAnswer":["domain: (-inf, 3), vertical asymptote: $$x=3$$"],"problemType":"MultipleChoice","stepTitle":"","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"domain: (-inf, 3), vertical asymptote: $$x=3$$","choices":["domain: (-inf ,inf), vertical asymptote: $$x=3$$","none of the above","domain: (-inf, 3), vertical asymptote: $$x=3$$","domain: (-inf, 3), vertical asymptote: $$x=3$$"],"hints":{"DefaultPathway":[{"id":"a98b1afgraphlog17a-h1","type":"hint","dependencies":[],"title":"Characteristics of the graph of the parent function","text":"The function $$f(x)=\\\\log_{b}\\\\left(x\\\\right)$$, $$b>0$$ and $$b$$ not equal to $$1$$, has the following properties: - one-to-one function\\\\n- vertical asymptote: $$x=0$$\\\\n- domain: $$(0,\\\\infty)$$\\\\n- range: $$(-\\\\infty,\\\\infty)$$\\\\n- x-intercept: $$(1,0)$$ and key point (b,1)\\\\n- y-intercept: none\\\\n- increasing if $$b>1$$\\\\n- decreasing if $$0<b<1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a98b1afgraphlog17a-h2","type":"hint","dependencies":[],"title":"Transformations of log functions","text":"To achieve the function above, the parent function is translated left by $$15$$ units, horizontally compressed by a factor of $$\\\\frac{1}{5}$$, reflected across the y-axis, and translated $$6$$ units up. How does this affect the domain and vertical asymptote?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a98b1afgraphlog18","title":"Finding the $$x$$ and $$y$$ intercepts of log functions","body":"What is the x-intercept of h(x)=log{4}(x-1)+1? Write as a coordinate pair. If it does not exist, write DNE.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Graphs of Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a98b1afgraphlog18a","stepAnswer":["(5/4, 0)"],"problemType":"TextBox","stepTitle":"","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a98b1afgraphlog18a-h1","type":"hint","dependencies":[],"title":"Definition of x-intercept","text":"The x-intercept describes the point where the function crosses the x-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a98b1afgraphlog18a-h2","type":"hint","dependencies":[],"title":"Setting up the equation","text":"Set the function equal to $$0$$. This represents the point in which the function crosses the x-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a98b1afgraphlog18a-h3","type":"hint","dependencies":[],"title":"Solving for $$x$$","text":"The last step is to solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a98b1afgraphlog19","title":"Finding the $$x$$ and $$y$$ intercepts of log functions","body":"What is the y-intercept of $$\\\\ln(5x+10)+3$$? Write as a coordinate pair. If it does not exist, write DNE.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Graphs of Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a98b1afgraphlog19a","stepAnswer":["(0,4)"],"problemType":"TextBox","stepTitle":"","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,4)$$","hints":{"DefaultPathway":[{"id":"a98b1afgraphlog19a-h1","type":"hint","dependencies":[],"title":"Definition of y-intercept","text":"The y-intercept describes the point in which the function crosses the y-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a98b1afgraphlog19a-h2","type":"hint","dependencies":[],"title":"Setting up the equation","text":"Let $$x=0$$. Solve the function f(x) $$=$$ $$\\\\ln(5x+10)+3$$. This represents the point in which the function crosses the y-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a98b1afgraphlog2","title":"Identifying the Domain of a Logarithmic Shift","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Graphs of Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a98b1afgraphlog2a","stepAnswer":["$$(-\\\\infty,\\\\frac{5}{2})$$"],"problemType":"MultipleChoice","stepTitle":"What is the domain for $$f(x)=\\\\ln(5-2x)$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\frac{5}{2})$$","choices":["$$(-\\\\infty,\\\\frac{5}{2})$$","$$(\\\\frac{5}{2},\\\\infty)$$","$$(-\\\\infty,\\\\infty)$$","$$(0,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"a98b1afgraphlog2a-h1","type":"hint","dependencies":[],"title":"Domain of log functions","text":"The domain of the log function $$f(x)=\\\\log_{a}\\\\left(x\\\\right)$$ is $$(0,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a98b1afgraphlog2a-h2","type":"hint","dependencies":[],"title":"Transformations","text":"$$f(x)=\\\\ln(5-2x)$$ translates the parent function $$5$$ units left, horizontally shrinks it by a factor of $$\\\\frac{1}{2}$$, and reflects it across the y-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a98b1afgraphlog20","title":"Finding the $$x$$ and $$y$$ intercepts of log functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Graphs of Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a98b1afgraphlog20a","stepAnswer":["DNE"],"problemType":"TextBox","stepTitle":"What is the y-intercept of $$ln(-x)-2$$? Write as a coordinate pair. If it doesn\'t exist, write DNE.","stepBody":"Write as a coordinate pair. If it doesn\'t exist, write DNE.","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a98b1afgraphlog20a-h1","type":"hint","dependencies":[],"title":"Characteristics of the graph of the parent function","text":"The function $$f(x)=\\\\log_{b}\\\\left(x\\\\right)$$, $$b>0$$ and $$b$$ not equal to $$1$$, has the following properties: - one-to-one function\\\\n- vertical asymptote: $$x=0$$\\\\n- domain: $$(0,\\\\infty)$$\\\\n- range: $$(-\\\\infty,\\\\infty)$$\\\\n- x-intercept: $$(1,0)$$ and key point (b,1)\\\\n- y-intercept: none\\\\n- increasing if $$b>1$$\\\\n- decreasing if $$0<b<1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a98b1afgraphlog20a-h2","type":"hint","dependencies":[],"title":"Definition of y-intercept","text":"The y-intercept describes the point in which the function crosses the y-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a98b1afgraphlog20a-h3","type":"hint","dependencies":[],"title":"Setting up the equation","text":"Let $$x=0$$. If possible, solve the function f(x) $$=$$ $$ln(-x)-2$$. This represents the point in which the function crosses the y-axis. If there is no solution to the equation, then the y-intercept does not exist.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a98b1afgraphlog3","title":"Domain, range, and asymptotes of log functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Graphs of Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a98b1afgraphlog3a","stepAnswer":["domain: $$(-4,\\\\infty)$$, range: $$(-\\\\infty,\\\\infty)$$, asymptote: $$x=-4$$"],"problemType":"MultipleChoice","stepTitle":"What is the domain, range, and asymptote of the function $$\\\\log_{3}\\\\left(x+4\\\\right)$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"domain: $$(-4,\\\\infty)$$, range: $$(-\\\\infty,\\\\infty)$$, asymptote: $$x=-4$$","choices":["domain: $$(-4,\\\\infty)$$, range: $$(-\\\\infty,\\\\infty)$$, asymptote: $$x=-4$$","domain: $$(-\\\\infty,\\\\infty)$$, $$range(-4,\\\\infty)$$, asymptote: $$x=-4$$","none"],"hints":{"DefaultPathway":[{"id":"a98b1afgraphlog3a-h1","type":"hint","dependencies":[],"title":"Characteristics of the graph of the parent function","text":"The function $$f(x)=\\\\log_{b}\\\\left(x\\\\right)$$, $$b>0$$ and $$b$$ not equal to $$1$$, has the following properties: - one-to-one function\\\\n- vertical asymptote: $$x=0$$\\\\n- domain: $$(0,\\\\infty)$$\\\\n- range: $$(-\\\\infty,\\\\infty)$$\\\\n- x-intercept: $$(1,0)$$ and key point (b,1)\\\\n- y-intercept: none\\\\n- increasing if $$b>1$$\\\\n- decreasing if $$0<b<1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a98b1afgraphlog3a-h2","type":"hint","dependencies":[],"title":"Transformations of log functions","text":"$$f(x)=\\\\log_{3}\\\\left(x+4\\\\right)$$ translates the parent function $$4$$ units to the left. How does this affect the domain, range, and asymptote?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a98b1afgraphlog4","title":"Domain and range of log functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Graphs of Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a98b1afgraphlog4a","stepAnswer":["domain: $$(-\\\\infty,\\\\frac{1}{2})$$, range: $$(-\\\\infty,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"What is the domain and range of $$h(x)=\\\\ln(\\\\frac{1}{2}-x)$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"domain: $$(-\\\\infty,\\\\frac{1}{2})$$, range: $$(-\\\\infty,\\\\infty)$$","choices":["domain: $$(-\\\\infty,\\\\frac{1}{2})$$, range: $$(-\\\\infty,\\\\infty)$$","domain: $$(\\\\frac{1}{2},\\\\infty)$$ range: $$(-\\\\infty,\\\\infty)$$","none of the above"],"hints":{"DefaultPathway":[{"id":"a98b1afgraphlog4a-h1","type":"hint","dependencies":[],"title":"Characteristics of the graph of the parent function","text":"The function $$f(x)=\\\\log_{b}\\\\left(x\\\\right)$$, $$b>0$$ and $$b$$ not equal to $$1$$, has the following properties: - one-to-one function\\\\n- vertical asymptote: $$x=0$$\\\\n- domain: $$(0,\\\\infty)$$\\\\n- range: $$(-\\\\infty,\\\\infty)$$\\\\n- x-intercept: $$(1,0)$$ and key point (b,1)\\\\n- y-intercept: none\\\\n- increasing if $$b>1$$\\\\n- decreasing if $$0<b<1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a98b1afgraphlog4a-h2","type":"hint","dependencies":[],"title":"Transformations of log functions","text":"The function shifts its parent function $$\\\\frac{1}{2}$$ units to the left and reflects it across the y-axis. How does this affect the domain and range?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a98b1afgraphlog5","title":"Domain and Range of log functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Graphs of Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a98b1afgraphlog5a","stepAnswer":["domain: $$(-5$$, inf), range: (-inf, inf)"],"problemType":"MultipleChoice","stepTitle":"What is the domain and range of $$\\\\log_{5}\\\\left(2x+9\\\\right)-2$$?","stepBody":"","answerType":"string","variabilization":{},"choices":["domain: $$(-5$$, inf), range: (-inf, inf)","none of the above","domain: $$(-5$$, inf), range: (-inf, inf)","domain: (-inf, 5), range: $$(-\\\\infty,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"a98b1afgraphlog5a-h1","type":"hint","dependencies":[],"title":"Characteristics of the graph of the parent function","text":"The function $$f(x)=\\\\log_{b}\\\\left(x\\\\right)$$, $$b>0$$ and $$b$$ not equal to $$1$$, has the following properties: - one-to-one function\\\\n- vertical asymptote: $$x=0$$\\\\n- domain: $$(0,\\\\infty)$$\\\\n- range: $$(-\\\\infty,\\\\infty)$$\\\\n- x-intercept: $$(1,0)$$ and key point (b,1)\\\\n- y-intercept: none\\\\n- increasing if $$b>1$$\\\\n- decreasing if $$0<b<1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a98b1afgraphlog5a-h2","type":"hint","dependencies":[],"title":"Transformations of log functions","text":"The function is translated $$2$$ units down, translated $$9$$ units to the left, and horizontally compressed by a factor of $$\\\\frac{1}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a98b1afgraphlog6","title":"Domain and range of log functions","body":"What is the domain and range of $$h(x)=\\\\ln(4x+17)-5$$?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Graphs of Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a98b1afgraphlog6a","stepAnswer":["domain: (-17/4, inf), range: (-inf, inf)"],"problemType":"MultipleChoice","stepTitle":"","stepBody":"","answerType":"string","variabilization":{},"choices":["domain: (-17/4, inf), range: (-inf, inf)","domain:(-inf, 17/4), range: (-inf, inf)","none of the above"],"hints":{"DefaultPathway":[{"id":"a98b1afgraphlog6a-h1","type":"hint","dependencies":[],"title":"Characteristics of the graph of the parent function","text":"The function $$f(x)=\\\\log_{b}\\\\left(x\\\\right)$$, $$b>0$$ and $$b$$ not equal to $$1$$, has the following properties: - one-to-one function\\\\n- vertical asymptote: $$x=0$$\\\\n- domain: $$(0,\\\\infty)$$\\\\n- range: $$(-\\\\infty,\\\\infty)$$\\\\n- x-intercept: $$(1,0)$$ and key point (b,1)\\\\n- y-intercept: none\\\\n- increasing if $$b>1$$\\\\n- decreasing if $$0<b<1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a98b1afgraphlog6a-h2","type":"hint","dependencies":[],"title":"Transformations of log functions","text":"The function is translated $$5$$ units down, translated $$17$$ units to the left, and horizonally compressed by a factor of $$\\\\frac{1}{4}$$. How does this affect the domain and range?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a98b1afgraphlog7","title":"Domain and range of log functions","body":"What is the domain and range of $$f(x)=\\\\log_{2}\\\\left(12-3x\\\\right)-3$$?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Graphs of Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a98b1afgraphlog7a","stepAnswer":["domain: (-inf, 4), range: (-inf, inf)"],"problemType":"MultipleChoice","stepTitle":"","stepBody":"","answerType":"string","variabilization":{},"choices":["domain: (-inf, 4), range: (-inf, inf)","domain: (-inf, inf), range: (-inf, 4)","none of the above"],"hints":{"DefaultPathway":[{"id":"a98b1afgraphlog7a-h1","type":"hint","dependencies":[],"title":"Characteristics of the graph of the parent function","text":"The function $$f(x)=\\\\log_{b}\\\\left(x\\\\right)$$, $$b>0$$ and $$b$$ not equal to $$1$$, has the following properties: - one-to-one function\\\\n- vertical asymptote: $$x=0$$\\\\n- domain: $$(0,\\\\infty)$$\\\\n- range: $$(-\\\\infty,\\\\infty)$$\\\\n- x-intercept: $$(1,0)$$ and key point (b,1)\\\\n- y-intercept: none\\\\n- increasing if $$b>1$$\\\\n- decreasing if $$0<b<1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a98b1afgraphlog7a-h2","type":"hint","dependencies":[],"title":"Transformations of log functions","text":"To achieve the function provided, the parent function is translated down $$3$$ units, translated $$12$$ units to the left, horizontally compressed by a factor of $$\\\\frac{1}{3}$$, and reflected across the y-axis. How does this affect the domain and range?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a98b1afgraphlog8","title":"Vertical asymptotes of log functions","body":"What is the vertical asymptote of f(x)=log{b}(x-5)? Write as an equation $$x=[number]$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Graphs of Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a98b1afgraphlog8a","stepAnswer":["$$x=5$$"],"problemType":"TextBox","stepTitle":"","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x=5$$","hints":{"DefaultPathway":[{"id":"a98b1afgraphlog8a-h1","type":"hint","dependencies":[],"title":"Characteristics of the graph of the parent function","text":"The function $$f(x)=\\\\log_{b}\\\\left(x\\\\right)$$, $$b>0$$ and $$b$$ not equal to $$1$$, has the following properties: - one-to-one function\\\\n- vertical asymptote: $$x=0$$\\\\n- domain: $$(0,\\\\infty)$$\\\\n- range: $$(-\\\\infty,\\\\infty)$$\\\\n- x-intercept: $$(1,0)$$ and key point (b,1)\\\\n- y-intercept: none\\\\n- increasing if $$b>1$$\\\\n- decreasing if $$0<b<1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a98b1afgraphlog8a-h2","type":"hint","dependencies":[],"title":"Transformations of log functions","text":"To achieve the given function, the parent function is shifted $$5$$ units to the right. How does this affect the vertical asymptote?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a98b1afgraphlog9","title":"Vertical asymptotes of log functions","body":"What is the vertical asymptote of $$g(x)=ln(3-x)$$? Write as an equation of $$x$$ (e.g. $$x=[number])$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Graphs of Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a98b1afgraphlog9a","stepAnswer":["$$x=3$$"],"problemType":"TextBox","stepTitle":"","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x=3$$","hints":{"DefaultPathway":[{"id":"a98b1afgraphlog9a-h1","type":"hint","dependencies":[],"title":"Characteristics of the graph of the parent function","text":"The function $$f(x)=\\\\log_{b}\\\\left(x\\\\right)$$, $$b>0$$ and $$b$$ not equal to $$1$$, has the following properties: - one-to-one function\\\\n- vertical asymptote: $$x=0$$\\\\n- domain: $$(0,\\\\infty)$$\\\\n- range: $$(-\\\\infty,\\\\infty)$$\\\\n- x-intercept: $$(1,0)$$ and key point (b,1)\\\\n- y-intercept: none\\\\n- increasing if $$b>1$$\\\\n- decreasing if $$0<b<1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a98b1afgraphlog9a-h2","type":"hint","dependencies":[],"title":"Transformations of log functions","text":"To achieve the function above, the parent function is shifted $$3$$ units to the left and then reflected across the y-axis. How does this affect the vertical asymptote?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9981ababs1","title":"Finding the Intercepts of an Absolute Value Function","body":"Find the $$x-$$ and y-intercepts of the given function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Absolute Value Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a9981ababs1a","stepAnswer":["There are no $$x-intercepts$$."],"problemType":"MultipleChoice","stepTitle":"Find the x-intercept of $$f(x)=4|x-3|+4$$","stepBody":"","answerType":"string","variabilization":{},"choices":["There are no $$x-intercepts$$.","$$2$$","$$4$$","$$3$$"],"hints":{"DefaultPathway":[{"id":"a9981ababs1a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$0$$ for f(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981ababs1a-h2","type":"hint","dependencies":["a9981ababs1a-h1"],"title":"Isolate","text":"Isolate the absolute value on one side of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981ababs1a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-1=|x-3|$$"],"dependencies":["a9981ababs1a-h2"],"title":"Isolate","text":"What do we get after the isolation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$1=|x-3|$$","$$-1=|x-3|$$","$$0=|x-3|$$"]},{"id":"a9981ababs1a-h4","type":"hint","dependencies":["a9981ababs1a-h3"],"title":"Absolute Value","text":"$$|x-3|$$ is always non-negative, so we cannot find x-intercepts in this case.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9981ababs1b","stepAnswer":["$$16$$"],"problemType":"TextBox","stepTitle":"Find the y-intercept of $$f(x)=4|x-3|+4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16$$","hints":{"DefaultPathway":[{"id":"a9981ababs1b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$0$$ for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Absolute Value Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a9981ababs10a","stepAnswer":["There are no $$x-intercepts$$."],"problemType":"MultipleChoice","stepTitle":"Find the x-intercept of $$y=|x|+1$$","stepBody":"","answerType":"string","variabilization":{},"choices":["There are no $$x-intercepts$$.","$$1$$","$$0$$","$$-1$$"],"hints":{"DefaultPathway":[{"id":"a9981ababs10a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$0$$ for $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981ababs10a-h2","type":"hint","dependencies":["a9981ababs10a-h1"],"title":"Isolate","text":"Isolate the absolute value on one side of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981ababs10a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-1=|x|$$"],"dependencies":["a9981ababs10a-h2"],"title":"Isolate","text":"What do we get after the isolation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$1=|x|$$","$$0=|x|$$","$$-1=|x|$$"]},{"id":"a9981ababs10a-h4","type":"hint","dependencies":["a9981ababs10a-h3"],"title":"Absolute Value","text":"$$|x|$$ is always non-negative, so we cannot find x-intercepts in this case.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9981ababs10b","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"Find the y-intercept of $$y=|x|+1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a9981ababs10b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$0$$ for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981ababs10b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a9981ababs10b-h1"],"title":"Solve the Equation","text":"What do we get for $$y$$ after solving $$y=|0|+1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9981ababs11","title":"Finding the Intercepts of an Absolute Value Function","body":"Find the $$x-$$ and y-intercepts of the given function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College 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4.0>"},{"id":"a9981ababs11a-h3","type":"hint","dependencies":["a9981ababs11a-h2"],"title":"Separate","text":"Break $$2=|x|$$ into two separate equations and solve.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981ababs11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a9981ababs11a-h3"],"title":"Separate","text":"What do we get for $$x$$ after solving $$2=x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981ababs11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a9981ababs11a-h4"],"title":"Separate","text":"What do we get for $$x$$ after solving $$-2=x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9981ababs11b","stepAnswer":["$$-2$$"],"problemType":"TextBox","stepTitle":"Find the y-intercept of $$y=|x|-2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-2$$","hints":{"DefaultPathway":[{"id":"a9981ababs11b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$0$$ for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981ababs11b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a9981ababs11b-h1"],"title":"Solve the Equation","text":"What do we get for $$y$$ after solving $$y=|0|-2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9981ababs12","title":"Finding the Intercepts of an Absolute Value Function","body":"Find the $$x-$$ and y-intercepts of the given 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$$0=-|x|$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9981ababs12b","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"Find the y-intercept of $$y=-|x|$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a9981ababs12b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$0$$ for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981ababs12b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a9981ababs12b-h1"],"title":"Solve the Equation","text":"What do we get for $$y$$ after solving $$y=-|0|$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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$$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981ababs13b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a9981ababs13b-h1"],"title":"Solve the Equation","text":"What do we get for $$y$$ after solving $$y=-|0|-2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9981ababs14","title":"Finding the Intercepts of an Absolute Value Function","body":"Find the $$x-$$ and y-intercepts of the given function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Absolute Value Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a9981ababs14a","stepAnswer":["There are no 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case.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9981ababs14b","stepAnswer":["$$-5$$"],"problemType":"TextBox","stepTitle":"Find the y-intercept of $$y=-|x-3|-2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-5$$","hints":{"DefaultPathway":[{"id":"a9981ababs14b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$0$$ for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981ababs14b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["a9981ababs14b-h1"],"title":"Solve the Equation","text":"What do we get for $$y$$ after solving $$y=-|0-3|-2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$3=|x-1|$$","$$-3=|x-1|$$","$$2=|x-1|$$"]},{"id":"a9981ababs5a-h4","type":"hint","dependencies":["a9981ababs5a-h3"],"title":"Separate","text":"Break the above equation into two separate equations and solve.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981ababs5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a9981ababs5a-h4"],"title":"Separate","text":"What do we get for $$x$$ after solving $$3=x-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981ababs5a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a9981ababs5a-h5"],"title":"Separate","text":"What do we get for $$x$$ after solving $$-3=x-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9981ababs5b","stepAnswer":["(0,-4)"],"problemType":"TextBox","stepTitle":"Find the y-intercept of $$f(x)=2|x-1|-6$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,-4)$$","hints":{"DefaultPathway":[{"id":"a9981ababs5b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$0$$ for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981ababs5b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$f(x)=2\\\\times1-6$$"],"dependencies":["a9981ababs5b-h1"],"title":"Substitute","text":"What do we get after the substitution?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$f(x)=2-5$$","$$f(x)=2\\\\left(-1\\\\right)-6$$","$$f(x)=2\\\\times1-6$$"]},{"id":"a9981ababs5b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["a9981ababs5b-h2"],"title":"Solve the Equation","text":"What do we get for $$y$$ after solving the above equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9981ababs6","title":"Finding the Intercepts of an Absolute Value Function","body":"Find the $$x-$$ and y-intercepts of the given function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Absolute Value Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a9981ababs6a","stepAnswer":["-6,7"],"problemType":"TextBox","stepTitle":"Find the x-intercept of $$f(x)=|-2x+1|-13$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a9981ababs6a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$0$$ for f(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981ababs6a-h2","type":"hint","dependencies":["a9981ababs6a-h1"],"title":"Isolate","text":"Isolate the absolute value on one side of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981ababs6a-h3","type":"hint","dependencies":["a9981ababs6a-h2"],"title":"Separate","text":"Break $$13=|-2x+1|$$ into two separate equations and solve.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981ababs6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6$$"],"dependencies":["a9981ababs6a-h3"],"title":"Separate","text":"What do we get for $$x$$ after solving $$13=-2x+1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981ababs6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a9981ababs6a-h4"],"title":"Separate","text":"What do we get for $$x$$ after solving $$-13=-2x+1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9981ababs6b","stepAnswer":["(0,-12)"],"problemType":"TextBox","stepTitle":"Find the y-intercept of $$f(x)=|-2x+1|-13$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,-12)$$","hints":{"DefaultPathway":[{"id":"a9981ababs6b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$0$$ for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981ababs6b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$f(x)=1-13$$"],"dependencies":["a9981ababs6b-h1"],"title":"Substitute","text":"What do we get after the substitution?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$f(x)=1-13$$","$$f(x)=-1-13$$","$$f(x)=3-13$$"]},{"id":"a9981ababs6b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-12$$"],"dependencies":["a9981ababs6b-h2"],"title":"Solve the Equation","text":"What do we get for $$y$$ after solving the above equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9981ababs7","title":"Finding the Intercepts of an Absolute Value Function","body":"Find the $$x-$$ and y-intercepts of the given function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Absolute Value Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a9981ababs7a","stepAnswer":["$$-7, 25$$"],"problemType":"MultipleChoice","stepTitle":"Find the x-intercept of $$f(x)=-|x-9|+16$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$-7, 25$$","$$7, -25$$"],"hints":{"DefaultPathway":[{"id":"a9981ababs7a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$0$$ for f(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981ababs7a-h2","type":"hint","dependencies":["a9981ababs7a-h1"],"title":"Isolate","text":"Isolate the absolute value on one side of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981ababs7a-h3","type":"hint","dependencies":["a9981ababs7a-h2"],"title":"Separate","text":"Break $$16=|x-9|$$ into two separate equations and solve.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981ababs7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["a9981ababs7a-h3"],"title":"Separate","text":"What do we get for $$x$$ after solving $$16=x-9$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981ababs7a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-7$$"],"dependencies":["a9981ababs7a-h4"],"title":"Separate","text":"What do we get for $$x$$ after solving $$-16=x-9$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9981ababs7b","stepAnswer":["$$7$$"],"problemType":"TextBox","stepTitle":"Find the y-intercept of $$f(x)=-|x-9|+16$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7$$","hints":{"DefaultPathway":[{"id":"a9981ababs7b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$0$$ for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Absolute Value Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a9981ababs8a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"Find the x-intercept of $$y=|x-1|$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a9981ababs8a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$0$$ for $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981ababs8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a9981ababs8a-h1"],"title":"Solve the Equation","text":"What do we get for $$x$$ after solving $$0=|x-1|$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9981ababs8b","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"Find the y-intercept of $$y=|x-1|$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a9981ababs8b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$0$$ for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981ababs8b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a9981ababs8b-h1"],"title":"Solve the Equation","text":"What do we get for $$y$$ after solving $$y=|0-1|$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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$$f(x)=|2x-4|-3$$. What is the $$x$$ value?","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{1}{2}$$, $$\\\\frac{7}{2}$$","choices":["$$\\\\frac{-1}{2}$$, $$\\\\frac{-7}{2}$$","$$\\\\frac{1}{2}$$, $$\\\\frac{7}{2}$$","$$-2$$, $$-4$$","$$2$$, $$4$$"],"hints":{"DefaultPathway":[{"id":"a9981abAbsolute1a-h1","type":"hint","dependencies":[],"title":"Setting $$y$$ to zero","text":"To find the x-intercept of a function, we start by setting the $$y$$ value to $$0$$, so we get the equation $$|2x-4|-3=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981abAbsolute1a-h2","type":"hint","dependencies":["a9981abAbsolute1a-h1"],"title":"Isolating the absoluate value","text":"Then, we should isolate the part with absolute value. We get the equation $$|2x-4|=3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981abAbsolute1a-h3","type":"hint","dependencies":["a9981abAbsolute1a-h2"],"title":"Solving equation with absoluate values","text":"We can break an equation with absolute values down into two equations that we can solve independently. We notice that the absolute value will be equal to $$3$$ if the quantity inside the absolute value is $$3$$ or $$-3$$. Therefore, the two equations we get are $$2x-4=3$$ and $$2x-4=-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981abAbsolute1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{7}{2}$$"],"dependencies":["a9981abAbsolute1a-h3"],"title":"Solving the first equation","text":"Solve the equation $$2x-4=3$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981abAbsolute1a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a9981abAbsolute1a-h3"],"title":"Solving the second equation","text":"Solve the equation $$2x-4=-3$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9981abAbsolute1b","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"Finding the y-intercept.","stepBody":"Find the y-intercept of the function $$f(x)=|2x-4|-3$$. What is the $$y$$ value?","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a9981abAbsolute1b-h1","type":"hint","dependencies":[],"title":"Setting $$x$$ to zero","text":"To find the y-intercept of a function, we start by setting the $$x$$ value to $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981abAbsolute1b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a9981abAbsolute1b-h1"],"title":"Solving for $$y$$","text":"What is $$|2\\\\times0-4|-3$$ equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9981abAbsolute2","title":"Solving for the Intercepts","body":"Find the $$x$$ and $$y$$ intercepts of the function $$f(x)=|3x+9|+2$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Absolute Value Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a9981abAbsolute2a","stepAnswer":["No solution"],"problemType":"MultipleChoice","stepTitle":"Finding the x-intercept.","stepBody":"Find the x-intercept of the function $$f(x)=|3x+9|+2$$. What is the $$x$$ value?","answerType":"string","variabilization":{},"choices":["$$-3$$","$$-2$$","$$2$$","No solution"],"hints":{"DefaultPathway":[{"id":"a9981abAbsolute2a-h1","type":"hint","dependencies":[],"title":"Setting $$y$$ to zero","text":"To find the x-intercept of a function, we start by setting the $$y$$ value to $$0$$, so we get the equation $$|3x+9|+2=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981abAbsolute2a-h2","type":"hint","dependencies":["a9981abAbsolute2a-h1"],"title":"Isolating the absoluate value","text":"Then, we should isolate the part with absolute value. We get the equation $$|3x+9|=-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981abAbsolute2a-h3","type":"hint","dependencies":["a9981abAbsolute2a-h2"],"title":"Solving equation with absoluate values","text":"Since an absolute value can never be negative, we say that this equation has no solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9981abAbsolute2b","stepAnswer":["$$11$$"],"problemType":"TextBox","stepTitle":"Finding the y-intercept.","stepBody":"Find the y-intercept of the function $$f(x)=|3x+9|+2$$. What is the $$y$$ value?","answerType":"arithmetic","variabilization":{},"answerLatex":"$$11$$","hints":{"DefaultPathway":[{"id":"a9981abAbsolute2b-h1","type":"hint","dependencies":[],"title":"Setting $$x$$ to zero","text":"To find the y-intercept of a function, we start by setting the $$x$$ value to $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981abAbsolute2b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11$$"],"dependencies":["a9981abAbsolute2b-h1"],"title":"Solving for $$y$$","text":"What is $$|3\\\\times0+9|+2$$ equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9981abAbsolute3","title":"Solving for the Intercepts","body":"Find the $$x$$ and $$y$$ intercepts of the function $$f(x)=-|x-1|-3$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Absolute Value Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a9981abAbsolute3a","stepAnswer":["No solution"],"problemType":"MultipleChoice","stepTitle":"Finding the x-intercept.","stepBody":"Find the x-intercept of the function $$f(x)=-|x-1|-3$$. What is the $$x$$ value?","answerType":"string","variabilization":{},"choices":["$$1$$","$$-1$$","$$3$$","No solution"],"hints":{"DefaultPathway":[{"id":"a9981abAbsolute3a-h1","type":"hint","dependencies":[],"title":"Setting $$y$$ to zero","text":"To find the x-intercept of a function, we start by setting the $$y$$ value to $$0$$, so we get the equation $$-|x-1|-3=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981abAbsolute3a-h2","type":"hint","dependencies":["a9981abAbsolute3a-h1"],"title":"Isolating the absoluate value","text":"Then, we should isolate the part with absolute value. We get the equation $$|x-1|=-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981abAbsolute3a-h3","type":"hint","dependencies":["a9981abAbsolute3a-h2"],"title":"Solving equation with absoluate values","text":"Since an absolute value can never be negative, we say that this equation has no solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9981abAbsolute3b","stepAnswer":["$$-4$$"],"problemType":"TextBox","stepTitle":"Finding the y-intercept.","stepBody":"Find the y-intercept of the function $$f(x)=-|x-1|-3$$. What is the $$y$$ value?","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-4$$","hints":{"DefaultPathway":[{"id":"a9981abAbsolute3b-h1","type":"hint","dependencies":[],"title":"Setting $$x$$ to zero","text":"To find the y-intercept of a function, we start by setting the $$x$$ value to $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981abAbsolute3b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["a9981abAbsolute3b-h1"],"title":"Solving for $$y$$","text":"What is $$-|0-1|-3$$ equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9981abAbsolute4","title":"Solving for the Intercepts","body":"Find the $$x$$ and $$y$$ intercepts of the function $$f(x)=-|x+4|-3$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Absolute Value Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a9981abAbsolute4a","stepAnswer":["No solution"],"problemType":"MultipleChoice","stepTitle":"Finding the x-intercept.","stepBody":"Find the x-intercept of the function $$f(x)=-|x+4|-3$$. What is the $$x$$ value?","answerType":"string","variabilization":{},"choices":["$$-4$$","$$4$$","$$3$$","No solution"],"hints":{"DefaultPathway":[{"id":"a9981abAbsolute4a-h1","type":"hint","dependencies":[],"title":"Setting $$y$$ to zero","text":"To find the x-intercept of a function, we start by setting the $$y$$ value to $$0$$, so we get the $$equation-|x+4|-3=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981abAbsolute4a-h2","type":"hint","dependencies":["a9981abAbsolute4a-h1"],"title":"Isolating the absoluate value","text":"Then, we should isolate the part with absolute value. We get the equation $$|x+4|=-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981abAbsolute4a-h3","type":"hint","dependencies":["a9981abAbsolute4a-h2"],"title":"Solving equation with absoluate values","text":"Since an absolute value can never be negative, we say that this equation has no solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9981abAbsolute4b","stepAnswer":["$$-7$$"],"problemType":"TextBox","stepTitle":"Finding the y-intercept.","stepBody":"Find the y-intercept of the function $$f(x)=-|x+4|-3$$. What is the $$y$$ value?","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-7$$","hints":{"DefaultPathway":[{"id":"a9981abAbsolute4b-h1","type":"hint","dependencies":[],"title":"Setting $$x$$ to zero","text":"To find the y-intercept of a function, we start by setting the $$x$$ value to $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981abAbsolute4b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-7$$"],"dependencies":["a9981abAbsolute4b-h1"],"title":"Solving for $$y$$","text":"What is $$-|0+4|-3$$ equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9981abAbsolute5","title":"Solving for the Intercepts","body":"Find the $$x$$ and $$y$$ intercepts of the function $$f(x)=\\\\frac{1}{2} |x+4|-3$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Absolute Value Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a9981abAbsolute5a","stepAnswer":["$$2, -10$$"],"problemType":"MultipleChoice","stepTitle":"Finding the x-intercept.","stepBody":"Find the x-intercept of the function $$f(x)=$$ $$\\\\frac{1}{2} |x+4|-3$$. What is the $$x$$ value?","answerType":"string","variabilization":{},"choices":["$$4, -4$$","$$2, -10$$","$$3, -4$$","No solution"],"hints":{"DefaultPathway":[{"id":"a9981abAbsolute5a-h1","type":"hint","dependencies":[],"title":"Setting $$y$$ to zero","text":"To find the x-intercept of a function, we start by setting the $$y$$ value to $$0$$, so we get the equation $$\\\\frac{1}{2} |x+4|-3=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981abAbsolute5a-h2","type":"hint","dependencies":["a9981abAbsolute5a-h1"],"title":"Isolating the absoluate value","text":"Then, we should isolate the part with absolute value. We get the equation $$\\\\frac{1}{2} |x+4|=3$$. Multiply $$2$$ to both sides, we get $$|x+4|=6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981abAbsolute5a-h3","type":"hint","dependencies":["a9981abAbsolute5a-h2"],"title":"Solving equation with absoluate values","text":"We can break an equation with absolute values down into two equations that we can solve independently. We notice that the absolute value will be equal to $$6$$ if the quantity inside the absolute value is $$6$$ or $$-6$$. Therefore, the two equations we get are $$x+4=6$$ and $$x+4=-6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981abAbsolute5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a9981abAbsolute5a-h3"],"title":"Solving the first equation","text":"Solve the equation $$x+4=6$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981abAbsolute5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-10$$"],"dependencies":["a9981abAbsolute5a-h3"],"title":"Solving the second equation","text":"Solve the equation $$x+4=-6$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9981abAbsolute5b","stepAnswer":["$$-1$$"],"problemType":"TextBox","stepTitle":"Finding the y-intercept.","stepBody":"Find the y-intercept of the function $$f(x)=$$ $$\\\\frac{1}{2} |x+4|-3$$. What is the $$y$$ value?","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1$$","hints":{"DefaultPathway":[{"id":"a9981abAbsolute5b-h1","type":"hint","dependencies":[],"title":"Setting $$x$$ to zero","text":"To find the y-intercept of a function, we start by setting the $$x$$ value to $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9981abAbsolute5b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a9981abAbsolute5b-h1"],"title":"Solving for $$y$$","text":"What is $$\\\\frac{1}{2} |0+4|-3$$ equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ad3f8applications1","title":"Translating a System of Equations","body":"Translate to a system of equations and solve.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Solve Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ad3f8applications1a","stepAnswer":["The wife earns $10,000."],"problemType":"MultipleChoice","stepTitle":"A married couple together earn $75,000. The husband earns $15,000 more than five times what his wife earns. What does the wife earn?","stepBody":"","answerType":"string","variabilization":{},"choices":["The wife earns $12,000.","The wife earns $8,000.","The wife earns $9,000.","The wife earns $10,000."],"hints":{"DefaultPathway":[{"id":"a9ad3f8applications1a-h1","type":"hint","dependencies":[],"title":"Identify","text":"We are looking for the amount the wife earns.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications1a-h2","type":"hint","dependencies":["a9ad3f8applications1a-h1"],"title":"First equation","text":"Since the couple earns $75,000 together, we can write the first equation as $$w+h=75, 000$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications1a-h3","type":"hint","dependencies":["a9ad3f8applications1a-h2"],"title":"Second equation","text":"Since the husband earns $15,000 more than five times what his wife earns, we can label the second equation as h=15,000+5w.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10000$$"],"dependencies":["a9ad3f8applications1a-h3"],"title":"Solve","text":"Next, using the system of equations just created, solve for w. What does it equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications1a-h5","type":"hint","dependencies":["a9ad3f8applications1a-h4"],"title":"Answer","text":"Therefore, the wife earns $10,000.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ad3f8applications10","title":"Translating a System of Equations","body":"Translate to a system of equations and solve.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Solve Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ad3f8applications10a","stepAnswer":["The measures are $$94$$ degrees and $$86$$ degrees"],"problemType":"MultipleChoice","stepTitle":"The difference of two supplementary angles is $$8$$ degrees. Find the measures of the angles.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"The measures are $$94$$ degrees and $$86$$ degrees","choices":["The measures are $$94$$ degrees and $$86$$ degrees","The measures are $$94$$ degrees and $$86$$ degreesIThe measures are $$24$$ degrees and $$35$$ degreesIThe measures are $$45$$ degrees and $$45$$ degreesIThe measures are $$22$$ degrees and $$78$$ degrees"],"hints":{"DefaultPathway":[{"id":"a9ad3f8applications10a-h1","type":"hint","dependencies":[],"title":"Supplementary","text":"When two angles are supplementary, it means their measures add up to $$180$$ degrees.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications10a-h2","type":"hint","dependencies":["a9ad3f8applications10a-h1"],"title":"First equation","text":"Since the problem identifies the angles as supplementary, we know that they add up to $$180$$ degrees. Therefore, the first equation should be $$a+b=180$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications10a-h3","type":"hint","dependencies":["a9ad3f8applications10a-h2"],"title":"Second equation","text":"The problem mentions the difference between the two angles is $$8$$ degrees. Because of this, we can write the second equation as $$a-b=8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$94$$"],"dependencies":["a9ad3f8applications10a-h3"],"title":"Solve","text":"Next, using the system of equations just created, solve for a. What does it equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$86$$"],"dependencies":["a9ad3f8applications10a-h4"],"title":"Solve","text":"What does $$t$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications10a-h6","type":"hint","dependencies":["a9ad3f8applications10a-h5"],"title":"Answer","text":"The two angle measures are $$94$$ degrees and $$84$$ degrees.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ad3f8applications11","title":"Translating a System of Equations","body":"Translate to a system of equations and solve.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Solve Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ad3f8applications11a","stepAnswer":["The measures are $$72.5$$ degrees and $$17.5$$ degrees."],"problemType":"MultipleChoice","stepTitle":"The difference of two complementary angles is $$55$$ degrees. Find the measures of the angles.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"The measures are $$72.5$$ degrees and $$17.5$$ degrees.","choices":["The measures are $$72.5$$ degrees and $$17.5$$ degrees.","The measures are $$72.5$$ degrees and $$17.5$$ degrees.IThe measures are $$20$$ degrees and $$30$$ degrees.IThe measures are $$22.5$$ degrees and $$70$$ degrees.IThe measures are $$20.5$$ degrees and $$160$$ degrees."],"hints":{"DefaultPathway":[{"id":"a9ad3f8applications11a-h1","type":"hint","dependencies":[],"title":"Complementary angles","text":"When two angles are complementary, it means their measures add up to $$90$$ degrees.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications11a-h2","type":"hint","dependencies":["a9ad3f8applications11a-h1"],"title":"First equation","text":"Since the problem identifies the angles as complementary, we know that they add up to $$90$$ degrees. Therefore, the first equation should be $$a+b=90$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications11a-h3","type":"hint","dependencies":["a9ad3f8applications11a-h2"],"title":"Second equation","text":"The problem mentions the difference between the two angles is $$55$$ degrees. Because of this, we can write the second equation as $$a-b=55$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$72.5$$"],"dependencies":["a9ad3f8applications11a-h3"],"title":"Solve","text":"Next, using the system of equations just created, solve for a. What does it equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$17.5$$"],"dependencies":["a9ad3f8applications11a-h4"],"title":"Solve","text":"What does $$t$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications11a-h6","type":"hint","dependencies":["a9ad3f8applications11a-h5"],"title":"Answer","text":"The two angle measures are $$72.5$$ degrees and $$17.5$$ degrees.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ad3f8applications12","title":"Translating a System of Equations","body":"Translate to a system of equations and solve.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Solve Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ad3f8applications12a","stepAnswer":["The measures are $$79$$ degrees and $$11$$ degrees."],"problemType":"MultipleChoice","stepTitle":"The difference of two complementary angles is $$68$$ degrees. Find the measures of the angles.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"The measures are $$79$$ degrees and $$11$$ degrees.","choices":["The measures are $$79$$ degrees and $$11$$ degrees.","The measures are $$20$$ degrees and $$30$$ degrees.","The measures are $$25$$ degrees and $$65$$ degrees.","The measures are $$20$$ degrees and $$160$$ degrees."],"hints":{"DefaultPathway":[{"id":"a9ad3f8applications12a-h1","type":"hint","dependencies":[],"title":"Complementary angles","text":"When two angles are complementary, it means their measures add up to $$90$$ degrees.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications12a-h2","type":"hint","dependencies":["a9ad3f8applications12a-h1"],"title":"First equation","text":"Since the problem identifies the angles as complementary, we know that they add up to $$90$$ degrees. Therefore, the first equation should be $$a+b=90$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications12a-h3","type":"hint","dependencies":["a9ad3f8applications12a-h2"],"title":"Second equation","text":"The problem mentions the difference between the two angles is $$68$$ degrees. Because of this, we can write the second equation as $$a-b=68$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$79$$"],"dependencies":["a9ad3f8applications12a-h3"],"title":"Solve","text":"Next, using the system of equations just created, solve for a. What does it equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications12a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11$$"],"dependencies":["a9ad3f8applications12a-h4"],"title":"Solve","text":"What does $$t$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications12a-h6","type":"hint","dependencies":["a9ad3f8applications12a-h5"],"title":"Answer","text":"The two angle measures are $$79$$ degrees and $$11$$ degrees.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ad3f8applications13","title":"Translating a System of Equations","body":"Translate to a system of equations and solve.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Solve Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ad3f8applications13a","stepAnswer":["The measures are $$102$$ degrees and $$78$$ degrees"],"problemType":"MultipleChoice","stepTitle":"The difference of two supplementary angles is $$24$$ degrees. Find the measure of the angles.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"The measures are $$102$$ degrees and $$78$$ degrees","choices":["The measures are $$102$$ degrees and $$78$$ degrees","The measures are $$22$$ degrees and $$17$$ degrees","The measures are $$94$$ degrees and $$68$$ degrees","The measures are $$11$$ degrees and $$58$$ degrees"],"hints":{"DefaultPathway":[{"id":"a9ad3f8applications13a-h1","type":"hint","dependencies":[],"title":"Supplementary","text":"When two angles are supplementary, it means their measures add up to $$180$$ degrees.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications13a-h2","type":"hint","dependencies":["a9ad3f8applications13a-h1"],"title":"First equation","text":"Since the problem identifies the angles as supplementary, we know that they add up to $$180$$ degrees. Therefore, the first equation should be $$a+b=180$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications13a-h3","type":"hint","dependencies":["a9ad3f8applications13a-h2"],"title":"Second equation","text":"The problem mentions the difference between the two angles is $$24$$ degrees. Because of this, we can write the second equation as $$a-b=24$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$102$$"],"dependencies":["a9ad3f8applications13a-h3"],"title":"Solve","text":"Next, using the system of equations just created, solve for a. What does it equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications13a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$78$$"],"dependencies":["a9ad3f8applications13a-h4"],"title":"Solve","text":"What does $$t$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications13a-h6","type":"hint","dependencies":["a9ad3f8applications13a-h5"],"title":"Answer","text":"The two angle measures are $$102$$ degrees and $$78$$ degrees.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ad3f8applications14","title":"Translating a System of Equations","body":"Translate to a system of equations and solve.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Solve Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ad3f8applications14a","stepAnswer":["The measures are $$134$$ degrees and $$46$$ degrees"],"problemType":"MultipleChoice","stepTitle":"The difference of two supplementary angles is $$88$$ degrees. Find the measures of the angles.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"The measures are $$134$$ degrees and $$46$$ degrees","choices":["The measures are $$134$$ degrees and $$46$$ degrees","The measures are $$22$$ degrees and $$17$$ degrees","The measures are $$94$$ degrees and $$68$$ degrees","The measures are $$11$$ degrees and $$58$$ degrees"],"hints":{"DefaultPathway":[{"id":"a9ad3f8applications14a-h1","type":"hint","dependencies":[],"title":"Supplementary","text":"When two angles are supplementary, it means their measures add up to $$180$$ degrees.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications14a-h2","type":"hint","dependencies":["a9ad3f8applications14a-h1"],"title":"First equation","text":"Since the problem identifies the angles as supplementary, we know that they add up to $$180$$ degrees. Therefore, the first equation should be $$a+b=180$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications14a-h3","type":"hint","dependencies":["a9ad3f8applications14a-h2"],"title":"Second equation","text":"The problem mentions the difference between the two angles is $$88$$ degrees. Because of this, we can write the second equation as $$a-b=88$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications14a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$134$$"],"dependencies":["a9ad3f8applications14a-h3"],"title":"Solve","text":"Next, using the system of equations just created, solve for a. What does it equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications14a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$46$$"],"dependencies":["a9ad3f8applications14a-h4"],"title":"Solve","text":"What does $$t$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications14a-h6","type":"hint","dependencies":["a9ad3f8applications14a-h5"],"title":"Answer","text":"The two angle measures are $$134$$ degrees and $$46$$ degrees.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ad3f8applications15","title":"Translating a System of Equations","body":"Translate to a system of equations and solve.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Solve Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ad3f8applications15a","stepAnswer":["The measures are $$53.5$$ degrees and $$36.5$$ degrees."],"problemType":"MultipleChoice","stepTitle":"The difference of two complementary angles is $$17$$ degrees. Find the measures of the angles.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"The measures are $$53.5$$ degrees and $$36.5$$ degrees.","choices":["The measures are $$53.5$$ degrees and $$36.5$$ degrees.","The measures are $$20$$ degrees and $$30$$ degrees.","The measures are $$22.5$$ degrees and $$70$$ degrees.","The measures are $$20.5$$ degrees and $$160$$ degrees."],"hints":{"DefaultPathway":[{"id":"a9ad3f8applications15a-h1","type":"hint","dependencies":[],"title":"Complementary angles","text":"When two angles are complementary, it means their measures add up to $$90$$ degrees.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications15a-h2","type":"hint","dependencies":["a9ad3f8applications15a-h1"],"title":"First equation","text":"Since the problem identifies the angles as complementary, we know that they add up to $$90$$ degrees. Therefore, the first equation should be $$a+b=90$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications15a-h3","type":"hint","dependencies":["a9ad3f8applications15a-h2"],"title":"Second equation","text":"The problem mentions the difference between the two angles is $$17$$ degrees. Because of this, we can write the second equation as $$a-b=17$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$53.5$$"],"dependencies":["a9ad3f8applications15a-h3"],"title":"Solve","text":"Next, using the system of equations just created, solve for a. What does it equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications15a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$36.5$$"],"dependencies":["a9ad3f8applications15a-h4"],"title":"Solve","text":"What does $$t$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications15a-h6","type":"hint","dependencies":["a9ad3f8applications15a-h5"],"title":"Answer","text":"The two angle measures are $$53.5$$ degrees and $$36.5$$ degrees.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ad3f8applications16","title":"System of Equations","body":"Translate the following problem to a systems of equation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Solve Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ad3f8applications16a","stepAnswer":["$$x+y=-14, x=y-4$$"],"problemType":"MultipleChoice","stepTitle":"The sum of two numbers is negative fourteen. One number is four less than the other.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x+y=-14, x=y-4$$","choices":["$$x-y=14, y=x+4$$","$$x+y=-14, x=y-4$$","$$2x+y=-7, y=x-2$$","$$x+y=14, x=y+10$$"],"hints":{"DefaultPathway":[{"id":"a9ad3f8applications16a-h1","type":"hint","dependencies":[],"title":"Identify the Numbers","text":"The problem specifies there are \\"two numbers.\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications16a-h2","type":"hint","dependencies":["a9ad3f8applications16a-h1"],"title":"Create Variables","text":"Name the two unknown numbers $$x$$ and $$y$$ to represent those quantities. Let $$x=one$$ number and $$y=second$$ number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications16a-h3","type":"hint","dependencies":["a9ad3f8applications16a-h2"],"title":"Translate Into System","text":"Since the sum of the two number is negative fourteen we can write $$x+y=-14$$. Additionally one number is four less than the other so we can write $$x=y-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications16a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x+y=-14, x=y-4$$"],"dependencies":["a9ad3f8applications16a-h3"],"title":"Translate Into System","text":"What is the systems of equation represented in the problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x-y=14, y=x+4$$","$$x+y=-14, x=y-4$$","$$2x+y=-7, y=x-2$$","$$x+y=14, x=y+10$$"]}]}}]},{"id":"a9ad3f8applications17","title":"System of Equations","body":"Translate the following problem to a systems of equation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Solve Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ad3f8applications17a","stepAnswer":["$$x+y=-23, x=y-7$$"],"problemType":"MultipleChoice","stepTitle":"The sum of two numbers is negative twenty-three. One number is $$7$$ less than the other.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x+y=-23, x=y-7$$","choices":["$$x+y=-23, x=y-7$$","$$x-y=23, x=y-4$$","$$x+y=-23, x=y+7$$","$$x+y=23, x=y+7$$"],"hints":{"DefaultPathway":[{"id":"a9ad3f8applications17a-h1","type":"hint","dependencies":[],"title":"Identify the Numbers","text":"The problem specifies there are \\"two numbers.\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications17a-h2","type":"hint","dependencies":["a9ad3f8applications17a-h1"],"title":"Create Variables","text":"Name the two unknown numbers $$x$$ and $$y$$ to represent those quantities. Let $$x=one$$ number and $$y=second$$ number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications17a-h3","type":"hint","dependencies":["a9ad3f8applications17a-h2"],"title":"Translate Into System","text":"Since the sum of the two number is negative twenty-three we can write $$x+y=-23$$. Additionally one number is seven less than the other so we can write $$x=y-7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications17a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x+y=-23, x=y-7$$"],"dependencies":["a9ad3f8applications17a-h3"],"title":"Translate Into System","text":"What is the systems of equation represented in the problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x+y=-23, x=y-7$$","$$x-y=23, x=y-4$$","$$x+y=-23, x=y+7$$","$$x+y=23, x=y+7$$"]}]}}]},{"id":"a9ad3f8applications18","title":"System of Equations","body":"Translate the following problem to a systems of equation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Solve Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ad3f8applications18a","stepAnswer":["$$x+y=-18, x=y+40$$"],"problemType":"MultipleChoice","stepTitle":"The sum of two numbers is negative eighteen. One number is $$40$$ more than the other.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x+y=-18, x=y+40$$","choices":["$$x+y=18, x=y+20$$","$$x-y=-18, x=y-10$$","$$x+y=-18, x=y-20$$","$$x+y=-18, x=y+40$$"],"hints":{"DefaultPathway":[{"id":"a9ad3f8applications18a-h1","type":"hint","dependencies":[],"title":"Identify the Numbers","text":"The problem specifies there are \\"two numbers.\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications18a-h2","type":"hint","dependencies":["a9ad3f8applications18a-h1"],"title":"Create Variables","text":"Name the two unknown numbers $$x$$ and $$y$$ to represent those quantities. Let $$x=one$$ number and $$y=second$$ number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications18a-h3","type":"hint","dependencies":["a9ad3f8applications18a-h2"],"title":"Translate Into System","text":"Since the sum of the two number is negative eighteen we can write $$x+y=-18$$. Additionally one number is fourty more than the other so we can write $$x=y+40$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications18a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x+y=-18, x=y+40$$"],"dependencies":["a9ad3f8applications18a-h3"],"title":"Translate Into System","text":"What is the systems of equation represented in the problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x+y=18, x=y+20$$","$$x-y=-18, x=y-10$$","$$x+y=-18, x=y-20$$","$$x+y=-18, x=y+40$$"]}]}}]},{"id":"a9ad3f8applications19","title":"System of Equations","body":"Translate the following problem to a systems of equation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Solve Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ad3f8applications19a","stepAnswer":["$$42000$$"],"problemType":"TextBox","stepTitle":"A married couple together earns $110,000 a year. The wife earns $16,000 less than twice what her husband earns. What does the husband earn?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$42000$$","hints":{"DefaultPathway":[{"id":"a9ad3f8applications19a-h1","type":"hint","dependencies":[],"title":"Create Variables","text":"Let $$h$$ represent the husband\'s earnings and w represent the wife\'s earning.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications19a-h2","type":"hint","dependencies":["a9ad3f8applications19a-h1"],"title":"Translate Into System","text":"Together they earn $110,000 translating into $$h+w=110000$$. The wife earns $16,000 less than twice her husband translating into $$w=2h-16000$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications19a-h3","type":"hint","dependencies":["a9ad3f8applications19a-h2"],"title":"Solve the System","text":"Using substitution we can substitute $$w=2h-16000$$ into $$h+w=110000$$ to solve for $$h$$, the husband\'s earnings.\\\\n$$h+w=110000$$\\\\n$$h+2h-16000=110000$$\\\\n$$3h-16000=110000$$\\\\n$$3h=126000$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications19a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$42000$$"],"dependencies":["a9ad3f8applications19a-h3"],"title":"Finding the Husband\'s Earnings","text":"How much does the husband earn?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ad3f8applications2","title":"Translating a System of Equations","body":"Translate to a system of equations and solve.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Solve Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ad3f8applications2a","stepAnswer":["She put $15,000 into a CD and $35,000 in bonds."],"problemType":"MultipleChoice","stepTitle":"Daniela invested a total of $50,000, some in a certificate of deposit (CD) and the remainder in bonds. The amount invested in bonds was $5000 more than twice the amount she put into the CD. How much did she invest in each account?","stepBody":"","answerType":"string","variabilization":{},"choices":["She put $15,000 into a CD and $35,000 in bonds.","She put $15,000 into a CD and $35,000 in bonds.IShe put $20,000 into a CD and $15,000 in bonds.IShe put $10,000 into a CD and $10,000 in bonds.IShe put $20,000 into a CD and $30,000 in bonds."],"hints":{"DefaultPathway":[{"id":"a9ad3f8applications2a-h1","type":"hint","dependencies":[],"title":"Identify","text":"We are looking for the amount Daniela put in each account.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications2a-h2","type":"hint","dependencies":["a9ad3f8applications2a-h1"],"title":"First equation","text":"Since the Daniela invested $50,000 in total we can write the first equation as $$c+b=50, 000$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications2a-h3","type":"hint","dependencies":["a9ad3f8applications2a-h2"],"title":"Second equation","text":"Since the amount invested in bonds was $5000 more than twice the amount she put into the CD, we can write the second equation as $$b=2c+5000$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15000$$"],"dependencies":["a9ad3f8applications2a-h3"],"title":"Solve","text":"Next, using the system of equations just created, solve for c. What does it equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications2a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$35000$$"],"dependencies":["a9ad3f8applications2a-h4"],"title":"Solve","text":"What does $$b$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications2a-h6","type":"hint","dependencies":["a9ad3f8applications2a-h5"],"title":"Answer","text":"Daniela invested $15,000 in the CD and $35,000 in bonds.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ad3f8applications20","title":"System of Equations","body":"Translate the following problem to a systems of equation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Solve Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ad3f8applications20a","stepAnswer":["$$34000$$"],"problemType":"TextBox","stepTitle":"A couple has a total household income of $84,000. The husband earns $18,000 less than twice what the wife earns. How much does the wife earn?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$34000$$","hints":{"DefaultPathway":[{"id":"a9ad3f8applications20a-h1","type":"hint","dependencies":[],"title":"Create Variables","text":"Let $$h$$ represent the husband\'s earnings and w represent the wife\'s earning.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications20a-h2","type":"hint","dependencies":["a9ad3f8applications20a-h1"],"title":"Translate Into System","text":"Together they earn $84,000 translating into $$h+w=84000$$. The husband earns $18,000 less than twice his wife translating into $$h=2h-18000$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications20a-h3","type":"hint","dependencies":["a9ad3f8applications20a-h2"],"title":"Solve the System","text":"Using substitution we can substitute $$h=2w-18000$$ into $$h+w=84000$$ to solve for w, the wife\'s earnings.\\\\n$$h+w=84000$$\\\\n$$2w-18000+w=84000$$\\\\n$$3w-18000=84000$$\\\\n$$3w=102000$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications20a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$34000$$"],"dependencies":["a9ad3f8applications20a-h3"],"title":"Finding the Wife\'s Earnings","text":"How much does the wife earn?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ad3f8applications21","title":"System of Equations","body":"Translate the following problem to a systems of equation and solve.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Solve Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ad3f8applications21a","stepAnswer":["$$Senior=\\\\$27, New=\\\\$16$$"],"problemType":"MultipleChoice","stepTitle":"A senior employee makes $5 less than twice what a new employee makes per hour. Together they make $43 per hour. How much does each employee make per hour?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$Senior=\\\\$27, New=\\\\$16$$","choices":["$$Senior=\\\\$27, New=\\\\$16$$","$$Senior=\\\\$27, New=\\\\$16ISenior=\\\\$27, New=\\\\$12ISenior=\\\\$15, New=\\\\$7ISenior=\\\\$43, New=\\\\$16$$"],"hints":{"DefaultPathway":[{"id":"a9ad3f8applications21a-h1","type":"hint","dependencies":[],"title":"Create Variables","text":"Let s represent the senior employee\'s earnings and $$n$$ represent the new employee\'s earning.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications21a-h2","type":"hint","dependencies":["a9ad3f8applications21a-h1"],"title":"Translate Into System","text":"Together they earn $43 per hour, translating into $$s+n=43$$. The senior employee earns $5 less than twice the new employee per hour translating into $$s=2n-5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications21a-h3","type":"hint","dependencies":["a9ad3f8applications21a-h2"],"title":"Solve the System","text":"Using substitution we can substitute $$s=2n-5$$ into $$s+n=43$$ to solve for $$n$$, the new employee\'s earnings.\\\\n$$s+n=43$$\\\\n$$2n-5+n=43$$\\\\n$$3n-5=43$$\\\\n$$3n=48$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications21a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["a9ad3f8applications21a-h3"],"title":"Finding the New Employee\'s Earnings","text":"How does the new employee earn per hour?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications21a-h5","type":"hint","dependencies":["a9ad3f8applications21a-h4"],"title":"Finding the Senior Employee\'s Earnings","text":"Given $$n=16$$, plug in the known value to solve for s.\\\\n$$s=2n-5$$\\\\n$$s=2\\\\times16-5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications21a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$27$$"],"dependencies":["a9ad3f8applications21a-h5"],"title":"Finding the Senior Employee\'s Earnings","text":"How does the senior employee earn per hour?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ad3f8applications22","title":"System of Equations","body":"Translate the following problem to a systems of equation and solve","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Solve Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ad3f8applications22a","stepAnswer":["$$Devon=38, Cooper=12$$"],"problemType":"MultipleChoice","stepTitle":"Devon is $$26$$ years older than his son Cooper. The sum of their ages is $$50$$. Find their ages.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$Devon=38, Cooper=12$$","choices":["$$Devon=32, Cooper=12IDevon=38, Cooper=12IDevon=24, Cooper=10IDevon=32, Cooper=10$$","$$Devon=38, Cooper=12$$"],"hints":{"DefaultPathway":[{"id":"a9ad3f8applications22a-h1","type":"hint","dependencies":[],"title":"Create Variables","text":"Let $$d$$ represent Devon\'s age and c represent Cooper\'s age.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications22a-h2","type":"hint","dependencies":["a9ad3f8applications22a-h1"],"title":"Translate to a System of Equations","text":"Devon is $$26$$ years older than Cooper, translating into $$d=c+26$$. The sum of their ages is $$50$$, translating into $$c+d=50$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications22a-h3","type":"hint","dependencies":["a9ad3f8applications22a-h2"],"title":"Solve the System","text":"Using substitution we can substitute $$d=c+26$$ into $$c+d=50$$ to solve for c, Cooper\'s age.\\\\n$$c+d=50$$\\\\n$$c+c+26=50$$\\\\n$$2c+26=50$$\\\\n$$2c=24$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications22a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a9ad3f8applications22a-h3"],"title":"Finding Cooper\'s Age","text":"What is Cooper\'s age?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications22a-h5","type":"hint","dependencies":["a9ad3f8applications22a-h4"],"title":"Finding Devon\'s Age","text":"Given $$c=12$$, plug the known value into $$d=c+26$$ to solve for $$d$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications22a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$38$$"],"dependencies":["a9ad3f8applications22a-h5"],"title":"Finding Devon\'s Age","text":"What is Devon\'s age?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ad3f8applications23","title":"System of Equations","body":"Translate the following problem to a systems of equation and solve.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Solve Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ad3f8applications23a","stepAnswer":["$$x=58, y=32$$"],"problemType":"MultipleChoice","stepTitle":"The difference of two complementary angles is $$26$$ degrees. Find the measures of the angles.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=58, y=32$$","choices":["$$x=40, y=50Ix=22$$, $$68Ix=58, y=32Ix=41, y=49$$","$$x=58, y=32$$"],"hints":{"DefaultPathway":[{"id":"a9ad3f8applications23a-h1","type":"hint","dependencies":[],"title":"Create Variables","text":"We are looking to find the measure of two angles. Let $$x=measure$$ of first angle and $$y=measure$$ of second angle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications23a-h2","type":"hint","dependencies":["a9ad3f8applications23a-h1"],"title":"Translate to a System of Equations","text":"The two angles are complementary, translating into $$x+y=90$$. The two angles have a difference of $$26$$, translating into $$x-y=26$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications23a-h3","type":"hint","dependencies":["a9ad3f8applications23a-h2"],"title":"Solve the System","text":"Using elimination we can add the two equations together.\\\\n$$x+y=90$$\\\\n$$x-y=26$$\\\\n$$y$$ is eliminated which allows us to solve for $$x$$.\\\\n$$2x=116$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications23a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$58$$"],"dependencies":["a9ad3f8applications23a-h3"],"title":"Finding Angle $$1$$","text":"What is the measure of angle $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications23a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$32$$"],"dependencies":["a9ad3f8applications23a-h4"],"title":"Finding Angle $$2$$","text":"Given $$x=58$$, plug the known value into one of the two systems to solve for $$y$$. What is the measure of angle $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ad3f8applications24","title":"System of Equations","body":"Translate the following problem to a systems of equation and solve.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Solve Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ad3f8applications24a","stepAnswer":["$$x=55, y=35$$"],"problemType":"MultipleChoice","stepTitle":"The difference of two complementary angles is $$20$$ degrees. Find the measures of the angles.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=55, y=35$$","choices":["$$x=50, y=40Ix=25, y=65Ix=30, y=60Ix=55, y=35$$","$$x=55, y=35$$"],"hints":{"DefaultPathway":[{"id":"a9ad3f8applications24a-h1","type":"hint","dependencies":[],"title":"Create Variables","text":"We are looking to find the measure of two angles. Let $$x=measure$$ of first angle and $$y=measure$$ of second angle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications24a-h2","type":"hint","dependencies":["a9ad3f8applications24a-h1"],"title":"Translate to a System of Equations","text":"The two angles are complementary, translating into $$x+y=90$$. The two angles have a difference of $$20$$, translating into $$x-y=20$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications24a-h3","type":"hint","dependencies":["a9ad3f8applications24a-h2"],"title":"Solve the System","text":"Using elimination we can add the two equations together.\\\\n$$x+y=90$$\\\\n$$x-y=20$$\\\\n$$y$$ is elimnated which allows us to solve for $$x$$.\\\\n$$2x=110$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications24a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$55$$"],"dependencies":["a9ad3f8applications24a-h3"],"title":"Finding Angle $$1$$","text":"What is the measure of angle $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications24a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$35$$"],"dependencies":["a9ad3f8applications24a-h4"],"title":"Finding Angle $$2$$","text":"Given $$x=55$$, plug the known value into one of the two systems to solve for $$y$$. What is the measure of angle $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ad3f8applications25","title":"System of Equations","body":"Translate the following problem to a systems of equation and solve.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Solve Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ad3f8applications25a","stepAnswer":["$$x=85, y=5$$"],"problemType":"MultipleChoice","stepTitle":"The difference of two complementary angles is $$80$$ degrees. Find the measures of the angles.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=85, y=5$$","choices":["$$x=85, y=5$$","$$x=85, y=5Ix=20, y=80Ix=10, y=70Ix=64, 26$$"],"hints":{"DefaultPathway":[{"id":"a9ad3f8applications25a-h1","type":"hint","dependencies":[],"title":"Create Variables","text":"We are looking to find the measure of two angles. Let $$x=measure$$ of first angle and $$y=measure$$ of second angle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications25a-h2","type":"hint","dependencies":["a9ad3f8applications25a-h1"],"title":"Translate to a System of Equations","text":"The two angles are complementary, translating into $$x+y=90$$. The two angles have a difference of $$80$$, translating into $$x-y=80$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications25a-h3","type":"hint","dependencies":["a9ad3f8applications25a-h2"],"title":"Solve the System","text":"Using elimination we can add the two equations together.\\\\n$$x+y=90$$\\\\n$$x-y=80$$\\\\n$$y$$ is eliminated which allows us to solve for $$x$$.\\\\n$$2x=170$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications25a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$85$$"],"dependencies":["a9ad3f8applications25a-h3"],"title":"Finding Angle $$1$$","text":"What is the measure of angle $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications25a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a9ad3f8applications25a-h4"],"title":"Finding Angle $$2$$","text":"Given $$x=85$$, plug the known value into one of the two systems to solve for $$y$$. What is the measure of angle $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ad3f8applications26","title":"System of Equations","body":"Translate the following problem to a systems of equation and solve.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Solve Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ad3f8applications26a","stepAnswer":["$$x=148, y=32$$"],"problemType":"MultipleChoice","stepTitle":"Two angles are supplementary. The measure of the larger angle is twelve degrees less than five times the measure of the smaller angle. Find the measures of both angles.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=148, y=32$$","choices":["$$x=148, y=32$$","$$x=150, y=30Ix=148, y=32Ix=122, y=53Ix=132, y=36$$"],"hints":{"DefaultPathway":[{"id":"a9ad3f8applications26a-h1","type":"hint","dependencies":[],"title":"Create Variables","text":"We are looking to find the measure of two angles. Let $$x=measure$$ of larger angle and $$y=measure$$ of smaller angle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications26a-h2","type":"hint","dependencies":["a9ad3f8applications26a-h1"],"title":"Translate to a System of Equations","text":"The two angles are supplementary, translating into $$x+y=180$$. The larger angle is twelve degrees less than five times the smaller angle, translating into $$x=5y-12$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications26a-h3","type":"hint","dependencies":["a9ad3f8applications26a-h2"],"title":"Solve the System","text":"Using substitution we can substitute $$x=5y-12$$ into $$x+y=180$$ to solve for $$y$$, the smaller angle.\\\\n$$x+y=180$$\\\\n$$5y-12+y=180$$\\\\n$$6y-12=180$$\\\\n$$6y=192$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications26a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$32$$"],"dependencies":["a9ad3f8applications26a-h3"],"title":"Finding Smaller Angle","text":"What is the measure of angle $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications26a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$148$$"],"dependencies":["a9ad3f8applications26a-h4"],"title":"Finding Larger Angle","text":"Given $$y=12$$, plug the known value into $$x=5y-12$$ to solve for $$x$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ad3f8applications27","title":"System of Equations","body":"Translate the following problem to a systems of equation and solve.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Solve Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ad3f8applications27a","stepAnswer":["$$x=138, y=42$$"],"problemType":"MultipleChoice","stepTitle":"Two angles are supplementary. The measure of the larger angle is $$12$$ degrees more than three times the smaller angle. Find the measures of the angles.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=138, y=42$$","choices":["$$x=138, y=42$$","$$x=138, y=42Ix=120, y=60Ix=102, y=78Ix=135, y=45$$"],"hints":{"DefaultPathway":[{"id":"a9ad3f8applications27a-h1","type":"hint","dependencies":[],"title":"Create Variables","text":"We are looking to find the measure of two angles. Let $$x=measure$$ of larger angle and $$y=measure$$ of smaller angle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications27a-h2","type":"hint","dependencies":["a9ad3f8applications27a-h1"],"title":"Translate to a System of Equations","text":"The two angles are supplementary, translating into $$x+y=180$$. The larger angle is $$12$$ degrees more than three times the smaller angle, translating into $$x=3y+12$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications27a-h3","type":"hint","dependencies":["a9ad3f8applications27a-h2"],"title":"Solve the System","text":"Using substitution we can substitute $$x=3y+12$$ into $$x+y=180$$ to solve for $$y$$, the smaller angle.\\\\n$$x+y=180$$\\\\n$$3y+12+y=180$$\\\\n$$4y+12=180$$\\\\n$$4y=168$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications27a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$42$$"],"dependencies":["a9ad3f8applications27a-h3"],"title":"Finding Smaller Angle","text":"What is the measure of angle $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications27a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$138$$"],"dependencies":["a9ad3f8applications27a-h4"],"title":"Finding Larger Angle","text":"Given $$y=42$$, plug the known value into one of the equations to solve for $$x$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ad3f8applications28","title":"System of Equations","body":"Translate the following problem to a systems of equation and solve.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Solve Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ad3f8applications28a","stepAnswer":["$$x=114, y=66$$"],"problemType":"MultipleChoice","stepTitle":"Two angles are supplementary. The measure of the larger angle is $$18$$ less than twice the measure of the smaller angle. Find the measures of the angles.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=114, y=66$$","choices":["$$x=114, y=66$$","$$x=82, y=60Ix=94, y=66Ix=121, y=79Ix=114, y=66$$"],"hints":{"DefaultPathway":[{"id":"a9ad3f8applications28a-h1","type":"hint","dependencies":[],"title":"Create Variables","text":"We are looking to find the measure of two angles. Let $$x=measure$$ of larger angle and $$y=measure$$ of smaller angle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications28a-h2","type":"hint","dependencies":["a9ad3f8applications28a-h1"],"title":"Translate to a System of Equations","text":"The two angles are supplementary, translating into $$x+y=180$$. The larger angle is $$18$$ less than twice the measure of the smaller angle, translating into $$x=2y-18$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications28a-h3","type":"hint","dependencies":["a9ad3f8applications28a-h2"],"title":"Solve the System","text":"Using substitution we can substitute $$x=2y-18$$ into $$x+y=180$$ to solve for $$y$$, the smaller angle.\\\\n$$x+y=180$$\\\\n$$2y-18+y$$ $$=180$$\\\\n$$3y-18=180$$\\\\n$$3y=198$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications28a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$66$$"],"dependencies":["a9ad3f8applications28a-h3"],"title":"Finding Smaller Angle","text":"What is the measure of angle $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications28a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$114$$"],"dependencies":["a9ad3f8applications28a-h4"],"title":"Finding Larger Angle","text":"Given $$y=66$$, plug the known value into one of the equations to solve for $$x$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ad3f8applications29","title":"System of Equations","body":"Translate the following problem to a systems of equation and solve.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Solve Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ad3f8applications29a","stepAnswer":["$$x=6, y=9$$"],"problemType":"MultipleChoice","stepTitle":"The sum of two numbers is fifteen. One number is three less than the other. Find the numbers.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=6, y=9$$","choices":["$$x=6, y=9$$","$$x=6, y=9Ix=2, y=10Ix=1, y=8Ix=3, y=12$$"],"hints":{"DefaultPathway":[{"id":"a9ad3f8applications29a-h1","type":"hint","dependencies":[],"title":"Identify the Numbers","text":"The problem specifies there are two numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications29a-h2","type":"hint","dependencies":["a9ad3f8applications29a-h1"],"title":"Create Variables","text":"Name the two unknown numbers $$x$$ and $$y$$ to represent those quantities. Let $$x=one$$ number and $$y=second$$ number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications29a-h3","type":"hint","dependencies":["a9ad3f8applications29a-h2"],"title":"Translate Into System","text":"Since the sum of the two number is fifteen we can write $$x+y=15$$. Additionally one number is three less than the other so we can write $$x=y-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications29a-h4","type":"hint","dependencies":["a9ad3f8applications29a-h3"],"title":"Solve the System","text":"Using substitution we can substitute $$x=y-3$$ into $$x+y=15$$ to solve for $$y$$.\\\\n$$x+y=15$$\\\\n$$y-3+y=15$$\\\\n$$2y-3=15$$\\\\n$$2y=18$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications29a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a9ad3f8applications29a-h4"],"title":"Solving for $$y$$","text":"What is $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications29a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a9ad3f8applications29a-h5"],"title":"Solving for $$x$$","text":"Given $$y=9$$, plug the known value into one of the equations to solve for $$x$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ad3f8applications3","title":"Translating a System of Equations","body":"Translate to a system of equations and solve.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Solve Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ad3f8applications3a","stepAnswer":["The amount of the first year\u2019s loan was $30,000 and the amount of the second year\u2019s loan was $12,000."],"problemType":"MultipleChoice","stepTitle":"In her last two years in college, Marlene received $42,000 in loans. The first year she received a loan that was $6,000 less than three times the amount of the second year\u2019s loan. What was the amount of her loan for each year?","stepBody":"","answerType":"string","variabilization":{},"choices":["The amount of the first year\u2019s loan was $30,000 and the amount of the second year\u2019s loan was $12,000.","The amount of the first year\u2019s loan was $30,000 and the amount of the second year\u2019s loan was $12,000.IThe amount of the first year\u2019s loan was $15,000 and the amount of the second year\u2019s loan was $2,000.IThe amount of the first year\u2019s loan was $12,000 and the amount of the second year\u2019s loan was $30,000.IThe amount of the first year\u2019s loan was $15,000 and the amount of the second year\u2019s loan was $20,000."],"hints":{"DefaultPathway":[{"id":"a9ad3f8applications3a-h1","type":"hint","dependencies":[],"title":"Identify","text":"We are looking for the amount of each loan.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications3a-h2","type":"hint","dependencies":["a9ad3f8applications3a-h1"],"title":"First equation","text":"Since Marlene recieved $42,000 in total we can write the first equation as $$a+b=42, 000$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications3a-h3","type":"hint","dependencies":["a9ad3f8applications3a-h2"],"title":"Second equation","text":"Since in the first year she received a loan, it was $6,000 less than three times the amount of the second year\u2019s loan, we can write the second equation as $$a=3b-6000$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$30000$$"],"dependencies":["a9ad3f8applications3a-h3"],"title":"Solve","text":"Next, using the system of equations just created, solve for a. What does it equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12000$$"],"dependencies":["a9ad3f8applications3a-h4"],"title":"Solve","text":"What does $$b$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications3a-h6","type":"hint","dependencies":["a9ad3f8applications3a-h5"],"title":"Answer","text":"Marlene recieved $30,000 for the first year loan and $12,000 for the second years loan.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ad3f8applications30","title":"System of Equations","body":"Translate the following problem to a systems of equation and solve.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Solve Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ad3f8applications30a","stepAnswer":["$$x=-25, y=-5$$"],"problemType":"MultipleChoice","stepTitle":"The sum of two numbers is negative thirty. One number is five times the other. Find the numbers.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=-25, y=-5$$","choices":["$$x=-25, y=-5$$","$$x=-25, y=-5Ix=-3, y=-27Ix=12, y=18Ix=6, y=-36$$"],"hints":{"DefaultPathway":[{"id":"a9ad3f8applications30a-h1","type":"hint","dependencies":[],"title":"Identify the Numbers","text":"The problem specifies there are two numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications30a-h2","type":"hint","dependencies":["a9ad3f8applications30a-h1"],"title":"Create Variables","text":"Name the two unknown numbers $$x$$ and $$y$$ to represent those quantities. Let $$x=one$$ number and $$y=second$$ number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications30a-h3","type":"hint","dependencies":["a9ad3f8applications30a-h2"],"title":"Translate Into System","text":"Since the sum of the two number is $$-30$$ we can write $$x+y=-30$$. Additionally one number is five times the other so we can write $$x=5y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications30a-h4","type":"hint","dependencies":["a9ad3f8applications30a-h3"],"title":"Solve the System","text":"Using substitution we can substitute $$x=5y$$ into $$x+y=-30$$ to solve for $$y$$.\\\\n$$x+y=-30$$\\\\n$$5y+y=-30$$\\\\n$$6y=-30$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications30a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["a9ad3f8applications30a-h4"],"title":"Solving for $$y$$","text":"What is $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications30a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-25$$"],"dependencies":["a9ad3f8applications30a-h5"],"title":"Solving for $$x$$","text":"Given $$y=-5$$, plug the known value into one of the equations to solve for $$x$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ad3f8applications4","title":"Translating a System of Equations","body":"Translate to a system of equations and solve.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Solve Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ad3f8applications4a","stepAnswer":["Bethany is $$16$$ years old and Alyssa is $$28$$ years old."],"problemType":"MultipleChoice","stepTitle":"Alyssa is twelve years older than her sister, Bethany. The sum of their ages is forty-four. Find their ages.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Bethany is $$16$$ years old and Alyssa is $$28$$ years old.","choices":["Bethany is $$16$$ years old and Alyssa is $$28$$ years old.","Bethany is $$16$$ years old and Alyssa is $$28$$ years old.IBethany is $$12$$ years old and Alyssa is $$32$$ years old.IBethany is $$16$$ years old and Alyssa is $$16$$ years old.IBethany is $$12$$ years old and Alyssa is $$15$$ years old."],"hints":{"DefaultPathway":[{"id":"a9ad3f8applications4a-h1","type":"hint","dependencies":[],"title":"Identify","text":"We are looking for the age of each sister.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications4a-h2","type":"hint","dependencies":["a9ad3f8applications4a-h1"],"title":"First equation","text":"Since the sisters ages together are $$44$$, we can write the first equation as $$a+b=44$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications4a-h3","type":"hint","dependencies":["a9ad3f8applications4a-h2"],"title":"Second equation","text":"Since Alyssa is twelve years older than her sister, we can write the second equatiion as $$a=b+12$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$28$$"],"dependencies":["a9ad3f8applications4a-h3"],"title":"Solve","text":"Next, using the system of equations just created, solve for a. What does it equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications4a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["a9ad3f8applications4a-h4"],"title":"Solve","text":"What does $$b$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications4a-h6","type":"hint","dependencies":["a9ad3f8applications4a-h5"],"title":"Answer","text":"Alyssa is $$28$$ while Bethany is $$16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ad3f8applications5","title":"Translating a System of Equations","body":"Translate to a system of equations and solve.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Solve Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ad3f8applications5a","stepAnswer":["Noelle is $$20$$ years old and her dad is $$54$$ years old."],"problemType":"MultipleChoice","stepTitle":"The age of Noelle\u2019s dad is six less than three times Noelle\u2019s age. The sum of their ages is seventy-four. Find their ages.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Noelle is $$20$$ years old and her dad is $$54$$ years old.","choices":["Noelle is $$20$$ years old and her dad is $$54$$ years old.","Noelle is $$20$$ years old and her dad is $$54$$ years old.INoelle is $$17$$ years old and her dad is $$44$$ years old.INoelle is $$17$$ years old and her dad is $$52$$ years old.INoelle is $$20$$ years old and her dad is $$34$$ years old."],"hints":{"DefaultPathway":[{"id":"a9ad3f8applications5a-h1","type":"hint","dependencies":[],"title":"Identify","text":"We are looking for the age of Noelle and her dad.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications5a-h2","type":"hint","dependencies":["a9ad3f8applications5a-h1"],"title":"First equation","text":"Since their ages together are $$74$$, we can write the first equation as $$n+d=74$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications5a-h3","type":"hint","dependencies":["a9ad3f8applications5a-h2"],"title":"Second equation","text":"Since the age of Noelle\u2019s dad is six less than three times Noelle\u2019s age, we can write the second equation as $$d=3n-6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a9ad3f8applications5a-h3"],"title":"Solve","text":"Next, using the system of equations just created, solve for $$n$$. What does it equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$54$$"],"dependencies":["a9ad3f8applications5a-h4"],"title":"Solve","text":"What does $$d$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications5a-h6","type":"hint","dependencies":["a9ad3f8applications5a-h5"],"title":"Answer","text":"Noelle is $$20$$ and her dad is $$54$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ad3f8applications6","title":"Translating a System of Equations","body":"Translate to a system of equations and solve.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Solve Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ad3f8applications6a","stepAnswer":["The small container holds $$20$$ gallons and the large container holds $$30$$ gallons."],"problemType":"MultipleChoice","stepTitle":"Two containers of gasoline hold a total of fifty gallons. The big container can hold ten gallons less than twice the small container. How many gallons does each container hold?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"The small container holds $$20$$ gallons and the large container holds $$30$$ gallons.","choices":["The small container holds $$20$$ gallons and the large container holds $$30$$ gallons.","The small container holds $$20$$ gallons and the large container holds $$30$$ gallons.IThe small container holds $$10$$ gallons and the large container holds $$40$$ gallons.IThe small container holds $$30$$ gallons and the large container holds $$40$$ gallons.IThe small container holds $$50$$ gallons and the large container holds $$10$$ gallons."],"hints":{"DefaultPathway":[{"id":"a9ad3f8applications6a-h1","type":"hint","dependencies":[],"title":"Identify","text":"We are looking for how many gallons of gas each container can hold.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications6a-h2","type":"hint","dependencies":["a9ad3f8applications6a-h1"],"title":"First equation","text":"Since their ages together are $$74$$, we can write the first equation as $$s+b=50$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications6a-h3","type":"hint","dependencies":["a9ad3f8applications6a-h2"],"title":"Second equation","text":"Since the big container can hold ten gallons less than twice the small container, we can write the second equation as $$b=2s-10$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a9ad3f8applications6a-h3"],"title":"Solve","text":"Next, using the system of equations just created, solve for s. What does it equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$30$$"],"dependencies":["a9ad3f8applications6a-h4"],"title":"Solve","text":"What does $$b$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications6a-h6","type":"hint","dependencies":["a9ad3f8applications6a-h5"],"title":"Answer","text":"Therefore, the small container holds $$20$$ gallons while the big container holds $$30$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ad3f8applications7","title":"Translating a System of Equations","body":"Translate to a system of equations and solve.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Solve Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ad3f8applications7a","stepAnswer":["Notebooks are $4 and thumb drives are $20."],"problemType":"MultipleChoice","stepTitle":"Troy and Lisa were shopping for school supplies. Each purchased different quantities of the same notebook and thumb drive. Troy bought four notebooks and five thumb drives for $116. Lisa bought two notebooks and three thumb dives for $68. 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What does it equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications7a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a9ad3f8applications7a-h4"],"title":"Solve","text":"What does $$t$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications7a-h6","type":"hint","dependencies":["a9ad3f8applications7a-h5"],"title":"Answer","text":"Therefore, the notebooks cost $4 while the thumb drives cost $20.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ad3f8applications8","title":"Translating a System of Equations","body":"Translate to a system of equations and solve.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Solve Applications with Systems of Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ad3f8applications8a","stepAnswer":["The measures are $$60$$ degrees and $$30$$ degrees."],"problemType":"MultipleChoice","stepTitle":"The difference of two complementary angles is $$30$$ degrees. 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What does it equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications9a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$55$$"],"dependencies":["a9ad3f8applications9a-h4"],"title":"Solve","text":"What does $$t$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ad3f8applications9a-h6","type":"hint","dependencies":["a9ad3f8applications9a-h5"],"title":"Answer","text":"The two angle measures are $$125$$ degrees and $$55$$ degrees.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ade3dRadicals1","title":"Simplify Radical Expressions","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate 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perfect square.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ade3dRadicals12a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$b^2 \\\\sqrt{b}$$"],"dependencies":["a9ade3dRadicals12a-h4"],"title":"Simplifying Expression","text":"If we factor the terms, what will our answer be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ade3dRadicals13","title":"Simplify Radical Expressions","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2  Simplify Radical Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9ade3dRadicals13a","stepAnswer":["$$p^4 \\\\sqrt{p}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{p^9}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$p^4 \\\\sqrt{p}$$","hints":{"DefaultPathway":[{"id":"a9ade3dRadicals13a-h1","type":"hint","dependencies":[],"title":"Largest Square Factor","text":"Find the largest square factor inside the square root and separate it from the other term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ade3dRadicals13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$p^8$$"],"dependencies":["a9ade3dRadicals13a-h1"],"title":"Largest Square Factor","text":"What is the largest square factor in the square root?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ade3dRadicals9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2$$"],"dependencies":["a9ade3dRadicals9a-h1"],"title":"Largest Square Factor","text":"What is the largest square factor in the square root?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ade3dRadicals9a-h3","type":"hint","dependencies":["a9ade3dRadicals9a-h2"],"title":"Factoring","text":"After we find the largest square factor, we can rewrite the radical as the product of two square roots.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ade3dRadicals9a-h4","type":"hint","dependencies":["a9ade3dRadicals9a-h3"],"title":"Factoring","text":"After we separate the two factors, we can take the square root of our perfect square.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ade3dRadicals9a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$|x| \\\\sqrt{x}$$"],"dependencies":["a9ade3dRadicals9a-h4"],"title":"Simplifying Expression","text":"If we factor the terms, what will our answer be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ae528add1","title":"Simplifying Expressions with Integers","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Add and Subtract Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ae528add1a","stepAnswer":["$$17$$"],"problemType":"TextBox","stepTitle":"$$24-|19-3\\\\left(6-2\\\\right)|$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$17$$","hints":{"DefaultPathway":[{"id":"a9ae528add1a-h1","type":"hint","dependencies":[],"title":"Working Inside the Parentheses","text":"The first step is to work on the expression inside the parentheses.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a9ae528add1a-h1"],"title":"Expression Inside Parentheses","text":"What is $$6-2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a9ae528add1a-h2"],"title":"Multiplying the Expression in the Parentheses","text":"The next step is to multiply the expression in the parentheses by the factor outside of it, which is $$3$$. What is $$3(6-2)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a9ae528add1a-h3"],"title":"Subtracting Inside the Absolute Value Bars","text":"Then, subtract within the absolute value bar. What is $$19-3(6-2)$$? Remember that you\'ve calculated the second term in the previous step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add1a-h5","type":"hint","dependencies":["a9ae528add1a-h4"],"title":"Getting Rid of the Absolute Value Bars","text":"The value $$19-3(6-2)$$, calculated in the previous step, is positive. so we can get rid of the absolute value bars. Then, the expression is $$24-7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add1a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$17$$"],"dependencies":["a9ae528add1a-h5"],"title":"Final Step","text":"What is $$24-7$$? The answer to this is the final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ae528add10","title":"Adding and Subtracting Integers","body":"Find the value of the following expressions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Add and Subtract Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ae528add10a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"$$2+4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"a9ae528add10a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":[],"title":"Seeing if the Signs Are Different","text":"Are the signs of the two terms different?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a9ae528add10a-h2","type":"hint","dependencies":["a9ae528add10a-h1"],"title":"First Step to Find the Value of the Expression","text":"Since $$2$$ and $$4$$ have the same sign, we add $$2$$ to $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add10a-h3","type":"hint","dependencies":["a9ae528add10a-h2"],"title":"Sign of the Answer","text":"The answer will be positive because there are only positives.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a9ae528add10a-h3"],"title":"Calculating the Value of the Expression","text":"What is $$2+4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add10b","stepAnswer":["$$-6$$"],"problemType":"TextBox","stepTitle":"$$-2+\\\\left(-4\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-6$$","hints":{"DefaultPathway":[{"id":"a9ae528add10b-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":[],"title":"Seeing if the Signs Are Different","text":"Are the signs of the two terms different?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a9ae528add10b-h2","type":"hint","dependencies":["a9ae528add10b-h1"],"title":"First Step to Find the Value of the Expression","text":"Since $$2$$ and $$4$$ have the same sign, we add $$2$$ to $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add10b-h3","type":"hint","dependencies":["a9ae528add10b-h2"],"title":"Sign of the Answer","text":"The answer will be negative because there are only negatives","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add10b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6$$"],"dependencies":["a9ae528add10b-h3"],"title":"Calculating the Value of the Expression","text":"What is $$-\\\\left(2+4\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ae528add11","title":"Adding and Subtracting Integers","body":"Find the value of the following expressions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Add and Subtract Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ae528add11a","stepAnswer":["$$7$$"],"problemType":"TextBox","stepTitle":"$$2+5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7$$","hints":{"DefaultPathway":[{"id":"a9ae528add11a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":[],"title":"Seeing if the Signs Are Different","text":"Are the signs of the two terms different?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a9ae528add11a-h2","type":"hint","dependencies":["a9ae528add11a-h1"],"title":"First Step to Find the Value of the Expression","text":"Since $$2$$ and $$5$$ have the same sign, we add $$2$$ to $$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add11a-h3","type":"hint","dependencies":["a9ae528add11a-h2"],"title":"Sign of the Answer","text":"The answer will be positive because there are only positives.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a9ae528add11a-h3"],"title":"Calculating the Value of the Expression","text":"What is $$2+5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add11b","stepAnswer":["$$-7$$"],"problemType":"TextBox","stepTitle":"$$-2+\\\\left(-5\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-7$$","hints":{"DefaultPathway":[{"id":"a9ae528add11b-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":[],"title":"Seeing if the Signs Are Different","text":"Are the signs of the two terms different?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a9ae528add11b-h2","type":"hint","dependencies":["a9ae528add11b-h1"],"title":"First Step to Find the Value of the Expression","text":"Since $$2$$ and $$5$$ have the same sign, we add $$2$$ to $$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add11b-h3","type":"hint","dependencies":["a9ae528add11b-h2"],"title":"Sign of the Answer","text":"The answer will be negative because there are only negatives","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add11b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-7$$"],"dependencies":["a9ae528add11b-h3"],"title":"Calculating the Value of the Expression","text":"What is $$-\\\\left(2+5\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ae528add12","title":"Adding and Subtracting Integers","body":"Find the value of the following expressions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Add and Subtract Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ae528add12a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"$$-2+4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a9ae528add12a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":[],"title":"Seeing if the Signs Are Different","text":"Are the signs of the two terms different?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a9ae528add12a-h2","type":"hint","dependencies":["a9ae528add12a-h1"],"title":"First Step to Find the Value of the Expression","text":"Since $$-2$$ and $$4$$ have different signs, we subtract $$2$$ from $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add12a-h3","type":"hint","dependencies":["a9ae528add12a-h2"],"title":"Sign of the Answer","text":"The answer will be positive because there are more positives than negatives.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add12b","stepAnswer":["$$-2$$"],"problemType":"TextBox","stepTitle":"$$2+\\\\left(-4\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-2$$","hints":{"DefaultPathway":[{"id":"a9ae528add12b-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":[],"title":"Seeing if the Signs Are Different","text":"Are the signs of the two terms different?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a9ae528add12b-h2","type":"hint","dependencies":["a9ae528add12b-h1"],"title":"First Step to Find the Value of the Expression","text":"Since $$2$$ and $$-4$$ have different signs, we subtract $$2$$ from $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add12b-h3","type":"hint","dependencies":["a9ae528add12b-h2"],"title":"Sign of the Answer","text":"The answer will be negative because there are more negatives than positives.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add12b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a9ae528add12b-h3"],"title":"Final Answer","text":"What is $$-(4-2)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ae528add13","title":"Adding and Subtracting Integers","body":"Find the value of the following expressions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Add and Subtract Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ae528add13a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"$$-2+5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a9ae528add13a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":[],"title":"Seeing if the Signs Are Different","text":"Are the signs of the two terms different?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a9ae528add13a-h2","type":"hint","dependencies":["a9ae528add13a-h1"],"title":"First Step to Find the Value of the Expression","text":"Since $$-2$$ and $$5$$ have different signs, we subtract $$2$$ from $$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add13a-h3","type":"hint","dependencies":["a9ae528add13a-h2"],"title":"Sign of the Answer","text":"The answer will be positive because there are more positives than negatives.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add13b","stepAnswer":["$$-3$$"],"problemType":"TextBox","stepTitle":"$$2+\\\\left(-5\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-3$$","hints":{"DefaultPathway":[{"id":"a9ae528add13b-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":[],"title":"Seeing if the Signs Are Different","text":"Are the signs of the two terms different?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a9ae528add13b-h2","type":"hint","dependencies":["a9ae528add13b-h1"],"title":"First Step to Find the Value of the Expression","text":"Since $$2$$ and $$-5$$ have different signs, we subtract $$2$$ from $$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add13b-h3","type":"hint","dependencies":["a9ae528add13b-h2"],"title":"Sign of the Answer","text":"The answer will be negative because there are more negatives than positives.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add13b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a9ae528add13b-h3"],"title":"Final Answer","text":"What is $$-(5-2)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ae528add14","title":"Adding and Subtracting Integers","body":"Find the value of the following expressions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Add and Subtract Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ae528add14a","stepAnswer":["$$-50$$"],"problemType":"TextBox","stepTitle":"$$-31+\\\\left(-19\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-50$$","hints":{"DefaultPathway":[{"id":"a9ae528add14a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":[],"title":"Seeing if the Signs Are Different","text":"Are the signs of the two terms different?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a9ae528add14a-h2","type":"hint","dependencies":["a9ae528add14a-h1"],"title":"First Step to Find the Value of the Expression","text":"Since $$31$$ and $$19$$ have the same sign, we add $$31$$ to $$19$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add14a-h3","type":"hint","dependencies":["a9ae528add14a-h2"],"title":"Sign of the Answer","text":"The answer will be negative because there are only negatives","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add14a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-50$$"],"dependencies":["a9ae528add14a-h3"],"title":"Calculating the Value of the Expression","text":"What is $$-\\\\left(31+19\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add14b","stepAnswer":["$$-17$$"],"problemType":"TextBox","stepTitle":"$$15+\\\\left(-32\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-17$$","hints":{"DefaultPathway":[{"id":"a9ae528add14b-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":[],"title":"Seeing if the Signs Are Different","text":"Are the signs of the two terms different?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a9ae528add14b-h2","type":"hint","dependencies":["a9ae528add14b-h1"],"title":"First Step to Find the Value of the Expression","text":"Since $$15$$ and $$32$$ have different signs, we subtract $$15$$ from $$32$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add14b-h3","type":"hint","dependencies":["a9ae528add14b-h2"],"title":"Sign of the Answer","text":"The answer will be negative because there are more negatives than positives.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add14b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-17$$"],"dependencies":["a9ae528add14b-h3"],"title":"Final Answer","text":"What is $$-(32-15)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ae528add15","title":"Adding and Subtracting Integers","body":"Find the value of the following expressions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Add and Subtract Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ae528add15a","stepAnswer":["$$-70$$"],"problemType":"TextBox","stepTitle":"$$-42+\\\\left(-28\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-70$$","hints":{"DefaultPathway":[{"id":"a9ae528add15a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":[],"title":"Seeing if the Signs Are Different","text":"Are the signs of the two terms different?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a9ae528add15a-h2","type":"hint","dependencies":["a9ae528add15a-h1"],"title":"First Step to Find the Value of the Expression","text":"Since $$42$$ and $$28$$ have the same sign, we add $$42$$ to $$28$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add15a-h3","type":"hint","dependencies":["a9ae528add15a-h2"],"title":"Sign of the Answer","text":"The answer will be negative because there are only negatives","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-70$$"],"dependencies":["a9ae528add15a-h3"],"title":"Calculating the Value of the Expression","text":"What is $$-\\\\left(42+28\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add15b","stepAnswer":["$$-36$$"],"problemType":"TextBox","stepTitle":"$$25+\\\\left(-61\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-36$$","hints":{"DefaultPathway":[{"id":"a9ae528add15b-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":[],"title":"Seeing if the Signs Are Different","text":"Are the signs of the two terms different?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a9ae528add15b-h2","type":"hint","dependencies":["a9ae528add15b-h1"],"title":"First Step to Find the Value of the Expression","text":"Since $$25$$ and $$61$$ have different signs, we subtract $$25$$ from $$61$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add15b-h3","type":"hint","dependencies":["a9ae528add15b-h2"],"title":"Sign of the Answer","text":"The answer will be negative because there are more negatives than positives.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add15b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-36$$"],"dependencies":["a9ae528add15b-h3"],"title":"Final Answer","text":"What is $$-(61-25)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ae528add16","title":"Order each of the following pairs of numbers, using < or >:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Add and Subtract Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ae528add16a","stepAnswer":[">"],"problemType":"MultipleChoice","stepTitle":"$$14$$ $$___$$ $$6$$","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">"],"hints":{"DefaultPathway":[{"id":"a9ae528add16a-h1","type":"hint","dependencies":[],"title":"Number Line","text":"To think about this question, you might want to locate $$14$$ and $$6$$ on a number line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add16a-h2","type":"hint","dependencies":["a9ae528add16a-h1"],"title":"Using Number Line to Reach Conclusion","text":"For numbers a and $$b$$, when a is to the right of $$b$$ on the number line, we say $$a>b$$ (read \\"a is greater than b\\").","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add16a-h3","type":"hint","dependencies":["a9ae528add16a-h2"],"title":"Answer","text":"Since $$14$$ is to the right of $$6$$ on the number line, we say $$14>6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add16b","stepAnswer":["<"],"problemType":"MultipleChoice","stepTitle":"$$-1$$ $$___$$ $$9$$","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">"],"hints":{"DefaultPathway":[{"id":"a9ae528add16b-h1","type":"hint","dependencies":[],"title":"Number Line","text":"To think about this question, you might want to locate $$-1$$ and $$9$$ on a number line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add16b-h2","type":"hint","dependencies":["a9ae528add16b-h1"],"title":"Using Number Line to Reach Conclusion","text":"For numbers a and $$b$$, when a is to the left of $$b$$ on the number line, we say $$a<b$$ (read \\"a is less than b\\").","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add16b-h3","type":"hint","dependencies":["a9ae528add16b-h2"],"title":"Answer","text":"Since $$-1$$ is to the left of $$9$$ on the number line, we say $$-1<9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add16c","stepAnswer":[">"],"problemType":"MultipleChoice","stepTitle":"$$-1$$ $$___$$ $$-4$$","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">"],"hints":{"DefaultPathway":[{"id":"a9ae528add16c-h1","type":"hint","dependencies":[],"title":"Number Line","text":"To think about this question, you might want to locate $$-1$$ and $$-4$$ on a number line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add16c-h2","type":"hint","dependencies":["a9ae528add16c-h1"],"title":"Using Number Line to Reach Conclusion","text":"For numbers a and $$b$$, when a is to the right of $$b$$ on the number line, we say $$a>b$$ (read \\"a is greater than b\\").","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add16c-h3","type":"hint","dependencies":["a9ae528add16c-h2"],"title":"Answer","text":"Since $$-1$$ is to the right of $$-4$$ on the number line, we say $$-1>-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add16d","stepAnswer":[">"],"problemType":"MultipleChoice","stepTitle":"$$2$$ $$___$$ $$-20$$","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">"],"hints":{"DefaultPathway":[{"id":"a9ae528add16d-h1","type":"hint","dependencies":[],"title":"Number Line","text":"To think about this question, you might want to locate $$2$$ and $$-20$$ on a number line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add16d-h2","type":"hint","dependencies":["a9ae528add16d-h1"],"title":"Using Number Line to Reach Conclusion","text":"For numbers a and $$b$$, when a is to the right of $$b$$ on the number line, we say $$a>b$$ (read \\"a is greater than b\\").","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add16d-h3","type":"hint","dependencies":["a9ae528add16d-h2"],"title":"Answer","text":"Since $$2$$ is to the right of $$-20$$ on the number line, we say $$2>-20$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ae528add17","title":"Find:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Add and Subtract Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ae528add17a","stepAnswer":["$$-7$$"],"problemType":"TextBox","stepTitle":"The opposite of $$7$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-7$$","hints":{"DefaultPathway":[{"id":"a9ae528add17a-h1","type":"hint","dependencies":[],"title":"Defining Opposite","text":"The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add17a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-7$$"],"dependencies":["a9ae528add17a-h1"],"title":"Answer","text":"Which number is on the opposite side of $$0$$, but is at the same distance from $$0$$ as 7?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add17b","stepAnswer":["$$10$$"],"problemType":"TextBox","stepTitle":"The opposite of $$-10$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10$$","hints":{"DefaultPathway":[{"id":"a9ae528add17b-h1","type":"hint","dependencies":[],"title":"Defining Opposite","text":"The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add17b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a9ae528add17b-h1"],"title":"Answer","text":"Which number is on the opposite side of $$0$$, but is at the same distance from $$0$$ as -10?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add17c","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"$$-(-6)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"a9ae528add17c-h1","type":"hint","dependencies":[],"title":"Opposite Notation","text":"The notation -a is read as \\"the opposite of a.\\" Therefore, in the problem, we are asked to find the opposite of $$-6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add17c-h2","type":"hint","dependencies":["a9ae528add17c-h1"],"title":"Defining Opposite","text":"The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add17c-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a9ae528add17c-h2"],"title":"Answer","text":"Which number is on the opposite side of $$0$$, but is at the same distance from $$0$$ as -6?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ae528add18","title":"Evaluate:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Add and Subtract Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ae528add18a","stepAnswer":["$$-8$$"],"problemType":"TextBox","stepTitle":"$$-x$$, when $$x=8$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-8$$","hints":{"DefaultPathway":[{"id":"a9ae528add18a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"To evaluate when $$x=8$$ means to substitute $$8$$ for $$x$$, so we get the expression $$-(8)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add18a-h2","type":"hint","dependencies":["a9ae528add18a-h1"],"title":"Evaluating","text":"$$-(8)$$ means the opposite of $$8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add18a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-8$$"],"dependencies":["a9ae528add18a-h2"],"title":"Answer","text":"What is the opposite of 8?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add18b","stepAnswer":["$$8$$"],"problemType":"TextBox","stepTitle":"$$-x$$, when $$x=-8$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8$$","hints":{"DefaultPathway":[{"id":"a9ae528add18b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"To evaluate when $$x=-8$$ means to substitute $$8$$ for $$x$$, so we get the expression $$-(-8)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add18b-h2","type":"hint","dependencies":["a9ae528add18b-h1"],"title":"Evaluating","text":"$$-(-8)$$ means the opposite of $$-8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add18b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a9ae528add18b-h2"],"title":"Answer","text":"What is the opposite of -8?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ae528add19","title":"Simplify:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Add and Subtract Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ae528add19a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"$$|3|$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a9ae528add19a-h1","type":"hint","dependencies":[],"title":"Defining Absolute Value","text":"The absolute value of a number is its distance from $$0$$ on the number line. Distance is never negative, so the absolute value is never negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a9ae528add19a-h1"],"title":"Distance from $$0$$","text":"What is the distance from $$3$$ to 0?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add19b","stepAnswer":["$$44$$"],"problemType":"TextBox","stepTitle":"$$|-44|$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$44$$","hints":{"DefaultPathway":[{"id":"a9ae528add19b-h1","type":"hint","dependencies":[],"title":"Defining Absolute Value","text":"The absolute value of a number is its distance from $$0$$ on the number line. Distance is never negative, so the absolute value is never negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add19b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$44$$"],"dependencies":["a9ae528add19b-h1"],"title":"Distance from $$0$$","text":"What is the distance from $$-44$$ to 0?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add19c","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"$$|0|$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a9ae528add19c-h1","type":"hint","dependencies":[],"title":"Defining Absolute Value","text":"The absolute value of a number is its distance from $$0$$ on the number line. Distance is never negative, so the absolute value is never negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add19c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a9ae528add19c-h1"],"title":"Distance from $$0$$","text":"What is the distance from $$0$$ to 0?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ae528add2","title":"Evaluating Absolute Value Expressions","body":"Evaluate the following absolute value expressions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Add and Subtract Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ae528add2a","stepAnswer":["$$35$$"],"problemType":"TextBox","stepTitle":"$$|x|$$ when $$x=-35$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$35$$","hints":{"DefaultPathway":[{"id":"a9ae528add2a-h1","type":"hint","dependencies":[],"title":"Subsituting in the Numeric Value","text":"The first step is to subsitute the value for $$x$$ in the expression $$|x|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add2a-h2","type":"hint","dependencies":["a9ae528add2a-h1"],"title":"Value of $$x$$","text":"$$x=-35$$, a negative number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add2a-h3","type":"hint","dependencies":["a9ae528add2a-h2"],"title":"Evaluating the Absolute Value of Negative Numbers","text":"If a is negative, then $$|a|=-a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add2b","stepAnswer":["$$20$$"],"problemType":"TextBox","stepTitle":"$$|-y|$$ when $$y=-20$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$20$$","hints":{"DefaultPathway":[{"id":"a9ae528add2b-h1","type":"hint","dependencies":[],"title":"Value of $$|-y|$$","text":"The first step is to find the value inside the absolute value bars.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add2b-h2","type":"hint","dependencies":["a9ae528add2b-h1"],"title":"Value of $$y$$","text":"$$y=-20$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add2b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a9ae528add2b-h2"],"title":"Calculating $$-y$$","text":"What is $$-(-20)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add2b-h4","type":"hint","dependencies":["a9ae528add2b-h3"],"title":"Absolute Value of a Positive Number","text":"The absolute value of a positive number is just that positive number. For example, $$|24|=24$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add2c","stepAnswer":["$$-12$$"],"problemType":"TextBox","stepTitle":"$$-|u|$$ when $$u=12$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-12$$","hints":{"DefaultPathway":[{"id":"a9ae528add2c-h1","type":"hint","dependencies":[],"title":"Finding the Value of $$|u|$$","text":"The first step is to find the value of $$|u|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add2c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a9ae528add2c-h1"],"title":"Calculating $$|u|$$","text":"$$u=12$$. What is $$|12|$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a9ae528add2c-h2-s1","type":"hint","dependencies":[],"title":"Absolute Value of a Positive Number","text":"The absolute value of a positive number is just that positive number. For example, $$|24|=24$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9ae528add2c-h3","type":"hint","dependencies":["a9ae528add2c-h2"],"title":"Finding the Value of $$-|u|$$","text":"Multiply $$|u|$$ by $$-1$$ to find $$-|u|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add2c-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-12$$"],"dependencies":["a9ae528add2c-h3"],"title":"Calculating $$-|u|$$ with the Substituted Value of u","text":"What is $$-1\\\\times12$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add2d","stepAnswer":["$$-14$$"],"problemType":"TextBox","stepTitle":"$$-|p|$$ when $$p=-14$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-14$$","hints":{"DefaultPathway":[{"id":"a9ae528add2d-h1","type":"hint","dependencies":[],"title":"Finding the Value of $$|p|$$","text":"The first step is to find the value of $$|p|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add2d-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$14$$"],"dependencies":["a9ae528add2d-h1"],"title":"Calculating $$|p|$$","text":"$$p=-14$$. What is $$|-14|$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a9ae528add2d-h2-s1","type":"hint","dependencies":[],"title":"Evaluating the Absolute Value of Negative Numbers","text":"If a is negative, then $$|a|=-a$$. For example, $$|-11|=11$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9ae528add2d-h3","type":"hint","dependencies":["a9ae528add2d-h2"],"title":"Finding the Value of $$-|p|$$","text":"Multiply $$|p|$$ by $$-1$$ to find $$-|p|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add2d-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-14$$"],"dependencies":["a9ae528add2d-h3"],"title":"Calculating $$-|p|$$ with the Substituted Value of $$p$$","text":"What is $$-1\\\\times14$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ae528add20","title":"Fill in <, >, or $$=$$ for each of the following pairs of numbers:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Add and Subtract Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ae528add20a","stepAnswer":[">"],"problemType":"MultipleChoice","stepTitle":"$$|-5|$$ $$___$$ $$-|-5|$$","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">","$$=$$"],"hints":{"DefaultPathway":[{"id":"a9ae528add20a-h1","type":"hint","dependencies":[],"title":"Simplify Both Sides","text":"We want to start by simplifying both sides and comparing the simplified value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a9ae528add20a-h1"],"title":"Left Hand Side","text":"What does $$|-5|$$ evaluate to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a9ae528add20a-h2-s1","type":"hint","dependencies":[],"title":"Left Hand Side","text":"The absolute value of a number is its distance from $$0$$ on the number line. Distance is never negative, so the absolute value is never negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9ae528add20a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["a9ae528add20a-h1"],"title":"Right Hand Side","text":"What does $$-|-5|$$ evaluate to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a9ae528add20a-h3-s1","type":"hint","dependencies":[],"title":"Right Hand Side","text":"We know that $$|-5|=5$$, so the opposite of $$|-5|$$ is $$-5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9ae528add20a-h4","type":"hint","dependencies":["a9ae528add20a-h2","a9ae528add20a-h3"],"title":"Comparing Numbers","text":"Now we are left to compare $$5$$ and $$-5$$. Since $$5$$ is to the right of $$-5$$ on the number line, we say $$5>-5$$. Therefore, $$|-5|>-|-5|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add20b","stepAnswer":[">"],"problemType":"MultipleChoice","stepTitle":"$$8$$ $$___$$ $$-|-8|$$","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">","$$=$$"],"hints":{"DefaultPathway":[{"id":"a9ae528add20b-h1","type":"hint","dependencies":[],"title":"Simplify Both Sides","text":"We want to start by simplifying both sides and comparing the simplified value. Since the left hand side is already simplified, all we need to do is to evaluate the right hand side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add20b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-8$$"],"dependencies":["a9ae528add20b-h1"],"title":"Right Hand Side","text":"What does $$-|-8|$$ evaluate to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a9ae528add20b-h2-s1","type":"hint","dependencies":[],"title":"Right Hand Side","text":"We know that $$|-8|=distance$$ from $$-8$$ to $$0=8$$, so the opposite of $$|-8|$$ is $$-8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9ae528add20b-h3","type":"hint","dependencies":["a9ae528add20b-h2"],"title":"Comparing Numbers","text":"Now we are left to compare $$8$$ and $$-8$$. Since $$8$$ is to the right of $$-8$$ on the number line, we say $$8>-8$$. Therefore, $$8>-|-8|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add20c","stepAnswer":["$$=$$"],"problemType":"MultipleChoice","stepTitle":"$$-9$$ $$___$$ $$-|-9|$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$=$$","choices":["<",">","$$=$$"],"hints":{"DefaultPathway":[{"id":"a9ae528add20c-h1","type":"hint","dependencies":[],"title":"Simplify Both Sides","text":"We want to start by simplifying both sides and comparing the simplified value. Since the left hand side is already simplified, all we need to do is to evaluate the right hand side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add20c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":["a9ae528add20c-h1"],"title":"Right Hand Side","text":"What does $$-|-9|$$ evaluate to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a9ae528add20c-h2-s1","type":"hint","dependencies":[],"title":"Right Hand Side","text":"We know that $$|-9|=distance$$ from $$-9$$ to $$0=9$$, so the opposite of $$|-9|$$ is $$-9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9ae528add20c-h3","type":"hint","dependencies":["a9ae528add20c-h2"],"title":"Comparing Numbers","text":"Now we are left to compare $$-9$$ and $$-9$$, and we know $$-9=-9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add20d","stepAnswer":[">"],"problemType":"MultipleChoice","stepTitle":"$$-(-16)$$ $$___$$ $$-|-16|$$","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">","$$=$$"],"hints":{"DefaultPathway":[{"id":"a9ae528add20d-h1","type":"hint","dependencies":[],"title":"Simplify Both Sides","text":"We want to start by simplifying both sides and comparing the simplified value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add20d-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["a9ae528add20d-h1"],"title":"Left Hand Side","text":"What does $$-(-16)$$ evaluate to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a9ae528add20d-h2-s1","type":"hint","dependencies":[],"title":"Left Hand Side","text":"The notation -a is read as \\"the opposite of a.\\" Therefore, in the problem, we are asked to find the opposite of $$-16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add20d-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["a9ae528add20d-h2-s1"],"title":"Left Hand Side","text":"What is the opposite of -16?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9ae528add20d-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-16$$"],"dependencies":["a9ae528add20d-h1"],"title":"Right Hand Side","text":"What does $$-|-16|$$ evaluate to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a9ae528add20d-h3-s1","type":"hint","dependencies":[],"title":"Right Hand Side","text":"We know that $$|-16|=distance$$ from $$-16$$ to $$0=16$$, so the opposite of $$|-16|$$ is $$-16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9ae528add20d-h4","type":"hint","dependencies":["a9ae528add20d-h2","a9ae528add20d-h3"],"title":"Comparing Numbers","text":"Now we are left to compare $$16$$ and $$-16$$. Since $$16$$ is to the right of $$-16$$ on the number line, we say $$16>-16$$. Therefore, $$-\\\\left(-16\\\\right)>-|-16|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ae528add21","title":"Order each of the following pairs of numbers, using < or >:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Add and Subtract Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ae528add21a","stepAnswer":[">"],"problemType":"MultipleChoice","stepTitle":"$$15$$ $$___$$ $$7$$","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">"],"hints":{"DefaultPathway":[{"id":"a9ae528add21a-h1","type":"hint","dependencies":[],"title":"Number Line","text":"To think about this question, you might want to locate $$15$$ and $$7$$ on a number line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add21a-h2","type":"hint","dependencies":["a9ae528add21a-h1"],"title":"Using Number Line to Reach Conclusion","text":"For numbers a and $$b$$, when a is to the right of $$b$$ on the number line, we say $$a>b$$ (read \\"a is greater than b\\").","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add21a-h3","type":"hint","dependencies":["a9ae528add21a-h2"],"title":"Answer","text":"Since $$15$$ is to the right of $$7$$ on the number line, we say $$15>7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add21b","stepAnswer":["<"],"problemType":"MultipleChoice","stepTitle":"$$-2$$ $$___$$ $$5$$","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">"],"hints":{"DefaultPathway":[{"id":"a9ae528add21b-h1","type":"hint","dependencies":[],"title":"Number Line","text":"To think about this question, you might want to locate $$-2$$ and $$5$$ on a number line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add21b-h2","type":"hint","dependencies":["a9ae528add21b-h1"],"title":"Using Number Line to Reach Conclusion","text":"For numbers a and $$b$$, when a is to the left of $$b$$ on the number line, we say $$a<b$$ (read \\"a is less than b\\").","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add21b-h3","type":"hint","dependencies":["a9ae528add21b-h2"],"title":"Answer","text":"Since $$-2$$ is to the left of $$5$$ on the number line, we say $$-2<5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add21c","stepAnswer":[">"],"problemType":"MultipleChoice","stepTitle":"$$-3$$ $$___$$ $$-7$$","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">"],"hints":{"DefaultPathway":[{"id":"a9ae528add21c-h1","type":"hint","dependencies":[],"title":"Number Line","text":"To think about this question, you might want to locate $$-3$$ and $$-7$$ on a number line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add21c-h2","type":"hint","dependencies":["a9ae528add21c-h1"],"title":"Using Number Line to Reach Conclusion","text":"For numbers a and $$b$$, when a is to the right of $$b$$ on the number line, we say $$a>b$$ (read \\"a is greater than b\\").","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add21c-h3","type":"hint","dependencies":["a9ae528add21c-h2"],"title":"Answer","text":"Since $$-3$$ is to the right of $$-7$$ on the number line, we say $$-3>-7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add21d","stepAnswer":[">"],"problemType":"MultipleChoice","stepTitle":"$$5$$ $$___$$ $$-17$$","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">"],"hints":{"DefaultPathway":[{"id":"a9ae528add21d-h1","type":"hint","dependencies":[],"title":"Number Line","text":"To think about this question, you might want to locate $$5$$ and $$-17$$ on a number line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add21d-h2","type":"hint","dependencies":["a9ae528add21d-h1"],"title":"Using Number Line to Reach Conclusion","text":"For numbers a and $$b$$, when a is to the right of $$b$$ on the number line, we say $$a>b$$ (read \\"a is greater than b\\").","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add21d-h3","type":"hint","dependencies":["a9ae528add21d-h2"],"title":"Answer","text":"Since $$5$$ is to the right of $$-17$$ on the number line, we say $$5>-17$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ae528add22","title":"Order each of the following pairs of numbers, using < or >:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Add and Subtract Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ae528add22a","stepAnswer":["<"],"problemType":"MultipleChoice","stepTitle":"$$8$$ $$___$$ $$13$$","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">"],"hints":{"DefaultPathway":[{"id":"a9ae528add22a-h1","type":"hint","dependencies":[],"title":"Number Line","text":"To think about this question, you might want to locate $$8$$ and $$13$$ on a number line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add22a-h2","type":"hint","dependencies":["a9ae528add22a-h1"],"title":"Using Number Line to Reach Conclusion","text":"For numbers a and $$b$$, when a is to the left of $$b$$ on the number line, we say $$a<b$$ (read \\"a is less than b\\").","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add22a-h3","type":"hint","dependencies":["a9ae528add22a-h2"],"title":"Answer","text":"Since $$8$$ is to the left of $$13$$ on the number line, we say $$8<13$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add22b","stepAnswer":[">"],"problemType":"MultipleChoice","stepTitle":"$$3$$ $$___$$ $$-4$$","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">"],"hints":{"DefaultPathway":[{"id":"a9ae528add22b-h1","type":"hint","dependencies":[],"title":"Number Line","text":"To think about this question, you might want to locate $$3$$ and $$-4$$ on a number line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add22b-h2","type":"hint","dependencies":["a9ae528add22b-h1"],"title":"Using Number Line to Reach Conclusion","text":"For numbers a and $$b$$, when a is to the right of $$b$$ on the number line, we say $$a>b$$ (read \\"a is greater than b\\").","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add22b-h3","type":"hint","dependencies":["a9ae528add22b-h2"],"title":"Answer","text":"Since $$3$$ is to the right of $$-4$$ on the number line, we say $$3>-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add22c","stepAnswer":["<"],"problemType":"MultipleChoice","stepTitle":"$$-5$$ $$___$$ $$-2$$","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">"],"hints":{"DefaultPathway":[{"id":"a9ae528add22c-h1","type":"hint","dependencies":[],"title":"Number Line","text":"To think about this question, you might want to locate $$-5$$ and $$-2$$ on a number line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add22c-h2","type":"hint","dependencies":["a9ae528add22c-h1"],"title":"Using Number Line to Reach Conclusion","text":"For numbers a and $$b$$, when a is to the left of $$b$$ on the number line, we say $$a<b$$ (read \\"a is less than b\\").","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add22c-h3","type":"hint","dependencies":["a9ae528add22c-h2"],"title":"Answer","text":"Since $$-5$$ is to the left of $$-2$$ on the number line, we say $$-5<-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add22d","stepAnswer":[">"],"problemType":"MultipleChoice","stepTitle":"$$9$$ $$___$$ $$-21$$","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">"],"hints":{"DefaultPathway":[{"id":"a9ae528add22d-h1","type":"hint","dependencies":[],"title":"Number Line","text":"To think about this question, you might want to locate $$9$$ and $$-21$$ on a number line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add22d-h2","type":"hint","dependencies":["a9ae528add22d-h1"],"title":"Using Number Line to Reach Conclusion","text":"For numbers a and $$b$$, when a is to the right of $$b$$ on the number line, we say $$a>b$$ (read \\"a is greater than b\\").","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add22d-h3","type":"hint","dependencies":["a9ae528add22d-h2"],"title":"Answer","text":"Since $$9$$ is to the right of $$-21$$ on the number line, we say $$9>-21$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ae528add23","title":"Find:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Add and Subtract Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ae528add23a","stepAnswer":["$$-4$$"],"problemType":"TextBox","stepTitle":"The opposite of $$4$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-4$$","hints":{"DefaultPathway":[{"id":"a9ae528add23a-h1","type":"hint","dependencies":[],"title":"Defining Opposite","text":"The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add23a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["a9ae528add23a-h1"],"title":"Answer","text":"Which number is on the opposite side of $$0$$, but is at the same distance from $$0$$ as 4?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add23b","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"The opposite of $$-3$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"a9ae528add23b-h1","type":"hint","dependencies":[],"title":"Defining Opposite","text":"The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add23b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a9ae528add23b-h1"],"title":"Answer","text":"Which number is on the opposite side of $$0$$, but is at the same distance from $$0$$ as -3?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add23c","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"$$-(-1)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a9ae528add23c-h1","type":"hint","dependencies":[],"title":"Opposite Notation","text":"The notation -a is read as \\"the opposite of a.\\" Therefore, in the problem, we are asked to find the opposite of $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add23c-h2","type":"hint","dependencies":["a9ae528add23c-h1"],"title":"Defining Opposite","text":"The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add23c-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a9ae528add23c-h2"],"title":"Answer","text":"Which number is on the opposite side of $$0$$, but is at the same distance from $$0$$ as -1?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ae528add24","title":"Find:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Add and Subtract Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ae528add24a","stepAnswer":["$$-8$$"],"problemType":"TextBox","stepTitle":"The opposite of $$8$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-8$$","hints":{"DefaultPathway":[{"id":"a9ae528add24a-h1","type":"hint","dependencies":[],"title":"Defining Opposite","text":"The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add24a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-8$$"],"dependencies":["a9ae528add24a-h1"],"title":"Answer","text":"Which number is on the opposite side of $$0$$, but is at the same distance from $$0$$ as 8?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add24b","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"The opposite of $$-5$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"a9ae528add24b-h1","type":"hint","dependencies":[],"title":"Defining Opposite","text":"The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add24b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a9ae528add24b-h1"],"title":"Answer","text":"Which number is on the opposite side of $$0$$, but is at the same distance from $$0$$ as -5?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add24c","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"$$-(-5)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"a9ae528add24c-h1","type":"hint","dependencies":[],"title":"Opposite Notation","text":"The notation -a is read as \\"the opposite of a.\\" Therefore, in the problem, we are asked to find the opposite of $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add24c-h2","type":"hint","dependencies":["a9ae528add24c-h1"],"title":"Defining Opposite","text":"The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add24c-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a9ae528add24c-h2"],"title":"Answer","text":"Which number is on the opposite side of $$0$$, but is at the same distance from $$0$$ as -5?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ae528add25","title":"Evaluate $$-n$$, when:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Add and Subtract Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ae528add25a","stepAnswer":["$$-4$$"],"problemType":"TextBox","stepTitle":"$$n=4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-4$$","hints":{"DefaultPathway":[{"id":"a9ae528add25a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"To evaluate when $$n=4$$ means to substitute $$4$$ for $$n$$, so we get the expression $$-(4)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add25a-h2","type":"hint","dependencies":["a9ae528add25a-h1"],"title":"Evaluating","text":"$$-(4)$$ means the opposite of $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add25a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["a9ae528add25a-h2"],"title":"Answer","text":"What is the opposite of 4?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add25b","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"$$n=-4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a9ae528add25b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"To evaluate when $$n=-4$$ means to substitute $$-4$$ for $$n$$, so we get the expression $$-(-4)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add25b-h2","type":"hint","dependencies":["a9ae528add25b-h1"],"title":"Evaluating","text":"$$-(-4)$$ means the opposite of $$-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add25b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a9ae528add25b-h2"],"title":"Answer","text":"What is the opposite of -4?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ae528add26","title":"Evaluate $$-m$$, when:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Add and Subtract Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ae528add26a","stepAnswer":["$$-11$$"],"problemType":"TextBox","stepTitle":"$$m=11$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-11$$","hints":{"DefaultPathway":[{"id":"a9ae528add26a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"To evaluate when $$m=11$$ means to substitute $$11$$ for $$m$$, so we get the expression $$-(11)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add26a-h2","type":"hint","dependencies":["a9ae528add26a-h1"],"title":"Evaluating","text":"$$-(11)$$ means the opposite of $$11$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add26a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-11$$"],"dependencies":["a9ae528add26a-h2"],"title":"Answer","text":"What is the opposite of 11?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add26b","stepAnswer":["$$11$$"],"problemType":"TextBox","stepTitle":"$$m=-11$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$11$$","hints":{"DefaultPathway":[{"id":"a9ae528add26b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"To evaluate when $$m=-11$$ means to substitute $$-11$$ for $$m$$, so we get the expression $$-(-11)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add26b-h2","type":"hint","dependencies":["a9ae528add26b-h1"],"title":"Evaluating","text":"$$-(-11)$$ means the opposite of $$-11$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add26b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11$$"],"dependencies":["a9ae528add26b-h2"],"title":"Answer","text":"What is the opposite of -11?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ae528add27","title":"Simplify:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Add and Subtract Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ae528add27a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"$$|4|$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a9ae528add27a-h1","type":"hint","dependencies":[],"title":"Defining Absolute Value","text":"The absolute value of a number is its distance from $$0$$ on the number line. Distance is never negative, so the absolute value is never negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add27a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a9ae528add27a-h1"],"title":"Distance from $$0$$","text":"What is the distance from $$4$$ to 0?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add27b","stepAnswer":["$$28$$"],"problemType":"TextBox","stepTitle":"$$|-28|$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$28$$","hints":{"DefaultPathway":[{"id":"a9ae528add27b-h1","type":"hint","dependencies":[],"title":"Defining Absolute Value","text":"The absolute value of a number is its distance from $$0$$ on the number line. Distance is never negative, so the absolute value is never negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add27b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$28$$"],"dependencies":["a9ae528add27b-h1"],"title":"Distance from $$0$$","text":"What is the distance from $$-28$$ to 0?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add27c","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"$$|0|$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a9ae528add27c-h1","type":"hint","dependencies":[],"title":"Defining Absolute Value","text":"The absolute value of a number is its distance from $$0$$ on the number line. Distance is never negative, so the absolute value is never negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add27c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a9ae528add27c-h1"],"title":"Distance from $$0$$","text":"What is the distance from $$0$$ to 0?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ae528add28","title":"Simplify:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Add and Subtract Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ae528add28a","stepAnswer":["$$13$$"],"problemType":"TextBox","stepTitle":"$$|-13|$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$13$$","hints":{"DefaultPathway":[{"id":"a9ae528add28a-h1","type":"hint","dependencies":[],"title":"Defining Absolute Value","text":"The absolute value of a number is its distance from $$0$$ on the number line. Distance is never negative, so the absolute value is never negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add28a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$13$$"],"dependencies":["a9ae528add28a-h1"],"title":"Distance from $$0$$","text":"What is the distance from $$-13$$ to 0?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add28b","stepAnswer":["$$47$$"],"problemType":"TextBox","stepTitle":"$$|47|$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$47$$","hints":{"DefaultPathway":[{"id":"a9ae528add28b-h1","type":"hint","dependencies":[],"title":"Defining Absolute Value","text":"The absolute value of a number is its distance from $$0$$ on the number line. Distance is never negative, so the absolute value is never negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add28b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$47$$"],"dependencies":["a9ae528add28b-h1"],"title":"Distance from $$0$$","text":"What is the distance from $$47$$ to 0?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add28c","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"$$|0|$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"a9ae528add28c-h1","type":"hint","dependencies":[],"title":"Defining Absolute Value","text":"The absolute value of a number is its distance from $$0$$ on the number line. Distance is never negative, so the absolute value is never negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add28c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a9ae528add28c-h1"],"title":"Distance from $$0$$","text":"What is the distance from $$0$$ to 0?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ae528add29","title":"Fill in <, >, or $$=$$ for each of the following pairs of numbers:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Add and Subtract Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ae528add29a","stepAnswer":[">"],"problemType":"MultipleChoice","stepTitle":"$$|-9|$$ $$___$$ $$-|-9|$$","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">","$$=$$"],"hints":{"DefaultPathway":[{"id":"a9ae528add29a-h1","type":"hint","dependencies":[],"title":"Simplify Both Sides","text":"We want to start by simplifying both sides and comparing the simplified value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add29a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a9ae528add29a-h1"],"title":"Left Hand Side","text":"What does $$|-9|$$ evaluate to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a9ae528add29a-h2-s1","type":"hint","dependencies":[],"title":"Left Hand Side","text":"The absolute value of a number is its distance from $$0$$ on the number line. Distance is never negative, so the absolute value is never negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9ae528add29a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":["a9ae528add29a-h1"],"title":"Right Hand Side","text":"What does $$-|-9|$$ evaluate to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a9ae528add29a-h3-s1","type":"hint","dependencies":[],"title":"Right Hand Side","text":"We know that $$|-9|=9$$, so the opposite of $$|-9|$$ is $$-9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9ae528add29a-h4","type":"hint","dependencies":["a9ae528add29a-h2","a9ae528add29a-h3"],"title":"Comparing Numbers","text":"Now we are left to compare $$9$$ and $$-9$$. Since $$9$$ is to the right of $$-9$$ on the number line, we say $$9>-9$$. Therefore, $$|-9|>-|-9|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add29b","stepAnswer":[">"],"problemType":"MultipleChoice","stepTitle":"$$2$$ $$___$$ $$-|-2|$$","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">","$$=$$"],"hints":{"DefaultPathway":[{"id":"a9ae528add29b-h1","type":"hint","dependencies":[],"title":"Simplify Both Sides","text":"We want to start by simplifying both sides and comparing the simplified value. Since the left hand side is already simplified, all we need to do is to evaluate the right hand side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add29b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["a9ae528add29b-h1"],"title":"Right Hand Side","text":"What does $$-|-2|$$ evaluate to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a9ae528add29b-h2-s1","type":"hint","dependencies":[],"title":"Right Hand Side","text":"We know that $$|-2|=distance$$ from $$-2$$ to $$0=2$$, so the opposite of $$|-2|$$ is $$-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9ae528add29b-h3","type":"hint","dependencies":["a9ae528add29b-h2"],"title":"Comparing Numbers","text":"Now we are left to compare $$2$$ and $$-2$$. Since $$2$$ is to the right of $$-2$$ on the number line, we say $$2>-2$$. Therefore, $$2>-|-2|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add29c","stepAnswer":["<"],"problemType":"MultipleChoice","stepTitle":"$$-8$$ $$___$$ $$|-8|$$","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">","$$=$$"],"hints":{"DefaultPathway":[{"id":"a9ae528add29c-h1","type":"hint","dependencies":[],"title":"Simplify Both Sides","text":"We want to start by simplifying both sides and comparing the simplified value. Since the left hand side is already simplified, all we need to do is to evaluate the right hand side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add29c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a9ae528add29c-h1"],"title":"Right Hand Side","text":"What does $$|-8|$$ evaluate to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a9ae528add29c-h2-s1","type":"hint","dependencies":[],"title":"Right Hand Side","text":"The absolute value of a number is its distance from $$0$$ on the number line. Distance is never negative, so the absolute value is never negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9ae528add29c-h3","type":"hint","dependencies":["a9ae528add29c-h2"],"title":"Comparing Numbers","text":"Now we are left to compare $$-8$$ and $$8$$. Since $$-8$$ is to the left of $$8$$ on the number line, we say $$-8<8$$. Therefore, $$-8<|-8|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add29d","stepAnswer":[">"],"problemType":"MultipleChoice","stepTitle":"$$-(-9)$$ $$___$$ $$-|-9|$$","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">","$$=$$"],"hints":{"DefaultPathway":[{"id":"a9ae528add29d-h1","type":"hint","dependencies":[],"title":"Simplify Both Sides","text":"We want to start by simplifying both sides and comparing the simplified value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add29d-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a9ae528add29d-h1"],"title":"Left Hand Side","text":"What does $$-(-9)$$ evaluate to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a9ae528add29d-h2-s1","type":"hint","dependencies":[],"title":"Left Hand Side","text":"The notation -a is read as \\"the opposite of a.\\" Therefore, in the problem, we are asked to find the opposite of $$-9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add29d-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a9ae528add29d-h2-s1"],"title":"Left Hand Side","text":"What is the opposite of -9?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9ae528add29d-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":["a9ae528add29d-h1"],"title":"Right Hand Side","text":"What does $$-|-9|$$ evaluate to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a9ae528add29d-h3-s1","type":"hint","dependencies":[],"title":"Right Hand Side","text":"We know that $$|-9|=distance$$ from $$-9$$ to $$0=9$$, so the opposite of $$|-9|$$ is $$-9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9ae528add29d-h4","type":"hint","dependencies":["a9ae528add29d-h2","a9ae528add29d-h3"],"title":"Comparing Numbers","text":"Now we are left to compare $$9$$ and $$-9$$. Since $$9$$ is to the right of $$-9$$ on the number line, we say $$9>-9$$. Therefore, $$-\\\\left(-9\\\\right)>-|-9|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ae528add3","title":"Adding and Subtracting Integers","body":"Find the value of the following expressions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Add and Subtract Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ae528add3a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"$$1+4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"a9ae528add3a-h1","type":"hint","dependencies":[],"title":"Seeing if the First Term is Positive or Negative","text":"The first term, which is $$1$$, is a positive number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add3a-h2","type":"hint","dependencies":["a9ae528add3a-h1"],"title":"Seeing if the Second Term is Positive or Negative","text":"The second term, which is $$4$$, is a positive number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add3a-h3","type":"hint","dependencies":["a9ae528add3a-h2"],"title":"Adding Two Positive Numbers","text":"One positive plus four positives is five positives. To visualize this, see the image attached to this hint.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add3b","stepAnswer":["$$-5$$"],"problemType":"TextBox","stepTitle":"$$-1+\\\\left(-4\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-5$$","hints":{"DefaultPathway":[{"id":"a9ae528add3b-h1","type":"hint","dependencies":[],"title":"Seeing if the First Term is Positive or Negative","text":"The first term, which is $$-1$$, is a negative number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add3b-h2","type":"hint","dependencies":["a9ae528add3b-h1"],"title":"Seeing if the Second Term is Positive or Negative","text":"The second term, which is $$-4$$, is a negative number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add3b-h3","type":"hint","dependencies":["a9ae528add3b-h2"],"title":"Adding Two Negative Numbers","text":"One negative plus four negatives equals five negatives. The visual representation of this is attached to this hint.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ae528add30","title":"Fill in <, >, or $$=$$ for each of the following pairs of numbers:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Add and Subtract Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ae528add30a","stepAnswer":[">"],"problemType":"MultipleChoice","stepTitle":"$$7$$ $$___$$ $$-|-7|$$","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">","$$=$$"],"hints":{"DefaultPathway":[{"id":"a9ae528add30a-h1","type":"hint","dependencies":[],"title":"Simplify Both Sides","text":"We want to start by simplifying both sides and comparing the simplified value. Since the left hand side is already simplified, all we need to do is to evaluate the right hand side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add30a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-7$$"],"dependencies":["a9ae528add30a-h1"],"title":"Right Hand Side","text":"What does $$-|-7|$$ evaluate to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a9ae528add30a-h2-s1","type":"hint","dependencies":[],"title":"Right Hand Side","text":"We know that $$|-7|=distance$$ from $$-7$$ to $$0=7$$, so the opposite of $$|-7|$$ is $$-7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9ae528add30a-h3","type":"hint","dependencies":["a9ae528add30a-h2"],"title":"Comparing Numbers","text":"Now we are left to compare $$7$$ and $$-7$$. Since $$7$$ is to the right of $$-7$$ on the number line, we say $$7>-7$$. Therefore, $$7>-|-7|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add30b","stepAnswer":[">"],"problemType":"MultipleChoice","stepTitle":"$$-(-10)$$ $$___$$ $$-|-10|$$","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">","$$=$$"],"hints":{"DefaultPathway":[{"id":"a9ae528add30b-h1","type":"hint","dependencies":[],"title":"Simplify Both Sides","text":"We want to start by simplifying both sides and comparing the simplified value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add30b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a9ae528add30b-h1"],"title":"Left Hand Side","text":"What does $$-(-10)$$ evaluate to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a9ae528add30b-h2-s1","type":"hint","dependencies":[],"title":"Left Hand Side","text":"The notation -a is read as \\"the opposite of a.\\" Therefore, in the problem, we are asked to find the opposite of $$-10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add30b-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a9ae528add30b-h2-s1"],"title":"Left Hand Side","text":"What is the opposite of -10?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9ae528add30b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-10$$"],"dependencies":["a9ae528add30b-h1"],"title":"Right Hand Side","text":"What does $$-|-10|$$ evaluate to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a9ae528add30b-h3-s1","type":"hint","dependencies":[],"title":"Right Hand Side","text":"We know that $$|-10|=distance$$ from $$-10$$ to $$0=10$$, so the opposite of $$|-10|$$ is $$-10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9ae528add30b-h4","type":"hint","dependencies":["a9ae528add30b-h2","a9ae528add30b-h3"],"title":"Comparing Numbers","text":"Now we are left to compare $$10$$ and $$-10$$. Since $$10$$ is to the right of $$-10$$ on the number line, we say $$10>-10$$. Therefore, $$-\\\\left(-10\\\\right)>-|-10|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add30c","stepAnswer":[">"],"problemType":"MultipleChoice","stepTitle":"$$|-4|$$ $$___$$ $$-|-4|$$","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">","$$=$$"],"hints":{"DefaultPathway":[{"id":"a9ae528add30c-h1","type":"hint","dependencies":[],"title":"Simplify Both Sides","text":"We want to start by simplifying both sides and comparing the simplified value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add30c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a9ae528add30c-h1"],"title":"Left Hand Side","text":"What does $$|-4|$$ evaluate to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a9ae528add30c-h2-s1","type":"hint","dependencies":[],"title":"Left Hand Side","text":"The absolute value of a number is its distance from $$0$$ on the number line. Distance is never negative, so the absolute value is never negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9ae528add30c-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["a9ae528add30c-h1"],"title":"Right Hand Side","text":"What does $$-|-4|$$ evaluate to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a9ae528add30c-h3-s1","type":"hint","dependencies":[],"title":"Right Hand Side","text":"We know that $$|-4|=4$$, so the opposite of $$|-4|$$ is $$--4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9ae528add30c-h4","type":"hint","dependencies":["a9ae528add30c-h2","a9ae528add30c-h3"],"title":"Comparing Numbers","text":"Now we are left to compare $$4$$ and $$-4$$. Since $$4$$ is to the right of $$-4$$ on the number line, we say $$4>-4$$. Therefore, $$|-4|>-|-4|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add30d","stepAnswer":["<"],"problemType":"MultipleChoice","stepTitle":"$$-1$$ $$___$$ $$|-1|$$","stepBody":"","answerType":"string","variabilization":{},"choices":["<",">","$$=$$"],"hints":{"DefaultPathway":[{"id":"a9ae528add30d-h1","type":"hint","dependencies":[],"title":"Simplify Both Sides","text":"We want to start by simplifying both sides and comparing the simplified value. Since the left hand side is already simplified, all we need to do is to evaluate the right hand side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add30d-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a9ae528add30d-h1"],"title":"Right Hand Side","text":"What does $$|-1|$$ evaluate to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a9ae528add30d-h2-s1","type":"hint","dependencies":[],"title":"Right Hand Side","text":"The absolute value of a number is its distance from $$0$$ on the number line. Distance is never negative, so the absolute value is never negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9ae528add30d-h3","type":"hint","dependencies":["a9ae528add30d-h2"],"title":"Comparing Numbers","text":"Now we are left to compare $$-1$$ and $$1$$. Since $$-1$$ is to the left of $$1$$ on the number line, we say $$-1<1$$. Therefore, $$-1<|-1|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ae528add4","title":"Adding and Subtracting Integers","body":"Find the value of the following expressions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Add and Subtract Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ae528add4a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"$$-1+5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a9ae528add4a-h1","type":"hint","dependencies":[],"title":"Seeing if the Sum Will be Negative or Positive","text":"There is one negative because of $$-1$$, and five positives because of $$5$$. Since five is more than one, there are more positives and the sum will be positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add4a-h2","type":"hint","dependencies":["a9ae528add4a-h1"],"title":"Visualizing the Problem","text":"The problem can be visualized with the image attached to this hint, where a red circle represents a negative and a blue circle represents a positive.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a9ae528add4a-h2"],"title":"Solving the Problem With the Image","text":"From the image, you can see that one negative and one positive cancel each other out, and together, they are in the purple circle. What is the value of the circles that are outside of the purple circle?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add4a-h4","type":"hint","dependencies":["a9ae528add4a-h3"],"title":"Final Answer","text":"$$4$$ positives are left after the red and blue circles in the purple circle are not counted, so the sum is a positive number, $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add4b","stepAnswer":["$$-4$$"],"problemType":"TextBox","stepTitle":"$$1+\\\\left(-5\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-4$$","hints":{"DefaultPathway":[{"id":"a9ae528add4b-h1","type":"hint","dependencies":[],"title":"Seeing if the Sum Will be Negative or Positive","text":"There is one positive because of $$1$$, and five negatives because of $$-5$$. Since five is more than one, there are more negatives and the sum will be negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add4b-h2","type":"hint","dependencies":["a9ae528add4b-h1"],"title":"Visualizing the Problem","text":"The problem can be visualized with the image attached to this hint, where a red circle represents a negative and a blue circle represents a positive.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add4b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["a9ae528add4b-h2"],"title":"Solving the Problem With the Image","text":"From the image, you can see that one negative and one positive cancel each other out, and together, they are in the purple circle. What is the value of the circles that are outside of the purple circle? Consider if they are negative of positive. Remember, one red circle represents $$-1$$ and one blue circle represents $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add4b-h4","type":"hint","dependencies":["a9ae528add4b-h3"],"title":"Final Answer","text":"$$4$$ negatives are left after the red and blue circles in the purple circle are not counted, so the sum is a positive number, $$-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ae528add5","title":"Adding and Subtracting Integers","body":"Find the value of the following expressions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Add and Subtract Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ae528add5a","stepAnswer":["$$-28$$"],"problemType":"TextBox","stepTitle":"$$19+\\\\left(-47\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-28$$","hints":{"DefaultPathway":[{"id":"a9ae528add5a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":[],"title":"Seeing if the Signs Are Different","text":"Are the signs of the two terms different?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a9ae528add5a-h2","type":"hint","dependencies":["a9ae528add5a-h1"],"title":"First Step to Find the Value of the Expression","text":"Since $$19$$ and $$47$$ have different signs, we subtract $$19$$ from $$47$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add5a-h3","type":"hint","dependencies":["a9ae528add5a-h2"],"title":"Sign of the Answer","text":"The answer will be negative because there are more negatives than positives.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-28$$"],"dependencies":["a9ae528add5a-h3"],"title":"Answer","text":"What is $$-(47-19)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add5b","stepAnswer":["$$-50$$"],"problemType":"TextBox","stepTitle":"$$-14+\\\\left(-36\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-50$$","hints":{"DefaultPathway":[{"id":"a9ae528add5b-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":[],"title":"Seeing if the Signs Are Different","text":"Are the signs of the two terms different?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a9ae528add5b-h2","type":"hint","dependencies":["a9ae528add5b-h1"],"title":"First Step to Find the Value of the Expression","text":"Since $$-14$$ and $$-36$$ have the same sign, we add $$14$$ to $$36$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add5b-h3","type":"hint","dependencies":["a9ae528add5b-h2"],"title":"Sign of the Answer","text":"The answer will be negative because there are only negatives.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add5b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-50$$"],"dependencies":["a9ae528add5b-h3"],"title":"Answer","text":"What is $$-\\\\left(14+36\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ae528add6","title":"Simplifying Expressions with Integers","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Add and Subtract Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ae528add6a","stepAnswer":["$$16$$"],"problemType":"TextBox","stepTitle":"$$19-|11-4\\\\left(3-1\\\\right)|$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16$$","hints":{"DefaultPathway":[{"id":"a9ae528add6a-h1","type":"hint","dependencies":[],"title":"Working Inside the Parentheses","text":"The first step is to work on the expression inside the parentheses.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a9ae528add6a-h1"],"title":"Expression Inside Parentheses","text":"What is $$3-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a9ae528add6a-h2"],"title":"Multiplying the Expression in the Parentheses","text":"The next step is to multiply the expression in the parentheses by the factor outside of it, which is $$4$$. What is $$4(3-1)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a9ae528add6a-h3"],"title":"Subtracting Inside the Absolute Value Bars","text":"Then, subtract within the absolute value bar. What is $$11-4(3-1)$$? Remember that you\'ve calculated the second term in the previous step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add6a-h5","type":"hint","dependencies":["a9ae528add6a-h4"],"title":"Getting Rid of the Absolute Value Bars","text":"The value $$11-4(3-1)$$, calculated in the previous step, is positive. so we can get rid of the absolute value bars. Then, the expression is $$19-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add6a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["a9ae528add6a-h5"],"title":"Final Step","text":"What is $$19-3$$? The answer to this is the final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ae528add7","title":"Simplifying Expressions with Integers","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Add and Subtract Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ae528add7a","stepAnswer":["$$9$$"],"problemType":"TextBox","stepTitle":"$$9-|8-4\\\\left(7-5\\\\right)|$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9$$","hints":{"DefaultPathway":[{"id":"a9ae528add7a-h1","type":"hint","dependencies":[],"title":"Working Inside the Parentheses","text":"The first step is to work on the expression inside the parentheses.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a9ae528add7a-h1"],"title":"Expression Inside Parentheses","text":"What is $$7-5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a9ae528add7a-h2"],"title":"Multiplying the Expression in the Parentheses","text":"The next step is to multiply the expression in the parentheses by the factor outside of it, which is $$4$$. What is $$4(7-5)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["a9ae528add7a-h3"],"title":"Subtracting Inside the Absolute Value Bars","text":"Then, subtract within the absolute value bar. What is $$8-4(7-5)$$? Remember that you\'ve calculated the second term in the previous step.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add7a-h5","type":"hint","dependencies":["a9ae528add7a-h4"],"title":"Final Answer","text":"The value $$8-4(7-5)$$, calculated in the previous step, is $$0$$, so we can rid of it. What remains in the expression is just $$9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ae528add8","title":"Evaluating Absolute Value Expressions","body":"Evaluate the following absolute value expressions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Add and Subtract Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ae528add8a","stepAnswer":["$$17$$"],"problemType":"TextBox","stepTitle":"$$|x|$$ when $$x=-17$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$17$$","hints":{"DefaultPathway":[{"id":"a9ae528add8a-h1","type":"hint","dependencies":[],"title":"Subsituting in the Numeric Value","text":"The first step is to subsitute the value for $$x$$ in the expression $$|x|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add8a-h2","type":"hint","dependencies":["a9ae528add8a-h1"],"title":"Value of $$x$$","text":"$$x=-17$$, a negative number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add8a-h3","type":"hint","dependencies":["a9ae528add8a-h2"],"title":"Evaluating the Absolute Value of Negative Numbers","text":"If a is negative, then $$|a|=-a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add8b","stepAnswer":["$$39$$"],"problemType":"TextBox","stepTitle":"$$|-y|$$ when $$y=-39$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$39$$","hints":{"DefaultPathway":[{"id":"a9ae528add8b-h1","type":"hint","dependencies":[],"title":"Value of $$|-y|$$","text":"The first step is to find the value inside the absolute value bars.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add8b-h2","type":"hint","dependencies":["a9ae528add8b-h1"],"title":"Value of $$y$$","text":"$$y=-39$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add8b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$39$$"],"dependencies":["a9ae528add8b-h2"],"title":"Calculating $$-y$$","text":"What is $$-(-39)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add8b-h4","type":"hint","dependencies":["a9ae528add8b-h3"],"title":"Absolute Value of a Positive Number","text":"The absolute value of a positive number is just that positive number. For example, $$|24|=24$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add8c","stepAnswer":["$$-22$$"],"problemType":"TextBox","stepTitle":"$$-|m|$$ when $$m=22$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-22$$","hints":{"DefaultPathway":[{"id":"a9ae528add8c-h1","type":"hint","dependencies":[],"title":"Finding the Value of $$|m|$$","text":"The first step is to find the value of $$|m|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add8c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$22$$"],"dependencies":["a9ae528add8c-h1"],"title":"Calculating $$|m|$$","text":"$$m=22$$. What is $$|22|$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a9ae528add8c-h2-s1","type":"hint","dependencies":[],"title":"Absolute Value of a Positive Number","text":"The absolute value of a positive number is just that positive number. For example, $$|24|=24$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9ae528add8c-h3","type":"hint","dependencies":["a9ae528add8c-h2"],"title":"Finding the Value of $$-|m|$$","text":"Multiply $$|m|$$ by $$-1$$ to find $$-|m|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add8c-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-22$$"],"dependencies":["a9ae528add8c-h3"],"title":"Calculating $$-|u|$$ with the Substituted Value of u","text":"What is $$-1\\\\times22$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add8d","stepAnswer":["$$-11$$"],"problemType":"TextBox","stepTitle":"$$-|p|$$ when $$p=-11$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-11$$","hints":{"DefaultPathway":[{"id":"a9ae528add8d-h1","type":"hint","dependencies":[],"title":"Finding the Value of $$|p|$$","text":"The first step is to find the value of $$|p|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add8d-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11$$"],"dependencies":["a9ae528add8d-h1"],"title":"Calculating $$|p|$$","text":"$$p=-11$$. What is $$|-11|$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a9ae528add8d-h2-s1","type":"hint","dependencies":[],"title":"Evaluating the Absolute Value of Negative Numbers","text":"If a is negative, then $$|a|=-a$$. For example, $$|-11|=11$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9ae528add8d-h3","type":"hint","dependencies":["a9ae528add8d-h2"],"title":"Finding the Value of $$-|p|$$","text":"Multiply $$|p|$$ by $$-1$$ to find $$-|p|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add8d-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-11$$"],"dependencies":["a9ae528add8d-h3"],"title":"Calculating $$-|p|$$ with the Substituted Value of $$p$$","text":"What is $$-1\\\\times11$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ae528add9","title":"Evaluating Absolute Value Expressions","body":"Evaluate the following absolute value expressions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Add and Subtract Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9ae528add9a","stepAnswer":["$$23$$"],"problemType":"TextBox","stepTitle":"$$|y|$$ when $$y=-23$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$23$$","hints":{"DefaultPathway":[{"id":"a9ae528add9a-h1","type":"hint","dependencies":[],"title":"Subsituting in the Numeric Value","text":"The first step is to subsitute the value for $$y$$ in the expression $$|y|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add9a-h2","type":"hint","dependencies":[],"title":"Value of $$y$$","text":"$$y=-23$$, a negative number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add9a-h3","type":"hint","dependencies":[],"title":"Evaluating the Absolute Value of Negative Numbers","text":"If a is negative, then $$|a|=-a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add9b","stepAnswer":["$$21$$"],"problemType":"TextBox","stepTitle":"$$|-y|$$ when $$y=-21$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$21$$","hints":{"DefaultPathway":[{"id":"a9ae528add9b-h1","type":"hint","dependencies":[],"title":"Value of $$|-y|$$","text":"The first step is to find the value inside the absolute value bars.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add9b-h2","type":"hint","dependencies":["a9ae528add9b-h1"],"title":"Value of $$y$$","text":"$$y=-21$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add9b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$21$$"],"dependencies":["a9ae528add9b-h2"],"title":"Calculating $$-y$$","text":"What is $$-(-21)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add9b-h4","type":"hint","dependencies":["a9ae528add9b-h3"],"title":"Absolute Value of a Positive Number","text":"The absolute value of a positive number is just that positive number. For example, $$|24|=24$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add9c","stepAnswer":["$$-37$$"],"problemType":"TextBox","stepTitle":"$$-|n|$$ when $$n=37$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-37$$","hints":{"DefaultPathway":[{"id":"a9ae528add9c-h1","type":"hint","dependencies":[],"title":"Finding the Value of $$|n|$$","text":"The first step is to find the value of $$|n|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add9c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$37$$"],"dependencies":["a9ae528add9c-h1"],"title":"Calculating $$|n|$$","text":"$$n=37$$. What is $$|37|$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a9ae528add9c-h2-s1","type":"hint","dependencies":[],"title":"Absolute Value of a Positive Number","text":"The absolute value of a positive number is just that positive number. For example, $$|24|=24$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9ae528add9c-h3","type":"hint","dependencies":["a9ae528add9c-h2"],"title":"Finding the Value of $$-|n|$$","text":"Multiply $$|n|$$ by $$-1$$ to find $$-|n|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add9c-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-37$$"],"dependencies":["a9ae528add9c-h3"],"title":"Calculating $$-|n|$$ with the Substituted Value of $$n$$","text":"What is $$-1\\\\times37$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ae528add9d","stepAnswer":["$$-49$$"],"problemType":"TextBox","stepTitle":"$$-|q|$$ when $$q=-49$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-49$$","hints":{"DefaultPathway":[{"id":"a9ae528add9d-h1","type":"hint","dependencies":[],"title":"Finding the Value of $$|q|$$","text":"The first step is to find the value of $$|q|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add9d-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$49$$"],"dependencies":["a9ae528add9d-h1"],"title":"Calculating $$|q|$$","text":"$$q=-49$$. What is $$|-49|$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a9ae528add9d-h2-s1","type":"hint","dependencies":[],"title":"Evaluating the Absolute Value of Negative Numbers","text":"If a is negative, then $$|a|=-a$$. For example, $$|-11|=11$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9ae528add9d-h3","type":"hint","dependencies":["a9ae528add9d-h2"],"title":"Finding the Value of $$-|q|$$","text":"Multiply $$|q|$$ by $$-1$$ to find $$-|q|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ae528add9d-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-49$$"],"dependencies":["a9ae528add9d-h3"],"title":"Calculating $$-|q|$$ with the Substituted Value of q","text":"What is $$-1\\\\times49$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9b0a01geo1","title":"Solve Applications Using Properties of Triangles","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9b0a01geo1a","stepAnswer":["$$43$$"],"problemType":"TextBox","stepTitle":"The measures of two angles of a triangle are $$55$$ and $$82$$ degrees. Find the measure of the third angle.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$43$$","hints":{"DefaultPathway":[{"id":"a9b0a01geo1a-h1","type":"hint","dependencies":[],"title":"Draw","text":"Draw the figure to try to better understand what the problem is asking","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo1a-h2","type":"hint","dependencies":["a9b0a01geo1a-h1"],"title":"Identify","text":"We are looking for the third angle of the triangle","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo1a-h3","type":"hint","dependencies":["a9b0a01geo1a-h2"],"title":"Name","text":"Name the variable we are looking for as \\"x\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo1a-h4","type":"hint","dependencies":["a9b0a01geo1a-h3"],"title":"Translate","text":"Each of the three angles added together equals $$180$$ degrees, $$x+82+55=180$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo1a-h5","type":"hint","dependencies":["a9b0a01geo1a-h4"],"title":"Solve","text":"Solve the equation using algebra","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo1a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$137$$"],"dependencies":["a9b0a01geo1a-h5"],"title":"Solve","text":"First let\'s simplify the left side of the equation. What is $$55+82$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo1a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$43$$"],"dependencies":["a9b0a01geo1a-h6"],"title":"Solve","text":"Now, let\'s subtract $$137$$ from both sides of the equation to isolate $$x$$ on the left side. What is $$180-137$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9b0a01geo10","title":"Solve Applications Using Properties of Triangles","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9b0a01geo10a","stepAnswer":["$$14$$"],"problemType":"TextBox","stepTitle":"The area of a triangular painting is $$126$$ square inches. The base is $$18$$ inches. What is the height?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$14$$","hints":{"DefaultPathway":[{"id":"a9b0a01geo10a-h1","type":"hint","dependencies":[],"title":"Draw","text":"Draw and label the sides of the triangle with the given information","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo10a-h2","type":"hint","dependencies":["a9b0a01geo10a-h1"],"title":"Identify and Name","text":"We are looking for the height of the triangle and we can name it \\"h\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo10a-h3","type":"hint","dependencies":["a9b0a01geo10a-h2"],"title":"Translate","text":"Using the formula $$a=0.5b h$$, we can substitute it to get $$126=18\\\\times0.5 h$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo10a-h4","type":"hint","dependencies":["a9b0a01geo10a-h3"],"title":"Solve","text":"Solve the equation by simplifying it first to get $$126=18\\\\times0.5 h$$, and then solving for $$h$$ by isolating the variable","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a9b0a01geo10a-h4"],"title":"Solve","text":"First let\'s simplify the right side of the equation. What is $$18\\\\times0.5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo10a-h6","type":"hint","dependencies":["a9b0a01geo10a-h5"],"title":"Solve","text":"Now, we can divide $$9$$ by both sides to isolate $$h$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9b0a01geo11","title":"Solve Applications Using Properties of Triangles","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9b0a01geo11a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"A triangular tent door has area $$15$$ square feet. The height is five feet. What is the base?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"a9b0a01geo11a-h1","type":"hint","dependencies":[],"title":"Draw","text":"Draw and label the sides of the triangle with the given information","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo11a-h2","type":"hint","dependencies":["a9b0a01geo11a-h1"],"title":"Identify and Name","text":"We are looking for the base of the triangle and we can name it \\"b\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo11a-h3","type":"hint","dependencies":["a9b0a01geo11a-h2"],"title":"Translate","text":"Using the formula $$a=0.5b h$$, we can substitute it to get $$15=0.5\\\\times5 b$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo11a-h4","type":"hint","dependencies":["a9b0a01geo11a-h3"],"title":"Solve","text":"Solve the equation by simplifying it first to get $$15=0.5\\\\times5 b$$, and then solving for $$b$$ by isolating the variable","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.5$$"],"dependencies":["a9b0a01geo11a-h4"],"title":"Solve","text":"First let\'s simplify the right side of the equation. What is $$0.5\\\\times5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo11a-h6","type":"hint","dependencies":["a9b0a01geo11a-h5"],"title":"Solve","text":"Now, we can divide $$2.5$$ by both sides to isolate $$b$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9b0a01geo12","title":"Solve Applications Using Properties of Triangles","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9b0a01geo12a","stepAnswer":["$$34$$"],"problemType":"TextBox","stepTitle":"One angle of a right triangle measures 56\xb0. What is the measure of the other small angle?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$34$$","hints":{"DefaultPathway":[{"id":"a9b0a01geo12a-h1","type":"hint","dependencies":[],"title":"Draw","text":"Draw the figure to try to better understand what the problem is asking","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo12a-h2","type":"hint","dependencies":["a9b0a01geo12a-h1"],"title":"Identify and Name","text":"We are looking for the third angle of the triangle and we can name it \\"x\\", we know the second side is $$90$$ because it is a right triangle","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo12a-h3","type":"hint","dependencies":["a9b0a01geo12a-h2"],"title":"Translate","text":"Each of the three angles added together equals $$180$$ degrees, $$x+90+56=180$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo12a-h4","type":"hint","dependencies":["a9b0a01geo12a-h3"],"title":"Solve","text":"Solve the equation using algebra","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo12a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$146$$"],"dependencies":["a9b0a01geo12a-h4"],"title":"Solve","text":"First let\'s simplify the left side of the equation. What is $$56+90$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo12a-h6","type":"hint","dependencies":["a9b0a01geo12a-h5"],"title":"Solve","text":"Now, we can subtract $$146$$ from both sides of the equation to find $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9b0a01geo13","title":"Solve Applications Using Properties of Triangles","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9b0a01geo13a","stepAnswer":["$$45$$"],"problemType":"TextBox","stepTitle":"One angle of a right triangle measures 45\xb0. What is the measure of the other small angle?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$45$$","hints":{"DefaultPathway":[{"id":"a9b0a01geo13a-h1","type":"hint","dependencies":[],"title":"Draw","text":"Draw the figure to try to better understand what the problem is asking","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo13a-h2","type":"hint","dependencies":["a9b0a01geo13a-h1"],"title":"Identify and Name","text":"We are looking for the third angle of the triangle and we can name it \\"x\\", we know the second side is $$90$$ because it is a right triangle","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo13a-h3","type":"hint","dependencies":["a9b0a01geo13a-h2"],"title":"Translate","text":"Each of the three angles added together equals $$180$$ degrees, $$x+90+45=180$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo13a-h4","type":"hint","dependencies":["a9b0a01geo13a-h3"],"title":"Solve","text":"Solve the equation using algebra","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo13a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$135$$"],"dependencies":["a9b0a01geo13a-h4"],"title":"Solve","text":"First let\'s simplify the left side of the equation. What is $$45+90$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo13a-h6","type":"hint","dependencies":["a9b0a01geo13a-h5"],"title":"Solve","text":"Now, we can subtract $$135$$ from both sides of the equation to find $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9b0a01geo14","title":"Solve Applications Using Properties of Triangles","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9b0a01geo14a","stepAnswer":["90,70,20"],"problemType":"TextBox","stepTitle":"The measure of one angle of a right triangle is 50\xb0 more than the measure of the smallest angle. Find the measures of all three angles. (Write them from largest to smallest using just a comma in the middle such as \\"80,34,26\\")","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a9b0a01geo14a-h1","type":"hint","dependencies":[],"title":"Draw","text":"Draw the figure to try to better understand what the problem is asking","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo14a-h2","type":"hint","dependencies":["a9b0a01geo14a-h1"],"title":"Identify and Name","text":"We are looking for all three angles and we can name them all. \\"x\\" is the first angle, \\"x+50\\" is the second angle, and $$90$$ is the third angle because it is a right triangle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo14a-h3","type":"hint","dependencies":["a9b0a01geo14a-h2"],"title":"Translate","text":"Each of the three angles added together equals $$180$$ degrees, $$x+x+50+90=180$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo14a-h4","type":"hint","dependencies":["a9b0a01geo14a-h3"],"title":"Solve","text":"Solve the equation using algebra","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo14a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x+140$$"],"dependencies":["a9b0a01geo14a-h4"],"title":"Solve","text":"First let\'s simplify the left side of the equation. What is $$x+x+50+90$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo14a-h6","type":"hint","dependencies":["a9b0a01geo14a-h5"],"title":"Solve","text":"Now, we can subtract $$140$$ from both sides of the equation to find $$2x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo14a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$40$$"],"dependencies":["a9b0a01geo14a-h6"],"title":"Solve","text":"What is $$180-140$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo14a-h8","type":"hint","dependencies":["a9b0a01geo14a-h7"],"title":"Solve","text":"Now, we have $$2x=40$$, so we can divide both sides by $$2$$ to get $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9b0a01geo15","title":"Solve Applications Using Properties of Triangles","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9b0a01geo15a","stepAnswer":["90,60,30"],"problemType":"TextBox","stepTitle":"The measure of one angle of a right triangle is 30\xb0 more than the measure of the smallest angle. Find the measures of all three angles. (Write them from largest to smallest using just a comma in the middle such as \\"80,34,26\\")","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a9b0a01geo15a-h1","type":"hint","dependencies":[],"title":"Draw","text":"Draw the figure to try to better understand what the problem is asking","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo15a-h2","type":"hint","dependencies":["a9b0a01geo15a-h1"],"title":"Identify and Name","text":"We are looking for all three angles and we can name them all. \\"x\\" is the first angle, \\"x+30\\" is the second angle, and $$90$$ is the third angle because it is a right triangle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo15a-h3","type":"hint","dependencies":["a9b0a01geo15a-h2"],"title":"Translate","text":"Each of the three angles added together equals $$180$$ degrees, $$x+x+30+90=180$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo15a-h4","type":"hint","dependencies":["a9b0a01geo15a-h3"],"title":"Solve","text":"Solve the equation using algebra","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo15a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x+120$$"],"dependencies":["a9b0a01geo15a-h4"],"title":"Solve","text":"First let\'s simplify the left side of the equation. What is $$x+x+30+90$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo15a-h6","type":"hint","dependencies":["a9b0a01geo15a-h5"],"title":"Solve","text":"Now, we can subtract $$120$$ from both sides of the equation to find $$2x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo15a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$60$$"],"dependencies":["a9b0a01geo15a-h6"],"title":"Solve","text":"What is $$180-120$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo15a-h8","type":"hint","dependencies":["a9b0a01geo15a-h7"],"title":"Solve","text":"Now, we have $$2x=60$$, so we can divide both sides by $$2$$ to get $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9b0a01geo2","title":"Solve Applications Using Properties of Triangles","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9b0a01geo2a","stepAnswer":["$$11$$"],"problemType":"TextBox","stepTitle":"The perimeter of a triangular garden is $$24$$ feet. The lengths of two sides are four feet and nine feet. How long is the third side?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$11$$","hints":{"DefaultPathway":[{"id":"a9b0a01geo2a-h1","type":"hint","dependencies":[],"title":"Draw","text":"Draw and label the sides of the triangle to get a better idea of what is going on.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo2a-h2","type":"hint","dependencies":["a9b0a01geo2a-h1"],"title":"Identify and Name","text":"We are looking for the third side of the triangle and we can name it \\"x\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo2a-h3","type":"hint","dependencies":["a9b0a01geo2a-h2"],"title":"Translate","text":"We know that all three sides of the triangle add up to $$24$$ so we have $$24=4+9+x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo2a-h4","type":"hint","dependencies":["a9b0a01geo2a-h3"],"title":"Solve","text":"Isolate $$x$$ by subtracting $$4$$ and $$9$$ from both sides of the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9b0a01geo3","title":"Solve Applications Using Properties of Triangles","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9b0a01geo3a","stepAnswer":["$$12$$"],"problemType":"TextBox","stepTitle":"The area of a triangular church window is $$90$$ square meters. The base of the window is $$15$$ meters. What is the window\u2019s height?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12$$","hints":{"DefaultPathway":[{"id":"a9b0a01geo3a-h1","type":"hint","dependencies":[],"title":"Draw","text":"Draw and label the sides of the triangle with the given information","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo3a-h2","type":"hint","dependencies":["a9b0a01geo3a-h1"],"title":"Identify and Name","text":"We are looking for the height of the triangle and we can name it \\"h\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo3a-h3","type":"hint","dependencies":["a9b0a01geo3a-h2"],"title":"Translate","text":"Using the formula $$a=0.5b h$$, we can substitute it to get $$90=0.5\\\\times15 h$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo3a-h4","type":"hint","dependencies":["a9b0a01geo3a-h3"],"title":"Solve","text":"Solve the equation by simplifying it first to get $$90=15\\\\times0.5 h$$, and then solving for $$h$$ by isolating the variable","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7.5$$"],"dependencies":["a9b0a01geo3a-h4"],"title":"Solve","text":"First let\'s simplify the right side of the equation. What is $$15\\\\times0.5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo3a-h6","type":"hint","dependencies":["a9b0a01geo3a-h5"],"title":"Solve","text":"Now, we can divide $$7.5$$ by both sides to isolate $$h$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9b0a01geo4","title":"Solve Applications Using Properties of Triangles","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9b0a01geo4a","stepAnswer":["$$62$$"],"problemType":"TextBox","stepTitle":"One angle of a right triangle measures 28\xb0. What is the measure of the third angle?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$62$$","hints":{"DefaultPathway":[{"id":"a9b0a01geo4a-h1","type":"hint","dependencies":[],"title":"Draw","text":"Draw the figure to try to better understand what the problem is asking","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo4a-h2","type":"hint","dependencies":["a9b0a01geo4a-h1"],"title":"Identify and Name","text":"We are looking for the third angle of the triangle and we can name it \\"x\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo4a-h3","type":"hint","dependencies":["a9b0a01geo4a-h2"],"title":"Translate","text":"Each of the three angles added together equals $$180$$ degrees, $$x+90+28=180$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo4a-h4","type":"hint","dependencies":["a9b0a01geo4a-h3"],"title":"Solve","text":"Solve the equation using algebra","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo4a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$118$$"],"dependencies":["a9b0a01geo4a-h4"],"title":"Solve","text":"First let\'s simplify the left side of the equation. What is $$28+90$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo4a-h6","type":"hint","dependencies":["a9b0a01geo4a-h5"],"title":"Solve","text":"Now, we can subtract $$118$$ from both sides of the equation to find $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9b0a01geo5","title":"Solve Applications Using Properties of Triangles","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9b0a01geo5a","stepAnswer":["90,55,35"],"problemType":"TextBox","stepTitle":"The measure of one angle of a right triangle is $$20$$ degrees more than the measure of the smallest angle. Find the measures of all three angles. (Write them from largest to smallest using just a comma in the middle such as \\"80,34,26\\")","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"a9b0a01geo5a-h1","type":"hint","dependencies":[],"title":"Draw","text":"Draw the figure to try to better understand what the problem is asking","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo5a-h2","type":"hint","dependencies":["a9b0a01geo5a-h1"],"title":"Identify and Name","text":"We are looking for all three angles and we can name them all. \\"x\\" is the first angle, \\"x+20\\" is the second angle, and $$90$$ is the third angle because it is a right triangle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo5a-h3","type":"hint","dependencies":["a9b0a01geo5a-h2"],"title":"Translate","text":"Each of the three angles added together equals $$180$$ degrees, $$x+x+20+90=180$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo5a-h4","type":"hint","dependencies":["a9b0a01geo5a-h3"],"title":"Solve","text":"Solve the equation using algebra","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x+110$$"],"dependencies":["a9b0a01geo5a-h4"],"title":"Solve","text":"First let\'s simplify the left side of the equation. What is $$x+x+20+90$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo5a-h6","type":"hint","dependencies":["a9b0a01geo5a-h5"],"title":"Solve","text":"Now, we can subtract $$110$$ from both sides of the equation to find $$2x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo5a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$70$$"],"dependencies":["a9b0a01geo5a-h6"],"title":"Solve","text":"What is $$180-110$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo5a-h8","type":"hint","dependencies":["a9b0a01geo5a-h7"],"title":"Solve","text":"Now, we have $$2x=70$$, so we can divide both sides by $$2$$ to get $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo5a-h9","type":"hint","dependencies":["a9b0a01geo5a-h8"],"title":"Solve","text":"After we found $$x$$, we know that the other angle is $$20+x$$, and the final angle is $$90$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9b0a01geo6","title":"Solve Applications Using Properties of Triangles","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9b0a01geo6a","stepAnswer":["$$21$$"],"problemType":"TextBox","stepTitle":"The measures of two angles of a triangle are $$31$$ and $$128$$ degrees. Find the measure of the third angle.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$21$$","hints":{"DefaultPathway":[{"id":"a9b0a01geo6a-h1","type":"hint","dependencies":[],"title":"Draw","text":"Draw the figure to try to better understand what the problem is asking","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo6a-h2","type":"hint","dependencies":["a9b0a01geo6a-h1"],"title":"Identify and Name","text":"We are looking for the third angle of the triangle and we can name it \\"x\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo6a-h3","type":"hint","dependencies":["a9b0a01geo6a-h2"],"title":"Translate","text":"Each of the three angles added together equals $$180$$ degrees, $$x+128+31=180$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo6a-h4","type":"hint","dependencies":["a9b0a01geo6a-h3"],"title":"Solve","text":"Solve the equation using algebra","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$159$$"],"dependencies":["a9b0a01geo6a-h4"],"title":"Solve","text":"First let\'s simplify the left side of the equation. What is $$128+31$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo6a-h6","type":"hint","dependencies":["a9b0a01geo6a-h5"],"title":"Solve","text":"Now, we can subtract $$159$$ from both sides of the equation to find $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9b0a01geo7","title":"Solve Applications Using Properties of Triangles","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9b0a01geo7a","stepAnswer":["$$56$$"],"problemType":"TextBox","stepTitle":"The measures of two angles of a triangle are $$49$$ and $$75$$ degrees. Find the measure of the third angle.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$56$$","hints":{"DefaultPathway":[{"id":"a9b0a01geo7a-h1","type":"hint","dependencies":[],"title":"Draw","text":"Draw the figure to try to better understand what the problem is asking","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo7a-h2","type":"hint","dependencies":["a9b0a01geo7a-h1"],"title":"Identify and Name","text":"We are looking for the third angle of the triangle and we can name it \\"x\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo7a-h3","type":"hint","dependencies":["a9b0a01geo7a-h2"],"title":"Translate","text":"Each of the three angles added together equals $$180$$ degrees, $$x+49+75=180$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo7a-h4","type":"hint","dependencies":["a9b0a01geo7a-h3"],"title":"Solve","text":"Solve the equation using algebra","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo7a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$124$$"],"dependencies":["a9b0a01geo7a-h4"],"title":"Solve","text":"First let\'s simplify the left side of the equation. What is $$49+75$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo7a-h6","type":"hint","dependencies":["a9b0a01geo7a-h5"],"title":"Solve","text":"Now, we can subtract $$124$$ from both sides of the equation to find $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9b0a01geo8","title":"Solve Applications Using Properties of Triangles","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9b0a01geo8a","stepAnswer":["$$8$$"],"problemType":"TextBox","stepTitle":"The perimeter of a triangular garden is $$48$$ feet. The lengths of two sides are $$18$$ feet and $$22$$ feet. How long is the third side?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8$$","hints":{"DefaultPathway":[{"id":"a9b0a01geo8a-h1","type":"hint","dependencies":[],"title":"Draw","text":"Draw and label the sides of the triangle to get a better idea of what is going on.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo8a-h2","type":"hint","dependencies":["a9b0a01geo8a-h1"],"title":"Identify and Name","text":"We are looking for the third side of the triangle and we can name it \\"x\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo8a-h3","type":"hint","dependencies":["a9b0a01geo8a-h2"],"title":"Translate","text":"We know that all three sides of the triangle add up to $$48$$ so we have $$48=18+22+x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo8a-h4","type":"hint","dependencies":["a9b0a01geo8a-h3"],"title":"Solve","text":"Isolate $$x$$ by subtracting $$18+22$$ from both sides of the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9b0a01geo9","title":"Solve Applications Using Properties of Triangles","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9b0a01geo9a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"The lengths of two sides of a triangular window are seven feet and five feet. The perimeter is $$18$$ feet. How long is the third side?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"a9b0a01geo9a-h1","type":"hint","dependencies":[],"title":"Draw","text":"Draw and label the sides of the triangle to get a better idea of what is going on.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo9a-h2","type":"hint","dependencies":["a9b0a01geo9a-h1"],"title":"Identify and Name","text":"We are looking for the third side of the triangle and we can name it \\"x\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo9a-h3","type":"hint","dependencies":["a9b0a01geo9a-h2"],"title":"Translate","text":"We know that all three sides of the triangle add up to $$18$$ so we have $$18=7+5+x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01geo9a-h4","type":"hint","dependencies":["a9b0a01geo9a-h3"],"title":"Solve","text":"Isolate $$x$$ by subtracting $$7+5$$ from both sides of the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9b0a01SolveGeometry1","title":"Solve Applications Using Rectangle Properties","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9b0a01SolveGeometry1a","stepAnswer":["$$104$$"],"problemType":"TextBox","stepTitle":"The length of a rectangle is $$32$$ meters and the width is $$20$$ meters. What is the perimeter?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$104$$","hints":{"DefaultPathway":[{"id":"a9b0a01SolveGeometry1a-h1","type":"hint","dependencies":[],"title":"Draw","text":"Try drawing out the geometric shape","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01SolveGeometry1a-h2","type":"hint","dependencies":["a9b0a01SolveGeometry1a-h1"],"title":"Definition","text":"Perimeter is defined as the length around the shape. In this case that is $$2length+2width$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9b0a01SolveGeometry2","title":"Solve Applications Using Rectangle Properties","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9b0a01SolveGeometry2a","stepAnswer":["$$340$$"],"problemType":"TextBox","stepTitle":"The length of a rectangle is $$120$$ yards and the width is $$50$$ yards. What is the perimeter?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$340$$","hints":{"DefaultPathway":[{"id":"a9b0a01SolveGeometry2a-h1","type":"hint","dependencies":[],"title":"Draw","text":"Try drawing out the geometric shape","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01SolveGeometry2a-h2","type":"hint","dependencies":["a9b0a01SolveGeometry2a-h1"],"title":"Definition","text":"Perimeter is defined as the length around the shape. In this case that is $$2length+2width$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9b0a01SolveGeometry3","title":"Solve Applications Using Rectangle Properties","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9b0a01SolveGeometry3a","stepAnswer":["$$220$$"],"problemType":"TextBox","stepTitle":"The length of a rectangle is $$62$$ feet and the width is $$48$$ feet. What is the perimeter?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$220$$","hints":{"DefaultPathway":[{"id":"a9b0a01SolveGeometry3a-h1","type":"hint","dependencies":[],"title":"Draw","text":"Try drawing out the geometric shape","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9b0a01SolveGeometry3a-h2","type":"hint","dependencies":["a9b0a01SolveGeometry3a-h1"],"title":"Definition","text":"Perimeter is defined as the length around the shape. In this case that is $$2length+2width$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9b0a01SolveGeometry4","title":"Solve Applications Using Rectangle Properties","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9b0a01SolveGeometry4a","stepAnswer":["$$12$$"],"problemType":"TextBox","stepTitle":"The area of a rectangular room is $$168$$ square feet. The length is $$14$$ feet. 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In this example, we choose the left side as the \\"variable\\" side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33f-h2","type":"hint","dependencies":["a9c142dVarCon33f-h1"],"title":"Subtraction property of equality","text":"When you subtract the same quantity from both sides of an equation, you still have equality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33f-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5p+14=6$$"],"dependencies":["a9c142dVarCon33f-h2"],"title":"Subtraction","text":"Subtract $$4p$$ from each side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33f-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5p=-8$$"],"dependencies":["a9c142dVarCon33f-h3"],"title":"Subtraction","text":"Subtract $$14$$ from each side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33f-h5","type":"hint","dependencies":["a9c142dVarCon33f-h4"],"title":"Division property of equality","text":"When you divide both sides of an equation by any non-zero number, you still have equality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33f-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$p=\\\\frac{-8}{5}$$"],"dependencies":["a9c142dVarCon33f-h5"],"title":"Division","text":"Divide $$5$$ from each side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33f-h7","type":"hint","dependencies":["a9c142dVarCon33f-h6"],"title":"Verification","text":"Check whether the result is a solution of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33f-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a9c142dVarCon33f-h7"],"title":"Verification","text":"Check whether $$9\\\\left(-\\\\frac{8}{5}\\\\right)+14$$ equals $$6+4\\\\left(-\\\\frac{8}{5}\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]}]}},{"id":"a9c142dVarCon33g","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"$$3y-4=12-y$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"a9c142dVarCon33g-h1","type":"hint","dependencies":[],"title":"Choosing side","text":"Choose a side to be the \\"variable\\" side and the other side will be the \\"constant\\" side. 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In this example, we choose the right side as the \\"variable\\" side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33i-h2","type":"hint","dependencies":["a9c142dVarCon33i-h1"],"title":"Addition property of equality","text":"When you add the same quantity to both sides of an equation, you still have equality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33i-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11=a+4$$"],"dependencies":["a9c142dVarCon33i-h2"],"title":"Addition","text":"Add $$\\\\frac{1}{5} a$$ to each side of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33i-h4","type":"hint","dependencies":["a9c142dVarCon33i-h3"],"title":"Subtraction property of equality","text":"When you subtract the same quantity from both sides of an equation, you still have equality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33i-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$a=7$$"],"dependencies":["a9c142dVarCon33i-h4"],"title":"Subtraction","text":"Subtract $$4$$ from each side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33i-h6","type":"hint","dependencies":["a9c142dVarCon33i-h5"],"title":"Verification","text":"Check whether the result is a solution of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33i-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a9c142dVarCon33i-h6"],"title":"Verification","text":"Check whether $$11-7\\\\frac{1}{5}$$ equals $$7\\\\frac{4}{5}+4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]}]}},{"id":"a9c142dVarCon33j","stepAnswer":["$$-40$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{5}{4} a+15=\\\\frac{3}{4} a-5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-40$$","hints":{"DefaultPathway":[{"id":"a9c142dVarCon33j-h1","type":"hint","dependencies":[],"title":"Choosing side","text":"Choose a side to be the \\"variable\\" side and the other side will be the \\"constant\\" side. In this example, we choose the left side as the \\"variable\\" side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33j-h2","type":"hint","dependencies":["a9c142dVarCon33j-h1"],"title":"Subtraction property of equality","text":"When you subtract the same quantity from both sides of an equation, you still have equality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33j-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$a+15=-5$$"],"dependencies":["a9c142dVarCon33j-h2"],"title":"Subtraction","text":"Subtract $$\\\\frac{3}{4} a$$ from each side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33j-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$a=-20$$"],"dependencies":["a9c142dVarCon33j-h3"],"title":"Subtraction","text":"Subtract $$15$$ from each side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33j-h5","type":"hint","dependencies":["a9c142dVarCon33j-h4"],"title":"Verification","text":"Check whether the result is a solution of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33j-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a9c142dVarCon33j-h5"],"title":"Verification","text":"Check whether $$\\\\frac{5}{4} \\\\left(-20\\\\right)+15$$ equals $$\\\\frac{3}{4} \\\\left(-20\\\\right)-5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]}]}},{"id":"a9c142dVarCon33k","stepAnswer":["$$15$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{3}{5} p+2=\\\\frac{4}{5} p-1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$15$$","hints":{"DefaultPathway":[{"id":"a9c142dVarCon33k-h1","type":"hint","dependencies":[],"title":"Choosing side","text":"Choose a side to be the \\"variable\\" side and the other side will be the \\"constant\\" side. In this example, we choose the right side as the \\"variable\\" side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33k-h2","type":"hint","dependencies":["a9c142dVarCon33k-h1"],"title":"Subtraction property of equality","text":"When you subtract the same quantity from both sides of an equation, you still have equality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33k-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2=\\\\frac{1}{5} p-1$$"],"dependencies":["a9c142dVarCon33k-h2"],"title":"Subtraction","text":"Subtract $$\\\\frac{3}{5} p$$ from each side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33k-h4","type":"hint","dependencies":["a9c142dVarCon33k-h3"],"title":"Addition property of equality","text":"When you add the same quantity to both sides of an equation, you still have equality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33k-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3=\\\\frac{1}{5} a$$"],"dependencies":["a9c142dVarCon33k-h4"],"title":"Addition","text":"Add $$1$$ to each side of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33k-h6","type":"hint","dependencies":["a9c142dVarCon33k-h5"],"title":"Multiplication property of equality","text":"If you multiply both sides of an equation by the same number, you still have equality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33k-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$a=15$$"],"dependencies":["a9c142dVarCon33k-h6"],"title":"Multiplication","text":"Multiple $$5$$ from each side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33k-h8","type":"hint","dependencies":["a9c142dVarCon33k-h7"],"title":"Verification","text":"Check whether the result is a solution of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33k-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a9c142dVarCon33k-h8"],"title":"Verification","text":"Check whether $$15\\\\frac{3}{5}+2$$ equals $$15\\\\frac{4}{5}-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]}]}},{"id":"a9c142dVarCon33l","stepAnswer":["$$3.46$$"],"problemType":"TextBox","stepTitle":"$$13z+6.45=8z+23.75$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3.46$$","hints":{"DefaultPathway":[{"id":"a9c142dVarCon33l-h1","type":"hint","dependencies":[],"title":"Choosing side","text":"Choose a side to be the \\"variable\\" side and the other side will be the \\"constant\\" side. In this example, we choose the left side as the \\"variable\\" side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33l-h2","type":"hint","dependencies":["a9c142dVarCon33l-h1"],"title":"Subtraction property of equality","text":"When you subtract the same quantity from both sides of an equation, you still have equality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33l-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5z+6.45=23.75$$"],"dependencies":["a9c142dVarCon33l-h2"],"title":"Subtraction","text":"Subtract $$8z$$ from each side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33l-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5z=17.3$$"],"dependencies":["a9c142dVarCon33l-h3"],"title":"Subtraction","text":"Subtract $$6.45$$ from each side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33l-h5","type":"hint","dependencies":["a9c142dVarCon33l-h4"],"title":"Division property of equality","text":"When you divide both sides of an equation by any non-zero number, you still have equality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33l-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$z=3.46$$"],"dependencies":["a9c142dVarCon33l-h5"],"title":"Division","text":"Divide $$5$$ from each side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33l-h7","type":"hint","dependencies":["a9c142dVarCon33l-h6"],"title":"Verification","text":"Check whether the result is a solution of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33l-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a9c142dVarCon33l-h7"],"title":"Verification","text":"Check whether $$13\\\\times3.46+6.45$$ equals $$8\\\\times2.46+23.75$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]}]}},{"id":"a9c142dVarCon33m","stepAnswer":["$$60$$"],"problemType":"TextBox","stepTitle":"$$2.7w-80=1.2w+10$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$60$$","hints":{"DefaultPathway":[{"id":"a9c142dVarCon33m-h1","type":"hint","dependencies":[],"title":"Choosing side","text":"Choose a side to be the \\"variable\\" side and the other side will be the \\"constant\\" side. In this example, we choose the left side as the \\"variable\\" side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33m-h2","type":"hint","dependencies":["a9c142dVarCon33m-h1"],"title":"Subtraction property of equality","text":"When you subtract the same quantity from both sides of an equation, you still have equality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33m-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.5w-80=10$$"],"dependencies":["a9c142dVarCon33m-h2"],"title":"Subtraction","text":"Subtract $$1.2w$$ from each side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33m-h4","type":"hint","dependencies":["a9c142dVarCon33m-h3"],"title":"Addition property of equality","text":"When you add the same quantity to both sides of an equation, you still have equality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33m-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.5w=90$$"],"dependencies":["a9c142dVarCon33m-h4"],"title":"Addition","text":"Add $$80$$ to each side of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33m-h6","type":"hint","dependencies":["a9c142dVarCon33m-h5"],"title":"Division property of equality","text":"When you divide both sides of an equation by any non-zero number, you still have equality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33m-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$w=60$$"],"dependencies":["a9c142dVarCon33m-h6"],"title":"Division","text":"Divide $$1.5$$ from each side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33m-h8","type":"hint","dependencies":["a9c142dVarCon33m-h7"],"title":"Verification","text":"Check whether the result is a solution of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33m-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a9c142dVarCon33m-h8"],"title":"Verification","text":"Check whether $$2.7\\\\times60-80$$ equals $$1.2\\\\times60+10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]}]}},{"id":"a9c142dVarCon33n","stepAnswer":["$$23$$"],"problemType":"TextBox","stepTitle":"$$6.6x-18.9=3.4x+54.7$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$23$$","hints":{"DefaultPathway":[{"id":"a9c142dVarCon33n-h1","type":"hint","dependencies":[],"title":"Choosing side","text":"Choose a side to be the \\"variable\\" side and the other side will be the \\"constant\\" side. In this example, we choose the left side as the \\"variable\\" side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33n-h2","type":"hint","dependencies":["a9c142dVarCon33n-h1"],"title":"Subtraction property of equality","text":"When you subtract the same quantity from both sides of an equation, you still have equality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon33n-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3.2x-18.9=54.7$$"],"dependencies":["a9c142dVarCon33n-h2"],"title":"Subtraction","text":"Subtract $$3.4x$$ from each side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon7f-h5","type":"hint","dependencies":["a9c142dVarCon7f-h4"],"title":"Verification","text":"Check whether the result is a solution of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon7f-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a9c142dVarCon7f-h5"],"title":"Verification","text":"Check whether $$-14\\\\left(-\\\\frac{9}{7}\\\\right)-2$$ equals $$16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]}]}}]},{"id":"a9c142dVarCon8","title":"Solve the equation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.3 Solve Equations with Variables and Constants on Both Sides","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9c142dVarCon8a","stepAnswer":["$$-6$$"],"problemType":"TextBox","stepTitle":"$$9x=8x-6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-6$$","hints":{"DefaultPathway":[{"id":"a9c142dVarCon8a-h1","type":"hint","dependencies":[],"title":"Subtraction property of equality","text":"When you subtract the same quantity from both sides of an equation, you still have equality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=-6$$"],"dependencies":["a9c142dVarCon8a-h1"],"title":"Subtraction","text":"Subtract $$8x$$ from each side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon8a-h3","type":"hint","dependencies":["a9c142dVarCon8a-h2"],"title":"Verification","text":"Check whether the result is a solution of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon8a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a9c142dVarCon8a-h3"],"title":"Verification","text":"Check whether $$9\\\\left(-6\\\\right)$$ equals $$8\\\\left(-6\\\\right)-6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]}]}}]},{"id":"a9c142dVarCon9","title":"Solve the equation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.3 Solve Equations with Variables and Constants on Both Sides","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"a9c142dVarCon9a","stepAnswer":["$$-10$$"],"problemType":"TextBox","stepTitle":"$$6n=5n-10$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-10$$","hints":{"DefaultPathway":[{"id":"a9c142dVarCon9a-h1","type":"hint","dependencies":[],"title":"Subtraction property of equality","text":"When you subtract the same quantity from both sides of an equation, you still have equality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$n=-10$$"],"dependencies":["a9c142dVarCon9a-h1"],"title":"Subtraction","text":"Subtract $$5n$$ from each side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon9a-h3","type":"hint","dependencies":["a9c142dVarCon9a-h2"],"title":"Verification","text":"Check whether the result is a solution of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9c142dVarCon9a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["a9c142dVarCon9a-h3"],"title":"Verification","text":"Check whether $$6\\\\left(-10\\\\right)$$ equals $$5\\\\left(-10\\\\right)-10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]}]}}]},{"id":"a9cf449complex1","title":"Expressing an Imaginary Number in Standard Form","body":"Express the following expression in standard form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Complex Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a9cf449complex1a","stepAnswer":["3i"],"problemType":"MultipleChoice","stepTitle":"$$\\\\sqrt{-9}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["3i","9i","i","4i"],"hints":{"DefaultPathway":[{"id":"a9cf449complex1a-h1","type":"hint","dependencies":[],"title":"Standard Form Definition","text":"The standard form of an imaginary number $$\\\\sqrt{-a}$$ is $$\\\\sqrt{a} i$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex1a-h2","type":"hint","dependencies":["a9cf449complex1a-h1"],"title":"Rewriting the Expression as a Product","text":"The first step is to rewrite the expression as a product of the square root of $$-1$$ and the square root of another value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex1a-h3","type":"hint","dependencies":["a9cf449complex1a-h2"],"title":"Rewriting the Expression as a Product","text":"The expression, $$\\\\sqrt{-9}$$, can be rewritten as $$\\\\sqrt{-1} \\\\sqrt{9}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex1a-h4","type":"hint","dependencies":[],"title":"Definition of i","text":"i represents $$\\\\sqrt{-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex1a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a9cf449complex1a-h3"],"title":"Square Root of $$9$$","text":"What is $$\\\\sqrt{9}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9cf449complex10","title":"Adding and Subtracting Complex Numbers","body":"For the following exercises, perform the indicated operation and express the result as a simplified complex number.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Complex Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a9cf449complex10a","stepAnswer":["$$-11+4i$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(-5+3i\\\\right)-6-i$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-11+4i$$","hints":{"DefaultPathway":[{"id":"a9cf449complex10a-h1","type":"hint","dependencies":[],"title":"Distributing the Negative Sign","text":"Because the second part of the expression is being subtracted, we can distribute the negative sign into that part. $$-(6-i)=-6+i$$. The expression is now $$\\\\left(-5+3i\\\\right)+\\\\left(-6+i\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex10a-h2","type":"hint","dependencies":["a9cf449complex10a-h1"],"title":"Associative Property","text":"The next step is to group the like terms. We can use the Associative Property to rewrite this expression as $$\\\\left(-5+\\\\left(-6\\\\right)\\\\right)+3i+i$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex10a-h3","type":"hint","dependencies":["a9cf449complex10a-h2"],"title":"Combining Like Terms","text":"Now, we can add numbers in the parentheses to combine like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-11$$"],"dependencies":["a9cf449complex10a-h3"],"title":"Combining Like Terms","text":"What is $$-5+\\\\left(-6\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["4i"],"dependencies":["a9cf449complex10a-h3"],"title":"Combining Like Terms","text":"What is $$3i+i$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex10a-h6","type":"hint","dependencies":["a9cf449complex10a-h4","a9cf449complex10a-h5"],"title":"Standard Form","text":"Finally, we can write the expression in $$a+bi$$ form: $$-11+4i$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9cf449complex11","title":"Adding and Subtracting Complex Numbers","body":"For the following exercises, perform the indicated operation and express the result as a simplified complex number.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Complex Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a9cf449complex11a","stepAnswer":["$$2-5i$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(-4+4i\\\\right)-\\\\left(-6+9i\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2-5i$$","hints":{"DefaultPathway":[{"id":"a9cf449complex11a-h1","type":"hint","dependencies":[],"title":"Distributing the Negative Sign","text":"Because the second part of the expression is being subtracted, we can distribute the negative sign into that part. $$-\\\\left(-6+9i\\\\right)=6-9i$$. The expression is now $$\\\\left(-4+4i\\\\right)+6-9i$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex11a-h2","type":"hint","dependencies":["a9cf449complex11a-h1"],"title":"Associative Property","text":"The next step is to group the like terms. We can use the Associative Property to rewrite this expression as $$\\\\left(-4+6\\\\right)+4i-9i$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex11a-h3","type":"hint","dependencies":["a9cf449complex11a-h2"],"title":"Combining Like Terms","text":"Now, we can add numbers in the parentheses to combine like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a9cf449complex11a-h3"],"title":"Combining Like Terms","text":"What is $$-4+6$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["-5i"],"dependencies":["a9cf449complex11a-h3"],"title":"Combining Like Terms","text":"What is 4i-9i?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex11a-h6","type":"hint","dependencies":["a9cf449complex11a-h4","a9cf449complex11a-h5"],"title":"Standard Form","text":"Finally, we can write the expression in $$a+bi$$ form: $$2-5i$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9cf449complex12","title":"Multiplying a Complex Number by a Complex Number","body":"For the following exercises, perform the indicated operation and express the result as a simplified complex number.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Complex Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a9cf449complex12a","stepAnswer":["$$6+15i$$"],"problemType":"TextBox","stepTitle":"$$(5-2i)(3i)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6+15i$$","hints":{"DefaultPathway":[{"id":"a9cf449complex12a-h1","type":"hint","dependencies":[],"title":"Distributive Property","text":"The first step is to apply the Distributive Property, which turns our expression into $$3i\\\\times5+3i \\\\left(-2i\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["15i"],"dependencies":["a9cf449complex12a-h1"],"title":"Simple Multiplication","text":"What is $$3i\\\\times5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex12a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a9cf449complex12a-h1"],"title":"Simple Multiplication","text":"What is $$3i \\\\left(-2i\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a9cf449complex12a-h3-s1","type":"hint","dependencies":[],"title":"Distributive Property","text":"Because i is the square root of $$-1$$, when we multiply $$(3i)(-2i)$$, the i is squared and becomes $$-1$$. So, $$(3i)(-2i)$$ becomes $$(-6)(-1)$$, or $$6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9cf449complex12a-h4","type":"hint","dependencies":["a9cf449complex12a-h2","a9cf449complex12a-h3"],"title":"Standard Form","text":"Finally, we can write the expression in $$a+bi$$ form: $$6+15i$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9cf449complex13","title":"Multiplying a Complex Number by a Real Number","body":"For the following exercises, perform the indicated operation and express the result as a simplified complex number.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Complex Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a9cf449complex13a","stepAnswer":["$$-16+32i$$"],"problemType":"TextBox","stepTitle":"$$8\\\\left(-2+4i\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-16+32i$$","hints":{"DefaultPathway":[{"id":"a9cf449complex13a-h1","type":"hint","dependencies":[],"title":"Distributive Property","text":"The first step is to apply the Distributive Property, which turns our expression into $$8\\\\left(-2\\\\right)+8\\\\times4i$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-16$$"],"dependencies":["a9cf449complex13a-h1"],"title":"Distributive Property","text":"What is $$8\\\\left(-2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex13a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["32i"],"dependencies":["a9cf449complex13a-h2"],"title":"Distributive Property","text":"What is $$8\\\\times4i$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex13a-h4","type":"hint","dependencies":["a9cf449complex13a-h3"],"title":"Standard Form","text":"Finally, we can write the expression in $$a+bi$$ form: $$-16+32i$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9cf449complex14","title":"Multiplying a Complex Number by a Complex Number","body":"For the following exercises, perform the indicated operation and express the result as a simplified complex number.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Complex Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a9cf449complex14a","stepAnswer":["$$-4-7i$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(-1+2i\\\\right) \\\\left(-2+3i\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-4-7i$$","hints":{"DefaultPathway":[{"id":"a9cf449complex14a-h1","type":"hint","dependencies":[],"title":"FOIL","text":"The first step is to FOIL the expressions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex14a-h2","type":"hint","dependencies":["a9cf449complex14a-h1"],"title":"FOIL - First","text":"Let\'s multiply the first terms in both expressions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex14a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a9cf449complex14a-h2"],"title":"FOIL - First","text":"What is $$\\\\left(-1\\\\right) \\\\left(-2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex14a-h4","type":"hint","dependencies":["a9cf449complex14a-h1"],"title":"FOIL - Outside","text":"Now, multiply the outside terms in both expressions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex14a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["-3i"],"dependencies":["a9cf449complex14a-h4"],"title":"FOIL - Outside","text":"What is $$\\\\left(-1\\\\right) 3i$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex14a-h6","type":"hint","dependencies":["a9cf449complex14a-h1"],"title":"FOIL - Inside","text":"Next, multiply the inside terms in both expressions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex14a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["-4i"],"dependencies":["a9cf449complex14a-h6"],"title":"FOIL - Inside","text":"What is $$2i \\\\left(-2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex14a-h8","type":"hint","dependencies":["a9cf449complex14a-h1"],"title":"FOIL - Last","text":"Finally, we can multiply the last terms in both expressions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex14a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6$$"],"dependencies":["a9cf449complex14a-h8"],"title":"FOIL - Last","text":"What is $$2i 3i$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a9cf449complex14a-h9-s1","type":"hint","dependencies":[],"title":"FOIL - Last","text":"Because i is the square root of $$-1$$, when we multiply (3i)(2i), the i is squared and becomes $$-1$$. So, (3i)(2i) becomes $$(6)(-1)$$, or $$-6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9cf449complex14a-h10","type":"hint","dependencies":["a9cf449complex14a-h3","a9cf449complex14a-h5","a9cf449complex14a-h7","a9cf449complex14a-h9"],"title":"Combining Like Terms","text":"Here, we can combine the like terms. Add the constants, $$2$$ and $$-6$$ together, and add the terms with i together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex14a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["a9cf449complex14a-h10"],"title":"Combining Like Terms","text":"What is $$(2-6)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex14a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["-7i"],"dependencies":["a9cf449complex14a-h10"],"title":"Combining Like Terms","text":"What is $$(-3i-4i)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex14a-h13","type":"hint","dependencies":["a9cf449complex14a-h11","a9cf449complex14a-h12"],"title":"Standard Form","text":"Our last step is to write the expression in $$a+bi$$ form: $$-4-7i$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9cf449complex15","title":"Multiplying a Complex Number by a Complex Number","body":"Perform the indicated operation and express the result as a simplified complex number.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Complex Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a9cf449complex15a","stepAnswer":["$$25$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(3+4i\\\\right) \\\\left(3-4i\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$25$$","hints":{"DefaultPathway":[{"id":"a9cf449complex15a-h1","type":"hint","dependencies":[],"title":"FOIL","text":"Use FOIL to find the product.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a9cf449complex15a-h1"],"title":"FOIL - First","text":"What is the product of the first two terms in each binomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["-12i"],"dependencies":["a9cf449complex15a-h1"],"title":"FOIL - Outside","text":"What is the product of the outer two terms in each binomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["12i"],"dependencies":["a9cf449complex15a-h1"],"title":"FOIL - Inside","text":"What is the product of the inner two terms in each binomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex15a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["a9cf449complex15a-h1"],"title":"FOIL","text":"What is the product of the last two terms in each binomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a9cf449complex15a-h5-s1","type":"hint","dependencies":[],"title":"Square of i","text":"By definition, i is the square root of $$-1$$. So $$i^2=-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9cf449complex15a-h6","type":"hint","dependencies":["a9cf449complex15a-h2","a9cf449complex15a-h3","a9cf449complex15a-h4","a9cf449complex15a-h5"],"title":"Combine like terms","text":"Combine all the like terms got from FOIL.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex15a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["a9cf449complex15a-h6"],"title":"Combine like terms","text":"What is the sum of $$9-12i+12i+16$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9cf449complex16","title":"Multiplying a Complex Number by a Real Number","body":"Perform the indicated operation and express the result as a simplified complex number.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Complex Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a9cf449complex16a","stepAnswer":["$$2-\\\\frac{2}{3} i$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{6-2i}{3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2-\\\\frac{2}{3} i$$","hints":{"DefaultPathway":[{"id":"a9cf449complex16a-h1","type":"hint","dependencies":[],"title":"Comverting Division into Multiplication","text":"Write the division as multiplying by a fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["a9cf449complex16a-h1"],"title":"Comverting Division into Multiplication","text":"What is the fraction such that multiplying by this fraction is equivalent to dividing by 3?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex16a-h3","type":"hint","dependencies":["a9cf449complex16a-h2"],"title":"Distributive Property","text":"Distribute the $$\\\\frac{1}{3}$$ to each term in the complex number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex16a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a9cf449complex16a-h3"],"title":"Distributive Property","text":"What is the first term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex16a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-\\\\left(\\\\frac{2}{3}\\\\right) i$$"],"dependencies":["a9cf449complex16a-h3"],"title":"Distributive Property","text":"What is the second term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex16a-h6","type":"hint","dependencies":["a9cf449complex16a-h4","a9cf449complex16a-h5"],"title":"Distributive Property","text":"Write the expression into standard form: $$2-\\\\frac{2}{3} i$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9cf449complex17","title":"Dividing Complex Numbers","body":"Perform the indicated operation and express the result as a simplified complex number.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Complex Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a9cf449complex17a","stepAnswer":["$$4-6i$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{6+4i}{i}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4-6i$$","hints":{"DefaultPathway":[{"id":"a9cf449complex17a-h1","type":"hint","dependencies":[],"title":"Complex Conjugate","text":"The first step is to find the complex conjugate of the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex17a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["-i"],"dependencies":["a9cf449complex17a-h1"],"title":"Complex Conjugate","text":"What is the complex conjugate of the denominator i?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex17a-h3","type":"hint","dependencies":["a9cf449complex17a-h2"],"title":"Multiplying Complex Conjugate","text":"Multiply the complex conjugate to the denominator and the numerator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex17a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["-6i"],"dependencies":["a9cf449complex17a-h3"],"title":"Multiplying Complex Numbers","text":"What is $$(6)(-i)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex17a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a9cf449complex17a-h3"],"title":"Multiplying Complex Numbers","text":"What is $$(4i)(-i)$$? Remember $$i^2=-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex17a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a9cf449complex17a-h3"],"title":"Multiplying Complex Numbers","text":"What is $$(i)(-i)$$? Remember $$i^2=-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex17a-h7","type":"hint","dependencies":["a9cf449complex17a-h4","a9cf449complex17a-h5","a9cf449complex17a-h6"],"title":"Combine Like Terms","text":"Add the terms in the numerator, we get $$4-6i$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9cf449complex18","title":"Dividing Complex Numbers","body":"Perform the indicated operation and express the result as a simplified complex number.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Complex Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a9cf449complex18a","stepAnswer":["$$\\\\frac{2}{5}+\\\\frac{11i}{5}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{3+4i}{2-i}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2}{5}+\\\\frac{11i}{5}$$","hints":{"DefaultPathway":[{"id":"a9cf449complex18a-h1","type":"hint","dependencies":[],"title":"Complex Conjugate","text":"The first step is to find the complex conjugate of the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2+i$$"],"dependencies":["a9cf449complex18a-h1"],"title":"Complex Conjugate","text":"What is the complex conjugate of the denominator 2-i?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex18a-h3","type":"hint","dependencies":["a9cf449complex18a-h2"],"title":"Multiplying Complex Conjugate","text":"Multiply $$2+i$$ to the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex18a-h4","type":"hint","dependencies":["a9cf449complex18a-h3"],"title":"Multiplying Complex Numbers in Numerator","text":"Use the FOIL method to find the product of $$\\\\left(3+4i\\\\right) \\\\left(2+i\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex18a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2+11i$$"],"dependencies":["a9cf449complex18a-h4"],"title":"Multiplying Complex Numbers in Numerator","text":"What is the product of $$\\\\left(3+4i\\\\right) \\\\left(2+i\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex18a-h6","type":"hint","dependencies":["a9cf449complex18a-h5"],"title":"Multiplying Complex Numbers in Demonimator","text":"Use the formula $$\\\\left(a+bi\\\\right) \\\\left(a-bi\\\\right)=a^2+b^2$$ to find the product of $$\\\\left(2-i\\\\right) \\\\left(2+i\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex18a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a9cf449complex18a-h6"],"title":"Multiplying Complex Numbers in Demonimator","text":"What is the product of $$\\\\left(2-i\\\\right) \\\\left(2+i\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex18a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{5}+\\\\frac{11i}{5}$$"],"dependencies":["a9cf449complex18a-h5","a9cf449complex18a-h7"],"title":"Dividing by a Real Number","text":"After dividing the numerator by the denominator, what is the remaining complex number?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9cf449complex19","title":"Expressing an Imaginary Number in Standard Form","body":"Perform the indicated operation and express the result as a simplified complex number.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Complex Numbers","courseName":"OpenStax: College 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4.0>"},{"id":"a9cf449complex19a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["12i"],"dependencies":["a9cf449complex19a-h2"],"title":"Standard Form Part $$2$$","text":"What is $$3\\\\sqrt{-16}$$ in standard form?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a9cf449complex19a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["12i"],"dependencies":[],"title":"Standard Form Part $$2$$","text":"What is (3)(4i)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9cf449complex19a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["15i"],"dependencies":["a9cf449complex19a-h3"],"title":"Common Terms","text":"Identify which terms are imaganary and which are real. Add common terms. What expression do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9cf449complex2","title":"Try it: Expressing an Imaginary Number in Standard Form","body":"Find the standard form of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Complex Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a9cf449complex2a","stepAnswer":["$$2\\\\sqrt{6} i$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\sqrt{-24}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2\\\\sqrt{6} i$$","choices":["$$\\\\sqrt{2} i$$","$$20\\\\sqrt{2} i$$","$$2\\\\sqrt{6} i$$","24i"],"hints":{"DefaultPathway":[{"id":"a9cf449complex2a-h1","type":"hint","dependencies":[],"title":"Standard Form Definition","text":"The standard form of an imaginary number $$\\\\sqrt{-a}$$ is $$\\\\sqrt{a} i$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex2a-h2","type":"hint","dependencies":["a9cf449complex2a-h1"],"title":"Rewriting the Expression as a Product","text":"The first step is to rewrite the expression as a product of the square root of $$-1$$ and the square root of another value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex2a-h3","type":"hint","dependencies":["a9cf449complex2a-h2"],"title":"Rewriting the Expression as a Product","text":"The expression, $$\\\\sqrt{-24}$$, can be rewritten as $$\\\\sqrt{-1} \\\\sqrt{24}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex2a-h4","type":"hint","dependencies":[],"title":"Definition of i","text":"i represents $$\\\\sqrt{-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex2a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2\\\\sqrt{6}$$"],"dependencies":["a9cf449complex2a-h3"],"title":"Square Root of $$24$$","text":"What is $$\\\\sqrt{24}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2\\\\sqrt{6}$$","$$2\\\\sqrt{3}$$","$$3\\\\sqrt{5}$$","$$6\\\\sqrt{6}$$"]}]}}]},{"id":"a9cf449complex20","title":"Multiplying a Complex Number by a Real Number","body":"Perform the indicated operation and express the result as a simplified complex number.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College 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fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex20a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["a9cf449complex20a-h2"],"title":"Comverting Division into Multiplication","text":"What is the fraction such that multiplying by this fraction is equivalent to dividing by 2?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex20a-h4","type":"hint","dependencies":["a9cf449complex20a-h3"],"title":"Distributive Property","text":"Distribute the $$\\\\frac{1}{2}$$ to each term in the complex number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex20a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a9cf449complex20a-h4"],"title":"Distributive Property","text":"What is $$2\\\\left(\\\\frac{1}{2}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex20a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$i\\\\left(\\\\sqrt{3}\\\\right)$$"],"dependencies":["a9cf449complex20a-h4"],"title":"Distributive Property","text":"What is $$2i \\\\sqrt{3} \\\\frac{1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex20a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1+i\\\\left(\\\\sqrt{3}\\\\right)$$"],"dependencies":["a9cf449complex20a-h5","a9cf449complex20a-h6"],"title":"Adding Complex Numbers","text":"Add the terms in the numerator. What expression do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9cf449complex21","title":"Simplifying Powers of i","body":"Perform the indicated operation and express the result as a simplified complex number.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Complex Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a9cf449complex21a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"$$i^8$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a9cf449complex21a-h1","type":"hint","dependencies":[],"title":"Writing $$i^8$$ in terms of $$i^4$$","text":"Simplify the problem by factoring out as many factors of $$i^4$$ as possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex21a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a9cf449complex21a-h1"],"title":"Writing $$i^8$$ in terms of $$i^4$$","text":"How many times does $$4$$ go into 8?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex21a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a9cf449complex21a-h2"],"title":"Writing $$i^8$$ in terms of $$i^4$$","text":"What is $$\\\\frac{8}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex21a-h4","type":"hint","dependencies":["a9cf449complex21a-h3"],"title":"Writing Expressions","text":"$$i^8$$ can be rewritten as $${\\\\left(i^4\\\\right)}^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex21a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a9cf449complex21a-h4"],"title":"Evaluating $$i^4$$","text":"What is $$i^4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex21a-h6","type":"hint","dependencies":["a9cf449complex21a-h5"],"title":"Evaluating $$i^4$$","text":"$$I^4$$ can be rewritten as $${\\\\left(i^2\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex21a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a9cf449complex21a-h6"],"title":"Evaluating $$i^2$$","text":"What is $$1^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9cf449complex22","title":"Simplifying Powers of i","body":"Perform the indicated operation and express the result as a simplified complex number.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Complex Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a9cf449complex22a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"$$i^{22}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a9cf449complex22a-h1","type":"hint","dependencies":[],"title":"Simplifying $$i^2$$ using $$i^4$$","text":"Simplify the problem by factoring out as many factors of $$i^4$$ as possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex22a-h2","type":"hint","dependencies":["a9cf449complex22a-h1"],"title":"Simplifying $$i^2$$ using $$i^4$$","text":"Determine how many times $$4$$ goes into $$22$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex22a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a9cf449complex22a-h2"],"title":"Division","text":"How many times does $$4$$ fully go into 22?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex22a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a9cf449complex22a-h3"],"title":"Division","text":"What is the remainder for $$\\\\frac{22}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex22a-h5","type":"hint","dependencies":["a9cf449complex22a-h4"],"title":"Writing Expressions","text":"Rewrite the expression as $${\\\\left(i^4\\\\right)}^5 i^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex22a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a9cf449complex22a-h5"],"title":"Evaluate $$i^4$$","text":"What is $$i^4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex22a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a9cf449complex22a-h6"],"title":"Exponents","text":"What is $$1^{25}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex22a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a9cf449complex22a-h7"],"title":"Evaluate $$i^2$$","text":"What is $$i^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex22a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a9cf449complex22a-h8"],"title":"Multiplication","text":"What is $$(1)(-1)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9cf449complex23","title":"Finding Complex Conjugates","body":"Find the complex conjugate of each number.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Complex Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a9cf449complex23a","stepAnswer":["2-i $$\\\\sqrt{5}$$"],"problemType":"TextBox","stepTitle":"$$2+i$$ $$\\\\sqrt{5}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"2-i $$\\\\sqrt{5}$$","hints":{"DefaultPathway":[{"id":"a9cf449complex23a-h1","type":"hint","dependencies":[],"title":"Conjugate","text":"If the number is in the form $$a+bi$$, then the conjugate is a-bi","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex23a-h2","type":"hint","dependencies":["a9cf449complex23a-h1"],"title":"Flip Sign","text":"So, just flip the sign if it is already in the form $$a+bi$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9cf449complex23b","stepAnswer":["$$\\\\frac{1}{2} i$$"],"problemType":"TextBox","stepTitle":"$$-\\\\left(\\\\frac{1}{2}\\\\right) i$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{2} i$$","hints":{"DefaultPathway":[{"id":"a9cf449complex23b-h1","type":"hint","dependencies":[],"title":"Conjugate","text":"If the number is in the form $$a+bi$$, then the conjugate is a-bi","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex23b-h2","type":"hint","dependencies":["a9cf449complex23b-h1"],"title":"What is a?","text":"Think of a to be $$0$$, so we get our expression as $$0-\\\\frac{1}{2} i$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex23b-h3","type":"hint","dependencies":["a9cf449complex23b-h2"],"title":"Flip Sign","text":"Now just flip the sign of the expression as it is already in the form $$a+bi$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9cf449complex23c","stepAnswer":["$$-3-4i$$"],"problemType":"TextBox","stepTitle":"$$-3+4i$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-3-4i$$","hints":{"DefaultPathway":[{"id":"a9cf449complex23c-h1","type":"hint","dependencies":[],"title":"Conjugate","text":"If the number is in the form $$a+bi$$, then the conjugate is a-bi","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex23c-h2","type":"hint","dependencies":["a9cf449complex23c-h1"],"title":"Flip Sign","text":"So, just flip the sign if as is already in the form $$a+bi$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9cf449complex24","title":"Dividing Complex Numbers","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Complex Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a9cf449complex24a","stepAnswer":["$$\\\\frac{3}{17}+\\\\frac{22}{17} i$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2+5i}{4-i}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{17}+\\\\frac{22}{17} i$$","hints":{"DefaultPathway":[{"id":"a9cf449complex24a-h1","type":"hint","dependencies":[],"title":"Conjugate Multiplication","text":"Multiply the top and the bottom by the complex conjugate of the denominator:((2+5i)(4+i))/((4-i)(4+i)).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex24a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$17$$"],"dependencies":["a9cf449complex24a-h1"],"title":"Denominator Result","text":"Here we should get $$\\\\left(a+bi\\\\right) \\\\left(a-bi\\\\right)$$ on the denominator which simplifies to $$a^2+b^2$$. What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex24a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3+22i$$"],"dependencies":["a9cf449complex24a-h1"],"title":"Numerator result","text":"What does the top simplify to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a9cf449complex24a-h3-s1","type":"hint","dependencies":[],"title":"Powers of i","text":"Remember: $$i^2$$ is $$-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9cf449complex24a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{17}+\\\\frac{22}{17} i$$"],"dependencies":["a9cf449complex24a-h2","a9cf449complex24a-h3"],"title":"Split","text":"Split the real and imaginary part. What do we get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9cf449complex25","title":"Simplifying Powers of i","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Complex Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a9cf449complex25a","stepAnswer":["-i"],"problemType":"TextBox","stepTitle":"Evaluate $$i^{35}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a9cf449complex25a-h1","type":"hint","dependencies":[],"title":"Factors","text":"Take out as many $$i^4$$ ths as you can because $$i^4$$ is just $$1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex25a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a9cf449complex25a-h1"],"title":"Divide","text":"How many times does $$4$$ go into 35?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex25a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$i^3$$"],"dependencies":["a9cf449complex25a-h2"],"title":"Simplify","text":"What does $$i^{35}$$ simplify to, taking accounr of the number of times $$4$$ goes into 35?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex25a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["-i"],"dependencies":["a9cf449complex25a-h3"],"title":"What does the simplified expression equal?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9cf449complex25b","stepAnswer":["$$-1$$"],"problemType":"TextBox","stepTitle":"Evaluate $$i^{18}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1$$","hints":{"DefaultPathway":[{"id":"a9cf449complex25b-h1","type":"hint","dependencies":[],"title":"Factors","text":"Take out as many $$i^4$$ ths as you can because $$i^4$$ is just $$1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex25b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a9cf449complex25b-h1"],"title":"Divide","text":"How many times does $$4$$ go into 18?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex25b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$i^2$$"],"dependencies":["a9cf449complex25b-h2"],"title":"Simplify","text":"What does $$i^{18}$$ simplify to, taking accounr of the number of times $$4$$ goes into 35?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex25b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["a9cf449complex25b-h3"],"title":"Simplify.","text":"What does the simplified expression equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9cf449complex26","title":"Evaluating Algebraic Equations with Complex Numbers","body":"For the following exercises, evaluate the algebraic expressions.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Complex Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a9cf449complex26a","stepAnswer":["$$-8+2i$$"],"problemType":"TextBox","stepTitle":"If $$y=x^2+x-4$$, evaluate $$y$$ given $$x=2i$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-8+2i$$","hints":{"DefaultPathway":[{"id":"a9cf449complex26a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute 2i wherever you see $$x$$, and we get the expression $${\\\\left(2i\\\\right)}^2+2i-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex26a-h2","type":"hint","dependencies":["a9cf449complex26a-h1"],"title":"Simplify","text":"We will then simplify $$y={\\\\left(2i\\\\right)}^2+2i-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex26a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["a9cf449complex26a-h2"],"title":"Squaring","text":"What is $${\\\\left(2i\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex26a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-8$$ + 2i"],"dependencies":["a9cf449complex26a-h3"],"title":"Combine like terms.","text":"Now combine the like terms (real and non-real). What expression do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9cf449complex27","title":"Evaluating Algebraic Equations with Complex Numbers","body":"For the following exercises, evaluate the algebraic expressions.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Complex Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a9cf449complex27a","stepAnswer":["$$14+7i$$"],"problemType":"TextBox","stepTitle":"If $$y=x^2+3x+5$$, evaluate $$y$$ given $$x=2+i$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$14+7i$$","hints":{"DefaultPathway":[{"id":"a9cf449complex27a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$2+i$$ wherever you see $$x$$, and we get the expression $$y={\\\\left(2+i\\\\right)}^2+3\\\\left(2+i\\\\right)+5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex27a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3+4i$$"],"dependencies":["a9cf449complex27a-h1"],"title":"Quadratic Term","text":"Evaluate the term $${\\\\left(2+i\\\\right)}^2$$. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a9cf449complex27a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3+4i$$"],"dependencies":[],"title":"Quadratic Term","text":"$${\\\\left(2+i\\\\right)}^2=\\\\left(2+i\\\\right) \\\\left(2+i\\\\right)=2^2+2i+2i+i^2$$. What does this evaluate to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9cf449complex27a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6+3i$$"],"dependencies":["a9cf449complex27a-h1"],"title":"X term","text":"Substitute $$2+i$$ for the $$3x$$ term. What is $$3\\\\left(2+i\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex27a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$14+7i$$"],"dependencies":["a9cf449complex27a-h2","a9cf449complex27a-h3"],"title":"Combine like terms.","text":"Combine the like terms (real and non real). What expression do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9cf449complex28","title":"Evaluating Algebraic Equations with Complex Numbers","body":"For the following exercises, evaluate the algebraic expressions.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Complex Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a9cf449complex28a","stepAnswer":["$$\\\\frac{-23}{29}+\\\\frac{15}{29} i$$"],"problemType":"TextBox","stepTitle":"If $$y=\\\\frac{x+1}{2-x}$$, evaluate $$y$$ given $$x=5i$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-23}{29}+\\\\frac{15}{29} i$$","hints":{"DefaultPathway":[{"id":"a9cf449complex28a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute 5i wherever you see $$x$$, and we get the expression $$y=\\\\frac{5i+1}{2-5i}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex28a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2+5i$$"],"dependencies":["a9cf449complex28a-h1"],"title":"Complex Conjugate of Denominator","text":"To start off, we will need to muliply both the numerator and the denominator by the complex conjugate of the denominator. What is the complex conjugate of the denominator $$2-5i$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex28a-h3","type":"hint","dependencies":["a9cf449complex28a-h2"],"title":"Multiply","text":"Multiply the numerator and denominator by the conjugate of the denominator: $$\\\\frac{\\\\left(5i+1\\\\right) \\\\left(2+5i\\\\right)}{\\\\left(2-5i\\\\right) \\\\left(2+5i\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex28a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$29$$"],"dependencies":["a9cf449complex28a-h3"],"title":"Denominator Result","text":"The $$\\\\left(a+bi\\\\right) \\\\left(a-bi\\\\right)$$ on the denominator simplifies to $$a^2+b^2$$. What is the value of the donominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex28a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-23+15i$$"],"dependencies":["a9cf449complex28a-h3"],"title":"Numerator result","text":"What does the top evaluate to? In other words, what is $$\\\\left(5i+1\\\\right) \\\\left(2+5i\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a9cf449complex28a-h5-s1","type":"hint","dependencies":[],"title":"Evaluating $$i^2$$","text":"Remember: $$i^2=-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex28a-h5-s2","type":"hint","dependencies":["a9cf449complex28a-h5-s1"],"title":"Numerator result","text":"When FOIL out, $$\\\\left(5i+1\\\\right) \\\\left(2+5i\\\\right)=10i+25i^2+2+5i$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9cf449complex28a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-23}{29}+\\\\frac{15}{29} i$$"],"dependencies":["a9cf449complex28a-h4","a9cf449complex28a-h5"],"title":"Split","text":"Split the real and imaginary parts. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9cf449complex3","title":"Adding and Subtracting Complex Numbers","body":"Simplify the expressions.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Complex Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a9cf449complex3a","stepAnswer":["$$5+i$$"],"problemType":"MultipleChoice","stepTitle":"$$3-4i+2+5i$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$5+i$$","choices":["$$4+i$$","$$6$$","$$5+i$$","i"],"hints":{"DefaultPathway":[{"id":"a9cf449complex3a-h1","type":"hint","dependencies":[],"title":"Adding Complex Numbers","text":"The first step to add complex numbers is to add real and imaginary parts separately.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex3a-h2","type":"hint","dependencies":["a9cf449complex3a-h1"],"title":"i Definition","text":"i represents the square root of $$-1$$, which is an imaginary number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex3a-h3","type":"hint","dependencies":["a9cf449complex3a-h2"],"title":"Real Parts in the Expression","text":"$$3$$ and $$2$$ are the real parts.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex3a-h4","type":"hint","dependencies":["a9cf449complex3a-h3"],"title":"Imaginary Parts in the Expression","text":"-4i and 5i are the imaginary parts.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a9cf449complex3a-h4"],"title":"Sum of the Real Parts","text":"What is the sum of the real parts?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex3a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["i"],"dependencies":["a9cf449complex3a-h5"],"title":"Sum of the Imaginary Parts","text":"What is the sum of the imaginary parts?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["9i","i","$$-i$$","$$1$$"]},{"id":"a9cf449complex3a-h7","type":"hint","dependencies":["a9cf449complex3a-h6"],"title":"Writing the Final Answer","text":"Next, write the expression as the sum of the real and imaginary numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9cf449complex3b","stepAnswer":["$$6+5i$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\left(-5+7i\\\\right)-\\\\left(-11+2i\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$6+5i$$","choices":["$$2+5i$$","$$4+3i$$","5i","$$6+5i$$"],"hints":{"DefaultPathway":[{"id":"a9cf449complex3b-h1","type":"hint","dependencies":[],"title":"Rewriting the Expression by Distributing the Negative Sign","text":"First, distribute the negative sign and rewrite the expression as an addition problem. For example, $$a+b-c+d=a+b-c-d$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex3b-h2","type":"hint","dependencies":["a9cf449complex3b-h1"],"title":"Adding Complex Numbers","text":"The next step to add complex numbers is to add real and imaginary parts separately.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex3b-h3","type":"hint","dependencies":["a9cf449complex3b-h2"],"title":"i Definition","text":"i represents the square root of $$-1$$, which is an imaginary number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex3b-h4","type":"hint","dependencies":["a9cf449complex3b-h3"],"title":"Real Parts in the Expression","text":"$$-5$$ and $$11$$ are the real parts.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex3b-h5","type":"hint","dependencies":["a9cf449complex3b-h4"],"title":"Imaginary Parts in the Expression","text":"7i and -2i are the imaginary parts.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex3b-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a9cf449complex3b-h5"],"title":"Sum of the Real Parts","text":"What is the sum of the real parts?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex3b-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["5i"],"dependencies":["a9cf449complex3b-h6"],"title":"Sum of the Imaginary Parts","text":"What is the sum of the imaginary parts?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex3b-h8","type":"hint","dependencies":["a9cf449complex3b-h7"],"title":"Writing the Final Answer","text":"Finally, write the expression as the sum of sums of the real and imaginary numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9cf449complex4","title":"Adding and Subtracting Complex Numbers","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Complex Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a9cf449complex4a","stepAnswer":["$$1-9i$$"],"problemType":"TextBox","stepTitle":"$$3-4i-2+5i$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1-9i$$","hints":{"DefaultPathway":[{"id":"a9cf449complex4a-h1","type":"hint","dependencies":[],"title":"Writing the Problem As an Expression","text":"First, write the problem as an expression: $$3-4i-2+5i$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex4a-h2","type":"hint","dependencies":["a9cf449complex4a-h1"],"title":"Rewriting the Expression by Distributing the Negative Sign","text":"Second, distribute the negative sign and rewrite the expression as an addition problem. For example, $$a+b-c+d=a+b-c-d$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex4a-h3","type":"hint","dependencies":["a9cf449complex4a-h2"],"title":"Adding Complex Numbers","text":"The next step to add complex numbers is to add real and imaginary parts separately.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex4a-h4","type":"hint","dependencies":["a9cf449complex4a-h3"],"title":"i Definition","text":"i represents the square root of $$-1$$, which is an imaginary number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex4a-h5","type":"hint","dependencies":["a9cf449complex4a-h3"],"title":"Real Numbers in the Expression","text":"$$3$$ and $$-2$$ are real numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex4a-h6","type":"hint","dependencies":["a9cf449complex4a-h3"],"title":"Imaginary Numbers in the Expression","text":"-4i and -5i are imaginary numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex4a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["a9cf449complex4a-h5"],"title":"Sum of the Real Numbers","text":"What is the sum of the real numbers?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex4a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["-9i"],"dependencies":["a9cf449complex4a-h6"],"title":"Sum of the Imaginary Numbers","text":"What is the sum of the imaginary numbers?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex4a-h9","type":"hint","dependencies":["a9cf449complex4a-h7","a9cf449complex4a-h8"],"title":"Writing the Final Answer","text":"Finally, write the expression as the sum of the real and imaginary numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9cf449complex5","title":"Multiplying a Complex Number by a Real Number","body":"Find the product.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Complex Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a9cf449complex5a","stepAnswer":["$$8+20i$$"],"problemType":"MultipleChoice","stepTitle":"$$4\\\\left(2+5i\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$8+20i$$","choices":["$$8+20i$$","$$12+4i$$","$$3+16i$$","$$8+10i$$"],"hints":{"DefaultPathway":[{"id":"a9cf449complex5a-h1","type":"hint","dependencies":[],"title":"Using the Distributive Property","text":"The first step is to use the Distributive Property, where $$a\\\\left(b+c\\\\right)$$ $$=$$ $$a b+b c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex5a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$4\\\\times2+4\\\\times5 i$$"],"dependencies":["a9cf449complex5a-h1"],"title":"Rewriting the Expression With the Distributive Property","text":"What does the expression look like after it is rewritten using the Distributive Property?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2+5i$$","$$4\\\\times2+4\\\\times5 i$$","$$4\\\\times2+5i$$","4i"]},{"id":"a9cf449complex5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a9cf449complex5a-h2"],"title":"Product of First Term","text":"What is the product of $$4\\\\times2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex5a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["20i"],"dependencies":["a9cf449complex5a-h2"],"title":"Product of Second Term","text":"What is the product of $$4\\\\times5 i$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["20i","i","10i","4i"]}]}}]},{"id":"a9cf449complex6","title":"Multiplying a Complex Number by a Real Number","body":"Find the product.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Complex Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a9cf449complex6a","stepAnswer":["$$60-24i$$"],"problemType":"MultipleChoice","stepTitle":"$$12(5-2i)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$60-24i$$","choices":["$$60$$","$$-24i$$","$$60-24i$$","$$50-24i$$"],"hints":{"DefaultPathway":[{"id":"a9cf449complex6a-h1","type":"hint","dependencies":[],"title":"Using the Distributive Property","text":"The first step is to use the Distributive Property, where $$a\\\\left(b+c\\\\right)$$ $$=$$ $$a b+b c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex6a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$12\\\\times5-12\\\\times2 i$$"],"dependencies":["a9cf449complex6a-h1"],"title":"Rewriting the Expression With the Distributive Property","text":"What does the expression look like after it is rewritten using the Distributive Property?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$5-2i$$","$$12\\\\times5-12\\\\times2 i$$","12i","5i"]},{"id":"a9cf449complex6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$60$$"],"dependencies":["a9cf449complex6a-h2"],"title":"Product of First Term","text":"What is the product of $$12\\\\times5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex6a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["24i"],"dependencies":["a9cf449complex6a-h2"],"title":"Product of Second Term","text":"What is the product of $$12\\\\times2 i$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["24i","12i","2i","16i"]}]}}]},{"id":"a9cf449complex7","title":"Multiplying a Complex Number by a Complex Number","body":"Find the product.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Complex Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a9cf449complex7a","stepAnswer":["$$23-14i$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\left(4+3i\\\\right) \\\\left(2-5i\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$23-14i$$","choices":["$$8+15i$$","$$8-15i$$","$$23-14i$$","$$41-7i$$"],"hints":{"DefaultPathway":[{"id":"a9cf449complex7a-h1","type":"hint","dependencies":[],"title":"Rewriting the Expression Using the Distributive Property: Part $$1$$","text":"Using the Distributive property, an expression with the form $$\\\\left(a+b\\\\right) \\\\left(c+d\\\\right)$$ can be rewritten as $$a\\\\left(c+d\\\\right)+b\\\\left(c+d\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex7a-h2","type":"hint","dependencies":["a9cf449complex7a-h1"],"title":"Rewriting the Expression Using the Distributive Property: Part $$2$$","text":"Again using the Distributive Property, $$a\\\\left(c+d\\\\right)+b\\\\left(c+d\\\\right)$$ can be simplified into $$a c+a d+b c+b d$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex7a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$8-20i+6i-15i^2$$"],"dependencies":["a9cf449complex7a-h2"],"title":"Rewritten Version of the Expression","text":"What does the expression look like after it has been rewritten using the Distributive Property?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$8-20i+6i-15i^2$$","$$8-20i+6i-15i$$","$$2+15i^2$$","$$5-20i+3i-15i^2$$"]},{"id":"a9cf449complex7a-h4","type":"hint","dependencies":["a9cf449complex7a-h3"],"title":"Value of $$i^2$$","text":"$$i^2$$ is the square of the square root of $$-1$$, so $$i^2=-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex7a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["a9cf449complex7a-h3"],"title":"Value of $$-15i^2$$","text":"What is the value of $$-15i^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex7a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$23$$"],"dependencies":["a9cf449complex7a-h3","a9cf449complex7a-h5"],"title":"Sum of Real Numbers","text":"What is the sum of the real numbers of the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a9cf449complex7a-h6-s1","type":"hint","dependencies":[],"title":"Real Numbers in the Expression","text":"$$8$$ and $$15$$ are real numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9cf449complex7a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-14i$$"],"dependencies":["a9cf449complex7a-h3"],"title":"Sum of Imaginary Numbers","text":"What is the sum of the imaginary numbers of the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$-14i$$","26i","14i","$$-26i$$"],"subHints":[{"id":"a9cf449complex7a-h7-s1","type":"hint","dependencies":[],"title":"Imaginary Numbers in the Expression","text":"-20i and 6i are imaginary numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9cf449complex7a-h8","type":"hint","dependencies":[],"title":"Final Step","text":"The final step is to add the sums of the real and imaginary numbers together for the final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9cf449complex8","title":"Multiplying a Complex Number by a Complex Number","body":"Find the product.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Complex Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a9cf449complex8a","stepAnswer":["$$18+i$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\left(3-4i\\\\right) \\\\left(2+3i\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$18+i$$","choices":["$$18+i$$","$$6-12i$$","$$18-12i$$","$$6-24i$$"],"hints":{"DefaultPathway":[{"id":"a9cf449complex8a-h1","type":"hint","dependencies":[],"title":"Rewriting the Expression Using the Distributive Property: Part $$1$$","text":"Using the Distributive property, an expression with the form $$\\\\left(a+b\\\\right) \\\\left(c+d\\\\right)$$ can be rewritten as $$a\\\\left(c+d\\\\right)+b\\\\left(c+d\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex8a-h2","type":"hint","dependencies":["a9cf449complex8a-h1"],"title":"Rewriting the Expression Using the Distributive Property: Part $$2$$","text":"Again using the Distributive Property, $$a\\\\left(c+d\\\\right)+b\\\\left(c+d\\\\right)$$ can be simplified into $$a c+a d+b c+b d$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex8a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$6+9i-8i-12i^2$$"],"dependencies":["a9cf449complex8a-h2"],"title":"Rewritten Version of the Expression","text":"What does the expression look like after it has been rewritten using the Distributive Property?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$6+9i-8i-12i^2$$","$$6+9i-8i-12i\\\\times2$$","$$2+9i-8i-12i\\\\times2$$","$$4-20i+3i-10i^2$$"]},{"id":"a9cf449complex8a-h4","type":"hint","dependencies":["a9cf449complex8a-h3"],"title":"Value of $$i^2$$","text":"$$i^2$$ is the square of the square root of $$-1$$, so $$i^2=-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex8a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a9cf449complex8a-h4"],"title":"Value of $$-12i^2$$","text":"What is the value of $$-12i^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex8a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18$$"],"dependencies":["a9cf449complex8a-h3","a9cf449complex8a-h5"],"title":"Sum of Real Numbers","text":"What is the sum of the real numbers of the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a9cf449complex8a-h6-s1","type":"hint","dependencies":[],"title":"Real Numbers in the Expression","text":"$$6$$ and $$12$$ are real numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9cf449complex8a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["i"],"dependencies":["a9cf449complex8a-h3"],"title":"Sum of Imaginary Numbers","text":"What is the sum of the imaginary numbers of the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$-17i$$","17i","i","$$-2i$$"],"subHints":[{"id":"a9cf449complex8a-h7-s1","type":"hint","dependencies":[],"title":"Imaginary Numbers in the Expression","text":"9i and -8i are imaginary numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"a9cf449complex8a-h8","type":"hint","dependencies":["a9cf449complex8a-h6","a9cf449complex8a-h7"],"title":"Final Step","text":"The final step is add the sums of the real and imaginary numbers together for the answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9cf449complex9","title":"Adding and Subtracting Complex Numbers","body":"For the following exercises, perform the indicated operation and express the result as a simplified complex number.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Complex Numbers","courseName":"OpenStax: College Algebra","steps":[{"id":"a9cf449complex9a","stepAnswer":["8-i"],"problemType":"TextBox","stepTitle":"$$3+2i+5-3i$$","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"a9cf449complex9a-h1","type":"hint","dependencies":[],"title":"Associative Property","text":"The first step is to group the like terms. We can use the Associative Property to rewrite this expression as $$3+5+2i-3i$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex9a-h2","type":"hint","dependencies":["a9cf449complex9a-h1"],"title":"Combining Like Terms","text":"Now, we can add numbers in the parentheses to combine like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["a9cf449complex9a-h2"],"title":"Combining Like Terms","text":"What is $$3+5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["-i"],"dependencies":["a9cf449complex9a-h2"],"title":"Combining Like Terms","text":"What is 2i-3i?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9cf449complex9a-h5","type":"hint","dependencies":["a9cf449complex9a-h3","a9cf449complex9a-h4"],"title":"Rewrite Answer","text":"Finally, we can write the expression in $$a+bi$$ form: 8-i.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9d3659treeandvenn16","title":"Balls Without Replacement","body":"Suppose there are four red (R) balls and three yellow (Y) balls in a box. Two balls are drawn from the box without replacement.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Tree and Venn Diagrams","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a9d3659treeandvenn16a","stepAnswer":["$$\\\\frac{24}{42}$$"],"problemType":"TextBox","stepTitle":"What is the probability that one ball of each coloring is selected?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{24}{42}$$","hints":{"DefaultPathway":[{"id":"a9d3659treeandvenn16a-h1","type":"hint","dependencies":[],"title":"How to Use Tree Diagrams","text":"When looking at a tree diagram, it consists of \\"branches\\", sometimes labeled with frequencies (whole numbers) or probabilities (fractions). Each branch of the tree represents a \\"choice\\" of sorts and each level represents the completion of a choice, and at the bottom of the tree--the leaf--houses the final outcomes. To get to the final outcome, you want to multiply all the probabilities going down the tree from the top to the end of the leaf.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3659treeandvenn16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{4}{7}$$"],"dependencies":["a9d3659treeandvenn16a-h1"],"title":"Using the Tree Diagram","text":"We want to calculate the probability of choosing one ball of each color, meaning P(RY OR YR). Let\'s start with P(RY). This mean we need to pick a red ball first. What is P(R)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3659treeandvenn16a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{6}$$"],"dependencies":["a9d3659treeandvenn16a-h2"],"title":"Using the Tree Diagram","text":"We want to calculate P(RY) first. That means the second ball is yellow. What is P(Y|R), remembering this is WITHOUT replacement?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3659treeandvenn16a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{12}{42}$$"],"dependencies":["a9d3659treeandvenn16a-h3"],"title":"Calculating Probability","text":"Now, multiply the two probabilities, $$P\\\\left(R\\\\right) P\\\\left(Y|R\\\\right)=P(RY)$$. What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3659treeandvenn16a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{7}$$"],"dependencies":["a9d3659treeandvenn16a-h4"],"title":"Using the Tree Diagram","text":"We want to calculate P(YR) second. That means the first ball is Y. What is P(Y)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3659treeandvenn16a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{4}{6}$$"],"dependencies":["a9d3659treeandvenn16a-h5"],"title":"Using the Tree Diagram","text":"We want to calculate P(YR) second. That means the second ball is red. What is P(R|Y), remembering this is WITHOUT replacement?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3659treeandvenn16a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{12}{42}$$"],"dependencies":["a9d3659treeandvenn16a-h6"],"title":"Calculating Probability","text":"Now, multiply the two probabilities, $$P\\\\left(Y\\\\right) P\\\\left(R|Y\\\\right)=P(YR)$$. What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3659treeandvenn16a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{24}{42}$$"],"dependencies":[],"title":"Summing the Probabilities","text":"Since we want to find P(RY OR YR), we use the operator, which means we add up the two probabilities. What is P(RY OR $$YR)=P\\\\left(RY\\\\right)+P\\\\left(YR\\\\right)=P(one$$ ball of each color is chosen)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9d3659treeandvenn21","title":"Probability in Real Life","body":"In a bookstore, the probability that the customer buys a novel is $$0.6$$, and the probability that the customer buys a non-fiction book is $$0.4$$. Suppose that the probability that the customer buys both is $$0.2$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Tree and Venn Diagrams","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a9d3659treeandvenn21a","stepAnswer":["$$0.8$$"],"problemType":"TextBox","stepTitle":"Find the probability that the customer buys either a novel or a non-fiction book.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.8$$","hints":{"DefaultPathway":[{"id":"a9d3659treeandvenn21a-h1","type":"hint","dependencies":[],"title":"Formula for OR","text":"Remember, the formula is P(A OR B)=P(A)+P(B)-P(A OR B). Let N $$=$$ the customer buys a novel and B $$=$$ the customer buys a non-fiction book. Substitute A for N and B for B to find the answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3659treeandvenn21a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.6$$"],"dependencies":["a9d3659treeandvenn21a-h1"],"title":"Determine P(N)","text":"What is P(N)? Hint: the answer is in the problem statement.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3659treeandvenn21a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.4$$"],"dependencies":["a9d3659treeandvenn21a-h2"],"title":"Determine P(B)","text":"What is P(B)? Hint: the answer is in the problem statement.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3659treeandvenn21a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.2$$"],"dependencies":["a9d3659treeandvenn21a-h3"],"title":"Determine P(N AND B)","text":"What is P(N AND B)? Hint: the answer is in the problem statement.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3659treeandvenn21a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.8$$"],"dependencies":["a9d3659treeandvenn21a-h4"],"title":"Determining P(N OR B)","text":"What is P(N)+P(B)-P(N AND B)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"a9d3659treeandvenn21a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.8$$"],"dependencies":[],"title":"Determining P(N OR B)","text":"What is $$0.6+0.4-0.2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a9d3659treeandvenn25","title":"Real World Probabilities","body":"A box of cookies contains three chocolate and seven butter cookies. Miguel randomly selects a cookie and eats it. Then he randomly selects another cookie and eats it.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Tree and Venn Diagrams","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a9d3659treeandvenn25a","stepAnswer":["$$\\\\frac{48}{90}$$"],"problemType":"TextBox","stepTitle":"Let S be the event that both cookies selected were the same flavor. Find P(S).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{48}{90}$$","hints":{"DefaultPathway":[{"id":"a9d3659treeandvenn25a-h1","type":"hint","dependencies":[],"title":"The OR Operator","text":"We want to know the probability of selecting two cookies of the same flavor. This means if we let C $$=$$ choosing a chocolate cookie and B $$=$$ choosing a butter cookie, we want to calculate P(CC OR BB).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3659treeandvenn25a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{10}$$"],"dependencies":["a9d3659treeandvenn25a-h1"],"title":"Determining P(CC)","text":"Let\'s start with P(CC). This mean we need to pick a chocolate cookie first. What is P(C)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3659treeandvenn25a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{9}$$"],"dependencies":["a9d3659treeandvenn25a-h2"],"title":"Determining P(CC)","text":"We want to calculate P(CC) first. That means the second cookie is chocolate. What\'s the probability that the second cookie is chocolate, given the first one was also chocolate and eaten (without replacement!)? In effect, what is P(C|C)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3659treeandvenn25a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{6}{90}$$"],"dependencies":["a9d3659treeandvenn25a-h3"],"title":"Calculating Probability","text":"Now, multiply the two probabilities, $$P\\\\left(C\\\\right) P\\\\left(C|C\\\\right)=P(CC)$$. What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3659treeandvenn25a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{7}{10}$$"],"dependencies":["a9d3659treeandvenn25a-h4"],"title":"Determining P(BB)","text":"We want to calculate P(BB) second. That means the first cookie is B. What is P(B)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3659treeandvenn25a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{6}{9}$$"],"dependencies":["a9d3659treeandvenn25a-h5"],"title":"Determining P(BB)","text":"We want to calculate P(BB) second. That means the second cookie is B. What\'s the probability that the second cookie is butter? In effect, what is P(B|B)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3659treeandvenn25a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{42}{90}$$"],"dependencies":["a9d3659treeandvenn25a-h6"],"title":"Calculating Probability","text":"Now, multiply the two probabilities, $$P\\\\left(B\\\\right) P\\\\left(B|B\\\\right)=P(BB)$$. What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3659treeandvenn25a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{48}{90}$$"],"dependencies":["a9d3659treeandvenn25a-h7"],"title":"Summing the Probabilities","text":"Since we want to find P(CC OR BB), we use the operator, which means we add up the two probabilities. What is P(CC OR $$BB)=P\\\\left(CC\\\\right)+P\\\\left(BB\\\\right)=P(two$$ cookies of the same type)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9d3aa3stat1","title":"Quantitative versus Qualitative","body":"Determine whether each of the following is quantitative data or qualitative. Indicate whether quantitative data are continuous or discrete. Hint: Data that are discrete often start with the words \\"the number of.\\"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Data, Sampling, and Variation in Data and Sampling","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a9d3aa3stat1a","stepAnswer":["Quantitative Continuous"],"problemType":"MultipleChoice","stepTitle":"The data are the areas of lawns in square feet. You sample five houses. The areas of the lawns are $$144$$ sq. feet, $$160$$ sq. feet, $$190$$ sq. feet, $$180$$ sq. feet, and $$210$$ sq. feet. What type of data is this?","stepBody":"","answerType":"string","variabilization":{},"choices":["Quantitative Discrete","Qualitative","Quantitative Continuous"],"hints":{"DefaultPathway":[{"id":"a9d3aa3stat1a-h1","type":"hint","dependencies":[],"title":"Definitions of Qualitative and Quantitative Data","text":"Qualitative data usually involves categorizing (working with categorical data) and therefore usually isn\'t numeric in context. Quantitative data, in contrast, always words with numbers whether that be counting or measuring.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3aa3stat1a-h2","type":"hint","dependencies":["a9d3aa3stat1a-h1"],"title":"Quantitative or Qualitative?","text":"We want to measure the area of the lawns. We can determine if this is qualitative or quantitative since qualitative data usually uses data while quantitative involves counting or measuring numerical values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3aa3stat1a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Quantitative"],"dependencies":["a9d3aa3stat1a-h2"],"title":"Quantitative or Qualitative?","text":"Is area of the lawns qualitative or quantitative data?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Quantitative","Qualitative"]},{"id":"a9d3aa3stat1a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Continuous"],"dependencies":["a9d3aa3stat1a-h3"],"title":"Continuous or Discrete?","text":"Is area of the lawns quantitative discrete or quantitative continuous data? Are we allowed to have $$2.6$$ sq. feet of lawns? If so, it is continuous quantitative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Discrete","Continuous"]}]}}]},{"id":"a9d3aa3stat10","title":"Identifying Qualitative and Quantitative Data","body":"A study was done to determine the age, number of times per week, and the duration (amount of time) of resident use of a local park in San Jose. The first house in the neighborhood around the park was selected randomly and then every 8th house in the neighborhood around the park was interviewed.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Data, Sampling, and Variation in Data and Sampling","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a9d3aa3stat10a","stepAnswer":["Quantitative Discrete"],"problemType":"MultipleChoice","stepTitle":"What type of data is \\"Number of times per week\\" ?","stepBody":"","answerType":"string","variabilization":{},"choices":["Qualitative","Quantitative Discrete","Quantitative Continuous"],"hints":{"DefaultPathway":[{"id":"a9d3aa3stat10a-h1","type":"hint","dependencies":[],"title":"Definitions of Qualitative and Quantitative Data","text":"Qualitative data usually involves categorizing (working with categorical data) and therefore usually isn\'t numeric in context. Quantitative data, in contrast, always words with numbers whether that be counting or measuring.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3aa3stat10a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Quantitative"],"dependencies":["a9d3aa3stat10a-h1"],"title":"Quantitative or Qualitative?","text":"Is number of times per week considered Quantitative or Qualitative?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Qualitative","Quantitative"]},{"id":"a9d3aa3stat10a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Discrete"],"dependencies":["a9d3aa3stat10a-h2"],"title":"Continuous or Discrete?","text":"Discrete data we know usually deals with integer or \\"discrete\\" values (in the sense that, for instance, you can\'t have half a person). Counting people would be an example of discrete data. On the other hand, continuous data can be represented as fractions or decimals. For instance, we can have $$1.2$$ miles $$(0.2$$ miles is valid, even though it\'s not a full mile). Is the number of times per week therefore discrete or continuous?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Discrete","Continuous"]}]}},{"id":"a9d3aa3stat10b","stepAnswer":["Quantitative Continuous"],"problemType":"MultipleChoice","stepTitle":"What type of data is \\"Duration (amount of time)\\"?","stepBody":"","answerType":"string","variabilization":{},"choices":["Qualitative","Quantitative Discrete","Quantitative Continuous"],"hints":{"DefaultPathway":[{"id":"a9d3aa3stat10b-h1","type":"hint","dependencies":[],"title":"Definitions of Qualitative and Quantitative Data","text":"Qualitative data usually involves categorizing (working with categorical data) and therefore usually isn\'t numeric in context. Quantitative data, in contrast, always words with numbers whether that be counting or measuring.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3aa3stat10b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Quantitative"],"dependencies":["a9d3aa3stat10b-h1"],"title":"Quantitative or Qualitative?","text":"We want to measure the duration (amount of time). Is this qualitative (uses categories) or quantitative (counting, measuring numerical values)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Quantitative","Qualitative"]},{"id":"a9d3aa3stat10b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Continuous"],"dependencies":["a9d3aa3stat10b-h2"],"title":"Continuous or Discrete?","text":"Discrete data we know usually deals with integer or \\"discrete\\" values (in the sense that, for instance, you can\'t have half a person). Counting people would be an example of discrete data. On the other hand, continuous data can be represented as fractions or decimals. For instance, we can have $$1.2$$ miles $$(0.2$$ miles is valid, even though it\'s not a full mile). Is the duration (amount of time) therefore discrete or continuous? Similarly, can you split up time into $$0.5$$ minutes or something of that nature?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Discrete","Continuous"]}]}}]},{"id":"a9d3aa3stat11","title":"Quantitative versus Qualitative","body":"For the following exercises, identify the type of data that would be used to describe a response (quantitative discrete, quantitative continuous, or qualitative).","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Data, Sampling, and Variation in Data and Sampling","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a9d3aa3stat11a","stepAnswer":["Quantitative Discrete"],"problemType":"MultipleChoice","stepTitle":"The number of tickets sold to a concert.","stepBody":"","answerType":"string","variabilization":{},"choices":["Qualitative","Quantitative Discrete","Quantitative Continuous"],"hints":{"DefaultPathway":[{"id":"a9d3aa3stat11a-h1","type":"hint","dependencies":[],"title":"Definitions of Qualitative and Quantitative Data","text":"Qualitative data usually involves categorizing (working with categorical data) and therefore usually isn\'t numeric in context. Quantitative data, in contrast, always words with numbers whether that be counting or measuring.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3aa3stat11a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Quantitative"],"dependencies":["a9d3aa3stat11a-h1"],"title":"Quantitative or Qualitative?","text":"We can count the number of tickets. Is the number of tickets qualitative or quantitative?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Qualitative","Quantitative"]},{"id":"a9d3aa3stat11a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Discrete"],"dependencies":["a9d3aa3stat11a-h2"],"title":"Continuous or Discrete","text":"To decide whether it is discrete or continuous, we must decide if we can have $$\\\\frac{fractions}{decimals}$$ of ticket. Are we allowed to have $$0.5$$ ticket? Is number of tickets continuous or discrete?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Discrete","Continuous"]}]}}]},{"id":"a9d3aa3stat12","title":"Quantitative versus Qualitative","body":"For the following exercises, identify the type of data that would be used to describe a response (quantitative discrete, quantitative continuous, or qualitative).","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Data, Sampling, and Variation in Data and Sampling","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a9d3aa3stat12a","stepAnswer":["Quantitative Discrete"],"problemType":"MultipleChoice","stepTitle":"The number of students enrolled at Evergreen Valley College.","stepBody":"","answerType":"string","variabilization":{},"choices":["Qualitative","Quantitative Discrete","Quantitative Continuous"],"hints":{"DefaultPathway":[{"id":"a9d3aa3stat12a-h1","type":"hint","dependencies":[],"title":"Definitions of Qualitative and Quantitative Data","text":"Qualitative data usually involves categorizing (working with categorical data) and therefore usually isn\'t numeric in context. Quantitative data, in contrast, always words with numbers whether that be counting or measuring.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3aa3stat12a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Quantitative"],"dependencies":["a9d3aa3stat12a-h1"],"title":"Quantitative or Qualitative?","text":"We can count the number of students. Is the number of students qualitative or quantitative?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Qualitative","Quantitative"]},{"id":"a9d3aa3stat12a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Discrete"],"dependencies":["a9d3aa3stat12a-h2"],"title":"Continuous or Discrete","text":"To decide whether it is discrete or continuous, we must decide if we can have $$\\\\frac{fractions}{decimals}$$ of students. Are we allowed to have $$0.5$$ student? Is number of students continuous or discrete?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Discrete","Continuous"]}]}}]},{"id":"a9d3aa3stat13","title":"Quantitative versus Qualitative","body":"For the following exercises, identify the type of data that would be used to describe a response (quantitative discrete, quantitative continuous, or qualitative).","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Data, Sampling, and Variation in Data and Sampling","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a9d3aa3stat13a","stepAnswer":["Quantitative Discrete"],"problemType":"MultipleChoice","stepTitle":"The number of competing computer spreadsheet software packages.","stepBody":"","answerType":"string","variabilization":{},"choices":["Qualitative","Quantitative Discrete","Quantitative Continuous"],"hints":{"DefaultPathway":[{"id":"a9d3aa3stat13a-h1","type":"hint","dependencies":[],"title":"Definitions of Qualitative and Quantitative Data","text":"Qualitative data usually involves categorizing (working with categorical data) and therefore usually isn\'t numeric in context. Quantitative data, in contrast, always words with numbers whether that be counting or measuring.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3aa3stat13a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Quantitative"],"dependencies":["a9d3aa3stat13a-h1"],"title":"Quantitative or Qualitative?","text":"We can count the number of software packages. Is the number of packages qualitative or quantitative?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Qualitative","Quantitative"]},{"id":"a9d3aa3stat13a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Discrete"],"dependencies":["a9d3aa3stat13a-h2"],"title":"Continuous or Discrete","text":"To decide whether it is discrete or continuous, we must decide if we can have $$\\\\frac{fractions}{decimals}$$ of packages. Are we allowed to have $$0.5$$ software package? Is number of software packages continuous or discrete?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Discrete","Continuous"]}]}}]},{"id":"a9d3aa3stat14","title":"Quantitative versus Qualitative","body":"For the following exercises, identify the type of data that would be used to describe a response (quantitative discrete, quantitative continuous, or qualitative).","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Data, Sampling, and Variation in Data and Sampling","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a9d3aa3stat14a","stepAnswer":["Qualitative"],"problemType":"MultipleChoice","stepTitle":"The brand of toothpaste.","stepBody":"","answerType":"string","variabilization":{},"choices":["Qualitative","Quantitative Discrete","Quantitative Continuous"],"hints":{"DefaultPathway":[{"id":"a9d3aa3stat14a-h1","type":"hint","dependencies":[],"title":"Definitions of Qualitative and Quantitative Data","text":"Qualitative data usually involves categorizing (working with categorical data) and therefore usually isn\'t numeric in context. Quantitative data, in contrast, always words with numbers whether that be counting or measuring.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3aa3stat14a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Qualitative"],"dependencies":["a9d3aa3stat14a-h1"],"title":"Quantitative or Qualitative?","text":"We can only name the brand of toothpaste and we can\'t count it. Is the brand of toothpaste qualitative or quantitative?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Qualitative","Quantitative"]}]}}]},{"id":"a9d3aa3stat15","title":"Quantitative versus Qualitative","body":"For the following exercises, identify the type of data that would be used to describe a response (quantitative discrete, quantitative continuous, or qualitative).","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Data, Sampling, and Variation in Data and Sampling","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a9d3aa3stat15a","stepAnswer":["Qualitative"],"problemType":"MultipleChoice","stepTitle":"Favorite baseball team.","stepBody":"","answerType":"string","variabilization":{},"choices":["Qualitative","Quantitative Discrete","Quantitative Continuous"],"hints":{"DefaultPathway":[{"id":"a9d3aa3stat15a-h1","type":"hint","dependencies":[],"title":"Definitions of Qualitative and Quantitative Data","text":"Qualitative data usually involves categorizing (working with categorical data) and therefore usually isn\'t numeric in context. Quantitative data, in contrast, always words with numbers whether that be counting or measuring.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3aa3stat15a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Qualitative"],"dependencies":["a9d3aa3stat15a-h1"],"title":"Quantitative or Qualitative?","text":"We can only name the favorite baseball team and we can\'t count it. Is the favorite baseball team qualitative or quantitative?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Qualitative","Quantitative"]}]}}]},{"id":"a9d3aa3stat2","title":"Quantitative versus Qualitative","body":"Determine whether each of the following is quantitative data or qualitative. Indicate whether quantitative data are continuous or discrete. Hint: Data that are discrete often start with the words \\"the number of.\\"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Data, Sampling, and Variation in Data and Sampling","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a9d3aa3stat2a","stepAnswer":["Quantitative Discrete"],"problemType":"MultipleChoice","stepTitle":"The number of pairs of shoes you own","stepBody":"","answerType":"string","variabilization":{},"choices":["Quantitative Discrete","Qualitative","Quantitative Continuous"],"hints":{"DefaultPathway":[{"id":"a9d3aa3stat2a-h1","type":"hint","dependencies":[],"title":"Definitions of Qualitative and Quantitative Data","text":"Qualitative data usually involves categorizing (working with categorical data) and therefore usually isn\'t numeric in context. Quantitative data, in contrast, always words with numbers whether that be counting or measuring.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3aa3stat2a-h2","type":"hint","dependencies":["a9d3aa3stat2a-h1"],"title":"Quantitative or Qualitative?","text":"We want to count the number of pairs of shoes you own. We can determine if this is qualitative or quantitative since qualitative data usually uses data while quantitative involves counting or measuring numerical values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3aa3stat2a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Quantitative"],"dependencies":["a9d3aa3stat2a-h2"],"title":"Quantitative or Qualitative?","text":"Is counting the number of pairs of shoes you own qualitative or quantitative data?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Quantitative","Qualitative"]},{"id":"a9d3aa3stat2a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Discrete"],"dependencies":["a9d3aa3stat2a-h3"],"title":"Continuous or Discrete?","text":"Look back to the hint about how discrete data deals with questions of the type \\"the number of\\". Is the number of pairs of shoes you own therefore discrete or continuous?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Discrete","Continuous"]}]}}]},{"id":"a9d3aa3stat3","title":"Quantitative versus Qualitative","body":"Determine whether each of the following is quantitative data or qualitative. Indicate whether quantitative data are continuous or discrete. Hint: Data that are discrete often start with the words \\"the number of.\\"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Data, Sampling, and Variation in Data and Sampling","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a9d3aa3stat3a","stepAnswer":["Qualitative"],"problemType":"MultipleChoice","stepTitle":"The type of car you drive","stepBody":"","answerType":"string","variabilization":{},"choices":["Quantitative Discrete","Qualitative","Quantitative Continuous"],"hints":{"DefaultPathway":[{"id":"a9d3aa3stat3a-h1","type":"hint","dependencies":[],"title":"Quantitative or Qualitative?","text":"We want to determine the type of car you drive. We can determine if this is qualitative or quantitative since qualitative data usually uses data while quantitative involves counting or measuring numerical values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3aa3stat3a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Qualitative"],"dependencies":["a9d3aa3stat3a-h1"],"title":"Quantitative or Qualitative?","text":"Is the type of car you drive qualitative or quantitative data?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Quantitative","Qualitative"]}]}}]},{"id":"a9d3aa3stat4","title":"Quantitative versus Qualitative","body":"Determine whether each of the following is quantitative data or qualitative. Indicate whether quantitative data are continuous or discrete. Hint: Data that are discrete often start with the words \\"the number of.\\"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Data, Sampling, and Variation in Data and Sampling","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a9d3aa3stat4a","stepAnswer":["Quantitative Continuous"],"problemType":"MultipleChoice","stepTitle":"The distance from your home to the nearest grocery store.","stepBody":"","answerType":"string","variabilization":{},"choices":["Quantitative Discrete","Qualitative","Quantitative Continuous"],"hints":{"DefaultPathway":[{"id":"a9d3aa3stat4a-h1","type":"hint","dependencies":[],"title":"Quantitative or Qualitative?","text":"We want to determine the distance it is from your home to the nearest grocery store. We can determine if this is qualitative or quantitative since qualitative data usually uses data while quantitative involves counting or measuring numerical values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3aa3stat4a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Quantitative"],"dependencies":["a9d3aa3stat4a-h1"],"title":"Quantitative or Qualitative?","text":"Is the distance from your home to the nearest grocery store qualitative or quantitative data?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Quantitative","Qualitative"]},{"id":"a9d3aa3stat4a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Continuous"],"dependencies":["a9d3aa3stat4a-h2"],"title":"Continuous or Discrete?","text":"Now, we know that distance is a quantitative value. Now, to decide whether it is discrete or continuous, we must decide if we can have $$\\\\frac{fractions}{decimals}$$ of a distance. For instance, is it valid to have a distance of $$1$$ mile? How about $$0.2$$ of a mile? If it is valid to include fractions or decimals of a $$\\\\frac{mile}{any}$$ distance, choose continuous, otherwise, choose discrete.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Discrete","Continuous"]}]}}]},{"id":"a9d3aa3stat5","title":"Quantitative versus Qualitative","body":"Determine whether each of the following is quantitative data or qualitative. Indicate whether quantitative data are continuous or discrete. Hint: Data that are discrete often start with the words \\"the number of.\\"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Data, Sampling, and Variation in Data and Sampling","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a9d3aa3stat5a","stepAnswer":["Quantitative Discrete"],"problemType":"MultipleChoice","stepTitle":"The number of classes you take per school year.","stepBody":"","answerType":"string","variabilization":{},"choices":["Quantitative Discrete","Qualitative","Quantitative Continuous"],"hints":{"DefaultPathway":[{"id":"a9d3aa3stat5a-h1","type":"hint","dependencies":[],"title":"Quantitative or Qualitative?","text":"We want to count the number of classes you take per school year. We can determine if this is qualitative or quantitative since qualitative data usually uses data while quantitative involves counting or measuring numerical values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3aa3stat5a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Quantitative"],"dependencies":["a9d3aa3stat5a-h1"],"title":"Quantitative or Qualitative?","text":"Is the number of classes you take per school year qualitative or quantitative data?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Quantitative","Qualitative"]},{"id":"a9d3aa3stat5a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Discrete"],"dependencies":["a9d3aa3stat5a-h2"],"title":"Continuous or Discrete?","text":"Look back to the hint about how discrete data deals with questions of the type \\"the number of\\". Is the number of classes you take per school year therefore discrete or continuous?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Discrete","Continuous"]}]}}]},{"id":"a9d3aa3stat6","title":"Quantitative versus Qualitative","body":"Determine whether each of the following is quantitative data or qualitative. Indicate whether quantitative data are continuous or discrete. Hint: Data that are discrete often start with the words \\"the number of.\\"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Data, Sampling, and Variation in Data and Sampling","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a9d3aa3stat6a","stepAnswer":["Qualitative"],"problemType":"MultipleChoice","stepTitle":"the type of calculator you use","stepBody":"","answerType":"string","variabilization":{},"choices":["Quantitative Discrete","Qualitative","Quantitative Continuous"],"hints":{"DefaultPathway":[{"id":"a9d3aa3stat6a-h1","type":"hint","dependencies":[],"title":"Quantitative or Qualitative?","text":"We want to determine the type of calculator you use. We know that qualitative data focuses on categorization while quantitative data focuses on numerical analysis. We can determine if this is qualitative or quantitative since qualitative data usually uses data while quantitative involves counting or measuring numerical values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3aa3stat6a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Qualitative"],"dependencies":["a9d3aa3stat6a-h1"],"title":"Quantitative or Qualitative?","text":"Is the type of calculator you use qualitative or quantitative data?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Quantitative","Qualitative"]}]}}]},{"id":"a9d3aa3stat7","title":"Quantitative versus Qualitative","body":"Determine whether each of the following is quantitative data or qualitative. Indicate whether quantitative data are continuous or discrete. Hint: Data that are discrete often start with the words \\"the number of.\\"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Data, Sampling, and Variation in Data and Sampling","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a9d3aa3stat7a","stepAnswer":["Quantitative Continuous"],"problemType":"MultipleChoice","stepTitle":"The weights of sumo wrestlers","stepBody":"","answerType":"string","variabilization":{},"choices":["Quantitative Discrete","Qualitative","Quantitative Continuous"],"hints":{"DefaultPathway":[{"id":"a9d3aa3stat7a-h1","type":"hint","dependencies":[],"title":"Quantitative or Qualitative?","text":"We want to determine the weights of sumo wrestlers. We know that qualitative data focuses on categorization while quantitative data focuses on numerical analysis. We can determine if this is qualitative or quantitative since qualitative data usually uses data while quantitative involves counting or measuring numerical values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3aa3stat7a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Quantitative"],"dependencies":["a9d3aa3stat7a-h1"],"title":"Quantitative or Qualitative?","text":"Is the weights of sumo wrestlers qualitative or quantitative data?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Quantitative","Qualitative"]},{"id":"a9d3aa3stat7a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Continuous"],"dependencies":["a9d3aa3stat7a-h2"],"title":"Continuous or Discrete?","text":"Now, we know that weight is a quantitative value. To decide whether it is discrete or continuous, we must decide if we can have $$\\\\frac{fractions}{decimals}$$ of a weight. For instance, is it valid to have a weight of $$100$$ pounds? How about $$100.2$$ pounds? If it is valid to include fractions or decimals of a $$\\\\frac{pound}{any}$$ unit of weight, choose continuous, otherwise, choose discrete.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Discrete","Continuous"]}]}}]},{"id":"a9d3aa3stat8","title":"Quantitative versus Qualitative","body":"Determine whether each of the following is quantitative data or qualitative. Indicate whether quantitative data are continuous or discrete. Hint: Data that are discrete often start with the words \\"the number of.\\"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Data, Sampling, and Variation in Data and Sampling","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a9d3aa3stat8a","stepAnswer":["Quantitative Discrete"],"problemType":"MultipleChoice","stepTitle":"The number of correct answers on a quiz.","stepBody":"","answerType":"string","variabilization":{},"choices":["Quantitative Discrete","Qualitative","Quantitative Continuous"],"hints":{"DefaultPathway":[{"id":"a9d3aa3stat8a-h1","type":"hint","dependencies":[],"title":"Quantitative or Qualitative?","text":"We want to count the number of correct answers on a quiz. We can determine if this is qualitative or quantitative since qualitative data usually uses data while quantitative involves counting or measuring numerical values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3aa3stat8a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Quantitative"],"dependencies":["a9d3aa3stat8a-h1"],"title":"Quantitative or Qualitative?","text":"Is counting the number of correct answers on a quiz qualitative or quantitative data?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Quantitative","Qualitative"]},{"id":"a9d3aa3stat8a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Discrete"],"dependencies":["a9d3aa3stat8a-h2"],"title":"Continuous or Discrete?","text":"Look back to the hint about how discrete data deals with questions of the type \\"the number of\\". Is the number of classes you take per school year therefore discrete or continuous?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Discrete","Continuous"]}]}}]},{"id":"a9d3aa3stat9","title":"Classification of Statistics Students","body":"A statistics professor collects information about the classification of her students as freshmen, sophomores, juniors, or seniors. The data she collects are summarized in the pie chart.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Data, Sampling, and Variation in Data and Sampling","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a9d3aa3stat9a","stepAnswer":["Qualitative (or Categorical)"],"problemType":"MultipleChoice","stepTitle":"What type of data does this graph show?","stepBody":"","answerType":"string","variabilization":{},"choices":["Qualitative (or Categorical)","Quantitative"],"hints":{"DefaultPathway":[{"id":"a9d3aa3stat9a-h1","type":"hint","dependencies":[],"title":"Definitions of Qualitative and Quantitative Data","text":"Qualitative data usually involves categorizing (working with categorical data) and therefore usually isn\'t numeric in context. Quantitative data, in contrast, always words with numbers whether that be counting or measuring.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9d3aa3stat9a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Categories"],"dependencies":["a9d3aa3stat9a-h1"],"title":"Qualitative or Quantitative?","text":"The data is separated out into groups in a pie chart. Is this an example of categorically separating out the data? Or is it numerically separating the data? In other words, are the labels \\"Freshman\\", \\"Sophomore\\", \\"Junior\\", \\"Senior\\" categories or numbers?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Categories","Numbers"]}]}}]},{"id":"a9e42b3CompInv1","title":"Do the pairs represent a function? If so, is it one-to-one?","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Finding Composite and Inverse Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9e42b3CompInv1a","stepAnswer":["Yes but not $$one-to-one$$"],"problemType":"MultipleChoice","stepTitle":"Do the pairs represent a function? 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For each ordered pair in the function, each y-value is matched with only one x-value. There are no repeated y-values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$27$$"],"dependencies":["a9e42b3CompInv1a-h3"],"title":"Determine if it is one-to-one","text":"What $$y$$ value has more than one $$x$$ paired with that $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9e42b3CompInv10","title":"Determine whether each graph is the graph of a function and, if so, whether it is one-to-one.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Finding Composite and Inverse Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9e42b3CompInv10a","stepAnswer":["Yes and $$one-to-one$$"],"problemType":"MultipleChoice","stepTitle":"Determine whether graph (b) is the graph of a function and, if so, whether it is one-to-one.","stepBody":"https://openstax.org/apps/archive/20220325.143432/resources/800d46efe8cb4fe050d5732e8b79efb8d3504280##figure1.gif## ","answerType":"string","variabilization":{},"choices":["No","Yes but not $$one-to-one$$","Yes and $$one-to-one$$"],"hints":{"DefaultPathway":[{"id":"a9e42b3CompInv10a-h1","type":"hint","dependencies":[],"title":"Check if it is a function","text":"Use vertical lines to determine if it is a function. 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If so, it is one-to-one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a9e42b3CompInv11","title":"Determine whether each graph is the graph of a function and, if so, whether it is one-to-one.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Finding Composite and Inverse Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9e42b3CompInv11a","stepAnswer":["Yes but not $$one-to-one$$"],"problemType":"MultipleChoice","stepTitle":"Determine whether graph (a) is the graph of a function and, if so, whether it is one-to-one.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["No","Yes but not $$one-to-one$$","Yes and $$one-to-one$$"],"hints":{"DefaultPathway":[{"id":"a9e42b3CompInv11a-h1","type":"hint","dependencies":[],"title":"Check if it is a function","text":"Use vertical lines to determine if it is a function. 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If so, it is one-to-one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a9e42b3CompInv12","title":"Determine whether each graph is the graph of a function and, if so, whether it is one-to-one.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Finding Composite and Inverse Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9e42b3CompInv12a","stepAnswer":["Yes and $$one-to-one$$"],"problemType":"MultipleChoice","stepTitle":"Determine whether graph (b) is the graph of a function and, if so, whether it is one-to-one.","stepBody":"https://openstax.org/apps/archive/20220325.143432/resources/f20d52396969a1e4b6677e62af4477bf9bde2900##figure1.gif## ","answerType":"string","variabilization":{},"choices":["No","Yes but not $$one-to-one$$","Yes and $$one-to-one$$"],"hints":{"DefaultPathway":[{"id":"a9e42b3CompInv12a-h1","type":"hint","dependencies":[],"title":"Check if it is a function","text":"Use vertical lines to determine if it is a function. 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If so, it is one-to-one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a9e42b3CompInv16","title":"Verify inverse functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Finding Composite and Inverse Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9e42b3CompInv16a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Verify if the following function are inverse functions","stepBody":"f(x) $$=$$ $$5x$$ - $$1$$, g(x) $$=$$ (x+1)/ $$5$$","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a9e42b3CompInv16a-h1","type":"hint","dependencies":[],"title":"inverse functions definition","text":"The functions are inverses of each other if $$g((f(x))=x$$ and $$g(f(x))=x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv16a-h2","type":"hint","dependencies":["a9e42b3CompInv16a-h1"],"title":"Substitute $$5x-1$$ for f(x)","text":"Check if $$g(5x-1)$$ $$=$$ $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv16a-h3","type":"hint","dependencies":["a9e42b3CompInv16a-h2"],"title":"Find $$g(5x-1)$$ where g(x) $$=$$ $$\\\\frac{x+1}{5}$$","text":"Check if $$\\\\frac{5x-1+1}{5}$$ $$=$$ $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv16a-h4","type":"hint","dependencies":["a9e42b3CompInv16a-h3"],"title":"Simplify","text":"$$\\\\frac{5x-1+1}{5}=\\\\frac{5x}{5}$$ $$=$$ $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv16a-h5","type":"hint","dependencies":["a9e42b3CompInv16a-h4"],"title":"substitute $$\\\\frac{x+1}{5}$$ for g(x)","text":"Check if $$f{\\\\left(\\\\frac{x+1}{5}\\\\right)}$$ $$=$$ $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv16a-h6","type":"hint","dependencies":["a9e42b3CompInv16a-h5"],"title":"Find f((x+1/5) where $$f(x)=5x-1$$","text":"$$f{\\\\left(\\\\frac{x+1}{5}\\\\right)}=5\\\\left(\\\\frac{x+1}{5}\\\\right)$$ - $$1=x+1-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv16a-h7","type":"hint","dependencies":["a9e42b3CompInv16a-h6"],"title":"Simplify","text":"$$x+1-1$$ $$=x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9e42b3CompInv17","title":"Verify inverse functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Finding Composite and Inverse Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9e42b3CompInv17a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Verify if the following function are inverse functions","stepBody":"f(x) $$=$$ $$4x$$ - $$3$$, g(x) $$=$$ (x+3)/ $$4$$","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a9e42b3CompInv17a-h1","type":"hint","dependencies":[],"title":"inverse functions definition","text":"The functions are inverses of each other if $$g((f(x))=x$$ and $$g(f(x))=x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv17a-h2","type":"hint","dependencies":["a9e42b3CompInv17a-h1"],"title":"Substitute $$4x-3$$ for f(x)","text":"Check if $$g(4x-3)$$ $$=$$ $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv17a-h3","type":"hint","dependencies":["a9e42b3CompInv17a-h2"],"title":"Find $$g(4x-3)$$ where g(x) $$=$$ $$\\\\frac{x+3}{4}$$","text":"$$\\\\frac{4x-3+3}{4}$$ $$=$$ $$\\\\frac{4x}{4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv17a-h4","type":"hint","dependencies":["a9e42b3CompInv17a-h3"],"title":"Simplify","text":"$$\\\\frac{4x}{4}$$ $$=$$ $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv17a-h5","type":"hint","dependencies":["a9e42b3CompInv17a-h4"],"title":"Substitute $$\\\\frac{x+3}{4}$$ for g(x)","text":"Check if $$f{\\\\left(\\\\frac{x+3}{4}\\\\right)}$$ $$=$$ $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv17a-h6","type":"hint","dependencies":["a9e42b3CompInv17a-h5"],"title":"Find $$f{\\\\left(\\\\frac{x+3}{4}\\\\right)}$$ where $$f(x)=4x-3$$","text":"$$4\\\\left(\\\\frac{x+3}{4}\\\\right)$$ - $$3=$$ $$x+3-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv17a-h7","type":"hint","dependencies":["a9e42b3CompInv17a-h6"],"title":"Simplify","text":"$$x+3-3$$ $$=x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9e42b3CompInv18","title":"Verify inverse functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Finding Composite and Inverse Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9e42b3CompInv18a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Verify if the following function are inverse functions","stepBody":"f(x) $$=$$ $$2x+6$$, g(x) $$=$$ (x-6)/ $$2$$","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"a9e42b3CompInv18a-h1","type":"hint","dependencies":[],"title":"Inverse functions definition","text":"The functions are inverses of each other if $$g((f(x))=x$$ and $$g(f(x))=x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv18a-h2","type":"hint","dependencies":["a9e42b3CompInv18a-h1"],"title":"Substitute $$2x+6$$ for f(x)","text":"Check if $$g{\\\\left(2x+6\\\\right)}$$ $$=$$ $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv18a-h3","type":"hint","dependencies":["a9e42b3CompInv18a-h2"],"title":"Find $$g{\\\\left(2x+6\\\\right)}$$ where g(x) $$=$$ (x-6)/ $$2$$","text":"$$\\\\frac{2x+6-6}{2}$$ $$=$$ $$\\\\frac{2x}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv18a-h4","type":"hint","dependencies":["a9e42b3CompInv18a-h3"],"title":"Simplify","text":"$$\\\\frac{2x}{x}$$ $$=$$ $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv18a-h5","type":"hint","dependencies":["a9e42b3CompInv18a-h4"],"title":"Substitute (x-6)/ $$2$$ for g(x)","text":"Check if $$f{\\\\left(\\\\frac{x-6}{2}\\\\right)}$$ $$=$$ $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv18a-h6","type":"hint","dependencies":["a9e42b3CompInv18a-h5"],"title":"Find $$f{\\\\left(\\\\frac{x-6}{2}\\\\right)}$$ where $$f(x)=2x+6$$","text":"$$f{\\\\left(\\\\frac{x-6}{2}\\\\right)}=2\\\\left(\\\\frac{x-6}{2}\\\\right)+6=$$ $$x+6-6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv18a-h7","type":"hint","dependencies":["a9e42b3CompInv18a-h6"],"title":"Simplify","text":"$$f{\\\\left(\\\\frac{x-6}{2}\\\\right)}=x+6-6$$ $$=x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9e42b3CompInv19","title":"Find the inverse of a One-to-One Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Finding Composite and Inverse Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9e42b3CompInv19a","stepAnswer":["$$y=\\\\frac{1}{4} x-\\\\frac{7}{4}$$"],"problemType":"TextBox","stepTitle":"Find the inverse of the following One-to-One Function. Please enter your answer as $$y=ax+b$$ where a and $$b$$ are real numbers.","stepBody":"$$y$$ $$=$$ $$4x+7$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=\\\\frac{1}{4} x-\\\\frac{7}{4}$$","hints":{"DefaultPathway":[{"id":"a9e42b3CompInv19a-h1","type":"hint","dependencies":[],"title":"Interchange the variables $$x$$ and $$y$$","text":"Replace $$x$$ with $$y$$ and then $$y$$ with $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv19a-h2","type":"hint","dependencies":["a9e42b3CompInv19a-h1"],"title":"What\'s the equation after replacing $$x$$ with $$y$$ and $$y$$ with $$x$$?","text":"$$x$$ $$=$$ $$4y+7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv19a-h3","type":"hint","dependencies":["a9e42b3CompInv19a-h2"],"title":"Solve for $$y$$","text":"Solve for $$y$$ and find the result","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv19a-h4","type":"hint","dependencies":["a9e42b3CompInv19a-h3"],"title":"Solve for $$y$$","text":"We can subtract $$7$$ from each side then divide by $$4$$. Then we will get $$y$$ $$=\\\\frac{x-7}{4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9e42b3CompInv2","title":"Do the pairs represent a function? If so, is it one-to-one?","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Finding Composite and Inverse Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9e42b3CompInv2a","stepAnswer":["Yes and $$one-to-one$$"],"problemType":"MultipleChoice","stepTitle":"Do the pairs represent a function? If so, is it one-to-one?","stepBody":"{$$(0,0),(1,1),(4,2),(9,3),(16,4)$$.}","answerType":"string","variabilization":{},"choices":["No","Yes but not $$one-to-one$$","Yes and $$one-to-one$$"],"hints":{"DefaultPathway":[{"id":"a9e42b3CompInv2a-h1","type":"hint","dependencies":[],"title":"Do the pairs represent a function?","text":"Think about the condition that makes some $$x-y$$ pairs a function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv2a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a9e42b3CompInv2a-h1"],"title":"If pairs represent a function, each x-value is matched with only one y-value.","text":"So, are the pairs in this question a function?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a9e42b3CompInv2a-h3","type":"hint","dependencies":["a9e42b3CompInv2a-h2"],"title":"Do the pairs represent a one-to-one function?","text":"A function is one-to-one if each value in the range corresponds to one element in the domain. For each ordered pair in the function, each y-value is matched with only one x-value. There are no repeated y-values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv2a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a9e42b3CompInv2a-h3"],"title":"Determine if it is one-to-one","text":"Is there a $$y$$ value that has more than one $$x$$ paired with that $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a9e42b3CompInv20","title":"Find the inverse of a One-to-One Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Finding Composite and Inverse Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9e42b3CompInv20a","stepAnswer":["$$y$$ $$=$$ $$\\\\frac{x}{5}+\\\\frac{3}{5}$$"],"problemType":"TextBox","stepTitle":"Find the inverse of the following One-to-One Function. Please enter your answer as $$y=ax+b$$ where a and $$b$$ are real numbers.","stepBody":"$$y$$ $$=$$ $$5x-3$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y$$ $$=$$ $$\\\\frac{x}{5}+\\\\frac{3}{5}$$","hints":{"DefaultPathway":[{"id":"a9e42b3CompInv20a-h1","type":"hint","dependencies":[],"title":"Interchange the variables $$x$$ and $$y$$","text":"Replace $$x$$ with $$y$$ and then $$y$$ with $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv20a-h2","type":"hint","dependencies":["a9e42b3CompInv20a-h1"],"title":"What\'s the equation after replacing $$x$$ with $$y$$ and $$y$$ with $$x$$","text":"$$x$$ $$=$$ $$5y-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv20a-h3","type":"hint","dependencies":["a9e42b3CompInv20a-h2"],"title":"Solve for $$y$$","text":"Solve for $$y$$ and find the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv20a-h4","type":"hint","dependencies":["a9e42b3CompInv20a-h3"],"title":"Solve for $$y$$","text":"We can add $$3$$ from each side then divide by $$5$$. It gives us $$y$$ $$=$$ $$\\\\frac{x-3}{5}$$. We can further simplify it into $$y$$ $$=$$ $$\\\\frac{x}{5}+\\\\frac{3}{5}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9e42b3CompInv21","title":"Find the inverse of a One-to-One Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Finding Composite and Inverse Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9e42b3CompInv21a","stepAnswer":["$$y$$ $$=\\\\frac{x}{8}-\\\\frac{5}{8}$$"],"problemType":"TextBox","stepTitle":"Find the inverse of the following One-to-One Function. Please enter your answer as $$y=ax+b$$ where a and $$b$$ are real numbers.","stepBody":"$$y$$ $$=$$ $$8x+5$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y$$ $$=\\\\frac{x}{8}-\\\\frac{5}{8}$$","hints":{"DefaultPathway":[{"id":"a9e42b3CompInv21a-h1","type":"hint","dependencies":[],"title":"Interchange the variables $$x$$ and $$y$$","text":"Replace $$x$$ with $$y$$ and then $$y$$ with $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv21a-h2","type":"hint","dependencies":["a9e42b3CompInv21a-h1"],"title":"What\'s the equation after replacing $$x$$ with $$y$$ and $$y$$ with $$x$$","text":"$$x$$ $$=$$ $$8y+5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv21a-h3","type":"hint","dependencies":["a9e42b3CompInv21a-h2"],"title":"Solve for $$y$$","text":"Solve for $$y$$ and find the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv21a-h4","type":"hint","dependencies":["a9e42b3CompInv21a-h3"],"title":"Solve for $$y$$","text":"We can subtract $$5$$ from each side then divide by $$8$$. It gives us $$y=\\\\frac{x-5}{8}$$. We can further simplify it into $$y$$ $$=\\\\frac{x}{8}-\\\\frac{5}{8}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9e42b3CompInv22","title":"Find the inverse of a One-to-One Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Finding Composite and Inverse Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9e42b3CompInv22a","stepAnswer":["$$y$$ $$=\\\\frac{1}{2} x^5$$ $$+\\\\frac{3}{2}$$"],"problemType":"TextBox","stepTitle":"Find the inverse of the following One-to-One Function. Please enter your answer as $$y={ax}^n+b$$ where $$n$$, a and $$b$$ are real numbers.","stepBody":"$$y$$ $$=$$ $${\\\\left(2x-3\\\\right)}^{\\\\frac{1}{5}}$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y$$ $$=\\\\frac{1}{2} x^5$$ $$+\\\\frac{3}{2}$$","hints":{"DefaultPathway":[{"id":"a9e42b3CompInv22a-h1","type":"hint","dependencies":[],"title":"Interchange the variables $$x$$ and $$y$$","text":"Replace $$x$$ with $$y$$ and then $$y$$ with $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv22a-h2","type":"hint","dependencies":["a9e42b3CompInv22a-h1"],"title":"What\'s the equation after replacing $$x$$ with $$y$$ and $$y$$ with $$x$$","text":"$$x$$ $$=$$ $${\\\\left(2y-3\\\\right)}^{\\\\frac{1}{5}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv22a-h3","type":"hint","dependencies":["a9e42b3CompInv22a-h2"],"title":"Solve for $$y$$","text":"We can rise both sides to the fifth power then add $$3$$ both sides. It gives $$x^5+3=2y$$. Then we can isolate $$y$$ on one side by dividing by $$2$$ both sides which gives $$\\\\frac{x^5+3}{2}=y$$. We can further simplify it into $$y$$ $$=\\\\frac{1}{2} x^5$$ $$+\\\\frac{3}{2}$$ which is the final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9e42b3CompInv23","title":"Find the inverse of a One-to-One Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Finding Composite and Inverse Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9e42b3CompInv23a","stepAnswer":["$$y$$ $$=\\\\frac{1}{3} x^5+\\\\frac{2}{3}$$"],"problemType":"TextBox","stepTitle":"Find the inverse of the following One-to-One Function. Please enter your answer as $$y={ax}^n+b$$ where $$n$$, a and $$b$$ are real numbers.","stepBody":"$$y$$ $$=$$ $${\\\\left(3x-2\\\\right)}^{\\\\frac{1}{5}}$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y$$ $$=\\\\frac{1}{3} x^5+\\\\frac{2}{3}$$","hints":{"DefaultPathway":[{"id":"a9e42b3CompInv23a-h1","type":"hint","dependencies":[],"title":"Interchange the variables $$x$$ and $$y$$","text":"Replace $$x$$ with $$y$$ and then $$y$$ with $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv23a-h2","type":"hint","dependencies":["a9e42b3CompInv23a-h1"],"title":"What\'s the equation after replacing $$x$$ with $$y$$ and $$y$$ with $$x$$","text":"$$x$$ $$=$$ $${\\\\left(3y-2\\\\right)}^{\\\\frac{1}{5}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv23a-h3","type":"hint","dependencies":["a9e42b3CompInv23a-h2"],"title":"Solve for $$y$$","text":"We can rise both sides to the fifth power then add $$2$$ both sides. It gives $$x^5+2=3y$$. We can isolate $$y$$ on one sides by dividing $$3$$ both sides which gives $$y$$ =(x**5 +2)/3. We can further simplify it into $$y$$ $$=\\\\frac{1}{3} x^5+\\\\frac{2}{3}$$ which is the final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9e42b3CompInv24","title":"Find the inverse of a One-to-One Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Finding Composite and Inverse Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9e42b3CompInv24a","stepAnswer":["$$y=\\\\frac{1}{6} x^4+\\\\frac{7}{6}$$"],"problemType":"TextBox","stepTitle":"Find the inverse of the following One-to-One Function. Please enter your answer as $$y={ax}^n+b$$ where $$n$$, a and $$b$$ are real numbers.","stepBody":"$$y$$ $$=$$ $${\\\\left(6x-7\\\\right)}^{\\\\frac{1}{4}}$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=\\\\frac{1}{6} x^4+\\\\frac{7}{6}$$","hints":{"DefaultPathway":[{"id":"a9e42b3CompInv24a-h1","type":"hint","dependencies":[],"title":"Interchange the variables $$x$$ and $$y$$","text":"Replace $$x$$ with $$y$$ and then $$y$$ with $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv24a-h2","type":"hint","dependencies":["a9e42b3CompInv24a-h1"],"title":"What\'s the equation after replacing $$x$$ with $$y$$ and $$y$$ with $$x$$?","text":"$$x$$ $$=$$ $${\\\\left(6y-7\\\\right)}^{\\\\frac{1}{4}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv24a-h3","type":"hint","dependencies":["a9e42b3CompInv24a-h2"],"title":"Solve for $$y$$","text":"Given $$x$$ $$=$$ $${\\\\left(6y-7\\\\right)}^{\\\\frac{1}{4}}$$, solve for $$y$$. We can raise both sides to the fourth power and get $$x^4=6y-7$$. Then we can isolate $$y$$ by adding $$7$$ both sides and dividing by $$6$$ both sides which give $$y$$ =(x**4 +7)/6. We can simplify it into $$y=\\\\frac{1}{6} x^4+\\\\frac{7}{6}$$ which is the final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9e42b3CompInv25","title":"Find the inverse of a One-to-One Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Finding Composite and Inverse Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9e42b3CompInv25a","stepAnswer":["$$y$$ $$=\\\\frac{1}{9} x$$"],"problemType":"TextBox","stepTitle":"Find the inverse of the following One-to-One Function. Please enter your answer as $$y=ax+b$$ where a and $$b$$ are real numbers.","stepBody":"$$y$$ $$=$$ $$9x$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y$$ $$=\\\\frac{1}{9} x$$","hints":{"DefaultPathway":[{"id":"a9e42b3CompInv25a-h1","type":"hint","dependencies":[],"title":"Interchange the variables $$x$$ and $$y$$","text":"Replace $$x$$ with $$y$$ and then $$y$$ with $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv25a-h2","type":"hint","dependencies":["a9e42b3CompInv25a-h1"],"title":"What\'s the equation after replacing $$x$$ with $$y$$ and $$y$$ with $$x$$","text":"$$x$$ $$=$$ $$9y$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv25a-h3","type":"hint","dependencies":["a9e42b3CompInv25a-h2"],"title":"Solve for $$y$$","text":"Given $$x$$ $$=$$ $$9y$$, we can solve for $$y$$. We can isolate $$y$$ on one side by dividing $$9$$ on both sides which gives $$y=\\\\frac{1}{9} x$$. Therefore the final answer is $$y=\\\\frac{1}{9} x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9e42b3CompInv26","title":"Find (f\xb0g)(x)","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Finding Composite and Inverse Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9e42b3CompInv26a","stepAnswer":["$$8x+23$$"],"problemType":"TextBox","stepTitle":"Find the (f\xb0g)(x) of the following functions. Please enter your answer as $$ax+b$$ where a and $$b$$ are real numbers.","stepBody":"f(x) $$=$$ $$4x+3$$, $$g(x)=2x+5$$.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8x+23$$","hints":{"DefaultPathway":[{"id":"a9e42b3CompInv26a-h1","type":"hint","dependencies":[],"title":"What does it mean by (f\xb0g)(x)?","text":"it means \\"evaluate g at $$x$$, then evaluate f at the result g(x)\\" which is f(g(x)).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv26a-h2","type":"hint","dependencies":["a9e42b3CompInv26a-h1"],"title":"Substitute $$2x+5$$ for f(x)","text":"Substitute $$2x+5$$ for $$x$$ in f(x) and get $$f{\\\\left(2x+5\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv26a-h3","type":"hint","dependencies":["a9e42b3CompInv26a-h2"],"title":"Calculate f(x) with new $$x$$","text":"Calculate f(x) $$=$$ $$4x+3$$ with $$x$$ replaced by $$2x+5$$, which is $$f(x)=4\\\\left(2x+5\\\\right)$$ + $$3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv26a-h4","type":"hint","dependencies":["a9e42b3CompInv26a-h3"],"title":"Simplify","text":"$$f(x)=4\\\\left(2x+5\\\\right)$$ + $$3=8x+23$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9e42b3CompInv27","title":"Find (g\xb0f)(x)","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Finding Composite and Inverse Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9e42b3CompInv27a","stepAnswer":["$$8x+11$$"],"problemType":"TextBox","stepTitle":"Find the (g\xb0f)(x) of the following functions. Please enter your answer as $$ax+b$$ where a and $$b$$ are real numbers.","stepBody":"f(x) $$=$$ $$4x+3$$, g(x) $$=$$ $$2x+5$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8x+11$$","hints":{"DefaultPathway":[{"id":"a9e42b3CompInv27a-h1","type":"hint","dependencies":[],"title":"What does it mean by (g\xb0f)(x)?","text":"it means \\"evaluate f at $$x$$, then evaluate g at the result f(x)\\" which is g(f(x)).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv27a-h2","type":"hint","dependencies":["a9e42b3CompInv27a-h1"],"title":"Substitute $$4x+3$$ for $$x$$ in g(x)","text":"Substitute $$4x+3$$ for f(x) and get $$g{\\\\left(4x+3\\\\right)}=g(f(x))$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv27a-h3","type":"hint","dependencies":["a9e42b3CompInv27a-h2"],"title":"Calculate g(x) with new $$x$$","text":"Calculate g(x) $$=$$ $$2x+5$$ with $$x$$ replaced by $$4x+3$$, which is $$g(x)=2\\\\left(4x+3\\\\right)$$ + $$5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv27a-h4","type":"hint","dependencies":["a9e42b3CompInv27a-h3"],"title":"Simplify","text":"$$g(x)=2\\\\left(4x+3\\\\right)$$ + $$5=8x+11$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9e42b3CompInv28","title":"Find $$f g x$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Finding Composite and Inverse Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9e42b3CompInv28a","stepAnswer":["$$8x^2$$ + $$26x$$ + $$15$$"],"problemType":"TextBox","stepTitle":"Find the $$f g x$$ of the following functions. Please enter your answer as $${ax}^n+bx+c$$ where $$n$$, c, a and $$b$$ are real numbers.","stepBody":"f(x) $$=$$ $$4x+3$$, g(x) $$=$$ $$2x+5$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8x^2$$ + $$26x$$ + $$15$$","hints":{"DefaultPathway":[{"id":"a9e42b3CompInv28a-h1","type":"hint","dependencies":[],"title":"What does it mean by $$f g x$$","text":"It means evaluate f and g at $$x$$ and multiply the results.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv28a-h2","type":"hint","dependencies":["a9e42b3CompInv28a-h1"],"title":"Multiply two functions","text":"What\'s $$\\\\left(4x+3\\\\right) \\\\left(2x+5\\\\right)$$? We can FOIL out the expression and get $$8x^2$$ + $$26x$$ + $$15$$ as the final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9e42b3CompInv29","title":"Find (f\xb0g)(2)","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Finding Composite and Inverse Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9e42b3CompInv29a","stepAnswer":["$$39$$"],"problemType":"TextBox","stepTitle":"Find the (f\xb0g)(2) of the following functions.","stepBody":"f(x) $$=$$ $$4x+3$$, g(x) $$=$$ $$2x+5$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$39$$","hints":{"DefaultPathway":[{"id":"a9e42b3CompInv29a-h1","type":"hint","dependencies":[],"title":"What does it mean by (f\xb0g)(x)?","text":"it means \\"evaluate g at $$x$$, then evaluate f at the result g(x)\\" which is f(g(x)).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv29a-h2","type":"hint","dependencies":["a9e42b3CompInv29a-h1"],"title":"Substitute $$2x+5$$ for $$x$$ in f(x)","text":"Substitute $$2x+5$$ for $$x$$ in f(x) and get $$f{\\\\left(2x+5\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv29a-h3","type":"hint","dependencies":["a9e42b3CompInv29a-h2"],"title":"Calculate f(x) with new $$x$$","text":"Calculate f(x) $$=$$ $$4x+3$$ with $$x$$ replaced by $$2x+5$$, which is $$f(x)=4\\\\left(2x+5\\\\right)$$ + $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv29a-h4","type":"hint","dependencies":["a9e42b3CompInv29a-h3"],"title":"Simplify","text":"$$f(x)=4\\\\left(2x+5\\\\right)$$ + $$3=8x+23$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv29a-h5","type":"hint","dependencies":["a9e42b3CompInv29a-h4"],"title":"Evaluate f(2)","text":"$$f(2)=8\\\\times2+23=39$$. Therefore $$f(g(2))=39$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9e42b3CompInv3","title":"Do the pairs represent a function? If so, is it one-to-one?","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Finding Composite and Inverse Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9e42b3CompInv3a","stepAnswer":["Yes and $$one-to-one$$"],"problemType":"MultipleChoice","stepTitle":"Do the pairs represent a function? If so, is it one-to-one?","stepBody":"{$$(-3,-6),(-2,-4),(-1,-2),(0,0),(1,2),(2,4),(3,6)$$}","answerType":"string","variabilization":{},"choices":["No","Yes but not $$one-to-one$$","Yes and $$one-to-one$$"],"hints":{"DefaultPathway":[{"id":"a9e42b3CompInv3a-h1","type":"hint","dependencies":[],"title":"Do the pairs represent a function?","text":"Think about the condition that makes some $$x-y$$ pairs a function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv3a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a9e42b3CompInv3a-h1"],"title":"If pairs represent a function, each x-value is matched with only one y-value.","text":"So, are the pairs in this question a function?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a9e42b3CompInv3a-h3","type":"hint","dependencies":["a9e42b3CompInv3a-h2"],"title":"Do the pairs represent a one-to-one function?","text":"A function is one-to-one if each value in the range corresponds to one element in the domain. For each ordered pair in the function, each y-value is matched with only one x-value. There are no repeated y-values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv3a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["a9e42b3CompInv3a-h3"],"title":"Determine if it is one-to-one","text":"Is there a $$y$$ value that has more than one $$x$$ paired with that $$y$$? If yes, then it is not one-to-one. If no, then it is one-to-one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a9e42b3CompInv30","title":"Find the (g\xb0f)(x) of the following functions.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Finding Composite and Inverse Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9e42b3CompInv30a","stepAnswer":["$$27$$"],"problemType":"TextBox","stepTitle":"Find (g\xb0f)(2)","stepBody":"f(x) $$=$$ $$4x+3$$, g(x) $$=$$ $$2x+5$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$27$$","hints":{"DefaultPathway":[{"id":"a9e42b3CompInv30a-h1","type":"hint","dependencies":[],"title":"What does it mean by (g\xb0f)(x)","text":"it means \\"evaluate f at $$x$$, then evaluate g at the result f(x)\\" which is g(f(x)).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv30a-h2","type":"hint","dependencies":["a9e42b3CompInv30a-h1"],"title":"Substitute $$4x+3$$ for $$x$$ in g(x)","text":"Substitute $$4x+3$$ for $$x$$ in g(x) and get $$g{\\\\left(4x+3\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv30a-h3","type":"hint","dependencies":["a9e42b3CompInv30a-h2"],"title":"Calculate g(x) with new $$x$$","text":"Calculate g(x) $$=$$ $$2x+5$$ with $$x$$ replaced by $$4x+3$$, which is $$g(x)=2\\\\left(4x+3\\\\right)$$ + $$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv30a-h4","type":"hint","dependencies":["a9e42b3CompInv30a-h3"],"title":"Simplify","text":"$$g(x)=2\\\\left(4x+3\\\\right)+5=8x+11$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv30a-h5","type":"hint","dependencies":["a9e42b3CompInv30a-h4"],"title":"Evaluate g(2)","text":"$$g(2)=8\\\\times2+11=27$$. Therefore $$g(f(2))=27$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9e42b3CompInv4","title":"Do the pairs represent a function? If so, is it one-to-one?","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Finding Composite and Inverse Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9e42b3CompInv4a","stepAnswer":["Yes but not $$one-to-one$$"],"problemType":"MultipleChoice","stepTitle":"Do the pairs represent a function? If so, is it one-to-one?","stepBody":"{$$(-4,8),(-2,4),(-1,2),(0,0),(1,2),(2,4),(4,8)$$}","answerType":"string","variabilization":{},"choices":["No","Yes but not $$one-to-one$$","Yes and $$one-to-one$$"],"hints":{"DefaultPathway":[{"id":"a9e42b3CompInv4a-h1","type":"hint","dependencies":[],"title":"Do the pairs represent a function?","text":"Think about the condition that makes some $$x-y$$ pairs a function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv4a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a9e42b3CompInv4a-h1"],"title":"If pairs represent a function, each x-value is matched with only one y-value.","text":"So, are the pairs in this question a function?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a9e42b3CompInv4a-h3","type":"hint","dependencies":["a9e42b3CompInv4a-h2"],"title":"Do the pairs represent a one-to-one function?","text":"A function is one-to-one if each value in the range corresponds to one element in the domain. For each ordered pair in the function, each y-value is matched with only one x-value. There are no repeated y-values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv4a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a9e42b3CompInv4a-h3"],"title":"Determine if it is one-to-one","text":"Is there a $$y$$ value that has more than one $$x$$ paired with that $$y$$? If yes, then it is not one-to-one. If no, then it is one-to-one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a9e42b3CompInv5","title":"Do the pairs represent a function? If so, is it one-to-one?","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Finding Composite and Inverse Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9e42b3CompInv5a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Do the pairs represent a function? 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If there is more than one of such $$x$$ value, please enter it as (a,b,c) where $$a<b<c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a9e42b3CompInv6","title":"Do the pairs represent a function? If so, is it one-to-one?","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Finding Composite and Inverse Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9e42b3CompInv6a","stepAnswer":["Yes but not $$one-to-one$$"],"problemType":"MultipleChoice","stepTitle":"Do the pairs represent a function? If so, is it one-to-one?","stepBody":"{$$(7,-3),(-5,-4),(8,0),(0,0),(-6,4),(-2,2),(-1,3)$$}","answerType":"string","variabilization":{},"choices":["No","Yes but not $$one-to-one$$","Yes and $$one-to-one$$"],"hints":{"DefaultPathway":[{"id":"a9e42b3CompInv6a-h1","type":"hint","dependencies":[],"title":"Do the pairs represent a function?","text":"Think about the condition that makes some $$x-y$$ pairs a function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv6a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a9e42b3CompInv6a-h1"],"title":"If pairs represent a function, each x-value is matched with only one y-value.","text":"So, are the pairs in this question a function?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"a9e42b3CompInv6a-h3","type":"hint","dependencies":["a9e42b3CompInv6a-h2"],"title":"Do the pairs represent a one-to-one function?","text":"A function is one-to-one if each value in the range corresponds to one element in the domain. For each ordered pair in the function, each y-value is matched with only one x-value. There are no repeated y-values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9e42b3CompInv6a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["a9e42b3CompInv6a-h3"],"title":"Determine if it is one-to-one","text":"Is there a $$y$$ value that has more than one $$x$$ paired with that $$y$$? If yes, then it is not one-to-one. If no, then it is one-to-one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"],"subHints":[{"id":"a9e42b3CompInv6a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":[],"title":"What $$y$$ value(s) have two or more $$x$$ values?","text":"What $$y$$ value(s) have two or more $$x$$ values?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"a9e42b3CompInv7","title":"Determine whether each graph is the graph of a function and, if so, whether it is one-to-one.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Finding Composite and Inverse Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9e42b3CompInv7a","stepAnswer":["Yes and $$one-to-one$$"],"problemType":"MultipleChoice","stepTitle":"Determine whether the left graph is the graph of a function and, if so, whether it is one-to-one.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["No","Yes but not $$one-to-one$$","Yes and $$one-to-one$$"],"hints":{"DefaultPathway":[{"id":"a9e42b3CompInv7a-h1","type":"hint","dependencies":[],"title":"Check if it is a function","text":"Use vertical lines to determine if it is a function. 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If so, it is one-to-one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a9e42b3CompInv9","title":"Determine whether each graph is the graph of a function and, if so, whether it is one-to-one.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Finding Composite and Inverse Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9e42b3CompInv9a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Determine whether the left graph (a) is the graph of a function and, if so, whether it is one-to-one.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["No","Yes but not $$one-to-one$$","Yes and $$one-to-one$$"],"hints":{"DefaultPathway":[{"id":"a9e42b3CompInv9a-h1","type":"hint","dependencies":[],"title":"Check if it is a function","text":"Use vertical lines to determine if it is a function. 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If so, it is a function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"a9ed39ahyperbola1","title":"","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 The Hyperbola","courseName":"OpenStax: College Algebra","steps":[{"id":"a9ed39ahyperbola1a","stepAnswer":["$$(0,7),(0,-7)$$"],"problemType":"MultipleChoice","stepTitle":"Find the vertices of $$\\\\frac{y^2}{49}-\\\\frac{x^2}{32}=1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,7),(0,-7)$$","choices":["$$(0,7),(0,-7)$$","(sqrt(32),0),(-sqrt(32),0)"],"hints":{"DefaultPathway":[{"id":"a9ed39ahyperbola1a-h1","type":"hint","dependencies":[],"title":"Standard Form of Hyperbola with Center $$(0,0)$$","text":"The standard form of the equation of a hyperbola with center $$(0,0)$$ and transverse axis on the x-axis is\\\\n$$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$\\\\nwhere\\\\nthe length of the transverse axis is 2a\\\\nthe coordinates of the vertices are $$(a,0),(-a,0)$$\\\\nthe length of the conjugate axis is $$2b$$\\\\nthe coordinates of the co-vertices are (0,b),(0,b)\\\\nthe distance between the foci is 2c, where $$c^2=a^2+b^2$$\\\\nthe coordinates of the foci are $$(c,0),(-c,0)$$\\\\nthe equations of the asymptotes are $$y=\\\\frac{b}{a} x$$, $$y=-\\\\left(\\\\frac{b}{a}\\\\right) x$$\\\\n(See the left image)\\\\n\\\\nThe standard form of the equation of a hyperbola with center $$(0,0)$$ and transverse axis on the y-axis is\\\\n$$\\\\frac{y^2}{a^2}-\\\\frac{x^2}{b^2}=1$$\\\\nwhere\\\\nthe length of the transverse axis is 2a\\\\nthe coordinates of the vertices are $$(0,a),(0,-a)$$\\\\nthe length of the conjugate axis is $$2b$$\\\\nthe coordinates of the co-vertices are $$(b,0),(-b,0)$$\\\\nthe distance between the foci is 2c, where $$c^2=a^2+b^2$$\\\\nthe coordinates of the foci are $$(0,c),(0,-c)$$\\\\nthe equations of the asymptotes are y=(a/b)*x,y=-(a/b)*x\\\\n(See the right image)\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola1a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y$$"],"dependencies":["a9ed39ahyperbola1a-h1"],"title":"Transverse Axis","text":"Which axis is transverse axis on?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x$$","$$y$$"]},{"id":"a9ed39ahyperbola1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["a9ed39ahyperbola1a-h2"],"title":"Finding the Vertices","text":"From the standard equation $$\\\\frac{y^2}{a^2}-\\\\frac{x^2}{b^2}=1$$, the vertices are a units away from the center of the hyperbola. What is a?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola1a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(0,7),(0,-7)$$"],"dependencies":["a9ed39ahyperbola1a-h3"],"title":"Finding the Vertices","text":"We add and subtract $$7$$ units to the center of the hyperbola along the transverse axis to find the two vertices. Given the center is $$(0,0)$$. What are the vertices?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(0,7),(0,-7)$$","(sqrt(32),0),(-sqrt(32),0)"]}]}},{"id":"a9ed39ahyperbola1b","stepAnswer":["$$(0,9),(0,-9)$$"],"problemType":"MultipleChoice","stepTitle":"Find the foci of $$\\\\frac{y^2}{49}-\\\\frac{x^2}{32}=1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,9),(0,-9)$$","choices":["$$(0,9),(0,-9)$$","$$(9,0),(-9,0)$$"],"hints":{"DefaultPathway":[{"id":"a9ed39ahyperbola1b-h1","type":"hint","dependencies":[],"title":"Standard Form of Hyperbola with Center $$(0,0)$$","text":"The standard form of the equation of a hyperbola with center $$(0,0)$$ and transverse axis on the x-axis is\\\\n$$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$\\\\nwhere\\\\nthe length of the transverse axis is 2a\\\\nthe coordinates of the vertices are $$(a,0),(-a,0)$$\\\\nthe length of the conjugate axis is $$2b$$\\\\nthe coordinates of the co-vertices are (0,b),(0,b)\\\\nthe distance between the foci is 2c, where $$c^2=a^2+b^2$$\\\\nthe coordinates of the foci are $$(c,0),(-c,0)$$\\\\nthe equations of the asymptotes are y=(b/a)*x,y=-(b/a)*x\\\\n(See the left image)\\\\n\\\\nThe standard form of the equation of a hyperbola with center $$(0,0)$$ and transverse axis on the y-axis is\\\\n$$\\\\frac{y^2}{a^2}-\\\\frac{x^2}{b^2}=1$$\\\\nwhere\\\\nthe length of the transverse axis is 2a\\\\nthe coordinates of the vertices are $$(0,a),(0,-a)$$\\\\nthe length of the conjugate axis is $$2b$$\\\\nthe coordinates of the co-vertices are $$(b,0),(-b,0)$$\\\\nthe distance between the foci is 2c, where $$c^2=a^2+b^2$$\\\\nthe coordinates of the foci are $$(0,c),(0,-c)$$\\\\nthe equations of the asymptotes are y=(a/b)*x,y=-(a/b)*x\\\\n(See the right image)\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola1b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y$$"],"dependencies":["a9ed39ahyperbola1b-h1"],"title":"Transverse Axis","text":"Which axis is transverse axis on?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x$$","$$y$$"]},{"id":"a9ed39ahyperbola1b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a9ed39ahyperbola1b-h2"],"title":"Finding the Foci","text":"From the standard equation $$\\\\frac{y^2}{a^2}-\\\\frac{x^2}{b^2}=1$$, the foci are c units away from the center of the hyperbola where $$c^2=a^2+b^2$$.\\\\nTo calculate c, we identify that in the given hyperbola, $$a=\\\\sqrt{49}$$ and $$b=\\\\sqrt{32}$$, thus $$c=\\\\sqrt{a^2+b^2}$$. What is c?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola1b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(0,9),(0,-9)$$"],"dependencies":["a9ed39ahyperbola1b-h3"],"title":"Finding the Foci","text":"We add and subtract $$9$$ units to the center of the hyperbola along the transverse axis to find the two vertices. Given the center is $$(0,0)$$. What are the foci?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(0,9),(0,-9)$$","$$(9,0),(-9,0)$$"]}]}}]},{"id":"a9ed39ahyperbola10","title":"The Hyperbola","body":"Determine whether the following equations represent hyperbolas. If so, write in standard form","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 The Hyperbola","courseName":"OpenStax: College Algebra","steps":[{"id":"a9ed39ahyperbola10a","stepAnswer":["$$\\\\frac{x^2}{6^2}-\\\\frac{y^2}{3^2}=1$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{x^2}{36}-\\\\frac{y^2}{9}=1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{x^2}{6^2}-\\\\frac{y^2}{3^2}=1$$","choices":["$$\\\\frac{x^2}{6^2}-\\\\frac{y^2}{3^2}=1$$","$$\\\\frac{x^2}{3^2}-\\\\frac{y^2}{6^2}=1$$","NA"],"hints":{"DefaultPathway":[{"id":"a9ed39ahyperbola10a-h1","type":"hint","dependencies":[],"title":"Standard Form of Hyperbola","text":"The standard form for hyperbola is either $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}-\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$\\\\nor $$\\\\frac{{\\\\left(y-k\\\\right)}^2}{a^2}-\\\\frac{{\\\\left(x-h\\\\right)}^2}{b^2}=1$$ depending on the transverse axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola10a-h2","type":"hint","dependencies":["a9ed39ahyperbola10a-h1"],"title":"Squaring","text":"We want to square the denominator to get a and $$b$$ so that we can express in the standard form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ed39ahyperbola11","title":"The Hyperbola","body":"Determine whether the following equations represent hyperbolas. If so, write in standard form","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 The Hyperbola","courseName":"OpenStax: College Algebra","steps":[{"id":"a9ed39ahyperbola11a","stepAnswer":["NA"],"problemType":"MultipleChoice","stepTitle":"$$5y^2+4x^2=6x$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$\\\\frac{y^2}{\\\\frac{1}{5}}-\\\\frac{{\\\\left(x-\\\\frac{3}{4}\\\\right)}^2}{\\\\frac{1}{4}}=1$$","$$\\\\frac{y^2}{5}-\\\\frac{{\\\\left(x-\\\\frac{3}{4}\\\\right)}^2}{4}=1$$","NA"],"hints":{"DefaultPathway":[{"id":"a9ed39ahyperbola11a-h1","type":"hint","dependencies":[],"title":"Standard Form of Hyperbola","text":"The standard form for hyperbola is either $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}-\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$\\\\nor $$\\\\frac{{\\\\left(y-k\\\\right)}^2}{a^2}-\\\\frac{{\\\\left(x-h\\\\right)}^2}{b^2}=1$$ depending on the transverse axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola11a-h2","type":"hint","dependencies":["a9ed39ahyperbola11a-h1"],"title":"Sign of Variable","text":"We observe that in the standard form of hyperbola that the two terms have opposing signs (the first term is positive in sign and the second term is negative). We observe that the coefficient of the $$x^2$$ and $$y^2$$ terms are both positive. We will not be able to manipulate the equation to obtain opposing signs. What does this tells us about whether the equation is a hyperbola.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ed39ahyperbola12","title":"The Hyperbola","body":"Determine whether the following equations represent hyperbolas. If so, write in standard form","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 The Hyperbola","courseName":"OpenStax: College Algebra","steps":[{"id":"a9ed39ahyperbola12a","stepAnswer":["$$\\\\frac{x^2}{16}-\\\\frac{y^2}{25}=1$$"],"problemType":"MultipleChoice","stepTitle":"$$25x^2-16y^2=400$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{x^2}{16}-\\\\frac{y^2}{25}=1$$","choices":["$$\\\\frac{x^2}{16}-\\\\frac{y^2}{25}=1$$","$$\\\\frac{x^2}{25}-\\\\frac{y^2}{16}=1$$","NA"],"hints":{"DefaultPathway":[{"id":"a9ed39ahyperbola12a-h1","type":"hint","dependencies":[],"title":"Standard Form of Hyperbola","text":"The standard form for hyperbola is either $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}-\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$\\\\nor $$\\\\frac{{\\\\left(y-k\\\\right)}^2}{a^2}-\\\\frac{{\\\\left(x-h\\\\right)}^2}{b^2}=1$$ depending on the transverse axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola12a-h2","type":"hint","dependencies":["a9ed39ahyperbola12a-h1"],"title":"Manipulating the Equation","text":"If we are not able to tell directly whether an equation is a hyperbola using properties of the equation, we can attempt to force out a similar form to check.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola12a-h3","type":"hint","dependencies":["a9ed39ahyperbola12a-h2"],"title":"Manipulating the Equation","text":"We would want to make the RHS $$1$$. We can do so by dividing the equation by $$400$$ on both side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola12a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{x^2}{16}-\\\\frac{y^2}{25}=1$$"],"dependencies":["a9ed39ahyperbola12a-h3"],"title":"Manipulating the Equation","text":"We observe that there is no coefficient on the numerator. This tells us we should remove the coefficient of $$x^2$$ and $$y^2$$. We can do so by dividing the denominator by the coefficient. We would thus divide $$400$$ by $$25$$ for the $$x^2$$ term and divide $$400$$ by $$16$$ for the $$y^2$$ term. What is the current expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{x^2}{16}-\\\\frac{y^2}{25}=1$$","$$\\\\frac{x^2}{25}-\\\\frac{y^2}{16}=1$$"]},{"id":"a9ed39ahyperbola12a-h5","type":"hint","dependencies":["a9ed39ahyperbola12a-h4"],"title":"Squaring","text":"We observe that the equation we found has the form of a hyperbola. We want to square the denominator to get a and $$b$$ so that we can express in the standard form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ed39ahyperbola13","title":"The Hyperbola","body":"Determine whether the following equations represent hyperbolas. If so, write in standard form","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 The Hyperbola","courseName":"OpenStax: College Algebra","steps":[{"id":"a9ed39ahyperbola13a","stepAnswer":["$$\\\\frac{{\\\\left(y+2\\\\right)}^2}{3^2}-\\\\frac{{\\\\left(x-1\\\\right)}^2}{1^2}=1$$"],"problemType":"MultipleChoice","stepTitle":"$$-9x^2+18x+y^2+4y-14=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{{\\\\left(y+2\\\\right)}^2}{3^2}-\\\\frac{{\\\\left(x-1\\\\right)}^2}{1^2}=1$$","choices":["$$\\\\frac{{\\\\left(y+2\\\\right)}^2}{3^2}-\\\\frac{{\\\\left(x-1\\\\right)}^2}{1^2}=1$$","$$\\\\frac{{\\\\left(y-1\\\\right)}^2}{3^2}-\\\\frac{{\\\\left(x+2\\\\right)}^2}{1^2}=1$$","NA"],"hints":{"DefaultPathway":[{"id":"a9ed39ahyperbola13a-h1","type":"hint","dependencies":[],"title":"Standard Form of Hyperbola","text":"The standard form for hyperbola is either $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}-\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$\\\\nor $$\\\\frac{{\\\\left(y-k\\\\right)}^2}{a^2}-\\\\frac{{\\\\left(x-h\\\\right)}^2}{b^2}=1$$ depending on the transverse axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola13a-h2","type":"hint","dependencies":["a9ed39ahyperbola13a-h1"],"title":"Completing the Square","text":"When there is an $$x^2$$ and $$x$$ term, we can complete the square to obtain an expression that is easier to work with. For $$a x^2+b x$$, we can re-express this as a*(x**2+b/a*x)=a*(x**2+b/a*x+(b/(2*a))**2-(b/(2*a))**2)=a*(x**2+b/a*x+(b/(2*a))**2)-a*(b/(2*a))**2)=a*(x+b/(2*a))**2-a*(b/(2*a))**2)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola13a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9{\\\\left(x-1\\\\right)}^2+9$$"],"dependencies":["a9ed39ahyperbola13a-h2"],"title":"Completing the Square for $$x$$","text":"In the equation, there is a $$-9x^2+18x$$. Note that $$a=-9$$ and $$b=18$$. What expression do we get after completing the square with the formula provided?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(y+2\\\\right)}^2-4$$"],"dependencies":["a9ed39ahyperbola13a-h3"],"title":"Completing the Square for $$y$$","text":"In the equation, there is a $$y^2+4y$$. Note that $$a=1$$ and $$b=4$$. What expression do we get after completing the square with the formula provided?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola13a-h5","type":"hint","dependencies":["a9ed39ahyperbola13a-h4"],"title":"Manipulating the Equation","text":"If we are not able to tell directly whether an equation is a hyperbola using properties of the equation, we can attempt to force out a similar form to check.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola13a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9{\\\\left(x-1\\\\right)}^2+9+{\\\\left(y+2\\\\right)}^2-4-14$$"],"dependencies":["a9ed39ahyperbola13a-h5"],"title":"Manipulating the Equation","text":"Simplifying the equation with the two expression we found by completing the square, what is the expression on the LHS?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola13a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a9ed39ahyperbola13a-h6"],"title":"Manipulating the Equation","text":"Next, we will move all the constant terms (9, $$-4$$, -14) to the RHS. What is value on the RHS?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola13a-h8","type":"hint","dependencies":["a9ed39ahyperbola13a-h7"],"title":"Manipulating the Equation","text":"We would want to make the RHS $$1$$. We can do so by dividing the equation by $$9$$ on both side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola13a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{{\\\\left(y+2\\\\right)}^2}{3^2}-\\\\frac{{\\\\left(x-1\\\\right)}^2}{1^2}=1$$"],"dependencies":["a9ed39ahyperbola13a-h8"],"title":"Manipulating the Equation","text":"We observe that there is no coefficient on the numerator. This tells us we should remove the coefficient of $$x^2$$ and $$y^2$$. We can do so by dividing the denominator by the coefficient. We would thus divide the denominator $$9$$ by $$9$$ for the $$x^2$$ term and divide the denominator $$9$$ by $$1$$ for the $$y^2$$ term. What is the current expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{{\\\\left(y+2\\\\right)}^2}{3^2}-\\\\frac{{\\\\left(x-1\\\\right)}^2}{1^2}=1$$","$$\\\\frac{{\\\\left(y-1\\\\right)}^2}{3^2}-\\\\frac{{\\\\left(x+2\\\\right)}^2}{1^2}=1$$"]}]}}]},{"id":"a9ed39ahyperbola14","title":"The Hyperbola","body":"$$\\\\frac{x^2}{25}-\\\\frac{y^2}{36}=1$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 The Hyperbola","courseName":"OpenStax: College Algebra","steps":[{"id":"a9ed39ahyperbola14a","stepAnswer":["$$\\\\frac{x^2}{5^2}-\\\\frac{y^2}{6^2}=1$$"],"problemType":"TextBox","stepTitle":"Writing Equations of Hyperbolas in Standard Form","stepBody":"Write the equation for the hyperbola in standard form if it not already.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{x^2}{5^2}-\\\\frac{y^2}{6^2}=1$$","hints":{"DefaultPathway":[{"id":"a9ed39ahyperbola14a-h1","type":"hint","dependencies":[],"title":"Hyperbola Standard Form Equation","text":"The standard form for a hyperbola not centered at $$0$$ is $$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{x^2}{5^2}-\\\\frac{y^2}{6^2}$$"],"dependencies":["a9ed39ahyperbola14a-h1"],"title":"Hyperbola Standard Form Equation","text":"Alter the denominator of the given hyperbola equation to match the standard form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ed39ahyperbola14b","stepAnswer":["$$(5,0)$$ and $$(-5,0)$$"],"problemType":"MultipleChoice","stepTitle":"Identify Vertices","stepBody":"Identify the vertices of the equation.","answerType":"string","variabilization":{},"answerLatex":"$$(5,0)$$ and $$(-5,0)$$","choices":["$$(5,0)$$ and $$(-5,0)$$","$$(4,0)$$","$$(0,4)$$","$$(0,5)$$ and $$(0, -5$$"],"hints":{"DefaultPathway":[{"id":"a9ed39ahyperbola14b-h1","type":"hint","dependencies":[],"title":"Transverse Axis","text":"The transverse axis is on the x-axis","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola14b-h2","type":"hint","dependencies":["a9ed39ahyperbola14b-h1"],"title":"Vertices in Standard Form","text":"Vertice coordinates can be found by identifying variable \\"a\\" which is located at the denominator under \\"x**2\\".","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola14b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a9ed39ahyperbola14b-h2"],"title":"Identifying \\"a\\"","text":"What is \\"a\\"?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ed39ahyperbola14c","stepAnswer":["$$(\\\\sqrt{61},0)$$ and $$(-\\\\sqrt{61},0)$$"],"problemType":"MultipleChoice","stepTitle":"Identify Foci","stepBody":"Identify the foci of the equation.","answerType":"string","variabilization":{},"answerLatex":"$$(\\\\sqrt{61},0)$$ and $$(-\\\\sqrt{61},0)$$","choices":["$$(\\\\sqrt{61},0)$$ and $$(-\\\\sqrt{61},0)$$","$$(\\\\sqrt{60},0)$$ and $$(-\\\\sqrt{60},0)$$","$$(\\\\sqrt{62},0)$$ and $$(-\\\\sqrt{62},0)$$"],"hints":{"DefaultPathway":[{"id":"a9ed39ahyperbola14c-h1","type":"hint","dependencies":[],"title":"Foci in Standard Form","text":"Foci coordinates can be found through $$c^2=a^2+b^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola14c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{61}$$"],"dependencies":["a9ed39ahyperbola14c-h1"],"title":"Identifying \\"c\\"","text":"What is \\"c\\"?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ed39ahyperbola14d","stepAnswer":["$$y=\\\\pm \\\\left(\\\\frac{6}{5}\\\\right) x$$"],"problemType":"MultipleChoice","stepTitle":"Identify Asymptotes","stepBody":"Write equations of asymptotes of the hyperbola.","answerType":"string","variabilization":{},"answerLatex":"$$y=\\\\pm \\\\left(\\\\frac{6}{5}\\\\right) x$$","choices":["$$y=\\\\pm \\\\left(\\\\frac{3}{5}\\\\right) x$$","$$y=\\\\frac{3}{5} x$$","$$y=\\\\pm \\\\left(\\\\frac{6}{5}\\\\right) x$$","$$y=\\\\frac{6}{5} x$$"],"hints":{"DefaultPathway":[{"id":"a9ed39ahyperbola14d-h1","type":"hint","dependencies":[],"title":"Asymptotes in Standard Form","text":"Equations of the asymptotes are $$y=\\\\pm \\\\left(\\\\frac{b}{a}\\\\right) x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola14d-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y=\\\\pm \\\\left(\\\\frac{6}{5}\\\\right) x$$"],"dependencies":["a9ed39ahyperbola14d-h1"],"title":"Equation for Asymptote","text":"What is the equation when plugging in known values?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y=\\\\pm \\\\left(\\\\frac{3}{5}\\\\right) x$$","$$y=\\\\frac{3}{5} x$$","$$y=\\\\pm \\\\left(\\\\frac{6}{5}\\\\right) x$$","$$y=\\\\frac{6}{5} x$$"]}]}}]},{"id":"a9ed39ahyperbola15","title":"The Hyperbola","body":"$$\\\\frac{x^2}{100}-\\\\frac{y^2}{9}=1$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 The Hyperbola","courseName":"OpenStax: College Algebra","steps":[{"id":"a9ed39ahyperbola15a","stepAnswer":["$$\\\\frac{x^2}{{10}^2}-\\\\frac{y^2}{3^2}=1$$"],"problemType":"TextBox","stepTitle":"Writing Equations of Hyperbolas in Standard Form","stepBody":"Write the equation for the hyperbola in standard form if it not already.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{x^2}{{10}^2}-\\\\frac{y^2}{3^2}=1$$","hints":{"DefaultPathway":[{"id":"a9ed39ahyperbola15a-h1","type":"hint","dependencies":[],"title":"Hyperbola Standard Form Equation","text":"The standard form for a hyperbola not centered at $$0$$ is $$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{x^2}{{10}^2}-\\\\frac{y^2}{3^2}=1$$"],"dependencies":["a9ed39ahyperbola15a-h1"],"title":"Hyperbola Standard Form Equation","text":"Alter the denominator of the given hyperbola equation to match the standard form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ed39ahyperbola15b","stepAnswer":["$$(10,0)$$ and $$(-10,0)$$"],"problemType":"MultipleChoice","stepTitle":"Identify Vertices","stepBody":"Identify the vertices of the equation.","answerType":"string","variabilization":{},"answerLatex":"$$(10,0)$$ and $$(-10,0)$$","choices":["$$(10,0)$$ and $$(-10,0)$$","$$(9,0)$$ and $$(-9,0)$$","$$(8,0)$$ and $$(-8,0)$$"],"hints":{"DefaultPathway":[{"id":"a9ed39ahyperbola15b-h1","type":"hint","dependencies":[],"title":"Transverse Axis","text":"The transverse axis is on the x-axis","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola15b-h2","type":"hint","dependencies":["a9ed39ahyperbola15b-h1"],"title":"Vertices in Standard Form","text":"Vertice coordinates can be found by identifying variable \\"a\\" which is located at the denominator under \\"x**2\\".","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola15b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["a9ed39ahyperbola15b-h2"],"title":"Identifying \\"a\\"","text":"What is \\"a\\"?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ed39ahyperbola15c","stepAnswer":["$$(\\\\sqrt{109},0)$$ and $$(-\\\\sqrt{109},0)$$"],"problemType":"MultipleChoice","stepTitle":"Identify Foci","stepBody":"Identify the foci of the equation.","answerType":"string","variabilization":{},"answerLatex":"$$(\\\\sqrt{109},0)$$ and $$(-\\\\sqrt{109},0)$$","choices":["$$(\\\\sqrt{109},0)$$ and $$(-\\\\sqrt{109},0)$$","$$(\\\\sqrt{10},0)$$ and $$(-\\\\sqrt{11},0)$$","$$(\\\\sqrt{18},0)$$ and $$(-\\\\sqrt{19},0)$$"],"hints":{"DefaultPathway":[{"id":"a9ed39ahyperbola15c-h1","type":"hint","dependencies":[],"title":"Foci in Standard Form","text":"Foci coordinates can be found through $$c^2=a^2+b^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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y-axis","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola16b-h2","type":"hint","dependencies":["a9ed39ahyperbola16b-h1"],"title":"Vertices in Standard Form","text":"Vertice coordinates can be found by identifying variable \\"a\\" which is located at the denominator under \\"y**2\\".","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola16b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a9ed39ahyperbola16b-h2"],"title":"Identifying \\"a\\"","text":"What is \\"a\\"?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ed39ahyperbola16c","stepAnswer":["(0, sqrt(85)) and (0, -sqrt(85))"],"problemType":"MultipleChoice","stepTitle":"Identify Foci","stepBody":"Identify the foci of the equation.","answerType":"string","variabilization":{},"choices":["(0, sqrt(85)) and (0, -sqrt(85))","(0, sqrt(8)) and (0, -sqrt(8))","(0, sqrt(65)) and (0, -sqrt(65))"],"hints":{"DefaultPathway":[{"id":"a9ed39ahyperbola16c-h1","type":"hint","dependencies":[],"title":"Foci in Standard Form","text":"Foci coordinates can be found through $$c^2=a^2+b^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola16c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{85}$$"],"dependencies":["a9ed39ahyperbola16c-h1"],"title":"Identifying \\"c\\"","text":"What is \\"c\\"?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ed39ahyperbola16d","stepAnswer":["$$y=\\\\pm \\\\left(\\\\frac{2}{9}\\\\right) x$$"],"problemType":"TextBox","stepTitle":"Identify Asymptotes","stepBody":"Write equations of asymptotes of the hyperbola.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=\\\\pm \\\\left(\\\\frac{2}{9}\\\\right) x$$","hints":{"DefaultPathway":[{"id":"a9ed39ahyperbola16d-h1","type":"hint","dependencies":[],"title":"Asymptotes in Standard Form","text":"Equations of the asymptotes are $$y=\\\\pm \\\\left(\\\\frac{a}{b}\\\\right) x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola16d-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y=\\\\pm \\\\left(\\\\frac{2}{9}\\\\right) x$$"],"dependencies":["a9ed39ahyperbola16d-h1"],"title":"Equation for Asymptote","text":"What is the equation when plugging in known values?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ed39ahyperbola17","title":"The Hyperbola","body":"$$9y^2-4x^2=1$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 The Hyperbola","courseName":"OpenStax: College Algebra","steps":[{"id":"a9ed39ahyperbola17a","stepAnswer":["$$\\\\frac{y^2}{{\\\\left(\\\\frac{1}{3}\\\\right)}^2}-\\\\frac{x^2}{{\\\\left(\\\\frac{1}{2}\\\\right)}^2}=1$$"],"problemType":"TextBox","stepTitle":"Writing Equations of Hyperbolas in Standard Form","stepBody":"Write the equation for the hyperbola in standard form if it not already.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{y^2}{{\\\\left(\\\\frac{1}{3}\\\\right)}^2}-\\\\frac{x^2}{{\\\\left(\\\\frac{1}{2}\\\\right)}^2}=1$$","hints":{"DefaultPathway":[{"id":"a9ed39ahyperbola17a-h1","type":"hint","dependencies":[],"title":"Hyperbola Standard Form 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exponent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ed39ahyperbola17b","stepAnswer":["$$(0,\\\\frac{1}{3})$$ and $$(0,\\\\frac{-1}{3})$$"],"problemType":"MultipleChoice","stepTitle":"Identify Vertices","stepBody":"Identify the vertices of the equation.","answerType":"string","variabilization":{},"answerLatex":"$$(0,\\\\frac{1}{3})$$ and $$(0,\\\\frac{-1}{3})$$","choices":["$$(0,\\\\frac{1}{3})$$ and $$(0,\\\\frac{-1}{3})$$","$$(0,3)$$ and $$(0,\\\\frac{-1}{3})$$"],"hints":{"DefaultPathway":[{"id":"a9ed39ahyperbola17b-h1","type":"hint","dependencies":[],"title":"Transverse Axis","text":"The transverse axis is on the y-axis","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola17b-h2","type":"hint","dependencies":["a9ed39ahyperbola17b-h1"],"title":"Vertices in Standard Form","text":"Vertice coordinates can be found by identifying variable \\"a\\" which is located at the denominator under \\"y**2\\".","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola17b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["a9ed39ahyperbola17b-h2"],"title":"Identifying \\"a\\"","text":"What is \\"a\\"?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ed39ahyperbola18","title":"The Hyperbola","body":"$$\\\\frac{y^2}{3^2}-\\\\frac{x^2}{3^2}=1$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 The Hyperbola","courseName":"OpenStax: College 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\\\\left(x+3\\\\right)+4$$"],"problemType":"MultipleChoice","stepTitle":"Finding Asymptotes","stepBody":"Find the equations of the asymptotes for each hyperbola.","answerType":"string","variabilization":{},"answerLatex":"$$y=\\\\pm \\\\left(\\\\frac{2}{5}\\\\right) \\\\left(x+3\\\\right)+4$$","choices":["$$y=\\\\pm \\\\left(\\\\frac{2}{5}\\\\right) \\\\left(x+3\\\\right)+4$$","$$y=\\\\pm \\\\left(\\\\frac{2}{5}\\\\right) x+4$$","$$y=\\\\pm \\\\left(\\\\frac{2}{5}\\\\right) \\\\left(x+2\\\\right)+4$$","$$y\\\\pm \\\\frac{2}{5} \\\\left(x+3\\\\right)+8$$"],"hints":{"DefaultPathway":[{"id":"a9ed39ahyperbola19a-h1","type":"hint","dependencies":[],"title":"Transverse Axis","text":"The transverse axis is on the x-axis","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola19a-h2","type":"hint","dependencies":["a9ed39ahyperbola19a-h1"],"title":"Asymptotes in Standard Form","text":"Equations of the asymptotes are $$y=\\\\pm \\\\left(\\\\frac{b}{a}\\\\right) \\\\left(x-h\\\\right)+k$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola19a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y=\\\\pm \\\\left(\\\\frac{2}{5}\\\\right) \\\\left(x+3\\\\right)+4$$"],"dependencies":["a9ed39ahyperbola19a-h2"],"title":"Equation for Asymptote","text":"What is the equation when plugging in known values?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y=\\\\pm \\\\left(\\\\frac{2}{5}\\\\right) \\\\left(x+3\\\\right)+4$$","$$y=\\\\pm \\\\left(\\\\frac{2}{5}\\\\right) x+4$$","$$y=\\\\pm \\\\left(\\\\frac{2}{5}\\\\right) \\\\left(x+2\\\\right)+4$$","$$y\\\\pm \\\\frac{2}{5} \\\\left(x+3\\\\right)+8$$"]}]}}]},{"id":"a9ed39ahyperbola2","title":"Locating a 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the vertices are $$(a,0),(-a,0)$$\\\\nthe length of the conjugate axis is $$2b$$\\\\nthe coordinates of the co-vertices are (0,b),(0,b)\\\\nthe distance between the foci is 2c, where $$c^2=a^2+b^2$$\\\\nthe coordinates of the foci are $$(c,0),(-c,0)$$\\\\nthe equations of the asymptotes are $$y=\\\\frac{b}{a} x$$, $$y=-\\\\left(\\\\frac{b}{a}\\\\right) x$$\\\\n(See the left image)\\\\n\\\\nThe standard form of the equation of a hyperbola with center $$(0,0)$$ and transverse axis on the y-axis is\\\\n$$\\\\frac{y^2}{a^2}-\\\\frac{x^2}{b^2}=1$$\\\\nwhere\\\\nthe length of the transverse axis is 2a\\\\nthe coordinates of the vertices are $$(0,a),(0,-a)$$\\\\nthe length of the conjugate axis is $$2b$$\\\\nthe coordinates of the co-vertices are $$(b,0),(-b,0)$$\\\\nthe distance between the foci is 2c, where $$c^2=a^2+b^2$$\\\\nthe coordinates of the foci are $$(0,c),(0,-c)$$\\\\nthe equations of the asymptotes are y=(a/b)*x,y=-(a/b)*x\\\\n(See the right image)\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola2a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x$$"],"dependencies":["a9ed39ahyperbola2a-h1"],"title":"Transverse Axis","text":"Which axis is transverse axis on?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x$$","$$y$$"]},{"id":"a9ed39ahyperbola2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a9ed39ahyperbola2a-h2"],"title":"Finding the Vertices","text":"From the standard equation $$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$, the vertices are a units away from the center of the hyperbola. What is a?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola2a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(3,0),(-3,0)$$"],"dependencies":["a9ed39ahyperbola2a-h3"],"title":"Finding the Vertices","text":"We add and subtract $$3$$ units to the center of the hyperbola along the transverse axis to find the two vertices. Given the center is $$(0,0)$$. What are the vertices?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(3,0),(-3,0)$$","$$(0,5),(0,-5)$$"]}]}},{"id":"a9ed39ahyperbola2b","stepAnswer":["(sqrt(34),0),(-sqrt(34),0)"],"problemType":"MultipleChoice","stepTitle":"Find the foci of $$\\\\frac{x^2}{9}-\\\\frac{y^2}{25}=1$$","stepBody":"","answerType":"string","variabilization":{},"choices":["(sqrt(34),0),(-sqrt(34),0)","(0,sqrt(34)),(0,-sqrt(34))"],"hints":{"DefaultPathway":[{"id":"a9ed39ahyperbola2b-h1","type":"hint","dependencies":[],"title":"Standard Form of Hyperbola with Center $$(0,0)$$","text":"The standard form of the equation of a hyperbola with center $$(0,0)$$ and transverse axis on the x-axis is\\\\n$$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$\\\\nwhere\\\\nthe length of the transverse axis is 2a\\\\nthe coordinates of the vertices are $$(a,0),(-a,0)$$\\\\nthe length of the conjugate axis is $$2b$$\\\\nthe coordinates of the co-vertices are (0,b),(0,b)\\\\nthe distance between the foci is 2c, where $$c^2=a^2+b^2$$\\\\nthe coordinates of the foci are $$(c,0),(-c,0)$$\\\\nthe equations of the asymptotes are $$y=\\\\frac{b}{a} x$$, $$y=-\\\\left(\\\\frac{b}{a}\\\\right) x$$\\\\n(See the left image)\\\\n\\\\nThe standard form of the equation of a hyperbola with center $$(0,0)$$ and transverse axis on the y-axis is\\\\n$$\\\\frac{y^2}{a^2}-\\\\frac{x^2}{b^2}=1$$\\\\nwhere\\\\nthe length of the transverse axis is 2a\\\\nthe coordinates of the vertices are $$(0,a),(0,-a)$$\\\\nthe length of the conjugate axis is $$2b$$\\\\nthe coordinates of the co-vertices are $$(b,0),(-b,0)$$\\\\nthe distance between the foci is 2c, where $$c^2=a^2+b^2$$\\\\nthe coordinates of the foci are $$(0,c),(0,-c)$$\\\\nthe equations of the asymptotes are y=(a/b)*x,y=-(a/b)*x\\\\n(See the right image)\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola2b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x$$"],"dependencies":["a9ed39ahyperbola2b-h1"],"title":"Transverse Axis","text":"Which axis is transverse axis on?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x$$","$$y$$"]},{"id":"a9ed39ahyperbola2b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{34}$$"],"dependencies":["a9ed39ahyperbola2b-h2"],"title":"Finding the Foci","text":"From the standard equation $$\\\\frac{y^2}{a^2}-\\\\frac{x^2}{b^2}=1$$, the foci are c units away from the center of the hyperbola where $$c^2=a^2+b^2$$.\\\\nTo calculate c, we identify that in the given hyperbola, $$a=\\\\sqrt{9}$$ and $$b=\\\\sqrt{25}$$, thus $$c=\\\\sqrt{a^2+b^2}$$. What is c?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola2b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["(sqrt(34),0),(-sqrt(34),0)"],"dependencies":["a9ed39ahyperbola2b-h3"],"title":"Finding the Foci","text":"We add and subtract $$\\\\sqrt{34}$$ units to the center of the hyperbola along the transverse axis to find the two vertices. Given the center is $$(0,0)$$. What are the foci?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["(sqrt(34),0),(-sqrt(34),0)","(0,sqrt(34)),(0,-sqrt(34))"]}]}}]},{"id":"a9ed39ahyperbola20","title":"The Hyperbola","body":"$$\\\\frac{{\\\\left(y-3\\\\right)}^2}{3^2}-\\\\frac{{\\\\left(x+5\\\\right)}^2}{6^2}=1$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 The Hyperbola","courseName":"OpenStax: College Algebra","steps":[{"id":"a9ed39ahyperbola20a","stepAnswer":["$$y=\\\\pm \\\\left(\\\\frac{1}{2}\\\\right) \\\\left(x+5\\\\right)+3$$"],"problemType":"MultipleChoice","stepTitle":"Finding Asymptotes","stepBody":"Find the equations of the asymptotes for each hyperbola.","answerType":"string","variabilization":{},"answerLatex":"$$y=\\\\pm \\\\left(\\\\frac{1}{2}\\\\right) \\\\left(x+5\\\\right)+3$$","choices":["$$y=\\\\pm \\\\left(\\\\frac{1}{2}\\\\right) \\\\left(x+5\\\\right)+3$$","$$y=\\\\pm \\\\left(\\\\frac{1}{4}\\\\right) \\\\left(x+5\\\\right)+3$$","$$y=\\\\frac{1}{4} \\\\left(x+5\\\\right)+3$$","$$y=\\\\frac{1}{2} \\\\left(x+5\\\\right)+3$$"],"hints":{"DefaultPathway":[{"id":"a9ed39ahyperbola20a-h1","type":"hint","dependencies":[],"title":"Transverse Axis","text":"The transverse axis is on the y-axis","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola20a-h2","type":"hint","dependencies":["a9ed39ahyperbola20a-h1"],"title":"Asymptotes in Standard Form","text":"Equations of the asymptotes are $$y=\\\\pm \\\\left(\\\\frac{a}{b}\\\\right) \\\\left(x-h\\\\right)+k$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola20a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y=\\\\pm \\\\left(\\\\frac{1}{2}\\\\right) \\\\left(x+5\\\\right)+3$$"],"dependencies":["a9ed39ahyperbola20a-h2"],"title":"Equation for Asymptote","text":"What is the equation when plugging in known values?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y=\\\\pm \\\\left(\\\\frac{1}{2}\\\\right) \\\\left(x+5\\\\right)+3$$","$$y=\\\\pm \\\\left(\\\\frac{1}{4}\\\\right) \\\\left(x+5\\\\right)+3$$","$$y=\\\\frac{1}{4} \\\\left(x+5\\\\right)+3$$","$$y=\\\\frac{1}{2} \\\\left(x+5\\\\right)+3$$"]}]}}]},{"id":"a9ed39ahyperbola21","title":"The Hyperbola","body":"$$9x^2-18x-16y^2+32y-151=0$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 The Hyperbola","courseName":"OpenStax: College Algebra","steps":[{"id":"a9ed39ahyperbola21a","stepAnswer":["$$\\\\frac{{\\\\left(x-1\\\\right)}^2}{4^2}-\\\\frac{{\\\\left(y-1\\\\right)}^2}{3^2}=1$$"],"problemType":"TextBox","stepTitle":"Writing Equations of Hyperbolas in Standard Form","stepBody":"Write the equation for the hyperbola in standard form if it not already.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{{\\\\left(x-1\\\\right)}^2}{4^2}-\\\\frac{{\\\\left(y-1\\\\right)}^2}{3^2}=1$$","hints":{"DefaultPathway":[{"id":"a9ed39ahyperbola21a-h1","type":"hint","dependencies":[],"title":"Grouping Terms","text":"Express the equation in standard form by first grouping the terms with the same variable so you get $$9x^2-18x-16y^2+32y=151$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola21a-h2","type":"hint","dependencies":["a9ed39ahyperbola21a-h1"],"title":"Factoring Out Coefficient","text":"Factor the leading coefficient of each expression so you get $$9\\\\left(x^2-2x\\\\right)-\\\\operatorname{16}\\\\left(y^2-2y\\\\right)=151$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola21a-h3","type":"hint","dependencies":["a9ed39ahyperbola21a-h2"],"title":"Complete the Square","text":"Complete the square twice and make sure to balance the equation by adding same constants on each side so you\'ll have $$9\\\\left(x^2-2x+1\\\\right)-\\\\operatorname{16}\\\\left(y^2-2y+1\\\\right)=151+9-16$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola21a-h4","type":"hint","dependencies":["a9ed39ahyperbola21a-h3"],"title":"Rewrite and Divide","text":"Rewrite as perfect squares and divide both sides by constants so that it\'s in standard form $$\\\\frac{{\\\\left(x-1\\\\right)}^2}{4^2}-\\\\frac{{\\\\left(y-1\\\\right)}^2}{3^2}=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9ed39ahyperbola21b","stepAnswer":["$$y=\\\\pm \\\\left(\\\\frac{3}{4}\\\\right) \\\\left(x-1\\\\right)+1$$"],"problemType":"MultipleChoice","stepTitle":"Finding Asymptotes","stepBody":"Find the equations of the asymptotes for each hyperbola.","answerType":"string","variabilization":{},"answerLatex":"$$y=\\\\pm \\\\left(\\\\frac{3}{4}\\\\right) \\\\left(x-1\\\\right)+1$$","choices":["$$y=\\\\pm \\\\left(\\\\frac{4}{3}\\\\right) \\\\left(x-1\\\\right)+1$$","$$y=\\\\pm \\\\left(\\\\frac{3}{4}\\\\right) \\\\left(x-1\\\\right)+1$$","$$y=\\\\frac{3}{4} \\\\left(x-1\\\\right)+1$$"],"hints":{"DefaultPathway":[{"id":"a9ed39ahyperbola21b-h1","type":"hint","dependencies":[],"title":"Transverse Axis","text":"The transverse axis is on the x-axis","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola21b-h2","type":"hint","dependencies":["a9ed39ahyperbola21b-h1"],"title":"Asymptotes in Standard Form","text":"Equations of the asymptotes are $$y=\\\\pm \\\\left(\\\\frac{b}{a}\\\\right) \\\\left(x-h\\\\right)+k$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola21b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y=\\\\pm \\\\left(\\\\frac{3}{4}\\\\right) \\\\left(x-1\\\\right)+1$$"],"dependencies":["a9ed39ahyperbola21b-h2"],"title":"Equation for Asymptote","text":"What is the equation when plugging in known values?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y=\\\\pm \\\\left(\\\\frac{4}{3}\\\\right) \\\\left(x-1\\\\right)+1$$","$$y=\\\\pm \\\\left(\\\\frac{3}{4}\\\\right) \\\\left(x-1\\\\right)+1$$","$$y=\\\\frac{3}{4} \\\\left(x-1\\\\right)+1$$"]}]}}]},{"id":"a9ed39ahyperbola22","title":"The Hyperbola","body":"Vertices at $$(3,0)$$ and $$(-3,0)$$ and one focus at $$(5,0)$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 The Hyperbola","courseName":"OpenStax: College Algebra","steps":[{"id":"a9ed39ahyperbola22a","stepAnswer":["$$\\\\frac{x^2}{9}-\\\\frac{y^2}{16}=1$$"],"problemType":"TextBox","stepTitle":"Writing Equations of Hyperbolas in Standard Form","stepBody":"Given information about the graph of the hyperbola, find its equation.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{x^2}{9}-\\\\frac{y^2}{16}=1$$","hints":{"DefaultPathway":[{"id":"a9ed39ahyperbola22a-h1","type":"hint","dependencies":[],"title":"Finding \\"a\\"","text":"The vertices are $$(3,0)$$ and $$(-3,0)$$, so $$a=3$$ and $$a^2=9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola22a-h2","type":"hint","dependencies":["a9ed39ahyperbola22a-h1"],"title":"Finding \\"c\\"","text":"The foci is $$(5,0)$$, so $$c=5$$ and $$c^2=25$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola22a-h3","type":"hint","dependencies":["a9ed39ahyperbola22a-h2"],"title":"Finding \\"b\\"","text":"To solve for $$b^2$$ use $$b^2=c^2-a^2$$ and plug in known values $$b^2=25-9=16$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola22a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{x^2}{9}-\\\\frac{y^2}{16}=1$$"],"dependencies":["a9ed39ahyperbola22a-h3"],"title":"Substitute All Values","text":"Substitute $$a^2$$ and $$b^2$$ into the standard equation form, $$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ed39ahyperbola23","title":"The Hyperbola","body":"Vertices at $$(0,6)$$ and $$(0,-6)$$ and one focus at $$(0,-8)$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 The Hyperbola","courseName":"OpenStax: College Algebra","steps":[{"id":"a9ed39ahyperbola23a","stepAnswer":["$$\\\\frac{x^2}{9}-\\\\frac{y^2}{16}=1$$"],"problemType":"TextBox","stepTitle":"Writing Equations of Hyperbolas in Standard Form","stepBody":"Given information about the graph of the hyperbola, find its equation.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{x^2}{9}-\\\\frac{y^2}{16}=1$$","hints":{"DefaultPathway":[{"id":"a9ed39ahyperbola23a-h1","type":"hint","dependencies":[],"title":"Finding \\"a\\"","text":"The vertices are $$(0,6)$$ and $$(0,-6)$$, so $$a=6$$ and $$a^2=36$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola23a-h2","type":"hint","dependencies":["a9ed39ahyperbola23a-h1"],"title":"Finding \\"c\\"","text":"The foci is $$(0,-8)$$, so $$c=-8$$ and $$c^2=64$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola23a-h3","type":"hint","dependencies":["a9ed39ahyperbola23a-h2"],"title":"Finding \\"b\\"","text":"To solve for $$b^2$$ use $$b^2=c^2-a^2$$ and plug in known values $$b^2=64-36=28$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola23a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{y^2}{36}-\\\\frac{x^2}{28}=1$$"],"dependencies":["a9ed39ahyperbola23a-h3"],"title":"Substitute All Values","text":"Substitute $$a^2$$ and $$b^2$$ into the standard equation form, $$\\\\frac{y^2}{a^2}-\\\\frac{x^2}{b^2}=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ed39ahyperbola24","title":"The Hyperbola","body":"Vertices at $$(1,1)$$ and $$(11,1)$$ and one focus at $$(12,1)$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 The Hyperbola","courseName":"OpenStax: College Algebra","steps":[{"id":"a9ed39ahyperbola24a","stepAnswer":["$$\\\\frac{{\\\\left(x-6\\\\right)}^2}{25}-\\\\frac{{\\\\left(y-1\\\\right)}^2}{11}=1$$"],"problemType":"TextBox","stepTitle":"Writing Equations of Hyperbolas in Standard Form","stepBody":"Given information about the graph of the hyperbola, find its equation.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{{\\\\left(x-6\\\\right)}^2}{25}-\\\\frac{{\\\\left(y-1\\\\right)}^2}{11}=1$$","hints":{"DefaultPathway":[{"id":"a9ed39ahyperbola24a-h1","type":"hint","dependencies":[],"title":"Finding Center","text":"Identify the center (h,k). The center is halfway between vertices $$(1,1)$$ and $$(11,1)$$. Applying the midpoint formular, we have $$(6,1)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola24a-h2","type":"hint","dependencies":["a9ed39ahyperbola24a-h1"],"title":"Solving for $$a^2$$","text":"Next, we find $$a^2$$. The length of the transverse axis, 2a, is bounded by ther vertices. So, we can find $$a^2$$ by finding the distance between the x-coordinates of ther vertices so $$a=5$$ and $$a^2=25$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola24a-h3","type":"hint","dependencies":["a9ed39ahyperbola24a-h2"],"title":"Solving for $$c^2$$","text":"Now we need to find $$c^2$$. The coordinates of the foci are (h~c,k). So $$(h+c,k)=(12,1)$$. We can use the x-coordinate from this point to solve for c using $$h=6$$ so $$c=6$$ and $$c^2=36$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola24a-h4","type":"hint","dependencies":["a9ed39ahyperbola24a-h3"],"title":"Solving for $$b^2$$","text":"Next, solve for $$b^2$$ using the equation $$b^2=c^2-a^2=36-25=11$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola24a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{{\\\\left(x-6\\\\right)}^2}{25}-\\\\frac{{\\\\left(y-1\\\\right)}^2}{11}=1$$"],"dependencies":["a9ed39ahyperbola24a-h4"],"title":"Substitute All Values","text":"Substitute h,k, $$a^2$$ and $$b^2$$ into the standard equation form","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ed39ahyperbola3","title":"Finding the Equation of a Hyperbola Centered at $$(0,0)$$ given its Foci and Vertices","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 The Hyperbola","courseName":"OpenStax: College Algebra","steps":[{"id":"a9ed39ahyperbola3a","stepAnswer":["$$\\\\frac{x^2}{6^2}-\\\\frac{y^2}{2^2}=1$$"],"problemType":"MultipleChoice","stepTitle":"What is the standard form equation of the hyperbola that has vertices $$(6,0),(-6,0)$$ and foci (2*sqrt(10),0),(-2*sqrt(10),0)?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{x^2}{6^2}-\\\\frac{y^2}{2^2}=1$$","choices":["$$\\\\frac{x^2}{6^2}-\\\\frac{y^2}{2^2}=1$$","$$\\\\frac{y^2}{6^2}-\\\\frac{x^2}{2^2}=1$$"],"hints":{"DefaultPathway":[{"id":"a9ed39ahyperbola3a-h1","type":"hint","dependencies":[],"title":"Standard Form of Hyperbola with Center $$(0,0)$$","text":"The standard form of the equation of a hyperbola with center $$(0,0)$$ and transverse axis on the x-axis is\\\\n$$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$\\\\nwhere\\\\nthe length of the transverse axis is 2a\\\\nthe coordinates of the vertices are $$(a,0),(-a,0)$$\\\\nthe length of the conjugate axis is $$2b$$\\\\nthe coordinates of the co-vertices are (0,b),(0,b)\\\\nthe distance between the foci is 2c, where $$c^2=a^2+b^2$$\\\\nthe coordinates of the foci are $$(c,0),(-c,0)$$\\\\nthe equations of the asymptotes are y=(b/a)*x,y=-(b/a)*x\\\\n(See the left image)\\\\n\\\\nThe standard form of the equation of a hyperbola with center $$(0,0)$$ and transverse axis on the y-axis is\\\\n$$\\\\frac{y^2}{a^2}-\\\\frac{x^2}{b^2}=1$$\\\\nwhere\\\\nthe length of the transverse axis is 2a\\\\nthe coordinates of the vertices are $$(0,a),(0,-a)$$\\\\nthe length of the conjugate axis is $$2b$$\\\\nthe coordinates of the co-vertices are $$(b,0),(-b,0)$$\\\\nthe distance between the foci is 2c, where $$c^2=a^2+b^2$$\\\\nthe coordinates of the foci are $$(0,c),(0,-c)$$\\\\nthe equations of the asymptotes are y=(a/b)*x,y=-(a/b)*x\\\\n(See the right image)\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola3a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x$$"],"dependencies":["a9ed39ahyperbola3a-h1"],"title":"Transverse Axis","text":"Which axis is transverse axis on? We can tell based on the coordinates of the vertices and foci.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x$$","$$y$$"]},{"id":"a9ed39ahyperbola3a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$"],"dependencies":["a9ed39ahyperbola3a-h2"],"title":"Appropriate Standard Form","text":"Since the transverse axis is on the x-axis, which standard form for hyperbola should we use?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$","$$\\\\frac{y^2}{a^2}-\\\\frac{x^2}{b^2}=1$$"]},{"id":"a9ed39ahyperbola3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a9ed39ahyperbola3a-h3"],"title":"Vertices","text":"It is given that vertices are $$(6,0)$$ and $$(-6,0)$$ with the center at $$(0,0)$$. Thus, what is a in the standard form of hyperbola?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2\\\\sqrt{10}$$"],"dependencies":["a9ed39ahyperbola3a-h4"],"title":"Foci","text":"It is given that foci are $$(2\\\\sqrt{10},0)$$ and $$(-2\\\\sqrt{10},0)$$ with the center at $$(0,0)$$. Thus, what is c associated with the hyperbola?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola3a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a9ed39ahyperbola3a-h5"],"title":"Finding $$b$$","text":"We know that $$c^2=a^2+b^2$$, we can thus rearrange the equation to $$b=\\\\sqrt{c^2-a^2}$$. With the a and c previously found, what is $$b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola3a-h7","type":"hint","dependencies":["a9ed39ahyperbola3a-h6"],"title":"Equation","text":"Having found a and $$b$$, we can now substitute them into the standard form equation $$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ed39ahyperbola4","title":"Finding the Equation of a Hyperbola Centered at $$(0,0)$$ given its Foci and Vertices","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 The Hyperbola","courseName":"OpenStax: College Algebra","steps":[{"id":"a9ed39ahyperbola4a","stepAnswer":["$$\\\\frac{y^2}{2^2}-\\\\frac{x^2}{4^2}=1$$"],"problemType":"MultipleChoice","stepTitle":"What is the standard form equation of the hyperbola that has vertices $$(0,2),(0,-2)$$ and foci (0,2*sqrt(5)),(0,-2*sqrt(5))?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{y^2}{2^2}-\\\\frac{x^2}{4^2}=1$$","choices":["$$\\\\frac{y^2}{2^2}-\\\\frac{x^2}{4^2}=1$$","$$\\\\frac{x^2}{2^2}-\\\\frac{y^2}{4^2}=1$$"],"hints":{"DefaultPathway":[{"id":"a9ed39ahyperbola4a-h1","type":"hint","dependencies":[],"title":"Standard Form of Hyperbola with Center $$(0,0)$$","text":"The standard form of the equation of a hyperbola with center $$(0,0)$$ and transverse axis on the x-axis is\\\\n$$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$\\\\nwhere\\\\nthe length of the transverse axis is 2a\\\\nthe coordinates of the vertices are $$(a,0),(-a,0)$$\\\\nthe length of the conjugate axis is $$2b$$\\\\nthe coordinates of the co-vertices are (0,b),(0,b)\\\\nthe distance between the foci is 2c, where $$c^2=a^2+b^2$$\\\\nthe coordinates of the foci are $$(c,0),(-c,0)$$\\\\nthe equations of the asymptotes are $$y=\\\\frac{b}{a} x$$, $$y=-\\\\left(\\\\frac{b}{a}\\\\right) x$$\\\\n(See the left image)\\\\n\\\\nThe standard form of the equation of a hyperbola with center $$(0,0)$$ and transverse axis on the y-axis is\\\\n$$\\\\frac{y^2}{a^2}-\\\\frac{x^2}{b^2}=1$$\\\\nwhere\\\\nthe length of the transverse axis is 2a\\\\nthe coordinates of the vertices are $$(0,a),(0,-a)$$\\\\nthe length of the conjugate axis is $$2b$$\\\\nthe coordinates of the co-vertices are $$(b,0),(-b,0)$$\\\\nthe distance between the foci is 2c, where $$c^2=a^2+b^2$$\\\\nthe coordinates of the foci are $$(0,c),(0,-c)$$\\\\nthe equations of the asymptotes are y=(a/b)*x,y=-(a/b)*x\\\\n(See the right image)\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola4a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y$$"],"dependencies":["a9ed39ahyperbola4a-h1"],"title":"Transverse Axis","text":"Which axis is transverse axis on? We can tell based on the coordinates of the vertices and foci.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x$$","$$y$$"]},{"id":"a9ed39ahyperbola4a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{y^2}{a^2}-\\\\frac{x^2}{b^2}=1$$"],"dependencies":["a9ed39ahyperbola4a-h2"],"title":"Appropriate Standard Form","text":"Since the transverse axis is on the y-axis, which standard form for hyperbola should we use?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$","$$\\\\frac{y^2}{a^2}-\\\\frac{x^2}{b^2}=1$$"]},{"id":"a9ed39ahyperbola4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["a9ed39ahyperbola4a-h3"],"title":"Vertices","text":"It is given that vertices are $$(0,2)$$ and $$(0,-2)$$ with the center at $$(0,0)$$. Thus, what is a in the standard form of hyperbola?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola4a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2\\\\sqrt{5}$$"],"dependencies":["a9ed39ahyperbola4a-h4"],"title":"Foci","text":"It is given that foci are $$(0,2\\\\sqrt{5})$$ and $$(0,-2\\\\sqrt{5})$$ with the center at $$(0,0)$$. Thus, what is c associated with the hyperbola?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola4a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a9ed39ahyperbola4a-h5"],"title":"Finding $$b$$","text":"We know that $$c^2=a^2+b^2$$, we can thus rearrange the equation to $$b=\\\\sqrt{c^2-a^2}$$. With the a and c previously found, what is $$b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola4a-h7","type":"hint","dependencies":["a9ed39ahyperbola4a-h6"],"title":"Equation","text":"Having found a and $$b$$, we can now substitute them into the standard form equation $$\\\\frac{y^2}{a^2}-\\\\frac{x^2}{b^2}=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ed39ahyperbola5","title":"Finding the Equation of a Hyperbola Centered at (h,k) given its Foci and Vertices","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 The Hyperbola","courseName":"OpenStax: College Algebra","steps":[{"id":"a9ed39ahyperbola5a","stepAnswer":["$$\\\\frac{{\\\\left(x-3\\\\right)}^2}{3^2}-\\\\frac{{\\\\left(y+2\\\\right)}^2}{4^2}=1$$"],"problemType":"MultipleChoice","stepTitle":"What is the standard form equation of the hyperbola that has vertices at $$(0,-2)$$ and $$(6,-2)$$ and foci at $$(-2,2)$$ and $$(8,-2)$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{{\\\\left(x-3\\\\right)}^2}{3^2}-\\\\frac{{\\\\left(y+2\\\\right)}^2}{4^2}=1$$","choices":["$$\\\\frac{{\\\\left(x-3\\\\right)}^2}{3^2}-\\\\frac{{\\\\left(y+2\\\\right)}^2}{4^2}=1$$","$$\\\\frac{{\\\\left(y-3\\\\right)}^2}{3^2}-\\\\frac{{\\\\left(x+2\\\\right)}^2}{4^2}=1$$"],"hints":{"DefaultPathway":[{"id":"a9ed39ahyperbola5a-h1","type":"hint","dependencies":[],"title":"Standard Form of Hyperbola with Center (h,k)","text":"The standard form of the equation of a hyperbola with center (h,k) and transverse axis parallel to the x-axis is\\\\n$$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}-\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$\\\\nwhere\\\\nthe length of the transverse axis is 2a\\\\nthe coordinates of the vertices are (h+a,k),(h-a,k)\\\\nthe length of the conjugate axis is $$2b$$\\\\nthe coordinates of the co-vertices are (h,k+b),(h,k-b)\\\\nthe distance between the foci is 2c, where $$c^2=a^2+b^2$$\\\\nthe coordinates of the foci are (h+c,k),(h-c,k)\\\\nThe asymptotes of the hyperbola coincide with the diagonals of the central rectangle. The length of the rectangle is 2a and its width is $$2b$$. The slopes of the diagonals are $$\\\\frac{b}{a}$$ and $$\\\\frac{-b}{a}$$, and each diagonal passes through the center (h,k). Using the point-slope formula, it is simple to show that the equations of the asymptotes are $$y=\\\\frac{b}{a} \\\\left(x-h\\\\right)+k$$ and $$y=\\\\frac{-b}{a} \\\\left(x-h\\\\right)+k$$.\\\\n(See the left image)\\\\n\\\\nThe standard form of the equation of a hyperbola with center (h,k) and transverse axis parallel to the y-axis is\\\\n$$\\\\frac{{\\\\left(y-k\\\\right)}^2}{a^2}-\\\\frac{{\\\\left(x-h\\\\right)}^2}{b^2}=1$$\\\\nwhere\\\\nthe length of the transverse axis is 2a\\\\nthe coordinates of the vertices are (h,k+a),(h,k-a)\\\\nthe length of the conjugate axis is $$2b$$\\\\nthe coordinates of the co-vertices are (h+b,k),(h-b,k)\\\\nthe distance between the foci is 2c, where $$c^2=a^2+b^2$$\\\\nthe coordinates of the foci are (h,k+c),(h,k-c)\\\\nUsing the reasoning above, the equations of the asymptotes are $$y=\\\\frac{a}{b} \\\\left(x-h\\\\right)+k$$ and $$y=\\\\frac{-a}{b} \\\\left(x-h\\\\right)+k$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola5a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x$$"],"dependencies":["a9ed39ahyperbola5a-h1"],"title":"Transverse Axis","text":"Which axis is transverse axis on? We can tell based on the coordinates of the vertices and foci.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x$$","$$y$$"]},{"id":"a9ed39ahyperbola5a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}-\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$"],"dependencies":["a9ed39ahyperbola5a-h2"],"title":"Appropriate Standard Form","text":"Since the transverse axis is on the x-axis, which standard form for hyperbola should we use?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}-\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$","$$\\\\frac{{\\\\left(y-k\\\\right)}^2}{a^2}-\\\\frac{{\\\\left(x-h\\\\right)}^2}{b^2}=1$$"]},{"id":"a9ed39ahyperbola5a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(3,-2)$$"],"dependencies":["a9ed39ahyperbola5a-h3"],"title":"Center","text":"We start by identifying the center. The center is halfway between the vertices $$(0,-2)$$ and $$(6,-2)$$. We can apply the midpoint formula $$(h,k)=(\\\\frac{x_1+x_2}{2},\\\\frac{y_1+y_2}{2})$$. What is the center (h,k)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(3,-2)$$","$$(-3,-2)$$"]},{"id":"a9ed39ahyperbola5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a9ed39ahyperbola5a-h4"],"title":"Vertices","text":"It is given that vertices are $$(0,-2)$$ and $$(6,-2)$$. The length of the transverse axis, 2a, is bounded by the vertices. We can thus find a by dividing the length between the two vertices by $$2$$. What is a in the standard form of hyperbola?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola5a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a9ed39ahyperbola5a-h5"],"title":"Foci","text":"It is given that foci are $$(-2,-2)$$ and $$(8,-2)$$. The coordinates of foci are also given by $$(h+c,k)$$ and $$(h-c,k)$$ where c is the distance of the foci from the center. We can use $$(h+c,k)=(8,-2)$$ with $$h=3$$ that we found earlier to find c. What is c associated with the hyperbola?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola5a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["a9ed39ahyperbola5a-h6"],"title":"Finding $$b$$","text":"We know that $$c^2=a^2+b^2$$, we can thus rearrange the equation to $$b=\\\\sqrt{c^2-a^2}$$. With the a and c previously found, what is $$b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola5a-h8","type":"hint","dependencies":["a9ed39ahyperbola5a-h7"],"title":"Equation","text":"Having found $$h$$, k, a and $$b$$, we can now substitute them into the standard form equation $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}-\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ed39ahyperbola6","title":"Finding the Equation of a Hyperbola Centered at (h,k) given its Foci and Vertices","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 The Hyperbola","courseName":"OpenStax: College Algebra","steps":[{"id":"a9ed39ahyperbola6a","stepAnswer":["$$\\\\frac{{\\\\left(y-3\\\\right)}^2}{5^2}-\\\\frac{{\\\\left(x-1\\\\right)}^2}{{12}^2}=1$$"],"problemType":"MultipleChoice","stepTitle":"What is the standard form equation of the hyperbola that has vertices at $$(1,-2)$$ and $$(1,8)$$ and foci at $$(1,-10)$$ and $$(1,16)$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{{\\\\left(y-3\\\\right)}^2}{5^2}-\\\\frac{{\\\\left(x-1\\\\right)}^2}{{12}^2}=1$$","choices":["$$\\\\frac{{\\\\left(y-3\\\\right)}^2}{5^2}-\\\\frac{{\\\\left(x-1\\\\right)}^2}{{12}^2}=1$$","$$\\\\frac{{\\\\left(x-3\\\\right)}^2}{5^2}-\\\\frac{{\\\\left(y-1\\\\right)}^2}{{12}^2}=1$$"],"hints":{"DefaultPathway":[{"id":"a9ed39ahyperbola6a-h1","type":"hint","dependencies":[],"title":"Standard Form of Hyperbola with Center (h,k)","text":"The standard form of the equation of a hyperbola with center (h,k) and transverse axis parallel to the x-axis is\\\\n$$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}-\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$\\\\nwhere\\\\nthe length of the transverse axis is 2a\\\\nthe coordinates of the vertices are (h+a,k),(h-a,k)\\\\nthe length of the conjugate axis is $$2b$$\\\\nthe coordinates of the co-vertices are (h,k+b),(h,k-b)\\\\nthe distance between the foci is 2c, where $$c^2=a^2+b^2$$\\\\nthe coordinates of the foci are (h+c,k),(h-c,k)\\\\nThe asymptotes of the hyperbola coincide with the diagonals of the central rectangle. The length of the rectangle is 2a and its width is $$2b$$. The slopes of the diagonals are $$\\\\frac{b}{a}$$ and $$\\\\frac{-b}{a}$$, and each diagonal passes through the center (h,k). Using the point-slope formula, it is simple to show that the equations of the asymptotes are $$y=\\\\frac{b}{a} \\\\left(x-h\\\\right)+k$$ and $$y=\\\\frac{-b}{a} \\\\left(x-h\\\\right)+k$$.\\\\n(See the left image)\\\\n\\\\nThe standard form of the equation of a hyperbola with center (h,k) and transverse axis parallel to the y-axis is\\\\n$$\\\\frac{{\\\\left(y-k\\\\right)}^2}{a^2}-\\\\frac{{\\\\left(x-h\\\\right)}^2}{b^2}=1$$\\\\nwhere\\\\nthe length of the transverse axis is 2a\\\\nthe coordinates of the vertices are (h,k+a),(h,k-a)\\\\nthe length of the conjugate axis is $$2b$$\\\\nthe coordinates of the co-vertices are (h+b,k),(h-b,k)\\\\nthe distance between the foci is 2c, where $$c^2=a^2+b^2$$\\\\nthe coordinates of the foci are (h,k+c),(h,k-c)\\\\nUsing the reasoning above, the equations of the asymptotes are $$y=\\\\frac{a}{b} \\\\left(x-h\\\\right)+k$$ and $$y=\\\\frac{-a}{b} \\\\left(x-h\\\\right)+k$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola6a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y$$"],"dependencies":["a9ed39ahyperbola6a-h1"],"title":"Transverse Axis","text":"Which axis is transverse axis on? We can tell based on the coordinates of the vertices and foci.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x$$","$$y$$"]},{"id":"a9ed39ahyperbola6a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{{\\\\left(y-k\\\\right)}^2}{a^2}-\\\\frac{{\\\\left(x-h\\\\right)}^2}{b^2}=1$$"],"dependencies":["a9ed39ahyperbola6a-h2"],"title":"Appropriate Standard Form","text":"Since the transverse axis is on the y-axis, which standard form for hyperbola should we use?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}-\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$","$$\\\\frac{{\\\\left(y-k\\\\right)}^2}{a^2}-\\\\frac{{\\\\left(x-h\\\\right)}^2}{b^2}=1$$"]},{"id":"a9ed39ahyperbola6a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(1,3)$$"],"dependencies":["a9ed39ahyperbola6a-h3"],"title":"Center","text":"We start by identifying the center. The center is halfway between the vertices $$(1,-2)$$ and $$(1,8)$$. We can apply the midpoint formula $$(h,k)=(\\\\frac{x_1+x_2}{2},\\\\frac{y_1+y_2}{2})$$. What is the center (h,k)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(1,3)$$","$$(1,5)$$"]},{"id":"a9ed39ahyperbola6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a9ed39ahyperbola6a-h4"],"title":"Vertices","text":"It is given that vertices are $$(1,-2)$$ and $$(1,8)$$. The length of the transverse axis, 2a, is bounded by the vertices. We can thus find a by dividing the length between the two vertices by $$2$$. What is a in the standard form of hyperbola?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola6a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$13$$"],"dependencies":["a9ed39ahyperbola6a-h5"],"title":"Foci","text":"It is given that foci are $$(1,-10)$$ and $$(1,16)$$. The coordinates of foci are also given by $$(h,k+c)$$ and $$(h,k-c)$$ where c is the distance of the foci from the center. We can use $$(h,k+c)=(1,16)$$ with $$k=3$$ that we found earlier to find c. What is c associated with the hyperbola?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola6a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["a9ed39ahyperbola6a-h6"],"title":"Finding $$b$$","text":"We know that $$c^2=a^2+b^2$$, we can thus rearrange the equation to $$b=\\\\sqrt{c^2-a^2}$$. With the a and c previously found, what is $$b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola6a-h8","type":"hint","dependencies":["a9ed39ahyperbola6a-h7"],"title":"Equation","text":"Having found $$h$$, k, a and $$b$$, we can now substitute them into the standard form equation $$\\\\frac{{\\\\left(y-k\\\\right)}^2}{a^2}-\\\\frac{{\\\\left(x-h\\\\right)}^2}{b^2}=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ed39ahyperbola7","title":"Solving Applied Problems Involving Hyperbolas","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 The Hyperbola","courseName":"OpenStax: College Algebra","steps":[{"id":"a9ed39ahyperbola7a","stepAnswer":["$$\\\\frac{x^2}{{30}^2}-\\\\frac{y^2}{{120.0015}^2}=1$$"],"problemType":"MultipleChoice","stepTitle":"The design layout of a cooling tower is shown in the picture. The tower stands $$179.6$$ meters tall. The diameter of the top is $$72$$ meters. At their closest, the sides of the tower are $$60$$ meters apart. Find the equation of the hyperbola that models the sides of the cooling tower. Assume that the center of the hyperbola\u2014indicated by the intersection of dashed perpendicular lines in the figure\u2014is the origin of the coordinate plane. Round final values to four decimal places.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{x^2}{{30}^2}-\\\\frac{y^2}{{120.0015}^2}=1$$","choices":["$$\\\\frac{x^2}{{30}^2}-\\\\frac{y^2}{{120.0015}^2}=1$$","$$\\\\frac{y^2}{{30}^2}-\\\\frac{x^2}{{120.0015}^2}=1$$"],"hints":{"DefaultPathway":[{"id":"a9ed39ahyperbola7a-h1","type":"hint","dependencies":[],"title":"Standard Form of Hyperbola with Center $$(0,0)$$","text":"The standard form of the equation of a hyperbola with center $$(0,0)$$ and transverse axis on the x-axis is\\\\n$$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$\\\\nwhere\\\\nthe length of the transverse axis is 2a\\\\nthe coordinates of the vertices are $$(a,0),(-a,0)$$\\\\nthe length of the conjugate axis is $$2b$$\\\\nthe coordinates of the co-vertices are (0,b),(0,b)\\\\nthe distance between the foci is 2c, where $$c^2=a^2+b^2$$\\\\nthe coordinates of the foci are $$(c,0),(-c,0)$$\\\\nthe equations of the asymptotes are y=(b/a)*x,y=-(b/a)*x\\\\n(See the left image)\\\\n\\\\nThe standard form of the equation of a hyperbola with center $$(0,0)$$ and transverse axis on the y-axis is\\\\n$$\\\\frac{y^2}{a^2}-\\\\frac{x^2}{b^2}=1$$\\\\nwhere\\\\nthe length of the transverse axis is 2a\\\\nthe coordinates of the vertices are $$(0,a),(0,-a)$$\\\\nthe length of the conjugate axis is $$2b$$\\\\nthe coordinates of the co-vertices are $$(b,0),(-b,0)$$\\\\nthe distance between the foci is 2c, where $$c^2=a^2+b^2$$\\\\nthe coordinates of the foci are $$(0,c),(0,-c)$$\\\\nthe equations of the asymptotes are y=(a/b)*x,y=-(a/b)*x\\\\n(See the right image)\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola7a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x$$"],"dependencies":["a9ed39ahyperbola7a-h1"],"title":"Transverse Axis","text":"Which axis is transverse axis on? We can tell based on vertices.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x$$","$$y$$"]},{"id":"a9ed39ahyperbola7a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$"],"dependencies":["a9ed39ahyperbola7a-h2"],"title":"Appropriate Standard Form","text":"Since the transverse axis is on the x-axis, which standard form for hyperbola should we use?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$","$$\\\\frac{y^2}{a^2}-\\\\frac{x^2}{b^2}=1$$"]},{"id":"a9ed39ahyperbola7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$30$$"],"dependencies":["a9ed39ahyperbola7a-h3"],"title":"Vertices","text":"We observe that the distance between the vertices (the closest distance between the two sides) is $$60m$$ with the center at $$(0,0)$$. The length of the transverse axis, 2a, is bounded by the vertices. We can thus find a by dividing the length between the two vertices by $$2$$. What is a in the standard form of hyperbola?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola7a-h5","type":"hint","dependencies":["a9ed39ahyperbola7a-h4"],"title":"Finding $$b$$","text":"Since there are no information about loci directly provided, we need to substitute for $$x$$ and $$y$$ in our equation using a known point.point. To do this, we can use the dimensions of the tower to find some point (x,y) that lies on the hyperbola. We will use the top right corner of the tower to represent that point. Since the y-axis bisects the tower, our x-value can be represented by the radius of the top, or $$36$$ meters. The y-value is represented by the distance from the origin to the top, which is given as $$79.6$$ meters. We can thus substitute $$(36, 79.6)$$ and $$a=30$$ into our equation to solve for $$b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola7a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{\\\\frac{y^2}{\\\\frac{x^2}{a^2}-1}}$$"],"dependencies":["a9ed39ahyperbola7a-h5"],"title":"Finding $$b$$","text":"We want to isolate $$b$$ in the standard form $$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$. Isolating $$b$$, what is the algebraic expression that $$b$$ is equals to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola7a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$120.0015$$"],"dependencies":["a9ed39ahyperbola7a-h6"],"title":"Finding $$b$$","text":"Having found an expression for $$b$$, we now substitute in $$x=36$$, $$y=79.6$$ and $$a=30$$ as previously explained. What is $$b$$? Round to four decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola7a-h8","type":"hint","dependencies":["a9ed39ahyperbola7a-h7"],"title":"Equation","text":"Having found a and $$b$$, we can now substitute them into the standard form equation $$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ed39ahyperbola8","title":"Solving Applied Problems Involving Hyperbolas","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 The Hyperbola","courseName":"OpenStax: College Algebra","steps":[{"id":"a9ed39ahyperbola8a","stepAnswer":["$$\\\\frac{x^2}{{20}^2}-\\\\frac{y^2}{{60}^2}=1$$"],"problemType":"MultipleChoice","stepTitle":"The design layout of a cooling tower is shown in the picture. The tower stands $$167.082$$ meters tall. The diameter of the top is $$60$$ meters. At their closest, the sides of the tower are $$40$$ meters apart. Find the equation of the hyperbola that models the sides of the cooling tower. Assume that the center of the hyperbola\u2014indicated by the intersection of dashed perpendicular lines in the figure\u2014is the origin of the coordinate plane. Round final values to four decimal places.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{x^2}{{20}^2}-\\\\frac{y^2}{{60}^2}=1$$","choices":["$$\\\\frac{x^2}{{20}^2}-\\\\frac{y^2}{{60}^2}=1$$","$$\\\\frac{y^2}{{20}^2}-\\\\frac{x^2}{{60}^2}=1$$"],"hints":{"DefaultPathway":[{"id":"a9ed39ahyperbola8a-h1","type":"hint","dependencies":[],"title":"Standard Form of Hyperbola with Center $$(0,0)$$","text":"The standard form of the equation of a hyperbola with center $$(0,0)$$ and transverse axis on the x-axis is\\\\n$$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$\\\\nwhere\\\\nthe length of the transverse axis is 2a\\\\nthe coordinates of the vertices are $$(a,0),(-a,0)$$\\\\nthe length of the conjugate axis is $$2b$$\\\\nthe coordinates of the co-vertices are (0,b),(0,b)\\\\nthe distance between the foci is 2c, where $$c^2=a^2+b^2$$\\\\nthe coordinates of the foci are $$(c,0),(-c,0)$$\\\\nthe equations of the asymptotes are y=(b/a)*x,y=-(b/a)*x\\\\n(See the left image)\\\\n\\\\nThe standard form of the equation of a hyperbola with center $$(0,0)$$ and transverse axis on the y-axis is\\\\n$$\\\\frac{y^2}{a^2}-\\\\frac{x^2}{b^2}=1$$\\\\nwhere\\\\nthe length of the transverse axis is 2a\\\\nthe coordinates of the vertices are $$(0,a),(0,-a)$$\\\\nthe length of the conjugate axis is $$2b$$\\\\nthe coordinates of the co-vertices are $$(b,0),(-b,0)$$\\\\nthe distance between the foci is 2c, where $$c^2=a^2+b^2$$\\\\nthe coordinates of the foci are $$(0,c),(0,-c)$$\\\\nthe equations of the asymptotes are y=(a/b)*x,y=-(a/b)*x\\\\n(See the right image)\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola8a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x$$"],"dependencies":["a9ed39ahyperbola8a-h1"],"title":"Transverse Axis","text":"Which axis is transverse axis on? We can tell based on vertices.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x$$","$$y$$"]},{"id":"a9ed39ahyperbola8a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$"],"dependencies":["a9ed39ahyperbola8a-h2"],"title":"Appropriate Standard Form","text":"Since the transverse axis is on the x-axis, which standard form for hyperbola should we use?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$","$$\\\\frac{y^2}{a^2}-\\\\frac{x^2}{b^2}=1$$"]},{"id":"a9ed39ahyperbola8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["a9ed39ahyperbola8a-h3"],"title":"Vertices","text":"We observe that the distance between the vertices (the closest distance between the two sides) is $$40m$$ with the center at $$(0,0)$$. The length of the transverse axis, 2a, is bounded by the vertices. We can thus find a by dividing the length between the two vertices by $$2$$. What is a in the standard form of hyperbola?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola8a-h5","type":"hint","dependencies":["a9ed39ahyperbola8a-h4"],"title":"Finding $$b$$","text":"Since there are no information about loci directly provided, we need to substitute for $$x$$ and $$y$$ in our equation using a known point.point. To do this, we can use the dimensions of the tower to find some point (x,y) that lies on the hyperbola. We will use the top right corner of the tower to represent that point. Since the y-axis bisects the tower, our x-value can be represented by the radius of the top, or $$30$$ meters. The y-value is represented by the distance from the origin to the top, which is given as $$67.082$$ meters. We can thus substitute $$(30, 67.082)$$ and $$a=20$$ into our equation to solve for $$b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola8a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{\\\\frac{y^2}{\\\\frac{x^2}{a^2}-1}}$$"],"dependencies":["a9ed39ahyperbola8a-h5"],"title":"Finding $$b$$","text":"We want to isolate $$b$$ in the standard form $$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$. Isolating $$b$$, what is the algebraic expression that $$b$$ is equals to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola8a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$60$$"],"dependencies":["a9ed39ahyperbola8a-h6"],"title":"Finding $$b$$","text":"Having found an expression for $$b$$, we now substitute in $$x=20$$, $$y=67.082$$ and $$a=20$$ as previously explained. What is $$b$$? Round to four decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola8a-h8","type":"hint","dependencies":["a9ed39ahyperbola8a-h7"],"title":"Equation","text":"Having found a and $$b$$, we can now substitute them into the standard form equation $$\\\\frac{x^2}{a^2}-\\\\frac{y^2}{b^2}=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ed39ahyperbola9","title":"The Hyperbola","body":"Determine whether the following equations represent hyperbolas. If so, write in standard form","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.2 The Hyperbola","courseName":"OpenStax: College Algebra","steps":[{"id":"a9ed39ahyperbola9a","stepAnswer":["NA"],"problemType":"MultipleChoice","stepTitle":"$$3y^2+2x=6$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$\\\\frac{y^2}{2}+\\\\frac{x}{3}=1$$","$$\\\\frac{y^2}{2^2}-\\\\frac{x^2}{3^3}=1$$","NA"],"hints":{"DefaultPathway":[{"id":"a9ed39ahyperbola9a-h1","type":"hint","dependencies":[],"title":"Standard Form of Hyperbola","text":"The standard form for hyperbola is either $$\\\\frac{{\\\\left(x-h\\\\right)}^2}{a^2}-\\\\frac{{\\\\left(y-k\\\\right)}^2}{b^2}=1$$\\\\nor $$\\\\frac{{\\\\left(y-k\\\\right)}^2}{a^2}-\\\\frac{{\\\\left(x-h\\\\right)}^2}{b^2}=1$$ depending on the transverse axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ed39ahyperbola9a-h2","type":"hint","dependencies":["a9ed39ahyperbola9a-h1"],"title":"Parabola","text":"Observe that only one of the variables, $$y$$, is of degree $$2$$, while $$x$$ is of degree $$1$$. Thus, this is a parabola.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ef311slicing1","title":"Deriving the Formula for the Volume of a Pyramid","body":"We know from geometry that the formula for the volume of a pyramid is $$V=\\\\frac{1}{3} A h$$. If the pyramid has a square base, this becomes $$V=\\\\frac{1}{3} a^2 h$$, where a denotes the length of one side of the base.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.2 Determining Volumes by Slicing","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a9ef311slicing1a","stepAnswer":["(a**2/h**2)*/int(x**2),0,h,x}"],"problemType":"TextBox","stepTitle":"Use the slicing method to derive the formula $$V=\\\\frac{1}{3} a^2 h$$ (What is the integral needed to derive the formula? Please include the integrand symbol. $$V=___?)$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(a**2/h**2)*/int(x**2),0,h,x}$$","hints":{"DefaultPathway":[{"id":"a9ef311slicing1a-h1","type":"hint","dependencies":[],"title":"Setting up the Integral: Consider the Figure","text":"We want to apply the slicing method to a pyramid with a square base. To set up the integral, consider figure (a) the pyramid shown in the figure, oriented along the x-axis, and figure (b) a two-dimentional view of the pyramid from the side.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a9ef311slicing1a-h2","type":"hint","dependencies":["a9ef311slicing1a-h1"],"title":"Setting up the Integral: Determine the Shape of a Cross-section of the Pyramid","text":"First, determine the shape of a cross-section of the pyramid. Since the base is a square, the cross-sections will also be squares.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a9ef311slicing1a-h3","type":"hint","dependencies":["a9ef311slicing1a-h2"],"title":"Setting up the Integral: Determine a Formula for the Area of the Cross-sectional Squares","text":"Next, determine a formula for the area of one of the cross-sectional squares. Looking at figure (b), and using a proportion, since these are similar triangles, we have $$\\\\frac{s}{a}=\\\\frac{x}{h}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a9ef311slicing1a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$s=\\\\frac{ax}{h}$$"],"dependencies":["a9ef311slicing1a-h3"],"title":"Setting up the Integral: Determine a Formula for the Area of the Cross-sectional Squares","text":"What is the formula in terms of s? Please write the full formula.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$s=\\\\frac{ax}{h}$$","$$s=a x h$$"]},{"id":"a9ef311slicing1a-h5","type":"hint","dependencies":["a9ef311slicing1a-h4"],"title":"Setting up the Integral: Area of one Cross-Sectional Square","text":"The area of one of the cross-sectional squares is $$A(x)=s^2={\\\\left(\\\\frac{ax}{h}\\\\right)}^2$$. The integral for the volume of the pyramid is $$V=\\\\int_{0}^{h} A(x) \\\\,dx=\\\\int_{0}^{h} {\\\\left(\\\\frac{a x}{h}\\\\right)}^2 \\\\,dx=(a**2/h**2)*/int(x**2),0,h,x}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a9ef311slicing1a-h6","type":"hint","dependencies":["a9ef311slicing1a-h5"],"title":"Find the Volume of the Pyramid","text":"Integrating from $$0$$ to $$h$$, we receive the formula we were looking for: $$V=\\\\frac{1}{3} a^2 h$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a9ef311slicing10","title":"Yogurt Containers","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.2 Determining Volumes by Slicing","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a9ef311slicing10a","stepAnswer":["((m**2*pi)/3)*(b**3-a**3)"],"problemType":"TextBox","stepTitle":"Yogurt containers can be shaped like frustums. Rotate the line $$y=\\\\frac{1}{m} x$$ around the y-axis to find the volume between $$y=a$$ and $$y=b$$. The answer will be in $${units}^3$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{m^2 \\\\pi}{3} \\\\left(b^3-a^3\\\\right)$$","hints":{"DefaultPathway":[]}}]},{"id":"a9ef311slicing11","title":"Bundt Cake","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.2 Determining Volumes by Slicing","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a9ef311slicing11a","stepAnswer":["2*pi**2"],"problemType":"TextBox","stepTitle":"What is the volume of the Bundt cake that comes from rotating $$y=sinx$$ around the y-axis from $$x=0$$ to $$x=pi$$. The answer will be in $${units}^3$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2{\\\\pi}^2$$","hints":{"DefaultPathway":[]}}]},{"id":"a9ef311slicing12","title":"Finding the Volume of the Solid Described","body":"Find the volume of the solid described. The answer will be in $${units}^3$$.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.2 Determining Volumes by Slicing","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a9ef311slicing12a","stepAnswer":["(2ab**2*pi)/3"],"problemType":"TextBox","stepTitle":"The base is the region enclosed by the generic ellipse $$\\\\frac{x^2}{a^2}+\\\\frac{y^2}{b^2}=1$$. Slices perpendicular to the x-axis are semicircles.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{2{ab}^2 \\\\pi}{3}$$","hints":{"DefaultPathway":[]}}]},{"id":"a9ef311slicing13","title":"Region Bounded by Curves","body":"Find the volume using the slicing method. The answer will be in $${units}^3$$.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.2 Determining Volumes by Slicing","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a9ef311slicing13a","stepAnswer":["2"],"problemType":"TextBox","stepTitle":"The base is region under the parabola $$y=1-x^2$$ and above the x-axis. Slices perpendicular to the y-axis are squares.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"a9ef311slicing13a-h1","type":"hint","dependencies":[],"title":"Determine the Shape of the Cross-Section","text":"Examine the solid and determine the shape of the cross-section of the solid. It is often helpful to draw a picture if one is not provided. In this problem, the drawing is already provided for you.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a9ef311slicing13a-h2","type":"hint","dependencies":["a9ef311slicing13a-h1"],"title":"Determine a Formula for the Cross-Section","text":"Determine a formula for the area of the cross-section.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a9ef311slicing13a-h3","type":"hint","dependencies":["a9ef311slicing13a-h2"],"title":"Integrate the Formula","text":"Integrate the area formula over the appropriate interval to get the volume.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a9ef311slicing14","title":"Region Bounded by Curves","body":"Use the disk method to find the volume when the region is rotated around the x-axis. The answer will be in units**3.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.2 Determining Volumes by Slicing","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a9ef311slicing14a","stepAnswer":["(4096*pi)/5"],"problemType":"TextBox","stepTitle":"$$y=2x^2$$, $$x=0$$, $$x=4$$, and $$y=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{4096\\\\pi}{5}$$","hints":{"DefaultPathway":[]}}]},{"id":"a9ef311slicing15","title":"Region Bounded by Curves","body":"Find the volume when the region is rotated around the y-axis. The answer will be in $${units}^3$$.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.2 Determining Volumes by Slicing","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a9ef311slicing15a","stepAnswer":["pi"],"problemType":"TextBox","stepTitle":"$$x=sec(y)$$ and $$y=\\\\frac{\\\\pi}{4}$$, $$y=0$$ and $$x=0$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[]}}]},{"id":"a9ef311slicing16","title":"Region Bounded by Curves","body":"Find the volume when the region is rotated around the x-axis. The answer will be in $${units}^3$$.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.2 Determining Volumes by Slicing","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a9ef311slicing16a","stepAnswer":["(3*pi)/10"],"problemType":"TextBox","stepTitle":"$$y=\\\\sqrt{x}$$ and $$y=x^2$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{3\\\\pi}{10}$$","hints":{"DefaultPathway":[]}}]},{"id":"a9ef311slicing17","title":"Region Bounded by Curves","body":"Use the washer method to find the volume when the region is revolved around the y-axis. The answer will be in $${units}^3$$.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.2 Determining Volumes by Slicing","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a9ef311slicing17a","stepAnswer":["9*pi"],"problemType":"TextBox","stepTitle":"$$y=x+2$$, $$y=2x-1$$, and $$x=0$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$9\\\\pi$$","hints":{"DefaultPathway":[]}}]},{"id":"a9ef311slicing18","title":"Volume Common to Spheres","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.2 Determining Volumes by Slicing","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a9ef311slicing18a","stepAnswer":["V=(2*pi/3)*(2r**3-3r**2h+h**3)"],"problemType":"TextBox","stepTitle":"Find the volume common to spheres of radius $$r$$ with centers that are $$2h$$ apart, as shown here.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$V=\\\\frac{2\\\\pi}{3} \\\\left(2r^3-3r^2 h+h^3\\\\right)$$","hints":{"DefaultPathway":[]}}]},{"id":"a9ef311slicing19","title":"Volume of a Football","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.2 Determining Volumes by Slicing","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a9ef311slicing19a","stepAnswer":["$$\\\\frac{\\\\pi}{3} \\\\left(h+R\\\\right) {\\\\left(h-2R\\\\right)}^2$$"],"problemType":"MultipleChoice","stepTitle":"Rotate the ellipse $$\\\\frac{x^2}{a^2}+\\\\frac{y^2}{b^2}=1$$ around the y-axis to approximate the volume of a football, as seen here. The answer will be in $${units}^3$$.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{\\\\pi}{3} \\\\left(h+R\\\\right) {\\\\left(h-2R\\\\right)}^2$$","choices":["$$\\\\frac{2\\\\pi}{3} \\\\left(h+2R\\\\right) {\\\\left(h-R\\\\right)}^2$$","$$\\\\frac{3\\\\pi}{3} \\\\left(2h+R\\\\right) {\\\\left(h-2R\\\\right)}^2$$","$$\\\\frac{\\\\pi}{3} \\\\left(h+R\\\\right) {\\\\left(h-2R\\\\right)}^2$$","$$\\\\frac{\\\\pi}{3} \\\\left(h-R\\\\right) {\\\\left(h+2R\\\\right)}^2$$","$$\\\\frac{\\\\pi}{3} \\\\left(h+R\\\\right) {\\\\left(h-2R\\\\right)}^2$$"],"hints":{"DefaultPathway":[]}}]},{"id":"a9ef311slicing2","title":"Using the Slicing Method to find the Volume of a Solid of Revolution","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.2 Determining Volumes by Slicing","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a9ef311slicing2a","stepAnswer":["(78/5)*pi"],"problemType":"TextBox","stepTitle":"Use the slicing method to find the volume of the solid of revolution bounded by the graphs of $$f(x)=x^2-4x+5$$, $$x=1$$, and $$x=4$$, and rotated abount the x-axis. The answer will be in $${units}^3$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{78}{5} \\\\pi$$","hints":{"DefaultPathway":[{"id":"a9ef311slicing2a-h1","type":"hint","dependencies":[],"title":"Sketch the Graph","text":"Using the problem-solving strategy, we first sketch the graph of the quadratic function over the interval [1,4], as shown in the following figure.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a9ef311slicing2a-h2","type":"hint","dependencies":["a9ef311slicing2a-h1"],"title":"Revolve the Region About the x-axis","text":"Next, revolve the region around the x-axis, as shown in the following figure.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a9ef311slicing2a-h3","type":"hint","dependencies":["a9ef311slicing2a-h2"],"title":"Formula for Area of the Circle","text":"Since the solid was formed by revolving the region around the x-axis, the cross-sections are circles. The area of the cross-section, then, is the area of a circle, and the radius of the circle is given by f(x). Use the formula for the area of the circle: $$A(x)=\\\\pi r^2=\\\\pi {f{\\\\left(x\\\\right)}}^2=\\\\pi {\\\\left(x^2-4x+5\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a9ef311slicing2a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$V=\\\\int_{1}^{4} {\\\\operatorname{\\\\pi}\\\\left(x^2-4x+5\\\\right)}^2 \\\\,dx$$"],"dependencies":["a9ef311slicing2a-h3"],"title":"Set Up the Integral","text":"Select the appropriate integral for finding the volume of the solid.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$V=\\\\int_{1}^{4} {\\\\operatorname{\\\\pi}\\\\left(x^2-4x+5\\\\right)}^2 \\\\,dx$$","$$V=\\\\int_{4}^{1} {\\\\operatorname{\\\\pi}\\\\left(x^2-4x+5\\\\right)}^2 \\\\,dx$$","$$V=\\\\int_{0}^{4} {\\\\operatorname{\\\\pi}\\\\left(x^2-4x+5\\\\right)}^2 \\\\,dx$$","$$V=\\\\int_{0}^{1} {\\\\operatorname{\\\\pi}\\\\left(x^2-4x+5\\\\right)}^2 \\\\,dx$$"]},{"id":"a9ef311slicing2a-h5","type":"hint","dependencies":["a9ef311slicing2a-h4"],"title":"Integrate the Formula","text":"Find the volume of the solid using the slicing method.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a9ef311slicing2a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{78\\\\pi}{5}$$"],"dependencies":["a9ef311slicing2a-h5"],"title":"Integrate the Formula","text":"What is the volume of the solid?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\frac{78\\\\pi}{5}$$","$$\\\\frac{78\\\\times5}{\\\\pi}$$"]}]}}]},{"id":"a9ef311slicing20","title":"Volume of Sphere with Cap Removed","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.2 Determining Volumes by Slicing","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a9ef311slicing20a","stepAnswer":["$$\\\\frac{\\\\pi}{3} \\\\left(h+R\\\\right) {\\\\left(h-2R\\\\right)}^2$$"],"problemType":"MultipleChoice","stepTitle":"Find the volume of a sphere of radius R with a cap of height $$h$$ removed from the top, as seen here. The answer will be in $${units}^3$$.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{\\\\pi}{3} \\\\left(h+R\\\\right) {\\\\left(h-2R\\\\right)}^2$$","choices":["$$\\\\frac{2\\\\pi}{3} \\\\left(h+2R\\\\right) {\\\\left(h-R\\\\right)}^2$$","$$\\\\frac{3\\\\pi}{3} \\\\left(2h+R\\\\right) {\\\\left(h-2R\\\\right)}^2$$","$$\\\\frac{\\\\pi}{3} \\\\left(h+R\\\\right) {\\\\left(h-2R\\\\right)}^2$$","$$\\\\frac{\\\\pi}{3} \\\\left(h-R\\\\right) {\\\\left(h+2R\\\\right)}^2$$","$$\\\\frac{\\\\pi}{3} \\\\left(h+R\\\\right) {\\\\left(h-2R\\\\right)}^2$$"],"hints":{"DefaultPathway":[]}}]},{"id":"a9ef311slicing3","title":"Using the Disk Method to find the Volume of a Solid of Revolution $$1$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.2 Determining Volumes by Slicing","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a9ef311slicing3a","stepAnswer":["(15*pi)/2"],"problemType":"TextBox","stepTitle":"Use the disk method to find the volume of the solid of revolution by rotating the region between the graph of $$f(x)=\\\\sqrt{x}$$ and the x-axis over the interval [1,4] around the x-axis. The answer will be in $${units}^3$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{15\\\\pi}{2}$$","hints":{"DefaultPathway":[{"id":"a9ef311slicing3a-h1","type":"hint","dependencies":[],"title":"Graph of the Function and Solid of Revolution","text":"The graphs of the function and the solid of revolution are shown in the following figure. Figure (a) displays the function $$f(x)=\\\\sqrt{x}$$ over the interval [1,4], and figure (b) displays the solid of revolution obtained by revolving the region under the graph of f(x) about the x-axis.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a9ef311slicing3a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$V=\\\\int_{1}^{4} \\\\pi {\\\\sqrt{x}}^2 \\\\,dx$$"],"dependencies":["a9ef311slicing3a-h1"],"title":"Set Up the Integral","text":"Select the appropriate integral for finding the volume of the solid.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$V=\\\\int_{1}^{4} \\\\pi {\\\\sqrt{x}}^2 \\\\,dx$$","$$V=\\\\int_{1}^{4} {\\\\sqrt{x}}^2 \\\\,dx$$","$$V=\\\\int_{4}^{1} \\\\pi {\\\\sqrt{x}}^2 \\\\,dx$$","$$V=\\\\int_{4}^{1} {\\\\sqrt{x}}^2 \\\\,dx$$"]},{"id":"a9ef311slicing3a-h3","type":"hint","dependencies":["a9ef311slicing3a-h2"],"title":"Integrate the Formula","text":"Find the volume of the solid using the disk method. The volume of the solid of revolution formed by revolving R around the x-axis using the disk method is given by V=/int{pi[f(x)]**2,a,b,x}.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a9ef311slicing3a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(15*pi)/2"],"dependencies":["a9ef311slicing3a-h3"],"title":"Integrate the Formula","text":"What is the volume of the solid of revolution?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a9ef311slicing4","title":"Using the Disk Method to find the Volume of a Solid of Revolution $$2$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.2 Determining Volumes by Slicing","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a9ef311slicing4a","stepAnswer":["8*pi"],"problemType":"TextBox","stepTitle":"Let R be the region bounded by the graph of $$g(y)=\\\\sqrt{4-y}$$ and the y-axis over the y-axis interval [0,4]. Use the disk method to find the volume of the solid of revolution generated by rotating R around the y-axis. The answer will be in $${units}^3$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$8\\\\pi$$","hints":{"DefaultPathway":[{"id":"a9ef311slicing4a-h1","type":"hint","dependencies":[],"title":"Representative Disk","text":"The following figure shows the function and a representative disk that can be used to estimate the volume. Notice that since we are revolving the function around the y-axis, the disks are horizontal, rather than vertical.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a9ef311slicing4a-h2","type":"hint","dependencies":["a9ef311slicing4a-h1"],"title":"Region and Full Solid of Revolution","text":"The region to be followed and the full solid of revolution are depicted in the following figure.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a9ef311slicing4a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$V=\\\\int_{0}^{4} \\\\pi {\\\\sqrt{4-y}}^2 \\\\,dy$$"],"dependencies":["a9ef311slicing4a-h2"],"title":"Set Up the Integral","text":"Select the appropriate integral for finding the volume of the solid.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$V=\\\\int_{0}^{4} \\\\pi {\\\\sqrt{4-y}}^2 \\\\,dy$$","$$V=\\\\int_{4}^{0} \\\\pi {\\\\sqrt{4-y}}^2 \\\\,dy$$","$$V=\\\\int_{0}^{4} \\\\sqrt{4-y} \\\\,dy$$","$$V=\\\\int_{4}^{0} \\\\sqrt{4-y} \\\\,dy$$"]},{"id":"a9ef311slicing4a-h4","type":"hint","dependencies":["a9ef311slicing4a-h3"],"title":"Integrate the Formula","text":"Find the volume of the solid using the disk method. The volume of the solid of revolution formed by revolving R around the y-axis using the disk method is given by $$V=\\\\int_{a}^{b} \\\\pi {f{\\\\left(y\\\\right)}}^2 \\\\,dy$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a9ef311slicing4a-h5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["8*pi"],"dependencies":["a9ef311slicing4a-h4"],"title":"Integrate the Formula","text":"What is the volume of the solid of revolution?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a9ef311slicing5","title":"Using the Washer Method","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.2 Determining Volumes by Slicing","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a9ef311slicing5a","stepAnswer":["(81*pi)/4"],"problemType":"TextBox","stepTitle":"Find the volume of a solid of revolution formed by revolving the region bounded above by the graph of $$f(x)=x$$ and below the graph of $$g(x)=\\\\frac{1}{x}$$ over the interval [1,4] around the x-axis. The answer will be in $${units}^3$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{81\\\\pi}{4}$$","hints":{"DefaultPathway":[{"id":"a9ef311slicing5a-h1","type":"hint","dependencies":[],"title":"Graph of the Function and Solid of Revolution","text":"The graph of the functions and the solid of revolution are shown in the following figure. (a) The region between the graphs of the functions $$f(x)=x$$ and $$g(x)=\\\\frac{1}{x}$$ over the interval [1,4]. (b) Revolving the region about the x-axis generates a solid of revolution with a cavity in the middle.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a9ef311slicing5a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$V=pi*\\\\int_{1}^{4} x^2-{\\\\left(\\\\frac{1}{x}\\\\right)}^2 \\\\,dx$$"],"dependencies":["a9ef311slicing5a-h1"],"title":"Set Up the Integral","text":"Select the appropriate integral for finding the volume of the solid.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$V=pi*\\\\int_{1}^{4} x^2-{\\\\left(\\\\frac{1}{x}\\\\right)}^2 \\\\,dx$$","$$V=pi*\\\\int_{4}^{1} x^2-{\\\\left(\\\\frac{1}{x}\\\\right)}^2 \\\\,dx$$","$$V=\\\\int_{1}^{4} \\\\pi x^2-\\\\frac{1}{x} \\\\,dx$$","$$V=\\\\int_{4}^{1} \\\\pi x^2-\\\\frac{1}{x} \\\\,dx$$"]},{"id":"a9ef311slicing5a-h3","type":"hint","dependencies":["a9ef311slicing5a-h2"],"title":"Integrate the Formula","text":"Find the volume of the solid using the washer method.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a9ef311slicing5a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{81\\\\pi}{4}$$"],"dependencies":["a9ef311slicing5a-h3"],"title":"Integrate the Formula","text":"What is the volume of the solid of revolution?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\frac{81\\\\pi}{4}$$","$$\\\\frac{\\\\left(-81\\\\pi\\\\right)}{4}$$"]}]}}]},{"id":"a9ef311slicing6","title":"The Washer Method with a Different Axis of Revolution","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.2 Determining Volumes by Slicing","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a9ef311slicing6a","stepAnswer":["(160*pi)/3"],"problemType":"TextBox","stepTitle":"Find the volume of a solid of revolution formed by revolving the region bounded above by $$f(x)=4-x$$ and below the x-axis over the interval [0,4] around the line $$y=-2$$. The answer will be in $${units}^3$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{160\\\\pi}{3}$$","hints":{"DefaultPathway":[{"id":"a9ef311slicing6a-h1","type":"hint","dependencies":[],"title":"Graph of the Function and Solid of Revolution","text":"The graph of the region and the solid of revolution are shown in the following figure. (a) The region between the graph of the function $$f(x)=4-x$$ and the x-axis over the interval [0,4]. (b) Revolving the region about the line $$y=-2$$ generates a solid of revolution with a cylindrical hole through its middle.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a9ef311slicing6a-h2","type":"hint","dependencies":["a9ef311slicing6a-h1"],"title":"Area of the Cross-Section","text":"We cannot apply the formula to this problem directly because the axis of revolution is not one of the coordinate axes. However, we still know that the area of the cross-section is the area of the outer circle less the area of the inner circle. Looking at the graph of the function, we see the radius of the outer circle is given by $$f{\\\\left(x\\\\right)}+2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a9ef311slicing6a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["6-x"],"dependencies":["a9ef311slicing6a-h2"],"title":"Area of the Cross-Section","text":"What is $$f{\\\\left(x\\\\right)}+2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a9ef311slicing6a-h4","type":"hint","dependencies":["a9ef311slicing6a-h3"],"title":"Radius of the Inner Circle","text":"The radius of the inner circle is $$g(x)=2$$. Therefore, we have $$V=\\\\int_{0}^{4} \\\\pi {\\\\left(6-x\\\\right)}^2-2^2 \\\\,dx$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a9ef311slicing6a-h5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(160*pi)/3"],"dependencies":["a9ef311slicing6a-h4"],"title":"Integrate the Formula","text":"What is the volume of the solid of revolution?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a9ef311slicing7","title":"Slicing Method","body":"Draw a typical slice and find the volume using the slicing method for the given volume. The answer will be in units**3.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.2 Determining Volumes by Slicing","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a9ef311slicing7a","stepAnswer":["$$8$$"],"problemType":"TextBox","stepTitle":"A pyramid with height $$6$$ units and square base of side $$2$$ units, as pictured here.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8$$","hints":{"DefaultPathway":[{"id":"a9ef311slicing7a-h1","type":"hint","dependencies":[],"title":"Determine the Shape of the Cross-Section","text":"Examine the solid and determine the shape of the cross-section of the solid. It is often helpful to draw a picture if one is not provided. In this problem, the drawing is already provided for you.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a9ef311slicing7a-h2","type":"hint","dependencies":["a9ef311slicing7a-h1"],"title":"Determine a Formula for the Cross-Section","text":"Determine a formula for the area of the cross-section.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a9ef311slicing7a-h3","type":"hint","dependencies":["a9ef311slicing7a-h2"],"title":"Integrate the Formula","text":"Integrate the area formula over the appropriate interval to get the volume.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a9ef311slicing8","title":"Slicing Method","body":"Draw a typical slice and find the volume using the slicing method for the given volume. The answer will be in units**3.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.2 Determining Volumes by Slicing","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a9ef311slicing8a","stepAnswer":["$$\\\\frac{32}{3\\\\sqrt{2}}$$"],"problemType":"TextBox","stepTitle":"A tetrahedron with a base side of $$4$$ units, as seen here.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{32}{3\\\\sqrt{2}}$$","hints":{"DefaultPathway":[{"id":"a9ef311slicing8a-h1","type":"hint","dependencies":[],"title":"Determine the Shape of the Cross-Section","text":"Examine the solid and determine the shape of the cross-section of the solid. It is often helpful to draw a picture if one is not provided. In this problem, the drawing is already provided for you.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a9ef311slicing8a-h2","type":"hint","dependencies":["a9ef311slicing8a-h1"],"title":"Determine a Formula for the Cross-Section","text":"Determine a formula for the area of the cross-section.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a9ef311slicing8a-h3","type":"hint","dependencies":["a9ef311slicing8a-h2"],"title":"Integrate the Formula","text":"Integrate the area formula over the appropriate interval to get the volume.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a9ef311slicing9","title":"Slicing Method","body":"Draw a typical slice and find the volume using the slicing method for the given volume. The answer will be in $${units}^3$$.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.2 Determining Volumes by Slicing","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"a9ef311slicing9a","stepAnswer":["(7/24)*pi*(r**2)*h"],"problemType":"TextBox","stepTitle":"A cone of radius $$r$$ and height $$h$$ has a smaller cone of radius $$\\\\frac{r}{2}$$ and height $$\\\\frac{h}{2}$$ removed from the top, as seen here. The resulting solid is called a frustum.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{7}{24} \\\\pi r^2 h$$","hints":{"DefaultPathway":[{"id":"a9ef311slicing9a-h1","type":"hint","dependencies":[],"title":"Determine the Shape of the Cross-Section","text":"Examine the solid and determine the shape of the cross-section of the solid. It is often helpful to draw a picture if one is not provided. In this problem, the drawing is already provided for you.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a9ef311slicing9a-h2","type":"hint","dependencies":["a9ef311slicing9a-h1"],"title":"Determine a Formula for the Cross-Section","text":"Determine a formula for the area of the cross-section.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"a9ef311slicing9a-h3","type":"hint","dependencies":["a9ef311slicing9a-h2"],"title":"Integrate the Formula","text":"Integrate the area formula over the appropriate interval to get the volume.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"a9f55eblinearmodels1","title":"Exercise 8: Finding Shapes Bounded By Lines","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Modeling with Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a9f55eblinearmodels1a","stepAnswer":["$$\\\\frac{4}{3}$$"],"problemType":"TextBox","stepTitle":"Find the area of a parallelogram bounded by the x-axis, the line g(x) $$=$$ $$2$$, the line f(x) $$=$$ $$3x$$, and the line parallel to f(x) passing through $$(6,1)$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{4}{3}$$","hints":{"DefaultPathway":[{"id":"a9f55eblinearmodels1a-h1","type":"hint","dependencies":[],"title":"Figuring out the Vertical Dimensions","text":"We have enough information to figure out the vertical dimensions-- the height-- of the parallelogram. We know that the parallelogram is bounded by $$g(x)=2$$ and the $$x$$ axis, so the height is $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels1a-h2","type":"hint","dependencies":["a9f55eblinearmodels1a-h1"],"title":"Figuring out the Horizontal Dimensions","text":"We can figure out the horizontal dimensions by first finding the equation of the line that passes through $$(6,1)$$, then finding its $$x$$ values when $$y$$ $$=$$ $$0$$ (meaning it intersects the $$x$$ axis) and when it intersects $$g(x)=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels1a-h3","type":"hint","dependencies":["a9f55eblinearmodels1a-h2"],"title":"Finding the Relevant Line","text":"We know that the line passing through $$(6,1)$$ is parallel to $$f(x)=3x$$. What this means is that the line we want to find also has a slope of $$3$$. Since we know the slope of the line and a point on the line, we can plug this into the equation $$y=mx+b$$ to find $$b$$. We know that the point $$(6,1)$$ means $$y=1$$ and $$x=6$$, and the slope, $$m$$, is $$3$$. Thus, $$1=3\\\\left(6\\\\right)+b$$. Solving for $$b$$, we get that $$b=-17$$. Therefore, the equation of the line is $$3x-17$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{17}{3}$$"],"dependencies":["a9f55eblinearmodels1a-h3"],"title":"Finding the Intersection of the Line With the X Axis","text":"Using the equation of the line $$y=3x-17$$, when $$y=0$$, what is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels1a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{19}{3}$$"],"dependencies":["a9f55eblinearmodels1a-h4"],"title":"Finding the Intersection of the Line With $$g(x)=2$$","text":"Using the equation of the line $$y=3x-17$$, when $$y=2$$, what is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels1a-h6","type":"hint","dependencies":["a9f55eblinearmodels1a-h5"],"title":"Finding the Horizontal Dimensions of the Parallelogram","text":"We now know that the horizontal base of the paralleogram is starts at $$x=\\\\frac{17}{3}$$ and ends at $$y=\\\\frac{19}{3}$$. We can find the length of the base by subtracting $$\\\\frac{17}{3}$$ from $$\\\\frac{19}{3}$$, which gives us $$\\\\frac{2}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels1a-h7","type":"hint","dependencies":["a9f55eblinearmodels1a-h6"],"title":"Formula For a Parallelogram\'s Area","text":"The area of a parallelogram is the height times the base. In equation form, $$A=bh$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels1a-h8","type":"hint","dependencies":["a9f55eblinearmodels1a-h7"],"title":"Using the Formula From the Problem\'s Conditions","text":"Input the dimensions created by the four lines into this formula,","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9f55eblinearmodels10","title":"Exercise 17: Using Linear Models to Predict Outcomes","body":"A town\'s population has an initial population of 75,000. It grows at a constant rate of 2,500 per year for five years.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Modeling with Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a9f55eblinearmodels10a","stepAnswer":["$$10$$"],"problemType":"TextBox","stepTitle":"What year number will the population reach 100,000? Enter your answer as a single number. For example, if the answer were the 6th year, the answer would be $$6$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10$$","hints":{"DefaultPathway":[{"id":"a9f55eblinearmodels10a-h1","type":"hint","dependencies":[],"title":"Subsituting in Relevant Values","text":"Enter 100,000 as the $$y$$ value and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9f55eblinearmodels11","title":"Exercise 18: Using Linear Models to Predict Outcomes","body":"A town\'s population has an initial population of 75,000. It grows at a constant rate of 2,500 per year for five years.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Modeling with Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a9f55eblinearmodels11a","stepAnswer":["$$105000$$"],"problemType":"TextBox","stepTitle":"What is the population $$12$$ years from the onset of the model?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$105000$$","hints":{"DefaultPathway":[{"id":"a9f55eblinearmodels11a-h1","type":"hint","dependencies":[],"title":"Subsituting a Value into the Trend Equation","text":"Input $$12$$ as the $$x$$ value. Then, calculate the equation using that value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels11a-h2","type":"hint","dependencies":["a9f55eblinearmodels11a-h1"],"title":"Answer","text":"$$75000+2500\\\\times12=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9f55eblinearmodels12","title":"Using a Linear Model to Investigate a Town\u2019s Population","body":"A town\u2019s population has been growing linearly. In $$2004$$, the population was 6,200. By $$2009$$, the population had grown to 8,100. Assume this trend continues.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Modeling with Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a9f55eblinearmodels12a","stepAnswer":["$$9620$$"],"problemType":"TextBox","stepTitle":"Predict the population in $$2013$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9620$$","hints":{"DefaultPathway":[{"id":"a9f55eblinearmodels12a-h1","type":"hint","dependencies":[],"title":"Identifying the Changing Quantities","text":"The first step is to identity the changing quantities. From the problem, we can see that the year town populations change, so our variables are \\"time\\" and \\"population.\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels12a-h2","type":"hint","dependencies":["a9f55eblinearmodels12a-h1"],"title":"Defining Input and Output","text":"The next step is to define input and output. Our input is \\"time,\\" and our output is \\"population.\\" To make the input nicer, instead of defining time with the year number, we will let time, represented by the variable $$t$$, be equal to the number of years after $$2004$$. Thus, in $$2004$$, $$t$$ $$=$$ $$0$$. Our output is P(t), which represents population as a function of time.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels12a-h3","type":"hint","dependencies":["a9f55eblinearmodels12a-h2"],"title":"Determining the Rate of Change","text":"We need to determine the rate of change-- that is, the change in output over the change in input. This is equivalent to the slope of the linear model, which is constant by the definition of linearity (a line.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels12a-h4","type":"hint","dependencies":["a9f55eblinearmodels12a-h3"],"title":"Solving for the Rate of Change","text":"In $$2004$$, $$t$$ $$=$$ $$0$$ and P(t) $$=$$ $$6200$$. We can represent this with the coordinate point $$(0,6200)$$. In $$2009$$, $$t$$ $$=$$ $$5$$ (since $$2009-2004=5)$$ and P(t) $$=$$ $$8100$$, giving us the point $$(5,8100)$$. The slope, $$m$$, is calculated as the change in $$y$$ over the change in $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels12a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1900$$"],"dependencies":["a9f55eblinearmodels12a-h4"],"title":"Solving for the Change in Y","text":"What is $$8100-6200$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels12a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["a9f55eblinearmodels12a-h5"],"title":"Solving for the Change in X","text":"What is $$5-0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels12a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$380$$"],"dependencies":["a9f55eblinearmodels12a-h6"],"title":"Dividing the Change in Y by the Change in X","text":"What is $$\\\\frac{1900}{5}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels12a-h8","type":"hint","dependencies":["a9f55eblinearmodels12a-h7"],"title":"Identifying the Y Intercept of the Line","text":"We have determined that the slope is $$380$$ $$\\\\frac{people}{year}$$ (since we dividing the change in $$y$$, which was in units of people, by the change in $$x$$, which was in units of years.) For the equation of the line, we need to know the y-intercept, which occurs when $$t$$ $$=$$ $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels12a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6200$$"],"dependencies":["a9f55eblinearmodels12a-h8"],"title":"Identifying the Value of the Y Intercept","text":"When $$t=0$$, $$P(t)=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels12a-h10","type":"hint","dependencies":["a9f55eblinearmodels12a-h9"],"title":"Writing the Equation of the Line","text":"The slope is $$380$$ and the $$y$$ intercept is $$6200$$, so we know the line $$y$$ $$=$$ mx + $$b$$, with $$y$$ being P(t), $$x$$ being $$t$$, $$m$$ being slope, and $$b$$ being the y-intercept, is $$P(t)=380t+6200$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels12a-h11","type":"hint","dependencies":["a9f55eblinearmodels12a-h10"],"title":"Predicting the Population in $$2013$$","text":"To predict the population in $$2013$$, we find the value of P(t) when $$t$$ $$=$$ $$9$$ (Since $$2013-2004=9)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels12a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$23.158$$"],"dependencies":["a9f55eblinearmodels12a-h11"],"title":"Predicting the Population in $$2013$$","text":"When $$t$$ $$=$$ $$9$$, and $$P(t)=380t+6200$$, what is the value of P(9)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$23.158$$","$$12.142$$","$$40.562$$","$$30.176$$","$$5.289$$"]}]}},{"id":"a9f55eblinearmodels12b","stepAnswer":["$$2027$$"],"problemType":"TextBox","stepTitle":"Identity the year in which the population will reach 15,000.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2027$$","hints":{"DefaultPathway":[{"id":"a9f55eblinearmodels12b-h1","type":"hint","dependencies":[],"title":"Setting the Output Equal to $$15000$$","text":"We are given that the population, P(t), is equal to $$15000$$, so now we need to solve for $$t$$. Using the equation obtained in part a, we get that $$15000=380t+6200$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels12b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$23.158$$"],"dependencies":["a9f55eblinearmodels12b-h1"],"title":"Solving the Equation When $$P(t)=15000$$","text":"$$15000=380t+6200$$, so $$15000-6200=380t$$. $$t$$ is equal to about?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$23.158$$","$$12.142$$","$$40.562$$","$$30.176$$","$$5.289$$"]}]}}]},{"id":"a9f55eblinearmodels13","title":"Using a Diagram to Model Distance Walked","body":"Anna and Emanuel start at the same intersection. Anna walks east at $$4$$ miles per hour while Emanuel walks south at $$3$$ miles per hour. They are communicating with a two-way radio that has a range of $$2$$ miles.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Modeling with Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a9f55eblinearmodels13a","stepAnswer":["$$0.4$$"],"problemType":"TextBox","stepTitle":"How long after they start walking will they fall out of radio contact? Enter your answer in hours.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.4$$","hints":{"DefaultPathway":[{"id":"a9f55eblinearmodels13a-h1","type":"hint","dependencies":[],"title":"Defining Input and Output","text":"The first step is to identify the changing variables and assigning them to input and output. Our input will be $$t$$, time in hours, and our output will be A(t), distance in miles, and E(t), distance in miles.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels13a-h2","type":"hint","dependencies":["a9f55eblinearmodels13a-h1"],"title":"Drawing a Diagram","text":"Because it is not obvious how to define our output variable, we\u2019ll start by drawing a picture such as in the image attached to this hint.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels13a-h3","type":"hint","dependencies":["a9f55eblinearmodels13a-h2"],"title":"Identifying the Initial Value","text":"They both start at the same intersection so when $$t=0$$, the distance traveled by each person should also be $$0$$. Thus the initial value for each is $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels13a-h4","type":"hint","dependencies":["a9f55eblinearmodels13a-h3"],"title":"Identifying the Rate of Change","text":"Anna is walking $$4$$ miles per hour and Emanuel is walking $$3$$ miles per hour, which are both rates of change. The slope for A is $$4$$ and the slope for E is $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels13a-h5","type":"hint","dependencies":["a9f55eblinearmodels13a-h4"],"title":"Defining a Coordinate System","text":"For this problem, the distances from the starting point are important. To notate these, we can define a coordinate system, identifying the \u201cstarting point\u201d at the intersection where they both started. Then we can use the variable, A, which we introduced above, to represent Anna\u2019s position, and define it to be a measurement from the starting point in the eastward direction. Likewise, can use the variable, E, to represent Emanuel\u2019s position, measured from the starting point in the southward direction. Note that in defining the coordinate system, we specified both the starting point of the measurement and the direction of measure.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels13a-h6","type":"hint","dependencies":["a9f55eblinearmodels13a-h5"],"title":"Defining a Third Variable, D","text":"We can then define a third variable, D, to be the measurement of the distance between Anna and Emanuel. Showing the variables on the diagram is often helpful, as we can see from the attached image. Recall that we need to know how long it takes for D, the distance between them, to equal $$2$$ miles. Notice that for any given input \ud835\udc61, the outputs A(t), E(t), and D(t) represent distances.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels13a-h7","type":"hint","dependencies":["a9f55eblinearmodels13a-h6"],"title":"Using the Pythagorean Theorem","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels13a-h8","type":"hint","dependencies":["a9f55eblinearmodels13a-h7"],"title":"Solving D(t) as a Linear Function","text":"$$D(t)=5t;$$ this means that the distance between Anna and Emanuel is also a linear function. Because D is a linear function, we can now answer the question of when the distance between them will reach $$2$$ miles. We will set the output $$D(t)=2D(t)=2$$ and solve for $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels13a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{5}$$"],"dependencies":["a9f55eblinearmodels13a-h8"],"title":"Solving For $$t$$ When D(t) $$=$$ $$2$$","text":"$$D(t)=5t$$, so $$5t=2$$. $$t=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9f55eblinearmodels14","title":"Using a Diagram to Model Distance Between Cities","body":"There is a straight road leading from the town of Westborough to Agritown $$30$$ miles east and $$10$$ miles north. Partway down this road, it junctions with a second road, perpendicular to the first, leading to the town of Eastborough.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Modeling with Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a9f55eblinearmodels14a","stepAnswer":["$$12.64$$"],"problemType":"TextBox","stepTitle":"If the town of Eastborough is located $$20$$ miles directly east of the town of Westborough, how far is the road junction from Westborough (round to the nearest hundredth mile)?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12.64$$","hints":{"DefaultPathway":[{"id":"a9f55eblinearmodels14a-h1","type":"hint","dependencies":[],"title":"Drawing a Diagram","text":"It might help here to draw a picture of the situation. See the attached image. It would then be helpful to introduce a coordinate system. While we could place the origin anywhere, placing it at Westborough seems convenient. This puts Agritown at coordinates $$(30,10),(30,10)$$, and Eastborough at $$(20,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels14a-h2","type":"hint","dependencies":["a9f55eblinearmodels14a-h1"],"title":"Determining the Slope","text":"Using the point $$(30,10)$$, we can find the slope of the line from Westborough to Agritown. Slope $$=$$ Change in $$\\\\frac{y}{Change}$$ in $$x$$ $$=$$ $$\\\\frac{30-0}{10-0}=3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels14a-h3","type":"hint","dependencies":["a9f55eblinearmodels14a-h2"],"title":"Writing The Equation of the Line","text":"Now we can write an equation to describe the road from Westborough to Agritown, $$W(x)=\\\\frac{1}{3} x$$. From this, we can determine the perpendicular road to Eastborough will have slope $$m=-3$$. Because the town of Eastborough is at the point $$(20,0)$$, we can find the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels14a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$60$$"],"dependencies":["a9f55eblinearmodels14a-h3"],"title":"Solving For $$b$$","text":"$$E(x)=-3x+b$$. When $$x=20$$, $$b=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels14a-h5","type":"hint","dependencies":["a9f55eblinearmodels14a-h4"],"title":"Setting The Lines Equal","text":"We can now find the coordinates of the junction of the roads by finding the intersection of these lines. Setting them equal, $$\\\\frac{1}{3} x=-3x+60$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels14a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18$$"],"dependencies":["a9f55eblinearmodels14a-h5"],"title":"Solving For $$x$$","text":"$$\\\\frac{1}{3} x=-3x+60$$, $$x=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels14a-h7","type":"hint","dependencies":["a9f55eblinearmodels14a-h6"],"title":"Subsituting $$x$$ Back into the Original Equation","text":"Next, subsitute $$x=18$$ back into the original equation. $$y=W(18)=\\\\frac{1}{3\\\\left(18\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels14a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["a9f55eblinearmodels14a-h7"],"title":"Solving for $$y$$","text":"$$y=\\\\frac{1}{3\\\\left(18\\\\right)}$$, $$y=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels14a-h9","type":"hint","dependencies":["a9f55eblinearmodels14a-h8"],"title":"Interpreting the Answer","text":"Thus, the roads intersect at $$(18,6)$$. Using the distance formula, we can now find the distance from Westborough to the junction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9f55eblinearmodels15","title":"Building a System of Linear Models to Choose a Truck Rental Company","body":"Jamal is choosing between two truck-rental companies. The first, Keep on Trucking, Inc., charges an up-front fee of $20, then $$59$$ cents a mile. The second, Move It Your Way, charges an up-front fee of $16, then $$63$$ cents a mile.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Modeling with Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a9f55eblinearmodels15a","stepAnswer":["$$100$$"],"problemType":"TextBox","stepTitle":"At how many miles will Keep on Trucking, Inc. be the better choice for Jamal?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$100$$","hints":{"DefaultPathway":[{"id":"a9f55eblinearmodels15a-h1","type":"hint","dependencies":[],"title":"Input and Output","text":"The first step is to indentify input and output. The input is $$d$$, distance driven in miles; the outputs are K(d): cost, in dollars, for renting from Keep on Trucking, and M(d): cost, in dollars, for renting from Move It Your Way.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels15a-h2","type":"hint","dependencies":["a9f55eblinearmodels15a-h1"],"title":"Initial Value and Rate of Change","text":"The initial value is the up-front fee: $$K(0)=20$$ and $$M(0)=16;$$ the rates of change are $$K(d)=\\\\$0.59$$ /mile and $$P(d)=\\\\$0.63$$ /mile.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels15a-h3","type":"hint","dependencies":["a9f55eblinearmodels15a-h2"],"title":"Writing Linear Functions","text":"A linear function is of the form $$f(x)=mx+b$$. Using the rates of change and initial charges, we can write the equations $$K(d)=0.59d+20$$ and $$M(d)=0.63d+16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels15a-h4","type":"hint","dependencies":["a9f55eblinearmodels15a-h3"],"title":"Graphing the Linear Functions to Find the Solution","text":"Using these equations, we can determine when Keep on Trucking, Inc., will be the better choice. Because all we have to make that decision from is the costs, we are looking for when Move It Your Way, will cost less, or when $$K\\\\left(d\\\\right)<M\\\\left(d\\\\right)$$. The solution pathway will lead us to find the equations for the two functions, find the intersection, and then see where the K(d) function is smaller. The attached image shows the graph of the two functions, with K(d) in blue.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels15a-h5","type":"hint","dependencies":["a9f55eblinearmodels15a-h4"],"title":"Setting the Intersections Equal And Solving","text":"To find the intersection, we set the equations equal and solve. $$K(d)=M(d)$$, so $$0.59d+20=0.63d+16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels15a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$100$$"],"dependencies":["a9f55eblinearmodels15a-h5"],"title":"Solving For the Intersection","text":"$$0.59d+20=0.63d+16$$, $$d=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels15a-h7","type":"hint","dependencies":["a9f55eblinearmodels15a-h6"],"title":"Answer","text":"The value of $$d$$ tells us that the cost from the two companies will be the same if $$100$$ miles are driven. Either by looking at the graph, or noting that K(d) is growing at a slower rate, we can conclude that Keep on Trucking, Inc. will be the cheaper price when more than $$100$$ miles are driven, that is $$d>100$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9f55eblinearmodels16","title":"Try It #1: Building Linear Models","body":"A company sells doughnuts. They incur a fixed cost of $25,000 for rent, insurance, and other expenses. It costs $$\\\\$0.25$$ to produce each doughnut.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Modeling with Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a9f55eblinearmodels16a","stepAnswer":["C=0.25x+25000"],"problemType":"TextBox","stepTitle":"Write a linear model to represent the cost C of the company as a function of $$x$$, the number of doughnuts produced.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$C=0.25x+25000$$","hints":{"DefaultPathway":[{"id":"a9f55eblinearmodels16a-h1","type":"hint","dependencies":[],"title":"Identifying Input and Output","text":"The first step is to identify the input and output of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels16a-h2","type":"hint","dependencies":["a9f55eblinearmodels16a-h1"],"title":"Input and Output of the Equation","text":"The input is donuts, $$x$$, and the output is cost, c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels16a-h3","type":"hint","dependencies":["a9f55eblinearmodels16a-h2"],"title":"Identifying Initial Value","text":"The initial value is the fixed cost of $$25000$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels16a-h4","type":"hint","dependencies":["a9f55eblinearmodels16a-h3"],"title":"Identifying Rate of Change","text":"The rate of change is the value of change in output over change in input. We know that the rate of change is $$0.25$$ because it takes $$0.25$$ units of output to produce a single unit of input, a donut.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels16a-h5","type":"hint","dependencies":["a9f55eblinearmodels16a-h4"],"title":"Writing the Equation With the Identified Values","text":"The equation of a linear model is $$y=mx+b$$, where $$y$$ is the output, $$x$$ is the input, $$m$$ is the rate of change, and $$b$$ is the initial value. Plug in the values we have found to obtain the final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9f55eblinearmodels16b","stepAnswer":["$$25000$$"],"problemType":"TextBox","stepTitle":"Find and interpret the y-intercept.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$25000$$","hints":{"DefaultPathway":[{"id":"a9f55eblinearmodels16b-h1","type":"hint","dependencies":[],"title":"Identifying the y-intercept.","text":"The y-intercept is the value of the output when the value of the input is $$0$$. Using the equation we derived in part a, when $$x=0$$, $$C=25000$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels16b-h2","type":"hint","dependencies":["a9f55eblinearmodels16b-h1"],"title":"Interpreting the y-intercept","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9f55eblinearmodels17","title":"Try It #2: Building Linear Models","body":"A city\u2019s population has been growing linearly. In $$2008$$, the population was 28,200. By $$2012$$, the population was 36,800. Assume this trend continues.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Modeling with Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a9f55eblinearmodels17a","stepAnswer":["$$41100$$"],"problemType":"TextBox","stepTitle":"Predict the population in $$2014$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$41100$$","hints":{"DefaultPathway":[{"id":"a9f55eblinearmodels17a-h1","type":"hint","dependencies":[],"title":"Building a Linear Model","text":"The first step is to build a linear model. For simplicity, we will define the input as years from $$2008$$, and the output as population. The initial value is $$28200$$ and the rate of change is $$\\\\frac{36800-28200}{4-0}$$, or the change in output over the change in input.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels17a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2150$$"],"dependencies":["a9f55eblinearmodels17a-h1"],"title":"Calculating the Rate of Change","text":"What is $$\\\\frac{36800-28200}{4-0}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels17a-h3","type":"hint","dependencies":["a9f55eblinearmodels17a-h2"],"title":"Final Linear Model","text":"The rate of change is $$2150$$ $$\\\\frac{people}{year}$$, so the equation is $$P(t)=2150t+28200$$, where $$t$$ is years since $$2008$$ and P(t) is the population.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels17a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$41100$$"],"dependencies":["a9f55eblinearmodels17a-h3"],"title":"Calculating the Final Answer","text":"$$P(t)=2150t+28200$$ when $$t$$ $$=$$ $$6$$ (since $$2014$$ is $$6$$ years after 2008), $$P(6)=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9f55eblinearmodels17b","stepAnswer":["$$2020$$"],"problemType":"TextBox","stepTitle":"Identify the year in which the population will reach $$54000$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2020$$","hints":{"DefaultPathway":[{"id":"a9f55eblinearmodels17b-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":[],"title":"Solving for $$t$$","text":"Solve for $$t$$ when $$P(t)=54000$$. $$54000=2150t+28200$$, $$t=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9f55eblinearmodels18","title":"Try It #3: Using a Diagram to Build a Model","body":"There is a straight road leading from the town of Timpson to Ashburn $$60$$ miles east and $$12$$ miles north. Partway down the road, it junctions with a second road, perpendicular to the first, leading to the town of Garrison.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Modeling with Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a9f55eblinearmodels18a","stepAnswer":["$$4.44$$ miles"],"problemType":"MultipleChoice","stepTitle":"If the town of Garrison is located $$22$$ miles directly east of the town of Timpson, how far is the road junction from Timpson?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$4.44$$ miles","choices":["$$5.55$$ miles","$$4.44$$ miles","$$3.33$$ miles","$$2.22$$ miles"],"hints":{"DefaultPathway":[{"id":"a9f55eblinearmodels18a-h1","type":"hint","dependencies":[],"title":"Graphing the Problem","text":"Graphing the problem is useful to understand the geometry we are working with. First, label a point T to represent the town of Timpson. We know that Ashburn is $$60$$ miles east and $$12$$ miles north of Timpson, so indicate a point A for Ashburn $$60$$ units right of Timpson and $$12$$ units up. This can be done by setting point T at the origin, $$(0,0)$$, and setting point A at $$(60,12)$$. We know that Garrison is perpendicular to the line connecting $$(0,0)$$ and $$(60,12)$$, based on the problem\'s given information. We can label Garrison as point G, at $$(20,0)$$ to represent being $$20$$ units east of point T. From the graph, we can tell that we are trying to find the point of the intersection between the line perpendicular to the line connecting T and A, that also goes through $$(20,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels18a-h2","type":"hint","dependencies":["a9f55eblinearmodels18a-h1"],"title":"Slope Between T and A","text":"The slope of the line between A and T is $$\\\\frac{60}{12}=5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels18a-h3","type":"hint","dependencies":["a9f55eblinearmodels18a-h2"],"title":"Finding the Relevant Line","text":"The slope of the perpendicular line is $$\\\\frac{-1}{5}$$. $$0=20\\\\frac{-1}{5}+b$$, so $$b=4$$. The line is $$y=\\\\frac{-1}{5} x+4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels18a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{10}{13}$$"],"dependencies":["a9f55eblinearmodels18a-h3"],"title":"Finding the Intersection","text":"$$\\\\frac{-1}{5} x+4=5x$$, $$x=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels18a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{50}{13}$$"],"dependencies":["a9f55eblinearmodels18a-h4"],"title":"Finding the Y Coordinate of the Intersection","text":"$$x=\\\\frac{10}{13}$$, $$f{\\\\left(\\\\frac{10}{13}\\\\right)}=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels18a-h6","type":"hint","dependencies":["a9f55eblinearmodels18a-h5"],"title":"Using the Pythagorean Theorem","text":"$$\\\\sqrt{{\\\\left(\\\\frac{50}{13}\\\\right)}^2+{\\\\left(\\\\frac{10}{13}\\\\right)}^2}=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9f55eblinearmodels19","title":"Exercise #5: Linear Models","body":"Find the area of a parallelogram bounded by the y-axis, the line $$x=3$$, the line $$f(x)=1+2x$$, and the line parallel to f(x) passing through $$(2,7)$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Modeling with Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a9f55eblinearmodels19a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"What is the area?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"a9f55eblinearmodels19a-h1","type":"hint","dependencies":[],"title":"Figuring out the Horizontal Dimensions","text":"We have enough information to figure out the length of the horizontal base. We know that the parallelogram is bounded by $$x=0$$ (the $$y$$ axis) and $$x=3$$, so it is $$3$$ units wide.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels19a-h2","type":"hint","dependencies":["a9f55eblinearmodels19a-h1"],"title":"Figuring out the Vertical Dimensions","text":"(It is useful to graph the information to better understand the geometry involved.) To find the vertical dimensions, first find the equation of the line that passes through $$(2,7)$$. Then, see where it intersects the $$y$$ axis, and alsto where f(x) intersects the $$y$$ axis. Parallelograms have equal vertical bases, so the difference in the $$y$$ coordinates will be the length of the vertical base.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels19a-h3","type":"hint","dependencies":["a9f55eblinearmodels19a-h2"],"title":"Finding the Relevant Line","text":"We know that the line passing through $$(2,7)$$ is parallel to $$f(x)=1+2x$$. What this means is that the line we want to find also has a slope of $$2$$. Since we know the slope of the line and a point on the line, we can plug this into the equation $$y=mx+b$$ to find $$b$$. We know that the point $$(2,7)$$ means $$y=7$$ and $$x=2$$, and the slope, $$m$$, is $$2$$. Thus, $$7=2\\\\left(2\\\\right)+b$$. Solving for $$b$$, we get that $$b=3$$. Therefore, the equation of the line is $$2x+3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels19a-h4","type":"hint","dependencies":["a9f55eblinearmodels19a-h3"],"title":"Finding Where the Relevant Line Intersects the Y Axis","text":"Next, we can find where this line intersects the $$y$$ axis. At the $$y$$ axis, $$x=0$$, so plug in $$0$$ to the equation. When $$x=0$$, $$y=2\\\\left(0\\\\right)+3=3$$. Thus, the coordinate point of one of the vertices is $$(0,3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels19a-h5","type":"hint","dependencies":["a9f55eblinearmodels19a-h4"],"title":"Finding Where f(x) Intersects the Y Axis","text":"We can then find where f(x) intersects the $$y$$ axis. Plugging in $$x=0$$, we get $$f(0)=1$$. Thus, the other vertice that we are looking for is $$(0,1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels19a-h6","type":"hint","dependencies":["a9f55eblinearmodels19a-h5"],"title":"Calculating the Length of the Vertical Base","text":"The difference of the $$y$$ coordinates of the two points we have found is $$3-1=3$$. Thus, the vertical base has a length of $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels19a-h7","type":"hint","dependencies":["a9f55eblinearmodels19a-h6"],"title":"Formula For a Parallelogram\'s Area","text":"The area of a parallelogram is the height times the base. In equation form, $$A=bh$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9f55eblinearmodels2","title":"Exercise 9: Using Linear Models to Predict Outcomes","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Modeling with Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a9f55eblinearmodels2a","stepAnswer":["$$2300$$"],"problemType":"TextBox","stepTitle":"For the following exercises, consider this scenario: A town\'s population has been decreasing at a constant rate. In $$2010$$ the population was 5,900. By $$2012$$ the population had dropped 4,700. Assume this trend continues. Predict the population in $$2016$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2300$$","hints":{"DefaultPathway":[{"id":"a9f55eblinearmodels2a-h1","type":"hint","dependencies":[],"title":"Finding a Trend Between Years","text":"Find the trend between the populations rom $$2010$$ and $$2012$$ based on how much their population dropped.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels2a-h2","type":"hint","dependencies":["a9f55eblinearmodels2a-h1"],"title":"Using an Equation to Represent Relationships","text":"Create an equation representing the relationship between population and the years.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels2a-h3","type":"hint","dependencies":["a9f55eblinearmodels2a-h2"],"title":"Subsituting a Value into the Trend Equation","text":"Insert $$2016$$ as $$x$$ in the equation created to get the population","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9f55eblinearmodels20","title":"Exercise #6: Linear Models","body":"Find the area of a triangle bounded by the x-axis, the line $$f(x)=12-\\\\frac{1}{3} x$$, and the line perpendicular to f(x) that passes through the origin.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Modeling with Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a9f55eblinearmodels20a","stepAnswer":["$$162$$"],"problemType":"TextBox","stepTitle":"What is the area?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$162$$","hints":{"DefaultPathway":[{"id":"a9f55eblinearmodels20a-h1","type":"hint","dependencies":[],"title":"Area Formula for a Triangle","text":"The area formula for a triangle is $$\\\\frac{1}{2} bh$$, where $$b$$ is the base and $$h$$ is the height.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels20a-h2","type":"hint","dependencies":["a9f55eblinearmodels20a-h1"],"title":"Drawing a Diagram","text":"First draw, a diagram. By graphing f(x) and the line perpendicular to it, we can tell that the height of the triangle is equal to the y-coordinate of the intersection between f(x) and the perpendicular line. We can also see that the horizontal base of the triangle is the distance from $$(0,0)$$ to the point where f(x) intersects the $$x$$ axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels20a-h3","type":"hint","dependencies":["a9f55eblinearmodels20a-h2"],"title":"Figuring Out the Equation of the Perpendicular Line","text":"The first step is to figure out the equation of the perpendicular line. Because it is perpendicular to $$f(x)=12-\\\\frac{1}{3} x$$, we know that its slope is equal to the opposite reciprocal of the slope of f(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels20a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["a9f55eblinearmodels20a-h3"],"title":"Figuring Out the Slope of the Relevant Line","text":"What is the opposite reciprocal of $$\\\\frac{-1}{3}$$? This is equal to the slope of the perpendicular line to f(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels20a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a9f55eblinearmodels20a-h4"],"title":"Figuring out the X Coordinate of the Intersection Between f(x) and its perpendicular line","text":"At what $$x$$ value does $$3x=12-\\\\frac{1}{3} x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels20a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["a9f55eblinearmodels20a-h5"],"title":"Figuring out the Y-Coordinate of the Intersection","text":"When $$x=9$$, $$f(9)=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels20a-h7","type":"hint","dependencies":["a9f55eblinearmodels20a-h6"],"title":"Interpreting the Intersection Between f(x) and its Reciprocal","text":"Through graphing, we know that the y-coordinate of the intersection is the height of the triangle, and the intersection has coordinates $$(9,9)$$. Thus, the height of the triangle is $$9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels20a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$36$$"],"dependencies":["a9f55eblinearmodels20a-h7"],"title":"Finding the Intersection of f(x) with the $$x$$ axis","text":"When $$y=0$$, f(x) intersects the $$x$$ axis. Plug in $$y=0$$ to the equation for f(x). What is $$x$$ equal to when $$y=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels20a-h9","type":"hint","dependencies":["a9f55eblinearmodels20a-h8"],"title":"Base of the Triangle","text":"The base of the triangle is equal to the x-coordinate of the intersection between f(x) and the $$x$$ axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9f55eblinearmodels21","title":"Exercise #7: Linear Models","body":"Find the area of a triangle bounded by the y-axis, the line $$f(x)=9-\\\\frac{6}{7} x$$, and the line perpendicular to f(x) that passes through the origin.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Modeling with Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a9f55eblinearmodels21a","stepAnswer":["$$\\\\frac{267}{7}$$"],"problemType":"TextBox","stepTitle":"What is the area?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{267}{7}$$","hints":{"DefaultPathway":[{"id":"a9f55eblinearmodels21a-h1","type":"hint","dependencies":[],"title":"Area Formula for a Triangle","text":"The area formula for a triangle is $$\\\\frac{1}{2} bh$$, where $$b$$ is the base and $$h$$ is the height.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels21a-h2","type":"hint","dependencies":["a9f55eblinearmodels21a-h1"],"title":"Drawing a Diagram","text":"First draw, a diagram. By graphing f(x) and the line perpendicular to it, we can tell that the height of the triangle is equal to the y-coordinate of the intersection between f(x) and the perpendicular line. We can also see that the horizontal base of the triangle is the distance from $$(0,0)$$ to the point where f(x) intersects the $$x$$ axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels21a-h3","type":"hint","dependencies":["a9f55eblinearmodels21a-h2"],"title":"Figuring Out the Equation of the Perpendicular Line","text":"The first step is to figure out the equation of the perpendicular line. Because it is perpendicular to $$f(x)=9-\\\\frac{6}{7} x$$, we know that its slope is equal to the opposite reciprocal of the slope of f(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels21a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{7}{6}$$"],"dependencies":["a9f55eblinearmodels21a-h3"],"title":"Figuring Out the Slope of the Relevant Line","text":"What is the opposite reciprocal of $$\\\\frac{-6}{7}$$? This is equal to the slope of the perpendicular line to f(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels21a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{85}{42}$$"],"dependencies":["a9f55eblinearmodels21a-h4"],"title":"Figuring out the X Coordinate of the Intersection Between f(x) and its perpendicular line","text":"At what $$x$$ value does $$\\\\frac{7}{6} x=9-\\\\frac{6}{7} x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels21a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{356}{49}$$"],"dependencies":["a9f55eblinearmodels21a-h5"],"title":"Figuring out the Y-Coordinate of the Intersection","text":"When $$x=\\\\frac{85}{42}$$, $$f{\\\\left(\\\\frac{85}{42}\\\\right)}=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels21a-h7","type":"hint","dependencies":["a9f55eblinearmodels21a-h6"],"title":"Interpreting the Intersection Between f(x) and its Reciprocal","text":"Through graphing, we know that the y-coordinate of the intersection is the height of the triangle, and the intersection has coordinates $$(\\\\frac{85}{42},\\\\frac{356}{49})$$. Thus, the height of the triangle is $$\\\\frac{356}{49}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels21a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{21}{2}$$"],"dependencies":["a9f55eblinearmodels21a-h7"],"title":"Finding the Intersection of f(x) with the $$x$$ axis","text":"When $$y=0$$, f(x) intersects the $$x$$ axis. Plug in $$y=0$$ to the equation for f(x). What is $$x$$ equal to when $$y=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels21a-h9","type":"hint","dependencies":["a9f55eblinearmodels21a-h8"],"title":"Base of the Triangle","text":"The base of the triangle is equal to the x-coordinate of the intersection between f(x) and the $$x$$ axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9f55eblinearmodels3","title":"Exercise 10: Using Linear Models to Predict Outcomes","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Modeling with Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a9f55eblinearmodels3a","stepAnswer":["$$2019$$"],"problemType":"TextBox","stepTitle":"A town\'s population has been decreasing at a constant rate. In $$2010$$ the population was 5,900. By $$2012$$ the population had dropped to 4,700. Assume this trend continues. Identify the year in which the population will reach $$0$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2019$$","hints":{"DefaultPathway":[{"id":"a9f55eblinearmodels3a-h1","type":"hint","dependencies":[],"title":"Finding a Trend Between Years","text":"Find the trend between the populations rom $$2010$$ and $$2012$$ based on how much their population dropped. How much does the population drop per year? (This is the slope of the equation.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels3a-h2","type":"hint","dependencies":["a9f55eblinearmodels3a-h1"],"title":"Using an Equation to Represent Relationships","text":"Create an equation representing the relationship between population and the years.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels3a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y=5900-600x$$"],"dependencies":["a9f55eblinearmodels3a-h2"],"title":"Verifying the Equation","text":"What is the equation that represents the town\'s population, with $$y$$ being population and $$x$$ being number of years since 2010?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y=5900-1200x$$","$$y=5900-1000x$$","$$y=5900-600x$$"]},{"id":"a9f55eblinearmodels3a-h4","type":"hint","dependencies":["a9f55eblinearmodels3a-h3"],"title":"Subsituting a Value into the Trend Equation","text":"Insert $$0$$ as $$y$$ in the equation to solve for $$x$$, the number of years passed when the population reaches $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9f55eblinearmodels4","title":"Exercise 11: Using Linear Models to Predict Outcomes","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Modeling with Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a9f55eblinearmodels4a","stepAnswer":["$$64170$$"],"problemType":"TextBox","stepTitle":"A town\'s population has been increased at a constant rate. In $$2010$$, the population was 46,020. By $$2012$$, the population had increased to 52,070. Assume this trend continues. Predict the population in $$2016$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$64170$$","hints":{"DefaultPathway":[{"id":"a9f55eblinearmodels4a-h1","type":"hint","dependencies":[],"title":"Finding a Trend Between Years","text":"Find the trend between the populations between the years.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels4a-h2","type":"hint","dependencies":["a9f55eblinearmodels4a-h1"],"title":"Using an Equation to Represent Relationships","text":"Create an equation representing the relationship between population and the years.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels4a-h3","type":"hint","dependencies":["a9f55eblinearmodels4a-h2"],"title":"Subsituting a Value into the Trend Equation","text":"Insert $$2016$$ as $$x$$ in the equation created to get the population","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9f55eblinearmodels5","title":"Exercise 12: Using Linear Models to Predict Outcomes","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Modeling with Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a9f55eblinearmodels5a","stepAnswer":["$$2019$$"],"problemType":"TextBox","stepTitle":"A town\'s population has been increased at a constant rate. In $$2010$$, the population was 46,020. By $$2012$$, the population had increased to 52,070. Assume this trend continues. Identify the year in which the population will reach 75,000.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2019$$","hints":{"DefaultPathway":[{"id":"a9f55eblinearmodels5a-h1","type":"hint","dependencies":[],"title":"Finding a Trend Between Years","text":"Find the trend between the populations between the years. How much does the population increase per year?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels5a-h2","type":"hint","dependencies":["a9f55eblinearmodels5a-h1"],"title":"Using an Equation to Represent Relationships","text":"Create an equation representing the relationship between population and the years.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y=46020+3025x$$"],"dependencies":["a9f55eblinearmodels5a-h2"],"title":"Verifying the Equation","text":"What is the equation that represents the town\'s population, with $$y$$ being population and $$x$$ being number of years since 2010?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y=46020+3025x$$","$$y=46020+6050x$$","$$y=46020-3025x$$"]},{"id":"a9f55eblinearmodels5a-h4","type":"hint","dependencies":["a9f55eblinearmodels5a-h3"],"title":"Subsituting a Value into the Trend Equation","text":"Insert 75,000 as $$y$$ in the equation created to get the population","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9f55eblinearmodels6","title":"Exercise 13: Using Linear Models to Predict Outcomes","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Modeling with Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a9f55eblinearmodels6a","stepAnswer":["$$P(t)=75000+2500t$$"],"problemType":"MultipleChoice","stepTitle":"A town\'s population has an initial population of 75,000. It grows at a constant rate of 2,500 per year for five years. Find the linear function that models the town\'s population P as a function of the year, $$t$$, where $$t$$ is the number of years since the model began.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$P(t)=75000+2500t$$","choices":["$$P(t)=2500+75000t$$","$$P(t)=75000+2500t$$","$$P(t)=750+2500t$$","$$P(t)=2500+75t$$"],"hints":{"DefaultPathway":[{"id":"a9f55eblinearmodels6a-h1","type":"hint","dependencies":[],"title":"Finding the Trend Between Populations","text":"Find the trend between the populations between the years.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels6a-h2","type":"hint","dependencies":["a9f55eblinearmodels6a-h1"],"title":"Creating a Linear Model","text":"Create an equation representing the relationship between population and the years.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9f55eblinearmodels7","title":"Exercise 14: Using Linear Models to Predict Outcomes","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Modeling with Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a9f55eblinearmodels7a","stepAnswer":["Domain: [0,5]; Range: [75000, 87500]"],"problemType":"MultipleChoice","stepTitle":"Find a reasonable domain and range for the function P.","stepBody":"","answerType":"string","variabilization":{},"choices":["Domain: $$[0,\\\\infty];$$ Range: [75000, inf]","Domain: [0,5]; Range: [75000, 87500]","Domain: $$[-\\\\infty,\\\\infty];$$ Range: [75000, inf]","Domain: [0,5]; Range: [75000, 85000]"],"hints":{"DefaultPathway":[{"id":"a9f55eblinearmodels7a-h1","type":"hint","dependencies":[],"title":"Considering Restrictions On Domain and Range","text":"Consider what makes sense in terms of domain and range. Will negative $$x$$ and negative $$y$$ values make sense in this equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels7a-h2","type":"hint","dependencies":["a9f55eblinearmodels7a-h1"],"title":"Considering Starts and Ends of Domain and Range","text":"The domain, years would start at $$0$$ and would end at five. Thus, the range would be from the population at the beginning to the population five years later.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9f55eblinearmodels8","title":"Exercise 15: Using Linear Models to Predict Outcomes","body":"A town\'s population has an initial population of 75,000. It grows at a constant rate of 2,500 per year for five years.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Modeling with Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a9f55eblinearmodels8a","stepAnswer":["$$x-intercept$$: $$(-30,0)$$, $$y-intercept$$: (0, 75,000)"],"problemType":"MultipleChoice","stepTitle":"If the function P is graphed, find and interpret the $$x-$$ and y-intercepts.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"x-intercept: $$(-30,0)$$, y-intercept: (0, 75,000)","choices":["$$x-intercept$$: $$(-30,0)$$, $$y-intercept$$: (0, 75,000)","$$x-intercept$$: (0, 75,000), $$y-intercept$$: $$(-30,0)$$","$$x-intercept$$: $$(-20,0)$$, $$y-intercept$$: (0, 85,000)","$$x-intercept$$: (0, 85,000), $$y-intercept$$: $$(-20,0)$$"],"hints":{"DefaultPathway":[{"id":"a9f55eblinearmodels8a-h1","type":"hint","dependencies":[],"title":"What do the different numbers represent?","text":"Thirty years before the start of this model, the town had no citizens. Initially, the town had a population of 75,000.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9f55eblinearmodels9","title":"Exercise 16: Using Linear Models to Predict Outcomes","body":"A town\'s population has an initial population of 75,000. It grows at a constant rate of 2,500 per year for five years.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Modeling with Linear Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"a9f55eblinearmodels9a","stepAnswer":["$$2500$$, change in population over change in time"],"problemType":"MultipleChoice","stepTitle":"If the function P is graphed, find and interpret the slope of the function.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2500$$, change in population over change in time","choices":["$$75000$$, change in population over change in time","$$2500$$, change in population over change in time","$$2500$$, change in time over change in population"],"hints":{"DefaultPathway":[{"id":"a9f55eblinearmodels9a-h1","type":"hint","dependencies":[],"title":"Definition of Slope","text":"Slope is the change in output over the change in input.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels9a-h2","type":"hint","dependencies":["a9f55eblinearmodels9a-h1"],"title":"Output and Input","text":"In this problem, the output is population and the input is time in years.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f55eblinearmodels9a-h3","type":"hint","dependencies":["a9f55eblinearmodels9a-h2"],"title":"Answer","text":"Every unit of change for the input, which is a year, corresponds to $$2500$$ units of change for the output, which is the population. Thus, the slope is $$2500$$, and it represents the change in population over change in time.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9f9bc74.1baker1","title":"","body":"Use the following information to answer the next six exercises: A baker is deciding how many batches of muffins to\\\\nmake to sell in his bakery. He wants to make enough to sell every one and no fewer. Through observation, the baker\\\\nhas established a probability distribution.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Probability Distribution Function (PDF) for a Discrete Random Variable","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a9f9bc74.1baker1a","stepAnswer":["$$0.85$$"],"problemType":"TextBox","stepTitle":"What is the probability the baker will sell more than one batch? $$P\\\\left(x>1\\\\right)=$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.85$$","hints":{"DefaultPathway":[{"id":"a9f9bc74.1baker1a-h1","type":"hint","dependencies":[],"title":"Probability Sum","text":"Add the probabilities when $$x$$ is greater than $$1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f9bc74.1baker1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.85$$"],"dependencies":["a9f9bc74.1baker1a-h1"],"title":"Addition","text":"$$0.35+0.4+0.1=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9f9bc74.1baker1b","stepAnswer":["$$0.15$$"],"problemType":"TextBox","stepTitle":"What is the probability the baker will sell exactly one one batch? $$P(x=1)=$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.15$$","hints":{"DefaultPathway":[{"id":"a9f9bc74.1baker1b-h1","type":"hint","dependencies":[],"title":"Interpretation","text":"The probability that the baker sells exactly one batch is the probability when random variable $$x$$ is $$1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f9bc74.1baker1b-h2","type":"hint","dependencies":["a9f9bc74.1baker1b-h1"],"title":"Interpretation","text":"The random variable $$x$$ is the number of batches of muffins that the baker sells, so the probability that the baker will sell exactly one batch is $$P(x=1)=0.15$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9f9bc74.1baker1c","stepAnswer":["$$2.45$$"],"problemType":"TextBox","stepTitle":"On average, how many batches should the baker make?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2.45$$","hints":{"DefaultPathway":[{"id":"a9f9bc74.1baker1c-h1","type":"hint","dependencies":[],"title":"Probability Sum","text":"Get the sum of the products of all x\'s and their respective P(x) to get the average number of batches that the baker sells.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f9bc74.1baker1c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.45$$"],"dependencies":["a9f9bc74.1baker1c-h1"],"title":"Probability Sum","text":"$$1\\\\left(0.15\\\\right)+2\\\\left(0.35\\\\right)+3\\\\left(0.4\\\\right)+4\\\\left(0.1\\\\right)=0.15+0.7+1.2+0.4=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f9bc74.1baker1c-h3","type":"hint","dependencies":["a9f9bc74.1baker1c-h2"],"title":"Interpretation","text":"$$2.45$$ is the value of the average number of batches that the baker sells according to the table","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9f9bc74.1basketball1","title":"","body":"Jeremiah has basketball practice two days a week. Ninety percent of the time, he attends both practices. Eight percent of the time, he attends one practice. Two percent of the time, he does not attend either practice.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Probability Distribution Function (PDF) for a Discrete Random Variable","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a9f9bc74.1basketball1a","stepAnswer":["Number of days Jeremiah has basketball practice weekly"],"problemType":"MultipleChoice","stepTitle":"","stepBody":"What is the random variable X?","answerType":"string","variabilization":{},"choices":["Number of days Jeremiah has basketball practice weekly","Number of basketballs Jeremiah owns","number of days in a week"],"hints":{"DefaultPathway":[{"id":"a9f9bc74.1basketball1a-h1","type":"hint","dependencies":[],"title":"","text":"The probabilities given concerns the amount of days taht Jeremiah has practice in a week.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9f9bc74.1basketball1b","stepAnswer":["$$0$$, $$1$$, and $$2$$"],"problemType":"MultipleChoice","stepTitle":"","stepBody":"What values does the random variable X take on?","answerType":"string","variabilization":{},"answerLatex":"$$0$$, $$1$$, and $$2$$","choices":["$$0$$, $$1$$, and $$2$$","$$2$$","1,2"],"hints":{"DefaultPathway":[{"id":"a9f9bc74.1basketball1b-h1","type":"hint","dependencies":[],"title":"","text":"Consider the given probabilities of each unique instance of days that Jeremiah attends basketball practice.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f9bc74.1basketball1b-h2","type":"hint","dependencies":["a9f9bc74.1basketball1b-h1"],"title":"","text":"Note that it is possible that Jeremiah doesn\'t attend practice at all, so $$0$$ is a possible value of X.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9f9bc74.1ellen1","title":"Ellen\'s Singing Practice","body":"Ellen has music practice three days a week. She practices for all of the three days 85% of the time, two days 8% of the time, one day 4% of the time, and no days 3% of the time. One week is selected at random. The random variable $$x$$ is a day that Ellen practices music.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Probability Distribution Function (PDF) for a Discrete Random Variable","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a9f9bc74.1ellen1a","stepAnswer":["$$0.03$$"],"problemType":"TextBox","stepTitle":"Given that the random variable $$x$$ is a day that Ellen practices music, what is $$P(X=0)=$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.03$$","hints":{"DefaultPathway":[{"id":"a9f9bc74.1ellen1a-h1","type":"hint","dependencies":[],"title":"Interpretation","text":"Ellen practices on \\"no days 3% of the time\\", therefore that is the probability that she practices when random variable $$x$$ is $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9f9bc74.1ellen1b","stepAnswer":["$$0.04$$"],"problemType":"TextBox","stepTitle":"$$P(X=1)=$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.04$$","hints":{"DefaultPathway":[{"id":"a9f9bc74.1ellen1b-h1","type":"hint","dependencies":[],"title":"Interpretation","text":"Ellen practices \\"one day 4% of the time\\".","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9f9bc74.1ellen1c","stepAnswer":["$$0.08$$"],"problemType":"TextBox","stepTitle":"$$P(X=2)=$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.08$$","hints":{"DefaultPathway":[{"id":"a9f9bc74.1ellen1c-h1","type":"hint","dependencies":[],"title":"Interpretation","text":"Ellen practices \\"two days 8% of the time\\".","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9f9bc74.1ellen1d","stepAnswer":["$$0.85$$"],"problemType":"TextBox","stepTitle":"Probability","stepBody":"What is the probability that Ellen practices $$3$$ days a week?","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.85$$","hints":{"DefaultPathway":[{"id":"a9f9bc74.1ellen1d-h1","type":"hint","dependencies":[],"title":"Interpretation","text":"It is stated that Ellen practices for 85% of the time for all three days.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9f9bc74.1employer1","title":"Probability Distribution Function (PDF) for a Discrete Random Variable","body":"Complete the table using the data provided. A company wants to evaluate its attrition rate, in other words, how long new hires stay with the company. Over the years, they have established the following probability distribution. Let X $$=$$ the number of years a new hire will stay with the company.\\\\nLet P(x) $$=$$ the probability that a new hire will stay with the company $$x$$ years.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Probability Distribution Function (PDF) for a Discrete Random Variable","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a9f9bc74.1employer1a","stepAnswer":["$$0.1$$"],"problemType":"TextBox","stepTitle":"$$x$$ $$=$$ 4; $$P(x=4)$$ $$=$$ ?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.1$$","hints":{"DefaultPathway":[{"id":"a9f9bc74.1employer1a-h1","type":"hint","dependencies":[],"title":"The Sum of a Probabillity Distribution","text":"The sum of all probabilities in a probability distribution is $$1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f9bc74.1employer1a-h2","type":"hint","dependencies":["a9f9bc74.1employer1a-h1"],"title":"The Sum of a Probabillity Distribution","text":"There is only one probability slot to complete. Subtract the sum of the goven probabilities by $$1$$ to get the probability of $$x=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f9bc74.1employer1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.9$$"],"dependencies":["a9f9bc74.1employer1a-h2"],"title":"Addition","text":"$$0.12+0.18+0.3+0.15+0.1+0.05=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f9bc74.1employer1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1$$"],"dependencies":["a9f9bc74.1employer1a-h3"],"title":"Subtraction","text":"$$1-0.9=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9f9bc74.1employer1b","stepAnswer":["$$0.1$$"],"problemType":"TextBox","stepTitle":"$$P(X=4)=$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.1$$","hints":{"DefaultPathway":[{"id":"a9f9bc74.1employer1b-h1","type":"hint","dependencies":[],"title":"Probability","text":"add the","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9f9bc74.1employer1c","stepAnswer":["$$0.15$$"],"problemType":"TextBox","stepTitle":"P(x $$ \\\\geq $$ $$5)=$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.15$$","hints":{"DefaultPathway":[{"id":"a9f9bc74.1employer1c-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.15$$"],"dependencies":[],"title":"Addition","text":"$$0.1+0.05=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9f9bc74.1employer1d","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"What does the column \\"P(x)\\" sum to?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a9f9bc74.1employer1d-h1","type":"hint","dependencies":[],"title":"Understanding Probability distriibutions","text":"Probability distributions always sum to $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9f9bc74.1javier1","title":"Probability","body":"Javier volunteers in community events each month.\\\\nHe does not do more than five events in a month. He attends exactly five events 35% of the time, four events 25%\\\\nof the time, three events 20% of the time, two events 10% of the time, one event 5% of the time, and no events 5%\\\\nof the time.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Probability Distribution Function (PDF) for a Discrete Random Variable","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a9f9bc74.1javier1a","stepAnswer":["the number of events Javier volunteers for each month"],"problemType":"MultipleChoice","stepTitle":"Random Variable","stepBody":"What is the random variable X ?","answerType":"string","variabilization":{},"choices":["the number of events Javier volunteers for each month","the number of months that Javier volunteers for events"],"hints":{"DefaultPathway":[{"id":"a9f9bc74.1javier1a-h1","type":"hint","dependencies":[],"title":"Interpretation","text":"The information provided concerns the probability of the number of times that Javier volunteers in a month.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9f9bc74.1javier1b","stepAnswer":["$$0.95$$"],"problemType":"TextBox","stepTitle":"Probability Density Functions","stepBody":"Find the probability that Javier volunteers for at least one event each month. $$P\\\\left(x>O\\\\right)=$$?","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.95$$","hints":{"DefaultPathway":[{"id":"a9f9bc74.1javier1b-h1","type":"hint","dependencies":[],"title":"Probability Density Functions","text":"Use the probability that Javier attends no events and subtract it from the complementary probability of $$P(X=0)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f9bc74.1javier1b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.95$$"],"dependencies":["a9f9bc74.1javier1b-h1"],"title":"Subtraction","text":"$$1-0.05=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9f9bc74.1javier1c","stepAnswer":["$$0.2$$"],"problemType":"TextBox","stepTitle":"Probability","stepBody":"Find the probability that Javier volunteers for less than three events each month. $$P\\\\left(x<3\\\\right)=$$?","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.2$$","hints":{"DefaultPathway":[{"id":"a9f9bc74.1javier1c-h1","type":"hint","dependencies":[],"title":"Interpretation","text":"The probability in this scenario is the sum of the probailities that Javier volunteers $$0$$, $$1$$, and $$2$$ times a week.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f9bc74.1javier1c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.2$$"],"dependencies":["a9f9bc74.1javier1c-h1"],"title":"Addition","text":"$$0.05+0.05+0.1=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9f9bc74.1nancy1","title":"","body":"Suppose Nancy has classes three days a week. She attends classes three days a week 80% of the time,\\\\ntwo days 15% of the time, one day 4% of the time, and no days 1% of the time. Suppose one week is\\\\nrandomly selected.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Probability Distribution Function (PDF) for a Discrete Random Variable","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a9f9bc74.1nancy1a","stepAnswer":["attends class per week"],"problemType":"MultipleChoice","stepTitle":"","stepBody":"Given the probabilities, let the random variable X $$=$$ the number of days Nancy $$_{}$$","answerType":"string","variabilization":{},"choices":["attends class per week","studies for class","skips classes","doesn\'t have a class"],"hints":{"DefaultPathway":[]}},{"id":"a9f9bc74.1nancy1b","stepAnswer":["$$0$$, $$1$$, $$2$$, and $$3$$"],"problemType":"MultipleChoice","stepTitle":"","stepBody":"X takes on what values?","answerType":"string","variabilization":{},"answerLatex":"$$0$$, $$1$$, $$2$$, and $$3$$","choices":["$$0$$, $$1$$, $$2$$, and $$3$$","$$3$$","$$1$$ and $$3$$"],"hints":{"DefaultPathway":[{"id":"a9f9bc74.1nancy1b-h1","type":"hint","dependencies":[],"title":"","text":"If it is given that Nancy has classes three days a week, then the possible values of X is three or less days a week. Only count discrete days.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9f9bc74.1nancy3","title":"","body":"Suppose one week is randomly chosen. Construct a probability distribution table.The table should have two columns labeled $$x$$ and P(x).","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.1 Probability Distribution Function (PDF) for a Discrete Random Variable","courseName":"OpenStax: Introductory Stats","steps":[{"id":"a9f9bc74.1nancy3a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"","stepBody":"What does the P(x) column sum to?","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"a9f9bc74.1nancy3a-h1","type":"hint","dependencies":[],"title":"","text":"Remember, a discrete probability distribution function has two characteristics:\\\\n\\\\n$$1$$. Each probability is between zero and one, inclusive.\\\\n$$2$$. The sum of the probabilities is one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"a9f9bc74.1nancy3b","stepAnswer":["$$0.01$$"],"problemType":"TextBox","stepTitle":"","stepBody":"$$P(X=0)=$$?","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.01$$","hints":{"DefaultPathway":[{"id":"a9f9bc74.1nancy3b-h1","type":"hint","dependencies":[],"title":"","text":"Note that Nancy attends \\"no days 1% of the time\\".","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9f9bc74.1nancy3b-h2","type":"hint","dependencies":["a9f9bc74.1nancy3b-h1"],"title":"","text":"To convert 1% back to number form, divide the percentage by $$100$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ff35apoly1","title":"Zero Product 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$$x-5=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"a9ff35apoly11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["a9ff35apoly11a-h4"],"title":"Solving for $$0$$","text":"Solve: $$x+3=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"a9ff35apoly12","title":"Factoring Polynomial Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.5 Polynomial Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"a9ff35apoly12a","stepAnswer":["$$-4$$, $$\\\\frac{2}{3}$$"],"problemType":"MultipleChoice","stepTitle":"Solve: 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$$\\\\frac{5}{6}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"aa27eearational1a-h1","type":"hint","dependencies":[],"title":"Find the LCD of the Fractions","text":"The LCD of $$\\\\frac{1}{x}$$, $$\\\\frac{1}{3}$$, and $$\\\\frac{5}{6}$$ is $$6x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational1a-h2","type":"hint","dependencies":["aa27eearational1a-h1"],"title":"Clear All Fractions","text":"We must multiply each fraction by the LCD. We get $$6+2x=5x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational1a-h3","type":"hint","dependencies":["aa27eearational1a-h2"],"title":"Solve The Equation","text":"Solving for $$x$$, we get $$3x=6$$. This simplifies to $$x=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa27eearational10","title":"Solving a Rational Equation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Solve Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aa27eearational10a","stepAnswer":["DNE"],"problemType":"MultipleChoice","stepTitle":"Solve: $$\\\\frac{m+11}{m^2-5m+4}=\\\\frac{5}{m-4}-\\\\frac{3}{m-1}$$. If there is no solution, choose DNE.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$-3, 5$$","$$-1, 2$$","7,6","DNE"],"hints":{"DefaultPathway":[{"id":"aa27eearational10a-h1","type":"hint","dependencies":[],"title":"Find the LCD of the Fractions","text":"LCD of $$\\\\frac{m+11}{m^2-5m+4}$$, $$\\\\frac{5}{m-4}$$, and $$\\\\frac{3}{m-1}$$ is $$(m-4)(m-1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational10a-h2","type":"hint","dependencies":["aa27eearational10a-h1"],"title":"Clear All Fractions","text":"We must multiply each fraction by the LCD. We get $$m+11=5(m-1)-3(m-4)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational10a-h3","type":"hint","dependencies":["aa27eearational10a-h2"],"title":"Simplify the Equation","text":"Let\'s simplify this equation: $$m+11=5m-5-3m+12$$. This simplifies to $$m=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational10a-h4","type":"hint","dependencies":["aa27eearational10a-h3"],"title":"Solve the Equation","text":"$$m=4$$ is our equation. $$4$$ is extraneous, however, since it is in the denominator of the original equation. DNE.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa27eearational11","title":"Solving a Rational Equation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Solve Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aa27eearational11a","stepAnswer":["DNE"],"problemType":"MultipleChoice","stepTitle":"Solve: $$\\\\frac{x+13}{x^2-7x+10}=\\\\frac{6}{x-5}+\\\\frac{4}{x-2}$$. If there is no solution, choose DNE.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$-3, 5$$","$$-1, 2$$","7,6","DNE"],"hints":{"DefaultPathway":[{"id":"aa27eearational11a-h1","type":"hint","dependencies":[],"title":"Find the LCD of the Fractions","text":"LCD of $$\\\\frac{x+13}{x^2-7x+10}$$, $$\\\\frac{6}{x-5}$$, and $$\\\\frac{4}{x-2}$$ is $$(x-5)(x-2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational11a-h2","type":"hint","dependencies":["aa27eearational11a-h1"],"title":"Clear All Fractions","text":"We must multiply each fraction by the LCD. We get $$x+13=6(x-2)-4(x-5)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational11a-h3","type":"hint","dependencies":["aa27eearational11a-h2"],"title":"Simplify the Equation","text":"Let\'s simplify this equation: $$x+13=6x-12-4x+20$$. This simplifies to $$x=5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational11a-h4","type":"hint","dependencies":["aa27eearational11a-h3"],"title":"Solve the Equation","text":"$$x=5$$ is our solution, but it is extraneous since it is in the denominator of the original equation. DNE.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa27eearational12","title":"Solving a Rational Equation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Solve Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aa27eearational12a","stepAnswer":["DNE"],"problemType":"MultipleChoice","stepTitle":"Solve: $$\\\\frac{y-6}{y^2+3y-4}=\\\\frac{2}{y+4}+\\\\frac{7}{y-1}$$. If there is no solution, choose DNE.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$-3, 5$$","$$-1, 2$$","7,6","DNE"],"hints":{"DefaultPathway":[{"id":"aa27eearational12a-h1","type":"hint","dependencies":[],"title":"Find the LCD of the Fractions","text":"LCD of $$\\\\frac{y-6}{y^2+3y-4}$$, $$\\\\frac{2}{y+4}$$, and $$\\\\frac{7}{y-1}$$ is $$\\\\left(y+4\\\\right) \\\\left(y-1\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational12a-h2","type":"hint","dependencies":["aa27eearational12a-h1"],"title":"Clear All Fractions","text":"We must multiply each fraction by the LCD. We get $$(y-6)=2\\\\left(y-1\\\\right)+7\\\\left(y+4\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational12a-h3","type":"hint","dependencies":["aa27eearational12a-h2"],"title":"Simplify the Equation","text":"Let\'s simplify this equation: $$y-6=2y-2+7y+28$$. This simplifies to $$8y=-32$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational12a-h4","type":"hint","dependencies":["aa27eearational12a-h3"],"title":"Solve the Equation","text":"$$-8y=32$$ becomes $$y=-4$$, which is our solution. It is extraneous, however, since it is in the denominator of the original equation. DNE.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa27eearational13","title":"Solving a Rational Equation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Solve Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aa27eearational13a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"Find the LCD of the Fractions","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"aa27eearational13a-h1","type":"hint","dependencies":[],"title":"Find the LCD","text":"LCD of $$\\\\frac{y}{y+6}$$, $$\\\\frac{72}{y^2-36}$$, and $$4$$ is $$\\\\left(y+6\\\\right) \\\\left(y-6\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational13a-h2","type":"hint","dependencies":["aa27eearational13a-h1"],"title":"Clear All Fractions","text":"We must multiply each fraction by the LCD. We get $$y(y-6)=72+4\\\\left(y^2-36\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational13a-h3","type":"hint","dependencies":["aa27eearational13a-h2"],"title":"Simplify the Equation","text":"Let\'s simplify this equation: $$y^2-6y=72+4y^2-144$$. This becomes $$y^2+2y-24=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational13a-h4","type":"hint","dependencies":["aa27eearational13a-h3"],"title":"Solve the Equation","text":"Let\'s factor to solve. $$0=\\\\left(y+6\\\\right) \\\\left(y-4\\\\right)$$. $$y=-6$$, $$y=4$$. $$y=-6$$ is extraneous since it is in the original denominator. So, our solution is $$y=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa27eearational14","title":"Solving a Rational Equation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Solve Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aa27eearational14a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"Solve: $$\\\\frac{x}{x+4}=\\\\frac{32}{x^2-6}+5$$. Enter solutions in increasing order, separated by a comma. If there is no solution, enter DNE.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"aa27eearational14a-h1","type":"hint","dependencies":[],"title":"Find the LCD of the Fractions","text":"LCD of $$\\\\frac{x}{x+4}$$, $$\\\\frac{32}{x^2-16}$$, $$5$$ is $$\\\\left(x+4\\\\right) \\\\left(x-4\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational14a-h2","type":"hint","dependencies":["aa27eearational14a-h1"],"title":"Clear All Fractions","text":"We must multiply each fraction by the LCD. We get $$x(x-4)=32+5\\\\left(x^2-16\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational14a-h3","type":"hint","dependencies":["aa27eearational14a-h2"],"title":"Simplify the Equation","text":"Let\'s simplify this equation: $$x^2-4x=32+5x^2-80$$. $$4x^2+4x-48=0$$ becomes $$x^2+x-12=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational14a-h4","type":"hint","dependencies":["aa27eearational14a-h3"],"title":"Solve the Equation","text":"Let\'s factor to solve. $$\\\\left(x+4\\\\right) \\\\left(x-3\\\\right)=0$$. $$x=-4$$ is not a solution since it is in the denominator. $$x=3$$ is a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa27eearational15","title":"Solving a Rational Equation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Solve Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aa27eearational15a","stepAnswer":["$$7$$"],"problemType":"TextBox","stepTitle":"Solve: $$\\\\frac{y}{y+8}=\\\\frac{128}{y^2-64}+9$$. Enter solutions in increasing order, separated by a comma. If there is no solution, enter DNE.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7$$","hints":{"DefaultPathway":[{"id":"aa27eearational15a-h1","type":"hint","dependencies":[],"title":"Find the LCD of the Fractions","text":"LCD of $$\\\\frac{y}{y+8}$$, $$\\\\frac{128}{y^2-64}$$, $$9$$ is $$\\\\left(y+8\\\\right) \\\\left(y-8\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational15a-h2","type":"hint","dependencies":["aa27eearational15a-h1"],"title":"Clear All Fractions","text":"We must multiply each fraction by the LCD. We get $$y(y-8)=128+9\\\\left(y^2-64\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational15a-h3","type":"hint","dependencies":["aa27eearational15a-h2"],"title":"Simplify the Equation","text":"Let\'s simplify this equation: $$y^2-8y=128+9y^2-576$$. This simplifies to $$8y^2+8y-448=0$$. $$y^2+y-56=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational15a-h4","type":"hint","dependencies":["aa27eearational15a-h3"],"title":"Solve the Equation","text":"Let\'s factor to solve. $$\\\\left(y+8\\\\right) \\\\left(y-7\\\\right)=0$$. $$y=-8$$ is not a solution since it is in the denominator. $$y=7$$ is a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa27eearational2","title":"Solving a Rational Equation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Solve Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aa27eearational2a","stepAnswer":["$$-7.5$$"],"problemType":"TextBox","stepTitle":"Solve: $$\\\\frac{1}{y}$$ + $$\\\\frac{2}{3}$$ $$=$$ $$\\\\frac{1}{5}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-7.5$$","hints":{"DefaultPathway":[{"id":"aa27eearational2a-h1","type":"hint","dependencies":[],"title":"Find the LCD of the Fractions","text":"The LCD of $$\\\\frac{1}{y}$$, $$\\\\frac{2}{3}$$, and $$\\\\frac{1}{5}$$ is $$15y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational2a-h2","type":"hint","dependencies":["aa27eearational2a-h1"],"title":"Clear All Fractions","text":"We must multiply each fraction by the LCD. We get $$15+10y=3y$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational2a-h3","type":"hint","dependencies":["aa27eearational2a-h2"],"title":"Solve The Equation","text":"Solving for $$y$$, we get $$7y=-15$$, which simplifies to $$y=-7.5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa27eearational3","title":"Solving a Rational Equation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Solve Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aa27eearational3a","stepAnswer":["$$-7.5$$"],"problemType":"TextBox","stepTitle":"Solve: $$\\\\frac{1}{y}$$ + $$\\\\frac{2}{3}$$ $$=$$ $$\\\\frac{1}{5}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-7.5$$","hints":{"DefaultPathway":[{"id":"aa27eearational3a-h1","type":"hint","dependencies":[],"title":"Find the LCD of the Fractions","text":"The LCD of $$\\\\frac{1}{y}$$, $$\\\\frac{2}{3}$$, and $$\\\\frac{1}{5}$$ is $$15y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational3a-h2","type":"hint","dependencies":["aa27eearational3a-h1"],"title":"Clear All Fractions","text":"We must multiply each fraction by the LCD. We get $$15+10y=3y$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational3a-h3","type":"hint","dependencies":["aa27eearational3a-h2"],"title":"Solve The Equation","text":"Solving for $$y$$, we get $$7y=-15$$, which simplifies to $$y=-7.5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa27eearational4","title":"Solving a Rational Equation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Solve Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aa27eearational4a","stepAnswer":["2,3"],"problemType":"MultipleChoice","stepTitle":"Solve: $$1-\\\\frac{5}{y}=\\\\frac{-6}{y^2}$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["2,3","$$-1, 2$$","7,6","4,5"],"hints":{"DefaultPathway":[{"id":"aa27eearational4a-h1","type":"hint","dependencies":[],"title":"Find the LCD of the Fractions","text":"The LCD of $$1$$, $$\\\\frac{5}{y}$$, and $$\\\\frac{6}{y^2}$$ is $$y^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational4a-h2","type":"hint","dependencies":["aa27eearational4a-h1"],"title":"Clear All Fractions","text":"We must multiply each fraction by the LCD. We get $$y^2-5y=-6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational4a-h3","type":"hint","dependencies":["aa27eearational4a-h2"],"title":"Solve The Equation","text":"$$y^2-5y+6=0$$ becomes $$(y-3)(y-2)$$. Our solutions are thus $$y=3$$ and $$y=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa27eearational5","title":"Solving a Rational Equation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Solve Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aa27eearational5a","stepAnswer":["$$-3, 5$$"],"problemType":"MultipleChoice","stepTitle":"Solve: $$1-\\\\frac{2}{x}=\\\\frac{15}{x^2}$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$-3, 5$$","$$-1, 2$$","7,6","4,5"],"hints":{"DefaultPathway":[{"id":"aa27eearational5a-h1","type":"hint","dependencies":[],"title":"Find the LCD of the Fractions","text":"The LCD of $$1$$, $$\\\\frac{2}{x}$$, and $$\\\\frac{15}{x^2}$$ is $$x^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational5a-h2","type":"hint","dependencies":["aa27eearational5a-h1"],"title":"Clear All Fractions","text":"We must multiply each fraction by the LCD. We get $$x^2-2x=15$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational5a-h3","type":"hint","dependencies":["aa27eearational5a-h2"],"title":"Solve The Equation","text":"$$x^2-2x-15=0$$ becomes $$\\\\left(x-5\\\\right) \\\\left(x+3\\\\right)$$. Our solutions are thus $$x=5$$ and $$x=-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa27eearational6","title":"Solving a Rational Equation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Solve Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aa27eearational6a","stepAnswer":["$$-2, 6$$"],"problemType":"MultipleChoice","stepTitle":"Solve: $$1-\\\\frac{4}{y}=\\\\frac{12}{y^2}$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$-3, 5$$","$$-1, 2$$","7,6","$$-2, 6$$"],"hints":{"DefaultPathway":[{"id":"aa27eearational6a-h1","type":"hint","dependencies":[],"title":"Find the LCD of the Fractions","text":"LCD of $$1$$, $$\\\\frac{4}{y}$$, and $$\\\\frac{12}{y^2}$$ is $$y^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational6a-h2","type":"hint","dependencies":["aa27eearational6a-h1"],"title":"Clear All Fractions","text":"We must multiply each fraction by the LCD. We get $$y^2-4y=12$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational6a-h3","type":"hint","dependencies":["aa27eearational6a-h2"],"title":"Solve The Equation","text":"$$y^2-4y=12$$ becomes $$y^2-4y-12$$ becomes $$\\\\left(y-6\\\\right) \\\\left(y+2\\\\right)$$. Our solutions are thus $$y=6$$ and $$y=-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa27eearational7","title":"Solving a Rational Equation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Solve Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aa27eearational7a","stepAnswer":["$$-1$$"],"problemType":"TextBox","stepTitle":"Solve: $$\\\\frac{2}{x+2}+\\\\frac{4}{x-2}=\\\\frac{x-1}{x^2-4}$$. Enter solutions in increasing order, separated by a comma.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1$$","hints":{"DefaultPathway":[{"id":"aa27eearational7a-h1","type":"hint","dependencies":[],"title":"Find the LCD of the Fractions","text":"The LCD of $$\\\\frac{2}{x+2}$$, $$\\\\frac{4}{x-2}$$, and $$\\\\frac{x-1}{x^2-4}$$ is $$\\\\left(x+2\\\\right) \\\\left(x-2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational7a-h2","type":"hint","dependencies":["aa27eearational7a-h1"],"title":"Clear All Fractions","text":"We must multiply each fraction by the LCD. We get $$2\\\\left(x-2\\\\right)+4\\\\left(x+2\\\\right)=(x-1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational7a-h3","type":"hint","dependencies":["aa27eearational7a-h2"],"title":"Simplify the Equation","text":"Let\'s simplify this equation: $$2x-4+4x+8=x-1$$ becomes $$5x=-5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational7a-h4","type":"hint","dependencies":["aa27eearational7a-h3"],"title":"Solve the Equation","text":"$$5x=-5$$ can be solved as $$x=-1$$. This solution is not extraneous.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa27eearational8","title":"Solving a Rational Equation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Solve Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aa27eearational8a","stepAnswer":["$$\\\\frac{2}{3}$$"],"problemType":"TextBox","stepTitle":"Solve: $$\\\\frac{2}{x+1}+\\\\frac{1}{x-1}=\\\\frac{1}{x^2-1}$$. Enter solutions in increasing order, separated by a comma.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2}{3}$$","hints":{"DefaultPathway":[{"id":"aa27eearational8a-h1","type":"hint","dependencies":[],"title":"Find the LCD of the Fractions","text":"LCD of $$\\\\frac{2}{x+1}$$, $$\\\\frac{1}{x-1}$$, and $$\\\\frac{1}{x^2-1}$$ is $$\\\\left(x+1\\\\right) \\\\left(x-1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational8a-h2","type":"hint","dependencies":["aa27eearational8a-h1"],"title":"Clear All Fractions","text":"We must multiply each fraction by the LCD. We get $$2\\\\left(x-1\\\\right)+x+1=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational8a-h3","type":"hint","dependencies":["aa27eearational8a-h2"],"title":"Simplify the Equation","text":"Let\'s simplify this equation: $$2x-2+x+1=1$$ simplifies to $$3x=2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational8a-h4","type":"hint","dependencies":["aa27eearational8a-h3"],"title":"Solve the Equation","text":"$$3x=2$$ can be solved as $$x=\\\\frac{2}{3}$$. This solution is not extraneous.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa27eearational9","title":"Solving a Rational Equation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.4 Solve Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aa27eearational9a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"Solve: $$\\\\frac{5}{y+3}+\\\\frac{2}{y-3}=\\\\frac{5}{y^2-9}$$. Enter solutions in increasing order, separated by a comma.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"aa27eearational9a-h1","type":"hint","dependencies":[],"title":"Find the LCD of the Fractions","text":"LCD of $$\\\\frac{5}{y+3}$$, $$\\\\frac{2}{y-3}$$, and $$\\\\frac{5}{y^2-9}$$ is $$\\\\left(y+3\\\\right) \\\\left(y-3\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational9a-h2","type":"hint","dependencies":["aa27eearational9a-h1"],"title":"Clear All Fractions","text":"We must multiply each fraction by the LCD. We get $$5\\\\left(y-3\\\\right)+2\\\\left(y+3\\\\right)=5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational9a-h3","type":"hint","dependencies":["aa27eearational9a-h2"],"title":"Simplify the Equation","text":"Let\'s simplify this equation: $$5y-15+2y+6=5$$. This simplifies to $$7y=14$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa27eearational9a-h4","type":"hint","dependencies":["aa27eearational9a-h3"],"title":"Solve the Equation","text":"$$7y=14$$ can be solved as $$y=2$$. This solution is not extraneous.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa30a09exponents1","title":"Simplifying Expressions With Integer Exponents","body":"Simplify.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Integer Exponents and Scientific Notation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa30a09exponents1a","stepAnswer":["$$\\\\frac{1}{16}$$"],"problemType":"TextBox","stepTitle":"$$4^{\\\\left(-2\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{16}$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents1a-h1","type":"hint","dependencies":[],"title":"Definition of Negative Exponent","text":"A negative exponent can be simplified using the following definition:a**(-n)=1/a**(n)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents1a-h2","type":"hint","dependencies":["aa30a09exponents1a-h1"],"title":"Answer","text":"The answer is $$\\\\frac{1}{16}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa30a09exponents10","title":"Simplifying Expressions With Integer Exponents","body":"Simplify.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Integer Exponents and Scientific Notation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa30a09exponents10a","stepAnswer":["$$\\\\frac{1}{16x^8}$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(-4x^4\\\\right)}^{\\\\left(-2\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{16x^8}$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents10a-h1","type":"hint","dependencies":[],"title":"Use the Product to a Power Property","text":"$${ab}^m=a^m b^m$$. Split the parts of the expression apart.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents10a-h2","type":"hint","dependencies":["aa30a09exponents10a-h1"],"title":"Use the Power Property","text":"$${\\\\left(a^m\\\\right)}^n=a^m n$$. Simplify the exponents under the $$x$$ variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents10a-h3","type":"hint","dependencies":["aa30a09exponents10a-h2"],"title":"Definition of Negative Exponent","text":"Use the definition of a negative exponent to take the reciprocals of the expressions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents10a-h4","type":"hint","dependencies":["aa30a09exponents10a-h3"],"title":"Simplify","text":"Simplify the expression by combining the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents10a-h5","type":"hint","dependencies":["aa30a09exponents10a-h4"],"title":"Answer","text":"The answer is $$\\\\frac{1}{16x^8}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa30a09exponents11","title":"Convert to Scientific Notation","body":"Write the integer in scientific notation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Integer Exponents and Scientific Notation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa30a09exponents11a","stepAnswer":["$$3.7{10}^4$$"],"problemType":"TextBox","stepTitle":"37,000","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3.7{10}^4$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents11a-h1","type":"hint","dependencies":[],"title":"Move","text":"Move the decimal point after the $$3$$, as $$3.7$$ is between $$1$$ and $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["aa30a09exponents11a-h1"],"title":"Count","text":"Count the number of times the decimal point was moved to the left. How many times was it moved?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aa30a09exponents11a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aa30a09exponents11a-h3","type":"hint","dependencies":["aa30a09exponents11a-h2"],"title":"Write Exponent","text":"The decimal point was moved $$4$$ times, so write the number as a product with the power $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents11a-h4","type":"hint","dependencies":["aa30a09exponents11a-h3"],"title":"Answer","text":"The answer is $$3.7{10}^4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa30a09exponents12","title":"Convert to Scientific Notation","body":"Write the integer in scientific notation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Integer Exponents and Scientific Notation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa30a09exponents12a","stepAnswer":["$$9.6{10}^4$$"],"problemType":"TextBox","stepTitle":"96,000","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9.6{10}^4$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents12a-h1","type":"hint","dependencies":[],"title":"Move","text":"Move the decimal point after the $$9$$, as $$9.6$$ is between $$1$$ and $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["aa30a09exponents12a-h1"],"title":"Count","text":"Count the number of times the decimal point was moved to the left. How many times was it moved?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aa30a09exponents12a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aa30a09exponents12a-h3","type":"hint","dependencies":["aa30a09exponents12a-h2"],"title":"Write Exponent","text":"The decimal point was moved $$4$$ times, so write the number as a product with the power $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents12a-h4","type":"hint","dependencies":["aa30a09exponents12a-h3"],"title":"Answer","text":"The answer is $$9.6{10}^4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa30a09exponents13","title":"Convert to Scientific Notation","body":"Write the integer in scientific notation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Integer Exponents and Scientific Notation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa30a09exponents13a","stepAnswer":["$$4.83\\\\times {10}^4$$"],"problemType":"TextBox","stepTitle":"48,300","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4.83\\\\times {10}^4$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents13a-h1","type":"hint","dependencies":[],"title":"Move","text":"Move the decimal point after the $$4$$, as $$4.8$$ is between $$1$$ and $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["aa30a09exponents13a-h1"],"title":"Count","text":"Count the number of times the decimal point was moved to the left. How many times was it moved?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aa30a09exponents13a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aa30a09exponents13a-h3","type":"hint","dependencies":["aa30a09exponents13a-h2"],"title":"Write Exponent","text":"The decimal point was moved $$4$$ times, so write the number as a product with the power $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents13a-h4","type":"hint","dependencies":["aa30a09exponents13a-h3"],"title":"Answer","text":"The answer is $$4.83\\\\times {10}^4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa30a09exponents14","title":"Convert to Scientific Notation","body":"Write the integer in scientific notation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Integer Exponents and Scientific Notation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa30a09exponents14a","stepAnswer":["$$5.2{10}^{\\\\left(-3\\\\right)}$$"],"problemType":"TextBox","stepTitle":"$$0.0052$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5.2{10}^{\\\\left(-3\\\\right)}$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents14a-h1","type":"hint","dependencies":[],"title":"Move","text":"Move the decimal point after the $$5$$, as $$5.2$$ is between $$1$$ and $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["aa30a09exponents14a-h1"],"title":"Count","text":"Count the number of times the decimal point was moved to the right. How many times was it moved?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aa30a09exponents14a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aa30a09exponents14a-h3","type":"hint","dependencies":["aa30a09exponents14a-h2"],"title":"Write Exponent","text":"The decimal point was moved $$3$$ times, so write the number as a product with the power $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents14a-h4","type":"hint","dependencies":["aa30a09exponents14a-h3"],"title":"Answer","text":"The answer is $$5.2{10}^{\\\\left(-3\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa30a09exponents15","title":"Convert to Scientific Notation","body":"Write the integer in scientific notation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Integer Exponents and Scientific Notation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa30a09exponents15a","stepAnswer":["$$7.8{10}^{\\\\left(-3\\\\right)}$$"],"problemType":"TextBox","stepTitle":"$$0.0078$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7.8{10}^{\\\\left(-3\\\\right)}$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents15a-h1","type":"hint","dependencies":[],"title":"Move","text":"Move the decimal point after the $$7$$, as $$7.8$$ is between $$1$$ and $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["aa30a09exponents15a-h1"],"title":"Count","text":"Count the number of times the decimal point was moved to the right. How many times was it moved?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aa30a09exponents15a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aa30a09exponents15a-h3","type":"hint","dependencies":["aa30a09exponents15a-h2"],"title":"Write Exponent","text":"The decimal point was moved $$3$$ times, so write the number as a product with the power $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents15a-h4","type":"hint","dependencies":["aa30a09exponents15a-h3"],"title":"Answer","text":"The answer is $$7.8{10}^{\\\\left(-3\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa30a09exponents16","title":"Use the Definition of a Negative Exponent","body":"Simplify the following","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Integer Exponents and Scientific Notation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa30a09exponents16a","stepAnswer":["$$\\\\frac{8}{343}$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(\\\\frac{7}{2}\\\\right)}^{-3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{8}{343}$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents16a-h1","type":"hint","dependencies":[],"title":"Quotient to a Negative Exponent Property","text":"Use the Quotient to a Negative Exponent Property: $${\\\\left(\\\\frac{a}{b}\\\\right)}^{-n}$$ $$=$$ $${\\\\left(\\\\frac{b}{a}\\\\right)}^n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents16a-h2","type":"hint","dependencies":["aa30a09exponents16a-h1"],"title":"Application of Quotient Property","text":"You should be left with the expreession $${\\\\left(\\\\frac{2}{7}\\\\right)}^3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents16a-h3","type":"hint","dependencies":["aa30a09exponents16a-h2"],"title":"Simplify","text":"Expand the exponent and simplify","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa30a09exponents16b","stepAnswer":["$$\\\\frac{-\\\\left(x^3 y^6\\\\right)}{27}$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(-\\\\frac{3}{x\\\\left(y^2\\\\right)}\\\\right)}^{\\\\left(-3\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-\\\\left(x^3 y^6\\\\right)}{27}$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents16b-h1","type":"hint","dependencies":[],"title":"Quotient to a Negative Exponent Property","text":"Use the Quotient to a Negative Exponent Property: $${\\\\left(\\\\frac{a}{b}\\\\right)}^{-n}$$ $$=$$ $${\\\\left(\\\\frac{b}{a}\\\\right)}^n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents16b-h2","type":"hint","dependencies":["aa30a09exponents16b-h1"],"title":"Application of Quotient Property","text":"You should be left with the expreession $${\\\\left(-\\\\frac{x\\\\left(y^2\\\\right)}{3}\\\\right)}^3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents16b-h3","type":"hint","dependencies":["aa30a09exponents16b-h2"],"title":"Simplify","text":"Expand the exponent and simplify","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa30a09exponents17","title":"Use the Definition of a Negative Exponent","body":"Simplify the following","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Integer Exponents and Scientific Notation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa30a09exponents17a","stepAnswer":["$$\\\\frac{1}{49}$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(-7\\\\right)}^{\\\\left(-2\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{49}$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents17a-h1","type":"hint","dependencies":[],"title":"Quotient to a Negative Exponent Property","text":"Use the Quotient to a Negative Exponent Property: $${\\\\left(\\\\frac{a}{b}\\\\right)}^{-n}$$ $$=$$ $${\\\\left(\\\\frac{b}{a}\\\\right)}^n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents17a-h2","type":"hint","dependencies":["aa30a09exponents17a-h1"],"title":"Application of Quotient Property","text":"You should be left with the expreession $${\\\\left(-\\\\frac{1}{7}\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents17a-h3","type":"hint","dependencies":["aa30a09exponents17a-h2"],"title":"Simplify","text":"Expand the exponent and simplify","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa30a09exponents17b","stepAnswer":["$$49$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(-\\\\frac{1}{7}\\\\right)}^{\\\\left(-2\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$49$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents17b-h1","type":"hint","dependencies":[],"title":"Quotient to a Negative Exponent Property","text":"Use the Quotient to a Negative Exponent Property: $${\\\\left(\\\\frac{a}{b}\\\\right)}^{-n}$$ $$=$$ $${\\\\left(\\\\frac{b}{a}\\\\right)}^n$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents17b-h2","type":"hint","dependencies":["aa30a09exponents17b-h1"],"title":"Application of Quotient Property","text":"You should be left with the expreession $${\\\\left(-7\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents17b-h3","type":"hint","dependencies":["aa30a09exponents17b-h2"],"title":"Simplify","text":"Expand the exponent and simplify","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa30a09exponents18","title":"Simplify Expressions with Integer Exponents","body":"Simplify the following","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Integer Exponents and Scientific Notation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa30a09exponents18a","stepAnswer":["$$\\\\frac{1}{s^4}$$"],"problemType":"TextBox","stepTitle":"$$s^3 s^{-7}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{s^4}$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents18a-h1","type":"hint","dependencies":[],"title":"Product Property","text":"Use the Product Property: $$a^m a^n$$ $$=$$ $$a^{m+n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["aa30a09exponents18a-h1"],"title":"Using the product Property","text":"What is $$3+\\\\left(-7\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents18a-h3","type":"hint","dependencies":["aa30a09exponents18a-h2"],"title":"Properties of Negative Exponents","text":"Use the definition of negative exponent, $$a^{\\\\left(-n\\\\right)}=\\\\frac{1}{a^n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa30a09exponents18b","stepAnswer":["$$\\\\frac{1}{q^5}$$"],"problemType":"TextBox","stepTitle":"$$q^{-8} q^3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{q^5}$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents18b-h1","type":"hint","dependencies":[],"title":"Product Property","text":"Use the Product Property: $$a^m a^n$$ $$=$$ $$a^{m+n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents18b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["aa30a09exponents18b-h1"],"title":"Using the product Property","text":"What is $$\\\\left(-8\\\\right)+3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents18b-h3","type":"hint","dependencies":["aa30a09exponents18b-h2"],"title":"Properties of Negative Exponents","text":"Use the definition of negative exponent, $$a^{\\\\left(-n\\\\right)}=\\\\frac{1}{a^n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa30a09exponents18c","stepAnswer":["$$\\\\frac{1}{y^7}$$"],"problemType":"TextBox","stepTitle":"$$y^{-2} y^{-5}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{y^7}$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents18c-h1","type":"hint","dependencies":[],"title":"Product Property","text":"Use the Product Property: $$a^m a^n$$ $$=$$ $$a^{m+n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents18c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-7$$"],"dependencies":["aa30a09exponents18c-h1"],"title":"Using the product Property","text":"What is $$\\\\left(-2\\\\right)+\\\\left(-5\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents18c-h3","type":"hint","dependencies":["aa30a09exponents18c-h2"],"title":"Properties of Negative Exponents","text":"Use the definition of negative exponent, $$a^{\\\\left(-n\\\\right)}=\\\\frac{1}{a^n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa30a09exponents19","title":"Simplify Expressions with Integer Exponents","body":"Simplify the following","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Integer Exponents and Scientific Notation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa30a09exponents19a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"$$y^5 y^{-5}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents19a-h1","type":"hint","dependencies":[],"title":"Product Property","text":"Use the Product Property: $$a^m a^n$$ $$=$$ $$a^{m+n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["aa30a09exponents19a-h1"],"title":"Using the product Property","text":"What is $$5+\\\\left(-5\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents19a-h3","type":"hint","dependencies":["aa30a09exponents19a-h2"],"title":"Zero Exponent Property","text":"Use the definition of zero exponent property, $$a^0=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa30a09exponents19b","stepAnswer":["$$y^6$$"],"problemType":"TextBox","stepTitle":"$$y y^5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y^6$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents19b-h1","type":"hint","dependencies":[],"title":"Product Property","text":"Use the Product Property: $$a^m a^n$$ $$=$$ $$a^{m+n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents19b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["aa30a09exponents19b-h1"],"title":"Using the product Property","text":"What is $$1+5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa30a09exponents19c","stepAnswer":["$$\\\\frac{1}{y^4}$$"],"problemType":"TextBox","stepTitle":"$$y y^{-5}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{y^4}$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents19c-h1","type":"hint","dependencies":[],"title":"Product Property","text":"Use the Product Property: $$a^m a^n$$ $$=$$ $$a^{m+n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents19c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["aa30a09exponents19c-h1"],"title":"Using the product Property","text":"What is $$1+\\\\left(-5\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents19c-h3","type":"hint","dependencies":["aa30a09exponents19c-h2"],"title":"Properties of Negative Exponents","text":"Use the definition of negative exponent, $$a^{\\\\left(-n\\\\right)}=\\\\frac{1}{a^n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa30a09exponents2","title":"Simplifying Expressions With Integer Exponents","body":"Simplify.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Integer Exponents and Scientific Notation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa30a09exponents2a","stepAnswer":["$$\\\\frac{1}{1000}$$"],"problemType":"TextBox","stepTitle":"$${10}^{\\\\left(-3\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{1000}$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents2a-h1","type":"hint","dependencies":[],"title":"Definition of Negative Exponent","text":"A negative exponent can be simplified using the following definition:a**(-n)=1/a**(n)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents2a-h2","type":"hint","dependencies":["aa30a09exponents2a-h1"],"title":"Answer","text":"The answer is $$\\\\frac{1}{1000}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa30a09exponents20","title":"Simplify Expressions with Integer Exponents","body":"Simplify the following","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Integer Exponents and Scientific Notation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa30a09exponents20a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"$$x^4 x^{-2} x^{-3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents20a-h1","type":"hint","dependencies":[],"title":"Product Property","text":"Use the Product Property: $$a^m a^n$$ $$=$$ $$a^{m+n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["aa30a09exponents20a-h1"],"title":"Using the product Property","text":"What is $$4+\\\\left(-2\\\\right)+\\\\left(-3\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents20a-h3","type":"hint","dependencies":["aa30a09exponents20a-h2"],"title":"Properties of Negative Exponents","text":"Use the definition of negative exponent, $$a^{\\\\left(-n\\\\right)}=\\\\frac{1}{a^n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa30a09exponents21","title":"Simplify Expressions with Integer Exponents","body":"Simplify the following","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Integer Exponents and Scientific Notation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa30a09exponents21a","stepAnswer":["$$\\\\frac{1}{m^2 n^4}$$"],"problemType":"TextBox","stepTitle":"$$m^3 n^{-3} m^{-5} n^{-1}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{m^2 n^4}$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents21a-h1","type":"hint","dependencies":[],"title":"Commutative Property","text":"Use the Commutative Property to get like bases together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents21a-h2","type":"hint","dependencies":["aa30a09exponents21a-h1"],"title":"Commutative Property Result","text":"You should be left with $$m^3 m^{-5} n^{-3} n^{-1}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents21a-h3","type":"hint","dependencies":["aa30a09exponents21a-h2"],"title":"Product Property","text":"Use the Product Property for each base: $$a^m a^n$$ $$=$$ $$a^{m+n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents21a-h4","type":"hint","dependencies":["aa30a09exponents21a-h3"],"title":"Properties of Negative Exponents","text":"Use the definition of negative exponent, $$a^{\\\\left(-n\\\\right)}=\\\\frac{1}{a^n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents21a-h5","type":"hint","dependencies":["aa30a09exponents21a-h4"],"title":"Simplify","text":"Simplify your ending expression $$m^{-2} n^{-4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa30a09exponents22","title":"Simplify Expressions with Integer Exponents","body":"Simplify the following","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Integer Exponents and Scientific Notation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa30a09exponents22a","stepAnswer":["$$\\\\frac{1}{p^5} q^7$$"],"problemType":"TextBox","stepTitle":"$$p q^{-4} p^{-6} q^{-3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{p^5} q^7$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents22a-h1","type":"hint","dependencies":[],"title":"Commutative Property","text":"Use the Commutative Property to get like bases together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents22a-h2","type":"hint","dependencies":["aa30a09exponents22a-h1"],"title":"Commutative Property Result","text":"You should be left with $$p p^{-6} q^{-4} q^{-3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents22a-h3","type":"hint","dependencies":["aa30a09exponents22a-h2"],"title":"Product Property","text":"Use the Product Property for each base: $$a^m a^n$$ $$=$$ $$a^{m+n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents22a-h4","type":"hint","dependencies":["aa30a09exponents22a-h3"],"title":"Properties of Negative Exponents","text":"Use the definition of negative exponent, $$a^{\\\\left(-n\\\\right)}=\\\\frac{1}{a^n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents22a-h5","type":"hint","dependencies":["aa30a09exponents22a-h4"],"title":"Simplify","text":"Simplify your ending expression $$p^{-5} q^{-7}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa30a09exponents23","title":"Simplify Expressions with Integer Exponents","body":"Simplify the following","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Integer Exponents and Scientific Notation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa30a09exponents23a","stepAnswer":["$$\\\\frac{-\\\\left(14k^5\\\\right)}{j^3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(-2j^{-5} k^8\\\\right) 7j^2 k^{-3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-\\\\left(14k^5\\\\right)}{j^3}$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents23a-h1","type":"hint","dependencies":[],"title":"Commutative Property","text":"Use the Commutative Property to get like bases together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents23a-h2","type":"hint","dependencies":["aa30a09exponents23a-h1"],"title":"Commutative Property Result","text":"You should be left with $$\\\\left(-2j^{-5} j^2\\\\right) 7k^8 k^{-3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents23a-h3","type":"hint","dependencies":["aa30a09exponents23a-h2"],"title":"Product Property","text":"Use the Product Property for each base: $$a^m a^n$$ $$=$$ $$a^{m+n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents23a-h4","type":"hint","dependencies":["aa30a09exponents23a-h3"],"title":"Properties of Negative Exponents","text":"Use the definition of negative exponent, $$a^{\\\\left(-n\\\\right)}=\\\\frac{1}{a^n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents23a-h5","type":"hint","dependencies":["aa30a09exponents23a-h4"],"title":"Simplify","text":"Simplify your ending expression $$-\\\\left(2j^{-3}\\\\right) 7k^5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa30a09exponents24","title":"Simplify Expressions with Integer Exponents","body":"Simplify the following","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Integer Exponents and Scientific Notation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa30a09exponents24a","stepAnswer":["$$\\\\frac{-\\\\left(40n^3\\\\right)}{m}$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(-5\\\\right) m^4 n^6 8m^{-5} n^{-3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-\\\\left(40n^3\\\\right)}{m}$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents24a-h1","type":"hint","dependencies":[],"title":"Commutative Property","text":"Use the Commutative Property to get like bases together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents24a-h2","type":"hint","dependencies":["aa30a09exponents24a-h1"],"title":"Commutative Property Result","text":"You should be left with $$\\\\left(-5\\\\right) m^4 m^{-5} 8n^6 n^{-3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents24a-h3","type":"hint","dependencies":["aa30a09exponents24a-h2"],"title":"Product Property","text":"Use the Product Property for each base: $$a^m a^n$$ $$=$$ $$a^{m+n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents24a-h4","type":"hint","dependencies":["aa30a09exponents24a-h3"],"title":"Properties of Negative Exponents","text":"Use the definition of negative exponent, $$a^{\\\\left(-n\\\\right)}=\\\\frac{1}{a^n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents24a-h5","type":"hint","dependencies":["aa30a09exponents24a-h4"],"title":"Simplify","text":"Simplify your ending expression $$\\\\left(-5\\\\right) m^{-1} 8n^3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa30a09exponents25","title":"Simplify Expressions with Integer Exponents","body":"Simplify the following","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Integer Exponents and Scientific Notation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa30a09exponents25a","stepAnswer":["$$\\\\frac{1}{64y^9}$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(4y^3\\\\right)}^{-3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{64y^9}$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents25a-h1","type":"hint","dependencies":[],"title":"Product to a Power","text":"Use the definition of product to power, $${ab}^m=a^m b^m$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents25a-h2","type":"hint","dependencies":["aa30a09exponents25a-h1"],"title":"Simplify","text":"Simplify $$4^{-3} y^{-9}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents25a-h3","type":"hint","dependencies":["aa30a09exponents25a-h2"],"title":"Properties of Negative Exponents","text":"Use the definition of negative exponent, $$a^{\\\\left(-n\\\\right)}=\\\\frac{1}{a^n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa30a09exponents26","title":"Simplify Expressions with Integer Exponents","body":"Simplify the following","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Integer Exponents and Scientific Notation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa30a09exponents26a","stepAnswer":["$$\\\\frac{4}{p^{10}}$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(2p^{-5}\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{4}{p^{10}}$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents26a-h1","type":"hint","dependencies":[],"title":"Product to a Power","text":"Use the definition of product to power, $${ab}^m=a^m b^m$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents26a-h2","type":"hint","dependencies":["aa30a09exponents26a-h1"],"title":"Simplify","text":"Simplify $$2^2 p^{-10}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents26a-h3","type":"hint","dependencies":["aa30a09exponents26a-h2"],"title":"Properties of Negative Exponents","text":"Use the definition of negative exponent, $$a^{\\\\left(-n\\\\right)}=\\\\frac{1}{a^n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa30a09exponents27","title":"Simplify Expressions with Integer Exponents","body":"Simplify the following","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Integer Exponents and Scientific Notation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa30a09exponents27a","stepAnswer":["$$n^7$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{n^5}{n^{-2}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$n^7$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents27a-h1","type":"hint","dependencies":[],"title":"Quotient Property","text":"Use the definition of the quotient property, $$\\\\frac{a^m}{a^n}=a^{m-n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents27a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["aa30a09exponents27a-h1"],"title":"Using the Quotient Property","text":"What is $$(5)-(-2)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa30a09exponents28","title":"Simplify Expressions with Integer Exponents","body":"Simplify the following","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Integer Exponents and Scientific Notation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa30a09exponents28a","stepAnswer":["$$y^5$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{y^{-5}}{y^{-10}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y^5$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents28a-h1","type":"hint","dependencies":[],"title":"Quotient Property","text":"Use the definition of the quotient property, $$\\\\frac{a^m}{a^n}=a^{m-n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents28a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["aa30a09exponents28a-h1"],"title":"Using the Quotient Property","text":"What is $$(-5)-(-10)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa30a09exponents29","title":"Convert from Decimal Notation to Scientific Notation","body":"Write the number in scientific notation","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Integer Exponents and Scientific Notation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa30a09exponents29a","stepAnswer":["$$3.4{10}^5$$"],"problemType":"TextBox","stepTitle":"$$340000$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3.4{10}^5$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents29a-h1","type":"hint","dependencies":[],"title":"Move decimal point","text":"Move decimal point so that the first factor is greater or equal to $$1$$ but less than $$10$$. You should have $$3.40000$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents29a-h2","type":"hint","dependencies":["aa30a09exponents29a-h1"],"title":"Count the number of decimal places","text":"Count the number of decimal places, $$n$$, that the decimal point was moved.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents29a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["aa30a09exponents29a-h2"],"title":"How many decimal places was the decimal point moved?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents29a-h4","type":"hint","dependencies":["aa30a09exponents29a-h3"],"title":"Write the number as a product with a power of $$10$$.","text":"The power of $$10$$ will have exponent $$5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents29a-h5","type":"hint","dependencies":["aa30a09exponents29a-h4"],"title":"Combine","text":"Combine both $$3.4$$ and $${10}^5$$ together","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa30a09exponents3","title":"Simplifying Expressions With Integer Exponents","body":"Simplify.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Integer Exponents and Scientific Notation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa30a09exponents3a","stepAnswer":["$$\\\\frac{1}{8}$$"],"problemType":"TextBox","stepTitle":"$$2^{\\\\left(-3\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{8}$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents3a-h1","type":"hint","dependencies":[],"title":"Definition of Negative Exponent","text":"A negative exponent can be simplified using the following definition:a**(-n)=1/a**(n)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents3a-h2","type":"hint","dependencies":["aa30a09exponents3a-h1"],"title":"Answer","text":"The answer is $$\\\\frac{1}{8}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa30a09exponents30","title":"Convert from Decimal Notation to Scientific Notation","body":"Write the number in scientific notation","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Integer Exponents and Scientific Notation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa30a09exponents30a","stepAnswer":["$$1.29\\\\times {10}^6$$"],"problemType":"TextBox","stepTitle":"$$1290000$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1.29\\\\times {10}^6$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents30a-h1","type":"hint","dependencies":[],"title":"Move decimal point","text":"Move decimal point so that the first factor is greater or equal to $$1$$ but less than $$10$$. You should have $$1.290000$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents30a-h2","type":"hint","dependencies":["aa30a09exponents30a-h1"],"title":"Count the number of decimal places","text":"Count the number of decimal places, $$n$$, that the decimal point was moved.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents30a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["aa30a09exponents30a-h2"],"title":"How many decimal places was the decimal point moved?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents30a-h4","type":"hint","dependencies":["aa30a09exponents30a-h3"],"title":"Write the number as a product with a power of $$10$$.","text":"The power of $$10$$ will have exponent $$6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents30a-h5","type":"hint","dependencies":["aa30a09exponents30a-h4"],"title":"Combine","text":"Combine both $$1.29$$ and $${10}^6$$ together","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa30a09exponents4","title":"Simplifying Expressions With Integer Exponents","body":"Simplify.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Integer Exponents and Scientific Notation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa30a09exponents4a","stepAnswer":["$$\\\\frac{1}{9}$$"],"problemType":"TextBox","stepTitle":"$$3^{\\\\left(-2\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{9}$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents4a-h1","type":"hint","dependencies":[],"title":"Definition of Negative Exponent","text":"A negative exponent can be simplified using the following definition:a**(-n)=1/a**(n)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents4a-h2","type":"hint","dependencies":["aa30a09exponents4a-h1"],"title":"Answer","text":"The answer is $$\\\\frac{1}{9}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa30a09exponents5","title":"Simplifying Expressions With Integer Exponents","body":"Simplify.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Integer Exponents and Scientific Notation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa30a09exponents5a","stepAnswer":["$$64$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{1}{4^{\\\\left(-3\\\\right)}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$64$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents5a-h1","type":"hint","dependencies":[],"title":"Definition of Negative Exponent","text":"A negative exponent can be simplified using the following definition:a**(-n)=1/a**(n)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents5a-h2","type":"hint","dependencies":["aa30a09exponents5a-h1"],"title":"Answer","text":"The answer is $$64$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa30a09exponents6","title":"Simplifying Expressions With Integer Exponents","body":"Simplify.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Integer Exponents and Scientific Notation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa30a09exponents6a","stepAnswer":["$$\\\\frac{1}{{mn}^5}$$"],"problemType":"TextBox","stepTitle":"$$m^4 n^{\\\\left(-3\\\\right)} m^{\\\\left(-5\\\\right)} n \\\\left(-2\\\\right)$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{{mn}^5}$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents6a-h1","type":"hint","dependencies":[],"title":"Use the Commutative Property","text":"Group like variables together. The commutative property states that multiplication is reversible, so rearranging the variables has no effect on the answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents6a-h2","type":"hint","dependencies":["aa30a09exponents6a-h1"],"title":"Add","text":"Add the exponents for each variable together using the Product Property.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents6a-h3","type":"hint","dependencies":["aa30a09exponents6a-h2"],"title":"Take Reciprocals","text":"Since the exponents are negative, simplify by taking the reciprocal of the variables and making the exponent positive. This will be $$\\\\frac{1}{m} \\\\frac{1}{n^5}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents6a-h4","type":"hint","dependencies":["aa30a09exponents6a-h3"],"title":"Simplify","text":"Combine the fractions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents6a-h5","type":"hint","dependencies":["aa30a09exponents6a-h4"],"title":"Answer","text":"The answer is $$\\\\frac{1}{{mn}^5}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa30a09exponents7","title":"Simplifying Expressions With Integer Exponents","body":"Simplify.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Integer Exponents and Scientific Notation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa30a09exponents7a","stepAnswer":["$$\\\\frac{1}{p^3 q^3}$$"],"problemType":"TextBox","stepTitle":"$$p^6 q^{\\\\left(-2\\\\right)} p^{\\\\left(-9\\\\right)} q^{\\\\left(-1\\\\right)}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{p^3 q^3}$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents7a-h1","type":"hint","dependencies":[],"title":"Use the Commutative Property","text":"Group like variables together. The commutative property states that multiplication is reversible, so rearranging the variables has no effect on the answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents7a-h2","type":"hint","dependencies":["aa30a09exponents7a-h1"],"title":"Add","text":"Add the exponents for each variable together using the Product Property.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents7a-h3","type":"hint","dependencies":["aa30a09exponents7a-h2"],"title":"Take Reciprocals","text":"Since the exponents are negative, simplify by taking the reciprocal of the variables and making the exponent positive. This will be $$\\\\frac{1}{p^3} \\\\frac{1}{q^3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents7a-h4","type":"hint","dependencies":["aa30a09exponents7a-h3"],"title":"Simplify","text":"Combine the fractions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents7a-h5","type":"hint","dependencies":["aa30a09exponents7a-h4"],"title":"Answer","text":"The answer is $$\\\\frac{1}{p^3 q^3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa30a09exponents8","title":"Simplifying Expressions With Integer Exponents","body":"Simplify.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Integer Exponents and Scientific Notation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa30a09exponents8a","stepAnswer":["$$\\\\frac{1}{r^2 s^8}$$"],"problemType":"TextBox","stepTitle":"$$r^5 s^{\\\\left(-3\\\\right)} r^{\\\\left(-7\\\\right)} s^{\\\\left(-5\\\\right)}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{r^2 s^8}$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents8a-h1","type":"hint","dependencies":[],"title":"Use the Commutative Property","text":"Group like variables together. The commutative property states that multiplication is reversible, so rearranging the variables has no effect on the answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents8a-h2","type":"hint","dependencies":["aa30a09exponents8a-h1"],"title":"Add","text":"Add the exponents for each variable together using the Product Property.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents8a-h3","type":"hint","dependencies":["aa30a09exponents8a-h2"],"title":"Take Reciprocals","text":"Since the exponents are negative, simplify by taking the reciprocal of the variables and making the exponent positive. This will be $$\\\\frac{1}{r^2} \\\\frac{1}{s^8}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents8a-h4","type":"hint","dependencies":["aa30a09exponents8a-h3"],"title":"Simplify","text":"Combine the fractions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents8a-h5","type":"hint","dependencies":["aa30a09exponents8a-h4"],"title":"Answer","text":"The answer is $$\\\\frac{1}{r^2 s^8}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa30a09exponents9","title":"Simplifying Expressions With Integer Exponents","body":"Simplify.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Integer Exponents and Scientific Notation","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa30a09exponents9a","stepAnswer":["$$\\\\frac{1}{36k^6}$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(6k^3\\\\right)}^{\\\\left(-2\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{36k^6}$$","hints":{"DefaultPathway":[{"id":"aa30a09exponents9a-h1","type":"hint","dependencies":[],"title":"Use the Product to a Power Property","text":"$${ab}^m=a^m b^m$$. Split the parts of the expression apart.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents9a-h2","type":"hint","dependencies":["aa30a09exponents9a-h1"],"title":"Use the Power Property","text":"$${\\\\left(a^m\\\\right)}^n=a^m n$$. Simplify the exponents under the k variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents9a-h3","type":"hint","dependencies":["aa30a09exponents9a-h2"],"title":"Definition of Negative Exponent","text":"Use the definition of a negative exponent to take the reciprocals of the expressions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents9a-h4","type":"hint","dependencies":["aa30a09exponents9a-h3"],"title":"Simplify","text":"Simplify the expression by combining the terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa30a09exponents9a-h5","type":"hint","dependencies":["aa30a09exponents9a-h4"],"title":"Answer","text":"The answer is $$\\\\frac{1}{36k^6}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa47f81type1","title":"Average Number of Sleep Time","body":"It is believed that Lake Tahoe Community College (LTCC) Intermediate Algebra students get less than seven hours of sleep per night, on average. A survey of $$22$$ LTCC Intermediate Algebra students generated a mean of $$7.24$$ hours with a standard deviation of $$1.93$$ hours. At a level of significance of 5%, do LTCC Intermediate Algebra students get less than seven hours of sleep per night, on average?","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Outcomes and the Type I and Type II Errors","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa47f81type1a","stepAnswer":["is less than seven hours."],"problemType":"MultipleChoice","stepTitle":"The Type II error is not to reject that the mean number of hours of sleep LTCC students get per night is at least seven when, in fact, the mean number of hours?","stepBody":"Choose the best answer from the following.","answerType":"string","variabilization":{},"choices":["is less than seven hours.","is less than seven hours.","is less than seven hours.","is less than seven hours."],"hints":{"DefaultPathway":[{"id":"aa47f81type1a-h1","type":"hint","dependencies":[],"title":"The Type Error","text":"What is the definition of a Type II Error?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa47f81type1a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["The decision is not to reject $$H_0$$ when, in fact, $$H_0$$ is false."],"dependencies":["aa47f81type1a-h1"],"title":"Type II error","text":"Choose the best answer from the following.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["The decision is to reject $$H_0$$ when $$H_0$$ is true.","The decision is not to reject $$H_0$$ when, in fact, $$H_0$$ is false."]},{"id":"aa47f81type1a-h3","type":"hint","dependencies":["aa47f81type1a-h2"],"title":"What is the Null Hypothesis in the context?","text":"Given in the problem, the null hypothesis is that the LTCC Intermediate Algebra students get less than seven hours of sleep per night, on average.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa47f81type10","title":"Type I vs. Type II Error","body":"Determine Type I| error for the following scenario: Assume a null hypothesis, $$H_0$$, that states the percentage of adults with jobs is at least 88%.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Outcomes and the Type I and Type II Errors","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa47f81type10a","stepAnswer":["Not to reject the null hypothesis that the percentage of adults who have jobs is at least 88% when that percentage is actually less than 88%"],"problemType":"MultipleChoice","stepTitle":"Identify the Type I| error from these four statements.","stepBody":"Choose the best answer from the following.","answerType":"string","variabilization":{},"choices":["Reject the null hypothesis that the percentage of adults who have jobs is at least 88% when the percentage is actually at least 88%.","Not to reject the null hypothesis that the percentage of adults who have jobs is at least 88% when that percentage is actually less than 88%","Not to reject the null hypothesis that the percentage of adults who have jobs is at least 88% when the percentage is actually at least 88%.","Reject the null hypothesis that the percentage of adults who have jobs is at least 88% when that percentage is actually less than 88%."],"hints":{"DefaultPathway":[{"id":"aa47f81type10a-h1","type":"hint","dependencies":[],"title":"The Type Error","text":"What is the definition of a Type II Error?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa47f81type10a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["The decision is not to reject $$H_0$$ when, in fact, $$H_0$$ is false."],"dependencies":["aa47f81type10a-h1"],"title":"Type II error","text":"Choose the best answer from the following.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["The decision is to reject $$H_0$$ when $$H_0$$ is true.","The decision is not to reject $$H_0$$ when, in fact, $$H_0$$ is false."]}]}}]},{"id":"aa47f81type11","title":"Automobile Accident","body":"Suppose the null hypothesis, $$H_0$$, is: The victim of an automobile accident is alive when he arrives at the emergency room of a hospital.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Outcomes and the Type I and Type II Errors","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa47f81type11a","stepAnswer":["Type I error"],"problemType":"MultipleChoice","stepTitle":"Which type of error has the greater consequence, Type I or Type II?","stepBody":"Choose the best answer from the following.","answerType":"string","variabilization":{},"choices":["Type I error","Type II error"],"hints":{"DefaultPathway":[{"id":"aa47f81type11a-h1","type":"hint","dependencies":[],"title":"The Type Error","text":"What is the definition of a Type I Error?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa47f81type11a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["The decision is to reject $$H_0$$ when $$H_0$$ is true."],"dependencies":["aa47f81type11a-h1"],"title":"Type I error","text":"Choose the best answer from the following.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["The decision is to reject $$H_0$$ when $$H_0$$ is true.","The decision is not to reject $$H_0$$ when, in fact, $$H_0$$ is false."]},{"id":"aa47f81type11a-h3","type":"hint","dependencies":["aa47f81type11a-h2"],"title":"Type I error under problem setting","text":"Type I error: The emergency crew thinks that the victim is dead when, in fact, the victim is alive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa47f81type11a-h4","type":"hint","dependencies":["aa47f81type11a-h3"],"title":"The Type Error","text":"What is the definition of a Type II Error?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa47f81type11a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["The decision is not to reject $$H_0$$ when, in fact, $$H_0$$ is false."],"dependencies":["aa47f81type11a-h4"],"title":"Type II error","text":"Choose the best answer from the following.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["The decision is to reject $$H_0$$ when $$H_0$$ is true.","The decision is not to reject $$H_0$$ when, in fact, $$H_0$$ is false."]},{"id":"aa47f81type11a-h6","type":"hint","dependencies":["aa47f81type11a-h5"],"title":"Type II error under problem setting","text":"Type II error: The emergency crew does not know if the victim is alive when, in fact, the victim is dead.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa47f81type11a-h7","type":"hint","dependencies":["aa47f81type11a-h6"],"title":"Type I vs. Type II error","text":"The error with the greater consequence is the Type I error. (If the emergency crew thinks the victim is dead, they will not treat him.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa47f81type12","title":"Frank\'s Rock Climbing Equipment","body":"Suppose the null hypothesis, H0, is: Frank\'s rock climbing equipment is safe.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Outcomes and the Type I and Type II Errors","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa47f81type12a","stepAnswer":["Type I| error"],"problemType":"MultipleChoice","stepTitle":"Which type of error has the greater consequence, Type I or Type II?","stepBody":"Choose the best answer from the following.","answerType":"string","variabilization":{},"choices":["Type I error","Type II error","Type I| error"],"hints":{"DefaultPathway":[{"id":"aa47f81type12a-h1","type":"hint","dependencies":[],"title":"The Type Error","text":"What is the definition of a Type I Error?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa47f81type12a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["The decision is to reject $$H_0$$ when $$H_0$$ is true."],"dependencies":["aa47f81type12a-h1"],"title":"Type I error","text":"Choose the best answer from the following.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["The decision is to reject $$H_0$$ when $$H_0$$ is true.","The decision is not to reject $$H_0$$ when, in fact, $$H_0$$ is false."]},{"id":"aa47f81type12a-h3","type":"hint","dependencies":["aa47f81type12a-h2"],"title":"Type I error under problem setting","text":"Type I error: Frank thinks that his rock climbing equipment may not be safe when, in fact, it really is safe.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa47f81type12a-h4","type":"hint","dependencies":["aa47f81type12a-h3"],"title":"The Type Error","text":"What is the definition of a Type II Error?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa47f81type12a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["The decision is not to reject $$H_0$$ when, in fact, $$H_0$$ is false."],"dependencies":["aa47f81type12a-h4"],"title":"Type II error","text":"Choose the best answer from the following.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["The decision is to reject $$H_0$$ when $$H_0$$ is true.","The decision is not to reject $$H_0$$ when, in fact, $$H_0$$ is false."]},{"id":"aa47f81type12a-h6","type":"hint","dependencies":["aa47f81type12a-h5"],"title":"Type II error under problem setting","text":"Type II error: Frank thinks that his rock climbing equipment may be safe when, in fact, it is not safe.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa47f81type12a-h7","type":"hint","dependencies":["aa47f81type12a-h6"],"title":"Type I vs. Type II error","text":"Notice that, in this case, the error with the greater consequence is the Type II error. (If Frank thinks his rock climbing equipment is safe, he will go ahead and use it.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa47f81type13","title":"Mean Price of Mid-Sized Cars","body":"The mean price of mid-sized cars in a region is $32,000. A test is conducted to see if the claim is true.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Outcomes and the Type I and Type II Errors","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa47f81type13a","stepAnswer":["The mean price of $$mid-sized$$ cars is $32,000, but we conclude that it is not $32,000."],"problemType":"MultipleChoice","stepTitle":"What is the Type I error in this case?","stepBody":"Choose the best answer from the following.","answerType":"string","variabilization":{},"choices":["The mean price of $$mid-sized$$ cars is $32,000, but we conclude that it is not $32,000.","The mean price of $$mid-sized$$ cars is not $32,000, but we conclude that it is $32,000."],"hints":{"DefaultPathway":[{"id":"aa47f81type13a-h1","type":"hint","dependencies":[],"title":"The Type Error","text":"What is the definition of a Type I Error?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa47f81type13a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["The decision is to reject $$H_0$$ when $$H_0$$ is true."],"dependencies":["aa47f81type13a-h1"],"title":"Type I error","text":"Choose the best answer from the following.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["The decision is to reject $$H_0$$ when $$H_0$$ is true.","The decision is not to reject $$H_0$$ when, in fact, $$H_0$$ is false."]}]}}]},{"id":"aa47f81type14","title":"Mean Price of Mid-Sized Cars","body":"The mean price of mid-sized cars in a region is $32,000. A test is conducted to see if the claim is true.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Outcomes and the Type I and Type II Errors","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa47f81type14a","stepAnswer":["The mean price of $$mid-sized$$ cars is not $32,000, but we conclude that it is $32,000."],"problemType":"MultipleChoice","stepTitle":"What is the Type I| error in this case?","stepBody":"Choose the best answer from the following.","answerType":"string","variabilization":{},"choices":["The mean price of $$mid-sized$$ cars is $32,000, but we conclude that it is not $32,000.","The mean price of $$mid-sized$$ cars is not $32,000, but we conclude that it is $32,000."],"hints":{"DefaultPathway":[{"id":"aa47f81type14a-h1","type":"hint","dependencies":[],"title":"The Type Error","text":"What is the definition of a Type II Error?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa47f81type14a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["The decision is not to reject $$H_0$$ when, in fact, $$H_0$$ is false."],"dependencies":["aa47f81type14a-h1"],"title":"Type II error","text":"Choose the best answer from the following.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["The decision is to reject $$H_0$$ when $$H_0$$ is true.","The decision is not to reject $$H_0$$ when, in fact, $$H_0$$ is false."]}]}}]},{"id":"aa47f81type15","title":"Perform an Operation","body":"A group of doctors is deciding whether or not to perform an operation. Suppose the null hypothesis, $$H_0$$, is: the surgical procedure will go well.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Outcomes and the Type I and Type II Errors","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa47f81type15a","stepAnswer":["The procedure will go well, but the doctors think it will not.|The procedure will not go well, but the doctors think it will."],"problemType":"MultipleChoice","stepTitle":"What is the Type I error in this case?","stepBody":"Choose the best answer from the following.","answerType":"string","variabilization":{},"choices":["The decision is not to reject $$H_0$$ when, in fact, $$H_0$$ is false.","The decision is to reject $$H_0$$ when $$H_0$$ is true.","The procedure will go well, but the doctors think it will not.|The procedure will not go well, but the doctors think it will."],"hints":{"DefaultPathway":[{"id":"aa47f81type15a-h1","type":"hint","dependencies":[],"title":"The Type Error","text":"What is the definition of a Type I Error?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa47f81type15a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["The decision is to reject $$H_0$$ when $$H_0$$ is true."],"dependencies":["aa47f81type15a-h1"],"title":"Type I error","text":"Choose the best answer from the following.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["The decision is to reject $$H_0$$ when $$H_0$$ is true.","The decision is not to reject $$H_0$$ when, in fact, $$H_0$$ is false."]}]}}]},{"id":"aa47f81type2","title":"New Drug","body":"When a new drug is created, the pharmaceutical company must subject it to testing before receiving the necessary permission from the Food and Drug Administration (FDA) to market the drug.","variabilization":{},"oer":"https://openstax.org","license":0,"lesson":"9.2 Outcomes and the Type I and Type II Errors","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa47f81type2a","stepAnswer":["Not to conclude the drug is safe when, in fact, it is safe."],"problemType":"MultipleChoice","stepTitle":"When a new drug is created, the pharmaceutical company must subject it to testing before receiving the necessary permission from the Food and Drug Administration (FDA) to market the drug.","stepBody":"Choose the best answer from the following.","answerType":"string","variabilization":{},"choices":["Not to conclude the drug is safe when, in fact, it is safe.","To conclude the drug is safe when in, fact, it is unsafe.","To conclude the drug is safe when in, fact, it is unsafe.","To conclude the drug is safe when in, fact, it is unsafe."],"hints":{"DefaultPathway":[{"id":"aa47f81type2a-h1","type":"hint","dependencies":[],"title":"The Type Error","text":"What is the definition of a Type II Error?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa47f81type2a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["The decision is not to reject $$H_0$$ when, in fact, $$H_0$$ is false."],"dependencies":["aa47f81type2a-h1"],"title":"Type II error","text":"Choose the best answer from the following.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["The decision is to reject $$H_0$$ when $$H_0$$ is true.","The decision is not to reject $$H_0$$ when, in fact, $$H_0$$ is false."]},{"id":"aa47f81type2a-h3","type":"hint","dependencies":["aa47f81type2a-h2"],"title":"What is the Null Hypothesis in the context?","text":"Given in the problem, the null hypothesis is \u201cthe drug is unsafe.\u201d","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa47f81type3","title":"EVC Students Attendences","body":"A statistics instructor believes that fewer than 20% of Evergreen Valley College (EVC) students attended the opening midnight showing of the latest Harry Potter movie. She surveys $$84$$ of her students and finds that $$11$$ of them attended the midnight showing.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Outcomes and the Type I and Type II Errors","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa47f81type3a","stepAnswer":["at least 20%, when in fact, it is less than 20%."],"problemType":"MultipleChoice","stepTitle":"The Type I error is to conclude that the percent of EVC students who attended is $$___$$ .","stepBody":"Choose the best answer from the following.","answerType":"string","variabilization":{},"choices":["at least 20%, when in fact, it is less than 20%.","20%, when in fact, it is 20%.","less than 20%, when in fact, it is at least 20%.","less than 20%, when in fact, it is less than 20%."],"hints":{"DefaultPathway":[{"id":"aa47f81type3a-h1","type":"hint","dependencies":[],"title":"The Type Error","text":"What is the definition of a Type I Error?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa47f81type3a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["The decision is to reject $$H_0$$ when $$H_0$$ is true."],"dependencies":["aa47f81type3a-h1"],"title":"Type I error","text":"Choose the best answer from the following.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["The decision is to reject $$H_0$$ when $$H_0$$ is true.","The decision is not to reject $$H_0$$ when, in fact, $$H_0$$ is false."]},{"id":"aa47f81type3a-h3","type":"hint","dependencies":["aa47f81type3a-h2"],"title":"What is the Null Hypothesis in the context?","text":"Combine with the context of the question, the null hypothesis is what the statistics instructor believes in, in which is less than 20% of the students attended the midnight showing.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa47f81type4","title":"Hours Spent on the Phone","body":"Previously, an organization reported that teenagers spent $$4.5$$ hours per week, on average, on the phone. The organization thinks that, currently, the mean is higher. Fifteen randomly chosen teenagers were asked how many hours per week they spend on the phone. The sample mean was $$4.75$$ hours with a sample standard deviation of $$2.0$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Outcomes and the Type I and Type II Errors","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa47f81type4a","stepAnswer":["to conclude that the mean hours per week currently is no higher than $$4.5$$, when in fact, it is not higher"],"problemType":"MultipleChoice","stepTitle":"Conduct a hypothesis test, the Type I error is:","stepBody":"Choose the best answer from the following.","answerType":"string","variabilization":{},"answerLatex":"to conclude that the mean hours per week currently is no higher than $$4.5$$, when in fact, it is not higher","choices":["to conclude that the mean hours per week currently is no higher than $$4.5$$, when in fact, it is not higher","to conclude that the current mean hours per week is higher than $$4.5$$, when in fact, it is higher","to conclude that the current mean hours per week is higher than $$4.5$$, when in fact, it is the same","to conclude that the mean hours per week currently is $$4.5$$, when in fact, it is higher"],"hints":{"DefaultPathway":[{"id":"aa47f81type4a-h1","type":"hint","dependencies":[],"title":"The Type Error","text":"What is the definition of a Type I Error?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa47f81type4a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["The decision is to reject $$H_0$$ when $$H_0$$ is true."],"dependencies":["aa47f81type4a-h1"],"title":"Type I error","text":"Choose the best answer from the following.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["The decision is to reject $$H_0$$ when $$H_0$$ is true.","The decision is not to reject $$H_0$$ when, in fact, $$H_0$$ is false."]},{"id":"aa47f81type4a-h3","type":"hint","dependencies":["aa47f81type4a-h2"],"title":"What is the Null Hypothesis in the context?","text":"Combine with the context of the question, the null hypothesis is that teenagers spent $$4.5$$ hours per week, on average, on the phone. Reject to $$H_0$$, in this context means to have a higher mean.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa47f81type5","title":"Type II Error","body":"The power of a test is $$0.981$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Outcomes and the Type I and Type II Errors","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa47f81type5a","stepAnswer":["$$0.019$$"],"problemType":"TextBox","stepTitle":"What is the probability of a Type II error?","stepBody":"Round your answer to third place after the decimal.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.019$$","hints":{"DefaultPathway":[{"id":"aa47f81type5a-h1","type":"hint","dependencies":[],"title":"The Type Error","text":"What is the definition of a Type II Error?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa47f81type5a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["The decision is not to reject $$H_0$$ when, in fact, $$H_0$$ is false."],"dependencies":["aa47f81type5a-h1"],"title":"Type II error","text":"Choose the best answer from the following.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["The decision is to reject $$H_0$$ when $$H_0$$ is true.","The decision is not to reject $$H_0$$ when, in fact, $$H_0$$ is false."]},{"id":"aa47f81type5a-h3","type":"hint","dependencies":["aa47f81type5a-h2"],"title":"The Probability of Type II Error","text":"The probability of Type II Error is $$1$$ - the power of the test.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa47f81type6","title":"Water Sample for E-coli","body":"A microbiologist is testing a water sample for E-coli. Suppose the null hypothesis, H0, is: the sample does not contain E-coli. The probability that the sample does not contain E-coli, but the microbiologist thinks it does is $$0.012$$. The probability that the sample does contain E-coli, but the microbiologist thinks it does not is $$0.002$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Outcomes and the Type I and Type II Errors","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa47f81type6a","stepAnswer":["$$0.998$$"],"problemType":"TextBox","stepTitle":"What is the power of this test?","stepBody":"Round your answer to $$3$$ decimal places.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.998$$","hints":{"DefaultPathway":[{"id":"aa47f81type6a-h1","type":"hint","dependencies":[],"title":"What is the relationship between the power of the test and type II error?","text":"The power of the test + the probability of type II error $$=$$ $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa47f81type6a-h2","type":"hint","dependencies":["aa47f81type6a-h1"],"title":"The Type Error","text":"What is the definition of a Type II Error?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa47f81type6a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["The decision is not to reject $$H_0$$ when, in fact, $$H_0$$ is false."],"dependencies":["aa47f81type6a-h2"],"title":"Type II error","text":"Choose the best answer from the following.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["The decision is to reject $$H_0$$ when $$H_0$$ is true.","The decision is not to reject $$H_0$$ when, in fact, $$H_0$$ is false."]},{"id":"aa47f81type6a-h4","type":"hint","dependencies":["aa47f81type6a-h3"],"title":"Type II error","text":"What is the Type II Error in this context?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa47f81type6a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$0.002$$"],"dependencies":["aa47f81type6a-h4"],"title":"The Type II Error is","text":"Choose the best answer from the following.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$0.002$$","$$0.012$$","$$0.998$$","$$0.988$$"]}]}}]},{"id":"aa47f81type7","title":"Boy Genetic Labs","body":"It\u2019s a Boy Genetic Labs claim to be able to increase the likelihood that a pregnancy will result in a boy being born. Statisticians want to test the claim. Suppose that the null hypothesis, H0, is: It\u2019s a Boy Genetic Labs has no effect on gender outcome.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Outcomes and the Type I and Type II Errors","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa47f81type7a","stepAnswer":["Type I error"],"problemType":"MultipleChoice","stepTitle":"Which type of error has the greater consequence, Type I or Type II?","stepBody":"Choose the best answer from the following.","answerType":"string","variabilization":{},"choices":["Type I error","Type II error"],"hints":{"DefaultPathway":[{"id":"aa47f81type7a-h1","type":"hint","dependencies":[],"title":"The Type Error","text":"What is the definition of a Type I Error?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa47f81type7a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["The decision is to reject $$H_0$$ when $$H_0$$ is true."],"dependencies":["aa47f81type7a-h1"],"title":"Type I error","text":"Choose the best answer from the following.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["The decision is to reject $$H_0$$ when $$H_0$$ is true.","The decision is not to reject $$H_0$$ when, in fact, $$H_0$$ is false."]},{"id":"aa47f81type7a-h3","type":"hint","dependencies":["aa47f81type7a-h2"],"title":"Type I error under problem setting","text":"Type I error: This results when a true null hypothesis is rejected. In the context of this scenario, we would state that we believe that It\u2019s a Boy Genetic Labs influences the gender outcome, when in fact it has no effect. The probability of this error occurring is denoted by the Greek letter alpha, \\\\alpha.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa47f81type7a-h4","type":"hint","dependencies":["aa47f81type7a-h3"],"title":"The Type Error","text":"What is the definition of a Type II Error?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa47f81type7a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["The decision is not to reject $$H_0$$ when, in fact, $$H_0$$ is false."],"dependencies":["aa47f81type7a-h4"],"title":"Type II error","text":"Choose the best answer from the following.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["The decision is to reject $$H_0$$ when $$H_0$$ is true.","The decision is not to reject $$H_0$$ when, in fact, $$H_0$$ is false."]},{"id":"aa47f81type7a-h6","type":"hint","dependencies":["aa47f81type7a-h5"],"title":"Type II error under problem setting","text":"Type II error: This results when we fail to reject a false null hypothesis. In context, we would state that It\u2019s a Boy Genetic Labs does not influence the gender outcome of a pregnancy when, in fact, it does. The probability of this error occurring is denoted by the Greek letter beta, \u03b2.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa47f81type7a-h7","type":"hint","dependencies":["aa47f81type7a-h6"],"title":"Type I vs. Type II error","text":"The error of greater consequence would be the Type I error since couples would use the It\u2019s a Boy Genetic Labs product in hopes of increasing the chances of having a boy.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa47f81type8","title":"Drug Cure Rate","body":"A certain experimental drug claims a cure rate of at least 75% for males with prostate cancer. Describe both the Type I and Type II errors in context.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Outcomes and the Type I and Type II Errors","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa47f81type8a","stepAnswer":["Type II error"],"problemType":"MultipleChoice","stepTitle":"Which error is the more serious? Type I or Type II?","stepBody":"Choose the best answer from the following.","answerType":"string","variabilization":{},"choices":["Type I error","Type II error"],"hints":{"DefaultPathway":[{"id":"aa47f81type8a-h1","type":"hint","dependencies":[],"title":"The Type Error","text":"What is the definition of a Type I Error?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa47f81type8a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["The decision is to reject $$H_0$$ when $$H_0$$ is true."],"dependencies":["aa47f81type8a-h1"],"title":"Type I error","text":"Choose the best answer from the following.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["The decision is to reject $$H_0$$ when $$H_0$$ is true.","The decision is not to reject $$H_0$$ when, in fact, $$H_0$$ is false."]},{"id":"aa47f81type8a-h3","type":"hint","dependencies":["aa47f81type8a-h2"],"title":"Type I error under problem setting","text":"Type I: A cancer patient believes the cure rate for the drug is less than 75% when it actually is at least 75%.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa47f81type8a-h4","type":"hint","dependencies":["aa47f81type8a-h3"],"title":"The Type Error","text":"What is the definition of a Type II Error?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa47f81type8a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["The decision is not to reject $$H_0$$ when, in fact, $$H_0$$ is false."],"dependencies":["aa47f81type8a-h4"],"title":"Type II error","text":"Choose the best answer from the following.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["The decision is to reject $$H_0$$ when $$H_0$$ is true.","The decision is not to reject $$H_0$$ when, in fact, $$H_0$$ is false."]},{"id":"aa47f81type8a-h6","type":"hint","dependencies":["aa47f81type8a-h5"],"title":"Type II error under problem setting","text":"Type II: A cancer patient believes the experimental drug has at least a 75% cure rate when it has a cure rate that is less than 75%.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa47f81type8a-h7","type":"hint","dependencies":["aa47f81type8a-h6"],"title":"Type I vs. Type II error","text":"In this scenario, the Type II error contains the more severe consequence. If a patient believes the drug works at least 75% of the time, this most likely will influence the patient\u2019s (and doctor\u2019s) choice about whether to use the drug as a treatment option.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa47f81type9","title":"Type I vs. Type II Error","body":"Determine Type I error for the following scenario: Assume a null hypothesis, $$H_0$$, that states the percentage of adults with jobs is at least 88%.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Outcomes and the Type I and Type II Errors","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa47f81type9a","stepAnswer":["Reject the null hypothesis that the percentage of adults who have jobs is at least 88% when the percentage is actually at least 88%."],"problemType":"MultipleChoice","stepTitle":"Identify the Type I error from these four statements.","stepBody":"Choose the best answer from the following.","answerType":"string","variabilization":{},"choices":["Reject the null hypothesis that the percentage of adults who have jobs is at least 88% when the percentage is actually at least 88%.","Not to reject the null hypothesis that the percentage of adults who have jobs is at least 88% when that percentage is actually less than 88%","Not to reject the null hypothesis that the percentage of adults who have jobs is at least 88% when the percentage is actually at least 88%.","Reject the null hypothesis that the percentage of adults who have jobs is at least 88% when that percentage is actually less than 88%."],"hints":{"DefaultPathway":[{"id":"aa47f81type9a-h1","type":"hint","dependencies":[],"title":"The Type Error","text":"What is the definition of a Type I Error?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa47f81type9a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["The decision is to reject $$H_0$$ when $$H_0$$ is true."],"dependencies":["aa47f81type9a-h1"],"title":"Type I error","text":"Choose the best answer from the following.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["The decision is to reject $$H_0$$ when $$H_0$$ is true.","The decision is not to reject $$H_0$$ when, in fact, $$H_0$$ is false."]}]}}]},{"id":"aa5ef86inv1","title":"Find the Multiplicative Inverse #1","body":"Find the inverse of the matrix using the formula to find the inverse of a 2x2 matrix.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.7 Solving Systems with Inverses","courseName":"OpenStax: College Algebra","steps":[{"id":"aa5ef86inv1a","stepAnswer":["$$\\\\begin{bmatrix} -3 & 2 \\\\\\\\ -2 & 1 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"$$\\\\begin{bmatrix} 1 & -2 \\\\\\\\ 2 & -3 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} -3 & 2 \\\\\\\\ -2 & 1 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"aa5ef86inv1a-h1","type":"hint","dependencies":[],"title":"Matrix Inverse Formula","text":"The inverse of a 2x2 matrix $$\\\\begin{bmatrix} a & b \\\\\\\\ c & d \\\\end{bmatrix}$$ is $$(1/(ad-bc))\\\\begin{bmatrix} d & -b \\\\\\\\ -c & a \\\\end{bmatrix}$$ where ad-bc is not $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa5ef86inv10","title":"Finding the Multiplicative Inverse of 3x3 Matrices #2","body":"Given the 3x3 matrix A, find the inverse.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.7 Solving Systems with Inverses","courseName":"OpenStax: College Algebra","steps":[{"id":"aa5ef86inv10a","stepAnswer":["$$\\\\begin{bmatrix} 1 & 1 & 2 \\\\\\\\ 2 & 4 & -3 \\\\\\\\ 3 & 6 & -5 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"$$A=\\\\begin{bmatrix} 2 & -17 & 11 \\\\\\\\ -1 & 11 & -7 \\\\\\\\ 0 & 3 & -2 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 1 & 1 & 2 \\\\\\\\ 2 & 4 & -3 \\\\\\\\ 3 & 6 & -5 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"aa5ef86inv10a-h1","type":"hint","dependencies":[],"title":"First Step","text":"Write the original matrix augmented with the identity matrix on the right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv10a-h2","type":"hint","dependencies":["aa5ef86inv10a-h1"],"title":"Second Step","text":"Use elementary row operations so that the identity appears on the left.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv10a-h3","type":"hint","dependencies":["aa5ef86inv10a-h2"],"title":"Third Step","text":"What is obtained on the right is the inverse of the original matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv10a-h4","type":"hint","dependencies":["aa5ef86inv10a-h3"],"title":"Fourth Step","text":"Use matrix multiplication to show that $$A A^{\\\\left(-1\\\\right)}=I$$ and $$A^{\\\\left(-1\\\\right)} A=I$$, where I is the identity matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa5ef86inv11","title":"Finding the Multiplicative Inverse of 3x3 Matrices #3","body":"Given the 3x3 matrix A, find the inverse.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.7 Solving Systems with Inverses","courseName":"OpenStax: College Algebra","steps":[{"id":"aa5ef86inv11a","stepAnswer":["$$\\\\begin{bmatrix} \\\\frac{-1}{8} & 0 & \\\\frac{3}{8} \\\\\\\\ \\\\frac{-25}{16} & 1 & \\\\frac{19}{16} \\\\\\\\ \\\\frac{3}{16} & 0 & \\\\frac{-1}{16} \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"$$A=\\\\begin{bmatrix} 1 & 0 & 6 \\\\\\\\ -2 & 1 & 7 \\\\\\\\ 3 & 0 & 2 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} \\\\frac{-1}{8} & 0 & \\\\frac{3}{8} \\\\\\\\ \\\\frac{-25}{16} & 1 & \\\\frac{19}{16} \\\\\\\\ \\\\frac{3}{16} & 0 & \\\\frac{-1}{16} \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"aa5ef86inv11a-h1","type":"hint","dependencies":[],"title":"First Step","text":"Write the original matrix augmented with the identity matrix on the right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv11a-h2","type":"hint","dependencies":["aa5ef86inv11a-h1"],"title":"Second Step","text":"Use elementary row operations so that the identity appears on the left.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv11a-h3","type":"hint","dependencies":["aa5ef86inv11a-h2"],"title":"Third Step","text":"What is obtained on the right is the inverse of the original matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv11a-h4","type":"hint","dependencies":["aa5ef86inv11a-h3"],"title":"Fourth Step","text":"Use matrix multiplication to show that $$A A^{\\\\left(-1\\\\right)}=I$$ and $$A^{\\\\left(-1\\\\right)} A=I$$, where I is the identity matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa5ef86inv12","title":"Finding the Multiplicative Inverse of 3x3 Matrices #3","body":"Given the 3x3 matrix A, find the inverse.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.7 Solving Systems with Inverses","courseName":"OpenStax: College Algebra","steps":[{"id":"aa5ef86inv12a","stepAnswer":["$$\\\\begin{bmatrix} \\\\frac{8}{25} & \\\\frac{-1}{5} & \\\\frac{-3}{35} \\\\\\\\ \\\\frac{17}{10} & \\\\frac{1}{10} & \\\\frac{-1}{35} \\\\\\\\ \\\\frac{-2}{7} & 0 & \\\\frac{-1}{7} \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"$$A=\\\\begin{bmatrix} 1 & 2 & -1 \\\\\\\\ -3 & 4 & 1 \\\\\\\\ -2 & -4 & -5 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} \\\\frac{8}{25} & \\\\frac{-1}{5} & \\\\frac{-3}{35} \\\\\\\\ \\\\frac{17}{10} & \\\\frac{1}{10} & \\\\frac{-1}{35} \\\\\\\\ \\\\frac{-2}{7} & 0 & \\\\frac{-1}{7} \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"aa5ef86inv12a-h1","type":"hint","dependencies":[],"title":"First Step","text":"Write the original matrix augmented with the identity matrix on the right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv12a-h2","type":"hint","dependencies":["aa5ef86inv12a-h1"],"title":"Second Step","text":"Use elementary row operations so that the identity appears on the left.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv12a-h3","type":"hint","dependencies":["aa5ef86inv12a-h2"],"title":"Third Step","text":"What is obtained on the right is the inverse of the original matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv12a-h4","type":"hint","dependencies":["aa5ef86inv12a-h3"],"title":"Fourth Step","text":"Use matrix multiplication to show that $$A A^{\\\\left(-1\\\\right)}=I$$ and $$A^{\\\\left(-1\\\\right)} A=I$$, where I is the identity matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa5ef86inv13","title":"Finding the Inverse of a 2x2 Matrix","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.7 Solving Systems with Inverses","courseName":"OpenStax: College Algebra","steps":[{"id":"aa5ef86inv13a","stepAnswer":["$$\\\\begin{bmatrix} 9 & 2 \\\\\\\\ -1 & 3 \\\\end{bmatrix}/29$$"],"problemType":"TextBox","stepTitle":"Find the inverse of the matrix $$\\\\begin{bmatrix} 3 & -2 \\\\\\\\ 1 & 9 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 9 & 2 \\\\\\\\ -1 & 3 \\\\end{bmatrix}/29$$","hints":{"DefaultPathway":[{"id":"aa5ef86inv13a-h1","type":"hint","dependencies":[],"title":"Finding the Determinant","text":"The determinant of the matrix is goven by $$a d-b c$$. This means that the determinant is (29).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv13a-h2","type":"hint","dependencies":["aa5ef86inv13a-h1"],"title":"Multiplying the Determinant with the Changed Matrix","text":"We must modify the matrix into the form $$\\\\begin{bmatrix} d & -b \\\\\\\\ -c & a \\\\end{bmatrix}$$. We now have $$\\\\begin{bmatrix} 9 & 2 \\\\\\\\ -1 & 3 \\\\end{bmatrix}$$. We now divide the matrix by $$29$$ to get $$\\\\begin{bmatrix} 9 & 2 \\\\\\\\ -1 & 3 \\\\end{bmatrix}/29$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa5ef86inv14","title":"Finding the Inverse of a 2x2 Matrix","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.7 Solving Systems with Inverses","courseName":"OpenStax: College Algebra","steps":[{"id":"aa5ef86inv14a","stepAnswer":["$$(-1/8)*\\\\begin{bmatrix} 1 & -2 \\\\\\\\ -3 & -2 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"Find the inverse of the matrix $$\\\\begin{bmatrix} -2 & 2 \\\\\\\\ 3 & 1 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$(-1/8)*\\\\begin{bmatrix} 1 & -2 \\\\\\\\ -3 & -2 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"aa5ef86inv14a-h1","type":"hint","dependencies":[],"title":"Finding the Determinant","text":"The determinant is given by $$a d-b c$$. For this matrix, the determinant is $$\\\\frac{1}{\\\\left(-2-6\\\\right)}$$ which is $$-8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv14a-h2","type":"hint","dependencies":["aa5ef86inv14a-h1"],"title":"Multiplying the Determinant with the Changed Matrix","text":"We must modify the matrix into the form $$\\\\begin{bmatrix} d & -b \\\\\\\\ -c & a \\\\end{bmatrix}$$. This means we now have $$\\\\begin{bmatrix} 1 & -2 \\\\\\\\ -3 & -2 \\\\end{bmatrix}$$. Dividing this by $$-8$$ gives us $$(-1/8)*\\\\begin{bmatrix} 1 & -2 \\\\\\\\ -3 & -2 \\\\end{bmatrix}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa5ef86inv15","title":"Finding the Inverse of a 2x2 Matrix","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.7 Solving Systems with Inverses","courseName":"OpenStax: College Algebra","steps":[{"id":"aa5ef86inv15a","stepAnswer":["$$\\\\begin{bmatrix} 2 & -7 \\\\\\\\ \\\\frac{-9}{-69} & \\\\frac{-3}{-69} \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"Find the inverse of the matrix $$\\\\begin{bmatrix} -3 & 7 \\\\\\\\ 9 & 2 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 2 & -7 \\\\\\\\ \\\\frac{-9}{-69} & \\\\frac{-3}{-69} \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"aa5ef86inv15a-h1","type":"hint","dependencies":[],"title":"Finding the Determinant","text":"The determinant is given by $$a d-b c$$. This evaluates to $$-6-63$$ which is $$-69$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv15a-h2","type":"hint","dependencies":["aa5ef86inv15a-h1"],"title":"Multiplying the Determinant with the Changed Matrix","text":"We must modify the matrix into the form $$\\\\begin{bmatrix} d & -b \\\\\\\\ -c & a \\\\end{bmatrix}$$. We now have $$/mat{(2,-7),(-9,-3)$$. Dividing this matrix by the determinant, we have $$/mat{(2,-7),(-9,-3)/(-69)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa5ef86inv16","title":"Finding the Inverse of a 2x2 Matrix","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.7 Solving Systems with Inverses","courseName":"OpenStax: College Algebra","steps":[{"id":"aa5ef86inv16a","stepAnswer":["$$\\\\begin{bmatrix} \\\\frac{-8}{47} & \\\\frac{3}{47} \\\\\\\\ \\\\frac{5}{47} & \\\\frac{-4}{47} \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"Find the inverse of the matrix $$\\\\begin{bmatrix} -4 & -3 \\\\\\\\ -5 & 8 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} \\\\frac{-8}{47} & \\\\frac{3}{47} \\\\\\\\ \\\\frac{5}{47} & \\\\frac{-4}{47} \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"aa5ef86inv16a-h1","type":"hint","dependencies":[],"title":"Finding the Determinant","text":"The determinant is given by $$a d-b c$$, which evaluates to $$(-32-15)$$, or $$-47$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv16a-h2","type":"hint","dependencies":["aa5ef86inv16a-h1"],"title":"Multiplying the Determinant with the Changed Matrix","text":"We must modify the matrix into the form $$\\\\begin{bmatrix} d & -b \\\\\\\\ -c & a \\\\end{bmatrix}$$. We now have $$\\\\begin{bmatrix} 8 & 3 \\\\\\\\ 5 & -4 \\\\end{bmatrix}$$. Dividing this by the determinant gives us $$\\\\begin{bmatrix} \\\\frac{-8}{47} & \\\\frac{3}{47} \\\\\\\\ \\\\frac{5}{47} & \\\\frac{-4}{47} \\\\end{bmatrix}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa5ef86inv17","title":"Finding the Inverse of a 2x2 Matrix","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.7 Solving Systems with Inverses","courseName":"OpenStax: College Algebra","steps":[{"id":"aa5ef86inv17a","stepAnswer":["NAN"],"problemType":"MultipleChoice","stepTitle":"Find the inverse of the matrix $$\\\\begin{bmatrix} 1 & 12 & 2 \\\\end{bmatrix}$$. Enter NAN if there is no inverse.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$\\\\begin{bmatrix} 0 & 1 \\\\\\\\ 1 & 0 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} 1 & 2 \\\\\\\\ 2 & 1 \\\\end{bmatrix}$$","NAN"],"hints":{"DefaultPathway":[{"id":"aa5ef86inv17a-h1","type":"hint","dependencies":[],"title":"Finding the Determinant","text":"The determinant is given by $$a d-b c$$, which evaluates to $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv17a-h2","type":"hint","dependencies":["aa5ef86inv17a-h1"],"title":"Determinant is $$0$$","text":"Since the determinant is $$0$$, there is no inverse.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa5ef86inv18","title":"Finding the Inverse of a 2x2 Matrix","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.7 Solving Systems with Inverses","courseName":"OpenStax: College Algebra","steps":[{"id":"aa5ef86inv18a","stepAnswer":["NAN"],"problemType":"MultipleChoice","stepTitle":"Find the inverse of the matrix $$\\\\begin{bmatrix} 0 & 1 \\\\\\\\ 1 & 0 \\\\end{bmatrix}$$. Enter NAN if there is no inverse.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$\\\\begin{bmatrix} 0 & 1 \\\\\\\\ 1 & 0 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} 1 & 0 \\\\\\\\ 1 & 0 \\\\end{bmatrix}$$","NAN"],"hints":{"DefaultPathway":[{"id":"aa5ef86inv18a-h1","type":"hint","dependencies":[],"title":"Finding the Determinant","text":"The determinant is given by $$a d-b c$$, which evaluates to $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv18a-h2","type":"hint","dependencies":["aa5ef86inv18a-h1"],"title":"Determinant is $$0$$","text":"Since the determinant is $$0$$, there is no inverse.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa5ef86inv19","title":"Finding the Inverse of a 2x2 Matrix","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.7 Solving Systems with Inverses","courseName":"OpenStax: College Algebra","steps":[{"id":"aa5ef86inv19a","stepAnswer":["$$(-4/7)\\\\begin{bmatrix} \\\\frac{2}{7} & \\\\frac{6}{7} \\\\\\\\ \\\\frac{4}{7} & \\\\frac{-2}{7} \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"Find the inverse of the matrix $$\\\\begin{bmatrix} \\\\frac{1}{2} & \\\\frac{3}{2} \\\\\\\\ 1 & \\\\frac{-1}{2} \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$(-4/7)\\\\begin{bmatrix} \\\\frac{2}{7} & \\\\frac{6}{7} \\\\\\\\ \\\\frac{4}{7} & \\\\frac{-2}{7} \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"aa5ef86inv19a-h1","type":"hint","dependencies":[],"title":"Finding the Determinant","text":"The determinant is given by $$a d-b c$$, which evaluates to $$\\\\left(-\\\\frac{1}{4}-\\\\frac{3}{2}\\\\right)$$, or $$\\\\frac{-7}{4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv19a-h2","type":"hint","dependencies":["aa5ef86inv19a-h1"],"title":"Multiplying the Determinant with the Changed Matrix","text":"We must modify the matrix into the form $$\\\\begin{bmatrix} d & -b \\\\\\\\ -c & a \\\\end{bmatrix}$$. We now have $$\\\\begin{bmatrix} \\\\frac{-1}{2} & \\\\frac{-3}{2} \\\\\\\\ -1 & \\\\frac{1}{2} \\\\end{bmatrix}$$. Dividing this by the determinant gives us $$(-4/7)\\\\begin{bmatrix} \\\\frac{-1}{2} & \\\\frac{-3}{2} \\\\\\\\ -1 & \\\\frac{1}{2} \\\\end{bmatrix}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa5ef86inv2","title":"Determining Multiplicative Inverses #1","body":"Are A and B multiplicative inverses of each other?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.7 Solving Systems with Inverses","courseName":"OpenStax: College Algebra","steps":[{"id":"aa5ef86inv2a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$A=\\\\begin{bmatrix} 1 & 5 \\\\\\\\ -2 & -9 \\\\end{bmatrix}$$ and $$B=\\\\begin{bmatrix} -9 & -5 \\\\\\\\ 2 & 1 \\\\end{bmatrix}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"aa5ef86inv2a-h1","type":"hint","dependencies":[],"title":"First Step","text":"Given matrix A of order nxn and matrix B of order nxn multiply AB.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv2a-h2","type":"hint","dependencies":["aa5ef86inv2a-h1"],"title":"Second Step","text":"If $$AB=I$$, where I is the identity matrix, then find the product BA. If $$BA=1$$, than $$B=A^{\\\\left(-1\\\\right)}$$ and $$A=B^{\\\\left(-1\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa5ef86inv20","title":"Solve a 2x2 system with the inverse of a matrix","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.7 Solving Systems with Inverses","courseName":"OpenStax: College Algebra","steps":[{"id":"aa5ef86inv20a","stepAnswer":["$$(-5,6)$$"],"problemType":"MultipleChoice","stepTitle":"Solve the following system with matrix inverses: 5x-6y=-61,4x+3y=-2","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-5,6)$$","choices":["$$(-5,7)$$","$$(-5,6)$$","$$(-5,5)$$","$$(-5,4)$$"],"hints":{"DefaultPathway":[{"id":"aa5ef86inv20a-h1","type":"hint","dependencies":[],"title":"Finding the Inverse Matrix","text":"We must first find the inverse of the coefficient matrix $$\\\\begin{bmatrix} 5 & -6 \\\\\\\\ 4 & 3 \\\\end{bmatrix}$$. This is equal to $$\\\\begin{bmatrix} \\\\frac{1}{13} & \\\\frac{2}{13} \\\\\\\\ \\\\frac{-4}{39} & \\\\frac{5}{39} \\\\end{bmatrix}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv20a-h2","type":"hint","dependencies":["aa5ef86inv20a-h1"],"title":"Multiplying the constant matrix and the coefficient matrix.","text":"We must now multiply $$\\\\begin{bmatrix} -61 \\\\\\\\ -2 \\\\end{bmatrix}$$ by $$\\\\begin{bmatrix} \\\\frac{1}{13} & \\\\frac{2}{13} \\\\\\\\ \\\\frac{-4}{39} & \\\\frac{5}{39} \\\\end{bmatrix}$$, which gives us $$\\\\begin{bmatrix} -5 \\\\\\\\ 6 \\\\end{bmatrix}$$. The answer is $$(-5,6)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa5ef86inv21","title":"Solve a 2x2 system with the inverse of a matrix","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.7 Solving Systems with Inverses","courseName":"OpenStax: College Algebra","steps":[{"id":"aa5ef86inv21a","stepAnswer":["$$(2,0)$$"],"problemType":"MultipleChoice","stepTitle":"Solve the following system with matrix inverses: $$3x-2y=6-x+5y=-2$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(2,0)$$","choices":["$$(2,0)$$","$$(3,0)$$","$$(4,0)$$"],"hints":{"DefaultPathway":[{"id":"aa5ef86inv21a-h1","type":"hint","dependencies":[],"title":"Finding the Inverse Matrix","text":"We must find the inverse of the coefficient matrix $$\\\\begin{bmatrix} 3 & -2 \\\\\\\\ -1 & 5 \\\\end{bmatrix}$$, which is $$\\\\begin{bmatrix} \\\\frac{5}{13} & \\\\frac{2}{13} \\\\\\\\ \\\\frac{1}{13} & \\\\frac{2}{13} \\\\end{bmatrix}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv21a-h2","type":"hint","dependencies":["aa5ef86inv21a-h1"],"title":"Multiplying the constant matrix and the coefficient matrix.","text":"Now, we must multiply $$\\\\begin{bmatrix} 6 \\\\\\\\ -2 \\\\end{bmatrix}$$ by $$\\\\begin{bmatrix} \\\\frac{5}{13} & \\\\frac{2}{13} \\\\\\\\ \\\\frac{1}{13} & \\\\frac{2}{13} \\\\end{bmatrix}$$ to get $$\\\\begin{bmatrix} 2 \\\\\\\\ 0 \\\\end{bmatrix}$$. $$(2,0)$$ is our answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa5ef86inv22","title":"Solve a 2x2 system with the inverse of a matrix","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.7 Solving Systems with Inverses","courseName":"OpenStax: College Algebra","steps":[{"id":"aa5ef86inv22a","stepAnswer":["$$(\\\\frac{1}{3},\\\\frac{-5}{2})$$"],"problemType":"MultipleChoice","stepTitle":"Solve the following system with matrix inverses: -3x-4y=9,12x+4y=-6","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(\\\\frac{1}{3},\\\\frac{-5}{2})$$","choices":["$$(\\\\frac{1}{6},\\\\frac{-5}{2})$$","$$(\\\\frac{1}{3},\\\\frac{-5}{2})$$","$$(\\\\frac{1}{5},\\\\frac{-5}{2})$$"],"hints":{"DefaultPathway":[{"id":"aa5ef86inv22a-h1","type":"hint","dependencies":[],"title":"Finding the Inverse Matrix","text":"We must find the inversee of the coefficient matrix $$\\\\begin{bmatrix} -3 & -4 \\\\\\\\ 12 & 4 \\\\end{bmatrix}$$, which is $$\\\\begin{bmatrix} \\\\frac{1}{9} & \\\\frac{1}{9} \\\\\\\\ \\\\frac{-1}{3} & \\\\frac{-1}{12} \\\\end{bmatrix}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv22a-h2","type":"hint","dependencies":["aa5ef86inv22a-h1"],"title":"Multiplying the constant matrix and the coefficient matrix.","text":"We must now multiply $$\\\\begin{bmatrix} 9 \\\\\\\\ -6 \\\\end{bmatrix}$$ by $$\\\\begin{bmatrix} \\\\frac{1}{9} & \\\\frac{1}{9} \\\\\\\\ \\\\frac{-1}{3} & \\\\frac{-1}{12} \\\\end{bmatrix}$$. This gives us $$\\\\begin{bmatrix} \\\\frac{1}{3} \\\\\\\\ \\\\frac{-5}{2} \\\\end{bmatrix}$$. Our answer is $$(\\\\frac{1}{3},\\\\frac{-5}{2})$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa5ef86inv23","title":"Solve a 2x2 system with the inverse of a matrix","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.7 Solving Systems with Inverses","courseName":"OpenStax: College Algebra","steps":[{"id":"aa5ef86inv23a","stepAnswer":["$$(\\\\frac{-2}{3},\\\\frac{-11}{6})$$"],"problemType":"MultipleChoice","stepTitle":"Solve the following system with matrix inverses: $$\\\\frac{-8}{5} x-\\\\frac{4}{5} y=\\\\frac{2}{5}-\\\\frac{8}{5} x+\\\\frac{1}{5} y=\\\\frac{7}{10}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(\\\\frac{-2}{3},\\\\frac{-11}{6})$$","choices":["$$(\\\\frac{-2}{3},\\\\frac{-1}{6})$$","$$(\\\\frac{2}{3},\\\\frac{-11}{6})$$","$$(\\\\frac{-2}{3},\\\\frac{-11}{6})$$"],"hints":{"DefaultPathway":[{"id":"aa5ef86inv23a-h1","type":"hint","dependencies":[],"title":"Finding the Inverse Matrix","text":"We must find the inverse of the matrix $$\\\\begin{bmatrix} \\\\frac{-8}{5} & \\\\frac{-4}{5} \\\\\\\\ \\\\frac{-8}{5} & \\\\frac{1}{5} \\\\end{bmatrix}$$, which is $$\\\\begin{bmatrix} \\\\frac{-1}{8} & \\\\frac{-1}{2} \\\\\\\\ -1 & 1 \\\\end{bmatrix}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv23a-h2","type":"hint","dependencies":["aa5ef86inv23a-h1"],"title":"Multiplying the constant matrix and the coefficient matrix.","text":"Now, we can multiply $$\\\\begin{bmatrix} \\\\frac{2}{5} \\\\\\\\ \\\\frac{7}{10} \\\\end{bmatrix}$$ by $$\\\\begin{bmatrix} \\\\frac{-1}{8} & \\\\frac{-1}{2} \\\\\\\\ -1 & 1 \\\\end{bmatrix}$$, giving us $$\\\\begin{bmatrix} \\\\frac{-2}{3} \\\\\\\\ \\\\frac{-11}{6} \\\\end{bmatrix}$$. Our answer is $$(\\\\frac{-2}{3},\\\\frac{-11}{6})$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa5ef86inv24","title":"Solve a 2x2 system with the inverse of a matrix","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.7 Solving Systems with Inverses","courseName":"OpenStax: College Algebra","steps":[{"id":"aa5ef86inv24a","stepAnswer":["$$(-9,-7)$$"],"problemType":"MultipleChoice","stepTitle":"Solve the following system with matrix inverses: $$8x+4y=-100, 3x-4y=1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-9,-7)$$","choices":["$$(-9,-7)$$","$$(-8,-7)$$","$$(-4,-6)$$"],"hints":{"DefaultPathway":[{"id":"aa5ef86inv24a-h1","type":"hint","dependencies":[],"title":"Finding the Inverse Matrix","text":"We must find the inverse of the matrix $$\\\\begin{bmatrix} 8 & 4 \\\\\\\\ 3 & -4 \\\\end{bmatrix}$$, which is $$\\\\begin{bmatrix} \\\\frac{1}{11} & \\\\frac{1}{11} \\\\\\\\ \\\\frac{3}{44} & \\\\frac{-2}{11} \\\\end{bmatrix}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv24a-h2","type":"hint","dependencies":["aa5ef86inv24a-h1"],"title":"Multiplying the constant matrix and the coefficient matrix.","text":"We must now multiply the matrix $$\\\\begin{bmatrix} -100 \\\\\\\\ 1 \\\\end{bmatrix}$$ by $$\\\\begin{bmatrix} \\\\frac{1}{11} & \\\\frac{1}{11} \\\\\\\\ \\\\frac{3}{44} & \\\\frac{-2}{11} \\\\end{bmatrix}$$, giving us $$\\\\begin{bmatrix} -9 \\\\\\\\ -7 \\\\end{bmatrix}$$. This means that our answer is $$(-9,-7)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa5ef86inv3","title":"Finding the Multiplicative Inverse of 3x3 Matrices #1","body":"Given the 3x3 matrix A, find the inverse.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.7 Solving Systems with Inverses","courseName":"OpenStax: College Algebra","steps":[{"id":"aa5ef86inv3a","stepAnswer":["$$\\\\begin{bmatrix} -1 & 1 & 0 \\\\\\\\ -1 & 0 & 1 \\\\\\\\ 6 & -2 & -3 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"$$A=\\\\begin{bmatrix} 2 & 3 & 1 \\\\\\\\ 3 & 3 & 1 \\\\\\\\ 2 & 4 & 1 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} -1 & 1 & 0 \\\\\\\\ -1 & 0 & 1 \\\\\\\\ 6 & -2 & -3 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"aa5ef86inv3a-h1","type":"hint","dependencies":[],"title":"First Step","text":"Write the original matrix augmented with the identity matrix on the right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv3a-h2","type":"hint","dependencies":["aa5ef86inv3a-h1"],"title":"Second Step","text":"Use elementary row operations so that the identity appears on the left.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv3a-h3","type":"hint","dependencies":["aa5ef86inv3a-h2"],"title":"Third Step","text":"What is obtained on the right is the inverse of the original matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv3a-h4","type":"hint","dependencies":["aa5ef86inv3a-h3"],"title":"Fourth Step","text":"Use matrix multiplication to show that $$A A^{\\\\left(-1\\\\right)}=I$$ and $$A^{\\\\left(-1\\\\right)} A=I$$, where I is the identity matrix.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa5ef86inv4","title":"Determining Multiplicative Inverses #2","body":"Are A and B multiplicative inverses of each other?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.7 Solving Systems with Inverses","courseName":"OpenStax: College Algebra","steps":[{"id":"aa5ef86inv4a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$A=\\\\begin{bmatrix} 1 & 4 \\\\\\\\ -1 & -3 \\\\end{bmatrix}$$ and $$B=\\\\begin{bmatrix} -3 & -4 \\\\\\\\ 1 & 1 \\\\end{bmatrix}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"aa5ef86inv4a-h1","type":"hint","dependencies":[],"title":"First Step","text":"Given matrix A of order nxn and matrix B of order nxn multiply AB.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv4a-h2","type":"hint","dependencies":["aa5ef86inv4a-h1"],"title":"Second Step","text":"If $$AB=I$$, where I is the identity matrix, then find the product BA. If $$BA=1$$, than $$B=A^{\\\\left(-1\\\\right)}$$ and $$A=B^{\\\\left(-1\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa5ef86inv5","title":"Multiplying By the Identity Matrix","body":"I repesents the identity matrix. Given matrix A, what is AI?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.7 Solving Systems with Inverses","courseName":"OpenStax: College Algebra","steps":[{"id":"aa5ef86inv5a","stepAnswer":["$$\\\\begin{bmatrix} 3 & 4 \\\\\\\\ -2 & 5 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"$$A=\\\\begin{bmatrix} 3 & 4 \\\\\\\\ -2 & 5 \\\\end{bmatrix}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 3 & 4 \\\\\\\\ -2 & 5 \\\\end{bmatrix}$$","choices":["$$\\\\begin{bmatrix} 4 & 3 \\\\\\\\ 5 & 2 \\\\end{bmatrix}$$","$$\\\\begin{bmatrix} 3 & 4 \\\\\\\\ -2 & 5 \\\\end{bmatrix}$$"],"hints":{"DefaultPathway":[{"id":"aa5ef86inv5a-h1","type":"hint","dependencies":[],"title":"Function of the Identity Matrix","text":"Multiplying by the identity matrix does not change the entries of the original matrix; it is like multiplying by one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa5ef86inv6","title":"Solving a 2x2 System Using the Inverse of a Matrix","body":"Solve the given system of equations using the inverse of a matrix.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.7 Solving Systems with Inverses","courseName":"OpenStax: College Algebra","steps":[{"id":"aa5ef86inv6a","stepAnswer":["$$(-1,1)$$"],"problemType":"MultipleChoice","stepTitle":"$$3x+8y=5$$, $$4x+11y=7$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-1,1)$$","choices":["$$(-1,1)$$","$$(1,1)$$","$$(1,-1)$$","$$(-1,-1)$$"],"hints":{"DefaultPathway":[{"id":"aa5ef86inv6a-h1","type":"hint","dependencies":[],"title":"Writing the System in Terms of Matrices","text":"Write the system in terms of a coefficient matrix, a variable matrix, and a constant matrix. $$A=\\\\begin{bmatrix} 3 & 8 \\\\\\\\ 4 & 11 \\\\end{bmatrix}$$, $$X=\\\\begin{bmatrix} x \\\\\\\\ y \\\\end{bmatrix}$$, $$B=\\\\begin{bmatrix} 5 \\\\\\\\ 7 \\\\end{bmatrix}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv6a-h2","type":"hint","dependencies":["aa5ef86inv6a-h1"],"title":"Writing The Equation","text":"Next, using the matrices we wrote above, $$AX=B$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv6a-h3","type":"hint","dependencies":["aa5ef86inv6a-h2"],"title":"Calculating $$A^{\\\\left(-1\\\\right)}$$","text":"Calculate $$A^{\\\\left(-1\\\\right)}$$. We can use the formula to find the inverse of 2x2 matrix: $$A^{\\\\left(-1\\\\right)}=1/(ad-bc)*\\\\begin{bmatrix} d & -b \\\\\\\\ -c & a \\\\end{bmatrix}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv6a-h4","type":"hint","dependencies":["aa5ef86inv6a-h3"],"title":"Multiplying Each Side of the Equation","text":"Multiply both sides of the equation by $$A^{\\\\left(-1\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa5ef86inv7","title":"Solving a 3x3 System Using the Inverse of a Matrix #1","body":"Solve the given system of equations using the inverse of a matrix.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.7 Solving Systems with Inverses","courseName":"OpenStax: College Algebra","steps":[{"id":"aa5ef86inv7a","stepAnswer":["(1,2,0)"],"problemType":"MultipleChoice","stepTitle":"$$5x+15y+56z=35$$, $$-4x-11y-41z=-26$$, $$-x-3y-11z=-7$$","stepBody":"","answerType":"string","variabilization":{},"choices":["(1,2,0)","$$(-1, 2, 0)$$","$$(-2, 1, 0)$$"],"hints":{"DefaultPathway":[{"id":"aa5ef86inv7a-h1","type":"hint","dependencies":[],"title":"Writing the Equation","text":"Write the solution $$AX=B$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv7a-h2","type":"hint","dependencies":["aa5ef86inv7a-h1"],"title":"Finding the Inverse","text":"Next, find the inverse of A by augmenting with the identity.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv7a-h3","type":"hint","dependencies":["aa5ef86inv7a-h2"],"title":"Multiplying Each Side of the Equation","text":"Multiply both sides of the equation with $$A^{\\\\left(-1\\\\right)}$$. We want $$A^{\\\\left(-1\\\\right)} AX=A^{\\\\left(-1\\\\right)} B$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa5ef86inv8","title":"Solving a 3x3 System Using the Inverse of a Matrix #2","body":"Solve the given system of equations using the inverse of a matrix.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.7 Solving Systems with Inverses","courseName":"OpenStax: College Algebra","steps":[{"id":"aa5ef86inv8a","stepAnswer":["(4,38,58)"],"problemType":"MultipleChoice","stepTitle":"$$2x-17y+11z=0$$, $$-x+11y-7z=8$$, $$3y-2z=-2$$","stepBody":"","answerType":"string","variabilization":{},"choices":["(41,62,39)","(4,38,58)","(49,62,58)"],"hints":{"DefaultPathway":[{"id":"aa5ef86inv8a-h1","type":"hint","dependencies":[],"title":"Writing the Equation","text":"Write the solution $$AX=B$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv8a-h2","type":"hint","dependencies":["aa5ef86inv8a-h1"],"title":"Finding the Inverse","text":"Next, find the inverse of A by augmenting with the identity.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv8a-h3","type":"hint","dependencies":["aa5ef86inv8a-h2"],"title":"Multiplying Each Side of the Equation","text":"Multiply both sides of the equation with $$A^{\\\\left(-1\\\\right)}$$. We want $$A^{\\\\left(-1\\\\right)} AX=A^{\\\\left(-1\\\\right)} B$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa5ef86inv9","title":"Solving a 3x3 System Using the Inverse of a Matrix #3","body":"Solve the given system of equations using the inverse of a matrix.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.7 Solving Systems with Inverses","courseName":"OpenStax: College Algebra","steps":[{"id":"aa5ef86inv9a","stepAnswer":["$$(-59, -24, 252)$$"],"problemType":"MultipleChoice","stepTitle":"$$2x+3y+z=32$$, $$3x+3y+z=-27$$, $$2x+4y+z=-2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-59, -24, 252)$$","choices":["$$(-57, -22, 252)$$","$$(-59, -22, 212)$$","$$(-59, -24, 252)$$"],"hints":{"DefaultPathway":[{"id":"aa5ef86inv9a-h1","type":"hint","dependencies":[],"title":"Writing the Equation","text":"Write the solution $$AX=B$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv9a-h2","type":"hint","dependencies":["aa5ef86inv9a-h1"],"title":"Finding the Inverse","text":"Next, find the inverse of A by augmenting with the identity.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa5ef86inv9a-h3","type":"hint","dependencies":["aa5ef86inv9a-h2"],"title":"Multiplying Each Side of the Equation","text":"Multiply both sides of the equation with $$A^{\\\\left(-1\\\\right)}$$. We want $$A^{\\\\left(-1\\\\right)} AX=A^{\\\\left(-1\\\\right)} B$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa60f04hist1","title":"Finding Bar Widths of Histograms","body":"The following data are the shoe sizes of $$50$$ male students. The sizes are discrete data since shoe size is measured in whole and half units only. Calculate the width of each bar or class interval.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Histograms, Frequency Polygons, and Time Series Graphs","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa60f04hist1a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"Suppose you choose six bars.\\\\n9; 9; $$9.5;$$ $$9.5;$$ 10; 10; 10; 10; 10; 10; $$10.5;$$ $$10.5;$$ $$10.5;$$ $$10.5;$$ $$10.5;$$ $$10.5;$$ $$10.5;$$ $$10.5$$\\\\n11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11; $$11.5;$$ $$11.5;$$ $$11.5;$$ $$11.5;$$ $$11.5;$$ $$11.5;$$ $$11.5$$\\\\n12; 12; 12; 12; 12; 12; 12; $$12.5;$$ $$12.5;$$ $$12.5;$$ $$12.5;$$ $$14$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"aa60f04hist1a-h1","type":"hint","dependencies":[],"title":"Calculating Bar Widths","text":"The first step in calculating Bar Widths is to find the highest number (14) and the lowest number (9).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa60f04hist1a-h2","type":"hint","dependencies":["aa60f04hist1a-h1"],"title":"Calculating Bar Widths","text":"The final step is to divide (highest - lowest)/number of bars then round up which gives us $$\\\\frac{14-9}{6}$$ $$=$$ $$.83$$ which rounds up to $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa60f04hist2","title":"Setting up a Histogram","body":"Create a histogram for the following data: the number of books bought by $$50$$ part-time college students at ABC College. The number of books is discrete data, since books are counted.\\\\n1; 1; 1; 1; 1; 1; 1; 1; 1; 1; $$1$$\\\\n2; 2; 2; 2; 2; 2; 2; 2; 2; $$2$$\\\\n3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; $$3$$\\\\n4; 4; 4; 4; 4; $$4$$\\\\n5; 5; 5; 5; $$5$$\\\\n6; $$6$$\\\\nEleven students buy one book. Ten students buy two books. Sixteen students buy three books. Six students buy four books. Five students buy five books. Two students buy six books.\\\\nBecause the data are integers, subtract $$0.5$$ from $$1$$, the smallest data value and add $$0.5$$ to $$6$$, the largest data value. Then the starting point is $$0.5$$ and the ending value is $$6.5$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Histograms, Frequency Polygons, and Time Series Graphs","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa60f04hist2a","stepAnswer":["$$3.5$$"],"problemType":"TextBox","stepTitle":"Next, calculate the width of each bar or class interval. If the data are discrete and there are not too many different values, a width that places the data values in the middle of the bar or class interval is the most convenient. Since the data consist of the numbers $$1$$, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, and the starting point is $$0.5$$, a width of one places the $$1$$ in the middle of the interval from $$0.5$$ to $$1.5$$, the $$2$$ in the middle of the interval from $$1.5$$ to $$2.5$$, the $$3$$ in the middle of the interval from $$2.5$$ to $$3.5$$, the $$4$$ in the middle of the interval from $$___$$ to $$___$$ (first blank only).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3.5$$","hints":{"DefaultPathway":[]}},{"id":"aa60f04hist2b","stepAnswer":["$$4.5$$"],"problemType":"TextBox","stepTitle":"Next, calculate the width of each bar or class interval. If the data are discrete and there are not too many different values, a width that places the data values in the middle of the bar or class interval is the most convenient. Since the data consist of the numbers $$1$$, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, and the starting point is $$0.5$$, a width of one places the $$1$$ in the middle of the interval from $$0.5$$ to $$1.5$$, the $$2$$ in the middle of the interval from $$1.5$$ to $$2.5$$, the $$3$$ in the middle of the interval from $$2.5$$ to $$3.5$$, the $$4$$ in the middle of the interval from $$___$$ to $$___$$ (second blank only).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4.5$$","hints":{"DefaultPathway":[]}},{"id":"aa60f04hist2c","stepAnswer":["$$4.5$$"],"problemType":"TextBox","stepTitle":"The $$5$$ is in the middle of the interval from $$___$$ to $$___$$ (first blank).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4.5$$","hints":{"DefaultPathway":[]}},{"id":"aa60f04hist2d","stepAnswer":["$$5.5$$"],"problemType":"TextBox","stepTitle":"The $$5$$ is in the middle of the interval from $$___$$ to $$___$$ (second blank).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5.5$$","hints":{"DefaultPathway":[]}},{"id":"aa60f04hist2e","stepAnswer":["$$5.5$$"],"problemType":"TextBox","stepTitle":"The $$6$$ in the middle of the interval from $$___$$ to $$___$$ (first blank).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5.5$$","hints":{"DefaultPathway":[]}},{"id":"aa60f04hist2f","stepAnswer":["$$6.6$$"],"problemType":"TextBox","stepTitle":"The $$6$$ is in the middle of the interval from $$___$$ to $$___$$ (second blank).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6.6$$","hints":{"DefaultPathway":[]}}]},{"id":"aa60f04hist3","title":"Setting up a Histogram","body":"The following data are the number of sports played by $$50$$ student athletes. The number of sports is discrete data since sports are counted.\\\\n1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; $$1$$\\\\n2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; $$2$$\\\\n3; 3; 3; 3; 3; 3; 3; $$3$$\\\\n$$20$$ student athletes play one sport. $$22$$ student athletes play two sports. Eight student athletes play three sports.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Histograms, Frequency Polygons, and Time Series Graphs","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa60f04hist3a","stepAnswer":["$$1.5$$"],"problemType":"TextBox","stepTitle":"Fill in the blanks for the following sentence. Since the data consist of the numbers $$1$$, $$2$$, $$3$$, and the starting point is $$0.5$$, a width of one places the $$1$$ in the middle of the interval $$0.5$$ to $$___$$ ,","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1.5$$","hints":{"DefaultPathway":[]}},{"id":"aa60f04hist3b","stepAnswer":["$$1.5$$"],"problemType":"TextBox","stepTitle":"the $$2$$ in the middle of the interval from $$___$$ to $$___$$ (first blank),","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1.5$$","hints":{"DefaultPathway":[]}},{"id":"aa60f04hist3c","stepAnswer":["$$2.5$$"],"problemType":"TextBox","stepTitle":"the $$2$$ in the middle of the interval from $$___$$ to $$___$$ (second blank),","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2.5$$","hints":{"DefaultPathway":[]}},{"id":"aa60f04hist3d","stepAnswer":["$$2.5$$"],"problemType":"TextBox","stepTitle":"and the $$3$$ in the middle of the interval from $$___$$ to $$___$$ (first blank).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2.5$$","hints":{"DefaultPathway":[]}},{"id":"aa60f04hist3e","stepAnswer":["$$3.5$$"],"problemType":"TextBox","stepTitle":"and the $$3$$ in the middle of the interval from $$___$$ to $$___$$ (second blank).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3.5$$","hints":{"DefaultPathway":[]}}]},{"id":"aa60f04hist4","title":"Interpreting a Histogram","body":"Use the following information to answer the next two exercises: Suppose one hundred eleven people who shopped in a special $$t-shirt$$ store were asked the number of $$t-shirts$$ they own costing more than $19 each.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Histograms, Frequency Polygons, and Time Series Graphs","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa60f04hist4a","stepAnswer":["$$45$$"],"problemType":"MultipleChoice","stepTitle":"The percentage of people who own at most three $$t-shirts$$ costing more than $19 each is approximately:","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$45$$","choices":["$$21$$","$$59$$","$$45$$","Cannot be determined"],"hints":{"DefaultPathway":[{"id":"aa60f04hist4a-h1","type":"hint","dependencies":[],"title":"The proportion above the bar represents the proportion of people (out of $$111$$ people) who own the corresponding number of T-shirts costing more than $19 each. For instance, $$\\\\frac{5}{111}$$ above the first bar represents there are $$5$$ people (out of $$111$$ people) own exactly one T-shirt costing more than $19. At most $$3$$ T-shirt means $$1$$ T-shirt, $$2$$ T-shirt, or $$3$$ T-shirt. Therefore, we should look at the sum of proportion of the first three bars which are $$\\\\frac{5}{111}$$, $$\\\\frac{17}{111}$$ and $$\\\\frac{23}{111}$$. The sum of numerator of these three fraction tells us how many people own at most $$3$$ T-shirts costing more than $19 each. What is the sum of numerator for the fractions $$\\\\frac{5}{111}$$, $$\\\\frac{17}{111}$$ and $$\\\\frac{23}{111}$$?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa60f04hist5","title":"Interpreting a Histogram","body":"The following data are the heights (in inches to the nearest half inch) of $$100$$ male semiprofessional soccer players. The heights are continuous data, since height is measured.\\\\n60; $$60.5;$$ 61; 61; $$61.5$$\\\\n$$63.5;$$ $$63.5;$$ $$63.5$$\\\\n64; 64; 64; 64; 64; 64; 64; $$64.5;$$ $$64.5;$$ $$64.5;$$ $$64.5;$$ $$64.5;$$ $$64.5;$$ $$64.5;$$ $$64.5$$\\\\n66; 66; 66; 66; 66; 66; 66; 66; 66; 66; $$66.5;$$ $$66.5;$$ $$66.5;$$ $$66.5;$$ $$66.5;$$ $$66.5;$$ $$66.5;$$ $$66.5;$$ $$66.5;$$ $$66.5;$$ $$66.5;$$ 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; $$67.5;$$ $$67.5;$$ $$67.5;$$ $$67.5;$$ $$67.5;$$ $$67.5;$$ $$67.5$$\\\\n68; 68; 69; 69; 69; 69; 69; 69; 69; 69; 69; 69; $$69.5;$$ $$69.5;$$ $$69.5;$$ $$69.5;$$ $$69.5$$\\\\n70; 70; 70; 70; 70; 70; $$70.5;$$ $$70.5;$$ $$70.5;$$ 71; 71; $$71$$\\\\n72; 72; 72; $$72.5;$$ $$72.5;$$ 73; $$73.5$$\\\\n$$74$$","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Histograms, Frequency Polygons, and Time Series Graphs","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa60f04hist5a","stepAnswer":["$$60$$"],"problemType":"MultipleChoice","stepTitle":"What is the lowest data value in the given data set?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$60$$","choices":["$$51$$","$$45$$","$$60$$","$$74$$"],"hints":{"DefaultPathway":[{"id":"aa60f04hist5a-h1","type":"hint","dependencies":[],"title":"Arrange the Data Set From Smallest to Biggest","text":"Since the given data set is ordered from smallest to biggest, we can just take the first number (which is the smallest). So the smallest value of this data set is $$60$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa60f04hist5b","stepAnswer":["$$59.95$$"],"problemType":"MultipleChoice","stepTitle":"What is the convenient starting point for the histogram that represent the data set (with height as the x-axis)? Note that this starting point is different from the smallest data value.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$59.95$$","choices":["$$59.95$$","$$45$$","$$60$$","$$74$$"],"hints":{"DefaultPathway":[{"id":"aa60f04hist5b-h1","type":"hint","dependencies":[],"title":"Find the Convenient Starting Point","text":"The smallest data value is $$60$$. Since the data with the most decimal places has one decimal (for instance, $$61.5)$$, we want our starting point to have two decimal places. Since the numbers $$0.5$$, $$0.05$$, $$0.005$$, etc. are convenient numbers, use $$0.05$$ and subtract it from $$60$$, the smallest value, for the convenient starting point.\\\\n$$60-0.05$$ $$=$$ $$59.95$$ which is more precise than, say, $$61.5$$ by one decimal place. The starting point is, then, $$59.95$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa60f04hist5c","stepAnswer":["$$74$$"],"problemType":"MultipleChoice","stepTitle":"What is the highest data value in the given data set?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$74$$","choices":["$$51$$","$$45$$","$$60$$","$$74$$"],"hints":{"DefaultPathway":[{"id":"aa60f04hist5c-h1","type":"hint","dependencies":[],"title":"Arrange the Data Set From Smallest to Biggest","text":"Since the given data set is ordered from smallest to biggest, we can just take the last number (which is the biggest). So the smallest value of this data set is $$74$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa60f04hist5d","stepAnswer":["$$74.05$$"],"problemType":"MultipleChoice","stepTitle":"What is the ending value of the histogram that represent this data set (with heights as the x-axis)?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$74.05$$","choices":["$$59.95$$","$$45$$","$$60$$","$$74.05$$"],"hints":{"DefaultPathway":[{"id":"aa60f04hist5d-h1","type":"hint","dependencies":[],"title":"Find the Convenient Ending Point","text":"Since the data with the most decimal places has one decimal, we want our ending point to have two decimal places. Since the numbers $$0.5$$, $$0.05$$, $$0.005$$, etc. are convenient numbers, use $$0.05$$ and add it to $$74$$, the smallest value, for the convenient ending point.The largest value is $$74$$, so $$74$$ + $$0.05$$ $$=$$ $$74.05$$ is the ending value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa60f04hist6","title":"Histogram Construction","body":"Using this data set, answer the questions below.\\\\n##figure2.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Histograms, Frequency Polygons, and Time Series Graphs","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa60f04hist6a","stepAnswer":["$$0$$"],"problemType":"MultipleChoice","stepTitle":"What is the starting point of the histogram","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$0$$","choices":["$$0$$","$$0.5$$","$$-0.5$$"],"hints":{"DefaultPathway":[{"id":"aa60f04hist6a-h1","type":"hint","dependencies":[],"title":"Find the Convenient Starting Point","text":"Since the lowest data point is zero and we can not have negative number of hours, the starting point of the histogram is $$0$$. Some values in this data set fall on boundaries for the class intervals. A value is counted in a class interval if it falls on the left boundary.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa60f04hist6b","stepAnswer":["Number of Hours"],"problemType":"MultipleChoice","stepTitle":"Checking x-axis and y-axis","stepBody":"What is the x-axis of the histogram that represents the given data points?","answerType":"string","variabilization":{},"choices":["Number of Hours","Number of Students"],"hints":{"DefaultPathway":[]}},{"id":"aa60f04hist6c","stepAnswer":["Number of Students"],"problemType":"MultipleChoice","stepTitle":"Checking x-axis and y-axis","stepBody":"What is the y-axis of the histogram that represents the given data points?","answerType":"string","variabilization":{},"choices":["Number of Hours","Number of Students"],"hints":{"DefaultPathway":[]}},{"id":"aa60f04hist6d","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Does the given histogram correctly represent the given data set?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"aa60f04hist6d-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":[],"title":"Check for Data Points","text":"The x-axis of the histogram represents the number of hours and the $$y$$ axis represents the number of students, we only need to check for the frequency of the corresponding categories. Does the histogram correctly represent the given data points?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"],"subHints":[{"id":"aa60f04hist6d-h1-s1","type":"hint","dependencies":[],"title":"Check for Data Points","text":"In other words, check if each category has the correct number of data points. For instance, there are only $$2$$ data points between $$0$$ and $$5$$ shown in the histogram. In the given data set, there are also $$2$$ data points between $$0$$ and $$5$$ which are $$0$$ and $$2.25$$. Check for each category on the histogram.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"aa60f04hist7","title":"Histogram Construction","body":"The following data represent the number of employees at various restaurants in New York City. Using this data, answer the questions below.\\\\n22; 35; 15; 26; 40; 28; 18; 20; 25; 34; 39; 42; 24; 22; 19; 27; 22; 34; 40; 20; 38; and $$28$$\\\\n\\\\n##figure2.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Histograms, Frequency Polygons, and Time Series Graphs","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa60f04hist7a","stepAnswer":["number of employees"],"problemType":"MultipleChoice","stepTitle":"Checking x-axis and y-axis","stepBody":"What is the x-axis of the histogram that represents the given data points?","answerType":"string","variabilization":{},"choices":["number of employees","number of restaurants"],"hints":{"DefaultPathway":[]}},{"id":"aa60f04hist7b","stepAnswer":["number of restaurants"],"problemType":"MultipleChoice","stepTitle":"Checking x-axis and y-axis","stepBody":"What is the y-axis of the histogram that represents the given data points?","answerType":"string","variabilization":{},"choices":["number of employees","number of restaurants"],"hints":{"DefaultPathway":[]}},{"id":"aa60f04hist7c","stepAnswer":["$$10$$"],"problemType":"MultipleChoice","stepTitle":"Starting point of the Histogram","stepBody":"Which one is the best starting point of the histogram that represents the given data points?","answerType":"string","variabilization":{},"answerLatex":"$$10$$","choices":["$$10$$","$$16$$","$$17$$","$$19$$"],"hints":{"DefaultPathway":[{"id":"aa60f04hist7c-h1","type":"hint","dependencies":[],"title":"Starting point of the Histogram","text":"$$10$$ is the best starting point amongst the four points. If $$16$$, $$17$$ or $$19$$ is the starting point, the data value of $$15$$ will be excluded from the histogram.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa60f04hist7d","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Does the given histogram correctly represent the given data set?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"aa60f04hist7d-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":[],"title":"Check for Data Points","text":"The x-axis of the histogram represents the number of employees and the $$y$$ axis represents the number of restaurants, we only need to check for the frequency of the corresponding categories. Does the histogram correctly represent the given data points?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"],"subHints":[{"id":"aa60f04hist7d-h1-s1","type":"hint","dependencies":[],"title":"Check for Data Points","text":"In other words, check if each category has the correct number of data points. For instance, there are only $$2$$ data points between $$10$$ and $$19$$ shown in the histogram. In the given data set, there are also $$2$$ data points between $$10$$ and $$19$$ which are $$15$$ and $$18$$. Check for each category on the histogram. Note that we use left endpoint convention on the histogram. In other words, the left endpoint of the category will be included and the right endpoint of the category is excluded. The category of $$19-28$$ contains the points between $$19$$ and $$28$$, including $$19$$ and excluding $$28$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"aa60f04hist8","title":"Finding Relative Frequency","body":"Often, cruise ships conduct all on-board transactions, with the exception of gambling, on a cashless basis. At the end of the cruise, guests pay one bill that covers all onboard transactions. Suppose that $$60$$ single travelers and $$70$$ couples were surveyed as to their on-board bills for a seven-day cruise from Los Angeles to the Mexican Riviera. Following is a summary of the bills for each group.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Histograms, Frequency Polygons, and Time Series Graphs","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa60f04hist8a","stepAnswer":["$$0.08$$"],"problemType":"MultipleChoice","stepTitle":"What is the relative frequency for the category amount $$51-100$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$0.08$$","choices":["$$0.08$$","$$0.17$$","$$0.25$$","$$5$$"],"hints":{"DefaultPathway":[{"id":"aa60f04hist8a-h1","type":"hint","dependencies":[],"title":"Find the Relative Frequency","text":"The relative frequency is equal to the frequency for an observed value of the data divided by the total number of data values in the sample. (Remember, frequency is defined as the number of times an answer occurs.) If:\\\\nf $$=$$ frequency\\\\n$$n$$ $$=$$ total number of data values (or the sum of the individual frequencies), and\\\\nRF $$=$$ relative frequency,\\\\nthen: $$RF=$$ $$\\\\frac{f}{n}$$\\\\nFor example, if three students in Mr. Ahab\'s English class of $$40$$ students received from 90% to 100%, then, f $$=$$ $$3$$, $$n$$ $$=$$ $$40$$, and $$RF=$$ $$\\\\frac{3}{40}=0.075$$\\\\n","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa60f04hist8b","stepAnswer":["$$0.17$$"],"problemType":"MultipleChoice","stepTitle":"What is the relative frequency for the category amount $$101-150$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$0.17$$","choices":["$$0.08$$","$$0.17$$","$$0.25$$","$$10$$"],"hints":{"DefaultPathway":[{"id":"aa60f04hist8b-h1","type":"hint","dependencies":[],"title":"Find the Relative Frequency","text":"The relative frequency is equal to the frequency for an observed value of the data divided by the total number of data values in the sample. (Remember, frequency is defined as the number of times an answer occurs.) If:\\\\nf $$=$$ frequency\\\\n$$n$$ $$=$$ total number of data values (or the sum of the individual frequencies), and\\\\nRF $$=$$ relative frequency,\\\\nthen: $$RF=$$ $$\\\\frac{f}{n}$$\\\\nFor example, if three students in Mr. Ahab\'s English class of $$40$$ students received from 90% to 100%, then, f $$=$$ $$3$$, $$n$$ $$=$$ $$40$$, and $$RF=$$ $$\\\\frac{3}{40}=0.075$$\\\\n","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa60f04hist8c","stepAnswer":["$$0.25$$"],"problemType":"MultipleChoice","stepTitle":"What is the relative frequency for the category amount $$151-200$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$0.25$$","choices":["$$0.08$$","$$0.17$$","$$0.25$$","$$15$$"],"hints":{"DefaultPathway":[{"id":"aa60f04hist8c-h1","type":"hint","dependencies":[],"title":"Find the Relative Frequency","text":"The relative frequency is equal to the frequency for an observed value of the data divided by the total number of data values in the sample. (Remember, frequency is defined as the number of times an answer occurs.) If:\\\\nf $$=$$ frequency\\\\n$$n$$ $$=$$ total number of data values (or the sum of the individual frequencies), and\\\\nRF $$=$$ relative frequency,\\\\nthen: $$RF=$$ $$\\\\frac{f}{n}$$\\\\nFor example, if three students in Mr. Ahab\'s English class of $$40$$ students received from 90% to 100%, then, f $$=$$ $$3$$, $$n$$ $$=$$ $$40$$, and $$RF=$$ $$\\\\frac{3}{40}=0.075$$\\\\n","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa60f04hist8d","stepAnswer":["$$0.25$$"],"problemType":"MultipleChoice","stepTitle":"What is the relative frequency for the category amount $$201-250$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$0.25$$","choices":["$$0.08$$","$$0.17$$","$$0.25$$","$$15$$"],"hints":{"DefaultPathway":[{"id":"aa60f04hist8d-h1","type":"hint","dependencies":[],"title":"Find the Relative Frequency","text":"The relative frequency is equal to the frequency for an observed value of the data divided by the total number of data values in the sample. (Remember, frequency is defined as the number of times an answer occurs.) If:\\\\nf $$=$$ frequency\\\\n$$n$$ $$=$$ total number of data values (or the sum of the individual frequencies), and\\\\nRF $$=$$ relative frequency,\\\\nthen: $$RF=$$ $$\\\\frac{f}{n}$$\\\\nFor example, if three students in Mr. Ahab\'s English class of $$40$$ students received from 90% to 100%, then, f $$=$$ $$3$$, $$n$$ $$=$$ $$40$$, and $$RF=$$ $$\\\\frac{3}{40}=0.075$$\\\\n","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa60f04hist8e","stepAnswer":["$$0.17$$"],"problemType":"MultipleChoice","stepTitle":"What is the relative frequency for the category amount $$251-300$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$0.17$$","choices":["$$0.08$$","$$0.17$$","$$0.25$$","$$10$$"],"hints":{"DefaultPathway":[{"id":"aa60f04hist8e-h1","type":"hint","dependencies":[],"title":"Find the Relative Frequency","text":"The relative frequency is equal to the frequency for an observed value of the data divided by the total number of data values in the sample. (Remember, frequency is defined as the number of times an answer occurs.) If:\\\\nf $$=$$ frequency\\\\n$$n$$ $$=$$ total number of data values (or the sum of the individual frequencies), and\\\\nRF $$=$$ relative frequency,\\\\nthen: $$RF=$$ $$\\\\frac{f}{n}$$\\\\nFor example, if three students in Mr. Ahab\'s English class of $$40$$ students received from 90% to 100%, then, f $$=$$ $$3$$, $$n$$ $$=$$ $$40$$, and $$RF=$$ $$\\\\frac{3}{40}=0.075$$\\\\n","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa60f04hist8f","stepAnswer":["$$0.08$$"],"problemType":"MultipleChoice","stepTitle":"What is the relative frequency for the category amount $$301-350$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$0.08$$","choices":["$$0.08$$","$$0.17$$","$$0.25$$","$$5$$"],"hints":{"DefaultPathway":[]}}]},{"id":"aa60f04hist9","title":"Finding Relative Frequency","body":"Suppose that three book publishers were interested in the number of fiction paperbacks adult consumers purchase per month. Each publisher conducted a survey. In the survey, adult consumers were asked the number of fiction paperbacks they had purchased the previous month. The results are as follows:\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Histograms, Frequency Polygons, and Time Series Graphs","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa60f04hist9a","stepAnswer":["$$0.15$$"],"problemType":"MultipleChoice","stepTitle":"What is the relative frequency for the category of $$0$$ book?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$0.15$$","choices":["$$0.15$$","$$0.18$$","$$0.24$$","$$0.12$$","$$0.088$$","$$0.03$$","$$10$$"],"hints":{"DefaultPathway":[{"id":"aa60f04hist9a-h1","type":"hint","dependencies":[],"title":"Find the Relative Frequency","text":"The relative frequency is equal to the frequency for an observed value of the data divided by the total number of data values in the sample. (Remember, frequency is defined as the number of times an answer occurs.) If:\\\\nf $$=$$ frequency\\\\n$$n$$ $$=$$ total number of data values (or the sum of the individual frequencies), and\\\\nRF $$=$$ relative frequency,\\\\nthen: $$RF=$$ $$\\\\frac{f}{n}$$\\\\nFor example, if three students in Mr. Ahab\'s English class of $$40$$ students received from 90% to 100%, then, f $$=$$ $$3$$, $$n$$ $$=$$ $$40$$, and $$RF=$$ $$\\\\frac{3}{40}=0.075$$\\\\n","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa60f04hist9b","stepAnswer":["$$0.18$$"],"problemType":"MultipleChoice","stepTitle":"What is the relative frequency for the category of $$1$$ book?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$0.18$$","choices":["$$0.15$$","$$0.18$$","$$0.24$$","$$0.12$$","$$0.088$$","$$0.03$$","$$12$$"],"hints":{"DefaultPathway":[{"id":"aa60f04hist9b-h1","type":"hint","dependencies":[],"title":"Find the Relative Frequency","text":"The relative frequency is equal to the frequency for an observed value of the data divided by the total number of data values in the sample. (Remember, frequency is defined as the number of times an answer occurs.) If:\\\\nf $$=$$ frequency\\\\n$$n$$ $$=$$ total number of data values (or the sum of the individual frequencies), and\\\\nRF $$=$$ relative frequency,\\\\nthen: $$RF=$$ $$\\\\frac{f}{n}$$\\\\nFor example, if three students in Mr. Ahab\'s English class of $$40$$ students received from 90% to 100%, then, f $$=$$ $$3$$, $$n$$ $$=$$ $$40$$, and $$RF=$$ $$\\\\frac{3}{40}=0.075$$\\\\n","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa60f04hist9c","stepAnswer":["$$0.24$$"],"problemType":"MultipleChoice","stepTitle":"What is the relative frequency for the category of $$2$$ book?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$0.24$$","choices":["$$0.15$$","$$0.18$$","$$0.24$$","$$0.12$$","$$0.088$$","$$0.03$$","$$16$$"],"hints":{"DefaultPathway":[{"id":"aa60f04hist9c-h1","type":"hint","dependencies":[],"title":"Find the Relative Frequency","text":"The relative frequency is equal to the frequency for an observed value of the data divided by the total number of data values in the sample. (Remember, frequency is defined as the number of times an answer occurs.) If:\\\\nf $$=$$ frequency\\\\n$$n$$ $$=$$ total number of data values (or the sum of the individual frequencies), and\\\\nRF $$=$$ relative frequency,\\\\nthen: $$RF=$$ $$\\\\frac{f}{n}$$\\\\nFor example, if three students in Mr. Ahab\'s English class of $$40$$ students received from 90% to 100%, then, f $$=$$ $$3$$, $$n$$ $$=$$ $$40$$, and $$RF=$$ $$\\\\frac{3}{40}=0.075$$\\\\n","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa60f04hist9d","stepAnswer":["$$0.18$$"],"problemType":"MultipleChoice","stepTitle":"What is the relative frequency for the category of $$3$$ book?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$0.18$$","choices":["$$0.15$$","$$0.18$$","$$0.24$$","$$0.12$$","$$0.088$$","$$0.03$$","$$12$$"],"hints":{"DefaultPathway":[{"id":"aa60f04hist9d-h1","type":"hint","dependencies":[],"title":"Find the Relative Frequency","text":"The relative frequency is equal to the frequency for an observed value of the data divided by the total number of data values in the sample. (Remember, frequency is defined as the number of times an answer occurs.) If:\\\\nf $$=$$ frequency\\\\n$$n$$ $$=$$ total number of data values (or the sum of the individual frequencies), and\\\\nRF $$=$$ relative frequency,\\\\nthen: $$RF=$$ $$\\\\frac{f}{n}$$\\\\nFor example, if three students in Mr. Ahab\'s English class of $$40$$ students received from 90% to 100%, then, f $$=$$ $$3$$, $$n$$ $$=$$ $$40$$, and $$RF=$$ $$\\\\frac{3}{40}=0.075$$\\\\n","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa60f04hist9e","stepAnswer":["$$0.12$$"],"problemType":"MultipleChoice","stepTitle":"What is the relative frequency for the category of $$4$$ book?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$0.12$$","choices":["$$0.15$$","$$0.18$$","$$0.24$$","$$0.12$$","$$0.088$$","$$0.03$$","$$8$$"],"hints":{"DefaultPathway":[{"id":"aa60f04hist9e-h1","type":"hint","dependencies":[],"title":"Find the Relative Frequency","text":"The relative frequency is equal to the frequency for an observed value of the data divided by the total number of data values in the sample. (Remember, frequency is defined as the number of times an answer occurs.) If:\\\\nf $$=$$ frequency\\\\n$$n$$ $$=$$ total number of data values (or the sum of the individual frequencies), and\\\\nRF $$=$$ relative frequency,\\\\nthen: $$RF=$$ $$\\\\frac{f}{n}$$\\\\nFor example, if three students in Mr. Ahab\'s English class of $$40$$ students received from 90% to 100%, then, f $$=$$ $$3$$, $$n$$ $$=$$ $$40$$, and $$RF=$$ $$\\\\frac{3}{40}=0.075$$\\\\n","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa60f04hist9f","stepAnswer":["$$0.088$$"],"problemType":"MultipleChoice","stepTitle":"What is the relative frequency for the category of $$5$$ book?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$0.088$$","choices":["$$0.15$$","$$0.18$$","$$0.24$$","$$0.12$$","$$0.088$$","$$0.03$$","$$6$$"],"hints":{"DefaultPathway":[{"id":"aa60f04hist9f-h1","type":"hint","dependencies":[],"title":"Find the Relative Frequency","text":"The relative frequency is equal to the frequency for an observed value of the data divided by the total number of data values in the sample. (Remember, frequency is defined as the number of times an answer occurs.) If:\\\\nf $$=$$ frequency\\\\n$$n$$ $$=$$ total number of data values (or the sum of the individual frequencies), and\\\\nRF $$=$$ relative frequency,\\\\nthen: $$RF=$$ $$\\\\frac{f}{n}$$\\\\nFor example, if three students in Mr. Ahab\'s English class of $$40$$ students received from 90% to 100%, then, f $$=$$ $$3$$, $$n$$ $$=$$ $$40$$, and $$RF=$$ $$\\\\frac{3}{40}=0.075$$\\\\n","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa60f04hist9g","stepAnswer":["$$0.03$$"],"problemType":"MultipleChoice","stepTitle":"What is the relative frequency for the category of $$6$$ book?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$0.03$$","choices":["$$0.15$$","$$0.18$$","$$0.24$$","$$0.12$$","$$0.088$$","$$0.03$$","$$2$$"],"hints":{"DefaultPathway":[{"id":"aa60f04hist9g-h1","type":"hint","dependencies":[],"title":"Find the Relative Frequency","text":"The relative frequency is equal to the frequency for an observed value of the data divided by the total number of data values in the sample. (Remember, frequency is defined as the number of times an answer occurs.) If:\\\\nf $$=$$ frequency\\\\n$$n$$ $$=$$ total number of data values (or the sum of the individual frequencies), and\\\\nRF $$=$$ relative frequency,\\\\nthen: $$RF=$$ $$\\\\frac{f}{n}$$\\\\nFor example, if three students in Mr. Ahab\'s English class of $$40$$ students received from 90% to 100%, then, f $$=$$ $$3$$, $$n$$ $$=$$ $$40$$, and $$RF=$$ $$\\\\frac{3}{40}=0.075$$\\\\n","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa60f04hist9h","stepAnswer":["$$0.03$$"],"problemType":"MultipleChoice","stepTitle":"What is the relative frequency for the category of $$8$$ book?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$0.03$$","choices":["$$0.15$$","$$0.18$$","$$0.24$$","$$0.12$$","$$0.088$$","$$0.03$$","$$2$$"],"hints":{"DefaultPathway":[{"id":"aa60f04hist9h-h1","type":"hint","dependencies":[],"title":"Find the Relative Frequency","text":"The relative frequency is equal to the frequency for an observed value of the data divided by the total number of data values in the sample. (Remember, frequency is defined as the number of times an answer occurs.) If:\\\\nf $$=$$ frequency\\\\n$$n$$ $$=$$ total number of data values (or the sum of the individual frequencies), and\\\\nRF $$=$$ relative frequency,\\\\nthen: $$RF=$$ $$\\\\frac{f}{n}$$\\\\nFor example, if three students in Mr. Ahab\'s English class of $$40$$ students received from 90% to 100%, then, f $$=$$ $$3$$, $$n$$ $$=$$ $$40$$, and $$RF=$$ $$\\\\frac{3}{40}=0.075$$\\\\n","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa7aa5dsampling1","title":"Understanding the Different Types of Sampling","body":"Determine the type of sampling used (simple random, stratified, systematic, cluster, or convenience).","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Sampling Experiment","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa7aa5dsampling1a","stepAnswer":["Stratified"],"problemType":"MultipleChoice","stepTitle":"A soccer coach selects six players from a group of boys aged eight to ten, seven players from a group of boys aged $$11$$ to $$12$$, and three players from a group of boys aged $$13$$ to $$14$$ to form a recreational soccer team.","stepBody":"","answerType":"string","variabilization":{},"choices":["Simple Random","Stratified","Systematic","Cluster","Convenience"],"hints":{"DefaultPathway":[{"id":"aa7aa5dsampling1a-h1","type":"hint","dependencies":[],"title":"Divide Up the Population","text":"There are two types of sampling that divide the population into groups of people: stratified and cluster. The differences between the two are that stratified sampling takes a proportionate number of subjects from each stratum to include into the sample while clustering usually means only a few clusters are selected to represent the entire population.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa7aa5dsampling1a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Stratified"],"dependencies":["aa7aa5dsampling1a-h1"],"title":"Stratified versus Cluster","text":"Does this soccer coach want to pick players from each group that he\'s created based on age? Or does he only want one or two groups to be represented. If the soccer coach wants people from each age group to be included in the recreational team, select stratified. If the coach only wants one or two groups to be represented, select cluster.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Stratified","Cluster"]}]}}]},{"id":"aa7aa5dsampling10","title":"Understanding the Different Types of Sampling","body":"Name the sampling method used in the following situation:","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Sampling Experiment","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa7aa5dsampling10a","stepAnswer":["Cluster"],"problemType":"MultipleChoice","stepTitle":"A teacher wants to know if her students are doing homework, so she randomly selects rows two and five and then calls on all students in row two and all students in row five to present the solutions to homework problems to the class.","stepBody":"","answerType":"string","variabilization":{},"choices":["Simple Random","Stratified","Systematic","Cluster","Convenience"],"hints":{"DefaultPathway":[{"id":"aa7aa5dsampling10a-h1","type":"hint","dependencies":[],"title":"Divide Up the Population","text":"There are two types of sampling that divide the population into groups of people: stratified and cluster. The difference between the two are that stratified sampling takes a proportionate number of subjects from each stratum to include into the sample while clustering usually means only a few clusters are selected to represent the entire population.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa7aa5dsampling10a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Cluster"],"dependencies":["aa7aa5dsampling10a-h1"],"title":"Stratified versus Cluster","text":"The teacher wants to call on all students in row $$5$$ and row $$2$$. Therefore, the different groups are the different rows. Note here that the teacher only call on students from row $$5$$ and row $$2$$. Does this mean that the teacher wants to call on proportion of students from each row (stratified) or that the teacher wants to call on all students in selected rows (cluster)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Stratified","Cluster"]}]}}]},{"id":"aa7aa5dsampling11","title":"Understanding the Different Types of Sampling","body":"Name the sampling method used in the following situation:","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Sampling Experiment","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa7aa5dsampling11a","stepAnswer":["Stratified"],"problemType":"MultipleChoice","stepTitle":"The marketing manager for an electronics chain store wants information about the ages of its customers. Over the next two weeks, at each store location, $$100$$ randomly selected customers are given questionnaires to fill out asking for information about age, as well as about other variables of interest.","stepBody":"","answerType":"string","variabilization":{},"choices":["Simple Random","Stratified","Systematic","Cluster","Convenience"],"hints":{"DefaultPathway":[{"id":"aa7aa5dsampling11a-h1","type":"hint","dependencies":[],"title":"Divide Up the Population","text":"There are two types of sampling that divide the population into groups of people: stratified and cluster. The differences between the two are that stratified sampling takes a proportionate number of subjects from each stratum to include into the sample while clustering usually means only a few clusters are selected to represent the entire population.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa7aa5dsampling11a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Stratified"],"dependencies":["aa7aa5dsampling11a-h1"],"title":"Stratified versus Cluster","text":"Does the manager want to pick customers from each store? Or does she only want customers from one store to be represented. If the manager wants customers from each store to be included in the sample, select stratified. If the manager only wants one or two stores to be represented, select cluster.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Stratified","Cluster"]}]}}]},{"id":"aa7aa5dsampling12","title":"Understanding the Different Types of Sampling","body":"Name the sampling method used in the following situation:","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Sampling Experiment","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa7aa5dsampling12a","stepAnswer":["Systematic"],"problemType":"MultipleChoice","stepTitle":"The librarian at a public library wants to determine what proportion of the library users are children. The librarian has a tally sheet on which she marks whether books are checked out by an adult or a child. She records this data for every fourth patron who checks out books.","stepBody":"","answerType":"string","variabilization":{},"choices":["Simple Random","Stratified","Systematic","Cluster","Convenience"],"hints":{"DefaultPathway":[{"id":"aa7aa5dsampling12a-h1","type":"hint","dependencies":[],"title":"Population is not Divided","text":"The population is not divided into different $$\\\\frac{groups}{clusters}$$. Therefore, we know that this is not stratified or clustered sampling and will now focus on simple random, systematic, and convenience sampling.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa7aa5dsampling12a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Systematic"],"dependencies":["aa7aa5dsampling12a-h1"],"title":"Choosing the nth patron","text":"Note that librarian records the data for every fourth patron who checks out books.We see that this is a methodology that is not based on convenience (the librarian doesn\'t just pick the most easily available data points). Therefore, it must either be systematic or simple random from process of elimination. The definition of a systematic sample is randomly selecting a starting point and take the nth piece of data from the population. The definition of simple random sampling is fairly selecting a group of size $$n$$ from the population so every individual has the same chance of being selected. Which one sounds the most like the librarian\'s selection method?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Systematic","Simple Random"]}]}}]},{"id":"aa7aa5dsampling13","title":"Understanding the Different Types of Sampling","body":"Name the sampling method used in the following situation:","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Sampling Experiment","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa7aa5dsampling13a","stepAnswer":["Simple Random"],"problemType":"MultipleChoice","stepTitle":"A political party wants to know the reaction of voters to a debate between the candidates. The day after the debate, the party\u2019s polling staff calls 1,200 randomly selected phone numbers. If a registered voter answers the phone or is available to come to the phone, that registered voter is asked whom he or she intends to vote for and whether the debate changed his or her opinion of the candidates.","stepBody":"","answerType":"string","variabilization":{},"choices":["Simple Random","Stratified","Systematic","Cluster","Convenience"],"hints":{"DefaultPathway":[{"id":"aa7aa5dsampling13a-h1","type":"hint","dependencies":[],"title":"Population is not Divided","text":"The population is not divided into different $$\\\\frac{groups}{clusters}$$. Therefore, we know that this is not stratified or clustered sampling and will now focus on simple random, systematic, and convenience sampling.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa7aa5dsampling13a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Simple Random"],"dependencies":["aa7aa5dsampling13a-h1"],"title":"Random Selection","text":"Note that the polling staff calls 1,200 randomly selected phone numbers. We see that this is a methodology that is not based on convenience (the polling staff doesn\'t just pick the most easily available data points). Therefore, it must either be systematic or simple random from process of elimination. The definition of a systematic sample is randomly selecting a starting point and take the nth piece of data from the population. The definition of simple random sampling is fairly selecting a group of size $$n$$ from the population so every individual has the same chance of being selected. Which of the two methods sounds the most like the high school counselor\'s selection method?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Systematic","Simple Random"]}]}}]},{"id":"aa7aa5dsampling2","title":"Understanding the Different Types of Sampling","body":"Determine the type of sampling used (simple random, stratified, systematic, cluster, or convenience).","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Sampling Experiment","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa7aa5dsampling2a","stepAnswer":["Cluster"],"problemType":"MultipleChoice","stepTitle":"A pollster interviews all human resource personnel in five different high tech companies.","stepBody":"","answerType":"string","variabilization":{},"choices":["Simple Random","Stratified","Systematic","Cluster","Convenience"],"hints":{"DefaultPathway":[{"id":"aa7aa5dsampling2a-h1","type":"hint","dependencies":[],"title":"Divide Up the Population","text":"There are two types of sampling that divide the population into groups of people: stratified and cluster. The difference between the two are that stratified sampling takes a proportionate number of subjects from each stratum to include into the sample while clustering usually means only a few clusters are selected to represent the entire population.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa7aa5dsampling2a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Cluster"],"dependencies":["aa7aa5dsampling2a-h1"],"title":"Stratified versus Cluster","text":"The pollster wants to interview all the human resource personnel in five different high tech companies. Therefore, the different groups are the human resource personnels in all the different high tech companies. Note here that the pollster only wants to interview personnel in five of the tech companies. Does this mean that the pollster wants to interview a proportion of human resource personnel from each of all the tech companies (stratified) or that the pollster wants to interview all the human resource personnel from a select number of companies (cluster)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Stratified","Cluster"]}]}}]},{"id":"aa7aa5dsampling3","title":"Understanding the Different Types of Sampling","body":"Determine the type of sampling used (simple random, stratified, systematic, cluster, or convenience).","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Sampling Experiment","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa7aa5dsampling3a","stepAnswer":["Stratified"],"problemType":"MultipleChoice","stepTitle":"A high school educational researcher interviews $$50$$ high school female teachers and $$50$$ high school male teachers.","stepBody":"","answerType":"string","variabilization":{},"choices":["Simple Random","Stratified","Systematic","Cluster","Convenience"],"hints":{"DefaultPathway":[{"id":"aa7aa5dsampling3a-h1","type":"hint","dependencies":[],"title":"Divide Up the Population","text":"There are two types of sampling that divide the population into groups of people: stratified and cluster. The difference between the two are that stratified sampling takes a proportionate number of subjects from each stratum to include into the sample while clustering usually means only a few clusters are selected to represent the entire population.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa7aa5dsampling3a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Stratified"],"dependencies":["aa7aa5dsampling3a-h1"],"title":"Stratified versus Cluster","text":"The high school educational researcher wants to interview $$50$$ high school female teachers and $$50$$ high school male teachers. Therefore, you know that the two separate groups are high school female teachers and high school male teachers. To decide whether this is stratified or cluster sampling, we must think about whether every group is represented in the sample. Choose STRATIFIED if a proportion of each separate group is included in the final sample; choose CLUSTER if not all the groups are included in the final sample, but every member of the selected groups is included.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Stratified","Cluster"]}]}}]},{"id":"aa7aa5dsampling4","title":"Understanding the Different Types of Sampling","body":"Determine the type of sampling used (simple random, stratified, systematic, cluster, or convenience).","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Sampling Experiment","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa7aa5dsampling4a","stepAnswer":["Systematic"],"problemType":"MultipleChoice","stepTitle":"A medical researcher interviews every third cancer patient from a list of cancer patients at a local hospital.","stepBody":"","answerType":"string","variabilization":{},"choices":["Simple Random","Stratified","Systematic","Cluster","Convenience"],"hints":{"DefaultPathway":[{"id":"aa7aa5dsampling4a-h1","type":"hint","dependencies":[],"title":"Population is not Divided","text":"The population is not divided into different $$\\\\frac{groups}{clusters}$$. Therefore, we know that this is not stratified or clustered sampling and will now focus on simple random, systematic, and convenience sampling.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa7aa5dsampling4a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Systematic"],"dependencies":["aa7aa5dsampling4a-h1"],"title":"Choosing the nth Person","text":"Note that the medical researcher interviews every third cancer patient from a list of cancer patients at a local hospital. We see that this is a methodology that is not based on convenience (the researcher doesn\'t just pick the most easily available data points). Therefore, it must either be systematic or simple random from process of elimination. The definition of a systematic sample is randomly selecting a starting point and take the nth piece of data from the population. The definition of simple random sampling is fairly selecting a group of size $$n$$ from the population so every individual has the same chance of being selected. Which one sounds the most like the medical researcher\'s selection method?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Systematic","Simple Random"]}]}}]},{"id":"aa7aa5dsampling5","title":"Understanding the Different Types of Sampling","body":"Determine the type of sampling used (simple random, stratified, systematic, cluster, or convenience).","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Sampling Experiment","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa7aa5dsampling5a","stepAnswer":["Simple Random"],"problemType":"MultipleChoice","stepTitle":"A high school counselor uses a computer to generate $$50$$ random numbers and then picks students whose names correspond to the numbers.","stepBody":"","answerType":"string","variabilization":{},"choices":["Simple Random","Stratified","Systematic","Cluster","Convenience"],"hints":{"DefaultPathway":[{"id":"aa7aa5dsampling5a-h1","type":"hint","dependencies":[],"title":"Population is not Divided","text":"The population is not divided into different $$\\\\frac{groups}{clusters}$$. Therefore, we know that this is not stratified or clustered sampling and will now focus on simple random, systematic, and convenience sampling.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa7aa5dsampling5a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Simple Random"],"dependencies":["aa7aa5dsampling5a-h1"],"title":"Random Selection","text":"Note that the high school counselor uses a computer to generate $$50$$ random numbers. We see that this is a methodology that is not based on convenience (the high school counselor doesn\'t just pick the most easily available data points). Therefore, it must either be systematic or simple random from process of elimination. The definition of a systematic sample is randomly selecting a starting point and take the nth piece of data from the population. The definition of simple random sampling is fairly selecting a group of size $$n$$ from the population so every individual has the same chance of being selected. We see that the high school counselor assigns students randomly to numbers and fairly randomly generated $$50$$ which each student having an equal chance. Which of the two methods sounds the most like the high school counselor\'s selection method?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Systematic","Simple Random"]}]}}]},{"id":"aa7aa5dsampling6","title":"Understanding the Different Types of Sampling","body":"Determine the type of sampling used (simple random, stratified, systematic, cluster, or convenience).","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Sampling Experiment","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa7aa5dsampling6a","stepAnswer":["Convenience"],"problemType":"MultipleChoice","stepTitle":"A student interviews classmates in his algebra class to determine how many pairs of jeans a student owns, on the average.","stepBody":"","answerType":"string","variabilization":{},"choices":["Simple Random","Stratified","Systematic","Cluster","Convenience"],"hints":{"DefaultPathway":[{"id":"aa7aa5dsampling6a-h1","type":"hint","dependencies":[],"title":"Population is not Divided","text":"The population is not divided into different $$\\\\frac{groups}{clusters}$$. Therefore, we know that this is not stratified or clustered sampling and will now focus on simple random, systematic, and convenience sampling.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa7aa5dsampling6a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["aa7aa5dsampling6a-h1"],"title":"Interviewing Classmates not at Random","text":"We see that the student interviews classmates in his algebra class. We know that this is not a simple random sample because not all students have an equal chance of being selected. Since the student doesn\'t have to perform much more work than just asking students in their class and is selecting students out of their proximity to them (in the same algebra class), is this a convenience sample?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"aa7aa5dsampling7","title":"Understanding the Different Types of Sampling","body":"Determine the type of sampling used (simple random, stratified, systematic, cluster, or convenience).","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Sampling Experiment","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa7aa5dsampling7a","stepAnswer":["Stratified"],"problemType":"MultipleChoice","stepTitle":"The instructor takes her sample by gathering data on five randomly selected students from each Lake Tahoe Community College math class.","stepBody":"","answerType":"string","variabilization":{},"choices":["Simple Random","Stratified","Systematic","Cluster","Convenience"],"hints":{"DefaultPathway":[{"id":"aa7aa5dsampling7a-h1","type":"hint","dependencies":[],"title":"Divide Up the Population","text":"There are two types of sampling that divide the population into groups of people: stratified and cluster. The differences between the two are that stratified sampling takes a proportionate number of subjects from each stratum to include into the sample while clustering usually means only a few clusters are selected to represent the entire population.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa7aa5dsampling7a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Stratified"],"dependencies":["aa7aa5dsampling7a-h1"],"title":"Stratified versus Cluster","text":"Does the instructor want to pick students from each Lake Tahoe Community College math class? Or does she only want one or two math classes to be represented. If the instructor wants student from each Lake Tahoe Community College math course to be included in the sample, select stratified. If the instructor only wants one or two math classes to be represented, select cluster.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Stratified","Cluster"]}]}}]},{"id":"aa7aa5dsampling8","title":"Understanding the Different Types of Sampling","body":"Determine the type of sampling used (simple random, stratified, systematic, cluster, or convenience).","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Sampling Experiment","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa7aa5dsampling8a","stepAnswer":["Systematic"],"problemType":"MultipleChoice","stepTitle":"A study was done to determine the age, number of times per week, and the duration (amount of time) of residents using a local park in San Jose. The first house in the neighborhood around the park was selected randomly and then every eighth house in the neighborhood around the park was interviewed. The sampling method was:","stepBody":"","answerType":"string","variabilization":{},"choices":["Simple Random","Stratified","Systematic","Cluster","Convenience"],"hints":{"DefaultPathway":[{"id":"aa7aa5dsampling8a-h1","type":"hint","dependencies":[],"title":"Population is not Divided","text":"The population is not divided into different $$\\\\frac{groups}{clusters}$$. Therefore, we know that this is not stratified or clustered sampling and will now focus on simple random, systematic, and convenience sampling.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa7aa5dsampling8a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Systematic"],"dependencies":["aa7aa5dsampling8a-h1"],"title":"Choosing the nth House","text":"Note that the researcher interviews every eighth house in the neighborhood around the park. We see that this is a methodology that is not based on convenience (the researcher doesn\'t just pick the most easily available data points). Therefore, it must either be systematic or simple random from process of elimination. The definition of a systematic sample is randomly selecting a starting point and take the nth piece of data from the population. The definition of simple random sampling is fairly selecting a group of size $$n$$ from the population so every individual has the same chance of being selected. Which one sounds the most like the researcher\'s selection method?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Systematic","Simple Random"]}]}}]},{"id":"aa7aa5dsampling9","title":"Understanding the Different Types of Sampling","body":"Name the sampling method used in the following situation:","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Sampling Experiment","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa7aa5dsampling9a","stepAnswer":["Convenience"],"problemType":"MultipleChoice","stepTitle":"A woman in the airport is handing out questionnaires to travelers asking them to evaluate the airport\u2019s service. She does not ask travelers who are hurrying through the airport with their hands full of luggage, but instead asks all travelers who are sitting near gates and not taking naps while they wait.","stepBody":"","answerType":"string","variabilization":{},"choices":["Simple Random","Stratified","Systematic","Cluster","Convenience"],"hints":{"DefaultPathway":[{"id":"aa7aa5dsampling9a-h1","type":"hint","dependencies":[],"title":"Population is not Divided","text":"The population is not divided into different $$\\\\frac{groups}{clusters}$$. Therefore, we know that this is not stratified or clustered sampling and will now focus on simple random, systematic, and convenience sampling.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa7aa5dsampling9a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Convenience"],"dependencies":["aa7aa5dsampling9a-h1"],"title":"Selecting the Most Convenient ones","text":"Note that the woman surveys the ones who are not in hurry or sleep. Convenience sampling is the researcher just pick the most easily available data points. The definition of a systematic sampling is one where you randomly select a starting point and take the nth piece of data from the population. The definition of simple random sampling is fairly selecting a group of size $$n$$ from the population so every individual has the same chance of being selected. Which one sounds the most like the woman\'s selection method?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Simple Random","Systematic","Convenience"]}]}}]},{"id":"aa822f8UniMotion1","title":"John\'s Airplane","body":"Find the speed of the airplane.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion1a","stepAnswer":["$$650$$"],"problemType":"TextBox","stepTitle":"John can fly his airplane $$2800$$ miles with a wind speed of $$50$$ mph in the same time he can travel $$2400$$ miles against the wind. If the speed of the wind is $$50$$ mph, find the speed of his airplane (in mph).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$650$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion1a-h1","type":"hint","dependencies":[],"title":"Setting up the Equation","text":"Set up the equation by setting the speed of John\'s airplane to $$r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion1a-h2","type":"hint","dependencies":["aa822f8UniMotion1a-h1"],"title":"Relating the Numbers","text":"We know that distance $$=$$ $$rate time$$ since we are told two different distances and rates have the same time, we can set these two expressions equal to each other through $$\\\\frac{distance1}{rate1}$$ $$=$$ $$\\\\frac{distance2}{rate2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2800}{r+50}=\\\\frac{2400}{r-50}$$"],"dependencies":["aa822f8UniMotion1a-h2"],"title":"Relating the Numbers","text":"What is the relationship between the two distances and rates?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$650$$"],"dependencies":["aa822f8UniMotion1a-h3"],"title":"Relating the Numbers","text":"What is $$r$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa822f8UniMotion10","title":"Brian\'s Concrete","body":"Find how long it takes them to lay the concrete together.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion10a","stepAnswer":["$$\\\\frac{12}{5}$$"],"problemType":"TextBox","stepTitle":"Brian can lay a slab of concrete in $$6$$ hours, while Greg can do it in $$4$$ hours. If Brian and Greg work together, how long will it take (in hours)?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{12}{5}$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion10a-h1","type":"hint","dependencies":[],"title":"Setting up the equation","text":"Set up the equation first by setting $$t$$ as the time it takes when the two parties work together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion10a-h2","type":"hint","dependencies":["aa822f8UniMotion10a-h1"],"title":"Relating the Numbers","text":"If two people take times t1 and t2 to accomplish a task, together it would take them time t; $$\\\\frac{1}{t1}$$ + $$\\\\frac{1}{t2}$$ $$=$$ $$\\\\frac{1}{t}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{6}$$ + $$\\\\frac{1}{4}$$ $$=$$ $$\\\\frac{1}{t}$$"],"dependencies":["aa822f8UniMotion10a-h2"],"title":"Setting up the equation","text":"What is the relationship between the two parties?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{12}{5}$$"],"dependencies":["aa822f8UniMotion10a-h3"],"title":"Solving the equation","text":"Solve for $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion10a-h5","type":"hint","dependencies":["aa822f8UniMotion10a-h4"],"title":"Double-Checking","text":"Double check that the answer makes sense in terms of the context of the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa822f8UniMotion11","title":"Leeson\'s Newspaper","body":"Find how long it takes Ryan to proofread alone.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion11a","stepAnswer":["$$12$$"],"problemType":"TextBox","stepTitle":"Leeson can proofread a newspaper copy in $$4$$ hours. If Ryan helps, they can do the job in $$3$$ hours. How long would it take for Ryan to do his job alone (in hours)?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion11a-h1","type":"hint","dependencies":[],"title":"Setting up the equation","text":"Set up the equation first by setting $$t$$ as the time it takes when the Ryan works alone.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion11a-h2","type":"hint","dependencies":["aa822f8UniMotion11a-h1"],"title":"Relating the Numbers","text":"If two people take times t1 and $$t$$ to accomplish a task individually, together it would take them time t2; $$\\\\frac{1}{t2}$$ - $$\\\\frac{1}{t1}$$ $$=$$ $$\\\\frac{1}{t}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion11a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$ - $$\\\\frac{1}{4}$$ $$=$$ $$\\\\frac{1}{t}$$"],"dependencies":["aa822f8UniMotion11a-h2"],"title":"Setting up the equation","text":"What is the relationship between the two parties?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["aa822f8UniMotion11a-h3"],"title":"Solving the equation","text":"Solve for $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion11a-h5","type":"hint","dependencies":["aa822f8UniMotion11a-h4"],"title":"Double-Checking","text":"Double check that the answer makes sense in terms of the context of the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa822f8UniMotion12","title":"Josephine\'s Students","body":"Find how long it takes the assistant to correct the papers alone.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion12a","stepAnswer":["$$7.5$$"],"problemType":"TextBox","stepTitle":"Josephine can correct her students\u2019 test papers in $$5$$ hours, but if her teacher\u2019s assistant helps, it would take them $$3$$ hours. How long would it take the assistant to do it alone (in hours)?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7.5$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion12a-h1","type":"hint","dependencies":[],"title":"Setting up the equation","text":"Set up the equation first by setting $$t$$ as the time it takes when the assistant works alone.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion12a-h2","type":"hint","dependencies":["aa822f8UniMotion12a-h1"],"title":"Relating the Numbers","text":"If two people take times t1 and $$t$$ to accomplish a task individually, together it would take them time t2; $$\\\\frac{1}{t2}$$ - $$\\\\frac{1}{t1}$$ $$=$$ $$\\\\frac{1}{t}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion12a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$ - $$\\\\frac{1}{5}$$ $$=\\\\frac{1}{t}$$"],"dependencies":["aa822f8UniMotion12a-h2"],"title":"Setting up the equation","text":"What is the relationship between the two parties?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7.5$$"],"dependencies":["aa822f8UniMotion12a-h3"],"title":"Solving the equation","text":"Solve for $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion12a-h5","type":"hint","dependencies":["aa822f8UniMotion12a-h4"],"title":"Double-Checking","text":"Double check that the answer makes sense in terms of the context of the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa822f8UniMotion13","title":"Jackson\'s House Shingles","body":"Find how long it takes them to remove the shingles together.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion13a","stepAnswer":["$$\\\\frac{35}{12}$$"],"problemType":"TextBox","stepTitle":"Jackson can remove the shingles off of a house in $$7$$ hours, while Martin can remove the shingles in $$5$$ hours. How long will it take them to remove the shingles if they work together (in hours)?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{35}{12}$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion13a-h1","type":"hint","dependencies":[],"title":"Setting up the equation","text":"Set up the equation first by setting $$t$$ as the time it takes when the two parties work together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion13a-h2","type":"hint","dependencies":["aa822f8UniMotion13a-h1"],"title":"Relating the Numbers","text":"If two people take times t1 and t2 to accomplish a task, together it would take them time t; $$\\\\frac{1}{t1}$$ + $$\\\\frac{1}{t2}$$ $$=$$ $$\\\\frac{1}{t}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion13a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{7}$$ + $$\\\\frac{1}{5}$$ $$=\\\\frac{1}{t}$$"],"dependencies":["aa822f8UniMotion13a-h2"],"title":"Setting up the equation","text":"What is the relationship between the two parties?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{35}{12}$$"],"dependencies":["aa822f8UniMotion13a-h3"],"title":"Solving the equation","text":"Solve for $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion13a-h5","type":"hint","dependencies":["aa822f8UniMotion13a-h4"],"title":"Double-Checking","text":"Double check that the answer makes sense in terms of the context of the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa822f8UniMotion14","title":"Ronald\'s Driveway","body":"Find how long is takes Donald to shovel the driveway alone.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion14a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"Ronald can shovel the driveway in $$4$$ hours, but if his brother Donald helps it would take $$2$$ hours. How long would it take Donald to shovel the driveway alone (in hours)?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion14a-h1","type":"hint","dependencies":[],"title":"Setting up the equation","text":"Set up the equation first by setting $$t$$ as the time it takes when the assistant works alone.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion14a-h2","type":"hint","dependencies":["aa822f8UniMotion14a-h1"],"title":"Relating the Numbers","text":"If two people take times t1 and $$t$$ to accomplish a task individually, together it would take them time t2; $$\\\\frac{1}{t2}$$ - $$\\\\frac{1}{t1}$$ $$=$$ $$\\\\frac{1}{t}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion14a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$ - $$\\\\frac{1}{4}$$ $$=$$ $$\\\\frac{1}{t}$$"],"dependencies":["aa822f8UniMotion14a-h2"],"title":"Setting up the equation","text":"What is the relationship between the two parties?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion14a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["aa822f8UniMotion14a-h3"],"title":"Solving the equation","text":"Solve for $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion14a-h5","type":"hint","dependencies":["aa822f8UniMotion14a-h4"],"title":"Double-Checking","text":"Double check that the answer makes sense in terms of the context of the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa822f8UniMotion15","title":"Dana\'s Dog","body":"Find Dana\'s walking speed.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion15a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"Dana enjoys taking her dog for a walk, but sometimes her dog gets away and she has to run after him. Dana walked her dog for $$7$$ miles but then had to run for $$1$$ mile, spending a total time of $$2.5$$ hours with her dog. Her running speed was $$3$$ mph faster than her walking speed. Find her walking speed (in mph).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion15a-h1","type":"hint","dependencies":[],"title":"Setting up the equation","text":"Set up the equation by setting the speed of Dana\'s walking to $$r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion15a-h2","type":"hint","dependencies":["aa822f8UniMotion15a-h1"],"title":"Relating the Numbers","text":"We know that distance $$=$$ $$rate time$$ since we are told two different distances and rates have related time, we can set these two expressions such that $$\\\\frac{distance1}{rate1}$$ + $$\\\\frac{distance2}{rate2}$$ $$=$$ $$2.5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{7}{r}$$ + $$\\\\frac{1}{r+3}=2.5$$"],"dependencies":["aa822f8UniMotion15a-h2"],"title":"Setting up the equation","text":"What is the relationship between the two distances and rates?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["aa822f8UniMotion15a-h3"],"title":"Solving the equation","text":"What is $$r$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion15a-h5","type":"hint","dependencies":["aa822f8UniMotion15a-h4"],"title":"Double-Checking","text":"Double check that the answer makes sense in terms of the context of the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa822f8UniMotion16","title":"Link\'s Bike","body":"Solve the following problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion16a","stepAnswer":["$$15$$"],"problemType":"TextBox","stepTitle":"Link can ride his bike $$20$$ miles into a $$3$$ mph headwind in the same amount of time he can ride $$30$$ miles with a $$3$$ mph tailwind. What is Link\u2019s biking speed?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$15$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion16a-h1","type":"hint","dependencies":[],"title":"Setting up the equation","text":"Set up the equation first by setting $$r$$ as the speed of Link\'s biking","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion16a-h2","type":"hint","dependencies":["aa822f8UniMotion16a-h1"],"title":"Setting up the equation","text":"Next, understand how the two speeds relate to each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion16a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{20}{r-3}=\\\\frac{30}{r+3}$$"],"dependencies":["aa822f8UniMotion16a-h2"],"title":"Setting up the equation","text":"What is the relationship between the two speeds?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion16a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["aa822f8UniMotion16a-h3"],"title":"Solving the equation","text":"What is $$r$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion16a-h5","type":"hint","dependencies":["aa822f8UniMotion16a-h4"],"title":"Double-Checking","text":"Double check that the answer makes sense in terms of the context of the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa822f8UniMotion17","title":"Judy\'s Boat","body":"Solve the following problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion17a","stepAnswer":["$$17$$"],"problemType":"TextBox","stepTitle":"Judy can sail her boat $$5$$ miles into a $$7$$ mph headwind in the same amount of time she can sail $$12$$ miles with a $$7$$ mph tailwind. What is the speed of Judy\u2019s boat without a wind?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$17$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion17a-h1","type":"hint","dependencies":[],"title":"Setting up the equation","text":"Set up the equation first by setting $$r$$ as the speed of Judy\'s Boat","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion17a-h2","type":"hint","dependencies":["aa822f8UniMotion17a-h1"],"title":"Setting up the equation","text":"Next, understand how the two speeds relate to each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion17a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{r-7}=\\\\frac{12}{r+7}$$"],"dependencies":["aa822f8UniMotion17a-h2"],"title":"Setting up the equation","text":"What is the relationship between the two speeds?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion17a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$17$$"],"dependencies":["aa822f8UniMotion17a-h3"],"title":"Solving the equation","text":"Solve for $$r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion17a-h5","type":"hint","dependencies":["aa822f8UniMotion17a-h4"],"title":"Double-Checking","text":"Double check that the answer makes sense in terms of the context of the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa822f8UniMotion18","title":"Dennis Skiing","body":"Solve the following problem. Please input the answer in the following form: Uphill skiing speed $$=$$ $$x$$, Downhill skiing speed $$=$$ $$y$$","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion18a","stepAnswer":["Uphill skiing speed = 5, Downhill skiing speed = 10"],"problemType":"TextBox","stepTitle":"Dennis went cross-country skiing for $$6$$ hours on Saturday. He skied $$20$$ miles uphill and then $$20$$ miles back downhill, returning to his starting point. His uphill speed was $$5$$ mph slower than his downhill speed. What was Dennis\u2019 speed going uphill and his speed going downhill?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Uphill skiing speed $$=$$ $$5$$, Downhill skiing speed $$=$$ $$10$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion18a-h1","type":"hint","dependencies":[],"title":"Setting up the equation","text":"Set up the equation first by setting $$r$$ as the speed of Dennis\' uphill skiing","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion18a-h2","type":"hint","dependencies":["aa822f8UniMotion18a-h1"],"title":"Setting up the equation","text":"Next, understand how the two speeds relate to each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion18a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{20}{r}+\\\\frac{20}{r+5}=6$$"],"dependencies":["aa822f8UniMotion18a-h2"],"title":"Setting up the equation","text":"What is the relationship between the two speeds?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion18a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["aa822f8UniMotion18a-h3"],"title":"Solving the equation","text":"Solve for $$r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion18a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["aa822f8UniMotion18a-h4"],"title":"Solving for the undefined speed","text":"If Dennis\' speed going uphill is 5mph then, looking at the context of the question, what is his speed going downhill?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion18a-h6","type":"hint","dependencies":["aa822f8UniMotion18a-h5"],"title":"Double-Checking","text":"Double check that the answer makes sense in terms of the context of the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa822f8UniMotion19","title":"Tony\'s Drive","body":"Solve the following problem. Please input the answer in the following form: Country road speed $$=$$ $$x$$, Interstate speed $$=$$ $$x$$","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion19a","stepAnswer":["Country road speed = 50, Interstate speed = 65"],"problemType":"TextBox","stepTitle":"Tony drove $$4$$ hours to his home, driving $$208$$ miles on the interstate and $$40$$ miles on country roads. If he drove $$15$$ mph faster on the interstate than on the country roads, what was his rate on the country roads?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Country road speed $$=$$ $$50$$, Interstate speed $$=$$ $$65$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion19a-h1","type":"hint","dependencies":[],"title":"Setting up the equation","text":"Set up the equation first by setting $$r$$ as the speed of Tony\'s driving on country roads","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion19a-h2","type":"hint","dependencies":["aa822f8UniMotion19a-h1"],"title":"Setting up the equation","text":"Next, understand how the two speeds relate to each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion19a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{208}{r+15}+\\\\frac{40}{r}=4$$"],"dependencies":["aa822f8UniMotion19a-h2"],"title":"Setting up the equation","text":"What is the relationship between the two speeds?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion19a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$50$$"],"dependencies":["aa822f8UniMotion19a-h3"],"title":"Solving the equation","text":"Solve for $$r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion19a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$65$$"],"dependencies":["aa822f8UniMotion19a-h4"],"title":"Solving for the undefined speed","text":"If Tony\'s drive on the country roads is 50mph, then what is his speed on the interstate, looking at the context of the qeustion?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion19a-h6","type":"hint","dependencies":["aa822f8UniMotion19a-h5"],"title":"Double-Checking","text":"Double check that the answer makes sense in terms of the context of the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa822f8UniMotion2","title":"Jim\'s Speedboat","body":"Find the speed of the boat.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion2a","stepAnswer":["$$63$$"],"problemType":"TextBox","stepTitle":"Jim\u2019s speedboat can travel $$20$$ miles upstream against a $$3$$ mph current in the same amount of time it travels $$22$$ miles downstream with a $$3$$ mph current speed. Find the speed of the Jim\u2019s boat (in mph).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$63$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion2a-h1","type":"hint","dependencies":[],"title":"Setting up the Equation","text":"Set up the equation by setting the speed of Jim\'s speedboat to $$r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion2a-h2","type":"hint","dependencies":["aa822f8UniMotion2a-h1"],"title":"Relating the Numbers","text":"We know that distance $$=$$ $$rate time$$ since we are told two different distances and rates have the same time, we can set these two expressions equal to each other through $$\\\\frac{distance1}{rate1}$$ $$=$$ $$\\\\frac{distance2}{rate2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{20}{r-3}=\\\\frac{22}{r+3}$$"],"dependencies":["aa822f8UniMotion2a-h2"],"title":"Relating the Numbers","text":"What is the relationship between the two distances and rates?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$63$$"],"dependencies":["aa822f8UniMotion2a-h3"],"title":"Relating the Numbers","text":"What is $$r$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa822f8UniMotion20","title":"Kayla\'s Bike","body":"Solve the following problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion20a","stepAnswer":["$$15$$"],"problemType":"TextBox","stepTitle":"Kayla rode her bike $$75$$ miles home from college one weekend and then rode the bus back to college. It took her $$2$$ hours less to ride back to college on the bus than it took her to ride home on her bike, and the average speed of the bus was $$10$$ miles per hour faster than Kayla\u2019s biking speed. Find Kayla\u2019s biking speed.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$15$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion20a-h1","type":"hint","dependencies":[],"title":"Setting up the equation","text":"Set up the equation first by setting $$r$$ as the speed of Kayla\'s biking","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion20a-h2","type":"hint","dependencies":["aa822f8UniMotion20a-h1"],"title":"Setting up the equation","text":"Next, understand how the two speeds relate to each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion20a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{75}{r}=\\\\frac{75}{r+10}+2$$"],"dependencies":["aa822f8UniMotion20a-h2"],"title":"Setting up the equation","text":"What is the relationship between the two speeds?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion20a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["aa822f8UniMotion20a-h3"],"title":"Solving the equation","text":"Solve for $$r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion20a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["aa822f8UniMotion20a-h4"],"title":"Solving for the undefined speed","text":"If Kayla bikes at a speed of 15mph, then what is her speed on the bus, looking at the context of the qeustion?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion20a-h6","type":"hint","dependencies":["aa822f8UniMotion20a-h5"],"title":"Double-Checking","text":"Double check that the answer makes sense in terms of the context of the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa822f8UniMotion21","title":"Victoria\'s Jog","body":"Solve the following problem","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion21a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"Victoria jogs $$12$$ miles to the park along a flat trail and then returns by jogging on a $$20$$ mile hilly trail. She jogs $$1$$ mile per hour slower on the hilly trail than on the flat trail, and her return trip takes her two hours longer. Find her rate of jogging on the flat trail.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion21a-h1","type":"hint","dependencies":[],"title":"Setting up the equation","text":"Set up the equation first by setting $$r$$ as the speed of Victoria\'s Jog on the flat trail","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion21a-h2","type":"hint","dependencies":["aa822f8UniMotion21a-h1"],"title":"Setting up the equation","text":"Next, understand how the two speeds relate to each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion21a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{12}{r}=\\\\frac{20}{r-1}-2$$"],"dependencies":["aa822f8UniMotion21a-h2"],"title":"Setting up the equation","text":"What is the relationship between the two speeds?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion21a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["aa822f8UniMotion21a-h3"],"title":"Solving the equation","text":"Solve for $$r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion21a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["aa822f8UniMotion21a-h4"],"title":"Solving for the undefined speed","text":"If Victoria jogs at a speed of 6mph on the flat trail, then what is her speed on the hilly tail, looking at the context of the qeustion?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion21a-h6","type":"hint","dependencies":["aa822f8UniMotion21a-h5"],"title":"Double-Checking","text":"Double check that the answer makes sense in terms of the context of the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa822f8UniMotion22","title":"Gardening","body":"Solve the following problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion22a","stepAnswer":["$$\\\\frac{12}{5}$$"],"problemType":"TextBox","stepTitle":"One gardener can mow a golf course in $$4$$ hours, while another gardener can mow the same golf course in $$6$$ hours. How long (in hours) would it take if the two gardeners worked together to mow the golf course?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{12}{5}$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion22a-h1","type":"hint","dependencies":[],"title":"Setting up the equation","text":"Set up the equation first by setting $$t$$ as the time it takes when the two parties work together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion22a-h2","type":"hint","dependencies":["aa822f8UniMotion22a-h1"],"title":"Setting up the equation","text":"Next, understand how the two parties relate to each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion22a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4}+\\\\frac{1}{6}=\\\\frac{1}{t}$$"],"dependencies":["aa822f8UniMotion22a-h2"],"title":"Setting up the equation","text":"What is the relationship between the two parties?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion22a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{12}{5}$$"],"dependencies":["aa822f8UniMotion22a-h3"],"title":"Solving the equation","text":"Solve for $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion22a-h5","type":"hint","dependencies":["aa822f8UniMotion22a-h4"],"title":"Double-Checking","text":"Double check that the answer makes sense in terms of the context of the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa822f8UniMotion23","title":"Weeding the Garden","body":"Solve the following problem","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion23a","stepAnswer":["$$\\\\frac{21}{10}$$"],"problemType":"TextBox","stepTitle":"Carrie can weed the garden in $$7$$ hours, while her mother can do it in $$3$$. How long will it take the two of them working together?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{21}{10}$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion23a-h1","type":"hint","dependencies":[],"title":"Setting up the equation","text":"Set up the equation first by setting $$t$$ as the time it takes when the two parties work together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion23a-h2","type":"hint","dependencies":["aa822f8UniMotion23a-h1"],"title":"Setting up the equation","text":"Next, understand how the two parties relate to each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion23a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{7}+\\\\frac{1}{3}=\\\\frac{1}{t}$$"],"dependencies":["aa822f8UniMotion23a-h2"],"title":"Setting up the equation","text":"What is the relationship between the two parties?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion23a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{21}{10}$$"],"dependencies":["aa822f8UniMotion23a-h3"],"title":"Solving the equation","text":"Solve for $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion23a-h5","type":"hint","dependencies":["aa822f8UniMotion23a-h4"],"title":"Double-Checking","text":"Double check that the answer makes sense in terms of the context of the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa822f8UniMotion24","title":"Hoses Filling the Pool","body":"Solve the following problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion24a","stepAnswer":["$$16.25$$"],"problemType":"TextBox","stepTitle":"Two hoses can fill a swimming pool in $$10$$ hours. It would take one hose $$26$$ hours to fill the pool by itself. How long would it take for the other hose, working alone, to fill the pool?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16.25$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion24a-h1","type":"hint","dependencies":[],"title":"Setting up the equation","text":"Set up the equation first by setting $$t$$ as the time it takes for the undefined party to work alone.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion24a-h2","type":"hint","dependencies":["aa822f8UniMotion24a-h1"],"title":"Setting up the equation","text":"Next, understand how the two parties relate to each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion24a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{26}+\\\\frac{1}{t}=\\\\frac{1}{10}$$"],"dependencies":["aa822f8UniMotion24a-h2"],"title":"Setting up the equation","text":"What is the relationship between the two parties?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion24a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16.25$$"],"dependencies":["aa822f8UniMotion24a-h3"],"title":"Solving the equation","text":"Solve for $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion24a-h5","type":"hint","dependencies":["aa822f8UniMotion24a-h4"],"title":"Double-Checking","text":"Double check that the answer makes sense in terms of the context of the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa822f8UniMotion25","title":"Cara and Cindy Raking","body":"Solve the following problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion25a","stepAnswer":["$$12$$"],"problemType":"TextBox","stepTitle":"Cara and Cindy, working together, can rake the yard in $$4$$ hours. Working alone, it takes Cindy $$6$$ hours to rake the yard. How long (in hours) would it take Cara to rake the yard alone?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion25a-h1","type":"hint","dependencies":[],"title":"Setting up the equation","text":"Set up the equation first by setting $$t$$ as the time it takes for the undefined party to work alone.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion25a-h2","type":"hint","dependencies":["aa822f8UniMotion25a-h1"],"title":"Setting up the equation","text":"Next, understand how the two parties relate to each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion25a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{6}+\\\\frac{1}{t}=\\\\frac{1}{4}$$"],"dependencies":["aa822f8UniMotion25a-h2"],"title":"Setting up the equation","text":"What is the relationship between the two parties?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion25a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["aa822f8UniMotion25a-h3"],"title":"Solving the equation","text":"Solve for $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion25a-h5","type":"hint","dependencies":["aa822f8UniMotion25a-h4"],"title":"Double-Checking","text":"Double check that the answer makes sense in terms of the context of the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa822f8UniMotion26","title":"Mary\'s Helicopter","body":"Solve the following problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion26a","stepAnswer":["$$160$$"],"problemType":"TextBox","stepTitle":"Mary takes a sightseeing tour on a helicopter that can fly $$450$$ miles against a $$35$$ mph headwind in the same amount of time it can travel $$702$$ miles with a $$35$$ mph tailwind. Find the speed of the helicopter in mph.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$160$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion26a-h1","type":"hint","dependencies":[],"title":"Setting up the equation","text":"Set up the equation first by setting $$r$$ as the speed of Mary\'s Helicopter.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion26a-h2","type":"hint","dependencies":["aa822f8UniMotion26a-h1"],"title":"Setting up the equation","text":"Next, understand how the two speeds relate to each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion26a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{450}{r-35}=\\\\frac{702}{r+35}$$"],"dependencies":["aa822f8UniMotion26a-h2"],"title":"Setting up the equation","text":"What is the relationship between the two speeds?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion26a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$160$$"],"dependencies":["aa822f8UniMotion26a-h3"],"title":"Solving the equation","text":"Solve for $$r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion26a-h5","type":"hint","dependencies":["aa822f8UniMotion26a-h4"],"title":"Double-Checking","text":"Double check that the answer makes sense in terms of the context of the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa822f8UniMotion27","title":"Private Jets","body":"Solve the following problem","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion27a","stepAnswer":["$$150$$"],"problemType":"TextBox","stepTitle":"A private jet can fly $$1210$$ miles against a $$25$$ mph headwind in the same amount of time it can fly $$1694$$ miles with a $$25$$ mph tailwind. Find the speed of the jet.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$150$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion27a-h1","type":"hint","dependencies":[],"title":"Setting up the equation","text":"Set up the equation first by setting $$r$$ as the speed of the jet.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion27a-h2","type":"hint","dependencies":["aa822f8UniMotion27a-h1"],"title":"Setting up the equation","text":"Next, understand how the two speeds relate to each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion27a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1210}{r-25}=\\\\frac{1694}{r+25}$$"],"dependencies":["aa822f8UniMotion27a-h2"],"title":"Setting up the equation","text":"What is the relationship between the two speeds?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion27a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$150$$"],"dependencies":["aa822f8UniMotion27a-h3"],"title":"Solving the equation","text":"Solve for $$r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion27a-h5","type":"hint","dependencies":["aa822f8UniMotion27a-h4"],"title":"Double-Checking","text":"Double check that the answer makes sense in terms of the context of the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa822f8UniMotion28","title":"A Boat\'s Journey","body":"Solve the following problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion28a","stepAnswer":["$$29$$"],"problemType":"TextBox","stepTitle":"A boat travels $$140$$ miles downstream in the same time as it travels $$92$$ miles upstream. The speed of the current is 6mph. What is the speed of the boat?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$29$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion28a-h1","type":"hint","dependencies":[],"title":"Setting up the equation","text":"Set up the equation first by setting $$r$$ as the speed of the boat.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion28a-h2","type":"hint","dependencies":["aa822f8UniMotion28a-h1"],"title":"Setting up the equation","text":"Next, understand how the two speeds relate to each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion28a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{140}{r+6}=\\\\frac{92}{r-6}$$"],"dependencies":["aa822f8UniMotion28a-h2"],"title":"Setting up the equation","text":"What is the relationship between the two speeds?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion28a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$29$$"],"dependencies":["aa822f8UniMotion28a-h3"],"title":"Solving the equation","text":"Solve for $$r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion28a-h5","type":"hint","dependencies":["aa822f8UniMotion28a-h4"],"title":"Double-Checking","text":"Double check that the answer makes sense in terms of the context of the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa822f8UniMotion29","title":"Darrin\'s Skateboard","body":"Solve the following problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion29a","stepAnswer":["$$8$$"],"problemType":"TextBox","stepTitle":"Darrin can skateboard $$2$$ miles against a $$4$$ mph wind in the same amount of time he skateboards $$6$$ miles with a $$4$$ mph wind. 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She travels $$900$$ miles by plane and $$250$$ miles by car. The plane travels $$250$$ mph faster than the car. If she drives the rental car for $$2$$ hours more than she rode the plane, find the speed of the car (in mph).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$50$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion3a-h1","type":"hint","dependencies":[],"title":"Setting up the Equation","text":"Set up the equation by setting the speed of Hazel\'s car to $$r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion3a-h2","type":"hint","dependencies":["aa822f8UniMotion3a-h1"],"title":"Relating the Numbers","text":"We know that distance $$=$$ $$rate time$$ since we are told two different distances and rates have related time, we can set these two expressions equal to each other through $$\\\\frac{distance1}{rate1}$$ $$=$$ $$\\\\frac{distance2}{rate2}$$ + $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{250}{r}=\\\\frac{900}{r+250}$$ + $$2$$"],"dependencies":["aa822f8UniMotion3a-h2"],"title":"Relating the Numbers","text":"What is the relationship between the two distances and rates?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$50$$"],"dependencies":["aa822f8UniMotion3a-h3"],"title":"Relating the Numbers","text":"What is $$r$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa822f8UniMotion30","title":"Jane\'s Exploration","body":"Solve the following problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion30a","stepAnswer":["$$30$$"],"problemType":"TextBox","stepTitle":"Jane spent $$2$$ hours exploring a mountain with a dirt bike. When she rode the $$40$$ miles uphill, she went $$5$$ mph slower than when she reached the peak and rode for $$12$$ miles along the summit. What was her rate (in mph) along the summit (if there are two possible answeres, choose the largest)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$30$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion30a-h1","type":"hint","dependencies":[],"title":"Setting up the equation","text":"Set up the equation first by setting $$r$$ as Jane\'s speed along the summit.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion30a-h2","type":"hint","dependencies":["aa822f8UniMotion30a-h1"],"title":"Setting up the equation","text":"Next, understand how the two speeds relate to each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion30a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{40}{r-5}+\\\\frac{12}{r}=2$$"],"dependencies":["aa822f8UniMotion30a-h2"],"title":"Setting up the equation","text":"What is the relationship between the two speeds?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion30a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$r$$ $$=$$ $$30$$ or $$1$$"],"dependencies":["aa822f8UniMotion30a-h3"],"title":"Solving the equation","text":"Solve for r; if there are two possible answers, choose the largest","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion30a-h5","type":"hint","dependencies":["aa822f8UniMotion30a-h4"],"title":"Double-Checking","text":"Double check that the answer makes sense in terms of the context of the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion30a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$30$$"],"dependencies":["aa822f8UniMotion30a-h5"],"title":"Double-Checking","text":"Which speed makes more sense in the context of the question?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa822f8UniMotion31","title":"Jill\'s Jog","body":"Solve the following problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion31a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"Jill wanted to lose some weight so she planned a day of exercising. She spent a total of $$2$$ hours riding her bike and jogging. She biked for $$12$$ miles and jogged for $$6$$ miles. Her rate for jogging was $$10$$ mph less than biking rate. What was her rate (in mph) when jogging?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion31a-h1","type":"hint","dependencies":[],"title":"Setting up the equation","text":"Set up the equation first by setting $$r$$ as Jill\'s jogging speed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion31a-h2","type":"hint","dependencies":["aa822f8UniMotion31a-h1"],"title":"Setting up the equation","text":"Next, understand how the two speeds relate to each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion31a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{6}{r}+\\\\frac{12}{r+10}=2$$"],"dependencies":["aa822f8UniMotion31a-h2"],"title":"Setting up the equation","text":"What is the relationship between the two speeds?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion31a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["aa822f8UniMotion31a-h3"],"title":"Solving the equation","text":"Solve for $$r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion31a-h5","type":"hint","dependencies":["aa822f8UniMotion31a-h4"],"title":"Double-Checking","text":"Double check that the answer makes sense in terms of the context of the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa822f8UniMotion32","title":"Bill on Water","body":"Solve the following problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion32a","stepAnswer":["$$20$$"],"problemType":"TextBox","stepTitle":"Bill wanted to try out different water craft. He went $$62$$ miles downstream in a motor boat and $$27$$ miles downstream on a jet ski. His speed on the jet ski was $$10$$ mph faster than in the motor boat. Bill spent a total of $$4$$ hours on the water. What was his rate of speed in the motor boat?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$20$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion32a-h1","type":"hint","dependencies":[],"title":"Setting up the equation","text":"Set up the equation first by setting $$r$$ as Bill\'s speed in the motorboat.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion32a-h2","type":"hint","dependencies":["aa822f8UniMotion32a-h1"],"title":"Setting up the equation","text":"Next, understand how the two speeds relate to each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion32a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{62}{r}+\\\\frac{27}{r+10}=4$$"],"dependencies":["aa822f8UniMotion32a-h2"],"title":"Setting up the equation","text":"What is the relationship between the two speeds?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion32a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["aa822f8UniMotion32a-h3"],"title":"Solving the equation","text":"Solve for $$r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion32a-h5","type":"hint","dependencies":["aa822f8UniMotion32a-h4"],"title":"Double-Checking","text":"Double check that the answer makes sense in terms of the context of the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa822f8UniMotion33","title":"Nancy\'s Drive","body":"Solve the following problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion33a","stepAnswer":["$$36$$"],"problemType":"TextBox","stepTitle":"Nancy took a $$3$$ hour drive. She went $$50$$ miles before she got caught in a storm. Then she drove $$68$$ miles at $$9$$ mph less than she had driven when the weather was good. What was her speed driving in the storm?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$36$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion33a-h1","type":"hint","dependencies":[],"title":"Setting up the equation","text":"Set up the equation first by setting $$r$$ as Nancy\'s speed in the storm.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion33a-h2","type":"hint","dependencies":["aa822f8UniMotion33a-h1"],"title":"Setting up the equation","text":"Next, understand how the two speeds relate to each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion33a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{50}{r}+9+\\\\frac{68}{r}=3$$"],"dependencies":["aa822f8UniMotion33a-h2"],"title":"Setting up the equation","text":"What is the relationship between the two speeds?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion33a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$36$$"],"dependencies":["aa822f8UniMotion33a-h3"],"title":"Solving the equation","text":"Solve for $$r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion33a-h5","type":"hint","dependencies":["aa822f8UniMotion33a-h4"],"title":"Double-Checking","text":"Double check that the answer makes sense in terms of the context of the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa822f8UniMotion34","title":"Chester\'s Bike","body":"Solve the following problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion34a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"Chester rode his bike uphill $$24$$ miles and then back downhill at $$2$$ mph faster than his uphill. If it took him $$2$$ hours longer to ride uphill than downhill, l, what was his uphill rate?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion34a-h1","type":"hint","dependencies":[],"title":"Setting up the equation","text":"Set up the equation first by setting $$r$$ as Chester\'s biking speed uphill.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion34a-h2","type":"hint","dependencies":["aa822f8UniMotion34a-h1"],"title":"Setting up the equation","text":"Next, understand how the two speeds relate to each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion34a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{24}{r}=\\\\frac{24}{r+2}+2$$"],"dependencies":["aa822f8UniMotion34a-h2"],"title":"Setting up the equation","text":"What is the relationship between the two speeds?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion34a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["aa822f8UniMotion34a-h3"],"title":"Solving the equation","text":"Solve for $$r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion34a-h5","type":"hint","dependencies":["aa822f8UniMotion34a-h4"],"title":"Double-Checking","text":"Double check that the answer makes sense in terms of the context of the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa822f8UniMotion35","title":"Matthew\'s Jog","body":"Solve the following problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion35a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"Matthew jogged to his friend\u2019s house $$12$$ miles away and then got a ride back home. It took him $$2$$ hours longer to jog there than ride back. His jogging rate was $$25$$ mph slower than the rate when he was riding. What was his jogging rate (in mph)?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion35a-h1","type":"hint","dependencies":[],"title":"Setting up the equation","text":"Set up the equation first by setting $$r$$ as Matthew\'s jogging speed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion35a-h2","type":"hint","dependencies":["aa822f8UniMotion35a-h1"],"title":"Setting up the equation","text":"Next, understand how the two speeds relate to each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion35a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{12}{r}=\\\\frac{12}{r+25}+2$$"],"dependencies":["aa822f8UniMotion35a-h2"],"title":"Setting up the equation","text":"What is the relationship between the two speeds?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion35a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["aa822f8UniMotion35a-h3"],"title":"Solving the equation","text":"Solve for $$r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion35a-h5","type":"hint","dependencies":["aa822f8UniMotion35a-h4"],"title":"Double-Checking","text":"Double check that the answer makes sense in terms of the context of the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa822f8UniMotion36","title":"Hudson\'s Car","body":"Solve the following problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion36a","stepAnswer":["$$60$$"],"problemType":"TextBox","stepTitle":"Hudson travels $$1080$$ miles in a jet and then $$240$$ miles by car to get to a business meeting. The jet goes $$300$$ mph faster than the rate of the car, and the car ride takes $$1$$ hour longer than the jet. What is the speed of the car?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$60$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion36a-h1","type":"hint","dependencies":[],"title":"Setting up the equation","text":"Set up the equation first by setting $$r$$ as Hudson\'s speed in the car","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion36a-h2","type":"hint","dependencies":["aa822f8UniMotion36a-h1"],"title":"Setting up the equation","text":"Next, understand how the two speeds relate to each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion36a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{240}{r}=\\\\frac{1080}{r+300}+1$$"],"dependencies":["aa822f8UniMotion36a-h2"],"title":"Setting up the equation","text":"What is the relationship between the two speeds?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion36a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$60$$"],"dependencies":["aa822f8UniMotion36a-h3"],"title":"Solving the equation","text":"Solve for $$r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion36a-h5","type":"hint","dependencies":["aa822f8UniMotion36a-h4"],"title":"Double-Checking","text":"Double check that the answer makes sense in terms of the context of the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa822f8UniMotion4","title":"Stu\'s Training","body":"Find Stu\'s running speed.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion4a","stepAnswer":["$$14$$"],"problemType":"TextBox","stepTitle":"Stu trained for $$3$$ hours yesterday. He ran $$14$$ miles and then biked $$40$$ miles. His biking speed is $$6$$ mph faster than his running speed. What is his running speed (in mph)?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$14$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion4a-h1","type":"hint","dependencies":[],"title":"Setting up the Equation","text":"Set up the equation by setting the speed of Stu\'s running to $$r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion4a-h2","type":"hint","dependencies":["aa822f8UniMotion4a-h1"],"title":"Relating the Numbers","text":"We know that distance $$=$$ $$rate time$$ since we are told two different distances and rates have related time, we can set these two expressions such that $$\\\\frac{distance1}{rate1}$$ + $$\\\\frac{distance2}{rate2}$$ $$=$$ $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{14}{r}+\\\\frac{40}{r+6}=3$$"],"dependencies":["aa822f8UniMotion4a-h2"],"title":"Relating the Numbers","text":"What is the relationship between the two distances and rates?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$14$$"],"dependencies":["aa822f8UniMotion4a-h3"],"title":"Relating the Numbers","text":"What is $$r$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion4a-h5","type":"hint","dependencies":["aa822f8UniMotion4a-h4"],"title":"Making Sense","text":"It\'s possible that, algebraically, you arrive at two different numbers for r; make sure that the answer you choose is reasonable (i.e. you can\'t have negative time).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa822f8UniMotion5","title":"Sharon\'s Driving","body":"Find Sharon\'s speed on country roads.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion5a","stepAnswer":["$$50$$"],"problemType":"TextBox","stepTitle":"When driving the $$9$$ hour trip home, Sharon drove $$390$$ miles on the interstate and $$150$$ miles on country roads. Her speed on the interstate was $$15$$ more than on country roads. What was her speed on country roads (in mph)?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$50$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion5a-h1","type":"hint","dependencies":[],"title":"Setting up the Equation","text":"Set up the equation by setting the speed of Sharon\'s driving on country roads to $$r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion5a-h2","type":"hint","dependencies":["aa822f8UniMotion5a-h1"],"title":"Relating the Numbers","text":"We know that distance $$=$$ $$rate time$$ since we are told two different distances and rates have related time, we can set these two expressions such that $$\\\\frac{distance1}{rate1}$$ + $$\\\\frac{distance2}{rate2}$$ $$=$$ $$9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{150}{r}+\\\\frac{390}{r+15}=9$$"],"dependencies":["aa822f8UniMotion5a-h2"],"title":"Relating the Numbers","text":"What is the relationship between the two distances and rates?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$50$$"],"dependencies":["aa822f8UniMotion5a-h3"],"title":"Relating the Numbers","text":"What is $$r$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion5a-h5","type":"hint","dependencies":["aa822f8UniMotion5a-h4"],"title":"Making Sense","text":"It\'s possible that, algebraically, you arrive at two different numbers for r; make sure that the answer you choose is reasonable (i.e. you can\'t have negative time).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa822f8UniMotion6","title":"Samantha\'s Bike","body":"Find Samantha\'s biking speed.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion6a","stepAnswer":["$$16$$"],"problemType":"TextBox","stepTitle":"Two sisters like to compete on their bike rides. Tamara can go $$4$$ mph faster than her sister, Samantha. If it takes Samantha $$1$$ hours longer than Tamara to go $$80$$ miles, how fast can Samantha ride her bike (in mph)?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion6a-h1","type":"hint","dependencies":[],"title":"Setting up the Equation","text":"Set up the equation by setting the speed of Samantha\'s biking to $$r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion6a-h2","type":"hint","dependencies":["aa822f8UniMotion6a-h1"],"title":"Relating the Numbers","text":"We know that distance $$=$$ $$rate time$$ since we are told two distances and rates have related time, we can set these two expressions such that $$\\\\frac{distance1}{rate1}$$ $$=$$ $$\\\\frac{distance2}{rate2}$$ + $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{80}{r}=\\\\frac{80}{r+4}$$ + $$1$$"],"dependencies":["aa822f8UniMotion6a-h2"],"title":"Relating the Numbers","text":"What is the relationship between the two distances and rates?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["aa822f8UniMotion6a-h3"],"title":"Relating the Numbers","text":"What is $$r$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion6a-h5","type":"hint","dependencies":["aa822f8UniMotion6a-h4"],"title":"Making Sense","text":"It\'s possible that, algebraically, you arrive at two different numbers for r; make sure that the answer you choose is reasonable (i.e. you can\'t have negative time).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa822f8UniMotion7","title":"Mike\'s Bricks","body":"Find how long it takes them to build the wall together.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion7a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"Mike, an experienced bricklayer, can build a wall in $$3$$ hours, while his son, who is learning, can do the job in $$6$$ hours. How long does it take for them to build a wall together (in hours)?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion7a-h1","type":"hint","dependencies":[],"title":"Setting up the equation","text":"Set up the equation first by setting $$t$$ as the time it takes when the two parties work together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion7a-h2","type":"hint","dependencies":["aa822f8UniMotion7a-h1"],"title":"Relating the Numbers","text":"If two people take times t1 and t2 to accomplish a task, together it would take them time t; $$\\\\frac{1}{t1}$$ + $$\\\\frac{1}{t2}$$ $$=$$ $$\\\\frac{1}{t}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$ + $$\\\\frac{1}{6}$$ $$=$$ $$\\\\frac{1}{t}$$"],"dependencies":["aa822f8UniMotion7a-h2"],"title":"Setting up the equation","text":"What is the relationship between the two parties?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["aa822f8UniMotion7a-h3"],"title":"Solving the equation","text":"Solve for $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion7a-h5","type":"hint","dependencies":["aa822f8UniMotion7a-h4"],"title":"Double-Checking","text":"Double check that the answer makes sense in terms of the context of the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa822f8UniMotion8","title":"Sam\'s Lawn","body":"Find how long it takes them to rake the lawn together.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion8a","stepAnswer":["$$\\\\frac{4}{3}$$"],"problemType":"TextBox","stepTitle":"It takes Sam $$4$$ hours to rake the front lawn while his brother, Dave, can rake the lawn in $$2$$ hours. How long will it take them to rake the lawn working together (in hours)?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{4}{3}$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion8a-h1","type":"hint","dependencies":[],"title":"Setting up the equation","text":"Set up the equation first by setting $$t$$ as the time it takes when the two parties work together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion8a-h2","type":"hint","dependencies":["aa822f8UniMotion8a-h1"],"title":"Relating the Numbers","text":"If two people take times t1 and t2 to accomplish a task, together it would take them time t; $$\\\\frac{1}{t1}$$ + $$\\\\frac{1}{t2}$$ $$=$$ $$\\\\frac{1}{t}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4}$$ + $$\\\\frac{1}{2}$$ $$=$$ $$\\\\frac{1}{t}$$"],"dependencies":["aa822f8UniMotion8a-h2"],"title":"Setting up the equation","text":"What is the relationship between the two parties?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{4}{3}$$"],"dependencies":["aa822f8UniMotion8a-h3"],"title":"Solving the equation","text":"Solve for $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion8a-h5","type":"hint","dependencies":["aa822f8UniMotion8a-h4"],"title":"Double-Checking","text":"Double check that the answer makes sense in terms of the context of the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa822f8UniMotion9","title":"Mary\'s Apartment","body":"Find how long it takes them to clean the apartment together.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Solve Uniform Motion and Work Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aa822f8UniMotion9a","stepAnswer":["$$\\\\frac{30}{11}$$"],"problemType":"TextBox","stepTitle":"Mary can clean her apartment in $$6$$ hours while her roommate can clean the apartment in $$5$$ hours. If they work together, how long would it take them to clean the apartment (in hours)?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{30}{11}$$","hints":{"DefaultPathway":[{"id":"aa822f8UniMotion9a-h1","type":"hint","dependencies":[],"title":"Setting up the equation","text":"Set up the equation first by setting $$t$$ as the time it takes when the two parties work together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion9a-h2","type":"hint","dependencies":["aa822f8UniMotion9a-h1"],"title":"Relating the Numbers","text":"If two people take times t1 and t2 to accomplish a task, together it would take them time t; $$\\\\frac{1}{t1}$$ + $$\\\\frac{1}{t2}$$ $$=$$ $$\\\\frac{1}{t}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{6}$$ + $$\\\\frac{1}{5}$$ $$=$$ $$\\\\frac{1}{t}$$"],"dependencies":["aa822f8UniMotion9a-h2"],"title":"Setting up the equation","text":"What is the relationship between the two parties?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{30}{11}$$"],"dependencies":["aa822f8UniMotion9a-h3"],"title":"Solving the equation","text":"Solve for $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa822f8UniMotion9a-h5","type":"hint","dependencies":["aa822f8UniMotion9a-h4"],"title":"Double-Checking","text":"Double check that the answer makes sense in terms of the context of the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa82aec5.4q1","title":"Continuous Distribution","body":"Collect the Data","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Continuous Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa82aec5.4q1a","stepAnswer":["$$100$$"],"problemType":"TextBox","stepTitle":"Take the $$10$$ numbers from $$0$$ to $$1$$ inclusive: $$0.7995$$, $$0.0112$$, $$0.0779$$, $$0.8041$$, $$0.5210$$, $$0.5889$$, $$0.4107$$, $$0.1378$$, $$0.6596$$, $$0.5464$$, List them in a table. Please enter the exact number \\"100\\" as answer when you complete the table.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$100$$","hints":{"DefaultPathway":[{"id":"aa82aec5.4q1a-h1","type":"hint","dependencies":[],"title":"Continuous Distribution","text":"Remember to list numbers in increasing order for your convenience, and write $$\\\\frac{title}{heading}$$ for the table!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa82aec5.4q10","title":"Theoretical Distribution: Uniform Distribution and its Third Quartile","body":"The theoretical distribution of X is X ~ $$ \\\\cup (0,1)$$","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Continuous Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa82aec5.4q10a","stepAnswer":["$$0.75$$"],"problemType":"TextBox","stepTitle":"Calculate the theoretical third quartile value for the uniform distribution, please round your answer to $$4$$ decimal places","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.75$$","hints":{"DefaultPathway":[{"id":"aa82aec5.4q10a-h1","type":"hint","dependencies":[],"title":"Theoretical Distribution: Uniform Distribution and its Third Quartile","text":"The formula for computing third quartile is 3rd Quartile $$=$$ $1-P\\\\left(X \\\\leq q_1\\\\right)=1-\\\\int_a**{q_1} \\\\frac{1}{b-a}$ dx for a uniform distribution X~U(a,b)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa82aec5.4q11","title":"Theoretical Distribution: Uniform Distribution and its Median","body":"The theoretical distribution of X is X ~ $$ \\\\cup (0,1)$$","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Continuous Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa82aec5.4q11a","stepAnswer":["$$0.5$$"],"problemType":"TextBox","stepTitle":"Calculate the theoretical median value for the uniform distribution, please round your answer to $$4$$ decimal places","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.5$$","hints":{"DefaultPathway":[{"id":"aa82aec5.4q11a-h1","type":"hint","dependencies":[],"title":"Theoretical Distribution: Uniform Distribution and its Median","text":"The formula for computing the median is $$=$$ (a + b) / $$2$$ where $$b-a$$ is the range of the data set","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa82aec5.4q12","title":"Compare and Contrast Theoretical and Empirical","body":"The theoretical distribution of X is X ~ $$ \\\\cup (0,1)$$, and the empirical value is from collected data in question one: $$0.7995$$, $$0.0112$$, $$0.0779$$, $$0.8041$$, $$0.5210$$, $$0.5889$$, $$0.4107$$, $$0.1378$$, $$0.6596$$, $$0.5464$$","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Continuous Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa82aec5.4q12a","stepAnswer":["They are similar"],"problemType":"MultipleChoice","stepTitle":"Are the empirical values (the data) in the section titled Collect the Data close to the corresponding theoretical values?","stepBody":"","answerType":"string","variabilization":{},"choices":["They are identical","They are similar","They are completely different"],"hints":{"DefaultPathway":[{"id":"aa82aec5.4q12a-h1","type":"hint","dependencies":[],"title":"Compare and Contrast Theoretical and Empirical","text":"Comparing the values from previous result, we can observe that the empirical values are off by a certain percent from the theoretical value, but most are very similar","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa82aec5.4q13","title":"Outliers","body":"Do you notice any potential outliers in our empirical data?","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Continuous Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa82aec5.4q13a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Please answer \\"Yes\\" or \\"No\\"","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"aa82aec5.4q13a-h1","type":"hint","dependencies":[],"title":"Outliers","text":"Recall that any DATA that are less than Q1 - $$1.5(IQR)$$ or more than Q3 + $$1.5(IQR)$$ are potential outliers. IQR means interquartile range.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa82aec5.4q13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-0.45$$"],"dependencies":["aa82aec5.4q13a-h1"],"title":"Outliers below the acceptable range","text":"What is the lower acceptable range? (answer for Q1 - $$1.5(IQR))$$ Please round to 2ed decimal places","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa82aec5.4q13a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.3$$"],"dependencies":["aa82aec5.4q13a-h2"],"title":"Outliers above the acceptable range","text":"What is the higher acceptable range? (answer for Q3 + $$1.5(IQR))$$ Please round to 2ed decimal places","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa82aec5.4q14","title":"Outliers","body":"Do you notice any potential outliers in our theoretical distribution?","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Continuous Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa82aec5.4q14a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Please answer \\"Yes\\" or \\"No\\"","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"aa82aec5.4q14a-h1","type":"hint","dependencies":[],"title":"Outliers","text":"Recall that any DATA that are less than Q1 - $$1.5(IQR)$$ or more than Q3 + $$1.5(IQR)$$ are potential outliers. IQR means interquartile range.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa82aec5.4q14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-0.5$$"],"dependencies":["aa82aec5.4q14a-h1"],"title":"Outliers below the acceptable range","text":"What is the lower acceptable range? (answer for Q1 - $$1.5(IQR))$$ Please round to 2ed decimal places","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa82aec5.4q14a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.5$$"],"dependencies":["aa82aec5.4q14a-h2"],"title":"Outliers above the acceptable range","text":"What is the higher acceptable range? (answer for Q3 + $$1.5(IQR))$$ Please round to 2ed decimal places","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa82aec5.4q15","title":"Sample size","body":"Suppose that the number of values generated was $$100$$, not $$10$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Continuous Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa82aec5.4q15a","stepAnswer":["sample mean gets closer to the population mean"],"problemType":"MultipleChoice","stepTitle":"How would that affect what you would expect the empirical data to be?","stepBody":"","answerType":"string","variabilization":{},"choices":["population mean increases","sample mean gets closer to the population mean","sample mean increase","sample mean gets closer to the population mean","standard deviation of the population decreases"],"hints":{"DefaultPathway":[{"id":"aa82aec5.4q15a-h1","type":"hint","dependencies":[],"title":"Theoretical Distribution: Uniform Distribution","text":"With the increase in sample size, the closer we get to the theoretical distribution","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa82aec5.4q2","title":"Continuous Distribution and its Mean","body":"From collected data in question one: $$0.7995$$, $$0.0112$$, $$0.0779$$, $$0.8041$$, $$0.5210$$, $$0.5889$$, $$0.4107$$, $$0.1378$$, $$0.6596$$, $$0.5464$$","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Continuous Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa82aec5.4q2a","stepAnswer":["$$0.4557$$"],"problemType":"TextBox","stepTitle":"Calculate the mean value for the data, please round your answer to $$4$$ decimal places","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.4557$$","hints":{"DefaultPathway":[{"id":"aa82aec5.4q2a-h1","type":"hint","dependencies":[],"title":"Continuous Distribution and its Mean","text":"The formula for computing $$\\\\bar{x$$} $$=$$ ($x_1,x_2,\\\\ldots,x_n$)/n$$, $$where$$ $$n$$ $$is$$ $$the$$ $$number$$ $$of$$ $$item$$ $$in$$ $$the$$ $$list$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa82aec5.4q3","title":"Continuous Distribution and its Standard Deviation","body":"From collected data in question one: $$0.7995$$, $$0.0112$$, $$0.0779$$, $$0.8041$$, $$0.5210$$, $$0.5889$$, $$0.4107$$, $$0.1378$$, $$0.6596$$, $$0.5464$$","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Continuous Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa82aec5.4q3a","stepAnswer":["$$0.2896$$"],"problemType":"TextBox","stepTitle":"Calculate the standard deviation value for the data, please round your answer to $$4$$ decimal places","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.2896$$","hints":{"DefaultPathway":[{"id":"aa82aec5.4q3a-h1","type":"hint","dependencies":[],"title":"Continuous Distribution and its Standard Deviation","text":"The formula for computing standard deviation is s $$=$$ \\\\sqrt{\\\\frac{1}{N-1} \\\\sum_{i=1}**N (x_i - \\\\overline{x})**2}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa82aec5.4q4","title":"Continuous Distribution and its First Quartile","body":"From collected data in question one: $$0.7995$$, $$0.0112$$, $$0.0779$$, $$0.8041$$, $$0.5210$$, $$0.5889$$, $$0.4107$$, $$0.1378$$, $$0.6596$$, $$0.5464$$","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Continuous Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa82aec5.4q4a","stepAnswer":["$$0.206$$"],"problemType":"TextBox","stepTitle":"Calculate the first quartile for the data, please round your answer to $$4$$ decimal places","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.206$$","hints":{"DefaultPathway":[{"id":"aa82aec5.4q4a-h1","type":"hint","dependencies":[],"title":"Continuous Distribution and its First Quartile","text":"The formula for computing first quartile is 1st Quartile $$=$$ $$n+1$$ * $$\\\\frac{1}{4}$$, where $$n$$ is the number of datas in the set","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa82aec5.4q5","title":"Continuous Distribution and its Third Quartile","body":"From collected data in question one: $$0.7995$$, $$0.0112$$, $$0.0779$$, $$0.8041$$, $$0.5210$$, $$0.5889$$, $$0.4107$$, $$0.1378$$, $$0.6596$$, $$0.5464$$","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Continuous Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa82aec5.4q5a","stepAnswer":["$$0.6419$$"],"problemType":"TextBox","stepTitle":"Calculate the third quartile for the data, please round your answer to $$4$$ decimal places","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.6419$$","hints":{"DefaultPathway":[{"id":"aa82aec5.4q5a-h1","type":"hint","dependencies":[],"title":"Continuous Distribution and its Third Quartile","text":"The formula for computing third quartile is 3rd Quartile $$=$$ $$n+1$$ * $$\\\\frac{3}{4}$$, where $$n$$ is the number of datas in the set","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa82aec5.4q6","title":"Continuous Distribution and its Median","body":"From collected data in question one: $$0.7995$$, $$0.0112$$, $$0.0779$$, $$0.8041$$, $$0.5210$$, $$0.5889$$, $$0.4107$$, $$0.1378$$, $$0.6596$$, $$0.5464$$","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Continuous Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa82aec5.4q6a","stepAnswer":["$$0.5337$$"],"problemType":"TextBox","stepTitle":"Calculate the median for the data, please round your answer to $$4$$ decimal places","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.5337$$","hints":{"DefaultPathway":[{"id":"aa82aec5.4q6a-h1","type":"hint","dependencies":[],"title":"Continuous Distribution and its Median","text":"The formula for computing the median is $$n+1$$ * $$\\\\frac{1}{2}$$, where $$n$$ is the number of datas in the set","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa82aec5.4q7","title":"Theoretical Distribution: Uniform Distribution and its Mean","body":"The theoretical distribution of X is X ~ $$ \\\\cup (0,1)$$","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Continuous Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa82aec5.4q7a","stepAnswer":["$$0.5$$"],"problemType":"TextBox","stepTitle":"Calculate the theoretical mean value for the uniform distribution, please round your answer to $$4$$ decimal places","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.5$$","hints":{"DefaultPathway":[{"id":"aa82aec5.4q7a-h1","type":"hint","dependencies":[],"title":"Theoretical Distribution: Uniform Distribution and its Mean","text":"The formula for computing the mean is \u03bc $$=$$ (a + b) / $$2$$ where $$b-a$$ is the range of the data set","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa82aec5.4q8","title":"Theoretical Distribution: Uniform Distribution and its Standard Deviation","body":"The theoretical distribution of X is X ~ $$ \\\\cup (0,1)$$","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Continuous Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa82aec5.4q8a","stepAnswer":["$$0.5$$"],"problemType":"TextBox","stepTitle":"Calculate the theoretical standard deviation value for the uniform distribution, please round your answer to $$4$$ decimal places","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.5$$","hints":{"DefaultPathway":[{"id":"aa82aec5.4q8a-h1","type":"hint","dependencies":[],"title":"Theoretical Distribution: Uniform Distribution and its Standard Deviation","text":"The formula for computing the variance of the distribution is $${\\\\sigma}^2$$ $$=$$ (b - a)**2 / $$12$$, where $$b-a$$ is the range of the data set","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa82aec5.4q8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{12}$$"],"dependencies":["aa82aec5.4q8a-h1"],"title":"Theoretical Distribution: Uniform Distribution and its Standard Deviation","text":"What is the variance for the distribution? Write your answer as a fraction","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa82aec5.4q8a-h3","type":"hint","dependencies":["aa82aec5.4q8a-h2"],"title":"Theoretical Distribution: Uniform Distribution and its Standard Deviation","text":"The formula for computing the standard deviation of the distribution is $$\u03c3=$$ $$\\\\sqrt{{\\\\sigma}^2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa82aec5.4q9","title":"Theoretical Distribution: Uniform Distribution and its First Quartile","body":"The theoretical distribution of X is X ~ $$ \\\\cup (0,1)$$","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.4 Continuous Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa82aec5.4q9a","stepAnswer":["$$0.25$$"],"problemType":"TextBox","stepTitle":"Calculate the theoretical first quartile value for the uniform distribution, please round your answer to $$4$$ decimal places","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.25$$","hints":{"DefaultPathway":[{"id":"aa82aec5.4q9a-h1","type":"hint","dependencies":[],"title":"Theoretical Distribution: Uniform Distribution and its First Quartile","text":"The formula for computing first quartile is 1st Quartile $$=$$ $P\\\\left(X \\\\leq q_1\\\\right)=\\\\int_a**{q_1} \\\\frac{1}{b-a}$ dx for a uniform distribution X~U(a,b)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa97d5brate1","title":"Finding Average Rate of Change","body":"For the following exercise, find the average rate of change of the function on the interval specified for real number $$h$$ in simplest form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Rates of Change and Behavior of Graphs","courseName":"OpenStax: College Algebra","steps":[{"id":"aa97d5brate1a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"$$p(x)=3x+4$$ on $$[2,2+h]$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"aa97d5brate1a-h1","type":"hint","dependencies":[],"title":"$$\\\\frac{Equation}{Substitute}$$","text":"The equation for calculating the rate of change is $$\\\\frac{p\\\\left(x_1\\\\right)-p\\\\left(x_2\\\\right)}{x_1-x_2}$$. So to find the answer, we can start by plugging in $$2+h$$ for $$x_1$$ and $$2$$ for $$x_2$$, and we get the expression $$\\\\frac{3\\\\left(2+h\\\\right)+4-3\\\\left(2\\\\right)+4}{2+h-2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["aa97d5brate1a-h1"],"title":"Simplify","text":"Simplify the expression $$\\\\frac{3\\\\left(2+h\\\\right)+4-3\\\\left(2\\\\right)+4}{2+h-2}$$. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa97d5brate10","title":"Finding Average Rate of Change","body":"For the following exercise, find the average rate of change of the function on the interval specified for real number $$h$$ in simplest form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Rates of Change and Behavior of Graphs","courseName":"OpenStax: College Algebra","steps":[{"id":"aa97d5brate10a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"$$f(x)=x^2$$ on [1, 5]","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"aa97d5brate10a-h1","type":"hint","dependencies":[],"title":"Equation/Substitute","text":"The equation for calculuating the rate of change is $$\\\\frac{f{\\\\left(x_1\\\\right)}-f{\\\\left(x_2\\\\right)}}{x_1-x_2}$$. So to find the answer, we can start by plugging in $$5$$ for $$x_1$$ and $$1$$ for $$x_2$$, and we get the expression $$\\\\frac{5^2-1^2}{5-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["aa97d5brate10a-h1"],"title":"Calculate","text":"Evaluate the expression $$\\\\frac{5^2-1^2}{5-1}$$. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa97d5brate11","title":"Finding an Average Rate of Change as an Expression","body":"Find the average rate of change of the function on the interval specified for real number $$b$$ in simplest form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Rates of Change and Behavior of Graphs","courseName":"OpenStax: College Algebra","steps":[{"id":"aa97d5brate11a","stepAnswer":["$$4\\\\left(b+1\\\\right)$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=4x^2-7$$ on [1,b]","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$4\\\\left(b+1\\\\right)$$","choices":["$$4\\\\left(b-3\\\\right)$$","$$4\\\\left(b-1\\\\right)$$","$$4\\\\left(b+1\\\\right)$$"],"hints":{"DefaultPathway":[{"id":"aa97d5brate11a-h1","type":"hint","dependencies":[],"title":"The difference $$y_2-y_1$$","text":"We should first calculate the difference of the function evaluated at the two endpoints.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate11a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$4b^2-4$$"],"dependencies":["aa97d5brate11a-h1"],"title":"The difference $$y_2-y_1$$","text":"What is the difference between f(b) and f(1)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$4b^2-10$$","$$4b^2-3$$","$$4b^2-4$$"]},{"id":"aa97d5brate11a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$4\\\\left(b+1\\\\right) \\\\left(b-1\\\\right)$$"],"dependencies":["aa97d5brate11a-h2"],"title":"Rearrangement","text":"How can we rearrange the above expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$4{\\\\left(b+1\\\\right)}^2$$","$$4{\\\\left(b-1\\\\right)}^2$$","$$4\\\\left(b+1\\\\right) \\\\left(b-1\\\\right)$$"]},{"id":"aa97d5brate11a-h4","type":"hint","dependencies":["aa97d5brate11a-h3"],"title":"The difference $$x_2-x_1$$","text":"We should then calculate the difference of the two endpoints $$x$$ values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$b-1$$"],"dependencies":["aa97d5brate11a-h4"],"title":"The difference $$x_2-x_1$$","text":"What is the difference between $$b$$ and 1?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate11a-h6","type":"hint","dependencies":["aa97d5brate11a-h5"],"title":"The ratio","text":"We should last find the ratio of the difference in $$y$$ and the difference in $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate11a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$4\\\\left(b+1\\\\right)$$"],"dependencies":["aa97d5brate11a-h6"],"title":"The ratio","text":"What is the ratio of the difference in $$y$$ and the difference in $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$4\\\\left(b-3\\\\right)$$","$$4\\\\left(b-1\\\\right)$$","$$4\\\\left(b+1\\\\right)$$"]}]}}]},{"id":"aa97d5brate12","title":"Finding an Average Rate of Change as an Expression","body":"Find the average rate of change of the function on the interval specified for real number $$b$$ in simplest form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Rates of Change and Behavior of Graphs","courseName":"OpenStax: College Algebra","steps":[{"id":"aa97d5brate12a","stepAnswer":["$$2\\\\left(b+4\\\\right)$$"],"problemType":"MultipleChoice","stepTitle":"$$g(x)=2x^2-9$$ on [4,b]","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2\\\\left(b+4\\\\right)$$","choices":["$$2\\\\left(b-6\\\\right)$$","$$2\\\\left(b-4\\\\right)$$","$$2\\\\left(b+4\\\\right)$$"],"hints":{"DefaultPathway":[{"id":"aa97d5brate12a-h1","type":"hint","dependencies":[],"title":"The difference $$y_2-y_1$$","text":"We should first calculate the difference of the function evaluated at the two endpoints.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate12a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2b^2-32$$"],"dependencies":["aa97d5brate12a-h1"],"title":"The difference $$y_2-y_1$$","text":"What is the difference between f(b) and f(4)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2b^2-32$$","$$2b^2+32$$","$$2b^2-14$$"]},{"id":"aa97d5brate12a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2\\\\left(b+4\\\\right) \\\\left(b-4\\\\right)$$"],"dependencies":["aa97d5brate12a-h2"],"title":"Rearrangement","text":"How can we rearrange the above expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2{\\\\left(b-4\\\\right)}^2$$","$$2\\\\left(b+4\\\\right) \\\\left(b-4\\\\right)$$","$$2{\\\\left(b+4\\\\right)}^2$$"]},{"id":"aa97d5brate12a-h4","type":"hint","dependencies":["aa97d5brate12a-h3"],"title":"The difference $$x_2-x_1$$","text":"We should then calculate the difference of the two endpoints $$x$$ values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate12a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$b-4$$"],"dependencies":["aa97d5brate12a-h4"],"title":"The difference $$x_2-x_1$$","text":"What is the difference between $$b$$ and 4?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate12a-h6","type":"hint","dependencies":["aa97d5brate12a-h5"],"title":"The ratio","text":"We should last find the ratio of the difference in $$y$$ and the difference in $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate12a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2\\\\left(b+4\\\\right)$$"],"dependencies":["aa97d5brate12a-h6"],"title":"The ratio","text":"What is the ratio of the difference in $$y$$ and the difference in $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2\\\\left(b-6\\\\right)$$","$$2\\\\left(b-4\\\\right)$$","$$2\\\\left(b+4\\\\right)$$"]}]}}]},{"id":"aa97d5brate13","title":"Computing Average Rate of Change for a Function Expressed as a Formula","body":"Find the average rate of change of the function on the interval specified.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Rates of Change and Behavior of Graphs","courseName":"OpenStax: College Algebra","steps":[{"id":"aa97d5brate13a","stepAnswer":["$$-4$$"],"problemType":"TextBox","stepTitle":"$$h(x)=5-2x^2$$ on [-2,4]","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-4$$","hints":{"DefaultPathway":[{"id":"aa97d5brate13a-h1","type":"hint","dependencies":[],"title":"The difference $$y_2-y_1$$","text":"We should first calculate the difference of the function evaluated at the two endpoints.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-24$$"],"dependencies":["aa97d5brate13a-h1"],"title":"The difference $$y_2-y_1$$","text":"What is the difference between h(4) and $$h(-2)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate13a-h3","type":"hint","dependencies":["aa97d5brate13a-h2"],"title":"The difference $$x_2-x_1$$","text":"We should then calculate the difference of the two endpoints $$x$$ values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["aa97d5brate13a-h3"],"title":"The difference $$x_2-x_1$$","text":"What is the difference between $$4$$ and -2?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate13a-h5","type":"hint","dependencies":["aa97d5brate13a-h4"],"title":"The ratio","text":"We should last find the ratio of the difference in $$y$$ and the difference in $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate13a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["aa97d5brate13a-h5"],"title":"The ratio","text":"What is the ratio of the difference in $$y$$ and the difference in $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa97d5brate14","title":"Computing Average Rate of Change for a Function Expressed as a Formula","body":"Find the average rate of change of the function on the interval specified.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Rates of Change and Behavior of Graphs","courseName":"OpenStax: College Algebra","steps":[{"id":"aa97d5brate14a","stepAnswer":["$$12$$"],"problemType":"TextBox","stepTitle":"$$q(x)=x^3$$ on [-4,2]","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12$$","hints":{"DefaultPathway":[{"id":"aa97d5brate14a-h1","type":"hint","dependencies":[],"title":"The difference $$y_2-y_1$$","text":"We should first calculate the difference of the function evaluated at the two endpoints.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$72$$"],"dependencies":["aa97d5brate14a-h1"],"title":"The difference $$y_2-y_1$$","text":"What is the difference between q(2) and $$q(-4)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate14a-h3","type":"hint","dependencies":["aa97d5brate14a-h2"],"title":"The difference $$x_2-x_1$$","text":"We should then calculate the difference of the two endpoints $$x$$ values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate14a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["aa97d5brate14a-h3"],"title":"The difference $$x_2-x_1$$","text":"What is the difference between $$2$$ and -4?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate14a-h5","type":"hint","dependencies":["aa97d5brate14a-h4"],"title":"The ratio","text":"We should last find the ratio of the difference in $$y$$ and the difference in $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate17a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-0.167$$"],"dependencies":["aa97d5brate17a-h5"],"title":"The ratio","text":"What is the ratio of the difference in $$y$$ and the difference in $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa97d5brate18","title":"Computing Average Rate of Change for a Function Expressed as a Formula","body":"Find the average rate of change of the function on the interval specified.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Rates of Change and Behavior of Graphs","courseName":"OpenStax: College 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So to find the answer, we can start by plugging in $$3+h$$ for $$x_1$$ and $$3$$ for $$x_2$$, and we get the expression $$\\\\frac{4\\\\left(3+h\\\\right)+2-4\\\\left(3\\\\right)+2}{3+h-3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["aa97d5brate2a-h1"],"title":"Simplify","text":"Simplify the expression $$\\\\frac{4\\\\left(3+h\\\\right)+2-4\\\\left(3\\\\right)+2}{3+h-3}$$. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa97d5brate20","title":"Finding an Average Rate of Change as an Expression","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Rates of Change and Behavior of Graphs","courseName":"OpenStax: College Algebra","steps":[{"id":"aa97d5brate20a","stepAnswer":["$$a+7$$"],"problemType":"TextBox","stepTitle":"Find the average rate of change of $$f(x)=x^2+2x-8$$ on the interval [5,a] in simplest forms in terms of a.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$a+7$$","hints":{"DefaultPathway":[{"id":"aa97d5brate20a-h1","type":"hint","dependencies":[],"title":"The difference $$y_2-y_1$$","text":"We should first calculate the difference of the function evaluated at the two endpoints.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate20a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$a^2+2a-35$$"],"dependencies":["aa97d5brate20a-h1"],"title":"The difference $$y_2-y_1$$","text":"What is the difference between f(a) and f(5)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$a^2+2a+19$$","$$a^2+2a-35$$","$$a^2+2a-49$$"]},{"id":"aa97d5brate20a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\left(a+7\\\\right) \\\\left(a-5\\\\right)$$"],"dependencies":["aa97d5brate20a-h2"],"title":"Rearrangement","text":"How can we rearrange the above expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\left(a+7\\\\right) \\\\left(a-5\\\\right)$$","$$\\\\left(a+5\\\\right) \\\\left(a-7\\\\right)$$","$$\\\\left(a+7\\\\right) \\\\left(a-7\\\\right)$$"]},{"id":"aa97d5brate20a-h4","type":"hint","dependencies":["aa97d5brate20a-h3"],"title":"The difference $$x_2-x_1$$","text":"We should then calculate the difference of the two endpoints $$x$$ values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate20a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["a-5"],"dependencies":["aa97d5brate20a-h4"],"title":"The difference $$x_2-x_1$$","text":"What is the difference between a and 5?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate20a-h6","type":"hint","dependencies":["aa97d5brate20a-h5"],"title":"The ratio","text":"We should last find the ratio of the difference in $$y$$ and the difference in $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate20a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$a+7$$"],"dependencies":["aa97d5brate20a-h6"],"title":"The ratio","text":"What is the ratio of the difference in $$y$$ and the difference in $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa97d5brate3","title":"Finding Average Rate of Change","body":"For the following exercise, find the average rate of change of the function on the interval specified for real number $$h$$ in simplest form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Rates of Change and Behavior of Graphs","courseName":"OpenStax: College Algebra","steps":[{"id":"aa97d5brate3a","stepAnswer":["$$4x+2h$$"],"problemType":"TextBox","stepTitle":"$$f(x)=2x^2+1$$ on $$[x,x+h]$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4x+2h$$","hints":{"DefaultPathway":[{"id":"aa97d5brate3a-h1","type":"hint","dependencies":[],"title":"$$\\\\frac{Equation}{Substitute}$$","text":"The equation for calculuating the rate of change is $$\\\\frac{f{\\\\left(x_1\\\\right)}-f{\\\\left(x_2\\\\right)}}{x_1-x_2}$$. So to find the answer, we can start by plugging in $$x+h$$ for $$x_1$$ and $$x$$ for $$x_2$$, and we get the expression $$\\\\frac{{2\\\\left(x+h\\\\right)}^2+1-2x^2+1}{x+h-x}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4x+2h$$"],"dependencies":["aa97d5brate3a-h1"],"title":"Simplify","text":"Simplify the expression $$\\\\frac{{2\\\\left(x+h\\\\right)}^2+1-2x^2+1}{x+h-x}$$. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa97d5brate4","title":"Finding Average Rate of Change","body":"For the following exercise, find the average rate of change of the function on the interval specified for real number $$h$$ in simplest form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Rates of Change and Behavior of Graphs","courseName":"OpenStax: College Algebra","steps":[{"id":"aa97d5brate4a","stepAnswer":["$$6x+3h$$"],"problemType":"TextBox","stepTitle":"$$f(x)=3x^2-2$$ on $$[x,x+h]$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6x+3h$$","hints":{"DefaultPathway":[{"id":"aa97d5brate4a-h1","type":"hint","dependencies":[],"title":"$$\\\\frac{Equation}{Substitute}$$","text":"The equation for calculuating the rate of change is $$\\\\frac{f{\\\\left(x_1\\\\right)}-f{\\\\left(x_2\\\\right)}}{x_1-x_2}$$. So to find the answer, we can start by plugging in $$x+h$$ for $$x_1$$ and $$x$$ for $$x_2$$, and we get the expression $$\\\\frac{{3\\\\left(x+h\\\\right)}^2-2-3x^2-2}{x+h-x}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6x+3h$$"],"dependencies":["aa97d5brate4a-h1"],"title":"Simplify","text":"Simplify the expression $$\\\\frac{{3\\\\left(x+h\\\\right)}^2-2-3x^2-2}{x+h-x}$$. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa97d5brate5","title":"Finding Average Rate of Change","body":"For the following exercise, find the average rate of change of the function on the interval specified for real number $$h$$ in simplest form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Rates of Change and Behavior of Graphs","courseName":"OpenStax: College Algebra","steps":[{"id":"aa97d5brate5a","stepAnswer":["$$\\\\frac{-1}{\\\\operatorname{13}\\\\left(13+h\\\\right)}$$"],"problemType":"MultipleChoice","stepTitle":"$$a(t)=\\\\frac{1}{t+4}$$ on $$[9,9+h]$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{-1}{\\\\operatorname{13}\\\\left(13+h\\\\right)}$$","choices":["$$\\\\frac{1}{\\\\operatorname{13}\\\\left(13+h\\\\right)}$$","$$\\\\frac{-1}{\\\\operatorname{13}\\\\left(13+h\\\\right)}$$","$$\\\\frac{1}{13h}$$","$$\\\\frac{-1}{13h}$$"],"hints":{"DefaultPathway":[{"id":"aa97d5brate5a-h1","type":"hint","dependencies":[],"title":"$$\\\\frac{Equation}{Substitute}$$","text":"The equation for calculuating the rate of change is $$\\\\frac{a\\\\left(x_1\\\\right)-a\\\\left(x_2\\\\right)}{x_1-x_2}$$. So to find the answer, we can start by plugging in $$9+h$$ for $$x_1$$ and $$9$$ for $$x_2$$, and we get the expression $$\\\\frac{\\\\frac{1}{9+h+4}-\\\\frac{1}{9+4}}{9+h-9}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate5a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{-1}{\\\\operatorname{13}\\\\left(13+h\\\\right)}$$"],"dependencies":["aa97d5brate5a-h1"],"title":"Simplify","text":"Simplify the expression $$\\\\frac{\\\\frac{1}{9+h+4}-\\\\frac{1}{9+4}}{9+h-9}$$. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{1}{\\\\operatorname{13}\\\\left(13+h\\\\right)}$$","$$\\\\frac{-1}{\\\\operatorname{13}\\\\left(13+h\\\\right)}$$","$$\\\\frac{1}{13h}$$","$$\\\\frac{-1}{13h}$$"]}]}}]},{"id":"aa97d5brate6","title":"Finding Average Rate of Change","body":"For the following exercise, find the average rate of change of the function on the interval specified for real number $$h$$ in simplest form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Rates of Change and Behavior of Graphs","courseName":"OpenStax: College Algebra","steps":[{"id":"aa97d5brate6a","stepAnswer":["$$\\\\frac{-1}{4h+16}$$"],"problemType":"MultipleChoice","stepTitle":"$$b(x)=\\\\frac{1}{x+3}$$ on $$[1,1+h]$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{-1}{4h+16}$$","choices":["$$\\\\frac{1}{4}$$","$$\\\\frac{1}{4h}$$","$$\\\\frac{1}{4h+16}$$","$$\\\\frac{-1}{4h+16}$$"],"hints":{"DefaultPathway":[{"id":"aa97d5brate6a-h1","type":"hint","dependencies":[],"title":"$$\\\\frac{Equation}{Substitute}$$","text":"The equation for calculuating the rate of change is $$\\\\frac{b\\\\left(x_1\\\\right)-b\\\\left(x_2\\\\right)}{x_1-x_2}$$. So to find the answer, we can start by plugging in $$1+h$$ for $$x_1$$ and $$1$$ for $$x_2$$, and we get the expression $$\\\\frac{\\\\frac{1}{1+h+3}-\\\\frac{1}{1+3}}{1+h-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate6a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{-1}{4h+16}$$"],"dependencies":["aa97d5brate6a-h1"],"title":"Simplify","text":"Simplify the expression $$\\\\frac{\\\\frac{1}{1+h+3}-\\\\frac{1}{1+3}}{1+h-1}$$. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{1}{4}$$","$$\\\\frac{1}{4h}$$","$$\\\\frac{1}{4h+16}$$","$$\\\\frac{-1}{4h+16}$$"]}]}}]},{"id":"aa97d5brate7","title":"Finding Average Rate of Change","body":"For the following exercise, find the average rate of change of the function on the interval specified for real number $$h$$ in simplest form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Rates of Change and Behavior of Graphs","courseName":"OpenStax: College Algebra","steps":[{"id":"aa97d5brate7a","stepAnswer":["$$9+9h+3h^2$$"],"problemType":"TextBox","stepTitle":"$$j(x)=3x^3$$ on $$[1,1+h]$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9+9h+3h^2$$","hints":{"DefaultPathway":[{"id":"aa97d5brate7a-h1","type":"hint","dependencies":[],"title":"Equation/Substitute","text":"The equation for calculuating the rate of change is $$\\\\frac{j\\\\left(x_1\\\\right)-j\\\\left(x_2\\\\right)}{x_1-x_2}$$. So to find the answer, we can start by plugging in $$1+h$$ for $$x_1$$ and $$1$$ for $$x_2$$, and we get the expression $$\\\\frac{{3\\\\left(1+h\\\\right)}^3-3\\\\times1^3}{1+h-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9+9h+3h^2$$"],"dependencies":["aa97d5brate7a-h1"],"title":"Simplify","text":"Simplify the expression $$\\\\frac{{3\\\\left(1+h\\\\right)}^3-3\\\\times1^3}{1+h-1}$$. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aa97d5brate7a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1+3h+3h^2+h^3$$"],"dependencies":[],"title":"Simplify","text":"What is $${\\\\left(1+h\\\\right)}^3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"aa97d5brate8","title":"Finding Average Rate of Change","body":"For the following exercise, find the average rate of change of the function on the interval specified for real number $$h$$ in simplest form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Rates of Change and Behavior of Graphs","courseName":"OpenStax: College Algebra","steps":[{"id":"aa97d5brate8a","stepAnswer":["$$48+24h+4h^2$$"],"problemType":"TextBox","stepTitle":"$$r(t)=4t^3$$ on $$[2,2+h]$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$48+24h+4h^2$$","hints":{"DefaultPathway":[{"id":"aa97d5brate8a-h1","type":"hint","dependencies":[],"title":"$$\\\\frac{Equation}{Substitute}$$","text":"The equation for calculuating the rate of change is $$\\\\frac{r\\\\left(x_1\\\\right)-r\\\\left(x_2\\\\right)}{x_1-x_2}$$. So to find the answer, we can start by plugging in $$2+h$$ for $$x_1$$ and $$2$$ for $$x_2$$, and we get the expression $$\\\\frac{{4\\\\left(2+h\\\\right)}^3-3\\\\times2^3}{2+h-2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$48+24h+4h^2$$"],"dependencies":[],"title":"Simplify","text":"Simplify the expression $$\\\\frac{{4\\\\left(2+h\\\\right)}^3-3\\\\times2^3}{2+h-2}$$. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aa97d5brate8a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8+12h+6h^2+h^3$$"],"dependencies":[],"title":"Simplify","text":"What is $${\\\\left(2+h\\\\right)}^3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate8a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$32+48h+24h^2+4h^3$$"],"dependencies":["aa97d5brate8a-h2-s1"],"title":"multiply","text":"What is $$4{\\\\left(2+h\\\\right)}^3$$, using the expression you get from the last part?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate8a-h2-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$48h+24h^2+4h^3$$"],"dependencies":["aa97d5brate8a-h2-s2"],"title":"Subtract","text":"Subtract $$4\\\\times2^3=32$$ from $$32+48h+24h^2+4h^3$$. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate8a-h2-s4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$48+24h+4h^2$$"],"dependencies":["aa97d5brate8a-h2-s3"],"title":"Divide","text":"The last step is to divide $$48h+24h^2+4h^3$$ by $$2+h-2$$. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"aa97d5brate9","title":"Finding Average Rate of Change","body":"For the following exercise, find the average rate of change of the function on the interval specified for real number $$h$$ in simplest form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Rates of Change and Behavior of Graphs","courseName":"OpenStax: College Algebra","steps":[{"id":"aa97d5brate9a","stepAnswer":["$$4x+2h-3$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{f{\\\\left(x+h\\\\right)}-f{\\\\left(x\\\\right)}}{h}$$ given $$f(x)=2x^2$$ - $$3x$$ on [x, x+h]","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4x+2h-3$$","hints":{"DefaultPathway":[{"id":"aa97d5brate9a-h1","type":"hint","dependencies":[],"title":"$$\\\\frac{Equation}{Substitute}$$","text":"The equation for calculuating the rate of change is $$\\\\frac{f{\\\\left(x_1\\\\right)}-f{\\\\left(x_2\\\\right)}}{x_1-x_2}$$. So to find the answer, we can start by plugging in $$x+h$$ for $$x_1$$ and $$x$$ for $$x_2$$, and we get the expression ((2(x+h)**2-3(x+h))-(2x**2 -3x))/(x+h - x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate9a-h2","type":"hint","dependencies":["aa97d5brate9a-h1"],"title":"Simplify","text":"Simplify the numerator, and we get the expression $$4xh+2h^2-3h$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brate9a-h3","type":"hint","dependencies":["aa97d5brate9a-h2"],"title":"Divide","text":"$$\\\\frac{Divide}{Cross}$$ out the $$h$$ terms in the numerator and denominator, and we get our answer $$4x+2h-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa97d5brates1","title":"Computing Average Rate of Change from a Table","body":"After picking up a friend who lives $$10$$ miles away and leaving on a trip, Anna records her distance from home over time. The values are shown in the table below.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Rates of Change and Behavior of Graphs","courseName":"OpenStax: College Algebra","steps":[{"id":"aa97d5brates1a","stepAnswer":["$$47$$"],"problemType":"TextBox","stepTitle":"Find her average speed over the first $$6$$ hours.","stepBody":"Ignore units when you write your answer.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$47$$","hints":{"DefaultPathway":[{"id":"aa97d5brates1a-h1","type":"hint","dependencies":[],"title":"Interpreting the Problem","text":"The average speed is the average rate of change in the first $$6$$ hours.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brates1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$282$$"],"dependencies":["aa97d5brates1a-h1"],"title":"Identifying distance","text":"What is the distance that Anna travels in the first $$6$$ hours?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brates1a-h3","type":"hint","dependencies":["aa97d5brates1a-h2"],"title":"Average rate of change","text":"To find the average rate of change, we calculate the ratio (change in y)/(change in x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brates1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$47$$"],"dependencies":["aa97d5brates1a-h3"],"title":"Average rate of change","text":"What is $$\\\\frac{282}{6}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa97d5brates2","title":"Computing Average Rate of Change for a Function Expressed as a Formula","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Rates of Change and Behavior of Graphs","courseName":"OpenStax: College Algebra","steps":[{"id":"aa97d5brates2a","stepAnswer":["$$\\\\frac{49}{8}$$"],"problemType":"TextBox","stepTitle":"Compute the average rate of change of $$f(x)=x^2-\\\\frac{1}{x}$$ on the interval[2,4].","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{49}{8}$$","hints":{"DefaultPathway":[{"id":"aa97d5brates2a-h1","type":"hint","dependencies":[],"title":"average rate of change","text":"To find the average rate of change, we calculate the change in $$y$$, the change in $$x$$, and the average rate of change is the ratio (change in y)/(change in x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brates2a-h2","type":"hint","dependencies":["aa97d5brates2a-h1"],"title":"Computing endpoints","text":"We can start by computing the function values at each endpoint of the interval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brates2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{7}{2}$$"],"dependencies":["aa97d5brates2a-h2"],"title":"Computing left endpoint","text":"What is f(2)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aa97d5brates2a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{7}{2}$$"],"dependencies":[],"title":"Computing left endpoint","text":"To find f(2), we plug in $$2$$ for every $$x$$ in the equation. What is $$2^2-\\\\frac{1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aa97d5brates2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{63}{4}$$"],"dependencies":["aa97d5brates2a-h2"],"title":"Computing right endpoint","text":"What is f(4)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aa97d5brates2a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{63}{4}$$"],"dependencies":[],"title":"Computing right endpoint","text":"To find f(4), we plug in $$4$$ for every $$x$$ in the equation. What is $$4^2-\\\\frac{1}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aa97d5brates2a-h5","type":"hint","dependencies":["aa97d5brates2a-h3","aa97d5brates2a-h4"],"title":"Average rate of change","text":"The average rate of change $$=\\\\frac{f{\\\\left(4\\\\right)}-f{\\\\left(2\\\\right)}}{4-2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brates2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{49}{8}$$"],"dependencies":["aa97d5brates2a-h5"],"title":"Computing average rate of change","text":"What is the average rate of change, plugging in $$f(2)=\\\\frac{7}{2}$$ and $$f(4)=\\\\frac{63}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa97d5brates3","title":"Finding the Average Rate of Change of a Force","body":"The electrostatic force F, measured in newtons, between two charged particles can be related to the distance between the particles $$d$$, in centimeters, by the formula $$F(d)=\\\\frac{2}{d^2}$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Rates of Change and Behavior of Graphs","courseName":"OpenStax: College Algebra","steps":[{"id":"aa97d5brates3a","stepAnswer":["$$\\\\frac{-1}{9}$$"],"problemType":"TextBox","stepTitle":"Find the average rate of change of force if the distance between the particles is increased from $$2$$ cm to $$6$$ cm.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-1}{9}$$","hints":{"DefaultPathway":[{"id":"aa97d5brates3a-h1","type":"hint","dependencies":[],"title":"Interpreting the problem","text":"We are computing the average rate of change of $$F(d)=\\\\frac{2}{d^2}$$ on the interval [2,6].","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brates3a-h2","type":"hint","dependencies":["aa97d5brates3a-h1"],"title":"average rate of change","text":"To find the average rate of change, we calculate the change in $$y$$, the change in $$x$$, and the average rate of change is the ratio (change in y)/(change in x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brates3a-h3","type":"hint","dependencies":["aa97d5brates3a-h2"],"title":"Computing endpoints","text":"We can start by computing the function values at each endpoint of the interval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brates3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["aa97d5brates3a-h3"],"title":"Computing left endpoint","text":"What is F(2)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aa97d5brates3a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":[],"title":"Computing left endpoint","text":"To find F(2), we plug in $$2$$ for every $$d$$ in the equation. What is $$\\\\frac{2}{2^2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aa97d5brates3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{18}$$"],"dependencies":["aa97d5brates3a-h3"],"title":"Computing right endpoint","text":"What is F(6)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aa97d5brates3a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{18}$$"],"dependencies":[],"title":"Computing right endpoint","text":"To find F(6), we plug in $$6$$ for every $$d$$ in the equation. What is $$\\\\frac{2}{6^2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aa97d5brates3a-h6","type":"hint","dependencies":["aa97d5brates3a-h4","aa97d5brates3a-h5"],"title":"Average rate of change","text":"The average rate of change $$=\\\\frac{F\\\\left(6\\\\right)-F\\\\left(2\\\\right)}{6-2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brates3a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{9}$$"],"dependencies":["aa97d5brates3a-h6"],"title":"Computing average rate of change","text":"What is the average rate of change, plugging in $$F(2)=\\\\frac{1}{2}$$ and $$F(6)=\\\\frac{1}{18}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa97d5brates4","title":"Finding an Average Rate of Change as an Expression","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Rates of Change and Behavior of Graphs","courseName":"OpenStax: College Algebra","steps":[{"id":"aa97d5brates4a","stepAnswer":["$$a+3$$"],"problemType":"TextBox","stepTitle":"Find the average rate of change of $$g(t)=t^2+3t+1$$ on the interval [0,a].","stepBody":"The answer will be an expression involving a in simplest form.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$a+3$$","hints":{"DefaultPathway":[{"id":"aa97d5brates4a-h1","type":"hint","dependencies":[],"title":"average rate of change","text":"To find the average rate of change, we calculate the change in $$y$$, the change in $$x$$, and the average rate of change is the ratio (change in y)/(change in x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brates4a-h2","type":"hint","dependencies":["aa97d5brates4a-h1"],"title":"Computing endpoints","text":"We can start by computing the function values at each endpoint of the interval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brates4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["aa97d5brates4a-h2"],"title":"Computing left endpoint","text":"What is g(0)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aa97d5brates4a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Computing left endpoint","text":"To find g(0), we plug in $$0$$ for every $$t$$ in the equation. What is $$0^2+3\\\\times0+1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aa97d5brates4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$a^2+3a+1$$"],"dependencies":["aa97d5brates4a-h2"],"title":"Computing right endpoint","text":"What is g(a)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aa97d5brates4a-h4-s1","type":"hint","dependencies":[],"title":"Computing right endpoint","text":"To find g(a), we plug in a for every $$t$$ in the equation. So we get the expression $$a^2+3a+1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aa97d5brates4a-h5","type":"hint","dependencies":["aa97d5brates4a-h3","aa97d5brates4a-h4"],"title":"Average rate of change","text":"The average rate of change =((g(a)-g(0))/(a-0).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brates4a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$a+3$$"],"dependencies":["aa97d5brates4a-h5"],"title":"Computing average rate of change","text":"What is the average rate of change, plugging in $$g(0)=1$$ and $$g(a)=a^2+3a+1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aa97d5brates4a-h6-s1","type":"hint","dependencies":[],"title":"Computing average rate of change","text":"g(0) and g(a), the average rate of $$change=\\\\frac{a^2+3a+1-1}{a-0}=\\\\frac{a^2+3a}{a}=\\\\frac{a\\\\left(a+3\\\\right)}{a}=a+3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"aa97d5brates5","title":"Finding Local Extrema from a Graph","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Rates of Change and Behavior of Graphs","courseName":"OpenStax: College Algebra","steps":[{"id":"aa97d5brates5a","stepAnswer":["local minimum: $$(2,3)$$ local maximum: $$(-3,-2)$$"],"problemType":"MultipleChoice","stepTitle":"Use the graph of the function $$f(x)=\\\\frac{2}{x}+\\\\frac{x}{3}$$ to estimate the local extrema.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"local minimum: $$(2,3)$$ local maximum: $$(-3,-2)$$","choices":["local minimum: $$(-1,0)$$ local maximum: $$(0,1)$$","local minimum: $$(-3,-2)$$ local maximum: $$(2,3)$$","local minimum: $$(0,1)$$ local maximum: $$(-1,0)$$","local minimum: $$(2,3)$$ local maximum: $$(-3,-2)$$","local minimum: $$(2,3)$$ local maximum: $$(-3,-2)$$"],"hints":{"DefaultPathway":[{"id":"aa97d5brates5a-h1","type":"hint","dependencies":[],"title":"Graph","text":"We start by graphing the function out using technology, and we get the following graph:\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brates5a-h2","type":"hint","dependencies":["aa97d5brates5a-h1"],"title":"Local mimimum","text":"A function f has a local minimum at $$x=b$$ if there exists an interval (a,c) with $$a<b<c$$ such that, for any $$x$$ in the interval (a,c), $$f(x) \\\\geq f(b)$$. In other words, a local minimum is a low point on the graph.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brates5a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(2,3)$$"],"dependencies":["aa97d5brates5a-h2"],"title":"Local mimimum","text":"In which interval is the low point of the graph located?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(-1,0)$$","$$(0,1)$$","$$(-3,-2)$$","$$(2,3)$$"]},{"id":"aa97d5brates5a-h4","type":"hint","dependencies":["aa97d5brates5a-h3"],"title":"Local Maximum","text":"A function f has a local maximum at $$x=b$$ if there exists an interval (a,c) with $$a<b<c$$ such that, for any $$x$$ in the interval (a,c), $$f(x) \\\\leq f(b)$$. In other words, a local minimum is a high point on the graph.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brates5a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(-3,-2)$$"],"dependencies":["aa97d5brates5a-h4"],"title":"Local Maximum","text":"In which interval is the high point of the graph located?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(-1,0)$$","$$(0,1)$$","$$(-3,-2)$$","$$(2,3)$$"]}]}}]},{"id":"aa97d5brates6","title":"Finding Local Maxima and Minima from a Graph","body":"For the function f whose graph is shown below, find all local maxima and minima.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Rates of Change and Behavior of Graphs","courseName":"OpenStax: College Algebra","steps":[{"id":"aa97d5brates6a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"Find the local maximum. Give the $$y$$ value.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"aa97d5brates6a-h1","type":"hint","dependencies":[],"title":"Defining Local Maximum","text":"A function f has a local maximum at $$x=b$$ if there exists an interval (a,c) with $$a<b<c$$ such that, for any $$x$$ in the interval (a,c), $$f(x) \\\\leq f(b)$$. In other words, a local minimum is a high point on the graph.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brates6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["aa97d5brates6a-h1"],"title":"Local Maximum","text":"At which $$x$$ does the graph attain a local maximum?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aa97d5brates6a-h2-s1","type":"hint","dependencies":[],"title":"Local Maximum","text":"The graph attains a local maximum at $$x=1$$ because it is the highest point in an open interval around $$x=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aa97d5brates6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["aa97d5brates6a-h2"],"title":"Local Maximum","text":"What is the corresponding y-coordinate at $$x=1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa97d5brates6b","stepAnswer":["$$-2$$"],"problemType":"TextBox","stepTitle":"Find the local minimum. Give the $$y$$ value.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-2$$","hints":{"DefaultPathway":[{"id":"aa97d5brates6b-h1","type":"hint","dependencies":[],"title":"Defining Local Minimum","text":"A function f has a local minimum at $$x=b$$ if there exists an interval (a,c) with $$a<b<c$$ such that, for any $$x$$ in the interval (a,c), $$f(x) \\\\geq f(b)$$. In other words, a local minimum is a low point on the graph.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brates6b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["aa97d5brates6b-h1"],"title":"Local Minimum","text":"At which $$x$$ does the graph attain a local minimum?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aa97d5brates6b-h2-s1","type":"hint","dependencies":[],"title":"Local Minimum","text":"The graph attains a local minimum at $$x=-1$$ because it is the lowest point in an open interval around $$x=-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aa97d5brates6b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["aa97d5brates6b-h2"],"title":"Local Minimum","text":"What is the corresponding $$y-doordnate$$ at $$x=-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa97d5brates7","title":"Finding Absolute Maxima and Minima from a Graph","body":"For the function f shown in the figure, find all absolute maxima and minima.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Rates of Change and Behavior of Graphs","courseName":"OpenStax: College Algebra","steps":[{"id":"aa97d5brates7a","stepAnswer":["$$16$$"],"problemType":"TextBox","stepTitle":"What is the $$y$$ value of the absolute maximum?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16$$","hints":{"DefaultPathway":[{"id":"aa97d5brates7a-h1","type":"hint","dependencies":[],"title":"Defining Absolute Maximum","text":"The absolute maximum of f at $$x=c$$ is f(c) where $$f(c) \\\\geq f(x)$$ for all $$x$$ in the domain of f.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brates7a-h2","type":"hint","dependencies":["aa97d5brates7a-h1"],"title":"Absolute Maximum","text":"The graph attains an absolute maximum in two locations, $$x=-2$$ and $$x=2$$, because at these locations, the graph attains its highest point on the domain of the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brates7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["aa97d5brates7a-h2"],"title":"Absolute Maximum","text":"What is the corresponding y-value at $$x=-2$$ and $$x=2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa97d5brates7b","stepAnswer":["$$-10$$"],"problemType":"TextBox","stepTitle":"What is the $$y$$ value of the absolute minimum?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-10$$","hints":{"DefaultPathway":[{"id":"aa97d5brates7b-h1","type":"hint","dependencies":[],"title":"Defining Absolute Minimum","text":"The absolute minimum of f at $$x=c$$ is f(c) where $$f(c) \\\\leq f(x)$$ for all $$x$$ in the domain of f.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brates7b-h2","type":"hint","dependencies":["aa97d5brates7b-h1"],"title":"Absolute Minimum","text":"The graph attains an absolute minimum at $$x=3$$, because it is the lowest point on the domain of the function\u2019s graph.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa97d5brates7b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-10$$"],"dependencies":["aa97d5brates7b-h2"],"title":"Absolute Minimum","text":"What is the corresponding y-value at $$x=3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa9eca4stat1","title":"Statistics Terminology","body":"Determine what the key terms (population, sample, parameter, statistic, variable, and data) refer to in the following study. We want to know the average (mean) amount of money first year college students spend at ABC College on school supplies that do not include books. We randomly surveyed $$100$$ first year students at the college. Three of those students spent $150, $200, and $225, respectively.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Definitions of Statistics, Probability, and Key Terms","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa9eca4stat1a","stepAnswer":["First year students attending ABC College this term"],"problemType":"MultipleChoice","stepTitle":"What is the population in this study?","stepBody":"","answerType":"string","variabilization":{},"choices":["First year students attending ABC College this term","Students enrolled in a beginning statistics course at ABC College","Average (mean) amount of money spent (excluding books) by first year college students at ABC College this term","Average (mean) amount of money spent (excluding books) by first year college students in the sample"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat1a-h1","type":"hint","dependencies":[],"title":"Definition of Population","text":"The population of a study is the entire group of persons, things, or objects under which you are observing for the study. In general terms, it is the group from which you want to generalize the study results to.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat1a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["First year students attending ABC College this term"],"dependencies":["aa9eca4stat1a-h1"],"title":"Identifying Populations","text":"What is the group of people that we\'re picking from to include in the study?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["First year students attending ABC College this term","Students enrolled in a beginning statistics course at ABC College","Average (mean) amount of money spent (excluding books) by first year college students at ABC College this term","Average (mean) amount of money spent (excluding books) by first year college students in the sample"]}]}},{"id":"aa9eca4stat1b","stepAnswer":["Students enrolled in a beginning statistics course at ABC College"],"problemType":"MultipleChoice","stepTitle":"What is a possible sample in this study?","stepBody":"","answerType":"string","variabilization":{},"choices":["First year students attending ABC College this term","Students enrolled in a beginning statistics course at ABC College","Average (mean) amount of money spent (excluding books) by first year college students at ABC College this term","Average (mean) amount of money spent (excluding books) by first year college students in the sample"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat1b-h1","type":"hint","dependencies":[],"title":"Definition of Sample","text":"The sample of study is a subset of the population (essentially, a part of the population) that is actually used to conduct the sample.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat1b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Students enrolled in a beginning statistics course at ABC College"],"dependencies":["aa9eca4stat1b-h1"],"title":"Identifying Possible Samples","text":"What is a possible subset of the population found in the first step that we can use to include in the study?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["First year students attending ABC College this term","Students enrolled in a beginning statistics course at ABC College","Average (mean) amount of money spent (excluding books) by first year college students at ABC College this term","Average (mean) amount of money spent (excluding books) by first year college students in the sample"]}]}},{"id":"aa9eca4stat1c","stepAnswer":["Average (mean) amount of money spent (excluding books) by first year college students at ABC College this term"],"problemType":"MultipleChoice","stepTitle":"What is the parameter in this study?","stepBody":"","answerType":"string","variabilization":{},"choices":["First year students attending ABC College this term","Students enrolled in a beginning statistics course at ABC College","Average (mean) amount of money spent (excluding books) by first year college students at ABC College this term","Average (mean) amount of money spent (excluding books) by first year college students in the sample"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat1c-h1","type":"hint","dependencies":[],"title":"Definition of Parameter","text":"The parameter is the numerical representation of the entire population that is estimated by a statistic. Essentially, the parameter is the data point that you desire to calculate from the entire population.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat1c-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Average (mean) amount of money spent (excluding books) by first year college students at ABC College this term"],"dependencies":["aa9eca4stat1c-h1"],"title":"Identifying Parameter","text":"What is the parameter of the study? Essentially, what is the calculated value that you want to determine for the entire population?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["First year students attending ABC College this term","Students enrolled in a beginning statistics course at ABC College","Average (mean) amount of money spent (excluding books) by first year college students at ABC College this term","Average (mean) amount of money spent (excluding books) by first year college students in the sample"]}]}},{"id":"aa9eca4stat1d","stepAnswer":["Average (mean) amount of money spent (excluding books) by first year college students in the sample"],"problemType":"MultipleChoice","stepTitle":"What is the statistic in this study?","stepBody":"","answerType":"string","variabilization":{},"choices":["First year students attending ABC College this term","Students enrolled in a beginning statistics course at ABC College","Average (mean) amount of money spent (excluding books) by first year college students at ABC College this term","Average (mean) amount of money spent (excluding books) by first year college students in the sample"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat1d-h1","type":"hint","dependencies":[],"title":"Definition of Statistic","text":"The statistic is the measured value from the sample. For instance, if the parameter was to measure the average hours a student studies at a specific college, the statistic would specify the average hours a student studies in the sample that was taken to survey. An easier way to think about statistic is that it measures exactly the same thing as the parameter does, just for a subset of the population.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat1d-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Average (mean) amount of money spent (excluding books) by first year college students in the sample"],"dependencies":["aa9eca4stat1d-h1"],"title":"Identifying Statistic","text":"What is the statistic in this study? In other words, what is the calculation that we will be performing on the $$100$$ first year students at the college?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Amount of money spent (excluding books) by one first year student","Dollar amount spent by first year students (examples included in the question are $150, $200, and $225)","Average (mean) amount of money spent (excluding books) by first year college students at ABC College this term","Average (mean) amount of money spent (excluding books) by first year college students in the sample"]}]}},{"id":"aa9eca4stat1e","stepAnswer":["Amount of money spent (excluding books) by one first year student"],"problemType":"MultipleChoice","stepTitle":"What is the variable in this study?","stepBody":"","answerType":"string","variabilization":{},"choices":["Amount of money spent (excluding books) by one first year student","Dollar amount spent by first year students (examples included in the question are $150, $200, and $225)","Average (mean) amount of money spent (excluding books) by first year college students at ABC College this term","Average (mean) amount of money spent (excluding books) by first year college students in the sample"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat1e-h1","type":"hint","dependencies":[],"title":"Definition of Variable","text":"The variable, usually denoted by a capital letter, commonly X or Y, is a numerical or categorical value that can be measured for each individual in a population. Therefore, variables usually refer to the value determined for one person, subject, or object.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat1e-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Amount of money spent (excluding books) by one first year student"],"dependencies":["aa9eca4stat1e-h1"],"title":"Identifying Variable","text":"What is the variable in this instance? Think about what the calculated value is and what one specific person, subject, or object represents in this data set (who/what are you sampling)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Amount of money spent (excluding books) by one first year student","Dollar amount spent by first year students (examples included in the question are $150, $200, and $225)","Average (mean) amount of money spent (excluding books) by first year college students at ABC College this term","Average (mean) amount of money spent (excluding books) by first year college students in the sample"]}]}},{"id":"aa9eca4stat1f","stepAnswer":["Dollar amount spent by first year students (examples included in the question are $150, $200, and $225)"],"problemType":"MultipleChoice","stepTitle":"What is the data in this study?","stepBody":"","answerType":"string","variabilization":{},"choices":["Amount of money spent (excluding books) by one first year student","Dollar amount spent by first year students (examples included in the question are $150, $200, and $225)","Average (mean) amount of money spent (excluding books) by first year college students at ABC College this term","Average (mean) amount of money spent (excluding books) by first year college students in the sample"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat1f-h1","type":"hint","dependencies":[],"title":"Definition of Data","text":"Data in a study is the values of the variables. Previously we determined that the variable in this study is the amount of money spent (excluding books) $$b$$ one first year student. What are the numerical values of this?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat1f-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Dollar amount spent by first year students (examples included in the question are $150, $200, and $225)"],"dependencies":["aa9eca4stat1f-h1"],"title":"Identifying Data","text":"What are the values associated with the variable defined above? Are there examples given in the problem of students (members of the population) and how much they spent on school supplies (excluding books)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Amount of money spent (excluding books) by one first year student","Dollar amount spent by first year students (examples included in the question are $150, $200, and $225)","Average (mean) amount of money spent (excluding books) by first year college students at ABC College this term","Average (mean) amount of money spent (excluding books) by first year college students in the sample"]}]}}]},{"id":"aa9eca4stat2","title":"Identifying Statistic Key Terms","body":"Determine what the key terms refer to in the following study. A study was conducted at a local college to analyze the average cumulative GPA\'s of students who graduated last year.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Definitions of Statistics, Probability, and Key Terms","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa9eca4stat2a","stepAnswer":["All students who graduated from the college last year"],"problemType":"MultipleChoice","stepTitle":"What represents the population?","stepBody":"","answerType":"string","variabilization":{},"choices":["All students who attended the college last year","The cumulative GPA of one student who graduated from the college last year","$$3.65$$, $$2.80$$, $$1.50$$, $$3.90$$","A group of students who graduated from the college last year, randomly selected","The average cumulative GPA of students who graduated from the college last year","All students who graduated from the college last year","The average cumulative GPA of students in the study who graduated from the college last year"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat2a-h1","type":"hint","dependencies":[],"title":"Definition of Population","text":"The population of a study is the entire group of persons, things, or objects under which you are observing for the study. In general terms, it is the group from which you want to generalize the study results to.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat2a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graduated"],"dependencies":["aa9eca4stat2a-h1"],"title":"Identifying Population","text":"Think about what the question says in terms of who they want to pull data from. Is it all students who GRADUATED last year, all students who ATTENDED last year, or ONE student who graduated last year?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graduated","Attended","One"]}]}},{"id":"aa9eca4stat2b","stepAnswer":["The average cumulative GPA of students in the study who graduated from the college last year"],"problemType":"MultipleChoice","stepTitle":"What represents the statistic?","stepBody":"","answerType":"string","variabilization":{},"choices":["All students who attended the college last year","The cumulative GPA of one student who graduated from the college last year","$$3.65$$, $$2.80$$, $$1.50$$, $$3.90$$","A group of students who graduated from the college last year, randomly selected","The average cumulative GPA of students who graduated from the college last year","All students who graduated from the college last year","The average cumulative GPA of students in the study who graduated from the college last year"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat2b-h1","type":"hint","dependencies":[],"title":"Definition of Statistic","text":"The statistic is the measured value from the sample. It measures exactly the same thing as the parameter does, just for a subset of the population.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat2b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Study"],"dependencies":["aa9eca4stat2b-h1"],"title":"Identifying Statistic","text":"What is the statistic in this study? As a value that measures a sample from the population, does the statistic measure the average cumulative GPA of all students who ATTENDED the college last year, all students who GRADUATED the college last year, or students from the STUDY who graduated the college last year?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Attended","Graduated","Study"]}]}},{"id":"aa9eca4stat2c","stepAnswer":["The average cumulative GPA of students who graduated from the college last year"],"problemType":"MultipleChoice","stepTitle":"What represents the parameter?","stepBody":"","answerType":"string","variabilization":{},"choices":["All students who attended the college last year","The cumulative GPA of one student who graduated from the college last year","$$3.65$$, $$2.80$$, $$1.50$$, $$3.90$$","A group of students who graduated from the college last year, randomly selected","The average cumulative GPA of students who graduated from the college last year","All students who graduated from the college last year","The average cumulative GPA of students in the study who graduated from the college last year"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat2c-h1","type":"hint","dependencies":[],"title":"Definition of Parameter","text":"The parameter is the numerical representation of the entire population that is estimated by a statistic. Essentially, the parameter is the data point that you desire to calculate from the entire population.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat2c-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graduated"],"dependencies":["aa9eca4stat2c-h1"],"title":"Identifying Parameter","text":"What is the parameter of the study? We know from the previous step that the statistic is the average cumulative GPA of students in the study who graduated from the college last year and now we want to broaden that to the population of all the students who graduated from the college last year. Is the parameter the average cumulative GPA of students in the STUDY who graduated from the college last year, of all students who ATTENDED the college last year, or of all students who GRADUATED from the college last year?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Attended","Graduated","Study"]}]}},{"id":"aa9eca4stat2d","stepAnswer":["A group of students who graduated from the college last year, randomly selected"],"problemType":"MultipleChoice","stepTitle":"What represents the sample?","stepBody":"","answerType":"string","variabilization":{},"choices":["All students who attended the college last year","The cumulative GPA of one student who graduated from the college last year","$$3.65$$, $$2.80$$, $$1.50$$, $$3.90$$","A group of students who graduated from the college last year, randomly selected","The average cumulative GPA of students who graduated from the college last year","All students who graduated from the college last year","The average cumulative GPA of students in the study who graduated from the college last year"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat2d-h1","type":"hint","dependencies":[],"title":"Definition of Sample","text":"The sample of study is a subset of the population (essentially, a part of the population) that is actually used to conduct the sample.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat2d-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Group"],"dependencies":["aa9eca4stat2d-h1"],"title":"Identifying Sample","text":"What is a subset of the population found in the first step that we can use to include in the study? Is it ALL the students who graduated last year, ONE student who graduated last year, or a GROUP of a students who graduated last year randomly selected?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["All","One","Group"]}]}},{"id":"aa9eca4stat2e","stepAnswer":["The cumulative GPA of one student who graduated from the college last year"],"problemType":"MultipleChoice","stepTitle":"What represents the variable?","stepBody":"","answerType":"string","variabilization":{},"choices":["All students who attended the college last year","The cumulative GPA of one student who graduated from the college last year","$$3.65$$, $$2.80$$, $$1.50$$, $$3.90$$","A group of students who graduated from the college last year, randomly selected","The average cumulative GPA of students who graduated from the college last year","All students who graduated from the college last year","The average cumulative GPA of students in the study who graduated from the college last year"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat2e-h1","type":"hint","dependencies":[],"title":"Definition of Variable","text":"The variable, usually denoted by a capital letter, commonly X or Y, is a numerical or categorical value that can be measured for each individual in a population. Therefore, variables usually refer to the value determined for one person, subject, or object.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat2e-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["One"],"dependencies":["aa9eca4stat2e-h1"],"title":"Identifying Variable","text":"What is the variable in this instance? Noting that the variable measures the statistic for one specific person, is the variable the cumulative GPA of ONE student who graduated from the college last year or the average cumulative GPA of ALL students who graduated from the college last year?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["All","One"]}]}},{"id":"aa9eca4stat2f","stepAnswer":["$$3.65$$, $$2.80$$, $$1.50$$, $$3.90$$"],"problemType":"MultipleChoice","stepTitle":"What represents the data?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$3.65$$, $$2.80$$, $$1.50$$, $$3.90$$","choices":["All students who attended the college last year","The cumulative GPA of one student who graduated from the college last year","$$3.65$$, $$2.80$$, $$1.50$$, $$3.90$$","A group of students who graduated from the college last year, randomly selected","The average cumulative GPA of students who graduated from the college last year","All students who graduated from the college last year","The average cumulative GPA of students in the study who graduated from the college last year"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat2f-h1","type":"hint","dependencies":[],"title":"Definition of data","text":"Data in a study is the values of the variables. Previously we determined that the variable in this study is the cumulative GPA of one student who graduated last year. The data just represents the variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat2f-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3.65$$, $$2.80$$, $$1.50$$, $$3.90$$"],"dependencies":["aa9eca4stat2f-h1"],"title":"Identifying data","text":"What are the values associated with the variable defined above? Which letter a-g from the beginning question represents the actual numerical values of cumulative GPA for different students who graduated last year?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["All students who attended the college last year","The cumulative GPA of one student who graduated from the college last year","$$3.65$$, $$2.80$$, $$1.50$$, $$3.90$$","A group of students who graduated from the college last year, randomly selected","The average cumulative GPA of students who graduated from the college last year","All students who graduated from the college last year","The average cumulative GPA of students in the study who graduated from the college last year"]}]}}]},{"id":"aa9eca4stat3","title":"Identifying Statistic Terms","body":"Identify the population, sample, parameter, statistic, variable, and data of the given statement. Ski resorts are interested in the mean age that children take their first ski and snowboard lessons. They need this information to plan their ski classes optimally.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Definitions of Statistics, Probability, and Key Terms","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa9eca4stat3a","stepAnswer":["All children who take ski or snowboard lessons"],"problemType":"MultipleChoice","stepTitle":"What is the population?","stepBody":"","answerType":"string","variabilization":{},"choices":["All children who take ski or snowboard lessons","A group of these children"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat3a-h1","type":"hint","dependencies":[],"title":"Definition of Population","text":"The population of a study is the entire group of persons, things, or objects under which you are observing for the study. In general terms, it is the group from which you want to generalize the study results to.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat3a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["All"],"dependencies":["aa9eca4stat3a-h1"],"title":"Identifying Population","text":"Think about what the question says in terms of the broader generalized population that they want to pull data from. Is it ALL the children who take ski or snowboard lessons or a SMALLER GROUP of these children?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["All","Smaller group"]}]}},{"id":"aa9eca4stat3b","stepAnswer":["A group of these children"],"problemType":"MultipleChoice","stepTitle":"What is the sample?","stepBody":"","answerType":"string","variabilization":{},"choices":["All children who take ski or snowboard lessons","A group of these children"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat3b-h1","type":"hint","dependencies":[],"title":"Definition of Sample","text":"The sample of study is a subset of the population (essentially, a part of the population) that is actually used to conduct the sample.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat3b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Smaller group"],"dependencies":["aa9eca4stat3b-h1"],"title":"Identifying Sample","text":"Think about what the question says in terms of the sample that they want to pull data from. Is it ALL the children who take ski or snowboard lessons or a SMALLER GROUP of these children?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["All","Smaller group"]}]}},{"id":"aa9eca4stat3c","stepAnswer":["Population mean age of children who take their first snowboard lesson"],"problemType":"MultipleChoice","stepTitle":"What is the parameter?","stepBody":"","answerType":"string","variabilization":{},"choices":["Population mean age of children who take their first snowboard lesson","Sample mean age of children who take their first snowboard lesson"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat3c-h1","type":"hint","dependencies":[],"title":"Definition of Parameter","text":"The parameter is the numerical representation of the entire population that is estimated by a statistic. Essentially, the parameter is the data point that you desire to calculate from the entire population.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat3c-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Population"],"dependencies":["aa9eca4stat3c-h1"],"title":"Identifying Parameter","text":"What is the parameter of the study? Essentially, what is the calculated value that you want to determine for the entire population as a whole? Is it the POPULATION mean age or the SAMPLE mean age?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Population","Sample"]}]}},{"id":"aa9eca4stat3d","stepAnswer":["Sample mean age of children who take their first snowboard lesson"],"problemType":"MultipleChoice","stepTitle":"What is the statistic?","stepBody":"","answerType":"string","variabilization":{},"choices":["Population mean age of children who take their first snowboard lesson","Sample mean age of children who take their first snowboard lesson"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat3d-h1","type":"hint","dependencies":[],"title":"Definition of Statistic","text":"The statistic is the measured value from the sample. It measures exactly the same thing as the parameter does, just for a subset of the population.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat3d-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Sample"],"dependencies":["aa9eca4stat3d-h1"],"title":"Identifying Statistic","text":"What is the statistic of the study? Essentially, what is the calculated value that you want to determine for the smaller sample? Is it the POPULATION mean age or the SAMPLE mean age?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Population","Sample"]}]}},{"id":"aa9eca4stat3e","stepAnswer":["$$X=The$$ age of one child who takes his or her first ski or snowboard lesson"],"problemType":"MultipleChoice","stepTitle":"What is the variable?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$X=The$$ age of one child who takes his or her first ski or snowboard lesson","choices":["$$X=The$$ age of one child who takes his or her first ski or snowboard lesson","Values for X, such as $$3$$, $$7$$, and so on"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat3e-h1","type":"hint","dependencies":[],"title":"Definition of Variable","text":"The variable, usually denoted by a capital letter, commonly X or Y, is a numerical or categorical value that can be measured for each individual in a population. Therefore, variables usually refer to the value determined for one person, subject, or object.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat3e-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$X=The$$ age of one child who takes his or her first ski or snowboard lesson"],"dependencies":["aa9eca4stat3e-h1"],"title":"Identifying Variable","text":"What is the variable in the study? Note that variables are usually defined as capital letters X or Y and calculate the value for one specific individual.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$X=The$$ age of one child who takes his or her first ski or snowboard lesson","Values for X, such as $$3$$, $$7$$, and so on"]}]}},{"id":"aa9eca4stat3f","stepAnswer":["Values for X, such as $$3$$, $$7$$, and so on"],"problemType":"MultipleChoice","stepTitle":"What is the data?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Values for X, such as $$3$$, $$7$$, and so on","choices":["$$X=The$$ age of one child who takes his or her first ski or snowboard lesson","Values for X, such as $$3$$, $$7$$, and so on"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat3f-h1","type":"hint","dependencies":[],"title":"Definition of data","text":"Data in a study is the values of the variables. Look at the previous determined variable: the data just represents the variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat3f-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Numerical"],"dependencies":["aa9eca4stat3f-h1"],"title":"Identifying data","text":"Are the data NUMERICAL values or defined variables like X?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Numerical","Variables"]}]}}]},{"id":"aa9eca4stat4","title":"Identifying Statistic Terms","body":"Identify the population, sample, parameter, statistic, variable, and data of the given statement. Insurance companies are interested in the mean health costs each year of their clients, so that they can determine the costs of health insurance.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Definitions of Statistics, Probability, and Key Terms","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa9eca4stat4a","stepAnswer":["The clients of the insurance companies"],"problemType":"MultipleChoice","stepTitle":"What is the population?","stepBody":"","answerType":"string","variabilization":{},"choices":["The clients of the insurance companies","a smaller group of the clients"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat4a-h1","type":"hint","dependencies":[],"title":"Definition of Population","text":"The population of a study is the entire group of persons, things, or objects under which you are observing for the study. In general terms, it is the group from which you want to generalize the study results to.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat4a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["All"],"dependencies":["aa9eca4stat4a-h1"],"title":"Identifying Population","text":"Think about what the question says in terms of who they want to pull data from. Is it ALL the clients of the insurance companies or a SMALLER GROUP of these clients?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["All","Smaller group"]}]}},{"id":"aa9eca4stat4b","stepAnswer":["A smaller group of the clients"],"problemType":"MultipleChoice","stepTitle":"What is the sample?","stepBody":"","answerType":"string","variabilization":{},"choices":["The clients of the insurance companies","A smaller group of the clients"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat4b-h1","type":"hint","dependencies":[],"title":"Definition of Sample","text":"The sample of study is a subset of the population (essentially, a part of the population) that is actually used to conduct the sample.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat4b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Smaller group"],"dependencies":["aa9eca4stat4b-h1"],"title":"Identifying Sample","text":"Think about how a sample pulls a smaller group of people from the population. Therefore, is the sample ALL the clients of the insurance companies or a SMALLER GROUP of these clients?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["All","Smaller group"]}]}},{"id":"aa9eca4stat4c","stepAnswer":["The mean health costs of all the clients"],"problemType":"MultipleChoice","stepTitle":"What is the parameter?","stepBody":"","answerType":"string","variabilization":{},"choices":["The mean health costs of all the clients","The mean health costs of the sample of the clients"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat4c-h1","type":"hint","dependencies":[],"title":"Definition of Parameter","text":"The parameter is the numerical representation of the entire population that is estimated by a statistic. Essentially, the parameter is the data point that you desire to calculate from the entire population.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat4c-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Population"],"dependencies":["aa9eca4stat4c-h1"],"title":"Identifying Parameter","text":"What is the parameter of the study? Essentially, what is the calculated value that you want to determine for the entire population as a whole? Is it the POPULATION mean health costs or the SAMPLE mean health costs?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Population","Sample"]}]}},{"id":"aa9eca4stat4d","stepAnswer":["The mean health costs of the sample of the clients"],"problemType":"MultipleChoice","stepTitle":"What is the statistic?","stepBody":"","answerType":"string","variabilization":{},"choices":["The mean health costs of all the clients","The mean health costs of the sample of the clients"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat4d-h1","type":"hint","dependencies":[],"title":"Definition of Statistic","text":"The statistic is the measured value from the sample. It measures exactly the same thing as the parameter does, just for a subset of the population.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat4d-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Sample"],"dependencies":["aa9eca4stat4d-h1"],"title":"Identifying Statistic","text":"What is the statistic of the study? Essentially, what is the calculated value that you want to determine for the smaller sample? Is it the POPULATION mean health costs or the SAMPLE mean health costs?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Population","Sample"]}]}},{"id":"aa9eca4stat4e","stepAnswer":["$$X=The$$ health costs of one client"],"problemType":"MultipleChoice","stepTitle":"What is the variable?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$X=The$$ health costs of one client","choices":["$$X=The$$ health costs of one client","Values for X, such as $$34$$, $$9$$, $$82$$, and so on"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat4e-h1","type":"hint","dependencies":[],"title":"Definition of Variable","text":"The variable, usually denoted by a capital letter, commonly X or Y, is a numerical or categorical value that can be measured for each individual in a population. Therefore, variables usually refer to the value determined for one person, subject, or object.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat4e-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$X=The$$ health costs of one client"],"dependencies":["aa9eca4stat4e-h1"],"title":"Identifying Variable","text":"What is the variable in the study? Note that variables are usually defined as capital letters X or Y and calculate the value for one specific individual.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$X=The$$ health costs of one client","Values for X, such as $$34$$, $$9$$, $$82$$, and so on"]}]}},{"id":"aa9eca4stat4f","stepAnswer":["Values for X, such as $$34$$, $$9$$, $$82$$, and so on"],"problemType":"MultipleChoice","stepTitle":"What is the data?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Values for X, such as $$34$$, $$9$$, $$82$$, and so on","choices":["$$X=The$$ health costs of one client","Values for X, such as $$34$$, $$9$$, $$82$$, and so on"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat4f-h1","type":"hint","dependencies":[],"title":"Definition of data","text":"Data in a study is the values of the variables. Look at the previous determined variable: the data just represents the variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat4f-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Numerical"],"dependencies":["aa9eca4stat4f-h1"],"title":"Identifying data","text":"Are the data NUMERICAL values or defined variables like X?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Numerical","Variables"]}]}}]},{"id":"aa9eca4stat5","title":"Identifying Statistic Terms","body":"Identify the population, sample, parameter, statistic, variable, and data of the given statement. A marketing company is interested in the proportion of people who will buy a particular product.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Definitions of Statistics, Probability, and Key Terms","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa9eca4stat5a","stepAnswer":["All people (maybe in a certain geographic area, such as the United States)"],"problemType":"MultipleChoice","stepTitle":"What is the population?","stepBody":"","answerType":"string","variabilization":{},"choices":["All people (maybe in a certain geographic area, such as the United States)","A smaller group of all the people"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat5a-h1","type":"hint","dependencies":[],"title":"Definition of Population","text":"The population of a study is the entire group of persons, things, or objects under which you are observing for the study. In general terms, it is the group from which you want to generalize the study results to.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat5a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["All"],"dependencies":["aa9eca4stat5a-h1"],"title":"Identifying Population","text":"Think about what the question says in terms of who they want to pull data from. Is it ALL the people who will buy a particular product or a SMALLER GROUP of these people?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["All","Smaller group"]}]}},{"id":"aa9eca4stat5b","stepAnswer":["A smaller group of all the people"],"problemType":"MultipleChoice","stepTitle":"What is the sample?","stepBody":"","answerType":"string","variabilization":{},"choices":["All people (maybe in a certain geographic area, such as the United States)","A smaller group of all the people"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat5b-h1","type":"hint","dependencies":[],"title":"Definition of Sample","text":"The sample of study is a subset of the population (essentially, a part of the population) that is actually used to conduct the sample.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat5b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Smaller group"],"dependencies":["aa9eca4stat5b-h1"],"title":"Identifying Sample","text":"Think about how a sample pulls a smaller group of people from the population. Therefore, is the sample ALL the people who will buy a particular product or a SMALLER GROUP of these clients?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["All","Smaller group"]}]}},{"id":"aa9eca4stat5c","stepAnswer":["The proportion of all the people who will buy the product"],"problemType":"MultipleChoice","stepTitle":"What is the parameter?","stepBody":"","answerType":"string","variabilization":{},"choices":["The proportion of all the people who will buy the product","The proportion of the sample of people who will buy the product"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat5c-h1","type":"hint","dependencies":[],"title":"Definition of Parameter","text":"The parameter is the numerical representation of the entire population that is estimated by a statistic. Essentially, the parameter is the data point that you desire to calculate from the entire population.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat5c-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Population"],"dependencies":["aa9eca4stat5c-h1"],"title":"Identifying Parameter","text":"What is the parameter of the study? Essentially, what is the calculated value that you want to determine for the entire population as a whole? Is it the POPULATION proportion of all people who will buy the product or the SAMPLE proportion of all people who will buy the product?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Population","Sample"]}]}},{"id":"aa9eca4stat5d","stepAnswer":["The proportion of the sample of people who will buy the product"],"problemType":"MultipleChoice","stepTitle":"What is the statistic?","stepBody":"","answerType":"string","variabilization":{},"choices":["The proportion of all the people who will buy the product","The proportion of the sample of people who will buy the product"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat5d-h1","type":"hint","dependencies":[],"title":"Definition of Statistic","text":"The statistic is the measured value from the sample. It measures exactly the same thing as the parameter does, just for a subset of the population.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat5d-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Sample"],"dependencies":["aa9eca4stat5d-h1"],"title":"Identifying Statistic","text":"What is the statistic of the study? Essentially, what is the calculated value that you want to determine for the smaller sample? Is it the POPULATION proportion of all people who will buy the product or the SAMPLE proportion of all people who will buy the product?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Population","Sample"]}]}},{"id":"aa9eca4stat5e","stepAnswer":["$$X=The$$ number of couples who stay married"],"problemType":"MultipleChoice","stepTitle":"What is the variable?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$X=The$$ number of couples who stay married","choices":["$$X=The$$ number of couples who stay married","Yes, No"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat5e-h1","type":"hint","dependencies":[],"title":"Definition of Variable","text":"The variable, usually denoted by a capital letter, commonly X or Y, is a numerical or categorical value that can be measured for each individual in a population. Therefore, variables usually refer to the value determined for one person, subject, or object.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat5e-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$X=The$$ number of couples who stay married"],"dependencies":["aa9eca4stat5e-h1"],"title":"Identifying Variable","text":"What is the variable in the study? Note that variables are usually defined as capital letters X or Y and calculate the value for one specific individual.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$X=The$$ number of couples who stay married","Yes, No"]}]}},{"id":"aa9eca4stat5f","stepAnswer":["Yes, No"],"problemType":"MultipleChoice","stepTitle":"What is the data?","stepBody":"","answerType":"string","variabilization":{},"choices":["$$X=The$$ number of couples who stay married","Yes, No"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat5f-h1","type":"hint","dependencies":[],"title":"Definition of data","text":"Data in a study is the values of the variables. Look at the previous determined variable: the data just represents the variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa9eca4stat6","title":"Identifying Statistic Terms","body":"Determine what the key terms refer to in the following study. We want to know the average (mean) amount of money spent on school uniforms each year by families with children at Knoll Academy. We randomly survey $$100$$ families with children in the school. Three of the families spent $65, $75, and $95, respectively.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Definitions of Statistics, Probability, and Key Terms","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa9eca4stat6a","stepAnswer":["All families with children at Knoll Academy"],"problemType":"MultipleChoice","stepTitle":"What is the population?","stepBody":"","answerType":"string","variabilization":{},"choices":["Average Amount of money spent on school uniforms each year by families with children at Knoll Academy","All families with children at Knoll Academy","Families with no children at Knoll Academy"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat6a-h1","type":"hint","dependencies":[],"title":"Definition of Population","text":"The population of a study is the entire group of persons, things, or objects under which you are observing for the study. In general terms, it is the group from which you want to generalize the study results to.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat6a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["All"],"dependencies":["aa9eca4stat6a-h1"],"title":"Identifying Population","text":"Think about what the question says in terms of who they want to pull data from. Is it ALL families with children at Knoll Academy or a SMALLER GROUP of these families?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["All","Smaller group"]}]}},{"id":"aa9eca4stat6b","stepAnswer":["A smaller group of all families with children at Knoll Academy"],"problemType":"MultipleChoice","stepTitle":"What is the sample?","stepBody":"","answerType":"string","variabilization":{},"choices":["A smaller group of all families with children at Knoll Academy","A smaller group of all the people","All people (maybe in a certain geographic area, such as the United States)"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat6b-h1","type":"hint","dependencies":[],"title":"Definition of Sample","text":"The sample of study is a subset of the population (essentially, a part of the population) that is actually used to conduct the sample.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat6b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Smaller group"],"dependencies":["aa9eca4stat6b-h1"],"title":"Identifying Sample","text":"Think about how a sample pulls a smaller group of people from the population. Therefore, is the sample ALL the families with children at Knoll Academy or a SMALLER GROUP of these families?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["All","Smaller group"]}]}},{"id":"aa9eca4stat6c","stepAnswer":["Average Amount of money spent on school uniforms each year by families with children at Knoll Academy"],"problemType":"MultipleChoice","stepTitle":"What is the parameter?","stepBody":"","answerType":"string","variabilization":{},"choices":["Average Amount of money spent on school uniforms each year by all families with children at Knoll Academy","Average Amount of money spent on school uniforms each year by families with children at Knoll Academy","Average Amount of money spent on school uniforms each year by small group of families with children at Knoll Academy"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat6c-h1","type":"hint","dependencies":[],"title":"Definition of Parameter","text":"The parameter is the numerical representation of the entire population that is estimated by a statistic. Essentially, the parameter is the data point that you desire to calculate from the entire population.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat6c-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Population"],"dependencies":["aa9eca4stat6c-h1"],"title":"Identifying Parameter","text":"What is the parameter of the study? Essentially, what is the calculated value that you want to determine for the entire population as a whole? Is it the POPULATION average amount of money spent on school uniforms each year by families with children at Knoll Academy or the SAMPLE average amount of money spent on school uniforms each year by families with children at Knoll Academy?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Population","Sample"]}]}},{"id":"aa9eca4stat6d","stepAnswer":["Average Amount of money spent on school uniforms each year by sample families with children at Knoll Academy"],"problemType":"MultipleChoice","stepTitle":"What is the statistic?","stepBody":"","answerType":"string","variabilization":{},"choices":["Average Amount of money spent on school uniforms each year by all families with children at Knoll Academy","Average Amount of money spent on school uniforms each year by sample families with children at Knoll Academy"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat6d-h1","type":"hint","dependencies":[],"title":"Definition of Statistic","text":"The statistic is the measured value from the sample. It measures exactly the same thing as the parameter does, just for a subset of the population.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat6d-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Sample"],"dependencies":["aa9eca4stat6d-h1"],"title":"Identifying Statistic","text":"What is the statistic of the study? Essentially, what is the calculated value that you want to determine for the smaller sample? Is it the POPULATION average amount of money spent on school uniforms each year by families with children at Knoll Academy or the SAMPLE average amount of money spent on school uniforms each year by families with children at Knoll Academy?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Population","Sample"]}]}},{"id":"aa9eca4stat6e","stepAnswer":["$$X=Average$$ Amount of money spent on school uniforms each year by one family with children at Knoll Academy"],"problemType":"MultipleChoice","stepTitle":"What is the variable?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$X=Average$$ Amount of money spent on school uniforms each year by one family with children at Knoll Academy","choices":["$$X=Average$$ Amount of money spent on school uniforms each year by one family with children at Knoll Academy","Values of X such as $$65$$, $$75$$, $$95$$ and so on"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat6e-h1","type":"hint","dependencies":[],"title":"Definition of Variable","text":"The variable, usually denoted by a capital letter, commonly X or Y, is a numerical or categorical value that can be measured for each individual in a population. Therefore, variables usually refer to the value determined for one person, subject, or object.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat6e-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$X=Average$$ Amount of money spent on school uniforms each year by one family with children at Knoll Academy"],"dependencies":["aa9eca4stat6e-h1"],"title":"Identifying Variable","text":"What is the variable in the study? Note that variables are usually defined as capital letters X or Y and calculate the value for one specific individual.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$X=Average$$ Amount of money spent on school uniforms each year by one family with children at Knoll Academy","Values of X such as $$65$$, $$75$$, $$95$$ and so on"]}]}},{"id":"aa9eca4stat6f","stepAnswer":["Values of X such as $$65$$, $$75$$, $$95$$ and so on"],"problemType":"MultipleChoice","stepTitle":"What is the data?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Values of X such as $$65$$, $$75$$, $$95$$ and so on","choices":["$$X=Average$$ Amount of money spent on school uniforms each year by one family with children at Knoll Academy","Values of X such as $$65$$, $$75$$, $$95$$ and so on"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat6f-h1","type":"hint","dependencies":[],"title":"Definition of data","text":"Data in a study is the values of the variables. Look at the previous determined variable: the data just represents the variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat6f-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Numerical"],"dependencies":["aa9eca4stat6f-h1"],"title":"Identifying data","text":"Are the data NUMERICAL values or defined variables like X?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Numerical","Variables"]}]}}]},{"id":"aa9eca4stat7","title":"Statistics Terminology","body":"Determine what the key terms (population, sample, parameter, statistic, variable, and data) refer to in the following study. Cars with dummies in the front seats were crashed into a wall at a speed of $$35$$ miles per hour. We want to know the proportion of dummies in the driver\u2019s seat that would have had head injuries, if they had been actual drivers. We start with a simple random sample of $$75$$ cars.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Definitions of Statistics, Probability, and Key Terms","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa9eca4stat7a","stepAnswer":["All cars containing dummies in the front seat"],"problemType":"MultipleChoice","stepTitle":"What is the population in this study?","stepBody":"","answerType":"string","variabilization":{},"choices":["All cars containing dummies in the front seat","The $$75$$ cars, selected by a simple random sample"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat7a-h1","type":"hint","dependencies":[],"title":"Definition of Population","text":"The population of a study is the entire group of persons, things, or objects under which you are observing for the study. In general terms, it is the group from which you want to generalize the study results to. In this study, the population is all cars containing dummies in the front seat.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa9eca4stat7b","stepAnswer":["The $$75$$ cars, selected by a simple random sample"],"problemType":"MultipleChoice","stepTitle":"What is a possible sample in this study?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"The $$75$$ cars, selected by a simple random sample","choices":["All cars containing dummies in the front seat","The $$75$$ cars, selected by a simple random sample"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat7b-h1","type":"hint","dependencies":[],"title":"Definition of Sample","text":"The sample of study is a subset of the population (essentially, a part of the population) that is actually used to conduct the sample. In this study, the sample is the $$75$$ cars, selected by a simple random sample.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa9eca4stat7c","stepAnswer":["The proportion of driver dummies (if they had been real people) who would have suffered head injuries in the population"],"problemType":"MultipleChoice","stepTitle":"What is the parameter in this study?","stepBody":"","answerType":"string","variabilization":{},"choices":["The proportion of driver dummies (if they had been real people) who would have suffered head injuries in the population","The proportion of driver dummies (if they had been real people) who would have suffered head injuries in the sample"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat7c-h1","type":"hint","dependencies":[],"title":"Definition of Parameter","text":"The parameter is the numerical representation of the entire population that is estimated by a statistic. Essentially, the parameter is the data point that you desire to calculate from the entire population. In this study, the parameter is the proportion of driver dummies (if they had been real people) who would have suffered head injuries in the population.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa9eca4stat7d","stepAnswer":["The proportion of driver dummies (if they had been real people) who would have suffered head injuries in the sample"],"problemType":"MultipleChoice","stepTitle":"What is the statistic in this study?","stepBody":"","answerType":"string","variabilization":{},"choices":["The proportion of driver dummies (if they had been real people) who would have suffered head injuries in the population","The proportion of driver dummies (if they had been real people) who would have suffered head injuries in the sample"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat7d-h1","type":"hint","dependencies":[],"title":"Definition of Statistic","text":"The statistic is the measured value from the sample. For instance, if the parameter was to measure the average hours a student studies at a specific college, the statistic would specify the average hours a student studies in the sample that was taken to survey. An easier way to think about statistic is that it measures exactly the same thing as the parameter does, just for a subset of the population. In this study, the statistic is the proportion of driver dummies (if they had been real people) who would have suffered head injuries in the sample.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa9eca4stat7e","stepAnswer":["X $$=$$ whether a dummy (if it had been a real person) who would have suffered head injuries"],"problemType":"MultipleChoice","stepTitle":"What is the variable in this study?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"X $$=$$ whether a dummy (if it had been a real person) who would have suffered head injuries","choices":["X $$=$$ whether a dummy (if it had been a real person) who would have suffered head injuries","yes, had head injury, or no, did not"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat7e-h1","type":"hint","dependencies":[],"title":"Definition of Variable","text":"The variable, usually denoted by a capital letter, commonly X or Y, is a numerical or categorical value that can be measured for each individual in a population. Therefore, variables usually refer to the value determined for one person, subject, or object. In this case, variable is X $$=$$ whether a dummy (if it had been a real person) who would have suffered head injuries.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa9eca4stat7f","stepAnswer":["yes, had head injury, or no, did not"],"problemType":"MultipleChoice","stepTitle":"What is the data in this study?","stepBody":"","answerType":"string","variabilization":{},"choices":["X $$=$$ whether a dummy (if it had been a real person) who would have suffered head injuries","yes, had head injury, or no, did not"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat7f-h1","type":"hint","dependencies":[],"title":"Definition of Data","text":"Data in a study is the values of the variables. In this case, variable is qualitative values which is either yes, had head injury, or no, did not.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa9eca4stat8","title":"Statistics Terminology","body":"Determine what the key terms (population, sample, parameter, statistic, variable, and data) refer to in the following study. An insurance company would like to determine the proportion of all medical doctors who have been involved in one or more malpractice lawsuits. The company selects $$500$$ doctors at random from a professional directory and determines the number in the sample who have been involved in a malpractice lawsuit.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Definitions of Statistics, Probability, and Key Terms","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa9eca4stat8a","stepAnswer":["All medical doctors listed in the professional directory"],"problemType":"MultipleChoice","stepTitle":"What is the population in this study?","stepBody":"","answerType":"string","variabilization":{},"choices":["All medical doctors listed in the professional directory","$$500$$ doctors selected at random from the professional directory"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat8a-h1","type":"hint","dependencies":[],"title":"Definition of Population","text":"The population of a study is the entire group of persons, things, or objects under which you are observing for the study. In general terms, it is the group from which you want to generalize the study results to. In this study, the population is all medical doctors listed in the professional directory.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa9eca4stat8b","stepAnswer":["$$500$$ doctors selected at random from the professional directory"],"problemType":"MultipleChoice","stepTitle":"What is a possible sample in this study?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$500$$ doctors selected at random from the professional directory","choices":["All medical doctors listed in the professional directory","$$500$$ doctors selected at random from the professional directory"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat8b-h1","type":"hint","dependencies":[],"title":"Definition of Sample","text":"The sample of study is a subset of the population (essentially, a part of the population) that is actually used to conduct the sample. In this study, the sample is the $$500$$ doctors selected at random from the professional directory.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa9eca4stat8c","stepAnswer":["The proportion of medical doctors who have been involved in one or more malpractice suits in the population."],"problemType":"MultipleChoice","stepTitle":"What is the parameter in this study?","stepBody":"","answerType":"string","variabilization":{},"choices":["The proportion of medical doctors who have been involved in one or more malpractice suits in the population","The proportion of medical doctors who have been involved in one or more malpractice suits in the population.","The proportion of medical doctors who have been involved in one or more malpractice suits in the sample"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat8c-h1","type":"hint","dependencies":[],"title":"Definition of Parameter","text":"The parameter is the numerical representation of the entire population that is estimated by a statistic. Essentially, the parameter is the data point that you desire to calculate from the entire population. In this study, the parameter is the proportion of medical doctors who have been involved in one or more malpractice suits in the population.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa9eca4stat8d","stepAnswer":["The proportion of medical doctors who have been involved in one or more malpractice suits in the sample"],"problemType":"MultipleChoice","stepTitle":"What is the statistic in this study?","stepBody":"","answerType":"string","variabilization":{},"choices":["The proportion of medical doctors who have been involved in one or more malpractice suits in the population","The proportion of medical doctors who have been involved in one or more malpractice suits in the sample"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat8d-h1","type":"hint","dependencies":[],"title":"Definition of Statistic","text":"The statistic is the measured value from the sample. For instance, if the parameter was to measure the average hours a student studies at a specific college, the statistic would specify the average hours a student studies in the sample that was taken to survey. An easier way to think about statistic is that it measures exactly the same thing as the parameter does, just for a subset of the population. In this study, the statistic is the proportion of medical doctors who have been involved in one or more malpractice suits in the sample.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa9eca4stat8e","stepAnswer":["X $$=$$ whether an individual doctor has been involved in a malpractice suit"],"problemType":"MultipleChoice","stepTitle":"What is the variable in this study?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"X $$=$$ whether an individual doctor has been involved in a malpractice suit","choices":["X $$=$$ whether an individual doctor has been involved in a malpractice suit","yes, was involved in one or more malpractice lawsuits, or no, was not"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat8e-h1","type":"hint","dependencies":[],"title":"Definition of Variable","text":"The variable, usually denoted by a capital letter, commonly X or Y, is a numerical or categorical value that can be measured for each individual in a population. Therefore, variables usually refer to the value determined for one person, subject, or object. In this case, variable is X $$=$$ whether an individual doctor has been involved in a malpractice suit.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa9eca4stat8f","stepAnswer":["yes, was involved in one or more malpractice lawsuits, or no, was not"],"problemType":"MultipleChoice","stepTitle":"What is the data in this study?","stepBody":"","answerType":"string","variabilization":{},"choices":["X $$=$$ whether an individual doctor has been involved in a malpractice suit","yes, was involved in one or more malpractice lawsuits, or no, was not"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat8f-h1","type":"hint","dependencies":[],"title":"Definition of Data","text":"Data in a study is the values of the variables. In this case, variable is qualitative values which is either yes, was involved in one or more malpractice lawsuits, or no, was not.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aa9eca4stat9","title":"Lake Tahoe Math Absences","body":"A Lake Tahoe Community College Instructor is interested in the mean number of days Lake Tahoe Community College math students are absent from class during a quarter.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Definitions of Statistics, Probability, and Key Terms","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aa9eca4stat9a","stepAnswer":["All Lake Tahoe Community College math students"],"problemType":"MultipleChoice","stepTitle":"What is the population she is interested in?","stepBody":"","answerType":"string","variabilization":{},"choices":["All Lake Tahoe Community College students","All Lake Tahoe Community College English students","All Lake Tahoe Community College students in her classes","All Lake Tahoe Community College math students"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat9a-h1","type":"hint","dependencies":[],"title":"Definition of Population","text":"The population of a study is the entire group of persons, things, or objects under which you are observing for the study. In general terms, it is the group from which you want to generalize the study results to.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat9a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Math"],"dependencies":["aa9eca4stat9a-h1"],"title":"Identifying Population","text":"We know that the population will be a subset of all Lake Tahoe Community College students. Which specific students does the instructor care about? ALL students, ENGLISH students, HER students, or MATH students?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["All","English","Her","Math"]}]}},{"id":"aa9eca4stat9b","stepAnswer":["Variable"],"problemType":"MultipleChoice","stepTitle":"Consider the following: $$X=number$$ of days a Lake Tahoe Community College math student is absent. In this case, X is an example of a:","stepBody":"","answerType":"string","variabilization":{},"choices":["Variable","Population","Statistic","Data"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat9b-h1","type":"hint","dependencies":[],"title":"Mathematical Notation of Variables","text":"In statistics, variables often are defined by a capital letter like X or Y to measure a value for a member of the population.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aa9eca4stat9c","stepAnswer":["Statistic"],"problemType":"MultipleChoice","stepTitle":"The instructor\'s sample produces a mean number of days absent of $$3.5$$ days. This value is an example of a:","stepBody":"","answerType":"string","variabilization":{},"choices":["Parameter","Statistic","Data","Variable"],"hints":{"DefaultPathway":[{"id":"aa9eca4stat9c-h1","type":"hint","dependencies":[],"title":"Definition of Data from a Sample","text":"The instructor wants to provide a statistical conclusion from a sample. A statistic is often a measure of a sample.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aa9eca4stat9c-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Statistic"],"dependencies":["aa9eca4stat9c-h1"],"title":"Identify the Term for Data from a Sample","text":"Which of the following terms finds the value of the desired data for a sample?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Parameter","Statistic","Data","Variable"]}]}}]},{"id":"aaa317eLinIneq1","title":"Using Interval Notation to Express All Real Numbers Greater Than or Equal to a","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Linear Inequalities and Absolute Value Inequalities","courseName":"OpenStax: College Algebra","steps":[{"id":"aaa317eLinIneq1a","stepAnswer":["$$[-2,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"Use interval notation to indicate all real numbers greater than or equal to-2.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[-2,\\\\infty)$$","choices":["$$[-2,\\\\infty)$$","$$(-2,\\\\infty]$$"],"hints":{"DefaultPathway":[{"id":"aaa317eLinIneq1a-h1","type":"hint","dependencies":[],"title":"Left side symbol","text":"$$2$$ is included in the interval","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq1a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["["],"dependencies":["aaa317eLinIneq1a-h1"],"title":"Left side symbol","text":"What symbol should we use?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["(","["]},{"id":"aaa317eLinIneq1a-h3","type":"hint","dependencies":[],"title":"Right side value","text":"We use the symbol $$\\\\infty$$ to represent $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq1a-h4","type":"hint","dependencies":["aaa317eLinIneq1a-h3"],"title":"Right side symbol","text":"We can\'t \\"equate\\" to infinity (or negative infinity).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq1a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":[")"],"dependencies":["aaa317eLinIneq1a-h4"],"title":"Right side symbol","text":"What symbol should we use?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":[")","]"]}]}}]},{"id":"aaa317eLinIneq10","title":"Solving an Absolute Value Inequality","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Linear Inequalities and Absolute Value Inequalities","courseName":"OpenStax: College Algebra","steps":[{"id":"aaa317eLinIneq10a","stepAnswer":["$$[-2,4]$$"],"problemType":"MultipleChoice","stepTitle":"Solve $$|x-1| \\\\leq 3$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$(-4,2)$$","$$[-4,2]$$","$$(-2,4)$$","$$[-2,4]$$"],"hints":{"DefaultPathway":[{"id":"aaa317eLinIneq10a-h1","type":"hint","dependencies":[],"title":"Absolute Value Inequalities","text":"$$|X| \\\\leq k$$ is equivalent to $$-k \\\\leq X \\\\leq k$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq10a-h2","type":"hint","dependencies":["aaa317eLinIneq10a-h1"],"title":"Applying the Formula for Absolute Value Inequalities","text":"The inequality can be rewritten as $$-3 \\\\leq x-1 \\\\leq 3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq10a-h3","type":"hint","dependencies":["aaa317eLinIneq10a-h2"],"title":"Solve Compound Inequality","text":"We can solve the compound inequality by simply add $$1$$ to all three parts.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq10a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-2 \\\\leq x \\\\leq 4$$"],"dependencies":["aaa317eLinIneq10a-h3"],"title":"Solve Compound Inequality","text":"What is the inequality?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$-4 \\\\leq x \\\\leq 2$$","$$-2 \\\\leq x \\\\leq 4$$"]},{"id":"aaa317eLinIneq10a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$[-2,4]$$"],"dependencies":["aaa317eLinIneq10a-h4"],"title":"Interval Notation","text":"What is $$-2 \\\\leq x \\\\leq 4$$ written in the interval notation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$[-2,4]$$","$$(-2,4)$$"]}]}}]},{"id":"aaa317eLinIneq11","title":"Using a Graphical Approach to Solve Absolute Value Inequalities","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Linear Inequalities and Absolute Value Inequalities","courseName":"OpenStax: College Algebra","steps":[{"id":"aaa317eLinIneq11a","stepAnswer":["$$(-\\\\infty,\\\\frac{-1}{4}) \\\\cup (\\\\frac{11}{4},\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"Given the equation y=-(1/2)abs(4x-5)+3,determine the x-values for which the y-values are negative.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\frac{-1}{4}) \\\\cup (\\\\frac{11}{4},\\\\infty)$$","choices":["$$(\\\\frac{-1}{4},\\\\frac{11}{4})$$","$$[\\\\frac{-1}{4},\\\\frac{11}{4}]$$","$$(-\\\\infty,\\\\frac{-1}{4}) \\\\cup (\\\\frac{11}{4},\\\\infty)$$","$$(-\\\\infty,\\\\frac{-1}{4}] \\\\cup [\\\\frac{11}{4},\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"aaa317eLinIneq11a-h1","type":"hint","dependencies":[],"title":"Interpreting the Problem","text":"We are trying to determine where $$y<0$$, which is when $$-\\\\left(\\\\frac{1}{2}\\\\right) |4x-5|+3<0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq11a-h2","type":"hint","dependencies":["aaa317eLinIneq11a-h1"],"title":"Isolating the Absolute Value","text":"We can subtract $$3$$ from both side, then multiply both sides by $$-2$$, which gives us $$|4x-5|>6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq11a-h3","type":"hint","dependencies":["aaa317eLinIneq11a-h2"],"title":"Solve Equality","text":"We proceed to solve for the equality $$|4x-5|=6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq11a-h4","type":"hint","dependencies":["aaa317eLinIneq11a-h3"],"title":"Solve Equality","text":"We can write out $$|4x-5|=6$$ as $$4x-5=6$$ and $$4x-5=-6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{11}{4}$$"],"dependencies":["aaa317eLinIneq11a-h4"],"title":"Solve Equality","text":"Solve $$4x-5=6$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq11a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{4}$$"],"dependencies":["aaa317eLinIneq11a-h4"],"title":"Solve Equality","text":"Solve $$4x-5=-6$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq11a-h7","type":"hint","dependencies":["aaa317eLinIneq11a-h6"],"title":"Examine Graph","text":"Now, we can examine the graph to observe where the y-values are negative. We observe where the branches are below the x-axis. Notice that it is not important exactly what the graph looks like, as long as we know that it crosses the horizontal axis at $$x=\\\\frac{-1}{4}$$ and x=11/4,and that the graph opens downward.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaa317eLinIneq12","title":"Solve the Inequality","body":"Write your final answer in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Linear Inequalities and Absolute Value Inequalities","courseName":"OpenStax: College Algebra","steps":[{"id":"aaa317eLinIneq12a","stepAnswer":["$$(-\\\\infty,4]$$"],"problemType":"MultipleChoice","stepTitle":"$$4x-7 \\\\leq 9$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,4]$$","choices":["$$[-\\\\infty,4]$$","$$(-\\\\infty,4]$$","$$[4,\\\\infty)$$","$$[4,\\\\infty]$$"],"hints":{"DefaultPathway":[{"id":"aaa317eLinIneq12a-h1","type":"hint","dependencies":[],"title":"Addition Property","text":"The addition property of inequality: if $$a \\\\leq b$$, then $$a+c \\\\leq b+c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["aaa317eLinIneq12a-h1"],"title":"Apply the Addition Property","text":"What number should we add to both sides of the inequality to isolate the variable term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq12a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4x$$"],"dependencies":["aaa317eLinIneq12a-h2"],"title":"Left side After Addition","text":"What is the left side after adding 7?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["aaa317eLinIneq12a-h2"],"title":"Right side After Addition","text":"What is the right side after adding 7?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq12a-h5","type":"hint","dependencies":["aaa317eLinIneq12a-h3","aaa317eLinIneq12a-h4"],"title":"Multiplication Property","text":"The multiplication property of inequality: If $$a \\\\leq b$$ and $$c>0$$, then $$ac \\\\leq bc;$$ if $$a \\\\leq b$$ and $$c<0$$, then $$ac \\\\geq bc$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq12a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4}$$"],"dependencies":["aaa317eLinIneq12a-h5"],"title":"Apply the Multiplication Property","text":"What number should we multiply to both sides of the inequality?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq12a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x$$"],"dependencies":["aaa317eLinIneq12a-h6"],"title":"Left Side","text":"What is the left side after multiplying $$\\\\frac{1}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq12a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["aaa317eLinIneq12a-h6"],"title":"Right Side","text":"What is the right side after multiplying by $$\\\\frac{1}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq12a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$ \\\\leq $$"],"dependencies":["aaa317eLinIneq12a-h6"],"title":"Sign","text":"What is the sign?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$ \\\\leq $$","$$ \\\\geq $$"]},{"id":"aaa317eLinIneq12a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(-\\\\infty,4]$$"],"dependencies":["aaa317eLinIneq12a-h7","aaa317eLinIneq12a-h8","aaa317eLinIneq12a-h9"],"title":"Interval Notation","text":"What is $$x \\\\leq 4$$ written in the interval notation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$[-\\\\infty,4]$$","$$(-\\\\infty,4]$$","$$[4,\\\\infty)$$","$$[4,\\\\infty]$$"]}]}}]},{"id":"aaa317eLinIneq13","title":"Solve the Inequality","body":"Write your final answer in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Linear Inequalities and Absolute Value Inequalities","courseName":"OpenStax: College Algebra","steps":[{"id":"aaa317eLinIneq13a","stepAnswer":["$$(-\\\\infty,\\\\frac{3}{4}]$$"],"problemType":"MultipleChoice","stepTitle":"$$3x+2 \\\\geq 7x-1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\frac{3}{4}]$$","choices":["$$(-\\\\infty,\\\\frac{3}{4}]$$","$$(-\\\\infty,\\\\frac{4}{3}]$$","$$[\\\\frac{3}{4},\\\\infty)$$","$$[\\\\frac{4}{3},\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"aaa317eLinIneq13a-h1","type":"hint","dependencies":[],"title":"Move Variable Terms","text":"We start by moving variable terms to one side of the inequality","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq13a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2 \\\\geq 4x-1$$"],"dependencies":["aaa317eLinIneq13a-h1"],"title":"Move Variable Terms","text":"What inequality do we get after moving the variable terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2 \\\\geq 10x-1$$","$$2 \\\\geq 4x-1$$"]},{"id":"aaa317eLinIneq13a-h3","type":"hint","dependencies":["aaa317eLinIneq13a-h2"],"title":"Isolate Variable Term","text":"We proceed by isolating the variable term $$4x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["aaa317eLinIneq13a-h3"],"title":"Isolate Variable Term","text":"What number should we add to both sides?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq13a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3 \\\\geq 4x$$"],"dependencies":["aaa317eLinIneq13a-h4"],"title":"Isolate Variable Term","text":"What inequality do we get after isolating the variable term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$1 \\\\geq 4x$$","$$3 \\\\geq 4x$$"]},{"id":"aaa317eLinIneq13a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4}$$"],"dependencies":["aaa317eLinIneq13a-h5"],"title":"Apply the Multiplication Property","text":"To make the coeffient of $$x$$ be $$1$$, what number should we multiply both sides of the inequality by?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq13a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{4}$$"],"dependencies":["aaa317eLinIneq13a-h6"],"title":"Left Side","text":"What is the left side after multiplying by $$\\\\frac{1}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq13a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x$$"],"dependencies":["aaa317eLinIneq13a-h6"],"title":"Right Side","text":"What is the right side after multiplying by $$\\\\frac{1}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq13a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$ \\\\geq $$"],"dependencies":["aaa317eLinIneq13a-h6"],"title":"Sign","text":"What is the sign of the inequality after multiplication?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$ \\\\leq $$","$$ \\\\geq $$"]},{"id":"aaa317eLinIneq13a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(-\\\\infty,\\\\frac{3}{4}]$$"],"dependencies":["aaa317eLinIneq13a-h7","aaa317eLinIneq13a-h8","aaa317eLinIneq13a-h9"],"title":"Interval Notation","text":"What is $$\\\\frac{3}{4} \\\\geq x$$ written in interval notion?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(-\\\\infty,\\\\frac{3}{4}]$$","$$[\\\\frac{3}{4},\\\\infty)$$"]}]}}]},{"id":"aaa317eLinIneq14","title":"Solve the Inequality","body":"Write your final answer in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Linear Inequalities and Absolute Value Inequalities","courseName":"OpenStax: College Algebra","steps":[{"id":"aaa317eLinIneq14a","stepAnswer":["$$(-\\\\infty,\\\\frac{3}{8})$$"],"problemType":"MultipleChoice","stepTitle":"$$-2x+3>x-5$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\frac{3}{8})$$","choices":["$$(-\\\\infty,\\\\frac{3}{8})$$","$$(-\\\\infty,\\\\frac{-3}{8})$$","$$(\\\\frac{3}{8},\\\\infty)$$","$$(\\\\frac{-3}{8},\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"aaa317eLinIneq14a-h1","type":"hint","dependencies":[],"title":"Move Variable Terms","text":"We start by moving variable terms to one side of the inequality","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq14a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3>3x-5$$"],"dependencies":["aaa317eLinIneq14a-h1"],"title":"Move Variable Terms","text":"What inequality do we get after moving the variable terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$3>3x-5$$","$$3>-x-5$$"]},{"id":"aaa317eLinIneq14a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["aaa317eLinIneq14a-h2"],"title":"Isolate Variable Term","text":"What number should we add to both sides?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq14a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$8>3x$$"],"dependencies":["aaa317eLinIneq14a-h3"],"title":"Isolate Variable Term","text":"What inequality do we get after isolating the variable term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$-2>3x$$","$$8>3x$$"]},{"id":"aaa317eLinIneq14a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["aaa317eLinIneq14a-h4"],"title":"Apply the Multiplication Property","text":"To make the coeffient of $$x$$ be $$1$$, what number should we multiply both sides of the inequality by?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq14a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{8}{3}$$"],"dependencies":["aaa317eLinIneq14a-h5"],"title":"Left Side","text":"What is the left side after multiplying by $$\\\\frac{1}{3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq14a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x$$"],"dependencies":["aaa317eLinIneq14a-h6"],"title":"Right Side","text":"What is the right side after multiplying by $$\\\\frac{1}{3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq14a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":[">"],"dependencies":["aaa317eLinIneq14a-h7"],"title":"Sign","text":"What is the sign of the inequality after multiplication?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["<",">"]},{"id":"aaa317eLinIneq14a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(-\\\\infty,\\\\frac{3}{8})$$"],"dependencies":["aaa317eLinIneq14a-h8"],"title":"Interval Notation","text":"What is $$\\\\frac{8}{3}>x$$ written in interval notion?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(-\\\\infty,\\\\frac{3}{8})$$","$$(\\\\frac{3}{8},\\\\infty)$$"]}]}}]},{"id":"aaa317eLinIneq15","title":"Solve the Inequality","body":"Write your final answer in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Linear Inequalities and Absolute Value Inequalities","courseName":"OpenStax: College Algebra","steps":[{"id":"aaa317eLinIneq15a","stepAnswer":["$$[\\\\frac{-13}{2},\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$4\\\\left(x+3\\\\right) \\\\geq 2x-1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[\\\\frac{-13}{2},\\\\infty)$$","choices":["$$[\\\\frac{11}{2},\\\\infty)$$","$$(\\\\frac{11}{2},\\\\infty)$$","$$[\\\\frac{-13}{2},\\\\infty)$$","$$(\\\\frac{-13}{2},\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"aaa317eLinIneq15a-h1","type":"hint","dependencies":[],"title":"Distributive Property","text":"We start by applying the distributive property of multiplication to the left side. The inequality becomes $$4x+12 \\\\geq 2x-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq15a-h2","type":"hint","dependencies":["aaa317eLinIneq15a-h1"],"title":"Move Variable Terms","text":"We then move the variable terms to one side of the inequality","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq15a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2x+12 \\\\geq -1$$"],"dependencies":["aaa317eLinIneq15a-h2"],"title":"Move Variable Terms","text":"What inequality do we get after moving the variable terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2x+12 \\\\geq -1$$","$$12 \\\\geq 2x-1$$"]},{"id":"aaa317eLinIneq15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-12$$"],"dependencies":["aaa317eLinIneq15a-h3"],"title":"Isolate Variable Term","text":"What number should we add to both sides?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq15a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2x \\\\geq -13$$"],"dependencies":["aaa317eLinIneq15a-h4"],"title":"Isolate Variable Term","text":"What inequality do we get after isolating the variable term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2x \\\\geq -13$$","$$2x \\\\geq 11$$"]},{"id":"aaa317eLinIneq15a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["aaa317eLinIneq15a-h5"],"title":"Apply the Multiplication Property","text":"To make the coeffient of $$x$$ be $$1$$, what number should we multiply both sides of the inequality by?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq15a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x$$"],"dependencies":["aaa317eLinIneq15a-h6"],"title":"Left Side","text":"What is the left side after multiplying by $$\\\\frac{1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq15a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-13}{2}$$"],"dependencies":["aaa317eLinIneq15a-h6"],"title":"Right Side","text":"What is the right side after multiplying by $$\\\\frac{1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq15a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$ \\\\geq $$"],"dependencies":["aaa317eLinIneq15a-h6"],"title":"Sign","text":"What is the sign of the inequality after multiplication?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$ \\\\leq $$","$$ \\\\geq $$"]},{"id":"aaa317eLinIneq15a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$[\\\\frac{-13}{2},\\\\infty)$$"],"dependencies":["aaa317eLinIneq15a-h7","aaa317eLinIneq15a-h8","aaa317eLinIneq15a-h9"],"title":"Interval Notation","text":"What is $$x \\\\geq \\\\frac{-13}{2}$$ written in interval notion?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$[\\\\frac{-13}{2},\\\\infty)$$","$$(\\\\frac{-13}{2},\\\\infty)$$"]}]}}]},{"id":"aaa317eLinIneq16","title":"Using Interval Notation to Express All Real Numbers Between a and $$b$$, Including a and $$b$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Linear Inequalities and Absolute Value Inequalities","courseName":"OpenStax: College Algebra","steps":[{"id":"aaa317eLinIneq16a","stepAnswer":["[-3,5]"],"problemType":"TextBox","stepTitle":"Use interval notation to indicate all real numbers between and including $$-3$$ and $$5$$.","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"aaa317eLinIneq16a-h1","type":"hint","dependencies":[],"title":"Bracket","text":"The bracket indicates that the number is included in the set.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq16a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["[a,b]"],"dependencies":["aaa317eLinIneq16a-h1"],"title":"Interval Notation","text":"How to represent all real numbers between a and $$b$$, including a and $$b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq16a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["[-3,5]"],"dependencies":["aaa317eLinIneq16a-h2"],"title":"Interval Notation","text":"Let\'s apply the above principle to our problem!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaa317eLinIneq17","title":"Using Interval Notation to Express All Real Numbers Less Than a or Greater Than or Equal to $$b$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Linear Inequalities and Absolute Value Inequalities","courseName":"OpenStax: College Algebra","steps":[{"id":"aaa317eLinIneq17a","stepAnswer":["$$(-\\\\infty,-2) \\\\cup [3,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"Express all real numbers less than $$-2$$ or greater than or equal to $$3$$ in interval notation.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,-2) \\\\cup [3,\\\\infty)$$","choices":["$$(-\\\\infty,-2] \\\\cup [3,\\\\infty)$$","$$(-\\\\infty,-2) \\\\cup (3,\\\\infty)$$","$$(-\\\\infty,-2) \\\\cup [3,\\\\infty)$$","$$(-\\\\infty,-2] \\\\cup (3,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"aaa317eLinIneq17a-h1","type":"hint","dependencies":[],"title":"Two Intervals","text":"We have to write two intervals for this problem.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq17a-h2","type":"hint","dependencies":["aaa317eLinIneq17a-h1"],"title":"Interval $$1$$","text":"The first interval must indicate all real numbers less than $$-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq17a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-inf,b)"],"dependencies":["aaa317eLinIneq17a-h2"],"title":"Interval Notation","text":"How to represent all real numbers less than $$b$$, but not including $$b$$? Remember that the parenthesis indicates that the number is not included in the set.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq17a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-inf,-2)"],"dependencies":["aaa317eLinIneq17a-h3"],"title":"Interval $$1$$","text":"Let\'s apply the above principle to our problem!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq17a-h5","type":"hint","dependencies":["aaa317eLinIneq17a-h4"],"title":"Interval $$2$$","text":"The second interval must indicate all real numbers greater than or equal to $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq17a-h6","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["[a,inf)"],"dependencies":["aaa317eLinIneq17a-h5"],"title":"Interval Notation","text":"How to represent all real numbers greater than a, including a? Remember that the bracket indicates that the number is included in the set.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq17a-h7","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["[3,inf)"],"dependencies":["aaa317eLinIneq17a-h6"],"title":"Interval $$2$$","text":"Let\'s apply the above principle to our problem!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq17a-h8","type":"hint","dependencies":["aaa317eLinIneq17a-h4","aaa317eLinIneq17a-h7"],"title":"Combination","text":"We want to combine these two sets and we can accomplish this by inserting the union symbol, U, between the two intervals.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaa317eLinIneq18","title":"Demonstrating the Addition and Multiplication Property","body":"Solve:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Linear Inequalities and Absolute Value Inequalities","courseName":"OpenStax: College Algebra","steps":[{"id":"aaa317eLinIneq18a","stepAnswer":["$$x<1$$"],"problemType":"MultipleChoice","stepTitle":"$$3x-2<1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x<1$$","choices":["$$x<-3$$","$$x<2$$","$$x>1$$","$$x<1$$"],"hints":{"DefaultPathway":[{"id":"aaa317eLinIneq18a-h1","type":"hint","dependencies":[],"title":"Addition Property","text":"The addition property states that if $$a<b$$, then $$a+c<b+c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["aaa317eLinIneq18a-h1"],"title":"Addition Property","text":"What number should we add on both sides in order to isolate the variable with its coefficient on the left?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq18a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x<3$$"],"dependencies":["aaa317eLinIneq18a-h2"],"title":"Addition Property","text":"What do we get when we add $$2$$ on both sides?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq18a-h4","type":"hint","dependencies":["aaa317eLinIneq18a-h3"],"title":"Multiplication Property","text":"The multiplication property states that if $$a<b$$ and $$c>0$$, then $$ac<bc$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq18a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["aaa317eLinIneq18a-h4"],"title":"Multiplication Property","text":"What number should we multiply on both sides in order to isolate the variable by itself on the left?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq18a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x<1$$"],"dependencies":["aaa317eLinIneq18a-h5"],"title":"Multiplication Property","text":"What do we get when we multiply $$\\\\frac{1}{3}$$ on both sides?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaa317eLinIneq19","title":"Demonstrating the Addition and Multiplication Property","body":"Solve:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Linear Inequalities and Absolute Value Inequalities","courseName":"OpenStax: College Algebra","steps":[{"id":"aaa317eLinIneq19a","stepAnswer":["$$x \\\\geq -5$$"],"problemType":"TextBox","stepTitle":"$$4x+7 \\\\geq 2x-3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x \\\\geq -5$$","hints":{"DefaultPathway":[{"id":"aaa317eLinIneq19a-h1","type":"hint","dependencies":[],"title":"Addition Property","text":"The addition property states that if $$a \\\\geq b$$, then $$a+c \\\\geq b+c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-7-2x$$"],"dependencies":["aaa317eLinIneq19a-h1"],"title":"Addition Property","text":"What should we add on both sides in order to isolate the variable with its coefficient on the left?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq19a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x \\\\geq -10$$"],"dependencies":["aaa317eLinIneq19a-h2"],"title":"Addition Property","text":"What do we get when we add $$-7-2x$$ on both sides?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq19a-h4","type":"hint","dependencies":["aaa317eLinIneq19a-h3"],"title":"Multiplication Property","text":"The multiplication property states that if $$a \\\\geq b$$ and $$c>0$$, then $$ac \\\\geq bc$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq19a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["aaa317eLinIneq19a-h4"],"title":"Multiplication Property","text":"What number should we multiply on both sides in order to isolate the variable by itself on the left?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq19a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x \\\\geq -5$$"],"dependencies":["aaa317eLinIneq19a-h5"],"title":"Multiplication Property","text":"What do we get when we multiply $$\\\\frac{1}{2}$$ on both sides?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaa317eLinIneq2","title":"Using Interval Notation to Express All Real Numbers Less Than or Equal to a or Greater Than or Equal to $$b$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Linear Inequalities and Absolute Value Inequalities","courseName":"OpenStax: College Algebra","steps":[{"id":"aaa317eLinIneq2a","stepAnswer":["$$(-\\\\infty,-1] \\\\cup [1,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"Write the interval expressing all real numbers less than or equal to $$-1$$ or greater than or equal to $$1$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,-1] \\\\cup [1,\\\\infty)$$","choices":["$$(-\\\\infty,-1] \\\\cap [1,\\\\infty)$$","$$(-\\\\infty,-1] \\\\cup [1,\\\\infty)$$","$$(-\\\\infty,-1] \\\\cup [1,\\\\infty)$$","[-1,-inf)U(inf, 1]","$$[-\\\\infty,-1) \\\\cup (1,\\\\infty]$$"],"hints":{"DefaultPathway":[{"id":"aaa317eLinIneq2a-h1","type":"hint","dependencies":[],"title":"Number of intervals","text":"We need two intervals, one for representing \\"less than or equal to -1\\", and another for \\"greater than or equal to 1.\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq2a-h2","type":"hint","dependencies":["aaa317eLinIneq2a-h1"],"title":"Or","text":"or means we should use the union symbol U (satisfy at least one OR the other condition)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq2a-h3","type":"hint","dependencies":["aaa317eLinIneq2a-h2"],"title":"First interval endpoints","text":"The two endpoints for the interval \\"less than or equal to -1\\" are $$-\\\\infty$$ and $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq2a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["("],"dependencies":["aaa317eLinIneq2a-h3"],"title":"First interval symbol","text":"Should we use parenthesis or bracket for the negative $$\\\\infty$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["(","["]},{"id":"aaa317eLinIneq2a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["]"],"dependencies":["aaa317eLinIneq2a-h4"],"title":"First interval symbol","text":"Should we use parenthesis or bracket for the -1?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":[")","]"]},{"id":"aaa317eLinIneq2a-h6","type":"hint","dependencies":["aaa317eLinIneq2a-h5"],"title":"Second interval endpoints","text":"The two endpoints for the interval \\"greater than or equal to b\\" are $$1$$ and $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq2a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["["],"dependencies":["aaa317eLinIneq2a-h6"],"title":"Second interval symbol","text":"Should we use parenthesis or bracket for the 1?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["(","["]},{"id":"aaa317eLinIneq2a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":[")"],"dependencies":["aaa317eLinIneq2a-h7"],"title":"Second interval symbol","text":"Should we use parenthesis or bracket for the $$\\\\infty$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":[")","]"]}]}}]},{"id":"aaa317eLinIneq20","title":"Solving an Inequality Algebraically","body":"Solve the inequality and write the answer using interval notation:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Linear Inequalities and Absolute Value Inequalities","courseName":"OpenStax: College Algebra","steps":[{"id":"aaa317eLinIneq20a","stepAnswer":["(2,inf)"],"problemType":"TextBox","stepTitle":"$$-x+4<\\\\frac{1}{2} x+1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(2,\\\\infty)$$","hints":{"DefaultPathway":[{"id":"aaa317eLinIneq20a-h1","type":"hint","dependencies":[],"title":"Isolation","text":"We can isolate the variable term by using the addition property.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x-1$$"],"dependencies":["aaa317eLinIneq20a-h1"],"title":"Addition","text":"What should we add in order to isolate the variable with its coefficient on the right?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq20a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3<\\\\frac{3}{2} x$$"],"dependencies":["aaa317eLinIneq20a-h2"],"title":"Addition","text":"What do we get after we add $$x-1$$ on both sides?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq20a-h4","type":"hint","dependencies":["aaa317eLinIneq20a-h3"],"title":"Isolation","text":"We can leave the variable term by itself on the right by using the multiplication property.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq20a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{3}$$"],"dependencies":["aaa317eLinIneq20a-h4"],"title":"Multiplication","text":"What number should we multiply in order to isolate the variable by itself?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq20a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x>2$$"],"dependencies":["aaa317eLinIneq20a-h5"],"title":"Multiplication","text":"What do we get after we multiply $$\\\\frac{2}{3}$$ on both sides?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq20a-h7","type":"hint","dependencies":["aaa317eLinIneq20a-h6"],"title":"Interval Notation","text":"$$(b,\\\\infty)$$ represents all real numbers greater than $$b$$, but not including $$b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaa317eLinIneq21","title":"Solving an Inequality with Fractions","body":"Solve the inequality and write the answer in interval notation:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Linear Inequalities and Absolute Value Inequalities","courseName":"OpenStax: College Algebra","steps":[{"id":"aaa317eLinIneq21a","stepAnswer":["[-3/14,inf)"],"problemType":"TextBox","stepTitle":"$$\\\\left(-\\\\frac{5}{6}\\\\right) x \\\\leq \\\\frac{3}{4}+\\\\frac{8}{3} x$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[\\\\frac{-3}{14},\\\\infty)$$","hints":{"DefaultPathway":[{"id":"aaa317eLinIneq21a-h1","type":"hint","dependencies":[],"title":"Isolation","text":"We need to first put the variable terms on one side by using the addition property.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq21a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{6} x-\\\\frac{3}{4}$$"],"dependencies":["aaa317eLinIneq21a-h1"],"title":"Addition","text":"What should we add in order to isolate the variable with its coefficient on the right?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq21a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{-3}{4} \\\\leq \\\\frac{5}{6} x+\\\\frac{8}{3} x$$"],"dependencies":["aaa317eLinIneq21a-h2"],"title":"Addition","text":"What do we get after we add $$\\\\frac{5}{6} x-\\\\frac{3}{4}$$ on both sides?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{-3}{4}+\\\\frac{5}{6} x \\\\leq \\\\frac{8}{3} x$$","$$\\\\frac{-3}{4} \\\\leq \\\\frac{5}{6} x+\\\\frac{8}{3} x$$","$$\\\\frac{3}{4} \\\\leq \\\\frac{5}{6} x+\\\\frac{8}{3} x$$"]},{"id":"aaa317eLinIneq21a-h4","type":"hint","dependencies":["aaa317eLinIneq21a-h3"],"title":"Common Denominator","text":"We then need to write fractions with common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq21a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["aaa317eLinIneq21a-h4"],"title":"Common Denominator","text":"What is the common denominator between $$3$$ and 6?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq21a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{-3}{4} \\\\leq \\\\frac{5}{6} x+\\\\frac{16}{6} x$$"],"dependencies":["aaa317eLinIneq21a-h5"],"title":"Common Denominator","text":"Let\'s rewrite the fractions with their common denominator $$6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{-3}{4} \\\\leq \\\\frac{5}{6} x+\\\\frac{16}{6} x$$","$$\\\\frac{-3}{4} \\\\leq \\\\frac{5}{6} x+\\\\frac{18}{6} x$$","$$\\\\frac{-3}{4} \\\\leq \\\\frac{5}{6} x+\\\\frac{8}{6} x$$"]},{"id":"aaa317eLinIneq21a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-3}{4} \\\\leq \\\\frac{21}{6} x$$"],"dependencies":["aaa317eLinIneq21a-h6"],"title":"Simplification","text":"Let\'s simplify the right-hand side!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq21a-h8","type":"hint","dependencies":["aaa317eLinIneq21a-h7"],"title":"Isolation","text":"We can leave the variable term by itself on the right by using the multiplication property.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq21a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{6}{21}$$"],"dependencies":["aaa317eLinIneq21a-h8"],"title":"Multiplication","text":"What number should we multiply in order to isolate the variable by itself?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq21a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x \\\\geq \\\\frac{-3}{14}$$"],"dependencies":["aaa317eLinIneq21a-h9"],"title":"Multiplication","text":"What do we get after we multiply $$\\\\frac{6}{21}$$ on both sides?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq21a-h11","type":"hint","dependencies":["aaa317eLinIneq21a-h10"],"title":"Interval Notation","text":"$$[b,\\\\infty)$$ represents all real numbers greater than $$b$$, including $$b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaa317eLinIneq22","title":"Solving a Compound Inequality","body":"Solve the compound inequality:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Linear Inequalities and Absolute Value Inequalities","courseName":"OpenStax: College Algebra","steps":[{"id":"aaa317eLinIneq22a","stepAnswer":["$$6<x \\\\leq 9$$"],"problemType":"MultipleChoice","stepTitle":"$$4<2x-8 \\\\leq 10$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$6<x \\\\leq 9$$","choices":["$$6<x \\\\leq 9$$","$$3<x \\\\leq 10$$","$$6<x \\\\leq 18$$","$$4<x \\\\leq 9$$"],"hints":{"DefaultPathway":[{"id":"aaa317eLinIneq22a-h1","type":"hint","dependencies":[],"title":"Isolation","text":"We can isolate the variable term by using the addition property.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq22a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["aaa317eLinIneq22a-h1"],"title":"Addition","text":"What number should we add in order to isolate the variable with its coefficient in the middle?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq22a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$12<2x \\\\leq 18$$"],"dependencies":["aaa317eLinIneq22a-h2"],"title":"Addition","text":"What do we get after we add 8?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$10<2x \\\\leq 18$$","$$12<2x \\\\leq 16$$","$$12<2x \\\\leq 18$$"]},{"id":"aaa317eLinIneq22a-h4","type":"hint","dependencies":["aaa317eLinIneq22a-h3"],"title":"Isolation","text":"We can leave the variable term by itself in the middle by using the multiplication property.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq22a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["aaa317eLinIneq22a-h4"],"title":"Multiplication","text":"What number should we multiply in order to isolate the variable by itself in the middle?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq22a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$6<x \\\\leq 9$$"],"dependencies":["aaa317eLinIneq22a-h5"],"title":"Multiplication","text":"What do we get after we multiply $$\\\\frac{1}{2}$$ through all three parts?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$6<x \\\\leq 9$$","$$3<x \\\\leq 10$$","$$6<x \\\\leq 18$$","$$4<x \\\\leq 9$$"]}]}}]},{"id":"aaa317eLinIneq23","title":"Solving a Compound Inequality with the Variable in All Three Parts","body":"Solve the compound inequality:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Linear Inequalities and Absolute Value Inequalities","courseName":"OpenStax: College Algebra","steps":[{"id":"aaa317eLinIneq23a","stepAnswer":["$$\\\\frac{-1}{8}<y<\\\\frac{1}{2}$$"],"problemType":"MultipleChoice","stepTitle":"$$3y<4-5y<5+3y$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{-1}{8}<y<\\\\frac{1}{2}$$","choices":["$$\\\\frac{1}{8}<y<\\\\frac{1}{2}$$","$$\\\\frac{-1}{8}<y<\\\\frac{1}{2}$$","$$\\\\frac{-1}{8}<y<\\\\frac{1}{6}$$","$$\\\\frac{-1}{2}<y<\\\\frac{-1}{8}$$"],"hints":{"DefaultPathway":[{"id":"aaa317eLinIneq23a-h1","type":"hint","dependencies":[],"title":"Separation","text":"We can separate the compound inequality into two inequalities.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq23a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3y<4-5y$$ and $$4-5y<5+3y$$"],"dependencies":["aaa317eLinIneq23a-h1"],"title":"Separation","text":"What two inequalities do we get after the separation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$3y<4-5y$$ and $$4-5y<5+3y$$","$$3y<4-5y$$ and $$4-5y>5+3y$$"]},{"id":"aaa317eLinIneq23a-h3","type":"hint","dependencies":["aaa317eLinIneq23a-h2"],"title":"Isolation for the first inequality","text":"We can isolate the variable term by using the addition property.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq23a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5y$$"],"dependencies":["aaa317eLinIneq23a-h3"],"title":"Addition","text":"What should we add in order to isolate the variable with its coefficient on the left?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq23a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8y<4$$"],"dependencies":["aaa317eLinIneq23a-h4"],"title":"Addition","text":"What do we get after we add $$5y$$ on both sides?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq23a-h6","type":"hint","dependencies":["aaa317eLinIneq23a-h5"],"title":"Isolation for the first inequality","text":"We can leave the variable term by itself on the left by using the multiplication property.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq23a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{8}$$"],"dependencies":["aaa317eLinIneq23a-h6"],"title":"Multiplication","text":"What number should we multiply on both sides in order to isolate the variable by itself on the left?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq23a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y<\\\\frac{1}{2}$$"],"dependencies":["aaa317eLinIneq23a-h7"],"title":"Multiplication","text":"What do we get after we multiply $$\\\\frac{1}{8}$$ on both sides?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq23a-h9","type":"hint","dependencies":["aaa317eLinIneq23a-h8"],"title":"Isolation for the second inequality","text":"We can isolate the variable term by using the addition property.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq23a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4-3y$$"],"dependencies":["aaa317eLinIneq23a-h9"],"title":"Addition","text":"What should we add in order to isolate the variable with its coefficient on the left?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq23a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-8y<1$$"],"dependencies":["aaa317eLinIneq23a-h10"],"title":"Addition","text":"What do we get after we add $$-4-3y$$ on both sides?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq23a-h12","type":"hint","dependencies":["aaa317eLinIneq23a-h11"],"title":"Isolation for the second inequality","text":"We can leave the variable term by itself on the left by using the multiplication property.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq23a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{8}$$"],"dependencies":["aaa317eLinIneq23a-h12"],"title":"Multiplication","text":"What number should we multiply on both sides in order to isolate the variable by itself on the left?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq23a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y>\\\\frac{-1}{8}$$"],"dependencies":["aaa317eLinIneq23a-h13"],"title":"Multiplication and Reverse","text":"What do we get after we multiply $$\\\\frac{-1}{8}$$ on both sides? Remember to reverse the inequality!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq23a-h15","type":"hint","dependencies":["aaa317eLinIneq23a-h8","aaa317eLinIneq23a-h14"],"title":"Putting two inequalities together","text":"We can draw the two inequalities on a number line to visualize their relationship.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq23a-h16","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{-1}{8}<y<\\\\frac{1}{2}$$"],"dependencies":["aaa317eLinIneq23a-h11"],"title":"Putting two inequalities together","text":"What do we get after we put the two inequalities above together?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{1}{8}<y<\\\\frac{1}{2}$$","$$\\\\frac{-1}{8}<y<\\\\frac{1}{2}$$","$$\\\\frac{-1}{8}<y<\\\\frac{1}{6}$$","$$\\\\frac{-1}{2}<y<\\\\frac{-1}{8}$$"]}]}}]},{"id":"aaa317eLinIneq24","title":"Determining a Number within a Prescribed Distance","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Linear Inequalities and Absolute Value Inequalities","courseName":"OpenStax: College Algebra","steps":[{"id":"aaa317eLinIneq24a","stepAnswer":["[-1,5]"],"problemType":"TextBox","stepTitle":"Describe all x-values within a distance of $$3$$ from the number $$2$$.","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"aaa317eLinIneq24a-h1","type":"hint","dependencies":[],"title":"Representing the distance","text":"The distance from $$x$$ to $$2$$ can be represented using an absolute value symbol.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq24a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$|x-2|$$"],"dependencies":["aaa317eLinIneq24a-h1"],"title":"Representing the distance","text":"How to represent the distance from $$x$$ to 2?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x-2$$","$$|x-2|$$","$$|x+2|$$"]},{"id":"aaa317eLinIneq24a-h3","type":"hint","dependencies":["aaa317eLinIneq24a-h2"],"title":"Inequality","text":"We can write the values of $$x$$ that satisfy the condition as an absolute value inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq24a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$|x-2| \\\\leq 3$$"],"dependencies":["aaa317eLinIneq24a-h3"],"title":"Taking into account the condition","text":"How to represent that the distance from $$x$$ to $$2$$ is less than or equal to 3?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$|x-2| \\\\leq 3$$","$$|x-2| \\\\geq 3$$","$$x-2 \\\\leq 3$$"]},{"id":"aaa317eLinIneq24a-h5","type":"hint","dependencies":["aaa317eLinIneq24a-h4"],"title":"Separation","text":"We can separate the absolute inequality into two inequalities by considering cases that $$x-2$$ is positive and $$x-2$$ is negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq24a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x-2 \\\\leq 3$$ and $$x-2 \\\\geq -3$$"],"dependencies":["aaa317eLinIneq24a-h5"],"title":"Separation","text":"What two inequalities do we get after the separation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x-2 \\\\leq 3$$ and $$x-2 \\\\geq -3$$","$$x-2 \\\\leq 3$$ and $$x-2 \\\\leq -3$$"]},{"id":"aaa317eLinIneq24a-h7","type":"hint","dependencies":["aaa317eLinIneq24a-h6"],"title":"Isolation","text":"We can isolate the variable term by using the addition property.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq24a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["aaa317eLinIneq24a-h7"],"title":"Addition","text":"What number should we add in order to isolate the variable on the left?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq24a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x \\\\leq 5$$ and $$x \\\\geq -1$$"],"dependencies":["aaa317eLinIneq24a-h8"],"title":"Addition","text":"What do we get after we add $$2$$ on both sides?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x \\\\leq 5$$ and $$x \\\\geq -5$$","$$x \\\\leq 5$$ and $$x \\\\geq -1$$"]},{"id":"aaa317eLinIneq24a-h10","type":"hint","dependencies":["aaa317eLinIneq24a-h9"],"title":"Putting two inequalities together","text":"We can draw the two inequalities on a number line to visualize their relationship.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq24a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-1 \\\\leq x \\\\leq 5$$"],"dependencies":["aaa317eLinIneq24a-h10"],"title":"Putting two inequalities together","text":"What do we get after we put the two inequalities above together?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$-1 \\\\leq x \\\\leq 5$$","$$1 \\\\leq x \\\\leq 5$$","$$-5 \\\\leq x \\\\leq -1$$"]},{"id":"aaa317eLinIneq24a-h12","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["[-1,5]"],"dependencies":["aaa317eLinIneq24a-h11"],"title":"Interval Notation","text":"How to represent the above inequality in interval notation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaa317eLinIneq25","title":"Solving an Absolute Value Inequality","body":"Solve:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Linear Inequalities and Absolute Value Inequalities","courseName":"OpenStax: College Algebra","steps":[{"id":"aaa317eLinIneq25a","stepAnswer":["$$k \\\\geq 7$$ or $$k \\\\leq 1$$"],"problemType":"MultipleChoice","stepTitle":"$$-2|k-4| \\\\leq -6$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$k \\\\geq 7$$ or $$k \\\\leq 1$$","choices":["$$k \\\\geq 7$$ and $$k \\\\leq 1$$","$$k \\\\geq 7$$ or $$k \\\\leq 1$$","$$k \\\\geq 8$$ or $$k \\\\leq -1$$","$$k \\\\geq 8$$ and $$k \\\\leq -1$$"],"hints":{"DefaultPathway":[{"id":"aaa317eLinIneq25a-h1","type":"hint","dependencies":[],"title":"Isolation","text":"We can first isolate the absolute value term by using the multiplication property.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq25a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{2}$$"],"dependencies":["aaa317eLinIneq25a-h1"],"title":"Multiplication","text":"What number should we multiply on both sides in order to isolate the absolute value term by itself on the left?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq25a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$|k-4| \\\\geq 3$$"],"dependencies":["aaa317eLinIneq25a-h2"],"title":"Multiplication","text":"What do we get after we multiply $$\\\\frac{-1}{2}$$ on both sides? Remember to reverse the inequality!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$|k-4| \\\\leq 3$$","$$|k-4| \\\\geq 3$$","$$|k+4| \\\\geq 3$$"]},{"id":"aaa317eLinIneq25a-h4","type":"hint","dependencies":["aaa317eLinIneq25a-h3"],"title":"Absolute Value Inequality","text":"For an algebraic expression X, and $$k>0$$, an absolute value inequality is an inequality of the form $$|X| \\\\geq k$$ is equivalent to $$X \\\\leq -k$$ or $$X \\\\geq k$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq25a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$k-4 \\\\geq 3$$ or $$k-4 \\\\leq -3$$"],"dependencies":["aaa317eLinIneq25a-h4"],"title":"Absolute Value Inequality","text":"How can we apply the above principle to our problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$k-4 \\\\geq 3$$ or $$k-4 \\\\leq -3$$","$$k-4 \\\\geq 3$$ or $$k+4 \\\\leq -3$$","$$k-4 \\\\geq 3$$ or $$-k+4 \\\\geq 3$$"]},{"id":"aaa317eLinIneq25a-h6","type":"hint","dependencies":["aaa317eLinIneq25a-h5"],"title":"Simplification","text":"Let\'s simplify the two inequalities by using the addition property.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq25a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["aaa317eLinIneq25a-h6"],"title":"Addition","text":"What number should we add in order to isolate the variable on the left?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq25a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$k \\\\geq 7$$ or $$k \\\\leq 1$$"],"dependencies":["aaa317eLinIneq25a-h7"],"title":"Addition","text":"What do we get after we add $$4$$ on both sides?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$k \\\\geq 7$$ and $$k \\\\leq 1$$","$$k \\\\geq 7$$ or $$k \\\\leq 1$$","$$k \\\\geq 8$$ or $$k \\\\leq -1$$","$$k \\\\geq 8$$ and $$k \\\\leq -1$$"]}]}}]},{"id":"aaa317eLinIneq26","title":"Solving an Absolute Value Inequality","body":"Solve the inequality involving absolute value. Write your final answer in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Linear Inequalities and Absolute Value Inequalities","courseName":"OpenStax: College Algebra","steps":[{"id":"aaa317eLinIneq26a","stepAnswer":["$$(-\\\\infty,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$|x+9| \\\\geq -6$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\infty)$$","choices":["$$[-\\\\infty,\\\\infty]$$","$$(-\\\\infty,\\\\infty)$$","$$(\\\\frac{-2}{3},9)$$","$$(6,15)$$"],"hints":{"DefaultPathway":[{"id":"aaa317eLinIneq26a-h1","type":"hint","dependencies":[],"title":"Absolute Value","text":"Remember that for expression with the absolute value sign, it\'s always greater than or equal to $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq26a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(-\\\\infty,\\\\infty)$$"],"dependencies":["aaa317eLinIneq26a-h1"],"title":"Absolute Value","text":"No matter what $$x$$ is, the expression on the left is always non-negative. It is always greater than a negative value. How do you represent $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$[-\\\\infty,\\\\infty]$$","$$(-\\\\infty,\\\\infty)$$","$$(\\\\frac{-2}{3},9)$$","$$(6,15)$$"]}]}}]},{"id":"aaa317eLinIneq27","title":"Solving an Absolute Value Inequality","body":"Solve the inequality involving absolute value. 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inequality by using the addition and multiplication property.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq29a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["aaa317eLinIneq29a-h6"],"title":"Addition","text":"What number should we add in order to isolate the variable with its coefficient in the middle?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq29a-h8","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["-6<=2x<=4"],"dependencies":["aaa317eLinIneq29a-h7"],"title":"Addition","text":"What do we get after we add $$-1$$ through all three parts?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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1?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq3b-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x$$"],"dependencies":["aaa317eLinIneq3b-h4"],"title":"Right side","text":"What is the right side after adding 1?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aaa317eLinIneq3c","stepAnswer":["$$x>2$$"],"problemType":"MultipleChoice","stepTitle":"$$x+7>9$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x>2$$","choices":["$$x>16$$","$$x>2$$"],"hints":{"DefaultPathway":[{"id":"aaa317eLinIneq3c-h1","type":"hint","dependencies":[],"title":"Addition Property","text":"The addition property of inequality: if $$a>b$$, then $$a+c>b+c$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq3c-h2","type":"hint","dependencies":["aaa317eLinIneq3c-h1"],"title":"Applying the Addition Property","text":"Our goal is to isolate $$x$$ (have $$x$$ on one side and numbers on the other side)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq3c-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-7$$"],"dependencies":["aaa317eLinIneq3c-h2"],"title":"Applying the Addition Property","text":"What number should we add to both sides of the inequality to isolate $$x$$? (hint: subtraction is the same as adding a negative number)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq3c-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x$$"],"dependencies":["aaa317eLinIneq3c-h3"],"title":"Left side","text":"What is the left side after adding -7?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq3c-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["aaa317eLinIneq3c-h4"],"title":"Right side","text":"What is the right side after adding -7?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaa317eLinIneq30","title":"Solving an Absolute Value Inequality","body":"Solve the inequality involving absolute value. Write your final answer in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Linear Inequalities and Absolute Value Inequalities","courseName":"OpenStax: College Algebra","steps":[{"id":"aaa317eLinIneq30a","stepAnswer":["$$(-\\\\infty,-4] \\\\cup [8,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$|x-2|+4 \\\\geq 10$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,-4] \\\\cup [8,\\\\infty)$$","choices":["$$(-\\\\infty,-6] \\\\cup (8,\\\\infty)$$","$$(-\\\\infty,-4) \\\\cup (8,\\\\infty)$$","$$(-\\\\infty,-4] \\\\cup [8,\\\\infty)$$","$$(-\\\\infty,-4) \\\\cup [10,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"aaa317eLinIneq30a-h1","type":"hint","dependencies":[],"title":"Isolation","text":"We can first isolate the absolute value term by using the addition 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10$$","$$x-2 \\\\geq 10$$"]},{"id":"aaa317eLinIneq30a-h4","type":"hint","dependencies":["aaa317eLinIneq30a-h3"],"title":"Absolute Value Inequality","text":"For an algebraic expression X, and $$k>0$$, an absolute value inequality is an inequality of the form $$|X| \\\\geq k$$ is equivalent to $$X \\\\leq -k$$ or $$X \\\\geq k$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq30a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x-2 \\\\leq -6$$ or $$x-2 \\\\geq 6$$"],"dependencies":["aaa317eLinIneq30a-h4"],"title":"Absolute Value Inequality","text":"How can we apply the above principle to our problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x+2 \\\\leq -6$$ or $$x-2 \\\\geq 6$$","$$x-2 \\\\leq -6$$ or $$-x+2 \\\\geq 6$$","$$x-2 \\\\leq -6$$ or $$x-2 \\\\geq 6$$"]},{"id":"aaa317eLinIneq30a-h6","type":"hint","dependencies":["aaa317eLinIneq30a-h5"],"title":"Simplification","text":"Let\'s simplify the above two inequalities by using the addition property.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq30a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["aaa317eLinIneq30a-h6"],"title":"Addition","text":"What number should we add in order to isolate the variable with its coefficient on the left?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq30a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x \\\\leq -4$$ or $$x \\\\geq 8$$"],"dependencies":["aaa317eLinIneq30a-h7"],"title":"Addition","text":"What do we get after we add 2?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x \\\\leq -4$$ or $$x \\\\geq -8$$","$$x \\\\leq -4$$ or $$x \\\\geq 4$$","$$x \\\\leq -4$$ or $$x \\\\geq 8$$"]},{"id":"aaa317eLinIneq30a-h9","type":"hint","dependencies":["aaa317eLinIneq30a-h8"],"title":"Interval Notation","text":"$$(-\\\\infty,b]$$ represents all real numbers less than $$b$$, including $$b$$, and $$[a,\\\\infty)srepresent$$ all real numbers greater than a, including a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaa317eLinIneq4","title":"Demonstrating the Multiplication Property","body":"Illustrate the multiplication property for inequalities by solving each of the following:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq4b-h2","type":"hint","dependencies":["aaa317eLinIneq4b-h1"],"title":"Addition Property","text":"The addition property of inequality: if $$a \\\\geq b$$, then $$a+c \\\\geq b+c$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq4b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["aaa317eLinIneq4b-h2"],"title":"Applying the Addition Property","text":"What number should we add to both sides of the inequality?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq4b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2x$$"],"dependencies":["aaa317eLinIneq4b-h3"],"title":"Left side After 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the right side after multiplying by $$\\\\frac{-1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq4b-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$ \\\\leq $$"],"dependencies":["aaa317eLinIneq4b-h9"],"title":"Sign","text":"What is the sign?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$ \\\\leq $$","$$ \\\\geq $$"]}]}},{"id":"aaa317eLinIneq4c","stepAnswer":["$$x<-5$$"],"problemType":"MultipleChoice","stepTitle":"$$5-x>10$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x<-5$$","choices":["$$x>5$$","$$x<5$$","$$x>-5$$","$$x<-5$$"],"hints":{"DefaultPathway":[{"id":"aaa317eLinIneq4c-h1","type":"hint","dependencies":[],"title":"Applying the Properties","text":"Our goal is to isolate $$x$$ (have $$x$$ on one side 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$$ac<bc$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq4c-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["aaa317eLinIneq4c-h6"],"title":"Applying the Multiplication Property","text":"What number should we multiply to both sides of the inequality?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq4c-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x$$"],"dependencies":["aaa317eLinIneq4c-h7"],"title":"Left Side","text":"What is the left side after multiplying -1?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq4c-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["aaa317eLinIneq4c-h8"],"title":"Right Side","text":"What is the right side after multiplying by -1?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq4c-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["<"],"dependencies":["aaa317eLinIneq4c-h9"],"title":"Sign","text":"What is the sign?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["<",">"]}]}}]},{"id":"aaa317eLinIneq5","title":"Solving an Inequality Algebraically","body":"Solve the inequality:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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do we get after grouping variable terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$13-17x \\\\geq -4$$","$$13+3x \\\\geq -4$$"]},{"id":"aaa317eLinIneq5a-h3","type":"hint","dependencies":["aaa317eLinIneq5a-h2"],"title":"Isolate","text":"We then isolate the variable term","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq5a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-17x \\\\geq -17$$"],"dependencies":["aaa317eLinIneq5a-h3"],"title":"Isolate","text":"What inequlity do we get after isolating the variable term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$-17x \\\\geq 9$$","$$-17x \\\\geq 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaa317eLinIneq6a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-17}{12}$$"],"dependencies":[],"title":"Combine Like Terms","text":"What is $$\\\\frac{-3}{4}-\\\\frac{3}{2}$$ (Hint: write fractions with common denominator)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aaa317eLinIneq6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-12}{17}$$"],"dependencies":["aaa317eLinIneq6a-h3"],"title":"Apply the Multiplication Property","text":"To make the coeffient of $$x$$ be $$1$$, what number should we multiply both sides of the inequality by?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaa317eLinIneq6a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-12}{17}$$"],"dependencies":[],"title":"Apply the Multiplication Property","text":"What is the inverse of $$\\\\frac{-17}{12}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aaa317eLinIneq6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x$$"],"dependencies":["aaa317eLinIneq6a-h4"],"title":"Left Side","text":"What is the left side after multiplying by $$\\\\frac{-12}{17}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq6a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{15}{34}$$"],"dependencies":["aaa317eLinIneq6a-h4"],"title":"Right Side","text":"What is the right side after multiplying by $$\\\\frac{-12}{17}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq6a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$ \\\\leq $$"],"dependencies":["aaa317eLinIneq6a-h4"],"title":"Sign","text":"What is the sign of the inequality after multiplication?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$ \\\\leq $$","$$ \\\\geq $$"]},{"id":"aaa317eLinIneq6a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(-\\\\infty,\\\\frac{15}{34}]$$"],"dependencies":["aaa317eLinIneq6a-h7"],"title":"Interval Notation","text":"What is $$x \\\\leq \\\\frac{15}{34}$$ written in interval notion?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$[\\\\frac{15}{34},\\\\infty)$$","$$(-\\\\infty,\\\\frac{15}{34}]$$"]}]}}]},{"id":"aaa317eLinIneq7","title":"Solving a Compound Inequality","body":"Solve the compound inequality:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Linear Inequalities and Absolute Value Inequalities","courseName":"OpenStax: College Algebra","steps":[{"id":"aaa317eLinIneq7a","stepAnswer":["$$[\\\\frac{1}{2},2)$$"],"problemType":"MultipleChoice","stepTitle":"$$3 \\\\leq 2x+2<6$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[\\\\frac{1}{2},2)$$","choices":["$$[\\\\frac{1}{2},2)$$","$$(\\\\frac{1}{2},2]$$","[1,2)","(1,2]"],"hints":{"DefaultPathway":[{"id":"aaa317eLinIneq7a-h1","type":"hint","dependencies":[],"title":"We can leave the compound inequality intact, and perform solving procedures on the three parts at the same time.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["aaa317eLinIneq7a-h1"],"title":"Applying the Addition Property","text":"What number should we add to all three parts to isolate the variable term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq7a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$1 \\\\leq 2x<4$$"],"dependencies":["aaa317eLinIneq7a-h2"],"title":"Applying the Addition Property","text":"What does the compound inequality become?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$5 \\\\leq 2x<4$$","$$1 \\\\leq 2x<4$$"]},{"id":"aaa317eLinIneq7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["aaa317eLinIneq7a-h3"],"title":"Applying the Multiplication Property","text":"What number should we multiply to all three parts to eliminate the coefficient of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq7a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1}{2} \\\\leq x<2$$"],"dependencies":["aaa317eLinIneq7a-h4"],"title":"Applying the Multiplication Property","text":"What does the compound inequality become?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{1}{2} \\\\leq x<2$$","$$2 \\\\leq x<8$$"]},{"id":"aaa317eLinIneq7a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$[\\\\frac{1}{2},2)$$"],"dependencies":["aaa317eLinIneq7a-h5"],"title":"Interval Notation","text":"What is this inequality written in the interval notation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$[\\\\frac{1}{2},2)$$","$$(\\\\frac{1}{2},2]$$"]}]}}]},{"id":"aaa317eLinIneq8","title":"Solving a Compound Inequality with the Variable in All Three Parts","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Linear Inequalities and Absolute Value Inequalities","courseName":"OpenStax: College Algebra","steps":[{"id":"aaa317eLinIneq8a","stepAnswer":["$$-4<x<\\\\frac{5}{6}$$"],"problemType":"MultipleChoice","stepTitle":"$$3+x>7x-2>5x-10$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-4<x<\\\\frac{5}{6}$$","choices":["$$x<-4$$ or $$x>\\\\frac{5}{6}$$","$$-4<x<\\\\frac{5}{6}$$"],"hints":{"DefaultPathway":[{"id":"aaa317eLinIneq8a-h1","type":"hint","dependencies":[],"title":"Break Down the Compound Inequality","text":"We can simplify the compound inequality by writing it as two separte inequalities: $$3+x>7x-2$$, $$7x-2>5x-10$$. The solution would be their intersection.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq8a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{5}{6}>x$$"],"dependencies":["aaa317eLinIneq8a-h1"],"title":"Solving Simple Inequality","text":"What does solving $$3+x>7x-1$$ give us?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{5}{6}>x$$","$$\\\\frac{5}{6}<x$$","$$\\\\frac{5}{8}>x$$","$$\\\\frac{5}{8}<x$$"],"subHints":[{"id":"aaa317eLinIneq8a-h2-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3>6x-2$$"],"dependencies":[],"title":"Combine Variable Terms","text":"What inequality do we get after combining the vairable terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$3>8x-2$$","$$3>6x-2$$"]},{"id":"aaa317eLinIneq8a-h2-s2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$5>6x$$"],"dependencies":["aaa317eLinIneq8a-h2-s1"],"title":"Isolate Variable Term","text":"What inequality do we get after isolating the vairable terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$5>6x$$","$$1>6x$$"]},{"id":"aaa317eLinIneq8a-h2-s3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{5}{6}>x$$"],"dependencies":["aaa317eLinIneq8a-h2-s2"],"title":"Eliminate Coefficient","text":"What inequality do we get after eliminating the coeffienct of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{5}{6}>x$$","$$\\\\frac{5}{6}<x$$"]}]},{"id":"aaa317eLinIneq8a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x>-4$$"],"dependencies":["aaa317eLinIneq8a-h1"],"title":"Solving Simple Inequality","text":"What does solving $$7x-2>5x-10$$ give us?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x>-6$$","$$x<-6$$","$$x>-4$$","$$x<-4$$"],"subHints":[{"id":"aaa317eLinIneq8a-h3-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2x-2>-10$$"],"dependencies":[],"title":"Combine Variable Terms","text":"What inequality do we get after combining the vairable terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2x-2>-10$$","$$2x-2<-10$$"]},{"id":"aaa317eLinIneq8a-h3-s2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2x>-8$$"],"dependencies":["aaa317eLinIneq8a-h3-s1"],"title":"Isolate Variable Term","text":"What inequality do we get after isolating the vairable terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2x>-8$$","$$2x>-12$$"]},{"id":"aaa317eLinIneq8a-h3-s3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x>-4$$"],"dependencies":["aaa317eLinIneq8a-h3-s2"],"title":"Eliminate Coefficient","text":"What inequality do we get after eliminating the coeffienct of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x>-4$$","$$x<-4$$"]}]},{"id":"aaa317eLinIneq8a-h4","type":"hint","dependencies":["aaa317eLinIneq8a-h2","aaa317eLinIneq8a-h3"],"title":"Intersection of the Two Simple Inequalities","text":"The intersection of the two simple inequalities is $$-4<x<\\\\frac{5}{6}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaa317eLinIneq9","title":"Determining a Number within a Perscibed Distance","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Linear Inequalities and Absolute Value Inequalities","courseName":"OpenStax: College Algebra","steps":[{"id":"aaa317eLinIneq9a","stepAnswer":["[1,9]"],"problemType":"MultipleChoice","stepTitle":"Describe all values $$x$$ within a distance of $$4$$ from the number $$5$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$(1,9)$$","[1,9]","$$(3,7)$$","[3,7]"],"hints":{"DefaultPathway":[{"id":"aaa317eLinIneq9a-h1","type":"hint","dependencies":[],"title":"Representing Distance with Algebraic Expression","text":"The distance from $$x$$ to $$5$$ can be represented using an absolute value symbol, $$|x-5|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq9a-h2","type":"hint","dependencies":["aaa317eLinIneq9a-h1"],"title":"Defining Inequality","text":"All values of $$x$$ that satisfy the condition can be represented as the inequality: $$|x-5| \\\\leq 4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq9a-h3","type":"hint","dependencies":["aaa317eLinIneq9a-h2"],"title":"Absolute Value Inequalities","text":"$$|X| \\\\leq k$$ is equivalent to $$-k \\\\leq X \\\\leq k$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq9a-h4","type":"hint","dependencies":["aaa317eLinIneq9a-h3"],"title":"Applying the Formula for Absolute Value Inequalities","text":"$$|x-5| \\\\leq 4$$ can be written as two inequalities: $$x-5 \\\\leq 4$$ and $$x-5 \\\\geq -4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq9a-h5","type":"hint","dependencies":["aaa317eLinIneq9a-h4"],"title":"Solution Set","text":"The two inequalities we get from solving $$x-5 \\\\leq 4$$ and $$x-5 \\\\geq -4$$ are: $$x \\\\leq 9$$ and $$x \\\\geq 1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaa317eLinIneq9a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["[1,9]"],"dependencies":["aaa317eLinIneq9a-h5"],"title":"Interval Notation","text":"What is $$x \\\\leq 9$$ and $$x \\\\geq 1$$ written in the interval notation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(1,9)$$","[1,9]"]}]}}]},{"id":"aaaec44conf1","title":"A Population Proportion","body":"Suppose that a market research firm is hired to estimate the percent of adults living in a large city who have cell phones. Five hundred randomly selected adult residents in this city are surveyed to determine whether they have cell phones. Of the $$500$$ people surveyed, $$421$$ responded yes - they own cell phones.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 A Population Proportion","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aaaec44conf1a","stepAnswer":["$$0.81$$"],"problemType":"TextBox","stepTitle":"Using a 95% confidence level, compute the lower confidence interval estimate for the true proportion of adult residents of this city who have cell phones.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.81$$","hints":{"DefaultPathway":[{"id":"aaaec44conf1a-h1","type":"hint","dependencies":[],"title":"p\', q\', and EBP","text":"Let X $$=$$ the number of people in the sample who have cell phones. X ~ B (500, 421/500). To calculate the confidence interval, you must find p\', q\', and EBP.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf1a-h2","type":"hint","dependencies":["aaaec44conf1a-h1"],"title":"p\' Equation","text":"p\', the estimated proportion of successes, is equal to the number of successes (x) divided by the size of the sample (n): p\' $$=$$ $$\\\\frac{x}{n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$500$$"],"dependencies":["aaaec44conf1a-h1","aaaec44conf1a-h2"],"title":"Sample Size","text":"What is the size of the sample (n)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$421$$"],"dependencies":["aaaec44conf1a-h3"],"title":"Number of Successes","text":"What is the number of successes (x)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf1a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.842$$"],"dependencies":["aaaec44conf1a-h4"],"title":"Solving for p\'","text":"What is p\' (or x/n)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaaec44conf1a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.842$$"],"dependencies":[],"title":"Plugging in $$x$$ and $$n$$ to solve for p\'","text":"What is $$\\\\frac{421}{500}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aaaec44conf1a-h6","type":"hint","dependencies":["aaaec44conf1a-h5"],"title":"q\' Equation","text":"q\' $$=$$ $$1$$ - p\'","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf1a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.158$$"],"dependencies":["aaaec44conf1a-h6"],"title":"Solving for q\'","text":"What is q\' (or $$1$$ - p\')?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaaec44conf1a-h7-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.158$$"],"dependencies":[],"title":"Plugging q\' into $$1$$ - p\' to solve for q\'","text":"What is $$1$$ - $$0.842$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aaaec44conf1a-h8","type":"hint","dependencies":["aaaec44conf1a-h7"],"title":"Confidence Level and $$\\\\frac{Z_\u03b1}{2}$$","text":"To solve for $$\\\\frac{z_\u03b1}{2}$$, or an alpha level\'s $$z-score$$ for a two tailed test, find \ud835\udefc where \ud835\udefc $$=$$ $$1$$ - CL, the confidence level.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf1a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.05$$"],"dependencies":["aaaec44conf1a-h8"],"title":"Solving for \ud835\udefc","text":"Given that CL $$=$$ $$0.95$$, what is $$1$$ - CL?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf1a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.025$$"],"dependencies":["aaaec44conf1a-h9"],"title":"Solving for $$\\\\frac{\u03b1}{2}$$","text":"What is $$\\\\frac{\u03b1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf1a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.96$$"],"dependencies":["aaaec44conf1a-h10"],"title":"Solving for $$\\\\frac{Z_\u03b1}{2}$$","text":"What is $$\\\\frac{Z_\u03b1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf1a-h12","type":"hint","dependencies":["aaaec44conf1a-h11"],"title":"EBP Equation","text":"EBP $$=$$ $$\\\\frac{Z_\u03b1}{2} \\\\sqrt{\\\\frac{p\'q\'}{n}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf1a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.032$$"],"dependencies":["aaaec44conf1a-h12"],"title":"Solving EBP","text":"What is EBP?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaaec44conf1a-h13-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.032$$"],"dependencies":[],"title":"Plugging Into the EBP Equation","text":"What is EBP $$=$$ $$1.96\\\\sqrt{\\\\frac{0.842\\\\times0.158}{500}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aaaec44conf1a-h14","type":"hint","dependencies":["aaaec44conf1a-h13"],"title":"Lower Confidence Interval","text":"To find the lower confidence interval estimate, solve for p\' - EBP.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf1a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.81$$"],"dependencies":["aaaec44conf1a-h14"],"title":"Solving for the Lower Confidence Interval Estimate","text":"What is p\' - EBP?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaaec44conf1a-h15-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.81$$"],"dependencies":[],"title":"Plugging Into p\' - EBP","text":"What is $$0.842$$ - $$0.032$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"aaaec44conf2","title":"A Population Proportion","body":"Suppose that a market research firm is hired to estimate the percent of adults living in a large city who have cell phones. Five hundred randomly selected adult residents in this city are surveyed to determine whether they have cell phones. Of the $$500$$ people surveyed, $$421$$ responded yes - they own cell phones.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 A Population Proportion","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aaaec44conf2a","stepAnswer":["$$0.874$$"],"problemType":"TextBox","stepTitle":"Using a 95% confidence level, compute the higher confidence interval estimate for the true proportion of adult residents of this city who have cell phones.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.874$$","hints":{"DefaultPathway":[{"id":"aaaec44conf2a-h1","type":"hint","dependencies":[],"title":"p\', q\', and EBP","text":"Let X $$=$$ the number of people in the sample who have cell phones. X ~ B (500, 421/500). To calculate the confidence interval, you must find p\', q\', and EBP.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf2a-h2","type":"hint","dependencies":["aaaec44conf2a-h1"],"title":"p\' Equation","text":"p\', the estimated proportion of successes, is equal to the number of successes (x) divided by the size of the sample (n): p\' $$=$$ $$\\\\frac{x}{n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$500$$"],"dependencies":["aaaec44conf2a-h1","aaaec44conf2a-h2"],"title":"Sample Size","text":"What is the size of the sample (n)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$421$$"],"dependencies":["aaaec44conf2a-h3"],"title":"Number of Successes","text":"What is the number of successes (x)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf2a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.842$$"],"dependencies":["aaaec44conf2a-h4"],"title":"Solving for p\'","text":"What is p\' (or x/n)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaaec44conf2a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.842$$"],"dependencies":[],"title":"Plugging in $$x$$ and $$n$$ to solve for p\'","text":"What is $$\\\\frac{421}{500}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aaaec44conf2a-h6","type":"hint","dependencies":["aaaec44conf2a-h5"],"title":"q\' Equation","text":"q\' $$=$$ $$1$$ - p\'","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf2a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.158$$"],"dependencies":["aaaec44conf2a-h6"],"title":"Solving for q\'","text":"What is q\' (or $$1$$ - p\')?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaaec44conf2a-h7-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.158$$"],"dependencies":[],"title":"Plugging q\' into $$1$$ - p\' to solve for q\'","text":"What is $$1$$ - $$0.842$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aaaec44conf2a-h8","type":"hint","dependencies":["aaaec44conf2a-h7"],"title":"Confidence Level and $$\\\\frac{z_\u03b1}{2}$$","text":"To solve for $$\\\\frac{Z_\u03b1}{2}$$, or an alpha level\'s $$z-score$$ for a two tailed test, find \ud835\udefc where \ud835\udefc $$=$$ $$1$$ - CL, the confidence level.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf2a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.05$$"],"dependencies":["aaaec44conf2a-h8"],"title":"Solving for \ud835\udefc","text":"Given that CL $$=$$ $$0.95$$, what is $$1$$ - CL?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf2a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.025$$"],"dependencies":["aaaec44conf2a-h9"],"title":"Solving for $$\\\\frac{\u03b1}{2}$$","text":"What is $$\\\\frac{\u03b1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf2a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.96$$"],"dependencies":["aaaec44conf2a-h10"],"title":"Solving for $$\\\\frac{Z_\u03b1}{2}$$","text":"What is $$\\\\frac{Z_\u03b1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf2a-h12","type":"hint","dependencies":["aaaec44conf2a-h11"],"title":"EBP Equation","text":"EBP $$=$$ $$\\\\frac{Z_\u03b1}{2} \\\\sqrt{\\\\frac{p\'q\'}{n}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf2a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.032$$"],"dependencies":["aaaec44conf2a-h12"],"title":"Solving EBP","text":"What is EBP?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaaec44conf2a-h13-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.032$$"],"dependencies":[],"title":"Plugging Into the EBP Equation","text":"What is EBP $$=$$ $$1.96\\\\sqrt{\\\\frac{0.842\\\\times0.158}{500}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aaaec44conf2a-h14","type":"hint","dependencies":["aaaec44conf2a-h13"],"title":"Higher Confidence Interval","text":"To find the higher confidence interval estimate, solve for p\' + EBP.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf2a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.874$$"],"dependencies":["aaaec44conf2a-h14"],"title":"Solving for the Higher Confidence Interval Estimate","text":"What is p\' + EBP?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaaec44conf2a-h15-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.874$$"],"dependencies":[],"title":"Plugging Into p\' + EBP","text":"What is $$0.842$$ + $$0.032$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"aaaec44conf3","title":"A Population Proportion","body":"For a class project, a political science student at a large university wants to estimate the percent of students who are registered voters. He surveys $$500$$ students and finds that $$300$$ are registered voters.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 A Population Proportion","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aaaec44conf3a","stepAnswer":["$$0.564$$"],"problemType":"TextBox","stepTitle":"Compute a 90% confidence interval for the lower true percent of students who are registered voters.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.564$$","hints":{"DefaultPathway":[{"id":"aaaec44conf3a-h1","type":"hint","dependencies":[],"title":"p\', q\', and EBP","text":"To calculate the confidence interval, you must find p\', q\', and EBP.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf3a-h2","type":"hint","dependencies":["aaaec44conf3a-h1"],"title":"p\' Equation","text":"p\', the estimated proportion of successes, is equal to the number of successes (x) divided by the size of the sample (n): p\' $$=$$ $$\\\\frac{x}{n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$500$$"],"dependencies":["aaaec44conf3a-h1","aaaec44conf3a-h2"],"title":"Sample Size","text":"What is the size of the sample (n)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$300$$"],"dependencies":["aaaec44conf3a-h3"],"title":"Number of Successes","text":"What is the number of successes (x)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.6$$"],"dependencies":["aaaec44conf3a-h4"],"title":"Solving for p\'","text":"What is p\' (or x/n)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaaec44conf3a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.6$$"],"dependencies":[],"title":"Plugging in $$x$$ and $$n$$ to solve for p\'","text":"What is $$\\\\frac{300}{500}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aaaec44conf3a-h6","type":"hint","dependencies":["aaaec44conf3a-h5"],"title":"q\' Equation","text":"q\' $$=$$ $$1$$ - p\'","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf3a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.4$$"],"dependencies":["aaaec44conf3a-h6"],"title":"Solving for q\'","text":"What is q\' (or $$1$$ - p\')?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaaec44conf3a-h7-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.4$$"],"dependencies":[],"title":"Plugging q\' into $$1$$ - p\' to solve for q\'","text":"What is $$1$$ - $$0.6$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aaaec44conf3a-h8","type":"hint","dependencies":["aaaec44conf3a-h7"],"title":"Confidence Level and $$\\\\frac{z_\u03b1}{2}$$","text":"To solve for $$\\\\frac{Z_\u03b1}{2}$$, or an alpha level\'s $$z-score$$ for a two tailed test, find \ud835\udefc where \ud835\udefc $$=$$ $$1$$ - CL, the confidence level.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf3a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1$$"],"dependencies":["aaaec44conf3a-h8"],"title":"Solving for \ud835\udefc","text":"Given that CL $$=$$ $$0.90$$, what is $$1$$ - CL?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf3a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.05$$"],"dependencies":["aaaec44conf3a-h9"],"title":"Solving for $$\\\\frac{\u03b1}{2}$$","text":"What is $$\\\\frac{\u03b1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf3a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.645$$"],"dependencies":["aaaec44conf3a-h10"],"title":"Solving for $$\\\\frac{Z_\u03b1}{2}$$","text":"What is $$\\\\frac{Z_\u03b1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf3a-h12","type":"hint","dependencies":["aaaec44conf3a-h11"],"title":"EBP Equation","text":"EBP $$=$$ $$\\\\frac{Z_\u03b1}{2} \\\\sqrt{\\\\frac{p\'q\'}{n}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf3a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.036$$"],"dependencies":["aaaec44conf3a-h12"],"title":"Solving EBP","text":"What is EBP?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaaec44conf3a-h13-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.036$$"],"dependencies":[],"title":"Plugging Into the EBP Equation","text":"What is EBP $$=$$ $$1.645\\\\sqrt{\\\\frac{0.6\\\\times0.4}{500}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aaaec44conf3a-h14","type":"hint","dependencies":["aaaec44conf3a-h13"],"title":"Lower Confidence Interval","text":"To find the lower confidence interval estimate, solve for p\' - EBP.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf3a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.564$$"],"dependencies":["aaaec44conf3a-h14"],"title":"Solving for the Lower Confidence Interval Estimate","text":"What is p\' - EBP?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaaec44conf3a-h15-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.564$$"],"dependencies":[],"title":"Plugging Into p\' - EBP","text":"What is $$0.60$$ - $$0.036$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"aaaec44conf4","title":"A Population Proportion","body":"For a class project, a political science student at a large university wants to estimate the percent of students who are registered voters. He surveys $$500$$ students and finds that $$300$$ are registered voters.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 A Population Proportion","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aaaec44conf4a","stepAnswer":["$$0.636$$"],"problemType":"TextBox","stepTitle":"Compute a 90% confidence interval for the higher true percent of students who are registered voters.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.636$$","hints":{"DefaultPathway":[{"id":"aaaec44conf4a-h1","type":"hint","dependencies":[],"title":"p\', q\', and EBP","text":"To calculate the confidence interval, you must find p\', q\', and EBP.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf4a-h2","type":"hint","dependencies":["aaaec44conf4a-h1"],"title":"p\' Equation","text":"p\', the estimated proportion of successes, is equal to the number of successes (x) divided by the size of the sample (n): p\' $$=$$ $$\\\\frac{x}{n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$500$$"],"dependencies":["aaaec44conf4a-h1","aaaec44conf4a-h2"],"title":"Sample Size","text":"What is the size of the sample (n)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$300$$"],"dependencies":["aaaec44conf4a-h3"],"title":"Number of Successes","text":"What is the number of successes (x)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf4a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.6$$"],"dependencies":["aaaec44conf4a-h4"],"title":"Solving for p\'","text":"What is p\' (or x/n)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaaec44conf4a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.6$$"],"dependencies":[],"title":"Plugging in $$x$$ and $$n$$ to solve for p\'","text":"What is $$\\\\frac{300}{500}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aaaec44conf4a-h6","type":"hint","dependencies":["aaaec44conf4a-h5"],"title":"q\' Equation","text":"q\' $$=$$ $$1$$ - p\'","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf4a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.4$$"],"dependencies":["aaaec44conf4a-h6"],"title":"Solving for q\'","text":"What is q\' (or $$1$$ - p\')?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaaec44conf4a-h7-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.4$$"],"dependencies":[],"title":"Plugging q\' into $$1$$ - p\' to solve for q\'","text":"What is $$1$$ - $$0.6$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aaaec44conf4a-h8","type":"hint","dependencies":["aaaec44conf4a-h7"],"title":"Confidence Level and $$\\\\frac{z_\u03b1}{2}$$","text":"To solve for $$\\\\frac{z_\u03b1}{2}$$, or an alpha level\'s $$z-score$$ for a two tailed test, find \ud835\udefc where \ud835\udefc $$=$$ $$1$$ - CL, the confidence level.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf4a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1$$"],"dependencies":["aaaec44conf4a-h8"],"title":"Solving for \ud835\udefc","text":"Given that CL $$=$$ $$0.90$$, what is $$1$$ - CL?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf4a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.05$$"],"dependencies":["aaaec44conf4a-h9"],"title":"Solving for $$\\\\frac{\u03b1}{2}$$","text":"What is $$\\\\frac{\u03b1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf4a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.645$$"],"dependencies":["aaaec44conf4a-h10"],"title":"Solving for $$\\\\frac{z_\u03b1}{2}$$","text":"What is $$\\\\frac{Z_\u03b1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf4a-h12","type":"hint","dependencies":["aaaec44conf4a-h11"],"title":"EBP Equation","text":"EBP $$=$$ $$\\\\frac{Z_\u03b1}{2} \\\\sqrt{\\\\frac{p\'q\'}{n}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf4a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.036$$"],"dependencies":["aaaec44conf4a-h12"],"title":"Solving EBP","text":"What is EBP?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaaec44conf4a-h13-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.036$$"],"dependencies":[],"title":"Plugging Into the EBP Equation","text":"What is EBP $$=$$ $$1.645\\\\sqrt{\\\\frac{0.6\\\\times0.4}{500}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aaaec44conf4a-h14","type":"hint","dependencies":["aaaec44conf4a-h13"],"title":"Higher Confidence Interval","text":"To find the higher confidence interval estimate, solve for p\' + EBP.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf4a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.636$$"],"dependencies":["aaaec44conf4a-h14"],"title":"Solving for the Higher Confidence Interval Estimate","text":"What is p\' + EBP?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaaec44conf4a-h15-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.636$$"],"dependencies":[],"title":"Plugging Into p\' + EBP","text":"What is $$0.60$$ + $$0.036$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"aaaec44conf5","title":"A Population Proportion","body":"A random sample of $$25$$ statistics students was asked: \u201cHave you smoked a cigarette in the past week?\u201d Six students reported smoking within the past week.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 A Population Proportion","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aaaec44conf5a","stepAnswer":["$$0.113$$"],"problemType":"TextBox","stepTitle":"Use the \u201cplus-four\u201d method to find a 95% confidence interval for the lower true proportion of statistics students who smoke.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.113$$","hints":{"DefaultPathway":[{"id":"aaaec44conf5a-h1","type":"hint","dependencies":[],"title":"Plus-Four Method","text":"To calculate the confidence interval using the \\"plus-four\\" method, pretend that we have four additional observations where two of these observations are successes and two are failures. The new sample size, then, is $$n$$ + $$4$$, and the new count of successes is $$x$$ + $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf5a-h2","type":"hint","dependencies":["aaaec44conf5a-h1"],"title":"p\' Equation","text":"p\', the estimated proportion of successes, is equal to the number of successes (x) divided by the size of the sample (n): p\' $$=$$ $$\\\\frac{x}{n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$29$$"],"dependencies":["aaaec44conf5a-h1","aaaec44conf5a-h2"],"title":"New Sample Size","text":"What is the new size of the sample (n + 4)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["aaaec44conf5a-h3"],"title":"New Number of Successes","text":"What is the new number of successes (x + 2)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.276$$"],"dependencies":["aaaec44conf5a-h4"],"title":"Solving for p\'","text":"What is p\' (or x/n) rounded to the nearest thousandths place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaaec44conf5a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.276$$"],"dependencies":[],"title":"Plugging in $$x$$ and $$n$$ to solve for p\'","text":"What is $$\\\\frac{8}{29}$$, rounded to the nearest thousandths place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aaaec44conf5a-h6","type":"hint","dependencies":["aaaec44conf5a-h5"],"title":"q\' Equation","text":"q\' $$=$$ $$1$$ - p\'","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf5a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.724$$"],"dependencies":["aaaec44conf5a-h6"],"title":"Solving for q\'","text":"What is q\' (or $$1$$ - p\')?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaaec44conf5a-h7-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.724$$"],"dependencies":[],"title":"Plugging q\' into $$1$$ - p\' to solve for q\'","text":"What is $$1$$ - $$0.276$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aaaec44conf5a-h8","type":"hint","dependencies":["aaaec44conf5a-h7"],"title":"Confidence Level and $$\\\\frac{z_\u03b1}{2}$$","text":"To solve for $$\\\\frac{z_\u03b1}{2}$$, or an alpha level\'s $$z-score$$ for a two tailed test, find \ud835\udefc where \ud835\udefc $$=$$ $$1$$ - CL, the confidence level.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf5a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.05$$"],"dependencies":["aaaec44conf5a-h8"],"title":"Solving for \ud835\udefc","text":"Given that CL $$=$$ $$0.95$$, what is $$1$$ - CL?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf5a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.025$$"],"dependencies":["aaaec44conf5a-h9"],"title":"Solving for $$\\\\frac{\u03b1}{2}$$","text":"What is $$\\\\frac{\u03b1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf5a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.96$$"],"dependencies":["aaaec44conf5a-h10"],"title":"Solving for $$\\\\frac{Z_\u03b1}{2}$$","text":"What is $$\\\\frac{Z_\u03b1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf5a-h12","type":"hint","dependencies":["aaaec44conf5a-h11"],"title":"EPB Equation","text":"EPB $$=$$ $$\\\\frac{Z_\u03b1}{2} \\\\sqrt{\\\\frac{p\'q\'}{n}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf5a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.163$$"],"dependencies":["aaaec44conf5a-h12"],"title":"Solving EPB","text":"What is EPB, rounded to the nearest thousandths place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaaec44conf5a-h13-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.163$$"],"dependencies":[],"title":"Plugging Into the EPB Equation","text":"What is EPB $$=$$ $$1.96\\\\sqrt{\\\\frac{0.276\\\\times0.724}{29}}$$, rounded to the nearest thousandths place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aaaec44conf5a-h14","type":"hint","dependencies":["aaaec44conf5a-h13"],"title":"Lower Confidence Interval","text":"To find the lower confidence interval estimate, solve for p\' - EPB.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf5a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.113$$"],"dependencies":["aaaec44conf5a-h14"],"title":"Solving for the Lower Confidence Interval Estimate","text":"What is p\' - EPB?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaaec44conf5a-h15-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.113$$"],"dependencies":[],"title":"Plugging Into p\' - EPB","text":"What is $$0.276$$ - $$0.163$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"aaaec44conf6","title":"A Population Proportion","body":"A random sample of $$25$$ statistics students was asked: \u201cHave you smoked a cigarette in the past week?\u201d Six students reported smoking within the past week.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 A Population Proportion","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aaaec44conf6a","stepAnswer":["$$0.439$$"],"problemType":"TextBox","stepTitle":"Use the \u201cplus-four\u201d method to find a 95% confidence interval for the higher true proportion of statistics students who smoke.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.439$$","hints":{"DefaultPathway":[{"id":"aaaec44conf6a-h1","type":"hint","dependencies":[],"title":"Plus-Four Method","text":"To calculate the confidence interval using the \\"plus-four\\" method, pretend that we have four additional observations where two of these observations are successes and two are failures. The new sample size, then, is $$n$$ + $$4$$, and the new count of successes is $$x$$ + $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf6a-h2","type":"hint","dependencies":["aaaec44conf6a-h1"],"title":"p\' Equation","text":"p\', the estimated proportion of successes, is equal to the number of successes (x) divided by the size of the sample (n): p\' $$=$$ $$\\\\frac{x}{n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$29$$"],"dependencies":["aaaec44conf6a-h1","aaaec44conf6a-h2"],"title":"New Sample Size","text":"What is the new size of the sample (n + 4)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["aaaec44conf6a-h3"],"title":"New Number of Successes","text":"What is the new number of successes (x + 2)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.276$$"],"dependencies":["aaaec44conf6a-h4"],"title":"Solving for p\'","text":"What is p\' (or x/n) rounded to the nearest thousandths place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaaec44conf6a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.276$$"],"dependencies":[],"title":"Plugging in $$x$$ and $$n$$ to solve for p\'","text":"What is $$\\\\frac{8}{29}$$, rounded to the nearest thousandths place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aaaec44conf6a-h6","type":"hint","dependencies":["aaaec44conf6a-h5"],"title":"q\' Equation","text":"q\' $$=$$ $$1$$ - p\'","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf6a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.724$$"],"dependencies":["aaaec44conf6a-h6"],"title":"Solving for q\'","text":"What is q\' (or $$1$$ - p\')?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaaec44conf6a-h7-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.724$$"],"dependencies":[],"title":"Plugging q\' into $$1$$ - p\' to solve for q\'","text":"What is $$1$$ - $$0.276$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aaaec44conf6a-h8","type":"hint","dependencies":["aaaec44conf6a-h7"],"title":"Confidence Level and $$\\\\frac{z_\u03b1}{2}$$","text":"To solve for $$\\\\frac{z_\u03b1}{2}$$, or an alpha level\'s $$z-score$$ for a two tailed test, find \ud835\udefc where \ud835\udefc $$=$$ $$1$$ - CL, the confidence level.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf6a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.05$$"],"dependencies":["aaaec44conf6a-h8"],"title":"Solving for \ud835\udefc","text":"Given that CL $$=$$ $$0.95$$, what is $$1$$ - CL?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf6a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.025$$"],"dependencies":["aaaec44conf6a-h9"],"title":"Solving for $$\\\\frac{\u03b1}{2}$$","text":"What is $$\\\\frac{\u03b1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf6a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.96$$"],"dependencies":["aaaec44conf6a-h10"],"title":"Solving for $$\\\\frac{Z_\u03b1}{2}$$","text":"What is $$\\\\frac{Z_\u03b1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf6a-h12","type":"hint","dependencies":["aaaec44conf6a-h11"],"title":"EPB Equation","text":"EPB $$=$$ $$\\\\frac{Z_\u03b1}{2} \\\\sqrt{\\\\frac{p\'q\'}{n}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf6a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.163$$"],"dependencies":["aaaec44conf6a-h12"],"title":"Solving EBP","text":"What is EPB, rounded to the nearest thousandths place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaaec44conf6a-h13-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.163$$"],"dependencies":[],"title":"Plugging Into the EPB Equation","text":"What is EPB $$=$$ $$1.96\\\\sqrt{\\\\frac{0.276\\\\times0.724}{29}}$$, rounded to the nearest thousandths place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aaaec44conf6a-h14","type":"hint","dependencies":["aaaec44conf6a-h13"],"title":"Higher Confidence Interval","text":"To find the higher confidence interval estimate, solve for p\' + EPB.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf6a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.439$$"],"dependencies":["aaaec44conf6a-h14"],"title":"Solving for the Higher Confidence Interval Estimate","text":"What is p\' + EPB?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaaec44conf6a-h15-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.439$$"],"dependencies":[],"title":"Plugging Into p\' + EPB","text":"What is $$0.276$$ + $$0.163$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"aaaec44conf7","title":"A Population Proportion","body":"The Berkman Center for Internet & Society at Harvard recently conducted a study analyzing the privacy management habits of teen internet users. In a group of $$50$$ teens, $$13$$ reported having more than $$500$$ friends on Facebook.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 A Population Proportion","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aaaec44conf7a","stepAnswer":["$$0.178$$"],"problemType":"TextBox","stepTitle":"Use the \\"plus four\\" method to find a 90% confidence interval for the lower true proportion of teens who would report having more than $$500$$ Facebook friends.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.178$$","hints":{"DefaultPathway":[{"id":"aaaec44conf7a-h1","type":"hint","dependencies":[],"title":"Plus-Four Method","text":"To calculate the confidence interval using the \\"plus-four\\" method, pretend that we have four additional observations where two of these observations are successes and two are failures. The new sample size, then, is $$n$$ + $$4$$, and the new count of successes is $$x$$ + $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf7a-h2","type":"hint","dependencies":["aaaec44conf7a-h1"],"title":"p\' Equation","text":"p\', the estimated proportion of successes, is equal to the number of successes (x) divided by the size of the sample (n): p\' $$=$$ $$\\\\frac{x}{n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$54$$"],"dependencies":["aaaec44conf7a-h1","aaaec44conf7a-h2"],"title":"New Sample Size","text":"What is the new size of the sample (n + 4)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["aaaec44conf7a-h3"],"title":"New Number of Successes","text":"What is the new number of successes (x + 2)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf7a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.278$$"],"dependencies":["aaaec44conf7a-h4"],"title":"Solving for p\'","text":"What is p\' (or x/n) rounded to the nearest thousandths place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaaec44conf7a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.278$$"],"dependencies":[],"title":"Plugging in $$x$$ and $$n$$ to solve for p\'","text":"What is $$\\\\frac{15}{54}$$, rounded to the nearest thousandths place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aaaec44conf7a-h6","type":"hint","dependencies":["aaaec44conf7a-h5"],"title":"q\' Equation","text":"q\' $$=$$ $$1$$ - p\'","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf7a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.722$$"],"dependencies":["aaaec44conf7a-h6"],"title":"Solving for q\'","text":"What is q\' (or $$1$$ - p\')?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaaec44conf7a-h7-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.722$$"],"dependencies":[],"title":"Plugging q\' into $$1$$ - p\' to solve for q\'","text":"What is $$1$$ - $$0.278$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aaaec44conf7a-h8","type":"hint","dependencies":["aaaec44conf7a-h7"],"title":"Confidence Level and $$\\\\frac{z_\u03b1}{2}$$","text":"To solve for $$\\\\frac{z_\u03b1}{2}$$, or an alpha level\'s $$z-score$$ for a two tailed test, find \ud835\udefc where \ud835\udefc $$=$$ $$1$$ - CL, the confidence level.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf7a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1$$"],"dependencies":["aaaec44conf7a-h8"],"title":"Solving for \ud835\udefc","text":"Given that CL $$=$$ $$0.90$$, what is $$1$$ - CL?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf7a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.05$$"],"dependencies":["aaaec44conf7a-h9"],"title":"Solving for $$\\\\frac{\u03b1}{2}$$","text":"What is $$\\\\frac{\u03b1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf7a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.645$$"],"dependencies":["aaaec44conf7a-h10"],"title":"Solving for $$\\\\frac{Z_\u03b1}{2}$$","text":"What is $$\\\\frac{Z_\u03b1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf7a-h12","type":"hint","dependencies":["aaaec44conf7a-h11"],"title":"EPB Equation","text":"EPB $$=$$ $$\\\\frac{Z_\u03b1}{2} \\\\sqrt{\\\\frac{p\'q\'}{n}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf7a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1$$"],"dependencies":["aaaec44conf7a-h12"],"title":"Solving EPB","text":"What is EPB, rounded to the nearest thousandths place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaaec44conf7a-h13-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1$$"],"dependencies":[],"title":"Plugging Into the EPB Equation","text":"What is EPB $$=$$ $$1.645\\\\sqrt{\\\\frac{0.278\\\\times0.722}{54}}$$, rounded to the nearest thousandths place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aaaec44conf7a-h14","type":"hint","dependencies":["aaaec44conf7a-h13"],"title":"Lower Confidence Interval","text":"To find the lower confidence interval estimate, solve for p\' - EPB.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf7a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.178$$"],"dependencies":["aaaec44conf7a-h14"],"title":"Solving for the Lower Confidence Interval Estimate","text":"What is p\' - EPB?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaaec44conf7a-h15-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.178$$"],"dependencies":[],"title":"Plugging Into p\' - EPB","text":"What is $$0.278$$ - $$0.100$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"aaaec44conf8","title":"A Population Proportion","body":"The Berkman Center for Internet & Society at Harvard recently conducted a study analyzing the privacy management habits of teen internet users. In a group of $$50$$ teens, $$13$$ reported having more than $$500$$ friends on Facebook.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 A Population Proportion","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aaaec44conf8a","stepAnswer":["$$0.378$$"],"problemType":"TextBox","stepTitle":"Use the \\"plus four\\" method to find a 90% confidence interval for the higher true proportion of teens who would report having more than $$500$$ Facebook friends.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.378$$","hints":{"DefaultPathway":[{"id":"aaaec44conf8a-h1","type":"hint","dependencies":[],"title":"Plus-Four Method","text":"To calculate the confidence interval using the \\"plus-four\\" method, pretend that we have four additional observations where two of these observations are successes and two are failures. The new sample size, then, is $$n$$ + $$4$$, and the new count of successes is $$x$$ + $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf8a-h2","type":"hint","dependencies":["aaaec44conf8a-h1"],"title":"p\' Equation","text":"p\', the estimated proportion of successes, is equal to the number of successes (x) divided by the size of the sample (n): p\' $$=$$ $$\\\\frac{x}{n}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$54$$"],"dependencies":["aaaec44conf8a-h1","aaaec44conf8a-h2"],"title":"New Sample Size","text":"What is the new size of the sample (n + 4)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["aaaec44conf8a-h3"],"title":"New Number of Successes","text":"What is the new number of successes (x + 2)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf8a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.278$$"],"dependencies":["aaaec44conf8a-h4"],"title":"Solving for p\'","text":"What is p\' (or x/n) rounded to the nearest thousandths place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaaec44conf8a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.278$$"],"dependencies":[],"title":"Plugging in $$x$$ and $$n$$ to solve for p\'","text":"What is $$\\\\frac{15}{54}$$, rounded to the nearest thousandths place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aaaec44conf8a-h6","type":"hint","dependencies":["aaaec44conf8a-h5"],"title":"q\' Equation","text":"q\' $$=$$ $$1$$ - p\'","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf8a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.722$$"],"dependencies":["aaaec44conf8a-h6"],"title":"Solving for q\'","text":"What is q\' (or $$1$$ - p\')?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaaec44conf8a-h7-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.722$$"],"dependencies":[],"title":"Plugging q\' into $$1$$ - p\' to solve for q\'","text":"What is $$1$$ - $$0.278$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aaaec44conf8a-h8","type":"hint","dependencies":["aaaec44conf8a-h7"],"title":"Confidence Level and $$\\\\frac{z_\u03b1}{2}$$","text":"To solve for $$\\\\frac{Z_\u03b1}{2}$$, or an alpha level\'s $$z-score$$ for a two tailed test, find \ud835\udefc where \ud835\udefc $$=$$ $$1$$ - CL, the confidence level.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf8a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1$$"],"dependencies":["aaaec44conf8a-h8"],"title":"Solving for \ud835\udefc","text":"Given that CL $$=$$ $$0.90$$, what is $$1$$ - CL?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf8a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.05$$"],"dependencies":["aaaec44conf8a-h9"],"title":"Solving for $$\\\\frac{\u03b1}{2}$$","text":"What is $$\\\\frac{\u03b1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf8a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.645$$"],"dependencies":["aaaec44conf8a-h10"],"title":"Solving for $$\\\\frac{Z_\u03b1}{2}$$","text":"What is $$\\\\frac{Z_\u03b1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf8a-h12","type":"hint","dependencies":["aaaec44conf8a-h11"],"title":"EPB Equation","text":"EPB $$=$$ $$\\\\frac{Z_\u03b1}{2} \\\\sqrt{\\\\frac{p\'q\'}{n}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf8a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1$$"],"dependencies":["aaaec44conf8a-h12"],"title":"Solving EPB","text":"What is EPB, rounded to the nearest thousandths place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaaec44conf8a-h13-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1$$"],"dependencies":[],"title":"Plugging Into the EPB Equation","text":"What is EPB $$=$$ $$1.645\\\\sqrt{\\\\frac{0.278\\\\times0.722}{54}}$$, rounded to the nearest thousandths place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aaaec44conf8a-h14","type":"hint","dependencies":["aaaec44conf8a-h13"],"title":"Higher Confidence Interval","text":"To find the higher confidence interval estimate, solve for p\' + EPB.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf8a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.378$$"],"dependencies":["aaaec44conf8a-h14"],"title":"Solving for the Higher Confidence Interval Estimate","text":"What is p\' + EPB?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaaec44conf8a-h15-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.378$$"],"dependencies":[],"title":"Plugging Into p\' + EPB","text":"What is $$0.278$$ + $$0.100$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"aaaec44conf9","title":"A Population Proportion","body":"Suppose a mobile phone company wants to determine the current percentage of customers aged 50+ who use text messaging on their cell phones.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 A Population Proportion","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aaaec44conf9a","stepAnswer":["$$752$$"],"problemType":"TextBox","stepTitle":"How many customers aged 50+ should the company survey in order to be 90% confident that the estimated (sample) proportion is within three percentage points of the true population proportion of customers aged 50+ who use text messaging on their cell phones.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$752$$","hints":{"DefaultPathway":[{"id":"aaaec44conf9a-h1","type":"hint","dependencies":[],"title":"What to Solve For","text":"Find $$n$$ where $$n$$ $$=$$ $$\\\\frac{z^2 p\'q\'}{{EBP}^2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf9a-h2","type":"hint","dependencies":["aaaec44conf9a-h1"],"title":"Z Equation","text":"To find Z, solve for $$\\\\frac{Z_\u03b1}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.645$$"],"dependencies":["aaaec44conf9a-h1","aaaec44conf9a-h2"],"title":"Solving for $$z$$","text":"What is $$\\\\frac{z_\u03b1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaaec44conf9a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1$$"],"dependencies":[],"title":"Solving for \ud835\udefc","text":"\ud835\udefc $$=$$ $$1$$ - CL. Given that CL $$=$$ $$0.90$$, what is \ud835\udefc?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf9a-h3-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.05$$"],"dependencies":[],"title":"Solving for $$\\\\frac{\u03b1}{2}$$","text":"What is $$\\\\frac{\u03b1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf9a-h3-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.645$$"],"dependencies":[],"title":"Solving for $$\\\\frac{Z_\u03b1}{2}$$","text":"What is Z_0.05?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aaaec44conf9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.03$$"],"dependencies":["aaaec44conf9a-h1"],"title":"EBP Equation","text":"What is EBP?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf9a-h5","type":"hint","dependencies":["aaaec44conf9a-h1"],"title":"Solving for p\' and q\'","text":"Test different products between p\' and q\' that results in the largest product.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf9a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.5$$"],"dependencies":["aaaec44conf9a-h5"],"title":"Testing Products","text":"What is the value of p\' and q\'?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf9a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.25$$"],"dependencies":["aaaec44conf9a-h6"],"title":"Testing Products Explanation","text":"Suppose we test the products of p\' and q\' as $$(0.6)(0.4)$$ $$=$$ $$0.24$$, $$(0.3)(0.7)$$ $$=$$ $$0.21$$, $$(0.2)(0.8)$$ $$=$$ $$0.16$$ and so on. What is the largest possible product?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaaec44conf9a-h7-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.25$$"],"dependencies":[],"title":"Largest Possible Product","text":"If p\' $$=$$ $$0.5$$ and q\' $$=$$ $$0.5$$, what is the product (p\' * q\')?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aaaec44conf9a-h8","type":"hint","dependencies":[],"title":"Plugging in Variables","text":"Plug $$z$$, p\', q\', and EBP into the equation $$n$$ $$=$$ (z\xb2p\'q\')/EBP\xb2.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf9a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$751.7$$"],"dependencies":["aaaec44conf9a-h8"],"title":"Solving for $$n$$","text":"What is $$n$$ $$=$$ $$\\\\frac{z^2 p\'q\'}{{EBP}^2}$$ where $$z$$ $$=$$ $$1.645$$, p\' $$=$$ $$0.5$$, q\' $$=$$ $$0.5$$, and EBP $$=$$ $$0.03$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaaec44conf9a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$752$$"],"dependencies":["aaaec44conf9a-h9"],"title":"Rounding $$n$$","text":"The answer is a sample size of the number of cell phone customers, meaning that it must be a whole number. What is $$n$$ rounded to the next whole number?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aab36429.3baker1","title":"Rare Events, The Sample, Decision, and Conclusion","body":"Suppose a baker claims that his bread height is more than $$15$$ cm, on average. Several of his customers do not believe him. To persuade his customers that he is right, the baker decides to do a hypothesis test. He bakes $$10$$ loaves of bread. The mean height of the sample loaves is $$17$$ cm. The baker knows from baking hundreds of loaves of bread that the standard deviation for the height is $$0.5$$ cm, and the distribution of heights is normal.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Distribution Needed for Hypothesis Testing","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aab36429.3baker1a","stepAnswer":["H_0:\u03bc<=15 H_a:\u03bc>15"],"problemType":"MultipleChoice","stepTitle":"Which one of the following hypothesis pairs is correct for this context? Note: $$H_0$$ is the null hypothesis and $$H_a$$ is the alternative hypothesis","stepBody":"","answerType":"string","variabilization":{},"choices":["H_0:\u03bc<=15 H_a:\u03bc>15","H_0:\u03bc>=15 H_a:\u03bc>15","H_0:\u03bc<17 H_a:\u03bc>=15","H_0:\u03bc<=15 H_a:\u03bc>17"],"hints":{"DefaultPathway":[{"id":"aab36429.3baker1a-h1","type":"hint","dependencies":[],"title":"Hypothesis Testing","text":"Regard the null hypothesis like the status quo.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aab36429.3baker1a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["The baker\'s bread height is less than 15cm on average"],"dependencies":["aab36429.3baker1a-h1"],"title":"Hypothesis Testing","text":"At the moment, the prevailing belief is that","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["The baker\'s bread height is less than 15cm on average","The baker\'s bread height is less than 15cm on average","The customers heights are more than 15cm on average","The standard deviation of bread heights is more than 15cm"]},{"id":"aab36429.3baker1a-h3","type":"hint","dependencies":["aab36429.3baker1a-h2"],"title":"Hypothesis Testing","text":"We know that the prevailing belief is that the bread height is less than 15cm. We also know that the null hypothesis is defined similarly to the status quo or the prevailing belief, therefore the null hypothesis is that the baker\'s bread is less than 15cm.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aab36429.3baker1a-h4","type":"hint","dependencies":["aab36429.3baker1a-h2"],"title":"Hypothesis Testing","text":"The baker making the claim forms the alternate hypothesis while the customers\' opinion of bread height averages forms the null hypothesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aab36429.3baker1b","stepAnswer":["yes"],"problemType":"MultipleChoice","stepTitle":"Suppose the null hypothesis is true (the mean height of the loaves is no more than 15cm). Using the provided normal distribution of mean heights, is the mean height (17 cm) calculated from the sample unexpectedly large?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["yes","no"],"hints":{"DefaultPathway":[{"id":"aab36429.3baker1b-h1","type":"hint","dependencies":[],"title":"Rare Events, The Sample, Decision, and Conclusion","text":"Given the normal distribution of means, it is indicated that the $$p-value$$ is close to $$0$$. We can interpret it as $$P\\\\left(X>17\\\\right)$$ is approximately zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aab36429.3baker1b-h2","type":"hint","dependencies":["aab36429.3baker1b-h1"],"title":"P-value","text":"A $$p-value$$ of approximately zero tells us that it is highly unlikely that a loaf of bread rises no more than $$15$$ cm, on average. That is, almost 0% of all loaves of bread would be at least as high as $$17$$ cm.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aab36429.3baker1b-h3","type":"hint","dependencies":["aab36429.3baker1b-h2"],"title":"P-value","text":"Since the outcome of 17cm is so unlikely we can conclude that the evidence is strongly against the null hypothesis (that the mean height is at most 15cm)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aab36429.3baker1b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["yes"],"dependencies":["aab36429.3baker1b-h3"],"title":"Hypothesis Testing","text":"Is there sufficient evidence that the true mean height for the population of the baker\'s loaves of bread is greater than 15cm?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["yes","no"]}]}}]},{"id":"aab36429.3decisionmaking1","title":"Rare Events, The Sample, Decision, and Conclusion","body":"Consider a as the significance level or preconceived alpha.","variabilization":{},"oer":"https://openstax.org/","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Distribution Needed for Hypothesis Testing","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aab36429.3decisionmaking1a","stepAnswer":["reject"],"problemType":"MultipleChoice","stepTitle":"If $$a>p-value$$, do we make the decision to reject or fail to reject H_0(the null hypothesis)","stepBody":"","answerType":"string","variabilization":{},"choices":["reject","fail to reject"],"hints":{"DefaultPathway":[{"id":"aab36429.3decisionmaking1a-h1","type":"hint","dependencies":[],"title":"P-value","text":"The results of the sample data are significant if \\\\alpha>p-value. There is significant evidence to conclude that $$H_0$$ is an incorrect belief and that the alternative hypothesis, $$H_a$$, may be correct.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aab36429.3decisionmaking1b","stepAnswer":["fail to reject"],"problemType":"MultipleChoice","stepTitle":"If $$a \\\\leq p-value$$ do we make the decision to reject or fail to reject the null hypothesis $$H_0$$?","stepBody":"","answerType":"string","variabilization":{},"choices":["reject","fail to reject"],"hints":{"DefaultPathway":[{"id":"aab36429.3decisionmaking1b-h1","type":"hint","dependencies":[],"title":"Given the results, the sample data are not significant since $$a \\\\leq p-value$$. There is not sufficient evidence to conclude that the alternative hypothesis, $$H_a$$, may be correct.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aab36429.3distributionchoice","title":"Hypothesis Testing Distributions","body":"Particular distributions are associated with hypothesis testing. Perform tests of a population mean using a normal distribution or a Student\'s $$t-distribution$$","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Distribution Needed for Hypothesis Testing","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aab36429.3distributionchoicea","stepAnswer":["Student\'s $$t-distribution$$"],"problemType":"MultipleChoice","stepTitle":"Suppose you want to perform hypothesis testing of a population mean and the population standard deviation is unknown and the distribution of the sample mean is approximately normal. The data collected is from simple random sampling. What distribution should you use?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Student\'s $$t-distribution$$","choices":["Student\'s $$t-distribution$$","Normal distribution","Exponential distribution","The distribution in this case doesn\'t matter"],"hints":{"DefaultPathway":[{"id":"aab36429.3distributionchoicea-h1","type":"hint","dependencies":[],"title":"Z-test","text":"Recall what decisions must be made to decide what distribution would be best to perform hypothesis tests. When you use a normal distribution for a hypothesis test (often called a $$z-test)$$, a simple random sample is taken from the population and the population that you are working with is normally distributed or the sample size is sufficiently large.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aab36429.3distributionchoicea-h2","type":"hint","dependencies":["aab36429.3distributionchoicea-h1"],"title":"T-test","text":"When you perform a hypothesis test of a single population mean \u03bc using a Student\'s $$t-distribution$$ (often called a $$t-test)$$, there are fundamental assumptions that need to be met in order for the test to work properly. Your data should be a simple random sample that comes from a population that is approximately normally distributed. You use the sample standard deviation to approximate the population standard deviation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aab36429.3distributionchoicea-h3","type":"hint","dependencies":["aab36429.3distributionchoicea-h2"],"title":"Exponential Distributions","text":"Exponential distributions involves a continuous random variable that appears when we are interested in the intervals of time between some random events. Take for example, the length of time between emergency arrivals at a hospital.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aab36429.3distributionchoicea-h4","type":"hint","dependencies":["aab36429.3distributionchoicea-h3"],"title":"Hypothesis Testing Distributions","text":"Exponential distributions cannot be used for hypothesis testing, the only eligible distributions are normal and $$t-test$$ distributions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aab36429.3distributionchoicea-h5","type":"hint","dependencies":["aab36429.3distributionchoicea-h4"],"title":"Hypothesis Testing Distributions","text":"We can either choose normal or $$t-test$$ distribution. Although the data in this context is approximately normal and we could use normal distribution, since we are unaware of the population standard deviation, the $$t-test$$ is the best choice for this hypothesis testing scenario.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aab36429.3distributionchoice2","title":"Hypothesis Testing Distributions","body":"Particular distributions are associated with hypothesis testing. Perform tests of a population mean using a normal distribution or a Student\'s $$t-distribution$$","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Distribution Needed for Hypothesis Testing","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aab36429.3distributionchoice2a","stepAnswer":["Normal distribution"],"problemType":"MultipleChoice","stepTitle":"Suppose you want to perform hypothesis testing of a population mean and the population standard deviation is known and the distribution of the sample mean is approximately normal. The data collected is from simple random sampling. What distribution should you use?","stepBody":"","answerType":"string","variabilization":{},"choices":["Student\'s $$t-distribution$$","Normal distribution","Exponential distribution","The distribution in this case doesn\'t matter"],"hints":{"DefaultPathway":[{"id":"aab36429.3distributionchoice2a-h1","type":"hint","dependencies":[],"title":"Z-test","text":"Recall what decisions must be made to decide what distribution would be best to perform hypothesis tests. When you use a normal distribution for a hypothesis test (often called a $$z-test)$$, a simple random sample is taken from the population and the population that you are working with is normally distributed or the sample size is sufficiently large.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aab36429.3distributionchoice2a-h2","type":"hint","dependencies":["aab36429.3distributionchoice2a-h1"],"title":"T-test","text":"When you perform a hypothesis test of a single population mean \u03bc using a Student\'s $$t-distribution$$ (often called a $$t-test)$$, there are fundamental assumptions that need to be met in order for the test to work properly. Your data should be a simple random sample that comes from a population that is approximately normally distributed. You use the sample standard deviation to approximate the population standard deviation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aab36429.3distributionchoice2a-h3","type":"hint","dependencies":["aab36429.3distributionchoice2a-h2"],"title":"Exponential Distributions","text":"Exponential distributions involves a continuous random variable that appears when we are interested in the intervals of time between some random events. Take for example, the length of time between emergency arrivals at a hospital.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aab36429.3distributionchoice2a-h4","type":"hint","dependencies":["aab36429.3distributionchoice2a-h3"],"title":"Hypothesis Testing Distributions","text":"Exponential distributions cannot be used for hypothesis testing, the only eligible distributions are normal and $$t-test$$ distributions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aab36429.3distributionchoice2a-h5","type":"hint","dependencies":["aab36429.3distributionchoice2a-h4"],"title":"Hypothesis Testing Distributions","text":"We can either choose normal or $$t-test$$ distribution. Although the data in this context is approximately normal and we could use $$t-testing$$ distribution, since we are aware of the population standard deviation, the $$t-test$$ is unnecessary and the normal distribution is the best choice for this hypothesis testing scenario.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aab36429.3distributionreview","title":"Hypothesis Testing Review","body":"Suppose you are performing a hypothesis test of a single population mean.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Distribution Needed for Hypothesis Testing","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aab36429.3distributionreviewa","stepAnswer":["mean"],"problemType":"MultipleChoice","stepTitle":"What statistic is used for the test?","stepBody":"","answerType":"string","variabilization":{},"choices":["mean","mode","range","median"],"hints":{"DefaultPathway":[{"id":"aab36429.3distributionreviewa-h1","type":"hint","dependencies":[],"title":"Hypothesis Testing Review","text":"We are testing for the mean of our data, so the components of the distribution formed should be the means.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aab36429.3distributionreviewb","stepAnswer":["sample mean"],"problemType":"MultipleChoice","stepTitle":"What is the estimated $$\\\\frac{value}{point}$$ estimate for the population parameter?","stepBody":"","answerType":"string","variabilization":{},"choices":["sample mean","population mean","sample standard deviation","sample median"],"hints":{"DefaultPathway":[{"id":"aab36429.3distributionreviewb-h1","type":"hint","dependencies":[],"title":"Estimate value","text":"The estimate value or point estimate is a single number computed from a sample and used to estimate a population parameter.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aab36429.3distributionreviewb-h2","type":"hint","dependencies":["aab36429.3distributionreviewb-h1"],"title":"Estimate value","text":"The population parameter in this case is the population mean, therefore the point estimate of our test should be the sample mean; we are using sample means to estimate the population mean in our test.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aab36429.3distributionreview2","title":"Hypothesis Testing Distributions","body":"Suppose you are performing a hypothesis test of a single population parameter $$p$$. $$x$$ is the number of successes, $$n$$ is the sample size, and s is the sample standard deviation.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Distribution Needed for Hypothesis Testing","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aab36429.3distributionreview2a","stepAnswer":["p\', where $$p\'=\\\\frac{x}{n}$$"],"problemType":"MultipleChoice","stepTitle":"What is the estimated value for $$p$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"p\', where $$p\'=\\\\frac{x}{n}$$","choices":["p\', where $$p\'=\\\\frac{x}{n}$$","p\', where $$p\'=\\\\frac{n}{x}$$","p\', where $$p\'=\\\\frac{x}{n} \\\\sqrt{s}$$","p\', where $$p\'=\\\\frac{x}{\\\\sqrt{n s}}$$"],"hints":{"DefaultPathway":[{"id":"aab36429.3distributionreview2a-h1","type":"hint","dependencies":[],"title":"Hypothesis Testing Distributions","text":"Recall that if you are testing a single population proportion, the distribution for the test is for proportions or percentages: P\'~N(p,sqrt((p*q)/n). In single population proportion testing, the estimated value is the number of successes divided by the sample size.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aab36429.3distributionreview2b","stepAnswer":["$$n p>5$$ and $$n q>5$$"],"problemType":"MultipleChoice","stepTitle":"When you perform this test in the right conditions, which of the following must be true?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$n p>5$$ and $$n q>5$$","choices":["$$n p>5$$ and $$n q>5$$","$$n p>30$$ and $$n q>30$$","The conditions for an exponential distribution are met","$$s>5$$"],"hints":{"DefaultPathway":[{"id":"aab36429.3distributionreview2b-h1","type":"hint","dependencies":[],"title":"Hypothesis Testing Distributions","text":"When conducting a hypothesis test of a single population proportion $$p$$, you must meet conditions for a binomial distribution, where the quantities $$n p$$ and $$n q$$ are greater than $$5$$. q is the complementary probably of succession $$(q=1-p)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aab36429.3distributionreview2b-h2","type":"hint","dependencies":["aab36429.3distributionreview2b-h1"],"title":"Hypothesis Testing Distributions","text":"The shape of the binomial distribution needs to be a similar shape to that of the normal distribution. In order to assess similarity, we ensure the product of $$n$$ and $$p$$ and the product of $$n$$ and q are both greater than $$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aab36429.3distributionreview2c","stepAnswer":["normal distribution"],"problemType":"MultipleChoice","stepTitle":"After ensuring that their are a certain number $$n$$ of independent trials, the outcomes of any trial are success or failure, and each trial has the same probability of a success $$p$$, and the shape of the binomial distribution form is confirmed to be similar to that of the normal distribution, we can now assume that the binomial distribution of the sample proportion can be approximated by the $$_{}$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["normal distribution","exponential distribution","$$t-test$$ distribution","poisson distribution"],"hints":{"DefaultPathway":[{"id":"aab36429.3distributionreview2c-h1","type":"hint","dependencies":[],"title":"Hypothesis Testing Distributions","text":"In hypothesis testing of the single population proportion, we ensure all of the axioms described before are true in order to \\"substitute\\" our binomial distribution with the normal distribution because it is more convenient to use it and it is an effort to take advantage of the normal distribution\'s properties that can be useful for hypothesis testing.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aab36429.3laketahoe","title":"Hypothesis Testing Distributions","body":"It is believed that Lake Tahoe Community College (LTCC) Intermediate Algebra students get less than seven hours of sleep per night, on average. A survey of $$22$$ LTCC Intermediate Algebra students generated a mean of $$7.24$$ hours with a standard deviation of $$1.93$$ hours. Consider the mean hours of sleep as $$m$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Distribution Needed for Hypothesis Testing","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aab36429.3laketahoea","stepAnswer":["$$t-distribution$$"],"problemType":"MultipleChoice","stepTitle":"What distribution should be used to assess if LTCC Intermediate Algebra students get less than seven hours of sleep per night, on average?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$t-distribution$$","choices":["$$t-distribution$$","Normal distribution","Poisson distribution","Exponential distribution"],"hints":{"DefaultPathway":[{"id":"aab36429.3laketahoea-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$H_0$$: $$m \\\\geq 7$$ H_a:m<7"],"dependencies":[],"title":"Hypothesis Testing Distributions","text":"State the null hypothesis and alternative hypothesis","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$H_0$$: $$m \\\\geq 7$$ H_a:m<7","$$H_0$$: $$m \\\\leq 7$$ H_a:m>7","$$H_0$$: $$m>7$$ H_a:m<=7","$$H_0$$: $$m \\\\geq 7.24$$ H_a:m<7"]},{"id":"aab36429.3laketahoea-h2","type":"hint","dependencies":["aab36429.3laketahoea-h1"],"title":"Hypothesis Testing Distributions","text":"After declaring the null and alternative hypothesis, we consider the random variable $$x$$ and the distribution it will make. First, we aren\'t given any information on the population standard deviation, therefore the $$t-distribution$$ should be used.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aab36429.3studenthours","title":"Hypothesis Testing Distributions","body":"It is believed that students at a local college get less than seven hours of sleep per night, on average. A survey of $$22$$ students generated a mean of $$7.24$$ hours with a standard deviation of $$1.93$$ hours. A campus wide required study reveals that the standard deviation of the college students\' sleep hours $$1.4$$ hours and that the data follows an approximately normal curve. Consider the mean hours of sleep as $$m$$ and the random variable $$x$$ as the college students\' sleep hours.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3  Distribution Needed for Hypothesis Testing","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aab36429.3studenthoursa","stepAnswer":["Normal distribution"],"problemType":"MultipleChoice","stepTitle":"What distribution should be used to assess if LTCC Intermediate Algebra students get less than seven hours of sleep per night, on average?","stepBody":"","answerType":"string","variabilization":{},"choices":["$$t-distribution$$","Normal distribution","Poisson distribution","Exponential distribution"],"hints":{"DefaultPathway":[{"id":"aab36429.3studenthoursa-h1","type":"hint","dependencies":[],"title":"T-test","text":"When you perform a hypothesis test of a single population mean \u03bc using a Student\'s $$t-distribution$$ (often called a $$t-test)$$, there are fundamental assumptions that need to be met in order for the test to work properly. Your data should be a simple random sample that comes from a population that is approximately normally distributed. You use the sample standard deviation to approximate the population standard deviation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aab36429.3studenthoursa-h2","type":"hint","dependencies":["aab36429.3studenthoursa-h1"],"title":"Z-test","text":"Recall what decisions must be made to decide what distribution would be best to perform hypothesis tests. When you use a normal distribution for a hypothesis test (often called a $$z-test)$$, a simple random sample is taken from the population and the population that you are working with is normally distributed or the sample size is sufficiently large.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aab36429.3studenthoursa-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["There is no sufficient information to obtain the population standard deviation"],"dependencies":["aab36429.3studenthoursa-h2"],"title":"Hypothesis Testing Distributions","text":"A key factor that can differentiate the viability of $$t-test$$ versus $$z-test$$ is the standard deviation. To be more exact, it must be the population standard deviation. What is the population standard deviation of student hours in hours? Round to the nearest four decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["There is no sufficient information to obtain the population standard deviation","$$7.2400$$","$$1.9300$$","$$6.2333$$"]},{"id":"aab36429.3studenthoursa-h4","type":"hint","dependencies":["aab36429.3studenthoursa-h3"],"title":"Hypothesis Testing Distributions","text":"We can either choose normal or $$t-test$$ distribution. Although the data in this context is approximately normal and we could use $$t-testing$$ distribution, since we are aware of the population standard deviation, the $$t-test$$ is unnecessary and the normal distribution is the best choice for this hypothesis testing scenario.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aab36429.3studenthoursb","stepAnswer":["$$t_{21}$$"],"problemType":"MultipleChoice","stepTitle":"The distribution to be used for this test is $$x\\\\pm _{}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$t_{21}$$","choices":["N(7.24,1.93/sqrt(22))","$$N(7.24, 1.93)$$","$$t_{22}$$","$$t_{21}$$"],"hints":{"DefaultPathway":[{"id":"aab36429.3studenthoursb-h1","type":"hint","dependencies":[],"title":"Hypothesis Testing Distributions","text":"Remember that degrees of freedom for $$t-test$$ distributions is $$n-1$$, where $$n$$ is the population","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aabf119factor1","title":"Factoring the Greatest Common Factor","body":"Factor the expression by pulling out the greatest common factor.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Factoring Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"aabf119factor1a","stepAnswer":["$$3\\\\operatorname{xy}\\\\left(2x^2 y^2+15xy+7\\\\right)$$"],"problemType":"MultipleChoice","stepTitle":"$$6x^3 y^3+45x^2 y^2+21xy$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$3\\\\operatorname{xy}\\\\left(2x^2 y^2+15xy+7\\\\right)$$","choices":["$$3\\\\operatorname{xy}\\\\left(2x^2 y^2+15xy+7\\\\right)$$","$$\\\\operatorname{xy}\\\\left(2x^2 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expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$3$$","$${xy}^2$$","$$y$$","3xy"]},{"id":"aabf119factor1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["aabf119factor1a-h3"],"title":"GCF of coefficients","text":"What is the Greatest Common Factor of the coefficients?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor1a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x$$"],"dependencies":["aabf119factor1a-h4"],"title":"GCF of $$x$$","text":"What is the Greatest Common Factor of the variable $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x^2$$","$$x$$","$$x^3$$"]},{"id":"aabf119factor1a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y$$"],"dependencies":["aabf119factor1a-h5"],"title":"GCF of $$y$$","text":"What is the Greatest Common Factor of the variable $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y^2$$","$$y$$","$$y^3$$"]},{"id":"aabf119factor1a-h7","type":"hint","dependencies":["aabf119factor1a-h6"],"title":"Finding the GCF","text":"The greatest common factor of the expression is the product of the GCF of its coefficients and variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aabf119factor10","title":"Factor the expression","body":"Factoring a Sum of Cubes.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Factoring Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"aabf119factor10a","stepAnswer":["$$\\\\left(x+8\\\\right) \\\\left(x^2-8x+64\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$x^3+512$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(x+8\\\\right) \\\\left(x^2-8x+64\\\\right)$$","hints":{"DefaultPathway":[{"id":"aabf119factor10a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x$$"],"dependencies":[],"title":"Cube Root","text":"What is the cube root of $$x^3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["aabf119factor10a-h1"],"title":"Cube Root","text":"What is the cube root of 512?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor10a-h3","type":"hint","dependencies":["aabf119factor10a-h2"],"title":"Sum of Cubes Formula","text":"$$\\\\left(a+b\\\\right) \\\\left(a^2-ab+b^2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor10a-h4","type":"hint","dependencies":["aabf119factor10a-h3"],"title":"Variable Values","text":"The value of a in the equation is $$x$$ and the value of $$b$$ is $$8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aabf119factor10b","stepAnswer":["$$\\\\left(6a+b\\\\right) \\\\left(36a^2+6ab+b^2\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$216a^3+b^3$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(6a+b\\\\right) \\\\left(36a^2+6ab+b^2\\\\right)$$","hints":{"DefaultPathway":[{"id":"aabf119factor10b-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["6a"],"dependencies":[],"title":"Cube Root","text":"What is the cube root of $$216a^3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor10b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$b$$"],"dependencies":["aabf119factor10b-h1"],"title":"Cube Root","text":"What is the cube root of $$b^3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor10b-h3","type":"hint","dependencies":["aabf119factor10b-h2"],"title":"Sum of Cubes Formula","text":"$$\\\\left(a+b\\\\right) \\\\left(a^2-ab+b^2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor10b-h4","type":"hint","dependencies":["aabf119factor10b-h3"],"title":"Variable Values","text":"The value of a in the equation is 6a and the value of $$b$$ is $$b$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aabf119factor11","title":"Factoring a Difference of Cubes","body":"Factor the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Factoring Polynomials","courseName":"OpenStax: College 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4.0>"},{"id":"aabf119factor11a-h3","type":"hint","dependencies":["aabf119factor11a-h2"],"title":"Difference of Cubes Formula","text":"$$\\\\left(a-b\\\\right) \\\\left(a^2+ab+b^2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor11a-h4","type":"hint","dependencies":["aabf119factor11a-h3"],"title":"Variable Values","text":"The value of a is $$2x$$ and the value of $$b$$ is $$5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aabf119factor11b","stepAnswer":["$$\\\\left(10x-1\\\\right) \\\\left(100x^2+10x+1\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$1000x^3-1$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(10x-1\\\\right) \\\\left(100x^2+10x+1\\\\right)$$","hints":{"DefaultPathway":[{"id":"aabf119factor11b-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10x$$"],"dependencies":[],"title":"Cube Root","text":"What is the cube root of $$1000x^3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor11b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["aabf119factor11b-h1"],"title":"Cube Root","text":"What is the cube root of 1?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor11b-h3","type":"hint","dependencies":["aabf119factor11b-h2"],"title":"Difference of Cubes Formula","text":"$$\\\\left(a-b\\\\right) \\\\left(a^2+ab+b^2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor11b-h4","type":"hint","dependencies":["aabf119factor11b-h3"],"title":"Variable Values","text":"The value of a is $$10x$$ and the value of $$b$$ is $$1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aabf119factor12","title":"Factoring an Expression with Fractional or Negative Exponents.","body":"Factor the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Factoring Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"aabf119factor12a","stepAnswer":["$$\\\\frac{{\\\\left(x+2\\\\right)}^{-1}}{3}$$ $$7x+8$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{3{x\\\\left(x+2\\\\right)}^{-1}}{3}+\\\\frac{{4\\\\left(x+2\\\\right)}^2}{3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{{\\\\left(x+2\\\\right)}^{-1}}{3}$$ $$7x+8$$","hints":{"DefaultPathway":[{"id":"aabf119factor12a-h1","type":"hint","dependencies":[],"title":"GCF","text":"Factor out the GCF $$\\\\frac{{\\\\left(x+2\\\\right)}^{-1}}{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor12a-h2","type":"hint","dependencies":["aabf119factor12a-h1"],"title":"Simplify","text":"Simplify the term other than the GCF","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aabf119factor12b","stepAnswer":["$$\\\\frac{{\\\\left(5a-1\\\\right)}^{-1}}{4} \\\\left(17a-2\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{{2\\\\left(5a-1\\\\right)}^3}{4}+\\\\frac{7{a\\\\left(5a-1\\\\right)}^{-1}}{4}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{{\\\\left(5a-1\\\\right)}^{-1}}{4} \\\\left(17a-2\\\\right)$$","hints":{"DefaultPathway":[{"id":"aabf119factor12b-h1","type":"hint","dependencies":[],"title":"GCF","text":"Factor out the GCF $$\\\\frac{{\\\\left(5a-4\\\\right)}^{-1}}{4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor12b-h2","type":"hint","dependencies":["aabf119factor12b-h1"],"title":"Simplify","text":"Simplify the term other than the GCF","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aabf119factor13","title":"Factoring Polynomials","body":"Find the greatest common factor.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Factoring Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"aabf119factor13a","stepAnswer":["$$7m$$"],"problemType":"TextBox","stepTitle":"$$49{mb}^2-35m^2 ba+77{ma}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7m$$","hints":{"DefaultPathway":[{"id":"aabf119factor13a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$49$$ $$=7\\\\times7$$, $$35$$ $$=$$ $$7\\\\times5$$, $$77$$ $$=$$ $$11\\\\times7$$"],"dependencies":[],"title":"Factor the expression","text":"What are the factors of $$49$$, $$35$$, 77?","variabilization":{},"oer":"","license":"","choices":["$$49$$ $$=7\\\\times7$$, $$35$$ $$=$$ $$7\\\\times5$$, $$77$$ $$=$$ $$11\\\\times7$$","$$49$$ $$=8\\\\times8$$, $$35$$ $$=$$ 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Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Factoring Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"aabf119factor4a","stepAnswer":["$$(x-1)(x-6)$$"],"problemType":"MultipleChoice","stepTitle":"$$x^2-7x+6$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(x-1)(x-6)$$","choices":["$$(x-2)(x-3)$$","$$\\\\left(x+2\\\\right) \\\\left(x-3\\\\right)$$","$$\\\\left(x+1\\\\right) \\\\left(x+6\\\\right)$$","$$(x-1)(x-6)$$"],"hints":{"DefaultPathway":[{"id":"aabf119factor4a-h1","type":"hint","dependencies":[],"title":"Factor the expression","text":"A trinomial of the form $$x^2+bx+c$$ can be written in factored form as $$\\\\left(x+p\\\\right) \\\\left(x+q\\\\right) \\\\left(x+p\\\\right) \\\\left(x+q\\\\right)$$ where $$pq=c$$ and $$p+q=b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor4a-h2","type":"hint","dependencies":["aabf119factor4a-h1"],"title":"Factor the expression","text":"To factor a polynomial $$x^2+bx+c$$, the first step is to find two numbers with a product of c and a sum of $$b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["aabf119factor4a-h2"],"title":"Product of Numbers","text":"What is $$\\\\left(-1\\\\right) \\\\left(-6\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-7$$"],"dependencies":["aabf119factor4a-h3"],"title":"Sum of Numbers","text":"What is $$\\\\left(-1\\\\right)+\\\\left(-6\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor4a-h5","type":"hint","dependencies":["aabf119factor4a-h4"],"title":"In a polynomial $$x^2+bx+c$$, if $$f g=c$$ and $$f+g=b$$, then the polynomial can be factored as $$\\\\left(x+f\\\\right) \\\\left(x+g\\\\right)$$.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aabf119factor5","title":"Factoring a Trionomial by Grouping.","body":"Factor the expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Factoring Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"aabf119factor5a","stepAnswer":["$$\\\\left(x+2\\\\right) \\\\left(5x-3\\\\right)$$"],"problemType":"MultipleChoice","stepTitle":"Factor $$5x^2+7x-6$$ by grouping.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\left(x+2\\\\right) \\\\left(5x-3\\\\right)$$","choices":["$$\\\\left(x+2\\\\right) \\\\left(x-3\\\\right)$$","$$\\\\left(5x+6\\\\right) \\\\left(x-1\\\\right)$$","$$\\\\left(x+2\\\\right) \\\\left(5x-3\\\\right)$$","$$\\\\left(x+3\\\\right) \\\\left(5x-2\\\\right)$$"],"hints":{"DefaultPathway":[{"id":"aabf119factor5a-h1","type":"hint","dependencies":[],"title":"Factor the expression","text":"To factor a trionomial $${ax}^2+bx+c$$, the first step is to find $$p$$ and q, a pair of factors of ac with a sum of $$b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-30$$"],"dependencies":["aabf119factor5a-h1"],"title":"Product of Numbers","text":"What is $$10\\\\left(-3\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["aabf119factor5a-h2"],"title":"Sum of Numbers","text":"What is $$\\\\left(-3\\\\right)+10$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor5a-h4","type":"hint","dependencies":["aabf119factor5a-h3"],"title":"Rewriting the Expression","text":"The second step is to rewrite the original expression as $${ax}^2+px+qx+c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor5a-h5","type":"hint","dependencies":["aabf119factor5a-h4"],"title":"Greatest Common Factor (GCF)","text":"Thirdly, pull out the GCF of $${ax}^2+px$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor5a-h6","type":"hint","dependencies":["aabf119factor5a-h5"],"title":"Greatest Common Factor (GCF)","text":"Then, pull out the GCF of $$qx+c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor5a-h7","type":"hint","dependencies":["aabf119factor5a-h6"],"title":"Factoring out the GCF","text":"Lastly, factor out the GCF of the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aabf119factor6","title":"Factoring a Trionomial by Grouping.","body":"Factor the expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Factoring Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"aabf119factor6a","stepAnswer":["$$\\\\left(x+3\\\\right) \\\\left(2x+3\\\\right)$$"],"problemType":"MultipleChoice","stepTitle":"$$2x^2+9x+9$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\left(x+3\\\\right) \\\\left(2x+3\\\\right)$$","choices":["$$\\\\left(x+3\\\\right) \\\\left(2x+3\\\\right)$$","$$\\\\left(x-3\\\\right) \\\\left(2x+3\\\\right)$$","$$\\\\left(x+1\\\\right) \\\\left(2x+3\\\\right)$$"],"hints":{"DefaultPathway":[{"id":"aabf119factor6a-h1","type":"hint","dependencies":[],"title":"Factor the expression","text":"To factor a trionomial $${ax}^2+bx+c$$, the first step is to find $$p$$ and q, a pair of factors of ac with a sum of $$b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18$$"],"dependencies":["aabf119factor6a-h1"],"title":"Product of Numbers","text":"What is $$3\\\\times6$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["aabf119factor6a-h2"],"title":"Sum of Numbers","text":"What is $$3+6$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor6a-h4","type":"hint","dependencies":["aabf119factor6a-h3"],"title":"Rewriting the Expression","text":"The second step is to rewrite the original expression as $${ax}^2+px+qx+c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor6a-h5","type":"hint","dependencies":["aabf119factor6a-h4"],"title":"Greatest Common Factor (GCF)","text":"Thirdly, pull out the GCF of $${ax}^2+px$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor6a-h6","type":"hint","dependencies":["aabf119factor6a-h5"],"title":"Greatest Common Factor (GCF)","text":"Then, pull out the GCF of $$qx+c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor6a-h7","type":"hint","dependencies":["aabf119factor6a-h6"],"title":"Factoring out the GCF","text":"Lastly, factor out the GCF of the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aabf119factor6b","stepAnswer":["$$\\\\left(3x-1\\\\right) \\\\left(2x+1\\\\right)$$"],"problemType":"MultipleChoice","stepTitle":"Factor $$6x^2+x-1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\left(3x-1\\\\right) \\\\left(2x+1\\\\right)$$","choices":["$$3x \\\\left(2x+1\\\\right)$$","$$\\\\left(3x-1\\\\right) \\\\left(2x+1\\\\right)$$","$$\\\\left(3x+1\\\\right) \\\\left(2x-1\\\\right)$$","$$\\\\left(3x-3\\\\right) \\\\left(2x+1\\\\right)$$"],"hints":{"DefaultPathway":[{"id":"aabf119factor6b-h1","type":"hint","dependencies":[],"title":"Factor the expression","text":"To factor a trionomial $${ax}^2+bx+c$$, the first step is to find $$p$$ and q, a pair of factors of ac with a sum of $$b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor6b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6$$"],"dependencies":["aabf119factor6b-h1"],"title":"Product of Numbers","text":"What is $$3\\\\left(-2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor6b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["aabf119factor6b-h2"],"title":"Sum of Numbers","text":"What is $$\\\\left(-2\\\\right)+3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor6b-h4","type":"hint","dependencies":["aabf119factor6b-h3"],"title":"Rewriting the Expression","text":"The second step is to rewrite the original expression as $${ax}^2+px+qx+c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor6b-h5","type":"hint","dependencies":["aabf119factor6b-h4"],"title":"Greatest Common Factor (GCF)","text":"Thirdly, pull out the GCF of $${ax}^2+px$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor6b-h6","type":"hint","dependencies":["aabf119factor6b-h5"],"title":"Greatest Common Factor (GCF)","text":"Then, pull out the GCF of $$qx+c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor6b-h7","type":"hint","dependencies":["aabf119factor6b-h6"],"title":"Factoring out the GCF","text":"Lastly, factor out the GCF of the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aabf119factor7","title":"Factoring a Perfect Square Trinomial","body":"What is the factored form of the following expression?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Factoring Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"aabf119factor7a","stepAnswer":["$${\\\\left(5x+2\\\\right)}^2$$"],"problemType":"MultipleChoice","stepTitle":"$$25x^2+20x+4$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$${\\\\left(5x+2\\\\right)}^2$$","choices":["$${\\\\left(5x+2\\\\right)}^2$$","$${\\\\left(3x+4\\\\right)}^2$$","$${\\\\left(25x+4\\\\right)}^2$$","$${\\\\left(x+2\\\\right)}^2$$"],"hints":{"DefaultPathway":[{"id":"aabf119factor7a-h1","type":"hint","dependencies":[],"title":"Perfect Square Trinomial Definition","text":"In a perfect square trinomial, the first and last terms are perfect squares and the middle term is twice the product of their roots.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor7a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$5x$$"],"dependencies":["aabf119factor7a-h1"],"title":"Square root of first term","text":"What is the square root of the first term, $$25x^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$5x$$","$$3x$$","$$5$$","$$3$$"]},{"id":"aabf119factor7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["aabf119factor7a-h2"],"title":"Square root of second term","text":"What is the square root of the second term, 4?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor7a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["aabf119factor7a-h3"],"title":"Middle Term of the Trinomial","text":"Is the middle term of the expression, $$20x$$, equal to $$2$$ times the square root of the first term and the square root of the second term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"aabf119factor7a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["aabf119factor7a-h4"],"title":"Perfect Square Trinomial","text":"Is the expression a Perfect Square Trinomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"aabf119factor7a-h6","type":"hint","dependencies":["aabf119factor7a-h5"],"title":"Perfect Square Trionmial","text":"A perfect square trinomial $${nx}^2$$ + 2nbx + $$b^2$$ can be factored as $${\\\\left(nx+b\\\\right)}^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aabf119factor8","title":"Factor the following expression","body":"Factoring a Perfect Square Trinomial.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Factoring Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"aabf119factor8a","stepAnswer":["$${\\\\left(7x-1\\\\right)}^2$$"],"problemType":"MultipleChoice","stepTitle":"$$49x^2-14x+1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$${\\\\left(7x-1\\\\right)}^2$$","choices":["$${\\\\left(7x-1\\\\right)}^2$$","$${\\\\left(7x-9\\\\right)}^2$$","$${\\\\left(7x-7\\\\right)}^2$$","$${\\\\left(3x-1\\\\right)}^2$$"],"hints":{"DefaultPathway":[{"id":"aabf119factor8a-h1","type":"hint","dependencies":[],"title":"Perfect Square Trinomial Definition","text":"In a perfect square trinomial, the first and last terms are perfect squares and the middle term is twice the product of their roots.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor8a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$7x$$"],"dependencies":["aabf119factor8a-h1"],"title":"Square root of first term","text":"What is the square root of the first term, $$25x^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$7x$$","$$2x$$","$$7$$","$$5x$$"]},{"id":"aabf119factor8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["aabf119factor8a-h2"],"title":"Square root of second term","text":"What is the square root of the second term, 4?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor8a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["aabf119factor8a-h3"],"title":"Middle Term of the Trinomial","text":"Is the middle term of the expression, $$20x$$, equal to $$2$$ times the square root of the first term and the square root of the second term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"aabf119factor8a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["aabf119factor8a-h4"],"title":"Perfect Square Trinomial","text":"Is the expression a Perfect Square Trinomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"aabf119factor8a-h6","type":"hint","dependencies":["aabf119factor8a-h5"],"title":"Perfect Square Trionmial","text":"A perfect square trinomial $${nx}^2$$ + 2nbx + $$b^2$$ can be factored as $${\\\\left(nx+b\\\\right)}^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aabf119factor9","title":"Factor the following expression","body":"Factoring a Difference of Squares.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Factoring Polynomials","courseName":"OpenStax: College Algebra","steps":[{"id":"aabf119factor9a","stepAnswer":["$$\\\\left(3x-5\\\\right) \\\\left(3x+5\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$9x^2-25$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(3x-5\\\\right) \\\\left(3x+5\\\\right)$$","hints":{"DefaultPathway":[{"id":"aabf119factor9a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x$$"],"dependencies":[],"title":"Square Root","text":"What is the square root of $$9x^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["aabf119factor9a-h1"],"title":"Square Root","text":"What is the square root of 25?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor9a-h3","type":"hint","dependencies":["aabf119factor9a-h2"],"title":"Difference of Squares","text":"Remember: The Difference of Squares formula is $$\\\\left(a-b\\\\right) \\\\left(a+b\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor9a-h4","type":"hint","dependencies":["aabf119factor9a-h3"],"title":"Variable Values","text":"The value of a is $$3x$$ and the value of $$b$$ is $$5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aabf119factor9b","stepAnswer":["$$\\\\left(9x-10\\\\right) \\\\left(9x+10\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$81x^2-100$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(9x-10\\\\right) \\\\left(9x+10\\\\right)$$","hints":{"DefaultPathway":[{"id":"aabf119factor9b-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9x$$"],"dependencies":[],"title":"Square Root","text":"What is the square root $$81x^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor9b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["aabf119factor9b-h1"],"title":"Square Root","text":"What is the square root of 100?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor9b-h3","type":"hint","dependencies":["aabf119factor9b-h2"],"title":"Difference of Squares","text":"Remember: The Difference of Squares formula is $$\\\\left(a-b\\\\right) \\\\left(a+b\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aabf119factor9b-h4","type":"hint","dependencies":["aabf119factor9b-h3"],"title":"Variable Values","text":"The value of a is $$9x$$ and the value of $$b$$ is $$10$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aac2a9fantiderivative1","title":"Finding Antiderivatives","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.10 Antiderivatives","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aac2a9fantiderivative1a","stepAnswer":["$$x^3+C$$"],"problemType":"MultipleChoice","stepTitle":"The antiderivative of $$f(x)=3x^2$$ is $$___$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x^3+C$$","choices":["$$x^3+C$$","$$x^3$$"],"hints":{"DefaultPathway":[{"id":"aac2a9fantiderivative1a-h1","type":"hint","dependencies":[],"title":"Adding Constants","text":"Because $$\\\\frac{d}{dx} x^3=3x^2$$, and there might be some constant. So you need to write \\"+C\\" afterwards.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"aac2a9fantiderivative2","title":"Finding Antiderivatives","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.10 Antiderivatives","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aac2a9fantiderivative2a","stepAnswer":["$$\\\\ln(|x|)+C$$"],"problemType":"MultipleChoice","stepTitle":"The antiderivative of $$f(x)=\\\\frac{1}{x}=___$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\ln(|x|)+C$$","choices":["$$\\\\ln(|x|)+C$$","$$\\\\ln(|x|)$$"],"hints":{"DefaultPathway":[{"id":"aac2a9fantiderivative2a-h1","type":"hint","dependencies":[],"title":"Natural Logarithm Antiderivatives","text":"Let $$f(x)=\\\\ln(|x|)$$. for $$x>0$$, $$f(x)=ln(x)$$ and $$\\\\frac{d}{dx} lnx=\\\\frac{1}{x}$$. For $$x<0$$, $$f(x)=ln(-x)$$ and $$\\\\frac{d}{dx} \\\\operatorname{lnnegneg}\\\\left(x\\\\right)=\\\\frac{-1}{-x}=\\\\frac{1}{x}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aac2a9fantiderivative2a-h2","type":"hint","dependencies":["aac2a9fantiderivative2a-h1"],"title":"Adding Constants","text":"Make sure to add \\"+C\\" after your intial antiderivative to account for the unknown constant.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"aac2a9fantiderivative3","title":"Finding Antiderivatives","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.10 Antiderivatives","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aac2a9fantiderivative3a","stepAnswer":["$$sin\\\\left(x\\\\right)+C$$"],"problemType":"MultipleChoice","stepTitle":"The antiderivative of $$f(x)=cos(x)=___$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$sin\\\\left(x\\\\right)+C$$","choices":["$$sin\\\\left(x\\\\right)+C$$","sin(x)"],"hints":{"DefaultPathway":[{"id":"aac2a9fantiderivative3a-h1","type":"hint","dependencies":[],"title":"Solving the Antiderivative","text":"We have $$\\\\frac{d}{dx} sin\\\\left(x\\\\right)=cos(x)$$. Thus, $$F(x)=sinx$$ is an antiderivative of cos $$x$$.","variabilization":{},"oer":"","license":""}]}}]},{"id":"aac2a9fantiderivative4","title":"Finding Antiderivatives","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.10 Antiderivatives","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aac2a9fantiderivative4a","stepAnswer":["$$e^x+C$$"],"problemType":"MultipleChoice","stepTitle":"The antiderivative of $$f(x)=e^x=___$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$e^x+C$$","choices":["$$e^x+C$$","$$e^x$$"],"hints":{"DefaultPathway":[{"id":"aac2a9fantiderivative4a-h1","type":"hint","dependencies":[],"title":"Solving the Antiderivative","text":"We have $$\\\\frac{d}{dx} e^x=e^x$$. Thus, $$F(x)=e^x$$ is an antiderivative of $$e^x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"aadff92AppQuad1","title":"Solve Applications Modeled by Quadratic Equations","body":"In the following exercises, solve using any method.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Solve Applications of Quadratic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aadff92AppQuad1a","stepAnswer":["(-17,-15),(15,17)"],"problemType":"TextBox","stepTitle":"The product of two consecutive odd numbers is $$255$$. Find the numbers.","stepBody":"Please enter your answer pair as (a,b) where $$a<b$$. If there is more than one pair of numbers that fulfill the condition, you may enter it as (a,b),(c,d) where $$a<c$$.","answerType":"string","variabilization":{},"answerLatex":"$$(-17,-15),(15,17)$$","hints":{"DefaultPathway":[{"id":"aadff92AppQuad1a-h1","type":"hint","dependencies":[],"title":"Set the Equation","text":"Let $$n$$ be the first odd number, and the second consecutive number will be $$n+2$$. The equation will be $$n\\\\left(n+2\\\\right)=255$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad1a-h2","type":"hint","dependencies":["aadff92AppQuad1a-h1"],"title":"Solve the Equation","text":"Solve $$n\\\\left(n+2\\\\right)=255$$. You foil out the equation as $$n^2+2n=255$$. $$n^2+2n-255=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad1a-h3","type":"hint","dependencies":["aadff92AppQuad1a-h2"],"title":"Apply Quadratic Equation to solve for $$n$$.","text":"Given $${ax}^2+bx+c=0$$, $$x=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4ac}\\\\right)}{2a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad1a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-17,15)"],"dependencies":["aadff92AppQuad1a-h3"],"title":"Apply Quadratic Equation to solve for $$n$$.","text":"Apply Quadratic Equation to solve for $$n$$ when $$n^2+2n-255=0$$. Enter your answer as (a,b) where $$a<b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad1a-h5","type":"hint","dependencies":["aadff92AppQuad1a-h4"],"title":"Find Answer Pairs","text":"When the first consecutive number is $$-17$$, the second consecutive number is $$-17+2=-15$$.\\\\nWhen the first consecutive number is $$15$$, the second consecutive number is $$15+2=17$$.\\\\nThe final answer is $$(-17,-15),(15,17)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aadff92AppQuad10","title":"Solve Applications Modeled by Quadratic Equations","body":"In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Solve Applications of Quadratic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aadff92AppQuad10a","stepAnswer":["(10,15)"],"problemType":"TextBox","stepTitle":"A triangular banner for the basketball championship hangs in the gym. It has an area of $$75$$ square feet. What is the length of the base and height , if the base is two-thirds of the height?","stepBody":"Please enter your answer as (a,b) where a is the base of triangle and $$b$$ is height of triangle.","answerType":"string","variabilization":{},"answerLatex":"$$(10,15)$$","hints":{"DefaultPathway":[{"id":"aadff92AppQuad10a-h1","type":"hint","dependencies":[],"title":"Set Up Equation For Problem","text":"Let the height of the triangle be $$h$$, we can write the base of the triangle in terms of $$h$$ as $$\\\\frac{2}{3} h$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad10a-h2","type":"hint","dependencies":["aadff92AppQuad10a-h1"],"title":"Equation For Triangle Area","text":"Area of $$triangle=\\\\frac{1}{2} base height$$. We can set up the equation using given condition as $$75=\\\\frac{1}{2} h \\\\frac{2}{3} h$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad10a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-15,15)"],"dependencies":["aadff92AppQuad10a-h2"],"title":"Solve For Unknown Using Quadratic Formula","text":"The quadratic formula states $${ax}^2+bx+c=0$$, then $$x=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4ac}\\\\right)}{2a}$$. Given $$\\\\frac{1}{2} h \\\\frac{2}{3} h=75$$, solve for $$h$$. Please enter your answer as (c,d) where $$c<d$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad10a-h4","type":"hint","dependencies":["aadff92AppQuad10a-h3"],"title":"Solve For Unknown Using Quadratic Formula","text":"Given that height of a triangle has to be positive, we can omit the negative answer. Therefore, the height of given triangle is $$15$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["aadff92AppQuad10a-h4"],"title":"Find the Height of Triangle","text":"Given that the height is $$15$$ and base of a triangle $$\\\\frac{2}{3}$$ of the height, what is the base?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aadff92AppQuad11","title":"Solve Applications Modeled by Quadratic Equations","body":"In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Solve Applications of Quadratic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aadff92AppQuad11a","stepAnswer":["(15,3.3)"],"problemType":"TextBox","stepTitle":"The length of a rectangular driveway is five feet more than three times the width. The area is $$50$$ square feet. Find the length and width of the driveway.","stepBody":"Please enter your answer as (a,b) where a is the length and $$b$$ is width.","answerType":"string","variabilization":{},"answerLatex":"$$(15, 3.3)$$","hints":{"DefaultPathway":[{"id":"aadff92AppQuad11a-h1","type":"hint","dependencies":[],"title":"Set Up Equation For Problem","text":"Let the width of the driveway be w, we can write the length in terms of w as $$5+3w$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad11a-h2","type":"hint","dependencies":["aadff92AppQuad11a-h1"],"title":"Equation For Rectangle Area","text":"Area of $$Rectangle=width length=w \\\\left(5+3w\\\\right)=50$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad11a-h3","type":"hint","dependencies":["aadff92AppQuad11a-h2"],"title":"Solve For Unknown Using Quadratic Formula","text":"The quadratic formula states $${ax}^2+bx+c=0$$, then $$x=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4ac}\\\\right)}{2a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad11a-h4","type":"hint","dependencies":["aadff92AppQuad11a-h3"],"title":"Solve For Unknown Using Quadratic Formula","text":"Given $$w \\\\left(5+3w\\\\right)=50$$, use quadratic formula to solve for w. We get the width $$w=\\\\frac{10}{3}$$ or $$-5$$. Given that width of rectangle has to be positive, we can omit the negative answer. Therefore, the width is $$\\\\frac{10}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["aadff92AppQuad11a-h4"],"title":"Find the Length","text":"Given the width is $$\\\\frac{10}{3}$$, the length is five feet more than three times the width. What is the length?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aadff92AppQuad12","title":"Solve Applications Modeled by Quadratic Equations","body":"In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Solve Applications of Quadratic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aadff92AppQuad12a","stepAnswer":["(10,14)"],"problemType":"TextBox","stepTitle":"A rectangular lawn has area $$140$$ square yards. Its width that is six less than twice the length. What are the length and width of the lawn?","stepBody":"Please enter your answer as (a,b) where a is the length and $$b$$ is width.","answerType":"string","variabilization":{},"answerLatex":"$$(10,14)$$","hints":{"DefaultPathway":[{"id":"aadff92AppQuad12a-h1","type":"hint","dependencies":[],"title":"Set Up Equation For Problem","text":"Let the length of the driveway be l, we can write the width in terms of l as 2l-6.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad12a-h2","type":"hint","dependencies":["aadff92AppQuad12a-h1"],"title":"Equation For Rectangle Area","text":"Area of $$Rectangle=width length=\\\\left(2l-6\\\\right) l=140$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad12a-h3","type":"hint","dependencies":["aadff92AppQuad12a-h2"],"title":"Solve For Unknown Using Quadratic Formula","text":"The quadratic formula states $${ax}^2+bx+c=0$$, then $$x=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4ac}\\\\right)}{2a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad12a-h4","type":"hint","dependencies":["aadff92AppQuad12a-h3"],"title":"Solve For Unknown Using Quadratic Formula","text":"Given $$\\\\left(2l-6\\\\right) l=140$$, use quadratic formula to solve for l. We get the length $$l=-7$$ or $$10$$. Given that length of rectangle has to be positive, we can omit the negative answer. Therefore, the length is $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad12a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$14$$"],"dependencies":["aadff92AppQuad12a-h4"],"title":"Find the Length","text":"Given the length is $$10$$, the width is six less than twice the length. What is the width?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aadff92AppQuad13","title":"Solve Applications Modeled by Quadratic Equations","body":"In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Solve Applications of Quadratic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aadff92AppQuad13a","stepAnswer":["(8,3)"],"problemType":"TextBox","stepTitle":"A rectangular table for the dining room has a surface area of $$24$$ square feet. The length is two more feet than twice the width of the table. Find the length and width of the table.","stepBody":"Please enter your answer as (a,b) where a is the length and $$b$$ is width.","answerType":"string","variabilization":{},"answerLatex":"$$(8,3)$$","hints":{"DefaultPathway":[{"id":"aadff92AppQuad13a-h1","type":"hint","dependencies":[],"title":"Set Up Equation For Problem","text":"Let the width be w, we can write the length in terms of w as $$2+2w$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad13a-h2","type":"hint","dependencies":["aadff92AppQuad13a-h1"],"title":"Equation For Rectangle Area","text":"Area of $$Rectangle=width length=w \\\\left(2+2w\\\\right)=24$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad13a-h3","type":"hint","dependencies":["aadff92AppQuad13a-h2"],"title":"Solve For Unknown Using Quadratic Formula","text":"The quadratic formula states $${ax}^2+bx+c=0$$, then $$x=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4ac}\\\\right)}{2a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad13a-h4","type":"hint","dependencies":["aadff92AppQuad13a-h3"],"title":"Solve For Unknown Using Quadratic Formula","text":"Given $$w \\\\left(2+2w\\\\right)=24$$, use quadratic formula to solve for w. We get the width $$w=-4$$ or $$3$$. Given that width of rectangle has to be positive, we can omit the negative answer. Therefore, the width is $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad13a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["aadff92AppQuad13a-h4"],"title":"Find the Length","text":"Given the width is $$3$$, the length is two more feet than twice the width. What is the length?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aadff92AppQuad14","title":"Solve Applications Modeled by Quadratic Equations","body":"In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Solve Applications of Quadratic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aadff92AppQuad14a","stepAnswer":["(16,10.5)"],"problemType":"TextBox","stepTitle":"The new computer has a surface area of $$168$$ square inches. If the the width is $$5.5$$ inches less that the length, what are the dimensions of the computer?","stepBody":"Please enter your answer as (a,b) where a is the length and $$b$$ is width.","answerType":"string","variabilization":{},"answerLatex":"$$(16, 10.5)$$","hints":{"DefaultPathway":[{"id":"aadff92AppQuad14a-h1","type":"hint","dependencies":[],"title":"Set Up Equation For Problem","text":"Let the length be l, we can write the width in terms of l as $$l-5.5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad14a-h2","type":"hint","dependencies":["aadff92AppQuad14a-h1"],"title":"Equation For Rectangle Area","text":"Area of $$Rectangle=width length=\\\\left(l-5.5\\\\right) l=168$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad14a-h3","type":"hint","dependencies":["aadff92AppQuad14a-h2"],"title":"Solve For Unknown Using Quadratic Formula","text":"The quadratic formula states $${ax}^2+bx+c=0$$, then $$x=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4ac}\\\\right)}{2a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad14a-h4","type":"hint","dependencies":["aadff92AppQuad14a-h3"],"title":"Solve For Unknown Using Quadratic Formula","text":"Given $$\\\\left(l-5.5\\\\right) l=168$$, use quadratic formula to solve for l. We get the length $$l=-10.5$$ or $$16$$. Given that length of rectangle has to be positive, we can omit the negative answer. Therefore, the length is $$16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad14a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10.5$$"],"dependencies":["aadff92AppQuad14a-h4"],"title":"Find the Length","text":"Given the length is $$16$$, width is $$5.5$$ inches less that the length. What is the width?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aadff92AppQuad15","title":"Solve Applications Modeled by Quadratic Equations","body":"In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Solve Applications of Quadratic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aadff92AppQuad15a","stepAnswer":["(1.7,3,3.5)"],"problemType":"TextBox","stepTitle":"The hypotenuse of a right triangle is twice the length of one of its legs. The length of the other leg is three feet. Find the lengths of the three sides of the triangle.","stepBody":"Please enter your answer as (a,b,c) where $$a<b<c$$.","answerType":"string","variabilization":{},"answerLatex":"$$(1.7, 3, 3.5)$$","hints":{"DefaultPathway":[{"id":"aadff92AppQuad15a-h1","type":"hint","dependencies":[],"title":"Set Up Equation For Problem","text":"Since it is a right triangle, we can employ pythagorean theorem. $$a^2+b^2=c^2$$ where a and $$b$$ are legs of triangles and c is the hypotenuse. Let $$x$$ be the length of unknown leg. We can set up the equation as $$3^2+x^2={\\\\left(2x\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad15a-h2","type":"hint","dependencies":["aadff92AppQuad15a-h1"],"title":"Solve For Unknown Using Quadratic Formula","text":"The quadratic formula states $${ax}^2+bx+c=0$$, then $$x=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4ac}\\\\right)}{2a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad15a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-1.73,1.73)"],"dependencies":["aadff92AppQuad15a-h2"],"title":"Solve For Unknown Using Quadratic Formula","text":"$$3^2+x^2={\\\\left(2x\\\\right)}^2$$ Solve for $$x$$ using the quadratic formula and round $$x$$ to the nearest hundredth. If there is more than one solution for $$x$$, please enter your answer as (a,b) where $$a<b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad15a-h4","type":"hint","dependencies":["aadff92AppQuad15a-h3"],"title":"Solve For Unknown Using Quadratic Formula","text":"Since the length of the side of a triangle has to be positive, we can omit the negative answer. Therefore the length of one side of the triangle is $$1.73$$. The hypotenuse is twice the length which is $$3.46$$. We need to round the final answer to the nearest tenth, so the length of the side is $$1.7$$ and the length of the hypotenuse is $$3.5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aadff92AppQuad16","title":"Solve Applications Modeled by Quadratic Equations","body":"In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Solve Applications of Quadratic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aadff92AppQuad16a","stepAnswer":["(3.2,9.5)"],"problemType":"TextBox","stepTitle":"The hypotenuse of a right triangle is $$10$$ cm long. One of the triangle\u2019s legs is three times the length of the other leg. Find the lengths of the two legs of the triangle. Round to the nearest tenth.","stepBody":"Please enter your answer as (a,b) where $$a<b$$.","answerType":"string","variabilization":{},"answerLatex":"$$(3.2, 9.5)$$","hints":{"DefaultPathway":[{"id":"aadff92AppQuad16a-h1","type":"hint","dependencies":[],"title":"Set Up Equation For Problem","text":"Since it is a right triangle, we can employ pythagorean theorem. $$a^2+b^2=c^2$$ where a and $$b$$ are legs of triangles and c is the hypotenuse. Let $$x$$ be the length of shortest leg. We can set up the equation as $${\\\\left(3x\\\\right)}^2+x^2={10}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad16a-h2","type":"hint","dependencies":["aadff92AppQuad16a-h1"],"title":"Solve For Unknown Using Quadratic Formula","text":"The quadratic formula states $${ax}^2+bx+c=0$$, then $$x=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4ac}\\\\right)}{2a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad16a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-3.16,3.16)"],"dependencies":["aadff92AppQuad16a-h2"],"title":"Solve For Unknown Using Quadratic Formula","text":"$${\\\\left(3x\\\\right)}^2+x^2={10}^2$$ Solve for $$x$$ using the quadratic formula and round $$x$$ to the nearest hundredth. If there is more than one solution for $$x$$, please enter your answer as (a,b) where $$a<b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad16a-h4","type":"hint","dependencies":["aadff92AppQuad16a-h3"],"title":"Solve For Unknown Using Quadratic Formula","text":"Since the length of the side of a triangle has to be positive, we can omit the negative answer. Therefore the length of the shortest side of the triangle is $$3.16$$. The other leg is three times the shortest leg which is $$9.48$$. We need to round the final answer to the nearest tenth, which are $$3.2$$ and $$9.5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aadff92AppQuad17","title":"Solve Applications Modeled by Quadratic Equations","body":"In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Solve Applications of Quadratic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aadff92AppQuad17a","stepAnswer":["$$7.3$$"],"problemType":"TextBox","stepTitle":"A rectangular garden will be divided into two plots by fencing it on the diagonal. The diagonal distance from one corner of the garden to the opposite corner is five yards longer than the width of the garden. The length of the garden is three times the width. Find the length of the diagonal of the garden.","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7.3$$","hints":{"DefaultPathway":[{"id":"aadff92AppQuad17a-h1","type":"hint","dependencies":[],"title":"Set Up Equation For Problem","text":"From the picture, we can see the diagonal, length and width of the rectangle form a right triangle, we can employ pythagorean theorem. $$a^2+b^2=c^2$$ where a and $$b$$ are legs of triangles and c is the hypotenuse. Let w be the width of rectangle. We can set up the equation as $${\\\\left(3w\\\\right)}^2+w^2={\\\\left(w+5\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad17a-h2","type":"hint","dependencies":["aadff92AppQuad17a-h1"],"title":"Solve For Unknown Using Quadratic Formula","text":"The quadratic formula states $${ax}^2+bx+c=0$$, then $$x=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4ac}\\\\right)}{2a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad17a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-1.20,2.31)"],"dependencies":["aadff92AppQuad17a-h2"],"title":"Solve For Unknown Using Quadratic Formula","text":"$${\\\\left(3w\\\\right)}^2+w^2={\\\\left(w+5\\\\right)}^2$$ can be simplified as $$9w^2-10w-25=0$$. Solve for w using the quadratic formula and round w to the nearest hundredth. If there is more than one solution for w, please enter your answer as (a,b) where $$a<b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad17a-h4","type":"hint","dependencies":["aadff92AppQuad17a-h3"],"title":"Solve For Unknown Using Quadratic Formula","text":"Since the widtth of a rectangle has to be positive, we can omit the negative answer. Therefore the $$w=2.31$$. The the diagonal of rectangle is $$w+5$$ which is $$2.31+5=7.31$$. We can round it to nearest tenth which gives $$7.3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aadff92AppQuad18","title":"Solve Applications Modeled by Quadratic Equations","body":"In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Solve Applications of Quadratic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aadff92AppQuad18a","stepAnswer":["$$10.2$$"],"problemType":"TextBox","stepTitle":"Nautical flags are used to represent letters of the alphabet. The flag for the letter, O consists of a yellow right triangle and a red right triangle which are sewn together along their hypotenuse to form a square. The hypotenuse of the two triangles is three inches longer than a side of the flag. Find the length of the side of the flag.","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10.2$$","hints":{"DefaultPathway":[{"id":"aadff92AppQuad18a-h1","type":"hint","dependencies":[],"title":"Set Up Equation For Problem","text":"From the picture, we can see the diagonal and the two sides of the square form a right triangle, we can employ pythagorean theorem. $$a^2+b^2=c^2$$ where a and $$b$$ are legs of triangles and c is the hypotenuse. Let s be the length of the side of square. We can set up the equation as $$s^2+s^2={\\\\left(s+3\\\\right)}^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad18a-h2","type":"hint","dependencies":["aadff92AppQuad18a-h1"],"title":"Solve For Unknown Using Quadratic Formula","text":"The quadratic formula states $${ax}^2+bx+c=0$$, then $$x=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4ac}\\\\right)}{2a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad18a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-1.24,7.24)"],"dependencies":["aadff92AppQuad18a-h2"],"title":"Solve For Unknown Using Quadratic Formula","text":"$$s^2+s^2={\\\\left(s+3\\\\right)}^2$$ can be simplified as $$9w^2-10w-25=0$$. Solve for $$x$$ using the quadratic formula and round $$x$$ to the nearest hundredth. If there is more than one solution for $$x$$, please enter your answer as (a,b) where $$a<b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad18a-h4","type":"hint","dependencies":["aadff92AppQuad18a-h3"],"title":"Solve For Unknown Using Quadratic Formula","text":"Since the length of the side of a sqaure has to be positive, we can omit the negative answer. Therefore the $$s=7.24$$. The the diagonal of rectangle is $$s+3$$ which is $$7.24+3=10.24$$. We can round it to nearest tenth which gives $$10.2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aadff92AppQuad19","title":"Solve Applications Modeled by Quadratic Equations","body":"In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Solve Applications of Quadratic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aadff92AppQuad19a","stepAnswer":["$$24.5$$"],"problemType":"TextBox","stepTitle":"Gerry plans to place a 25-foot ladder against the side of his house to clean his gutters. The bottom of the ladder will be $$5$$ feet from the house.How far up the side of the house will the ladder reach?","stepBody":"Please enter your answer as a real number without a unit.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$24.5$$","hints":{"DefaultPathway":[{"id":"aadff92AppQuad19a-h1","type":"hint","dependencies":[],"title":"Set Up Equation For Problem","text":"We can employ pythagorean theorem. $$a^2+b^2=c^2$$ where a and $$b$$ are legs of triangles and c is the hypotenuse. The length of ladder will be the hypotenuse. The distance from bottom of ladder to the house and the verticle distance that ladder reach are two legs of the triangle. Let $$x$$ be the verticle distance that the ladder can reach. We can set up the equation as $$5^2+x^2={25}^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad19a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-24.50,24.50)"],"dependencies":["aadff92AppQuad19a-h1"],"title":"Solve For Unknown Using Quadratic Formula","text":"$$5^2+x^2={25}^2$$ Solve for $$x$$ and round $$x$$ to the nearest hundredth. If there is more than one solution for $$x$$, please enter your answer as (a,b) where $$a<b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad19a-h3","type":"hint","dependencies":["aadff92AppQuad19a-h2"],"title":"Solve For Unknown Using Quadratic Formula","text":"Since the length of the side of a triangle has to be positive, we can omit the negative answer. Therefore the height ladder can reach is $$24.5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aadff92AppQuad2","title":"Solve Applications Modeled by Quadratic Equations","body":"In the following exercises, solve using any method.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Solve Applications of Quadratic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aadff92AppQuad2a","stepAnswer":["(-20,-18),(18,20)"],"problemType":"TextBox","stepTitle":"The product of two consecutive even numbers is $$360$$. Find the numbers.","stepBody":"Please enter your answer pair as (a,b) where $$a<b$$. If there is more than one pair of numbers that fulfill the condition, you may enter it as (a,b),(c,d) where $$a<c$$.","answerType":"string","variabilization":{},"answerLatex":"$$(-20,-18),(18,20)$$","hints":{"DefaultPathway":[{"id":"aadff92AppQuad2a-h1","type":"hint","dependencies":[],"title":"Set the Equation","text":"Let $$n$$ be the first even number, and the second consecutive number will be $$n+2$$. The equation will be $$n\\\\left(n+2\\\\right)=360$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad2a-h2","type":"hint","dependencies":["aadff92AppQuad2a-h1"],"title":"Solve the Equation","text":"Solve $$n\\\\left(n+2\\\\right)=360$$. You foil out the equation as $$n^2+2n=360$$. $$n^2+2n-360=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad2a-h3","type":"hint","dependencies":["aadff92AppQuad2a-h2"],"title":"Apply Quadratic Equation to solve for $$n$$.","text":"Given $${ax}^2+bx+c=0$$, $$x=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4ac}\\\\right)}{2a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad2a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-20,18)"],"dependencies":["aadff92AppQuad2a-h3"],"title":"Apply Quadratic Equation to solve for $$n$$.","text":"Apply Quadratic Equation to solve for $$n$$ when $$n^2+2n-360=0$$. Enter your answer as (a,b) where $$a<b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad2a-h5","type":"hint","dependencies":["aadff92AppQuad2a-h4"],"title":"Find Answer Pairs","text":"When the first consecutive number is $$-20$$, the second consecutive number is $$-20+2=-18$$.\\\\nWhen the first consecutive number is $$18$$, the second consecutive number is $$18+2=20$$.\\\\nThe final answer is $$(-20,-18),(18,20)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aadff92AppQuad20","title":"Solve Applications Modeled by Quadratic Equations","body":"In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Solve Applications of Quadratic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aadff92AppQuad20a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"John has a 10-foot piece of rope that he wants to use to support his 8-foot tree. How far from the base of the tree should he secure the rope?","stepBody":"Please enter your answer as a real number without a unit.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"aadff92AppQuad20a-h1","type":"hint","dependencies":[],"title":"Set Up Equation For Problem","text":"We can employ pythagorean theorem. $$a^2+b^2=c^2$$ where a and $$b$$ are legs of triangles and c is the hypotenuse. The length of the rope will be the hypotenuse. The height of the tree and the horizontal distance from the base of the tree to the place he secure the rope are two legs of the triangle. Let $$x$$ be the horizontal distance from the base of the tree to the place he secure the rope . We can set up the equation as $$8^2+x^2={10}^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad20a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-6,6)"],"dependencies":["aadff92AppQuad20a-h1"],"title":"Solve For Unknown Using Quadratic Formula","text":"$$5^2+x^2={25}^2$$ Solve for $$x$$ using quadratic formula. If there is more than one solution for $$x$$, please enter your answer as (a,b) where $$a<b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad20a-h3","type":"hint","dependencies":["aadff92AppQuad20a-h2"],"title":"Solve For Unknown Using Quadratic Formula","text":"Since the length of the side of a triangle has to be positive, we can omit the negative answer. Therefore the final answer is $$6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aadff92AppQuad21","title":"Solve Applications Modeled by Quadratic Equations","body":"In the following exercises, solve using any method. Round your answers to the nearest hundredth, if needed.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Solve Applications of Quadratic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aadff92AppQuad21a","stepAnswer":["(1.97,38.03)"],"problemType":"TextBox","stepTitle":"A firework rocket is shot upward at a rate of $$640$$ $$\\\\frac{ft}{sec}$$. Use the projectile formula $$h$$ $$=$$ $$-16t^2$$ + vt to determine when the height of the firework rocket will be $$1200$$ feet.","stepBody":"Please enter your answer as a real number without a unit. If there is more than one answer, please enter it as (a,b) where $$a<b$$.","answerType":"string","variabilization":{},"answerLatex":"$$(1.97, 38.03)$$","hints":{"DefaultPathway":[{"id":"aadff92AppQuad21a-h1","type":"hint","dependencies":[],"title":"Set Up Equation For Problem","text":"We know that the initial velocity $$v=640$$ and the desire height is $$1200$$. We can set up the equation as $$-16t^2+640t=1200$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad21a-h2","type":"hint","dependencies":["aadff92AppQuad21a-h1"],"title":"Solve For Unknown Using Quadratic Formula","text":"The quadratic formula states $${ax}^2+bx+c=0$$, then $$x=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4ac}\\\\right)}{2a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad21a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(1.97,38.03)"],"dependencies":["aadff92AppQuad21a-h2"],"title":"Solve For Unknown Using Quadratic Formula","text":"$$-16t^2+640t=1200$$ Solve for $$t$$ using the quadratic formula. Round your answer to the nearest hundredth. If there is more than one answer, please enter it as (a,b) where $$a<b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad21a-h4","type":"hint","dependencies":["aadff92AppQuad21a-h3"],"title":"Final Answer","text":"Since both $$1.97$$ and $$38.03$$ are positive, they are valid answer for the problem. Therefore, at $$1.97$$ second and $$38.03$$ second, the height of the firework rocket will be $$1200$$ feet.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aadff92AppQuad22","title":"Solve Applications Modeled by Quadratic Equations","body":"In the following exercises, solve using any method. Round your answers to the nearest hundredth, if needed.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Solve Applications of Quadratic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aadff92AppQuad22a","stepAnswer":["(2.16,11.59)"],"problemType":"TextBox","stepTitle":"An arrow is shot vertically upward at a rate of $$220$$ feet per second. Use the projectile formula $$h$$ $$=$$ $$-16t^2$$ + vt, to determine when height of the arrow will be $$400$$ feet.","stepBody":"Please enter your answer as a real number without a unit. If there is more than one answer, please enter it as (a,b) where $$a<b$$.","answerType":"string","variabilization":{},"answerLatex":"$$(2.16, 11.59)$$","hints":{"DefaultPathway":[{"id":"aadff92AppQuad22a-h1","type":"hint","dependencies":[],"title":"Set Up Equation For Problem","text":"We know that the initial velocity $$v=220$$ and the desire height is $$400$$. We can set up the equation as $$-16t^2+220t=400$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad22a-h2","type":"hint","dependencies":["aadff92AppQuad22a-h1"],"title":"Solve For Unknown Using Quadratic Formula","text":"The quadratic formula states $${ax}^2+bx+c=0$$, then $$x=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4ac}\\\\right)}{2a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad22a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(2.16,11.59)"],"dependencies":["aadff92AppQuad22a-h2"],"title":"Solve For Unknown Using Quadratic Formula","text":"$$-16t^2+220t=400$$ Solve for $$t$$ using the quadratic formula. Round your answer to the nearest hundredth. If there is more than one answer, please enter it as (a,b) where $$a<b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad22a-h4","type":"hint","dependencies":["aadff92AppQuad22a-h3"],"title":"Final Answer","text":"Since both $$2.16$$ and $$11.59$$ are positive, they are valid answer for the problem. Therefore, at $$2.16$$ second and $$11.59$$ second, the height of the arrow will be $$400$$ feet.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aadff92AppQuad23","title":"Solve Applications Modeled by Quadratic Equations","body":"In the following exercises, solve using any method. Round your answers to the nearest hundredth, if needed.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Solve Applications of Quadratic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aadff92AppQuad23a","stepAnswer":["$$70.01$$"],"problemType":"TextBox","stepTitle":"A bullet is fired straight up from a BB gun with initial velocity $$1120$$ feet per second at an initial height of $$8$$ feet. Use the formula $$h$$ $$=$$ $$-16t^2$$ + vt + $$8$$ to determine how many seconds it will take for the bullet to hit the ground. (That is, when will $$h$$ $$=$$ 0?)","stepBody":"Please enter your answer as a real number without a unit. If there is more than one answer, please enter it as (a,b) where $$a<b$$.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$70.01$$","hints":{"DefaultPathway":[{"id":"aadff92AppQuad23a-h1","type":"hint","dependencies":[],"title":"Set Up Equation For Problem","text":"We know that the initial velocity $$v=1120$$ and the desire height is $$0$$. We can set up the equation as $$-16t^2+1120t+8=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad23a-h2","type":"hint","dependencies":["aadff92AppQuad23a-h1"],"title":"Solve For Unknown Using Quadratic Formula","text":"The quadratic formula states $${ax}^2+bx+c=0$$, then $$x=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4ac}\\\\right)}{2a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad23a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-0.01,70.01)"],"dependencies":["aadff92AppQuad23a-h2"],"title":"Solve For Unknown Using Quadratic Formula","text":"$$-16t^2+1120t+8=0$$. Solve for $$t$$ using the quadratic formula. Round your answer to the nearest hundredth. If there is more than one answer, please enter it as (a,b) where $$a<b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad23a-h4","type":"hint","dependencies":["aadff92AppQuad23a-h3"],"title":"Final Answer","text":"Since time can not be negative, $$70.01$$ is the only valid answer for the problem. Therefore, at $$70.01$$ second, the bullet hits the ground.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aadff92AppQuad24","title":"Solve Applications Modeled by Quadratic Equations","body":"In the following exercises, solve using any method. Round your answers to the nearest hundredth, if needed.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Solve Applications of Quadratic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aadff92AppQuad24a","stepAnswer":["$$3.5$$"],"problemType":"TextBox","stepTitle":"A stone is dropped from a 196-foot platform. Use the formula $$h$$ $$=-16t^2$$ + vt + $$196$$ to determine how many seconds it will take for the stone to hit the ground. (Since the stone is dropped, $$v=$$ 0.)","stepBody":"Please enter your answer as a real number without a unit. If there is more than one answer, please enter it as (a,b) where $$a<b$$.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3.5$$","hints":{"DefaultPathway":[{"id":"aadff92AppQuad24a-h1","type":"hint","dependencies":[],"title":"Set Up Equation For Problem","text":"We know that the velocity is $$0$$ and the desire height is $$0$$. We can set up the equation as $$0$$ =-16t**2+ $$196$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad24a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-3.5,3.5)"],"dependencies":["aadff92AppQuad24a-h1"],"title":"Solve For Unknown Using Quadratic Formula","text":"$$0$$ =-16t**2+ $$196$$. Solve for $$t$$ and round your answer to the nearest hundredth, if needed. If there is more than one answer, please enter it as (a,b) where $$a<b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad24a-h3","type":"hint","dependencies":["aadff92AppQuad24a-h2"],"title":"Final Answer","text":"Since time can not be negative, $$3.5$$ is the only valid answer for the problem. Therefore, at $$3.5$$ second, the stone hit the ground.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aadff92AppQuad25","title":"Solve Applications Modeled by Quadratic Equations","body":"In the following exercises, solve using any method. Round your answers to the nearest hundredth, if needed.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Solve Applications of Quadratic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aadff92AppQuad25a","stepAnswer":["$$49$$"],"problemType":"TextBox","stepTitle":"The businessman took a small airplane for a quick flight up the coast for a lunch meeting and then returned home. The plane flew a total of $$4$$ hours and each way the trip was $$200$$ miles. What was the speed of the wind that affected the plane which was flying at a speed of $$120$$ mph?","stepBody":"Please enter your answer as a real number without the unit. Round it to the nearest tenth if needed.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$49$$","hints":{"DefaultPathway":[{"id":"aadff92AppQuad25a-h1","type":"hint","dependencies":[],"title":"Interpret the Conditions","text":"When the plane flies with the wind, the wind increases its speed and so the rate is $$120$$ + $$r$$.\\\\nWhen the plane flies against the wind, the wind decreases its speed and the rate is $$120$$ - $$r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad25a-h2","type":"hint","dependencies":["aadff92AppQuad25a-h1"],"title":"Set Up Equation For Problem","text":"$$Distance=Rate Time$$. We can solve for Time and get $$Time=\\\\frac{Distance}{Rate}$$. When flies with the wind, it takes $$\\\\frac{200}{120+r}$$. When flies against the wind, it takes $$\\\\frac{200}{120-r}$$. In total it takes $$4$$ hours. We can set up the equation as $$\\\\frac{200}{120+r}+\\\\frac{200}{120-r}=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad25a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-49,49)"],"dependencies":["aadff92AppQuad25a-h2"],"title":"Solve For Unknown Using Quadratic Formula","text":"$$\\\\frac{200}{120+r}+\\\\frac{200}{120-r}=4$$. Multiply both sides by the LCD which is $$\\\\left(120+r\\\\right) \\\\left(120-r\\\\right)$$. It turns the equation into $$200\\\\left(120-r\\\\right)+200\\\\left(120+r\\\\right)=4\\\\left(120-r\\\\right) \\\\left(120+r\\\\right)$$. We can further simplify it into $$4r^2=9600$$. Solve for $$r$$ and round it to the nearest tenth. If there is more than one answer, please enter it as (a,b) where $$a<b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad25a-h4","type":"hint","dependencies":["aadff92AppQuad25a-h3"],"title":"Final Answer","text":"Since the wind speed can not be negative, we omit the negative answer. So the speed of the wind was $$49$$ mph.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aadff92AppQuad26","title":"Solve Applications Modeled by Quadratic Equations","body":"In the following exercises, solve using any method. Round your answers to the nearest hundredth, if needed.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Solve Applications of Quadratic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aadff92AppQuad26a","stepAnswer":["$$25$$"],"problemType":"TextBox","stepTitle":"The couple took a small airplane for a quick flight up to the wine country for a romantic dinner and then returned home. The plane flew a total of $$5$$ hours and each way the trip was $$300$$ miles. If the plane was flying at $$125$$ mph, what was the speed of the wind that affected the plane?","stepBody":"Please enter your answer as a real number without the unit.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$25$$","hints":{"DefaultPathway":[{"id":"aadff92AppQuad26a-h1","type":"hint","dependencies":[],"title":"Interpret the Conditions","text":"When the plane flies with the wind, the wind increases its speed and so the rate is $$125$$ + $$r$$.\\\\nWhen the plane flies against the wind, the wind decreases its speed and the rate is $$125$$ - $$r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad26a-h2","type":"hint","dependencies":["aadff92AppQuad26a-h1"],"title":"Set Up Equation For Problem","text":"$$Distance=Rate Time$$. We can solve for Time and get $$Time=\\\\frac{Distance}{Rate}$$. When flies with the wind, it takes $$\\\\frac{300}{125+r}$$. When flies against the wind, it takes $$\\\\frac{300}{125-r}$$. In total it takes $$5$$ hours. We can set up the equation as $$\\\\frac{300}{125+r}+\\\\frac{300}{125-r}=5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad26a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-25,25)"],"dependencies":["aadff92AppQuad26a-h2"],"title":"Solve For Unknown Using Quadratic Formula","text":"$$\\\\frac{300}{125+r}+\\\\frac{300}{125-r}=5$$. Multiply both sides by the LCD which is $$\\\\left(125+r\\\\right) \\\\left(125-r\\\\right)$$. It turns the equation into $$300\\\\left(125-r\\\\right)+300\\\\left(125+r\\\\right)=5\\\\left(125-r\\\\right) \\\\left(125+r\\\\right)$$. We can further simplify it into $$5r^2=3125$$. Solve for $$r$$. If there is more than one answer, please enter it as (a,b) where $$a<b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad26a-h4","type":"hint","dependencies":["aadff92AppQuad26a-h3"],"title":"Final Answer","text":"Since the wind speed can not be negative, we omit the negative answer. So the speed of the wind was $$25$$ mph.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aadff92AppQuad27","title":"Solve Applications Modeled by Quadratic Equations","body":"In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Solve Applications of Quadratic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aadff92AppQuad27a","stepAnswer":["$$4.3$$"],"problemType":"TextBox","stepTitle":"Roy kayaked up the river and then back in a total time of $$6$$ hours. The trip was $$4$$ miles each way and the current was difficult. If Roy kayaked at a speed of $$5$$ mph, what was the speed of the current?","stepBody":"Please enter your answer as a real number without the unit round your answer to the nearest tenth if applicable.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4.3$$","hints":{"DefaultPathway":[{"id":"aadff92AppQuad27a-h1","type":"hint","dependencies":[],"title":"Interpret the Conditions","text":"When Roy kayaked along the current, the current increases the speed and so the rate is $$5$$ + $$r$$.\\\\nWhen Roy kayaked against the current, the current decreases the speed and so the rate is $$5$$ - $$r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad27a-h2","type":"hint","dependencies":["aadff92AppQuad27a-h1"],"title":"Set Up Equation For Problem","text":"$$Distance=Rate Time$$. We can solve for Time and get $$Time=\\\\frac{Distance}{Rate}$$. When Roy kayaked along the current, it takes $$\\\\frac{4}{5+r}$$. When Roy kayaked against the current, it takes $$\\\\frac{4}{5-r}$$. In total it takes $$6$$ hours. We can set up the equation as $$\\\\frac{4}{5+r}+\\\\frac{4}{5-r}=6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad27a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-4.3,4.3)"],"dependencies":["aadff92AppQuad27a-h2"],"title":"Solve For Unknown Using Quadratic Formula","text":"$$\\\\frac{4}{5+r}+\\\\frac{4}{5-r}=6$$. Multiply both sides by the LCD which is $$\\\\left(5+r\\\\right) \\\\left(5-r\\\\right)$$. It turns the equation into $$4\\\\left(5-r\\\\right)+4\\\\left(5+r\\\\right)=6\\\\left(25-r^2\\\\right)$$. We can further simplify it into $$6r^2=110$$. Solve for $$r$$. If there is more than one answer, please enter it as (a,b) where $$a<b$$. Please round your answer to the nearest tenth, if applicable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad27a-h4","type":"hint","dependencies":["aadff92AppQuad27a-h3"],"title":"Final Answer","text":"Since the current speed can not be negative, we omit the negative answer. So the speed of the current was $$4.3$$ mph.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aadff92AppQuad28","title":"Solve Applications Modeled by Quadratic Equations","body":"In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Solve Applications of Quadratic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aadff92AppQuad28a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"Rick paddled up the river, spent the night camping, and and then paddled back. He spent $$10$$ hours paddling and the campground was $$24$$ miles away. If Rick kayaked at a speed of $$5$$ mph, what was the speed of the current?","stepBody":"Please enter your answer as a real number without the unit round your answer to the nearest tenth if applicable.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"aadff92AppQuad28a-h1","type":"hint","dependencies":[],"title":"Interpret the Conditions","text":"When Rick kayaked along the current, the current increases the speed and so the rate is $$5$$ + $$r$$.\\\\nWhen Rick kayaked against the current, the current decreases the speed and so the rate is $$5$$ - $$r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad28a-h2","type":"hint","dependencies":["aadff92AppQuad28a-h1"],"title":"Set Up Equation For Problem","text":"$$Distance=Rate Time$$. We can solve for Time and get $$Time=\\\\frac{Distance}{Rate}$$. When Rick kayaked along the current, it takes $$\\\\frac{24}{5+r}$$. When Roy kayaked against the current, it takes $$\\\\frac{24}{5-r}$$. In total it takes $$10$$ hours. We can set up the equation as $$\\\\frac{24}{5+r}+\\\\frac{24}{5-r}=10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad28a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-1,1)"],"dependencies":["aadff92AppQuad28a-h2"],"title":"Solve For Unknown Using Quadratic Formula","text":"$$\\\\frac{24}{5+r}+\\\\frac{24}{5-r}=10$$. Multiply both sides by the LCD which is $$\\\\left(5+r\\\\right) \\\\left(5-r\\\\right)$$. It turns the equation into $$24\\\\left(5-r\\\\right)+24\\\\left(5+r\\\\right)=10\\\\left(25-r^2\\\\right)$$. We can further simplify it into $$6r^2=110$$. Solve for $$r$$. If there is more than one answer, please enter it as (a,b) where $$a<b$$. Please round your answer to the nearest tenth, if applicable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad28a-h4","type":"hint","dependencies":["aadff92AppQuad28a-h3"],"title":"Final Answer","text":"Since the current speed can not be negative, we omit the negative answer. So the speed of the current was $$1$$ mph.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aadff92AppQuad29","title":"Solve Applications Modeled by Quadratic Equations","body":"In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Solve Applications of Quadratic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aadff92AppQuad29a","stepAnswer":["(3,6)"],"problemType":"TextBox","stepTitle":"Two painters can paint a room in $$2$$ hours if they work together. The less experienced painter takes $$3$$ hours more than the more experienced painter to finish the job. How long does it take for each painter to paint the room individually?","stepBody":"Please enter your answer as (a,b) where $$a<b$$.","answerType":"string","variabilization":{},"answerLatex":"$$(3,6)$$","hints":{"DefaultPathway":[{"id":"aadff92AppQuad29a-h1","type":"hint","dependencies":[],"title":"Interpret the Conditions","text":"Let $$x=$$ the number of hours that the more experienced painter takes to paint the room. Then $$x+3$$ will be the number of hours that the less experienced painter takes to paint the room. In one hour, the less experienced painter gets $$\\\\frac{1}{x+3}$$ part of the job done and the more experienced painter gets $$\\\\frac{1}{x}$$ part of the job done. Together, in one hour, if both painters work together, they will get $$\\\\frac{1}{x}+\\\\frac{1}{x+3}$$ part of the job done.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad29a-h2","type":"hint","dependencies":["aadff92AppQuad29a-h1"],"title":"Set Up Equation For Problem","text":"We know that, in one hour, if both the painters work together they will get $$\\\\frac{1}{x}+\\\\frac{1}{x+3}$$ part of the job done. Given that if both painters work together, it takes $$2$$ hours to paint the room. In other words, they get $$\\\\frac{1}{2}$$ the job done in one hour. We can set the equation as $$\\\\frac{1}{x}+\\\\frac{1}{x+3}=\\\\frac{1}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad29a-h3","type":"hint","dependencies":["aadff92AppQuad29a-h2"],"title":"Solve for Unknown Using Quadratic Formula","text":"$$\\\\frac{1}{x}+\\\\frac{1}{x+3}=\\\\frac{1}{2}$$ Solve for $$x$$. We can multiply both sides by the LCD which is $$x \\\\left(x+3\\\\right)$$. We get $$x+3+x=\\\\frac{1}{2} x \\\\left(x+3\\\\right)$$. We can futher simplify it into $$\\\\frac{1}{2} x^2-\\\\frac{1}{2} x-3=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad29a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-2,3)"],"dependencies":["aadff92AppQuad29a-h3"],"title":"Solve for Unknown Using Quadratic Formula","text":"The quadratic formula states $${ax}^2+bx+c=0$$, then $$x=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4ac}\\\\right)}{2a}$$. Given $$\\\\frac{1}{2} x^2-\\\\frac{1}{2} x-3=0$$, solve for $$x$$. If there are two answers, please enter them as (a,b) where $$a<b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad29a-h5","type":"hint","dependencies":["aadff92AppQuad29a-h4"],"title":"Final Answer","text":"Time can not be negative, so we can omit the negative answer. Therefore, the more experienced painter takes $$3$$ hours to get the job done alone. The less experienced painter takes $$3+3=6$$ hours to get the job done alone.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aadff92AppQuad3","title":"Solve Applications Modeled by Quadratic Equations","body":"In the following exercises, solve using any method.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Solve Applications of Quadratic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aadff92AppQuad3a","stepAnswer":["(-26,-24),(24,26)"],"problemType":"TextBox","stepTitle":"The product of two consecutive even numbers is $$624$$. Find the numbers.","stepBody":"Please enter your answer pair as (a,b) where $$a<b$$. If there is more than one pair of numbers that fulfill the condition, you may enter it as (a,b),(c,d) where $$a<c$$.","answerType":"string","variabilization":{},"answerLatex":"$$(-26,-24),(24,26)$$","hints":{"DefaultPathway":[{"id":"aadff92AppQuad3a-h1","type":"hint","dependencies":[],"title":"Set the Equation","text":"Let $$n$$ be the first even number, and the second consecutive number will be $$n+2$$. The equation will be $$n\\\\left(n+2\\\\right)=624$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad3a-h2","type":"hint","dependencies":["aadff92AppQuad3a-h1"],"title":"Solve the Equation","text":"Solve $$n\\\\left(n+2\\\\right)=624$$. You foil out the equation as $$n^2+2n=624$$. $$n^2+2n-624=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad3a-h3","type":"hint","dependencies":["aadff92AppQuad3a-h2"],"title":"Apply Quadratic Equation to solve for $$n$$.","text":"Given $${ax}^2+bx+c=0$$, $$x=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4ac}\\\\right)}{2a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad3a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-26,24)"],"dependencies":["aadff92AppQuad3a-h3"],"title":"Apply Quadratic Equation to solve for $$n$$.","text":"Apply Quadratic Equation to solve for $$n$$ when $$n^2+2n-624=0$$. Enter your answer as (a,b) where $$a<b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad3a-h5","type":"hint","dependencies":["aadff92AppQuad3a-h4"],"title":"Find Answer Pairs","text":"When the first consecutive number is $$-26$$, the second consecutive number is $$-26+2=-24$$.\\\\nWhen the first consecutive number is $$24$$, the second consecutive number is $$24+2=26$$.\\\\nThe final answer is $$(-26,-24),(24,26)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aadff92AppQuad30","title":"Solve Applications Modeled by Quadratic Equations","body":"In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Solve Applications of Quadratic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aadff92AppQuad30a","stepAnswer":["(12,24)"],"problemType":"TextBox","stepTitle":"Two gardeners can do the weekly yard maintenance in $$8$$ minutes if they work together. The older gardener takes $$12$$ minutes more than the younger gardener to finish the job by himself. How long does it take for each gardener to do the weekly yard maintainence individually?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(12,24)$$","hints":{"DefaultPathway":[{"id":"aadff92AppQuad30a-h1","type":"hint","dependencies":[],"title":"Interpret the Conditions","text":"Let $$x=$$ the minutes that the younger gardener takes to get the job done alone. Then $$x+12$$ will be the number of minutes that the older gardener takes to get the job done alone. In one hour, the younger gardener gets $$\\\\frac{1}{x}$$ part of the job done and the older gardener gets $$\\\\frac{1}{12+x}$$ part of the job done. Together, in one minute, if both gardeners work together, they will get $$\\\\frac{1}{x}+\\\\frac{1}{x+12}$$ part of the job done.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad30a-h2","type":"hint","dependencies":["aadff92AppQuad30a-h1"],"title":"Set Up Equation For Problem","text":"We know that, in one minute, if both gardeners work together, they will get $$\\\\frac{1}{x}+\\\\frac{1}{x+12}$$ part of the job done. Given that if both painters work together, it takes $$8$$ minutes to finish the job. In other words, they get $$\\\\frac{1}{8}$$ the job done in one minute. We can set the equation as $$\\\\frac{1}{x}+\\\\frac{1}{x+12}=\\\\frac{1}{8}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad30a-h3","type":"hint","dependencies":["aadff92AppQuad30a-h2"],"title":"Solve for Unknown Using Quadratic Formula","text":"$$\\\\frac{1}{x}+\\\\frac{1}{x+12}=\\\\frac{1}{8}$$ Solve for $$x$$. We can multiply both sides by the LCD which is $$x \\\\left(x+12\\\\right)$$. We get $$x+12+x=\\\\frac{1}{8} x \\\\left(x+12\\\\right)$$. We can futher simplify it into $$\\\\frac{1}{8} x^2-\\\\frac{1}{2} x-12=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad30a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-8,12)"],"dependencies":["aadff92AppQuad30a-h3"],"title":"Solve for Unknown Using Quadratic Formula","text":"The quadratic formula states $${ax}^2+bx+c=0$$, then $$x=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4ac}\\\\right)}{2a}$$. Given $$\\\\frac{1}{8} x^2-\\\\frac{1}{2} x-12=0$$, solve for $$x$$. If there are two answers, please enter them as (a,b) where $$a<b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad30a-h5","type":"hint","dependencies":["aadff92AppQuad30a-h4"],"title":"Final Answer","text":"Time can not be negative, so we can omit the negative answer. Therefore, the younger gardener takes $$12$$ minutes to get the job done alone. The older gardener takes $$12+12=24$$ minutes to get the job done alone.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aadff92AppQuad4","title":"Solve Applications Modeled by Quadratic Equations","body":"In the following exercises, solve using any method.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Solve Applications of Quadratic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aadff92AppQuad4a","stepAnswer":["(-33,-31),(31,33)"],"problemType":"TextBox","stepTitle":"The product of two consecutive odd numbers is 1,023. Find the numbers.","stepBody":"Please enter your answer pair as (a,b) where $$a<b$$. If there is more than one pair of numbers that fulfill the condition, you may enter it as (a,b),(c,d) where $$a<c$$.","answerType":"string","variabilization":{},"answerLatex":"$$(-33,-31),(31,33)$$","hints":{"DefaultPathway":[{"id":"aadff92AppQuad4a-h1","type":"hint","dependencies":[],"title":"Set the Equation","text":"Let $$n$$ be the first odd number, and the second consecutive number will be $$n+2$$. The equation will be $$n\\\\left(n+2\\\\right)=1023$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad4a-h2","type":"hint","dependencies":["aadff92AppQuad4a-h1"],"title":"Solve the Equation","text":"Solve $$n\\\\left(n+2\\\\right)=1023$$. You foil out the equation as $$n^2+2n=1023$$. $$n^2+2n-1023=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad4a-h3","type":"hint","dependencies":["aadff92AppQuad4a-h2"],"title":"Apply Quadratic Equation to solve for $$n$$.","text":"Given $${ax}^2+bx+c=0$$, $$x=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4ac}\\\\right)}{2a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad4a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-33,31)"],"dependencies":["aadff92AppQuad4a-h3"],"title":"Apply Quadratic Equation to solve for $$n$$.","text":"Apply Quadratic Equation to solve for $$n$$ when $$n^2+2n-1023=0$$. Enter your answer as (a,b) where $$a<b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad4a-h5","type":"hint","dependencies":["aadff92AppQuad4a-h4"],"title":"Find Answer Pairs","text":"When the first consecutive number is $$-33$$, the second consecutive number is $$-33+2=-31$$.\\\\nWhen the first consecutive number is $$31$$, the second consecutive number is $$31+2=33$$.\\\\nThe final answer is $$(-33,-31),(31,33)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aadff92AppQuad5","title":"Solve Applications Modeled by Quadratic Equations","body":"In the following exercises, solve using any method.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Solve Applications of Quadratic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aadff92AppQuad5a","stepAnswer":["(-23,-21),(21,23)"],"problemType":"TextBox","stepTitle":"The product of two consecutive odd numbers is $$483$$. Find the numbers.","stepBody":"Please enter your answer pair as (a,b) where $$a<b$$. If there is more than one pair of numbers that fulfill the condition, you may enter it as (a,b),(c,d) where $$a<c$$.","answerType":"string","variabilization":{},"answerLatex":"$$(-23,-21),(21,23)$$","hints":{"DefaultPathway":[{"id":"aadff92AppQuad5a-h1","type":"hint","dependencies":[],"title":"Set the Equation","text":"Let $$n$$ be the first odd number, and the second consecutive number will be $$n+2$$. The equation will be $$n\\\\left(n+2\\\\right)=483$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad5a-h2","type":"hint","dependencies":["aadff92AppQuad5a-h1"],"title":"Solve the Equation","text":"Solve $$n\\\\left(n+2\\\\right)=483$$. You foil out the equation as $$n^2+2n=483$$. $$n^2+2n-483=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad5a-h3","type":"hint","dependencies":["aadff92AppQuad5a-h2"],"title":"Apply Quadratic Equation to solve for $$n$$.","text":"Given $${ax}^2+bx+c=0$$, $$x=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4ac}\\\\right)}{2a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad5a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-23,21)"],"dependencies":["aadff92AppQuad5a-h3"],"title":"Apply Quadratic Equation to solve for $$n$$.","text":"Apply Quadratic Equation to solve for $$n$$ when $$n^2+2n-483=0$$. Enter your answer as (a,b) where $$a<b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad5a-h5","type":"hint","dependencies":["aadff92AppQuad5a-h4"],"title":"Find Answer Pairs","text":"When the first consecutive number is $$-23$$, the second consecutive number is $$-23+2=-21$$.\\\\nWhen the first consecutive number is $$21$$, the second consecutive number is $$21+2=23$$.\\\\nThe final answer is $$(-23,-21),(21,23)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aadff92AppQuad6","title":"Solve Applications Modeled by Quadratic Equations","body":"In the following exercises, solve using any method.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Solve Applications of Quadratic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aadff92AppQuad6a","stepAnswer":["(-24,-22),(22,24)"],"problemType":"TextBox","stepTitle":"The product of two consecutive even numbers is $$528$$. Find the numbers.","stepBody":"Please enter your answer pair as (a,b) where $$a<b$$. If there is more than one pair of numbers that fulfill the condition, you may enter it as (a,b),(c,d) where $$a<c$$.","answerType":"string","variabilization":{},"answerLatex":"$$(-24,-22),(22,24)$$","hints":{"DefaultPathway":[{"id":"aadff92AppQuad6a-h1","type":"hint","dependencies":[],"title":"Set the Equation","text":"Let $$n$$ be the first even number, and the second consecutive number will be $$n+2$$. The equation will be $$n\\\\left(n+2\\\\right)=528$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad6a-h2","type":"hint","dependencies":["aadff92AppQuad6a-h1"],"title":"Solve the Equation","text":"Solve $$n\\\\left(n+2\\\\right)=528$$. You foil out the equation as $$n^2+2n=528$$. $$n^2+2n-528=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad6a-h3","type":"hint","dependencies":["aadff92AppQuad6a-h2"],"title":"Apply Quadratic Equation to solve for $$n$$.","text":"Given $${ax}^2+bx+c=0$$, $$x=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4ac}\\\\right)}{2a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad6a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-24,22)"],"dependencies":["aadff92AppQuad6a-h3"],"title":"Apply Quadratic Equation to solve for $$n$$.","text":"Apply Quadratic Equation to solve for $$n$$ when $$n^2+2n-528=0$$. Enter your answer as (a,b) where $$a<b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad6a-h5","type":"hint","dependencies":["aadff92AppQuad6a-h4"],"title":"Find Answer Pairs","text":"When the first consecutive number is $$-24$$, the second consecutive number is $$-24+2=-22$$.\\\\nWhen the first consecutive number is $$22$$, the second consecutive number is $$22+2=24$$.\\\\nThe final answer is $$(-24,-22),(22,24)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aadff92AppQuad7","title":"Solve Applications Modeled by Quadratic Equations","body":"In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Solve Applications of Quadratic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aadff92AppQuad7a","stepAnswer":["(5,18)"],"problemType":"TextBox","stepTitle":"A triangle with area $$45$$ square inches has a height that is two less than four times the base. Find the base and height of the triangle.","stepBody":"Please enter your answer as (a,b) where a is the width of triangle and $$b$$ is height of triangle.","answerType":"string","variabilization":{},"answerLatex":"$$(5,18)$$","hints":{"DefaultPathway":[{"id":"aadff92AppQuad7a-h1","type":"hint","dependencies":[],"title":"Set Up Equation For Problem","text":"Let the bases of the triangle be $$b$$, we can write the height of the triangle in terms of $$b$$ as $$4b-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad7a-h2","type":"hint","dependencies":["aadff92AppQuad7a-h1"],"title":"Equation For Triangle Area","text":"Area of $$triangle=\\\\frac{1}{2} base height$$. We can set up the equation using given condition as $$45=\\\\frac{1}{2} b \\\\left(4b-2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad7a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-4.5,5)"],"dependencies":["aadff92AppQuad7a-h2"],"title":"Solve For Unknown Using Quadratic Formula","text":"The quadratic formula states $${ax}^2+bx+c=0$$, then $$x=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4ac}\\\\right)}{2a}$$. Given $$45=\\\\frac{1}{2} b \\\\left(4b-2\\\\right)$$, solve for $$b$$. Please enter your answer as (c,d) where $$c<d$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad7a-h4","type":"hint","dependencies":["aadff92AppQuad7a-h3"],"title":"Solve For Unknown Using Quadratic Formula","text":"Given that base of a triangle has to be positive, we can omit the negative answer. Therefore, the base of given triangle is $$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad7a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18$$"],"dependencies":["aadff92AppQuad7a-h4"],"title":"Find the Height of Triangle","text":"Given that the base is $$5$$ and the height is $$2$$ less than $$4$$ times the base, what is the height?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aadff92AppQuad8","title":"Solve Applications Modeled by Quadratic Equations","body":"In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Solve Applications of Quadratic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aadff92AppQuad8a","stepAnswer":["(22,8)"],"problemType":"TextBox","stepTitle":"The base of a triangle is six more than twice the height. The area of the triangle is $$88$$ square yards. Find the base and height of the triangle.","stepBody":"Please enter your answer as (a,b) where a is the base of triangle and $$b$$ is height of triangle.","answerType":"string","variabilization":{},"answerLatex":"$$(22,8)$$","hints":{"DefaultPathway":[{"id":"aadff92AppQuad8a-h1","type":"hint","dependencies":[],"title":"Set Up Equation For Problem","text":"Let the height of the triangle be $$h$$, we can write the base of the triangle in terms of $$h$$ as $$6+2h$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad8a-h2","type":"hint","dependencies":["aadff92AppQuad8a-h1"],"title":"Equation For Triangle Area","text":"Area of $$triangle=\\\\frac{1}{2} base height$$. We can set up the equation using given condition as $$88=\\\\frac{1}{2} h \\\\left(6+2h\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad8a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-11,8)"],"dependencies":["aadff92AppQuad8a-h2"],"title":"Solve For Unknown Using Quadratic Formula","text":"The quadratic formula states $${ax}^2+bx+c=0$$, then $$x=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4ac}\\\\right)}{2a}$$. Given $$\\\\frac{1}{2} h \\\\left(6+2h\\\\right)=88$$, solve for $$h$$. Please enter your answer as (c,d) where $$c<d$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad8a-h4","type":"hint","dependencies":["aadff92AppQuad8a-h3"],"title":"Solve For Unknown Using Quadratic Formula","text":"Given that height of a triangle has to be positive, we can omit the negative answer. Therefore, the height of given triangle is $$8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad8a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$22$$"],"dependencies":["aadff92AppQuad8a-h4"],"title":"Find the Height of Triangle","text":"Given that the height is $$8$$ and base of a triangle is six more than twice the height, what is the base?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aadff92AppQuad9","title":"Solve Applications Modeled by Quadratic Equations","body":"In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Solve Applications of Quadratic Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aadff92AppQuad9a","stepAnswer":["(24,10)"],"problemType":"TextBox","stepTitle":"The area of a triangular flower bed in the park has an area of $$120$$ square feet. The base is $$4$$ feet longer that twice the height. What are the base and height of the triangle?","stepBody":"Please enter your answer as (a,b) where a is the base of triangle and $$b$$ is height of triangle.","answerType":"string","variabilization":{},"answerLatex":"$$(24,10)$$","hints":{"DefaultPathway":[{"id":"aadff92AppQuad9a-h1","type":"hint","dependencies":[],"title":"Set Up Equation For Problem","text":"Let the height of the triangle be $$h$$, we can write the base of the triangle in terms of $$h$$ as $$4+2h$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad9a-h2","type":"hint","dependencies":["aadff92AppQuad9a-h1"],"title":"Equation For Triangle Area","text":"Area of $$triangle=\\\\frac{1}{2} base height$$. We can set up the equation using given condition as $$120=\\\\frac{1}{2} h \\\\left(4+2h\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad9a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["(-12,10)"],"dependencies":["aadff92AppQuad9a-h2"],"title":"Solve For Unknown Using Quadratic Formula","text":"The quadratic formula states $${ax}^2+bx+c=0$$, then $$x=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4ac}\\\\right)}{2a}$$. Given $$\\\\frac{1}{2} h \\\\left(4+2h\\\\right)=120$$, solve for $$h$$. Please enter your answer as (c,d) where $$c<d$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad9a-h4","type":"hint","dependencies":["aadff92AppQuad9a-h3"],"title":"Solve For Unknown Using Quadratic Formula","text":"Given that height of a triangle has to be positive, we can omit the negative answer. Therefore, the height of given triangle is $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aadff92AppQuad9a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$24$$"],"dependencies":["aadff92AppQuad9a-h4"],"title":"Find the Height of Triangle","text":"Given that the height is $$10$$ and base of a triangle is $$4$$ more than twice the height, what is the base?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aae6efeSolving1","title":"Use a Problem Solving Strategy for Word Problems","body":"Solve the following word problem","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Use a Problem Solving Strategy","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aae6efeSolving1a","stepAnswer":["$$25$$"],"problemType":"TextBox","stepTitle":"Normal yearly snowfall at the local ski resort is $$12$$ inches more than twice the amount it received last season. The normal yearly snowfall is $$62$$ inches. What was the snowfall last season at the ski resort?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$25$$","hints":{"DefaultPathway":[{"id":"aae6efeSolving1a-h1","type":"hint","dependencies":[],"title":"Setup","text":"Assume the snowfall last season is $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving1a-h2","type":"hint","dependencies":["aae6efeSolving1a-h1"],"title":"Setup","text":"Write an equation based on the question: $$2x+12=62$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving1a-h3","type":"hint","dependencies":["aae6efeSolving1a-h2"],"title":"Organizing","text":"Rewrite the equation as $$2x=62-12$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving1a-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$50$$"],"dependencies":["aae6efeSolving1a-h3"],"title":"Subtraction","text":"What is $$62-12$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving1a-h4","type":"hint","dependencies":["aae6efeSolving1a-h3"],"title":"Division","text":"Dividing the coefficient on $$x$$ to get the answer","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aae6efeSolving10","title":"Solve Number Word Problems","body":"Solve the following word problem","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Use a Problem Solving Strategy","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aae6efeSolving10a","stepAnswer":["$$-31, -32, -33$$"],"problemType":"MultipleChoice","stepTitle":"Find three consecutive integers whose sum is $$-96$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$-31, -32, -33$$","$$-32, -33, -34$$","$$-33, -34, -35$$"],"hints":{"DefaultPathway":[{"id":"aae6efeSolving10a-h1","type":"hint","dependencies":[],"title":"Setup","text":"Assume numbers are $$x$$, $$x+1$$, $$x+2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving10a-h2","type":"hint","dependencies":["aae6efeSolving10a-h1"],"title":"Setup","text":"Write an equation based on the condition: $$x+x+1+x+2=-96$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving10a-h3","type":"hint","dependencies":["aae6efeSolving10a-h2"],"title":"Organizing","text":"Rewrite the equation as $$3x=-96-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving10a-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-99$$"],"dependencies":["aae6efeSolving10a-h3"],"title":"Subtraction","text":"What is $$-96-3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving10a-h4","type":"hint","dependencies":["aae6efeSolving10a-h3"],"title":"Division","text":"Dividing the coefficient on $$x$$ to get the answer","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aae6efeSolving11","title":"Solve Number Word Problems","body":"Solve the following word problem","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Use a Problem Solving Strategy","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aae6efeSolving11a","stepAnswer":["32,34,36"],"problemType":"MultipleChoice","stepTitle":"Find three consecutive even integers whose sum is $$102$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["30,32,34","32,34,36",",34,36,38"],"hints":{"DefaultPathway":[{"id":"aae6efeSolving11a-h1","type":"hint","dependencies":[],"title":"Setup","text":"Assume numbers are $$x$$, $$x+2$$, $$x+4$$ (x is an even integer)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving11a-h2","type":"hint","dependencies":["aae6efeSolving11a-h1"],"title":"Setup","text":"Write an equation based on the condition: $$x+x+2+x+4=102$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving11a-h3","type":"hint","dependencies":["aae6efeSolving11a-h2"],"title":"Organizing","text":"Rewrite the equation as $$3x=102-6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving11a-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$96$$"],"dependencies":["aae6efeSolving11a-h3"],"title":"Subtraction","text":"What is $$102-6$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving11a-h4","type":"hint","dependencies":["aae6efeSolving11a-h3"],"title":"Division","text":"Dividing the coefficient on $$x$$ to get the answer","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aae6efeSolving12","title":"Solve Number Word Problems","body":"A married couple together earns $110,000 a year. The wife earns $16,000 less than twice what her husband earns.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Use a Problem Solving Strategy","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aae6efeSolving12a","stepAnswer":["$$42000$$"],"problemType":"TextBox","stepTitle":"What does the husband earn?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$42000$$","hints":{"DefaultPathway":[{"id":"aae6efeSolving12a-h1","type":"hint","dependencies":[],"title":"Setup","text":"Assume husband earns $$x$$ dollar and wife earns $$2x-16000$$, based on the second condition","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving12a-h2","type":"hint","dependencies":["aae6efeSolving12a-h1"],"title":"Setup","text":"Write an equation based on the first condition: $$x+2x-16000=110000$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving12a-h3","type":"hint","dependencies":["aae6efeSolving12a-h2"],"title":"Organizing","text":"Rewrite the equation as $$3x=110000+16000$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving12a-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$126000$$"],"dependencies":["aae6efeSolving12a-h3"],"title":"Subtraction","text":"What is $$110000+16000$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving12a-h4","type":"hint","dependencies":["aae6efeSolving12a-h3"],"title":"Division","text":"Dividing the coefficient on $$x$$ to get the answer","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aae6efeSolving16","title":"Cub Scouts","body":"Find the number of adult leaders.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Use a Problem Solving Strategy","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aae6efeSolving16a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"There are $$18$$ Cub Scouts in Troop $$645$$. The number of scouts is three more than five times the number of adult leaders. Find the number of adult leaders.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"aae6efeSolving16a-h1","type":"hint","dependencies":[],"title":"Organizing","text":"Determine the desire outcome and set it as $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving16a-h2","type":"hint","dependencies":["aae6efeSolving16a-h1"],"title":"Assumption","text":"Assume the number of adult leaders is $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving16a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5x+3$$"],"dependencies":["aae6efeSolving16a-h2"],"title":"Calculation","text":"What is the number of scouts in terms of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving16a-h4","type":"hint","dependencies":["aae6efeSolving16a-h3"],"title":"Converting","text":"Recall: the number of scouts is three more than five times the number of adult leader","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving16a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18$$"],"dependencies":["aae6efeSolving16a-h4"],"title":"Recall","text":"How many scouts are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving16a-h6","type":"hint","dependencies":["aae6efeSolving16a-h5"],"title":"Principle","text":"The number of scout in term of $$x$$ should equal to $$18$$. Write an equation as $$5x+3=18$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving16a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["aae6efeSolving16a-h6"],"title":"Calculation","text":"What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aae6efeSolving17","title":"Jeff\'s Bicycles","body":"Find the number of children\'s bicycles.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Use a Problem Solving Strategy","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aae6efeSolving17a","stepAnswer":["$$117$$"],"problemType":"TextBox","stepTitle":"Jeff is lining up children\'s and adult bicycles at the bike shop where he works. The number of children\'s bicycles is nine less than three times the number of adult bicycles. There are $$42$$ adult bicycles. How many children\'s bicycles are there?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$117$$","hints":{"DefaultPathway":[{"id":"aae6efeSolving17a-h1","type":"hint","dependencies":[],"title":"Organizing","text":"Determine the desire outcome and set it as $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving17a-h2","type":"hint","dependencies":["aae6efeSolving17a-h1"],"title":"Assumption","text":"Assume the number of adult bicycles is $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving17a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x-9$$"],"dependencies":["aae6efeSolving17a-h2"],"title":"Calculation","text":"What is the number of children\'s bicycles in terms of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving17a-h4","type":"hint","dependencies":["aae6efeSolving17a-h3"],"title":"Converting","text":"Recall: the number of children\'s bicycles is nine less than three times the number of adult bicycles","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving17a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$42$$"],"dependencies":["aae6efeSolving17a-h4"],"title":"Recall","text":"How many adult bicycles are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving17a-h6","type":"hint","dependencies":["aae6efeSolving17a-h5"],"title":"Principle","text":"The numbers of adult bicycles is $$x=42$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving17a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$117$$"],"dependencies":["aae6efeSolving17a-h6"],"title":"Calculation","text":"What is the value of $$3x-9$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aae6efeSolving18","title":"Solve Number Word Problems","body":"Find the number.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Use a Problem Solving Strategy","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aae6efeSolving18a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"The sum of twice a number and six is $$14$$. Find the number.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"aae6efeSolving18a-h1","type":"hint","dependencies":[],"title":"Setup","text":"Assume the number we want to find is $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving18a-h2","type":"hint","dependencies":["aae6efeSolving18a-h1"],"title":"Rewritting","text":"Rewrite the question in equation form with variable $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving18a-h3","type":"hint","dependencies":["aae6efeSolving18a-h2"],"title":"Setup","text":"Equation: $$2x+6=14$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving18a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["aae6efeSolving18a-h3"],"title":"Calculation","text":"What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aae6efeSolving19","title":"Solve Number Word Problems","body":"Find the bigger number.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Use a Problem Solving Strategy","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aae6efeSolving19a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"The sum of two numbers is zero. One number is nine less than twice the other. Find the bigger number.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"aae6efeSolving19a-h1","type":"hint","dependencies":[],"title":"Setup","text":"Assume one number as $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving19a-h2","type":"hint","dependencies":["aae6efeSolving19a-h1"],"title":"Setup","text":"The value of the other number can be determined by the given conditions","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving19a-h3","type":"hint","dependencies":["aae6efeSolving19a-h2"],"title":"Principle","text":"The two results in the same number, so the equation can be set as $$2x-9+x=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving19a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["aae6efeSolving19a-h3"],"title":"Calculation","text":"What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aae6efeSolving2","title":"Use a Problem Solving Strategy for Word Problems","body":"Solve the following word problem","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Use a Problem Solving Strategy","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aae6efeSolving2a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"Guillermo bought textbooks and notebooks at the bookstore. The number of textbooks was three more than twice the number of notebooks. He bought seven textbooks. How many notebooks did he buy?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"aae6efeSolving2a-h1","type":"hint","dependencies":[],"title":"Setup","text":"Assume the number of notebook is $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving2a-h2","type":"hint","dependencies":["aae6efeSolving2a-h1"],"title":"Setup","text":"Write an equation based on the question: $$2x+3=7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving2a-h3","type":"hint","dependencies":["aae6efeSolving2a-h2"],"title":"Organizing","text":"Rewrite the equation as $$2x=7-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving2a-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["aae6efeSolving2a-h3"],"title":"Subtraction","text":"What is $$7-3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving2a-h4","type":"hint","dependencies":["aae6efeSolving2a-h3"],"title":"Division","text":"Dividing the coefficient on $$x$$ to get the answer","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aae6efeSolving20","title":"Solve Number Word Problems","body":"Find the smaller number.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Use a Problem Solving Strategy","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aae6efeSolving20a","stepAnswer":["$$44$$"],"problemType":"TextBox","stepTitle":"The sum of two consecutive numbers is $$89$$. Find the smaller number.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$44$$","hints":{"DefaultPathway":[{"id":"aae6efeSolving20a-h1","type":"hint","dependencies":[],"title":"Setup","text":"Assume one number is $$x$$, the other is $$x+1$$ because they are consecutive","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving20a-h2","type":"hint","dependencies":["aae6efeSolving20a-h1"],"title":"Organizing","text":"According to the question, the equation can be set as $$x+x+1=89$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving20a-h3","type":"hint","dependencies":["aae6efeSolving20a-h2"],"title":"Simplify","text":"The equation can be combined as $$2x+1=89$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving20a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$44$$"],"dependencies":["aae6efeSolving20a-h3"],"title":"Calculation","text":"What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aae6efeSolving21","title":"Solve Number Word Problems","body":"Find the smallest such number.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Use a Problem Solving Strategy","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aae6efeSolving21a","stepAnswer":["$$72$$"],"problemType":"TextBox","stepTitle":"There are three consecutive even integers whose sum is $$222$$. Find the smallest.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$72$$","hints":{"DefaultPathway":[{"id":"aae6efeSolving21a-h1","type":"hint","dependencies":[],"title":"Setup","text":"Assume the second number is $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving21a-h2","type":"hint","dependencies":["aae6efeSolving21a-h1"],"title":"Principle","text":"The difference between two consecutive even integers is two. Therefore, the other numbers are $$x+2$$ and $$x-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving21a-h3","type":"hint","dependencies":["aae6efeSolving21a-h2"],"title":"Substitution","text":"Rewrite the sum in the equation as $$x+x+2+x-2=222$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving21a-h4","type":"hint","dependencies":["aae6efeSolving21a-h3"],"title":"Simplify","text":"The equation can be combined as $$3x=222$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving21a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$74$$"],"dependencies":["aae6efeSolving21a-h4"],"title":"Calculation","text":"What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving21a-h6","type":"hint","dependencies":["aae6efeSolving21a-h5"],"title":"Substitution","text":"Use the value of $$x$$ to find the other numbers","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aae6efeSolving22","title":"Marc\'s Car","body":"Find the price of his wife\'s car.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Use a Problem Solving Strategy","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aae6efeSolving22a","stepAnswer":["$$30700$$"],"problemType":"TextBox","stepTitle":"Marc just bought an SUV for $54,000. This is $7,400 less than twice what his wife paid for her car last year. How much did his wife pay for her car?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$30700$$","hints":{"DefaultPathway":[{"id":"aae6efeSolving22a-h1","type":"hint","dependencies":[],"title":"Setup","text":"Assume Marc\'s wife paid $$x$$ dollars","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving22a-h2","type":"hint","dependencies":["aae6efeSolving22a-h1"],"title":"Organizing","text":"Marc paid 54,000 dollars for the SUV","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving22a-h3","type":"hint","dependencies":["aae6efeSolving22a-h2"],"title":"Organizing","text":"According to the second statement, Marc paid $$2x-7400$$ dollars","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving22a-h4","type":"hint","dependencies":["aae6efeSolving22a-h3"],"title":"Combining","text":"The equations represent the same amount of value, so it can be combined as $$54000=2x-7400$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving22a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$30700$$"],"dependencies":["aae6efeSolving22a-h4"],"title":"Calculation","text":"What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aae6efeSolving23","title":"Erica\'s Jobs","body":"Find how much Erica makes from her college job.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Use a Problem Solving Strategy","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aae6efeSolving23a","stepAnswer":["$$12300$$"],"problemType":"TextBox","stepTitle":"Erica earned a total of $50,450 last last year from her two jobs. The amount she earned from her job at the store was $1,250 more than three times the amount she earned from her job at the college. How much did she earn from her job at the college?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12300$$","hints":{"DefaultPathway":[{"id":"aae6efeSolving23a-h1","type":"hint","dependencies":[],"title":"Setup","text":"Assume Erica earned $$x$$ dollars from her job at college","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving23a-h2","type":"hint","dependencies":["aae6efeSolving23a-h1"],"title":"Organizing","text":"According to her total income, she earned $$50450-x$$ from her job at the store","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving23a-h3","type":"hint","dependencies":["aae6efeSolving23a-h2"],"title":"Organizing","text":"According to the second statement, she earned $$3x+1250$$ from her job at the store","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving23a-h4","type":"hint","dependencies":["aae6efeSolving23a-h3"],"title":"Combining","text":"The income from her job at the store is constant, so the equation can be set as $$50450-x=3x+1250$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving23a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12300$$"],"dependencies":["aae6efeSolving23a-h4"],"title":"Calculation","text":"What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aae6efeSolving24","title":"Solve Percent Application","body":"Solve the following percentage questions.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Use a Problem Solving Strategy","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aae6efeSolving24a","stepAnswer":["$$54$$"],"problemType":"TextBox","stepTitle":"What number is 45% of 120?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$54$$","hints":{"DefaultPathway":[{"id":"aae6efeSolving24a-h1","type":"hint","dependencies":[],"title":"Setup","text":"Assume the number is $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving24a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.45$$"],"dependencies":["aae6efeSolving24a-h1"],"title":"Conversion","text":"What is 45% in numeric form?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving24a-h3","type":"hint","dependencies":["aae6efeSolving24a-h2"],"title":"Translation","text":"Transform the question in equation form as $$120\\\\times0.45=x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving24a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$54$$"],"dependencies":["aae6efeSolving24a-h3"],"title":"Calculation","text":"What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aae6efeSolving24b","stepAnswer":["$$108$$"],"problemType":"TextBox","stepTitle":"$$81$$ is 75% of what number?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$108$$","hints":{"DefaultPathway":[{"id":"aae6efeSolving24b-h1","type":"hint","dependencies":[],"title":"Setup","text":"Assume the number is $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving24b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.75$$"],"dependencies":["aae6efeSolving24b-h1"],"title":"Conversion","text":"What is 75% in numeric form?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving24b-h3","type":"hint","dependencies":["aae6efeSolving24b-h2"],"title":"Translation","text":"Transform the question in equation form as $$0.75x=81$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving24b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$108$$"],"dependencies":["aae6efeSolving24b-h3"],"title":"Calculation","text":"What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aae6efeSolving24c","stepAnswer":["$$30$$"],"problemType":"TextBox","stepTitle":"What percent of $$260$$ is 78?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$30$$","hints":{"DefaultPathway":[{"id":"aae6efeSolving24c-h1","type":"hint","dependencies":[],"title":"Setup","text":"Assume the fraction is $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving24c-h2","type":"hint","dependencies":["aae6efeSolving24c-h1"],"title":"Translation","text":"Transform the question in equation form as $$\\\\frac{78}{260}=x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving24c-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.3$$"],"dependencies":["aae6efeSolving24c-h2"],"title":"Calculation","text":"What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving24c-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$30$$"],"dependencies":["aae6efeSolving24c-h3"],"title":"Conversion","text":"Tanslate $$0.3$$ to a percentage (omit the percent symbol in your answer)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aae6efeSolving25","title":"Solve Percent Application","body":"Solve the following percentage questions.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Use a Problem Solving Strategy","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aae6efeSolving25a","stepAnswer":["$$135$$"],"problemType":"TextBox","stepTitle":"150% of $$90$$ is what number?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$135$$","hints":{"DefaultPathway":[{"id":"aae6efeSolving25a-h1","type":"hint","dependencies":[],"title":"Setup","text":"Assume the number is $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving25a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.5$$"],"dependencies":["aae6efeSolving25a-h1"],"title":"Conversion","text":"What is 150% in numeric form?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving25a-h3","type":"hint","dependencies":["aae6efeSolving25a-h2"],"title":"Translation","text":"Transform the question in equation form as $$90\\\\times1.5=x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving25a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$135$$"],"dependencies":["aae6efeSolving25a-h3"],"title":"Calculation","text":"What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aae6efeSolving25b","stepAnswer":["$$45$$"],"problemType":"TextBox","stepTitle":"$$6.4\\\\%$$ of what amount is $$\\\\$2.88$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$45$$","hints":{"DefaultPathway":[{"id":"aae6efeSolving25b-h1","type":"hint","dependencies":[],"title":"Setup","text":"Assume the number is $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving25b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.064$$"],"dependencies":["aae6efeSolving25b-h1"],"title":"Conversion","text":"What is $$6.4\\\\%$$ in numeric form?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving25b-h3","type":"hint","dependencies":["aae6efeSolving25b-h2"],"title":"Translation","text":"Transform the question in equation form as $$0.064x=2.88$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving25b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$45$$"],"dependencies":["aae6efeSolving25b-h3"],"title":"Calculation","text":"What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aae6efeSolving25c","stepAnswer":["$$125$$"],"problemType":"TextBox","stepTitle":"$$50$$ is what percent of 40?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$125$$","hints":{"DefaultPathway":[{"id":"aae6efeSolving25c-h1","type":"hint","dependencies":[],"title":"Setup","text":"Assume the fraction is $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving25c-h2","type":"hint","dependencies":["aae6efeSolving25c-h1"],"title":"Translation","text":"Transform the question in equation form as $$\\\\frac{50}{40}=x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving25c-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.25$$"],"dependencies":["aae6efeSolving25c-h2"],"title":"Calculation","text":"What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving25c-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$125$$"],"dependencies":["aae6efeSolving25c-h3"],"title":"Conversion","text":"Translate $$1.25$$ to a percentage (omit the percent sign in your answer)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aae6efeSolving26","title":"Hiro\'s Lunch","body":"Find the tip amount.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Use a Problem Solving Strategy","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aae6efeSolving26a","stepAnswer":["$$16.29$$"],"problemType":"TextBox","stepTitle":"When Hiro and his co-workers had lunch at a restaurant near their work, the bill was $$\\\\$90.50$$. They want to leave 18% of the total bill as a tip. How much should the tip be?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16.29$$","hints":{"DefaultPathway":[{"id":"aae6efeSolving26a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.18$$"],"dependencies":[],"title":"Conversion","text":"What is the numeric form of 18%?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving26a-h2","type":"hint","dependencies":["aae6efeSolving26a-h1"],"title":"Setup","text":"Assume the amount of tip as $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving26a-h3","type":"hint","dependencies":["aae6efeSolving26a-h2"],"title":"Setup","text":"The equation can be translated in the equation form as $$90.5\\\\times0.18=x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving26a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16.29$$"],"dependencies":["aae6efeSolving26a-h3"],"title":"Calculation","text":"What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aae6efeSolving27","title":"Salad Nutrition","body":"Fine the total recommended daily amount of sodium.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Use a Problem Solving Strategy","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aae6efeSolving27a","stepAnswer":["$$2407$$"],"problemType":"TextBox","stepTitle":"A grilled chicken salad at a popular fast food restaurant contains $$650$$ milligrams (mg) of sodium, which is 27% of the recommended daily amount. What is the total recommended daily amount of sodium (rounded to the nearest whole number)?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2407$$","hints":{"DefaultPathway":[{"id":"aae6efeSolving27a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.27$$"],"dependencies":[],"title":"Conversion","text":"What is the numeric form of 27%?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving27a-h2","type":"hint","dependencies":["aae6efeSolving27a-h1"],"title":"Setup","text":"Assume the total recommended daily amount of sodium as $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving27a-h3","type":"hint","dependencies":["aae6efeSolving27a-h2"],"title":"Setup","text":"The equation can be translated in the equation form as $$650=0.27x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving27a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2407$$"],"dependencies":["aae6efeSolving27a-h3"],"title":"Calculation","text":"What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aae6efeSolving28","title":"Errol\'s Helmet","body":"Find the helmet\'s sale price.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Use a Problem Solving Strategy","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aae6efeSolving28a","stepAnswer":["$$29.97$$"],"problemType":"TextBox","stepTitle":"Errol bought a skateboard helmet on sale at 40% off. The original price was $$\\\\$49.95$$. Find the sale price","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$29.97$$","hints":{"DefaultPathway":[{"id":"aae6efeSolving28a-h1","type":"hint","dependencies":[],"title":"Setup","text":"40% off means the sale price is $$(100\\\\%-40\\\\%)$$ off the original price","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving28a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.4$$"],"dependencies":["aae6efeSolving28a-h1"],"title":"Conversion","text":"What is the numeric form of 40%?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving28a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.6$$"],"dependencies":["aae6efeSolving28a-h2"],"title":"Calculation","text":"What is the numeric form of $$(100\\\\%-40\\\\%)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving28a-h4","type":"hint","dependencies":["aae6efeSolving28a-h3"],"title":"Organizing","text":"The amount of discount can be set as $$price discount$$, $$49.95\\\\times0.4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving28a-h5","type":"hint","dependencies":["aae6efeSolving28a-h4"],"title":"Organizing","text":"The sale price in equation form: $$49.95\\\\times0.6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving28a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$29.97$$"],"dependencies":["aae6efeSolving28a-h5"],"title":"Calculation","text":"What is the sale price as $$49.95\\\\times0.6$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aae6efeSolving29","title":"Casey\'s Bank","body":"Find the amount of interest earned in two years.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Use a Problem Solving Strategy","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aae6efeSolving29a","stepAnswer":["$$116$$"],"problemType":"TextBox","stepTitle":"Casey deposited $1,450 in a bank account that earned simple interest at an interest rate of 4%. How much interest was earned in two years?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$116$$","hints":{"DefaultPathway":[{"id":"aae6efeSolving29a-h1","type":"hint","dependencies":[],"title":"Conversion","text":"Interest rate of 4% can be regarded as gaining $$0.04$$ of the original amount per year","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving29a-h2","type":"hint","dependencies":["aae6efeSolving29a-h1"],"title":"Multiplying","text":"After two years, the gained interest is $$0.04\\\\times2$$ of original amount","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving29a-h3","type":"hint","dependencies":["aae6efeSolving29a-h2"],"title":"Setup","text":"The question can be set in the equation form as $$1450\\\\times0.08$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving29a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$116$$"],"dependencies":["aae6efeSolving29a-h3"],"title":"Calculation","text":"What is the interest earned after two year?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aae6efeSolving3","title":"Solve Number Word Problems","body":"Solve the following word problem","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Use a Problem Solving Strategy","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aae6efeSolving3a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"The sum of seven times a number and eight is thirty-six. Find the number.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"aae6efeSolving3a-h1","type":"hint","dependencies":[],"title":"Setup","text":"Assume the number is $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving3a-h2","type":"hint","dependencies":["aae6efeSolving3a-h1"],"title":"Setup","text":"Write an equation based on the question: $$7x+8=36$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving3a-h3","type":"hint","dependencies":["aae6efeSolving3a-h2"],"title":"Organizing","text":"Rewrite the equation as $$7x=36-8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving3a-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$28$$"],"dependencies":["aae6efeSolving3a-h3"],"title":"Subtraction","text":"What is $$36-8$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving3a-h4","type":"hint","dependencies":["aae6efeSolving3a-h3"],"title":"Division","text":"Dividing the coefficient on $$x$$ to get the answer","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aae6efeSolving30","title":"Margaret\'s Car","body":"Find how much Margaret had to borrow.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Use a Problem Solving Strategy","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aae6efeSolving30a","stepAnswer":["$$15680$$"],"problemType":"TextBox","stepTitle":"Margaret\'s car loan statement said she would pay $$\\\\$7, 683.20$$ in simple interest for a five-year loan at $$9.8\\\\%$$. How much did Margaret borrow to buy the car (rounded to the nearest whole number)?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$15680$$","hints":{"DefaultPathway":[{"id":"aae6efeSolving30a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.098$$"],"dependencies":[],"title":"Conversion","text":"What is the numeric form of $$9.8\\\\%$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving30a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.49$$"],"dependencies":["aae6efeSolving30a-h1"],"title":"Multiplication","text":"The interest value is $$0.098$$ per year, what is the total interest after $$5$$ years?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving30a-h3","type":"hint","dependencies":["aae6efeSolving30a-h2"],"title":"Setup","text":"Assume the total amount of loan is $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving30a-h4","type":"hint","dependencies":["aae6efeSolving30a-h3"],"title":"Setup","text":"The total interest is $$7, 683.20$$, $$0.49$$ of the total loan $$x$$. The equation can be structured as $$7683.20=0.49x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving30a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15680$$"],"dependencies":["aae6efeSolving30a-h4"],"title":"Calculation","text":"What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aae6efeSolving4","title":"Solve Number Word Problems","body":"Solve the following word problem","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Use a Problem Solving Strategy","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aae6efeSolving4a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"The sum of three times a number and seven is twenty-five. Find the number.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"aae6efeSolving4a-h1","type":"hint","dependencies":[],"title":"Setup","text":"Assume the number is $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving4a-h2","type":"hint","dependencies":["aae6efeSolving4a-h1"],"title":"Setup","text":"Write an equation based on the question: $$3x+7=25$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving4a-h3","type":"hint","dependencies":["aae6efeSolving4a-h2"],"title":"Organizing","text":"Rewrite the equation as $$3x=25-7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving4a-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18$$"],"dependencies":["aae6efeSolving4a-h3"],"title":"Subtraction","text":"What is $$25-7$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving4a-h4","type":"hint","dependencies":["aae6efeSolving4a-h3"],"title":"Division","text":"Dividing the coefficient on $$x$$ to get the answer","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aae6efeSolving5","title":"Solve Number Word Problems","body":"Solve the following word problem","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Use a Problem Solving Strategy","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aae6efeSolving5a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"The sum of four times a number and two is fourteen. Find the number.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"aae6efeSolving5a-h1","type":"hint","dependencies":[],"title":"Setup","text":"Assume the number is $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving5a-h2","type":"hint","dependencies":["aae6efeSolving5a-h1"],"title":"Setup","text":"Write an equation based on the question: $$4x+2=14$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving5a-h3","type":"hint","dependencies":["aae6efeSolving5a-h2"],"title":"Organizing","text":"Rewrite the equation as $$4x=14-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving5a-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["aae6efeSolving5a-h3"],"title":"Subtraction","text":"What is $$14-2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving5a-h4","type":"hint","dependencies":["aae6efeSolving5a-h3"],"title":"Division","text":"Dividing the coefficient on $$x$$ to get the answer","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aae6efeSolving6","title":"Solve Number Word Problems","body":"Solve the following word problem","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Use a Problem Solving Strategy","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aae6efeSolving6a","stepAnswer":["$$-3, -12$$"],"problemType":"MultipleChoice","stepTitle":"The sum of two numbers is negative fifteen. One number is nine less than the other. Find the numbers.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$-3, -12$$","$$-8, -17$$","$$-6, -15$$"],"hints":{"DefaultPathway":[{"id":"aae6efeSolving6a-h1","type":"hint","dependencies":[],"title":"Setup","text":"Assume one number is $$x$$ and another is $$x-9$$, based on the second condition","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving6a-h2","type":"hint","dependencies":["aae6efeSolving6a-h1"],"title":"Setup","text":"Write an equation based on the first condition: $$x+x-9=-15$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving6a-h3","type":"hint","dependencies":["aae6efeSolving6a-h2"],"title":"Organizing","text":"Rewrite the equation as $$2x=-15-(-9)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving6a-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6$$"],"dependencies":["aae6efeSolving6a-h3"],"title":"Subtraction","text":"What is $$-15-(-9)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving6a-h4","type":"hint","dependencies":["aae6efeSolving6a-h3"],"title":"Division","text":"Dividing the coefficient on $$x$$ to get the answer","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aae6efeSolving7","title":"Solve Number Word Problems","body":"Solve the following word problem","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Use a Problem Solving Strategy","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aae6efeSolving7a","stepAnswer":["$$-8, -15$$"],"problemType":"MultipleChoice","stepTitle":"The sum of two numbers is negative twenty-three. One number is seven less than the other. Find the numbers.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$-8, -15$$","$$-9, -16$$","$$-10, -15$$"],"hints":{"DefaultPathway":[{"id":"aae6efeSolving7a-h1","type":"hint","dependencies":[],"title":"Setup","text":"Assume one number is $$x$$ and another is $$x-7$$, based on the second condition","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving7a-h2","type":"hint","dependencies":["aae6efeSolving7a-h1"],"title":"Setup","text":"Write an equation based on the first condition: $$x+x-7=-23$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving7a-h3","type":"hint","dependencies":["aae6efeSolving7a-h2"],"title":"Organizing","text":"Rewrite the equation as $$2x=-23-(-7)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving7a-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-16$$"],"dependencies":["aae6efeSolving7a-h3"],"title":"Subtraction","text":"What is $$-23-(-7)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving7a-h4","type":"hint","dependencies":["aae6efeSolving7a-h3"],"title":"Division","text":"Dividing the coefficient on $$x$$ to get the answer","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aae6efeSolving8","title":"Solve Number Word Problems","body":"Solve the following word problem","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Use a Problem Solving Strategy","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aae6efeSolving8a","stepAnswer":["$$11, -29$$"],"problemType":"MultipleChoice","stepTitle":"The sum of two numbers is negative eighteen. One number is forty more than the other. Find the numbers.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$18, -22$$","$$11, -29$$","$$9, -31$$"],"hints":{"DefaultPathway":[{"id":"aae6efeSolving8a-h1","type":"hint","dependencies":[],"title":"Setup","text":"Assume one number is $$x$$ and another is $$x+40$$, based on the second condition","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving8a-h2","type":"hint","dependencies":["aae6efeSolving8a-h1"],"title":"Setup","text":"Write an equation based on the first condition: $$x+x+40=-18$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving8a-h3","type":"hint","dependencies":["aae6efeSolving8a-h2"],"title":"Organizing","text":"Rewrite the equation as $$2x=-18-40$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving8a-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-58$$"],"dependencies":["aae6efeSolving8a-h3"],"title":"Subtraction","text":"What is $$-18-40$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving8a-h4","type":"hint","dependencies":["aae6efeSolving8a-h3"],"title":"Division","text":"Dividing the coefficient on $$x$$ to get the answer","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aae6efeSolving9","title":"Solve Number Word Problems","body":"Solve the following word problem","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.2 Use a Problem Solving Strategy","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aae6efeSolving9a","stepAnswer":["$$-17, -18, -19$$"],"problemType":"MultipleChoice","stepTitle":"Find three consecutive integers whose sum is $$-54$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$-19, -20, -21$$","$$-15, -16, -17$$","$$-17, -18, -19$$"],"hints":{"DefaultPathway":[{"id":"aae6efeSolving9a-h1","type":"hint","dependencies":[],"title":"Setup","text":"Assume numbers are $$x$$, $$x+1$$, $$x+2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving9a-h2","type":"hint","dependencies":["aae6efeSolving9a-h1"],"title":"Setup","text":"Write an equation based on the condition: $$x+x+1+x+2=-54$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving9a-h3","type":"hint","dependencies":["aae6efeSolving9a-h2"],"title":"Organizing","text":"Rewrite the equation as $$3x=-54-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving9a-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-57$$"],"dependencies":["aae6efeSolving9a-h3"],"title":"Subtraction","text":"What is $$-54-3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aae6efeSolving9a-h4","type":"hint","dependencies":["aae6efeSolving9a-h3"],"title":"Division","text":"Dividing the coefficient on $$x$$ to get the answer","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaebddaPolynomial1","title":"Degree of a polynomial","body":"Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of the following polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Add and Subtract Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aaebddaPolynomial1a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"a) $$7y^2-5y+3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"aaebddaPolynomial1a-h1","type":"hint","dependencies":[],"title":"Analysis","text":"The variable in this equation is $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":[],"title":"Counting","text":"How many different degrees are on $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Principle","text":"What is the highest degree on $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aaebddaPolynomial1b","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"b) $$-2a^4 b^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"aaebddaPolynomial1b-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Counting","text":"How many terms are in this polynomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial1b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":[],"title":"Principle","text":"What is the highest degree in this polynomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaebddaPolynomial1b-h2-h3","type":"hint","dependencies":["aaebddaPolynomial1b-h2"],"title":"Principle","text":"The degrees on different variables can be combined if they are in one term","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}},{"id":"aaebddaPolynomial1c","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"c) $$3x^5-4x^3-6x^2+x-8$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"aaebddaPolynomial1c-h1","type":"hint","dependencies":[],"title":"Analysis","text":"The variable in this equation is $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial1c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":[],"title":"Counting","text":"How many different degrees are on $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial1c-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":[],"title":"Principle","text":"What is the highest degree on $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aaebddaPolynomial1d","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"d) $$2y-8{xy}^3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"aaebddaPolynomial1d-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Counting","text":"How many terms are in this polynomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial1d-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":[],"title":"Principle","text":"What is the highest degree in this polynomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aaebddaPolynomial1d-h2-h3","type":"hint","dependencies":["aaebddaPolynomial1d-h2"],"title":"Principle","text":"The degrees on different variables can be combined if they are in one term","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}},{"id":"aaebddaPolynomial1e","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"e) $$15$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"aaebddaPolynomial1e-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Counting","text":"How many terms are in this polynomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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Algebra","steps":[{"id":"aaebddaPolynomial10a","stepAnswer":["$$8x^2-11x+8$$"],"problemType":"TextBox","stepTitle":"$$7x^2-4x+5+x^2-7x+3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8x^2-11x+8$$","hints":{"DefaultPathway":[{"id":"aaebddaPolynomial10a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The addition and subtraction can only happen in the term with same degree","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial10a-h2","type":"hint","dependencies":[],"title":"Organizing","text":"It has the common terms of $$x^2$$, $$x$$, and constant","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":[],"title":"Addition","text":"What is $$7+1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-11$$"],"dependencies":[],"title":"Addition","text":"What is $$\\\\left(-4\\\\right)+\\\\left(-7\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":[],"title":"Addition","text":"What is $$5+3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial10a-h6","type":"hint","dependencies":[],"title":"Addition","text":"Add all the terms up","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaebddaPolynomial11","title":"Add and Subtract Polynomial","body":"Find the difference of the following polynomial:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Add and Subtract Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aaebddaPolynomial11a","stepAnswer":["$$11w^2-7w+1$$"],"problemType":"TextBox","stepTitle":"$$9w^2-7w+5+2w^2-4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$11w^2-7w+1$$","hints":{"DefaultPathway":[{"id":"aaebddaPolynomial11a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The addition and subtraction can only happen in the term with same degree","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial11a-h2","type":"hint","dependencies":[],"title":"Organizing","text":"It has the common terms of $$w^2$$ and constant","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial11a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11$$"],"dependencies":[],"title":"Addition","text":"What is $$9+2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Addition","text":"What is $$5-4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial11a-h5","type":"hint","dependencies":[],"title":"Addition","text":"Add all the terms up","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaebddaPolynomial12","title":"Add and Subtract Polynomial","body":"Find the difference of the following polynomial:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Add and Subtract Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aaebddaPolynomial12a","stepAnswer":["$$-10pq+3q^2$$"],"problemType":"TextBox","stepTitle":"Subtract $$p^2+10pq-2q^2$$ from $$p^2+q^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-10pq+3q^2$$","hints":{"DefaultPathway":[{"id":"aaebddaPolynomial12a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The addition and subtraction can only happen in the term with same degree","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial12a-h2","type":"hint","dependencies":[],"title":"Organizing","text":"It has the common terms of $$p^2$$ and $$q^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial12a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":[],"title":"Addition","text":"What is $$1-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC 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Intermediate Algebra","steps":[{"id":"aaebddaPolynomial13a","stepAnswer":["$$52$$"],"problemType":"TextBox","stepTitle":"a) f(4)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$52$$","hints":{"DefaultPathway":[{"id":"aaebddaPolynomial13a-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Substitute $$4$$ for $$x$$ into the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$52$$"],"dependencies":[],"title":"Substitution","text":"What is the value of $$5\\\\times4^2-8\\\\times4+4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial13a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$52$$"],"dependencies":[],"title":"Calculation","text":"What is the value of $$80-32+4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aaebddaPolynomial13b","stepAnswer":["$$40$$"],"problemType":"TextBox","stepTitle":"b) $$f(-2)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$40$$","hints":{"DefaultPathway":[{"id":"aaebddaPolynomial13b-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Substitute $$-2$$ for $$x$$ into the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial13b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$40$$"],"dependencies":[],"title":"Substitution","text":"What is the value of $$5{\\\\left(-2\\\\right)}^2-8\\\\left(-2\\\\right)+4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial13b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$40$$"],"dependencies":[],"title":"Calculation","text":"What is the value of $$20-\\\\left(-16\\\\right)+4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aaebddaPolynomial13c","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"c) f(0)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"aaebddaPolynomial13c-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Substitute $$0$$ for $$x$$ into the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial13c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":[],"title":"Substitution","text":"What is the value of $$5\\\\times0^2-8\\\\times0+4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial13c-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":[],"title":"Calculation","text":"What is the value of $$0+0+4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaebddaPolynomial14","title":"Add and Subtract Polynomial Functions","body":"For functions $$f(x)=3x^2-5x+7$$ and $$g(x)=x^2-4x-3$$, find the following values:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Add and Subtract Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aaebddaPolynomial14a","stepAnswer":["$$4x^2-9x+4$$"],"problemType":"TextBox","stepTitle":"a) $$\\\\left(f+g\\\\right) x$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4x^2-9x+4$$","hints":{"DefaultPathway":[{"id":"aaebddaPolynomial14a-h1","type":"hint","dependencies":[],"title":"Formula","text":"Recall that the formula of $$\\\\left(f+g\\\\right) 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$$4\\\\times3^2-9\\\\times3+4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial14b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$13$$"],"dependencies":[],"title":"Substitution","text":"What is the value of $$36-27+4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aaebddaPolynomial14c","stepAnswer":["$$2x^2-x+10$$"],"problemType":"TextBox","stepTitle":"c) $$(f-g)(x)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2x^2-x+10$$","hints":{"DefaultPathway":[{"id":"aaebddaPolynomial14c-h1","type":"hint","dependencies":[],"title":"Formula","text":"Recall that the formula of $$(f-g)(x)=f(x)-g(x)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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$$3+\\\\left(-6\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial15a-h7","type":"hint","dependencies":[],"title":"Addition","text":"Add all the terms up","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aaebddaPolynomial15b","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"b) $$3\\\\left(f+g\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"aaebddaPolynomial15b-h1","type":"hint","dependencies":[],"title":"Formula","text":"Recall the polynomial calculated in part a","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial15b-h2","type":"hint","dependencies":[],"title":"Substitution","text":"Substitute 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4.0>"}]}}]},{"id":"aaebddaPolynomial6","title":"Add and Subtract Polynomial","body":"Simplify the following polynomial:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Add and Subtract Polynomials","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aaebddaPolynomial6a","stepAnswer":["$$-5a^2+7b^2$$"],"problemType":"TextBox","stepTitle":"a) $$a^2+7b^2-6a^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-5a^2+7b^2$$","hints":{"DefaultPathway":[{"id":"aaebddaPolynomial6a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The addition and subtraction can only happen in the term with same degree","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$a^2$$"],"dependencies":[],"title":"Organizing","text":"What is the common term in this polynomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":[],"title":"Addition","text":"What is $$1-6$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aaebddaPolynomial6b","stepAnswer":["$$u^2 v+5u^2-3v^2$$"],"problemType":"TextBox","stepTitle":"b) $$u^2 v+5u^2-3v^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$u^2 v+5u^2-3v^2$$","hints":{"DefaultPathway":[{"id":"aaebddaPolynomial6b-h1","type":"hint","dependencies":[],"title":"Principle","text":"The addition and subtraction can only happen in the term with same degree","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial6b-h2","type":"hint","dependencies":[],"title":"Organizing","text":"There is no common term","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaebddaPolynomial7","title":"Add and Subtract Polynomial","body":"Simplify the following polynomial:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.1 Add and Subtract Polynomials","courseName":"OpenStax: Intermediate 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4.0>"},{"id":"aaebddaPolynomial7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":[],"title":"Addition","text":"What is $$8-3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aaebddaPolynomial7b","stepAnswer":["$$m^2 n^2-8m^2+4n^2$$"],"problemType":"TextBox","stepTitle":"b) $$m^2 n^2-8m^2+4n^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$m^2 n^2-8m^2+4n^2$$","hints":{"DefaultPathway":[{"id":"aaebddaPolynomial7b-h1","type":"hint","dependencies":[],"title":"Principle","text":"The addition and subtraction can only happen in the term with same degree","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial7b-h2","type":"hint","dependencies":[],"title":"Organizing","text":"There is no common 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$m^2$$"],"dependencies":[],"title":"Organizing","text":"What is the common term in this polynomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":[],"title":"Addition","text":"What is $$3-7$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aaebddaPolynomial8b","stepAnswer":["$${pq}^2-6p-5q^2$$"],"problemType":"TextBox","stepTitle":"b) $${pq}^2-6p-5q^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${pq}^2-6p-5q^2$$","hints":{"DefaultPathway":[{"id":"aaebddaPolynomial8b-h1","type":"hint","dependencies":[],"title":"Principle","text":"The addition and subtraction can only happen in the term with same degree","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial8b-h2","type":"hint","dependencies":[],"title":"Organizing","text":"There is no common term","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaebddaPolynomial9","title":"Add and Subtract Polynomial","body":"Find the sum of the following polynomial:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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4.0>"},{"id":"aaebddaPolynomial9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11$$"],"dependencies":[],"title":"Addition","text":"What is $$7+4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-10$$"],"dependencies":[],"title":"Addition","text":"What is $$\\\\left(-2\\\\right)+\\\\left(-8\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaebddaPolynomial9a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Addition","text":"What is $$9+\\\\left(-7\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC 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a?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(4,0)$$","choices":["$$(4,0)$$","$$(4,0)I(0,4)I(2,0)I(0,2)$$"],"hints":{"DefaultPathway":[{"id":"aafbadeintercepts1a-h1","type":"hint","dependencies":[],"title":"Graph crosses x-axis where?","text":"Find where the line in the graph crosses the x-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts1a-h2","type":"hint","dependencies":["aafbadeintercepts1a-h1"],"title":"Answer","text":"The answer is $$(4,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aafbadeintercepts1b","stepAnswer":["$$(0,2)$$"],"problemType":"MultipleChoice","stepTitle":"What is the $$y$$ intercept of the line in the graph labeled a?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,2)$$","choices":["$$(0,2)$$","$$(4,0)I(0,4)I(2,0)I(0,2)$$"],"hints":{"DefaultPathway":[{"id":"aafbadeintercepts1b-h1","type":"hint","dependencies":[],"title":"Graph crosses y-axis where?","text":"Find where the line in the graph crosses the y-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts1b-h2","type":"hint","dependencies":["aafbadeintercepts1b-h1"],"title":"Answer","text":"The answer is $$(0,2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aafbadeintercepts1c","stepAnswer":["$$(2,0)$$"],"problemType":"MultipleChoice","stepTitle":"What is the $$x$$ intercept of the line in the graph labeled $$b$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(2,0)$$","choices":["$$(2,0)$$","$$(6,0)I(0,6)I(2,0)I(0,2)$$"],"hints":{"DefaultPathway":[{"id":"aafbadeintercepts1c-h1","type":"hint","dependencies":[],"title":"Graph crosses x-axis where?","text":"Find where the line in the graph crosses the x-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts1c-h2","type":"hint","dependencies":["aafbadeintercepts1c-h1"],"title":"Answer","text":"The answer is $$(2,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aafbadeintercepts1d","stepAnswer":["$$(0,6)$$"],"problemType":"MultipleChoice","stepTitle":"What is the $$y$$ intercept of the line in the graph labeled $$b$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,6)$$","choices":["$$(0,6)$$","$$(6,0)I(0,6)I(2,0)I(0,2)$$"],"hints":{"DefaultPathway":[{"id":"aafbadeintercepts1d-h1","type":"hint","dependencies":[],"title":"Graph crosses y-axis where?","text":"Find where the line in the graph crosses the y-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts1d-h2","type":"hint","dependencies":["aafbadeintercepts1d-h1"],"title":"Answer","text":"The answer is $$(0,6)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aafbadeintercepts1e","stepAnswer":["$$(-5,0)$$"],"problemType":"MultipleChoice","stepTitle":"What is the $$x$$ intercept of the line in the graph labeled c?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-5,0)$$","choices":["$$(-5,0)$$","$$(0,-5)I(-5,0)I(-3,0)I(0,-3)$$"],"hints":{"DefaultPathway":[{"id":"aafbadeintercepts1e-h1","type":"hint","dependencies":[],"title":"Graph crosses x-axis where?","text":"Find where the line in the graph crosses the x-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts1e-h2","type":"hint","dependencies":["aafbadeintercepts1e-h1"],"title":"Answer","text":"The answer is $$(-5,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aafbadeintercepts1f","stepAnswer":["$$(0,-5)$$"],"problemType":"MultipleChoice","stepTitle":"What is the $$y$$ intercept of the line in the graph labeled c?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,-5)$$","choices":["$$(0,-5)$$","$$(6,0)I(0,6)I(2,0)I(0,2)$$"],"hints":{"DefaultPathway":[{"id":"aafbadeintercepts1f-h1","type":"hint","dependencies":[],"title":"Graph crosses y-axis where?","text":"Find where the line in the graph crosses the y-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts1f-h2","type":"hint","dependencies":["aafbadeintercepts1f-h1"],"title":"Answer","text":"The answer is $$(0,-5)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafbadeintercepts10","title":"Intercepts","body":"Find $$x$$ and $$y$$ intercepts from an equation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Graph with Intercepts","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafbadeintercepts10a","stepAnswer":["$$(0,-3)$$"],"problemType":"MultipleChoice","stepTitle":"Find the $$y$$ intercept of $$3x-4y=12$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,-3)$$","choices":["$$(0,-3)$$","$$(0,2)I(0,-4)I(0,-3)I(0,-7)$$"],"hints":{"DefaultPathway":[{"id":"aafbadeintercepts10a-h1","type":"hint","dependencies":[],"title":"Let $$x=0$$","text":"Set $$x=0$$ and begin solving the equation. The remaining portion of the equation is $$0-4y=12$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts10a-h2","type":"hint","dependencies":["aafbadeintercepts10a-h1"],"title":"Solve for $$y$$","text":"Divide both sides by $$-4$$ to get $$y=-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts10a-h3","type":"hint","dependencies":["aafbadeintercepts10a-h2"],"title":"Interpret","text":"After substituting $$x$$, we find that $$y=-3$$. Therefore, the $$x$$ coordinate when $$x=0$$ is $$y=-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts10a-h4","type":"hint","dependencies":["aafbadeintercepts10a-h3"],"title":"Answer","text":"The answer is $$(0,-3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafbadeintercepts11","title":"Intercepts","body":"Find $$x$$ and $$y$$ intercepts from an equation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Graph with Intercepts","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafbadeintercepts11a","stepAnswer":["$$(4,0)$$"],"problemType":"MultipleChoice","stepTitle":"Find the $$x$$ intercept of $$2x-4y=8$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(4,0)$$","choices":["$$(4,0)$$","$$(8,0)I(4,0)I(3,0)I(0,-3)$$"],"hints":{"DefaultPathway":[{"id":"aafbadeintercepts11a-h1","type":"hint","dependencies":[],"title":"Let $$y=0$$","text":"Set $$y=0$$ and begin solving the equation. The remaining portion of the equation is $$2x-0=8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts11a-h2","type":"hint","dependencies":["aafbadeintercepts11a-h1"],"title":"Interpret","text":"After substituting $$x$$, we find that $$x=4$$. Therefore, the $$x$$ coordinate when $$y=0$$ is $$x=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts11a-h3","type":"hint","dependencies":["aafbadeintercepts11a-h2"],"title":"Answer","text":"The answer is $$(4,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafbadeintercepts12","title":"Intercepts","body":"Find $$x$$ and $$y$$ intercepts from an equation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Graph with Intercepts","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafbadeintercepts12a","stepAnswer":["$$(0,-2)$$"],"problemType":"MultipleChoice","stepTitle":"Find the $$y$$ intercept of $$2x-4y=8$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,-2)$$","choices":["$$(0,-2)$$","$$(0,1)I(0,-4)I(0,-2)I(0,-1)$$"],"hints":{"DefaultPathway":[{"id":"aafbadeintercepts12a-h1","type":"hint","dependencies":[],"title":"Let $$x=0$$","text":"Set $$x=0$$ and begin solving the equation. The remaining portion of the equation is $$0-4y=8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts12a-h2","type":"hint","dependencies":["aafbadeintercepts12a-h1"],"title":"Solve for $$y$$","text":"Divide both sides by $$-4$$ to get $$y=-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts12a-h3","type":"hint","dependencies":["aafbadeintercepts12a-h2"],"title":"Interpret","text":"After substituting $$x$$, we find that $$y=-2$$. Therefore, the $$x$$ coordinate when $$x=0$$ is $$y=-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts12a-h4","type":"hint","dependencies":["aafbadeintercepts12a-h3"],"title":"Answer","text":"The answer is $$(0,-2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafbadeintercepts13","title":"Intercepts","body":"Find $$x$$ and $$y$$ intercepts from an equation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Graph with Intercepts","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafbadeintercepts13a","stepAnswer":["$$(-6,0)$$"],"problemType":"MultipleChoice","stepTitle":"Find the $$x$$ intercept of $$-x+2y=6$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-6,0)$$","choices":["$$(-6,0)$$","$$(-6,0)I(-1,0)I(2,0)I(0,-3)$$"],"hints":{"DefaultPathway":[{"id":"aafbadeintercepts13a-h1","type":"hint","dependencies":[],"title":"Let $$y=0$$","text":"Set $$y=0$$ and begin solving the equation. The remaining portion of the equation is $$-x+0=6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts13a-h2","type":"hint","dependencies":["aafbadeintercepts13a-h1"],"title":"Interpret","text":"After substituting $$x$$, we find that $$x=-6$$. Therefore, the $$x$$ coordinate when $$y=0$$ is $$x=-6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts13a-h3","type":"hint","dependencies":["aafbadeintercepts13a-h2"],"title":"Answer","text":"The answer is $$(-6,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafbadeintercepts14","title":"Intercepts","body":"Find $$x$$ and $$y$$ intercepts from an equation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Graph with Intercepts","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafbadeintercepts14a","stepAnswer":["$$(0,3)$$"],"problemType":"MultipleChoice","stepTitle":"Find the $$y$$ intercept of $$-x+2y=6$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,3)$$","choices":["$$(0,2)I(0,1)I(0,3)I(0,-1)$$","$$(0,3)$$"],"hints":{"DefaultPathway":[{"id":"aafbadeintercepts14a-h1","type":"hint","dependencies":[],"title":"Let $$x=0$$","text":"Set $$x=0$$ and begin solving the equation. The remaining portion of the equation is $$0+2y=6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts14a-h2","type":"hint","dependencies":["aafbadeintercepts14a-h1"],"title":"Interpret","text":"After substituting $$x$$, we find that $$y=3$$. Therefore, the $$x$$ coordinate when $$x=0$$ is $$y=3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts14a-h3","type":"hint","dependencies":["aafbadeintercepts14a-h2"],"title":"Answer","text":"The answer is $$(0,3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafbadeintercepts15","title":"Intercepts","body":"Find $$x$$ and $$y$$ intercepts from an equation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Graph with Intercepts","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafbadeintercepts15a","stepAnswer":["$$(4,0)$$"],"problemType":"MultipleChoice","stepTitle":"Find the $$x$$ intercept of $$x-2y=4$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(4,0)$$","choices":["$$(4,0)$$","$$(4,0)I(-1,0)I(1,0)I(0,-3)$$"],"hints":{"DefaultPathway":[{"id":"aafbadeintercepts15a-h1","type":"hint","dependencies":[],"title":"Let $$y=0$$","text":"Set $$y=0$$ and begin solving the equation. The remaining portion of the equation is $$x-0=4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts15a-h2","type":"hint","dependencies":["aafbadeintercepts15a-h1"],"title":"Interpret","text":"After substituting $$x$$, we find that $$x=4$$. Therefore, the $$x$$ coordinate when $$y=0$$ is $$x=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts15a-h3","type":"hint","dependencies":["aafbadeintercepts15a-h2"],"title":"Answer","text":"The answer is $$(4,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafbadeintercepts16","title":"Finding the Intercepts of an Equation","body":"Identify the $$x$$ and $$y$$ intercepts to solve the given problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Graph with Intercepts","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafbadeintercepts16a","stepAnswer":["$$8$$"],"problemType":"TextBox","stepTitle":"Find the sum of the x-value of the x-intercept and the y-value of the y-intercept of the following equation: $$x+y=4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8$$","hints":{"DefaultPathway":[{"id":"aafbadeintercepts16a-h1","type":"hint","dependencies":[],"title":"Finding the X- and Y- Intercepts","text":"$$x+y=4$$. The x-intercept can be calculated by the equation $$x=4$$, and the y-intercept can be calculated by the equation $$y=4$$. Solving these equations, we get the x-intercept as $$4$$ and the y-intercept as $$4$$. $$4+4=8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafbadeintercepts17","title":"Finding the Intercepts of an Equation","body":"Identify the $$x$$ and $$y$$ intercepts to solve the given problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Graph with Intercepts","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafbadeintercepts17a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"Find the sum of the x-value of the x-intercept and the y-value of the y-intercept of the following equation: $$x+y=3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"aafbadeintercepts17a-h1","type":"hint","dependencies":[],"title":"Finding the X- and Y- Intercepts","text":"$$x+y=3$$. The x-intercept can be calculated by the equation $$x=3$$, and the y-intercept can be calculated by the equation $$y=3$$. Solving these equations, we get the x-intercept as $$3$$ and the y-intercept as $$3$$. $$3+3=6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafbadeintercepts18","title":"Finding the Intercepts of an Equation","body":"Identify the $$x$$ and $$y$$ intercepts to solve the given problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Graph with Intercepts","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafbadeintercepts18a","stepAnswer":["$$-4$$"],"problemType":"TextBox","stepTitle":"Find the sum of the x-value of the x-intercept and the y-value of the y-intercept of the following equation: $$x+y=-2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-4$$","hints":{"DefaultPathway":[{"id":"aafbadeintercepts18a-h1","type":"hint","dependencies":[],"title":"Finding the X- and Y- Intercepts","text":"$$x+y=-2$$. The x-intercept can be calculated by the equation $$x=-2$$, and the y-intercept can be calculated by the equation $$y=-2$$. Solving these equations, we get the x-intercept as $$-2$$ and the y-intercept as $$-2$$. $$-2+\\\\left(-2\\\\right)=-4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafbadeintercepts2","title":"Intercepts","body":"Find $$x$$ and $$y$$ intercepts of the line.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Graph with Intercepts","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafbadeintercepts2a","stepAnswer":["$$(2,0)$$"],"problemType":"MultipleChoice","stepTitle":"What is the $$x$$ intercept of the line in the graph?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(2,0)$$","choices":["$$(-2,0)I(0,-2)I(2,0)I(0,2)$$","$$(2,0)$$"],"hints":{"DefaultPathway":[{"id":"aafbadeintercepts2a-h1","type":"hint","dependencies":[],"title":"Graph crosses x-axis where?","text":"Find where the line in the graph crosses the x-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts2a-h2","type":"hint","dependencies":["aafbadeintercepts2a-h1"],"title":"Answer","text":"The answer is $$(2,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aafbadeintercepts2b","stepAnswer":["$$(0,-2)$$"],"problemType":"MultipleChoice","stepTitle":"What is the $$y$$ intercept of the line in the graph?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,-2)$$","choices":["$$(-2,0)I(0,-2)I(2,0)I(0,2)$$","$$(0,-2)$$"],"hints":{"DefaultPathway":[{"id":"aafbadeintercepts2b-h1","type":"hint","dependencies":[],"title":"Graph crosses y-axis where?","text":"Find where the line in the graph crosses the y-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts2b-h2","type":"hint","dependencies":["aafbadeintercepts2b-h1"],"title":"Answer","text":"The answer is $$(0,-2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafbadeintercepts20","title":"Finding the Intercepts of an Equation","body":"Identify the $$x$$ and $$y$$ intercepts to solve the given problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Graph with Intercepts","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafbadeintercepts20a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"Find the sum of the x-value of the x-intercept and the y-value of the y-intercept of the following equation: $$x-y=5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"aafbadeintercepts20a-h1","type":"hint","dependencies":[],"title":"Finding the X- and Y- Intercepts","text":"$$x-y=5$$. The x-intercept can be calculated by the equation $$x=5$$, and the y-intercept can be calculated by the equation $$-y=5$$. Solving these equations, we get the x-intercept as $$5$$ and the y-intercept as $$-5$$. $$5+\\\\left(-5\\\\right)=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafbadeintercepts21","title":"Finding the Intercepts of an Equation","body":"Identify the $$x$$ and $$y$$ intercepts to solve the given problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Graph with Intercepts","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafbadeintercepts21a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"Find the sum of the x-value of the x-intercept and the y-value of the y-intercept of the following equation: $$x-y=1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"aafbadeintercepts21a-h1","type":"hint","dependencies":[],"title":"Finding the X- and Y- Intercepts","text":"$$x-y=1$$. The x-intercept can be calculated by the equation $$x=1$$, and the y-intercept can be calculated by the equation $$-y=1$$. Solving these equations, we get the x-intercept as $$1$$ and the y-intercept as $$-1$$. $$1+\\\\left(-1\\\\right)=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafbadeintercepts22","title":"Finding the Intercepts of an Equation","body":"Identify the $$x$$ and $$y$$ intercepts to solve the given problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Graph with Intercepts","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafbadeintercepts22a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"Find the sum of the x-value of the x-intercept and the y-value of the y-intercept of the following equation: $$x-y=-3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"aafbadeintercepts22a-h1","type":"hint","dependencies":[],"title":"Finding the X- and Y- Intercepts","text":"$$x-y=-3$$. The x-intercept can be calculated by the equation $$x=-3$$, and the y-intercept can be calculated by the equation $$-y=-3$$. Solving these equations, we get the x-intercept as $$-3$$ and the y-intercept as $$3$$. $$-3+3=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafbadeintercepts23","title":"Finding the Intercepts of an Equation","body":"Identify the $$x$$ and $$y$$ intercepts to solve the given problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Graph with Intercepts","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafbadeintercepts23a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"Find the sum of the x-value of the x-intercept and the y-value of the y-intercept of the following equation: $$x-y=-4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"aafbadeintercepts23a-h1","type":"hint","dependencies":[],"title":"Finding the X- and Y- Intercepts","text":"$$x-y=-4$$. The x-intercept can be calculated by the equation $$x=-4$$, and the y-intercept can be calculated by the equation $$-y=-4$$. Solving these equations, we get the x-intercept as $$-4$$ and the y-intercept as $$4$$. $$-4+4=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafbadeintercepts24","title":"Finding the Intercepts of an Equation","body":"Identify the $$x$$ and $$y$$ intercepts to solve the given problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Graph with Intercepts","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafbadeintercepts24a","stepAnswer":["$$12$$"],"problemType":"TextBox","stepTitle":"Find the sum of the x-value of the x-intercept and the y-value of the y-intercept of the following equation: $$x+2y=8$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12$$","hints":{"DefaultPathway":[{"id":"aafbadeintercepts24a-h1","type":"hint","dependencies":[],"title":"Finding the X- and Y- Intercepts","text":"$$x+2y=8$$. The x-intercept can be calculated by the equation $$x=8$$, and the y-intercept can be calculated by the equation $$2y=8$$. Solving these equations, we get the x-intercept as $$8$$ and the y-intercept as $$4$$. $$8+4=12$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafbadeintercepts25","title":"Finding the Intercepts of an Equation","body":"Identify the $$x$$ and $$y$$ intercepts to solve the given problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Graph with Intercepts","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafbadeintercepts25a","stepAnswer":["$$15$$"],"problemType":"TextBox","stepTitle":"Find the sum of the x-value of the x-intercept and the y-value of the y-intercept of the following equation: $$x+2y=10$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$15$$","hints":{"DefaultPathway":[{"id":"aafbadeintercepts25a-h1","type":"hint","dependencies":[],"title":"Finding the X- and Y- Intercepts","text":"$$x+2y=10$$. The x-intercept can be calculated by the equation $$x=10$$, and the y-intercept can be calculated by the equation $$2y=10$$. Solving these equations, we get the x-intercept as $$10$$ and the y-intercept as $$5$$. $$10+5=15$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafbadeintercepts26","title":"Finding the Intercepts of an Equation","body":"Identify the $$x$$ and $$y$$ intercepts to solve the given problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Graph with Intercepts","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafbadeintercepts26a","stepAnswer":["$$8$$"],"problemType":"TextBox","stepTitle":"Find the sum of the x-value of the x-intercept and the y-value of the y-intercept of the following equation: $$3x+y=6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8$$","hints":{"DefaultPathway":[{"id":"aafbadeintercepts26a-h1","type":"hint","dependencies":[],"title":"Finding the X- and Y- Intercepts","text":"$$3x+y=6$$. The x-intercept can be calculated by the equation $$3x=6$$, and the y-intercept can be calculated by the equation $$y=6$$. Solving these equations, we get the x-intercept as $$2$$ and the y-intercept as $$6$$. $$2+6=8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafbadeintercepts27","title":"Finding the Intercepts of an Equation","body":"Identify the $$x$$ and $$y$$ intercepts to solve the given problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Graph with Intercepts","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafbadeintercepts27a","stepAnswer":["$$12$$"],"problemType":"TextBox","stepTitle":"Find the sum of the x-value of the x-intercept and the y-value of the y-intercept of the following equation: $$3x+y=9$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12$$","hints":{"DefaultPathway":[{"id":"aafbadeintercepts27a-h1","type":"hint","dependencies":[],"title":"Finding the X- and Y- Intercepts","text":"$$x-y=1$$. The x-intercept can be calculated by the equation $$x=1$$, and the y-intercept can be calculated by the equation $$-y=1$$. Solving these equations, we get the x-intercept as $$1$$ and the y-intercept as $$-1$$. $$1+\\\\left(-1\\\\right)=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafbadeintercepts28","title":"Finding the Intercepts of an Equation","body":"Identify the $$x$$ and $$y$$ intercepts to solve the given problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Graph with Intercepts","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafbadeintercepts28a","stepAnswer":["$$8$$"],"problemType":"TextBox","stepTitle":"Find the sum of the x-value of the x-intercept and the y-value of the y-intercept of the following equation: $$x-3y=12$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8$$","hints":{"DefaultPathway":[{"id":"aafbadeintercepts28a-h1","type":"hint","dependencies":[],"title":"Finding the X- and Y- Intercepts","text":"$$x-y=-3$$. The x-intercept can be calculated by the equation $$x=-3$$, and the y-intercept can be calculated by the equation $$-y=-3$$. Solving these equations, we get the x-intercept as $$-3$$ and the y-intercept as $$3$$. $$-3+3=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafbadeintercepts29","title":"Finding the Intercepts of an Equation","body":"Identify the $$x$$ and $$y$$ intercepts to solve the given problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Graph with Intercepts","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafbadeintercepts29a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"Find the sum of the x-value of the x-intercept and the y-value of the y-intercept of the following equation: $$x-2y=8$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"aafbadeintercepts29a-h1","type":"hint","dependencies":[],"title":"Finding the X- and Y- Intercepts","text":"$$x-y=-4$$. The x-intercept can be calculated by the equation $$x=-4$$, and the y-intercept can be calculated by the equation $$-y=-4$$. Solving these equations, we get the x-intercept as $$-4$$ and the y-intercept as $$4$$. $$-4+4=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafbadeintercepts3","title":"Intercepts","body":"Find $$x$$ and $$y$$ intercepts of the line.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Graph with Intercepts","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafbadeintercepts3a","stepAnswer":["$$(3,0)$$"],"problemType":"MultipleChoice","stepTitle":"What is the $$x$$ intercept of the line in the graph?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(3,0)$$","choices":["$$(2,0)I(0,2)I(3,0)I(0,3)$$","$$(3,0)$$"],"hints":{"DefaultPathway":[{"id":"aafbadeintercepts3a-h1","type":"hint","dependencies":[],"title":"Graph crosses x-axis where?","text":"Find where the line in the graph crosses the x-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts3a-h2","type":"hint","dependencies":["aafbadeintercepts3a-h1"],"title":"Answer","text":"The answer is $$(3,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aafbadeintercepts3b","stepAnswer":["$$(0,2)$$"],"problemType":"MultipleChoice","stepTitle":"What is the $$y$$ intercept of the line in the graph?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,2)$$","choices":["$$(0,2)$$","$$(2,0)I(0,2)I(3,0)I(0,3)$$"],"hints":{"DefaultPathway":[{"id":"aafbadeintercepts3b-h1","type":"hint","dependencies":[],"title":"Graph crosses y-axis where?","text":"Find where the line in the graph crosses the y-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts3b-h2","type":"hint","dependencies":["aafbadeintercepts3b-h1"],"title":"Answer","text":"The answer is $$(0,2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafbadeintercepts30","title":"Finding the Intercepts of an Equation","body":"Identify the $$x$$ and $$y$$ intercepts to solve the given problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Graph with Intercepts","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafbadeintercepts30a","stepAnswer":["$$-6$$"],"problemType":"TextBox","stepTitle":"Find the sum of the x-value of the x-intercept and the y-value of the y-intercept of the following equation: $$4x-y=8$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-6$$","hints":{"DefaultPathway":[{"id":"aafbadeintercepts30a-h1","type":"hint","dependencies":[],"title":"Finding the X- and Y- Intercepts","text":"$$x+2y=8$$. The x-intercept can be calculated by the equation $$x=8$$, and the y-intercept can be calculated by the equation $$2y=8$$. Solving these equations, we get the x-intercept as $$8$$ and the y-intercept as $$4$$. $$8+4=12$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafbadeintercepts31","title":"Finding the Intercepts of an Equation","body":"Identify the $$x$$ and $$y$$ intercepts to solve the given problem.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Graph with Intercepts","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafbadeintercepts31a","stepAnswer":["$$-4$$"],"problemType":"TextBox","stepTitle":"Find the sum of the x-value of the x-intercept and the y-value of the y-intercept of the following equation: $$5x-y=5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-4$$","hints":{"DefaultPathway":[{"id":"aafbadeintercepts31a-h1","type":"hint","dependencies":[],"title":"Finding the X- and Y- Intercepts","text":"$$x+2y=10$$. The x-intercept can be calculated by the equation $$x=10$$, and the y-intercept can be calculated by the equation $$2y=10$$. Solving these equations, we get the x-intercept as $$10$$ and the y-intercept as $$5$$. $$10+5=15$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafbadeintercepts4","title":"Intercepts","body":"Find $$x$$ and $$y$$ intercepts from an equation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Graph with Intercepts","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafbadeintercepts4a","stepAnswer":["$$(3,0)$$"],"problemType":"MultipleChoice","stepTitle":"Find the $$x$$ intercept of $$2x+y=6$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(3,0)$$","choices":["$$(3,0)$$","$$(3,0)I(6,0)I(-5,0)I(0,-2)$$"],"hints":{"DefaultPathway":[{"id":"aafbadeintercepts4a-h1","type":"hint","dependencies":[],"title":"Let $$y=0$$","text":"Set $$y=0$$ and begin solving the equation. The remaining portion of the equation is $$2x+0=6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts4a-h2","type":"hint","dependencies":["aafbadeintercepts4a-h1"],"title":"Solve for $$x$$","text":"Divide both sides by $$2$$ to isolate $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts4a-h3","type":"hint","dependencies":["aafbadeintercepts4a-h2"],"title":"Interpret","text":"After isolating $$x$$, we find that $$x=3$$. Therefore, the $$x$$ coordinate when $$y=0$$ is $$x=3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts4a-h4","type":"hint","dependencies":["aafbadeintercepts4a-h3"],"title":"Answer","text":"The answer is $$(3,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aafbadeintercepts4b","stepAnswer":["$$(0,6)$$"],"problemType":"MultipleChoice","stepTitle":"Find the $$y$$ intercept of $$2x+y=6$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,6)$$","choices":["$$(0,3)I(0,6)I(0,-5)I(0,-2)$$","$$(0,6)$$"],"hints":{"DefaultPathway":[{"id":"aafbadeintercepts4b-h1","type":"hint","dependencies":[],"title":"Let $$x=0$$","text":"Set $$x=0$$ and begin solving the equation. The remaining portion of the equation is $$0+y=6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts4b-h2","type":"hint","dependencies":["aafbadeintercepts4b-h1"],"title":"Interpret","text":"After isolating $$y$$, we find that $$y=3$$. Therefore, the $$x$$ coordinate when $$x=0$$ is $$y=6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts4b-h3","type":"hint","dependencies":["aafbadeintercepts4b-h2"],"title":"Answer","text":"The answer is $$(0,6)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafbadeintercepts5","title":"Intercepts","body":"Find $$x$$ and $$y$$ intercepts from an equation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Graph with Intercepts","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafbadeintercepts5a","stepAnswer":["$$(4,0)$$"],"problemType":"MultipleChoice","stepTitle":"Find the $$x$$ intercept of $$3x+y=12$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(4,0)$$","choices":["$$(2,0)I(4,0)I(-6,0)I(0,-3)$$","$$(4,0)$$"],"hints":{"DefaultPathway":[{"id":"aafbadeintercepts5a-h1","type":"hint","dependencies":[],"title":"Let $$y=0$$","text":"Set $$y=0$$ and begin solving the equation. The remaining portion of the equation is $$3x+0=12$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts5a-h2","type":"hint","dependencies":["aafbadeintercepts5a-h1"],"title":"Solve for $$x$$","text":"Divide both sides by $$4$$ to isolate $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts5a-h3","type":"hint","dependencies":["aafbadeintercepts5a-h2"],"title":"Interpret","text":"After isolating $$x$$, we find that $$x=4$$. Therefore, the $$x$$ coordinate when $$y=0$$ is $$x=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts5a-h4","type":"hint","dependencies":["aafbadeintercepts5a-h3"],"title":"Answer","text":"The answer is $$(4,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aafbadeintercepts5b","stepAnswer":["$$(0,12)$$"],"problemType":"MultipleChoice","stepTitle":"Find the $$y$$ intercept of $$2x+y=6$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,12)$$","choices":["$$(0,12)$$","$$(0,12)I(0,3)I(0,-2)I(0,-7)$$"],"hints":{"DefaultPathway":[{"id":"aafbadeintercepts5b-h1","type":"hint","dependencies":[],"title":"Let $$x=0$$","text":"Set $$x=0$$ and begin solving the equation. The remaining portion of the equation is $$y=12$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts5b-h2","type":"hint","dependencies":["aafbadeintercepts5b-h1"],"title":"Interpret","text":"After substituting $$x$$, we find that $$y=12$$. Therefore, the $$x$$ coordinate when $$x=0$$ is $$y=12$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts5b-h3","type":"hint","dependencies":["aafbadeintercepts5b-h2"],"title":"Answer","text":"The answer is $$(0,12)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafbadeintercepts6","title":"Intercepts","body":"Find $$x$$ and $$y$$ intercepts from an equation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Graph with Intercepts","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafbadeintercepts6a","stepAnswer":["$$(8,0)$$"],"problemType":"MultipleChoice","stepTitle":"Find the $$x$$ intercept of $$x+4y=8$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(8,0)$$","choices":["$$(8,0)$$","$$(8,0)I(4,0)I(-6,0)I(0,-3)$$"],"hints":{"DefaultPathway":[{"id":"aafbadeintercepts6a-h1","type":"hint","dependencies":[],"title":"Let $$y=0$$","text":"Set $$y=0$$ and begin solving the equation. The remaining portion of the equation is $$x+0=8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts6a-h2","type":"hint","dependencies":["aafbadeintercepts6a-h1"],"title":"Interpret","text":"After isolating $$x$$, we find that $$x=8$$. Therefore, the $$x$$ coordinate when $$y=0$$ is $$x=8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts6a-h3","type":"hint","dependencies":["aafbadeintercepts6a-h2"],"title":"Answer","text":"The answer is $$(8,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aafbadeintercepts6b","stepAnswer":["$$(0,2)$$"],"problemType":"MultipleChoice","stepTitle":"Find the $$y$$ intercept of $$x+4y=8$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,2)$$","choices":["$$(0,2)$$","$$(0,2)I(0,3)I(0,-2)I(0,-7)$$"],"hints":{"DefaultPathway":[{"id":"aafbadeintercepts6b-h1","type":"hint","dependencies":[],"title":"Let $$x=0$$","text":"Set $$x=0$$ and begin solving the equation. The remaining portion of the equation is $$4y=8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts6b-h2","type":"hint","dependencies":["aafbadeintercepts6b-h1"],"title":"Solve for $$y$$","text":"Divide both sides by $$4$$ to get $$y=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts6b-h3","type":"hint","dependencies":["aafbadeintercepts6b-h2"],"title":"Interpret","text":"After substituting $$x$$, we find that $$y=2$$. Therefore, the $$x$$ coordinate when $$x=0$$ is $$y=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts6b-h4","type":"hint","dependencies":["aafbadeintercepts6b-h3"],"title":"Answer","text":"The answer is $$(0,2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafbadeintercepts7","title":"Intercepts","body":"Find $$x$$ and $$y$$ intercepts from an equation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Graph with Intercepts","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafbadeintercepts7a","stepAnswer":["$$(3,0)$$"],"problemType":"MultipleChoice","stepTitle":"Find the $$x$$ intercept of $$4x-3y=12$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(3,0)$$","choices":["$$(2,0)I(4,0)I(3,0)I(0,-3)$$","$$(3,0)$$"],"hints":{"DefaultPathway":[{"id":"aafbadeintercepts7a-h1","type":"hint","dependencies":[],"title":"Let $$y=0$$","text":"Set $$y=0$$ and begin solving the equation. The remaining portion of the equation is $$4x-0=12$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts7a-h2","type":"hint","dependencies":["aafbadeintercepts7a-h1"],"title":"Interpret","text":"After substituting $$x$$, we find that $$x=3$$. Therefore, the $$x$$ coordinate when $$y=0$$ is $$x=3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts7a-h3","type":"hint","dependencies":["aafbadeintercepts7a-h2"],"title":"Answer","text":"The answer is $$(3,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafbadeintercepts8","title":"Intercepts","body":"Find $$x$$ and $$y$$ intercepts from an equation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Graph with Intercepts","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafbadeintercepts8a","stepAnswer":["$$(0,-4)$$"],"problemType":"MultipleChoice","stepTitle":"Find the $$y$$ intercept of $$4x-3y=12$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,-4)$$","choices":["$$(0,-4)$$","$$(0,2)I(0,-4)I(0,-2)I(0,-7)$$"],"hints":{"DefaultPathway":[{"id":"aafbadeintercepts8a-h1","type":"hint","dependencies":[],"title":"Let $$x=0$$","text":"Set $$x=0$$ and begin solving the equation. The remaining portion of the equation is $$0-3y=12$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts8a-h2","type":"hint","dependencies":["aafbadeintercepts8a-h1"],"title":"Solve for $$y$$","text":"Divide both sides by $$-3$$ to get $$y=-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts8a-h3","type":"hint","dependencies":["aafbadeintercepts8a-h2"],"title":"Interpret","text":"After substituting $$x$$, we find that $$y=-4$$. Therefore, the $$x$$ coordinate when $$x=0$$ is $$y=-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts8a-h4","type":"hint","dependencies":["aafbadeintercepts8a-h3"],"title":"Answer","text":"The answer is $$(0,-4)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafbadeintercepts9","title":"Intercepts","body":"Find $$x$$ and $$y$$ intercepts from an equation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Graph with Intercepts","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafbadeintercepts9a","stepAnswer":["$$(4,0)$$"],"problemType":"MultipleChoice","stepTitle":"Find the $$x$$ intercept of $$3x-4y=12$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(4,0)$$","choices":["$$(1,0)I(4,0)I(-2,0)I(0,-3)$$","$$(4,0)$$"],"hints":{"DefaultPathway":[{"id":"aafbadeintercepts9a-h1","type":"hint","dependencies":[],"title":"Let $$y=0$$","text":"Set $$y=0$$ and begin solving the equation. The remaining portion of the equation is $$3x-0=12$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts9a-h2","type":"hint","dependencies":["aafbadeintercepts9a-h1"],"title":"Interpret","text":"After substituting $$x$$, we find that $$x=4$$. Therefore, the $$x$$ coordinate when $$y=0$$ is $$x=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafbadeintercepts9a-h3","type":"hint","dependencies":["aafbadeintercepts9a-h2"],"title":"Answer","text":"The answer is $$(4,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafc2dcMultiply1","title":"Multiply","body":"Calculate the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Multiply and Divide Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafc2dcMultiply1a","stepAnswer":["$$-27$$"],"problemType":"TextBox","stepTitle":"$$-9\\\\times3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-27$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply1a-h1","type":"hint","dependencies":[],"title":"Multiplication Definition","text":"$$a b$$ means add a, $$b$$ times.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$27$$"],"dependencies":["aafc2dcMultiply1a-h1"],"title":"Multiplying Positive Parts","text":"Let\'s ignore the negative sign first. What is $$9\\\\times3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aafc2dcMultiply1a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$27$$"],"dependencies":[],"title":"Multiplying Positive Parts","text":"To calculate $$9\\\\times3$$, we can use the definition of multiplication: $$9\\\\times3=9+9+9$$. What is $$9+9+9$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aafc2dcMultiply1a-h3","type":"hint","dependencies":["aafc2dcMultiply1a-h2"],"title":"Negative Sign","text":"For multiplication of two signed numbers, when the signs are different, the product is negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-27$$"],"dependencies":["aafc2dcMultiply1a-h3"],"title":"Negative Sign","text":"If the sign for our final answer is negative, what should our final answer be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafc2dcMultiply10","title":"Translate and Simplify","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Multiply and Divide Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafc2dcMultiply10a","stepAnswer":["$$-1$$"],"problemType":"TextBox","stepTitle":"Translate and simplify: the sum of $$8$$ and $$-12$$, increased by $$3$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply10a-h1","type":"hint","dependencies":[],"title":"Translating into math expression","text":"The phrase translates into [8+(-12)]+3","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["aafc2dcMultiply10a-h1"],"title":"Simplify","text":"By the orders of operation, we evaluate $$8+\\\\left(-12\\\\right)$$ first. What is $$8+\\\\left(-12\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["aafc2dcMultiply10a-h2"],"title":"Simplify","text":"The last step: what is $$-4+3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafc2dcMultiply11","title":"Translate and Simplify","body":"Translate and then simplify the following:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Multiply and Divide Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafc2dcMultiply11a","stepAnswer":["$$34$$"],"problemType":"TextBox","stepTitle":"The difference of $$13$$ and $$-21$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$34$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply11a-h1","type":"hint","dependencies":[],"title":"Translating into math expression","text":"The phrase translates into $$13-(-21)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$34$$"],"dependencies":["aafc2dcMultiply11a-h1"],"title":"Simplify","text":"What is $$13-(-21)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aafc2dcMultiply11b","stepAnswer":["$$-43$$"],"problemType":"TextBox","stepTitle":"subtract $$24$$ from $$-19$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-43$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply11b-h1","type":"hint","dependencies":[],"title":"Translating into math expression","text":"Remember, \\"subtract $$b$$ from a\\" means $$a-b$$. So the phrase translates to $$-19-24$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply11b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-43$$"],"dependencies":["aafc2dcMultiply11b-h1"],"title":"Simplify","text":"What is $$-19-24$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafc2dcMultiply12","title":"Translate and Simplify","body":"Translate to an algebraic expression and simplify if possible:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Multiply and Divide Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafc2dcMultiply12a","stepAnswer":["$$-28$$"],"problemType":"TextBox","stepTitle":"The product of $$-2$$ and $$14$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-28$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply12a-h1","type":"hint","dependencies":[],"title":"Translating into math expression","text":"The phrase translates into $$14\\\\left(-2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-28$$"],"dependencies":["aafc2dcMultiply12a-h1"],"title":"Simplify","text":"What is $$14\\\\left(-2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafc2dcMultiply13","title":"Translate and Simplify","body":"Translate to an algebraic expression and simplify if possible:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Multiply and Divide Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafc2dcMultiply13a","stepAnswer":["$$8$$"],"problemType":"TextBox","stepTitle":"The quotient of $$-56$$ and $$-7$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply13a-h1","type":"hint","dependencies":[],"title":"Translating into math expression","text":"The phrase translates into $$\\\\frac{-56}{\\\\left(-7\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["aafc2dcMultiply13a-h1"],"title":"Simplify","text":"What is $$\\\\frac{-56}{\\\\left(-7\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply13a-h3","type":"hint","dependencies":["aafc2dcMultiply13a-h2"],"title":"Simplify","text":"Since $$-56$$ and $$-7$$ have the same sign, the answer is positive: $$\\\\frac{-56}{\\\\left(-7\\\\right)}=8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafc2dcMultiply14","title":"How to Apply a Strategy to Solve Applications with Integers","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Multiply and Divide Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafc2dcMultiply14a","stepAnswer":["$$20$$"],"problemType":"TextBox","stepTitle":"In the morning, the temperature in Urbana, Illinois was $$11$$ degrees. By mid-afternoon, the temperature had dropped to $$-9$$ degrees. What was the difference of the morning and afternoon temperatures?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$20$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply14a-h1","type":"hint","dependencies":[],"title":"Read the Problem","text":"The first step is to make sure all the words and ideas are understood.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply14a-h2","type":"hint","dependencies":["aafc2dcMultiply14a-h1"],"title":"Identify the Unknown","text":"What we are asked to find is the difference of the morning and afternoon temperatures.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply14a-h3","type":"hint","dependencies":["aafc2dcMultiply14a-h2"],"title":"Rephrase the Question","text":"To rephrase the questions, we are finding the different of $$11$$ and $$-9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply14a-h4","type":"hint","dependencies":["aafc2dcMultiply14a-h3"],"title":"Translate into Math","text":"Translate the phrase into mathematical expression, we get $$11-(-9)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply14a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["aafc2dcMultiply14a-h4"],"title":"Simplify","text":"What is $$11-(-9)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafc2dcMultiply15","title":"How to Apply a Strategy to Solve Applications with Integers","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Multiply and Divide Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafc2dcMultiply15a","stepAnswer":["$$45$$"],"problemType":"TextBox","stepTitle":"The Mustangs football team received three penalties in the third quarter. Each penalty gave them a loss of fifteen yards. What is the number of yards lost?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$45$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply15a-h1","type":"hint","dependencies":[],"title":"Read the Problem","text":"The first step is to make sure all the words and ideas are understood.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply15a-h2","type":"hint","dependencies":["aafc2dcMultiply15a-h1"],"title":"Identify the Unknown","text":"What we are asked to find is the number of yards lost.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply15a-h3","type":"hint","dependencies":["aafc2dcMultiply15a-h2"],"title":"Rephrase the Question","text":"To rephrase the questions, we are finding the value of three times a 15-yard penalty.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply15a-h4","type":"hint","dependencies":["aafc2dcMultiply15a-h3"],"title":"Translate into Math","text":"Translate the phrase into mathematical expression, we get $$3\\\\times15$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply15a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$45$$"],"dependencies":["aafc2dcMultiply15a-h4"],"title":"Simplify","text":"What is $$3\\\\times15$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafc2dcMultiply16","title":"Multiply Integers","body":"Multiply:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Multiply and Divide Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafc2dcMultiply16a","stepAnswer":["$$-32$$"],"problemType":"TextBox","stepTitle":"$$-4\\\\times8$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-32$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply16a-h1","type":"hint","dependencies":[],"title":"Interpret","text":"We are adding $$-4$$, $$8$$ times","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply16a-h2","type":"hint","dependencies":["aafc2dcMultiply16a-h1"],"title":"Signs","text":"The signs are different(one is negative and one is positive), so the product is negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply16a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-32$$"],"dependencies":["aafc2dcMultiply16a-h2"],"title":"Multiply","text":"What\'s the result of this multiplication?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafc2dcMultiply17","title":"Multiply Integers","body":"Multiply:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Multiply and Divide Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafc2dcMultiply17a","stepAnswer":["$$14$$"],"problemType":"TextBox","stepTitle":"$$-1\\\\left(-14\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$14$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply17a-h1","type":"hint","dependencies":[],"title":"Multiply","text":"Each time we multiply a number by $$-1$$, we get its opposite!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply17a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$14$$"],"dependencies":["aafc2dcMultiply17a-h1"],"title":"Opposite","text":"What is the opposite of -14?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafc2dcMultiply18","title":"Divide Integers","body":"Divide:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Multiply and Divide Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafc2dcMultiply18a","stepAnswer":["$$13$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{-52}{\\\\left(-4\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$13$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply18a-h1","type":"hint","dependencies":[],"title":"Division Rule","text":"With signs that are the same(both are negative), the quotient is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$13$$"],"dependencies":["aafc2dcMultiply18a-h1"],"title":"Divide","text":"What\'s the result of this division?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafc2dcMultiply19","title":"Simplify Expressions with Integers","body":"Simplify the expression:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Multiply and Divide Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafc2dcMultiply19a","stepAnswer":["$$-47$$"],"problemType":"TextBox","stepTitle":"$$5\\\\left(-6\\\\right)+7\\\\left(-2\\\\right)-3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-47$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply19a-h1","type":"hint","dependencies":[],"title":"Multiply","text":"We need to multiply first.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-30$$"],"dependencies":["aafc2dcMultiply19a-h1"],"title":"Multiply","text":"What do we get for $$5\\\\left(-6\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply19a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-14$$"],"dependencies":["aafc2dcMultiply19a-h2"],"title":"Multiply","text":"What do we get for $$7\\\\left(-2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply19a-h4","type":"hint","dependencies":["aafc2dcMultiply19a-h3"],"title":"Add","text":"We then need to add $$-30$$ and $$-14$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply19a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-44$$"],"dependencies":["aafc2dcMultiply19a-h4"],"title":"Add","text":"What do we get after the addition?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply19a-h6","type":"hint","dependencies":["aafc2dcMultiply19a-h5"],"title":"Subtract","text":"We then need to subtract $$3$$ from $$-44$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply19a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-47$$"],"dependencies":["aafc2dcMultiply19a-h6"],"title":"Subtract","text":"What do we get after the subtraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafc2dcMultiply2","title":"Multiply by $$-1$$","body":"Calculate the following expressions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Multiply and Divide Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafc2dcMultiply2a","stepAnswer":["$$-7$$"],"problemType":"TextBox","stepTitle":"$$-1\\\\times7$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-7$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply2a-h1","type":"hint","dependencies":[],"title":"Multiplication by $$-1$$","text":"Multiplying a number by $$-1$$ gives its opposite: $$-1a=-a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-7$$"],"dependencies":["aafc2dcMultiply2a-h1"],"title":"Answer","text":"Since multiiplying $$7$$ by $$-1$$ gives the opposite of $$7$$, our answer is the opposite of $$7$$. What is that?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aafc2dcMultiply2b","stepAnswer":["$$11$$"],"problemType":"TextBox","stepTitle":"$$-1\\\\left(-11\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$11$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply2b-h1","type":"hint","dependencies":[],"title":"Multiplication by $$-1$$","text":"Multiplying a number by $$-1$$ gives its opposite: $$-1a=-a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply2b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11$$"],"dependencies":["aafc2dcMultiply2b-h1"],"title":"Answer","text":"Since multiiplying $$-11$$ by $$-1$$ gives the opposite of $$-11$$, our answer is the opposite of $$-11$$. What is that?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafc2dcMultiply20","title":"Simplify Expressions with Integers","body":"Simplify the expression:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Multiply and Divide Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafc2dcMultiply20a","stepAnswer":["$$64$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(-2\\\\right)}^6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$64$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply20a-h1","type":"hint","dependencies":[],"title":"Expanded Form","text":"We need to first write the expression in expanded form $$\\\\left(-2\\\\right) \\\\left(-2\\\\right) \\\\left(-2\\\\right) \\\\left(-2\\\\right) \\\\left(-2\\\\right) \\\\left(-2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["aafc2dcMultiply20a-h1"],"title":"Multiply","text":"What do we get for $$\\\\left(-2\\\\right) \\\\left(-2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply20a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-8$$"],"dependencies":["aafc2dcMultiply20a-h2"],"title":"Multiply","text":"What do we get for $$4\\\\left(-2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply20a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["aafc2dcMultiply20a-h3"],"title":"Multiply","text":"What do we get for $$\\\\left(-8\\\\right) \\\\left(-2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply20a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-32$$"],"dependencies":["aafc2dcMultiply20a-h4"],"title":"Multiply","text":"What do we get for $$16\\\\left(-2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply20a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$64$$"],"dependencies":["aafc2dcMultiply20a-h5"],"title":"Multiply","text":"What do we get for $$\\\\left(-32\\\\right) \\\\left(-2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafc2dcMultiply21","title":"Simplify Expressions with Integers","body":"Simplify the expression:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Multiply and Divide Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafc2dcMultiply21a","stepAnswer":["$$41$$"],"problemType":"TextBox","stepTitle":"$$26-3\\\\left(2-7\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$41$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply21a-h1","type":"hint","dependencies":[],"title":"Parentheses","text":"We need to subtract in parentheses first.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply21a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["aafc2dcMultiply21a-h1"],"title":"Subtract","text":"What do we get for $$2-7$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply21a-h3","type":"hint","dependencies":["aafc2dcMultiply21a-h2"],"title":"Multiply","text":"We then need to multiply $$3$$ with $$-5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply21a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-15$$"],"dependencies":["aafc2dcMultiply21a-h3"],"title":"Multiply","text":"What do we get after the multiplication?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply21a-h5","type":"hint","dependencies":["aafc2dcMultiply21a-h4"],"title":"Subtract","text":"We then need to subtract $$-15$$ from $$26$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply21a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$41$$"],"dependencies":["aafc2dcMultiply21a-h5"],"title":"Subtract","text":"What do we get after the subtraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafc2dcMultiply22","title":"Simplify Expressions with Integers","body":"Simplify the expression:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Multiply and Divide Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafc2dcMultiply22a","stepAnswer":["$$-9$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{65}{\\\\left(-5\\\\right)}+\\\\frac{\\\\left(-28\\\\right)}{\\\\left(-7\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-9$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply22a-h1","type":"hint","dependencies":[],"title":"Divide","text":"We need to divide first.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply22a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-13$$"],"dependencies":["aafc2dcMultiply22a-h1"],"title":"Divide","text":"What do we get for $$\\\\frac{65}{\\\\left(-5\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply22a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["aafc2dcMultiply22a-h2"],"title":"Divide","text":"What do we get for $$\\\\frac{\\\\left(-28\\\\right)}{\\\\left(-7\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply22a-h4","type":"hint","dependencies":["aafc2dcMultiply22a-h3"],"title":"Add","text":"We then need to add the two quotients.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply22a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":["aafc2dcMultiply22a-h4"],"title":"Add","text":"What do we get for $$\\\\left(-13\\\\right)+4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafc2dcMultiply23","title":"Simplify Expressions with Integers","body":"Simplify the expression:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Multiply and Divide Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafc2dcMultiply23a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(-3\\\\right)}^2-\\\\frac{24}{8-2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply23a-h1","type":"hint","dependencies":[],"title":"Exponents","text":"We need to evaluate the exponentiation term first.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply23a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["aafc2dcMultiply23a-h1"],"title":"Exponents","text":"What do we get for $${\\\\left(-3\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply23a-h3","type":"hint","dependencies":["aafc2dcMultiply23a-h2"],"title":"Parentheses","text":"We then need to subtract in parentheses.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply23a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["aafc2dcMultiply23a-h3"],"title":"Subtract","text":"What do we get for $$8-2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply23a-h5","type":"hint","dependencies":["aafc2dcMultiply23a-h4"],"title":"Divide","text":"We then need to divide.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply23a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["aafc2dcMultiply23a-h5"],"title":"Divide","text":"What do we get for $$\\\\frac{24}{6}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply23a-h7","type":"hint","dependencies":["aafc2dcMultiply23a-h6"],"title":"Subtract","text":"Lastly, we need to subtract the quotient from the exponentiation term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply23a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["aafc2dcMultiply23a-h7"],"title":"Subtract","text":"What do we get for $$9-4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafc2dcMultiply24","title":"Evaluate Variable Expressions with Integers","body":"Evaluate the expression:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Multiply and Divide Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafc2dcMultiply24a","stepAnswer":["$$121$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(x+y\\\\right)}^2$$ when $$x=-3$$, $$y=14$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$121$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply24a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We first need to substitute $$-3$$ for $$x$$ and $$14$$ for $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply24a-h2","type":"hint","dependencies":["aafc2dcMultiply24a-h1"],"title":"Add","text":"We then need to add inside parentheses.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply24a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11$$"],"dependencies":["aafc2dcMultiply24a-h2"],"title":"Add","text":"What do we get for $$-3+14$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply24a-h4","type":"hint","dependencies":["aafc2dcMultiply24a-h3"],"title":"Exponents","text":"We then need to evaluate the exponentiation term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply24a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$121$$"],"dependencies":["aafc2dcMultiply24a-h4"],"title":"Exponents","text":"What do we get for $${11}^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafc2dcMultiply25","title":"Translate Phrases to Expressions with Integers","body":"In the following exercises, translate to an algebraic expression and simplify if possible.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Multiply and Divide Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafc2dcMultiply25a","stepAnswer":["$$-5$$"],"problemType":"TextBox","stepTitle":"The sum of $$3$$ and $$-15$$, increased by $$7$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-5$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply25a-h1","type":"hint","dependencies":[],"title":"Translate","text":"We first need to translate the phrase to expressions with integers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply25a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3+\\\\left(-15\\\\right)+7$$"],"dependencies":["aafc2dcMultiply25a-h1"],"title":"Translate","text":"What do we get after translating?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$3-\\\\left(-15\\\\right)+7$$","$$3+\\\\left(-15\\\\right)+7$$","$$3+\\\\left(-15\\\\right)-7$$","$$(3-(-15))-7$$"]},{"id":"aafc2dcMultiply25a-h3","type":"hint","dependencies":["aafc2dcMultiply25a-h2"],"title":"Simplify","text":"Now, we want to simplify the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply25a-h4","type":"hint","dependencies":["aafc2dcMultiply25a-h3"],"title":"Simplify","text":"We first need to add inside parentheses.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply25a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-12$$"],"dependencies":["aafc2dcMultiply25a-h4"],"title":"Add","text":"What do we get for $$3+\\\\left(-15\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply25a-h6","type":"hint","dependencies":["aafc2dcMultiply25a-h5"],"title":"Add","text":"Lastly, we need to add $$-12$$ with $$7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply25a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["aafc2dcMultiply25a-h6"],"title":"Add","text":"What do we get after the addition?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafc2dcMultiply26","title":"Checking Account","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Multiply and Divide Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafc2dcMultiply26a","stepAnswer":["$$-28$$"],"problemType":"TextBox","stepTitle":"Mayra has $124 in her checking account. She writes a check for $152. What is the new balance in her checking account?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-28$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply26a-h1","type":"hint","dependencies":[],"title":"Identify","text":"We are asked to find the new balance in her checking account.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply26a-h2","type":"hint","dependencies":["aafc2dcMultiply26a-h1"],"title":"Write","text":"Let\'s write a phrase that gives the information to find it - the difference of $$124$$ and $$152$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply26a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$124-152$$"],"dependencies":["aafc2dcMultiply26a-h2"],"title":"Translate","text":"What do we get after translating the phrase to an expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$152-124$$","$$124+152$$","$$124-152$$"]},{"id":"aafc2dcMultiply26a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-28$$"],"dependencies":["aafc2dcMultiply26a-h3"],"title":"Simplify","text":"What do we get after the subtraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafc2dcMultiply27","title":"Evaluate Variable Expressions with Integers","body":"Evaluate the expression:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Multiply and Divide Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafc2dcMultiply27a","stepAnswer":["$$9$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(8-11\\\\right) \\\\left(9-12\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply27a-h1","type":"hint","dependencies":[],"title":"Parentheses","text":"We first need to subtract in parentheses.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply27a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["aafc2dcMultiply27a-h1"],"title":"Subtract","text":"What do we get for $$8-11$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply27a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["aafc2dcMultiply27a-h2"],"title":"Subtract","text":"What do we get for $$9-12$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply27a-h4","type":"hint","dependencies":["aafc2dcMultiply27a-h3"],"title":"Multiply","text":"We then need to multiply the two differences.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply27a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["aafc2dcMultiply27a-h4"],"title":"Multiply","text":"What do we get for $$-3\\\\left(-3\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafc2dcMultiply28","title":"Evaluate Variable Expressions with Integers","body":"Evaluate the expression:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Multiply and Divide Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafc2dcMultiply28a","stepAnswer":["$$-29$$"],"problemType":"TextBox","stepTitle":"$$9-2\\\\left(3-8\\\\left(-2\\\\right)\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-29$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply28a-h1","type":"hint","dependencies":[],"title":"Parentheses","text":"We first need to evaluate the expression in parentheses.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply28a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-16$$"],"dependencies":["aafc2dcMultiply28a-h1"],"title":"Multiply","text":"What do we get for $$8\\\\left(-2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply28a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$19$$"],"dependencies":["aafc2dcMultiply28a-h2"],"title":"Subtract","text":"What do we get for $$3-(-16)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply28a-h4","type":"hint","dependencies":["aafc2dcMultiply28a-h3"],"title":"Multiply","text":"We then need to multiply $$19$$ with $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply28a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$38$$"],"dependencies":["aafc2dcMultiply28a-h4"],"title":"Multiply","text":"What do we get after the multiplication?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply28a-h6","type":"hint","dependencies":["aafc2dcMultiply28a-h5"],"title":"Subtract","text":"Lastly, we need to find the difference of $$9$$ and $$38$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply28a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-29$$"],"dependencies":["aafc2dcMultiply28a-h6"],"title":"Subtract","text":"What do we get after the subtraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafc2dcMultiply29","title":"Divide Integers","body":"Divide:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Multiply and Divide Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafc2dcMultiply29a","stepAnswer":["$$-12$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{-180}{15}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-12$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply29a-h1","type":"hint","dependencies":[],"title":"Division Rule","text":"With different signs, the quotient is negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply29a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-12$$"],"dependencies":["aafc2dcMultiply29a-h1"],"title":"Divide","text":"What\'s the result of this division?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafc2dcMultiply3","title":"Divide Intergers","body":"Calculate the following expressions.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Multiply and Divide Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafc2dcMultiply3a","stepAnswer":["$$-9$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{-27}{3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-9$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply3a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":[],"title":"Dividing Positive Parts","text":"Let\'s ignore the negative sign first. What is $$\\\\frac{27}{3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply3a-h2","type":"hint","dependencies":["aafc2dcMultiply3a-h1"],"title":"Negative Sign","text":"For division of two signed numbers, when the signs are different, the quotient is negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":["aafc2dcMultiply3a-h2"],"title":"Negative Sign","text":"If the sign for our final answer is negative, what should our final answer be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aafc2dcMultiply3b","stepAnswer":["$$25$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{-100}{\\\\left(-4\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$25$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply3b-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":[],"title":"Dividing Positive Parts","text":"Let\'s ignore the negative sign first. What is $$\\\\frac{100}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply3b-h2","type":"hint","dependencies":["aafc2dcMultiply3b-h1"],"title":"Negative Sign","text":"For division of two signed numbers, when the signs are the same, the quotient is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply3b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["aafc2dcMultiply3b-h2"],"title":"Negative Sign","text":"If the sign for our final answer is positive, what should our final answer be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafc2dcMultiply30","title":"Evaluate Variable Expressions with Integers","body":"Evaluate the expression:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Multiply and Divide Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafc2dcMultiply30a","stepAnswer":["$$90$$"],"problemType":"TextBox","stepTitle":"$$-3\\\\left(-5\\\\right)\\\\times6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$90$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply30a-h1","type":"hint","dependencies":[],"title":"Multiplication Rule","text":"With signs that are the same, the product is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply30a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["aafc2dcMultiply30a-h1"],"title":"Multiply","text":"What do we get for $$-3\\\\left(-5\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply30a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$90$$"],"dependencies":["aafc2dcMultiply30a-h2"],"title":"Multiply","text":"What do we get for $$15\\\\times6$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafc2dcMultiply4","title":"Simplify Expressions","body":"Simplify the following integer expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Multiply and Divide Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafc2dcMultiply4a","stepAnswer":["$$-48$$"],"problemType":"TextBox","stepTitle":"$$7\\\\left(-2\\\\right)+4\\\\left(-7\\\\right)-6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-48$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply4a-h1","type":"hint","dependencies":[],"title":"Multiplying","text":"First, by the order of operations, we should do the multiplications first.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-14$$"],"dependencies":["aafc2dcMultiply4a-h1"],"title":"Multiplying","text":"What is $$7\\\\left(-2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply4a-h3","type":"hint","dependencies":["aafc2dcMultiply4a-h2"],"title":"Multiplying","text":"$$7\\\\times2=14$$, and since $$7$$ and $$-2$$ have different signs, our final answer is $$-14$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-28$$"],"dependencies":["aafc2dcMultiply4a-h3"],"title":"Multiplying","text":"What is $$4\\\\left(-7\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply4a-h5","type":"hint","dependencies":["aafc2dcMultiply4a-h4"],"title":"Multiplying","text":"$$4\\\\times7=28$$, and since $$4$$ and $$-7$$ have different signs, our final answer is $$-28$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply4a-h6","type":"hint","dependencies":["aafc2dcMultiply4a-h5"],"title":"Expression after Multiplication","text":"After the multiplications, our expression now turns into $$-14+\\\\left(-28\\\\right)-6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply4a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-42$$"],"dependencies":["aafc2dcMultiply4a-h6"],"title":"Adding","text":"By the order of operation, we then proceed by adding the first two terms together. What is $$-14+\\\\left(-28\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply4a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-48$$"],"dependencies":["aafc2dcMultiply4a-h7"],"title":"Subtracting","text":"We are now left with the expression $$-42-6$$. What does this evaluate to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafc2dcMultiply5","title":"Simplify Expressions","body":"Simplify the following integer expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Multiply and Divide Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafc2dcMultiply5a","stepAnswer":["$$16$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(-2\\\\right)}^4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply5a-h1","type":"hint","dependencies":[],"title":"Write in Expanded Form","text":"We start by writing out the expression in its expanded form: $${\\\\left(-2\\\\right)}^4=\\\\left(-2\\\\right) \\\\left(-2\\\\right) \\\\left(-2\\\\right) \\\\left(-2\\\\right)$$, and we will simplify $$\\\\left(-2\\\\right) \\\\left(-2\\\\right) \\\\left(-2\\\\right) \\\\left(-2\\\\right)$$ by performing multiplication three times.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["aafc2dcMultiply5a-h1"],"title":"First Multiplication","text":"We now multiply the first $$\\\\left(-2\\\\right) \\\\left(-2\\\\right)$$ pair together and substitute the value back into the expression. What is $$\\\\left(-2\\\\right) \\\\left(-2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply5a-h3","type":"hint","dependencies":["aafc2dcMultiply5a-h2"],"title":"First Multiplication","text":"$$2\\\\times2=4$$, and since $$-2$$ and $$-2$$ have the same signs, our final answer is positive: $$\\\\left(-2\\\\right) \\\\left(-2\\\\right)=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply5a-h4","type":"hint","dependencies":["aafc2dcMultiply5a-h3"],"title":"Current Expression","text":"Substituting $$4$$ back, the orignal expression now becomes $$4\\\\left(-2\\\\right) \\\\left(-2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-8$$"],"dependencies":["aafc2dcMultiply5a-h4"],"title":"Second Multiplication","text":"What is $$4\\\\left(-2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply5a-h6","type":"hint","dependencies":["aafc2dcMultiply5a-h5"],"title":"Second Multiplication","text":"$$4\\\\times2=8$$, and since $$4$$ and $$-2$$ have different signs, our final answer is negative: $$4\\\\left(-2\\\\right)=-8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply5a-h7","type":"hint","dependencies":["aafc2dcMultiply5a-h6"],"title":"Current Expression","text":"Substituting $$-8$$ back, the orignal expression now becomes $$-8\\\\left(-2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply5a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["aafc2dcMultiply5a-h7"],"title":"Third Multiplication","text":"What is $$-8\\\\left(-2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply5a-h9","type":"hint","dependencies":["aafc2dcMultiply5a-h8"],"title":"Third Multiplication","text":"$$8\\\\times2=16$$, and since $$-8$$ and $$-2$$ have the same signs, our final answer is positive: $$-8\\\\left(-2\\\\right)=16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aafc2dcMultiply5b","stepAnswer":["$$-16$$"],"problemType":"TextBox","stepTitle":"$$-\\\\left(2^4\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-16$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply5b-h1","type":"hint","dependencies":[],"title":"Write in Expanded Form","text":"We start by writing out the expression in its expanded form: $$-\\\\left(2^4\\\\right)=-\\\\left(2\\\\times2\\\\times2\\\\times2\\\\right)$$, and we will simplify $$-\\\\left(2\\\\times2\\\\times2\\\\times2\\\\right)$$ by performing multiplication four times.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply5b-h2","type":"hint","dependencies":["aafc2dcMultiply5b-h1"],"title":"Order of Operations","text":"By the order of operations, we simplify what\'s in the parenthesis first.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply5b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["aafc2dcMultiply5b-h2"],"title":"First Multiplication","text":"We now multiply the first $$2\\\\times2$$ pair together and substitute the value back into the expression. What is $$2\\\\times2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply5b-h4","type":"hint","dependencies":["aafc2dcMultiply5b-h3"],"title":"First Multiplication","text":"$$2\\\\times2=4$$, and since $$2$$ and $$2$$ have the same signs, our final answer is positive: $$2\\\\times2=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply5b-h5","type":"hint","dependencies":["aafc2dcMultiply5b-h4"],"title":"Current Expression","text":"Substituting $$4$$ back, the orignal expression now becomes $$-\\\\left(4\\\\times2\\\\times2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply5b-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["aafc2dcMultiply5b-h5"],"title":"Second Multiplication","text":"What is $$4\\\\times2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply5b-h7","type":"hint","dependencies":["aafc2dcMultiply5b-h6"],"title":"Second Multiplication","text":"$$4\\\\times2=8$$, and since $$4$$ and $$2$$ have the same signs, our final answer is positive: $$4\\\\times2=8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply5b-h8","type":"hint","dependencies":["aafc2dcMultiply5b-h7"],"title":"Current Expression","text":"Substituting $$8$$ back, the orignal expression now becomes $$-\\\\left(8\\\\times2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply5b-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["aafc2dcMultiply5b-h8"],"title":"Third Multiplication","text":"What is $$8\\\\times2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply5b-h10","type":"hint","dependencies":["aafc2dcMultiply5b-h9"],"title":"Third Multiplication","text":"$$8\\\\times2=16$$, and since $$8$$ and $$2$$ have the same signs, our final answer is positive: $$8\\\\times2=16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply5b-h11","type":"hint","dependencies":["aafc2dcMultiply5b-h10"],"title":"Final Answer","text":"Substituting $$16$$ back in, we get our final answer $$-16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafc2dcMultiply6","title":"Simplify Expression","body":"Simply the following expression","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Multiply and Divide Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafc2dcMultiply6a","stepAnswer":["$$9$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{8\\\\left(-9\\\\right)}{{\\\\left(-2\\\\right)}^3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply6a-h1","type":"hint","dependencies":[],"title":"Exponent","text":"By the order of operation, we calculate the exponents in the expression first.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-8$$"],"dependencies":["aafc2dcMultiply6a-h1"],"title":"Exponent","text":"What is $${\\\\left(-2\\\\right)}^3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply6a-h3","type":"hint","dependencies":["aafc2dcMultiply6a-h2"],"title":"Exponent","text":"To calculate $${\\\\left(-2\\\\right)}^3$$, we can write it out as $$\\\\left(-2\\\\right) \\\\left(-2\\\\right) \\\\left(-2\\\\right)$$. We simplify it as $$\\\\left(-2\\\\right) \\\\left(-2\\\\right) \\\\left(-2\\\\right)=4\\\\left(-2\\\\right)=-8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply6a-h4","type":"hint","dependencies":["aafc2dcMultiply6a-h3"],"title":"Multiply","text":"Substituting $$-8$$ back for $${\\\\left(-2\\\\right)}^3$$, we get the expression $$\\\\frac{8\\\\left(-9\\\\right)}{\\\\left(-8\\\\right)}$$. We proceed by multiplying.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-72$$"],"dependencies":["aafc2dcMultiply6a-h4"],"title":"Multiply","text":"What is $$8\\\\left(-9\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply6a-h6","type":"hint","dependencies":["aafc2dcMultiply6a-h5"],"title":"Divide","text":"Substituting $$-72$$ back for $$8\\\\left(-9\\\\right)$$, we get the expression $$\\\\frac{-72}{\\\\left(-8\\\\right)}$$. We proceed by dividing.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply6a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["aafc2dcMultiply6a-h6"],"title":"Divide","text":"What is $$\\\\frac{-72}{\\\\left(-8\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafc2dcMultiply7","title":"Evaluate Variable Equation with Integers","body":"When $$y=-9$$, evaluate:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Multiply and Divide Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafc2dcMultiply7a","stepAnswer":["$$-1$$"],"problemType":"TextBox","stepTitle":"$$y+8$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply7a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Our first step is to substitute $$-9$$ for $$y$$ into the expression, and we get the expression $$-9+8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["aafc2dcMultiply7a-h1"],"title":"Evaluate","text":"What is $$-9+8$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aafc2dcMultiply7b","stepAnswer":["$$17$$"],"problemType":"TextBox","stepTitle":"$$-y+8$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$17$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply7b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Our first step is to substitute $$-9$$ for $$y$$ into the expression, and we get the expression $$-\\\\left(-9\\\\right)+8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply7b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["aafc2dcMultiply7b-h1"],"title":"Evaluate $$-y$$","text":"By the order of operation, the next step is to evaluate $$-(-9)$$. What is $$-(-9)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply7b-h3","type":"hint","dependencies":["aafc2dcMultiply7b-h2"],"title":"Evaluate $$-y$$","text":"The opposite of $$-9$$ is $$9$$, so $$-(-9)=9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply7b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$17$$"],"dependencies":["aafc2dcMultiply7b-h3"],"title":"Final Answer","text":"The expressin now becomes $$9+8$$. What is $$9+8$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafc2dcMultiply8","title":"Evaluate Variable Equation with Integers","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Multiply and Divide Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafc2dcMultiply8a","stepAnswer":["$$36$$"],"problemType":"TextBox","stepTitle":"Evaluate $${\\\\left(x+y\\\\right)}^2$$ when $$x=-18$$ and $$y=24$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$36$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply8a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Our first step is to substitute $$-18$$ for $$x$$ and $$24$$ for $$y$$ into the expression, and we get the expression $${\\\\left(-18+24\\\\right)}^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply8a-h2","type":"hint","dependencies":["aafc2dcMultiply8a-h1"],"title":"Order of Operations","text":"By the order of operations, we should evaluate what\'s inside the parenthesis first.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["aafc2dcMultiply8a-h2"],"title":"Addition","text":"What is $$-18+24$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply8a-h4","type":"hint","dependencies":["aafc2dcMultiply8a-h3"],"title":"Substitute","text":"Substitute $$6$$ back for $$-18+24$$, the expression now becomes $$6^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply8a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$36$$"],"dependencies":["aafc2dcMultiply8a-h4"],"title":"Exponent","text":"What is $$6^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply8a-h6","type":"hint","dependencies":["aafc2dcMultiply8a-h5"],"title":"Exponent","text":"$$6^2$$ is the same as $$6\\\\times6$$, which evaluates to $$36$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aafc2dcMultiply9","title":"Evaluate Variable Equation with Integers","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.4 Multiply and Divide Integers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aafc2dcMultiply9a","stepAnswer":["$$39$$"],"problemType":"TextBox","stepTitle":"Evaluate: $$3x^2-2x+6$$ when $$x=-3$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$39$$","hints":{"DefaultPathway":[{"id":"aafc2dcMultiply9a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Our first step is to substitute $$-3$$ for $$x$$ into the expression, and we get the $$3{\\\\left(-3\\\\right)}^2-2\\\\left(-3\\\\right)+6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["aafc2dcMultiply9a-h1"],"title":"Exponent","text":"By the order of operations, we should first evaluate the exponents. What is $${\\\\left(-3\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply9a-h3","type":"hint","dependencies":["aafc2dcMultiply9a-h2"],"title":"Exponent","text":"$${\\\\left(-3\\\\right)}^2$$ is the same as $$\\\\left(-3\\\\right) \\\\left(-3\\\\right)$$, which is $$9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply9a-h4","type":"hint","dependencies":["aafc2dcMultiply9a-h3"],"title":"Multiplication","text":"Substituting $$9$$ back into the expression, we get $$3\\\\times9-2\\\\left(-3\\\\right)+6$$. The next step is to evaluate the two multiplications.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply9a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$27$$"],"dependencies":["aafc2dcMultiply9a-h4"],"title":"First Multiplication","text":"What is $$3\\\\times9$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply9a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6$$"],"dependencies":["aafc2dcMultiply9a-h5"],"title":"Second Multiplication","text":"What is $$2\\\\left(-3\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply9a-h7","type":"hint","dependencies":["aafc2dcMultiply9a-h6"],"title":"$$\\\\frac{Addition}{Subtraction}$$","text":"Substituting $$27$$ and $$-6$$ back into the expression, we get $$27-\\\\left(-6\\\\right)+6$$. We need to evaluate this expression, from left to right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aafc2dcMultiply9a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$39$$"],"dependencies":["aafc2dcMultiply9a-h7"],"title":"$$\\\\frac{Addition}{Subtraction}$$","text":"What is $$27-\\\\left(-6\\\\right)+6$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaff50dradical1","title":"Simplifying Square Roots","body":"Simplify the following square roots.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Simplify Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aaff50dradical1a","stepAnswer":["$$4\\\\sqrt{3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{48}$$","stepBody":"Write your answer in radical form.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4\\\\sqrt{3}$$","hints":{"DefaultPathway":[{"id":"aaff50dradical1a-h1","type":"hint","dependencies":[],"title":"Finding the Largest Perfect Square","text":"You must first identify the largest perfect square factor of the number inside the radical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["aaff50dradical1a-h1"],"title":"Finding the Largest Perfect Square","text":"What is the largest perfect square factor of the number inside the radical?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical1a-h3","type":"hint","dependencies":["aaff50dradical1a-h2"],"title":"Rewriting the Radical","text":"You must now rewrite the radical as the product of two radicals. One of these radicals must include the perfect square. This is possible because of the product rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{16} \\\\sqrt{3}$$"],"dependencies":["aaff50dradical1a-h3"],"title":"Rewriting the Radical","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical1a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4\\\\sqrt{3}$$"],"dependencies":["aaff50dradical1a-h4"],"title":"Simplfying the Radicals","text":"What is the result after simplifying the square root with the perfect square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaff50dradical10","title":"Simplifying Square Roots","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Simplify Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aaff50dradical10a","stepAnswer":["$$5a^4 \\\\sqrt{3a}$$"],"problemType":"TextBox","stepTitle":"Simplify $$\\\\sqrt{75a^9}$$","stepBody":"Write your answer in radical form.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5a^4 \\\\sqrt{3a}$$","hints":{"DefaultPathway":[{"id":"aaff50dradical10a-h1","type":"hint","dependencies":[],"title":"Finding the Largest Perfect Square","text":"You must first identify the largest perfect square factor of the variable inside the radical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25a^8$$"],"dependencies":["aaff50dradical10a-h1"],"title":"Finding the Largest Perfect Square","text":"What is the largest perfect square factor of the variable inside the radical?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical10a-h3","type":"hint","dependencies":["aaff50dradical10a-h2"],"title":"Rewriting the Radical","text":"You must now rewrite the radical as the product of two radicals. One of these radicals must include the perfect square. This is possible because of the product rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{25a^8} \\\\sqrt{3a}$$"],"dependencies":["aaff50dradical10a-h3"],"title":"Rewriting the Radical","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5a^4 \\\\sqrt{3a}$$"],"dependencies":["aaff50dradical10a-h4"],"title":"Simplfying the Radicals","text":"What is the result after simplifying the square root with the perfect square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaff50dradical11","title":"Simplifying Square Roots","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Simplify Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aaff50dradical11a","stepAnswer":["$$5+5\\\\sqrt{3}$$"],"problemType":"TextBox","stepTitle":"Simplify $$5+\\\\sqrt{75}$$","stepBody":"Write your answer in radical form.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5+5\\\\sqrt{3}$$","hints":{"DefaultPathway":[{"id":"aaff50dradical11a-h1","type":"hint","dependencies":[],"title":"Finding the Largest Perfect Square","text":"You must first identify the largest perfect square factor of the number inside the radical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["aaff50dradical11a-h1"],"title":"Finding the Largest Perfect Square","text":"What is the largest perfect square factor of the number inside the radical?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical11a-h3","type":"hint","dependencies":["aaff50dradical11a-h2"],"title":"Rewriting the Radical","text":"You must now rewrite the radical as the product of two radicals. One of these radicals must include the perfect square. This is possible because of the product rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5+\\\\sqrt{25} \\\\sqrt{3}$$"],"dependencies":["aaff50dradical11a-h3"],"title":"Rewriting the Radical","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5+5\\\\sqrt{3}$$"],"dependencies":["aaff50dradical11a-h4"],"title":"Simplfying the Radicals","text":"What is the result after simplifying the square root with the perfect square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaff50dradical12","title":"Simplifying Square Roots","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Simplify Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aaff50dradical12a","stepAnswer":["$$2+7\\\\sqrt{2}$$"],"problemType":"TextBox","stepTitle":"Simplify $$2+\\\\sqrt{98}$$","stepBody":"Write your answer in radical form.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2+7\\\\sqrt{2}$$","hints":{"DefaultPathway":[{"id":"aaff50dradical12a-h1","type":"hint","dependencies":[],"title":"Finding the Largest Perfect Square","text":"You must first identify the largest perfect square factor of the number inside the radical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$49$$"],"dependencies":["aaff50dradical12a-h1"],"title":"Finding the Largest Perfect Square","text":"What is the largest perfect square factor of the number inside the radical?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical12a-h3","type":"hint","dependencies":["aaff50dradical12a-h2"],"title":"Rewriting the Radical","text":"You must now rewrite the radical as the product of two radicals. One of these radicals must include the perfect square. This is possible because of the product rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2+\\\\sqrt{49} \\\\sqrt{2}$$"],"dependencies":["aaff50dradical12a-h3"],"title":"Rewriting the Radical","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical12a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2+7\\\\sqrt{2}$$"],"dependencies":["aaff50dradical12a-h4"],"title":"Simplfying the Radicals","text":"What is the result after simplifying the square root with the perfect square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaff50dradical13","title":"Simplifying Square Roots","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Simplify Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aaff50dradical13a","stepAnswer":["$$2-\\\\sqrt{3}$$"],"problemType":"TextBox","stepTitle":"Simplify (10-sqrt(75))/5)","stepBody":"Write your answer in radical form.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2-\\\\sqrt{3}$$","hints":{"DefaultPathway":[{"id":"aaff50dradical13a-h1","type":"hint","dependencies":[],"title":"Finding the Largest Perfect Square","text":"You must first identify the largest perfect square factor of the number inside the radical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["aaff50dradical13a-h1"],"title":"Finding the Largest Perfect Square","text":"What is the largest perfect square factor of the number inside the radical?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical13a-h3","type":"hint","dependencies":["aaff50dradical13a-h2"],"title":"Rewriting the Radical","text":"You must now rewrite the radical as the product of two radicals. One of these radicals must include the perfect square. This is possible because of the product rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{10-\\\\sqrt{25} \\\\sqrt{3}}{5}$$"],"dependencies":["aaff50dradical13a-h3"],"title":"Rewriting the Radical","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical13a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2-\\\\sqrt{3}$$"],"dependencies":["aaff50dradical13a-h4"],"title":"Simplfying the Radicals","text":"What is the result after simplifying the square root with the perfect square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaff50dradical14","title":"Simplifying Square Roots","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Simplify Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aaff50dradical14a","stepAnswer":["$$2-\\\\sqrt{5}$$"],"problemType":"TextBox","stepTitle":"Simplify $$\\\\frac{6-\\\\sqrt{45}}{3}$$","stepBody":"Write your answer in radical form.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2-\\\\sqrt{5}$$","hints":{"DefaultPathway":[{"id":"aaff50dradical14a-h1","type":"hint","dependencies":[],"title":"Finding the Largest Perfect Square","text":"You must first identify the largest perfect square factor of the number inside the radical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["aaff50dradical14a-h1"],"title":"Finding the Largest Perfect Square","text":"What is the largest perfect square factor of the number inside the radical?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical14a-h3","type":"hint","dependencies":["aaff50dradical14a-h2"],"title":"Rewriting the Radical","text":"You must now rewrite the radical as the product of two radicals. One of these radicals must include the perfect square. This is possible because of the product rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical14a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{6-\\\\sqrt{9} \\\\sqrt{5}}{3}$$"],"dependencies":["aaff50dradical14a-h3"],"title":"Rewriting the Radical","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical14a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2-\\\\sqrt{5}$$"],"dependencies":["aaff50dradical14a-h4"],"title":"Simplfying the Radicals","text":"What is the result after simplifying the square root with the perfect square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaff50dradical15","title":"Simplifying Square Roots","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Simplify Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aaff50dradical15a","stepAnswer":["$$\\\\frac{5}{4}$$"],"problemType":"TextBox","stepTitle":"Simplify $$\\\\sqrt{\\\\frac{25}{16}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5}{4}$$","hints":{"DefaultPathway":[{"id":"aaff50dradical15a-h1","type":"hint","dependencies":[],"title":"Finding the Largest Perfect Square","text":"You must first identify the largest perfect square factor of the numbers inside the radicals.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["aaff50dradical15a-h1"],"title":"Finding the Largest Perfect Square","text":"What is the largest perfect square factor of the number inside the radical in the numerator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["aaff50dradical15a-h2"],"title":"Finding the Largest Perfect Square","text":"What is the largest perfect square factor of the number inside the radical in the denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical15a-h4","type":"hint","dependencies":["aaff50dradical15a-h3"],"title":"Rewriting the Radical","text":"You must now rewrite the radical as the quotient of two radicals. Both of these radicals must include the perfect squares.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical15a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\sqrt{25}}{\\\\sqrt{16}}$$"],"dependencies":["aaff50dradical15a-h4"],"title":"Rewriting the Radical","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical15a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{4}$$"],"dependencies":["aaff50dradical15a-h5"],"title":"Simplfying the Radicals","text":"What is the result after simplifying both square roots?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaff50dradical16","title":"Simplifying Square Roots","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Simplify Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aaff50dradical16a","stepAnswer":["$$3\\\\sqrt{3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{27}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3\\\\sqrt{3}$$","hints":{"DefaultPathway":[{"id":"aaff50dradical16a-h1","type":"hint","dependencies":[],"title":"Finding the Greatest Square Factor","text":"The greatest square factor of $$27$$ is $$9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical16a-h2","type":"hint","dependencies":["aaff50dradical16a-h1"],"title":"Simplifying the Root","text":"$$\\\\sqrt{27}=\\\\sqrt{9\\\\times3}=3\\\\sqrt{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaff50dradical17","title":"Simplifying Square Roots","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Simplify Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aaff50dradical17a","stepAnswer":["$$4\\\\sqrt{5}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{80}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4\\\\sqrt{5}$$","hints":{"DefaultPathway":[{"id":"aaff50dradical17a-h1","type":"hint","dependencies":[],"title":"Finding the Greatest Square Factor","text":"The greatest square factor of $$80$$ is $$16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical17a-h2","type":"hint","dependencies":["aaff50dradical17a-h1"],"title":"Simplifying the Root","text":"$$\\\\sqrt{80}=\\\\sqrt{16\\\\times6}=4\\\\sqrt{5}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaff50dradical18","title":"Simplifying Square Roots","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Simplify Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aaff50dradical18a","stepAnswer":["$$5\\\\sqrt{5}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{125}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5\\\\sqrt{5}$$","hints":{"DefaultPathway":[{"id":"aaff50dradical18a-h1","type":"hint","dependencies":[],"title":"Finding the Greatest Square Factor","text":"The greatest square factor of $$125$$ is $$25$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical18a-h2","type":"hint","dependencies":["aaff50dradical18a-h1"],"title":"Simplifying the Root","text":"$$\\\\sqrt{125}=\\\\sqrt{25\\\\times5}=5\\\\sqrt{5}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaff50dradical19","title":"Simplifying Square Roots","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Simplify Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aaff50dradical19a","stepAnswer":["$$4\\\\sqrt{6}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{96}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4\\\\sqrt{6}$$","hints":{"DefaultPathway":[{"id":"aaff50dradical19a-h1","type":"hint","dependencies":[],"title":"Finding the Greatest Square Factor","text":"The greatest square factor of $$96$$ is $$16$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical19a-h2","type":"hint","dependencies":["aaff50dradical19a-h1"],"title":"Simplifying the Root","text":"$$\\\\sqrt{96}=\\\\sqrt{16\\\\times6}=4\\\\sqrt{6}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaff50dradical2","title":"Simplifying Square Roots","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Simplify Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aaff50dradical2a","stepAnswer":["$$3\\\\sqrt{5}$$"],"problemType":"TextBox","stepTitle":"Simplify $$\\\\sqrt{45}$$","stepBody":"Write your answer in radical form.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3\\\\sqrt{5}$$","hints":{"DefaultPathway":[{"id":"aaff50dradical2a-h1","type":"hint","dependencies":[],"title":"Finding the Largest Perfect Square","text":"You must first identify the largest perfect square factor of the number inside the radical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["aaff50dradical2a-h1"],"title":"Finding the Largest Perfect Square","text":"What is the largest perfect square factor of the number inside the radical?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical2a-h3","type":"hint","dependencies":["aaff50dradical2a-h2"],"title":"Rewriting the Radical","text":"You must now rewrite the radical as the product of two radicals. One of these radicals must include the perfect square. This is possible because of the product rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{9} \\\\sqrt{5}$$"],"dependencies":["aaff50dradical2a-h3"],"title":"Rewriting the Radical","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical2a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3\\\\sqrt{5}$$"],"dependencies":["aaff50dradical2a-h4"],"title":"Simplfying the Radicals","text":"What is the result after simplifying the square root with the perfect square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaff50dradical20","title":"Simplifying Square Roots","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Simplify Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aaff50dradical20a","stepAnswer":["$$10\\\\sqrt{2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{200}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10\\\\sqrt{2}$$","hints":{"DefaultPathway":[{"id":"aaff50dradical20a-h1","type":"hint","dependencies":[],"title":"Finding the Greatest Square Factor","text":"The greatest square factor of $$200$$ is $$100$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical20a-h2","type":"hint","dependencies":["aaff50dradical20a-h1"],"title":"Simplifying the Root","text":"$$\\\\sqrt{200}=\\\\sqrt{2\\\\times100}=10\\\\sqrt{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaff50dradical21","title":"Simplifying Square Roots","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Simplify Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aaff50dradical21a","stepAnswer":["$$7\\\\sqrt{3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{147}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7\\\\sqrt{3}$$","hints":{"DefaultPathway":[{"id":"aaff50dradical21a-h1","type":"hint","dependencies":[],"title":"Finding the Greatest Square Factor","text":"The greatest square factor of $$147$$ is $$49$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical21a-h2","type":"hint","dependencies":["aaff50dradical21a-h1"],"title":"Simplifying the Root","text":"$$\\\\sqrt{147}=\\\\sqrt{49\\\\times3}=7\\\\sqrt{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaff50dradical22","title":"Simplifying Square Roots","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Simplify Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aaff50dradical22a","stepAnswer":["$$15\\\\sqrt{2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{450}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$15\\\\sqrt{2}$$","hints":{"DefaultPathway":[{"id":"aaff50dradical22a-h1","type":"hint","dependencies":[],"title":"Finding the Greatest Square Factor","text":"The greatest square factior of $$450$$ is $$225$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical22a-h2","type":"hint","dependencies":["aaff50dradical22a-h1"],"title":"Simplifying the Root","text":"$$\\\\sqrt{450}=\\\\sqrt{225\\\\times2}=15\\\\sqrt{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaff50dradical23","title":"Simplifying Square Roots","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Simplify Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aaff50dradical23a","stepAnswer":["$$6\\\\sqrt{7}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{252}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6\\\\sqrt{7}$$","hints":{"DefaultPathway":[{"id":"aaff50dradical23a-h1","type":"hint","dependencies":[],"title":"Finding the Greatest Square Factor","text":"The greatest square factor of $$252$$ is $$36$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical23a-h2","type":"hint","dependencies":["aaff50dradical23a-h1"],"title":"Simplifying the Root","text":"$$\\\\sqrt{252}=\\\\sqrt{36\\\\times7}=6\\\\sqrt{7}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaff50dradical24","title":"Simplifying Square Roots","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Simplify Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aaff50dradical24a","stepAnswer":["$$20\\\\sqrt{2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{800}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$20\\\\sqrt{2}$$","hints":{"DefaultPathway":[{"id":"aaff50dradical24a-h1","type":"hint","dependencies":[],"title":"Finding the Greatest Square Factor","text":"The greatest square factor of $$800$$ is $$400$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical24a-h2","type":"hint","dependencies":["aaff50dradical24a-h1"],"title":"Simplifying the Root","text":"$$\\\\sqrt{800}=\\\\sqrt{400\\\\times2}=20\\\\sqrt{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaff50dradical25","title":"Simplifying Square Roots","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Simplify Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aaff50dradical25a","stepAnswer":["$$12\\\\sqrt{2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{288}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12\\\\sqrt{2}$$","hints":{"DefaultPathway":[{"id":"aaff50dradical25a-h1","type":"hint","dependencies":[],"title":"Finding the Greatest Square Factor","text":"The greatest square factor of $$288$$ is $$144$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical25a-h2","type":"hint","dependencies":["aaff50dradical25a-h1"],"title":"Simplifying the Root","text":"$$\\\\sqrt{288}=\\\\sqrt{144\\\\times2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaff50dradical26","title":"Simplifying Square Roots","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Simplify Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aaff50dradical26a","stepAnswer":["$$15\\\\sqrt{3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{675}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$15\\\\sqrt{3}$$","hints":{"DefaultPathway":[{"id":"aaff50dradical26a-h1","type":"hint","dependencies":[],"title":"Finding the Greatest Square Factor","text":"The greatest square factor of $$675$$ is $$225$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical26a-h2","type":"hint","dependencies":["aaff50dradical26a-h1"],"title":"Simplifying the Root","text":"$$\\\\sqrt{675}=\\\\sqrt{225\\\\times3}=15\\\\sqrt{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaff50dradical27","title":"Simplifying Square Roots","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Simplify Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aaff50dradical27a","stepAnswer":["$$25\\\\sqrt{2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{1250}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$25\\\\sqrt{2}$$","hints":{"DefaultPathway":[{"id":"aaff50dradical27a-h1","type":"hint","dependencies":[],"title":"Finding the Greatest Square Factor","text":"The greatest square factor of $$1250$$ is $$625$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical27a-h2","type":"hint","dependencies":["aaff50dradical27a-h1"],"title":"Simplifying the Root","text":"$$\\\\sqrt{1250}=\\\\sqrt{625\\\\times2}=25\\\\sqrt{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaff50dradical28","title":"Simplifying Square Roots","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Simplify Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aaff50dradical28a","stepAnswer":["$$x^3 \\\\sqrt{x}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{x^7}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^3 \\\\sqrt{x}$$","hints":{"DefaultPathway":[{"id":"aaff50dradical28a-h1","type":"hint","dependencies":[],"title":"Finding the Greatest Square Factor","text":"The greatest square factor of $$x^7$$ is $$x^6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical28a-h2","type":"hint","dependencies":["aaff50dradical28a-h1"],"title":"Simplifying the Root","text":"$$\\\\sqrt{x^7}=\\\\sqrt{x^6 x^1}=x^3 \\\\sqrt{x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaff50dradical29","title":"Simplifying Square Roots","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Simplify Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aaff50dradical29a","stepAnswer":["$$y^5 \\\\sqrt{y}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{y^{11}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y^5 \\\\sqrt{y}$$","hints":{"DefaultPathway":[{"id":"aaff50dradical29a-h1","type":"hint","dependencies":[],"title":"Finding the Greatest Square Factor","text":"The greatest square factor of $$y^{11}$$ is $$y^{10}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical29a-h2","type":"hint","dependencies":["aaff50dradical29a-h1"],"title":"Simplifying the Root","text":"$$\\\\sqrt{y^{11}}=\\\\sqrt{y^{10} y}=y^5 \\\\sqrt{y}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaff50dradical3","title":"Simplifying Square Roots","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Simplify Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aaff50dradical3a","stepAnswer":["$$12\\\\sqrt{2}$$"],"problemType":"TextBox","stepTitle":"Simplify $$\\\\sqrt{288}$$","stepBody":"Write your answer in radical form.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12\\\\sqrt{2}$$","hints":{"DefaultPathway":[{"id":"aaff50dradical3a-h1","type":"hint","dependencies":[],"title":"Finding the Largest Perfect Square","text":"You must first identify the largest perfect square factor of the number inside the radical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["aaff50dradical3a-h1"],"title":"Finding the Largest Perfect Square","text":"What is the largest perfect square factor of the number inside the radical?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical3a-h3","type":"hint","dependencies":["aaff50dradical3a-h2"],"title":"Rewriting the Radical","text":"You must now rewrite the radical as the product of two radicals. One of these radicals must include the perfect square. This is possible because of the product rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{144} \\\\sqrt{2}$$"],"dependencies":["aaff50dradical3a-h3"],"title":"Rewriting the Radical","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12\\\\sqrt{2}$$"],"dependencies":["aaff50dradical3a-h4"],"title":"Simplfying the Radicals","text":"What is the result after simplifying the square root with the perfect square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaff50dradical30","title":"Simplifying Square Roots","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Simplify Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aaff50dradical30a","stepAnswer":["$$p \\\\sqrt{p}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{p^3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$p \\\\sqrt{p}$$","hints":{"DefaultPathway":[{"id":"aaff50dradical30a-h1","type":"hint","dependencies":[],"title":"Finding the Greatest Square Factor","text":"The greatest square factor of $$p^3$$ is $$p^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical30a-h2","type":"hint","dependencies":["aaff50dradical30a-h1"],"title":"Simplifying the Root","text":"$$\\\\sqrt{p^3}=\\\\sqrt{p^2 p}=\\\\operatorname{psqrt}\\\\left(p\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaff50dradical31","title":"Simplifying Square Roots","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Simplify Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aaff50dradical31a","stepAnswer":["$$q^2 \\\\sqrt{q}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{q^5}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$q^2 \\\\sqrt{q}$$","hints":{"DefaultPathway":[{"id":"aaff50dradical31a-h1","type":"hint","dependencies":[],"title":"Finding the Greatest Square Factor","text":"The greatest square factor of $$q^5$$ is $$q^4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical31a-h2","type":"hint","dependencies":["aaff50dradical31a-h1"],"title":"Simplifying the Root","text":"$$\\\\sqrt{q^5}=\\\\sqrt{q^4 q}=q^2 \\\\sqrt{q}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaff50dradical4","title":"Simplifying Square Roots","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Simplify Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aaff50dradical4a","stepAnswer":["$$4\\\\sqrt{27}$$"],"problemType":"TextBox","stepTitle":"Simplify $$\\\\sqrt{432}$$","stepBody":"Write your answer in radical form.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4\\\\sqrt{27}$$","hints":{"DefaultPathway":[{"id":"aaff50dradical4a-h1","type":"hint","dependencies":[],"title":"Finding the Largest Perfect Square","text":"You must first identify the largest perfect square factor of the number inside the radical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["aaff50dradical4a-h1"],"title":"Finding the Largest Perfect Square","text":"What is the largest perfect square factor of the number inside the radical?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical4a-h3","type":"hint","dependencies":["aaff50dradical4a-h2"],"title":"Rewriting the Radical","text":"You must now rewrite the radical as the product of two radicals. One of these radicals must include the perfect square. This is possible because of the product rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{16} \\\\sqrt{27}$$"],"dependencies":["aaff50dradical4a-h3"],"title":"Rewriting the Radical","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical4a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4\\\\sqrt{27}$$"],"dependencies":["aaff50dradical4a-h4"],"title":"Simplfying the Radicals","text":"What is the result after simplifying the square root with the perfect square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaff50dradical5","title":"Simplifying Square Roots","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Simplify Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aaff50dradical5a","stepAnswer":["$$b^2 \\\\sqrt{b}$$"],"problemType":"TextBox","stepTitle":"Simplify $$\\\\sqrt{b^5}$$","stepBody":"Write your answer in radical form.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$b^2 \\\\sqrt{b}$$","hints":{"DefaultPathway":[{"id":"aaff50dradical5a-h1","type":"hint","dependencies":[],"title":"Finding the Largest Perfect Square","text":"You must first identify the largest perfect square factor of the variable inside the radical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$b^4$$"],"dependencies":["aaff50dradical5a-h1"],"title":"Finding the Largest Perfect Square","text":"What is the largest perfect square factor of the variable inside the radical?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical5a-h3","type":"hint","dependencies":["aaff50dradical5a-h2"],"title":"Rewriting the Radical","text":"You must now rewrite the radical as the product of two radicals. One of these radicals must include the perfect square. This is possible because of the product rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{b^4} \\\\sqrt{b}$$"],"dependencies":["aaff50dradical5a-h3"],"title":"Rewriting the Radical","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$b^2 \\\\sqrt{b}$$"],"dependencies":["aaff50dradical5a-h4"],"title":"Simplfying the Radicals","text":"What is the result after simplifying the square root with the perfect square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaff50dradical6","title":"Simplifying Square Roots","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Simplify Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aaff50dradical6a","stepAnswer":["$$p^4 \\\\sqrt{p}$$"],"problemType":"TextBox","stepTitle":"Simplify $$\\\\sqrt{p^9}$$","stepBody":"Write your answer in radical form.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$p^4 \\\\sqrt{p}$$","hints":{"DefaultPathway":[{"id":"aaff50dradical6a-h1","type":"hint","dependencies":[],"title":"Finding the Largest Perfect Square","text":"You must first identify the largest perfect square factor of the variable inside the radical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$p^8$$"],"dependencies":["aaff50dradical6a-h1"],"title":"Finding the Largest Perfect Square","text":"What is the largest perfect square factor of the variable inside the radical?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical6a-h3","type":"hint","dependencies":["aaff50dradical6a-h2"],"title":"Rewriting the Radical","text":"You must now rewrite the radical as the product of two radicals. One of these radicals must include the perfect square. This is possible because of the product rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{p^8} \\\\sqrt{p}$$"],"dependencies":["aaff50dradical6a-h3"],"title":"Rewriting the Radical","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$p^4 \\\\sqrt{p}$$"],"dependencies":["aaff50dradical6a-h4"],"title":"Simplfying the Radicals","text":"What is the result after simplifying the square root with the perfect square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaff50dradical7","title":"Simplifying Square Roots","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Simplify Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aaff50dradical7a","stepAnswer":["$$4x^3 \\\\sqrt{x}$$"],"problemType":"TextBox","stepTitle":"Simplify $$\\\\sqrt{16x^7}$$","stepBody":"Write your answer in radical form.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4x^3 \\\\sqrt{x}$$","hints":{"DefaultPathway":[{"id":"aaff50dradical7a-h1","type":"hint","dependencies":[],"title":"Finding the Largest Perfect Square","text":"You must first identify the largest perfect square factor of the variable inside the radical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16x^6$$"],"dependencies":["aaff50dradical7a-h1"],"title":"Finding the Largest Perfect Square","text":"What is the largest perfect square factor of the variable inside the radical?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical7a-h3","type":"hint","dependencies":["aaff50dradical7a-h2"],"title":"Rewriting the Radical","text":"You must now rewrite the radical as the product of two radicals. One of these radicals must include the perfect square. This is possible because of the product rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{16x^6} \\\\sqrt{x}$$"],"dependencies":["aaff50dradical7a-h3"],"title":"Rewriting the Radical","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical7a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4x^3 \\\\sqrt{x}$$"],"dependencies":["aaff50dradical7a-h4"],"title":"Simplfying the Radicals","text":"What is the result after simplifying the square root with the perfect square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaff50dradical8","title":"Simplifying Square Roots","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Simplify Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aaff50dradical8a","stepAnswer":["$$7v^4 \\\\sqrt{v}$$"],"problemType":"TextBox","stepTitle":"Simplify $$\\\\sqrt{49v^9}$$","stepBody":"Write your answer in radical form.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7v^4 \\\\sqrt{v}$$","hints":{"DefaultPathway":[{"id":"aaff50dradical8a-h1","type":"hint","dependencies":[],"title":"Finding the Largest Perfect Square","text":"You must first identify the largest perfect square factor of the variable inside the radical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$49v^8$$"],"dependencies":["aaff50dradical8a-h1"],"title":"Finding the Largest Perfect Square","text":"What is the largest perfect square factor of the variable inside the radical?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical8a-h3","type":"hint","dependencies":["aaff50dradical8a-h2"],"title":"Rewriting the Radical","text":"You must now rewrite the radical as the product of two radicals. One of these radicals must include the perfect square. This is possible because of the product rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{49v^8} \\\\sqrt{v}$$"],"dependencies":["aaff50dradical8a-h3"],"title":"Rewriting the Radical","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical8a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7v^4 \\\\sqrt{v}$$"],"dependencies":["aaff50dradical8a-h4"],"title":"Simplfying the Radicals","text":"What is the result after simplifying the square root with the perfect square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aaff50dradical9","title":"Simplifying Square Roots","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.2 Simplify Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aaff50dradical9a","stepAnswer":["$$4y^2 \\\\sqrt{2y}$$"],"problemType":"TextBox","stepTitle":"Simplify $$\\\\sqrt{32y^5}$$","stepBody":"Write your answer in radical form.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4y^2 \\\\sqrt{2y}$$","hints":{"DefaultPathway":[{"id":"aaff50dradical9a-h1","type":"hint","dependencies":[],"title":"Finding the Largest Perfect Square","text":"You must first identify the largest perfect square factor of the variable inside the radical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16y^4$$"],"dependencies":["aaff50dradical9a-h1"],"title":"Finding the Largest Perfect Square","text":"What is the largest perfect square factor of the variable inside the radical?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical9a-h3","type":"hint","dependencies":["aaff50dradical9a-h2"],"title":"Rewriting the Radical","text":"You must now rewrite the radical as the product of two radicals. One of these radicals must include the perfect square. This is possible because of the product rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{16y^4} \\\\sqrt{2y}$$"],"dependencies":["aaff50dradical9a-h3"],"title":"Rewriting the Radical","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aaff50dradical9a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4y^2 \\\\sqrt{2y}$$"],"dependencies":["aaff50dradical9a-h4"],"title":"Simplfying the Radicals","text":"What is the result after simplifying the square root with the perfect square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab19046LimitLaw1","title":"The Limit Laws","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.3 The Limit Laws","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ab19046LimitLaw1a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"Evaluate $$\\\\lim_{x\\\\to0} 4x^2-2x+3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"ab19046LimitLaw1a-h1","type":"hint","dependencies":[],"title":"Breaking down the question","text":"We can separate the functions into $$4x^2$$, $$-2x$$ and $$3$$ as $$x$$ approaches $$0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ab19046LimitLaw1a-h2","type":"hint","dependencies":["ab19046LimitLaw1a-h1"],"title":"Find Limits","text":"What is $$\\\\lim_{x\\\\to0} 4x^2,\\\\lim_{x\\\\to0} -2x$$ and $$\\\\lim_{x\\\\to0} 3\uff1f$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ab19046LimitLaw1a-h3","type":"hint","dependencies":["ab19046LimitLaw1a-h2"],"title":"Using the above info, add all three limits and get the answer.","text":"$$0-0+3=3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ab19046LimitLaw10","title":"The Limit Laws","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.3 The Limit Laws","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ab19046LimitLaw10a","stepAnswer":["$$9$$"],"problemType":"TextBox","stepTitle":"Evaluate $$\\\\lim_{x\\\\to3} \\\\ln(e^{3x})$$ by direct substitution","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9$$","hints":{"DefaultPathway":[{"id":"ab19046LimitLaw10a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$e^9$$"],"dependencies":[],"title":"Sub-questions","text":"Attempt to plug in the $$x$$ value into the equation, what do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$e^3$$","$$e^9$$","$$e^6$$"]},{"id":"ab19046LimitLaw10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["ab19046LimitLaw10a-h1"],"title":"Sub-questions","text":"What\'s $$\\\\ln(e^9)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ab19046LimitLaw2","title":"The Limit Laws","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.3 The Limit Laws","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ab19046LimitLaw2a","stepAnswer":["$$-3$$"],"problemType":"TextBox","stepTitle":"Evaluate $$\\\\lim_{x\\\\to1} \\\\frac{x^3+3x^2+5}{4-7x}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-3$$","hints":{"DefaultPathway":[{"id":"ab19046LimitLaw2a-h1","type":"hint","dependencies":[],"title":"Breaking down the question","text":"Is the denominator $$0$$ after we substitute 1?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ab19046LimitLaw2a-h2","type":"hint","dependencies":["ab19046LimitLaw2a-h1"],"title":"Breaking down the question","text":"Since denominaotor is not $$0$$, we can substitute $$1$$ directly.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ab19046LimitLaw2a-h3","type":"hint","dependencies":["ab19046LimitLaw2a-h2"],"title":"Substitute $$1$$ and get the answer","text":"$$\\\\frac{1+3+5}{4-7}=-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ab19046LimitLaw3","title":"The Limit Laws","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.3 The Limit Laws","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ab19046LimitLaw3a","stepAnswer":["$$\\\\sqrt{19}$$"],"problemType":"MultipleChoice","stepTitle":"Evaluate $$\\\\lim_{x\\\\to-2} \\\\sqrt{x^2-6x+3}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\sqrt{19}$$","choices":["$$\\\\sqrt{19}$$","$$\\\\sqrt{3}$$","$$0$$"],"hints":{"DefaultPathway":[{"id":"ab19046LimitLaw3a-h1","type":"hint","dependencies":[],"title":"Breaking down the question","text":"After we substitute $$-2$$, is the function inside the square root less than 0?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ab19046LimitLaw3a-h2","type":"hint","dependencies":["ab19046LimitLaw3a-h1"],"title":"Ability to plug in","text":"Since the value inside the square root is not negative, we can plug in the value directly.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ab19046LimitLaw3a-h3","type":"hint","dependencies":["ab19046LimitLaw3a-h2"],"title":"Substitute $$-2$$ and get the answer","text":"$$\\\\sqrt{4+12+3}=\\\\sqrt{19}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ab19046LimitLaw4","title":"The Limit Laws","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.3 The Limit Laws","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ab19046LimitLaw4a","stepAnswer":["$$64$$"],"problemType":"TextBox","stepTitle":"Evaluate $$\\\\lim_{x\\\\to-1} {\\\\left(9x+1\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$64$$","hints":{"DefaultPathway":[{"id":"ab19046LimitLaw4a-h1","type":"hint","dependencies":[],"title":"Breaking down the question","text":"$${\\\\left(9x+1\\\\right)}^2=\\\\left(9x+1\\\\right) \\\\left(9x+1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ab19046LimitLaw4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-8$$"],"dependencies":["ab19046LimitLaw4a-h1"],"title":"Sub-questions","text":"What\'s $$\\\\lim_{x\\\\to-1} 9x+1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ab19046LimitLaw4a-h3","type":"hint","dependencies":["ab19046LimitLaw4a-h2"],"title":"Breaking down the question","text":"Square your results to get the answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ab19046LimitLaw5","title":"The Limit Laws","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.3 The Limit Laws","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ab19046LimitLaw5a","stepAnswer":["$$49$$"],"problemType":"TextBox","stepTitle":"Evaluate $$\\\\lim_{x\\\\to7} x^2$$ by direct substitution","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$49$$","hints":{"DefaultPathway":[{"id":"ab19046LimitLaw5a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$49$$"],"dependencies":[],"title":"Sub-questions","text":"Attempt to plug in the $$x$$ value into the equation, what do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ab19046LimitLaw6","title":"The Limit Laws","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.3 The Limit Laws","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ab19046LimitLaw6a","stepAnswer":["$$15$$"],"problemType":"TextBox","stepTitle":"Evaluate $$\\\\lim_{x\\\\to-2} 4x^2-1$$ by direct substitution","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$15$$","hints":{"DefaultPathway":[{"id":"ab19046LimitLaw6a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":[],"title":"Sub-questions","text":"Attempt to plug in the $$x$$ value into the equation, what do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ab19046LimitLaw7","title":"The Limit Laws","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.3 The Limit Laws","courseName":"OpenStax: Calculus Volume 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$$\\\\lim_{x\\\\to0} e^{2x-x^2}$$ by direct subtituttion","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"ab19046LimitLaw8a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Sub-questions","text":"Attempt to plug in the $$x$$ value into the equation, what do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ab19046LimitLaw8a-h1-s1","type":"hint","dependencies":[],"title":"Explanation","text":"$$e^0=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}]}}]},{"id":"ab19046LimitLaw9","title":"The Limit Laws","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.3 The Limit Laws","courseName":"OpenStax: Calculus Volume 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equality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab1ad7fGenStr9a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$n=2$$"],"dependencies":["ab1ad7fGenStr9a-h7"],"title":"Division","text":"Divide $$7$$ from each side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab1ad7fGenStr9a-h9","type":"hint","dependencies":["ab1ad7fGenStr9a-h8"],"title":"Verification","text":"Check whether the result is a solution of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab1ad7fGenStr9a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["ab1ad7fGenStr9a-h9"],"title":"Verification","text":"Check whether $$7\\\\left(2-3\\\\right)-8$$ equals $$-15$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]}]}}]},{"id":"ab2e2b210.4AIDS","title":"Hypothesis Testing with Two Samples","body":"A new AIDS prevention drug was tried on a group of $$224$$ HIV positive patients. Forty-five patients developed AIDS after four years. In a control group of $$224$$ HIV positive patients, $$68$$ developed AIDS after four years. We want to test whether the method of treatment reduces the proportion of patients that develop AIDS after four years or if the proportions of the treated group and the untreated group stay the same.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Matched or Paired Samples","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab2e2b210.4AIDSa","stepAnswer":["H_0:$$p_t$$=$$p_ut$$ and H_a:$$p_t$$<$$p_ut$$"],"problemType":"MultipleChoice","stepTitle":"Hypothesis Testing with Two Samples","stepBody":"","answerType":"string","variabilization":{},"choices":["H_0:$$p_t$$=$$p_ut$$ and H_a:$$p_t$$<$$p_ut$$","H_0:$$p_t$$<$$p_ut$$ and H_a:$$p_t$$>=$$p_ut$$","H_0:$$p_t$$<=$$p_ut$$ and H_a:$$p_t$$>$$p_ut$$","H_0:$$p_t$$=$$p_ut$$ and H_a:$$p_t>=p_ut$$"],"hints":{"DefaultPathway":[{"id":"ab2e2b210.4AIDSa-h1","type":"hint","dependencies":[],"title":"Hypothesis Testing with Two Samples","text":"Consider the context: a new drug is being tested, therefore the null hypothesis is that the proportion is greater or equal to assume the worst in its effectiveness.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab2e2b210.4AIDSa-h2","type":"hint","dependencies":["ab2e2b210.4AIDSa-h1"],"title":"Hypothesis Testing with Two Samples","text":"We know that the null assumption proportion is equal to the control or greater if it is an option. The alternative claim, or attribute that is being tested is if the drug reduced the proportion of people that developed AIDS in four years since the aim is to hinder or stop AIDS development.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab2e2b210.4biofeedback","title":"Hypothesis Testing with Two Samples","body":"An experiment is conducted to show that blood pressure can be consciously reduced in people trained in a \u201cbiofeedback exercise program.\u201d Six subjects were randomly selected and blood pressure measurements were recorded before and after the training. The difference between blood pressures was calculated (after - before) producing the following results:x_d=-10.2, $$s_d=8.4$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Matched or Paired Samples","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab2e2b210.4biofeedbacka","stepAnswer":["$$0.0155;$$ There is sufficient evidence to conclude that the blood pressure decreased after the training."],"problemType":"MultipleChoice","stepTitle":"If $$a=0.05$$, the $$p-value$$ and the conclusion are $$_{}$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$0.0155;$$ There is sufficient evidence to conclude that the blood pressure decreased after the training.","choices":["$$0.0155;$$ There is sufficient evidence to conclude that the blood pressure decreased after the training.","$$0.1550;$$ There is sufficient evidence to conclude that the blood pressure did not decrease after the training.","$$0.003400;$$ There is sufficient evidence to conclude that the blood pressure decreased after the training.","$$0.3400;$$ There is sufficient evidence to conclude that the blood pressure increased after the training."],"hints":{"DefaultPathway":[{"id":"ab2e2b210.4biofeedbacka-h1","type":"hint","dependencies":[],"title":"Hypothesis Testing with Two Samples","text":"There is no need to calculate differences data as the sample mean and standard deviation are given.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab2e2b210.4biofeedbacka-h2","type":"hint","dependencies":["ab2e2b210.4biofeedbacka-h1"],"title":"Hypothesis Testing with Two Samples","text":"Methods to calculate the $$p-value$$ will vary by calculator, make sure to use a $$t-test$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab2e2b210.4biofeedbacka-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$H_0$$: $$u \\\\geq 0$$, $$H_a$$: $$u<0$$"],"dependencies":["ab2e2b210.4biofeedbacka-h2"],"title":"Hypothesis Testing with Two Samples","text":"The key testing feature is that blood pressure is decreased after training. Suppose u denotes blood pressure level change, which of the following claims are correct?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$H_0$$: $$u \\\\geq 0$$, $$H_a$$: $$u<0$$","$$H_0$$: $$u \\\\geq 5$$, $$H_a$$: $$u<5$$","$$H_0$$: $$u=0$$, $$H_a$$: $$u \\\\leq 0$$","$$H_0$$: $$u>s$$, $$H_a$$: $$u \\\\leq s$$"]},{"id":"ab2e2b210.4biofeedbacka-h4","type":"hint","dependencies":["ab2e2b210.4biofeedbacka-h3"],"title":"Hypothesis Testing with Two Samples","text":"Using $$x=-10.2$$, $$s=8.4$$, and $$n=6$$, the test statistic can be found through the formula test statistic $$t=\\\\frac{x-0}{\\\\frac{s}{\\\\sqrt{n}}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab2e2b210.4biofeedbacka-h5","type":"hint","dependencies":["ab2e2b210.4biofeedbacka-h4"],"title":"Hypothesis Testing with Two Samples","text":"The computed test statistic is $$\\\\frac{-10.2}{\\\\frac{8.4}{\\\\sqrt{6}}}$$, the $$p-value$$ can be found in the form P(t_5<=-10.2/(8.4/sqrt(6)))","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab2e2b210.4biofeedbacka-h6","type":"hint","dependencies":["ab2e2b210.4biofeedbacka-h5"],"title":"Hypothesis Testing with Two Samples","text":"It is found that $$P-value=0.0155$$ approximately, since the $$p-value$$ is less than $$0.05$$, our significance level, the null hypothesis is rejected and there is suffcient evidence to support the alternative hypothesis that blood pressure decreased after the special exercise protocol.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab2e2b210.4bloodpressure","title":"Hypothesis Testing with Two Samples","body":"An experiment is conducted to show that blood pressure can be consciously reduced in people trained in a \u201cbiofeedback exercise program.\u201d Six subjects were randomly selected and blood pressure measurements were recorded before and after the training. The difference between blood pressures was calculated (after - before) producing the following results:x_d=-10.2, $$s_d=8.4$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Matched or Paired Samples","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab2e2b210.4bloodpressurea","stepAnswer":["Reject the null hypothesis that there is no difference."],"problemType":"MultipleChoice","stepTitle":"Using the data, test the hypothesis that the blood pressure has decreased after the training.","stepBody":"","answerType":"string","variabilization":{},"choices":["Reject the null hypothesis that there is no difference.","Fail to reject the null hypothesis that there is no difference."],"hints":{"DefaultPathway":[{"id":"ab2e2b210.4bloodpressurea-h1","type":"hint","dependencies":[],"title":"T-test","text":"When you perform a hypothesis test of a single population mean \u03bc using a Student\'s $$t-distribution$$ (often called a $$t-test)$$, there are fundamental assumptions that need to be met in order for the test to work properly. Your data should be a simple random sample that comes from a population that is approximately normally distributed. You use the sample standard deviation to approximate the population standard deviation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab2e2b210.4bloodpressurea-h2","type":"hint","dependencies":["ab2e2b210.4bloodpressurea-h1"],"title":"Z-test","text":"Recall what decisions must be made to decide what distribution would be best to perform hypothesis tests. When you use a normal distribution for a hypothesis test (often called a $$z-test)$$, a simple random sample is taken from the population and the population that you are working with is normally distributed or the sample size is sufficiently large.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab2e2b210.4bloodpressurea-h3","type":"hint","dependencies":["ab2e2b210.4bloodpressurea-h2"],"title":"Hypothesis Distribution","text":"Remember that degrees of freedom for $$t-test$$ distributions is $$n-1$$, where $$n$$ is the population. $$n$$ is $$6$$ in this case.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab2e2b210.4coach","title":"A college football coach was interested in whether the college\'s strength development class increased his players\' maximum lift (in pounds) on the bench press exercise. He asked four of his players to participate in a study. The amount of weight they could each lift was recorded before they took the strength development class. After completing the class, the amount of weight they could each lift was again measured. The data are as follows:\\\\n\\\\n","body":"\\\\n##figure2.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Matched or Paired Samples","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab2e2b210.4coacha","stepAnswer":["{90, $$11$$, $$-8$$, $$-8$$}"],"problemType":"MultipleChoice","stepTitle":"The coach wants to know if the strength development class makes his players stronger, on average. Record the differences data of the amount of weight lifted prior to the class from the weight lifted after completing the class. The data for the differences are:","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"{90, $$11$$, $$-8$$, -8}","choices":["{90, $$11$$, $$-8$$, $$-8$$}","{32, $$11$$, $$-8$$, $$-8$$}","{500, $$493$$, $$638$$, 728}","{8,4}"],"hints":{"DefaultPathway":[{"id":"ab2e2b210.4coacha-h1","type":"hint","dependencies":[],"title":"Hypothesis Testing with Two Samples","text":"Calculate the differences between the corresponding players when weight was lifted prior and after the class to get an ordered list of differences.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab2e2b210.4coacha-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$90$$"],"dependencies":["ab2e2b210.4coacha-h1"],"title":"Hypothesis Testing with Two Samples","text":"What is the calculated difference of Player 1?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab2e2b210.4coacha-h3","type":"hint","dependencies":["ab2e2b210.4coacha-h2"],"title":"Hypothesis Testing with Two Samples","text":"The difference of Player 1\'s weight prior and after the class is can be found by subtracting $$295$$ and $$205$$. Do the same for the rest of the set of players.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab2e2b210.4coachb","stepAnswer":["$$21.3$$"],"problemType":"TextBox","stepTitle":"Assume the differences have a normal distribution. Using the differences data, calculate the sample mean. Round to the first decimal place.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$21.3$$","hints":{"DefaultPathway":[{"id":"ab2e2b210.4coachb-h1","type":"hint","dependencies":[],"title":"Gathering sample statistics","text":"You may use the new list of differences to find the mean differences by adding all the terms and dividing by the number of terms in the list. You may alternatively put the differences data on a table on your graphing calculator to find the sample mean.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab2e2b210.4coachc","stepAnswer":["$$46.7$$"],"problemType":"TextBox","stepTitle":"Calculate the sample standard deviation. Round to the first decimal place.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$46.7$$","hints":{"DefaultPathway":[{"id":"ab2e2b210.4coachc-h1","type":"hint","dependencies":[],"title":"You may use the list of differences gathered on a graphing calculator to find the standard deviation or follow the formula provided. Remember that we are calculating the statistic, not the parameter, so the formula for sample standard deviation is different.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab2e2b210.4coachd","stepAnswer":["$$X=Mean$$ difference in the maximum lift per player"],"problemType":"MultipleChoice","stepTitle":"Random variable","stepBody":"Define the random variable.","answerType":"string","variabilization":{},"answerLatex":"$$X=Mean$$ difference in the maximum lift per player","choices":["$$X=Mean$$ difference in the maximum lift per player","$$X=Mean$$ difference in lifts between players","$$X=Mean$$ difference in the height of each player","$$X=Mean$$ difference in the time to lift maximum weight in class"],"hints":{"DefaultPathway":[{"id":"ab2e2b210.4coachd-h1","type":"hint","dependencies":[],"title":"Random variable","text":"The choices provided are really similar. It is best to choose the random variable that best represents our sample. In this case, the maximum weight lifted difference per player is the random variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab2e2b210.4coache","stepAnswer":["There is not sufficient evidence to conclude that the strength development class helped to make the players stronger, on average."],"problemType":"MultipleChoice","stepTitle":"A hypothesis test was performed given that the null hypothesis is that the mean differences is less than or equal to $$0$$, and the alternate hypothesis is greater than $$0$$. A graph of the $$p-value$$ found is given. What is the conclusion if the level of significance is 5%?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["There is not sufficient evidence to conclude that the strength development class helped to make the players stronger, on average.","There is sufficient evidence to conclude that the strength development class helped to make the players stronger, on average."],"hints":{"DefaultPathway":[{"id":"ab2e2b210.4coache-h1","type":"hint","dependencies":[],"title":"Hypothesis Testing with Two Samples","text":"If the level of significance is 5%, the decision is not to reject the null hypothesis, because the significance value $$0.05<p-value$$, therefore, given our sample data, there is not sufficient evidence to conclude that the strength development class helped to make the players stronger, on average.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab2e2b210.4drugsP-value","title":"Hypothesis Testing with Two Samples","body":"A new AIDS prevention drug was tried on a group of $$224$$ HIV positive patients. Forty-five patients developed AIDS after four years. In a control group of $$224$$ HIV positive patients, $$68$$ developed AIDS after four years. We want to test whether the method of treatment reduces the proportion of patients that develop AIDS after four years or if the proportions of the treated group and the untreated group stay the same.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Matched or Paired Samples","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab2e2b210.4drugsP-valuea","stepAnswer":["There is sufficient evidence to conclude that the method reduces the proportion of HIV positive patients who develop AIDS after four years."],"problemType":"MultipleChoice","stepTitle":"If the $$p-value$$ is $$0.0062$$ what is the conclusion (use significance level $$=$$ $$0.05)$$?","stepBody":"","answerType":"string","variabilization":{},"choices":["There is sufficient evidence to conclude that the method reduces the proportion of HIV positive patients who develop AIDS after four years.","The method has no effect.","There is sufficient evidence to conclude that the method increases the proportion of HIV positive patients who develop AIDS after four years.","There is insufficient evidence to conclude that the method reduces the proportion of HIV positive patients who develop AIDS after four years."],"hints":{"DefaultPathway":[{"id":"ab2e2b210.4drugsP-valuea-h1","type":"hint","dependencies":[],"title":"P-value","text":"Recall that the $$p-value$$ is the probability that an event will happen purely by chance assuming the null hypothesis is true. The smaller the $$p-value$$, the stronger the evidence is against the null hypothesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab2e2b210.4drugsP-valuea-h2","type":"hint","dependencies":["ab2e2b210.4drugsP-valuea-h1"],"title":"Level of significance","text":"The level of significance in this instance is $$0.05$$. Our $$p-value$$ is less than that boundary value, therefore there is statistical significance of a lower proportion of patients that developed AIDS in four years when taking the new drug compared to the control group.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab2e2b210.4golf","title":"Hypothesis Testing with Two Samples","body":"A golf instructor is interested in determining if her new technique for improving players\u2019 golf scores is effective. She takes four new students. She records their 18-hole scores before learning the technique and then after having taken her class. She conducts a hypothesis test. The data are as follows.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Matched or Paired Samples","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab2e2b210.4golfa","stepAnswer":["Do not reject the $$H_0$$"],"problemType":"MultipleChoice","stepTitle":"The correct decision is:","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Do not reject the $$H_0$$","choices":["Do not reject the $$H_0$$","Reject $$H_0$$","Insufficient data is given"],"hints":{"DefaultPathway":[{"id":"ab2e2b210.4golfa-h1","type":"hint","dependencies":[],"title":"Hypothesis Testing with Two Samples","text":"Input the differences data into a calculator to obtain the mean differences and standard deviation to get the test statistic.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab2e2b210.4golfa-h2","type":"hint","dependencies":["ab2e2b210.4golfa-h1"],"title":"Hypothesis Testing with Two Samples","text":"This is a paired $$t-test$$ as the before and after data is provided as we are testing the mean difference between pairs of measurements if they are zero or not.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab2e2b210.4golfa-h3","type":"hint","dependencies":["ab2e2b210.4golfa-h2"],"title":"Hypothesis Testing with Two Samples","text":"The test statistic can be found through the formula test statistic $$t=\\\\frac{x-0}{\\\\frac{s}{\\\\sqrt{n}}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab2e2b210.4golfa-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.841$$"],"dependencies":["ab2e2b210.4golfa-h3"],"title":"Hypothesis Testing with Two Samples","text":"The probability of obtaining our test statistic is $$0.841$$, and the test statistic obtained is $$-1.19$$, what is the $$p-value$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab2e2b210.4golfa-h5","type":"hint","dependencies":["ab2e2b210.4golfa-h4"],"title":"Hypothesis Testing with Two Samples","text":"Since $$0.841>0.05$$, the null hypothesis is rejected and there is sufficient evidence that the new technique taught by the golf instructor can improve golf players\' scores.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab2e2b210.4hypnotism","title":"A study was conducted to investigate the effectiveness of hypnotism in reducing pain. Results for randomly selected subjects are shown. A lower score indicates less pain. The \'before\' value is matched to an \'after\' value and the differences are calculated. The differences have a normal distribution.","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Matched or Paired Samples","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab2e2b210.4hypnotisma","stepAnswer":["yes"],"problemType":"MultipleChoice","stepTitle":"Are the sensory measurements, on average, lower after hypnotism? Test at a 5% significance level.","stepBody":"","answerType":"string","variabilization":{},"choices":["yes","no"],"hints":{"DefaultPathway":[{"id":"ab2e2b210.4hypnotisma-h1","type":"hint","dependencies":[],"title":"Using Technology","text":"For the TI-83+ and TI-84 calculators, you can either calculate the differences ahead of time (after - before) and put the differences into a list or you can put the after data into a first list and the before data into a second list. Then go to a third list and arrow up to the name. Enter 1st list name - 2nd list name. The calculator will do the subtraction, and you will have the differences in the third list.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab2e2b210.4hypnotisma-h2","type":"hint","dependencies":["ab2e2b210.4hypnotisma-h1"],"title":"Use your list of differences as the data. On your calculator go to the STATS section and $$2$$ sample T-test. Arrow over to Data and press ENTER. Enter $$0$$ for \\\\mu_0, the name of the list where you put the data, and $$1$$ for frequency. Go to \\\\mu and arrow over to <\\\\mu_0. Calculate the differences.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab2e2b210.4hypnotisma-h3","type":"hint","dependencies":["ab2e2b210.4hypnotisma-h2"],"title":"Average Difference","text":"We are testing if the measurements, on average, are lower after hypnotism.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab2e2b210.4hypnotisma-h4","type":"hint","dependencies":["ab2e2b210.4hypnotisma-h3"],"title":"Hypothesis Testing with Two Samples","text":"We can now start formulating the claims. We have before and after data and have a null assumption that there is a mean zero difference between the data, therefore we are conducting a paired two sample test.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab2e2b210.4hypnotisma-h5","type":"hint","dependencies":["ab2e2b210.4hypnotisma-h4"],"title":"Hypothesis Testing with Two Samples","text":"We must assume that there are no effects post hypnotism. Let $$\\\\mu_d$$ denote the difference in before and after pain measurements.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab2e2b210.4hypnotisma-h6","type":"hint","dependencies":["ab2e2b210.4hypnotisma-h5"],"title":"Hypothesis Testing with Two Samples","text":"Let $$\u03bc_d$$ denote the difference in pain measurements.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab2e2b210.4hypnotisma-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$H_0$$: $$\\\\mu_d$$>=0, $$H_a$$: $$\\\\mu_d$$<0"],"dependencies":["ab2e2b210.4hypnotisma-h6"],"title":"Hypothesis Testing with Two Samples","text":"Which of the following claim pairs matches the situation described?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$H_0$$: $$\\\\mu_d$$>=0, $$H_a$$: $$\\\\mu_d$$<0","$$H_0$$: $$\\\\mu_d$$<=0, $$H_a$$: $$\\\\mu_d$$<0","$$H_0$$: $$\\\\mu_d$$<0, $$H_a$$: $$\\\\mu_d$$<0","$$H_0$$: $$\\\\mu_d$$>0, $$H_a$$: $$\\\\mu_d$$<=0"]},{"id":"ab2e2b210.4hypnotisma-h8","type":"hint","dependencies":["ab2e2b210.4hypnotisma-h7"],"title":"Hypothesis Testing with Two Samples","text":"We must assume that the treatment offered is worse for pain or has no effect, therefore the mean differences in pain measurements by default is greater than or equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab2e2b210.4hypnotisma-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3.0359$$"],"dependencies":["ab2e2b210.4hypnotisma-h8"],"title":"Compute the test statistic. Round to the nearest four decimal places.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab2e2b210.4hypnotisma-h10","type":"hint","dependencies":["ab2e2b210.4hypnotisma-h8"],"title":"Test statistic","text":"test statistic $$t=x_d-\\\\mu_d/(s_d/sqrt(n))$$, $$otherwise$$ $$known$$ $$as$$ $$the$$ $$quantity$$ $$mean$$ $$sample$$ $$difference$$ $$subtracted$$ $$by$$ $$the$$ $$population$$ $$mean$$ $$for$$ $$the$$ $$differences$$ $$divided$$ $$by$$ $$the$$ $$sample$$ $$standard$$ $$deviation$$ $$divided$$ $$by$$ $$the$$ $$square$$ $$root$$ $$of$$ $$n$$. $$Note$$ $$that$$ $$this$$ $$is$$ $$a$$ $$t$$ $$distribution$$ $$with$$ $$degrees$$ $$of$$ $$freedom$$ $$n-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab2e2b210.4hypnotisma-h11","type":"hint","dependencies":["ab2e2b210.4hypnotisma-h10"],"title":"Hypothesis Testing with Two Samples","text":"We must know the mean ($$\\\\bar x_d$$) and standard deviation differences($$s_d$$). Using a graphing calculator, the mean differences and the standard deviation of differences can be found in the STAT menu. The mean difference is found to be $$3.125$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab2e2b210.4hypnotisma-h12","type":"hint","dependencies":["ab2e2b210.4hypnotisma-h11"],"title":"Hypothesis Testing with Two Samples","text":"With mean difference equalling $$3.125$$, $$s_d$$=2.91143 $$(the$$ $$standard$$ $$deviation$$ $$of$$ $$differences)$$, $$and$$ $$n=8$$, $$test$$ $$statistic$$ $$t=3.03591$$. $$Round$$ $$to$$ $$four$$ $$decimal$$ $$places$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab2e2b210.4hypnotisma-h13","type":"hint","dependencies":["ab2e2b210.4hypnotisma-h12"],"title":"P-value","text":"Under the P-value table associated with $$t=3.0359$$ and degrees of freedom equaling $$8+8-2=14$$, the $$p-value$$ is found to be $$0.00948$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab2e2b210.4hypnotisma-h14","type":"hint","dependencies":["ab2e2b210.4hypnotisma-h13"],"title":"Hypothesis Testing with Two Samples","text":"Since $$0.00948$$, the $$p-value$$, is less than our significance level of $$0.05$$, the result is significant and we reject the null hypothesis that there is no difference.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab2ed79implicit10","title":"Finding Tangent Lines","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.8 Implicit Differentiation","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ab2ed79implicit10a","stepAnswer":["$$y=-2x+\\\\frac{\\\\pi}{2}+1$$"],"problemType":"MultipleChoice","stepTitle":"For the equation $$tan(xy)=y$$, find the equation of the tangent line at $$(\\\\frac{\\\\pi}{4},1)$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=-2x+\\\\frac{\\\\pi}{2}+1$$","choices":["$$y=-x+\\\\frac{\\\\pi}{4}$$","$$y=-2x+\\\\frac{\\\\pi}{2}+1$$","$$y={sec}^{2\\\\left(\\\\frac{\\\\pi}{4}\\\\right)}-2x+1$$","$$y=-2x+\\\\frac{\\\\pi}{4}$$"],"hints":{"DefaultPathway":[{"id":"ab2ed79implicit10a-h1","type":"hint","dependencies":[],"title":"Implicit Differentiation","text":"First, use implicit differentiation to find $$\\\\frac{dy}{dx}$$. Differentiate both sides of the equation, then simplify.","variabilization":{},"oer":"","license":""},{"id":"ab2ed79implicit10a-h2","type":"hint","dependencies":["ab2ed79implicit10a-h1"],"title":"$$\\\\frac{dy}{dx}$$","text":"$$\\\\frac{dy}{dx}$$ is $$\\\\frac{-y {sec}^2 x y}{x {sec}^2 x y-1}$$","variabilization":{},"oer":"","license":""},{"id":"ab2ed79implicit10a-h3","type":"hint","dependencies":["ab2ed79implicit10a-h2"],"title":"Finding the Slope","text":"To find the slope, plug $$(\\\\frac{\\\\pi}{4},1)$$ into $$\\\\frac{dy}{dx}$$.","variabilization":{},"oer":"","license":""},{"id":"ab2ed79implicit10a-h4","type":"hint","dependencies":["ab2ed79implicit10a-h3"],"title":"Point-Slope Form","text":"Substitute into the point-slope equation of the line to obtain the equation.","variabilization":{},"oer":"","license":""}]}}]},{"id":"ab2ed79implicit20","title":"Cell Phone Production","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.8 Implicit Differentiation","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ab2ed79implicit20a","stepAnswer":["$$-0.593$$"],"problemType":"TextBox","stepTitle":"The number of cell phones produced when $$x$$ dollars is spent on labor and $$y$$ dollars is spent on capital invested by a manufacturer can be modeled by the equation $$60x^{\\\\frac{3}{4}} y^{\\\\frac{1}{4}}=3240$$. Find $$\\\\frac{dy}{dx}$$ and evaluate the point at $$(81,16)$$. Round to $$3$$ decimal places.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-0.593$$","hints":{"DefaultPathway":[{"id":"ab2ed79implicit20a-h1","type":"hint","dependencies":[],"title":"Implicit Differentiation","text":"Use implicit differentiation to find $$\\\\frac{dy}{dx}$$. Differentiate both sides of the equation, then simplify.","variabilization":{},"oer":"","license":""},{"id":"ab2ed79implicit20a-h2","type":"hint","dependencies":["ab2ed79implicit20a-h1"],"title":"Point-Slope Form","text":"Plug $$(81,16)$$ into the $$\\\\frac{dy}{dx}$$ equation for the slope of the equation at $$(81,16)$$.","variabilization":{},"oer":"","license":""}]}}]},{"id":"ab2ed79implicit21","title":"Volume of A Cone","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.8 Implicit Differentiation","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ab2ed79implicit21a","stepAnswer":["$$-0.593$$"],"problemType":"TextBox","stepTitle":"The number of cell phones produced when $$x$$ dollars is spent on labor and $$y$$ dollars is spent on capital invested by a manufacturer can be modeled by the equation $$60x^{\\\\frac{3}{4}} y^{\\\\frac{1}{4}}=3240$$. Find $$\\\\frac{dy}{dx}$$ and evaluate the point at $$(81,16)$$. Round to $$3$$ decimal places.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-0.593$$","hints":{"DefaultPathway":[{"id":"ab2ed79implicit21a-h1","type":"hint","dependencies":[],"title":"Implicit Differentiation","text":"Use implicit differentiation to find $$\\\\frac{dy}{dx}$$. Differentiate both sides of the equation, then simplify.","variabilization":{},"oer":"","license":""},{"id":"ab2ed79implicit21a-h2","type":"hint","dependencies":["ab2ed79implicit21a-h1"],"title":"Point-Slope Form","text":"Plug $$(81,16)$$ into the $$\\\\frac{dy}{dx}$$ equation for the slope of the equation at $$(81,16)$$.","variabilization":{},"oer":"","license":""}]}}]},{"id":"ab2ed79implicit22","title":"Surface Area of a Box","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.8 Implicit Differentiation","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ab2ed79implicit22a","stepAnswer":["$$4xy+2x^2$$"],"problemType":"MultipleChoice","stepTitle":"Consider a box with a square base with sides $$x$$ and height $$y$$. Find an equation for the surface area of the rectangular box, S(x,y).","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$4xy+2x^2$$","choices":["$$4xy+2x^2$$","$$2xy+4x^2$$","$$8x^2 y-2y$$"],"hints":{"DefaultPathway":[{"id":"ab2ed79implicit22a-h1","type":"hint","dependencies":[],"title":"Sides of a Box","text":"A rectangular box has six sides. The base is square, and there are $$4$$ rectangles that create the sides of the box.","variabilization":{},"oer":"","license":""}]}}]},{"id":"ab2ed79implicit23","title":"Surface Area of a Box","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.8 Implicit Differentiation","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ab2ed79implicit23a","stepAnswer":["$$-2.67$$"],"problemType":"TextBox","stepTitle":"If the surface area of the rectangular box is $$78$$ square feet, find $$\\\\frac{dy}{dx}$$ when $$x=3$$ feet and $$y=5$$\\\\nfeet. The surface area equation of the box is $$4xy+2x^2$$. Round to $$2$$ decimal places.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-2.67$$","hints":{"DefaultPathway":[{"id":"ab2ed79implicit23a-h1","type":"hint","dependencies":[],"title":"Implicit Differentiation","text":"Set the equation of the SA of a box to $$78$$. Then, use implicit differentiation to find $$\\\\frac{dy}{dx}$$ by differentiating both sides of the equation and simplifying.","variabilization":{},"oer":"","license":""},{"id":"ab2ed79implicit23a-h2","type":"hint","dependencies":["ab2ed79implicit23a-h1"],"title":"Point-Slope Form","text":"Plug $$x=3$$ and $$y=5$$ into the $$\\\\frac{dy}{dx}$$ equation.","variabilization":{},"oer":"","license":""}]}}]},{"id":"ab2ed79implicit24","title":"Implicit Differentiation Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.8 Implicit Differentiation","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ab2ed79implicit24a","stepAnswer":["$$\\\\frac{-1}{\\\\sqrt{1-x^2}}$$"],"problemType":"MultipleChoice","stepTitle":"For $$x=cos(y)$$, use implicit differentiation to determine y\u2032. Think about wheher the answer agrees with the formulas previously determined.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{-1}{\\\\sqrt{1-x^2}}$$","choices":["$$\\\\frac{-1}{\\\\sqrt{1-x^2}}$$","$$1-y^2$$","$$\\\\sqrt{1-x^2}$$","$$-x$$"],"hints":{"DefaultPathway":[{"id":"ab2ed79implicit24a-h1","type":"hint","dependencies":[],"title":"Implicit Differentiation","text":"Use implicit differentiation to find $$\\\\frac{dy}{dx}$$. Differentiate both sides of the equation, then simplify.","variabilization":{},"oer":"","license":""},{"id":"ab2ed79implicit24a-h2","type":"hint","dependencies":["ab2ed79implicit24a-h1"],"title":"Derivative Rules","text":"Use the product rule and the chain rule. The product rule is: $$\\\\frac{d\\\\left(uv\\\\right)}{dx}=u\\\\left(\\\\frac{dv}{dx}\\\\right)+v\\\\left(\\\\frac{du}{dx}\\\\right)$$. The chain rule is: $$[f(g(x))]\'=\\\\operatorname{f\'}\\\\left(g{\\\\left(x\\\\right)}\\\\right) \\\\operatorname{g\'}\\\\left(x\\\\right)$$.","variabilization":{},"oer":"","license":""}]}}]},{"id":"ab2ed79implicit25","title":"Implicit Differentiation Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.8 Implicit Differentiation","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ab2ed79implicit25a","stepAnswer":["$$\\\\frac{1}{{sec}^2} y$$"],"problemType":"MultipleChoice","stepTitle":"For $$x=tan(y)$$, use implicit differentiation to determine y\u2032. Think about wheher the answer agrees with the formulas previously determined.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{1}{{sec}^2} y$$","choices":["$$\\\\frac{1}{{sec}^2} y$$","$${tan}^2 y$$","$$2{cos}^{-1} y$$","$${sec}^2 x$$"],"hints":{"DefaultPathway":[{"id":"ab2ed79implicit25a-h1","type":"hint","dependencies":[],"title":"Implicit Differentiation","text":"Use implicit differentiation to find $$\\\\frac{dy}{dx}$$. Differentiate both sides of the equation, then simplify.","variabilization":{},"oer":"","license":""},{"id":"ab2ed79implicit25a-h2","type":"hint","dependencies":["ab2ed79implicit25a-h1"],"title":"Derivative Rules","text":"Use the product rule and the chain rule. The product rule is: $$\\\\frac{d\\\\left(uv\\\\right)}{dx}=u\\\\left(\\\\frac{dv}{dx}\\\\right)+v\\\\left(\\\\frac{du}{dx}\\\\right)$$. The chain rule is: $$[f(g(x))]\'=\\\\operatorname{f\'}\\\\left(g{\\\\left(x\\\\right)}\\\\right) \\\\operatorname{g\'}\\\\left(x\\\\right)$$.","variabilization":{},"oer":"","license":""}]}}]},{"id":"ab2ed79implicit9","title":"Finding Tangent Lines","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.8 Implicit Differentiation","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ab2ed79implicit9a","stepAnswer":["$$y=-0.5x+2$$"],"problemType":"MultipleChoice","stepTitle":"For the equation $$x^2 y^2+5xy=14$$, find the equation of the tangent line at $$(2,1)$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=-0.5x+2$$","choices":["$$y=-0.5x+2$$","$$y=2x-1$$","$$y=-1.5x+3$$","$$y=2x+14$$"],"hints":{"DefaultPathway":[{"id":"ab2ed79implicit9a-h1","type":"hint","dependencies":[],"title":"Implicit Differentiation","text":"First, use implicit differentiation to find $$\\\\frac{dy}{dx}$$. Differentiate both sides of the equation, then simplify.","variabilization":{},"oer":"","license":""},{"id":"ab2ed79implicit9a-h2","type":"hint","dependencies":["ab2ed79implicit9a-h1"],"title":"$$\\\\frac{dy}{dx}$$","text":"$$\\\\frac{dy}{dx}$$ is $$\\\\frac{-y}{x}$$","variabilization":{},"oer":"","license":""},{"id":"ab2ed79implicit9a-h3","type":"hint","dependencies":["ab2ed79implicit9a-h2"],"title":"Finding the Slope","text":"To find the slope, plug $$(2,1)$$ into $$\\\\frac{dy}{dx}$$.","variabilization":{},"oer":"","license":""},{"id":"ab2ed79implicit9a-h4","type":"hint","dependencies":["ab2ed79implicit9a-h3"],"title":"Point-Slope Form","text":"Substitute into the point-slope equation of the line to obtain the equation.","variabilization":{},"oer":"","license":""}]}}]},{"id":"ab303f8factoring1","title":"Factor Perfect Square Trinomials","body":"Factor the following perfect square trinomials","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Factor Special Products","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ab303f8factoring1a","stepAnswer":["$${\\\\left(3x+2\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$9x^2+12x+4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(3x+2\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"ab303f8factoring1a-h1","type":"hint","dependencies":[],"title":"Goal","text":"Determine if the trinomial is a perfect square.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring1a-h2","type":"hint","dependencies":["ab303f8factoring1a-h1"],"title":"Standard form of a trinomial","text":"Check if the trinomial follows the pattern of $$a^2+2ab+b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x$$"],"dependencies":["ab303f8factoring1a-h2"],"title":"Square Root","text":"What is the square root of $$9x^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ab303f8factoring1a-h3"],"title":"Square Root","text":"What is the square root of 4?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring1a-h5","type":"hint","dependencies":["ab303f8factoring1a-h4"],"title":"Manipulate","text":"Organize the term into $${\\\\left(3x+2\\\\right)}^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab303f8factoring1b","stepAnswer":["$$\\\\left(5x+1\\\\right) \\\\left(5x-1\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$25x^2-1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(5x+1\\\\right) \\\\left(5x-1\\\\right)$$","hints":{"DefaultPathway":[{"id":"ab303f8factoring1b-h1","type":"hint","dependencies":[],"title":"Difference of Squares Formula","text":"Notice that the problem is a difference of squares and can be factored into the $$\\\\operatorname{form}\\\\left(k a+b\\\\right) \\\\left(k a-b\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring14a-h3","type":"hint","dependencies":["ab303f8factoring14a-h2"],"title":"Goal","text":"Determine if the trinomial is a perfect square.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring14a-h4","type":"hint","dependencies":["ab303f8factoring14a-h3"],"title":"Standard form of a trinomial","text":"Check if the trinomial follow the pattern of $$a^2-b^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring14a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2$$"],"dependencies":["ab303f8factoring14a-h4"],"title":"Square Roots","text":"What is the square root of $$64y^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring14a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ab303f8factoring14a-h5"],"title":"Square Roots","text":"What is the square root of 9?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring14a-h7","type":"hint","dependencies":["ab303f8factoring14a-h6"],"title":"Manipulate","text":"Organize the term into $$2y^{2\\\\left(x^2-4\\\\right)} \\\\left(x^2+4\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab303f8factoring14b","stepAnswer":["$$\\\\left(16x+8\\\\right) \\\\left(16x-8\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor completely: 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4.0>"},{"id":"ab303f8factoring14b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$64$$"],"dependencies":["ab303f8factoring14b-h2"],"title":"Finding $$b^2$$","text":"What is $$b^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring14b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16x$$"],"dependencies":["ab303f8factoring14b-h3"],"title":"Finding a","text":"What is a?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring14b-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["ab303f8factoring14b-h4"],"title":"Finding $$b$$","text":"What is $$b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring14b-h6","type":"hint","dependencies":["ab303f8factoring14b-h5","ab303f8factoring14b-h4"],"title":"Use formula","text":"Put a and $$b$$ in the form $$\\\\left(a+b\\\\right) \\\\left(a-b\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab303f8factoring15","title":"Sum and Difference of Cubes Pattern","body":"Factor the following polynomial","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Factor Special Products","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ab303f8factoring15a","stepAnswer":["$$\\\\left(x+4\\\\right) 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring15b-h6","type":"hint","dependencies":["ab303f8factoring15b-h5","ab303f8factoring15b-h4"],"title":"Use formula","text":"Put a and $$b$$ in the form $$\\\\left(a+b\\\\right) \\\\left(a-b\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab303f8factoring2","title":"Factor Perfect Square Trinomials","body":"Factor the following perfect square trinomials","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Factor Special Products","courseName":"OpenStax: Intermediate 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab303f8factoring2b","stepAnswer":["$$\\\\left(13x+1\\\\right) \\\\left(13x-1\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor completely: $$169x^2-1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(13x+1\\\\right) \\\\left(13x-1\\\\right)$$","hints":{"DefaultPathway":[{"id":"ab303f8factoring2b-h1","type":"hint","dependencies":[],"title":"Difference of Squares Formula","text":"Notice that the problem is a difference of squares and can be factored into the $$\\\\operatorname{form}\\\\left(k a+b\\\\right) \\\\left(k a-b\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring2b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$169x^2$$"],"dependencies":["ab303f8factoring2b-h1"],"title":"Finding 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<CC BY 4.0>","lesson":"6.3 Factor Special Products","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ab303f8factoring3a","stepAnswer":["$${\\\\left(9y-4\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$81y^2-72y+16$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(9y-4\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"ab303f8factoring3a-h1","type":"hint","dependencies":[],"title":"Goal","text":"Determine if the trinomial is a perfect square","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring3a-h2","type":"hint","dependencies":["ab303f8factoring3a-h1"],"title":"Standard form of a trinomial","text":"Check if the trinomial follows the pattern of $$a^2+2ab+b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab303f8factoring3b","stepAnswer":["$$\\\\left(7x+2\\\\right) \\\\left(7x-2\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor completely: $$49x^2-4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(7x+2\\\\right) \\\\left(7x-2\\\\right)$$","hints":{"DefaultPathway":[{"id":"ab303f8factoring3b-h1","type":"hint","dependencies":[],"title":"Difference of Squares Formula","text":"Notice that the problem is a difference of squares and can be factored into the $$\\\\operatorname{form}\\\\left(k a+b\\\\right) \\\\left(k a-b\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring3b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$49x^2$$"],"dependencies":["ab303f8factoring3b-h1"],"title":"Finding $$a^2$$","text":"What is $$a^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring3b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ab303f8factoring3b-h2"],"title":"Finding $$b^2$$","text":"What is $$b^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring3b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7x$$"],"dependencies":["ab303f8factoring3b-h3"],"title":"Finding a","text":"What is a?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring3b-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ab303f8factoring3b-h4"],"title":"Finding $$b$$","text":"What is $$b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring3b-h6","type":"hint","dependencies":["ab303f8factoring3b-h5","ab303f8factoring3b-h4"],"title":"Use formula","text":"Put a and $$b$$ in the form $$\\\\left(a+b\\\\right) \\\\left(a-b\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab303f8factoring4","title":"Factor Perfect Square Trinomials","body":"Factor the following perfect square trinomials","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Factor Special Products","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ab303f8factoring4a","stepAnswer":["$${\\\\left(8y-5\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$64y^2-80y+25$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(8y-5\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"ab303f8factoring4a-h1","type":"hint","dependencies":[],"title":"Goal","text":"Determine if the trinomial is a perfect square.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring4a-h2","type":"hint","dependencies":["ab303f8factoring4a-h1"],"title":"Standard form of a trinomial","text":"Check if the trinomial follows the pattern of $$a^2+2ab+b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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<CC BY 4.0>","lesson":"6.3 Factor Special Products","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ab303f8factoring6a","stepAnswer":["$${\\\\left(8m+7n\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$64m^2+112mn+49n^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(8m+7n\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"ab303f8factoring6a-h1","type":"hint","dependencies":[],"title":"Goal","text":"Determine if the trinomial is a perfect square","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring6a-h2","type":"hint","dependencies":["ab303f8factoring6a-h1"],"title":"Standard form of a trinomial","text":"Check if the trinomial follows the pattern of $$a^2+2ab+b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab303f8factoring6b","stepAnswer":["$$\\\\left(5x+2\\\\right) \\\\left(5x-2\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor completely: $$25x^2-4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(5x+2\\\\right) \\\\left(5x-2\\\\right)$$","hints":{"DefaultPathway":[{"id":"ab303f8factoring6b-h1","type":"hint","dependencies":[],"title":"Difference of Squares Formula","text":"Notice that the problem is a difference of squares and can be factored into the $$\\\\operatorname{form}\\\\left(k a+b\\\\right) \\\\left(k a-b\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring6b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25x^2$$"],"dependencies":["ab303f8factoring6b-h1"],"title":"Finding 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$$b^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring7b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5x$$"],"dependencies":["ab303f8factoring7b-h3"],"title":"Finding a","text":"What is a?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring7b-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["ab303f8factoring7b-h4"],"title":"Finding $$b$$","text":"What is $$b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring7b-h6","type":"hint","dependencies":["ab303f8factoring7b-h5","ab303f8factoring7b-h4"],"title":"Use formula","text":"Put a and $$b$$ in the form $$\\\\left(a+b\\\\right) \\\\left(a-b\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab303f8factoring8","title":"Factor Perfect Square Trinomials","body":"Factor the following perfect square trinomials","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Factor Special Products","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ab303f8factoring8a","stepAnswer":["$$2{y\\\\left(2x-3\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$8x^2 y-24xy+18y$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2{y\\\\left(2x-3\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"ab303f8factoring8a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2y$$"],"dependencies":[],"title":"Common 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring8a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x$$"],"dependencies":["ab303f8factoring8a-h4"],"title":"Square Roots","text":"What is the square root of $$4x^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring8a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ab303f8factoring8a-h5"],"title":"Square Roots","text":"What is the square root of 9?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring8a-h7","type":"hint","dependencies":["ab303f8factoring8a-h6"],"title":"Manipulate","text":"Organize the term into $$2{y\\\\left(2x-3\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab303f8factoring8b","stepAnswer":["$$\\\\left(4x+1\\\\right) \\\\left(4x-1\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor completely: $$16x^2-1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(4x+1\\\\right) \\\\left(4x-1\\\\right)$$","hints":{"DefaultPathway":[{"id":"ab303f8factoring8b-h1","type":"hint","dependencies":[],"title":"Difference of Squares Formula","text":"Notice that the problem is a difference of squares and can be factored into the $$\\\\operatorname{form}\\\\left(k a+b\\\\right) \\\\left(k a-b\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring8b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16x^2$$"],"dependencies":["ab303f8factoring8b-h1"],"title":"Finding $$a^2$$","text":"What is $$a^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring8b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ab303f8factoring8b-h2"],"title":"Finding $$b^2$$","text":"What is $$b^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring8b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4x$$"],"dependencies":["ab303f8factoring8b-h3"],"title":"Finding a","text":"What is a?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring8b-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ab303f8factoring8b-h4"],"title":"Finding $$b$$","text":"What is $$b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring8b-h6","type":"hint","dependencies":["ab303f8factoring8b-h5","ab303f8factoring8b-h4"],"title":"Use formula","text":"Put a and $$b$$ in the form $$\\\\left(a+b\\\\right) \\\\left(a-b\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab303f8factoring9","title":"Factor differences of squares.","body":"Factor the following binomial","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Factor Special Products","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ab303f8factoring9a","stepAnswer":["$$\\\\left(8y-1\\\\right) \\\\left(8y+1\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$64y^2-1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(8y-1\\\\right) \\\\left(8y+1\\\\right)$$","hints":{"DefaultPathway":[{"id":"ab303f8factoring9a-h1","type":"hint","dependencies":[],"title":"Goal","text":"Determine if the trinomial is a perfect square.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8y$$"],"dependencies":["ab303f8factoring9a-h1"],"title":"Square Roots","text":"What is the square root of $$64y^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ab303f8factoring9a-h2"],"title":"Square Roots","text":"What is the square root of 1?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring9a-h4","type":"hint","dependencies":["ab303f8factoring9a-h3"],"title":"Manipulate","text":"Organize the term into $$\\\\left(8y-1\\\\right) \\\\left(8y+1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab303f8factoring9b","stepAnswer":["$$\\\\left(8x+9\\\\right) \\\\left(8x-9\\\\right)$$"],"problemType":"TextBox","stepTitle":"Factor completely: $$64x^2-81$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(8x+9\\\\right) \\\\left(8x-9\\\\right)$$","hints":{"DefaultPathway":[{"id":"ab303f8factoring9b-h1","type":"hint","dependencies":[],"title":"Difference of Squares Formula","text":"Notice that the problem is a difference of squares and can be factored into the $$\\\\operatorname{form}\\\\left(k a+b\\\\right) \\\\left(k a-b\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring9b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$64x^2$$"],"dependencies":["ab303f8factoring9b-h1"],"title":"Finding $$a^2$$","text":"What is $$a^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring9b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$81$$"],"dependencies":["ab303f8factoring9b-h2"],"title":"Finding $$b^2$$","text":"What is $$b^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring9b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8x$$"],"dependencies":["ab303f8factoring9b-h3"],"title":"Finding a","text":"What is a?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring9b-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["ab303f8factoring9b-h4"],"title":"Finding $$b$$","text":"What is $$b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab303f8factoring9b-h6","type":"hint","dependencies":["ab303f8factoring9b-h5","ab303f8factoring9b-h4"],"title":"Use formula","text":"Put a and $$b$$ in the form $$\\\\left(a+b\\\\right) \\\\left(a-b\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab3c11fVisualize1","title":"Simplify:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Visualize Fractions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab3c11fVisualize1a","stepAnswer":["$$\\\\frac{-4}{7}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{-32}{56}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-4}{7}$$","hints":{"DefaultPathway":[{"id":"ab3c11fVisualize1a-h1","type":"hint","dependencies":[],"title":"Defining Simplified Fraction","text":"A fraction is considered simplified if there are no common factors in its numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize1a-h2","type":"hint","dependencies":["ab3c11fVisualize1a-h1"],"title":"Finding Common Factors","text":"If we find the common factor of the numerator and the denominator, we can easily use the equivalent fractions property to simplify the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["ab3c11fVisualize1a-h2"],"title":"Finding Common Factors","text":"What is the largerst common factor between $$32$$ and 56?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab3c11fVisualize1a-h3-s1","type":"hint","dependencies":[],"title":"Finding Common Factors","text":"We can rewrite $$32$$ as $$4\\\\times8$$ and $$56$$ as $$7\\\\times8$$, which tells us that the largest common factor is $$8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ab3c11fVisualize1a-h4","type":"hint","dependencies":["ab3c11fVisualize1a-h3"],"title":"Equivalent Fractions Property","text":"We can then use the equivalent fractions property to simpily the fraction, which states that if a, $$b$$, c are numbers where $$b \\\\neq 0$$, $$c \\\\neq 0$$, then $$\\\\frac{a}{b}=\\\\frac{a c}{b c}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize1a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-4}{7}$$"],"dependencies":["ab3c11fVisualize1a-h4"],"title":"Final Answer","text":"In the previous steps, we rewrite $$\\\\frac{-32}{56}$$ as $$\\\\frac{-\\\\left(4\\\\times8\\\\right)}{7\\\\times8}$$. Observe this fraction. What can we simplify it to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab3c11fVisualize10","title":"Divide:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Visualize Fractions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab3c11fVisualize10a","stepAnswer":["$$\\\\frac{-15}{8q}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{-\\\\left(\\\\frac{5}{8}\\\\right)}{\\\\frac{q}{3}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-15}{8q}$$","hints":{"DefaultPathway":[{"id":"ab3c11fVisualize10a-h1","type":"hint","dependencies":[],"title":"Fraction Division","text":"To divide fractions, we multiply the first fraction by the reciprocal of the second.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize10a-h2","type":"hint","dependencies":["ab3c11fVisualize10a-h1"],"title":"Reciprocal","text":"The reciprocal of $$\\\\frac{a}{b}$$ is $$\\\\frac{b}{a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{q}$$"],"dependencies":["ab3c11fVisualize10a-h2"],"title":"Reciprocal of the Second Fraction","text":"What is the reciprocal of $$\\\\frac{q}{3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize10a-h4","type":"hint","dependencies":["ab3c11fVisualize10a-h3"],"title":"Multiplying Fractions","text":"The next step is to multiply the first fraction by the reciprocal of the second.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize10a-h5","type":"hint","dependencies":["ab3c11fVisualize10a-h4"],"title":"Determining the Sign","text":"Since the signs are different, the product is negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize10a-h6","type":"hint","dependencies":["ab3c11fVisualize10a-h5"],"title":"Multiplying Fractions","text":"To multiply fractions, we multiply the numerators and the denominators.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize10a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["ab3c11fVisualize10a-h6"],"title":"Multiplying the Numerators","text":"What is $$5\\\\times3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize10a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["8q"],"dependencies":["ab3c11fVisualize10a-h7"],"title":"Multiplying the Denominators","text":"What is $$8q$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize10a-h9","type":"hint","dependencies":["ab3c11fVisualize10a-h8"],"title":"Putting Top and Bottom Together","text":"Putting our answers to the last two steps together, we get the fraction $$\\\\frac{-15}{8q}$$. The next step is to simplify our answer if possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize10a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ab3c11fVisualize10a-h9"],"title":"Checking for Common Factors","text":"Is $$\\\\frac{-15}{8q}$$ already in its simplest form?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"ab3c11fVisualize11","title":"Find the quotient:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Visualize Fractions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab3c11fVisualize11a","stepAnswer":["$$\\\\frac{3}{4}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{-\\\\left(\\\\frac{7}{18}\\\\right)}{\\\\left(-\\\\frac{14}{27}\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{4}$$","hints":{"DefaultPathway":[{"id":"ab3c11fVisualize11a-h1","type":"hint","dependencies":[],"title":"Fraction Division","text":"To divide fractions, we multiply the first fraction by the reciprocal of the second.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize11a-h2","type":"hint","dependencies":["ab3c11fVisualize11a-h1"],"title":"Reciprocal","text":"The reciprocal of $$\\\\frac{a}{b}$$ is $$\\\\frac{b}{a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize11a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-27}{14}$$"],"dependencies":["ab3c11fVisualize11a-h2"],"title":"Reciprocal of the Second Fraction","text":"What is the reciprocal of $$\\\\frac{-14}{27}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize11a-h4","type":"hint","dependencies":["ab3c11fVisualize11a-h3"],"title":"Multiplying Fractions","text":"The next step is to multiply the first fraction by the reciprocal of the second.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize11a-h5","type":"hint","dependencies":["ab3c11fVisualize11a-h4"],"title":"Determining the Sign","text":"Since the signs are the same, the product is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize11a-h6","type":"hint","dependencies":["ab3c11fVisualize11a-h5"],"title":"Multiplying Fractions","text":"To multiply fractions, we multiply the numerators and the denominators. So the numerator is $$7\\\\times27$$, and the denominator is $$18\\\\times14$$. The next step is to simplify the fraction if possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize11a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ab3c11fVisualize11a-h6"],"title":"Checking for Common Factors","text":"Do $$7\\\\times27$$ and $$18\\\\times14$$ have any common factors?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ab3c11fVisualize11a-h8","type":"hint","dependencies":["ab3c11fVisualize11a-h7"],"title":"Common Factors","text":"We will rewrite $$7\\\\times27$$ and $$18\\\\times14$$ as the product of primes and divide out the common factors from there.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize11a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3\\\\times3\\\\times3\\\\times7$$"],"dependencies":["ab3c11fVisualize11a-h8"],"title":"Rewriting the Numerator","text":"What is $$7\\\\times27$$ written as the product of primes?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$3\\\\times3\\\\times7$$","$$7\\\\times27$$","$$3\\\\times3\\\\times3\\\\times7$$","$$3\\\\times3\\\\times7\\\\times7$$"]},{"id":"ab3c11fVisualize11a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2\\\\times2\\\\times3\\\\times3\\\\times7$$"],"dependencies":["ab3c11fVisualize11a-h8"],"title":"Rewriting the Denominator","text":"What is $$18\\\\times14$$ written as the product of primes?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2\\\\times3\\\\times7$$","$$2\\\\times3\\\\times3\\\\times3\\\\times7$$","$$2\\\\times2\\\\times3\\\\times3\\\\times7$$","$$2\\\\times5\\\\times5\\\\times7\\\\times7$$"]},{"id":"ab3c11fVisualize11a-h11","type":"hint","dependencies":["ab3c11fVisualize11a-h9","ab3c11fVisualize11a-h10"],"title":"Dividing Common Factors","text":"We can now divide out the common factor $$3\\\\times3\\\\times7$$ from both the top and the bottom, which gives us the fraction $$2\\\\frac{3}{2}=\\\\frac{3}{4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab3c11fVisualize12","title":"Find the quotient:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Visualize Fractions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab3c11fVisualize12a","stepAnswer":["$$\\\\frac{4}{15}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{-\\\\left(\\\\frac{7}{27}\\\\right)}{\\\\left(-\\\\frac{35}{36}\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{4}{15}$$","hints":{"DefaultPathway":[{"id":"ab3c11fVisualize12a-h1","type":"hint","dependencies":[],"title":"Fraction Division","text":"To divide fractions, we multiply the first fraction by the reciprocal of the second.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize12a-h2","type":"hint","dependencies":["ab3c11fVisualize12a-h1"],"title":"Reciprocal","text":"The reciprocal of $$\\\\frac{a}{b}$$ is $$\\\\frac{b}{a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize12a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-36}{35}$$"],"dependencies":["ab3c11fVisualize12a-h2"],"title":"Reciprocal of the Second Fraction","text":"What is the reciprocal of $$\\\\frac{-35}{36}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize12a-h4","type":"hint","dependencies":["ab3c11fVisualize12a-h3"],"title":"Multiplying Fractions","text":"The next step is to multiply the first fraction by the reciprocal of the second.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize12a-h5","type":"hint","dependencies":["ab3c11fVisualize12a-h4"],"title":"Determining the Sign","text":"Since the signs are the same, the product is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize12a-h6","type":"hint","dependencies":["ab3c11fVisualize12a-h5"],"title":"Multiplying Fractions","text":"To multiply fractions, we multiply the numerators and the denominators. So the numerator is $$7\\\\times36$$, and the denominator is $$27\\\\times35$$. The next step is to simplify the fraction if possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize12a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ab3c11fVisualize12a-h6"],"title":"Checking for Common Factors","text":"Do $$7\\\\times36$$ and $$27\\\\times35$$ have any common factors?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ab3c11fVisualize12a-h8","type":"hint","dependencies":["ab3c11fVisualize12a-h7"],"title":"Common Factors","text":"We will rewrite $$7\\\\times36$$ and $$27\\\\times35$$ as the product of primes and divide out the common factors from there.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize12a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2\\\\times2\\\\times3\\\\times3\\\\times7$$"],"dependencies":["ab3c11fVisualize12a-h8"],"title":"Rewriting the Numerator","text":"What is $$7\\\\times36$$ written as the product of primes?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2\\\\times3\\\\times7$$","$$2\\\\times3\\\\times3\\\\times3\\\\times7$$","$$2\\\\times2\\\\times3\\\\times3\\\\times7$$","$$2\\\\times5\\\\times5\\\\times7\\\\times7$$"]},{"id":"ab3c11fVisualize12a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3\\\\times3\\\\times3\\\\times5\\\\times7$$"],"dependencies":["ab3c11fVisualize12a-h8"],"title":"Rewriting the Denominator","text":"What is $$27\\\\times35$$ written as the product of primes?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$3\\\\times3\\\\times5\\\\times7$$","$$3\\\\times3\\\\times3\\\\times5\\\\times7$$","$$3\\\\times3\\\\times3\\\\times3\\\\times5\\\\times7$$","$$2\\\\times3\\\\times3\\\\times3\\\\times5\\\\times7$$"]},{"id":"ab3c11fVisualize12a-h11","type":"hint","dependencies":["ab3c11fVisualize12a-h9","ab3c11fVisualize12a-h10"],"title":"Dividing Common Factors","text":"We can now divide out the common factor $$3\\\\times3\\\\times7$$ from both the top and the bottom, which gives us the fraction $$5\\\\frac{2\\\\times2}{3}=\\\\frac{4}{15}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab3c11fVisualize13","title":"Find the quotient:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Visualize Fractions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab3c11fVisualize13a","stepAnswer":["$$\\\\frac{6}{5}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\frac{3}{4}}{\\\\frac{5}{8}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{6}{5}$$","hints":{"DefaultPathway":[{"id":"ab3c11fVisualize13a-h1","type":"hint","dependencies":[],"title":"Fraction Division","text":"To divide fractions, we multiply the first fraction by the reciprocal of the second.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize13a-h2","type":"hint","dependencies":["ab3c11fVisualize13a-h1"],"title":"Reciprocal","text":"The reciprocal of $$\\\\frac{a}{b}$$ is $$\\\\frac{b}{a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize13a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{8}{5}$$"],"dependencies":["ab3c11fVisualize13a-h2"],"title":"Reciprocal of the Second Fraction","text":"What is the reciprocal of $$\\\\frac{5}{8}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize13a-h4","type":"hint","dependencies":["ab3c11fVisualize13a-h3"],"title":"Multiplying Fractions","text":"The next step is to multiply the first fraction by the reciprocal of the second.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize13a-h5","type":"hint","dependencies":["ab3c11fVisualize13a-h4"],"title":"Determining the Sign","text":"Since the signs are the same, the product is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize13a-h6","type":"hint","dependencies":["ab3c11fVisualize13a-h5"],"title":"Multiplying Fractions","text":"To multiply fractions, we multiply the numerators and the denominators. So the numerator is $$3\\\\times8$$ and the denominator is $$4\\\\times5$$. The next step is to simplify the fraction if possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize13a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ab3c11fVisualize13a-h6"],"title":"Checking for Common Factors","text":"Do $$3\\\\times8$$ and $$4\\\\times5$$ have any common factors?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ab3c11fVisualize13a-h8","type":"hint","dependencies":["ab3c11fVisualize13a-h7"],"title":"Common Factors","text":"We will rewrite $$3\\\\times8$$ and $$4\\\\times5$$ as the product of primes and divide out the common factors from there.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize13a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2\\\\times2\\\\times2\\\\times3$$"],"dependencies":["ab3c11fVisualize13a-h8"],"title":"Rewriting the Numerator","text":"What is $$3\\\\times8$$ written as the product of primes?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2\\\\times3\\\\times3$$","$$2\\\\times3\\\\times3\\\\times5$$","$$2\\\\times2\\\\times2\\\\times3$$","$$2\\\\times3\\\\times5\\\\times7\\\\times7$$"]},{"id":"ab3c11fVisualize13a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2\\\\times2\\\\times5$$"],"dependencies":["ab3c11fVisualize13a-h8"],"title":"Rewriting the Denominator","text":"What is $$4\\\\times5$$ written as the product of primes?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2\\\\times2\\\\times5$$","$$2\\\\times3\\\\times5$$","$$3\\\\times5$$","$$2\\\\times2\\\\times2\\\\times5$$"]},{"id":"ab3c11fVisualize13a-h11","type":"hint","dependencies":["ab3c11fVisualize13a-h9","ab3c11fVisualize13a-h10"],"title":"Dividing Common Factors","text":"We can now divide out the common factor $$2\\\\times2$$ from both the top and the bottom, which gives us the fraction $$\\\\frac{2\\\\times3}{5}=\\\\frac{6}{5}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab3c11fVisualize14","title":"Find the quotient:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Visualize Fractions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab3c11fVisualize14a","stepAnswer":["$$\\\\frac{4}{5}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\frac{2}{3}}{\\\\frac{5}{6}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{4}{5}$$","hints":{"DefaultPathway":[{"id":"ab3c11fVisualize14a-h1","type":"hint","dependencies":[],"title":"Fraction Division","text":"To divide fractions, we multiply the first fraction by the reciprocal of the second.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize14a-h2","type":"hint","dependencies":["ab3c11fVisualize14a-h1"],"title":"Reciprocal","text":"The reciprocal of $$\\\\frac{a}{b}$$ is $$\\\\frac{b}{a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize14a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{6}{5}$$"],"dependencies":["ab3c11fVisualize14a-h2"],"title":"Reciprocal of the Second Fraction","text":"What is the reciprocal of $$\\\\frac{5}{6}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize14a-h4","type":"hint","dependencies":["ab3c11fVisualize14a-h3"],"title":"Multiplying Fractions","text":"The next step is to multiply the first fraction by the reciprocal of the second.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize14a-h5","type":"hint","dependencies":["ab3c11fVisualize14a-h4"],"title":"Determining the Sign","text":"Since the signs are the same, the product is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize14a-h6","type":"hint","dependencies":["ab3c11fVisualize14a-h5"],"title":"Multiplying Fractions","text":"To multiply fractions, we multiply the numerators and the denominators. So the numerator is $$2\\\\times6$$ and the denominator is $$3\\\\times5$$. The next step is to simplify the fraction if possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize14a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ab3c11fVisualize14a-h6"],"title":"Checking for Common Factors","text":"Do $$2\\\\times6$$ and $$3\\\\times5$$ have any common factors?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ab3c11fVisualize14a-h8","type":"hint","dependencies":["ab3c11fVisualize14a-h7"],"title":"Common Factors","text":"We will rewrite $$2\\\\times6$$ and $$3\\\\times5$$ as the product of primes and divide out the common factors from there.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize14a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2\\\\times2\\\\times3$$"],"dependencies":["ab3c11fVisualize14a-h8"],"title":"Rewriting the Numerator","text":"What is $$2\\\\times6$$ written as the product of primes?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2\\\\times2\\\\times3$$","$$2\\\\times3\\\\times3$$","$$2\\\\times2\\\\times2\\\\times3$$","$$2\\\\times3\\\\times5$$"]},{"id":"ab3c11fVisualize14a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3\\\\times5$$"],"dependencies":["ab3c11fVisualize14a-h8"],"title":"Rewriting the Denominator","text":"What is $$3\\\\times5$$ written as the product of primes?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2\\\\times2\\\\times5$$","$$2\\\\times3\\\\times5$$","$$3\\\\times5$$","$$2\\\\times2.5$$"]},{"id":"ab3c11fVisualize14a-h11","type":"hint","dependencies":["ab3c11fVisualize14a-h9","ab3c11fVisualize14a-h10"],"title":"Dividing Common Factors","text":"We can now divide out the common factor $$3$$ from both the top and the bottom, which gives us the fraction $$\\\\frac{2\\\\times2}{5}=\\\\frac{4}{5}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab3c11fVisualize15","title":"Find the quotient:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Visualize Fractions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab3c11fVisualize15a","stepAnswer":["$$\\\\frac{3}{y}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\frac{x}{2}}{\\\\frac{xy}{6}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{y}$$","hints":{"DefaultPathway":[{"id":"ab3c11fVisualize15a-h1","type":"hint","dependencies":[],"title":"Fraction Division","text":"To divide fractions, we multiply the first fraction by the reciprocal of the second.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize15a-h2","type":"hint","dependencies":["ab3c11fVisualize15a-h1"],"title":"Reciprocal","text":"The reciprocal of $$\\\\frac{a}{b}$$ is $$\\\\frac{b}{a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{6}{xy}$$"],"dependencies":["ab3c11fVisualize15a-h2"],"title":"Reciprocal of the Second Fraction","text":"What is the reciprocal of $$\\\\frac{xy}{6}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize15a-h4","type":"hint","dependencies":["ab3c11fVisualize15a-h3"],"title":"Multiplying Fractions","text":"The next step is to multiply the first fraction by the reciprocal of the second.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize15a-h5","type":"hint","dependencies":["ab3c11fVisualize15a-h4"],"title":"Determining the Sign","text":"Since the signs are the same, the product is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize15a-h6","type":"hint","dependencies":["ab3c11fVisualize15a-h5"],"title":"Multiplying Fractions","text":"To multiply fractions, we multiply the numerators and the denominators. So the numerator is $$6x$$ and the denominator is $$2xy$$. The next step is to simplify the fraction if possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize15a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ab3c11fVisualize15a-h6"],"title":"Checking for Common Factors","text":"Do $$6x$$ and $$2xy$$ have any common factors?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ab3c11fVisualize15a-h8","type":"hint","dependencies":["ab3c11fVisualize15a-h7"],"title":"Common Factors","text":"We will rewrite $$6x$$ and $$2xy$$ as the product of primes and divide out the common factors from there.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize15a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2\\\\times3 x$$"],"dependencies":["ab3c11fVisualize15a-h8"],"title":"Rewriting the Numerator","text":"What is $$6x$$ written as the product of primes and variable(s)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2\\\\times2 x$$","$$2\\\\times3 x$$","$$2\\\\times2\\\\times3 x$$","$$2\\\\times5 x$$"]},{"id":"ab3c11fVisualize15a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2x y$$"],"dependencies":["ab3c11fVisualize15a-h8"],"title":"Rewriting the Denominator","text":"What is $$2xy$$ written as the product of primes and variable(s)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2x y$$","$$2x x y$$","$$x y$$"]},{"id":"ab3c11fVisualize15a-h11","type":"hint","dependencies":["ab3c11fVisualize15a-h9","ab3c11fVisualize15a-h10"],"title":"Dividing Common Factors","text":"We can now divide out the common factor $$2x$$ from both the top and the bottom, which gives us the fraction $$\\\\frac{3}{y}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab3c11fVisualize16","title":"Find the quotient:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Visualize Fractions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab3c11fVisualize16a","stepAnswer":["$$\\\\frac{4}{q}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\frac{p}{2}}{\\\\frac{pq}{8}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{4}{q}$$","hints":{"DefaultPathway":[{"id":"ab3c11fVisualize16a-h1","type":"hint","dependencies":[],"title":"Fraction Division","text":"To divide fractions, we multiply the first fraction by the reciprocal of the second.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize16a-h2","type":"hint","dependencies":["ab3c11fVisualize16a-h1"],"title":"Reciprocal","text":"The reciprocal of $$\\\\frac{a}{b}$$ is $$\\\\frac{b}{a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize16a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{8}{pq}$$"],"dependencies":["ab3c11fVisualize16a-h2"],"title":"Reciprocal of the Second Fraction","text":"What is the reciprocal of $$\\\\frac{pq}{8}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize16a-h4","type":"hint","dependencies":["ab3c11fVisualize16a-h3"],"title":"Multiplying Fractions","text":"The next step is to multiply the first fraction by the reciprocal of the second.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize16a-h5","type":"hint","dependencies":["ab3c11fVisualize16a-h4"],"title":"Determining the Sign","text":"Since the signs are the same, the product is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize16a-h6","type":"hint","dependencies":["ab3c11fVisualize16a-h5"],"title":"Multiplying Fractions","text":"To multiply fractions, we multiply the numerators and the denominators. So the numerator is $$8p$$ and the denominator is $$2pq$$. The next step is to simplify the fraction if possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize16a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ab3c11fVisualize16a-h6"],"title":"Checking for Common Factors","text":"Do $$8p$$ and $$2pq$$ have any common factors?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ab3c11fVisualize16a-h8","type":"hint","dependencies":["ab3c11fVisualize16a-h7"],"title":"Common Factors","text":"We will rewrite $$8q$$ and $$2pq$$ as the product of primes and divide out the common factors from there.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize16a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2\\\\times2\\\\times2 p$$"],"dependencies":["ab3c11fVisualize16a-h8"],"title":"Rewriting the Numerator","text":"What is $$8p$$ written as the product of primes and variable(s)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2\\\\times2 p$$","$$2\\\\times2 q$$","$$2\\\\times2\\\\times2 p$$","$$2\\\\times2\\\\times3 p$$"]},{"id":"ab3c11fVisualize16a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2p q$$"],"dependencies":["ab3c11fVisualize16a-h8"],"title":"Rewriting the Denominator","text":"What is $$2pq$$ written as the product of primes and variable(s)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2p q$$","$$2p p q$$","$$p q$$"]},{"id":"ab3c11fVisualize16a-h11","type":"hint","dependencies":["ab3c11fVisualize16a-h9","ab3c11fVisualize16a-h10"],"title":"Dividing Common Factors","text":"We can now divide out the common factor $$2p$$ from both the top and the bottom, which gives us the fraction $$\\\\frac{2\\\\times2}{q}=\\\\frac{4}{q}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab3c11fVisualize17","title":"Simplify:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Visualize Fractions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab3c11fVisualize17a","stepAnswer":["$$\\\\frac{-1}{3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{4-2\\\\left(3\\\\right)}{2^2+2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-1}{3}$$","hints":{"DefaultPathway":[{"id":"ab3c11fVisualize17a-h1","type":"hint","dependencies":[],"title":"General Approach","text":"To simplify an expression with a fraction bar, we first simplify the expression in the numerator and the expression in the denominator. Second, we simplify the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize17a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["ab3c11fVisualize17a-h1"],"title":"Simplifying Numerator","text":"What is $$4-2(3)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize17a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["ab3c11fVisualize17a-h1"],"title":"Simplifying Denominator","text":"What is $$2^2+2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize17a-h4","type":"hint","dependencies":["ab3c11fVisualize17a-h2","ab3c11fVisualize17a-h3"],"title":"Putting Top and Bottom Together","text":"Putting the simplified expressions together, we get the fraction $$\\\\frac{-2}{6}$$, which we will then try to simplify if possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize17a-h5","type":"hint","dependencies":["ab3c11fVisualize17a-h4"],"title":"Determining the Sign","text":"A negative divided by a positive is negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize17a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ab3c11fVisualize17a-h5"],"title":"Checking for Common Factors","text":"Do $$2$$ and $$6$$ have any common factors?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ab3c11fVisualize17a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ab3c11fVisualize17a-h6"],"title":"Finding Largest Common Factor","text":"What is the largest common factor between the numerator and the denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize17a-h8","type":"hint","dependencies":["ab3c11fVisualize17a-h7"],"title":"Simplifying","text":"To simplify the fraction, we can divide both the numerator and the denominator by their largest common factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize17a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ab3c11fVisualize17a-h8"],"title":"Dividing Numerator","text":"What is $$\\\\frac{2}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize17a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ab3c11fVisualize17a-h8"],"title":"Dividing Denominator","text":"What is $$\\\\frac{6}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize17a-h11","type":"hint","dependencies":["ab3c11fVisualize17a-h9","ab3c11fVisualize17a-h10"],"title":"Final Answer","text":"So $$\\\\frac{-1}{3}$$ is our final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab3c11fVisualize18","title":"Simplify:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Visualize Fractions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab3c11fVisualize18a","stepAnswer":["$$\\\\frac{-3}{4}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{6-3\\\\left(5\\\\right)}{3^2+3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-3}{4}$$","hints":{"DefaultPathway":[{"id":"ab3c11fVisualize18a-h1","type":"hint","dependencies":[],"title":"General Approach","text":"To simplify an expression with a fraction bar, we first simplify the expression in the numerator and the expression in the denominator. Second, we simplify the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":["ab3c11fVisualize18a-h1"],"title":"Simplifying Numerator","text":"What is $$6-3(5)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize18a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["ab3c11fVisualize18a-h1"],"title":"Simplifying Denominator","text":"What is $$3^2+3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize18a-h4","type":"hint","dependencies":["ab3c11fVisualize18a-h2","ab3c11fVisualize18a-h3"],"title":"Putting Top and Bottom Together","text":"Putting the simplified expressions together, we get the fraction $$\\\\frac{-9}{12}$$, which we will then try to simplify if possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize18a-h5","type":"hint","dependencies":["ab3c11fVisualize18a-h4"],"title":"Determining the Sign","text":"A negative divided by a positive is negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize18a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ab3c11fVisualize18a-h5"],"title":"Checking for Common Factors","text":"Do $$9$$ and $$12$$ have any common factors?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ab3c11fVisualize18a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ab3c11fVisualize18a-h6"],"title":"Finding Largest Common Factor","text":"What is the largest common factor between the numerator and the denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize18a-h8","type":"hint","dependencies":["ab3c11fVisualize18a-h7"],"title":"Simplifying","text":"To simplify the fraction, we can divide both the numerator and the denominator by their largest common factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize18a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ab3c11fVisualize18a-h8"],"title":"Dividing Numerator","text":"What is $$\\\\frac{9}{3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize18a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ab3c11fVisualize18a-h8"],"title":"Dividing Denominator","text":"What is $$\\\\frac{12}{3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize18a-h11","type":"hint","dependencies":["ab3c11fVisualize18a-h9","ab3c11fVisualize18a-h10"],"title":"Final Answer","text":"So $$\\\\frac{-3}{4}$$ is our final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab3c11fVisualize19","title":"Simplify:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Visualize Fractions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab3c11fVisualize19a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"(4(-3)+6(-2))/(-3(2)-2)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"ab3c11fVisualize19a-h1","type":"hint","dependencies":[],"title":"General Approach","text":"To simplify an expression with a fraction bar, we first simplify the expression in the numerator and the expression in the denominator. Second, we simplify the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-24$$"],"dependencies":["ab3c11fVisualize19a-h1"],"title":"Simplifying Numerator","text":"What is 4(-3)+6(-2)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize19a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-8$$"],"dependencies":["ab3c11fVisualize19a-h1"],"title":"Simplifying Denominator","text":"What is $$-3(2)-2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize19a-h4","type":"hint","dependencies":["ab3c11fVisualize19a-h2","ab3c11fVisualize19a-h3"],"title":"Putting Top and Bottom Together","text":"Putting the simplified expressions together, we get the fraction $$\\\\frac{-24}{\\\\left(-8\\\\right)}$$, which we will then try to simplify if possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize19a-h5","type":"hint","dependencies":["ab3c11fVisualize19a-h4"],"title":"Determining the Sign","text":"A negative divided by a negative is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize19a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ab3c11fVisualize19a-h5"],"title":"Checking for Common Factors","text":"Do $$24$$ and $$8$$ have any common factors?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ab3c11fVisualize19a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["ab3c11fVisualize19a-h6"],"title":"Finding Largest Common Factor","text":"What is the largest common factor between the numerator and the denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize19a-h8","type":"hint","dependencies":["ab3c11fVisualize19a-h7"],"title":"Simplifying","text":"To simplify the fraction, we can divide both the numerator and the denominator by their largest common factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize19a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ab3c11fVisualize19a-h8"],"title":"Dividing Numerator","text":"What is $$\\\\frac{24}{8}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize19a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ab3c11fVisualize19a-h8"],"title":"Dividing Denominator","text":"What is $$\\\\frac{8}{8}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize19a-h11","type":"hint","dependencies":["ab3c11fVisualize19a-h9","ab3c11fVisualize19a-h10"],"title":"Final Answer","text":"So $$\\\\frac{3}{1}=3$$ is our final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab3c11fVisualize2","title":"Simplify:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Visualize Fractions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab3c11fVisualize2a","stepAnswer":["$$\\\\frac{-7}{9}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{-42}{54}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-7}{9}$$","hints":{"DefaultPathway":[{"id":"ab3c11fVisualize2a-h1","type":"hint","dependencies":[],"title":"Defining Simplified Fraction","text":"A fraction is considered simplified if there are no common factors in its numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize2a-h2","type":"hint","dependencies":["ab3c11fVisualize2a-h1"],"title":"Finding Common Factors","text":"If we find the common factor of the numerator and the denominator, we can easily use the equivalent fractions property to simplify the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["ab3c11fVisualize2a-h2"],"title":"Finding Common Factors","text":"What is the largerst common factor between $$42$$ and 54?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab3c11fVisualize2a-h3-s1","type":"hint","dependencies":[],"title":"Finding Common Factors","text":"We can rewrite $$42$$ as $$6\\\\times7$$ and $$54$$ as $$6\\\\times9$$, which tells us that the largest common factor is $$6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ab3c11fVisualize2a-h4","type":"hint","dependencies":["ab3c11fVisualize2a-h3"],"title":"Equivalent Fractions Property","text":"We can then use the equivalent fractions property to simpily the fraction, which states that if a, $$b$$, c are numbers where $$b \\\\neq 0$$, $$c \\\\neq 0$$, then $$\\\\frac{a}{b}=\\\\frac{a c}{b c}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize2a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-7}{9}$$"],"dependencies":["ab3c11fVisualize2a-h4"],"title":"Final Answer","text":"In the previous steps, we rewrite $$\\\\frac{-42}{54}$$ as $$\\\\frac{-\\\\left(6\\\\times7\\\\right)}{6\\\\times9}$$. Observe this fraction. What can we simplify it to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab3c11fVisualize20","title":"Simplify:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Visualize Fractions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab3c11fVisualize20a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"(7(-1)+9(-3))/(-5(3)-2)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"ab3c11fVisualize20a-h1","type":"hint","dependencies":[],"title":"General Approach","text":"To simplify an expression with a fraction bar, we first simplify the expression in the numerator and the expression in the denominator. Second, we simplify the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-34$$"],"dependencies":["ab3c11fVisualize20a-h1"],"title":"Simplifying Numerator","text":"What is 7(-1)+9(-3)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize20a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-17$$"],"dependencies":["ab3c11fVisualize20a-h1"],"title":"Simplifying Denominator","text":"What is $$-5(3)-2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize20a-h4","type":"hint","dependencies":["ab3c11fVisualize20a-h2","ab3c11fVisualize20a-h3"],"title":"Putting Top and Bottom Together","text":"Putting the simplified expressions together, we get the fraction $$\\\\frac{-34}{\\\\left(-17\\\\right)}$$, which we will then try to simplify if possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize20a-h5","type":"hint","dependencies":["ab3c11fVisualize20a-h4"],"title":"Determining the Sign","text":"A negative divided by a negative is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize20a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ab3c11fVisualize20a-h5"],"title":"Checking for Common Factors","text":"Do $$34$$ and $$17$$ have any common factors?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ab3c11fVisualize20a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$17$$"],"dependencies":["ab3c11fVisualize20a-h6"],"title":"Finding Largest Common Factor","text":"What is the largest common factor between the numerator and the denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize20a-h8","type":"hint","dependencies":["ab3c11fVisualize20a-h7"],"title":"Simplifying","text":"To simplify the fraction, we can divide both the numerator and the denominator by their largest common factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize20a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ab3c11fVisualize20a-h8"],"title":"Dividing Numerator","text":"What is $$\\\\frac{34}{17}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize20a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ab3c11fVisualize20a-h8"],"title":"Dividing Denominator","text":"What is $$\\\\frac{17}{17}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize20a-h11","type":"hint","dependencies":["ab3c11fVisualize20a-h9","ab3c11fVisualize20a-h10"],"title":"Final Answer","text":"So $$\\\\frac{2}{1}=2$$ is our final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab3c11fVisualize21","title":"Translate the English phrase into an algebraic expression:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Visualize Fractions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab3c11fVisualize21a","stepAnswer":["$$\\\\frac{m-n}{p}$$"],"problemType":"MultipleChoice","stepTitle":"The quotient of the difference of $$m$$ and $$n$$, and $$p$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{m-n}{p}$$","choices":["$$\\\\frac{m}{p}-n$$","$$m-\\\\frac{n}{p}$$","$$\\\\frac{m}{n}-p$$","$$\\\\frac{m-n}{p}$$"],"hints":{"DefaultPathway":[{"id":"ab3c11fVisualize21a-h1","type":"hint","dependencies":[],"title":"Interpreting the Problem","text":"We are looking for the quotient of the difference of $$m$$ and $$n$$, and $$p$$. This means we want to divide the difference of $$m$$ and $$n$$ by $$p$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize21a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$m-n$$"],"dependencies":["ab3c11fVisualize21a-h1"],"title":"Difference","text":"What is the algebraic expression for the difference of $$m$$ and $$n$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$m-n$$","$$n-m$$","$$\\\\frac{m}{n}$$","$$\\\\frac{n}{m}$$"]},{"id":"ab3c11fVisualize21a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{m-n}{p}$$"],"dependencies":["ab3c11fVisualize21a-h2"],"title":"Quotient","text":"What is the algebraic expression for the quotient of $$m-n$$ and $$p$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(m-n)-p$$","$$p-(m-n)$$","$$\\\\frac{m-n}{p}$$","$$\\\\frac{p}{m-n}$$"]}]}}]},{"id":"ab3c11fVisualize3","title":"How to Simplify a Fraction","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Visualize Fractions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab3c11fVisualize3a","stepAnswer":["$$\\\\frac{-6}{11}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{-210}{385}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-6}{11}$$","hints":{"DefaultPathway":[{"id":"ab3c11fVisualize3a-h1","type":"hint","dependencies":[],"title":"Defining Simplified Fraction","text":"A fraction is considered simplified if there are no common factors in its numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize3a-h2","type":"hint","dependencies":["ab3c11fVisualize3a-h1"],"title":"General Approach","text":"Since it is hard to find all the common factors between the numerator and the denominator by observation, we will rewrite $$210$$ and $$385$$ as the product of primes and divide out the common factors from there.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize3a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2\\\\times3\\\\times5\\\\times7$$"],"dependencies":["ab3c11fVisualize3a-h2"],"title":"Rewriting $$210$$","text":"What is $$210$$ written as the product of primes?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$10\\\\times21$$","$$2\\\\times5\\\\times21$$","$$2\\\\times3\\\\times5\\\\times7$$","$$2\\\\times3\\\\times3\\\\times5\\\\times7$$"]},{"id":"ab3c11fVisualize3a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$5\\\\times7\\\\times11$$"],"dependencies":["ab3c11fVisualize3a-h2"],"title":"Rewriting $$385$$","text":"What is $$385$$ written as the product of primes?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$5\\\\times77$$","$$5\\\\times7\\\\times11$$","$$3\\\\times5\\\\times8$$","$$3\\\\times5\\\\times7$$"]},{"id":"ab3c11fVisualize3a-h5","type":"hint","dependencies":["ab3c11fVisualize3a-h3","ab3c11fVisualize3a-h4"],"title":"Rewriting Fraction","text":"Plug in the prime factoriazation of both the numerator and the denominator, we can rewrite the fraction as $$\\\\frac{-\\\\left(2\\\\times3\\\\times5\\\\times7\\\\right)}{5\\\\times7\\\\times11}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize3a-h6","type":"hint","dependencies":["ab3c11fVisualize3a-h5"],"title":"Dividing Common Factors","text":"We can now divide out the common factors $$5$$ and $$7$$ from both the top and the bottom, which gives us the fraction $$\\\\frac{-\\\\left(2\\\\times3\\\\right)}{11}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize3a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-6}{11}$$"],"dependencies":["ab3c11fVisualize3a-h6"],"title":"Final Answer","text":"Multiply the top. What simplified fraction do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab3c11fVisualize4","title":"How to Simplify a Fraction","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Visualize Fractions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab3c11fVisualize4a","stepAnswer":["$$\\\\frac{-23}{40}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{-69}{120}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-23}{40}$$","hints":{"DefaultPathway":[{"id":"ab3c11fVisualize4a-h1","type":"hint","dependencies":[],"title":"Defining Simplified Fraction","text":"A fraction is considered simplified if there are no common factors in its numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize4a-h2","type":"hint","dependencies":["ab3c11fVisualize4a-h1"],"title":"General Approach","text":"Since it is hard to find all the common factors between the numerator and the denominator by observation, we will rewrite $$69$$ and $$120$$ as the product of primes and divide out the common factors from there.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize4a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3\\\\times23$$"],"dependencies":["ab3c11fVisualize4a-h2"],"title":"Rewriting $$69$$","text":"What is $$69$$ written as the product of primes?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$69$$ is a prime","$$6\\\\times9$$","$$3\\\\times23$$","$$3\\\\times13$$"]},{"id":"ab3c11fVisualize4a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2\\\\times2\\\\times2\\\\times3\\\\times5$$"],"dependencies":["ab3c11fVisualize4a-h2"],"title":"Rewriting $$120$$","text":"What is $$120$$ written as the product of primes?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2\\\\times5\\\\times12$$","$$2\\\\times3\\\\times3\\\\times4\\\\times5$$","$$2\\\\times2\\\\times2\\\\times3\\\\times5$$","$$2\\\\times2\\\\times3\\\\times3\\\\times5$$"]},{"id":"ab3c11fVisualize4a-h5","type":"hint","dependencies":["ab3c11fVisualize4a-h3","ab3c11fVisualize4a-h4"],"title":"Rewriting Fraction","text":"Plug in the prime factoriazation of both the numerator and the denominator, we can rewrite the fraction as $$\\\\frac{-\\\\left(3\\\\times23\\\\right)}{2\\\\times2\\\\times2\\\\times3\\\\times5}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize4a-h6","type":"hint","dependencies":["ab3c11fVisualize4a-h5"],"title":"Dividing Common Factors","text":"We can now divide out the common factor $$3$$ from both the top and the bottom, so we are left with the fraction $$\\\\frac{-23}{2\\\\times2\\\\times2\\\\times5}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize4a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-23}{40}$$"],"dependencies":["ab3c11fVisualize4a-h6"],"title":"Final Answer","text":"Multiply the bottom. What simplified fraction do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab3c11fVisualize5","title":"How to Simplify a Fraction","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Visualize Fractions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab3c11fVisualize5a","stepAnswer":["$$\\\\frac{x}{y}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{5x}{5y}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{x}{y}$$","hints":{"DefaultPathway":[{"id":"ab3c11fVisualize5a-h1","type":"hint","dependencies":[],"title":"Defining Simplified Fraction","text":"A fraction is considered simplified if there are no common factors in its numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize5a-h2","type":"hint","dependencies":["ab3c11fVisualize5a-h1"],"title":"Finding Common Factors","text":"If we find the common factor of the numerator and the denominator, we can easily use the equivalent fractions property to simplify the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize5a-h3","type":"hint","dependencies":["ab3c11fVisualize5a-h2"],"title":"Finding Common Factors","text":"By observing the numerator and the denominator, we find the common factor to be $$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{x}{y}$$"],"dependencies":["ab3c11fVisualize5a-h3"],"title":"Dividing Common Factors","text":"Dividing out $$5$$ from both the top and the bottom. What simplified fraction do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab3c11fVisualize6","title":"Multiply:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Visualize Fractions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab3c11fVisualize6a","stepAnswer":["$$\\\\frac{-55}{84}$$"],"problemType":"TextBox","stepTitle":"$$-\\\\left(\\\\frac{11}{12}\\\\right) \\\\frac{5}{7}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-55}{84}$$","hints":{"DefaultPathway":[{"id":"ab3c11fVisualize6a-h1","type":"hint","dependencies":[],"title":"Determining the Sign","text":"The first step is to find the sign of the product. Since the signs are the different, the product is negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize6a-h2","type":"hint","dependencies":["ab3c11fVisualize6a-h1"],"title":"Multiplying","text":"The next step is to multiply the two numerators and the two demoninators.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$55$$"],"dependencies":["ab3c11fVisualize6a-h2"],"title":"Multiplying the Numerators","text":"What is $$11\\\\times5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$84$$"],"dependencies":["ab3c11fVisualize6a-h3"],"title":"Multiplying the Denominators","text":"What is $$12\\\\times7$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize6a-h5","type":"hint","dependencies":["ab3c11fVisualize6a-h4"],"title":"Putting Top and Bottom Together","text":"Putting our answers to the last two steps together, we get the fraction $$\\\\frac{-55}{84}$$. The next step is to simplify our answer if possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize6a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ab3c11fVisualize6a-h5"],"title":"Checking for Common Factors","text":"Is $$\\\\frac{-55}{84}$$ already in its simplest form?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"ab3c11fVisualize7","title":"Multiply:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Visualize Fractions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab3c11fVisualize7a","stepAnswer":["$$\\\\frac{-4}{21}$$"],"problemType":"TextBox","stepTitle":"$$-\\\\left(\\\\frac{10}{28}\\\\right) \\\\frac{8}{15}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-4}{21}$$","hints":{"DefaultPathway":[{"id":"ab3c11fVisualize7a-h1","type":"hint","dependencies":[],"title":"Determining the Sign","text":"The first step is to find the sign of the product. Since the signs are the different, the product is negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize7a-h2","type":"hint","dependencies":["ab3c11fVisualize7a-h1"],"title":"Multiplying","text":"The next step is to multiply the two numerators and the two demoninators.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$80$$"],"dependencies":["ab3c11fVisualize7a-h2"],"title":"Multiplying the Numerators","text":"What is $$10\\\\times8$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$420$$"],"dependencies":["ab3c11fVisualize7a-h2"],"title":"Multiplying the Denominators","text":"What is $$28\\\\times15$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize7a-h5","type":"hint","dependencies":["ab3c11fVisualize7a-h3","ab3c11fVisualize7a-h4"],"title":"Putting Top and Bottom Together","text":"Putting our answers to the last two steps together, we get the fraction $$\\\\frac{-80}{420}$$. The next step is to simplify our answer if possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize7a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ab3c11fVisualize7a-h5"],"title":"Checking for Common Factors","text":"Is $$\\\\frac{-80}{420}$$ already in its simplest form?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ab3c11fVisualize7a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["ab3c11fVisualize7a-h6"],"title":"Finding Largest Common Factor","text":"What is the largest common factor between the numerator and the denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize7a-h8","type":"hint","dependencies":["ab3c11fVisualize7a-h7"],"title":"Simplifying","text":"To simplify the fraction, we can divide both the numerator and the denominator by their largest common factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize7a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ab3c11fVisualize7a-h8"],"title":"Dividing Numerator","text":"What is $$\\\\frac{80}{20}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize7a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$21$$"],"dependencies":["ab3c11fVisualize7a-h8"],"title":"Dividing Denominator","text":"What is $$\\\\frac{420}{20}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize7a-h11","type":"hint","dependencies":["ab3c11fVisualize7a-h9","ab3c11fVisualize7a-h10"],"title":"Final Answer","text":"So $$\\\\frac{-4}{21}$$ is our final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab3c11fVisualize8","title":"Multiply:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Visualize Fractions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab3c11fVisualize8a","stepAnswer":["$$48x$$"],"problemType":"TextBox","stepTitle":"$$-\\\\left(\\\\frac{12}{5}\\\\right) \\\\left(-20x\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$48x$$","hints":{"DefaultPathway":[{"id":"ab3c11fVisualize8a-h1","type":"hint","dependencies":[],"title":"Determining the Sign","text":"The first step is to find the sign of the product. Since the signs are the same, the product is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize8a-h2","type":"hint","dependencies":["ab3c11fVisualize8a-h1"],"title":"Multiplying by an Integer","text":"When multiplying a fraction by an integer, it may be helpful to write the integer as a fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize8a-h3","type":"hint","dependencies":["ab3c11fVisualize8a-h2"],"title":"Rewriting Integer","text":"Rewrite $$20x$$ as $$\\\\frac{20x}{1}$$, so we can multiply $$\\\\frac{12}{5}$$ and $$\\\\frac{20x}{1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize8a-h4","type":"hint","dependencies":["ab3c11fVisualize8a-h3"],"title":"Multiplying Fractions","text":"The next step is to multiply the two numerators and the two demoninators.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize8a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$240x$$"],"dependencies":["ab3c11fVisualize8a-h4"],"title":"Multiplying the Numerators","text":"What is $$12\\\\times20 x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize8a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["ab3c11fVisualize8a-h4"],"title":"Multiplying the Denominators","text":"What is $$5\\\\times1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize8a-h7","type":"hint","dependencies":["ab3c11fVisualize8a-h5","ab3c11fVisualize8a-h6"],"title":"Putting Top and Bottom Together","text":"Putting our answers to the last two steps together, we get the fraction $$\\\\frac{240x}{5}$$. The next step is to simplify our answer if possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize8a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ab3c11fVisualize8a-h7"],"title":"Checking for Common Factors","text":"Is $$\\\\frac{240x}{5}$$ not in its simplest form?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ab3c11fVisualize8a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["ab3c11fVisualize8a-h8"],"title":"Finding Largest Common Factor","text":"What is the largest common factor between the numerator and the denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize8a-h10","type":"hint","dependencies":["ab3c11fVisualize8a-h9"],"title":"Simplifying","text":"To simplify the fraction, we can divide both the numerator and the denominator by their largest common factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize8a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$48x$$"],"dependencies":["ab3c11fVisualize8a-h10"],"title":"Dividing Numerator","text":"What is $$\\\\frac{240x}{5}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize8a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ab3c11fVisualize8a-h11"],"title":"Dividing Denominator","text":"What is $$\\\\frac{5}{5}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize8a-h13","type":"hint","dependencies":["ab3c11fVisualize8a-h12"],"title":"Final Answer","text":"So $$48x$$ is our final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab3c11fVisualize9","title":"Divide:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.5 Visualize Fractions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab3c11fVisualize9a","stepAnswer":["$$\\\\frac{-10}{3n}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{-\\\\left(\\\\frac{2}{3}\\\\right)}{\\\\frac{n}{5}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-10}{3n}$$","hints":{"DefaultPathway":[{"id":"ab3c11fVisualize9a-h1","type":"hint","dependencies":[],"title":"Fraction Division","text":"To divide fractions, we multiply the first fraction by the reciprocal of the second.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize9a-h2","type":"hint","dependencies":["ab3c11fVisualize9a-h1"],"title":"Reciprocal","text":"The reciprocal of $$\\\\frac{a}{b}$$ is $$\\\\frac{b}{a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{n}$$"],"dependencies":["ab3c11fVisualize9a-h2"],"title":"Reciprocal of the Second Fraction","text":"What is the reciprocal of $$\\\\frac{n}{5}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize9a-h4","type":"hint","dependencies":["ab3c11fVisualize9a-h3"],"title":"Multiplying Fractions","text":"The next step is to multiply the first fraction by the reciprocal of the second.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize9a-h5","type":"hint","dependencies":["ab3c11fVisualize9a-h4"],"title":"Determining the Sign","text":"Since the signs are different, the product is negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize9a-h6","type":"hint","dependencies":["ab3c11fVisualize9a-h5"],"title":"Multiplying Fractions","text":"To multiply fractions, we multiply the numerators and the denominators.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize9a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["ab3c11fVisualize9a-h6"],"title":"Multiplying the Numerators","text":"What is $$2\\\\times5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize9a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3n$$"],"dependencies":["ab3c11fVisualize9a-h7"],"title":"Multiplying the Denominators","text":"What is $$3n$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize9a-h9","type":"hint","dependencies":["ab3c11fVisualize9a-h8"],"title":"Putting Top and Bottom Together","text":"Putting our answers to the last two steps together, we get the fraction $$\\\\frac{-10}{3n}$$. The next step is to simplify our answer if possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab3c11fVisualize9a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ab3c11fVisualize9a-h9"],"title":"Checking for Common Factors","text":"Is $$\\\\frac{-10}{3n}$$ already in its simplest form?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"ab50a95log1","title":"Converting from Logarithmic Form to Exponential Form","body":"Write the following logarithmic equations in exponential form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ab50a95log1a","stepAnswer":["$$6^{\\\\frac{1}{2}}=\\\\sqrt{6}$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\log_{6}\\\\left(\\\\sqrt{6}\\\\right)=\\\\frac{1}{2}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$6^{\\\\frac{1}{2}}=\\\\sqrt{6}$$","choices":["$$6^{\\\\frac{1}{2}}=\\\\sqrt{6}$$","$${\\\\left(\\\\frac{1}{2}\\\\right)}^6={\\\\left(\\\\frac{1}{2}\\\\right)}^6$$","$$6^2=\\\\sqrt{6}$$","None of the above"],"hints":{"DefaultPathway":[{"id":"ab50a95log1a-h1","type":"hint","dependencies":[],"title":"Rewrite.","text":"$$\\\\log_{a}\\\\left(x\\\\right)=b$$ is equivalent to $$a^b=x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab50a95log1b","stepAnswer":["$$3^2=9$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\log_{3}\\\\left(9\\\\right)=2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$3^2=9$$","choices":["$$2^3=9$$","$$3^2=9$$","$$\\\\sqrt{9}=3$$","None of the above"],"hints":{"DefaultPathway":[{"id":"ab50a95log1b-h1","type":"hint","dependencies":[],"title":"Rewrite.","text":"$$\\\\log_{a}\\\\left(x\\\\right)=b$$ is equivalent to $$a^b=x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab50a95log10","title":"Finding the Value of a Common Logarithm Mentally","body":"Solve the expression without a calculator","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ab50a95log10a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"$$y=\\\\ln(1000000)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"ab50a95log10a-h1","type":"hint","dependencies":[],"title":"Natural Log","text":"Whenever there is no base, assume the base is $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab50a95log10a-h2","type":"hint","dependencies":["ab50a95log10a-h1"],"title":"Rethink","text":"Think of it like $${10}^y=1000000$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab50a95log11","title":"Finding the Value of a Common Logarithm Using a Calculator","body":"Solve the expression with a calculator to $$4$$ decimal places.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ab50a95log11a","stepAnswer":["$$2.5065$$"],"problemType":"TextBox","stepTitle":"$$y=\\\\ln(321)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2.5065$$","hints":{"DefaultPathway":[{"id":"ab50a95log11a-h1","type":"hint","dependencies":[],"title":"Calculator","text":"Type the expression into a calculator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab50a95log12","title":"Finding the Value of a Common Logarithm Using a Calculator","body":"Solve the expression with a calculator to $$4$$ decimal places.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ab50a95log12a","stepAnswer":["$$2.09$$"],"problemType":"TextBox","stepTitle":"$$y=\\\\ln(123)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2.09$$","hints":{"DefaultPathway":[{"id":"ab50a95log12a-h1","type":"hint","dependencies":[],"title":"Calculator","text":"Type the expression into a calculator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab50a95log13","title":"Rewriting and Solving a Real-World Exponential Model","body":"Solve the expression using logarithmic and exponential expressions.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ab50a95log13a","stepAnswer":["$$2.699$$"],"problemType":"TextBox","stepTitle":"The amount of energy released from one earthquake was $$500$$ times greater than the amount of energy released from another. The equation $$10x=500$$ represents this situation, where $$x$$ is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2.699$$","hints":{"DefaultPathway":[{"id":"ab50a95log13a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\ln(500)=x$$"],"dependencies":[],"title":"Conversion","text":"What is the expression as a logarithmic one?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\ln(500)=x$$","$$\\\\ln(x)=500$$","$$\\\\log_{x}\\\\left(500\\\\right)=10$$","None of the above"]},{"id":"ab50a95log13a-h2","type":"hint","dependencies":["ab50a95log13a-h1"],"title":"Calculator","text":"Evaluate the log using a calculator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab50a95log14","title":"Rewriting and Solving a Real-World Exponential Model","body":"Solve the expression using logarithmic and exponential expressions.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ab50a95log14a","stepAnswer":["$$3.929$$"],"problemType":"TextBox","stepTitle":"The amount of energy released from one earthquake was 8,500 times greater than the amount of energy released from another. The equation $$10x=8500$$ represents this situation, where $$x$$ is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3.929$$","hints":{"DefaultPathway":[{"id":"ab50a95log14a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\ln(8500)=x$$"],"dependencies":[],"title":"Conversion","text":"What is the expression as a logarithmic one?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\ln(8500)=x$$","$$\\\\log_{x}\\\\left(8500\\\\right)=10$$","$$\\\\ln(x)=8500$$","None of the above"]},{"id":"ab50a95log14a-h2","type":"hint","dependencies":["ab50a95log14a-h1"],"title":"Calculator","text":"Evaluate the log using a calculator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab50a95log15","title":"Evaluating a Natural Logarithm Using a Calculator","body":"Use a calculator to find the value of the expression to four decimal places.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ab50a95log15a","stepAnswer":["$$6.2146$$"],"problemType":"TextBox","stepTitle":"$$y=ln(500)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6.2146$$","hints":{"DefaultPathway":[{"id":"ab50a95log15a-h1","type":"hint","dependencies":[],"title":"Natural Log","text":"Evaluate the natural log using a calculator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab50a95log16","title":"Evaluating a Natural Logarithm Using a Calculator","body":"Use a calculator to find the value of the expression to four decimal places.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ab50a95log16a","stepAnswer":["$$y=undefined$$"],"problemType":"MultipleChoice","stepTitle":"$$y=ln(-500)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=undefined$$","choices":["$$y=8.867$$","$$y=7.9888$$","$$y=undefined$$","$$y=0.9867$$"],"hints":{"DefaultPathway":[{"id":"ab50a95log16a-h1","type":"hint","dependencies":[],"title":"Natural Log of Negative Numbers","text":"You can never take the natural log of negative numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab50a95log17","title":"Rewriting Equations in Exponential Form: Exercise #1","body":"For the following exercise, rewrite the equation in exponential form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ab50a95log17a","stepAnswer":["$$4^m=q$$"],"problemType":"TextBox","stepTitle":"log base $$4$$ of q $$=$$ $$m$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4^m=q$$","hints":{"DefaultPathway":[{"id":"ab50a95log17a-h1","type":"hint","dependencies":[],"title":"Identifying $$y$$, $$b$$, and $$x$$","text":"The first step is to examine the equation $$y=log$$ base $$b$$ of $$x$$ and identify $$y$$, $$b$$, and $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab50a95log17a-h2","type":"hint","dependencies":["ab50a95log17a-h1"],"title":"Rewriting the Equation","text":"Next, rewrite the equation $$\\"y=log$$ base $$b$$ of x\\" as $$b^y=x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab50a95log18","title":"Rewriting Equations in Exponential Form: Exercise #2","body":"For the following exercise, rewrite the equation in exponential form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ab50a95log18a","stepAnswer":["$$a^c=b$$"],"problemType":"TextBox","stepTitle":"log base a of $$b$$ $$=$$ c","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$a^c=b$$","hints":{"DefaultPathway":[{"id":"ab50a95log18a-h1","type":"hint","dependencies":[],"title":"Identifying $$y$$, $$b$$, and $$x$$","text":"The first step is to examine the equation $$y=log$$ base $$b$$ of $$x$$ and identify $$y$$, $$b$$, and $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab50a95log18a-h2","type":"hint","dependencies":["ab50a95log18a-h1"],"title":"Rewriting the Equation","text":"Next, rewrite the equation $$\\"y=log$$ base $$b$$ of x\\" as $$b^y=x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab50a95log19","title":"Rewriting Equations in Exponential Form: Exercise #3","body":"For the following exercise, rewrite the equation in exponential form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ab50a95log19a","stepAnswer":["$${16}^x=y$$"],"problemType":"TextBox","stepTitle":"log base $$16$$ of $$y$$ $$=$$ $$x$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${16}^x=y$$","hints":{"DefaultPathway":[{"id":"ab50a95log19a-h1","type":"hint","dependencies":[],"title":"Identifying $$y$$, $$b$$, and $$x$$","text":"The first step is to examine the equation $$y=log$$ base $$b$$ of $$x$$ and identify $$y$$, $$b$$, and $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab50a95log19a-h2","type":"hint","dependencies":["ab50a95log19a-h1"],"title":"Rewriting the Equation","text":"Next, rewrite the equation $$\\"y=log$$ base $$b$$ of x\\" as $$b^y=x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab50a95log2","title":"Converting from Logarithmic Form to Exponential Form","body":"Write the following logarithmic equations in exponential form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ab50a95log2a","stepAnswer":["$${10}^6=1000000$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\log_{10}\\\\left(1000000\\\\right)=6$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$${10}^6=1000000$$","choices":["$$6^{10}=100000$$","$${10}^6=1000000$$","$${1000000}^{\\\\frac{1}{6}}=10$$","None of the above"],"hints":{"DefaultPathway":[{"id":"ab50a95log2a-h1","type":"hint","dependencies":[],"title":"Rewrite.","text":"$$\\\\log_{a}\\\\left(x\\\\right)=b$$ is equivalent to $$a^b=x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab50a95log2b","stepAnswer":["$$5^2=25$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\log_{5}\\\\left(25\\\\right)$$ $$=$$ $$2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$5^2=25$$","choices":["$$5^2=25$$","$${25}^{\\\\frac{1}{2}}=5$$","$$2^5=25$$","None of the above"],"hints":{"DefaultPathway":[{"id":"ab50a95log2b-h1","type":"hint","dependencies":[],"title":"Rewrite.","text":"$$\\\\log_{a}\\\\left(x\\\\right)=b$$ is equivalent to $$a^b=x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab50a95log20","title":"Rewriting Equations in Exponential Form: Exercise #4","body":"For the following exercise, rewrite the equation in exponential form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ab50a95log20a","stepAnswer":["$$a^c=b$$"],"problemType":"TextBox","stepTitle":"log base a of $$b$$ $$=$$ c","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$a^c=b$$","hints":{"DefaultPathway":[{"id":"ab50a95log20a-h1","type":"hint","dependencies":[],"title":"Identifying $$y$$, $$b$$, and $$x$$","text":"The first step is to examine the equation $$y=log$$ base $$b$$ of $$x$$ and identify $$y$$, $$b$$, and $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab50a95log20a-h2","type":"hint","dependencies":["ab50a95log20a-h1"],"title":"Rewriting the Equation","text":"Next, rewrite the equation $$\\"y=log$$ base $$b$$ of x\\" as $$b^y=x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab50a95log21","title":"Rewriting Equations in Exponential Form: Exercise #5","body":"For the following exercise, rewrite the equation in exponential form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ab50a95log21a","stepAnswer":["$$x^y=64$$"],"problemType":"TextBox","stepTitle":"log base $$x$$ of $$64$$ $$=$$ $$y$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^y=64$$","hints":{"DefaultPathway":[{"id":"ab50a95log21a-h1","type":"hint","dependencies":[],"title":"Identifying $$y$$, $$b$$, and $$x$$","text":"The first step is to examine the equation $$y=log$$ base $$b$$ of $$x$$ and identify $$y$$, $$b$$, and $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab50a95log21a-h2","type":"hint","dependencies":["ab50a95log21a-h1"],"title":"Rewriting the Equation","text":"Next, rewrite the equation $$\\"y=log$$ base $$b$$ of x\\" as $$b^y=x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab50a95log22","title":"Rewriting Equations in Exponential Form: Exercise #6","body":"For the following exercise, rewrite the equation in exponential form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ab50a95log22a","stepAnswer":["$$y^{\\\\left(-11\\\\right)}=x$$"],"problemType":"TextBox","stepTitle":"log base $$y$$ of $$x$$ $$=$$ $$-11$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y^{\\\\left(-11\\\\right)}=x$$","hints":{"DefaultPathway":[{"id":"ab50a95log22a-h1","type":"hint","dependencies":[],"title":"Identifying $$y$$, $$b$$, and $$x$$","text":"The first step is to examine the equation $$y=log$$ base $$b$$ of $$x$$ and identify $$y$$, $$b$$, and $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab50a95log22a-h2","type":"hint","dependencies":["ab50a95log22a-h1"],"title":"Rewriting the Equation","text":"Next, rewrite the equation $$\\"y=log$$ base $$b$$ of x\\" as $$b^y=x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab50a95log22b","stepAnswer":["$${13}^a=142$$"],"problemType":"TextBox","stepTitle":"log base $$13$$ of $$142$$ $$=$$ a","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${13}^a=142$$","hints":{"DefaultPathway":[{"id":"ab50a95log22b-h1","type":"hint","dependencies":[],"title":"Identifying $$y$$, $$b$$, and $$x$$","text":"The first step is to examine the equation $$y=log$$ base $$b$$ of $$x$$ and identify $$y$$, $$b$$, and $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab50a95log22b-h2","type":"hint","dependencies":["ab50a95log22b-h1"],"title":"Rewriting the Equation","text":"Next, rewrite the equation $$\\"y=log$$ base $$b$$ of x\\" as $$b^y=x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab50a95log23","title":"Rewriting Equations in Exponential Form: Exercise #8","body":"For the following exercise, rewrite the equation in exponential form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ab50a95log23a","stepAnswer":["$$e^n=w$$"],"problemType":"TextBox","stepTitle":"$$ln(w)=n$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$e^n=w$$","hints":{"DefaultPathway":[{"id":"ab50a95log23a-h1","type":"hint","dependencies":[],"title":"Identifying $$y$$, $$b$$, and $$x$$","text":"The first step is to examine the equation $$y=log$$ base $$b$$ of $$x$$ and identify $$y$$, $$b$$, and $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab50a95log23a-h2","type":"hint","dependencies":["ab50a95log23a-h1"],"title":"Rewriting the Equation","text":"Next, rewrite the equation $$\\"y=log$$ base $$b$$ of x\\" as $$b^y=x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab50a95log23a-h3","type":"hint","dependencies":["ab50a95log23a-h2"],"title":"Meaning of ln","text":"ln means \\"log base e.\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab50a95log24","title":"Rewriting Equations in Exponential Form: Exercise #9","body":"For the following exercise, rewrite the equation in exponential form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ab50a95log24a","stepAnswer":["$${10}^t=v$$"],"problemType":"TextBox","stepTitle":"$$\\\\ln(v)=t$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${10}^t=v$$","hints":{"DefaultPathway":[{"id":"ab50a95log24a-h1","type":"hint","dependencies":[],"title":"Identifying $$y$$, $$b$$, and $$x$$","text":"The first step is to examine the equation $$y=log$$ base $$b$$ of $$x$$ and identify $$y$$, $$b$$, and $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab50a95log24a-h2","type":"hint","dependencies":["ab50a95log24a-h1"],"title":"Rewriting the Equation","text":"Next, rewrite the equation $$\\"y=log$$ base $$b$$ of x\\" as $$b^y=x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab50a95log24a-h3","type":"hint","dependencies":["ab50a95log24a-h2"],"title":"Meaning of log","text":"When there is no base indicated, log means \\"log base 10.\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab50a95log25","title":"Rewriting Equations in Logarithmic Form: Exercise #1","body":"For the following exercise, rewrite the equation in logarithmic form, such as $$\\"y=log$$ base $$b$$ of x.\\"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ab50a95log25a","stepAnswer":["x=log base 4 of y"],"problemType":"TextBox","stepTitle":"$$4^x=y$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=log$$ base $$4$$ of $$y$$","hints":{"DefaultPathway":[{"id":"ab50a95log25a-h1","type":"hint","dependencies":[],"title":"Identifying $$y$$, $$b$$, and $$x$$","text":"The first step is to examine the equation $$b^y=x$$ and identify $$y$$, $$b$$, and $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab50a95log25a-h2","type":"hint","dependencies":["ab50a95log25a-h1"],"title":"Rewriting the Equation","text":"Next, rewrite it as $$y=log$$ base $$b$$ of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab50a95log26","title":"Rewriting Equations in Exponential Form: Exercise #10","body":"For the following exercise, rewrite the equation in exponential form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ab50a95log26a","stepAnswer":["$${15}^b=a$$"],"problemType":"TextBox","stepTitle":"log base $$15$$ of $$a=b$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${15}^b=a$$","hints":{"DefaultPathway":[{"id":"ab50a95log26a-h1","type":"hint","dependencies":[],"title":"Identifying $$y$$, $$b$$, and $$x$$","text":"The first step is to examine the equation $$y=log$$ base $$b$$ of $$x$$ and identify $$y$$, $$b$$, and $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab50a95log26a-h2","type":"hint","dependencies":["ab50a95log26a-h1"],"title":"Rewriting the Equation","text":"Next, rewrite the equation $$\\"y=log$$ base $$b$$ of x\\" as $$b^y=x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab50a95log27","title":"Evaluating Logarithms: Exercise #1","body":"For the following exercise, solve for $$x$$ by converting the logarithmic equation to exponential form. $$x=$$?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ab50a95log27a","stepAnswer":["$$9$$"],"problemType":"TextBox","stepTitle":"log base $$3$$ of $$x=2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9$$","hints":{"DefaultPathway":[{"id":"ab50a95log27a-h1","type":"hint","dependencies":[],"title":"Identifying $$y$$, $$b$$, and $$x$$","text":"The first step is to examine the equation $$y=log$$ base $$b$$ of $$x$$ and identify $$y$$, $$b$$, and $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab50a95log27a-h2","type":"hint","dependencies":["ab50a95log27a-h1"],"title":"Rewriting the Equation","text":"Next, rewrite the equation $$\\"y=log$$ base $$b$$ of x\\" as $$b^y=x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab50a95log28","title":"Evaluating Logarithms: Exercise #2","body":"For the following exercise, solve for $$x$$ by converting the logarithmic equation to exponential form. $$x=$$?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ab50a95log28a","stepAnswer":["$$\\\\frac{1}{8}$$"],"problemType":"TextBox","stepTitle":"log base $$2$$ of $$x=-3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{8}$$","hints":{"DefaultPathway":[{"id":"ab50a95log28a-h1","type":"hint","dependencies":[],"title":"Identifying $$y$$, $$b$$, and $$x$$","text":"The first step is to examine the equation $$y=log$$ base $$b$$ of $$x$$ and identify $$y$$, $$b$$, and $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab50a95log28a-h2","type":"hint","dependencies":["ab50a95log28a-h1"],"title":"Rewriting the Equation","text":"Next, rewrite the equation $$\\"y=log$$ base $$b$$ of x\\" as $$b^y=x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab50a95log28b","stepAnswer":["$$27$$"],"problemType":"TextBox","stepTitle":"log base $$3$$ of $$x=3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$27$$","hints":{"DefaultPathway":[{"id":"ab50a95log28b-h1","type":"hint","dependencies":[],"title":"Identifying $$y$$, $$b$$, and $$x$$","text":"The first step is to examine the equation $$y=log$$ base $$b$$ of $$x$$ and identify $$y$$, $$b$$, and $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab50a95log28b-h2","type":"hint","dependencies":["ab50a95log28b-h1"],"title":"Rewriting the Equation","text":"Next, rewrite the equation $$\\"y=log$$ base $$b$$ of x\\" as $$b^y=x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab50a95log29","title":"Evaluating Logarithms: Exercise #4","body":"For the following exercise, solve for $$x$$ by converting the logarithmic equation to exponential form. $$x=$$?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ab50a95log29a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"log base $$9$$ of $$x=\\\\frac{1}{2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"ab50a95log29a-h1","type":"hint","dependencies":[],"title":"Identifying $$y$$, $$b$$, and $$x$$","text":"The first step is to examine the equation $$y=log$$ base $$b$$ of $$x$$ and identify $$y$$, $$b$$, and $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab50a95log29a-h2","type":"hint","dependencies":["ab50a95log29a-h1"],"title":"Rewriting the Equation","text":"Next, rewrite the equation $$\\"y=log$$ base $$b$$ of x\\" as $$b^y=x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab50a95log3","title":"Converting from Exponential Form to Logarithmic Form","body":"Write the following exponential equations in logarithmic form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ab50a95log3a","stepAnswer":["$$\\\\log_{2}\\\\left(8\\\\right)=3$$"],"problemType":"MultipleChoice","stepTitle":"$$2^3=8$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\log_{2}\\\\left(8\\\\right)=3$$","choices":["$$\\\\log_{2}\\\\left(8\\\\right)=3$$","$$\\\\log_{3}\\\\left(8\\\\right)=2$$","$$\\\\log_{2}\\\\left(3\\\\right)=8$$","None of the above"],"hints":{"DefaultPathway":[{"id":"ab50a95log3a-h1","type":"hint","dependencies":[],"title":"Rewrite.","text":"$$\\\\log_{a}\\\\left(x\\\\right)=b$$ is equivalent to $$a^b=x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab50a95log3b","stepAnswer":["$$\\\\log_{5}\\\\left(25\\\\right)=2$$"],"problemType":"MultipleChoice","stepTitle":"$$5^2=25$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\log_{5}\\\\left(25\\\\right)=2$$","choices":["$$\\\\log_{5}\\\\left(25\\\\right)=2$$","$$\\\\log_{2}\\\\left(25\\\\right)=5$$","$$\\\\log_{5}\\\\left(2\\\\right)=25$$","None of the above"],"hints":{"DefaultPathway":[{"id":"ab50a95log3b-h1","type":"hint","dependencies":[],"title":"Rewrite.","text":"log{a}(x)=b is equivalent to $$a^b=x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab50a95log3c","stepAnswer":["$$\\\\log_{10}\\\\left(\\\\frac{1}{10000}\\\\right)=-4$$"],"problemType":"MultipleChoice","stepTitle":"$${10}^{\\\\left(-4\\\\right)}=\\\\frac{1}{10000}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\log_{10}\\\\left(\\\\frac{1}{10000}\\\\right)=-4$$","choices":["$$\\\\log_{10}\\\\left(\\\\frac{1}{10000}\\\\right)=-4$$","$$\\\\log_{-4}\\\\left(10\\\\right)=\\\\frac{1}{10000}$$","$$\\\\log_{10}\\\\left(\\\\frac{1}{10000}\\\\right)=4$$","None of the above"],"hints":{"DefaultPathway":[{"id":"ab50a95log3c-h1","type":"hint","dependencies":[],"title":"Rewrite.","text":"$$\\\\log_{a}\\\\left(x\\\\right)=b$$ is equivalent to $$a^b=x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab50a95log30","title":"Evaluating Logarithms: Exercise #5","body":"For the following exercise, solve for $$x$$ by converting the logarithmic equation to exponential form. $$x=$$?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ab50a95log30a","stepAnswer":["$$1000$$"],"problemType":"TextBox","stepTitle":"$$\\\\ln(x)=3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1000$$","hints":{"DefaultPathway":[{"id":"ab50a95log30a-h1","type":"hint","dependencies":[],"title":"Identifying $$y$$, $$b$$, and $$x$$","text":"The first step is to examine the equation $$y=log$$ base $$b$$ of $$x$$ and identify $$y$$, $$b$$, and $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab50a95log30a-h2","type":"hint","dependencies":["ab50a95log30a-h1"],"title":"Rewriting the Equation","text":"Next, rewrite the equation $$\\"y=log$$ base $$b$$ of x\\" as $$b^y=x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab50a95log30a-h3","type":"hint","dependencies":["ab50a95log30a-h2"],"title":"Meaning of log","text":"When there is no base indicated, log means \\"log base 10.\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab50a95log31","title":"Evaluating Logarithms: Exercise #6","body":"For the following exercise, solve for $$x$$ by converting the logarithmic equation to exponential form. $$x=$$?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ab50a95log31a","stepAnswer":["$$324$$"],"problemType":"TextBox","stepTitle":"log base $$18$$ of $$x=2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$324$$","hints":{"DefaultPathway":[{"id":"ab50a95log31a-h1","type":"hint","dependencies":[],"title":"Identifying $$y$$, $$b$$, and $$x$$","text":"The first step is to examine the equation $$y=log$$ base $$b$$ of $$x$$ and identify $$y$$, $$b$$, and $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab50a95log31a-h2","type":"hint","dependencies":["ab50a95log31a-h1"],"title":"Rewriting the Equation","text":"Next, rewrite the equation $$\\"y=log$$ base $$b$$ of x\\" as $$b^y=x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab50a95log32","title":"Evaluating Logarithms: Exercise #7","body":"For the following exercise, solve for $$x$$ by converting the logarithmic equation to exponential form. $$x=$$?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ab50a95log32a","stepAnswer":["$$\\\\frac{1}{216}$$"],"problemType":"TextBox","stepTitle":"log base $$6$$ of $$x=-3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{216}$$","hints":{"DefaultPathway":[{"id":"ab50a95log32a-h1","type":"hint","dependencies":[],"title":"Identifying $$y$$, $$b$$, and $$x$$","text":"The first step is to examine the equation $$y=log$$ base $$b$$ of $$x$$ and identify $$y$$, $$b$$, and $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab50a95log32a-h2","type":"hint","dependencies":["ab50a95log32a-h1"],"title":"Rewriting the Equation","text":"Next, rewrite the equation $$\\"y=log$$ base $$b$$ of x\\" as $$b^y=x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab50a95log4","title":"Converting from Exponential Form to Logarithmic Form","body":"Write the following exponential equations in logarithmic form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ab50a95log4a","stepAnswer":["$$\\\\log_{3}\\\\left(9\\\\right)=2$$"],"problemType":"MultipleChoice","stepTitle":"$$3^2=9$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\log_{3}\\\\left(9\\\\right)=2$$","choices":["$$\\\\log_{9}\\\\left(3\\\\right)=2$$","$$\\\\log_{3}\\\\left(9\\\\right)=2$$","$$\\\\log_{2}\\\\left(3\\\\right)=9$$","None of the above"],"hints":{"DefaultPathway":[{"id":"ab50a95log4a-h1","type":"hint","dependencies":[],"title":"Rewrite.","text":"$$\\\\log_{a}\\\\left(x\\\\right)=b$$ is equivalent to $$a^b=x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab50a95log4b","stepAnswer":["$$\\\\log_{5}\\\\left(125\\\\right)=3$$"],"problemType":"MultipleChoice","stepTitle":"$$5^3=125$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\log_{5}\\\\left(125\\\\right)=3$$","choices":["$$\\\\log_{5}\\\\left(125\\\\right)=3$$","$$\\\\log_{125}\\\\left(5\\\\right)=3$$","$$\\\\log_{5}\\\\left(3\\\\right)=125$$","None of the above"],"hints":{"DefaultPathway":[{"id":"ab50a95log4b-h1","type":"hint","dependencies":[],"title":"Rewrite.","text":"$$\\\\log_{a}\\\\left(x\\\\right)=b$$ is equivalent to $$a^b=x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab50a95log4c","stepAnswer":["$$\\\\log_{2}\\\\left(\\\\frac{1}{2}\\\\right)=-1$$"],"problemType":"MultipleChoice","stepTitle":"$$2^{\\\\left(-1\\\\right)}=\\\\frac{1}{2}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\log_{2}\\\\left(\\\\frac{1}{2}\\\\right)=-1$$","choices":["$$\\\\log_{2}\\\\left(\\\\frac{1}{2}\\\\right)=-1$$","$$\\\\log_{2}\\\\left(-1\\\\right)=\\\\frac{1}{2}$$","$$\\\\log_{1/2}\\\\left(-1\\\\right)=2$$","None of the above"],"hints":{"DefaultPathway":[{"id":"ab50a95log4c-h1","type":"hint","dependencies":[],"title":"Rewrite.","text":"$$\\\\log_{a}\\\\left(x\\\\right)=b$$ is equivalent to $$a^b=x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab50a95log5","title":"Solving Logarithms Mentally","body":"Solve the expression without a calculator","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ab50a95log5a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"$$y=\\\\log_{4}\\\\left(64\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"ab50a95log5a-h1","type":"hint","dependencies":[],"title":"Rethink","text":"Think of it like $$4^y=64$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab50a95log6","title":"Solving Logarithms Mentally","body":"Solve the expression without a calculator.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ab50a95log6a","stepAnswer":["$$\\\\frac{1}{2}$$"],"problemType":"TextBox","stepTitle":"$$y=\\\\log_{121}\\\\left(11\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{2}$$","hints":{"DefaultPathway":[{"id":"ab50a95log6a-h1","type":"hint","dependencies":[],"title":"Rethink","text":"Think of it like $${121}^y=11$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab50a95log7","title":"Evaluating the Logarithm of a Reciprocal","body":"Solve the expression without a calculator.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ab50a95log7a","stepAnswer":["$$-3$$"],"problemType":"TextBox","stepTitle":"$$y=\\\\log_{3}\\\\left(\\\\frac{1}{27}\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-3$$","hints":{"DefaultPathway":[{"id":"ab50a95log7a-h1","type":"hint","dependencies":[],"title":"Rethink","text":"Think of it like $$3^y=\\\\frac{1}{27}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab50a95log8","title":"Evaluating the Logarithm of a Reciprocal","body":"Solve the expression without a calculator.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ab50a95log8a","stepAnswer":["$$-6$$"],"problemType":"TextBox","stepTitle":"$$y=\\\\log_{2}\\\\left(\\\\frac{1}{32}\\\\right)$$","stepBody":"$$y=\\\\log_{2}\\\\left(\\\\frac{1}{32}\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-6$$","hints":{"DefaultPathway":[{"id":"ab50a95log8a-h1","type":"hint","dependencies":[],"title":"Rethink","text":"Think of it like $$2^y=\\\\frac{1}{32}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab50a95log9","title":"Finding the Value of a Common Logarithm Mentally","body":"Solve the expression without a calculator.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.3 Logarithmic Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ab50a95log9a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"$$y=\\\\ln(1000)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"ab50a95log9a-h1","type":"hint","dependencies":[],"title":"Natural Log","text":"Whenever there is no base, assume the base is $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab50a95log9a-h2","type":"hint","dependencies":["ab50a95log9a-h1"],"title":"Rethink","text":"Think of it like $${10}^y=1000$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab62b81rational1","title":"Adding Rational Expressions.","body":"Find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 Add and Subtract Rational Expressions with a Common Denominator","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab62b81rational1a","stepAnswer":["$$\\\\frac{2}{3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{5}{18}+\\\\frac{7}{18}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2}{3}$$","hints":{"DefaultPathway":[{"id":"ab62b81rational1a-h1","type":"hint","dependencies":[],"title":"Common Denominator","text":"The fractions have a common denominator, so add the numerators and place the sum over the common denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["ab62b81rational1a-h1"],"title":"Adding the Numerator","text":"What is $$5+7$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational1a-h3","type":"hint","dependencies":["ab62b81rational1a-h2"],"title":"Factor","text":"Factor the numerator and denominator to show the common factors. Remove the common factors and simplify if possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab62b81rational10","title":"Adding Rational Expressions.","body":"Find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 Add and Subtract Rational Expressions with a Common Denominator","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab62b81rational10a","stepAnswer":["$$\\\\frac{x+1}{x-2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{x}{x-2}+\\\\frac{1}{x-2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{x+1}{x-2}$$","hints":{"DefaultPathway":[{"id":"ab62b81rational10a-h1","type":"hint","dependencies":[],"title":"Common Denominator","text":"The fractions have a common denominator, so add the numerators and place the sum over the common denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+1$$"],"dependencies":["ab62b81rational10a-h1"],"title":"Adding the Numerator","text":"What is $$x+1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational10a-h3","type":"hint","dependencies":["ab62b81rational10a-h2"],"title":"Factor","text":"The denominator and numerator cannot be factored.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab62b81rational11","title":"Adding Rational Expressions.","body":"Find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 Add and Subtract Rational Expressions with a Common Denominator","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab62b81rational11a","stepAnswer":["$$x+4$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{7x+12}{x+3}+\\\\frac{x^2}{x+3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x+4$$","hints":{"DefaultPathway":[{"id":"ab62b81rational11a-h1","type":"hint","dependencies":[],"title":"Common Denominator","text":"The fractions have a common denominator, so add the numerators and place the sum over the common denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2+7x+12$$"],"dependencies":["ab62b81rational11a-h1"],"title":"Adding the Numerator","text":"What is $$7x+12+x^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational11a-h3","type":"hint","dependencies":["ab62b81rational11a-h2"],"title":"Factor","text":"Factor the numerator and simplify by removing the common factors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational11a-h4","type":"hint","dependencies":["ab62b81rational11a-h3"],"title":"Numerator","text":"The numerator can be factored into $$\\\\left(x+3\\\\right) \\\\left(x+4\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab62b81rational12","title":"Subtracting Rational Expressions.","body":"Find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 Add and Subtract Rational Expressions with a Common Denominator","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab62b81rational12a","stepAnswer":["$$n+10$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{n^2}{n-10}-\\\\frac{100}{n-10}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$n+10$$","hints":{"DefaultPathway":[{"id":"ab62b81rational12a-h1","type":"hint","dependencies":[],"title":"Common Denominator","text":"The fractions have a common denominator, so subtract the numerators and place the sum over the common denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational12a-h2","type":"hint","dependencies":["ab62b81rational12a-h1"],"title":"Factor","text":"Factor the numerator and simplify by removing the common factors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational12a-h3","type":"hint","dependencies":["ab62b81rational12a-h2"],"title":"Numerator","text":"The numerator can be factored into $$\\\\left(n+10\\\\right) \\\\left(n-10\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab62b81rational13","title":"Subtracting Rational Expressions.","body":"Find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 Add and Subtract Rational Expressions with a Common Denominator","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab62b81rational13a","stepAnswer":["$$x-3$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{x^2}{x+3}-\\\\frac{9}{x+3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x-3$$","hints":{"DefaultPathway":[{"id":"ab62b81rational13a-h1","type":"hint","dependencies":[],"title":"Common Denominator","text":"The fractions have a common denominator, so subtract the numerators and place the sum over the common denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational13a-h2","type":"hint","dependencies":["ab62b81rational13a-h1"],"title":"Factor","text":"Factor the numerator and simplify by removing the common factors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational13a-h3","type":"hint","dependencies":["ab62b81rational13a-h2"],"title":"Numerator","text":"The numerator can be factored into $$\\\\left(x+3\\\\right) \\\\left(x-3\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab62b81rational14","title":"Subtracting Rational Expressions.","body":"Find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 Add and Subtract Rational Expressions with a Common Denominator","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab62b81rational14a","stepAnswer":["$$2x+5$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{4x^2}{2x-5}-\\\\frac{25}{2x-5}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2x+5$$","hints":{"DefaultPathway":[{"id":"ab62b81rational14a-h1","type":"hint","dependencies":[],"title":"Common Denominator","text":"The fractions have a common denominator, so subtract the numerators and place the sum over the common denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational14a-h2","type":"hint","dependencies":["ab62b81rational14a-h1"],"title":"Factor","text":"Factor the numerator and simplify by removing the common factors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational14a-h3","type":"hint","dependencies":["ab62b81rational14a-h2"],"title":"Numerator","text":"The numerator can be factored into $$\\\\left(2x+5\\\\right) \\\\left(2x-5\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab62b81rational15","title":"Subtracting Rational Expressions.","body":"Find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 Add and Subtract Rational Expressions with a Common Denominator","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab62b81rational15a","stepAnswer":["$$y+4$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{y^2}{y-6}-\\\\frac{2y+24}{y-6}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y+4$$","hints":{"DefaultPathway":[{"id":"ab62b81rational15a-h1","type":"hint","dependencies":[],"title":"Common Denominator","text":"The fractions have a common denominator, so subtract the numerators and place the sum over the common denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational15a-h2","type":"hint","dependencies":["ab62b81rational15a-h1"],"title":"Factor","text":"Factor the numerator and simplify by removing the common factors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational15a-h3","type":"hint","dependencies":["ab62b81rational15a-h2"],"title":"Numerator","text":"The numerator can be factored into $$\\\\left(y-6\\\\right) \\\\left(y+4\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab62b81rational16","title":"Add Rational Expressions with a Common Denominator","body":"In the following exercise, add:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 Add and Subtract Rational Expressions with a Common Denominator","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab62b81rational16a","stepAnswer":["$$8t$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{8t^2}{t+4}+\\\\frac{32t}{t+4}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8t$$","hints":{"DefaultPathway":[{"id":"ab62b81rational16a-h1","type":"hint","dependencies":[],"title":"Identifying Common Denominator","text":"The fractions have a common denominator, so add the numerators and place the sum over the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational16a-h2","type":"hint","dependencies":["ab62b81rational16a-h1"],"title":"Factoring the Numerator","text":"The numerator can be factored to produce $$8t \\\\left(t+4\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational16a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8t$$"],"dependencies":["ab62b81rational16a-h2"],"title":"Simplifying Fraction","text":"After replacing the factored parts, what is the simplified form of the fraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab62b81rational17","title":"Add Rational Expressions with a Common Denominator","body":"In the folloing exercise, add:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 Add and Subtract Rational Expressions with a Common Denominator","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab62b81rational17a","stepAnswer":["$$s+5$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{3s^2}{3s-2}+\\\\frac{13s-10}{3s-2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$s+5$$","hints":{"DefaultPathway":[{"id":"ab62b81rational17a-h1","type":"hint","dependencies":[],"title":"Identifying Common Denominator","text":"The fractions have a common denominator, so add the numerators and place the sum over the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational17a-h2","type":"hint","dependencies":["ab62b81rational17a-h1"],"title":"Factoring the Numerator","text":"The numerator can be factored to produce $$\\\\left(3s-2\\\\right) \\\\left(s+5\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational17a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$s+5$$"],"dependencies":["ab62b81rational17a-h2"],"title":"Simplifying Fraction","text":"After replacing the factored parts, what is the simplified form of the fraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab62b81rational18","title":"Add Rational Expressions with a Common Denominator","body":"In the folloing exercise, add:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 Add and Subtract Rational Expressions with a Common Denominator","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab62b81rational18a","stepAnswer":["$$6v$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{6v^2}{v+5}+\\\\frac{30v}{v+5}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6v$$","hints":{"DefaultPathway":[{"id":"ab62b81rational18a-h1","type":"hint","dependencies":[],"title":"Identifying Common Denominator","text":"The fractions have a common denominator, so add the numerators and place the sum over the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational18a-h2","type":"hint","dependencies":["ab62b81rational18a-h1"],"title":"Factoring the Numerator","text":"The numerator can be factored to produce $$6v \\\\left(v+5\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational18a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6v$$"],"dependencies":["ab62b81rational18a-h2"],"title":"Simplifying Fraction","text":"After replacing the factored parts, what is the simplified form of the fraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab62b81rational19","title":"Add Rational Expressions with a Common Denominator","body":"In the folloing exercise, add:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 Add and Subtract Rational Expressions with a Common Denominator","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab62b81rational19a","stepAnswer":["$$\\\\frac{2w}{w-4}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2w^2}{w^2-16}+\\\\frac{8w}{w^2-16}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2w}{w-4}$$","hints":{"DefaultPathway":[{"id":"ab62b81rational19a-h1","type":"hint","dependencies":[],"title":"Identifying Common Denominator","text":"The fractions have a common denominator, so add the numerators and place the sum over the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational19a-h2","type":"hint","dependencies":["ab62b81rational19a-h1"],"title":"Factoring the Numerator","text":"The numerator can be factored into $$2w \\\\left(w+4\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational19a-h3","type":"hint","dependencies":["ab62b81rational19a-h2"],"title":"Factoring the Denominator","text":"The denominator can be factored into $$\\\\left(w+4\\\\right) \\\\left(w-4\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational19a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2w}{w-4}$$"],"dependencies":["ab62b81rational19a-h3"],"title":"Simplifying Fraction","text":"After replacing the factored parts, what is the simplified form of the fraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab62b81rational2","title":"Adding Rational Expressions.","body":"Find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 Add and Subtract Rational Expressions with a Common Denominator","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab62b81rational2a","stepAnswer":["$$\\\\frac{3}{4}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{7}{16}+\\\\frac{5}{16}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{4}$$","hints":{"DefaultPathway":[{"id":"ab62b81rational2a-h1","type":"hint","dependencies":[],"title":"Common Denominator","text":"The fractions have a common denominator, so add the numerators and place the sum over the common denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["ab62b81rational2a-h1"],"title":"Adding the Numerator","text":"What is $$7+5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational2a-h3","type":"hint","dependencies":["ab62b81rational2a-h2"],"title":"Factor","text":"Factor the numerator and denominator to show the common factors. Remove the common factors and simplify if possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab62b81rational20","title":"Add Rational Expressions with a Common Denominator","body":"In the folloing exercise, add:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 Add and Subtract Rational Expressions with a Common Denominator","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab62b81rational20a","stepAnswer":["$$\\\\frac{7x}{x-3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{7x^2}{x^2-9}+\\\\frac{21x}{x^2-9}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{7x}{x-3}$$","hints":{"DefaultPathway":[{"id":"ab62b81rational20a-h1","type":"hint","dependencies":[],"title":"Identifying Common Denominator","text":"The fractions have a common denominator, so add the numerators and place the sum over the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational20a-h2","type":"hint","dependencies":["ab62b81rational20a-h1"],"title":"Factoring the Numerator","text":"The numerator can be factored into $$7x \\\\left(x+3\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational20a-h3","type":"hint","dependencies":["ab62b81rational20a-h2"],"title":"Factoring the Denominator","text":"The denominator can be factored into $$\\\\left(x+3\\\\right) \\\\left(x-3\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational20a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2w}{w-4}$$"],"dependencies":["ab62b81rational20a-h3"],"title":"Simplifying Fraction","text":"After replacing the factored parts, what is the simplified form of the fraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab62b81rational21","title":"Subtract Rational Expressions with a Common Denominator","body":"In the folloing exercise, subtract:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 Add and Subtract Rational Expressions with a Common Denominator","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab62b81rational21a","stepAnswer":["$$y-8$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{y^2}{y+8}-\\\\frac{64}{y+8}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y-8$$","hints":{"DefaultPathway":[{"id":"ab62b81rational21a-h1","type":"hint","dependencies":[],"title":"Identifying Common Denominator","text":"The fractions have a common denominator, so subtract the numerators and place the sum over the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational21a-h2","type":"hint","dependencies":["ab62b81rational21a-h1"],"title":"Factoring the Numerator","text":"The numerator can be factored to produce $$\\\\left(y+8\\\\right) \\\\left(y-8\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational21a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y-8$$"],"dependencies":["ab62b81rational21a-h2"],"title":"Simplifying Fraction","text":"After replacing the factored parts, what is the simplified form of the fraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab62b81rational22","title":"Subtract Rational Expressions with a Common Denominator","body":"In the folloing exercise, subtract:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 Add and Subtract Rational Expressions with a Common Denominator","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab62b81rational22a","stepAnswer":["$$z-2$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{z^2}{z+2}-\\\\frac{4}{z+2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$z-2$$","hints":{"DefaultPathway":[{"id":"ab62b81rational22a-h1","type":"hint","dependencies":[],"title":"Identifying Common Denominator","text":"The fractions have a common denominator, so subtract the numerators and place the sum over the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational22a-h2","type":"hint","dependencies":["ab62b81rational22a-h1"],"title":"Factoring the Numerator","text":"The numerator can be factored to produce $$\\\\left(z+2\\\\right) \\\\left(z-2\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational22a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$z-2$$"],"dependencies":["ab62b81rational22a-h2"],"title":"Simplifying Fraction","text":"After replacing the factored parts, what is the simplified form of the fraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab62b81rational23","title":"Subtract Rational Expressions with a Common Denominator","body":"In the folloing exercise, subtract:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 Add and Subtract Rational Expressions with a Common Denominator","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab62b81rational23a","stepAnswer":["$$3a+7$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{9a^2}{3a-7}-\\\\frac{49}{3a-7}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3a+7$$","hints":{"DefaultPathway":[{"id":"ab62b81rational23a-h1","type":"hint","dependencies":[],"title":"Identifying Common Denominator","text":"The fractions have a common denominator, so subtract the numerators and place the sum over the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational23a-h2","type":"hint","dependencies":["ab62b81rational23a-h1"],"title":"Factoring the Numerator","text":"The numerator can be factored to produce $$\\\\left(3a-7\\\\right) \\\\left(3a+7\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational23a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3a+7$$"],"dependencies":["ab62b81rational23a-h2"],"title":"Simplifying Fraction","text":"After replacing the factored parts, what is the simplified form of the fraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab62b81rational24","title":"Subtract Rational Expressions with a Common Denominator","body":"In the folloing exercise, subtract:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 Add and Subtract Rational Expressions with a Common Denominator","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab62b81rational24a","stepAnswer":["$$5b+6$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{25b^2}{5b-6}-\\\\frac{36}{5b-6}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5b+6$$","hints":{"DefaultPathway":[{"id":"ab62b81rational24a-h1","type":"hint","dependencies":[],"title":"Identifying Common Denominator","text":"The fractions have a common denominator, so subtract the numerators and place the sum over the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational24a-h2","type":"hint","dependencies":["ab62b81rational24a-h1"],"title":"Factoring the Numerator","text":"The numerator can be factored to produce $$\\\\left(5b+6\\\\right) \\\\left(5b-6\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational24a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5b+6$$"],"dependencies":["ab62b81rational24a-h2"],"title":"Simplifying Fraction","text":"After replacing the factored parts, what is the simplified form of the fraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab62b81rational25","title":"Subtract Rational Expressions with a Common Denominator","body":"In the folloing exercise, subtract:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 Add and Subtract Rational Expressions with a Common Denominator","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab62b81rational25a","stepAnswer":["$$c+2$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{c^2}{c-8}-\\\\frac{6c+16}{c-8}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$c+2$$","hints":{"DefaultPathway":[{"id":"ab62b81rational25a-h1","type":"hint","dependencies":[],"title":"Identifying Common Denominator","text":"The fractions have a common denominator, so subtract the numerators and place the sum over the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational25a-h2","type":"hint","dependencies":["ab62b81rational25a-h1"],"title":"Factoring the Numerator","text":"The numerator can be factored to produce $$\\\\left(c-8\\\\right) \\\\left(c+2\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational25a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$c+2$$"],"dependencies":["ab62b81rational25a-h2"],"title":"Simplifying Fraction","text":"After replacing the factored parts, what is the simplified form of the fraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab62b81rational26","title":"Subtract Rational Expressions with a Common Denominator","body":"In the folloing exercise, subtract:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 Add and Subtract Rational Expressions with a Common Denominator","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab62b81rational26a","stepAnswer":["$$d+3$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{d^2}{d-9}-\\\\frac{6d+27}{d-9}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$d+3$$","hints":{"DefaultPathway":[{"id":"ab62b81rational26a-h1","type":"hint","dependencies":[],"title":"Identifying Common Denominator","text":"The fractions have a common denominator, so subtract the numerators and place the sum over the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational26a-h2","type":"hint","dependencies":["ab62b81rational26a-h1"],"title":"Factoring the Numerator","text":"The numerator can be factored to produce $$\\\\left(d-9\\\\right) \\\\left(d+3\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational26a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$d+3$$"],"dependencies":["ab62b81rational26a-h2"],"title":"Simplifying Fraction","text":"After replacing the factored parts, what is the simplified form of the fraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab62b81rational27","title":"Subtract Rational Expressions with a Common Denominator","body":"In the folloing exercise, subtract:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 Add and Subtract Rational Expressions with a Common Denominator","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab62b81rational27a","stepAnswer":["$$\\\\frac{m-2}{2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{3m^2}{6m-30}-\\\\frac{21m-30}{6m-30}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{m-2}{2}$$","hints":{"DefaultPathway":[{"id":"ab62b81rational27a-h1","type":"hint","dependencies":[],"title":"Identifying Common Denominator","text":"The fractions have a common denominator, so subtract the numerators and place the sum over the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational27a-h2","type":"hint","dependencies":["ab62b81rational27a-h1"],"title":"Factoring the Numerator","text":"The numerator can be factored into $$3\\\\left(m-5\\\\right) \\\\left(m-2\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational27a-h3","type":"hint","dependencies":["ab62b81rational27a-h2"],"title":"Factoring the Denominator","text":"The denominator can be factored into $$6\\\\left(m-5\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational27a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{m-2}{2}$$"],"dependencies":["ab62b81rational27a-h3"],"title":"Simplifying Fraction","text":"After replacing the factored parts, what is the simplified form of the fraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab62b81rational28","title":"Subtract Rational Expressions with a Common Denominator","body":"In the folloing exercise, subtract:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 Add and Subtract Rational Expressions with a Common Denominator","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab62b81rational28a","stepAnswer":["$$\\\\frac{n-1}{2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2n^2}{4n-32}-\\\\frac{18n-16}{4n-32}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{n-1}{2}$$","hints":{"DefaultPathway":[{"id":"ab62b81rational28a-h1","type":"hint","dependencies":[],"title":"Identifying Common Denominator","text":"The fractions have a common denominator, so subtract the numerators and place the sum over the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational28a-h2","type":"hint","dependencies":["ab62b81rational28a-h1"],"title":"Factoring the Numerator","text":"The numerator can be factored into $$2\\\\left(n-8\\\\right) \\\\left(n-1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational28a-h3","type":"hint","dependencies":["ab62b81rational28a-h2"],"title":"Factoring the Denominator","text":"The denominator can be factored into $$4\\\\left(n-8\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational28a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{n-1}{2}$$"],"dependencies":["ab62b81rational28a-h3"],"title":"Simplifying Fraction","text":"After replacing the factored parts, what is the simplified form of the fraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab62b81rational29","title":"Subtract Rational Expressions with a Common Denominator","body":"In the folloing exercise, subtract:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 Add and Subtract Rational Expressions with a Common Denominator","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab62b81rational29a","stepAnswer":["$$\\\\frac{q-8}{q+4}$$"],"problemType":"TextBox","stepTitle":"(((5*q**2)+(3*q)-9)/((q**2)+(6*q)+8))-(((4*q**1)+(9*q)+7)/((q**2)+(6*q)+8)))","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{q-8}{q+4}$$","hints":{"DefaultPathway":[{"id":"ab62b81rational29a-h1","type":"hint","dependencies":[],"title":"Identifying Common Denominator","text":"The fractions have a common denominator, so subtract the numerators and place the sum over the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational29a-h2","type":"hint","dependencies":["ab62b81rational29a-h1"],"title":"Factoring the Numerator","text":"The numerator can be factored into $$\\\\left(q-8\\\\right) \\\\left(q+2\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational29a-h3","type":"hint","dependencies":["ab62b81rational29a-h2"],"title":"Factoring the Denominator","text":"The denominator can be factored into $$\\\\left(q+4\\\\right) \\\\left(q+2\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational29a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{q-8}{q+4}$$"],"dependencies":["ab62b81rational29a-h3"],"title":"Simplifying Fraction","text":"After replacing the factored parts, what is the simplified form of the fraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab62b81rational3","title":"Adding Rational Expressions.","body":"Find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 Add and Subtract Rational Expressions with a Common Denominator","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab62b81rational3a","stepAnswer":["$$\\\\frac{2}{5}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{3}{10}+\\\\frac{1}{10}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2}{5}$$","hints":{"DefaultPathway":[{"id":"ab62b81rational3a-h1","type":"hint","dependencies":[],"title":"Common Denominator","text":"The fractions have a common denominator, so add the numerators and place the sum over the common denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ab62b81rational3a-h1"],"title":"Adding the Numerator","text":"What is $$3+1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational3a-h3","type":"hint","dependencies":["ab62b81rational3a-h2"],"title":"Factor","text":"Factor the numerator and denominator to show the common factors. Remove the common factors and simplify if possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab62b81rational30","title":"Subtract Rational Expressions with a Common Denominator","body":"In the folloing exercise, subtract:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 Add and Subtract Rational Expressions with a Common Denominator","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab62b81rational30a","stepAnswer":["$$\\\\frac{p+3}{p+5}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{6p^2+3p+4}{p^2+4p-5}-\\\\frac{5p^2+p+7}{p^2+4p-5}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{p+3}{p+5}$$","hints":{"DefaultPathway":[{"id":"ab62b81rational30a-h1","type":"hint","dependencies":[],"title":"Identifying Common Denominator","text":"The fractions have a common denominator, so subtract the numerators and place the sum over the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational30a-h2","type":"hint","dependencies":["ab62b81rational30a-h1"],"title":"Factoring the Numerator","text":"The numerator can be factored into $$\\\\left(p+3\\\\right) \\\\left(p-1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational30a-h3","type":"hint","dependencies":["ab62b81rational30a-h2"],"title":"Factoring the Denominator","text":"The denominator can be factored into $$\\\\left(p+5\\\\right) \\\\left(p-1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational30a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{p+3}{p+5}$$"],"dependencies":["ab62b81rational30a-h3"],"title":"Simplifying Fraction","text":"After replacing the factored parts, what is the simplified form of the fraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab62b81rational4","title":"Adding Rational Expressions.","body":"Find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 Add and Subtract Rational Expressions with a Common Denominator","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab62b81rational4a","stepAnswer":["$$\\\\frac{3}{5}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2}{15}+\\\\frac{7}{15}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{5}$$","hints":{"DefaultPathway":[{"id":"ab62b81rational4a-h1","type":"hint","dependencies":[],"title":"Common Denominator","text":"The fractions have a common denominator, so add the numerators and place the sum over the common denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["ab62b81rational4a-h1"],"title":"Adding the Numerator","text":"What is $$2+7$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational4a-h3","type":"hint","dependencies":["ab62b81rational4a-h2"],"title":"Factor","text":"Factor the numerator and denominator to show the common factors. Remove the common factors and simplify if possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab62b81rational5","title":"Adding Rational Expressions.","body":"Find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 Add and Subtract Rational Expressions with a Common Denominator","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab62b81rational5a","stepAnswer":["$$\\\\frac{1}{3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{4}{21}+\\\\frac{3}{21}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{3}$$","hints":{"DefaultPathway":[{"id":"ab62b81rational5a-h1","type":"hint","dependencies":[],"title":"Common Denominator","text":"The fractions have a common denominator, so add the numerators and place the sum over the common denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["ab62b81rational5a-h1"],"title":"Adding the Numerator","text":"What is $$4+3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational5a-h3","type":"hint","dependencies":["ab62b81rational5a-h2"],"title":"Factor","text":"Factor the numerator and denominator to show the common factors. Remove the common factors and simplify if possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab62b81rational6","title":"Adding Rational Expressions.","body":"Find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 Add and Subtract Rational Expressions with a Common Denominator","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab62b81rational6a","stepAnswer":["$$\\\\frac{3}{4}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{7}{24}+\\\\frac{11}{24}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{4}$$","hints":{"DefaultPathway":[{"id":"ab62b81rational6a-h1","type":"hint","dependencies":[],"title":"Common Denominator","text":"The fractions have a common denominator, so add the numerators and place the sum over the common denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18$$"],"dependencies":["ab62b81rational6a-h1"],"title":"Adding the Numerator","text":"What is $$7+11$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational6a-h3","type":"hint","dependencies":["ab62b81rational6a-h2"],"title":"Factor","text":"Factor the numerator and denominator to show the common factors. Remove the common factors and simplify if possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab62b81rational7","title":"Adding Rational Expressions.","body":"Find the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.3 Add and Subtract Rational Expressions with a Common Denominator","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ab62b81rational7a","stepAnswer":["$$\\\\frac{5}{9}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{7}{36}+\\\\frac{13}{36}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5}{9}$$","hints":{"DefaultPathway":[{"id":"ab62b81rational7a-h1","type":"hint","dependencies":[],"title":"Common Denominator","text":"The fractions have a common denominator, so add the numerators and place the sum over the common denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["ab62b81rational7a-h1"],"title":"Adding the Numerator","text":"What is $$7+13$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab62b81rational7a-h3","type":"hint","dependencies":["ab62b81rational7a-h2"],"title":"Factor","text":"Factor the numerator and denominator to show the common factors. 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The sample average for the same data set is $$10.53$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Measures of the Spread of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab63d62spread1a","stepAnswer":["$$11.25$$"],"problemType":"TextBox","stepTitle":"Find the value that is one standard deviation above the mean.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$11.25$$","hints":{"DefaultPathway":[{"id":"ab63d62spread1a-h1","type":"hint","dependencies":[],"title":"Definition of Differences Using Standard Deviation","text":"Standard deviation measures, essentially, the average deviations a sample is from its sample mean. 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Pay careful attention to signs when comparing and interpreting the answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab63d62spread10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-0.21$$"],"dependencies":["ab63d62spread10a-h1"],"title":"John\'s Standard Deviation","text":"How many standard deviations away from the mean is John? Pay careful attention to sign. Round to the nearest hundredth.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab63d62spread10a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-0.21$$"],"dependencies":[],"title":"John\'s Standard Deviation","text":"What is $$\\\\frac{2.85-3}{0.7}$$? 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Round to the nearest hundredth.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ab63d62spread10a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["John"],"dependencies":["ab63d62spread10a-h3"],"title":"Determining Which Student\'s GPA is Better","text":"Which student had a higher GPA when compared to their school? Effectively, which student had a higher standard deviation score?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["John","Ali"]}]}}]},{"id":"ab63d62spread11","title":"Comparing Values from Different Data Sets","body":"Two swimmers, Angie and Beth, from different teams, wanted to find out who had the fastest time for the $$50$$ meter freestyle when compared to her team.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Measures of the Spread of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab63d62spread11a","stepAnswer":["Beth"],"problemType":"MultipleChoice","stepTitle":"Which swimmer had the fastest time when compared to her team?","stepBody":"","answerType":"string","variabilization":{},"choices":["Angie","Beth"],"hints":{"DefaultPathway":[{"id":"ab63d62spread11a-h1","type":"hint","dependencies":[],"title":"Determine Swimmers\' Standard Deviations","text":"For each swimmer, determine how many standard deviations (#ofSTDEVs) her time is away from the average, for her team. 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Round to the nearest hundredth.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ab63d62spread11a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Beth"],"dependencies":["ab63d62spread11a-h3"],"title":"Determining Which Swimmer\'s Time is Better","text":"Which swimmer had a faster time when compared to their team? Effectively, which swimmer had a lower standard deviation score?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Angie","Beth"]}]}}]},{"id":"ab63d62spread12","title":"Calculating Mean and Standard Deviation","body":"The following data are the distances between $$20$$ retail stores and a large distribution center. The distances are in miles. 29; 37; 38; 40; 58; 67; 68; 69; 76; 86; 87; 95; 96; 96; 99; 106; 112; 127; 145; $$150$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Measures of the Spread of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab63d62spread12a","stepAnswer":["$$84.05$$"],"problemType":"TextBox","stepTitle":"Calculate the mean.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$84.05$$","hints":{"DefaultPathway":[{"id":"ab63d62spread12a-h1","type":"hint","dependencies":[],"title":"Definition of the Mean","text":"The mean serves as the average of the data: summing up all the different data points and then dividing by the total number of data points that exist in the data.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab63d62spread12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$84.05$$"],"dependencies":["ab63d62spread12a-h1"],"title":"Determine the Mean","text":"What is the mean of the data?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab63d62spread12b","stepAnswer":["$$34.52$$"],"problemType":"TextBox","stepTitle":"Using your calculator or computer, calculate the sample standard deviation of the data set. 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The sample average for the same data set is $$84.05$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Measures of the Spread of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab63d62spread13a","stepAnswer":["$$49.53$$"],"problemType":"TextBox","stepTitle":"Find the value that is one standard deviation below the mean.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$49.53$$","hints":{"DefaultPathway":[{"id":"ab63d62spread13a-h1","type":"hint","dependencies":[],"title":"Definition of Differences Using Standard Deviation","text":"Standard deviation measures, essentially, the average deviations a sample is from its sample mean. To calculate the value that is one standard deviation below the mean then, therefore, we must calculate the value that is the mean minus the standard deviation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab63d62spread13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$49.53$$"],"dependencies":["ab63d62spread13a-h1"],"title":"Finding the Value Desired","text":"What is the value that is one standard deviation below the mean?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab63d62spread13a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$49.53$$"],"dependencies":[],"title":"Finding the Value Desired","text":"What is $$84.05-1\\\\times34.52$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"ab63d62spread14","title":"Comparing Values from Different Data Sets","body":"Two baseball players, Fredo and Karl, on different teams wanted to find out who had the higher batting average when compared to his team.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Measures of the Spread of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab63d62spread14a","stepAnswer":["Fredo"],"problemType":"MultipleChoice","stepTitle":"Which baseball player had the higher batting average when compared to his team?","stepBody":"","answerType":"string","variabilization":{},"choices":["Fredo","Karl"],"hints":{"DefaultPathway":[{"id":"ab63d62spread14a-h1","type":"hint","dependencies":[],"title":"Determine Baseball Players\' Standard Deviations","text":"For each baseball player, determine how many standard deviations (#ofSTDEVs) his batting average is away from the average, for his team. 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Round to the nearest hundredth.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ab63d62spread14a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Fredo"],"dependencies":["ab63d62spread14a-h3"],"title":"Determining Which Student\'s GPA is Better","text":"Which baseball player had a higher batting average when compared to their school? Effectively, which baseball player had a higher standard deviation score?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Fredo","Karl"]}]}}]},{"id":"ab63d62spread15","title":"Calculate Values Given Standard Deviation","body":"The sample standard deviation of a data set is $$s=0.012$$. The sample average for the same data set is $$0.166$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Measures of the Spread of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab63d62spread15a","stepAnswer":["$$0.13$$"],"problemType":"TextBox","stepTitle":"Find the value that is three standard deviations below the mean.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.13$$","hints":{"DefaultPathway":[{"id":"ab63d62spread15a-h1","type":"hint","dependencies":[],"title":"Definition of Differences Using Standard Deviation","text":"Standard deviation measures, essentially, the average deviations a sample is from its sample mean. 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To calculate the value that is three standard deviations above the mean then, therefore, we must calculate the value that is the mean plus three times the standard deviation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab63d62spread15b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.202$$"],"dependencies":["ab63d62spread15b-h3"],"title":"Finding the Value Desired","text":"What is the value that is three standard deviations above the mean?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab63d62spread15b-h4-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.202$$"],"dependencies":[],"title":"Finding the Value Desired","text":"What is $$0.166+3\\\\times0.012$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"ab63d62spread16","title":"Calculate Values Given Standard Deviation","body":"The sample standard deviation of a data set is $$s=0.015$$. The sample average for the same data set is $$0.189$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Measures of the Spread of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab63d62spread16a","stepAnswer":["$$0.144$$"],"problemType":"TextBox","stepTitle":"Find the value that is three standard deviations below the mean.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.144$$","hints":{"DefaultPathway":[{"id":"ab63d62spread16a-h1","type":"hint","dependencies":[],"title":"Definition of Differences Using Standard Deviation","text":"Standard deviation measures, essentially, the average deviations a sample is from its sample mean. To calculate the value that is three standard deviations below the mean then, therefore, we must calculate the value that is the mean minus three times the standard deviation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab63d62spread16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.144$$"],"dependencies":["ab63d62spread16a-h1"],"title":"Finding the Value Desired","text":"What is the value that is three standard deviations below the mean?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab63d62spread16a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.144$$"],"dependencies":[],"title":"Finding the Value Desired","text":"What is $$0.189-3\\\\times0.015$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}},{"id":"ab63d62spread16b","stepAnswer":["$$0.234$$"],"problemType":"TextBox","stepTitle":"Find the value that is three standard deviations above the mean.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.234$$","hints":{"DefaultPathway":[{"id":"ab63d62spread16b-h3","type":"hint","dependencies":["ab63d62spread16a-h2"],"title":"Definition of Differences Using Standard Deviation","text":"Standard deviation measures, essentially, the average deviations a sample is from its sample mean. To calculate the value that is three standard deviations above the mean then, therefore, we must calculate the value that is the mean plus three times the standard deviation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab63d62spread16b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.234$$"],"dependencies":["ab63d62spread16b-h3"],"title":"Finding the Value Desired","text":"What is the value that is three standard deviations above the mean?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab63d62spread16b-h4-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.234$$"],"dependencies":[],"title":"Finding the Value Desired","text":"What is $$0.189+3\\\\times0.015$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"ab63d62spread17","title":"Calculating Standard Deviation","body":"Use the table provided.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Measures of the Spread of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab63d62spread17a","stepAnswer":["$$10.88$$"],"problemType":"TextBox","stepTitle":"Find the standard deviation for the data provided in the table using a calculator or computer. 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Round to the nearest hundredth.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab63d62spread17a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$193157.45$$"],"dependencies":[],"title":"Determine sum{}{}{f*m**2}","text":"What is $$2{54.5}^2+3{64.5}^2+8{74.5}^2+12{84.5}^2+5{94.5}^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ab63d62spread17a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10.88$$"],"dependencies":["ab63d62spread17a-h4"],"title":"Determine the Sample Standard Deviation","text":"What is the sample standard deviation? Plug in the sum{}{}{f*m**2}, $$n$$, and $$x$$ that we calculated into the formula. Round to the nearest hundredth.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab63d62spread17a-h5-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10.88$$"],"dependencies":[],"title":"Determine the Sample Standard Deviation","text":"What is $$\\\\sqrt{\\\\frac{193157.45}{30}-{79.5}^2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"ab63d62spread18","title":"Calculating Standard Deviation","body":"Use the table provided.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Measures of the Spread of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab63d62spread18a","stepAnswer":["$$7.62$$"],"problemType":"TextBox","stepTitle":"Find the standard deviation for the data provided in the table using a calculator or computer. 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Round to the nearest tenth.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab63d62spread18a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$380945.3$$"],"dependencies":[],"title":"Determine sum{}{}{f*m**2}","text":"What is $$53{54.5}^2+32{64.5}^2+15{74.5}^2+1{84.5}^2+0{94.5}^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ab63d62spread18a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7.62$$"],"dependencies":["ab63d62spread18a-h4"],"title":"Determine the Sample Standard Deviation","text":"What is the sample standard deviation? Plug in the sum{}{}{f*m**2}, $$n$$, and $$x$$ that we calculated into the formula. Round to the nearest hundredth.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab63d62spread18a-h5-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7.62$$"],"dependencies":[],"title":"Determine the Sample Standard Deviation","text":"What is $$\\\\sqrt{\\\\frac{380945.3}{101}-{60.94}^2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"ab63d62spread19","title":"Calculating Standard Deviation","body":"Use the table provided.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Measures of the Spread of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab63d62spread19a","stepAnswer":["$$11.14$$"],"problemType":"TextBox","stepTitle":"Find the standard deviation for the data provided in the table using a calculator or computer. 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Round to the nearest tenth.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab63d62spread19a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$440051.5$$"],"dependencies":[],"title":"Determine sum{}{}{f*m**2}","text":"What is $$14{54.5}^2+32{64.5}^2+15{74.5}^2+23{84.5}^2+2{94.5}^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ab63d62spread19a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11.14$$"],"dependencies":["ab63d62spread19a-h4"],"title":"Determine the Sample Standard Deviation","text":"What is the sample standard deviation? Plug in the sum{}{}{f*m**2}, $$n$$, and $$x$$ that we calculated into the formula. Round to the nearest hundredth.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab63d62spread19a-h5-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11.14$$"],"dependencies":[],"title":"Determine the Sample Standard Deviation","text":"What is $$\\\\sqrt{\\\\frac{440051.5}{86}-{70.66}^2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"ab63d62spread2","title":"Calculate Values Given Standard Deviation","body":"The sample standard deviation of a data set is $$s=0.72$$. The sample average for the same data set is $$10.53$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Measures of the Spread of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab63d62spread2a","stepAnswer":["$$9.09$$"],"problemType":"TextBox","stepTitle":"Find the value that is two standard deviations below the mean.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9.09$$","hints":{"DefaultPathway":[{"id":"ab63d62spread2a-h1","type":"hint","dependencies":[],"title":"Definition of Differences Using Standard Deviation","text":"Standard deviation measures, essentially, the average deviations a sample is from its sample mean. 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The average is $$1000$$ FTES. The median is $$1014$$ FTES. The standard deviation is $$474$$ FTES. The first quartile is $$528.5$$ FTES. The third quartile is $$1447.5$$ FTES. The total number of years sampled was $$29$$ years.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Measures of the Spread of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab63d62spread20a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"A sample of $$11$$ years is taken. About how many are expected to have a FTES of $$1014$$ or above?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"ab63d62spread20a-h1","type":"hint","dependencies":[],"title":"Comparing $$1014$$ to the Population","text":"We can use the value that we were given in the question (1014) and compare it to the population statistics that we were given to see where in the distribution it lies.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab63d62spread20a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Median"],"dependencies":["ab63d62spread20a-h1"],"title":"Comparing $$1014$$ to the Population","text":"$$1014$$ is the value given to us that we are to compare to the sample. What is $$1014$$ to the sample?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Median","Mean","Range","Standard Deviation"]},{"id":"ab63d62spread20a-h3","type":"hint","dependencies":["ab63d62spread20a-h2"],"title":"Definition of a Median","text":"Since $$1014$$ FTES is the median of the population, note that the definition of a median means that half of the data values of the population will be less than the median and half of the data values of the population will be greater than the median.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab63d62spread20a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["ab63d62spread20a-h3"],"title":"Determining Expected Years","text":"How many years are expected to be at or below the median?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab63d62spread20a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":[],"title":"Determining Expected Years","text":"Effectively, what is one half of $$11$$, rounded up?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"ab63d62spread21","title":"Calculate Values Given Standard Deviation","body":"The population parameters below describe the full-time equivalent number of students (FTES) each year at Lake Tahoe Community College from $$1976-1977$$ through $$2004-2005$$. The average is $$1000$$ FTES. The median is $$1014$$ FTES. The standard deviation is $$474$$ FTES. The first quartile is $$528.5$$ FTES. The third quartile is $$1447.5$$ FTES. The total number of years sampled was $$29$$ years.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Measures of the Spread of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab63d62spread21a","stepAnswer":["$$1948$$"],"problemType":"TextBox","stepTitle":"75% of all years have an FTES at or below what?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1948$$","hints":{"DefaultPathway":[{"id":"ab63d62spread21a-h1","type":"hint","dependencies":[],"title":"Chebyshev\'s Rule","text":"By Chebyshev\'s Rule, at least 75% of the data is within two standard deviations of the mean. Therefore, to determine the value that 75% of all years have an FTES at or below is to calculate the mean plus two times the standard deviation since this will give us an upper bound.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab63d62spread21a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1948$$"],"dependencies":["ab63d62spread21a-h1"],"title":"Determining Two Standard Deviations Above the Mean","text":"What is two standard deviations above the mean?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab63d62spread21b","stepAnswer":["$$52$$"],"problemType":"TextBox","stepTitle":"75% of all years have an FTES at or above what?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$52$$","hints":{"DefaultPathway":[{"id":"ab63d62spread21b-h3","type":"hint","dependencies":["ab63d62spread21a-h2"],"title":"Chebyshev\'s Rule","text":"By Chebyshev\'s Rule, at least 75% of the data is within two standard deviations of the mean. Therefore, to determine the value that 75% of all years have an FTES at or above is to calculate the mean minus two times the standard deviation since this will give us a lower bound.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab63d62spread21b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$52$$"],"dependencies":["ab63d62spread21b-h3"],"title":"Determining Two Standard Deviations Below the Mean","text":"What is two standard deviations below the mean?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab63d62spread22","title":"Calculating Proportions Using Summary Statistics","body":"The population parameters below describe the full-time equivalent number of students (FTES) each year at Lake Tahoe Community College from $$1976-1977$$ through $$2004-2005$$. The average is $$1000$$ FTES. The median is $$1014$$ FTES. The standard deviation is $$474$$ FTES. The first quartile is $$528.5$$ FTES. The third quartile is $$1447.5$$ FTES. The total number of years sampled was $$29$$ years.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Measures of the Spread of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab63d62spread22a","stepAnswer":["$$50$$"],"problemType":"TextBox","stepTitle":"What percent of the FTES were from $$528.5$$ to $$1447.5$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$50$$","hints":{"DefaultPathway":[{"id":"ab63d62spread22a-h1","type":"hint","dependencies":[],"title":"Comparing $$528.5$$ to the Population","text":"We can use the value that we were given in the question $$(528.5)$$ and compare it to the population statistics that we were given to see where in the distribution it lies.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab63d62spread22a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["First Quartile"],"dependencies":["ab63d62spread22a-h1"],"title":"Comparing $$528.5$$ to the Population","text":"$$528.5$$ is the value given to us that we are to compare to the sample. What is $$528.5$$ to the sample? Is it one of the population statistics?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["First Quartile","Third Quartile","Median","Mean"]},{"id":"ab63d62spread22a-h3","type":"hint","dependencies":["ab63d62spread22a-h2"],"title":"Comparing $$1447.5$$ to the Population","text":"We can use the value that we were given in the question $$(1447.5)$$ and compare it to the population statistics that we were given to see where in the distribution it lies.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab63d62spread22a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Third Quartile"],"dependencies":["ab63d62spread22a-h3"],"title":"Comparing $$1447.5$$ to the Population","text":"$$1447.5$$ is the value given to us that we are to compare to the sample. What is $$1447.5$$ to the sample? Is it one of the population statistics?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["First Quartile","Third Quartile","Median","Mean"]},{"id":"ab63d62spread22a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["IQR"],"dependencies":["ab63d62spread22a-h4"],"title":"Determining Percentage of the FTES","text":"Since $$1447.5$$ is the third quartile and $$528.5$$ is the first quartile, if we subtract the first quartile from the third, what summary statistic do we get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["IQR","Median","Mean","Range"]},{"id":"ab63d62spread22a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$50$$"],"dependencies":["ab63d62spread22a-h5"],"title":"Determining Percentage of the FTES","text":"What percentage of the data resides in the IQR?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab63d62spread23","title":"Determining Standard Deviations Away From Mean","body":"The population parameters below describe the full-time equivalent number of students (FTES) each year at Lake Tahoe Community College from $$1976-1977$$ through $$2004-2005$$. The average is $$1000$$ FTES. The median is $$1014$$ FTES. The standard deviation is $$474$$ FTES. The first quartile is $$528.5$$ FTES. The third quartile is $$1447.5$$ FTES. The total number of years sampled was $$29$$ years.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Measures of the Spread of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab63d62spread23a","stepAnswer":["$$0.0295$$"],"problemType":"TextBox","stepTitle":"How many standard deviations away from the mean is the median? 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Use the formula given.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab63d62spread23a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.0295$$"],"dependencies":[],"title":"Determining Standard Deviations Away From Mean","text":"What is $$1014-\\\\frac{1000}{474}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"ab63d62spread24","title":"Comparing Values from Different Data Sets","body":"A music school has budgeted to purchase three musical instruments. They plan to purchase a piano costing $3,000, a guitar costing $550, and a drum set costing $600. The mean cost for a piano is $4,000 with a standard deviation of $2,500. The mean cost for a guitar is $500 with a standard deviation of $200. The mean cost for drums is $700 with a standard deviation of $100.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Measures of the Spread of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab63d62spread24a","stepAnswer":["Drums"],"problemType":"MultipleChoice","stepTitle":"Which cost is the lowest, when compared to other instruments of the same type?","stepBody":"","answerType":"string","variabilization":{},"choices":["Drums","Piano","Guitar"],"hints":{"DefaultPathway":[{"id":"ab63d62spread24a-h1","type":"hint","dependencies":[],"title":"Determine Instrument Costs\' Standard Deviations","text":"For each instrument, determine how many standard deviations (#ofSTDEVs) their cost is away from the average, for that type of instrument. Pay careful attention to signs when comparing and interpreting the answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab63d62spread24a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-0.4$$"],"dependencies":["ab63d62spread24a-h1"],"title":"Piano\'s Standard Deviation","text":"How many standard deviations away from the mean is the piano\'s cost? Pay careful attention to sign. Round to the nearest hundredth.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab63d62spread24a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-0.4$$"],"dependencies":[],"title":"Piano\'s Standard Deviation","text":"What is $$\\\\frac{3000-4000}{2500}$$? Round to the nearest hundredth.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ab63d62spread24a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.25$$"],"dependencies":["ab63d62spread24a-h2"],"title":"Guitar\'s Standard Deviation","text":"How many standard deviations away from the mean is the guitar\'s cost? Pay careful attention to sign. Round to the nearest hundredth.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab63d62spread24a-h3-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.25$$"],"dependencies":[],"title":"Guitar\'s Standard Deviation","text":"What is $$\\\\frac{550-500}{200}$$? 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Round to the nearest hundredth.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ab63d62spread24a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Drums"],"dependencies":["ab63d62spread24a-h4"],"title":"Determining Which Instrument\'s Cost is Better","text":"Which instrument\'s cost is the lowest in comparison to its instrument type? Effectively, which instrument that the school wanted to purchase had the lowest standard deviation score?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Drums","Piano","Guitar"]}]}}]},{"id":"ab63d62spread25","title":"Comparing Values from Different Data Sets","body":"Three students were applying to the same graduate school. They came from schools with different grading systems.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Measures of the Spread of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab63d62spread25a","stepAnswer":["Kamala"],"problemType":"MultipleChoice","stepTitle":"Which student had the best GPA when compared to other students at his school?","stepBody":"","answerType":"string","variabilization":{},"choices":["Thuy","Vichet","Kamala"],"hints":{"DefaultPathway":[{"id":"ab63d62spread25a-h1","type":"hint","dependencies":[],"title":"Determine Student\'s GPAs\' Standard Deviations","text":"For each student, determine how many standard deviations (#ofSTDEVs) their GPA is away from the average, for their school. Pay careful attention to signs when comparing and interpreting the answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab63d62spread25a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-0.625$$"],"dependencies":["ab63d62spread25a-h1"],"title":"Thuy\'s Standard Deviation","text":"How many standard deviations away from the mean is Thuy\'s GPA? Pay careful attention to sign. Round to the nearest hundredth.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab63d62spread25a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-0.625$$"],"dependencies":[],"title":"Thuy\'s Standard Deviation","text":"What is $$\\\\frac{2.7-3.2}{0.8}$$? 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Round to the nearest hundredth.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ab63d62spread25a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.5$$"],"dependencies":["ab63d62spread25a-h3"],"title":"Kamala\'s Standard Deviation","text":"How many standard deviations away from the mean is Kamala\'s GPA? Pay careful attention to sign. Round to the nearest hundredth.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab63d62spread25a-h4-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.5$$"],"dependencies":[],"title":"Kamala\'s Standard Deviation","text":"What is $$\\\\frac{8.6-8}{0.4}$$? 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The sample average for the same data set is $$10.53$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Measures of the Spread of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab63d62spread3a","stepAnswer":["$$9.45$$"],"problemType":"TextBox","stepTitle":"Find the value that is $$1.5$$ standard deviations below the mean.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9.45$$","hints":{"DefaultPathway":[{"id":"ab63d62spread3a-h1","type":"hint","dependencies":[],"title":"Definition of Differences Using Standard Deviation","text":"Standard deviation measures, essentially, the average deviations a sample is from its sample mean. To calculate the value that is $$1.5$$ standard deviations below the mean then, therefore, we must calculate the value that is the mean minus $$1.5$$ times the standard deviation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab63d62spread3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9.45$$"],"dependencies":["ab63d62spread3a-h1"],"title":"Finding the Value Desired","text":"What is the value that is two standard deviations below the mean?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab63d62spread3a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9.45$$"],"dependencies":[],"title":"Finding the Value Desired","text":"What is $$10.53-1.5\\\\times0.72$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}},{"id":"ab63d62spread3b","stepAnswer":["$$9.45$$"],"problemType":"TextBox","stepTitle":"Find the value that is $$1.5$$ standard deviations above the mean.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9.45$$","hints":{"DefaultPathway":[{"id":"ab63d62spread3b-h3","type":"hint","dependencies":["ab63d62spread3a-h2"],"title":"Definition of Differences Using Standard Deviation","text":"Standard deviation measures, essentially, the average deviations a sample is from its sample mean. 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The sample average for the same data set is $$30.68$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Measures of the Spread of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab63d62spread5a","stepAnswer":["$$18.5$$"],"problemType":"TextBox","stepTitle":"Find the value that is two standard deviations below the mean.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$18.5$$","hints":{"DefaultPathway":[{"id":"ab63d62spread5a-h1","type":"hint","dependencies":[],"title":"Definition of Differences Using Standard Deviation","text":"Standard deviation measures, essentially, the average deviations a sample is from its sample mean. 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(512=a_1*r**9)/(64=a_1*r**6)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab6a46ageo1a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Solve for $$r$$","text":"What is $$r$$ if $$r=\\\\sqrt[3]{8}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ab6a46ageo1a-h6","type":"hint","dependencies":["ab6a46ageo1a-h5"],"title":"Solve for $$a_1$$","text":"Since $$r=2$$, substitute $$2$$ into $$64=a_1 r^6$$ to find $$a_1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo1a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ab6a46ageo1a-h6"],"title":"Solve for $$a_1$$","text":"What is $$a_1=\\\\frac{64}{2^6}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo1a-h8","type":"hint","dependencies":["ab6a46ageo1a-h5","ab6a46ageo1a-h7"],"title":"Explicit Formula for a Geometric Sequence","text":"Since $$r=2$$ and $$a_1=1$$, substitute $$2$$ and $$1$$ into $$a_n=a_1 r^{n-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo1a-h9","type":"hint","dependencies":["ab6a46ageo1a-h8"],"title":"Explicit Formula for a Geometric Sequence","text":"The formula for this geometric sequence is $$a_n=1\\\\times2^{n-1}$$ which simplifies to $$a_n=2^{n-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo1a-h10","type":"hint","dependencies":["ab6a46ageo1a-h9"],"title":"Find $$a_2$$","text":"To find the second term, let $$n=2$$ and plug in $$2$$ into $$a_n=2^{n-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo1a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ab6a46ageo1a-h10"],"title":"Find $$a_2$$","text":"What is $$a_2=2^{2-1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo1a-h12","type":"hint","dependencies":["ab6a46ageo1a-h9"],"title":"Find $$a_3$$","text":"To find the third term, let $$n=3$$ and plug in $$3$$ into $$a_n=2^{n-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo1a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ab6a46ageo1a-h12"],"title":"Find $$a_3$$","text":"What is $$a_3=2^{3-1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo1a-h14","type":"hint","dependencies":["ab6a46ageo1a-h9"],"title":"Find $$a_4$$","text":"To find the fourth term, let $$n=4$$ and plug in $$4$$ into $$a_n=2^{n-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo1a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["ab6a46ageo1a-h14"],"title":"Find $$a_4$$","text":"What is $$a_4=2^{4-1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo1a-h16","type":"hint","dependencies":["ab6a46ageo1a-h9"],"title":"Find $$a_5$$","text":"To find the fifth term, let $$n=5$$ and plug in $$5$$ into $$a_n=2^{n-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo1a-h17","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["ab6a46ageo1a-h16"],"title":"Find $$a_5$$","text":"What is $$a_5=2^{5-1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo1a-h18","type":"hint","dependencies":["ab6a46ageo1a-h7","ab6a46ageo1a-h11","ab6a46ageo1a-h13","ab6a46ageo1a-h15","ab6a46ageo1a-h17"],"title":"First $$5$$ terms","text":"The first five terms are 1,2,4,8,16.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab6a46ageo10","title":"Write a recursive formula for each geometric sequence.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Geometric Sequences","courseName":"OpenStax: College Algebra","steps":[{"id":"ab6a46ageo10a","stepAnswer":["$$a_1=-32$$, $$a_n=\\\\frac{1}{2} a\\\\left(n-1\\\\right)$$"],"problemType":"MultipleChoice","stepTitle":"$$a_n=$$ {-32,-16,-8,-4,...}","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$a_1=-32$$, $$a_n=\\\\frac{1}{2} a\\\\left(n-1\\\\right)$$","choices":["$$a_1=\\\\frac{1}{2}$$, $$a_n=-32a_n-1$$","$$a_1=-32$$, $$a_n=\\\\frac{1}{2} a\\\\left(n-1\\\\right)$$","$$a_n=\\\\frac{1}{2} a_n-1$$"],"hints":{"DefaultPathway":[{"id":"ab6a46ageo10a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-31$$"],"dependencies":[],"title":"Identify $$a_1$$","text":"What is the first term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo10a-h2","type":"hint","dependencies":["ab6a46ageo10a-h1"],"title":"Find $$r$$","text":"The common ratio can be found by dividing the second term by the first term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["ab6a46ageo10a-h2"],"title":"Find $$r$$","text":"What is $$\\\\frac{-16}{-32}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo10a-h4","type":"hint","dependencies":["ab6a46ageo10a-h3"],"title":"Recursive Formula for Geometric Sequences","text":"Define $$a_1$$ and then substitute the common ratio into the recursive formula: $$a_n=r a_n-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo10a-h5","type":"hint","dependencies":["ab6a46ageo10a-h4"],"title":"Recursive Formula for Geometric Sequences","text":"The recursive formula for this geometric sequence is $$a_1=-32$$, $$a_n=\\\\frac{1}{2} a_n-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab6a46ageo2","title":"Writing the First Five Terms of the Geometric Sequence","body":"Write the first five terms of the geometric sequence, given any two terms.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Geometric Sequences","courseName":"OpenStax: College Algebra","steps":[{"id":"ab6a46ageo2a","stepAnswer":["800,400,200,100,50"],"problemType":"MultipleChoice","stepTitle":"$$a_6=25$$, $$a_8=6.25$$","stepBody":"","answerType":"string","variabilization":{},"choices":["800,400,200,100,50","50,100,200,400,800","400,200,100,50,0"],"hints":{"DefaultPathway":[{"id":"ab6a46ageo2a-h1","type":"hint","dependencies":[],"title":"Explicit Formula for a Geometric Sequence\\\\n","text":"Plug in the two terms into the explicit formula of a geometric sequence: $$a_n=a_1 r^{n-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo2a-h2","type":"hint","dependencies":["ab6a46ageo2a-h1"],"title":"Explicit Formula for a Geometric Sequence\\\\n","text":"Plug in $$a_6=25$$ into the formula to obtain $$a_6=a_1 r^{6-1}$$, which simplifies to $$25=a_1 r^5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo2a-h3","type":"hint","dependencies":["ab6a46ageo2a-h1"],"title":"Explicit Formula for a Geometric Sequence\\\\n","text":"Plug in $$a_8=6.25$$ into the formula to obtain $$a_8=a_1 r^{8-1}$$, which simplifies to $$6.25=a_1 r^7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo2a-h4","type":"hint","dependencies":["ab6a46ageo2a-h2","ab6a46ageo2a-h3"],"title":"Systems of Equations","text":"Since both equations contain $$a_1$$, divide the equations from one another to solve for $$r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo2a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.25=r^2$$"],"dependencies":["ab6a46ageo2a-h4"],"title":"Systems of Equations","text":"What is the equation obtained for (512=a_1*r**9)/(64=a_1*r**6)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab6a46ageo2a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.5$$"],"dependencies":[],"title":"Solve for $$r$$","text":"What is $$r=\\\\sqrt{0.25}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ab6a46ageo2a-h6","type":"hint","dependencies":["ab6a46ageo2a-h5"],"title":"Solve for $$a_1$$","text":"Since $$r=0.5$$, substitute $$0.5$$ into $$25=a_1 r^5$$ to find $$a_1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo2a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$800$$"],"dependencies":["ab6a46ageo2a-h6"],"title":"Solve for $$a_1$$","text":"What is $$a_1=\\\\frac{25}{{0.5}^5}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo2a-h8","type":"hint","dependencies":["ab6a46ageo2a-h5","ab6a46ageo2a-h7"],"title":"Explicit Formula for a Geometric Sequence","text":"Since $$r=0.5$$ and $$a_1=800$$, substitute $$0.5$$ and $$800$$ into $$a_n=a_1 r^{n-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo2a-h9","type":"hint","dependencies":["ab6a46ageo2a-h8"],"title":"Explicit Formula for a Geometric Sequence","text":"The formula for this geometric sequence is $$a_n=800{0.5}^{n-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo2a-h10","type":"hint","dependencies":["ab6a46ageo2a-h9"],"title":"Find $$a_2$$","text":"To find the second term, let $$n=2$$ and plug in $$2$$ into $$a_n=800{0.5}^{n-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo2a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$400$$"],"dependencies":["ab6a46ageo2a-h10"],"title":"Find $$a_2$$","text":"What is $$a_2=800{0.5}^{2-1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo2a-h12","type":"hint","dependencies":["ab6a46ageo2a-h9"],"title":"Find $$a_3$$","text":"To find the third term, let $$n=3$$ and plug in $$3$$ into $$a_n=800{0.5}^{n-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo2a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$200$$"],"dependencies":["ab6a46ageo2a-h12"],"title":"Find $$a_3$$","text":"What is $$a_3=800{0.5}^{3-1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo2a-h14","type":"hint","dependencies":["ab6a46ageo2a-h9"],"title":"Find $$a_4$$","text":"To find the fourth term, let $$n=4$$ and plug in $$4$$ into $$a_n=800{0.5}^{n-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo2a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$100$$"],"dependencies":["ab6a46ageo2a-h14"],"title":"Find $$a_4$$","text":"What is $$a_4=800{0.5}^{4-1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo2a-h16","type":"hint","dependencies":["ab6a46ageo2a-h9"],"title":"Find $$a_5$$","text":"To find the fifth term, let $$n=5$$ and plug in $$5$$ into $$a_n=800{0.5}^{n-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo2a-h17","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$50$$"],"dependencies":["ab6a46ageo2a-h16"],"title":"Find $$a_5$$","text":"What is $$a_5=800{0.5}^{5-1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo2a-h18","type":"hint","dependencies":["ab6a46ageo2a-h7","ab6a46ageo2a-h11","ab6a46ageo2a-h13","ab6a46ageo2a-h15","ab6a46ageo2a-h17"],"title":"First $$5$$ terms","text":"The first five terms are 800,400,200,100,50.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab6a46ageo3","title":"Finding the Specified Term for the Geometric Sequence","body":"Find the specified term for the geometric sequence, given the first term and common ratio.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Geometric Sequences","courseName":"OpenStax: College Algebra","steps":[{"id":"ab6a46ageo3a","stepAnswer":["$$162$$"],"problemType":"TextBox","stepTitle":"The first term is $$2$$, and the common ratio is $$3$$. Find the fifth term.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$162$$","hints":{"DefaultPathway":[{"id":"ab6a46ageo3a-h1","type":"hint","dependencies":[],"title":"Explicit Formula for a Geometric Sequence\\\\n","text":"Plug in the given variables into the explicit formula of a geometric sequence: $$a_n=a_1 r^{n-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo3a-h2","type":"hint","dependencies":["ab6a46ageo3a-h1"],"title":"Initializing $$a_1$$","text":"Since the first term is $$2$$, $$a_1=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo3a-h3","type":"hint","dependencies":["ab6a46ageo3a-h1"],"title":"Initializing $$r$$","text":"Since the common ratio is $$3$$, $$r=3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo3a-h4","type":"hint","dependencies":["ab6a46ageo3a-h2","ab6a46ageo3a-h3"],"title":"Explicit Formula for a Geometric Sequence\\\\n","text":"Plug in the given variables into formula to obtain $$a_n=2\\\\times3^{n-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo3a-h5","type":"hint","dependencies":["ab6a46ageo3a-h4"],"title":"Find $$a_5$$","text":"Since we want to find the 5th term, $$n=5$$. Substitute $$5$$ into $$a_n=2\\\\times3^{n-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo3a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$162$$"],"dependencies":["ab6a46ageo3a-h5"],"title":"Find $$a_5$$","text":"What is $$a_5=2\\\\times3^{5-1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo3a-h7","type":"hint","dependencies":["ab6a46ageo3a-h6"],"title":"Fifth Term","text":"The fifth term is $$162$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab6a46ageo4","title":"Finding the Specified Term for the Geometric Sequence","body":"Find the specified term for the geometric sequence, given the first term and common ratio.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Geometric Sequences","courseName":"OpenStax: College Algebra","steps":[{"id":"ab6a46ageo4a","stepAnswer":["$$\\\\frac{-16}{27}$$"],"problemType":"TextBox","stepTitle":"The first term is $$16$$ and the common ratio is $$\\\\frac{-1}{3}$$. Find the fourth term.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-16}{27}$$","hints":{"DefaultPathway":[{"id":"ab6a46ageo4a-h1","type":"hint","dependencies":[],"title":"Explicit Formula for a Geometric Sequence\\\\n","text":"Plug in the given variables into the explicit formula of a geometric sequence: $$a_n=a_1 r^{n-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo4a-h2","type":"hint","dependencies":["ab6a46ageo4a-h1"],"title":"Initializing $$a_1$$","text":"Since the first term is $$16$$, $$a_1=16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo4a-h3","type":"hint","dependencies":["ab6a46ageo4a-h1"],"title":"Initializing $$r$$","text":"Since the common ratio is $$\\\\frac{-1}{3}$$, $$r=\\\\frac{-1}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo4a-h4","type":"hint","dependencies":["ab6a46ageo4a-h2","ab6a46ageo4a-h3"],"title":"Explicit Formula for a Geometric Sequence\\\\n","text":"Plug in the given variables into formula to obtain $$a_n=16{\\\\left(-\\\\frac{1}{3}\\\\right)}^{n-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo4a-h5","type":"hint","dependencies":["ab6a46ageo4a-h4"],"title":"Find $$a_5$$","text":"Since we want to find the 4th term, $$n=4$$. Substitute $$4$$ into $$a_n=16{\\\\left(-\\\\frac{1}{3}\\\\right)}^{n-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo4a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-16}{27}$$"],"dependencies":["ab6a46ageo4a-h5"],"title":"Find $$a_5$$","text":"What is $$a_4=16{\\\\left(-\\\\frac{1}{3}\\\\right)}^{4-1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo4a-h7","type":"hint","dependencies":["ab6a46ageo4a-h6"],"title":"Fifth Term","text":"The fourth term is $$\\\\frac{-16}{27}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab6a46ageo5","title":"Finding the Specified Term for the Geometric Sequence","body":"Find the specified term for the geometric sequence, given the first four terms.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Geometric Sequences","courseName":"OpenStax: College Algebra","steps":[{"id":"ab6a46ageo5a","stepAnswer":["$$2048$$"],"problemType":"TextBox","stepTitle":"$$a_n=$$ {-1,2,-4,8,...}. Find $$a_{12}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2048$$","hints":{"DefaultPathway":[{"id":"ab6a46ageo5a-h1","type":"hint","dependencies":[],"title":"Find $$a_1$$","text":"$$a_1$$ is the first term of the sequence.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["ab6a46ageo5a-h1"],"title":"Find $$a_1$$","text":"What is $$a_1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo5a-h3","type":"hint","dependencies":["ab6a46ageo5a-h2"],"title":"Find $$r$$","text":"The common ratio can be found by dividing the second term by the first term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["ab6a46ageo5a-h3"],"title":"Find $$r$$","text":"What is $$\\\\frac{2}{-1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo5a-h5","type":"hint","dependencies":["ab6a46ageo5a-h2","ab6a46ageo5a-h4"],"title":"Explicit Formula for a Geometric Sequence\\\\n","text":"Since $$a_1=-1$$ and $$r=-2$$, substitute $$-1$$ and $$-2$$ into the formula: $$a_n=a_1 r^{n-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo5a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$a_n=-1-\\\\left(2^{n-1}\\\\right)$$"],"dependencies":["ab6a46ageo5a-h5"],"title":"Explicit Formula for a Geometric Sequence\\\\n","text":"What is the explicit formula for this geometric sequence?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo5a-h7","type":"hint","dependencies":["ab6a46ageo5a-h6"],"title":"Find $$a_{12}$$","text":"Since we want to find the 12th term, $$n=12$$. Substitute $$12$$ into $$a_n=-1{\\\\left(-2\\\\right)}^{n-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo5a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2048$$"],"dependencies":["ab6a46ageo5a-h7"],"title":"Solve for $$a_{12}$$","text":"What is $$-1{\\\\left(-2\\\\right)}^{12-1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo5a-h9","type":"hint","dependencies":["ab6a46ageo5a-h8"],"title":"12th term","text":"$$a_{12}=2048$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab6a46ageo6","title":"Finding the Specified Term for the Geometric Sequence","body":"Find the specified term for the geometric sequence, given the first four terms.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Geometric Sequences","courseName":"OpenStax: College Algebra","steps":[{"id":"ab6a46ageo6a","stepAnswer":["$$\\\\frac{-2}{729}$$"],"problemType":"TextBox","stepTitle":"a_n={-2,2/3,-2/9,2/27,...}. Find $$a_7$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-2}{729}$$","hints":{"DefaultPathway":[{"id":"ab6a46ageo6a-h1","type":"hint","dependencies":[],"title":"Find $$a_1$$","text":"$$a_1$$ is the first term of the sequence.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["ab6a46ageo6a-h1"],"title":"Find $$a_1$$","text":"What is $$a_1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo6a-h3","type":"hint","dependencies":["ab6a46ageo6a-h2"],"title":"Find $$r$$","text":"The common ratio can be found by dividing the second term by the first term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{3}$$"],"dependencies":["ab6a46ageo6a-h3"],"title":"Find $$r$$","text":"What is $$\\\\frac{\\\\frac{2}{3}}{-2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo6a-h5","type":"hint","dependencies":["ab6a46ageo6a-h2","ab6a46ageo6a-h4"],"title":"Explicit Formula for a Geometric Sequence\\\\n","text":"Since $$a_1=-2$$ and $$r=\\\\frac{-1}{3}$$, substitute $$-2$$ and $$\\\\frac{-1}{3}$$ into the formula: $$a_n=a_1 r^{n-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo6a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$a_n=-2{\\\\left(-\\\\frac{1}{3}\\\\right)}^{n-1}$$"],"dependencies":["ab6a46ageo6a-h5"],"title":"Explicit Formula for a Geometric Sequence\\\\n","text":"What is the explicit formula for this geometric sequence?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo6a-h7","type":"hint","dependencies":["ab6a46ageo6a-h6"],"title":"Find $$a_7$$","text":"Since we want to find the 7th term, $$n=7$$. Substitute $$12$$ into $$a_n=-1{\\\\left(-2\\\\right)}^{n-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo6a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-2}{729}$$"],"dependencies":["ab6a46ageo6a-h7"],"title":"Solve for $$a_7$$","text":"What is $$-2{\\\\left(-\\\\frac{1}{3}\\\\right)}^{7-1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo6a-h9","type":"hint","dependencies":["ab6a46ageo6a-h8"],"title":"7th Term","text":"$$a_7=\\\\frac{-2}{729}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab6a46ageo9","title":"Write a recursive formula for each geometric sequence.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Geometric Sequences","courseName":"OpenStax: College Algebra","steps":[{"id":"ab6a46ageo9a","stepAnswer":["$$a_1=-1$$, $$a_n=-5a_n-1$$"],"problemType":"MultipleChoice","stepTitle":"$$a_n$$ $$=$$ {-1,5,-25,125,...}","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$a_1=-1$$, $$a_n=-5a_n-1$$","choices":["$$a_1=-1$$, $$a_n=-5a_n-1$$","$$a_1=-1$$, $$a_n=5a_n-1$$","$$a_n=-5a_n-1$$"],"hints":{"DefaultPathway":[{"id":"ab6a46ageo9a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":[],"title":"Identify $$a_1$$","text":"What is the first term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo9a-h2","type":"hint","dependencies":["ab6a46ageo9a-h1"],"title":"Find $$r$$","text":"The common ratio can be found by dividing the second term by the first term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["ab6a46ageo9a-h2"],"title":"Find $$r$$","text":"What is $$\\\\frac{5}{-1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo9a-h4","type":"hint","dependencies":["ab6a46ageo9a-h3"],"title":"Recursive Formula for Geometric Sequences","text":"Define $$a_1$$ and then substitute the common ratio into the recursive formula: $$a_n=r a_n-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageo9a-h5","type":"hint","dependencies":["ab6a46ageo9a-h4"],"title":"Recursive Formula for Geometric Sequences","text":"The recursive formula for this geometric sequence is $$a_1=-1$$, $$a_n=-5a_n-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab6a46ageometric1","title":"Finding the Common Ratio of the Sequence","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Geometric Sequences","courseName":"OpenStax: College Algebra","steps":[{"id":"ab6a46ageometric1a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"1,2,4,8,16,...","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"ab6a46ageometric1a-h1","type":"hint","dependencies":[],"title":"Divide","text":"First, divide each term by its previous term. So for the first term, you would divide $$2$$ by $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageometric1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ab6a46ageometric1a-h1"],"title":"Divide all terms","text":"Divide all of the terms by their previous one. What does the division equal for each term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab6a46ageometric1a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ab6a46ageometric1a-h3","type":"hint","dependencies":["ab6a46ageometric1a-h2"],"title":"Answer","text":"Therefore, the answer is $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab6a46ageometric10","title":"Finding the Common Ratio","body":"Find the common ratio for the geometric sequence.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Geometric Sequences","courseName":"OpenStax: College Algebra","steps":[{"id":"ab6a46ageometric10a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"1,3,9,27,81,...","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"ab6a46ageometric10a-h1","type":"hint","dependencies":[],"title":"Divide","text":"First, divide the second term by the first term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageometric10a-h2","type":"hint","dependencies":["ab6a46ageometric10a-h1"],"title":"Answer","text":"When you divide the second term by the first, $$r$$ becomes $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab6a46ageometric11","title":"Writing the Terms of a Geometric Sequence","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Geometric Sequences","courseName":"OpenStax: College Algebra","steps":[{"id":"ab6a46ageometric11a","stepAnswer":["$$8, 2.4, 0.72, 0.21$$"],"problemType":"MultipleChoice","stepTitle":"List the first four terms of the geometric sequence with $$a_1=8$$ and $$r=0.3$$","stepBody":"Enter your answer in the form: a_1,a_2,a_3,a_4","answerType":"string","variabilization":{},"answerLatex":"$$8, 2.4, 0.72, 0.21$$","choices":["$$8, 2.4, 0.62, 0.21$$","$$8, 2.4, 0.72, 0.21$$","$$8, 2.4, 0.72, 0.11$$"],"hints":{"DefaultPathway":[{"id":"ab6a46ageometric11a-h1","type":"hint","dependencies":[],"title":"Finding the first term","text":"Multiply $$a_1$$ by $$r$$ to find $$a_2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageometric11a-h2","type":"hint","dependencies":["ab6a46ageometric11a-h1"],"title":"Continue","text":"Continue this process to find all terms, multiplying each term by $$r$$ to get the next one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageometric11a-h3","type":"hint","dependencies":["ab6a46ageometric11a-h2"],"title":"Answer","text":"Therefore, the answer is: $$8, 2.4, 0.72, 0.21$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab6a46ageometric12","title":"Writing the Terms of a Geometric Sequence","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Geometric Sequences","courseName":"OpenStax: College Algebra","steps":[{"id":"ab6a46ageometric12a","stepAnswer":["5,1,1/5,1/25"],"problemType":"MultipleChoice","stepTitle":"List the first four terms of the geometric sequence with $$a_1=5$$ and $$r=\\\\frac{1}{5}$$","stepBody":"Enter your answer in the form: a_1,a_2,a_3,a_4","answerType":"string","variabilization":{},"choices":["5,1/5,1/25,1/125","5,1/5,1/75,1/125","5,1,1/5,1/25"],"hints":{"DefaultPathway":[{"id":"ab6a46ageometric12a-h1","type":"hint","dependencies":[],"title":"Finding the first term","text":"Multiply $$a_1$$ by $$r$$ to find $$a_2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageometric12a-h2","type":"hint","dependencies":["ab6a46ageometric12a-h1"],"title":"Continue","text":"Continue this process to find all terms, multiplying each term by $$r$$ to get the next one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageometric12a-h3","type":"hint","dependencies":["ab6a46ageometric12a-h2"],"title":"Answer","text":"Therefore, the answer is: 5,1/5,1/25,1/125","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab6a46ageometric13","title":"Finding the Common Ratio of the Sequence","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Geometric Sequences","courseName":"OpenStax: College Algebra","steps":[{"id":"ab6a46ageometric13a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"1,3,9,27,81,...","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"ab6a46ageometric13a-h1","type":"hint","dependencies":[],"title":"Divide","text":"First, divide each term by its previous term. So for the first term, you would divide $$2$$ by $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageometric13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ab6a46ageometric13a-h1"],"title":"Divide all terms","text":"Divide all of the terms by their previous one. What does the division equal for each term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab6a46ageometric13a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ab6a46ageometric13a-h3","type":"hint","dependencies":["ab6a46ageometric13a-h2"],"title":"Answer","text":"Therefore, the answer is $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab6a46ageometric14","title":"Finding the Common Ratio of the Sequence","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Geometric Sequences","courseName":"OpenStax: College Algebra","steps":[{"id":"ab6a46ageometric14a","stepAnswer":["$$-2$$"],"problemType":"TextBox","stepTitle":"$$-0.125, 0.25, -0.5, 1, -2, ..$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-2$$","hints":{"DefaultPathway":[{"id":"ab6a46ageometric14a-h1","type":"hint","dependencies":[],"title":"Divide","text":"First, divide each term by its previous term. So for the first term, you would divide $$2$$ by $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageometric14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["ab6a46ageometric14a-h1"],"title":"Divide all terms","text":"Divide all of the terms by their previous one. What does the division equal for each term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab6a46ageometric14a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ab6a46ageometric14a-h3","type":"hint","dependencies":["ab6a46ageometric14a-h2"],"title":"Answer","text":"Therefore, the answer is $$-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab6a46ageometric15","title":"Finding the Common Ratio of the Sequence","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Geometric Sequences","courseName":"OpenStax: College Algebra","steps":[{"id":"ab6a46ageometric15a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"-2,-12,-72, $$-432..$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"ab6a46ageometric15a-h1","type":"hint","dependencies":[],"title":"Divide","text":"First, divide each term by its previous term. So for the first term, you would divide $$-12$$ by $$-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageometric15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["ab6a46ageometric15a-h1"],"title":"Divide all terms","text":"Divide all of the terms by their previous one. What does the division equal for each term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab6a46ageometric15a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ab6a46ageometric15a-h3","type":"hint","dependencies":["ab6a46ageometric15a-h2"],"title":"Answer","text":"Therefore, the answer is $$6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab6a46ageometric2","title":"Finding the Common Ratio of the Sequence","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Geometric Sequences","courseName":"OpenStax: College Algebra","steps":[{"id":"ab6a46ageometric2a","stepAnswer":["$$\\\\frac{1}{2}$$"],"problemType":"TextBox","stepTitle":"48,12,4, $$2, ..$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{2}$$","hints":{"DefaultPathway":[{"id":"ab6a46ageometric2a-h1","type":"hint","dependencies":[],"title":"Divide","text":"First, divide each term by its previous term. So for the first term, you would divide $$2$$ by $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageometric2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["ab6a46ageometric2a-h1"],"title":"Divide all terms","text":"Divide all of the terms by their previous one. What does the division equal for each term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab6a46ageometric2a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$\\\\frac{1}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ab6a46ageometric2a-h3","type":"hint","dependencies":["ab6a46ageometric2a-h2"],"title":"Answer","text":"Therefore, the answer is $$\\\\frac{1}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab6a46ageometric3","title":"Finding the Common Ratio of the Sequence","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Geometric Sequences","courseName":"OpenStax: College Algebra","steps":[{"id":"ab6a46ageometric3a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"5,10,20,...","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"ab6a46ageometric3a-h1","type":"hint","dependencies":[],"title":"Divide","text":"First, divide each term by its previous term. So for the first term, you would divide $$2$$ by $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageometric3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ab6a46ageometric3a-h1"],"title":"Divide all terms","text":"Divide all of the terms by their previous one. What does the division equal for each term?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab6a46ageometric3a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ab6a46ageometric3a-h3","type":"hint","dependencies":["ab6a46ageometric3a-h2"],"title":"Answer","text":"Therefore, the answer is $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab6a46ageometric4","title":"Writing the Terms of a Geometric Sequence","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Geometric Sequences","courseName":"OpenStax: College Algebra","steps":[{"id":"ab6a46ageometric4a","stepAnswer":["$$5, -10, 20, -40$$"],"problemType":"MultipleChoice","stepTitle":"List the first four terms of the geometric sequence with $$a_1=5$$ and $$r=-2$$.","stepBody":"Enter your answer in the form: $$a_1$$, $$a_2$$, $$a_3$$, $$a_4$$","answerType":"string","variabilization":{},"choices":["$$5, -10, 20, -40$$","$$5, -11, 21, -40$$","$$5, -10, 25, -40$$"],"hints":{"DefaultPathway":[{"id":"ab6a46ageometric4a-h1","type":"hint","dependencies":[],"title":"Finding the first term","text":"Multiply $$a_1$$ by $$r$$ to find $$a_2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageometric4a-h2","type":"hint","dependencies":["ab6a46ageometric4a-h1"],"title":"Continue","text":"Continue this process to find all terms, multiplying each term by $$r$$ to get the next one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageometric4a-h3","type":"hint","dependencies":["ab6a46ageometric4a-h2"],"title":"Answer","text":"Therefore, the answer is: 5,-10,20,-40","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab6a46ageometric5","title":"Writing the Terms of a Geometric Sequence","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Geometric Sequences","courseName":"OpenStax: College Algebra","steps":[{"id":"ab6a46ageometric5a","stepAnswer":["$$18$$, $$6$$, $$2$$, $$\\\\frac{2}{3}$$"],"problemType":"MultipleChoice","stepTitle":"List the first four terms of the geometric sequence with $$a_1=18$$ and $$r=\\\\frac{1}{3}$$.","stepBody":"Enter your answer in the form: $$a_1$$, $$a_2$$, $$a_3$$, $$a_4$$","answerType":"string","variabilization":{},"answerLatex":"$$18$$, $$6$$, $$2$$, $$\\\\frac{2}{3}$$","choices":["$$15$$, $$6$$, $$2$$, $$\\\\frac{2}{3}$$","$$18$$, $$6$$, $$1$$, $$\\\\frac{2}{3}$$","$$18$$, $$6$$, $$2$$, $$\\\\frac{2}{3}$$"],"hints":{"DefaultPathway":[{"id":"ab6a46ageometric5a-h1","type":"hint","dependencies":[],"title":"Finding the first term","text":"Multiply $$a_1$$ by $$r$$ to find $$a_2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageometric5a-h2","type":"hint","dependencies":["ab6a46ageometric5a-h1"],"title":"Continue","text":"Continue this process to find all terms, multiplying each term by $$r$$ to get the next one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageometric5a-h3","type":"hint","dependencies":["ab6a46ageometric5a-h2"],"title":"Answer","text":"Therefore, the answer is: 18,6,2,2/3","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab6a46ageometric6","title":"Writing Terms of Geometric Sequences Using the Explicit Formula","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Geometric Sequences","courseName":"OpenStax: College Algebra","steps":[{"id":"ab6a46ageometric6a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"Given a geometric sequence with $$a_1=3$$ and $$a_4=24$$, find $$a_2$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"ab6a46ageometric6a-h1","type":"hint","dependencies":[],"title":"Use the explicit formula","text":"The explicit formula for a geometric sequence is $$a_n=a_1 r^{n-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageometric6a-h2","type":"hint","dependencies":["ab6a46ageometric6a-h1"],"title":"Use formula to find $$r$$","text":"The explicit formula can be used to find the second term. First, find $$r$$. To do so, substitute an for $$a_4$$ and $$a_1$$ for $$3$$. Additionally, substitute $$4$$ into $$n$$, so that the power is $$r^3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageometric6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$a_4=3r^3$$"],"dependencies":["ab6a46ageometric6a-h2"],"title":"Plug in","text":"After plugging in the variables, what equation remains?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab6a46ageometric6a-h3-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$a_4=3r^3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ab6a46ageometric6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ab6a46ageometric6a-h3"],"title":"Solve","text":"Next, solve for $$r$$. What does $$r$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab6a46ageometric6a-h4-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ab6a46ageometric6a-h5","type":"hint","dependencies":["ab6a46ageometric6a-h4"],"title":"Multiply","text":"Now, multiply the first term by $$r$$ to find the second term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageometric6a-h6","type":"hint","dependencies":["ab6a46ageometric6a-h5"],"title":"Answer","text":"The answer is $$6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab6a46ageometric7","title":"Writing Terms of Geometric Sequences Using the Explicit Formula","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Geometric Sequences","courseName":"OpenStax: College Algebra","steps":[{"id":"ab6a46ageometric7a","stepAnswer":["$$16384$$"],"problemType":"TextBox","stepTitle":"Given a geometric sequence with $$a_2=4$$ and $$a_3=32$$ , find $$a_6$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16384$$","hints":{"DefaultPathway":[{"id":"ab6a46ageometric7a-h1","type":"hint","dependencies":[],"title":"Find $$r$$","text":"Since the two terms are sequential to each other, you can divide the third term by the second to find $$r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageometric7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["ab6a46ageometric7a-h1"],"title":"Find $$r$$","text":"What is $$r$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab6a46ageometric7a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ab6a46ageometric7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["ab6a46ageometric7a-h2"],"title":"Find $$a_1$$","text":"Next, find $$a_1$$ by dividing $$a_2$$ by $$r$$. What is $$a_1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab6a46ageometric7a-h3-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$\\\\frac{1}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ab6a46ageometric7a-h4","type":"hint","dependencies":["ab6a46ageometric7a-h3"],"title":"Use recursive formula.","text":"The explicit formula for a geometric sequence is $$a_n=a_1 r^{n-1}$$. Plug in $$a_6$$ for an, $$a_1$$, $$r$$, and $$6$$ for $$n$$. Now simplify and solve.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageometric7a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16384$$"],"dependencies":["ab6a46ageometric7a-h4"],"title":"Simplify and solve","text":"What does $$a_6$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab6a46ageometric7a-h5-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$16384$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ab6a46ageometric7a-h6","type":"hint","dependencies":["ab6a46ageometric7a-h5"],"title":"Answer","text":"Therefore, the sixth term of the sequence is $$16384$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab6a46ageometric8","title":"Writing an Explicit Formula for the $$n$$ th Term of a Geometric Sequence","body":"Write an explicit formula for the nth term of the following geometric sequence.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Geometric Sequences","courseName":"OpenStax: College Algebra","steps":[{"id":"ab6a46ageometric8a","stepAnswer":["$$a_n=2\\\\times5^{n-1}$$"],"problemType":"MultipleChoice","stepTitle":"$$2$$, $$10$$, $$50$$, $$250$$, ...","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$a_n=2\\\\times5^{n-1}$$","choices":["$$a_n=2\\\\times5^{n-1}$$","$$a_n=2\\\\times3^{n-1}$$","$$a_n=10\\\\times5^{n-1}$$","$$a_n=10\\\\times3^{n-1}$$"],"hints":{"DefaultPathway":[{"id":"ab6a46ageometric8a-h1","type":"hint","dependencies":[],"title":"Find $$r$$","text":"First, find the common ratio. Divide the second term by the first.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageometric8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["ab6a46ageometric8a-h1"],"title":"Find $$r$$","text":"What does $$r$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab6a46ageometric8a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ab6a46ageometric8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ab6a46ageometric8a-h2"],"title":"$$a_1$$","text":"What is $$a_1$$? It is the first term of the sequence.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageometric8a-h4","type":"hint","dependencies":["ab6a46ageometric8a-h3"],"title":"Plug in","text":"Plug in $$a_1$$ and $$r$$ into the explicit formula.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageometric8a-h5","type":"hint","dependencies":["ab6a46ageometric8a-h4"],"title":"Answer","text":"Therefore, the answer is $$a_n=2\\\\times5^{n-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab6a46ageometric9","title":"Writing an Explicit Formula for the $$n$$ th Term of a Geometric Sequence","body":"Write an explicit formula for the nth term of the following geometric sequence.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.3 Geometric Sequences","courseName":"OpenStax: College Algebra","steps":[{"id":"ab6a46ageometric9a","stepAnswer":["$$a_n=-1-\\\\left(3^{n-1}\\\\right)$$"],"problemType":"MultipleChoice","stepTitle":"$$-1$$, $$3$$, $$-9$$, $$27$$, ...","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$a_n=-1-\\\\left(3^{n-1}\\\\right)$$","choices":["$$a_n=-1-\\\\left(3^{n-1}\\\\right)$$","$$a_n=1-\\\\left(3^{n-1}\\\\right)$$","$$a_n=-1\\\\times2^{n-1}$$","$$a_n=-1\\\\times2^{n-1}$$"],"hints":{"DefaultPathway":[{"id":"ab6a46ageometric9a-h1","type":"hint","dependencies":[],"title":"Find $$r$$","text":"First, find the common ratio. Divide the second term by the first.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageometric9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["ab6a46ageometric9a-h1"],"title":"Find $$r$$","text":"What does $$r$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab6a46ageometric9a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ab6a46ageometric9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["ab6a46ageometric9a-h2"],"title":"$$a_1$$","text":"What is $$a_1$$? It is the first term of the sequence.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageometric9a-h4","type":"hint","dependencies":["ab6a46ageometric9a-h3"],"title":"Plug in","text":"Plug in $$a_1$$ and $$r$$ into the explicit formula.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab6a46ageometric9a-h5","type":"hint","dependencies":["ab6a46ageometric9a-h4"],"title":"Answer","text":"Therefore, the answer is $$a_n=-1-\\\\left(3^{n-1}\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab846b4standard1","title":"Z-scores","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 The Standard Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab846b4standard1a","stepAnswer":["$$\\\\frac{-11}{3}$$"],"problemType":"TextBox","stepTitle":"What is the $$z-score$$ of $$x$$, when $$x=1$$ and $$X~N(12,3)$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-11}{3}$$","hints":{"DefaultPathway":[{"id":"ab846b4standard1a-h1","type":"hint","dependencies":[],"title":"Identify \u03bc and \u03c3","text":"We know $$X~N(\u03bc,\u03c3)$$. In this problem, $$X~N(12,3)$$. The first step would be to identify what is \u03bc and \u03c3.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["ab846b4standard1a-h1"],"title":"Identify \u03bc","text":"The first argument in $$N(12,3)$$ is \u03bc. What is \u03bc?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ab846b4standard1a-h1"],"title":"Identify \u03c3","text":"The second argument in $$N(12,3)$$ is \u03c3. What is \u03c3?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard1a-h4","type":"hint","dependencies":["ab846b4standard1a-h2","ab846b4standard1a-h3"],"title":"Calculate $$z-score$$","text":"The formula for the $$z-score$$ is $$z=\\\\frac{x-\u03bc}{\\\\sigma}$$. Substitute the variables in the formula for numerical values, and solve for $$z$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard1a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$z=\\\\frac{1-12}{3}$$"],"dependencies":["ab846b4standard1a-h4"],"title":"Substitution","text":"We know $$x=1$$, $$\u03bc=12$$, and $$\u03c3=3$$. What is $$z$$ after substituting these variables?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$z=\\\\frac{1-12}{3}$$","$$z=\\\\frac{3-12}{1}$$","$$z=\\\\frac{12-1}{3}$$","$$z=\\\\frac{1-3}{12}$$"]},{"id":"ab846b4standard1a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-11}{3}$$"],"dependencies":["ab846b4standard1a-h5"],"title":"Solve for $$z$$","text":"What is $$\\\\frac{1-12}{3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab846b4standard10","title":"Z-scores","body":"The mean height of $$15$$ to 18-year-old males from Chile from $$2009$$ to $$2010$$ was $$170$$ cm with a standard deviation of $$6.28$$ cm. Male heights are known to follow a normal distribution. Let $$X=the$$ height of a $$15$$ to 18-year-old male from Chile in $$2009$$ to $$2010$$. Then X~N(170, $$6.28)$$. Suppose that the height of a $$15$$ to 18-year-old male from Chile from $$2009$$ to $$2010$$ has a $$z-score$$ of $$z=1.27$$. What is the male\u2019s height? The $$z-score$$ $$(z=1.27)$$ tells you that the male\u2019s height is $$___$$ standard deviations to the $$___$$ (right or left) of the mean.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 The Standard Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab846b4standard10a","stepAnswer":["$$177.98$$"],"problemType":"TextBox","stepTitle":"What is the male\'s height?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$177.98$$","hints":{"DefaultPathway":[{"id":"ab846b4standard10a-h1","type":"hint","dependencies":[],"title":"Standard Deviation","text":"We know that the male\'s height has a $$z-score$$ of $$1.27$$. The absolute value of the $$z-score$$ is how many standard deviations our $$x$$ value is from the mean. Identify the absolute value of the $$z-score$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.27$$"],"dependencies":["ab846b4standard10a-h1"],"title":"Standard Deviation","text":"What is the absolute value of $$1.27$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard10a-h3","type":"hint","dependencies":["ab846b4standard10a-h2"],"title":"Values Within the Standard Deviation","text":"Find what values are within $$1.27$$ standard deviations of the mean.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7.98$$"],"dependencies":["ab846b4standard10a-h3"],"title":"Standard Deviation","text":"What is $$1.27$$ standard deviations?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6.28$$"],"dependencies":["ab846b4standard10a-h4"],"title":"Identify Value of Standard Deviation","text":"We know $$X~N(\u03bc,\u03c3)$$. In this problem, X~N(170,6.28). What is the standard deviation, \u03c3?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard10a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7.98$$"],"dependencies":["ab846b4standard10a-h5"],"title":"$$1.27$$ Standard Deviations","text":"Find what $$1.27$$ standard deviations from the mean is. What is $$1.27\\\\times6.28$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard10a-h7","type":"hint","dependencies":["ab846b4standard10a-h6"],"title":"Direction","text":"When $$z$$ is positive, $$x$$, the male\'s height, is to the right of the mean, \u03bc. When $$z$$ is negative, $$x$$ is to the left of \u03bc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard10a-h8","type":"hint","dependencies":["ab846b4standard10a-h7"],"title":"Direction","text":"Since $$1.27$$ is positive, $$x$$ is right of the mean. What is $$1.27$$ standard deviations to the right of the mean?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard10a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$177.98$$"],"dependencies":["ab846b4standard10a-h8"],"title":"$$1.27$$ standard deviations to the right of the mean","text":"What is $$170+7.98$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab846b4standard10b","stepAnswer":["$$1.27$$"],"problemType":"TextBox","stepTitle":"The $$z-score$$ $$(z=1.27)$$ tells you that the male\u2019s height is $$___$$ standard deviations from the mean. Fill in the blank.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1.27$$","hints":{"DefaultPathway":[{"id":"ab846b4standard10b-h1","type":"hint","dependencies":[],"title":"Identify the $$z-score$$.","text":"The absolute value of the $$z-score$$ is how many standard deviations our $$x$$ value is from the mean. Identify the absolute value of the $$z-score$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard10b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.27$$"],"dependencies":["ab846b4standard10a-h2"],"title":"Z-score","text":"In the problem, the $$z-score$$ is given. What is the $$z-score$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard10b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.27$$"],"dependencies":["ab846b4standard10a-h3"],"title":"Absolute Value","text":"What is the absolute value of the $$z-score$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab846b4standard10c","stepAnswer":["right"],"problemType":"MultipleChoice","stepTitle":"The $$z-score$$ $$(z=1.27)$$ tells you that the male\u2019s height is $$___$$ standard deviations to the $$___$$ (right or left) of the mean. Fill in the second blank.","stepBody":"","answerType":"string","variabilization":{},"choices":["Left","Right","right"],"hints":{"DefaultPathway":[{"id":"ab846b4standard10c-h1","type":"hint","dependencies":[],"title":"Determining right or left","text":"When $$z$$ is positive, $$x$$, the male\'s height, is to the right of the mean, \u03bc. When $$z$$ is negative, $$x$$ is to the left of \u03bc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard10c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.27$$"],"dependencies":["ab846b4standard10c-h1"],"title":"Z-score","text":"What is the value of $$z$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard10c-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["positive"],"dependencies":["ab846b4standard10c-h2"],"title":"Positive or negative","text":"Is this positive or negative?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Negative","Positive"]}]}}]},{"id":"ab846b4standard11","title":"Z-scores","body":"The scores on a college entrance exam have an approximate normal distribution with mean, $$\\\\mu=52$$ points and a standard deviation, $$\u03c3=11$$ points.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 The Standard Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab846b4standard11a","stepAnswer":["$$41$$ and $$63$$"],"problemType":"MultipleChoice","stepTitle":"About 68% of the $$y$$ values lie between what two values? These values are $$___$$ . Fill in the blank.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$41$$ and $$63$$","choices":["$$41$$ and $$63$$","$$32$$ and $$45$$","$$45$$ and $$83$$","$$34$$ and $$61$$"],"hints":{"DefaultPathway":[{"id":"ab846b4standard11a-h1","type":"hint","dependencies":[],"title":"Standard Deviation","text":"68% of the values should be within $$1$$ standard deviation of the mean. Find what values are within $$1$$ standard deviation of the mean.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11$$"],"dependencies":["ab846b4standard11a-h1"],"title":"Standard Deviation","text":"What is $$1$$ standard deviation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard11a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11$$"],"dependencies":["ab846b4standard11a-h2"],"title":"Identify Value of Standard Deviation","text":"Identify what is the standard deviation, \u03c3.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard11a-h4","type":"hint","dependencies":["ab846b4standard11a-h3"],"title":"Range","text":"Add and subtract this amount from the mean to get the values that fall within $$1$$ standard deviation of the mean.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$63$$"],"dependencies":["ab846b4standard11a-h4"],"title":"Add for Upper Bound","text":"What is $$52+11$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard11a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$41$$"],"dependencies":["ab846b4standard11a-h4"],"title":"Subtract for Lower Bound","text":"What is $$52-11$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab846b4standard11b","stepAnswer":["$$-1$$ and $$1$$"],"problemType":"MultipleChoice","stepTitle":"About 68% of the $$y$$ values lie between what two values? These values are $$166.02$$ and $$178.7$$. The $$z-scores$$ are $$___$$ , respectively. Fill in the blank.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-1$$ and $$1$$","choices":["$$-1$$ and $$1$$","$$-2$$ and $$2$$","$$-3$$ and $$3$$","$$-4$$ and $$4$$"],"hints":{"DefaultPathway":[{"id":"ab846b4standard11b-h1","type":"hint","dependencies":[],"title":"Standard Deviation","text":"68% of the values should be within $$1$$ standard deviation of the mean.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard11b-h2","type":"hint","dependencies":["ab846b4standard11b-h1"],"title":"Z-scores","text":"The $$z-scores$$ that correspond to being within $$1$$ standard deviation of the mean are $$-1$$ and $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab846b4standard12","title":"Z-scores","body":"The scores on a college entrance exam have an approximate normal distribution with mean, $$\\\\mu=52$$ points and a standard deviation, $$\u03c3=11$$ points.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 The Standard Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab846b4standard12a","stepAnswer":["$$30$$ and $$74$$"],"problemType":"MultipleChoice","stepTitle":"About 95% of the $$y$$ values lie between what two values? These values are $$___$$ . Fill in the blank.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$30$$ and $$74$$","choices":["$$30$$ and $$74$$","$$32$$ and $$82$$","$$21$$ and $$45$$","$$46$$ and $$91$$"],"hints":{"DefaultPathway":[{"id":"ab846b4standard12a-h1","type":"hint","dependencies":[],"title":"Standard Deviation","text":"95% of the values should be within $$2$$ standard deviations of the mean. Find what values are within $$2$$ standard deviations of the mean.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$22$$"],"dependencies":["ab846b4standard12a-h1"],"title":"Standard Deviation","text":"What is $$2$$ standard deviations?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard12a-h3","type":"hint","dependencies":["ab846b4standard12a-h2"],"title":"Identify Value of Standard Deviation","text":"Identify what is the standard deviation, \u03c3.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$22$$"],"dependencies":["ab846b4standard12a-h3"],"title":"$$2$$ Standard Deviations","text":"We need to know what $$2$$ standard deviations is. What is $$2\\\\times11$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard12a-h5","type":"hint","dependencies":["ab846b4standard12a-h4"],"title":"Range","text":"Add and subtract this amount from the mean to get the values that fall within $$2$$ standard deviations of the mean.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard12a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$74$$"],"dependencies":["ab846b4standard12a-h5"],"title":"Add for Upper Bound","text":"What is $$52+22$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard12a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$30$$"],"dependencies":["ab846b4standard12a-h5"],"title":"Subtract for Lower Bound","text":"What is $$52-22$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab846b4standard12b","stepAnswer":["$$-2$$ and $$2$$"],"problemType":"MultipleChoice","stepTitle":"About 95% of the $$y$$ values lie between what two values? These values are $$159.68$$ and $$185.04$$. The $$z-scores$$ are $$___$$ , respectively. Fill in the blank.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-2$$ and $$2$$","choices":["$$-1$$ and $$1$$","$$-2$$ and $$2$$","$$-3$$ and $$3$$","$$-4$$ and $$4$$"],"hints":{"DefaultPathway":[{"id":"ab846b4standard12b-h1","type":"hint","dependencies":[],"title":"Standard Deviation","text":"95% of the values should be within $$2$$ standard deviations of the mean.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard12b-h2","type":"hint","dependencies":["ab846b4standard12b-h1"],"title":"Z-scores","text":"The $$z-scores$$ that correspond to being within $$2$$ standard deviations of the mean are $$-2$$ and $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab846b4standard13","title":"Z-scores","body":"The scores on a college entrance exam have an approximate normal distribution with mean, $$\\\\mu=52$$ points and a standard deviation, $$\u03c3=11$$ points.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 The Standard Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab846b4standard13a","stepAnswer":["$$19$$ and $$85$$"],"problemType":"MultipleChoice","stepTitle":"About $$99.7\\\\%\\\\%$$ of the $$y$$ values lie between what two values? These values are $$___$$ . Fill in the blank.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$19$$ and $$85$$","choices":["$$19$$ and $$85$$","$$12$$ and $$71$$","$$23$$ and $$91$$","$$34$$ and $$91$$"],"hints":{"DefaultPathway":[{"id":"ab846b4standard13a-h1","type":"hint","dependencies":[],"title":"Standard Deviation","text":"$$99.7\\\\%\\\\%$$ of the values should be within $$3$$ standard deviations of the mean. Find what values are within $$3$$ standard deviations of the mean.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$33$$"],"dependencies":["ab846b4standard13a-h1"],"title":"Standard Deviation","text":"What is $$3$$ standard deviations?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab846b4standard13a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11$$"],"dependencies":[],"title":"Identify Value of Standard Deviation","text":"What is the standard deviation, \u03c3?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard13a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$33$$"],"dependencies":[],"title":"$$3$$ Standard Deviations","text":"What is $$3\\\\times11$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ab846b4standard13a-h3","type":"hint","dependencies":["ab846b4standard13a-h2"],"title":"Range","text":"Add and subtract this amount from the mean to get the values that fall within $$3$$ standard deviations of the mean.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$85$$"],"dependencies":["ab846b4standard13a-h3"],"title":"Add for Upper Bound","text":"What is $$52+33$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard13a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$19$$"],"dependencies":["ab846b4standard13a-h3"],"title":"Subtract for Lower Bound","text":"What is $$52-33$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab846b4standard13b","stepAnswer":["$$-3$$ and $$3$$"],"problemType":"MultipleChoice","stepTitle":"About $$99.7\\\\%$$ of the $$y$$ values lie between what two values? These values are $$153.34$$ and $$191.38$$. The $$z-scores$$ are $$___$$ , respectively. Fill in the blank.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-3$$ and $$3$$","choices":["$$-1$$ and $$1$$","$$-2$$ and $$2$$","$$-3$$ and $$3$$","$$-4$$ and $$4$$"],"hints":{"DefaultPathway":[{"id":"ab846b4standard13b-h1","type":"hint","dependencies":[],"title":"Standard Deviation","text":"$$99.7\\\\%$$ of the values should be within $$3$$ standard deviations of the mean.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard13b-h2","type":"hint","dependencies":["ab846b4standard13b-h1"],"title":"Z-scores","text":"The $$z-scores$$ that correspond to being within $$3$$ standard deviations of the mean are $$-3$$ and $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab846b4standard14","title":"Z-scores","body":"The mean height of $$15$$ to 18-year-old males from Chile from $$2009$$ to $$2010$$ was $$170$$ cm with a standard deviation of $$6.28$$ cm. Male heights are known to follow a normal distribution. Let $$X=the$$ height of a $$15$$ to 18-year-old male from Chile in $$2009$$ to $$2010$$. Then X~N(170,6.28). Suppose a $$15$$ to 18-year-old male from Chile was $$176$$ cm tall from $$2009$$ to $$2010$$. The $$z-score$$ when $$x=176$$ cm is $$z=$$ $$___$$ . This $$z-score$$ tells you that $$x=176$$ cm is $$___$$ standard deviations to the $$___$$ (right or left) of the mean $$___$$ (What is the mean?).","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 The Standard Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab846b4standard14a","stepAnswer":["$$0.96$$"],"problemType":"TextBox","stepTitle":"The $$z-score$$ when $$x=176$$ cm is $$z=$$ $$___$$ . Fill in the blank.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.96$$","hints":{"DefaultPathway":[{"id":"ab846b4standard14a-h1","type":"hint","dependencies":[],"title":"Formula","text":"The formula for the $$z-score$$ is $$z=\\\\frac{x-\u03bc}{\\\\sigma}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$176$$"],"dependencies":["ab846b4standard14a-h1"],"title":"Identify $$x$$","text":"What is $$x$$ in this scenario?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard14a-h3","type":"hint","dependencies":["ab846b4standard14a-h2"],"title":"Z-score","text":"Find the $$z-score$$ of $$x$$, when $$x=176$$ and X~N(170,6.28) as given in the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard14a-h4","type":"hint","dependencies":["ab846b4standard14a-h3"],"title":"Identify \u03bc and \u03c3","text":"We know X~N(\u03bc, \u03c3). In this problem, X~N(170,6.28). We need to identify what is \u03bc and \u03c3.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard14a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$170$$"],"dependencies":["ab846b4standard14a-h4"],"title":"Identify \u03bc","text":"The first argument in X~N(170,6.28) is \u03bc. What is \u03bc?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard14a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6.28$$"],"dependencies":["ab846b4standard14a-h4"],"title":"Identify \u03c3","text":"The second argument in X~N(170,6.28) is \u03c3. What is \u03c3?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard14a-h7","type":"hint","dependencies":["ab846b4standard14a-h5","ab846b4standard14a-h6"],"title":"Calculate $$z-score$$","text":"The formula for the $$z-score$$ is $$z=\\\\frac{x-\u03bc}{\\\\sigma}$$. Substitute the variables in the formula for numerical values, and solve for $$z$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard14a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$z=\\\\frac{176-170}{6.28}$$"],"dependencies":["ab846b4standard14a-h7"],"title":"Substitution","text":"We know $$x=176$$, $$\u03bc=170$$, and $$\u03c3=6.28$$. What is $$z$$ after substituting these variables?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$z=\\\\frac{176-170}{6.28}$$","$$z=\\\\frac{170-6.28}{176}$$","$$z=\\\\frac{176-6.28}{170}$$","$$z=\\\\frac{170-176}{6.28}$$"]},{"id":"ab846b4standard14a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.96$$"],"dependencies":["ab846b4standard14a-h8"],"title":"Solve for $$z$$","text":"What is $$\\\\frac{176-170}{6.28}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab846b4standard14b","stepAnswer":["$$0.32$$"],"problemType":"TextBox","stepTitle":"This $$z-score$$ tells you that $$x=176$$ is $$___$$ standard deviations from the mean. Fill in the blank.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.32$$","hints":{"DefaultPathway":[{"id":"ab846b4standard14b-h1","type":"hint","dependencies":[],"title":"Identify the $$z-score$$.","text":"The absolute value of the $$z-score$$ is how many standard deviations our $$x$$ value is from the mean. Identify the absolute value of the $$z-score$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard14b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.96$$"],"dependencies":["ab846b4standard14b-h1"],"title":"Z-score","text":"In the problem, we solved for the $$z-score$$ when $$x=176$$. What is the $$z-score$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard14b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.96$$"],"dependencies":["ab846b4standard14b-h2"],"title":"Absolute Value","text":"What is the absolute value of the $$z-score$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab846b4standard14c","stepAnswer":["right"],"problemType":"MultipleChoice","stepTitle":"This $$z-score$$ tells you that $$x=176$$ is $$___$$ standard deviations to the $$___$$ (right or left) of the mean. Fill in the second blank.","stepBody":"","answerType":"string","variabilization":{},"choices":["Left","Right","right"],"hints":{"DefaultPathway":[{"id":"ab846b4standard14c-h1","type":"hint","dependencies":[],"title":"Determining right or left","text":"When $$z$$ is positive, $$x$$ is to the right of the mean, \u03bc. When $$z$$ is negative, $$x$$ is to the left of \u03bc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard14c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.96$$"],"dependencies":["ab846b4standard14c-h1"],"title":"Z-score","text":"What is the value of $$z$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard14c-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["positive"],"dependencies":["ab846b4standard14c-h2"],"title":"Positive or negative","text":"Is this positive or negative?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Negative","Positive"]}]}},{"id":"ab846b4standard14d","stepAnswer":["$$170$$"],"problemType":"TextBox","stepTitle":"Suppose a $$15$$ to 18-year-old male from Chile was $$168$$ cm tall from $$2009$$ to $$2010$$. This $$z-score$$ tells you that $$x=168$$ is $$___$$ standard deviations to the $$___$$ (right or left) of the mean $$___$$ (What is the mean?). Fill in the third blank.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$170$$","hints":{"DefaultPathway":[{"id":"ab846b4standard14d-h1","type":"hint","dependencies":[],"title":"Identify \u03bc","text":"We know X~N(\u03bc, \u03c3). In this problem, X~N(170,6.28). Identify what is the mean, \u03bc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard14d-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$170$$"],"dependencies":["ab846b4standard14d-h1"],"title":"Identify \u03bc","text":"The first argument in X~N(170,6.28) is \u03bc. What is \u03bc?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab846b4standard15","title":"Z-scores","body":"The mean height of $$15$$ to 18-year-old males from Chile from $$2009$$ to $$2010$$ was $$170$$ cm with a standard deviation of $$6.28$$ cm. Male heights are known to follow a normal distribution. Let $$X=the$$ height of a $$15$$ to 18-year-old male from Chile in $$2009$$ to $$2010$$. Then X~N(170, $$6.28)$$. Suppose that the height of a $$15$$ to 18-year-old male from Chile from $$2009$$ to $$2010$$ has a $$z-score$$ of $$z=-2$$. What is the male\u2019s height? The $$z-score$$ $$(z=-2)$$ tells you that the male\u2019s height is $$___$$ standard deviations to the $$___$$ (right or left) of the mean.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 The Standard Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab846b4standard15a","stepAnswer":["$$157.44$$"],"problemType":"TextBox","stepTitle":"What is the male\'s height?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$157.44$$","hints":{"DefaultPathway":[{"id":"ab846b4standard15a-h1","type":"hint","dependencies":[],"title":"Standard Deviation","text":"We know that the male\'s height has a $$z-score$$ of $$-2$$. The absolute value of the $$z-score$$ is how many standard deviations our $$x$$ value is from the mean. Identify the absolute value of the $$z-score$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ab846b4standard15a-h1"],"title":"Standard Deviation","text":"What is the absolute value of -2?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard15a-h3","type":"hint","dependencies":["ab846b4standard15a-h2"],"title":"Values Within the Standard Deviation","text":"Find what values are within $$2$$ standard deviations of the mean.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12.56$$"],"dependencies":["ab846b4standard15a-h3"],"title":"Standard Deviation","text":"What is $$2$$ standard deviations?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard15a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6.28$$"],"dependencies":["ab846b4standard15a-h4"],"title":"Identify Value of Standard Deviation","text":"We know X~N(\u03bc, \u03c3). In this problem, X~N(170, $$6.28)$$. What is the standard deviation, \u03c3?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard15a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12.56$$"],"dependencies":["ab846b4standard15a-h5"],"title":"$$1.27$$ Standard Deviations","text":"Find what $$2$$ standard deviations from the mean is. What is $$2\\\\times6.28$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard15a-h7","type":"hint","dependencies":["ab846b4standard15a-h6"],"title":"Direction","text":"When $$z$$ is positive, $$x$$, the male\'s height, is to the right of the mean, \u03bc. When $$z$$ is negative, $$x$$ is to the left of \u03bc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard15a-h8","type":"hint","dependencies":["ab846b4standard15a-h7"],"title":"Direction","text":"Since $$-2$$ is negative, $$x$$ is left of the mean. What is $$2$$ standard deviations to the left of the mean?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard15a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$157.44$$"],"dependencies":["ab846b4standard15a-h8"],"title":"$$2$$ standard deviations left of the mean","text":"What is $$170-12.56$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab846b4standard15b","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"The $$z-score$$ $$(z=-2)$$ tells you that the male\u2019s height is $$___$$ standard deviations from the mean. 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Fill in the second blank.","stepBody":"","answerType":"string","variabilization":{},"choices":["Left","Right","left"],"hints":{"DefaultPathway":[{"id":"ab846b4standard15c-h1","type":"hint","dependencies":[],"title":"Determining right or left","text":"When $$z$$ is positive, $$x$$, the male\'s height, is to the right of the mean, \u03bc. When $$z$$ is negative, $$x$$ is to the left of \u03bc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard15c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["ab846b4standard15c-h1"],"title":"Z-score","text":"What is the value of $$z$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard15c-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["negative"],"dependencies":["ab846b4standard15c-h2"],"title":"Positive or negative","text":"Is this positive or negative?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Negative","Positive"]}]}}]},{"id":"ab846b4standard2","title":"Z-scores","body":"Some doctors believe that a person can lose five pounds, on the average, in a month by reducing his or her fat intake and by exercising consistently. Suppose weight loss has a normal distribution. Let $$X=the$$ amount of weight lost (in pounds) by a person in a month. Use a standard deviation of two pounds. $$X~N(5,2)$$. Fill in the blanks. Suppose a person lost ten pounds in a month. The $$z-score$$ when $$x=10$$ pounds is $$z=2.5$$ (verify). This $$z-score$$ tells you that $$x=10$$ is $$___$$ standard deviations to the $$___$$ (right or left) of the mean $$___$$ (What is the mean?).","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 The Standard Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab846b4standard2a","stepAnswer":["$$2.5$$"],"problemType":"TextBox","stepTitle":"According to the $$z-score$$, $$x=10$$ is $$___$$ standard deviations from the mean. What is the answer to the blank?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2.5$$","hints":{"DefaultPathway":[{"id":"ab846b4standard2a-h1","type":"hint","dependencies":[],"title":"Identify the absolute value of the $$z-score$$.","text":"The absolute value of the $$z-score$$ is how many standard deviations our $$x$$ value is from the mean. Identify the absolute value of the $$z-score$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.5$$"],"dependencies":["ab846b4standard2a-h1"],"title":"Z-score","text":"In the problem, we are told the $$z-score$$ when $$x=10$$ pounds. What is the $$z-score$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.5$$"],"dependencies":["ab846b4standard2a-h2"],"title":"Absolute Value","text":"What is the absolute value of the $$z-score$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab846b4standard2b","stepAnswer":["Right"],"problemType":"MultipleChoice","stepTitle":"This $$z-score$$ tells you that $$x=10$$ is $$___$$ standard deviations to the $$___$$ (right or left) of the mean $$___$$ (What is the mean?). What goes into the second blank?","stepBody":"","answerType":"string","variabilization":{},"choices":["Right","Left"],"hints":{"DefaultPathway":[{"id":"ab846b4standard2b-h1","type":"hint","dependencies":[],"title":"Determining right or left","text":"When $$z$$ is positive, $$x$$ is to the right of the mean, \u03bc. When $$z$$ is negative, $$x$$ is to the left of \u03bc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard2b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.5$$"],"dependencies":["ab846b4standard2b-h1"],"title":"Z-score","text":"What is the value of $$z$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab846b4standard2b-h2-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["positive"],"dependencies":[],"title":"Positive or negative","text":"Is this positive or negative?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Negative","Positive"]}]}]}},{"id":"ab846b4standard2c","stepAnswer":["five"],"problemType":"MultipleChoice","stepTitle":"This $$z-score$$ tells you that $$x=10$$ is $$___$$ standard deviations to the $$___$$ (right or left) of the mean $$___$$ (What is the mean?). Fill in the third blank.","stepBody":"","answerType":"string","variabilization":{},"choices":["Five","Seven","Three","Two","five"],"hints":{"DefaultPathway":[{"id":"ab846b4standard2c-h1","type":"hint","dependencies":[],"title":"Identify \u03bc","text":"We know $$X~N(\u03bc,\u03c3)$$. In this problem, $$X~N(5,2)$$. Identify what is the mean, \u03bc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard2c-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["five"],"dependencies":["ab846b4standard2c-h1"],"title":"Identify \u03bc","text":"The first argument in $$N(5,2)$$ is \u03bc. What is \u03bc?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Two","Three","Five","Seven"]}]}}]},{"id":"ab846b4standard3","title":"Z-scores","body":"Some doctors believe that a person can lose five pounds, on the average, in a month by reducing his or her fat intake and by exercising consistently. Suppose weight loss has a normal distribution. Let $$X=the$$ amount of weight lost (in pounds) by a person in a month. Use a standard deviation of two pounds. $$X~N(5,2)$$. Fill in the blanks. Suppose a person lost ten pounds in a month. Suppose a person gained three pounds (a negative weight loss). Then $$z=$$ $$___$$ . This $$z-score$$ tells you that $$x=-3$$ is $$___$$ standard deviations to the $$___$$ (right or left) of the mean.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 The Standard Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab846b4standard3a","stepAnswer":["$$-3$$"],"problemType":"TextBox","stepTitle":"Suppose a person gained three pounds (a negative weight loss). Then $$z=$$ $$___$$ .","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-3$$","hints":{"DefaultPathway":[{"id":"ab846b4standard3a-h1","type":"hint","dependencies":[],"title":"Formula","text":"The formula for the $$z-score$$ is $$z=\\\\frac{x-\u03bc}{\\\\sigma}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["ab846b4standard3a-h1"],"title":"Identify $$x$$","text":"What is $$x$$ in this scenario?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab846b4standard3a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":[],"title":"Identify $$x$$","text":"How many pounds was the weight loss?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ab846b4standard3a-h3","type":"hint","dependencies":["ab846b4standard3a-h2"],"title":"Z-score","text":"Find the $$z-score$$ of $$x$$, when $$x=-3$$ and $$X~N(5,2)$$ as given in the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard3a-h4","type":"hint","dependencies":["ab846b4standard3a-h3"],"title":"Identify \u03bc and \u03c3","text":"We know $$X~N(\u03bc,\u03c3)$$. In this problem, $$X~N(5,2)$$. We need to identify what is \u03bc and \u03c3.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["ab846b4standard3a-h4"],"title":"Identify \u03bc","text":"The first argument in $$N(5,2)$$ is \u03bc. What is \u03bc?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard3a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ab846b4standard3a-h4"],"title":"Identify \u03c3","text":"The second argument in $$N(5,2)$$ is \u03c3. What is \u03c3?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard3a-h7","type":"hint","dependencies":["ab846b4standard3a-h5","ab846b4standard3a-h6"],"title":"Calculate $$z-score$$","text":"The formula for the $$z-score$$ is $$z=\\\\frac{x-\u03bc}{\\\\sigma}$$. Substitute the variables in the formula for numerical values, and solve for $$z$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard3a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$z=\\\\frac{\\\\left(-3-5\\\\right)}{2}$$"],"dependencies":["ab846b4standard3a-h7"],"title":"Substitution","text":"We know $$x=-3$$, $$\u03bc=5$$, and $$\u03c3=2$$. What is $$z$$ after substituting these variables?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$z=\\\\frac{\\\\left(-3-5\\\\right)}{2}$$","$$z=\\\\frac{5-3}{2}$$","$$z=\\\\frac{\\\\left(-5-2\\\\right)}{3}$$","$$z=\\\\frac{2-5}{3}$$"]},{"id":"ab846b4standard3a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["ab846b4standard3a-h8"],"title":"Solve for $$z$$","text":"What is $$\\\\frac{\\\\left(-3-5\\\\right)}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab846b4standard3b","stepAnswer":["four"],"problemType":"MultipleChoice","stepTitle":"Suppose a person gained three pounds (a negative weight loss). Then $$z=$$ $$___$$ . This $$z-score$$ tells you that $$x=-3$$ is $$___$$ standard deviations to the $$___$$ (right or left) of the mean. Fill in the second blank.","stepBody":"","answerType":"string","variabilization":{},"choices":["Five","Four","One","Two","four"],"hints":{"DefaultPathway":[{"id":"ab846b4standard3b-h1","type":"hint","dependencies":[],"title":"Identify the $$z-score$$.","text":"The absolute value of the $$z-score$$ is how many standard deviations our $$x$$ value is from the mean. Identify the absolute value of the $$z-score$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard3b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["ab846b4standard3b-h1"],"title":"Z-score","text":"In the problem, we solved for the $$z-score$$ when $$x=-3$$ pounds. What is the $$z-score$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard3b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["four"],"dependencies":["ab846b4standard3b-h2"],"title":"Absolute Value","text":"What is the absolute value of the $$z-score$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["One","Two","Four","Five"]}]}},{"id":"ab846b4standard3c","stepAnswer":["left"],"problemType":"MultipleChoice","stepTitle":"Suppose a person gained three pounds (a negative weight loss). Then $$z=$$ $$___$$ . This $$z-score$$ tells you that $$x=-3$$ is $$___$$ standard deviations to the $$___$$ (right or left) of the mean. Fill in the second blank. Fill in the third blank.","stepBody":"","answerType":"string","variabilization":{},"choices":["Left","Right","left"],"hints":{"DefaultPathway":[{"id":"ab846b4standard3c-h1","type":"hint","dependencies":[],"title":"Determining right or left","text":"When $$z$$ is positive, $$x$$ is to the right of the mean, \u03bc. When $$z$$ is negative, $$x$$ is to the left of \u03bc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard3c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["ab846b4standard3c-h1"],"title":"Z-score","text":"What is the value of $$z$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard3c-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["negative"],"dependencies":["ab846b4standard3c-h2"],"title":"Positive or negative","text":"Is this positive or negative?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Negative","Positive"]}]}}]},{"id":"ab846b4standard4","title":"Z-scores","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 The Standard Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab846b4standard4a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"Suppose the random variables X and Y have the following normal distributions: $$X~N(5,6)$$ and $$Y~N(2,1)$$. If $$x=17$$, then $$z=2$$. (This was previously shown.) If $$y=4$$, what is $$z$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"ab846b4standard4a-h1","type":"hint","dependencies":[],"title":"Identify \u03bc and \u03c3","text":"We know $$Y~N(\u03bc,\u03c3)$$. In this problem, $$Y~N(2,1)$$. The first step would be to identify what is \u03bc and \u03c3.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ab846b4standard4a-h1"],"title":"Identify \u03bc","text":"The first argument in $$N(2,1)$$ is \u03bc. What is \u03bc?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ab846b4standard4a-h1"],"title":"Identify \u03c3","text":"The second argument in $$N(2,1)$$ is \u03c3. What is \u03c3?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard4a-h4","type":"hint","dependencies":["ab846b4standard4a-h2","ab846b4standard4a-h3"],"title":"Calculate $$z-score$$","text":"The formula for the $$z-score$$ is $$z=\\\\frac{x-\u03bc}{\\\\sigma}$$. Substitute the variables in the formula for numerical values, and solve for $$z$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard4a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$z=\\\\frac{4-2}{1}$$"],"dependencies":["ab846b4standard4a-h4"],"title":"Substitution","text":"We know $$y=4$$, $$\u03bc=2$$, and $$\u03c3=1$$. What is $$z$$ after substituting these variables?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$z=\\\\frac{4-2}{1}$$","$$z=\\\\frac{4-1}{2}$$","$$z=\\\\frac{2-1}{4}$$","$$z=\\\\frac{1-4}{2}$$"]},{"id":"ab846b4standard4a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ab846b4standard4a-h5"],"title":"Solve for $$z$$","text":"What is $$\\\\frac{4-2}{1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab846b4standard5","title":"Z-scores","body":"From $$1984$$ to $$1985$$, the mean height of $$15$$ to 18-year-old males from Chile was $$172.36$$ cm, and the standard deviation was $$6.34$$ cm. Let $$Y=the$$ height of $$15$$ to 18-year-old males in $$1984$$ to $$1985$$. Then Y~N(172.36,6.34).","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 The Standard Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab846b4standard5a","stepAnswer":["$$166.02$$ and $$178.7$$"],"problemType":"MultipleChoice","stepTitle":"About 68% of the $$y$$ values lie between what two values? These values are $$___$$ . Fill in the blank.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$166.02$$ and $$178.7$$","choices":["$$166.02$$ and $$178.7$$","$$163.76$$ and $$172.1$$","$$112.2$$ and $$183.21$$","$$170.42$$ and $$198.82$$"],"hints":{"DefaultPathway":[{"id":"ab846b4standard5a-h1","type":"hint","dependencies":[],"title":"Standard Deviation","text":"68% of the values should be within $$1$$ standard deviation of the mean. Find what values are within $$1$$ standard deviation of the mean.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6.34$$"],"dependencies":["ab846b4standard5a-h1"],"title":"Standard Deviation","text":"What is $$1$$ standard deviation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab846b4standard5a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6.34$$"],"dependencies":[],"title":"Identify Value of Standard Deviation","text":"We know $$Y~N(\u03bc,\u03c3)$$. In this problem, Y~N(172.36,6.34). Identify what is the standard deviation, \u03c3.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ab846b4standard5a-h3","type":"hint","dependencies":["ab846b4standard5a-h2"],"title":"Range","text":"Add and subtract this amount from the mean to get the values that fall within $$1$$ standard deviation of the mean.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$178.7$$"],"dependencies":["ab846b4standard5a-h3"],"title":"Add for Upper Bound","text":"What is $$172.36+6.34$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$166.02$$"],"dependencies":["ab846b4standard5a-h3"],"title":"Subtract for Lower Bound","text":"What is $$172.36-6.34$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab846b4standard5b","stepAnswer":["$$-1$$ and $$1$$"],"problemType":"MultipleChoice","stepTitle":"About 68% of the $$y$$ values lie between what two values? These values are $$166.02$$ and $$178.7$$. The $$z-scores$$ are $$___$$ , respectively. Fill in the blank.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-1$$ and $$1$$","choices":["$$-1$$ and $$1$$","$$-2$$ and $$2$$","$$-3$$ and $$3$$","$$-4$$ and $$4$$"],"hints":{"DefaultPathway":[{"id":"ab846b4standard5b-h1","type":"hint","dependencies":[],"title":"Standard Deviation","text":"68% of the values should be within $$1$$ standard deviation of the mean.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard5b-h2","type":"hint","dependencies":["ab846b4standard5b-h1"],"title":"Z-scores","text":"The $$z-scores$$ that correspond to being within $$1$$ standard deviation of the mean are $$-1$$ and $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab846b4standard6","title":"Z-scores","body":"From $$1984$$ to $$1985$$, the mean height of $$15$$ to 18-year-old males from Chile was $$172.36$$ cm, and the standard deviation was $$6.34$$ cm. Let $$Y=the$$ height of $$15$$ to 18-year-old males in $$1984$$ to $$1985$$. Then Y~N(172.36,6.34).","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 The Standard Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab846b4standard6a","stepAnswer":["$$159.68$$ and $$185.04$$"],"problemType":"MultipleChoice","stepTitle":"About 95% of the $$y$$ values lie between what two values? These values are $$___$$ . Fill in the blank.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$159.68$$ and $$185.04$$","choices":["$$159.68$$ and $$185.04$$","$$134.12$$ and $$187.81$$","$$125.3$$ and $$192.45$$","$$156.34$$ and $$183.46$$"],"hints":{"DefaultPathway":[{"id":"ab846b4standard6a-h1","type":"hint","dependencies":[],"title":"Standard Deviation","text":"95% of the values should be within $$2$$ standard deviations of the mean. Find what values are within $$2$$ standard deviations of the mean.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12.68$$"],"dependencies":["ab846b4standard6a-h1"],"title":"Standard Deviation","text":"What is $$2$$ standard deviations?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab846b4standard6a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6.34$$"],"dependencies":[],"title":"Identify Value of Standard Deviation","text":"We know Y~N(\u03bc, \u03c3). In this problem, Y~N(172.36,6.34). Identify what is the standard deviation, \u03c3.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard6a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12.68$$"],"dependencies":[],"title":"$$2$$ Standard Deviations","text":"What is $$2\\\\times6.34$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ab846b4standard6a-h3","type":"hint","dependencies":["ab846b4standard6a-h2"],"title":"Range","text":"Add and subtract this amount from the mean to get the values that fall within $$2$$ standard deviations of the mean.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$159.68$$"],"dependencies":["ab846b4standard6a-h3"],"title":"Add for Upper Bound","text":"What is $$172.36+12.68$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$185.04$$"],"dependencies":["ab846b4standard6a-h3"],"title":"Subtract for Lower Bound","text":"What is $$172.36-12.68$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab846b4standard6b","stepAnswer":["$$-2$$ and $$2$$"],"problemType":"MultipleChoice","stepTitle":"About 95% of the $$y$$ values lie between what two values? These values are $$159.68$$ and $$185.04$$. The $$z-scores$$ are $$___$$ , respectively. Fill in the blank.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-2$$ and $$2$$","choices":["$$-1$$ and $$1$$","$$-2$$ and $$2$$","$$-3$$ and $$3$$","$$-4$$ and $$4$$"],"hints":{"DefaultPathway":[{"id":"ab846b4standard6b-h1","type":"hint","dependencies":[],"title":"Standard Deviation","text":"95% of the values should be within $$2$$ standard deviations of the mean.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard6b-h2","type":"hint","dependencies":["ab846b4standard6b-h1"],"title":"Z-scores","text":"The $$z-scores$$ that correspond to being within $$2$$ standard deviations of the mean are $$-2$$ and $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab846b4standard7","title":"Z-scores","body":"From $$1984$$ to $$1985$$, the mean height of $$15$$ to 18-year-old males from Chile was $$172.36$$ cm, and the standard deviation was $$6.34$$ cm. Let $$Y=the$$ height of $$15$$ to 18-year-old males in $$1984$$ to $$1985$$. Then Y~N(172.36,6.34).","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 The Standard Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab846b4standard7a","stepAnswer":["$$153.34$$ and $$191.38$$"],"problemType":"MultipleChoice","stepTitle":"About $$99.7\\\\%\\\\%$$ of the $$y$$ values lie between what two values? These values are $$___$$ . Fill in the blank.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$153.34$$ and $$191.38$$","choices":["$$153.34$$ and $$191.38$$","$$123.4$$ and $$187.3$$","$$154.82$$ and $$174.82$$","$$142.54$$ and $$187.39$$"],"hints":{"DefaultPathway":[{"id":"ab846b4standard7a-h1","type":"hint","dependencies":[],"title":"Standard Deviation","text":"$$99.7\\\\%\\\\%$$ of the values should be within $$3$$ standard deviations of the mean. Find what values are within $$3$$ standard deviations of the mean.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$19.02$$"],"dependencies":["ab846b4standard7a-h1"],"title":"Standard Deviation","text":"What is $$3$$ standard deviations?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ab846b4standard7a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6.34$$"],"dependencies":[],"title":"Identify Value of Standard Deviation","text":"We know $$Y~N(\u03bc,\u03c3)$$. In this problem, Y~N(172.36,6.34). Identify what is the standard deviation, \u03c3.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard7a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$19.02$$"],"dependencies":[],"title":"$$3$$ Standard Deviations","text":"What is $$3\\\\times6.34$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ab846b4standard7a-h3","type":"hint","dependencies":["ab846b4standard7a-h2"],"title":"Range","text":"Add and subtract this amount from the mean to get the values that fall within $$3$$ standard deviations of the mean.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$153.34$$"],"dependencies":["ab846b4standard7a-h3"],"title":"Add for Upper Bound","text":"What is $$172.36+19.02$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard7a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$191.38$$"],"dependencies":["ab846b4standard7a-h3"],"title":"Subtract for Lower Bound","text":"What is $$172.36-19.02$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab846b4standard7b","stepAnswer":["$$-3$$ and $$3$$"],"problemType":"MultipleChoice","stepTitle":"About $$99.7\\\\%$$ of the $$y$$ values lie between what two values? These values are $$153.34$$ and $$191.38$$. The $$z-scores$$ are $$___$$ , respectively. Fill in the blank.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-3$$ and $$3$$","choices":["$$-1$$ and $$1$$","$$-2$$ and $$2$$","$$-3$$ and $$3$$","$$-4$$ and $$4$$"],"hints":{"DefaultPathway":[{"id":"ab846b4standard7b-h1","type":"hint","dependencies":[],"title":"Standard Deviation","text":"$$99.7\\\\%$$ of the values should be within $$3$$ standard deviations of the mean.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard7b-h2","type":"hint","dependencies":["ab846b4standard7b-h1"],"title":"Z-scores","text":"The $$z-scores$$ that correspond to being within $$3$$ standard deviations of the mean are $$-3$$ and $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab846b4standard8","title":"Z-scores","body":"Jerome averages $$16$$ points a game with a standard deviation of four points. $$X~N(16,4)$$. Suppose Jerome scores ten points in a game. The $$z-score$$ when $$x=10$$ is $$-1.5$$. This score tells you that $$x=10$$ is $$___$$ standard deviations to the $$___$$ (right or left) of the mean $$___$$ (What is the mean?).","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 The Standard Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab846b4standard8a","stepAnswer":["$$1.5$$"],"problemType":"TextBox","stepTitle":"The $$z-score$$ when $$x=10$$ is $$-1.5$$. This score tells you that $$x=10$$ is $$___$$ standard deviations. What is the answer to the blank?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1.5$$","hints":{"DefaultPathway":[{"id":"ab846b4standard8a-h1","type":"hint","dependencies":[],"title":"Identify the absolute value of the $$z-score$$.","text":"The absolute value of the $$z-score$$ is how many standard deviations our $$x$$ value is from the mean. Identify the absolute value of the $$z-score$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1.5$$"],"dependencies":["ab846b4standard8a-h1"],"title":"Z-score","text":"In the problem, we are told the $$z-score$$ when $$x=10$$. What is the $$z-score$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.5$$"],"dependencies":["ab846b4standard8a-h2"],"title":"Absolute Value","text":"What is the absolute value of the $$z-score$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab846b4standard8b","stepAnswer":["left"],"problemType":"MultipleChoice","stepTitle":"The $$z-score$$ when $$x=10$$ is $$-1.5$$. This score tells you that $$x=10$$ is $$___$$ standard deviations to the $$___$$ (right or left) of the mean $$___$$ (What is the mean?). What goes into the second blank?","stepBody":"","answerType":"string","variabilization":{},"choices":["Left","Right","left"],"hints":{"DefaultPathway":[{"id":"ab846b4standard8b-h1","type":"hint","dependencies":[],"title":"Determining right or left","text":"When $$z$$ is positive, $$x$$ is to the right of the mean, \u03bc. When $$z$$ is negative, $$x$$ is to the left of \u03bc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard8b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1.5$$"],"dependencies":["ab846b4standard8b-h1"],"title":"Z-score","text":"What is the value of $$z$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard8b-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["negative"],"dependencies":[],"title":"Positive or negative","text":"Is this positive or negative?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Negative","Positive"]}]}},{"id":"ab846b4standard8c","stepAnswer":["$$16$$"],"problemType":"TextBox","stepTitle":"The $$z-score$$ when $$x=10$$ is $$-1.5$$. This score tells you that $$x=10$$ is $$___$$ standard deviations to the $$___$$ (right or left) of the mean $$___$$ (What is the mean?). Fill in the third blank.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16$$","hints":{"DefaultPathway":[{"id":"ab846b4standard8c-h1","type":"hint","dependencies":[],"title":"Identify \u03bc","text":"We know $$X~N(\u03bc,\u03c3)$$. In this problem, $$X~N(16,4)$$. Identify what is the mean, \u03bc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard8c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["ab846b4standard8c-h1"],"title":"Identify \u03bc","text":"The first argument in $$N(16,4)$$ is \u03bc. What is \u03bc?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab846b4standard9","title":"Z-scores","body":"The mean height of $$15$$ to 18-year-old males from Chile from $$2009$$ to $$2010$$ was $$170$$ cm with a standard deviation of $$6.28$$ cm. Male heights are known to follow a normal distribution. Let $$X=the$$ height of a $$15$$ to 18-year-old male from Chile in $$2009$$ to $$2010$$. Then X~N(170, $$6.28)$$. Suppose a $$15$$ to 18-year-old male from Chile was $$168$$ cm tall from $$2009$$ to $$2010$$. The $$z-score$$ when $$x=168$$ cm is $$z=$$ $$___$$ . This $$z-score$$ tells you that $$x=168$$ is $$___$$ standard deviations to the $$___$$ (right or left) of the mean $$___$$ (What is the mean?).","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 The Standard Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ab846b4standard9a","stepAnswer":["$$-0.32$$"],"problemType":"TextBox","stepTitle":"The $$z-score$$ when $$x=168$$ cm is $$z=$$ $$___$$ . Fill in the blank.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-0.32$$","hints":{"DefaultPathway":[{"id":"ab846b4standard9a-h1","type":"hint","dependencies":[],"title":"Formula","text":"The formula for the $$z-score$$ is $$z=\\\\frac{x-\u03bc}{\\\\sigma}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$168$$"],"dependencies":["ab846b4standard9a-h1"],"title":"Identify $$x$$","text":"What is $$x$$ in this scenario?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard9a-h3","type":"hint","dependencies":["ab846b4standard9a-h2"],"title":"Z-score","text":"Find the $$z-score$$ of $$x$$, when $$x=168$$ and X~N(170,6.28) as given in the question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard9a-h4","type":"hint","dependencies":["ab846b4standard9a-h3"],"title":"Identify \u03bc and \u03c3","text":"We know X~N(\u03bc, \u03c3). In this problem, X~N(170,6.28). We need to identify what is \u03bc and \u03c3.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard9a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$170$$"],"dependencies":["ab846b4standard9a-h4"],"title":"Identify \u03bc","text":"The first argument in X~N(170,6.28) is \u03bc. What is \u03bc?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard9a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6.28$$"],"dependencies":["ab846b4standard9a-h4"],"title":"Identify \u03c3","text":"The second argument in X~N(170,6.28) is \u03c3. What is \u03c3?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard9a-h7","type":"hint","dependencies":["ab846b4standard9a-h5","ab846b4standard9a-h6"],"title":"Calculate $$z-score$$","text":"The formula for the $$z-score$$ is $$z=\\\\frac{x-\u03bc}{\\\\sigma}$$. Substitute the variables in the formula for numerical values, and solve for $$z$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard9a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$z=\\\\frac{168-170}{6.28}$$"],"dependencies":["ab846b4standard9a-h7"],"title":"Substitution","text":"We know $$x=168$$, $$\u03bc=170$$, and $$\u03c3=6.28$$. What is $$z$$ after substituting these variables?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$z=\\\\frac{168-170}{6.28}$$","$$z=\\\\frac{170-6.28}{168}$$","$$z=\\\\frac{168-6.28}{170}$$","$$z=\\\\frac{170-168}{6.28}$$"]},{"id":"ab846b4standard9a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-0.32$$"],"dependencies":["ab846b4standard9a-h8"],"title":"Solve for $$z$$","text":"What is $$\\\\frac{168-170}{6.28}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab846b4standard9b","stepAnswer":["$$0.32$$"],"problemType":"TextBox","stepTitle":"The $$z-score$$ when $$x=168$$ cm is $$z=$$ $$___$$ . This $$z-score$$ tells you that $$x=168$$ is $$___$$ standard deviations to the $$___$$ (right or left) of the mean $$___$$ (What is the mean?). Fill in the second blank.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.32$$","hints":{"DefaultPathway":[{"id":"ab846b4standard9b-h1","type":"hint","dependencies":[],"title":"Identify the $$z-score$$.","text":"The absolute value of the $$z-score$$ is how many standard deviations our $$x$$ value is from the mean. Identify the absolute value of the $$z-score$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard9b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-0.32$$"],"dependencies":["ab846b4standard9b-h1"],"title":"Z-score","text":"In the problem, we solved for the $$z-score$$ when $$x=168$$. What is the $$z-score$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard9b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.32$$"],"dependencies":["ab846b4standard9b-h2"],"title":"Absolute Value","text":"What is the absolute value of the $$z-score$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab846b4standard9c","stepAnswer":["left"],"problemType":"MultipleChoice","stepTitle":"The $$z-score$$ when $$x=168$$ cm is $$z=$$ $$___$$ . This $$z-score$$ tells you that $$x=168$$ is $$___$$ standard deviations to the $$___$$ (right or left) of the mean $$___$$ (What is the mean?). Fill in the third blank.","stepBody":"","answerType":"string","variabilization":{},"choices":["Left","Right","left"],"hints":{"DefaultPathway":[{"id":"ab846b4standard9c-h1","type":"hint","dependencies":[],"title":"Determining right or left","text":"When $$z$$ is positive, $$x$$ is to the right of the mean, \u03bc. When $$z$$ is negative, $$x$$ is to the left of \u03bc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard9c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-0.32$$"],"dependencies":["ab846b4standard9c-h1"],"title":"Z-score","text":"What is the value of $$z$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard9c-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["negative"],"dependencies":["ab846b4standard9c-h2"],"title":"Positive or negative","text":"Is this positive or negative?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Negative","Positive"]}]}},{"id":"ab846b4standard9d","stepAnswer":["$$170$$"],"problemType":"TextBox","stepTitle":"The $$z-score$$ when $$x=168$$ cm is $$z=$$ $$___$$ . This $$z-score$$ tells you that $$x=168$$ is $$___$$ standard deviations to the $$___$$ (right or left) of the mean $$___$$ (What is the mean?). Fill in the fourth blank.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$170$$","hints":{"DefaultPathway":[{"id":"ab846b4standard9d-h1","type":"hint","dependencies":[],"title":"Identify \u03bc","text":"We know X~N(\u03bc, \u03c3). In this problem, X~N(170,6.28). Identify what is the mean, \u03bc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab846b4standard9d-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$170$$"],"dependencies":["ab846b4standard9d-h1"],"title":"Identify \u03bc","text":"The first argument in X~N(170,6.28) is \u03bc. What is \u03bc?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8934ethreevar10","title":"Solving a System of Three Equations in Three Variables by Gaussian Elimination","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Systems of Linear Equations: Three Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8934ethreevar10a","stepAnswer":["$$\\\\frac{4}{7}-\\\\frac{1}{7}-\\\\frac{3}{7}$$"],"problemType":"MultipleChoice","stepTitle":"Solve the system of equations by elimination: $$x+y+z=0$$, $$2x-y+3z=0$$, $$x-z=1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{4}{7}-\\\\frac{1}{7}-\\\\frac{3}{7}$$","choices":["(1,1,1)","$$\\\\frac{4}{7}-\\\\frac{1}{7}-\\\\frac{3}{7}$$","(0,0,0)","$$\\\\frac{4}{7}-\\\\frac{1}{7}-\\\\frac{3}{7}$$"],"hints":{"DefaultPathway":[{"id":"ab8934ethreevar10a-h1","type":"hint","dependencies":[],"title":"Solving for one variable","text":"The first step is to pick two equations and solve for one variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar10a-h2","type":"hint","dependencies":["ab8934ethreevar10a-h1"],"title":"Solving for one variable","text":"The next step is to pick another pair of equations and solve for the same variable","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar10a-h3","type":"hint","dependencies":["ab8934ethreevar10a-h2"],"title":"Solving the resulting system of two equations","text":"Using the resulting equations, sovle the system of two equations to find the values of those variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar10a-h4","type":"hint","dependencies":["ab8934ethreevar10a-h3"],"title":"Back-substitution","text":"Back-substitute the known variables into any one of the original equations and solve for the missing variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8934ethreevar11","title":"Determining Whether an Ordered Triple Is a Solution to a System","body":"Determine whether $$(3, -2, 1)$$ is a solution to the following system","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Systems of Linear Equations: Three Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8934ethreevar11a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"x+y+z=2,6x-4y+5z=31,5x+2y+2x=13.","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ab8934ethreevar11a-h1","type":"hint","dependencies":[],"title":"Plugging in the Ordered Triplet","text":"We must plug in the values $$x=3, y=-2$$, and $$z=1$$ into the system of equations.\\\\n$$3-2+1=2$$. This is true.\\\\n6(3)-4(-2)+5(1)=31. This is also true.\\\\n5(3)+2(-2)+2(1)=13. This is also true. This meaens that the ordered pair is a solution and the answer is \'y\'.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8934ethreevar12","title":"Solving a System of Three Equations in Three Variables by Elimination","body":"Find a solution to the following system.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Systems of Linear Equations: Three Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8934ethreevar12a","stepAnswer":["$$(1, -1, 2)$$"],"problemType":"MultipleChoice","stepTitle":"x-2y+3z=9,-x+3y-z=-6,2x-5y+5z=17","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(1, -1, 2)$$","choices":["$$(1, -1, 3)$$","(1,1,2)","$$(1, -1, 2)$$"],"hints":{"DefaultPathway":[{"id":"ab8934ethreevar12a-h1","type":"hint","dependencies":[],"title":"Eliminate the X Variable","text":"We can eliminate $$x$$ to add the first two equations to get $$y+2z=3$$. We must however, find another equation without $$x$$ in it. To do this, we must multiply equation $$1$$ by $$-2$$ and add the result to equation $$3$$. This will result in the equation $$-y-z=-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar12a-h2","type":"hint","dependencies":["ab8934ethreevar12a-h1"],"title":"Solving for $$x$$, $$y$$, and $$z$$","text":"We can solve for $$z$$ by adding the two equations derived in the previous step: $$y+2z=3$$ and $$-y-z=-1$$. The sum is $$z=2$$. We can now back-substitute $$z$$ into $$y+2z=3$$ to get $$y=-1$$. Finally, this can all be plugged back into equation $$1$$ to yield the solution for $$x$$. $$x+2+6=9$$, $$x=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8934ethreevar13","title":"Solving a Real-World Problen Using a System of Three Equations in Three Variables","body":"In the problem posed at the beginning of the section, John invested his inheritance of $12,000 in three different funds: part in a money-market fund paying 3% interest annually; part in municipal bonds paying 4% annually; and the rest in mutual funds paying 7% annually. John invested $4,000 more in mutual funds than he invested in municipal bonds. The total interest earned in one year was $670.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Systems of Linear Equations: Three Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8934ethreevar13a","stepAnswer":["(2000, $$3000$$, 7000)"],"problemType":"MultipleChoice","stepTitle":"How much did he invest in each type of fund?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"(2000, $$3000$$, 7000)","choices":["(2000, $$4000$$, 6000)","(1000, $$3000$$, 7000)","(2000, $$3000$$, 7000)"],"hints":{"DefaultPathway":[{"id":"ab8934ethreevar13a-h1","type":"hint","dependencies":[],"title":"Setting up Equations","text":"To solve this problem, we use all of the information given and set up three equations. First, we assign a variable to each of the three investment amounts: $$x=money_{market}$$ fund, $$y=municipal$$ bonds, $$z=mutual$$ funds. With these variables, we can create the three following equations based on the information given in the word problem: x+y+z=12000,z=y+4000,0.03x+0.04y+0.07z=670. These three equations make a system. To make the caculations easier, the last equation can be multipled by $$100$$, giving us $$3x+4y+7z=67000$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar13a-h2","type":"hint","dependencies":["ab8934ethreevar13a-h1"],"title":"Solving for $$x$$, $$y$$, and $$z$$","text":"To solve this equation, we must first multiply equation $$1$$ by $$-3$$, giving us $$y+4z=31000$$. We can now add equation $$2$$ to that derived in the eprevious step to give us $$5z=35000$$, $$z=7000$$. With $$z$$, we can solve for y: we plug back in to equation $$2$$ and solve for $$y$$. $$-y+7000=4000, y=3000$$. Finally, we plug $$z$$ and $$y$$ into equation $$1$$ to get $$x+3000+7000=12000$$, leading to $$x=2000$$. Thus, we have the soluttion nas (2000,3000,7000)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8934ethreevar14","title":"Solving a System of Three Equations in Three Variables by Elimination","body":"Solve the system of equations in three variables.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Systems of Linear Equations: Three Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8934ethreevar14a","stepAnswer":["$$(1, 1, -1)$$"],"problemType":"MultipleChoice","stepTitle":"2x+y-2z=-1,3x-3y-z=5,x-2y+3z=6","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(1, 1, -1)$$","choices":["$$(-1, 2, -1)$$","$$(1, 1, -1)$$","$$(-1, 1, -1)$$"],"hints":{"DefaultPathway":[{"id":"ab8934ethreevar14a-h1","type":"hint","dependencies":[],"title":"Eliminate the X Variable","text":"We can multiply equation $$3$$ by $$-3$$ and add to the second equation to get $$3y=10z=-13$$. This eliminates the $$x-variablee$$ in one equation, but we must do it in another as well. So, we can multiply equation $$3$$ by $$-2$$ and add to the third equation to get $$5y-8z=-13$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar14a-h2","type":"hint","dependencies":["ab8934ethreevar14a-h1"],"title":"Solving for $$x$$, $$y$$, and $$z$$","text":"With the equations derived in the previous steps, we can solve for $$z$$. We can multiply $$5y-8z=-13$$ by $$\\\\frac{-5}{3}$$ and add to $$3y-10z=-13$$ to get $$z=1$$. This can be substituted back into $$5y-8z=-13$$ to get $$y=-1$$. All this can be substituted into the first equation to get $$x=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8934ethreevar15","title":"Solving an Inconsistent System of Three Equations in Three Variables","body":"Solve the following system.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Systems of Linear Equations: Three Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8934ethreevar15a","stepAnswer":["DNE"],"problemType":"MultipleChoice","stepTitle":"x-3y+z=4,-x+2y-5z=3,5x-13y+13z=8. Enter \'DNE\' if the system is inconsistent.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$x=2, y=1, z=-1$$","$$x=1, y=1, z=-1$$","DNE"],"hints":{"DefaultPathway":[{"id":"ab8934ethreevar15a-h1","type":"hint","dependencies":[],"title":"Eliminate the X Variable","text":"We can eliminate $$x$$ by adding equation $$1$$ to equation $$2$$ to get $$-y-4z=7$$. We can then multiply equation $$1$$ by $$-5$$ and add to equation $$3$$ to get $$2y+8z=-12$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar15a-h2","type":"hint","dependencies":["ab8934ethreevar15a-h1"],"title":"Solving for $$x$$, $$y$$, and $$z$$","text":"Equation $$4$$ can be multiplied by $$2$$ and added to equation $$5$$ to try and solve for $$z$$. However, this leads to both variables $$z$$ and $$y$$ to be cancelled out, leading to the equation $$0=2$$. Since this is false, the system is inconsistent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8934ethreevar16","title":"Solving a System of Three Equations in Three Variables by Elimination","body":"Solve the following system.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Systems of Linear Equations: Three Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8934ethreevar16a","stepAnswer":["DNE"],"problemType":"MultipleChoice","stepTitle":"x+y+z=2,y=3z=1,2x+y+5z=0. Enter \'DNE\' if the system is inconsistent.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$x=3, y=1, z=-1$$","$$x=4, y=2, z=-1$$","DNE"],"hints":{"DefaultPathway":[{"id":"ab8934ethreevar16a-h1","type":"hint","dependencies":[],"title":"Eliminate the X Variable","text":"To eliminate $$x$$, multiply the first equation by $$-2$$ and add the result to the third equation. We get $$-y+3z=-4$$. If we add this to the second equation, we get $$0=-3$$, which is false. So, the system is inconsistent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8934ethreevar17","title":"Finding the Solution to a Dependent System of Equations","body":"Find the solution to the given system.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Systems of Linear Equations: Three Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8934ethreevar17a","stepAnswer":["(x,5/2x,3/2x)"],"problemType":"MultipleChoice","stepTitle":"2x+y-3z=0,4x+2y-6z=0,x-y+z=0","stepBody":"","answerType":"string","variabilization":{},"choices":["(x,2x,3x)","(x,5/2x,3/2x)","(x,2/5x,3/2x)"],"hints":{"DefaultPathway":[{"id":"ab8934ethreevar17a-h1","type":"hint","dependencies":[],"title":"Eliminating X","text":"First, we can multiply equation $$1$$ by $$-2$$ and add it to equation $$2$$, but this gives us $$0=0$$ which is always true. S","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar17a-h2","type":"hint","dependencies":["ab8934ethreevar17a-h1"],"title":"Finding the General Solution","text":"This means that we must find the general solution, or the solution in terms of $$x$$. Add equations $$1$$ and $$2$$ to get $$3x-2z=0$$. $$z=\\\\frac{3}{2} x$$. Back-substitute the expressionn for $$z$$ and solve for $$y$$. $$2x+y-\\\\frac{9}{2} x=0, y=\\\\frac{5}{2} x$$. So, the solution is (x,5/2x,3/2x)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8934ethreevar18","title":"Finding the Solution to a Dependent System of Equations","body":"Solve the following system.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Systems of Linear Equations: Three Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8934ethreevar18a","stepAnswer":["(-1/5z+18/5,-4/5z+17/5,z)"],"problemType":"MultipleChoice","stepTitle":"x+y+z=7,3x-2y-z=4,x+6y+5z=24. Write solution in terms of $$z$$ if necessary.","stepBody":"","answerType":"string","variabilization":{},"choices":["(-1/5z+18/7,-4/5z+17/5,z)","(-5z+18/5,-4/5z+17/5,z)","(-1/5z+18/5,-4/5z+17/5,z)"],"hints":{"DefaultPathway":[{"id":"ab8934ethreevar18a-h1","type":"hint","dependencies":[],"title":"Eliminating $$x$$","text":"Multiply the first equation by $$-3$$ and add the result to the second equation. This gives us $$-5y-4z=-17$$. Now, multiply thee first equation by $$-1$$ and add to the third equation to get $$5y+4z=17$$. Add these two equations together to get $$0=0$$. This system is dependent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar18a-h2","type":"hint","dependencies":["ab8934ethreevar18a-h1"],"title":"Finding a Solution in terms of Z","text":"Solve for $$y$$ in the equation $$5y+4z-17$$ to get $$y=\\\\frac{-4}{5} t+\\\\frac{17}{5}$$. Substitute this into the first equation to find $$x$$. $$x+\\\\left(-\\\\frac{4}{5} z+\\\\frac{17}{5}\\\\right)+z=7, x=\\\\frac{-1}{5} z+\\\\frac{18}{5}$$. The solution is (-1/5z+18/5,-4/5z+17/5,z)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8934ethreevar19","title":"Systems of Linear Equations","body":"For the following exercise, solve the system for $$x$$, $$y$$, and $$z$$ (indicate in coordinates)","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Systems of Linear Equations: Three Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8934ethreevar19a","stepAnswer":["(128/557, $$\\\\frac{23}{557}$$, 28/557)"],"problemType":"MultipleChoice","stepTitle":"$$5x-3y-\\\\frac{z+1}{2}=\\\\frac{1}{2}$$ | $$6x+\\\\frac{y-9}{2}+2z=-3$$ | $$\\\\frac{x+8}{2}-4y+z=4$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"(128/557, $$\\\\frac{23}{557}$$, 28/557)","choices":["(128/557, $$\\\\frac{23}{557}$$, 28/557)","(128/557, $$\\\\frac{23}{57}$$, 28/557)","(128/557, $$\\\\frac{23}{557}$$, 228/557)"],"hints":{"DefaultPathway":[{"id":"ab8934ethreevar19a-h1","type":"hint","dependencies":[],"title":"Simplifying the System","text":"Multiply all $$3$$ equations by $$2$$ to remove the fraction. Then rearrange the equation so all variables are on one side and constants on the other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$32x-11y=7$$"],"dependencies":["ab8934ethreevar19a-h1"],"title":"Breakding Down the Equations","text":"Multiply equation (1) by $$2$$ and add it to equation (2).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar19a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11x+9y=3$$"],"dependencies":["ab8934ethreevar19a-h2"],"title":"Breakding Down the Equations","text":"Subtract equation (2) from (3)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar19a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y=\\\\frac{3-11x}{9}$$"],"dependencies":["ab8934ethreevar19a-h3"],"title":"Simplified Variable Expression","text":"Using the last expression, rearrange to solve for an expression for $$y$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar19a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{128}{557}$$"],"dependencies":["ab8934ethreevar19a-h4"],"title":"Solving for $$x$$","text":"Plug-in the expression for $$y$$ into one of the two broken down equations. Then solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar19a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{23}{557}$$"],"dependencies":["ab8934ethreevar19a-h5"],"title":"Solving for $$y$$","text":"Plug-in the value for $$x$$ into one of the two broken down equations and solve for $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar19a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{28}{557}$$"],"dependencies":["ab8934ethreevar19a-h6"],"title":"Solveing for $$x$$","text":"Plug in the values for $$x$$ and $$y$$ into any of the $$3$$ original equations and solve for $$z$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8934ethreevar2","title":"Determining Whether an Ordered Triple Is a Solution to a System","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Systems of Linear Equations: Three Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8934ethreevar2a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Determine whether $$(4, 4, -1)$$ is the solution to the system of equations: $$x-y=0$$, $$x-z=5$$, $$x-y-z=-1$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ab8934ethreevar2a-h1","type":"hint","dependencies":[],"title":"Plugging in solution into the equations","text":"The first step is to plug the given point into the equations. If the equality that results is true for all the equations, then the point is the solution to the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["ab8934ethreevar2a-h1"],"title":"Evaluating the equation","text":"What is $$4-4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar2a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ab8934ethreevar2a-h2"],"title":"Determining the equality","text":"Does $$0=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ab8934ethreevar2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["ab8934ethreevar2a-h3"],"title":"Evaluating the equation","text":"What is $$4-(-1)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar2a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ab8934ethreevar2a-h4"],"title":"Determining the equality","text":"Does $$5=5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ab8934ethreevar2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["ab8934ethreevar2a-h5"],"title":"Evaluating the equation","text":"What is $$4-4+\\\\left(-1\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar2a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ab8934ethreevar2a-h6"],"title":"Determining the equality","text":"Does $$-1=-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"ab8934ethreevar20","title":"Systems of Linear Equations","body":"For the following exercise, solve the system for $$x$$, $$y$$, and $$z$$ (indicate in coordinates)","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Systems of Linear Equations: Three Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8934ethreevar20a","stepAnswer":["$$(6, -1, 0)$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{x-3}{6}+\\\\frac{y+2}{2}-\\\\frac{z-3}{3}=2$$ | $$\\\\frac{x+2}{4}+\\\\frac{y-5}{2}+\\\\frac{z+4}{2}=1$$ | $$\\\\frac{x+6}{2}-\\\\frac{y-3}{2}+z+1=9$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(6, -1, 0)$$","choices":["$$(3, -1, 0)$$","$$(6, -1, 0)$$","(2,5,1)"],"hints":{"DefaultPathway":[{"id":"ab8934ethreevar20a-h1","type":"hint","dependencies":[],"title":"Simplifying the System","text":"Multiply equation (1) by $$6$$, equation (2) by $$4$$, and equation (3) by $$12$$ to remove the fractions. Then rearrange the equation so all variables are on one side and constants on the other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar20a-h2","type":"hint","dependencies":["ab8934ethreevar20a-h1"],"title":"Breaking Down the Equations","text":"Add equation (1) and (2)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar20a-h3","type":"hint","dependencies":["ab8934ethreevar20a-h2"],"title":"Breaking Down the Equations","text":"Subtract equation (2) from (3)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar20a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["ab8934ethreevar20a-h3"],"title":"Solving for $$y$$","text":"Rearrange the last expression to isolate $$y$$. Plug $$y$$ into the first expression and solve for $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar20a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["ab8934ethreevar20a-h4"],"title":"Solving for $$x$$","text":"Taking any of the two broken down equations, plug in $$y$$ and solve for $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar20a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["ab8934ethreevar20a-h5"],"title":"Solving for $$z$$","text":"Plug in the values for $$x$$ and $$y$$ into any of the three original equations and solve for $$z$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8934ethreevar21","title":"Systems of Linear Equations","body":"Three even numbers sum up to $$108$$. The smaller is half the larger and the middle number is $$\\\\frac{3}{4}$$ the larger.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Systems of Linear Equations: Three Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8934ethreevar21a","stepAnswer":["$$24$$, $$36$$, $$48$$"],"problemType":"MultipleChoice","stepTitle":"What are the numbers?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$24$$, $$36$$, $$48$$","choices":["$$10$$, $$20$$, $$30$$","$$14$$, $$28$$, $$42$$","$$24$$, $$36$$, $$48$$"],"hints":{"DefaultPathway":[{"id":"ab8934ethreevar21a-h1","type":"hint","dependencies":[],"title":"Setting Up","text":"Set up three equations in variable form to represent the situation. Assuming $$x$$ is smaller, $$y$$ is middle and $$z$$ is larger we have $$x=0.5z$$, $$y=0.75z$$, and $$x+y+z=108$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar21a-h2","type":"hint","dependencies":["ab8934ethreevar21a-h1"],"title":"Simplifying to $$1$$ Variable","text":"Plug in the expressions for $$x$$ and $$y$$ into the equation $$x+y+z=108$$ and combines the $$z-values$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar21a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$48$$"],"dependencies":["ab8934ethreevar21a-h2"],"title":"Solving for $$z$$","text":"Using division solve for $$z$$ from the previous expression","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar21a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$36$$"],"dependencies":["ab8934ethreevar21a-h3"],"title":"Solvingn for $$y$$","text":"Plug in the value of $$z$$ into the original expression for $$y$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar21a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$24$$"],"dependencies":["ab8934ethreevar21a-h4"],"title":"Solvingn for $$x$$","text":"Plug in the value of $$z$$ into the original expression for $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8934ethreevar22","title":"Systems of Linear Equations","body":"At a family reunion, there were only blood relatives, consisting of children, parents, and grandparents, in attendance. There were $$400$$ people total. There were twice as many parents as grandparents, and $$50$$ more children than parents.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Systems of Linear Equations: Three Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8934ethreevar22a","stepAnswer":["$$190$$, $$140$$, $$70$$"],"problemType":"MultipleChoice","stepTitle":"How many children, parents, and grandparents were in attendance? (separate answer with commas in the order stated)","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$190$$, $$140$$, $$70$$","choices":["$$170$$, $$140$$, $$70$$","$$190$$, $$120$$, $$70$$","$$190$$, $$140$$, $$70$$"],"hints":{"DefaultPathway":[{"id":"ab8934ethreevar22a-h1","type":"hint","dependencies":[],"title":"Setting Up","text":"Set up three equatinos in variable form to represent the situation $$(c=children$$, $$p=parent$$, and $$g=grandparent)$$. We know that $$c+p+g=400$$, $$2g=p$$, and $$p+50=c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar22a-h2","type":"hint","dependencies":["ab8934ethreevar22a-h1"],"title":"Simplifying to $$1$$ Variable","text":"Plug in the expressions $$c=p+50$$ and $$g=0.5p$$ into $$c+p+g=400$$ so that only the variable $$p$$ is present.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar22a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$140$$"],"dependencies":["ab8934ethreevar22a-h2"],"title":"Solving for $$p$$","text":"Combine the $$p$$ variables and solve for the number of parents using algebra","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar22a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$190$$"],"dependencies":["ab8934ethreevar22a-h3"],"title":"Solving for c","text":"Plug in the value of $$p$$ into the expression $$c=p+50$$ to solve for the number of children","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar22a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$70$$"],"dependencies":["ab8934ethreevar22a-h4"],"title":"Solving for g","text":"Plug in the value of $$p$$ into the expression $$g=0.5p$$ to solve for the number of grandparents","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8934ethreevar23","title":"Systems of Linear Equations","body":"Your roommate, Sarah, offered to buy groceries for you and your other roommate. The total bill was $82. She forgot to save the individual receipts but remembered that your groceries were $$\\\\$0.05$$ cheaper than half of her groceries, and that your other roommate\u2019s groceries were $$\\\\$2.10$$ more than your groceries.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Systems of Linear Equations: Three Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8934ethreevar23a","stepAnswer":["$$\\\\$19.95$$, $40, $$\\\\$22.05$$"],"problemType":"MultipleChoice","stepTitle":"How much was each of your share of the groceries? (list your answer starting with your salary, Sarah\'s slaray and then your other roomate\'s salary)","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\$19.95$$, $40, $$\\\\$22.05$$","choices":["$$\\\\$19.85$$, $40, $$\\\\$22.05$$","$$\\\\$19.95$$, $40, $$\\\\$22.05$$","$$\\\\$19.95$$, $45, $$\\\\$22.00$$"],"hints":{"DefaultPathway":[{"id":"ab8934ethreevar23a-h1","type":"hint","dependencies":[],"title":"Setting Up","text":"Set up three equatinos in variable form to represent the situation $$(x=your$$ share, $$y=Sarah\'s$$ share, and $$z=your$$ other roommate\'s share). We have $$x+y+z=82$$, $$y=2x+0.1$$, and $$z=x+2.1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar23a-h2","type":"hint","dependencies":["ab8934ethreevar23a-h1"],"title":"Simplifying to $$1$$ Variable","text":"Plug in the expressions for $$y$$ and $$z$$ into $$x+y+z=82$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar23a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$19.95$$"],"dependencies":["ab8934ethreevar23a-h2"],"title":"Solving for $$x$$","text":"Combine the $$x$$ values and use algebra to solve for your share","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar23a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$40$$"],"dependencies":["ab8934ethreevar23a-h3"],"title":"Solving for $$y$$","text":"Plug in the value of $$x$$ into $$y=2x+0.1$$ to solve for Sarah\'s share","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar23a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$22$$"],"dependencies":["ab8934ethreevar23a-h4"],"title":"Solving for $$z$$","text":"Plug in the value of $$x$$ into $$z=x+2.1$$ to solve for your other roommate\'s share","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8934ethreevar24","title":"Systems of Linear Equations","body":"Three coworkers work for the same employer. Their jobs are warehouse manager, office manager, and truck driver. The sum of the annual salaries of the warehouse manager and office manager is $82,000. The office manager makes $4,000 more than the truck driver annually. The annual salaries of the warehouse manager and the truck driver total $78,000.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Systems of Linear Equations: Three Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8934ethreevar24a","stepAnswer":["Infinite"],"problemType":"MultipleChoice","stepTitle":"What is the annual salary of each of the co-workers? (if there are infinitely many solutions choose \\"Infinite\\"; if, there is no solution, choose \\"No Solution\\".)","stepBody":"","answerType":"string","variabilization":{},"choices":["Infinite","No Solution","Warehouse Manager: $40,000; Office Manager: $42,000; Truck Driver: $38,000"],"hints":{"DefaultPathway":[{"id":"ab8934ethreevar24a-h1","type":"hint","dependencies":[],"title":"Setting Up","text":"Set up three equations in variable form to represent the situation $$(w=warehouse$$, $$o=office$$, and $$t=truck)$$. We have $$w+o=82000$$, $$o=t+4000$$, and $$w+t=78000$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar24a-h2","type":"hint","dependencies":["ab8934ethreevar24a-h1"],"title":"Simplifying to $$1$$ Variable","text":"Plug in $$w=78000-t$$ and $$o=t+4000$$ into $$w+o=82000$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar24a-h3","type":"hint","dependencies":["ab8934ethreevar24a-h2"],"title":"Understanding the Equation","text":"Notice that the $$t$$ cancels out leaving us with $$82000=82000$$. This means there are infinitely many solutions and that more information is needed to answer the problem.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8934ethreevar25","title":"Systems of Linear Equations","body":"A local band sells out for their concert. They sell all 1,175 tickets for a total purse of $$\\\\$28, 112.50$$. The tickets were priced at $20 for student tickets, $$\\\\$22.50$$ for children, and $29 for adult tickets.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Systems of Linear Equations: Three Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8934ethreevar25a","stepAnswer":["$$500$$, $$225$$, $$250$$"],"problemType":"MultipleChoice","stepTitle":"If the band sold twice as many adult as children tickets, how many of each type was sold? (list answer in the order of students, children, and adults)","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$500$$, $$225$$, $$250$$","choices":["$$300$$, $$225$$, $$250$$","$$500$$, $$265$$, $$250$$","$$500$$, $$225$$, $$250$$"],"hints":{"DefaultPathway":[{"id":"ab8934ethreevar25a-h1","type":"hint","dependencies":[],"title":"Setting Up","text":"Set up three equatinos in variable form to represent the situation $$(c=children$$ tickets, $$s=student$$ tickets, and $$a=adult$$ tickets). We have $$c+s+a=1175$$, $$22.5c+20s+29a=28112.5$$, and $$2c=a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar25a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3c+s=1175$$"],"dependencies":["ab8934ethreevar25a-h1"],"title":"Rearrange Equation","text":"Plug in $$a=2c$$ into $$c+s+a=1175$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar25a-h3","type":"hint","dependencies":["ab8934ethreevar25a-h2"],"title":"Simplifying to $$1$$ Variable","text":"Rearrange $$3c+s=1175$$ to isolate s and plug in the expresion for s into the equation containing the total price of tickets. Plug in $$a=2c$$ into the total price of tickets equation so that the final equation only has one variable c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar25a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$500$$"],"dependencies":["ab8934ethreevar25a-h3"],"title":"Solve for c","text":"Using algebra solve for c, the number of childrens tickets sold","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar25a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$225$$"],"dependencies":["ab8934ethreevar25a-h4"],"title":"Solve for s","text":"Plug in the value of c into $$3c+s=1175$$ and solve for s, number of student tickets sold","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar25a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$250$$"],"dependencies":["ab8934ethreevar25a-h5"],"title":"Solve for $$p$$","text":"Plug in the value of c and s into any equatino containing $$p$$ to solve for the number of parent tickets sold","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8934ethreevar26","title":"Systems of Linear Equations","body":"At one time, in the United States, $$398$$ species of animals were on the endangered species list. The top groups were mammals, birds, and fish, which comprised 55% of the endangered species. Birds accounted for $$0.7\\\\%$$ more than fish, and fish accounted for $$1.5\\\\%$$ more than mammals.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Systems of Linear Equations: Three Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8934ethreevar26a","stepAnswer":["$$19.3$$, $$18.6$$, $$17.1$$"],"problemType":"MultipleChoice","stepTitle":"What percent of the endangered species came from mammals, birds, and fish? (list in order by birds, fish, and mammals in percent form)","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$19.3$$, $$18.6$$, $$17.1$$","choices":["$$19.3$$, $$18.5$$, $$17.1$$","$$19.1$$, $$18.6$$, $$17.1$$","$$19.3$$, $$18.6$$, $$17.1$$"],"hints":{"DefaultPathway":[{"id":"ab8934ethreevar26a-h1","type":"hint","dependencies":[],"title":"Setting Up","text":"Set up three equatinos in variable form to represent the situation $$(b=birds$$, $$m=mammals$$, and $$f=fishes)$$. We have $$b+f+m=398(0.05)$$, $$b=\\\\operatorname{0.07}\\\\left(398\\\\right)+f$$, and $$f=\\\\operatorname{0.015}\\\\left(398\\\\right)+m$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar26a-h2","type":"hint","dependencies":["ab8934ethreevar26a-h1"],"title":"Simplifying to $$1$$ Variable","text":"Plug in $$m=f-0.015(398)$$ and $$b=\\\\operatorname{0.07}\\\\left(398\\\\right)+f$$ into the equation containing all three variables so that only variable f is present.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar26a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18.6$$"],"dependencies":["ab8934ethreevar26a-h2"],"title":"Solving for f","text":"Combine the f variables and solve for f using algebra","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar26a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$17.1$$"],"dependencies":["ab8934ethreevar26a-h3"],"title":"Solving for $$m$$","text":"Plug in the value of f into $$m=f-0.015(398)$$ and solve for $$m$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar26a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$19.3$$"],"dependencies":["ab8934ethreevar26a-h4"],"title":"Solving for $$b$$","text":"Plug in the value of f into $$b=\\\\operatorname{0.07}\\\\left(398\\\\right)+f$$ and solve for $$b$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8934ethreevar27","title":"Systems of Linear Equations","body":"The top three sources of oil imports for the United States in the same year were Saudi Arabia, Mexico, and Canada. The three top countries accounted for 47% of oil imports. The United States imported $$1.8\\\\%$$ more from Saudi Arabia than they did from Mexico, and $$1.7\\\\%$$ more from Saudi Arabia than they did from Canada.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Systems of Linear Equations: Three Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8934ethreevar27a","stepAnswer":["$$16.8$$, $$15.1$$, $$15$$"],"problemType":"MultipleChoice","stepTitle":"What percent of the United States oil imports were from these three countries? (list your answers in the order of Saudi Arabi, Canada, and Mexico in percent form)","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$16.8$$, $$15.1$$, $$15$$","choices":["$$16.8$$, $$15.1$$, $$15$$","$$16.8$$, $$15.1$$, $$14$$","$$16.8$$, $$15.2$$, $$15$$"],"hints":{"DefaultPathway":[{"id":"ab8934ethreevar27a-h1","type":"hint","dependencies":[],"title":"Setting Up","text":"Set up three equatinos in variable form to represent the situation $$(s=Saudi$$ Arabi, $$m=Mexico$$, and $$c=Canada)$$. We have $$s+m+c=47$$, $$m+1.8=s$$, and $$c+1.7=s$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar27a-h2","type":"hint","dependencies":["ab8934ethreevar27a-h1"],"title":"Simplifying to $$1$$ Variable","text":"Plug in $$m=s-1.8$$ and $$c=s-1.7$$ into the equation containing all three variables so that only one variable s is present","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar27a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16.8$$"],"dependencies":["ab8934ethreevar27a-h2"],"title":"Solving for s","text":"Combine the s values and solve for s using algebra","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar27a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["ab8934ethreevar27a-h3"],"title":"Solving for","text":"Plug in the value of s into $$m=s-1.8$$ to solve for $$m$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar27a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15.1$$"],"dependencies":["ab8934ethreevar27a-h4"],"title":"Solving for","text":"Plug in the value of s into $$c=s-1.7$$ to solve for c","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8934ethreevar28","title":"Systems of Linear Equations","body":"Last year, at Haven\u2019s Pond Car Dealership, for a particular model of BMW, Jeep, and Toyota, one could purchase all three cars for a total of $140,000. This year, due to $$inflation$$, the same cars would cost $151,830. The cost of the BMW increased by 8%, the Jeep by 5%, and the Toyota by 12%.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Systems of Linear Equations: Three Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8934ethreevar28a","stepAnswer":["$$49636$$, $$42636$$, $$47747$$"],"problemType":"MultipleChoice","stepTitle":"If the price of last year\u2019s Jeep was $7,000 less than the price of last year\u2019s BMW, what was the price of each of the three cars last year? (list answer in order of BMW, Jeep, and Toyota)","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$49636$$, $$42636$$, $$47747$$","choices":["$$49236$$, $$42756$$, $$47747$$","$$49646$$, $$42636$$, $$47647$$","$$49636$$, $$42636$$, $$47747$$"],"hints":{"DefaultPathway":[{"id":"ab8934ethreevar28a-h1","type":"hint","dependencies":[],"title":"Setting Up","text":"Set up three equations in variable form to represent the situation $$(b=BMW$$, $$j=Jeep$$, and $$t=Toyota)$$. We have $$b+j+t=140000$$, $$b=j+7000$$, and $$1.08b+1.05j+1.12t=151830$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar28a-h2","type":"hint","dependencies":["ab8934ethreevar28a-h1"],"title":"Plugging In","text":"Substitute $$b=j+7000$$ into both of the equations containing all three variables so that only the variables j and $$t$$ are present","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar28a-h3","type":"hint","dependencies":["ab8934ethreevar28a-h2"],"title":"Rearranging Equation","text":"Rearrange one of the two equations to represent an expression for j","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar28a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$42636$$"],"dependencies":["ab8934ethreevar28a-h3"],"title":"Solving for j","text":"Substitute the expression for j into the other equation not rearranged and solve for j","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar28a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$49636$$"],"dependencies":["ab8934ethreevar28a-h4"],"title":"Solving for $$b$$","text":"Substitute the value for j into $$b=j+7000$$ to solve for $$b$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar28a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$47747$$"],"dependencies":["ab8934ethreevar28a-h5"],"title":"Solving for $$t$$","text":"Use any of the three variable equations to plug in the values of $$b$$ and j to solve for $$t$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8934ethreevar3","title":"Solving a System of Three Equations in Three Variables by Elimination","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Systems of Linear Equations: Three Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8934ethreevar3a","stepAnswer":["$$(-1, 4, 2)$$"],"problemType":"MultipleChoice","stepTitle":"Solve the system of equations by elimination: $$3x-4y+2z=-15$$, $$2x+4y+z=16$$, $$2x+3y+5z=20$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-1, 4, 2)$$","choices":["$$(-1, 4, 2)$$","(1,2,4)","(2,1,4)","None of the above"],"hints":{"DefaultPathway":[{"id":"ab8934ethreevar3a-h1","type":"hint","dependencies":[],"title":"Using Two Equations","text":"The first step is to select two out of the three equations to find the relationship between two of the variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar3a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y-4z=-4$$"],"dependencies":["ab8934ethreevar3a-h1"],"title":"Using Two Equations","text":"What is the result of subtracting equations (2) and (3)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$-y+4z=4$$","$$y-4z=-4$$","None of the above"]},{"id":"ab8934ethreevar3a-h3","type":"hint","dependencies":["ab8934ethreevar3a-h2"],"title":"Using Two Equations","text":"The next step is to pick another pair of equations and solve for the same variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar3a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-20y+3z=-93$$"],"dependencies":["ab8934ethreevar3a-h3"],"title":"Using Two Equations","text":"What is the resulting equation that comes from multiplying equation (1) by $$2$$, multiplying equation (2) by $$3$$, and subtracting the two equations?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$-20y+3z=-93$$","$$20y-3z=93$$","None of the above"]},{"id":"ab8934ethreevar3a-h5","type":"hint","dependencies":["ab8934ethreevar3a-h4"],"title":"Solving a System of Two Equations","text":"The next step is to solve the resulting system of two equations","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar3a-h6","type":"hint","dependencies":["ab8934ethreevar3a-h5"],"title":"Back-Substitution","text":"The last step is to back-substitute the known variables into any one of the original equations and solve for the missing variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8934ethreevar4","title":"Determining Whether an Ordered Triple Is a Solution to a System","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Systems of Linear Equations: Three Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8934ethreevar4a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Determine whether $$(0, 1, -1)$$ is the solution to the system of equations: $$2x-6y+6z=-12$$, $$x+4y+5z=-1$$, $$-x+2y+3z=-1$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ab8934ethreevar4a-h1","type":"hint","dependencies":[],"title":"Plugging in solution into the equations","text":"The first step is to plug the given point into the equations. If the equality that results is true for all the equations, then the point is the solution to the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-12$$"],"dependencies":["ab8934ethreevar4a-h1"],"title":"Evaluating the equation","text":"What is 0-6(1)+6(-1)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar4a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ab8934ethreevar4a-h2"],"title":"Determining the equality","text":"Does $$-12=-12$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ab8934ethreevar4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["ab8934ethreevar4a-h3"],"title":"Evaluating the equation","text":"What is 0+4(1)+5(-1)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar4a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ab8934ethreevar4a-h4"],"title":"Determining the equality","text":"Does $$-1=-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ab8934ethreevar4a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["ab8934ethreevar4a-h5"],"title":"Evaluating the equation","text":"What is -0+2(1)+3(-1)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar4a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ab8934ethreevar4a-h6"],"title":"Determining the equality","text":"Does $$-1=-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"ab8934ethreevar5","title":"Determining Whether an Ordered Triple Is a Solution to a System","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Systems of Linear Equations: Three Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8934ethreevar5a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Determine whether $$(4, 2, -6)$$ is the solution to the system of equations: $$6x-7y+z=2$$, $$-x-y+3z=4$$, $$2x+y-z=1$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ab8934ethreevar5a-h1","type":"hint","dependencies":[],"title":"Plugging in solution into the equations","text":"The first step is to plug the given point into the equations. If the equality that results is true for all the equations, then the point is the solution to the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ab8934ethreevar5a-h1"],"title":"Evaluating the equation","text":"What is $$6\\\\left(4\\\\right)-7\\\\left(2\\\\right)+\\\\left(-6\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar5a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ab8934ethreevar5a-h2"],"title":"Determining the equality","text":"Does $$4=6$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"ab8934ethreevar6","title":"Determining Whether an Ordered Triple Is a Solution to a System","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Systems of Linear Equations: Three Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8934ethreevar6a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Determine whether $$(4, 1, -7)$$ is the solution to the system of equations: $$-x-y+2z=3$$, $$5x+8y-3z=4$$, $$-x+3y-5z=-5$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ab8934ethreevar6a-h1","type":"hint","dependencies":[],"title":"Plugging in solution into the equations","text":"The first step is to plug the given point into the equations. If the equality that results is true for all the equations, then the point is the solution to the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-19$$"],"dependencies":["ab8934ethreevar6a-h1"],"title":"Evaluating the equation","text":"What is -4-1+2(-7)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar6a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ab8934ethreevar6a-h2"],"title":"Determining the equality","text":"Does $$-19=3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"ab8934ethreevar7","title":"Solving a System of Three Equations in Three Variables by Gaussian Elimination","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Systems of Linear Equations: Three Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8934ethreevar7a","stepAnswer":["$$(4, -6, 1)$$"],"problemType":"MultipleChoice","stepTitle":"Solve the system of equations by elimination: $$2x-y+3z=17$$, $$-5x+4y-2z=-46$$, $$2y+5z=-7$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(4, -6, 1)$$","choices":["$$(4, -6, 1)$$","$$(1, -6, 4)$$","None of the above"],"hints":{"DefaultPathway":[{"id":"ab8934ethreevar7a-h1","type":"hint","dependencies":[],"title":"Reducing to a system of two equations","text":"The first step is to pick two equations and solve for one variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar7a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3y+11z=-7$$"],"dependencies":["ab8934ethreevar7a-h1"],"title":"Solving for one variable","text":"What is the resulting equation that comes from multiplying equation (1) by $$5$$, multiplying equation (2) by $$2$$, and adding the two equations together?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$3y+11z=-7$$","$$-3y-11z=7$$","None of the above"]},{"id":"ab8934ethreevar7a-h3","type":"hint","dependencies":["ab8934ethreevar7a-h2"],"title":"Solving the system of two equations","text":"The next step is to use the resulting equation and equation $$3$$ to solve the system of two equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar7a-h4","type":"hint","dependencies":["ab8934ethreevar7a-h3"],"title":"Finding the value of the last variable","text":"The last step is to back-substitute the known variables into any one of the original equations and solve for the missing variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8934ethreevar8","title":"Solving a System of Three Equations in Three Variables by Gaussian Elimination","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Systems of Linear Equations: Three Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8934ethreevar8a","stepAnswer":["No Solution"],"problemType":"MultipleChoice","stepTitle":"Solve the system of equations by elimination: $$x+y+z=14$$, $$2y+3z=-14$$, $$-16y-24z=-112$$","stepBody":"","answerType":"string","variabilization":{},"choices":["No Solution","(1,5,2)","$$(4, 7, -8)$$"],"hints":{"DefaultPathway":[{"id":"ab8934ethreevar8a-h1","type":"hint","dependencies":[],"title":"Reducing the Problem to a system of two equations","text":"The first step is to pick two equations and solve for one variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar8a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$0=-224$$"],"dependencies":["ab8934ethreevar8a-h1"],"title":"Solving for one variable","text":"What is the resulting equation that comes from multiplying equation (2) by $$8$$ and adding it to equation (3)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$0=-224$$","$$0=-235$$"]},{"id":"ab8934ethreevar8a-h3","type":"hint","dependencies":["ab8934ethreevar8a-h2"],"title":"Definition of no solution","text":"If the equation that results will always be false, then there is no solution to the system of equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8934ethreevar9","title":"Solving a System of Three Equations in Three Variables by Gaussian Elimination","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.2 Systems of Linear Equations: Three Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8934ethreevar9a","stepAnswer":["(0,0,0)"],"problemType":"MultipleChoice","stepTitle":"Solve the system of equations by elimination: $$x+y+z=0$$, $$2x-y+3z=0$$, $$x-z=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["(1,1,1)","(2,3,4)","(0,0,0)","(0,0,0)"],"hints":{"DefaultPathway":[{"id":"ab8934ethreevar9a-h1","type":"hint","dependencies":[],"title":"Solving for one variable","text":"The first step is to pick two equations and solve for one variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar9a-h2","type":"hint","dependencies":["ab8934ethreevar9a-h1"],"title":"Solving for one variable","text":"The next step is to pick another pair of equations and solve for the same variable","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar9a-h3","type":"hint","dependencies":["ab8934ethreevar9a-h2"],"title":"Solving the resulting system of two equations","text":"Using the resulting equations, sovle the system of two equations to find the values of those variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8934ethreevar9a-h4","type":"hint","dependencies":["ab8934ethreevar9a-h3"],"title":"Back-substitution","text":"Back-substitute the known variables into any one of the original equations and solve for the missing variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8b840systemeq1","title":"Solving Systems of Linear Equations","body":"Solve the system of linear equations.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Systems of Linear Equations: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8b840systemeq1a","stepAnswer":["No solution"],"problemType":"MultipleChoice","stepTitle":"$$x-0.2y=1$$, $$-10x+2y=5$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$x=5$$, $$y=1$$","$$x=2, y=5$$","No solution"],"hints":{"DefaultPathway":[{"id":"ab8b840systemeq1a-h1","type":"hint","dependencies":[],"title":"Clearing the Decimal","text":"The first step is to multiply the first equation by a number in order to change the $$0.2$$ into a whole number. This will make the equation easier to solve.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systemeq1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ab8b840systemeq1a-h1"],"title":"Multiplying decimals","text":"What is $$10\\\\times0.2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systemeq1a-h3","type":"hint","dependencies":[],"title":"Using substitution","text":"The next step is to rewrite one of the equations in terms of one of the variables and plug it into the other equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systemeq1a-h4","type":"hint","dependencies":[],"title":"Definition of inconsistent systems of equations","text":"An inconsistent system of equations consists of parallel lines that never intersect. Algebraically, that means that the system of equation will result in a false equality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8b840systemeq10","title":"Solving systems of linear equations","body":"Solve the system:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Systems of Linear Equations: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8b840systemeq10a","stepAnswer":["No solution"],"problemType":"MultipleChoice","stepTitle":"$$-x+2y=-1$$, $$5x-10y=6$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Infinitely many solutions","$$(4,6)$$","$$(17,29)$$","No solution"],"hints":{"DefaultPathway":[{"id":"ab8b840systemeq10a-h1","type":"hint","dependencies":[],"title":"Multiplying equations by the least common multiple","text":"The first step is to multiply the equation(s) by the least common multiple of all the coefficients.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systemeq10a-h2","type":"hint","dependencies":["ab8b840systemeq10a-h1"],"title":"Definition of the least common multiple","text":"The least common multiple is the smallest quantity that is a multiple of the given set of numbers","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systemeq10a-h3","type":"hint","dependencies":["ab8b840systemeq10a-h2"],"title":"Adding(or subtracting) the equations","text":"The next step is to add or subtract the equations to eliminate one of the variables. This will let us solve for the other variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systemeq10a-h4","type":"hint","dependencies":["ab8b840systemeq10a-h3"],"title":"Solving the System","text":"The last step is to solve for the variable of the resulting equation. Then, plug that value back into one of the original equations in the system to solve for the other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8b840systemeq2","title":"Solving Systems of Linear Equations","body":"Solve the system of equations:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Systems of Linear Equations: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8b840systemeq2a","stepAnswer":["No solution"],"problemType":"MultipleChoice","stepTitle":"$$3x+5y=9$$, $$30x+50y=-90$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$x=10, y=4$$","$$x=17, y=15$$","No solution"],"hints":{"DefaultPathway":[{"id":"ab8b840systemeq2a-h1","type":"hint","dependencies":[],"title":"Rewriting the equations","text":"The first step is to multiply the equation(s) by a number so that the coefficients of at least one of the variables is the same. This will make the equation easier to solve. Make sure you multiply all the terms in the equation by the same number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systemeq2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$30$$"],"dependencies":["ab8b840systemeq2a-h1"],"title":"Finding the common denominator","text":"What is $$10\\\\times3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systemeq2a-h3","type":"hint","dependencies":["ab8b840systemeq2a-h2"],"title":"Using substitution","text":"The next step is to rewrite one of the equations in terms of one of the variables and plug it into the other equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systemeq2a-h4","type":"hint","dependencies":["ab8b840systemeq2a-h3"],"title":"Definition of inconsistent systems of equations","text":"An inconsistent system of equations consists of parallel lines that never intersect. Algebraically, that means that the system of equation will result in a false equality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8b840systemeq3","title":"Solving Systems of Linear Equations","body":"Solve the system:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Systems of Linear Equations: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8b840systemeq3a","stepAnswer":["infinitely many solutions"],"problemType":"MultipleChoice","stepTitle":"$$-3x+y=2$$, $$12x-4y=-8$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$x=2$$, $$y=8$$","$$x=4$$, $$y=5$$","infinitely many solutions","no solution"],"hints":{"DefaultPathway":[{"id":"ab8b840systemeq3a-h1","type":"hint","dependencies":[],"title":"Using substitution","text":"The first step is to write one of the equations in terms of one of the variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systemeq3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y=2+3x$$"],"dependencies":["ab8b840systemeq3a-h1"],"title":"Writing an equation in terms of one of the variables","text":"Write the first equation in terms of $$y$$ (solve for x). What is the resulting equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systemeq3a-h3","type":"hint","dependencies":["ab8b840systemeq3a-h2"],"title":"Solving the system","text":"The next step is to plug in the rewritten equation into the other equation and solve for the remaining variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systemeq3a-h4","type":"hint","dependencies":["ab8b840systemeq3a-h3"],"title":"Definition of a system of dependent solutions","text":"A system of dependent equations consist of two equations that represent the same line. Algebraically, this means that the resulting equation will be an identity and therefore always true.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8b840systemeq4","title":"Solving Systems of Linear Equations","body":"Solve the system of linear equations:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Systems of Linear Equations: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8b840systemeq4a","stepAnswer":["$$x=\\\\frac{72}{5}$$, $$y=\\\\frac{132}{5}$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{1}{2} x+\\\\frac{1}{3} y=16$$, $$\\\\frac{1}{6} x+\\\\frac{1}{4} y=9$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=\\\\frac{72}{5}$$, $$y=\\\\frac{132}{5}$$","choices":["$$x=72$$, $$y=132$$","$$x=\\\\frac{72}{5}$$, $$y=\\\\frac{132}{5}$$","$$x=27$$, $$y=35$$","none of the above"],"hints":{"DefaultPathway":[{"id":"ab8b840systemeq4a-h1","type":"hint","dependencies":[],"title":"Using substitution","text":"The first step is to write one of the equations in terms of one of the variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systemeq4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=32-\\\\frac{2}{3} y$$"],"dependencies":["ab8b840systemeq4a-h1"],"title":"Writing one of the equations in terms of one of the variables","text":"Write the first equation in terms of $$y$$ (solve for x). What is the resulting equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systemeq4a-h3","type":"hint","dependencies":["ab8b840systemeq4a-h2"],"title":"Solving the system","text":"The next step is to plug in the rewritten equation into the other equation and solve for the remaining variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systemeq4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{6} \\\\left(32-\\\\frac{2}{3} y\\\\right)+\\\\frac{1}{4} y=9$$"],"dependencies":["ab8b840systemeq4a-h3"],"title":"Solving for $$y$$","text":"What is the resulting equation after you plug the first one in? Do not simplify.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systemeq4a-h5","type":"hint","dependencies":["ab8b840systemeq4a-h4"],"title":"Solving for $$x$$","text":"The last step is to plug in the value you got for $$y$$ into one of the equations to solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8b840systemeq5","title":"Solving Systems of Linear Equations","body":"Solve the system:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Systems of Linear Equations: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8b840systemeq5a","stepAnswer":["$$x=-8$$, $$y=6$$"],"problemType":"MultipleChoice","stepTitle":"$$-\\\\left(\\\\frac{1}{4}\\\\right) x+\\\\frac{3}{2} y=11$$, $$-\\\\left(\\\\frac{1}{8}\\\\right) x+\\\\frac{1}{3} y=3$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=-8$$, $$y=6$$","choices":["$$x=4, y=3$$","$$x=7$$, $$y=-10$$","$$x=-8$$, $$y=6$$"],"hints":{"DefaultPathway":[{"id":"ab8b840systemeq5a-h1","type":"hint","dependencies":[],"title":"Using substitution","text":"The first step is to write one of the equations in terms of one of the variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systemeq5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=6y-44$$"],"dependencies":["ab8b840systemeq5a-h1"],"title":"Writing one of the equations in terms of one of the variables","text":"Write the first equation in terms of $$y$$ (solve for x). What is the resulting equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systemeq5a-h3","type":"hint","dependencies":["ab8b840systemeq5a-h2"],"title":"Solving the system","text":"The next step is to plug in the rewritten equation into the other equation and solve for the remaining variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systemeq5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-\\\\left(\\\\frac{1}{8}\\\\right) \\\\left(6y-44\\\\right)+\\\\frac{1}{3} y=3$$"],"dependencies":["ab8b840systemeq5a-h3"],"title":"Solving for $$y$$","text":"What is the resulting equation after you plug the first one in? Do not simplify.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systemeq5a-h5","type":"hint","dependencies":["ab8b840systemeq5a-h4"],"title":"Solving for $$x$$","text":"The last step is to plug in the value you got for $$y$$ into one of the equations to solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8b840systemeq6","title":"Solving Systems of Linear Equations","body":"Solve the system:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Systems of Linear Equations: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8b840systemeq6a","stepAnswer":["$$(6,-6)$$"],"problemType":"MultipleChoice","stepTitle":"$$-2x+5y=-42$$, $$7x+2y=30$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(6,-6)$$","choices":["$$(6,-6)$$","$$(-6,-6)$$","$$(6,6)$$","$$(-6,6)$$"],"hints":{"DefaultPathway":[{"id":"ab8b840systemeq6a-h1","type":"hint","dependencies":[],"title":"Multiplying equations by the least common multiple","text":"The first step is to multiply the equation(s) by the least common multiple of all the coefficients.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systemeq6a-h2","type":"hint","dependencies":["ab8b840systemeq6a-h1"],"title":"Definition of the least common multiple","text":"The least common multiple is the smallest quantity that is a multiple of the given set of numbers","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systemeq6a-h3","type":"hint","dependencies":["ab8b840systemeq6a-h2"],"title":"Adding(or subtracting) the equations","text":"The next step is to add or subtract the equations to eliminate one of the variables. This will let us solve for the other variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systemeq6a-h4","type":"hint","dependencies":["ab8b840systemeq6a-h3"],"title":"Solving the System","text":"The last step is to solve for the variable of the resulting equation. Then, plug that value back into one of the original equations in the system to solve for the other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8b840systemeq7","title":"Solving Systems of Linear Equations","body":"Solve the system:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Systems of Linear Equations: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8b840systemeq7a","stepAnswer":["$$(-4,2)$$"],"problemType":"MultipleChoice","stepTitle":"$$6x-5y=-34$$, $$7x+2y=30$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-4,2)$$","choices":["$$(-4,2)$$","$$(4,2)$$","$$(4,-2)$$","$$(-4,-2)$$"],"hints":{"DefaultPathway":[{"id":"ab8b840systemeq7a-h1","type":"hint","dependencies":[],"title":"Multiplying equations by the least common multiple","text":"The first step is to multiply the equation(s) by the least common multiple of all the coefficients.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systemeq7a-h2","type":"hint","dependencies":["ab8b840systemeq7a-h1"],"title":"Definition of the least common multiple","text":"The least common multiple is the smallest quantity that is a multiple of the given set of numbers","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systemeq7a-h3","type":"hint","dependencies":["ab8b840systemeq7a-h2"],"title":"Adding(or subtracting) the equations","text":"The next step is to add or subtract the equations to eliminate one of the variables. This will let us solve for the other variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systemeq7a-h4","type":"hint","dependencies":["ab8b840systemeq7a-h3"],"title":"Solving the System","text":"The last step is to solve for the variable of the resulting equation. Then, plug that value back into one of the original equations in the system to solve for the other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8b840systemeq8","title":"Solving Systems of Linear Equations","body":"Solve the system:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Systems of Linear Equations: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8b840systemeq8a","stepAnswer":["$$(\\\\frac{-1}{2},\\\\frac{1}{10})$$"],"problemType":"MultipleChoice","stepTitle":"$$5x-y=-2.6$$, $$-4x-6y=1.4$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(\\\\frac{-1}{2},\\\\frac{1}{10})$$","choices":["$$(\\\\frac{-1}{2},\\\\frac{1}{10})$$","$$(2,10)$$","(1/2, 1/10)","None of the above"],"hints":{"DefaultPathway":[{"id":"ab8b840systemeq8a-h1","type":"hint","dependencies":[],"title":"Rewriting equations","text":"The first step is to multiply the equation(s) by a factor of $$10$$ so that the terms are all whole numbers. This will make the equation easier to solve.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systemeq8a-h2","type":"hint","dependencies":["ab8b840systemeq8a-h1"],"title":"Multiplying equations by the least common multiple","text":"The next step is to multiply the equation(s) by the least common multiple of all the coefficients.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systemeq8a-h3","type":"hint","dependencies":["ab8b840systemeq8a-h2"],"title":"Definition of the least common multiple","text":"The least common multiple is the smallest quantity that is a multiple of the given set of numbers","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systemeq8a-h4","type":"hint","dependencies":["ab8b840systemeq8a-h3"],"title":"Adding(or subtracting) the equations","text":"The next step is to add or subtract the equations to eliminate one of the variables. This will let us solve for the other variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systemeq8a-h5","type":"hint","dependencies":["ab8b840systemeq8a-h4"],"title":"Solving the System","text":"The last step is to solve for the variable of the resulting equation. Then, plug that value back into one of the original equations in the system to solve for the other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8b840systemeq9","title":"Solving Systems of Linear Equations","body":"Solve the system:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Systems of Linear Equations: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8b840systemeq9a","stepAnswer":["$$(\\\\frac{1}{2},\\\\frac{1}{4})$$"],"problemType":"MultipleChoice","stepTitle":"$$7x-2y=3$$, $$4x+5y=3.25$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(\\\\frac{1}{2},\\\\frac{1}{4})$$","choices":["$$(\\\\frac{1}{2},\\\\frac{1}{4})$$","(1/4, 1/2)","(-1/2, -1/4)","(1/4, 1/2)"],"hints":{"DefaultPathway":[{"id":"ab8b840systemeq9a-h1","type":"hint","dependencies":[],"title":"Rewriting the equation(s)","text":"The first step is to convert the decimal terms to fractions. Then, multiply the equation by the fraction\'s denominator. This will make the system easier to solve.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systemeq9a-h2","type":"hint","dependencies":["ab8b840systemeq9a-h1"],"title":"Multiplying equations by the least common multiple","text":"The next step is to multiply the equation(s) by the least common multiple of all the coefficients.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systemeq9a-h3","type":"hint","dependencies":["ab8b840systemeq9a-h2"],"title":"Definition of the least common multiple","text":"The least common multiple is the smallest quantity that is a multiple of the given set of numbers","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systemeq9a-h4","type":"hint","dependencies":["ab8b840systemeq9a-h3"],"title":"Adding(or subtracting) the equations","text":"The next step is to add or subtract the equations to eliminate one of the variables. This will let us solve for the other variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systemeq9a-h5","type":"hint","dependencies":["ab8b840systemeq9a-h4"],"title":"Solving the System","text":"The last step is to solve for the variable of the resulting equation. Then, plug that value back into one of the original equations in the system to solve for the other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8b840systems1","title":"Systems of Linear Equations: Two Variables","body":"Determine whether the ordered pair $$(5,1)$$ is a solution to the given system of equations.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Systems of Linear Equations: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8b840systems1a","stepAnswer":["TRUE"],"problemType":"MultipleChoice","stepTitle":"$$x+3y=8$$ $$2x-9=y$$","stepBody":"","answerType":"string","variabilization":{},"choices":["TRUE","FALSE"],"hints":{"DefaultPathway":[{"id":"ab8b840systems1a-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Substitute the ordered pair $$(5,1)$$ into the 1st equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["ab8b840systems1a-h1"],"title":"Substitution","text":"What does $$5+3\\\\left(1\\\\right)$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems1a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["ab8b840systems1a-h2"],"title":"Substitution","text":"Does $$8$$ $$=$$ 8?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"ab8b840systems1a-h4","type":"hint","dependencies":[],"title":"Substitution","text":"Substitute the ordered pair $$(5,1)$$ into 2nd equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems1a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ab8b840systems1a-h1"],"title":"Substitution","text":"What does $$2(5)-9$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems1a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["ab8b840systems1a-h2"],"title":"Substitution","text":"Does $$1$$ $$=$$ 1?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]}]}}]},{"id":"ab8b840systems10","title":"Writing and Solving a System of Equations in Two Variables","body":"The cost of a ticket to the circus is $$\\\\$25.00$$ for children and $$\\\\$50.00$$ for adults. On a certain day, attendance at the circus is 2,000 and the total gate revenue is $70,000.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Systems of Linear Equations: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8b840systems10a","stepAnswer":["$$1200$$ children $$800$$ adults"],"problemType":"MultipleChoice","stepTitle":"How many children and how many adults bought tickets?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$1200$$ children $$800$$ adults","choices":["$$25$$ children, $$50$$ adults","$$40$$ children, $$50$$ adults","$$800$$ children, $$1200$$ adults","$$1200$$ children $$800$$ adults","$$1200$$ children, $$800$$ adults"],"hints":{"DefaultPathway":[{"id":"ab8b840systems10a-h1","type":"hint","dependencies":[],"title":"Writing a System of Equations","text":"Let c $$=$$ the number of children and a $$=$$ the number of adults in attendance. The total number of people is 2,000. Use this to write an equation for the number of people at the circus that day.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$c+a=2000$$"],"dependencies":["ab8b840systems10a-h1"],"title":"Writing a System of Equations","text":"What is the equation to represent the number of people at the circus that day?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems10a-h3","type":"hint","dependencies":["ab8b840systems10a-h2"],"title":"Writing a System of Equations","text":"The revenue from all children can be found by multiplying $$\\\\$25.00$$ by the number of children, 25c. The revenue from all adults can be found by multiplying $$\\\\$50.00$$ by the number of adults, 50a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$70000$$"],"dependencies":["ab8b840systems10a-h3"],"title":"Writing a System of Equations","text":"What is the total revenue?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems10a-h5","type":"hint","dependencies":["ab8b840systems10a-h4"],"title":"Writing a System of Equations","text":"We now have a system of linear equations in two variables. In the first equation, the coefficient of both variables is $$1$$. We can quickly solve the first equation for either c or a. We will solve for a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems10a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["2000-c"],"dependencies":["ab8b840systems10a-h5"],"title":"Writing a System of Equations","text":"Rewrite the first equation. What is a equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems10a-h7","type":"hint","dependencies":["ab8b840systems10a-h6"],"title":"Substitution","text":"Substitute the expression 2,000-c in the second equation for a and solve for c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems10a-h8","type":"hint","dependencies":["ab8b840systems10a-h7"],"title":"Isolating Variables","text":"Solve for c in $$25c+\\\\operatorname{50}\\\\left(2000-c\\\\right)=70000$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems10a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12000$$"],"dependencies":["ab8b840systems10a-h8"],"title":"Isolating Variables","text":"What is c equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems10a-h10","type":"hint","dependencies":["ab8b840systems10a-h9"],"title":"Substitution","text":"Substitute $$c=1, 200$$ into the first equation to solve for a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems10a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$800$$"],"dependencies":["ab8b840systems10a-h10"],"title":"Isolating Variables","text":"What is a equal to in $$1200+a=2000$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems10a-h12","type":"hint","dependencies":["ab8b840systems10a-h11"],"title":"Solving a System of Equations in Two Variables by Substitution","text":"We find that 1,200 children and $$800$$ adults bought tickets to the circus that day.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab8b840systems10b","stepAnswer":["$$(-1,1)$$ is a solution to the system of equations."],"problemType":"MultipleChoice","stepTitle":"$$-2x+5y=7$$, $$2x+9y=7$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-1,1)$$ is a solution to the system of equations.","choices":["$$(-1,1)$$ is a solution to the system of equations.","$$(-1,1)$$ is a solution to the system of equations.","$$(-1,1)$$ is not a solution to the system of equations."],"hints":{"DefaultPathway":[{"id":"ab8b840systems10b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"First, substitute the given point into the equation. If the equality that results is true, then the point is a solution to the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems10b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ab8b840systems10b-h1"],"title":"Substitute","text":"When the point is substituted into the equation $$-2x+5y=7$$, do both sides equal each other?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"],"subHints":[{"id":"ab8b840systems10b-h2-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":[],"title":"Substitute","text":"When the point is substituted into the equation $$2x+9y=7$$, do both sides equal each other?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]},{"id":"ab8b840systems10b-h3","type":"hint","dependencies":["ab8b840systems10b-h2"],"title":"Interpret","text":"Therefore, since the equation is equal on both sides for both equations, the point satisfies the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8b840systems11","title":"Solving systems of Equations","body":"Determine whether the ordered pair $$(3,5)$$ is a solution to the system of equations:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Systems of Linear Equations: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8b840systems11a","stepAnswer":["$$(3,5)$$ is not a solution to the system of equations."],"problemType":"MultipleChoice","stepTitle":"$$-x+8y=43$$, $$3x-2y=-1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(3,5)$$ is not a solution to the system of equations.","choices":["$$(3,5)$$ is a solution to the system of equations.","$$(3,5)$$ is not a solution to the system of equations.","$$(3,5)$$ is not a solution to the system of equations."],"hints":{"DefaultPathway":[{"id":"ab8b840systems11a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"First, substitute the given point into the equation. If the equality that results is true, then the point is a solution to the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems11a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ab8b840systems11a-h1"],"title":"Substitute","text":"When the point is substituted into the equation $$x+8y=43$$, do both sides equal each other?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"],"subHints":[{"id":"ab8b840systems11a-h2-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":[],"title":"Substitute","text":"When the point is substituted into the equation $$3x-2y=-1$$, do both sides equal each other?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]},{"id":"ab8b840systems11a-h3","type":"hint","dependencies":["ab8b840systems11a-h2-s1"],"title":"Interpret","text":"Therefore, since the equation is equal on both sides for both equations, the point satisfies the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8b840systems12","title":"Solving systems of Equations by Substitution","body":"Solve the System of Equations using substitution:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Systems of Linear Equations: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8b840systems12a","stepAnswer":["$$(-1,2)$$"],"problemType":"MultipleChoice","stepTitle":"$$x+3y=5$$, $$2x+3y=4$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-1,2)$$","choices":["$$(-1,2)$$","$$(2,1)$$","$$(1,-2)$$","$$(-2,-1)$$"],"hints":{"DefaultPathway":[{"id":"ab8b840systems12a-h1","type":"hint","dependencies":[],"title":"Rewriting equations","text":"The first step is to rewrite one of the equations in terms of one of the variables. For example, write $$x+3y=5$$ in terms of $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=-3y+5$$"],"dependencies":["ab8b840systems12a-h1"],"title":"Rewriting equations","text":"Write the first equation in terms of $$y$$. What is the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems12a-h3","type":"hint","dependencies":["ab8b840systems12a-h2"],"title":"Substitute","text":"Next, substitute the first equation into the second. Use the x-value you found in the previous hint, and simplify until you find the value of $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ab8b840systems12a-h3"],"title":"Substitute","text":"What is the value of $$y$$ after you substitute the first equation into the second?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems12a-h5","type":"hint","dependencies":["ab8b840systems12a-h4"],"title":"Substitute","text":"Next, substitute the value of $$y$$ that you now know into either or the original equations. Solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems12a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["ab8b840systems12a-h5"],"title":"Substitute","text":"What is the value of $$x$$ when you substitute $$y$$ into either of the original equations?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems12a-h7","type":"hint","dependencies":["ab8b840systems12a-h6"],"title":"Interpret","text":"Knowing that $$x=-1$$ and $$y=2$$, we can finalize our answer to be $$(-1,2)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8b840systems13","title":"Solving systems of Equations by Substitution","body":"Solve the System of Equations using substitution:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Systems of Linear Equations: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8b840systems13a","stepAnswer":["$$(4,-3)$$"],"problemType":"MultipleChoice","stepTitle":"$$3x-2y=18$$, $$5x+10y=-10$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(4,-3)$$","choices":["$$(4,-3)$$","$$(3,-4)$$","$$(-3,-4)$$","$$(4,3)$$"],"hints":{"DefaultPathway":[{"id":"ab8b840systems13a-h1","type":"hint","dependencies":[],"title":"Solve","text":"The first step is to rewrite one of the equations in terms of one of the variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=-2y-2$$"],"dependencies":["ab8b840systems13a-h1"],"title":"Solve","text":"Rewrite the second equation in terms of $$y$$. What is the equation you end up with?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems13a-h3","type":"hint","dependencies":["ab8b840systems13a-h2"],"title":"Substitute","text":"Next, substitute the second equation into the first. Use the x-value you found in the previous hint, and simplify until you find the value of $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["ab8b840systems13a-h3"],"title":"Solve","text":"Solve for $$y$$ with the new substituted equation. What is the value of $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems13a-h5","type":"hint","dependencies":["ab8b840systems13a-h4"],"title":"Substitute","text":"Next, substitute the value of $$y$$ that you now know into either or the original equations. Solve for $$x$$ to find the numeric solution the system of equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems13a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ab8b840systems13a-h5"],"title":"Solve","text":"What does $$x$$ equal when the $$y$$ value is substituted into either of the original equations?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems13a-h7","type":"hint","dependencies":["ab8b840systems13a-h6"],"title":"Interpret","text":"Knowing that $$x=4$$ and $$y=-3$$, we can finalize our answer to be $$(4,-3)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8b840systems14","title":"Solving systems of Equations by Substitution","body":"Solve the System of Equations using substitution:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Systems of Linear Equations: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8b840systems14a","stepAnswer":["(-3,1)"],"problemType":"TextBox","stepTitle":"$$4x+2y=-10$$, $$3x+9y=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-3,1)$$","hints":{"DefaultPathway":[{"id":"ab8b840systems14a-h1","type":"hint","dependencies":[],"title":"Solve","text":"The first step is to rewrite one of the equations in terms of one of the variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y=-2x-5$$"],"dependencies":["ab8b840systems14a-h1"],"title":"Solve","text":"Rewrite the first equation in terms of $$x$$. What is the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems14a-h3","type":"hint","dependencies":["ab8b840systems14a-h2"],"title":"Substitute","text":"Next, substitute the first equation into the second. Use the y-value you found in the previous hint, and simplify until you find the value of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems14a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["ab8b840systems14a-h3"],"title":"Solve","text":"Solve for $$x$$ with the new substituted equation. What is the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems14a-h5","type":"hint","dependencies":["ab8b840systems14a-h4"],"title":"Substitute","text":"Next, substitute the numeric value of $$x$$ that you now know into either or the original equations. Solve for $$y$$ to find the numeric solution the system of equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems14a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ab8b840systems14a-h5"],"title":"Solve","text":"What does $$y$$ equal when the $$x$$ value is substituted into either of the original equations?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems14a-h7","type":"hint","dependencies":["ab8b840systems14a-h6"],"title":"Interpret","text":"Knowing that $$x=-3$$ and $$y=-1$$, we can finalize our answer to be $$(-3,1)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8b840systems15","title":"Solving systems of Equations by Substitution","body":"Solve the System of Equations using substitution:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Systems of Linear Equations: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8b840systems15a","stepAnswer":["$$(-0.3, -0.8)$$"],"problemType":"MultipleChoice","stepTitle":"$$2x+4y=-3.8$$, $$9x-5y=1.3$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-0.3, -0.8)$$","choices":["$$(0.3$$, $$0.8)$$","$$(-0.3, -0.8)$$","$$(-0.3$$, $$0.8)$$","$$(0.3, -0.8)$$"],"hints":{"DefaultPathway":[{"id":"ab8b840systems15a-h1","type":"hint","dependencies":[],"title":"Solve","text":"The first step is to rewrite one of the equations in terms of one of the variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=-2y-1.9$$"],"dependencies":["ab8b840systems15a-h1"],"title":"Solve","text":"Rewrite the first equation in terms of $$y$$. What is the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems15a-h3","type":"hint","dependencies":["ab8b840systems15a-h2"],"title":"Substitute","text":"Next, substitute the first equation into the second. Use the x-value you found in the previous hint, and simplify until you find the value of $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-0.8$$"],"dependencies":["ab8b840systems15a-h3"],"title":"Solve","text":"Solve for $$y$$ with the new substituted equation. What is the value of $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems15a-h5","type":"hint","dependencies":["ab8b840systems15a-h4"],"title":"Substitute","text":"Next, substitute the numeric value of $$y$$ that you now know into either or the original equations. Solve for $$x$$ to find the numeric solution the system of equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems15a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-0.3$$"],"dependencies":["ab8b840systems15a-h5"],"title":"Solve","text":"What does $$x$$ equal when the $$y$$ value is substituted into either of the original equations?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems15a-h7","type":"hint","dependencies":["ab8b840systems15a-h6"],"title":"Interpret","text":"Knowing that $$x=-0.3$$ and $$y=-0.8$$, we can finalize our answer to be $$(-0.3, -0.8)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8b840systems16","title":"Solving systems of Equations by Substitution","body":"Solve the System of Equations using substitution:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Systems of Linear Equations: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8b840systems16a","stepAnswer":["(-0.6,0)"],"problemType":"TextBox","stepTitle":"$$-2x+3y=1.2$$, $$-3x-6y=1.8$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-0.6, 0)$$","hints":{"DefaultPathway":[{"id":"ab8b840systems16a-h1","type":"hint","dependencies":[],"title":"Solve","text":"The first step is to rewrite one of the equations in terms of one of the variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=\\\\frac{3y}{2}-0.6$$"],"dependencies":["ab8b840systems16a-h1"],"title":"Solve","text":"Rewrite the first equation in terms of $$y$$ What is the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems16a-h3","type":"hint","dependencies":["ab8b840systems16a-h2"],"title":"Substitute","text":"Next, substitute the first equation into the second. Use the x-value you found in the previous hint, and simplify until you find the value of $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems16a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["ab8b840systems16a-h3"],"title":"Solve","text":"Solve for $$y$$ with the new substituted equation. What is the value of $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems16a-h5","type":"hint","dependencies":["ab8b840systems16a-h4"],"title":"Substitute","text":"Next, substitute the numeric value of $$y$$ that you now know into either or the original equations. Solve for $$x$$ to find the numeric solution the system of equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems16a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-0.6$$"],"dependencies":["ab8b840systems16a-h5"],"title":"Solve","text":"What does $$x$$ equal when the $$y$$ value is substituted into either of the original equations?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems16a-h7","type":"hint","dependencies":["ab8b840systems16a-h6"],"title":"Interpret","text":"Knowing that $$x=-0.6$$ and $$y=0$$, we can finalize our answer to be $$(-0.6, 0)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8b840systems2","title":"Systems of Linear Equations: Two Variables","body":"Determine whether the ordered pair $$(8,5)$$ is a solution to the given system of equations.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Systems of Linear Equations: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8b840systems2a","stepAnswer":["FALSE"],"problemType":"MultipleChoice","stepTitle":"$$5x-4y=20$$ $$2x+1=3y$$","stepBody":"","answerType":"string","variabilization":{},"choices":["TRUE","FALSE"],"hints":{"DefaultPathway":[{"id":"ab8b840systems2a-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Substitute the ordered pair $$(8,5)$$ into the 1st equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["ab8b840systems2a-h1"],"title":"Substitution","text":"What does $$5(8)-4(5)$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems2a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["ab8b840systems2a-h2"],"title":"Substitution","text":"Does $$20$$ $$=$$ 20?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"ab8b840systems2a-h4","type":"hint","dependencies":["ab8b840systems2a-h3"],"title":"Substitution","text":"Substitute the ordered pair $$(8,5)$$ into 2nd equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems2a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$17$$"],"dependencies":["ab8b840systems2a-h4"],"title":"Substitution","text":"What does $$2\\\\left(8\\\\right)+1$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems2a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["FALSE"],"dependencies":["ab8b840systems2a-h5"],"title":"Substitution","text":"Does 3(5) $$=$$ 17?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]}]}}]},{"id":"ab8b840systems3","title":"Solving a System of Equations in Two Variables by Substitution","body":"Solve the following system of equations by substitution.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Systems of Linear Equations: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8b840systems3a","stepAnswer":["$$(8,3)$$"],"problemType":"MultipleChoice","stepTitle":"$$-x+y=-5$$ $$2x-5y=1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(8,3)$$","choices":["$$(3,8)$$","$$(4,6)$$","$$(6,4)$$","$$(8,3)$$","$$(8,3)$$"],"hints":{"DefaultPathway":[{"id":"ab8b840systems3a-h1","type":"hint","dependencies":[],"title":"Isolating Variables","text":"Solve the first equation for $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x-5$$"],"dependencies":["ab8b840systems3a-h1"],"title":"Isolating Variables","text":"What does $$y$$ equal? Write $$y$$ in terms of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems3a-h3","type":"hint","dependencies":["ab8b840systems3a-h2"],"title":"Substitution","text":"Substitute the expression $$x-5$$ for $$y$$ in the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["ab8b840systems3a-h3"],"title":"Isolating Variables","text":"What does $$x$$ equal in $$2x-5(x-5)=1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems3a-h5","type":"hint","dependencies":["ab8b840systems3a-h4"],"title":"Substitution","text":"Substitute $$x=8$$ into the first equation and solve for $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems3a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ab8b840systems3a-h5"],"title":"Substitution","text":"What does $$y$$ equal in $$-8+y=-5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems3a-h7","type":"hint","dependencies":["ab8b840systems3a-h6"],"title":"Checking the Solution","text":"Check the solution by substituting $$(8,3)$$ into both equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems3a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["ab8b840systems3a-h7"],"title":"Checking the Solution","text":"Does $$-8+3=-5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]},{"id":"ab8b840systems3a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["ab8b840systems3a-h8"],"title":"Checking the Solution","text":"Does $$2(8)-5(3)=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]}]}}]},{"id":"ab8b840systems4","title":"Solving Systems of Equations in Two Variables by the Addition Method","body":"Solve the given system of equations by addition.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Systems of Linear Equations: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8b840systems4a","stepAnswer":["(-7/3, 2/3)"],"problemType":"MultipleChoice","stepTitle":"$$x+2y=-1$$ $$-x+y=3$$","stepBody":"","answerType":"string","variabilization":{},"choices":["(-7/3, 2/3)","$$(\\\\frac{-7}{3},\\\\frac{2}{3})$$","$$(2,-7)$$","$$(\\\\frac{2}{3},\\\\frac{-7}{3})$$","$$(7,2)$$"],"hints":{"DefaultPathway":[{"id":"ab8b840systems4a-h1","type":"hint","dependencies":[],"title":"Addition Method","text":"Since both equations with $$x-$$ and $$y-variables$$ are on the left side of the equal sign and constants on the right, write one equation above the other, lining up corresponding variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems4a-h2","type":"hint","dependencies":["ab8b840systems4a-h1"],"title":"Addition Method","text":"Notice that the coefficient of $$x$$ in the second equation, $$-1$$, is the opposite of the coefficient of $$x$$ in the first equation, $$1$$. Add the two equations to eliminate $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3y=2$$"],"dependencies":["ab8b840systems4a-h2"],"title":"Addition Method","text":"What is the resulting equation when you add the first and second equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{2}$$"],"dependencies":["ab8b840systems4a-h3"],"title":"Isolating Variables","text":"What is $$y$$ equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems4a-h5","type":"hint","dependencies":["ab8b840systems4a-h4"],"title":"Substitution","text":"Substitute this value for $$y$$ into one of the original equations and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems4a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-7}{3}$$"],"dependencies":["ab8b840systems4a-h5"],"title":"Isolating Variables","text":"What is $$x$$ equal to in $$-x+\\\\frac{2}{3}=3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems4a-h7","type":"hint","dependencies":["ab8b840systems4a-h6"],"title":"Checking the Solution","text":"Check the solution by substituting (-7/3, 2/3) into the first equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems4a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["TRUE"],"dependencies":["ab8b840systems4a-h7"],"title":"Checking the Solution","text":"Does $$\\\\left(-\\\\frac{7}{3}\\\\right)+2\\\\frac{2}{3}=-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["TRUE","FALSE"]}]}}]},{"id":"ab8b840systems5","title":"Solving Systems of Equations in Two Variables by the Addition Method","body":"Solve the given system of equations by addition.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Systems of Linear Equations: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8b840systems5a","stepAnswer":["$$(3,-4)$$"],"problemType":"MultipleChoice","stepTitle":"$$3x+5y=-11$$ $$x-2y=11$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(3,-4)$$","choices":["$$(4,-3)$$","$$(3,-4)$$","$$(-3,4)$$","$$(-4,3)$$"],"hints":{"DefaultPathway":[{"id":"ab8b840systems5a-h1","type":"hint","dependencies":[],"title":"Addition Method","text":"Adding these equations as presented will not eliminate a variable. However, we see that the first equation has $$3x$$ in it and the second equation has $$x$$. So, multiply the second equation by $$-3$$, the $$x-terms$$ will add to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3x+6y=-33$$"],"dependencies":["ab8b840systems5a-h1"],"title":"Multiplying Equations","text":"After you multiply the first equation by $$-3$$, what is the new equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems5a-h3","type":"hint","dependencies":["ab8b840systems5a-h2"],"title":"Addition Method","text":"Now add the first and second equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11y=-44$$"],"dependencies":["ab8b840systems5a-h3"],"title":"Addition Method","text":"What is the resulting equation when you add the first and second equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["ab8b840systems5a-h4"],"title":"Isolating Variables","text":"What is $$y$$ equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems5a-h6","type":"hint","dependencies":["ab8b840systems5a-h5"],"title":"Substitution","text":"Substitute $$y=-4$$ into one of the original equations and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems5a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ab8b840systems5a-h6"],"title":"Isolating Variables","text":"What is $$x$$ equal to in 3x+5(-4)=-11","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems5a-h8","type":"hint","dependencies":["ab8b840systems5a-h7"],"title":"Checking the Solution","text":"Our solution is the ordered pair $$(3,-4)$$. Check the solution in the original second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems5a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11$$"],"dependencies":["ab8b840systems5a-h8"],"title":"Checking the Solution","text":"What does $$(3)-2(-4)$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8b840systems6","title":"Solving Systems of Equations in Two Variables by the Addition Method","body":"Solve the given system of equations by addition.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Systems of Linear Equations: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8b840systems6a","stepAnswer":["$$(-2,-4)$$"],"problemType":"MultipleChoice","stepTitle":"$$2x+3y=-16$$ $$5x-10y=30$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-2,-4)$$","choices":["$$(-2,-4)$$","$$(2,4)$$","$$(-4,-2)$$","$$(4,2)$$"],"hints":{"DefaultPathway":[{"id":"ab8b840systems6a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10x$$"],"dependencies":[],"title":"Common Multiple","text":"One equation has $$2x$$ and the other has $$5x$$. What is the common multiple between $$2x$$ and $$5x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems6a-h2","type":"hint","dependencies":["ab8b840systems6a-h1"],"title":"Multiplying Equations","text":"Multiply both equations by a constant in order to eliminate one variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["ab8b840systems6a-h2"],"title":"Multiplying Equations","text":"What should you mutliply to the first equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ab8b840systems6a-h3"],"title":"Multiplying Equations","text":"What should you mutliply to the second equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems6a-h5","type":"hint","dependencies":["ab8b840systems6a-h4"],"title":"Addition Method","text":"Add the two equations together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems6a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-35y=140$$"],"dependencies":["ab8b840systems6a-h5"],"title":"Addition Method","text":"What is the resulting equation after you add both equations together?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems6a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["ab8b840systems6a-h6"],"title":"Isolating Variables","text":"What is $$y$$ equal in $$-35y=140$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems6a-h8","type":"hint","dependencies":["ab8b840systems6a-h7"],"title":"Substitution","text":"Substitute $$y=-4$$ into the original first equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems6a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["ab8b840systems6a-h8"],"title":"Isolating Variables","text":"What is $$x$$ equal to in 2x+3(-4)=-16?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems6a-h10","type":"hint","dependencies":["ab8b840systems6a-h9"],"title":"Checking the Solution","text":"Our solution is the ordered pair $$(-2,-4)$$. Check the solution in the original second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems6a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$30$$"],"dependencies":["ab8b840systems6a-h10"],"title":"Checking the Solution","text":"What does $$5(-2)-10(-4)$$ equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8b840systems7","title":"Solving Systems of Equations in Two Variables","body":"Solve the given system of equations.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Systems of Linear Equations: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8b840systems7a","stepAnswer":["No Solution"],"problemType":"MultipleChoice","stepTitle":"$$x=9-2y$$ $$x+2y=13$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$(3,5)$$","$$(6,8)$$","Infinite Solutions","No Solution"],"hints":{"DefaultPathway":[{"id":"ab8b840systems7a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["substitution method"],"dependencies":[],"title":"Determining the Method","text":"Because one equation is already solved for $$x$$, which method should you use to solve the system?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["substitution method","subtraction method"]},{"id":"ab8b840systems7a-h2","type":"hint","dependencies":["ab8b840systems7a-h1"],"title":"Substitution","text":"Substitute the first equation into the second one.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["ab8b840systems7a-h2"],"title":"Substitution","text":"What does $$9-2y+2y$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems7a-h4","type":"hint","dependencies":["ab8b840systems7a-h3"],"title":"No Solution Equations","text":"Clearly, this statement is a contradiction because $$9 \\\\neq 13$$. Therefore, the system has no solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems7a-h5","type":"hint","dependencies":["ab8b840systems7a-h4"],"title":"Slope-Intercept Form","text":"We can take a different approach to this problem. Manipulate the equations so that they are both in slope-intercept form. $$y=mx+b$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems7a-h6","type":"hint","dependencies":["ab8b840systems7a-h5"],"title":"Slope-Intercept Form","text":"Isolate the $$y-variable$$ in the first equation to find the slope-intercept form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems7a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(-\\\\frac{1}{2}\\\\right) x+\\\\frac{9}{2}$$"],"dependencies":["ab8b840systems7a-h6"],"title":"Slope-Intercept Form","text":"What does $$y$$ equal when you manipulate the first equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems7a-h8","type":"hint","dependencies":["ab8b840systems7a-h7"],"title":"Slope-Intercept Form","text":"Isolate the $$y-variable$$ in the second equation to find the slope-intercept form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems7a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(-\\\\frac{1}{2}\\\\right) x+\\\\frac{13}{2}$$"],"dependencies":["ab8b840systems7a-h8"],"title":"Slope-Intercept Form","text":"What does $$y$$ equal when you manipulate the second equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems7a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{2}$$"],"dependencies":["ab8b840systems7a-h9"],"title":"Slope-Intercept Form","text":"What is the value of the slope for both equations?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems7a-h11","type":"hint","dependencies":["ab8b840systems7a-h10"],"title":"Slope-Intercept Form","text":"Comparing the equations, we see that they have the same slope but different y-intercepts. Therefore, the lines are parallel and do not intersect.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab8b840systems7b","stepAnswer":["$$(4,2)$$ is not a solution to the system of equations."],"problemType":"MultipleChoice","stepTitle":"$$5x-y=4$$, $$x+6y=2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(4,2)$$ is not a solution to the system of equations.","choices":["$$(4,2)$$ is a solution to the system of equations.","$$(4,2)$$ is not a solution to the system of equations.","$$(4,2)$$ is not a solution to the system of equations."],"hints":{"DefaultPathway":[{"id":"ab8b840systems7b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"First, substitute the given point into the equation. If the equality that results is true, then the point is a solution to the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems7b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ab8b840systems7b-h1"],"title":"Substitute","text":"When the point is substituted into the equation $$5x-y=4$$, do both sides equal each other?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ab8b840systems7b-h3","type":"hint","dependencies":["ab8b840systems7b-h2"],"title":"Interpret","text":"Since the equation is not equal on both $$sides(18=4)$$, the ordered pair is not a solution to the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8b840systems8","title":"Finding a Solution to a Dependent System of Linear Equations","body":"Find a solution to the system of equations using the addition method.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Systems of Linear Equations: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8b840systems8a","stepAnswer":["$$(x,\\\\left(-\\\\frac{1}{3}\\\\right) x+\\\\frac{2}{3})$$"],"problemType":"MultipleChoice","stepTitle":"$$x+3y=2$$, $$3x+9y=6$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(x,\\\\left(-\\\\frac{1}{3}\\\\right) x+\\\\frac{2}{3})$$","choices":["((-1/3)*x+(2/3), y)","None of the above","$$(x,\\\\left(-\\\\frac{1}{3}\\\\right) x+\\\\frac{2}{3})$$","$$(x,\\\\left(-\\\\frac{1}{3}\\\\right) x+\\\\frac{2}{3})$$"],"hints":{"DefaultPathway":[{"id":"ab8b840systems8a-h1","type":"hint","dependencies":[],"title":"Slope-Intercept Form","text":"If we rewrote both equations in the slope-intercept form, we might know what the solution would look like before adding. Rewrite both equations in slope-intercept form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(-\\\\frac{1}{3}\\\\right) x+\\\\frac{2}{3}$$"],"dependencies":["ab8b840systems8a-h1"],"title":"Slope-Intercept Form","text":"What does $$y$$ equal when you manipulate the first equation to be in slope-intercept form?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(-\\\\frac{1}{3}\\\\right) x+\\\\frac{2}{3}$$"],"dependencies":["ab8b840systems8a-h2"],"title":"Slope-Intercept Form","text":"What does $$y$$ equal when you manipulate the second equation to be in slope-intercept form?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems8a-h4","type":"hint","dependencies":["ab8b840systems8a-h3"],"title":"General Solution","text":"Notice that the results are the same. Find the general solution to the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems8a-h5","type":"hint","dependencies":["ab8b840systems8a-h4"],"title":"General Solution","text":"Since $$y$$ is equal to the same thing in both equations, $$y$$ is equal to $$\\\\left(-\\\\frac{1}{3}\\\\right) x+\\\\frac{2}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab8b840systems8b","stepAnswer":["$$(-6,1)$$ is a solution to the system of equations."],"problemType":"MultipleChoice","stepTitle":"$$-3x-5y=13$$, $$-x+4y=10$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-6,1)$$ is a solution to the system of equations.","choices":["$$(-6,1)$$ is a solution to the system of equations.","$$(-6,1)$$ is a solution to the system of equations.","$$(-6,1)$$ is not a solution to the system of equations."],"hints":{"DefaultPathway":[{"id":"ab8b840systems8b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"First, substitute the given point into the equation. If the equality that results is true, then the point is a solution to the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems8b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ab8b840systems8b-h1"],"title":"Substitute","text":"When the point is substituted into the equation $$-3x-5y=13$$, do both sides equal each other?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"],"subHints":[{"id":"ab8b840systems8b-h2-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":[],"title":"Substitute","text":"When the point is substituted into the equation $$-x+4y=10$$, do both sides equal each other?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]},{"id":"ab8b840systems8b-h3","type":"hint","dependencies":["ab8b840systems8b-h2"],"title":"Interpret","text":"Therefore, since the equation is equal on both sides for both equations, the point satisfies the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ab8b840systems9","title":"Finding the Break-Even Point and the Profit Function Using Substitution","body":"Given the cost function \ud835\udc36(x)=0.85x+35,000 and the revenue function $$\ud835\udc45(x)=1.55x$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Systems of Linear Equations: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ab8b840systems9a","stepAnswer":["$$P(x)=0.7x-35000$$"],"problemType":"MultipleChoice","stepTitle":"Find the break-even point and the profit function.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$P(x)=0.7x-35000$$","choices":["$$P(x)=-3500x-0.7$$","$$P(x)=0.7x-35000$$","$$P(x)=3500x$$ + $$0.7$$","$$P(x)=-0.7x+3500$$","$$P(x)=0.7x-35000$$"],"hints":{"DefaultPathway":[{"id":"ab8b840systems9a-h1","type":"hint","dependencies":[],"title":"Function Notation","text":"Write the system of equations using $$y$$ to replace function notation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems9a-h2","type":"hint","dependencies":["ab8b840systems9a-h1"],"title":"Substitution","text":"Substitute the expression 0.85x+35,000 from the first equation into the second equation and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$50000$$"],"dependencies":["ab8b840systems9a-h2"],"title":"Isolating Variables","text":"What is $$x$$ equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems9a-h4","type":"hint","dependencies":["ab8b840systems9a-h3"],"title":"Substitution","text":"Substitute $$x=50, 000$$ into either the cost function or the revenue function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems9a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$77500$$"],"dependencies":["ab8b840systems9a-h4"],"title":"Isolating Variables","text":"What is $$y$$ equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems9a-h6","type":"hint","dependencies":["ab8b840systems9a-h5"],"title":"Break-Even Point","text":"The break-even point is (50,000, 77,500).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems9a-h7","type":"hint","dependencies":["ab8b840systems9a-h6"],"title":"Profit Function Formula","text":"The profit function is found using the formula $$P(x)=R(x)-C(x)$$. Plug in the $$x$$ and $$y$$ values to find the profit function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems9a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$P(x)=0.7x-35000$$"],"dependencies":["ab8b840systems9a-h7"],"title":"Profit Function Formula","text":"What is the profit function?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ab8b840systems9b","stepAnswer":["$$(2,3)$$ is not a solution to the system of equations."],"problemType":"MultipleChoice","stepTitle":"$$3x+7y=1$$, $$2x+4y=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(2,3)$$ is not a solution to the system of equations.","choices":["$$(2,3)$$ is a solution to the system of equations.","$$(2,3)$$ is not a solution to the system of equations."],"hints":{"DefaultPathway":[{"id":"ab8b840systems9b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"First, substitute the given point into the equation. If the equality that results is true, then the point is a solution to the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ab8b840systems9b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ab8b840systems9b-h1"],"title":"Substitute","text":"When the point is substituted into the equation $$3x+7y=1$$, do both sides equal each other?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ab8b840systems9b-h3","type":"hint","dependencies":["ab8b840systems9b-h2"],"title":"Interpret","text":"Since the equation is not equal on both $$sides(27=1)$$, the ordered pair is not a solution to the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abc1682approximating1","title":"Using Sigma Notation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.1  Approximating Areas","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"abc1682approximating1a","stepAnswer":["$$120$$"],"problemType":"TextBox","stepTitle":"Write in sigma notation and evaluate the sum of terms $$2^i$$ for $$i=3, 4, 5, 6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$120$$","hints":{"DefaultPathway":[{"id":"abc1682approximating1a-h1","type":"hint","dependencies":[],"title":"Sigma Notation","text":"sum{i\\\\=3}{6}{2**i}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["abc1682approximating1a-h1"],"title":"Lower bound","text":"What is the lower bound of the summation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"abc1682approximating1a-h2-s1","type":"hint","dependencies":[],"title":"Lower bound","text":"Lower bound indicates the first term or initial term of the sequence. We begin to create a first term by substituting this value into the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"abc1682approximating1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["abc1682approximating1a-h1"],"title":"Upper bound","text":"What is the upper bound of the summation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"abc1682approximating1a-h3-s1","type":"hint","dependencies":[],"title":"Upper bound","text":"Upper bound indicates the last term or ending term of the sequence. We stop substituting after this value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"abc1682approximating1a-h4","type":"hint","dependencies":["abc1682approximating1a-h1","abc1682approximating1a-h2","abc1682approximating1a-h3"],"title":"Evaluate the sum","text":"We substitute $$i=3, i=4, i=5, i=6$$ into the expression and sum all the resulting terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating1a-h5","type":"hint","dependencies":["abc1682approximating1a-h4"],"title":"Evaluate the sum","text":"sum{i\\\\=3}{6}{2**i}=(2**3)+(2**4)+(2**5)+(2**6)=8+16+32+64=120","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"abc1682approximating10","title":"In the following exercises, use summation properties and formulas to rewrite and evaluate the sums.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.1  Approximating Areas","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"abc1682approximating10a","stepAnswer":["$$40375$$"],"problemType":"TextBox","stepTitle":"sum{j\\\\=1}{50}{((j**2)-2*j)}","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$40375$$","hints":{"DefaultPathway":[{"id":"abc1682approximating10a-h1","type":"hint","dependencies":[],"title":"Properties of Sigma Notation","text":"We firstly separate $$2$$ terms In the sum then use the Properties of Sigma Notation to calculate them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating10a-h2","type":"hint","dependencies":["abc1682approximating10a-h1"],"title":"Properties of Sigma Notation","text":"We can rewrite the sum as sum{j\\\\=1}{50}{j**2)}-2*sum{j\\\\=1}{50}{j}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$42925$$"],"dependencies":["abc1682approximating10a-h2"],"title":"Evaluate the sum","text":"What is the answer for the term sum{j\\\\=1}{50}{j**2)} ?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating10a-h4","type":"hint","dependencies":["abc1682approximating10a-h3"],"title":"Evaluate the sum","text":"Using Sums and Powers of Integers to obtain $$\\\\frac{50\\\\times51\\\\times101}{6}=42925$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2550$$"],"dependencies":["abc1682approximating10a-h4"],"title":"Evaluate the sum","text":"What is the answer for th term 2*sum{j\\\\=1}{50}{j} ?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating10a-h6","type":"hint","dependencies":["abc1682approximating10a-h5"],"title":"Evaluate the sum","text":"Using Sums and Powers of Integers to obtain $$\\\\frac{50\\\\times51\\\\times2}{2}=2550$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating10a-h7","type":"hint","dependencies":["abc1682approximating10a-h6"],"title":"Evaluate the sum","text":"$$42925-2550=40375$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"abc1682approximating11","title":"Let $$L_n$$ denote the left-endpoint sum using $$n$$ subintervals and let $$R_n$$ denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and right sums for the given functions on the indicated interval.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.1  Approximating Areas","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"abc1682approximating11a","stepAnswer":["$$-0.25$$"],"problemType":"TextBox","stepTitle":"$$R_4$$ for $$g(x)=cos\\\\left(\\\\pi x\\\\right)$$ on [0,1]","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-0.25$$","hints":{"DefaultPathway":[{"id":"abc1682approximating11a-h1","type":"hint","dependencies":[],"title":"Subinterval","text":"The first step is to divide the interval from $$0$$ to $$1$$ into $$n$$ equal subintervals. These subintervals can be seen as rectangles.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["abc1682approximating11a-h1"],"title":"Subinterval","text":"How many subinterval or rectangle can we create in this problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"abc1682approximating11a-h2-s1","type":"hint","dependencies":[],"title":"Subinterval","text":"Since $$n=4$$ is given in this problem, we can divide the interval from $$x=0$$ to $$x=1$$ into $$4$$ rectangles.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"abc1682approximating11a-h3","type":"hint","dependencies":["abc1682approximating11a-h2"],"title":"Find the step size","text":"The step size or width of the rectangles can be found using $$\u0394x=\\\\frac{b-a}{n}$$ where $$b$$ is the upper bound, a is the lower bound and $$n$$ is the number of rectangles.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["abc1682approximating11a-h3"],"title":"Find the step size","text":"What is the upper bound of the interval?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["abc1682approximating11a-h3"],"title":"Find the step size","text":"What is the lower bound of the interval?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating11a-h6","type":"hint","dependencies":["abc1682approximating11a-h3"],"title":"Find the step size","text":"For the $$4$$ rectangles, each rectangle has a width of $$\u0394x=\\\\frac{1-0}{4}=0.25$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating11a-h7","type":"hint","dependencies":["abc1682approximating11a-h6"],"title":"Right-endpoint approximation","text":"The right rectangle approximation is when you make the right hand points of the pieces the height of the rectangles. In other words, the height of each rectangle is equal to the value of the function at the right endpoint of its base.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating11a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["abc1682approximating11a-h7"],"title":"Right-endpoint approximation","text":"Using the right-endpoint approximation, what is the height of a rectangle in the interval $$[0.25, 0.5]$$ ?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating11a-h9","type":"hint","dependencies":["abc1682approximating11a-h8"],"title":"Right-endpoint approximation","text":"Using a right-endpoint approximation, the heights are $$f(0.25)=\\\\frac{\\\\sqrt{2}}{2}$$, $$f(0.5)=0$$, $$f(0.75)=\\\\frac{-\\\\sqrt{2}}{2}$$, $$f(1)=-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating11a-h10","type":"hint","dependencies":["abc1682approximating11a-h9"],"title":"Find the area of the rectangles","text":"With the width and height just found, we can easily find the area of each rectangle using area of a rectangle formula $$A=w h$$ then add all of them up to get an approximation for the area under the curve.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating11a-h11","type":"hint","dependencies":["abc1682approximating11a-h10"],"title":"Approximating the Area","text":"Starting with $$x=0.25$$ from the righ-hand point of the first rectangles, we have $$A=f{\\\\left(x_1\\\\right)} \\\\Delta x+f{\\\\left(x_2\\\\right)} \\\\Delta x+f{\\\\left(x_3\\\\right)} \\\\Delta x+f{\\\\left(x_4\\\\right)} \\\\Delta x=\\\\Delta x \\\\left(f{\\\\left(x_1\\\\right)}+f{\\\\left(x_2\\\\right)}+f{\\\\left(x_3\\\\right)}+f{\\\\left(x_4\\\\right)}\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating11a-h12","type":"hint","dependencies":["abc1682approximating11a-h11"],"title":"Approximating the Area","text":"$$0.25\\\\left(\\\\frac{\\\\sqrt{2}}{2}+0-\\\\frac{\\\\sqrt{2}}{2}-1\\\\right)=-0.25$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"abc1682approximating12","title":"Let $$L_n$$ denote the left-endpoint sum using $$n$$ subintervals and let $$R_n$$ denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and right sums for the given functions on the indicated interval.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.1  Approximating Areas","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"abc1682approximating12a","stepAnswer":["$$0.6875$$"],"problemType":"TextBox","stepTitle":"$$L_8$$ for $$g(x)=x^2-2x+1$$ on [0,2]","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.6875$$","hints":{"DefaultPathway":[{"id":"abc1682approximating12a-h1","type":"hint","dependencies":[],"title":"Subinterval","text":"The first step is to divide the interval from $$0$$ to $$1$$ into $$n$$ equal subintervals. These subintervals can be seen as rectangles.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["abc1682approximating12a-h1"],"title":"Subinterval","text":"How many subinterval or rectangle can we create in this problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"abc1682approximating12a-h2-s1","type":"hint","dependencies":[],"title":"Subinterval","text":"Since $$n=8$$ is given in this problem, we can divide the interval from $$x=0$$ to $$x=2$$ into $$8$$ rectangles.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"abc1682approximating12a-h3","type":"hint","dependencies":["abc1682approximating12a-h2"],"title":"Find the step size","text":"The step size or width of the rectangles can be found using $$\u0394x=\\\\frac{b-a}{n}$$ where $$b$$ is the upper bound, a is the lower bound and $$n$$ is the number of rectangles.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["abc1682approximating12a-h3"],"title":"Find the step size","text":"What is the upper bound of the interval?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating12a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["abc1682approximating12a-h3"],"title":"Find the step size","text":"What is the lower bound of the interval?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating12a-h6","type":"hint","dependencies":["abc1682approximating12a-h3"],"title":"Find the step size","text":"For the $$8$$ rectangles, each rectangle has a width of $$\u0394x=\\\\frac{2-0}{8}=0.25$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating12a-h7","type":"hint","dependencies":["abc1682approximating12a-h6"],"title":"Left-endpoint approximation","text":"The left rectangle approximation is when you make the left hand points of the pieces the height of the rectangles. In other words, the height of each rectangle is equal to the value of the function at the left endpoint of its base.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating12a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.5625$$"],"dependencies":["abc1682approximating12a-h7"],"title":"Left-endpoint approximation","text":"Using the left-endpoint approximation, what is the height of a rectangle in the interval $$[1.75, 2]$$ ?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating12a-h9","type":"hint","dependencies":["abc1682approximating12a-h8"],"title":"Left-endpoint approximation","text":"Using a left-endpoint approximation, the heights of each rectangle are $$f(0)=1, f(0.25)=0.5625, f(0.5)=0.25, f(0.75)=0.0625, f(1)=0, f(1.25)=0.0625, f(1.5)=0.25, f(1.75)=0.5625$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating12a-h10","type":"hint","dependencies":["abc1682approximating12a-h9"],"title":"Find the area of the rectangles","text":"With the width and height just found, we can easily find the area of each rectangle using area of a rectangle formula $$A=w h$$ then add all of them up to get an approximation for the area under the curve $$0.6875$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating12a-h11","type":"hint","dependencies":["abc1682approximating12a-h10"],"title":"Approximating the Area","text":"Starting with $$x=0$$ from the left-hand point of the first rectangles, we have $$A=f{\\\\left(x_1\\\\right)} \\\\Delta x+f{\\\\left(x_2\\\\right)} \\\\Delta x+f{\\\\left(x_3\\\\right)} \\\\Delta x+f{\\\\left(x_4\\\\right)} \\\\Delta x+f{\\\\left(x_5\\\\right)} \\\\Delta x+f{\\\\left(x_6\\\\right)} \\\\Delta x+f{\\\\left(x_7\\\\right)} \\\\Delta x+f{\\\\left(x_8\\\\right)} \\\\Delta x=\\\\Delta x \\\\left(f{\\\\left(x_1\\\\right)}+f{\\\\left(x_2\\\\right)}+f{\\\\left(x_3\\\\right)}+f{\\\\left(x_4\\\\right)}+f{\\\\left(x_5\\\\right)}+f{\\\\left(x_6\\\\right)}+f{\\\\left(x_7\\\\right)}+f{\\\\left(x_8\\\\right)}\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating12a-h12","type":"hint","dependencies":["abc1682approximating12a-h11"],"title":"Approximating the Area","text":"$$0.25\\\\left(1+0.5625+0.25+0.0625+0+0.0625+0.25+0.5625\\\\right)=0.6875$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"abc1682approximating13","title":"Section $$5.1$$ Exercises","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.1  Approximating Areas","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"abc1682approximating13a","stepAnswer":["$$24$$"],"problemType":"TextBox","stepTitle":"In the following exercises, estimate the areas under the curves by computing the left Riemann sums, $$L_8$$.","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$24$$","hints":{"DefaultPathway":[{"id":"abc1682approximating13a-h1","type":"hint","dependencies":[],"title":"Subinterval","text":"The first step is to divide the interval from $$0$$ to $$1$$ into $$n$$ equal subintervals. These subintervals can be seen as rectangles.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["abc1682approximating13a-h1"],"title":"Subinterval","text":"How many subinterval or rectangle can we create in this problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"abc1682approximating13a-h2-s1","type":"hint","dependencies":[],"title":"Subinterval","text":"Since $$n=8$$ is given in this problem, we can divide the interval from $$x=0$$ to $$x=8$$ into $$8$$ rectangles.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"abc1682approximating13a-h3","type":"hint","dependencies":["abc1682approximating13a-h2"],"title":"Find the step size","text":"The step size or width of the rectangles can be found using $$\u0394x=\\\\frac{b-a}{n}$$ where $$b$$ is the upper bound, a is the lower bound and $$n$$ is the number of rectangles.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["abc1682approximating13a-h3"],"title":"Find the step size","text":"What is the upper bound of the interval?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating13a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["abc1682approximating13a-h3","abc1682approximating13a-h4"],"title":"Find the step size","text":"What is the lower bound of the interval?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating13a-h6","type":"hint","dependencies":["abc1682approximating13a-h3","abc1682approximating13a-h4","abc1682approximating13a-h5"],"title":"Find the step size","text":"For the $$8$$ rectangles, each rectangle has a width of $$\u0394x=\\\\frac{8-0}{8}=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating13a-h7","type":"hint","dependencies":["abc1682approximating13a-h6"],"title":"Left-endpoint approximation","text":"The left rectangle approximation is when you make the left hand points of the pieces the height of the rectangles. In other words, the height of each rectangle is equal to the value of the function at the left endpoint of its base.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating13a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["abc1682approximating13a-h7"],"title":"Left-endpoint approximation","text":"Using the left-endpoint approximation, what is the height of a rectangle in the interval [4,5] ?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating13a-h9","type":"hint","dependencies":["abc1682approximating13a-h8"],"title":"Left-endpoint approximation","text":"Using a left-endpoint approximation, the heights of each rectangle are $$f(0)=3, f(1)=2, f(2)=1, f(3)=2, f(4)=3, f(5)=4, f(6)=5, f(7)=4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating13a-h10","type":"hint","dependencies":["abc1682approximating13a-h9"],"title":"Find the area of the rectangles","text":"With the width and height just found, we can easily find the area of each rectangle using area of a rectangle formula $$A=w h$$ then add all of them up to get an approximation for the area under the curve.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating13a-h11","type":"hint","dependencies":["abc1682approximating13a-h10"],"title":"Approximating the Area","text":"Starting with $$x=0$$ from the left-hand point of the first rectangles, we have $$A=f{\\\\left(x_1\\\\right)} \\\\Delta x+f{\\\\left(x_2\\\\right)} \\\\Delta x+f{\\\\left(x_3\\\\right)} \\\\Delta x+f{\\\\left(x_4\\\\right)} \\\\Delta x+f{\\\\left(x_5\\\\right)} \\\\Delta x+f{\\\\left(x_6\\\\right)} \\\\Delta x+f{\\\\left(x_7\\\\right)} \\\\Delta x+f{\\\\left(x_8\\\\right)} \\\\Delta x=\\\\Delta x \\\\left(f{\\\\left(x_1\\\\right)}+f{\\\\left(x_2\\\\right)}+f{\\\\left(x_3\\\\right)}+f{\\\\left(x_4\\\\right)}+f{\\\\left(x_5\\\\right)}+f{\\\\left(x_6\\\\right)}+f{\\\\left(x_7\\\\right)}+f{\\\\left(x_8\\\\right)}\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating13a-h12","type":"hint","dependencies":["abc1682approximating13a-h11"],"title":"Approximating the Area","text":"$$1\\\\left(3+2+1+2+3+4+5+4\\\\right)=24$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"abc1682approximating14","title":"Suppose that sum{i\\\\=1}{100}{a_1)}=15 and sum{i\\\\=1}{100}{b_i)}=-12. In the following exercises,compute the sums.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.1  Approximating Areas","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"abc1682approximating14a","stepAnswer":["$$27$$"],"problemType":"TextBox","stepTitle":"sum{i\\\\=1}{100}{(5a_i)+(4b_i)}","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$27$$","hints":{"DefaultPathway":[{"id":"abc1682approximating14a-h1","type":"hint","dependencies":[],"title":"Properties of Sigma Notation","text":"We use the properties of sigma notation to separate the $$2$$ terms of the sum then substitue each term with the given information to calculate the sum.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating14a-h2","type":"hint","dependencies":["abc1682approximating14a-h1"],"title":"Properties of Sigma Notation","text":"sum{i\\\\=1}{100}{(5a_i)}+sum{i\\\\=1}{100}{(4b_i)}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating14a-h3","type":"hint","dependencies":["abc1682approximating14a-h2"],"title":"Substitution","text":"sum{i\\\\=1}{100}{(5a_i)}+sum{i\\\\=1}{100}{(4b_i)}=5(15)+4(-12)=27","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"abc1682approximating15","title":"In the following exercises, use summation properties and formulas to rewrite and evaluate the sums.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.1  Approximating Areas","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"abc1682approximating15a","stepAnswer":["$$-10400$$"],"problemType":"TextBox","stepTitle":"sum{k\\\\=1}{25}{(((2*k)**2)-100*k)}","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-10400$$","hints":{"DefaultPathway":[{"id":"abc1682approximating15a-h1","type":"hint","dependencies":[],"title":"Properties of Sigma Notation","text":"We firstly separate $$2$$ terms In the sum then use the Properties of Sigma Notation to calculate them.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating15a-h2","type":"hint","dependencies":["abc1682approximating15a-h1"],"title":"Properties of Sigma Notation","text":"We can rewrite the sum as 2**2*sum{k\\\\=1}{25}{k**2)}-100*sum{j\\\\=1}{25}{k}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$22100$$"],"dependencies":["abc1682approximating15a-h2"],"title":"Evaluate the sum","text":"What is the answer for the term 4*sum{k\\\\=1}{25}{k**2)} ?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating15a-h4","type":"hint","dependencies":["abc1682approximating15a-h3"],"title":"Evaluate the sum","text":"Using Sums and Powers of Integers to obtain $$\\\\frac{50\\\\times51\\\\times101}{6}=42925$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating15a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$32500$$"],"dependencies":["abc1682approximating15a-h4"],"title":"Evaluate the sum","text":"What is the answer for th term 100*sum{k\\\\=1}{25}{k} ?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating15a-h6","type":"hint","dependencies":["abc1682approximating15a-h5"],"title":"Evaluate the sum","text":"Using Sums and Powers of Integers to obtain $$\\\\frac{50\\\\times51\\\\times2}{2}=2550$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating15a-h7","type":"hint","dependencies":["abc1682approximating15a-h6"],"title":"Evaluate the sum","text":"$$22100-32500=-10400$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"abc1682approximating2","title":"Evaluation Using Sigma Notation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.1  Approximating Areas","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"abc1682approximating2a","stepAnswer":["$$2567900$$"],"problemType":"TextBox","stepTitle":"Write using sigma notation and evaluate:","stepBody":"a. The sum of the terms $${\\\\left(i-3\\\\right)}^2$$ for $$i=1, 2, \u2026, 200$$.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2567900$$","hints":{"DefaultPathway":[{"id":"abc1682approximating2a-h1","type":"hint","dependencies":[],"title":"Sigma Notation","text":"We express a sigma notation using the initial number as a lower bound and the ending number as an upper bound. The expression is given in this case sum{i\\\\=1}{200}{(i-3)**2}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating2a-h2","type":"hint","dependencies":["abc1682approximating2a-h1"],"title":"Expand","text":"According to the expansion of the perfect square formula, the summation can be written as sum{i\\\\=1}{200}{(i**2)-(6*i)+9}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating2a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["sum{i\\\\=1}{200}{(i**2)}-sum{i\\\\=1}{200}{(6*i)}+sum{i\\\\=1}{200}{9}"],"dependencies":["abc1682approximating2a-h2"],"title":"Properties of Sigma Notation","text":"How can we separate the resulting three terms using the properties of sigma notation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["sum{i\\\\=1}{200}{(i**2)}-sum{i\\\\=1}{200}{(6*i)}+sum{i\\\\=1}{200}{9}","sum{i\\\\=1}{200}{(i**2)}-sum{i\\\\=1}{200}{(6*i)}"],"subHints":[{"id":"abc1682approximating2a-h3-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["6*sum{i\\\\=1}{200}{i}"],"dependencies":[],"title":"Properties of Sigma Notation","text":"How can we isolate the constant in the second term sum{i\\\\=1}{200}{(6*i)} using the properties of sigma notation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["6*sum{i\\\\=1}{200}{i}","sum{i\\\\=1}{200}{i}"]}]},{"id":"abc1682approximating2a-h4","type":"hint","dependencies":["abc1682approximating2a-h3"],"title":"Summation Rules","text":"Applying summation rules to obtain $$\\\\frac{200\\\\left(200+1\\\\right) \\\\left(400+1\\\\right)}{6}-\\\\frac{6\\\\left(200\\\\left(200+1\\\\right)\\\\right)}{2}+9\\\\times200$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating2a-h5","type":"hint","dependencies":["abc1682approximating2a-h4"],"title":"Simplify","text":"2,686,700-120,600+1800=2567900","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"abc1682approximating3","title":"Finding the Sum of the Function Values","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.1  Approximating Areas","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"abc1682approximating3a","stepAnswer":["$$440$$"],"problemType":"TextBox","stepTitle":"Evaluate the sum indicated by the notation sum{k\\\\=1}{20}{(2*k+1)}","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$440$$","hints":{"DefaultPathway":[{"id":"abc1682approximating3a-h1","type":"hint","dependencies":[],"title":"General guide","text":"We begin to solve the problem by separating the summation into $$2$$ terms then applying the Property of Sigma Notation to each separated term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating3a-h2","type":"hint","dependencies":["abc1682approximating3a-h1"],"title":"Seperate the terms","text":"sum{k\\\\=1}{20}{(2*k)}+sum{k\\\\=1}{20}{(1)}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating3a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$420$$"],"dependencies":["abc1682approximating3a-h1","abc1682approximating3a-h2"],"title":"Properties of Sigma Notation","text":"What is the answer of the first term after applying Properties of Sigma Notation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$40$$","$$840$$","$$420$$","$$400$$"],"subHints":[{"id":"abc1682approximating3a-h3-s1","type":"hint","dependencies":[],"title":"Properties of Sigma Notation","text":"$$\\\\frac{2\\\\times20\\\\left(20+1\\\\right)}{2}=20\\\\times21=420$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"abc1682approximating3a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$20$$"],"dependencies":["abc1682approximating3a-h1","abc1682approximating3a-h2"],"title":"Properties of Sigma Notation","text":"What is the answer of the second term after applying Properties of Sigma Notation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$20$$","$$40$$","$$400$$","$$10$$"],"subHints":[{"id":"abc1682approximating3a-h4-s1","type":"hint","dependencies":[],"title":"Properties of Sigma Notation","text":"$$20\\\\times1=20$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating3a-h4-s2","type":"hint","dependencies":["abc1682approximating3a-h4-s1"],"title":"Properties of Sigma Notation","text":"Using the formulas: sum{k\\\\=1}{n}{(k)}=k*c","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"abc1682approximating3a-h5","type":"hint","dependencies":["abc1682approximating3a-h3","abc1682approximating3a-h4"],"title":"Combining $$2$$ resulted terms","text":"$$\\\\frac{2\\\\times20\\\\left(20+1\\\\right)}{2}+20\\\\times1=420+20=440$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"abc1682approximating4","title":"Approximating the Area Under a Curve","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.1  Approximating Areas","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"abc1682approximating4a","stepAnswer":["$$1.75$$"],"problemType":"TextBox","stepTitle":"Use left-endpoint approximations to approximate the area under the curve of $$f(x)=x^2$$ on the interval $$[0.2];$$ use $$n=4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1.75$$","hints":{"DefaultPathway":[{"id":"abc1682approximating4a-h1","type":"hint","dependencies":[],"title":"Subinterval","text":"The first step is to divide the interval from $$0$$ to $$2$$ into $$n$$ equal subintervals. These subintervals can be seen as rectangles.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["abc1682approximating4a-h1"],"title":"Subinterval","text":"How many subinterval or rectangle can we create in this problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"abc1682approximating4a-h2-s1","type":"hint","dependencies":[],"title":"Subinterval","text":"Since $$n=4$$ is given in this problem, we can divide the interval from $$x=0$$ to $$x=2$$ into $$4$$ rectangles.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"abc1682approximating4a-h3","type":"hint","dependencies":["abc1682approximating4a-h2"],"title":"Find the step size","text":"The step size or width of the rectangles can be found using $$\u0394x=\\\\frac{b-a}{n}$$ where $$b$$ is the upper bound, a is the lower bound and $$n$$ is the number of rectangles.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["abc1682approximating4a-h3"],"title":"Find the step size","text":"What is the upper bound of the interval?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating4a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["abc1682approximating4a-h3"],"title":"Find the step size","text":"What is the lower bound of the interval?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating4a-h6","type":"hint","dependencies":["abc1682approximating4a-h3"],"title":"Find the step size","text":"For the $$4$$ rectangles, each rectangle has a width of $$\u0394x=\\\\frac{2-0}{4}=0.5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating4a-h7","type":"hint","dependencies":["abc1682approximating4a-h6"],"title":"Find the start and end points of each rectangle","text":"We begin with the first rectangle whose starting point of $$x=0$$ and a width of $$0.5$$. The rectangle will end at the point $$x=0.5$$ and that endpoint of the current rectangle is also the starting point of the next rectangle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating4a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$[1, 1.5]$$"],"dependencies":["abc1682approximating4a-h7"],"title":"Find the start and end points of each rectangle","text":"Express in interval notation, what are the starting and ending points of the 3rd rectangle?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$(1, 1.5)$$","$$[1, 1.5]$$"]},{"id":"abc1682approximating4a-h9","type":"hint","dependencies":["abc1682approximating4a-h8"],"title":"Find the start and end points of each rectangle","text":"Continuing the process, we obtain $$4$$ intervals which are $$[0, 0.5], [0.5, 1], [1, 1.5]$$ and $$[1.5, 2]$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating4a-h10","type":"hint","dependencies":["abc1682approximating4a-h9"],"title":"Left-endpoint approximation","text":"The left rectangle approximation is when you make the left hand points of the pieces the height of the rectangles. In other words, the height of each rectangle is equal to the value of the function at the left endpoint of its base.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating4a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$0.25$$"],"dependencies":["abc1682approximating4a-h10"],"title":"Left-endpoint approximation","text":"Using the left-endpoint approximation, what is the height of a rectangle in the interval $$[0.5, 1]$$ ?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$0.5$$","$$1$$","$$0.25$$","$$1.75$$"]},{"id":"abc1682approximating4a-h12","type":"hint","dependencies":["abc1682approximating4a-h11"],"title":"Left-endpoint approximation","text":"Using a left-endpoint approximation, the heights are $$f(0)=0, f(0.5)=0.25, f(1)=1, f(1.5)=2.25$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating4a-h13","type":"hint","dependencies":["abc1682approximating4a-h12"],"title":"Find the area of the rectangles","text":"With the width and height just found, we can easily find the area of each rectangle using area of a rectangle formula $$A=w h$$ then add all of them up to get an approximation for the area under the curve.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating4a-h14","type":"hint","dependencies":["abc1682approximating4a-h13"],"title":"Approximating the Area","text":"Starting with $$x=0$$ from the left-hand point of the first rectangles, we have $$A=f{\\\\left(x_0\\\\right)} \\\\Delta x+f{\\\\left(x_1\\\\right)} \\\\Delta x+f{\\\\left(x_2\\\\right)} \\\\Delta x+f{\\\\left(x_3\\\\right)} \\\\Delta x=\\\\Delta x \\\\left(f{\\\\left(x_0\\\\right)}+f{\\\\left(x_1\\\\right)}+f{\\\\left(x_2\\\\right)}+f{\\\\left(x_3\\\\right)}\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating4a-h15","type":"hint","dependencies":["abc1682approximating4a-h14"],"title":"Approximating the Area","text":"$$0.5\\\\left(0+0.25+1+2.25\\\\right)=1.75$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"abc1682approximating5","title":"Approximating the Area Under a Curve","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.1  Approximating Areas","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"abc1682approximating5a","stepAnswer":["$$3.75$$"],"problemType":"TextBox","stepTitle":"Use right-endpoint approximations to approximate the area under the curve of $$f(x)=x^2$$ on the interval $$[0.2];$$ use $$n=4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3.75$$","hints":{"DefaultPathway":[{"id":"abc1682approximating5a-h1","type":"hint","dependencies":[],"title":"Subinterval","text":"The first step is to divide the interval from $$0$$ to $$2$$ into $$n$$ equal subintervals. These subintervals can be seen as rectangles.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["abc1682approximating5a-h1"],"title":"Subinterval","text":"How many subinterval or rectangle can we create in this problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"abc1682approximating5a-h2-s1","type":"hint","dependencies":[],"title":"Subinterval","text":"Since $$n=4$$ is given in this problem, we can divide the interval from $$x=0$$ to $$x=2$$ into $$4$$ rectangles.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"abc1682approximating5a-h3","type":"hint","dependencies":["abc1682approximating5a-h2"],"title":"Find the step size","text":"The step size or width of the rectangles can be found using $$\u0394x=\\\\frac{b-a}{n}$$ where $$b$$ is the upper bound, a is the lower bound and $$n$$ is the number of rectangles.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["abc1682approximating5a-h3"],"title":"Find the step size","text":"What is the upper bound of the interval?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["abc1682approximating5a-h3"],"title":"Find the step size","text":"What is the lower bound of the interval?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating5a-h6","type":"hint","dependencies":["abc1682approximating5a-h3"],"title":"Find the step size","text":"For the $$4$$ rectangles, each rectangle has a width of $$\u0394x=\\\\frac{2-0}{4}=0.5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating5a-h7","type":"hint","dependencies":["abc1682approximating5a-h6"],"title":"Find the start and end points of each rectangle","text":"Since we are asked to use the right-hand point approximation, we begin with the first rectangle whose right-hand point of $$x=0.5$$ and a width of $$0.5$$. Based on the illustration, we can see that the right-hand point of the next rectangles with the same width will be $$1, 1.5, 2$$ respectively.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating5a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$[1, 1.5]$$"],"dependencies":["abc1682approximating5a-h7"],"title":"Find the start and end points of each rectangle","text":"Express in interval notation, what are the starting and ending points of the third rectangle?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$(1, 1.5)$$","$$[1, 1.5]$$"]},{"id":"abc1682approximating5a-h9","type":"hint","dependencies":["abc1682approximating5a-h8"],"title":"Find the start and end points of each rectangle","text":"Continuing the process, we obtain $$4$$ intervals which are $$[0, 0.5], [0.5, 1], [1, 1.5]$$ and $$[1.5, 2]$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating5a-h10","type":"hint","dependencies":["abc1682approximating5a-h9"],"title":"Right-endpoint approximation","text":"The right rectangle approximation is when you make the right hand points of the pieces the height of the rectangles. In other words, the height of each rectangle is equal to the value of the function at the right endpoint of its base.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating5a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["abc1682approximating5a-h10"],"title":"Right-endpoint approximation","text":"Using the right-endpoint approximation, what is the height of a rectangle in the interval $$[0.5, 1]$$ ?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$0.5$$","$$1$$","$$0.25$$","$$1.75$$"]},{"id":"abc1682approximating5a-h12","type":"hint","dependencies":["abc1682approximating5a-h11"],"title":"Right-endpoint approximation","text":"Using a right-endpoint approximation, the heights are $$f(0.5)=0.25, f(1)=1, f(1.5)=2.25, f(2)=4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating5a-h13","type":"hint","dependencies":["abc1682approximating5a-h12"],"title":"Find the area of the rectangles","text":"With the width and height just found, we can easily find the area of each rectangle using area of a rectangle formula $$A=w h$$ then add all of them up to get an approximation for the area under the curve.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating5a-h14","type":"hint","dependencies":["abc1682approximating5a-h13"],"title":"Approximating the Area","text":"Starting with $$x=0.5$$ from the right-hand point of the first rectangles, we have $$A=f{\\\\left(x_1\\\\right)} \\\\Delta x+f{\\\\left(x_2\\\\right)} \\\\Delta x+f{\\\\left(x_3\\\\right)} \\\\Delta x+f{\\\\left(x_4\\\\right)} \\\\Delta x=\\\\Delta x \\\\left(f{\\\\left(x_1\\\\right)}+f{\\\\left(x_2\\\\right)}+f{\\\\left(x_3\\\\right)}+f{\\\\left(x_4\\\\right)}\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating5a-h15","type":"hint","dependencies":["abc1682approximating5a-h14"],"title":"Approximating the Area","text":"$$0.5\\\\left(0.25+1+2.25+4\\\\right)=3.75$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"abc1682approximating6","title":"Find Lower and Upper Sums","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.1  Approximating Areas","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"abc1682approximating6a","stepAnswer":["$$7.28$$"],"problemType":"TextBox","stepTitle":"Find a lower sum for $$f(x)=10-x^2$$ on [1,2]; let $$n=4$$ subintervals.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7.28$$","hints":{"DefaultPathway":[{"id":"abc1682approximating6a-h1","type":"hint","dependencies":[],"title":"Subinterval","text":"The first step is to divide the interval from $$1$$ to $$2$$ into $$n$$ equal subintervals. These subintervals can be seen as rectangles.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["abc1682approximating6a-h1"],"title":"Subinterval","text":"How many subinterval or rectangle can we create in this problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"abc1682approximating6a-h2-s1","type":"hint","dependencies":[],"title":"Subinterval","text":"Since $$n=4$$ is given in this problem, we can divide the interval from $$x=1$$ to $$x=2$$ into $$4$$ rectangles.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"abc1682approximating6a-h3","type":"hint","dependencies":["abc1682approximating6a-h2"],"title":"Find the step size","text":"The step size or width of the rectangles can be found using $$\u0394x=\\\\frac{b-a}{n}$$ where $$b$$ is the upper bound, a is the lower bound and $$n$$ is the number of rectangles.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["abc1682approximating6a-h3"],"title":"Find the step size","text":"What is the upper bound of the interval?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["abc1682approximating6a-h3"],"title":"Find the step size","text":"What is the lower bound of the interval?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating6a-h6","type":"hint","dependencies":["abc1682approximating6a-h3"],"title":"Find the step size","text":"For the $$4$$ rectangles, each rectangle has a width of $$\u0394x=\\\\frac{2-1}{4}=0.25$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating6a-h7","type":"hint","dependencies":["abc1682approximating6a-h6"],"title":"Find the start and end points of each rectangle","text":"We begin with the first rectangle whose left starting point of $$x=1$$ and a width of $$0.25$$. The rectangle will end at the point $$x=1.25$$ so the first interval will be $$[1, 1.25]$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating6a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$[1.5, 1.75]$$"],"dependencies":["abc1682approximating6a-h7"],"title":"Find the start and end points of each rectangle","text":"Express in interval notation, what are the starting and ending points of the 3rd rectangle in the illustration?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$[1.5, 1.75]$$","$$[1.5, 1.75)$$"]},{"id":"abc1682approximating6a-h9","type":"hint","dependencies":["abc1682approximating6a-h8"],"title":"Find the start and end points of each rectangle","text":"Continuing the process, we obtain $$4$$ intervals which are $$[1, 1.25], [1.25, 1.5], [1.5, 1.75]$$ and $$[1.75, 2]$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating6a-h10","type":"hint","dependencies":["abc1682approximating6a-h9"],"title":"Right-endpoint approximation","text":"Because the function is decreasing over the interval [1,2], a lower sum is obtained by using the right endpoints. In addition, the graph of $$f(x)=10-x^2$$ is set up for a right-endpoint approximation of the area bounded by the curve and the x-axis on [1,2], and it shows a lower sum.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating6a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7.28$$"],"dependencies":["abc1682approximating6a-h10"],"title":"Riemann sum","text":"What is the area under the curve of the given function using the Riemann sum?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"abc1682approximating6a-h11-s1","type":"hint","dependencies":[],"title":"Riemann sum","text":"The Riemann sum can be expressed as: (sum{k\\\\=1}{4}{10-(x_k)**2})*0.25","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating6a-h11-s2","type":"hint","dependencies":["abc1682approximating6a-h11-s1"],"title":"Expand","text":"$$0.25\\\\left(10-{1.25}^2+10-{1.5}^2+10-{1.75}^2+10-2^2\\\\right)=0.25\\\\left(8.43575+7.75+6.9375+6\\\\right)=7.28$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"abc1682approximating6a-h12","type":"hint","dependencies":["abc1682approximating6a-h11"],"title":"Why using the right-endpoint approximation","text":"Since we are asked to find a lower sum of the area of the function under the curve, the righ-endpoint approximation would be the most suitable method to use because it gives us an underestimate.Instead, approximating the left endpoint will give us an overestimation, which is not necessary in this case.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"abc1682approximating7","title":"Finding Lower and Upper Sums for $$f(x)=sinx$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.1  Approximating Areas","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"abc1682approximating7a","stepAnswer":["$$0.863$$"],"problemType":"TextBox","stepTitle":"Finding Lower and Upper Sums for $$f(x)=sinx$$ over the interval $$[a,b]=[0,\\\\frac{\\\\pi}{2}];$$ let $$n=6$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.863$$","hints":{"DefaultPathway":[{"id":"abc1682approximating7a-h1","type":"hint","dependencies":[],"title":"Subinterval","text":"The first step is to divide the interval from $$0$$ to $$\\\\frac{\\\\pi}{2}$$ into $$n$$ equal subintervals. These subintervals can be seen as rectangles.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["abc1682approximating7a-h1"],"title":"Subinterval","text":"How many subinterval or rectangle can we create in this problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\pi}{12}$$"],"dependencies":["abc1682approximating7a-h2"],"title":"Subinterval","text":"What is the \u0394x of each rectangle?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating7a-h4","type":"hint","dependencies":["abc1682approximating7a-h3"],"title":"Subinterval","text":"The graph of $$y=sinx$$ is divided into six rectangle: $$\u0394x=\\\\frac{\\\\frac{\\\\pi}{2}}{6}=\\\\frac{\\\\pi}{12}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating7a-h5","type":"hint","dependencies":["abc1682approximating7a-h4"],"title":"Find the start and end points of each rectangle","text":"As the graphs increases in the intervals $$[0,\\\\frac{\\\\pi}{2}]$$, we will use left-endpoint approximation to calculate the lower sum. We begin with the first rectangle whose starting point of $$x=0$$ and a width of $$\\\\frac{\\\\pi}{12}$$. The rectangle will end at the point $$x=\\\\frac{\\\\pi}{12}$$ and that endpoint of the current rectangle is also the starting point of the next rectangle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating7a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$[\\\\frac{\\\\pi}{12},\\\\frac{\\\\pi}{6}]$$"],"dependencies":["abc1682approximating7a-h5"],"title":"Find the start and end points of each rectangle","text":"Express in interval notation, what are the starting and ending points of the next rectangle?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$[\\\\frac{\\\\pi}{12},\\\\frac{\\\\pi}{6}]$$","$$(\\\\frac{\\\\pi}{12},\\\\frac{\\\\pi}{6})$$"]},{"id":"abc1682approximating7a-h7","type":"hint","dependencies":["abc1682approximating7a-h6"],"title":"Find the start and end points of each rectangle","text":"Continuing the process, we obtain $$6$$ intervals which are $$[0,\\\\frac{\\\\pi}{12}]$$, $$[\\\\frac{\\\\pi}{12},\\\\frac{\\\\pi}{6}]$$, $$[\\\\frac{\\\\pi}{6},\\\\frac{\\\\pi}{4}]$$, $$[\\\\frac{\\\\pi}{4},\\\\frac{\\\\pi}{3}]$$, $$[\\\\frac{\\\\pi}{3},\\\\frac{5\\\\pi}{12}]$$ and $$[\\\\frac{5\\\\pi}{12},\\\\frac{\\\\pi}{2}]$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating7a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.863$$"],"dependencies":["abc1682approximating7a-h7"],"title":"Riemann sum","text":"What is the area under the curve of the given function using the Riemann sum? (Three decimal places)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating7a-h9","type":"hint","dependencies":["abc1682approximating7a-h8"],"title":"Riemann sum","text":"sum{i\\\\=0}{5}{sin(x_i)*(pi/12)}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating7a-h10","type":"hint","dependencies":["abc1682approximating7a-h9"],"title":"Riemann sum","text":"$$A=sin\\\\left(0\\\\right) \\\\frac{\\\\pi}{12}+sin\\\\left(\\\\frac{\\\\pi}{12}\\\\right) \\\\frac{\\\\pi}{12}+sin\\\\left(\\\\frac{\\\\pi}{6}\\\\right) \\\\frac{\\\\pi}{12}+sin\\\\left(\\\\frac{\\\\pi}{4}\\\\right) \\\\frac{\\\\pi}{12}+sin\\\\left(\\\\frac{\\\\pi}{3}\\\\right) \\\\frac{\\\\pi}{12}+sin\\\\left(\\\\frac{5\\\\pi}{12}\\\\right) \\\\frac{\\\\pi}{12}=0.863$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"abc1682approximating8","title":"In the following exercises, use the rules for sums of powers of integers to compute the sums.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.1  Approximating Areas","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"abc1682approximating8a","stepAnswer":["$$355$$"],"problemType":"TextBox","stepTitle":"sum{i\\\\=5}{10}{i**2}","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$355$$","hints":{"DefaultPathway":[{"id":"abc1682approximating8a-h1","type":"hint","dependencies":[],"title":"Expand the sum","text":"In order to calculate the sum, we need to define the starting value which is a number below the summation sign. If it appears as $$i=1$$, we can easily subtitute $$n$$ into the sum according to the consecutive integers squared and compute the sum. Otherwise, we have to manipulate it by expanding the sum.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["abc1682approximating8a-h1"],"title":"Expand the sum","text":"What is the starting value of this sum?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating8a-h3","type":"hint","dependencies":["abc1682approximating8a-h2"],"title":"Expand the sum","text":"Since this sum starts from the fifth term, we have to calculate the sum of the tenth term minus the sum of the first $$4$$ terms","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating8a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$30$$"],"dependencies":["abc1682approximating8a-h3"],"title":"Expand the sum","text":"What is the sum of the first four terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$30$$","$$16$$","$$360$$","$$9$$"],"subHints":[{"id":"abc1682approximating8a-h4-s1","type":"hint","dependencies":[],"title":"Expand the sum","text":"Expanding the first four terms, we obtain: $$1^2+2^2+3^2+4^2=30$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"abc1682approximating8a-h5","type":"hint","dependencies":["abc1682approximating8a-h3"],"title":"Rearranging the sum","text":"We can rewrite the sum of the first four terms as:sum{i\\\\=1}{4}{i**2)}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating8a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$385$$"],"dependencies":["abc1682approximating8a-h5"],"title":"Compute the sum","text":"What is the sum of the whole ten terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating8a-h7","type":"hint","dependencies":["abc1682approximating8a-h6"],"title":"Sums of integers","text":"Using the sum of consecutive integers squared, we obtain:sum{i\\\\=1}{10}{i**2}=10*(10+1)*(2*10+1)/6=385","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating8a-h8","type":"hint","dependencies":["abc1682approximating8a-h4","abc1682approximating8a-h7"],"title":"Sums of last five terms","text":"The summation with the starting value of $$5$$ can be calculated $$as:385-30=355$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"abc1682approximating9","title":"Suppose that sum{i\\\\=1}{100}{a_1)}=15 and sum{i\\\\=1}{100}{b_i)}=-12. In the following exercises,compute the sums.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.1  Approximating Areas","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"abc1682approximating9a","stepAnswer":["$$27$$"],"problemType":"TextBox","stepTitle":"sum{i\\\\=1}{100}{(a_i)-(b_i)}","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$27$$","hints":{"DefaultPathway":[{"id":"abc1682approximating9a-h1","type":"hint","dependencies":[],"title":"Properties of Sigma Notation","text":"We use the properties of sigma notation to separate the $$2$$ terms of the sum then substitue each term with the given information to calculate the sum.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating9a-h2","type":"hint","dependencies":["abc1682approximating9a-h1"],"title":"Properties of Sigma Notation","text":"sum{i\\\\=1}{100}{(a_i)}-sum{i\\\\=1}{100}{(b_i)}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"abc1682approximating9a-h3","type":"hint","dependencies":["abc1682approximating9a-h2"],"title":"Substitution","text":"sum{i\\\\=1}{100}{(a_i)}-sum{i\\\\=1}{100}{(b_i)}=15-(-12)=27","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"abcc0f7chi1","title":"Type of Test","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Comparison of the Chi-Square Tests","courseName":"OpenStax: Introductory Stats","steps":[{"id":"abcc0f7chi1a","stepAnswer":["Goodness of Fit"],"problemType":"MultipleChoice","stepTitle":"Which test do you use to decide whether an observed distribution is the same as an expected distribution?","stepBody":"","answerType":"string","variabilization":{},"choices":["Goodness of Fit","Independence","Homogeneity"],"hints":{"DefaultPathway":[{"id":"abcc0f7chi1a-h1","type":"hint","dependencies":[],"title":"Goodness of Fit Test","text":"The goodness-of-fit test is used to decide whether a population with an unknown distribution \\"fits\\" a known distribution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abcc0f7chi1a-h2","type":"hint","dependencies":[],"title":"Independence Test","text":"The test for independence is used to decide whether two variables (factors) are independent or dependent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abcc0f7chi1a-h3","type":"hint","dependencies":[],"title":"Homogeneity","text":"The test for homogeneity is used to decide if two populations with unknown distributions have the same distribution as each other","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abcc0f7chi10","title":"Lifting Weights","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Comparison of the Chi-Square Tests","courseName":"OpenStax: Introductory Stats","steps":[{"id":"abcc0f7chi10a","stepAnswer":["Goodness of Fit"],"problemType":"MultipleChoice","stepTitle":"A personal trainer is putting together a weight-lifting program for her clients. For a 90-day program, she expects each client to lift a specific maximum weight each week. As she goes along, she records the actual maximum weights her clients lifted. She wants to know how well her expectations met with what was observed. What type of test should be used?","stepBody":"","answerType":"string","variabilization":{},"choices":["Goodness of Fit","Independence","Homogeneity"],"hints":{"DefaultPathway":[{"id":"abcc0f7chi10a-h1","type":"hint","dependencies":[],"title":"Goodness of Fit Test","text":"The goodness-of-fit test is used to decide whether a population with an unknown distribution \\"fits\\" a known distribution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abcc0f7chi10a-h2","type":"hint","dependencies":[],"title":"Independence Test","text":"The test for independence is used to decide whether two variables (factors) are independent or dependent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abcc0f7chi10a-h3","type":"hint","dependencies":[],"title":"Homogeneity","text":"The test for homogeneity is used to decide if two populations with unknown distributions have the same distribution as each other","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abcc0f7chi11","title":"Baseball Team Salaries","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Comparison of the Chi-Square Tests","courseName":"OpenStax: Introductory Stats","steps":[{"id":"abcc0f7chi11a","stepAnswer":["Independence"],"problemType":"MultipleChoice","stepTitle":"The owner of a baseball team is interested in the relationship between player salaries and team winning percentage. He takes a random sample of $$100$$ players from different organizations. What type of test should be used?","stepBody":"","answerType":"string","variabilization":{},"choices":["Goodness of Fit","Independence","Homogeneity"],"hints":{"DefaultPathway":[{"id":"abcc0f7chi11a-h1","type":"hint","dependencies":[],"title":"Goodness of Fit Test","text":"The goodness-of-fit test is used to decide whether a population with an unknown distribution \\"fits\\" a known distribution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abcc0f7chi11a-h2","type":"hint","dependencies":[],"title":"Independence Test","text":"The test for independence is used to decide whether two variables (factors) are independent or dependent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abcc0f7chi11a-h3","type":"hint","dependencies":[],"title":"Homogeneity","text":"The test for homogeneity is used to decide if two populations with unknown distributions have the same distribution as each other","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abcc0f7chi12","title":"College Demographics","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Comparison of the Chi-Square Tests","courseName":"OpenStax: Introductory Stats","steps":[{"id":"abcc0f7chi12a","stepAnswer":["Goodness of Fit"],"problemType":"MultipleChoice","stepTitle":"The local university wants to know if their student body accurately represents the demographics of the surrounding area. What type of test should be conducted?","stepBody":"","answerType":"string","variabilization":{},"choices":["Goodness of Fit","Independence","Homogeneity"],"hints":{"DefaultPathway":[{"id":"abcc0f7chi12a-h1","type":"hint","dependencies":[],"title":"Goodness of Fit Test","text":"The goodness-of-fit test is used to decide whether a population with an unknown distribution \\"fits\\" a known distribution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abcc0f7chi12a-h2","type":"hint","dependencies":[],"title":"Independence Test","text":"The test for independence is used to decide whether two variables (factors) are independent or dependent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abcc0f7chi12a-h3","type":"hint","dependencies":[],"title":"Homogeneity","text":"The test for homogeneity is used to decide if two populations with unknown distributions have the same distribution as each other","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abcc0f7chi13","title":"Election After Earthquake","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Comparison of the Chi-Square Tests","courseName":"OpenStax: Introductory Stats","steps":[{"id":"abcc0f7chi13a","stepAnswer":["Homogeneity"],"problemType":"MultipleChoice","stepTitle":"Both before and after a recent earthquake, surveys were conducted asking voters which of the three candidates they planned on voting for in the upcoming city council election. What type of test should be used to determine if there has been a change since the earthquake?","stepBody":"","answerType":"string","variabilization":{},"choices":["Goodness of Fit","Independence","Homogeneity"],"hints":{"DefaultPathway":[{"id":"abcc0f7chi13a-h1","type":"hint","dependencies":[],"title":"Goodness of Fit Test","text":"The goodness-of-fit test is used to decide whether a population with an unknown distribution \\"fits\\" a known distribution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abcc0f7chi13a-h2","type":"hint","dependencies":[],"title":"Independence Test","text":"The test for independence is used to decide whether two variables (factors) are independent or dependent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abcc0f7chi13a-h3","type":"hint","dependencies":[],"title":"Homogeneity","text":"The test for homogeneity is used to decide if two populations with unknown distributions have the same distribution as each other","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abcc0f7chi14","title":"Employment Statistics","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Comparison of the Chi-Square Tests","courseName":"OpenStax: Introductory Stats","steps":[{"id":"abcc0f7chi14a","stepAnswer":["Independence"],"problemType":"MultipleChoice","stepTitle":"The Bureau of Labor Statistics gathers data about employment in the United States. A sample is taken to calculate the number of U.S. citizens working in one of several industry sectors over time. What test should be conducted to determine the relationship between the change in number of jobs and years","stepBody":"","answerType":"string","variabilization":{},"choices":["Goodness of Fit","Independence","Homogeneity"],"hints":{"DefaultPathway":[{"id":"abcc0f7chi14a-h1","type":"hint","dependencies":[],"title":"Goodness of Fit Test","text":"The goodness-of-fit test is used to decide whether a population with an unknown distribution \\"fits\\" a known distribution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abcc0f7chi14a-h2","type":"hint","dependencies":[],"title":"Independence Test","text":"The test for independence is used to decide whether two variables (factors) are independent or dependent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abcc0f7chi14a-h3","type":"hint","dependencies":[],"title":"Homogeneity","text":"The test for homogeneity is used to decide if two populations with unknown distributions have the same distribution as each other","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abcc0f7chi15","title":"Doing Homework","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Comparison of the Chi-Square Tests","courseName":"OpenStax: Introductory Stats","steps":[{"id":"abcc0f7chi15a","stepAnswer":["Goodness of Fit"],"problemType":"MultipleChoice","stepTitle":"Teachers want to know which night each week their students are doing most of their homework. Most teachers think that students do homework equally throughout the week. Suppose a random sample of $$56$$ students were asked on which night of the week they did the most homework. What type of test should be conducted to determine if the nights with the highest number of students doing a majority of the homework occur with equal frequencies during a week?","stepBody":"","answerType":"string","variabilization":{},"choices":["Goodness of Fit","Independence","Homogeneity"],"hints":{"DefaultPathway":[{"id":"abcc0f7chi15a-h1","type":"hint","dependencies":[],"title":"Goodness of Fit Test","text":"The goodness-of-fit test is used to decide whether a population with an unknown distribution \\"fits\\" a known distribution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abcc0f7chi15a-h2","type":"hint","dependencies":[],"title":"Independence Test","text":"The test for independence is used to decide whether two variables (factors) are independent or dependent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abcc0f7chi15a-h3","type":"hint","dependencies":[],"title":"Homogeneity","text":"The test for homogeneity is used to decide if two populations with unknown distributions have the same distribution as each other","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abcc0f7chi2","title":"Type of Test","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Comparison of the Chi-Square Tests","courseName":"OpenStax: Introductory Stats","steps":[{"id":"abcc0f7chi2a","stepAnswer":["Independence"],"problemType":"MultipleChoice","stepTitle":"Which test would you use to decide whether two factors have a relationship?","stepBody":"","answerType":"string","variabilization":{},"choices":["Goodness of Fit","Independence","Homogeneity"],"hints":{"DefaultPathway":[{"id":"abcc0f7chi2a-h1","type":"hint","dependencies":[],"title":"Goodness of Fit Test","text":"The goodness-of-fit test is used to decide whether a population with an unknown distribution \\"fits\\" a known distribution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abcc0f7chi2a-h2","type":"hint","dependencies":[],"title":"Independence Test","text":"The test for independence is used to decide whether two variables (factors) are independent or dependent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abcc0f7chi2a-h3","type":"hint","dependencies":[],"title":"Homogeneity","text":"The test for homogeneity is used to decide if two populations with unknown distributions have the same distribution as each other","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abcc0f7chi3","title":"Type of Test","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Comparison of the Chi-Square Tests","courseName":"OpenStax: Introductory Stats","steps":[{"id":"abcc0f7chi3a","stepAnswer":["Homogeneity"],"problemType":"MultipleChoice","stepTitle":"Which test would you use to decide whether two factors have the same distribution as each other?","stepBody":"","answerType":"string","variabilization":{},"choices":["Goodness of Fit","Independence","Homogeneity"],"hints":{"DefaultPathway":[{"id":"abcc0f7chi3a-h1","type":"hint","dependencies":[],"title":"Goodness of Fit Test","text":"The goodness-of-fit test is used to decide whether a population with an unknown distribution \\"fits\\" a known distribution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abcc0f7chi3a-h2","type":"hint","dependencies":[],"title":"Independence Test","text":"The test for independence is used to decide whether two variables (factors) are independent or dependent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abcc0f7chi3a-h3","type":"hint","dependencies":[],"title":"Homogeneity","text":"The test for homogeneity is used to decide if two populations with unknown distributions have the same distribution as each other","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abcc0f7chi4","title":"Math Test Scores","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Comparison of the Chi-Square Tests","courseName":"OpenStax: Introductory Stats","steps":[{"id":"abcc0f7chi4a","stepAnswer":["Homogeneity"],"problemType":"MultipleChoice","stepTitle":"A math teacher wants to see if two of her classes have the same distribution of test scores. What test should she use?","stepBody":"","answerType":"string","variabilization":{},"choices":["Goodness of Fit","Independence","Homogeneity"],"hints":{"DefaultPathway":[{"id":"abcc0f7chi4a-h1","type":"hint","dependencies":[],"title":"Goodness of Fit Test","text":"The goodness-of-fit test is used to decide whether a population with an unknown distribution \\"fits\\" a known distribution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abcc0f7chi4a-h2","type":"hint","dependencies":[],"title":"Independence Test","text":"The test for independence is used to decide whether two variables (factors) are independent or dependent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abcc0f7chi4a-h3","type":"hint","dependencies":[],"title":"Homogeneity","text":"The test for homogeneity is used to decide if two populations with unknown distributions have the same distribution as each other","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abcc0f7chi5","title":"Store Sales Figures","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Comparison of the Chi-Square Tests","courseName":"OpenStax: Introductory Stats","steps":[{"id":"abcc0f7chi5a","stepAnswer":["Homogeneity"],"problemType":"MultipleChoice","stepTitle":"A market researcher wants to see if two different stores have the same distribution of sales throughout the year. What type of test should he use?","stepBody":"","answerType":"string","variabilization":{},"choices":["Goodness of Fit","Independence","Homogeneity"],"hints":{"DefaultPathway":[{"id":"abcc0f7chi5a-h1","type":"hint","dependencies":[],"title":"Goodness of Fit Test","text":"The goodness-of-fit test is used to decide whether a population with an unknown distribution \\"fits\\" a known distribution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abcc0f7chi5a-h2","type":"hint","dependencies":[],"title":"Independence Test","text":"The test for independence is used to decide whether two variables (factors) are independent or dependent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abcc0f7chi5a-h3","type":"hint","dependencies":[],"title":"Homogeneity","text":"The test for homogeneity is used to decide if two populations with unknown distributions have the same distribution as each other","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abcc0f7chi6","title":"Viral Symptoms","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Comparison of the Chi-Square Tests","courseName":"OpenStax: Introductory Stats","steps":[{"id":"abcc0f7chi6a","stepAnswer":["Independence"],"problemType":"MultipleChoice","stepTitle":"A pharmaceutical company is interested in the relationship between age and presentation of symptoms for a common viral $$infection$$. A random sample is taken of $$500$$ people with the infection across different age groups. What type of test should be used?","stepBody":"","answerType":"string","variabilization":{},"choices":["Goodness of Fit","Independence","Homogeneity"],"hints":{"DefaultPathway":[{"id":"abcc0f7chi6a-h1","type":"hint","dependencies":[],"title":"Goodness of Fit Test","text":"The goodness-of-fit test is used to decide whether a population with an unknown distribution \\"fits\\" a known distribution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abcc0f7chi6a-h2","type":"hint","dependencies":[],"title":"Independence Test","text":"The test for independence is used to decide whether two variables (factors) are independent or dependent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abcc0f7chi6a-h3","type":"hint","dependencies":[],"title":"Homogeneity","text":"The test for homogeneity is used to decide if two populations with unknown distributions have the same distribution as each other","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abcc0f7chi7","title":"Running Shoe Brands","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Comparison of the Chi-Square Tests","courseName":"OpenStax: Introductory Stats","steps":[{"id":"abcc0f7chi7a","stepAnswer":["Independence"],"problemType":"MultipleChoice","stepTitle":"A marathon runner is interested in the relationship between the brand of shoes runners wear and their run times. She takes a random sample of $$50$$ runners and records their run times as well as the brand of shoes they were wearing. What type of test should be used?","stepBody":"","answerType":"string","variabilization":{},"choices":["Goodness of Fit","Independence","Homogeneity"],"hints":{"DefaultPathway":[{"id":"abcc0f7chi7a-h1","type":"hint","dependencies":[],"title":"Goodness of Fit Test","text":"The goodness-of-fit test is used to decide whether a population with an unknown distribution \\"fits\\" a known distribution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abcc0f7chi7a-h2","type":"hint","dependencies":[],"title":"Independence Test","text":"The test for independence is used to decide whether two variables (factors) are independent or dependent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abcc0f7chi7a-h3","type":"hint","dependencies":[],"title":"Homogeneity","text":"The test for homogeneity is used to decide if two populations with unknown distributions have the same distribution as each other","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abcc0f7chi8","title":"Digging for Artifacts","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Comparison of the Chi-Square Tests","courseName":"OpenStax: Introductory Stats","steps":[{"id":"abcc0f7chi8a","stepAnswer":["Goodness of Fit"],"problemType":"MultipleChoice","stepTitle":"An archeologist is calculating the distribution of the frequency of the number of artifacts she finds in a dig site. Based on previous digs, the archeologist creates an expected distribution broken down by grid sections in the dig site. Once the site has been fully excavated, she compares the actual number of artifacts found in each grid section to see if her expectation was accurate. What type of test should be used?","stepBody":"","answerType":"string","variabilization":{},"choices":["Goodness of Fit","Independence","Homogeneity"],"hints":{"DefaultPathway":[{"id":"abcc0f7chi8a-h1","type":"hint","dependencies":[],"title":"Goodness of Fit Test","text":"The goodness-of-fit test is used to decide whether a population with an unknown distribution \\"fits\\" a known distribution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abcc0f7chi8a-h2","type":"hint","dependencies":[],"title":"Independence Test","text":"The test for independence is used to decide whether two variables (factors) are independent or dependent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abcc0f7chi8a-h3","type":"hint","dependencies":[],"title":"Homogeneity","text":"The test for homogeneity is used to decide if two populations with unknown distributions have the same distribution as each other","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abcc0f7chi9","title":"Stock Market Predictions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.5 Comparison of the Chi-Square Tests","courseName":"OpenStax: Introductory Stats","steps":[{"id":"abcc0f7chi9a","stepAnswer":["Goodness of Fit"],"problemType":"MultipleChoice","stepTitle":"An economist is deriving a model to predict outcomes on the stock market. He creates a list of expected points on the stock market index for the next two weeks. At the close of each day\u2019s trading, he records the actual points on the index. He wants to see how well his model matched what actually happened. What type of test should be used?","stepBody":"","answerType":"string","variabilization":{},"choices":["Goodness of Fit","Independence","Homogeneity"],"hints":{"DefaultPathway":[{"id":"abcc0f7chi9a-h1","type":"hint","dependencies":[],"title":"Goodness of Fit Test","text":"The goodness-of-fit test is used to decide whether a population with an unknown distribution \\"fits\\" a known distribution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abcc0f7chi9a-h2","type":"hint","dependencies":[],"title":"Independence Test","text":"The test for independence is used to decide whether two variables (factors) are independent or dependent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abcc0f7chi9a-h3","type":"hint","dependencies":[],"title":"Homogeneity","text":"The test for homogeneity is used to decide if two populations with unknown distributions have the same distribution as each other","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abd4034poly1","title":"Divide Polynomials","body":"Divide each polynomial by the monomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Divide Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"abd4034poly1a","stepAnswer":["$$17y^2+14$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{51y^4+42y^2}{3y^2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$17y^2+14$$","hints":{"DefaultPathway":[{"id":"abd4034poly1a-h1","type":"hint","dependencies":[],"title":"Separating Terms","text":"Separate the terms so that it forms multiple fractions with the same denominator. It will result in the following: $$\\\\frac{51y^4}{3y^2}+\\\\frac{42y^2}{3y^2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abd4034poly1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$17y^2+14$$"],"dependencies":["abd4034poly1a-h1"],"title":"Simplify Terms","text":"What is the result once each fraction term is simplified?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abd4034poly10","title":"Divide Polynomials","body":"Divide each polynomial by the monomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Divide Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"abd4034poly10a","stepAnswer":["$$4q+1-\\\\frac{1}{3q}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{12q^2+3q-1}{3q}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4q+1-\\\\frac{1}{3q}$$","hints":{"DefaultPathway":[{"id":"abd4034poly10a-h1","type":"hint","dependencies":[],"title":"Separating Terms","text":"Separate the terms so that it forms multiple fractions with the same denominator. It will result in the following: $$\\\\frac{12q^2}{3q}+\\\\frac{3q}{3q}-\\\\frac{1}{3q}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abd4034poly10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4q+1-\\\\frac{1}{3q}$$"],"dependencies":["abd4034poly10a-h1"],"title":"Simplify Terms","text":"What is the result once each fraction term is simplified?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abd4034poly11","title":"Divide Polynomials","body":"Divide each polynomial by the monomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Divide Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"abd4034poly11a","stepAnswer":["$$\\\\left(-2x\\\\right)-1+\\\\frac{4}{5x}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{10x^2+5x-4}{\\\\left(-5x\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(-2x\\\\right)-1+\\\\frac{4}{5x}$$","hints":{"DefaultPathway":[{"id":"abd4034poly11a-h1","type":"hint","dependencies":[],"title":"Separating Terms","text":"Separate the terms so that it forms multiple fractions with the same denominator. It will result in the following: $$\\\\frac{10x^2}{\\\\left(-5x\\\\right)}+\\\\frac{5x}{\\\\left(-5x\\\\right)}-\\\\frac{4}{\\\\left(-5x\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abd4034poly11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(-2x\\\\right)-1+\\\\frac{4}{5x}$$"],"dependencies":["abd4034poly11a-h1"],"title":"Simplify Terms","text":"What is the result once each fraction term is simplified?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abd4034poly12","title":"Divide Polynomials","body":"Divide each polynomial by the monomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Divide Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"abd4034poly12a","stepAnswer":["$$\\\\left(-5y\\\\right)-3+\\\\frac{1}{4y}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{20y^2+12y-1}{\\\\left(-4y\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(-5y\\\\right)-3+\\\\frac{1}{4y}$$","hints":{"DefaultPathway":[{"id":"abd4034poly12a-h1","type":"hint","dependencies":[],"title":"Separating Terms","text":"Separate the terms so that it forms multiple fractions with the same denominator. It will result in the following: $$\\\\frac{20y^2}{\\\\left(-4y\\\\right)}+\\\\frac{12y}{\\\\left(-4y\\\\right)}-\\\\frac{1}{\\\\left(-4y\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abd4034poly12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(-5y\\\\right)-3+\\\\frac{1}{4y}$$"],"dependencies":["abd4034poly12a-h1"],"title":"Simplify Terms","text":"What is the result once each fraction term is simplified?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abd4034poly13","title":"Divide Polynomials","body":"Divide each polynomial by the monomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Divide Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"abd4034poly13a","stepAnswer":["$$6p+3-\\\\frac{2}{p}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{36p^3+18p^2-12p}{6p^2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6p+3-\\\\frac{2}{p}$$","hints":{"DefaultPathway":[{"id":"abd4034poly13a-h1","type":"hint","dependencies":[],"title":"Separating Terms","text":"Separate the terms so that it forms multiple fractions with the same denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abd4034poly13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6p+3-\\\\frac{2}{p}$$"],"dependencies":["abd4034poly13a-h1"],"title":"Simplify Terms","text":"What is the result once each fraction term is simplified?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abd4034poly14","title":"Divide Polynomials","body":"Divide each polynomial by the monomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Divide Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"abd4034poly14a","stepAnswer":["$$7a-12+\\\\frac{11}{a}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{63a^3+108a^2+99a}{9a^2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7a-12+\\\\frac{11}{a}$$","hints":{"DefaultPathway":[{"id":"abd4034poly14a-h1","type":"hint","dependencies":[],"title":"Separating Terms","text":"Separate the terms so that it forms multiple fractions with the same denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abd4034poly14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7a-12+\\\\frac{11}{a}$$"],"dependencies":["abd4034poly14a-h1"],"title":"Simplify Terms","text":"What is the result once each fraction term is simplified?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abd4034poly15","title":"Divide Polynomials","body":"Divide each polynomial by the binomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Divide Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"abd4034poly15a","stepAnswer":["$$d+6$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{d^2+8d+12}{d+2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$d+6$$","hints":{"DefaultPathway":[{"id":"abd4034poly15a-h1","type":"hint","dependencies":[],"title":"Perform Long Division","text":"Write it as a long division problem.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abd4034poly15a-h2","type":"hint","dependencies":["abd4034poly15a-h1"],"title":"Executing Long Division","text":"Divide $$d^2$$ by $$d$$. Put the answer, $$d$$, in the quotient over the $$d$$ term. Multiply $$d$$ times $$d+2$$. Line up the like terms. Subtract and then bring down the next term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abd4034poly15a-h3","type":"hint","dependencies":["abd4034poly15a-h2"],"title":"Finishing Long Division","text":"Divide $$6d$$ by $$d$$. Put the answer, $$6$$, in the quotient over the $$12$$ term. Multiply $$6$$ times $$d+2$$. Line up like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abd4034poly15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$d+6$$"],"dependencies":["abd4034poly15a-h3"],"title":"Answer to Long Division","text":"What is the result of the long division?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abd4034poly16","title":"Division of a Polynomial by a Monomial","body":"Find the quotient by dividing the polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Divide Polynomials","courseName":"OpenStax: Elementary 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It will result in the following: $$\\\\frac{46x^3}{2x^2}+\\\\frac{38x^2}{2x^2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abd4034poly2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$23x+19$$"],"dependencies":["abd4034poly2a-h1"],"title":"Separating Terms","text":"What is the result once each fraction term is simplified?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abd4034poly20","title":"Division of a Polynomial by a Monomial","body":"Find the quotient by dividing the polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Divide Polynomials","courseName":"OpenStax: Elementary 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Algebra","steps":[{"id":"abd4034poly3a","stepAnswer":["$$\\\\left(-8p\\\\right)+11$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{24p^2-33p}{\\\\left(-3p\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(-8p\\\\right)+11$$","hints":{"DefaultPathway":[{"id":"abd4034poly3a-h1","type":"hint","dependencies":[],"title":"Separating Terms","text":"Separate the terms so that it forms multiple fractions with the same denominator. It will result in the following: $$\\\\frac{24p^2}{\\\\left(-3p\\\\right)}-\\\\frac{33p}{\\\\left(-3p\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abd4034poly3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(-8p\\\\right)+11$$"],"dependencies":["abd4034poly3a-h1"],"title":"Simplify Terms","text":"What is the result once each fraction term is simplified?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abd4034poly30","title":"Division of a Polynomial by a Monomial","body":"Find the quotient by dividing the polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Divide Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"abd4034poly30a","stepAnswer":["$$8a^5 b+6a^3 b^2$$"],"problemType":"TextBox","stepTitle":"Find the quotient: $$\\\\frac{\\\\left(-48a^8 b^4-36a^6 b^5\\\\right)}{\\\\left(-6a^3 b^3\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8a^5 b+6a^3 b^2$$","hints":{"DefaultPathway":[{"id":"abd4034poly30a-h1","type":"hint","dependencies":[],"title":"We must divide each term in the numerator by $$-6a^3 b^3$$, so we get $$8a^5 b+6a^3 b^2$$","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abd4034poly4","title":"Divide Polynomials","body":"Divide each polynomial by the monomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Divide Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"abd4034poly4a","stepAnswer":["$$\\\\left(-5x^3\\\\right)+3$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{35x^4-21x}{\\\\left(-7x\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(-5x^3\\\\right)+3$$","hints":{"DefaultPathway":[{"id":"abd4034poly4a-h1","type":"hint","dependencies":[],"title":"Separating Terms","text":"Separate the terms so that it forms multiple fractions with the same denominator. It will result in the following: $$\\\\frac{35x^4}{\\\\left(-7x\\\\right)}-\\\\frac{21x}{\\\\left(-7x\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abd4034poly4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(-5x^3\\\\right)+3$$"],"dependencies":["abd4034poly4a-h1"],"title":"Simplify Terms","text":"What is the result once each fraction term is simplified?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abd4034poly5","title":"Divide Polynomials","body":"Divide each polynomial by the monomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Divide Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"abd4034poly5a","stepAnswer":["$$\\\\left(-9m^2\\\\right)+6m$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{63m^4-42m^3}{\\\\left(-7m^2\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(-9m^2\\\\right)+6m$$","hints":{"DefaultPathway":[{"id":"abd4034poly5a-h1","type":"hint","dependencies":[],"title":"Separating Terms","text":"Separate the terms so that it forms multiple fractions with the same denominator. It will result in the following: $$\\\\frac{63m^4}{\\\\left(-7m^2\\\\right)}-\\\\frac{42m^3}{\\\\left(-7m^2\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abd4034poly5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(-9m^2\\\\right)+6m$$"],"dependencies":["abd4034poly5a-h1"],"title":"Simplify Terms","text":"What is the result once each fraction term is simplified?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abd4034poly6","title":"Divide Polynomials","body":"Divide each polynomial by the monomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Divide Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"abd4034poly6a","stepAnswer":["$$\\\\left(-6y^2\\\\right)+3y$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{48y^4-24y^3}{\\\\left(-8y^2\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(-6y^2\\\\right)+3y$$","hints":{"DefaultPathway":[{"id":"abd4034poly6a-h1","type":"hint","dependencies":[],"title":"Separating Terms","text":"Separate the terms so that it forms multiple fractions with the same denominator. It will result in the following: $$\\\\frac{48y^4}{\\\\left(-8y^2\\\\right)}-\\\\frac{24y^3}{\\\\left(-8y^2\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abd4034poly6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(-6y^2\\\\right)+3y$$"],"dependencies":["abd4034poly6a-h1"],"title":"Simplify Terms","text":"What is the result once each fraction term is simplified?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abd4034poly7","title":"Divide Polynomials","body":"Divide each polynomial by the monomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Divide Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"abd4034poly7a","stepAnswer":["$$7a b^2+8b^3$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{63a^2 b^3+72a b^4}{9a b}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7a b^2+8b^3$$","hints":{"DefaultPathway":[{"id":"abd4034poly7a-h1","type":"hint","dependencies":[],"title":"Separating Terms","text":"Separate the terms so that it forms multiple fractions with the same denominator. It will result in the following: ((63*(a**2)*(b**3)/(9*a*b))+((72*a*b**4)/(9*a*b))","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abd4034poly7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7a b^2+8b^3$$"],"dependencies":["abd4034poly7a-h1"],"title":"Simplify Terms","text":"What is the result once each fraction term is simplified?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abd4034poly8","title":"Divide Polynomials","body":"Divide each polynomial by the monomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Divide Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"abd4034poly8a","stepAnswer":["$$9x^2 y^3+12y$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{45x^3 y^4+60x y^2}{5x y}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9x^2 y^3+12y$$","hints":{"DefaultPathway":[{"id":"abd4034poly8a-h1","type":"hint","dependencies":[],"title":"Separating Terms","text":"Separate the terms so that it forms multiple fractions with the same denominator. It will result in the following: $$\\\\frac{45x^3 y^4}{5x y}+\\\\frac{60x y^2}{5x y}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abd4034poly8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9x^2 y^3+12y$$"],"dependencies":["abd4034poly8a-h1"],"title":"Simplify Terms","text":"What is the result once each fraction term is simplified?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abd4034poly9","title":"Divide Polynomials","body":"Divide each polynomial by the monomial.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.6 Divide Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"abd4034poly9a","stepAnswer":["$$2w+1-\\\\frac{5}{2w}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{4w^2+2w-5}{2w}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2w+1-\\\\frac{5}{2w}$$","hints":{"DefaultPathway":[{"id":"abd4034poly9a-h1","type":"hint","dependencies":[],"title":"Separating Terms","text":"Separate the terms so that it forms multiple fractions with the same denominator. It will result in the following: $$\\\\frac{4w^2}{2w}+\\\\frac{2w}{2w}-\\\\frac{5}{2w}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abd4034poly9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2w+1-\\\\frac{5}{2w}$$"],"dependencies":["abd4034poly9a-h1"],"title":"Simplify Terms","text":"What is the result once each fraction term is simplified?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abe88c3log1","title":"Solving Logorithms","body":"Evaluate using the properties of logarithms","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Use the Properties of Logarithms","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"abe88c3log1a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"Solving Logorithms","stepBody":"$$\\\\log_{8}\\\\left(1\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"abe88c3log1a-h1","type":"hint","dependencies":[],"title":"Properties of logorithms","text":"Use the property $$\\\\log_{a}\\\\left(1\\\\right)$$ $$=$$ $$0$$. Thus, $$\\\\log_{8}\\\\left(1\\\\right)=$$ $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abe88c3log10","title":"Solving Logorithms","body":"Evaluate using the properties of logarithms","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Use the Properties of Logarithms","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"abe88c3log10a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"Solving Logorithms","stepBody":"$$\\\\log_{7}\\\\left(7^4\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"abe88c3log10a-h1","type":"hint","dependencies":[],"title":"Properties of Logorithms","text":"Use the property $$a**\\\\log_{a}\\\\left(x\\\\right)$$ $$=$$ $$x$$. Thus, $$\\\\log_{7}\\\\left(7^4\\\\right)$$ $$=$$ $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abe88c3log11","title":"Solving Logorithms","body":"Evaluate using the properties of logarithms","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Use the Properties of Logarithms","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"abe88c3log11a","stepAnswer":["$$8$$"],"problemType":"TextBox","stepTitle":"Solving Logorithms","stepBody":"$$2**\\\\log_{2}\\\\left(8\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8$$","hints":{"DefaultPathway":[{"id":"abe88c3log11a-h1","type":"hint","dependencies":[],"title":"Properties of Logorithms","text":"Use the property $$a**\\\\log_{a}\\\\left(x\\\\right)$$ $$=$$ $$x$$. Thus, $$2**\\\\log_{2}\\\\left(8\\\\right)$$ $$=$$ $$8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abe88c3log12","title":"Solving Logorithms","body":"Evaluate using the properties of logarithms","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Use the Properties of Logarithms","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"abe88c3log12a","stepAnswer":["$$15$$"],"problemType":"TextBox","stepTitle":"Solving Logorithms","stepBody":"$$\\\\log_{2}\\\\left(2^{15}\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$15$$","hints":{"DefaultPathway":[{"id":"abe88c3log12a-h1","type":"hint","dependencies":[],"title":"Properties of Logorithms","text":"Use the property $$a**\\\\log_{a}\\\\left(x\\\\right)$$ $$=$$ $$x$$. Thus, $$\\\\log_{2}\\\\left(2^{15}\\\\right)$$ $$=$$ $$15$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abe88c3log13","title":"Rewriting Logorithms with Product Property of Logorithms.","body":"Use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Use the Properties of Logarithms","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"abe88c3log13a","stepAnswer":["$$\\\\log_{3}\\\\left(7\\\\right)+\\\\log_{3}\\\\left(x\\\\right)$$"],"problemType":"TextBox","stepTitle":"Rewriting Logorithms with Product Property of Logorithms.","stepBody":"$$\\\\log_{3}\\\\left(7x\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\log_{3}\\\\left(7\\\\right)+\\\\log_{3}\\\\left(x\\\\right)$$","hints":{"DefaultPathway":[{"id":"abe88c3log13a-h1","type":"hint","dependencies":[],"title":"Using the Product Property of Logarithms.","text":"Use the Product Property: $$\\\\log_{a}\\\\left(M N\\\\right)$$ $$=$$ $$\\\\log_{a}\\\\left(M\\\\right)$$ + $$\\\\log_{a}\\\\left(N\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abe88c3log13a-h2","type":"hint","dependencies":["abe88c3log13a-h1"],"title":"Breaking down this Question.","text":"$$7x$$ $$=$$ $$7x$$ so $$\\\\log_{3}\\\\left(7x\\\\right)$$ $$=$$ $$\\\\log_{3}\\\\left(7\\\\right)$$ + $$\\\\log_{3}\\\\left(x\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abe88c3log14","title":"Rewriting Logorithms with Product Property of Logorithms","body":"Use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Use the Properties of Logarithms","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"abe88c3log14a","stepAnswer":["$$3+\\\\log_{4}\\\\left(x\\\\right)+\\\\log_{4}\\\\left(y\\\\right)$$"],"problemType":"TextBox","stepTitle":"Rewriting Logorithms with Product Property of Logorithms","stepBody":"$$\\\\log_{4}\\\\left(64xy\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3+\\\\log_{4}\\\\left(x\\\\right)+\\\\log_{4}\\\\left(y\\\\right)$$","hints":{"DefaultPathway":[{"id":"abe88c3log14a-h1","type":"hint","dependencies":[],"title":"Using the Product Property of Logarithms","text":"Use the Product Property: $$\\\\log_{a}\\\\left(M N\\\\right)$$ $$=$$ $$\\\\log_{a}\\\\left(M\\\\right)$$ + $$\\\\log_{a}\\\\left(N\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abe88c3log14a-h2","type":"hint","dependencies":["abe88c3log14a-h1"],"title":"Breaking down this Question.","text":"64xy $$=$$ $$64x y$$ so $$\\\\log_{4}\\\\left(64xy\\\\right)$$ $$=$$ $$\\\\log_{4}\\\\left(64\\\\right)$$ + $$\\\\log_{4}\\\\left(x\\\\right)$$ + $$\\\\log_{4}\\\\left(y\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abe88c3log14a-h3","type":"hint","dependencies":["abe88c3log14a-h2"],"title":"Simplifying","text":"$$\\\\log_{4}\\\\left(64\\\\right)$$ $$=$$ $$3$$ so $$\\\\log_{4}\\\\left(64xy\\\\right)$$ $$=$$ $$3$$ + $$\\\\log_{4}\\\\left(x\\\\right)$$ + $$\\\\log_{4}\\\\left(y\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abe88c3log15","title":"Rewriting Logorithms with Product Property of Logorithms.","body":"Use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Use the Properties of Logarithms","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"abe88c3log15a","stepAnswer":["$$1+\\\\log_{3}\\\\left(x\\\\right)$$"],"problemType":"TextBox","stepTitle":"Rewriting Logorithms with Product Property of Logorithms.","stepBody":"$$\\\\log_{3}\\\\left(3x\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1+\\\\log_{3}\\\\left(x\\\\right)$$","hints":{"DefaultPathway":[{"id":"abe88c3log15a-h1","type":"hint","dependencies":[],"title":"Using the Product Property of Logarithms.","text":"Use the Product Property: $$\\\\log_{a}\\\\left(M N\\\\right)$$ $$=$$ $$\\\\log_{a}\\\\left(M\\\\right)$$ + $$\\\\log_{a}\\\\left(N\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abe88c3log15a-h2","type":"hint","dependencies":["abe88c3log15a-h1"],"title":"Breaking down this Question.","text":"$$3x$$ $$=$$ $$3x$$ so $$\\\\log_{3}\\\\left(3x\\\\right)$$ $$=$$ $$\\\\log_{3}\\\\left(3\\\\right)$$ + $$\\\\log_{3}\\\\left(x\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abe88c3log15a-h3","type":"hint","dependencies":["abe88c3log15a-h2"],"title":"Simplifying","text":"$$\\\\log_{3}\\\\left(3\\\\right)$$ $$=$$ $$1$$ so $$\\\\log_{3}\\\\left(3x\\\\right)$$ $$=$$ $$1$$ + $$\\\\log_{3}\\\\left(x\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abe88c3log16","title":"Rewriting Logorithms with Product Property of Logorithms.","body":"Use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Use the Properties of Logarithms","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"abe88c3log16a","stepAnswer":["$$3+\\\\log_{2}\\\\left(x\\\\right)+\\\\log_{2}\\\\left(y\\\\right)$$"],"problemType":"TextBox","stepTitle":"Rewriting Logorithms with Product Property of Logorithms.","stepBody":"$$\\\\log_{2}\\\\left(8xy\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3+\\\\log_{2}\\\\left(x\\\\right)+\\\\log_{2}\\\\left(y\\\\right)$$","hints":{"DefaultPathway":[{"id":"abe88c3log16a-h1","type":"hint","dependencies":[],"title":"Using the Product Property of Logarithms.","text":"Use the Product Property: $$\\\\log_{a}\\\\left(M N\\\\right)$$ $$=$$ $$\\\\log_{a}\\\\left(M\\\\right)$$ + $$\\\\log_{a}\\\\left(N\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abe88c3log16a-h2","type":"hint","dependencies":["abe88c3log16a-h1"],"title":"Breaking down this Question.","text":"8xy $$=$$ $$8x y$$ so $$\\\\log_{2}\\\\left(8xy\\\\right)$$ $$=$$ $$\\\\log_{2}\\\\left(8\\\\right)$$ + $$\\\\log_{2}\\\\left(x\\\\right)$$ + $$\\\\log_{2}\\\\left(y\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abe88c3log16a-h3","type":"hint","dependencies":["abe88c3log16a-h2"],"title":"Simplifying","text":"$$\\\\log_{2}\\\\left(8\\\\right)$$ $$=$$ $$3$$ so $$\\\\log_{2}\\\\left(8xy\\\\right)$$ $$=$$ $$3$$ + $$\\\\log_{2}\\\\left(x\\\\right)$$ + $$\\\\log_{2}\\\\left(y\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abe88c3log17","title":"Rewriting Logorithms with the Quotient Property.","body":"Use the Quotient Property of Logarithms to write each logarithm as a difference of logarithms. Simplify, if possible.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Use the Properties of Logarithms","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"abe88c3log17a","stepAnswer":["$$1-\\\\log_{5}\\\\left(7\\\\right)$$"],"problemType":"TextBox","stepTitle":"Rewriting Logorithms with the Quotient Property.","stepBody":"$$\\\\log_{5}\\\\left(\\\\frac{5}{7}\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1-\\\\log_{5}\\\\left(7\\\\right)$$","hints":{"DefaultPathway":[{"id":"abe88c3log17a-h1","type":"hint","dependencies":[],"title":"Using the Quotient Property of Logorithms.","text":"Use the Quotient Property of Logarithms: $$\\\\log_{a}\\\\left(\\\\frac{M}{N}\\\\right)$$ $$=$$ $$\\\\log_{a}\\\\left(M\\\\right)$$ - $$\\\\log_{a}\\\\left(N\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abe88c3log17a-h2","type":"hint","dependencies":["abe88c3log17a-h1"],"title":"Applying the rule to this question.","text":"In this case, $$\\\\log_{5}\\\\left(\\\\frac{5}{7}\\\\right)$$ $$=$$ $$\\\\log_{5}\\\\left(5\\\\right)$$ - $$\\\\log_{5}\\\\left(7\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abe88c3log17a-h3","type":"hint","dependencies":["abe88c3log17a-h2"],"title":"Simplify.","text":"$$\\\\log_{5}\\\\left(5\\\\right)$$ $$=$$ $$1$$ so $$\\\\log_{5}\\\\left(\\\\frac{5}{7}\\\\right)$$ $$=$$ $$1$$ - $$\\\\log_{5}\\\\left(7\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abe88c3log18","title":"Rewriting Logorithms with the Quotient Property.","body":"Use the Quotient Property of Logarithms to write each logarithm as a difference of logarithms. Simplify, if possible.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Use the Properties of Logarithms","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"abe88c3log18a","stepAnswer":["$$\\\\log_{10}\\\\left(x\\\\right)-10$$"],"problemType":"TextBox","stepTitle":"Rewriting Logorithms with the Quotient Property.","stepBody":"$$\\\\log_{10}\\\\left(\\\\frac{x}{100}\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\log_{10}\\\\left(x\\\\right)-10$$","hints":{"DefaultPathway":[{"id":"abe88c3log18a-h1","type":"hint","dependencies":[],"title":"Using the Quotient Property of Logorithms.","text":"Use the Quotient Property of Logarithms: $$\\\\log_{a}\\\\left(\\\\frac{M}{N}\\\\right)$$ $$=$$ $$\\\\log_{a}\\\\left(M\\\\right)$$ - $$\\\\log_{a}\\\\left(N\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abe88c3log18a-h2","type":"hint","dependencies":["abe88c3log18a-h1"],"title":"Applying the rule to this question.","text":"In this case, $$\\\\log_{10}\\\\left(\\\\frac{x}{100}\\\\right)$$ $$=$$ $$\\\\log_{10}\\\\left(x\\\\right)$$ - $$\\\\log_{10}\\\\left(100\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abe88c3log18a-h3","type":"hint","dependencies":["abe88c3log18a-h2"],"title":"Simplify.","text":"$$\\\\log_{10}\\\\left(100\\\\right)$$ $$=$$ $$10$$ so $$\\\\log_{10}\\\\left(\\\\frac{x}{100}\\\\right)$$ $$=$$ $$\\\\log_{10}\\\\left(x\\\\right)$$ - $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abe88c3log19","title":"Rewriting Logorithms with the Quotient Property.","body":"Use the Quotient Property of Logarithms to write each logarithm as a difference of logarithms. Simplify, if possible.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Use the Properties of Logarithms","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"abe88c3log19a","stepAnswer":["$$\\\\log_{4}\\\\left(3\\\\right)-1$$"],"problemType":"TextBox","stepTitle":"Rewriting Logorithms with the Quotient Property.","stepBody":"$$\\\\log_{4}\\\\left(\\\\frac{3}{4}\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\log_{4}\\\\left(3\\\\right)-1$$","hints":{"DefaultPathway":[{"id":"abe88c3log19a-h1","type":"hint","dependencies":[],"title":"Using the Quotient Property of Logorithms.","text":"Use the Quotient Property of Logarithms: $$\\\\log_{a}\\\\left(\\\\frac{M}{N}\\\\right)$$ $$=$$ $$\\\\log_{a}\\\\left(M\\\\right)$$ - $$\\\\log_{a}\\\\left(N\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abe88c3log19a-h2","type":"hint","dependencies":["abe88c3log19a-h1"],"title":"Applying the rule to this question.","text":"In this case, $$\\\\log_{4}\\\\left(\\\\frac{3}{4}\\\\right)=$$ $$\\\\log_{4}\\\\left(3\\\\right)$$ - $$\\\\log_{4}\\\\left(4\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abe88c3log19a-h3","type":"hint","dependencies":["abe88c3log19a-h2"],"title":"Simplify.","text":"$$\\\\log_{4}\\\\left(4\\\\right)$$ $$=$$ $$1$$ so $$\\\\log_{4}\\\\left(\\\\frac{3}{4}\\\\right)$$ $$=$$ $$\\\\log_{4}\\\\left(3\\\\right)$$ - $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abe88c3log2","title":"Solving Logorithms","body":"Evaluate using the properties of logarithms","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Use the Properties of Logarithms","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"abe88c3log2a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"Solving Logorithms","stepBody":"$$\\\\log_{6}\\\\left(6\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"abe88c3log2a-h1","type":"hint","dependencies":[],"title":"Properties of logorithms","text":"Use the property $$\\\\log_{a}\\\\left(a\\\\right)$$ $$=$$ $$1$$. Thus, $$\\\\log_{6}\\\\left(6\\\\right)$$ $$=$$ $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abe88c3log20","title":"Rewriting Logorithms with the Quotient Property.","body":"Use the Quotient Property of Logarithms to write each logarithm as a difference of logarithms. Simplify, if possible.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Use the Properties of Logarithms","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"abe88c3log20a","stepAnswer":["$$\\\\log_{10}\\\\left(x\\\\right)-100$$"],"problemType":"TextBox","stepTitle":"Rewriting Logorithms with the Quotient Property.","stepBody":"$$\\\\log_{10}\\\\left(\\\\frac{x}{1000}\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\log_{10}\\\\left(x\\\\right)-100$$","hints":{"DefaultPathway":[{"id":"abe88c3log20a-h1","type":"hint","dependencies":[],"title":"Using the Quotient Property of Logorithms.","text":"Use the Quotient Property of Logarithms: $$\\\\log_{a}\\\\left(\\\\frac{M}{N}\\\\right)$$ $$=$$ $$\\\\log_{a}\\\\left(M\\\\right)$$ - $$\\\\log_{a}\\\\left(N\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abe88c3log20a-h2","type":"hint","dependencies":["abe88c3log20a-h1"],"title":"Applying the rule to this question.","text":"In this case, $$\\\\log_{10}\\\\left(\\\\frac{x}{1000}\\\\right)$$ $$=$$ $$\\\\log_{10}\\\\left(x\\\\right)$$ - $$\\\\log_{10}\\\\left(1000\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abe88c3log20a-h3","type":"hint","dependencies":["abe88c3log20a-h2"],"title":"Simplify.","text":"$$\\\\log_{10}\\\\left(1000\\\\right)$$ $$=$$ $$100$$ so $$\\\\log_{10}\\\\left(\\\\frac{x}{100}\\\\right)$$ $$=$$ $$\\\\log_{10}\\\\left(x\\\\right)$$ - $$100$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abe88c3log21","title":"Rewriting Logorithms with the Quotient Property.","body":"Use the Quotient Property of Logarithms to write each logarithm as a difference of logarithms. Simplify, if possible.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Use the Properties of Logarithms","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"abe88c3log21a","stepAnswer":["$$\\\\log_{2}\\\\left(5\\\\right)-2$$"],"problemType":"TextBox","stepTitle":"Rewriting Logorithms with the Quotient Property.","stepBody":"$$\\\\log_{2}\\\\left(\\\\frac{5}{4}\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\log_{2}\\\\left(5\\\\right)-2$$","hints":{"DefaultPathway":[{"id":"abe88c3log21a-h1","type":"hint","dependencies":[],"title":"Using the Quotient Property of Logorithms.","text":"Use the Quotient Property of Logarithms: $$\\\\log_{a}\\\\left(\\\\frac{M}{N}\\\\right)$$ $$=$$ $$\\\\log_{a}\\\\left(M\\\\right)$$ - $$\\\\log_{a}\\\\left(N\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abe88c3log21a-h2","type":"hint","dependencies":["abe88c3log21a-h1"],"title":"Applying the rule to this question.","text":"In this case, $$\\\\log_{2}\\\\left(\\\\frac{5}{4}\\\\right)=$$ $$\\\\log_{2}\\\\left(5\\\\right)$$ - $$\\\\log_{2}\\\\left(4\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abe88c3log21a-h3","type":"hint","dependencies":["abe88c3log21a-h2"],"title":"Simplify.","text":"$$\\\\log_{2}\\\\left(4\\\\right)$$ $$=$$ $$2$$ so $$\\\\log_{2}\\\\left(\\\\frac{5}{4}\\\\right)$$ $$=$$ $$\\\\log_{2}\\\\left(5\\\\right)$$ - $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abe88c3log22","title":"Rewriting Logorithms with the Quotient Property.","body":"Use the Quotient Property of Logarithms to write each logarithm as a difference of logarithms. Simplify, if possible.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Use the Properties of Logarithms","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"abe88c3log22a","stepAnswer":["$$1-\\\\log_{10}\\\\left(y\\\\right)$$"],"problemType":"TextBox","stepTitle":"Rewriting Logorithms with the Quotient Property.","stepBody":"$$\\\\log_{10}\\\\left(\\\\frac{10}{y}\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1-\\\\log_{10}\\\\left(y\\\\right)$$","hints":{"DefaultPathway":[{"id":"abe88c3log22a-h1","type":"hint","dependencies":[],"title":"Using the Quotient Property of Logorithms.","text":"Use the Quotient Property of Logarithms: $$\\\\log_{a}\\\\left(\\\\frac{M}{N}\\\\right)$$ $$=$$ $$\\\\log_{a}\\\\left(M\\\\right)$$ - $$\\\\log_{a}\\\\left(N\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abe88c3log22a-h2","type":"hint","dependencies":["abe88c3log22a-h1"],"title":"Applying the rule to this question.","text":"In this case, $$\\\\log_{10}\\\\left(\\\\frac{10}{y}\\\\right)$$ $$=$$ $$\\\\log_{10}\\\\left(10\\\\right)$$ - $$\\\\log_{10}\\\\left(y\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abe88c3log22a-h3","type":"hint","dependencies":["abe88c3log22a-h2"],"title":"Simplify.","text":"$$\\\\log_{10}\\\\left(10\\\\right)$$ $$=$$ $$1$$ so $$\\\\log_{10}\\\\left(\\\\frac{10}{y}\\\\right)$$ $$=$$ $$1$$ - $$\\\\log_{10}\\\\left(y\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abe88c3log23","title":"Rewriting Logorithms with the Power Property.","body":"Use the Power Property of Logarithms to write each logarithm as a product of logarithms. Simplify, if possible.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Use the Properties of Logarithms","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"abe88c3log23a","stepAnswer":["$$3*\\\\log_{5}\\\\left(4\\\\right)$$"],"problemType":"TextBox","stepTitle":"Rewriting Logorithms with the Power Property.","stepBody":"$$\\\\log_{5}\\\\left(4^3\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3*\\\\log_{5}\\\\left(4\\\\right)$$","hints":{"DefaultPathway":[{"id":"abe88c3log23a-h1","type":"hint","dependencies":[],"title":"Using the Power Property of Logarithms.","text":"Use the Power Property of Logarithms: $$\\\\log_{a}\\\\left(M^p\\\\right)$$ $$=$$ $$p*\\\\log_{a}\\\\left(M\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abe88c3log23a-h2","type":"hint","dependencies":["abe88c3log23a-h1"],"title":"Applying the rule to this question.","text":"In this case, $$\\\\log_{5}\\\\left(4^3\\\\right)$$ $$=$$ $$3*\\\\log_{5}\\\\left(4\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abe88c3log24","title":"Rewriting Logorithms with the Power Property.","body":"Use the Power Property of Logarithms to write each logarithm as a product of logarithms. Simplify, if possible.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Use the Properties of Logarithms","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"abe88c3log24a","stepAnswer":["$$10*\\\\log_{10}\\\\left(x\\\\right)$$"],"problemType":"TextBox","stepTitle":"Rewriting Logorithms with the Power Property.","stepBody":"$$\\\\log_{10}\\\\left(x^{10}\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10*\\\\log_{10}\\\\left(x\\\\right)$$","hints":{"DefaultPathway":[{"id":"abe88c3log24a-h1","type":"hint","dependencies":[],"title":"Using the Power Property of Logarithms.","text":"Use the Power Property of Logarithms: $$\\\\log_{a}\\\\left(M^p\\\\right)$$ $$=$$ $$p*\\\\log_{a}\\\\left(M\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abe88c3log24a-h2","type":"hint","dependencies":["abe88c3log24a-h1"],"title":"Applying the rule to this question.","text":"In this case, $$\\\\log_{10}\\\\left(x^{10}\\\\right)$$ $$=$$ $$10*\\\\log_{10}\\\\left(x\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abe88c3log25","title":"Rewriting Logorithms with the Power Property.","body":"Use the Power Property of Logarithms to write each logarithm as a product of logarithms. Simplify, if possible.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Use the Properties of Logarithms","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"abe88c3log25a","stepAnswer":["$$4*\\\\log_{7}\\\\left(5\\\\right)$$"],"problemType":"TextBox","stepTitle":"Rewriting Logorithms with the Power Property.","stepBody":"$$\\\\log_{7}\\\\left(5^4\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4*\\\\log_{7}\\\\left(5\\\\right)$$","hints":{"DefaultPathway":[{"id":"abe88c3log25a-h1","type":"hint","dependencies":[],"title":"Using the Power Property of Logarithms.","text":"Use the Power Property of Logarithms: $$\\\\log_{a}\\\\left(M^p\\\\right)$$ $$=$$ $$p*\\\\log_{a}\\\\left(M\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abe88c3log25a-h2","type":"hint","dependencies":["abe88c3log25a-h1"],"title":"Applying the rule to this question.","text":"In this case, $$\\\\log_{7}\\\\left(5^4\\\\right)$$ $$=$$ $$4*\\\\log_{7}\\\\left(5\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abe88c3log26","title":"Rewriting Logorithms with the Power Property.","body":"Use the Power Property of Logarithms to write each logarithm as a product of logarithms. Simplify, if possible.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Use the Properties of Logarithms","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"abe88c3log26a","stepAnswer":["$$100*\\\\log_{10}\\\\left(x\\\\right)$$"],"problemType":"TextBox","stepTitle":"Rewriting Logorithms with the Power Property.","stepBody":"$$\\\\log_{10}\\\\left(x^{100}\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$100*\\\\log_{10}\\\\left(x\\\\right)$$","hints":{"DefaultPathway":[{"id":"abe88c3log26a-h1","type":"hint","dependencies":[],"title":"Using the Power Property of Logarithms.","text":"Use the Power Property of Logarithms: $$\\\\log_{a}\\\\left(M^p\\\\right)$$ $$=$$ $$p*\\\\log_{a}\\\\left(M\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abe88c3log26a-h2","type":"hint","dependencies":["abe88c3log26a-h1"],"title":"Applying the rule to this question.","text":"In this case, $$\\\\log_{10}\\\\left(x^{100}\\\\right)$$ $$=$$ $$100*\\\\log_{10}\\\\left(x\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abe88c3log27","title":"Rewriting Logorithms with the Power Property.","body":"Use the Power Property of Logarithms to write each logarithm as a product of logarithms. Simplify, if possible.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Use the Properties of Logarithms","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"abe88c3log27a","stepAnswer":["$$7*\\\\log_{2}\\\\left(3\\\\right)$$"],"problemType":"TextBox","stepTitle":"Rewriting Logorithms with the Power Property.","stepBody":"$$\\\\log_{2}\\\\left(3^7\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7*\\\\log_{2}\\\\left(3\\\\right)$$","hints":{"DefaultPathway":[{"id":"abe88c3log27a-h1","type":"hint","dependencies":[],"title":"Using the Power Property of Logarithms.","text":"Use the Power Property of Logarithms: $$\\\\log_{a}\\\\left(M^p\\\\right)$$ $$=$$ $$p*\\\\log_{a}\\\\left(M\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abe88c3log27a-h2","type":"hint","dependencies":["abe88c3log27a-h1"],"title":"Applying the rule to this question.","text":"In this case, $$\\\\log_{2}\\\\left(3^7\\\\right)$$ $$=$$ $$7*\\\\log_{2}\\\\left(3\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abe88c3log28","title":"Rewriting Logorithms with the Power Property.","body":"Use the Power Property of Logarithms to write each logarithm as a product of logarithms. Simplify, if possible.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Use the Properties of Logarithms","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"abe88c3log28a","stepAnswer":["$$20*\\\\log_{10}\\\\left(x\\\\right)$$"],"problemType":"TextBox","stepTitle":"Rewriting Logorithms with the Power Property.","stepBody":"$$\\\\log_{10}\\\\left(x^{20}\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$20*\\\\log_{10}\\\\left(x\\\\right)$$","hints":{"DefaultPathway":[{"id":"abe88c3log28a-h1","type":"hint","dependencies":[],"title":"Using the Power Property of Logarithms.","text":"Use the Power Property of Logarithms: $$\\\\log_{a}\\\\left(M^p\\\\right)$$ $$=$$ $$p*\\\\log_{a}\\\\left(M\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abe88c3log28a-h2","type":"hint","dependencies":["abe88c3log28a-h1"],"title":"Applying the rule to this question.","text":"In this case, $$\\\\log_{10}\\\\left(x^{20}\\\\right)$$ $$=$$ $$20*\\\\log_{10}\\\\left(x\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abe88c3log29","title":"Using the Properties of Logorithms.","body":"Use the Properties of Logarithms to expand the following logarithm. Simplify, if possible.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Use the Properties of Logarithms","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"abe88c3log29a","stepAnswer":["$$1/2+3*\\\\log_{4}\\\\left(x\\\\right)+2*\\\\log_{4}\\\\left(y\\\\right)$$"],"problemType":"TextBox","stepTitle":"Using the Properties of Logorithms.","stepBody":"$$\\\\log_{4}\\\\left(2x^3 y^2\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1/2+3*\\\\log_{4}\\\\left(x\\\\right)+2*\\\\log_{4}\\\\left(y\\\\right)$$","hints":{"DefaultPathway":[{"id":"abe88c3log29a-h1","type":"hint","dependencies":[],"title":"Using the Product Property.","text":"Use the Product Property first: $$\\\\log_{a}\\\\left(M N\\\\right)$$ $$=$$ $$\\\\log_{a}\\\\left(M\\\\right)$$ + $$\\\\log_{a}\\\\left(N\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abe88c3log29a-h2","type":"hint","dependencies":["abe88c3log29a-h1"],"title":"Applying the Product Property to this problem.","text":"Here, the Product Property gives $$\\\\log_{4}\\\\left(2\\\\right)$$ + $$\\\\log_{4}\\\\left(x^3\\\\right)$$ + $$\\\\log_{4}\\\\left(y^2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abe88c3log29a-h3","type":"hint","dependencies":["abe88c3log29a-h2"],"title":"Using the Power Property.","text":"Use the Power Property next: $$\\\\log_{a}\\\\left(M^p\\\\right)$$ $$=$$ $$p*\\\\log_{a}\\\\left(M\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abe88c3log29a-h4","type":"hint","dependencies":["abe88c3log29a-h3"],"title":"Applying the Power Property to this problem.","text":"Here, the Power Property on $$\\\\log_{4}\\\\left(x^3\\\\right)$$ + $$\\\\log_{4}\\\\left(y^2\\\\right)$$ gives $$3*\\\\log_{4}\\\\left(x\\\\right)$$ + $$2*\\\\log_{4}\\\\left(y\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abe88c3log29a-h5","type":"hint","dependencies":["abe88c3log29a-h4"],"title":"Simplifying.","text":"Simplify the elements that you can. Here, $$\\\\log_{4}\\\\left(2\\\\right)$$ $$=$$ $$\\\\frac{1}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abe88c3log29a-h6","type":"hint","dependencies":["abe88c3log29a-h5"],"title":"Putting it all together.","text":"The previous steps leave the equation $$1/2+3*\\\\log_{4}\\\\left(x\\\\right)+2*\\\\log_{4}\\\\left(y\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abe88c3log3","title":"Solving Logorithms","body":"Evaluate using the properties of logarithms","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Use the Properties of Logarithms","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"abe88c3log3a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"Solving Logorithms","stepBody":"$$\\\\log_{9}\\\\left(9\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"abe88c3log3a-h1","type":"hint","dependencies":[],"title":"Properties of logorithms","text":"Use the property $$\\\\log_{a}\\\\left(a\\\\right)$$ $$=$$ $$1$$. Thus, $$\\\\log_{9}\\\\left(9\\\\right)$$ $$=$$ $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abe88c3log30","title":"Using the Properties of Logorithms.","body":"Use the Properties of Logarithms to expand the following logarithm. Simplify, if possible.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Use the Properties of Logarithms","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"abe88c3log30a","stepAnswer":["$$\\\\log_{2}\\\\left(5\\\\right)+4*\\\\log_{2}\\\\left(x\\\\right)+2*\\\\log_{2}\\\\left(y\\\\right)$$"],"problemType":"TextBox","stepTitle":"Using the Properties of Logorithms.","stepBody":"$$\\\\log_{2}\\\\left(5x^4 y^2\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\log_{2}\\\\left(5\\\\right)+4*\\\\log_{2}\\\\left(x\\\\right)+2*\\\\log_{2}\\\\left(y\\\\right)$$","hints":{"DefaultPathway":[{"id":"abe88c3log30a-h1","type":"hint","dependencies":[],"title":"Using the Product Property.","text":"Use the Product Property first: $$\\\\log_{a}\\\\left(M N\\\\right)$$ $$=$$ $$\\\\log_{a}\\\\left(M\\\\right)$$ + $$\\\\log_{a}\\\\left(N\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abe88c3log30a-h2","type":"hint","dependencies":["abe88c3log30a-h1"],"title":"Applying the Product Property to this problem.","text":"Here, the Product Property gives $$\\\\log_{2}\\\\left(5\\\\right)$$ + $$\\\\log_{2}\\\\left(x^4\\\\right)$$ + $$\\\\log_{2}\\\\left(y^2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abe88c3log30a-h3","type":"hint","dependencies":["abe88c3log30a-h2"],"title":"Using the Power Property.","text":"Use the Power Property next: $$\\\\log_{a}\\\\left(M^p\\\\right)$$ $$=$$ $$p*\\\\log_{a}\\\\left(M\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abe88c3log30a-h4","type":"hint","dependencies":["abe88c3log30a-h3"],"title":"Applying the Power Property to this problem.","text":"Here, the Power Property on $$\\\\log_{2}\\\\left(x^4\\\\right)$$ + $$\\\\log_{2}\\\\left(y^2\\\\right)$$ gives $$4*\\\\log_{4}\\\\left(x\\\\right)$$ + $$2*\\\\log_{4}\\\\left(y\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abe88c3log30a-h5","type":"hint","dependencies":["abe88c3log30a-h4"],"title":"Putting it all together.","text":"The previous steps leave the equation $$\\\\log_{2}\\\\left(5\\\\right)+4*\\\\log_{2}\\\\left(x\\\\right)+2*\\\\log_{2}\\\\left(y\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abe88c3log4","title":"Solving Logorithms","body":"Evaluate using the properties of logarithms","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Use the Properties of Logarithms","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"abe88c3log4a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"Solving Logorithms","stepBody":"$$\\\\log_{13}\\\\left(1\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"abe88c3log4a-h1","type":"hint","dependencies":[],"title":"Properties of logorithms","text":"Use the property $$\\\\log_{a}\\\\left(1\\\\right)$$ $$=$$ $$0$$. Thus, $$\\\\log_{13}\\\\left(1\\\\right)$$ $$=$$ $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abe88c3log5","title":"Solving Logorithms","body":"Evaluate using the properties of logarithms","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Use the Properties of Logarithms","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"abe88c3log5a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"Solving Logorithms","stepBody":"$$\\\\log_{5}\\\\left(1\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"abe88c3log5a-h1","type":"hint","dependencies":[],"title":"Properties of logorithms","text":"Use the property $$\\\\log_{a}\\\\left(1\\\\right)$$ $$=$$ $$0$$. Thus, $$\\\\log_{5}\\\\left(1\\\\right)$$ $$=$$ $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abe88c3log6","title":"Solving Logorithms","body":"Evaluate using the properties of logarithms","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Use the Properties of Logarithms","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"abe88c3log6a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"Solving Logorithms","stepBody":"$$\\\\log_{7}\\\\left(7\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"abe88c3log6a-h1","type":"hint","dependencies":[],"title":"Properties of logorithms","text":"Use the property $$\\\\log_{a}\\\\left(a\\\\right)$$ $$=$$ $$1$$. Thus, $$\\\\log_{7}\\\\left(7\\\\right)$$ $$=$$ $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abe88c3log7","title":"Solving Logorithms","body":"Evaluate using the properties of logarithms","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Use the Properties of Logarithms","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"abe88c3log7a","stepAnswer":["$$9$$"],"problemType":"TextBox","stepTitle":"Solving Logorithms","stepBody":"$$4**\\\\log_{4}\\\\left(9\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9$$","hints":{"DefaultPathway":[{"id":"abe88c3log7a-h1","type":"hint","dependencies":[],"title":"Properties of Logorithms","text":"Use the property $$a**\\\\log_{a}\\\\left(x\\\\right)$$ $$=$$ $$x$$. Thus, $$4**\\\\log_{4}\\\\left(9\\\\right)$$ $$=$$ $$9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abe88c3log8","title":"Solving Logorithms","body":"Evaluate using the properties of logarithms","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Use the Properties of Logarithms","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"abe88c3log8a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"Solving Logorithms","stepBody":"$$\\\\log_{3}\\\\left(3^5\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"abe88c3log8a-h1","type":"hint","dependencies":[],"title":"Properties of Logorithms","text":"Use the property $$a**\\\\log_{a}\\\\left(x\\\\right)$$ $$=$$ $$x$$. Thus, $$\\\\log_{3}\\\\left(3^5\\\\right)$$ $$=$$ $$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abe88c3log9","title":"Solving Logorithms","body":"Evaluate using the properties of logarithms","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.4 Use the Properties of Logarithms","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"abe88c3log9a","stepAnswer":["$$15$$"],"problemType":"TextBox","stepTitle":"Solving Logorithms","stepBody":"$$5**\\\\log_{5}\\\\left(15\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$15$$","hints":{"DefaultPathway":[{"id":"abe88c3log9a-h1","type":"hint","dependencies":[],"title":"Properties of Logorithms","text":"Use the property $$a**\\\\log_{a}\\\\left(x\\\\right)$$ $$=$$ $$x$$. Thus, $$5**\\\\log_{5}\\\\left(15\\\\right)$$ $$=$$ $$15$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abf1b34stem1","title":"Exam Score Stem-and-Leaf Graph","body":"Susan Dean compiled a list of exam scores for her spring pre-calculus class. Look at the data and identify which stem-and-leaf plot it relates to.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs","courseName":"OpenStax: Introductory Stats","steps":[{"id":"abf1b34stem1a","stepAnswer":["Graph A"],"problemType":"MultipleChoice","stepTitle":"The data points are 33; 42; 49; 49; 53; 55; 55; 61; 63; 67; 68; 68; 69; 69; 72; 73; 74; 78; 80; 83; 88; 88; 88; 90; 92; 94; 94; 94; 94; 96; $$100$$.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Graph A","Graph B","Graph C","Graph D"],"hints":{"DefaultPathway":[{"id":"abf1b34stem1a-h1","type":"hint","dependencies":[],"title":"Definition of a Stem-and-Leaf Plot","text":"A stem-and-leaf graph (also known as a stemplot) is used for exploratory data analysis. On the \\"leaf\\" side of the table, we will list out the final significant digit (the last digit important to the data set). All other digits go in the \\"stem\\" side of the table. For instance, $$23$$ has stem $$2$$ and leaf 3; $$5432$$ has stem $$543$$ and leaf 2; $$9.3$$ has stem $$9$$ and leaf $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem1a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph A, Graph D"],"dependencies":["abf1b34stem1a-h1"],"title":"Identifying the Correct Stems","text":"Which two stem-and-leaf graphs include all the stems required for this question?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A, Graph D","Graph A, Graph C","Graph A, Graph B","Graph B, Graph C"],"subHints":[{"id":"abf1b34stem1a-h2-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph A, Graph D"],"dependencies":[],"title":"Identifying the Correct Stems","text":"Which two stem-and-leaf graphs include stems from $$3$$ (for data points in the 30s) to $$10$$ (for data points in the 100s)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A, Graph D","Graph A, Graph C","Graph A, Graph B","Graph B, Graph C"]}]},{"id":"abf1b34stem1a-h3","type":"hint","dependencies":["abf1b34stem1a-h2"],"title":"Duplicates in Stem-and-Leaf","text":"All data points will be visible in the stem-and-leaf graph, even duplicates. For instance, if $$94$$ appears twice in the given data set, it will appear twice in the stem-and-leaf graph as well.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem1a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph A"],"dependencies":["abf1b34stem1a-h3"],"title":"Identifying the Correct Stem-and-Leaf Graph","text":"Which graph includes all the correct data values for the given data?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph D"]},{"id":"abf1b34stem1a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph A"],"dependencies":["abf1b34stem1a-h4"],"title":"Identifying the Correct Stem-and-Leaf Graph","text":"Which graph includes all the duplicates? For instance, $$49$$, $$55$$, $$68$$, and $$69$$ should appear twice in the graph while $$88$$ should appear three times and $$94$$ four times.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph D"]}]}}]},{"id":"abf1b34stem10","title":"Creating a Bar Graph","body":"By the end of $$2011$$, Facebook had over $$146$$ million users in the United States. The table shows three age groups, the number of users in each age group, and the proportion (%) of users in each age group.\\\\n##figure2.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs","courseName":"OpenStax: Introductory Stats","steps":[{"id":"abf1b34stem10a","stepAnswer":["Graph A"],"problemType":"MultipleChoice","stepTitle":"Which bar graph uses the data provided in the table? The bar graphs use age groups and proportion (%) of Facebook users.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Graph A","Graph B","Graph C","Graph D"],"hints":{"DefaultPathway":[{"id":"abf1b34stem10a-h1","type":"hint","dependencies":[],"title":"Definition of a Bar Graph","text":"Bar graphs show different bars that are separated from one another, separating categorical data into their categories. The bars are rectangles that can be horizontal or vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem10a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Ages"],"dependencies":["abf1b34stem10a-h1"],"title":"Determining the Categories for the Bars","text":"Since we\'re using vertical bars, our horizontal axis will list out the categories. Are the categories the ages or the proportions? Think about how we\'re separating the data into different groups. What are those groups?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Ages","Proportions"]},{"id":"abf1b34stem10a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph C"],"dependencies":["abf1b34stem10a-h2"],"title":"Determining the Horizontal Axis","text":"Knowing that the categories are ages, which graph can we EXCLUDE?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph B","Graph C","Graph D"],"subHints":[{"id":"abf1b34stem10a-h3-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph C"],"dependencies":[],"title":"Determining the Horizontal Axis","text":"Which graph does not have ages on the horizontal axis?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph B","Graph C","Graph D"]}]},{"id":"abf1b34stem10a-h4","type":"hint","dependencies":["abf1b34stem10a-h3"],"title":"Determining Correctness of Bins","text":"From the three remaining graphs, Graph A, B, and D, we can determine next which ones use the accurate \\"bins\\" for the ages.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem10a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph D"],"dependencies":["abf1b34stem10a-h4"],"title":"Determining Correctness of Bins","text":"Which graph does not use the correct bins? In effect, which graph\'s age groups is incorrect?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph B","Graph D"]},{"id":"abf1b34stem10a-h6","type":"hint","dependencies":["abf1b34stem10a-h5"],"title":"Correctness of Data Values for Each Bin","text":"Now we\'re down to the final two: Graph A and Graph B. To determine which bar graph is correct, we must determine which one uses the correct data points. For instance, the proportion for the $$13-25$$ age group should be 45%, for the $$26-44$$ age group should be 36%, and the proportion for the $$45-64$$ age group should be 19%.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem10a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph A"],"dependencies":["abf1b34stem10a-h6"],"title":"Correctness of Data Values for Each Bin","text":"Which of Graph A and Graph B is the correct graph? Which graph uses the correct data points listed above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph B"]}]}}]},{"id":"abf1b34stem11","title":"Creating a Bar Graph","body":"The population in Park City is made up of children, working-age adults, and retirees. The table provided shows the three age groups, the number of people in the town from each age group, and the proportion (%) of people in each age group.\\\\n##figure2.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs","courseName":"OpenStax: Introductory Stats","steps":[{"id":"abf1b34stem11a","stepAnswer":["Graph C"],"problemType":"MultipleChoice","stepTitle":"Which bar graph uses the data provided in the table? The bar graphs use age groups and proportion (%) of the population.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Graph A","Graph B","Graph C","Graph D"],"hints":{"DefaultPathway":[{"id":"abf1b34stem11a-h1","type":"hint","dependencies":[],"title":"Definition of a Bar Graph","text":"Bar graphs show different bars that are separated from one another, separating categorical data into their categories. The bars are rectangles that can be horizontal or vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem11a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Ages"],"dependencies":["abf1b34stem11a-h1"],"title":"Determining the Categories for the Bars","text":"Since we\'re using vertical bars, our horizontal axis will list out the categories. Are the categories the ages or the proportions? Think about how we\'re separating the data into different groups. What are those groups?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Ages","Proportions"]},{"id":"abf1b34stem11a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph B"],"dependencies":["abf1b34stem11a-h2"],"title":"Determining the Horizontal Axis","text":"Knowing that the categories are ages, which graph can we EXCLUDE?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph B","Graph C","Graph D"],"subHints":[{"id":"abf1b34stem11a-h3-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph B"],"dependencies":[],"title":"Determining the Horizontal Axis","text":"Which graph does not have ages on the horizontal axis?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph B","Graph C","Graph D"]}]},{"id":"abf1b34stem11a-h4","type":"hint","dependencies":["abf1b34stem11a-h3"],"title":"Determining Correctness of Bins","text":"From the three remaining graphs, Graph A, C, and D, we can determine which ones use the accurate \\"bins\\" for the ages.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem11a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph D"],"dependencies":["abf1b34stem11a-h4"],"title":"Determining Correctness of Bins","text":"Which graph does not use the correct bins? In other words, which age categories of which graph is incorrect?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph C","Graph D"]},{"id":"abf1b34stem11a-h6","type":"hint","dependencies":["abf1b34stem11a-h5"],"title":"Correctness of Data Values for Each Bin","text":"Now we\'re down to the final two: Graph A and Graph C. To determine which bar graph is correct, we must determine which one uses the correct data points. For instance, the proportion for the Children age group should be 19%, the Working-age adults group should be 43%, and the proportion for the Retirees group should be 38%.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem11a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph C"],"dependencies":["abf1b34stem11a-h6"],"title":"Correctness of Data Values for Each Bin","text":"Which of Graph A and Graph C is the correct graph? Which graph uses the correct data points listed above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph C"]}]}}]},{"id":"abf1b34stem12","title":"Creating a Bar Graph","body":"The table contains the $$2010$$ obesity rates in U.S. states and Washington, DC.\\\\n##figure2.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs","courseName":"OpenStax: Introductory Stats","steps":[{"id":"abf1b34stem12a","stepAnswer":["Graph C"],"problemType":"MultipleChoice","stepTitle":"Which bar graph uses the data provided in the table for all the states beginning with the letter \\"A\\"? The bar graphs use state and percent (%) of the population.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Graph A","Graph B","Graph C","Graph D"],"hints":{"DefaultPathway":[{"id":"abf1b34stem12a-h1","type":"hint","dependencies":[],"title":"Definition of a Bar Graph","text":"Bar graphs show different bars that are separated from one another, separating categorical data into their categories. The bars are rectangles that can be horizontal or vertical. Here the four states that begin with an \\"A\\" are Alabama, Alaska, Arizona, and Arkansas.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem12a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["States"],"dependencies":["abf1b34stem12a-h1"],"title":"Determining the Categories for the Bars","text":"Since we\'re using vertical bars, our horizontal axis will list out the categories. Are the categories the states or the percents? Think about how we\'re separating the data into different groups. What are those groups?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["States","Percents"]},{"id":"abf1b34stem12a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph D"],"dependencies":["abf1b34stem12a-h2"],"title":"Determining the Horizontal Axis","text":"Knowing that the categories are states, which graph can we EXCLUDE?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph B","Graph C","Graph D"],"subHints":[{"id":"abf1b34stem12a-h3-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph D"],"dependencies":[],"title":"Determining the Horizontal Axis","text":"Which graph does not have states on the horizontal axis?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph B","Graph C","Graph D"]}]},{"id":"abf1b34stem12a-h4","type":"hint","dependencies":["abf1b34stem12a-h3"],"title":"Determining Correctness of Bins","text":"From the three remaining graphs, Graph A, B, and C, we can determine next which ones use the accurate names for the states.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem12a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph A"],"dependencies":["abf1b34stem12a-h4"],"title":"Determining Correctness of Bins","text":"Which graph does not use the correct names? In other words, which graph\'s chosen states are not the ones we want (all the states beginning with the letter \\"A\\")?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph B","Graph C"]},{"id":"abf1b34stem12a-h6","type":"hint","dependencies":["abf1b34stem12a-h5"],"title":"Correctness of Data Values for Each Bin","text":"Now we\'re down to the final two: Graph B and Graph C. To determine which bar graph is correct, we must determine which one uses the correct data points. For instance, the percent for Alabama should be $$32.2\\\\%$$, for Alaska should be $$24.5\\\\%$$, for Arizona should be $$24.3\\\\%$$, and for Arkansas should be $$30.1\\\\%$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem12a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph C"],"dependencies":["abf1b34stem12a-h6"],"title":"Correctness of Data Values for Each Bin","text":"Which of Graph B and Graph C is the correct graph? Which graph uses the correct data points listed above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph B","Graph C"]}]}}]},{"id":"abf1b34stem13","title":"Creating a Bar Graph","body":"The table contains the $$2010$$ obesity rates in U.S. states and Washington, DC.\\\\n##figure2.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs","courseName":"OpenStax: Introductory Stats","steps":[{"id":"abf1b34stem13a","stepAnswer":["Graph D"],"problemType":"MultipleChoice","stepTitle":"Which bar graph uses the data provided in the table for all the states beginning with the letter \\"W\\"? The bar graphs use state and percent (%) of the population.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Graph A","Graph B","Graph C","Graph D"],"hints":{"DefaultPathway":[{"id":"abf1b34stem13a-h1","type":"hint","dependencies":[],"title":"Definition of a Bar Graph","text":"Bar graphs show different bars that are separated from one another, separating categorical data into their categories. The bars are rectangles that can be horizontal or vertical. The four states that begin with a \\"W\\" are Washington, West Virginia, Wisconsin, and Wyoming.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem13a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["States"],"dependencies":["abf1b34stem13a-h1"],"title":"Determining the Categories for the Bars","text":"Since we\'re using vertical bars, our horizontal axis will list out the categories. Are the categories the states or the percents? Think about how we\'re separating the data into different groups. What are those groups?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["States","Percents"]},{"id":"abf1b34stem13a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph A"],"dependencies":["abf1b34stem13a-h2"],"title":"Determining the Horizontal Axis","text":"Knowing that the categories are states, which graph can we EXCLUDE?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph B","Graph C","Graph D"],"subHints":[{"id":"abf1b34stem13a-h3-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph A"],"dependencies":[],"title":"Determining the Horizontal Axis","text":"Which graph does not have states on the horizontal axis?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph B","Graph C","Graph D"]}]},{"id":"abf1b34stem13a-h4","type":"hint","dependencies":["abf1b34stem13a-h3"],"title":"Correctness of Bins","text":"From the three remaining graphs, Graph B, C, and D, we can determine next which ones use the accurate names for the states.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem13a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph C"],"dependencies":["abf1b34stem13a-h4"],"title":"Correctness of Bins","text":"Which graph does not use the correct names? In other words, which graph\'s chosen states are not the ones we want (all the states beginning with the letter \\"W\\")?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph B","Graph C","Graph D"]},{"id":"abf1b34stem13a-h6","type":"hint","dependencies":["abf1b34stem13a-h5"],"title":"Correctness of Data Values for Each Bin","text":"Now we\'re down to the final two: Graph B and Graph D. To determine which bar graph is correct, we must determine which one uses the correct data points. For instance, the percent for Washington should be $$25.5\\\\%$$, for West Virginia should be $$32.5\\\\%$$, for Wisconsin should be $$26.3\\\\%$$, and for Wyoming should be $$25.1\\\\%$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem13a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph D"],"dependencies":["abf1b34stem13a-h6"],"title":"Correctness of Data Values for Each Bin","text":"Which of Graph B and Graph D is the correct graph? Which graph uses the correct data points listed above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph B","Graph D"]}]}}]},{"id":"abf1b34stem14","title":"Creating a Bar Graph","body":"The columns in the provided table contain: the race or ethnicity of students in U.S. Public Schools for the class of $$2011$$, percentages for the Advanced Placement examine population for that class, and percentages for the overall student population.\\\\n##figure2.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs","courseName":"OpenStax: Introductory Stats","steps":[{"id":"abf1b34stem14a","stepAnswer":["Graph D"],"problemType":"MultipleChoice","stepTitle":"Listed below are four bar graphs. Identify the bar graph with the student race or ethnicity (qualitative data) on the x-axis, and the Advanced Placement examinee population percentages on the y-axis. Note that for brevity\'s sake, the $$\\\\frac{race}{ethnicity}$$ categories will be listed as the numerical equivalents from the table.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Graph A","Graph B","Graph C","Graph D"],"hints":{"DefaultPathway":[{"id":"abf1b34stem14a-h1","type":"hint","dependencies":[],"title":"Definition of a Bar Graph","text":"Bar graphs show different bars that are separated from one another, separating categorical data into their categories. The bars are rectangles that can be horizontal or vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem14a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{race}{ethnicity}$$"],"dependencies":["abf1b34stem14a-h1"],"title":"Determining the Categories for the Bins","text":"Since we\'re using vertical bars, our horizontal axis will list out the categories. Are the categories the $$\\\\frac{race}{ethnicity}$$ or the percents? Think about how we\'re separating the data into different groups. What are those groups?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{race}{ethnicity}$$","percentages"]},{"id":"abf1b34stem14a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph B"],"dependencies":["abf1b34stem14a-h2"],"title":"Determining the Horizontal Axis","text":"Knowing that the categories are $$\\\\frac{race}{ethnicity}$$, which graph can we EXCLUDE?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph B","Graph C","Graph D"],"subHints":[{"id":"abf1b34stem14a-h3-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph B"],"dependencies":[],"title":"Determining the Horizontal Axis","text":"Which graph does not have $$\\\\frac{race}{ethnicity}$$ on the horizontal axis?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph B","Graph C","Graph D"]}]},{"id":"abf1b34stem14a-h4","type":"hint","dependencies":["abf1b34stem14a-h3"],"title":"Correctness of Bins","text":"From the three remaining graphs, Graph A, C, and D, we can determine next which one does not correspond to any data set from the table.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem14a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph C"],"dependencies":["abf1b34stem14a-h4"],"title":"Correctness of Bins","text":"Which graph does not use data from the table? Specifically, which table doesn\'t use either the AP Examinee Population nor the Overall Student Population?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph C","Graph D"],"subHints":[{"id":"abf1b34stem14a-h5-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph C"],"dependencies":[],"title":"Correctness of Bins","text":"For $$\\\\frac{race}{ethnicity}$$ $$1$$ (Asian, Asian American, or Pacific Islander), which graph of Graph A, C, or D does not have a data point of $$10.3\\\\%$$ or $$5.7\\\\%$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph C","Graph D"]}]},{"id":"abf1b34stem14a-h6","type":"hint","dependencies":["abf1b34stem14a-h5"],"title":"Correctness of Data Values for Each Bin","text":"Now we\'re down to the final two: Graph A and Graph D. To determine which bar graph is correct, we must determine which one uses the correct data points. We know that one graph uses the AP Examinee Population percentages while the other uses the overall Student Population percentages.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem14a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph D"],"dependencies":["abf1b34stem14a-h6"],"title":"Correctness of Data Values for Each Bin","text":"We specifically want the graph of the Advanced Placement examinee population percentages. Which of Graph A or D uses this?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph D"],"subHints":[{"id":"abf1b34stem14a-h7-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph D"],"dependencies":[],"title":"Correctness of Data Values for Each Bin","text":"For $$\\\\frac{race}{ethnicity}$$ $$1$$ (Asian, Asian American, or Pacific Islander), which graph of Graph A or D has a data point of $$10.3\\\\%$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph D"]}]}]}}]},{"id":"abf1b34stem15","title":"Creating a Bar Graph","body":"Park city is broken down into six voting districts. The table shows the percent of the total registered voter population that lives in each district as well as the percent total of the entire population that lives in each district.\\\\n##figure2.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs","courseName":"OpenStax: Introductory Stats","steps":[{"id":"abf1b34stem15a","stepAnswer":["Graph A"],"problemType":"MultipleChoice","stepTitle":"Listed below are four bar graphs. Identify the bar graph with the district (qualitative data) on the x-axis, and the registered voter population percentages on the y-axis.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Graph A","Graph B","Graph C","Graph D"],"hints":{"DefaultPathway":[{"id":"abf1b34stem15a-h1","type":"hint","dependencies":[],"title":"Definition of a Bar Graph","text":"Bar graphs show different bars that are separated from one another, separating categorical data into their categories. The bars are rectangles that can be horizontal or vertical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem15a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Districts"],"dependencies":["abf1b34stem15a-h1"],"title":"Determining the Categories for the Bins","text":"Since we\'re using vertical bars, our horizontal axis will list out the categories. Are the categories the district or the percents? Think about how we\'re separating the data into different groups. What are those groups?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Districts","Percents"]},{"id":"abf1b34stem15a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph B"],"dependencies":["abf1b34stem15a-h2"],"title":"Determining the Horizontal Axis","text":"Knowing that the categories are distincts, which graph can we EXCLUDE?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph B","Graph C","Graph D"],"subHints":[{"id":"abf1b34stem15a-h3-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph B"],"dependencies":[],"title":"Determining the Horizontal Axis","text":"Which graph does not have district on the horizontal axis?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph B","Graph C","Graph D"]}]},{"id":"abf1b34stem15a-h4","type":"hint","dependencies":["abf1b34stem15a-h3"],"title":"Correctness of Bins","text":"From the three remaining graphs, Graph A, C, and D, we can determine next which one does not correspond to any data set from the table.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem15a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph D"],"dependencies":["abf1b34stem15a-h4"],"title":"Correctness of Bins","text":"Which graph does not use data from the table? Specifically, which table doesn\'t use either the Registered voter population or Overall city population?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph C","Graph D"],"subHints":[{"id":"abf1b34stem15a-h5-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph D"],"dependencies":[],"title":"Correctness of Bins","text":"For district $$1$$, which graph of Graph A, C, or D does not have a data point of $$15.5\\\\%$$ or $$19.4\\\\%\\\\%$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph C","Graph D"]}]},{"id":"abf1b34stem15a-h6","type":"hint","dependencies":["abf1b34stem15a-h5"],"title":"Correctness of Data Values for Each Bin","text":"Now we\'re down to the final two: Graph A and Graph C. To determine which bar graph is correct, we must determine which one uses the correct data points. We know that one graph uses the Registered voter population percentages while the other uses the overall city population percentages.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem15a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph A"],"dependencies":["abf1b34stem15a-h6"],"title":"Correctness of Data Values for Each Bin","text":"We specifically want the graph of the registered voter population percentages. Which of Graph A or C uses this?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph C"],"subHints":[{"id":"abf1b34stem15a-h7-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph A"],"dependencies":[],"title":"Correctness of Data Values for Each Bin","text":"For district $$1$$, which graph of Graph A or C has a data point of $$15.5\\\\%$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph C"]}]}]}}]},{"id":"abf1b34stem2","title":"Reading Stem-and-Leaf Plots","body":"Look at the stem-and-leaf plot provided. These are the exam scores for Susan Dean\'s spring pre-calculus class. You can read the stem-and-leaf plot as the Stem houses the tens digit and the Leaf houses the ones digit. For instance, the data point with Stem $$3$$ and Leaf $$3$$ is an exam score of $$33$$ while the data point with Stem $$10$$ and Leaf $$0$$ is an exam score of 100.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs","courseName":"OpenStax: Introductory Stats","steps":[{"id":"abf1b34stem2a","stepAnswer":["$$\\\\frac{8}{31}$$"],"problemType":"TextBox","stepTitle":"What proportion of students got an A on the test? In other words, what proportion of students scored in the 90s or 100? Please do not estimate, give the exact fraction.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{8}{31}$$","hints":{"DefaultPathway":[{"id":"abf1b34stem2a-h1","type":"hint","dependencies":[],"title":"Definition of a Stem-and-Leaf Plot","text":"A stem-and-leaf graph (also known as a stemplot) is used for exploratory data analysis. On the \\"leaf\\" side of the table, we will list out the final significant digit (the last digit important to the data set). All other digits go in the \\"stem\\" side of the table. For instance, $$23$$ has stem $$2$$ and leaf 3; $$5432$$ has stem $$543$$ and leaf 2; $$9.3$$ has stem $$9$$ and leaf $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem2a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$9$$, $$10$$"],"dependencies":["abf1b34stem2a-h1"],"title":"Identifying the Correct Stems","text":"Now, think about what it means for a student to get a score in the 90s or $$100$$. What would be the correct stems to look at from the graph?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$9$$, $$10$$","$$3$$, $$4$$","$$3$$, $$9$$","$$5$$, $$10$$"]},{"id":"abf1b34stem2a-h3","type":"hint","dependencies":["abf1b34stem2a-h2"],"title":"Identifying Number of Data Points","text":"Each leaf represents one data point in the chart. For instance, since there are four leaves labeled $$4$$ in the $$9$$ stem, we know that four people got 94s on the exam.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["abf1b34stem2a-h3"],"title":"Identifying Correct Data Points","text":"How many data points are there total in the $$9$$ and $$10$$ stems?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"abf1b34stem2a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":[],"title":"Identifying Correct Data Points","text":"How many leaves are there for the $$9$$ and $$10$$ stem? Specifically, how many numbers are there in total in the data set $$0$$, $$2$$, $$4$$, $$4$$, $$4$$, $$4$$, $$6$$, 0?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"abf1b34stem2a-h5","type":"hint","dependencies":["abf1b34stem2a-h4"],"title":"Definition of a Proportion","text":"The definition of a proportion is a part of the whole. Knowing that the part we\'re specifying is $$8$$, we can now determine the size of the whole.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$31$$"],"dependencies":["abf1b34stem2a-h5"],"title":"Determining Total Number of Overall Data Points","text":"How many data points are there in total?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"abf1b34stem2a-h6-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$31$$"],"dependencies":[],"title":"Determining Total Number of Overall Data Points","text":"How many numbers are there in all the leaves?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"abf1b34stem2a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["abf1b34stem2a-h6"],"title":"Determine the Final Proportion","text":"What is the part divided by the whole? In essence, what is the number of students who got an A (90 and above) over all the students in the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem2a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{8}{31}$$"],"dependencies":["abf1b34stem2a-h7"],"title":"Determine the Final Proportion","text":"We know that $$8$$ people score $$90$$ or above, and there are $$31$$ people total. Therefore, the proportion is $$\\\\frac{8}{31}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"abf1b34stem3","title":"Creating Stem-and-Leaf Plots","body":"The Park City basketball team wants to analyze data from its past $$30$$ games. It has compiled the list of scores from its last $$30$$ games and wants to create a stem-and-leaf plot from the data provided.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs","courseName":"OpenStax: Introductory Stats","steps":[{"id":"abf1b34stem3a","stepAnswer":["Graph D"],"problemType":"MultipleChoice","stepTitle":"What is the stem-and-leaf plot of the data provided next? 32; 32; 33; 34; 38; 40; 42; 42; 43; 44; 46; 47; 47; 48; 48; 48; 49; 50; 50; 51; 52; 52; 52; 53; 54; 56; 57; 57; 60; $$61$$.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Graph A","Graph B","Graph C","Graph D"],"hints":{"DefaultPathway":[{"id":"abf1b34stem3a-h1","type":"hint","dependencies":[],"title":"Definition of a Stem-and-Leaf Plot","text":"A stem-and-leaf graph (also known as a stemplot) is used for exploratory data analysis. On the \\"leaf\\" side of the table, we will list out the final significant digit (the last digit important to the data set). All other digits go in the \\"stem\\" side of the table. For instance, $$23$$ has stem $$2$$ and leaf 3; $$5432$$ has stem $$543$$ and leaf 2; $$9.3$$ has stem $$9$$ and leaf $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem3a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph A, Graph D"],"dependencies":["abf1b34stem3a-h1"],"title":"Identifying the Correct Stems","text":"Which two stem-and-leaf graphs include all the stems required for this question?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A, Graph D","Graph A, Graph C","Graph A, Graph B","Graph B, Graph C"],"subHints":[{"id":"abf1b34stem3a-h2-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph A, Graph D"],"dependencies":[],"title":"Identifying the Correct Stems","text":"Which two stem-and-leaf graphs include stems from $$3$$ (for data points in the 30s) to $$6$$ (for data points in the 60s)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A, Graph D","Graph A, Graph C","Graph A, Graph B","Graph B, Graph C"]}]},{"id":"abf1b34stem3a-h3","type":"hint","dependencies":["abf1b34stem3a-h2"],"title":"Determining Duplicates in Stem-and-Leaf Plots","text":"All data points will be visible in the stem-and-leaf graph, even duplicates. For instance, if $$32$$ appears twice in the given data set, it will appear twice in the stem-and-leaf graph as well.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem3a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph D"],"dependencies":["abf1b34stem3a-h3"],"title":"Identifying the Correct Stem-and-Leaf Graph","text":"Which graph includes all the correct data values for the given data?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph D"],"subHints":[{"id":"abf1b34stem3a-h4-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph D"],"dependencies":[],"title":"Identifying the Correct Stem-and-Leaf Graph","text":"Which graph includes all the duplicates? For instance, $$32$$, $$42$$, $$47$$, $$50$$, and $$57$$ should appear twice in the graph while $$48$$ and $$52$$ should appear three times.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph D"]}]}]}}]},{"id":"abf1b34stem4","title":"Identifying Concentrations and Outliers","body":"The data here represents distances (in kilometers) from a home to local supermarkets. The stems are the ones places and the leaves are the tenths places. For instance, the data point with stem $$1$$ and leaf $$1$$ represents $$1.1$$ kilometer.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs","courseName":"OpenStax: Introductory Stats","steps":[{"id":"abf1b34stem4a","stepAnswer":["$$3$$ and $$4$$ kilometers"],"problemType":"MultipleChoice","stepTitle":"Do the data seem to have any concentration of values? If so, where?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$3$$ and $$4$$ kilometers","choices":["$$1$$ and $$2$$ kilometers","$$3$$ and $$4$$ kilometers","$$5$$ and $$6$$ kilometers","$$4$$ and $$5$$ kilometers"],"hints":{"DefaultPathway":[{"id":"abf1b34stem4a-h1","type":"hint","dependencies":[],"title":"Definition of Clustering","text":"Concentration of data points occur where the most data are.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem4a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3$$ and $$4$$ kilometers"],"dependencies":["abf1b34stem4a-h1"],"title":"Identifying Clusters in the Data","text":"Where are the clusters of data?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$1$$ and $$2$$ kilometers","$$3$$ and $$4$$ kilometers","$$5$$ and $$6$$ kilometers","$$4$$ and $$5$$ kilometers"],"subHints":[{"id":"abf1b34stem4a-h2-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3$$ and $$4$$ kilometers"],"dependencies":[],"title":"Identifying Clusters in the Leaves","text":"Which two stems have the most leaves? For instance, Stem $$1$$ has two leaves: $$1$$ and $$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$1$$ and $$2$$ kilometers","$$3$$ and $$4$$ kilometers","$$5$$ and $$6$$ kilometers","$$4$$ and $$5$$ kilometers"]}]}]}},{"id":"abf1b34stem4b","stepAnswer":["$$12.3$$"],"problemType":"MultipleChoice","stepTitle":"Which point is an outlier?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$12.3$$","choices":["$$1.1$$","$$12.3$$","$$2.3$$","$$4.0$$"],"hints":{"DefaultPathway":[{"id":"abf1b34stem4b-h1","type":"hint","dependencies":[],"title":"Definition of Outliers","text":"An outlier is a piece of data that doesn\'t fit in with the rest of the data. From a visual standpoint, an outlier is extreme in the sense that it sits far away from the other data points.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem4b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$12.3$$"],"dependencies":["abf1b34stem4b-h1"],"title":"Identifying the Outlier","text":"Which of the four given data points is far away from all the other data points? That point is the outlier.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$1.1$$","$$12.3$$","$$2.3$$","$$4.0$$"]}]}}]},{"id":"abf1b34stem5","title":"Creating a Stem-and-Leaf Graph","body":"The following data show the distance (in miles) from the homes of off-campus statistics students to the college. $$0.5;$$ $$0.7;$$ $$1.1;$$ $$1.2;$$ $$1.2;$$ $$1.3;$$ $$1.3;$$ $$1.5;$$ $$1.5;$$ $$1.7;$$ $$1.7;$$ $$1.8;$$ $$1.9;$$ $$2.0;$$ $$2.2;$$ $$2.5;$$ $$2.6;$$ $$2.8;$$ $$2.8;$$ $$2.8;$$ $$3.5;$$ $$3.8;$$ $$4.4;$$ $$4.8;$$ $$4.9;$$ $$5.2;$$ $$5.5;$$ $$5.7;$$ $$5.8;$$ $$8.0$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs","courseName":"OpenStax: Introductory Stats","steps":[{"id":"abf1b34stem5a","stepAnswer":["Graph C"],"problemType":"MultipleChoice","stepTitle":"Identify the stem plot that uses the data from above.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Graph A","Graph B","Graph C","Graph D"],"hints":{"DefaultPathway":[{"id":"abf1b34stem5a-h1","type":"hint","dependencies":[],"title":"Definition of a Stem-and-Leaf Plot","text":"A stem-and-leaf graph (also known as a stemplot) is used for exploratory data analysis. On the \\"leaf\\" side of the table, we will list out the final significant digit (the last digit important to the data set). All other digits go in the \\"stem\\" side of the table. For instance, $$23$$ has stem $$2$$ and leaf 3; $$5432$$ has stem $$543$$ and leaf 2; $$9.3$$ has stem $$9$$ and leaf $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem5a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph A, Graph C"],"dependencies":["abf1b34stem5a-h1"],"title":"Identifying the Correct Stems","text":"Which two stem-and-leaf graphs include all the stems required for this question?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A, Graph D","Graph A, Graph C","Graph A, Graph B","Graph B, Graph C"],"subHints":[{"id":"abf1b34stem5a-h2-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph A, Graph C"],"dependencies":[],"title":"Identifying the Correct Stems","text":"Which two stem-and-leaf graphs include stems from $$0$$ (for data points in the 0s) to $$8$$ (for data points in the 8s)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A, Graph D","Graph A, Graph C","Graph A, Graph B","Graph B, Graph C"]}]},{"id":"abf1b34stem5a-h3","type":"hint","dependencies":["abf1b34stem5a-h2"],"title":"Determining Duplicates in Stem-and-Leaf","text":"All data points will be visible on the stem-and-leaf graph, even duplicates. For instance, if $$1.2$$ appears twice in the given data set, it will appear twice in the stem-and-leaf graph as well.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem5a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph C"],"dependencies":["abf1b34stem5a-h3"],"title":"Identifying the Correct Stem-and-Leaf Graph","text":"Which graph includes all the correct data values for the given data?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph C"],"subHints":[{"id":"abf1b34stem5a-h4-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph C"],"dependencies":[],"title":"Identifying the Correct Stem-and-Leaf Graph","text":"Which graph includes all the duplicates? For instance, $$1.2$$, $$1.3$$, $$1.5$$, and $$1.7$$ should appear twice in the graph while $$2.8$$ should appear three times.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph C"]}]}]}}]},{"id":"abf1b34stem6","title":"Creating a Stem-and-Leaf Graph","body":"Student grades on a chemistry exam were $$77$$, $$78$$, $$76$$, $$81$$, $$86$$, $$51$$, $$79$$, $$82$$, $$84$$, $$99$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs","courseName":"OpenStax: Introductory Stats","steps":[{"id":"abf1b34stem6a","stepAnswer":["Graph B"],"problemType":"MultipleChoice","stepTitle":"Which of the graphs given is the stem-and-leaf plot for the given data set?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Graph A","Graph B","Graph C","Graph D"],"hints":{"DefaultPathway":[{"id":"abf1b34stem6a-h1","type":"hint","dependencies":[],"title":"Reorder the Data from Least to Greatest","text":"The first tip in creating a stem-and-leaf plot is to reorder the data from smallest to greatest in order to determine what stems to include in the data. Organization is key.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem6a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$51$$, $$76$$, $$77$$, $$78$$, $$79$$, $$81$$, $$82$$, $$84$$, $$86$$, $$99$$"],"dependencies":["abf1b34stem6a-h1"],"title":"Choose the Correct Reordering of the Data","text":"What is the correct ordering, from least to greatest, of the data?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$51$$, $$76$$, $$77$$, $$78$$, $$79$$, $$81$$, $$82$$, $$84$$, $$86$$, $$99$$","$$51$$, $$66$$, $$67$$, $$68$$, $$69$$, $$81$$, $$82$$, $$84$$, $$86$$, $$99$$","$$76$$, $$77$$, $$78$$, $$79$$, $$81$$, $$82$$, $$84$$, $$86$$, $$99$$","$$51$$, $$76$$, $$77$$, $$78$$, $$79$$, $$89$$, $$91$$, $$92$$, $$94$$, $$96$$"]},{"id":"abf1b34stem6a-h3","type":"hint","dependencies":["abf1b34stem6a-h2"],"title":"Definition of a Stem-and-Leaf Plot","text":"Now that you reordered the data, we can create the stem-and-leaf plot. A stem-and-leaf graph (also known as a stemplot) is used for exploratory data analysis. On the \\"leaf\\" side of the table, we will list out the final significant digit (the last digit important to the data set). All other digits go in the \\"stem\\" side of the table. For instance, $$23$$ has stem $$2$$ and leaf 3; $$5432$$ has stem $$543$$ and leaf 2; $$9.3$$ has stem $$9$$ and leaf $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem6a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph B, Graph D"],"dependencies":["abf1b34stem6a-h3"],"title":"Identifying the Correct Stems","text":"Which two stem-and-leaf graphs include all the stems required for this question?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A, Graph D","Graph A, Graph C","Graph A, Graph B","Graph B, Graph D"],"subHints":[{"id":"abf1b34stem6a-h4-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph B, Graph D"],"dependencies":[],"title":"Identifying the Correct Stems","text":"Which two stem-and-leaf graphs include stems $$5$$, $$7$$, $$8$$, 9? They could include more than just those stems, but all four of those stems must be present.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A, Graph D","Graph A, Graph C","Graph A, Graph B","Graph B, Graph D"]}]},{"id":"abf1b34stem6a-h5","type":"hint","dependencies":["abf1b34stem6a-h4"],"title":"Checking Stem and Leaf Alignments and Data Points","text":"All data points for leaves must align with their stems. For instance, the data point at $$99$$ should have a stem at $$9$$ and a leaf at $$9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem6a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph B"],"dependencies":["abf1b34stem6a-h5"],"title":"Identifying the Correct Stem-and-Leaf Graph","text":"Which graph includes all the correct leaves paired with their stems for the given data?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph B","Graph D"],"subHints":[{"id":"abf1b34stem6a-h6-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph B"],"dependencies":[],"title":"Identifying the Correct Stem-and-Leaf Graph","text":"Which graph does not flip data points around? For instance, which graph has a stem of $$9$$ that aligns with a leaf of $$9$$ for the data point 99? As a sanity check, this graph should also have a stem of $$8$$ that aligns with a leaf of $$2$$ for the data point $$82$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph B","Graph D"]}]}]}}]},{"id":"abf1b34stem7","title":"Identifying Outliers","body":"The data here represents student grades on a chemistry exam. The stems are the tens places and the leaves are the ones places. For instance, the data point with stem $$7$$ and leaf $$7$$ represents a score of 77.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs","courseName":"OpenStax: Introductory Stats","steps":[{"id":"abf1b34stem7a","stepAnswer":["$$51$$"],"problemType":"TextBox","stepTitle":"Which point is an outlier?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$51$$","hints":{"DefaultPathway":[{"id":"abf1b34stem7a-h1","type":"hint","dependencies":[],"title":"Definition of Outliers","text":"An outlier is a piece of data that doesn\'t fit in with the rest of the data. From a visual standpoint, an outlier is extreme in the sense that it sits far away from the other data points.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem7a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$51$$"],"dependencies":["abf1b34stem7a-h1"],"title":"Identifying the Outlier","text":"Which of the four given data points is far away from all the other data points? This is the outlier.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$77$$","$$78$$","$$99$$","$$51$$"],"subHints":[{"id":"abf1b34stem7a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":[],"title":"Identifying Breaks in the Data","text":"One method of determining an outlier is finding if there is a \\"space\\" in the data where there are no data points. What is the stem in the graph for which there is no leaf?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem7a-h2-s2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Greater than $$6$$"],"dependencies":[],"title":"Identifying the Correct Side of the Outlier","text":"Now, knowing that there are no data points at $$6$$, we can determine where the outlier is. Is there more data on the side less than $$6$$ or the side greater than 6?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Greater than $$6$$","Less than $$6$$"]},{"id":"abf1b34stem7a-h2-s3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$51$$"],"dependencies":[],"title":"Determining the Overall Outlier","text":"Knowing now that there is less data on the side less than $$6$$, what is the data point that represents the outlier? It would be on the side less than $$6$$ and as a reminder, all the data points are student grades. For instance, a stem of $$7$$ with a leaf of $$6$$ would combine to form a test score of $$76$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$77$$","$$78$$","$$99$$","$$51$$"]}]}]}}]},{"id":"abf1b34stem8","title":"Creating a Line Graph","body":"In a survey, $$40$$ mothers were asked how many times per week a teenager must be reminded to do his or her chores.\\\\n##figure2.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs","courseName":"OpenStax: Introductory Stats","steps":[{"id":"abf1b34stem8a","stepAnswer":["Graph C"],"problemType":"MultipleChoice","stepTitle":"Which line graph is the correct line graph of the given data?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Graph A","Graph B","Graph C","Graph D"],"hints":{"DefaultPathway":[{"id":"abf1b34stem8a-h1","type":"hint","dependencies":[],"title":"Definition of a Line Graph","text":"Line graphs show an x-axis (the horizontal axis) of data values and a y-axis (vertical axis) of frequency points.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem8a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph D"],"dependencies":["abf1b34stem8a-h1"],"title":"Determining the Correct Axes","text":"Which line graph does NOT follow the definition of a line graph listed above? Specifically, which line graph uses data values (number of times teenager is reminded) as the y-axis (vertical axis) and the frequency points (frequency) as the x-axis (horizontal axis)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph B","Graph C","Graph D"]},{"id":"abf1b34stem8a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph B"],"dependencies":["abf1b34stem8a-h2"],"title":"Using the Correct Data Points to Determine the Graph","text":"Which is the remaining three graphs (Graph A, B, and C) uses the wrong data points? In other words, which of the three graphs does not have an x-axis (horizontal axis) that uses the data points $$0$$, $$1$$, $$2$$, $$3$$, $$4$$, and 5?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph B","Graph C"]},{"id":"abf1b34stem8a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph C"],"dependencies":["abf1b34stem8a-h3"],"title":"Determining the Correct Frequency for Each Data Point","text":"Which of the remaining two graphs (Graph A and Graph C) uses the correct frequencies for each of the data points in the table?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph C"],"subHints":[{"id":"abf1b34stem8a-h4-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph C"],"dependencies":[],"title":"Determining the Correct Frequency for Each Data Point","text":"Which of the remaining two graphs does not flip the data for $$2$$ and $$3$$ (number of times teenager is reminded). Specifically, which graph correctly uses the data points $$(2,8)$$ and $$(3,14)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph C"]}]}]}}]},{"id":"abf1b34stem9","title":"Creating a Line Graph","body":"In a survey, $$40$$ people were asked how many times per year they had their car in the shop for repairs. The results are shown in the table provided.\\\\n##figure2.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs","courseName":"OpenStax: Introductory Stats","steps":[{"id":"abf1b34stem9a","stepAnswer":["Graph D"],"problemType":"MultipleChoice","stepTitle":"Which line graph is the correct line graph of the given data?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Graph A","Graph B","Graph C","Graph D"],"hints":{"DefaultPathway":[{"id":"abf1b34stem9a-h1","type":"hint","dependencies":[],"title":"Definition of a Line Graph","text":"Line graphs show an x-axis (the horizontal axis) of data values and a y-axis (vertical axis) of frequency points.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"abf1b34stem9a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph B"],"dependencies":["abf1b34stem9a-h1"],"title":"Determining the Correct Axes","text":"Which line graph does NOT follow the definition of a line graph listed above? Specifically, which line graph uses data values (number of times in shop) as the y-axis (vertical axis) and the frequency points (frequency) as the x-axis (horizontal axis)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph B","Graph C","Graph D"]},{"id":"abf1b34stem9a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph C"],"dependencies":["abf1b34stem9a-h2"],"title":"Using the Correct Data Points to Determine the Graph","text":"Which is the remaining three graphs (Graph A, C, and D) uses the wrong data points? In other words, which of the three graphs does not have an x-axis (horizontal axis) that uses the data points from $$0$$ to $$3$$ and instead uses a different range?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph C","Graph D"]},{"id":"abf1b34stem9a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph D"],"dependencies":["abf1b34stem9a-h3"],"title":"Determining the Correct Frequency for Each Data Point","text":"Which of the remaining two graphs (Graph A and Graph D) uses the correct frequencies for each of the data points in the table?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph D"],"subHints":[{"id":"abf1b34stem9a-h4-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph D"],"dependencies":[],"title":"Determining the Correct Frequency for Each Data Point","text":"Which of the remaining two graphs does not flip the data for $$2$$ and $$3$$ (number of times in shop). Specifically, which graph correctly uses the data points $$(1,10)$$ and $$(2,14)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph D"]}]}]}}]},{"id":"ac0c1d3spe1","title":"Binomial Squares Pattern","body":"Square each binomial using the Binomial Squares Pattern.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac0c1d3spe1a","stepAnswer":["$$x^2+10x+25$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(x+5\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^2+10x+25$$","hints":{"DefaultPathway":[{"id":"ac0c1d3spe1a-h1","type":"hint","dependencies":[],"title":"Addition Binomial Square Formula","text":"We compare our expression to the addition binomial square formula: $${\\\\left(a+b\\\\right)}^2=a^2+2a b+b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe1a-h2","type":"hint","dependencies":["ac0c1d3spe1a-h1"],"title":"Compare the Binomial\\\\n","text":"$${\\\\left(a+b\\\\right)}^2$$\\\\n$${\\\\left(x+5\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x$$"],"dependencies":["ac0c1d3spe1a-h2"],"title":"Identify a","text":"What is a in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["ac0c1d3spe1a-h3"],"title":"Identify $$b$$","text":"What is $$b$$ in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe1a-h5","type":"hint","dependencies":["ac0c1d3spe1a-h3","ac0c1d3spe1a-h4"],"title":"Plug in the Terms","text":"Substitute $$a=x$$ and $$b=5$$ into the addition binomial square formula:\\\\n$${\\\\left(a+b\\\\right)}^2=a^2+2a b+b^2$$\\\\n$${\\\\left(x+5\\\\right)}^2=x^2+2x\\\\times5+5^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe1a-h6","type":"hint","dependencies":["ac0c1d3spe1a-h5"],"title":"Simplify","text":"$$x^2+10x+25$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac0c1d3spe10","title":"Binomial Squares Pattern","body":"Square each binomial using the Binomial Squares Pattern.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac0c1d3spe10a","stepAnswer":["$$p^2-26p+169$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(p-13\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$p^2-26p+169$$","hints":{"DefaultPathway":[{"id":"ac0c1d3spe10a-h1","type":"hint","dependencies":[],"title":"Subtraction Binomial Square Formula","text":"We compare our expression to the subtraction binomial square formula: $${\\\\left(a-b\\\\right)}^2=a^2-2a b+b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe10a-h2","type":"hint","dependencies":["ac0c1d3spe10a-h1"],"title":"Compare the Binomial\\\\n","text":"$${\\\\left(a-b\\\\right)}^2$$\\\\n$${\\\\left(p-13\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$p$$"],"dependencies":["ac0c1d3spe10a-h2"],"title":"Identify a","text":"What is a in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$13$$"],"dependencies":["ac0c1d3spe10a-h3"],"title":"Identify $$b$$","text":"What is $$b$$ in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe10a-h5","type":"hint","dependencies":["ac0c1d3spe10a-h3","ac0c1d3spe10a-h4"],"title":"Plug in the Terms","text":"Substitute $$a=p$$ and $$b=13$$ into the subtraction binomial square formula:\\\\n$${\\\\left(a-b\\\\right)}^2=a^2+2a b+b^2$$\\\\n$${\\\\left(p-13\\\\right)}^2=p^2-2p\\\\times13+{13}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe10a-h6","type":"hint","dependencies":["ac0c1d3spe10a-h5"],"title":"Simplify","text":"$$p^2-26p+169$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac0c1d3spe11","title":"Binomial Squares Pattern","body":"Square each binomial using the Binomial Squares Pattern.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac0c1d3spe11a","stepAnswer":["$$16x^2+48x+36$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(4x+6\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16x^2+48x+36$$","hints":{"DefaultPathway":[{"id":"ac0c1d3spe11a-h1","type":"hint","dependencies":[],"title":"Addition Binomial Square Formula","text":"We compare our expression to the addition binomial square formula: $${\\\\left(a+b\\\\right)}^2=a^2+2a b+b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe11a-h2","type":"hint","dependencies":["ac0c1d3spe11a-h1"],"title":"Compare the Binomial\\\\n","text":"$${\\\\left(a+b\\\\right)}^2$$\\\\n$${\\\\left(4x+6\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe11a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4x$$"],"dependencies":["ac0c1d3spe11a-h2"],"title":"Identify a","text":"What is a in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["ac0c1d3spe11a-h3"],"title":"Identify $$b$$","text":"What is $$b$$ in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe11a-h5","type":"hint","dependencies":["ac0c1d3spe11a-h3","ac0c1d3spe11a-h4"],"title":"Plug in the Terms","text":"Substitute $$a=4x$$ and $$b=6$$ into the addition binomial square formula:\\\\n$${\\\\left(a+b\\\\right)}^2=a^2+2a b+b^2$$\\\\n$${\\\\left(4x+6\\\\right)}^2={\\\\left(4x\\\\right)}^2+2\\\\times4x\\\\times6+6^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe11a-h6","type":"hint","dependencies":["ac0c1d3spe11a-h5"],"title":"Simplify","text":"$$16x^2+48x+36$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac0c1d3spe12","title":"Binomial Squares Pattern","body":"Square each binomial using the Binomial Squares Pattern.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac0c1d3spe12a","stepAnswer":["$$9d^2+6d+1$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(3d+1\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9d^2+6d+1$$","hints":{"DefaultPathway":[{"id":"ac0c1d3spe12a-h1","type":"hint","dependencies":[],"title":"Addition Binomial Square Formula","text":"We compare our expression to the addition binomial square formula: $${\\\\left(a+b\\\\right)}^2=a^2+2a b+b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe12a-h2","type":"hint","dependencies":["ac0c1d3spe12a-h1"],"title":"Compare the Binomial\\\\n","text":"$${\\\\left(a+b\\\\right)}^2$$\\\\n$${\\\\left(3d+1\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe12a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3d$$"],"dependencies":["ac0c1d3spe12a-h2"],"title":"Identify a","text":"What is a in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ac0c1d3spe12a-h3"],"title":"Identify $$b$$","text":"What is $$b$$ in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe12a-h5","type":"hint","dependencies":["ac0c1d3spe12a-h3","ac0c1d3spe12a-h4"],"title":"Plug in the Terms","text":"Substitute $$a=3d$$ and $$b=1$$ into the addition binomial square formula:\\\\n$${\\\\left(a+b\\\\right)}^2=a^2+2a b+b^2$$\\\\n$${\\\\left(3d+1\\\\right)}^2={\\\\left(3d\\\\right)}^2+2\\\\times3d\\\\times1+1^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe12a-h6","type":"hint","dependencies":["ac0c1d3spe12a-h5"],"title":"Simplify","text":"$$9d^2+6d+1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac0c1d3spe13","title":"Binomial Squares Pattern","body":"Square each binomial using the Binomial Squares Pattern.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac0c1d3spe13a","stepAnswer":["$$16a^2+80a+100$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(4a+10\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16a^2+80a+100$$","hints":{"DefaultPathway":[{"id":"ac0c1d3spe13a-h1","type":"hint","dependencies":[],"title":"Addition Binomial Square Formula","text":"We compare our expression to the addition binomial square formula: $${\\\\left(a+b\\\\right)}^2=a^2+2a b+b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe13a-h2","type":"hint","dependencies":["ac0c1d3spe13a-h1"],"title":"Compare the Binomial\\\\n","text":"$${\\\\left(a+b\\\\right)}^2$$\\\\n$${\\\\left(4a+10\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe13a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4a$$"],"dependencies":["ac0c1d3spe13a-h2"],"title":"Identify a","text":"What is a in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["ac0c1d3spe13a-h3"],"title":"Identify $$b$$","text":"What is $$b$$ in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe13a-h5","type":"hint","dependencies":["ac0c1d3spe13a-h3","ac0c1d3spe13a-h4"],"title":"Plug in the Terms","text":"Substitute $$a=4a$$ and $$b=10$$ into the addition binomial square formula:\\\\n$${\\\\left(a+b\\\\right)}^2=a^2+2a b+b^2$$\\\\n$${\\\\left(4a+10\\\\right)}^2={\\\\left(4a\\\\right)}^2+2\\\\times4a\\\\times10+{10}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe13a-h6","type":"hint","dependencies":["ac0c1d3spe13a-h5"],"title":"Simplify","text":"$$16a^2+80a+100$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac0c1d3spe14","title":"Binomial Squares Pattern","body":"Square each binomial using the Binomial Squares Pattern.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac0c1d3spe14a","stepAnswer":["$$4x^2-12x y+9y^2$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(2q+\\\\frac{1}{3}\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4x^2-12x y+9y^2$$","hints":{"DefaultPathway":[{"id":"ac0c1d3spe14a-h1","type":"hint","dependencies":[],"title":"Addition Binomial Square Formula","text":"We compare our expression to the addition binomial square formula: $${\\\\left(a+b\\\\right)}^2=a^2+2a b+b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe14a-h2","type":"hint","dependencies":["ac0c1d3spe14a-h1"],"title":"Compare the Binomial\\\\n","text":"$${\\\\left(a+b\\\\right)}^2$$\\\\n$${\\\\left(2q+\\\\frac{1}{3}\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe14a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2q$$"],"dependencies":["ac0c1d3spe14a-h2"],"title":"Identify a","text":"What is a in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe14a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["ac0c1d3spe14a-h3"],"title":"Identify $$b$$","text":"What is $$b$$ in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe14a-h5","type":"hint","dependencies":["ac0c1d3spe14a-h3","ac0c1d3spe14a-h4"],"title":"Plug in the Terms","text":"Substitute $$a=2q$$ and $$b=\\\\frac{1}{3}$$ into the addition binomial square formula:\\\\n$${\\\\left(a+b\\\\right)}^2=a^2+2a b+b^2$$\\\\n$${\\\\left(2q+\\\\frac{1}{3}\\\\right)}^2={\\\\left(2q\\\\right)}^2+\\\\frac{2\\\\times2q\\\\times1}{3}+{\\\\left(\\\\frac{1}{3}\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe14a-h6","type":"hint","dependencies":["ac0c1d3spe14a-h5"],"title":"Simplify","text":"$$4q^2+\\\\frac{4}{3} q+\\\\frac{1}{9}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac0c1d3spe15","title":"Binomial Squares Pattern","body":"Square each binomial using the Binomial Squares Pattern.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac0c1d3spe15a","stepAnswer":["$$9z^2+\\\\frac{6}{5} z+\\\\frac{1}{25}$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(3z+\\\\frac{1}{5}\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9z^2+\\\\frac{6}{5} z+\\\\frac{1}{25}$$","hints":{"DefaultPathway":[{"id":"ac0c1d3spe15a-h1","type":"hint","dependencies":[],"title":"Addition Binomial Square Formula","text":"We compare our expression to the addition binomial square formula: $${\\\\left(a+b\\\\right)}^2=a^2+2a b+b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe15a-h2","type":"hint","dependencies":["ac0c1d3spe15a-h1"],"title":"Compare the Binomial\\\\n","text":"$${\\\\left(a+b\\\\right)}^2$$\\\\n$${\\\\left(3z+\\\\frac{1}{5}\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3z$$"],"dependencies":["ac0c1d3spe15a-h2"],"title":"Identify a","text":"What is a in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{5}$$"],"dependencies":["ac0c1d3spe15a-h3"],"title":"Identify $$b$$","text":"What is $$b$$ in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe15a-h5","type":"hint","dependencies":["ac0c1d3spe15a-h3","ac0c1d3spe15a-h4"],"title":"Plug in the Terms","text":"Substitute $$a=3z$$ and $$b=\\\\frac{1}{5}$$ into the addition binomial square formula:\\\\n$${\\\\left(a+b\\\\right)}^2=a^2+2a b+b^2$$\\\\n$${\\\\left(3z+\\\\frac{1}{5}\\\\right)}^2={\\\\left(3z\\\\right)}^2+\\\\frac{2\\\\times3z\\\\times1}{5}+{\\\\left(\\\\frac{1}{5}\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe15a-h6","type":"hint","dependencies":["ac0c1d3spe15a-h5"],"title":"Simplify","text":"$$9z^2+\\\\frac{6}{5} z+\\\\frac{1}{25}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac0c1d3spe16","title":"Binomial Squares Pattern","body":"Square each binomial using the Binomial Squares Pattern.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac0c1d3spe16a","stepAnswer":["$$4x^2-12x y+9y^2$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(2x-3y\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4x^2-12x y+9y^2$$","hints":{"DefaultPathway":[{"id":"ac0c1d3spe16a-h1","type":"hint","dependencies":[],"title":"Subtraction Binomial Square Formula","text":"We compare our expression to the subtraction binomial square formula: $${\\\\left(a-b\\\\right)}^2=a^2-2a b+b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe16a-h2","type":"hint","dependencies":["ac0c1d3spe16a-h1"],"title":"Compare the Binomial\\\\n","text":"$${\\\\left(a-b\\\\right)}^2$$\\\\n$${\\\\left(2x-3y\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe16a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x$$"],"dependencies":["ac0c1d3spe16a-h2"],"title":"Identify a","text":"What is a in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe16a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3y$$"],"dependencies":["ac0c1d3spe16a-h3"],"title":"Identify $$b$$","text":"What is $$b$$ in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe16a-h5","type":"hint","dependencies":["ac0c1d3spe16a-h3","ac0c1d3spe16a-h4"],"title":"Plug in the Terms","text":"Substitute $$a=2x$$ and $$b=3y$$ into the subtraction binomial square formula:\\\\n$${\\\\left(a-b\\\\right)}^2=a^2+2a b+b^2$$\\\\n$${\\\\left(2x-3y\\\\right)}^2={\\\\left(2x\\\\right)}^2-2\\\\times2x 3y+{\\\\left(3y\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe16a-h6","type":"hint","dependencies":["ac0c1d3spe16a-h5"],"title":"Simplify","text":"$$4x^2-12x y+9y^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac0c1d3spe17","title":"Binomial Squares Pattern","body":"Square each binomial using the Binomial Squares Pattern.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac0c1d3spe17a","stepAnswer":["$$9x^2-6x y+y^2$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(3x-y\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9x^2-6x y+y^2$$","hints":{"DefaultPathway":[{"id":"ac0c1d3spe17a-h1","type":"hint","dependencies":[],"title":"Subtraction Binomial Square Formula","text":"We compare our expression to the subtraction binomial square formula: $${\\\\left(a-b\\\\right)}^2=a^2-2a b+b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe17a-h2","type":"hint","dependencies":["ac0c1d3spe17a-h1"],"title":"Compare the Binomial\\\\n","text":"$${\\\\left(a-b\\\\right)}^2$$\\\\n$${\\\\left(3x-y\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe17a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x$$"],"dependencies":["ac0c1d3spe17a-h2"],"title":"Identify a","text":"What is a in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe17a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y$$"],"dependencies":["ac0c1d3spe17a-h3"],"title":"Identify $$b$$","text":"What is $$b$$ in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe17a-h5","type":"hint","dependencies":["ac0c1d3spe17a-h3","ac0c1d3spe17a-h4"],"title":"Plug in the Terms","text":"Substitute $$a=3x$$ and $$b=y$$ into the subtraction binomial square formula:\\\\n$${\\\\left(a-b\\\\right)}^2=a^2+2a b+b^2$$\\\\n$${\\\\left(3x-y\\\\right)}^2={\\\\left(3x\\\\right)}^2-2\\\\times3x y+y^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe17a-h6","type":"hint","dependencies":["ac0c1d3spe17a-h5"],"title":"Simplify","text":"$$9x^2-6x y+y^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac0c1d3spe18","title":"Binomial Squares Pattern","body":"Square each binomial using the Binomial Squares Pattern.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac0c1d3spe18a","stepAnswer":["$$4y^2-12y z+9z^2$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(2y-3z\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4y^2-12y z+9z^2$$","hints":{"DefaultPathway":[{"id":"ac0c1d3spe18a-h1","type":"hint","dependencies":[],"title":"Subtraction Binomial Square Formula","text":"We compare our expression to the subtraction binomial square formula: $${\\\\left(a-b\\\\right)}^2=a^2-2a b+b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe18a-h2","type":"hint","dependencies":["ac0c1d3spe18a-h1"],"title":"Compare the Binomial\\\\n","text":"$${\\\\left(a-b\\\\right)}^2$$\\\\n$${\\\\left(2y-3z\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe18a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2y$$"],"dependencies":["ac0c1d3spe18a-h2"],"title":"Identify a","text":"What is a in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe18a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3z$$"],"dependencies":["ac0c1d3spe18a-h3"],"title":"Identify $$b$$","text":"What is $$b$$ in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe18a-h5","type":"hint","dependencies":["ac0c1d3spe18a-h3","ac0c1d3spe18a-h4"],"title":"Plug in the Terms","text":"Substitute $$a=2y$$ and $$b=3z$$ into the subtraction binomial square formula:\\\\n$${\\\\left(a-b\\\\right)}^2=a^2+2a b+b^2$$\\\\n$${\\\\left(3x-y\\\\right)}^2={\\\\left(2y\\\\right)}^2-2\\\\times2y 3z+{\\\\left(3z\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe18a-h6","type":"hint","dependencies":["ac0c1d3spe18a-h5"],"title":"Simplify","text":"$$4y^2-12y z+9z^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac0c1d3spe19","title":"Binomial Squares Pattern","body":"Square each binomial using the Binomial Squares Pattern.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac0c1d3spe19a","stepAnswer":["$$\\\\frac{1}{25} x^2-\\\\frac{2}{35} x y+\\\\frac{1}{49} y^2$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(\\\\frac{1}{5} x-\\\\frac{1}{7} y\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{25} x^2-\\\\frac{2}{35} x y+\\\\frac{1}{49} y^2$$","hints":{"DefaultPathway":[{"id":"ac0c1d3spe19a-h1","type":"hint","dependencies":[],"title":"Subtraction Binomial Square Formula","text":"We compare our expression to the subtraction binomial square formula: $${\\\\left(a-b\\\\right)}^2=a^2-2a b+b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe19a-h2","type":"hint","dependencies":["ac0c1d3spe19a-h1"],"title":"Compare the Binomial\\\\n","text":"$${\\\\left(a-b\\\\right)}^2$$\\\\n$${\\\\left(\\\\frac{1}{5} x-\\\\frac{1}{7} y\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe19a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{5} x$$"],"dependencies":["ac0c1d3spe19a-h2"],"title":"Identify a","text":"What is a in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe19a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{7} y$$"],"dependencies":["ac0c1d3spe19a-h3"],"title":"Identify $$b$$","text":"What is $$b$$ in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe19a-h5","type":"hint","dependencies":["ac0c1d3spe19a-h3","ac0c1d3spe19a-h4"],"title":"Plug in the Terms","text":"Substitute $$a=\\\\frac{1}{5} x$$ and $$b=\\\\frac{1}{7} y$$ into the subtraction binomial square formula:\\\\n$${\\\\left(a-b\\\\right)}^2=a^2+2a b+b^2$$\\\\n$${\\\\left(\\\\frac{1}{5} x-\\\\frac{1}{7} y\\\\right)}^2={\\\\left(\\\\frac{1}{5} x\\\\right)}^2-2\\\\frac{1}{5} x \\\\frac{1}{7} y+{\\\\left(\\\\frac{1}{7} y\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe19a-h6","type":"hint","dependencies":["ac0c1d3spe19a-h5"],"title":"Simplify","text":"$$\\\\frac{1}{25} x^2-\\\\frac{2}{35} x y+\\\\frac{1}{49} y^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac0c1d3spe2","title":"Binomial Squares Pattern","body":"Square each binomial using the Binomial Squares Pattern.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac0c1d3spe2a","stepAnswer":["$$w^2+8w+16$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(w+4\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$w^2+8w+16$$","hints":{"DefaultPathway":[{"id":"ac0c1d3spe2a-h1","type":"hint","dependencies":[],"title":"Addition Binomial Square Formula","text":"We compare our expression to the addition binomial square formula: $${\\\\left(a+b\\\\right)}^2=a^2+2a b+b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe2a-h2","type":"hint","dependencies":["ac0c1d3spe2a-h1"],"title":"Compare the Binomial\\\\n","text":"$${\\\\left(a+b\\\\right)}^2$$\\\\n$${\\\\left(w+4\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["w"],"dependencies":["ac0c1d3spe2a-h2"],"title":"Identify a","text":"What is a in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ac0c1d3spe2a-h3"],"title":"Identify $$b$$","text":"What is $$b$$ in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe2a-h5","type":"hint","dependencies":["ac0c1d3spe2a-h3","ac0c1d3spe2a-h4"],"title":"Plug in the Terms","text":"Substitute $$a=w$$ and $$b=4$$ into the addition binomial square formula:\\\\n$${\\\\left(a+b\\\\right)}^2=a^2+2a b+b^2$$\\\\n$${\\\\left(w+4\\\\right)}^2=w^2+2w\\\\times4+4^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe2a-h6","type":"hint","dependencies":["ac0c1d3spe2a-h5"],"title":"Simplify","text":"$$w^2+8w+16$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac0c1d3spe20","title":"Binomial Squares Pattern","body":"Square each binomial using the Binomial Squares Pattern.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac0c1d3spe20a","stepAnswer":["$$\\\\frac{1}{64} x^2-\\\\frac{2}{72} x y+\\\\frac{1}{81} y^2$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(\\\\frac{1}{8} x-\\\\frac{1}{9} y\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{64} x^2-\\\\frac{2}{72} x y+\\\\frac{1}{81} y^2$$","hints":{"DefaultPathway":[{"id":"ac0c1d3spe20a-h1","type":"hint","dependencies":[],"title":"Subtraction Binomial Square Formula","text":"We compare our expression to the subtraction binomial square formula: $${\\\\left(a-b\\\\right)}^2=a^2-2a b+b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe20a-h2","type":"hint","dependencies":["ac0c1d3spe20a-h1"],"title":"Compare the Binomial\\\\n","text":"$${\\\\left(a-b\\\\right)}^2$$\\\\n$${\\\\left(\\\\frac{1}{8} x-\\\\frac{1}{9} y\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe20a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{8} x$$"],"dependencies":["ac0c1d3spe20a-h2"],"title":"Identify a","text":"What is a in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe20a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{9} y$$"],"dependencies":["ac0c1d3spe20a-h3"],"title":"Identify $$b$$","text":"What is $$b$$ in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe20a-h5","type":"hint","dependencies":["ac0c1d3spe20a-h3","ac0c1d3spe20a-h4"],"title":"Plug in the Terms","text":"Substitute $$a=\\\\frac{1}{8} x$$ and $$b=\\\\frac{1}{9} y$$ into the subtraction binomial square formula:\\\\n$${\\\\left(a-b\\\\right)}^2=a^2+2a b+b^2$$\\\\n$${\\\\left(\\\\frac{1}{8} x-\\\\frac{1}{9} y\\\\right)}^2={\\\\left(\\\\frac{1}{8} x\\\\right)}^2-2\\\\frac{1}{8} x \\\\frac{1}{9} y+{\\\\left(\\\\frac{1}{9} y\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe20a-h6","type":"hint","dependencies":["ac0c1d3spe20a-h5"],"title":"Simplify","text":"$$\\\\frac{1}{64} x^2-\\\\frac{2}{72} x y+\\\\frac{1}{81} y^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac0c1d3spe21","title":"Binomial Squares Pattern","body":"Square each binomial using the Binomial Squares Pattern.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac0c1d3spe21a","stepAnswer":["$$16u^6+8u^3+1$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(4u^3+1\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16u^6+8u^3+1$$","hints":{"DefaultPathway":[{"id":"ac0c1d3spe21a-h1","type":"hint","dependencies":[],"title":"Addition Binomial Square Formula","text":"We compare our expression to the addition binomial square formula: $${\\\\left(a+b\\\\right)}^2=a^2+2a b+b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe21a-h2","type":"hint","dependencies":["ac0c1d3spe21a-h1"],"title":"Compare the Binomial\\\\n","text":"$${\\\\left(a+b\\\\right)}^2$$\\\\n$${\\\\left(4u^3+1\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe21a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4u^3$$"],"dependencies":["ac0c1d3spe21a-h2"],"title":"Identify a","text":"What is a in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe21a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ac0c1d3spe21a-h3"],"title":"Identify $$b$$","text":"What is $$b$$ in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe21a-h5","type":"hint","dependencies":["ac0c1d3spe21a-h3","ac0c1d3spe21a-h4"],"title":"Plug in the Terms","text":"Substitute $$a=4u^3$$ and $$b=1$$ into the addition binomial square formula:\\\\n$${\\\\left(a+b\\\\right)}^2=a^2+2a b+b^2$$\\\\n$${\\\\left(4u^3+1\\\\right)}^2={\\\\left(4u^3\\\\right)}^2+2\\\\times4u^3\\\\times1+1^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe21a-h6","type":"hint","dependencies":["ac0c1d3spe21a-h5"],"title":"Simplify","text":"$$16u^6+8u^3+1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac0c1d3spe22","title":"Binomial Squares Pattern","body":"Square each binomial using the Binomial Squares Pattern.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac0c1d3spe22a","stepAnswer":["$$9x^4+12x^2+4$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(3x^2+2\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9x^4+12x^2+4$$","hints":{"DefaultPathway":[{"id":"ac0c1d3spe22a-h1","type":"hint","dependencies":[],"title":"Addition Binomial Square Formula","text":"We compare our expression to the addition binomial square formula: $${\\\\left(a+b\\\\right)}^2=a^2+2a b+b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe22a-h2","type":"hint","dependencies":["ac0c1d3spe22a-h1"],"title":"Compare the Binomial\\\\n","text":"$${\\\\left(a+b\\\\right)}^2$$\\\\n$${\\\\left(3x^2+2\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe22a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x^2$$"],"dependencies":["ac0c1d3spe22a-h2"],"title":"Identify a","text":"What is a in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe22a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ac0c1d3spe22a-h3"],"title":"Identify $$b$$","text":"What is $$b$$ in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe22a-h5","type":"hint","dependencies":["ac0c1d3spe22a-h3","ac0c1d3spe22a-h4"],"title":"Plug in the Terms","text":"Substitute $$a=3x^2$$ and $$b=2$$ into the addition binomial square formula:\\\\n$${\\\\left(a+b\\\\right)}^2=a^2+2a b+b^2$$\\\\n$${\\\\left(3x^2+2\\\\right)}^2={\\\\left(3x^2\\\\right)}^2+2\\\\times3x^2\\\\times2+2^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe22a-h6","type":"hint","dependencies":["ac0c1d3spe22a-h5"],"title":"Simplify","text":"$$9x^4+12x^2+4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac0c1d3spe23","title":"Binomial Squares Pattern","body":"Square each binomial using the Binomial Squares Pattern.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac0c1d3spe23a","stepAnswer":["$$25u^4+90u^2+81$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(5u^2+9\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$25u^4+90u^2+81$$","hints":{"DefaultPathway":[{"id":"ac0c1d3spe23a-h1","type":"hint","dependencies":[],"title":"Addition Binomial Square Formula","text":"We compare our expression to the addition binomial square formula: $${\\\\left(a+b\\\\right)}^2=a^2+2a b+b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe23a-h2","type":"hint","dependencies":["ac0c1d3spe23a-h1"],"title":"Compare the Binomial\\\\n","text":"$${\\\\left(a+b\\\\right)}^2$$\\\\n$${\\\\left(5u^2+9\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe23a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5u^2$$"],"dependencies":["ac0c1d3spe23a-h2"],"title":"Identify a","text":"What is a in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe23a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["ac0c1d3spe23a-h3"],"title":"Identify $$b$$","text":"What is $$b$$ in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe23a-h5","type":"hint","dependencies":["ac0c1d3spe23a-h3","ac0c1d3spe23a-h4"],"title":"Plug in the Terms","text":"Substitute $$a=5u^2$$ and $$b=9$$ into the addition binomial square formula:\\\\n$${\\\\left(a+b\\\\right)}^2=a^2+2a b+b^2$$\\\\n$${\\\\left(5u^2+9\\\\right)}^2={\\\\left(5u^2\\\\right)}^2+2\\\\times5u^2\\\\times9+9^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe23a-h6","type":"hint","dependencies":["ac0c1d3spe23a-h5"],"title":"Simplify","text":"$$25u^4+90u^2+81$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac0c1d3spe24","title":"Binomial Squares Pattern","body":"Square each binomial using the Binomial Squares Pattern.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac0c1d3spe24a","stepAnswer":["$$16y^6-16y^3+4$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(4y^3-2\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16y^6-16y^3+4$$","hints":{"DefaultPathway":[{"id":"ac0c1d3spe24a-h1","type":"hint","dependencies":[],"title":"Subtraction Binomial Square Formula","text":"We compare our expression to the subtraction binomial square formula: $${\\\\left(a-b\\\\right)}^2=a^2-2a b+b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe24a-h2","type":"hint","dependencies":["ac0c1d3spe24a-h1"],"title":"Compare the Binomial\\\\n","text":"$${\\\\left(a-b\\\\right)}^2$$\\\\n$${\\\\left(4y^3-2\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe24a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4y^3$$"],"dependencies":["ac0c1d3spe24a-h2"],"title":"Identify a","text":"What is a in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe24a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ac0c1d3spe24a-h3"],"title":"Identify $$b$$","text":"What is $$b$$ in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe24a-h5","type":"hint","dependencies":["ac0c1d3spe24a-h3","ac0c1d3spe24a-h4"],"title":"Plug in the Terms","text":"Substitute $$a=4y^3$$ and $$b=2$$ into the subtraction binomial square formula:\\\\n$${\\\\left(a-b\\\\right)}^2=a^2+2a b+b^2$$\\\\n$${\\\\left(4y^3-2\\\\right)}^2={\\\\left(4y^3\\\\right)}^2-2\\\\times4y^3\\\\times2+2^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe24a-h6","type":"hint","dependencies":["ac0c1d3spe24a-h5"],"title":"Simplify","text":"$$16y^6-16y^3+4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac0c1d3spe25","title":"Binomial Squares Pattern","body":"Square each binomial using the Binomial Squares Pattern.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac0c1d3spe25a","stepAnswer":["$$64p^6-48p^3+9$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(8p^3-3\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$64p^6-48p^3+9$$","hints":{"DefaultPathway":[{"id":"ac0c1d3spe25a-h1","type":"hint","dependencies":[],"title":"Subtraction Binomial Square Formula","text":"We compare our expression to the subtraction binomial square formula: $${\\\\left(a-b\\\\right)}^2=a^2-2a b+b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe25a-h2","type":"hint","dependencies":["ac0c1d3spe25a-h1"],"title":"Compare the Binomial\\\\n","text":"$${\\\\left(a-b\\\\right)}^2$$\\\\n$${\\\\left(8p^3-3\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe25a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8p^3$$"],"dependencies":["ac0c1d3spe25a-h2"],"title":"Identify a","text":"What is a in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe25a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ac0c1d3spe25a-h3"],"title":"Identify $$b$$","text":"What is $$b$$ in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe25a-h5","type":"hint","dependencies":["ac0c1d3spe25a-h3","ac0c1d3spe25a-h4"],"title":"Plug in the Terms","text":"Substitute $$a=8p^3$$ and $$b=3$$ into the subtraction binomial square formula:\\\\n$${\\\\left(a-b\\\\right)}^2=a^2+2a b+b^2$$\\\\n$${\\\\left(8p^3-3\\\\right)}^2={\\\\left(8p^3\\\\right)}^2-2\\\\times8p^3\\\\times3+3^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe25a-h6","type":"hint","dependencies":["ac0c1d3spe25a-h5"],"title":"Simplify","text":"$$64p^6-48p^3+9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac0c1d3spe26","title":"Product of Conjugates Pattern","body":"Multiply each pair of conjugates using the Product of Conjugates Pattern.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac0c1d3spe26a","stepAnswer":["$$x^2-64$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(x-8\\\\right) \\\\left(x+8\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^2-64$$","hints":{"DefaultPathway":[{"id":"ac0c1d3spe26a-h1","type":"hint","dependencies":[],"title":"Product of Conjugates Pattern","text":"We compare our expression to the product of conjugates pattern formula: $$\\\\left(a-b\\\\right) \\\\left(a+b\\\\right)=a^2-b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe26a-h2","type":"hint","dependencies":["ac0c1d3spe26a-h1"],"title":"Compare the Binomials\\\\n","text":"$$\\\\left(a-b\\\\right) \\\\left(a+b\\\\right)$$\\\\n$$\\\\left(x-8\\\\right) \\\\left(x+8\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe26a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x$$"],"dependencies":["ac0c1d3spe26a-h2"],"title":"Identify a","text":"What is a in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe26a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["ac0c1d3spe26a-h3"],"title":"Identify $$b$$","text":"What is $$b$$ in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe26a-h5","type":"hint","dependencies":["ac0c1d3spe26a-h3","ac0c1d3spe26a-h4"],"title":"Plug in the Terms","text":"Substitute $$a=x$$ and $$b=8$$ into the product of conjugates pattern formula:\\\\n$$\\\\left(a-b\\\\right) \\\\left(a+b\\\\right)=a^2-b^2$$\\\\n$$\\\\left(x-8\\\\right) \\\\left(x+8\\\\right)=x^2-8^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe26a-h6","type":"hint","dependencies":["ac0c1d3spe26a-h5"],"title":"Simplify","text":"$$x^2-64$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac0c1d3spe27","title":"Product of Conjugates Pattern","body":"Multiply each pair of conjugates using the Product of Conjugates Pattern.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac0c1d3spe27a","stepAnswer":["$$m^2-49$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(m-7\\\\right) \\\\left(m+7\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$m^2-49$$","hints":{"DefaultPathway":[{"id":"ac0c1d3spe27a-h1","type":"hint","dependencies":[],"title":"Product of Conjugates Pattern","text":"We compare our expression to the product of conjugates pattern formula: $$\\\\left(a-b\\\\right) \\\\left(a+b\\\\right)=a^2-b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe27a-h2","type":"hint","dependencies":["ac0c1d3spe27a-h1"],"title":"Compare the Binomials\\\\n","text":"$$\\\\left(a-b\\\\right) \\\\left(a+b\\\\right)$$\\\\n$$\\\\left(m-7\\\\right) \\\\left(m+7\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe27a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$m$$"],"dependencies":["ac0c1d3spe27a-h2"],"title":"Identify a","text":"What is a in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe27a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["ac0c1d3spe27a-h3"],"title":"Identify $$b$$","text":"What is $$b$$ in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe27a-h5","type":"hint","dependencies":["ac0c1d3spe27a-h3","ac0c1d3spe27a-h4"],"title":"Plug in the Terms","text":"Substitute $$a=m$$ and $$b=7$$ into the product of conjugates pattern formula:\\\\n$$\\\\left(a-b\\\\right) \\\\left(a+b\\\\right)=a^2-b^2$$\\\\n$$\\\\left(m-7\\\\right) \\\\left(m+7\\\\right)=m^2-7^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe27a-h6","type":"hint","dependencies":["ac0c1d3spe27a-h5"],"title":"Simplify","text":"$$m^2-49$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac0c1d3spe28","title":"Product of Conjugates Pattern","body":"Multiply each pair of conjugates using the Product of Conjugates Pattern.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac0c1d3spe28a","stepAnswer":["$$c^2-25$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(c-5\\\\right) \\\\left(c+5\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$c^2-25$$","hints":{"DefaultPathway":[{"id":"ac0c1d3spe28a-h1","type":"hint","dependencies":[],"title":"Product of Conjugates Pattern","text":"We compare our expression to the product of conjugates pattern formula: $$\\\\left(a-b\\\\right) \\\\left(a+b\\\\right)=a^2-b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe28a-h2","type":"hint","dependencies":["ac0c1d3spe28a-h1"],"title":"Compare the Binomials\\\\n","text":"$$\\\\left(a-b\\\\right) \\\\left(a+b\\\\right)$$\\\\n$$\\\\left(c-5\\\\right) \\\\left(c+5\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe28a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["c"],"dependencies":["ac0c1d3spe28a-h2"],"title":"Identify a","text":"What is a in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe28a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["ac0c1d3spe28a-h3"],"title":"Identify $$b$$","text":"What is $$b$$ in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe28a-h5","type":"hint","dependencies":["ac0c1d3spe28a-h3","ac0c1d3spe28a-h4"],"title":"Plug in the Terms","text":"Substitute $$a=c$$ and $$b=5$$ into the product of conjugates pattern formula:\\\\n$$\\\\left(a-b\\\\right) \\\\left(a+b\\\\right)=a^2-b^2$$\\\\n$$\\\\left(c-5\\\\right) \\\\left(c+5\\\\right)=c^2-5^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe28a-h6","type":"hint","dependencies":["ac0c1d3spe28a-h5"],"title":"Simplify","text":"$$c^2-25$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac0c1d3spe29","title":"Product of Conjugates Pattern","body":"Multiply each pair of conjugates using the Product of Conjugates Pattern.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac0c1d3spe29a","stepAnswer":["$$x^2-\\\\frac{9}{16}$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(x+\\\\frac{3}{4}\\\\right) \\\\left(x-\\\\frac{3}{4}\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^2-\\\\frac{9}{16}$$","hints":{"DefaultPathway":[{"id":"ac0c1d3spe29a-h1","type":"hint","dependencies":[],"title":"Product of Conjugates Pattern","text":"We compare our expression to the product of conjugates pattern formula: $$\\\\left(a-b\\\\right) \\\\left(a+b\\\\right)=a^2-b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe29a-h2","type":"hint","dependencies":["ac0c1d3spe29a-h1"],"title":"Compare the Binomials\\\\n","text":"In a conjugate pair, it does not matter which binomial comes first, as long as one binomial is a sum and the other is a difference.\\\\n$$\\\\left(a-b\\\\right) \\\\left(a+b\\\\right)$$\\\\n$$\\\\left(a+b\\\\right) \\\\left(a-b\\\\right)$$\\\\n$$\\\\left(x+\\\\frac{3}{4}\\\\right) \\\\left(x-\\\\frac{3}{4}\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe29a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x$$"],"dependencies":["ac0c1d3spe29a-h2"],"title":"Identify a","text":"What is a in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe29a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{4}$$"],"dependencies":["ac0c1d3spe29a-h3"],"title":"Identify $$b$$","text":"What is $$b$$ in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe29a-h5","type":"hint","dependencies":["ac0c1d3spe29a-h3","ac0c1d3spe29a-h4"],"title":"Plug in the Terms","text":"Substitute $$a=x$$ and $$b=\\\\frac{3}{4}$$ into the product of conjugates pattern formula:\\\\n$$\\\\left(a+b\\\\right) \\\\left(a-b\\\\right)=a^2-b^2$$\\\\n$$\\\\left(x+\\\\frac{3}{4}\\\\right) \\\\left(x-\\\\frac{3}{4}\\\\right)=x^2-{\\\\left(\\\\frac{3}{4}\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe29a-h6","type":"hint","dependencies":["ac0c1d3spe29a-h5"],"title":"Simplify","text":"$$x^2-\\\\frac{9}{16}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac0c1d3spe3","title":"Binomial Squares Pattern","body":"Square each binomial using the Binomial Squares Pattern.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac0c1d3spe3a","stepAnswer":["$$q^2+24q+144$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(q+12\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$q^2+24q+144$$","hints":{"DefaultPathway":[{"id":"ac0c1d3spe3a-h1","type":"hint","dependencies":[],"title":"Addition Binomial Square Formula","text":"We compare our expression to the addition binomial square formula: $${\\\\left(a+b\\\\right)}^2=a^2+2a b+b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe3a-h2","type":"hint","dependencies":["ac0c1d3spe3a-h1"],"title":"Compare the Binomial\\\\n","text":"$${\\\\left(a+b\\\\right)}^2$$\\\\n$${\\\\left(q+12\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["q"],"dependencies":["ac0c1d3spe3a-h2"],"title":"Identify a","text":"What is a in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["ac0c1d3spe3a-h3"],"title":"Identify $$b$$","text":"What is $$b$$ in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe3a-h5","type":"hint","dependencies":["ac0c1d3spe3a-h3","ac0c1d3spe3a-h4"],"title":"Plug in the Terms","text":"Substitute $$a=q$$ and $$b=12$$ into the addition binomial square formula:\\\\n$${\\\\left(a+b\\\\right)}^2=a^2+2a b+b^2$$\\\\n$${\\\\left(q+12\\\\right)}^2=q^2+2q\\\\times12+{12}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe3a-h6","type":"hint","dependencies":["ac0c1d3spe3a-h5"],"title":"Simplify","text":"$$q^2+24q+144$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac0c1d3spe30","title":"Product of Conjugates Pattern","body":"Multiply each pair of conjugates using the Product of Conjugates Pattern.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac0c1d3spe30a","stepAnswer":["$$b^2-\\\\frac{36}{49}$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(b+\\\\frac{6}{7}\\\\right) \\\\left(b-\\\\frac{6}{7}\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$b^2-\\\\frac{36}{49}$$","hints":{"DefaultPathway":[{"id":"ac0c1d3spe30a-h1","type":"hint","dependencies":[],"title":"Product of Conjugates Pattern","text":"We compare our expression to the product of conjugates pattern formula: $$\\\\left(a-b\\\\right) \\\\left(a+b\\\\right)=a^2-b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe30a-h2","type":"hint","dependencies":["ac0c1d3spe30a-h1"],"title":"Compare the Binomials\\\\n","text":"In a conjugate pair, it does not matter which binomial comes first, as long as one binomial is a sum and the other is a difference.\\\\n$$\\\\left(a-b\\\\right) \\\\left(a+b\\\\right)$$\\\\n$$\\\\left(a+b\\\\right) \\\\left(a-b\\\\right)$$\\\\n$$\\\\left(b+\\\\frac{6}{7}\\\\right) \\\\left(b-\\\\frac{6}{7}\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe30a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$b$$"],"dependencies":["ac0c1d3spe30a-h2"],"title":"Identify a","text":"What is a in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe30a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{6}{7}$$"],"dependencies":["ac0c1d3spe30a-h3"],"title":"Identify $$b$$","text":"What is $$b$$ in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe30a-h5","type":"hint","dependencies":["ac0c1d3spe30a-h3","ac0c1d3spe30a-h4"],"title":"Plug in the Terms","text":"Substitute $$a=b$$ and $$b=\\\\frac{6}{7}$$ into the product of conjugates pattern formula:\\\\n$$\\\\left(a+b\\\\right) \\\\left(a-b\\\\right)=a^2-b^2$$\\\\n$$\\\\left(b+\\\\frac{6}{7}\\\\right) \\\\left(b-\\\\frac{6}{7}\\\\right)=b^2-{\\\\left(\\\\frac{6}{7}\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe30a-h6","type":"hint","dependencies":["ac0c1d3spe30a-h5"],"title":"Simplify","text":"$$b^2-\\\\frac{36}{49}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac0c1d3spe4","title":"Binomial Squares Pattern","body":"Square each binomial using the Binomial Squares Pattern.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac0c1d3spe4a","stepAnswer":["$$y^2+\\\\frac{1}{2} y+\\\\frac{1}{16}$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(y+\\\\frac{1}{4}\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y^2+\\\\frac{1}{2} y+\\\\frac{1}{16}$$","hints":{"DefaultPathway":[{"id":"ac0c1d3spe4a-h1","type":"hint","dependencies":[],"title":"Addition Binomial Square Formula","text":"We compare our expression to the addition binomial square formula: $${\\\\left(a+b\\\\right)}^2=a^2+2a b+b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe4a-h2","type":"hint","dependencies":["ac0c1d3spe4a-h1"],"title":"Compare the Binomial\\\\n","text":"$${\\\\left(a+b\\\\right)}^2$$\\\\n$${\\\\left(y+\\\\frac{1}{4}\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y$$"],"dependencies":["ac0c1d3spe4a-h2"],"title":"Identify a","text":"What is a in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4}$$"],"dependencies":["ac0c1d3spe4a-h3"],"title":"Identify $$b$$","text":"What is $$b$$ in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe4a-h5","type":"hint","dependencies":["ac0c1d3spe4a-h3","ac0c1d3spe4a-h4"],"title":"Plug in the Terms","text":"Substitute $$a=y$$ and $$b=\\\\frac{1}{4}$$ into the addition binomial square formula:\\\\n$${\\\\left(a+b\\\\right)}^2=a^2+2a b+b^2$$\\\\n$${\\\\left(y+\\\\frac{1}{4}\\\\right)}^2=y^2+\\\\frac{2y\\\\times1}{4}+{\\\\left(\\\\frac{1}{4}\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe4a-h6","type":"hint","dependencies":["ac0c1d3spe4a-h5"],"title":"Simplify","text":"$$y^2+\\\\frac{1}{2} y+\\\\frac{1}{16}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac0c1d3spe5","title":"Binomial Squares Pattern","body":"Square each binomial using the Binomial Squares Pattern.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac0c1d3spe5a","stepAnswer":["$$x^2+\\\\frac{4}{3} x+\\\\frac{4}{9}$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(x+\\\\frac{2}{3}\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x^2+\\\\frac{4}{3} x+\\\\frac{4}{9}$$","hints":{"DefaultPathway":[{"id":"ac0c1d3spe5a-h1","type":"hint","dependencies":[],"title":"Addition Binomial Square Formula","text":"We compare our expression to the addition binomial square formula: $${\\\\left(a+b\\\\right)}^2=a^2+2a b+b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe5a-h2","type":"hint","dependencies":["ac0c1d3spe5a-h1"],"title":"Compare the Binomial\\\\n","text":"$${\\\\left(a+b\\\\right)}^2$$\\\\n$${\\\\left(x+\\\\frac{2}{3}\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x$$"],"dependencies":["ac0c1d3spe5a-h2"],"title":"Identify a","text":"What is a in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{3}$$"],"dependencies":["ac0c1d3spe5a-h3"],"title":"Identify $$b$$","text":"What is $$b$$ in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe5a-h5","type":"hint","dependencies":["ac0c1d3spe5a-h3","ac0c1d3spe5a-h4"],"title":"Plug in the Terms","text":"Substitute $$a=x$$ and $$b=\\\\frac{2}{3}$$ into the addition binomial square formula:\\\\n$${\\\\left(a+b\\\\right)}^2=a^2+2a b+b^2$$\\\\n$${\\\\left(x+\\\\frac{2}{3}\\\\right)}^2=x^2+\\\\frac{2y\\\\times2}{3}+{\\\\left(\\\\frac{2}{3}\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe5a-h6","type":"hint","dependencies":["ac0c1d3spe5a-h5"],"title":"Simplify","text":"$$x^2+\\\\frac{4}{3} x+\\\\frac{4}{9}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac0c1d3spe6","title":"Binomial Squares Pattern","body":"Square each binomial using the Binomial Squares Pattern.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac0c1d3spe6a","stepAnswer":["$$y^2-6y+9$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(y-3\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y^2-6y+9$$","hints":{"DefaultPathway":[{"id":"ac0c1d3spe6a-h1","type":"hint","dependencies":[],"title":"Subtraction Binomial Square Formula","text":"We compare our expression to the subtraction binomial square formula: $${\\\\left(a-b\\\\right)}^2=a^2-2a b+b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe6a-h2","type":"hint","dependencies":["ac0c1d3spe6a-h1"],"title":"Compare the Binomial\\\\n","text":"$${\\\\left(a+b\\\\right)}^2$$\\\\n$${\\\\left(y-3\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y$$"],"dependencies":["ac0c1d3spe6a-h2"],"title":"Identify a","text":"What is a in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ac0c1d3spe6a-h3"],"title":"Identify $$b$$","text":"What is $$b$$ in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe6a-h5","type":"hint","dependencies":["ac0c1d3spe6a-h3","ac0c1d3spe6a-h4"],"title":"Plug in the Terms","text":"Substitute $$a=y$$ and $$b=3$$ into the addition binomial square formula:\\\\n$${\\\\left(a-b\\\\right)}^2=a^2+2a b+b^2$$\\\\n$${\\\\left(y-3\\\\right)}^2=y^2-2y\\\\times3+3^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe6a-h6","type":"hint","dependencies":["ac0c1d3spe6a-h5"],"title":"Simplify","text":"$$y^2-6y+9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac0c1d3spe7","title":"Binomial Squares Pattern","body":"Square each binomial using the Binomial Squares Pattern.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac0c1d3spe7a","stepAnswer":["$$b^2-14y+49$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(b-7\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$b^2-14y+49$$","hints":{"DefaultPathway":[{"id":"ac0c1d3spe7a-h1","type":"hint","dependencies":[],"title":"Subtraction Binomial Square Formula","text":"We compare our expression to the subtraction binomial square formula: $${\\\\left(a-b\\\\right)}^2=a^2-2a b+b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe7a-h2","type":"hint","dependencies":["ac0c1d3spe7a-h1"],"title":"Compare the Binomial\\\\n","text":"$${\\\\left(a-b\\\\right)}^2$$\\\\n$${\\\\left(b-7\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$b$$"],"dependencies":["ac0c1d3spe7a-h2"],"title":"Identify a","text":"What is a in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["ac0c1d3spe7a-h3"],"title":"Identify $$b$$","text":"What is $$b$$ in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe7a-h5","type":"hint","dependencies":["ac0c1d3spe7a-h3","ac0c1d3spe7a-h4"],"title":"Plug in the Terms","text":"Substitute $$a=b$$ and $$b=7$$ into the subtraction binomial square formula:\\\\n$${\\\\left(a-b\\\\right)}^2=a^2+2a b+b^2$$\\\\n$${\\\\left(b-7\\\\right)}^2=b^2-2b\\\\times7+7^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe7a-h6","type":"hint","dependencies":["ac0c1d3spe7a-h5"],"title":"Simplify","text":"$$b^2-14y+49$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac0c1d3spe8","title":"Binomial Squares Pattern","body":"Square each binomial using the Binomial Squares Pattern.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac0c1d3spe8a","stepAnswer":["$$y^2-12y+36$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(y-6\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y^2-12y+36$$","hints":{"DefaultPathway":[{"id":"ac0c1d3spe8a-h1","type":"hint","dependencies":[],"title":"Subtraction Binomial Square Formula","text":"We compare our expression to the subtraction binomial square formula: $${\\\\left(a-b\\\\right)}^2=a^2-2a b+b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe8a-h2","type":"hint","dependencies":["ac0c1d3spe8a-h1"],"title":"Compare the Binomial\\\\n","text":"$${\\\\left(a-b\\\\right)}^2$$\\\\n$${\\\\left(y-6\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y$$"],"dependencies":["ac0c1d3spe8a-h2"],"title":"Identify a","text":"What is a in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["ac0c1d3spe8a-h3"],"title":"Identify $$b$$","text":"What is $$b$$ in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe8a-h5","type":"hint","dependencies":["ac0c1d3spe8a-h3","ac0c1d3spe8a-h4"],"title":"Plug in the Terms","text":"Substitute $$a=y$$ and $$b=6$$ into the subtraction binomial square formula:\\\\n$${\\\\left(a-b\\\\right)}^2=a^2+2a b+b^2$$\\\\n$${\\\\left(y-6\\\\right)}^2=y^2-2y\\\\times6+6^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe8a-h6","type":"hint","dependencies":["ac0c1d3spe8a-h5"],"title":"Simplify","text":"$$y^2-12y+36$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac0c1d3spe9","title":"Binomial Squares Pattern","body":"Square each binomial using the Binomial Squares Pattern.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.4 Special Products","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac0c1d3spe9a","stepAnswer":["$$m^2-30y+225$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(m-15\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$m^2-30y+225$$","hints":{"DefaultPathway":[{"id":"ac0c1d3spe9a-h1","type":"hint","dependencies":[],"title":"Subtraction Binomial Square Formula","text":"We compare our expression to the subtraction binomial square formula: $${\\\\left(a-b\\\\right)}^2=a^2-2a b+b^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe9a-h2","type":"hint","dependencies":["ac0c1d3spe9a-h1"],"title":"Compare the Binomial\\\\n","text":"$${\\\\left(a-b\\\\right)}^2$$\\\\n$${\\\\left(m-15\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$m$$"],"dependencies":["ac0c1d3spe9a-h2"],"title":"Identify a","text":"What is a in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["ac0c1d3spe9a-h3"],"title":"Identify $$b$$","text":"What is $$b$$ in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe9a-h5","type":"hint","dependencies":["ac0c1d3spe9a-h3","ac0c1d3spe9a-h4"],"title":"Plug in the Terms","text":"Substitute $$a=m$$ and $$b=15$$ into the subtraction binomial square formula:\\\\n$${\\\\left(a-b\\\\right)}^2=a^2+2a b+b^2$$\\\\n$${\\\\left(m-15\\\\right)}^2=m^2-2m\\\\times15+{15}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac0c1d3spe9a-h6","type":"hint","dependencies":["ac0c1d3spe9a-h5"],"title":"Simplify","text":"$$m^2-30y+225$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac1df14related1","title":"Quantities","body":"For the following exercises, find the quantities for the given equation.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.1 Related Rates","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ac1df14related1a","stepAnswer":["$$8$$"],"problemType":"TextBox","stepTitle":"Find $$\\\\frac{dy}{dt}$$ at $$x=1$$ and $$y=x^2+3$$ if $$\\\\frac{dx}{dt}=4$$. Write in the form of an integer.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8$$","hints":{"DefaultPathway":[{"id":"ac1df14related1a-h1","type":"hint","dependencies":[],"title":"Partial derivatives","text":"Take the partial derivatives to this equation in relation to the variable $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ac1df14related1a-h2","type":"hint","dependencies":["ac1df14related1a-h1"],"title":"Partial derivatives","text":"When taking partial derivatives, take the derivative of every value as normal, and make sure to include $$\\\\frac{du}{dx}$$ (with u being the variable involved) to each term with a variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ac1df14related1a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{dy}{dt}=\\\\frac{2x dx}{dt}$$"],"dependencies":["ac1df14related1a-h2"],"title":"Derive","text":"Derive $$y=x^2+3$$ in terms of $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\frac{dy}{dt}=\\\\frac{2x dx}{dt}$$","$$\\\\frac{dy}{dt}=\\\\frac{4x dx}{dt}$$","$$\\\\frac{dy}{dt}=\\\\frac{\\\\left(-2x\\\\right) dx}{dt}$$","$$\\\\frac{dy}{dt}=\\\\frac{x dx}{dt}$$"],"subHints":[{"id":"ac1df14related1a-h3-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1dy}{dt}$$"],"dependencies":[],"title":"Derive","text":"What is the derivative of $$y$$ with respect to $$t$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$1\\\\left(\\\\frac{dy}{dt}\\\\right)$$","$$y\\\\left(\\\\frac{dy}{dt}\\\\right)$$","$$-1\\\\left(\\\\frac{dy}{dt}\\\\right)$$","$$y^2 \\\\frac{dy}{dt}$$"]},{"id":"ac1df14related1a-h3-s2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{2x dy}{dt}$$"],"dependencies":[],"title":"Derive","text":"What is the derivative of $$x^2$$ with respect to $$t$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\frac{2x dx}{dt}$$","$$\\\\frac{x^2 dx}{dt}$$","$$\\\\frac{2x^2 dx}{dt}$$","$$\\\\frac{\\\\left(-2x\\\\right) dx}{dt}$$"]},{"id":"ac1df14related1a-h3-s3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$0$$"],"dependencies":[],"title":"Derive","text":"What is the derivative of $$3$$ with respect to $$t$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$0$$","$$\\\\frac{3x dx}{dt}$$","$$1$$","$$3$$"]}]},{"id":"ac1df14related1a-h4","type":"hint","dependencies":["ac1df14related1a-h3"],"title":"Use $$\\\\frac{dx}{dt}$$ and $$x$$","text":"Now, the $$\\\\frac{dx}{dt}$$ and $$x$$ that were given in the problem statement can be plugged in to solve for the solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ac1df14related10","title":"Related Rates: Triangles","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.1 Related Rates","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ac1df14related10a","stepAnswer":["$$\\\\frac{3\\\\sqrt{3}}{8}$$ $$\\\\frac{{ft}^2}{sec}$$"],"problemType":"MultipleChoice","stepTitle":"A triangle has two constant sides of length $$3$$ ft and $$5$$ ft. The angle between these two sides is increasing at a rate of $$0.1$$ $$\\\\frac{rad}{sec}$$.","stepBody":"Find the rate at which the area of the triangle is changing when the angle between the two sides is $$\\\\frac{\\\\pi}{6}$$.","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{3\\\\sqrt{3}}{8}$$ $$\\\\frac{{ft}^2}{sec}$$","choices":["$$\\\\frac{3\\\\sqrt{3}}{8}$$ $$\\\\frac{{ft}^2}{sec}$$","$$\\\\frac{-\\\\left(3\\\\sqrt{3}\\\\right)}{8}$$ $$\\\\frac{{ft}^2}{sec}$$","$$\\\\frac{-\\\\left(3\\\\sqrt{3}\\\\right)}{2}$$ $$\\\\frac{{ft}^2}{sec}$$","$$\\\\frac{\\\\sqrt{3}}{8}$$ $$\\\\frac{{ft}^2}{sec}$$"],"hints":{"DefaultPathway":[{"id":"ac1df14related10a-h1","type":"hint","dependencies":[],"title":"Strategy","text":"We can use the equation for area of triangle from two sides and differentiate it with respect to time","variabilization":{},"oer":"","license":""},{"id":"ac1df14related10a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$A=\\\\frac{1}{2} a b sin\\\\left(C\\\\right)$$"],"dependencies":["ac1df14related10a-h1"],"title":"Area of triangle","text":"Let\'s set area of triangle as A, two sides as a and $$b$$, and angle between a and be as C. What is the equation for computing A?","variabilization":{},"oer":"","license":"","choices":["$$A=a b cos\\\\left(C\\\\right)$$","$$A=\\\\frac{1}{2} a b cos\\\\left(C\\\\right)$$","$$A=\\\\frac{1}{2} a b sin\\\\left(C\\\\right)$$"]},{"id":"ac1df14related10a-h3","type":"hint","dependencies":["ac1df14related10a-h2"],"title":"Differentiate","text":"Next step is to differentiate both sides of the equation with respect to time (t)","variabilization":{},"oer":"","license":""},{"id":"ac1df14related10a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{dA}{dt}=\\\\frac{1}{2} a b cos\\\\left(C\\\\right) \\\\frac{dC}{dt}$$"],"dependencies":["ac1df14related10a-h3"],"title":"Differentiate","text":"What is differentiated equation?","variabilization":{},"oer":"","license":"","choices":["$$\\\\frac{dA}{dt}=a b cos\\\\left(C\\\\right) \\\\frac{dC}{dt}$$","$$\\\\frac{dA}{dt}=\\\\frac{1}{2} a b cos\\\\left(C\\\\right) \\\\frac{dC}{dt}$$","$$\\\\frac{dA}{dt}=\\\\frac{1}{2} a b sin\\\\left(C\\\\right) \\\\frac{dC}{dt}$$"]},{"id":"ac1df14related10a-h5","type":"hint","dependencies":["ac1df14related10a-h4"],"title":"Strategy","text":"We want to find $$\\\\frac{dA}{dt}$$ when $$\\\\frac{dC}{dt}=0.1$$ $$\\\\frac{rad}{sec}$$, $$a=3ft$$, $$b=5ft$$, and $$C=\\\\frac{\\\\pi}{6}$$","variabilization":{},"oer":"","license":""},{"id":"ac1df14related10a-h6","type":"hint","dependencies":["ac1df14related10a-h5"],"title":"Plug in","text":"Final step is to plug in the values of $$\\\\frac{dC}{dt}$$, a,b,and C into the differentiated equation to find $$\\\\frac{dA}{dt}$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"ac1df14related11","title":"Related Rates: Water $$1$$","body":"The dimensions of the conical tank are a height of $$16$$ ft and a radius of $$5$$ ft.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.1 Related Rates","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ac1df14related11a","stepAnswer":["$$\\\\frac{128}{125} \\\\pi$$ $$\\\\frac{ft}{min}$$"],"problemType":"MultipleChoice","stepTitle":"How fast does the depth of the water change when the water is $$10$$ ft high if the cone leaks water at a rate of $$10$$ $$\\\\frac{{ft}^3}{min}$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{128}{125} \\\\pi$$ $$\\\\frac{ft}{min}$$","choices":["$$\\\\frac{128}{125} \\\\pi$$ $$\\\\frac{ft}{min}$$","$$\\\\frac{2}{5} \\\\pi$$ $$\\\\frac{ft}{min}$$","$$-\\\\left(\\\\frac{128}{125} \\\\pi\\\\right)$$ $$\\\\frac{ft}{min}$$","$$\\\\frac{128}{125}$$ $$\\\\frac{ft}{min}$$"],"hints":{"DefaultPathway":[{"id":"ac1df14related11a-h1","type":"hint","dependencies":[],"title":"Strategy","text":"We can use similar triangle and the equation for volume of right cone to differentiate it with respect to time","variabilization":{},"oer":"","license":""},{"id":"ac1df14related11a-h2","type":"hint","dependencies":["ac1df14related11a-h1"],"title":"Similar triangle","text":"We can find radius when height is 10ft, initial radius, and initial height because $$r$$ and $$h$$ relate by similar triangle","variabilization":{},"oer":"","license":""},{"id":"ac1df14related11a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$r=\\\\frac{R}{H} h$$"],"dependencies":["ac1df14related11a-h2"],"title":"Similar triangle","text":"Let\'s set initial radius as R , inital height as H, asked radius as $$r$$, and asked height (10ft) as $$h$$. What is $$r$$?","variabilization":{},"oer":"","license":"","choices":["$$r=\\\\frac{H}{R} h$$","$$r=R H h$$","$$r=\\\\frac{R}{H} h$$"]},{"id":"ac1df14related11a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$V=\\\\frac{1}{3} \\\\pi {\\\\left(\\\\frac{R}{H}\\\\right)}^2 h^3$$"],"dependencies":["ac1df14related11a-h3"],"title":"Area of triangle","text":"Let\'s set volume of right cone as V. What is the equation for computing V using $$r$$ from previous and $$h$$?","variabilization":{},"oer":"","license":"","choices":["$$V=\\\\frac{1}{3} \\\\pi \\\\frac{R}{H} h^2$$","$$V=\\\\frac{1}{3} \\\\pi {\\\\left(\\\\frac{R}{H}\\\\right)}^2 h^3$$","$$V=\\\\pi {\\\\left(\\\\frac{R}{H}\\\\right)}^2 h$$"]},{"id":"ac1df14related11a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$V=\\\\frac{1}{3} \\\\pi \\\\frac{25}{256} h^3$$"],"dependencies":["ac1df14related11a-h4"],"title":"Area of triangle","text":"What is equation after plugging R and H?","variabilization":{},"oer":"","license":"","choices":["$$V=\\\\frac{1}{3} \\\\pi \\\\frac{25}{256} h^3$$","$$V=\\\\frac{1}{3} \\\\frac{25}{256} h^3$$"]},{"id":"ac1df14related11a-h6","type":"hint","dependencies":["ac1df14related11a-h5"],"title":"Differentiate","text":"Next step is to differentiate both sides of the equation with respect to time (t) to relate $$\\\\frac{dV}{dt}$$ and $$\\\\frac{dh}{dt}$$","variabilization":{},"oer":"","license":""},{"id":"ac1df14related11a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{dV}{dt}=\\\\frac{1}{3} \\\\pi \\\\frac{25}{256} 3\\\\left(h^2\\\\right) \\\\frac{dh}{dt}$$"],"dependencies":["ac1df14related11a-h6"],"title":"Differentiate","text":"What is differentiated equation?","variabilization":{},"oer":"","license":"","choices":["dV/dt=(1/3)*pi*(25/256)* 3(h**2)(dh/dt))","$$\\\\frac{dV}{dt}=\\\\frac{1}{3} \\\\pi \\\\frac{25}{256} h^3 \\\\frac{dh}{dt}$$","$$\\\\frac{dV}{dt}=3\\\\frac{1}{3} \\\\pi \\\\frac{25}{256} h{\\\\left(\\\\frac{dh}{dt}\\\\right)}$$"]},{"id":"ac1df14related11a-h8","type":"hint","dependencies":["ac1df14related11a-h7"],"title":"Strategy","text":"We want to find $$\\\\frac{dh}{dt}$$ when $$\\\\frac{dV}{dt}=$$ $$-10$$ $$\\\\frac{{ft}^3}{min}$$, $$h=10ft$$","variabilization":{},"oer":"","license":""},{"id":"ac1df14related11a-h9","type":"hint","dependencies":["ac1df14related11a-h8"],"title":"Plug in","text":"Final step is to plug in the values of $$\\\\frac{dV}{dt}$$ and $$h$$ into the differentiated equation to find $$\\\\frac{dh}{dt}$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"ac1df14related12","title":"Related Rates: Water $$2$$","body":"The dimensions of the conical tank are a height of $$16$$ ft and a radius of $$5$$ ft.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.1 Related Rates","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ac1df14related12a","stepAnswer":["$$\\\\frac{25\\\\pi}{16}$$ $$\\\\frac{{ft}^3}{min}$$"],"problemType":"MultipleChoice","stepTitle":"If the water level is decreasing at a rate of $$3$$ in/min when the depth of the water is $$8$$ ft, determine the rate at which water is leaking out of the cone.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{25\\\\pi}{16}$$ $$\\\\frac{{ft}^3}{min}$$","choices":["$$\\\\frac{25\\\\pi}{16}$$ $$\\\\frac{{ft}^3}{min}$$","$$\\\\frac{5\\\\pi}{4}$$ $$\\\\frac{{ft}^3}{min}$$","$$\\\\frac{5}{4}$$ $$\\\\frac{{ft}^3}{min}$$","$$\\\\frac{25\\\\pi}{4}$$ $$\\\\frac{{ft}^3}{min}$$"],"hints":{"DefaultPathway":[{"id":"ac1df14related12a-h1","type":"hint","dependencies":[],"title":"Strategy","text":"We can use similar triangle and the equation for volume of right cone to differentiate it with respect to time","variabilization":{},"oer":"","license":""},{"id":"ac1df14related12a-h2","type":"hint","dependencies":["ac1df14related12a-h1"],"title":"Similar triangle","text":"We can find radius when height is 8ft, initial radius, and initial height because $$r$$ and $$h$$ relate by similar triangle","variabilization":{},"oer":"","license":""},{"id":"ac1df14related12a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$r=\\\\frac{R}{H} h$$"],"dependencies":["ac1df14related12a-h2"],"title":"Similar triangle","text":"Let\'s set initial radius as R , inital height as H, asked radius as $$r$$, and asked height (8ft) as $$h$$. What is $$r$$?","variabilization":{},"oer":"","license":"","choices":["$$r=\\\\frac{H}{R} h$$","$$r=R H h$$","$$r=\\\\frac{R}{H} h$$"]},{"id":"ac1df14related12a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$V=\\\\frac{1}{3} \\\\pi {\\\\left(\\\\frac{R}{H}\\\\right)}^2 h^3$$"],"dependencies":["ac1df14related12a-h3"],"title":"Area of triangle","text":"Let\'s set volume of right cone as V. What is the equation for computing V using $$r$$ from previous and $$h$$?","variabilization":{},"oer":"","license":"","choices":["$$V=\\\\frac{1}{3} \\\\pi \\\\frac{R}{H} h^2$$","$$V=\\\\frac{1}{3} \\\\pi {\\\\left(\\\\frac{R}{H}\\\\right)}^2 h^3$$","$$V=\\\\pi {\\\\left(\\\\frac{R}{H}\\\\right)}^2 h$$"]},{"id":"ac1df14related12a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$V=\\\\frac{1}{3} \\\\pi \\\\frac{25}{256} h^3$$"],"dependencies":["ac1df14related12a-h4"],"title":"Area of triangle","text":"What is equation after plugging R and H?","variabilization":{},"oer":"","license":"","choices":["$$V=\\\\frac{1}{3} \\\\pi \\\\frac{25}{256} h^3$$","$$V=\\\\frac{1}{3} \\\\frac{25}{256} h^3$$"]},{"id":"ac1df14related12a-h6","type":"hint","dependencies":["ac1df14related12a-h5"],"title":"Differentiate","text":"Next step is to differentiate both sides of the equation with respect to time (t) to relate $$\\\\frac{dV}{dt}$$ and $$\\\\frac{dh}{dt}$$","variabilization":{},"oer":"","license":""},{"id":"ac1df14related12a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{dV}{dt}=\\\\frac{1}{3} \\\\pi \\\\frac{25}{256} 3\\\\left(h^2\\\\right) \\\\frac{dh}{dt}$$"],"dependencies":["ac1df14related12a-h6"],"title":"Differentiate","text":"What is differentiated equation?","variabilization":{},"oer":"","license":"","choices":["dV/dt=(1/3)*pi*(25/256)* 3(h**2)(dh/dt))","$$\\\\frac{dV}{dt}=\\\\frac{1}{3} \\\\pi \\\\frac{25}{256} h^3 \\\\frac{dh}{dt}$$","$$\\\\frac{dV}{dt}=3\\\\frac{1}{3} \\\\pi \\\\frac{25}{256} h{\\\\left(\\\\frac{dh}{dt}\\\\right)}$$"]},{"id":"ac1df14related12a-h8","type":"hint","dependencies":["ac1df14related12a-h7"],"title":"Strategy","text":"We want to find $$\\\\frac{dV}{dt}$$ when $$\\\\frac{dh}{dt}=$$ $$-3$$ in/min, $$h=8ft$$","variabilization":{},"oer":"","license":""},{"id":"ac1df14related12a-h9","type":"hint","dependencies":["ac1df14related12a-h8"],"title":"Plug in","text":"Final step is to plug in the values of $$\\\\frac{dh}{dt}$$ and $$h$$ into the differentiated equation to find $$\\\\frac{dV}{dt}$$","variabilization":{},"oer":"","license":""},{"id":"ac1df14related12a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{-1}{4}$$ $$\\\\frac{ft}{min}$$"],"dependencies":["ac1df14related12a-h9"],"title":"$$\\\\frac{dh}{dt}$$","text":"What is $$\\\\frac{dh}{dx}$$ in unit of $$\\\\frac{ft}{min}$$?","variabilization":{},"oer":"","license":"","choices":["$$\\\\frac{1}{4}$$ $$\\\\frac{ft}{min}$$","$$\\\\frac{-1}{4}$$ $$\\\\frac{ft}{min}$$"]}]}}]},{"id":"ac1df14related13","title":"Related Rates: Water $$2$$","body":"The dimensions of the conical tank are a height of $$16$$ ft and a radius of $$5$$ ft.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.1 Related Rates","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ac1df14related13a","stepAnswer":["$$\\\\frac{2\\\\pi}{5}$$ $$\\\\frac{m^3}{min}$$"],"problemType":"MultipleChoice","stepTitle":"A cylinder is leaking water but you are unable to determine at what rate. The cylinder has a height of $$2$$ $$m$$ and a radius of $$2$$ $$m$$.","stepBody":"Find the rate at which the water is leaking out of the cylinder if the rate at which the height is decreasing is $$10$$ $$\\\\frac{cm}{min}$$ when the height is $$1$$ $$m$$.","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{2\\\\pi}{5}$$ $$\\\\frac{m^3}{min}$$","choices":["$$\\\\frac{2\\\\pi}{5}$$ $$\\\\frac{m^3}{min}$$","$$\\\\frac{4\\\\pi}{25}$$ $$\\\\frac{m^3}{min}$$","$$\\\\frac{2}{5}$$ $$\\\\frac{m^3}{min}$$","$$\\\\frac{2\\\\pi}{25}$$ $$\\\\frac{m^3}{min}$$"],"hints":{"DefaultPathway":[{"id":"ac1df14related13a-h1","type":"hint","dependencies":[],"title":"Strategy","text":"We can use the equation for volume of cylinder and differentiate it with respect to time","variabilization":{},"oer":"","license":""},{"id":"ac1df14related13a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$V=\\\\pi r^2 h$$"],"dependencies":["ac1df14related13a-h1"],"title":"Surface area of sphere","text":"Let\'s set volume of cylinder as V, height as $$h$$ and radius as $$r$$. What is the equation for computing V?","variabilization":{},"oer":"","license":"","choices":["$$V=\\\\pi r h$$","$$V=\\\\pi r^2 h$$","$$V=\\\\pi r^3 h$$"]},{"id":"ac1df14related13a-h3","type":"hint","dependencies":["ac1df14related13a-h2"],"title":"Differentiate","text":"Next step is to differentiate both sides of the equation with respect to time (t) to relate $$\\\\frac{dh}{dt}$$","variabilization":{},"oer":"","license":""},{"id":"ac1df14related13a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{dV}{dt}=\\\\pi r^2 \\\\frac{dh}{dt}$$"],"dependencies":["ac1df14related13a-h3"],"title":"Differentiate","text":"What is differentiated equation?","variabilization":{},"oer":"","license":"","choices":["$$\\\\frac{dV}{dt}=2\\\\pi r \\\\frac{dh}{dt}$$","$$\\\\frac{dV}{dt}=\\\\pi r^2 h \\\\frac{dh}{dt}$$","$$\\\\frac{dV}{dt}=\\\\pi r^2 \\\\frac{dh}{dt}$$"]},{"id":"ac1df14related13a-h5","type":"hint","dependencies":["ac1df14related13a-h4"],"title":"Strategy","text":"We want to find $$\\\\frac{dV}{dt}$$ when $$\\\\frac{dh}{dt}$$ is $$\\\\frac{-10cm}{min}$$, and $$r=2m$$","variabilization":{},"oer":"","license":""},{"id":"ac1df14related13a-h6","type":"hint","dependencies":["ac1df14related13a-h5"],"title":"Plug in","text":"Final step is to plug in the values of $$h$$, $$\\\\frac{dh}{dt}$$ , into the differentiated equation to find $$\\\\frac{dV}{dt}$$","variabilization":{},"oer":"","license":""},{"id":"ac1df14related13a-h7","type":"hint","dependencies":["ac1df14related13a-h6"],"title":"$${cm}^3$$ to $$m^3$$","text":"We need to convert $${cm}^3$$ to $$m^3$$: $$1$$ $${cm}^3={10}^{\\\\left(-6\\\\right)}$$ $$m^3$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"ac1df14related14","title":"Related Rates: Water $$4$$","body":"The dimensions of the conical tank are a height of $$16$$ ft and a radius of $$5$$ ft.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.1 Related Rates","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ac1df14related14a","stepAnswer":["$$\\\\frac{3}{2}$$ $$\\\\frac{m}{sec}$$"],"problemType":"MultipleChoice","stepTitle":"A tank is shaped like an upside-down square pyramid, with base of $$4$$ $$m$$ by $$4$$ $$m$$ and a height of $$12$$ $$m$$ (see the following figure).","stepBody":"How fast does the height increase when the water is $$2$$ $$m$$ deep if water is being pumped in at a rate of $$\\\\frac{2}{3}$$ $$\\\\frac{m^3}{sec}$$?##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{3}{2}$$ $$\\\\frac{m}{sec}$$","choices":["$$\\\\frac{2}{3}$$ $$\\\\frac{m}{sec}$$","$$\\\\frac{1}{3}$$ $$\\\\frac{m}{sec}$$","$$\\\\frac{3}{2}$$ $$\\\\frac{m}{sec}$$","$$\\\\frac{1}{2}$$ $$\\\\frac{m}{sec}$$"],"hints":{"DefaultPathway":[{"id":"ac1df14related14a-h1","type":"hint","dependencies":[],"title":"Strategy","text":"We can use the eqation for volumen of square pyramid $$V=\\\\frac{1}{3} A h$$ (\'A\' represents the area of the square) and similar triangle to differentiate","variabilization":{},"oer":"","license":""}]}}]},{"id":"ac1df14related15","title":"Related Rates: Cones","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.1 Related Rates","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ac1df14related15a","stepAnswer":["$$\\\\frac{2}{45} \\\\pi$$ $$\\\\frac{ft}{min}$$"],"problemType":"MultipleChoice","stepTitle":"Gravel is being unloaded from a truck and falls into a pile shaped like a cone at a rate of $$10$$ $$\\\\frac{ft3}{min}$$. The radius of the cone base is three times the height of the cone.","stepBody":"Find the rate at which the height of the gravel changes when the pile has a height of $$5$$ ft.","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{2}{45} \\\\pi$$ $$\\\\frac{ft}{min}$$","choices":["$$\\\\frac{2}{45} \\\\pi$$ $$\\\\frac{ft}{min}$$","$$\\\\frac{2}{45}$$ $$\\\\frac{ft}{min}$$","$$\\\\frac{4}{45} \\\\pi$$ $$\\\\frac{ft}{min}$$","$$\\\\frac{4}{45}$$ $$\\\\frac{ft}{min}$$"],"hints":{"DefaultPathway":[{"id":"ac1df14related15a-h1","type":"hint","dependencies":[],"title":"Strategy","text":"We can use the equation for volume of cone and differentiate it with respect to time","variabilization":{},"oer":"","license":""},{"id":"ac1df14related15a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$V=\\\\frac{1}{3} \\\\pi r^2 h$$"],"dependencies":["ac1df14related15a-h1"],"title":"volume of cone","text":"Let\'s set volume of cone as V, height as $$h$$ and radius as $$r$$. What is the equation for computing V?","variabilization":{},"oer":"","license":"","choices":["$$V=\\\\frac{1}{3} \\\\pi r h$$","$$V=\\\\frac{1}{3} \\\\pi r^2 h$$","$$V=\\\\frac{1}{3} \\\\pi r^3 h$$"]},{"id":"ac1df14related15a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$r=3h$$"],"dependencies":["ac1df14related15a-h2"],"title":"$$r$$ and $$h$$","text":"What is $$r$$ using $$h$$?","variabilization":{},"oer":"","license":"","choices":["$$r=h$$","$$r=2h$$","$$r=3h$$"]},{"id":"ac1df14related15a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$V=3\\\\pi h^3$$"],"dependencies":["ac1df14related15a-h3"],"title":"Plug in","text":"What is the equation for volume after plugging in $$r=3h$$?","variabilization":{},"oer":"","license":"","choices":["$$V=3\\\\pi h$$","$$V=3\\\\pi h^2$$","$$V=3\\\\pi h^3$$"]},{"id":"ac1df14related15a-h5","type":"hint","dependencies":["ac1df14related15a-h4"],"title":"Differentiate","text":"Next step is to differentiate both sides of the equation with respect to time (t) to relate $$\\\\frac{dh}{dt}$$","variabilization":{},"oer":"","license":""},{"id":"ac1df14related15a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{dV}{dt}=9\\\\pi h^2 \\\\frac{dh}{dt}$$"],"dependencies":["ac1df14related15a-h5"],"title":"Differentiate","text":"What is differentiated equation?","variabilization":{},"oer":"","license":"","choices":["$$\\\\frac{dV}{dt}=3\\\\operatorname{\\\\pi}\\\\left(h^2\\\\right) \\\\frac{dh}{dt}$$","$$\\\\frac{dV}{dt}=9\\\\pi h \\\\frac{dh}{dt}$$","$$\\\\frac{dV}{dt}=9\\\\pi h^2 \\\\frac{dh}{dt}$$"]},{"id":"ac1df14related15a-h7","type":"hint","dependencies":["ac1df14related15a-h6"],"title":"Strategy","text":"We want to find $$\\\\frac{dh}{dt}$$ when $$\\\\frac{dV}{dt}$$ is $$\\\\frac{\\\\operatorname{10}\\\\left({ft}^3\\\\right)}{min}$$, and $$h=5ft$$","variabilization":{},"oer":"","license":""},{"id":"ac1df14related15a-h8","type":"hint","dependencies":["ac1df14related15a-h7"],"title":"Plug in","text":"Final step is to plug in the values of $$h$$, $$\\\\frac{dV}{dt}$$ , into the differentiated equation to find $$\\\\frac{dh}{dt}$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"ac1df14related16","title":"Related Rates: Angle of Elevation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.1 Related Rates","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ac1df14related16a","stepAnswer":["$$\\\\frac{400}{1681}$$ $$\\\\frac{rad}{sec}$$"],"problemType":"MultipleChoice","stepTitle":"You are stationary on the ground and are watching a bird fly horizontally at a rate of $$10$$ $$\\\\frac{m}{sec}$$. The bird is located $$40$$ $$m$$ above your head.","stepBody":"How fast does the angle of elevation decrease when the horizontal distance between you and the bird is $$9$$ $$m$$?","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{400}{1681}$$ $$\\\\frac{rad}{sec}$$","choices":["$$\\\\frac{400}{1681}$$ $$\\\\frac{rad}{sec}$$","$$\\\\frac{20}{41}$$ $$\\\\frac{rad}{sec}$$","$$\\\\frac{400}{41}$$ $$\\\\frac{rad}{sec}$$","$$\\\\frac{20}{1681}$$ $$\\\\frac{rad}{sec}$$"],"hints":{"DefaultPathway":[{"id":"ac1df14related16a-h1","type":"hint","dependencies":[],"title":"Strategy","text":"We can use the trigonometory and differentiate it with respect to time","variabilization":{},"oer":"","license":""},{"id":"ac1df14related16a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$tan(\\\\theta)=\\\\frac{h}{x}$$"],"dependencies":["ac1df14related16a-h1"],"title":"trigonometry","text":"Let\'s set the horizontal distance between you and bird as $$x$$, height the bird fly above you as $$h$$, and the angle of elevation as \u03b8, what equation can you use from trigonometry?","variabilization":{},"oer":"","license":"","choices":["$$tan(\\\\theta)=\\\\frac{x}{h}$$","$$cos(\\\\theta)=\\\\frac{h}{x}$$","$$sin(\\\\theta)=\\\\frac{x}{h}$$","$$tan(\\\\theta)=\\\\frac{h}{x}$$"]},{"id":"ac1df14related16a-h3","type":"hint","dependencies":["ac1df14related16a-h2"],"title":"Differentiate","text":"Next step is to differentiate both sides of the equation with respect to time (t)","variabilization":{},"oer":"","license":""},{"id":"ac1df14related16a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${sec}^{2\\\\left(\\\\theta\\\\right)} \\\\frac{dtheta}{dt}=$$ $$\\\\frac{1}{x} \\\\frac{dh}{dt}-\\\\frac{h}{x^2} \\\\frac{dx}{dt}$$"],"dependencies":["ac1df14related16a-h3"],"title":"Differentiate","text":"What is differentiated equation?","variabilization":{},"oer":"","license":"","choices":["$${sec}^{2\\\\left(\\\\theta\\\\right)} \\\\frac{dtheta}{dt}=\\\\frac{1}{x} \\\\frac{dh}{dt}$$","$${sec}^{2\\\\left(\\\\theta\\\\right)} \\\\frac{dtheta}{dt}=$$ $$\\\\frac{1}{x} \\\\frac{dh}{dt}-\\\\frac{h}{x^2} \\\\frac{dx}{dt}$$","$${sec}^{2\\\\left(\\\\theta\\\\right)} \\\\frac{dtheta}{dt}=\\\\frac{h}{x^2} \\\\frac{dx}{dt}$$"]},{"id":"ac1df14related16a-h5","type":"hint","dependencies":["ac1df14related16a-h4"],"title":"$$\\\\frac{dh}{dt}$$","text":"Because the bird is flying horizontally, the $$h$$ won\'t change and $$\\\\frac{dh}{dt}=0$$","variabilization":{},"oer":"","license":""},{"id":"ac1df14related16a-h6","type":"hint","dependencies":["ac1df14related16a-h5"],"title":"Strategy","text":"We want to find $$\\\\frac{dtheta}{dt}$$ when $$\\\\frac{dh}{dt}$$ is $$0$$, $$\\\\frac{dx}{dt}=\\\\frac{10m}{sec}$$, and $$h=40m$$","variabilization":{},"oer":"","license":""},{"id":"ac1df14related16a-h7","type":"hint","dependencies":["ac1df14related16a-h6"],"title":"Plug in","text":"Final step is to plug in the values of $$h$$, $$\\\\frac{dh}{dt}$$. $$\\\\frac{dx}{dt}$$ , into the differentiated equation to find $$\\\\frac{dtheta}{dt}$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"ac1df14related17","title":"Related Rates: Speed","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.1 Related Rates","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ac1df14related17a","stepAnswer":["$$100\\\\pi$$ $$\\\\frac{mi}{min}$$"],"problemType":"MultipleChoice","stepTitle":"A lighthouse, L, is on an island $$4$$ mi away from the closest point, P, on the beach.","stepBody":"How fast does the angle of elevation change when the horizontal distance between you and the bird is $$9$$ $$m$$?##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$100\\\\pi$$ $$\\\\frac{mi}{min}$$","choices":["$$100\\\\pi$$ $$\\\\frac{mi}{min}$$","$$10\\\\pi$$ $$\\\\frac{mi}{min}$$","$$1\\\\pi$$ $$\\\\frac{mi}{min}$$","$$1000\\\\pi$$ $$\\\\frac{mi}{min}$$"],"hints":{"DefaultPathway":[{"id":"ac1df14related17a-h1","type":"hint","dependencies":[],"title":"Strategy","text":"We can use the trigonometory and differentiate it with respect to time","variabilization":{},"oer":"","license":""},{"id":"ac1df14related17a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$tan(\\\\theta)=\\\\frac{h}{x}$$"],"dependencies":["ac1df14related17a-h1"],"title":"trigonometry","text":"Let\'s set the horizontal distance between L and P as $$x$$, distance between P and the point the light hit in the beach as $$h$$, and the angle between PL and light as \u03b8, what equation can you use from trigonometry?","variabilization":{},"oer":"","license":"","choices":["$$tan(\\\\theta)=\\\\frac{x}{h}$$","$$cos(\\\\theta)=\\\\frac{h}{x}$$","$$sin(\\\\theta)=\\\\frac{x}{h}$$","$$tan(\\\\theta)=\\\\frac{h}{x}$$"]},{"id":"ac1df14related17a-h3","type":"hint","dependencies":["ac1df14related17a-h2"],"title":"Differentiate","text":"Next step is to differentiate both sides of the equation with respect to time (t)","variabilization":{},"oer":"","license":""},{"id":"ac1df14related17a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Let\'s set the horizontal distance between L and P as $$x$$, distance between P and the point the light hit in the beach as $$h$$, and the angle between PL and light as \u03b8, what equation can you use from trigonometry?"],"dependencies":["ac1df14related17a-h3"],"title":"Differentiate","text":"What is differentiated equation?","variabilization":{},"oer":"","license":"","choices":["$${sec}^{2\\\\left(\\\\theta\\\\right)} \\\\frac{dtheta}{dt}=\\\\frac{1}{x} \\\\frac{dh}{dt}$$","$${sec}^{2\\\\left(\\\\theta\\\\right)} \\\\frac{dtheta}{dt}=$$ $$\\\\frac{1}{x} \\\\frac{dh}{dt}-\\\\frac{h}{x^2} \\\\frac{dx}{dt}$$","$${sec}^{2\\\\left(\\\\theta\\\\right)} \\\\frac{dtheta}{dt}=\\\\frac{h}{x^2} \\\\frac{dx}{dt}$$"]},{"id":"ac1df14related17a-h5","type":"hint","dependencies":["ac1df14related17a-h4"],"title":"$$\\\\frac{dx}{dt}$$","text":"Because distance between P and L won\'t chnge, $$\\\\frac{dx}{dt}=0$$","variabilization":{},"oer":"","license":""},{"id":"ac1df14related17a-h6","type":"hint","dependencies":["ac1df14related17a-h5"],"title":"Strategy","text":"We want to find $$\\\\frac{dh}{dt}$$ when $$\\\\frac{dx}{dt}$$ is $$0$$, $$\\\\frac{dtheta}{dt}=1o$$ $$\\\\frac{revolution}{min}$$, $$x=4mi$$, and $$h=2mi$$","variabilization":{},"oer":"","license":""},{"id":"ac1df14related17a-h7","type":"hint","dependencies":["ac1df14related17a-h6"],"title":"Plug in","text":"Final step is to plug in the values of $$x$$, $$h$$, $$\\\\frac{dtheta}{dt}$$. $$\\\\frac{dx}{dt}$$ , into the differentiated equation to find $$\\\\frac{dh}{dt}$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"ac1df14related18","title":"Related Rates: Angle Change","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.1 Related Rates","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ac1df14related18a","stepAnswer":["$$\\\\frac{11}{25}$$ $$\\\\frac{rad}{sec}$$"],"problemType":"MultipleChoice","stepTitle":"You are walking to a bus stop at a right-angle corner. You move north at a rate of $$2$$ $$\\\\frac{m}{sec}$$ and are $$20$$ $$m$$ south of the intersection. The bus travels west at a rate of $$10$$ $$\\\\frac{m}{sec}$$ away from the intersection - you have missed the bus!","stepBody":"What is the rate at which the angle between you and the bus is changing when you are $$20$$ $$m$$ south of the intersection and the bus is $$10$$ $$m$$ west of the intersection?","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{11}{25}$$ $$\\\\frac{rad}{sec}$$","choices":["$$\\\\frac{11}{25}$$ $$\\\\frac{rad}{sec}$$","$$\\\\frac{11}{5}$$ $$\\\\frac{rad}{sec}$$","$$\\\\frac{121}{25}$$ $$\\\\frac{rad}{sec}$$","$$\\\\frac{121}{5}$$ $$\\\\frac{rad}{sec}$$"],"hints":{"DefaultPathway":[{"id":"ac1df14related18a-h1","type":"hint","dependencies":[],"title":"Strategy","text":"We can use the trigonometory and differentiate it with respect to time","variabilization":{},"oer":"","license":""},{"id":"ac1df14related18a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$tan(\\\\theta)=\\\\frac{y}{x}$$"],"dependencies":["ac1df14related18a-h1"],"title":"trigonometry","text":"Let\'s set the distance you traveled north as $$x$$, distance the bus traveled west as $$y$$ , and the angle between you and bus as \u03b8, what equation can you use from trigonometry?","variabilization":{},"oer":"","license":"","choices":["$$tan(\\\\theta)=\\\\frac{x}{y}$$","$$cos(\\\\theta)=\\\\frac{y}{x}$$","$$sin(\\\\theta)=\\\\frac{x}{y}$$","$$tan(\\\\theta)=\\\\frac{y}{x}$$"]},{"id":"ac1df14related18a-h3","type":"hint","dependencies":["ac1df14related18a-h2"],"title":"Differentiate","text":"Next step is to differentiate both sides of the equation with respect to time (t)","variabilization":{},"oer":"","license":""},{"id":"ac1df14related18a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${sec}^2 \\\\theta \\\\frac{dtheta}{dt}=$$ $$\\\\frac{1}{x} \\\\frac{dy}{dt}-\\\\frac{y}{x^2} \\\\frac{dx}{dt}$$"],"dependencies":["ac1df14related18a-h3"],"title":"Differentiate","text":"What is differentiated equation?","variabilization":{},"oer":"","license":"","choices":["$${sec}^2 \\\\theta \\\\frac{dtheta}{dt}=\\\\frac{1}{x} \\\\frac{dy}{dt}$$","$${sec}^{2\\\\left(\\\\theta\\\\right)} \\\\frac{dtheta}{dt}=$$ $$\\\\frac{1}{x} \\\\frac{dy}{dt}-\\\\frac{y}{x^2} \\\\frac{dx}{dt}$$","$${sec}^{2\\\\left(\\\\theta\\\\right)} \\\\frac{dtheta}{dt}=\\\\frac{u}{x^2} \\\\frac{dx}{dt}$$"]},{"id":"ac1df14related18a-h5","type":"hint","dependencies":["ac1df14related18a-h4"],"title":"Strategy","text":"We want to find $$\\\\frac{dtheta}{dt}$$ when $$\\\\frac{dx}{dt}$$ is $$\\\\frac{2m}{sec}$$, $$\\\\frac{dy}{dt}=10$$ $$\\\\frac{m}{sec}$$, $$x=20mi$$, and $$y=10mi$$","variabilization":{},"oer":"","license":""},{"id":"ac1df14related18a-h6","type":"hint","dependencies":["ac1df14related18a-h5"],"title":"Plug in","text":"Final step is to plug in the values of $$x$$, $$y$$, $$\\\\frac{dy}{dt}$$. $$\\\\frac{dx}{dt}$$ , into the differentiated equation to find $$\\\\frac{dtheta}{dt}$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"ac1df14related2","title":"Quantities","body":"For the following exercises, find the quantities for the given equation.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.1 Related Rates","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ac1df14related2a","stepAnswer":["$$8$$"],"problemType":"TextBox","stepTitle":"Find $$\\\\frac{dx}{dt}$$ at $$x=-2$$ and $$y=2x^2+1$$ if $$\\\\frac{dy}{dt}=-1$$. Write in the form of an integer.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8$$","hints":{"DefaultPathway":[{"id":"ac1df14related2a-h1","type":"hint","dependencies":[],"title":"Partial derivatives","text":"Take the partial derivatives to this equation in relation to the variable $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ac1df14related2a-h2","type":"hint","dependencies":["ac1df14related2a-h1"],"title":"Partial derivatives","text":"When taking partial derivatives, take the derivative of every value as normal, and make sure to include $$\\\\frac{du}{dx}$$ (with u being the variable involved) to each term with a variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ac1df14related2a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{dy}{dt}=\\\\frac{4x dx}{dt}$$"],"dependencies":["ac1df14related2a-h2"],"title":"Derive","text":"Derive $$y=2x^2+1$$ in terms of $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\frac{dy}{dt}=\\\\frac{2x dx}{dt}$$","$$\\\\frac{dy}{dt}=\\\\frac{4x dx}{dt}$$","$$\\\\frac{dy}{dt}=\\\\frac{\\\\left(-2x\\\\right) dx}{dt}$$","$$\\\\frac{dy}{dt}=\\\\frac{x dx}{dt}$$"],"subHints":[{"id":"ac1df14related2a-h3-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1dy}{dt}$$"],"dependencies":[],"title":"Derive","text":"What is the derivative of $$y$$ with respect to $$t$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$1\\\\left(\\\\frac{dy}{dt}\\\\right)$$","$$y\\\\left(\\\\frac{dy}{dt}\\\\right)$$","$$-1\\\\left(\\\\frac{dy}{dt}\\\\right)$$","$$y^2 \\\\frac{dy}{dt}$$"]},{"id":"ac1df14related2a-h3-s2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{4x dx}{dt}$$"],"dependencies":[],"title":"Derive","text":"What is the derivative of $$2x^2$$ with respect to $$t$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\frac{4x dx}{dt}$$","$$\\\\frac{x^2 dx}{dt}$$","$$\\\\frac{4x^2 dx}{dt}$$","$$\\\\frac{\\\\left(-4x\\\\right) dx}{dt}$$"]},{"id":"ac1df14related2a-h3-s3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$0$$"],"dependencies":[],"title":"Derive","text":"What is the derivative of $$1$$ with respect to $$t$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$0$$","$$\\\\frac{2x dx}{dt}$$","$$1$$","$$2$$"]}]},{"id":"ac1df14related2a-h4","type":"hint","dependencies":["ac1df14related2a-h3"],"title":"Use $$\\\\frac{dx}{dt}$$ and $$x$$","text":"Now, the $$\\\\frac{dx}{dt}$$ and $$x$$ that were given in the problem statement can be plugged in to solve for the solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ac1df14related3","title":"Quantities","body":"For the following exercises, find the quantities for the given equation.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.1 Related Rates","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ac1df14related3a","stepAnswer":["$$\\\\frac{\\\\pm 13}{\\\\sqrt{10}}$$"],"problemType":"MultipleChoice","stepTitle":"Find $$\\\\frac{dz}{dt}$$ at $$(x,y)=(1,3)$$ and $$z^2=x^2+y^2$$ if $$\\\\frac{dx}{dt}=4$$ and $$\\\\frac{dy}{dt}=3$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{\\\\pm 13}{\\\\sqrt{10}}$$","choices":["$$\\\\frac{\\\\pm 13}{\\\\sqrt{10}}$$","$$\\\\frac{\\\\pm \\\\sqrt{13}}{10}$$","$$\\\\frac{\\\\pm 13}{10}$$","$$\\\\frac{13}{10}$$"],"hints":{"DefaultPathway":[{"id":"ac1df14related3a-h1","type":"hint","dependencies":[],"title":"Partial derivatives","text":"Take the partial derivatives to this equation in relation to the variable $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ac1df14related3a-h2","type":"hint","dependencies":["ac1df14related3a-h1"],"title":"Partial derivatives","text":"When taking partial derivatives, take the derivative of every value as normal, and make sure to include $$\\\\frac{du}{dx}$$ (with u being the variable involved) to each term with a variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ac1df14related3a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{2z dz}{dt}=\\\\frac{2x dx}{dt}+\\\\frac{2y dy}{dt}$$"],"dependencies":["ac1df14related3a-h2"],"title":"Derive","text":"Derive $$z^2=x^2+y^2$$ in terms of $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\frac{2z dz}{dt}=\\\\frac{2x dx}{dt}+\\\\frac{2y dy}{dt}$$","$$\\\\frac{z dz}{dt}=\\\\frac{x dx}{dt}+\\\\frac{y dy}{dt}$$","$$\\\\frac{2z^2 dz}{dt}=\\\\frac{2x^2 dx}{dt}+\\\\frac{2y^2 dy}{dt}$$","$$\\\\frac{2dz}{dt}=\\\\frac{2dx}{dt}+\\\\frac{2dy}{dt}$$"],"subHints":[{"id":"ac1df14related3a-h3-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{2z dz}{dt}$$"],"dependencies":[],"title":"Derive","text":"What is the derivative of $$z^2$$ with respect to $$t$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\frac{2z dz}{dt}$$","$$\\\\frac{2z^2 dz}{dt}$$","$$\\\\frac{z^2 dz}{dt}$$","$$\\\\frac{2dz}{dt}$$"]},{"id":"ac1df14related3a-h3-s2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{2x dx}{dt}$$"],"dependencies":[],"title":"Derive","text":"What is the derivative of $$x^2$$ with respect to $$t$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\frac{2x dx}{dt}$$","$$\\\\frac{2x^2 dx}{dt}$$","$$\\\\frac{x^2 dx}{dt}$$","$$\\\\frac{2dx}{dt}$$"]},{"id":"ac1df14related3a-h3-s3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{2y dx}{dt}$$"],"dependencies":[],"title":"Derive","text":"What is the derivative of $$y^2$$ with respect to $$t$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\frac{2y dy}{dt}$$","$$\\\\frac{2y^2 dy}{dt}$$","$$\\\\frac{y^2 dy}{dt}$$","$$\\\\frac{2dy}{dt}$$"]}]},{"id":"ac1df14related3a-h4","type":"hint","dependencies":["ac1df14related3a-h3"],"title":"Use $$\\\\frac{dx}{dt}$$ and $$x$$","text":"Now, the $$\\\\frac{dx}{dt}$$ and $$x$$ that were given in the problem statement can be plugged in to solve for the solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ac1df14related4","title":"Related Rates: Ladders","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.1 Related Rates","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ac1df14related4a","stepAnswer":["$$2\\\\sqrt{3}$$"],"problemType":"MultipleChoice","stepTitle":"A 10-ft ladder is leaning against a wall.","stepBody":"If the top of the ladder slides down the wall at a rate of $$2$$ $$\\\\frac{ft}{sec}$$, how fast is the bottom moving along the ground when the bottom of the ladder is $$5$$ ft from the wall? All answers are in $$\\\\frac{ft}{sec}$$.##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"$$2\\\\sqrt{3}$$","choices":["$$2\\\\sqrt{3}$$","$$-2\\\\sqrt{3}$$","$$3\\\\sqrt{2}$$","$$5\\\\sqrt{2}$$"],"hints":{"DefaultPathway":[{"id":"ac1df14related4a-h1","type":"hint","dependencies":[],"title":"Strategy","text":"we can use the relationship between the sides of a right triangle and differentiate it with respect to time","variabilization":{},"oer":"","license":""},{"id":"ac1df14related4a-h2","type":"hint","dependencies":["ac1df14related4a-h1"],"title":"Pythagorean theory","text":"Let\'s set the distance from tha wall to bottom of ladder as $$x$$, and the distance from the bottom to top of ladder as $$y$$. Then use Pythagorean theory to make equation to differentiate with respect to time","variabilization":{},"oer":"","license":""},{"id":"ac1df14related4a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x^2+y^2={10}^2$$"],"dependencies":["ac1df14related4a-h2"],"title":"Pythagorean theory","text":"What is the equation by plugging in $$x$$, $$y$$, and the length of ladder into pythagorean theory?","variabilization":{},"oer":"","license":"","choices":["$$x+y=10$$","$$x^2+y^2=10$$","$$x^2+y^2={10}^2$$"]},{"id":"ac1df14related4a-h4","type":"hint","dependencies":["ac1df14related4a-h3"],"title":"Differentiate","text":"Next step is to differentiate both sides of the equation with respect to time (t) to relate $$\\\\frac{dx}{dt}$$ and $$\\\\frac{dy}{dt}$$","variabilization":{},"oer":"","license":""},{"id":"ac1df14related4a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2x\\\\left(\\\\frac{dx}{dt}\\\\right)+2y\\\\left(\\\\frac{dy}{dt}\\\\right)=0$$"],"dependencies":["ac1df14related4a-h4"],"title":"Differentiate","text":"What is differentiated equation?","variabilization":{},"oer":"","license":"","choices":["$$2x\\\\left(\\\\frac{dx}{dt}\\\\right)+2y\\\\left(\\\\frac{dy}{dt}\\\\right)=0$$","$$2\\\\left(\\\\frac{dx}{dt}\\\\right)+2\\\\left(\\\\frac{dy}{dt}\\\\right)=0$$","$$x^2 \\\\frac{dx}{dt}+y^2 \\\\frac{dy}{dt}=0$$"]},{"id":"ac1df14related4a-h6","type":"hint","dependencies":["ac1df14related4a-h5"],"title":"Find $$y$$","text":"We need to find $$y$$ when $$x=5ft$$","variabilization":{},"oer":"","license":""},{"id":"ac1df14related4a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5\\\\left(\\\\sqrt{3}\\\\right)$$"],"dependencies":["ac1df14related4a-h6"],"title":"Find $$y$$","text":"What is $$y$$ when $$x$$ is $$5$$ ft?","variabilization":{},"oer":"","license":""},{"id":"ac1df14related4a-h8","type":"hint","dependencies":["ac1df14related4a-h7"],"title":"Plug in","text":"Final step is to plug in the values of $$x$$, $$y$$, $$\\\\frac{dy}{dt}$$ , into the differentiated equation to find $$\\\\frac{dx}{dt}$$","variabilization":{},"oer":"","license":""},{"id":"ac1df14related4a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{-2ft}{sec}$$"],"dependencies":["ac1df14related4a-h8"],"title":"Plug in","text":"What is dy/dt?(we are given this value)","variabilization":{},"oer":"","license":"","choices":["$$\\\\frac{2ft}{sec}$$","$$\\\\frac{-2ft}{sec}$$"]}]}}]},{"id":"ac1df14related5","title":"Related Rates: Airplanes","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.1 Related Rates","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ac1df14related5a","stepAnswer":["$$-390$$"],"problemType":"TextBox","stepTitle":"Two airplanes are flying in the air at the same height: airplane A is flying east at $$250$$ $$\\\\frac{mi}{h}$$ and airplane B is flying north at $$\\\\frac{300mi}{h}$$.","stepBody":"If they are both heading to the same airport, located $$30$$ miles east of airplane A and $$40$$ miles north of airplane B, at what rate is the distance between the airplanes changing? Write in the form of an integer, in miles per hour.##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-390$$","hints":{"DefaultPathway":[{"id":"ac1df14related5a-h1","type":"hint","dependencies":[],"title":"Strategy","text":"we can use the relationship between the sides of a right triangle and differentiate it with respect to time","variabilization":{},"oer":"","license":""},{"id":"ac1df14related5a-h2","type":"hint","dependencies":["ac1df14related5a-h1"],"title":"Pythagorean theory","text":"Let\'s set the distance from airplane A to destination as $$x$$, the distance from airplane B to destination as $$y$$, and the distance between two airplanes as $$z$$. Then use Pythagorean theory to make equation to differentiate with respect to time","variabilization":{},"oer":"","license":""},{"id":"ac1df14related5a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x^2+y^2=z^2$$"],"dependencies":["ac1df14related5a-h2"],"title":"Pythagorean theory","text":"What is the equation by plugging in $$x$$, $$y$$, $$z$$ into pythagorean theory?","variabilization":{},"oer":"","license":"","choices":["$$x+y=z^2$$","$$x^2+y^2=z^2$$","$$x^2+y^2=z$$"]},{"id":"ac1df14related5a-h4","type":"hint","dependencies":["ac1df14related5a-h3"],"title":"Differentiate","text":"Next step is to differentiate both sides of the equation with respect to time (t) to relate $$\\\\frac{dx}{dt}$$, $$\\\\frac{dy}{dt}$$, and $$\\\\frac{dz}{dt}$$","variabilization":{},"oer":"","license":""},{"id":"ac1df14related5a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2x\\\\left(\\\\frac{dx}{dt}\\\\right)+2y\\\\left(\\\\frac{dy}{dt}\\\\right)=2z\\\\left(\\\\frac{dz}{dt}\\\\right)$$"],"dependencies":["ac1df14related5a-h4"],"title":"Differentiate","text":"What is differentiated equation?","variabilization":{},"oer":"","license":"","choices":["$$2x\\\\left(\\\\frac{dx}{dt}\\\\right)+2y\\\\left(\\\\frac{dy}{dt}\\\\right)=2z$$","$$2\\\\left(\\\\frac{dx}{dt}\\\\right)+2\\\\left(\\\\frac{dy}{dt}\\\\right)=2\\\\left(\\\\frac{dy}{dt}\\\\right)$$","$$x^2 \\\\frac{dx}{dt}+y^2 \\\\frac{dy}{dt}=z^2 \\\\frac{dz}{dt}$$","$$2x\\\\left(\\\\frac{dx}{dt}\\\\right)+2y\\\\left(\\\\frac{dy}{dt}\\\\right)=2z\\\\left(\\\\frac{dz}{dt}\\\\right)$$"]},{"id":"ac1df14related5a-h6","type":"hint","dependencies":["ac1df14related5a-h5"],"title":"Find $$z$$","text":"We need to find $$z$$ when $$x=30mi$$ and $$y=40mi$$","variabilization":{},"oer":"","license":""},{"id":"ac1df14related5a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$50$$"],"dependencies":["ac1df14related5a-h6"],"title":"Find $$y$$","text":"What is $$z$$ when $$x=30$$ and $$y=40$$?","variabilization":{},"oer":"","license":""},{"id":"ac1df14related5a-h8","type":"hint","dependencies":["ac1df14related5a-h7"],"title":"Plug in","text":"Final step is to plug in the values of $$x$$, $$y$$, $$z$$, $$\\\\frac{dx}{dt}$$, $$\\\\frac{dy}{dt}$$, into the differentiated equation to find $$\\\\frac{dz}{dt}$$","variabilization":{},"oer":"","license":""},{"id":"ac1df14related5a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{-250mi}{h}$$"],"dependencies":["ac1df14related5a-h8"],"title":"Plug in","text":"What is dx/dt?(we are given)","variabilization":{},"oer":"","license":"","choices":["$$\\\\frac{250mi}{h}$$","$$\\\\frac{-250mi}{h}$$","$$\\\\frac{300mi}{h}$$","$$\\\\frac{-300mi}{h}$$"]},{"id":"ac1df14related5a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{-300mi}{h}$$"],"dependencies":["ac1df14related5a-h8"],"title":"Plug in","text":"What is dy/dt?(we are given this value)","variabilization":{},"oer":"","license":"","choices":["$$\\\\frac{250mi}{h}$$","$$\\\\frac{-250mi}{h}$$","$$\\\\frac{300mi}{h}$$","$$\\\\frac{-300mi}{h}$$"]}]}}]},{"id":"ac1df14related6","title":"Related Rates: Buses","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.1 Related Rates","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ac1df14related6a","stepAnswer":["$$101.5$$"],"problemType":"MultipleChoice","stepTitle":"Two buses are driving along parallel freeways that are 5mi apart, one heading east and the other heading west.","stepBody":"Assuming that each bus drives a constant 55mph, find the rate at which the distance between the buses is changing when they are 13mi apart, heading toward each other.","answerType":"string","variabilization":{},"answerLatex":"$$101.5$$","choices":["$$101.5$$","$$-101.5$$","$$-55.75$$","$$55.75$$"],"hints":{"DefaultPathway":[{"id":"ac1df14related6a-h1","type":"hint","dependencies":[],"title":"Strategy","text":"we can use the relationship between the sides of a right triangle and differentiate it with respect to time","variabilization":{},"oer":"","license":""},{"id":"ac1df14related6a-h2","type":"hint","dependencies":["ac1df14related6a-h1"],"title":"Pythagorean theory","text":"Let\'s set the distance between two buses as $$x$$, the horizontal distance between two buses as $$y$$. Then use Pythagorean theory to make equation to differentiate with respect to time","variabilization":{},"oer":"","license":""},{"id":"ac1df14related6a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y^2+5^2=x^2$$"],"dependencies":["ac1df14related6a-h2"],"title":"Pythagorean theory","text":"What is the equation by plugging in $$x$$, $$y$$, and the distance between parallel freeways into pythagorean theory?","variabilization":{},"oer":"","license":"","choices":["$$y+5=x^2$$","$$y^2+5^2=x^2$$","$$y^2+5^2=x$$"]},{"id":"ac1df14related6a-h4","type":"hint","dependencies":["ac1df14related6a-h3"],"title":"Differentiate","text":"Next step is to differentiate both sides of the equation with respect to time (t) to relate $$\\\\frac{dx}{dt}$$ and $$\\\\frac{dy}{dt}$$","variabilization":{},"oer":"","license":""},{"id":"ac1df14related6a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2y\\\\left(\\\\frac{dy}{dt}\\\\right)=2x\\\\left(\\\\frac{dx}{dt}\\\\right)$$"],"dependencies":["ac1df14related6a-h4"],"title":"Differentiate","text":"What is differentiated equation?","variabilization":{},"oer":"","license":"","choices":["$$2y\\\\left(\\\\frac{dy}{dt}\\\\right)=2x\\\\left(\\\\frac{dx}{dt}\\\\right)$$","$$2\\\\left(\\\\frac{dy}{dt}\\\\right)+5=2\\\\left(\\\\frac{dx}{dt}\\\\right)$$","$$y^2 \\\\frac{dy}{dt}+5^2=x^2 \\\\frac{dx}{dt}$$"]},{"id":"ac1df14related6a-h6","type":"hint","dependencies":["ac1df14related6a-h5"],"title":"Find $$y$$","text":"We need to find $$y$$ when $$x=13mi$$","variabilization":{},"oer":"","license":""},{"id":"ac1df14related6a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["ac1df14related6a-h6"],"title":"Find $$y$$","text":"What is $$y$$ when $$x$$ $$=13$$?","variabilization":{},"oer":"","license":""},{"id":"ac1df14related6a-h8","type":"hint","dependencies":["ac1df14related6a-h7"],"title":"Plug in","text":"Final step is to plug in the values of $$x$$, $$y$$, $$\\\\frac{dy}{dt}$$ , into the differentiated equation to find $$\\\\frac{dx}{dt}$$","variabilization":{},"oer":"","license":""},{"id":"ac1df14related6a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["110mph"],"dependencies":["ac1df14related6a-h8"],"title":"Plug in","text":"What is $$\\\\frac{dy}{dt}$$?","variabilization":{},"oer":"","license":"","choices":["110mph","55mph"]}]}}]},{"id":"ac1df14related7","title":"Related Rates: Cubes","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.1 Related Rates","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ac1df14related7a","stepAnswer":["$$\\\\frac{-5}{6}$$ $$\\\\frac{m}{s}$$"],"problemType":"MultipleChoice","stepTitle":"The volume of a cube decreases at a rate of $$10$$ $$\\\\frac{m^3}{s}$$.","stepBody":"Find the rate at which the side of the cube changes when the side of the cube is $$2$$ $$m$$.","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{-5}{6}$$ $$\\\\frac{m}{s}$$","choices":["$$\\\\frac{-5}{6}$$ $$\\\\frac{m}{s}$$","$$\\\\frac{5}{6}$$ $$\\\\frac{m}{s}$$","$$\\\\frac{6}{5}$$ $$\\\\frac{m}{s}$$","$$\\\\frac{-6}{5}$$ $$\\\\frac{m}{s}$$"],"hints":{"DefaultPathway":[{"id":"ac1df14related7a-h1","type":"hint","dependencies":[],"title":"Strategy","text":"we can use the equation for Volume of cube and differentiate it with respect to time","variabilization":{},"oer":"","license":""},{"id":"ac1df14related7a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$V=s^3$$"],"dependencies":["ac1df14related7a-h1"],"title":"Volume of cube","text":"Let\'s set volume of cube as V, and length of side of cube as s. What is the equation for computing Volume of cube?","variabilization":{},"oer":"","license":"","choices":["$$V=s$$","$$V=s^2$$","$$V=s^3$$"]},{"id":"ac1df14related7a-h3","type":"hint","dependencies":["ac1df14related7a-h2"],"title":"Differentiate","text":"Next step is to differentiate both sides of the equation with respect to time (t)","variabilization":{},"oer":"","license":""},{"id":"ac1df14related7a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{dV}{dt}=3s^2 \\\\frac{ds}{dt}$$"],"dependencies":["ac1df14related7a-h3"],"title":"Differentiate","text":"What is differentiated equation?","variabilization":{},"oer":"","license":"","choices":["$$\\\\frac{dV}{dt}=3s \\\\frac{ds}{dt}$$","$$\\\\frac{dV}{dt}=3s^2 \\\\frac{ds}{dt}$$","$$\\\\frac{dV}{dt}=s^3 \\\\frac{ds}{dt}$$"]},{"id":"ac1df14related7a-h5","type":"hint","dependencies":["ac1df14related7a-h4"],"title":"Strategy","text":"We want to find $$\\\\frac{ds}{dt}$$ when $$\\\\frac{dV}{dt}$$ is $$\\\\frac{-10m^3}{s}$$, and when s $$=2m$$","variabilization":{},"oer":"","license":""},{"id":"ac1df14related7a-h6","type":"hint","dependencies":["ac1df14related7a-h5"],"title":"Plug in","text":"Final step is to plug in the values of s, $$\\\\frac{dV}{dt}$$ , into the differentiated equation to find $$\\\\frac{ds}{dt}$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"ac1df14related8","title":"Related Rates: Spheres $$1$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.1 Related Rates","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ac1df14related8a","stepAnswer":["$$240\\\\pi$$ $$\\\\frac{m^2}{sec}$$"],"problemType":"MultipleChoice","stepTitle":"The radius of a sphere decreases at a rate of $$3$$ $$\\\\frac{m}{sec}$$.","stepBody":"Find the rate at which the surface area decreases when the radius is $$10$$ $$m$$.","answerType":"string","variabilization":{},"answerLatex":"$$240\\\\pi$$ $$\\\\frac{m^2}{sec}$$","choices":["$$240\\\\pi$$ $$\\\\frac{m^2}{sec}$$","$$-120\\\\pi$$ $$\\\\frac{m^2}{sec}$$","$$120\\\\pi$$ $$\\\\frac{m^2}{sec}$$","$$-240\\\\pi$$ $$\\\\frac{m^2}{sec}$$"],"hints":{"DefaultPathway":[{"id":"ac1df14related8a-h1","type":"hint","dependencies":[],"title":"Strategy","text":"we can use the equation for surface area of sphere and differentiate it with respect to time","variabilization":{},"oer":"","license":""},{"id":"ac1df14related8a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$A=4\\\\pi r^2$$"],"dependencies":["ac1df14related8a-h1"],"title":"Surface area of sphere","text":"Let\'s set surface area of sphere as A, and radius as $$r$$. What is the equation for computing A?","variabilization":{},"oer":"","license":"","choices":["$$A=4\\\\pi r$$","$$A=3\\\\pi r^2$$","$$A=4\\\\pi r^2$$"]},{"id":"ac1df14related8a-h3","type":"hint","dependencies":["ac1df14related8a-h2"],"title":"Differentiate","text":"Next step is to differentiate both sides of the equation with respect to time (t)","variabilization":{},"oer":"","license":""},{"id":"ac1df14related8a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{dA}{dt}=8\\\\pi r\\\\left(\\\\frac{dr}{dt}\\\\right)$$"],"dependencies":["ac1df14related8a-h3"],"title":"Differentiate","text":"What is differentiated equation?","variabilization":{},"oer":"","license":"","choices":["$$\\\\frac{dA}{dt}=8\\\\pi r\\\\left(\\\\frac{dr}{dt}\\\\right)$$","$$\\\\frac{dA}{dt}=4\\\\pi r^2 \\\\frac{dr}{dt}$$","$$\\\\frac{dA}{dt}=4\\\\pi r\\\\left(\\\\frac{dr}{dt}\\\\right)$$"]},{"id":"ac1df14related8a-h5","type":"hint","dependencies":["ac1df14related8a-h4"],"title":"Strategy","text":"We want to find $$\\\\frac{dA}{dt}$$ when $$\\\\frac{dr}{dt}$$ is $$\\\\frac{-3m}{sec}$$, and $$r=10m$$","variabilization":{},"oer":"","license":""},{"id":"ac1df14related8a-h6","type":"hint","dependencies":["ac1df14related8a-h5"],"title":"Plug in","text":"Final step is to plug in the values of $$r$$, $$\\\\frac{dr}{dt}$$ , into the differentiated equation to find $$\\\\frac{dA}{dt}$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"ac1df14related9","title":"Related Rates: Spheres $$2$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.1 Related Rates","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ac1df14related9a","stepAnswer":["$$\\\\frac{1}{2\\\\sqrt{\\\\pi}}$$"],"problemType":"MultipleChoice","stepTitle":"The radius of a sphere is increasing at a rate of $$9$$ $$\\\\frac{cm}{sec}$$.","stepBody":"Find the radius of the sphere when the volume and the radius of the sphere are increasing at the same numerical rate.","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{1}{2\\\\sqrt{\\\\pi}}$$","choices":["$$\\\\frac{1}{2\\\\sqrt{\\\\pi}}$$","$$\\\\frac{-1}{2\\\\sqrt{\\\\pi}}$$","$$2\\\\sqrt{\\\\pi}$$","$$\\\\pi \\\\sqrt{\\\\frac{1}{2}}$$"],"hints":{"DefaultPathway":[{"id":"ac1df14related9a-h1","type":"hint","dependencies":[],"title":"Strategy","text":"we can use the equation for volume of sphere and differentiate it with respect to time","variabilization":{},"oer":"","license":""},{"id":"ac1df14related9a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$V=\\\\frac{4}{3} \\\\pi r^3$$"],"dependencies":["ac1df14related9a-h1"],"title":"Volume of sphere","text":"Let\'s set volume of sphere as V, and radius as $$r$$. What is the equation for computing V?","variabilization":{},"oer":"","license":"","choices":["$$V=\\\\frac{4}{3} \\\\pi r$$","$$V=\\\\frac{4}{3} \\\\pi r^3$$","$$V=4\\\\pi r^3$$"]},{"id":"ac1df14related9a-h3","type":"hint","dependencies":["ac1df14related9a-h2"],"title":"Differentiate","text":"Next step is to differentiate both sides of the equation with respect to time (t)","variabilization":{},"oer":"","license":""},{"id":"ac1df14related9a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{dV}{dt}=4\\\\pi r^2 \\\\frac{dr}{dt}$$"],"dependencies":["ac1df14related9a-h3"],"title":"Differentiate","text":"What is differentiated equation?","variabilization":{},"oer":"","license":"","choices":["$$\\\\frac{dV}{dt}=\\\\frac{4}{3} \\\\pi r\\\\left(\\\\frac{dr}{dt}\\\\right)$$","$$\\\\frac{dV}{dt}=4\\\\pi r^2 \\\\frac{dr}{dt}$$","$$\\\\frac{dV}{dt}=4\\\\pi r\\\\left(\\\\frac{dr}{dt}\\\\right)$$"]},{"id":"ac1df14related9a-h5","type":"hint","dependencies":["ac1df14related9a-h4"],"title":"Strategy","text":"We want to find $$r$$ when $$\\\\frac{dV}{dt}=\\\\frac{dr}{dt}=\\\\frac{9cm}{sec}$$","variabilization":{},"oer":"","license":""},{"id":"ac1df14related9a-h6","type":"hint","dependencies":["ac1df14related9a-h5"],"title":"Plug in","text":"Final step is to plug in the values of $$\\\\frac{dV}{dt}$$, $$\\\\frac{dr}{dt}$$ into the differentiated equation to find $$r$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"ac26263equations1","title":"Is the ordered triple a solution to the system?","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Solve Systems of Equations with Three Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac26263equations1a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$-3x+y+z=-4$$, $$-x+2y-2z=1$$, $$2x-y-z=-1$$, $$(-5, -7, 4)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ac26263equations1a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute the ordered pair into all $$3$$ systems. If both sides are made true in all $$3$$ equations, the ordered triple is a solution to the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac26263equations10","title":"Solve the system","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Solve Systems of Equations with Three Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac26263equations10a","stepAnswer":["$$(3, -4, -2)$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{x}{3}-y+\\\\frac{z}{2}=4$$, $$\\\\frac{2x}{3}+\\\\frac{5y}{2}-4z=0$$, $$x-\\\\frac{y}{2}+\\\\frac{3z}{2}=2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(3, -4, -2)$$","choices":["(203/16,-25/16, -231/16)","$$(-3, -5, 4)$$","$$(2, -3, -2)$$","$$(3, -4, -2)$$"],"hints":{"DefaultPathway":[{"id":"ac26263equations10a-h1","type":"hint","dependencies":[],"title":"Choose","text":"Choose $$2$$ pairs of equations, both which will be used to eliminate the same variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263equations10a-h2","type":"hint","dependencies":["ac26263equations10a-h1"],"title":"Eliminate","text":"Eliminate a variable in the two pairs of 3-variable equations. Do this by multiplying the equation by a constant so that once you subtract or add the two equations, the variable is eliminated.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263equations10a-h3","type":"hint","dependencies":["ac26263equations10a-h2"],"title":"Eliminate again","text":"Now, you should have two 2-variable equations. Using the same process, eliminate the variable in this set of equations so that you are left with one variable. Solve the equation for this variable, if necessary.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263equations10a-h4","type":"hint","dependencies":["ac26263equations10a-h3"],"title":"Plug in","text":"Plug in the variable you just found into one of the 2-variable equations to find the other variable. Then plug in those two variables into the one of the 3-variable equations to find the final variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac26263equations11","title":"Solve the system","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Solve Systems of Equations with Three Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac26263equations11a","stepAnswer":["$$(-3, 2, 3)$$"],"problemType":"MultipleChoice","stepTitle":"$$2x+5y=4$$, $$3y-z=3$$, $$4x+3z=-3$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-3, 2, 3)$$","choices":["(203/16,-25/16, -231/16)","$$(-3, -5, 4)$$","$$(2, -3, -2)$$","$$(-3, 2, 3)$$"],"hints":{"DefaultPathway":[{"id":"ac26263equations11a-h1","type":"hint","dependencies":[],"title":"Choose","text":"Choose a variable to eliminate. Then choose $$2$$ equations that have that variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263equations11a-h2","type":"hint","dependencies":["ac26263equations11a-h1"],"title":"Eliminate","text":"Multiply equations by constants as needed to eliminate variables. Then, subtract the two equations you chose. This should eliminate the one common variable, and leave you with two other variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263equations11a-h3","type":"hint","dependencies":["ac26263equations11a-h2"],"title":"Eliminate again","text":"Now, you should have a 2-variable equation. Use one of the $$3$$ given equations that has the same two variables as the one you just solved for, eliminate another variable. This should leave you with one variable equaling a value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263equations11a-h4","type":"hint","dependencies":["ac26263equations11a-h3"],"title":"Plug in","text":"Plug in the variable value you just found into one of the original $$3$$ equations. This will give you the value of another variable. Now, with the two variables you know the value of, find the last value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac26263equations12","title":"Solve the system","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Solve Systems of Equations with Three Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac26263equations12a","stepAnswer":["$$(-2, 0, -3)$$"],"problemType":"MultipleChoice","stepTitle":"$$3x-z=-3$$, $$5y+2z=-6$$, $$4x+3y=-8$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-2, 0, -3)$$","choices":["$$(-2, 0, -3)$$","No Solution","$$(2, -3, -2)$$","$$(-3, 2, 3)$$"],"hints":{"DefaultPathway":[{"id":"ac26263equations12a-h1","type":"hint","dependencies":[],"title":"Choose","text":"Choose a variable to eliminate. Then choose $$2$$ equations that have that variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263equations12a-h2","type":"hint","dependencies":["ac26263equations12a-h1"],"title":"Eliminate","text":"Multiply equations by constants as needed to eliminate variables. Then, subtract the two equations you chose. This should eliminate the one common variable, and leave you with two other variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263equations12a-h3","type":"hint","dependencies":["ac26263equations12a-h2"],"title":"Eliminate again","text":"Now, you should have a 2-variable equation. Use one of the $$3$$ given equations that has the same two variables as the one you just solved for, eliminate another variable. This should leave you with one variable equaling a value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263equations12a-h4","type":"hint","dependencies":["ac26263equations12a-h3"],"title":"Plug in","text":"Plug in the variable value you just found into one of the original $$3$$ equations. This will give you the value of another variable. Now, with the two variables you know the value of, find the last value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac26263equations13","title":"Solve the system","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Solve Systems of Equations with Three Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac26263equations13a","stepAnswer":["No Solution"],"problemType":"MultipleChoice","stepTitle":"$$x-2y+2z=1$$, $$-2x+y-z=2$$, $$x-y+z=5$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$(-2, 0, -3)$$","No Solution","$$(2, -3, -2)$$","$$(-3, 2, 3)$$"],"hints":{"DefaultPathway":[{"id":"ac26263equations13a-h1","type":"hint","dependencies":[],"title":"Choose","text":"Choose a variable to eliminate. Then choose $$2$$ equations that have that variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263equations13a-h2","type":"hint","dependencies":["ac26263equations13a-h1"],"title":"Eliminate","text":"Multiply equations by constants as needed to eliminate variables. Then, subtract the two equations you chose. This should eliminate the one common variable, and leave you with two other variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263equations13a-h3","type":"hint","dependencies":["ac26263equations13a-h2"],"title":"Eliminate again","text":"Now, you should have a 2-variable equation. Use one of the $$3$$ given equations that has the same two variables as the one you just solved for, eliminate another variable. This should leave you with one variable equaling a value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263equations13a-h4","type":"hint","dependencies":["ac26263equations13a-h3"],"title":"Plug in","text":"Plug in the variable value you just found into one of the original $$3$$ equations. This will give you the value of another variable. Now, with the two variables you know the value of, find the last value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac26263equations14","title":"Solve the system","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Solve Systems of Equations with Three Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac26263equations14a","stepAnswer":["(203/16,-25/16, -231/16)"],"problemType":"MultipleChoice","stepTitle":"$$x+4y+z=-8$$, $$4x-y+3z=9$$, $$2x+7y+z=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$(-2, 0, -3)$$","No Solution","(203/16,-25/16, -231/16)","$$(-3, 2, 3)$$"],"hints":{"DefaultPathway":[{"id":"ac26263equations14a-h1","type":"hint","dependencies":[],"title":"Choose","text":"Choose a variable to eliminate. Then choose $$2$$ equations that have that variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263equations14a-h2","type":"hint","dependencies":["ac26263equations14a-h1"],"title":"Eliminate","text":"Multiply equations by constants as needed to eliminate variables. Then, subtract the two equations you chose. This should eliminate the one common variable, and leave you with two other variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263equations14a-h3","type":"hint","dependencies":["ac26263equations14a-h2"],"title":"Eliminate again","text":"Now, you should have a 2-variable equation. Use one of the $$3$$ given equations that has the same two variables as the one you just solved for, eliminate another variable. This should leave you with one variable equaling a value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263equations14a-h4","type":"hint","dependencies":["ac26263equations14a-h3"],"title":"Plug in","text":"Plug in the variable value you just found into one of the original $$3$$ equations. This will give you the value of another variable. Now, with the two variables you know the value of, find the last value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac26263equations15","title":"Solve the system","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Solve Systems of Equations with Three Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac26263equations15a","stepAnswer":["(5z+2, $$-3z+1$$, any real number)"],"problemType":"MultipleChoice","stepTitle":"$$x+y-2z=3$$, $$-2x-3y+z=-7$$, $$x+2y+z=4$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"(5z+2, $$-3z+1$$, any real number)","choices":["$$(-2, 0, -3)$$","No Solution","(203/16,-25/16, -231/16)","(5z+2, $$-3z+1$$, any real number)"],"hints":{"DefaultPathway":[{"id":"ac26263equations15a-h1","type":"hint","dependencies":[],"title":"Choose","text":"Choose a variable to eliminate. Then choose $$2$$ equations that have that variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263equations15a-h2","type":"hint","dependencies":["ac26263equations15a-h1"],"title":"Eliminate","text":"Multiply equations by constants as needed to eliminate variables. Then, subtract the two equations you chose. This should eliminate the one common variable, and leave you with two other variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263equations15a-h3","type":"hint","dependencies":["ac26263equations15a-h2"],"title":"Eliminate again","text":"Now, you should have a 2-variable equation. Use one of the $$3$$ given equations that has the same two variables as the one you just solved for, eliminate another variable. This should leave you with one variable equaling a value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263equations15a-h4","type":"hint","dependencies":["ac26263equations15a-h3"],"title":"Plug in","text":"Plug in the variable value you just found into one of the original $$3$$ equations. This will give you the value of another variable. Now, with the two variables you know the value of, find the last value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac26263equations2","title":"Is the ordered triple a solution to the system?","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Solve Systems of Equations with Three Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac26263equations2a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$-3x+y+z=-4$$, $$-x+2y-2z=1$$, $$2x-y-z=-1$$, (5,7,4)","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ac26263equations2a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute the ordered pair into all $$3$$ systems. If both sides are made true in all $$3$$ equations, the ordered triple is a solution to the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac26263equations3","title":"Is the ordered triple a solution to the system?","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Solve Systems of Equations with Three Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac26263equations3a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$x+3y-z=15$$, $$y=23x-2$$, $$x-3y+z=-2$$, (-6,5,1/2)","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ac26263equations3a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute the ordered pair into all $$3$$ systems. If both sides are made true in all $$3$$ equations, the ordered triple is a solution to the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac26263equations4","title":"Is the ordered triple a solution to the system?","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Solve Systems of Equations with Three Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac26263equations4a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$x+3y-z=15$$, $$y=23x-2$$, $$x-3y+z=-2$$, (5,4/3,-3)","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ac26263equations4a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute the ordered pair into all $$3$$ systems. If both sides are made true in all $$3$$ equations, the ordered triple is a solution to the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac26263equations5","title":"Is the ordered triple a solution to the system?","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Solve Systems of Equations with Three Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac26263equations5a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$y-10z=-8$$, $$2x-y=2$$, $$x-5z=3$$, (7,12,2)","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ac26263equations5a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute the ordered pair into all $$3$$ systems. If both sides are made true in all $$3$$ equations, the ordered triple is a solution to the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac26263equations6","title":"Solve the system","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Solve Systems of Equations with Three Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac26263equations6a","stepAnswer":["(4,5,2)"],"problemType":"MultipleChoice","stepTitle":"$$6x-5y+2z=3$$, $$2x+y-4z=5$$, $$3x-3y+z=-1$$","stepBody":"","answerType":"string","variabilization":{},"choices":["(4,5,2)","$$(-6, 7, 4)$$","No Solution)","$$(-2, 1, 3)$$"],"hints":{"DefaultPathway":[{"id":"ac26263equations6a-h1","type":"hint","dependencies":[],"title":"Choose","text":"Choose $$2$$ pairs of equations, both which will be used to eliminate the same variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263equations6a-h2","type":"hint","dependencies":["ac26263equations6a-h1"],"title":"Eliminate","text":"Eliminate a variable in the two pairs of 3-variable equations. Do this by multiplying the equation by a constant so that once you subtract or add the two equations, the variable is eliminated.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263equations6a-h3","type":"hint","dependencies":["ac26263equations6a-h2"],"title":"Eliminate again","text":"Now, you should have two 2-variable equations. Using the same process, eliminate the variable in this set of equations so that you are left with one variable. Solve the equation for this variable, if necessary.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263equations6a-h4","type":"hint","dependencies":["ac26263equations6a-h3"],"title":"Plug in","text":"Plug in the variable you just found into one of the 2-variable equations to find the other variable. Then plug in those two variables into the one of the 3-variable equations to find the final variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac26263equations7","title":"Solve the system","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Solve Systems of Equations with Three Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac26263equations7a","stepAnswer":["$$(7, 12, -2)$$"],"problemType":"MultipleChoice","stepTitle":"$$5x-3y+2z=-5$$, $$2x-y-z=4$$, $$3x-2y+2z=-7$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(7, 12, -2)$$","choices":["$$(7, 12, -2)$$","$$(-6, 7, 4)$$","(0,3,7)","No Solution"],"hints":{"DefaultPathway":[{"id":"ac26263equations7a-h1","type":"hint","dependencies":[],"title":"Choose","text":"Choose $$2$$ pairs of equations, both which will be used to eliminate the same variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263equations7a-h2","type":"hint","dependencies":["ac26263equations7a-h1"],"title":"Eliminate","text":"Eliminate a variable in the two pairs of 3-variable equations. Do this by multiplying the equation by a constant so that once you subtract or add the two equations, the variable is eliminated.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263equations7a-h3","type":"hint","dependencies":["ac26263equations7a-h2"],"title":"Eliminate again","text":"Now, you should have two 2-variable equations. Using the same process, eliminate the variable in this set of equations so that you are left with one variable. Solve the equation for this variable, if necessary.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263equations7a-h4","type":"hint","dependencies":["ac26263equations7a-h3"],"title":"Plug in","text":"Plug in the variable you just found into one of the 2-variable equations to find the other variable. Then plug in those two variables into the one of the 3-variable equations to find the final variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac26263equations8","title":"Solve the system","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Solve Systems of Equations with Three Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac26263equations8a","stepAnswer":["$$(-3, -5, 4)$$"],"problemType":"MultipleChoice","stepTitle":"$$4x-3y+z=7$$, $$2x-5y-4z=3$$, $$3x-2y-2z=-7$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-3, -5, 4)$$","choices":["No Solution","$$(-3, -5, 4)$$","(0,3,7)","$$(-2, 1, 3)$$"],"hints":{"DefaultPathway":[{"id":"ac26263equations8a-h1","type":"hint","dependencies":[],"title":"Choose","text":"Choose $$2$$ pairs of equations, both which will be used to eliminate the same variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263equations8a-h2","type":"hint","dependencies":["ac26263equations8a-h1"],"title":"Eliminate","text":"Eliminate a variable in the two pairs of 3-variable equations. Do this by multiplying the equation by a constant so that once you subtract or add the two equations, the variable is eliminated.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263equations8a-h3","type":"hint","dependencies":["ac26263equations8a-h2"],"title":"Eliminate again","text":"Now, you should have two 2-variable equations. Using the same process, eliminate the variable in this set of equations so that you are left with one variable. Solve the equation for this variable, if necessary.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263equations8a-h4","type":"hint","dependencies":["ac26263equations8a-h3"],"title":"Plug in","text":"Plug in the variable you just found into one of the 2-variable equations to find the other variable. Then plug in those two variables into the one of the 3-variable equations to find the final variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac26263equations9","title":"Solve the system","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Solve Systems of Equations with Three Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac26263equations9a","stepAnswer":["$$(2, -3, -2)$$"],"problemType":"MultipleChoice","stepTitle":"$$11x+9y+2z=-9$$, $$7x+5y+3z=-7$$, $$4x+3y+z=-3$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(2, -3, -2)$$","choices":["$$(7, 12, -2)$$","No Solution","$$(2, -3, -2)$$","$$(-2, 1, 3)$$"],"hints":{"DefaultPathway":[{"id":"ac26263equations9a-h1","type":"hint","dependencies":[],"title":"Choose","text":"Choose $$2$$ pairs of equations, both which will be used to eliminate the same variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263equations9a-h2","type":"hint","dependencies":["ac26263equations9a-h1"],"title":"Eliminate","text":"Eliminate a variable in the two pairs of 3-variable equations. Do this by multiplying the equation by a constant so that once you subtract or add the two equations, the variable is eliminated.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263equations9a-h3","type":"hint","dependencies":["ac26263equations9a-h2"],"title":"Eliminate again","text":"Now, you should have two 2-variable equations. Using the same process, eliminate the variable in this set of equations so that you are left with one variable. Solve the equation for this variable, if necessary.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263equations9a-h4","type":"hint","dependencies":["ac26263equations9a-h3"],"title":"Plug in","text":"Plug in the variable you just found into one of the 2-variable equations to find the other variable. Then plug in those two variables into the one of the 3-variable equations to find the final variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac26263ThreeVariables1","title":"Linear Equation in Three Variables","body":"Determine whether the ordered triple is a solution to the system: $$x-y+z=2;$$ $$2x-y-z=-6;$$ $$2x+2y+z=-3$$","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Solve Systems of Equations with Three Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac26263ThreeVariables1a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"a) $$(-2, -1, 3)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ac26263ThreeVariables1a-h1","type":"hint","dependencies":[],"title":"Principle","text":"Translate $$(-2, -1, 3)$$ to $$x=-2$$, $$y=-1$$, $$z=3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables1a-h2","type":"hint","dependencies":["ac26263ThreeVariables1a-h1"],"title":"Substitution","text":"Substitute the given value to the first equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ac26263ThreeVariables1a-h2"],"title":"Substitute","text":"What is $$-2-\\\\left(-1\\\\right)+3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables1a-h4","type":"hint","dependencies":["ac26263ThreeVariables1a-h3"],"title":"Comparison","text":"The substituted answer matches the given answer so it is the solution to the first equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables1a-h5","type":"hint","dependencies":["ac26263ThreeVariables1a-h4"],"title":"Substitution","text":"Substitute the given value to the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables1a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6$$"],"dependencies":["ac26263ThreeVariables1a-h5"],"title":"Substitute","text":"What is $$2\\\\left(-2\\\\right)-\\\\left(-1\\\\right)-3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables1a-h7","type":"hint","dependencies":["ac26263ThreeVariables1a-h6"],"title":"Comparison","text":"The substituted answer matches the given answer so it is the solution to the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables1a-h8","type":"hint","dependencies":["ac26263ThreeVariables1a-h7"],"title":"Substitution","text":"Substitute the given value to the third equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables1a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["ac26263ThreeVariables1a-h8"],"title":"Substitute","text":"What is $$2\\\\left(-2\\\\right)+2\\\\left(-1\\\\right)+3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables1a-h10","type":"hint","dependencies":["ac26263ThreeVariables1a-h9"],"title":"Comparison","text":"The substituted answer matches the given answer so it is the solution to the third equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables1a-h11","type":"hint","dependencies":["ac26263ThreeVariables1a-h4","ac26263ThreeVariables1a-h7","ac26263ThreeVariables1a-h10"],"title":"Conclusion","text":"$$(-2, -1, 3)$$ is the solution to all three equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ac26263ThreeVariables1b","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"b) $$(-4, -3, 4)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ac26263ThreeVariables1b-h1","type":"hint","dependencies":[],"title":"Principle","text":"Translate $$(-4, 3, 4)$$ to $$x=-4$$, $$y=3$$, $$z=4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables1b-h2","type":"hint","dependencies":["ac26263ThreeVariables1b-h1"],"title":"Substitution","text":"Substitute the given value to the first equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables1b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ac26263ThreeVariables1b-h2"],"title":"Substitute","text":"What is $$-4-\\\\left(-3\\\\right)+4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables1b-h4","type":"hint","dependencies":["ac26263ThreeVariables1b-h3"],"title":"Comparison","text":"The substituted answer does not match the given answer so it is the solution to the first equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables1b-h5","type":"hint","dependencies":["ac26263ThreeVariables1b-h4"],"title":"Substitution","text":"Substitute the given value to the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables1b-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":["ac26263ThreeVariables1b-h5"],"title":"Substitute","text":"What is $$2\\\\left(-4\\\\right)-\\\\left(-3\\\\right)-4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables1b-h7","type":"hint","dependencies":["ac26263ThreeVariables1b-h6"],"title":"Comparison","text":"The substituted answer does not match the given answer so it is the solution to the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables1b-h8","type":"hint","dependencies":["ac26263ThreeVariables1b-h7"],"title":"Substitution","text":"Substitute the given value to the third equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables1b-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-10$$"],"dependencies":["ac26263ThreeVariables1b-h8"],"title":"Substitute","text":"What is $$2\\\\left(-4\\\\right)+2\\\\left(-3\\\\right)+4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables1b-h10","type":"hint","dependencies":["ac26263ThreeVariables1b-h9"],"title":"Comparison","text":"The substituted answer does not match the given answer so it is the solution to the third equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables1b-h11","type":"hint","dependencies":["ac26263ThreeVariables1b-h4","ac26263ThreeVariables1b-h7","ac26263ThreeVariables1b-h10"],"title":"Conclusion","text":"$$(-2, -1, 3)$$ is not the solution to all three equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac26263ThreeVariables10","title":"How to Solve a System of Equations With Three Variables by Elimination","body":"Solve the following system","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Solve Systems of Equations with Three Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac26263ThreeVariables10a","stepAnswer":["no solution"],"problemType":"MultipleChoice","stepTitle":"Solve the system of equations: $$x+2y+6z=5;$$ $$-x+y-2z=3;$$ $$x-4y-2z=1$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$(-4, 9, 3)$$","no solution","$$(2, -6, 1)$$"],"hints":{"DefaultPathway":[{"id":"ac26263ThreeVariables10a-h1","type":"hint","dependencies":[],"title":"Calculation","text":"Add the first equation to the second equation: $$x+2y+6z+\\\\left(-x+y-2z\\\\right)=5+3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables10a-h2","type":"hint","dependencies":["ac26263ThreeVariables10a-h1"],"title":"Calculation","text":"The equation becomes $$3y+4z=8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables10a-h3","type":"hint","dependencies":["ac26263ThreeVariables10a-h2"],"title":"Calculation","text":"Add the second equation to the third equation: $$\\\\left(-x+y-2z\\\\right)+x-4y-2z=3+1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables10a-h4","type":"hint","dependencies":["ac26263ThreeVariables10a-h3"],"title":"Calculation","text":"The equation becomes $$-3y-4z=4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables10a-h5","type":"hint","dependencies":["ac26263ThreeVariables10a-h4"],"title":"Subtraction","text":"Add $$3y+4z=8$$ to $$-3y-4z=4$$. $$3y+4z+\\\\left(-3y-4z\\\\right)=8+4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables10a-h6","type":"hint","dependencies":["ac26263ThreeVariables10a-h5"],"title":"Subtraction","text":"The equation becomes $$0=12$$, which is a false statement","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac26263ThreeVariables11","title":"How to Solve a System of Equations With Three Variables by Elimination","body":"Solve the following system","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Solve Systems of Equations with Three Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac26263ThreeVariables11a","stepAnswer":["no solution"],"problemType":"MultipleChoice","stepTitle":"Solve the system of equations: $$2x-2y+3z=6;$$ $$4x-3y+2z=0;$$ $$-2x+3y-7z=1$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$(-8, 4, 2)$$","$$(2, 4, -8)$$","no solution"],"hints":{"DefaultPathway":[{"id":"ac26263ThreeVariables11a-h1","type":"hint","dependencies":[],"title":"Calculation","text":"Multiply the first equation by $$2$$ and subtract with the second equation: $$2\\\\left(2x-2y+3z\\\\right)-4x-3y+2z=2\\\\times6-0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables11a-h2","type":"hint","dependencies":["ac26263ThreeVariables11a-h1"],"title":"Calculation","text":"The equation becomes $$-y+4z=12$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables11a-h3","type":"hint","dependencies":["ac26263ThreeVariables11a-h2"],"title":"Calculation","text":"Add the first equation to the third equation: $$2x-2y+3z+\\\\left(-2x+3y-7z\\\\right)=6+1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables11a-h4","type":"hint","dependencies":["ac26263ThreeVariables11a-h3"],"title":"Calculation","text":"The equation becomes $$y-4z=7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables11a-h5","type":"hint","dependencies":["ac26263ThreeVariables11a-h4"],"title":"Subtraction","text":"Add $$-y+4z=12$$ to $$y-4z=7$$. $$\\\\left(-y+4z\\\\right)+\\\\left(-y-4z\\\\right)=12+7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables11a-h6","type":"hint","dependencies":["ac26263ThreeVariables11a-h5"],"title":"Subtraction","text":"The equation becomes $$0=19$$, which is a false statement","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac26263ThreeVariables12","title":"How to Solve a System of Equations With Three Variables by Elimination","body":"Solve the following system","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Solve Systems of Equations with Three Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac26263ThreeVariables12a","stepAnswer":["infinite solutions"],"problemType":"MultipleChoice","stepTitle":"Solve the system of equations: $$x+2y-z=1;$$ $$2x+7y+4z=11;$$ $$x+3y+z=4$$","stepBody":"","answerType":"string","variabilization":{},"choices":["infinite solutions","$$(3, 10, -5)$$","no solution"],"hints":{"DefaultPathway":[{"id":"ac26263ThreeVariables12a-h1","type":"hint","dependencies":[],"title":"Calculation","text":"Multiply the first equation by $$2$$ and subtract with the second equation: $$2\\\\left(x+2y-z\\\\right)-2x+7y+4z=2\\\\times1-11$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables12a-h2","type":"hint","dependencies":["ac26263ThreeVariables12a-h1"],"title":"Calculation","text":"The equation becomes $$-3y-6z=-9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables12a-h3","type":"hint","dependencies":["ac26263ThreeVariables12a-h2"],"title":"Calculation","text":"Subtract the first equation with the third equation: $$x+2y-z+x+3y+z=1-4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables12a-h4","type":"hint","dependencies":["ac26263ThreeVariables12a-h3"],"title":"Calculation","text":"The equation becomes $$-y-2z=-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables12a-h5","type":"hint","dependencies":["ac26263ThreeVariables12a-h4"],"title":"Subtraction","text":"Subtract $$-3y-6z=-9$$ to $$3\\\\left(-y-2z\\\\right)=3\\\\left(-3\\\\right)$$. $$\\\\left(-3y-6z\\\\right)+3\\\\left(-y-2z\\\\right)=-9-(-9)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables12a-h6","type":"hint","dependencies":["ac26263ThreeVariables12a-h5"],"title":"Subtraction","text":"The equation becomes $$0=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables12a-h7","type":"hint","dependencies":["ac26263ThreeVariables12a-h6"],"title":"Representation","text":"Rewrite $$y$$ in the variable of $$z$$ with $$-y-2z=-3$$, so $$y=-2z+3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables12a-h8","type":"hint","dependencies":["ac26263ThreeVariables12a-h7"],"title":"Substitution","text":"Substitute $$y=-2z+3$$ into the first equation to get $$x+2\\\\left(-2z+3\\\\right)-z=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables12a-h9","type":"hint","dependencies":["ac26263ThreeVariables12a-h8"],"title":"Calculation","text":"Rewrite $$x$$ in the variable of $$z$$ with the equation to get $$x=5z-5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables12a-h10","type":"hint","dependencies":["ac26263ThreeVariables12a-h9"],"title":"Principle","text":"$$z$$ can be any number, so there are infinitely many solutions","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac26263ThreeVariables13","title":"How to Solve a System of Equations With Three Variables by Elimination","body":"Solve the following system","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Solve Systems of Equations with Three Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac26263ThreeVariables13a","stepAnswer":["infinite solutions"],"problemType":"MultipleChoice","stepTitle":"Solve the system of equations: $$x+y-z=0;$$ $$2x+4y-2z=6;$$ $$3x+6y-3z=9$$","stepBody":"","answerType":"string","variabilization":{},"choices":["(1,3,4)","infinite solutions","no solution"],"hints":{"DefaultPathway":[{"id":"ac26263ThreeVariables13a-h1","type":"hint","dependencies":[],"title":"Calculation","text":"Multiply the first equation by $$2$$ and subtract with the second equation: $$2\\\\left(x+y-z\\\\right)-2x+4y-2z=2\\\\times0-6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables13a-h2","type":"hint","dependencies":["ac26263ThreeVariables13a-h1"],"title":"Calculation","text":"The equation becomes $$-2y=-6;$$ $$y=3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables13a-h3","type":"hint","dependencies":["ac26263ThreeVariables13a-h2"],"title":"Substitution","text":"Substitute $$y=3$$ into the first equation to get $$x-z=-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables13a-h4","type":"hint","dependencies":["ac26263ThreeVariables13a-h3"],"title":"Substitution","text":"Substitute $$y=3$$ into the third equation to get $$3x-3z=-9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables13a-h5","type":"hint","dependencies":["ac26263ThreeVariables13a-h4"],"title":"Calculation","text":"Subtract $$3x-3z=-9$$ with $$3\\\\left(x-z\\\\right)=3\\\\left(-3\\\\right)$$. $$3x-3z-3\\\\left(x-z\\\\right)=-9-(-9)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables13a-h6","type":"hint","dependencies":["ac26263ThreeVariables13a-h5"],"title":"Subtraction","text":"The equation becomes $$0=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables13a-h7","type":"hint","dependencies":["ac26263ThreeVariables13a-h6"],"title":"Representation","text":"Rewrite $$x$$ in the variable of $$z$$ with $$x-z=-3$$, so $$x=z-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables13a-h8","type":"hint","dependencies":["ac26263ThreeVariables13a-h7"],"title":"Principle","text":"$$z$$ can be any number, so there are infinitely many solutions","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac26263ThreeVariables14","title":"How to Solve a System of Equations With Three Variables by Elimination","body":"Solve the following system","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Solve Systems of Equations with Three Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac26263ThreeVariables14a","stepAnswer":["infinite solutions"],"problemType":"MultipleChoice","stepTitle":"Solve the system of equations: $$x-y-z=1;$$ $$-x+2y-3z=-4;$$ $$3x-2y-7z=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["(1,3,4)","infinite solutions","no solution"],"hints":{"DefaultPathway":[{"id":"ac26263ThreeVariables14a-h1","type":"hint","dependencies":[],"title":"Calculation","text":"Multiply the first equation by $$3$$ and subtract with the third equation: $$3\\\\left(x-y-z\\\\right)-3x-2y-7z=2\\\\times1-0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables14a-h2","type":"hint","dependencies":["ac26263ThreeVariables14a-h1"],"title":"Calculation","text":"The equation becomes $$-y+4z=3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables14a-h3","type":"hint","dependencies":["ac26263ThreeVariables14a-h2"],"title":"Addition","text":"Add the first equation to the second equation: $$x-y-z+\\\\left(-x+2y-3z\\\\right)=1+\\\\left(-4\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables14a-h4","type":"hint","dependencies":["ac26263ThreeVariables14a-h3"],"title":"Calculation","text":"The equation becomes $$y-4z=-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables14a-h5","type":"hint","dependencies":["ac26263ThreeVariables14a-h4"],"title":"Calculation","text":"Add $$-y+4z=3$$ to $$y-4z=-3$$. $$\\\\left(-y+4z\\\\right)+y-4z=-3+\\\\left(-3\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables14a-h6","type":"hint","dependencies":["ac26263ThreeVariables14a-h5"],"title":"Subtraction","text":"The equation becomes $$0=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables14a-h7","type":"hint","dependencies":["ac26263ThreeVariables14a-h6"],"title":"Representation","text":"Rewrite $$y$$ in the variable of $$z$$ with $$y-4z=-3$$, so $$y=4z-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables14a-h8","type":"hint","dependencies":["ac26263ThreeVariables14a-h7"],"title":"Substitution","text":"Substitute $$y=4z-3$$ into the first equation to get $$x-(4z-3)-z=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables14a-h9","type":"hint","dependencies":["ac26263ThreeVariables14a-h8"],"title":"Calculation","text":"Rewrite $$x$$ in the variable of $$z$$ with the equation to get $$x=5z-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables14a-h10","type":"hint","dependencies":["ac26263ThreeVariables14a-h9"],"title":"Principle","text":"$$z$$ can be any number, so there are infinitely many solutions","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac26263ThreeVariables15","title":"Solve Applications using Systems of Linear Equations with Three Variables","body":"Solve the following word problem:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Solve Systems of Equations with Three Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac26263ThreeVariables15a","stepAnswer":["$$75$$ adult tickets, $$150$$ student tickets, $$25$$ child tickets"],"problemType":"MultipleChoice","stepTitle":"The community college theater department sold three kinds of tickets to its latest play production. The adult tickets sold for $15, the student tickets for $10 and the child tickets for $8. The theater department was thrilled to have sold $$250$$ tickets and brought in $2,825 in one night. The number of student tickets sold is twice the number of adult tickets sold. How many of each type did the department sell?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$75$$ adult tickets, $$150$$ student tickets, $$25$$ child tickets","choices":["$$80$$ adult tickets, $$150$$ student tickets, $$20$$ child tickets","$$75$$ adult tickets, $$150$$ student tickets, $$25$$ child tickets","$$65$$ adult tickets, $$145$$ student tickets, $$35$$ child tickets"],"hints":{"DefaultPathway":[{"id":"ac26263ThreeVariables15a-h1","type":"hint","dependencies":[],"title":"Setup","text":"Assume the number of adult tickets is x; the number of student ticket is y; the number of child ticket is $$z$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables15a-h2","type":"hint","dependencies":["ac26263ThreeVariables15a-h1"],"title":"Setup","text":"The theater had sold $$250$$ tickets in total, so $$x+y+z=250$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables15a-h3","type":"hint","dependencies":["ac26263ThreeVariables15a-h2"],"title":"Setup","text":"The total income is 2,825, so $$15x+10y+8z=2825$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables15a-h4","type":"hint","dependencies":["ac26263ThreeVariables15a-h3"],"title":"Setup","text":"The number of student tickets sold is twice the number of adult ticket sold, so $$y=2x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables15a-h5","type":"hint","dependencies":["ac26263ThreeVariables15a-h4"],"title":"Substitution","text":"Substitute $$y=2x$$ into the other two equations to get $$x+2x+z=250$$ and $$15x+10\\\\times2x+8z=2825$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables15a-h6","type":"hint","dependencies":["ac26263ThreeVariables15a-h5"],"title":"Subtraction","text":"Multiply the substituted first equation by $$8$$ and subtract with the substituted second equation: $$8\\\\left(3x+z\\\\right)-35x+8z=8\\\\times250-2825$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables15a-h7","type":"hint","dependencies":["ac26263ThreeVariables15a-h6"],"title":"Simplifying the equation","text":"The equation becomes $$-11x=-825;$$ $$x=75$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables15a-h8","type":"hint","dependencies":["ac26263ThreeVariables15a-h7"],"title":"Finding values","text":"Find the value of $$y$$ from the equation of $$y=2x$$ and $$x=75$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables15a-h9","type":"hint","dependencies":["ac26263ThreeVariables15a-h8"],"title":"Finding values","text":"Find the value of $$z$$ from the equation of $$x+y+z=250$$ with the known values of $$x$$ and $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac26263ThreeVariables2","title":"Linear Equation in Three Variables","body":"Determine whether the ordered triple is a solution to the system: $$3x+y+z=2;$$ $$x+2y+z=-3;$$ $$3x+y+2z=4$$","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Solve Systems of Equations with Three Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac26263ThreeVariables2a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"a) $$(1, -3, 2)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ac26263ThreeVariables2a-h1","type":"hint","dependencies":[],"title":"Principle","text":"Translate $$(1, -3, 2)$$ to $$x=1$$, $$y=-3$$, $$z=2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables2a-h2","type":"hint","dependencies":["ac26263ThreeVariables2a-h1"],"title":"Substitution","text":"Substitute the given value to the first equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ac26263ThreeVariables2a-h2"],"title":"Substitute","text":"What is $$3\\\\times1+\\\\left(-3\\\\right)+2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables2a-h4","type":"hint","dependencies":["ac26263ThreeVariables2a-h3"],"title":"Comparison","text":"The substituted answer matches the given answer so it is the solution to the first equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables2a-h5","type":"hint","dependencies":["ac26263ThreeVariables2a-h4"],"title":"Substitution","text":"Substitute the given value to the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["ac26263ThreeVariables2a-h5"],"title":"Substitute","text":"What is $$1+2\\\\left(-3\\\\right)+2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables2a-h7","type":"hint","dependencies":["ac26263ThreeVariables2a-h6"],"title":"Comparison","text":"The substituted answer matches the given answer so it is the solution to the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables2a-h8","type":"hint","dependencies":["ac26263ThreeVariables2a-h7"],"title":"Substitution","text":"Substitute the given value to the third equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables2a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ac26263ThreeVariables2a-h8"],"title":"Substitute","text":"What is $$3\\\\times1+\\\\left(-3\\\\right)+2\\\\times2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables2a-h10","type":"hint","dependencies":["ac26263ThreeVariables2a-h9"],"title":"Comparison","text":"The substituted answer matches the given answer so it is the solution to the third equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables2a-h11","type":"hint","dependencies":["ac26263ThreeVariables2a-h4","ac26263ThreeVariables2a-h7","ac26263ThreeVariables2a-h10"],"title":"Conclusion","text":"$$(1, -3, 2)$$ is the solution to all three equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ac26263ThreeVariables2b","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"b) $$(4, -1, -5)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ac26263ThreeVariables2b-h1","type":"hint","dependencies":[],"title":"Principle","text":"Translate $$(4, -1, -5)$$ to $$x=4$$, $$y=-1$$, $$z=-5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables2b-h2","type":"hint","dependencies":["ac26263ThreeVariables2b-h1"],"title":"Substitution","text":"Substitute the given value to the first equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables2b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["ac26263ThreeVariables2b-h2"],"title":"Substitute","text":"What is $$3\\\\times4+\\\\left(-1\\\\right)+\\\\left(-5\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables2b-h4","type":"hint","dependencies":["ac26263ThreeVariables2b-h3"],"title":"Comparison","text":"The substituted answer does not match the given answer so it is the solution to the first equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables2b-h5","type":"hint","dependencies":["ac26263ThreeVariables2b-h4"],"title":"Substitution","text":"Substitute the given value to the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables2b-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["ac26263ThreeVariables2b-h5"],"title":"Substitute","text":"What is $$4+2\\\\left(-1\\\\right)+\\\\left(-5\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables2b-h7","type":"hint","dependencies":["ac26263ThreeVariables2b-h6"],"title":"Comparison","text":"The substituted answer matches the given answer so it is the solution to the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables2b-h8","type":"hint","dependencies":["ac26263ThreeVariables2b-h7"],"title":"Substitution","text":"Substitute the given value to the third equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables2b-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ac26263ThreeVariables2b-h8"],"title":"Substitute","text":"What is $$3\\\\times4+\\\\left(-1\\\\right)+2\\\\left(-5\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables2b-h10","type":"hint","dependencies":["ac26263ThreeVariables2b-h9"],"title":"Comparison","text":"The substituted answer does not match the given answer so it is the solution to the third equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables2b-h11","type":"hint","dependencies":["ac26263ThreeVariables2b-h4","ac26263ThreeVariables2b-h7","ac26263ThreeVariables2b-h10"],"title":"Conclusion","text":"$$(-2, -1, 3)$$ is not the solution to all three equations","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac26263ThreeVariables3","title":"Linear Equation in Three Variables","body":"Determine whether the ordered triple is a solution to the system: $$x-3y+z=-5;$$ $$-3x-y-z=1;$$ $$2x-2y+3z=1$$","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Solve Systems of Equations with Three Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac26263ThreeVariables3a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"a) $$(2, -2, 3)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ac26263ThreeVariables3a-h1","type":"hint","dependencies":[],"title":"Principle","text":"Translate $$(2, -2, 3)$$ to $$x=2$$, $$y=-2$$, $$z=3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables3a-h2","type":"hint","dependencies":["ac26263ThreeVariables3a-h1"],"title":"Substitution","text":"Substitute the given value to the first equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11$$"],"dependencies":["ac26263ThreeVariables3a-h2"],"title":"Substitute","text":"What is $$2-3\\\\left(-2\\\\right)+3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables3a-h4","type":"hint","dependencies":["ac26263ThreeVariables3a-h3"],"title":"Comparison","text":"The substituted answer does not match the given answer so it is the solution to the first equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables3a-h5","type":"hint","dependencies":["ac26263ThreeVariables3a-h4"],"title":"Substitution","text":"Substitute the given value to the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables3a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-7$$"],"dependencies":["ac26263ThreeVariables3a-h5"],"title":"Substitute","text":"What is $$-3\\\\times2-\\\\left(-2\\\\right)-3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables3a-h7","type":"hint","dependencies":["ac26263ThreeVariables3a-h6"],"title":"Comparison","text":"The substituted answer does not match the given answer so it is the solution to the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables3a-h8","type":"hint","dependencies":["ac26263ThreeVariables3a-h7"],"title":"Substitution","text":"Substitute the given value to the third equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables3a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$17$$"],"dependencies":["ac26263ThreeVariables3a-h8"],"title":"Substitute","text":"What is $$2\\\\times2-2\\\\left(-2\\\\right)+3\\\\times3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables3a-h10","type":"hint","dependencies":["ac26263ThreeVariables3a-h9"],"title":"Comparison","text":"The substituted answer does not match the given answer so it is the solution to the third equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables3a-h11","type":"hint","dependencies":["ac26263ThreeVariables3a-h4","ac26263ThreeVariables3a-h7","ac26263ThreeVariables3a-h10"],"title":"Conclusion","text":"$$(2, -2, 3)$$ is not the solution to all three equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ac26263ThreeVariables3b","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"b) $$(-2, 2, 3)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ac26263ThreeVariables3b-h1","type":"hint","dependencies":[],"title":"Principle","text":"Translate $$(-2, 2, 3)$$ to $$x=-2$$, $$y=2$$, $$z=3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables3b-h2","type":"hint","dependencies":["ac26263ThreeVariables3b-h1"],"title":"Substitution","text":"Substitute the given value to the first equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables3b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["ac26263ThreeVariables3b-h2"],"title":"Substitute","text":"What is $$-2-3\\\\times2+3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables3b-h4","type":"hint","dependencies":["ac26263ThreeVariables3b-h3"],"title":"Comparison","text":"The substituted answer matches the given answer so it is the solution to the first equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables3b-h5","type":"hint","dependencies":["ac26263ThreeVariables3b-h4"],"title":"Substitution","text":"Substitute the given value to the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables3b-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["ac26263ThreeVariables3b-h5"],"title":"Substitute","text":"What is $$-3\\\\left(-2\\\\right)-2-3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables3b-h7","type":"hint","dependencies":["ac26263ThreeVariables3b-h6"],"title":"Comparison","text":"The substituted answer matches the given answer so it is the solution to the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables3b-h8","type":"hint","dependencies":["ac26263ThreeVariables3b-h7"],"title":"Substitution","text":"Substitute the given value to the third equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables3b-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ac26263ThreeVariables3b-h8"],"title":"Substitute","text":"What is $$2\\\\left(-2\\\\right)-2\\\\times2+3\\\\times3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables3b-h10","type":"hint","dependencies":["ac26263ThreeVariables3b-h9"],"title":"Comparison","text":"The substituted answer matches the given answer so it is the solution to the third equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables3b-h11","type":"hint","dependencies":["ac26263ThreeVariables3b-h4","ac26263ThreeVariables3b-h7","ac26263ThreeVariables3b-h10"],"title":"Conclusion","text":"$$(-2, 2, 3)$$ is not the solution to all three equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac26263ThreeVariables4","title":"How to Solve a System of Equations With Three Variables by Elimination","body":"Solve the following system.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Solve Systems of Equations with Three Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac26263ThreeVariables4a","stepAnswer":["$$(4, -1, -3)$$"],"problemType":"MultipleChoice","stepTitle":"Solve the system by elimination: $$x-2y+z=3;$$ $$2x+y+z=4;$$ $$3x+4y+3z=-1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(4, -1, -3)$$","choices":["$$(-1, 4, -3)$$","$$(4, -1, -3)$$","$$(-3, -1, 4)$$","$$(-1, -3, 4)$$"],"hints":{"DefaultPathway":[{"id":"ac26263ThreeVariables4a-h1","type":"hint","dependencies":[],"title":"Multiplying","text":"Multiply the first equation by $$2$$ and subtract the second equation: $$2\\\\left(x-2y+z\\\\right)-2x+y+z=2\\\\times3-4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables4a-h2","type":"hint","dependencies":["ac26263ThreeVariables4a-h1"],"title":"Calculating","text":"The equation becomes $$-5y+z=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables4a-h3","type":"hint","dependencies":["ac26263ThreeVariables4a-h2"],"title":"Multiplying","text":"Multiply the first equation by $$3$$ and subtract with the third equation: $$3\\\\left(x-2y+z\\\\right)-3x+4y+3z=3\\\\times3-\\\\left(-1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables4a-h4","type":"hint","dependencies":["ac26263ThreeVariables4a-h3"],"title":"Simplifying the equation","text":"The equation becomes $$-10y=10$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables4a-h5","type":"hint","dependencies":["ac26263ThreeVariables4a-h4"],"title":"Division","text":"$$y=-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables4a-h6","type":"hint","dependencies":["ac26263ThreeVariables4a-h5"],"title":"Substituting","text":"Substitute $$y=-1$$ into $$-5y+2z=2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables4a-h7","type":"hint","dependencies":["ac26263ThreeVariables4a-h6"],"title":"Simplifying the equation","text":"The equation becomes $$z=-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables4a-h8","type":"hint","dependencies":["ac26263ThreeVariables4a-h5","ac26263ThreeVariables4a-h7"],"title":"Substituting","text":"Substitute $$y=-1$$ and $$z=-3$$ into the first equation to get $$x-2\\\\left(-1\\\\right)-3=3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables4a-h9","type":"hint","dependencies":["ac26263ThreeVariables4a-h8"],"title":"Conclusion","text":"The equation becomes $$x=4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac26263ThreeVariables5","title":"How to Solve a System of Equations With Three Variables by Elimination","body":"Solve the following system:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Solve Systems of Equations with Three Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac26263ThreeVariables5a","stepAnswer":["$$(-2, 3, 4)$$"],"problemType":"MultipleChoice","stepTitle":"Solve the system by elimination: $$4x+y+z=-1;$$ $$-2x-2y+z=2;$$ $$2x+3y-z=1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-2, 3, 4)$$","choices":["$$(-2, 4, 3)$$","$$(4, -2, 3)$$","$$(-2, 3, 4)$$","$$(3, -2, 4)$$"],"hints":{"DefaultPathway":[{"id":"ac26263ThreeVariables5a-h1","type":"hint","dependencies":[],"title":"Calculation","text":"Subtract the first equation with the second equation: $$4x+y+z-\\\\left(-2x-2y+z\\\\right)=-1-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables5a-h2","type":"hint","dependencies":["ac26263ThreeVariables5a-h1"],"title":"Calculating","text":"The equation becomes $$6x+3y=-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables5a-h3","type":"hint","dependencies":["ac26263ThreeVariables5a-h2"],"title":"Calculating","text":"Combine the first equation with the third equation: $$4x+y+z+2x+3y-z=\\\\left(-1\\\\right)+1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables5a-h4","type":"hint","dependencies":["ac26263ThreeVariables5a-h3"],"title":"Calculating","text":"The equation becomes $$6x+4y=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables5a-h5","type":"hint","dependencies":["ac26263ThreeVariables5a-h4"],"title":"Subtracting","text":"Subtract $$6x+4y=0$$ with $$6x+3y=-3$$. $$6x+4y-6x+3y=0-(-3)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables5a-h6","type":"hint","dependencies":["ac26263ThreeVariables5a-h5"],"title":"Simplifying the equation","text":"The equation becomes $$y=3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables5a-h7","type":"hint","dependencies":["ac26263ThreeVariables5a-h6"],"title":"Simplifying the equation","text":"Substitute $$y=3$$ into $$6x+3y=-3$$ and get $$x=-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables5a-h8","type":"hint","dependencies":["ac26263ThreeVariables5a-h7"],"title":"Simplifying the equation","text":"Substitute $$x=-2$$ and $$y=3$$ into the first equation to get $$z=4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac26263ThreeVariables6","title":"Solve a system of linear equations with three variables","body":"Solve the following system","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Solve Systems of Equations with Three Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac26263ThreeVariables6a","stepAnswer":["$$(-4, 1, -3)$$"],"problemType":"MultipleChoice","stepTitle":"Solve $$3x-4z=0;$$ $$3y+2z=-3;$$ $$2x+3y=-5$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-4, 1, -3)$$","choices":["$$(-4, 1, -3)$$","$$(-3, 1, -4)$$","$$(3, 1, -4)$$"],"hints":{"DefaultPathway":[{"id":"ac26263ThreeVariables6a-h1","type":"hint","dependencies":[],"title":"Calculation","text":"Multiply the second equation by $$2$$ and add the first equation to it: $$2\\\\left(3y+2z\\\\right)+3x-4z=2\\\\left(-3\\\\right)+0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables6a-h2","type":"hint","dependencies":["ac26263ThreeVariables6a-h1"],"title":"Calculating","text":"The equation becomes $$3x+6y=-6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables6a-h3","type":"hint","dependencies":["ac26263ThreeVariables6a-h2"],"title":"Calculating","text":"Multiply the third equation by $$2$$ and subtract with $$3x+6y=6$$: $$2\\\\left(2x+3y\\\\right)-3x+6y=2\\\\left(-5\\\\right)-\\\\left(-6\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables6a-h4","type":"hint","dependencies":["ac26263ThreeVariables6a-h3"],"title":"Calculating","text":"The equation becomes $$x=-4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables6a-h5","type":"hint","dependencies":["ac26263ThreeVariables6a-h4"],"title":"Substituting","text":"Substitute $$x=-4$$ to the first equation and get $$z=-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables6a-h6","type":"hint","dependencies":["ac26263ThreeVariables6a-h5"],"title":"Substitution","text":"Substitute $$x=-4$$ to the third equation and get $$y=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac26263ThreeVariables7","title":"Solve a system of linear equations with three variables","body":"Solve the following system","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Solve Systems of Equations with Three Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac26263ThreeVariables7a","stepAnswer":["$$(-2, 3, -1)$$"],"problemType":"MultipleChoice","stepTitle":"Solve $$4x-3z=-5;$$ $$3y+2z=7;$$ $$3x+4y=6$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-2, 3, -1)$$","choices":["$$(-1, 3, -2)$$","$$(-2, 3, -1)$$","$$(3, 1, -2)$$"],"hints":{"DefaultPathway":[{"id":"ac26263ThreeVariables7a-h1","type":"hint","dependencies":[],"title":"Calculating","text":"Multiply the first equation by $$2$$, multiply the second equation by $$3$$, and add them together: $$2\\\\left(4x-3z\\\\right)+3\\\\left(3y+2z\\\\right)=2\\\\left(-5\\\\right)+3\\\\times7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables7a-h2","type":"hint","dependencies":["ac26263ThreeVariables7a-h1"],"title":"Calculating","text":"The equation becomes $$8x+9y=11$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables7a-h3","type":"hint","dependencies":["ac26263ThreeVariables7a-h2"],"title":"Calculating","text":"Multiply the third equation by $$8$$ and subtract with $$3\\\\left(8x+9y\\\\right)=3\\\\times11$$: $$8\\\\left(3x+4y\\\\right)-3\\\\left(8x+9y\\\\right)=8\\\\times6-3\\\\times11$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables7a-h4","type":"hint","dependencies":["ac26263ThreeVariables7a-h3"],"title":"Calculation","text":"The equation becomes $$5y=15;$$ $$y=3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables7a-h5","type":"hint","dependencies":["ac26263ThreeVariables7a-h4"],"title":"Substitution","text":"Substitute $$y=3$$ to the second equation and get $$z=-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables7a-h6","type":"hint","dependencies":["ac26263ThreeVariables7a-h5"],"title":"Substitution","text":"Substitute $$y=3$$ to the third equation and get $$x=-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac26263ThreeVariables8","title":"Solve a system of linear equations with three variables","body":"Solve the following system","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Solve Systems of Equations with Three Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac26263ThreeVariables8a","stepAnswer":["$$(-3, 4, -2)$$"],"problemType":"MultipleChoice","stepTitle":"Solve $$3x-4z=-1;$$ $$2y+3z=2;$$ $$2x+3y=6$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-3, 4, -2)$$","choices":["$$(-3, 4, -2)$$","$$(4, -2, -3)$$","$$(3, -2, 4)$$"],"hints":{"DefaultPathway":[{"id":"ac26263ThreeVariables8a-h1","type":"hint","dependencies":[],"title":"Calculation","text":"Multiply the first equation by $$3$$, multiply the second equation by $$4$$, and add them together: $$3\\\\left(3x-4z\\\\right)+4\\\\left(2y+3z\\\\right)=3\\\\left(-1\\\\right)+4\\\\times2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables8a-h2","type":"hint","dependencies":["ac26263ThreeVariables8a-h1"],"title":"Calculation","text":"The equation becomes $$9x+8y=5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables8a-h3","type":"hint","dependencies":["ac26263ThreeVariables8a-h2"],"title":"Calculation","text":"Multiply the third equation by $$9$$ and subtract with $$2\\\\left(9x+8y\\\\right)=2\\\\times5$$: $$9\\\\left(2x+3y\\\\right)-2\\\\left(9x+8y\\\\right)=9\\\\times6-2\\\\times5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables8a-h4","type":"hint","dependencies":["ac26263ThreeVariables8a-h3"],"title":"Calculation","text":"The equation becomes $$11y=44;$$ $$y=4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables8a-h5","type":"hint","dependencies":["ac26263ThreeVariables8a-h4"],"title":"Substitution","text":"Substitute $$y=4$$ to the second equation and get $$z=-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables8a-h6","type":"hint","dependencies":["ac26263ThreeVariables8a-h5"],"title":"Substitution","text":"Substitute $$y=4$$ to the third equation and get $$x=-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac26263ThreeVariables9","title":"How to Solve a System of Equations With Three Variables by Elimination","body":"Solve the following system","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.4 Solve Systems of Equations with Three Variables","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac26263ThreeVariables9a","stepAnswer":["no solution"],"problemType":"MultipleChoice","stepTitle":"Solve the system of equations: $$x+2y-3z=-1;$$ $$x-3y+z=1;$$ $$2x-y-2z=2$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$(-2, 4, 3)$$","$$(4, -6, 5)$$","no solution"],"hints":{"DefaultPathway":[{"id":"ac26263ThreeVariables9a-h1","type":"hint","dependencies":[],"title":"Calculation","text":"Multiply the second equation by $$3$$ and add the first equation to it: $$3\\\\left(x-3y+z\\\\right)+x+2y-3z=3\\\\times1-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables9a-h2","type":"hint","dependencies":["ac26263ThreeVariables9a-h1"],"title":"Calculation","text":"The equation becomes $$4x-7y=2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables9a-h3","type":"hint","dependencies":["ac26263ThreeVariables9a-h2"],"title":"Calculation","text":"Multiply the second equation by $$2$$ and add the third equation to it: $$2\\\\left(x-3y+z\\\\right)+2x+y-2z=2\\\\times1+2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables9a-h4","type":"hint","dependencies":["ac26263ThreeVariables9a-h3"],"title":"Calculation","text":"The equation becomes $$4x-7y=4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables9a-h5","type":"hint","dependencies":["ac26263ThreeVariables9a-h4"],"title":"Subtraction","text":"Subtract $$4x-7y=2$$ with $$4x-7y=4$$. $$(4x-7y)-(4x-7y)=2-4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac26263ThreeVariables9a-h6","type":"hint","dependencies":["ac26263ThreeVariables9a-h5"],"title":"Subtraction","text":"The equation becomes $$0=-2$$, which is a false statement","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac3d94dProperties1","title":"In the following exercise, use the associative property to simplify.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.9 Properties of Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac3d94dProperties1a","stepAnswer":["$$12x$$"],"problemType":"TextBox","stepTitle":"$$3\\\\times4x$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12x$$","hints":{"DefaultPathway":[{"id":"ac3d94dProperties1a-h1","type":"hint","dependencies":[],"title":"Associative Property of Multiplication","text":"The associative property of multiplication states that if a, $$b$$, c are real numbers, then $$a b c=a b c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties1a-h2","type":"hint","dependencies":["ac3d94dProperties1a-h1"],"title":"Regroup","text":"Using the associative property, we can regroup the terms in the expression into $$3\\\\times4 x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["ac3d94dProperties1a-h2"],"title":"Multiplication","text":"What is $$3\\\\times4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties1a-h4","type":"hint","dependencies":["ac3d94dProperties1a-h3"],"title":"Final Answer","text":"So the final answer is $$12x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac3d94dProperties10","title":"Use the Properties of Zero","body":"Simplify the following exercise.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.9 Properties of Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac3d94dProperties10a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"$$0\\\\left(-3.14\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"ac3d94dProperties10a-h1","type":"hint","dependencies":[],"title":"Multiplication by zero","text":"Recall that for any real number a, $$0a=0$$, $$0a=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties10a-h2","type":"hint","dependencies":["ac3d94dProperties10a-h1"],"title":"Answer","text":"Therefore, $$0\\\\left(-3.14\\\\right)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac3d94dProperties11","title":"Use the Properties of Zero","body":"Simplify the following exercise.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.9 Properties of Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac3d94dProperties11a","stepAnswer":["undefined"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{32-5a}{0}$$, where $$32-5a \\\\neq 0$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$0$$","undefined","cannot solve for the answer without knowing a"],"hints":{"DefaultPathway":[{"id":"ac3d94dProperties11a-h1","type":"hint","dependencies":[],"title":"Division by zero","text":"Recall that for any real number a, except $$0$$, $$\\\\frac{a}{0}$$ is undefined.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties11a-h2","type":"hint","dependencies":["ac3d94dProperties11a-h1"],"title":"Answer","text":"Since $$32-5a$$ is a nonzero real number, we can apply this property and say that the answer is undefined.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac3d94dProperties12","title":"Simplify the following expression.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.9 Properties of Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac3d94dProperties12a","stepAnswer":["$$36d+90$$"],"problemType":"TextBox","stepTitle":"$$15\\\\frac{3}{5} \\\\left(4d+10\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$36d+90$$","hints":{"DefaultPathway":[{"id":"ac3d94dProperties12a-h1","type":"hint","dependencies":[],"title":"Multiplying numbers","text":"The first step is to multiply $$15$$ and $$\\\\frac{3}{5}$$ so that later on we can distribute this number into the parenthesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["ac3d94dProperties12a-h1"],"title":"Multiplying numbers","text":"What is $$15\\\\frac{3}{5}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties12a-h3","type":"hint","dependencies":["ac3d94dProperties12a-h2"],"title":"Distributive Property","text":"The distributive property states that if a, $$b$$, c are real numbers, then $$a \\\\left(b+c\\\\right)=ab+ac$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties12a-h4","type":"hint","dependencies":["ac3d94dProperties12a-h3"],"title":"Applying the Distributive Property","text":"Applying the distributive property on $$9\\\\left(4d+10\\\\right)$$, we get the expression $$9\\\\times4 d+9\\\\times10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties12a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$36d$$"],"dependencies":["ac3d94dProperties12a-h4"],"title":"Multiplying","text":"What is $$9\\\\times4 d$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties12a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$90$$"],"dependencies":["ac3d94dProperties12a-h4"],"title":"Multiplying","text":"What is $$9\\\\times10$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties12a-h7","type":"hint","dependencies":["ac3d94dProperties12a-h5","ac3d94dProperties12a-h6"],"title":"Final Answer","text":"Therefore, our final answer is $$36d+90$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac3d94dProperties13","title":"Simplify the following expression using the distributive property.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.9 Properties of Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac3d94dProperties13a","stepAnswer":["$$32y+72$$"],"problemType":"TextBox","stepTitle":"$$8\\\\left(4y+9\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$32y+72$$","hints":{"DefaultPathway":[{"id":"ac3d94dProperties13a-h1","type":"hint","dependencies":[],"title":"Distributive Property","text":"The distributive property states that if a, $$b$$, c are real numbers, then $$a \\\\left(b+c\\\\right)=ab+ac$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties13a-h2","type":"hint","dependencies":["ac3d94dProperties13a-h1"],"title":"Applying the Distributive Property","text":"Applying the distributive property, we get the expression $$8\\\\times4 y+8\\\\times9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties13a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$32y$$"],"dependencies":["ac3d94dProperties13a-h2"],"title":"Multiplying","text":"What is $$8\\\\times4 y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$72$$"],"dependencies":["ac3d94dProperties13a-h2"],"title":"Multiplying","text":"What is $$8\\\\times9$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties13a-h5","type":"hint","dependencies":["ac3d94dProperties13a-h3","ac3d94dProperties13a-h4"],"title":"Final Answer","text":"Putting them together, our final answer is $$32y+72$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac3d94dProperties14","title":"Simplify the following expression using the distributive property.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.9 Properties of Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac3d94dProperties14a","stepAnswer":["$$22n+9$$"],"problemType":"TextBox","stepTitle":"$$5\\\\left(2n+9\\\\right)+12\\\\left(n-3\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$22n+9$$","hints":{"DefaultPathway":[{"id":"ac3d94dProperties14a-h1","type":"hint","dependencies":[],"title":"Distributive Property","text":"Observing this question, we find that we will need to apply the distributive property twice, once for each of the parentheses.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties14a-h2","type":"hint","dependencies":["ac3d94dProperties14a-h1"],"title":"Distributive Property","text":"The distributive property states that if a, $$b$$, c are real numbers, then $$a \\\\left(b+c\\\\right)=ab+ac$$, and $$a \\\\left(b-c\\\\right)=ab-ac$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties14a-h3","type":"hint","dependencies":["ac3d94dProperties14a-h2"],"title":"Applying the Distributive Property","text":"Applying the distributive property to $$5\\\\left(2n+9\\\\right)$$, we get the expression $$5\\\\times2 n+5\\\\times9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties14a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10n$$"],"dependencies":["ac3d94dProperties14a-h3"],"title":"Multiplying","text":"What is $$5\\\\times2 n$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties14a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$45$$"],"dependencies":["ac3d94dProperties14a-h3"],"title":"Multiplying","text":"What is $$5\\\\times9$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties14a-h6","type":"hint","dependencies":["ac3d94dProperties14a-h4","ac3d94dProperties14a-h5"],"title":"First Expression","text":"Putting together the previous two parts, we get $$5\\\\left(2n+9\\\\right)=10n+45$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties14a-h7","type":"hint","dependencies":["ac3d94dProperties14a-h2"],"title":"Applying the Distributive Property","text":"Applying the distributive property to $$12\\\\left(n-3\\\\right)$$, we get the expression $$12n-12\\\\times3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties14a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$36$$"],"dependencies":["ac3d94dProperties14a-h7"],"title":"Multiplying","text":"What is $$12\\\\times3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties14a-h9","type":"hint","dependencies":["ac3d94dProperties14a-h8"],"title":"Second Expression","text":"Putting together the previous parts, we get $$12\\\\left(n-3\\\\right)=12n-36$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties14a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$22n+9$$"],"dependencies":["ac3d94dProperties14a-h6","ac3d94dProperties14a-h9"],"title":"Combine Like Terms","text":"After applying the distributive property, the expression becomes $$10n+45+12n-36$$. Combine like terms, what do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac3d94dProperties15","title":"Simplify the following expression using the distributive property.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.9 Properties of Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac3d94dProperties15a","stepAnswer":["$$12y+63$$"],"problemType":"TextBox","stepTitle":"$$6\\\\left(7y+8\\\\right)-30y-15$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12y+63$$","hints":{"DefaultPathway":[{"id":"ac3d94dProperties15a-h1","type":"hint","dependencies":[],"title":"Distributive Property","text":"Observing this question, we find that we will need to apply the distributive property twice, once for each of the parentheses.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties15a-h2","type":"hint","dependencies":["ac3d94dProperties15a-h1"],"title":"Distributive Property","text":"The distributive property states that if a, $$b$$, c are real numbers, then $$a \\\\left(b+c\\\\right)=ab+ac$$, and $$a \\\\left(b-c\\\\right)=ab-ac$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties15a-h3","type":"hint","dependencies":["ac3d94dProperties15a-h2"],"title":"Applying the Distributive Property","text":"Applying the distributive property to $$6\\\\left(7y+8\\\\right)$$, we get the expression $$6\\\\times7 y+6\\\\times8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$42y$$"],"dependencies":["ac3d94dProperties15a-h3"],"title":"Multiplying","text":"What is $$6\\\\times7 y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties15a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$48$$"],"dependencies":["ac3d94dProperties15a-h3"],"title":"Multiplying","text":"What is $$6\\\\times8$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties15a-h6","type":"hint","dependencies":["ac3d94dProperties15a-h4","ac3d94dProperties15a-h5"],"title":"First Expression","text":"Putting together the previous two parts, we get $$6\\\\left(7y+8\\\\right)=42y+48$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties15a-h7","type":"hint","dependencies":["ac3d94dProperties15a-h2"],"title":"Applying the Distributive Property","text":"Applying the distributive property to $$-(30y-15)$$, we get the expression $$-1\\\\times30 y-15\\\\left(-1\\\\right)$$ (remember that $$-(30y-15)$$ can be written as (-1)*(30y-15)).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties15a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-30y$$"],"dependencies":["ac3d94dProperties15a-h7"],"title":"Multiplying","text":"What is $$30\\\\left(-1\\\\right) y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties15a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-15$$"],"dependencies":["ac3d94dProperties15a-h7"],"title":"Multiplying","text":"What is $$15\\\\left(-1\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties15a-h10","type":"hint","dependencies":["ac3d94dProperties15a-h8","ac3d94dProperties15a-h9"],"title":"Second Expression","text":"Putting together the previous two parts, we get $$-(30y-15)=-30y-(-15)=-30y+15$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties15a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12y+63$$"],"dependencies":["ac3d94dProperties15a-h6","ac3d94dProperties15a-h10"],"title":"Combine Like Terms","text":"After applying the distributive property, the expression becomes $$42y+48-30y+15$$. Combine like terms, what do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac3d94dProperties16","title":"Use the Commutative Property","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.9 Properties of Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac3d94dProperties16a","stepAnswer":["$$33p+11q$$"],"problemType":"TextBox","stepTitle":"$$18p+6q+15p+5q$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$33p+11q$$","hints":{"DefaultPathway":[{"id":"ac3d94dProperties16a-h1","type":"hint","dependencies":[],"title":"Commutative Property of Addition","text":"We need to use the commutative property of addition to re-order so that like terms are together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18p+15p+6q+5q$$"],"dependencies":["ac3d94dProperties16a-h1"],"title":"Re-order","text":"What do we get after re-ordering?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties16a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$33p+11q$$"],"dependencies":["ac3d94dProperties16a-h2"],"title":"Addition","text":"What do we get after adding like terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac3d94dProperties17","title":"Use the Associative Property","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.9 Properties of Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac3d94dProperties17a","stepAnswer":["$$\\\\frac{18}{13}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{5}{13}+\\\\frac{3}{4}+\\\\frac{1}{4}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{18}{13}$$","hints":{"DefaultPathway":[{"id":"ac3d94dProperties17a-h1","type":"hint","dependencies":[],"title":"Change the Grouping","text":"We can notice that the last $$2$$ terms have a common denominator, so change the grouping to $$\\\\frac{5}{13}+\\\\frac{3}{4}+\\\\frac{1}{4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties17a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{13}+1$$"],"dependencies":["ac3d94dProperties17a-h1"],"title":"Addition","text":"What do we get after adding in parentheses first?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties17a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{18}{13}$$"],"dependencies":["ac3d94dProperties17a-h2"],"title":"Addition","text":"What do we get after adding the two numbers?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac3d94dProperties18","title":"Use the Associative Property","body":"Use the associative property to simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.9 Properties of Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac3d94dProperties18a","stepAnswer":["$$18x$$"],"problemType":"TextBox","stepTitle":"$$6\\\\times3x$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$18x$$","hints":{"DefaultPathway":[{"id":"ac3d94dProperties18a-h1","type":"hint","dependencies":[],"title":"Change the Grouping","text":"We should change the grouping to $$6\\\\times3 x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18x$$"],"dependencies":["ac3d94dProperties18a-h1"],"title":"Multiply","text":"What do we get after multiplying in the parentheses?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac3d94dProperties19","title":"Use the Identity and Inverse Properties of Addition","body":"Find the additive inverse of","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.9 Properties of Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac3d94dProperties19a","stepAnswer":["$$\\\\frac{-5}{8}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{5}{8}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-5}{8}$$","hints":{"DefaultPathway":[{"id":"ac3d94dProperties19a-h1","type":"hint","dependencies":[],"title":"Inverse Property of Addition","text":"For any real number a, -a is the additive inverse of a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-5}{8}$$"],"dependencies":["ac3d94dProperties19a-h1"],"title":"Additive Inverse","text":"What is the additive inverse of $$\\\\frac{5}{8}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac3d94dProperties2","title":"In the following exercise, use the associative property to simplify.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.9 Properties of Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac3d94dProperties2a","stepAnswer":["$$y+40$$"],"problemType":"TextBox","stepTitle":"$$y+12+28$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y+40$$","hints":{"DefaultPathway":[{"id":"ac3d94dProperties2a-h1","type":"hint","dependencies":[],"title":"Associative Property of Addition","text":"The associative property of addition states that if a, $$b$$, c are real numbers, then $$a+b+c=a+b+c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties2a-h2","type":"hint","dependencies":["ac3d94dProperties2a-h1"],"title":"Regroup","text":"Using the associative property, we can regroup the numeric terms together: $$y+12+28$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$40$$"],"dependencies":["ac3d94dProperties2a-h2"],"title":"Addition","text":"What is $$12+28$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties2a-h4","type":"hint","dependencies":["ac3d94dProperties2a-h3"],"title":"Final Answer","text":"So the final answer is $$y+40$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac3d94dProperties20","title":"Use the Identity and Inverse Properties of Multiplication","body":"Find the multiplicative inverse of","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.9 Properties of Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac3d94dProperties20a","stepAnswer":["$$-9$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{-1}{9}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-9$$","hints":{"DefaultPathway":[{"id":"ac3d94dProperties20a-h1","type":"hint","dependencies":[],"title":"Inverse Property of Multiplication","text":"For any real number a, $$\\\\frac{1}{a}$$ is the multiplicative inverse of a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":["ac3d94dProperties20a-h1"],"title":"Multiplicative Inverse","text":"What is the multiplicative inverse of $$\\\\frac{-1}{9}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac3d94dProperties21","title":"Use the Properties of Zero","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.9 Properties of Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac3d94dProperties21a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"$$-8\\\\times0$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"ac3d94dProperties21a-h1","type":"hint","dependencies":[],"title":"Multiplication by Zero","text":"The product of any real number and $$0$$ is $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties21a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["ac3d94dProperties21a-h1"],"title":"Multiplication by Zero","text":"What is $$-8\\\\times0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac3d94dProperties22","title":"Use the Properties of Zero","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.9 Properties of Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac3d94dProperties22a","stepAnswer":["undefined"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{10-3p}{0}$$, where $$10-3p$$ is not equal to $$0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["undefined","$$0$$"],"hints":{"DefaultPathway":[{"id":"ac3d94dProperties22a-h1","type":"hint","dependencies":[],"title":"Division by Zero","text":"Division by $$0$$ is undefined.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac3d94dProperties23","title":"Use the Commutative Property","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.9 Properties of Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac3d94dProperties23a","stepAnswer":["$$\\\\frac{8}{23}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{7}{15} \\\\frac{8}{23} \\\\frac{15}{7}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{8}{23}$$","hints":{"DefaultPathway":[{"id":"ac3d94dProperties23a-h1","type":"hint","dependencies":[],"title":"Commutative Property of Multiplication","text":"Notice that the first and third terms are reciprocals, so use the commutative property of multiplication to re-order the factors. We get $$\\\\frac{7}{15} \\\\frac{15}{7} \\\\frac{8}{23}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties23a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{8}{23}$$"],"dependencies":["ac3d94dProperties23a-h1"],"title":"Multiply","text":"What do we get after multiplying from left to right?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac3d94dProperties24","title":"Use the Identity and Inverse Properties of Multiplication","body":"\\bSimplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.9 Properties of Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac3d94dProperties24a","stepAnswer":["$$6x+12$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{3}{4} \\\\frac{4}{3} \\\\left(6x+12\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6x+12$$","hints":{"DefaultPathway":[{"id":"ac3d94dProperties24a-h1","type":"hint","dependencies":[],"title":"Multiply","text":"There is nothing to do in the parentheses, so multiply the two fractions first\u2014notice, they are reciprocals.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties24a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1\\\\left(6x+12\\\\right)$$"],"dependencies":["ac3d94dProperties24a-h1"],"title":"Multiply","text":"What do we get after multiplying the two reciprocals?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties24a-h3","type":"hint","dependencies":["ac3d94dProperties24a-h2"],"title":"Multiplicative Identity","text":"For any real number a: $$1$$ is the multiplicative identity. $$1a=a$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties24a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6x+12$$"],"dependencies":["ac3d94dProperties24a-h3"],"title":"Multiplicative Identity","text":"What will the expression be simplified to after we recognize the multiplicative identity?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac3d94dProperties25","title":"Simplify Expressions Using the Distributive Property","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.9 Properties of Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac3d94dProperties25a","stepAnswer":["$$3x+12$$"],"problemType":"TextBox","stepTitle":"$$3\\\\left(x+4\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3x+12$$","hints":{"DefaultPathway":[{"id":"ac3d94dProperties25a-h1","type":"hint","dependencies":[],"title":"Distributive Property","text":"If a, $$b$$, c are real numbers, then $$a \\\\left(b+c\\\\right)=a b+a c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties25a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x+3\\\\times4$$"],"dependencies":["ac3d94dProperties25a-h1"],"title":"Distribute","text":"What do we get after distributing?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties25a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x+12$$"],"dependencies":["ac3d94dProperties25a-h2"],"title":"Multiply","text":"What do we get after simplifying the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac3d94dProperties26","title":"Simplify Expressions Using the Distributive Property","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.9 Properties of Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac3d94dProperties26a","stepAnswer":["$$-8y-2$$"],"problemType":"TextBox","stepTitle":"$$-2\\\\left(4y+1\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-8y-2$$","hints":{"DefaultPathway":[{"id":"ac3d94dProperties26a-h1","type":"hint","dependencies":[],"title":"Distributive Property","text":"If a, $$b$$, c are real numbers, then $$a \\\\left(b+c\\\\right)=a b+a c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties26a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-2\\\\times4 y+1\\\\left(-2\\\\right)$$"],"dependencies":["ac3d94dProperties26a-h1"],"title":"Distribute","text":"What do we get after distributing?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$-2\\\\times4 y+2\\\\times1$$","$$-2\\\\times4 y+1\\\\left(-2\\\\right)$$","$$2\\\\times4 y+2\\\\times1$$"]},{"id":"ac3d94dProperties26a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-8y-2$$"],"dependencies":["ac3d94dProperties26a-h2"],"title":"Multiply","text":"What do we get after simplifying the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac3d94dProperties27","title":"Simplify Expressions Using the Distributive Property","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.9 Properties of Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac3d94dProperties27a","stepAnswer":["$$-2x+2$$"],"problemType":"TextBox","stepTitle":"$$8-2\\\\left(x+3\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-2x+2$$","hints":{"DefaultPathway":[{"id":"ac3d94dProperties27a-h1","type":"hint","dependencies":[],"title":"Order of Operations","text":"Be sure to follow the order of operations. Multiplication comes before subtraction, so we will distribute the $$2$$ first and then subtract.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties27a-h2","type":"hint","dependencies":["ac3d94dProperties27a-h1"],"title":"Distributive Property","text":"If a, $$b$$, c are real numbers, then $$a \\\\left(b+c\\\\right)=a b+a c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties27a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$8-2x-2\\\\times3$$"],"dependencies":["ac3d94dProperties27a-h2"],"title":"Distribute","text":"What do we get after distributing?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$8-2x-2\\\\times3$$","$$8-2-2\\\\times3$$","$$8-2x+2\\\\times3$$"]},{"id":"ac3d94dProperties27a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2x+2$$"],"dependencies":["ac3d94dProperties27a-h3"],"title":"Simplify","text":"What do we get after simplifying the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac3d94dProperties28","title":"Simplify Expressions Using the Distributive Property","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.9 Properties of Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac3d94dProperties28a","stepAnswer":["$$3x-29$$"],"problemType":"TextBox","stepTitle":"$$4\\\\left(x-8\\\\right)-x+3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3x-29$$","hints":{"DefaultPathway":[{"id":"ac3d94dProperties28a-h1","type":"hint","dependencies":[],"title":"Distributive Property","text":"If a, $$b$$, c are real numbers, then $$a \\\\left(b+c\\\\right)=a b+a c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties28a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$4x-32-x-3$$"],"dependencies":["ac3d94dProperties28a-h1"],"title":"Distribute","text":"What do we get after distributing?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$4x-32-x+3$$","$$4x-32-x-3$$","$$4x-32+x-3$$"]},{"id":"ac3d94dProperties28a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x-29$$"],"dependencies":["ac3d94dProperties28a-h2"],"title":"Simplify","text":"What do we get after simplifying the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac3d94dProperties29","title":"Use the Properties of Zero","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.9 Properties of Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac3d94dProperties29a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{0}{\\\\left(-2\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"ac3d94dProperties29a-h1","type":"hint","dependencies":[],"title":"Division of Zero","text":"Zero divided by any real number except itself is zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac3d94dProperties3","title":"Simplify the following expression.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.9 Properties of Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac3d94dProperties3a","stepAnswer":["$$\\\\frac{7}{8}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{1}{2}+\\\\frac{7}{8}+\\\\left(-\\\\frac{1}{2}\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{7}{8}$$","hints":{"DefaultPathway":[{"id":"ac3d94dProperties3a-h1","type":"hint","dependencies":[],"title":"Commutative Property of Addition","text":"The commutative property of addition states that if a, $$b$$ are real numbers, then $$a+b=b+a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties3a-h2","type":"hint","dependencies":["ac3d94dProperties3a-h1"],"title":"Re-ordering","text":"Using the commutative property of addition on the last two terms, we can reorder the expression into $$\\\\frac{1}{2}+\\\\left(-\\\\frac{1}{2}\\\\right)+\\\\frac{7}{8}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["ac3d94dProperties3a-h2"],"title":"Addition","text":"What is $$\\\\frac{1}{2}+\\\\left(-\\\\frac{1}{2}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties3a-h4","type":"hint","dependencies":["ac3d94dProperties3a-h3"],"title":"Final Answer","text":"So our final answer is $$0+\\\\frac{7}{8}=\\\\frac{7}{8}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac3d94dProperties30","title":"Simplify Expressions Using the Distributive Property","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.9 Properties of Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac3d94dProperties30a","stepAnswer":["$$-y-5$$"],"problemType":"TextBox","stepTitle":"$$-\\\\left(y+5\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-y-5$$","hints":{"DefaultPathway":[{"id":"ac3d94dProperties30a-h1","type":"hint","dependencies":[],"title":"Multiply","text":"Multiplying by $$-1$$ results in the opposite.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties30a-h2","type":"hint","dependencies":["ac3d94dProperties30a-h1"],"title":"Distributive Property","text":"If a, $$b$$, c are real numbers, then $$a \\\\left(b+c\\\\right)=a b+a c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties30a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-y-5$$"],"dependencies":["ac3d94dProperties30a-h2"],"title":"Distribute","text":"What do we get after distributing?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac3d94dProperties4","title":"Simplify the following expression.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.9 Properties of Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac3d94dProperties4a","stepAnswer":["$$\\\\frac{49}{11}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{3}{20} \\\\frac{49}{11} \\\\frac{20}{3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{49}{11}$$","hints":{"DefaultPathway":[{"id":"ac3d94dProperties4a-h1","type":"hint","dependencies":[],"title":"Commutative Property of Multiplication","text":"The commutative property of multiplication states that if a, $$b$$ are real numbers, then $$a b=b a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties4a-h2","type":"hint","dependencies":["ac3d94dProperties4a-h1"],"title":"Re-ordering","text":"Using the commutative property of multiplication on the last two terms, we can reorder the expression as $$\\\\frac{3}{20} \\\\frac{20}{3} \\\\frac{49}{11}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ac3d94dProperties4a-h2"],"title":"Multiplication","text":"What is $$\\\\frac{3}{20} \\\\frac{20}{3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties4a-h4","type":"hint","dependencies":["ac3d94dProperties4a-h3"],"title":"Multiplication","text":"Since $$\\\\frac{3}{20}$$ and $$\\\\frac{20}{3}$$ are reciprocal of each other, and a number and its reciprocal multiply to one, $$\\\\frac{3}{20} \\\\frac{20}{3}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties4a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{49}{11}$$"],"dependencies":["ac3d94dProperties4a-h4"],"title":"Multiplication","text":"The expression turns into $$1\\\\frac{49}{11}$$. What is this value?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac3d94dProperties5","title":"Simplify the following expression.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.9 Properties of Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac3d94dProperties5a","stepAnswer":["$$17$$"],"problemType":"TextBox","stepTitle":"$$17\\\\times0.25\\\\times4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$17$$","hints":{"DefaultPathway":[{"id":"ac3d94dProperties5a-h1","type":"hint","dependencies":[],"title":"Associative Property of Multiplication","text":"The associative property of multiplication states that if a, $$b$$, c are real numbers, then $$a b c=a b c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties5a-h2","type":"hint","dependencies":["ac3d94dProperties5a-h1"],"title":"Regroup","text":"Using the associative property, we can regroup the terms in the expression into $$17((0.25)(4))$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties5a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ac3d94dProperties5a-h2"],"title":"Multiplication","text":"What is $$0.25\\\\times4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$17$$"],"dependencies":["ac3d94dProperties5a-h3"],"title":"Multiplication","text":"The expression now becomes 17(1). What is this value?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac3d94dProperties6","title":"Simplify the following expression.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.9 Properties of Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac3d94dProperties6a","stepAnswer":["$$27m-21n$$"],"problemType":"TextBox","stepTitle":"$$43m+\\\\left(-12n\\\\right)+\\\\left(-16m\\\\right)+\\\\left(-9n\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$27m-21n$$","hints":{"DefaultPathway":[{"id":"ac3d94dProperties6a-h1","type":"hint","dependencies":[],"title":"Commutative Property of Addition","text":"The commutative property of addition states that if a, $$b$$ are real numbers, then $$a+b=b+a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties6a-h2","type":"hint","dependencies":["ac3d94dProperties6a-h1"],"title":"Re-ordering","text":"Using the commutative property of addition on the middle two terms, we can reorder the expression into $$43m+\\\\left(-16m\\\\right)+\\\\left(-12n\\\\right)+\\\\left(-9n\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties6a-h3","type":"hint","dependencies":["ac3d94dProperties6a-h2"],"title":"Add like-temrs","text":"The next step is to add the like terms together. In other words, we need to evaluate $$43m+\\\\left(-16m\\\\right)+\\\\left(-12n\\\\right)+\\\\left(-9n\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$27m$$"],"dependencies":["ac3d94dProperties6a-h3"],"title":"$$m$$ terms","text":"What is $$43m+\\\\left(-16m\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-21n$$"],"dependencies":["ac3d94dProperties6a-h3"],"title":"$$n$$ terms","text":"What is $$\\\\left(-12n\\\\right)+\\\\left(-9n\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties6a-h6","type":"hint","dependencies":["ac3d94dProperties6a-h4","ac3d94dProperties6a-h5"],"title":"Final Answer","text":"The final answer is putting the $$m$$ term and the $$n$$ term together, which is $$27m-21n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac3d94dProperties7","title":"Use the Inverse Property of Addition","body":"In the following exercises, find the additive inverse of each number.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.9 Properties of Real Numbers","courseName":"OpenStax: Elementary 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4.0>"}]}},{"id":"ac3d94dProperties7b","stepAnswer":["$$-4.3$$"],"problemType":"TextBox","stepTitle":"$$4.3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-4.3$$","hints":{"DefaultPathway":[{"id":"ac3d94dProperties7b-h1","type":"hint","dependencies":[],"title":"Additive inverse","text":"For any real number a, -a is the additive inverse of a. $$a+\\\\left(-a\\\\right)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties7b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4.3$$"],"dependencies":["ac3d94dProperties7b-h1"],"title":"Opposite of $$4.3$$","text":"What is the opposite of $$4.3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ac3d94dProperties7c","stepAnswer":["$$8$$"],"problemType":"TextBox","stepTitle":"$$-8$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8$$","hints":{"DefaultPathway":[{"id":"ac3d94dProperties7c-h1","type":"hint","dependencies":[],"title":"Additive inverse","text":"For any real number a, -a is the additive inverse of a. $$a+\\\\left(-a\\\\right)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties7c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["ac3d94dProperties7c-h1"],"title":"Opposite of $$-8$$","text":"What is the opposite of -8?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ac3d94dProperties7d","stepAnswer":["$$\\\\frac{10}{3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{-10}{3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{10}{3}$$","hints":{"DefaultPathway":[{"id":"ac3d94dProperties7d-h1","type":"hint","dependencies":[],"title":"Additive inverse","text":"For any real number a, -a is the additive inverse of a. $$a+\\\\left(-a\\\\right)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties7d-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{10}{3}$$"],"dependencies":["ac3d94dProperties7d-h1"],"title":"Opposite of $$\\\\frac{-10}{3}$$","text":"What is the opposite of $$\\\\frac{-10}{3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac3d94dProperties8","title":"Use the Inverse Property of Multiplication","body":"In the following exercises, find the multiplicative inverse of each number.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.9 Properties of Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac3d94dProperties8a","stepAnswer":["$$\\\\frac{12}{11}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{11}{12}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{12}{11}$$","hints":{"DefaultPathway":[{"id":"ac3d94dProperties8a-h1","type":"hint","dependencies":[],"title":"Multiplicative inverse","text":"For any real number a, $$\\\\frac{1}{a}$$ is the multiplicative inverse of a. $$a \\\\frac{1}{a}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{12}{11}$$"],"dependencies":["ac3d94dProperties8a-h1"],"title":"Reciprocal of $$\\\\frac{11}{12}$$","text":"What is $$\\\\frac{1}{\\\\frac{11}{12}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ac3d94dProperties8b","stepAnswer":["$$\\\\frac{-10}{11}$$"],"problemType":"TextBox","stepTitle":"$$-1.1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-10}{11}$$","hints":{"DefaultPathway":[{"id":"ac3d94dProperties8b-h1","type":"hint","dependencies":[],"title":"Multiplicative inverse","text":"For any real number a, $$\\\\frac{1}{a}$$ is the multiplicative inverse of a. $$a \\\\frac{1}{a}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties8b-h2","type":"hint","dependencies":["ac3d94dProperties8b-h1"],"title":"Reciprocal of $$-1.1$$","text":"To find $$\\\\frac{1}{\\\\left(-1.1\\\\right)}$$, it is easier to write $$-1.1$$ in fractional form first, and then reverse the fraction. $$-1.1$$ written as a fracion is $$\\\\frac{-11}{10}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties8b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-10}{11}$$"],"dependencies":["ac3d94dProperties8b-h2"],"title":"Reciprocal of $$-1.1$$","text":"What is $$\\\\frac{1}{\\\\left(-\\\\frac{11}{10}\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ac3d94dProperties8c","stepAnswer":["$$\\\\frac{-1}{4}$$"],"problemType":"TextBox","stepTitle":"$$-4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-1}{4}$$","hints":{"DefaultPathway":[{"id":"ac3d94dProperties8c-h1","type":"hint","dependencies":[],"title":"Multiplicative inverse","text":"For any real number a, $$\\\\frac{1}{a}$$ is the multiplicative inverse of a. $$a \\\\frac{1}{a}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties8c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{4}$$"],"dependencies":["ac3d94dProperties8c-h1"],"title":"Reciprocal of $$-4$$","text":"What is the reciprocal of -4?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties8c-h3","type":"hint","dependencies":["ac3d94dProperties8c-h2"],"title":"Reciprocal of $$-4$$","text":"$$\\\\frac{1}{\\\\left(-4\\\\right)}=\\\\frac{-1}{4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac3d94dProperties9","title":"Use the Properties of Zero","body":"Simplify the following exercise.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.9 Properties of Real Numbers","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac3d94dProperties9a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{0}{6}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"ac3d94dProperties9a-h1","type":"hint","dependencies":[],"title":"Division of zero","text":"Recall that for any real number a, except $$0$$, $$\\\\frac{0}{a}=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac3d94dProperties9a-h2","type":"hint","dependencies":["ac3d94dProperties9a-h1"],"title":"Answer","text":"Therefore, $$\\\\frac{0}{6}=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac48862calc2_10","title":"Finding the Slope of the Secant Line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.1 A Preview of Calculus","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ac48862calc2_10a","stepAnswer":["$$2$$"],"problemType":"MultipleChoice","stepTitle":"$$P(1,2)$$ and $$Q(1.0001$$, $$2.0002)$$ are on the graph of the function $$f(x)=x^2+1$$. What is the slope of the secant line that passing through $$P(1,2)$$ and $$Q(1.0001$$, $$2.0002)$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2$$","choices":["$$2.1$$","$$2.01$$","$$2$$","$$2.001$$"],"hints":{"DefaultPathway":[{"id":"ac48862calc2_10a-h1","type":"hint","dependencies":[],"title":"Secant Line Slope","text":"The secant to the function f(x) through the points (a, f(a)) and (x, f(x)) is the line passing through these points. Its slope is given by: $$m_{sec}=\\\\frac{f{\\\\left(x\\\\right)}-f{\\\\left(a\\\\right)}}{x-a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ac48862calc2_10a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2$$"],"dependencies":["ac48862calc2_10a-h1"],"title":"Finding the Slope","text":"Given the equation to the secant line slope, what is the slope?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$2.1$$","$$2.01$$","$$2$$","$$2.001$$"],"subHints":[{"id":"ac48862calc2_10a-h2-s1","type":"hint","dependencies":[],"title":"Finding the Slope","text":"$$\\\\frac{2.0002-2}{1.0001-1}=2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}]}}]},{"id":"ac48862calc2_11","title":"Find the Average Velocity","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.1 A Preview of Calculus","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ac48862calc2_11a","stepAnswer":["$$-9.849$$"],"problemType":"MultipleChoice","stepTitle":"Consider a rocket shot into the air that then returns to Earth. The height of the rocket in meters is given by $$h(t)=600+78.4t-4.9t^2$$, where $$t$$ is measured in seconds. Compute the average velocity of the rocket over the time interval: [9, $$9.01]$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-9.849$$","choices":["$$166.649$$","$$-166.649$$","$$-9.849$$","$$9.849$$"],"hints":{"DefaultPathway":[{"id":"ac48862calc2_11a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$908.7$$"],"dependencies":[],"title":"Compute h(9)","text":"What is the value of h(9)?","variabilization":{},"oer":"","license":"","choices":["$$908.7$$","$$900$$"]},{"id":"ac48862calc2_11a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$908.60151$$"],"dependencies":["ac48862calc2_11a-h1"],"title":"Compute $$h(9.01)$$","text":"What is the value of $$h(9.01)$$?","variabilization":{},"oer":"","license":"","choices":["$$908.60151$$","$$906.60151$$"]},{"id":"ac48862calc2_11a-h3","type":"hint","dependencies":["ac48862calc2_11a-h2"],"title":"Average Velocity","text":"Let f(x) be the position of an object moving along a coordinate axis at time $$t$$. The average velocity of the object over a time interval [a,t] is: $$v_{avg}=\\\\frac{f{\\\\left(t\\\\right)}-f{\\\\left(a\\\\right)}}{t-a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ac48862calc2_11a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-9.849$$"],"dependencies":["ac48862calc2_11a-h3"],"title":"Plug-in and Solve","text":"Given the equation, plug in the given interval into the equations to solve for the average velocity. What is the answer of the average velocity?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$166.649$$","$$-166.649$$","$$-9.849$$","$$9.849$$"],"subHints":[{"id":"ac48862calc2_11a-h4-s1","type":"hint","dependencies":[],"title":"Plug-in and Solve","text":"$$\\\\frac{908.60151-908.7}{9.01-9}=-9.849$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}]}}]},{"id":"ac48862calc2_12","title":"Find the Average Velocity","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.1 A Preview of Calculus","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ac48862calc2_12a","stepAnswer":["$$-9.751$$"],"problemType":"MultipleChoice","stepTitle":"Consider a rocket shot into the air that then returns to Earth. The height of the rocket in meters is given by $$h(t)=600+78.4t-4.9t^2$$, where $$t$$ is measured in seconds. Compute the average velocity of the rocket over the time interval: $$[8.99$$, 9].","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-9.751$$","choices":["$$166.551$$","$$-166.551$$","$$-9.751$$","$$9.751$$"],"hints":{"DefaultPathway":[{"id":"ac48862calc2_12a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$908.79751$$"],"dependencies":[],"title":"Compute $$h(8.99)$$","text":"What is the value of $$h(8.99)$$?","variabilization":{},"oer":"","license":"","choices":["$$908.79751$$","$$906.6$$"]},{"id":"ac48862calc2_12a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$908.7$$"],"dependencies":["ac48862calc2_12a-h1"],"title":"Compute h(9)","text":"What is the value of h(9)?","variabilization":{},"oer":"","license":"","choices":["$$908.7$$","$$900$$"]},{"id":"ac48862calc2_12a-h3","type":"hint","dependencies":["ac48862calc2_12a-h2"],"title":"Average Velocity","text":"Let f(x) be the position of an object moving along a coordinate axis at time $$t$$. The average velocity of the object over a time interval [a,t] is: $$v_{avg}=\\\\frac{f{\\\\left(t\\\\right)}-f{\\\\left(a\\\\right)}}{t-a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ac48862calc2_12a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-9.751$$"],"dependencies":["ac48862calc2_12a-h3"],"title":"Plug-in and Solve","text":"Given the equation, plug in the given interval into the equations to solve for the average velocity. What is the answer of the average velocity?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$166.551$$","$$-166.551$$","$$-9.751$$","$$9.751$$"],"subHints":[{"id":"ac48862calc2_12a-h4-s1","type":"hint","dependencies":[],"title":"Plug-in and Solve","text":"$$\\\\frac{908.7-908.79751}{9-8.99}=-9.751$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}]}}]},{"id":"ac48862calc2_13","title":"Find the Average Velocity","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.1 A Preview of Calculus","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ac48862calc2_13a","stepAnswer":["$$-9.8049$$"],"problemType":"MultipleChoice","stepTitle":"Consider a rocket shot into the air that then returns to Earth. The height of the rocket in meters is given by $$h(t)=600+78.4t-4.9t^2$$, where $$t$$ is measured in seconds. Compute the average velocity of the rocket over the time interval: [9, $$9.001]$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-9.8049$$","choices":["$$166.6049$$","$$-166.6049$$","$$-9.8049$$","$$9.8049$$"],"hints":{"DefaultPathway":[{"id":"ac48862calc2_13a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$908.7$$"],"dependencies":[],"title":"Compute h(9)","text":"What is the value of h(9)?","variabilization":{},"oer":"","license":"","choices":["$$908.7$$","$$900$$"]},{"id":"ac48862calc2_13a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$908.6901951$$"],"dependencies":["ac48862calc2_13a-h1"],"title":"Compute $$h(9.001)$$","text":"What is the value of $$h(9.001)$$?","variabilization":{},"oer":"","license":"","choices":["$$908.6901951$$","$$906.6901951$$"]},{"id":"ac48862calc2_13a-h3","type":"hint","dependencies":["ac48862calc2_13a-h2"],"title":"Average Velocity","text":"Let f(x) be the position of an object moving along a coordinate axis at time $$t$$. The average velocity of the object over a time interval [a,t] is: $$v_{avg}=\\\\frac{f{\\\\left(t\\\\right)}-f{\\\\left(a\\\\right)}}{t-a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ac48862calc2_13a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-9.8049$$"],"dependencies":["ac48862calc2_13a-h3"],"title":"Plug-in and Solve","text":"Given the equation, plug in the given interval into the equations to solve for the average velocity. What is the answer of the average velocity?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$166.6049$$","$$-166.6049$$","$$-9.8049$$","$$9.8049$$"],"subHints":[{"id":"ac48862calc2_13a-h4-s1","type":"hint","dependencies":[],"title":"Plug-in and Solve","text":"$$\\\\frac{908.6901951-908.7}{9.001-9}=-9.8049$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}]}}]},{"id":"ac48862calc2_14","title":"Find the Average Velocity","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.1 A Preview of Calculus","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ac48862calc2_14a","stepAnswer":["$$-9.7951$$"],"problemType":"MultipleChoice","stepTitle":"Consider a rocket shot into the air that then returns to Earth. The height of the rocket in meters is given by $$h(t)=600+78.4t-4.9t^2$$, where $$t$$ is measured in seconds. Compute the average velocity of the rocket over the time interval: $$[8.999, 9]$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-9.7951$$","choices":["$$166.5951$$","$$-166.5951$$","$$-9.7951$$","$$9.7951$$"],"hints":{"DefaultPathway":[{"id":"ac48862calc2_14a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$908.7097951$$"],"dependencies":[],"title":"Compute $$h(8.999)$$","text":"What is the value of $$h(8.999)$$?","variabilization":{},"oer":"","license":"","choices":["$$908.7097951$$","$$906.7097951$$"]},{"id":"ac48862calc2_14a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$908.7$$"],"dependencies":["ac48862calc2_14a-h1"],"title":"Compute h(9)","text":"What is the value of h(9)?","variabilization":{},"oer":"","license":"","choices":["$$908.7$$","$$900$$"]},{"id":"ac48862calc2_14a-h3","type":"hint","dependencies":["ac48862calc2_14a-h2"],"title":"Average Velocity","text":"Let f(x) be the position of an object moving along a coordinate axis at time $$t$$. The average velocity of the object over a time interval [a,t] is: $$v_{avg}=\\\\frac{f{\\\\left(t\\\\right)}-f{\\\\left(a\\\\right)}}{t-a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ac48862calc2_14a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-9.7951$$"],"dependencies":["ac48862calc2_14a-h3"],"title":"Plug-in and Solve","text":"Given the equation, plug in the given interval into the equations to solve for the average velocity. What is the answer of the average velocity?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$166.5951$$","$$-166.5951$$","$$-9.7951$$","$$9.7951$$"],"subHints":[{"id":"ac48862calc2_14a-h4-s1","type":"hint","dependencies":[],"title":"Plug-in and Solve","text":"$$\\\\frac{908.7-908.7097951}{9-8.999}=-9.7951$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}]}}]},{"id":"ac48862calc2_15","title":"Find the Average Velocity","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.1 A Preview of Calculus","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ac48862calc2_15a","stepAnswer":["$$-15.84$$"],"problemType":"MultipleChoice","stepTitle":"A rock is dropped from a height of $$64$$ ft. It is determined that its height (in feet) above ground $$t$$ seconds later (for 0<t<2) is given by s(t) $$=$$ $$-16t^2+64$$. Find the average velocity of the rock over the time intervals $$[0.49$$, $$0.5]$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-15.84$$","choices":["$$-15.84$$","$$-15.88$$","$$-13.84$$","$$-13.88$$"],"hints":{"DefaultPathway":[{"id":"ac48862calc2_15a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$60.1584$$"],"dependencies":[],"title":"Compute $$S(0.49)$$","text":"What is the value of $$S(0.49)$$?","variabilization":{},"oer":"","license":"","choices":["$$60.1584$$","$$62.38$$","$$60$$"]},{"id":"ac48862calc2_15a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$60$$"],"dependencies":["ac48862calc2_15a-h1"],"title":"Compute $$S(0.5)$$","text":"What is the value of $$S(0.5)$$?","variabilization":{},"oer":"","license":"","choices":["$$60.1584$$","$$62.38$$","$$60$$"]},{"id":"ac48862calc2_15a-h3","type":"hint","dependencies":["ac48862calc2_15a-h2"],"title":"Average Velocity","text":"Let f(x) be the position of an object moving along a coordinate axis at time $$t$$. The average velocity of the object over a time interval [a,t] is: $$v_{avg}=\\\\frac{f{\\\\left(t\\\\right)}-f{\\\\left(a\\\\right)}}{t-a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ac48862calc2_15a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-15.84$$"],"dependencies":["ac48862calc2_15a-h3"],"title":"Plug-in and Solve","text":"Given the equation, plug in the given interval into the equations to solve for the average velocity. What is the answer of the average velocity?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$-15.84$$","$$-15.88$$","$$-13.84$$","$$-13.88$$"],"subHints":[{"id":"ac48862calc2_15a-h4-s1","type":"hint","dependencies":[],"title":"Plug-in and Solve","text":"$$\\\\frac{60-60.1584}{0.5-0.49}=-15.84$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}]}}]},{"id":"ac48862calc2_16","title":"Find the Average Velocity","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.1 A Preview of Calculus","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ac48862calc2_16a","stepAnswer":["$$-16.16$$"],"problemType":"MultipleChoice","stepTitle":"A rock is dropped from a height of $$64$$ ft. It is determined that its height (in feet) above ground $$t$$ seconds later (for 0<t<2) is given by s(t) $$=$$ $$-16t^2+64$$. Find the average velocity of the rock over the time intervals $$[0.5$$, $$0.51]$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-16.16$$","choices":["$$-16.88$$","$$-16.16$$","$$-13.88$$","$$-13.16$$"],"hints":{"DefaultPathway":[{"id":"ac48862calc2_16a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$60$$"],"dependencies":[],"title":"Compute $$S(0.5)$$","text":"What is the value of $$S(0.5)$$?","variabilization":{},"oer":"","license":"","choices":["$$60.1584$$","$$62.38$$","$$60$$","$$59.8384$$"]},{"id":"ac48862calc2_16a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$59.8384$$"],"dependencies":["ac48862calc2_16a-h1"],"title":"Compute $$S(0.51)$$","text":"What is the value of $$S(0.51)$$?","variabilization":{},"oer":"","license":"","choices":["$$60.1584$$","$$62.38$$","$$60$$","$$59.8384$$"]},{"id":"ac48862calc2_16a-h3","type":"hint","dependencies":["ac48862calc2_16a-h2"],"title":"Average Velocity","text":"Let f(x) be the position of an object moving along a coordinate axis at time $$t$$. The average velocity of the object over a time interval [a,t] is: $$v_{avg}=\\\\frac{f{\\\\left(t\\\\right)}-f{\\\\left(a\\\\right)}}{t-a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ac48862calc2_16a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-16.16$$"],"dependencies":["ac48862calc2_16a-h3"],"title":"Plug-in and Solve","text":"Given the equation, plug in the given interval into the equations to solve for the average velocity. What is the answer of the average velocity?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$-16.88$$","$$-16.16$$","$$-13.88$$","$$-13.16$$"],"subHints":[{"id":"ac48862calc2_16a-h4-s1","type":"hint","dependencies":[],"title":"Plug-in and Solve","text":"$$\\\\frac{59.8384-60}{0.51-0.5}=-16.16$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}]}}]},{"id":"ac48862calc2_1_1","title":"Finding Points on Quadratic Graphs","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.1 A Preview of Calculus","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ac48862calc2_1_1a","stepAnswer":["$$2.21$$"],"problemType":"MultipleChoice","stepTitle":"$$P(1,2)$$ and $$Q(x,y)$$ are on the graph of the function $$f(x)=x^2+1$$. Let $$x=1.1$$. What is $$y$$ in $$Q(x,y)$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2.21$$","choices":["$$2.21$$","$$2.021$$","$$2.0021$$","$$2.00021$$"],"hints":{"DefaultPathway":[{"id":"ac48862calc2_1_1a-h1","type":"hint","dependencies":[],"title":"Finding $$y$$","text":"To find $$y$$, remove all variable instances of $$x$$ (make sure that only numbers are present on the right side)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ac48862calc2_1_1a-h2","type":"hint","dependencies":["ac48862calc2_1_1a-h1"],"title":"Plug-in values","text":"Plug in the value of $$x=1.1$$ on the right side to determine $$y$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ac48862calc2_2","title":"Finding Points on Quadratic Graphs","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.1 A Preview of Calculus","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ac48862calc2_2a","stepAnswer":["$$(1.1$$, $$2.21)$$"],"problemType":"MultipleChoice","stepTitle":"$$P(1,2)$$ and $$Q(x,y)$$ are on the graph of the function $$f(x)=x^2+1$$. Let $$x=1.1$$. What is $$Q(x, y)$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(1.1$$, $$2.21)$$","choices":["$$(1.1$$, $$2.21)$$","$$(1.1$$, $$2.021)$$","$$(1.1$$, $$2.0021)$$","$$(1.01$$, $$2.21)$$"],"hints":{"DefaultPathway":[{"id":"ac48862calc2_2a-h1","type":"hint","dependencies":[],"title":"Finding Y","text":"To find the value of $$y$$, plug in the value of $$x$$ into f(x)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ac48862calc2_2a-h2","type":"hint","dependencies":["ac48862calc2_2a-h1"],"title":"Ordered Pair","text":"Put $$x$$ and $$y$$ into the form of an ordered pair (x, y).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ac48862calc2_3","title":"Finding the Slope of the Secant Line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.1 A Preview of Calculus","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ac48862calc2_3a","stepAnswer":["$$2.1$$"],"problemType":"MultipleChoice","stepTitle":"$$P(1,2)$$ and $$Q(1.1$$, $$2.21)$$ are on the graph of the function $$f(x)=x^2+1$$. What is the slope of the secant line that passing through $$P(1,2)$$ and $$Q(1.1$$, $$2.21)$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2.1$$","choices":["$$2.1$$","$$2.01$$","$$2$$","$$2.001$$"],"hints":{"DefaultPathway":[{"id":"ac48862calc2_3a-h1","type":"hint","dependencies":[],"title":"Secant Line Slope","text":"The secant line to the function f(x) through the points (a, f(a)) and (x, f(x)) is the line passing through these points. Its slope is given by: $$m_{sec}=\\\\frac{f{\\\\left(x\\\\right)}-f{\\\\left(a\\\\right)}}{x-a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ac48862calc2_3a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2.1$$"],"dependencies":["ac48862calc2_3a-h1"],"title":"Finding the Slope","text":"Given the equation to the secant line slope, what is the slope?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$2.1$$","$$2.01$$","$$2$$","$$2.001$$"],"subHints":[{"id":"ac48862calc2_3a-h2-s1","type":"hint","dependencies":[],"title":"Finding the Slope","text":"$$\\\\frac{2.21-2}{1.1-1}=2.1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}]}}]},{"id":"ac48862calc2_4","title":"Finding Points on Quadratic Graphs","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.1 A Preview of Calculus","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ac48862calc2_4a","stepAnswer":["$$2.0201$$"],"problemType":"MultipleChoice","stepTitle":"$$P(1,2)$$ and $$Q(x,y)$$ are on the graph of the function $$f(x)=x^2+1$$. Let $$x=1.01$$. What is $$y$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2.0201$$","choices":["$$2.0201$$","$$2.021$$","$$2.0021$$","$$2.21$$"],"hints":{"DefaultPathway":[{"id":"ac48862calc2_4a-h1","type":"hint","dependencies":[],"title":"Finding $$y$$","text":"To find $$y$$, remove all variable instances of $$x$$ (make sure that only numbers are present on the right side)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ac48862calc2_4a-h2","type":"hint","dependencies":["ac48862calc2_4a-h1"],"title":"Plug-in values","text":"Plug in the value of $$x=1.01$$ on the right side to determine $$y$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ac48862calc2_5","title":"Finding Points on Quadratic Graphs","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.1 A Preview of Calculus","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ac48862calc2_5a","stepAnswer":["$$(1.01$$, $$2.0201)$$"],"problemType":"MultipleChoice","stepTitle":"$$P(1,2)$$ and $$Q(x,y)$$ are on the graph of the function $$f(x)=x^2+1$$. Let $$x=1.01$$. What is $$Q(x, y)$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(1.01$$, $$2.0201)$$","choices":["$$(1.01$$, $$2.0201)$$","$$(1.01$$, $$2.00201)$$","$$(1.1$$, $$2.21)$$","$$(1.01$$, $$2.002001)$$"],"hints":{"DefaultPathway":[{"id":"ac48862calc2_5a-h1","type":"hint","dependencies":[],"title":"Finding Y","text":"To find the value of $$y$$, plug in the value of $$x$$ into f(x)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ac48862calc2_5a-h2","type":"hint","dependencies":["ac48862calc2_5a-h1"],"title":"Ordered Pair","text":"Put $$x$$ and $$y$$ into the form of an ordered pair (x, y).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ac48862calc2_6","title":"Finding the Slope of the Secant Line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.1 A Preview of Calculus","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ac48862calc2_6a","stepAnswer":["$$2.01$$"],"problemType":"MultipleChoice","stepTitle":"$$P(1,2)$$ and $$Q(1.01$$, $$2.0201)$$ are on the graph of the function $$f(x)=x^2+1$$. What is the slope of the secant line that passing through $$P(1,2)$$ and $$Q(1.01$$, $$2.0201)$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2.01$$","choices":["$$2.1$$","$$2.01$$","$$2$$","$$2.001$$"],"hints":{"DefaultPathway":[{"id":"ac48862calc2_6a-h1","type":"hint","dependencies":[],"title":"Secant Line Slope","text":"The secant to the function f(x) through the points (a, f(a)) and (x, f(x)) is the line passing through these points. Its slope is given by: $$m_{sec}=\\\\frac{f{\\\\left(x\\\\right)}-f{\\\\left(a\\\\right)}}{x-a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ac48862calc2_6a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2.01$$"],"dependencies":["ac48862calc2_6a-h1"],"title":"Finding the Slope","text":"Given the equation to the secant line slope, what is the slope?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$2.1$$","$$2.01$$","$$2$$","$$2.001$$"],"subHints":[{"id":"ac48862calc2_6a-h2-s1","type":"hint","dependencies":[],"title":"Finding the Slope","text":"$$\\\\frac{2.0201-2}{1.01-1}=2.01$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}]}}]},{"id":"ac48862calc2_7","title":"Finding Points on Quadratic Graphs","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.1 A Preview of Calculus","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ac48862calc2_7a","stepAnswer":["$$2.002001$$"],"problemType":"MultipleChoice","stepTitle":"$$P(1,2)$$ and $$Q(x,y)$$ are on the graph of the function $$f(x)=x^2+1$$. Let $$x=1.001$$. What is $$y$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2.002001$$","choices":["$$2.0201$$","$$2.021$$","$$2.0021$$","$$2.002001$$"],"hints":{"DefaultPathway":[{"id":"ac48862calc2_7a-h1","type":"hint","dependencies":[],"title":"Finding $$y$$","text":"To find $$y$$, remove all variable instances of $$x$$ (make sure that only numbers are present on the right side)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ac48862calc2_7a-h2","type":"hint","dependencies":["ac48862calc2_7a-h1"],"title":"Plug-in values","text":"Plug in the value of $$x=1.001$$ on the right side to determine $$y$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ac48862calc2_8","title":"Finding Points on Quadratic Graphs","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.1 A Preview of Calculus","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ac48862calc2_8a","stepAnswer":["$$(1.001$$, $$2.002001)$$"],"problemType":"MultipleChoice","stepTitle":"$$P(1,2)$$ and $$Q(x,y)$$ are on the graph of the function $$f(x)=x^2+1$$. Let $$x=1.001$$. What is $$Q(x, y)$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(1.001$$, $$2.002001)$$","choices":["$$(1.001$$, $$2.002001)$$","$$(1.01$$, $$2.00201)$$","$$(1.001$$, $$2.0201)$$","$$(1.001$$, $$2.21)$$"],"hints":{"DefaultPathway":[{"id":"ac48862calc2_8a-h1","type":"hint","dependencies":[],"title":"Finding Y","text":"To find the value of $$y$$, plug in the value of $$x$$ into f(x)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ac48862calc2_8a-h2","type":"hint","dependencies":["ac48862calc2_8a-h1"],"title":"Ordered Pair","text":"Put $$x$$ and $$y$$ into the form of an ordered pair (x, y).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}},{"id":"ac48862calc2_8b","stepAnswer":["$$2.001$$"],"problemType":"MultipleChoice","stepTitle":"$$P(1,2)$$ and $$Q(1.001$$, $$2.002001)$$ are on the graph of the function $$f(x)=x^2+1$$. What is the slope of the secant line that passing through $$P(1,2)$$ and $$Q(1.001$$, $$2.002001)$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2.001$$","choices":["$$2.1$$","$$2.01$$","$$2$$","$$2.001$$"],"hints":{"DefaultPathway":[{"id":"ac48862calc2_8b-h1","type":"hint","dependencies":[],"title":"Secant Line Slope","text":"The secant to the function f(x) through the points (a, f(a)) and (x, f(x)) is the line passing through these points. Its slope is given by: $$m_{sec}=\\\\frac{f{\\\\left(x\\\\right)}-f{\\\\left(a\\\\right)}}{x-a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ac48862calc2_8b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2.001$$"],"dependencies":["ac48862calc2_8b-h1"],"title":"Finding the Slope","text":"Given the equation to the secant line slope, what is the slope?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$2.1$$","$$2.01$$","$$2$$","$$2.001$$"],"subHints":[{"id":"ac48862calc2_8b-h2-s1","type":"hint","dependencies":[],"title":"Finding the Slope","text":"$$\\\\frac{2.002001-2}{1.001-1}=2.001$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}]}}]},{"id":"ac48862calc2_9","title":"Finding Points on Quadratic Graphs","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"2.1 A Preview of Calculus","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ac48862calc2_9a","stepAnswer":["$$2.0002$$"],"problemType":"MultipleChoice","stepTitle":"$$P(1,2)$$ and $$Q(x,y)$$ are on the graph of the function $$f(x)=x^2+1$$. Let $$x=1.0001$$. What is $$y$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2.0002$$","choices":["$$2.0002$$","$$2.02$$","$$2.0021$$","$$2.00002$$"],"hints":{"DefaultPathway":[{"id":"ac48862calc2_9a-h1","type":"hint","dependencies":[],"title":"Finding $$y$$","text":"To find $$y$$, remove all variable instances of $$x$$ (make sure that only numbers are present on the right side)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ac48862calc2_9a-h2","type":"hint","dependencies":["ac48862calc2_9a-h1"],"title":"Plug-in values","text":"Plug in the value of $$x=1.0001$$ on the right side to determine $$y$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ac561a1systems1","title":"Solving a System of Nonlinear Equations Using Elimination","body":"Solve the system of nonlinear equations using elimination.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Systems of Nonlinear Equations and Inequalities: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ac561a1systems1a","stepAnswer":["$$(-3,0),(3,0)$$"],"problemType":"MultipleChoice","stepTitle":"$$4x^2-9y^2=36$$,\\\\n$$4x^2+9y^2=36$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-3,0),(3,0)$$","choices":["$$(-3,0),(3,0)$$","$$(-3,3),(3,-3)$$","$$(0,-3),(0,3)$$","No Solution"],"hints":{"DefaultPathway":[{"id":"ac561a1systems1a-h1","type":"hint","dependencies":[],"title":"Eliminating Variables","text":"When both equations in the system have like variables of the second degree, solving them using elimination by addition is often easier than substitution. Generally, elimination is a far simpler method when the system involves only two equations in two variables (a two-by-two system), rather than a three-by-three system, as there are fewer steps.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems1a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$8x^2=72$$"],"dependencies":["ac561a1systems1a-h1"],"title":"Adding Equations","text":"We observe that by adding the two equations together, we can remove the $$y^2$$ term. What is the equation obtained after adding the two equations?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$8x^2=72$$","$$18x^2=72$$","$$18y^2=72$$"]},{"id":"ac561a1systems1a-h3","type":"hint","dependencies":["ac561a1systems1a-h2"],"title":"Solving for $$x$$","text":"With the equation $$8x^2=72$$, we can solve for $$x$$. The x-coordinate found indicates a possible solution where an intersection may occur if the system of equations yield a feasible solution later on.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems1a-h4","type":"hint","dependencies":["ac561a1systems1a-h3"],"title":"Solving for $$x$$","text":"Divide $$8$$ on boths sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems1a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-3, 3$$"],"dependencies":["ac561a1systems1a-h4"],"title":"Solving for $$x$$","text":"Take the square root on both sides. What are the possible $$x$$ coordinates?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$-3, 3$$","$$0$$","$$9$$"]},{"id":"ac561a1systems1a-h6","type":"hint","dependencies":["ac561a1systems1a-h5"],"title":"Substitution","text":"We substitute all possible $$x$$ coordinates back into one of the two equations to find the corresponding $$y$$ coordinate to each $$x$$ coordinate. Using any of the two equations work because the $$x$$ coordinate found is a solution to the system of equations. We usually use the equation that is easier to calculate the other coordinate with.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems1a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["ac561a1systems1a-h6"],"title":"Substitution #1","text":"Substituting $$x=-3$$ into $$4x^2-9y^2=36$$. What is the corresponding $$y$$ coordinate?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems1a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["ac561a1systems1a-h7"],"title":"Substitution #2","text":"Substituting $$x=3$$ into $$4x^2-9y^2=36$$. What is the corresponding $$y$$ coordinate?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems1a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ac561a1systems1a-h8"],"title":"Intersection","text":"We have found the coordinates of all intersections. How many are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac561a1systems10","title":"Solving a System of Nonlinear Equations Using Substitution","body":"Solve the system of nonlinear equations using substitution.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Systems of Nonlinear Equations and Inequalities: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ac561a1systems10a","stepAnswer":["$$(\\\\frac{1}{2},0)$$"],"problemType":"MultipleChoice","stepTitle":"$$2x^3-x^2=y$$,\\\\n$$y=\\\\frac{1}{2}-x$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(\\\\frac{1}{2},0)$$","choices":["$$(\\\\frac{1}{2},0)$$","$$(1,1)$$","(1/2,0),(1,1)","No Solution"],"hints":{"DefaultPathway":[{"id":"ac561a1systems10a-h1","type":"hint","dependencies":[],"title":"Subsituting Possible Values","text":"The substitution method we used for linear systems is the same method we will use for nonlinear systems. We solve one equation for one variable and then substitute the result into the second equation to solve for another variable, and so on.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y$$"],"dependencies":["ac561a1systems10a-h1"],"title":"Common Terms between Equations","text":"What is the common term between the two equations that can be used for substitution?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems10a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2x^3-x^2+x-\\\\frac{1}{2}=0$$"],"dependencies":["ac561a1systems10a-h2"],"title":"Substitution","text":"What is the equation that we are trying to solve for after substituting the second equation into the first?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2x^3-x^2+x-\\\\frac{1}{2}=0$$","$$2x^3-x^2-x+\\\\frac{1}{2}=0$$"]},{"id":"ac561a1systems10a-h4","type":"hint","dependencies":["ac561a1systems10a-h3"],"title":"Factorization","text":"Factorize $$2x^3-x^2+x-\\\\frac{1}{2}=0$$ so that we can find the $$x$$ coordinates.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(x-\\\\frac{1}{2}\\\\right) \\\\left(2x^2+1\\\\right)$$"],"dependencies":["ac561a1systems10a-h4"],"title":"Factorization","text":"One of the zeros to $$f(x)=2x^3-x^2+x-\\\\frac{1}{2}$$ is $$\\\\frac{1}{2}$$. We can use synthetic division to divide $$x-\\\\frac{1}{2}$$ from the expression. What is the factored expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems10a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ac561a1systems10a-h5"],"title":"Factorization","text":"Does $$f(x)=2x^2+1$$ have any zeros? Consider that it is a quadratic curve above the x-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ac561a1systems10a-h7","type":"hint","dependencies":["ac561a1systems10a-h6"],"title":"Solving for $$y$$","text":"There is only one real solution to the equation, $$2x^3-x^2+x-\\\\frac{1}{2}=0$$. We now want to find the $$y$$ coordinate that correspond to this $$x$$ coordinate.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems10a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["ac561a1systems10a-h7"],"title":"Solving for $$y$$","text":"Substituting $$x=\\\\frac{1}{2}$$ into y=1/2-x/ What is the corresponding $$y$$ coordinate?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems10a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ac561a1systems10a-h8"],"title":"Verifying the Number Of Intersections","text":"We have found the coordinates of all intersections. How many are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac561a1systems11","title":"Solving a System of Nonlinear Equations Representing a Parabola and a Line #1","body":"Solve the system of equations.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Systems of Nonlinear Equations and Inequalities: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ac561a1systems11a","stepAnswer":["$$(1,2)$$ and $$(0,1)$$"],"problemType":"MultipleChoice","stepTitle":"$$x-y=-1$$, $$y=x^2+1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(1,2)$$ and $$(0,1)$$","choices":["$$(0,2)$$ and $$(1,0)$$","$$(1,2)$$ and $$(2,0)$$","$$(1,2)$$ and $$(0,1)$$"],"hints":{"DefaultPathway":[{"id":"ac561a1systems11a-h1","type":"hint","dependencies":[],"title":"Solving for $$x$$","text":"Solve the first equation for $$x$$ and then substitute the resulting expression into the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems11a-h2","type":"hint","dependencies":["ac561a1systems11a-h1"],"title":"Solving for Zeroes","text":"Expand the equation and set it equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems11a-h3","type":"hint","dependencies":["ac561a1systems11a-h2"],"title":"Subsituting Each Y Value","text":"Next, substitute each value for $$y$$ into the first equation to solve for $$x$$. Always substitute the value into the linear equation to check for extraneous solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems11a-h4","type":"hint","dependencies":["ac561a1systems11a-h3"],"title":"Verifying the Solution","text":"The solutions can be verified by subsituting these (x,y) values into both of the original equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac561a1systems12","title":"Finding the Intersection of a Circle and a Line by Substitution #1","body":"Find the intersection of the given circle and the given line by substitution.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Systems of Nonlinear Equations and Inequalities: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ac561a1systems12a","stepAnswer":["$$(2,1)$$ and $$(1,-2)$$"],"problemType":"MultipleChoice","stepTitle":"$$x^2+y^2=5$$, $$y=3x-5$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(2,1)$$ and $$(1,-2)$$","choices":["$$(2,1)$$ and $$(1,-2)$$","$$(2,3)$$ and $$(1,-2)$$","$$(2,0)$$ and $$(1,-2)$$"],"hints":{"DefaultPathway":[{"id":"ac561a1systems12a-h1","type":"hint","dependencies":[],"title":"Subsituting Y","text":"One of the equations has already been solved for $$y$$. We will substitute $$y=3x-5$$ into the equation for the circle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems12a-h2","type":"hint","dependencies":["ac561a1systems12a-h1"],"title":"Factoring the Equation","text":"Then, factor and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems12a-h3","type":"hint","dependencies":["ac561a1systems12a-h2"],"title":"Solving For Y","text":"Next, substitute the two x-values into the original linear equation to solve for $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems12a-h4","type":"hint","dependencies":["ac561a1systems12a-h3"],"title":"Verifying the Solution","text":"The solutions can be verified by subsituting these (x,y) values into both of the original equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac561a1systems13","title":"Finding the Intersection of a Circle and a Line by Substitution #2","body":"Solve the system of nonlinear equations using subsitution.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Systems of Nonlinear Equations and Inequalities: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ac561a1systems13a","stepAnswer":["((4-sqrt(2))/2, (4+sqrt(2))/2), ((4+sqrt(2))/2, (4-sqrt(2))/2)"],"problemType":"MultipleChoice","stepTitle":"$$x+y=4$$, $$x^2+y^2=9$$","stepBody":"","answerType":"string","variabilization":{},"choices":["((3-sqrt(2))/2, (3+sqrt(2))/2), ((3+sqrt(2))/2, (3-sqrt(2))/2)","((3-sqrt(3))/2, (3+sqrt(3))/2), ((3+sqrt(3))/2, (3-sqrt(3))/2)","((4-sqrt(2))/2, (4+sqrt(2))/2), ((4+sqrt(2))/2, (4-sqrt(2))/2)"],"hints":{"DefaultPathway":[{"id":"ac561a1systems13a-h1","type":"hint","dependencies":[],"title":"Subsituting Y","text":"Solve for $$y$$ in the first equation. Next, subsitute it into the equation for the circle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems13a-h2","type":"hint","dependencies":["ac561a1systems13a-h1"],"title":"Factoring the Equation","text":"Then, factor and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems13a-h3","type":"hint","dependencies":["ac561a1systems13a-h2"],"title":"Solving For Y","text":"Next, substitute the two x-values into the original linear equation to solve for $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems13a-h4","type":"hint","dependencies":["ac561a1systems13a-h3"],"title":"Verifying the Solution","text":"The solutions can be verified by subsituting these (x,y) values into both of the original equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac561a1systems14","title":"Finding the Intersection of a Circle and a Line by Substitution #3","body":"Solve the system of nonlinear equations using subsitution.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Systems of Nonlinear Equations and Inequalities: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ac561a1systems14a","stepAnswer":["$$(0,-3)$$ and $$(3,0)$$"],"problemType":"MultipleChoice","stepTitle":"$$y=x-3$$, $$x^2+y^2=9$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,-3)$$ and $$(3,0)$$","choices":["$$(0,-3)$$ and $$(3,0)$$","$$(1,-3)$$ and $$(3,1)$$","$$(2,-3)$$ and $$(3,2)$$"],"hints":{"DefaultPathway":[{"id":"ac561a1systems14a-h1","type":"hint","dependencies":[],"title":"Subsituting Y","text":"One of the equations has already been solved for $$y$$. We will substitute $$y=x-3$$ into the equation for the circle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems14a-h2","type":"hint","dependencies":["ac561a1systems14a-h1"],"title":"Factoring the Equation","text":"Then, factor and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems14a-h3","type":"hint","dependencies":["ac561a1systems14a-h2"],"title":"Solving For Y","text":"Next, substitute the two x-values into the original linear equation to solve for $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems14a-h4","type":"hint","dependencies":["ac561a1systems14a-h3"],"title":"Verifying the Solution","text":"The solutions can be verified by subsituting these (x,y) values into both of the original equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac561a1systems15","title":"Finding the Intersection of a Circle and a Line by Substitution #4","body":"Solve the system of nonlinear equations using subsitution.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Systems of Nonlinear Equations and Inequalities: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ac561a1systems15a","stepAnswer":["(3sqrt(2)/2,3sqrt(2)/2),(-3sqrt(2)/2,-3sqrt(2)/2)"],"problemType":"MultipleChoice","stepTitle":"$$y=x$$, $$x^2+y^2=9$$","stepBody":"","answerType":"string","variabilization":{},"choices":["(2sqrt(2)/2,2sqrt(2)/2),(-3sqrt(2)/2,-3sqrt(2)/2)","(3sqrt(2)/2,3sqrt(2)/2),(-3sqrt(2)/2,-3sqrt(2)/2)","(2sqrt(2)/2,2sqrt(2)/2),(-2sqrt(2)/2,-2sqrt(2)/2)"],"hints":{"DefaultPathway":[{"id":"ac561a1systems15a-h1","type":"hint","dependencies":[],"title":"Subsituting Y","text":"One of the equations has already been solved for $$y$$. We will substitute $$y=x$$ into the equation for the circle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems15a-h2","type":"hint","dependencies":["ac561a1systems15a-h1"],"title":"Factoring the Equation","text":"Then, factor and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems15a-h3","type":"hint","dependencies":["ac561a1systems15a-h2"],"title":"Solving For Y","text":"Next, substitute the two x-values into the original linear equation to solve for $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems15a-h4","type":"hint","dependencies":["ac561a1systems15a-h3"],"title":"Verifying the Solution","text":"The solutions can be verified by subsituting these (x,y) values into both of the original equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac561a1systems16","title":"Finding the Intersection of a Circle and a Line by Substitution #5","body":"Solve the system of nonlinear equations using subsitution.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Systems of Nonlinear Equations and Inequalities: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ac561a1systems16a","stepAnswer":["(3sqrt(2)/2,-3sqrt(2)/2),(-3sqrt(2)/2,3sqrt(2)/2)"],"problemType":"MultipleChoice","stepTitle":"$$y=-x$$, $$x^2+y^2=9$$","stepBody":"","answerType":"string","variabilization":{},"choices":["(3sqrt(2)/2,-3sqrt(2)/2),(-3sqrt(2)/2,3sqrt(3)/2)","(2sqrt(2)/2,-3sqrt(2)/2),(-2sqrt(2)/2,3sqrt(2)/2)","(3sqrt(2)/2,-3sqrt(2)/2),(-3sqrt(2)/2,3sqrt(2)/2)"],"hints":{"DefaultPathway":[{"id":"ac561a1systems16a-h1","type":"hint","dependencies":[],"title":"Subsituting Y","text":"One of the equations has already been solved for $$y$$. We will substitute $$y=-x$$ into the equation for the circle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems16a-h2","type":"hint","dependencies":["ac561a1systems16a-h1"],"title":"Factoring the Equation","text":"Then, factor and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems16a-h3","type":"hint","dependencies":["ac561a1systems16a-h2"],"title":"Solving For Y","text":"Next, substitute the two x-values into the original linear equation to solve for $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems16a-h4","type":"hint","dependencies":["ac561a1systems16a-h3"],"title":"Verifying the Solution","text":"The solutions can be verified by subsituting these (x,y) values into both of the original equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac561a1systems17","title":"Solving a System of Nonlinear Equations Representing a Circle and an Ellipse #1","body":"Solve the system of nonlinear equations.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Systems of Nonlinear Equations and Inequalities: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ac561a1systems17a","stepAnswer":["$$(5,1),(5,-1),(-1,5),(-5,-1)$$"],"problemType":"MultipleChoice","stepTitle":"$$x^2+y^2=26$$, $$3x^2+25y^2=100$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(5,1),(5,-1),(-1,5),(-5,-1)$$","choices":["$$(4,1),(4,-1),(-1,4),(-4,-1)$$","$$(5,1),(5,-1),(-1,5),(-5,-1)$$","$$(3,2),(3,-2),(-2,3),(-3,-2)$$"],"hints":{"DefaultPathway":[{"id":"ac561a1systems17a-h1","type":"hint","dependencies":[],"title":"Multiplying and Adding Equations","text":"First, multiply the first equation by $$-3$$, and add it to the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems17a-h2","type":"hint","dependencies":["ac561a1systems17a-h1"],"title":"Solving For $$y$$","text":"After adding the two equations together, solve for $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems17a-h3","type":"hint","dependencies":["ac561a1systems17a-h2"],"title":"Solving For $$x$$","text":"Subsitute $$y=-1$$ and $$y=1$$ into one of the equations and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems17a-h4","type":"hint","dependencies":["ac561a1systems17a-h3"],"title":"Number of Solutions","text":"There are four solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac561a1systems18","title":"Solving a System of Nonlinear Equations Representing a Parabola and a Line #2","body":"Solve the system of equations.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Systems of Nonlinear Equations and Inequalities: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ac561a1systems18a","stepAnswer":["(-1/2,1/2),(2,8)"],"problemType":"MultipleChoice","stepTitle":"$$3x-y=-2$$, $$2x^2-y=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["(-1,2),(2/3,8)","(-1/3,1/3),(3,5)","(-1/2,1/2),(2,8)"],"hints":{"DefaultPathway":[{"id":"ac561a1systems18a-h1","type":"hint","dependencies":[],"title":"Solving for $$x$$","text":"Solve the first equation for $$x$$ and then substitute the resulting expression into the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems18a-h2","type":"hint","dependencies":["ac561a1systems18a-h1"],"title":"Solving for Zeroes","text":"Expand the equation and set it equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems18a-h3","type":"hint","dependencies":["ac561a1systems18a-h2"],"title":"Subsituting Each Y Value","text":"Next, substitute each value for $$y$$ into the first equation to solve for $$x$$. Always substitute the value into the linear equation to check for extraneous solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems18a-h4","type":"hint","dependencies":["ac561a1systems18a-h3"],"title":"Verifying the Solution","text":"The solutions can be verified by subsituting these (x,y) values into both of the original equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac561a1systems19","title":"Finding the Intersection of a Circle and a Line by Substitution #6","body":"Solve the system of nonlinear equations using subsitution.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Systems of Nonlinear Equations and Inequalities: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ac561a1systems19a","stepAnswer":["$$(-1,3)$$"],"problemType":"MultipleChoice","stepTitle":"$$x^2+y^2=10$$, $$x-3y=-10$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-1,3)$$","choices":["$$(-1,3)$$","$$(-4,6)$$","$$(-5,6)$$"],"hints":{"DefaultPathway":[{"id":"ac561a1systems19a-h1","type":"hint","dependencies":[],"title":"Subsituting Y","text":"Solve for $$y$$ in the second equation, and subsitute it into the equation for the circle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems19a-h2","type":"hint","dependencies":["ac561a1systems19a-h1"],"title":"Factoring the Equation","text":"Then, factor and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems19a-h3","type":"hint","dependencies":["ac561a1systems19a-h2"],"title":"Solving For Y","text":"Next, substitute the two x-values into the original linear equation to solve for $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems19a-h4","type":"hint","dependencies":["ac561a1systems19a-h3"],"title":"Verifying the Solution","text":"The solutions can be verified by subsituting these (x,y) values into both of the original equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac561a1systems2","title":"Solving a System of Nonlinear Equations Using Elimination","body":"Solve the system of nonlinear equations using elimination.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Systems of Nonlinear Equations and Inequalities: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ac561a1systems2a","stepAnswer":["$$(-\\\\sqrt{13},-\\\\sqrt{12})$$, $$(-\\\\sqrt{13},\\\\sqrt{12})$$, $$(\\\\sqrt{13},-\\\\sqrt{12})$$, $$(\\\\sqrt{13},\\\\sqrt{12})$$"],"problemType":"MultipleChoice","stepTitle":"$$x^2+y^2=25$$,\\\\n$$x^2-y^2=1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\sqrt{13},-\\\\sqrt{12})$$, $$(-\\\\sqrt{13},\\\\sqrt{12})$$, $$(\\\\sqrt{13},-\\\\sqrt{12})$$, $$(\\\\sqrt{13},\\\\sqrt{12})$$","choices":["$$(-\\\\sqrt{13},-\\\\sqrt{12})$$, $$(-\\\\sqrt{13},\\\\sqrt{12})$$, $$(\\\\sqrt{13},-\\\\sqrt{12})$$, $$(\\\\sqrt{13},\\\\sqrt{12})$$","$$(-\\\\sqrt{13},-\\\\sqrt{12})$$, $$(-\\\\sqrt{13},\\\\sqrt{12})$$","$$(\\\\sqrt{13},-\\\\sqrt{12})$$, $$(\\\\sqrt{13},\\\\sqrt{12})$$","No Solution"],"hints":{"DefaultPathway":[{"id":"ac561a1systems2a-h1","type":"hint","dependencies":[],"title":"Eliminating Variables","text":"When both equations in the system have like variables of the second degree, solving them using elimination by addition is often easier than substitution. Generally, elimination is a far simpler method when the system involves only two equations in two variables (a two-by-two system), rather than a three-by-three system, as there are fewer steps.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems2a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2x^2=26$$"],"dependencies":["ac561a1systems2a-h1"],"title":"Adding Equations","text":"We observe that by adding the two equations together, we can remove the $$y^2$$ term. What is the equation obtained after adding the two equations?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2x^2=26$$","$$2x^2=24$$","$$2y^2=26$$"]},{"id":"ac561a1systems2a-h3","type":"hint","dependencies":["ac561a1systems2a-h2"],"title":"Solving for $$x$$","text":"With the equation $$2x^2=26$$, we can solve for $$x$$. The x-coordinate found indicates a possible solution where an intersection may occur if the system of equations yield a feasible solution later on.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems2a-h4","type":"hint","dependencies":["ac561a1systems2a-h3"],"title":"Solving for $$x$$","text":"Divide $$2$$ on boths sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems2a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["-sqrt(13),sqrt(13)"],"dependencies":["ac561a1systems2a-h4"],"title":"Solving for $$x$$","text":"Take the square root on both sides. What are all the possible $$x$$ coordinates?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["-sqrt(13),sqrt(13)","$$-13, 13$$","-sqrt(12),sqrt(12)"]},{"id":"ac561a1systems2a-h6","type":"hint","dependencies":["ac561a1systems2a-h5"],"title":"Substitution","text":"We substitute all possible $$x$$ coordinates back into one of the two equations to find the corresponding $$y$$ coordinate to each $$x$$ coordinate. Using any of the two equations work because the $$x$$ coordinate found is a solution to the system of equations. We usually use the equation that is easier to calculate the other coordinate with.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems2a-h7","type":"hint","dependencies":["ac561a1systems2a-h6"],"title":"Substitution #1","text":"Substituting $$x=-\\\\sqrt{13}$$ into $$x^2+y^2=25$$. Find all possible corresponding $$y$$ coordinates.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems2a-h8","type":"hint","dependencies":["ac561a1systems2a-h7"],"title":"Substitution #1","text":"After substituting in $$-\\\\sqrt{13}$$ and squaring it, subtract $$13$$ from both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems2a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["-sqrt(12),sqrt(12)"],"dependencies":["ac561a1systems2a-h8"],"title":"Substitution #1","text":"Take the square root on both sides. What are all the possible $$y$$ coordinates?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["-sqrt(12),sqrt(12)","$$-12, 12$$"]},{"id":"ac561a1systems2a-h10","type":"hint","dependencies":["ac561a1systems2a-h9"],"title":"Substitution #2","text":"Substituting $$x=\\\\sqrt{13}$$ into $$x^2+y^2=25$$. Find all possible corresponding $$y$$ coordinates.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems2a-h11","type":"hint","dependencies":["ac561a1systems2a-h10"],"title":"Substitution #2","text":"After substituting in $$\\\\sqrt{13}$$ and squaring it, subtract $$13$$ from both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems2a-h12","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["-sqrt(12),sqrt(12)"],"dependencies":["ac561a1systems2a-h11"],"title":"Substitution #2","text":"Take the square root on both sides. What are all the possible $$y$$ coordinates?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["-sqrt(12),sqrt(12)","$$-12, 12$$"]},{"id":"ac561a1systems2a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ac561a1systems2a-h12"],"title":"Verifying the Number Of Intersections","text":"We have found the coordinates of all intersections. How many are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac561a1systems20","title":"Solving a System of Nonlinear Equations Representing a Circle and an Ellipse #1","body":"Solve the system of nonlinear equations.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Systems of Nonlinear Equations and Inequalities: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ac561a1systems20a","stepAnswer":["$$(1,3),(1,-3),(-1,3),(-1,-3)$$"],"problemType":"MultipleChoice","stepTitle":"$$4x^2+y^2=13$$, $$x^2+y^2=10$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(1,3),(1,-3),(-1,3),(-1,-3)$$","choices":["$$(1,4),(1,-4),(-1,4),(-1,-4)$$","$$(1,3),(1,-3),(-1,3),(-1,-3)$$","$$(1,2),(1,-2),(-1,2),(-1,-2)$$"],"hints":{"DefaultPathway":[{"id":"ac561a1systems20a-h1","type":"hint","dependencies":[],"title":"Multiplying and Adding Equations","text":"First, multiply the second equation by $$-4$$, and add it to the first equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems20a-h2","type":"hint","dependencies":["ac561a1systems20a-h1"],"title":"Solving For $$y$$","text":"After adding the two equations together, solve for $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems20a-h3","type":"hint","dependencies":["ac561a1systems20a-h2"],"title":"Solving For $$x$$","text":"Subsitute the $$y$$ values into one of the equations and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems20a-h4","type":"hint","dependencies":["ac561a1systems20a-h3"],"title":"Number of Solutions","text":"There are four solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac561a1systems21","title":"Solving Systems","body":"Use any method to solve the system of nonlinear equations.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Systems of Nonlinear Equations and Inequalities: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ac561a1systems21a","stepAnswer":["$$(5,0)$$"],"problemType":"MultipleChoice","stepTitle":"$$9x^2+25y^2=225$$\\\\n$${\\\\left(x-6\\\\right)}^2+y^2=1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(5,0)$$","choices":["$$(5,0)$$","$$(0,5)$$","$$(-5,0)$$","$$(0,-5)$$"],"hints":{"DefaultPathway":[{"id":"ac561a1systems21a-h1","type":"hint","dependencies":[],"title":"Multiply","text":"First, multiply the second equation by $$25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems21a-h2","type":"hint","dependencies":["ac561a1systems21a-h1"],"title":"Subtract","text":"Next, subtract the two equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems21a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$9x^2-{\\\\operatorname{25}\\\\left(x-6\\\\right)}^2=200$$"],"dependencies":["ac561a1systems21a-h2"],"title":"Subtract","text":"What equation do you get after subtracting the two original equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$-9x^2+{\\\\operatorname{25}\\\\left(x-6\\\\right)}^2=-200$$","$$9x^2-{\\\\operatorname{25}\\\\left(x-6\\\\right)}^2=200$$","$$18x^2-{2\\\\left(x-6\\\\right)}^2=400$$"]},{"id":"ac561a1systems21a-h4","type":"hint","dependencies":["ac561a1systems21a-h3"],"title":"Expand","text":"Now, expand the quantity of $${\\\\left(x-6\\\\right)}^2$$ and multiply by $$-25$$ to simplify.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems21a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$9x^2-25x^2+300x-900$$"],"dependencies":["ac561a1systems21a-h4"],"title":"Expand and multiply","text":"What is your new equation after expanding and multiplying $$-\\\\left({\\\\operatorname{25}\\\\left(x-6\\\\right)}^2\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$9x^2-25x^2+300x-900$$","$$-9x^2+25x^2-300x+900$$","$$18x^2-5x^2+300x-900$$"]},{"id":"ac561a1systems21a-h6","type":"hint","dependencies":["ac561a1systems21a-h5"],"title":"Solve","text":"Now, solve for $$x$$ by combining like terms and factoring.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems21a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$5$$ and $$\\\\frac{55}{4}$$"],"dependencies":["ac561a1systems21a-h6"],"title":"Solve","text":"What are the two solutions for $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$5$$ and $$\\\\frac{55}{4}$$","$$-5$$ and $$\\\\frac{55}{4}$$"]},{"id":"ac561a1systems21a-h8","type":"hint","dependencies":["ac561a1systems21a-h7"],"title":"Substitute","text":"Now, plug in both values for $$x$$ in seperate equations. $$x=\\\\frac{55}{4}$$ yields an equation with the imaginary number i, so you can disregard it, as it is not a real solution. Plug $$5$$ into either of the original equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems21a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["ac561a1systems21a-h8"],"title":"Substitute","text":"What does $$y$$ equal when you plug in $$x=5$$ into either of the equations","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems21a-h10","type":"hint","dependencies":["ac561a1systems21a-h9"],"title":"Answer","text":"The final solution for the system of nonlinear equations is $$(5,0)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac561a1systems22","title":"Solving Systems","body":"Use any method to solve the system of nonlinear equations.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Systems of Nonlinear Equations and Inequalities: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ac561a1systems22a","stepAnswer":["$$(0,0)$$"],"problemType":"MultipleChoice","stepTitle":"$$x^4-x^2=y$$\\\\n$$x^2+y=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,0)$$","choices":["$$(4,9)$$","$$(0,0)$$","$$(2,10)$$","$$(3,6)$$"],"hints":{"DefaultPathway":[{"id":"ac561a1systems22a-h1","type":"hint","dependencies":[],"title":"Solve","text":"First, solve for $$y$$ in the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems22a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y=-\\\\left(x^2\\\\right)$$"],"dependencies":["ac561a1systems22a-h1"],"title":"Solve","text":"What does $$y$$ equal when it is solved in terms of $$x$$ in the second equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y=-\\\\left(x^2\\\\right)$$","$$y=x^2$$","$$y=2x$$","$$y=-2x$$"]},{"id":"ac561a1systems22a-h3","type":"hint","dependencies":["ac561a1systems22a-h1"],"title":"Substitute","text":"Next, plug $$y=-\\\\left(x^2\\\\right)$$ into the first equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems22a-h4","type":"hint","dependencies":["ac561a1systems22a-h3"],"title":"Simplify","text":"Next, simplify the first equation so that you have one $$x$$ term. You can add $$x^2$$ to both sides to eliminate it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems22a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["ac561a1systems22a-h4"],"title":"Solve","text":"Solve for $$x$$. What does $$x$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems22a-h6","type":"hint","dependencies":["ac561a1systems22a-h5"],"title":"Substitute","text":"Now, substitute $$x=0$$ into either of the original equations to find $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems22a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["ac561a1systems22a-h6"],"title":"Solve","text":"What does $$y$$ equal when $$x=0$$ is plugged in?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems22a-h8","type":"hint","dependencies":["ac561a1systems22a-h7"],"title":"Answer","text":"The final answer is therefore $$(0,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac561a1systems23","title":"Solving Systems","body":"Use any method to solve the system of nonlinear equations.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Systems of Nonlinear Equations and Inequalities: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ac561a1systems23a","stepAnswer":["$$(0,0)$$"],"problemType":"MultipleChoice","stepTitle":"$$2x^3-x^2=y$$\\\\n$$x^2+y=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,0)$$","choices":["$$(0,0)$$","$$(4,9)$$","$$(3,6)$$","$$(2,10)$$"],"hints":{"DefaultPathway":[{"id":"ac561a1systems23a-h1","type":"hint","dependencies":[],"title":"Solve","text":"First, solve for $$y$$ in the second equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems23a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y=-\\\\left(x^2\\\\right)$$"],"dependencies":["ac561a1systems23a-h1"],"title":"Solve","text":"What does $$y$$ equal when it is solved in terms of $$x$$ in the second equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y=-\\\\left(x^2\\\\right)$$","$$y=x^2$$","$$y=2x$$","$$y=-2x$$"]},{"id":"ac561a1systems23a-h3","type":"hint","dependencies":["ac561a1systems23a-h2"],"title":"Substitute","text":"Next, plug $$y=-\\\\left(x^2\\\\right)$$ into the first equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems23a-h4","type":"hint","dependencies":["ac561a1systems23a-h3"],"title":"Simplify","text":"Next, simplify the first equation so that you have one $$x$$ term. You can add $$x^2$$ to both sides to eliminate it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems23a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["ac561a1systems23a-h4"],"title":"Solve","text":"Solve for $$x$$. What does $$x$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems23a-h6","type":"hint","dependencies":["ac561a1systems23a-h5"],"title":"Substitute","text":"Now, substitute $$x=0$$ into either of the original equations to find $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems23a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["ac561a1systems23a-h6"],"title":"Solve","text":"What does $$y$$ equal when $$x=0$$ is plugged in?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems23a-h8","type":"hint","dependencies":["ac561a1systems23a-h7"],"title":"Answer","text":"The final answer is therefore $$(0,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac561a1systems24","title":"Solving Systems","body":"Use any method to solve the system of nonlinear equations.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Systems of Nonlinear Equations and Inequalities: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ac561a1systems24a","stepAnswer":["$$(0,3)$$, $$(\\\\sqrt{5},-2)$$, $$(-\\\\sqrt{5},-2)$$"],"problemType":"MultipleChoice","stepTitle":"$$x^2+y^2=9$$\\\\n$$y=3-x^2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,3)$$, $$(\\\\sqrt{5},-2)$$, $$(-\\\\sqrt{5},-2)$$","choices":["$$(0,3)$$, $$(\\\\sqrt{5},-2)$$","$$(0,3)$$","$$(0,3)$$, $$(\\\\sqrt{5},-2)$$, $$(-\\\\sqrt{5},-2)$$"],"hints":{"DefaultPathway":[{"id":"ac561a1systems24a-h1","type":"hint","dependencies":[],"title":"Subtract","text":"First, subtract the two equations to eliminate $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems24a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["x**2+y**2-y=6+x**2,y**2-y=6"],"dependencies":["ac561a1systems24a-h1"],"title":"Subtract","text":"After subtracting, what equation are you left with?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems24a-h3","type":"hint","dependencies":["ac561a1systems24a-h2"],"title":"Simplify","text":"Now, simplfiy the equation to eliminate $$x^2$$ by subtracting it from each side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems24a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y^2-y=6$$"],"dependencies":["ac561a1systems24a-h3"],"title":"Simplify","text":"What are you left with after subtracting $$x^2$$ from both sides?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y^2-y=6$$","$$x^2-x=6$$","$$y^2-x=6$$","$$x^2-y=6$$"]},{"id":"ac561a1systems24a-h5","type":"hint","dependencies":["ac561a1systems24a-h4"],"title":"Simplfiy","text":"Subtract $$6$$ from both sides to get $$y^2-y-6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems24a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\left(y-3\\\\right) \\\\left(y+2\\\\right)$$"],"dependencies":["ac561a1systems24a-h5"],"title":"Factor","text":"Now, factor the equation. Your answer should be in the form $$\\\\left(ay+b\\\\right) \\\\left(cy+d\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\left(y-3\\\\right) \\\\left(y+2\\\\right)$$","$$\\\\left(y+3\\\\right) \\\\left(y-2\\\\right)$$","$$\\\\left(x+3\\\\right) \\\\left(x-2\\\\right)$$","$$\\\\left(x-3\\\\right) \\\\left(x+2\\\\right)$$"]},{"id":"ac561a1systems24a-h7","type":"hint","dependencies":["ac561a1systems24a-h6"],"title":"Find solutions","text":"This means $$y=3$$ and $$y=-2$$. Plug both of these values into either of the original equations and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems24a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["ac561a1systems24a-h7"],"title":"Substitute and Solve","text":"Plug $$y=3$$ into either of the original equations. What does $$x$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems24a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{5}$$"],"dependencies":["ac561a1systems24a-h8"],"title":"Substitute and Solve","text":"Plug $$y=-2$$ into either of the original equations. What does $$x$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems24a-h10","type":"hint","dependencies":["ac561a1systems24a-h9"],"title":"Answer","text":"Therefore, the final answers are three coordinate points: (0,3)(sqrt(5),-2)(-sqrt(5),-2).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac561a1systems25","title":"Solving Systems","body":"Use any method to solve the system of nonlinear equations.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Systems of Nonlinear Equations and Inequalities: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ac561a1systems25a","stepAnswer":["$$(3,0)$$"],"problemType":"MultipleChoice","stepTitle":"$$x^2-y^2=9$$\\\\n$$x=3$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(3,0)$$","choices":["$$(0,3)$$","$$(0,-3)$$","$$(-3,0)$$","$$(3,0)$$"],"hints":{"DefaultPathway":[{"id":"ac561a1systems25a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"First, substitute $$x=3$$ into the first equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems25a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["ac561a1systems25a-h1"],"title":"Simplify and solve","text":"Now, simplify the first equation until $$y$$ is on one side. Then solve for $$y$$. What does $$y$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems25a-h3","type":"hint","dependencies":["ac561a1systems25a-h2"],"title":"Answer","text":"$$x=3$$ is given, and the y-coordinate is $$0$$, so the solution to the system of equations is $$(3,0)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac561a1systems26","title":"Solving Systems","body":"Use any method to solve the system of nonlinear equations.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Systems of Nonlinear Equations and Inequalities: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ac561a1systems26a","stepAnswer":["$$(18,3)$$"],"problemType":"MultipleChoice","stepTitle":"$$x^2-y^2=9$$\\\\n$$y=3$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(18,3)$$","choices":["$$(3,18)$$","$$(-3,18)$$","$$(18,3)$$","$$(-18,3)$$"],"hints":{"DefaultPathway":[{"id":"ac561a1systems26a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"First, substitute $$y=3$$ into the first equation and simplify.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems26a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18$$"],"dependencies":["ac561a1systems26a-h1"],"title":"Solve","text":"Next, solve for $$x$$. What does $$x$$ equal?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ac561a1systems26a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$18$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ac561a1systems26a-h3","type":"hint","dependencies":["ac561a1systems26a-h2"],"title":"Answer","text":"$$y=3$$ is given, and the x-coordinate is $$18$$, so the solution to the system of equations is $$(18,3))$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac561a1systems27","title":"Solving Systems","body":"Use any method to solve the system of nonlinear equations.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Systems of Nonlinear Equations and Inequalities: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ac561a1systems27a","stepAnswer":["No solutions exist."],"problemType":"MultipleChoice","stepTitle":"$$x^2-y^2=9$$\\\\n$$x-y=0$$","stepBody":"Are there solutions for this system of equations?","answerType":"string","variabilization":{},"choices":["No solutions exist.","Solutions exist"],"hints":{"DefaultPathway":[{"id":"ac561a1systems27a-h1","type":"hint","dependencies":[],"title":"Evaluate","text":"First, solve the second equation to receieve $$x=y$$. Try plugging this into the first equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems27a-h2","type":"hint","dependencies":["ac561a1systems27a-h1"],"title":"Answer","text":"When plugging this in, the equation will have a $$constant=9$$. This implies that no solutions exist.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ac561a1systems27b","stepAnswer":["$$(1,3),(3,11)$$"],"problemType":"MultipleChoice","stepTitle":"$$-\\\\left(x^2\\\\right)+y=2$$\\\\n$$-4x+y=-1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(1,3),(3,11)$$","choices":["$$(11,3)$$","$$(1,3)$$","$$(1,3),(3,11)$$","None of the Above"],"hints":{"DefaultPathway":[{"id":"ac561a1systems27b-h1","type":"hint","dependencies":[],"title":"Subtract","text":"The first step is to subtract both equations from each other, eliminating $$y$$, then moving all terms to the left side of the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems27b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-\\\\left(x^2\\\\right)+4x-3$$"],"dependencies":["ac561a1systems27b-h1"],"title":"Subtract and Simplify","text":"After subtracting the two equations and bringing all terms to the left side, what equation do you have?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$-\\\\left(x^2\\\\right)+4x-3$$","$$x^2-4x+3$$","$$2x^2+6x+1$$","None of the Above"]},{"id":"ac561a1systems27b-h3","type":"hint","dependencies":["ac561a1systems27b-h2"],"title":"Factor","text":"The next step is to factor the equation and set it equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems27b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(x-1)(x-3)$$"],"dependencies":["ac561a1systems27b-h3"],"title":"Factor","text":"What is the factored form of the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(x-1)(x-3)$$","$$\\\\left(x+1\\\\right) \\\\left(x-3\\\\right)$$","$$\\\\left(x-1\\\\right) \\\\left(x+3\\\\right)$$","None of the Above"]},{"id":"ac561a1systems27b-h5","type":"hint","dependencies":["ac561a1systems27b-h4"],"title":"Solve","text":"Next, solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems27b-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y=3, 11$$"],"dependencies":["ac561a1systems27b-h5"],"title":"Substitute and Solve","text":"Next, plug $$x=1$$ and $$x=3$$ into either of the original equations. Solve for $$y$$ in both cases. What are the values of $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y=3, 11$$","$$y=-3, 11$$","$$y=3, -11$$","$$y=-3, -11$$"]},{"id":"ac561a1systems27b-h7","type":"hint","dependencies":["ac561a1systems27b-h6"],"title":"Answer","text":"Therefore, the solutions to the nonlinear sysytem are $$(1,3)(3,11)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac561a1systems28","title":"Solving Systems","body":"Use any method to solve the system of nonlinear equations.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Systems of Nonlinear Equations and Inequalities: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ac561a1systems28a","stepAnswer":["No solutions exist."],"problemType":"MultipleChoice","stepTitle":"Does the following system have a solution?","stepBody":"$$-\\\\left(x^2\\\\right)+y=2$$\\\\n$$2y=-x$$","answerType":"string","variabilization":{},"choices":["No solutions exist.","Solutions exist"],"hints":{"DefaultPathway":[{"id":"ac561a1systems28a-h1","type":"hint","dependencies":[],"title":"Subsititute","text":"First, multiply by $$-1$$ on both sides of the second equation, then substiute $$-2y$$ for $$x$$ in the first equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems28a-h2","type":"hint","dependencies":["ac561a1systems28a-h1"],"title":"Simplify and Solve","text":"Next, simplify the equation so that all terms are squared properly and all terms are on the left side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems28a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ac561a1systems28a-h2"],"title":"Quadratic equation","text":"Now, use the quadratic equation to find the intercepts of the function. Are there any real solutions?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ac561a1systems28a-h4","type":"hint","dependencies":["ac561a1systems28a-h3"],"title":"Therefore, since there are no real solutions for $$y$$, the system of nonlinear equations has no solutions.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac561a1systems29","title":"Solving Systems","body":"Use any method to solve the system of nonlinear equations.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Systems of Nonlinear Equations and Inequalities: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ac561a1systems29a","stepAnswer":["No solutions exist."],"problemType":"MultipleChoice","stepTitle":"Does the following system have a solution?","stepBody":"$$x^2+y^2=25$$\\\\n$$x^2-y^2=36$$","answerType":"string","variabilization":{},"choices":["No solutions exist.","Solutions exist"],"hints":{"DefaultPathway":[{"id":"ac561a1systems29a-h1","type":"hint","dependencies":[],"title":"Subtract","text":"First, subtract the two equations from each other to eliminate $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems29a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2y^2=-11$$"],"dependencies":["ac561a1systems29a-h1"],"title":"Simplify","text":"Now, what are you left with after subtracting?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ac561a1systems29a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$2y^2=-11$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ac561a1systems29a-h3","type":"hint","dependencies":["ac561a1systems29a-h2"],"title":"Answer","text":"When solving the previous equation for $$y$$, you are left with an imaginary $$\\\\operatorname{number}\\\\left(\\\\sqrt{\\\\frac{-11}{2}}\\\\right)$$. This means there are no real solutions to the system of equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac561a1systems3","title":"Solving a System of Nonlinear Equations Using Elimination","body":"Solve the system of nonlinear equations using elimination.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Systems of Nonlinear Equations and Inequalities: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ac561a1systems3a","stepAnswer":["(1/4,-sqrt(62)/8),(1/4,sqrt(62)/8)"],"problemType":"MultipleChoice","stepTitle":"$$2x^2+4y^2=4$$,\\\\n$$2x^2-4y^2=25x-10$$","stepBody":"","answerType":"string","variabilization":{},"choices":["(1/4,-sqrt(62)/8),(1/4,sqrt(62)/8)","(-1/4,-sqrt(62)/8),(-1/4,sqrt(62)/8)","(-1/4,sqrt(62)/8),(1/4,sqrt(62)/8)","No Solution"],"hints":{"DefaultPathway":[{"id":"ac561a1systems3a-h1","type":"hint","dependencies":[],"title":"Eliminating Variables","text":"When both equations in the system have like variables of the second degree, solving them using elimination by addition is often easier than substitution. Generally, elimination is a far simpler method when the system involves only two equations in two variables (a two-by-two system), rather than a three-by-three system, as there are fewer steps.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems3a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$4x^2-25x+6=0$$"],"dependencies":["ac561a1systems3a-h1"],"title":"Adding Equations","text":"We observe that by adding the two equations together, we can remove the $$y^2$$ term. What is the equation obtained after adding the two equations?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$4x^2-25x+6=0$$","$$4x^2+25x-6=0$$"]},{"id":"ac561a1systems3a-h3","type":"hint","dependencies":["ac561a1systems3a-h2"],"title":"Solving for $$x$$","text":"With the equation $$4x^2-25x+6=0$$, we can solve for $$x$$. The x-coordinate found indicates a possible solution where an intersection may occur if the system of equations yield a feasible solution later on.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems3a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\left(4x-1\\\\right) \\\\left(x-6\\\\right)$$"],"dependencies":["ac561a1systems3a-h3"],"title":"Solving for $$x$$","text":"Factorize $$4x^2-25x+6$$, what are the factors?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\left(4x-1\\\\right) \\\\left(x-6\\\\right)$$","$$\\\\left(4x+1\\\\right) \\\\left(x+6\\\\right)$$","$$\\\\left(4x+1\\\\right) \\\\left(x-6\\\\right)$$"]},{"id":"ac561a1systems3a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["1/4,6"],"dependencies":["ac561a1systems3a-h4"],"title":"Solving for $$x$$","text":"What are all the possible $$x$$ coordinates?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["1/4,6","$$-6-\\\\frac{1}{4}$$","-1/4,6","-6,1/4"]},{"id":"ac561a1systems3a-h6","type":"hint","dependencies":["ac561a1systems3a-h5"],"title":"Subsituting Possible Values","text":"We substitute all possible $$x$$ coordinates back into one of the two equations to find the corresponding $$y$$ coordinate to each $$x$$ coordinate. Using any of the two equations work because the $$x$$ coordinate found is a solution to the system of equations. We usually use the equation that is easier to calculate the other coordinate with.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems3a-h7","type":"hint","dependencies":["ac561a1systems3a-h6"],"title":"Substitution #1","text":"Substituting $$x=\\\\frac{1}{4}$$ into $$2x^2+4y^2=4$$. Find all possible corresponding $$y$$ coordinates.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems3a-h8","type":"hint","dependencies":["ac561a1systems3a-h7"],"title":"Substitution #1","text":"After substituting in $$\\\\frac{1}{4}$$ and squaring it, subtract $$\\\\frac{1}{8}$$ from both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems3a-h9","type":"hint","dependencies":["ac561a1systems3a-h8"],"title":"Substitution #1","text":"Divide by $$4$$ on both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems3a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["-sqrt(62)/8,sqrt(62)/8"],"dependencies":["ac561a1systems3a-h9"],"title":"Substitution #1","text":"Take the square root on both sides. What are all the possible $$y$$ coordinates? You should rationalize the square roots in the denominator and simplify the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["-sqrt(62)/8,sqrt(62)/8","-sqrt(31)/8,sqrt(31)/8","-sqrt(31)/4,sqrt(31)/4"]},{"id":"ac561a1systems3a-h11","type":"hint","dependencies":["ac561a1systems3a-h10"],"title":"Substitution #2","text":"Substituting $$x=6$$ into $$2x^2+4y^2=4$$. Find all possible corresponding $$y$$ coordinates.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems3a-h12","type":"hint","dependencies":["ac561a1systems3a-h11"],"title":"Substitution #2","text":"After substituting in $$6$$ and squaring it, subtract $$72$$ from both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems3a-h13","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ac561a1systems3a-h12"],"title":"Substitution #2","text":"Divide by $$4$$ on both sides. We observe that $$y^2$$ is equal to a negative number. Is there a real solution for $$y$$ to this equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ac561a1systems3a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ac561a1systems3a-h13"],"title":"Verifying the Number Of Intersections","text":"We have found the coordinates of all intersections. How many are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac561a1systems4","title":"Solving a System of Nonlinear Equations Using Elimination","body":"Solve the system of nonlinear equations using elimination.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Systems of Nonlinear Equations and Inequalities: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ac561a1systems4a","stepAnswer":["No Solution"],"problemType":"MultipleChoice","stepTitle":"$$y^2-x^2=9$$,\\\\n$$3x^2+2y^2=8$$","stepBody":"","answerType":"string","variabilization":{},"choices":["(-sqrt(2),sqrt(7)),(sqrt(2),sqrt(7))","(sqrt(2),-sqrt(7)),(sqrt(2),sqrt(7))","(-sqrt(2),sqrt(7)),(sqrt(2),sqrt(7)),(sqrt(2),-sqrt(7)),(sqrt(2),sqrt(7))","No Solution"],"hints":{"DefaultPathway":[{"id":"ac561a1systems4a-h1","type":"hint","dependencies":[],"title":"Eliminating Variables","text":"When both equations in the system have like variables of the second degree, solving them using elimination by addition is often easier than substitution. Generally, elimination is a far simpler method when the system involves only two equations in two variables (a two-by-two system), rather than a three-by-three system, as there are fewer steps.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems4a-h2","type":"hint","dependencies":["ac561a1systems4a-h1"],"title":"Common Terms between Equations","text":"We observe that we can multiply $$3$$ to the first equation, so that $$3x^2$$ becomes a common term in both equations. This would allow us to solve the system by elimination.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems4a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$5y^2=35$$"],"dependencies":["ac561a1systems4a-h2"],"title":"Adding Equations","text":"After multiplying $$3$$ to the first equation, we observe that by adding the two equations together, we can remove the $$x^2$$ term. What is the equation obtained after adding the two equations?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$5y^2=35$$","$$y^2=19$$"]},{"id":"ac561a1systems4a-h4","type":"hint","dependencies":["ac561a1systems4a-h3"],"title":"Solving for $$y$$","text":"With the equation $$5y^2=35$$, we can solve for $$y$$. The y-coordinate found indicates a possible solution where an intersection may occur if the system of equations yield a feasible solution later on.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems4a-h5","type":"hint","dependencies":["ac561a1systems4a-h4"],"title":"Solving for $$y$$","text":"Divide by $$5$$ on both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems4a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["-sqrt(7),sqrt(7)"],"dependencies":["ac561a1systems4a-h5"],"title":"Solving for $$y$$","text":"Take the square root on both sides. What are all the possible $$y$$ coordinates?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["-sqrt(7),sqrt(7)","$$-7, 7$$"]},{"id":"ac561a1systems4a-h7","type":"hint","dependencies":["ac561a1systems4a-h6"],"title":"Subsituting Possible Values","text":"We substitute all possible $$y$$ coordinates back into one of the two equations to find the corresponding $$x$$ coordinate to each $$y$$ coordinate. Using any of the two equations work because the $$y$$ coordinate found is a solution to the system of equations. We usually use the equation that is easier to calculate the other coordinate with.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems4a-h8","type":"hint","dependencies":["ac561a1systems4a-h7"],"title":"Substitution #1","text":"Substituting $$y=-\\\\sqrt{7}$$ into $$y^2-x^2=9$$. Find all possible corresponding $$x$$ coordinates.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems4a-h9","type":"hint","dependencies":["ac561a1systems4a-h8"],"title":"Substitution #1","text":"After substituting in $$-\\\\sqrt{7}$$ and squaring it, subtract $$7$$ from both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems4a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ac561a1systems4a-h9"],"title":"Substitution #1","text":"Multiply by $$-1$$ on both sides. We observe that $$x^2$$ is equal to a negative number. Is there a real solution for $$x$$ to this equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ac561a1systems4a-h11","type":"hint","dependencies":["ac561a1systems4a-h10"],"title":"Substitution #2","text":"Substituting $$y=\\\\sqrt{7}$$ into $$y^2-x^2=9$$. Find all possible corresponding $$x$$ coordinates.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems4a-h12","type":"hint","dependencies":["ac561a1systems4a-h11"],"title":"Substitution #2","text":"After substituting in $$\\\\sqrt{7}$$ and squaring it, subtract $$7$$ from both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems4a-h13","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ac561a1systems4a-h12"],"title":"Substitution #2","text":"Multiply by $$-1$$ on both sides. We observe that $$x^2$$ is equal to a negative number. Is there a real solution for $$x$$ to this equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ac561a1systems4a-h14","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ac561a1systems4a-h13"],"title":"Verifying the Number Of Intersections","text":"Were we able to find any intersection?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"ac561a1systems5","title":"Solving a System of Nonlinear Equations Using Elimination","body":"Solve the system of nonlinear equations using elimination.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Systems of Nonlinear Equations and Inequalities: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ac561a1systems5a","stepAnswer":["(-sqrt(398)/4,199/4),(sqrt(398)/4,199/4)"],"problemType":"MultipleChoice","stepTitle":"$$x^2+y^2+\\\\frac{1}{16}=2500$$,\\\\n$$y=2x^2$$","stepBody":"","answerType":"string","variabilization":{},"choices":["(-sqrt(398)/4,199/4),(sqrt(398)/4,199/4)","(sqrt(398)/4,-199/4),(sqrt(398)/4,199/4)","No Solution"],"hints":{"DefaultPathway":[{"id":"ac561a1systems5a-h1","type":"hint","dependencies":[],"title":"Eliminating Variables","text":"When both equations in the system have like variables of the second degree, solving them using elimination by addition is often easier than substitution. Generally, elimination is a far simpler method when the system involves only two equations in two variables (a two-by-two system), rather than a three-by-three system, as there are fewer steps.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems5a-h2","type":"hint","dependencies":["ac561a1systems5a-h1"],"title":"Common Terms between Equations","text":"We observe that we can multiply $$2$$ to the first equation, so that $$2x^2$$ becomes a common term in both equations. This would allow us to solve the system by elimination.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems5a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2y^2+y+\\\\frac{1}{8}-5000=0$$"],"dependencies":["ac561a1systems5a-h2"],"title":"Adding Equations","text":"We observe that by adding the two equations together, we can remove the $$x^2$$ term. What is the equation obtained after adding the two equations?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2y^2+y+\\\\frac{1}{8}-5000=0$$","$$y^2+2y+\\\\frac{1}{16}-5000=0$$"]},{"id":"ac561a1systems5a-h4","type":"hint","dependencies":["ac561a1systems5a-h3"],"title":"Solving for $$y$$","text":"With the equation $$2y^2+y+\\\\frac{1}{8}-5000=0$$, we can solve for $$y$$. The y-coordinate found indicates a possible solution where an intersection may occur if the system of equations yield a feasible solution later on.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems5a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["-201/4,199/4"],"dependencies":["ac561a1systems5a-h4"],"title":"Solving for $$y$$","text":"We can use the quadratic formula $$y=\\\\frac{\\\\left(-b+-\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$ to obtain the solutions. What are the $$y$$ coordinates?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["-201/4,199/4","-199/4,201/4"]},{"id":"ac561a1systems5a-h6","type":"hint","dependencies":["ac561a1systems5a-h5"],"title":"Subsituting Possible Values","text":"We substitute all possible $$y$$ coordinates back into one of the two equations to find the corresponding $$x$$ coordinate to each $$y$$ coordinate. Using any of the two equations work because the $$y$$ coordinate found is a solution to the system of equations. We usually use the equation that is easier to calculate the other coordinate with.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems5a-h7","type":"hint","dependencies":["ac561a1systems5a-h6"],"title":"Substitution #1","text":"Substituting $$y=\\\\frac{-201}{4}$$ into $$y=2x^2$$. Find all possible corresponding $$x$$ coordinates.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems5a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ac561a1systems5a-h7"],"title":"Substitution #1","text":"Divide $$2$$ on both sides. We observe that $$x^2$$ is equal to a negative number. Is there a real solution for $$x$$ to this equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ac561a1systems5a-h9","type":"hint","dependencies":["ac561a1systems5a-h8"],"title":"Substitution #2","text":"Substituting $$y=\\\\frac{199}{4}$$ into $$y=2x^2$$. Find all possible corresponding $$x$$ coordinates.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems5a-h10","type":"hint","dependencies":["ac561a1systems5a-h9"],"title":"Substitution #2","text":"Divide by $$2$$ on both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems5a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["-sqrt(398)/4,sqrt(398)/4"],"dependencies":["ac561a1systems5a-h10"],"title":"Substitution #2","text":"Take the square root on both sides. What are all the possible $$x$$ coordinates? You should rationalize the square roots in the denominator and simplify the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["-sqrt(398)/4,sqrt(398)/4","$$\\\\frac{199}{8}$$","-sqrt(199)/4,sqrt(199)/4"]},{"id":"ac561a1systems5a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ac561a1systems5a-h11"],"title":"Verifying the Number Of Intersections","text":"We have found the coordinates of all intersections. How many are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac561a1systems6","title":"Solving a System of Nonlinear Equations Using Elimination","body":"Solve the system of nonlinear equations using elimination.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Systems of Nonlinear Equations and Inequalities: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ac561a1systems6a","stepAnswer":["$$(1,-3),(2,3)$$"],"problemType":"MultipleChoice","stepTitle":"$$-2x^2+y=-5$$,\\\\n$$6x-y=9$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(1,-3),(2,3)$$","choices":["$$(1,-3),(2,3)$$","$$(1,3),(2,3)$$","$$(-1,3),(-2,3)$$","No Solution"],"hints":{"DefaultPathway":[{"id":"ac561a1systems6a-h1","type":"hint","dependencies":[],"title":"Eliminating Variables","text":"When both equations in the system have like variables of the second degree, solving them using elimination by addition is often easier than substitution. Generally, elimination is a far simpler method when the system involves only two equations in two variables (a two-by-two system), rather than a three-by-three system, as there are fewer steps.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems6a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-2x^2+6x=4$$"],"dependencies":["ac561a1systems6a-h1"],"title":"Adding Equations","text":"We observe that by adding the two equations together, we can remove the $$y$$ term. What is the equation obtained after adding the two equations?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$-2x^2+6x=4$$","$$4x^2=4$$","$$2x^2+6x=4$$"]},{"id":"ac561a1systems6a-h3","type":"hint","dependencies":["ac561a1systems6a-h2"],"title":"Solving for $$x$$","text":"With the equation $$-2x^2+6x=4$$, we can solve for $$x$$. The x-coordinate found indicates a possible solution where an intersection may occur if the system of equations yield a feasible solution later on.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems6a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["1,2"],"dependencies":["ac561a1systems6a-h3"],"title":"Solving for $$x$$","text":"We can use the quadratic formula $$x=\\\\frac{\\\\left(-b+-\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$ to obtain the solutions. What are the $$x$$ coordinates?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["1,2","$$-2, -1$$","$$-2, 1$$"]},{"id":"ac561a1systems6a-h5","type":"hint","dependencies":["ac561a1systems6a-h4"],"title":"Subsituting Possible Values","text":"We substitute all possible $$x$$ coordinates back into one of the two equations to find the corresponding $$y$$ coordinate to each $$x$$ coordinate. Using any of the two equations work because the $$x$$ coordinate found is a solution to the system of equations. We usually use the equation that is easier to calculate the other coordinate with.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems6a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["ac561a1systems6a-h5"],"title":"Substitution #1","text":"Substituting $$x=1$$ into $$6x-y=9$$. What is the corresponding $$y$$ coordinate?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems6a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ac561a1systems6a-h6"],"title":"Substitution #2","text":"Substituting $$x=2$$ into $$6x-y=9$$. What is the corresponding $$y$$ coordinate?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems6a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ac561a1systems6a-h7"],"title":"Verifying the Number Of Intersections","text":"We have found the coordinates of all intersections. How many are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac561a1systems7","title":"Solving a System of Nonlinear Equations Using Elimination","body":"Solve the system of nonlinear equations using elimination.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Systems of Nonlinear Equations and Inequalities: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ac561a1systems7a","stepAnswer":["$$(0,2),(1,3)$$"],"problemType":"MultipleChoice","stepTitle":"$$-\\\\left(x^2\\\\right)+y=2$$,\\\\n$$-x+y=2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,2),(1,3)$$","choices":["$$(0,2),(1,3)$$","$$(-1,1),(0,2)$$","No Solution"],"hints":{"DefaultPathway":[{"id":"ac561a1systems7a-h1","type":"hint","dependencies":[],"title":"Eliminating Variables","text":"When both equations in the system have like variables of the second degree, solving them using elimination by addition is often easier than substitution. Generally, elimination is a far simpler method when the system involves only two equations in two variables (a two-by-two system), rather than a three-by-three system, as there are fewer steps.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems7a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-\\\\left(x^2\\\\right)+x=0$$"],"dependencies":["ac561a1systems7a-h1"],"title":"Subtracting Equations","text":"We observe that by subtracting the second equation from the first, we can remove the $$y$$ term. What is the equation obtained after subtracting the second equation from the first?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$-\\\\left(x^2\\\\right)+x=0$$","$$-\\\\left(x^2\\\\right)-x=0$$"]},{"id":"ac561a1systems7a-h3","type":"hint","dependencies":["ac561a1systems7a-h2"],"title":"Solving for $$x$$","text":"With the equation $$-\\\\left(x^2\\\\right)+x=0$$, we can solve for $$x$$. The x-coordinate found indicates a possible solution where an intersection may occur if the system of equations yield a feasible solution later on.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems7a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["(-x+1),x"],"dependencies":["ac561a1systems7a-h3"],"title":"Solving for $$x$$","text":"We observe that we can factorize out $$x$$ from the left side of the equation. What are the factors?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["(-x+1),x","$$(-x-1), x$$"]},{"id":"ac561a1systems7a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["0,1"],"dependencies":["ac561a1systems7a-h4"],"title":"Solving for $$x$$","text":"What are the solution to $$-\\\\left(x^2\\\\right)+x=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["0,1","$$-1, 0$$"]},{"id":"ac561a1systems7a-h6","type":"hint","dependencies":["ac561a1systems7a-h5"],"title":"Subsituting Possible Values","text":"We substitute all possible $$x$$ coordinates back into one of the two equations to find the corresponding $$y$$ coordinate to each $$x$$ coordinate. Using any of the two equations work because the $$x$$ coordinate found is a solution to the system of equations. We usually use the equation that is easier to calculate the other coordinate with.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems7a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ac561a1systems7a-h6"],"title":"Substitution #1","text":"Substituting $$x=0$$ into $$-x+y=2$$. What is the corresponding $$y$$ coordinate?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems7a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ac561a1systems7a-h7"],"title":"Substitution #2","text":"Substituting $$x=1$$ into $$-x+y=2$$. What is the corresponding $$y$$ coordinate?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems7a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ac561a1systems7a-h8"],"title":"Verifying the Number Of Intersections","text":"We have found the coordinates of all intersections. How many are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac561a1systems8","title":"Solving a System of Nonlinear Equations Using Elimination","body":"Solve the system of nonlinear equations using elimination.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Systems of Nonlinear Equations and Inequalities: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ac561a1systems8a","stepAnswer":["(-sqrt(39)/20,19/20),(0,-1),(sqrt(30)/20,19/20)"],"problemType":"MultipleChoice","stepTitle":"$$x^2+y^2=1$$,\\\\n$$y=20x^2-1$$","stepBody":"","answerType":"string","variabilization":{},"choices":["(-sqrt(39)/20,19/20),(0,-1),(sqrt(30)/20,19/20)","(-sqrt(39)/20,19/20),(sqrt(30)/20,19/20)","No Solution"],"hints":{"DefaultPathway":[{"id":"ac561a1systems8a-h1","type":"hint","dependencies":[],"title":"Eliminating Variables","text":"When both equations in the system have like variables of the second degree, solving them using elimination by addition is often easier than substitution. Generally, elimination is a far simpler method when the system involves only two equations in two variables (a two-by-two system), rather than a three-by-three system, as there are fewer steps.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems8a-h2","type":"hint","dependencies":["ac561a1systems8a-h1"],"title":"Common Terms between Equations","text":"We observe that we can multiply $$20$$ to the first equation, so that $$20x^2$$ becomes a common term in both equations. This would allow us to solve the system by elimination.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems8a-h3","type":"hint","dependencies":["ac561a1systems8a-h2"],"title":"Shifting Terms","text":"We can shift all the terms in the equations to the left hand side to make the equations tidier and to make future steps easier.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems8a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$20y^2+y-19=0$$"],"dependencies":["ac561a1systems8a-h3"],"title":"Adding Equations","text":"We observe that by adding the two equations together, we can remove the $$x^2$$ term. What is the equation obtained after adding the two equations?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$20y^2+y-19=0$$","$$20y^2-y+21=0$$","$$y^2+20y-19=0$$"]},{"id":"ac561a1systems8a-h5","type":"hint","dependencies":["ac561a1systems8a-h4"],"title":"Solving for $$y$$","text":"With the equation $$20y^2+y-19=0$$, we can solve for $$y$$. The y-coordinate found indicates a possible solution where an intersection may occur if the system of equations yield a feasible solution later on.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems8a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["-1,19/20"],"dependencies":["ac561a1systems8a-h5"],"title":"Solving for $$y$$","text":"We can use the quadratic formula $$y=\\\\frac{\\\\left(-b+-\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$ to obtain the solutions. What are the $$y$$ coordinates?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$-1-\\\\frac{19}{20}$$","-1,19/20","$$1-\\\\frac{19}{20}$$","1,19/20"]},{"id":"ac561a1systems8a-h7","type":"hint","dependencies":["ac561a1systems8a-h6"],"title":"Subsituting Possible Values","text":"We substitute all possible $$y$$ coordinates back into one of the two equations to find the corresponding $$x$$ coordinate to each $$y$$ coordinate. Using any of the two equations work because the $$y$$ coordinate found is a solution to the system of equations. We usually use the equation that is easier to calculate the other coordinate with.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems8a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["ac561a1systems8a-h7"],"title":"Substitution #1","text":"Substituting $$y=-1$$ into $$x^2+y^2=1$$. What is the corresponding $$x$$ coordinate?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems8a-h9","type":"hint","dependencies":["ac561a1systems8a-h8"],"title":"Substitution #2","text":"Substituting $$y=\\\\frac{19}{20}$$ into $$y=20x^2-1$$. Find all possible corresponding $$x$$ coordinates.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems8a-h10","type":"hint","dependencies":["ac561a1systems8a-h9"],"title":"Substitution #2","text":"Add $$1$$ to both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems8a-h11","type":"hint","dependencies":["ac561a1systems8a-h10"],"title":"Substitution #2","text":"Divide by $$20$$ to both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems8a-h12","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["-sqrt(39)/20,sqrt(39)/20"],"dependencies":["ac561a1systems8a-h11"],"title":"Substitution #2","text":"Take square root on both sides. What are the corresponding $$x$$ coordinates?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["-sqrt(39)/20,sqrt(39)/20","-sqrt(39)/40,sqrt(39)/40"]},{"id":"ac561a1systems8a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ac561a1systems8a-h12"],"title":"Verifying the Number Of Intersections","text":"We have found the coordinates of all intersections. How many are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac561a1systems9","title":"Solving a System of Nonlinear Equations Using Elimination","body":"Solve the system of nonlinear equations using elimination.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.3 Systems of Nonlinear Equations and Inequalities: Two Variables","courseName":"OpenStax: College Algebra","steps":[{"id":"ac561a1systems9a","stepAnswer":["(-sqrt((sqrt(5)-1)/2),(1-sqrt(5))/2),(sqrt((sqrt(5)-1)/2),(1-sqrt(5))/2)"],"problemType":"MultipleChoice","stepTitle":"$$x^2+y^2=1$$,\\\\n$$y=-\\\\left(x^2\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["(-sqrt((sqrt(5)-1)/2),(1-sqrt(5))/2),(sqrt((sqrt(5)-1)/2),(1-sqrt(5))/2)","(-sqrt((1-sqrt(5))/2),(1-sqrt(5))/2),(sqrt((1-sqrt(5))/2),(1-sqrt(5))/2)","No solution"],"hints":{"DefaultPathway":[{"id":"ac561a1systems9a-h1","type":"hint","dependencies":[],"title":"Eliminating Variables","text":"When both equations in the system have like variables of the second degree, solving them using elimination by addition is often easier than substitution. Generally, elimination is a far simpler method when the system involves only two equations in two variables (a two-by-two system), rather than a three-by-three system, as there are fewer steps.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems9a-h2","type":"hint","dependencies":["ac561a1systems9a-h1"],"title":"Shifting Terms","text":"We can shift all the terms in the equations to the left hand side to make the equations tidier and to make future steps easier.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems9a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y^2-y-1=0$$"],"dependencies":["ac561a1systems9a-h2"],"title":"Subtracting Equations","text":"We observe that by subtracting the second equation from the first, we can remove the $$x^2$$ term. What is the equation obtained after subtracting the second equation from the first?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y^2-y-1=0$$","$$y^2+y+1=0$$","$$y^2-y+1=0$$"]},{"id":"ac561a1systems9a-h4","type":"hint","dependencies":["ac561a1systems9a-h3"],"title":"Solving for $$y$$","text":"With the equation $$y^2-y-1=0$$, we can solve for $$y$$. The y-coordinate found indicates a possible solution where an intersection may occur if the system of equations yield a feasible solution later on.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems9a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["(1-sqrt(5))/2,(1+sqrt(5))/2"],"dependencies":["ac561a1systems9a-h4"],"title":"Solving for $$y$$","text":"We can use the quadratic formula $$y=\\\\frac{\\\\left(-b+-\\\\sqrt{b^2-4a c}\\\\right)}{2a}$$ to obtain the solutions. What are the $$y$$ coordinates?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["(1-sqrt(5))/2,(1+sqrt(5))/2","(sqrt(5)-1)/2,(1+sqrt(5))/2"]},{"id":"ac561a1systems9a-h6","type":"hint","dependencies":["ac561a1systems9a-h5"],"title":"Subsituting Possible Values","text":"We substitute all possible $$y$$ coordinates back into one of the two equations to find the corresponding $$x$$ coordinate to each $$y$$ coordinate. Using any of the two equations work because the $$y$$ coordinate found is a solution to the system of equations. We usually use the equation that is easier to calculate the other coordinate with.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems9a-h7","type":"hint","dependencies":["ac561a1systems9a-h6"],"title":"Substitution #1","text":"Substituting $$y=\\\\frac{1-\\\\sqrt{5}}{2}$$ into $$y=-\\\\left(x^2\\\\right)$$. What is the corresponding $$x$$ coordinate?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems9a-h8","type":"hint","dependencies":["ac561a1systems9a-h7"],"title":"Substitution #1","text":"Multiply by $$-1$$ on both sides then take square root on both sides. The corresponding $$x$$ coordinates are $$-\\\\sqrt{\\\\frac{\\\\sqrt{5}-1}{2}}$$, $$\\\\sqrt{\\\\frac{\\\\sqrt{5}-1}{2}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems9a-h9","type":"hint","dependencies":["ac561a1systems9a-h8"],"title":"Substitution #2","text":"Substituting $$y=\\\\frac{19}{20}$$ into $$y=20x^2-1$$. Find all possible corresponding $$x$$ coordinates.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems9a-h10","type":"hint","dependencies":["ac561a1systems9a-h9"],"title":"Substitution #2","text":"Add $$1$$ to both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems9a-h11","type":"hint","dependencies":["ac561a1systems9a-h10"],"title":"Substitution #2","text":"Divide by $$20$$ to both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac561a1systems9a-h12","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["-sqrt(39)/20,sqrt(39)/20"],"dependencies":["ac561a1systems9a-h11"],"title":"Substitution #2","text":"Take square root on both sides. What are the corresponding $$x$$ coordinates?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["-sqrt(39)/20,sqrt(39)/20","-sqrt(39)/40,sqrt(39)/40"]},{"id":"ac561a1systems9a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ac561a1systems9a-h12"],"title":"Verifying the Number Of Intersections","text":"We have found the coordinates of all intersections. How many are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac5ce8413.1assumption1","title":"One-way ANOVA","body":"There are five basic assumptions that must be fulfilled in order to perform a one-way ANOVA test.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"13.1 One-Way ANOVA","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac5ce8413.1assumption1a","stepAnswer":["Each population from which a sample is taken is assumed to be normal."],"problemType":"MultipleChoice","stepTitle":"Which of the assumptions are true?","stepBody":"","answerType":"string","variabilization":{},"choices":["Each population from which a sample is taken is assumed to be normal.","Each population from which a sample is taken is assumed to form left skewed distribution.","All the group means are not aprroximately equal.","The response is a categorical variable."],"hints":{"DefaultPathway":[{"id":"ac5ce8413.1assumption1a-h1","type":"hint","dependencies":[],"title":"One-way ANOVA","text":"Consider the option that the response is a categorical variable. This is the dependent variable measurement that we are testing, therefore the variable must be a numerical variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac5ce8413.1assumption1a-h2","type":"hint","dependencies":["ac5ce8413.1assumption1a-h1"],"title":"One-way ANOVA","text":"Consider the option that all the group means are not approximately equal to one another. In an ANOVA test, the alternative hypothesis is that out of all the group means considered, at least two of the group means are different, while the null prevailing belief is that the group means are approximately the same.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac5ce8413.1assumption1a-h3","type":"hint","dependencies":["ac5ce8413.1assumption1a-h2"],"title":"One-way ANOVA","text":"Consider the option that each population from which a sample is taken is assumed to form left skewed distribution. ANOVA only operates with normal group data and there are no assumptions related to the skewness of the data.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac5ce8413.1assumption1a-h4","type":"hint","dependencies":["ac5ce8413.1assumption1a-h3"],"title":"One-way ANOVA","text":"Analysis of variance extends the comparison of two groups to several, each a level of a categorical variable (factor). Samples from each group are independent, and must be randomly selected from normal populations with equal variances. We test the null hypothesis of equal means of the response in every group versus the alternative hypothesis of one or more group means being different from the others. A one-way ANOVA hypothesis test determines if several population means are equal. The distribution for the test is the F distribution with two different degrees of freedom.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac5ce8413.1assumption2","title":"One-way ANOVA","body":"There are five basic assumptions that must be fulfilled in order to perform a one-way ANOVA test.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"13.1 One-Way ANOVA","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac5ce8413.1assumption2a","stepAnswer":["All samples are randomly selected and independent."],"problemType":"MultipleChoice","stepTitle":"Which of the assumptions are true?","stepBody":"","answerType":"string","variabilization":{},"choices":["All samples are randomly selected and independent.","Each population from which a sample is taken is assumed to form left skewed distribution.","All the group means are not aprroximately equal.","The response is a categorical variable."],"hints":{"DefaultPathway":[{"id":"ac5ce8413.1assumption2a-h1","type":"hint","dependencies":[],"title":"Variable types","text":"Consider the option that the response is a categorical variable. This is the dependent variable measurement that we are testing, therefore the variable must be a numerical variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac5ce8413.1assumption2a-h2","type":"hint","dependencies":["ac5ce8413.1assumption2a-h1"],"title":"One-way ANOVA","text":"Consider the option that all the group means are not approximately equal to one another. In an ANOVA test, the alternative hypothesis is that out of all the group means considered, at least two of the group means are different, while the null prevailing belief is that the group means are approximately the same.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac5ce8413.1assumption2a-h3","type":"hint","dependencies":["ac5ce8413.1assumption2a-h2"],"title":"Distribution shape","text":"Consider the option that each population from which a sample is taken is assumed to form left skewed distribution. ANOVA only operates with normal group data and there are no assumptions related to the skewness of the data.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac5ce8413.1assumption2a-h4","type":"hint","dependencies":["ac5ce8413.1assumption2a-h3"],"title":"One-way ANOVA","text":"Analysis of variance extends the comparison of two groups to several, each a level of a categorical variable (factor). Samples from each group are independent, and must be randomly selected from normal populations with equal variances. We test the null hypothesis of equal means of the response in every group versus the alternative hypothesis of one or more group means being different from the others. A one-way ANOVA hypothesis test determines if several population means are equal. The distribution for the test is the F distribution with two different degrees of freedom.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac5ce8413.1assumption3","title":"One-way ANOVA","body":"There are five basic assumptions that must be fulfilled in order to perform a one-way ANOVA test.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"13.1 One-Way ANOVA","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac5ce8413.1assumption3a","stepAnswer":["The populations are assumed to have equal standard deviations (or variances)."],"problemType":"MultipleChoice","stepTitle":"Which of the assumptions are true?","stepBody":"","answerType":"string","variabilization":{},"choices":["The populations are assumed to have equal standard deviations (or variances).","Each population from which a sample is taken is assumed to form left skewed distribution.","All the group means are not aprroximately equal.","The response is a categorical variable."],"hints":{"DefaultPathway":[{"id":"ac5ce8413.1assumption3a-h1","type":"hint","dependencies":[],"title":"Distribution shape","text":"Consider the option that each population from which a sample is taken is assumed to form left skewed distribution. ANOVA only operates with normal group data and there are no assumptions related to the skewness of the data.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac5ce8413.1assumption3a-h2","type":"hint","dependencies":["ac5ce8413.1assumption3a-h1"],"title":"Variable types","text":"Consider the option that the response is a categorical variable. This is the dependent variable measurement that we are testing, therefore the variable must be a numerical variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac5ce8413.1assumption3a-h3","type":"hint","dependencies":["ac5ce8413.1assumption3a-h2"],"title":"One-way ANOVA","text":"Consider the option that all the group means are not approximately equal to one another. In an ANOVA test, the alternative hypothesis is that out of all the group means considered, at least two of the group means are different, while the null prevailing belief is that the group means are approximately the same.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac5ce8413.1assumption3a-h4","type":"hint","dependencies":["ac5ce8413.1assumption3a-h3"],"title":"One-way ANOVA","text":"Analysis of variance extends the comparison of two groups to several, each a level of a categorical variable (factor). Samples from each group are independent, and must be randomly selected from normal populations with equal variances. We test the null hypothesis of equal means of the response in every group versus the alternative hypothesis of one or more group means being different from the others. A one-way ANOVA hypothesis test determines if several population means are equal. The distribution for the test is the F distribution with two different degrees of freedom.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac5ce8413.1assumption4","title":"One-way ANOVA","body":"There are five basic assumptions that must be fulfilled in order to perform a one-way ANOVA test.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"13.1 One-Way ANOVA","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac5ce8413.1assumption4a","stepAnswer":["The factor is a categorical variable."],"problemType":"MultipleChoice","stepTitle":"Which of the assumptions are true?","stepBody":"","answerType":"string","variabilization":{},"choices":["The factor is a categorical variable.","The standard deviations are different across the population.","All the group means are not aprroximately equal.","The response is a categorical variable."],"hints":{"DefaultPathway":[{"id":"ac5ce8413.1assumption4a-h1","type":"hint","dependencies":[],"title":"One-way ANOVA","text":"Consider the option that all the group means are not approximately equal to one another. In an ANOVA test, the alternative hypothesis is that out of all the group means considered, at least two of the group means are different, while the null prevailing belief is that the group means are approximately the same.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac5ce8413.1assumption4a-h2","type":"hint","dependencies":["ac5ce8413.1assumption4a-h1"],"title":"One-way ANOVA","text":"In a one-way ANOVA test, the standard deviation in the groups should be equal to one another.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac5ce8413.1assumption4a-h3","type":"hint","dependencies":["ac5ce8413.1assumption4a-h2"],"title":"Factor variable","text":"Consider the factor. This is the independent variable or predictor, and the factor is another way of reffering to a categorical variable. Take for example, in an experiment, the different planned treatments denote the different factors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac5ce8413.1assumption4a-h4","type":"hint","dependencies":["ac5ce8413.1assumption4a-h3"],"title":"One-way ANOVA","text":"Analysis of variance extends the comparison of two groups to several, each a level of a categorical variable (factor). Samples from each group are independent, and must be randomly selected from normal populations with equal variances. We test the null hypothesis of equal means of the response in every group versus the alternative hypothesis of one or more group means being different from the others. A one-way ANOVA hypothesis test determines if several population means are equal. The distribution for the test is the F distribution with two different degrees of freedom.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac5ce8413.1assumption5","title":"One-way ANOVA","body":"There are five basic assumptions that must be fulfilled in order to perform a one-way ANOVA test.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"13.1 One-Way ANOVA","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac5ce8413.1assumption5a","stepAnswer":["The response is a numerical variable."],"problemType":"MultipleChoice","stepTitle":"Which of the assumptions are true?","stepBody":"","answerType":"string","variabilization":{},"choices":["The response is a numerical variable.","The standard deviations are different across the population.","All the group means are not aprroximately equal.","The response is a categorical variable."],"hints":{"DefaultPathway":[{"id":"ac5ce8413.1assumption5a-h1","type":"hint","dependencies":[],"title":"One-way ANOVA","text":"In a one-way ANOVA test, the standard deviation in the groups should be equal to one another.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac5ce8413.1assumption5a-h2","type":"hint","dependencies":["ac5ce8413.1assumption5a-h1"],"title":"One-way ANOVA","text":"Consider the option that all the group means are not approximately equal to one another. In an ANOVA test, the alternative hypothesis is that out of all the group means considered, at least two of the group means are different, while the null prevailing belief is that the group means are approximately the same.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac5ce8413.1assumption5a-h3","type":"hint","dependencies":["ac5ce8413.1assumption5a-h2"],"title":"Variable types","text":"Consider the option that the response is a categorical variable. This is the dependent variable measurement that we are testing, therefore the variable must be a numerical variable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac5ce8413.1assumption5a-h4","type":"hint","dependencies":["ac5ce8413.1assumption5a-h3"],"title":"One-way ANOVA","text":"Analysis of variance extends the comparison of two groups to several, each a level of a categorical variable (factor). Samples from each group are independent, and must be randomly selected from normal populations with equal variances. We test the null hypothesis of equal means of the response in every group versus the alternative hypothesis of one or more group means being different from the others. A one-way ANOVA hypothesis test determines if several population means are equal. The distribution for the test is the F distribution with two different degrees of freedom.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac5ce8413.1assumption6","title":"One-way ANOVA","body":"One-way ANOVA","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"13.1 One-Way ANOVA","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac5ce8413.1assumption6a","stepAnswer":["At least two of the group means out of the three group means are not equal."],"problemType":"MultipleChoice","stepTitle":"State the alternative hypothesis for a one-way ANOVA test if there are three groups.","stepBody":"","answerType":"string","variabilization":{},"choices":["At least two of the group means out of the three group means are not equal.","At least one of the group means out of the three group means are not equal.","All the group means are the same.","The group means are less than or equal to $$1$$."],"hints":{"DefaultPathway":[{"id":"ac5ce8413.1assumption6a-h1","type":"hint","dependencies":[],"title":"One-way ANOVA","text":"In a one-way ANOVA test, the default prevailing belief is that the group means are equal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac5ce8413.1assumption6a-h2","type":"hint","dependencies":["ac5ce8413.1assumption6a-h1"],"title":"One-way ANOVA","text":"Since the prevailing belief is that the group means are equal, then the alternative hypothesis would be that at least two of the group means aren\'t the same to each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac5ce8413.1assumption7","title":"One-way ANOVA","body":"One-way ANOVA","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"13.1 One-Way ANOVA","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac5ce8413.1assumption7a","stepAnswer":["All the group means are approximately the same."],"problemType":"MultipleChoice","stepTitle":"One-way ANOVA","stepBody":"State the null hypothesis for a one-way ANOVA test if there are four groups.","answerType":"string","variabilization":{},"choices":["All the group means are approximately the same.","At least two of the group means are the same","At least two of the group means aren\'t the same","All the group means add up to $$1$$."],"hints":{"DefaultPathway":[{"id":"ac5ce8413.1assumption7a-h1","type":"hint","dependencies":[],"title":"One-way ANOVA","text":"In a one-way ANOVA test, the default prevailing belief is that the group means are equal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac5ce8413.1assumption7a-h2","type":"hint","dependencies":["ac5ce8413.1assumption7a-h1"],"title":"One-way ANOVA","text":"The alternative claim is that at least two of the group means aren\'t equal. The prevailing belief is that the group means are approximately equal to each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac5ce8413.1driverslicense","title":"One-way ANOVA","body":"Suppose a group is interested in determining whether teenagers obtain their drivers licenses at approximately the same average age across the country. Suppose that the following data are randomly collected from five teenagers in each region of the country. The numbers represent the age at which teenagers obtained their drivers licenses.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"13.1 One-Way ANOVA","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac5ce8413.1driverslicensea","stepAnswer":["$$H_0$$: $$\\\\mu_1=\\\\mu_2=\\\\mu_3=\\\\mu_4=\\\\mu_5 $$H_a$$: $$At$$ $$least$$ $$two$$ $$of$$ $$the$$ $$group$$ $$means$$ $$are$$ $$not$$ $$equal$$."],"problemType":"MultipleChoice","stepTitle":"State the hypotheses.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$H_0$$: $$\\\\mu_1=\\\\mu_2=\\\\mu_3=\\\\mu_4=\\\\mu_5 $$H_a$$: $$At$$ $$least$$ $$two$$ $$of$$ $$the$$ $$group$$ $$means$$ $$are$$ $$not$$ $$equal$$.","choices":["$$H_0$$: $$\\\\mu_1=\\\\mu_2=\\\\mu_3=\\\\mu_4=\\\\mu_5 $$H_a$$: $$At$$ $$least$$ $$two$$ $$of$$ $$the$$ $$group$$ $$means$$ $$are$$ $$not$$ $$equal$$.","$$H_0$$: $$\\\\mu_1=\\\\mu_2H_a: $$At$$ $$least$$ $$two$$ $$of$$ $$the$$ $$group$$ $$means$$ $$are$$ $$not$$ $$equal$$.","$$H_0$$: Teenagers obtain their drivers licenses at approximately the same age in the country. $$H_a$$: At least two of the group means are not equal.","$$H_0$$: Teenagers obtain their drivers licenses at approximately the same average age in the country. $$H_a$$: At least two of the group means are not equal."],"hints":{"DefaultPathway":[{"id":"ac5ce8413.1driverslicensea-h1","type":"hint","dependencies":[],"title":"One-way ANOVA","text":"We are interested to see if teenagers obtain their licenses in the same average ages across the country.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac5ce8413.1driverslicensea-h2","type":"hint","dependencies":["ac5ce8413.1driverslicensea-h1"],"title":"One-way ANOVA","text":"The null hypothesis is that all the group population means are the same. We can see from the table that the values are also relatively similar to one another across the regions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac5ce8413.1driverslicensea-h3","type":"hint","dependencies":["ac5ce8413.1driverslicensea-h1"],"title":"One-way ANOVA","text":"We are testing to see if the average age significantly varies in at least one of the five groups because we want to see if there really is a same average age across the regions in the country.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac5ce8413.1traffic","title":"One-way ANOVA","body":"Three different traffic routes are tested for mean driving time. The entries in the Table $$13.18$$ are the driving times in minutes on the three different routes.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"13.1 One-Way ANOVA","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac5ce8413.1traffica","stepAnswer":["$$26$$"],"problemType":"TextBox","stepTitle":"State $$SS_{between}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$26$$","hints":{"DefaultPathway":[{"id":"ac5ce8413.1traffica-h1","type":"hint","dependencies":[],"title":"One-way ANOVA","text":"First, calculate the sum of square errors and the mean square errors. Make sure you have use $$3$$ collumns to record your data onto your graphing calculator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac5ce8413.1traffica-h2","type":"hint","dependencies":["ac5ce8413.1traffica-h1"],"title":"One-way ANOVA","text":"After inputting your data onto your graphing table, use the ANOVA function on the calculator. ANOVA function typically requires that you specify which collumns to use (e.g. $$3$$ collumns L1,L2, and L3 contain data. 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What is the probability that in $$10$$ trials, the dolphin performs the trick twice?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.1757$$","hints":{"DefaultPathway":[{"id":"ac6025fBinomial11a-h1","type":"hint","dependencies":[],"title":"Success?","text":"What would we consider as a \\"success\\" in $$1$$ trial of this situation? By the way the situation is presented, it is fairly clear that the dolphin doing the trick should be considered a win. This is because the probability of our dolphin doing a trick was brought up before the probability of our dolphin failing. In addition, this means that a \\"failure\\" would be the dolphin failing the trick.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial11a-h2","type":"hint","dependencies":["ac6025fBinomial11a-h1"],"title":"Binomial Equations","text":"Keep in mind that the equation to calculate binomial probabilities is: (n choose x) * $$p^x$$ * $$q^{n-x}$$, where $$n$$ $$=$$ total trials, $$x$$ $$=$$ trials with a success, $$p$$ $$=$$ probabilty of success, and q $$=$$ probability of failure. You can also use most calculators nowadays to calculate binomial probabilities, in addition to the internet.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial11a-h3","type":"hint","dependencies":["ac6025fBinomial11a-h2"],"title":"Variables","text":"Now that we have our equation, what will all our relevant variables be? $$n$$ $$=$$ $$10$$, $$x$$ $$=$$ $$2$$, $$p$$ $$=$$ $$.35$$, and q $$=$$ $$.65$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1757$$"],"dependencies":["ac6025fBinomial11a-h3"],"title":"Answer","text":"Knowing about what we consider a \\"success\\", in addition to our probability equation, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac6025fBinomial12","title":"Binomial Distributions","body":"A trainer is teaching a dolphin to do tricks.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Binomial Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac6025fBinomial12a","stepAnswer":["$$0.1812$$"],"problemType":"TextBox","stepTitle":"The probability that the dolphin successfully performs the trick is 35%, and the probability that the dolphin does not successfully perform the trick is 65%. What is the probability that in $$5$$ trials, the dolphin fails the trick twice?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.1812$$","hints":{"DefaultPathway":[{"id":"ac6025fBinomial12a-h1","type":"hint","dependencies":[],"title":"Success?","text":"What would we consider as a \\"success\\" in $$1$$ trial of this situation? By the way the situation is presented, it is fairly clear that the dolphin doing the trick should be considered a win. This is because the probability of our dolphin doing a trick was brought up before the probability of our dolphin failing. In addition, this means that a \\"failure\\" would be the dolphin failing the trick.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial12a-h2","type":"hint","dependencies":["ac6025fBinomial12a-h1"],"title":"Binomial Equations","text":"Keep in mind that the equation to calculate binomial probabilities is: (n choose x) * $$p^x$$ * $$q^{n-x}$$, where $$n$$ $$=$$ total trials, $$x$$ $$=$$ trials with a success, $$p$$ $$=$$ probabilty of success, and q $$=$$ probability of failure. You can also use most calculators nowadays to calculate binomial probabilities, in addition to the internet.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial12a-h3","type":"hint","dependencies":["ac6025fBinomial12a-h2"],"title":"Variables","text":"Now that we have our equation, what will all our relevant variables be? $$n$$ $$=$$ $$5$$, $$x$$ $$=$$ $$2$$, $$p$$ $$=$$ $$.65$$, and q $$=$$ $$.35$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1812$$"],"dependencies":["ac6025fBinomial12a-h3"],"title":"Answer","text":"Knowing about what we consider a \\"success\\", in addition to our probability equation, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac6025fBinomial13","title":"Binomial Distributions","body":"Suppose you play a game that you can only either win or lose.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Binomial Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac6025fBinomial13a","stepAnswer":["$$0.3341$$"],"problemType":"TextBox","stepTitle":"The probability that you win any game is 55%, and the probability that you lose is 45%. Each game you play is independent. What is the probability that in $$3$$ trials, you win the game once?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.3341$$","hints":{"DefaultPathway":[{"id":"ac6025fBinomial13a-h1","type":"hint","dependencies":[],"title":"Success?","text":"What would we consider as a \\"success\\" in $$1$$ trial of this situation? By the way the situation is presented, it is fairly clear that winning the game is considered a success. This is because the probability of winning was brought up before the probability of losing. In addition, this means that a \\"failure\\" would be losing the game.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial13a-h2","type":"hint","dependencies":["ac6025fBinomial13a-h1"],"title":"Binomial Equations","text":"Keep in mind that the equation to calculate binomial probabilities is: (n choose x) * $$p^x$$ * $$q^{n-x}$$, where $$n$$ $$=$$ total trials, $$x$$ $$=$$ trials with a success, $$p$$ $$=$$ probabilty of success, and q $$=$$ probability of failure. You can also use most calculators nowadays to calculate binomial probabilities, in addition to the internet.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial13a-h3","type":"hint","dependencies":["ac6025fBinomial13a-h2"],"title":"Variables","text":"Now that we have our equation, what will all our relevant variables be? $$n$$ $$=$$ $$3$$, $$x$$ $$=$$ $$1$$, $$p$$ $$=$$ $$.55$$, and q $$=$$ $$.45$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.3341$$"],"dependencies":["ac6025fBinomial13a-h3"],"title":"Answer","text":"Knowing about what we consider a \\"success\\", in addition to our probability equation, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac6025fBinomial14","title":"Binomial Distributions","body":"Suppose you play a game that you can only either win or lose.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Binomial Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac6025fBinomial14a","stepAnswer":["$$0.234$$"],"problemType":"TextBox","stepTitle":"The probability that you win any game is 55%, and the probability that you lose is 45%. Each game you play is independent. What is the probability that in $$10$$ trials, you win the game $$5$$ times?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.234$$","hints":{"DefaultPathway":[{"id":"ac6025fBinomial14a-h1","type":"hint","dependencies":[],"title":"Success?","text":"What would we consider as a \\"success\\" in $$1$$ trial of this situation? By the way the situation is presented, it is fairly clear that winning the game is considered a success. This is because the probability of winning was brought up before the probability of losing. In addition, this means that a \\"failure\\" would be losing the game.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial14a-h2","type":"hint","dependencies":["ac6025fBinomial14a-h1"],"title":"Binomial Equations","text":"Keep in mind that the equation to calculate binomial probabilities is: (n choose x) * $$p^x$$ * $$q^{n-x}$$, where $$n$$ $$=$$ total trials, $$x$$ $$=$$ trials with a success, $$p$$ $$=$$ probabilty of success, and q $$=$$ probability of failure. You can also use most calculators nowadays to calculate binomial probabilities, in addition to the internet.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial14a-h3","type":"hint","dependencies":["ac6025fBinomial14a-h2"],"title":"Variables","text":"Now that we have our equation, what will all our relevant variables be? $$n$$ $$=$$ $$10$$, $$x$$ $$=$$ $$5$$, $$p$$ $$=$$ $$.55$$, and q $$=$$ $$.45$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial14a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.234$$"],"dependencies":["ac6025fBinomial14a-h3"],"title":"Answer","text":"Knowing about what we consider a \\"success\\", in addition to our probability equation, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac6025fBinomial15","title":"Binomial Distributions","body":"Suppose you play a game that you can only either win or lose.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Binomial Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac6025fBinomial15a","stepAnswer":["$$0.3369$$"],"problemType":"TextBox","stepTitle":"The probability that you win any game is 55%, and the probability that you lose is 45%. Each game you play is independent. What is the probability that in $$5$$ trials, you lose the game twice?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.3369$$","hints":{"DefaultPathway":[{"id":"ac6025fBinomial15a-h1","type":"hint","dependencies":[],"title":"Success?","text":"What would we consider as a \\"success\\" in $$1$$ trial of this situation? By the way the situation is presented, it is fairly clear that winning the game is considered a success. This is because the probability of winning was brought up before the probability of losing. In addition, this means that a \\"failure\\" would be losing the game.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial15a-h2","type":"hint","dependencies":["ac6025fBinomial15a-h1"],"title":"Binomial Equations","text":"Keep in mind that the equation to calculate binomial probabilities is: (n choose x) * $$p^x$$ * $$q^{n-x}$$, where $$n$$ $$=$$ total trials, $$x$$ $$=$$ trials with a success, $$p$$ $$=$$ probabilty of success, and q $$=$$ probability of failure. You can also use most calculators nowadays to calculate binomial probabilities, in addition to the internet.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial15a-h3","type":"hint","dependencies":["ac6025fBinomial15a-h2"],"title":"Variables","text":"Now that we have our equation, what will all our relevant variables be? $$n$$ $$=$$ $$5$$, $$x$$ $$=$$ $$2$$, $$p$$ $$=$$ $$.45$$, and q $$=$$ $$.55$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.3369$$"],"dependencies":["ac6025fBinomial15a-h3"],"title":"Answer","text":"Knowing about what we consider a \\"success\\", in addition to our probability equation, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac6025fBinomial16","title":"Binomial Distributions","body":"Suppose we take a poll about American spending habits and record the data.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Binomial Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac6025fBinomial16a","stepAnswer":["$$0.216$$"],"problemType":"TextBox","stepTitle":"According to our data, 60% of American adults prefer saving over spending. This also implies that 40% of American adults prefer spending over saving. What is the probability that out of three Americans, all of them prefer to save their money rather than spend it?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.216$$","hints":{"DefaultPathway":[{"id":"ac6025fBinomial16a-h1","type":"hint","dependencies":[],"title":"Success?","text":"What would we consider as a \\"success\\" in $$1$$ trial of this situation? By the way the situation is presented, it is fairly clear that saving money is considered a success. This is because the probability of saving money was brought up before the probability of spending money. In addition, this means that a \\"failure\\" would be an American spending their money.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial16a-h2","type":"hint","dependencies":["ac6025fBinomial16a-h1"],"title":"Binomial Equations","text":"Keep in mind that the equation to calculate binomial probabilities is: (n choose x) * $$p^x$$ * $$q^{n-x}$$, where $$n$$ $$=$$ total trials, $$x$$ $$=$$ trials with a success, $$p$$ $$=$$ probabilty of success, and q $$=$$ probability of failure. You can also use most calculators nowadays to calculate binomial probabilities, in addition to the internet.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial16a-h3","type":"hint","dependencies":["ac6025fBinomial16a-h2"],"title":"Variables","text":"Now that we have our equation, what will all our relevant variables be? $$n$$ $$=$$ $$3$$, $$x$$ $$=$$ $$3$$, $$p$$ $$=$$ $$.6$$, and q $$=$$ $$.4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial16a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.216$$"],"dependencies":["ac6025fBinomial16a-h3"],"title":"Answer","text":"Knowing about what we consider a \\"success\\", in addition to our probability equation, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac6025fBinomial17","title":"Binomial Distributions","body":"Suppose we take a poll about American spending habits and record the data.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Binomial Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac6025fBinomial17a","stepAnswer":["$$0.006$$"],"problemType":"TextBox","stepTitle":"According to our data, 60% of American adults prefer saving over spending. This also implies that 40% of American adults prefer spending over saving. What is the probability that out of ten Americans, all of them prefer to save their money rather than spend it?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.006$$","hints":{"DefaultPathway":[{"id":"ac6025fBinomial17a-h1","type":"hint","dependencies":[],"title":"Success?","text":"What would we consider as a \\"success\\" in $$1$$ trial of this situation? By the way the situation is presented, it is fairly clear that saving money is considered a success. This is because the probability of saving money was brought up before the probability of spending money. In addition, this means that a \\"failure\\" would be an American spending their money.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial17a-h2","type":"hint","dependencies":["ac6025fBinomial17a-h1"],"title":"Binomial Equations","text":"Keep in mind that the equation to calculate binomial probabilities is: (n choose x) * $$p^x$$ * $$q^{n-x}$$, where $$n$$ $$=$$ total trials, $$x$$ $$=$$ trials with a success, $$p$$ $$=$$ probabilty of success, and q $$=$$ probability of failure. You can also use most calculators nowadays to calculate binomial probabilities, in addition to the internet.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial17a-h3","type":"hint","dependencies":["ac6025fBinomial17a-h2"],"title":"Variables","text":"Now that we have our equation, what will all our relevant variables be? $$n$$ $$=$$ $$10$$, $$x$$ $$=$$ $$10$$, $$p$$ $$=$$ $$.6$$, and q $$=$$ $$.4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial17a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.006$$"],"dependencies":["ac6025fBinomial17a-h3"],"title":"Answer","text":"Knowing about what we consider a \\"success\\", in addition to our probability equation, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac6025fBinomial18","title":"Binomial Distributions","body":"Suppose we take a poll about American spending habits and record the data.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Binomial Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac6025fBinomial18a","stepAnswer":["$$0.3456$$"],"problemType":"TextBox","stepTitle":"According to our data, 60% of American adults prefer saving over spending. This also implies that 40% of American adults prefer spending over saving. What is the probability that out of five Americans, two of them prefer to spend money rather than save?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.3456$$","hints":{"DefaultPathway":[{"id":"ac6025fBinomial18a-h1","type":"hint","dependencies":[],"title":"Failure?","text":"What would we consider as a \\"success\\" in $$1$$ trial of this situation? By the way the situation is presented, it is fairly clear that saving money is considered a success. This is because the probability of saving money was brought up before the probability of spending money. In addition, this means that a \\"failure\\" would be an American spending their money.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial18a-h2","type":"hint","dependencies":["ac6025fBinomial18a-h1"],"title":"Binomial Equations","text":"Keep in mind that the equation to calculate binomial probabilities is: (n choose x) * $$p^x$$ * $$q^{n-x}$$, where $$n$$ $$=$$ total trials, $$x$$ $$=$$ trials with a success, $$p$$ $$=$$ probabilty of success, and q $$=$$ probability of failure. You can also use most calculators nowadays to calculate binomial probabilities, in addition to the internet.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial18a-h3","type":"hint","dependencies":["ac6025fBinomial18a-h2"],"title":"Variables","text":"Now that we have our equation, what will all our relevant variables be? $$n$$ $$=$$ $$5$$, $$x$$ $$=$$ $$2$$, $$p$$ $$=$$ $$.4$$, and q $$=$$ $$.6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial18a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.3456$$"],"dependencies":["ac6025fBinomial18a-h3"],"title":"Answer","text":"Knowing about what we consider a \\"success\\", in addition to our probability equation, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac6025fBinomial19","title":"Binomial Distributions","body":"Assume we perform a study about pancreatic cancer and save our results to a data set that we can examine.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Binomial Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac6025fBinomial19a","stepAnswer":["$$2.56$$"],"problemType":"TextBox","stepTitle":"The lifetime risk of developing pancreatic cancer is about one in $$78$$ $$(1.28\\\\%)$$. Suppose we randomly sample $$200$$ people. Let X $$=$$ the number of people who will develop pancreatic cancer. What is the mean of X?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2.56$$","hints":{"DefaultPathway":[{"id":"ac6025fBinomial19a-h1","type":"hint","dependencies":[],"title":"Formula","text":"Keep in mind that the formula to calculate the mean of a binomial distribution is Mean $$=$$ np, where $$n$$ $$=$$ Total Trials and $$p$$ $$=$$ The Probability of Success.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial19a-h2","type":"hint","dependencies":["ac6025fBinomial19a-h1"],"title":"Variables","text":"What is $$n$$ (total trials) going to be for this problem? What about $$p$$ (probability of success)? As it turns out, $$n$$ $$=$$ $$200$$ and $$p$$ $$=$$ $$.0156$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial19a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.56$$"],"dependencies":["ac6025fBinomial19a-h2"],"title":"Answer","text":"Knowing our equation for calculating our answer along with our variables, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac6025fBinomial2","title":"Binomial Distributions","body":"A trainer is teaching a dolphin to do tricks.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Binomial Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac6025fBinomial2a","stepAnswer":["The dolphin failing the trick."],"problemType":"MultipleChoice","stepTitle":"The probability that the dolphin successfully performs the trick is 35%, and the probability that the dolphin does not successfully perform the trick is 65%. What would a \\"failure\\" be in this case?","stepBody":"","answerType":"string","variabilization":{},"choices":["The dolphin performing the trick.","The dolphin failing the trick."],"hints":{"DefaultPathway":[{"id":"ac6025fBinomial2a-h1","type":"hint","dependencies":[],"title":"Failure Example","text":"At ABC College, the withdrawal rate from an elementary physics course is 30% for any given term. This implies that, for any given term, 70% of the students stay in the class for the entire term. A \\"failure\\" could be defined as an individual who stayed in class for the entire term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial2a-h2","type":"hint","dependencies":["ac6025fBinomial2a-h1"],"title":"Binomial Distributions","text":"Since our distribution is Binomial, we can think of our distribution as being a series of Bernoulli trials. What would a failure look like in a Bernoulli trial? Is it the same as a failure in our Binomial trial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial2a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["The dolphin failing the trick."],"dependencies":["ac6025fBinomial2a-h2"],"title":"Answer","text":"With the example fresh in our minds, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["The dolphin performing the trick.","The dolphin failing the trick."]}]}}]},{"id":"ac6025fBinomial20","title":"Binomial Distributions","body":"Assume we perform a study about pancreatic cancer and save our results to a data set that we can examine.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Binomial Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac6025fBinomial20a","stepAnswer":["$$2.527$$"],"problemType":"TextBox","stepTitle":"The lifetime risk of developing pancreatic cancer is about one in $$78$$ $$(1.28\\\\%)$$. Suppose we randomly sample $$200$$ people. Let X $$=$$ the number of people who will develop pancreatic cancer. What is the variance of X?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2.527$$","hints":{"DefaultPathway":[{"id":"ac6025fBinomial20a-h1","type":"hint","dependencies":[],"title":"Formula","text":"Keep in mind that the formula to calculate the variance of a binomial distribution is Variance $$=$$ npq, where $$n$$ $$=$$ The Total # of Trials, $$p$$ $$=$$ Probability of Success, and q $$=$$ Probability of Failure.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial20a-h2","type":"hint","dependencies":["ac6025fBinomial20a-h1"],"title":"Variables","text":"What is $$n$$ (total trials) going to be for this problem? What about $$p$$ (probability of success)? As it turns out, $$n$$ $$=$$ $$200$$, $$p$$ $$=$$ $$.0156$$, and q $$=$$ $$.9844$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial20a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.527$$"],"dependencies":["ac6025fBinomial20a-h2"],"title":"Answer","text":"Knowing our equation for calculating our answer along with our variables, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac6025fBinomial21","title":"Binomial Distributions","body":"Assume we perform a study about pancreatic cancer and save our results to a data set that we can examine.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Binomial Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac6025fBinomial21a","stepAnswer":["$$1.5897$$"],"problemType":"TextBox","stepTitle":"The lifetime risk of developing pancreatic cancer is about one in $$78$$ $$(1.28\\\\%)$$. Suppose we randomly sample $$200$$ people. Let X $$=$$ the number of people who will develop pancreatic cancer. What is the standard deviation of X?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1.5897$$","hints":{"DefaultPathway":[{"id":"ac6025fBinomial21a-h1","type":"hint","dependencies":[],"title":"Formula","text":"Keep in mind that the formula to calculate the variance of a binomial distribution is Variance $$=$$ npq, where $$n$$ $$=$$ The Total # of Trials, $$p$$ $$=$$ Probability of Success, and q $$=$$ Probability of Failure. To then calculate our standard deviation, we can take the square root of our variance.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial21a-h2","type":"hint","dependencies":["ac6025fBinomial21a-h1"],"title":"Variables","text":"What is $$n$$ (total trials) going to be for this problem? What about $$p$$ (probability of success)? As it turns out, $$n$$ $$=$$ $$200$$, $$p$$ $$=$$ $$.0156$$, and q $$=$$ $$.9844$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial21a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.5897$$"],"dependencies":["ac6025fBinomial21a-h2"],"title":"Answer","text":"Knowing our equation for calculating our answer along with our variables, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac6025fBinomial22","title":"Binomial Distributions","body":"During the $$2013$$ regular NBA season, DeAndre Jordan of the Los Angeles Clippers had the highest field goal completion rate in the league.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Binomial Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac6025fBinomial22a","stepAnswer":["$$49.04$$"],"problemType":"TextBox","stepTitle":"DeAndre scored with $$61.3\\\\%$$ of his shots. Suppose you choose a random sample of $$80$$ shots made by DeAndre during the $$2013$$ season. Let X $$=$$ the number of shots that scored points. What is the mean of X?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$49.04$$","hints":{"DefaultPathway":[{"id":"ac6025fBinomial22a-h1","type":"hint","dependencies":[],"title":"Formula","text":"Keep in mind that the formula to calculate the mean of a binomial distribution is Mean $$=$$ np, where $$n$$ $$=$$ Total Trials and $$p$$ $$=$$ The Probability of Success.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial22a-h2","type":"hint","dependencies":["ac6025fBinomial22a-h1"],"title":"Variables","text":"What is $$n$$ (total trials) going to be for this problem? What about $$p$$ (probability of success)? As it turns out, $$n$$ $$=$$ $$80$$ and $$p$$ $$=$$ $$.613$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial22a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$49.04$$"],"dependencies":["ac6025fBinomial22a-h2"],"title":"Answer","text":"Knowing our equation for calculating our answer along with our variables, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac6025fBinomial23","title":"Binomial Distributions","body":"During the $$2013$$ regular NBA season, DeAndre Jordan of the Los Angeles Clippers had the highest field goal completion rate in the league.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Binomial Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac6025fBinomial23a","stepAnswer":["$$18.98$$"],"problemType":"TextBox","stepTitle":"DeAndre scored with $$61.3\\\\%$$ of his shots. Suppose you choose a random sample of $$80$$ shots made by DeAndre during the $$2013$$ season. Let X $$=$$ the number of shots that scored points. What is the variance of X?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$18.98$$","hints":{"DefaultPathway":[{"id":"ac6025fBinomial23a-h1","type":"hint","dependencies":[],"title":"Formula","text":"Keep in mind that the formula to calculate the variance of a binomial distribution is Variance $$=$$ npq, where $$n$$ $$=$$ The Total # of Trials, $$p$$ $$=$$ Probability of Success, and q $$=$$ Probability of Failure.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial23a-h2","type":"hint","dependencies":["ac6025fBinomial23a-h1"],"title":"Variables","text":"What is $$n$$ (total trials) going to be for this problem? What about $$p$$ (probability of success)? As it turns out, $$n$$ $$=$$ $$80$$, $$p$$ $$=$$ $$.613$$, and q $$=$$ $$.387$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial23a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18.98$$"],"dependencies":["ac6025fBinomial23a-h2"],"title":"Answer","text":"Knowing our equation for calculating our answer along with our variables, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac6025fBinomial24","title":"Binomial Distributions","body":"During the $$2013$$ regular NBA season, DeAndre Jordan of the Los Angeles Clippers had the highest field goal completion rate in the league.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Binomial Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac6025fBinomial24a","stepAnswer":["$$4.356$$"],"problemType":"TextBox","stepTitle":"DeAndre scored with $$61.3\\\\%$$ of his shots. Suppose you choose a random sample of $$80$$ shots made by DeAndre during the $$2013$$ season. Let X $$=$$ the number of shots that scored points. What is the standard deviation of X?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4.356$$","hints":{"DefaultPathway":[{"id":"ac6025fBinomial24a-h1","type":"hint","dependencies":[],"title":"Formula","text":"Keep in mind that the formula to calculate the variance of a binomial distribution is Variance $$=$$ npq, where $$n$$ $$=$$ The Total # of Trials, $$p$$ $$=$$ Probability of Success, and q $$=$$ Probability of Failure. To then calculate our standard deviation, we can take the square root of our variance.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial24a-h2","type":"hint","dependencies":["ac6025fBinomial24a-h1"],"title":"Variables","text":"What is $$n$$ (total trials) going to be for this problem? What about $$p$$ (probability of success)? As it turns out, $$n$$ $$=$$ $$80$$, $$p$$ $$=$$ $$.613$$, and q $$=$$ $$.387$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial24a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4.356$$"],"dependencies":["ac6025fBinomial24a-h2"],"title":"Answer","text":"Knowing our equation for calculating our answer along with our variables, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac6025fBinomial25","title":"Binomial Distributions","body":"During the $$2013$$ regular NBA season, DeAndre Jordan of the Los Angeles Clippers had the highest field goal completion rate in the league.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Binomial Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac6025fBinomial25a","stepAnswer":["$$0.25$$"],"problemType":"TextBox","stepTitle":"DeAndre scored with $$61.3\\\\%$$ of his shots. If DeAndre takes $$10$$ shots, what is the probability that he makes $$6$$ out of $$10$$ shots?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.25$$","hints":{"DefaultPathway":[{"id":"ac6025fBinomial25a-h1","type":"hint","dependencies":[],"title":"Success?","text":"What would we consider as a \\"success\\" in $$1$$ trial of this situation? By the way the situation is presented, it is fairly clear that DeAndre scoring is considered a success. This is because the probability of scoring was brought up before the probability of missing. In addition, this means that a \\"failure\\" would be DeAndre missing the shot.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial25a-h2","type":"hint","dependencies":["ac6025fBinomial25a-h1"],"title":"Binomial Equations","text":"Keep in mind that the equation to calculate binomial probabilities is: (n choose x) * $$p^x$$ * $$q^{n-x}$$, where $$n$$ $$=$$ total trials, $$x$$ $$=$$ trials with a success, $$p$$ $$=$$ probabilty of success, and q $$=$$ probability of failure. You can also use most calculators nowadays to calculate binomial probabilities, in addition to the internet.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial25a-h3","type":"hint","dependencies":["ac6025fBinomial25a-h2"],"title":"Variables","text":"Now that we have our equation, what will all our relevant variables be? $$n$$ $$=$$ $$10$$, $$x$$ $$=$$ $$6$$, $$p$$ $$=$$ $$.613$$, and q $$=$$ $$.387$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial25a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.25$$"],"dependencies":["ac6025fBinomial25a-h3"],"title":"Answer","text":"Knowing about what we consider a \\"success\\", in addition to our probability equation, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac6025fBinomial3","title":"Binomial Distributions","body":"A trainer is teaching a dolphin to do tricks.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Binomial Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac6025fBinomial3a","stepAnswer":["The dolphin performing the trick."],"problemType":"MultipleChoice","stepTitle":"The probability that the dolphin successfully performs the trick is 35%, and the probability that the dolphin does not successfully perform the trick is 65%. What would a \\"success\\" be in this case?","stepBody":"","answerType":"string","variabilization":{},"choices":["The dolphin performing the trick.","The dolphin failing the trick."],"hints":{"DefaultPathway":[{"id":"ac6025fBinomial3a-h1","type":"hint","dependencies":[],"title":"Success Example","text":"At ABC College, the withdrawal rate from an elementary physics course is 30% for any given term. This implies that, for any given term, 70% of the students stay in the class for the entire term. A \\"success\\" could be defined as an individual who withdrew.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial3a-h2","type":"hint","dependencies":["ac6025fBinomial3a-h1"],"title":"Binomial Distributions","text":"Since our distribution is Binomial, we can think of our distribution as being a series of Bernoulli trials. What would a success look like in a Bernoulli trial? Is it the same as a success in our Binomial trial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial3a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["The dolphin performing the trick."],"dependencies":["ac6025fBinomial3a-h2"],"title":"Answer","text":"With the example fresh in our minds, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["The dolphin performing the trick.","The dolphin failing the trick."]}]}}]},{"id":"ac6025fBinomial4","title":"Binomial Distributions","body":"The state health board is concerned about the amount of fruit available in school lunches.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Binomial Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac6025fBinomial4a","stepAnswer":["A school lunch not having fruit."],"problemType":"MultipleChoice","stepTitle":"Forty-eight percent of schools in the state offer fruit in their lunches every day. This implies that 52% do not. What would a \\"success\\" be in this case?","stepBody":"","answerType":"string","variabilization":{},"choices":["A school lunch having fruit.","A school lunch not having fruit."],"hints":{"DefaultPathway":[{"id":"ac6025fBinomial4a-h1","type":"hint","dependencies":[],"title":"Failure Example","text":"At ABC College, the withdrawal rate from an elementary physics course is 30% for any given term. This implies that, for any given term, 70% of the students stay in the class for the entire term. A \\"failure\\" could be defined as an individual who stayed in class for the entire term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial4a-h2","type":"hint","dependencies":["ac6025fBinomial4a-h1"],"title":"Binomial Distributions","text":"Since our distribution is Binomial, we can think of our distribution as being a series of Bernoulli trials. What would a failure look like in a Bernoulli trial? Is it the same as a failure in our Binomial trial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial4a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["A school lunch not having fruit."],"dependencies":["ac6025fBinomial4a-h2"],"title":"Answer","text":"With the example fresh in our minds, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["A school lunch having fruit.","A school lunch not having fruit."]}]}}]},{"id":"ac6025fBinomial5","title":"Binomial Distributions","body":"Suppose you play a game that you can only either win or lose.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Binomial Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac6025fBinomial5a","stepAnswer":["Winning the game."],"problemType":"MultipleChoice","stepTitle":"The probability that you win any game is 55%, and the probability that you lose is 45%. Each game you play is independent. What would a \\"success\\" be in this case?","stepBody":"","answerType":"string","variabilization":{},"choices":["Winning the game.","Losing the game."],"hints":{"DefaultPathway":[{"id":"ac6025fBinomial5a-h1","type":"hint","dependencies":[],"title":"Success Example","text":"At ABC College, the withdrawal rate from an elementary physics course is 30% for any given term. This implies that, for any given term, 70% of the students stay in the class for the entire term. A \\"success\\" could be defined as an individual who withdrew.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial5a-h2","type":"hint","dependencies":["ac6025fBinomial5a-h1"],"title":"Binomial Distributions","text":"Since our distribution is Binomial, we can think of our distribution as being a series of Bernoulli trials. What would a success look like in a Bernoulli trial? Is it the same as a success in our Binomial trial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial5a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Winning the game."],"dependencies":["ac6025fBinomial5a-h2"],"title":"Answer","text":"With the example fresh in our minds, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Winning the game.","Losing the game."]}]}}]},{"id":"ac6025fBinomial6","title":"Binomial Distributions","body":"Suppose you play a game that you can only either win or lose.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Binomial Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac6025fBinomial6a","stepAnswer":["Losing the game."],"problemType":"MultipleChoice","stepTitle":"The probability that you win any game is 55%, and the probability that you lose is 45%. Each game you play is independent. What would a \\"failure\\" be in this case?","stepBody":"","answerType":"string","variabilization":{},"choices":["Winning the game.","Losing the game."],"hints":{"DefaultPathway":[{"id":"ac6025fBinomial6a-h1","type":"hint","dependencies":[],"title":"Failure Example","text":"At ABC College, the withdrawal rate from an elementary physics course is 30% for any given term. This implies that, for any given term, 70% of the students stay in the class for the entire term. A \\"failure\\" could be defined as an individual who stayed in class for the entire term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial6a-h2","type":"hint","dependencies":["ac6025fBinomial6a-h1"],"title":"Binomial Distributions","text":"Since our distribution is Binomial, we can think of our distribution as being a series of Bernoulli trials. What would a failure look like in a Bernoulli trial? Is it the same as a failure in our Binomial trial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial6a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Losing the game."],"dependencies":["ac6025fBinomial6a-h2"],"title":"Answer","text":"With the example fresh in our minds, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Winning the game.","Losing the game."]}]}}]},{"id":"ac6025fBinomial7","title":"Binomial Distributions","body":"The state health board is concerned about the amount of fruit available in school lunches.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Binomial Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac6025fBinomial7a","stepAnswer":["$$0.4992$$"],"problemType":"TextBox","stepTitle":"Forty-eight percent of schools in the state offer fruit in their lunches every day. This implies that 52% do not. What is the probability that we get $$1$$ \\"success\\" in two trials?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.4992$$","hints":{"DefaultPathway":[{"id":"ac6025fBinomial7a-h1","type":"hint","dependencies":[],"title":"Success?","text":"What would we consider as a \\"success\\" in $$1$$ trial of this situation? By the way the situation is presented, it is fairly clear that a school offering fruit in their lunch should be considered a win. This is because the probability of having fruit in school lunches was brought up before the probability of not having fruit in school lunches.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial7a-h2","type":"hint","dependencies":["ac6025fBinomial7a-h1"],"title":"Binomial Equations","text":"Keep in mind that the equation to calculate binomial probabilities is: (n choose x) * $$p^x$$ * $$q^{n-x}$$, where $$n$$ $$=$$ total trials, $$x$$ $$=$$ trials with a success, $$p$$ $$=$$ probabilty of success, and q $$=$$ probability of failure. You can also use most calculators nowadays to calculate binomial probabilities, in addition to the internet.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial7a-h3","type":"hint","dependencies":["ac6025fBinomial7a-h2"],"title":"Variables","text":"Now that we have our equation, what will all our relevant variables be? $$n$$ $$=$$ $$2$$, $$x$$ $$=$$ $$1$$, $$p$$ $$=$$ $$.48$$, and q $$=$$ $$.52$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.4992$$"],"dependencies":["ac6025fBinomial7a-h3"],"title":"Answer","text":"Knowing about what we consider a \\"success\\", in addition to our probability equation, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac6025fBinomial8","title":"Binomial Distributions","body":"The state health board is concerned about the amount of fruit available in school lunches.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Binomial Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac6025fBinomial8a","stepAnswer":["$$0.299$$"],"problemType":"TextBox","stepTitle":"Forty-eight percent of schools in the state offer fruit in their lunches every day. This implies that 52% do not. What is the probability that we get $$3$$ \\"successes\\" in five trials?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.299$$","hints":{"DefaultPathway":[{"id":"ac6025fBinomial8a-h1","type":"hint","dependencies":[],"title":"Success?","text":"What would we consider as a \\"success\\" in $$1$$ trial of this situation? By the way the situation is presented, it is fairly clear that a school offering fruit in their lunch should be considered a win. This is because the probability of having fruit in school lunches was brought up before the probability of not having fruit in school lunches.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial8a-h2","type":"hint","dependencies":["ac6025fBinomial8a-h1"],"title":"Binomial Equations","text":"Keep in mind that the equation to calculate binomial probabilities is: (n choose x) * $$p^x$$ * $$q^{n-x}$$, where $$n$$ $$=$$ total trials, $$x$$ $$=$$ trials with a success, $$p$$ $$=$$ probabilty of success, and q $$=$$ probability of failure. You can also use most calculators nowadays to calculate binomial probabilities, in addition to the internet.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial8a-h3","type":"hint","dependencies":["ac6025fBinomial8a-h2"],"title":"Variables","text":"Now that we have our equation, what will all our relevant variables be? $$n$$ $$=$$ $$5$$, $$x$$ $$=$$ $$3$$, $$p$$ $$=$$ $$.48$$, and q $$=$$ $$.52$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.299$$"],"dependencies":["ac6025fBinomial8a-h3"],"title":"Answer","text":"Knowing about what we consider a \\"success\\", in addition to our probability equation, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac6025fBinomial9","title":"Binomial Distributions","body":"The state health board is concerned about the amount of fruit available in school lunches.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.3 Binomial Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac6025fBinomial9a","stepAnswer":["$$0.324$$"],"problemType":"TextBox","stepTitle":"Forty-eight percent of schools in the state offer fruit in their lunches every day. This implies that 52% do not. What is the probability that we get $$3$$ \\"failures\\" in five trials?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.324$$","hints":{"DefaultPathway":[{"id":"ac6025fBinomial9a-h1","type":"hint","dependencies":[],"title":"Success?","text":"What would we consider as a \\"success\\" in $$1$$ trial of this situation? By the way the situation is presented, it is fairly clear that a school offering fruit in their lunch should be considered a win. This is because the probability of having fruit in school lunches was brought up before the probability of not having fruit in school lunches. In addition, this means that a \\"failure\\" would be a school lunch not having fruit.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial9a-h2","type":"hint","dependencies":["ac6025fBinomial9a-h1"],"title":"Binomial Equations","text":"Keep in mind that the equation to calculate binomial probabilities is: (n choose x) * $$p^x$$ * $$q^{n-x}$$, where $$n$$ $$=$$ total trials, $$x$$ $$=$$ trials with a success, $$p$$ $$=$$ probabilty of success, and q $$=$$ probability of failure. You can also use most calculators nowadays to calculate binomial probabilities, in addition to the internet.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial9a-h3","type":"hint","dependencies":["ac6025fBinomial9a-h2"],"title":"Variables","text":"Now that we have our equation, what will all our relevant variables be? $$n$$ $$=$$ $$5$$, $$x$$ $$=$$ $$3$$, $$p$$ $$=$$ $$.52$$, and q $$=$$ $$.48$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6025fBinomial9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.324$$"],"dependencies":["ac6025fBinomial9a-h3"],"title":"Answer","text":"Knowing about what we consider a \\"success\\", in addition to our probability equation, what is our final answer going to be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac613aahypergeometric1","title":"Hypergeometric Probability Statements","body":"A candy dish contains $$100$$ jelly beans and $$80$$ gumdrops. Fifty candies are picked at random. We want to find the probability that $$35$$ of the $$50$$ are gumdrops. The two groups are jelly beans and gumdrops.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Hypergeometric Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac613aahypergeometric1a","stepAnswer":["Gumdrops"],"problemType":"MultipleChoice","stepTitle":"What is the group of interest?","stepBody":"","answerType":"string","variabilization":{},"choices":["Gumdrops","Jelly beans"],"hints":{"DefaultPathway":[{"id":"ac613aahypergeometric1a-h1","type":"hint","dependencies":[],"title":"Reading the Question In-Depth","text":"We know that we want to find the probability that $$35$$ of the $$50$$ are gumdrops. Think about how gumdrops is a group within the greater population within the candy dish.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric1a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Gumdrops"],"dependencies":["ac613aahypergeometric1a-h1"],"title":"Determining the group of interest","text":"Since we want to find the probability that $$35$$ of the $$50$$ are gumdrops, what is the group of interest?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Gumdrops","Jelly beans"]}]}},{"id":"ac613aahypergeometric1b","stepAnswer":["$$P(x=35)$$"],"problemType":"MultipleChoice","stepTitle":"Let X $$=$$ the number of gumdrops in the sample of $$50$$. X takes on the values $$x=0$$, $$1$$, $$2$$, ..., $$50$$. What is the probability statement written mathematically?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$P(x=35)$$","choices":["$$P(x=35)$$","$$P(y=35)$$","$$P(x=15)$$","$$P(x=80)$$"],"hints":{"DefaultPathway":[{"id":"ac613aahypergeometric1b-h3","type":"hint","dependencies":["ac613aahypergeometric1a-h2"],"title":"Structure of Probability Statements","text":"The structure of a probability statement is $$P(x=C)$$ where P stands for the probability, $$x$$ is the variable we are testing against and C is the constant value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric1b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x$$"],"dependencies":["ac613aahypergeometric1b-h3"],"title":"Determining the variable","text":"What is the variable that we are using?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x$$","$$y$$","$$z$$","c"]},{"id":"ac613aahypergeometric1b-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$35$$"],"dependencies":["ac613aahypergeometric1b-h4"],"title":"Determining the constant","text":"What is the number of gumdrops we want in the sample?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric1b-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$P(x=35)$$"],"dependencies":["ac613aahypergeometric1b-h5"],"title":"Putting it All Together","text":"What is the probability statement written mathematically, using the variable and constant we found?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$P(x=35)$$","$$P(y=35)$$","$$P(x=15)$$","$$P(x=80)$$"]}]}}]},{"id":"ac613aahypergeometric10","title":"Hypergeometric Basics","body":"Suppose that a group of statistics students is divided into two groups: business majors and non-business majors. There are $$16$$ business majors in the group and seven non-business majors in the group. A random sample of nine students is taken. We are interested in the number of business majors in the sample. Let X $$=$$ the number of business majors in the sample.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Hypergeometric Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac613aahypergeometric10a","stepAnswer":["X ~ H(16, $$7$$, 9)"],"problemType":"MultipleChoice","stepTitle":"What is the hypergeometric distribution notation for this question?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"X ~ H(16, $$7$$, 9)","choices":["X ~ H(16, $$7$$, 9)","X ~ H(7, $$16$$, 9)","X ~ H(9, $$16$$, 7)","X ~ H(6, $$9$$, 16)"],"hints":{"DefaultPathway":[{"id":"ac613aahypergeometric10a-h1","type":"hint","dependencies":[],"title":"Structure of Hypergeometric Notation","text":"Note that the notation for a Hypergeometric distribution is X ~ H(r, $$b$$, n) that reads as \\"X is a random variable with a hypergeometric distribution\\". The parameters are $$r$$, $$b$$, and $$n$$ where $$r$$ is the size of the group of interest (first group), $$b$$ is the size of the second group, and $$n$$ is the size of the chosen sample.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["ac613aahypergeometric10a-h1"],"title":"Determining the Size of the Group of Interest","text":"What is the size of the group of interest? In other words, how many business majors are in the pool to decide the sample?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["ac613aahypergeometric10a-h2"],"title":"Determining the Size of the Second Group","text":"What is the size of the second group? In other words, how many non-business majors are in the pool to determine the sample?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["ac613aahypergeometric10a-h3"],"title":"Determining the Size of the Chosen Sample","text":"What is the size of the sample? In other words, how many statistics students will be sample?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric10a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["X ~ H(16, $$7$$, 9)"],"dependencies":["ac613aahypergeometric10a-h4"],"title":"Putting it All Together","text":"Knowing that the size of the group of interest is $$16$$, the size of the second group is $$7$$, and the size of the chosen sample is $$9$$, plug this into the formula to determine the hypergeometric notation of this question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["X ~ H(16, $$7$$, 9)","X ~ H(7, $$16$$, 9)","X ~ H(9, $$16$$, 7)","X ~ H(6, $$9$$, 16)"]}]}}]},{"id":"ac613aahypergeometric11","title":"Hypergeometric Basics","body":"Suppose that a group of statistics students is divided into two groups: business majors and non-business majors. There are $$16$$ business majors in the group and seven non-business majors in the group. A random sample of nine students is taken. We are interested in the number of business majors in the sample. Let X $$=$$ the number of business majors in the sample.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Hypergeometric Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac613aahypergeometric11a","stepAnswer":["$$X=2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$, $$9$$"],"problemType":"MultipleChoice","stepTitle":"What values does X take on?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$X=2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$, $$9$$","choices":["$$X=2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$, $$9$$","$$X=0$$, $$1$$, $$2$$, $$3$$, ..., $$9$$","$$X=0$$, $$1$$, $$2$$, $$3$$, ..., $$16$$","$$X=1$$, $$2$$, $$3$$, ..., $$16$$"],"hints":{"DefaultPathway":[{"id":"ac613aahypergeometric11a-h1","type":"hint","dependencies":[],"title":"Determining All Possible Values of X","text":"We want to know all the total possible values of X, not just the values we want. Also, note that we have $$9$$ students that will be sampled and $$16$$ business majors and $$7$$ non-business majors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric11a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ac613aahypergeometric11a-h1"],"title":"Determining Possibilities","text":"Is it possible to have $$0$$ business majors?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"],"subHints":[{"id":"ac613aahypergeometric11a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Having all $$7$$ Non-Business Majors","text":"If we have all $$7$$ non-business majors be part of the sample, we still need $$9-7=2$$ students to be part of the group. Therefore, what is the minimum number of business majors that must be part of the group?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ac613aahypergeometric11a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$X=2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$, $$9$$"],"dependencies":["ac613aahypergeometric11a-h2"],"title":"Determining All Possible Values of X","text":"Understanding that the values that X takes on includes the entire set of possible number of business majors in the sample of $$9$$, what values does X take on? In other words, what are the possible number of business majors in the sample?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$X=2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$, $$9$$","$$X=0$$, $$1$$, $$2$$, $$3$$, ..., $$9$$","$$X=0$$, $$1$$, $$2$$, $$3$$, ..., $$16$$","$$X=1$$, $$2$$, $$3$$, ..., $$16$$"]}]}}]},{"id":"ac613aahypergeometric12","title":"Hypergeometric Basics","body":"Suppose that a group of statistics students is divided into two groups: business majors and non-business majors. There are $$16$$ business majors in the group and seven non-business majors in the group. A random sample of nine students is taken. We are interested in the number of business majors in the sample. Let X $$=$$ the number of business majors in the sample.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Hypergeometric Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac613aahypergeometric12a","stepAnswer":["$$6.26$$"],"problemType":"TextBox","stepTitle":"On average, how many would you expect to be business majors rounded to the hundredths place?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6.26$$","hints":{"DefaultPathway":[{"id":"ac613aahypergeometric12a-h1","type":"hint","dependencies":[],"title":"Formula to Determine Mean","text":"The formula for the mean of a hypergeometric distribution is $$\\\\frac{nr}{r+b}$$ where $$n$$ $$=$$ the size of the chosen sample, $$r$$ $$=$$ the size of the group of interest (first group), and $$b$$ $$=$$ the size of the second group.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["ac613aahypergeometric12a-h1"],"title":"Determining the Size of the Chosen Sample","text":"What is the size of the sample? In other words, how many statistics students will be sample?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric12a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["ac613aahypergeometric12a-h2"],"title":"Determining the Size of the Group of Interest","text":"What is the size of the group of interest? In other words, how many business majors are in the pool to decide the sample?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["ac613aahypergeometric12a-h3"],"title":"Determining the Size of the Second Group","text":"What is the size of the second group? In other words, how many non-business majors are in the pool to determine the sample?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric12a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6.26$$"],"dependencies":["ac613aahypergeometric12a-h4"],"title":"Use the Formula","text":"Using the formula, what is the expected value rounded to the hundredths place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac613aahypergeometric13","title":"Hypergeometric Notation","body":"A group of Martial Arts students is planning on participating in an upcoming demonstration. Six are students of Tae Kwon Do; seven are students of Shotokan Karate. Suppose that eight students are randomly picked to be in the first demonstration. We are interested in the number of Shotokan Karate students in the first demonstration. Let X $$=$$ the number of Shotokan Karate students in the first demonstration.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Hypergeometric Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac613aahypergeometric13a","stepAnswer":["$$X=2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$"],"problemType":"MultipleChoice","stepTitle":"What values can X take on?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$X=2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$","choices":["$$X=2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$","$$X=0$$, $$1$$, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$","$$X=1$$, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$","$$X=0$$, $$1$$, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$"],"hints":{"DefaultPathway":[{"id":"ac613aahypergeometric13a-h1","type":"hint","dependencies":[],"title":"Determining All Possible Values of X","text":"We want to know all the total possible values of X, not just the values we want. Also, note that we have $$8$$ students that in the first demonstration total and $$6$$ Tae Kwon Do students and $$7$$ Shotokan Karate students.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric13a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ac613aahypergeometric13a-h1"],"title":"Determining Possibilities","text":"Is it possible to have $$0$$ Shotokan Karate students in the first demonstration?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ac613aahypergeometric13a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ac613aahypergeometric13a-h2"],"title":"Having all $$6$$ Tae Kwon Do","text":"If we have all $$6$$ Tae Kwon Do students be part of the first demonstration, we still need $$8-6=2$$ students to be part of the first demonstration. Therefore, what is the minimum number of Shotokan Karate that must be part of the group?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric13a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$X=2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$"],"dependencies":["ac613aahypergeometric13a-h3"],"title":"Determining All Possible Values of X","text":"Understanding that the values that X takes on includes the entire set of possible number of Shotokan Karate students in the sample of $$8$$, what values does X take on? In other words, what are the possible number of Shotokan Karate students in the first demonstration?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$X=2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$","$$X=0$$, $$1$$, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$","$$X=1$$, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$","$$X=0$$, $$1$$, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$"]}]}},{"id":"ac613aahypergeometric13b","stepAnswer":["X ~ H(7, $$6$$, 8)"],"problemType":"MultipleChoice","stepTitle":"What is the hypergeometric distribution notation for this problem?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"X ~ H(7, $$6$$, 8)","choices":["X ~ H(7, $$6$$, 8)","X ~ H(6, $$8$$, 7)","X ~ H(6, $$7$$, 8)","X ~ H(7, $$8$$, 6)"],"hints":{"DefaultPathway":[{"id":"ac613aahypergeometric13b-h5","type":"hint","dependencies":["ac613aahypergeometric13a-h4"],"title":"Structure of Hypergeometric Notation","text":"Note that the notation for a Hypergeometric distribution is X ~ H(r, $$b$$, n) that reads as \\"X is a random variable with a hypergeometric distribution\\". The parameters are $$r$$, $$b$$, and $$n$$ where $$r$$ is the size of the group of interest (first group), $$b$$ is the size of the second group, and $$n$$ is the size of the chosen sample.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric13b-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["ac613aahypergeometric13b-h5"],"title":"Determining the Size of the Group of Interest","text":"What is the size of the group of interest? In other words, how many Shotokan Karate students can be chosen for the first demonstration?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric13b-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["ac613aahypergeometric13b-h6"],"title":"Determining the Size of the Second Group","text":"What is the size of the second group? In other words, how many Tae Kwon Do students can be chosen for the first demonstration?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric13b-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["ac613aahypergeometric13b-h7"],"title":"Determining the Size of the Chosen Sample","text":"What is the size of the sample? In other words, how many students are in the first demonstration?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric13b-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["X ~ H(7, $$6$$, 8)"],"dependencies":["ac613aahypergeometric13b-h8"],"title":"Putting it All Together","text":"Knowing that the size of the group of interest is $$7$$, the size of the second group is $$6$$, and the size of the chosen sample is $$8$$, plug this into the formula to determine the hypergeometric notation of this question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["X ~ H(7, $$6$$, 8)","X ~ H(6, $$8$$, 7)","X ~ H(6, $$7$$, 8)","X ~ H(7, $$8$$, 6)"]}]}}]},{"id":"ac613aahypergeometric14","title":"Means of Hypergeometric Distributions","body":"A group of Martial Arts students is planning on participating in an upcoming demonstration. Six are students of Tae Kwon Do; seven are students of Shotokan Karate. Suppose that eight students are randomly picked to be in the first demonstration. We are interested in the number of Shotokan Karate students in the first demonstration. Let X $$=$$ the number of Shotokan Karate students in the first demonstration.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Hypergeometric Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac613aahypergeometric14a","stepAnswer":["$$4.31$$"],"problemType":"TextBox","stepTitle":"How many Shotokan Karate students do we expect to be in the first demonstration? What is the expected $$\\\\frac{value}{mean}$$? Round to the nearest hundredths place.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4.31$$","hints":{"DefaultPathway":[{"id":"ac613aahypergeometric14a-h1","type":"hint","dependencies":[],"title":"Formula to Determine Mean","text":"The formula for the mean of a hypergeometric distribution is $$\\\\frac{nr}{r+b}$$ where $$n$$ $$=$$ the size of the chosen sample, $$r$$ $$=$$ the size of the group of interest (first group), and $$b$$ $$=$$ the size of the second group.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["ac613aahypergeometric14a-h1"],"title":"Determining the Size of the Chosen Sample","text":"What is the size of the sample? In other words, how many students will be in the first demonstration?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric14a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["ac613aahypergeometric14a-h2"],"title":"Determining the Size of the Group of Interest","text":"What is the size of the group of interest? In other words, how many Shotokan Karate students are there in total that can perform in the first demonstration?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric14a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["ac613aahypergeometric14a-h3"],"title":"Determining the Size of the Second Group","text":"What is the size of the second group? In other words, how many Tae Kwon Do students are there in total that can perform in the first demonstration?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric14a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4.31$$"],"dependencies":["ac613aahypergeometric14a-h4"],"title":"Use the Formula","text":"Using the formula, what is the expected value rounded to the hundredths place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac613aahypergeometric15","title":"Hypergeometric Notation","body":"In one of its Spring catalogs, L.L. Bean\xae advertised footwear on $$29$$ of its $$192$$ catalog pages. Suppose we randomly survey $$20$$ pages. We are interested in the number of pages that advertise footwear. Each page may be picked at most once. Let X $$=$$ the number of pages that advertise footwear.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Hypergeometric Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac613aahypergeometric15a","stepAnswer":["$$X=0$$, $$1$$, $$2$$, $$3$$, ..., $$20$$"],"problemType":"MultipleChoice","stepTitle":"What values can X take on?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$X=0$$, $$1$$, $$2$$, $$3$$, ..., $$20$$","choices":["$$X=0$$, $$1$$, $$2$$, $$3$$, ..., $$20$$","$$X=0$$, $$1$$, $$2$$, $$3$$, ..., $$29$$","$$X=1$$, $$2$$, $$3$$, ..., $$20$$","$$X=1$$, $$2$$, $$3$$, ..., $$29$$"],"hints":{"DefaultPathway":[{"id":"ac613aahypergeometric15a-h1","type":"hint","dependencies":[],"title":"Determining All Possible Values of X","text":"We want to know all the total possible values of X, not just the values we want.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric15a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$X=0$$, $$1$$, $$2$$, $$3$$, ..., $$20$$"],"dependencies":["ac613aahypergeometric15a-h1"],"title":"Determining All Possible Values of X","text":"Understanding that the values that X takes on includes the entire set of possible number of footwear pages that can be selected, what values does X take on? In other words, what are the possible number of footwear pages selected in the random survey?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$X=2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$","$$X=0$$, $$1$$, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$","$$X=1$$, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$","$$X=0$$, $$1$$, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$"]}]}},{"id":"ac613aahypergeometric15b","stepAnswer":["X ~ H(29, $$163$$, 20)"],"problemType":"MultipleChoice","stepTitle":"What is the hypergeometric distribution notation for this problem?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"X ~ H(29, $$163$$, 20)","choices":["X ~ H(29, $$163$$, 20)","X ~ H(20, $$29$$, 163)","X ~ H(29, $$20$$, 163)","X ~ H(20, $$163$$, 29)"],"hints":{"DefaultPathway":[{"id":"ac613aahypergeometric15b-h3","type":"hint","dependencies":["ac613aahypergeometric15a-h2"],"title":"Structure of Hypergeometric Notation","text":"Note that the notation for a Hypergeometric distribution is X ~ H(r, $$b$$, n) that reads as \\"X is a random variable with a hypergeometric distribution\\". The parameters are $$r$$, $$b$$, and $$n$$ where $$r$$ is the size of the group of interest (first group), $$b$$ is the size of the second group, and $$n$$ is the size of the chosen sample.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric15b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$29$$"],"dependencies":["ac613aahypergeometric15b-h3"],"title":"Determining the Size of the Group of Interest","text":"What is the size of the group of interest? In other words, how many footwear pages are there total?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric15b-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$163$$"],"dependencies":["ac613aahypergeometric15b-h4"],"title":"Determining the Size of the Second Group","text":"What is the size of the second group? In other words, how many non-footwear pages are there total?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric15b-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["ac613aahypergeometric15b-h5"],"title":"Determining the Size of the Chosen Sample","text":"What is the size of the sample? In other words, how many pages will be surveyed?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric15b-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["X ~ H(29, $$163$$, 20)"],"dependencies":["ac613aahypergeometric15b-h6"],"title":"Putting it All Together","text":"Knowing that the size of the group of interest is $$29$$, the size of the second group is $$163$$, and the size of the chosen sample is $$20$$, plug this into the formula to determine the hypergeometric notation of this question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["X ~ H(29, $$163$$, 20)","X ~ H(20, $$29$$, 163)","X ~ H(29, $$20$$, 163)","X ~ H(20, $$163$$, 29)"]}]}}]},{"id":"ac613aahypergeometric16","title":"Means of Hypergeometric Distributions","body":"In one of its Spring catalogs, L.L. Bean\xae advertised footwear on $$29$$ of its $$192$$ catalog pages. Suppose we randomly survey $$20$$ pages. We are interested in the number of pages that advertise footwear. Each page may be picked at most once. Let X $$=$$ the number of pages that advertise footwear.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Hypergeometric Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac613aahypergeometric16a","stepAnswer":["$$3.03$$"],"problemType":"TextBox","stepTitle":"How many footwear pages do we expect to be in the surveyed pages? What is the expected $$\\\\frac{value}{mean}$$? Round to the nearest hundredths place.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3.03$$","hints":{"DefaultPathway":[{"id":"ac613aahypergeometric16a-h1","type":"hint","dependencies":[],"title":"Formula to Determine Mean","text":"The formula for the mean of a hypergeometric distribution is $$\\\\frac{nr}{r+b}$$ where $$n$$ $$=$$ the size of the chosen sample, $$r$$ $$=$$ the size of the group of interest (first group), and $$b$$ $$=$$ the size of the second group.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["ac613aahypergeometric16a-h1"],"title":"Determining the Size of the Chosen Sample","text":"What is the size of the sample? In other words, how many pages will be surveyed?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric16a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$29$$"],"dependencies":["ac613aahypergeometric16a-h2"],"title":"Determining the Size of the Group of Interest","text":"What is the size of the group of interest? In other words, how many footwear pages are there in total?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric16a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$163$$"],"dependencies":["ac613aahypergeometric16a-h3"],"title":"Determining the Size of the Second Group","text":"What is the size of the second group? In other words, how many non-footwear pages are there total?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric16a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3.03$$"],"dependencies":["ac613aahypergeometric16a-h4"],"title":"Use the Formula","text":"Using the formula, what is the expected value rounded to the hundredths place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac613aahypergeometric17","title":"Hypergeometric Notation","body":"Suppose that a technology task force is being formed to study technology awareness among instructors. Assume that ten people will be randomly chosen to be on the committee from a group of $$28$$ volunteers, $$20$$ of whom are technically proficient and $$8$$ of whom are not. We are interested in the number on the committee who are not technically proficient. Let X $$=$$ the number of people on the committee who are not technically proficient.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Hypergeometric Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac613aahypergeometric17a","stepAnswer":["$$X=0$$, $$1$$, $$2$$, $$3$$, ..., $$8$$"],"problemType":"MultipleChoice","stepTitle":"What values can X take on?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$X=0$$, $$1$$, $$2$$, $$3$$, ..., $$8$$","choices":["$$X=0$$, $$1$$, $$2$$, $$3$$, ..., $$8$$","$$X=1$$, $$2$$, $$3$$, ..., $$8$$","$$X=0$$, $$1$$, $$2$$, $$3$$, ..., $$10$$","$$X=1$$, $$2$$, $$3$$, ..., $$10$$"],"hints":{"DefaultPathway":[{"id":"ac613aahypergeometric17a-h1","type":"hint","dependencies":[],"title":"Determining All Possible Values of X","text":"We want to know all the total possible values of X, not just the values we want.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric17a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$X=0$$, $$1$$, $$2$$, $$3$$, ..., $$8$$"],"dependencies":["ac613aahypergeometric17a-h1"],"title":"Determining All Possible Values of X","text":"Understanding that the values that X takes on includes the entire set of possible number of not technically proficient members that can be selected, what values does X take on? In other words, what are the possible number of not technically proficient members selected in the technology task force?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$X=0$$, $$1$$, $$2$$, $$3$$, ..., $$8$$","$$X=1$$, $$2$$, $$3$$, ..., $$8$$","$$X=0$$, $$1$$, $$2$$, $$3$$, ..., $$10$$","$$X=1$$, $$2$$, $$3$$, ..., $$10$$"]}]}},{"id":"ac613aahypergeometric17b","stepAnswer":["X ~ H(8, $$20$$, 10)"],"problemType":"MultipleChoice","stepTitle":"What is the hypergeometric distribution notation for this problem?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"X ~ H(8, $$20$$, 10)","choices":["X ~ H(8, $$20$$, 10)","X ~ H(8, $$10$$, 20)","X ~ H(10, $$8$$, 20)","X ~ H(20, $$10$$, 8)"],"hints":{"DefaultPathway":[{"id":"ac613aahypergeometric17b-h3","type":"hint","dependencies":["ac613aahypergeometric17a-h2"],"title":"Structure of Hypergeometric Notation","text":"Note that the notation for a Hypergeometric distribution is X ~ H(r, $$b$$, n) that reads as \\"X is a random variable with a hypergeometric distribution\\". The parameters are $$r$$, $$b$$, and $$n$$ where $$r$$ is the size of the group of interest (first group), $$b$$ is the size of the second group, and $$n$$ is the size of the chosen sample.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric17b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["ac613aahypergeometric17b-h3"],"title":"Determining the Size of the Group of Interest","text":"What is the size of the group of interest? In other words, how many not technically proficient volunteers are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric17b-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["ac613aahypergeometric17b-h4"],"title":"Determining the Size of the Second Group","text":"What is the size of the second group? In other words, how many technically proficient volunteers are there?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric17b-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["ac613aahypergeometric17b-h5"],"title":"Determining the Size of the Chosen Sample","text":"What is the size of the sample? In other words, how many people will be chosen to be part of the technology task force committee?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric17b-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["X ~ H(8, $$20$$, 10)"],"dependencies":["ac613aahypergeometric17b-h6"],"title":"Putting it All Together","text":"Knowing that the size of the group of interest is $$8$$, the size of the second group is $$20$$, and the size of the chosen sample is $$10$$, plug this into the formula to determine the hypergeometric notation of this question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["X ~ H(8, $$20$$, 10)","X ~ H(8, $$10$$, 20)","X ~ H(10, $$8$$, 20)","X ~ H(20, $$10$$, 8)"]}]}}]},{"id":"ac613aahypergeometric18","title":"Means of Hypergeometric Distributions","body":"Suppose that a technology task force is being formed to study technology awareness among instructors. Assume that ten people will be randomly chosen to be on the committee from a group of $$28$$ volunteers, $$20$$ of whom are technically proficient and $$8$$ of whom are not. We are interested in the number on the committee who are not technically proficient. Let X $$=$$ the number of people on the committee who are not technically proficient.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Hypergeometric Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac613aahypergeometric18a","stepAnswer":["$$2.86$$"],"problemType":"TextBox","stepTitle":"How many not technically proficient volunteers do we expect to be in the technology task force committee? What is the expected $$\\\\frac{value}{mean}$$? Round to the nearest hundredths place.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2.86$$","hints":{"DefaultPathway":[{"id":"ac613aahypergeometric18a-h1","type":"hint","dependencies":[],"title":"Formula to Determine Mean","text":"The formula for the mean of a hypergeometric distribution is $$\\\\frac{nr}{r+b}$$ where $$n$$ $$=$$ the size of the chosen sample, $$r$$ $$=$$ the size of the group of interest (first group), and $$b$$ $$=$$ the size of the second group.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["ac613aahypergeometric18a-h1"],"title":"Determining the Size of the Chosen Sample","text":"What is the size of the sample? In other words, how many people will be selected to be on the committee?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric18a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["ac613aahypergeometric18a-h2"],"title":"Determining the Size of the Group of Interest","text":"What is the size of the group of interest? In other words, how many not technically proficient volunteers are there in total?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric18a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["ac613aahypergeometric18a-h3"],"title":"Determining the Size of the Second Group","text":"What is the size of the second group? In other words, how many technically proficient volunteers are there in total?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric18a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.86$$"],"dependencies":["ac613aahypergeometric18a-h4"],"title":"Use the Formula","text":"Using the formula, what is the expected value rounded to the hundredths place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac613aahypergeometric19","title":"Hypergeometric Notation","body":"Suppose that nine Massachusetts athletes are scheduled to appear at a charity benefit. The nine are randomly chosen from eight volunteers from the Boston Celtics and four volunteers are from the New England Patriots. We are interested in the number of Patriots picked. Let X $$=$$ the number of New England Patriots picked to appear at the charity benefit.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Hypergeometric Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac613aahypergeometric19a","stepAnswer":["$$X=1$$, $$2$$, $$3$$, $$4$$"],"problemType":"MultipleChoice","stepTitle":"What values can X take on?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$X=1$$, $$2$$, $$3$$, $$4$$","choices":["$$X=1$$, $$2$$, $$3$$, $$4$$","$$X=0$$, $$1$$, $$2$$, $$3$$, $$4$$","$$X=0$$, $$1$$, $$2$$, $$3$$, ..., $$9$$","$$X=1$$, $$2$$, $$3$$, ..., $$9$$"],"hints":{"DefaultPathway":[{"id":"ac613aahypergeometric19a-h1","type":"hint","dependencies":[],"title":"Determining All Possible Values of X","text":"We want to know all the total possible values of X, not just the values we want. Also, note that we have $$9$$ randomly chosen athletes total, chosen from $$8$$ Boston Celtics and $$4$$ New England Patriots.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric19a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ac613aahypergeometric19a-h1"],"title":"Determining Possibilities","text":"Is it possible to have $$0$$ New England Patriots chosen?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ac613aahypergeometric19a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ac613aahypergeometric19a-h2"],"title":"Having all $$6$$ Tae Kwon Do","text":"If we have all $$8$$ Boston Celtics chosen for the charity benefit, we still need $$9-8=1$$ Massachusetts athletes to appear at the charity benefit. Therefore, what is the minimum number of New England Patriots that must be part of the group?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric19a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$X=1$$, $$2$$, $$3$$, $$4$$"],"dependencies":["ac613aahypergeometric19a-h3"],"title":"Determining All Possible Values of X","text":"Understanding that the values that X takes on includes the entire set of possible number of New England Patriots in the sample of $$9$$, what values does X take on? In other words, what are the possible number of New England Patriots that can be chosen?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$X=1$$, $$2$$, $$3$$, $$4$$","$$X=0$$, $$1$$, $$2$$, $$3$$, $$4$$","$$X=0$$, $$1$$, $$2$$, $$3$$, ..., $$9$$","$$X=1$$, $$2$$, $$3$$, ..., $$9$$"]}]}},{"id":"ac613aahypergeometric19b","stepAnswer":["X ~ H(4, $$8$$, 9)"],"problemType":"MultipleChoice","stepTitle":"What is the hypergeometric distribution notation for this problem?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"X ~ H(4, $$8$$, 9)","choices":["X ~ H(4, $$8$$, 9)","X ~ H(4, $$9$$, 8)","X ~ H(8, $$4$$, 9)","X ~ H(8, $$9$$, 4)"],"hints":{"DefaultPathway":[{"id":"ac613aahypergeometric19b-h5","type":"hint","dependencies":["ac613aahypergeometric19a-h4"],"title":"Structure of Hypergeometric Notation","text":"Note that the notation for a Hypergeometric distribution is X ~ H(r, $$b$$, n) that reads as \\"X is a random variable with a hypergeometric distribution\\". The parameters are $$r$$, $$b$$, and $$n$$ where $$r$$ is the size of the group of interest (first group), $$b$$ is the size of the second group, and $$n$$ is the size of the chosen sample.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric19b-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ac613aahypergeometric19b-h5"],"title":"Determining the Size of the Group of Interest","text":"What is the size of the group of interest? In other words, how many New England Patriots are there total?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric19b-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["ac613aahypergeometric19b-h6"],"title":"Determining the Size of the Second Group","text":"What is the size of the second group? In other words, how many Boston Celtics (non-new England Patriots) are there total?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric19b-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["ac613aahypergeometric19b-h7"],"title":"Determining the Size of the Chosen Sample","text":"What is the size of the sample? In other words, how many total Massachusetts athletes will be selected to show up at the charity event?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric19b-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["X ~ H(4, $$8$$, 9)"],"dependencies":["ac613aahypergeometric19b-h8"],"title":"Putting it All Together","text":"Knowing that the size of the group of interest is $$7$$, the size of the second group is $$6$$, and the size of the chosen sample is $$8$$, plug this into the formula to determine the hypergeometric notation of this question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["X ~ H(4, $$8$$, 9)","X ~ H(4, $$9$$, 8)","X ~ H(8, $$4$$, 9)","X ~ H(8, $$9$$, 4)"]}]}}]},{"id":"ac613aahypergeometric2","title":"Reading Hypergeometric Questions","body":"A bag contains letter tiles. Forty-four of the tiles are vowels, and $$56$$ are consonants. Seven tiles are picked at random. You want to know the probability that four of the seven tiles are vowels.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Hypergeometric Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac613aahypergeometric2a","stepAnswer":["Vowels"],"problemType":"MultipleChoice","stepTitle":"What is the group of interest?","stepBody":"","answerType":"string","variabilization":{},"choices":["Vowels","Consonants","Tiles","Letters"],"hints":{"DefaultPathway":[{"id":"ac613aahypergeometric2a-h1","type":"hint","dependencies":[],"title":"Reading the Question","text":"The probability question asks for the probability that four of the seven tiles that are picked at random are vowels.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric2a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Vowels"],"dependencies":["ac613aahypergeometric2a-h1"],"title":"Determining the Group of Interest","text":"Knowing that the probability question asks for the probability that four of the seven tiles are vowels, what is the group of interest?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Vowels","Consonants","Tiles","Letters"]}]}},{"id":"ac613aahypergeometric2b","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"What is the size of the group of interest?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"ac613aahypergeometric2b-h3","type":"hint","dependencies":["ac613aahypergeometric2a-h2"],"title":"Reading the Question","text":"The probability question asks for the probability that four of the seven tiles that are picked at random are vowels. Remember that the group of interest is vowels. Also remember that the size of the group of interest is different than the size of the sample.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric2b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ac613aahypergeometric2b-h3"],"title":"Determining the Size of the Group of Interest","text":"Knowing that we want to determine the probability that four of the seven tiles are vowels, what is the size of the group of interest?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ac613aahypergeometric2c","stepAnswer":["$$7$$"],"problemType":"TextBox","stepTitle":"What is the size of the sample?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7$$","hints":{"DefaultPathway":[{"id":"ac613aahypergeometric2c-h5","type":"hint","dependencies":["ac613aahypergeometric2b-h4"],"title":"Reading the Question","text":"The probability question asks for the probability that four of the seven tiles that are picked at random are vowels. Remember that the size of the sample is the number of tiles that we pick at random.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric2c-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["ac613aahypergeometric2c-h5"],"title":"Determining the Size of the Sample","text":"Knowing that the size of a sample is the number of tiles that is picked at random, what is the size of the sample of letter tiles?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac613aahypergeometric20","title":"Means of Hypergeometric Distributions","body":"Suppose that nine Massachusetts athletes are scheduled to appear at a charity benefit. The nine are randomly chosen from eight volunteers from the Boston Celtics and four volunteers are from the New England Patriots. We are interested in the number of Patriots picked. Let X $$=$$ the number of New England Patriots picked to appear at the charity benefit.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Hypergeometric Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac613aahypergeometric20a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"How many New England Patriots volunteers do we expect to be selected to appear at the charity benefit? What is the expected $$\\\\frac{value}{mean}$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"ac613aahypergeometric20a-h1","type":"hint","dependencies":[],"title":"Formula to Determine Mean","text":"The formula for the mean of a hypergeometric distribution is $$\\\\frac{nr}{r+b}$$ where $$n$$ $$=$$ the size of the chosen sample, $$r$$ $$=$$ the size of the group of interest (first group), and $$b$$ $$=$$ the size of the second group.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["ac613aahypergeometric20a-h1"],"title":"Determining the Size of the Chosen Sample","text":"What is the size of the sample? In other words, how many Massachusetts athletes will be selected to show up for the charity benefit?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric20a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ac613aahypergeometric20a-h2"],"title":"Determining the Size of the Group of Interest","text":"What is the size of the group of interest? In other words, how many New England Patriots are there total?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric20a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["ac613aahypergeometric20a-h3"],"title":"Determining the Size of the Second Group","text":"What is the size of the second group? In other words, how many Boston Celtics are there total?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric20a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ac613aahypergeometric20a-h4"],"title":"Use the Formula","text":"Using the formula, what is the expected value?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac613aahypergeometric3","title":"Hypergeometric Probability Statements","body":"A bag contains letter tiles. Forty-four of the tiles are vowels, and $$56$$ are consonants. Seven tiles are picked at random. You want to know the probability that four of the seven tiles are vowels.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Hypergeometric Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac613aahypergeometric3a","stepAnswer":["$$P(x=4)$$"],"problemType":"MultipleChoice","stepTitle":"Let X $$=$$ the number of vowel tiles in the sample of $$7$$. X takes on the values $$x=1$$, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$. What is the probability statement written mathematically?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$P(x=4)$$","choices":["$$P(x=4)$$","$$P(y=4)$$","$$P(x=3)$$","$$P(x=7)$$"],"hints":{"DefaultPathway":[{"id":"ac613aahypergeometric3a-h1","type":"hint","dependencies":[],"title":"Structure of Probability Statements","text":"The structure of a probability statement is $$P(x=C)$$ where P stands for the probability, $$x$$ is the variable we are testing against and C is the constant value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric3a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x$$"],"dependencies":["ac613aahypergeometric3a-h1"],"title":"Determining the variable","text":"What is the variable that we are using?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x$$","$$y$$","$$z$$","c"]},{"id":"ac613aahypergeometric3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ac613aahypergeometric3a-h2"],"title":"Determining the constant","text":"What is the number of vowel tiles we want in the sample?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric3a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$P(x=4)$$"],"dependencies":["ac613aahypergeometric3a-h3"],"title":"Putting it All Together","text":"What is the probability statement written mathematically, using the variable and constant we found?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$P(x=4)$$","$$P(y=4)$$","$$P(x=3)$$","$$P(x=7)$$"]}]}}]},{"id":"ac613aahypergeometric4","title":"Hypergeometric Probability Statements","body":"Suppose a shipment of $$100$$ DVD players is known to have ten defective players. An inspector randomly chooses $$12$$ for inspection. He is interested in determining the probability that, among the $$12$$ players, at most two are defective. The two groups are the $$90$$ non-defective DVD players and the $$10$$ defective DVD players. The group of interest (first group) is the defective group because the probability question asks for the probability of at most two defective DVD players. The size of the sample is $$12$$ DVD players. (They may be non-defective or defective.) Let X $$=$$ the number of defective DVD players in the sample of $$12$$. X takes on the values $$0$$, $$1$$, $$2$$, ..., $$10$$. X may not take on the values $$11$$ or $$12$$. The sample size is $$12$$, but there are only $$10$$ defective DVD players.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Hypergeometric Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac613aahypergeometric4a","stepAnswer":["$$P(x \\\\leq 2)$$"],"problemType":"MultipleChoice","stepTitle":"What is the probability statement written mathematically?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$P(x \\\\leq 2)$$","choices":["$$P(x \\\\leq 2)$$","$$P\\\\left(x<2\\\\right)$$","$$P\\\\left(x>2\\\\right)$$","$$P(x \\\\geq 2)$$"],"hints":{"DefaultPathway":[{"id":"ac613aahypergeometric4a-h1","type":"hint","dependencies":[],"title":"Structure of Probability Statements","text":"The structure of a probability statement is P(xOC) where P stands for the probability, $$x$$ is the variable we are testing against, O is the operation (equal to, less than, less than or equal to, greater than, greater than or equal to) and C is the constant value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric4a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x$$"],"dependencies":["ac613aahypergeometric4a-h1"],"title":"Determining the variable","text":"What is the variable that we are using?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x$$","$$y$$","$$z$$","c"]},{"id":"ac613aahypergeometric4a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Less than or equal to"],"dependencies":["ac613aahypergeometric4a-h2"],"title":"Determining the operation","text":"What is the operation? The inspector wants at most $$2$$ of the $$12$$ players to be defective. What operation best represents a ceiling when compared to a constant, or \\"at most\\", which is inclusive?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Less than or equal to","Less than","Greater than","Greater than or equal to"]},{"id":"ac613aahypergeometric4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ac613aahypergeometric4a-h3"],"title":"Determining the constant","text":"What is the number of defective players we want at most in the sample?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric4a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$P(x \\\\leq 2)$$"],"dependencies":["ac613aahypergeometric4a-h4"],"title":"Putting it All Together","text":"What is the probability statement written mathematically, using the variable and constant we found?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$P(x \\\\leq 2)$$","$$P\\\\left(x<2\\\\right)$$","$$P\\\\left(x>2\\\\right)$$","$$P(x \\\\geq 2)$$"]}]}}]},{"id":"ac613aahypergeometric5","title":"Hypergeometric Random Variables","body":"A gross of eggs contains $$144$$ eggs. A particular gross is known to have $$12$$ cracked eggs. An inspector randomly chooses $$15$$ for inspection. She wants to know the probability that, among the $$15$$, at most three are cracked.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Hypergeometric Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac613aahypergeometric5a","stepAnswer":["$$X=1$$, $$2$$, $$3$$, $$4$$, ..., $$15$$"],"problemType":"MultipleChoice","stepTitle":"Let X be the nmber of eggs within the sample of $$15$$ that are cracked. What values does X take on?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$X=1$$, $$2$$, $$3$$, $$4$$, ..., $$15$$","choices":["$$X=1$$, $$2$$, $$3$$, $$4$$, ..., $$15$$","$$X=1$$, $$2$$, $$3$$","$$X=1$$, $$2$$, $$3$$, $$4$$, ..., $$12$$","$$X=1$$, $$2$$, $$3$$, $$4$$, $$5$$"],"hints":{"DefaultPathway":[{"id":"ac613aahypergeometric5a-h1","type":"hint","dependencies":[],"title":"Determining All Possible Values of X","text":"We want to know all the total possible values of X, not just the values we want.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric5a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$X=1$$, $$2$$, $$3$$, $$4$$, ..., $$15$$"],"dependencies":["ac613aahypergeometric5a-h1"],"title":"Determining All Possible Values of X","text":"Understanding that the values that X takes on includes the entire set of possible number of cracked eggs within $$15$$, what values does X take on? In other words, how many possible cracked eggs could there be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$X=1$$, $$2$$, $$3$$, $$4$$, ..., $$15$$","$$X=1$$, $$2$$, $$3$$","$$X=1$$, $$2$$, $$3$$, $$4$$, ..., $$12$$","$$X=1$$, $$2$$, $$3$$, $$4$$, $$5$$"]}]}},{"id":"ac613aahypergeometric5b","stepAnswer":["$$P(x \\\\leq 3)$$"],"problemType":"MultipleChoice","stepTitle":"What is the probability statement, written mathematically?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$P(x \\\\leq 3)$$","choices":["$$P(x \\\\leq 3)$$","$$P\\\\left(x<3\\\\right)$$","$$P(x \\\\geq 3)$$","$$P\\\\left(x>3\\\\right)$$"],"hints":{"DefaultPathway":[{"id":"ac613aahypergeometric5b-h1","type":"hint","dependencies":[],"title":"Structure of Probability Statements","text":"The structure of a probability statement is P(xOC) where P stands for the probability, $$x$$ is the variable we are testing against, O is the operation (equal to, less than, less than or equal to, greater than, greater than or equal to) and C is the constant value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric5b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x$$"],"dependencies":["ac613aahypergeometric5b-h1"],"title":"Determining the variable","text":"What is the variable that we are using?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x$$","$$y$$","$$z$$","c"]},{"id":"ac613aahypergeometric5b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Less than or equal to"],"dependencies":["ac613aahypergeometric5b-h2"],"title":"Determining the operation","text":"What is the operation? The inspector wants at most $$3$$ of the $$15$$ eggs to be defective. What operation best represents a ceiling when compared to a constant, or \\"at most\\", which is inclusive?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Less than or equal to","Less than","Greater than","Greater than or equal to"]},{"id":"ac613aahypergeometric5b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ac613aahypergeometric5b-h3"],"title":"Determining the constant","text":"What is the number of cracked eggs we want at most in the sample?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric5b-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$P(x \\\\leq 3)$$"],"dependencies":["ac613aahypergeometric5b-h4"],"title":"Putting it All Together","text":"What is the probability statement written mathematically, using the variable and constant we found?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$P(x \\\\leq 3)$$","$$P\\\\left(x<3\\\\right)$$","$$P(x \\\\geq 3)$$","$$P\\\\left(x>3\\\\right)$$"]}]}}]},{"id":"ac613aahypergeometric6","title":"Hypergeometric Basics","body":"You are president of an on-campus special events organization. You need a committee of seven students to plan a special birthday party for the president of the college. Your organization consists of $$18$$ women and $$15$$ men. You are interested in the number of men on your committee, particularly, the probability that the committee has more than four men. This is a hypergeometric problem because you are choosing your committee from two groups (men and women).","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Hypergeometric Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac613aahypergeometric6a","stepAnswer":["Without"],"problemType":"MultipleChoice","stepTitle":"Are you choosing with or without replacement?","stepBody":"","answerType":"string","variabilization":{},"choices":["Without","With"],"hints":{"DefaultPathway":[{"id":"ac613aahypergeometric6a-h1","type":"hint","dependencies":[],"title":"Characteristics of Hypergeometric Distributions","text":"Remember characteristic (3) of hypergeometric distributions. We know that because this is a hypergeometric distribution, you must sample without replacement from the combined groups.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric6a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Without"],"dependencies":["ac613aahypergeometric6a-h1"],"title":"With or Without Replacement","text":"Knowing that a characteristic of hypergeometric distribution is sampling without replacement, is this experiment performed with or without replacement?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Without","With"]}]}},{"id":"ac613aahypergeometric6b","stepAnswer":["The men"],"problemType":"MultipleChoice","stepTitle":"What is the group of interest?","stepBody":"","answerType":"string","variabilization":{},"choices":["The men","The women","The committee","The president"],"hints":{"DefaultPathway":[{"id":"ac613aahypergeometric6b-h3","type":"hint","dependencies":["ac613aahypergeometric6a-h2"],"title":"Reading the Question","text":"The probability question asks for the probability that the committee has more than four men. Remember that the group of interest doesn\'t focus on the broader sample or the formation of a committee, but rather the smaller group that we want to check the probability of out of a larger group.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric6b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["The men"],"dependencies":["ac613aahypergeometric6b-h3"],"title":"Determining the Group of Interest","text":"Knowing that the probability question asks for the probability that the committe has more than four men, what is the group of interest?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["The men","The women","The committee","The president"]}]}},{"id":"ac613aahypergeometric6c","stepAnswer":["$$15$$"],"problemType":"TextBox","stepTitle":"How many are in the group of interest?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$15$$","hints":{"DefaultPathway":[{"id":"ac613aahypergeometric6c-h5","type":"hint","dependencies":["ac613aahypergeometric6b-h4"],"title":"Reading the Question","text":"The probability question asks for the probability that the committe has more than four men. Remember that the group of interest is the men. Also remember that the size of the group of interest is different than the size of the sample.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric6c-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["ac613aahypergeometric6c-h5"],"title":"Determining the Size of the Group of Interest","text":"Knowing that the group of interest is the men and that the size of the group of interest (what we want to determine) is the overall size of the group, not just what we expect or the number we\'re looking for. In other words, how many men are in the organization?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac613aahypergeometric7","title":"Hypergeometric Basics","body":"You are president of an on-campus special events organization. You need a committee of seven students to plan a special birthday party for the president of the college. Your organization consists of $$18$$ women and $$15$$ men. You are interested in the number of men on your committee, particularly, the probability that the committee has more than four men. This is a hypergeometric problem because you are choosing your committee from two groups (men and women).","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Hypergeometric Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac613aahypergeometric7a","stepAnswer":["$$X=0$$, $$1$$, $$2$$, ..., $$7$$"],"problemType":"MultipleChoice","stepTitle":"Let X $$=$$ the number of men on the committee. What values does X take on?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$X=0$$, $$1$$, $$2$$, ..., $$7$$","choices":["$$X=0$$, $$1$$, $$2$$, ..., $$7$$","$$X=0$$, $$1$$, $$2$$, ..., $$15$$","$$X=0$$, $$1$$, $$2$$, ..., $$33$$","$$X=0$$, $$1$$, $$2$$, $$3$$, $$4$$"],"hints":{"DefaultPathway":[{"id":"ac613aahypergeometric7a-h1","type":"hint","dependencies":[],"title":"Determining All Possible Values of X","text":"We want to know all the total possible values of X, not just the values we want.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric7a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$X=0$$, $$1$$, $$2$$, ..., $$7$$"],"dependencies":["ac613aahypergeometric7a-h1"],"title":"Determining All Possible Values of X","text":"Understanding that the values that X takes on includes the entire set of possible number of men on the committee of $$7$$, what values does X take on? In other words, how many possible men could be on the committee?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$X=0$$, $$1$$, $$2$$, ..., $$7$$","$$X=0$$, $$1$$, $$2$$, ..., $$15$$","$$X=0$$, $$1$$, $$2$$, ..., $$33$$","$$X=0$$, $$1$$, $$2$$, $$3$$, $$4$$"]}]}},{"id":"ac613aahypergeometric7b","stepAnswer":["$$P\\\\left(x>4\\\\right)$$"],"problemType":"MultipleChoice","stepTitle":"What is the probability statement, written mathematically?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$P\\\\left(x>4\\\\right)$$","choices":["$$P(x \\\\leq 4)$$","$$P\\\\left(x<4\\\\right)$$","$$P\\\\left(x>4\\\\right)$$","$$P(x \\\\geq 4)$$"],"hints":{"DefaultPathway":[{"id":"ac613aahypergeometric7b-h3","type":"hint","dependencies":["ac613aahypergeometric7a-h2"],"title":"Structure of Probability Statements","text":"The structure of a probability statement is P(xOC) where P stands for the probability, $$x$$ is the variable we are testing against, O is the operation (equal to, less than, less than or equal to, greater than, greater than or equal to) and C is the constant value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric7b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x$$"],"dependencies":["ac613aahypergeometric7b-h3"],"title":"Determining the variable","text":"What is the variable that we are using?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x$$","$$y$$","$$z$$","c"]},{"id":"ac613aahypergeometric7b-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Greater than"],"dependencies":["ac613aahypergeometric7b-h4"],"title":"Determining the operation","text":"What is the operation? We want to determine the probability that the committee has more than four men. What operation best represents a floor when compared to a constant, or \\"more than\\", which is exclusive?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Less than or equal to","Less than","Greater than","Greater than or equal to"]},{"id":"ac613aahypergeometric7b-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ac613aahypergeometric7b-h5"],"title":"Determining the constant","text":"What is the constant that regards to men in the committee? In effect, what is the number that we want the number of men to be greater than?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric7b-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$P\\\\left(x>4\\\\right)$$"],"dependencies":["ac613aahypergeometric7b-h6"],"title":"Putting it All Together","text":"What is the probability statement written mathematically, using the variable and constant we found?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$P(x \\\\leq 4)$$","$$P\\\\left(x<4\\\\right)$$","$$P\\\\left(x>4\\\\right)$$","$$P(x \\\\geq 4)$$"]}]}}]},{"id":"ac613aahypergeometric8","title":"Hypergeometric Probability Distribution Notation","body":"A school site committee is to be chosen randomly from six men and five women. For the question, let the committee consists of four members chosen randomly and men are the group of interest. Let X $$=$$ the number of men on the committee of four.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Hypergeometric Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac613aahypergeometric8a","stepAnswer":["$$X=0$$, $$1$$, $$2$$, $$3$$, $$4$$"],"problemType":"MultipleChoice","stepTitle":"What values can X take on?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$X=0$$, $$1$$, $$2$$, $$3$$, $$4$$","choices":["$$X=0$$, $$1$$, $$2$$, $$3$$, $$4$$","$$X=1$$, $$2$$, $$3$$, $$4$$","$$X=1$$, $$2$$, $$3$$, $$4$$, ..., $$11$$","$$X=0$$, $$1$$, $$2$$, $$3$$, $$4$$, $$5$$"],"hints":{"DefaultPathway":[{"id":"ac613aahypergeometric8a-h1","type":"hint","dependencies":[],"title":"Determining All Possible Values of X","text":"We want to know all the total possible values of X, not just the values we want.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric8a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$X=0$$, $$1$$, $$2$$, $$3$$, $$4$$"],"dependencies":["ac613aahypergeometric8a-h1"],"title":"Determining All Possible Values of X","text":"Understanding that the values that X takes on includes the entire set of possible number of men on the committee of $$4$$, what values does X take on? In other words, how many possible men could be on the committee?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$X=0$$, $$1$$, $$2$$, $$3$$, $$4$$","$$X=1$$, $$2$$, $$3$$, $$4$$","$$X=1$$, $$2$$, $$3$$, $$4$$, ..., $$11$$","$$X=0$$, $$1$$, $$2$$, $$3$$, $$4$$, $$5$$"]}]}},{"id":"ac613aahypergeometric8b","stepAnswer":["X ~ H(6, $$5$$, 4)"],"problemType":"MultipleChoice","stepTitle":"What is the hypergeometric representation of the problem?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"X ~ H(6, $$5$$, 4)","choices":["X ~ H(6, $$5$$, 4)","X ~ H(6, $$4$$, 5)","X ~ H(5, $$6$$, 4)","X ~ H(6, $$5$$, 11)"],"hints":{"DefaultPathway":[{"id":"ac613aahypergeometric8b-h3","type":"hint","dependencies":["ac613aahypergeometric8a-h2"],"title":"Structure of Hypergeometric Notation","text":"Note that the notation for a Hypergeometric distribution is X ~ H(r, $$b$$, n) that reads as \\"X is a random variable with a hypergeometric distribution\\". The parameters are $$r$$, $$b$$, and $$n$$ where $$r$$ is the size of the group of interest (first group), $$b$$ is the size of the second group, and $$n$$ is the size of the chosen sample.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric8b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["ac613aahypergeometric8b-h3"],"title":"Determining the Size of the Group of Interest","text":"What is the size of the group of interest? In other words, how many men are in the pool of candidates for the school site committee?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric8b-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["ac613aahypergeometric8b-h4"],"title":"Determining the Size of the Second Group","text":"What is the size of the second group? In other words, how many women are in the pool of candidates for the school site committee?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric8b-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ac613aahypergeometric8b-h5"],"title":"Determining the Size of the Chosen Sample","text":"What is the size of the sample? In other words, how many members will be selected to form the committee?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric8b-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["X ~ H(6, $$5$$, 4)"],"dependencies":["ac613aahypergeometric8b-h6"],"title":"Putting it All Together","text":"Knowing that the size of the group of interest is $$6$$, the size of the second group is $$5$$, and the size of the chosen sample is $$4$$, plug this into the formula to determine the hypergeometric notation of this question.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["X ~ H(6, $$5$$, 4)","X ~ H(6, $$4$$, 5)","X ~ H(5, $$6$$, 4)","X ~ H(6, $$5$$, 11)"]}]}}]},{"id":"ac613aahypergeometric9","title":"Hypergeometric Group of Interest","body":"An intramural basketball team is to be chosen randomly from $$15$$ boys and $$12$$ girls. The team has ten slots. You want to know the probability that eight of the players will be boys.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Hypergeometric Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac613aahypergeometric9a","stepAnswer":["The boys"],"problemType":"MultipleChoice","stepTitle":"What is the group of interest?","stepBody":"","answerType":"string","variabilization":{},"choices":["The boys","The girls","The basketballs","The team"],"hints":{"DefaultPathway":[{"id":"ac613aahypergeometric9a-h1","type":"hint","dependencies":[],"title":"Reading the Question","text":"The probability question asks for the probability that eight of the players will be boys. Remember that the group of interest doesn\'t focus on the broader sample or the formation of a committee, but rather the smaller group that we want to check the probability of out of a larger group.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric9a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["The boys"],"dependencies":["ac613aahypergeometric9a-h1"],"title":"Determining the Group of Interest","text":"Knowing that the probability question asks for the probability that eight of the players will be boys, what is the group of interest?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["The boys","The girls","The basketballs","The team"]}]}},{"id":"ac613aahypergeometric9b","stepAnswer":["$$15$$"],"problemType":"TextBox","stepTitle":"How many are in the group of interest?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$15$$","hints":{"DefaultPathway":[{"id":"ac613aahypergeometric9b-h3","type":"hint","dependencies":["ac613aahypergeometric9a-h2"],"title":"Reading the Question","text":"The probability question asks for the probability that eight of the players will be boys. Remember that the group of interest is the boys. Also remember that the size of the group of interest is different than the size of the sample.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac613aahypergeometric9b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["ac613aahypergeometric9b-h3"],"title":"Determining the Size of the Group of Interest","text":"Remember that the group of interest is the boys and that the size of the group of interest (what we want to determine) is the overall size of that group, not just what we expect or the number we\'re looking for. In other words, how many boys are candidates for the intramural basketball team?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac6628d13.4collegeinstructor1","title":"Test of Two Variances","body":"Two college instructors are interested in whether or not there is any variation in the way they grade math exams. They each grade the same set of $$30$$ exams. The first instructor\'s grades have a variance of $$52.3$$. The second instructor\'s grades have a variance of $$89.9$$. Test the claim that the first instructor\'s variance is smaller. (In most colleges, it is desirable for the variances of exam grades to be nearly the same among instructors.) The level of significance is 10%.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"13.4 Test of Two Variances","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac6628d13.4collegeinstructor1a","stepAnswer":["H_0:$$(\\\\sigma_1)**2=(\\\\sigma_2)**2, $$H_a$$: (\\\\sigma_1)**2<(\\\\sigma_2)**2"],"problemType":"MultipleChoice","stepTitle":"State the null and alternative hypotheses.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"H_0:$$(\\\\sigma_1)**2=(\\\\sigma_2)**2, $$H_a$$: (\\\\sigma_1)**2<(\\\\sigma_2)**2","choices":["H_0:$$(\\\\sigma_1)**2=(\\\\sigma_2)**2, $$H_a$$: (\\\\sigma_1)**2<(\\\\sigma_2)**2","H_0:$$(\\\\sigma_1)**2>(\\\\sigma_2)**2, $$H_a$$: (\\\\sigma_1)**2<(\\\\sigma_2)**2","H_0:$$(\\\\sigma_1)**2<(\\\\sigma_2)**2, $$H_a$$: (\\\\sigma_1)**2<(\\\\sigma_2)**2","H_0:$$(\\\\sigma_1)**2=(\\\\sigma_2)**2, $$H_a$$: (\\\\sigma_1)**2>=(\\\\sigma_2)**2"],"hints":{"DefaultPathway":[{"id":"ac6628d13.4collegeinstructor1a-h1","type":"hint","dependencies":[],"title":"Test of Two Variances","text":"Since we are interested in comparing the two sample variances, we use the F ratio. A test of two variances may be left, right, or two-tailed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6628d13.4collegeinstructor1a-h2","type":"hint","dependencies":["ac6628d13.4collegeinstructor1a-h1"],"title":"Test of Two Variances","text":"The statistical value could be variance or standard deviation. Let the variance of instructors $$1$$ and $$2$$ equal $$\\\\sigma_1 $$and$$ $$\\\\sigma_2 $$respectively$$. $$The$$ $$instructors$$ $$are$$ $$concerned$$ $$with$$ $$variation$$ $$of$$ $$test$$ $$results$$, $$therefore$$ $$a$$ $$test$$ $$of$$ $$two$$ $$variances$$ $$will$$ $$be$$ $$performed$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6628d13.4collegeinstructor1a-h3","type":"hint","dependencies":["ac6628d13.4collegeinstructor1a-h2"],"title":"Test of Two Variances","text":"Test the claim that the first instructor\'s variance is smaller is a key indicator of the alternative hypothesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6628d13.4collegeinstructor1a-h4","type":"hint","dependencies":["ac6628d13.4collegeinstructor1a-h3"],"title":"Test of Two Variances","text":"By default, the variances are expected to be the same among college instructors, so let that be the null hypothesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ac6628d13.4collegeinstructor1b","stepAnswer":["$$0.5818$$"],"problemType":"TextBox","stepTitle":"Calculate the test statistic. Round to four decimal places.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.5818$$","hints":{"DefaultPathway":[{"id":"ac6628d13.4collegeinstructor1b-h1","type":"hint","dependencies":[],"title":"Test of Two Variances","text":"By the null hypothesis that the variances are the same, the F statistic is $$\\\\frac{{s_1}^2}{{s_2}^2}$$, where s would denote the sample variance.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6628d13.4collegeinstructor1b-h2","type":"hint","dependencies":["ac6628d13.4collegeinstructor1b-h1"],"title":"Test of Two Variances","text":"Instructor 1\'s variance is $$52.3$$, and instructor 2\'s variance is $$89.9$$. divide instructor 1\'s variance with instructor 2\'s variance.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6628d13.4collegeinstructor1b-h3","type":"hint","dependencies":["ac6628d13.4collegeinstructor1b-h2"],"title":"Test of Two Variances","text":"$$F=\\\\frac{52.3}{89.9}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ac6628d13.4collegeinstructor1c","stepAnswer":["$$left-tailed$$"],"problemType":"MultipleChoice","stepTitle":"","stepBody":"What tail test type will be performed?","answerType":"string","variabilization":{},"choices":["$$left-tailed$$","$$right-tailed$$","$$two-tailed$$","none of the options listed is valid."],"hints":{"DefaultPathway":[{"id":"ac6628d13.4collegeinstructor1c-h1","type":"hint","dependencies":[],"title":"Test of Two Variances","text":"Since the alternative claim is that instructor 1\'s grade variances are less than that of instructor $$2$$, the test will be left tailed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ac6628d13.4collegeinstructor1d","stepAnswer":["Since our alpha level of $$0.10$$ is greater than $$0.0753$$, the null hypothesis is rejected."],"problemType":"MultipleChoice","stepTitle":"What conclusion can be made based off of the $$p-value$$ shown?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"Since our alpha level of $$0.10$$ is greater than $$0.0753$$, the null hypothesis is rejected.","choices":["Since our alpha level of $$0.10$$ is greater than $$0.0753$$, the null hypothesis is rejected.","Since our alpha level of $$0.10$$ is greater than $$0.0753$$, we fail to reject the null hypothesis.","Since our alpha level of $$10$$ is greater than $$7.53$$, no conclusion can be made.","Since our alpha level of $$0.0010$$ is less than $$0.0753$$, we fail to reject the null hypothesis."],"hints":{"DefaultPathway":[{"id":"ac6628d13.4collegeinstructor1d-h1","type":"hint","dependencies":[],"title":"Test of Two Variances","text":"In a hypothesis test, a $$p-value$$ lower than the significance level means that the null hypothesis is rejected. With a 10% of significance converted to $$0.10$$, the significance level is higher than the $$p-value$$ of $$0.0753$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac6628d13.4cyclist","title":"Test of Two Variables","body":"Two cyclists are comparing the variances of their overall paces going uphill. Each cyclist records his or her speeds going up $$35$$ hills. The first cyclist has a variance of $$23.8$$ and the second cyclist has a variance of $$32.1$$. The cyclists want to see if their variances are the same or different. Assume that commute times are normally distributed.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"13.4 Test of Two Variances","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac6628d13.4cyclista","stepAnswer":["H_0:$$(\\\\sigma_1)**2=(\\\\sigma_2)**2, $$H_a$$: (\\\\sigma_1)**\\\\neq(\\\\sigma_2)**2"],"problemType":"MultipleChoice","stepTitle":"State the null and alternate hypotheses","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"H_0:$$(\\\\sigma_1)**2=(\\\\sigma_2)**2, $$H_a$$: (\\\\sigma_1)**\\\\neq(\\\\sigma_2)**2","choices":["H_0:$$(\\\\sigma_1)**2=(\\\\sigma_2)**2, $$H_a$$: (\\\\sigma_1)**\\\\neq(\\\\sigma_2)**2","H_0:$$(\\\\sigma_1)**2>(\\\\sigma_2)**2, $$H_a$$: (\\\\sigma_1)**\\\\neq(\\\\sigma_2)**2","H_0:$$(\\\\sigma_1)**2>=(\\\\sigma_2)**2, $$H_a$$: (\\\\sigma_1)**\\\\neq(\\\\sigma_2)**2","H_0:$$(\\\\sigma_1)**2<(\\\\sigma_2)**2, $$H_a$$: (\\\\sigma_1)**2>(\\\\sigma_2)**2"],"hints":{"DefaultPathway":[{"id":"ac6628d13.4cyclista-h1","type":"hint","dependencies":[],"title":"Test of Two Variables","text":"The bikers are interested in whether or not the variances are the same. This indicates that a two tailed test is being performed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6628d13.4cyclista-h2","type":"hint","dependencies":["ac6628d13.4cyclista-h1"],"title":"Test of Two Variables","text":"Variances will be compared. Variances are squares of the standard deviation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ac6628d13.4cyclistb","stepAnswer":["$$0.7414$$"],"problemType":"TextBox","stepTitle":"What is the F statistic? Round to four decimal places.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.7414$$","hints":{"DefaultPathway":[{"id":"ac6628d13.4cyclistb-h1","type":"hint","dependencies":[],"title":"F statistic","text":"By the null hypothesis that the variances are the same, let s denote the standard deviations. The F statistic is $$\\\\frac{{s_1}^2}{{s_2}^2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6628d13.4cyclistb-h2","type":"hint","dependencies":["ac6628d13.4cyclistb-h1"],"title":"F statistic","text":"Suppose we sample randomly from two independent normal populations. Let $$\\\\sigma**{2}_{1} $$and$$ $$\\\\sigma**{2}_{2} $$be$$ $$the$$ $$population$$ $$variances$$ $$and$$ $$s**{2}_{1} $$and$$ $$s**{2}_{1} $$be$$ $$the$$ $$sample$$ $$variances$$. $$Let$$ $$the$$ $$sample$$ $$sizes$$ $$be$$ $$n_1$$ $$and$$. $$_2$$. $$Since$$ $$we$$ $$are$$ $$interested$$ $$in$$ $$comparing$$ $$the$$ $$two$$ $$sample$$ $$variances$$, $$we$$ $$use$$ $$the$$ $$F$$ $$ratio$$: $$F$$ $$=$$ \\\\dfrac{\\\\left[\\\\dfrac{(s_{1})**{2}}{(\\\\sigma_{1})**{2}}\\\\right]}{\\\\left[\\\\dfrac{(s_{2})**{2}}{(\\\\sigma_{2})**{2}}\\\\right]}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6628d13.4cyclistb-h3","type":"hint","dependencies":["ac6628d13.4cyclistb-h2"],"title":"F statistic","text":"Given the variances, we can take the square root of each of them to find the standard deviation, and thereafterwards divide them by each other to get our F statistic.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac6628d13.4Fstatmath","title":"Test of Two Variables","body":"Two students are interested in whether or not there is variation in their test scores for math class. There are $$15$$ total math tests they have taken so far. The first student\u2019s grades have a standard deviation of $$38.1$$. The second student\u2019s grades have a standard deviation of $$22.5$$. The second student thinks his scores are more consistent.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"13.4 Test of Two Variances","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac6628d13.4Fstatmatha","stepAnswer":["$$2.8674$$"],"problemType":"TextBox","stepTitle":"What is the F statistic? Round to four decimal places.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2.8674$$","hints":{"DefaultPathway":[{"id":"ac6628d13.4Fstatmatha-h1","type":"hint","dependencies":[],"title":"Hypothesis Testing","text":"By the null hypothesis that the variances are the same. Let s denote the standard deviations. The F statistic is $$\\\\frac{{s_1}^2}{{s_2}^2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6628d13.4Fstatmatha-h2","type":"hint","dependencies":["ac6628d13.4Fstatmatha-h1"],"title":"Hypothesis Testing","text":"By $$\\\\frac{{38.1}^2}{{22.5}^2}$$, the F statistic is $$2.8674$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6628d13.4Fstatmatha-h3","type":"hint","dependencies":["ac6628d13.4Fstatmatha-h1"],"title":"Test of Two Variables","text":"Student 1\'s standard deviation is $$38.1$$, and student 2\'s standard deviation is $$22.5$$. Divide student 1\'s variance with student 2\'s variance.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac6628d13.4mathstudents","title":"Test of Two Variables","body":"Two students are interested in whether or not there is variation in their test scores for math class. There are $$15$$ total math tests they have taken so far. The first student\u2019s grades have a standard deviation of $$38.1$$. The second student\u2019s grades have a standard deviation of $$22.5$$. The second student thinks his scores are more consistent.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"13.4 Test of Two Variances","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac6628d13.4mathstudentsa","stepAnswer":["H_0:s_1=s_2, H_a:s_1<s_2"],"problemType":"MultipleChoice","stepTitle":"State the null and alternative hypotheses.","stepBody":"","answerType":"string","variabilization":{},"choices":["H_0:s_1=s_2, H_a:s_1<s_2","H_0:s_1<=s_2, H_a:s_1>s_2","H_0:s_1>s_2, H_a:s_1<s_2","H_0:s_1<s_2, H_a:s_1>s_2"],"hints":{"DefaultPathway":[{"id":"ac6628d13.4mathstudentsa-h1","type":"hint","dependencies":[],"title":"Hypothesis Testing","text":"Let the variance of students $$1$$ and $$2$$ equal $$\\\\sigma_1 $$and$$ $$\\\\sigma_2 $$respectively$$. $$The$$ $$students$$ $$are$$ $$concerned$$ $$with$$ $$variation$$ $$of$$ $$grade$$ $$results$$, $$therefore$$ $$a$$ $$test$$ $$of$$ $$two$$ $$variances$$ $$will$$ $$be$$ $$performed$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6628d13.4mathstudentsa-h2","type":"hint","dependencies":["ac6628d13.4mathstudentsa-h1"],"title":"Hypothesis Testing","text":"If the second student thinks that he is more consistent, then that signals that his variance in grades is smaller than the first student. Let that claim be the alternative hypothesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6628d13.4mathstudentsa-h3","type":"hint","dependencies":["ac6628d13.4mathstudentsa-h2"],"title":"Hypothesis Testing","text":"The students are interested if there is variation between their grades or not, therefore the null hypothesis is that their variations are the same approximately.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac6628d13.4nycsingers","title":"Hypothesis Testing","body":"The New York Choral Society divides male singers up into four categories from highest voices to lowest: Tenor1, Tenor2, Bass1, Bass2. In the table are heights of the men in the Tenor1 and Bass2 groups. One suspects that taller men will have lower voices, and that the variance of height may go up with the lower voices as well.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"13.4 Test of Two Variances","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac6628d13.4nycsingersa","stepAnswer":["We have no good evidence from the data that the heights of Tenor1 and Bass2 singers have different variances."],"problemType":"MultipleChoice","stepTitle":"Do we have good evidence that the variance of the heights of singers in each of these two groups (Tenor1 and Bass2) are different?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["We have good evidence from the data that the heights of Tenor1 and Bass2 singers have $$1$$ mean standard deviation from each other.","We have good evidence from the data that the heights of Tenor1 and Bass2 singers have $$1$$ mean standard deviation from each other.","We have good evidence from the data that the heights of Tenor1 and Bass2 singers have different.","We have no good evidence from the data that the heights of Tenor1 and Bass2 singers have different variances","We have no good evidence from the data that the heights of Tenor1 and Bass2 singers have different variances."],"hints":{"DefaultPathway":[{"id":"ac6628d13.4nycsingersa-h1","type":"hint","dependencies":[],"title":"Test of Two Variances","text":"Let T1 $$=$$ tenor $$1$$, and B2 $$=$$ bass $$2$$. The standard deviations of the samples are $$s_{T1}=3.3302$$ and $$s_{B2}=2.7208$$ after inputting the data into separate groups of Bass and Tenor men.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6628d13.4nycsingersa-h2","type":"hint","dependencies":["ac6628d13.4nycsingersa-h1"],"title":"Test of Two Variances","text":"Since we are interested in comparing the two sample variances, we use the F ratio. A test of two variances may be left, right, or two-tailed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6628d13.4nycsingersa-h3","type":"hint","dependencies":["ac6628d13.4nycsingersa-h2"],"title":"Test of Two Variances","text":"By default, the height variances are equal to each other, and the challenging statement that motivates the test is that the variance of heights go up with lower voices. It\'s hard to form an alternative hypothesis inequality statement off of this, so formulate the hypotheses based off of the base question if the variance of the heights of the singers of the two groups are different from each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6628d13.4nycsingersa-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["H_{0}: \\\\sigma**{2}_{\\\\text{T1}} $$=$$ \\\\sigma**{2}_{\\\\text{B2}}"],"dependencies":["ac6628d13.4nycsingersa-h3"],"title":"Test of Two Variances","text":"State the null and alternative hypotheses. Let $$\\\\sigma**{2}_{\\\\text{T1}} $$denote$$ $$Tenor$$ $$1$$ $$singers$$ $$and$$ $$\\\\sigma**{2}_{\\\\text{B2}} $$denote$$ $$Bass$$ $$2$$ $$players$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["H_{0}: \\\\sigma**{2}_{\\\\text{T1}} $$=$$ \\\\sigma**{2}_{\\\\text{B2}}, H_{a}: \\\\sigma**{2}_{\\\\text{T1}} \\\\neq \\\\sigma**{2}_{\\\\text{B2}}","H_{0}: \\\\sigma**{2}_{\\\\text{T1}} $$ \\\\leq $$ \\\\sigma**{2}_{\\\\text{B2}}, H_{a}: \\\\sigma**{2}_{\\\\text{T1}} \\\\neq \\\\sigma**{2}_{\\\\text{B2}}","H_{0}: \\\\sigma**{2}_{\\\\text{T1}} $$ \\\\geq $$ \\\\sigma**{2}_{\\\\text{B2}}, H_{a}: \\\\sigma**{2}_{\\\\text{T1}} \\\\neq \\\\sigma**{2}_{\\\\text{B2}}","H_{0}: \\\\sigma**{2}_{\\\\text{T1}} $$ \\\\geq $$ \\\\sigma**{2}_{\\\\text{B2}}, H_{a}: \\\\sigma**{2}_{\\\\text{T1}} < \\\\sigma**{2}_{\\\\text{B2}}"]},{"id":"ac6628d13.4nycsingersa-h5","type":"hint","dependencies":["ac6628d13.4nycsingersa-h4"],"title":"Test of Two Variances","text":"If the key motivation behind the test is finding a difference in variances, then the alternative hypothesis is that the variances of the two groups concerned aren\'t equal to each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6628d13.4nycsingersa-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.4894$$"],"dependencies":["ac6628d13.4nycsingersa-h5"],"title":"F statistic","text":"What is the F statistic? Round to four decimal places/","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6628d13.4nycsingersa-h7","type":"hint","dependencies":["ac6628d13.4nycsingersa-h6"],"title":"F statistic","text":"Suppose we sample randomly from two independent normal populations. Let $$\\\\sigma**{2}_{1} $$and$$ $$\\\\sigma**{2}_{2} $$be$$ $$the$$ $$population$$ $$variances$$ $$and$$ $$s**{2}_{1} $$and$$ $$s**{2}_{1} $$be$$ $$the$$ $$sample$$ $$variances$$. $$Let$$ $$the$$ $$sample$$ $$sizes$$ $$be$$ $$n_1$$ $$and$$. $$_2$$. $$Since$$ $$we$$ $$are$$ $$interested$$ $$in$$ $$comparing$$ $$the$$ $$two$$ $$sample$$ $$variances$$, $$we$$ $$use$$ $$the$$ $$F$$ $$ratio$$: $$F$$ $$=$$ \\\\dfrac{\\\\left[\\\\dfrac{(s_{1})**{2}}{(\\\\sigma_{1})**{2}}\\\\right]}{\\\\left[\\\\dfrac{(s_{2})**{2}}{(\\\\sigma_{2})**{2}}\\\\right]}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6628d13.4nycsingersa-h8","type":"hint","dependencies":["ac6628d13.4nycsingersa-h7"],"title":"F statistic","text":"Calculate the F statistic by squaring each of the standard deviations and dividing the tenor group by the bass group squared standard deviations. Remember to round to four decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6628d13.4nycsingersa-h9","type":"hint","dependencies":["ac6628d13.4nycsingersa-h8"],"title":"F statistic","text":"The F statistic is $$1.4894$$ with $$20$$ and $$25$$ degrees of freedom for Tenor $$1$$ and Bass $$2$$ respectively.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac6628dFassumption1","title":"Test of Two Variances","body":"There are two assumptions that must be true in order to perform an F test of two variances.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"13.4 Test of Two Variances","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac6628dFassumption1a","stepAnswer":["The populations from which the two samples are drawn are normally distributed."],"problemType":"MultipleChoice","stepTitle":"Name one assumption that must be true.","stepBody":"","answerType":"string","variabilization":{},"choices":["The populations from which the two samples are drawn are normally distributed.","At least one of the populations from which the two samples are drawn have an $$n$$ population greater than or equal to $$29$$.","One of the group statistics aren\'t equal to each other.","The two populations being tested are dependent upon one another."],"hints":{"DefaultPathway":[{"id":"ac6628dFassumption1a-h1","type":"hint","dependencies":[],"title":"Test of Two Variances","text":"Another of the uses of the F distribution is testing two variances. It is often desirable to compare two variances rather than two averages. For instance, college administrators would like two college professors grading exams to have the same variation in their grading. In order for a lid to fit a container, the variation in the lid and the container should be the same. A supermarket might be interested in the variability of check-out times for two checkers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6628dFassumption1a-h2","type":"hint","dependencies":["ac6628dFassumption1a-h1"],"title":"Test of Two Variances","text":"In order to perform a F test of two variances, since the test in equality is very sensitive to deviation compared to other tests used, the populations must be consistent.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6628dFassumption1a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["The populations from which the two samples are drawn are normally distributed."],"dependencies":["ac6628dFassumption1a-h2"],"title":"Test of Two Variances","text":"Since a test in two variances is very sensitive to deviations from normality, what would be a good assumption to make?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["The populations from which the two samples are drawn are normally distributed.","At least one of the populations from which the two samples are drawn have an $$n$$ population greater than or equal to $$29$$.","One of the group statistics aren\'t equal to each other.","The two populations being tested are dependent upon one another."]},{"id":"ac6628dFassumption1a-h4","type":"hint","dependencies":["ac6628dFassumption1a-h2"],"title":"Test of Two Variances","text":"If the test is sensitive to deviations from normality, then the two populations being tested from which the two samples will be drawn should be normally distributed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac6628dFassumption2","title":"Test of Two Variances","body":"There are two assumptions that must be true in order to perform an F test of two variances.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"13.4 Test of Two Variances","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac6628dFassumption2a","stepAnswer":["The two populations are independent of each other."],"problemType":"MultipleChoice","stepTitle":"Name one assumption that must be true.","stepBody":"","answerType":"string","variabilization":{},"choices":["The two populations are independent of each other.","At least one of the populations from which the two samples are drawn have an $$n$$ population greater than or equal to $$29$$.","One of the group statistics aren\'t equal to each other.","The two populations are independent of each other."],"hints":{"DefaultPathway":[{"id":"ac6628dFassumption2a-h1","type":"hint","dependencies":[],"title":"Test of Two Variances","text":"Another of the uses of the F distribution is testing two variances. It is often desirable to compare two variances rather than two averages. For instance, college administrators would like two college professors grading exams to have the same variation in their grading. In order for a lid to fit a container, the variation in the lid and the container should be the same. A supermarket might be interested in the variability of check-out times for two checkers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6628dFassumption2a-h2","type":"hint","dependencies":["ac6628dFassumption2a-h1"],"title":"Test of Two Variances","text":"When performing a test of two variances (or frankly most tests), we don\'t want our results to be affected by any other confounding variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6628dFassumption2a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["The two populations are independent of each other."],"dependencies":["ac6628dFassumption2a-h2"],"title":"Test of Two Variances","text":"Since we don\'t want to get our results messed up by external factors or data relation to each other, what would be a good assumption to ensure an efficient test of two variances?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["The two populations are independent of each other.","At least one of the populations from which the two samples are drawn have an $$n$$ population greater than or equal to $$29$$.","One of the group statistics aren\'t equal to each other.","The two populations are independent of each other."]},{"id":"ac6628dFassumption2a-h4","type":"hint","dependencies":["ac6628dFassumption2a-h2"],"title":"Test of Two Variances","text":"Given the options listed, the best way to reduce confounding factors is to ensure that the two populations are independent of each other so that misleading correlations are avoided.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac6628dFassumption3","title":"Test of Two Variances","body":"Two coworkers commute from the same building. They are interested in whether or not there is any variation in the time it takes them to drive to work. They each record their times for $$20$$ commutes. The first worker\u2019s times have a variance of $$12.1$$. The second worker\u2019s times have a variance of $$16.9$$. The first worker thinks that he is more consistent with his commute times. Test the claim at the 10% level. Assume that commute times are normally distributed.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"13.4 Test of Two Variances","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac6628dFassumption3a","stepAnswer":["H_0:$$\\\\sigma_1=\\\\sigma_2, $$H_a$$: \\\\sigma_1<\\\\sigma_2"],"problemType":"MultipleChoice","stepTitle":"State the null and alternative hypotheses.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"H_0:$$\\\\sigma_1=\\\\sigma_2, $$H_a$$: \\\\sigma_1<\\\\sigma_2","choices":["H_0:$$\\\\sigma_1=\\\\sigma_2, $$H_a$$: \\\\sigma_1<\\\\sigma_2","H_0:$$\\\\sigma_1>=\\\\sigma_2, $$H_a$$: \\\\sigma_1>\\\\sigma_2","H_0:$$\\\\sigma_1>\\\\sigma_2, $$H_a$$: \\\\sigma_1>=\\\\sigma_2","H_0:$$\\\\mu_1>=\\\\mu_2,H_a:\\\\mu_1<\\\\mu_2"],"hints":{"DefaultPathway":[{"id":"ac6628dFassumption3a-h1","type":"hint","dependencies":[],"title":"Test of Two Variances","text":"Let the variance of workers $$1$$ and $$2$$ equal $$\\\\sigma_1 $$and$$ $$\\\\sigma_2 $$respectively$$. $$The$$ $$workers$$ $$are$$ $$concerned$$ $$with$$ $$variation$$ $$of$$ $$commute$$ $$times$$, $$therefore$$ $$a$$ $$test$$ $$of$$ $$two$$ $$variances$$ $$will$$ $$be$$ $$performed$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6628dFassumption3a-h2","type":"hint","dependencies":["ac6628dFassumption3a-h1"],"title":"Test of Two Variances","text":"If the first worker thinks that he is more consistent, then that signals that his variance in time to commute to work is smaller than the second worker. Let that claim be the alternative hypothesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6628dFassumption3a-h3","type":"hint","dependencies":["ac6628dFassumption3a-h2"],"title":"Test of Two Variances","text":"By default, since the workers commute at the same place in the same distances, the variances will be equal to each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6628dFassumption3a-h4","type":"hint","dependencies":["ac6628dFassumption3a-h3"],"title":"Test of Two Variances","text":"Since we are interested in comparing the two sample variances, we use the F ratio. A test of two variances may be left, right, or two-tailed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac6628dFassumption4","title":"Test of Two Variances","body":"Two coworkers commute from the same building. They are interested in whether or not there is any variation in the time it takes them to drive to work. They each record their times for $$20$$ commutes. The first worker\u2019s times have a variance of $$12.1$$. The second worker\u2019s times have a variance of $$16.9$$. The first worker thinks that he is more consistent with his commute times. Test the claim at the 10% level. Assume that commute times are normally distributed.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"13.4 Test of Two Variances","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac6628dFassumption4a","stepAnswer":["$$3.48$$"],"problemType":"TextBox","stepTitle":"Test of Two Variances","stepBody":"Let s denote sample standard deviation. What is $$s_1$$ in this problem? Round to two decimal places.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3.48$$","hints":{"DefaultPathway":[{"id":"ac6628dFassumption4a-h1","type":"hint","dependencies":[],"title":"Test of Two Variances","text":"We are given the sample variances. The variances are the standard deviations squared.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6628dFassumption4a-h2","type":"hint","dependencies":["ac6628dFassumption4a-h1"],"title":"Test of Two Variances","text":"With the variance of worker $$1$$ being $$12.1$$, the sample standard deviation would be the square root of $$12.1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ac6628dFassumption4b","stepAnswer":["$$4.11$$"],"problemType":"TextBox","stepTitle":"Test of Two Variances","stepBody":"Let s denote sample standard deviation. What is $$s_1$$ in this problem? Round to two decimal places.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4.11$$","hints":{"DefaultPathway":[{"id":"ac6628dFassumption4b-h1","type":"hint","dependencies":[],"title":"Test of Two Variances","text":"We are given the sample variances. The variances are the standard deviations squared.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6628dFassumption4b-h2","type":"hint","dependencies":["ac6628dFassumption4b-h1"],"title":"Test of Two Variances","text":"With the variance of worker $$2$$ being $$16.9$$, the sample standard deviation would be the square root of $$16.9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac6628dFassumption5","title":"Test of Two Variances","body":"Two coworkers commute from the same building. They are interested in whether or not there is any variation in the time it takes them to drive to work. They each record their times for $$20$$ commutes. The first worker\u2019s times have a variance of $$12.1$$. The second worker\u2019s times have a variance of $$16.9$$. The first worker thinks that he is more consistent with his commute times. Test the claim at the 10% level. Assume that commute times are normally distributed.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"13.4 Test of Two Variances","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac6628dFassumption5a","stepAnswer":["$$20$$"],"problemType":"TextBox","stepTitle":"Enter the value of $$n$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$20$$","hints":{"DefaultPathway":[{"id":"ac6628dFassumption5a-h1","type":"hint","dependencies":[],"title":"Hypothesis Testing","text":"$$n$$ is the sample size.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6628dFassumption5a-h2","type":"hint","dependencies":["ac6628dFassumption5a-h1"],"title":"Test of Two Variances","text":"In this test of two variances, $$n$$ is expressed as the populations of each of the samples. The workers happened to have the same $$n$$ commutes in our case.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ac6628dFassumption5b","stepAnswer":["$$0.7159$$"],"problemType":"TextBox","stepTitle":"What is the F statistic? Round to four decimal places.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.7159$$","hints":{"DefaultPathway":[{"id":"ac6628dFassumption5b-h1","type":"hint","dependencies":[],"title":"Hypothesis Testing","text":"By the null hypothesis that the variances are the same, the F statistic is $$\\\\frac{{s_1}^2}{{s_2}^2}$$, where s would denote the sample variance.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac6628dFassumption5b-h2","type":"hint","dependencies":["ac6628dFassumption5b-h1"],"title":"Test of Two Variances","text":"Worker 1\'s variance is $$12.1$$, and worker 2\'s variance is $$16.9$$. divide worker 1\'s variance with worker 2\'s variance.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac726d3location1","title":"Calculate Interquartile Range","body":"Use the following $$13$$ real estate prices (in dollars) for this question. 389,950; 230,500; 158,000; 479,000; 639,000; 114,950; 5,500,000; 387,000; 659,000; 529,000; 575,000; 488,800; 1,095,000","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.3 Measures of the Location of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac726d3location1a","stepAnswer":["$$340250$$"],"problemType":"TextBox","stepTitle":"Calculate the IQR (InterQuartile Range).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$340250$$","hints":{"DefaultPathway":[{"id":"ac726d3location1a-h1","type":"hint","dependencies":[],"title":"Order the Data","text":"First order the data from smallest to largest. Here is the ordering: 114,950; 158,000; 230,500; 387,000; 389,950; 479,000; 488,800; 529,000; 575,000; 639,000; 659,000; 1,095,000; 5,500,000.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location1a-h2","type":"hint","dependencies":["ac726d3location1a-h1"],"title":"Find the Median","text":"Remember that the IQR is the formula Quartile $$3$$ - Quartile $$1$$. Therefore, to determine the quartiles, we\'ll first determine the median.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$488800$$"],"dependencies":["ac726d3location1a-h2"],"title":"Determine the Median","text":"What is the median of the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location1a-h4","type":"hint","dependencies":["ac726d3location1a-h3"],"title":"Find Quartiles","text":"The definition for Quartile $$1$$ is the middle value of the lower half of the data. The definition for Quartile $$3$$ is the middle value for the upper half of the data. Remember that the halfpoint of the data is determined by the median, so we now need to find the middle points of the remaining data.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location1a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$308750$$"],"dependencies":["ac726d3location1a-h4"],"title":"Find Quartile $$1$$","text":"What is Quartile 1? What is the middle value of the lower half of the data?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location1a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$649000$$"],"dependencies":["ac726d3location1a-h5"],"title":"Find Quartile $$3$$","text":"What is Quartile 3? What is the middle value of the upper half of the data?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location1a-h7","type":"hint","dependencies":["ac726d3location1a-h6"],"title":"Definition of IQR","text":"The definition of IQR (interquartile range) is the number that indicates the spread of the middle half or the middle 50% of the data. It is the difference between the third quartile and the first quartile.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location1a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$340250$$"],"dependencies":["ac726d3location1a-h7"],"title":"Determine the IQR","text":"What is the IQR?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ac726d3location1a-h8-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$340250$$"],"dependencies":[],"title":"Determine the IQR","text":"What is $$649000-308750$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}},{"id":"ac726d3location1b","stepAnswer":["$$5500000$$"],"problemType":"TextBox","stepTitle":"Determine the potential outlier.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5500000$$","hints":{"DefaultPathway":[{"id":"ac726d3location1b-h9","type":"hint","dependencies":["ac726d3location1a-h8"],"title":"Using IQR to Determine Outliers","text":"The IQR can help to determine potential outliers. A value is a potential outlier if it is less than $$(1.5)(IQR)$$ below the first quartile or more than $$(1.5)(IQR)$$ above the third quartile.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location1b-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$510375$$"],"dependencies":["ac726d3location1b-h9"],"title":"Determine $$(1.5)(IQR)$$","text":"What is $$(1.5)(IQR)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ac726d3location1b-h10-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$510375$$"],"dependencies":[],"title":"Determine $$(1.5)(IQR)$$","text":"What is $$(1.5)(340250)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ac726d3location1b-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-201625$$"],"dependencies":["ac726d3location1b-h10"],"title":"Determine Quartile $$1-(1.5)(IQR)$$","text":"What is Quartile $$1-(1.5)(IQR)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ac726d3location1b-h11-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-201625$$"],"dependencies":[],"title":"Determine Quartile $$1-(1.5)(IQR)$$","text":"What is $$308750-510375$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ac726d3location1b-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1159375$$"],"dependencies":["ac726d3location1b-h11"],"title":"Determine Quartile $$3+1.5IQR$$","text":"What is Quartile $$3+1.5IQR$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ac726d3location1b-h12-s4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1159375$$"],"dependencies":[],"title":"Determine Quartile $$3+1.5IQR$$","text":"What is $$649000+510375$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ac726d3location1b-h13","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ac726d3location1b-h12"],"title":"Determining Outliers","text":"Are there any values in the data set that are less than $$(1.5)(IQR)$$ below the first quartile?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ac726d3location1b-h14","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ac726d3location1b-h13"],"title":"Determining Outliers","text":"Are there any values in the data set that are greater than $$(1.5)(IQR)$$ above the third quartile?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ac726d3location1b-h15","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$5500000$$"],"dependencies":["ac726d3location1b-h14"],"title":"Determine the Potential Outlier","text":"What is the potential outlier that is greater than $$(1.5)(IQR)$$ above Quartile 3?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$5500000$$","$$1095000$$","$$659000$$","$$639000$$"]}]}}]},{"id":"ac726d3location10","title":"Finding Percentiles","body":"Listed are $$29$$ ages for Academy Award winning best actors in order from smallest to largest. 18; 21; 22; 25; 26; 27; 29; 30; 31; 33; 36; 37; 41; 42; 47; 52; 55; 57; 58; 62; 64; 67; 69; 71; 72; 73; 74; 76; $$77$$","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.3 Measures of the Location of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac726d3location10a","stepAnswer":["$$12$$"],"problemType":"TextBox","stepTitle":"Find the percentile for $$25$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12$$","hints":{"DefaultPathway":[{"id":"ac726d3location10a-h1","type":"hint","dependencies":[],"title":"Formula for Finding the Percentile of a Value in a Data Set","text":"Remember the formula for finding the Percentile of a Value in a Data Set. We want to know three values once the list is ordered (which is already is from least to greatest): $$x=the$$ number of data values counting from the bottom of the data list up to but not including the data value for which you want to find the percentile; $$y=the$$ number of data values equal to the data value for which you want to find the percentile; $$n=the$$ total number of data. Then, we can use the formula to calculate the percentile as $$100\\\\frac{x+0.5y}{n}$$. If the percentile is not an integer, round to the nearest integer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ac726d3location10a-h1"],"title":"Determining $$x$$ (Number of Points Below Desired Value)","text":"What is $$x$$? In other words, how many data values are there counting from the bottom of the data list up to but not including the data value for which you want to find the percentile?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ac726d3location10a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":[],"title":"Determining $$x$$ (Number of Points Below Desired Value)","text":"How many values in the data set are less than 58?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ac726d3location10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ac726d3location10a-h2"],"title":"Determining $$y$$ (Repetition of Desired Value)","text":"How many times does $$58$$ show up in the data?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$29$$"],"dependencies":["ac726d3location10a-h3"],"title":"Determining $$n$$ (Total Number of Data)","text":"What is $$n$$, the total number of data in the set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location10a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$12.07$$"],"dependencies":["ac726d3location10a-h4"],"title":"Determining the Percentile","text":"Calculate the percentile using the Formula for Finding the Percentile of a Value in a Data Set. Round your answer to the nearest hundredth.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$3.5$$","$$0.1207$$","$$12.07$$","$$12$$"],"subHints":[{"id":"ac726d3location10a-h5-s2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$12.07$$"],"dependencies":[],"title":"Determining the Percentile","text":"What is $$100\\\\frac{3+0.5\\\\times1}{29}$$ rounded to the nearest hundredth?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$3.5$$","$$0.1207$$","$$12.07$$","$$12$$"]}]},{"id":"ac726d3location10a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ac726d3location10a-h5"],"title":"Rounding the Percentile","text":"Is $$12.07$$ a whole number?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ac726d3location10a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["ac726d3location10a-h6"],"title":"Rounding the Percentile","text":"What is $$12.07$$ rounded to the nearest whole number? This is the final percentile.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac726d3location11","title":"Interpreting Quartiles","body":"On a timed math test, the first quartile for time it took to finish the exam was $$35$$ minutes.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.3 Measures of the Location of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac726d3location11a","stepAnswer":["$$Twenty-five$$ percent of students finished the exam in $$35$$ minutes or less."],"problemType":"MultipleChoice","stepTitle":"Interpret the first quartile in the context of this situation.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Twenty-five percent of students finished the exam in $$35$$ minutes or less.","choices":["$$Twenty-five$$ percent of students finished the exam in $$35$$ minutes or less.","$$Twenty-five$$ percent of students finished the exam in $$35$$ minutes or more.","$$Seventy-five$$ percent of students finished the exam in $$35$$ minutes or less.","Fifty percent of students finished the exam in $$35$$ minutes or less."],"hints":{"DefaultPathway":[{"id":"ac726d3location11a-h1","type":"hint","dependencies":[],"title":"Understanding the First Quartile","text":"The first quartile references the 25th percentile. We note that the percentage of data values that are less than or equal to the 25th percentile is 25%.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location11a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$35$$ minutes or less"],"dependencies":["ac726d3location11a-h1"],"title":"Determining Directionality","text":"Based on this connection between the first quartile and the 25th percentile, since the first quartile for time it took to finish the exam was $$35$$ minutes, does that equate to 25% of people finishing the exam in $$35$$ minutes or less OR $$35$$ minutes more?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$35$$ minutes or less","$$35$$ minutes or more"]}]}},{"id":"ac726d3location11b","stepAnswer":["Good"],"problemType":"MultipleChoice","stepTitle":"Is scoring a low percentile (such as the 25th $$\\\\frac{percentile}{first}$$ quartile) a \\"good\\" or a \\"bad\\" thing here?","stepBody":"","answerType":"string","variabilization":{},"choices":["Good","Bad"],"hints":{"DefaultPathway":[{"id":"ac726d3location11b-h3","type":"hint","dependencies":["ac726d3location11a-h2"],"title":"Good or Bad in Context","text":"A low percentile means that it took less time to complete the test while a high percentile means it took more time to complete the test.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location11b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Lower Percentile"],"dependencies":["ac726d3location11b-h3"],"title":"Good or Bad in Context","text":"In context, since this is a timed math test, is it better to have a lower percentile (finishing quickly) or have a higher percentile (finishing slower, risking not being able to finish)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Lower Percentile","Higher Percentile"]}]}}]},{"id":"ac726d3location12","title":"Interpreting Quartiles","body":"For the 100-meter dash, the third quartile for times for finishing the race was $$11.5$$ seconds.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.3 Measures of the Location of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac726d3location12a","stepAnswer":["$$Seventy-five$$ percent of students finished the race in $$11.5$$ seconds or less."],"problemType":"MultipleChoice","stepTitle":"Interpret the third quartile in the context of this situation.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Seventy-five percent of students finished the race in $$11.5$$ seconds or less.","choices":["$$Seventy-five$$ percent of students finished the race in $$11.5$$ seconds or less.","$$Twenty-five$$ percent of students finished the race in $$11.5$$ seconds or less.","Fifty percent of students finished the race in $$11.5$$ seconds or less.","$$Seventy-five$$ percent of students finished the race in $$11.5$$ seconds exactly."],"hints":{"DefaultPathway":[{"id":"ac726d3location12a-h1","type":"hint","dependencies":[],"title":"Understanding the Third Quartile","text":"The third quartile references the 75th percentile. We note that the percentage of data values that are less than or equal to the 75th percentile is 75%.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location12a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$11.5$$ seconds or less"],"dependencies":["ac726d3location12a-h1"],"title":"Determining Directionality","text":"Based on this connection between the third quartile and the 75th percentile, since the third quartile for time it took to finish the race was $$11.5$$ seconds, does that equate to 75% of people finishing the race in $$11.5$$ seconds or less OR $$11.5$$ seconds or more?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$11.5$$ seconds or less","$$11.5$$ seconds or more"]}]}},{"id":"ac726d3location12b","stepAnswer":["Bad"],"problemType":"MultipleChoice","stepTitle":"Is scoring a high percentile (such as the 75th $$\\\\frac{percentile}{third}$$ quartile) a \\"good\\" or a \\"bad\\" thing here?","stepBody":"","answerType":"string","variabilization":{},"choices":["Good","Bad"],"hints":{"DefaultPathway":[{"id":"ac726d3location12b-h3","type":"hint","dependencies":["ac726d3location12a-h2"],"title":"Good or Bad in Context","text":"A low percentile means that it took less time to complete the race while a high percentile means it took more time to complete the race.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location12b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Lower Percentile"],"dependencies":["ac726d3location12b-h3"],"title":"Good or Bad in Context","text":"In context, since this is a 100-meter dash, is it better to have a lower percentile (finishing quickly and having a lower time) or have a higher percentile (finishing slower, taking more time, falling behind competitors)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Lower Percentile","Higher Percentile"]},{"id":"ac726d3location12b-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Bad"],"dependencies":["ac726d3location12b-h4"],"title":"Determining if 75th Percentile is Good or Bad","text":"Since it is better to have a lower percentile (running faster), is having a high percentile such as $$75$$ (greater than 50) relatively a good thing or a bad thing?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Good","Bad"]}]}}]},{"id":"ac726d3location13","title":"Interpreting Percentiles in Context","body":"On a $$20$$ question math test, the 70th percentile for number of correct answers was $$16$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":0,"lesson":"2.3 Measures of the Location of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac726d3location13a","stepAnswer":["Seventy percent of students finished the test with $$16$$ questions correct or less."],"problemType":"MultipleChoice","stepTitle":"Interpret the 70th percentile in the context of this situation.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Seventy percent of students finished the test with $$16$$ questions correct or less.","choices":["Seventy percent of students finished the test with $$16$$ questions correct or less.","Seventy percent of students finished the test with $$16$$ questions correct or more.","Seventy percent of students finished the test with $$16$$ questions correct exactly.","Thirty percent of students only finished the test with $$16$$ questions correct or less."],"hints":{"DefaultPathway":[{"id":"ac726d3location13a-h1","type":"hint","dependencies":[],"title":"Understanding Percentiles","text":"The percentage of data values that are less than or equal to the 70th percentile is 70%.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location13a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$16$$ correct answers or less"],"dependencies":["ac726d3location13a-h1"],"title":"Determining Directionality","text":"Since the 70th percentile for the number of correct answers was $$16$$, does that equate to 70% of people finishing the test with $$16$$ correct answers or less OR $$16$$ correct answers or more?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$16$$ correct answers or less","$$16$$ correct answers or more"]}]}},{"id":"ac726d3location13b","stepAnswer":["Good"],"problemType":"MultipleChoice","stepTitle":"Is scoring a high percentile (such as the 75th $$\\\\frac{percentile}{third}$$ quartile) a \\"good\\" or a \\"bad\\" thing here?","stepBody":"","answerType":"string","variabilization":{},"choices":["Good","Bad"],"hints":{"DefaultPathway":[{"id":"ac726d3location13b-h3","type":"hint","dependencies":["ac726d3location13a-h2"],"title":"Good or Bad in Context","text":"A low percentile means that that a student got less answers correct while a high percentile means that a student got more answers correct.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location13b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Higher Percentile"],"dependencies":["ac726d3location13b-h3"],"title":"Good or Bad in Context","text":"In context, since this assesses accuracy, is it better to have a lower percentile (having a smaller number of correct answers) or have a higher percentile (having a higher number of correct answers)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Lower Percentile","Higher Percentile"]},{"id":"ac726d3location13b-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Good"],"dependencies":["ac726d3location13b-h4"],"title":"Determining if 70th Percentile is Good or Bad","text":"Since it is better to have a higher percentile (scoring more answers correct), is having a high percentile such as $$70$$ (greater than 50) relatively a good thing or a bad thing?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Good","Bad"]}]}}]},{"id":"ac726d3location14","title":"Interpreting Percentiles in Context","body":"On a $$60$$ point written assignment, the 80th percentile for the number of points earned was $$49$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":0,"lesson":"2.3 Measures of the Location of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac726d3location14a","stepAnswer":["Eighty percent of students finished the test with $$49$$ points earned or less."],"problemType":"MultipleChoice","stepTitle":"Interpret the 80th percentile in context of this situation.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Eighty percent of students finished the test with $$49$$ points earned or less.","choices":["Eighty percent of students finished the test with $$49$$ points earned or less.","Eighty percent of students finished the test with $$49$$ points earned or more.","Eighty percent of students finished the test with $$49$$ points earned exactly.","Twenty percent of students only finished the test with $$49$$ points earned or less."],"hints":{"DefaultPathway":[{"id":"ac726d3location14a-h1","type":"hint","dependencies":[],"title":"Understanding Percentiles","text":"The percentage of data values that are less than or equal to the 80th percentile is 80%.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location14a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$49$$ points or less"],"dependencies":["ac726d3location14a-h1"],"title":"Determining Directionality","text":"Since the 80th percentile for the number of points earned was $$49$$, does that equate to 80% of people finishing the written assignment with $$49$$ points or less OR $$49$$ points or more?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$49$$ points or less","$$49$$ points or more"]}]}},{"id":"ac726d3location14b","stepAnswer":["Good"],"problemType":"MultipleChoice","stepTitle":"Is scoring a high percentile (such as the 80th $$\\\\frac{percentile}{third}$$ quartile) a \\"good\\" or a \\"bad\\" thing here?","stepBody":"","answerType":"string","variabilization":{},"choices":["Good","Bad"],"hints":{"DefaultPathway":[{"id":"ac726d3location14b-h3","type":"hint","dependencies":["ac726d3location14a-h2"],"title":"Good or Bad in Context","text":"A low percentile means that that a student got less points while a high percentile means that a student got more points.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location14b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Higher Percentile"],"dependencies":["ac726d3location14b-h3"],"title":"Good or Bad in Context","text":"In context, since this assesses accuracy, is it better to have a lower percentile (getting less points on the assignment) or have a higher percentile (getting more points on the assignment)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Lower Percentile","Higher Percentile"]},{"id":"ac726d3location14b-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Good"],"dependencies":["ac726d3location14b-h4"],"title":"Determining if 80th Percentile is Good or Bad","text":"Since it is better to have a higher percentile (scoring more answers correct), is having a high percentile such as $$80$$ (greater than 50) relatively a good thing or a bad thing?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Good","Bad"]}]}}]},{"id":"ac726d3location15","title":"Interpreting Percentiles in Context","body":"During a season, the 40th percentile for points scored per player in a game is eight.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":0,"lesson":"2.3 Measures of the Location of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac726d3location15a","stepAnswer":["Forty percent of scored eight points per game or less."],"problemType":"MultipleChoice","stepTitle":"Interpret the 40th percentile in context of this situation.","stepBody":"","answerType":"string","variabilization":{},"choices":["Forty percent of scored eight points per game or less.","Forty percent of scored eight points per game or more.","Forty percent of scored eight points per game exactly.","Sixty percent of scored eight points per game or less."],"hints":{"DefaultPathway":[{"id":"ac726d3location15a-h1","type":"hint","dependencies":[],"title":"Understanding Percentiles","text":"The percentage of data values that are less than or equal to the 40th percentile is 40%.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location15a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Eight points per game or less"],"dependencies":["ac726d3location15a-h1"],"title":"Determining Directionality","text":"Since the 40th percentile for the number of points scored per game for players was $$8$$, does that equate to 40% of people finishing with eight points per game or less OR eight points per game or more?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Eight points per game or less","Eight points per game or more"]}]}},{"id":"ac726d3location15b","stepAnswer":["Bad"],"problemType":"MultipleChoice","stepTitle":"Is scoring a low percentile (such as the 40th $$\\\\frac{percentile}{third}$$ quartile) a \\"good\\" or a \\"bad\\" thing here? Assume this is like basketball, a game where high scores are better than low scores.","stepBody":"","answerType":"string","variabilization":{},"choices":["Good","Bad"],"hints":{"DefaultPathway":[{"id":"ac726d3location15b-h3","type":"hint","dependencies":["ac726d3location15a-h2"],"title":"Good or Bad in Context","text":"A low percentile means that that a player scored less points per game while a high percentile means that a player scored more points per game.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location15b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Higher Percentile"],"dependencies":["ac726d3location15b-h3"],"title":"Good or Bad in Context","text":"In context, since this assesses accuracy, is it better to have a lower percentile (scoring less points per game) or have a higher percentile (scoring more points per game)? Assume this is like basketball, a game where high scores are better than low scores.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Lower Percentile","Higher Percentile"]},{"id":"ac726d3location15b-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Bad"],"dependencies":["ac726d3location15b-h4"],"title":"Determining if 40th Percentile is Good or Bad","text":"Since it is better to have a higher percentile (scoring more answers correct), is having a low percentile such as $$40$$ (less than 50) relatively a good thing or a bad thing?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Good","Bad"]}]}}]},{"id":"ac726d3location16","title":"Finding Percentiles","body":"Listed are $$29$$ ages for Academy Award winning best actors in order from smallest to largest: 18; 21; 22; 25; 26; 27; 29; 30; 31; 33; 36; 37; 41; 42; 47; 52; 55; 57; 58; 62; 64; 67; 69; 71; 72; 73; 74; 76; $$77$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":0,"lesson":"2.3 Measures of the Location of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac726d3location16a","stepAnswer":["$$37$$"],"problemType":"TextBox","stepTitle":"Find the 40th Percentile.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$37$$","hints":{"DefaultPathway":[{"id":"ac726d3location16a-h1","type":"hint","dependencies":[],"title":"Formula for Finding kth Percentile","text":"Remember the formula for finding the kth Percentile. We want to know three values: $$k=the$$ kth percentile, usually given to you; $$i=the$$ index (ranking or position of the data value), what we normally want to find; $$n=the$$ total number of data. Then, we can use the formula to calculate $$i=\\\\frac{k}{100} \\\\left(n+1\\\\right)$$. If i is an integer, we use the data at the ith position in the ordered data. If i is not an integer, we then round i down and up to the nearest integers and take the average of these two.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$40$$"],"dependencies":["ac726d3location16a-h1"],"title":"Determining k (Percentile)","text":"What is the kth percentile we want to determine?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location16a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$29$$"],"dependencies":["ac726d3location16a-h2"],"title":"Determining $$n$$ (Total Number of Data)","text":"What is $$n$$, the total number of data in the set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location16a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["ac726d3location16a-h3"],"title":"Determining i (Desired Index)","text":"Using the Formula for Finding the kth Percentile, what is i?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ac726d3location16a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":[],"title":"Determining i (Desired Index)","text":"What is $$\\\\frac{40}{100} \\\\left(29+1\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ac726d3location16a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ac726d3location16a-h4"],"title":"Determining the position of the data","text":"Is i an integer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"],"subHints":[{"id":"ac726d3location16a-h5-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$37$$"],"dependencies":[],"title":"Determining the 83rd Percentile","text":"What is the ith data value in the data set? What is the 12th data point?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}},{"id":"ac726d3location16b","stepAnswer":["$$70$$"],"problemType":"TextBox","stepTitle":"Find the 78th Percentile.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$70$$","hints":{"DefaultPathway":[{"id":"ac726d3location16b-h6","type":"hint","dependencies":["ac726d3location16a-h5"],"title":"Formula for Finding kth Percentile","text":"Remember the formula for finding the kth Percentile. We want to know three values: $$k=the$$ kth percentile, usually given to you; $$i=the$$ index (ranking or position of the data value), what we normally want to find; $$n=the$$ total number of data. Then, we can use the formula to calculate $$i=\\\\frac{k}{100} \\\\left(n+1\\\\right)$$. If i is an integer, we use the data at the ith position in the ordered data. If i is not an integer, we then round i down and up to the nearest integers and take the average of these two.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location16b-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$78$$"],"dependencies":["ac726d3location16b-h6"],"title":"Determining k (Percentile)","text":"What is the kth percentile we want to determine?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location16b-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$29$$"],"dependencies":["ac726d3location16b-h7"],"title":"Determining $$n$$ (Total Number of Data)","text":"What is $$n$$, the total number of data in the set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location16b-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$23.4$$"],"dependencies":["ac726d3location16b-h8"],"title":"Determining i (Desired Index)","text":"Using the Formula for Finding the kth Percentile, what is i?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ac726d3location16b-h9-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$23.4$$"],"dependencies":[],"title":"Determining i (Desired Index)","text":"What is $$\\\\frac{78}{100} \\\\left(29+1\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ac726d3location16b-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ac726d3location16b-h9"],"title":"Determining the position of the data","text":"Is i an integer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ac726d3location16b-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$23$$"],"dependencies":["ac726d3location16b-h10"],"title":"Determining the Lower Index","text":"What is i rounded down to a whole number?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location16b-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$24$$"],"dependencies":["ac726d3location16b-h11"],"title":"Determining the Upper Index","text":"What is i rounded up to a whole number?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location16b-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$69$$"],"dependencies":["ac726d3location16b-h12"],"title":"Determining the Lower Index Data Point","text":"What is the data value in the 23rd position of the ordered data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location16b-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$71$$"],"dependencies":["ac726d3location16b-h13"],"title":"Determining the Upper Index Data Point","text":"What is the data value in the 24th position of the ordered data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location16b-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$70$$"],"dependencies":["ac726d3location16b-h14"],"title":"Calculating the 83rd Percentile","text":"What is the average between the data values at the 23rd and 24th positions? This is the 78th percentile.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ac726d3location16b-h15-s4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$70$$"],"dependencies":[],"title":"Calculating the 83rd Percentile","text":"What is $$\\\\frac{69+71}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"ac726d3location17","title":"Percentiles in Context","body":"Jesse was ranked 37th in his graduating class of $$180$$ students.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":0,"lesson":"2.3 Measures of the Location of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac726d3location17a","stepAnswer":["$$80$$"],"problemType":"TextBox","stepTitle":"At what percentile is Jesse\'s ranking?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$80$$","hints":{"DefaultPathway":[{"id":"ac726d3location17a-h1","type":"hint","dependencies":[],"title":"Formula for Finding the Percentile of a Value in a Data Set","text":"Remember the formula for finding the Percentile of a Value in a Data Set. We want to know three values once the list is ordered (which is already is from least to greatest): $$x=the$$ number of data values counting from the bottom of the data list up to but not including the data value for which you want to find the percentile; $$y=the$$ number of data values equal to the data value for which you want to find the percentile; $$n=the$$ total number of data. Then, we can use the formula to calculate the percentile as $$100\\\\frac{x+0.5y}{n}$$. If the percentile is not an integer, round to the nearest integer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location17a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$143$$"],"dependencies":["ac726d3location17a-h1"],"title":"Determining $$x$$ (Number of Points Below Desired Value)","text":"What is $$x$$? How many students ranked below Jesse?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ac726d3location17a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$143$$"],"dependencies":[],"title":"Determining $$x$$ (Number of Points Below Desired Value)","text":"What is $$180-37$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ac726d3location17a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ac726d3location17a-h2"],"title":"Determining $$y$$ (Repetition of Desired Value)","text":"How many times does $$37$$ show up in the data? How many students can be ranked 37th?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location17a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ac726d3location17a-h3"],"title":"Determining $$y$$ (Repetition of Desired Value)","text":"Assuming it is not possible for two students to tie in rank, how many students can be ranked 37th?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location17a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$180$$"],"dependencies":["ac726d3location17a-h4"],"title":"Determining $$n$$ (Total Number of Data)","text":"What is $$n$$, the total number of data in the set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location17a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$79.72$$"],"dependencies":["ac726d3location17a-h5"],"title":"Calculating the Percentile","text":"Calculate the percentile using the Formula for Finding the Percentile of a Value in a Data Set. Round your answer to the nearest hundredth.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$3.5$$","$$0.1207$$","$$12.07$$","$$12$$"],"subHints":[{"id":"ac726d3location17a-h6-s2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$79.72$$"],"dependencies":[],"title":"Calculating the Percentile","text":"What is $$100\\\\frac{143+0.5\\\\times1}{180}$$ rounded to the nearest hundredth?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$3.5$$","$$0.1207$$","$$12.07$$","$$12$$"]}]},{"id":"ac726d3location17a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ac726d3location17a-h6"],"title":"Rounding the Percentile","text":"Is $$79.72$$ a whole number?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ac726d3location17a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$80$$"],"dependencies":["ac726d3location17a-h7"],"title":"Rounding the Percentile","text":"What is $$79.72$$ rounded to the nearest whole number? This is the final percentile.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac726d3location18","title":"Interpreting Percentiles","body":"For runners in a race, a higher speed means a faster run.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":0,"lesson":"2.3 Measures of the Location of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac726d3location18a","stepAnswer":["High Percentile"],"problemType":"MultipleChoice","stepTitle":"Is it more desirable to have a speed with a high or low percentile when running a race?","stepBody":"","answerType":"string","variabilization":{},"choices":["High Percentile","Higher Percentile","Lower Percentile"],"hints":{"DefaultPathway":[{"id":"ac726d3location18a-h1","type":"hint","dependencies":[],"title":"Good or Bad in Context","text":"A low percentile for speed means that the runner has a slower speed through the race while a high percentile for speed means that the runner has a faster speed throughout the race.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location18a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Higher Percentile"],"dependencies":["ac726d3location18a-h1"],"title":"Good or Bad in Context","text":"In context, since this is a race, is it better to have a lower percentile (finishing slower due to slower speed) or have a higher percentile (finishing faster due to faster speed)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Lower Percentile","Higher Percentile"]}]}},{"id":"ac726d3location18b","stepAnswer":["40% of runners ran at speeds of $$7.5$$ miles per hour or less (slower). 60% of runners ran at speeds of $$7.5$$ miles per hour or more (faster)."],"problemType":"MultipleChoice","stepTitle":"The 40th percentile of speeds in a particular race is $$7.5$$ miles per hour. How can we interpret the 40th percentile in the context of the situation?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"40% of runners ran at speeds of $$7.5$$ miles per hour or less (slower). 60% of runners ran at speeds of $$7.5$$ miles per hour or more (faster).","choices":["40% of runners ran at speeds of $$7.5$$ miles per hour or less (slower). 60% of runners ran at speeds of $$7.5$$ miles per hour or more (faster).","60% of runners ran at speeds of $$7.5$$ miles per hour or less (slower). 40% of runners ran at speeds of $$7.5$$ miles per hour or more (faster).","40% of runners ran at exactly $$7.5$$ miles per hour.","60% of runners ran exactly at $$7.5$$ miles per hour."],"hints":{"DefaultPathway":[{"id":"ac726d3location18b-h3","type":"hint","dependencies":["ac726d3location18a-h2"],"title":"Understanding the 40th Percentile","text":"The percentage of data values that are less than or equal to the 40th percentile is 40%.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location18b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$7.5$$ miles per hour or less"],"dependencies":["ac726d3location18b-h3"],"title":"Determining Directionality","text":"Based on this, since the 40th percentile for speed in a particular race is $$7.5$$ miles per hour, does that equate to 40% of runners finishing the race with a speed of $$7.5$$ miles per hour or less OR $$7.5$$ miles per hour or more?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$7.5$$ miles per hour or less","$$7.5$$ miles per hour or more"]}]}}]},{"id":"ac726d3location19","title":"Interpreting Percentiles","body":"Mina is waiting in line at the Department of Motor Vehicles (DMV). Her wait time of $$32$$ minutes is the 85th percentile of wait times.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":0,"lesson":"2.3 Measures of the Location of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac726d3location19a","stepAnswer":["85% of people at the DMV waited $$32$$ minutes or less. 15% of people at the DMV waited $$32$$ minutes or longer."],"problemType":"MultipleChoice","stepTitle":"How can we interpret the 85th percentile in the context of the situation?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"85% of people at the DMV waited $$32$$ minutes or less. 15% of people at the DMV waited $$32$$ minutes or longer.","choices":["85% of people at the DMV waited $$32$$ minutes or less. 15% of people at the DMV waited $$32$$ minutes or longer.","15% of people at the DMV waited $$32$$ minutes or less. 85% of people at the DMV waited $$32$$ minutes or longer.","85% of people at the DMV waited exactly $$32$$ minutes.","15% of people at the DMV waited exactly $$32$$ minutes."],"hints":{"DefaultPathway":[{"id":"ac726d3location19a-h1","type":"hint","dependencies":[],"title":"Understanding the 85th Percentile","text":"The percentage of data values that are less than or equal to the 85th percentile is 85%.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location19a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$32$$ minutes or less"],"dependencies":["ac726d3location19a-h1"],"title":"Determining Directionality","text":"Based on this, since the 85th percentile for time spend waiting at the DMV is $$32$$ minutes, does that equate to 85% of people waiting at the DMV for $$32$$ minutes or less OR $$32$$ minutes or more?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$32$$ minutes or less","$$32$$ minutes or more"]}]}},{"id":"ac726d3location19b","stepAnswer":["Lower Percentile"],"problemType":"MultipleChoice","stepTitle":"Is it more desirable to have a higher or lower percentile in this situation? Assume this person is busy and does not like standing in line.","stepBody":"","answerType":"string","variabilization":{},"choices":["Lower Percentile","Higher Percentile"],"hints":{"DefaultPathway":[{"id":"ac726d3location19b-h1","type":"hint","dependencies":[],"title":"Good or Bad in Context","text":"A low percentile for wait time means that the person has to wait less minutes at the DMV, which means spending less time of their day in a line while a high percentile for wait time means the person has to wait more minutes at the DMV, which means spending more time of their day in a line. Assume this person is busy and does not like standing in line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location19b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Lower Percentile"],"dependencies":["ac726d3location19b-h1"],"title":"Good or Bad in Context","text":"In context, since this person is busy, is it better to have a lower percentile (shorter wait times) or have a higher percentile (longer wait times)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Lower Percentile","Higher Percentile"]}]}}]},{"id":"ac726d3location2","title":"Calculate Interquartile Range","body":"Use the following $$11$$ salaries for the question below. The salaries are in dollars. $33,000; $64,500; $28,000; $54,000; $72,000; $68,500; $69,000; $42,000; $54,000; $120,000; $40,500","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.3 Measures of the Location of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac726d3location2a","stepAnswer":["$$28500$$"],"problemType":"TextBox","stepTitle":"Calculate the IQR (InterQuartile Range).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$28500$$","hints":{"DefaultPathway":[{"id":"ac726d3location2a-h1","type":"hint","dependencies":[],"title":"Order the Data","text":"First order the data from smallest to largest. Here is the ordering: $28,000; $33,000; $40,500; $42,000; $54,000; $54,000; $64,500; $68,500; $69,000; $72,000; $120,000.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location2a-h2","type":"hint","dependencies":["ac726d3location2a-h1"],"title":"Find the Median","text":"Remember that the IQR is the formula Quartile $$3$$ - Quartile $$1$$. Therefore, to determine the quartiles, we\'ll first determine the median.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$54000$$"],"dependencies":["ac726d3location2a-h2"],"title":"Determine the Median","text":"What is the median of the data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location2a-h4","type":"hint","dependencies":["ac726d3location2a-h3"],"title":"Find Quartiles","text":"The definition for Quartile $$1$$ is the middle value of the lower half of the data. The definition for Quartile $$3$$ is the middle value for the upper half of the data. Remember that the halfpoint of the data is determined by the median, so we now need to find the middle points of the remaining data.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location2a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$40500$$"],"dependencies":["ac726d3location2a-h4"],"title":"Find Quartile $$1$$","text":"What is Quartile 1? What is the middle value of the lower half of the data?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$69000$$"],"dependencies":["ac726d3location2a-h5"],"title":"Find Quartile $$3$$","text":"What is Quartile 3? What is the middle value of the upper half of the data?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location2a-h7","type":"hint","dependencies":["ac726d3location2a-h6"],"title":"Definition of IQR","text":"The definition of IQR (interquartile range) is the number that indicates the spread of the middle half or the middle 50% of the data. It is the difference between the third quartile and the first quartile.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location2a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$28500$$"],"dependencies":["ac726d3location2a-h7"],"title":"Determine the IQR","text":"What is the IQR?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ac726d3location2a-h8-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$28500$$"],"dependencies":[],"title":"Determine the IQR","text":"What is $$69000-40500$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}},{"id":"ac726d3location2b","stepAnswer":["$$120000$$"],"problemType":"TextBox","stepTitle":"Determine the potential outlier.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$120000$$","hints":{"DefaultPathway":[{"id":"ac726d3location2b-h9","type":"hint","dependencies":["ac726d3location2a-h8"],"title":"Using IQR to Determine Outliers","text":"The IQR can help to determine potential outliers. A value is a potential outlier if it is less than $$(1.5)(IQR)$$ below the first quartile or more than $$(1.5)(IQR)$$ above the third quartile.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location2b-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$42750$$"],"dependencies":["ac726d3location2b-h9"],"title":"Determine $$(1.5)(IQR)$$","text":"What is $$(1.5)(IQR)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ac726d3location2b-h10-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$42750$$"],"dependencies":[],"title":"Determine $$(1.5)(IQR)$$","text":"What is $$(1.5)(28500)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ac726d3location2b-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2250$$"],"dependencies":["ac726d3location2b-h10"],"title":"Determine Quartile $$1-(1.5)(IQR)$$","text":"What is Quartile $$1-(1.5)(IQR)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ac726d3location2b-h11-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-201625$$"],"dependencies":[],"title":"Determine Quartile $$1-(1.5)(IQR)$$","text":"What is $$40500-42750$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ac726d3location2b-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$111750$$"],"dependencies":["ac726d3location2b-h11"],"title":"Determine Quartile $$3+1.5IQR$$","text":"What is Quartile $$3+1.5IQR$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ac726d3location2b-h12-s4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$111750$$"],"dependencies":[],"title":"Determine Quartile $$3+1.5IQR$$","text":"What is $$69000+42750$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ac726d3location2b-h13","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ac726d3location2b-h12"],"title":"Determining Outliers","text":"Are there any values in the data set that are less than $$(1.5)(IQR)$$ below the first quartile?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ac726d3location2b-h14","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ac726d3location2b-h13"],"title":"Determining Outliers","text":"Are there any values in the data set that are greater than $$(1.5)(IQR)$$ above the third quartile?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ac726d3location2b-h15","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$120000$$"],"dependencies":["ac726d3location2b-h14"],"title":"Determine the Potential Outlier","text":"What is the potential outlier that is greater than $$(1.5)(IQR)$$ above Quartile 3?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$69000$$","$$72000$$","$$120000$$","$$685000$$"]}]}}]},{"id":"ac726d3location20","title":"Interpreting Percentiles","body":"Suppose that you are buying a house. You and your realtor have determined that the most expensive house you can afford is the 34th percentile. The 34th percentile of housing prices is $240,000 in the town you want to move to.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":0,"lesson":"2.3 Measures of the Location of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac726d3location20a","stepAnswer":["34% of houses cost $240,000 or less. 66% of houses cost $240,000 or more."],"problemType":"MultipleChoice","stepTitle":"How can we interpret the 34th percentile in this context?","stepBody":"","answerType":"string","variabilization":{},"choices":["34% of houses cost $240,000 or less. 66% of houses cost $240,000 or more.","66% of houses cost $240,000 or less. 34% of houses cost $240,000 or more.","34% of houses cost $240,000 exactly.","66% of houses cost $240,000 exactly."],"hints":{"DefaultPathway":[{"id":"ac726d3location20a-h1","type":"hint","dependencies":[],"title":"Understanding the 34th Percentile","text":"The percentage of data values that are less than or equal to the 34th percentile is 34%.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location20a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$240,000 or less"],"dependencies":["ac726d3location20a-h1"],"title":"Determining Directionality","text":"Based on this, since the 34th percentile of housing prices in the town you want to move to is $240,000, does that equate to 34% of housing prices to be $240,000 or less OR $240,000 or more?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$240,000 or less","$240,000 or more"]}]}}]},{"id":"ac726d3location3","title":"Determining Median","body":"Fifty statistics students were asked how much sleep they get per school night (rounded to the nearest hour). The results are listed in the table.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.3 Measures of the Location of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac726d3location3a","stepAnswer":["$$7$$"],"problemType":"TextBox","stepTitle":"Find the median.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7$$","hints":{"DefaultPathway":[{"id":"ac726d3location3a-h1","type":"hint","dependencies":[],"title":"Median and Frequency","text":"Note that in the \\"cumulative relative frequency\\" column we see the value $$0.52$$. The median is the 50th percentile or the second quartile. Therefore, we can count of the \\"frequency\\" column and find the midpoint.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["ac726d3location3a-h1"],"title":"Looking at the Frequency Column","text":"There are $$50$$ total data points. How many data points will be less than the median?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["ac726d3location3a-h2"],"title":"Determining Median from the Frequency Column","text":"Where is the 25th data point based on frequency?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ac726d3location3a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$14$$"],"dependencies":[],"title":"Determining Median from the Frequency Column","text":"How many data points (what is the sum of the frequencies) up to and including $$6$$ hours of sleep per school night?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location3a-h3-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$26$$"],"dependencies":[],"title":"Determining Median from the Frequency Column","text":"How many data points (what is the sum of the frequencies) up to and including $$7$$ hours of sleep per school night?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ac726d3location3a-h4","type":"hint","dependencies":["ac726d3location3a-h3"],"title":"Median Between 25th and 26th Data Point","text":"Because there are an even number of total data points, we note that there will be $$\\\\frac{50}{2}=25$$ data points below the median and $$\\\\frac{50}{2}=25$$ data points above the median. Therefore, the median will be the middle point between the 25th and 26th data points.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["ac726d3location3a-h4"],"title":"25th Data Point","text":"What is the 25th data point?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ac726d3location3a-h5-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":[],"title":"25th Data Point","text":"The first $$14$$ data points reside in $$4$$ hours to $$6$$ hours of sleep per night. The first $$26$$ data points reside in $$4$$ hours to $$7$$ hours of sleep per night. This means that data points $$15$$ through $$26$$ reside in the $$7$$ hours of sleep per night category. Therefore, what is the 25th data point?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ac726d3location3a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["ac726d3location3a-h5"],"title":"26th Data Point","text":"What is the 26th data point?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ac726d3location3a-h6-s4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":[],"title":"26th Data Point","text":"The first $$14$$ data points reside in $$4$$ hours to $$6$$ hours of sleep per night. The first $$26$$ data points reside in $$4$$ hours to $$7$$ hours of sleep per night. This means that data points $$15$$ through $$26$$ reside in the $$7$$ hours of sleep per night category. Therefore, what is the 26th data point?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ac726d3location3a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["ac726d3location3a-h6"],"title":"Determining the Median","text":"What is the middle point between the 25th and 26th data points?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ac726d3location3a-h7-s5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":[],"title":"Determining the Median","text":"What is $$\\\\frac{7+7}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"ac726d3location4","title":"Determining Median","body":"Fifty statistics students were asked how much sleep they get per school night (rounded to the nearest hour). The results are listed in the table.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.3 Measures of the Location of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac726d3location4a","stepAnswer":["$$9$$"],"problemType":"TextBox","stepTitle":"Find the 90th percentile.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9$$","hints":{"DefaultPathway":[{"id":"ac726d3location4a-h1","type":"hint","dependencies":[],"title":"Percentiles and Cumulative Frequency","text":"Note that the 90th percentile will be the data point (90% of 50)th data point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$45$$"],"dependencies":["ac726d3location4a-h1"],"title":"Determining the Correct Data Point","text":"What is $$0.90(50)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location4a-h3","type":"hint","dependencies":["ac726d3location4a-h2"],"title":"Percentiles and Data Points","text":"Therefore, the 90th percentile will reside between the 45th and 46th data points in this data set.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["ac726d3location4a-h3"],"title":"Determining the 9th Bucket","text":"What is the 45th data point? Look at the frequency column (sums up to 50) and determine which amount of sleep the 45th data point resides in.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ac726d3location4a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":[],"title":"Determining the 9th Bucket","text":"Since the \\"frequency\\" column sums up to $$50$$, we note we can go backwards. The $$10$$ \\"hours per school night\\" column has $$3$$ values so it contains the 48th to 50th data points. How many values does the $$9$$ \\"hours per school night\\" column have?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ac726d3location4a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$41$$"],"dependencies":["ac726d3location4a-h4"],"title":"Determining the Lower Limit of the $$9$$ Bucket","text":"This means that the $$9$$ \\"hours per school night\\" amount houses values from the Xth value to the 47th value. We note that there are $$7$$ total items in the $$9$$ \\"hours per school night\\" bin. What is X?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location4a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ac726d3location4a-h5"],"title":"45th Data Point","text":"Is $$45$$ greater than $$41$$ and less than 48?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ac726d3location4a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["ac726d3location4a-h6"],"title":"45th Data Point","text":"What is the value of the 45th data point?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location4a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ac726d3location4a-h7"],"title":"46th Data Point","text":"Is $$46$$ greater than $$41$$ and less than 48?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ac726d3location4a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["ac726d3location4a-h8"],"title":"46th Data Point","text":"What is the value of the 46th data point?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location4a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["ac726d3location4a-h9"],"title":"Average of 45th and 46th Data Point","text":"What is the average of the 45th and 46th data point?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac726d3location5","title":"Determining Data Points from a Box-and-Whisker Plot","body":"The following box plot shows the U.S. population for $$1990$$, the latest available year at the time.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.3 Measures of the Location of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac726d3location5a","stepAnswer":["More Children"],"problemType":"MultipleChoice","stepTitle":"Are there fewer or more children (age $$17$$ and under) than senior citizens (age $$65$$ and over)?","stepBody":"","answerType":"string","variabilization":{},"choices":["More Children","Fewer Children"],"hints":{"DefaultPathway":[{"id":"ac726d3location5a-h1","type":"hint","dependencies":[],"title":"Reading a Box-and-Whisker Plot","text":"For a Box-and-Whisker Plot, we note that there are two boxes and two whiskers (two lines extending from the boxes). The boxes represent the quartiles: the first box goes from the first quartile to the second quartile and the second box goes from the second quartile to the third quartile. The whiskers represent the remaining data (the 25% of the data that is less than the first quartile and the 25% of the data greater than the third quartile).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location5a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Whiskers"],"dependencies":["ac726d3location5a-h1"],"title":"Boxes or Whiskers?","text":"As children are defined as age $$17$$ and under while senior citizens are age $$65$$ and over, are children and seniors represented as boxes or whiskers?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Boxes","Whiskers"]},{"id":"ac726d3location5a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["25%"],"dependencies":["ac726d3location5a-h2"],"title":"Determining Whisker Length","text":"How much data does each of the whiskers represent?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["25%","50%","33%","5%"]},{"id":"ac726d3location5a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Left"],"dependencies":["ac726d3location5a-h3"],"title":"Determining Whisker Length","text":"Is the length of the whisker on the left shorter or the length of the whisker on the right shorter?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Left","Right"]},{"id":"ac726d3location5a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ac726d3location5a-h4"],"title":"Determining Percentage of Children","text":"Does the left whisker (age $$17$$ and under) include other age groups except children?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ac726d3location5a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["25%"],"dependencies":["ac726d3location5a-h5"],"title":"Determining Percentage of Children","text":"What percentage of the U.S. population is children?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["25%","50%","<25%",">25%"]},{"id":"ac726d3location5a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ac726d3location5a-h6"],"title":"Determining Percentage of Seniors","text":"Does the right whisker (age $$50$$ and older) include other age groups except seniors?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"],"subHints":[{"id":"ac726d3location5a-h7-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":[],"title":"Determining Percentage of Seniors","text":"Seniors are age $$65$$ and over. The right whisker starts at age $$50$$. Are people age $$50$$ to $$64$$ considered seniors?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]},{"id":"ac726d3location5a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["<25%"],"dependencies":["ac726d3location5a-h7"],"title":"Determining Percentage of Seniors","text":"What percentage of the U.S. population is seniors?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["25%","50%","<25%",">25%"]},{"id":"ac726d3location5a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["<25%"],"dependencies":["ac726d3location5a-h8"],"title":"Determining Percentage of Seniors","text":"If there are other people also part of the whisker, which contains 25% of the data point, this means that the population of seniors does not take up the entire whisker. If a whisker contains 25% of the data, does this mean that seniors make up 25% of the data, <25% of the data, or >25% of the data?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["25%","<25%",">25%"]},{"id":"ac726d3location5a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["More"],"dependencies":["ac726d3location5a-h9"],"title":"Which Percentage is Greater?","text":"Are there fewer or more children than senior citizens?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Fewer","More"]},{"id":"ac726d3location5a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ac726d3location5a-h10"],"title":"Which Percentage is Greater?","text":"Is \\"25%\\" > \\"less than 25%\\"?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"ac726d3location6","title":"Determining Percentages of Data","body":"Six hundred adult Americans were asked by telephone poll, \\"What do you think constitutes a middle-class income?\\" The results are in the table provided. Also, include the left endpoint for each row of the table, but not the right endpoint.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.3 Measures of the Location of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac726d3location6a","stepAnswer":["$$0.06$$"],"problemType":"TextBox","stepTitle":"What proportion of the survey answered \\"not sure\\"?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.06$$","hints":{"DefaultPathway":[{"id":"ac726d3location6a-h1","type":"hint","dependencies":[],"title":"The Complement Rule","text":"The complement rule essentially explains that the proportion of the survey that answered \\"not sure\\" is equivalent to $$1$$ (or 100%) minus the sum of all the relative frequencies of those who answered in the table. Essentially, the proportion that answered \\"not sure\\" should be $$1$$ minus the sum of all the relative frequencies of those who answered <20,000; $$20, 000-25, 000;$$ $$25, 000-30, 000;$$ $$30, 000-40, 000;$$ $$40, 000-50, 000;$$ $$50, 000-75, 000;$$ $$75, 000-99, 999;$$ and 100,000+.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.94$$"],"dependencies":["ac726d3location6a-h1"],"title":"Sum of the Relative Frequencies","text":"What is the sum of the relative frequencies for those who responded <20,000; $$20, 000-25, 000;$$ $$25, 000-30, 000;$$ $$30, 000-40, 000;$$ $$40, 000-50, 000;$$ $$50, 000-75, 000;$$ $$75, 000-99, 999;$$ or 100,000+?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ac726d3location6a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.94$$"],"dependencies":[],"title":"Sum of the Relative Frequencies","text":"What is $$0.02+0.09+0.19+0.26+0.18+0.17+0.02+0.01$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ac726d3location6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.06$$"],"dependencies":["ac726d3location6a-h2"],"title":"Use the Complement Rule","text":"What is $$1$$ minus the sum of the relative frequencies?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.06$$"],"dependencies":["ac726d3location6a-h3"],"title":"Use the Complement Rule","text":"What is $$1-0.94$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ac726d3location6b","stepAnswer":["$$0.63$$"],"problemType":"TextBox","stepTitle":"What proportion think that middle-class is from $25,000 to $50,000?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.63$$","hints":{"DefaultPathway":[{"id":"ac726d3location6b-h5","type":"hint","dependencies":["ac726d3location6a-h4"],"title":"Sum of Relative Frequencies","text":"The proportion that think that middle-class is from $25,000 to $50,000 is just the sum of the relative frequencies of the columns that include that range.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location6b-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$25, 000-30, 000;$$ $$30, 000-40, 000;$$ $$40, 000-50, 000$$"],"dependencies":["ac726d3location6b-h5"],"title":"Determining Which Columns","text":"Which columns include the full range 20,000 to 50,000? Remember that for each row, we only include the left endpoint and not the right (so $$20, 000-25, 000$$ includes 20,000 but doesn\'t include 25,000).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$25, 000-30, 000;$$ $$30, 000-40, 000;$$ $$40, 000-50, 000$$","$$20, 000-25, 000;$$ $$30, 000-40, 000;$$ $$40, 000-50, 000$$","$$25, 000-30, 000;$$ $$40, 000-50, 000$$","$$25, 000-30, 000$$"]},{"id":"ac726d3location6b-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.63$$"],"dependencies":["ac726d3location6b-h6"],"title":"Summing the Relative Frequencies","text":"What is the sum of the relative frequencies of the columns you determined?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ac726d3location6b-h7-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.63$$"],"dependencies":[],"title":"Summing the Relative Frequencies","text":"What is $$0.19+0.26+0.18$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"ac726d3location7","title":"Finding Percentiles","body":"Listed are $$29$$ ages for Academy Award winning best actors in order from smallest to largest. 18; 21; 22; 25; 26; 27; 29; 30; 31; 33; 36; 37; 41; 42; 47; 52; 55; 57; 58; 62; 64; 67; 69; 71; 72; 73; 74; 76; $$77$$","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.3 Measures of the Location of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac726d3location7a","stepAnswer":["$$64$$"],"problemType":"TextBox","stepTitle":"Find the 70th percentile.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$64$$","hints":{"DefaultPathway":[{"id":"ac726d3location7a-h1","type":"hint","dependencies":[],"title":"Formula for Finding kth Percentile","text":"Remember the formula for finding the kth Percentile. We want to know three values: $$k=the$$ kth percentile, usually given to you; $$i=the$$ index (ranking or position of the data value), what we normally want to find; $$n=the$$ total number of data. Then, we can use the formula to calculate $$i=\\\\frac{k}{100} \\\\left(n+1\\\\right)$$. If i is an integer, we use the data at the ith position in the ordered data. If i is not an integer, we then round i down and up to the nearest integers and take the average of these two.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$70$$"],"dependencies":["ac726d3location7a-h1"],"title":"Determining k (Percentile)","text":"What is the kth percentile we want to determine?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$29$$"],"dependencies":["ac726d3location7a-h2"],"title":"Determining $$n$$ (Total Number of Data)","text":"What is $$n$$, the total number of data in the set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$21$$"],"dependencies":["ac726d3location7a-h3"],"title":"Determining i (Desired Index)","text":"Using the Formula for Finding the kth Percentile, what is i?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ac726d3location7a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$21$$"],"dependencies":[],"title":"Determining i (Desired Index)","text":"What is $$\\\\frac{70}{100} \\\\left(29+1\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ac726d3location7a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ac726d3location7a-h4"],"title":"Determining the position of the data","text":"Is i an integer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ac726d3location7a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$64$$"],"dependencies":["ac726d3location7a-h5"],"title":"Determining the 70th Percentile","text":"What is the data value in the 21st position of the ordered data set? This is the 70th percentile.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac726d3location8","title":"Finding Percentiles","body":"Listed are $$29$$ ages for Academy Award winning best actors in order from smallest to largest. 18; 21; 22; 25; 26; 27; 29; 30; 31; 33; 36; 37; 41; 42; 47; 52; 55; 57; 58; 62; 64; 67; 69; 71; 72; 73; 74; 76; $$77$$","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.3 Measures of the Location of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac726d3location8a","stepAnswer":["$$71.5$$"],"problemType":"TextBox","stepTitle":"Find the 83rd percentile.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$71.5$$","hints":{"DefaultPathway":[{"id":"ac726d3location8a-h1","type":"hint","dependencies":[],"title":"Formula for Finding kth Percentile","text":"Remember the formula for finding the kth Percentile. We want to know three values: $$k=the$$ kth percentile, usually given to you; $$i=the$$ index (ranking or position of the data value), what we normally want to find; $$n=the$$ total number of data. Then, we can use the formula to calculate $$i=\\\\frac{k}{100} \\\\left(n+1\\\\right)$$. If i is an integer, we use the data at the ith position in the ordered data. If i is not an integer, we then round i down and up to the nearest integers and take the average of these two.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$83$$"],"dependencies":["ac726d3location8a-h1"],"title":"Determining k (Percentile)","text":"What is the kth percentile we want to determine?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$29$$"],"dependencies":["ac726d3location8a-h2"],"title":"Determining $$n$$ (Total Number of Data)","text":"What is $$n$$, the total number of data in the set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$24.9$$"],"dependencies":["ac726d3location8a-h3"],"title":"Determining i (Desired Index)","text":"Using the Formula for Finding the kth Percentile, what is i?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ac726d3location8a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$24.9$$"],"dependencies":[],"title":"Determining i (Desired Index)","text":"What is $$\\\\frac{83}{100} \\\\left(29+1\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ac726d3location8a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ac726d3location8a-h4"],"title":"Determining the position of the data","text":"Is i an integer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ac726d3location8a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$24$$"],"dependencies":["ac726d3location8a-h5"],"title":"Determining the Lower Index","text":"What is i rounded down to a whole number?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location8a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["ac726d3location8a-h6"],"title":"Determining the Upper Index","text":"What is i rounded up to a whole number?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location8a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$71$$"],"dependencies":["ac726d3location8a-h7"],"title":"Determining the Lower Index Data Point","text":"What is the data value in the 24th position of the ordered data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location8a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$72$$"],"dependencies":["ac726d3location8a-h8"],"title":"Determining the Upper Index Data Point","text":"What is the data value in the 25th position of the ordered data set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location8a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$71.5$$"],"dependencies":["ac726d3location8a-h9"],"title":"Determining the 83rd Percentile","text":"What is the average between the data values at the 24th and 25th positions? This is the 83rd percentile.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ac726d3location8a-h10-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$71.5$$"],"dependencies":[],"title":"Determining the 83rd Percentile","text":"What is $$\\\\frac{71+72}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"ac726d3location9","title":"Finding Percentiles","body":"Listed are $$29$$ ages for Academy Award winning best actors in order from smallest to largest. 18; 21; 22; 25; 26; 27; 29; 30; 31; 33; 36; 37; 41; 42; 47; 52; 55; 57; 58; 62; 64; 67; 69; 71; 72; 73; 74; 76; $$77$$","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.3 Measures of the Location of the Data","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac726d3location9a","stepAnswer":["$$64$$"],"problemType":"TextBox","stepTitle":"Find the percentile for $$58$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$64$$","hints":{"DefaultPathway":[{"id":"ac726d3location9a-h1","type":"hint","dependencies":[],"title":"Formula for Finding the Percentile of a Value in a Data Set","text":"Remember the formula for finding the Percentile of a Value in a Data Set. We want to know three values once the list is ordered (which is already is from least to greatest): $$x=the$$ number of data values counting from the bottom of the data list up to but not including the data value for which you want to find the percentile; $$y=the$$ number of data values equal to the data value for which you want to find the percentile; $$n=the$$ total number of data. Then, we can use the formula to calculate the percentile as $$100\\\\frac{x+0.5y}{n}$$. If the percentile is not an integer, round to the nearest integer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18$$"],"dependencies":["ac726d3location9a-h1"],"title":"Determining $$x$$ (Number of Points Below Desired Value)","text":"What is $$x$$? In other words, how many data values are there counting from the bottom of the data list up to but not including the data value for which you want to find the percentile?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ac726d3location9a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18$$"],"dependencies":[],"title":"Determining $$x$$ (Number of Points Below Desired Value)","text":"How many values in the data set are less than 58?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ac726d3location9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ac726d3location9a-h2"],"title":"Determining $$y$$ (Repetition of Desired Value)","text":"How many times does $$58$$ show up in the data?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$29$$"],"dependencies":["ac726d3location9a-h3"],"title":"Determining $$n$$ (Total Number of Data)","text":"What is $$n$$, the total number of data in the set?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac726d3location9a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$63.79$$"],"dependencies":["ac726d3location9a-h4"],"title":"Determining the Percentile","text":"Calculate the percentile using the Formula for Finding the Percentile of a Value in a Data Set. Round your answer to the nearest hundredth.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$63.79$$","$$18$$","$$29$$","$$0.64$$"],"subHints":[{"id":"ac726d3location9a-h5-s2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$63.79$$"],"dependencies":[],"title":"Determining the Percentile","text":"What is $$100\\\\frac{18+0.5\\\\times1}{29}$$ rounded to the nearest hundredth?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$63.79$$","$$18$$","$$29$$","$$0.64$$"]}]},{"id":"ac726d3location9a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ac726d3location9a-h5"],"title":"Rounding the Percentile","text":"Is $$63.79$$ a whole number?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ac726d3location9a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$64$$"],"dependencies":["ac726d3location9a-h6"],"title":"Rounding the Percentile","text":"What is $$63.79$$ rounded to the nearest whole number? This is the final percentile.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac7990clinear1","title":"Linear Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.1 Linear Equations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac7990clinear1a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Is $$y=-0.125-3.5x$$ a linear equation?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ac7990clinear1a-h1","type":"hint","dependencies":[],"title":"Linear Equation","text":"A linear equation is in the format $$y=a+bx$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-0.125$$"],"dependencies":["ac7990clinear1a-h1"],"title":"A value","text":"Can you identify the a value in $$y=-0.125-3.5x$$? What is it?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3.5$$"],"dependencies":["ac7990clinear1a-h1"],"title":"B value","text":"Can you identify the $$b$$ value in $$y=-0.125-3.5x$$? What is it?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear1a-h4","type":"hint","dependencies":["ac7990clinear1a-h2","ac7990clinear1a-h3"],"title":"Interpretation","text":"This means that $$y=-0.125-3.5x$$ is in $$y=a+bx$$ format.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac7990clinear10","title":"Linear Equations","body":"Svetlana tutors to make extra money for college. For each tutoring session, she charges a one-time fee of $25 plus $15 per hour of tutoring. A linear equation that expresses the total amount of money Svetlana earns for each session she tutors is $$y=25+15x$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.1 Linear Equations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac7990clinear10a","stepAnswer":["$$y=the$$ total cost of each tutoring session"],"problemType":"MultipleChoice","stepTitle":"What is the dependent variable?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=the$$ total cost of each tutoring session","choices":["$$x=the$$ cost per hour of tutoring","$$x=the$$ $$one-time$$ fee","$$y=the$$ cost per hour of tutoring","$$y=the$$ total cost of each tutoring session"],"hints":{"DefaultPathway":[{"id":"ac7990clinear10a-h1","type":"hint","dependencies":[],"title":"Interpret","text":"The dependent variable is the $$y$$. Interpret what $$y$$ could be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear10a-h2","type":"hint","dependencies":["ac7990clinear10a-h1"],"title":"Meaning","text":"$$y=25+15x$$. This mean we are adding the one-time fee and the cost of the tutoring.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear10a-h3","type":"hint","dependencies":["ac7990clinear10a-h2"],"title":"Meaning","text":"The one-time fee and the cost of tutoring is the total cost of the tutoring session.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac7990clinear11","title":"Linear Equations","body":"Svetlana tutors to make extra money for college. For each tutoring session, she charges a one-time fee of $25 plus $15 per hour of tutoring. A linear equation that expresses the total amount of money Svetlana earns for each session she tutors is $$y=25+15x$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.1 Linear Equations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac7990clinear11a","stepAnswer":["$25. This represents the $$one-time$$ fee, which would be charged even for $$0$$ hours of tutoring."],"problemType":"MultipleChoice","stepTitle":"What is the y-intercept? Interpret this.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$25. This represents the one-time fee, which would be charged even for $$0$$ hours of tutoring.","choices":["$25. This represents the $$one-time$$ fee, which would be charged even for $$0$$ hours of tutoring.","$25. This represents the charge per hour of tutoring.","$15. This represents the $$one-time$$ fee, which would be charged even for $$0$$ hours of tutoring.","$15. This represents the charge per hour of tutoring."],"hints":{"DefaultPathway":[{"id":"ac7990clinear11a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"The y-intercept is when $$x=0$$. Substitute this into the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["ac7990clinear11a-h1"],"title":"Substitute","text":"$$y=25+\\\\operatorname{15}\\\\left(0\\\\right)$$. What is this equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear11a-h3","type":"hint","dependencies":["ac7990clinear11a-h2"],"title":"Meaning","text":"The $25 represents the one-time fee.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac7990clinear12","title":"Linear Equations","body":"Svetlana tutors to make extra money for college. For each tutoring session, she charges a one-time fee of $25 plus $15 per hour of tutoring. A linear equation that expresses the total amount of money Svetlana earns for each session she tutors is $$y=25+15x$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.1 Linear Equations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac7990clinear12a","stepAnswer":["$15. This represents the charge per hour of tutoring."],"problemType":"MultipleChoice","stepTitle":"What is the slope? Interpret this.","stepBody":"","answerType":"string","variabilization":{},"choices":["$25. This represents the $$one-time$$ fee, which would be charged even for $$0$$ hours of tutoring.","$25. This represents the charge per hour of tutoring.","$15. This represents the $$one-time$$ fee, which would be charged even for $$0$$ hours of tutoring.","$15. This represents the charge per hour of tutoring."],"hints":{"DefaultPathway":[{"id":"ac7990clinear12a-h1","type":"hint","dependencies":[],"title":"Form","text":"In $$y=a+bx$$, $$b$$ is the slope.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["ac7990clinear12a-h1"],"title":"Find $$b$$","text":"What is $$b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear12a-h3","type":"hint","dependencies":["ac7990clinear12a-h2"],"title":"Meaning","text":"$15 is the cost per hour of tutoring.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac7990clinear13","title":"Linear Equations","body":"Ethan repairs household appliances like dishwashers and refrigerators. For each visit, he charges $25 plus $20 per hour of work. A linear equation that expresses the total amount of money Ethan earns per visit is $$y=25+20x$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.1 Linear Equations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac7990clinear13a","stepAnswer":["$$x=the$$ cost per hour of work"],"problemType":"MultipleChoice","stepTitle":"What is the independent variable?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=the$$ cost per hour of work","choices":["$$x=the$$ cost per hour of work","$$x=the$$ $$one-time$$ charge","$$y=the$$ cost per hour of work","$$y=the$$ total cost for the work"],"hints":{"DefaultPathway":[{"id":"ac7990clinear13a-h1","type":"hint","dependencies":[],"title":"Interpret","text":"The independent variable is the $$x$$. Interpret what $$x$$ could be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["ac7990clinear13a-h1"],"title":"Coefficient","text":"What number is in front of the $$x$$ in $$y=25+20x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear13a-h3","type":"hint","dependencies":["ac7990clinear13a-h2"],"title":"Meaning","text":"The $$20$$ represents the cost per hour of work. Therefore, $$x$$ would represent this cost.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac7990clinear14","title":"Linear Equations","body":"Ethan repairs household appliances like dishwashers and refrigerators. For each visit, he charges $25 plus $20 per hour of work. A linear equation that expresses the total amount of money Ethan earns per visit is $$y=25+20x$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.1 Linear Equations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac7990clinear14a","stepAnswer":["$$y=the$$ total cost for the work"],"problemType":"MultipleChoice","stepTitle":"What is the dependent variable?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=the$$ total cost for the work","choices":["$$x=the$$ cost per hour of work","$$x=the$$ $$one-time$$ charge","$$y=the$$ cost per hour of work","$$y=the$$ total cost for the work"],"hints":{"DefaultPathway":[{"id":"ac7990clinear14a-h1","type":"hint","dependencies":[],"title":"Interpret","text":"The dependent variable is the $$y$$. Interpret what $$y$$ could be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear14a-h2","type":"hint","dependencies":["ac7990clinear14a-h1"],"title":"Meaning","text":"$$y=25+20x$$. This mean we are adding the one-time charge and the cost of the work.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear14a-h3","type":"hint","dependencies":["ac7990clinear14a-h2"],"title":"Meaning","text":"The one-time charge and the cost of the work is the total cost of the work.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac7990clinear15","title":"Linear Equations","body":"Ethan repairs household appliances like dishwashers and refrigerators. For each visit, he charges $25 plus $20 per hour of work. A linear equation that expresses the total amount of money Ethan earns per visit is $$y=25+20x$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.1 Linear Equations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac7990clinear15a","stepAnswer":["$25. This represents the $$one-time$$ charge, which would be charged even for $$0$$ hours of work."],"problemType":"MultipleChoice","stepTitle":"What is the y-intercept? Interpret this.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$25. This represents the one-time charge, which would be charged even for $$0$$ hours of work.","choices":["$25. This represents the $$one-time$$ charge, which would be charged even for $$0$$ hours of work.","$25. This represents the charge per hour of work.","$20. This represents the $$one-time$$ charge, which would be charged even for $$0$$ hours of work.","$20. This represents the charge per hour of work."],"hints":{"DefaultPathway":[{"id":"ac7990clinear15a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"The y-intercept is when $$x=0$$. Substitute this into the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["ac7990clinear15a-h1"],"title":"Substitute","text":"$$y=25+\\\\operatorname{20}\\\\left(0\\\\right)$$. What is this equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear15a-h3","type":"hint","dependencies":["ac7990clinear15a-h2"],"title":"Meaning","text":"The $25 represents the one-time charge.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac7990clinear16","title":"Linear Equations","body":"Ethan repairs household appliances like dishwashers and refrigerators. For each visit, he charges $25 plus $20 per hour of work. A linear equation that expresses the total amount of money Ethan earns per visit is $$y=25+20x$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.1 Linear Equations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac7990clinear16a","stepAnswer":["$20. This represents the charge per hour of work."],"problemType":"MultipleChoice","stepTitle":"What is the slope? Interpret this.","stepBody":"","answerType":"string","variabilization":{},"choices":["$25. This represents the $$one-time$$ charge, which would be charged even for $$0$$ hours of work.","$25. This represents the charge per hour of work.","$20. This represents the $$one-time$$ charge, which would be charged even for $$0$$ hours of work.","$20. This represents the charge per hour of work."],"hints":{"DefaultPathway":[{"id":"ac7990clinear16a-h1","type":"hint","dependencies":[],"title":"Form","text":"In $$y=a+bx$$, $$b$$ is the slope.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["ac7990clinear16a-h1"],"title":"Find $$b$$","text":"What is $$b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear16a-h3","type":"hint","dependencies":["ac7990clinear16a-h2"],"title":"Meaning","text":"$20 is the cost per hour of work.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac7990clinear2","title":"Linear Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.1 Linear Equations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac7990clinear2a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Is $$y=3+2x$$ a linear equation?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ac7990clinear2a-h1","type":"hint","dependencies":[],"title":"Linear Equation","text":"A linear equation is in the format $$y=a+bx$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ac7990clinear2a-h1"],"title":"A value","text":"Can you identify the a value in $$y=3+2x$$? What is it?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ac7990clinear2a-h1"],"title":"B value","text":"Can you identify the $$b$$ value in $$y=3+2x$$? What is it?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear2a-h4","type":"hint","dependencies":["ac7990clinear2a-h2","ac7990clinear2a-h3"],"title":"Interpretation","text":"This means that $$y=3+2x$$ is in $$y=a+bx$$ format.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac7990clinear3","title":"Linear Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.1 Linear Equations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac7990clinear3a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Is $$y=-0.01+1.2x$$ a linear equation?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ac7990clinear3a-h1","type":"hint","dependencies":[],"title":"Linear Equation","text":"A linear equation is in the format $$y=a+bx$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-0.01$$"],"dependencies":["ac7990clinear3a-h1"],"title":"A value","text":"Can you identify the a value in $$y=-0.01+1.2x$$? What is it?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.2$$"],"dependencies":["ac7990clinear3a-h1"],"title":"B value","text":"Can you identify the $$b$$ value in $$y=-0.01+1.2x$$? What is it?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear3a-h4","type":"hint","dependencies":["ac7990clinear3a-h2","ac7990clinear3a-h3"],"title":"Interpretation","text":"This means that $$y=-0.01+1.2x$$ is in $$y=a+bx$$ format.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac7990clinear4","title":"Linear Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.1 Linear Equations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac7990clinear4a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Is $$y=-1+2x$$ a linear equation?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ac7990clinear4a-h1","type":"hint","dependencies":[],"title":"Linear Equation","text":"A linear equation is in the format $$y=a+bx$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["ac7990clinear4a-h1"],"title":"A value","text":"Can you identify the a value in $$y=-1+2x$$? What is it?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ac7990clinear4a-h1"],"title":"B value","text":"Can you identify the $$b$$ value in $$y=-1+2x$$? What is it?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear4a-h4","type":"hint","dependencies":["ac7990clinear4a-h2","ac7990clinear4a-h3"],"title":"Interpretation","text":"This means that $$y=-1+2x$$ is in $$y=a+bx$$ format.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac7990clinear5","title":"Linear Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.1 Linear Equations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac7990clinear5a","stepAnswer":["The graph is linear because it is a straight line."],"problemType":"MultipleChoice","stepTitle":"Prove that $$y=-1+2x$$ is a linear equation through graphing.","stepBody":"","answerType":"string","variabilization":{},"choices":["The graph is linear because it is a straight line.","The graph is linear because it is not a straight line."],"hints":{"DefaultPathway":[{"id":"ac7990clinear5a-h1","type":"hint","dependencies":[],"title":"Y-intercept","text":"When graphing $$y=-1+2x$$, the y-intercept would be $$(0,-1)$$ since that represents when $$x=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear5a-h2","type":"hint","dependencies":["ac7990clinear5a-h1"],"title":"Slope","text":"When graphing $$y=-1+2x$$, the slope would be $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear5a-h3","type":"hint","dependencies":["ac7990clinear5a-h2"],"title":"Line","text":"Examine if the line is straight or curvy.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear5a-h4","type":"hint","dependencies":["ac7990clinear5a-h3"],"title":"Meaning","text":"A straight line represents a linear relationship.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac7990clinear6","title":"Linear Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.1 Linear Equations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac7990clinear6a","stepAnswer":["The graph is linear because it is a straight line."],"problemType":"MultipleChoice","stepTitle":"Is this an example of a linear equation? Why or why not?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["No, this is not a linear equation since the line is curvy.","No, this is not a linear equation since the line is straight.","The graph is linear because it is a straight line.","Yes, this is a linear equation since the line is curvy.","Yes, this is a linear equation since the line is straight."],"hints":{"DefaultPathway":[{"id":"ac7990clinear6a-h1","type":"hint","dependencies":[],"title":"A straight line represents a linear relationship while a curvy line does not.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear6a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Straight"],"dependencies":["ac7990clinear6a-h1"],"title":"Is the line straight or curvy?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Straight","Curvy"]}]}}]},{"id":"ac7990clinear7","title":"Linear Equations","body":"Aaron\'s Word Processing Service (AWPS) does word processing. The rate for services is $32 per hour plus a $$\\\\$31.50$$ one-time charge. The total cost to a customer depends on the number of hours it takes to complete the job.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.1 Linear Equations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac7990clinear7a","stepAnswer":["$$y=31.5+32x$$"],"problemType":"MultipleChoice","stepTitle":"Find the equation that expresses the total cost in terms of the number of hours required to complete the job.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=31.5+32x$$","choices":["$$y=31.5+32x$$","$$y=32+31.5x$$","$$y=0.5+31.5x$$","$$y=0.5+32x$$"],"hints":{"DefaultPathway":[{"id":"ac7990clinear7a-h1","type":"hint","dependencies":[],"title":"Establish Variables","text":"First, establish your variables. Let\'s say $$x=the$$ number of hours it takes to get the job done and $$y=the$$ total cost to the customer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear7a-h2","type":"hint","dependencies":["ac7990clinear7a-h1"],"title":"Cost","text":"If we were to say that $$x$$ hours were taken for the job, the cost would be $32x.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear7a-h3","type":"hint","dependencies":["ac7990clinear7a-h2"],"title":"Equation","text":"In our $$y=a+bx$$ equation, $$bx=\\\\$32x$$. Now, find a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\$31.50$$"],"dependencies":["ac7990clinear7a-h3"],"title":"Find a","text":"What is our fixed cost?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear7a-h5","type":"hint","dependencies":["ac7990clinear7a-h4"],"title":"Put it Together","text":"This means our equation becomes $$y=31.5+32x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac7990clinear8","title":"Linear Equations","body":"Emma\u2019s Extreme Sports hires hang-gliding instructors and pays them a fee of $50 per class as well as $20 per student in the class. The total cost Emma pays depends on the number of students in a class.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.1 Linear Equations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac7990clinear8a","stepAnswer":["$$y=50+20x$$"],"problemType":"MultipleChoice","stepTitle":"Find the equation that expresses the total cost in terms of the number of students in a class.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=50+20x$$","choices":["$$y=50+20x$$","$$y=20+50x$$","$$y=30+20x$$","$$y=30+50x$$"],"hints":{"DefaultPathway":[{"id":"ac7990clinear8a-h1","type":"hint","dependencies":[],"title":"Establish Variables","text":"First, establish your variables. Let\'s say $$x=the$$ number of students in a class and $$y=the$$ total cost.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear8a-h2","type":"hint","dependencies":["ac7990clinear8a-h1"],"title":"Cost","text":"If we were to say that $$x$$ students are in a class, the cost would be $20x.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear8a-h3","type":"hint","dependencies":["ac7990clinear8a-h2"],"title":"Equation","text":"In our $$y=a+bx$$ equation, $$bx=\\\\$20x$$. Now, find a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\$50.00$$"],"dependencies":["ac7990clinear8a-h3"],"title":"Find a","text":"What is our fixed cost?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear8a-h5","type":"hint","dependencies":["ac7990clinear8a-h4"],"title":"Put it Together","text":"This means our equation becomes $$y=50+20x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac7990clinear9","title":"Linear Equations","body":"Svetlana tutors to make extra money for college. For each tutoring session, she charges a one-time fee of $25 plus $15 per hour of tutoring. A linear equation that expresses the total amount of money Svetlana earns for each session she tutors is $$y=25+15x$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.1 Linear Equations","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ac7990clinear9a","stepAnswer":["$$x=the$$ cost per hour of tutoring"],"problemType":"MultipleChoice","stepTitle":"What is the independent variable?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=the$$ cost per hour of tutoring","choices":["$$x=the$$ cost per hour of tutoring","$$x=the$$ $$one-time$$ fee","$$y=the$$ cost per hour of tutoring","$$y=the$$ total cost of each tutoring session"],"hints":{"DefaultPathway":[{"id":"ac7990clinear9a-h1","type":"hint","dependencies":[],"title":"Interpret","text":"The independent variable is the $$x$$. Interpret what $$x$$ could be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["ac7990clinear9a-h1"],"title":"Coefficient","text":"What number is in front of the $$x$$ in $$y=25+15x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac7990clinear9a-h3","type":"hint","dependencies":["ac7990clinear9a-h2"],"title":"Meaning","text":"The $$15$$ represents the cost per hour of tutoring. Therefore, $$x$$ would represent this cost.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac879b3graph1","title":"Relationship between the Solutions of an Equation and its Graph","body":"The graph of $$y=2x-3$$ is shown. For each ordered pair, decide whether the ordered pair is a solution to the equation.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Graph Linear Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac879b3graph1a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$(0,-3)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ac879b3graph1a-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Substitute the $$x-$$ and $$y-$$ values into the equation to check if the ordered pair is a solution to the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac879b3graph1a-h2","type":"hint","dependencies":["ac879b3graph1a-h1"],"title":"Application","text":"Substitute $$y$$ for $$-3$$ and $$x$$ for $$0$$. Are $$-3$$ and $$2(0)-3$$ equal? If so, the ordered pair is a solution. If not, the ordered pair is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ac879b3graph1b","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$(3,3)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ac879b3graph1b-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Substitute the $$x-$$ and $$y-$$ values into the equation to check if the ordered pair is a solution to the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac879b3graph1b-h2","type":"hint","dependencies":["ac879b3graph1b-h1"],"title":"Application","text":"Substitute $$y$$ for $$3$$ and $$x$$ for $$3$$. Are $$3$$ and $$2(3)-3$$ equal? If so, the ordered pair is a solution. If not, the ordered pair is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ac879b3graph1c","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$(2,-3)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ac879b3graph1c-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Substitute the $$x-$$ and $$y-$$ values into the equation to check if the ordered pair is a solution to the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac879b3graph1c-h2","type":"hint","dependencies":["ac879b3graph1c-h1"],"title":"Application","text":"Substitute $$y$$ for $$-3$$ and $$x$$ for $$2$$. Are $$-3$$ and $$2(2)-3$$ equal? If so, the ordered pair is a solution. If not, the ordered pair is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ac879b3graph1d","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$(-1,-5)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ac879b3graph1d-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Substitute the $$x-$$ and $$y-$$ values into the equation to check if the ordered pair is a solution to the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac879b3graph1d-h2","type":"hint","dependencies":["ac879b3graph1d-h1"],"title":"Application","text":"Substitute $$y$$ for $$-5$$ and $$x$$ for $$-1$$. Are $$-5$$ and $$2(-1)-3$$ equal? If so, the ordered pair is a solution. If not, the ordered pair is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac879b3graph10","title":"Determining the X- and Y-Intercepts to the Equation $$3x+y=1$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Graph Linear Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac879b3graph10a","stepAnswer":["$$\\\\frac{1}{3}$$"],"problemType":"TextBox","stepTitle":"What is the x-value for the x-intercept to the equation $$3x+y=1$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{3}$$","hints":{"DefaultPathway":[{"id":"ac879b3graph10a-h1","type":"hint","dependencies":[],"title":"Substitution","text":"To determine the x-intercept, set $$y$$ equal to $$0$$ and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac879b3graph10a-h2","type":"hint","dependencies":["ac879b3graph10a-h1"],"title":"Solve for $$x$$","text":"What should the value of $$x$$ be to set the equation \\"3x+(0)=1\\" true? Rearrange the equation to solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ac879b3graph10b","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"What is the y-value for the y-intercept to the equation $$3x+y=1$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"ac879b3graph10b-h1","type":"hint","dependencies":[],"title":"Substitution","text":"To determine the y-intercept, set $$x$$ equal to $$0$$ and solve for $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac879b3graph10b-h2","type":"hint","dependencies":["ac879b3graph10b-h1"],"title":"Solve for $$y$$","text":"What should the value of $$y$$ be to set the equation \\"3(0)+y=1\\" true? Rearrange the equation to solve for $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac879b3graph11","title":"Determining the X- and Y-Intercepts to the Equation $$4x+y=-3$$.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Graph Linear Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac879b3graph11a","stepAnswer":["$$\\\\frac{-3}{4}$$"],"problemType":"TextBox","stepTitle":"What is the x-value for the x-intercept to the equation $$4x+y=-3$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-3}{4}$$","hints":{"DefaultPathway":[{"id":"ac879b3graph11a-h1","type":"hint","dependencies":[],"title":"Substitution","text":"To determine the x-intercept, set $$y$$ equal to $$0$$ and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac879b3graph11a-h2","type":"hint","dependencies":["ac879b3graph11a-h1"],"title":"Solve for $$x$$","text":"What should the value of $$x$$ be to set the equation \\"4x+(0)=-3\\" true? Rearrange the equation to solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ac879b3graph11b","stepAnswer":["$$-3$$"],"problemType":"TextBox","stepTitle":"What is the y-value for the y-intercept to the equation $$4x+y=-3$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-3$$","hints":{"DefaultPathway":[{"id":"ac879b3graph11b-h1","type":"hint","dependencies":[],"title":"Substitution","text":"To determine the y-intercept, set $$x$$ equal to $$0$$ and solve for $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac879b3graph11b-h2","type":"hint","dependencies":["ac879b3graph11b-h1"],"title":"Solve for $$y$$","text":"What should the value of $$y$$ be to set the equation \\"4(0)+y=-3\\" true? Rearrange the equation to solve for $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac879b3graph12","title":"Determining the X- and Y-Intercepts to the Equation $$2x-4y=8$$.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Graph Linear Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac879b3graph12a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"What is the x-value for the x-intercept to the equation $$2x-4y=8$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"ac879b3graph12a-h1","type":"hint","dependencies":[],"title":"Substitution","text":"To determine the x-intercept, set $$y$$ equal to $$0$$ and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac879b3graph12a-h2","type":"hint","dependencies":["ac879b3graph12a-h1"],"title":"Solve for $$x$$","text":"What should the value of $$x$$ be to set the equation $$\\"2x-4(0)=8\\"$$ true? Rearrange the equation to solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ac879b3graph12b","stepAnswer":["$$-2$$"],"problemType":"TextBox","stepTitle":"What is the y-value for the y-intercept to the equation $$2x-4y=8$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-2$$","hints":{"DefaultPathway":[{"id":"ac879b3graph12b-h1","type":"hint","dependencies":[],"title":"Substitution","text":"To determine the y-intercept, set $$x$$ equal to $$0$$ and solve for $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac879b3graph12b-h2","type":"hint","dependencies":["ac879b3graph12b-h1"],"title":"Solve for $$y$$","text":"What should the value of $$y$$ be to set the equation $$\\"2(0)-4y=8\\"$$ true? Rearrange the equation to solve for $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac879b3graph13","title":"Identifying whether the expression $$x=2$$ represents a vertical or horizontal line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Graph Linear Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac879b3graph13a","stepAnswer":["Vertical"],"problemType":"MultipleChoice","stepTitle":"Is $$x=2$$ a vertical or horizontal line?","stepBody":"","answerType":"string","variabilization":{},"choices":["Vertical","Horizontal"],"hints":{"DefaultPathway":[{"id":"ac879b3graph13a-h1","type":"hint","dependencies":[],"title":"A vertical line is the graph of an equation of the form $$x=a$$. A horizontal line is the graph of an equation of the form $$y=b$$.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac879b3graph13a-h2","type":"hint","dependencies":["ac879b3graph13a-h1"],"title":"The expression $$x=2$$ does not contain the variable $$y$$. This equation cannot be a horizontal line.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac879b3graph14","title":"Identifying whether the expression $$y=-4$$ represents a vertical or horizontal line","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Graph Linear Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac879b3graph14a","stepAnswer":["Horizontal"],"problemType":"MultipleChoice","stepTitle":"Is $$y=-4$$ a vertical or horizontal line?","stepBody":"","answerType":"string","variabilization":{},"choices":["Vertical","Horizontal"],"hints":{"DefaultPathway":[{"id":"ac879b3graph14a-h1","type":"hint","dependencies":[],"title":"A vertical line is the graph of an equation of the form $$x=a$$. A horizontal line is the graph of an equation of the form $$y=b$$.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac879b3graph14a-h2","type":"hint","dependencies":["ac879b3graph14a-h1"],"title":"The expression $$y=-4$$ does not contain the variable $$x$$. This equation cannot be a vertical line.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac879b3graph15","title":"Identifying solutions to the expression $$y=\\\\frac{1}{3} x+2$$.","body":"The graph of $$y=\\\\frac{1}{3} x+2$$ is shown. For each ordered pair, decide whether the ordered pair is a solution to the equation.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Graph Linear Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac879b3graph15a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$(0,2)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ac879b3graph15a-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Substitute the $$x-$$ and $$y-$$ values into the equation to check if the ordered pair is a solution to the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac879b3graph15a-h2","type":"hint","dependencies":["ac879b3graph15a-h1"],"title":"Application","text":"Substitute $$y$$ for $$2$$ and $$x$$ for $$0$$. Are $$2$$ and $$\\\\frac{1}{3\\\\left(0\\\\right)}+2$$ equal? If so, the ordered pair is a solution. If not, the ordered pair is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ac879b3graph15b","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$(3,3)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ac879b3graph15b-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Substitute the $$x-$$ and $$y-$$ values into the equation to check if the ordered pair is a solution to the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac879b3graph15b-h2","type":"hint","dependencies":["ac879b3graph15b-h1"],"title":"Application","text":"Substitute $$y$$ for $$3$$ and $$x$$ for $$3$$. Are $$3$$ and $$\\\\frac{1}{3\\\\left(3\\\\right)}+2$$ equal? If so, the ordered pair is a solution. If not, the ordered pair is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ac879b3graph15c","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$(-3,2)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ac879b3graph15c-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Substitute the $$x-$$ and $$y-$$ values into the equation to check if the ordered pair is a solution to the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac879b3graph15c-h2","type":"hint","dependencies":["ac879b3graph15c-h1"],"title":"Application","text":"Substitute $$y$$ for $$2$$ and $$x$$ for $$-3$$. Are $$2$$ and 1/3(-3)+2 equal? If so, the ordered pair is a solution. If not, the ordered pair is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ac879b3graph15d","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$(-6,0)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ac879b3graph15d-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Substitute the $$x-$$ and $$y-$$ values into the equation to check if the ordered pair is a solution to the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac879b3graph15d-h2","type":"hint","dependencies":["ac879b3graph15d-h1"],"title":"Application","text":"Substitute $$y$$ for $$0$$ and $$x$$ for $$-6$$. Are $$0$$ and 1/3(-6)+2 equal? If so, the ordered pair is a solution. If not, the ordered pair is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac879b3graph2","title":"Identifying which Ordered Pairs are part of the Solution","body":"The graph of $$y=3x-1$$ is shown. For each ordered pair, decide whether the ordered pair is a solution to the equation.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Graph Linear Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac879b3graph2a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$(0,-1)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ac879b3graph2a-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Substitute the $$x-$$ and $$y-$$ values into the equation to check if the ordered pair is a solution to the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac879b3graph2a-h2","type":"hint","dependencies":["ac879b3graph2a-h1"],"title":"Application","text":"Substitute $$y$$ for $$-1$$ and $$x$$ for $$0$$. Are $$-1$$ and $$3(0)-1$$ equal? If so, the ordered pair is a solution. If not, the ordered pair is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ac879b3graph2b","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$(2,5)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ac879b3graph2b-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Substitute the $$x-$$ and $$y-$$ values into the equation to check if the ordered pair is a solution to the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac879b3graph2b-h2","type":"hint","dependencies":["ac879b3graph2b-h1"],"title":"Application","text":"Substitute $$y$$ for $$5$$ and $$x$$ for $$2$$. Are $$5$$ and $$3(2)-1$$ equal? If so, the ordered pair is a solution. If not, the ordered pair is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac879b3graph3","title":"More Practice Identifying Solutions to Graphs","body":"The graph of $$y=3x-1$$ is shown. For each ordered pair, decide whether the ordered pair is a solution to the equation.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Graph Linear Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac879b3graph3a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$(3,-1)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ac879b3graph3a-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Substitute the $$x-$$ and $$y-$$ values into the equation to check if the ordered pair is a solution to the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac879b3graph3a-h2","type":"hint","dependencies":["ac879b3graph3a-h1"],"title":"Application","text":"Substitute $$y$$ for $$-1$$ and $$x$$ for $$3$$. Are $$-1$$ and $$3(3)-1$$ equal? If so, the ordered pair is a solution. If not, the ordered pair is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ac879b3graph3b","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$(-1,-4)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ac879b3graph3b-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Substitute the $$x-$$ and $$y-$$ values into the equation to check if the ordered pair is a solution to the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac879b3graph3b-h2","type":"hint","dependencies":["ac879b3graph3b-h1"],"title":"Application","text":"Substitute $$y$$ for $$-4$$ and $$x$$ for $$-1$$. Are $$-4$$ and $$3(-1)-1$$ equal? If so, the ordered pair is a solution. If not, the ordered pair is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac879b3graph4","title":"Determining the X- and Y-Intercepts to an Equation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.2 Graph Linear Equations in Two Variables","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac879b3graph4a","stepAnswer":["$$\\\\frac{-1}{2}$$"],"problemType":"TextBox","stepTitle":"What is the x-value for the x-intercept to the equation $$y=2x+1$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-1}{2}$$","hints":{"DefaultPathway":[{"id":"ac879b3graph4a-h1","type":"hint","dependencies":[],"title":"Substitution","text":"To determine the x-intercept, set $$y$$ equal to $$0$$ and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac879b3graph4a-h2","type":"hint","dependencies":["ac879b3graph4a-h1"],"title":"Solve for $$x$$","text":"What should the value of $$x$$ be to set the equation \\"0=2x+1\\" true? Rearrange the equation to solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ac879b3graph4b","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"What is the y-value for the y-intercept to the equation $$y=2x+1$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"ac879b3graph4b-h1","type":"hint","dependencies":[],"title":"Substitution","text":"To determine the y-intercept, set $$x$$ equal to $$0$$ and solve for $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac879b3graph4b-h2","type":"hint","dependencies":["ac879b3graph4b-h1"],"title":"Solve for $$y$$","text":"What should the value of $$y$$ be to set the equation \\"y=2(0)+1\\" true? 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<CC BY 4.0>","lesson":"8.1  Simplify Expressions with Roots","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac9802fsimproots15a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt[3]{27}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"ac9802fsimproots15a-h1","type":"hint","dependencies":[],"title":"Identifying the Root","text":"How many times do we have to multiply a number by itself?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9802fsimproots15a-h2","type":"hint","dependencies":["ac9802fsimproots15a-h1"],"title":"Finding Factors","text":"What are some factors that can be multiplied to get to 27?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9802fsimproots17a-h2","type":"hint","dependencies":["ac9802fsimproots17a-h1"],"title":"Finding Factors","text":"What are some factors that can be multiplied to get to 243?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9802fsimproots17a-h3","type":"hint","dependencies":["ac9802fsimproots17a-h2"],"title":"Fifth Root","text":"A number multiplied by itself $$5$$ times is to the fifth power.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac9802fsimproots18","title":"Simplifying Expressions with Roots","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1  Simplify Expressions with Roots","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac9802fsimproots18a","stepAnswer":["$$|x|$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{x^2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$|x|$$","hints":{"DefaultPathway":[{"id":"ac9802fsimproots18a-h1","type":"hint","dependencies":[],"title":"Even or Odd Root?","text":"Is the root even or odd? 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root.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9802fsimproots25a-h5","type":"hint","dependencies":["ac9802fsimproots25a-h4"],"title":"Simplifying","text":"What is $$\\\\frac{12}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac9802fsimproots3","title":"Simplifying Expressions with Roots","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1  Simplify Expressions with Roots","courseName":"OpenStax: Intermediate 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9802fsimproots4a-h2","type":"hint","dependencies":["ac9802fsimproots4a-h1"],"title":"Finding Factors","text":"What are some factors that can be multiplied to get to 64?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9802fsimproots4a-h3","type":"hint","dependencies":["ac9802fsimproots4a-h2"],"title":"Squaring","text":"A number multiplied by itself is squared.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac9802fsimproots5","title":"Simplifying Expressions with Roots","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1  Simplify Expressions with Roots","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac9802fsimproots5a","stepAnswer":["$$10$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{100}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10$$","hints":{"DefaultPathway":[{"id":"ac9802fsimproots5a-h1","type":"hint","dependencies":[],"title":"Is this a real number?","text":"Is the inside of the square root a positive or negative number?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9802fsimproots5a-h2","type":"hint","dependencies":["ac9802fsimproots5a-h1"],"title":"Finding Factors","text":"What are some factors that can be multiplied to get to 100?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9802fsimproots5a-h3","type":"hint","dependencies":["ac9802fsimproots5a-h2"],"title":"Squaring","text":"A number multiplied by itself is squared.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac9802fsimproots6","title":"Simplifying Expressions with Roots","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1  Simplify Expressions with Roots","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac9802fsimproots6a","stepAnswer":["$$-11$$"],"problemType":"TextBox","stepTitle":"$$-\\\\sqrt{121}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-11$$","hints":{"DefaultPathway":[{"id":"ac9802fsimproots6a-h1","type":"hint","dependencies":[],"title":"Is this a real number?","text":"Is the inside of the square root a positive or negative number?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9802fsimproots6a-h2","type":"hint","dependencies":["ac9802fsimproots6a-h1"],"title":"Finding Factors","text":"What are some factors that can be multiplied to get to 121?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9802fsimproots6a-h3","type":"hint","dependencies":["ac9802fsimproots6a-h2"],"title":"Squaring","text":"A number multiplied by itself is squared.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac9802fsimproots7","title":"Simplifying Expressions with Roots","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1  Simplify Expressions with Roots","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac9802fsimproots7a","stepAnswer":["$$\\\\sqrt{-196}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{-196}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\sqrt{-196}$$","hints":{"DefaultPathway":[{"id":"ac9802fsimproots7a-h1","type":"hint","dependencies":[],"title":"Is this a real number?","text":"Is the inside of the square root a positive or negative number?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9802fsimproots7a-h2","type":"hint","dependencies":["ac9802fsimproots7a-h1"],"title":"Real Number","text":"Since it is an imaginary number, can it be further simplified?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac9802fsimproots8","title":"Simplifying Expressions with Roots","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1  Simplify Expressions with Roots","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac9802fsimproots8a","stepAnswer":["$$\\\\sqrt{-169}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{-169}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\sqrt{-169}$$","hints":{"DefaultPathway":[{"id":"ac9802fsimproots8a-h1","type":"hint","dependencies":[],"title":"Is this a real number?","text":"Is the inside of the square root a positive or negative number?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9802fsimproots8a-h2","type":"hint","dependencies":["ac9802fsimproots8a-h1"],"title":"Real Number","text":"Since it is an imaginary number, can it be further simplified?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac9802fsimproots9","title":"Simplifying Expressions with Roots","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1  Simplify Expressions with Roots","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac9802fsimproots9a","stepAnswer":["$$-9$$"],"problemType":"TextBox","stepTitle":"$$-\\\\sqrt{81}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-9$$","hints":{"DefaultPathway":[{"id":"ac9802fsimproots9a-h1","type":"hint","dependencies":[],"title":"Is this a real number?","text":"Is the inside of the square root a positive or negative number?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9802fsimproots9a-h2","type":"hint","dependencies":["ac9802fsimproots9a-h1"],"title":"Finding Factors","text":"What are some factors that can be multiplied to get to 81?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9802fsimproots9a-h3","type":"hint","dependencies":["ac9802fsimproots9a-h2"],"title":"Squaring","text":"A number multiplied by itself is squared.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac98d98matrices1","title":"Write the Augmented Matrix for a System of Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Solve Systems of Equations Using Matrices","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac98d98matrices1a","stepAnswer":["$$\\\\begin{bmatrix} 5 & -3 \\\\\\\\ -2 & 1 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"Write the system as an augmented matrix: $$5x-3y=-1$$, $$y=2x-2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 5 & -3 \\\\\\\\ -2 & 1 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"ac98d98matrices1a-h1","type":"hint","dependencies":[],"title":"Convert to Standard Form","text":"The second equation must be in standard form: $$-2x+y=-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac98d98matrices1a-h2","type":"hint","dependencies":["ac98d98matrices1a-h1"],"title":"Writing the Matrix","text":"Each equation goes in one row of the matrix. We now have: $$mat({5, -3}, ({-2, 1})$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac98d98matrices10","title":"Use Row Operations on a Matrix","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Solve Systems of Equations Using Matrices","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac98d98matrices10a","stepAnswer":["$$\\\\begin{bmatrix} 6 & -5 & 2 & 3 \\\\\\\\ 3 & -3 & 1 & -1 \\\\\\\\ 2 & 1 & -4 & 5 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"$$mat({6, -5, 2, 3}, {2, 1, -4, 5}, {3, -3, 1, -1})$$. Interchange rows $$2$$ and $$3$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 6 & -5 & 2 & 3 \\\\\\\\ 3 & -3 & 1 & -1 \\\\\\\\ 2 & 1 & -4 & 5 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"ac98d98matrices10a-h1","type":"hint","dependencies":[],"title":"We must interchange rows $$2$$ and $$3$$, as shown in the image:","text":"\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac98d98matrices11","title":"Use Row Operations on a Matrix","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Solve Systems of Equations Using Matrices","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac98d98matrices11a","stepAnswer":["$$\\\\begin{bmatrix} 6 & -5 & 2 & 3 \\\\\\\\ 10 & 5 & -20 & 25 \\\\\\\\ 3 & -3 & 1 & -1 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"$$mat({6, -5, 2, 3}, {2, 1, -4, 5}, {3, -3, 1, -1})$$. Multiply row $$2$$ by $$5$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 6 & -5 & 2 & 3 \\\\\\\\ 10 & 5 & -20 & 25 \\\\\\\\ 3 & -3 & 1 & -1 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"ac98d98matrices11a-h1","type":"hint","dependencies":[],"title":"We must multiply row $$2$$ by $$5$$, as shown in the image:","text":"\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac98d98matrices12","title":"Use Row Operations on a Matrix","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Solve Systems of Equations Using Matrices","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac98d98matrices12a","stepAnswer":["$$\\\\begin{bmatrix} 0 & 1 & 0 & 5 \\\\\\\\ 2 & 1 & -4 & 5 \\\\\\\\ 3 & -3 & 1 & -1 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"$$mat({6, -5, 2, 3}, {2, 1, -4, 5}, {3, -3, 1, -1})$$. Multiply row $$3$$ by $$-2$$ and add to row $$1$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 0 & 1 & 0 & 5 \\\\\\\\ 2 & 1 & -4 & 5 \\\\\\\\ 3 & -3 & 1 & -1 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"ac98d98matrices12a-h1","type":"hint","dependencies":[],"title":"We multiply row $$3$$ by $$-2$$ and add to row $$1$$, as shown in the image:","text":"\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac98d98matrices13","title":"Use Row Operations on a Matrix","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Solve Systems of Equations Using Matrices","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac98d98matrices13a","stepAnswer":["$$\\\\begin{bmatrix} -2 & 3 & 0 & -1 \\\\\\\\ 4 & -1 & -4 & 4 \\\\\\\\ 5 & -2 & -2 & -2 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"$$mat({5, -2, -2, -2}, {4, -1, -4, 4}, {-2, 3, 0, -1})$$. Interchange rows $$1$$ and $$3$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} -2 & 3 & 0 & -1 \\\\\\\\ 4 & -1 & -4 & 4 \\\\\\\\ 5 & -2 & -2 & -2 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"ac98d98matrices13a-h1","type":"hint","dependencies":[],"title":"We must interchange the first and third rows. Doing this, we get: mat({-2, $$3$$, $$0$$, -1}, {4, $$-1$$, $$-4$$, 4}, {5, $$-2$$, $$-2$$, -2})","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac98d98matrices14","title":"Use Row Operations on a Matrix","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Solve Systems of Equations Using Matrices","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac98d98matrices14a","stepAnswer":["$$\\\\begin{bmatrix} 5 & -2 & -2 & -2 \\\\\\\\ 4 & -1 & -4 & 4 \\\\\\\\ -6 & 9 & 0 & -3 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"$$mat({5, -2, -2, -2}, {4, -1, -4, 4}, {-2, 3, 0, -1})$$. Multiply row $$3$$ by $$3$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 5 & -2 & -2 & -2 \\\\\\\\ 4 & -1 & -4 & 4 \\\\\\\\ -6 & 9 & 0 & -3 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"ac98d98matrices14a-h1","type":"hint","dependencies":[],"title":"We can multiply all the elements in row $$3$$ by $$3$$, and leave all else the same. We get mat({5, $$-2$$, $$-2$$, -2}, {4, $$-1$$, $$-4$$, 4}, {-6, $$9$$, $$0$$, -3})","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac98d98matrices15","title":"Use Row Operations on a Matrix","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Solve Systems of Equations Using Matrices","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac98d98matrices15a","stepAnswer":["$$\\\\begin{bmatrix} 4 & -6 & -4 & -8 \\\\\\\\ 4 & 1 & -3 & 2 \\\\\\\\ 5 & 0 & 4 & -1 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"$$mat({2, -3, -2, -4}, {4, 1, -3, 2}, {5, 0, 4, -1})$$. Multiply row $$1$$ by $$2$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 4 & -6 & -4 & -8 \\\\\\\\ 4 & 1 & -3 & 2 \\\\\\\\ 5 & 0 & 4 & -1 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"ac98d98matrices15a-h1","type":"hint","dependencies":[],"title":"We can multiply all the elements in row $$1$$ by $$2$$, and leave all else the same. We get mat({4, $$-6$$, $$-4$$, -8}, {4, $$1$$, $$-3$$, 2}, {5, $$0$$, $$4$$, -1})","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac98d98matrices2","title":"Write the Augmented Matrix for a System of Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Solve Systems of Equations Using Matrices","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac98d98matrices2a","stepAnswer":["$$\\\\begin{bmatrix} 6 & -5 & 2 & 3 \\\\\\\\ 2 & 1 & -4 & 5 \\\\\\\\ 3 & -3 & 1 & -1 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"Write the system as an augmented matrix: $$6x-5y+2z=3$$, $$2x+y-4z=5$$, $$3x-3y+z=-1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 6 & -5 & 2 & 3 \\\\\\\\ 2 & 1 & -4 & 5 \\\\\\\\ 3 & -3 & 1 & -1 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"ac98d98matrices2a-h1","type":"hint","dependencies":[],"title":"Convert to Standard Form","text":"All equations are in standard form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac98d98matrices2a-h2","type":"hint","dependencies":["ac98d98matrices2a-h1"],"title":"Writing the Matrix","text":"We create a matrix such that one row has one equation. $$mat({6, -5, 2, 3}, {2, 1, -4, 5}, {3, -3, 1, -1})$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac98d98matrices3","title":"Write the Augmented Matrix for a System of Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Solve Systems of Equations Using Matrices","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac98d98matrices3a","stepAnswer":["$$\\\\begin{bmatrix} 11 & -9 & -5 \\\\\\\\ 7 & 5 & -1 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"Write the system as an augmented matrix: $$11x=-9y-5$$, $$7x+5y=-1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 11 & -9 & -5 \\\\\\\\ 7 & 5 & -1 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"ac98d98matrices3a-h1","type":"hint","dependencies":[],"title":"Convert to Standard Form","text":"We must convert the first equation into standard form: $$11x+9y=-5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac98d98matrices3a-h2","type":"hint","dependencies":["ac98d98matrices3a-h1"],"title":"Writing the Matrix","text":"We create a matrix such that one row has one equation. mat({11, $$-9$$, -5}, {7, $$5$$, -1}).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac98d98matrices4","title":"Write the Augmented Matrix for a System of Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Solve Systems of Equations Using Matrices","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac98d98matrices4a","stepAnswer":["$$\\\\begin{bmatrix} 5 & -3 & 2 & -5 \\\\\\\\ 2 & -1 & -1 & 4 \\\\\\\\ 3 & -2 & 2 & -8 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"Write the system as an augmented matrix: $$5x-3y+2z=-5$$, $$2x-y-z=4$$, $$3x-2y+2z=-7$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 5 & -3 & 2 & -5 \\\\\\\\ 2 & -1 & -1 & 4 \\\\\\\\ 3 & -2 & 2 & -8 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"ac98d98matrices4a-h1","type":"hint","dependencies":[],"title":"Convert to Standard Form","text":"All equations are in standard form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac98d98matrices4a-h2","type":"hint","dependencies":["ac98d98matrices4a-h1"],"title":"Writing the Matrix","text":"We create a matrix such that one row has one equation. mat({5, $$-3$$, $$2$$, -5}, {2, $$-1$$, $$-1$$, 4}, {3, $$-2$$, $$2$$, -8})","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac98d98matrices5","title":"Write the Systems of Equations for the Augmented Matrix","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Solve Systems of Equations Using Matrices","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac98d98matrices5a","stepAnswer":["4x-3y+3z=-1, x+2y-z=2, -2x-y+3z=-4"],"problemType":"TextBox","stepTitle":"Write the matrix as a system. Enter the equations in the order that they appear in the matrix. $$mat({4, -3, -3, -1}, {1, 2, -1, 2}, {-2, -1, 3, -4})$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$4x-3y+3z=-1$$, $$x+2y-z=2$$, $$-2x-y+3z=-4$$","hints":{"DefaultPathway":[{"id":"ac98d98matrices5a-h1","type":"hint","dependencies":[],"title":"Writing the Matrix","text":"Each row corresponds to an equation, and the fourth column is separated by the equal sign. We have: $$4x-3y+3z=-1$$, $$x+2y-z=2$$, $$-2x-y+3z=-4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac98d98matrices6","title":"Write the Augmented Matrix for a System of Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Solve Systems of Equations Using Matrices","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac98d98matrices6a","stepAnswer":["$$\\\\begin{bmatrix} 3 & 8 & -3 \\\\\\\\ 2 & 5 & -3 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"Write the system as an augmented matrix: $$3x+8y=-3$$, $$2x=-5y-3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 3 & 8 & -3 \\\\\\\\ 2 & 5 & -3 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"ac98d98matrices6a-h1","type":"hint","dependencies":[],"title":"Convert to Standard Form","text":"The second equation must be converted to standard form: $$2x+5y=-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac98d98matrices6a-h2","type":"hint","dependencies":["ac98d98matrices6a-h1"],"title":"Writing the Matrix","text":"We create a matrix such that one row has one equation. mat({3, $$8$$, -3}, {2, $$5$$, -3})","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac98d98matrices7","title":"Write the Augmented Matrix for a System of Equations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Solve Systems of Equations Using Matrices","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac98d98matrices7a","stepAnswer":["$$\\\\begin{bmatrix} 2 & -5 & 3 & 8 \\\\\\\\ 3 & -1 & 4 & 7 \\\\\\\\ 1 & 3 & 2 & -3 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"Write the system as an augmented matrix: $$2x-5y+3z=8$$, $$3x-y+4z=7$$, $$x+3y+2z=-3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 2 & -5 & 3 & 8 \\\\\\\\ 3 & -1 & 4 & 7 \\\\\\\\ 1 & 3 & 2 & -3 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"ac98d98matrices7a-h1","type":"hint","dependencies":[],"title":"Convert to Standard Form","text":"All equations are in standard form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac98d98matrices7a-h2","type":"hint","dependencies":["ac98d98matrices7a-h1"],"title":"Writing the Matrix","text":"We create a matrix such that one row has one equation. mat({2, $$-5$$, $$3$$, 8}, {3, $$-1$$, $$4$$, 7}, {1, $$3$$, $$2$$, -3})","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac98d98matrices8","title":"Write the Systems of Equations for the Augmented Matrix","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Solve Systems of Equations Using Matrices","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac98d98matrices8a","stepAnswer":["x-y+2z=3, 2x+y-2z=1, 4x-y+2z=0"],"problemType":"TextBox","stepTitle":"Write the matrix as a system. Enter the equations in the order that they appear in the matrix. $$mat({1, -2, 2, 3}, {2, 1, -2, 1}, {4, -1, 2, 0})$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x-y+2z=3$$, $$2x+y-2z=1$$, $$4x-y+2z=0$$","hints":{"DefaultPathway":[{"id":"ac98d98matrices8a-h1","type":"hint","dependencies":[],"title":"Writing the Matrix","text":"Each row corresponds to an equation, and the fourth column is separated by the equal sign. We have: $$x-y+2z=3$$, $$2x+y-2z=1$$, $$4x-y+2z=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac98d98matrices9","title":"Write the Systems of Equations for the Augmented Matrix","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Solve Systems of Equations Using Matrices","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac98d98matrices9a","stepAnswer":["x+y+z=4, 2x+3z-y=8, x+y-z=3"],"problemType":"TextBox","stepTitle":"Write the matrix as a system. Enter the equations in the order that they appear in the matrix. mat({1, $$1$$, $$1$$, 4}, {2, $$3$$, $$-1$$, 8}, {1, $$1$$, $$-1$$, 3})","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x+y+z=4$$, $$2x+3z-y=8$$, $$x+y-z=3$$","hints":{"DefaultPathway":[{"id":"ac98d98matrices9a-h1","type":"hint","dependencies":[],"title":"Writing the Matrix","text":"Each row corresponds to an equation, and the fourth column is separated by the equal sign. We have: $$x+y+z=4$$, $$2x+3z-y=8$$, $$x+y-z=3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac98d98systemMat1","title":"Writing the Augmented Matrix for a System of Equations","body":"Write the augmented matrix for the given system of equations.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Solve Systems of Equations Using Matrices","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac98d98systemMat1a","stepAnswer":["$$\\\\begin{bmatrix} 3 & -1 & -1 \\\\\\\\ 2 & 2 & 5 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"$$3x-y=-1$$, $$2y=2x+5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 3 & -1 & -1 \\\\\\\\ 2 & 2 & 5 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"ac98d98systemMat1a-h1","type":"hint","dependencies":[],"title":"Definition of an Augmented Matrix","text":"The augmented matrix displays the coefficients of the variables, and an additional column for the constants.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac98d98systemMat1a-h2","type":"hint","dependencies":["ac98d98systemMat1a-h1"],"title":"Writing the Coefficients","text":"Example: The corresponding entries in the matrix for an equation $$2x-9x=1$$ would be 2,-9 and $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac98d98systemMat10","title":"Writing the System of Equations","body":"Write the system of equations that corresponds to the augmented matrix. In the matrix, use $$x$$ for the first column, $$y$$ for the second, and $$z$$ for the third. Format your answer such as \\"3x+y=-2,x-7y=1\\" without the quotes. If a variable has a coefficient of zero, don\'t include it in the equation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Solve Systems of Equations Using Matrices","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac98d98systemMat10a","stepAnswer":["2x-4y=-2,3x-3y=-1"],"problemType":"TextBox","stepTitle":"$$\\\\begin{bmatrix} 2 & -4 & -2 \\\\\\\\ 3 & -3 & -1 \\\\end{bmatrix}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2x-4y=-2, 3x-3y=-1$$","hints":{"DefaultPathway":[{"id":"ac98d98systemMat10a-h1","type":"hint","dependencies":[],"title":"Definition of the Augmented Matrix","text":"The augmented matrix displays the coefficients of the variables, and an additional column for the constants.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac98d98systemMat10a-h2","type":"hint","dependencies":["ac98d98systemMat10a-h1"],"title":"Rightmost Column of an Augmented Matrix","text":"The rightmost columns of augmented matrices are constants. For example, in the equation $$2x+y=3$$, $$3$$ would be in the column at the very right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac98d98systemMat11","title":"Writing the System of Equations","body":"Write the system of equations that corresponds to the augmented matrix. In the matrix, use $$x$$ for the first column, $$y$$ for the second, and $$z$$ for the third. Format your answer such as \\"3x+y=-2,x-7y=1\\" without the quotes. If a variable has a coefficient of zero, don\'t include it in the equation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Solve Systems of Equations Using Matrices","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac98d98systemMat11a","stepAnswer":["x-3z=-1,x-2y=-2,-y+2z=3"],"problemType":"TextBox","stepTitle":"$$\\\\begin{bmatrix} 1 & 0 & -3 & -1 \\\\\\\\ 1 & -2 & 0 & -2 \\\\\\\\ 0 & -1 & 2 & 3 \\\\end{bmatrix}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x-3z=-1, x-2y=-2-y+2z=3$$","hints":{"DefaultPathway":[{"id":"ac98d98systemMat11a-h1","type":"hint","dependencies":[],"title":"Definition of the Augmented Matrix","text":"The augmented matrix displays the coefficients of the variables, and an additional column for the constants.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac98d98systemMat11a-h2","type":"hint","dependencies":["ac98d98systemMat11a-h1"],"title":"Rightmost Column of an Augmented Matrix","text":"The rightmost columns of augmented matrices are constants. For example, in the equation $$2x+y=3$$, $$3$$ would be in the column at the very right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac98d98systemMat12","title":"Writing the System of Equations","body":"Write the system of equations that corresponds to the augmented matrix. In the matrix, use $$x$$ for the first column, $$y$$ for the second, and $$z$$ for the third. Format your answer such as \\"3x+y=-2,x-7y=1\\" without the quotes. If a variable has a coefficient of zero, don\'t include it in the equation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Solve Systems of Equations Using Matrices","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac98d98systemMat12a","stepAnswer":["2x-2y=-1,2y-z=2,3x-z=-2"],"problemType":"TextBox","stepTitle":"$$\\\\begin{bmatrix} 2 & -2 & 0 & -1 \\\\\\\\ 0 & 2 & -1 & 2 \\\\\\\\ 3 & 0 & -1 & -2 \\\\end{bmatrix}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2x-2y=-1, 2y-z=2, 3x-z=-2$$","hints":{"DefaultPathway":[{"id":"ac98d98systemMat12a-h1","type":"hint","dependencies":[],"title":"Definition of the Augmented Matrix","text":"The augmented matrix displays the coefficients of the variables, and an additional column for the constants.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac98d98systemMat12a-h2","type":"hint","dependencies":["ac98d98systemMat12a-h1"],"title":"Rightmost Column of an Augmented Matrix","text":"The rightmost columns of augmented matrices are constants. For example, in the equation $$2x+y=3$$, $$3$$ would be in the column at the very right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac98d98systemMat13","title":"Using Row Operations on a Matrix","body":"Perform the indicated operations on the augmented matrix $$\\\\begin{bmatrix} 6 & -4 & 3 \\\\\\\\ 3 & -2 & 1 \\\\end{bmatrix}$$","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Solve Systems of Equations Using Matrices","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac98d98systemMat13a","stepAnswer":["$$\\\\begin{bmatrix} 3 & -2 & 1 \\\\\\\\ 6 & -4 & 3 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"Interchange rows $$1$$ and $$2$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 3 & -2 & 1 \\\\\\\\ 6 & -4 & 3 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"ac98d98systemMat13a-h1","type":"hint","dependencies":[],"title":"Interchanging Rows in a Matrix","text":"Swap rows $$1$$ and $$2$$ with each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ac98d98systemMat13b","stepAnswer":["$$\\\\begin{bmatrix} 6 & -4 & 3 \\\\\\\\ 9 & -6 & 3 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"Multiply row $$2$$ by $$3$$. Perform this on the original matrix.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 6 & -4 & 3 \\\\\\\\ 9 & -6 & 3 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"ac98d98systemMat13b-h1","type":"hint","dependencies":[],"title":"Multiplying the Row","text":"(Do this on the original matrix). Multiply each entry of row $$2$$ by $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ac98d98systemMat13c","stepAnswer":["$$\\\\begin{bmatrix} 6 & -4 & 3 \\\\\\\\ 0 & 0 & 1 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"Multiply row $$2$$ by $$-2$$ and add row $$1$$ to it. Perform this on the original matrix.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 6 & -4 & 3 \\\\\\\\ 0 & 0 & 1 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"ac98d98systemMat13c-h1","type":"hint","dependencies":[],"title":"First Step","text":"First, multiply each entry of row $$2$$ by $$-2$$. The result is $$\\\\begin{bmatrix} 6 & -4 & 3 \\\\\\\\ -6 & 4 & -2 \\\\end{bmatrix}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac98d98systemMat13c-h2","type":"hint","dependencies":["ac98d98systemMat13c-h1"],"title":"Second Step","text":"Then, add each entry of row $$1$$ to the entry in the same column of row $$2$$. (Row $$1$$, meanwhile, remains unchanged.) The answer is $$\\\\begin{bmatrix} 6 & -4 & 3 \\\\\\\\ 0 & 0 & 1 \\\\end{bmatrix}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac98d98systemMat14","title":"Using Row Operations on a Matrix","body":"Perform the indicated operations on the augmented matrix $$\\\\begin{bmatrix} 4 & -6 & -3 \\\\\\\\ 3 & 2 & 1 \\\\end{bmatrix}$$","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Solve Systems of Equations Using Matrices","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac98d98systemMat14a","stepAnswer":["$$\\\\begin{bmatrix} 3 & 2 & 1 \\\\\\\\ 4 & -6 & -3 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"Interchange rows $$1$$ and $$2$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 3 & 2 & 1 \\\\\\\\ 4 & -6 & -3 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"ac98d98systemMat14a-h1","type":"hint","dependencies":[],"title":"Interchanging Rows in a Matrix","text":"Swap rows $$1$$ and $$2$$ with each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ac98d98systemMat14b","stepAnswer":["$$\\\\begin{bmatrix} 16 & -24 & -12 \\\\\\\\ 3 & 2 & 1 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"Multiply row $$1$$ by $$4$$. Perform this on the original matrix.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 16 & -24 & -12 \\\\\\\\ 3 & 2 & 1 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"ac98d98systemMat14b-h1","type":"hint","dependencies":[],"title":"Multiplying the Row","text":"(Do this on the original matrix). Multiply each entry of row $$2$$ by $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ac98d98systemMat14c","stepAnswer":["$$\\\\begin{bmatrix} 4 & -6 & -3 \\\\\\\\ 13 & 0 & 0 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"Multiply row $$2$$ by $$3$$ and add row $$1$$ to it. Perform this on the original matrix.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 4 & -6 & -3 \\\\\\\\ 13 & 0 & 0 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"ac98d98systemMat14c-h1","type":"hint","dependencies":[],"title":"First Step","text":"First, multiply each entry of row $$2$$ by $$3$$. The result is $$\\\\begin{bmatrix} 4 & -6 & -3 \\\\\\\\ 9 & 6 & 3 \\\\end{bmatrix}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac98d98systemMat14c-h2","type":"hint","dependencies":["ac98d98systemMat14c-h1"],"title":"Second Step","text":"Then, add each entry of row $$1$$ to the entry in the same column of row $$2$$. (Row $$1$$, meanwhile, remains unchanged.) The answer is $$\\\\begin{bmatrix} 4 & -6 & -3 \\\\\\\\ 13 & 0 & 0 \\\\end{bmatrix}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac98d98systemMat15","title":"Using Row Operations on a Matrix","body":"Perform the indicated operations on the augmented matrix $$\\\\begin{bmatrix} 4 & -12 & -8 & 16 \\\\\\\\ 4 & -2 & -3 & -1 \\\\\\\\ -6 & 2 & -1 & -1 \\\\end{bmatrix}$$","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Solve Systems of Equations Using Matrices","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac98d98systemMat15a","stepAnswer":["$$\\\\begin{bmatrix} 4 & -2 & -3 & -1 \\\\\\\\ 4 & -12 & -8 & 16 \\\\\\\\ -6 & 2 & -1 & -1 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"Interchange rows $$1$$ and $$2$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 4 & -2 & -3 & -1 \\\\\\\\ 4 & -12 & -8 & 16 \\\\\\\\ -6 & 2 & -1 & -1 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"ac98d98systemMat15a-h1","type":"hint","dependencies":[],"title":"Interchanging Rows in a Matrix","text":"Swap rows $$1$$ and $$2$$ with each other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ac98d98systemMat15b","stepAnswer":["$$\\\\begin{bmatrix} 16 & -48 & -32 & 64 \\\\\\\\ 4 & -2 & -3 & -1 \\\\\\\\ -6 & 2 & -1 & -1 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"Multiply row $$1$$ by $$4$$. Perform this on the original matrix.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 16 & -48 & -32 & 64 \\\\\\\\ 4 & -2 & -3 & -1 \\\\\\\\ -6 & 2 & -1 & -1 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"ac98d98systemMat15b-h1","type":"hint","dependencies":[],"title":"Multiplying the Row","text":"(Do this on the original matrix). Multiply each entry of row $$2$$ by $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ac98d98systemMat15c","stepAnswer":["$$\\\\begin{bmatrix} 4 & -12 & -8 & 16 \\\\\\\\ -4 & -8 & -2 & 18 \\\\\\\\ -6 & 2 & -1 & -1 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"Multiply row $$2$$ by $$-2$$ and add row $$1$$ to it. Perform this on the original matrix.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 4 & -12 & -8 & 16 \\\\\\\\ -4 & -8 & -2 & 18 \\\\\\\\ -6 & 2 & -1 & -1 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"ac98d98systemMat15c-h1","type":"hint","dependencies":[],"title":"First Step","text":"First, multiply each entry of row $$2$$ by $$-2$$. The result is $$\\\\begin{bmatrix} 4 & -12 & -8 & 16 \\\\\\\\ -8 & 4 & 6 & 2 \\\\\\\\ -6 & 2 & -1 & -1 \\\\end{bmatrix}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac98d98systemMat15c-h2","type":"hint","dependencies":["ac98d98systemMat15c-h1"],"title":"Second Step","text":"Then, add each entry of row $$1$$ to the entry in the same column of row $$2$$. (Row $$1$$, meanwhile, remains unchanged.) The answer is $$\\\\begin{bmatrix} 4 & -12 & -8 & 16 \\\\\\\\ -4 & -8 & -2 & 18 \\\\\\\\ -6 & 2 & -1 & -1 \\\\end{bmatrix}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac98d98systemMat2","title":"Writing the Augmented Matrix for a System of Equations","body":"Write the augmented matrix for the given system of equations.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Solve Systems of Equations Using Matrices","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac98d98systemMat2a","stepAnswer":["$$\\\\begin{bmatrix} 4 & 3 & 0 & -2 \\\\\\\\ 1 & -2 & -3 & 7 \\\\\\\\ 2 & -1 & 2 & 6 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"4x+3y=-2,x-2y-3z=7,2x-y+2z=-6","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 4 & 3 & 0 & -2 \\\\\\\\ 1 & -2 & -3 & 7 \\\\\\\\ 2 & -1 & 2 & 6 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"ac98d98systemMat2a-h1","type":"hint","dependencies":[],"title":"Definition of an Augmented Matrix","text":"The augmented matrix displays the coefficients of the variables, and an additional column for the constants.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac98d98systemMat2a-h2","type":"hint","dependencies":["ac98d98systemMat2a-h1"],"title":"Writing the Coefficients","text":"Example: The corresponding entries in the matrix for an equation $$2x-9y=1$$ would be 2,-9 and $$1$$ (This is if $$x$$ and $$y$$ were the only variables in the system. If there was another variable, for example $$z$$, then $$z$$ would have a coefficient of $$0$$ and its coefficient would be written to the right of the column of answers.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac98d98systemMat3","title":"Writing the Augmented Matrix for a System of Equations","body":"Write the augmented matrix for the given system of equations.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Solve Systems of Equations Using Matrices","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac98d98systemMat3a","stepAnswer":["$$\\\\begin{bmatrix} 2 & 4 & -5 \\\\\\\\ 3 & -2 & 2 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"$$2x+4y=-5, 3x-2y=2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 2 & 4 & -5 \\\\\\\\ 3 & -2 & 2 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"ac98d98systemMat3a-h1","type":"hint","dependencies":[],"title":"Definition of an Augmented Matrix","text":"The augmented matrix displays the coefficients of the variables, and an additional column for the constants.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac98d98systemMat3a-h2","type":"hint","dependencies":["ac98d98systemMat3a-h1"],"title":"Writing the Coefficients","text":"Example: The corresponding entries in the matrix for an equation $$2x-9y=1$$ would be 2,-9 and $$1$$ (This is if $$x$$ and $$y$$ were the only variables in the system. If there was another variable, for example $$z$$, then $$z$$ would have a coefficient of $$0$$ and its coefficient would be written to the right of the column of answers.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac98d98systemMat4","title":"Writing the Augmented Matrix for a System of Equations","body":"Write the augmented matrix for the given system of equations.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Solve Systems of Equations Using Matrices","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac98d98systemMat4a","stepAnswer":["$$\\\\begin{bmatrix} 3 & -2 & -1 & -2 \\\\\\\\ -2 & 1 & 0 & 5 \\\\\\\\ 5 & 4 & 1 & -1 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"3x-2y-z=-2,-2x+y=5,5x+4y+z=-1","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 3 & -2 & -1 & -2 \\\\\\\\ -2 & 1 & 0 & 5 \\\\\\\\ 5 & 4 & 1 & -1 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"ac98d98systemMat4a-h1","type":"hint","dependencies":[],"title":"Definition of an Augmented Matrix","text":"The augmented matrix displays the coefficients of the variables, and an additional column for the constants.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac98d98systemMat4a-h2","type":"hint","dependencies":["ac98d98systemMat4a-h1"],"title":"Writing the Coefficients","text":"Example: The corresponding entries in the matrix for an equation $$2x-9y=1$$ would be 2,-9 and $$1$$ (This is if $$x$$ and $$y$$ were the only variables in the system. If there was another variable, for example $$z$$, then $$z$$ would have a coefficient of $$0$$ and its coefficient would be written to the right of the column of answers.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac98d98systemMat5","title":"Writing the Augmented Matrix for a System of Equations","body":"Write the augmented matrix for the given system of equations.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Solve Systems of Equations Using Matrices","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac98d98systemMat5a","stepAnswer":["$$\\\\begin{bmatrix} 3 & -1 & -4 \\\\\\\\ 2 & -1 & 2 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"$$3x-y=-4, 2=y+2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 3 & -1 & -4 \\\\\\\\ 2 & -1 & 2 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"ac98d98systemMat5a-h1","type":"hint","dependencies":[],"title":"Definition of an Augmented Matrix","text":"The augmented matrix displays the coefficients of the variables, and an additional column for the constants.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac98d98systemMat5a-h2","type":"hint","dependencies":["ac98d98systemMat5a-h1"],"title":"Writing the Coefficients","text":"Example: The corresponding entries in the matrix for an equation $$2x-9y=1$$ would be 2,-9 and $$1$$ (This is if $$x$$ and $$y$$ were the only variables in the system. If there was another variable, for example $$z$$, then $$z$$ would have a coefficient of $$0$$ and its coefficient would be written to the right of the column of answers.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac98d98systemMat6","title":"Writing the Augmented Matrix for a System of Equations","body":"Write the augmented matrix for the given system of equations.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Solve Systems of Equations Using Matrices","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac98d98systemMat6a","stepAnswer":["$$\\\\begin{bmatrix} 1 & -3 & -4 & -2 \\\\\\\\ 4 & 2 & 2 & 5 \\\\\\\\ 2 & -5 & 7 & -8 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"x-3y-4z=-2,4x+2y+2z=5,2x-5y+7z=-8","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 1 & -3 & -4 & -2 \\\\\\\\ 4 & 2 & 2 & 5 \\\\\\\\ 2 & -5 & 7 & -8 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"ac98d98systemMat6a-h1","type":"hint","dependencies":[],"title":"Definition of an Augmented Matrix","text":"The augmented matrix displays the coefficients of the variables, and an additional column for the constants.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac98d98systemMat6a-h2","type":"hint","dependencies":["ac98d98systemMat6a-h1"],"title":"Writing the Coefficients","text":"Example: The corresponding entries in the matrix for an equation $$2x-9y=1$$ would be 2,-9 and $$1$$ (This is if $$x$$ and $$y$$ were the only variables in the system. If there was another variable, for example $$z$$, then $$z$$ would have a coefficient of $$0$$ and its coefficient would be written to the right of the column of answers.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac98d98systemMat7","title":"Writing the Augmented Matrix for a System of Equations","body":"Write the augmented matrix for the given system of equations.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Solve Systems of Equations Using Matrices","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac98d98systemMat7a","stepAnswer":["$$\\\\begin{bmatrix} 2 & -5 & -3 \\\\\\\\ 4 & -3 & -1 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"$$2x-5y=-3, 4x=3y-1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 2 & -5 & -3 \\\\\\\\ 4 & -3 & -1 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"ac98d98systemMat7a-h1","type":"hint","dependencies":[],"title":"Definition of an Augmented Matrix","text":"The augmented matrix displays the coefficients of the variables, and an additional column for the constants.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac98d98systemMat7a-h2","type":"hint","dependencies":["ac98d98systemMat7a-h1"],"title":"Writing the Coefficients","text":"Example: The corresponding entries in the matrix for an equation $$2x-9y=1$$ would be 2,-9 and $$1$$ (This is if $$x$$ and $$y$$ were the only variables in the system. If there was another variable, for example $$z$$, then $$z$$ would have a coefficient of $$0$$ and its coefficient would be written to the right of the column of answers.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac98d98systemMat8","title":"Writing the Augmented Matrix for a System of Equations","body":"Write the augmented matrix for the given system of equations.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Solve Systems of Equations Using Matrices","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac98d98systemMat8a","stepAnswer":["$$\\\\begin{bmatrix} 4 & 3 & -2 & -3 \\\\\\\\ -2 & 1 & -3 & 4 \\\\\\\\ -1 & -4 & 5 & -2 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"$$4x+3y-2z=-3-2x+y-3z=4-x-4y+5z=-2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 4 & 3 & -2 & -3 \\\\\\\\ -2 & 1 & -3 & 4 \\\\\\\\ -1 & -4 & 5 & -2 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"ac98d98systemMat8a-h1","type":"hint","dependencies":[],"title":"Definition of an Augmented Matrix","text":"The augmented matrix displays the coefficients of the variables, and an additional column for the constants.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac98d98systemMat8a-h2","type":"hint","dependencies":["ac98d98systemMat8a-h1"],"title":"Writing the Coefficients","text":"Example: The corresponding entries in the matrix for an equation $$2x-9y=1$$ would be 2,-9 and $$1$$ (This is if $$x$$ and $$y$$ were the only variables in the system. If there was another variable, for example $$z$$, then $$z$$ would have a coefficient of $$0$$ and its coefficient would be written to the right of the column of answers.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac98d98systemMat9","title":"Writing the System of Equations","body":"Write the system of equations that corresponds to the augmented matrix. In the matrix, use $$x$$ for the first column, $$y$$ for the second, and $$z$$ for the third. Format your answer such as \\"3x+y=-2,x-7y=1\\" without the quotes. If a variable has a coefficient of zero, don\'t include it in the equation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.5 Solve Systems of Equations Using Matrices","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ac98d98systemMat9a","stepAnswer":["2x-y=4,x-3=2"],"problemType":"TextBox","stepTitle":"$$\\\\begin{bmatrix} 2 & -1 & 4 \\\\\\\\ 1 & -3 & 2 \\\\end{bmatrix}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2x-y=4, x-3=2$$","hints":{"DefaultPathway":[{"id":"ac98d98systemMat9a-h1","type":"hint","dependencies":[],"title":"Definition of the Augmented Matrix","text":"The augmented matrix displays the coefficients of the variables, and an additional column for the constants.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac98d98systemMat9a-h2","type":"hint","dependencies":["ac98d98systemMat9a-h1"],"title":"Rightmost Column of an Augmented Matrix","text":"The rightmost columns of augmented matrices are constants. For example, in the equation $$2x+y=3$$, $$3$$ would be in the column at the very right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac9c764addand1","title":"Add Fractions with a Common Denominator","body":"Find the sum:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.6 Add and Subtract Fractions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac9c764addand1a","stepAnswer":["$$\\\\frac{x+2}{3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{x}{3}+\\\\frac{2}{3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{x+2}{3}$$","hints":{"DefaultPathway":[{"id":"ac9c764addand1a-h1","type":"hint","dependencies":[],"title":"Add","text":"Add the numerators and place the sum over the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+2$$"],"dependencies":["ac9c764addand1a-h1"],"title":"Numerator","text":"What do we get for the numerator after adding the two together?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{x+2}{3}$$"],"dependencies":["ac9c764addand1a-h2"],"title":"Final Answer","text":"What do we get after placing the sum over the common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac9c764addand10","title":"How to Simplify Complex Fractions","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.6 Add and Subtract Fractions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac9c764addand10a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\frac{1}{2}+\\\\frac{2}{3}}{\\\\frac{3}{4}-\\\\frac{1}{6}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"ac9c764addand10a-h1","type":"hint","dependencies":[],"title":"Numerator","text":"We first need to simplify the numerator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["ac9c764addand10a-h1"],"title":"Numerator","text":"What is the LCD of $$2$$ and 3?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{6}$$"],"dependencies":["ac9c764addand10a-h2"],"title":"Numerator","text":"What should $$\\\\frac{1}{2}$$ be changed into with the LCD, 6?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{4}{6}$$"],"dependencies":["ac9c764addand10a-h3"],"title":"Numerator","text":"What should $$\\\\frac{2}{3}$$ be changed into with the LCD, 6?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{7}{6}$$"],"dependencies":["ac9c764addand10a-h4"],"title":"Numerator","text":"What do we get after simplifying the numerator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand10a-h6","type":"hint","dependencies":["ac9c764addand10a-h5"],"title":"Denominator","text":"We then need to simplify the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand10a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["ac9c764addand10a-h6"],"title":"Denominator","text":"What is the LCD of $$4$$ and 6?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand10a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{9}{12}$$"],"dependencies":["ac9c764addand10a-h7"],"title":"Denominator","text":"What should $$\\\\frac{3}{4}$$ be changed into with the LCD, 12?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand10a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{12}$$"],"dependencies":["ac9c764addand10a-h8"],"title":"Denominator","text":"What should $$\\\\frac{1}{6}$$ be changed into with the LCD, 12?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand10a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{7}{12}$$"],"dependencies":["ac9c764addand10a-h9"],"title":"Denominator","text":"What do we get after simplifying the denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand10a-h11","type":"hint","dependencies":["ac9c764addand10a-h10"],"title":"Divide","text":"Finally, we need to divide the numerator by the denominator. Simplify if possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand10a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ac9c764addand10a-h11"],"title":"Divide","text":"What do we get after the division?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac9c764addand11","title":"Evaluate Variable Expressions with Fractions","body":"Evaluate","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.6 Add and Subtract Fractions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac9c764addand11a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"$$x+\\\\frac{1}{3}$$ when $$x=\\\\frac{-1}{3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"ac9c764addand11a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$\\\\frac{-1}{3}$$ for $$x$$ in the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["ac9c764addand11a-h1"],"title":"Substitute","text":"What do we get for $$\\\\frac{-1}{3}+\\\\frac{1}{3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac9c764addand12","title":"Evaluate Variable Expressions with Fractions","body":"Evaluate","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.6 Add and Subtract Fractions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac9c764addand12a","stepAnswer":["$$\\\\frac{-1}{6}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{-5}{6}-y$$ when $$y=\\\\frac{-2}{3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-1}{6}$$","hints":{"DefaultPathway":[{"id":"ac9c764addand12a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$\\\\frac{-2}{3}$$ for $$y$$ in the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand12a-h2","type":"hint","dependencies":["ac9c764addand12a-h1"],"title":"Substitute","text":"We get $$\\\\frac{-5}{6}+\\\\frac{2}{3}$$ after the substitution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand12a-h3","type":"hint","dependencies":["ac9c764addand12a-h2"],"title":"LCD","text":"We then need to rewrite each fraction with the least common denominator because they don\'t have a common denominator to start with.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["ac9c764addand12a-h3"],"title":"LCD","text":"What is the LCD of $$3$$ and 6?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand12a-h5","type":"hint","dependencies":["ac9c764addand12a-h4"],"title":"Rewrite","text":"We then need to change into equivalent fractions with the LCD, $$6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand12a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-5}{6}$$"],"dependencies":["ac9c764addand12a-h5"],"title":"Rewrite","text":"What should $$\\\\frac{-5}{6}$$ be changed into with the LCD, 6?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand12a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{4}{6}$$"],"dependencies":["ac9c764addand12a-h6"],"title":"Rewrite","text":"What should $$\\\\frac{2}{3}$$ be changed into with the LCD, 6?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand12a-h8","type":"hint","dependencies":["ac9c764addand12a-h7"],"title":"Add","text":"Finally, we need to add the two fractions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand12a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{6}$$"],"dependencies":["ac9c764addand12a-h8"],"title":"Add","text":"What do we get after the addition?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac9c764addand13","title":"Evaluate Variable Expressions with Fractions","body":"Evaluate","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.6 Add and Subtract Fractions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac9c764addand13a","stepAnswer":["$$\\\\frac{-1}{12}$$"],"problemType":"TextBox","stepTitle":"$$2x^2 y$$ when $$x=\\\\frac{1}{4}$$ and $$y=\\\\frac{-2}{3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-1}{12}$$","hints":{"DefaultPathway":[{"id":"ac9c764addand13a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$\\\\frac{1}{4}$$ for $$x$$ and $$\\\\frac{-2}{3}$$ for $$y$$ in the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand13a-h2","type":"hint","dependencies":["ac9c764addand13a-h1"],"title":"Substitute","text":"We get $$2{\\\\left(\\\\frac{1}{4}\\\\right)}^2 \\\\left(-\\\\frac{2}{3}\\\\right)$$ after the substitution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand13a-h3","type":"hint","dependencies":["ac9c764addand13a-h2"],"title":"Exponents","text":"Simplify exponents first.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{16}$$"],"dependencies":["ac9c764addand13a-h3"],"title":"Exponents","text":"What do we get for $${\\\\left(\\\\frac{1}{4}\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand13a-h5","type":"hint","dependencies":["ac9c764addand13a-h4"],"title":"Multiply","text":"Multiply. Divide out the common factors. Notice we write $$16$$ as $$2\\\\times2\\\\times4$$ to make it easy to remove common factors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand13a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{12}$$"],"dependencies":["ac9c764addand13a-h5"],"title":"Multiply","text":"What do we get after the multiplication?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac9c764addand14","title":"Evaluate Variable Expressions with Fractions","body":"Evaluate","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.6 Add and Subtract Fractions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac9c764addand14a","stepAnswer":["$$\\\\frac{-3}{4}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{p+q}{r}$$ when $$p=-4$$, $$q=-2$$, and $$r=8$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-3}{4}$$","hints":{"DefaultPathway":[{"id":"ac9c764addand14a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$-4$$ for $$p$$, $$-2$$ for q, and $$8$$ for $$r$$ in the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand14a-h2","type":"hint","dependencies":["ac9c764addand14a-h1"],"title":"Substitute","text":"We get $$\\\\frac{\\\\left(-4+\\\\left(-2\\\\right)\\\\right)}{8}$$ after the substitution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand14a-h3","type":"hint","dependencies":["ac9c764addand14a-h2"],"title":"Numerator","text":"Add in the numerator first.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand14a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6$$"],"dependencies":["ac9c764addand14a-h3"],"title":"Numerator","text":"What do we get for the numerator after adding the two together?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand14a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-3}{4}$$"],"dependencies":["ac9c764addand14a-h4"],"title":"Simplify","text":"What do we get after simplifying the fraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac9c764addand15","title":"Evaluate Variable Expressions with Fractions","body":"Evaluate","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.6 Add and Subtract Fractions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac9c764addand15a","stepAnswer":["$$\\\\frac{-1}{4}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{-1}{2}-y$$ when $$y=\\\\frac{-1}{4}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-1}{4}$$","hints":{"DefaultPathway":[{"id":"ac9c764addand15a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$\\\\frac{-1}{4}$$ for $$y$$ in the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand15a-h2","type":"hint","dependencies":["ac9c764addand15a-h1"],"title":"Substitute","text":"We get $$\\\\frac{-1}{2}+\\\\frac{1}{4}$$ after the substitution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand15a-h3","type":"hint","dependencies":["ac9c764addand15a-h2"],"title":"LCD","text":"We then need to rewrite each fraction with the least common denominator because they don\'t have a common denominator to start with.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ac9c764addand15a-h3"],"title":"LCD","text":"What is the LCD of $$2$$ and 4?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand15a-h5","type":"hint","dependencies":["ac9c764addand15a-h4"],"title":"Rewrite","text":"We then need to change into equivalent fractions with the LCD, $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand15a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-2}{4}$$"],"dependencies":["ac9c764addand15a-h5"],"title":"Rewrite","text":"What should $$\\\\frac{-1}{2}$$ be changed into with the LCD, 4?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand15a-h7","type":"hint","dependencies":["ac9c764addand15a-h6"],"title":"Add","text":"Finally, we need to add the two fractions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand15a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{4}$$"],"dependencies":["ac9c764addand15a-h7"],"title":"Add","text":"What do we get after the addition?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac9c764addand16","title":"Evaluate Variable Expressions with Fractions","body":"Evaluate","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.6 Add and Subtract Fractions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac9c764addand16a","stepAnswer":["$$\\\\frac{2}{3}$$"],"problemType":"TextBox","stepTitle":"$$4c^3 d$$ when $$c=\\\\frac{-1}{2}$$ and $$d=\\\\frac{-4}{3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2}{3}$$","hints":{"DefaultPathway":[{"id":"ac9c764addand16a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$\\\\frac{-1}{2}$$ for c and $$\\\\frac{-4}{3}$$ for $$d$$ in the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand16a-h2","type":"hint","dependencies":["ac9c764addand16a-h1"],"title":"Substitute","text":"We get $$4{\\\\left(-\\\\frac{1}{2}\\\\right)}^3 \\\\left(-\\\\frac{4}{3}\\\\right)$$ after the substitution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand16a-h3","type":"hint","dependencies":["ac9c764addand16a-h2"],"title":"Exponents","text":"Simplify exponents first.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand16a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{8}$$"],"dependencies":["ac9c764addand16a-h3"],"title":"Exponents","text":"What do we get for $${\\\\left(-\\\\frac{1}{2}\\\\right)}^3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand16a-h5","type":"hint","dependencies":["ac9c764addand16a-h4"],"title":"Multiply","text":"Multiply. Divide out the common factors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand16a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{3}$$"],"dependencies":["ac9c764addand16a-h5"],"title":"Multiply","text":"What do we get after the multiplication?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac9c764addand17","title":"Evaluate Variable Expressions with Fractions","body":"Evaluate","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.6 Add and Subtract Fractions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac9c764addand17a","stepAnswer":["$$\\\\frac{3}{2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{x+y}{z}$$ when $$x=9$$, $$y=-18$$, and $$z=-6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{2}$$","hints":{"DefaultPathway":[{"id":"ac9c764addand17a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$9$$ for $$x$$, $$-18$$ for $$y$$, and $$-6$$ for $$z$$ in the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand17a-h2","type":"hint","dependencies":["ac9c764addand17a-h1"],"title":"Substitute","text":"We get $$\\\\frac{9+\\\\left(-18\\\\right)}{\\\\left(-6\\\\right)}$$ after the substitution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand17a-h3","type":"hint","dependencies":["ac9c764addand17a-h2"],"title":"Numerator","text":"Add in the numerator first.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand17a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":["ac9c764addand17a-h3"],"title":"Numerator","text":"What do we get for the numerator after adding the two together?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand17a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{2}$$"],"dependencies":["ac9c764addand17a-h4"],"title":"Simplify","text":"What do we get after simplifying the fraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac9c764addand18","title":"Add Fractions with a Common Denominator","body":"Find the sum:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.6 Add and Subtract Fractions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac9c764addand18a","stepAnswer":["$$\\\\frac{3+x}{4}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{x}{4}+\\\\frac{3}{4}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3+x}{4}$$","hints":{"DefaultPathway":[{"id":"ac9c764addand18a-h1","type":"hint","dependencies":[],"title":"Add","text":"Add the numerators and place the sum over the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3+x$$"],"dependencies":["ac9c764addand18a-h1"],"title":"Numerator","text":"What do we get for the numerator after adding the two together?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand18a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3+x}{4}$$"],"dependencies":["ac9c764addand18a-h2"],"title":"Final Answer","text":"What do we get after placing the sum over the common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac9c764addand19","title":"How to Add or Subtract Fractions","body":"Subtract:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.6 Add and Subtract Fractions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac9c764addand19a","stepAnswer":["$$\\\\frac{1}{48}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{7}{12}-\\\\frac{9}{16}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{48}$$","hints":{"DefaultPathway":[{"id":"ac9c764addand19a-h1","type":"hint","dependencies":[],"title":"LCD","text":"We need to rewrite each fraction with the least common denominator because they don\'t have a common denominator to start with.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$48$$"],"dependencies":["ac9c764addand19a-h1"],"title":"LCD","text":"What is the LCD of $$12$$ and $$16$$ $$(12=2\\\\times2\\\\times3$$ and 16=2*2*2*2)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand19a-h3","type":"hint","dependencies":["ac9c764addand19a-h2"],"title":"Rewrite","text":"We then need to change into equivalent fractions with the LCD, $$48$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand19a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{28}{48}$$"],"dependencies":["ac9c764addand19a-h3"],"title":"Rewrite","text":"What should $$\\\\frac{7}{12}$$ be changed into with the LCD, 48?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand19a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{27}{48}$$"],"dependencies":["ac9c764addand19a-h4"],"title":"Rewrite","text":"What should $$\\\\frac{9}{16}$$ be changed into with the LCD, 48?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand19a-h6","type":"hint","dependencies":["ac9c764addand19a-h5"],"title":"Subtract","text":"Subtract the numerators and place the difference over the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand19a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ac9c764addand19a-h6"],"title":"Numerator","text":"What do we get for the numerator after the subtraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand19a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{48}$$"],"dependencies":["ac9c764addand19a-h7"],"title":"Final Answer","text":"What do we get after placing the difference over the common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac9c764addand2","title":"Subtract Fractions with a Common Denominator","body":"Find the difference:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.6 Add and Subtract Fractions","courseName":"OpenStax: Elementary 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7?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand20a-h3","type":"hint","dependencies":["ac9c764addand20a-h2"],"title":"Rewrite","text":"We then need to change into equivalent fractions with the LCD, $$14$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand20a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{7}{14}$$"],"dependencies":["ac9c764addand20a-h3"],"title":"Rewrite","text":"What should $$\\\\frac{1}{2}$$ be changed into?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand20a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{9}{14}$$"],"dependencies":["ac9c764addand20a-h7"],"title":"Final Answer","text":"What do we get after placing the sum over the common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac9c764addand3","title":"Subtract Fractions with a Common Denominator","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.6 Add and Subtract Fractions","courseName":"OpenStax: Elementary 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$$(12=2\\\\times2\\\\times3$$, 18=2*3*3)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand5a-h3","type":"hint","dependencies":["ac9c764addand5a-h2"],"title":"Rewrite","text":"We then need to change into equivalent fractions with the LCD, $$36$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{21}{36}$$"],"dependencies":["ac9c764addand5a-h3"],"title":"Rewrite","text":"What should $$\\\\frac{7}{12}$$ be changed into?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand5a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{31}{36}$$"],"dependencies":["ac9c764addand5a-h7"],"title":"Final Answer","text":"What do we get after placing the sum over the common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac9c764addand6","title":"How to Add or Subtract Fractions","body":"Subtract:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.6 Add and Subtract Fractions","courseName":"OpenStax: Elementary 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand6a-h6","type":"hint","dependencies":["ac9c764addand6a-h5"],"title":"Subtract","text":"Subtract the numerators and place the difference over the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand6a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-39$$"],"dependencies":["ac9c764addand6a-h6"],"title":"Numerator","text":"What do we get for the numerator after the subtraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand6a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-39}{120}$$"],"dependencies":["ac9c764addand6a-h7"],"title":"Denominator","text":"What do we get after placing the difference over the common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand6a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-13}{40}$$"],"dependencies":["ac9c764addand6a-h8"],"title":"Simplify","text":"What do we get after simplifying the above expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac9c764addand7","title":"How to Add or Subtract Fractions","body":"Add:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.6 Add and Subtract Fractions","courseName":"OpenStax: Elementary 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand7a-h6","type":"hint","dependencies":["ac9c764addand7a-h5"],"title":"Add","text":"Add the numerators and place the sum over the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand7a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$24+5x$$"],"dependencies":["ac9c764addand7a-h6"],"title":"Numerator","text":"What do we get for the numerator after adding the two together?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand7a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{24+5x}{40}$$"],"dependencies":["ac9c764addand7a-h7"],"title":"Final Answer","text":"What do we get after placing the sum over the common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac9c764addand8","title":"How to Add or Subtract Fractions","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.6 Add and Subtract Fractions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac9c764addand8a","stepAnswer":["$$\\\\frac{25x-9}{30}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{5x}{6}-\\\\frac{3}{10}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{25x-9}{30}$$","hints":{"DefaultPathway":[{"id":"ac9c764addand8a-h1","type":"hint","dependencies":[],"title":"LCD","text":"We need to first rewrite each fraction with the least common denominator because they don\'t have a common denominator to start with.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$30$$"],"dependencies":["ac9c764addand8a-h1"],"title":"LCD","text":"What is the LCD of $$6$$ and $$10$$ $$(6=2\\\\times3$$, 10=2*5)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand8a-h3","type":"hint","dependencies":["ac9c764addand8a-h2"],"title":"Rewrite","text":"We then need to change into equivalent fractions with the LCD, $$30$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{25x}{30}$$"],"dependencies":["ac9c764addand8a-h3"],"title":"Rewrite","text":"What should $$\\\\frac{5x}{6}$$ be changed into?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand8a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{9}{30}$$"],"dependencies":["ac9c764addand8a-h4"],"title":"Rewrite","text":"What should $$\\\\frac{3}{10}$$ be changed into?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand8a-h6","type":"hint","dependencies":["ac9c764addand8a-h5"],"title":"Subtract","text":"Subtract the numerators and place the difference over the common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand8a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25x-9$$"],"dependencies":["ac9c764addand8a-h6"],"title":"Numerator","text":"What do we get for the numerator after the subtraction?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand8a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{25x-9}{30}$$"],"dependencies":["ac9c764addand8a-h7"],"title":"Final Answer","text":"What do we get after placing the difference over the common denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ac9c764addand9","title":"How to Simplify Complex Fractions","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.6 Add and Subtract Fractions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ac9c764addand9a","stepAnswer":["$$\\\\frac{1}{52}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{{\\\\left(\\\\frac{1}{2}\\\\right)}^2}{4+3^2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{52}$$","hints":{"DefaultPathway":[{"id":"ac9c764addand9a-h1","type":"hint","dependencies":[],"title":"Numerator","text":"Remember $${\\\\left(\\\\frac{1}{2}\\\\right)}^2$$ means $$\\\\frac{1}{2} \\\\frac{1}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4}$$"],"dependencies":["ac9c764addand9a-h1"],"title":"Numerator","text":"What do we get after simplifying the numerator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand9a-h3","type":"hint","dependencies":["ac9c764addand9a-h2"],"title":"Denominator","text":"We then need to simplify the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$13$$"],"dependencies":["ac9c764addand9a-h3"],"title":"Denominator","text":"What do we get after simplifying the denominator?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand9a-h5","type":"hint","dependencies":["ac9c764addand9a-h4"],"title":"Divide","text":"Finally, we need to divide the numerator by the denominator. Simplify if possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ac9c764addand9a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{52}$$"],"dependencies":["ac9c764addand9a-h5"],"title":"Divide","text":"What do we get after the division?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aca1a7dderivative1","title":"","body":"Find f\u2032(x).","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.3 Differentiation Rules","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aca1a7dderivative1a","stepAnswer":["$$7x^6$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=x^7+10$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$7x^6$$","choices":["$$7x^6$$","$$x^7$$","$$7x^7$$","$$7x$$"],"hints":{"DefaultPathway":[{"id":"aca1a7dderivative1a-h1","type":"hint","dependencies":[],"title":"The Power Rule","text":"If $$f(x)=x^n$$, then $$f\'(x)={nx}^{n-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aca1a7dderivative1a-h2","type":"hint","dependencies":["aca1a7dderivative1a-h1"],"title":"Application of Power Rule $$1$$","text":"To get the derivative of $$x^7$$, we multiply our constant (1) by our exponent (7) and subtract the exponent power by $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aca1a7dderivative1a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$7x^6$$"],"dependencies":["aca1a7dderivative1a-h2"],"title":"Application of Power Rule $$2$$","text":"What is the derivative of $$x^7$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$7x^6$$","$$x^7$$","$$7x^7$$","$$7x$$"]},{"id":"aca1a7dderivative1a-h4","type":"hint","dependencies":["aca1a7dderivative1a-h3"],"title":"The Constant Rule","text":"If f(x) is a constant c, $$f\'(x)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aca1a7dderivative1a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$0$$"],"dependencies":["aca1a7dderivative1a-h4"],"title":"Application of Constant Rule","text":"What is the derivative of 10?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$0$$","$$1$$","$$-1$$","$$10$$"]},{"id":"aca1a7dderivative1a-h6","type":"hint","dependencies":["aca1a7dderivative1a-h5"],"title":"The Sum Rule $$1$$","text":"The derivative of the sum of a function f and a function g is the same as the sum of the derivative of f and the derivative of g.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aca1a7dderivative1a-h7","type":"hint","dependencies":["aca1a7dderivative1a-h6"],"title":"The Sum Rule $$2$$","text":"You can add the two separate derivatives that we just found to find our answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"aca1a7dderivative10","title":"","body":"Find f\u2032(x).","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.3 Differentiation Rules","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aca1a7dderivative10a","stepAnswer":["$$\\\\frac{4x^4+2x^2-2x}{x^4}$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{4x^3-2x+1}{x^2}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{4x^4+2x^2-2x}{x^4}$$","choices":["$$\\\\frac{4x^4+2x^2-2x}{x^4}$$","$$\\\\frac{-\\\\left(4x^4+2x^2-2x\\\\right)}{x^4}$$","$$4x-\\\\frac{2}{x}+\\\\frac{1}{x^2}$$","$$\\\\frac{4x^4+2x^2-2x}{x^2}$$"],"hints":{"DefaultPathway":[]}}]},{"id":"aca1a7dderivative11","title":"","body":"Find f\u2032(x).","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.3 Differentiation Rules","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aca1a7dderivative11a","stepAnswer":["$$\\\\frac{-16x}{{\\\\left(x^2-4\\\\right)}^2}$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{x^2+4}{x^2-4}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{-16x}{{\\\\left(x^2-4\\\\right)}^2}$$","choices":["$$\\\\frac{-16x}{{\\\\left(x^2-4\\\\right)}^2}$$","$$\\\\frac{16x}{{\\\\left(x^2-4\\\\right)}^2}$$","$$\\\\frac{-8x}{x^2-4}$$","$$\\\\frac{-16x}{x^2-4}$$"],"hints":{"DefaultPathway":[]}}]},{"id":"aca1a7dderivative12","title":"","body":"Find f\u2032(x).","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.3 Differentiation Rules","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aca1a7dderivative12a","stepAnswer":["$$\\\\frac{\\\\left(-x^2-18x+64\\\\right)}{{\\\\left(x^2-7x+1\\\\right)}^2}$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{x+9}{x^2-7x+1}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{\\\\left(-x^2-18x+64\\\\right)}{{\\\\left(x^2-7x+1\\\\right)}^2}$$","choices":["$$\\\\frac{\\\\left(-x^2-18x+64\\\\right)}{{\\\\left(x^2-7x+1\\\\right)}^2}$$","$$\\\\frac{\\\\left(-x^2-18x+64\\\\right)}{x^2-7x+1}$$","$$\\\\frac{x^2+18x-64}{{\\\\left(x^2-7x+1\\\\right)}^2}$$","$$\\\\frac{x^2+18x-64}{x^2-7x+1}$$"],"hints":{"DefaultPathway":[]}}]},{"id":"aca1a7dderivative2","title":"","body":"Find f\u2032(x).","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.3 Differentiation Rules","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aca1a7dderivative2a","stepAnswer":["$$15x^2-1$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=5x^3-x+1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$15x^2-1$$","choices":["$$15x^2-1$$","$$5x^3-x$$","$$15x^2-x$$","$$4x^2+1$$"],"hints":{"DefaultPathway":[{"id":"aca1a7dderivative2a-h1","type":"hint","dependencies":[],"title":"The Power Rule","text":"If $$f(x)=x^n$$, then $$f\'(x)={nx}^{n-1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aca1a7dderivative2a-h2","type":"hint","dependencies":["aca1a7dderivative2a-h1"],"title":"Application of Power Rule $$1$$","text":"To get the derivative of $$5x^3$$, we multiply our constant (5) by our exponent (3) and subtract the exponent power by $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aca1a7dderivative2a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$15x^2$$"],"dependencies":["aca1a7dderivative2a-h2"],"title":"Application of Power Rule $$2$$","text":"What is the derivative of $$5x^3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$15x^2-1$$","$$5x^3-x$$","$$15x^2-x$$","$$4x^2+1$$","$$15x^2$$"]},{"id":"aca1a7dderivative2a-h4","type":"hint","dependencies":["aca1a7dderivative2a-h3"],"title":"Application of Power Rule $$3$$","text":"To get the derivative of $$-x$$, we multiply our constant $$(-1)$$ by our exponent (1) and subtract the exponent power by $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aca1a7dderivative2a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-1$$"],"dependencies":["aca1a7dderivative2a-h4"],"title":"Application of Power Rule $$2$$","text":"What is the derivative of $$-x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$0$$","$$1$$","$$-1$$","$$x$$"]},{"id":"aca1a7dderivative2a-h6","type":"hint","dependencies":["aca1a7dderivative2a-h5"],"title":"The Constant Rule","text":"If f(x) is a constant c, $$f\'(x)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aca1a7dderivative2a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$0$$"],"dependencies":["aca1a7dderivative2a-h6"],"title":"Application of Constant Rule","text":"What is the derivative of 1?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$0$$","$$1$$","$$-1$$","$$10$$"]},{"id":"aca1a7dderivative2a-h8","type":"hint","dependencies":["aca1a7dderivative2a-h7"],"title":"The Sum Rule","text":"You can add the two separate derivatives that we just found to find our answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"aca1a7dderivative3","title":"","body":"Find f\u2032(x).","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.3 Differentiation Rules","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aca1a7dderivative3a","stepAnswer":["$$8x-7$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=4x^2-7x$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$8x-7$$","choices":["$$8x-7$$","$$8x^2-7$$","$$4x-7$$","$$-3x^2$$"],"hints":{"DefaultPathway":[]}}]},{"id":"aca1a7dderivative4","title":"","body":"Find f\u2032(x).","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.3 Differentiation Rules","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aca1a7dderivative4a","stepAnswer":["$$32x^3+18x$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=8x^4+9x^2-1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$32x^3+18x$$","choices":["$$32x^3+18x$$","$$12x^3+11x$$","$$36x^4+18x^2$$","$$32x^3+18x-1$$"],"hints":{"DefaultPathway":[]}}]},{"id":"aca1a7dderivative5","title":"","body":"Find f\u2032(x).","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.3 Differentiation Rules","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aca1a7dderivative5a","stepAnswer":["$$4x^3-\\\\frac{2}{x^2}$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=x^4+\\\\frac{2}{x}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$4x^3-\\\\frac{2}{x^2}$$","choices":["$$4x^3-\\\\frac{2}{x^2}$$","$$x^3+\\\\frac{2}{x^2}$$","$$4x^3-\\\\frac{2}{x}$$","$$4x^3-\\\\frac{4}{x^2}$$"],"hints":{"DefaultPathway":[]}}]},{"id":"aca1a7dderivative6","title":"","body":"Find f\u2032(x).","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.3 Differentiation Rules","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aca1a7dderivative6a","stepAnswer":["$$270x^4+\\\\frac{39}{{\\\\left(x+1\\\\right)}^2}$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=3x\\\\left(18x^4+\\\\frac{13}{x+1}\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$270x^4+\\\\frac{39}{{\\\\left(x+1\\\\right)}^2}$$","choices":["$$270x^4+\\\\frac{39}{{\\\\left(x+1\\\\right)}^2}$$","$$\\\\frac{54x^6+54x^5+39x}{x+1}$$","$$39x^4+\\\\frac{270}{{\\\\left(x+1\\\\right)}^2}$$","$$270x^5+\\\\frac{39}{{\\\\left(x+1\\\\right)}^3}$$"],"hints":{"DefaultPathway":[]}}]},{"id":"aca1a7dderivative7","title":"","body":"Find f\u2032(x).","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.3 Differentiation Rules","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aca1a7dderivative7a","stepAnswer":["$$6x^2+8x-3$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\left(x+2\\\\right) \\\\left(2x^2-3\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$6x^2+8x-3$$","choices":["$$6x^2+8x-3$$","$$12x+8$$","$$\\\\frac{x+2}{2x^2-3}$$","$$2x^3+4x^2-3x-6$$"],"hints":{"DefaultPathway":[]}}]},{"id":"aca1a7dderivative8","title":"","body":"Find f\u2032(x).","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.3 Differentiation Rules","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aca1a7dderivative8a","stepAnswer":["$$\\\\frac{-5}{x^2}$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=x^2 \\\\left(\\\\frac{2}{x^2}+\\\\frac{5}{x^3}\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{-5}{x^2}$$","choices":["$$\\\\frac{-5}{x^2}$$","$$1+\\\\frac{5}{x}$$","$$\\\\frac{5}{x^2}$$","$$-1-\\\\frac{5}{x}$$"],"hints":{"DefaultPathway":[]}}]},{"id":"aca1a7dderivative9","title":"","body":"Find f\u2032(x).","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.3 Differentiation Rules","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aca1a7dderivative9a","stepAnswer":["$$\\\\frac{3x^2+4x}{3}$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{x^3+2x^2-4}{3}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{3x^2+4x}{3}$$","choices":["$$\\\\frac{3x^2+4x}{3}$$","$$\\\\frac{-\\\\left(3x^2+4x\\\\right)}{3}$$","$$3x^2+4x$$","$$-\\\\left(3x^2+4x\\\\right)$$"],"hints":{"DefaultPathway":[]}}]},{"id":"aca1a7dderivativetangent1","title":"","body":"Evaluate f\'(a), and find the equation of the tangent line f(x) at $$x$$ $$=$$ a.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.3 Differentiation Rules","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aca1a7dderivativetangent1a","stepAnswer":["$$f\'(a)=23;$$ $$y=23x-28$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=2x^3+3x-x^2$$ $$a=2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$f\'(a)=23;$$ $$y=23x-28$$","choices":["$$f\'(a)=23;$$ $$y=23x-28$$","$$f\'(a)=2;$$ $$y=2x-2$$","$$f\'(a)=-23;$$ $$y=-23x-28$$","$$f\'(a)=23;$$ $$y=23x-28$$"],"hints":{"DefaultPathway":[]}}]},{"id":"aca1a7dderivativetangent2","title":"","body":"Evaluate f\'(a), and find the equation of the tangent line f(x) at $$x$$ $$=$$ a.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.3 Differentiation Rules","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aca1a7dderivativetangent2a","stepAnswer":["$$f\'(a)=-3;$$ $$y=-3x+3$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{1}{x}-x^2$$ $$a=1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$f\'(a)=-3;$$ $$y=-3x+3$$","choices":["$$f\'(a)=-3;$$ $$y=-3x+3$$","$$f\'(a)=3;$$ $$y=3x+3$$","$$f\'(a)=-3;$$ $$y=-3x-3$$","$$f\'(a)=-3;$$ $$y=3x-3$$"],"hints":{"DefaultPathway":[]}}]},{"id":"aca1a7dderivativetangent3","title":"","body":"Evaluate f\'(a), and find the equation of the tangent line f(x) at $$x$$ $$=$$ a.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.3 Differentiation Rules","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aca1a7dderivativetangent3a","stepAnswer":["$$f\'(a)=3;$$ $$y=3x+2$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=x^2-x^{12}+3x+2$$ $$a=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$f\'(a)=3;$$ $$y=3x+2$$","choices":["$$f\'(a)=3;$$ $$y=3x+2$$","$$f\'(a)=-3;$$ $$y=-3x+2$$","$$f\'(a)=3;$$ $$y=3x-2$$","$$f\'(a)=-3;$$ $$y=-3x-2$$"],"hints":{"DefaultPathway":[]}}]},{"id":"aca1a7dderivativetangent4","title":"","body":"Evaluate f\'(a), and find the equation of the tangent line f(x) at $$x$$ $$=$$ a.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.3 Differentiation Rules","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aca1a7dderivativetangent4a","stepAnswer":["$$f\'(a)=1;$$ $$y=x-1$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{1}{x}-x^2$$ $$a=-1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$f\'(a)=1;$$ $$y=x-1$$","choices":["$$f\'(a)=1;$$ $$y=x-1$$","$$f\'(a)=-1;$$ $$y=-x+1$$","$$f\'(a)=1;$$ $$y=x+1$$","$$f\'(a)=-1;$$ $$y=-x-1$$"],"hints":{"DefaultPathway":[]}}]},{"id":"aca1a7dtangentline1","title":"","body":"Find the equation of the tangent line T(x) to the graph of the given function at the indicated point.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.3 Differentiation Rules","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aca1a7dtangentline1a","stepAnswer":["$$y=4x+1$$"],"problemType":"MultipleChoice","stepTitle":"$$y=3x^2+4x+1$$ at $$(0,1)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=4x+1$$","choices":["$$y=4x+1$$","$$y=\\\\frac{1}{4} x+1$$","$$y=6x+4$$","$$y=\\\\frac{-1}{4} x+4$$"],"hints":{"DefaultPathway":[]}}]},{"id":"aca1a7dtangentline2","title":"","body":"Find the equation of the tangent line T(x) to the graph of the given function at the indicated point.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.3 Differentiation Rules","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aca1a7dtangentline2a","stepAnswer":["$$y=-4x+7$$"],"problemType":"MultipleChoice","stepTitle":"$$y=\\\\frac{2x}{x^2+1}$$ at $$(1,3)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=-4x+7$$","choices":["$$y=-4x+7$$","$$y=\\\\frac{-7}{4} x+4$$","$$y=-2x+\\\\frac{7}{4}$$","$$y=7x-4$$"],"hints":{"DefaultPathway":[]}}]},{"id":"aca1a7dtangentline3","title":"","body":"Find the equation of the tangent line T(x) to the graph of the given function at the indicated point.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.3 Differentiation Rules","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aca1a7dtangentline3a","stepAnswer":["$$y=\\\\frac{-1}{2} x+\\\\frac{1}{2}$$"],"problemType":"MultipleChoice","stepTitle":"$$y=\\\\frac{2x}{x-1}$$ at $$(-1,1)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=\\\\frac{-1}{2} x+\\\\frac{1}{2}$$","choices":["$$y=\\\\frac{-1}{2} x+\\\\frac{1}{2}$$","$$y=-x+1$$","$$y=-2x+2$$","$$y=\\\\frac{1}{2} x-\\\\frac{1}{2}$$"],"hints":{"DefaultPathway":[]}}]},{"id":"aca1a7dtangentline4","title":"","body":"Find the equation of the tangent line T(x) to the graph of the given function at the indicated point.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.3 Differentiation Rules","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aca1a7dtangentline4a","stepAnswer":["$$y=4x-5$$"],"problemType":"MultipleChoice","stepTitle":"$$y=\\\\frac{2}{x}-\\\\frac{3}{x^2}$$ ar $$(1,-1)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=4x-5$$","choices":["$$y=4x-5$$","$$y=5x-4$$","$$y=4x-\\\\frac{5}{4}$$","$$y=4x+5$$"],"hints":{"DefaultPathway":[]}}]},{"id":"aca1a7dtangentline5","title":"","body":"Find the equation of the tangent line to the graph of the given function at the indicated x-value.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.3 Differentiation Rules","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aca1a7dtangentline5a","stepAnswer":["$$y=-7x-3$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=2x^3+4x^2-5x-3$$ at $$x=-1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=-7x-3$$","choices":["$$y=-7x-3$$","$$y=7x+3$$","$$y=7x-3$$","$$y=-7x+3$$"],"hints":{"DefaultPathway":[]}}]},{"id":"aca1a7dtangentline6","title":"","body":"Find the equation of the tangent line to the graph of the given function at the indicated x-value.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.3 Differentiation Rules","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aca1a7dtangentline6a","stepAnswer":["$$y=\\\\frac{255}{16} x-73$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=x^2+\\\\frac{4}{x}-10$$ at $$x=8$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=\\\\frac{255}{16} x-73$$","choices":["$$y=\\\\frac{255}{16} x-73$$","$$y=-\\\\left(\\\\frac{255}{16}\\\\right) x+73$$","$$y=\\\\frac{15}{4} x-73$$","$$y=\\\\frac{255}{16} x+73$$"],"hints":{"DefaultPathway":[]}}]},{"id":"aca1a7dtangentline7","title":"","body":"Find the equation of the tangent line to the graph of the given function at the indicated x-value.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.3 Differentiation Rules","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aca1a7dtangentline7a","stepAnswer":["$$y=-5x+7$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\left(3x-x^2\\\\right) \\\\left(3-x-x^2\\\\right)$$ at $$x=1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=-5x+7$$","choices":["$$y=-5x+7$$","$$y=5x+7$$","$$y=5x-7$$","$$y=-5x-7$$"],"hints":{"DefaultPathway":[]}}]},{"id":"aca1a7dtangentline8","title":"","body":"Find the equation of the line passing through the point $$P(3,3)$$ and tangent to the graph of the given function","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.3 Differentiation Rules","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aca1a7dtangentline8a","stepAnswer":["$$y=\\\\frac{-3}{2} x+\\\\frac{15}{2}$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{6}{x-1}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=\\\\frac{-3}{2} x+\\\\frac{15}{2}$$","choices":["$$y=\\\\frac{-3}{2} x+\\\\frac{15}{2}$$","$$y=\\\\frac{-3}{2} x-\\\\frac{15}{2}$$","$$y=\\\\frac{3}{2} x+\\\\frac{15}{2}$$","$$y=\\\\frac{3}{2} x-\\\\frac{15}{2}$$"],"hints":{"DefaultPathway":[]}}]},{"id":"aca1a7dvelocity1","title":"","body":"A herring swimming along a straight line has traveled $$s(t)=\\\\frac{t^2}{t^2+2}$$ feet in $$t$$ seconds.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.3 Differentiation Rules","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aca1a7dvelocity1a","stepAnswer":["$$\\\\frac{12}{121}$$"],"problemType":"MultipleChoice","stepTitle":"Determine the velocity of the herring when it has traveled $$3$$ seconds.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{12}{121}$$","choices":["$$\\\\frac{12}{121}$$","$$\\\\frac{9}{11}$$","$$\\\\frac{3}{121}$$","$$\\\\frac{6}{121}$$"],"hints":{"DefaultPathway":[]}}]},{"id":"aca598bexp1","title":"Algebra with Exponents and Logarithms","body":"These questions test your knowledge of the core concepts.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Algebra with Exponents and Logarithms","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"aca598bexp1a","stepAnswer":["$$P(x)=\\\\left(x^3-2x^2-x-1\\\\right) \\\\left(x-1\\\\right)$$ and $$Q(x)=3{\\\\left(x-2\\\\right)}^2$$ for $$x \\\\neq 1$$ or $$2$$"],"problemType":"MultipleChoice","stepTitle":"Find polynomials P(x) and Q(x) so that the equation below is satisfied. Be sure to note any values of $$x$$ for which this is impossible. $$\\\\frac{P\\\\left(x\\\\right)}{Q\\\\left(x\\\\right)} \\\\frac{3}{x-1}=\\\\frac{x^2}{x-2}-\\\\frac{x+1}{{\\\\left(x-2\\\\right)}^2}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$P(x)=\\\\left(x^3-2x^2-x-1\\\\right) \\\\left(x-1\\\\right)$$ and $$Q(x)=3{\\\\left(x-2\\\\right)}^2$$ for $$x \\\\neq 1$$ or $$2$$","choices":["$$P(x)=\\\\left(x^3-2x^2-x-1\\\\right) \\\\left(x-1\\\\right)$$ and $$Q(x)=3{\\\\left(x-2\\\\right)}^2$$ for $$x \\\\neq 1$$ or $$2$$","$$P(x)=x^3-2x^2-x-1$$ and $$Q(x)=3\\\\left(x-1\\\\right) {\\\\left(x-2\\\\right)}^2$$ for $$x \\\\neq 1$$ or $$2$$","$$P(x)=\\\\left(x^3-2x^2-x+1\\\\right) \\\\left(x-1\\\\right)$$ and $$Q(x)=3{\\\\left(x-2\\\\right)}^2$$ for $$x \\\\neq 1$$ or $$2$$"],"hints":{"DefaultPathway":[{"id":"aca598bexp1a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["1&2"],"dependencies":[],"title":"Can\'t divide by $$0$$","text":"$$x$$ cannot equal to which values in this case?","variabilization":{},"oer":"","license":"","choices":["1&2","$$-1&2$$","$$-1&-2$$","$$1&-2$$"]},{"id":"aca598bexp1a-h2","type":"hint","dependencies":["aca598bexp1a-h1"],"title":"RHS","text":"Using the property $$\\\\frac{a}{b}=\\\\frac{a c}{b c}$$, RHS can equal to $$\\\\frac{\\\\left(x-2\\\\right) x^2}{{\\\\left(x-2\\\\right)}^2}-\\\\frac{x+1}{{\\\\left(x-2\\\\right)}^2}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp1a-h3","type":"hint","dependencies":["aca598bexp1a-h2"],"title":"RHS","text":"Using the Distributive Law, $$x^2 \\\\left(x-2\\\\right)=x^3-2x^2$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp1a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{x^3-2x^2-x-1}{{\\\\left(x-2\\\\right)}^2}$$"],"dependencies":["aca598bexp1a-h3"],"title":"RHS","text":"Note $$\\\\frac{a}{b}-\\\\frac{c}{b}=\\\\frac{a-c}{b}$$. Which option is the correct simplification of $$x^3-\\\\frac{2x^2}{{\\\\left(x-2\\\\right)}^2}-\\\\frac{x+1}{{\\\\left(x-2\\\\right)}^2}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{x^3-2x^2-x-1}{{\\\\left(x-2\\\\right)}^2}$$","$$\\\\frac{x^3-2x^2-x+1}{{\\\\left(x-2\\\\right)}^2}$$"]},{"id":"aca598bexp1a-h5","type":"hint","dependencies":["aca598bexp1a-h4"],"title":"$$LHS=RHS$$","text":"$$\\\\frac{P\\\\left(x\\\\right)}{Q\\\\left(x\\\\right)} \\\\frac{3}{x-1}=\\\\frac{x^3-2x^2-x-1}{{\\\\left(x-2\\\\right)}^2}$$ since $$LHS=RHS$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp1a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{x^3-2x^2-x-1}{{\\\\left(x-2\\\\right)}^2} \\\\frac{x-1}{3}$$"],"dependencies":["aca598bexp1a-h5"],"title":"$$LHS=RHS$$","text":"What does $$\\\\frac{P\\\\left(x\\\\right)}{Q\\\\left(x\\\\right)}$$ equal to?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{x^3-2x^2-x-1}{{\\\\left(x-2\\\\right)}^2} \\\\frac{x-1}{3}$$","$$\\\\frac{x^3-2x^2-x-1}{{\\\\left(x-2\\\\right)}^2}$$"]},{"id":"aca598bexp1a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{\\\\left(x^3-2x^2-x-1\\\\right) \\\\left(x-1\\\\right)}{3{\\\\left(x-2\\\\right)}^2}$$"],"dependencies":["aca598bexp1a-h6"],"title":"Simplification","text":"Using $$\\\\frac{a}{b} \\\\frac{c}{d}=\\\\frac{a c}{b d}$$, what value can $$\\\\frac{P\\\\left(x\\\\right)}{Q\\\\left(x\\\\right)}$$ be simplified?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{\\\\left(x^3-2x^2-x-1\\\\right) \\\\left(x-1\\\\right)}{3{\\\\left(x-2\\\\right)}^2}$$","$$\\\\frac{x^3-2x^2-x-1}{3{\\\\left(x-2\\\\right)}^2 \\\\left(x-1\\\\right)}$$","$$\\\\frac{\\\\left(x^3-2x^2-x+1\\\\right) \\\\left(x-1\\\\right)}{3{\\\\left(x-2\\\\right)}^2}$$"]},{"id":"aca598bexp1a-h8","type":"hint","dependencies":["aca598bexp1a-h7"],"title":"Conclusion","text":"So we can choose $$P(x)=\\\\left(x^3-2x^2-x-1\\\\right) \\\\left(x-1\\\\right)$$ and $$Q(x)=3{\\\\left(x-2\\\\right)}^2$$ for $$x \\\\neq 1$$ or $$2$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"aca598bexp2","title":"Algebra with Exponents and Logarithms","body":"These questions test your knowledge of the core concepts.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Algebra with Exponents and Logarithms","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"aca598bexp2a","stepAnswer":["$$2x^2+2x-4$$"],"problemType":"MultipleChoice","stepTitle":"Use the laws of logarithms to rewrite the following expression as a polynomial. $$\\\\log_{3}\\\\left(\\\\frac{3^{2x-1} 9^{x^2}}{27}\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2x^2+2x-4$$","choices":["$$2x^2+2x-4$$","$$2x^2+2x+4$$","$$2x^2+2x-2$$"],"hints":{"DefaultPathway":[{"id":"aca598bexp2a-h1","type":"hint","dependencies":[],"title":"Logarithmic Laws","text":"Note $$\\\\log_{c}\\\\left(\\\\frac{a}{b}\\\\right)=\\\\log_{c}\\\\left(a\\\\right)-\\\\log_{c}\\\\left(b\\\\right)$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp2a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\log_{3}\\\\left(3^{2x-1} 9^{x^2}\\\\right)-\\\\log_{3}\\\\left(27\\\\right)$$"],"dependencies":["aca598bexp2a-h1"],"title":"Logarithmic Laws","text":"Which option can $$\\\\log_{3}\\\\left(\\\\frac{3^{2x-1} 9^{x^2}}{27}\\\\right)$$ be simplified using the given property?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\log_{3}\\\\left(3^{2x-1} 9^{x^2}\\\\right)-\\\\log_{3}\\\\left(27\\\\right)$$","$$\\\\log_{3}\\\\left(3^{2x-1} 9^{x^2}\\\\right)+\\\\log_{3}\\\\left(27\\\\right)$$","$$\\\\log_{3}\\\\left(27\\\\right)-\\\\log_{3}\\\\left(3^{2x-1} 9^{x^2}\\\\right)$$"]},{"id":"aca598bexp2a-h3","type":"hint","dependencies":["aca598bexp2a-h2"],"title":"Logarithmic Laws","text":"Note $$\\\\log_{c}\\\\left(a b\\\\right)=\\\\log_{c}\\\\left(a\\\\right)+\\\\log_{c}\\\\left(b\\\\right)$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp2a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\log_{3}\\\\left(3^{2x-1}\\\\right)+\\\\log_{3}\\\\left(9^{x^2}\\\\right)-\\\\log_{3}\\\\left(27\\\\right)$$"],"dependencies":["aca598bexp2a-h3"],"title":"Logarithmic Laws","text":"Which option can $$\\\\log_{3}\\\\left(3^{2x-1} 9^{x^2}\\\\right)-\\\\log_{3}\\\\left(27\\\\right)$$ be simplified using the given property?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\log_{3}\\\\left(3^{2x-1}\\\\right)+\\\\log_{3}\\\\left(9^{x^2}\\\\right)-\\\\log_{3}\\\\left(27\\\\right)$$","$$\\\\log_{3}\\\\left(3^{2x-1}\\\\right)-\\\\log_{3}\\\\left(9^{x^2}\\\\right)-\\\\log_{3}\\\\left(27\\\\right)$$"]},{"id":"aca598bexp2a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["aca598bexp2a-h4"],"title":"Simplification","text":"Is $$9^{x^2}$$ equivalent to $${\\\\left(3^2\\\\right)}^{x^2}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"aca598bexp2a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["aca598bexp2a-h5"],"title":"Simplification","text":"Is $${\\\\left(3^2\\\\right)}^{x^2}$$ equivalent to $$3^{2x^2}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"],"subHints":[{"id":"aca598bexp2a-h6-s1","type":"hint","dependencies":[],"title":"Simplification","text":"Note $${\\\\left(a^b\\\\right)}^c=a^{b c}$$","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"aca598bexp2a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["aca598bexp2a-h6"],"title":"Simplification","text":"Does $$27$$ equal to $$3^3$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"aca598bexp2a-h8","type":"hint","dependencies":["aca598bexp2a-h7"],"title":"Simplification","text":"Now we have $$\\\\log_{3}\\\\left(3^{2x-1}\\\\right)+\\\\log_{3}\\\\left(3^{2x^2}\\\\right)-\\\\log_{3}\\\\left(3^3\\\\right)$$. Note that we can use an important fact that $$\\\\log_{b}\\\\left(b^a\\\\right)=a$$ to simplify further.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp2a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2x-1+2x^2-3$$"],"dependencies":["aca598bexp2a-h8"],"title":"Simplification","text":"Which option is the result of simplification of $$\\\\log_{3}\\\\left(3^{2x-1}\\\\right)+\\\\log_{3}\\\\left(3^{2x^2}\\\\right)-\\\\log_{3}\\\\left(3^3\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$2x-1+2x^2-3$$","$$2x-1-2x^2-3$$"]},{"id":"aca598bexp2a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2x^2+2x-4$$"],"dependencies":["aca598bexp2a-h9"],"title":"Conclusion","text":"What is the simplified answer of $$\\\\log_{3}\\\\left(\\\\frac{3^{2x-1} 9^{x^2}}{27}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$2x^2+2x-4$$","$$2x^2+2x+4$$","$$2x^2+2x-2$$"],"subHints":[{"id":"aca598bexp2a-h10-s1","type":"hint","dependencies":[],"title":"Conclusion","text":"$$2x-1+2x^2-3=2x^2+2x-4$$","variabilization":{},"oer":"https://OATutor.io","license":""}]}]}}]},{"id":"aca598bexp3","title":"Algebra with Exponents and Logarithms","body":"Using the laws of exponents, write each expression below in the specified form.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Algebra with Exponents and Logarithms","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"aca598bexp3a","stepAnswer":["$${\\\\left(x^2+4\\\\right)}^{\\\\frac{8}{15}}$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\sqrt[5]{x^2+4} \\\\sqrt[3]{x^2+4}$$ in the form of $${\\\\left(x^2+4\\\\right)}^A$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$${\\\\left(x^2+4\\\\right)}^{\\\\frac{8}{15}}$$","choices":["$${\\\\left(x^2+4\\\\right)}^{\\\\frac{8}{15}}$$","$${\\\\left(x^2+4\\\\right)}^{\\\\frac{1}{15}}$$"],"hints":{"DefaultPathway":[{"id":"aca598bexp3a-h1","type":"hint","dependencies":[],"title":"Conversion","text":"Convert radicals to exponents using the fact that $$\\\\sqrt[n]{y}=y^{\\\\frac{1}{n}}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp3a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${\\\\left(x^2+4\\\\right)}^{\\\\frac{1}{5}} {\\\\left(x^2+4\\\\right)}^{\\\\frac{1}{3}}$$"],"dependencies":["aca598bexp3a-h1"],"title":"Conversion","text":"What is the result of $$\\\\sqrt[5]{x^2+4} \\\\sqrt[3]{x^2+4}$$ after conversion?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$${\\\\left(x^2+4\\\\right)}^{\\\\frac{1}{5}} {\\\\left(x^2+4\\\\right)}^{\\\\frac{1}{3}}$$","$${\\\\left(x^2+4\\\\right)}^5 {\\\\left(x^2+4\\\\right)}^3$$"]},{"id":"aca598bexp3a-h3","type":"hint","dependencies":["aca598bexp3a-h2"],"title":"Property","text":"$$a^b a^c=a^{b+c}$$, so $${\\\\left(x^2+4\\\\right)}^{\\\\frac{1}{5}} {\\\\left(x^2+4\\\\right)}^{\\\\frac{1}{3}}={\\\\left(x^2+4\\\\right)}^{\\\\frac{1}{5}+\\\\frac{1}{3}}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{8}{15}$$"],"dependencies":["aca598bexp3a-h3"],"title":"Simplification","text":"What is the result of $$\\\\frac{1}{5}+\\\\frac{1}{3}$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp3a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${\\\\left(x^2+4\\\\right)}^{\\\\frac{8}{15}}$$"],"dependencies":["aca598bexp3a-h4"],"title":"Conclusion","text":"What is the simplified answer of this question?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$${\\\\left(x^2+4\\\\right)}^{\\\\frac{8}{15}}$$","$${\\\\left(x^2+4\\\\right)}^{\\\\frac{1}{15}}$$"]}]}},{"id":"aca598bexp3b","stepAnswer":["$${\\\\left(x^8+7\\\\right)}^{\\\\frac{2}{3}}$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{x^8+7}{\\\\sqrt[3]{x^8+7}}$$ in the form of $${\\\\left(x^8+7\\\\right)}^A$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$${\\\\left(x^8+7\\\\right)}^{\\\\frac{2}{3}}$$","choices":["$${\\\\left(x^8+7\\\\right)}^{\\\\frac{2}{3}}$$","$${\\\\left(x^8+7\\\\right)}^{\\\\frac{1}{3}}$$","$${\\\\left(x^8+7\\\\right)}^{\\\\left(-\\\\frac{2}{3}\\\\right)}$$"],"hints":{"DefaultPathway":[{"id":"aca598bexp3b-h1","type":"hint","dependencies":[],"title":"Conversion","text":"Convert radicals to exponents using the fact that $$\\\\sqrt[n]{y}=y^{\\\\frac{1}{n}}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp3b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{x^8+7}{{\\\\left(x^8+7\\\\right)}^{\\\\frac{1}{3}}}$$"],"dependencies":["aca598bexp3b-h1"],"title":"Conversion","text":"What is the result of $$\\\\frac{x^8+7}{\\\\sqrt[3]{x^8+7}}$$ after conversion?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{x^8+7}{{\\\\left(x^8+7\\\\right)}^{\\\\frac{1}{3}}}$$","$$\\\\frac{x^8+7}{{\\\\left(x^8+7\\\\right)}^3}$$"]},{"id":"aca598bexp3b-h3","type":"hint","dependencies":["aca598bexp3b-h2"],"title":"Property","text":"Use the property $$\\\\frac{a^b}{a^c}=a^{b-c}$$ to simplify $$\\\\frac{x^8+7}{{\\\\left(x^8+7\\\\right)}^{\\\\frac{1}{3}}}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp3b-h4","type":"hint","dependencies":["aca598bexp3b-h3"],"title":"Property","text":"You can treat $$x^8+7$$ as $${\\\\left(x^8+7\\\\right)}^1$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp3b-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${\\\\left(x^8+7\\\\right)}^{\\\\frac{2}{3}}$$"],"dependencies":["aca598bexp3b-h4"],"title":"Property","text":"What is the result using the given property?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$${\\\\left(x^8+7\\\\right)}^{\\\\frac{2}{3}}$$","$${\\\\left(x^8+7\\\\right)}^{\\\\frac{1}{3}}$$","$${\\\\left(x^8+7\\\\right)}^{\\\\left(-\\\\frac{2}{3}\\\\right)}$$"],"subHints":[{"id":"aca598bexp3b-h5-s1","type":"hint","dependencies":[],"title":"Property","text":"$${\\\\left(x^8+7\\\\right)}^{1-\\\\frac{1}{3}}$$ is equivalent to $${\\\\left(x^8+7\\\\right)}^{\\\\frac{2}{3}}$$ since $$1-\\\\frac{1}{3}=\\\\frac{2}{3}$$","variabilization":{},"oer":"https://OATutor.io","license":""}]}]}},{"id":"aca598bexp3c","stepAnswer":["$$16{\\\\left(x^6+1\\\\right)}^{\\\\frac{4}{7}}$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\sqrt[7]{{\\\\left(128x^6+128\\\\right)}^4}$$ in the form of $$B {\\\\left(x^6+1\\\\right)}^A$$, where B is an integer.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$16{\\\\left(x^6+1\\\\right)}^{\\\\frac{4}{7}}$$","choices":["$$16{\\\\left(x^6+1\\\\right)}^{\\\\frac{4}{7}}$$","$$128{\\\\left(x^6+1\\\\right)}^{\\\\frac{4}{7}}$$","$$16{\\\\left(x^6+1\\\\right)}^{\\\\frac{1}{7}}$$"],"hints":{"DefaultPathway":[{"id":"aca598bexp3c-h1","type":"hint","dependencies":[],"title":"Conversion","text":"Convert radicals to exponents using the fact that $$\\\\sqrt[n]{y}=y^{\\\\frac{1}{n}}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp3c-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${\\\\left({\\\\left(128x^6+128\\\\right)}^4\\\\right)}^{\\\\frac{1}{7}}$$"],"dependencies":["aca598bexp3c-h1"],"title":"Conversion","text":"What is the result of $$\\\\sqrt[7]{{\\\\left(128x^6+128\\\\right)}^4}$$ after conversion?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$${\\\\left({\\\\left(128x^6+128\\\\right)}^4\\\\right)}^{\\\\frac{1}{7}}$$","$${\\\\left(128x^6+128\\\\right)}^{\\\\frac{1}{7}}$$"]},{"id":"aca598bexp3c-h3","type":"hint","dependencies":["aca598bexp3c-h2"],"title":"Property","text":"Use the property $${\\\\left(a^b\\\\right)}^c=a^{b c}$$ to simplify $${\\\\left({\\\\left(128x^6+128\\\\right)}^4\\\\right)}^{\\\\frac{1}{7}}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp3c-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${\\\\left(128x^6+128\\\\right)}^{\\\\frac{4}{7}}$$"],"dependencies":["aca598bexp3c-h3"],"title":"Property","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$${\\\\left(128x^6+128\\\\right)}^{\\\\frac{4}{7}}$$","$${\\\\left(128x^6+128\\\\right)}^{\\\\frac{1}{28}}$$","$${\\\\left(128x^6+128\\\\right)}^{\\\\frac{1}{7}}$$"]},{"id":"aca598bexp3c-h5","type":"hint","dependencies":["aca598bexp3c-h4"],"title":"Distributive Law","text":"Using the distributive law, we can change $${\\\\left(128x^6+128\\\\right)}^{\\\\frac{4}{7}}$$ to $${128}^{\\\\frac{4}{7}} {\\\\left(x^6+1\\\\right)}^{\\\\frac{4}{7}}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp3c-h6","type":"hint","dependencies":["aca598bexp3c-h5"],"title":"Simplification","text":"$$128=2^7$$, so $${128}^{\\\\frac{4}{7}} {\\\\left(x^6+1\\\\right)}^{\\\\frac{4}{7}}$$ is equivalent to $${\\\\left(2^7\\\\right)}^{{\\\\left(\\\\frac{1}{7}\\\\right)}^4} {\\\\left(x^6+1\\\\right)}^{\\\\frac{4}{7}}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp3c-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["aca598bexp3c-h6"],"title":"Simplification","text":"Is $${\\\\left(2^7\\\\right)}^{{\\\\left(\\\\frac{1}{7}\\\\right)}^4}$$ equivalent to $$2^4$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"aca598bexp3c-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["aca598bexp3c-h7"],"title":"Simplification","text":"What is the value of $$2^4$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp3c-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$16{\\\\left(x^6+1\\\\right)}^{\\\\frac{4}{7}}$$"],"dependencies":["aca598bexp3c-h8"],"title":"Simplification","text":"What is the answer in the form of $$B {\\\\left(x^6+1\\\\right)}^A$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$16{\\\\left(x^6+1\\\\right)}^{\\\\frac{4}{7}}$$","$$128{\\\\left(x^6+1\\\\right)}^{\\\\frac{4}{7}}$$","$$16{\\\\left(x^6+1\\\\right)}^{\\\\frac{1}{7}}$$"]}]}}]},{"id":"aca598bexp4","title":"Algebra with Exponents and Logarithms","body":"These problems are harder, often highlighting an important subtlety","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Algebra with Exponents and Logarithms","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"aca598bexp4a","stepAnswer":["$$\\\\frac{-1}{2}$$"],"problemType":"TextBox","stepTitle":"Find all values of $$x$$ such that $$\\\\frac{30}{x^2-9}+2=$$ $$\\\\frac{5}{x-3}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-1}{2}$$","hints":{"DefaultPathway":[{"id":"aca598bexp4a-h1","type":"hint","dependencies":[],"title":"Can\'t divide by $$0$$","text":"$$x^2-9 \\\\neq 0$$, $$x-3 \\\\neq 0$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp4a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3&-3$$"],"dependencies":["aca598bexp4a-h1"],"title":"Can\'t divide by $$0$$","text":"What values $$x$$ cannot be?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$3&-3$$","$$3$$","$$-3$$"]},{"id":"aca598bexp4a-h3","type":"hint","dependencies":["aca598bexp4a-h2"],"title":"LHS","text":"$$a=\\\\frac{a b}{b}$$, so $$\\\\frac{30}{x^2-9}+2=\\\\frac{30}{x^2-9}+\\\\frac{2\\\\left(x^2-9\\\\right)}{x^2-9}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp4a-h4","type":"hint","dependencies":["aca598bexp4a-h3"],"title":"LHS","text":"$$\\\\frac{a}{c}+\\\\frac{b}{c}=\\\\frac{a+b}{c}$$, so $$\\\\frac{30}{x^2-9}+\\\\frac{2\\\\left(x^2-9\\\\right)}{x^2-9}=\\\\frac{30+2\\\\left(x^2-9\\\\right)}{x^2-9}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp4a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2x^2+12$$"],"dependencies":["aca598bexp4a-h4"],"title":"LHS","text":"What is $$30+2\\\\left(x^2-9\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$2x^2+12$$","$$2x^2+42$$","$$2x^2-12$$"]},{"id":"aca598bexp4a-h6","type":"hint","dependencies":["aca598bexp4a-h5"],"title":"LHS","text":"$$LHS=\\\\frac{2x^2+12}{x^2-9}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp4a-h7","type":"hint","dependencies":["aca598bexp4a-h6"],"title":"RHS","text":"Multiply $$\\\\frac{x+3}{x+3}$$ to make the denominator of $$LHS=x^{-9}$$ also.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp4a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$5x+15$$"],"dependencies":["aca598bexp4a-h7"],"title":"RHS","text":"$$\\\\frac{5}{x-3} \\\\frac{x+3}{x+3}$$. In the numerator, what is $$5\\\\left(x+3\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$5x+15$$","$$5x-15$$"]},{"id":"aca598bexp4a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x^2-9$$"],"dependencies":["aca598bexp4a-h8"],"title":"RHS","text":"$$\\\\frac{5}{x-3} \\\\frac{x+3}{x+3}$$. In the denominator, what is $$\\\\left(x-3\\\\right) \\\\left(x+3\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$x^2-9$$","$$x^2+9$$"]},{"id":"aca598bexp4a-h10","type":"hint","dependencies":["aca598bexp4a-h9"],"title":"RHS","text":"$$RHS=\\\\frac{5x+15}{x^2-9}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp4a-h11","type":"hint","dependencies":["aca598bexp4a-h10"],"title":"$$LHS=RHS$$","text":"For $$x \\\\neq \\\\pm 3$$, $$\\\\frac{2x^2+12}{x^2-9}=\\\\frac{5x+15}{x^2-9}$$ means that $$2x^2+12=5x+15$$, which is $$2x^2-5x-3=0$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp4a-h12","type":"hint","dependencies":["aca598bexp4a-h11"],"title":"Solve the roots","text":"To solve $$2x^2-5x-3=0$$, use the formula $$x=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4a c}\\\\right)}{2a}$$ where $$a=2$$, $$b=-5$$, $$c=-3$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp4a-h13","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3$$ or $$\\\\frac{-1}{2}$$"],"dependencies":["aca598bexp4a-h12"],"title":"Solve the roots","text":"What is $$x$$ after solving the equation?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$3$$ or $$\\\\frac{-1}{2}$$","$$-3$$ or $$\\\\frac{-1}{2}$$","$$3$$ or $$\\\\frac{1}{2}$$","$$-3$$ or $$\\\\frac{1}{2}$$"],"subHints":[{"id":"aca598bexp4a-h13-s1","type":"hint","dependencies":[],"title":"Solve the roots","text":"$$\\\\frac{5\\\\pm \\\\sqrt{25+24}}{4}=\\\\frac{5\\\\pm \\\\sqrt{49}}{4}=\\\\frac{5\\\\pm 7}{4}$$","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"aca598bexp4a-h14","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["aca598bexp4a-h13"],"title":"Solve the roots","text":"Can we choose $$x=3$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"aca598bexp4a-h15","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{-1}{2}$$"],"dependencies":["aca598bexp4a-h14"],"title":"Conclusion","text":"What is $$x$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{-1}{2}$$","$$\\\\frac{1}{2}$$","$$3$$"]}]}}]},{"id":"aca598bexp5","title":"Algebra with Exponents and Logarithms","body":"These problems are harder, often highlighting an important subtlety","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Algebra with Exponents and Logarithms","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"aca598bexp5a","stepAnswer":["$$67$$"],"problemType":"TextBox","stepTitle":"Find all values of $$x$$ such that $$\\\\frac{{\\\\left(2x-6\\\\right)}^2}{\\\\sqrt[3]{x-3}}=4096$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$67$$","hints":{"DefaultPathway":[{"id":"aca598bexp5a-h1","type":"hint","dependencies":[],"title":"Can\'t divide by $$0$$","text":"$$\\\\sqrt[3]{x-3} \\\\neq 0$$, so $$x \\\\neq 3$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp5a-h2","type":"hint","dependencies":["aca598bexp5a-h1"],"title":"Distributive Law","text":"Using distributive law for LHS numerator, $${\\\\left(2x-6\\\\right)}^2=2^2 {\\\\left(x-3\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp5a-h3","type":"hint","dependencies":["aca598bexp5a-h2"],"title":"Conversion","text":"Convert radicals to exponents for LHS denominator using the fact that $$\\\\sqrt[n]{y}=y^{\\\\frac{1}{n}}$$, so $$\\\\sqrt[3]{x-3}={\\\\left(x-3\\\\right)}^{\\\\frac{1}{3}}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp5a-h4","type":"hint","dependencies":["aca598bexp5a-h3"],"title":"LHS","text":"We have $$\\\\frac{2^2 {\\\\left(x-3\\\\right)}^2}{{\\\\left(x-3\\\\right)}^{\\\\frac{1}{3}}}$$, which is equivalent to $$2^2 \\\\frac{{\\\\left(x-3\\\\right)}^2}{{\\\\left(x-3\\\\right)}^{\\\\frac{1}{3}}}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp5a-h5","type":"hint","dependencies":["aca598bexp5a-h4"],"title":"Property","text":"$$\\\\frac{a^b}{a^c}=a^{b-c}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp5a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2^2 {\\\\left(x-3\\\\right)}^{\\\\frac{5}{3}}$$"],"dependencies":["aca598bexp5a-h5"],"title":"Property","text":"Which option is equivalent to $$2^2 \\\\frac{{\\\\left(x-3\\\\right)}^2}{{\\\\left(x-3\\\\right)}^{\\\\frac{1}{3}}}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$2^2 {\\\\left(x-3\\\\right)}^{\\\\frac{5}{3}}$$","$$2^2 {\\\\left(x-3\\\\right)}^{\\\\frac{7}{3}}$$"]},{"id":"aca598bexp5a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["aca598bexp5a-h6"],"title":"RHS","text":"$$2^a=4096$$. What is the value of a?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp5a-h8","type":"hint","dependencies":["aca598bexp5a-h7"],"title":"Simplification","text":"We have $$2^2 {\\\\left(x-3\\\\right)}^{\\\\frac{5}{3}}=2^{12}$$. Divide both sides by $$2^2$$, we get $${\\\\left(x-3\\\\right)}^{\\\\frac{5}{3}}=2^{10}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp5a-h9","type":"hint","dependencies":["aca598bexp5a-h8"],"title":"Simplification","text":"To transform $${\\\\left(x-3\\\\right)}^{\\\\frac{5}{3}}$$ to $$x-3$$, we can raise both sides of the equation to the power of $$\\\\frac{3}{5}$$ to get $${\\\\left({\\\\left(x-3\\\\right)}^{\\\\frac{5}{3}}\\\\right)}^{\\\\frac{3}{5}}={\\\\left(2^{10}\\\\right)}^{\\\\frac{3}{5}}$$, which can be simplified to $$(x-3)=2^6$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp5a-h10","type":"hint","dependencies":["aca598bexp5a-h9"],"title":"Simplification","text":"Since $${\\\\left(a^b\\\\right)}^c=a^{b c}$$, $${\\\\left({\\\\left(x-3\\\\right)}^{\\\\frac{5}{3}}\\\\right)}^{\\\\frac{3}{5}}=\\\\frac{{\\\\left(2^{10}\\\\right)}^3}{5}$$ can be simplified to $$(x-3)=2^6$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp5a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$64$$"],"dependencies":["aca598bexp5a-h10"],"title":"Simplification","text":"What is $$2^6$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp5a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$67$$"],"dependencies":["aca598bexp5a-h11"],"title":"Solve the roots","text":"$$x-3=64$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"aca598bexp6","title":"Algebra with Exponents and Logarithms","body":"These problems are harder, often highlighting an important subtlety","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Algebra with Exponents and Logarithms","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"aca598bexp6a","stepAnswer":["$$\\\\log_{9}\\\\left(10\\\\right)$$"],"problemType":"MultipleChoice","stepTitle":"Find all values of $$x$$ such that $$3^{2x+2}$$ + $$9^x$$ $$=$$ $${81}^x$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\log_{9}\\\\left(10\\\\right)$$","choices":["$$\\\\log_{9}\\\\left(10\\\\right)$$","$$(1/2)*\\\\log_{9}\\\\left(10\\\\right)$$","$$0$$"],"hints":{"DefaultPathway":[{"id":"aca598bexp6a-h1","type":"hint","dependencies":[],"title":"LHS","text":"$$a^{b+c}=a^b a^c$$, so $$3^{2x+2}=3^2 3^{2x}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp6a-h2","type":"hint","dependencies":["aca598bexp6a-h1"],"title":"LHS","text":"$$a^{b c}={\\\\left(a^b\\\\right)}^c$$, so $$3^{2x}={\\\\left(3^2\\\\right)}^x=9^x$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp6a-h3","type":"hint","dependencies":["aca598bexp6a-h2"],"title":"LHS","text":"Therefore, $$3^{2x+2}$$ + $$9^x=3^2 3^{2x}+9^x=9{\\\\left(3^2\\\\right)}^x+9x=9\\\\times9^x+9^x=\\\\left(9+1\\\\right) 9^x=10\\\\times9^x$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp6a-h4","type":"hint","dependencies":["aca598bexp6a-h3"],"title":"RHS","text":"$${81}^x={\\\\left(9^2\\\\right)}^x=9^{2x}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp6a-h5","type":"hint","dependencies":["aca598bexp6a-h4"],"title":"$$LHS=RHS$$","text":"$$10\\\\times9^x=9^{2x}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp6a-h6","type":"hint","dependencies":["aca598bexp6a-h5"],"title":"Log Property","text":"If a, $$b$$, $$c>0$$, $$a=b$$ implies $$\\\\log_{c}\\\\left(a\\\\right)=\\\\log_{c}\\\\left(b\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp6a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["aca598bexp6a-h6"],"title":"Log Property","text":"Can $$10\\\\times9^x=9^{2x}$$ imply $$\\\\log_{9}\\\\left(10\\\\times9^x\\\\right)=\\\\log_{9}\\\\left(9^{2x}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"aca598bexp6a-h8","type":"hint","dependencies":["aca598bexp6a-h7"],"title":"Log Property","text":"$$\\\\log_{c}\\\\left(a b\\\\right)=\\\\log_{c}\\\\left(a\\\\right)+\\\\log_{c}\\\\left(b\\\\right)$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp6a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["aca598bexp6a-h8"],"title":"Log Property","text":"Is $$\\\\log_{9}\\\\left(10\\\\times9^x\\\\right)$$ equivalent to $$\\\\log_{9}\\\\left(10\\\\right)+\\\\log_{9}\\\\left(9^x\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"aca598bexp6a-h10","type":"hint","dependencies":["aca598bexp6a-h9"],"title":"Log Property","text":"$$\\\\log_{b}\\\\left(b^a\\\\right)=a$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp6a-h11","type":"hint","dependencies":["aca598bexp6a-h10"],"title":"Log Property","text":"Therefore, $$\\\\log_{9}\\\\left(9^x\\\\right)=x$$, $$\\\\log_{9}\\\\left(9^{2x}\\\\right)=2x$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp6a-h12","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\log_{9}\\\\left(10\\\\right)$$"],"dependencies":["aca598bexp6a-h11"],"title":"Simplification","text":"$$\\\\log_{9}\\\\left(10\\\\right)+x=2x$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\log_{9}\\\\left(10\\\\right)$$","$$(1/2)*\\\\log_{9}\\\\left(10\\\\right)$$","$$0$$"]}]}}]},{"id":"aca598bexp7","title":"Algebra with Exponents and Logarithms","body":"These questions are challenging, requiring mastery of each concept and their interrelations.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Algebra with Exponents and Logarithms","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"aca598bexp7a","stepAnswer":["$$-\\\\left(\\\\frac{1}{3}\\\\right)$$"],"problemType":"MultipleChoice","stepTitle":"Find all values of $$x$$ satisfying the equation $$\\\\frac{\\\\sqrt{7x^2+1}}{2x}=$$ $$-2$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-\\\\left(\\\\frac{1}{3}\\\\right)$$","choices":["$$-\\\\left(\\\\frac{1}{3}\\\\right)$$","$$\\\\frac{1}{3}$$","$$\\\\pm \\\\left(\\\\frac{1}{3}\\\\right)$$"],"hints":{"DefaultPathway":[{"id":"aca598bexp7a-h1","type":"hint","dependencies":[],"title":"Can\'t divide by $$0$$","text":"Since the denominator cannot be $$0$$, $$2x$$ cannot be $$0$$, which implies $$x$$ cannot be $$0$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp7a-h2","type":"hint","dependencies":["aca598bexp7a-h1"],"title":"Simplification","text":"To remove the denominator, we can multiply $$2x$$ on both sides of the equation.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp7a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\sqrt{7x^2+1}=-4x$$"],"dependencies":["aca598bexp7a-h2"],"title":"Simplification","text":"What is the result after multiplying $$2x$$ on both sides?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\sqrt{7x^2+1}=-4x$$","$$\\\\sqrt{7x^2+1}=4x$$"]},{"id":"aca598bexp7a-h4","type":"hint","dependencies":["aca598bexp7a-h3"],"title":"Simplification","text":"$$\\\\sqrt{7x^2+1} \\\\geq 0$$, so $$-4x \\\\geq 0$$ also.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp7a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x \\\\leq 0$$"],"dependencies":["aca598bexp7a-h4"],"title":"Simplification","text":"Which option can be implied from $$-4x \\\\geq 0$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$x \\\\leq 0$$","$$x \\\\geq 0$$"]},{"id":"aca598bexp7a-h6","type":"hint","dependencies":["aca598bexp7a-h5"],"title":"Simplification","text":"To remove the square root for the equation $$\\\\sqrt{7x^2+1}=-4x$$, we can square on both sides.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp7a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$7x^2+1=16x^2$$"],"dependencies":["aca598bexp7a-h6"],"title":"Simplification","text":"What is $${\\\\sqrt{7x^2+1}}^2={\\\\left(-4x\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$7x^2+1=16x^2$$","$$7x^2+1=-16x^2$$"]},{"id":"aca598bexp7a-h8","type":"hint","dependencies":["aca598bexp7a-h7"],"title":"Simplification","text":"$$7x^2+1=16x^2$$ is equivalent to $$9x^2=1$$, so $$x^2=\\\\frac{1}{9}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp7a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\pm \\\\left(\\\\frac{1}{3}\\\\right)$$"],"dependencies":["aca598bexp7a-h8"],"title":"Simplification","text":"Solve $$x^2=\\\\frac{1}{9}$$. Which option is correct for the value of $$x$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\pm \\\\left(\\\\frac{1}{3}\\\\right)$$","$$\\\\pm \\\\left(\\\\frac{1}{9}\\\\right)$$"]},{"id":"aca598bexp7a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-\\\\left(\\\\frac{1}{3}\\\\right)$$"],"dependencies":["aca598bexp7a-h9"],"title":"Simplification","text":"Since $$x<0$$, what is $$x$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$-\\\\left(\\\\frac{1}{3}\\\\right)$$","$$\\\\frac{1}{3}$$","$$\\\\pm \\\\left(\\\\frac{1}{3}\\\\right)$$"]}]}}]},{"id":"aca598bexp8","title":"Algebra with Exponents and Logarithms","body":"These questions are challenging, requiring mastery of each concept and their interrelations.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Algebra with Exponents and Logarithms","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"aca598bexp8a","stepAnswer":["$$A=1$$, $$R=\\\\frac{5}{2}$$"],"problemType":"MultipleChoice","stepTitle":"If $$x$$ < $$1$$, determine constants A and $$r$$ such that $$\\\\frac{{|x-1|}^3}{\\\\sqrt{1-x}}=A {\\\\left(1-x\\\\right)}^r$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$A=1$$, $$R=\\\\frac{5}{2}$$","choices":["$$A=1$$, $$R=\\\\frac{5}{2}$$","$$A=\\\\frac{5}{2}$$, $$R=1$$"],"hints":{"DefaultPathway":[{"id":"aca598bexp8a-h1","type":"hint","dependencies":[],"title":"Absolute value","text":"Since $$x<1$$, then $$x-1<0$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp8a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$1-x$$"],"dependencies":["aca598bexp8a-h1"],"title":"Absolute value","text":"What does $$|x-1|$$ equal to in this case?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$1-x$$","$$x-1$$"]},{"id":"aca598bexp8a-h3","type":"hint","dependencies":["aca598bexp8a-h2"],"title":"LHS","text":"Now we have $$\\\\frac{{\\\\left(1-x\\\\right)}^3}{\\\\sqrt{1-x}}$$, which is equivalent to $$\\\\frac{{\\\\left(1-x\\\\right)}^3}{{\\\\left(1-x\\\\right)}^{\\\\frac{1}{2}}}$$ after converting radicals to exponents.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp8a-h4","type":"hint","dependencies":["aca598bexp8a-h3"],"title":"LHS","text":"Since $$\\\\frac{a^b}{a^c}=a^{b-c}$$, $$\\\\frac{{\\\\left(1-x\\\\right)}^3}{{\\\\left(1-x\\\\right)}^{\\\\frac{1}{2}}}={\\\\left(1-x\\\\right)}^{3-\\\\frac{1}{2}}={\\\\left(1-x\\\\right)}^{\\\\frac{5}{2}}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp8a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$A=1$$, $$R=\\\\frac{5}{2}$$"],"dependencies":["aca598bexp8a-h4"],"title":"Simplification","text":"For our simplified expression $${\\\\left(1-x\\\\right)}^{\\\\frac{5}{2}}$$, what are values of A and R respectively?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$A=1$$, $$R=\\\\frac{5}{2}$$","$$A=\\\\frac{5}{2}$$, $$R=1$$"]}]}}]},{"id":"aca598bexp9","title":"Algebra with Exponents and Logarithms","body":"These questions are challenging, requiring mastery of each concept and their interrelations.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Algebra with Exponents and Logarithms","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"aca598bexp9a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"Find all values of $$x$$ satisfying the equation $$\\\\log_{7}(\\\\frac{49(x-1)**3}{x**3-3x-10})=2.$$","stepBody":"Hint: Use the laws of logarithms. Remember that for a positive base, one cannot take the logarithm of a negative number.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"aca598bexp9a-h1","type":"hint","dependencies":[],"title":"Hint","text":"By the hint, we know $$x-1>0$$, so $$x>1$$, and $$x^3-3x-10>0$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp9a-h2","type":"hint","dependencies":["aca598bexp9a-h1"],"title":"LHS","text":"$$b*(\\\\log_{c}\\\\left(a\\\\right))=\\\\log_{c}\\\\left(a^b\\\\right)$$, so $$\\\\log_{7}\\\\left(49\\\\right)+3*(\\\\log_{7}\\\\left(x-1\\\\right))-\\\\log_{7}\\\\left(x^3-3x-10\\\\right)=\\\\log_{7}\\\\left(49\\\\right)+\\\\log_{7}\\\\left({\\\\left(x-1\\\\right)}^3\\\\right)-\\\\log_{7}\\\\left(x^3-3x-10\\\\right)$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp9a-h3","type":"hint","dependencies":["aca598bexp9a-h2"],"title":"LHS","text":"Since $$\\\\log_{c}\\\\left(a b\\\\right)=\\\\log_{c}\\\\left(a\\\\right)+\\\\log_{c}\\\\left(b\\\\right)$$ and $$\\\\log_{c}\\\\left(\\\\frac{a}{b}\\\\right)=\\\\log_{c}\\\\left(a\\\\right)-\\\\log_{c}\\\\left(b\\\\right)$$, so $$\\\\log_{7}\\\\left(49\\\\right)+\\\\log_{7}\\\\left({\\\\left(x-1\\\\right)}^3\\\\right)-\\\\log_{7}\\\\left(x^3-3x-10\\\\right)=\\\\log_{7}\\\\left(\\\\frac{49{\\\\left(x-1\\\\right)}^3}{x^3-3x-10}\\\\right)$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp9a-h4","type":"hint","dependencies":["aca598bexp9a-h3"],"title":"$$LHS=RHS$$","text":"For $$a>0$$, $$\\\\log_{c}\\\\left(a\\\\right)=b$$ implies $$a=c^b$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp9a-h5","type":"hint","dependencies":["aca598bexp9a-h4"],"title":"Simplification","text":"$$\\\\log_{7}\\\\left(\\\\frac{49{\\\\left(x-1\\\\right)}^3}{x^3-3x-10}\\\\right)=2$$ is equivalent to $$\\\\frac{49{\\\\left(x-1\\\\right)}^3}{x^3-3x-10}=7^2=49$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp9a-h6","type":"hint","dependencies":["aca598bexp9a-h5"],"title":"Simplification","text":"Divide by $$49$$ on both sides, we get $$\\\\frac{{\\\\left(x-1\\\\right)}^3}{x^3-3x-10}=1$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp9a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${\\\\left(x-1\\\\right)}^3=x^3-3x-10$$"],"dependencies":["aca598bexp9a-h6"],"title":"Simplification","text":"Since we\'ve showed $$x^3-3x-10>0$$, which one is equivalent to $$\\\\frac{{\\\\left(x-1\\\\right)}^3}{x^3-3x-10}=1$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$${\\\\left(x-1\\\\right)}^3=x^3-3x-10$$","$$-\\\\left({\\\\left(x-1\\\\right)}^3\\\\right)=x^3-3x-10$$"]},{"id":"aca598bexp9a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x^3-3x^2+3x-1$$"],"dependencies":["aca598bexp9a-h7"],"title":"Simplification","text":"Which one is equivalent to $${\\\\left(x-1\\\\right)}^3$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$x^3-3x^2+3x-1$$","$$x^3+3x^2+3x-1$$","$$x^3+3x^2-3x-1$$","$$x^3-3x^2-3x-1$$"]},{"id":"aca598bexp9a-h9","type":"hint","dependencies":["aca598bexp9a-h8"],"title":"Simplification","text":"$$x^3-3x^2+3x-1=x^3-3x-10$$, so $$3x^2-6x-9=0$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp9a-h10","type":"hint","dependencies":["aca598bexp9a-h9"],"title":"Simplification","text":"Divide by $$3$$ on both sides, we get $$x^2-2x-3=0$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp9a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\left(x-3\\\\right) \\\\left(x+1\\\\right)$$"],"dependencies":["aca598bexp9a-h10"],"title":"Simplification","text":"$$x^2-2x-3$$ is equivalent to which option?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\left(x-3\\\\right) \\\\left(x+1\\\\right)$$","$$\\\\left(x+3\\\\right) \\\\left(x+1\\\\right)$$","$$\\\\left(x-3\\\\right) \\\\left(x-1\\\\right)$$","$$\\\\left(x+3\\\\right) \\\\left(x-1\\\\right)$$"]},{"id":"aca598bexp9a-h12","type":"hint","dependencies":["aca598bexp9a-h11"],"title":"Simplification","text":"$$\\\\left(x-3\\\\right) \\\\left(x+1\\\\right)=0$$, so $$x=3$$ or $$-1$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aca598bexp9a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["aca598bexp9a-h12"],"title":"Simplification","text":"Since $$x>1$$, what can $$x$$ be?","variabilization":{},"oer":"https://OATutor.io","license":"","subHints":[{"id":"aca598bexp9a-h13-s1","type":"hint","dependencies":[],"title":"Double Check","text":"Substitute $$x=3$$ to $$x^3-3x-10$$, we get $$3^3-3\\\\times3-10=8>0$$. So $$x=3$$ is a valid solution.","variabilization":{},"oer":"https://OATutor.io","license":""}]}]}}]},{"id":"acc652fPlotCorrelation","title":"Scatter Plots","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.2 Scatter Plots","courseName":"OpenStax: Introductory Stats","steps":[{"id":"acc652fPlotCorrelationa","stepAnswer":["Strong"],"problemType":"MultipleChoice","stepTitle":"Does the scatter plot have a strong or weak relation?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Strong","Weak"],"hints":{"DefaultPathway":[{"id":"acc652fPlotCorrelationa-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Straight"],"dependencies":[],"title":"Is the line straight or curved?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Straight","Curvy"]},{"id":"acc652fPlotCorrelationa-h2","type":"hint","dependencies":[],"title":"Correlation","text":"When examine the graph, it is clear that majority of the points are close to a linear line with equation $$y=\\\\frac{7}{9}$$ $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acc652fPlotDirection","title":"Scatter Plots","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.2 Scatter Plots","courseName":"OpenStax: Introductory Stats","steps":[{"id":"acc652fPlotDirectiona","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Does the scatter plot appear positive or negative??","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"acc652fPlotDirectiona-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Straight"],"dependencies":[],"title":"Is the line straight or curved?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Straight","Curvy"]}]}}]},{"id":"acc652fPlotShape","title":"Scatter Plots","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.2 Scatter Plots","courseName":"OpenStax: Introductory Stats","steps":[{"id":"acc652fPlotShapea","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Does the scatter plot appear linear?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"acc652fPlotShapea-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Straight"],"dependencies":[],"title":"Is the line straight or curved?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Straight","Curvy"]}]}}]},{"id":"acc652fregr1","title":"Best-Fit Line","body":"The idea behind finding the best-fit line is based on the assumption that the data are scattered about a straight line.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.2 Scatter Plots","courseName":"OpenStax: Introductory Stats","steps":[{"id":"acc652fregr1a","stepAnswer":["Linear Regression"],"problemType":"MultipleChoice","stepTitle":"What is the process of fitting the best-fit line is called?","stepBody":"","answerType":"string","variabilization":{},"choices":["Linear Regression","Linear Approximation","Linear Interpolation","Linear Correlation"],"hints":{"DefaultPathway":[{"id":"acc652fregr1a-h1","type":"hint","dependencies":[],"title":"Linear Regression","text":"Linear Regression is the process of finding the line that best fits the data on the plot.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acc652fregr10","title":"The Coefficient of Determination","body":"It represents the percent of variation in the dependent (predicted) variable $$y$$ that can be explained by variation in the independent (explanatory) variable $$x$$ using the regression $$(best-fit)$$ line.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.2 Scatter Plots","courseName":"OpenStax: Introductory Stats","steps":[{"id":"acc652fregr10a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Can a coefficient of determination be negative?","stepBody":"Choose the answer from the following.","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"acc652fregr10a-h1","type":"hint","dependencies":[],"title":"The variable $$r^2$$ is called the coefficient of determination and is the square of the correlation coefficient.","text":"The number to the second power is always a positive numebr.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acc652fregr11","title":"The Regression Equation","body":"The correlation coefficient, $$r$$, developed by Karl Pearson in the early 1900s, is numerical and provides a measure of strength and direction of the linear association between the independent variable $$x$$ and the dependent variable $$y$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.2 Scatter Plots","courseName":"OpenStax: Introductory Stats","steps":[{"id":"acc652fregr11a","stepAnswer":["Yes, there are enough data points and the value of $$r$$ is strong enough to show that there is a strong negative correlation between the data sets."],"problemType":"MultipleChoice","stepTitle":"When $$n$$ $$=$$ $$100$$ and $$r$$ $$=$$ $$-0.89$$, is there a significant correlation? Explain.","stepBody":"Choose the answer from the following.","answerType":"string","variabilization":{},"answerLatex":"Yes, there are enough data points and the value of $$r$$ is strong enough to show that there is a strong negative correlation between the data sets.","choices":["Yes, there are enough data points and the value of $$r$$ is strong enough to show that there is a strong negative correlation between the data sets.","No, there are not enough data points and the value of $$r$$ is not strong enough to show that there is a not strong negative correlation between the data sets.","No, there are enough data points but the value of $$r$$ is not strong enough to show that there is a strong negative correlation between the data sets.","Yes, there are not enough data points and the value of $$r$$ is small enough to show that there is a strong negative correlation between the data sets."],"hints":{"DefaultPathway":[{"id":"acc652fregr11a-h1","type":"hint","dependencies":[],"title":"What the value of $$r$$ tells us:","text":"If $$r$$ $$=$$ $$-1$$, there is perfect negative correlation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acc652fregr12","title":"Third Exam vs. Final Exam Example","body":"Slope: The slope of the line is $$b$$ $$=$$ $$4.83$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.2 Scatter Plots","courseName":"OpenStax: Introductory Stats","steps":[{"id":"acc652fregr12a","stepAnswer":["For a $$one-point$$ increase in the score on the third exam, the final exam score increases by $$4.83$$ points, on average."],"problemType":"MultipleChoice","stepTitle":"Interpret the slope.","stepBody":"Choose the best answer from the following.","answerType":"string","variabilization":{},"answerLatex":"For a one-point increase in the score on the third exam, the final exam score increases by $$4.83$$ points, on average.","choices":["For $$4.83$$ points increase in the score on the third exam, the final exam score increases by $$4.83$$ points, on average.","For $$4.83$$ points increase in the score on the third exam, the final exam score increases by one point, on average.","For a one point increase in the score on the third exam, the final exam score increases by $$4.83$$ points, on average.","For a one point increase in the score on the third exam, the final exam score increases by one point, on average.","For a $$one-point$$ increase in the score on the third exam, the final exam score increases by $$4.83$$ points, on average."],"hints":{"DefaultPathway":[{"id":"acc652fregr12a-h1","type":"hint","dependencies":[],"title":"Interpretation of the Slope","text":"The slope of the best-fit line tells us how the dependent variable (y) changes for every one unit increase in the independent (x) variable, on average.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acc652fregr13","title":"Third Exam vs. Final Exam Example","body":"The coefficient of determination is $$r^2$$ $$=$$ $$0.66312$$ $$=$$ $$0.4397$$","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.2 Scatter Plots","courseName":"OpenStax: Introductory Stats","steps":[{"id":"acc652fregr13a","stepAnswer":["Approximately 44% of the variation $$(0.4397$$ is approximately $$0.44)$$ in the $$final-exam$$ grades can be explained by the variation in the grades on the third exam, using the $$best-fit$$ regression line."],"problemType":"MultipleChoice","stepTitle":"Interpret of $$r^2$$","stepBody":"Choose the best answer from the following.","answerType":"string","variabilization":{},"answerLatex":"Approximately 44% of the variation $$(0.4397$$ is approximately $$0.44)$$ in the final-exam grades can be explained by the variation in the grades on the third exam, using the best-fit regression line.","choices":["Approximately 44% of the variation in the $$final-exam$$ grades can be explained by the variation in the grades on the third exam, using the $$best-fit$$ regression line.","Approximately 44% of the variation $$(0.4397$$ is approximately $$0.44)$$ in the $$final-exam$$ grades can be explained by the variation in the grades on the third exam, using the $$best-fit$$ regression line."],"hints":{"DefaultPathway":[{"id":"acc652fregr13a-h1","type":"hint","dependencies":[],"title":"The Coefficient of Determination","text":"$$r^2$$, when expressed as a percent, represents the percent of variation in the dependent (predicted) variable $$y$$ that can be explained by variation in the independent (explanatory) variable $$x$$ using the regression $$(best-fit)$$ line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acc652fregr14","title":"The Regression Equation","body":"The correlation coefficient, $$r$$, developed by Karl Pearson in the early 1900s, is numerical and provides a measure of strength and direction of the linear association between the independent variable $$x$$ and the dependent variable $$y$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.2 Scatter Plots","courseName":"OpenStax: Introductory Stats","steps":[{"id":"acc652fregr14a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"When $$n$$ $$=$$ $$2$$ and $$r$$ $$=$$ $$1$$, are the data significant?","stepBody":"Choose the answer from the following.","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"acc652fregr14a-h1","type":"hint","dependencies":[],"title":"What the value of $$r$$ tells us:","text":"If $$r$$ $$=$$ $$1$$, there is perfect positive correlation. However, there is not enough of data points to support to show that there is a strong positive correlation between the data sets.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acc652fregr15","title":"Least Squares Criteria for Best Fit","body":"The process of fitting the best-fit line is called linear regression.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.2 Scatter Plots","courseName":"OpenStax: Introductory Stats","steps":[{"id":"acc652fregr15a","stepAnswer":["$$Least-squares$$ Regression Line"],"problemType":"MultipleChoice","stepTitle":"What is the best fit line called when use the criteria of minimizing the sum of the squared errors (SSE)?","stepBody":"Choose the answer from the following.","answerType":"string","variabilization":{},"choices":["$$Least-squares$$ Regression Line","LASSO Regression Line","Linear Regression Line","Logistic Regression Line","Ridge Regression Line"],"hints":{"DefaultPathway":[{"id":"acc652fregr15a-h1","type":"hint","dependencies":[],"title":"Sum of the Squared Error","text":"The sum of the squared error is the sum of absolute value of a residual errors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acc652fregr16","title":"Least Squares Criteria for Best Fit","body":"The process of fitting the best-fit line is called linear regression.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.2 Scatter Plots","courseName":"OpenStax: Introductory Stats","steps":[{"id":"acc652fregr16a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Any other line you might choose would have a higher SSE than the best-fit line. Is this true?","stepBody":"Choose the best answer from the following.","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"acc652fregr16a-h1","type":"hint","dependencies":[],"title":"Sum of the Squared Error","text":"The criteria for the best fit line is that the sum of the squared errors (SSE) is minimized, that is, made as small as possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acc652fregr2","title":"Linear Regression","body":"The correlation coefficient, $$r$$, developed by Karl Pearson in the early 1900s, is numerical and provides a measure of strength and direction of the linear association between the independent variable $$x$$ and the dependent variable $$y$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.2 Scatter Plots","courseName":"OpenStax: Introductory Stats","steps":[{"id":"acc652fregr2a","stepAnswer":["It means that 72% of the variation in the dependent variable (y) can be explained by the variation in the independent variable (x)."],"problemType":"MultipleChoice","stepTitle":"Explain what it means when a correlation has an r2 of $$0.72$$.","stepBody":"Choose the answer from the following.","answerType":"string","variabilization":{},"choices":["It means that 72% of the variation in the dependent variable (y) can be explained by the variation in the independent variable (x).","It means that 72% of the distribution in the dependent variable (y) can be explained by the variation in the independent variable (x)."],"hints":{"DefaultPathway":[{"id":"acc652fregr2a-h1","type":"hint","dependencies":[],"title":"The Correlation Coefficient $$r$$","text":"The correlation coefficient is calculated as below, where $$n$$ $$=$$ the number of data points.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acc652fregr2a-h2","type":"hint","dependencies":["acc652fregr2a-h1"],"title":"Linear Relationship","text":"If you suspect a linear relationship between $$x$$ and $$y$$, then $$r$$ can measure how strong the linear relationship is.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acc652fregr3","title":"The Correlation Coefficient $$r$$","body":"The size of the correlation $$r$$ indicates the strength of the linear relationship between $$x$$ and $$y$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.2 Scatter Plots","courseName":"OpenStax: Introductory Stats","steps":[{"id":"acc652fregr3a","stepAnswer":["$$1$$"],"problemType":"MultipleChoice","stepTitle":"What is the upper bound of the correlation $$r$$? Choose the answer from the following.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$1$$","choices":["$$1$$","$$0$$","$$-1$$","\u221e","$$10$$"],"hints":{"DefaultPathway":[{"id":"acc652fregr3a-h1","type":"hint","dependencies":[],"title":"What the VALUE of $$r$$ tells us:","text":"If $$r$$ $$=$$ $$1$$, there is perfect positive correlation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acc652fregr4","title":"The Correlation Coefficient $$r$$","body":"The size of the correlation $$r$$ indicates the strength of the linear relationship between $$x$$ and $$y$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.2 Scatter Plots","courseName":"OpenStax: Introductory Stats","steps":[{"id":"acc652fregr4a","stepAnswer":["$$-1$$"],"problemType":"MultipleChoice","stepTitle":"What is the lower bound of the correlation $$r$$? Choose the answer from the following.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-1$$","choices":["$$1$$","$$0$$","$$-1$$","\u221e","$$10$$"],"hints":{"DefaultPathway":[{"id":"acc652fregr4a-h1","type":"hint","dependencies":[],"title":"What the VALUE of $$r$$ tells us:","text":"If $$r$$ $$=$$ $$-1$$, there is perfect negative correlation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acc652fregr5","title":"The Correlation Coefficient $$r$$","body":"If you suspect a linear relationship between $$x$$ and $$y$$, then $$r$$ can measure how strong the linear relationship is.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.2 Scatter Plots","courseName":"OpenStax: Introductory Stats","steps":[{"id":"acc652fregr5a","stepAnswer":["A positive value of $$r$$ means that when $$x$$ increases, $$y$$ tends to increase and when $$x$$ decreases, $$y$$ tends to decrease."],"problemType":"MultipleChoice","stepTitle":"What does the positive sign of $$r$$ tells us?","stepBody":"Choose the answer from the following.","answerType":"string","variabilization":{},"answerLatex":"A positive value of $$r$$ means that when $$x$$ increases, $$y$$ tends to increase and when $$x$$ decreases, $$y$$ tends to decrease.","choices":["A positive value of $$r$$ means that when $$x$$ increases, $$y$$ tends to increase and when $$x$$ decreases, $$y$$ tends to decrease.","A positive value of $$r$$ means that when $$x$$ increases, $$y$$ tends to decrease and when $$x$$ decreases, $$y$$ tends to increase."],"hints":{"DefaultPathway":[{"id":"acc652fregr5a-h1","type":"hint","dependencies":[],"title":"Correlation $$r$$","text":"The sign of $$r$$ is the same as the sign of the slope, $$b$$, of the best-fit line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acc652fregr6","title":"The Correlation Coefficient $$r$$","body":"If you suspect a linear relationship between $$x$$ and $$y$$, then $$r$$ can measure how strong the linear relationship is.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.2 Scatter Plots","courseName":"OpenStax: Introductory Stats","steps":[{"id":"acc652fregr6a","stepAnswer":["A negative value of $$r$$ means that when $$x$$ increases, $$y$$ tends to decrease and when $$x$$ decreases, $$y$$ tends to increase."],"problemType":"MultipleChoice","stepTitle":"What does the negative sign of $$r$$ tells us?","stepBody":"Choose the answer from the following.","answerType":"string","variabilization":{},"answerLatex":"A negative value of $$r$$ means that when $$x$$ increases, $$y$$ tends to decrease and when $$x$$ decreases, $$y$$ tends to increase.","choices":["A negative value of $$r$$ means that when $$x$$ increases, $$y$$ tends to increase and when $$x$$ decreases, $$y$$ tends to decrease.","A negative value of $$r$$ means that when $$x$$ increases, $$y$$ tends to decrease and when $$x$$ decreases, $$y$$ tends to increase."],"hints":{"DefaultPathway":[{"id":"acc652fregr6a-h1","type":"hint","dependencies":[],"title":"Correlation $$r$$","text":"The sign of $$r$$ is the same as the sign of the slope, $$b$$, of the best-fit line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acc652fregr7","title":"The Correlation Coefficient and Causation","body":"If you suspect a linear relationship between $$x$$ and $$y$$, then $$r$$ can measure how strong the linear relationship is.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.2 Scatter Plots","courseName":"OpenStax: Introductory Stats","steps":[{"id":"acc652fregr7a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"With a correlation coefficient of $$0.998$$, does this suggest that $$x$$ causes $$y$$ or $$y$$ causes $$x$$?","stepBody":"Choose the answer from the following.","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"acc652fregr7a-h1","type":"hint","dependencies":[],"title":"Relationship between correlation and causation","text":"Strong correlation does not suggest that $$x$$ causes $$y$$ or $$y$$ causes $$x$$. We say \\"correlation does not imply causation.\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acc652fregr8","title":"The Coefficient of Determination","body":"The variable r2 is called the coefficient of determination and is the square of the correlation coefficient.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.2 Scatter Plots","courseName":"OpenStax: Introductory Stats","steps":[{"id":"acc652fregr8a","stepAnswer":["It means that 72% of the variation in the dependent variable (y) can be explained by the variation in the independent variable (x)."],"problemType":"MultipleChoice","stepTitle":"Explain what it means when a correlation has an $$r^2$$ of $$0.72$$.","stepBody":"Choose the answer from the following.","answerType":"string","variabilization":{},"choices":["It means that 72% of the variation in the dependent variable (y) can be explained by the variation in the independent variable (x).","It means that 72% of the variation in the dependent variable (y) can not be explained by the variation in the independent variable (x)."],"hints":{"DefaultPathway":[{"id":"acc652fregr8a-h1","type":"hint","dependencies":[],"title":"$$r^2$$","text":"$$r^2$$ ythat can be explained by variation in the independent (explanatory) variable $$x$$ using the regression $$(best-fit)$$ line.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acc652fregr9","title":"The Correlation Coefficient $$r$$","body":"If you suspect a linear relationship between $$x$$ and $$y$$, then $$r$$ can measure how strong the linear relationship is.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"12.2 Scatter Plots","courseName":"OpenStax: Introductory Stats","steps":[{"id":"acc652fregr9a","stepAnswer":["It means that there is no correlation between the data sets."],"problemType":"MultipleChoice","stepTitle":"What does an $$r$$ value of zero mean?","stepBody":"Choose the answer from the following.","answerType":"string","variabilization":{},"choices":["It means that there is no correlation between the data sets.","It means that there is a positive correlation between the data sets.","It means that there is negative correlation between the data sets."],"hints":{"DefaultPathway":[{"id":"acc652fregr9a-h1","type":"hint","dependencies":[],"title":"What the value of $$r$$ tells us:","text":"If $$r$$ $$=$$ $$0$$ there is likely no linear correlation. It is important to view the scatterplot, however, because data that exhibit a curved or horizontal pattern may have a correlation of $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acccc99hypo1","title":"Calculate null hypothesis","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Null and Alternative Hypotheses","courseName":"OpenStax: Introductory Stats","steps":[{"id":"acccc99hypo1a","stepAnswer":["equal to $$2.0$$"],"problemType":"MultipleChoice","stepTitle":"We want to test whether the mean GPA of students in American colleges is different from $$2.0$$ (out of $$4.0)$$. What is the null hypothesis for this statement? Choose the most relevant choice","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"equal to $$2.0$$","choices":["not equal to $$2.0$$","equal to $$2.0$$"],"hints":{"DefaultPathway":[{"id":"acccc99hypo1a-h1","type":"hint","dependencies":[],"title":"Definition of Null Hypothesis","text":"It is a statement of no difference between the variables\u2014they are not related.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acccc99hypo2","title":"Calculate alternate hypothesis","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Null and Alternative Hypotheses","courseName":"OpenStax: Introductory Stats","steps":[{"id":"acccc99hypo2a","stepAnswer":["not equal to $$2.0$$"],"problemType":"MultipleChoice","stepTitle":"We want to test whether the mean GPA of students in American colleges is different from $$2.0$$ (out of $$4.0)$$. What is the alternate hypothesis for this statement? Choose the most relevant choice","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"not equal to $$2.0$$","choices":["not equal to $$2.0$$","equal to $$2.0$$"],"hints":{"DefaultPathway":[{"id":"acccc99hypo2a-h1","type":"hint","dependencies":[],"title":"Definition of Alternate Hypothesis","text":"It is a claim about the population that is contradictory to H0 and what we conclude when we reject H0.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acccc99hypo3","title":"Calculate null and alternate hypothesis","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Null and Alternative Hypotheses","courseName":"OpenStax: Introductory Stats","steps":[{"id":"acccc99hypo3a","stepAnswer":["$$(p \\\\leq 0.066$$, p>0.066)"],"problemType":"MultipleChoice","stepTitle":"In an issue of U.S. News and World Report, an article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third pass. The same article stated that $$6.6\\\\%$$ of U.S. students take advanced placement exams and $$4.4\\\\%$$ pass. Test if the percentage of U.S. students who take advanced placement exams is more than $$6.6\\\\%$$. State the null and alternative hypotheses. (p)","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(p \\\\leq 0.066$$, p>0.066)","choices":["$$(p \\\\leq 0.066$$, p>0.066)","(p<0.066, $$p \\\\geq 0.066)$$","(p>0.066, $$p \\\\geq 0.066)$$","$$(p \\\\geq 0.066$$, $$p \\\\geq 0.066)$$"],"hints":{"DefaultPathway":[{"id":"acccc99hypo3a-h1","type":"hint","dependencies":[],"title":"Definition of Null and Alternate Hypothesis","text":"Null: It is a statement of no difference between the variables\u2014they are not related; Alternate: It is a claim about the population that is contradictory to H0 and what we conclude when we reject H0.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acccc99hypo4","title":"Calculate null and alternate hypothesis","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Null and Alternative Hypotheses","courseName":"OpenStax: Introductory Stats","steps":[{"id":"acccc99hypo4a","stepAnswer":["$$(p \\\\geq 5$$, p<5)"],"problemType":"MultipleChoice","stepTitle":"We want to test if college students take less than five years to graduate from college, on the average. State the null and alternative hypotheses. (p)","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(p \\\\geq 5$$, p<5)","choices":["$$(p \\\\geq 5$$, p<5)","$$(p \\\\leq 5$$, p<5)","$$(p \\\\leq 5$$, $$p \\\\geq 5)$$","$$(p \\\\geq 5$$, $$p \\\\leq 5)$$"],"hints":{"DefaultPathway":[{"id":"acccc99hypo4a-h1","type":"hint","dependencies":[],"title":"Definition of Null and Alternate Hypothesis","text":"Null: It is a statement of no difference between the variables\u2014they are not related; Alternate: It is a claim about the population that is contradictory to H0 and what we conclude when we reject H0.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acccc99hypo5","title":"Calculate alternate hypothesis","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Null and Alternative Hypotheses","courseName":"OpenStax: Introductory Stats","steps":[{"id":"acccc99hypo5a","stepAnswer":["> $$0.30$$"],"problemType":"MultipleChoice","stepTitle":"Over the past few decades, public health officials have examined the link between weight concerns and teen girls\' smoking. Researchers surveyed a group of $$273$$ randomly selected teen girls living in Massachusetts (between $$12$$ and $$15$$ years old). After four years the girls were surveyed again. Sixty-three said they smoked to stay thin. Is there good evidence that more than thirty percent of the teen girls smoke to stay thin? What is the alternate hypothesis for this statement? (p) Choose the most relevant choice","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"> $$0.30$$","choices":["< $$0.30$$","$$ \\\\leq $$ $$0.30$$","> $$0.30$$","$$ \\\\geq $$ $$0.30$$"],"hints":{"DefaultPathway":[{"id":"acccc99hypo5a-h1","type":"hint","dependencies":[],"title":"Definition of Alternate Hypothesis","text":"It is a claim about the population that is contradictory to H0 and what we conclude when we reject H0.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acccc99hypo6","title":"Calculate alternate hypothesis","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Null and Alternative Hypotheses","courseName":"OpenStax: Introductory Stats","steps":[{"id":"acccc99hypo6a","stepAnswer":["(< $$0.20)$$"],"problemType":"MultipleChoice","stepTitle":"A statistics instructor believes that fewer than 20% of Evergreen Valley College (EVC) students attended the opening night midnight showing of the latest Harry Potter movie. She surveys $$84$$ of her students and finds that $$11$$ attended the midnight showing. What is the alternate hypothesis for this statement? (p) Choose the most relevant choice","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"(< $$0.20)$$","choices":["$$(=$$ $$0.20)$$","(> $$0.20)$$","(< $$0.20)$$","$$( \\\\leq $$ $$0.20)$$"],"hints":{"DefaultPathway":[{"id":"acccc99hypo6a-h1","type":"hint","dependencies":[],"title":"Definition of Alternate Hypothesis","text":"It is a claim about the population that is contradictory to H0 and what we conclude when we reject H0.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acebb4ddomain1","title":"Finding the Domain and Range of a Relation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Relations and Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"acebb4ddomain1a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"Find the highest domain of the relation {$$(1,1),(2,4),(3,9),(4,16),(5,25)$$}","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"acebb4ddomain1a-h1","type":"hint","dependencies":[],"title":"What is Domain?","text":"Domain is the set of x-values of a relation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acebb4ddomain1a-h2","type":"hint","dependencies":["acebb4ddomain1a-h1"],"title":"Finding the Highest Number in the Domain","text":"The highest x-value is $$5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acebb4ddomain10","title":"Finding the Domain and Range of a Relation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Relations and Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"acebb4ddomain10a","stepAnswer":["{(Khan Nguyen, kn68413), (Abigail Brown, ab56781), (Sumantha Mishal, sm32479), (Jose Hernandez, jh47983)}"],"problemType":"TextBox","stepTitle":"For the mapping shown, list the ordered pairs.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"acebb4ddomain10a-h1","type":"hint","dependencies":[],"title":"Mapping Notation","text":"Each arrow matches the person to their Student ID.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acebb4ddomain10a-h2","type":"hint","dependencies":["acebb4ddomain10a-h1"],"title":"Creating Ordered Pairs","text":"We create ordered pairs with the person\u2019s name as the x-value and their Student ID as the y-value. We get {(Khan Nguyen, kn68413), (Abigail Brown, ab56781), (Sumantha Mishal, sm32479), (Jose Hernandez, jh47983)}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acebb4ddomain11","title":"Finding the Domain and Range of a Relation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Relations and Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"acebb4ddomain11a","stepAnswer":["{Khan Nguyen, Abigail Brown, Sumantha Mishal, Jose Hernandez}"],"problemType":"TextBox","stepTitle":"For the mapping shown, find the domain.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"acebb4ddomain11a-h1","type":"hint","dependencies":[],"title":"What is Domain?","text":"The domain is the set of all x-values of the relation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acebb4ddomain11a-h2","type":"hint","dependencies":["acebb4ddomain11a-h1"],"title":"Listing the Domain","text":"We can list the domain as follows: {Khan Nguyen, Abigail Brown, Sumantha Mishal, Jose Hernandez}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acebb4ddomain12","title":"Finding the Domain and Range of a Relation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Relations and Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"acebb4ddomain12a","stepAnswer":["{ab56781, jh47983, kn68413, sm32479}"],"problemType":"TextBox","stepTitle":"For the mapping shown, find the range.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"acebb4ddomain12a-h1","type":"hint","dependencies":[],"title":"What is Range?","text":"The range is the set of all y-values of the relation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acebb4ddomain12a-h2","type":"hint","dependencies":["acebb4ddomain12a-h1"],"title":"Listing the Range","text":"We can list the range as follows: {ab56781, jh47983, kn68413, sm32479}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acebb4ddomain13","title":"Finding the Domain and Range of a Relation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Relations and Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"acebb4ddomain13a","stepAnswer":["{(Maria, November 6), (Armando, January 18), (Cynthia, December 8), (Kelly, March 15), (Rachel, November 6)}"],"problemType":"TextBox","stepTitle":"For the mapping shown, list the ordered pairs.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"acebb4ddomain13a-h1","type":"hint","dependencies":[],"title":"Mapping Notation","text":"Each arrow matches the person to their birthday.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acebb4ddomain13a-h2","type":"hint","dependencies":["acebb4ddomain13a-h1"],"title":"Creating Ordered Pairs","text":"We create ordered pairs with the person\u2019s name as the x-value and their birthday as the y-value. We get {(Maria, November 6), (Armando, January 18), (Cynthia, December 8), (Kelly, March 15), (Rachel, November 6)}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acebb4ddomain14","title":"Finding the Domain and Range of a Relation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Relations and Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"acebb4ddomain14a","stepAnswer":["{Maria, Armando, Cynthia, Kelly, Rachel}"],"problemType":"TextBox","stepTitle":"For the mapping shown, find the domain.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"acebb4ddomain14a-h1","type":"hint","dependencies":[],"title":"What is Domain?","text":"The domain is the set of all x-values of the relation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acebb4ddomain14a-h2","type":"hint","dependencies":["acebb4ddomain14a-h1"],"title":"Listing the Domain","text":"We can list the domain as follows: {Maria, Armando, Cynthia, Kelly, Rachel}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acebb4ddomain15","title":"Finding the Domain and Range of a Relation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Relations and Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"acebb4ddomain15a","stepAnswer":["{January 18, March 15, November 6, December 8}"],"problemType":"TextBox","stepTitle":"For the mapping shown, find the range.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"{January $$18$$, March $$15$$, November $$6$$, December 8}","hints":{"DefaultPathway":[{"id":"acebb4ddomain15a-h1","type":"hint","dependencies":[],"title":"What is Range?","text":"The range is the set of all y-values of the relation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acebb4ddomain15a-h2","type":"hint","dependencies":["acebb4ddomain15a-h1"],"title":"Listing the Range","text":"We can list the range as follows: {January $$18$$, March $$15$$, November $$6$$, December 8}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acebb4ddomain2","title":"Finding the Domain and Range of a Relation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Relations and Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"acebb4ddomain2a","stepAnswer":["$$25$$"],"problemType":"TextBox","stepTitle":"Find the highest range of the relation {$$(1,1),(2,4),(3,9),(4,16),(5,25)$$}","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$25$$","hints":{"DefaultPathway":[{"id":"acebb4ddomain2a-h1","type":"hint","dependencies":[],"title":"What is Range?","text":"Range is the set of y-values of a relation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acebb4ddomain2a-h2","type":"hint","dependencies":["acebb4ddomain2a-h1"],"title":"Finding the Highest Number in the Range","text":"The highest y-value is $$25$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acebb4ddomain3","title":"Finding the Domain and Range of a Relation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Relations and Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"acebb4ddomain3a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"Find the highest domain of the relation {$$(1,1),(2,8),(3,27),(4,64),(5,125)$$}","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"acebb4ddomain3a-h1","type":"hint","dependencies":[],"title":"What is Domain?","text":"Domain is the set of x-values of a relation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acebb4ddomain3a-h2","type":"hint","dependencies":["acebb4ddomain3a-h1"],"title":"Finding the Highest Number in the Domain","text":"The highest x-value is $$5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acebb4ddomain4","title":"Finding the Domain and Range of a Relation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Relations and Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"acebb4ddomain4a","stepAnswer":["$$125$$"],"problemType":"TextBox","stepTitle":"Find the highest range of the relation {$$(1,1),(2,8),(3,27),(4,64),(5,125)$$}","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$125$$","hints":{"DefaultPathway":[{"id":"acebb4ddomain4a-h1","type":"hint","dependencies":[],"title":"What is Range?","text":"Range is the set of y-values of a relation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acebb4ddomain4a-h2","type":"hint","dependencies":["acebb4ddomain4a-h1"],"title":"Finding the Highest Number in the Range","text":"The highest y-value is $$125$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acebb4ddomain5","title":"Finding the Domain and Range of a Relation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Relations and Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"acebb4ddomain5a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"Find the highest domain of the relation {$$(1,3),(2,6),(3,9),(4,12),(5,15)$$}","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"acebb4ddomain5a-h1","type":"hint","dependencies":[],"title":"What is Domain?","text":"Domain is the set of x-values of a relation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acebb4ddomain5a-h2","type":"hint","dependencies":["acebb4ddomain5a-h1"],"title":"Finding the Highest Number in the Domain","text":"The highest x-value is $$5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acebb4ddomain6","title":"Finding the Domain and Range of a Relation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Relations and Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"acebb4ddomain6a","stepAnswer":["$$15$$"],"problemType":"TextBox","stepTitle":"Find the highest range of the relation {$$(1,3),(2,6),(3,9),(4,12),(5,15)$$}","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$15$$","hints":{"DefaultPathway":[{"id":"acebb4ddomain6a-h1","type":"hint","dependencies":[],"title":"What is Range?","text":"Range is the set of y-values of a relation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acebb4ddomain6a-h2","type":"hint","dependencies":["acebb4ddomain6a-h1"],"title":"Finding the Highest Number in the Range","text":"The highest y-value is $$15$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acebb4ddomain7","title":"Finding the Domain and Range of a Relation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Relations and Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"acebb4ddomain7a","stepAnswer":["{(Alison, April 25), (Penelope, May 23), (June, August 2), (Gregory, September 15), (Geoffrey, January 12), (Lauren, May 10), (Stephen, July 24), (Alice, February 3), (Liz, August 2), (Danny, July 24)}"],"problemType":"TextBox","stepTitle":"For the mapping shown, list the ordered pairs.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"acebb4ddomain7a-h1","type":"hint","dependencies":[],"title":"Mapping Notation","text":"Each arrow matches the person to their birthday.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acebb4ddomain7a-h2","type":"hint","dependencies":["acebb4ddomain7a-h1"],"title":"Creating Ordered Pairs","text":"We create ordered pairs with the person\u2019s name as the x-value and their birthday as the y-value. We get {(Alison, April 25), (Penelope, May 23), (June, August 2), (Gregory, September 15), (Geoffrey, January 12), (Lauren, May 10), (Stephen, July 24), (Alice, February 3), (Liz, August 2), (Danny, July 24)}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acebb4ddomain8","title":"Finding the Domain and Range of a Relation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Relations and Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"acebb4ddomain8a","stepAnswer":["{Alison, Penelope, June, Gregory, Geoffrey, Lauren, Stephen, Alice, Liz, Danny}"],"problemType":"TextBox","stepTitle":"For the mapping shown, find the domain.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"acebb4ddomain8a-h1","type":"hint","dependencies":[],"title":"What is Domain?","text":"The domain is the set of all x-values of the relation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acebb4ddomain8a-h2","type":"hint","dependencies":["acebb4ddomain8a-h1"],"title":"Listing the Domain","text":"We can list the domain as follows: {Alison, Penelope, June, Gregory, Geoffrey, Lauren, Stephen, Alice, Liz, Danny}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acebb4ddomain9","title":"Finding the Domain and Range of a Relation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Relations and Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"acebb4ddomain9a","stepAnswer":["{January 12, February 3, April 25, May 10, May 23, July 24, August 2, September 15}"],"problemType":"TextBox","stepTitle":"For the mapping shown, find the range.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"answerLatex":"{January $$12$$, February $$3$$, April $$25$$, May $$10$$, May $$23$$, July $$24$$, August $$2$$, September 15}","hints":{"DefaultPathway":[{"id":"acebb4ddomain9a-h1","type":"hint","dependencies":[],"title":"What is Range?","text":"The range is the set of all y-values of the relation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acebb4ddomain9a-h2","type":"hint","dependencies":["acebb4ddomain9a-h1"],"title":"Listing the Range","text":"We can list the range as follows: {January $$12$$, February $$3$$, April $$25$$, May $$10$$, May $$23$$, July $$24$$, August $$2$$, September 15}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acebb4dfunctions1","title":"Evaluate at f(2)","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Relations and Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"acebb4dfunctions1a","stepAnswer":["$$7$$"],"problemType":"TextBox","stepTitle":"$$f(x)=5x-3$$","stepBody":"What is f(2)?","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7$$","hints":{"DefaultPathway":[{"id":"acebb4dfunctions1a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$2$$ into f(x). Do this by replacing every $$x$$ in the function with $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acebb4dfunctions1a-h2","type":"hint","dependencies":["acebb4dfunctions1a-h1"],"title":"Simplify","text":"Simplify the equation into one value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acebb4dfunctions10","title":"Evaluate at $$g{\\\\left(x+2\\\\right)}$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Relations and Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"acebb4dfunctions10a","stepAnswer":["$$-3x-8$$"],"problemType":"TextBox","stepTitle":"$$g(x)=-3x-2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-3x-8$$","hints":{"DefaultPathway":[{"id":"acebb4dfunctions10a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$x+2$$ into g(x). Do this by replacing every $$x$$ in the function with $$x+2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acebb4dfunctions10a-h2","type":"hint","dependencies":["acebb4dfunctions10a-h1"],"title":"Simplify","text":"Simplify the equation as much as possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acebb4dfunctions11","title":"Evaluate at $$g{\\\\left(x+2\\\\right)}$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Relations and Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"acebb4dfunctions11a","stepAnswer":["$$-x+1$$"],"problemType":"TextBox","stepTitle":"$$g(x)=3-x$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-x+1$$","hints":{"DefaultPathway":[{"id":"acebb4dfunctions11a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$x+2$$ into g(x). Do this by replacing every $$x$$ in the function with $$x+2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acebb4dfunctions11a-h2","type":"hint","dependencies":["acebb4dfunctions11a-h1"],"title":"Simplify","text":"Simplify the equation as much as possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acebb4dfunctions12","title":"Evaluate $$g{\\\\left(x\\\\right)}+g{\\\\left(2\\\\right)}$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Relations and Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"acebb4dfunctions12a","stepAnswer":["$$2x+6$$"],"problemType":"TextBox","stepTitle":"$$g(x)=2x+1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2x+6$$","hints":{"DefaultPathway":[{"id":"acebb4dfunctions12a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Add the function given to g(2). For g(2), replace every $$x$$ in the function with $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acebb4dfunctions12a-h2","type":"hint","dependencies":["acebb4dfunctions12a-h1"],"title":"Simplify","text":"Simplify the equation as much as possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acebb4dfunctions13","title":"Evaluate $$g{\\\\left(x\\\\right)}+g{\\\\left(2\\\\right)}$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Relations and Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"acebb4dfunctions13a","stepAnswer":["$$-3x-10$$"],"problemType":"TextBox","stepTitle":"$$g(x)=-3x-2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-3x-10$$","hints":{"DefaultPathway":[{"id":"acebb4dfunctions13a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Add the function given to g(2). For g(2), replace every $$x$$ in the function with $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acebb4dfunctions13a-h2","type":"hint","dependencies":["acebb4dfunctions13a-h1"],"title":"Simplify","text":"Simplify the equation as much as possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acebb4dfunctions14","title":"Evaluate $$g{\\\\left(x\\\\right)}+g{\\\\left(2\\\\right)}$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Relations and Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"acebb4dfunctions14a","stepAnswer":["$$4-x$$"],"problemType":"TextBox","stepTitle":"$$g(x)=3-x$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4-x$$","hints":{"DefaultPathway":[{"id":"acebb4dfunctions14a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Add the function given to g(2). For g(2), replace every $$x$$ in the function with $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acebb4dfunctions14a-h2","type":"hint","dependencies":["acebb4dfunctions14a-h1"],"title":"Simplify","text":"Simplify the equation as much as possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acebb4dfunctions15","title":"Evaluate $$g{\\\\left(x\\\\right)}+g{\\\\left(2\\\\right)}$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Relations and Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"acebb4dfunctions15a","stepAnswer":["$$4-5x$$"],"problemType":"TextBox","stepTitle":"$$g(x)=7-5x$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4-5x$$","hints":{"DefaultPathway":[{"id":"acebb4dfunctions15a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Add the function given to g(2). For g(2), replace every $$x$$ in the function with $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acebb4dfunctions15a-h2","type":"hint","dependencies":["acebb4dfunctions15a-h1"],"title":"Simplify","text":"Simplify the equation as much as possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acebb4dfunctions2","title":"Evaluate at f(2)","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Relations and Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"acebb4dfunctions2a","stepAnswer":["$$-6$$"],"problemType":"TextBox","stepTitle":"$$f(x)=-4x+2$$","stepBody":"What is f(2)?","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-6$$","hints":{"DefaultPathway":[{"id":"acebb4dfunctions2a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$2$$ into f(x). Do this by replacing every $$x$$ in the function with $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acebb4dfunctions2a-h2","type":"hint","dependencies":["acebb4dfunctions2a-h1"],"title":"Simplify","text":"Simplify the equation into one value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acebb4dfunctions3","title":"Evaluate at f(2)","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Relations and Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"acebb4dfunctions3a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"$$f(x)=x^2-x+3$$","stepBody":"What is f(2)?","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"acebb4dfunctions3a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$2$$ into f(x). Do this by replacing every $$x$$ in the function with $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acebb4dfunctions3a-h2","type":"hint","dependencies":["acebb4dfunctions3a-h1"],"title":"Simplify","text":"Simplify the equation into one value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acebb4dfunctions4","title":"Evaluate at f(2)","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Relations and Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"acebb4dfunctions4a","stepAnswer":["$$9$$"],"problemType":"TextBox","stepTitle":"$$2x^2-x+3$$","stepBody":"What is f(2)?","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9$$","hints":{"DefaultPathway":[{"id":"acebb4dfunctions4a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$2$$ into f(x). Do this by replacing every $$x$$ in the function with $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acebb4dfunctions4a-h2","type":"hint","dependencies":["acebb4dfunctions4a-h1"],"title":"Simplify","text":"Simplify the equation into one value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acebb4dfunctions5","title":"Evaluate at $$f(-1)$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Relations and Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"acebb4dfunctions5a","stepAnswer":["$$-8$$"],"problemType":"TextBox","stepTitle":"$$f(x)=5x-3$$","stepBody":"What is $$f(-1)$$?","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-8$$","hints":{"DefaultPathway":[{"id":"acebb4dfunctions5a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$-1$$ into f(x). Do this by replacing every $$x$$ in the function with $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acebb4dfunctions5a-h2","type":"hint","dependencies":["acebb4dfunctions5a-h1"],"title":"Simplify","text":"Simplify the equation into one value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acebb4dfunctions6","title":"Evaluate at $$g{\\\\left(h^2\\\\right)}$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Relations and Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"acebb4dfunctions6a","stepAnswer":["$$2h^2+1$$"],"problemType":"TextBox","stepTitle":"$$g(x)=2x+1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2h^2+1$$","hints":{"DefaultPathway":[{"id":"acebb4dfunctions6a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$h^2$$ into g(x). Do this by replacing every $$x$$ in the function with $$h^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acebb4dfunctions6a-h2","type":"hint","dependencies":["acebb4dfunctions6a-h1"],"title":"Simplify","text":"Simplify the equation as much as possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acebb4dfunctions7","title":"Evaluate at $$g{\\\\left(h^2\\\\right)}$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Relations and Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"acebb4dfunctions7a","stepAnswer":["$$-3h^2-2$$"],"problemType":"TextBox","stepTitle":"$$g(x)=-3x-2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-3h^2-2$$","hints":{"DefaultPathway":[{"id":"acebb4dfunctions7a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$h^2$$ into g(x). Do this by replacing every $$x$$ in the function with $$h^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acebb4dfunctions7a-h2","type":"hint","dependencies":["acebb4dfunctions7a-h1"],"title":"Simplify","text":"Simplify the equation as much as possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acebb4dfunctions8","title":"Evaluate at $$g{\\\\left(h^2\\\\right)}$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Relations and Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"acebb4dfunctions8a","stepAnswer":["$$3-h^2$$"],"problemType":"TextBox","stepTitle":"$$g(x)=3-x$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3-h^2$$","hints":{"DefaultPathway":[{"id":"acebb4dfunctions8a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$h^2$$ into g(x). Do this by replacing every $$x$$ in the function with $$h^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acebb4dfunctions8a-h2","type":"hint","dependencies":["acebb4dfunctions8a-h1"],"title":"Simplify","text":"Simplify the equation as much as possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acebb4dfunctions9","title":"Evaluate at $$g{\\\\left(x+2\\\\right)}$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.5 Relations and Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"acebb4dfunctions9a","stepAnswer":["$$2x+5$$"],"problemType":"TextBox","stepTitle":"$$g(x)=2x+1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2x+5$$","hints":{"DefaultPathway":[{"id":"acebb4dfunctions9a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$x+2$$ into g(x). Do this by replacing every $$x$$ in the function with $$x+2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acebb4dfunctions9a-h2","type":"hint","dependencies":["acebb4dfunctions9a-h1"],"title":"Simplify","text":"Simplify the equation as much as possible.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acf3da9gaussian1","title":"Writing the Augmented Matrix for a System of Equations","body":"Write the augmented matrix for the given system of equations.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acf3da9gaussian1a","stepAnswer":["$$\\\\begin{bmatrix} 1 & 2 & -1 & 3 \\\\\\\\ 2 & -1 & 2 & 6 \\\\\\\\ 1 & -3 & 3 & 4 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"$$x+2y-z=3$$ $$2x-y+2z=6$$ $$x-3y+3z=4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 1 & 2 & -1 & 3 \\\\\\\\ 2 & -1 & 2 & 6 \\\\\\\\ 1 & -3 & 3 & 4 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"acf3da9gaussian1a-h1","type":"hint","dependencies":[],"title":"Definition","text":"The augmented matrix displays the coefficients of the variables, and an additional column for the constants.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian1a-h2","type":"hint","dependencies":["acf3da9gaussian1a-h1"],"title":"Write","text":"The first row has the values 1,2,-1,3.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian1a-h3","type":"hint","dependencies":["acf3da9gaussian1a-h2"],"title":"Answer","text":"The answer is $$\\\\begin{bmatrix} 1 & 2 & -1 & 3 \\\\\\\\ 2 & -1 & 2 & 6 \\\\\\\\ 1 & -3 & 3 & 4 \\\\end{bmatrix}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acf3da9gaussian13","title":"Solving Quadratic Equations by Factoring","body":"Find the sum of the following equation\'s solutions by factoring:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acf3da9gaussian13a","stepAnswer":["$$-7$$"],"problemType":"TextBox","stepTitle":"$$x^2+7x+12$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-7$$","hints":{"DefaultPathway":[{"id":"acf3da9gaussian13a-h1","type":"hint","dependencies":[],"title":"The Zero Product Property","text":"The zero product property says if the product of two quantities is zero, it must be that at least one of the quantities is zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian13a-h2","type":"hint","dependencies":["acf3da9gaussian13a-h1"],"title":"Factoring the Equation","text":"We can factor $$x^2+7x+12$$ to use the zero product property. $$x^2+7x+12=\\\\left(x+4\\\\right) \\\\left(x+3\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian13a-h3","type":"hint","dependencies":["acf3da9gaussian13a-h2"],"title":"Using the Zero Product Property","text":"As per the zero product property, if $$\\\\left(x+4\\\\right) \\\\left(x+3\\\\right)=0$$, then either $$x+4$$ or $$x+3$$ is equal to $$0$$. Setting both of these to $$0$$, we get $$x=-4$$ and $$x=-3$$. Summing these values, we get $$-4-3=-7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acf3da9gaussian14","title":"Solving Quadratic Equations by Factoring","body":"Find the sum of the following equation\'s solutions by factoring:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acf3da9gaussian14a","stepAnswer":["$$8$$"],"problemType":"TextBox","stepTitle":"$$y^2-8y+15=0$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8$$","hints":{"DefaultPathway":[{"id":"acf3da9gaussian14a-h1","type":"hint","dependencies":[],"title":"The Zero Product Property","text":"The zero product property says if the product of two quantities is zero, it must be that at least one of the quantities is zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian14a-h2","type":"hint","dependencies":["acf3da9gaussian14a-h1"],"title":"Factoring the Equation","text":"We can factor $$y^2-8y+15=0$$ to use the zero product property. $$y^2-8y+15=(y-5)(y-3)=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian14a-h3","type":"hint","dependencies":["acf3da9gaussian14a-h2"],"title":"Using the Zero Product Property","text":"As per the zero product property, if $$(y-5)(y-3)$$, then either $$y-5$$ or $$y-3$$ is equal to $$0$$. Setting both of these to $$0$$, we get $$y=5$$ and $$x=3$$. Summing these values, we get $$8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acf3da9gaussian15","title":"Solving Quadratic Equations by Factoring","body":"Find the sum of the following equation\'s solutions by factoring:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acf3da9gaussian15a","stepAnswer":["$$5.2$$"],"problemType":"TextBox","stepTitle":"$$5a^2-26a=24$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5.2$$","hints":{"DefaultPathway":[{"id":"acf3da9gaussian15a-h1","type":"hint","dependencies":[],"title":"The Zero Product Property","text":"The zero product property says if the product of two quantities is zero, it must be that at least one of the quantities is zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian15a-h2","type":"hint","dependencies":["acf3da9gaussian15a-h1"],"title":"Factoring the Equation","text":"We can factor $$5a^2-26-24=0$$ to use the zero product property. $$5a^2-26a-24=\\\\left(5a+4\\\\right) \\\\left(a-6\\\\right)=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian15a-h3","type":"hint","dependencies":["acf3da9gaussian15a-h2"],"title":"Using the Zero Product Property","text":"As per the zero product property, if $$\\\\left(5a+4\\\\right) \\\\left(a-6\\\\right)$$, then either $$5a+4$$ or a-6 is equal to $$0$$. Setting both of these to $$0$$, we get $$a=\\\\frac{-4}{5}$$ and $$a=6$$. Summing these values, we get $$5.2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acf3da9gaussian16","title":"Solving Quadratic Equations by Factoring","body":"Find the sum of the following equation\'s solutions by factoring:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acf3da9gaussian16a","stepAnswer":["$$-1.75$$"],"problemType":"TextBox","stepTitle":"$$4b^2+7b=-3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1.75$$","hints":{"DefaultPathway":[{"id":"acf3da9gaussian16a-h1","type":"hint","dependencies":[],"title":"The Zero Product Property","text":"The zero product property says if the product of two quantities is zero, it must be that at least one of the quantities is zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian16a-h2","type":"hint","dependencies":["acf3da9gaussian16a-h1"],"title":"Factoring the Equation","text":"We can factor $$4b^2+7b+3$$ to use the zero product property. $$4b^2+7b+3=\\\\left(4b+3\\\\right) \\\\left(b+1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian16a-h3","type":"hint","dependencies":["acf3da9gaussian16a-h2"],"title":"Using the Zero Product Property","text":"As per the zero product property, if $$\\\\left(4b+3\\\\right) \\\\left(b+1\\\\right)$$, then either $$4b+3$$ or $$b+1$$ is equal to $$0$$. Setting both of these to $$0$$, we get $$b=\\\\frac{-3}{4}$$ and $$b=-1$$. Summing these values, we get $$-1.75$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acf3da9gaussian17","title":"Solving Quadratic Equations by Factoring","body":"Find the sum of the following equation\'s solutions by factoring:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acf3da9gaussian17a","stepAnswer":["$$4.25$$"],"problemType":"TextBox","stepTitle":"$$4m^2=17m-15$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4.25$$","hints":{"DefaultPathway":[{"id":"acf3da9gaussian17a-h1","type":"hint","dependencies":[],"title":"The Zero Product Property","text":"The zero product property says if the product of two quantities is zero, it must be that at least one of the quantities is zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian17a-h2","type":"hint","dependencies":["acf3da9gaussian17a-h1"],"title":"Factoring the Equation","text":"We can factor $$4m^2-17m+15=0$$ to use the zero product property. $$4m^2-17m+15=(4m-5)(m-3)=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian17a-h3","type":"hint","dependencies":["acf3da9gaussian17a-h2"],"title":"Using the Zero Product Property","text":"As per the zero product property, if $$(4m-5)(m-3)=0$$, then either $$4m-5$$ or $$m-3$$ is equal to $$0$$. Setting both of these to $$0$$, we get $$m=\\\\frac{5}{4}$$ and $$m=3$$. Summing these values, we get $$4.25$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acf3da9gaussian18","title":"Solving Quadratic Equations by Factoring","body":"Find the sum of the following equation\'s solutions by factoring:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acf3da9gaussian18a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"$$n^2=5n-6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"acf3da9gaussian18a-h1","type":"hint","dependencies":[],"title":"The Zero Product Property","text":"The zero product property says if the product of two quantities is zero, it must be that at least one of the quantities is zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian18a-h2","type":"hint","dependencies":["acf3da9gaussian18a-h1"],"title":"Factoring the Equation","text":"We can factor $$n^2-5n+6=0$$ to use the zero product property. $$n^2-5n+6=\\\\left(n-6\\\\right) \\\\left(n+1\\\\right)=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian18a-h3","type":"hint","dependencies":["acf3da9gaussian18a-h2"],"title":"Using the Zero Product Property","text":"As per the zero product property, if $$\\\\left(n-6\\\\right) \\\\left(n+1\\\\right)=0$$, then either $$n-6$$ or $$n+1$$ is equal to $$0$$. Setting both of these to $$0$$, we get $$n=6$$ or $$n=-1$$. Summing these values, we get $$5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acf3da9gaussian19","title":"Solving Quadratic Equations by Factoring","body":"Find the sum of the following equation\'s solutions by factoring:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acf3da9gaussian19a","stepAnswer":["$$-1$$"],"problemType":"TextBox","stepTitle":"$$7a^2+14a=7a$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1$$","hints":{"DefaultPathway":[{"id":"acf3da9gaussian19a-h1","type":"hint","dependencies":[],"title":"The Zero Product Property","text":"The zero product property says if the product of two quantities is zero, it must be that at least one of the quantities is zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian19a-h2","type":"hint","dependencies":["acf3da9gaussian19a-h1"],"title":"Factoring the Equation","text":"We can factor $$7a^2+14a-7a=0$$ to use the zero product property. $$n^2-5n+6=7a^2+7a=a\\\\left(7a+7\\\\right)=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian19a-h3","type":"hint","dependencies":["acf3da9gaussian19a-h2"],"title":"Using the Zero Product Property","text":"As per the zero product property, if $$a\\\\left(7a+7\\\\right)=0$$, then either a or $$7a+7$$ is equal to $$0$$. Setting both of these to $$0$$, we get $$a=0$$ or $$a=-1$$. Summing these values, we get $$-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acf3da9gaussian2","title":"Writing the Augmented Matrix for a System of Equations","body":"Write the augmented matrix of the given system of equations.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acf3da9gaussian2a","stepAnswer":["$$\\\\begin{bmatrix} 4 & -3 & 11 \\\\\\\\ 3 & 2 & 4 \\\\end{bmatrix}$$"],"problemType":"TextBox","stepTitle":"4x-3y=11,3x+2y=4","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\begin{bmatrix} 4 & -3 & 11 \\\\\\\\ 3 & 2 & 4 \\\\end{bmatrix}$$","hints":{"DefaultPathway":[{"id":"acf3da9gaussian2a-h1","type":"hint","dependencies":[],"title":"Definition","text":"The augmented matrix displays the coefficients of the variables, and an additional column for the constants.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian2a-h2","type":"hint","dependencies":["acf3da9gaussian2a-h1"],"title":"Write","text":"The first row has the values 4,-3,11.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian2a-h3","type":"hint","dependencies":["acf3da9gaussian2a-h2"],"title":"Answer","text":"The answer is $$\\\\begin{bmatrix} 4 & -3 & 11 \\\\\\\\ 3 & 2 & 4 \\\\end{bmatrix}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acf3da9gaussian20","title":"Solving Quadratic Equations by Factoring","body":"Find the sum of the following equation\'s solutions by factoring:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acf3da9gaussian20a","stepAnswer":["$$\\\\frac{1}{2}$$"],"problemType":"TextBox","stepTitle":"$$12b^2-15b=-9b$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{2}$$","hints":{"DefaultPathway":[{"id":"acf3da9gaussian20a-h1","type":"hint","dependencies":[],"title":"The Zero Product Property","text":"The zero product property says if the product of two quantities is zero, it must be that at least one of the quantities is zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian20a-h2","type":"hint","dependencies":["acf3da9gaussian20a-h1"],"title":"Factoring the Equation","text":"We can factor $$12b^2-6b=0$$ to use the zero product property. $$12b^2-6b=6b(2b-1)=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian20a-h3","type":"hint","dependencies":["acf3da9gaussian20a-h2"],"title":"Using the Zero Product Property","text":"As per the zero product property, if $$6b(2b-1)$$, then either $$6b$$ or $$2b-1$$ is equal to $$0$$. Setting both of these to $$0$$, we get $$b=0$$ or $$b=\\\\frac{1}{2}$$. Summing these values, we get $$\\\\frac{1}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acf3da9gaussian21","title":"Solving Quadratic Equations by Factoring","body":"Find the sum of the following equation\'s solutions by factoring:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acf3da9gaussian21a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"$$49m^2=144$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"acf3da9gaussian21a-h1","type":"hint","dependencies":[],"title":"The Zero Product Property","text":"The zero product property says if the product of two quantities is zero, it must be that at least one of the quantities is zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian21a-h2","type":"hint","dependencies":["acf3da9gaussian21a-h1"],"title":"Factoring the Equation","text":"We can factor $$49m^2-144$$ to use the zero product property. $$49m^2-144=\\\\left(7m+12\\\\right) \\\\left(7m-12\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian21a-h3","type":"hint","dependencies":["acf3da9gaussian21a-h2"],"title":"Using the Zero Product Property","text":"As per the zero product property, if $$\\\\left(7m+12\\\\right) \\\\left(7m-12\\\\right)=0$$, then either $$7m+12$$ or $$7m-12$$ is equal to $$0$$. Setting both of these to $$0$$, $$m=\\\\frac{-12}{7}$$ or $$m=\\\\frac{12}{7}$$. Summing these values, we get $$0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acf3da9gaussian22","title":"Solving Quadratic Equations by Factoring","body":"Find the sum of the following equation\'s solutions by factoring:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acf3da9gaussian22a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"$$625=x^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"acf3da9gaussian22a-h1","type":"hint","dependencies":[],"title":"The Zero Product Property","text":"The zero product property says if the product of two quantities is zero, it must be that at least one of the quantities is zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian22a-h2","type":"hint","dependencies":["acf3da9gaussian22a-h1"],"title":"Factoring the Equation","text":"We can factor $$x^2-625$$ to use the zero product property. $$x^2-625=\\\\left(x+25\\\\right) \\\\left(x-25\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian22a-h3","type":"hint","dependencies":["acf3da9gaussian22a-h2"],"title":"Using the Zero Product Property","text":"As per the zero product property, if $$\\\\left(x+25\\\\right) \\\\left(x-25\\\\right)=0$$, then either $$x+25$$ or $$x-25$$ is equal to $$0$$. Setting both of these to $$0$$, $$x=-25$$ and $$x=25$$. Summing these values, we get $$0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acf3da9gaussian23","title":"Solving Quadratic Equations by Factoring","body":"Find the sum of the following equation\'s solutions by factoring:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acf3da9gaussian23a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(y-3\\\\right) \\\\left(y+2\\\\right)=4y$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"acf3da9gaussian23a-h1","type":"hint","dependencies":[],"title":"The Zero Product Property","text":"The zero product property says if the product of two quantities is zero, it must be that at least one of the quantities is zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian23a-h2","type":"hint","dependencies":["acf3da9gaussian23a-h1"],"title":"Factoring the Equation","text":"To factor this equation, we must expand and re-factor. $$\\\\left(y-3\\\\right) \\\\left(y+2\\\\right)=y^2-y-6$$. So, we have $$y^2-5y-6=0$$. Factoring this, we get $$\\\\left(y-6\\\\right) \\\\left(y+1\\\\right)=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian23a-h3","type":"hint","dependencies":["acf3da9gaussian23a-h2"],"title":"Using the Zero Product Property","text":"As per the zero product property, if $$\\\\left(y-6\\\\right) \\\\left(y+1\\\\right)=0$$, then either $$y-6$$ or $$y+1$$ is equal to $$0$$. Setting both of these to $$0$$, $$y=6$$, $$y=-1$$. Summing these values, we get $$5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acf3da9gaussian24","title":"Solving Quadratic Equations by Factoring","body":"Find the sum of the following equation\'s solutions by factoring:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acf3da9gaussian24a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(p-5\\\\right) \\\\left(p+3\\\\right)=-7$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"acf3da9gaussian24a-h1","type":"hint","dependencies":[],"title":"The Zero Product Property","text":"The zero product property says if the product of two quantities is zero, it must be that at least one of the quantities is zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian24a-h2","type":"hint","dependencies":["acf3da9gaussian24a-h1"],"title":"Factoring the Equation","text":"To factor this equation, we must expand and re-factor. $$\\\\left(p-5\\\\right) \\\\left(p+3\\\\right)=p^2-2p-15$$. So, we have $$p^2-2p-8=0$$. Factoring this, we get $$\\\\left(p-4\\\\right) \\\\left(p+2\\\\right)=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian24a-h3","type":"hint","dependencies":["acf3da9gaussian24a-h2"],"title":"Using the Zero Product Property","text":"As per the zero product property, if $$\\\\left(p-4\\\\right) \\\\left(p+2\\\\right)=0$$, then either $$p-4$$ or $$p+2$$ is equal to $$0$$. Setting both of these to $$0$$, $$p=4$$ and $$p=-2$$. Summing these values, we get $$2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acf3da9gaussian28","title":"Solving Quadratic Equations by Factoring","body":"Find the sum of the following equation\'s solutions by factoring:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acf3da9gaussian28a","stepAnswer":["$$\\\\frac{1}{2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(2x+1\\\\right) \\\\left(x-3\\\\right)=-4x$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{2}$$","hints":{"DefaultPathway":[{"id":"acf3da9gaussian28a-h1","type":"hint","dependencies":[],"title":"The Zero Product Property","text":"The zero product property says if the product of two quantities is zero, it must be that at least one of the quantities is zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian28a-h2","type":"hint","dependencies":["acf3da9gaussian28a-h1"],"title":"Factoring the Equation","text":"To factor this equation, we must expand and re-factor. $$\\\\left(2x+1\\\\right) \\\\left(x-3\\\\right)=2x^2-5x-3$$. So, we have $$2x^2-x-3=0$$. Factoring this, we get $$\\\\left(x+1\\\\right) \\\\left(2x-3\\\\right)=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian28a-h3","type":"hint","dependencies":["acf3da9gaussian28a-h2"],"title":"Using the Zero Product Property","text":"As per the zero product property, if $$\\\\left(x+1\\\\right) \\\\left(2x-3\\\\right)=0$$, then either $$x+1$$ or $$2x-3$$ is equal to $$0$$. Setting both of these to $$0$$, $$x=-1$$, $$x=\\\\frac{3}{2}$$. Summing these values, we get $$\\\\frac{1}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acf3da9gaussian29","title":"Solving Quadratic Equations by Factoring","body":"Find the sum of the following equation\'s solutions by factoring:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acf3da9gaussian29a","stepAnswer":["$$-3$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(x+6\\\\right) \\\\left(x-3\\\\right)=-8$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-3$$","hints":{"DefaultPathway":[{"id":"acf3da9gaussian29a-h1","type":"hint","dependencies":[],"title":"The Zero Product Property","text":"The zero product property says if the product of two quantities is zero, it must be that at least one of the quantities is zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian29a-h2","type":"hint","dependencies":["acf3da9gaussian29a-h1"],"title":"Factoring the Equation","text":"To factor this equation, we must expand and re-factor. $$\\\\left(x+6\\\\right) \\\\left(x-3\\\\right)=x^2+3x-18$$. So, we have $$x^2+3x-10$$. Factoring this, we get $$\\\\left(x+5\\\\right) \\\\left(x-2\\\\right)=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian29a-h3","type":"hint","dependencies":["acf3da9gaussian29a-h2"],"title":"Using the Zero Product Property","text":"As per the zero product property, if $$\\\\left(x+5\\\\right) \\\\left(x-2\\\\right)=0$$, then either $$x+5$$ or $$x-2$$ is equal to $$0$$. Setting both of these to $$0$$, $$x=-5$$ and $$x=2$$. Summing these values, we get $$-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acf3da9gaussian3","title":"Writing a System of Equations from an Augmented Matrix Form","body":"Find the system of equations from the augmented matrix when the variables are $$x$$, $$y$$, and $$z$$.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acf3da9gaussian3a","stepAnswer":["x-3y-5z=-2 2x-5y-4z=5 -3x+5y+4z=6"],"problemType":"TextBox","stepTitle":"$$\\\\begin{bmatrix} 1 & -3 & -5 & -2 \\\\\\\\ 2 & -5 & -4 & 5 \\\\\\\\ -3 & 5 & 4 & 6 \\\\end{bmatrix}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x-3y-5z=-2$$ $$2x-5y-4z=5$$ $$-3x+5y+4z=6$$","hints":{"DefaultPathway":[{"id":"acf3da9gaussian3a-h1","type":"hint","dependencies":[],"title":"Rows","text":"Rows represent a single equation. There are $$3$$ rows, so you know there are $$3$$ equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian3a-h2","type":"hint","dependencies":["acf3da9gaussian3a-h1"],"title":"Coefficients","text":"The numbers in the matrix represent coefficients of variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian3a-h3","type":"hint","dependencies":["acf3da9gaussian3a-h2"],"title":"Answer","text":"The answer is $$x-3y-5z=-2$$ $$2x-5y-4z=5$$ $$-3x+5y+4z=6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acf3da9gaussian30","title":"Solving Quadratic Equations by Factoring","body":"Find the sum of the following equation\'s solutions by factoring:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acf3da9gaussian30a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"$$20x^2-60x=-45$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"acf3da9gaussian30a-h1","type":"hint","dependencies":[],"title":"The Zero Product Property","text":"The zero product property says if the product of two quantities is zero, it must be that at least one of the quantities is zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian30a-h2","type":"hint","dependencies":["acf3da9gaussian30a-h1"],"title":"Factoring the Equation","text":"We can factor $$20x^2-60x+45$$ to use the zero product property. $$20x^2-60x+45=5(2x-3)(2x-3)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian30a-h3","type":"hint","dependencies":["acf3da9gaussian30a-h2"],"title":"Using the Zero Product Property","text":"As per the zero product property, if $$5(2x-3)(2x-3)=0$$, then either $$2x-3=0$$. We get $$x=\\\\frac{3}{2}$$. Since $$(2x-3)$$ is squared, we double $$\\\\frac{3}{2}$$ to get $$3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acf3da9gaussian4","title":"Writing a System of Equations from an Augmented Matrix Form","body":"Find the system of equations from the augmented matrix when the variables are $$x$$, $$y$$, and $$z$$ (enter them in parentheses, separated by commas).","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.6 Quadratic Equations","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acf3da9gaussian4a","stepAnswer":["(x-y+z=5),(2x-y+3z=1),(y+z=-9)"],"problemType":"TextBox","stepTitle":"$$\\\\begin{bmatrix} 1 & -1 & 1 & 5 \\\\\\\\ 2 & -1 & 3 & 1 \\\\\\\\ 0 & 1 & 1 & -9 \\\\end{bmatrix}$$","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"acf3da9gaussian4a-h1","type":"hint","dependencies":[],"title":"Rows","text":"Rows represent a single equation. There are $$3$$ rows, so you know there are $$3$$ equations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian4a-h2","type":"hint","dependencies":["acf3da9gaussian4a-h1"],"title":"Coefficients","text":"The numbers in the matrix represent coefficients of variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acf3da9gaussian4a-h3","type":"hint","dependencies":["acf3da9gaussian4a-h2"],"title":"Answer","text":"The answer is $$x-y+z=5$$ $$2x-y+3z=1$$ $$y+z=-9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acf60f3DerivativeFunction1","title":"Finding the Derivative of a Square-root function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.2 The Derivative as a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"acf60f3DerivativeFunction1a","stepAnswer":["$$\\\\frac{1}{2} \\\\sqrt{x}$$"],"problemType":"TextBox","stepTitle":"Find the derivative of $$f(x)=\\\\sqrt{x}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{2} \\\\sqrt{x}$$","hints":{"DefaultPathway":[{"id":"acf60f3DerivativeFunction1a-h1","type":"hint","dependencies":[],"title":"Derivative function","text":"Start directly with the definition of the derivative function. $$f\u2032(x)=\\\\lim_{h\\\\to0} \\\\frac{f{\\\\left(x+h\\\\right)}-f{\\\\left(x\\\\right)}}{h}$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction1a-h2","type":"hint","dependencies":["acf60f3DerivativeFunction1a-h1"],"title":"Derivative function","text":"We will get $$f\'(x)=\\\\lim_{h\\\\to0} \\\\frac{\\\\sqrt{x+h}-\\\\sqrt{x}}{h}$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction1a-h3","type":"hint","dependencies":["acf60f3DerivativeFunction1a-h2"],"title":"Simplify","text":"To simplify, Multiply numerator and denominator by $$\\\\sqrt{h+x}+\\\\sqrt{x}$$ without distributing in the denominator","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction1a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1}{\\\\sqrt{x+h}+\\\\sqrt{x}}$$"],"dependencies":["acf60f3DerivativeFunction1a-h3"],"title":"Simplify","text":"What will get after multipling?","variabilization":{},"oer":"","license":"","choices":["$$\\\\frac{1}{\\\\sqrt{x+h}+\\\\sqrt{x}}$$","$$\\\\sqrt{x+h}+\\\\sqrt{x}$$"],"subHints":[{"id":"acf60f3DerivativeFunction1a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["h"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\left(\\\\sqrt{x+h}-\\\\sqrt{x}\\\\right) \\\\left(\\\\sqrt{x+h}+\\\\sqrt{x}\\\\right)$$?","variabilization":{},"oer":"","license":""}]},{"id":"acf60f3DerivativeFunction1a-h5","type":"hint","dependencies":["acf60f3DerivativeFunction1a-h4"],"title":"Evaluate the limit","text":"To get a final answer, evaluate the limit by inserting h--\x3e0 $$=$$ $$h=0$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"acf60f3DerivativeFunction10","title":"Finding a Second Derivative","body":"For $$f(x)=2-3x$$, find f\'\'(x).","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.2 The Derivative as a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"acf60f3DerivativeFunction10a","stepAnswer":["$$-3$$"],"problemType":"TextBox","stepTitle":"First derivative","stepBody":"First, find the f\'(x) using derivative function","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-3$$","hints":{"DefaultPathway":[{"id":"acf60f3DerivativeFunction10a-h1","type":"hint","dependencies":[],"title":"Derivative function","text":"Start directly with the definition of the derivative function. $$f\u2032(x)=\\\\lim_{h\\\\to0} \\\\frac{f{\\\\left(x+h\\\\right)}-f{\\\\left(x\\\\right)}}{h}$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction10a-h2","type":"hint","dependencies":["acf60f3DerivativeFunction10a-h1"],"title":"Derivative function","text":"We will get $$f\'(x)=\\\\lim_{h\\\\to0} \\\\frac{2-3\\\\left(x+h\\\\right)-2-3x}{h}$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction10a-h3","type":"hint","dependencies":["acf60f3DerivativeFunction10a-h2"],"title":"Expand","text":"Expand $$\\\\frac{2-3\\\\left(x+h\\\\right)-2-3x}{h}$$ and Simplify","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction10a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{h\\\\to0} -3$$"],"dependencies":["acf60f3DerivativeFunction10a-h3"],"title":"Simplify","text":"What will get?","variabilization":{},"oer":"","license":"","choices":["$$\\\\lim_{h\\\\to0} -3x$$","$$\\\\lim_{h\\\\to0} -3$$","$$\\\\lim_{h\\\\to0} -3h$$"]},{"id":"acf60f3DerivativeFunction10a-h5","type":"hint","dependencies":["acf60f3DerivativeFunction10a-h4"],"title":"Evaluate the limit","text":"Evaluate the limit by inserting h--\x3e0 $$=$$ $$h=0$$","variabilization":{},"oer":"","license":""}]}},{"id":"acf60f3DerivativeFunction10b","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"Finding a Second Derivative","stepBody":"Next, find f\'\'(x) by taking the derivative of $$f\u2032(x)=-3$$.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"acf60f3DerivativeFunction10b-h1","type":"hint","dependencies":[],"title":"Second derivative","text":"Derivative of number (whether negative or positive) is zero","variabilization":{},"oer":"","license":""}]}}]},{"id":"acf60f3DerivativeFunction11","title":"Finding a Second Derivative","body":"For $$f(x)=x+\\\\frac{1}{x}$$, find f\'\'(x).","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.2 The Derivative as a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"acf60f3DerivativeFunction11a","stepAnswer":["$$1-\\\\frac{1}{x^2}$$"],"problemType":"TextBox","stepTitle":"First derivative","stepBody":"First, find the f\'(x) using derivative function","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1-\\\\frac{1}{x^2}$$","hints":{"DefaultPathway":[{"id":"acf60f3DerivativeFunction11a-h1","type":"hint","dependencies":[],"title":"Derivative function","text":"Start directly with the definition of the derivative function. $$f\u2032(x)=\\\\lim_{h\\\\to0} \\\\frac{f{\\\\left(x+h\\\\right)}-f{\\\\left(x\\\\right)}}{h}$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction11a-h2","type":"hint","dependencies":["acf60f3DerivativeFunction11a-h1"],"title":"Derivative function","text":"We will get $$f\'(x)=\\\\lim_{h\\\\to0} \\\\frac{x+h+\\\\frac{1}{x+h}-x+\\\\frac{1}{x}}{h}$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction11a-h3","type":"hint","dependencies":["acf60f3DerivativeFunction11a-h2"],"title":"Expand","text":"Expand $$\\\\frac{x+h+\\\\frac{1}{x+h}-x+\\\\frac{1}{x}}{h}$$ and Simplify","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction11a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{h\\\\to0} 1-\\\\frac{1}{x^2+xh}$$"],"dependencies":["acf60f3DerivativeFunction11a-h3"],"title":"Simplify","text":"What will get?","variabilization":{},"oer":"","license":"","choices":["$$\\\\lim_{h\\\\to0} \\\\frac{1}{x^2+xh}$$","$$\\\\lim_{h\\\\to0} 1-\\\\frac{1}{x^2+xh}$$","$$\\\\lim_{h\\\\to0} 1+\\\\frac{1}{x^2+xh}$$"]},{"id":"acf60f3DerivativeFunction11a-h5","type":"hint","dependencies":["acf60f3DerivativeFunction11a-h4"],"title":"Evaluate the limit","text":"Evaluate the limit by inserting h--\x3e0 $$=$$ $$h=0$$","variabilization":{},"oer":"","license":""}]}},{"id":"acf60f3DerivativeFunction11b","stepAnswer":["$$\\\\frac{2}{x^3}$$"],"problemType":"TextBox","stepTitle":"Finding a Second Derivative","stepBody":"Next, find f\'\'(x) by taking the derivative of $$f\u2032(x)=2x$$.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2}{x^3}$$","hints":{"DefaultPathway":[{"id":"acf60f3DerivativeFunction11b-h1","type":"hint","dependencies":[],"title":"Second derivative","text":"Use $$f\'\'(x)=\\\\lim_{h\\\\to0} \\\\frac{\\\\operatorname{f\'}\\\\left(x+h\\\\right)-\\\\operatorname{f\'}\\\\left(x\\\\right)}{h}$$ to find second derivative","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction11b-h2","type":"hint","dependencies":["acf60f3DerivativeFunction11b-h1"],"title":"Derivative function","text":"We will get $$f\'\'(x)=\\\\lim_{h\\\\to0} \\\\frac{1-\\\\frac{1}{{\\\\left(x+h\\\\right)}^2}-1-\\\\frac{1}{x^2}}{h}$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction11b-h3","type":"hint","dependencies":["acf60f3DerivativeFunction11b-h2"],"title":"Expand","text":"Expand $$1-\\\\frac{1}{{\\\\left(x+h\\\\right)}^2}-\\\\frac{1-\\\\frac{1}{x^2}}{h}$$ and Simplify","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction11b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{h\\\\to0} \\\\frac{2x+h}{x^4+2x^3 h+h^2}$$"],"dependencies":["acf60f3DerivativeFunction11b-h3"],"title":"Simplify","text":"What will get?","variabilization":{},"oer":"","license":"","choices":["$$\\\\lim_{h\\\\to0} \\\\frac{2x-h}{x^4+2x^3 h+h^2}$$","$$\\\\lim_{h\\\\to0} \\\\frac{2x+h}{x^2+2xh+h^2}$$","$$\\\\lim_{h\\\\to0} \\\\frac{2x+h}{x^4+2x^3 h+h^2}$$"]},{"id":"acf60f3DerivativeFunction11b-h5","type":"hint","dependencies":["acf60f3DerivativeFunction11b-h4"],"title":"Evaluate the limit","text":"To get a final answer, evaluate the limit by inserting h--\x3e0 $$=$$ $$h=0$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"acf60f3DerivativeFunction12","title":"Finding Acceleration","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.2 The Derivative as a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"acf60f3DerivativeFunction12a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"The position of a particle along a coordinate axis at time $$t$$ (in seconds) is given by $$s(t)=3t^2-4t+1$$ (in meters). Find the function that describes its acceleration at time $$t$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"acf60f3DerivativeFunction12a-h1","type":"hint","dependencies":[],"title":"Derivative function","text":"Since $$v(t)=s\u2032(t)$$ and $$a(t)=v\u2032(t)=s\'\'(t)$$, we begin by finding the derivative of s(t): $$s\'(t)=\\\\lim_{h\\\\to0} \\\\frac{s\\\\left(t+h\\\\right)-s\\\\left(t\\\\right)}{h}$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction12a-h2","type":"hint","dependencies":["acf60f3DerivativeFunction12a-h1"],"title":"Derivative function","text":"We will get $$s\'(t)=\\\\lim_{h\\\\to0} \\\\frac{3{\\\\left(t+h\\\\right)}^2-4\\\\left(t+h\\\\right)+1-3t^2-4t+1}{h}$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction12a-h3","type":"hint","dependencies":["acf60f3DerivativeFunction12a-h2"],"title":"Expand","text":"Expand $$\\\\frac{3{\\\\left(t+h\\\\right)}^2-4\\\\left(t+h\\\\right)+1-3t^2-4t+1}{h}$$ and Simplify","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction12a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{h\\\\to0} 6t+3h-4$$"],"dependencies":["acf60f3DerivativeFunction12a-h3"],"title":"Simplify","text":"What will get?","variabilization":{},"oer":"","license":"","choices":["$$\\\\lim_{h\\\\to0} 6t+3th-4h$$","$$\\\\lim_{h\\\\to0} 6t+3h-4$$","$$\\\\lim_{h\\\\to0} 6th+3h-4$$"]},{"id":"acf60f3DerivativeFunction12a-h5","type":"hint","dependencies":["acf60f3DerivativeFunction12a-h4"],"title":"Evaluate the limit","text":"Evaluate the limit by inserting h--\x3e0 $$=$$ $$h=0$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction12a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6t-4$$"],"dependencies":["acf60f3DerivativeFunction12a-h5"],"title":"Derivative of s(t)","text":"What is s\'(t)?","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction12a-h7","type":"hint","dependencies":["acf60f3DerivativeFunction12a-h6"],"title":"Second derivative","text":"Next, need to find s\'\'(t) from $$s\'(t)=6t-4$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction12a-h8","type":"hint","dependencies":["acf60f3DerivativeFunction12a-h7"],"title":"Second derivative","text":"Use $$s\'\'(t)=\\\\lim_{h\\\\to0} \\\\frac{\\\\operatorname{s\'}\\\\left(t+h\\\\right)-\\\\operatorname{s\'}\\\\left(t\\\\right)}{h}$$ to find second derivative","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction12a-h9","type":"hint","dependencies":["acf60f3DerivativeFunction12a-h8"],"title":"Derivative function","text":"We will get $$s\'\'(t)=\\\\lim_{h\\\\to0} \\\\frac{6\\\\left(t+h\\\\right)-4-6t+4}{h}$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction12a-h10","type":"hint","dependencies":["acf60f3DerivativeFunction12a-h9"],"title":"Expand","text":"Expand $$\\\\frac{6\\\\left(t+h\\\\right)-4-6t+4}{h}$$ and Simplify","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction12a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{h\\\\to0} 6$$"],"dependencies":["acf60f3DerivativeFunction12a-h10"],"title":"Simplify","text":"What will get?","variabilization":{},"oer":"","license":"","choices":["$$\\\\lim_{h\\\\to0} 6$$","$$\\\\lim_{h\\\\to0} 6h$$","$$\\\\lim_{h\\\\to0} 6x$$"]},{"id":"acf60f3DerivativeFunction12a-h12","type":"hint","dependencies":["acf60f3DerivativeFunction12a-h11"],"title":"Evaluate the limit","text":"To get a final answer, evaluate the limit by inserting h--\x3e0 $$=$$ $$h=0$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"acf60f3DerivativeFunction13","title":"Finding Acceleration","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.2 The Derivative as a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"acf60f3DerivativeFunction13a","stepAnswer":["$$6t$$"],"problemType":"TextBox","stepTitle":"For $$s(t)=t^3$$, find a(t).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6t$$","hints":{"DefaultPathway":[{"id":"acf60f3DerivativeFunction13a-h1","type":"hint","dependencies":[],"title":"Derivative function","text":"Since $$v(t)=s\u2032(t)$$ and $$a(t)=v\u2032(t)=s\'\'(t)$$, we begin by finding the derivative of s(t): $$s\'(t)=\\\\lim_{h\\\\to0} \\\\frac{s\\\\left(t+h\\\\right)-s\\\\left(t\\\\right)}{h}$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction13a-h2","type":"hint","dependencies":["acf60f3DerivativeFunction13a-h1"],"title":"Derivative function","text":"We will get $$s\'(t)=\\\\lim_{h\\\\to0} \\\\frac{{\\\\left(t+h\\\\right)}^3-t^3}{h}$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction13a-h3","type":"hint","dependencies":["acf60f3DerivativeFunction13a-h2"],"title":"Expand","text":"Expand $$\\\\frac{{\\\\left(t+h\\\\right)}^3-t^3}{h}$$ and Simplify","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction13a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{h\\\\to0} 3t^3+3th+h^2$$"],"dependencies":["acf60f3DerivativeFunction13a-h3"],"title":"Simplify","text":"What will get?","variabilization":{},"oer":"","license":"","choices":["$$\\\\lim_{h\\\\to0} 3t^3+3th+h^2$$","$$\\\\lim_{h\\\\to0} 3t^2+3th+h$$","$$\\\\lim_{h\\\\to0} 3t^3+3th+3h^2$$"]},{"id":"acf60f3DerivativeFunction13a-h5","type":"hint","dependencies":["acf60f3DerivativeFunction13a-h4"],"title":"Evaluate the limit","text":"Evaluate the limit by inserting h--\x3e0 $$=$$ $$h=0$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction13a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3t^2$$"],"dependencies":["acf60f3DerivativeFunction13a-h5"],"title":"Derivative of s(t)","text":"What is s\'(t)?","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction13a-h7","type":"hint","dependencies":["acf60f3DerivativeFunction13a-h6"],"title":"Second derivative","text":"Next, need to find s\'\'(t) from $$s\'(t)=3t^2$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction13a-h8","type":"hint","dependencies":["acf60f3DerivativeFunction13a-h7"],"title":"Second derivative","text":"Use $$s\'\'(t)=\\\\lim_{h\\\\to0} \\\\frac{\\\\operatorname{s\'}\\\\left(t+h\\\\right)-\\\\operatorname{s\'}\\\\left(t\\\\right)}{h}$$ to find second derivative","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction13a-h9","type":"hint","dependencies":["acf60f3DerivativeFunction13a-h8"],"title":"Derivative function","text":"We will get $$s\'\'(t)=\\\\lim_{h\\\\to0} \\\\frac{3{\\\\left(t+h\\\\right)}^2-3t^2}{h}$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction13a-h10","type":"hint","dependencies":["acf60f3DerivativeFunction13a-h9"],"title":"Expand","text":"Expand $$\\\\frac{3{\\\\left(t+h\\\\right)}^2-3t^2}{h}$$ and Simplify","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction13a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{h\\\\to0} 6t+3h$$"],"dependencies":["acf60f3DerivativeFunction13a-h10"],"title":"Simplify","text":"What will get?","variabilization":{},"oer":"","license":"","choices":["$$\\\\lim_{h\\\\to0} 6+3h$$","$$\\\\lim_{h\\\\to0} 6th+3$$","$$\\\\lim_{h\\\\to0} 6t+3h$$"]},{"id":"acf60f3DerivativeFunction13a-h12","type":"hint","dependencies":["acf60f3DerivativeFunction13a-h11"],"title":"Evaluate the limit","text":"To get a final answer, evaluate the limit by inserting h--\x3e0 $$=$$ $$h=0$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"acf60f3DerivativeFunction14","title":"Derivative function","body":"For the following exercises, the given limit represents the derivative of a function $$y=f(x)$$ at $$x=a$$. Find f(x) and a.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.2 The Derivative as a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"acf60f3DerivativeFunction14a","stepAnswer":["$$a=2$$, $$f(x)=3x^2+2$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\lim_{h\\\\to0} \\\\frac{3{\\\\left(2+h\\\\right)}^2+2-14}{h}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$a=2$$, $$f(x)=3x^2+2$$","choices":["$$a=12$$, $$f(x)=3x^2$$","$$a=14$$, $$f(x)=x^2+2$$","$$a=2$$, $$f(x)=3x^2+2$$"],"hints":{"DefaultPathway":[{"id":"acf60f3DerivativeFunction14a-h1","type":"hint","dependencies":[],"title":"Derivative function","text":"Acording to derivative function, the given limit represents $$f\'(a)=\\\\lim_{h\\\\to0} \\\\frac{f{\\\\left(a+h\\\\right)}-f{\\\\left(a\\\\right)}}{h}$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction14a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$14$$"],"dependencies":["acf60f3DerivativeFunction14a-h1"],"title":"Find f(a)","text":"Reflecting derivative function, what is f(a)?","variabilization":{},"oer":"","license":"","choices":["$$14$$","$$3{\\\\left(2+h\\\\right)}^2+2$$"]},{"id":"acf60f3DerivativeFunction14a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3{\\\\left(2+h\\\\right)}^2+2$$"],"dependencies":["acf60f3DerivativeFunction14a-h2"],"title":"Find $$f{\\\\left(a+h\\\\right)}$$","text":"Reflecting derivative function, what is $$f{\\\\left(a+h\\\\right)}$$?","variabilization":{},"oer":"","license":"","choices":["$$14$$","$$3{\\\\left(2+h\\\\right)}^2+2$$"]},{"id":"acf60f3DerivativeFunction14a-h4","type":"hint","dependencies":["acf60f3DerivativeFunction14a-h3"],"title":"Find a and f(x)","text":"From $$f{\\\\left(a+h\\\\right)}=3{\\\\left(2+h\\\\right)}^2+2$$, we can find a, and f(x) by substituting $$x$$ into $$x$$ and ignoring $$h$$ too","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction14a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2$$"],"dependencies":["acf60f3DerivativeFunction14a-h4"],"title":"Find a","text":"what is a?","variabilization":{},"oer":"","license":"","choices":["$$2$$","$$3$$","$$6$$"]},{"id":"acf60f3DerivativeFunction14a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3x^2+2$$"],"dependencies":["acf60f3DerivativeFunction14a-h5"],"title":"Find f(x)","text":"what is f(x)?","variabilization":{},"oer":"","license":"","choices":["$${\\\\left(2+x\\\\right)}^2$$","$$3\\\\left(2+x\\\\right)+2$$","$$3x^2+2$$"]}]}}]},{"id":"acf60f3DerivativeFunction15","title":"Derivative function","body":"For the following exercises, the given limit represents the derivative of a function $$y=f(x)$$ at $$x=a$$. Find f(x) and a.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.2 The Derivative as a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"acf60f3DerivativeFunction15a","stepAnswer":["$$a=2$$, $$f(x)=x^4$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\lim_{h\\\\to0} \\\\frac{{\\\\left(2+h\\\\right)}^4-16}{h}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$a=2$$, $$f(x)=x^4$$","choices":["$$a=16$$, $$f(x)=x^4$$","$$a=2$$, $$f(x)={\\\\left(x+2\\\\right)}^4$$","$$a=2$$, $$f(x)=x^4$$"],"hints":{"DefaultPathway":[{"id":"acf60f3DerivativeFunction15a-h1","type":"hint","dependencies":[],"title":"Derivative function","text":"Acording to derivative function, the given limit represents $$f\'(a)=\\\\lim_{h\\\\to0} \\\\frac{f{\\\\left(a+h\\\\right)}-f{\\\\left(a\\\\right)}}{h}$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction15a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$16$$"],"dependencies":["acf60f3DerivativeFunction15a-h1"],"title":"Find f(a)","text":"Reflecting derivative function, what is f(a)?","variabilization":{},"oer":"","license":"","choices":["$${\\\\left(2+h\\\\right)}^4$$","$$16$$","$$-16$$"]},{"id":"acf60f3DerivativeFunction15a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${\\\\left(2+h\\\\right)}^4$$"],"dependencies":["acf60f3DerivativeFunction15a-h2"],"title":"Find $$f{\\\\left(a+h\\\\right)}$$","text":"Reflecting derivative function, what is $$f{\\\\left(a+h\\\\right)}$$?","variabilization":{},"oer":"","license":"","choices":["$${\\\\left(2+h\\\\right)}^4$$","$$16$$","$$-16$$"]},{"id":"acf60f3DerivativeFunction15a-h4","type":"hint","dependencies":["acf60f3DerivativeFunction15a-h3"],"title":"Find a and f(x)","text":"From $$f{\\\\left(a+h\\\\right)}={\\\\left(2+h\\\\right)}^4$$, we can find a, and f(x) by substituting $$x$$ into $$x$$ and ignoring $$h$$ too","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction15a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2$$"],"dependencies":["acf60f3DerivativeFunction15a-h4"],"title":"Find a","text":"what is a?","variabilization":{},"oer":"","license":"","choices":["$$16$$","$$2$$"]},{"id":"acf60f3DerivativeFunction15a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x^4$$"],"dependencies":["acf60f3DerivativeFunction15a-h5"],"title":"Find f(x)","text":"what is f(x)?","variabilization":{},"oer":"","license":"","choices":["$$x^4$$","$${\\\\left(2+x\\\\right)}^4$$"]}]}}]},{"id":"acf60f3DerivativeFunction2","title":"Finding the Derivative of a Quadratic Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.2 The Derivative as a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"acf60f3DerivativeFunction2a","stepAnswer":["$$2x-2$$"],"problemType":"TextBox","stepTitle":"Find the derivative of the function $$f(x)=x^2-2x$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2x-2$$","hints":{"DefaultPathway":[{"id":"acf60f3DerivativeFunction2a-h1","type":"hint","dependencies":[],"title":"Derivative function","text":"Start directly with the definition of the derivative function. $$f\u2032(x)=\\\\lim_{h\\\\to0} \\\\frac{f{\\\\left(x+h\\\\right)}-f{\\\\left(x\\\\right)}}{h}$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction2a-h2","type":"hint","dependencies":["acf60f3DerivativeFunction2a-h1"],"title":"Derivative function","text":"We will get $$f\'(x)=\\\\lim_{h\\\\to0} \\\\frac{{\\\\left(x+h\\\\right)}^2-2\\\\left(x+h\\\\right)-x^2-2x}{h}$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction2a-h3","type":"hint","dependencies":["acf60f3DerivativeFunction2a-h2"],"title":"Expand","text":"Expand $${\\\\left(x+h\\\\right)}^2-2\\\\left(x+h\\\\right)$$ and simplify","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction2a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{h\\\\to0} \\\\frac{2xh-2h+h^2}{h}$$"],"dependencies":["acf60f3DerivativeFunction2a-h3"],"title":"Simplify","text":"What will you get?","variabilization":{},"oer":"","license":"","choices":["$$\\\\lim_{h\\\\to0} \\\\frac{2xh-2h+h^2}{h}$$","$$\\\\lim_{h\\\\to0} \\\\frac{xh-h+h^2}{h}$$","$$\\\\lim_{h\\\\to0} \\\\frac{xh-3h}{h}$$"]},{"id":"acf60f3DerivativeFunction2a-h5","type":"hint","dependencies":["acf60f3DerivativeFunction2a-h4"],"title":"Simplify","text":"Factor out $$h$$ from numerator and Cancel $$h$$ out","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction2a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$/lim{h,0$$, $$2x-2+h$$}"],"dependencies":["acf60f3DerivativeFunction2a-h5"],"title":"Simplify","text":"What will you get?","variabilization":{},"oer":"","license":"","choices":["$$/lim{h,0$$, $$x+h$$}","$$/lim{h,0$$, $$(2x-h)$$}","$$/lim{h,0$$, $$2x-2+h$$}"]},{"id":"acf60f3DerivativeFunction2a-h7","type":"hint","dependencies":["acf60f3DerivativeFunction2a-h6"],"title":"Evaluate the limit","text":"To get a final answer, evaluate the limit by inserting h--\x3e0 $$=$$ $$h=0$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"acf60f3DerivativeFunction3","title":"Finding derivative","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.2 The Derivative as a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"acf60f3DerivativeFunction3a","stepAnswer":["$$2x$$"],"problemType":"TextBox","stepTitle":"Find the derivative of $$f(x)=x^2$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2x$$","hints":{"DefaultPathway":[{"id":"acf60f3DerivativeFunction3a-h1","type":"hint","dependencies":[],"title":"Derivative function","text":"Start directly with the definition of the derivative function. $$f\u2032(x)=\\\\lim_{h\\\\to0} \\\\frac{f{\\\\left(x+h\\\\right)}-f{\\\\left(x\\\\right)}}{h}$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction3a-h2","type":"hint","dependencies":["acf60f3DerivativeFunction3a-h1"],"title":"Derivative function","text":"We will get $$f\'(x)=\\\\lim_{h\\\\to0} \\\\frac{{\\\\left(x+h\\\\right)}^2-x^2}{h}$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction3a-h3","type":"hint","dependencies":["acf60f3DerivativeFunction3a-h2"],"title":"Expand","text":"Expand $$\\\\frac{{\\\\left(x+h\\\\right)}^2-x^2}{h}$$ and Simplify","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction3a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{h\\\\to0} \\\\frac{2xh+h^2}{h}$$"],"dependencies":["acf60f3DerivativeFunction3a-h3"],"title":"Simplify","text":"What will get?","variabilization":{},"oer":"","license":"","choices":["$$\\\\lim_{h\\\\to0} \\\\frac{xh+h}{h}$$","$$\\\\lim_{h\\\\to0} \\\\frac{2xh+h^2}{h}$$","$$\\\\lim_{h\\\\to0} \\\\frac{2xh+h}{h}$$"]},{"id":"acf60f3DerivativeFunction3a-h5","type":"hint","dependencies":["acf60f3DerivativeFunction3a-h4"],"title":"Simplify","text":"Cancel the common factor $$h$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction3a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{h\\\\to0} 2x+h$$"],"dependencies":["acf60f3DerivativeFunction3a-h5"],"title":"Simplify","text":"What will get ?","variabilization":{},"oer":"","license":"","choices":["$$\\\\lim_{h\\\\to0} x+h$$","$$\\\\lim_{h\\\\to0} 2+h$$","$$\\\\lim_{h\\\\to0} 2x+h$$"]},{"id":"acf60f3DerivativeFunction3a-h7","type":"hint","dependencies":["acf60f3DerivativeFunction3a-h6"],"title":"Evaluate the limit","text":"To get a final answer, evaluate the limit by inserting h--\x3e0 $$=$$ $$h=0$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"acf60f3DerivativeFunction4","title":"Finding derivative","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.2 The Derivative as a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"acf60f3DerivativeFunction4a","stepAnswer":["$$-3$$"],"problemType":"TextBox","stepTitle":"Use the definition of a derivative to find f\u2032(x).","stepBody":"$$f(x)=2-3x$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-3$$","hints":{"DefaultPathway":[{"id":"acf60f3DerivativeFunction4a-h1","type":"hint","dependencies":[],"title":"Derivative function","text":"Start directly with the definition of the derivative function. $$f\u2032(x)=\\\\lim_{h\\\\to0} \\\\frac{f{\\\\left(x+h\\\\right)}-f{\\\\left(x\\\\right)}}{h}$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction4a-h2","type":"hint","dependencies":["acf60f3DerivativeFunction4a-h1"],"title":"Derivative function","text":"We will get $$f\'(x)=\\\\lim_{h\\\\to0} \\\\frac{2-3\\\\left(x+h\\\\right)-2-3x}{h}$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction4a-h3","type":"hint","dependencies":["acf60f3DerivativeFunction4a-h2"],"title":"Expand","text":"Expand $$\\\\frac{2-3\\\\left(x+h\\\\right)-2-3x}{h}$$ and Simplify","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction4a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{h\\\\to0} \\\\frac{\\\\left(-3h\\\\right)}{h}$$"],"dependencies":["acf60f3DerivativeFunction4a-h3"],"title":"Simplify","text":"What will get?","variabilization":{},"oer":"","license":"","choices":["$$\\\\lim_{h\\\\to0} \\\\frac{\\\\left(-3h\\\\right)}{h}$$","$$\\\\lim_{h\\\\to0} \\\\frac{\\\\left(-3h+x\\\\right)}{h}$$","$$\\\\lim_{h\\\\to0} \\\\frac{3h}{h}$$"]},{"id":"acf60f3DerivativeFunction4a-h5","type":"hint","dependencies":["acf60f3DerivativeFunction4a-h4"],"title":"Simplify","text":"Cancel the common factor $$h$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction4a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{h\\\\to0} -3$$"],"dependencies":["acf60f3DerivativeFunction4a-h5"],"title":"Simplify","text":"What will get ?","variabilization":{},"oer":"","license":"","choices":["$$\\\\lim_{h\\\\to0} 3$$","$$\\\\lim_{h\\\\to0} -3h$$","$$\\\\lim_{h\\\\to0} -3$$"]},{"id":"acf60f3DerivativeFunction4a-h7","type":"hint","dependencies":["acf60f3DerivativeFunction4a-h6"],"title":"Evaluate the limit","text":"To get a final answer, evaluate the limit by inserting h--\x3e0 $$=$$ $$h=0$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"acf60f3DerivativeFunction5","title":"Finding derivative","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.2 The Derivative as a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"acf60f3DerivativeFunction5a","stepAnswer":["$$8x$$"],"problemType":"TextBox","stepTitle":"Use the definition of a derivative to find f\u2032(x).","stepBody":"$$f(x)=4x^2$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8x$$","hints":{"DefaultPathway":[{"id":"acf60f3DerivativeFunction5a-h1","type":"hint","dependencies":[],"title":"Derivative function","text":"Start directly with the definition of the derivative function. $$f\u2032(x)=\\\\lim_{h\\\\to0} \\\\frac{f{\\\\left(x+h\\\\right)}-f{\\\\left(x\\\\right)}}{h}$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction5a-h2","type":"hint","dependencies":["acf60f3DerivativeFunction5a-h1"],"title":"Derivative function","text":"We will get $$f\'(x)=\\\\lim_{h\\\\to0} \\\\frac{4{\\\\left(x+h\\\\right)}^2-4x^2}{h}$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction5a-h3","type":"hint","dependencies":["acf60f3DerivativeFunction5a-h2"],"title":"Expand","text":"Expand $$\\\\frac{4{\\\\left(x+h\\\\right)}^2-4x^2}{h}$$ and Simplify","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction5a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{h\\\\to0} \\\\frac{8xh+4h^2}{h}$$"],"dependencies":["acf60f3DerivativeFunction5a-h3"],"title":"Simplify","text":"What will get?","variabilization":{},"oer":"","license":"","choices":["$$\\\\lim_{h\\\\to0} \\\\frac{8xh+4h}{h}$$","$$\\\\lim_{h\\\\to0} \\\\frac{xh+h^2}{h}$$","$$\\\\lim_{h\\\\to0} \\\\frac{8xh+4h^2}{h}$$"]},{"id":"acf60f3DerivativeFunction5a-h5","type":"hint","dependencies":["acf60f3DerivativeFunction5a-h4"],"title":"Simplify","text":"Cancel the common factor $$h$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction5a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{h\\\\to0} 8x+4h$$"],"dependencies":["acf60f3DerivativeFunction5a-h5"],"title":"Simplify","text":"What will get ?","variabilization":{},"oer":"","license":"","choices":["$$\\\\lim_{h\\\\to0} 8x+4h$$","$$\\\\lim_{h\\\\to0} x+h$$","$$\\\\lim_{h\\\\to0} 8x+4$$"]},{"id":"acf60f3DerivativeFunction5a-h7","type":"hint","dependencies":["acf60f3DerivativeFunction5a-h6"],"title":"Evaluate the limit","text":"To get a final answer, evaluate the limit by inserting h--\x3e0 $$=$$ $$h=0$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"acf60f3DerivativeFunction6","title":"Finding derivative","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.2 The Derivative as a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"acf60f3DerivativeFunction6a","stepAnswer":["$$\\\\frac{1}{\\\\sqrt{2x}}$$"],"problemType":"TextBox","stepTitle":"Use the definition of a derivative to find f\u2032(x).","stepBody":"$$f(x)=\\\\sqrt{2x}$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{\\\\sqrt{2x}}$$","hints":{"DefaultPathway":[{"id":"acf60f3DerivativeFunction6a-h1","type":"hint","dependencies":[],"title":"Derivative function","text":"Start directly with the definition of the derivative function. $$f\u2032(x)=\\\\lim_{h\\\\to0} \\\\frac{f{\\\\left(x+h\\\\right)}-f{\\\\left(x\\\\right)}}{h}$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction6a-h2","type":"hint","dependencies":["acf60f3DerivativeFunction6a-h1"],"title":"Derivative function","text":"We will get $$f\'(x)=\\\\lim_{h\\\\to0} \\\\frac{\\\\sqrt{2\\\\left(x+h\\\\right)}-\\\\sqrt{2x}}{h}$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction6a-h3","type":"hint","dependencies":["acf60f3DerivativeFunction6a-h2"],"title":"Simplify","text":"To simplify, Multiply numerator and denominator by $$\\\\sqrt{2\\\\left(h+x\\\\right)}+\\\\sqrt{x}$$ without distributing in the denominator","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction6a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{h\\\\to0} \\\\frac{2h}{h \\\\sqrt{2x+2h}+h \\\\sqrt{2x}}$$"],"dependencies":["acf60f3DerivativeFunction6a-h3"],"title":"Simplify","text":"What will get after multipling?","variabilization":{},"oer":"","license":"","choices":["$$\\\\lim_{h\\\\to0} \\\\frac{2h}{h \\\\sqrt{2x+2h}+h \\\\sqrt{2x}}$$","$$\\\\lim_{h\\\\to0} \\\\frac{2\\\\left(x+h\\\\right)}{h \\\\sqrt{2x+2h}+h \\\\sqrt{2x}}$$","$$\\\\lim_{h\\\\to0} 2h+\\\\frac{x}{h \\\\sqrt{2x+2h}+h \\\\sqrt{2x}}$$"],"subHints":[{"id":"acf60f3DerivativeFunction6a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2h$$"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\left(\\\\sqrt{2\\\\left(x+h\\\\right)}-\\\\sqrt{x}\\\\right) \\\\left(\\\\sqrt{2\\\\left(x+h\\\\right)}+\\\\sqrt{x}\\\\right)$$?","variabilization":{},"oer":"","license":""}]},{"id":"acf60f3DerivativeFunction6a-h5","type":"hint","dependencies":["acf60f3DerivativeFunction6a-h4"],"title":"Simplify","text":"Cancel the common factor $$h$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction6a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$/lim{h,0,2/(sqrt(2x+2h)+sqrt(2x))$$"],"dependencies":["acf60f3DerivativeFunction6a-h5"],"title":"Simplify","text":"What will get?","variabilization":{},"oer":"","license":"","choices":["$$/lim{h,0,2/(sqrt(x+h)+sqrt(x))$$","$$/lim{h,0,2/(sqrt(2x+2h)+sqrt(2x))$$","$$/lim{h,0,1/(sqrt(2x+2h)+sqrt(2x))$$"]},{"id":"acf60f3DerivativeFunction6a-h7","type":"hint","dependencies":["acf60f3DerivativeFunction6a-h6"],"title":"Evaluate the limit","text":"To get a final answer, evaluate the limit by inserting h--\x3e0 $$=$$ $$h=0$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"acf60f3DerivativeFunction7","title":"Finding derivative","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.2 The Derivative as a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"acf60f3DerivativeFunction7a","stepAnswer":["$$\\\\frac{-9}{x^2}$$"],"problemType":"MultipleChoice","stepTitle":"Use the definition of a derivative to find f\u2032(x).","stepBody":"$$f(x)=\\\\frac{9}{x}$$","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{-9}{x^2}$$","choices":["$$\\\\frac{-9}{x^2}$$","$$\\\\frac{9}{x^2}$$"],"hints":{"DefaultPathway":[{"id":"acf60f3DerivativeFunction7a-h1","type":"hint","dependencies":[],"title":"Derivative function","text":"Start directly with the definition of the derivative function. $$f\u2032(x)=\\\\lim_{h\\\\to0} \\\\frac{f{\\\\left(x+h\\\\right)}-f{\\\\left(x\\\\right)}}{h}$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction7a-h2","type":"hint","dependencies":["acf60f3DerivativeFunction7a-h1"],"title":"Derivative function","text":"We will get $$f\'(x)=\\\\lim_{h\\\\to0} \\\\frac{\\\\frac{9}{x+h}-\\\\frac{9}{x}}{h}$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction7a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{-9h}{x^2+xh}$$"],"dependencies":["acf60f3DerivativeFunction7a-h2"],"title":"Simplify numerator","text":"What is the simplified numerator $$\\\\frac{9}{x+h}-\\\\frac{9}{x}$$?","variabilization":{},"oer":"","license":"","choices":["$$\\\\frac{-9h}{x^2+xh}$$","$$\\\\frac{9h}{x^2+xh}$$"]},{"id":"acf60f3DerivativeFunction7a-h4","type":"hint","dependencies":["acf60f3DerivativeFunction7a-h3"],"title":"Simplify","text":"$$\\\\frac{\\\\left(-\\\\frac{9}{x^2+xh}\\\\right)}{h}$$ equals to $$\\\\frac{1}{h} \\\\left(-\\\\frac{9}{x^2+xh}\\\\right)$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction7a-h5","type":"hint","dependencies":["acf60f3DerivativeFunction7a-h4"],"title":"Simplify","text":"Cancel the common factor $$h$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction7a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$/lim{h,0$$, $$\\\\frac{-9}{x^2+xh}$$}"],"dependencies":["acf60f3DerivativeFunction7a-h5"],"title":"Simplify","text":"What will get ?","variabilization":{},"oer":"","license":"","choices":["$$/lim{h,0$$, $$\\\\frac{-9h}{x^2+xh}$$}","$$/lim{h,0$$, $$\\\\frac{-9}{x^2+xh}$$}","$$/lim{h,0$$, $$\\\\frac{-9}{h{\\\\left(x^2+xh\\\\right)}}$$}"]},{"id":"acf60f3DerivativeFunction7a-h7","type":"hint","dependencies":["acf60f3DerivativeFunction7a-h6"],"title":"Evaluate the limit","text":"To get a final answer, evaluate the limit by inserting h--\x3e0 $$=$$ $$h=0$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"acf60f3DerivativeFunction8","title":"Finding a Second Derivative","body":"For $$f(x)=2x^2-3x+1$$, find f\'\'(x).","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.2 The Derivative as a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"acf60f3DerivativeFunction8a","stepAnswer":["$$4x-3$$"],"problemType":"TextBox","stepTitle":"First derivative","stepBody":"First, find the f\'(x) using derivative function","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4x-3$$","hints":{"DefaultPathway":[{"id":"acf60f3DerivativeFunction8a-h1","type":"hint","dependencies":[],"title":"Derivative function","text":"Start directly with the definition of the derivative function. $$f\u2032(x)=\\\\lim_{h\\\\to0} \\\\frac{f{\\\\left(x+h\\\\right)}-f{\\\\left(x\\\\right)}}{h}$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction8a-h2","type":"hint","dependencies":["acf60f3DerivativeFunction8a-h1"],"title":"Derivative function","text":"We will get $$f\'(x)=\\\\lim_{h\\\\to0} \\\\frac{2{\\\\left(x+h\\\\right)}^2-3\\\\left(x+h\\\\right)+1-2x^2-3x+1}{h}$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction8a-h3","type":"hint","dependencies":["acf60f3DerivativeFunction8a-h2"],"title":"Expand","text":"Expand $$\\\\frac{2{\\\\left(x+h\\\\right)}^2-3\\\\left(x+h\\\\right)+1-2x^2-3x+1}{h}$$ and Simplify","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction8a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{h\\\\to0} \\\\frac{4xh+2h^2-3h}{h}$$"],"dependencies":["acf60f3DerivativeFunction8a-h3"],"title":"Simplify","text":"What will get?","variabilization":{},"oer":"","license":"","choices":["$$\\\\lim_{h\\\\to0} \\\\frac{4xh+2h^2-3h}{h}$$","$$\\\\lim_{h\\\\to0} \\\\frac{xh+h^2-h}{h}$$","$$\\\\lim_{h\\\\to0} \\\\frac{4xh+2h-3}{h}$$"]},{"id":"acf60f3DerivativeFunction8a-h5","type":"hint","dependencies":["acf60f3DerivativeFunction8a-h4"],"title":"Simplify","text":"Cancel the common factor $$h$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction8a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{h\\\\to0} 4x+2h-3$$"],"dependencies":["acf60f3DerivativeFunction8a-h5"],"title":"Simplify","text":"What will get ?","variabilization":{},"oer":"","license":"","choices":["$$\\\\lim_{h\\\\to0} 4x+2-3h$$","$$\\\\lim_{h\\\\to0} 4xh+2h-3$$","$$\\\\lim_{h\\\\to0} 4x+2h-3$$"]},{"id":"acf60f3DerivativeFunction8a-h7","type":"hint","dependencies":["acf60f3DerivativeFunction8a-h6"],"title":"Evaluate the limit","text":"Evaluate the limit by inserting h--\x3e0 $$=$$ $$h=0$$","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":""}]}},{"id":"acf60f3DerivativeFunction8b","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"Finding a Second Derivative","stepBody":"Next, find f\'\'(x) by taking the derivative of $$f\u2032(x)=4x-3$$.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"acf60f3DerivativeFunction8b-h1","type":"hint","dependencies":[],"title":"Second derivative","text":"Use $$f\'\'(x)=\\\\lim_{h\\\\to0} \\\\frac{\\\\operatorname{f\'}\\\\left(x+h\\\\right)-\\\\operatorname{f\'}\\\\left(x\\\\right)}{h}$$ to find second derivative","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction8b-h2","type":"hint","dependencies":["acf60f3DerivativeFunction8b-h1"],"title":"Derivative function","text":"We will get $$f\'\'(x)=\\\\lim_{h\\\\to0} \\\\frac{4\\\\left(x+h\\\\right)-3-4x-3}{h}$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction8b-h3","type":"hint","dependencies":["acf60f3DerivativeFunction8b-h2"],"title":"Expand","text":"Expand $$\\\\frac{4\\\\left(x+h\\\\right)-3-4x-3}{h}$$ and Simplify","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction8b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{h\\\\to0} 4$$"],"dependencies":["acf60f3DerivativeFunction8b-h3"],"title":"Simplify","text":"What will get?","variabilization":{},"oer":"","license":"","choices":["$$\\\\lim_{h\\\\to0} 4$$","$$\\\\lim_{h\\\\to0} 4h$$","$$\\\\lim_{h\\\\to0} \\\\frac{4}{h}$$"]},{"id":"acf60f3DerivativeFunction8b-h5","type":"hint","dependencies":["acf60f3DerivativeFunction8b-h4"],"title":"Evaluate the limit","text":"To get a final answer, evaluate the limit by inserting h--\x3e0 $$=$$ $$h=0$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"acf60f3DerivativeFunction9","title":"Finding a Second Derivative","body":"For $$f(x)=x^2$$, find f\'\'(x).","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"3.2 The Derivative as a Function","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"acf60f3DerivativeFunction9a","stepAnswer":["$$2x$$"],"problemType":"TextBox","stepTitle":"First derivative","stepBody":"First, find the f\'(x) using derivative function","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2x$$","hints":{"DefaultPathway":[{"id":"acf60f3DerivativeFunction9a-h1","type":"hint","dependencies":[],"title":"Derivative function","text":"Start directly with the definition of the derivative function. $$f\u2032(x)=\\\\lim_{h\\\\to0} \\\\frac{f{\\\\left(x+h\\\\right)}-f{\\\\left(x\\\\right)}}{h}$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction9a-h2","type":"hint","dependencies":["acf60f3DerivativeFunction9a-h1"],"title":"Derivative function","text":"We will get $$f\'(x)=\\\\lim_{h\\\\to0} \\\\frac{{\\\\left(x+h\\\\right)}^2-x^2}{h}$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction9a-h3","type":"hint","dependencies":["acf60f3DerivativeFunction9a-h2"],"title":"Expand","text":"Expand $$\\\\frac{{\\\\left(x+h\\\\right)}^2-x^2}{h}$$ and Simplify","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction9a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{h\\\\to0} \\\\frac{2xh+h^2}{h}$$"],"dependencies":["acf60f3DerivativeFunction9a-h3"],"title":"Simplify","text":"What will get?","variabilization":{},"oer":"","license":"","choices":["$$\\\\lim_{h\\\\to0} \\\\frac{2xh+h^2+h}{h}$$","$$\\\\lim_{h\\\\to0} \\\\frac{2xh+h}{h}$$","$$\\\\lim_{h\\\\to0} \\\\frac{2xh+h^2}{h}$$"]},{"id":"acf60f3DerivativeFunction9a-h5","type":"hint","dependencies":["acf60f3DerivativeFunction9a-h4"],"title":"Simplify","text":"Cancel the common factor $$h$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction9a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{h\\\\to0} 2x+h$$"],"dependencies":["acf60f3DerivativeFunction9a-h5"],"title":"Simplify","text":"What will get ?","variabilization":{},"oer":"","license":"","choices":["$$\\\\lim_{h\\\\to0} x+h$$","$$\\\\lim_{h\\\\to0} 2x+h$$","$$\\\\lim_{h\\\\to0} 2x$$"]},{"id":"acf60f3DerivativeFunction9a-h7","type":"hint","dependencies":["acf60f3DerivativeFunction9a-h6"],"title":"Evaluate the limit","text":"Evaluate the limit by inserting h--\x3e0 $$=$$ $$h=0$$","variabilization":{},"oer":"","license":""}]}},{"id":"acf60f3DerivativeFunction9b","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"Finding a Second Derivative","stepBody":"Next, find f\'\'(x) by taking the derivative of $$f\u2032(x)=2x$$.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"acf60f3DerivativeFunction9b-h1","type":"hint","dependencies":[],"title":"Second derivative","text":"Use $$f\'\'(x)=\\\\lim_{h\\\\to0} \\\\frac{\\\\operatorname{f\'}\\\\left(x+h\\\\right)-\\\\operatorname{f\'}\\\\left(x\\\\right)}{h}$$ to find second derivative","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction9b-h2","type":"hint","dependencies":["acf60f3DerivativeFunction9b-h1"],"title":"Derivative function","text":"We will get $$f\'\'(x)=\\\\lim_{h\\\\to0} \\\\frac{2\\\\left(x+h\\\\right)-2x}{h}$$","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction9b-h3","type":"hint","dependencies":["acf60f3DerivativeFunction9b-h2"],"title":"Expand","text":"Expand $$\\\\frac{2\\\\left(x+h\\\\right)-2x}{h}$$ and Simplify","variabilization":{},"oer":"","license":""},{"id":"acf60f3DerivativeFunction9b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\lim_{h\\\\to0} 2$$"],"dependencies":["acf60f3DerivativeFunction9b-h3"],"title":"Simplify","text":"What will get?","variabilization":{},"oer":"","license":"","choices":["$$\\\\lim_{h\\\\to0} 2x$$","$$\\\\lim_{h\\\\to0} 2+h$$","$$\\\\lim_{h\\\\to0} 2$$"]},{"id":"acf60f3DerivativeFunction9b-h5","type":"hint","dependencies":["acf60f3DerivativeFunction9b-h4"],"title":"Evaluate the limit","text":"To get a final answer, evaluate the limit by inserting h--\x3e0 $$=$$ $$h=0$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"acff542gra1","title":"System of Linear Inequalities","body":"Determine whether the ordered pair is a solution to the system.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acff542gra1a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$(-2,4)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"acff542gra1a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitute $$x=-2$$ and $$y=4$$ into both inequalities.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra1a-h2","type":"hint","dependencies":["acff542gra1a-h1"],"title":"Substitute into First Equation","text":"$$x+4y \\\\geq 10$$\\\\n$$-2+4\\\\times4 \\\\geq 10$$\\\\n$$14 \\\\geq 10$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra1a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["T"],"dependencies":["acff542gra1a-h2"],"title":"Substitute into First Equation","text":"Is the inequality above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"acff542gra1a-h4","type":"hint","dependencies":["acff542gra1a-h3"],"title":"Substitute into Second Equation","text":"$$3x-2y<12$$\\\\n$$3\\\\left(-2\\\\right)-2\\\\times4<12$$\\\\n$$-14<12$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra1a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["T"],"dependencies":["acff542gra1a-h4"],"title":"Substitute into Second Equation","text":"Is the inequality above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"acff542gra1a-h6","type":"hint","dependencies":["acff542gra1a-h5"],"title":"Solutions of a System of Equations","text":"$$(-2,4)$$ does make both inequalities true. $$(-2,4)$$ is a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"acff542gra1b","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$(3,1)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"acff542gra1b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitute $$x=3$$ and $$y=1$$ into both inequalities.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra1b-h2","type":"hint","dependencies":["acff542gra1b-h1"],"title":"Substitute into First Equation","text":"$$x+4y \\\\geq 10$$\\\\n$$3+4\\\\times1 \\\\geq 10$$\\\\n$$7 \\\\geq 10$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra1b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["F"],"dependencies":["acff542gra1b-h2"],"title":"Substitute into First Equation","text":"Is the inequality above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"acff542gra1b-h4","type":"hint","dependencies":["acff542gra1b-h3"],"title":"Solutions of a System of Equations","text":"Since $$(3,1)$$ does not satisfy the first inequality, then it cannot make both inequalities true. $$(3,1)$$ is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acff542gra10","title":"System of Linear Inequalities","body":"Solve the system by graphing. Which graph represents the solution?\\\\n##figure6.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acff542gra10a","stepAnswer":["A"],"problemType":"MultipleChoice","stepTitle":"Refer to the first image for the system.","stepBody":"##figure1.gif## ##figure2.gif## ##figure3.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C"],"hints":{"DefaultPathway":[{"id":"acff542gra10a-h1","type":"hint","dependencies":[],"title":"Graph First Inequality","text":"Graph $$y \\\\geq 2x-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra10a-h2","type":"hint","dependencies":["acff542gra10a-h1"],"title":"Graph the Boundary Line","text":"It is a solid line because the inequality sign is $$ \\\\geq $$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra10a-h3","type":"hint","dependencies":["acff542gra10a-h2"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y \\\\geq 2x-1$$\\\\n$$0 \\\\geq 2\\\\times0-1$$\\\\n$$0 \\\\geq -1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra10a-h4","type":"hint","dependencies":["acff542gra10a-h3"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is a solution to $$y \\\\geq 2x-1$$. Since $$(0,0)$$ is on the left side of the line, so we shade the left side of the boundary line in red.\\\\n##figure4.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra10a-h5","type":"hint","dependencies":["acff542gra10a-h4"],"title":"Graph Second Inequality","text":"Graph $$y<x+1$$ on the same grid.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra10a-h6","type":"hint","dependencies":["acff542gra10a-h5"],"title":"Graph the Boundary Line","text":"It is a dashed line because the inequality sign is <.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra10a-h7","type":"hint","dependencies":["acff542gra10a-h6"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y<x+1$$\\\\n$$0<0+1$$\\\\n$$0<1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra10a-h8","type":"hint","dependencies":["acff542gra10a-h7"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is a solution to $$y<x+1$$. Since $$(0,0)$$ is on the right side of the line, so we shade in the right side of the line $$y=x+1$$ in blue.\\\\n##figure5.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra10a-h9","type":"hint","dependencies":["acff542gra10a-h8"],"title":"Overlap Shading","text":"The solution is all points in the darker shaded region.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acff542gra11","title":"System of Linear Inequalities","body":"Solve the system by graphing. Which graph represents the solution?\\\\n##figure7.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acff542gra11a","stepAnswer":["B"],"problemType":"MultipleChoice","stepTitle":"","stepBody":"##figure1.gif## ##figure2.gif## ##figure3.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C"],"hints":{"DefaultPathway":[{"id":"acff542gra11a-h1","type":"hint","dependencies":[],"title":"Graph First Inequality","text":"Graph $$y \\\\leq 3x+2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra11a-h2","type":"hint","dependencies":["acff542gra11a-h1"],"title":"Graph the Boundary Line","text":"It is a solid line because the inequality sign is $$ \\\\leq $$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra11a-h3","type":"hint","dependencies":["acff542gra11a-h2"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y \\\\leq 3x+2$$\\\\n$$0 \\\\leq 3\\\\times0+2$$\\\\n$$0 \\\\leq 2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra11a-h4","type":"hint","dependencies":["acff542gra11a-h3"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is a solution to $$y \\\\leq 3x+2$$. Since $$(0,0)$$ is on the right side of the line, so we shade the right side of the boundary line in red.\\\\n##figure4.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra11a-h5","type":"hint","dependencies":["acff542gra11a-h4"],"title":"Graph Second Inequality","text":"Graph $$y>x-1$$ on the same grid.\\\\n##figure5.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra11a-h6","type":"hint","dependencies":["acff542gra11a-h5"],"title":"Graph the Boundary Line","text":"It is a dashed line because the inequality sign is >.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra11a-h7","type":"hint","dependencies":["acff542gra11a-h6"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y>x-1$$\\\\n$$0>0-1$$\\\\n$$0>-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra11a-h8","type":"hint","dependencies":["acff542gra11a-h7"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is a solution to $$y>x-1$$. Since $$(0,0)$$ is on the left side of the line, so we shade in the left side of the line $$y=x-1$$ in green.\\\\n##figure6.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra11a-h9","type":"hint","dependencies":["acff542gra11a-h8"],"title":"Overlap Shading","text":"The solution is all points in the darker shaded region.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acff542gra12","title":"System of Linear Inequalities","body":"Solve the system by graphing. Which graph represents the solution?\\\\n##figure6.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acff542gra12a","stepAnswer":["C"],"problemType":"MultipleChoice","stepTitle":"The graphs are in the order A, B, and C.","stepBody":"##figure1.gif## ##figure2.gif## ##figure3.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C"],"hints":{"DefaultPathway":[{"id":"acff542gra12a-h1","type":"hint","dependencies":[],"title":"Graph First Inequality","text":"Graph $$y<-2x+2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra12a-h2","type":"hint","dependencies":["acff542gra12a-h1"],"title":"Graph the Boundary Line","text":"It is a dashed line because the inequality sign is <.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra12a-h3","type":"hint","dependencies":["acff542gra12a-h2"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y<-2x+2$$\\\\n$$0<-2\\\\times0+2$$\\\\n$$0<2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra12a-h4","type":"hint","dependencies":["acff542gra12a-h3"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is a solution to $$y<-2x+2$$. Since $$(0,0)$$ is on the left side of the line, we shade the left side of the boundary line in red.\\\\n##figure4.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra12a-h5","type":"hint","dependencies":["acff542gra12a-h4"],"title":"Graph Second Inequality","text":"Graph $$y \\\\geq -x-1$$ on the same grid.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra12a-h6","type":"hint","dependencies":["acff542gra12a-h5"],"title":"Graph the Boundary Line","text":"It is a solid line because the inequality sign is $$ \\\\geq $$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra12a-h7","type":"hint","dependencies":["acff542gra12a-h6"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y \\\\geq -x-1$$\\\\n$$0 \\\\geq -0-1$$\\\\n$$0 \\\\geq -1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra12a-h8","type":"hint","dependencies":["acff542gra12a-h7"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is a solution to $$y \\\\geq -x-1$$. Since $$(0,0)$$ is on the right side of the line, so we shade in the right side of the line $$y=-x-1$$ in green.\\\\n##figure5.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra12a-h9","type":"hint","dependencies":["acff542gra12a-h8"],"title":"Overlap Shading","text":"The solution is all points in the darker shaded region.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acff542gra13","title":"System of Linear Inequalities","body":"Solve the system by graphing. Which graph represents the solution?\\\\n##figure6.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acff542gra13a","stepAnswer":["A"],"problemType":"MultipleChoice","stepTitle":"The graphs are in the order A, B, and C.","stepBody":"##figure1.gif## ##figure2.gif## ##figure3.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C"],"hints":{"DefaultPathway":[{"id":"acff542gra13a-h1","type":"hint","dependencies":[],"title":"Graph First Inequality","text":"Graph $$y<2x-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra13a-h2","type":"hint","dependencies":["acff542gra13a-h1"],"title":"Graph the Boundary Line","text":"It is a dashed line because the inequality sign is <.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra13a-h3","type":"hint","dependencies":["acff542gra13a-h2"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y<2x-1$$\\\\n$$0<2\\\\times0-1$$\\\\n$$0<-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra13a-h4","type":"hint","dependencies":["acff542gra13a-h3"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is not a solution to $$y<2x-1$$. Since $$(0,0)$$ is on the left side of the line, we shade the right side of the boundary line in red.\\\\n##figure4.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra13a-h5","type":"hint","dependencies":["acff542gra13a-h4"],"title":"Graph Second Inequality","text":"Graph $$y \\\\leq \\\\frac{-1}{2} x+4$$ on the same grid.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra13a-h6","type":"hint","dependencies":["acff542gra13a-h5"],"title":"Graph the Boundary Line","text":"It is a solid line because the inequality sign is $$ \\\\leq $$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra13a-h7","type":"hint","dependencies":["acff542gra13a-h6"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y \\\\leq \\\\frac{-1}{2} x+4$$\\\\n$$0 \\\\leq 0\\\\frac{-1}{2}+4$$\\\\n$$0 \\\\leq 4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra13a-h8","type":"hint","dependencies":["acff542gra13a-h7"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is a solution to $$y \\\\leq \\\\frac{-1}{2} x+4$$. Since $$(0,0)$$ is on the left side of the line, so we shade in the left side of the line $$y=\\\\frac{-1}{2} x+4$$ in green.\\\\n##figure5.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra13a-h9","type":"hint","dependencies":["acff542gra13a-h8"],"title":"Overlap Shading","text":"The solution is all points in the darker shaded region.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acff542gra14","title":"System of Linear Inequalities","body":"Solve the system by graphing. Which graph represents the solution?\\\\n##figure6.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acff542gra14a","stepAnswer":["B"],"problemType":"MultipleChoice","stepTitle":"The graphs are in the order A, B, and C.","stepBody":"##figure1.gif## ##figure2.gif## ##figure3.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C"],"hints":{"DefaultPathway":[{"id":"acff542gra14a-h1","type":"hint","dependencies":[],"title":"Graph First Inequality","text":"Graph $$y \\\\geq \\\\frac{-2}{3} x+2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra14a-h2","type":"hint","dependencies":["acff542gra14a-h1"],"title":"Graph the Boundary Line","text":"It is a solid line because the inequality sign is $$ \\\\geq $$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra14a-h3","type":"hint","dependencies":["acff542gra14a-h2"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y \\\\geq \\\\frac{-2}{3} x+2$$\\\\n$$0 \\\\geq 0\\\\frac{-2}{3}+2$$\\\\n$$0 \\\\geq 2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra14a-h4","type":"hint","dependencies":["acff542gra14a-h3"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is not a solution to $$y \\\\geq \\\\frac{-2}{3} x+2$$. Since $$(0,0)$$ is on the left side of the line, we shade the right side of the boundary line in red.\\\\n##figure4.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra14a-h5","type":"hint","dependencies":["acff542gra14a-h4"],"title":"Graph Second Inequality","text":"Graph $$y>2x-3$$ on the same grid.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra14a-h6","type":"hint","dependencies":["acff542gra14a-h5"],"title":"Graph the Boundary Line","text":"It is a dashed line because the inequality sign is >.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra14a-h7","type":"hint","dependencies":["acff542gra14a-h6"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y>2x-3$$\\\\n$$0>2\\\\times0-3$$\\\\n$$0>-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra14a-h8","type":"hint","dependencies":["acff542gra14a-h7"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is a solution to $$y \\\\geq \\\\frac{-2}{3} x+2$$. Since $$(0,0)$$ is on the left side of the line, so we shade in the left side of the line $$y=\\\\frac{-2}{3} x+2$$ in green.\\\\n##figure5.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra14a-h9","type":"hint","dependencies":["acff542gra14a-h8"],"title":"Overlap Shading","text":"The solution is all points in the darker shaded region.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acff542gra15","title":"System of Linear Inequalities","body":"Solve the system by graphing. Which graph represents the solution?\\\\n##figure6.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acff542gra15a","stepAnswer":["C"],"problemType":"MultipleChoice","stepTitle":"The graphs are in the order A, B, and C.","stepBody":"##figure1.gif## ##figure2.gif## ##figure3.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C"],"hints":{"DefaultPathway":[{"id":"acff542gra15a-h1","type":"hint","dependencies":[],"title":"Graph First Inequality","text":"Graph $$x-y>3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra15a-h2","type":"hint","dependencies":["acff542gra15a-h1"],"title":"Change to Slope-Intercept Form","text":"$$x-y>3$$\\\\n$$-y>-x+3$$\\\\n$$y<x-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra15a-h3","type":"hint","dependencies":["acff542gra15a-h2"],"title":"Graph the Boundary Line","text":"It is a dashed line because the inequality sign is <.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra15a-h4","type":"hint","dependencies":["acff542gra15a-h3"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y<x-3$$\\\\n$$0<0-3$$\\\\n$$0<-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra15a-h5","type":"hint","dependencies":["acff542gra15a-h4"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is not a solution to $$y<x-3$$. Since $$(0,0)$$ is on the left side of the line, we shade the right side of the boundary line in red.\\\\n##figure4.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra15a-h6","type":"hint","dependencies":["acff542gra15a-h5"],"title":"Graph Second Inequality","text":"Graph $$y<\\\\frac{-1}{5} x+4$$ on the same grid.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra15a-h7","type":"hint","dependencies":["acff542gra15a-h6"],"title":"Graph the Boundary Line","text":"It is a dashed line because the inequality sign is <.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra15a-h8","type":"hint","dependencies":["acff542gra15a-h7"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y<\\\\frac{-1}{5} x+4$$\\\\n$$0<0\\\\frac{-1}{5}+4$$\\\\n$$0<4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra15a-h9","type":"hint","dependencies":["acff542gra15a-h8"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is a solution to $$y<\\\\frac{-1}{5} x+4$$. Since $$(0,0)$$ is on the left side of the line, so we shade in the left side of the line $$y=\\\\frac{-1}{5} x+4$$ in green.\\\\n##figure5.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra15a-h10","type":"hint","dependencies":["acff542gra15a-h9"],"title":"Overlap Shading","text":"The solution is all points in the darker shaded region.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acff542gra16","title":"System of Linear Inequalities","body":"Solve the system by graphing. Which graph represents the solution?\\\\n##figure6.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acff542gra16a","stepAnswer":["A"],"problemType":"MultipleChoice","stepTitle":"The graphs are in the order A, B, and C.","stepBody":"##figure1.gif## ##figure2.gif## ##figure3.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C"],"hints":{"DefaultPathway":[{"id":"acff542gra16a-h1","type":"hint","dependencies":[],"title":"Graph First Inequality","text":"Graph $$x-y>1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra16a-h2","type":"hint","dependencies":["acff542gra16a-h1"],"title":"Change to Slope-Intercept Form","text":"$$x-y>1$$\\\\n$$-y>-x+1$$\\\\n$$y<x-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra16a-h3","type":"hint","dependencies":["acff542gra16a-h2"],"title":"Graph the Boundary Line","text":"It is a dashed line because the inequality sign is <.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra16a-h4","type":"hint","dependencies":["acff542gra16a-h3"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y<x-1$$\\\\n$$0<0-1$$\\\\n$$0<-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra16a-h5","type":"hint","dependencies":["acff542gra16a-h4"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is not a solution to $$y<x-1$$. Since $$(0,0)$$ is on the left side of the line, we shade the right side of the boundary line in red.\\\\n##figure4.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra16a-h6","type":"hint","dependencies":["acff542gra16a-h5"],"title":"Graph Second Inequality","text":"Graph $$y<\\\\frac{-1}{4} x+3$$ on the same grid.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra16a-h7","type":"hint","dependencies":["acff542gra16a-h6"],"title":"Graph the Boundary Line","text":"It is a dashed line because the inequality sign is <.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra16a-h8","type":"hint","dependencies":["acff542gra16a-h7"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y<\\\\frac{-1}{4} x+3$$\\\\n$$0<0\\\\frac{-1}{4}+3$$\\\\n$$0<3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra16a-h9","type":"hint","dependencies":["acff542gra16a-h8"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is a solution to $$y<\\\\frac{-1}{4} x+3$$. Since $$(0,0)$$ is on the left side of the line, so we shade in the left side of the line $$y=\\\\frac{-1}{4} x+3$$ in green.\\\\n##figure5.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra16a-h10","type":"hint","dependencies":["acff542gra16a-h9"],"title":"Overlap Shading","text":"The solution is all points in the darker shaded region.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acff542gra2","title":"System of Linear Inequalities","body":"Determine whether the ordered pair is a solution to the system.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acff542gra2a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$(3,-3)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"acff542gra2a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitute $$x=3$$ and $$y=-3$$ into both inequalities.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra2a-h2","type":"hint","dependencies":["acff542gra2a-h1"],"title":"Substitute into First Equation","text":"$$3x+y>5$$\\\\n$$3\\\\times3-3>5$$\\\\n$$6>5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra2a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["T"],"dependencies":["acff542gra2a-h2"],"title":"Substitute into First Equation","text":"Is the inequality above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"acff542gra2a-h4","type":"hint","dependencies":["acff542gra2a-h3"],"title":"Substitute into Second Equation","text":"$$2x-y \\\\leq 10$$\\\\n$$2\\\\times3-\\\\left(-3\\\\right) \\\\leq 10$$\\\\n$$9 \\\\leq 10$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra2a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["T"],"dependencies":["acff542gra2a-h4"],"title":"Substitute into Second Equation","text":"Is the inequality above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"acff542gra2a-h6","type":"hint","dependencies":["acff542gra2a-h5"],"title":"Solutions of a System of Equations","text":"$$(3,-3)$$ makes both inequalities true. $$(3,-3)$$ is a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"acff542gra2b","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$(7,1)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"acff542gra2b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitute $$x=7$$ and $$y=1$$ into both inequalities.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra2b-h2","type":"hint","dependencies":["acff542gra2b-h1"],"title":"Substitute into First Equation","text":"$$3x+y>5$$\\\\n$$3\\\\times7+1>5$$\\\\n$$22>5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra2b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["T"],"dependencies":["acff542gra2b-h2"],"title":"Substitute into First Equation","text":"Is the inequality above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"acff542gra2b-h4","type":"hint","dependencies":["acff542gra2b-h3"],"title":"Substitute into Second Equation","text":"$$2x-y \\\\leq 10$$\\\\n$$2\\\\times7-1 \\\\leq 10$$\\\\n$$13 \\\\leq 10$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra2b-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["F"],"dependencies":["acff542gra2b-h4"],"title":"Substitute into Second Equation","text":"Is the inequality above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"acff542gra2b-h6","type":"hint","dependencies":["acff542gra2b-h5"],"title":"Solutions of a System of Equations","text":"$$(7,1)$$ does not make both inequalities true. $$(7,1)$$ is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acff542gra22","title":"System of Linear Inequalities","body":"Solve the system by graphing. Which graph represents the solution?\\\\n##figure6.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acff542gra22a","stepAnswer":["A"],"problemType":"MultipleChoice","stepTitle":"The graphs are in the order A, B, and C.","stepBody":"##figure1.gif## ##figure2.gif## ##figure3.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C"],"hints":{"DefaultPathway":[{"id":"acff542gra22a-h1","type":"hint","dependencies":[],"title":"Graph First Inequality","text":"Graph $$x-3y>4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra22a-h2","type":"hint","dependencies":["acff542gra22a-h1"],"title":"Change to Slope-Intercept Form","text":"$$x-3y>4$$\\\\n$$-3y>-x+4$$\\\\n$$\\\\frac{-3y}{-3}>\\\\frac{\\\\left(-x+4\\\\right)}{-3}$$\\\\n$$y<\\\\frac{1}{3} x-\\\\frac{4}{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra22a-h3","type":"hint","dependencies":["acff542gra22a-h2"],"title":"Graph the Boundary Line","text":"It is a dashed line because the inequality sign is <.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra22a-h4","type":"hint","dependencies":["acff542gra22a-h3"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y<\\\\frac{1}{3} x-\\\\frac{4}{3}$$\\\\n$$0<0\\\\frac{1}{3}-\\\\frac{4}{3}$$\\\\n$$0<\\\\frac{-4}{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra22a-h5","type":"hint","dependencies":["acff542gra22a-h4"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is not a solution to $$y<\\\\frac{1}{3} x-\\\\frac{4}{3}$$. Since $$(0,0)$$ is on the left side of the line, we shade the right side of the boundary line in red.\\\\n##figure4.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra22a-h6","type":"hint","dependencies":["acff542gra22a-h5"],"title":"Graph Second Inequality","text":"Graph $$y \\\\leq -1$$ on the same grid. Recognize that it is a horizontal line through $$y=-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra22a-h7","type":"hint","dependencies":["acff542gra22a-h6"],"title":"Graph the Boundary Line","text":"It is a solid line because the inequality sign is $$ \\\\leq $$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra22a-h8","type":"hint","dependencies":["acff542gra22a-h7"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y \\\\leq -1$$\\\\n$$0 \\\\leq -1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra22a-h9","type":"hint","dependencies":["acff542gra22a-h8"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is not a solution to $$y \\\\leq -1$$. Since $$(0,0)$$ is on the top side of the line, so we shade in the bottom side of the line $$y=-1$$ in green.\\\\n##figure5.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra22a-h10","type":"hint","dependencies":["acff542gra22a-h9"],"title":"Overlap Shading","text":"The solution is all points in the darker shaded region.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acff542gra23","title":"System of Linear Inequalities","body":"Solve the system by graphing. Which graph represents the solution?\\\\n##figure4.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acff542gra23a","stepAnswer":["B"],"problemType":"MultipleChoice","stepTitle":"The graphs are in the order A, B, and C.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C"],"hints":{"DefaultPathway":[{"id":"acff542gra23a-h1","type":"hint","dependencies":[],"title":"Graph First Inequality","text":"Graph $$y \\\\geq \\\\frac{-1}{2} x-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra23a-h2","type":"hint","dependencies":["acff542gra23a-h1"],"title":"Graph the Boundary Line","text":"It is a solid line because the inequality sign is $$ \\\\geq $$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra23a-h3","type":"hint","dependencies":["acff542gra23a-h2"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y \\\\geq \\\\frac{-1}{2} x-3$$\\\\n$$0 \\\\geq 0\\\\frac{-1}{2}-3$$\\\\n$$0 \\\\geq -3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra23a-h4","type":"hint","dependencies":["acff542gra23a-h3"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is a solution to $$y \\\\geq \\\\frac{-1}{2} x-3$$. Since $$(0,0)$$ is on the right side of the line, we shade the right side of the boundary line in red.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra23a-h5","type":"hint","dependencies":["acff542gra23a-h4"],"title":"Graph Second Inequality","text":"Graph $$x \\\\leq 2$$ on the same grid. Recognize that it is a vertical line through $$x=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra23a-h6","type":"hint","dependencies":["acff542gra23a-h5"],"title":"Graph the Boundary Line","text":"It is a solid line because the inequality sign is $$ \\\\leq $$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra23a-h7","type":"hint","dependencies":["acff542gra23a-h6"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$x \\\\leq 2$$\\\\n$$0 \\\\leq 2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra23a-h8","type":"hint","dependencies":["acff542gra23a-h7"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is a solution to $$x \\\\leq 2$$. Since $$(0,0)$$ is on the left side of the line, so we shade in the left side of the line $$x=2$$ in green.\\\\n##figure3.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra23a-h9","type":"hint","dependencies":["acff542gra23a-h8"],"title":"Overlap Shading","text":"The solution is all points in the darker shaded region.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acff542gra24","title":"System of Linear Inequalities","body":"Solve the system by graphing. Which graph represents the solution?\\\\n##figure6.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acff542gra24a","stepAnswer":["C"],"problemType":"MultipleChoice","stepTitle":"The graphs are in the order A, B, and C.","stepBody":"##figure1.gif## ##figure2.gif## ##figure3.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C"],"hints":{"DefaultPathway":[{"id":"acff542gra24a-h1","type":"hint","dependencies":[],"title":"Graph First Inequality","text":"Graph $$y \\\\leq \\\\frac{-2}{3} x+5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra24a-h2","type":"hint","dependencies":["acff542gra24a-h1"],"title":"Graph the Boundary Line","text":"It is a solid line because the inequality sign is $$ \\\\leq $$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra24a-h3","type":"hint","dependencies":["acff542gra24a-h2"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y \\\\leq \\\\frac{-2}{3} x+5$$\\\\n$$0 \\\\leq 0\\\\frac{-2}{3}+5$$\\\\n$$0 \\\\leq 5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra24a-h4","type":"hint","dependencies":["acff542gra24a-h3"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is a solution to $$y \\\\leq \\\\frac{-2}{3} x+5$$. Since $$(0,0)$$ is on the left side of the line, we shade the left side of the boundary line in red.\\\\n##figure4.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra24a-h5","type":"hint","dependencies":["acff542gra24a-h4"],"title":"Graph Second Inequality","text":"Graph $$x \\\\geq 3$$ on the same grid. Recognize that it is a vertical line through $$x=3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra24a-h6","type":"hint","dependencies":["acff542gra24a-h5"],"title":"Graph the Boundary Line","text":"It is a solid line because the inequality sign is $$ \\\\geq $$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra24a-h7","type":"hint","dependencies":["acff542gra24a-h6"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$x \\\\geq 3$$\\\\n$$0 \\\\geq 3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra24a-h8","type":"hint","dependencies":["acff542gra24a-h7"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is not a solution to $$x \\\\geq 3$$. Since $$(0,0)$$ is on the left side of the line, so we shade in the right side of the line $$x=3$$ in green.\\\\n##figure5.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra24a-h9","type":"hint","dependencies":["acff542gra24a-h8"],"title":"Overlap Shading","text":"The solution is all points in the darker shaded region.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acff542gra25","title":"System of Linear Inequalities","body":"Solve the system by graphing. Which graph represents the solution?\\\\n##figure6.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acff542gra25a","stepAnswer":["A"],"problemType":"MultipleChoice","stepTitle":"The graphs are in the order A, B, and C.","stepBody":"##figure1.gif## ##figure2.gif## ##figure3.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C"],"hints":{"DefaultPathway":[{"id":"acff542gra25a-h1","type":"hint","dependencies":[],"title":"Graph First Inequality","text":"Graph $$y \\\\geq \\\\frac{3}{4} x-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra25a-h2","type":"hint","dependencies":["acff542gra25a-h1"],"title":"Graph the Boundary Line","text":"It is a solid line because the inequality sign is $$ \\\\geq $$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra25a-h3","type":"hint","dependencies":["acff542gra25a-h2"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y \\\\geq \\\\frac{3}{4} x-2$$\\\\n$$0 \\\\geq 0\\\\frac{3}{4}-2$$\\\\n$$0 \\\\geq -2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra25a-h4","type":"hint","dependencies":["acff542gra25a-h3"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is a solution to $$y \\\\geq \\\\frac{3}{4} x-2$$. Since $$(0,0)$$ is on the left side of the line, we shade the left side of the boundary line in red.\\\\n##figure4.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra25a-h5","type":"hint","dependencies":["acff542gra25a-h4"],"title":"Graph Second Inequality","text":"Graph $$y<2$$ on the same grid. Recognize that it is a horizontal line through $$y=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra25a-h6","type":"hint","dependencies":["acff542gra25a-h5"],"title":"Graph the Boundary Line","text":"It is a dashed line because the inequality sign is <.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra25a-h7","type":"hint","dependencies":["acff542gra25a-h6"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y<2$$\\\\n$$0<2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra25a-h8","type":"hint","dependencies":["acff542gra25a-h7"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is a solution to $$y<2$$. Since $$(0,0)$$ is on the bottom side of the line, so we shade in the bottom side of the line $$y=2$$ in green.\\\\n##figure5.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra25a-h9","type":"hint","dependencies":["acff542gra25a-h8"],"title":"Overlap Shading","text":"The solution is all points in the darker shaded region.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acff542gra26","title":"System of Linear Inequalities","body":"Solve the system by graphing. Which graph represents the solution?\\\\n##figure4.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acff542gra26a","stepAnswer":["B"],"problemType":"MultipleChoice","stepTitle":"The graphs are in the order A, B, and C.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C"],"hints":{"DefaultPathway":[{"id":"acff542gra26a-h1","type":"hint","dependencies":[],"title":"Graph First Inequality","text":"Graph $$y \\\\leq \\\\frac{-1}{2} x+3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra26a-h2","type":"hint","dependencies":["acff542gra26a-h1"],"title":"Graph the Boundary Line","text":"It is a solid line because the inequality sign is $$ \\\\leq $$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra26a-h3","type":"hint","dependencies":["acff542gra26a-h2"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y \\\\leq \\\\frac{-1}{2} x+3$$\\\\n$$0 \\\\leq 0\\\\frac{-1}{2}+3$$\\\\n$$0 \\\\leq 3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra26a-h4","type":"hint","dependencies":["acff542gra26a-h3"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is a solution to $$y \\\\leq \\\\frac{-1}{2} x+3$$. Since $$(0,0)$$ is on the left side of the line, we shade the left side of the boundary line in red.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra26a-h5","type":"hint","dependencies":["acff542gra26a-h4"],"title":"Graph Second Inequality","text":"Graph $$y<1$$ on the same grid. Recognize that it is a horizontal line through $$y=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra26a-h6","type":"hint","dependencies":["acff542gra26a-h5"],"title":"Graph the Boundary Line","text":"It is a dashed line because the inequality sign is <.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra26a-h7","type":"hint","dependencies":["acff542gra26a-h6"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y<1$$\\\\n$$0<1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra26a-h8","type":"hint","dependencies":["acff542gra26a-h7"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is a solution to $$y<1$$. Since $$(0,0)$$ is on the bottom side of the line, so we shade in the bottom side of the line $$y=1$$ in green.\\\\n##figure3.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra26a-h9","type":"hint","dependencies":["acff542gra26a-h8"],"title":"Overlap Shading","text":"The solution is all points in the darker shaded region.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acff542gra27","title":"System of Linear Inequalities","body":"Solve the system by graphing. Which graph represents the solution?\\\\n##figure4.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acff542gra27a","stepAnswer":["C"],"problemType":"MultipleChoice","stepTitle":"The graphs are in the order A, B, and C.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C"],"hints":{"DefaultPathway":[{"id":"acff542gra27a-h1","type":"hint","dependencies":[],"title":"Graph First Inequality","text":"Graph $$x \\\\geq 3$$. Recognize that it is a vertical line through $$x=3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra27a-h2","type":"hint","dependencies":["acff542gra27a-h1"],"title":"Graph the Boundary Line","text":"It is a solid line because the inequality sign is $$ \\\\geq $$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra27a-h3","type":"hint","dependencies":["acff542gra27a-h2"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$x \\\\geq 3$$\\\\n$$0 \\\\geq 3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra27a-h4","type":"hint","dependencies":["acff542gra27a-h3"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is not a solution to $$x \\\\geq 3$$. Since $$(0,0)$$ is on the left side of the line, so we shade in the right side of the line $$x=3$$ in green.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra27a-h5","type":"hint","dependencies":["acff542gra27a-h4"],"title":"Graph Second Inequality","text":"Graph $$y \\\\leq 2$$ on the same grid. Recognize that it is a horizontal line through $$y=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra27a-h6","type":"hint","dependencies":["acff542gra27a-h5"],"title":"Graph the Boundary Line","text":"It is a solid line because the inequality sign is $$ \\\\leq $$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra27a-h7","type":"hint","dependencies":["acff542gra27a-h6"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y \\\\leq 2$$\\\\n$$0 \\\\leq 2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra27a-h8","type":"hint","dependencies":["acff542gra27a-h7"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is a solution to $$y \\\\leq 2$$. Since $$(0,0)$$ is on the bottom side of the line, so we shade in the bottom side of the line $$y=2$$ in green.\\\\n##figure3.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra27a-h9","type":"hint","dependencies":["acff542gra27a-h8"],"title":"Overlap Shading","text":"The solution is all points in the darker shaded region.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acff542gra28","title":"System of Linear Inequalities","body":"Solve the system by graphing. Which graph represents the solution?\\\\n##figure4.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acff542gra28a","stepAnswer":["A"],"problemType":"MultipleChoice","stepTitle":"The graphs are in the order A, B, and C.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C"],"hints":{"DefaultPathway":[{"id":"acff542gra28a-h1","type":"hint","dependencies":[],"title":"Graph First Inequality","text":"Graph $$x \\\\leq -1$$. Recognize that it is a vertical line through $$x=-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra28a-h2","type":"hint","dependencies":["acff542gra28a-h1"],"title":"Graph the Boundary Line","text":"It is a solid line because the inequality sign is $$ \\\\leq $$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra28a-h3","type":"hint","dependencies":["acff542gra28a-h2"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$x \\\\leq -1$$\\\\n$$0 \\\\leq -1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra28a-h4","type":"hint","dependencies":["acff542gra28a-h3"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is not a solution to $$x \\\\leq -1$$. Since $$(0,0)$$ is on the right side of the line, so we shade in the left side of the line $$x=-1$$ in green.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra28a-h5","type":"hint","dependencies":["acff542gra28a-h4"],"title":"Graph Second Inequality","text":"Graph $$y \\\\geq 3$$ on the same grid. Recognize that it is a horizontal line through $$y=3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra28a-h6","type":"hint","dependencies":["acff542gra28a-h5"],"title":"Graph the Boundary Line","text":"It is a solid line because the inequality sign is $$ \\\\geq $$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra28a-h7","type":"hint","dependencies":["acff542gra28a-h6"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y \\\\geq 3$$\\\\n$$0 \\\\geq 3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra28a-h8","type":"hint","dependencies":["acff542gra28a-h7"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is not a solution to $$y \\\\geq 3$$. Since $$(0,0)$$ is on the bottom side of the line, so we shade in the top side of the line $$y=3$$ in green.\\\\n##figure3.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra28a-h9","type":"hint","dependencies":["acff542gra28a-h8"],"title":"Overlap Shading","text":"The solution is all points in the darker shaded region.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acff542gra29","title":"System of Linear Inequalities","body":"Solve the system by graphing. Which graph represents the solution?\\\\n##figure4.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acff542gra29a","stepAnswer":["B"],"problemType":"MultipleChoice","stepTitle":"The graphs are in the order A, B, and C.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C"],"hints":{"DefaultPathway":[{"id":"acff542gra29a-h1","type":"hint","dependencies":[],"title":"Graph First Inequality","text":"Graph $$y \\\\geq 3x-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra29a-h2","type":"hint","dependencies":["acff542gra29a-h1"],"title":"Graph the Boundary Line","text":"It is a solid line because the inequality sign is $$ \\\\geq $$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra29a-h3","type":"hint","dependencies":["acff542gra29a-h2"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y \\\\geq 3x-2$$\\\\n$$0 \\\\geq 3\\\\times0-2$$\\\\n$$0 \\\\geq -2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra29a-h4","type":"hint","dependencies":["acff542gra29a-h3"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is a solution to $$y \\\\geq 3x-2$$. Since $$(0,0)$$ is on the left side of the line, we shade the left side of the boundary line in red.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra29a-h5","type":"hint","dependencies":["acff542gra29a-h4"],"title":"Graph Second Inequality","text":"Graph $$y<-1$$ on the same grid. Recognize that it is a horizontal line through $$y=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra29a-h6","type":"hint","dependencies":["acff542gra29a-h5"],"title":"Graph the Boundary Line","text":"It is a dashed line because the inequality sign is <.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra29a-h7","type":"hint","dependencies":["acff542gra29a-h6"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y<-1$$\\\\n$$0<-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra29a-h8","type":"hint","dependencies":["acff542gra29a-h7"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is not a solution to $$y<-1$$. Since $$(0,0)$$ is on the top side of the line, so we shade in the bottom side of the line $$y=-1$$ in green.\\\\n##figure3.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra29a-h9","type":"hint","dependencies":["acff542gra29a-h8"],"title":"Overlap Shading","text":"The solution is all points in the darker shaded region.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acff542gra3","title":"System of Linear Inequalities","body":"Determine whether the ordered pair is a solution to the system.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acff542gra3a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$(5,-2)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"acff542gra3a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitute $$x=5$$ and $$y=-2$$ into both inequalities.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra3a-h2","type":"hint","dependencies":["acff542gra3a-h1"],"title":"Substitute into First Equation","text":"$$4x-y<10$$\\\\n$$4\\\\times5-\\\\left(-2\\\\right)<10$$\\\\n$$22<10$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra3a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["F"],"dependencies":["acff542gra3a-h2"],"title":"Substitute into First Equation","text":"Is the inequality above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"acff542gra3a-h4","type":"hint","dependencies":["acff542gra3a-h3"],"title":"Solutions of a System of Equations","text":"Since $$(5,2)$$ does not satisfy the first inequality, then it cannot make both inequalities true. $$(5,2)$$ is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"acff542gra3b","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$(-1,3)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"acff542gra3b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitute $$x=-1$$ and $$y=3$$ into both inequalities.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra3b-h2","type":"hint","dependencies":["acff542gra3b-h1"],"title":"Substitute into First Equation","text":"$$4x-y<10$$\\\\n$$4\\\\left(-1\\\\right)-3<10$$\\\\n$$-7<10$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra3b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["T"],"dependencies":["acff542gra3b-h2"],"title":"Substitute into First Equation","text":"Is the inequality above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"acff542gra3b-h4","type":"hint","dependencies":["acff542gra3b-h3"],"title":"Substitute into Second Equation","text":"$$-2x+2y>-8$$\\\\n$$-2\\\\left(-1\\\\right)+2\\\\times3>-8$$\\\\n$$8>-8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra3b-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["T"],"dependencies":["acff542gra3b-h4"],"title":"Substitute into Second Equation","text":"Is the inequality above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"acff542gra3b-h6","type":"hint","dependencies":["acff542gra3b-h5"],"title":"Solutions of a System of Equations","text":"$$(-1,3)$$ makes both inequalities true. $$(-1,3)$$ is a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acff542gra30","title":"System of Linear Inequalities","body":"Solve the system by graphing. The first image is the problem and the next three are the three possible graphs of it.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acff542gra30a","stepAnswer":["C"],"problemType":"MultipleChoice","stepTitle":"Which graph represents the solution?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C"],"hints":{"DefaultPathway":[{"id":"acff542gra30a-h1","type":"hint","dependencies":[],"title":"Graph First Inequality","text":"Graph $$y<3x+2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra30a-h2","type":"hint","dependencies":["acff542gra30a-h1"],"title":"Graph the Boundary Line","text":"It is a dashed line because the inequality sign is <.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra30a-h3","type":"hint","dependencies":["acff542gra30a-h2"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y<3x+2$$\\\\n$$0<3\\\\times0+2$$\\\\n$$0<2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra30a-h4","type":"hint","dependencies":["acff542gra30a-h3"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is a solution to $$y<3x+2$$. Since $$(0,0)$$ is on the right side of the line, so we shade the right side of the boundary line in red.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra30a-h5","type":"hint","dependencies":["acff542gra30a-h4"],"title":"Graph Second Inequality","text":"Graph $$y>-x-1$$ on the same grid.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra30a-h6","type":"hint","dependencies":["acff542gra30a-h5"],"title":"Graph the Boundary Line","text":"It is a dashed line because the inequality sign is >.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra30a-h7","type":"hint","dependencies":["acff542gra30a-h6"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y>-x-1$$\\\\n$$0>-0-1$$\\\\n$$0>-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra30a-h8","type":"hint","dependencies":["acff542gra30a-h7"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is a solution to $$y>-x-1$$. Since $$(0,0)$$ is on the right side of the line, so we shade in the right side of the line $$y=-x-1$$ in green.\\\\n##figure3.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra30a-h9","type":"hint","dependencies":["acff542gra30a-h8"],"title":"Overlap Shading","text":"The solution is all points in the darker shaded region.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acff542gra4","title":"System of Linear Inequalities","body":"Determine whether the ordered pair is a solution to the system.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acff542gra4a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$(6,-4)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"acff542gra4a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitute $$x=6$$ and $$y=-4$$ into both inequalities.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra4a-h2","type":"hint","dependencies":["acff542gra4a-h1"],"title":"Substitute into First Equation","text":"$$y>\\\\frac{2}{3} x-5$$\\\\n$$-4>6\\\\frac{2}{3}-5$$\\\\n$$-4>-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra4a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["F"],"dependencies":["acff542gra4a-h2"],"title":"Substitute into First Equation","text":"Is the inequality above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"acff542gra4a-h4","type":"hint","dependencies":["acff542gra4a-h3"],"title":"Solutions of a System of Equations","text":"Since $$(6,-4)$$ does not satisfy the first inequality, then it cannot make both inequalities true. $$(6,-4)$$ is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"acff542gra4b","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$(3,0)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"acff542gra4b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitute $$x=3$$ and $$y=0$$ into both inequalities.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra4b-h2","type":"hint","dependencies":["acff542gra4b-h1"],"title":"Substitute into First Equation","text":"$$y>\\\\frac{2}{3} x-5$$\\\\n$$0>3\\\\frac{2}{3}-5$$\\\\n$$0>-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra4b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["T"],"dependencies":["acff542gra4b-h2"],"title":"Substitute into First Equation","text":"Is the inequality above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"acff542gra4b-h4","type":"hint","dependencies":["acff542gra4b-h3"],"title":"Substitute into Second Equation","text":"$$x+\\\\frac{1}{2} y \\\\leq 4$$\\\\n$$3+0\\\\frac{1}{2} \\\\leq 4$$\\\\n$$3 \\\\leq 4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra4b-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["T"],"dependencies":["acff542gra4b-h4"],"title":"Substitute into Second Equation","text":"Is the inequality above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"acff542gra4b-h6","type":"hint","dependencies":["acff542gra4b-h5"],"title":"Solutions of a System of Equations","text":"$$(3,0)$$ makes both inequalities true. $$(3,0)$$ is a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acff542gra5","title":"System of Linear Inequalities","body":"Determine whether the ordered pair is a solution to the system.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acff542gra5a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$(-4,-1)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"acff542gra5a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitute $$x=-4$$ and $$y=-1$$ into both inequalities.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra5a-h2","type":"hint","dependencies":["acff542gra5a-h1"],"title":"Substitute into First Equation","text":"$$y>\\\\frac{3}{2} x+3$$\\\\n$$-1>-4\\\\frac{3}{2}+3$$\\\\n$$-1>-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra5a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["F"],"dependencies":["acff542gra5a-h2"],"title":"Substitute into First Equation","text":"Is the inequality above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"acff542gra5a-h4","type":"hint","dependencies":["acff542gra5a-h3"],"title":"Solutions of a System of Equations","text":"Since $$(-4,-1)$$ does not satisfy the first inequality, then it cannot make both inequalities true. $$(-4,-1)$$ is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"acff542gra5b","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$(8,3)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"acff542gra5b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitute $$x=8$$ and $$y=3$$ into both inequalities.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra5b-h2","type":"hint","dependencies":["acff542gra5b-h1"],"title":"Substitute into First Equation","text":"$$y>\\\\frac{3}{2} x+3$$\\\\n$$3>8\\\\frac{3}{2}+3$$\\\\n$$3>15$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra5b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["T"],"dependencies":["acff542gra5b-h2"],"title":"Substitute into First Equation","text":"Is the inequality above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"acff542gra5b-h4","type":"hint","dependencies":["acff542gra5b-h3"],"title":"Substitute into Second Equation","text":"$$\\\\frac{3}{4} x-2y<5$$\\\\n$$8\\\\frac{3}{4}-2\\\\times3<5$$\\\\n$$0<5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra5b-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["T"],"dependencies":["acff542gra5b-h4"],"title":"Substitute into Second Equation","text":"Is the inequality above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"acff542gra5b-h6","type":"hint","dependencies":["acff542gra5b-h5"],"title":"Solutions of a System of Equations","text":"$$(8,3)$$ makes both inequalities true. $$(8,3)$$ is a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acff542gra6","title":"System of Linear Inequalities","body":"Determine whether the ordered pair is a solution to the system.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acff542gra6a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$(2,3)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"acff542gra6a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitute $$x=2$$ and $$y=3$$ into both inequalities.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra6a-h2","type":"hint","dependencies":["acff542gra6a-h1"],"title":"Substitute into First Equation","text":"$$7x+2y>14$$\\\\n$$7\\\\times2+2\\\\times3>14$$\\\\n$$20>14$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra6a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["T"],"dependencies":["acff542gra6a-h2"],"title":"Substitute into First Equation","text":"Is the inequality above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"acff542gra6a-h4","type":"hint","dependencies":["acff542gra6a-h3"],"title":"Substitute into Second Equation","text":"$$5x-y \\\\leq 8$$\\\\n$$5\\\\times2-3 \\\\leq 8$$\\\\n$$7 \\\\leq 8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra6a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["T"],"dependencies":["acff542gra6a-h4"],"title":"Substitute into Second Equation","text":"Is the inequality above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"acff542gra6a-h6","type":"hint","dependencies":["acff542gra6a-h5"],"title":"Solutions of a System of Equations","text":"$$(2,3)$$ makes both inequalities true. $$(2,3)$$ is a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"acff542gra6b","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$(7,-1)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"acff542gra6b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitute $$x=7$$ and $$y=-1$$ into both inequalities.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra6b-h2","type":"hint","dependencies":["acff542gra6b-h1"],"title":"Substitute into First Equation","text":"$$7x+2y>14$$\\\\n$$7\\\\times7+2\\\\left(-1\\\\right)>14$$\\\\n$$47>14$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra6b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["T"],"dependencies":["acff542gra6b-h2"],"title":"Substitute into First Equation","text":"Is the inequality above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"acff542gra6b-h4","type":"hint","dependencies":["acff542gra6b-h3"],"title":"Substitute into Second Equation","text":"$$5x-y \\\\leq 8$$\\\\n$$5\\\\times7-\\\\left(-1\\\\right) \\\\leq 8$$\\\\n$$36 \\\\leq 8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra6b-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["F"],"dependencies":["acff542gra6b-h4"],"title":"Substitute into Second Equation","text":"Is the inequality above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"acff542gra6b-h6","type":"hint","dependencies":["acff542gra6b-h5"],"title":"Solutions of a System of Equations","text":"$$(7,-1)$$ does not make both inequalities true. $$(7,-1)$$ is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acff542gra7","title":"System of Linear Inequalities","body":"Determine whether the ordered pair is a solution to the system.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acff542gra7a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$(1,-3)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"acff542gra7a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitute $$x=1$$ and $$y=-3$$ into both inequalities.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra7a-h2","type":"hint","dependencies":["acff542gra7a-h1"],"title":"Substitute into First Equation","text":"$$6x-5y<20$$\\\\n$$6\\\\times1-5\\\\left(-3\\\\right)<20$$\\\\n$$21<20$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra7a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["F"],"dependencies":["acff542gra7a-h2"],"title":"Substitute into First Equation","text":"Is the inequality above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"acff542gra7a-h4","type":"hint","dependencies":["acff542gra7a-h3"],"title":"Solutions of a System of Equations","text":"Since $$(1,-3)$$ does not satisfy the first inequality, then it cannot make both inequalities true. $$(1,-3)$$ is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"acff542gra7b","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$(-4,4)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"acff542gra7b-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitute $$x=-4$$ and $$y=4$$ into both inequalities.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra7b-h2","type":"hint","dependencies":["acff542gra7b-h1"],"title":"Substitute into First Equation","text":"$$6x-5y<20$$\\\\n$$6\\\\left(-4\\\\right)-5\\\\times4<20$$\\\\n$$-44<20$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra7b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["T"],"dependencies":["acff542gra7b-h2"],"title":"Substitute into First Equation","text":"Is the inequality above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"acff542gra7b-h4","type":"hint","dependencies":["acff542gra7b-h3"],"title":"Substitute into Second Equation","text":"$$-2x+7y>-8$$\\\\n$$-2\\\\left(-4\\\\right)+7\\\\times4>-8$$\\\\n$$36>-8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra7b-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["T"],"dependencies":["acff542gra7b-h4"],"title":"Substitute into Second Equation","text":"Is the inequality above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"acff542gra7b-h6","type":"hint","dependencies":["acff542gra7b-h5"],"title":"Solutions of a System of Equations","text":"$$(-4,4)$$ makes both inequalities true. $$(-4,4)$$ is a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acff542gra8","title":"System of Linear Inequalities","body":"Determine whether the ordered pair is a solution to the system.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acff542gra8a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$(\\\\frac{1}{4},\\\\frac{7}{6})$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"acff542gra8a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitute $$x=\\\\frac{1}{4}$$ and $$y=\\\\frac{7}{6}$$ into both inequalities.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra8a-h2","type":"hint","dependencies":["acff542gra8a-h1"],"title":"Substitute into First Equation","text":"$$2x+3y \\\\geq 2$$\\\\n$$2\\\\frac{1}{4}+3\\\\frac{7}{6} \\\\geq 2$$\\\\n$$4 \\\\geq 2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra8a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["T"],"dependencies":["acff542gra8a-h2"],"title":"Substitute into First Equation","text":"Is the inequality above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"acff542gra8a-h4","type":"hint","dependencies":["acff542gra8a-h3"],"title":"Substitute into Second Equation","text":"$$4x-6y<-1$$\\\\n$$4\\\\frac{1}{4}-6\\\\frac{7}{6}<-1$$\\\\n$$-7<-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra8a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["T"],"dependencies":["acff542gra8a-h4"],"title":"Substitute into Second Equation","text":"Is the inequality above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"acff542gra8a-h6","type":"hint","dependencies":["acff542gra8a-h5"],"title":"Solutions of a System of Equations","text":"$$(\\\\frac{1}{4},\\\\frac{7}{6})$$ does make both inequalities true. $$(\\\\frac{1}{4},\\\\frac{7}{6})$$ is a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acff542gra9","title":"System of Linear Inequalities","body":"Determine whether the ordered pair is a solution to the system.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acff542gra9a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$(\\\\frac{-3}{10},\\\\frac{7}{6})$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"acff542gra9a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"We substitute $$x=\\\\frac{-3}{10}$$ and $$y=\\\\frac{7}{6}$$ into both inequalities.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra9a-h2","type":"hint","dependencies":["acff542gra9a-h1"],"title":"Substitute into First Equation","text":"$$5x-3y<-2$$\\\\n$$5\\\\left(-\\\\frac{3}{10}\\\\right)-3\\\\frac{7}{6}<-2$$\\\\n$$-5<-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra9a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["T"],"dependencies":["acff542gra9a-h2"],"title":"Substitute into First Equation","text":"Is the inequality above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"acff542gra9a-h4","type":"hint","dependencies":["acff542gra9a-h3"],"title":"Substitute into Second Equation","text":"$$10x+6y>4$$\\\\n$$10\\\\left(-\\\\frac{3}{10}\\\\right)+6\\\\frac{7}{6}>4$$\\\\n$$4>4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542gra9a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["F"],"dependencies":["acff542gra9a-h4"],"title":"Substitute into Second Equation","text":"Is the inequality above true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["T","F"]},{"id":"acff542gra9a-h6","type":"hint","dependencies":["acff542gra9a-h5"],"title":"Solutions of a System of Equations","text":"$$(\\\\frac{-3}{10},\\\\frac{7}{6})$$ does not makes both inequalities true. $$(\\\\frac{-3}{10},\\\\frac{7}{6})$$ is not a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acff542grap17","title":"System of Linear Inequalities","body":"Solve the system by graphing. Which graph represents the solution?\\\\n##figure6.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acff542grap17a","stepAnswer":["B"],"problemType":"MultipleChoice","stepTitle":"The graphs are in the order A, B, and C.","stepBody":"##figure1.gif## ##figure2.gif## ##figure3.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C"],"hints":{"DefaultPathway":[{"id":"acff542grap17a-h1","type":"hint","dependencies":[],"title":"Graph First Inequality","text":"Graph $$x+2y<4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap17a-h2","type":"hint","dependencies":["acff542grap17a-h1"],"title":"Change to Slope-Intercept Form","text":"$$x+2y<4$$\\\\n$$2y<-x+4$$\\\\n$$\\\\frac{2y}{2}<\\\\frac{\\\\left(-x+4\\\\right)}{2}$$\\\\n$$y<\\\\frac{-1}{2} x+2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap17a-h3","type":"hint","dependencies":["acff542grap17a-h2"],"title":"Graph the Boundary Line","text":"It is a dashed line because the inequality sign is <.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap17a-h4","type":"hint","dependencies":["acff542grap17a-h3"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y<\\\\frac{-1}{2} x+2$$\\\\n$$0<0\\\\frac{-1}{2}+2$$\\\\n$$0<2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap17a-h5","type":"hint","dependencies":["acff542grap17a-h4"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is a solution to $$y<\\\\frac{-1}{2} x+2$$. Since $$(0,0)$$ is on the left side of the line, we shade the left side of the boundary line in red.\\\\n##figure4.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap17a-h6","type":"hint","dependencies":["acff542grap17a-h5"],"title":"Graph Second Inequality","text":"Graph $$y<x-2$$ on the same grid.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap17a-h7","type":"hint","dependencies":["acff542grap17a-h6"],"title":"Graph the Boundary Line","text":"It is a dashed line because the inequality sign is <.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap17a-h8","type":"hint","dependencies":["acff542grap17a-h7"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y<x-2$$\\\\n$$0<0-2$$\\\\n$$0<-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap17a-h9","type":"hint","dependencies":["acff542grap17a-h8"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is not a solution to $$y<x-2$$. Since $$(0,0)$$ is on the left side of the line, so we shade in the right side of the line $$y=x-2$$ in green.\\\\n##figure5.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap17a-h10","type":"hint","dependencies":["acff542grap17a-h9"],"title":"Overlap Shading","text":"The solution is all points in the darker shaded region.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acff542grap18","title":"System of Linear Inequalities","body":"Solve the system by graphing. Which graph represents the solution?\\\\n##figure6.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acff542grap18a","stepAnswer":["C"],"problemType":"MultipleChoice","stepTitle":"The graphs are in the order A, B, and C.","stepBody":"##figure1.gif## ##figure2.gif## ##figure3.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C"],"hints":{"DefaultPathway":[{"id":"acff542grap18a-h1","type":"hint","dependencies":[],"title":"Graph First Inequality","text":"Graph $$x-2y<5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap18a-h2","type":"hint","dependencies":["acff542grap18a-h1"],"title":"Change to Slope-Intercept Form","text":"$$x-2y<5$$\\\\n$$-2y<-x+5$$\\\\n$$\\\\frac{-2y}{-2}<\\\\frac{\\\\left(-x+5\\\\right)}{-2}$$\\\\n$$y>\\\\frac{1}{2} x-\\\\frac{5}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap18a-h3","type":"hint","dependencies":["acff542grap18a-h2"],"title":"Graph the Boundary Line","text":"It is a dashed line because the inequality sign is >.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap18a-h4","type":"hint","dependencies":["acff542grap18a-h3"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y>\\\\frac{1}{2} x-\\\\frac{5}{2}$$\\\\n$$0>0\\\\frac{1}{2}-\\\\frac{5}{2}$$\\\\n$$0>\\\\frac{-5}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap18a-h5","type":"hint","dependencies":["acff542grap18a-h4"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is a solution to $$y<\\\\frac{1}{2} x-\\\\frac{5}{2}$$. Since $$(0,0)$$ is on the left side of the line, we shade the left side of the boundary line in red.\\\\n##figure4.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap18a-h6","type":"hint","dependencies":["acff542grap18a-h5"],"title":"Graph Second Inequality","text":"Graph $$y>-4$$ on the same grid. Recognize that it is a horizontal line through $$y=-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap18a-h7","type":"hint","dependencies":["acff542grap18a-h6"],"title":"Graph the Boundary Line","text":"It is a dashed line because the inequality sign is >.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap18a-h8","type":"hint","dependencies":["acff542grap18a-h7"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y>-4$$\\\\n$$0>-4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap18a-h9","type":"hint","dependencies":["acff542grap18a-h8"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is a solution to $$y>-4$$. Since $$(0,0)$$ is on the top side of the line, so we shade in the top side of the line $$y=-4$$ in green.\\\\n##figure5.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap18a-h10","type":"hint","dependencies":["acff542grap18a-h9"],"title":"Overlap Shading","text":"The solution is all points in the darker shaded region.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acff542grap19","title":"System of Linear Inequalities","body":"Solve the system by graphing. Which graph represents the solution?\\\\n##figure6.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acff542grap19a","stepAnswer":["A"],"problemType":"MultipleChoice","stepTitle":"The graphs are in the order A, B, and C.","stepBody":"##figure1.gif## ##figure2.gif## ##figure3.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C"],"hints":{"DefaultPathway":[{"id":"acff542grap19a-h1","type":"hint","dependencies":[],"title":"Graph First Inequality","text":"Graph $$3x-y \\\\leq 6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap19a-h2","type":"hint","dependencies":["acff542grap19a-h1"],"title":"Change to Slope-Intercept Form","text":"$$3x-y \\\\leq 6$$\\\\n$$-y \\\\leq -3x+6$$\\\\n$$y \\\\geq 3x-6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap19a-h3","type":"hint","dependencies":["acff542grap19a-h2"],"title":"Graph the Boundary Line","text":"It is a solid line because the inequality sign is $$ \\\\geq $$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap19a-h4","type":"hint","dependencies":["acff542grap19a-h3"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y \\\\geq 3x-6$$\\\\n$$0 \\\\geq 3\\\\times0-6$$\\\\n$$0 \\\\geq -6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap19a-h5","type":"hint","dependencies":["acff542grap19a-h4"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is a solution to $$y \\\\geq 3x-6$$. Since $$(0,0)$$ is on the left side of the line, we shade the left side of the boundary line in red.\\\\n##figure4.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap19a-h6","type":"hint","dependencies":["acff542grap19a-h5"],"title":"Graph Second Inequality","text":"Graph $$y \\\\geq \\\\frac{-1}{2} x$$ on the same grid.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap19a-h7","type":"hint","dependencies":["acff542grap19a-h6"],"title":"Graph the Boundary Line","text":"It is a solid line because the inequality sign is $$ \\\\geq $$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap19a-h8","type":"hint","dependencies":["acff542grap19a-h7"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(1,0)$$ as a test point to see if it is a solution.\\\\n$$y \\\\geq \\\\frac{-1}{2} x$$\\\\n$$0 \\\\geq 1\\\\frac{-1}{2}$$\\\\n$$0 \\\\geq \\\\frac{-1}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap19a-h9","type":"hint","dependencies":["acff542grap19a-h8"],"title":"Shade in the Side of the Boundary Line","text":"$$(1,0)$$ is a solution to $$y \\\\geq \\\\frac{-1}{2} x$$. Since $$(1,0)$$ is on the right side of the line, so we shade in the rigjt side of the line $$y=\\\\frac{-1}{2} x$$ in green.\\\\n##figure5.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap19a-h10","type":"hint","dependencies":["acff542grap19a-h9"],"title":"Overlap Shading","text":"The solution is all points in the darker shaded region.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acff542grap20","title":"System of Linear Inequalities","body":"Solve the system by graphing. Which graph represents the solution?\\\\n##figure6.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acff542grap20a","stepAnswer":["B"],"problemType":"MultipleChoice","stepTitle":"The graphs are in the order A, B, and C.","stepBody":"##figure1.gif## ##figure2.gif## ##figure3.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C"],"hints":{"DefaultPathway":[{"id":"acff542grap20a-h1","type":"hint","dependencies":[],"title":"Graph First Inequality","text":"Graph $$2x+4y \\\\geq 8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap20a-h2","type":"hint","dependencies":["acff542grap20a-h1"],"title":"Change to Slope-Intercept Form","text":"$$2x+4y \\\\geq 8$$\\\\n$$4y \\\\geq -2x+8$$\\\\n$$\\\\frac{4y}{4} \\\\geq \\\\frac{\\\\left(-2x+8\\\\right)}{4}$$\\\\n$$y \\\\geq \\\\frac{-1}{2} x+2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap20a-h3","type":"hint","dependencies":["acff542grap20a-h2"],"title":"Graph the Boundary Line","text":"It is a solid line because the inequality sign is $$ \\\\geq $$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap20a-h4","type":"hint","dependencies":["acff542grap20a-h3"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y \\\\geq \\\\frac{-1}{2} x+2$$\\\\n$$0 \\\\geq 0\\\\frac{-1}{2}+2$$\\\\n$$0 \\\\geq 2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap20a-h5","type":"hint","dependencies":["acff542grap20a-h4"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is not a solution to $$y \\\\geq \\\\frac{-1}{2} x+2$$. Since $$(0,0)$$ is on the left side of the line, we shade the right side of the boundary line in red.\\\\n##figure4.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap20a-h6","type":"hint","dependencies":["acff542grap20a-h5"],"title":"Graph Second Inequality","text":"Graph $$y \\\\leq \\\\frac{3}{4} x$$ on the same grid.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap20a-h7","type":"hint","dependencies":["acff542grap20a-h6"],"title":"Graph the Boundary Line","text":"It is a solid line because the inequality sign is $$ \\\\leq $$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap20a-h8","type":"hint","dependencies":["acff542grap20a-h7"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(1,0)$$ as a test point to see if it is a solution.\\\\n$$y \\\\leq \\\\frac{3}{4} x$$\\\\n$$0 \\\\leq 1\\\\frac{3}{4}$$\\\\n$$0 \\\\leq \\\\frac{3}{4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap20a-h9","type":"hint","dependencies":["acff542grap20a-h8"],"title":"Shade in the Side of the Boundary Line","text":"$$(1,0)$$ is a solution to $$y \\\\leq \\\\frac{3}{4} x$$. Since $$(1,0)$$ is on the right side of the line, so we shade in the rigjt side of the line $$y=\\\\frac{3}{4} x$$ in green.\\\\n##figure5.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap20a-h10","type":"hint","dependencies":["acff542grap20a-h9"],"title":"Overlap Shading","text":"The solution is all points in the darker shaded region.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"acff542grap21","title":"System of Linear Inequalities","body":"Solve the system by graphing. Which graph represents the solution?\\\\n##figure6.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.6 Graphing Systems of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"acff542grap21a","stepAnswer":["C"],"problemType":"MultipleChoice","stepTitle":"The graphs are in the order A, B, and C.","stepBody":"##figure1.gif## ##figure2.gif## ##figure3.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C"],"hints":{"DefaultPathway":[{"id":"acff542grap21a-h1","type":"hint","dependencies":[],"title":"Graph First Inequality","text":"Graph $$x-2y<3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap21a-h2","type":"hint","dependencies":["acff542grap21a-h1"],"title":"Change to Slope-Intercept Form","text":"$$x-2y<3$$\\\\n$$-2y<-x+3$$\\\\n$$\\\\frac{-2y}{-2}<\\\\frac{\\\\left(-x+3\\\\right)}{-2}$$\\\\n$$y>\\\\frac{1}{2} x-\\\\frac{3}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap21a-h3","type":"hint","dependencies":["acff542grap21a-h2"],"title":"Graph the Boundary Line","text":"It is a dashed line because the inequality sign is >.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap21a-h4","type":"hint","dependencies":["acff542grap21a-h3"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y>\\\\frac{1}{2} x-\\\\frac{3}{2}$$\\\\n$$0>0\\\\frac{1}{2}-\\\\frac{3}{2}$$\\\\n$$0>\\\\frac{-3}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap21a-h5","type":"hint","dependencies":["acff542grap21a-h4"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is a solution to $$y<\\\\frac{1}{2} x-\\\\frac{3}{2}$$. Since $$(0,0)$$ is on the left side of the line, we shade the left side of the boundary line in red.\\\\n##figure4.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap21a-h6","type":"hint","dependencies":["acff542grap21a-h5"],"title":"Graph Second Inequality","text":"Graph $$y \\\\leq 1$$ on the same grid. Recognize that it is a horizontal line through $$y=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap21a-h7","type":"hint","dependencies":["acff542grap21a-h6"],"title":"Graph the Boundary Line","text":"It is a solid line because the inequality sign is $$ \\\\leq $$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap21a-h8","type":"hint","dependencies":["acff542grap21a-h7"],"title":"Shade in the Side of the Boundary Line","text":"A test point should be a point not on the line. Choose $$(0,0)$$ as a test point to see if it is a solution.\\\\n$$y \\\\leq 1$$\\\\n$$0 \\\\leq 1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap21a-h9","type":"hint","dependencies":["acff542grap21a-h8"],"title":"Shade in the Side of the Boundary Line","text":"$$(0,0)$$ is a solution to $$y \\\\leq 1$$. Since $$(0,0)$$ is on the bottom side of the line, so we shade in the bottom side of the line $$y=1$$ in green.\\\\n##figure5.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"acff542grap21a-h10","type":"hint","dependencies":["acff542grap21a-h9"],"title":"Overlap Shading","text":"The solution is all points in the darker shaded region.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad0aaf6linear1","title":"Linear Approximations and Differentials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.2 Linear Approximations and Differentials","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad0aaf6linear1a","stepAnswer":["$$3.0167$$"],"problemType":"TextBox","stepTitle":"Find the linear approximation of $$f(x)=\\\\sqrt{x}$$ at $$x=9$$ and use the approximation to estimate $$\\\\sqrt{9.1}$$. Round your answer to four decimal places.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3.0167$$","hints":{"DefaultPathway":[{"id":"ad0aaf6linear1a-h1","type":"hint","dependencies":[],"title":"Linear Approximation Function","text":"Use the following equation L(x)=f(a)+f\'(a)*(x - a) to make a linear approximation, where f(a) is the function and f\'(a) is the derivative of the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear1a-h2","type":"hint","dependencies":["ad0aaf6linear1a-h1"],"title":"Plugging Into the Formula","text":"Since we are looking for the linear approximation at $$x$$ $$=$$ $$9$$, we know that the formula is given by L(x) $$=$$ f(9) + f\'(9)(x - 9).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear1a-h3","type":"hint","dependencies":["ad0aaf6linear1a-h2"],"title":"Solving for f(a) and f\'(a)","text":"Now, we must determine the values of f(a) and f\'(a) to find L(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ad0aaf6linear1a-h3"],"title":"Determining f(a)","text":"Given that $$x=9$$, what is f(a)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ad0aaf6linear1a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":[],"title":"Plugging Into f(9)","text":"$$f(x)=\\\\sqrt{x}$$ such that $$f(9)=\\\\sqrt{9}$$. What is $$\\\\sqrt{9}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"ad0aaf6linear1a-h5","type":"hint","dependencies":["ad0aaf6linear1a-h4"],"title":"Taking the Derivative of f(x) to Get f\'(x)","text":"The next step is finding f\'(x) by taking the derivative of f(x) $$=$$ $$\\\\sqrt{x}$$:\\\\n$$f(x)=\\\\sqrt{x}=x^{\\\\frac{1}{2}}$$\\\\nf\'(x)=(1/2)*x**(1/2 - 2/2)\\\\n$$f\'(x)=\\\\frac{1}{2} x^{\\\\left(-\\\\frac{1}{2}\\\\right)}$$\\\\n$$f\'(x)=\\\\frac{1}{2\\\\sqrt{x}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear1a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{6}$$"],"dependencies":["ad0aaf6linear1a-h5"],"title":"Determining f\'(a)","text":"Given that $$x$$ $$=$$ $$9$$ and f\'(x) $$=$$ $$\\\\frac{1}{2\\\\sqrt{x}}$$, what is f\'(a)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ad0aaf6linear1a-h6-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{6}$$"],"dependencies":[],"title":"Plugging Into f\'(9)","text":"f\'(9) $$=$$ $$\\\\frac{1}{2\\\\sqrt{9}}$$ $$=$$ $$\\\\frac{1}{2\\\\times3}$$. What is f\'(9)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"ad0aaf6linear1a-h7","type":"hint","dependencies":["ad0aaf6linear1a-h6"],"title":"Putting it Together","text":"Now that we have found the values of f(9) and f\'(9), we know that L(x) $$=$$ 3+(1/6)*(x - 9).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear1a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3.0167$$"],"dependencies":["ad0aaf6linear1a-h7"],"title":"Estimating $$\\\\sqrt{9.1}$$","text":"Solve for $$\\\\sqrt{9.1}$$ $$=$$ $$f(9.1)$$ \u2248 $$L(9.1)$$ where L(x)=3+(1/6)*(x - 9). What is the approximation to estimate $$\\\\sqrt{9.1}$$? Round your answer to four decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear1a-h9","type":"hint","dependencies":["ad0aaf6linear1a-h8"],"title":"Final Answer","text":"$$L(9.1)$$ $$=$$ $$3$$ + (1/6)(9.1 - 9) $$=$$ $$3$$ + $$0.1\\\\frac{1}{6}$$ $$=$$ $$3$$ + $$0.0166666$$ $$=$$ $$3.0167$$\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ad0aaf6linear10","title":"Linear Approximations and Differentials","body":"Suppose the side length of a cube is measured to be $$5$$ cm with an accuracy of $$0.1$$ cm.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.2 Linear Approximations and Differentials","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad0aaf6linear10a","stepAnswer":["$$7.651$$"],"problemType":"TextBox","stepTitle":"Given two side lengths, the higher side length is $$5.1$$ cm. Compute the volume of the cube to compare the estimated error with the actual potential error. What is the error in the computed volume?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7.651$$","hints":{"DefaultPathway":[{"id":"ad0aaf6linear10a-h1","type":"hint","dependencies":[],"title":"Volume Formula","text":"The volume of a cube is given by V $$=$$ $$x^3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$132.651$$"],"dependencies":["ad0aaf6linear10a-h1"],"title":"Calculating $$V(5.1)$$","text":"If the side length is actually $$5.1$$ cm, then the volume of the cube is $$V(5.1)$$ $$=$$ $${5.1}^3$$. What is $$V(5.1)$$ in $${cm}^3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear10a-h3","type":"hint","dependencies":["ad0aaf6linear10a-h2"],"title":"Understanding the Problem","text":"From what we calculated, the actual volume of the cube will be between V(the lower side length) and $$132.651$$. Since the side length is measured to be $$5$$ cm, the computed volume is V(5).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$125$$"],"dependencies":["ad0aaf6linear10a-h3"],"title":"Solving for V(5)","text":"What is V(5)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ad0aaf6linear10a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$125$$"],"dependencies":[],"title":"Plugging into V(5)","text":"V(5) $$=$$ $$5^3$$. What is V(5)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"ad0aaf6linear10a-h5","type":"hint","dependencies":["ad0aaf6linear10a-h4"],"title":"Calculating the Difference","text":"To find the error in the computed volume, we must find the difference between the actual volume, $$V(5.1)$$, and the calculated volume, V(5).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear10a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7.651$$"],"dependencies":["ad0aaf6linear10a-h5"],"title":"Error in the Computed Volume","text":"What is the error in the computed volume?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ad0aaf6linear10a-h6-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7.651$$"],"dependencies":[],"title":"Plugging in the Difference","text":"What is $$V(5.1)$$ - V(5)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear10a-h6-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7.651$$"],"dependencies":[],"title":"Final Difference","text":"What is $$132.651$$ - 125?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"ad0aaf6linear10a-h7","type":"hint","dependencies":["ad0aaf6linear10a-h6"],"title":"Interpretation","text":"From our answer, we know that the error in the computed volume is the following: the difference of the lower side length $$ \\\\leq $$ \u0394V $$ \\\\leq $$ $$7.651$$. The estimated error dV is relatively close to the actual potential error in the computed volume.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ad0aaf6linear11","title":"Linear Approximations and Differentials","body":"An astronaut using a camera measures the radius of Earth as $$4000$$ mi with an error of \xb180 mi.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.2 Linear Approximations and Differentials","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad0aaf6linear11a","stepAnswer":["$$0.06$$"],"problemType":"TextBox","stepTitle":"Use differentials to estimate the relative error of using this radius measurement to calculate the volume of Earth, assuming the planet is a perfect sphere.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.06$$","hints":{"DefaultPathway":[{"id":"ad0aaf6linear11a-h1","type":"hint","dependencies":[],"title":"Understanding the Problem","text":"If the measurement of the radius is accurate to within \xb180, we have $$-80$$ $$ \\\\leq $$ dr $$ \\\\leq $$ $$80$$. Solve for dV.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear11a-h2","type":"hint","dependencies":["ad0aaf6linear11a-h1"],"title":"Volume Formula","text":"The volume of a sphere is given by V $$=$$ $$\\\\frac{4}{3} {pir}^3$$. Take the derivative of V to get the dV equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear11a-h3","type":"hint","dependencies":["ad0aaf6linear11a-h2"],"title":"dV Equation","text":"dV=(3*(4/3)*pi*r**(3 - 1))*dr=(4*pi*r**2)*dr","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear11a-h4","type":"hint","dependencies":["ad0aaf6linear11a-h3"],"title":"Plugging Into dV","text":"Using the measured radius of $$4000$$ mi, we can estimate by plugging $$r$$ $$=$$ $$4000$$ and dr $$=$$ $$80$$ into $$dV=4\\\\pi r^2 dr$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4\\\\pi {4000}^2\\\\times80$$"],"dependencies":["ad0aaf6linear11a-h4"],"title":"Solving for dV","text":"What is dV (\xb1)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear11a-h6","type":"hint","dependencies":["ad0aaf6linear11a-h5"],"title":"dV","text":"$$-4\\\\pi {4000}^2\\\\times80$$ $$ \\\\leq $$ dV $$ \\\\leq $$ $$4\\\\pi {4000}^2\\\\times80$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear11a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.06$$"],"dependencies":["ad0aaf6linear11a-h6"],"title":"Estimating the Relative Error","text":"To estimate the relative error, consider $$\\\\frac{dV}{V}$$. Since we do not know the exact value of the volume V, use the measured radius $$r$$ $$=$$ $$4000$$ mi to estimate V. We obtain V \u2248 $$\\\\frac{4}{3} {\\\\operatorname{\\\\pi}\\\\left(4000\\\\right)}^3$$. Therefore, the relative error satisfies $$\\\\frac{\\\\left(-4\\\\pi {4000}^2\\\\times80\\\\right)}{\\\\frac{4\\\\pi {4000}^3}{3}}$$ $$ \\\\leq $$ $$\\\\frac{dV}{V}$$ $$ \\\\leq $$ $$\\\\frac{4\\\\pi {4000}^2\\\\times80}{\\\\frac{4\\\\pi {4000}^3}{3}}$$. What does $$\\\\frac{4\\\\pi {4000}^2\\\\times80}{\\\\frac{4\\\\pi {4000}^3}{3}}$$ simplify to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear11a-h8","type":"hint","dependencies":["ad0aaf6linear11a-h7"],"title":"Simplification & Answer","text":"$$-0.06$$ $$ \\\\leq $$ $$\\\\frac{dV}{V}$$ $$ \\\\leq $$ $$0.06$$. Thus, the relative error is $$0.06$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ad0aaf6linear12","title":"Linear Approximations and Differentials","body":"An astronaut using a camera measures the radius of Earth as $$4000$$ mi with an error of \xb180 mi.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.2 Linear Approximations and Differentials","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad0aaf6linear12a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"Use differentials to estimate the percentage error of using this radius measurement to calculate the volume of Earth, assuming the planet is a perfect sphere.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"ad0aaf6linear12a-h1","type":"hint","dependencies":[],"title":"Understanding the Problem","text":"If the measurement of the radius is accurate to within \xb180, we have $$-80$$ $$ \\\\leq $$ dr $$ \\\\leq $$ $$80$$. Solve for dV.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear12a-h2","type":"hint","dependencies":["ad0aaf6linear12a-h1"],"title":"Volume Formula","text":"The volume of a sphere is given by $$V=\\\\frac{4}{3} \\\\pi r^3$$. Take the derivative of V to get the dV equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear12a-h3","type":"hint","dependencies":["ad0aaf6linear12a-h2"],"title":"dV Equation","text":"dV=(3*(4/3)*pi*r**(3 - 1))*dr=(4*pi*r**2)*dr","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear12a-h4","type":"hint","dependencies":["ad0aaf6linear12a-h3"],"title":"Plugging Into dV","text":"Using the measured radius of $$4000$$ mi, we can estimate by plugging $$r$$ $$=$$ $$4000$$ and dr $$=$$ $$80$$ into $$dV=4\\\\pi r^2 dr$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear12a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4\\\\pi {4000}^2\\\\times80$$"],"dependencies":["ad0aaf6linear12a-h4"],"title":"Solving for dV","text":"What is dV (\xb1)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear12a-h6","type":"hint","dependencies":["ad0aaf6linear12a-h5"],"title":"dV","text":"$$-4\\\\pi {4000}^2\\\\times80$$ $$ \\\\leq $$ dV $$ \\\\leq $$ $$4\\\\pi {4000}^2\\\\times80$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear12a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.06$$"],"dependencies":["ad0aaf6linear12a-h6"],"title":"Estimating the Relative Error","text":"To estimate the relative error, consider $$\\\\frac{dV}{V}$$. Since we do not know the exact value of the volume V, use the measured radius $$r$$ $$=$$ $$4000$$ mi to estimate V. We obtain V \u2248 $$\\\\frac{4}{3} {\\\\operatorname{\\\\pi}\\\\left(4000\\\\right)}^3$$. Therefore, the relative error satisfies $$\\\\frac{\\\\left(-4\\\\pi {4000}^2\\\\times80\\\\right)}{\\\\frac{4\\\\pi {4000}^3}{3}}$$ $$ \\\\leq $$ $$\\\\frac{dV}{V}$$ $$ \\\\leq $$ $$\\\\frac{4\\\\pi {4000}^2\\\\times80}{\\\\frac{4\\\\pi {4000}^3}{3}}$$. What does $$\\\\frac{4\\\\pi {4000}^2\\\\times80}{\\\\frac{4\\\\pi {4000}^3}{3}}$$ simplify to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear12a-h8","type":"hint","dependencies":["ad0aaf6linear12a-h7"],"title":"Simplification","text":"$$-0.06$$ $$ \\\\leq $$ $$\\\\frac{dV}{V}$$ $$ \\\\leq $$ $$0.06$$. Thus, the relative error is $$0.06$$. Convert $$0.06$$ to a percentage to get the percentage error.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear12a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["ad0aaf6linear12a-h8"],"title":"Converting to a Percentage","text":"Solve for $$0.06\\\\times100$$ to convert the relative error, $$0.06$$, to a percentage error. What answer do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear12a-h10","type":"hint","dependencies":["ad0aaf6linear12a-h9"],"title":"Final Answer","text":"$$0.06\\\\times100$$ $$=$$ $$6$$. Thus, the percentage error is 6%.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ad0aaf6linear13","title":"Linear Approximations and Differentials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.2 Linear Approximations and Differentials","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad0aaf6linear13a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"What value must f\'(a) be such that the linear approximation function is constant?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"ad0aaf6linear13a-h1","type":"hint","dependencies":[],"title":"Understanding the Problem","text":"The linear function is defined by $$L(x)=f{\\\\left(a\\\\right)}+\\\\operatorname{f\'}\\\\left(a\\\\right) \\\\left(x-a\\\\right)$$. This function L is also known as the linearization of f at $$x=a$$. When the linear approximation function is constant, $$\\\\operatorname{f\'}\\\\left(a\\\\right) \\\\left(x-a\\\\right)=0$$ such that $$f\'(a)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ad0aaf6linear14","title":"Linear Approximations and Differentials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.2 Linear Approximations and Differentials","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad0aaf6linear14a","stepAnswer":["$$y$$ $$=$$ f(x) is linear or constant"],"problemType":"MultipleChoice","stepTitle":"When is the linear approximation exact?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y$$ $$=$$ f(x) is linear or constant","choices":["Both $$x$$ and $$y$$ $$=$$ $$0$$","$$y$$ $$=$$ f(x) is linear or constant","$$y$$ $$=$$ $$0$$","$$x$$ $$=$$ $$0$$"],"hints":{"DefaultPathway":[{"id":"ad0aaf6linear14a-h1","type":"hint","dependencies":[],"title":"Understanding the Problem","text":"The linear approximation exact when $$y$$ $$=$$ f(x) is linear or constant.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ad0aaf6linear15","title":"Linear Approximations and Differentials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.2 Linear Approximations and Differentials","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad0aaf6linear15a","stepAnswer":["$$\\\\frac{1}{2}-\\\\frac{1}{4\\\\left(x-2\\\\right)}$$"],"problemType":"TextBox","stepTitle":"Find the linear approximation L(x) to $$y$$ $$=$$ f(x) near $$x$$ $$=$$ a for the function f(x) $$=$$ $$\\\\frac{1}{x}$$, a $$=$$ $$2$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{2}-\\\\frac{1}{4\\\\left(x-2\\\\right)}$$","hints":{"DefaultPathway":[{"id":"ad0aaf6linear15a-h1","type":"hint","dependencies":[],"title":"Linear Approximation Function","text":"Use the following equation $$L(x)=f{\\\\left(a\\\\right)}+\\\\operatorname{f\'}\\\\left(a\\\\right) \\\\left(x-a\\\\right)$$ to make a linear approximation, where f(a) is the function and f\'(a) is the derivative of the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear15a-h2","type":"hint","dependencies":["ad0aaf6linear15a-h1"],"title":"Plugging Into the Formula","text":"Since we are looking for the linear approximation at a $$=$$ $$2$$, we know that the formula is given by $$L(x)=f{\\\\left(2\\\\right)}+\\\\operatorname{f\'}\\\\left(2\\\\right) \\\\left(x-2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear15a-h3","type":"hint","dependencies":["ad0aaf6linear15a-h2"],"title":"Solving for f(a) and f\'(a)","text":"Now, we must determine the values of f(a) and f\'(a) to find L(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["ad0aaf6linear15a-h3"],"title":"Determining f(a)","text":"Given that a $$=$$ $$2$$, what is f(a)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear15a-h5","type":"hint","dependencies":["ad0aaf6linear15a-h4"],"title":"Plugging Into f(2)","text":"f(x) $$=$$ $$\\\\frac{1}{x}$$ such that f(2) $$=$$ $$\\\\frac{1}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear15a-h6","type":"hint","dependencies":["ad0aaf6linear15a-h5"],"title":"Taking the Derivative of f(x) to Get f\'(x)","text":"The next step is finding f\'(x) by taking the derivative of $$f(x)=\\\\frac{1}{x}$$:\\\\n$$f(x)=\\\\frac{1}{x}=x^{-1}$$\\\\n$$f\'(x)=-1x^{\\\\left(-1-1\\\\right)}=-\\\\left(x^{-2}\\\\right)=\\\\frac{-1}{x^2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear15a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{4}$$"],"dependencies":["ad0aaf6linear15a-h6"],"title":"Determining f\'(a)","text":"What is f\'(2)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear15a-h8","type":"hint","dependencies":["ad0aaf6linear15a-h7"],"title":"Solving for f\'(2)","text":"$$f\'(2)=\\\\frac{-1}{2^2}=\\\\frac{-1}{4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear15a-h9","type":"hint","dependencies":["ad0aaf6linear15a-h8"],"title":"Putting it Together","text":"Now that we have found the values of f(2) and f\'(2), plug them into $$L(x)=f{\\\\left(a\\\\right)}+\\\\operatorname{f\'}\\\\left(a\\\\right) \\\\left(x-a\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear15a-h10","type":"hint","dependencies":["ad0aaf6linear15a-h9"],"title":"Final Answer","text":"$$L(x)=\\\\frac{1}{2}+\\\\left(-\\\\frac{1}{4}\\\\right) \\\\left(x-2\\\\right)=\\\\frac{1}{2}-\\\\frac{1}{4} \\\\left(x-2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ad0aaf6linear16","title":"Linear Approximations and Differentials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.2 Linear Approximations and Differentials","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad0aaf6linear16a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"Find the linear approximation L(x) to $$y$$ $$=$$ f(x) near $$x$$ $$=$$ a for the function f(x) $$=$$ sinx, a $$=$$ $$\\\\frac{\\\\pi}{2}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"ad0aaf6linear16a-h1","type":"hint","dependencies":[],"title":"Linear Approximation Function","text":"Use the following equation $$L(x)=f{\\\\left(a\\\\right)}+\\\\operatorname{f\'}\\\\left(a\\\\right) \\\\left(x-a\\\\right)$$ to make a linear approximation, where f(a) is the function and f\'(a) is the derivative of the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear16a-h2","type":"hint","dependencies":["ad0aaf6linear16a-h1"],"title":"Plugging Into the Formula","text":"Since we are looking for the linear approximation at a $$=$$ $$\\\\frac{\\\\pi}{2}$$, we know that the formula is given by $$L(x)=f{\\\\left(\\\\frac{\\\\pi}{2}\\\\right)}+\\\\operatorname{f\'}\\\\left(\\\\frac{\\\\pi}{2}\\\\right) \\\\left(x-\\\\frac{\\\\pi}{2}\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear16a-h3","type":"hint","dependencies":["ad0aaf6linear16a-h2"],"title":"Solving for f(a) and f\'(a)","text":"Now, we must determine the values of f(a) and f\'(a) to find L(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear16a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ad0aaf6linear16a-h3"],"title":"Determining f(a)","text":"Given that a $$=$$ $$\\\\frac{\\\\pi}{2}$$, what is f(a)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear16a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ad0aaf6linear16a-h4"],"title":"Plugging Into $$f{\\\\left(\\\\frac{\\\\pi}{2}\\\\right)}$$","text":"f(x) $$=$$ sinx such that $$f{\\\\left(\\\\frac{\\\\pi}{2}\\\\right)}$$ $$=$$ $$sin\\\\left(\\\\frac{\\\\pi}{2}\\\\right)$$. What is $$sin\\\\left(\\\\frac{\\\\pi}{2}\\\\right)$$? Use the graph to guide your answer, where $$x$$ $$=$$ cos(x) and $$y$$ $$=$$ sin(x).\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear16a-h6","type":"hint","dependencies":["ad0aaf6linear16a-h5"],"title":"Taking the Derivative of f(x) to Get f\'(x)","text":"The next step is finding f\'(x) by taking the derivative of f(x) $$=$$ sinx where f\'(x) $$=$$ cosx.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear16a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["ad0aaf6linear16a-h6"],"title":"Determining f\'(a)","text":"What is $$\\\\operatorname{f\'}\\\\left(\\\\frac{\\\\pi}{2}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear16a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["ad0aaf6linear16a-h7"],"title":"Solving for f\'(2)","text":"$$\\\\operatorname{f\'}\\\\left(\\\\frac{\\\\pi}{2}\\\\right)$$ $$=$$ $$cos\\\\left(\\\\frac{\\\\pi}{2}\\\\right)$$. What is $$cos\\\\left(\\\\frac{\\\\pi}{2}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear16a-h9","type":"hint","dependencies":["ad0aaf6linear16a-h8"],"title":"Putting it Together","text":"Now that we have found the values of $$f{\\\\left(\\\\frac{\\\\pi}{2}\\\\right)}$$ and $$\\\\operatorname{f\'}\\\\left(\\\\frac{\\\\pi}{2}\\\\right)$$, plug them into $$L(x)=f{\\\\left(a\\\\right)}+\\\\operatorname{f\'}\\\\left(a\\\\right) \\\\left(x-a\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear16a-h10","type":"hint","dependencies":["ad0aaf6linear16a-h9"],"title":"Final Answer","text":"$$L(x)=1+0\\\\left(x-\\\\frac{\\\\pi}{2}\\\\right)=1+0=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ad0aaf6linear17","title":"Linear Approximations and Differentials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.2 Linear Approximations and Differentials","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad0aaf6linear17a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"Find the linear approximation L(x) to $$y$$ $$=$$ f(x) near $$x$$ $$=$$ a for the function f(x) $$=$$ $${sin}^2 x$$, a $$=$$ $$0$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"ad0aaf6linear17a-h1","type":"hint","dependencies":[],"title":"Linear Approximation Function","text":"Use the following equation $$L(x)=f{\\\\left(a\\\\right)}+\\\\operatorname{f\'}\\\\left(a\\\\right) \\\\left(x-a\\\\right)$$ to make a linear approximation, where f(a) is the function and f\'(a) is the derivative of the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear17a-h2","type":"hint","dependencies":["ad0aaf6linear17a-h1"],"title":"Plugging Into the Formula","text":"Since we are looking for the linear approximation at a $$=$$ $$0$$, we know that the formula is given by L(x) $$=$$ f(0) + f\'(0)(x - 0).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear17a-h3","type":"hint","dependencies":["ad0aaf6linear17a-h2"],"title":"Solving for f(a) and f\'(a)","text":"Now, we must determine the values of f(a) and f\'(a) to find L(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear17a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["ad0aaf6linear17a-h3"],"title":"Determining f(a)","text":"Given that a $$=$$ $$0$$, what is f(a)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear17a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["ad0aaf6linear17a-h4"],"title":"Plugging Into $$f{\\\\left(\\\\frac{\\\\pi}{2}\\\\right)}$$","text":"f(x) $$=$$ $${sin}^2 x$$ such that f(0) $$=$$ $${sin}^{2\\\\left(0\\\\right)}$$. What is $${sin}^{2\\\\left(0\\\\right)}$$? Use the graph to guide your answer, where $$x$$ $$=$$ cos(x) and $$y$$ $$=$$ sin(x). Remember that $${sin}^2 x$$ $$=$$ $$sin\\\\left(x\\\\right) sin\\\\left(x\\\\right)$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear17a-h6","type":"hint","dependencies":["ad0aaf6linear17a-h5"],"title":"Taking the Derivative of f(x) to Get f\'(x)","text":"The next step is finding f\'(x) by taking the derivative of f(x) $$=$$ $${sin}^2 x$$. By using the chain rule $$F\'(x)=f\'(g(x))(g\'(x))$$, we first take the derivative of the outer function such that $$\\\\frac{d}{dx} u^2$$ $$=$$ 2u. Next, we take the derivative of the inner function where $$\\\\frac{d}{dx} \\\\operatorname{sinx}\\\\left(x\\\\right)$$ $$=$$ cos(x). Lastly, we combine the two steps through multiplication to get $$\\\\frac{d}{dx} {sin}^2 x$$ $$=$$ f\'(x) $$=$$ $$2u cos\\\\left(x\\\\right)$$ $$=$$ 2sin(x)cos(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear17a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["ad0aaf6linear17a-h6"],"title":"Determining f\'(a)","text":"Given that f\'(a) $$=$$ 2sin(x)cos(x), what is f\'(0)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear17a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["ad0aaf6linear17a-h7"],"title":"Solving for f\'(0)","text":"f\'(0) $$=$$ $$2sin\\\\left(0\\\\right) cos\\\\left(0\\\\right)$$ $$=$$ $$2\\\\times0\\\\times1$$. What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear17a-h9","type":"hint","dependencies":["ad0aaf6linear17a-h8"],"title":"Putting it Together","text":"Now that we have found the values of f(0) and f\'(0), plug them into $$L(x)=f{\\\\left(a\\\\right)}+\\\\operatorname{f\'}\\\\left(a\\\\right) \\\\left(x-a\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear17a-h10","type":"hint","dependencies":["ad0aaf6linear17a-h9"],"title":"Final Answer","text":"$$L(x)=0+0\\\\left(x-0\\\\right)=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ad0aaf6linear18","title":"Linear Approximations and Differentials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.2 Linear Approximations and Differentials","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad0aaf6linear18a","stepAnswer":["(cosx - xsinx)dx"],"problemType":"TextBox","stepTitle":"Find the differential of the function $$y$$ $$=$$ xcosx.","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"ad0aaf6linear18a-h1","type":"hint","dependencies":[],"title":"Obtaining dy","text":"In order to determine dy, we must take the derivative of $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["(cosx - xsinx)dx"],"dependencies":["ad0aaf6linear18a-h1"],"title":"Solving for y\'","text":"Given that $$y$$ $$=$$ xcosx, what is y\'?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear18a-h3","type":"hint","dependencies":["ad0aaf6linear18a-h2"],"title":"Taking the Derivative","text":"y\' $$=$$ 1(cosx) + $$x(-sinx)$$ $$=$$ cosx - xsinx. Thus, dy $$=$$ (cosx - xsinx)dx.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ad0aaf6linear19","title":"Linear Approximations and Differentials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.2 Linear Approximations and Differentials","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad0aaf6linear19a","stepAnswer":["((x**2 - $$2x$$ - 2)/((x - 1)**2))dx"],"problemType":"TextBox","stepTitle":"Find the differential of the function $$y=\\\\frac{x^2+2}{x-1}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"((x**2 - $$2x$$ - 2)/((x - 1)**2))dx","hints":{"DefaultPathway":[{"id":"ad0aaf6linear19a-h1","type":"hint","dependencies":[],"title":"Obtaining dy","text":"First, we must determine dy by taking the derivative of $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["((x**2 - $$2x$$ - 2)/((x - 1)**2))dx"],"dependencies":["ad0aaf6linear19a-h1"],"title":"Solving for y\'","text":"Given that $$y=\\\\frac{x^2+2}{x-1}$$, what is y\'?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ad0aaf6linear19a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["((x**2 - $$2x$$ - 2)/((x - 1)**2))dx"],"dependencies":[],"title":"Taking the Derivative","text":"Use the quotient rule such that (d/dx)[f(x)/g(x)] $$=$$ (g(x)(d/dx)[f(x)] - f(x)(d/dx)[g(x)])/(g(x)**2). What is dy?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear19a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["((x**2 - $$2x$$ - 2)/((x - 1)**2))dx"],"dependencies":[],"title":"Final Answer","text":"dy $$=$$ ((x**2 - $$2x$$ - 2)/((x - 1)**2))dx. This is your answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}]}}]},{"id":"ad0aaf6linear2","title":"Linear Approximations and Differentials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.2 Linear Approximations and Differentials","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad0aaf6linear2a","stepAnswer":["$$0.88348$$"],"problemType":"TextBox","stepTitle":"Find the linear approximation of f(x) $$=$$ sin(x) at $$x$$ $$=$$ $$\\\\frac{\\\\pi}{3}$$ and use it to approximate sin(62\xb0). Round your answer to five decimal places.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.88348$$","hints":{"DefaultPathway":[{"id":"ad0aaf6linear2a-h1","type":"hint","dependencies":[],"title":"Linear Approximation Function","text":"Use the following equation L(x)=f(a)+f\'(a)*(x - a) to make a linear approximation, where f(a) is the function and f\'(a) is the derivative of the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear2a-h2","type":"hint","dependencies":["ad0aaf6linear2a-h1"],"title":"Plugging Into the Formula","text":"$$\\\\frac{\\\\pi}{3}$$ rad is equivalent to 60\xb0. Since we are looking for the linear approximation at $$x=\\\\frac{\\\\pi}{3}$$, the formula is given by $$L(x)=f{\\\\left(\\\\frac{\\\\pi}{3}\\\\right)}+\\\\operatorname{f\'}\\\\left(\\\\frac{\\\\pi}{3}\\\\right) \\\\left(x-\\\\frac{\\\\pi}{3}\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear2a-h3","type":"hint","dependencies":["ad0aaf6linear2a-h2"],"title":"Solving for f(a) and f\'(a)","text":"Now, we must determine the values of f(a) and f\'(a) to find L(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\sqrt{3}}{2}$$"],"dependencies":["ad0aaf6linear2a-h3"],"title":"Determining f(a)","text":"Given that $$x$$ $$=$$ $$\\\\frac{\\\\pi}{3}$$, what is f(a)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ad0aaf6linear2a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\sqrt{3}}{2}$$"],"dependencies":[],"title":"Plugging Into $$f{\\\\left(\\\\frac{\\\\pi}{3}\\\\right)}$$","text":"$$f(x)=sin(x)$$ such that $$f{\\\\left(\\\\frac{\\\\pi}{3}\\\\right)}=sin\\\\left(\\\\frac{\\\\pi}{3}\\\\right)$$. What is $$sin\\\\left(\\\\frac{\\\\pi}{3}\\\\right)$$? Remember that $$x=cos(x)$$ and $$y=sin(x)$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"ad0aaf6linear2a-h5","type":"hint","dependencies":["ad0aaf6linear2a-h4"],"title":"Taking the Derivative of f(x)","text":"The next step is finding f\'(x) by taking the derivative of $$f(x)=sin(x)$$ where $$f\'(x)=cos(x)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["ad0aaf6linear2a-h5"],"title":"Determining f\'(a)","text":"Given that $$x=\\\\frac{\\\\pi}{3}$$ and $$f\'(x)=cos(x)$$, what is f\'(a)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ad0aaf6linear2a-h6-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":[],"title":"Plugging Into $$\\\\operatorname{f\'}\\\\left(\\\\frac{\\\\pi}{3}\\\\right)$$","text":"$$\\\\operatorname{f\'}\\\\left(\\\\frac{\\\\pi}{3}\\\\right)=cos\\\\left(\\\\frac{\\\\pi}{3}\\\\right)$$. What is $$cos\\\\left(\\\\frac{\\\\pi}{3}\\\\right)$$? Remember that $$x=cos(x)$$ and $$y=sin(x)$$.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"ad0aaf6linear2a-h7","type":"hint","dependencies":["ad0aaf6linear2a-h6"],"title":"Putting it Together","text":"Now that we have found the values of $$f{\\\\left(\\\\frac{\\\\pi}{3}\\\\right)}$$ and $$\\\\operatorname{f\'}\\\\left(\\\\frac{\\\\pi}{3}\\\\right)$$, we know that $$L(x)=\\\\frac{\\\\sqrt{3}}{2}+\\\\frac{1}{2} \\\\left(x-\\\\frac{\\\\pi}{3}\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear2a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.88348$$"],"dependencies":["ad0aaf6linear2a-h7"],"title":"Estimating sin(62\xb0)","text":"Solve for $$sin(62\\\\degree)=f{\\\\left(\\\\frac{62\\\\pi}{180}\\\\right)}$$ \u2248 $$L\\\\left(\\\\frac{62\\\\pi}{180}\\\\right)$$ where $$L(x)=\\\\frac{\\\\sqrt{3}}{2}+\\\\frac{1}{2} \\\\left(x-\\\\frac{\\\\pi}{3}\\\\right)$$. What is the approximation to estimate sin(62\xb0)? Round your answer to five decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear2a-h9","type":"hint","dependencies":["ad0aaf6linear2a-h8"],"title":"Final Answer","text":"$$sin(62\\\\degree)=\\\\frac{\\\\sqrt{3}}{2}+\\\\frac{1}{2} \\\\left(\\\\frac{60\\\\pi}{180}-\\\\frac{\\\\pi}{3}\\\\right)=\\\\frac{\\\\sqrt{3}}{2}+\\\\frac{1}{2} \\\\frac{2\\\\pi}{180}=\\\\frac{\\\\sqrt{3}}{2}+\\\\frac{\\\\pi}{180}=0.88348$$\\\\n##figure3.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ad0aaf6linear20","title":"Linear Approximations and Differentials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.2 Linear Approximations and Differentials","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad0aaf6linear20a","stepAnswer":["$$\\\\frac{-1}{16}$$"],"problemType":"TextBox","stepTitle":"Given that $$y$$ $$=$$ $$\\\\frac{1}{x+1}$$, $$x$$ $$=$$ $$1$$, dx $$=$$ $$0.25$$, find the differential and evaluate for the given $$x$$ and dx to get dy.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-1}{16}$$","hints":{"DefaultPathway":[{"id":"ad0aaf6linear20a-h1","type":"hint","dependencies":[],"title":"Obtaining dy","text":"In order to determine dy, we must take the derivative of $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear20a-h2","type":"hint","dependencies":["ad0aaf6linear20a-h1"],"title":"Solving for y\'","text":"$$y$$ $$=$$ $$\\\\frac{1}{x+1}$$ $$=$$ $${\\\\left(x+1\\\\right)}^{-1}$$ such that y\' $$=$$ $$-\\\\left({\\\\left(x+1\\\\right)}^{\\\\left(-1-1\\\\right)}\\\\right)$$ $$=$$ $$\\\\frac{-1}{{\\\\left(x+1\\\\right)}^2}$$. Thus, dy $$=$$ $$\\\\left(-\\\\frac{1}{{\\\\left(x+1\\\\right)}^2}\\\\right) dx$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear20a-h3","type":"hint","dependencies":["ad0aaf6linear20a-h2"],"title":"Plugging Into y\'","text":"Plug $$x$$ $$=$$ $$1$$ and dx $$=$$ $$0.25$$ into dy $$=$$ $$\\\\left(-\\\\frac{1}{{\\\\left(x+1\\\\right)}^2}\\\\right) dx$$ to solve for dy.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear20a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{16}$$"],"dependencies":["ad0aaf6linear20a-h3"],"title":"Determining dy","text":"What is dy?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ad0aaf6linear20a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{16}$$"],"dependencies":[],"title":"Substitution and Simplification to Getting dy","text":"dy $$=$$ $$\\\\left(-\\\\frac{1}{{\\\\left(x+1\\\\right)}^2}\\\\right) dx$$ $$=$$ $$0.25\\\\left(-\\\\frac{1}{{\\\\left(1+1\\\\right)}^2}\\\\right)$$ $$=$$ $$0.25\\\\left(-\\\\frac{1}{4}\\\\right)$$. What is dy?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}]}}]},{"id":"ad0aaf6linear21","title":"Linear Approximations and Differentials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.2 Linear Approximations and Differentials","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad0aaf6linear21a","stepAnswer":["$$-0.1$$"],"problemType":"TextBox","stepTitle":"Given that $$y=\\\\frac{3x^2+2}{\\\\sqrt{x+1}}$$, $$x$$ $$=$$ $$0$$, dx $$=$$ $$0.1$$, find the differential and evaluate for the given $$x$$ and dx to get dy.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-0.1$$","hints":{"DefaultPathway":[{"id":"ad0aaf6linear21a-h1","type":"hint","dependencies":[],"title":"Obtaining dy","text":"In order to determine dy, we must take the derivative of $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear21a-h2","type":"hint","dependencies":["ad0aaf6linear21a-h1"],"title":"Solving for y\'","text":"$$y=\\\\frac{3x^2+2}{\\\\sqrt{x+1}}$$ such that $$y\'=\\\\frac{9x^2+12x-2}{2{\\\\left(x+1\\\\right)}^{\\\\frac{3}{2}}}$$. Thus, $$dy=\\\\frac{9x^2+12x-2}{2{\\\\left(x+1\\\\right)}^{\\\\frac{3}{2}}} dx$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear21a-h3","type":"hint","dependencies":["ad0aaf6linear21a-h2"],"title":"Plugging Into y\'","text":"Plug $$x$$ $$=$$ $$0$$ and dx $$=$$ $$0.1$$ into $$dy=\\\\frac{9x^2+12x-2}{2{\\\\left(x+1\\\\right)}^{\\\\frac{3}{2}}} dx$$ to solve for dy.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear21a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-0.1$$"],"dependencies":["ad0aaf6linear21a-h3"],"title":"Determining dy","text":"What is dy?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ad0aaf6linear21a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-0.1$$"],"dependencies":[],"title":"Substitution and Simplification to Getting dy","text":"dy=((9*x**2+12*x-2)/(2*(x+1)**(3/2)))dx=((9*(0**2)+12*0-2)/(2*(0+1)**(3/2))*(0.1)=(-2/(2*1**(3/2)))*(0.1)=(-2/2)*(0.1)=-1*0.1. What is dy?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}]}}]},{"id":"ad0aaf6linear3","title":"Linear Approximations and Differentials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.2 Linear Approximations and Differentials","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad0aaf6linear3a","stepAnswer":["$$1.03$$"],"problemType":"TextBox","stepTitle":"Find the linear approximation of f(x) $$=$$ (1 + x)**n at $$x$$ $$=$$ $$0$$. Use this approximation to estimate $${1.01}^3$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1.03$$","hints":{"DefaultPathway":[{"id":"ad0aaf6linear3a-h1","type":"hint","dependencies":[],"title":"Linear Approximation Function","text":"Use the following equation L(x)=f(a)+f\'(a)*(x - a) to make a linear approximation, where f(a) is the function and f\'(a) is the derivative of the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear3a-h2","type":"hint","dependencies":["ad0aaf6linear3a-h1"],"title":"Plugging Into the Formula","text":"The linear approximation at $$x$$ $$=$$ $$0$$ is given by L(x) $$=$$ f(0)+f\'(0)*(x - 0).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear3a-h3","type":"hint","dependencies":["ad0aaf6linear3a-h2"],"title":"Solving for f(a) and f\'(a)","text":"Now, we must determine the values of f(a) and f\'(a) to find L(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ad0aaf6linear3a-h3"],"title":"Determining f(a)","text":"Given that $$x$$ $$=$$ $$0$$, what is f(a)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ad0aaf6linear3a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Plugging Into $$f{\\\\left(\\\\frac{\\\\pi}{3}\\\\right)}$$","text":"$$f(x)=(1$$ + x)**n such that $$f(0)=(1$$ + 0)**n. What is $$1^n$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"ad0aaf6linear3a-h5","type":"hint","dependencies":["ad0aaf6linear3a-h4"],"title":"Taking the Derivative of f(x)","text":"The next step is finding f\'(x) by taking the derivative of $$f(x)=(1$$ + x)**n where f\'(x)=n*(1 + x)**(n - 1).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear3a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$n$$"],"dependencies":["ad0aaf6linear3a-h5"],"title":"Determining f\'(a)","text":"Given that $$x=0$$ and f\'(x)=n*(1 + x)**(n - 1), what is f\'(a)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ad0aaf6linear3a-h6-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$n$$"],"dependencies":[],"title":"Plugging Into f\'(0)","text":"$$f\'(0)=n {\\\\left(1+0\\\\right)}^{n-1}$$ $$=$$ $$n 1^{n-1}$$. What is $$n^{n-1}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"ad0aaf6linear3a-h7","type":"hint","dependencies":["ad0aaf6linear3a-h6"],"title":"Putting it Together","text":"Now that we have found the values of f(0) and f\'(0), we know that $$L(x)=1+n \\\\left(x-0\\\\right)=1+n x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear3a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.03$$"],"dependencies":["ad0aaf6linear3a-h7"],"title":"Estimating $${1.01}^3$$","text":"Evaluate $$L(0.01)$$ when $$n$$ $$=$$ $$3$$ such that $${1.01}^3$$ $$=$$ $$f(1.01)$$ \u2248 $$L(0.01)$$ $$=$$ $$1$$ + $$3x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear3a-h9","type":"hint","dependencies":["ad0aaf6linear3a-h8"],"title":"Final Answer","text":"$$f(1.01)$$ \u2248 $$L(0.01)$$ $$=$$ $$1$$ + $$3(0.01)$$ $$=$$ $$1.03$$\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ad0aaf6linear4","title":"Linear Approximations and Differentials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.2 Linear Approximations and Differentials","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad0aaf6linear4a","stepAnswer":["$$0.8$$"],"problemType":"TextBox","stepTitle":"Given the function $$y=x^2+2x$$, find dy and evaluate when $$x=3$$ and $$dx=0.1$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.8$$","hints":{"DefaultPathway":[{"id":"ad0aaf6linear4a-h1","type":"hint","dependencies":[],"title":"Obtaining dy","text":"In order to determine dy, we must take the derivative of $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear4a-h2","type":"hint","dependencies":["ad0aaf6linear4a-h1"],"title":"Solving for y\'","text":"$$y=x^2+2x$$ such that $$y\'=2x^{2-1}+2=2x+2$$. Thus, $$dy=\\\\left(2x+2\\\\right) dx$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear4a-h3","type":"hint","dependencies":["ad0aaf6linear4a-h2"],"title":"Plugging Into y\'","text":"Plug $$x$$ $$=$$ $$3$$ and dx $$=$$ $$0.1$$ into $$dy=\\\\left(2x+2\\\\right) dx$$ to solve for dy.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.8$$"],"dependencies":["ad0aaf6linear4a-h3"],"title":"Determining dy","text":"What is dy?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ad0aaf6linear4a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.8$$"],"dependencies":[],"title":"Substitution and Simplification to Getting dy","text":"$$dy=\\\\left(2x+2\\\\right) dx=0.1\\\\left(2\\\\times3+2\\\\right)=0.1\\\\left(6+2\\\\right)=$$ $$8\\\\times0.1$$. What is dy?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}]}}]},{"id":"ad0aaf6linear5","title":"Linear Approximations and Differentials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.2 Linear Approximations and Differentials","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad0aaf6linear5a","stepAnswer":["$$-0.1sin(3)$$"],"problemType":"TextBox","stepTitle":"Given the function $$y$$ $$=$$ cos(x), find dy and evaluate when $$x$$ $$=$$ $$3$$ and dx $$=$$ $$0.1$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-0.1sin(3)$$","hints":{"DefaultPathway":[{"id":"ad0aaf6linear5a-h1","type":"hint","dependencies":[],"title":"Obtaining dy","text":"In order to determine dy, we must take the derivative of $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear5a-h2","type":"hint","dependencies":["ad0aaf6linear5a-h1"],"title":"Solving for y\'","text":"$$y$$ $$=$$ cos(x) such that y\' $$=$$ $$-sin(x)$$. Thus, dy $$=$$ $$(-sin(x))dx$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear5a-h3","type":"hint","dependencies":["ad0aaf6linear5a-h2"],"title":"Plugging Into y\'","text":"Plug $$x$$ $$=$$ $$3$$ and dx $$=$$ $$0.1$$ into dy $$=$$ $$(-sin(x))dx$$ to solve for dy.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-0.1sin(3)$$"],"dependencies":["ad0aaf6linear5a-h3"],"title":"Determining dy","text":"What is dy?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ad0aaf6linear5a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-0.1sin(3)$$"],"dependencies":[],"title":"Substitution and Simplification to Getting dy","text":"dy $$=$$ $$(-sin(x))dx$$ $$=$$ $$(-sin(3))(0.1)$$. What is dy?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}]}}]},{"id":"ad0aaf6linear6","title":"Linear Approximations and Differentials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.2 Linear Approximations and Differentials","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad0aaf6linear6a","stepAnswer":["$$0.81$$"],"problemType":"TextBox","stepTitle":"Let $$y$$ $$=$$ $$x^2$$ + $$2x$$. Compute \u0394y at $$x$$ $$=$$ $$3$$ and dx $$=$$ $$0.1$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.81$$","hints":{"DefaultPathway":[{"id":"ad0aaf6linear6a-h1","type":"hint","dependencies":[],"title":"\u0394y Formula","text":"\u0394y $$=$$ f(a + dx) - f(a)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3.1$$"],"dependencies":["ad0aaf6linear6a-h1"],"title":"First Step: $$x_1$$","text":"dx is the difference between two values of $$x$$. Given that dx $$=$$ $$0.1$$ and the initial value of $$x$$ is $$3$$, what is the new value, $$x_1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ad0aaf6linear6a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3.1$$"],"dependencies":[],"title":"Determining $$x_1$$","text":"$$x_1$$ $$=$$ a + dx. What is $$x_1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear6a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3.1$$"],"dependencies":[],"title":"Plugging Into $$x_1$$","text":"$$x_1$$ $$=$$ $$3$$ + $$0.1$$. What is $$x_1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"ad0aaf6linear6a-h3","type":"hint","dependencies":["ad0aaf6linear6a-h2"],"title":"Applying Solved Values to the \u0394y Formula","text":"Plug $$x$$ $$=$$ $$3$$ and $$x_1$$ $$=$$ $$3.1$$ into \u0394y $$=$$ $$f{\\\\left(x_1\\\\right)}$$ - f(x) to get \u0394y $$=$$ $$f(3.1)$$ - f(3).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["ad0aaf6linear6a-h3"],"title":"Solving for f(x)","text":"What is f(x)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ad0aaf6linear6a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":[],"title":"Plugging $$x$$ Into the Equation","text":"Given that $$x$$ $$=$$ $$3$$, plug this x-value into the equation $$y$$ $$=$$ $$x^2$$ + $$2x$$ to solve for f(x). What answer do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear6a-h4-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":[],"title":"Next Steps of Finding $$x$$","text":"Solve for $$y$$ $$=$$ $$3^2$$ + 2(3). What is f(x)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear6a-h4-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":[],"title":"Simplification of the Equation","text":"$$y$$ $$=$$ $$9$$ + $$6$$. What is f(x)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"ad0aaf6linear6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15.81$$"],"dependencies":["ad0aaf6linear6a-h3"],"title":"Solving for $$f{\\\\left(x_1\\\\right)}$$","text":"What is $$f{\\\\left(x_1\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ad0aaf6linear6a-h5-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15.81$$"],"dependencies":[],"title":"Plugging $$x_1$$ Into the Equation","text":"Given that $$x_1$$ $$=$$ $$3.1$$, plug this x-value into the equation $$y$$ $$=$$ $$x^2$$ + $$2x$$ to solve for $$f{\\\\left(x_1\\\\right)}$$. What answer do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear6a-h5-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15.81$$"],"dependencies":[],"title":"Next Steps of Finding $$x_1$$","text":"Solve for $$y$$ $$=$$ $${3.1}^2$$ + $$2(3.1)$$. What is $$f{\\\\left(x_1\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear6a-h5-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15.81$$"],"dependencies":[],"title":"Simplification of the Equation","text":"$$y$$ $$=$$ $$9.61$$ + $$6.2$$. What is $$f{\\\\left(x_1\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"ad0aaf6linear6a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.81$$"],"dependencies":["ad0aaf6linear6a-h4","ad0aaf6linear6a-h5"],"title":"Putting it Together","text":"Now that we know that f(x) $$=$$ $$15$$ and $$f{\\\\left(x_1\\\\right)}$$ $$=$$ $$15.81$$, what is \u0394y?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear6a-h7","type":"hint","dependencies":["ad0aaf6linear6a-h6"],"title":"Final Answer","text":"\u0394y $$=$$ $$f{\\\\left(x_1\\\\right)}$$ - f(x) $$=$$ $$15.81$$ - $$15$$ $$=$$ $$0.81$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ad0aaf6linear7","title":"Linear Approximations and Differentials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.2 Linear Approximations and Differentials","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad0aaf6linear7a","stepAnswer":["$$0.8$$"],"problemType":"TextBox","stepTitle":"Let $$y$$ $$=$$ $$x^2$$ + $$2x$$. Compute \u0394y and dy at $$x$$ $$=$$ $$3$$ and dx $$=$$ $$0.1$$. First, what is dy?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.8$$","hints":{"DefaultPathway":[{"id":"ad0aaf6linear7a-h1","type":"hint","dependencies":[],"title":"Obtaining dy","text":"In order to determine dy, we must take the derivative of $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear7a-h2","type":"hint","dependencies":["ad0aaf6linear7a-h1"],"title":"Solving for y\'","text":"$$y$$ $$=$$ $$x^2$$ + $$2x$$ such that y\' $$=$$ $$2x^{2-1}$$ + $$2$$ $$=$$ $$2x$$ + $$2$$. Thus, dy $$=$$ (2x + 2)dx.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear7a-h3","type":"hint","dependencies":["ad0aaf6linear7a-h2"],"title":"Plugging Into y\'","text":"Plug $$x$$ $$=$$ $$3$$ and dx $$=$$ $$0.1$$ into dy $$=$$ (2x + 2)dx to solve for dy.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.8$$"],"dependencies":["ad0aaf6linear7a-h3"],"title":"Determining dy","text":"What is dy?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ad0aaf6linear7a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.8$$"],"dependencies":[],"title":"Substitution and Simplification to Getting dy","text":"dy $$=$$ (2x + 2)dx $$=$$ (2(3) + $$2)(0.1)$$ $$=$$ (6 + $$2)(0.1)$$ $$=$$ $$8\\\\times0.1$$. What is dy?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}]}}]},{"id":"ad0aaf6linear8","title":"Linear Approximations and Differentials","body":"Suppose the side length of a cube is measured to be $$5$$ cm with an accuracy of $$0.1$$ cm.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.2 Linear Approximations and Differentials","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad0aaf6linear8a","stepAnswer":["$$7.5$$"],"problemType":"TextBox","stepTitle":"Use differentials to estimate the error in the computed volume of the cube. Your answer should be a number that fills in this equation: $$-___$$ $$ \\\\leq $$ dV $$ \\\\leq $$ $$___$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7.5$$","hints":{"DefaultPathway":[{"id":"ad0aaf6linear8a-h1","type":"hint","dependencies":[],"title":"Understanding the Problem","text":"The measurement of the side length is accurate to within $$\xb10.1$$ cm such that $$-0.1$$ $$ \\\\leq $$ dx $$ \\\\leq $$ $$0.1$$. Solve for dV.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear8a-h2","type":"hint","dependencies":["ad0aaf6linear8a-h1"],"title":"Volume Formula","text":"The volume of a cube is given by V $$=$$ $$x^3$$. Take the derivative of V to get the dV equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear8a-h3","type":"hint","dependencies":["ad0aaf6linear8a-h2"],"title":"dV Equation","text":"dV=(3*x**(3 - $$1))dx=3x^2 dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear8a-h4","type":"hint","dependencies":["ad0aaf6linear8a-h3"],"title":"Plugging Into dV","text":"Using the measured side length of $$5$$ cm, we can estimate the error by plugging $$x$$ $$=$$ $$5$$ and dx $$=$$ $$0.1$$ into dV $$=$$ $$3x^2 dx$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear8a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7.5$$"],"dependencies":["ad0aaf6linear8a-h4"],"title":"Solving for dV","text":"What is dV?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear8a-h6","type":"hint","dependencies":["ad0aaf6linear8a-h5"],"title":"Final Answer","text":"dV $$=$$ $$0.1{3\\\\left(5\\\\right)}^2$$ $$=$$ $$3\\\\times35\\\\times0.1$$ $$=$$ $$7.5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ad0aaf6linear9","title":"Linear Approximations and Differentials","body":"Suppose the side length of a cube is measured to be $$5$$ cm with an accuracy of $$0.1$$ cm.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4.2 Linear Approximations and Differentials","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad0aaf6linear9a","stepAnswer":["$$-7.351$$"],"problemType":"TextBox","stepTitle":"Given two side lengths, the lower side length is $$4.9$$ cm. Compute the volume of the cube to compare the estimated error with the actual potential error. What is the error in the computed volume?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-7.351$$","hints":{"DefaultPathway":[{"id":"ad0aaf6linear9a-h1","type":"hint","dependencies":[],"title":"Volume Formula","text":"The volume of a cube is given by V $$=$$ $$x^3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$117.649$$"],"dependencies":["ad0aaf6linear9a-h1"],"title":"Calculating $$V(4.9)$$","text":"If the side length is actually $$4.9$$ cm, then the volume of the cube is $$V(4.9)$$ $$=$$ $${4.9}^3$$. What is $$V(4.9)$$ in $${cm}^3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear9a-h3","type":"hint","dependencies":["ad0aaf6linear9a-h2"],"title":"Understanding the Problem","text":"From what we calculated, the actual volume of the cube will be between $$117.649$$ and V(the higher side length). Since the side length is measured to be $$5$$ cm, the computed volume is V(5).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$125$$"],"dependencies":["ad0aaf6linear9a-h3"],"title":"Solving for V(5)","text":"What is V(5)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ad0aaf6linear9a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$125$$"],"dependencies":[],"title":"Plugging into V(5)","text":"V(5) $$=$$ $$5^3$$. What is V(5)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"ad0aaf6linear9a-h5","type":"hint","dependencies":["ad0aaf6linear9a-h4"],"title":"Calculating the Difference","text":"To find the error in the computed volume, we must find the difference between the actual volume, $$V(4.9)$$, and the calculated volume, V(5).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear9a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-7.351$$"],"dependencies":["ad0aaf6linear9a-h5"],"title":"Error in the Computed Volume","text":"What is the error in the computed volume?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ad0aaf6linear9a-h6-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-7.351$$"],"dependencies":[],"title":"Plugging in the Difference","text":"What is $$V(4.9)$$ - V(5)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad0aaf6linear9a-h6-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-7.351$$"],"dependencies":[],"title":"Final Difference","text":"What is $$117.649$$ - 125?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"ad0aaf6linear9a-h7","type":"hint","dependencies":["ad0aaf6linear9a-h6"],"title":"Interpretation","text":"From our answer, we know that the error in the computed volume is $$-7.351$$ $$ \\\\leq $$ \u0394V $$ \\\\leq $$ the difference of the higher side length. The estimated error dV is relatively close to the actual potential error in the computed volume.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ad16e82class1","title":"Finding Slope","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.2 Basic Classes of Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad16e82class1a","stepAnswer":["$$-1$$"],"problemType":"TextBox","stepTitle":"Find the slope of the line passing through the points $$(-2,4)$$ and $$(1,1)$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1$$","hints":{"DefaultPathway":[{"id":"ad16e82class1a-h1","type":"hint","dependencies":[],"title":"Equation for 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Slope","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.2 Basic Classes of Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad16e82class12a","stepAnswer":["$$-6$$"],"problemType":"TextBox","stepTitle":"Find the slope for the linear equation $$f(x)=-6x$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-6$$","hints":{"DefaultPathway":[{"id":"ad16e82class12a-h1","type":"hint","dependencies":[],"title":"Equation for Slope","text":"For $$y=m x+b$$, the coefficient of $$x$$, which is $$m$$, represents the slope.","variabilization":{},"oer":"","license":""},{"id":"ad16e82class12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6$$"],"dependencies":["ad16e82class12a-h1"],"title":"Finding Slope","text":"What is the slope of this line?","variabilization":{},"oer":"","license":""}]}}]},{"id":"ad16e82class13","title":"Finding Slope","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.2 Basic Classes of Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad16e82class13a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"Find the slope for the linear equation $$4y$$ + $$24$$ $$=$$ $$0$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"ad16e82class13a-h1","type":"hint","dependencies":[],"title":"Equation for Slope","text":"For $$y=m x+b$$, the coefficient of $$x$$, which is $$m$$, represents the slope.","variabilization":{},"oer":"","license":""},{"id":"ad16e82class13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["ad16e82class13a-h1"],"title":"Finding Slope","text":"What is the slope of this line?","variabilization":{},"oer":"","license":""}]}}]},{"id":"ad16e82class14","title":"Finding Slope","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.2 Basic Classes of Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad16e82class14a","stepAnswer":["$$\\\\frac{-2}{3}$$"],"problemType":"TextBox","stepTitle":"Find the slope for the linear equation $$2x+3y$$ $$=$$ $$6$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-2}{3}$$","hints":{"DefaultPathway":[{"id":"ad16e82class14a-h1","type":"hint","dependencies":[],"title":"Equation for Slope","text":"For $$y=m x+b$$, the coefficient of $$x$$, which is $$m$$, represents the slope.","variabilization":{},"oer":"","license":""},{"id":"ad16e82class14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-2}{3}$$"],"dependencies":["ad16e82class14a-h1"],"title":"Finding Slope","text":"What is the slope of this line?","variabilization":{},"oer":"","license":"","subHints":[{"id":"ad16e82class14a-h2-s1","type":"hint","dependencies":[],"title":"Finding Slope","text":"$$2x+3y$$ $$=$$ $$6$$ is equivalent to $$y=2-\\\\frac{2}{3} x$$","variabilization":{},"oer":"","license":""}]}]}}]},{"id":"ad16e82class15","title":"Finding Slope","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.2 Basic Classes of Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad16e82class15a","stepAnswer":["$$\\\\frac{6}{5}$$"],"problemType":"TextBox","stepTitle":"Find the slope for the linear equation $$6x-5y+15=0$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{6}{5}$$","hints":{"DefaultPathway":[{"id":"ad16e82class15a-h1","type":"hint","dependencies":[],"title":"Equation for Slope","text":"For $$y=m x+b$$, the coefficient of $$x$$, which is $$m$$, represents the slope.","variabilization":{},"oer":"","license":""},{"id":"ad16e82class15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{6}{5}$$"],"dependencies":["ad16e82class15a-h1"],"title":"Finding Slope","text":"What is the slope of this line?","variabilization":{},"oer":"","license":"","subHints":[{"id":"ad16e82class15a-h2-s1","type":"hint","dependencies":[],"title":"Finding Slope","text":"$$6x-5y+15=0$$ is equivalent to $$y=\\\\frac{6}{5} x+3$$","variabilization":{},"oer":"","license":""}]}]}}]},{"id":"ad16e82class16","title":"Degree of Polynomial","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.2 Basic Classes of Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad16e82class16a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"Find the degree of the polynomial f(x) $$=$$ $$2x^2$$ - $$3x$$ $$-5$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"ad16e82class16a-h1","type":"hint","dependencies":[],"title":"Definition of Degree","text":"The degree of a polynomial is the highest power of the variable in the polynomial, with non-zero coefficient.","variabilization":{},"oer":"","license":""}]}}]},{"id":"ad16e82class17","title":"Degree of Polynomial","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.2 Basic Classes of Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad16e82class17a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"Find the degree of the polynomial f(x) $$=$$ $$-3x^2$$ + $$6x$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"ad16e82class17a-h1","type":"hint","dependencies":[],"title":"Definition of Degree","text":"The degree of a polynomial is the highest power of the variable in the polynomial, with non-zero coefficient.","variabilization":{},"oer":"","license":""}]}}]},{"id":"ad16e82class18","title":"Degree of Polynomial","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.2 Basic Classes of Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad16e82class18a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"Find the degree of the polynomial f(x) $$=$$ $$\\\\frac{1}{2}$$ $$x^2$$ - $$1$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"ad16e82class18a-h1","type":"hint","dependencies":[],"title":"Definition of Degree","text":"The degree of a polynomial is the highest power of the variable in the polynomial, with non-zero coefficient.","variabilization":{},"oer":"","license":""}]}}]},{"id":"ad16e82class19","title":"Degree of Polynomial","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.2 Basic Classes of Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad16e82class19a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"Find the degree of the polynomial f(x) $$=$$ $$x^3$$ + $$3x^2$$ - $$x$$ - $$3$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"ad16e82class19a-h1","type":"hint","dependencies":[],"title":"Definition of Degree","text":"The degree of a polynomial is the highest power of the variable in the polynomial, with non-zero coefficient.","variabilization":{},"oer":"","license":""}]}}]},{"id":"ad16e82class2","title":"Finding Slope","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 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line?","variabilization":{},"oer":"","license":"","subHints":[{"id":"ad16e82class2a-h2-s1","type":"hint","dependencies":[],"title":"Calculate the slope","text":"$$m=\\\\frac{5-2}{3+1}=\\\\frac{3}{4}$$","variabilization":{},"oer":"","license":""}]}]}}]},{"id":"ad16e82class20","title":"Degree of Polynomial","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.2 Basic Classes of Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad16e82class20a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"Find the degree of the polynomial f(x) $$=$$ $$3x$$ - $$x^2$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"ad16e82class20a-h1","type":"hint","dependencies":[],"title":"Definition of Degree","text":"The degree of a polynomial is the highest power 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At the end of a 3-year period, the value of the equipment has decreased linearly to $12,300. 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In $$2012$$ $$(t=0)$$, total online holiday sales were $$\\\\$42.3$$ billion, whereas in $$2013$$ they were $$\\\\$48.1$$ billion. Find the slope of the function that models this scenario.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5.8$$","hints":{"DefaultPathway":[{"id":"ad16e82class22a-h1","type":"hint","dependencies":[],"title":"Explanation","text":"We want to find a line that passes through $$(0, 42.3)$$ and $$(1, 48.1)$$.","variabilization":{},"oer":"","license":""},{"id":"ad16e82class22a-h2","type":"hint","dependencies":["ad16e82class22a-h1"],"title":"Equation for Slope","text":"$$m=\\\\frac{y1-y2}{x1-x2}$$","variabilization":{},"oer":"","license":""},{"id":"ad16e82class22a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5.8$$"],"dependencies":["ad16e82class22a-h2"],"title":"Solve the Value","text":"Following the given equation, what is the slope of the function?","variabilization":{},"oer":"","license":"","subHints":[{"id":"ad16e82class22a-h3-s1","type":"hint","dependencies":[],"title":"Solve the Value","text":"$$\\\\frac{48.1-42.3}{1-0}=5.8$$","variabilization":{},"oer":"","license":""}]}]}}]},{"id":"ad16e82class23","title":"Cupcake Stand","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.2 Basic Classes of Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad16e82class23a","stepAnswer":["$$0.75$$"],"problemType":"TextBox","stepTitle":"A family bakery makes cupcakes and sells them at local outdoor festivals. For a music festival, there is a fixed cost of $125 to set up a cupcake stand. The owner estimates that it costs $$\\\\$0.75$$ to make each cupcake. The owner is interested in determining the total cost (C) as a function of number of cupcakes made (x). Find the slope of the function that models this scenario.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.75$$","hints":{"DefaultPathway":[{"id":"ad16e82class23a-h1","type":"hint","dependencies":[],"title":"Explanation","text":"Given the conditions in the question, we can conclude that the function C in terms of $$x$$ is $$C=0.75x+125$$.","variabilization":{},"oer":"","license":""},{"id":"ad16e82class23a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.75$$"],"dependencies":["ad16e82class23a-h1"],"title":"Solve the Value","text":"Following the given equation, what is the slope of the function?","variabilization":{},"oer":"","license":""}]}}]},{"id":"ad16e82class24","title":"Value of House","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.2 Basic Classes of Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad16e82class24a","stepAnswer":["$$13888.89$$"],"problemType":"MultipleChoice","stepTitle":"A house purchased for $250,000 is expected to be worth twice its purchase price in $$18$$ years. Assume the price increases by a fixed value each year; what is the fixed value?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$13888.89$$","choices":["$$13888.89$$","$$13800$$","$$15888.89$$"],"hints":{"DefaultPathway":[{"id":"ad16e82class24a-h1","type":"hint","dependencies":[],"title":"Equation for Slope","text":"$$m=\\\\frac{y1-y2}{x1-x2}$$","variabilization":{},"oer":"","license":""},{"id":"ad16e82class24a-h2","type":"hint","dependencies":["ad16e82class24a-h1"],"title":"Explanation","text":"After eighteen years, the price will be $$250000\\\\times2=500000$$.","variabilization":{},"oer":"","license":""},{"id":"ad16e82class24a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$13888.89$$"],"dependencies":["ad16e82class24a-h2"],"title":"Solve the Value","text":"What is the fixed value?","variabilization":{},"oer":"","license":"","choices":["$$13888.89$$","$$13800$$","$$15888.89$$"],"subHints":[{"id":"ad16e82class24a-h3-s1","type":"hint","dependencies":[],"title":"Solve the Value","text":"$$\\\\frac{500000-250000}{18}=13888.89$$","variabilization":{},"oer":"","license":""}]}]}}]},{"id":"ad16e82class25","title":"Car Value","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.2 Basic Classes of Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad16e82class25a","stepAnswer":["$$-1500$$"],"problemType":"TextBox","stepTitle":"A car was purchased for $26,000. After one year, the value of the car decreased to $24,500. Calculate the annual rate of depreciation of the car\'s value.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1500$$","hints":{"DefaultPathway":[{"id":"ad16e82class25a-h1","type":"hint","dependencies":[],"title":"Equation for Slope","text":"$$m=\\\\frac{y1-y2}{x1-x2}$$","variabilization":{},"oer":"","license":""},{"id":"ad16e82class25a-h2","type":"hint","dependencies":["ad16e82class25a-h1"],"title":"Solve the Value","text":"$$\\\\frac{24500-26000}{1-0}=-1500$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"ad16e82class3","title":"Finding Slope","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.2 Basic Classes of Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad16e82class3a","stepAnswer":["$$\\\\frac{4}{3}$$"],"problemType":"TextBox","stepTitle":"Find the slope of the line passing through the points $$(2,3)$$ and $$(5,7)$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{4}{3}$$","hints":{"DefaultPathway":[{"id":"ad16e82class3a-h1","type":"hint","dependencies":[],"title":"Equation for Slope","text":"$$m=\\\\frac{y1-y2}{x1-x2}$$","variabilization":{},"oer":"","license":""},{"id":"ad16e82class3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{4}{3}$$"],"dependencies":["ad16e82class3a-h1"],"title":"Calculate the slope","text":"What is the slope of this line?","variabilization":{},"oer":"","license":"","subHints":[{"id":"ad16e82class3a-h2-s1","type":"hint","dependencies":[],"title":"Calculate the slope","text":"$$m=\\\\frac{3-7}{2-5}=\\\\frac{-4}{-3}=\\\\frac{4}{3}$$","variabilization":{},"oer":"","license":""}]}]}}]},{"id":"ad16e82class4","title":"Finding Slope","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.2 Basic Classes of Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad16e82class4a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"Find the slope of the line passing through the points $$(2,4)$$ and $$(1,4)$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"ad16e82class4a-h1","type":"hint","dependencies":[],"title":"Equation for Slope","text":"$$m=\\\\frac{y1-y2}{x1-x2}$$","variabilization":{},"oer":"","license":""},{"id":"ad16e82class4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["ad16e82class4a-h1"],"title":"Calculate the slope","text":"What is the slope of this line?","variabilization":{},"oer":"","license":"","subHints":[{"id":"ad16e82class4a-h2-s1","type":"hint","dependencies":[],"title":"Calculate the slope","text":"$$m=\\\\frac{4-4}{2-1}=\\\\frac{0}{1}=0$$","variabilization":{},"oer":"","license":""}]}]}}]},{"id":"ad16e82class5","title":"Finding Slope","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.2 Basic Classes of Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad16e82class5a","stepAnswer":["Decreasing"],"problemType":"MultipleChoice","stepTitle":"Indicate whether the line passing through the points $$(-1,4)$$ and $$(3,-1)$$ is increasing, decreasing, horizontal, or vertical.","stepBody":"","answerType":"string","variabilization":{},"choices":["Increasing","Decreasing","Horizontal","Vertical"],"hints":{"DefaultPathway":[{"id":"ad16e82class5a-h1","type":"hint","dependencies":[],"title":"Equation for Slope","text":"$$m=\\\\frac{y1-y2}{x1-x2}$$","variabilization":{},"oer":"","license":""},{"id":"ad16e82class5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{\\\\left(-4\\\\right)}$$"],"dependencies":["ad16e82class5a-h1"],"title":"Calculate the slope","text":"What is the slope of this line?","variabilization":{},"oer":"","license":"","subHints":[{"id":"ad16e82class5a-h2-s1","type":"hint","dependencies":[],"title":"Calculate the slope","text":"$$m=\\\\frac{4+1}{\\\\left(-1-3\\\\right)}=\\\\frac{5}{\\\\left(-4\\\\right)}$$","variabilization":{},"oer":"","license":""}]},{"id":"ad16e82class5a-h3","type":"hint","dependencies":["ad16e82class5a-h2"],"title":"Explanation","text":"Since $$m<0$$, the line is decreasing.","variabilization":{},"oer":"","license":""}]}}]},{"id":"ad16e82class6","title":"Finding Slope","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.2 Basic Classes of Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad16e82class6a","stepAnswer":["Increasing"],"problemType":"MultipleChoice","stepTitle":"Indicate whether the line passing through the points $$(6,4)$$ and $$(4,-3)$$ is increasing, decreasing, horizontal, or vertical.","stepBody":"","answerType":"string","variabilization":{},"choices":["Increasing","Decreasing","Horizontal","Vertical"],"hints":{"DefaultPathway":[{"id":"ad16e82class6a-h1","type":"hint","dependencies":[],"title":"Equation for Slope","text":"$$m=\\\\frac{y1-y2}{x1-x2}$$","variabilization":{},"oer":"","license":""},{"id":"ad16e82class6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{7}{2}$$"],"dependencies":["ad16e82class6a-h1"],"title":"Calculate the slope","text":"What is the slope of this line?","variabilization":{},"oer":"","license":"","subHints":[{"id":"ad16e82class6a-h2-s1","type":"hint","dependencies":[],"title":"Calculate the slope","text":"$$m=\\\\frac{4+3}{6-4}=\\\\frac{7}{2}$$","variabilization":{},"oer":"","license":""}]},{"id":"ad16e82class6a-h3","type":"hint","dependencies":["ad16e82class6a-h2"],"title":"Explanation","text":"Since $$m>0$$, the line is increasing.","variabilization":{},"oer":"","license":""}]}}]},{"id":"ad16e82class7","title":"Finding Slope","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.2 Basic Classes of Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad16e82class7a","stepAnswer":["Increasing"],"problemType":"MultipleChoice","stepTitle":"Indicate whether the line passing through the points $$(1,9)$$ and $$(-8,5)$$ is increasing, decreasing, horizontal, or vertical.","stepBody":"","answerType":"string","variabilization":{},"choices":["Increasing","Decreasing","Horizontal","Vertical"],"hints":{"DefaultPathway":[{"id":"ad16e82class7a-h1","type":"hint","dependencies":[],"title":"Equation for Slope","text":"$$m=\\\\frac{y1-y2}{x1-x2}$$","variabilization":{},"oer":"","license":""},{"id":"ad16e82class7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{4}{9}$$"],"dependencies":["ad16e82class7a-h1"],"title":"Calculate the slope","text":"What is the slope of this line?","variabilization":{},"oer":"","license":"","subHints":[{"id":"ad16e82class7a-h2-s1","type":"hint","dependencies":[],"title":"Calculate the slope","text":"$$m=\\\\frac{9-5}{1+8}=\\\\frac{4}{9}$$","variabilization":{},"oer":"","license":""}]},{"id":"ad16e82class7a-h3","type":"hint","dependencies":["ad16e82class7a-h2"],"title":"Explanation","text":"Since $$m>0$$, the line is increasing.","variabilization":{},"oer":"","license":""}]}}]},{"id":"ad16e82class8","title":"Finding Slope","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.2 Basic Classes of Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad16e82class8a","stepAnswer":["Vertical"],"problemType":"MultipleChoice","stepTitle":"Indicate whether the line passing through the points $$(1,4)$$ and $$(1,0)$$ is increasing, decreasing, horizontal, or vertical.","stepBody":"","answerType":"string","variabilization":{},"choices":["Increasing","Decreasing","Horizontal","Vertical"],"hints":{"DefaultPathway":[{"id":"ad16e82class8a-h1","type":"hint","dependencies":[],"title":"Equation for Slope","text":"$$m=\\\\frac{y1-y2}{x1-x2}$$","variabilization":{},"oer":"","license":""},{"id":"ad16e82class8a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ad16e82class8a-h1"],"title":"Calculate the slope","text":"What is the slope of this line? Is it defined?","variabilization":{},"oer":"","license":"","choices":["Yes","No"],"subHints":[{"id":"ad16e82class8a-h2-s1","type":"hint","dependencies":[],"title":"Calculate the slope","text":"$$m=\\\\frac{4-0}{1-1}=\\\\frac{4}{0}$$ (undefined)","variabilization":{},"oer":"","license":""}]},{"id":"ad16e82class8a-h3","type":"hint","dependencies":["ad16e82class8a-h2"],"title":"Explanation","text":"Since the denominator of $$m$$ is $$0$$, the line is vertical.","variabilization":{},"oer":"","license":""}]}}]},{"id":"ad16e82class9","title":"Finding Slope","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.2 Basic Classes of Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad16e82class9a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"Find the slope for the linear equation $$y=2x-3$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"ad16e82class9a-h1","type":"hint","dependencies":[],"title":"Equation for Slope","text":"For $$y=m x+b$$, the coefficient of $$x$$, which is $$m$$, represents the slope.","variabilization":{},"oer":"","license":""},{"id":"ad16e82class9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ad16e82class9a-h1"],"title":"Finding Slope","text":"What is the slope of this line?","variabilization":{},"oer":"","license":""}]}}]},{"id":"ad1b49dc1","title":"Determining Continuity at a Point","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1","license":0,"lesson":"2.4 Continuity","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad1b49dc1a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Using the definition, determine whether the function $$f(x)=\\\\frac{x^2-4}{x-2}$$ is continuous at $$x=2$$. Justify the conclusion.","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ad1b49dc1a-h1","type":"hint","dependencies":[],"title":"Problem-Solving Strategy: Determining Continuity at a Point","text":"Step 1: check to see if f(a) is defined. If f(a) is undefined, we need go no further. The function is not continuous at a. If f(a) is defined, continue to step $$2$$. Step 2: Compute $$\\\\lim_{x\\\\toa} f(x)$$. In some cases, we may need to do this by first computing $$\\\\lim_{x\\\\toa-} f(x)$$ and /lim{x,a+,f(x)}. If $$\\\\lim_{x\\\\toa} f(x)$$ does not exist (that is, it is not a real number), then the function is not continuous at a and the problem is solved. If $$\\\\lim_{x\\\\toa} f(x)$$ exists, then continue to step $$3$$. Step 3: Compare f(a) and $$\\\\lim_{x\\\\toa} f(x)$$. If $$\\\\lim_{x\\\\toa} f(x)$$ does not equal to f(a), then the function is not continuous at a. If $$\\\\lim_{x\\\\toa} f(x)=f(a)$$, then the function is continuous at a.","variabilization":{},"oer":"","license":""},{"id":"ad1b49dc1a-h2","type":"hint","dependencies":["ad1b49dc1a-h1"],"title":"Determine f(2)","text":"Following the definitions, let\u2019s begin by trying to calculate f(2). We can see that $$f(2)=\\\\frac{0}{0}$$.","variabilization":{},"oer":"","license":""},{"id":"ad1b49dc1a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ad1b49dc1a-h2"],"title":"Determine f(2)","text":"Is it defined?","variabilization":{},"oer":"","license":"","choices":["Yes","No"]},{"id":"ad1b49dc1a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ad1b49dc1a-h3"],"title":"Determine Continuity","text":"Is the function continuous at $$x=2$$?","variabilization":{},"oer":"","license":"","choices":["Yes","No"],"subHints":[{"id":"ad1b49dc1a-h4-s1","type":"hint","dependencies":[],"title":"Determine Continuity","text":"The function is discontinuous at $$2$$ because f(2) is undefined.","variabilization":{},"oer":"","license":""}]}]}}]},{"id":"ad1b49dc10","title":"When Can You Apply the Intermediate Value Theorem?","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1","license":0,"lesson":"2.4 Continuity","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad1b49dc10a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"For $$f(x)=\\\\frac{1}{x}$$, $$f(-1)=-1<0$$ and $$f(1)=1>0$$. Can we conclude that f(x) has a zero in the interval [-1,1]?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ad1b49dc10a-h1","type":"hint","dependencies":[],"title":"The Intermediate Value Theorem","text":"The Intermediate Value Theorem: Let f be continuous over a closed, bounded interval [a,b]. If $$z$$ is any real number between f(a) and f(b), then there is a number c in [a,b] satisfying $$f(c)=z$$.\\\\n##figure1.gif##","variabilization":{},"oer":"","license":""},{"id":"ad1b49dc10a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ad1b49dc10a-h1"],"title":"Determine the continuity","text":"Is the function continuous over [-1,1]?","variabilization":{},"oer":"","license":"","choices":["Yes","No"],"subHints":[{"id":"ad1b49dc10a-h2-s1","type":"hint","dependencies":[],"title":"Determine the continuity","text":"f(x) is undefined at $$x=0$$.","variabilization":{},"oer":"","license":""}]},{"id":"ad1b49dc10a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ad1b49dc10a-h2"],"title":"The Intermediate Value Theorem","text":"Therefore, can we apply the Intermediate Value Theorem?","variabilization":{},"oer":"","license":"","choices":["Yes","No"]}]}}]},{"id":"ad1b49dc2","title":"Determining Continuity at a Point","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1","license":0,"lesson":"2.4 Continuity","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad1b49dc2a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Using the definition, determine whether the function $$f(x)=-\\\\left(x^2\\\\right)+4$$ if $$x \\\\leq 3;$$ $$4x-8$$ if $$x>3$$ is continuous at $$x=3$$. Justify the conclusion.","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ad1b49dc2a-h1","type":"hint","dependencies":[],"title":"Problem-Solving Strategy: Determining Continuity at a Point","text":"Step 1: check to see if f(a) is defined. If f(a) is undefined, we need go no further. The function is not continuous at a. If f(a) is defined, continue to step $$2$$. Step 2: Compute $$\\\\lim_{x\\\\toa} f(x)$$. In some cases, we may need to do this by first computing $$\\\\lim_{x\\\\toa-} f(x)$$ and /lim{x,a+,f(x)}. If $$\\\\lim_{x\\\\toa} f(x)$$ does not exist (that is, it is not a real number), then the function is not continuous at a and the problem is solved. If $$\\\\lim_{x\\\\toa} f(x)$$ exists, then continue to step $$3$$. Step 3: Compare f(a) and $$\\\\lim_{x\\\\toa} f(x)$$. If $$\\\\lim_{x\\\\toa} f(x)$$ does not equal to f(a), then the function is not continuous at a. If $$\\\\lim_{x\\\\toa} f(x)=f(a)$$, then the function is continuous at a.","variabilization":{},"oer":"","license":""},{"id":"ad1b49dc2a-h2","type":"hint","dependencies":["ad1b49dc2a-h1"],"title":"Determine f(3)","text":"Following the definitions, let\u2019s begin by trying to calculate f(3).","variabilization":{},"oer":"","license":""},{"id":"ad1b49dc2a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ad1b49dc2a-h2"],"title":"Determine f(3)","text":"Is it defined?","variabilization":{},"oer":"","license":"","choices":["Yes","No"],"subHints":[{"id":"ad1b49dc2a-h3-s1","type":"hint","dependencies":[],"title":"Determine f(3)","text":"$$f(3)=-9+4=-5$$, so f(3) is defined.","variabilization":{},"oer":"","license":""}]},{"id":"ad1b49dc2a-h4","type":"hint","dependencies":["ad1b49dc2a-h3"],"title":"Determine the limit","text":"Next, we calculate $$\\\\lim_{x\\\\to3} f(x)$$. To do this, we must compute $$\\\\lim_{x\\\\to3-} f(x)$$ and /lim{x,3+,f(x)}.","variabilization":{},"oer":"","license":""},{"id":"ad1b49dc2a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["ad1b49dc2a-h4"],"title":"Determine the limit","text":"What is the value of $$\\\\lim_{x\\\\to3-} f(x)$$?","variabilization":{},"oer":"","license":"","subHints":[{"id":"ad1b49dc2a-h5-s1","type":"hint","dependencies":[],"title":"Determine the limit","text":"$$\\\\lim_{x\\\\to3-} f(x)=-\\\\left(3^2\\\\right)+4=-9+4=-5$$","variabilization":{},"oer":"","license":""}]},{"id":"ad1b49dc2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ad1b49dc2a-h5"],"title":"Determine the limit","text":"What is the value of /lim{x,3+,f(x)}?","variabilization":{},"oer":"","license":"","subHints":[{"id":"ad1b49dc2a-h6-s1","type":"hint","dependencies":[],"title":"Determine the limit","text":"/lim{x,3+,f(x)}=4*3-8=4","variabilization":{},"oer":"","license":""}]},{"id":"ad1b49dc2a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ad1b49dc2a-h6"],"title":"Determine the limit","text":"Does the limit exist?","variabilization":{},"oer":"","license":"","choices":["Yes","No"],"subHints":[{"id":"ad1b49dc2a-h7-s1","type":"hint","dependencies":[],"title":"Determine the limit","text":"The limit does not exist since $$\\\\lim_{x\\\\to3-} f(x)$$ does not equal to /lim{x,3+,f(x)}","variabilization":{},"oer":"","license":""}]},{"id":"ad1b49dc2a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ad1b49dc2a-h7"],"title":"Determine Continuity","text":"Is the function continuous at $$x=3$$?","variabilization":{},"oer":"","license":"","choices":["Yes","No"]}]}}]},{"id":"ad1b49dc3","title":"Determining Continuity at a Point","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1","license":0,"lesson":"2.4 Continuity","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad1b49dc3a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Using the definition, determine whether the function $$f(x)=\\\\fracsin^x\\\\left(x\\\\right)}$$ if $$x \\\\neq 0;$$ $$1$$ if $$x=0$$ is continuous at $$x=0$$. Justify the conclusion.","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ad1b49dc3a-h1","type":"hint","dependencies":[],"title":"Problem-Solving Strategy: Determining Continuity at a Point","text":"Step 1: check to see if f(a) is defined. If f(a) is undefined, we need go no further. The function is not continuous at a. If f(a) is defined, continue to step $$2$$. Step 2: Compute $$\\\\lim_{x\\\\toa} f(x)$$. In some cases, we may need to do this by first computing $$\\\\lim_{x\\\\toa-} f(x)$$ and /lim{x,a+,f(x)}. If $$\\\\lim_{x\\\\toa} f(x)$$ does not exist (that is, it is not a real number), then the function is not continuous at a and the problem is solved. If $$\\\\lim_{x\\\\toa} f(x)$$ exists, then continue to step $$3$$. Step 3: Compare f(a) and $$\\\\lim_{x\\\\toa} f(x)$$. If $$\\\\lim_{x\\\\toa} f(x)$$ does not equal to f(a), then the function is not continuous at a. If $$\\\\lim_{x\\\\toa} f(x)=f(a)$$, then the function is continuous at a.","variabilization":{},"oer":"","license":""},{"id":"ad1b49dc3a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ad1b49dc3a-h1"],"title":"Determine f(0)","text":"Following the definitions, let\u2019s begin by trying to calculate f(0). Is f(0) defined?","variabilization":{},"oer":"","license":"","choices":["Yes","No"],"subHints":[{"id":"ad1b49dc3a-h2-s1","type":"hint","dependencies":[],"title":"Determine f(0)","text":"$$f(0)=1$$, so f(0) is defined.","variabilization":{},"oer":"","license":""}]},{"id":"ad1b49dc3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ad1b49dc3a-h2"],"title":"Determine the limit","text":"Next, calculate $$\\\\lim_{x\\\\to0} f(x)$$. What is the value of the limit?","variabilization":{},"oer":"","license":"","subHints":[{"id":"ad1b49dc3a-h3-s1","type":"hint","dependencies":[],"title":"Determine the limit","text":"$$\\\\lim_{x\\\\to0} f(x)=\\\\lim_{x\\\\to0} \\\\fracsin^x\\\\left(x\\\\right)}=1$$","variabilization":{},"oer":"","license":""}]},{"id":"ad1b49dc3a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ad1b49dc3a-h3"],"title":"Determine the limit","text":"Last, compare f(0) and $$\\\\lim_{x\\\\to0} f(x)$$. Are they equal to each other?","variabilization":{},"oer":"","license":"","choices":["Yes","No"],"subHints":[{"id":"ad1b49dc3a-h4-s1","type":"hint","dependencies":[],"title":"Determine the limit","text":"$$f(0)=\\\\lim_{x\\\\to0} f(x)=1$$","variabilization":{},"oer":"","license":""}]},{"id":"ad1b49dc3a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ad1b49dc3a-h4"],"title":"Determine Continuity","text":"Since all three of the conditions in the definition of continuity are satisfied, is f(x) continuous at $$x=0$$?","variabilization":{},"oer":"","license":"","choices":["Yes","No"]}]}}]},{"id":"ad1b49dc4","title":"Classifying a Discontinuity","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1","license":0,"lesson":"2.4 Continuity","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad1b49dc4a","stepAnswer":["removable"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{x^2-4}{x-2}$$ is discontinuous at $$x=2$$. Classify this discontinuity as removable, jump, or infinite.","stepBody":"","answerType":"string","variabilization":{},"choices":["Infinite","Jump","Removable","removable"],"hints":{"DefaultPathway":[{"id":"ad1b49dc4a-h1","type":"hint","dependencies":[],"title":"Definition","text":"If f(x) is discontinuous at a, then $$1$$. f has a removable discontinuity at a if $$\\\\lim_{x\\\\toa} f(x)$$ exists $$(=L$$, where L is a real number). $$2$$. f has a jump discontinuity at a if $$\\\\lim_{x\\\\toa-} f(x)$$ and /lim{x,a+,f(x)} both exist, but /lim{x,a-,f(x)}!=/lim{x,a+,f(x)}. (Note: when we say \'exist\', we mean that both limits are real-valued and that neither take on the values ~inf.) $$3$$. f has an infinite discontinuity at a if $$\\\\lim_{x\\\\toa-} f(x)=\\\\pm \\\\infty$$ and/or /lim{x,a+,f(x)}=~inf.","variabilization":{},"oer":"","license":""},{"id":"ad1b49dc4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ad1b49dc4a-h1"],"title":"Evaluate the limit","text":"What is the value of $$\\\\lim_{x\\\\to2} f(x)$$?","variabilization":{},"oer":"","license":"","subHints":[{"id":"ad1b49dc4a-h2-s1","type":"hint","dependencies":[],"title":"Evaluate the limit","text":"$$\\\\lim_{x\\\\to2} f(x)=\\\\lim_{x\\\\to2} \\\\frac{x^2-4}{x-2}=\\\\lim_{x\\\\to2} x+2=4$$","variabilization":{},"oer":"","license":""}]},{"id":"ad1b49dc4a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["removable"],"dependencies":["ad1b49dc4a-h2"],"title":"Determine the type of discontinuity","text":"Since f is discontinuous at $$2$$ and $$\\\\lim_{x\\\\to2} f(x)$$ exists, what is the type of discontinuity at $$x=2$$?","variabilization":{},"oer":"","license":"","choices":["Removable","Jump","Infinite"]}]}}]},{"id":"ad1b49dc5","title":"Classifying a Discontinuity","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1","license":0,"lesson":"2.4 Continuity","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad1b49dc5a","stepAnswer":["Infinite"],"problemType":"MultipleChoice","stepTitle":"Determine whether $$\\\\frac{x+2}{x+1}$$ is continuous at $$-1$$. If the function is discontinuous at $$-1$$, classify the discontinuity as removable, jump, or infinite.","stepBody":"","answerType":"string","variabilization":{},"choices":["Removable","Jump","Infinite"],"hints":{"DefaultPathway":[{"id":"ad1b49dc5a-h1","type":"hint","dependencies":[],"title":"Definition","text":"If f(x) is discontinuous at a, then $$1$$. f has a removable discontinuity at a if $$\\\\lim_{x\\\\toa} f(x)$$ exists $$(=L$$, where L is a real number). $$2$$. f has a jump discontinuity at a if $$\\\\lim_{x\\\\toa-} f(x)$$ and /lim{x,a+,f(x)} both exist, but /lim{x,a-,f(x)}!=/lim{x,a+,f(x)}. (Note: when we say \'exist\', we mean that both limits are real-valued and that neither take on the values ~inf.) $$3$$. f has an infinite discontinuity at a if $$\\\\lim_{x\\\\toa-} f(x)=\\\\pm \\\\infty$$ and/or /lim{x,a+,f(x)}=~inf.","variabilization":{},"oer":"","license":""},{"id":"ad1b49dc5a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ad1b49dc5a-h1"],"title":"Determine $$f(-1)$$","text":"Is $$f(-1)$$ defined?","variabilization":{},"oer":"","license":"","choices":["Yes","No"],"subHints":[{"id":"ad1b49dc5a-h2-s1","type":"hint","dependencies":[],"title":"Determine $$f(-1)$$","text":"$$f(-1)$$ is not defined because denominator cannot be zero.","variabilization":{},"oer":"","license":""}]},{"id":"ad1b49dc5a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ad1b49dc5a-h2"],"title":"Determine continuity","text":"Therefore, is the function continuous at $$x=-1$$?","variabilization":{},"oer":"","license":"","choices":["Yes","No"]},{"id":"ad1b49dc5a-h4","type":"hint","dependencies":["ad1b49dc5a-h3"],"title":"Determine the limit","text":"To determine the type of discontinuity, we must determine the limit at $$-1$$.","variabilization":{},"oer":"","license":""},{"id":"ad1b49dc5a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-\\\\infty$$"],"dependencies":["ad1b49dc5a-h4"],"title":"Determine the limit","text":"What is the value of $$\\\\lim_{x\\\\to-1-} \\\\frac{x+2}{x+1}$$?","variabilization":{},"oer":"","license":"","choices":["$$-\\\\infty$$","$$\\\\infty$$","$$0$$"]},{"id":"ad1b49dc5a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\infty$$"],"dependencies":["ad1b49dc5a-h5"],"title":"Determine the limit","text":"What is the value of /lim{x,-1+,(x+2)/(x+1)}?","variabilization":{},"oer":"","license":"","choices":["$$-\\\\infty$$","$$\\\\infty$$","$$0$$"]},{"id":"ad1b49dc5a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Infinite"],"dependencies":["ad1b49dc5a-h6"],"title":"Determine the type of discontinuity","text":"What is the type of discontinuity at $$x=-1$$?","variabilization":{},"oer":"","license":"","choices":["Removable","Jump","Infinite"]}]}}]},{"id":"ad1b49dc6","title":"Classifying a Discontinuity","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1","license":0,"lesson":"2.4 Continuity","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad1b49dc6a","stepAnswer":["jump"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=-\\\\left(x^2\\\\right)+4$$ if $$x \\\\leq 3;$$ $$4x-8$$ if $$x>3$$ is discontinuous at $$x=3$$. Classify this discontinuity as removable, jump, or infinite.","stepBody":"","answerType":"string","variabilization":{},"choices":["Infinite","Jump","Removable","jump"],"hints":{"DefaultPathway":[{"id":"ad1b49dc6a-h1","type":"hint","dependencies":[],"title":"Definition","text":"If f(x) is discontinuous at a, then $$1$$. f has a removable discontinuity at a if $$\\\\lim_{x\\\\toa} f(x)$$ exists $$(=L$$, where L is a real number). $$2$$. f has a jump discontinuity at a if $$\\\\lim_{x\\\\toa-} f(x)$$ and /lim{x,a+,f(x)} both exist, but /lim{x,a-,f(x)}!=/lim{x,a+,f(x)}. (Note: when we say \'exist\', we mean that both limits are real-valued and that neither take on the values ~inf.) $$3$$. f has an infinite discontinuity at a if $$\\\\lim_{x\\\\toa-} f(x)=\\\\pm \\\\infty$$ and/or /lim{x,a+,f(x)}=~inf.","variabilization":{},"oer":"","license":""},{"id":"ad1b49dc6a-h2","type":"hint","dependencies":["ad1b49dc6a-h1"],"title":"Determine the limit","text":"f is discontinuous at $$3$$ because $$\\\\lim_{x\\\\to3} f(x)$$ does not exist.","variabilization":{},"oer":"","license":""},{"id":"ad1b49dc6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["ad1b49dc6a-h2"],"title":"Determine the limit","text":"What is the value of $$\\\\lim_{x\\\\to3-} f(x)$$?","variabilization":{},"oer":"","license":"","subHints":[{"id":"ad1b49dc6a-h3-s1","type":"hint","dependencies":[],"title":"Determine the limit","text":"$$-\\\\left(3^2\\\\right)+4=-5$$","variabilization":{},"oer":"","license":""}]},{"id":"ad1b49dc6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ad1b49dc6a-h3"],"title":"Determine the limit","text":"What is the value of /lim{x,3+,f(x)}?","variabilization":{},"oer":"","license":"","subHints":[{"id":"ad1b49dc6a-h4-s1","type":"hint","dependencies":[],"title":"Determine the limit","text":"$$4\\\\times3-8=4$$","variabilization":{},"oer":"","license":""}]},{"id":"ad1b49dc6a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["jump"],"dependencies":["ad1b49dc6a-h4"],"title":"Determine the limit","text":"Since $$\\\\lim_{x\\\\to3-} f(x)$$ and /lim{x,3+,f(x)} both exist, what is the type of discontinuity at $$x=3$$?","variabilization":{},"oer":"","license":"","choices":["Removable","Jump","Infinite"]}]}}]},{"id":"ad1b49dc7","title":"Continuity over an Interval","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1","license":0,"lesson":"2.4 Continuity","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad1b49dc7a","stepAnswer":["$$[-2,2]$$"],"problemType":"MultipleChoice","stepTitle":"State the interval(s) over which the function $$f(x)=\\\\sqrt{4-x^2}$$ is continuous.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$[-2,2]$$","$$(-2,2)$$","$$(-\\\\infty,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"ad1b49dc7a-h1","type":"hint","dependencies":[],"title":"The limit laws","text":"From the limit laws, we know that $$\\\\lim_{x\\\\toa} \\\\sqrt{4-x^2}=\\\\sqrt{4-a^2}$$ for all values of a in $$(-2,2)$$.","variabilization":{},"oer":"","license":""},{"id":"ad1b49dc7a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ad1b49dc7a-h1"],"title":"Determine the limit","text":"Does /lim{x,-2+,sqrt(4-x**2)} exist?","variabilization":{},"oer":"","license":"","choices":["Yes","No"],"subHints":[{"id":"ad1b49dc7a-h2-s1","type":"hint","dependencies":[],"title":"Determine the limit","text":"/lim{x,-2+,sqrt(4-x**2)}=0","variabilization":{},"oer":"","license":""}]},{"id":"ad1b49dc7a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ad1b49dc7a-h2"],"title":"Determine the limit","text":"Does $$\\\\lim_{x\\\\to2-} \\\\sqrt{4-x^2}$$ exist?","variabilization":{},"oer":"","license":"","choices":["Yes","No"],"subHints":[{"id":"ad1b49dc7a-h3-s1","type":"hint","dependencies":[],"title":"Determine the limit","text":"$$\\\\lim_{x\\\\to2-} \\\\sqrt{4-x^2}=0$$","variabilization":{},"oer":"","license":""}]},{"id":"ad1b49dc7a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$[-2,2]$$"],"dependencies":["ad1b49dc7a-h3"],"title":"Determine the interval","text":"Therefore, f(x) is continuous over which interval?","variabilization":{},"oer":"","license":"","choices":["$$[-2,2]$$","$$(-2,2)$$","$$(-\\\\infty,\\\\infty)$$"]}]}}]},{"id":"ad1b49dc8","title":"Limit of a Composite Cosine Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1","license":0,"lesson":"2.4 Continuity","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad1b49dc8a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"Evaluate $$\\\\lim_{x\\\\to\\\\frac{\\\\pi}{2}} cos\\\\left(x-\\\\frac{\\\\pi}{2}\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"ad1b49dc8a-h1","type":"hint","dependencies":[],"title":"Composite Function","text":"The given function is a composite of cos(x) and $$x-\\\\frac{\\\\pi}{2}$$.","variabilization":{},"oer":"","license":""},{"id":"ad1b49dc8a-h2","type":"hint","dependencies":["ad1b49dc8a-h1"],"title":"Composite Function Theorem","text":"If f(x) is continuous at L and $$\\\\lim_{x\\\\toa} g(x)=L, then$$ $$\\\\lim_{x\\\\toa} f(g(x))=f(\\\\lim_{x\\\\toa} g(x))=f(L)$$.","variabilization":{},"oer":"","license":""},{"id":"ad1b49dc8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["ad1b49dc8a-h2"],"title":"Determine the limit","text":"What is the value of $$\\\\lim_{x\\\\to\\\\frac{\\\\pi}{2}} x-\\\\frac{\\\\pi}{2}$$?","variabilization":{},"oer":"","license":""},{"id":"ad1b49dc8a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ad1b49dc8a-h3"],"title":"Determine the continuity","text":"Is cos(x) continuous at $$x=0$$?","variabilization":{},"oer":"","license":"","choices":["Yes","No"]},{"id":"ad1b49dc8a-h5","type":"hint","dependencies":["ad1b49dc8a-h4"],"title":"Composite Function Theorem","text":"Since $$\\\\lim_{x\\\\to\\\\frac{\\\\pi}{2}} x-\\\\frac{\\\\pi}{2}=0$$ (a real number) and cos(x)is continuous at $$x=0$$, we may apply the composite function theorem.","variabilization":{},"oer":"","license":""},{"id":"ad1b49dc8a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ad1b49dc8a-h5"],"title":"Determine the limit","text":"What is the value of $$\\\\lim_{x\\\\to\\\\frac{\\\\pi}{2}} cos\\\\left(x-\\\\frac{\\\\pi}{2}\\\\right)$$?","variabilization":{},"oer":"","license":"","subHints":[{"id":"ad1b49dc8a-h6-s1","type":"hint","dependencies":[],"title":"Determine the limit","text":"$$\\\\lim_{x\\\\to\\\\frac{\\\\pi}{2}} cos\\\\left(x-\\\\frac{\\\\pi}{2}\\\\right)=cos(\\\\lim_{x\\\\to\\\\frac{\\\\pi}{2}} x-\\\\frac{\\\\pi}{2})=cos(0)=1$$","variabilization":{},"oer":"","license":""}]}]}}]},{"id":"ad1b49dc9","title":"Application of the Intermediate Value Theorem","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1","license":0,"lesson":"2.4 Continuity","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad1b49dc9a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Show that $$f(x)=x-cos(x)$$ has at least one zero. Can we apply the Intermediate Value Theorem?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ad1b49dc9a-h1","type":"hint","dependencies":[],"title":"The Intermediate Value Theorem","text":"The Intermediate Value Theorem: Let f be continuous over a closed, bounded interval [a,b]. If $$z$$ is any real number between f(a) and f(b), then there is a number c in [a,b] satisfying $$f(c)=z$$.\\\\n##figure1.gif##","variabilization":{},"oer":"","license":""},{"id":"ad1b49dc9a-h2","type":"hint","dependencies":["ad1b49dc9a-h1"],"title":"The Intermediate Value Theorem","text":"Since $$f(x)=x-cos(x)$$ is continuous over $$(-\\\\infty,\\\\infty)$$, it is continuous over any closed interval of the form [a,b].","variabilization":{},"oer":"","license":""},{"id":"ad1b49dc9a-h3","type":"hint","dependencies":["ad1b49dc9a-h2"],"title":"The Intermediate Value Theorem","text":"If you can find an interval [a,b] such that f(a) and f(b) have opposite signs, you can use the Intermediate Value Theorem to conclude there must be a real number c in (a,b) that satisfies $$f(c)=0$$.","variabilization":{},"oer":"","license":""},{"id":"ad1b49dc9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["ad1b49dc9a-h3"],"title":"The Intermediate Value Theorem","text":"What is the value of f(0)?","variabilization":{},"oer":"","license":""},{"id":"ad1b49dc9a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\pi}{2}$$"],"dependencies":["ad1b49dc9a-h4"],"title":"The Intermediate Value Theorem","text":"What is the value of $$f{\\\\left(\\\\frac{\\\\pi}{2}\\\\right)}$$?","variabilization":{},"oer":"","license":""},{"id":"ad1b49dc9a-h6","type":"hint","dependencies":["ad1b49dc9a-h5"],"title":"The Intermediate Value Theorem","text":"Since we find an interval [a,b] such that f(a) and f(b) have opposite signs, we can apply the Intermediate Value Theorem. we can see that there must be a real number c in $$[0,\\\\frac{\u03c0}{2}]$$ that satisfies $$\ud835\udc53(\ud835\udc50)=0$$. Therefore, $$\ud835\udc53(x)=x-cosx$$ has at least one zero.","variabilization":{},"oer":"","license":""},{"id":"ad1b49dc9a-h7","type":"hint","dependencies":["ad1b49dc9a-h6"],"title":"The Intermediate Value Theorem","text":"We can see that there must be a real number c in $$[0,\\\\frac{\\\\pi}{2}]$$ that satisfies $$f(c)=0$$. Therefore, $$f(x)=x-cos(x)$$ has at least one zero.","variabilization":{},"oer":"","license":""}]}}]},{"id":"ad220b3GraphQuadratic1","title":"Recognize the Graph of a Quadratic Function","body":"For each of the following exercises, determine if the parabola opens up or down.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Graph Quadratic Functions Using Properties","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad220b3GraphQuadratic1a","stepAnswer":["Down"],"problemType":"TextBox","stepTitle":"$$f(x)=-2x^2-6x-7$$","stepBody":"If the parabola opens up, please enter \\"Up\\". If the parabola opens down, please enter \\"Down\\".","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic1a-h1","type":"hint","dependencies":[],"title":"Check the Leading Coefficient","text":"Given any parabola in the form $${ax}^2+bx+c$$, a is called the leading coefficient of the parabola. If a is negative, the parabola opens downward. If a is positive, the parabola opens upward.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic1a-h2","type":"hint","dependencies":["ad220b3GraphQuadratic1a-h1"],"title":"Check the Leading Coefficient","text":"In the given question, $$a=-2$$ which is negative, so the parabola opens downward.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ad220b3GraphQuadratic1b","stepAnswer":["Up"],"problemType":"TextBox","stepTitle":"$$f(x)=6x^2+2x+3$$","stepBody":"If the parabola opens up, please enter \\"Up\\". If the parabola opens down, please enter \\"Down\\".","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic1b-h1","type":"hint","dependencies":[],"title":"Check the Leading Coefficient","text":"Given any parabola in the form $${ax}^2+bx+c$$, a is called the leading coefficient of the parabola. If a is negative, the parabola opens downward. If a is positive, the parabola opens upward.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic1b-h2","type":"hint","dependencies":["ad220b3GraphQuadratic1b-h1"],"title":"Check the Leading Coefficient","text":"In the given question, $$a=6$$ which is positive, so the parabola opens upward.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad220b3GraphQuadratic10","title":"Find the Intercepts of a Parabola","body":"In the following exercises, find the intercepts of the parabola whose function is given.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Graph Quadratic Functions Using Properties","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad220b3GraphQuadratic10a","stepAnswer":["(-11,0),(1,0)"],"problemType":"TextBox","stepTitle":"$$f(x)=x^2+10x-11$$","stepBody":"Find the $$x-intercept(s)$$ of the given parabola. Please enter your answer as \\"(x,y)\\" where $$x$$ is the x-coordinate of the intercept and $$y$$ is the y-coordinate of the intercept. If there are more than one x-intercept, you can enter it as \\"(x1,y1),(x2,y2)\\" where x1 is small than x2.","answerType":"string","variabilization":{},"answerLatex":"$$(-11,0),(1,0)$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic10a-h1","type":"hint","dependencies":[],"title":"Set $$f(x)=0$$","text":"To find the x-intercept, let $$f(x)=0$$ and and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic10a-h2","type":"hint","dependencies":["ad220b3GraphQuadratic10a-h1"],"title":"Solve For The Parabola","text":"Solve for $$x^2+10x-11=0$$. We can factor it as $$\\\\left(x+11\\\\right) \\\\left(x-1\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic10a-h3","type":"hint","dependencies":["ad220b3GraphQuadratic10a-h2"],"title":"Solve For The Parabola","text":"$$\\\\left(x+11\\\\right) \\\\left(x-1\\\\right)=0$$\\\\nUse zero product property, we know $$x+11=0$$ or $$x-1=0$$ which gives the answer $$x=-11$$ or $$x=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic10a-h4","type":"hint","dependencies":["ad220b3GraphQuadratic10a-h3"],"title":"Put The Answer Into (x,y)","text":"We know that $$y=0$$ in x-intercepts and we get $$x=-11$$, $$1$$ through the calculations above. Put together, we get two x-intercepts-- $$(-11,0)$$ and $$(1,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ad220b3GraphQuadratic10b","stepAnswer":["(0,-11)"],"problemType":"TextBox","stepTitle":"$$f(x)=x^2+10x-11$$","stepBody":"Find the y-intercept of the given parabola. Please enter your answer as \\"(x,y)\\" where $$x$$ is the x-coordinate of the intercept and $$y$$ is the y-coordinate of the intercept.","answerType":"string","variabilization":{},"answerLatex":"$$(0,-11)$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic10b-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-11$$"],"dependencies":[],"title":"Set f(0) As the y-coordinate","text":"We know that $$x=0$$ in the y-intercept. We can compute f(0) to find the $$y$$ coordinate if y-intercept. What is f(0)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic10b-h2","type":"hint","dependencies":["ad220b3GraphQuadratic10b-h1"],"title":"Put The Answer Into (x,y)","text":"$$f(0)=0^2+10\\\\times0-11=-11$$. We get $$x=0$$ and $$y=-11$$ for y-intercept. Put together, we have $$(0,-11)$$ as y-intercept.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad220b3GraphQuadratic11","title":"Find the Intercepts of a Parabola","body":"In the following exercises, find the intercepts of the parabola whose function is given.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Graph Quadratic Functions Using Properties","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad220b3GraphQuadratic11a","stepAnswer":["(-6,0),(-2,0)"],"problemType":"TextBox","stepTitle":"$$f(x)=x^2+8x+12$$","stepBody":"Find the $$x-intercept(s)$$ of the given parabola. Please enter your answer as \\"(x,y)\\" where $$x$$ is the x-coordinate of the intercept and $$y$$ is the y-coordinate of the intercept. If there are more than one x-intercept, you can enter it as \\"(x1,y1),(x2,y2)\\" where x1 is small than x2.","answerType":"string","variabilization":{},"answerLatex":"$$(-6,0),(-2,0)$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic11a-h1","type":"hint","dependencies":[],"title":"Set $$f(x)=0$$","text":"To find the x-intercept, let $$f(x)=0$$ and and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic11a-h2","type":"hint","dependencies":["ad220b3GraphQuadratic11a-h1"],"title":"Solve For The Parabola","text":"Solve for $$x^2+8x+12=0$$. We can factor it as $$\\\\left(x+6\\\\right) \\\\left(x+2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic11a-h3","type":"hint","dependencies":["ad220b3GraphQuadratic11a-h2"],"title":"Solve For The Parabola","text":"$$\\\\left(x+6\\\\right) \\\\left(x+2\\\\right)=0$$\\\\nUse zero product property, we know $$x+6=0$$ or $$x+2=0$$ which gives the answer $$x=-6$$ or $$x=-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic11a-h4","type":"hint","dependencies":["ad220b3GraphQuadratic11a-h3"],"title":"Put The Answer Into (x,y)","text":"We know that $$y=0$$ in x-intercepts and we get $$x=-6$$, $$-2$$ through the calculations above. Put together, we get two x-intercepts-- $$(-6,0),(-2,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ad220b3GraphQuadratic11b","stepAnswer":["(0,12)"],"problemType":"TextBox","stepTitle":"$$f(x)=x^2+8x+12$$","stepBody":"Find the y-intercept of the given parabola. Please enter your answer as \\"(x,y)\\" where $$x$$ is the x-coordinate of the intercept and $$y$$ is the y-coordinate of the intercept.","answerType":"string","variabilization":{},"answerLatex":"$$(0,12)$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic11b-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":[],"title":"Set f(0) As the y-coordinate","text":"We know that $$x=0$$ in the y-intercept. We can compute f(0) to find the $$y$$ coordinate if y-intercept. What is f(0)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic11b-h2","type":"hint","dependencies":["ad220b3GraphQuadratic11b-h1"],"title":"Put The Answer Into (x,y)","text":"$$f(0)=0^2+8\\\\times0+12=12$$. We get $$x=0$$ and $$y=12$$ for y-intercept. Put together, we have $$(0,12)$$ as y-intercept.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad220b3GraphQuadratic12","title":"Find the Intercepts of a Parabola","body":"In the following exercises, find the intercepts of the parabola whose function is given.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Graph Quadratic Functions Using Properties","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad220b3GraphQuadratic12a","stepAnswer":["(-3,0),(-2,0)"],"problemType":"TextBox","stepTitle":"$$f(x)=x^2+5x+6$$","stepBody":"Find the $$x-intercept(s)$$ of the given parabola. Please enter your answer as \\"(x,y)\\" where $$x$$ is the x-coordinate of the intercept and $$y$$ is the y-coordinate of the intercept. If there are more than one x-intercept, you can enter it as \\"(x1,y1),(x2,y2)\\" where x1 is small than x2.","answerType":"string","variabilization":{},"answerLatex":"$$(-3,0),(-2,0)$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic12a-h1","type":"hint","dependencies":[],"title":"Set $$f(x)=0$$","text":"To find the x-intercept, let $$f(x)=0$$ and and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic12a-h2","type":"hint","dependencies":["ad220b3GraphQuadratic12a-h1"],"title":"Solve For The Parabola","text":"Solve for $$f(x)=x^2+5x+6=0$$. We can factor it as $$\\\\left(x+3\\\\right) \\\\left(x+2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic12a-h3","type":"hint","dependencies":["ad220b3GraphQuadratic12a-h2"],"title":"Solve For The Parabola","text":"$$\\\\left(x+3\\\\right) \\\\left(x+2\\\\right)=0$$\\\\nUse zero product property, we know $$x+3=0$$ or $$x+2=0$$ which gives the answer $$x=-3$$ or $$x=-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic12a-h4","type":"hint","dependencies":["ad220b3GraphQuadratic12a-h3"],"title":"Put The Answer Into (x,y)","text":"We know that $$y=0$$ in x-intercepts and we get $$x=-3$$, $$-2$$ through the calculations above. Put together, we get two x-intercepts-- $$(-3,0)$$ and $$(-2,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ad220b3GraphQuadratic12b","stepAnswer":["(0,6)"],"problemType":"TextBox","stepTitle":"$$f(x)=x^2+5x+6$$","stepBody":"Find the y-intercept of the given parabola. Please enter your answer as \\"(x,y)\\" where $$x$$ is the x-coordinate of the intercept and $$y$$ is the y-coordinate of the intercept.","answerType":"string","variabilization":{},"answerLatex":"$$(0,6)$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic12b-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["ad220b3GraphQuadratic12a-h2"],"title":"Set f(0) As the y-coordinate","text":"We know that $$x=0$$ in the y-intercept. We can compute f(0) to find the $$y$$ coordinate if y-intercept. What is f(0)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic12b-h2","type":"hint","dependencies":["ad220b3GraphQuadratic12a-h3"],"title":"Put The Answer Into (x,y)","text":"$$f(0)=0^2+5\\\\times0+6=6$$. We get $$x=0$$ and $$y=6$$ for y-intercept. Put together, we have $$(0,6)$$ as y-intercept.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad220b3GraphQuadratic13","title":"Find the Intercepts of a Parabola","body":"In the following exercises, find the intercepts of the parabola whose function is given.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Graph Quadratic Functions Using Properties","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad220b3GraphQuadratic13a","stepAnswer":["N"],"problemType":"TextBox","stepTitle":"$$f(x)=-\\\\left(x^2\\\\right)+8x-19$$","stepBody":"Find the $$x-intercept(s)$$ of the given parabola. Please enter your answer as \\"(x,y)\\" where $$x$$ is the x-coordinate of the intercept and $$y$$ is the y-coordinate of the intercept. If there are more than one x-intercept, you can enter it as \\"(x1,y1),(x2,y2)\\" where x1 is small than x2. If there is no x-intercept for this function, please enter \\"N\\".","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic13a-h1","type":"hint","dependencies":[],"title":"Set $$f(x)=0$$","text":"$$f(x)=-\\\\left(x^2\\\\right)+8x-19=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic13a-h2","type":"hint","dependencies":["ad220b3GraphQuadratic13a-h1"],"title":"Check for Discriminant","text":"For any given parabolar $${ax}^2+bx+c$$, the discriminant is $$b^2-4ac$$. If the discriminant is greater than $$0$$, there are two x-intercepts for the parabola. If the discriminant equals zero, there is only one x-intercept for the parabola. If the discriminant is less than zero, there is no x-intercept for the parabola.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic13a-h3","type":"hint","dependencies":["ad220b3GraphQuadratic13a-h2"],"title":"Check for Discriminant","text":"In this given parabola, $$a=-1$$, $$b=8$$, and $$c=-19$$. $$b^2-4ac=8^2-4\\\\left(-1\\\\right) \\\\left(-19\\\\right)=-12$$ which is less than zero, so there is no x-intercept for this parabola.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ad220b3GraphQuadratic13b","stepAnswer":["(0,-19)"],"problemType":"TextBox","stepTitle":"$$f(x)=-\\\\left(x^2\\\\right)+8x-19$$","stepBody":"Find the y-intercept of the given parabola. Please enter your answer as \\"(x,y)\\" where $$x$$ is the x-coordinate of the intercept and $$y$$ is the y-coordinate of the intercept.","answerType":"string","variabilization":{},"answerLatex":"$$(0,-19)$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic13b-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-19$$"],"dependencies":["ad220b3GraphQuadratic13a-h2"],"title":"Set f(0) As the y-coordinate","text":"We know that $$x=0$$ in the y-intercept. We can compute f(0) to find the $$y$$ coordinate if y-intercept. What is f(0)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic13b-h2","type":"hint","dependencies":["ad220b3GraphQuadratic13a-h3"],"title":"Put The Answer Into (x,y)","text":"$$f(0)=-\\\\left(0^2\\\\right)+8\\\\times0-19=-19$$. We get $$x=0$$ and $$y=-19$$ for y-intercept. Put together, we have $$(0,-19)$$ as y-intercept.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad220b3GraphQuadratic14","title":"Find the Intercepts of a Parabola","body":"In the following exercises, find the intercepts of the parabola whose function is given.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Graph Quadratic Functions Using Properties","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad220b3GraphQuadratic14a","stepAnswer":["N"],"problemType":"TextBox","stepTitle":"$$f(x)=-3x^2+x-1$$","stepBody":"Find the $$x-intercept(s)$$ of the given parabola. Please enter your answer as \\"(x,y)\\" where $$x$$ is the x-coordinate of the intercept and $$y$$ is the y-coordinate of the intercept. If there are more than one x-intercept, you can enter it as \\"(x1,y1),(x2,y2)\\" where x1 is small than x2. If there is no x-intercept for this function, please enter \\"N\\".","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic14a-h1","type":"hint","dependencies":[],"title":"Set $$f(x)=0$$","text":"$$f(x)=-3x^2+x-1=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic14a-h2","type":"hint","dependencies":["ad220b3GraphQuadratic14a-h1"],"title":"Check for Discriminant","text":"For any given parabolar $${ax}^2+bx+c$$, the discriminant is $$b^2-4ac$$. If the discriminant is greater than $$0$$, there are two x-intercepts for the parabola. If the discriminant equals zero, there is only one x-intercept for the parabola. If the discriminant is less than zero, there is no x-intercept for the parabola.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic14a-h3","type":"hint","dependencies":["ad220b3GraphQuadratic14a-h2"],"title":"Check for Discriminant","text":"In this given parabola, $$a=-3$$, $$b=1$$, and $$c=-1$$. $$b^2-4ac=1^2-4\\\\left(-3\\\\right) \\\\left(-1\\\\right)=-11$$ which is less than zero, so there is no x-intercept for this parabola.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ad220b3GraphQuadratic14b","stepAnswer":["(0,-1)"],"problemType":"TextBox","stepTitle":"$$f(x)=-3x^2+x-1$$","stepBody":"Find the y-intercept of the given parabola. Please enter your answer as \\"(x,y)\\" where $$x$$ is the x-coordinate of the intercept and $$y$$ is the y-coordinate of the intercept.","answerType":"string","variabilization":{},"answerLatex":"$$(0,-1)$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic14b-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["ad220b3GraphQuadratic14a-h2"],"title":"Set f(0) As the y-coordinate","text":"We know that $$x=0$$ in the y-intercept. We can compute f(0) to find the $$y$$ coordinate if y-intercept. What is f(0)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic14b-h2","type":"hint","dependencies":["ad220b3GraphQuadratic14a-h3"],"title":"Put The Answer Into (x,y)","text":"$$f(0)=-3\\\\times0^2+0-1=-1$$. We get $$x=0$$ and $$y=-1$$ for y-intercept. Put together, we have $$(0,-1)$$ as y-intercept.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad220b3GraphQuadratic15","title":"Find the Intercepts of a Parabola","body":"In the following exercises, find the intercepts of the parabola whose function is given.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Graph Quadratic Functions Using Properties","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad220b3GraphQuadratic15a","stepAnswer":["N"],"problemType":"TextBox","stepTitle":"$$f(x)=x^2+6x+13$$","stepBody":"Find the $$x-intercept(s)$$ of the given parabola. Please enter your answer as \\"(x,y)\\" where $$x$$ is the x-coordinate of the intercept and $$y$$ is the y-coordinate of the intercept. If there are more than one x-intercept, you can enter it as \\"(x1,y1),(x2,y2)\\" where x1 is small than x2. If there is no x-intercept for this function, please enter \\"N\\".","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic15a-h1","type":"hint","dependencies":[],"title":"Set $$f(x)=0$$","text":"$$f(x)=x^2+6x+13=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic15a-h2","type":"hint","dependencies":["ad220b3GraphQuadratic15a-h1"],"title":"Check for Discriminant","text":"For any given parabolar $${ax}^2+bx+c$$, the discriminant is $$b^2-4ac$$. If the discriminant is greater than $$0$$, there are two x-intercepts for the parabola. If the discriminant equals zero, there is only one x-intercept for the parabola. If the discriminant is less than zero, there is no x-intercept for the parabola.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic15a-h3","type":"hint","dependencies":["ad220b3GraphQuadratic15a-h2"],"title":"Check for Discriminant","text":"In this given parabola, $$a=1$$, $$b=6$$, and $$c=13$$. $$b^2-4ac=6^2-4\\\\times1\\\\times13=-16$$ which is less than zero, so there is no x-intercept for this parabola.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ad220b3GraphQuadratic15b","stepAnswer":["(0,13)"],"problemType":"TextBox","stepTitle":"$$f(x)=x^2+6x+13$$","stepBody":"Find the y-intercept of the given parabola. Please enter your answer as \\"(x,y)\\" where $$x$$ is the x-coordinate of the intercept and $$y$$ is the y-coordinate of the intercept.","answerType":"string","variabilization":{},"answerLatex":"$$(0,13)$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic15b-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$13$$"],"dependencies":["ad220b3GraphQuadratic15a-h2"],"title":"Set f(0) As the y-coordinate","text":"We know that $$x=0$$ in the y-intercept. We can compute f(0) to find the $$y$$ coordinate if y-intercept. What is f(0)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic15b-h2","type":"hint","dependencies":["ad220b3GraphQuadratic15a-h3"],"title":"Put The Answer Into (x,y)","text":"$$f(0)=0^2+6\\\\times0+13=13$$. We get $$x=0$$ and $$y=13$$ for y-intercept. Put together, we have $$(0,13)$$ as y-intercept.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad220b3GraphQuadratic16","title":"Find the Intercepts of a Parabola","body":"In the following exercises, find the intercepts of the parabola whose function is given.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Graph Quadratic Functions Using Properties","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad220b3GraphQuadratic16a","stepAnswer":["(2.5,0)"],"problemType":"TextBox","stepTitle":"$$f(x)=4x^2-20x+25$$","stepBody":"Find the $$x-intercept(s)$$ of the given parabola. Please enter your answer as \\"(x,y)\\" where $$x$$ is the x-coordinate of the intercept and $$y$$ is the y-coordinate of the intercept. Please convert $$x$$ and $$y$$ value into decimal form. If there are more than one x-intercept, you can enter it as \\"(x1,y1),(x2,y2)\\" where x1 is small than x2. If there is no x-intercept for this function, please enter \\"N\\".","answerType":"string","variabilization":{},"answerLatex":"$$(2.5, 0)$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic16a-h1","type":"hint","dependencies":[],"title":"Set $$f(x)=0$$","text":"To find the x-intercept, let $$f(x)=0$$ and and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic16a-h2","type":"hint","dependencies":["ad220b3GraphQuadratic16a-h1"],"title":"Solve For The Parabola","text":"Given $$a x^2+bx+c=0$$. We can use quadratic formula to get $$x=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. In the given parabola, $$a=4$$, $$b=-20$$ and $$c=25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic16a-h3","type":"hint","dependencies":["ad220b3GraphQuadratic16a-h2"],"title":"Solve For The Parabola","text":"Through apply the quadratic formula, we get the solution for $$f(x)=4x^2-20x+25=0$$ is $$x=2.5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic16a-h4","type":"hint","dependencies":["ad220b3GraphQuadratic16a-h3"],"title":"Put The Answer Into (x,y)","text":"We know that $$y=0$$ in x-intercepts and we get $$x=-2.5$$ through the calculations above. Put together, we get one x-intercept $$(2.5, 0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ad220b3GraphQuadratic16b","stepAnswer":["(0,25)"],"problemType":"TextBox","stepTitle":"$$f(x)=4x^2-20x+25$$","stepBody":"Find the y-intercept of the given parabola. Please enter your answer as \\"(x,y)\\" where $$x$$ is the x-coordinate of the intercept and $$y$$ is the y-coordinate of the intercept.","answerType":"string","variabilization":{},"answerLatex":"$$(0,25)$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic16b-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":[],"title":"Set f(0) As the y-coordinate","text":"We know that $$x=0$$ in the y-intercept. We can compute f(0) to find the $$y$$ coordinate if y-intercept. What is f(0)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic16b-h2","type":"hint","dependencies":["ad220b3GraphQuadratic16b-h1"],"title":"Put The Answer Into (x,y)","text":"$$f(0)=4\\\\times0^2-20\\\\times0+25=25$$. We get $$x=0$$ and $$y=25$$ for y-intercept. Put together, we have $$(0,25)$$ as y-intercept.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad220b3GraphQuadratic17","title":"Find the Intercepts of a Parabola","body":"In the following exercises, find the intercepts of the parabola whose function is given.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Graph Quadratic Functions Using Properties","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad220b3GraphQuadratic17a","stepAnswer":["(-7,0)"],"problemType":"TextBox","stepTitle":"$$f(x)=-\\\\left(x^2\\\\right)-14x-49$$","stepBody":"Find the $$x-intercept(s)$$ of the given parabola. Please enter your answer as \\"(x,y)\\" where $$x$$ is the x-coordinate of the intercept and $$y$$ is the y-coordinate of the intercept. Please convert $$x$$ and $$y$$ value into decimal form. If there are more than one x-intercept, you can enter it as \\"(x1,y1),(x2,y2)\\" where x1 is small than x2. If there is no x-intercept for this function, please enter \\"N\\".","answerType":"string","variabilization":{},"answerLatex":"$$(-7,0)$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic17a-h1","type":"hint","dependencies":[],"title":"Set $$f(x)=0$$","text":"To find the x-intercept, let $$f(x)=0$$ and and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic17a-h2","type":"hint","dependencies":["ad220b3GraphQuadratic17a-h1"],"title":"Solve For The Parabola","text":"Given $$a x^2+bx+c=0$$. We can use quadratic formula to get $$x=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. In the given parabola, $$a=-1$$, $$b=-14$$ and $$c=-49$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic17a-h3","type":"hint","dependencies":["ad220b3GraphQuadratic17a-h2"],"title":"Solve For The Parabola","text":"Through apply the quadratic formula, we get the solution for $$f(x)=-\\\\left(x^2\\\\right)-14x-49=0$$ is $$x=-7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic17a-h4","type":"hint","dependencies":["ad220b3GraphQuadratic17a-h3"],"title":"Put The Answer Into (x,y)","text":"We know that $$y=0$$ in x-intercepts and we get $$x=-7$$ through the calculations above. Put together, we get one x-intercept $$(-7,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ad220b3GraphQuadratic17b","stepAnswer":["(0,-49)"],"problemType":"TextBox","stepTitle":"$$f(x)=-\\\\left(x^2\\\\right)-14x-49$$","stepBody":"Find the y-intercept of the given parabola. Please enter your answer as \\"(x,y)\\" where $$x$$ is the x-coordinate of the intercept and $$y$$ is the y-coordinate of the intercept.","answerType":"string","variabilization":{},"answerLatex":"$$(0,-49)$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic17b-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-49$$"],"dependencies":[],"title":"Set f(0) As the y-coordinate","text":"We know that $$x=0$$ in the y-intercept. We can compute f(0) to find the $$y$$ coordinate if y-intercept. What is f(0)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic17b-h2","type":"hint","dependencies":["ad220b3GraphQuadratic17b-h1"],"title":"Put The Answer Into (x,y)","text":"$$f(0)=-1\\\\times0^2-14\\\\times0-49=-49$$. We get $$x=0$$ and $$y=-49$$ for y-intercept. Put together, we have $$(0,-49)$$ as y-intercept.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad220b3GraphQuadratic18","title":"Find the Intercepts of a Parabola","body":"In the following exercises, find the intercepts of the parabola whose function is given.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Graph Quadratic Functions Using Properties","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad220b3GraphQuadratic18a","stepAnswer":["(-3,0)"],"problemType":"TextBox","stepTitle":"$$f(x)=-\\\\left(x^2\\\\right)-6x-9$$","stepBody":"Find the $$x-intercept(s)$$ of the given parabola. Please enter your answer as \\"(x,y)\\" where $$x$$ is the x-coordinate of the intercept and $$y$$ is the y-coordinate of the intercept. Please convert $$x$$ and $$y$$ value into decimal form. If there are more than one x-intercept, you can enter it as \\"(x1,y1),(x2,y2)\\" where x1 is small than x2. If there is no x-intercept for this function, please enter \\"N\\".","answerType":"string","variabilization":{},"answerLatex":"$$(-3,0)$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic18a-h1","type":"hint","dependencies":[],"title":"Set $$f(x)=0$$","text":"To find the x-intercept, let $$f(x)=0$$ and and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic18a-h2","type":"hint","dependencies":["ad220b3GraphQuadratic18a-h1"],"title":"Solve For The Parabola","text":"Given $$a x^2+bx+c=0$$. We can use quadratic formula to get $$x=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. In the given parabola, $$a=-1$$, $$b=-6$$ and $$c=-9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic18a-h3","type":"hint","dependencies":["ad220b3GraphQuadratic18a-h2"],"title":"Solve For The Parabola","text":"Through apply the quadratic formula, we get the solution for $$f(x)=-\\\\left(x^2\\\\right)-6x-9=0$$ is $$x=-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic18a-h4","type":"hint","dependencies":["ad220b3GraphQuadratic18a-h3"],"title":"Put The Answer Into (x,y)","text":"We know that $$y=0$$ in x-intercepts and we get $$x=-3$$ through the calculations above. Put together, we get one x-intercept $$(-3,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ad220b3GraphQuadratic18b","stepAnswer":["(0,-9)"],"problemType":"TextBox","stepTitle":"$$f(x)=-\\\\left(x^2\\\\right)-14x-49$$","stepBody":"Find the y-intercept of the given parabola. Please enter your answer as \\"(x,y)\\" where $$x$$ is the x-coordinate of the intercept and $$y$$ is the y-coordinate of the intercept.","answerType":"string","variabilization":{},"answerLatex":"$$(0,-9)$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic18b-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":[],"title":"Set f(0) As the y-coordinate","text":"We know that $$x=0$$ in the y-intercept. We can compute f(0) to find the $$y$$ coordinate if y-intercept. What is f(0)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic18b-h2","type":"hint","dependencies":["ad220b3GraphQuadratic18b-h1"],"title":"Put The Answer Into (x,y)","text":"$$f(0)=-1\\\\times0^2-6\\\\times0-9=-9$$. We get $$x=0$$ and $$y=-9$$ for y-intercept. Put together, we have $$(0,-9)$$ as y-intercept.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad220b3GraphQuadratic19","title":"Find the Intercepts of a Parabola","body":"In the following exercises, find the intercepts of the parabola whose function is given.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Graph Quadratic Functions Using Properties","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad220b3GraphQuadratic19a","stepAnswer":["(-0.5,0)"],"problemType":"TextBox","stepTitle":"$$f(x)=4x^2+4x+1$$","stepBody":"Find the $$x-intercept(s)$$ of the given parabola. Please enter your answer as \\"(x,y)\\" where $$x$$ is the x-coordinate of the intercept and $$y$$ is the y-coordinate of the intercept. Please convert $$x$$ and $$y$$ value into decimal form. If there are more than one x-intercept, you can enter it as \\"(x1,y1),(x2,y2)\\" where x1 is small than x2. If there is no x-intercept for this function, please enter \\"N\\".","answerType":"string","variabilization":{},"answerLatex":"$$(-0.5, 0)$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic19a-h1","type":"hint","dependencies":[],"title":"Set $$f(x)=0$$","text":"To find the x-intercept, let $$f(x)=0$$ and and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic19a-h2","type":"hint","dependencies":["ad220b3GraphQuadratic19a-h1"],"title":"Solve For The Parabola","text":"Given $$a x^2+bx+c=0$$. We can use quadratic formula to get $$x=\\\\frac{\\\\left(-b\\\\pm \\\\sqrt{b^2-4ac}\\\\right)}{2} a$$. In the given parabola, $$a=4$$, $$b=4$$ and $$c=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic19a-h3","type":"hint","dependencies":["ad220b3GraphQuadratic19a-h2"],"title":"Solve For The Parabola","text":"Through apply the quadratic formula, we get the solution for $$f(x)=4x^2+4x+1=0$$ is $$x=-0.5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic19a-h4","type":"hint","dependencies":["ad220b3GraphQuadratic19a-h3"],"title":"Put The Answer Into (x,y)","text":"We know that $$y=0$$ in x-intercepts and we get $$x=-0.5$$ through the calculations above. Put together, we get one x-intercept $$(-0.5, 0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ad220b3GraphQuadratic19b","stepAnswer":["(0,1)"],"problemType":"TextBox","stepTitle":"$$f(x)=4x^2+4x+1$$","stepBody":"Find the y-intercept of the given parabola. Please enter your answer as \\"(x,y)\\" where $$x$$ is the x-coordinate of the intercept and $$y$$ is the y-coordinate of the intercept.","answerType":"string","variabilization":{},"answerLatex":"$$(0,1)$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic19b-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":[],"title":"Set f(0) As the y-coordinate","text":"We know that $$x=0$$ in the y-intercept. We can compute f(0) to find the $$y$$ coordinate if y-intercept. What is f(0)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic19b-h2","type":"hint","dependencies":["ad220b3GraphQuadratic19b-h1"],"title":"Put The Answer Into (x,y)","text":"$$f(0)=4\\\\times0^2+4\\\\times0+1=1$$. We get $$x=0$$ and $$y=1$$ for y-intercept. Put together, we have $$(0,1)$$ as y-intercept.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad220b3GraphQuadratic2","title":"Recognize the Graph of a Quadratic Function","body":"For each of the following exercises, determine if the parabola opens up or down.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Graph Quadratic Functions Using Properties","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad220b3GraphQuadratic2a","stepAnswer":["Up"],"problemType":"TextBox","stepTitle":"$$f(x)=4x^2+x-4$$","stepBody":"If the parabola opens up, please enter \\"Up\\". If the parabola opens down, please enter \\"Down\\".","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic2a-h1","type":"hint","dependencies":[],"title":"Check the Leading Coefficient","text":"Given any parabola in the form $${ax}^2+bx+c$$, a is called the leading coefficient of the parabola. If a is negative, the parabola opens downward. If a is positive, the parabola opens upward.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic2a-h2","type":"hint","dependencies":["ad220b3GraphQuadratic2a-h1"],"title":"Check the Leading Coefficient","text":"In the given question, $$a=4$$ which is positive, so the parabola opens upward.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ad220b3GraphQuadratic2b","stepAnswer":["Down"],"problemType":"TextBox","stepTitle":"$$f(x)=-9x^2-24x-16$$","stepBody":"If the parabola opens up, please enter \\"Up\\". If the parabola opens down, please enter \\"Down\\".","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic2b-h1","type":"hint","dependencies":[],"title":"Check the Leading Coefficient","text":"Given any parabola in the form $${ax}^2+bx+c$$, a is called the leading coefficient of the parabola. If a is negative, the parabola opens downward. If a is positive, the parabola opens upward.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic2b-h2","type":"hint","dependencies":["ad220b3GraphQuadratic2b-h1"],"title":"Check the Leading Coefficient","text":"In the given question, $$a=-9$$ which is positive, so the parabola opens downward.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad220b3GraphQuadratic20","title":"Solve Maximum and Minimum Applications","body":"In the following exercises, find the maximum or minimum value of each function.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Graph Quadratic Functions Using Properties","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad220b3GraphQuadratic20a","stepAnswer":["(-0.25,-1.125)"],"problemType":"TextBox","stepTitle":"$$f(x)=2x^2+x-1$$","stepBody":"Find the minimum or maximumof the given parabola. Please enter you answer as \\"(x1,y1)\\" where (x,y) is the minimum or maximum point of the function. Please convert $$x$$ and $$y$$ values into decimal form.","answerType":"string","variabilization":{},"answerLatex":"$$(-0.25, -1.125)$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic20a-h1","type":"hint","dependencies":[],"title":"Decide the Shape of Parabola","text":"Given any parabola in the form $${ax}^2+bx+c$$, a is called the leading coefficient of the parabola. If a is negative, the parabola opens downward. If a is positive, the parabola opens upward.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic20a-h2","type":"hint","dependencies":["ad220b3GraphQuadratic20a-h1"],"title":"Decide the Shape of Parabola","text":"Since a is positive, the parabola opens upward. So the quadratic equation has a minimum.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic20a-h3","type":"hint","dependencies":["ad220b3GraphQuadratic20a-h2"],"title":"Find the Equation For Axis of Symmetry","text":"Given $${ax}^2+bx+c$$, the axis of symmetry for the parabola is $$x=\\\\frac{-b}{2a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic20a-h4","type":"hint","dependencies":["ad220b3GraphQuadratic20a-h3"],"title":"Find the Equation For Axis of Symmetry","text":"In the given parabola, $$a=2$$, $$b=1$$. The axis of symmetry is $$x=\\\\frac{-1}{4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic20a-h5","type":"hint","dependencies":["ad220b3GraphQuadratic20a-h4"],"title":"Find the $$y$$ coordinate of Vertex","text":"The maximum or minimum point of the parabola is located on the axis of symmetry. Given that $$x=a$$ is the axis of symmetry, we can find the $$y$$ coordinate of the maximum or minimum point by computing f(a).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic20a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1.125$$"],"dependencies":["ad220b3GraphQuadratic20a-h5"],"title":"Find the $$y$$ coordinate of Vertex","text":"Subsitute $$x=\\\\frac{-1}{4}$$ into $$f(x)=2x^2+x-1$$. $$f(x)=$$? Please enter your answer in decimal form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic20a-h7","type":"hint","dependencies":["ad220b3GraphQuadratic20a-h6"],"title":"Put The Answer Into (x,y)","text":"We know the maximum point of the parabola exists at $$x=-0.25$$ and $$f(-0.25)=-1.125$$. So the minimum point in xy-coordinate is $$(-0.25, -1.125)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad220b3GraphQuadratic21","title":"Solve Maximum and Minimum Applications","body":"In the following exercises, find the maximum or minimum value of each function.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Graph Quadratic Functions Using Properties","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad220b3GraphQuadratic21a","stepAnswer":["(1.5,4)"],"problemType":"TextBox","stepTitle":"$$f(x)=-4x^2+12x-5$$","stepBody":"Find the minimum or maximumof the given parabola. Please enter you answer as \\"(x1,y1)\\" where (x,y) is the minimum or maximum point of the function. Please convert $$x$$ and $$y$$ values into decimal form.","answerType":"string","variabilization":{},"answerLatex":"$$(1.5, 4)$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic21a-h1","type":"hint","dependencies":[],"title":"Decide the Shape of Parabola","text":"Given any parabola in the form $${ax}^2+bx+c$$, a is called the leading coefficient of the parabola. If a is negative, the parabola opens downward. If a is positive, the parabola opens upward.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic21a-h2","type":"hint","dependencies":["ad220b3GraphQuadratic21a-h1"],"title":"Decide the Shape of Parabola","text":"Since a is negative, the parabola opens downward. So the quadratic equation has a maximum.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic21a-h3","type":"hint","dependencies":["ad220b3GraphQuadratic21a-h2"],"title":"Find the Equation For Axis of Symmetry","text":"Given $${ax}^2+bx+c$$, the axis of symmetry for the parabola is $$x=\\\\frac{-b}{2a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic21a-h4","type":"hint","dependencies":["ad220b3GraphQuadratic21a-h3"],"title":"Find the Equation For Axis of Symmetry","text":"In the given parabola, $$a=-4$$, $$b=12$$. The axis of symmetry is $$x=1.5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic21a-h5","type":"hint","dependencies":["ad220b3GraphQuadratic21a-h4"],"title":"Find the $$y$$ coordinate of Vertex","text":"The maximum or minimum point of the parabola is located on the axis of symmetry. Given that $$x=a$$ is the axis of symmetry, we can find the $$y$$ coordinate of the maximum or minimum point by computing f(a).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic21a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ad220b3GraphQuadratic21a-h5"],"title":"Find the $$y$$ coordinate of Vertex","text":"Subsitute $$x=-1.5$$ into $$f(x)=-4x^2+12x-5$$. $$f(x)=$$? Please enter your answer in decimal form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic21a-h7","type":"hint","dependencies":["ad220b3GraphQuadratic21a-h6"],"title":"Put The Answer Into (x,y)","text":"We know the minimum point of the parabola exists at $$x=1.5$$ and $$f(1.5)=4$$. So the minimum point in xy-coordinate is $$(1.5, 4)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad220b3GraphQuadratic22","title":"Solve Maximum and Minimum Applications","body":"In the following exercises, find the maximum or minimum value of each function.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Graph Quadratic Functions Using Properties","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad220b3GraphQuadratic22a","stepAnswer":["(3,6)"],"problemType":"TextBox","stepTitle":"$$f(x)=x^2-6x+15$$","stepBody":"Find the minimum or maximumof the given parabola. Please enter you answer as \\"(x1,y1)\\" where (x,y) is the minimum or maximum point of the function. Please convert $$x$$ and $$y$$ values into decimal form.","answerType":"string","variabilization":{},"answerLatex":"$$(3,6)$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic22a-h1","type":"hint","dependencies":[],"title":"Decide the Shape of Parabola","text":"Given any parabola in the form $${ax}^2+bx+c$$, a is called the leading coefficient of the parabola. If a is negative, the parabola opens downward. If a is positive, the parabola opens upward.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic22a-h2","type":"hint","dependencies":["ad220b3GraphQuadratic22a-h1"],"title":"Decide the Shape of Parabola","text":"Since a is positive, the parabola opens upward. So the quadratic equation has a minimum.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic22a-h3","type":"hint","dependencies":["ad220b3GraphQuadratic22a-h2"],"title":"Find the Equation For Axis of Symmetry","text":"Given $${ax}^2+bx+c$$, the axis of symmetry for the parabola is $$x=\\\\frac{-b}{2a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic22a-h4","type":"hint","dependencies":["ad220b3GraphQuadratic22a-h3"],"title":"Find the Equation For Axis of Symmetry","text":"In the given parabola, $$a=1$$, $$b=-6$$. The axis of symmetry is $$x=3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic22a-h5","type":"hint","dependencies":["ad220b3GraphQuadratic22a-h4"],"title":"Find the $$y$$ coordinate of Vertex","text":"The maximum or minimum point of the parabola is located on the axis of symmetry. Given that $$x=a$$ is the axis of symmetry, we can find the $$y$$ coordinate of the maximum or minimum point by computing f(a).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic22a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["ad220b3GraphQuadratic22a-h5"],"title":"Find the $$y$$ coordinate of Vertex","text":"Subsitute $$x=3$$ into $$f(x)=x^2-6x+15$$. $$f(x)=$$? Please enter your answer in decimal form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic22a-h7","type":"hint","dependencies":["ad220b3GraphQuadratic22a-h6"],"title":"Put The Answer Into (x,y)","text":"We know the maximum point of the parabola exists at $$x=3$$ and $$f(3)=6$$. So the minimum point in xy-coordinate is $$(3,6)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad220b3GraphQuadratic23","title":"Solve Maximum and Minimum Applications","body":"In the following exercises, find the maximum or minimum value of each function.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Graph Quadratic Functions Using Properties","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad220b3GraphQuadratic23a","stepAnswer":["(2,-1)"],"problemType":"TextBox","stepTitle":"$$f(x)=-\\\\left(x^2\\\\right)+4x-5$$","stepBody":"Find the minimum or maximumof the given parabola. Please enter you answer as \\"(x1,y1)\\" where (x,y) is the minimum or maximum point of the function. Please convert $$x$$ and $$y$$ values into decimal form.","answerType":"string","variabilization":{},"answerLatex":"$$(2,-1)$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic23a-h1","type":"hint","dependencies":[],"title":"Decide the Shape of Parabola","text":"Given any parabola in the form $${ax}^2+bx+c$$, a is called the leading coefficient of the parabola. If a is negative, the parabola opens downward. If a is positive, the parabola opens upward.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic23a-h2","type":"hint","dependencies":["ad220b3GraphQuadratic23a-h1"],"title":"Decide the Shape of Parabola","text":"Since a is negative, the parabola opens downward. So the quadratic equation has a maximum.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic23a-h3","type":"hint","dependencies":["ad220b3GraphQuadratic23a-h2"],"title":"Find the Equation For Axis of Symmetry","text":"Given $${ax}^2+bx+c$$, the axis of symmetry for the parabola is $$x=\\\\frac{-b}{2a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic23a-h4","type":"hint","dependencies":["ad220b3GraphQuadratic23a-h3"],"title":"Find the Equation For Axis of Symmetry","text":"In the given parabola, $$a=-1$$, $$b=4$$. The axis of symmetry is $$x=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic23a-h5","type":"hint","dependencies":["ad220b3GraphQuadratic23a-h4"],"title":"Find the $$y$$ coordinate of Vertex","text":"The maximum or minimum point of the parabola is located on the axis of symmetry. Given that $$x=a$$ is the axis of symmetry, we can find the $$y$$ coordinate of the maximum or minimum point by computing f(a).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic23a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["ad220b3GraphQuadratic23a-h5"],"title":"Find the $$y$$ coordinate of Vertex","text":"Subsitute $$x=2$$ into $$f(x)=-\\\\left(x^2\\\\right)+4x-5$$. $$f(x)=$$? Please enter your answer in decimal form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic23a-h7","type":"hint","dependencies":["ad220b3GraphQuadratic23a-h6"],"title":"Put The Answer Into (x,y)","text":"We know the minimum point of the parabola exists at $$x=2$$ and $$f(2)=-1$$. So the minimum point in xy-coordinate is $$(2,-1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad220b3GraphQuadratic24","title":"Solve Maximum and Minimum Applications","body":"In the following exercises, find the maximum or minimum value of each function.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Graph Quadratic Functions Using Properties","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad220b3GraphQuadratic24a","stepAnswer":["(0,16)"],"problemType":"TextBox","stepTitle":"$$f(x)=-9x^2+16$$","stepBody":"Find the minimum or maximumof the given parabola. Please enter you answer as \\"(x1,y1)\\" where (x,y) is the minimum or maximum point of the function. Please convert $$x$$ and $$y$$ values into decimal form.","answerType":"string","variabilization":{},"answerLatex":"$$(0,16)$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic24a-h1","type":"hint","dependencies":[],"title":"Decide the Shape of Parabola","text":"Given any parabola in the form $${ax}^2+bx+c$$, a is called the leading coefficient of the parabola. If a is negative, the parabola opens downward. If a is positive, the parabola opens upward.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic24a-h2","type":"hint","dependencies":["ad220b3GraphQuadratic24a-h1"],"title":"Decide the Shape of Parabola","text":"Since a is negative, the parabola opens downward. So the quadratic equation has a maximum.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic24a-h3","type":"hint","dependencies":["ad220b3GraphQuadratic24a-h2"],"title":"Find the Equation For Axis of Symmetry","text":"Given $${ax}^2+bx+c$$, the axis of symmetry for the parabola is $$x=\\\\frac{-b}{2a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic24a-h4","type":"hint","dependencies":["ad220b3GraphQuadratic24a-h3"],"title":"Find the Equation For Axis of Symmetry","text":"In the given parabola, $$a=-9$$, $$b=0$$. The axis of symmetry is $$x=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic24a-h5","type":"hint","dependencies":["ad220b3GraphQuadratic24a-h4"],"title":"Find the $$y$$ coordinate of Vertex","text":"The maximum or minimum point of the parabola is located on the axis of symmetry. Given that $$x=a$$ is the axis of symmetry, we can find the $$y$$ coordinate of the maximum or minimum point by computing f(a).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic24a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["ad220b3GraphQuadratic24a-h5"],"title":"Find the $$y$$ coordinate of Vertex","text":"Subsitute $$x=0$$ into $$f(x)=-9x^2+16$$. $$f(x)=$$? Please enter your answer in decimal form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic24a-h7","type":"hint","dependencies":["ad220b3GraphQuadratic24a-h6"],"title":"Put The Answer Into (x,y)","text":"We know the maximum point of the parabola exists at $$x=0$$ and $$f(0)=16$$. So the minimum point in xy-coordinate is $$(0,16)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad220b3GraphQuadratic25","title":"Solve Maximum and Minimum Applications","body":"In the following exercises, find the maximum or minimum value of each function.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Graph Quadratic Functions Using Properties","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad220b3GraphQuadratic25a","stepAnswer":["(0,-49)"],"problemType":"TextBox","stepTitle":"$$f(x)=4x^2-49$$","stepBody":"Find the minimum or maximumof the given parabola. Please enter you answer as \\"(x1,y1)\\" where (x,y) is the minimum or maximum point of the function. Please convert $$x$$ and $$y$$ values into decimal form.","answerType":"string","variabilization":{},"answerLatex":"$$(0,-49)$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic25a-h1","type":"hint","dependencies":[],"title":"Decide the Shape of Parabola","text":"Given any parabola in the form $${ax}^2+bx+c$$, a is called the leading coefficient of the parabola. If a is negative, the parabola opens downward. If a is positive, the parabola opens upward.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic25a-h2","type":"hint","dependencies":["ad220b3GraphQuadratic25a-h1"],"title":"Decide the Shape of Parabola","text":"Since a is positive, the parabola opens upward. So the quadratic equation has a minimum.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic25a-h3","type":"hint","dependencies":["ad220b3GraphQuadratic25a-h2"],"title":"Find the Equation For Axis of Symmetry","text":"Given $${ax}^2+bx+c$$, the axis of symmetry for the parabola is $$x=\\\\frac{-b}{2a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic25a-h4","type":"hint","dependencies":["ad220b3GraphQuadratic25a-h3"],"title":"Find the Equation For Axis of Symmetry","text":"In the given parabola, $$a=4$$, $$b=0$$. The axis of symmetry is $$x=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic25a-h5","type":"hint","dependencies":["ad220b3GraphQuadratic25a-h4"],"title":"Find the $$y$$ coordinate of Vertex","text":"The maximum or minimum point of the parabola is located on the axis of symmetry. Given that $$x=a$$ is the axis of symmetry, we can find the $$y$$ coordinate of the maximum or minimum point by computing f(a).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic25a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-49$$"],"dependencies":["ad220b3GraphQuadratic25a-h5"],"title":"Find the $$y$$ coordinate of Vertex","text":"Subsitute $$x=0$$ into $$f(x)=4x^2-49$$. $$f(x)=$$? Please enter your answer in decimal form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic25a-h7","type":"hint","dependencies":["ad220b3GraphQuadratic25a-h6"],"title":"Put The Answer Into (x,y)","text":"We know the minimum point of the parabola exists at $$x=0$$ and $$f(0)=-49$$. So the minimum point in xy-coordinate is $$(0,-49)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic25a-h2","type":"hint","dependencies":["ad220b3GraphQuadratic25a-h1"],"title":"Decide the Shape of Parabola","text":"Given any parabola in the form $${ax}^2+bx+c$$, a is called the leading coefficient of the parabola. If a is negative, the parabola opens downward. If a is positive, the parabola opens upward.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic25a-h3","type":"hint","dependencies":["ad220b3GraphQuadratic25a-h2"],"title":"Decide the Shape of Parabola","text":"Since a is negative, the parabola opens downward. So the quadratic equation has a maximum.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad220b3GraphQuadratic27","title":"In the following exercises, solve.","body":"Round answers to the nearest tenth.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Graph Quadratic Functions Using Properties","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad220b3GraphQuadratic27a","stepAnswer":["(-5,420)"],"problemType":"TextBox","stepTitle":"A stone is thrown vertically upward from a platform that is $$20$$ feet height at a rate of $$160$$ $$\\\\frac{ft}{sec}$$. Use the quadratic function $$h(t)=-16t^2+160t+20$$ to find how long it will take the stone to reach its maximum height, and then find the maximum height.","stepBody":"Enter your answer as (x,y) where $$x$$ is the time that arrow reaches its maximum height and $$y$$ is the maximum height.","answerType":"string","variabilization":{},"answerLatex":"$$(-5,420)$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic27a-h1","type":"hint","dependencies":[],"title":"Interpret the Meaning of h(t)","text":"The given h(t) describes the relationship between time (t) and height. At time $$t$$, the stone will be h(t) feet above the platform. So finding the maximum height and the time that the stone reach the maximum height is the same as finding the maximum point of the function h(t).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic27a-h2","type":"hint","dependencies":["ad220b3GraphQuadratic27a-h1"],"title":"Decide the Shape of Parabola","text":"Given any parabola in the form $${ax}^2+bx+c$$, a is called the leading coefficient of the parabola. If a is negative, the parabola opens downward. If a is positive, the parabola opens upward.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic27a-h3","type":"hint","dependencies":["ad220b3GraphQuadratic27a-h2"],"title":"Decide the Shape of Parabola","text":"Since a is negative, the parabola opens downward. So the quadratic equation has a maximum.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic27a-h4","type":"hint","dependencies":["ad220b3GraphQuadratic27a-h3"],"title":"Find the Equation For Axis of Symmetry","text":"Given $${ax}^2+bx+c$$, the axis of symmetry for the parabola is $$x=\\\\frac{-b}{2a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic27a-h5","type":"hint","dependencies":["ad220b3GraphQuadratic27a-h4"],"title":"Find the Equation For Axis of Symmetry","text":"In the given parabola, $$a=-16$$, $$b=160$$. The axis of symmetry is $$t=5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic27a-h6","type":"hint","dependencies":["ad220b3GraphQuadratic27a-h5"],"title":"Find the $$y$$ coordinate of Vertex","text":"The maximum or minimum point of the parabola is located on the axis of symmetry. Given that $$x=a$$ is the axis of symmetry, we can find the $$y$$ coordinate of the maximum or minimum point by computing f(a).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic27a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$420$$"],"dependencies":["ad220b3GraphQuadratic27a-h6"],"title":"Find the $$y$$ coordinate of Vertex","text":"Subsitute $$t=5$$ into $$h(t)=-16t^2+160t+20$$, $$h(5)=$$? Please enter your answer in decimal form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic27a-h8","type":"hint","dependencies":["ad220b3GraphQuadratic27a-h7"],"title":"Put The Answer Into (x,y)","text":"We know the maximum point of the parabola exists at $$t=5$$ and $$h(5)=420$$. $$t=5$$ is the time that the arrow reaches it maximum height of $$h(5)=420$$. So we have the final answer as $$(5,420)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad220b3GraphQuadratic28","title":"In the following exercises, solve.","body":"Round answers to the nearest tenth.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Graph Quadratic Functions Using Properties","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad220b3GraphQuadratic28a","stepAnswer":["(3.4,185.6)"],"problemType":"TextBox","stepTitle":"A ball is thrown vertically upward from the ground with an initial velocity of $$109$$ $$\\\\frac{ft}{sec}$$. Use the quadratic function $$h(t)=-16t^2+109t$$ to find how long it will take for the ball to reach its maximum height, and then find the maximum height.","stepBody":"Enter your answer as (x,y) where $$x$$ is the time that arrow reaches its maximum height and $$y$$ is the maximum height.","answerType":"string","variabilization":{},"answerLatex":"$$(3.4, 185.6)$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic28a-h1","type":"hint","dependencies":[],"title":"Interpret the Meaning of h(t)","text":"The given h(t) describes the relationship between time (t) and height. At time $$t$$, the ball will be h(t) feet above the ground. So finding the maximum height and the time that the ball reach the maximum height is the same as finding the maximum point of the function h(t).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic28a-h2","type":"hint","dependencies":["ad220b3GraphQuadratic28a-h1"],"title":"Decide the Shape of Parabola","text":"Given any parabola in the form $${ax}^2+bx+c$$, a is called the leading coefficient of the parabola. If a is negative, the parabola opens downward. If a is positive, the parabola opens upward.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic28a-h3","type":"hint","dependencies":["ad220b3GraphQuadratic28a-h2"],"title":"Decide the Shape of Parabola","text":"Since a is negative, the parabola opens downward. So the quadratic equation has a maximum.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic28a-h4","type":"hint","dependencies":["ad220b3GraphQuadratic28a-h3"],"title":"Find the Equation For Axis of Symmetry","text":"Given $${ax}^2+bx+c$$, the axis of symmetry for the parabola is $$x=\\\\frac{-b}{2a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic28a-h5","type":"hint","dependencies":["ad220b3GraphQuadratic28a-h4"],"title":"Find the Equation For Axis of Symmetry","text":"In the given parabola, $$a=-16$$, $$b=109$$. The axis of symmetry is $$t=3.406$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic28a-h6","type":"hint","dependencies":["ad220b3GraphQuadratic28a-h5"],"title":"Find the $$y$$ coordinate of Vertex","text":"The maximum or minimum point of the parabola is located on the axis of symmetry. Given that $$x=a$$ is the axis of symmetry, we can find the $$y$$ coordinate of the maximum or minimum point by computing f(a).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic28a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$185.6$$"],"dependencies":["ad220b3GraphQuadratic28a-h6"],"title":"Find the $$y$$ coordinate of Vertex","text":"Subsitute $$t=3.406$$ into $$h(t)=-16t^2+109t$$, $$h(3.406)=$$? Please enter your answer in decimal form and round it to nearest tenth.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic28a-h8","type":"hint","dependencies":["ad220b3GraphQuadratic28a-h7"],"title":"Put The Answer Into (x,y)","text":"We know the maximum point of the parabola exists at $$t=3.406$$ and $$h(3.406)=185.6$$. $$t=3.406$$ is the time that the arrow reaches it maximum height of $$h(3.406)=185.6$$. So we have the final answer as $$(3.4, 185.6)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad220b3GraphQuadratic29","title":"In the following exercises, solve.","body":"Round answers to the nearest tenth.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Graph Quadratic Functions Using Properties","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad220b3GraphQuadratic29a","stepAnswer":["(3.4,185.6)"],"problemType":"TextBox","stepTitle":"A ball is thrown vertically upward from the ground with an initial velocity of $$122$$ $$\\\\frac{ft}{sec}$$. Use the quadratic function $$h(t)=-16t^2+122t$$ to find how long it will take for the ball to reach its maximum height, and then find the maximum height.","stepBody":"Enter your answer as (x,y) where $$x$$ is the time that arrow reaches its maximum height and $$y$$ is the maximum height.","answerType":"string","variabilization":{},"answerLatex":"$$(3.4, 185.6)$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic29a-h1","type":"hint","dependencies":[],"title":"Interpret the Meaning of h(t)","text":"The given h(t) describes the relationship between time (t) and height. At time $$t$$, the ball will be h(t) feet above the ground. So finding the maximum height and the time that the ball reach the maximum height is the same as finding the maximum point of the function h(t).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic29a-h2","type":"hint","dependencies":["ad220b3GraphQuadratic29a-h1"],"title":"Decide the Shape of Parabola","text":"Given any parabola in the form $${ax}^2+bx+c$$, a is called the leading coefficient of the parabola. If a is negative, the parabola opens downward. If a is positive, the parabola opens upward.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic29a-h3","type":"hint","dependencies":["ad220b3GraphQuadratic29a-h2"],"title":"Decide the Shape of Parabola","text":"Since a is negative, the parabola opens downward. So the quadratic equation has a maximum.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic29a-h4","type":"hint","dependencies":["ad220b3GraphQuadratic29a-h3"],"title":"Find the Equation For Axis of Symmetry","text":"Given $${ax}^2+bx+c$$, the axis of symmetry for the parabola is $$x=\\\\frac{-b}{2a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic29a-h5","type":"hint","dependencies":["ad220b3GraphQuadratic29a-h4"],"title":"Find the Equation For Axis of Symmetry","text":"In the given parabola, $$a=-16$$, $$b=122$$. The axis of symmetry is $$t=3.813$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic29a-h6","type":"hint","dependencies":["ad220b3GraphQuadratic29a-h5"],"title":"Find the $$y$$ coordinate of Vertex","text":"The maximum or minimum point of the parabola is located on the axis of symmetry. Given that $$x=a$$ is the axis of symmetry, we can find the $$y$$ coordinate of the maximum or minimum point by computing f(a).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic29a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$232.6$$"],"dependencies":["ad220b3GraphQuadratic29a-h6"],"title":"Find the $$y$$ coordinate of Vertex","text":"Subsitute $$t=3.813$$ into $$h(t)=-16t^2+122t$$ , $$h(3.813)=$$? Please enter your answer in decimal form and round it to nearest tenth.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic29a-h8","type":"hint","dependencies":["ad220b3GraphQuadratic29a-h7"],"title":"Put The Answer Into (x,y)","text":"We know the maximum point of the parabola exists at $$t=3.813$$ and $$h(3.813)=232.6$$. $$t=3.813$$ is the time that the arrow reaches it maximum height of $$h(3.813)=232.6$$. We need to around the answer to nearest tenth. So we have the final answer as $$(3.8$$, $$232.6)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad220b3GraphQuadratic3","title":"Recognize the Graph of a Quadratic Function","body":"For each of the following exercises, determine if the parabola opens up or down.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Graph Quadratic Functions Using Properties","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad220b3GraphQuadratic3a","stepAnswer":["Down"],"problemType":"TextBox","stepTitle":"$$f(x)=-3x^2+5x-1$$","stepBody":"If the parabola opens up, please enter \\"Up\\". If the parabola opens down, please enter \\"Down\\".","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic3a-h1","type":"hint","dependencies":[],"title":"Check the Leading Coefficient","text":"Given any parabola in the form $${ax}^2+bx+c$$, a is called the leading coefficient of the parabola. If a is negative, the parabola opens downward. If a is positive, the parabola opens upward.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic3a-h2","type":"hint","dependencies":["ad220b3GraphQuadratic3a-h1"],"title":"Check the Leading Coefficient","text":"In the given question, $$a=-3$$ which is positive, so the parabola opens downward.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ad220b3GraphQuadratic3b","stepAnswer":["Up"],"problemType":"TextBox","stepTitle":"$$f(x)=2x^2-4x+5$$","stepBody":"If the parabola opens up, please enter \\"Up\\". If the parabola opens down, please enter \\"Down\\".","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic3b-h1","type":"hint","dependencies":[],"title":"Check the Leading Coefficient","text":"Given any parabola in the form $${ax}^2+bx+c$$, a is called the leading coefficient of the parabola. If a is negative, the parabola opens downward. If a is positive, the parabola opens upward.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic3b-h2","type":"hint","dependencies":["ad220b3GraphQuadratic3b-h1"],"title":"Check the Leading Coefficient","text":"In the given question, $$a=2$$ which is positive, so the parabola opens upward.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad220b3GraphQuadratic30","title":"In the following exercises, solve.","body":"Round answers to the nearest tenth.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Graph Quadratic Functions Using Properties","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad220b3GraphQuadratic30a","stepAnswer":["$$800$$"],"problemType":"TextBox","stepTitle":"A land owner is planning to build a fenced in rectangular patio behind his garage, using his garage as one of the \u201cwalls.\u201d He wants to maximize the area using $$80$$ feet of fencing. The quadratic function $$A(x)=x \\\\left(80-2x\\\\right)$$ gives the area of the patio, where $$x$$ is the width of one side. Find the maximum area of the patio.","stepBody":"Please only enter the number of maximum areas and do not include any unit in your answer.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$800$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic30a-h1","type":"hint","dependencies":[],"title":"Interpret the Meaning of A(x)","text":"We can find the maximum area of the patio by finding the maximum point of the function A(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic30a-h2","type":"hint","dependencies":["ad220b3GraphQuadratic30a-h1"],"title":"Put The Parabola Into Standard Form","text":"We need to foil out the function A(x) and put it into standard form before finding the maximum. $$A(x)=x \\\\left(80-2x\\\\right)=-2x^2+80x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic30a-h3","type":"hint","dependencies":["ad220b3GraphQuadratic30a-h2"],"title":"Decide the Shape of Parabola","text":"Given any parabola in the form $${ax}^2+bx+c$$, a is called the leading coefficient of the parabola. If a is negative, the parabola opens downward. If a is positive, the parabola opens upward.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic30a-h4","type":"hint","dependencies":["ad220b3GraphQuadratic30a-h3"],"title":"Decide the Shape of Parabola","text":"Since a is negative, the parabola opens downward. So the quadratic equation has a maximum.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic30a-h5","type":"hint","dependencies":["ad220b3GraphQuadratic30a-h4"],"title":"Find the Equation For Axis of Symmetry","text":"Given $${ax}^2+bx+c$$, the axis of symmetry for the parabola is $$x=\\\\frac{-b}{2a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic30a-h6","type":"hint","dependencies":["ad220b3GraphQuadratic30a-h5"],"title":"Find the Equation For Axis of Symmetry","text":"In the given parabola, $$a=-2$$, $$b=80$$. The axis of symmetry is $$x=20$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic30a-h7","type":"hint","dependencies":["ad220b3GraphQuadratic30a-h6"],"title":"Find the $$y$$ coordinate of Vertex","text":"The maximum or minimum point of the parabola is located on the axis of symmetry. Given that $$x=a$$ is the axis of symmetry, we can find the $$y$$ coordinate of the maximum or minimum point by computing f(a).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic30a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$800$$"],"dependencies":["ad220b3GraphQuadratic30a-h7"],"title":"Find the $$y$$ coordinate of Vertex","text":"Subsitute $$x=20$$ into $$A(x)=-2x^2+80x$$ , $$A(20)=$$? Please enter your answer in decimal form and round it to nearest tenth.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic30a-h9","type":"hint","dependencies":["ad220b3GraphQuadratic30a-h8"],"title":"Put The Answer Into (x,y)","text":"We know the maximum point of the parabola exists at $$x=20$$ and $$A(20)=800$$. So the maximum area of the patio is $$800$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad220b3GraphQuadratic4","title":"Recognize the Graph of a Quadratic Function","body":"For each of the following exercises, determine if the parabola opens up or down.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Graph Quadratic Functions Using Properties","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad220b3GraphQuadratic4a","stepAnswer":["Up"],"problemType":"TextBox","stepTitle":"$$f(x)=x^2+3x-4$$","stepBody":"If the parabola opens up, please enter \\"Up\\". If the parabola opens down, please enter \\"Down\\".","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic4a-h1","type":"hint","dependencies":[],"title":"Check the Leading Coefficient","text":"Given any parabola in the form $${ax}^2+bx+c$$, a is called the leading coefficient of the parabola. If a is negative, the parabola opens downward. If a is positive, the parabola opens upward.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic4a-h2","type":"hint","dependencies":["ad220b3GraphQuadratic4a-h1"],"title":"Check the Leading Coefficient","text":"In the given question, $$a=1$$ which is positive, so the parabola opens upward.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ad220b3GraphQuadratic4b","stepAnswer":["Down"],"problemType":"TextBox","stepTitle":"$$f(x)=-4x^2-12x-9$$","stepBody":"If the parabola opens up, please enter \\"Up\\". If the parabola opens down, please enter \\"Down\\".","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic4b-h1","type":"hint","dependencies":[],"title":"Check the Leading Coefficient","text":"Given any parabola in the form $${ax}^2+bx+c$$, a is called the leading coefficient of the parabola. If a is negative, the parabola opens downward. If a is positive, the parabola opens upward.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic4b-h2","type":"hint","dependencies":["ad220b3GraphQuadratic4b-h1"],"title":"Check the Leading Coefficient","text":"In the given question, $$a=-4$$ which is negative, so the parabola opens downward.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad220b3GraphQuadratic5","title":"Find the Axis of Symmetry and Vertex of a Parabola","body":"In the following functions, find the equation of the axis of symmetry and the vertex of its graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Graph Quadratic Functions Using Properties","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad220b3GraphQuadratic5a","stepAnswer":["x=-4"],"problemType":"TextBox","stepTitle":"$$f(x)=x^2+8x-1$$","stepBody":"Find the axis of symmetry for the equation. Please enter your answer as $$\\"x=...\\"$$ or $$\\"y=...\\"$$","answerType":"string","variabilization":{},"answerLatex":"$$x=-4$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic5a-h1","type":"hint","dependencies":[],"title":"Use The Formula to Compute","text":"Given any parabola in the form $${ax}^2+bx+c$$, the axis of symmetry is the vertical line $$x=\\\\frac{-b}{2} a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic5a-h2","type":"hint","dependencies":["ad220b3GraphQuadratic5a-h1"],"title":"Substitute The Value Into the Equation","text":"$$a=1$$, $$b=8$$, so $$x=\\\\frac{-b}{2a}=\\\\frac{-8}{2\\\\times1}=-4$$. $$x=-4$$ is the axis of symmetry.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ad220b3GraphQuadratic5b","stepAnswer":["(-4,-17)"],"problemType":"TextBox","stepTitle":"$$f(x)=x^2+8x-1$$","stepBody":"Find the vertex of the parabola graph. Please enter your answer as \\"(x,y)\\" where $$x$$ is the $$x$$ coordinate of vertex and $$y$$ is the $$y$$ coordinate of the vertex.","answerType":"string","variabilization":{},"answerLatex":"$$(-4,-17)$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic5b-h1","type":"hint","dependencies":[],"title":"Vertex Locates On The Axis of Symmetry","text":"The vertex is a point on the line of symmetry, so its x-coordinate will be $$x=-4$$. We only need to compute $$f(-4)$$ to get the $$y$$ coordinate.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic5b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-17$$"],"dependencies":["ad220b3GraphQuadratic5b-h1"],"title":"Compute The y-coordinate","text":"Plug $$x=-4$$ into the equation $$f(x)=x^2+8x-1$$. What is $$f(-4)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic5b-h3","type":"hint","dependencies":["ad220b3GraphQuadratic5b-h2"],"title":"Put The Answer Into (x,y)","text":"As the final answer, we can put together the $$x$$ and $$y$$ coordinates which gives $$(-4,-17)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad220b3GraphQuadratic6","title":"Find the Axis of Symmetry and Vertex of a Parabola","body":"In the following functions, find the equation of the axis of symmetry and the vertex of its graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Graph Quadratic Functions Using Properties","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad220b3GraphQuadratic6a","stepAnswer":["x=-5"],"problemType":"TextBox","stepTitle":"$$f(x)=x^2+10x+25$$","stepBody":"Find the axis of symmetry for the equation. Please enter your answer as $$\\"x=...\\"$$ or $$\\"y=...\\"$$","answerType":"string","variabilization":{},"answerLatex":"$$x=-5$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic6a-h1","type":"hint","dependencies":[],"title":"Use The Formula to Compute","text":"Given any parabola in the form $${ax}^2+bx+c$$, the axis of symmetry is the vertical line $$x=\\\\frac{-b}{2} a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic6a-h2","type":"hint","dependencies":["ad220b3GraphQuadratic6a-h1"],"title":"Substitute The Value Into the Equation","text":"$$a=1$$, $$b=10$$, so $$x=\\\\frac{-b}{2a}=\\\\frac{-10}{2\\\\times1}=-5$$. $$x=-5$$ is the axis of symmetry.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ad220b3GraphQuadratic6b","stepAnswer":["(-5,0)"],"problemType":"TextBox","stepTitle":"$$f(x)=x^2+10x+25$$","stepBody":"Find the vertex of the parabola graph. Please enter your answer as \\"(x,y)\\" where $$x$$ is the $$x$$ coordinate of vertex and $$y$$ is the $$y$$ coordinate of the vertex.","answerType":"string","variabilization":{},"answerLatex":"$$(-5,0)$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic6b-h1","type":"hint","dependencies":[],"title":"Vertex Locates On The Axis of Symmetry","text":"The vertex is a point on the line of symmetry, so its x-coordinate will be $$x=-5$$. We only need to compute $$f(-5)$$ to get the $$y$$ coordinate.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic6b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["ad220b3GraphQuadratic6b-h1"],"title":"Compute The y-coordinate","text":"Plug $$x=-5$$ into the equation $$f(x)=x^2+10x+25$$. What is $$f(-5)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic6b-h3","type":"hint","dependencies":["ad220b3GraphQuadratic6b-h2"],"title":"Put The Answer Into (x,y)","text":"As the final answer, we can put together the $$x$$ and $$y$$ coordinates which gives $$(-5,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad220b3GraphQuadratic7","title":"Find the Axis of Symmetry and Vertex of a Parabola","body":"In the following functions, find the equation of the axis of symmetry and the vertex of its graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Graph Quadratic Functions Using Properties","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad220b3GraphQuadratic7a","stepAnswer":["x=1"],"problemType":"TextBox","stepTitle":"$$f(x)=-\\\\left(x^2\\\\right)+2x+5$$","stepBody":"Find the axis of symmetry for the equation. Please enter your answer as $$\\"x=...\\"$$ or $$\\"y=...\\"$$","answerType":"string","variabilization":{},"answerLatex":"$$x=1$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic7a-h1","type":"hint","dependencies":[],"title":"Use The Formula to Compute","text":"Given any parabola in the form $${ax}^2+bx+c$$, the axis of symmetry is the vertical line $$x=\\\\frac{-b}{2} a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic7a-h2","type":"hint","dependencies":["ad220b3GraphQuadratic7a-h1"],"title":"Substitute The Value Into the Equation","text":"$$a=-1$$, $$b=2$$, so $$x=\\\\frac{-b}{2a}=\\\\frac{-2}{2\\\\left(-1\\\\right)}=1$$. $$x=1$$ is the axis of symmetry.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ad220b3GraphQuadratic7b","stepAnswer":["(1,6)"],"problemType":"TextBox","stepTitle":"$$f(x)=-\\\\left(x^2\\\\right)+2x+5$$","stepBody":"Find the vertex of the parabola graph. Please enter your answer as \\"(x,y)\\" where $$x$$ is the $$x$$ coordinate of vertex and $$y$$ is the $$y$$ coordinate of the vertex.","answerType":"string","variabilization":{},"answerLatex":"$$(1,6)$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic7b-h1","type":"hint","dependencies":[],"title":"Vertex Locates On The Axis of Symmetry","text":"The vertex is a point on the line of symmetry, so its x-coordinate will be $$x=1$$. We only need to compute f(1) to get the $$y$$ coordinate.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic7b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["ad220b3GraphQuadratic7b-h1"],"title":"Compute The y-coordinate","text":"Plug $$x=1$$ into the equation $$f(x)=-\\\\left(x^2\\\\right)+2x+5$$. What is f(1)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic7b-h3","type":"hint","dependencies":["ad220b3GraphQuadratic7b-h2"],"title":"Put The Answer Into (x,y)","text":"As the final answer, we can put together the $$x$$ and $$y$$ coordinates which gives $$(1,6)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad220b3GraphQuadratic8","title":"Find the Axis of Symmetry and Vertex of a Parabola","body":"In the following functions, find the equation of the axis of symmetry and the vertex of its graph.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Graph Quadratic Functions Using Properties","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad220b3GraphQuadratic8a","stepAnswer":["x=-2"],"problemType":"TextBox","stepTitle":"$$f(x)=-2x^2-8x-3$$","stepBody":"Find the axis of symmetry for the equation. Please enter your answer as $$\\"x=...\\"$$ or $$\\"y=...\\"$$","answerType":"string","variabilization":{},"answerLatex":"$$x=-2$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic8a-h1","type":"hint","dependencies":[],"title":"Use The Formula to Compute","text":"Given any parabola in the form $${ax}^2+bx+c$$, the axis of symmetry is the vertical line $$x=\\\\frac{-b}{2} a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic8a-h2","type":"hint","dependencies":["ad220b3GraphQuadratic8a-h1"],"title":"Substitute The Value Into the Equation","text":"$$a=-2$$, $$b=-8$$, so $$x=\\\\frac{-b}{2a}=\\\\frac{8}{2\\\\left(-2\\\\right)}=-2$$. $$x=-2$$ is the axis of symmetry.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ad220b3GraphQuadratic8b","stepAnswer":["(-2,5)"],"problemType":"TextBox","stepTitle":"$$f(x)=-2x^2-8x-3$$","stepBody":"Find the vertex of the parabola graph. Please enter your answer as \\"(x,y)\\" where $$x$$ is the $$x$$ coordinate of vertex and $$y$$ is the $$y$$ coordinate of the vertex.","answerType":"string","variabilization":{},"answerLatex":"$$(-2,5)$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic8b-h1","type":"hint","dependencies":[],"title":"Vertex Locates On The Axis of Symmetry","text":"The vertex is a point on the line of symmetry, so its x-coordinate will be $$x=-2$$. We only need to compute $$f(-2)$$ to get the $$y$$ coordinate.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic8b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["ad220b3GraphQuadratic8b-h1"],"title":"Compute The y-coordinate","text":"Plug $$x=-2$$ into the equation $$f(x)=-2x^2-8x-3$$. What is $$f(-2)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic8b-h3","type":"hint","dependencies":["ad220b3GraphQuadratic8b-h2"],"title":"Put The Answer Into (x,y)","text":"As the final answer, we can put together the $$x$$ and $$y$$ coordinates which gives $$(-2,5)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad220b3GraphQuadratic9","title":"Find the Intercepts of a Parabola","body":"In the following exercises, find the intercepts of the parabola whose function is given.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.6 Graph Quadratic Functions Using Properties","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad220b3GraphQuadratic9a","stepAnswer":["(-6,0),(-1,0)"],"problemType":"TextBox","stepTitle":"$$f(x)=x^2+7x+6$$","stepBody":"Find the $$x-intercept(s)$$ of the given parabola. Please enter your answer as \\"(x,y)\\" where $$x$$ is the x-coordinate of the intercept and $$y$$ is the y-coordinate of the intercept. If there are more than one x-intercept, you can enter it as \\"(x1,y1),(x2,y2)\\" where x1 is small than x2.","answerType":"string","variabilization":{},"answerLatex":"$$(-6,0),(-1,0)$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic9a-h1","type":"hint","dependencies":[],"title":"Set $$f(x)=0$$","text":"To find the x-intercept, let $$f(x)=0$$ and and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic9a-h2","type":"hint","dependencies":["ad220b3GraphQuadratic9a-h1"],"title":"Solve For The Parabola","text":"Solve for $$x^2+7x+6=0$$. We can factor it as $$\\\\left(x+1\\\\right) \\\\left(x+6\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic9a-h3","type":"hint","dependencies":["ad220b3GraphQuadratic9a-h2"],"title":"Solve For The Parabola","text":"$$\\\\left(x+1\\\\right) \\\\left(x+6\\\\right)=0$$ Use zero product property, we know $$x+6=0$$ or $$x+1=0$$ which gives the answer $$x=-6$$ or $$x=-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic9a-h4","type":"hint","dependencies":["ad220b3GraphQuadratic9a-h3"],"title":"Put The Answer Into (x,y)","text":"We know that $$y=0$$ in x-intercepts and we get $$x=-6, -1$$ through the calculations above. Put together, we get two x-intercepts-- $$(-6,0)$$ and $$(-1,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ad220b3GraphQuadratic9b","stepAnswer":["(0,6)"],"problemType":"TextBox","stepTitle":"$$f(x)=x^2+7x+6$$","stepBody":"Find the y-intercept of the given parabola. Please enter your answer as \\"(x,y)\\" where $$x$$ is the x-coordinate of the intercept and $$y$$ is the y-coordinate of the intercept.","answerType":"string","variabilization":{},"answerLatex":"$$(0,6)$$","hints":{"DefaultPathway":[{"id":"ad220b3GraphQuadratic9b-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["ad220b3GraphQuadratic9a-h2"],"title":"Set f(0) As the y-coordinate","text":"We know that $$x=0$$ in the y-intercept. We can compute f(0) to find the $$y$$ coordinate if y-intercept. What is f(0)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad220b3GraphQuadratic9b-h2","type":"hint","dependencies":["ad220b3GraphQuadratic9a-h3"],"title":"Put The Answer Into (x,y)","text":"$$f(0)=0^2+7\\\\times0+6=6$$. We get $$x=0$$ and $$y=6$$ for y-intercept. Put together, we have $$(0,6)$$ as y-intercept.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad291beRadRat1","title":"Evaluating Principal Square Roots","body":"Evaluate each expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Radicals and Rational Exponents","courseName":"OpenStax: College Algebra","steps":[{"id":"ad291beRadRat1a","stepAnswer":["$$10$$"],"problemType":"TextBox","stepTitle":"What is the $$\\\\sqrt{100}$$?","stepBody":"$$cos^{\\\\frac{1}{3}}\\\\left(\\\\frac{1}{2} x\\\\right)$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10$$","hints":{"DefaultPathway":[{"id":"ad291beRadRat1a-h1","type":"hint","dependencies":[],"title":"Principal Square Root","text":"The principal square root of a is the nonnegative number that, when multiplied by itself, equals a. It is written as a radical expression, with a symbol called a radical over the term called the radicand: $$\\\\sqrt{a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["ad291beRadRat1a-h1"],"title":"Principal Square Root","text":"What number multiplied by itself equals 100?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ad291beRadRat1b","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"What is $$\\\\sqrt{\\\\sqrt{16}}$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"ad291beRadRat1b-h1","type":"hint","dependencies":[],"title":"Inner Radical","text":"You can start of by simplifying the inner radical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat1b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ad291beRadRat1b-h1"],"title":"Principal Square Root","text":"What is $$\\\\sqrt{16}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ad291beRadRat1b-h2-s1","type":"hint","dependencies":[],"title":"Principal Square Root","text":"The principal square root of a is the nonnegative number that, when multiplied by itself, equals a. It is written as a radical expression, with a symbol called a radical over the term called the radicand: $$\\\\sqrt{a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat1b-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ad291beRadRat1b-h2-s1"],"title":"Principal Square Root","text":"What number multiplied by itself equals 16?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ad291beRadRat1b-h3","type":"hint","dependencies":["ad291beRadRat1b-h2"],"title":"Inner Radical","text":"Simlify the expression: $$\\\\sqrt{\\\\sqrt{16}}=\\\\sqrt{4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat1b-h4","type":"hint","dependencies":["ad291beRadRat1b-h3"],"title":"Principal Square Root","text":"The principal square root of a is the nonnegative number that, when multiplied by itself, equals a. It is written as a radical expression, with a symbol called a radical over the term called the radicand: $$\\\\sqrt{a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat1b-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ad291beRadRat1b-h4"],"title":"Principal Square Root","text":"What number multiplied by itself equals 4?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ad291beRadRat1c","stepAnswer":["$$13$$"],"problemType":"TextBox","stepTitle":"What is $$\\\\sqrt{25+144}$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$13$$","hints":{"DefaultPathway":[{"id":"ad291beRadRat1c-h1","type":"hint","dependencies":[],"title":"Parentheses","text":"The first step is to simplify the parentheses.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat1c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$169$$"],"dependencies":["ad291beRadRat1c-h1"],"title":"Addition","text":"What is $$25+144$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat1c-h3","type":"hint","dependencies":["ad291beRadRat1c-h2"],"title":"Simplify","text":"Simlify the expression: $$\\\\sqrt{25+144}=\\\\sqrt{169}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat1c-h4","type":"hint","dependencies":["ad291beRadRat1c-h3"],"title":"Principal Square Root","text":"The principal square root of a is the nonnegative number that, when multiplied by itself, equals a. It is written as a radical expression, with a symbol called a radical over the term called the radicand: $$\\\\sqrt{a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat1c-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$13$$"],"dependencies":["ad291beRadRat1c-h4"],"title":"Principal Square Root","text":"What number multiplied by itself equals 169?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad291beRadRat10","title":"Addition of Square Roots","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Radicals and Rational Exponents","courseName":"OpenStax: College Algebra","steps":[{"id":"ad291beRadRat10a","stepAnswer":["$$13\\\\sqrt{5}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{5}+6\\\\sqrt{20}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$13\\\\sqrt{5}$$","hints":{"DefaultPathway":[{"id":"ad291beRadRat10a-h1","type":"hint","dependencies":[],"title":"Rewriting Square Roots","text":"$$6\\\\sqrt{20}$$ can be rewritten as $$6\\\\sqrt{5\\\\times4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat10a-h2","type":"hint","dependencies":["ad291beRadRat10a-h1"],"title":"The Product Rule for Simplifying Square Roots","text":"If a and $$b$$ are nonnegative, the square root of the product ab is equal to the product of the square roots of a and b: $$\\\\sqrt{ab}=\\\\sqrt{a} \\\\sqrt{b}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat10a-h3","type":"hint","dependencies":["ad291beRadRat10a-h2"],"title":"The Product Rule for Simplifying Square Roots","text":"Use the product rule to simplify the expression: $$\\\\sqrt{5\\\\times4}=\\\\sqrt{5} \\\\sqrt{4}=2\\\\sqrt{5}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat10a-h4","type":"hint","dependencies":["ad291beRadRat10a-h3"],"title":"Simplify the Square Roots","text":"$$\\\\sqrt{5}+6\\\\sqrt{20}=\\\\sqrt{5}+6\\\\times2 \\\\sqrt{5}=13\\\\sqrt{5}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad291beRadRat11","title":"Rationalizing a Denominator Containing a Single Term","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Radicals and Rational Exponents","courseName":"OpenStax: College Algebra","steps":[{"id":"ad291beRadRat11a","stepAnswer":["$$\\\\frac{\\\\sqrt{30}}{15}$$"],"problemType":"TextBox","stepTitle":"Write $$\\\\frac{2\\\\sqrt{3}}{3\\\\sqrt{10}}$$ in simplest form.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{\\\\sqrt{30}}{15}$$","hints":{"DefaultPathway":[{"id":"ad291beRadRat11a-h1","type":"hint","dependencies":[],"title":"Simplifying Single Square Root Denominators","text":"Given an expression with a single square root radical term in the denominator, to rationalize the denominator multiply the numerator and denominator by the radical in the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat11a-h2","type":"hint","dependencies":["ad291beRadRat11a-h1"],"title":"Simplifying Single Square Root Denominators","text":"The radical in the denominator is $$\\\\sqrt{10}$$. So multiply the fraction by $$\\\\frac{\\\\sqrt{10}}{\\\\sqrt{10}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat11a-h3","type":"hint","dependencies":["ad291beRadRat11a-h2"],"title":"The Product Rule for Simplifying Square Roots","text":"If a and $$b$$ are nonnegative, the square root of the product ab is equal to the product of the square roots of a and b: $$\\\\sqrt{ab}=\\\\sqrt{a} \\\\sqrt{b}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat11a-h4","type":"hint","dependencies":["ad291beRadRat11a-h3"],"title":"The Product Rule for Simplifying Square Roots","text":"Use the product rule to simplify the expression: $$\\\\frac{2\\\\sqrt{3}}{3\\\\sqrt{10}} \\\\frac{\\\\sqrt{10}}{\\\\sqrt{10}}=\\\\frac{2\\\\sqrt{30}}{30}=\\\\frac{\\\\sqrt{30}}{15}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad291beRadRat12","title":"Rationalizing a Denominator Containing a Single Term","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Radicals and Rational Exponents","courseName":"OpenStax: College Algebra","steps":[{"id":"ad291beRadRat12a","stepAnswer":["$$6\\\\sqrt{6}$$"],"problemType":"TextBox","stepTitle":"Write $$\\\\frac{12\\\\sqrt{3}}{\\\\sqrt{2}}$$ in simplest form.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6\\\\sqrt{6}$$","hints":{"DefaultPathway":[{"id":"ad291beRadRat12a-h1","type":"hint","dependencies":[],"title":"Simplifying Single Square Root Denominators","text":"Given an expression with a single square root radical term in the denominator, to rationalize the denominator multiply the numerator and denominator by the radical in the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat12a-h2","type":"hint","dependencies":["ad291beRadRat12a-h1"],"title":"Simplifying Single Square Root Denominators","text":"The radical in the denominator is $$\\\\sqrt{2}$$. So multiply the fraction by $$\\\\frac{\\\\sqrt{2}}{\\\\sqrt{2}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat12a-h3","type":"hint","dependencies":["ad291beRadRat12a-h2"],"title":"The Product Rule for Simplifying Square Roots","text":"If a and $$b$$ are nonnegative, the square root of the product ab is equal to the product of the square roots of a and b: $$\\\\sqrt{ab}=\\\\sqrt{a} \\\\sqrt{b}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat12a-h4","type":"hint","dependencies":["ad291beRadRat12a-h3"],"title":"The Product Rule for Simplifying Square Roots","text":"Use the product rule to simplify the expression: $$\\\\frac{12\\\\sqrt{3}}{\\\\sqrt{2}} \\\\frac{\\\\sqrt{2}}{\\\\sqrt{2}}=\\\\frac{12\\\\sqrt{6}}{2}=6\\\\sqrt{6}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad291beRadRat13","title":"Rationalizing a Denominator Containing Two Terms","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Radicals and Rational Exponents","courseName":"OpenStax: College Algebra","steps":[{"id":"ad291beRadRat13a","stepAnswer":["$$\\\\sqrt{5}-1$$"],"problemType":"TextBox","stepTitle":"Write $$\\\\frac{4}{1+\\\\sqrt{5}}$$ in simplest form.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\sqrt{5}-1$$","hints":{"DefaultPathway":[{"id":"ad291beRadRat13a-h1","type":"hint","dependencies":[],"title":"Simplifying Denominators Containing Two terms","text":"Given an expression with a radical term and a constant in the denominator, to rationalize the denominator find the conjugate of the denominator and then multiply the numerator and denominator by the conjugate.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat13a-h2","type":"hint","dependencies":["ad291beRadRat13a-h1"],"title":"Simplifying Denominators Containing Two terms","text":"The denominator is $$1+\\\\sqrt{5}$$. So multiply the fraction by $$\\\\frac{1-\\\\sqrt{5}}{1-\\\\sqrt{5}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat13a-h3","type":"hint","dependencies":["ad291beRadRat13a-h2"],"title":"The Distributive Property","text":"Use the distributive property to simplify: $$\\\\frac{\\\\frac{4}{1+\\\\sqrt{5}} \\\\left(1-\\\\sqrt{5}\\\\right)}{1-\\\\sqrt{5}}=\\\\frac{4-4\\\\sqrt{5}}{-4}=\\\\sqrt{5}-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad291beRadRat14","title":"Rationalizing a Denominator Containing Two Terms","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Radicals and Rational Exponents","courseName":"OpenStax: College Algebra","steps":[{"id":"ad291beRadRat14a","stepAnswer":["$$\\\\frac{3\\\\sqrt{x}-\\\\sqrt{3x}}{2}$$"],"problemType":"TextBox","stepTitle":"Write $$\\\\frac{\\\\sqrt{12x}}{2+\\\\sqrt{23}}$$ in simplest form.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3\\\\sqrt{x}-\\\\sqrt{3x}}{2}$$","hints":{"DefaultPathway":[{"id":"ad291beRadRat14a-h1","type":"hint","dependencies":[],"title":"Simplifying Denominators Containing Two terms","text":"Given an expression with a radical term and a constant in the denominator, to rationalize the denominator find the conjugate of the denominator and then multiply the numerator and denominator by the conjugate.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat14a-h2","type":"hint","dependencies":["ad291beRadRat14a-h1"],"title":"Simplifying Denominators Containing Two terms","text":"The denominator is $$2+\\\\sqrt{23}$$. So multiply the fraction by $$\\\\frac{2-\\\\sqrt{23}}{2-\\\\sqrt{23}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat14a-h3","type":"hint","dependencies":["ad291beRadRat14a-h2"],"title":"The Distributive Property","text":"Use the distributive property to simplify: $$\\\\frac{\\\\frac{\\\\sqrt{12x}}{2+\\\\sqrt{23}} \\\\left(2-\\\\sqrt{23}\\\\right)}{2-\\\\sqrt{23}}=\\\\frac{3\\\\sqrt{x}-\\\\sqrt{3x}}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad291beRadRat15","title":"Simplifying nth Roots","body":"Simplify each of the following roots.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Radicals and Rational Exponents","courseName":"OpenStax: College Algebra","steps":[{"id":"ad291beRadRat15a","stepAnswer":["$$-2$$"],"problemType":"TextBox","stepTitle":"What is $$\\\\sqrt[5]{-32}$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-2$$","hints":{"DefaultPathway":[{"id":"ad291beRadRat15a-h1","type":"hint","dependencies":[],"title":"Principal nth Root","text":"If a is a real number with at least one nth root, then the principal nth root of a, written as $$\\\\sqrt[n]{a}$$ , is the number with the same sign as a that, when raised to the nth power, equals a. The index of the radical is $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["ad291beRadRat15a-h1"],"title":"Principal nth Root","text":"What number multiplied by itself $$5$$ times equals -32?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ad291beRadRat15b","stepAnswer":["$$6\\\\sqrt[4]{3}$$"],"problemType":"TextBox","stepTitle":"What is $$8\\\\sqrt[4]{3}$$ - $$\\\\sqrt[4]{48}$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6\\\\sqrt[4]{3}$$","hints":{"DefaultPathway":[{"id":"ad291beRadRat15b-h1","type":"hint","dependencies":[],"title":"Simplify The Radicals","text":"You can simply the expression to get the form $$8\\\\sqrt[4]{3}-2\\\\sqrt[4]{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat15b-h2","type":"hint","dependencies":["ad291beRadRat15b-h1"],"title":"Subtracting Radicals","text":"Simplify the expression: $$8\\\\sqrt[4]{3}-2\\\\sqrt[4]{3}=6\\\\sqrt[4]{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad291beRadRat16","title":"Simplification of nth Roots","body":"Simplify each of the following:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Radicals and Rational Exponents","courseName":"OpenStax: College Algebra","steps":[{"id":"ad291beRadRat16a","stepAnswer":["$$-6$$"],"problemType":"TextBox","stepTitle":"What is $$\\\\sqrt[3]{-216}$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-6$$","hints":{"DefaultPathway":[{"id":"ad291beRadRat16a-h1","type":"hint","dependencies":[],"title":"Principal nth Root","text":"If a is a real number with at least one nth root, then the principal nth root of a, written as $$\\\\sqrt[n]{a}$$ , is the number with the same sign as a that, when raised to the nth power, equals a. The index of the radical is $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6$$"],"dependencies":["ad291beRadRat16a-h1"],"title":"Principal nth Root","text":"What number multiplied by itself $$3$$ times equals -216?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ad291beRadRat16b","stepAnswer":["$$183$$"],"problemType":"TextBox","stepTitle":"What is $$6\\\\sqrt[3]{9000}+7\\\\sqrt[3]{576}$$? (Rounded to the nearest whole number)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$183$$","hints":{"DefaultPathway":[{"id":"ad291beRadRat16b-h1","type":"hint","dependencies":[],"title":"Principal nth Root","text":"If a is a real number with at least one nth root, then the principal nth root of a, written as $$\\\\sqrt[n]{a}$$ , is the number with the same sign as a that, when raised to the nth power, equals a. The index of the radical is $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat16b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20.8$$"],"dependencies":["ad291beRadRat16b-h1"],"title":"Principal nth Root","text":"What number multiplied by itself $$3$$ times equals 9000?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat16b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8.3$$"],"dependencies":["ad291beRadRat16b-h2"],"title":"Principal nth Root","text":"What number multiplied by itself $$3$$ times equals 576?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat16b-h4","type":"hint","dependencies":["ad291beRadRat16b-h3"],"title":"Simplify","text":"$$6\\\\times20.8+7\\\\times8.3=183$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad291beRadRat17","title":"Writing Rational Exponents as Radicals","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Radicals and Rational Exponents","courseName":"OpenStax: College Algebra","steps":[{"id":"ad291beRadRat17a","stepAnswer":["$$49$$"],"problemType":"TextBox","stepTitle":"What is $${343}^{\\\\frac{2}{3}}$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$49$$","hints":{"DefaultPathway":[{"id":"ad291beRadRat17a-h1","type":"hint","dependencies":[],"title":"Radical Form","text":"Rewrite the expression as a radical: $${343}^{\\\\frac{2}{3}}={\\\\sqrt[3]{343}}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat17a-h2","type":"hint","dependencies":["ad291beRadRat17a-h1"],"title":"Principal nth Root","text":"If a is a real number with at least one nth root, then the principal nth root of a, written as $$\\\\sqrt[n]{a}$$ , is the number with the same sign as a that, when raised to the nth power, equals a. The index of the radical is $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat17a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["ad291beRadRat17a-h2"],"title":"Principal nth Root","text":"What number multiplied by itself $$3$$ times equals 343?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat17a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$49$$"],"dependencies":["ad291beRadRat17a-h3"],"title":"Simplify","text":"What is $$7^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad291beRadRat18","title":"Simplifying 4th Roots","body":"Simplify each of the following roots.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Radicals and Rational Exponents","courseName":"OpenStax: College Algebra","steps":[{"id":"ad291beRadRat18a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"What is $$\\\\sqrt[4]{16}$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"ad291beRadRat18a-h1","type":"hint","dependencies":[],"title":"Principal nth Root","text":"If a is a real number with at least one nth root, then the principal nth root of a, written as $$\\\\sqrt[n]{a}$$ , is the number with the same sign as a that, when raised to the nth power, equals a. The index of the radical is $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ad291beRadRat18a-h1"],"title":"Principal nth Root","text":"What number multiplied by itself $$4$$ times equals 16?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad291beRadRat19","title":"Simplifying 3rd Roots","body":"Simplify each of the following roots.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Radicals and Rational Exponents","courseName":"OpenStax: College Algebra","steps":[{"id":"ad291beRadRat19a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"What is $$\\\\sqrt[3]{64}$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"ad291beRadRat19a-h1","type":"hint","dependencies":[],"title":"Principal nth Root","text":"If a is a real number with at least one nth root, then the principal nth root of a, written as $$\\\\sqrt[n]{a}$$ , is the number with the same sign as a that, when raised to the nth power, equals a. The index of the radical is $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ad291beRadRat19a-h1"],"title":"Principal nth Root","text":"What number multiplied by itself $$3$$ times equals 64?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad291beRadRat2","title":"Evaluating Principal Square Roots","body":"Evaluate each expression.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Radicals and Rational Exponents","courseName":"OpenStax: College Algebra","steps":[{"id":"ad291beRadRat2a","stepAnswer":["$$15$$"],"problemType":"TextBox","stepTitle":"What is the $$\\\\sqrt{225}$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$15$$","hints":{"DefaultPathway":[{"id":"ad291beRadRat2a-h1","type":"hint","dependencies":[],"title":"Principal Square Root","text":"The principal square root of a is the nonnegative number that, when multiplied by itself, equals a. It is written as a radical expression, with a symbol called a radical over the term called the radicand: $$\\\\sqrt{a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["ad291beRadRat2a-h1"],"title":"Principal Square Root","text":"What number multiplied by itself equals 225?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ad291beRadRat2b","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"What is $$\\\\sqrt{\\\\sqrt{81}}$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"ad291beRadRat2b-h1","type":"hint","dependencies":[],"title":"Inner Radical","text":"You can start of by simplifying the inner radical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat2b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["ad291beRadRat2b-h1"],"title":"Principal Square Root","text":"What is $$\\\\sqrt{81}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ad291beRadRat2b-h2-s1","type":"hint","dependencies":[],"title":"Principal Square Root","text":"The principal square root of a is the nonnegative number that, when multiplied by itself, equals a. It is written as a radical expression, with a symbol called a radical over the term called the radicand: $$\\\\sqrt{a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat2b-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["ad291beRadRat2b-h2-s1"],"title":"Principal Square Root","text":"What number multiplied by itself equals 81?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ad291beRadRat2b-h3","type":"hint","dependencies":["ad291beRadRat2b-h2"],"title":"Inner Radical","text":"Simlify the expression: $$\\\\sqrt{\\\\sqrt{81}}=\\\\sqrt{9}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat2b-h4","type":"hint","dependencies":["ad291beRadRat2b-h3"],"title":"Principal Square Root","text":"The principal square root of a is the nonnegative number that, when multiplied by itself, equals a. It is written as a radical expression, with a symbol called a radical over the term called the radicand: $$\\\\sqrt{a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat2b-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ad291beRadRat2b-h4"],"title":"Principal Square Root","text":"What number multiplied by itself equals 9?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ad291beRadRat2c","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"What is $$\\\\sqrt{25-9}$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"ad291beRadRat2c-h1","type":"hint","dependencies":[],"title":"Parentheses","text":"The first step is to simplify the parentheses.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat2c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["ad291beRadRat2c-h1"],"title":"Subtraction","text":"What is $$25-9$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat2c-h3","type":"hint","dependencies":["ad291beRadRat2c-h2"],"title":"Simplify","text":"Simlify the expression: $$\\\\sqrt{25-9}=\\\\sqrt{16}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat2c-h4","type":"hint","dependencies":["ad291beRadRat2c-h3"],"title":"Principal Square Root","text":"The principal square root of a is the nonnegative number that, when multiplied by itself, equals a. It is written as a radical expression, with a symbol called a radical over the term called the radicand: $$\\\\sqrt{a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat2c-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ad291beRadRat2c-h4"],"title":"Principal Square Root","text":"What number multiplied by itself equals 16?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad291beRadRat20","title":"Simplifying 5th Roots","body":"Simplify each of the following roots.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Radicals and Rational Exponents","courseName":"OpenStax: College Algebra","steps":[{"id":"ad291beRadRat20a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"What is $$\\\\sqrt[5]{243}$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"ad291beRadRat20a-h1","type":"hint","dependencies":[],"title":"Principal nth Root","text":"If a is a real number with at least one nth root, then the principal nth root of a, written as $$\\\\sqrt[n]{a}$$ , is the number with the same sign as a that, when raised to the nth power, equals a. The index of the radical is $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ad291beRadRat20a-h1"],"title":"Principal nth Root","text":"What number multiplied by itself $$5$$ times equals 243?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad291beRadRat21","title":"Simplifying 4th Roots","body":"Simplify each of the following roots.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Radicals and Rational Exponents","courseName":"OpenStax: College Algebra","steps":[{"id":"ad291beRadRat21a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"What is $$\\\\sqrt[4]{625}$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"ad291beRadRat21a-h1","type":"hint","dependencies":[],"title":"Principal nth Root","text":"If a is a real number with at least one nth root, then the principal nth root of a, written as $$\\\\sqrt[n]{a}$$ , is the number with the same sign as a that, when raised to the nth power, equals a. The index of the radical is $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat21a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["ad291beRadRat21a-h1"],"title":"Principal nth Root","text":"What number multiplied by itself $$4$$ times equals 625?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad291beRadRat3","title":"Using the Product Rule to Simplify Square Roots","body":"Simplify the radical expressions.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Radicals and Rational Exponents","courseName":"OpenStax: College Algebra","steps":[{"id":"ad291beRadRat3a","stepAnswer":["$$10\\\\sqrt{3}$$"],"problemType":"TextBox","stepTitle":"What is $$\\\\sqrt{300}$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10\\\\sqrt{3}$$","hints":{"DefaultPathway":[{"id":"ad291beRadRat3a-h1","type":"hint","dependencies":[],"title":"Perfect Square","text":"Factor a perfect square from the radicand to simplify the expression: $$\\\\sqrt{300}=\\\\sqrt{100\\\\times3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat3a-h2","type":"hint","dependencies":["ad291beRadRat3a-h1"],"title":"The Product Rule for Simplifying Square Roots","text":"If a and $$b$$ are nonnegative, the square root of the product ab is equal to the product of the square roots of a and b: $$\\\\sqrt{ab}=\\\\sqrt{a} \\\\sqrt{b}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat3a-h3","type":"hint","dependencies":["ad291beRadRat3a-h2"],"title":"The Product Rule for Simplifying Square Roots","text":"Use the product rule to simplify the expression: $$\\\\sqrt{100\\\\times3}=\\\\sqrt{100} \\\\sqrt{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat3a-h4","type":"hint","dependencies":["ad291beRadRat3a-h3"],"title":"Principal Square Root","text":"Simplify the expression: $$\\\\sqrt{100} \\\\sqrt{3}=10\\\\sqrt{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ad291beRadRat3b","stepAnswer":["$$9\\\\left(a^2\\\\right) b^2 \\\\sqrt{2a}$$"],"problemType":"TextBox","stepTitle":"What is $$\\\\sqrt{162a^5 b^4}$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9\\\\left(a^2\\\\right) b^2 \\\\sqrt{2a}$$","hints":{"DefaultPathway":[{"id":"ad291beRadRat3b-h1","type":"hint","dependencies":[],"title":"Perfect Square","text":"Factor a perfect square from the radicand to simplify the expression: $$\\\\sqrt{\\\\operatorname{162}\\\\left(a^5\\\\right) b^4}=\\\\sqrt{2\\\\operatorname{81}\\\\left(a^4\\\\right) b^4 a}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat3b-h2","type":"hint","dependencies":["ad291beRadRat3b-h1"],"title":"The Product Rule for Simplifying Square Roots","text":"If a and $$b$$ are nonnegative, the square root of the product ab is equal to the product of the square roots of a and b: $$\\\\sqrt{ab}=\\\\sqrt{a} \\\\sqrt{b}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat3b-h3","type":"hint","dependencies":["ad291beRadRat3b-h2"],"title":"The Product Rule for Simplifying Square Roots","text":"Use the product rule to simplify the expression: $$\\\\sqrt{2\\\\operatorname{81}\\\\left(a^4\\\\right) b^4 a}=\\\\sqrt{\\\\operatorname{81}\\\\left(a^4\\\\right) b^4} \\\\sqrt{2a}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat3b-h4","type":"hint","dependencies":["ad291beRadRat3b-h3"],"title":"Principal Square Root","text":"Simplify the expression: $$\\\\sqrt{81a^4 b^4} \\\\sqrt{2a}=9a^2 b^2 \\\\sqrt{2a}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad291beRadRat4","title":"Using the Product Rule to Simplify the Product of Multiple Square Roots","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Radicals and Rational Exponents","courseName":"OpenStax: College Algebra","steps":[{"id":"ad291beRadRat4a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"Simplify the radical expression: $$\\\\sqrt{12} \\\\sqrt{3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"ad291beRadRat4a-h1","type":"hint","dependencies":[],"title":"The Product Rule for Simplifying Square Roots","text":"If a and $$b$$ are nonnegative, the square root of the product ab is equal to the product of the square roots of a and b: $$\\\\sqrt{a b}=\\\\sqrt{a} \\\\sqrt{b}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat4a-h2","type":"hint","dependencies":["ad291beRadRat4a-h1"],"title":"The Product Rule for Simplifying Square Roots","text":"Use the product rule to simplify the expression: $$\\\\sqrt{12} \\\\sqrt{3}=\\\\sqrt{12\\\\times3}=\\\\sqrt{36}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat4a-h3","type":"hint","dependencies":["ad291beRadRat4a-h2"],"title":"Principal Square Root","text":"The principal square root of a is the nonnegative number that, when multiplied by itself, equals a. It is written as a radical expression, with a symbol called a radical over the term called the radicand: $$\\\\sqrt{a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["ad291beRadRat4a-h3"],"title":"Principal Square Root","text":"What number multiplied by itself equals 36?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad291beRadRat5","title":"Using the Quotient Rule to Simplify Square Roots","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Radicals and Rational Exponents","courseName":"OpenStax: College Algebra","steps":[{"id":"ad291beRadRat5a","stepAnswer":["$$\\\\frac{\\\\sqrt{5}}{6}$$"],"problemType":"TextBox","stepTitle":"Simplify the radical expression: $$\\\\sqrt{\\\\frac{5}{36}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{\\\\sqrt{5}}{6}$$","hints":{"DefaultPathway":[{"id":"ad291beRadRat5a-h1","type":"hint","dependencies":[],"title":"The Quotient Rule for Simplifying Square Roots","text":"The square root of the quotient $$\\\\frac{a}{b}$$ is equal to the quotient of the square roots of a and $$b$$, where $$b \\\\neq 0$$: $$\\\\sqrt{\\\\frac{a}{b}}=\\\\frac{\\\\sqrt{a}}{\\\\sqrt{b}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat5a-h2","type":"hint","dependencies":["ad291beRadRat5a-h1"],"title":"The Quotient Rule for Simplifying Square Roots","text":"Use the quotient rule to simplify the expression: $$\\\\sqrt{\\\\frac{5}{36}}=\\\\frac{\\\\sqrt{5}}{\\\\sqrt{36}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat5a-h3","type":"hint","dependencies":["ad291beRadRat5a-h2"],"title":"Principal Square Root","text":"The principal square root of a is the nonnegative number that, when multiplied by itself, equals a. It is written as a radical expression, with a symbol called a radical over the term called the radicand: $$\\\\sqrt{a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["ad291beRadRat5a-h3"],"title":"Principal Square Root","text":"What number multiplied by itself equals 36?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad291beRadRat6","title":"Using the Product Rule to Simplify the Product of Multiple Square Roots","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Radicals and Rational Exponents","courseName":"OpenStax: College Algebra","steps":[{"id":"ad291beRadRat6a","stepAnswer":["$$10x$$"],"problemType":"TextBox","stepTitle":"Simplify $$\\\\sqrt{50x} \\\\sqrt{2x}$$ assuming $$x>0$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10x$$","hints":{"DefaultPathway":[{"id":"ad291beRadRat6a-h1","type":"hint","dependencies":[],"title":"The Product Rule for Simplifying Square Roots","text":"If a and $$b$$ are nonnegative, the square root of the product ab is equal to the product of the square roots of a and b: $$\\\\sqrt{ab}=\\\\sqrt{a} \\\\sqrt{b}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat6a-h2","type":"hint","dependencies":["ad291beRadRat6a-h1"],"title":"The Product Rule for Simplifying Square Roots","text":"Use the product rule to simplify the expression: $$\\\\sqrt{50x} \\\\sqrt{2x}=$$ $$\\\\sqrt{50x\\\\times2 x}=$$ $$\\\\sqrt{100x^2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat6a-h3","type":"hint","dependencies":["ad291beRadRat6a-h2"],"title":"Principal Square Root","text":"The principal square root of a is the nonnegative number that, when multiplied by itself, equals a. It is written as a radical expression, with a symbol called a radical over the term called the radicand: $$\\\\sqrt{a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10x$$"],"dependencies":["ad291beRadRat6a-h3"],"title":"Principal Square Root","text":"What number multiplied by itself equals $$100x^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad291beRadRat7","title":"Using the Quotient Rule to Simplify Square Roots","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Radicals and Rational Exponents","courseName":"OpenStax: College Algebra","steps":[{"id":"ad291beRadRat7a","stepAnswer":["$$\\\\frac{x \\\\sqrt{2}}{3} y^2$$"],"problemType":"TextBox","stepTitle":"Simplify the radical expression: $$\\\\sqrt{\\\\frac{2x^2}{9y^4}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{x \\\\sqrt{2}}{3} y^2$$","hints":{"DefaultPathway":[{"id":"ad291beRadRat7a-h1","type":"hint","dependencies":[],"title":"The Quotient Rule for Simplifying Square Roots","text":"The square root of the quotient $$\\\\frac{a}{b}$$ is equal to the quotient of the square roots of a and $$b$$, where $$b \\\\neq 0$$: $$\\\\sqrt{\\\\frac{a}{b}}=\\\\frac{\\\\sqrt{a}}{\\\\sqrt{b}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat7a-h2","type":"hint","dependencies":["ad291beRadRat7a-h1"],"title":"The Quotient Rule for Simplifying Square Roots","text":"Use the quotient rule to simplify the expression: $$\\\\sqrt{\\\\frac{2x^2}{9y^4}}=\\\\frac{\\\\sqrt{2x^2}}{\\\\sqrt{9y^4}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat7a-h3","type":"hint","dependencies":["ad291beRadRat7a-h2"],"title":"Principal Square Root","text":"The principal square root of a is the nonnegative number that, when multiplied by itself, equals a. It is written as a radical expression, with a symbol called a radical over the term called the radicand: $$\\\\sqrt{a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3y^2$$"],"dependencies":["ad291beRadRat7a-h3"],"title":"Principal Square Root","text":"What number multiplied by itself equals $$9y^4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat7a-h5","type":"hint","dependencies":["ad291beRadRat7a-h4"],"title":"Principal Square Root","text":"Simplify the expression: $$\\\\frac{\\\\sqrt{2x^2}}{\\\\sqrt{9y^4}}=\\\\frac{x \\\\sqrt{2}}{3} y^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad291beRadRat8","title":"Using the Quotient Rule to Simplify an Expression with Two Square Roots","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Radicals and Rational Exponents","courseName":"OpenStax: College Algebra","steps":[{"id":"ad291beRadRat8a","stepAnswer":["$$3x^2$$"],"problemType":"TextBox","stepTitle":"Simplify the radical expression: $$\\\\frac{\\\\sqrt{23\\\\times4 x^{11} y}}{\\\\sqrt{26x^7 y}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3x^2$$","hints":{"DefaultPathway":[{"id":"ad291beRadRat8a-h1","type":"hint","dependencies":[],"title":"The Quotient Rule for Simplifying Square Roots","text":"The square root of the quotient $$\\\\frac{a}{b}$$ is equal to the quotient of the square roots of a and $$b$$, where $$b \\\\neq 0$$: $$\\\\sqrt{\\\\frac{a}{b}}=\\\\frac{\\\\sqrt{a}}{\\\\sqrt{b}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat8a-h2","type":"hint","dependencies":["ad291beRadRat8a-h1"],"title":"The Quotient Rule for Simplifying Square Roots","text":"Use the quotient rule to simplify the expression: $$\\\\frac{\\\\sqrt{23\\\\times4 x^{11} y}}{\\\\sqrt{26x^7 y}}=\\\\sqrt{\\\\frac{23\\\\times4 x^{11} y}{26x^7 y}}=\\\\sqrt{9x^4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat8a-h3","type":"hint","dependencies":["ad291beRadRat8a-h2"],"title":"Principal Square Root","text":"The principal square root of a is the nonnegative number that, when multiplied by itself, equals a. It is written as a radical expression, with a symbol called a radical over the term called the radicand: $$\\\\sqrt{a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x^2$$"],"dependencies":["ad291beRadRat8a-h3"],"title":"Principal Square Root","text":"What number multiplied by itself equals $$9x^4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad291beRadRat9","title":"Adding Square Roots","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.3 Radicals and Rational Exponents","courseName":"OpenStax: College Algebra","steps":[{"id":"ad291beRadRat9a","stepAnswer":["$$12\\\\sqrt{3}$$"],"problemType":"TextBox","stepTitle":"Add $$5\\\\sqrt{12}+2\\\\sqrt{3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12\\\\sqrt{3}$$","hints":{"DefaultPathway":[{"id":"ad291beRadRat9a-h1","type":"hint","dependencies":[],"title":"Rewriting Square Roots","text":"$$5\\\\sqrt{12}$$ can be rewritten as $$5\\\\sqrt{3\\\\times4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat9a-h2","type":"hint","dependencies":["ad291beRadRat9a-h1"],"title":"The Product Rule for Simplifying Square Roots","text":"If a and $$b$$ are nonnegative, the square root of the product ab is equal to the product of the square roots of a and b: $$\\\\sqrt{ab}=\\\\sqrt{a} \\\\sqrt{b}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat9a-h3","type":"hint","dependencies":["ad291beRadRat9a-h2"],"title":"The Product Rule for Simplifying Square Roots","text":"Use the product rule to simplify the expression: $$\\\\sqrt{3\\\\times4}=\\\\sqrt{3} \\\\sqrt{4}=2\\\\sqrt{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad291beRadRat9a-h4","type":"hint","dependencies":["ad291beRadRat9a-h3"],"title":"Simplify the Square Roots","text":"$$5\\\\sqrt{12}+2\\\\sqrt{3}=5\\\\times2 \\\\sqrt{3}+2\\\\sqrt{3}=12\\\\sqrt{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad33ee8parabolas1","title":"Graphing Vertical Parabolas","body":"Graph the vertical parabola using its properties.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Parabolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad33ee8parabolas1a","stepAnswer":["D"],"problemType":"MultipleChoice","stepTitle":"$$y=\\\\left(-x^2\\\\right)+6x-8$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"ad33ee8parabolas1a-h1","type":"hint","dependencies":[],"title":"Direction","text":"We can tell whether the parabola goes up or down based on the sign of a in the equation, $$y=a x^2+b x+c$$. If a is negative, then the parabola goes downwards, if it is positive, it goes upwards.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas1a-h2","type":"hint","dependencies":["ad33ee8parabolas1a-h1"],"title":"Direction","text":"Since our a is negative, what does this tell us about our parabola?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas1a-h3","type":"hint","dependencies":["ad33ee8parabolas1a-h2"],"title":"Downwards Direction","text":"Since our a is negative, we know that our parabola will open up downwards.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas1a-h4","type":"hint","dependencies":["ad33ee8parabolas1a-h3"],"title":"Finding the Vertex","text":"To find the vertex, we need to find the axis is symmetry, which is found using the formula, $$x=\\\\frac{-b}{2} a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas1a-h5","type":"hint","dependencies":["ad33ee8parabolas1a-h4"],"title":"Finding the Axis of Symmetry","text":"To find the axis of symmetry, we have to find the $$b$$ and a from our equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas1a-h6","type":"hint","dependencies":["ad33ee8parabolas1a-h5"],"title":"Finding the Axis of Symmetry","text":"Our $$b$$ and a are $$6$$ and $$-1$$ respectively. Our axis of symmetry is $$x=-1\\\\frac{-6}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas1a-h8","type":"hint","dependencies":["ad33ee8parabolas1a-h6"],"title":"Finding the Vertex","text":"Now that we have our axis of symmetry, we want to plug it back into the original equation to find the $$y$$ value of our vertex.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas1a-h8","type":"hint","dependencies":["ad33ee8parabolas1a-h8"],"title":"Finding the Vertex","text":"Since our $$x$$ is $$3$$, we plug it into our equation as, $$y=-\\\\left(3^2\\\\right)+6\\\\times3-8$$. This gives us $$y=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas1a-h9","type":"hint","dependencies":["ad33ee8parabolas1a-h8"],"title":"Vertex","text":"Since we know that our $$x$$ is $$3$$ and our $$y$$ is $$1$$, our vertex is at the coordinates, $$(3,1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas1a-h10","type":"hint","dependencies":["ad33ee8parabolas1a-h9"],"title":"Finding Intercepts","text":"To graph our parabola, we need to know our $$x$$ and $$y$$ intercepts.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas1a-h11","type":"hint","dependencies":["ad33ee8parabolas1a-h10"],"title":"Finding the y-intercepts","text":"Our y-intercept occurs when $$x=0$$. Knowing this, we substitute $$0$$ into our equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas1a-h12","type":"hint","dependencies":["ad33ee8parabolas1a-h11"],"title":"Finding the y-intercepts","text":"After plugging in $$x=0$$ to our equation, we see that a y-intercept occurs when $$y=-8$$, so our y-intercept is $$(0,-8)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas1a-h13","type":"hint","dependencies":["ad33ee8parabolas1a-h12"],"title":"Finding the x-intercepts","text":"Our $$x$$ intercepts occur when $$y=0$$. We plug in $$y=0$$ to our equation and solve for the x\'s.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas1a-h14","type":"hint","dependencies":["ad33ee8parabolas1a-h13"],"title":"Finding the x-intercepts","text":"We must factor the GCF from our equation and factor the trinomial. After this, we solve for $$x$$, which are $$4$$ and $$2$$ in this case, so the x-intercepts are $$(4,0)$$ and $$(2,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas1a-h15","type":"hint","dependencies":["ad33ee8parabolas1a-h14"],"title":"Graphing","text":"Now that we have all of our properties, we are ready to graph our parabolas.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad33ee8parabolas10","title":"Rewriting the equation in Standard Form.","body":"Rewrite the given equation into standard form.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Parabolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad33ee8parabolas10a","stepAnswer":["$$2{\\\\left(x+1\\\\right)}^2+3$$"],"problemType":"TextBox","stepTitle":"$$y=2x^2+4x+5$$ (Note: Do not include $$\\"y=\\"$$ in answer.)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2{\\\\left(x+1\\\\right)}^2+3$$","hints":{"DefaultPathway":[{"id":"ad33ee8parabolas10a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"The standard form is $$y={a\\\\left(x-h\\\\right)}^2+k$$. We are given the equation in $$y=a x^2+b x+c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas10a-h2","type":"hint","dependencies":["ad33ee8parabolas10a-h1"],"title":"Finding a","text":"To find a, we must divide our $$a x^2$$ term by a by factoring by grouping.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas10a-h3","type":"hint","dependencies":["ad33ee8parabolas10a-h2"],"title":"Completing the Square","text":"After we find a, our equation now looks like, $$a \\\\left(x^2+\\\\frac{b}{a} x\\\\right)+c$$. We must complete the square by adding $${\\\\left(\\\\frac{\\\\frac{b}{a}}{2}\\\\right)}^2$$ into the parenthesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas10a-h4","type":"hint","dependencies":["ad33ee8parabolas10a-h3"],"title":"Completing the Square","text":"Since we can\'t simply add a number out of thin air, we must take our equation and subtract it by what we added into the parenthesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas10a-h5","type":"hint","dependencies":["ad33ee8parabolas10a-h4"],"title":"Completing the Square","text":"After doing so, our equation should now be in standard form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad33ee8parabolas11","title":"Rewriting the equation in Standard Form.","body":"Rewrite the given equation into standard form.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Parabolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad33ee8parabolas11a","stepAnswer":["$$-2{\\\\left(x-2\\\\right)}^2+1$$"],"problemType":"TextBox","stepTitle":"$$y=-2x^2+8x-7$$ (Note: Do not include $$\\"y=\\"$$ in answer.)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-2{\\\\left(x-2\\\\right)}^2+1$$","hints":{"DefaultPathway":[{"id":"ad33ee8parabolas11a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"The standard form is $$y={a\\\\left(x-h\\\\right)}^2+k$$. We are given the equation in $$y=a x^2+b x+c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas11a-h2","type":"hint","dependencies":["ad33ee8parabolas11a-h1"],"title":"Finding a","text":"To find a, we must divide our $$a x^2$$ term by a by factoring by grouping.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas11a-h3","type":"hint","dependencies":["ad33ee8parabolas11a-h2"],"title":"Completing the Square","text":"After we find a, our equation now looks like, $$a \\\\left(x^2+\\\\frac{b}{a} x\\\\right)+c$$. We must complete the square by adding $${\\\\left(\\\\frac{\\\\frac{b}{a}}{2}\\\\right)}^2$$ into the parenthesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas11a-h4","type":"hint","dependencies":["ad33ee8parabolas11a-h3"],"title":"Completing the Square","text":"Since we can\'t simply add a number out of thin air, we must take our equation and subtract it by what we added into the parenthesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas11a-h5","type":"hint","dependencies":["ad33ee8parabolas11a-h4"],"title":"Completing the Square","text":"After doing so, our equation should now be in standard form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad33ee8parabolas12","title":"Rewriting the equation in Standard Form.","body":"Rewrite the given equation into standard form.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Parabolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad33ee8parabolas12a","stepAnswer":["$$2{\\\\left(x+1\\\\right)}^2+4$$"],"problemType":"TextBox","stepTitle":"$$y=2x^2+4x-6$$ (Note: Do not include $$\\"y=\\"$$ in answer.)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2{\\\\left(x+1\\\\right)}^2+4$$","hints":{"DefaultPathway":[{"id":"ad33ee8parabolas12a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"The standard form is $$y={a\\\\left(x-h\\\\right)}^2+k$$. We are given the equation in $$y=a x^2+b x+c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas12a-h2","type":"hint","dependencies":["ad33ee8parabolas12a-h1"],"title":"Finding a","text":"To find a, we must divide our $$a x^2$$ term by a by factoring by grouping.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas12a-h3","type":"hint","dependencies":["ad33ee8parabolas12a-h2"],"title":"Completing the Square","text":"After we find a, our equation now looks like, $$a \\\\left(x^2+\\\\frac{b}{a} x\\\\right)+c$$. We must complete the square by adding $${\\\\left(\\\\frac{\\\\frac{b}{a}}{2}\\\\right)}^2$$ into the parenthesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas12a-h4","type":"hint","dependencies":["ad33ee8parabolas12a-h3"],"title":"Completing the Square","text":"Since we can\'t simply add a number out of thin air, we must take our equation and subtract it by what we added into the parenthesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas12a-h5","type":"hint","dependencies":["ad33ee8parabolas12a-h4"],"title":"Completing the Square","text":"After doing so, our equation should now be in standard form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad33ee8parabolas13","title":"Rewriting the equation in Standard Form.","body":"Rewrite the given equation into standard form.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Parabolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad33ee8parabolas13a","stepAnswer":["$$-2{\\\\left(x+1\\\\right)}^2-3$$"],"problemType":"TextBox","stepTitle":"$$y=-2x^2-4x-5$$ (Note: Do not include $$\\"y=\\"$$ in answer.)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-2{\\\\left(x+1\\\\right)}^2-3$$","hints":{"DefaultPathway":[{"id":"ad33ee8parabolas13a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"The standard form is $$y={a\\\\left(x-h\\\\right)}^2+k$$. We are given the equation in $$y=a x^2+b x+c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas13a-h2","type":"hint","dependencies":["ad33ee8parabolas13a-h1"],"title":"Finding a","text":"To find a, we must divide our $$a x^2$$ term by a by factoring by grouping.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas13a-h3","type":"hint","dependencies":["ad33ee8parabolas13a-h2"],"title":"Completing the Square","text":"After we find a, our equation now looks like, $$a \\\\left(x^2+\\\\frac{b}{a} x\\\\right)+c$$. We must complete the square by adding $${\\\\left(\\\\frac{\\\\frac{b}{a}}{2}\\\\right)}^2$$ into the parenthesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas13a-h4","type":"hint","dependencies":["ad33ee8parabolas13a-h3"],"title":"Completing the Square","text":"Since we can\'t simply add a number out of thin air, we must take our equation and subtract it by what we added into the parenthesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas13a-h5","type":"hint","dependencies":["ad33ee8parabolas13a-h4"],"title":"Completing the Square","text":"After doing so, our equation should now be in standard form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad33ee8parabolas14","title":"Rewriting the equation in Standard Form.","body":"Rewrite the given equation into standard form.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Parabolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad33ee8parabolas14a","stepAnswer":["$$3{\\\\left(x-2\\\\right)}^2-5$$"],"problemType":"TextBox","stepTitle":"$$y=3x^2-12x+7$$ (Note: Do not include $$\\"y=\\"$$ in answer.)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3{\\\\left(x-2\\\\right)}^2-5$$","hints":{"DefaultPathway":[{"id":"ad33ee8parabolas14a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"The standard form is $$y={a\\\\left(x-h\\\\right)}^2+k$$. We are given the equation in $$y=a x^2+b x+c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas14a-h2","type":"hint","dependencies":["ad33ee8parabolas14a-h1"],"title":"Finding a","text":"To find a, we must divide our $$a x^2$$ term by a by factoring by grouping.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas14a-h3","type":"hint","dependencies":["ad33ee8parabolas14a-h2"],"title":"Completing the Square","text":"After we find a, our equation now looks like, $$a \\\\left(x^2+\\\\frac{b}{a} x\\\\right)+c$$. We must complete the square by adding $${\\\\left(\\\\frac{\\\\frac{b}{a}}{2}\\\\right)}^2$$ into the parenthesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas14a-h4","type":"hint","dependencies":["ad33ee8parabolas14a-h3"],"title":"Completing the Square","text":"Since we can\'t simply add a number out of thin air, we must take our equation and subtract it by what we added into the parenthesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas14a-h5","type":"hint","dependencies":["ad33ee8parabolas14a-h4"],"title":"Completing the Square","text":"After doing so, our equation should now be in standard form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad33ee8parabolas15","title":"Graphing Horizontal Parabolas","body":"Graph the horizontal parabola given its properties.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Parabolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad33ee8parabolas15a","stepAnswer":["D"],"problemType":"MultipleChoice","stepTitle":"$$x=-\\\\left(y^2\\\\right)+2y+8$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"ad33ee8parabolas15a-h1","type":"hint","dependencies":[],"title":"Direction","text":"We can tell whether the parabola goes left or right based on the sign of a in the equation, $$x=a y^2+b y+c$$. If a is negative, then the parabola goes left, if it is positive, it goes right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas15a-h2","type":"hint","dependencies":["ad33ee8parabolas15a-h1"],"title":"Direction","text":"Since our a is negative, what does this tell us about our parabola?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas15a-h3","type":"hint","dependencies":["ad33ee8parabolas15a-h2"],"title":"Left Direction","text":"Since our a is negative, we know that our parabola will open up towards the left.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas15a-h4","type":"hint","dependencies":["ad33ee8parabolas15a-h3"],"title":"Finding the Vertex","text":"To find the vertex, we need to find the axis is symmetry, which is found using the formula, $$y=\\\\frac{-b}{2} a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas15a-h5","type":"hint","dependencies":["ad33ee8parabolas15a-h4"],"title":"Finding the Axis of Symmetry","text":"To find the axis of symmetry, we have to find the $$b$$ and a from our equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas15a-h6","type":"hint","dependencies":["ad33ee8parabolas15a-h5"],"title":"Finding the Axis of Symmetry","text":"Our $$b$$ and a are $$2$$ and $$-1$$ respectively. Our axis of symmetry is $$y=-1\\\\frac{-2}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas15a-h8","type":"hint","dependencies":["ad33ee8parabolas15a-h6"],"title":"Finding the Vertex","text":"Now that we have our axis of symmetry, we want to plug it back into the original equation to find the $$x$$ value of our vertex.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas15a-h8","type":"hint","dependencies":["ad33ee8parabolas15a-h8"],"title":"Finding the Vertex","text":"Since our $$y$$ is $$1$$, we plug it into our equation as, $$x=-\\\\left(1^2\\\\right)+2\\\\times1+8$$. This gives us $$x=9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas15a-h9","type":"hint","dependencies":["ad33ee8parabolas15a-h8"],"title":"Vertex","text":"Since we know that our $$y$$ is $$1$$ and our $$x$$ is $$9$$, our vertex is at the coordinates, $$(9,1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas15a-h10","type":"hint","dependencies":["ad33ee8parabolas15a-h9"],"title":"Finding Intercepts","text":"To graph our parabola, we need to know our $$x$$ and $$y$$ intercepts.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas15a-h11","type":"hint","dependencies":["ad33ee8parabolas15a-h10"],"title":"Finding the x-intercepts","text":"Our x-intercept occurs when $$y=0$$. Knowing this, we substitute $$0$$ into our equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas15a-h12","type":"hint","dependencies":["ad33ee8parabolas15a-h11"],"title":"Finding the x-intercepts","text":"After plugging in $$y=0$$ to our equation, we see that a x-intercept occurs when $$x=8$$, so our x-intercept is $$(8,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas15a-h13","type":"hint","dependencies":["ad33ee8parabolas15a-h12"],"title":"Finding the y-intercepts","text":"Our $$y$$ intercepts occur when $$x=0$$. We plug in $$x=0$$ to our equation and solve for the y\'s.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas15a-h14","type":"hint","dependencies":["ad33ee8parabolas15a-h13"],"title":"Finding the y-intercepts","text":"We must factor the GCF from our equation and factor the trinomial. After this, we solve for $$y$$, which are $$-2$$ and $$4$$ in this case, so the x-intercepts are $$(0,-2)$$ and $$(0,4)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas15a-h15","type":"hint","dependencies":["ad33ee8parabolas15a-h14"],"title":"Graphing","text":"Now that we have all of our properties, we are ready to graph our parabolas.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad33ee8parabolas16","title":"Graphing Horizontal Parabolas","body":"Graph the horizontal parabola given its properties.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Parabolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad33ee8parabolas16a","stepAnswer":["B"],"problemType":"MultipleChoice","stepTitle":"$$x=-\\\\left(y^2\\\\right)-4y+12$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"ad33ee8parabolas16a-h1","type":"hint","dependencies":[],"title":"Direction","text":"We can tell whether the parabola goes left or right based on the sign of a in the equation, $$x=a y^2+b y+c$$. If a is negative, then the parabola goes left, if it is positive, it goes right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas16a-h2","type":"hint","dependencies":["ad33ee8parabolas16a-h1"],"title":"Direction","text":"Since our a is negative, what does this tell us about our parabola?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas16a-h3","type":"hint","dependencies":["ad33ee8parabolas16a-h2"],"title":"Left Direction","text":"Since our a is negative, we know that our parabola will open up towards the left.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas16a-h4","type":"hint","dependencies":["ad33ee8parabolas16a-h3"],"title":"Finding the Vertex","text":"To find the vertex, we need to find the axis is symmetry, which is found using the formula, $$y=\\\\frac{-b}{2} a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas16a-h5","type":"hint","dependencies":["ad33ee8parabolas16a-h4"],"title":"Finding the Axis of Symmetry","text":"To find the axis of symmetry, we have to find the $$b$$ and a from our equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas16a-h6","type":"hint","dependencies":["ad33ee8parabolas16a-h5"],"title":"Finding the Axis of Symmetry","text":"Our $$b$$ and a are $$-4$$ and $$-1$$ respectively. Our axis of symmetry is $$y=-1\\\\frac{4}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas16a-h8","type":"hint","dependencies":["ad33ee8parabolas16a-h6"],"title":"Finding the Vertex","text":"Now that we have our axis of symmetry, we want to plug it back into the original equation to find the $$x$$ value of our vertex.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas16a-h8","type":"hint","dependencies":["ad33ee8parabolas16a-h8"],"title":"Finding the Vertex","text":"Since our $$y$$ is $$-2$$, we plug it into our equation as, $$x=2^2-4\\\\times-2+12$$. This gives us $$x=16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas16a-h9","type":"hint","dependencies":["ad33ee8parabolas16a-h8"],"title":"Vertex","text":"Since we know that our $$y$$ is $$-2$$ and our $$x$$ is $$16$$, our vertex is at the coordinates, $$(16,-2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas16a-h10","type":"hint","dependencies":["ad33ee8parabolas16a-h9"],"title":"Finding Intercepts","text":"To graph our parabola, we need to know our $$x$$ and $$y$$ intercepts.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas16a-h11","type":"hint","dependencies":["ad33ee8parabolas16a-h10"],"title":"Finding the x-intercepts","text":"Our x-intercept occurs when $$y=0$$. Knowing this, we substitute $$0$$ into our equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas16a-h12","type":"hint","dependencies":["ad33ee8parabolas16a-h11"],"title":"Finding the x-intercepts","text":"After plugging in $$y=0$$ to our equation, we see that a x-intercept occurs when $$x=12$$, so our x-intercept is $$(12,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas16a-h13","type":"hint","dependencies":["ad33ee8parabolas16a-h12"],"title":"Finding the y-intercepts","text":"Our $$y$$ intercepts occur when $$x=0$$. We plug in $$x=0$$ to our equation and solve for the y\'s.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas16a-h14","type":"hint","dependencies":["ad33ee8parabolas16a-h13"],"title":"Finding the y-intercepts","text":"We must factor the GCF from our equation and factor the trinomial. After this, we solve for $$y$$, which are $$-6$$ and $$2$$ in this case, so the x-intercepts are $$(0,-6)$$ and $$(0,2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas16a-h15","type":"hint","dependencies":["ad33ee8parabolas16a-h14"],"title":"Graphing","text":"Now that we have all of our properties, we are ready to graph our parabolas.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad33ee8parabolas17","title":"Graphing Horizontal Parabolas","body":"Graph the horizontal parabola given its properties.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Parabolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad33ee8parabolas17a","stepAnswer":["C"],"problemType":"MultipleChoice","stepTitle":"$$x=-\\\\left(y^2\\\\right)+2y-3$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"ad33ee8parabolas17a-h1","type":"hint","dependencies":[],"title":"Direction","text":"We can tell whether the parabola goes left or right based on the sign of a in the equation, $$x=a y^2+b y+c$$. If a is negative, then the parabola goes left, if it is positive, it goes right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas17a-h2","type":"hint","dependencies":["ad33ee8parabolas17a-h1"],"title":"Direction","text":"Since our a is negative, what does this tell us about our parabola?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas17a-h3","type":"hint","dependencies":["ad33ee8parabolas17a-h2"],"title":"Left Direction","text":"Since our a is negative, we know that our parabola will open up towards the left.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas17a-h4","type":"hint","dependencies":["ad33ee8parabolas17a-h3"],"title":"Finding the Vertex","text":"To find the vertex, we need to find the axis is symmetry, which is found using the formula, $$y=\\\\frac{-b}{2} a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas17a-h5","type":"hint","dependencies":["ad33ee8parabolas17a-h4"],"title":"Finding the Axis of Symmetry","text":"To find the axis of symmetry, we have to find the $$b$$ and a from our equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas17a-h6","type":"hint","dependencies":["ad33ee8parabolas17a-h5"],"title":"Finding the Axis of Symmetry","text":"Our $$b$$ and a are $$2$$ and $$-1$$ respectively. Our axis of symmetry is $$y=-1\\\\frac{-2}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas17a-h8","type":"hint","dependencies":["ad33ee8parabolas17a-h6"],"title":"Finding the Vertex","text":"Now that we have our axis of symmetry, we want to plug it back into the original equation to find the $$x$$ value of our vertex.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas17a-h8","type":"hint","dependencies":["ad33ee8parabolas17a-h8"],"title":"Finding the Vertex","text":"Since our $$y$$ is $$1$$, we plug it into our equation as, $$x=-\\\\left(1^2\\\\right)+2\\\\times1-3$$. This gives us $$x=-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas17a-h9","type":"hint","dependencies":["ad33ee8parabolas17a-h8"],"title":"Vertex","text":"Since we know that our $$y$$ is $$1$$ and our $$x$$ is $$-2$$, our vertex is at the coordinates, $$(-2,1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas17a-h10","type":"hint","dependencies":["ad33ee8parabolas17a-h9"],"title":"Finding Intercepts","text":"To graph our parabola, we need to know our $$x$$ and $$y$$ intercepts.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas17a-h11","type":"hint","dependencies":["ad33ee8parabolas17a-h10"],"title":"Finding the x-intercepts","text":"Our x-intercept occurs when $$y=0$$. Knowing this, we substitute $$0$$ into our equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas17a-h12","type":"hint","dependencies":["ad33ee8parabolas17a-h11"],"title":"Finding the x-intercepts","text":"After plugging in $$y=0$$ to our equation, we see that a x-intercept occurs when $$x=-3$$, so our x-intercept is $$(-3,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas17a-h13","type":"hint","dependencies":["ad33ee8parabolas17a-h12"],"title":"Finding the y-intercepts","text":"Our $$y$$ intercepts occur when $$x=0$$. We plug in $$x=0$$ to our equation and solve for the y\'s.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas17a-h14","type":"hint","dependencies":["ad33ee8parabolas17a-h13"],"title":"Finding the y-intercepts","text":"We must factor the GCF from our equation and factor the trinomial. After this, we find that we don\'t have any y-intercepts.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas17a-h15","type":"hint","dependencies":["ad33ee8parabolas17a-h14"],"title":"Graphing","text":"Now that we have all of our properties, we are ready to graph our parabolas.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad33ee8parabolas18","title":"Graphing Horizontal Parabolas","body":"Graph the horizontal parabola given its properties.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Parabolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad33ee8parabolas18a","stepAnswer":["A"],"problemType":"MultipleChoice","stepTitle":"$$x=-\\\\left(y^2\\\\right)-2y+3$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"ad33ee8parabolas18a-h1","type":"hint","dependencies":[],"title":"Direction","text":"We can tell whether the parabola goes left or right based on the sign of a in the equation, $$x=a y^2+b y+c$$. If a is negative, then the parabola goes left, if it is positive, it goes right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas18a-h2","type":"hint","dependencies":["ad33ee8parabolas18a-h1"],"title":"Direction","text":"Since our a is negative, what does this tell us about our parabola?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas18a-h3","type":"hint","dependencies":["ad33ee8parabolas18a-h2"],"title":"Left Direction","text":"Since our a is negative, we know that our parabola will open up towards the left.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas18a-h4","type":"hint","dependencies":["ad33ee8parabolas18a-h3"],"title":"Finding the Vertex","text":"To find the vertex, we need to find the axis is symmetry, which is found using the formula, $$y=\\\\frac{-b}{2} a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas18a-h5","type":"hint","dependencies":["ad33ee8parabolas18a-h4"],"title":"Finding the Axis of Symmetry","text":"To find the axis of symmetry, we have to find the $$b$$ and a from our equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas18a-h6","type":"hint","dependencies":["ad33ee8parabolas18a-h5"],"title":"Finding the Axis of Symmetry","text":"Our $$b$$ and a are $$-2$$ and $$-1$$ respectively. Our axis of symmetry is $$y=-1\\\\frac{2}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas18a-h8","type":"hint","dependencies":["ad33ee8parabolas18a-h6"],"title":"Finding the Vertex","text":"Now that we have our axis of symmetry, we want to plug it back into the original equation to find the $$x$$ value of our vertex.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas18a-h8","type":"hint","dependencies":["ad33ee8parabolas18a-h8"],"title":"Finding the Vertex","text":"Since our $$y$$ is $$-1$$, we plug it into our equation as, $$x=-\\\\left(1^2\\\\right)-2\\\\times-1+3$$. This gives us $$x=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas18a-h9","type":"hint","dependencies":["ad33ee8parabolas18a-h8"],"title":"Vertex","text":"Since we know that our $$y$$ is $$-1$$ and our $$x$$ is $$4$$, our vertex is at the coordinates, $$(4,-1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas18a-h10","type":"hint","dependencies":["ad33ee8parabolas18a-h9"],"title":"Finding Intercepts","text":"To graph our parabola, we need to know our $$x$$ and $$y$$ intercepts.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas18a-h11","type":"hint","dependencies":["ad33ee8parabolas18a-h10"],"title":"Finding the x-intercepts","text":"Our x-intercept occurs when $$y=0$$. Knowing this, we substitute $$0$$ into our equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas18a-h12","type":"hint","dependencies":["ad33ee8parabolas18a-h11"],"title":"Finding the x-intercepts","text":"After plugging in $$y=0$$ to our equation, we see that a x-intercept occurs when $$x=3$$, so our x-intercept is $$(3,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas18a-h13","type":"hint","dependencies":["ad33ee8parabolas18a-h12"],"title":"Finding the y-intercepts","text":"Our $$y$$ intercepts occur when $$x=0$$. We plug in $$x=0$$ to our equation and solve for the y\'s.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas18a-h14","type":"hint","dependencies":["ad33ee8parabolas18a-h13"],"title":"Finding the y-intercepts","text":"We must factor the GCF from our equation and factor the trinomial. After this, we solve for $$y$$, which are $$-3$$ and $$1$$ in this case, so the x-intercepts are $$(0,-3)$$ and $$(0,1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas18a-h15","type":"hint","dependencies":["ad33ee8parabolas18a-h14"],"title":"Graphing","text":"Now that we have all of our properties, we are ready to graph our parabolas.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad33ee8parabolas2","title":"Graphing Vertical Parabolas","body":"Graph the vertical parabola using its properties.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Parabolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad33ee8parabolas2a","stepAnswer":["C"],"problemType":"MultipleChoice","stepTitle":"$$y=\\\\left(-x^2\\\\right)+5x-6$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"ad33ee8parabolas2a-h1","type":"hint","dependencies":[],"title":"Direction","text":"We can tell whether the parabola goes up or down based on the sign of a in the equation, $$y=a x^2+b x+c$$. If a is negative, then the parabola goes downwards, if it is positive, it goes upwards.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas2a-h2","type":"hint","dependencies":["ad33ee8parabolas2a-h1"],"title":"Direction","text":"Since our a is negative, what does this tell us about our parabola?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas2a-h3","type":"hint","dependencies":["ad33ee8parabolas2a-h2"],"title":"Downwards Direction","text":"Since our a is negative, we know that our parabola will open up downwards.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas2a-h4","type":"hint","dependencies":["ad33ee8parabolas2a-h3"],"title":"Finding the Vertex","text":"To find the vertex, we need to find the axis is symmetry, which is found using the formula, $$x=\\\\frac{-b}{2} a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas2a-h5","type":"hint","dependencies":["ad33ee8parabolas2a-h4"],"title":"Finding the Axis of Symmetry","text":"To find the axis of symmetry, we have to find the $$b$$ and a from our equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas2a-h6","type":"hint","dependencies":["ad33ee8parabolas2a-h5"],"title":"Finding the Axis of Symmetry","text":"Our $$b$$ and a are $$5$$ and $$-1$$ respectively. Our axis of symmetry is $$x=-1\\\\frac{-5}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas2a-h8","type":"hint","dependencies":["ad33ee8parabolas2a-h6"],"title":"Finding the Vertex","text":"Now that we have our axis of symmetry, we want to plug it back into the original equation to find the $$y$$ value of our vertex.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas2a-h8","type":"hint","dependencies":["ad33ee8parabolas2a-h8"],"title":"Finding the Vertex","text":"Since our $$x$$ is $$2.5$$, we plug it into our equation as, $$y=-\\\\left({2.5}^2\\\\right)+5\\\\times2.5-6$$. This gives us $$y=0.25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas2a-h9","type":"hint","dependencies":["ad33ee8parabolas2a-h8"],"title":"Vertex","text":"Since we know that our $$x$$ is $$2.5$$ and our $$y$$ is $$0.25$$, our vertex is at the coordinates, $$(2.5, 0.25)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas2a-h10","type":"hint","dependencies":["ad33ee8parabolas2a-h9"],"title":"Finding Intercepts","text":"To graph our parabola, we need to know our $$x$$ and $$y$$ intercepts.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas2a-h11","type":"hint","dependencies":["ad33ee8parabolas2a-h10"],"title":"Finding the y-intercepts","text":"Our y-intercept occurs when $$x=0$$. Knowing this, we substitute $$0$$ into our equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas2a-h12","type":"hint","dependencies":["ad33ee8parabolas2a-h11"],"title":"Finding the y-intercepts","text":"After plugging in $$x=0$$ to our equation, we see that a y-intercept occurs when $$y=-6$$, so our y-intercept is $$(0,-6)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas2a-h13","type":"hint","dependencies":["ad33ee8parabolas2a-h12"],"title":"Finding the x-intercepts","text":"Our $$x$$ intercepts occur when $$y=0$$. We plug in $$y=0$$ to our equation and solve for the x\'s.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas2a-h14","type":"hint","dependencies":["ad33ee8parabolas2a-h13"],"title":"Finding the x-intercepts","text":"We must factor the GCF from our equation and factor the trinomial. After this, we solve for $$x$$, which are $$2$$ and $$3$$ in this case, so the x-intercepts are $$(2,0)$$ and $$(3,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas2a-h15","type":"hint","dependencies":["ad33ee8parabolas2a-h14"],"title":"Graphing","text":"Now that we have all of our properties, we are ready to graph our parabolas.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad33ee8parabolas23","title":"Graphing Horizontal Parabolas from the Origin","body":"Graph the horizontal parabola given its properties.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Parabolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad33ee8parabolas23a","stepAnswer":["B"],"problemType":"MultipleChoice","stepTitle":"$$x=-3y^2$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"ad33ee8parabolas23a-h1","type":"hint","dependencies":[],"title":"Direction","text":"We can tell whether the parabola goes left or right based on the sign of a in the equation, $$x=a y^2+b y+c$$. If a is negative, then the parabola goes left, if it is positive, it goes right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas23a-h2","type":"hint","dependencies":["ad33ee8parabolas23a-h1"],"title":"Direction","text":"Since our a is negative, what does this tell us about our parabola?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas23a-h3","type":"hint","dependencies":["ad33ee8parabolas23a-h2"],"title":"Left Direction","text":"Since our a is negative, we know that our parabola will open up towards the left.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas23a-h4","type":"hint","dependencies":["ad33ee8parabolas23a-h3"],"title":"Finding the Vertex","text":"To find the vertex, we need to find the axis is symmetry, which is found using the formula, $$y=\\\\frac{-b}{2} a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas23a-h5","type":"hint","dependencies":["ad33ee8parabolas23a-h4"],"title":"Finding the Axis of Symmetry","text":"To find the axis of symmetry, we have to find the $$b$$ and a from our equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas23a-h6","type":"hint","dependencies":["ad33ee8parabolas23a-h5"],"title":"Finding the Axis of Symmetry","text":"Our $$b$$ and a are $$0$$ and $$3$$ respectively. Our axis of symmetry is $$y=-3\\\\frac{0}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas23a-h8","type":"hint","dependencies":["ad33ee8parabolas23a-h6"],"title":"Finding the Vertex","text":"Now that we have our axis of symmetry, we want to plug it back into the original equation to find the $$x$$ value of our vertex.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas23a-h8","type":"hint","dependencies":["ad33ee8parabolas23a-h8"],"title":"Finding the Vertex","text":"Since our $$y$$ is $$0$$, we plug it into our equation as, $$x=3\\\\times0^2$$. This gives us $$x=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas23a-h9","type":"hint","dependencies":["ad33ee8parabolas23a-h8"],"title":"Vertex","text":"Since we know that our $$y$$ is $$0$$ and our $$x$$ is $$0$$, our vertex is at the coordinates, $$(0,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas23a-h10","type":"hint","dependencies":["ad33ee8parabolas23a-h9"],"title":"Intercepts","text":"Since the vertex is $$(0,0)$$, the intercepts are also both $$(0,0)$$, so we need to plot points to graph it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas23a-h11","type":"hint","dependencies":["ad33ee8parabolas23a-h10"],"title":"Graphing","text":"Note that points are symmetric to each other across the x-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad33ee8parabolas24","title":"Graphing Horizontal Parabolas from the Origin","body":"Graph the horizontal parabola given its properties.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Parabolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad33ee8parabolas24a","stepAnswer":["A"],"problemType":"MultipleChoice","stepTitle":"$$x=-4y^2$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"ad33ee8parabolas24a-h1","type":"hint","dependencies":[],"title":"Direction","text":"We can tell whether the parabola goes left or right based on the sign of a in the equation, $$x=a y^2+b y+c$$. If a is negative, then the parabola goes left, if it is positive, it goes right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas24a-h2","type":"hint","dependencies":["ad33ee8parabolas24a-h1"],"title":"Direction","text":"Since our a is negative, what does this tell us about our parabola?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas24a-h3","type":"hint","dependencies":["ad33ee8parabolas24a-h2"],"title":"Left Direction","text":"Since our a is negative, we know that our parabola will open up towards the left.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas24a-h4","type":"hint","dependencies":["ad33ee8parabolas24a-h3"],"title":"Finding the Vertex","text":"To find the vertex, we need to find the axis is symmetry, which is found using the formula, $$y=\\\\frac{-b}{2} a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas24a-h5","type":"hint","dependencies":["ad33ee8parabolas24a-h4"],"title":"Finding the Axis of Symmetry","text":"To find the axis of symmetry, we have to find the $$b$$ and a from our equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas24a-h6","type":"hint","dependencies":["ad33ee8parabolas24a-h5"],"title":"Finding the Axis of Symmetry","text":"Our $$b$$ and a are $$0$$ and $$-4$$ respectively. Our axis of symmetry is $$y=-4\\\\frac{0}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas24a-h8","type":"hint","dependencies":["ad33ee8parabolas24a-h6"],"title":"Finding the Vertex","text":"Now that we have our axis of symmetry, we want to plug it back into the original equation to find the $$x$$ value of our vertex.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas24a-h8","type":"hint","dependencies":["ad33ee8parabolas24a-h8"],"title":"Finding the Vertex","text":"Since our $$y$$ is $$0$$, we plug it into our equation as, $$x=3\\\\times0^2$$. This gives us $$x=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas24a-h9","type":"hint","dependencies":["ad33ee8parabolas24a-h8"],"title":"Vertex","text":"Since we know that our $$y$$ is $$0$$ and our $$x$$ is $$0$$, our vertex is at the coordinates, $$(0,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas24a-h10","type":"hint","dependencies":["ad33ee8parabolas24a-h9"],"title":"Intercepts","text":"Since the vertex is $$(0,0)$$, the intercepts are also both $$(0,0)$$, so we need to plot points to graph it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas24a-h11","type":"hint","dependencies":["ad33ee8parabolas24a-h10"],"title":"Graphing","text":"Note that points are symmetric to each other across the x-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad33ee8parabolas25","title":"Graphing Horizontal Parabolas from the Origin","body":"Graph the horizontal parabola given its properties.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Parabolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad33ee8parabolas25a","stepAnswer":["C"],"problemType":"MultipleChoice","stepTitle":"$$x=4y^2$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"ad33ee8parabolas25a-h1","type":"hint","dependencies":[],"title":"Direction","text":"We can tell whether the parabola goes left or right based on the sign of a in the equation, $$x=a y^2+b y+c$$. If a is negative, then the parabola goes left, if it is positive, it goes right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas25a-h2","type":"hint","dependencies":["ad33ee8parabolas25a-h1"],"title":"Direction","text":"Since our a is positive, what does this tell us about our parabola?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas25a-h3","type":"hint","dependencies":["ad33ee8parabolas25a-h2"],"title":"Right Direction","text":"Since our a is postitive, we know that our parabola will open up towards the right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas25a-h4","type":"hint","dependencies":["ad33ee8parabolas25a-h3"],"title":"Finding the Vertex","text":"To find the vertex, we need to find the axis is symmetry, which is found using the formula, $$y=\\\\frac{-b}{2} a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas25a-h5","type":"hint","dependencies":["ad33ee8parabolas25a-h4"],"title":"Finding the Axis of Symmetry","text":"To find the axis of symmetry, we have to find the $$b$$ and a from our equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas25a-h6","type":"hint","dependencies":["ad33ee8parabolas25a-h5"],"title":"Finding the Axis of Symmetry","text":"Our $$b$$ and a are $$0$$ and $$4$$ respectively. Our axis of symmetry is $$y=4\\\\frac{0}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas25a-h8","type":"hint","dependencies":["ad33ee8parabolas25a-h6"],"title":"Finding the Vertex","text":"Now that we have our axis of symmetry, we want to plug it back into the original equation to find the $$x$$ value of our vertex.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas25a-h8","type":"hint","dependencies":["ad33ee8parabolas25a-h8"],"title":"Finding the Vertex","text":"Since our $$y$$ is $$0$$, we plug it into our equation as, $$x=3\\\\times0^2$$. This gives us $$x=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas25a-h9","type":"hint","dependencies":["ad33ee8parabolas25a-h8"],"title":"Vertex","text":"Since we know that our $$y$$ is $$0$$ and our $$x$$ is $$0$$, our vertex is at the coordinates, $$(0,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas25a-h10","type":"hint","dependencies":["ad33ee8parabolas25a-h9"],"title":"Intercepts","text":"Since the vertex is $$(0,0)$$, the intercepts are also both $$(0,0)$$, so we need to plot points to graph it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas25a-h11","type":"hint","dependencies":["ad33ee8parabolas25a-h10"],"title":"Graphing","text":"Note that points are symmetric to each other across the x-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad33ee8parabolas26","title":"Graphing Horizontal Parabolas from the Origin","body":"Graph the horizontal parabola given its properties.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Parabolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad33ee8parabolas26a","stepAnswer":["D"],"problemType":"MultipleChoice","stepTitle":"$$x=3y^2$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"ad33ee8parabolas26a-h1","type":"hint","dependencies":[],"title":"Direction","text":"We can tell whether the parabola goes left or right based on the sign of a in the equation, $$x=a y^2+b y+c$$. If a is negative, then the parabola goes left, if it is positive, it goes right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas26a-h2","type":"hint","dependencies":["ad33ee8parabolas26a-h1"],"title":"Direction","text":"Since our a is positive, what does this tell us about our parabola?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas26a-h3","type":"hint","dependencies":["ad33ee8parabolas26a-h2"],"title":"Right Direction","text":"Since our a is postitive, we know that our parabola will open up towards the right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas26a-h4","type":"hint","dependencies":["ad33ee8parabolas26a-h3"],"title":"Finding the Vertex","text":"To find the vertex, we need to find the axis is symmetry, which is found using the formula, $$y=\\\\frac{-b}{2} a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas26a-h5","type":"hint","dependencies":["ad33ee8parabolas26a-h4"],"title":"Finding the Axis of Symmetry","text":"To find the axis of symmetry, we have to find the $$b$$ and a from our equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas26a-h6","type":"hint","dependencies":["ad33ee8parabolas26a-h5"],"title":"Finding the Axis of Symmetry","text":"Our $$b$$ and a are $$0$$ and $$3$$ respectively. Our axis of symmetry is $$y=3\\\\frac{0}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas26a-h8","type":"hint","dependencies":["ad33ee8parabolas26a-h6"],"title":"Finding the Vertex","text":"Now that we have our axis of symmetry, we want to plug it back into the original equation to find the $$x$$ value of our vertex.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas26a-h8","type":"hint","dependencies":["ad33ee8parabolas26a-h8"],"title":"Finding the Vertex","text":"Since our $$y$$ is $$0$$, we plug it into our equation as, $$x=3\\\\times0^2$$. This gives us $$x=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas26a-h9","type":"hint","dependencies":["ad33ee8parabolas26a-h8"],"title":"Vertex","text":"Since we know that our $$y$$ is $$0$$ and our $$x$$ is $$0$$, our vertex is at the coordinates, $$(0,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas26a-h10","type":"hint","dependencies":["ad33ee8parabolas26a-h9"],"title":"Intercepts","text":"Since the vertex is $$(0,0)$$, the intercepts are also both $$(0,0)$$, so we need to plot points to graph it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas26a-h11","type":"hint","dependencies":["ad33ee8parabolas26a-h10"],"title":"Graphing","text":"Note that points are symmetric to each other across the x-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad33ee8parabolas27","title":"Rewriting the equation in Standard Form.","body":"Rewrite the given equation into standard form.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Parabolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad33ee8parabolas27a","stepAnswer":["$$2{\\\\left(y+3\\\\right)}^2-1$$"],"problemType":"TextBox","stepTitle":"$$x=2y^2+12y+17$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2{\\\\left(y+3\\\\right)}^2-1$$","hints":{"DefaultPathway":[{"id":"ad33ee8parabolas27a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"The standard form is $$x={a\\\\left(y-h\\\\right)}^2+k$$. We are given the equation in $$x=a y^2+b y+c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas27a-h2","type":"hint","dependencies":["ad33ee8parabolas27a-h1"],"title":"Finding a","text":"To find a, we must divide our $$a y^2$$ term by a by factoring by grouping.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas27a-h3","type":"hint","dependencies":["ad33ee8parabolas27a-h2"],"title":"Completing the Square","text":"After we find a, our equation now looks like, $$a \\\\left(y^2+\\\\frac{b}{a} x\\\\right)+c$$. We must complete the square by adding $${\\\\left(\\\\frac{\\\\frac{b}{a}}{2}\\\\right)}^2$$ into the parenthesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas27a-h4","type":"hint","dependencies":["ad33ee8parabolas27a-h3"],"title":"Completing the Square","text":"Since we can\'t simply add a number out of thin air, we must take our equation and subtract it by what we added into the parenthesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas27a-h5","type":"hint","dependencies":["ad33ee8parabolas27a-h4"],"title":"Completing the Square","text":"After doing so, our equation should now be in standard form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad33ee8parabolas28","title":"Rewriting the equation in Standard Form.","body":"Rewrite the given equation into standard form.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Parabolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad33ee8parabolas28a","stepAnswer":["$$3{\\\\left(y+1\\\\right)}^2+4$$"],"problemType":"TextBox","stepTitle":"$$x=3y^2+6y+7$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3{\\\\left(y+1\\\\right)}^2+4$$","hints":{"DefaultPathway":[{"id":"ad33ee8parabolas28a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"The standard form is $$x={a\\\\left(y-h\\\\right)}^2+k$$. We are given the equation in $$x=a y^2+b y+c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas28a-h2","type":"hint","dependencies":["ad33ee8parabolas28a-h1"],"title":"Finding a","text":"To find a, we must divide our $$a y^2$$ term by a by factoring by grouping.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas28a-h3","type":"hint","dependencies":["ad33ee8parabolas28a-h2"],"title":"Completing the Square","text":"After we find a, our equation now looks like, $$a \\\\left(y^2+\\\\frac{b}{a} x\\\\right)+c$$. We must complete the square by adding $${\\\\left(\\\\frac{\\\\frac{b}{a}}{2}\\\\right)}^2$$ into the parenthesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas28a-h4","type":"hint","dependencies":["ad33ee8parabolas28a-h3"],"title":"Completing the Square","text":"Since we can\'t simply add a number out of thin air, we must take our equation and subtract it by what we added into the parenthesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas28a-h5","type":"hint","dependencies":["ad33ee8parabolas28a-h4"],"title":"Completing the Square","text":"After doing so, our equation should now be in standard form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad33ee8parabolas29","title":"Rewriting the equation in Standard Form.","body":"Rewrite the given equation into standard form.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Parabolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad33ee8parabolas29a","stepAnswer":["$$-4{\\\\left(y+2\\\\right)}^2+4$$"],"problemType":"TextBox","stepTitle":"$$x=-4y^2-16y-12$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-4{\\\\left(y+2\\\\right)}^2+4$$","hints":{"DefaultPathway":[{"id":"ad33ee8parabolas29a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"The standard form is $$x={a\\\\left(y-h\\\\right)}^2+k$$. We are given the equation in $$x=a y^2+b y+c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas29a-h2","type":"hint","dependencies":["ad33ee8parabolas29a-h1"],"title":"Finding a","text":"To find a, we must divide our $$a y^2$$ term by a by factoring by grouping.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas29a-h3","type":"hint","dependencies":["ad33ee8parabolas29a-h2"],"title":"Completing the Square","text":"After we find a, our equation now looks like, $$a \\\\left(y^2+\\\\frac{b}{a} x\\\\right)+c$$. We must complete the square by adding $${\\\\left(\\\\frac{\\\\frac{b}{a}}{2}\\\\right)}^2$$ into the parenthesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas29a-h4","type":"hint","dependencies":["ad33ee8parabolas29a-h3"],"title":"Completing the Square","text":"Since we can\'t simply add a number out of thin air, we must take our equation and subtract it by what we added into the parenthesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas29a-h5","type":"hint","dependencies":["ad33ee8parabolas29a-h4"],"title":"Completing the Square","text":"After doing so, our equation should now be in standard form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad33ee8parabolas3","title":"Graphing Vertical Parabolas","body":"Graph the vertical parabola using its properties.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Parabolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad33ee8parabolas3a","stepAnswer":["A"],"problemType":"MultipleChoice","stepTitle":"$$y=\\\\left(-x^2\\\\right)+8x-12$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"ad33ee8parabolas3a-h1","type":"hint","dependencies":[],"title":"Direction","text":"We can tell whether the parabola goes up or down based on the sign of a in the equation, $$y=a x^2+b x+c$$. If a is negative, then the parabola goes downwards, if it is positive, it goes upwards.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas3a-h2","type":"hint","dependencies":["ad33ee8parabolas3a-h1"],"title":"Direction","text":"Since our a is negative, what does this tell us about our parabola?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas3a-h3","type":"hint","dependencies":["ad33ee8parabolas3a-h2"],"title":"Downwards Direction","text":"Since our a is negative, we know that our parabola will open up downwards.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas3a-h4","type":"hint","dependencies":["ad33ee8parabolas3a-h3"],"title":"Finding the Vertex","text":"To find the vertex, we need to find the axis is symmetry, which is found using the formula, $$x=\\\\frac{-b}{2} a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas3a-h5","type":"hint","dependencies":["ad33ee8parabolas3a-h4"],"title":"Finding the Axis of Symmetry","text":"To find the axis of symmetry, we have to find the $$b$$ and a from our equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas3a-h6","type":"hint","dependencies":["ad33ee8parabolas3a-h5"],"title":"Finding the Axis of Symmetry","text":"Our $$b$$ and a are $$8$$ and $$-1$$ respectively. Our axis of symmetry is $$x=-1\\\\frac{-8}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas3a-h8","type":"hint","dependencies":["ad33ee8parabolas3a-h6"],"title":"Finding the Vertex","text":"Now that we have our axis of symmetry, we want to plug it back into the original equation to find the $$y$$ value of our vertex.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas3a-h8","type":"hint","dependencies":["ad33ee8parabolas3a-h8"],"title":"Finding the Vertex","text":"Since our $$x$$ is $$4$$, we plug it into our equation as, $$y=-\\\\left(4^2\\\\right)+8\\\\times4-12$$. This gives us $$y=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas3a-h9","type":"hint","dependencies":["ad33ee8parabolas3a-h8"],"title":"Vertex","text":"Since we know that our $$x$$ is $$4$$ and our $$y$$ is $$4$$, our vertex is at the coordinates, $$(4,4)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas3a-h10","type":"hint","dependencies":["ad33ee8parabolas3a-h9"],"title":"Finding Intercepts","text":"To graph our parabola, we need to know our $$x$$ and $$y$$ intercepts.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas3a-h11","type":"hint","dependencies":["ad33ee8parabolas3a-h10"],"title":"Finding the y-intercepts","text":"Our y-intercept occurs when $$x=0$$. Knowing this, we substitute $$0$$ into our equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas3a-h12","type":"hint","dependencies":["ad33ee8parabolas3a-h11"],"title":"Finding the y-intercepts","text":"After plugging in $$x=0$$ to our equation, we see that a y-intercept occurs when $$y=-12$$, so our y-intercept is $$(0,-12)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas3a-h13","type":"hint","dependencies":["ad33ee8parabolas3a-h12"],"title":"Finding the x-intercepts","text":"Our $$x$$ intercepts occur when $$y=0$$. We plug in $$y=0$$ to our equation and solve for the x\'s.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas3a-h14","type":"hint","dependencies":["ad33ee8parabolas3a-h13"],"title":"Finding the x-intercepts","text":"We must factor the GCF from our equation and factor the trinomial. After this, we solve for $$x$$, which are $$2$$ and $$6$$ in this case, so the x-intercepts are $$(2,0)$$ and $$(6,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas3a-h15","type":"hint","dependencies":["ad33ee8parabolas3a-h14"],"title":"Graphing","text":"Now that we have all of our properties, we are ready to graph our parabolas.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad33ee8parabolas30","title":"Rewriting the equation in Standard Form.","body":"Rewrite the given equation into standard form.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Parabolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad33ee8parabolas30a","stepAnswer":["$$-3{\\\\left(y+1\\\\right)}^2-2$$"],"problemType":"TextBox","stepTitle":"$$x=-3y^2-6y-5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-3{\\\\left(y+1\\\\right)}^2-2$$","hints":{"DefaultPathway":[{"id":"ad33ee8parabolas30a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"The standard form is $$x={a\\\\left(y-h\\\\right)}^2+k$$. We are given the equation in $$x=a y^2+b y+c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas30a-h2","type":"hint","dependencies":["ad33ee8parabolas30a-h1"],"title":"Finding a","text":"To find a, we must divide our $$a y^2$$ term by a by factoring by grouping.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas30a-h3","type":"hint","dependencies":["ad33ee8parabolas30a-h2"],"title":"Completing the Square","text":"After we find a, our equation now looks like, $$a \\\\left(y^2+\\\\frac{b}{a} x\\\\right)+c$$. We must complete the square by adding $${\\\\left(\\\\frac{\\\\frac{b}{a}}{2}\\\\right)}^2$$ into the parenthesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas30a-h4","type":"hint","dependencies":["ad33ee8parabolas30a-h3"],"title":"Completing the Square","text":"Since we can\'t simply add a number out of thin air, we must take our equation and subtract it by what we added into the parenthesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas30a-h5","type":"hint","dependencies":["ad33ee8parabolas30a-h4"],"title":"Completing the Square","text":"After doing so, our equation should now be in standard form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad33ee8parabolas4","title":"Graphing Vertical Parabolas","body":"Graph the vertical parabola using its properties.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Parabolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad33ee8parabolas4a","stepAnswer":["B"],"problemType":"MultipleChoice","stepTitle":"$$y=\\\\left(-x^2\\\\right)+4x-3$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"ad33ee8parabolas4a-h1","type":"hint","dependencies":[],"title":"Direction","text":"We can tell whether the parabola goes up or down based on the sign of a in the equation, $$y=a x^2+b x+c$$. If a is negative, then the parabola goes downwards, if it is positive, it goes upwards.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas4a-h2","type":"hint","dependencies":["ad33ee8parabolas4a-h1"],"title":"Direction","text":"Since our a is negative, what does this tell us about our parabola?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas4a-h3","type":"hint","dependencies":["ad33ee8parabolas4a-h2"],"title":"Downwards Direction","text":"Since our a is negative, we know that our parabola will open up downwards.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas4a-h4","type":"hint","dependencies":["ad33ee8parabolas4a-h3"],"title":"Finding the Vertex","text":"To find the vertex, we need to find the axis is symmetry, which is found using the formula, $$x=\\\\frac{-b}{2} a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas4a-h5","type":"hint","dependencies":["ad33ee8parabolas4a-h4"],"title":"Finding the Axis of Symmetry","text":"To find the axis of symmetry, we have to find the $$b$$ and a from our equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas4a-h6","type":"hint","dependencies":["ad33ee8parabolas4a-h5"],"title":"Finding the Axis of Symmetry","text":"Our $$b$$ and a are $$4$$ and $$-1$$ respectively. Our axis of symmetry is $$x=-1\\\\frac{-4}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas4a-h8","type":"hint","dependencies":["ad33ee8parabolas4a-h6"],"title":"Finding the Vertex","text":"Now that we have our axis of symmetry, we want to plug it back into the original equation to find the $$y$$ value of our vertex.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas4a-h8","type":"hint","dependencies":["ad33ee8parabolas4a-h8"],"title":"Finding the Vertex","text":"Since our $$x$$ is $$2$$, we plug it into our equation as, $$y=-\\\\left(2^2\\\\right)+4\\\\times2-3$$. This gives us $$y=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas4a-h9","type":"hint","dependencies":["ad33ee8parabolas4a-h8"],"title":"Vertex","text":"Since we know that our $$x$$ is $$2$$ and our $$y$$ is $$1$$, our vertex is at the coordinates, $$(2,1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas4a-h10","type":"hint","dependencies":["ad33ee8parabolas4a-h9"],"title":"Finding Intercepts","text":"To graph our parabola, we need to know our $$x$$ and $$y$$ intercepts.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas4a-h11","type":"hint","dependencies":["ad33ee8parabolas4a-h10"],"title":"Finding the y-intercepts","text":"Our y-intercept occurs when $$x=0$$. Knowing this, we substitute $$0$$ into our equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas4a-h12","type":"hint","dependencies":["ad33ee8parabolas4a-h11"],"title":"Finding the y-intercepts","text":"After plugging in $$x=0$$ to our equation, we see that a y-intercept occurs when $$y=-3$$, so our y-intercept is $$(0,-3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas4a-h13","type":"hint","dependencies":["ad33ee8parabolas4a-h12"],"title":"Finding the x-intercepts","text":"Our $$x$$ intercepts occur when $$y=0$$. We plug in $$y=0$$ to our equation and solve for the x\'s.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas4a-h14","type":"hint","dependencies":["ad33ee8parabolas4a-h13"],"title":"Finding the x-intercepts","text":"We must factor the GCF from our equation and factor the trinomial. After this, we solve for $$x$$, which are $$1$$ and $$3$$ in this case, so the x-intercepts are $$(1,0)$$ and $$(3,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas4a-h15","type":"hint","dependencies":["ad33ee8parabolas4a-h14"],"title":"Graphing","text":"Now that we have all of our properties, we are ready to graph our parabolas.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad33ee8parabolas5","title":"Graphing Vertical Parabolas","body":"Graph the vertical parabola using its properties.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Parabolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad33ee8parabolas5a","stepAnswer":["D"],"problemType":"MultipleChoice","stepTitle":"$$y=\\\\left(-x^2\\\\right)+8x-15$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"ad33ee8parabolas5a-h1","type":"hint","dependencies":[],"title":"Direction","text":"We can tell whether the parabola goes up or down based on the sign of a in the equation, $$y=a x^2+b x+c$$. If a is negative, then the parabola goes downwards, if it is positive, it goes upwards.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas5a-h2","type":"hint","dependencies":["ad33ee8parabolas5a-h1"],"title":"Direction","text":"Since our a is negative, what does this tell us about our parabola?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas5a-h3","type":"hint","dependencies":["ad33ee8parabolas5a-h2"],"title":"Downwards Direction","text":"Since our a is negative, we know that our parabola will open up downwards.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas5a-h4","type":"hint","dependencies":["ad33ee8parabolas5a-h3"],"title":"Finding the Vertex","text":"To find the vertex, we need to find the axis is symmetry, which is found using the formula, $$x=\\\\frac{-b}{2} a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas5a-h5","type":"hint","dependencies":["ad33ee8parabolas5a-h4"],"title":"Finding the Axis of Symmetry","text":"To find the axis of symmetry, we have to find the $$b$$ and a from our equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas5a-h6","type":"hint","dependencies":["ad33ee8parabolas5a-h5"],"title":"Finding the Axis of Symmetry","text":"Our $$b$$ and a are $$8$$ and $$-1$$ respectively. Our axis of symmetry is $$x=-1\\\\frac{-8}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas5a-h8","type":"hint","dependencies":["ad33ee8parabolas5a-h6"],"title":"Finding the Vertex","text":"Now that we have our axis of symmetry, we want to plug it back into the original equation to find the $$y$$ value of our vertex.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas5a-h8","type":"hint","dependencies":["ad33ee8parabolas5a-h8"],"title":"Finding the Vertex","text":"Since our $$x$$ is $$4$$, we plug it into our equation as, $$y=-\\\\left(4^2\\\\right)+8\\\\times4-3$$. This gives us $$y=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas5a-h9","type":"hint","dependencies":["ad33ee8parabolas5a-h8"],"title":"Vertex","text":"Since we know that our $$x$$ is $$4$$ and our $$y$$ is $$1$$, our vertex is at the coordinates, $$(4,1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas5a-h10","type":"hint","dependencies":["ad33ee8parabolas5a-h9"],"title":"Finding Intercepts","text":"To graph our parabola, we need to know our $$x$$ and $$y$$ intercepts.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas5a-h11","type":"hint","dependencies":["ad33ee8parabolas5a-h10"],"title":"Finding the y-intercepts","text":"Our y-intercept occurs when $$x=0$$. Knowing this, we substitute $$0$$ into our equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas5a-h12","type":"hint","dependencies":["ad33ee8parabolas5a-h11"],"title":"Finding the y-intercepts","text":"After plugging in $$x=0$$ to our equation, we see that a y-intercept occurs when $$y=-15$$, so our y-intercept is $$(0,-15)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas5a-h13","type":"hint","dependencies":["ad33ee8parabolas5a-h12"],"title":"Finding the x-intercepts","text":"Our $$x$$ intercepts occur when $$y=0$$. We plug in $$y=0$$ to our equation and solve for the x\'s.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas5a-h14","type":"hint","dependencies":["ad33ee8parabolas5a-h13"],"title":"Finding the x-intercepts","text":"We must factor the GCF from our equation and factor the trinomial. After this, we solve for $$x$$, which are $$3$$ and $$5$$ in this case, so the x-intercepts are $$(3,0)$$ and $$(5,0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas5a-h15","type":"hint","dependencies":["ad33ee8parabolas5a-h14"],"title":"Graphing","text":"Now that we have all of our properties, we are ready to graph our parabolas.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad33ee8parabolas6","title":"Graphing Vertical Parabolas","body":"Graph the vertical parabola using its properties.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Parabolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad33ee8parabolas6a","stepAnswer":["C"],"problemType":"MultipleChoice","stepTitle":"$$y=6x^2+2x-1$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"ad33ee8parabolas6a-h1","type":"hint","dependencies":[],"title":"Direction","text":"We can tell whether the parabola goes up or down based on the sign of a in the equation, $$y=a x^2+b x+c$$. If a is negative, then the parabola goes downwards, if it is positive, it goes upwards.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas6a-h2","type":"hint","dependencies":["ad33ee8parabolas6a-h1"],"title":"Direction","text":"Since our a is positive, what does this tell us about our parabola?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas6a-h3","type":"hint","dependencies":["ad33ee8parabolas6a-h2"],"title":"Upwards Direction","text":"Since our a is positive, we know that our parabola will open up upwards.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas6a-h4","type":"hint","dependencies":["ad33ee8parabolas6a-h3"],"title":"Finding the Vertex","text":"To find the vertex, we need to find the axis is symmetry, which is found using the formula, $$x=\\\\frac{-b}{2} a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas6a-h5","type":"hint","dependencies":["ad33ee8parabolas6a-h4"],"title":"Finding the Axis of Symmetry","text":"To find the axis of symmetry, we have to find the $$b$$ and a from our equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas6a-h6","type":"hint","dependencies":["ad33ee8parabolas6a-h5"],"title":"Finding the Axis of Symmetry","text":"Our $$b$$ and a are $$2$$ and $$6$$ respectively. Our axis of symmetry is $$x=6\\\\frac{-2}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas6a-h7","type":"hint","dependencies":["ad33ee8parabolas6a-h6"],"title":"Finding the Vertex","text":"Now that we have our axis of symmetry, we want to plug it back into the original equation to find the $$y$$ value of our vertex.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas6a-h8","type":"hint","dependencies":["ad33ee8parabolas6a-h7"],"title":"Finding the Vertex","text":"Since our $$x$$ is $$\\\\frac{-1}{6}$$, we plug it into our equation as, $$y=-\\\\left({\\\\left(-\\\\frac{1}{6}\\\\right)}^2\\\\right)+2\\\\left(-\\\\frac{1}{6}\\\\right)-1$$. This gives us $$y=\\\\frac{-7}{6}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas6a-h9","type":"hint","dependencies":["ad33ee8parabolas6a-h8"],"title":"Vertex","text":"Since we know that our $$x$$ is $$\\\\frac{-1}{6}$$ and our $$y$$ is $$\\\\frac{-7}{6}$$, our vertex is at the coordinates, $$(\\\\frac{-1}{6},\\\\frac{-7}{6})$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas6a-h10","type":"hint","dependencies":["ad33ee8parabolas6a-h9"],"title":"Finding Intercepts","text":"To graph our parabola, we need to know our $$x$$ and $$y$$ intercepts.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas6a-h11","type":"hint","dependencies":["ad33ee8parabolas6a-h10"],"title":"Finding the y-intercepts","text":"Our y-intercept occurs when $$x=0$$. Knowing this, we substitute $$0$$ into our equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas6a-h12","type":"hint","dependencies":["ad33ee8parabolas6a-h11"],"title":"Finding the y-intercepts","text":"After plugging in $$x=0$$ to our equation, we see that a y-intercept occurs when $$y=-1$$, so our y-intercept is $$(0,-1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas6a-h13","type":"hint","dependencies":["ad33ee8parabolas6a-h12"],"title":"Finding the x-intercepts","text":"Our $$x$$ intercepts occur when $$y=0$$. We plug in $$y=0$$ to our equation and solve for the x\'s.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas6a-h14","type":"hint","dependencies":["ad33ee8parabolas6a-h13"],"title":"Finding the x-intercepts","text":"We must factor the GCF from our equation and factor the trinomial. After this, we solve for $$x$$, which are $$-0.608$$ and $$0.274$$ in this case, so the x-intercepts are $$(-0.608, 0)$$ and $$(0.274, 0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas6a-h15","type":"hint","dependencies":["ad33ee8parabolas6a-h14"],"title":"Graphing","text":"Now that we have all of our properties, we are ready to graph our parabolas.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad33ee8parabolas7","title":"Graphing Vertical Parabolas","body":"Graph the vertical parabola using its properties.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Parabolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad33ee8parabolas7a","stepAnswer":["B"],"problemType":"MultipleChoice","stepTitle":"$$y=8x^2-10x+3$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"ad33ee8parabolas7a-h1","type":"hint","dependencies":[],"title":"Direction","text":"We can tell whether the parabola goes up or down based on the sign of a in the equation, $$y=a x^2+b x+c$$. If a is negative, then the parabola goes downwards, if it is positive, it goes upwards.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas7a-h2","type":"hint","dependencies":["ad33ee8parabolas7a-h1"],"title":"Direction","text":"Since our a is positive, what does this tell us about our parabola?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas7a-h3","type":"hint","dependencies":["ad33ee8parabolas7a-h2"],"title":"Upwards Direction","text":"Since our a is positive, we know that our parabola will open up upwards.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas7a-h4","type":"hint","dependencies":["ad33ee8parabolas7a-h3"],"title":"Finding the Vertex","text":"To find the vertex, we need to find the axis is symmetry, which is found using the formula, $$x=\\\\frac{-b}{2} a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas7a-h5","type":"hint","dependencies":["ad33ee8parabolas7a-h4"],"title":"Finding the Axis of Symmetry","text":"To find the axis of symmetry, we have to find the $$b$$ and a from our equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas7a-h6","type":"hint","dependencies":["ad33ee8parabolas7a-h5"],"title":"Finding the Axis of Symmetry","text":"Our $$b$$ and a are $$-10$$ and $$8$$ respectively. Our axis of symmetry is $$x=8\\\\frac{10}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas7a-h8","type":"hint","dependencies":["ad33ee8parabolas7a-h6"],"title":"Finding the Vertex","text":"Now that we have our axis of symmetry, we want to plug it back into the original equation to find the $$y$$ value of our vertex.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas7a-h8","type":"hint","dependencies":["ad33ee8parabolas7a-h8"],"title":"Finding the Vertex","text":"Since our $$x$$ is $$\\\\frac{5}{8}$$, we plug it into our equation as, $$y=-\\\\left({\\\\left(-\\\\frac{5}{8}\\\\right)}^2\\\\right)-10\\\\frac{5}{8}+3$$. This gives us $$y=\\\\frac{-1}{8}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas7a-h9","type":"hint","dependencies":["ad33ee8parabolas7a-h8"],"title":"Vertex","text":"Since we know that our $$x$$ is $$\\\\frac{5}{8}$$ and our $$y$$ is $$\\\\frac{-1}{8}$$, our vertex is at the coordinates, $$(\\\\frac{5}{8},\\\\frac{-1}{8})$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas7a-h10","type":"hint","dependencies":["ad33ee8parabolas7a-h9"],"title":"Finding Intercepts","text":"To graph our parabola, we need to know our $$x$$ and $$y$$ intercepts.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas7a-h11","type":"hint","dependencies":["ad33ee8parabolas7a-h10"],"title":"Finding the y-intercepts","text":"Our y-intercept occurs when $$x=0$$. Knowing this, we substitute $$0$$ into our equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas7a-h12","type":"hint","dependencies":["ad33ee8parabolas7a-h11"],"title":"Finding the y-intercepts","text":"After plugging in $$x=0$$ to our equation, we see that a y-intercept occurs when $$y=3$$, so our y-intercept is $$(0,3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas7a-h13","type":"hint","dependencies":["ad33ee8parabolas7a-h12"],"title":"Finding the x-intercepts","text":"Our $$x$$ intercepts occur when $$y=0$$. We plug in $$y=0$$ to our equation and solve for the x\'s.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas7a-h14","type":"hint","dependencies":["ad33ee8parabolas7a-h13"],"title":"Finding the x-intercepts","text":"We must factor the GCF from our equation and factor the trinomial. After this, we solve for $$x$$, which are $$\\\\frac{1}{2}$$ and $$\\\\frac{3}{4}$$ in this case, so the x-intercepts are $$(\\\\frac{1}{2},0)$$ and $$(\\\\frac{3}{4},0)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas7a-h15","type":"hint","dependencies":["ad33ee8parabolas7a-h14"],"title":"Graphing","text":"Now that we have all of our properties, we are ready to graph our parabolas.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad33ee8parabolas8","title":"Graphing Vertical Parabolas","body":"Graph the vertical parabola using its properties.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Parabolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad33ee8parabolas8a","stepAnswer":["A"],"problemType":"MultipleChoice","stepTitle":"$$y=\\\\left(-x^2\\\\right)+2x-4$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"ad33ee8parabolas8a-h1","type":"hint","dependencies":[],"title":"Direction","text":"We can tell whether the parabola goes up or down based on the sign of a in the equation, $$y=a x^2+b x+c$$. If a is negative, then the parabola goes downwards, if it is positive, it goes upwards.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas8a-h2","type":"hint","dependencies":["ad33ee8parabolas8a-h1"],"title":"Direction","text":"Since our a is negative, what does this tell us about our parabola?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas8a-h3","type":"hint","dependencies":["ad33ee8parabolas8a-h2"],"title":"Downwards Direction","text":"Since our a is negative, we know that our parabola will open up downwards.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas8a-h4","type":"hint","dependencies":["ad33ee8parabolas8a-h3"],"title":"Finding the Vertex","text":"To find the vertex, we need to find the axis is symmetry, which is found using the formula, $$x=\\\\frac{-b}{2} a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas8a-h5","type":"hint","dependencies":["ad33ee8parabolas8a-h4"],"title":"Finding the Axis of Symmetry","text":"To find the axis of symmetry, we have to find the $$b$$ and a from our equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas8a-h6","type":"hint","dependencies":["ad33ee8parabolas8a-h5"],"title":"Finding the Axis of Symmetry","text":"Our $$b$$ and a are $$2$$ and $$-1$$ respectively. Our axis of symmetry is $$x=-1\\\\frac{-2}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas8a-h8","type":"hint","dependencies":["ad33ee8parabolas8a-h6"],"title":"Finding the Vertex","text":"Now that we have our axis of symmetry, we want to plug it back into the original equation to find the $$y$$ value of our vertex.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas8a-h8","type":"hint","dependencies":["ad33ee8parabolas8a-h8"],"title":"Finding the Vertex","text":"Since our $$x$$ is $$1$$, we plug it into our equation as, $$y=-\\\\left(1^2\\\\right)+2\\\\times1-4$$. This gives us $$y=-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas8a-h9","type":"hint","dependencies":["ad33ee8parabolas8a-h8"],"title":"Vertex","text":"Since we know that our $$x$$ is $$1$$ and our $$y$$ is $$-3$$, our vertex is at the coordinates, $$(1,-3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas8a-h10","type":"hint","dependencies":["ad33ee8parabolas8a-h9"],"title":"Finding Intercepts","text":"To graph our parabola, we need to know our $$x$$ and $$y$$ intercepts.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas8a-h11","type":"hint","dependencies":["ad33ee8parabolas8a-h10"],"title":"Finding the y-intercepts","text":"Our y-intercept occurs when $$x=0$$. Knowing this, we substitute $$0$$ into our equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas8a-h12","type":"hint","dependencies":["ad33ee8parabolas8a-h11"],"title":"Finding the y-intercepts","text":"After plugging in $$x=0$$ to our equation, we see that a y-intercept occurs when $$y=-4$$, so our y-intercept is $$(0,-4)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas8a-h13","type":"hint","dependencies":["ad33ee8parabolas8a-h12"],"title":"Finding the x-intercepts","text":"Our $$x$$ intercepts occur when $$y=0$$. We plug in $$y=0$$ to our equation and solve for the x\'s.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas8a-h14","type":"hint","dependencies":["ad33ee8parabolas8a-h13"],"title":"Finding the x-intercepts","text":"We must factor the GCF from our equation and factor the trinomial. After trying to do this, we find that there are no x-intercepts.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas8a-h15","type":"hint","dependencies":["ad33ee8parabolas8a-h14"],"title":"Graphing","text":"Now that we have all of our properties, we are ready to graph our parabolas.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad33ee8parabolas9","title":"Rewriting the equation in Standard Form.","body":"Rewrite the given equation into standard form.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.2 Parabolas","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad33ee8parabolas9a","stepAnswer":["$$3{\\\\left(x-1\\\\right)}^2+2$$"],"problemType":"TextBox","stepTitle":"$$y=3x^2-6x+5$$ (Note: Do not include $$\\"y=\\"$$ in answer.)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3{\\\\left(x-1\\\\right)}^2+2$$","hints":{"DefaultPathway":[{"id":"ad33ee8parabolas9a-h1","type":"hint","dependencies":[],"title":"Standard Form","text":"The standard form is $$y={a\\\\left(x-h\\\\right)}^2+k$$. We are given the equation in $$y=a x^2+b x+c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas9a-h2","type":"hint","dependencies":["ad33ee8parabolas9a-h1"],"title":"Finding a","text":"To find a, we must divide our $$a x^2$$ term by a by factoring by grouping.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas9a-h3","type":"hint","dependencies":["ad33ee8parabolas9a-h2"],"title":"Completing the Square","text":"After we find a, our equation now looks like, $$a \\\\left(x^2+\\\\frac{b}{a} x\\\\right)+c$$. We must complete the square by adding $${\\\\left(\\\\frac{\\\\frac{b}{a}}{2}\\\\right)}^2$$ into the parenthesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas9a-h4","type":"hint","dependencies":["ad33ee8parabolas9a-h3"],"title":"Completing the Square","text":"Since we can\'t simply add a number out of thin air, we must take our equation and subtract it by what we added into the parenthesis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad33ee8parabolas9a-h5","type":"hint","dependencies":["ad33ee8parabolas9a-h4"],"title":"Completing the Square","text":"After doing so, our equation should now be in standard form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad37b31probability1","title":"Computing the Probability of an Event with Equally Likely Outcomes","body":"A six-sided number cube is rolled.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Probability","courseName":"OpenStax: College Algebra","steps":[{"id":"ad37b31probability1a","stepAnswer":["$$\\\\frac{1}{2}$$"],"problemType":"TextBox","stepTitle":"Find the probability of rolling an odd number.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{2}$$","hints":{"DefaultPathway":[{"id":"ad37b31probability1a-h1","type":"hint","dependencies":[],"title":"Rolling an odd number","text":"There are $$3$$ separate ways to roll an odd number. Either you roll a $$1$$, $$3$$, or $$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability1a-h2","type":"hint","dependencies":["ad37b31probability1a-h1"],"title":"Total possible outcomes","text":"There are a total of $$6$$ possible outcomes when rolling a 6-sided cube. Either you roll a $$1$$, $$2$$, $$3$$, $$4$$, $$5$$ or $$6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability1a-h3","type":"hint","dependencies":["ad37b31probability1a-h2"],"title":"Divide","text":"Divide the number of outcomes for rolling an odd number by the total number of outcomes for your answer","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad37b31probability10","title":"Computing the Probability of an Event","body":"Refer to the image.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Probability","courseName":"OpenStax: College Algebra","steps":[{"id":"ad37b31probability10a","stepAnswer":["$$\\\\frac{1}{2}$$"],"problemType":"TextBox","stepTitle":"Find the probability of landing on a vowel.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{2}$$","hints":{"DefaultPathway":[{"id":"ad37b31probability10a-h1","type":"hint","dependencies":[],"title":"Sample Space","text":"There are $$8$$ total areas where the arrow can land on.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability10a-h2","type":"hint","dependencies":["ad37b31probability10a-h1"],"title":"Vowel Slice","text":"There are $$4$$ slices in the circle that the arrow can land on: the A, I, E, O.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability10a-h3","type":"hint","dependencies":["ad37b31probability10a-h2"],"title":"Divide","text":"Divide the number of ways the arrow can land on a vowel by the total number of slices the arrow can land on.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad37b31probability11","title":"Computing the Probability of an Event","body":"Refer to the image.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Probability","courseName":"OpenStax: College Algebra","steps":[{"id":"ad37b31probability11a","stepAnswer":["$$\\\\frac{3}{4}$$"],"problemType":"TextBox","stepTitle":"Find the probability of not landing on green.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{4}$$","hints":{"DefaultPathway":[{"id":"ad37b31probability11a-h1","type":"hint","dependencies":[],"title":"Sample Space","text":"There are $$8$$ total areas where the arrow can land on.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability11a-h2","type":"hint","dependencies":["ad37b31probability11a-h1"],"title":"Landing on Green","text":"There are $$2$$ slices in the circle that the arrow can land on: the I and the F.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability11a-h3","type":"hint","dependencies":["ad37b31probability11a-h2"],"title":"Complement Rule","text":"We can simply subtract $$\\\\frac{2}{8}$$ from $$1$$ to find the probability of not landing on green.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad37b31probability12","title":"Computing the Probability of an Event","body":"Refer to the image.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Probability","courseName":"OpenStax: College Algebra","steps":[{"id":"ad37b31probability12a","stepAnswer":["$$\\\\frac{5}{8}$$"],"problemType":"TextBox","stepTitle":"Find the probability of landing on green or a vowel.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5}{8}$$","hints":{"DefaultPathway":[{"id":"ad37b31probability12a-h1","type":"hint","dependencies":[],"title":"Sample Space","text":"There are $$8$$ total areas where the arrow can land on.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability12a-h2","type":"hint","dependencies":["ad37b31probability12a-h1"],"title":"Landing on Green","text":"There are $$2$$ slices in the circle that the arrow can land on: the I and the F.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability12a-h3","type":"hint","dependencies":["ad37b31probability12a-h2"],"title":"Landing on a Vowel","text":"There are $$4$$ slices in the circle that the arrow can land on: the A, I, E, O.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability12a-h4","type":"hint","dependencies":["ad37b31probability12a-h3"],"title":"Landing on a Green Vowel","text":"There is only $$1$$ slice in the circle that the arrow can land on: the green I.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability12a-h5","type":"hint","dependencies":["ad37b31probability12a-h4"],"title":"Probability of the Union of Two Events","text":"Add the probablity of landing on a green and the probablity of landing on a vowel. Subtract the probability of landing on a slice that is both green and a vowel.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad37b31probability13","title":"Computing the Probability of an Event","body":"Refer to the image.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Probability","courseName":"OpenStax: College Algebra","steps":[{"id":"ad37b31probability13a","stepAnswer":["$$\\\\frac{3}{8}$$"],"problemType":"TextBox","stepTitle":"Find the probability of landing on green or on red.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{8}$$","hints":{"DefaultPathway":[{"id":"ad37b31probability13a-h1","type":"hint","dependencies":[],"title":"Mutually Exclusive Events","text":"The events \u201clanding on green\u201d and \u201clanding on red\u201d are mutually exclusive because they cannot occur at the same time.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability13a-h2","type":"hint","dependencies":["ad37b31probability13a-h1"],"title":"Sample Space","text":"There are $$8$$ total areas where the arrow can land on.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability13a-h3","type":"hint","dependencies":["ad37b31probability13a-h2"],"title":"Landing on Green","text":"There are $$2$$ slices in the circle that the arrow can land on: the I and the F.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability13a-h4","type":"hint","dependencies":["ad37b31probability13a-h3"],"title":"Landing on Red","text":"There is only $$1$$ slice in the circle that the arrow can land on.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability13a-h5","type":"hint","dependencies":["ad37b31probability13a-h4"],"title":"Landing on a Green Vowel","text":"There is only $$1$$ slice in the circle that the arrow can land on: the green I.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability13a-h6","type":"hint","dependencies":["ad37b31probability13a-h5"],"title":"Addition","text":"Add the probability of landing on green with the probability of landing on red.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad37b31probability14","title":"Computing the Probability of an Event","body":"Refer to the image.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Probability","courseName":"OpenStax: College Algebra","steps":[{"id":"ad37b31probability14a","stepAnswer":["$$\\\\frac{5}{8}$$"],"problemType":"TextBox","stepTitle":"Find the probability of landing on yellow or a consonant.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5}{8}$$","hints":{"DefaultPathway":[{"id":"ad37b31probability14a-h1","type":"hint","dependencies":[],"title":"Sample Space","text":"There are $$8$$ total areas where the arrow can land on.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability14a-h2","type":"hint","dependencies":["ad37b31probability14a-h1"],"title":"Landing on Yellow","text":"There is only $$1$$ slice in the circle that the arrow can land on.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability14a-h3","type":"hint","dependencies":["ad37b31probability14a-h2"],"title":"Landing on a Consonant","text":"There are $$4$$ slices in the circle that the arrow can land on: the B, D, C, F.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability14a-h4","type":"hint","dependencies":["ad37b31probability14a-h3"],"title":"Landing on a Yellow Consonant","text":"There isn\'t any way to land on both a consonant and a yellow slice.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability14a-h5","type":"hint","dependencies":["ad37b31probability14a-h4"],"title":"Probability of the Union of Two Events","text":"Add the probablity of landing on a yellow and the probablity of landing on a consonant.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad37b31probability15","title":"Computing the Probability of an Event","body":"Refer to the image.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Probability","courseName":"OpenStax: College Algebra","steps":[{"id":"ad37b31probability15a","stepAnswer":["$$\\\\frac{3}{8}$$"],"problemType":"TextBox","stepTitle":"Find the probability of not landing on yellow or a consonant.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{8}$$","hints":{"DefaultPathway":[{"id":"ad37b31probability15a-h1","type":"hint","dependencies":[],"title":"Sample Space","text":"There are $$8$$ total areas where the arrow can land on.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability15a-h2","type":"hint","dependencies":["ad37b31probability15a-h1"],"title":"Landing on Yellow or a Consonant","text":"There are $$5$$ slices in the circle that the arrow can land on.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability15a-h3","type":"hint","dependencies":["ad37b31probability15a-h2"],"title":"Complement Rule","text":"We can simply subtract $$\\\\frac{5}{8}$$ from $$1$$ to find the probability of not landing on yellow or a consonant.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad37b31probability16","title":"Construction Probability Models","body":"For the following exercises, two coins are tossed.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Probability","courseName":"OpenStax: College Algebra","steps":[{"id":"ad37b31probability16a","stepAnswer":["HH, HT, TT, TH"],"problemType":"MultipleChoice","stepTitle":"What is the sample space?","stepBody":"","answerType":"string","variabilization":{},"choices":["HH, HT, TT, TH","HT, TH","TT, HH, HT","HH,TT"],"hints":{"DefaultPathway":[{"id":"ad37b31probability16a-h1","type":"hint","dependencies":[],"title":"Defining Sample Space","text":"The set of all possible outcomes of an experiment is called the sample space of the experiment.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability16a-h2","type":"hint","dependencies":["ad37b31probability16a-h1"],"title":"Format of the Outcomes","text":"Given there are two coins, each outcome is represented by two letters one for the first coin and the other for the second coin.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability16a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["HH, HT, TT, TH"],"dependencies":["ad37b31probability16a-h2"],"title":"Finding Sample Space","text":"List all the possible outcomes of the two coin flips using H for heads and T for tails.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["HH, HT, TT, TH","HT, TH","TT, HH, HT","HH,TT"]}]}}]},{"id":"ad37b31probability17","title":"Computing Probabilities of Equally Likely Outcomes","body":"For the following exercises, two coins are tossed.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Probability","courseName":"OpenStax: College Algebra","steps":[{"id":"ad37b31probability17a","stepAnswer":["$$\\\\frac{1}{4}$$"],"problemType":"TextBox","stepTitle":"Find the probability of tossing two heads.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{4}$$","hints":{"DefaultPathway":[{"id":"ad37b31probability17a-h1","type":"hint","dependencies":[],"title":"Proability of Events with Equally Likely Outcomes","text":"The probability of an event E in an experiment with sample space S with equally likely outcomes is given by: P(E) $$=$$ (# of elements in E)/(# of elements in S)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability17a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ad37b31probability17a-h1"],"title":"Number of Event Elements","text":"How many elements result in two heads?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability17a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ad37b31probability17a-h2"],"title":"Number of Sample Space Elements","text":"How many elements are there in the sample space?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability17a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4}$$"],"dependencies":["ad37b31probability17a-h3"],"title":"Finding the Probability","text":"What is the probaility of getting two heads?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad37b31probability18","title":"Computing Probabilities of Equally Likely Outcomes","body":"For the following exercises, two coins are tossed.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Probability","courseName":"OpenStax: College Algebra","steps":[{"id":"ad37b31probability18a","stepAnswer":["$$\\\\frac{1}{2}$$"],"problemType":"TextBox","stepTitle":"Find the probability of tossing exactly one tail.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{2}$$","hints":{"DefaultPathway":[{"id":"ad37b31probability18a-h1","type":"hint","dependencies":[],"title":"Proability of Events with Equally Likely Outcomes","text":"The probability of an event E in an experiment with sample space S with equally likely outcomes is given by: P(E) $$=$$ (# of elements in E)/(# of elements in S)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ad37b31probability18a-h1"],"title":"Number of Event Elements","text":"How many elements result in only one tail?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability18a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ad37b31probability18a-h2"],"title":"Number of Sample Space Elements","text":"How many elements are there in the sample space?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability18a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["ad37b31probability18a-h3"],"title":"Finding the Probability","text":"What is the probaility of getting two heads?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad37b31probability19","title":"Computing Probabilities of Equally Likely Outcomes","body":"For the following exercises, two coins are tossed.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Probability","courseName":"OpenStax: College Algebra","steps":[{"id":"ad37b31probability19a","stepAnswer":["$$\\\\frac{3}{4}$$"],"problemType":"TextBox","stepTitle":"Find the probability of tossing at least one tail.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{4}$$","hints":{"DefaultPathway":[{"id":"ad37b31probability19a-h1","type":"hint","dependencies":[],"title":"Proability of Events with Equally Likely Outcomes","text":"The probability of an event E in an experiment with sample space S with equally likely outcomes is given by: P(E) $$=$$ (# of elements in E)/(# of elements in S)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ad37b31probability19a-h1"],"title":"Number of Event Elements","text":"How many elements result in at least one tail?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability19a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ad37b31probability19a-h2"],"title":"Number of Sample Space Elements","text":"How many elements are there in the sample space?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability19a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{4}$$"],"dependencies":["ad37b31probability19a-h3"],"title":"Finding the 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So the probability of drawing a heart is $$\\\\frac{1}{4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability3a-h2","type":"hint","dependencies":["ad37b31probability3a-h1"],"title":"Drawing a $$7$$","text":"There are four 7s in a standard deck, and there are a total of $$52$$ cards. So the probability of drawing a $$7$$ is $$\\\\frac{4}{52}$$ or $$\\\\frac{1}{13}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability3a-h3","type":"hint","dependencies":["ad37b31probability3a-h2"],"title":"Drawing a heart and a $$7$$","text":"The only card in the deck that is both a heart and a $$7$$ is the $$7$$ of hearts, so the probability of drawing both a heart and a $$7$$ is $$\\\\frac{1}{52}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability3a-h4","type":"hint","dependencies":["ad37b31probability3a-h3"],"title":"Probability of the Union of Two Events","text":"Add the probablity of drawing a heart and the probablity of drawing a $$7$$. Subtract the probability of drawing a card that is both a heart and a $$7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad37b31probability30","title":"Computing Probabilities of Equally Likely Outcomes","body":"For the following exercises, one card is drawn from a standard deck of $$52$$ cards.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Probability","courseName":"OpenStax: College Algebra","steps":[{"id":"ad37b31probability30a","stepAnswer":["$$\\\\frac{12}{13}$$"],"problemType":"TextBox","stepTitle":"Find the probability of drawing a heart or a non-jack.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{12}{13}$$","hints":{"DefaultPathway":[{"id":"ad37b31probability30a-h1","type":"hint","dependencies":[],"title":"Proability of Events with Equally Likely Outcomes","text":"The probability of an event E in an experiment with sample space S with equally likely outcomes is given by: P(E) $$=$$ (# of elements in E)/(# of elements in S)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability30a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$48$$"],"dependencies":["ad37b31probability30a-h1"],"title":"Number of Event Elements","text":"How many elements result in drawing a heart or a non-jack?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability30a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$52$$"],"dependencies":["ad37b31probability30a-h2"],"title":"Number of Sample Space Elements","text":"How many elements are there in the sample space?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability30a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{12}{13}$$"],"dependencies":["ad37b31probability30a-h3"],"title":"Finding the Probability","text":"What is the probaility of getting a heart or a non-jack?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad37b31probability4","title":"Computing the Probability of the Union of Two Events","body":"A card is drawn from a standard deck.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Probability","courseName":"OpenStax: College Algebra","steps":[{"id":"ad37b31probability4a","stepAnswer":["$$\\\\frac{7}{13}$$"],"problemType":"TextBox","stepTitle":"Find the probability of drawing a red card or an ace.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{7}{13}$$","hints":{"DefaultPathway":[{"id":"ad37b31probability4a-h1","type":"hint","dependencies":[],"title":"Drawing a red card","text":"A standard deck contains an equal number of hearts, diamonds, clubs, and spades. Hearts and diamonds are both the only red cards. So the probability of drawing a red card is $$\\\\frac{1}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability4a-h2","type":"hint","dependencies":["ad37b31probability4a-h1"],"title":"Drawing an ace","text":"There are four aces in a standard deck, and there are a total of $$52$$ cards. So the probability of drawing an ace is $$\\\\frac{4}{52}$$ or $$\\\\frac{1}{13}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability4a-h3","type":"hint","dependencies":["ad37b31probability4a-h2"],"title":"Drawing a red card and an ace","text":"The only two cards in the deck that is both a red card and an ace is the ace of hearts and the ace of diamonds, so the probability of drawing both a red card and an ace is $$\\\\frac{2}{52}$$ or $$\\\\frac{1}{26}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability4a-h4","type":"hint","dependencies":["ad37b31probability4a-h3"],"title":"Probability of the Union of Two Events","text":"Add the probablity of drawing a red card and the probablity of drawing an ace. Subtract the probability of drawing a card that is both a red card and an ace.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad37b31probability5","title":"Computing the Probability of the Union of Mutually Exclusive Events","body":"A card is drawn from a standard deck.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Probability","courseName":"OpenStax: College Algebra","steps":[{"id":"ad37b31probability5a","stepAnswer":["$$\\\\frac{1}{2}$$"],"problemType":"TextBox","stepTitle":"Find the probability of drawing a heart or a spade.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{2}$$","hints":{"DefaultPathway":[{"id":"ad37b31probability5a-h1","type":"hint","dependencies":[],"title":"Mutually Exclusive Events","text":"The events \u201cdrawing a heart\u201d and \u201cdrawing a spade\u201d are mutually exclusive because they cannot occur at the same time.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability5a-h2","type":"hint","dependencies":["ad37b31probability5a-h1"],"title":"Drawing a Heart","text":"The probability of drawing a heart is $$\\\\frac{1}{4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability5a-h3","type":"hint","dependencies":["ad37b31probability5a-h2"],"title":"Drawing a Spade","text":"the probability of drawing a spade is also $$\\\\frac{1}{4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability5a-h4","type":"hint","dependencies":["ad37b31probability5a-h3"],"title":"Addition","text":"Add the probability of drawing a heart with the probability of drawing a spade.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad37b31probability6","title":"Computing the Probability of the Union of Mutually Exclusive Events","body":"A card is drawn from a standard deck.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Probability","courseName":"OpenStax: College Algebra","steps":[{"id":"ad37b31probability6a","stepAnswer":["$$\\\\frac{2}{13}$$"],"problemType":"TextBox","stepTitle":"Find the probability of drawing an ace or a king.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2}{13}$$","hints":{"DefaultPathway":[{"id":"ad37b31probability6a-h1","type":"hint","dependencies":[],"title":"Mutually Exclusive Events","text":"The events \u201cdrawing an ace\u201d and \u201cdrawing a king\u201d are mutually exclusive because they cannot occur at the same time.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability6a-h2","type":"hint","dependencies":["ad37b31probability6a-h1"],"title":"Drawing an Ace","text":"The probability of drawing an ace is $$\\\\frac{1}{13}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability6a-h3","type":"hint","dependencies":["ad37b31probability6a-h2"],"title":"Drawing a King","text":"the probability of drawing a spade is also $$\\\\frac{1}{13}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability6a-h4","type":"hint","dependencies":["ad37b31probability6a-h3"],"title":"Addition","text":"Add the probability of drawing an ace with the probability of drawing a king.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad37b31probability7","title":"Using the Complement Rule to Calculate Probabilities","body":"Two six-sided number cubes are rolled.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Probability","courseName":"OpenStax: College Algebra","steps":[{"id":"ad37b31probability7a","stepAnswer":["$$\\\\frac{1}{12}$$"],"problemType":"TextBox","stepTitle":"Find the probability that the sum of the numbers rolled is less than or equal to $$3$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{12}$$","hints":{"DefaultPathway":[{"id":"ad37b31probability7a-h1","type":"hint","dependencies":[],"title":"Sample Space","text":"There are two number cubes, and each number cube has six possible outcomes. Using the Multiplication Principle, we find that there are $$6\\\\times6$$, or $$36$$ total possible outcomes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability7a-h2","type":"hint","dependencies":["ad37b31probability7a-h1"],"title":"Rolling a Sum of $$3$$","text":"We need to count the number of ways to roll a sum of $$3$$ or less. These would include the following outcomes: $$1-1$$, $$1-2$$, and $$2-1$$. $$(1-1$$ represents a $$1$$ rolled on each number cube.) So there are only three ways to roll a sum of $$3$$ or less.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability7a-h3","type":"hint","dependencies":["ad37b31probability7a-h2"],"title":"Divide","text":"Divide the number of outcomes for a sum less than $$3$$ by the total number of outcomes for your answer","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ad37b31probability7b","stepAnswer":["$$\\\\frac{11}{12}$$"],"problemType":"TextBox","stepTitle":"Find the / that the sum of the numbers rolled is greater than $$3$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{11}{12}$$","hints":{"DefaultPathway":[{"id":"ad37b31probability7b-h1","type":"hint","dependencies":[],"title":"Complement Rule","text":"We have already found the probability of the complement of this event. Use the complement rule to calculate your answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability7b-h2","type":"hint","dependencies":["ad37b31probability7b-h1"],"title":"Complement Rule","text":"We can simply subtract $$\\\\frac{1}{12}$$ from $$1$$ to find the probability that the sum of the numbers rolled is greater than $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad37b31probability8","title":"Using the Complement Rule to Calculate Probabilities","body":"Two number cubes are rolled.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Probability","courseName":"OpenStax: College Algebra","steps":[{"id":"ad37b31probability8a","stepAnswer":["$$\\\\frac{5}{6}$$"],"problemType":"TextBox","stepTitle":"Use the Complement Rule to find the probability that the sum is less than $$10$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5}{6}$$","hints":{"DefaultPathway":[{"id":"ad37b31probability8a-h1","type":"hint","dependencies":[],"title":"Sample Space","text":"There are two number cubes, and each number cube has six possible outcomes. Using the Multiplication Principle, we find that there are $$6\\\\times6$$, or $$36$$ total possible outcomes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability8a-h2","type":"hint","dependencies":["ad37b31probability8a-h1"],"title":"Complement Event","text":"It is easier to start the problem by finding the possible ways of rolling a sum of $$10$$ or more.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability8a-h3","type":"hint","dependencies":["ad37b31probability8a-h2"],"title":"Complement Event","text":"We need to count the number of ways to roll a sum of $$10$$ or more. These would include the following outcomes: $$6-6$$, $$6-5$$, $$5-6$$, $$6-4$$, $$5-5$$, $$4-6$$. $$(1-1$$ represents a $$1$$ rolled on each number cube.) So there are only six ways to roll a sum of $$10$$ or more.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability8a-h4","type":"hint","dependencies":["ad37b31probability8a-h3"],"title":"Complement Rule","text":"We can simply subtract $$\\\\frac{6}{36}$$ from $$1$$ to find the probability that the sum of the numbers rolled is less than $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad37b31probability9","title":"Computing the Probability of an Event","body":"Refer to the image.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.7 Probability","courseName":"OpenStax: College Algebra","steps":[{"id":"ad37b31probability9a","stepAnswer":["$$\\\\frac{1}{8}$$"],"problemType":"TextBox","stepTitle":"Find the probability of landing on red.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{8}$$","hints":{"DefaultPathway":[{"id":"ad37b31probability9a-h1","type":"hint","dependencies":[],"title":"Sample Space","text":"There are $$8$$ total areas where the arrow can land on.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability9a-h2","type":"hint","dependencies":["ad37b31probability9a-h1"],"title":"Red Slice","text":"There is only $$1$$ red slice in the circle that the arrow can land on.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad37b31probability9a-h3","type":"hint","dependencies":["ad37b31probability9a-h2"],"title":"Divide","text":"Divide the number of ways the arrow can land on red by the total number of slices the arrow can land on.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad3a8eemixapp1","title":"Solve Mixture Word Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Solve Mixture Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad3a8eemixapp1a","stepAnswer":["$$2$$ pounds of raisins, $$8$$ pounds of nuts"],"problemType":"MultipleChoice","stepTitle":"Henning is mixing raisins and nuts to make $$10$$ pounds of trail mix. Raisins cost $2 a pound and nuts cost $6 a pound. If Henning wants his cost for the trail mix to be $$\\\\$5.20$$ a pound, how many pounds of raisins and how many pounds of nuts should he use?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2$$ pounds of raisins, $$8$$ pounds of nuts","choices":["$$2$$ pounds of raisins, $$8$$ pounds of nuts","$$8$$ pounds of raisins, $$2$$ pounds of nuts","$$4$$ pounds raisins, $$16$$ pounds nuts","$$16$$ pounds raisins, $$4$$ pounds nuts"],"hints":{"DefaultPathway":[{"id":"ad3a8eemixapp1a-h1","type":"hint","dependencies":[],"title":"Write a mathematical equation","text":"The first step is to express the number of pounds of raisins and nuts in the $$10$$ pounds of trail mix as a mathematical expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eemixapp1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10-x$$"],"dependencies":["ad3a8eemixapp1a-h1"],"title":"Assigning variables","text":"Let the number of pounds of raisins be $$x$$. How can you express the number of pounds of nuts as a mathematical expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eemixapp1a-h3","type":"hint","dependencies":["ad3a8eemixapp1a-h2"],"title":"Writing a mathematical expression","text":"The next step is to express the total price as a mathematical expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eemixapp1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x$$"],"dependencies":["ad3a8eemixapp1a-h3"],"title":"Total value for raisins","text":"How would you express the total value for raisins?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eemixapp1a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6\\\\left(10-x\\\\right)$$"],"dependencies":["ad3a8eemixapp1a-h4"],"title":"Total value for nuts","text":"How would you express the total value for nuts?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eemixapp1a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10\\\\times5.2$$"],"dependencies":["ad3a8eemixapp1a-h5"],"title":"Total value of trail mix","text":"How would you express the total value of the trail mix?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eemixapp1a-h7","type":"hint","dependencies":["ad3a8eemixapp1a-h6"],"title":"Combining the values","text":"We can combine our mathematical expressions into $$2x+6\\\\left(10-x\\\\right)=10\\\\times5.2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eemixapp1a-h8","type":"hint","dependencies":["ad3a8eemixapp1a-h7"],"title":"Solve for $$x$$","text":"The next step is to solve for $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eemixapp1a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x$$"],"dependencies":["ad3a8eemixapp1a-h8"],"title":"Finding pounds for raisins","text":"What did variable did we use for raisins?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eemixapp1a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10-x$$"],"dependencies":["ad3a8eemixapp1a-h9"],"title":"Finding pounds for nuts","text":"What mathematical expression did we use for nuts?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad3a8eemixapp10","title":"Solving Mixture Word Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Solve Mixture Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad3a8eemixapp10a","stepAnswer":["$$7$$ $5, $$12$$ $1"],"problemType":"MultipleChoice","stepTitle":"Joe\u2019s wallet contains $1 and $5 bills worth $47. The number of $1 bills is five more than the number of $5 bills. How many of each bill does he have?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$7$$ $5, $$12$$ $1","choices":["$$7$$ $5, $$12$$ $1","$$12$$ $1, $$7$$ $5","$$10$$ $1, $$8$$ $5"],"hints":{"DefaultPathway":[{"id":"ad3a8eemixapp10a-h1","type":"hint","dependencies":[],"title":"Expressing values as mathematical expressions","text":"The first step is to express the values as mathematical expressions","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eemixapp10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+5$$"],"dependencies":["ad3a8eemixapp10a-h1"],"title":"Assigning variables","text":"Let $$x=the$$ number of $5. How many $1 bills are there? 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Write the answer as a mathematical expression","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eemixapp10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$47$$"],"dependencies":["ad3a8eemixapp10a-h4"],"title":"Total cost","text":"What is the total cost?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad3a8eemixapp11","title":"Solving Mixture Word Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Solve Mixture Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad3a8eemixapp11a","stepAnswer":["$$18$$ quarters, $$36$$ nickels"],"problemType":"MultipleChoice","stepTitle":"Rachelle has $$\\\\$6.30$$ in nickels and quarters in her coin purse. The number of nickels is twice the number of quarters. How many coins of each type does she have?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$18$$ quarters, $$36$$ nickels","choices":["$$18$$ quarters, $$36$$ nickels","$$18$$ quarters, $$36$$ nickels","$$36$$ nickels, $$18$$ quarters"],"hints":{"DefaultPathway":[{"id":"ad3a8eemixapp11a-h1","type":"hint","dependencies":[],"title":"Expressing values as mathematical expressions","text":"The first step is to express the values as mathematical expressions","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eemixapp11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x$$"],"dependencies":["ad3a8eemixapp11a-h1"],"title":"Assigning variables","text":"Let $$x=the$$ number of quarters. How many nickels are there? 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Write the answer as a mathematical expression","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eemixapp11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6.3$$"],"dependencies":["ad3a8eemixapp11a-h4"],"title":"Total cost","text":"What is the total cost?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad3a8eemixapp12","title":"Solving Mixture Word Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Solve Mixture Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad3a8eemixapp12a","stepAnswer":["$$45$$ pennies, $$15$$ nickels"],"problemType":"MultipleChoice","stepTitle":"Deloise has $$\\\\$1.20$$ in pennies and nickels in a jar on her desk. 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Write the answer as a mathematical expression","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eemixapp12a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.2$$"],"dependencies":["ad3a8eemixapp12a-h4"],"title":"Total cost","text":"What is the total cost?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad3a8eemixapp2","title":"Solve Mixture Word Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Solve Mixture Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad3a8eemixapp2a","stepAnswer":["$5,000 in the accound that earns 3% and $15,000 in the account that earns 5%"],"problemType":"MultipleChoice","stepTitle":"Stacey has $20,000 to invest in two different bank accounts. One account pays interest at 3% per year and the other account pays interest at 5% per year. How much should she invest in each account if she wants to earn $$4.5\\\\%$$ interest per year on the total amount?","stepBody":"","answerType":"string","variabilization":{},"choices":["$15,000 in the account that earns 3% and $5,000 in the account that earns 5%","$5,000 in the accound that earns 3% and $15,000 in the account that earns 5%","$15,000","$5,000"],"hints":{"DefaultPathway":[{"id":"ad3a8eemixapp2a-h1","type":"hint","dependencies":[],"title":"Interest formula","text":"The simple interest formula is $$I=Prt$$, where I is interest, P is principal (amount invested), $$r$$ is rate, and $$t$$ is time","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eemixapp2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20000-x$$"],"dependencies":["ad3a8eemixapp2a-h1"],"title":"Assigning variables","text":"Let $$x=the$$ amount invested at 3%. 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Express as a mathematical expression","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eemixapp2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.045\\\\times20000$$"],"dependencies":["ad3a8eemixapp2a-h5"],"title":"Amount of interest total","text":"What is the total amount of interest?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eemixapp2a-h7","type":"hint","dependencies":["ad3a8eemixapp2a-h6"],"title":"Write the mathematical equation","text":"We can use these values to write a mathematical equation, $$0.03x+0.05\\\\left(20000-x\\\\right)=0.045\\\\times20000$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eemixapp2a-h8","type":"hint","dependencies":["ad3a8eemixapp2a-h7"],"title":"Solve the equation","text":"The next step is to solve for $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eemixapp2a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x$$"],"dependencies":["ad3a8eemixapp2a-h8"],"title":"Finding the amount to invest in the 3% account","text":"What variable did we assign to the amount invested in the 3% account?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eemixapp2a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20000-x$$"],"dependencies":["ad3a8eemixapp2a-h9"],"title":"Finding the amount to invest in the 5% account","text":"What variable did we assign to the amount invested in the 5% account?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad3a8eemixapp3","title":"Solve Ticket and Stamp Word Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Solve Mixture Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad3a8eemixapp3a","stepAnswer":["$$10$$ $$21-cent$$ stamps, $$26$$ $$41-cent$$ stamps"],"problemType":"MultipleChoice","stepTitle":"Kailee paid $$\\\\$12.66$$ for stamps. 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Orlando wants to make $$30$$ pounds of party mix at a cost of $$\\\\$6.50$$ a pound, how many pounds of nuts and how many pounds of cereal squares should he use?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$25$$ pounds of nuts, $$5$$ pounds of cereal squares","choices":["$$5$$ pounds of nuts, $$25$$ pounds of cereal squares","$$25$$ pounds of nuts, $$5$$ pounds of cereal squares","$$10$$ pounds of nuts, $$20$$ pounds of cereal squares","$$20$$ pounds of nuts, $$10$$ pounds of cereal squares"],"hints":{"DefaultPathway":[{"id":"ad3a8eemixapp4a-h1","type":"hint","dependencies":[],"title":"Writing mathematical expressions","text":"The first step is to express the total cost of nuts and cereal bars in terms of a mathematical expression","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eemixapp4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$30-x$$"],"dependencies":["ad3a8eemixapp4a-h1"],"title":"Assigning variables","text":"Let $$x=the$$ number of pounds of nuts. Given that we have $$30$$ pounds total, how can we express the number of pounds of cereal bars in terms of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eemixapp4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7x$$"],"dependencies":["ad3a8eemixapp4a-h2"],"title":"Total cost of nuts","text":"What is the total cost for $$x$$ pounds of nuts? Express as a mathematical expression in terms of $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eemixapp4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4\\\\left(30-x\\\\right)$$"],"dependencies":["ad3a8eemixapp4a-h3"],"title":"Total cost of cereal bars","text":"What is the total cost of cereal bars? 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How many student tickets and how many adult tickets were sold (enter the asnwers as student,adult)?","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"ad3a8eeSolveMixture1a-h1","type":"hint","dependencies":[],"title":"Define variables","text":"Assign the number of student tickets a varible (like x) and the number of teacher tickets another variable (like y)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture1a-h2","type":"hint","dependencies":["ad3a8eeSolveMixture1a-h1"],"title":"Setting up equations","text":"Based on the question if $$x$$ is the number of student tickets and $$y$$ is the number of teacher tickets then you seperate all the information into two different equations","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture1a-h3","type":"hint","dependencies":["ad3a8eeSolveMixture1a-h2"],"title":"Cost Equation","text":"One of the equations can represent the total cost of the event which we know to be $1560. Since each student ticket costs $6 we can multiply that with number of tickets to represent the total revenue of students. You can do the same with teachers and their sum would be equal to the total cost.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6x+9y=1506$$"],"dependencies":["ad3a8eeSolveMixture1a-h3"],"title":"Cost Equation","text":"What is the equation equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture1a-h5","type":"hint","dependencies":["ad3a8eeSolveMixture1a-h4"],"title":"Tickets equation","text":"You can represent the relationship between student tickets and adult tickets using the information given.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture1a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y=3x-5$$"],"dependencies":["ad3a8eeSolveMixture1a-h5"],"title":"Tickets equation","text":"What is the relationship","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture1a-h7","type":"hint","dependencies":["ad3a8eeSolveMixture1a-h6"],"title":"Combining the equations","text":"Seperate one variable in one of the equations and plug it in the second equation. In this case it would be easier to plug in $$y=3x-5$$ into the equation $$6x+9y=1506$$ which would equal $$6x+9\\\\left(3x-5\\\\right)$$ $$=$$ $$1506$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture1a-h8","type":"hint","dependencies":["ad3a8eeSolveMixture1a-h7"],"title":"Solve for the missing variable","text":"Use algebra to simplify the equation to solve for the missing variable (x)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture1a-h9","type":"hint","dependencies":["ad3a8eeSolveMixture1a-h8"],"title":"Plug in for $$y$$","text":"Once you have a value for $$x$$ plug that value one of the two equations to get the value of $$y$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad3a8eeSolveMixture10","title":"Maria\'s Quarters and Pennies","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Solve Mixture Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad3a8eeSolveMixture10a","stepAnswer":["$$9$$ quarters and $$18$$ pennies"],"problemType":"MultipleChoice","stepTitle":"Maria has $$\\\\$2.43$$ in quarters and pennies in her wallet. She has twice as many pennies as quarters. How many coins for each type does she have?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$9$$ quarters and $$18$$ pennies","choices":["$$18$$ quarters and $$9$$ pennies","$$14$$ quarters and $$7$$ pennies","$$9$$ quarters and $$18$$ pennies"],"hints":{"DefaultPathway":[{"id":"ad3a8eeSolveMixture10a-h1","type":"hint","dependencies":[],"title":"Read the Problem.","text":"Make sure all the word and ideas are understood. Determine the types of coins involved. Think about the strategy we used to find the value of the handful of coins. The first thing we need is to notice what type of coins are involved. Maria has quarters and pennies. Create a table to organize the information where the columns indicate the type of coin and the rows include the number of coins, their value, and their total value when multiplied. We can work this problem all in cents or in dollars. Here we\'ll do it in dollars and put in the dollar sign in the tail as a reminder. The value of a quarter is $$\\\\$0.25$$ and the value of a penny is $$\\\\$0.01$$. The total value of all the coins is $$\\\\$2.43$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture10a-h2","type":"hint","dependencies":["ad3a8eeSolveMixture10a-h1"],"title":"Identify and Name the Unknowns","text":"Choose a variable to represent the quantity for which we are looking for. So, let $$q=number$$ of quarters. $$2q=number$$ of pennies.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture10a-h3","type":"hint","dependencies":["ad3a8eeSolveMixture10a-h2"],"title":"Total Value of Each Type","text":"To get the total value of each type of coin, we can use the relationship: total value of a $$type=number$$ of coins of this $$type value$$ of this type.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture10a-h4","type":"hint","dependencies":["ad3a8eeSolveMixture10a-h3"],"title":"Translate into an Equation","text":"We know that value of $$quarters+value$$ of $$pennies=total$$ value of coins. Plug in the numbers and variables, we get the equation: $$0.25q+\\\\operatorname{0.01}\\\\left(2\\\\right) q=2.43$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture10a-h5","type":"hint","dependencies":["ad3a8eeSolveMixture10a-h4"],"title":"Solve the Equation","text":"To solve $$0.25q+\\\\operatorname{0.01}\\\\left(2\\\\right) q=2.43$$, we follow the following steps: 0.25q+0.02q=2.43--\x3e0.27q=2.43--\x3eq=9. Therefore, the number of quarters is $$9$$, and the number of pennies is $$2\\\\times9=18$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad3a8eeSolveMixture11","title":"Sumanta\'s Nickels and Dimes","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Solve Mixture Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad3a8eeSolveMixture11a","stepAnswer":["$$21$$ dimes and $$42$$ nickels"],"problemType":"MultipleChoice","stepTitle":"Sumanta has $$\\\\$4.20$$ in nickels and dimes in her piggy bank. She has twice as many nickels as dimes. How many coins of each type does she have?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$21$$ dimes and $$42$$ nickels","choices":["$$15$$ dimes and $$30$$ nickels","$$21$$ dimes and $$42$$ nickels","$$16$$ nickels and $$32$$ dimes"],"hints":{"DefaultPathway":[{"id":"ad3a8eeSolveMixture11a-h1","type":"hint","dependencies":[],"title":"Read the Problem.","text":"Make sure all the word and ideas are understood. Determine the types of coins involved. Think about the strategy we used to find the value of the handful of coins. The first thing we need is to notice what type of coins are involved. Sumanta has dimes and nickels. Create a table to organize the information where the columns indicate the type of coin and the rows include the number of coins, their value, and their total value when multiplied. We can work this problem all in cents or in dollars. Here we\'ll do it in dollars and put in the dollar sign in the tail as a reminder. The value of a dime is $$\\\\$0.10$$ and the value of a nickel is $$\\\\$0.05$$. The total value of all the coins is $$\\\\$4.20$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture11a-h2","type":"hint","dependencies":["ad3a8eeSolveMixture11a-h1"],"title":"Identify and Name the Unknowns","text":"Choose a variable to represent the quantity for which we are looking for. So, let $$d=number$$ of dimes. $$2d=number$$ of nickels.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture11a-h3","type":"hint","dependencies":["ad3a8eeSolveMixture11a-h2"],"title":"Total Value of Each Type","text":"To get the total value of each type of coin, we can use the relationship: total value of a $$type=number$$ of coins of this $$type value$$ of this type.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture11a-h4","type":"hint","dependencies":["ad3a8eeSolveMixture11a-h3"],"title":"Translate into an Equation","text":"We know that value of $$dimes+value$$ of $$nickels=total$$ value of coins. Plug in the numbers and variables, we get the equation: $$0.1d+\\\\operatorname{0.05}\\\\left(2\\\\right) d=4.20$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture11a-h5","type":"hint","dependencies":["ad3a8eeSolveMixture11a-h4"],"title":"Solve the Equation","text":"To solve $$0.1d+\\\\operatorname{0.05}\\\\left(2\\\\right) d=4.20$$, we follow the following steps: 0.1d+0.1d=4.20--\x3e0.2d=4.20--\x3ed=21. Therefore, the number of dimes is $$21$$, and the number of nickels is $$2\\\\times21=42$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad3a8eeSolveMixture12","title":"Alison\'s Dimes and Quarters","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Solve Mixture Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad3a8eeSolveMixture12a","stepAnswer":["$$51$$ dimes and $$17$$ quarters"],"problemType":"MultipleChoice","stepTitle":"Alison has three times as many dimes as quarters in her purse. She has $$\\\\$9.35$$ altogether. How many coins of each type does she have?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$51$$ dimes and $$17$$ quarters","choices":["$$51$$ dimes and $$17$$ quarters","$$48$$ dimes and $$16$$ quarters","$$52$$ dimes and $$17$$ quarters"],"hints":{"DefaultPathway":[{"id":"ad3a8eeSolveMixture12a-h1","type":"hint","dependencies":[],"title":"Read the Problem.","text":"Make sure all the word and ideas are understood. Determine the types of coins involved. Think about the strategy we used to find the value of the handful of coins. The first thing we need is to notice what type of coins are involved. Alison has dimes and quarters. Create a table to organize the information where the columns indicate the type of coin and the rows include the number of coins, their value, and their total value when multiplied. We can work this problem all in cents or in dollars. Here we\'ll do it in dollars and put in the dollar sign in the tail as a reminder. The value of a dime is $$\\\\$0.10$$ and the value of a quarter is $$\\\\$0.25$$. The total value of all the coins is $$\\\\$9.35$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture12a-h2","type":"hint","dependencies":["ad3a8eeSolveMixture12a-h1"],"title":"Identify and Name the Unknowns","text":"Choose a variable to represent the quantity for which we are looking for. So, let $$q=number$$ of quarters. $$3q=number$$ of dimes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture12a-h3","type":"hint","dependencies":["ad3a8eeSolveMixture12a-h2"],"title":"Total Value of Each Type","text":"To get the total value of each type of coin, we can use the relationship: total value of a $$type=number$$ of coins of this $$type value$$ of this type.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture12a-h4","type":"hint","dependencies":["ad3a8eeSolveMixture12a-h3"],"title":"Translate into an Equation","text":"We know that value of $$dimes+value$$ of $$quarters=total$$ value of coins. Plug in the numbers and variables, we get the equation: $$0.25q+\\\\operatorname{0.1}\\\\left(3q\\\\right)=9.35$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture12a-h5","type":"hint","dependencies":["ad3a8eeSolveMixture12a-h4"],"title":"Solve the Equation","text":"To solve $$0.25q+\\\\operatorname{0.1}\\\\left(3q\\\\right)=9.35$$, we follow the following steps: 0.25q+0.3q=9.35--\x3e0.55q=9.35--\x3eq=17. Therefore, the number of quarters is $$17$$, and the number of dimes is $$3\\\\times17=51$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad3a8eeSolveMixture13","title":"Danny\'s Pennies and Nickels","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Solve Mixture Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad3a8eeSolveMixture13a","stepAnswer":["$$4$$ pennies and $$42$$ nickels"],"problemType":"MultipleChoice","stepTitle":"Danny has $$\\\\$2.14$$ worth of pennies and nickels in his piggy bank. The number of nickels is two more than ten times the number of pennies. How many nickels and how many pennies does Danny have?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$4$$ pennies and $$42$$ nickels","choices":["$$6$$ pennines and $$40$$ nickels","$$4$$ pennies and $$42$$ nickels","$$40$$ pennies and $$5$$ nickels"],"hints":{"DefaultPathway":[{"id":"ad3a8eeSolveMixture13a-h1","type":"hint","dependencies":[],"title":"Read the Problem.","text":"Make sure all the word and ideas are understood. Determine the types of coins involved. Think about the strategy we used to find the value of the handful of coins. The first thing we need is to notice what type of coins are involved. Danny has pennies and nickels. Create a table to organize the information where the columns indicate the type of coin and the rows include the number of coins, their value, and their total value when multiplied. We can work this problem all in cents or in dollars. Here we\'ll do it in dollars and put in the dollar sign in the tail as a reminder. The value of a penny is $$\\\\$0.01$$ and the value of a nickel is $$\\\\$0.05$$. The total value of all the coins is $$\\\\$2.14$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture13a-h2","type":"hint","dependencies":["ad3a8eeSolveMixture13a-h1"],"title":"Identify and Name the Unknowns","text":"Choose a variable to represent the quantity for which we are looking for. So, let $$p=number$$ of pennies. $$10p+2=number$$ of nickels","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture13a-h3","type":"hint","dependencies":["ad3a8eeSolveMixture13a-h2"],"title":"Total Value of Each Type","text":"To get the total value of each type of coin, we can use the relationship: total value of a $$type=number$$ of coins of this $$type value$$ of this type.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture13a-h4","type":"hint","dependencies":["ad3a8eeSolveMixture13a-h3"],"title":"Translate into an Equation","text":"We know that value of $$pennies+value$$ of $$nickels=total$$ value of coins. Plug in the numbers and variables, we get the equation: $$0.01p+\\\\operatorname{0.05}\\\\left(10p+2\\\\right)=2.14$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture13a-h5","type":"hint","dependencies":["ad3a8eeSolveMixture13a-h4"],"title":"Solve the Equation","text":"To solve $$0.01p+\\\\operatorname{0.05}\\\\left(10p+2\\\\right)=2.14$$, we follow the following steps: 0.01p+0.05*10p+0.05*2=2.14--\x3e0.51p=2.04--\x3ep=4. Therefore, the number of pennies is $$4$$, and the number of nickels is $$10\\\\times4+2=42$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad3a8eeSolveMixture14","title":"Jesse\'s Quarters and Nickels","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Solve Mixture Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad3a8eeSolveMixture14a","stepAnswer":["$$41$$ nickels and $$18$$ quarters"],"problemType":"MultipleChoice","stepTitle":"Jesse has $$\\\\$6.55$$ worth of quarters and nickels in his pocket. The number of nickels is five more than two times the number of quarters. How many nickels and how many quarts does Jesse have?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$41$$ nickels and $$18$$ quarters","choices":["$$41$$ nickels and $$18$$ quarters","$$41$$ quarters and $$18$$ nickel","$$36$$ nickels and $$24$$ quarters"],"hints":{"DefaultPathway":[{"id":"ad3a8eeSolveMixture14a-h1","type":"hint","dependencies":[],"title":"Read the Problem.","text":"Make sure all the word and ideas are understood. Determine the types of coins involved. Think about the strategy we used to find the value of the handful of coins. The first thing we need is to notice what type of coins are involved. Jesse has quarters and nickels. Create a table to organize the information where the columns indicate the type of coin and the rows include the number of coins, their value, and their total value when multiplied. We can work this problem all in cents or in dollars. Here we\'ll do it in dollars and put in the dollar sign in the tail as a reminder. The value of a quarter is $$\\\\$0.25$$ and the value of a nickel is $$\\\\$0.05$$. The total value of all the coins is $$\\\\$6.55$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture14a-h2","type":"hint","dependencies":["ad3a8eeSolveMixture14a-h1"],"title":"Identify and Name the Unknowns","text":"Choose a variable to represent the quantity for which we are looking for. So, let $$q=number$$ of quarters. $$2p+5=number$$ of nickels","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture14a-h3","type":"hint","dependencies":["ad3a8eeSolveMixture14a-h2"],"title":"Total Value of Each Type","text":"To get the total value of each type of coin, we can use the relationship: total value of a $$type=number$$ of coins of this $$type value$$ of this type.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture14a-h4","type":"hint","dependencies":["ad3a8eeSolveMixture14a-h3"],"title":"Translate into an Equation","text":"We know that value of $$quarters+value$$ of $$nickels=total$$ value of coins. Plug in the numbers and variables, we get the equation: $$0.25q+\\\\operatorname{0.05}\\\\left(5+2q\\\\right)=6.55$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture14a-h5","type":"hint","dependencies":["ad3a8eeSolveMixture14a-h4"],"title":"Solve the Equation","text":"To solve $$0.25q+\\\\operatorname{0.05}\\\\left(5+2q\\\\right)=6.55$$, we follow the following steps: 0.25q+0.05*5+0.05*2q=6.55--\x3e0.35q=6.3--\x3e18. Therefore, the number of quarters is $$18$$, and the number of nickels is $$2\\\\times18+5=41$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad3a8eeSolveMixture15","title":"Elane\'s Dimes and Nickels","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Solve Mixture Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad3a8eeSolveMixture15a","stepAnswer":["$$22$$ nickels and $$59$$ dimes"],"problemType":"MultipleChoice","stepTitle":"Elane has $$\\\\$7.00$$ total in dimes and nickels in her coin jar. The number of dimes that Elena has is seven less than three times the number of nickels. How many of each coin does Elaine have?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$22$$ nickels and $$59$$ dimes","choices":["$$12$$ nickels and $$60$$ dimes","$$22$$ nickels and $$59$$ dimes","$$16$$ nickels and $$66$$ dimes"],"hints":{"DefaultPathway":[{"id":"ad3a8eeSolveMixture15a-h1","type":"hint","dependencies":[],"title":"Read the Problem.","text":"Make sure all the word and ideas are understood. Determine the types of coins involved. Think about the strategy we used to find the value of the handful of coins. The first thing we need is to notice what type of coins are involved. Elane has dimes and nickels. Create a table to organize the information where the columns indicate the type of coin and the rows include the number of coins, their value, and their total value when multiplied. We can work this problem all in cents or in dollars. Here we\'ll do it in dollars and put in the dollar sign in the tail as a reminder. The value of a dime is $$\\\\$0.10$$ and the value of a nickel is $$\\\\$0.05$$. The total value of all the coins is $$\\\\$7.00$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture15a-h2","type":"hint","dependencies":["ad3a8eeSolveMixture15a-h1"],"title":"Identify and Name the Unknowns","text":"Choose a variable to represent the quantity for which we are looking for. So, let $$n=number$$ of nickels. $$3n-7=number$$ of dimes","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture15a-h3","type":"hint","dependencies":["ad3a8eeSolveMixture15a-h2"],"title":"Total Value of Each Type","text":"To get the total value of each type of coin, we can use the relationship: total value of a $$type=number$$ of coins of this $$type value$$ of this type.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture15a-h4","type":"hint","dependencies":["ad3a8eeSolveMixture15a-h3"],"title":"Translate into an Equation","text":"We know that value of $$dimes+value$$ of $$nickels=total$$ value of coins. Plug in the numbers and variables, we get the equation: $$\\\\operatorname{0.1}\\\\left(3n-7\\\\right)+\\\\operatorname{0.05}\\\\left(n\\\\right)=7.00$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture15a-h5","type":"hint","dependencies":["ad3a8eeSolveMixture15a-h4"],"title":"Solve the Equation","text":"To solve $$\\\\operatorname{0.1}\\\\left(3n-7\\\\right)+\\\\operatorname{0.05}\\\\left(n\\\\right)=7.00$$, we follow the following steps: 0.1*3n-0.1*7+0.05n=7.00--\x3e0.35n=7.7--\x3en=22. Therefore, the number of nickels is $$22$$, and the number of dimes is $$3\\\\times22-7=59$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad3a8eeSolveMixture2","title":"Solve Ticket and Stamp Word Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Solve Mixture Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad3a8eeSolveMixture2a","stepAnswer":["330,367"],"problemType":"TextBox","stepTitle":"The first day of a water polo tournament the total value of tickets sold was $17,610. One-day passes sold for $20 and tournament passes sold for $30. The number of tournament passes sold was $$37$$ more than the number of day passes sold. How many day passes and how many tournament passes were sold (entere the answer as days,passes)?","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"ad3a8eeSolveMixture2a-h1","type":"hint","dependencies":[],"title":"Define variables","text":"Assign the number of one day passes a varible (like x) and the number of tournament passes another variable (like y)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture2a-h2","type":"hint","dependencies":["ad3a8eeSolveMixture2a-h1"],"title":"Setting up equations","text":"Based on the question if $$x$$ is the number of day passes and $$y$$ is the number of tournament passes then you seperate all the information into two different equations","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture2a-h3","type":"hint","dependencies":["ad3a8eeSolveMixture2a-h2"],"title":"Cost Equation","text":"One of the equations can represent the total cost of the event which we know to be $1761. Since each day pass costs $20 we can multiply that with number of day passes to represent the total revenue of day passes. You can do the same with tournament passes and their sum would be equal to the total cost.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20x+30y=17610$$"],"dependencies":["ad3a8eeSolveMixture2a-h3"],"title":"Cost Equation","text":"What is the equation equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture2a-h5","type":"hint","dependencies":["ad3a8eeSolveMixture2a-h4"],"title":"Tickets equation","text":"You can represent the relationship between day passes and tournament passes using the information given.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y=37+x$$"],"dependencies":["ad3a8eeSolveMixture2a-h5"],"title":"Tickets equation","text":"What is the relationship","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture2a-h7","type":"hint","dependencies":["ad3a8eeSolveMixture2a-h6"],"title":"Combining the equations","text":"Seperate one variable in one of the equations and plug it in the second equation. In this case it would be easier to plug in $$y=x+37$$ into the equation $$20x+30y$$ $$=$$ $$17610$$ which would equal $$20x+\\\\operatorname{30}\\\\left(x+37\\\\right)$$ $$=$$ $$17610$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture2a-h8","type":"hint","dependencies":["ad3a8eeSolveMixture2a-h7"],"title":"Solve for the missing variable","text":"Use algebra to simplify the equation to solve for the missing variable (x)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture2a-h9","type":"hint","dependencies":["ad3a8eeSolveMixture2a-h8"],"title":"Plug in for $$y$$","text":"Once you have a value for $$x$$ plug that value one of the two equations to get the value of $$y$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad3a8eeSolveMixture3","title":"Solve Ticket and Stamp Word Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Solve Mixture Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad3a8eeSolveMixture3a","stepAnswer":["80,135"],"problemType":"TextBox","stepTitle":"At the movie theater, the total value of tickets sold was $$\\\\$2, 612.50$$. Adult tickets sold for $10 each and $$\\\\frac{senior}{child}$$ tickets sold for $$\\\\$7.50$$ each. The number of $$\\\\frac{senior}{child}$$ tickets sold was $$25$$ less than twice the number of adult tickets sold. How many $$\\\\frac{senior}{child}$$ tickets and how many adult tickets were sold (enter the answer as child tickets,adult tickets)?","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"ad3a8eeSolveMixture3a-h1","type":"hint","dependencies":[],"title":"Define variables","text":"Assign the number of child tickets a varible (like x) and the number of adult tickets another variable (like y)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture3a-h2","type":"hint","dependencies":["ad3a8eeSolveMixture3a-h1"],"title":"Setting up equations","text":"Based on the question if $$x$$ is the number of child tickets and $$y$$ is the number of adult tickets then you seperate all the information into two different equations","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture3a-h3","type":"hint","dependencies":["ad3a8eeSolveMixture3a-h2"],"title":"Cost Equation","text":"One of the equations can represent the total cost of the event which we know to be $$\\\\$2612.5$$. Since each child ticket costs $$\\\\$7.5$$ we can multiply that with number of tickets to represent the total revenue of students. You can do the same with adult tickets and their sum would be equal to the total cost.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10y+7.5x=2612.5$$"],"dependencies":["ad3a8eeSolveMixture3a-h3"],"title":"Cost Equation","text":"What is the equation equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture3a-h5","type":"hint","dependencies":["ad3a8eeSolveMixture3a-h4"],"title":"Tickets equation","text":"You can represent the relationship between child tickets and adult tickets using the information given.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture3a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=2y-25$$"],"dependencies":["ad3a8eeSolveMixture3a-h5"],"title":"Tickets equation","text":"What is the relationship","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture3a-h7","type":"hint","dependencies":["ad3a8eeSolveMixture3a-h6"],"title":"Combining the equations","text":"Seperate one variable in one of the equations and plug it in the second equation. In this case it would be easier to plug in $$x$$ $$=$$ $$2y-25$$ into the equation $$10y+7.5x=2612.5$$ which would equal $$10y+\\\\operatorname{7.5}\\\\left(2y-25\\\\right)=2612.5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture3a-h8","type":"hint","dependencies":["ad3a8eeSolveMixture3a-h7"],"title":"Solve for the missing variable","text":"Use algebra to simplify the equation to solve for the missing variable (x)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture3a-h9","type":"hint","dependencies":["ad3a8eeSolveMixture3a-h8"],"title":"Plug in for $$y$$","text":"Once you have a value for $$x$$ plug that value one of the two equations to get the value of $$y$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad3a8eeSolveMixture4","title":"Solve Ticket and Stamp Word Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Solve Mixture Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad3a8eeSolveMixture4a","stepAnswer":["615,195"],"problemType":"TextBox","stepTitle":"Galen sold $$810$$ tickets for his church\u2019s carnival for a total of $2,820. Children\u2019s tickets cost $3 each and adult tickets cost $5 each. How many children\u2019s tickets and how many adult tickets did he sell (enter the answer as child tickets,adult tickets)?","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"ad3a8eeSolveMixture4a-h1","type":"hint","dependencies":[],"title":"Define variables","text":"Assign the number of child tickets a varible (like x) and the number of adult tickets another variable (like y)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture4a-h2","type":"hint","dependencies":["ad3a8eeSolveMixture4a-h1"],"title":"Setting up equations","text":"Based on the question if $$x$$ is the number of child tickets and $$y$$ is the number of adult tickets then you seperate all the information into two different equations","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture4a-h3","type":"hint","dependencies":["ad3a8eeSolveMixture4a-h2"],"title":"Cost Equation","text":"One of the equations can represent the total cost of the event which we know to be $2820. Since each child ticket costs $3 we can multiply that with number of tickets to represent the total revenue of students. You can do the same with adult tickets and their sum would be equal to the total cost.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x+5y=2820$$"],"dependencies":["ad3a8eeSolveMixture4a-h3"],"title":"Cost Equation","text":"What is the equation equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture4a-h5","type":"hint","dependencies":["ad3a8eeSolveMixture4a-h4"],"title":"Tickets equation","text":"You can represent the relationship between child tickets and adult tickets using the information given.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture4a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y=810-x$$"],"dependencies":["ad3a8eeSolveMixture4a-h5"],"title":"Tickets equation","text":"What is the relationship","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture4a-h7","type":"hint","dependencies":["ad3a8eeSolveMixture4a-h6"],"title":"Combining the equations","text":"Seperate one variable in one of the equations and plug it in the second equation. In this case it would be easier to plug in $$y=810-x$$ into the equation $$3x+5y=2820$$ which would equal $$3x+5\\\\left(810-x\\\\right)=2820$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture4a-h8","type":"hint","dependencies":["ad3a8eeSolveMixture4a-h7"],"title":"Solve for the missing variable","text":"Use algebra to simplify the equation to solve for the missing variable (x)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture4a-h9","type":"hint","dependencies":["ad3a8eeSolveMixture4a-h8"],"title":"Plug in for $$y$$","text":"Once you have a value for $$x$$ plug that value one of the two equations to get the value of $$y$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad3a8eeSolveMixture5","title":"Solve Ticket and Stamp Word Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Solve Mixture Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad3a8eeSolveMixture5a","stepAnswer":["$$3184$$"],"problemType":"TextBox","stepTitle":"During her shift at the museum ticket booth, Leah sold $$115$$ tickets for a total of $1,163. Adult tickets cost $12 and student tickets cost $5. How many adult tickets and how many student tickets did Leah sell?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3184$$","hints":{"DefaultPathway":[{"id":"ad3a8eeSolveMixture5a-h1","type":"hint","dependencies":[],"title":"Define variables","text":"Assign the number of child tickets a varible (like x) and the number of adult tickets another variable (like y)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture5a-h2","type":"hint","dependencies":["ad3a8eeSolveMixture5a-h1"],"title":"Setting up equations","text":"Based on the question if $$x$$ is the number of child tickets and $$y$$ is the number of adult tickets then you seperate all the information into two different equations","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture5a-h3","type":"hint","dependencies":["ad3a8eeSolveMixture5a-h2"],"title":"Cost Equation","text":"One of the equations can represent the total cost of the event which we know to be $1163. Since each child ticket costs $5 we can multiply that with number of tickets to represent the total revenue of students. You can do the same with adult tickets and their sum would be equal to the total cost.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5x+12y=1163$$"],"dependencies":["ad3a8eeSolveMixture5a-h3"],"title":"Cost Equation","text":"What is the equation equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture5a-h5","type":"hint","dependencies":["ad3a8eeSolveMixture5a-h4"],"title":"Tickets equation","text":"You can represent the relationship between child tickets and adult tickets using the information given.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture5a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=115-y$$"],"dependencies":["ad3a8eeSolveMixture5a-h5"],"title":"Tickets equation","text":"What is the relationship","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture5a-h7","type":"hint","dependencies":["ad3a8eeSolveMixture5a-h6"],"title":"Combining the equations","text":"Seperate one variable in one of the equations and plug it in the second equation. In this case it would be easier to plug in $$x=115-y$$ into the equation $$5x+12y=1163$$ which would equal $$5(115-y)$$ $$\\\\left(+12\\\\right) y=1163$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture5a-h8","type":"hint","dependencies":["ad3a8eeSolveMixture5a-h7"],"title":"Solve for the missing variable","text":"Use algebra to simplify the equation to solve for the missing variable (x)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture5a-h9","type":"hint","dependencies":["ad3a8eeSolveMixture5a-h8"],"title":"Plug in for $$y$$","text":"Once you have a value for $$x$$ plug that value one of the two equations to get the value of $$y$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad3a8eeSolveMixture6","title":"Solve Ticket and Stamp Word Problems","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Solve Mixture Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad3a8eeSolveMixture6a","stepAnswer":["$$2614$$"],"problemType":"TextBox","stepTitle":"A whale-watching ship had $$40$$ paying passengers on board. The total collected from tickets was $1,196. Full-fare passengers paid $32 each and reduced-fare passengers paid $26 each. How many full-fare passengers and how many reduced-fare passengers were on the ship?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2614$$","hints":{"DefaultPathway":[{"id":"ad3a8eeSolveMixture6a-h1","type":"hint","dependencies":[],"title":"Define variables","text":"Assign the number of full-fare passengers a varible (like x) and the number of reduced-fare passengers another variable (like y)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture6a-h2","type":"hint","dependencies":["ad3a8eeSolveMixture6a-h1"],"title":"Setting up equations","text":"Based on the question if $$x$$ is the number of full-fare passengers and $$y$$ is the number of reduced-fare passengers then you seperate all the information into two different equations","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture6a-h3","type":"hint","dependencies":["ad3a8eeSolveMixture6a-h2"],"title":"Cost Equation","text":"One of the equations can represent the total cost of the event which we know to be $1196. Since each child ticket costs $32 we can multiply that with number of tickets to represent the total revenue of students. You can do the same with adult tickets and their sum would be equal to the total cost.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$32x+26y=1196$$"],"dependencies":["ad3a8eeSolveMixture6a-h3"],"title":"Cost Equation","text":"What is the equation equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture6a-h5","type":"hint","dependencies":["ad3a8eeSolveMixture6a-h4"],"title":"Tickets equation","text":"You can represent the relationship between full-fare passengers and reduced-fare passengers using the information given.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture6a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x=40-y$$"],"dependencies":["ad3a8eeSolveMixture6a-h5"],"title":"Tickets equation","text":"What is the relationship","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture6a-h7","type":"hint","dependencies":["ad3a8eeSolveMixture6a-h6"],"title":"Combining the equations","text":"Seperate one variable in one of the equations and plug it in the second equation. In this case it would be easier to plug in $$x=40-y$$ into the equation $$32x+26y$$ $$=$$ $$1196$$ which would equal $$32(40-y)$$ $$\\\\left(+26\\\\right) y$$ $$=$$ $$1196$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture6a-h8","type":"hint","dependencies":["ad3a8eeSolveMixture6a-h7"],"title":"Solve for the missing variable","text":"Use algebra to simplify the equation to solve for the missing variable (x)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture6a-h9","type":"hint","dependencies":["ad3a8eeSolveMixture6a-h8"],"title":"Plug in for $$y$$","text":"Once you have a value for $$x$$ plug that value one of the two equations to get the value of $$y$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad3a8eeSolveMixture7","title":"Adalberto\'s Dimes and Nickels","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Solve Mixture Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad3a8eeSolveMixture7a","stepAnswer":["$$12$$ dimes and $$21$$ nickels"],"problemType":"MultipleChoice","stepTitle":"Adalberto has $$\\\\$2.25$$ in dimes and nickels in his pocket. He has nine more nickels than dimes. How many of each type of coin does he have?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$12$$ dimes and $$21$$ nickels","choices":["$$6$$ dimes and $$14$$ nickels","$$12$$ dimes and $$7$$ nickels","$$4$$ dimes and $$20$$ nickels","$$12$$ dimes and $$21$$ nickels"],"hints":{"DefaultPathway":[{"id":"ad3a8eeSolveMixture7a-h1","type":"hint","dependencies":[],"title":"Read the Problem.","text":"Make sure all the word and ideas are understood. Determine the types of coins involved. Think about the strategy we used to find the value of the handful of coins. The first thing we need is to notice what type of coins are involved. Adalberto has dimes and nickels. Create a table to organize the information where the columns indicate the type of coin and the rows include the number of coins, their value, and their total value when multiplied. We can work this problem all in cents or in dollars. Here we\'ll do it in dollars and put in the dollar sign in the tail as a reminder. The value of a dime is $$\\\\$0.10$$ and the value of a nickel is $$\\\\$0.05$$. The total value of all the coins is $$\\\\$2.25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture7a-h2","type":"hint","dependencies":["ad3a8eeSolveMixture7a-h1"],"title":"Identify and Name the Unknowns","text":"Choose a variable to represent the quantity for which we are looking for. So, let $$d=number$$ of dimes. $$d+9=number$$ of nickels","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture7a-h3","type":"hint","dependencies":["ad3a8eeSolveMixture7a-h2"],"title":"Total Value of Each Type","text":"To get the total value of each type of coin, we can use the relationship: total value of a $$type=number$$ of coins of this $$type value$$ of this type.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture7a-h4","type":"hint","dependencies":["ad3a8eeSolveMixture7a-h3"],"title":"Translate into an Equation","text":"We know that value of $$dimes+value$$ of $$nickels=total$$ value of coins. Plug in the numbers and variables, we get the equation $$0.1d+\\\\operatorname{0.05}\\\\left(d+9\\\\right)=2.25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture7a-h5","type":"hint","dependencies":["ad3a8eeSolveMixture7a-h4"],"title":"Solve the Equation","text":"To solve $$0.1d+\\\\operatorname{0.05}\\\\left(d+9\\\\right)=2.25$$, we follow the following steps: 0.10d+0.05d+0.05*9=2.25--\x3e0.15d=1.8--\x3ed=12. Therefore, the number of dimes is $$12$$, and the number of nickels is $$12+9=21$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture7a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ad3a8eeSolveMixture7a-h5"],"title":"Quick Check","text":"With $$12$$ dimes and $$21$$ nickels, does the total value equal $$\\\\$2.25$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"ad3a8eeSolveMixture8","title":"Michaela\'s Dimes and Nickels","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Solve Mixture Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad3a8eeSolveMixture8a","stepAnswer":["$$9$$ nickels and $$16$$ dimes"],"problemType":"MultipleChoice","stepTitle":"Michaela has $$\\\\$2.05$$ in dimes and nickels in her change purse. She has seven more dimes than nickels. How many coins for each type does she have?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$9$$ nickels and $$16$$ dimes","choices":["$$11$$ dimes and $$18$$ nickels","$$10$$ dimes and $$17$$ dimes","$$9$$ nickels and $$16$$ dimes"],"hints":{"DefaultPathway":[{"id":"ad3a8eeSolveMixture8a-h1","type":"hint","dependencies":[],"title":"Read the Problem.","text":"Make sure all the word and ideas are understood. Determine the types of coins involved. Think about the strategy we used to find the value of the handful of coins. The first thing we need is to notice what type of coins are involved. Michaela has dimes and nickels. Create a table to organize the information where the columns indicate the type of coin and the rows include the number of coins, their value, and their total value when multiplied. We can work this problem all in cents or in dollars. Here we\'ll do it in dollars and put in the dollar sign in the tail as a reminder. The value of a dime is $$\\\\$0.10$$ and the value of a nickel is $$\\\\$0.05$$. The total value of all the coins is $$\\\\$2.05$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture8a-h2","type":"hint","dependencies":["ad3a8eeSolveMixture8a-h1"],"title":"Identify and Name the Unknowns","text":"Choose a variable to represent the quantity for which we are looking for. So, let $$n=number$$ of nickels. $$n+7=number$$ of dimes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture8a-h3","type":"hint","dependencies":["ad3a8eeSolveMixture8a-h2"],"title":"Total Value of Each Type","text":"To get the total value of each type of coin, we can use the relationship: total value of a $$type=number$$ of coins of this $$type value$$ of this type.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture8a-h4","type":"hint","dependencies":["ad3a8eeSolveMixture8a-h3"],"title":"Translate into an Equation","text":"We know that value of $$dimes+value$$ of $$nickels=total$$ value of coins. Plug in the numbers and variables, we get the equation $$\\\\operatorname{0.1}\\\\left(n+7\\\\right)+0.05n=2.05$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture8a-h5","type":"hint","dependencies":["ad3a8eeSolveMixture8a-h4"],"title":"Solve the Equation","text":"To solve $$\\\\operatorname{0.1}\\\\left(n+7\\\\right)+0.05n=2.05$$, we follow the following steps: 0.10n+0.10*7+0.05n=2.05--\x3e0.15n=1.35--\x3en=9. Therefore, the number of nickels is $$9$$, and the number of dimes is $$9+7=16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad3a8eeSolveMixture9","title":"Liliana\'s Nickels and Quarters","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.3 Solve Mixture Applications","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad3a8eeSolveMixture9a","stepAnswer":["$$5$$ quarters and $$17$$ nickels"],"problemType":"MultipleChoice","stepTitle":"Liliana has $$\\\\$2.10$$ in nickels and quarters in her backpack. She has $$12$$ more nickels than quarters. How many coins for each type does she have?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$5$$ quarters and $$17$$ nickels","choices":["$$18$$ quarters and $$4$$ nickels","$$5$$ quarters and $$17$$ nickels","$$16$$ nickels and $$4$$ quarters"],"hints":{"DefaultPathway":[{"id":"ad3a8eeSolveMixture9a-h1","type":"hint","dependencies":[],"title":"Read the Problem.","text":"Make sure all the word and ideas are understood. Determine the types of coins involved. Think about the strategy we used to find the value of the handful of coins. The first thing we need is to notice what type of coins are involved. Liliana has nickels and quarters. Create a table to organize the information where the columns indicate the type of coin and the rows include the number of coins, their value, and their total value when multiplied. We can work this problem all in cents or in dollars. Here we\'ll do it in dollars and put in the dollar sign in the tail as a reminder. The value of a quarter is $$\\\\$0.25$$ and the value of a nickel is $$\\\\$0.05$$. The total value of all the coins is $$\\\\$2.10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture9a-h2","type":"hint","dependencies":["ad3a8eeSolveMixture9a-h1"],"title":"Identify and Name the Unknowns","text":"Choose a variable to represent the quantity for which we are looking for. So, let $$q=number$$ of quarters. $$q+12=number$$ of nickels.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture9a-h3","type":"hint","dependencies":["ad3a8eeSolveMixture9a-h2"],"title":"Total Value of Each Type","text":"To get the total value of each type of coin, we can use the relationship: total value of a $$type=number$$ of coins of this $$type value$$ of this type.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture9a-h4","type":"hint","dependencies":["ad3a8eeSolveMixture9a-h3"],"title":"Translate into an Equation","text":"We know that value of $$quarters+value$$ of $$nickels=total$$ value of coins. Plug in the numbers and variables, we get the equation $$0.25q+\\\\operatorname{0.05}\\\\left(q+12\\\\right)=2.10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad3a8eeSolveMixture9a-h5","type":"hint","dependencies":["ad3a8eeSolveMixture9a-h4"],"title":"Solve the Equation","text":"To solve $$0.25q+\\\\operatorname{0.05}\\\\left(q+12\\\\right)=2.10$$, we follow the following steps: 0.25q+0.05q+0.05*12=2.10--\x3e0.3q=1.5--\x3eq-5. Therefore, the number of quarters is $$5$$, and the number of nickels is $$5+12=17$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad4e7e2decimals1","title":"How to Name Decimals","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.7 Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad4e7e2decimals1a","stepAnswer":["four and three tenths"],"problemType":"MultipleChoice","stepTitle":"Name the decimal $$4.3$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["forty theee tenths","$$forty-theee$$","four and theee hundredths","four and theee tenths","four and three tenths"],"hints":{"DefaultPathway":[{"id":"ad4e7e2decimals1a-h1","type":"hint","dependencies":[],"title":"Left Value","text":"Name the number to the left of the decimal: four","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals1a-h2","type":"hint","dependencies":["ad4e7e2decimals1a-h1"],"title":"Decimal Point","text":"Write \\"and\\" for the decimal point","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals1a-h3","type":"hint","dependencies":["ad4e7e2decimals1a-h2"],"title":"Right Value","text":"Name the number to the right of the decimal: three","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals1a-h4","type":"hint","dependencies":["ad4e7e2decimals1a-h3"],"title":"Place Value","text":"Name the place of the decimal: tenth","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad4e7e2decimals10","title":"How to Round Decimals","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.7 Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad4e7e2decimals10a","stepAnswer":["$$18.38$$"],"problemType":"TextBox","stepTitle":"Round $$18.379$$ to the nearest hundredth.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$18.38$$","hints":{"DefaultPathway":[{"id":"ad4e7e2decimals10a-h1","type":"hint","dependencies":[],"title":"Locate","text":"Locate the given place value and mark it or highlight it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals10a-h2","type":"hint","dependencies":["ad4e7e2decimals10a-h1"],"title":"Digits to the Right","text":"Underline the value to the right of the highlighted or marked value","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals10a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["yes"],"dependencies":["ad4e7e2decimals10a-h2"],"title":"Value","text":"Is this value greater than or equal to 5?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["yes","no"]},{"id":"ad4e7e2decimals10a-h4","type":"hint","dependencies":["ad4e7e2decimals10a-h3"],"title":"Add","text":"If the underlined digit is greater than or equal to $$5$$, we will need to add one to the digit in the given place value","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals10a-h5","type":"hint","dependencies":["ad4e7e2decimals10a-h4"],"title":"Remove","text":"Rewrite the number, removing all digits to the right of the rounding digit.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad4e7e2decimals11","title":"How to Round Decimals","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.7 Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad4e7e2decimals11a","stepAnswer":["$$1.05$$"],"problemType":"TextBox","stepTitle":"Round to the nearest hundredth: $$1.047$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1.05$$","hints":{"DefaultPathway":[{"id":"ad4e7e2decimals11a-h1","type":"hint","dependencies":[],"title":"Locate","text":"Locate the given place value and mark it or highlight it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals11a-h2","type":"hint","dependencies":["ad4e7e2decimals11a-h1"],"title":"Digits to the Right","text":"Underline the value to the right of the highlighted or marked value","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals11a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["yes"],"dependencies":["ad4e7e2decimals11a-h2"],"title":"Value","text":"Is this value greater than or equal to 5?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["yes","no"]},{"id":"ad4e7e2decimals11a-h4","type":"hint","dependencies":["ad4e7e2decimals11a-h3"],"title":"Add","text":"If the underlined digit is greater than or equal to $$5$$, we will need to add one to the digit in the given place value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals11a-h5","type":"hint","dependencies":["ad4e7e2decimals11a-h4"],"title":"Remove","text":"Rewrite the number, removing all digits to the right of the rounding digit.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad4e7e2decimals12","title":"How to Round Decimals","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.7 Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad4e7e2decimals12a","stepAnswer":["$$9.17$$"],"problemType":"TextBox","stepTitle":"Round to the nearest hundredth: $$9.173$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9.17$$","hints":{"DefaultPathway":[{"id":"ad4e7e2decimals12a-h1","type":"hint","dependencies":[],"title":"Locate","text":"Locate the given place value and mark it or highlight it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals12a-h2","type":"hint","dependencies":["ad4e7e2decimals12a-h1"],"title":"Digits to the Right","text":"Underline the value to the right of the highlighted or marked value","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals12a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["no"],"dependencies":["ad4e7e2decimals12a-h2"],"title":"Value","text":"Is this value greater than or equal to 5?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["yes","no"]},{"id":"ad4e7e2decimals12a-h4","type":"hint","dependencies":["ad4e7e2decimals12a-h3"],"title":"Add","text":"If the underlined digit is not greater than or equal to $$5$$, we will not need to add one to the digit in the given place value","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals12a-h5","type":"hint","dependencies":["ad4e7e2decimals12a-h4"],"title":"Remove","text":"Rewrite the number, removing all digits to the right of the rounding digit.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad4e7e2decimals13","title":"How to Round Decimals","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.7 Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad4e7e2decimals13a","stepAnswer":["$$18.4$$"],"problemType":"TextBox","stepTitle":"Round $$18.379$$ to the nearest tenth","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$18.4$$","hints":{"DefaultPathway":[{"id":"ad4e7e2decimals13a-h1","type":"hint","dependencies":[],"title":"Locate","text":"Locate the given place value and mark it or highlight it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals13a-h2","type":"hint","dependencies":["ad4e7e2decimals13a-h1"],"title":"Digits to the Right","text":"Underline the value to the right of the highlighted or marked value","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals13a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["yes"],"dependencies":["ad4e7e2decimals13a-h2"],"title":"Value","text":"Is this value greater than or equal to 5?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["yes","no"]},{"id":"ad4e7e2decimals13a-h4","type":"hint","dependencies":["ad4e7e2decimals13a-h3"],"title":"Add","text":"If the underlined digit is greater than or equal to $$5$$, we will need to add one to the digit in the given place value","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals13a-h5","type":"hint","dependencies":["ad4e7e2decimals13a-h4"],"title":"Remove","text":"Rewrite the number, removing all digits to the right of the rounding digit.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ad4e7e2decimals13b","stepAnswer":["$$18$$"],"problemType":"TextBox","stepTitle":"Round $$18.379$$ to the nearest whole number","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$18$$","hints":{"DefaultPathway":[{"id":"ad4e7e2decimals13b-h1","type":"hint","dependencies":[],"title":"Locate","text":"Locate the given place value and mark it or highlight it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals13b-h2","type":"hint","dependencies":["ad4e7e2decimals13b-h1"],"title":"Digits to the Right","text":"Underline the value to the right of the highlighted or marked value","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals13b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["no"],"dependencies":["ad4e7e2decimals13b-h2"],"title":"Value","text":"Is this value greater than or equal to 5?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["yes","no"]},{"id":"ad4e7e2decimals13b-h4","type":"hint","dependencies":["ad4e7e2decimals13b-h3"],"title":"Add","text":"If the underlined digit is not greater than or equal to $$5$$, we will not need to add one to the digit in the given place value","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals13b-h5","type":"hint","dependencies":["ad4e7e2decimals13b-h4"],"title":"Remove","text":"Rewrite the number, removing all digits to the right of the rounding digit.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad4e7e2decimals14","title":"Add:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.7 Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad4e7e2decimals14a","stepAnswer":["$$64.88$$"],"problemType":"TextBox","stepTitle":"$$23.5+41.38$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$64.88$$","hints":{"DefaultPathway":[{"id":"ad4e7e2decimals14a-h1","type":"hint","dependencies":[],"title":"Match Up","text":"Match up the decimal points, adding zeroes as placeholders. In this case, we write the expression as $$23.5+41.38$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals14a-h2","type":"hint","dependencies":["ad4e7e2decimals14a-h1"],"title":"Add","text":"Add the numbers as if they were whole numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals14a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6488$$"],"dependencies":["ad4e7e2decimals14a-h2"],"title":"Add","text":"What is $$2350+4138$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals14a-h4","type":"hint","dependencies":["ad4e7e2decimals14a-h3"],"title":"Putting Decimal Point","text":"Place the decimal point in the sum at the same place as where it is in the given numbers, which gives us $$64.88$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad4e7e2decimals15","title":"Add:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.7 Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad4e7e2decimals15a","stepAnswer":["$$16.49$$"],"problemType":"TextBox","stepTitle":"$$4.8+11.69$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16.49$$","hints":{"DefaultPathway":[{"id":"ad4e7e2decimals15a-h1","type":"hint","dependencies":[],"title":"Match Up","text":"Match up the decimal points, adding zeroes as placeholders. In this case, we write the expression as $$4.8+11.69$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals15a-h2","type":"hint","dependencies":["ad4e7e2decimals15a-h1"],"title":"Add","text":"Add the numbers as if they were whole numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1649$$"],"dependencies":["ad4e7e2decimals15a-h2"],"title":"Add","text":"What is $$480+1169$$? Remember to carry when needed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals15a-h4","type":"hint","dependencies":["ad4e7e2decimals15a-h3"],"title":"Putting Decimal Point","text":"Place the decimal point in the sum at the same place as where it is in the given numbers, which gives us $$16.49$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad4e7e2decimals16","title":"Add:","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.7 Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad4e7e2decimals16a","stepAnswer":["$$23.593$$"],"problemType":"TextBox","stepTitle":"$$5.123+18.47$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$23.593$$","hints":{"DefaultPathway":[{"id":"ad4e7e2decimals16a-h1","type":"hint","dependencies":[],"title":"Match Up","text":"Match up the decimal points, adding zeroes as placeholders. In this case, we write the expression as $$5.123+18.47$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals16a-h2","type":"hint","dependencies":["ad4e7e2decimals16a-h1"],"title":"Add","text":"Add the numbers as if they were whole numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals16a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$23593$$"],"dependencies":["ad4e7e2decimals16a-h2"],"title":"Add","text":"What is $$5123+18470$$? Remember to carry when needed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals16a-h4","type":"hint","dependencies":["ad4e7e2decimals16a-h3"],"title":"Putting Decimal Point","text":"Place the decimal point in the sum at the same place as where it is in the given numbers, which gives us $$23.593$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad4e7e2decimals17","title":"Multiply $$5.63$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.7 Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad4e7e2decimals17a","stepAnswer":["$$56.3$$"],"problemType":"TextBox","stepTitle":"by $$10$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$56.3$$","hints":{"DefaultPathway":[{"id":"ad4e7e2decimals17a-h1","type":"hint","dependencies":[],"title":"Multiply a Decimal by a Power of Ten.","text":"Move the decimal point to the right the same number of places as the number of zeros in the power of $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals17a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ad4e7e2decimals17a-h1"],"title":"Number of Zeros","text":"How many zeros are there in 10?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals17a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$56.3$$"],"dependencies":["ad4e7e2decimals17a-h2"],"title":"Answer","text":"What number do you get when you move the decimal point in $$5.63$$ to the right by one decimal place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ad4e7e2decimals17b","stepAnswer":["$$563$$"],"problemType":"TextBox","stepTitle":"by $$100$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$563$$","hints":{"DefaultPathway":[{"id":"ad4e7e2decimals17b-h1","type":"hint","dependencies":[],"title":"Multiply a Decimal by a Power of Ten.","text":"Move the decimal point to the right the same number of places as the number of zeros in the power of $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals17b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ad4e7e2decimals17b-h1"],"title":"Number of Zeros","text":"How many zeros are there in 100?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals17b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$563$$"],"dependencies":["ad4e7e2decimals17b-h2"],"title":"Answer","text":"What number do you get when you move the decimal point in $$5.63$$ to the right by two decimal places?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ad4e7e2decimals17c","stepAnswer":["$$5630$$"],"problemType":"TextBox","stepTitle":"by $$1000$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5630$$","hints":{"DefaultPathway":[{"id":"ad4e7e2decimals17c-h1","type":"hint","dependencies":[],"title":"Multiply a Decimal by a Power of Ten.","text":"Move the decimal point to the right the same number of places as the number of zeros in the power of $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals17c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ad4e7e2decimals17c-h1"],"title":"Number of Zeros","text":"How many zeros are there in 1000?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals17c-h3","type":"hint","dependencies":["ad4e7e2decimals17c-h2"],"title":"Adding Zeros","text":"We can add zeros to the end of the number as needed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals17c-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5630$$"],"dependencies":["ad4e7e2decimals17c-h3"],"title":"Answer","text":"What number do you get when you move the decimal point in $$5.63$$ to the right by three decimal places and adding zero(s) as needed?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad4e7e2decimals18","title":"Multiply $$2.58$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.7 Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad4e7e2decimals18a","stepAnswer":["$$25.8$$"],"problemType":"TextBox","stepTitle":"by $$10$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$25.8$$","hints":{"DefaultPathway":[{"id":"ad4e7e2decimals18a-h1","type":"hint","dependencies":[],"title":"Multiply a Decimal by a Power of Ten.","text":"Move the decimal point to the right the same number of places as the number of zeros in the power of $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ad4e7e2decimals18a-h1"],"title":"Number of Zeros","text":"How many zeros are there in 10?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals18a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25.8$$"],"dependencies":["ad4e7e2decimals18a-h2"],"title":"Answer","text":"What number do you get when you move the decimal point in $$2.58$$ to the right by one decimal place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ad4e7e2decimals18b","stepAnswer":["$$258$$"],"problemType":"TextBox","stepTitle":"by $$100$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$258$$","hints":{"DefaultPathway":[{"id":"ad4e7e2decimals18b-h1","type":"hint","dependencies":[],"title":"Multiply a Decimal by a Power of Ten.","text":"Move the decimal point to the right the same number of places as the number of zeros in the power of $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals18b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ad4e7e2decimals18b-h1"],"title":"Number of Zeros","text":"How many zeros are there in 100?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals18b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$258$$"],"dependencies":["ad4e7e2decimals18b-h2"],"title":"Answer","text":"What number do you get when you move the decimal point in $$5.63$$ to the right by two decimal places?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ad4e7e2decimals18c","stepAnswer":["$$2580$$"],"problemType":"TextBox","stepTitle":"by $$1000$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2580$$","hints":{"DefaultPathway":[{"id":"ad4e7e2decimals18c-h1","type":"hint","dependencies":[],"title":"Multiply a Decimal by a Power of Ten.","text":"Move the decimal point to the right the same number of places as the number of zeros in the power of $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals18c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ad4e7e2decimals18c-h1"],"title":"Number of Zeros","text":"How many zeros are there in 1000?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals18c-h3","type":"hint","dependencies":["ad4e7e2decimals18c-h2"],"title":"Adding Zeros","text":"We can add zeros to the end of the number as needed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals18c-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2580$$"],"dependencies":["ad4e7e2decimals18c-h3"],"title":"Answer","text":"What number do you get when you move the decimal point in $$2.58$$ to the right by three decimal places and adding zero(s) as needed?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad4e7e2decimals19","title":"Multiply $$14.2$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.7 Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad4e7e2decimals19a","stepAnswer":["$$142$$"],"problemType":"TextBox","stepTitle":"by $$10$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$142$$","hints":{"DefaultPathway":[{"id":"ad4e7e2decimals19a-h1","type":"hint","dependencies":[],"title":"Multiply a Decimal by a Power of Ten.","text":"Move the decimal point to the right the same number of places as the number of zeros in the power of $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ad4e7e2decimals19a-h1"],"title":"Number of Zeros","text":"How many zeros are there in 10?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals19a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$142$$"],"dependencies":["ad4e7e2decimals19a-h2"],"title":"Answer","text":"What number do you get when you move the decimal point in $$14.2$$ to the right by one decimal place?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ad4e7e2decimals19b","stepAnswer":["$$1420$$"],"problemType":"TextBox","stepTitle":"by $$100$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1420$$","hints":{"DefaultPathway":[{"id":"ad4e7e2decimals19b-h1","type":"hint","dependencies":[],"title":"Multiply a Decimal by a Power of Ten.","text":"Move the decimal point to the right the same number of places as the number of zeros in the power of $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad4e7e2decimals28","title":"Round Decimals","body":"In the following exercises, round each number to the nearest tenth.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.7 Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad4e7e2decimals28a","stepAnswer":["$$2.8$$"],"problemType":"TextBox","stepTitle":"$$2.84$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2.8$$","hints":{"DefaultPathway":[{"id":"ad4e7e2decimals28a-h1","type":"hint","dependencies":[],"title":"Locate","text":"Locate the given place value and mark it or highlight it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad4e7e2decimals3","title":"How to Name Decimals","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.7 Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad4e7e2decimals3a","stepAnswer":["five and eight tenths"],"problemType":"MultipleChoice","stepTitle":"Name the decimal $$5.8$$","stepBody":"","answerType":"string","variabilization":{},"choices":["five and eight tenths","five and eight hundredths","$$fifty-eight$$","fifty and eight and tenths"],"hints":{"DefaultPathway":[{"id":"ad4e7e2decimals3a-h1","type":"hint","dependencies":[],"title":"Left Value","text":"Name the number to the left of the decimal: five","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals3a-h2","type":"hint","dependencies":["ad4e7e2decimals3a-h1"],"title":"Decimal Point","text":"Write \\"and\\" for the decimal point","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals3a-h3","type":"hint","dependencies":["ad4e7e2decimals3a-h2"],"title":"Right Value","text":"Name the number to the right of the decimal: eight","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals3a-h4","type":"hint","dependencies":["ad4e7e2decimals3a-h3"],"title":"Place Value","text":"Name the place of the decimal","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad4e7e2decimals31","title":"Round Decimals","body":"In the following exercises, round each number to the nearest hundredth.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.7 Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad4e7e2decimals31a","stepAnswer":["$$0.76$$"],"problemType":"TextBox","stepTitle":"$$0.761$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.76$$","hints":{"DefaultPathway":[{"id":"ad4e7e2decimals31a-h1","type":"hint","dependencies":[],"title":"Locate","text":"Locate the given place value and mark it or highlight it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals31a-h5","type":"hint","dependencies":["ad4e7e2decimals31a-h4"],"title":"Remove","text":"Rewrite the number, removing all digits to the right of the rounding digit.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad4e7e2decimals4","title":"How to Name Decimals","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.7 Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad4e7e2decimals4a","stepAnswer":["negative fifteen and five hundred seventy one thousandths"],"problemType":"MultipleChoice","stepTitle":"Name the decimal $$-15.571$$","stepBody":"","answerType":"string","variabilization":{},"choices":["negative fifteen and five 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals4a-h4","type":"hint","dependencies":["ad4e7e2decimals4a-h3"],"title":"Right Value","text":"Name the number to the right of the decimal: five hundred seventy one","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals4a-h5","type":"hint","dependencies":["ad4e7e2decimals4a-h4"],"title":"Place Value","text":"Name the place of the decimal","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals4a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["thousandths"],"dependencies":["ad4e7e2decimals4a-h5"],"title":"Place Value","text":"What is the place value of the decimal","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["tenth","hundredths","thousandths","ones"]}]}}]},{"id":"ad4e7e2decimals5","title":"How to Name Decimals","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.7 Decimals","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ad4e7e2decimals5a","stepAnswer":["negative thirteen and $$four-hundred-sixty-one$$ thousandths"],"problemType":"MultipleChoice","stepTitle":"Name the decimal: $$-13.461$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["negative thirteen and $$four-hundred-sixty-one$$ tenths","negative thirteen and $$four-hundred-sixty-one$$","negative thirteen and $$four-hundred-sixty-one$$ hundreths","negative thirteen and $$four-hundred-sixty-one$$ 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the number to the right of the decimal: four-hundred-sixty-one","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals5a-h5","type":"hint","dependencies":["ad4e7e2decimals5a-h4"],"title":"Place Value","text":"Name the place of the decimal","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals5a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["thousandths"],"dependencies":["ad4e7e2decimals5a-h5"],"title":"Place Value","text":"What is the place value of the decimal","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["tenth","hundredths","thousandths","ones"]}]}}]},{"id":"ad4e7e2decimals6","title":"How to Name 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals6a-h5","type":"hint","dependencies":["ad4e7e2decimals6a-h4"],"title":"Place Value","text":"Name the place of the decimal","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad4e7e2decimals6a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$ten-thousandths$$"],"dependencies":["ad4e7e2decimals6a-h5"],"title":"Place Value","text":"What is the place value of the decimal","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["tenth","hundredths","thousandths","$$ten-thousandths$$"]}]}}]},{"id":"ad4e7e2decimals7","title":"How to Write Decimals","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary 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Find the moment of the system with respect to the origin and find the center of mass of the system.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.4$$","hints":{"DefaultPathway":[{"id":"ad4fd60center2a-h1","type":"hint","dependencies":[],"title":"Formula for the moment of a system.","text":"M=sum{i\\\\=1}{n}{m_i*x_i}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad4fd60center2a-h2","type":"hint","dependencies":["ad4fd60center2a-h1"],"title":"Filling in $$n$$ and executing the summation.","text":"M=sum{i\\\\=1}{4}{m_i*x_i}=m_1*x_1+m_2*x_2+m_3*x_3+m_4*x_4","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad4fd60center2a-h3","type":"hint","dependencies":["ad4fd60center2a-h2"],"title":"Replace the variables with their given values and solve.","text":"$$M=12\\\\left(-4\\\\right)+12\\\\times4+30\\\\times2+6\\\\left(-6\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad4fd60center2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$24$$"],"dependencies":["ad4fd60center2a-h3"],"title":"Find the moment of the system M.","text":"What is the value of M?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad4fd60center2a-h5","type":"hint","dependencies":["ad4fd60center2a-h4"],"title":"Find the center of mass.","text":"To find the center of mass, we need the total mass (m) of the system.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad4fd60center2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$60$$"],"dependencies":["ad4fd60center2a-h5"],"title":"Find the total mass.","text":"What is the value of total mass?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ad4fd60center2a-h6-s1","type":"hint","dependencies":[],"title":"Find the total mass.","text":"$$12+12+30+6=60kg$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"ad4fd60center2a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.4$$"],"dependencies":["ad4fd60center2a-h6"],"title":"Find the center of mass.","text":"What is the value of the center of mass?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ad4fd60center2a-h7-s1","type":"hint","dependencies":[],"title":"Find the center of mass.","text":"center of $$mass=\\\\frac{M}{m}=\\\\frac{24}{60}=0.4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}]}}]},{"id":"ad4fd60center3","title":"Finding the Center of Mass of Objects in a Plane.","body":"","variabilization":{},"oer":"https://openstax.org/books/calculus-volume-1/pages/6-6-moments-and-centers-of-mass","license":0,"lesson":"6.6 Moments and Centers of Mass","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad4fd60center3a","stepAnswer":["$$(1,\\\\frac{1}{3})$$"],"problemType":"MultipleChoice","stepTitle":"Suppose three point masses are placed in the xy-plane as follows (assume coordinates are given in meters): $$m_1=2kg$$, placed at $$(-1,3)$$, $$m_2=6kg$$, placed at $$(1,1)$$, and $$m_3=4kg$$, placed at $$(2,-2)$$. Find the center of mass of the system.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(1,\\\\frac{1}{3})$$","choices":["$$(1,\\\\frac{1}{3})$$","$$(-1,\\\\frac{1}{3})$$","$$(1,\\\\frac{-1}{3})$$","$$(-1,\\\\frac{-1}{3})$$"],"hints":{"DefaultPathway":[{"id":"ad4fd60center3a-h1","type":"hint","dependencies":[],"title":"Calculate the total mass of the system.","text":"Add the mass values together, the sum is written as such: m=sum{i\\\\=1}{3}{m_i}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad4fd60center3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["ad4fd60center3a-h1"],"title":"Find the total mass.","text":"What is the value of the total mass (m)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ad4fd60center3a-h2-s1","type":"hint","dependencies":[],"title":"Find the total mass.","text":"$$2+6+4=12kg$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"ad4fd60center3a-h3","type":"hint","dependencies":["ad4fd60center3a-h2"],"title":"Find the moments with respect to the $$x-$$ and $$y-axes$$.","text":"Find both M_y=sum{i\\\\=1}{3}{m_i*x_i} and M_x=sum{i\\\\=1}{3}{m_i*y_i}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad4fd60center3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["ad4fd60center3a-h3"],"title":"Find the moment with respect to the x-axis.","text":"What is the value of $$M_y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ad4fd60center3a-h4-s1","type":"hint","dependencies":[],"title":"Find the moment with respect to the x-axis.","text":"$$-2+6+8=12$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"ad4fd60center3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ad4fd60center3a-h4"],"title":"Find the moment with respect to the y-axis.","text":"What is the value of $$M_x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ad4fd60center3a-h5-s1","type":"hint","dependencies":[],"title":"Find the moment with respect to the y-axis.","text":"$$6+6-8=4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"ad4fd60center3a-h6","type":"hint","dependencies":["ad4fd60center3a-h5"],"title":"Find the center of mass x\u0304 and \u0233.","text":"Then we have $$x\u0304=\\\\frac{M_y}{m}=\\\\frac{12}{12}=1$$ and $$\u0233=\\\\frac{M_x}{m}=\\\\frac{4}{12}=\\\\frac{1}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad4fd60center3a-h7","type":"hint","dependencies":["ad4fd60center3a-h6"],"title":"How to write and interpret the result.","text":"The center of mass of the system is $$(1,\\\\frac{1}{3})$$, in meters.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ad4fd60center4","title":"Finding the Center of Mass of Objects in a Plane.","body":"","variabilization":{},"oer":"https://openstax.org/books/calculus-volume-1/pages/6-6-moments-and-centers-of-mass","license":0,"lesson":"6.6 Moments and Centers of Mass","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad4fd60center4a","stepAnswer":["$$(-1,-1)$$"],"problemType":"MultipleChoice","stepTitle":"Suppose three point masses are placed in the xy-plane as follows (assume coordinates are given in meters): $$m_1=5kg$$, placed at $$(-2,-3)$$, $$m_2=3kg$$, placed at $$(2,3)$$, and $$m_3=2kg$$, placed at $$(-3,-2)$$. Find the center of mass of the system.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-1,-1)$$","choices":["$$(-1,-1)$$","$$(-1,1)$$","$$(1,-1)$$","$$(1,1)$$"],"hints":{"DefaultPathway":[{"id":"ad4fd60center4a-h1","type":"hint","dependencies":[],"title":"Calculate the total mass of the system.","text":"Add the mass values together, the sum is written as such: m=sum{i\\\\=1}{3}{m_i}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad4fd60center4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["ad4fd60center4a-h1"],"title":"Find the total mass.","text":"What is the value of the total mass (m)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ad4fd60center4a-h2-s1","type":"hint","dependencies":[],"title":"Find the total mass.","text":"$$5+3+2=10kg$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"ad4fd60center4a-h3","type":"hint","dependencies":["ad4fd60center4a-h2"],"title":"Find the moments with respect to the $$x-$$ and $$y-axes$$.","text":"Find both M_y=sum{i\\\\=1}{3}{m_i*x_i} and M_x=sum{i\\\\=1}{3}{m_i*y_i}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad4fd60center4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-10$$"],"dependencies":["ad4fd60center4a-h3"],"title":"Find the moment with respect to the x-axis.","text":"What is the value of $$M_y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ad4fd60center4a-h4-s1","type":"hint","dependencies":[],"title":"Find the moment with respect to the x-axis.","text":"$$-10+6-6=-10$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"ad4fd60center4a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-10$$"],"dependencies":["ad4fd60center4a-h4"],"title":"Find the moment with respect to the y-axis.","text":"What is the value of $$M_x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ad4fd60center4a-h5-s1","type":"hint","dependencies":[],"title":"Find the moment with respect to the y-axis.","text":"$$-15+9-4=-10$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"ad4fd60center4a-h6","type":"hint","dependencies":["ad4fd60center4a-h5"],"title":"Calculate the center of mass.","text":"$$x\u0304=\\\\frac{M_y}{m}=\\\\frac{-10}{10}=-1$$ $$\u0233=\\\\frac{M_x}{m}=\\\\frac{-10}{10}=-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad4fd60center4a-h7","type":"hint","dependencies":["ad4fd60center4a-h6"],"title":"How to write and interpret the result.","text":"The center of mass of the system is $$(-1,-1)$$, in meters.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ad4fd60center5","title":"Finding the Center of Mass of a Lamina.","body":"","variabilization":{},"oer":"https://openstax.org/books/calculus-volume-1/pages/6-6-moments-and-centers-of-mass","license":0,"lesson":"6.6 Moments and Centers of Mass","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad4fd60center5a","stepAnswer":["$$(\\\\frac{12}{5},\\\\frac{3}{4})$$"],"problemType":"MultipleChoice","stepTitle":"Let R be the region bounded above by the graph of the function $$f(x)=\\\\sqrt{x}$$ and below by the x-axis over the interval [0,4]. Find the centroid of the region.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(\\\\frac{12}{5},\\\\frac{3}{4})$$","choices":["$$(\\\\frac{12}{5},\\\\frac{3}{4})$$","$$(\\\\frac{3}{4},\\\\frac{3}{4})$$","$$(\\\\frac{12}{5},\\\\frac{12}{5})$$","$$(\\\\frac{3}{4},\\\\frac{12}{5})$$"],"hints":{"DefaultPathway":[{"id":"ad4fd60center5a-h1","type":"hint","dependencies":[],"title":"Image of the Region","text":"The region is depicted in the following figure.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ad4fd60center5a-h2","type":"hint","dependencies":["ad4fd60center5a-h1"],"title":"The value of \ud835\udf0c","text":"Since we are only asked for the centroid of the region, rather than the mass or moments of the associated lamina, we know the density constant \ud835\udf0c cancels out of the calculations eventually. Therefore, for the sake of convenience, let\u2019s assume $$\ud835\udf0c=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ad4fd60center6","title":"Finding the Centroid of a Symmetric Region","body":"","variabilization":{},"oer":"https://openstax.org/books/calculus-volume-1/pages/6-6-moments-and-centers-of-mass","license":0,"lesson":"6.6 Moments and Centers of Mass","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad4fd60center6a","stepAnswer":["$$(0,\\\\frac{8}{5})$$"],"problemType":"MultipleChoice","stepTitle":"Let R be the region bounded above by the graph of the function $$f(x)=4-x^2$$ and below by the x-axis. Find the centroid of the region.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,\\\\frac{8}{5})$$","choices":["$$(0,\\\\frac{8}{5})$$","$$(\\\\frac{8}{5},0)$$","$$(\\\\frac{8}{5},\\\\frac{8}{5})$$"],"hints":{"DefaultPathway":[{"id":"ad4fd60center6a-h1","type":"hint","dependencies":[],"title":"The depicted region","text":"The region is depicted in the following figure.\\\\n##figure1.gif##","variabilization":{},"oer":"","license":""},{"id":"ad4fd60center6a-h2","type":"hint","dependencies":["ad4fd60center6a-h1"],"title":"The symmetric principle","text":"If a region R is symmetric about a line l, then the centroid of R lies on l.","variabilization":{},"oer":"","license":""},{"id":"ad4fd60center6a-h3","type":"hint","dependencies":["ad4fd60center6a-h2"],"title":"The symmetric principle","text":"We can use the symmetry principle to help find the centroid of a symmetric region.","variabilization":{},"oer":"","license":""},{"id":"ad4fd60center6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["ad4fd60center6a-h3"],"title":"The x-coordinate of the centroid","text":"The region is symmetric with respect to the y-axis. Therefore, what is the the x-coordinate of the centroid?","variabilization":{},"oer":"","license":""},{"id":"ad4fd60center6a-h5","type":"hint","dependencies":["ad4fd60center6a-h4"],"title":"Explanation","text":"Therefore, we only need to calculate \u0233 . For the sake of convenience, assume $$\ud835\udf0c=1$$.","variabilization":{},"oer":"","license":""},{"id":"ad4fd60center6a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{32}{3}$$"],"dependencies":["ad4fd60center6a-h5"],"title":"Total mass","text":"First, we calculate the total mass: $$\ud835\udf0c*\\\\int_{a}^{b} f(x) \\\\,dx=\\\\int_{-2}^{2} 4-x^2 \\\\,dx$$. What is the value of the total mass?","variabilization":{},"oer":"","license":"","choices":["$$\\\\frac{32}{3}$$","$$\\\\frac{-32}{3}$$","$$0$$"]},{"id":"ad4fd60center6a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{256}{15}$$"],"dependencies":["ad4fd60center6a-h6"],"title":"The moments","text":"Next, we calculate the moments. We only need \ud835\udc40x: \ud835\udc40x=\ud835\udf0c*/int{f(x))**2/2,a,b,x}=(1/2)*/int{(4-x**2)**2,-2,2,x}=(1/2)*/int{16-8*x**2+x**4,-2,2,x}. What is the value of the moments?","variabilization":{},"oer":"","license":"","choices":["$$\\\\frac{256}{15}$$","$$\\\\frac{-256}{15}$$","$$8$$","$$0$$"]},{"id":"ad4fd60center6a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{8}{5}$$"],"dependencies":["ad4fd60center6a-h7"],"title":"The centroid of the region","text":"Using the formula $$\u0233=\\\\frac{Mx}{m}$$, what is the value of \u0233?","variabilization":{},"oer":"","license":"","subHints":[{"id":"ad4fd60center6a-h8-s1","type":"hint","dependencies":[],"title":"The centroid of the region","text":"$$\\\\frac{256}{15} \\\\frac{3}{32}=\\\\frac{8}{5}$$","variabilization":{},"oer":"","license":""}]},{"id":"ad4fd60center6a-h9","type":"hint","dependencies":["ad4fd60center6a-h8"],"title":"Conclusion","text":"The centroid of the region is $$(0,\\\\frac{8}{5})$$.","variabilization":{},"oer":"","license":""}]}}]},{"id":"ad4fd60center7","title":"Using the Theorem of Pappus for Volume","body":"","variabilization":{},"oer":"https://openstax.org/books/calculus-volume-1/pages/6-6-moments-and-centers-of-mass","license":0,"lesson":"6.6 Moments and Centers of Mass","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad4fd60center7a","stepAnswer":["$$32{\\\\pi}^2$$"],"problemType":"TextBox","stepTitle":"Let R be a circle of radius $$2$$ centered at $$(4,0)$$. Use the theorem of Pappus for volume to find the volume of the torus generated by revolving R around the y-axis.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$32{\\\\pi}^2$$","hints":{"DefaultPathway":[{"id":"ad4fd60center7a-h1","type":"hint","dependencies":[],"title":"The depicted region","text":"The region and torus are depicted in the following figure. Determining the volume of a torus by using the theorem of Pappus. (a) A circular region R in the plane; (b) the torus generated by revolving R about the y-axis.\\\\n##figure1.gif##","variabilization":{},"oer":"","license":""},{"id":"ad4fd60center7a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$4\\\\pi$$"],"dependencies":["ad4fd60center7a-h1"],"title":"The area of R","text":"What is the area of R?","variabilization":{},"oer":"","license":"","choices":["$$8\\\\pi$$","$$4\\\\pi$$","$$2\\\\pi$$"]},{"id":"ad4fd60center7a-h3","type":"hint","dependencies":["ad4fd60center7a-h2"],"title":"The symmetric principle","text":"By the symmetry principle, the centroid of R is the center of the circle. The centroid travels around the y-axis in a circular path of radius $$4$$, so the centroid travels $$d=8\\\\pi$$ units.","variabilization":{},"oer":"","license":""},{"id":"ad4fd60center7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$32{\\\\pi}^2$$"],"dependencies":["ad4fd60center7a-h3"],"title":"The volume of the torus","text":"What is the volume of the torus?","variabilization":{},"oer":"","license":""}]}}]},{"id":"ad4fd60center8","title":"Finding the Centroid of a Region Bounded by Two Functions","body":"","variabilization":{},"oer":"https://openstax.org/books/calculus-volume-1/pages/6-6-moments-and-centers-of-mass","license":0,"lesson":"6.6 Moments and Centers of Mass","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ad4fd60center8a","stepAnswer":["$$(-\\\\left(\\\\frac{1}{2}\\\\right),-\\\\left(\\\\frac{3}{5}\\\\right))$$"],"problemType":"MultipleChoice","stepTitle":"Let R be the region bounded above by the graph of the function $$f(x)=1-x^2$$ and below by the graph of the function $$g(x)=x-1$$. Find the centroid of the region.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\left(\\\\frac{1}{2}\\\\right),-\\\\left(\\\\frac{3}{5}\\\\right))$$","choices":["$$(-\\\\left(\\\\frac{1}{2}\\\\right),-\\\\left(\\\\frac{3}{5}\\\\right))$$","$$(\\\\frac{1}{2},-\\\\left(\\\\frac{3}{5}\\\\right))$$","$$(-\\\\left(\\\\frac{1}{2}\\\\right),\\\\frac{3}{5})$$","$$(\\\\frac{1}{2},\\\\frac{3}{5})$$"],"hints":{"DefaultPathway":[{"id":"ad4fd60center8a-h1","type":"hint","dependencies":[],"title":"The depicted region","text":"The graphs of the functions intersect at $$(-2,-3)$$ and $$(1,0)$$, so we integrate from $$-2$$ to $$1$$. For the sake of convenience, assume $$\ud835\udf0c=1$$.\\\\n##figure1.gif##","variabilization":{},"oer":"","license":""},{"id":"ad4fd60center8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{9}{2}$$"],"dependencies":["ad4fd60center8a-h1"],"title":"Total mass","text":"First, we calculate the total mass: $$\ud835\udf0c*\\\\int_{a}^{b} f(x)-g(x) \\\\,dx=\\\\int_{-2}^{1} 1-x^2-x-1 \\\\,dx$$. What is the value of the total mass?","variabilization":{},"oer":"","license":""},{"id":"ad4fd60center8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-27}{10}$$"],"dependencies":["ad4fd60center8a-h2"],"title":"The moments","text":"Next, we calculate the moments. $$\ud835\udc40x=\ud835\udf0c*\\\\int_{a}^{b} \\\\frac{1}{2} \\\\left({f{\\\\left(x\\\\right)}}^2-{g{\\\\left(x\\\\right)}}^2\\\\right) \\\\,dx=(1/2)*\\\\int_{-2}^{1} {\\\\left(1-x^2\\\\right)}^2-{\\\\left(x-1\\\\right)}^2 \\\\,dx$$. What is the value of\ud835\udc40x?","variabilization":{},"oer":"","license":""},{"id":"ad4fd60center8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-9}{4}$$"],"dependencies":["ad4fd60center8a-h3"],"title":"The moments","text":"$$M_y=\ud835\udf0c*\\\\int_{a}^{b} x \\\\left(f{\\\\left(x\\\\right)}-g{\\\\left(x\\\\right)}\\\\right) \\\\,dx=\\\\int_{-2}^{1} x \\\\left(1-x^2-x-1\\\\right) \\\\,dx$$","variabilization":{},"oer":"","license":""},{"id":"ad4fd60center8a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{2}$$"],"dependencies":["ad4fd60center8a-h4"],"title":"The centroid of the region","text":"What is the value of x\u0304?","variabilization":{},"oer":"","license":"","subHints":[{"id":"ad4fd60center8a-h5-s1","type":"hint","dependencies":[],"title":"The centroid of the region","text":"$$x\u0304=\\\\frac{M_y}{m}=\\\\left(-\\\\frac{9}{4}\\\\right) \\\\frac{2}{9}=\\\\frac{-1}{2}$$","variabilization":{},"oer":"","license":""}]},{"id":"ad4fd60center8a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-3}{5}$$"],"dependencies":["ad4fd60center8a-h5"],"title":"The centroid of the region","text":"What is the value of \u0233?","variabilization":{},"oer":"","license":"","subHints":[{"id":"ad4fd60center8a-h6-s1","type":"hint","dependencies":["ad4fd60center8a-h6"],"title":"The centroid of the region","text":"$$\u0233=\\\\frac{M_x}{m}=-\\\\left(\\\\frac{27}{10}\\\\right) \\\\frac{2}{9}=\\\\frac{-3}{5}$$","variabilization":{},"oer":"","license":""}]}]}}]},{"id":"ad4fd60center9","title":"The center of mass","body":"","variabilization":{},"oer":"https://openstax.org/books/calculus-volume-1/pages/6-6-moments-and-centers-of-mass","license":0,"lesson":"6.6 Moments and Centers of Mass","courseName":"OpenStax: Calculus Volume 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b^5\\\\right)}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-\\\\left(\\\\frac{9a}{b^2}\\\\right)$$","hints":{"DefaultPathway":[{"id":"ad51073poly1a-h1","type":"hint","dependencies":[],"title":"Decompose","text":"Use Fraction Multiplication: $$\\\\frac{54}{-6} \\\\frac{a^2}{a} \\\\frac{b^3}{b^5}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-\\\\left(\\\\frac{9a}{b^2}\\\\right)$$"],"dependencies":["ad51073poly1a-h1"],"title":"Simplify","text":"What is $$\\\\frac{54}{-6} \\\\frac{a^2}{a} \\\\frac{b^3}{b^5}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly1a-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\frac{54}{-6}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly1a-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["a"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\frac{a^2}{a}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly1a-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{b^2}$$"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\frac{b^3}{b^5}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad51073poly10","title":"Factor Theorem","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add, Subtract, and Multiply Radical Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad51073poly10a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Is $$x-4$$ is a factor of $$f(x)=x^3-64$$?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ad51073poly10a-h1","type":"hint","dependencies":[],"title":"Factor Theorem","text":"The Factor Theorem states that if a polynomial function f(x) is divided by $$x-c$$, then the remainder is f(c).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly10a-h2","type":"hint","dependencies":["ad51073poly10a-h1"],"title":"Factor Theorem","text":"The Factor Theorem tells us that $$x-4$$ is a factor of $$f(x)=x^3-64$$ if $$f(4)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":[],"title":"Solve","text":"What is f(4)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad51073poly2","title":"Division of a Polynomial by a Monomial","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add, Subtract, and Multiply Radical Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad51073poly2a","stepAnswer":["$$-6x^2+12y$$"],"problemType":"TextBox","stepTitle":"Find the quotient: $$\\\\frac{18x^3 y-36x y^2}{\\\\left(-3x y\\\\right)}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-6x^2+12y$$","hints":{"DefaultPathway":[{"id":"ad51073poly2a-h1","type":"hint","dependencies":[],"title":"Decompose","text":"Divide each term by the divisor. Be careful with the signs: $$\\\\frac{18x^3 y}{\\\\left(-3x y\\\\right)}$$ - $$\\\\frac{36x y^2}{\\\\left(-3x y\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6x^2$$"],"dependencies":["ad51073poly2a-h1"],"title":"Simplify","text":"What is $$\\\\frac{18x^3 y}{\\\\left(-3x y\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-12y$$"],"dependencies":["ad51073poly2a-h2"],"title":"Simplify","text":"What is $$\\\\frac{36x y^2}{\\\\left(-3x y\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6x^2+12y$$"],"dependencies":["ad51073poly2a-h3"],"title":"Simplify","text":"What is $$\\\\left(-6x^2\\\\right)-\\\\left(-12y\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad51073poly3","title":"Divide Polynomials Using Long Division","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add, Subtract, and Multiply Radical Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad51073poly3a","stepAnswer":["$$x+4$$"],"problemType":"TextBox","stepTitle":"Find the quotient: $$\\\\frac{x^2+9x+20}{x+5}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x+4$$","hints":{"DefaultPathway":[{"id":"ad51073poly3a-h1","type":"hint","dependencies":[],"title":"Long Division","text":"Write it as a long division problem.\\\\nBe sure the dividend is in standard form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly3a-h2","type":"hint","dependencies":["ad51073poly3a-h1"],"title":"Long Division","text":"Divide $$x^2$$ by $$x$$. It may help to ask yourself, \u201cWhat do I need\\\\nto multiply $$x$$ by to get x**2?\u201d","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly3a-h3","type":"hint","dependencies":["ad51073poly3a-h2"],"title":"Long Division","text":"Put the answer, $$x$$, in the quotient over the $$x$$ term.\\\\nMultiply $$x$$ times $$x+5$$. Line up the like terms under the dividend.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly3a-h4","type":"hint","dependencies":["ad51073poly3a-h3"],"title":"Long Division","text":"Subtract $$x^2+5x$$ from $$x^2+9x$$.\\\\nYou may find it easier to change the signs and then add.\\\\nThen bring down the last term, $$20$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly3a-h5","type":"hint","dependencies":["ad51073poly3a-h4"],"title":"Long Division","text":"Divide $$4x$$ by $$x$$. It may help to ask yourself, \u201cWhat do I\\\\nneed to multiply $$x$$ by to get $$4x$$ ?\u201d\\\\nPut the answer, $$4$$ , in the quotient over the constant term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly3a-h6","type":"hint","dependencies":["ad51073poly3a-h5"],"title":"Long Division","text":"Multiply $$4$$ times $$x+5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly3a-h7","type":"hint","dependencies":["ad51073poly3a-h6"],"title":"Long Division","text":"Subtract $$4x+20$$ from $$4x+20$$. We get $$0$$ so there is no remainder.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad51073poly4","title":"Dividing Monomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add, Subtract, and Multiply Radical Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad51073poly4a","stepAnswer":["$$\\\\frac{-9}{a^5 b}$$"],"problemType":"TextBox","stepTitle":"Find the quotient: $$\\\\frac{\\\\left(-72a^7 b^3\\\\right)}{8a^{12} b^4}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-9}{a^5 b}$$","hints":{"DefaultPathway":[{"id":"ad51073poly4a-h1","type":"hint","dependencies":[],"title":"Decompose","text":"Use Fraction Multiplication: $$\\\\left(-\\\\frac{72}{8}\\\\right) \\\\frac{a^7}{a^{12}} \\\\frac{b^3}{b^4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-9}{a^5 b}$$"],"dependencies":["ad51073poly4a-h1"],"title":"Simplify","text":"What is $$\\\\left(-\\\\frac{72}{8}\\\\right) \\\\frac{a^7}{a^{12}} \\\\frac{b^3}{b^4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly4a-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\left(-\\\\frac{72}{8}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly4a-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{a^5}$$"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\frac{a^7}{a^{12}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly4a-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{b}$$"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\frac{b^3}{b^4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad51073poly5","title":"Dividing Monomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add, Subtract, and Multiply Radical Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad51073poly5a","stepAnswer":["$$\\\\frac{\\\\left(-9b\\\\right)}{a^4}$$"],"problemType":"TextBox","stepTitle":"Find the quotient: $$\\\\frac{\\\\left(-63a^8 b^3\\\\right)}{7a^{12} b^2}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{\\\\left(-9b\\\\right)}{a^4}$$","hints":{"DefaultPathway":[{"id":"ad51073poly5a-h1","type":"hint","dependencies":[],"title":"Decompose","text":"Use Fraction Multiplication: $$\\\\left(-\\\\frac{63}{7}\\\\right) \\\\frac{a^8}{a^{12}} \\\\frac{b^3}{b^2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\left(-9b\\\\right)}{a^4}$$"],"dependencies":["ad51073poly5a-h1"],"title":"Simplify","text":"What is $$\\\\left(-\\\\frac{63}{7}\\\\right) \\\\frac{a^8}{a^{12}} \\\\frac{b^3}{b^2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly5a-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-9$$"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\left(-\\\\frac{63}{7}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly5a-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{a^4}$$"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\frac{a^8}{a^{12}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly5a-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$b$$"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\frac{b^3}{b^2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad51073poly6","title":"Dividing Monomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add, Subtract, and Multiply Radical Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad51073poly6a","stepAnswer":["$$\\\\frac{2b^6}{3a^4}$$"],"problemType":"TextBox","stepTitle":"Find the quotient: $$\\\\frac{14a^7 b^{12}}{21a^{11} b^6}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2b^6}{3a^4}$$","hints":{"DefaultPathway":[{"id":"ad51073poly6a-h1","type":"hint","dependencies":[],"title":"Decompose","text":"Use Fraction Multiplication: $$\\\\frac{14}{21} \\\\frac{a^7}{a^{11}} \\\\frac{b^{12}}{b^6}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2b^6}{3a^4}$$"],"dependencies":["ad51073poly6a-h1"],"title":"Simplify","text":"What is $$\\\\frac{14}{21}$$ * $$\\\\frac{a^7}{a^{11}}$$ * $$\\\\frac{b^{12}}{b^6}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly6a-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{3}$$"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\frac{14}{21}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly6a-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{a^4}$$"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\frac{a^7}{a^{11}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly6a-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$b^6$$"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\frac{b^{12}}{b^6}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad51073poly7","title":"Dividing Monomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add, Subtract, and Multiply Radical Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad51073poly7a","stepAnswer":["$$\\\\frac{4b^2}{7a^4}$$"],"problemType":"TextBox","stepTitle":"Find the quotient: $$\\\\frac{28a^5 b^{14}}{49a^9 b^{12}}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{4b^2}{7a^4}$$","hints":{"DefaultPathway":[{"id":"ad51073poly7a-h1","type":"hint","dependencies":[],"title":"Decompose","text":"Use Fraction Multiplication: $$\\\\frac{28}{49} \\\\frac{a^5}{a^9} \\\\frac{b^{14}}{b^{12}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{4b^2}{7a^4}$$"],"dependencies":["ad51073poly7a-h1"],"title":"Simplify","text":"What is $$\\\\frac{28}{49}$$ * $$\\\\frac{a^5}{a^9}$$ * $$\\\\frac{b^{14}}{b^{12}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly7a-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{4}{7}$$"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\frac{28}{49}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly7a-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{a^4}$$"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\frac{a^5}{a^9}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly7a-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$b^2$$"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\frac{b^{14}}{b^{12}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad51073poly8","title":"Dividing Monomials","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add, Subtract, and Multiply Radical Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad51073poly8a","stepAnswer":["$$\\\\frac{5}{8a^5 b^3}$$"],"problemType":"TextBox","stepTitle":"Find the quotient: $$\\\\frac{30a^5 b^{11}}{48a^{10} b^{14}}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5}{8a^5 b^3}$$","hints":{"DefaultPathway":[{"id":"ad51073poly8a-h1","type":"hint","dependencies":[],"title":"Decompose","text":"Use Fraction Multiplication: $$\\\\frac{30}{48} \\\\frac{a^5}{a^{10}} \\\\frac{b^{11}}{b^{14}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{8a^5 b^3}$$"],"dependencies":["ad51073poly8a-h1"],"title":"Simplify","text":"What is $$\\\\frac{30}{48} \\\\frac{a^5}{a^{10}} \\\\frac{b^{11}}{b^{14}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly8a-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{8}$$"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\frac{30}{48}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly8a-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{a^5}$$"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\frac{a^5}{a^{10}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ad51073poly8a-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{b^3}$$"],"dependencies":[],"title":"Simplify","text":"What is $$\\\\frac{b^{11}}{b^{14}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ad51073poly9","title":"Division of a Polynomial by a Monomial","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add, Subtract, and Multiply Radical Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ad51073poly9a","stepAnswer":["$$-4x+2y$$"],"problemType":"TextBox","stepTitle":"Find the quotient: $$\\\\frac{32x^2 y-16x y^2}{\\\\left(-8x y\\\\right)}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-4x+2y$$","hints":{"DefaultPathway":[{"id":"ad51073poly9a-h1","type":"hint","dependencies":[],"title":"Decompose","text":"Divide each term by the divisor. 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What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr15a-h3","type":"hint","dependencies":["ada10e6CompletingSqr15a-h2"],"title":"Creating a Binomial Squared","text":"Rewrite as a binomial squared.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(m-12\\\\right)}^2$$"],"dependencies":["ada10e6CompletingSqr15a-h3"],"title":"Creating a Binomial Squared","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ada10e6CompletingSqr16","title":"Complete a Square","body":"Complete the square to make a perfect square trinomial. 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr19a-h5","type":"hint","dependencies":["ada10e6CompletingSqr19a-h4"],"title":"Substitution","text":"With the known a and 2ab, $$b$$ can be calculated","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr19a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["ada10e6CompletingSqr19a-h5"],"title":"Calculation","text":"$$b=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr19a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["ada10e6CompletingSqr19a-h6"],"title":"Calculation","text":"$$2b b=-4b$$, what is $$b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr19a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["ada10e6CompletingSqr19a-h7"],"title":"Substitution","text":"What is the third term $$b^2$$ in the perfect square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr19a-h9","type":"hint","dependencies":["ada10e6CompletingSqr19a-h8"],"title":"Principle","text":"Remember that $${\\\\left(-x\\\\right)}^2=\\\\left(-x\\\\right) \\\\left(-x\\\\right)=x^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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Trinomial","text":"Take the coefficient, divide it by $$2$$, then take the answer and square it. This will complete the square. What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr2a-h3","type":"hint","dependencies":["ada10e6CompletingSqr2a-h2"],"title":"Creating a Binomial Squared","text":"Rewrite as a binomial squared.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(b+6\\\\right)}^2$$"],"dependencies":["ada10e6CompletingSqr2a-h3"],"title":"Creating a Binomial Squared","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ada10e6CompletingSqr20","title":"Complete a Square","body":"Complete the square to make a perfect square trinomial. Then, write the result as a binomial square.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Solve Quadratic Equations by Completing the Square","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ada10e6CompletingSqr20a","stepAnswer":["$${\\\\left(p+\\\\frac{1}{8}\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$p^2+\\\\frac{1}{4} p$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(p+\\\\frac{1}{8}\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"ada10e6CompletingSqr20a-h1","type":"hint","dependencies":[],"title":"Formula","text":"The pattern of perfect square $$=$$ $$a^2+2ab+b^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$p^2$$"],"dependencies":["ada10e6CompletingSqr20a-h1"],"title":"Identification","text":"What is $$a^2$$ $$=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr20a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$p$$"],"dependencies":["ada10e6CompletingSqr20a-h2"],"title":"Identification","text":"$$a=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr20a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{p}{4}$$"],"dependencies":["ada10e6CompletingSqr20a-h3"],"title":"Identification","text":"What is 2ab?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr20a-h5","type":"hint","dependencies":["ada10e6CompletingSqr20a-h4"],"title":"Substitution","text":"With the known a and 2ab, $$b$$ can be calculated","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr20a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{8}$$"],"dependencies":["ada10e6CompletingSqr20a-h5"],"title":"Calculation","text":"$$b=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr20a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{8}$$"],"dependencies":["ada10e6CompletingSqr20a-h6"],"title":"Calculation","text":"$$2p b=\\\\frac{1}{4} p$$, what is $$b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr20a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{64}$$"],"dependencies":["ada10e6CompletingSqr20a-h7"],"title":"Substitution","text":"What is the third term $$b^2$$ in the perfect square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr20a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{64}$$"],"dependencies":["ada10e6CompletingSqr20a-h8"],"title":"Squaring","text":"What is $${\\\\left(\\\\frac{1}{8}\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr20a-h10","type":"hint","dependencies":["ada10e6CompletingSqr20a-h9"],"title":"Principle","text":"Recall that $$a^2+2ab+b^2={\\\\left(a+b\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ada10e6CompletingSqr21","title":"Complete a Square","body":"Complete the square to make a perfect square trinomial. Then, write the result as a binomial square.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Solve Quadratic Equations by Completing the Square","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ada10e6CompletingSqr21a","stepAnswer":["$${\\\\left(q+\\\\frac{1}{3}\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$q^2+\\\\frac{2}{3} q$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(q+\\\\frac{1}{3}\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"ada10e6CompletingSqr21a-h1","type":"hint","dependencies":[],"title":"Formula","text":"The pattern of perfect square $$=$$ $$a^2+2ab+b^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr21a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$q^2$$"],"dependencies":["ada10e6CompletingSqr21a-h1"],"title":"Identification","text":"What is $$a^2$$ $$=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr21a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["q"],"dependencies":["ada10e6CompletingSqr21a-h2"],"title":"Identification","text":"$$a=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr21a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2q}{3}$$"],"dependencies":["ada10e6CompletingSqr21a-h3"],"title":"Identification","text":"What is 2ab?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr21a-h5","type":"hint","dependencies":["ada10e6CompletingSqr21a-h4"],"title":"Substitution","text":"With the known a and 2ab, $$b$$ can be calculated","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr21a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["ada10e6CompletingSqr21a-h5"],"title":"Calculation","text":"$$b=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr21a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["ada10e6CompletingSqr21a-h6"],"title":"Calculation","text":"$$2q b=\\\\frac{2q}{3}$$, what is $$b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr21a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{9}$$"],"dependencies":["ada10e6CompletingSqr21a-h7"],"title":"Substitution","text":"What is the third term $$b^2$$ in the perfect square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr21a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{9}$$"],"dependencies":["ada10e6CompletingSqr21a-h8"],"title":"Squaring","text":"What is $${\\\\left(\\\\frac{1}{3}\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr21a-h10","type":"hint","dependencies":["ada10e6CompletingSqr21a-h9"],"title":"Principle","text":"Recall that $$a^2+2ab+b^2={\\\\left(a+b\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ada10e6CompletingSqr22","title":"Solve Quadratic Equations of the Form $$x^2+bx+c=0$$ by Completing the Square","body":"Solve the following equation by completing the square","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Solve Quadratic Equations by Completing the Square","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ada10e6CompletingSqr22a","stepAnswer":["$$4$$ or $$-12$$"],"problemType":"MultipleChoice","stepTitle":"$$x^2+8x=48$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$4$$ or $$-12$$","choices":["$$4$$ or $$-12$$","$$4$$ or 12I4 or $$-12I-4$$ or $$-12$$"],"hints":{"DefaultPathway":[{"id":"ada10e6CompletingSqr22a-h1","type":"hint","dependencies":[],"title":"Formula","text":"$${\\\\left(a+b\\\\right)}^2$$ $$=$$ $$a^2+2ab+b^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr22a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2$$"],"dependencies":["ada10e6CompletingSqr22a-h1"],"title":"Identification","text":"What is $$a^2$$ $$=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr22a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x$$"],"dependencies":["ada10e6CompletingSqr22a-h2"],"title":"Identification","text":"$$a=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr22a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8x$$"],"dependencies":["ada10e6CompletingSqr22a-h3"],"title":"Identification","text":"What is 2ab?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr22a-h5","type":"hint","dependencies":["ada10e6CompletingSqr22a-h4"],"title":"Substitution","text":"With the known a and 2ab, $$b$$ can be calculated","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr22a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ada10e6CompletingSqr22a-h5"],"title":"Calculation","text":"$$b=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr22a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ada10e6CompletingSqr22a-h6"],"title":"Calculation","text":"$$2x b=8x$$, what is $$b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr22a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["ada10e6CompletingSqr22a-h7"],"title":"Substitution","text":"What is the third term $$b^2$$ in the perfect square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr22a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["ada10e6CompletingSqr22a-h8"],"title":"Squaring","text":"What is $$4^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr22a-h10","type":"hint","dependencies":["ada10e6CompletingSqr22a-h9"],"title":"Organizing","text":"In order to form a perfect square on left side, $$b^2$$ has to be added on both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr22a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$64$$"],"dependencies":["ada10e6CompletingSqr22a-h10"],"title":"Addition","text":"What will be the number on right side after adding 16?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr22a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$64$$"],"dependencies":["ada10e6CompletingSqr22a-h11"],"title":"Addition","text":"What is $$48+16$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr22a-h13","type":"hint","dependencies":["ada10e6CompletingSqr22a-h12"],"title":"Organizing","text":"The equation becomes $$x^2+8x+16=64$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr22a-h14","type":"hint","dependencies":["ada10e6CompletingSqr22a-h13"],"title":"Simplifying","text":"It can be simplified to $${\\\\left(x+4\\\\right)}^2=64$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr22a-h15","type":"hint","dependencies":["ada10e6CompletingSqr22a-h14"],"title":"Square Root","text":"The square can be removed by square root","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr22a-h16","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+4$$"],"dependencies":["ada10e6CompletingSqr22a-h15"],"title":"Square Root","text":"What is the sqrt[(x+4)**2]?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr22a-h17","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\pm 8$$"],"dependencies":["ada10e6CompletingSqr22a-h16"],"title":"Square Root","text":"What is $$\\\\sqrt{64}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr22a-h18","type":"hint","dependencies":["ada10e6CompletingSqr22a-h17"],"title":"Organizing","text":"The equation becomes $$x+4=8$$ or $$x+4=-8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr22a-h19","type":"hint","dependencies":["ada10e6CompletingSqr22a-h18"],"title":"Subtraction","text":"$$x=4$$ or $$-12$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ada10e6CompletingSqr23","title":"Solve Quadratic Equations of the Form $$x^2+bx+c=0$$ by Completing the Square","body":"Solve the following equation by completing the square","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Solve Quadratic Equations by Completing the Square","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ada10e6CompletingSqr23a","stepAnswer":["$$1$$ or $$-5$$"],"problemType":"MultipleChoice","stepTitle":"$$c^2+4c=5$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$1$$ or $$-5$$","choices":["$$1$$ or $$-5$$","$$1$$ or $$-5I1$$ or $$5I-1$$ or $$-5$$"],"hints":{"DefaultPathway":[{"id":"ada10e6CompletingSqr23a-h1","type":"hint","dependencies":[],"title":"Formula","text":"$${\\\\left(a+b\\\\right)}^2$$ $$=$$ $$a^2+2ab+b^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr23a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$c^2$$"],"dependencies":["ada10e6CompletingSqr23a-h1"],"title":"Identification","text":"What is $$a^2$$ $$=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr23a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["c"],"dependencies":["ada10e6CompletingSqr23a-h2"],"title":"Identification","text":"$$a=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr23a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["4c"],"dependencies":["ada10e6CompletingSqr23a-h3"],"title":"Identification","text":"What is 2ab?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr23a-h5","type":"hint","dependencies":["ada10e6CompletingSqr23a-h4"],"title":"Substitution","text":"With the known a and 2ab, $$b$$ can be calculated","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr23a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ada10e6CompletingSqr23a-h5"],"title":"Calculation","text":"$$b=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr23a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ada10e6CompletingSqr23a-h6"],"title":"Calculation","text":"$$2c b=4c$$, what is $$b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr23a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ada10e6CompletingSqr23a-h7"],"title":"Substitution","text":"What is the third term $$b^2$$ in the perfect square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr23a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ada10e6CompletingSqr23a-h8"],"title":"Squaring","text":"What is $$2^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr23a-h10","type":"hint","dependencies":["ada10e6CompletingSqr23a-h9"],"title":"Organizing","text":"In order to form a perfect square on left side, $$b^2$$ has to be added on both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr23a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["ada10e6CompletingSqr23a-h10"],"title":"Addition","text":"What will be the number on right side after adding 4?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr23a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["ada10e6CompletingSqr23a-h11"],"title":"Addition","text":"What is $$5+4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr23a-h13","type":"hint","dependencies":["ada10e6CompletingSqr23a-h12"],"title":"Organizing","text":"The equation becomes $$c^2+4c+4=9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr23a-h14","type":"hint","dependencies":["ada10e6CompletingSqr23a-h13"],"title":"Simplifying","text":"It can be simplified to $${\\\\left(c+2\\\\right)}^2=9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr23a-h15","type":"hint","dependencies":["ada10e6CompletingSqr23a-h14"],"title":"Square Root","text":"The square can be removed by square root","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr23a-h16","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$c+2$$"],"dependencies":["ada10e6CompletingSqr23a-h15"],"title":"Square Root","text":"What is the sqrt[(c+2)**2]?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr23a-h17","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\pm 3$$"],"dependencies":["ada10e6CompletingSqr23a-h16"],"title":"Square Root","text":"What is $$\\\\sqrt{9}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr23a-h18","type":"hint","dependencies":["ada10e6CompletingSqr23a-h17"],"title":"Organizing","text":"The equation becomes $$x+2=3$$ or $$x+2=-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr23a-h19","type":"hint","dependencies":["ada10e6CompletingSqr23a-h18"],"title":"Subtraction","text":"$$c=1$$ or $$-5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ada10e6CompletingSqr24","title":"Solve Quadratic Equations of the Form $$x^2+bx+c=0$$ by Completing the Square","body":"Solve the following equation by completing the square","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Solve Quadratic Equations by Completing the Square","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ada10e6CompletingSqr24a","stepAnswer":["$$8$$ or $$-2$$"],"problemType":"MultipleChoice","stepTitle":"$$y^2-6y=16$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$8$$ or $$-2$$","choices":["$$-8$$ or 2I $$-2$$ or $$-8I$$ $$8$$ or $$-2$$","$$8$$ or $$-2$$"],"hints":{"DefaultPathway":[{"id":"ada10e6CompletingSqr24a-h1","type":"hint","dependencies":[],"title":"Formula","text":"$${\\\\left(a+b\\\\right)}^2$$ $$=$$ $$a^2+2ab+b^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr24a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y^2$$"],"dependencies":["ada10e6CompletingSqr24a-h1"],"title":"Identification","text":"What is $$a^2$$ $$=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr24a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y$$"],"dependencies":["ada10e6CompletingSqr24a-h2"],"title":"Identification","text":"$$a=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr24a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-6y$$"],"dependencies":["ada10e6CompletingSqr24a-h3"],"title":"Identification","text":"What is 2ab?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr24a-h5","type":"hint","dependencies":["ada10e6CompletingSqr24a-h4"],"title":"Substitution","text":"With the known a and 2ab, $$b$$ can be calculated","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr24a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["ada10e6CompletingSqr24a-h5"],"title":"Calculation","text":"$$b=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr24a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["ada10e6CompletingSqr24a-h6"],"title":"Calculation","text":"$$2y b=-6y$$, what is $$b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr24a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["ada10e6CompletingSqr24a-h7"],"title":"Substitution","text":"What is the third term $$b^2$$ in the perfect square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr24a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["ada10e6CompletingSqr24a-h8"],"title":"Squaring","text":"What is $${\\\\left(-3\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr24a-h10","type":"hint","dependencies":["ada10e6CompletingSqr24a-h9"],"title":"Organizing","text":"In order to form a perfect square on left side, $$b^2$$ has to be added on both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr24a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["ada10e6CompletingSqr24a-h10"],"title":"Addition","text":"What will be the number on right side after adding 9?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr24a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["ada10e6CompletingSqr24a-h11"],"title":"Addition","text":"What is $$16+9$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr24a-h13","type":"hint","dependencies":["ada10e6CompletingSqr24a-h12"],"title":"Organizing","text":"The equation becomes $$y^2-6y+9=25$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr24a-h14","type":"hint","dependencies":["ada10e6CompletingSqr24a-h13"],"title":"Simplifying","text":"It can be simplified to $${\\\\left(y-3\\\\right)}^2=25$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr24a-h15","type":"hint","dependencies":["ada10e6CompletingSqr24a-h14"],"title":"Square Root","text":"The square can be removed by square root","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr24a-h16","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$y-3$$"],"dependencies":["ada10e6CompletingSqr24a-h15"],"title":"Square Root","text":"What is the sqrt[(y-3)**2]?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr24a-h17","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\pm 5$$"],"dependencies":["ada10e6CompletingSqr24a-h16"],"title":"Square Root","text":"What is $$\\\\sqrt{25}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr24a-h18","type":"hint","dependencies":["ada10e6CompletingSqr24a-h17"],"title":"Organizing","text":"The equation becomes $$y-3=5$$ or $$y-3=-5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr24a-h19","type":"hint","dependencies":["ada10e6CompletingSqr24a-h18"],"title":"Subtraction","text":"$$y=8$$ or $$-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ada10e6CompletingSqr25","title":"Solve Quadratic Equations of the Form $$x^2+bx+c=0$$ by Completing the Square","body":"Solve the following equation by completing the square","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Solve Quadratic Equations by Completing the Square","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ada10e6CompletingSqr25a","stepAnswer":["$$11$$ or $$-1$$"],"problemType":"MultipleChoice","stepTitle":"$$t^2-10t=11$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$11$$ or $$-1$$","choices":["$$-11$$ or $$1I-11$$ or $$-1I11$$ or $$-1$$","$$11$$ or $$-1$$"],"hints":{"DefaultPathway":[{"id":"ada10e6CompletingSqr25a-h1","type":"hint","dependencies":[],"title":"Formula","text":"$${\\\\left(a+b\\\\right)}^2$$ $$=$$ $$a^2+2ab+b^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr25a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$t^2$$"],"dependencies":["ada10e6CompletingSqr25a-h1"],"title":"Identification","text":"What is $$a^2$$ $$=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr25a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$t$$"],"dependencies":["ada10e6CompletingSqr25a-h2"],"title":"Identification","text":"$$a=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr25a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-10t$$"],"dependencies":["ada10e6CompletingSqr25a-h3"],"title":"Identification","text":"What is 2ab?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr25a-h5","type":"hint","dependencies":["ada10e6CompletingSqr25a-h4"],"title":"Substitution","text":"With the known a and 2ab, $$b$$ can be calculated","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr25a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["ada10e6CompletingSqr25a-h5"],"title":"Calculation","text":"$$b=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr25a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["ada10e6CompletingSqr25a-h6"],"title":"Calculation","text":"$$2t b=-10t$$, what is $$b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr25a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["ada10e6CompletingSqr25a-h7"],"title":"Substitution","text":"What is the third term $$b^2$$ in the perfect square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr25a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["ada10e6CompletingSqr25a-h8"],"title":"Squaring","text":"What is $${\\\\left(-5\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr25a-h10","type":"hint","dependencies":["ada10e6CompletingSqr25a-h9"],"title":"Organizing","text":"In order to form a perfect square on left side, $$b^2$$ has to be added on both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr25a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$36$$"],"dependencies":["ada10e6CompletingSqr25a-h10"],"title":"Addition","text":"What will be the number on right side after adding 25?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr25a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$36$$"],"dependencies":["ada10e6CompletingSqr25a-h11"],"title":"Addition","text":"What is $$11+25$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr25a-h13","type":"hint","dependencies":["ada10e6CompletingSqr25a-h12"],"title":"Organizing","text":"The equation becomes $$t^2-10t+25=36$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr25a-h14","type":"hint","dependencies":["ada10e6CompletingSqr25a-h13"],"title":"Simplifying","text":"It can be simplified to $${\\\\left(t-5\\\\right)}^2=36$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr25a-h15","type":"hint","dependencies":["ada10e6CompletingSqr25a-h14"],"title":"Square Root","text":"The square can be removed by square root","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr25a-h16","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$t-5$$"],"dependencies":["ada10e6CompletingSqr25a-h15"],"title":"Square Root","text":"What is the sqrt[(t-5)**2]?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr25a-h17","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\pm 6$$"],"dependencies":["ada10e6CompletingSqr25a-h16"],"title":"Square Root","text":"What is $$\\\\sqrt{36}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr25a-h18","type":"hint","dependencies":["ada10e6CompletingSqr25a-h17"],"title":"Organizing","text":"The equation becomes $$t-5=6$$ or $$t-5=-6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr25a-h19","type":"hint","dependencies":["ada10e6CompletingSqr25a-h18"],"title":"Subtraction","text":"$$t=11$$ or $$-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ada10e6CompletingSqr26","title":"Solve Quadratic Equations of the Form $$x^2+bx+c=0$$ by Completing the Square","body":"Solve the following equation by completing the square","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Solve Quadratic Equations by Completing the Square","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ada10e6CompletingSqr26a","stepAnswer":["No Real Solution"],"problemType":"MultipleChoice","stepTitle":"$$x^2+4x=-21$$","stepBody":"","answerType":"string","variabilization":{},"choices":["No Real Solution","$$\\\\sqrt{21}$$","$$-\\\\sqrt{21}$$"],"hints":{"DefaultPathway":[{"id":"ada10e6CompletingSqr26a-h1","type":"hint","dependencies":[],"title":"Formula","text":"$${\\\\left(a+b\\\\right)}^2$$ $$=$$ $$a^2+2ab+b^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr26a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2$$"],"dependencies":["ada10e6CompletingSqr26a-h1"],"title":"Identification","text":"What is $$a^2$$ $$=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr26a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x$$"],"dependencies":["ada10e6CompletingSqr26a-h2"],"title":"Identification","text":"$$a=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr26a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4x$$"],"dependencies":["ada10e6CompletingSqr26a-h3"],"title":"Identification","text":"What is 2ab?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr26a-h5","type":"hint","dependencies":["ada10e6CompletingSqr26a-h4"],"title":"Substitution","text":"With the known a and 2ab, $$b$$ can be calculated","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr26a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ada10e6CompletingSqr26a-h5"],"title":"Calculation","text":"$$b=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr26a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ada10e6CompletingSqr26a-h6"],"title":"Calculation","text":"$$2x b=4x$$, what is $$b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr26a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ada10e6CompletingSqr26a-h7"],"title":"Substitution","text":"What is the third term $$b^2$$ in the perfect square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr26a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ada10e6CompletingSqr26a-h8"],"title":"Squaring","text":"What is $$2^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr26a-h10","type":"hint","dependencies":["ada10e6CompletingSqr26a-h9"],"title":"Organizing","text":"In order to form a perfect square on left side, $$b^2$$ has to be added on both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr26a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-17$$"],"dependencies":["ada10e6CompletingSqr26a-h10"],"title":"Addition","text":"What will be the number on right side after adding 4?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr26a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-17$$"],"dependencies":["ada10e6CompletingSqr26a-h11"],"title":"Addition","text":"What is $$-21+4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr26a-h13","type":"hint","dependencies":["ada10e6CompletingSqr26a-h12"],"title":"Organizing","text":"The equation becomes $$x^2+4x+4=-21$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr26a-h14","type":"hint","dependencies":["ada10e6CompletingSqr26a-h13"],"title":"Simplifying","text":"It can be simplified to $${\\\\left(x+2\\\\right)}^2=-21$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr26a-h15","type":"hint","dependencies":["ada10e6CompletingSqr26a-h14"],"title":"Square Root","text":"The square can be removed by square root","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr26a-h16","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+2$$"],"dependencies":["ada10e6CompletingSqr26a-h15"],"title":"Square Root","text":"What is the sqrt[(x+2)**2]?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr26a-h17","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No Real Solution"],"dependencies":["ada10e6CompletingSqr26a-h16"],"title":"Square Root","text":"What is $$\\\\sqrt{-21}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["No Real Solution","$$\\\\sqrt{21}$$","$$-\\\\sqrt{21}$$"]}]}}]},{"id":"ada10e6CompletingSqr28","title":"Solve Quadratic Equations of the Form $$x^2+bx+c=0$$ by Completing the Square","body":"Solve the following equation by completing the square","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Solve Quadratic Equations by Completing the Square","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ada10e6CompletingSqr28a","stepAnswer":["$$4$$ or $$-1$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\left(x-3\\\\right) \\\\left(x+5\\\\right)=9$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$4$$ or $$-1$$","choices":["$$-1$$ or $$4$$","$$4$$ or $$-1$$","$$-1$$ or $$-4$$"],"hints":{"DefaultPathway":[{"id":"ada10e6CompletingSqr28a-h1","type":"hint","dependencies":[],"title":"Organizing","text":"Expand the equation on the left","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr28a-h2","type":"hint","dependencies":["ada10e6CompletingSqr28a-h1"],"title":"Organizing","text":"The equation becomes $$x^2+2x-15=9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr28a-h3","type":"hint","dependencies":["ada10e6CompletingSqr28a-h2"],"title":"Formula","text":"$${\\\\left(a+b\\\\right)}^2$$ $$=$$ $$a^2+2ab+b^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr28a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2$$"],"dependencies":["ada10e6CompletingSqr28a-h3"],"title":"Identification","text":"What is $$a^2$$ $$=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr28a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x$$"],"dependencies":["ada10e6CompletingSqr28a-h4"],"title":"Identification","text":"$$a=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr28a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x$$"],"dependencies":["ada10e6CompletingSqr28a-h5"],"title":"Identification","text":"What is 2ab?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr28a-h7","type":"hint","dependencies":["ada10e6CompletingSqr28a-h6"],"title":"Substitution","text":"With the known a and 2ab, $$b$$ can be calculated","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr28a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ada10e6CompletingSqr28a-h7"],"title":"Calculation","text":"$$b=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr28a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ada10e6CompletingSqr28a-h8"],"title":"Calculation","text":"$$2x b=2x$$, what is $$b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr28a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ada10e6CompletingSqr28a-h9"],"title":"Substitution","text":"What is the third term $$b^2$$ in the perfect square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr28a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ada10e6CompletingSqr28a-h10"],"title":"Squaring","text":"What is $$1^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr28a-h12","type":"hint","dependencies":["ada10e6CompletingSqr28a-h11"],"title":"Organizing","text":"In order to form a perfect square on left side, $$16$$ has to be added on both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr28a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["ada10e6CompletingSqr28a-h12"],"title":"Addition","text":"What will be the number on right side after adding 16?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr28a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["ada10e6CompletingSqr28a-h13"],"title":"Addition","text":"What is $$16+9$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr28a-h15","type":"hint","dependencies":["ada10e6CompletingSqr28a-h14"],"title":"Organizing","text":"The equation becomes $$x^2+2x+1=25$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr28a-h16","type":"hint","dependencies":["ada10e6CompletingSqr28a-h15"],"title":"Simplifying","text":"It can be simplified to $${\\\\left(x+1\\\\right)}^2=25$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr28a-h17","type":"hint","dependencies":["ada10e6CompletingSqr28a-h16"],"title":"Square Root","text":"The square can be removed by square root","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr28a-h18","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+1$$"],"dependencies":["ada10e6CompletingSqr28a-h17"],"title":"Square Root","text":"What is the sqrt[(x+1)**2]?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr28a-h19","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\pm 5$$"],"dependencies":["ada10e6CompletingSqr28a-h18"],"title":"Square Root","text":"What is $$\\\\sqrt{25}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$5$$","$$-5$$","$$\\\\pm 5$$"]},{"id":"ada10e6CompletingSqr28a-h20","type":"hint","dependencies":["ada10e6CompletingSqr28a-h19"],"title":"Organizing","text":"The equation becomes $$x+1=5$$ or $$x+1=-5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr28a-h21","type":"hint","dependencies":["ada10e6CompletingSqr28a-h20"],"title":"Subtraction","text":"$$x=4$$ or $$-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ada10e6CompletingSqr29","title":"Solve Quadratic Equations of the Form $${ax}^2+bx+c=0$$ by Completing the Square","body":"Solve the following equation by completing the square","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Solve Quadratic Equations by Completing the Square","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ada10e6CompletingSqr29a","stepAnswer":["$$4$$ or $$\\\\frac{-10}{4}$$"],"problemType":"MultipleChoice","stepTitle":"$$2x^2-3x=20$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$4$$ or $$\\\\frac{-10}{4}$$","choices":["$$-4$$ or $$\\\\frac{10}{4}$$","$$4$$ or $$\\\\frac{10}{4}$$","$$4$$ or $$\\\\frac{-10}{4}$$"],"hints":{"DefaultPathway":[{"id":"ada10e6CompletingSqr29a-h1","type":"hint","dependencies":[],"title":"Planning","text":"Turn the equation to $$x^2+bx+c=d$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr29a-h2","type":"hint","dependencies":["ada10e6CompletingSqr29a-h1"],"title":"Dividing","text":"Divid the whole equation by $$2$$ to get $$x^2-\\\\frac{3}{2} x=10$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr29a-h3","type":"hint","dependencies":["ada10e6CompletingSqr29a-h2"],"title":"Formula","text":"$${\\\\left(a+b\\\\right)}^2$$ $$=$$ $$a^2+2ab+b^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr29a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2$$"],"dependencies":["ada10e6CompletingSqr29a-h3"],"title":"Identification","text":"What is $$a^2$$ $$=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr29a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x$$"],"dependencies":["ada10e6CompletingSqr29a-h4"],"title":"Identification","text":"$$a=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr29a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-3x}{2}$$"],"dependencies":["ada10e6CompletingSqr29a-h5"],"title":"Identification","text":"What is 2ab?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr29a-h7","type":"hint","dependencies":["ada10e6CompletingSqr29a-h6"],"title":"Substitution","text":"With the known a and 2ab, $$b$$ can be calculated","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr29a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-3}{4}$$"],"dependencies":["ada10e6CompletingSqr29a-h7"],"title":"Calculation","text":"$$b=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr29a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-3}{4}$$"],"dependencies":["ada10e6CompletingSqr29a-h8"],"title":"Calculation","text":"$$2x b=\\\\frac{-3x}{2}$$, what is $$b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr29a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{9}{16}$$"],"dependencies":["ada10e6CompletingSqr29a-h9"],"title":"Substitution","text":"What is the third term $$b^2$$ in the perfect square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr29a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{9}{16}$$"],"dependencies":["ada10e6CompletingSqr29a-h10"],"title":"Squaring","text":"What is $${\\\\left(\\\\frac{3}{4}\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr29a-h12","type":"hint","dependencies":["ada10e6CompletingSqr29a-h11"],"title":"Organizing","text":"In order to form a perfect square on left side, $$b^2$$ has to be added on both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr29a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{169}{16}$$"],"dependencies":["ada10e6CompletingSqr29a-h12"],"title":"Addition","text":"What will be the number on right side after adding $$\\\\frac{9}{16}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr29a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{169}{16}$$"],"dependencies":["ada10e6CompletingSqr29a-h13"],"title":"Addition","text":"What is $$\\\\frac{160+9}{16}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr29a-h15","type":"hint","dependencies":["ada10e6CompletingSqr29a-h14"],"title":"Organizing","text":"The equation becomes $$x^2-\\\\frac{3x}{2}+\\\\frac{9}{16}=\\\\frac{169}{16}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr29a-h16","type":"hint","dependencies":["ada10e6CompletingSqr29a-h15"],"title":"Simplifying","text":"It can be simplified to $${\\\\left(x-\\\\frac{3}{4}\\\\right)}^2=\\\\frac{169}{16}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr29a-h17","type":"hint","dependencies":["ada10e6CompletingSqr29a-h16"],"title":"Square Root","text":"The square can be removed by square root","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr29a-h18","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x-\\\\frac{3}{4}$$"],"dependencies":["ada10e6CompletingSqr29a-h17"],"title":"Square Root","text":"What is the sqrt[(x-3/4)**2]?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr29a-h19","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{\\\\pm 13}{4}$$"],"dependencies":["ada10e6CompletingSqr29a-h18"],"title":"Square Root","text":"What is $$\\\\sqrt{\\\\frac{169}{16}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{13}{4}$$","$$\\\\frac{-13}{4}$$","$$\\\\frac{\\\\pm 13}{4}$$"]},{"id":"ada10e6CompletingSqr29a-h20","type":"hint","dependencies":["ada10e6CompletingSqr29a-h19"],"title":"Organizing","text":"The equation becomes $$x-\\\\frac{3}{4}=\\\\frac{13}{4}$$ or $$x-\\\\frac{3}{4}=\\\\frac{-13}{4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr29a-h21","type":"hint","dependencies":["ada10e6CompletingSqr29a-h20"],"title":"Subtraction","text":"$$x=4$$ or $$\\\\frac{-10}{4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ada10e6CompletingSqr3","title":"Completing the Square","body":"First complete the following square by creating a perfect square trinomial then convert the result into a binomial squared.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Solve Quadratic Equations by Completing the Square","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ada10e6CompletingSqr3a","stepAnswer":["$${\\\\left(m+9\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$m^2+18m$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(m+9\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"ada10e6CompletingSqr3a-h1","type":"hint","dependencies":[],"title":"Creating a Perfect Square Trinomial","text":"First identify the coefficient of the term without the square.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$81$$"],"dependencies":["ada10e6CompletingSqr3a-h1"],"title":"Creating a Perfect Square Trinomial","text":"Take the coefficient, divide it by $$2$$, then take the answer and square it. This will complete the square. What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr3a-h3","type":"hint","dependencies":["ada10e6CompletingSqr3a-h2"],"title":"Creating a Binomial Squared","text":"Rewrite as a binomial squared.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(m+9\\\\right)}^2$$"],"dependencies":["ada10e6CompletingSqr3a-h3"],"title":"Creating a Binomial Squared","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ada10e6CompletingSqr30","title":"Solve Quadratic Equations of the Form $${ax}^2+bx+c=0$$ by Completing the Square","body":"Solve the following equation by completing the square","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Solve Quadratic Equations by Completing the Square","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ada10e6CompletingSqr30a","stepAnswer":["$$\\\\frac{-1}{3}\\\\pm \\\\frac{\\\\sqrt{13}}{3}$$"],"problemType":"TextBox","stepTitle":"$$3x^2+2x=4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-1}{3}\\\\pm \\\\frac{\\\\sqrt{13}}{3}$$","hints":{"DefaultPathway":[{"id":"ada10e6CompletingSqr30a-h1","type":"hint","dependencies":[],"title":"Planning","text":"Turn the equation to $$x^2+bx+c=d$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr30a-h2","type":"hint","dependencies":["ada10e6CompletingSqr30a-h1"],"title":"Dividing","text":"Divid the whole equation by $$3$$ to get $$x^2+\\\\frac{2}{3} x=\\\\frac{4}{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr30a-h3","type":"hint","dependencies":["ada10e6CompletingSqr30a-h2"],"title":"Formula","text":"$${\\\\left(a+b\\\\right)}^2$$ $$=$$ $$a^2+2ab+b^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr30a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2$$"],"dependencies":["ada10e6CompletingSqr30a-h3"],"title":"Identification","text":"What is $$a^2$$ $$=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr30a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x$$"],"dependencies":["ada10e6CompletingSqr30a-h4"],"title":"Identification","text":"$$a=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr30a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2x}{3}$$"],"dependencies":["ada10e6CompletingSqr30a-h5"],"title":"Identification","text":"What is 2ab?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr30a-h7","type":"hint","dependencies":["ada10e6CompletingSqr30a-h6"],"title":"Substitution","text":"With the known a and 2ab, $$b$$ can be calculated","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr30a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["ada10e6CompletingSqr30a-h7"],"title":"Calculation","text":"$$b=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr30a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["ada10e6CompletingSqr30a-h8"],"title":"Calculation","text":"$$2x b=\\\\frac{2x}{3}$$, what is $$b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr30a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{9}$$"],"dependencies":["ada10e6CompletingSqr30a-h9"],"title":"Substitution","text":"What is the third term $$b^2$$ in the perfect square?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr30a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{9}$$"],"dependencies":["ada10e6CompletingSqr30a-h10"],"title":"Squaring","text":"What is $${\\\\left(\\\\frac{1}{3}\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr30a-h12","type":"hint","dependencies":["ada10e6CompletingSqr30a-h11"],"title":"Organizing","text":"In order to form a perfect square on left side, $$b^2$$ has to be added on both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr30a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{13}{9}$$"],"dependencies":["ada10e6CompletingSqr30a-h12"],"title":"Addition","text":"What will be the number on right side after adding $$\\\\frac{9}{16}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr30a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{13}{9}$$"],"dependencies":["ada10e6CompletingSqr30a-h13"],"title":"Addition","text":"What is $$\\\\frac{4\\\\times3+1}{9}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr30a-h15","type":"hint","dependencies":["ada10e6CompletingSqr30a-h14"],"title":"Organizing","text":"The equation becomes $$x^2+\\\\frac{2x}{3}+\\\\frac{1}{9}=\\\\frac{13}{9}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr30a-h16","type":"hint","dependencies":["ada10e6CompletingSqr30a-h15"],"title":"Simplifying","text":"It can be simplified to $${\\\\left(x+\\\\frac{1}{3}\\\\right)}^2=\\\\frac{13}{9}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr30a-h17","type":"hint","dependencies":["ada10e6CompletingSqr30a-h16"],"title":"Square Root","text":"The square can be removed by square root","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr30a-h18","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+\\\\frac{1}{3}$$"],"dependencies":["ada10e6CompletingSqr30a-h17"],"title":"Square Root","text":"What is the sqrt[(x+1/3)**2]?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr30a-h19","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{\\\\pm \\\\sqrt{13}}{3}$$"],"dependencies":["ada10e6CompletingSqr30a-h18"],"title":"Square Root","text":"What is $$\\\\sqrt{\\\\frac{13}{9}}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{\\\\pm \\\\sqrt{13}}{3}$$","$$\\\\frac{\\\\sqrt{13}}{3}$$","$$\\\\frac{-\\\\sqrt{13}}{3}$$"]},{"id":"ada10e6CompletingSqr30a-h20","type":"hint","dependencies":["ada10e6CompletingSqr30a-h19"],"title":"Organizing","text":"The equation becomes $$x+\\\\frac{1}{3}=\\\\frac{\\\\sqrt{13}}{3}$$ or $$x+\\\\frac{1}{3}=\\\\frac{-\\\\sqrt{13}}{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr30a-h21","type":"hint","dependencies":["ada10e6CompletingSqr30a-h20"],"title":"Subtraction","text":"$$x=\\\\frac{-1}{3}\\\\pm \\\\frac{\\\\sqrt{13}}{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ada10e6CompletingSqr4","title":"Completing the Square","body":"First complete the following square by creating a perfect square trinomial then convert the result into a binomial 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This will complete the square. What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr4a-h3","type":"hint","dependencies":["ada10e6CompletingSqr4a-h2"],"title":"Creating a Binomial Squared","text":"Rewrite as a binomial squared.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(n-8\\\\right)}^2$$"],"dependencies":["ada10e6CompletingSqr4a-h3"],"title":"Creating a Binomial Squared","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ada10e6CompletingSqr5","title":"Completing the Square","body":"First complete the following square by creating a perfect square trinomial then convert the result into a binomial squared.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Solve Quadratic Equations by Completing the Square","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ada10e6CompletingSqr5a","stepAnswer":["$${\\\\left(p-11\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$p^2-22p$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(p-11\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"ada10e6CompletingSqr5a-h1","type":"hint","dependencies":[],"title":"Creating a Perfect Square Trinomial","text":"First identify the coefficient of the term without the square.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$121$$"],"dependencies":["ada10e6CompletingSqr5a-h1"],"title":"Creating a Perfect Square Trinomial","text":"Take the coefficient, divide it by $$2$$, then take the answer and square it. This will complete the square. What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr5a-h3","type":"hint","dependencies":["ada10e6CompletingSqr5a-h2"],"title":"Creating a Binomial Squared","text":"Rewrite as a binomial squared.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(p-11\\\\right)}^2$$"],"dependencies":["ada10e6CompletingSqr5a-h3"],"title":"Creating a Binomial Squared","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ada10e6CompletingSqr6","title":"Completing the Square","body":"First complete the following square by creating a perfect square trinomial then convert the result into a binomial squared.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Solve Quadratic Equations by Completing the Square","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ada10e6CompletingSqr6a","stepAnswer":["$${\\\\left(q-3\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$q^2-6q$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(q-3\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"ada10e6CompletingSqr6a-h1","type":"hint","dependencies":[],"title":"Creating a Perfect Square Trinomial","text":"First identify the coefficient of the term without the square.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["ada10e6CompletingSqr6a-h1"],"title":"Creating a Perfect Square Trinomial","text":"Take the coefficient, divide it by $$2$$, then take the answer and square it. This will complete the square. What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr6a-h3","type":"hint","dependencies":["ada10e6CompletingSqr6a-h2"],"title":"Creating a Binomial Squared","text":"Rewrite as a binomial squared.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(q-3\\\\right)}^2$$"],"dependencies":["ada10e6CompletingSqr6a-h3"],"title":"Creating a Binomial Squared","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ada10e6CompletingSqr7","title":"Completing the Square","body":"First complete the following square by creating a perfect square trinomial then convert the result into a binomial squared.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Solve Quadratic Equations by Completing the Square","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ada10e6CompletingSqr7a","stepAnswer":["$${\\\\left(x-4.5\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$x^2-9x$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(x-4.5\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"ada10e6CompletingSqr7a-h1","type":"hint","dependencies":[],"title":"Creating a Perfect Square Trinomial","text":"First identify the coefficient of the term without the square.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20.25$$"],"dependencies":["ada10e6CompletingSqr7a-h1"],"title":"Creating a Perfect Square Trinomial","text":"Take the coefficient, divide it by $$2$$, then take the answer and square it. This will complete the square. What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr7a-h3","type":"hint","dependencies":["ada10e6CompletingSqr7a-h2"],"title":"Creating a Binomial Squared","text":"Rewrite as a binomial squared.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(x-4.5\\\\right)}^2$$"],"dependencies":["ada10e6CompletingSqr7a-h3"],"title":"Creating a Binomial Squared","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ada10e6CompletingSqr8","title":"Completing the Square","body":"First complete the following square by creating a perfect square trinomial then convert the result into a binomial squared.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Solve Quadratic Equations by Completing the Square","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ada10e6CompletingSqr8a","stepAnswer":["$${\\\\left(y+5.5\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$y^2-11y$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(y+5.5\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"ada10e6CompletingSqr8a-h1","type":"hint","dependencies":[],"title":"Creating a Perfect Square Trinomial","text":"First identify the coefficient of the term without the square.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$30.25$$"],"dependencies":["ada10e6CompletingSqr8a-h1"],"title":"Creating a Perfect Square Trinomial","text":"Take the coefficient, divide it by $$2$$, then take the answer and square it. This will complete the square. What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr8a-h3","type":"hint","dependencies":["ada10e6CompletingSqr8a-h2"],"title":"Creating a Binomial Squared","text":"Rewrite as a binomial squared.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(y+5.5\\\\right)}^2$$"],"dependencies":["ada10e6CompletingSqr8a-h3"],"title":"Creating a Binomial Squared","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ada10e6CompletingSqr9","title":"Completing the Square","body":"First complete the following square by creating a perfect square trinomial then convert the result into a binomial squared. Round to three decinal places","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.2 Solve Quadratic Equations by Completing the Square","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ada10e6CompletingSqr9a","stepAnswer":["$${\\\\left(p-0.167\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$p^2-\\\\frac{p}{3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(p-0.167\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"ada10e6CompletingSqr9a-h1","type":"hint","dependencies":[],"title":"Creating a Perfect Square Trinomial","text":"First identify the coefficient of the term without the square.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.028$$"],"dependencies":["ada10e6CompletingSqr9a-h1"],"title":"Creating a Perfect Square Trinomial","text":"Take the coefficient, divide it by $$2$$, then take the answer and square it. This will complete the square. What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr9a-h3","type":"hint","dependencies":["ada10e6CompletingSqr9a-h2"],"title":"Creating a Binomial Squared","text":"Rewrite as a binomial squared.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ada10e6CompletingSqr9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(p-0.167\\\\right)}^2$$"],"dependencies":["ada10e6CompletingSqr9a-h3"],"title":"Creating a Binomial Squared","text":"What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add17b3fractions1","title":"Simplifying Rational Expressions","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Simplify Complex Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add17b3fractions1a","stepAnswer":["$$\\\\frac{y+3}{2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\frac{4}{y-3}}{\\\\frac{8}{y^2-9}}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{y+3}{2}$$","hints":{"DefaultPathway":[{"id":"add17b3fractions1a-h1","type":"hint","dependencies":[],"title":"Reciprocal","text":"Rewrite the second fraction as a reciprocal and change the sign to multiplication.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions1a-h2","type":"hint","dependencies":["add17b3fractions1a-h1"],"title":"Multiply","text":"Multiply the two fractions together, such that the product is one fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions1a-h3","type":"hint","dependencies":["add17b3fractions1a-h2"],"title":"Simplify","text":"Factor the equation, looking for common terms in the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions1a-h4","type":"hint","dependencies":["add17b3fractions1a-h3"],"title":"Answer","text":"The answer is $$\\\\frac{y+3}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add17b3fractions10","title":"Simplifying Rational Expressions","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Simplify Complex Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add17b3fractions10a","stepAnswer":["$$\\\\frac{\\\\left(b^2+2b\\\\right) \\\\left(b-5\\\\right)}{3b-5}$$"],"problemType":"TextBox","stepTitle":"(b-(3b/(b+5)))/((2/(b+5))+(1/(b-5))","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{\\\\left(b^2+2b\\\\right) \\\\left(b-5\\\\right)}{3b-5}$$","hints":{"DefaultPathway":[{"id":"add17b3fractions10a-h1","type":"hint","dependencies":[],"title":"LCD","text":"Find the common denominator in both the numerator and denominator. Add the two fractions together. You should now have one rational expression each in the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions10a-h2","type":"hint","dependencies":["add17b3fractions10a-h1"],"title":"Seperate","text":"Seperate the numerator and denominator, preparing to divide the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions10a-h3","type":"hint","dependencies":["add17b3fractions10a-h2"],"title":"Divide","text":"Divide the fractions, taking the reciprocal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions10a-h4","type":"hint","dependencies":["add17b3fractions10a-h3"],"title":"Factor and simplify","text":"Factor the expression, then eliminate any common factors in the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions10a-h5","type":"hint","dependencies":["add17b3fractions10a-h4"],"title":"Answer","text":"The answer is $$\\\\frac{\\\\left(x^2+2x\\\\right) \\\\left(x-5\\\\right)}{3x-5}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add17b3fractions11","title":"Simplifying Rational Expressions","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Simplify Complex Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add17b3fractions11a","stepAnswer":["$$\\\\frac{3}{c+3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{1-\\\\frac{3}{c+4}}{\\\\frac{1}{c+4}+\\\\frac{c}{3}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{c+3}$$","hints":{"DefaultPathway":[{"id":"add17b3fractions11a-h1","type":"hint","dependencies":[],"title":"LCD","text":"Find the common denominator in both the numerator and denominator. Add the two fractions together. You should now have one rational expression each in the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions11a-h2","type":"hint","dependencies":["add17b3fractions11a-h1"],"title":"Seperate","text":"Seperate the numerator and denominator, preparing to divide the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions11a-h3","type":"hint","dependencies":["add17b3fractions11a-h2"],"title":"Divide","text":"Divide the fractions, taking the reciprocal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions11a-h4","type":"hint","dependencies":["add17b3fractions11a-h3"],"title":"Factor and simplify","text":"Factor the expression, then eliminate any common factors in the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions11a-h5","type":"hint","dependencies":["add17b3fractions11a-h4"],"title":"Answer","text":"The answer is $$\\\\frac{3}{c+3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add17b3fractions12","title":"Simplifying Rational Expressions","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Simplify Complex Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add17b3fractions12a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\frac{1}{3}+\\\\frac{1}{6}}{\\\\frac{1}{2}-\\\\frac{1}{3}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"add17b3fractions12a-h1","type":"hint","dependencies":[],"title":"LCD","text":"Find the common denominator in both the numerator and denominator. Multiply both the numerator and denominator by the LCD. Simplify. You should now have one rational expression each in the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions12a-h2","type":"hint","dependencies":["add17b3fractions12a-h1"],"title":"Simplify","text":"Simplify the expression, finding one number in the numerator and one in the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions12a-h3","type":"hint","dependencies":["add17b3fractions12a-h2"],"title":"Answer","text":"The answer is $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add17b3fractions13","title":"Simplifying Rational Expressions","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Simplify Complex Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add17b3fractions13a","stepAnswer":["$$\\\\frac{7}{3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\frac{1}{2}+\\\\frac{1}{5}}{\\\\frac{1}{10}+\\\\frac{1}{5}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{7}{3}$$","hints":{"DefaultPathway":[{"id":"add17b3fractions13a-h1","type":"hint","dependencies":[],"title":"LCD","text":"Find the common denominator in both the numerator and denominator. 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You should now have one rational expression each in the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions13a-h2","type":"hint","dependencies":["add17b3fractions13a-h1"],"title":"Simplify","text":"Simplify the expression, finding one number in the numerator and one in the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions13a-h3","type":"hint","dependencies":["add17b3fractions13a-h2"],"title":"Answer","text":"The answer is $$\\\\frac{7}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add17b3fractions14","title":"Simplifying Rational Expressions","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Simplify Complex Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add17b3fractions14a","stepAnswer":["$$\\\\frac{10}{3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\frac{1}{4}+\\\\frac{3}{8}}{\\\\frac{1}{2}-\\\\frac{5}{16}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{10}{3}$$","hints":{"DefaultPathway":[{"id":"add17b3fractions14a-h1","type":"hint","dependencies":[],"title":"LCD","text":"Find the common denominator in both the numerator and denominator. Multiply both the numerator and denominator by the LCD. Simplify. You should now have one rational expression each in the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions14a-h2","type":"hint","dependencies":["add17b3fractions14a-h1"],"title":"Simplify","text":"Simplify the expression, finding one number in the numerator and one in the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions14a-h3","type":"hint","dependencies":["add17b3fractions14a-h2"],"title":"Answer","text":"The answer is $$\\\\frac{10}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add17b3fractions15","title":"Simplifying Rational Expressions","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Simplify Complex Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add17b3fractions15a","stepAnswer":["$$\\\\frac{1}{x-y}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\frac{1}{x}+\\\\frac{1}{y}}{\\\\frac{x}{y}-\\\\frac{y}{x}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{x-y}$$","hints":{"DefaultPathway":[{"id":"add17b3fractions15a-h1","type":"hint","dependencies":[],"title":"LCD","text":"Find the common denominator in both the numerator and denominator. Multiply both the numerator and denominator by the LCD.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions15a-h2","type":"hint","dependencies":["add17b3fractions15a-h1"],"title":"Simplify","text":"Simplify the expression, finding one number in the numerator and one in the denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions15a-h3","type":"hint","dependencies":["add17b3fractions15a-h2"],"title":"Answer","text":"The answer is $$\\\\frac{1}{x-y}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add17b3fractions16","title":"Simplify a Complex Rational Expression by Writing It as Division","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Simplify Complex Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add17b3fractions16a","stepAnswer":["$$\\\\frac{20}{57}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\frac{2}{3}-\\\\frac{1}{9}}{\\\\frac{3}{4}+\\\\frac{5}{6}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{20}{57}$$","hints":{"DefaultPathway":[{"id":"add17b3fractions16a-h1","type":"hint","dependencies":[],"title":"Rewrite fraction as a multiplication","text":"Rewrite as the product of the first times the reciprocal of the second","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions16a-h2","type":"hint","dependencies":["add17b3fractions16a-h1"],"title":"Multiply and simplify","text":"Multiply and look to factor out common factors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add17b3fractions17","title":"Simplify a Complex Rational Expression by Writing It as Division","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Simplify Complex Rational Expressions","courseName":"OpenStax: Elementary 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4.0>"},{"id":"add17b3fractions17a-h3","type":"hint","dependencies":["add17b3fractions17a-h2"],"title":"Simplify","text":"Simplify the answer if applicable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add17b3fractions18","title":"Simplify a Complex Rational Expression by Writing It as Division","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Simplify Complex Rational Expressions","courseName":"OpenStax: Elementary 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4.0>"},{"id":"add17b3fractions18a-h3","type":"hint","dependencies":["add17b3fractions18a-h2"],"title":"Simplify","text":"Simplify the answer if applicable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add17b3fractions19","title":"Simplify a Complex Rational Expression by Writing It as Division","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Simplify Complex Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add17b3fractions19a","stepAnswer":["$$\\\\frac{\\\\left(x+1\\\\right) \\\\left(x-3\\\\right)}{2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{x-\\\\frac{2x}{x+3}}{\\\\frac{1}{x}+3+\\\\frac{1}{x}-3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{\\\\left(x+1\\\\right) \\\\left(x-3\\\\right)}{2}$$","hints":{"DefaultPathway":[{"id":"add17b3fractions19a-h1","type":"hint","dependencies":[],"title":"Rewrite fraction as a multiplication","text":"Rewrite as the product of the first times the reciprocal of the second","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions19a-h2","type":"hint","dependencies":["add17b3fractions19a-h1"],"title":"Multiply and simplify","text":"Multiply and look to factor out common factors.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions19a-h3","type":"hint","dependencies":["add17b3fractions19a-h2"],"title":"Simplify","text":"Simplify the answer if applicable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add17b3fractions2","title":"Simplifying Rational Expressions","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Simplify Complex Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add17b3fractions2a","stepAnswer":["$$\\\\frac{\\\\frac{2}{x^2-1}}{\\\\frac{3}{x+1}}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\frac{2}{x^2-1}}{\\\\frac{3}{x+1}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{\\\\frac{2}{x^2-1}}{\\\\frac{3}{x+1}}$$","hints":{"DefaultPathway":[{"id":"add17b3fractions2a-h1","type":"hint","dependencies":[],"title":"Reciprocal","text":"Rewrite the second fraction as a reciprocal and change the sign to multiplication.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions2a-h2","type":"hint","dependencies":["add17b3fractions2a-h1"],"title":"Multiply","text":"Multiply the two fractions together, such that the product is one fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions2a-h3","type":"hint","dependencies":["add17b3fractions2a-h2"],"title":"Simplify","text":"Factor the equation, looking for common terms in the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions2a-h4","type":"hint","dependencies":["add17b3fractions2a-h3"],"title":"Answer","text":"The answer is $$\\\\frac{\\\\frac{2}{x^2-1}}{\\\\frac{3}{x+1}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add17b3fractions20","title":"Simplify a Complex Rational Expression by Writing It as Division","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Simplify Complex 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4.0>"},{"id":"add17b3fractions4a-h3","type":"hint","dependencies":["add17b3fractions4a-h2"],"title":"Simplify","text":"Multiply and then reduce the two fractions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions4a-h4","type":"hint","dependencies":["add17b3fractions4a-h3"],"title":"Answer","text":"The answer is $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add17b3fractions5","title":"Simplifying Rational Expressions","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Simplify Complex Rational Expressions","courseName":"OpenStax: Elementary 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4.0>"},{"id":"add17b3fractions5a-h3","type":"hint","dependencies":["add17b3fractions5a-h2"],"title":"Simplify","text":"Multiply and then reduce the two fractions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions5a-h4","type":"hint","dependencies":["add17b3fractions5a-h3"],"title":"Answer","text":"The answer is $$\\\\frac{14}{11}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add17b3fractions6","title":"Simplifying Rational Expressions","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Simplify Complex Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add17b3fractions6a","stepAnswer":["$$\\\\frac{1}{x-y}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\frac{1}{x}+\\\\frac{1}{y}}{\\\\frac{x}{y}-\\\\frac{y}{x}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{x-y}$$","hints":{"DefaultPathway":[{"id":"add17b3fractions6a-h1","type":"hint","dependencies":[],"title":"LCD","text":"Find the common denominator in both the numerator and denominator. Add the two fractions together. You should now have one rational expression each in the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions6a-h2","type":"hint","dependencies":["add17b3fractions6a-h1"],"title":"Seperate","text":"Seperate the numerator and denominator, preparing to divide the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions6a-h3","type":"hint","dependencies":["add17b3fractions6a-h2"],"title":"Divide","text":"Divide the fractions, taking the reciprocal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions6a-h4","type":"hint","dependencies":["add17b3fractions6a-h3"],"title":"Factor and simplify","text":"Factor the expression, then eliminate any common factors in the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions6a-h5","type":"hint","dependencies":["add17b3fractions6a-h4"],"title":"Answer","text":"The answer is $$\\\\frac{1}{x-y}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add17b3fractions7","title":"Simplifying Rational Expressions","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Simplify Complex Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add17b3fractions7a","stepAnswer":["$$\\\\frac{y+x}{y-x}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\frac{1}{x}+\\\\frac{1}{y}}{\\\\frac{1}{x}-\\\\frac{1}{y}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{y+x}{y-x}$$","hints":{"DefaultPathway":[{"id":"add17b3fractions7a-h1","type":"hint","dependencies":[],"title":"LCD","text":"Find the common denominator in both the numerator and denominator. Add the two fractions together. You should now have one rational expression each in the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions7a-h2","type":"hint","dependencies":["add17b3fractions7a-h1"],"title":"Seperate","text":"Seperate the numerator and denominator, preparing to divide the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions7a-h3","type":"hint","dependencies":["add17b3fractions7a-h2"],"title":"Divide","text":"Divide the fractions, taking the reciprocal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions7a-h4","type":"hint","dependencies":["add17b3fractions7a-h3"],"title":"Factor and simplify","text":"Factor the expression, then eliminate any common factors in the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions7a-h5","type":"hint","dependencies":["add17b3fractions7a-h4"],"title":"Answer","text":"The answer is $$\\\\frac{y+x}{y-x}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add17b3fractions8","title":"Simplifying Rational Expressions","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Simplify Complex Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add17b3fractions8a","stepAnswer":["$$\\\\frac{ab}{b-a}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\frac{1}{a}+\\\\frac{1}{b}}{\\\\frac{1}{a^2}-\\\\frac{1}{b^2}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{ab}{b-a}$$","hints":{"DefaultPathway":[{"id":"add17b3fractions8a-h1","type":"hint","dependencies":[],"title":"LCD","text":"Find the common denominator in both the numerator and denominator. Add the two fractions together. You should now have one rational expression each in the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions8a-h2","type":"hint","dependencies":["add17b3fractions8a-h1"],"title":"Seperate","text":"Seperate the numerator and denominator, preparing to divide the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions8a-h3","type":"hint","dependencies":["add17b3fractions8a-h2"],"title":"Divide","text":"Divide the fractions, taking the reciprocal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions8a-h4","type":"hint","dependencies":["add17b3fractions8a-h3"],"title":"Factor and simplify","text":"Factor the expression, then eliminate any common factors in the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions8a-h5","type":"hint","dependencies":["add17b3fractions8a-h4"],"title":"Answer","text":"The answer is $$\\\\frac{ab}{b-a}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add17b3fractions9","title":"Simplifying Rational Expressions","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Simplify Complex Rational Expressions","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add17b3fractions9a","stepAnswer":["$$\\\\frac{\\\\left(n+1\\\\right) \\\\left(n-5\\\\right)}{2} n$$"],"problemType":"TextBox","stepTitle":"(n-(4n/(n+5)))/((1/(n+5))-(1/(n-5))","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{\\\\left(n+1\\\\right) \\\\left(n-5\\\\right)}{2} n$$","hints":{"DefaultPathway":[{"id":"add17b3fractions9a-h1","type":"hint","dependencies":[],"title":"LCD","text":"Find the common denominator in both the numerator and denominator. Add the two fractions together. You should now have one rational expression each in the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions9a-h2","type":"hint","dependencies":["add17b3fractions9a-h1"],"title":"Seperate","text":"Seperate the numerator and denominator, preparing to divide the fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions9a-h3","type":"hint","dependencies":["add17b3fractions9a-h2"],"title":"Divide","text":"Divide the fractions, taking the reciprocal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions9a-h4","type":"hint","dependencies":["add17b3fractions9a-h3"],"title":"Factor and simplify","text":"Factor the expression, then eliminate any common factors in the numerator and denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add17b3fractions9a-h5","type":"hint","dependencies":["add17b3fractions9a-h4"],"title":"Answer","text":"The answer is $$\\\\frac{\\\\left(n+1\\\\right) \\\\left(n-5\\\\right)}{2} n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add2dfegeneral1","title":"Recognize and Use the Appropriate Method to Factor a Polynomial Completely","body":"Factor the following expression completely","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 General Strategy for Factoring Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add2dfegeneral1a","stepAnswer":["$$8\\\\left(m-2\\\\right) \\\\left(m+2\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$8m^2-32$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8\\\\left(m-2\\\\right) \\\\left(m+2\\\\right)$$","hints":{"DefaultPathway":[{"id":"add2dfegeneral1a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":[],"title":"Is there a greatest common factor?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"add2dfegeneral1a-h2","type":"hint","dependencies":["add2dfegeneral1a-h1"],"title":"GCF","text":"Factor out the greatest common factor","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral1a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Binomial"],"dependencies":["add2dfegeneral1a-h2"],"title":"Is the polynomial a binomial, trinomial, or are there more than three terms?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Binomial","Trinomial","Other"]},{"id":"add2dfegeneral1a-h4","type":"hint","dependencies":["add2dfegeneral1a-h3"],"title":"Binomial","text":"Since it\'s a binomial, check to see whether the expression is a sum or difference.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral1a-h5","type":"hint","dependencies":["add2dfegeneral1a-h4"],"title":"Difference of squares","text":"Factor as the product of conjugates. $$a^2-b^2=\\\\left(a-b\\\\right) \\\\left(a+b\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral1a-h6","type":"hint","dependencies":["add2dfegeneral1a-h5"],"title":"Check","text":"Is it factored completely? Do the factors multiply back to the original polynomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add2dfegeneral10","title":"Recognize and Use the Appropriate Method to Factor a Polynomial Completely","body":"Factor the following expression completely","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 General Strategy for Factoring Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add2dfegeneral10a","stepAnswer":["$$3\\\\left(x-3\\\\right) \\\\left(x^2+3x+9\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$5t^3-40$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3\\\\left(x-3\\\\right) \\\\left(x^2+3x+9\\\\right)$$","hints":{"DefaultPathway":[{"id":"add2dfegeneral10a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":[],"title":"Is there a greatest common factor?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"add2dfegeneral10a-h2","type":"hint","dependencies":["add2dfegeneral10a-h1"],"title":"GCF","text":"Factor out the greatest common factor $$5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral10a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Binomial"],"dependencies":["add2dfegeneral10a-h2"],"title":"Is the polynomial a binomial, trinomial, or are there more than three terms?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Binomial","Trinomial","Other"]},{"id":"add2dfegeneral10a-h4","type":"hint","dependencies":["add2dfegeneral10a-h3"],"title":"Binomial","text":"Since it\'s a binomial, check to see whether the expression is a sum or difference.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral10a-h5","type":"hint","dependencies":["add2dfegeneral10a-h4"],"title":"Difference of cubes","text":"$$a^3-b^3=\\\\left(a-b\\\\right) \\\\left(a^2+ab+b^2\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral10a-h6","type":"hint","dependencies":["add2dfegeneral10a-h5"],"title":"Check","text":"Is it factored completely? Do the factors multiply back to the original polynomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add2dfegeneral11","title":"Recognize and Use the Appropriate Method to Factor a Polynomial Completely","body":"Factor the following expression completely","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 General Strategy for Factoring Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add2dfegeneral11a","stepAnswer":["$$\\\\left(k-2\\\\right) \\\\left(k+2\\\\right) \\\\left(k^2+4\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$k^4-16$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(k-2\\\\right) \\\\left(k+2\\\\right) \\\\left(k^2+4\\\\right)$$","hints":{"DefaultPathway":[{"id":"add2dfegeneral11a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":[],"title":"Is there a greatest common factor?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"add2dfegeneral11a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Binomial"],"dependencies":["add2dfegeneral11a-h1"],"title":"Is the polynomial a binomial, trinomial, or are there more than three terms?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Binomial","Trinomial","Other"]},{"id":"add2dfegeneral11a-h3","type":"hint","dependencies":["add2dfegeneral11a-h2"],"title":"Binomial","text":"Since it\'s a binomial, check to see whether the expression is a sum or difference.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral11a-h4","type":"hint","dependencies":["add2dfegeneral11a-h3"],"title":"Difference of squares","text":"$$a^2-b^2=\\\\left(a-b\\\\right) \\\\left(a+b\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral11a-h5","type":"hint","dependencies":["add2dfegeneral11a-h4"],"title":"Check","text":"If you are left with $$\\\\left(k^2+4\\\\right) \\\\left(k^2-4\\\\right)$$, look at the expression to see if you can further factor the expression. $$k^2-4$$ can be further factored!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral11a-h6","type":"hint","dependencies":["add2dfegeneral11a-h5"],"title":"Difference of squares again","text":"$$k^2-4$$ is a difference of squares and can be furthered factored into $$\\\\left(k+2\\\\right) \\\\left(k-2\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add2dfegeneral12","title":"Recognize and Use the Appropriate Method to Factor a Polynomial Completely","body":"Factor the following expression completely","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 General Strategy for Factoring Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add2dfegeneral12a","stepAnswer":["$$\\\\left(m-3\\\\right) \\\\left(m+3\\\\right) \\\\left(m^2+9\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$m^4-81$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(m-3\\\\right) \\\\left(m+3\\\\right) \\\\left(m^2+9\\\\right)$$","hints":{"DefaultPathway":[{"id":"add2dfegeneral12a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":[],"title":"Is there a greatest common factor?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"add2dfegeneral12a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Binomial"],"dependencies":["add2dfegeneral12a-h1"],"title":"Is the polynomial a binomial, trinomial, or are there more than three terms?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Binomial","Trinomial","Other"]},{"id":"add2dfegeneral12a-h3","type":"hint","dependencies":["add2dfegeneral12a-h2"],"title":"Binomial","text":"Since it\'s a binomial, check to see whether the expression is a sum or difference.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral12a-h4","type":"hint","dependencies":["add2dfegeneral12a-h3"],"title":"Difference of squares","text":"$$a^2-b^2=\\\\left(a-b\\\\right) \\\\left(a+b\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral12a-h5","type":"hint","dependencies":["add2dfegeneral12a-h4"],"title":"Check","text":"If you are left with $$\\\\left(m^2+9\\\\right) \\\\left(m^2-9\\\\right)$$, look at the expression to see if you can further factor the expression. $$m^2-9$$ can be further factored!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral12a-h6","type":"hint","dependencies":["add2dfegeneral12a-h5"],"title":"Difference of squares again","text":"$$m^2-9$$ is a difference of squares and can be furthered factored into $$\\\\left(m+3\\\\right) \\\\left(m-3\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add2dfegeneral13","title":"Recognize and Use the Appropriate Method to Factor a Polynomial Completely","body":"Factor the following expression completely","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 General Strategy for Factoring Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add2dfegeneral13a","stepAnswer":["$$3\\\\left(5p+4\\\\right) \\\\left(q-1\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$15pq-15p+12q-12$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3\\\\left(5p+4\\\\right) \\\\left(q-1\\\\right)$$","hints":{"DefaultPathway":[{"id":"add2dfegeneral13a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":[],"title":"Is there a greatest common factor?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"add2dfegeneral13a-h2","type":"hint","dependencies":["add2dfegeneral13a-h1"],"title":"GCF","text":"Factor out the greatest common factor $$3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral13a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Other"],"dependencies":["add2dfegeneral13a-h2"],"title":"Is the polynomial a binomial, trinomial, or are there more than three terms?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Binomial","Trinomial","Other"]},{"id":"add2dfegeneral13a-h4","type":"hint","dependencies":["add2dfegeneral13a-h3"],"title":"More than three terms","text":"Since there are more than three terms, factor by grouping.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral13a-h5","type":"hint","dependencies":["add2dfegeneral13a-h4"],"title":"Grouping","text":"By grouping, you should be left have 3(5p(q-1)+4(q-1). Group together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral13a-h6","type":"hint","dependencies":["add2dfegeneral13a-h5"],"title":"Check","text":"Is it factored completely? Do the factors multiply back to the original polynomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add2dfegeneral14","title":"Recognize and Use the Appropriate Method to Factor a Polynomial Completely","body":"Factor the following expression completely","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 General Strategy for Factoring Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add2dfegeneral14a","stepAnswer":["$$\\\\left(2b-1\\\\right) \\\\left(6a+5\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$12ab-6a+10b-5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\left(2b-1\\\\right) \\\\left(6a+5\\\\right)$$","hints":{"DefaultPathway":[{"id":"add2dfegeneral14a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["no"],"dependencies":[],"title":"Is there a greatest common factor?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"add2dfegeneral14a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Other"],"dependencies":["add2dfegeneral14a-h1"],"title":"Is the polynomial a binomial, trinomial, or are there more than three terms?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Binomial","Trinomial","Other"]},{"id":"add2dfegeneral14a-h3","type":"hint","dependencies":["add2dfegeneral14a-h2"],"title":"More than three terms","text":"Since there are more than three terms, factor by grouping.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral14a-h4","type":"hint","dependencies":["add2dfegeneral14a-h3"],"title":"Grouping","text":"By grouping, you should be left have $$6a \\\\left(2b-1\\\\right)+5\\\\left(2b-1\\\\right)$$. Group together.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral14a-h5","type":"hint","dependencies":["add2dfegeneral14a-h4"],"title":"Check","text":"Is it factored completely? Do the factors multiply back to the original polynomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add2dfegeneral15","title":"Recognize and Use the Appropriate Method to Factor a Polynomial Completely","body":"Factor the following expression completely","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 General Strategy for Factoring Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add2dfegeneral15a","stepAnswer":["$$4\\\\left(x+3\\\\right) \\\\left(x+7\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$4x^2+40x+84$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4\\\\left(x+3\\\\right) \\\\left(x+7\\\\right)$$","hints":{"DefaultPathway":[{"id":"add2dfegeneral15a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":[],"title":"Is there a greatest common factor?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"add2dfegeneral15a-h2","type":"hint","dependencies":["add2dfegeneral15a-h1"],"title":"GCF","text":"Factor out the greatest common factor $$4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral15a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Trinomial"],"dependencies":["add2dfegeneral15a-h2"],"title":"Is the polynomial a binomial, trinomial, or are there more than three terms?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Binomial","Trinomial","Other"]},{"id":"add2dfegeneral15a-h4","type":"hint","dependencies":["add2dfegeneral15a-h3"],"title":"Trinomial","text":"Since it\'s a trinomial, check to see whether the expression is in the form of $$x^2+bx+x$$ or $${ax}^2+bx+c$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral15a-h5","type":"hint","dependencies":["add2dfegeneral15a-h4"],"title":"$$x^2+bx+c$$","text":"Undo using FOIL","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral15a-h6","type":"hint","dependencies":["add2dfegeneral15a-h5"],"title":"Check","text":"Is it factored completely? Do the factors multiply back to the original polynomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add2dfegeneral16","title":"General Strategy for Factoring Polynomials","body":"Factor the following expression completely.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 General Strategy for Factoring Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add2dfegeneral16a","stepAnswer":["$$4x^4 \\\\left(x+3\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$4x^5+12x^4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4x^4 \\\\left(x+3\\\\right)$$","hints":{"DefaultPathway":[{"id":"add2dfegeneral16a-h1","type":"hint","dependencies":[],"title":"Identifying the GCF","text":"The expression contains a GCF (greatest common factor) which is $$4x^4$$. Factor out the GCF.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral16a-h2","type":"hint","dependencies":["add2dfegeneral16a-h1"],"title":"Identifying Polynomial Type","text":"Since there are two terms in the parantheses, the polynomial type is a binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral16a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4x^4 \\\\left(x+3\\\\right)$$"],"dependencies":["add2dfegeneral16a-h2"],"title":"Factor Completely","text":"What is the expression when factored completely? Make sure to multiply and check.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add2dfegeneral17","title":"General Strategy for Factoring Polynomials","body":"Factor the following expression completely.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 General Strategy for Factoring Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add2dfegeneral17a","stepAnswer":["$$3a^3 \\\\left(a+6\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$3a^4+18a^3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3a^3 \\\\left(a+6\\\\right)$$","hints":{"DefaultPathway":[{"id":"add2dfegeneral17a-h1","type":"hint","dependencies":[],"title":"Identifying the GCF","text":"The expression contains a GCF (greatest common factor) which is $$3a^3$$. Factor out the GCF.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral17a-h2","type":"hint","dependencies":["add2dfegeneral17a-h1"],"title":"Identifying Polynomial Type","text":"Since there are two terms in the parantheses, the polynomial type is a binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral17a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3a^3 \\\\left(a+6\\\\right)$$"],"dependencies":["add2dfegeneral17a-h2"],"title":"Factor Completely","text":"What is the expression when factored completely? Make sure to multiply and check.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add2dfegeneral18","title":"General Strategy for Factoring Polynomials","body":"Factor the following expression completely.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 General Strategy for Factoring Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add2dfegeneral18a","stepAnswer":["$$9b^5 \\\\left(5b+3\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$45b^6+27b^5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9b^5 \\\\left(5b+3\\\\right)$$","hints":{"DefaultPathway":[{"id":"add2dfegeneral18a-h1","type":"hint","dependencies":[],"title":"Identifying the GCF","text":"The expression contains a GCF (greatest common factor) which is $$9b^5$$. Factor out the GCF.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral18a-h2","type":"hint","dependencies":["add2dfegeneral18a-h1"],"title":"Identifying Polynomial Type","text":"Since there are two terms in the parantheses, the polynomial type is a binomial.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral18a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9b^5 \\\\left(5b+3\\\\right)$$"],"dependencies":["add2dfegeneral18a-h2"],"title":"Factor Completely","text":"What is the expression when factored completely? Make sure to multiply and check.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add2dfegeneral19","title":"General Strategy for Factoring Polynomials","body":"Factor the following expression completely.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 General Strategy for Factoring Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add2dfegeneral19a","stepAnswer":["$$(3x-2)(4x-1)$$"],"problemType":"TextBox","stepTitle":"$$12x^2+11x+2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$(3x-2)(4x-1)$$","hints":{"DefaultPathway":[{"id":"add2dfegeneral19a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":[],"title":"Identifying the GCF","text":"Is there a GCF?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"add2dfegeneral19a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Trinomial"],"dependencies":["add2dfegeneral19a-h1"],"title":"Identifying Polynomial Type","text":"Is it a binomial, trinomial, or are there more than three terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Binomial","Trinomial","More than $$3$$"]},{"id":"add2dfegeneral19a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["add2dfegeneral19a-h2"],"title":"Checking for Perfect Squares","text":"Are a and c perfect squares?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"add2dfegeneral19a-h4","type":"hint","dependencies":["add2dfegeneral19a-h3"],"title":"Trial and Error","text":"Since there is no GCF, use trial and error or the \\"ac\\" method to factor the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral19a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$(3x-2)(4x-1)$$"],"dependencies":["add2dfegeneral19a-h4"],"title":"Factor Completely","text":"What is the expression when factored completely? Make sure to multiply and check.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add2dfegeneral2","title":"Recognize and Use the Appropriate Method to Factor a Polynomial Completely","body":"Factor the following expression completely","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 General Strategy for Factoring Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add2dfegeneral2a","stepAnswer":["$$4\\\\left(3q-5\\\\right) \\\\left(3q+5\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$36q^2-100$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4\\\\left(3q-5\\\\right) \\\\left(3q+5\\\\right)$$","hints":{"DefaultPathway":[{"id":"add2dfegeneral2a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":[],"title":"Is there a greatest common factor?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"add2dfegeneral2a-h2","type":"hint","dependencies":["add2dfegeneral2a-h1"],"title":"GCF","text":"Factor out the greatest common factor","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral2a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Binomial"],"dependencies":["add2dfegeneral2a-h2"],"title":"Is the polynomial a binomial, trinomial, or are there more than three terms?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Binomial","Trinomial","Other"]},{"id":"add2dfegeneral2a-h4","type":"hint","dependencies":["add2dfegeneral2a-h3"],"title":"Binomial","text":"Since it\'s a binomial, check to see whether the expression is a sum or difference.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral2a-h5","type":"hint","dependencies":["add2dfegeneral2a-h4"],"title":"Difference of squares","text":"Factor as the product of conjugates. $$a^2-b^2=\\\\left(a-b\\\\right) \\\\left(a+b\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral2a-h6","type":"hint","dependencies":["add2dfegeneral2a-h5"],"title":"Check","text":"Is it factored completely? Do the factors multiply back to the original polynomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add2dfegeneral20","title":"General Strategy for Factoring Polynomials","body":"Factor the following expression completely.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 General Strategy for Factoring Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add2dfegeneral20a","stepAnswer":["$$(5a-6)(2a-1)$$"],"problemType":"TextBox","stepTitle":"$$10a^2-17a+6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$(5a-6)(2a-1)$$","hints":{"DefaultPathway":[{"id":"add2dfegeneral20a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":[],"title":"Identifying the GCF","text":"Is there a GCF?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"add2dfegeneral20a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Trinomial"],"dependencies":["add2dfegeneral20a-h1"],"title":"Identifying Polynomial Type","text":"Is it a binomial, trinomial, or are there more than three terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Binomial","Trinomial","More than $$3$$"]},{"id":"add2dfegeneral20a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["add2dfegeneral20a-h2"],"title":"Checking for Perfect Squares","text":"Are a and c perfect squares?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"add2dfegeneral20a-h4","type":"hint","dependencies":["add2dfegeneral20a-h3"],"title":"Trial and Error","text":"Since there is no GCF, use trial and error or the \\"ac\\" method to factor the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral20a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$(5a-6)(2a-1)$$"],"dependencies":["add2dfegeneral20a-h4"],"title":"Factor Completely","text":"What is the expression when factored completely? Make sure to multiply and check.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add2dfegeneral21","title":"General Strategy for Factoring Polynomials","body":"Factor the following expression completely.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 General Strategy for Factoring Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add2dfegeneral21a","stepAnswer":["$$(4x-3)(2x-3)$$"],"problemType":"TextBox","stepTitle":"$$8x^2-18x+9$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$(4x-3)(2x-3)$$","hints":{"DefaultPathway":[{"id":"add2dfegeneral21a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":[],"title":"Identifying the GCF","text":"Is there a GCF?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"add2dfegeneral21a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Trinomial"],"dependencies":["add2dfegeneral21a-h1"],"title":"Identifying Polynomial Type","text":"Is it a binomial, trinomial, or are there more than three terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Binomial","Trinomial","More than $$3$$"]},{"id":"add2dfegeneral21a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["add2dfegeneral21a-h2"],"title":"Checking for Perfect Squares","text":"Are a and c perfect squares?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"add2dfegeneral21a-h4","type":"hint","dependencies":["add2dfegeneral21a-h3"],"title":"Trial and Error","text":"Since there is no GCF, use trial and error or the \\"ac\\" method to factor the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral21a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$(4x-3)(2x-3)$$"],"dependencies":["add2dfegeneral21a-h4"],"title":"Factor Completely","text":"What is the expression when factored completely? 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Do the factors multiply back to the original polynomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add2dfegeneral30","title":"General Strategy for Factoring Polynomials","body":"Factor the following expression completely.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 General Strategy for Factoring Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add2dfegeneral30a","stepAnswer":["$$a^3 \\\\left(a^2+9\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$a^5+9a^3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$a^3 \\\\left(a^2+9\\\\right)$$","hints":{"DefaultPathway":[{"id":"add2dfegeneral30a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":[],"title":"Identifying the GCF","text":"Is there a GCF?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"],"subHints":[{"id":"add2dfegeneral30a-h1-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$a^3$$"],"dependencies":[],"title":"Identifying the GCF","text":"What is the GCF? Make sure to factor it out.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"add2dfegeneral30a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Binomial"],"dependencies":["add2dfegeneral30a-h1"],"title":"Identifying Polynomial Type","text":"Is it a binomial, trinomial, or are there more than three terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Binomial","Trinomial","More than $$3$$"]},{"id":"add2dfegeneral30a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$a^3 \\\\left(a^2+9\\\\right)$$"],"dependencies":["add2dfegeneral30a-h2"],"title":"Factor Completely","text":"What is the expression when factored completely? Make sure to multiply and check.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add2dfegeneral4","title":"Recognize and Use the Appropriate Method to Factor a Polynomial Completely","body":"Factor the following expression completely","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 General Strategy for Factoring Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add2dfegeneral4a","stepAnswer":["$${\\\\left(7b-8\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$49b^2-112b+64$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(7b-8\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"add2dfegeneral4a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":[],"title":"Is there a greatest common factor?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"add2dfegeneral4a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Trionomial"],"dependencies":["add2dfegeneral4a-h1"],"title":"Is the polynomial a binomial, trinomial, or are there more than three terms?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Binomial","Trinomial","Other"]},{"id":"add2dfegeneral4a-h3","type":"hint","dependencies":["add2dfegeneral4a-h2"],"title":"Trinomial","text":"Since it\'s a trinomial, check to see whether the expression is in the form of $$x^2+bx+x$$ or $${ax}^2+bx+c$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral4a-h4","type":"hint","dependencies":["add2dfegeneral4a-h3"],"title":"$${ax}^2+bx+c$$","text":"Since a and c are squares, check if it fits the trinomial square pattern","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral4a-h5","type":"hint","dependencies":["add2dfegeneral4a-h4"],"title":"Trinomial square pattern","text":"The expression fits the square pattern $${\\\\left(a-b\\\\right)}^2=a^2-2ab+b^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral4a-h6","type":"hint","dependencies":["add2dfegeneral4a-h5"],"title":"Check","text":"Is it factored completely? Do the factors multiply back to the original polynomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add2dfegeneral5","title":"Recognize and Use the Appropriate Method to Factor a Polynomial Completely","body":"Factor the following expression completely","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 General Strategy for Factoring Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add2dfegeneral5a","stepAnswer":["$${\\\\left(m+7n\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$m^2+14mn+49n^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(m+7n\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"add2dfegeneral5a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":[],"title":"Is there a greatest common factor?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"add2dfegeneral5a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Trionomial"],"dependencies":["add2dfegeneral5a-h1"],"title":"Is the polynomial a binomial, trinomial, or are there more than three terms?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Binomial","Trinomial","Other"]},{"id":"add2dfegeneral5a-h3","type":"hint","dependencies":["add2dfegeneral5a-h2"],"title":"Trinomial","text":"Since it\'s a trinomial, check to see whether the expression is in the form of $$x^2+bx+x$$ or $${ax}^2+bx+c$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral5a-h4","type":"hint","dependencies":["add2dfegeneral5a-h3"],"title":"$$x^2+bx+c$$","text":"Undo using FOIL","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral5a-h5","type":"hint","dependencies":["add2dfegeneral5a-h4"],"title":"Check","text":"Is it factored completely? Do the factors multiply back to the original polynomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add2dfegeneral6","title":"Recognize and Use the Appropriate Method to Factor a Polynomial Completely","body":"Factor the following expression completely","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 General Strategy for Factoring Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add2dfegeneral6a","stepAnswer":["$${\\\\left(8x+y\\\\right)}^2$$"],"problemType":"TextBox","stepTitle":"$$64x^2+16xy+y^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$${\\\\left(8x+y\\\\right)}^2$$","hints":{"DefaultPathway":[{"id":"add2dfegeneral6a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":[],"title":"Is there a greatest common factor?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"add2dfegeneral6a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Trionomial"],"dependencies":["add2dfegeneral6a-h1"],"title":"Is the polynomial a binomial, trinomial, or are there more than three terms?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Binomial","Trinomial","Other"]},{"id":"add2dfegeneral6a-h3","type":"hint","dependencies":["add2dfegeneral6a-h2"],"title":"Trinomial","text":"Since it\'s a trinomial, check to see whether the expression is in the form of $$x^2+bx+x$$ or $${ax}^2+bx+c$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral6a-h4","type":"hint","dependencies":["add2dfegeneral6a-h3"],"title":"$${ax}^2+bx+c$$","text":"Since a and c are squares, check if it fits the trinomial square pattern","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral6a-h5","type":"hint","dependencies":["add2dfegeneral6a-h4"],"title":"Trinomial square pattern","text":"The expression fits the square pattern $${\\\\left(a+b\\\\right)}^2=a^2+2ab+b^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral6a-h6","type":"hint","dependencies":["add2dfegeneral6a-h5"],"title":"Check","text":"Is it factored completely? Do the factors multiply back to the original polynomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add2dfegeneral7","title":"Recognize and Use the Appropriate Method to Factor a Polynomial Completely","body":"Factor the following expression completely","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 General Strategy for Factoring Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add2dfegeneral7a","stepAnswer":["$$7\\\\left(b+3\\\\right) \\\\left(b-2\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$7b^2+7b-42$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7\\\\left(b+3\\\\right) \\\\left(b-2\\\\right)$$","hints":{"DefaultPathway":[{"id":"add2dfegeneral7a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":[],"title":"Is there a greatest common factor?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"add2dfegeneral7a-h2","type":"hint","dependencies":["add2dfegeneral7a-h1"],"title":"GCF","text":"Factor out the greatest common factor $$7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral7a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Trionomial"],"dependencies":["add2dfegeneral7a-h2"],"title":"Is the polynomial a binomial, trinomial, or are there more than three terms?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Binomial","Trinomial","Other"]},{"id":"add2dfegeneral7a-h4","type":"hint","dependencies":["add2dfegeneral7a-h3"],"title":"Trinomial","text":"Since it\'s a trinomial, check to see whether the expression is in the form of $$x^2+bx+x$$ or $${ax}^2+bx+c$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral7a-h5","type":"hint","dependencies":["add2dfegeneral7a-h4"],"title":"$$x^2+bx+c$$","text":"Undo using FOIL","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral7a-h6","type":"hint","dependencies":["add2dfegeneral7a-h5"],"title":"Check","text":"Is it factored completely? Do the factors multiply back to the original polynomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add2dfegeneral8","title":"Recognize and Use the Appropriate Method to Factor a Polynomial Completely","body":"Factor the following expression completely","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 General Strategy for Factoring Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add2dfegeneral8a","stepAnswer":["$$3\\\\left(n+4\\\\right) \\\\left(n+6\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$3n^2+30n+72$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3\\\\left(n+4\\\\right) \\\\left(n+6\\\\right)$$","hints":{"DefaultPathway":[{"id":"add2dfegeneral8a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":[],"title":"Is there a greatest common factor?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"add2dfegeneral8a-h2","type":"hint","dependencies":["add2dfegeneral8a-h1"],"title":"GCF","text":"Factor out the greatest common factor $$3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral8a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Trionomial"],"dependencies":["add2dfegeneral8a-h2"],"title":"Is the polynomial a binomial, trinomial, or are there more than three terms?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Binomial","Trinomial","Other"]},{"id":"add2dfegeneral8a-h4","type":"hint","dependencies":["add2dfegeneral8a-h3"],"title":"Trinomial","text":"Since it\'s a trinomial, check to see whether the expression is in the form of $$x^2+bx+x$$ or $${ax}^2+bx+c$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral8a-h5","type":"hint","dependencies":["add2dfegeneral8a-h4"],"title":"$$x^2+bx+c$$","text":"Undo using FOIL","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral8a-h6","type":"hint","dependencies":["add2dfegeneral8a-h5"],"title":"Check","text":"Is it factored completely? Do the factors multiply back to the original polynomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add2dfegeneral9","title":"Recognize and Use the Appropriate Method to Factor a Polynomial Completely","body":"Factor the following expression completely","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 General Strategy for Factoring Polynomials","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"add2dfegeneral9a","stepAnswer":["$$3\\\\left(x-3\\\\right) \\\\left(x^2+3x+9\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$3x^3-81$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3\\\\left(x-3\\\\right) \\\\left(x^2+3x+9\\\\right)$$","hints":{"DefaultPathway":[{"id":"add2dfegeneral9a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":[],"title":"Is there a greatest common factor?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"add2dfegeneral9a-h2","type":"hint","dependencies":["add2dfegeneral9a-h1"],"title":"GCF","text":"Factor out the greatest common factor $$3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral9a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Binomial"],"dependencies":["add2dfegeneral9a-h2"],"title":"Is the polynomial a binomial, trinomial, or are there more than three terms?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Binomial","Trinomial","Other"]},{"id":"add2dfegeneral9a-h4","type":"hint","dependencies":["add2dfegeneral9a-h3"],"title":"Binomial","text":"Since it\'s a binomial, check to see whether the expression is a sum or difference.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral9a-h5","type":"hint","dependencies":["add2dfegeneral9a-h4"],"title":"Difference of cubes","text":"$$a^3-b^3=\\\\left(a-b\\\\right) \\\\left(a^2+ab+b^2\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add2dfegeneral9a-h6","type":"hint","dependencies":["add2dfegeneral9a-h5"],"title":"Check","text":"Is it factored completely? Do the factors multiply back to the original polynomial?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add685ddesc-stat1","title":"Discrete vs Continuous","body":"Determine if the following variable is continuous or discrete:","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.8 Descriptive Statistics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"add685ddesc-stat1a","stepAnswer":["Discrete"],"problemType":"MultipleChoice","stepTitle":"The number of people in a family.","stepBody":"","answerType":"string","variabilization":{},"choices":["Discrete","Continuous","Both"],"hints":{"DefaultPathway":[{"id":"add685ddesc-stat1a-h1","type":"hint","dependencies":[],"title":"Definition of Discrete","text":"A discrete variable is a variable that can only take on a finite amount of values. For example, the sizes of pizzas at a certain restaurant would be a discrete variable, since they can only have lengths of 8\\", 12\\", and 16\\".","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add685ddesc-stat1a-h2","type":"hint","dependencies":["add685ddesc-stat1a-h1"],"title":"Definition of Continuous","text":"A continuous variable is a variable that can take on an infinite amount of values. For example, the temperature outside your front door is continuous since it can have almost any value. Whether its $$23.5$$, $$20.111...$$, or $$25$$ degrees outside, the temperature can take on an infinite amount of values since we can have an infinite amount of decimal points.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add685ddesc-stat1a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Discrete"],"dependencies":["add685ddesc-stat1a-h2"],"title":"Solution","text":"Knowing the definition of discrete and continuous variables, is the answer to our initial question continuous or discrete?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Discrete","Continuous","Both"]}]}}]},{"id":"add685ddesc-stat10","title":"What is this?","body":"Determine what the following definition is describing.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.8 Descriptive Statistics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"add685ddesc-stat10a","stepAnswer":["Mean"],"problemType":"MultipleChoice","stepTitle":"The average of all our values, AKA the value we get when we add all our values and divide that by the total number of data points we have.","stepBody":"","answerType":"string","variabilization":{},"choices":["Maximum","Minimum","Mean","Median","IQR","Quartile $$1$$","Quartile $$3$$"],"hints":{"DefaultPathway":[{"id":"add685ddesc-stat10a-h1","type":"hint","dependencies":[],"title":"Definitions","text":"We know that we are talking about a singular value with this definition, so we can immediately rule out the IQR, Quartile $$1$$, and Quartile $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add685ddesc-stat10a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Mean"],"dependencies":["add685ddesc-stat10a-h1"],"title":"Solution","text":"Given that we are looking for a \\"middle\\" value, we can rule out Maximum and Minimum since they describe \\"extreme\\" values. Knowing this, what word are we describing with our definition?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Maximum","Minimum","Mean","Median","IQR","Quartile $$1$$","Quartile $$3$$"]}]}}]},{"id":"add685ddesc-stat11","title":"What is this?","body":"Determine what the following definition is describing.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.8 Descriptive Statistics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"add685ddesc-stat11a","stepAnswer":["Maximum"],"problemType":"MultipleChoice","stepTitle":"The largest quantity in a given dataset.","stepBody":"","answerType":"string","variabilization":{},"choices":["Maximum","Minimum","Mean","Median","IQR","Quartile $$1$$","Quartile $$3$$"],"hints":{"DefaultPathway":[{"id":"add685ddesc-stat11a-h1","type":"hint","dependencies":[],"title":"Definitions","text":"We know that we are talking about a singular value with this definition, so we can immediately rule out the IQR, Quartile $$1$$, and Quartile $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add685ddesc-stat11a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Maximum"],"dependencies":["add685ddesc-stat11a-h1"],"title":"Solution","text":"Given that we are looking for the name of the largest value, we can rule out Mean and Median since they represent middle values. Knowing this, what are we currently describing?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Maximum","Minimum","Mean","Median","IQR","Quartile $$1$$","Quartile $$3$$"]}]}}]},{"id":"add685ddesc-stat12","title":"What is this?","body":"Determine what the following definition is describing.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.8 Descriptive Statistics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"add685ddesc-stat12a","stepAnswer":["Median"],"problemType":"MultipleChoice","stepTitle":"If we have $$n$$ values in a dataset, this value would be the $$\\\\frac{n}{2}$$ value, or the value right in the middle of an ordered dataset.","stepBody":"","answerType":"string","variabilization":{},"choices":["Maximum","Minimum","Mean","Median","IQR","Quartile $$1$$","Quartile $$3$$"],"hints":{"DefaultPathway":[{"id":"add685ddesc-stat12a-h1","type":"hint","dependencies":[],"title":"Definitions","text":"We know that we are talking about a singular value with this definition, so we can immediately rule out the IQR, Quartile $$1$$, and Quartile $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add685ddesc-stat12a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Median"],"dependencies":["add685ddesc-stat12a-h1"],"title":"Solution","text":"Given that we are looking for a \\"middle\\" value, we can rule out Maximum and Minimum since they look for the largest and smallest value respectively. Therefore, what are we describing?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Maximum","Minimum","Mean","Median","IQR","Quartile $$1$$","Quartile $$3$$"]}]}}]},{"id":"add685ddesc-stat13","title":"What is this?","body":"Determine what the following definition is describing.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.8 Descriptive Statistics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"add685ddesc-stat13a","stepAnswer":["Quartile $$1$$"],"problemType":"MultipleChoice","stepTitle":"The value which 25% of our data points are found underneath this point when we arrange our dataset in ascending order.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Quartile $$1$$","choices":["Maximum","Minimum","Mean","Median","IQR","Quartile $$1$$","Quartile $$3$$"],"hints":{"DefaultPathway":[{"id":"add685ddesc-stat13a-h1","type":"hint","dependencies":[],"title":"Definitions","text":"If we are looking for a value at 25%, we know that we aren\'t looking for the smallest or largest values, AKA the Minimum or Maximum.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add685ddesc-stat13a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Quartile $$1$$"],"dependencies":["add685ddesc-stat13a-h1"],"title":"Solution","text":"We also know that we are not looking for any of the \\"middle\\" values, so we can rule out the Mean and Median as well. Knowing this, what are we describing?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Maximum","Minimum","Mean","Median","IQR","Quartile $$1$$","Quartile $$3$$"]}]}}]},{"id":"add685ddesc-stat14","title":"What is this?","body":"Determine what the following definition is describing.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.8 Descriptive Statistics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"add685ddesc-stat14a","stepAnswer":["IQR"],"problemType":"MultipleChoice","stepTitle":"The difference between the third and first quartiles in a dataset.","stepBody":"","answerType":"string","variabilization":{},"choices":["Maximum","Minimum","Mean","Median","IQR","Quartile $$1$$","Quartile $$3$$"],"hints":{"DefaultPathway":[{"id":"add685ddesc-stat14a-h1","type":"hint","dependencies":[],"title":"Definitions","text":"If we are looking at the differences between our third and first quartiles, we know that we are not describing Quartile $$1$$ or Quartile $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add685ddesc-stat14a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["IQR"],"dependencies":["add685ddesc-stat14a-h1"],"title":"Solution","text":"Additionally, by the definition provided, we know that we are not looking for the Maximum or Minimum, since we are looking at a \\"range\\" rather than an exact value. Knowing this, what are we currently describing?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Maximum","Minimum","Mean","Median","IQR","Quartile $$1$$","Quartile $$3$$"]}]}}]},{"id":"add685ddesc-stat15","title":"What is this?","body":"Determine what the following definition is describing.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.8 Descriptive Statistics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"add685ddesc-stat15a","stepAnswer":["Quartile $$3$$"],"problemType":"MultipleChoice","stepTitle":"The value which 75% of our data points are found underneath this point when we arrange our dataset in ascending order.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Quartile $$3$$","choices":["Maximum","Minimum","Mean","Median","IQR","Quartile $$1$$","Quartile $$3$$"],"hints":{"DefaultPathway":[{"id":"add685ddesc-stat15a-h1","type":"hint","dependencies":[],"title":"Definitions","text":"If we are looking for a value at 75%, we know that we aren\'t looking for the smallest or largest values, AKA the Minimum or Maximum.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add685ddesc-stat15a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Quartile $$3$$"],"dependencies":["add685ddesc-stat15a-h1"],"title":"Solution","text":"We also know that we are not looking for any of the \\"middle\\" values, so we can rule out the Mean and Median as well. Knowing this, what are we describing?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Maximum","Minimum","Mean","Median","IQR","Quartile $$1$$","Quartile $$3$$"]}]}}]},{"id":"add685ddesc-stat16","title":"What is this?","body":"Determine what the following definition is describing.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.8 Descriptive Statistics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"add685ddesc-stat16a","stepAnswer":["Histogram"],"problemType":"MultipleChoice","stepTitle":"A diagram made of rectangles where the area is equal to the frequency of a variable, while the rectangle width is equal to the class interval.","stepBody":"","answerType":"string","variabilization":{},"choices":["Box Plot","Histogram","Bar Chart","Outlier","Standard Deviation"],"hints":{"DefaultPathway":[{"id":"add685ddesc-stat16a-h1","type":"hint","dependencies":[],"title":"Definitions","text":"Since we are looking for a plot, we can immediately rule out Outliers and Standard Deviations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add685ddesc-stat16a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Histogram"],"dependencies":["add685ddesc-stat16a-h1"],"title":"Solution","text":"Given our three charts, which most closely resembles our definition?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Box Plot","Histogram","Bar Chart","Outlier","Standard Deviation"]}]}}]},{"id":"add685ddesc-stat17","title":"What is this?","body":"Determine what the following definition is describing.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.8 Descriptive Statistics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"add685ddesc-stat17a","stepAnswer":["Box Plot"],"problemType":"MultipleChoice","stepTitle":"A plot which can show us the median, IQR, outliers, etc., shaped like a rectangle with two lines on the left and right.","stepBody":"","answerType":"string","variabilization":{},"choices":["Box Plot","Histogram","Bar Chart","Outlier","Standard Deviation"],"hints":{"DefaultPathway":[{"id":"add685ddesc-stat17a-h1","type":"hint","dependencies":[],"title":"Definitions","text":"Since we are looking for a plot, we can immediately rule out Outliers and Standard Deviations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add685ddesc-stat17a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Box Plot"],"dependencies":["add685ddesc-stat17a-h1"],"title":"Solution","text":"Given our three charts, which most closely resembles our definition?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Box Plot","Histogram","Bar Chart","Outlier","Standard Deviation"]}]}}]},{"id":"add685ddesc-stat18","title":"What is this?","body":"Determine what the following definition is describing.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.8 Descriptive Statistics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"add685ddesc-stat18a","stepAnswer":["Outlier"],"problemType":"MultipleChoice","stepTitle":"A data point which differs significantly from the other points in a dataset.","stepBody":"","answerType":"string","variabilization":{},"choices":["Box Plot","Histogram","Bar Chart","Outlier","Standard Deviation"],"hints":{"DefaultPathway":[{"id":"add685ddesc-stat18a-h1","type":"hint","dependencies":[],"title":"Definitions","text":"We know that we are looking for a single value rather than a plot, therefore we can rule out Box Plot, Histogram, and Bar Chart.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add685ddesc-stat18a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Outlier"],"dependencies":["add685ddesc-stat18a-h1"],"title":"Solution","text":"Given the option of Standard Deviation or Outlier, which more closely resembles our definition?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Box Plot","Histogram","Bar Chart","Outlier","Standard Deviation"]}]}}]},{"id":"add685ddesc-stat19","title":"What is this?","body":"Determine what the following definition is describing.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.8 Descriptive Statistics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"add685ddesc-stat19a","stepAnswer":["Bar Chart"],"problemType":"MultipleChoice","stepTitle":"A diagram in which variables are represented by rectangles, with the values of each variable being associated with the height of each rectangle.","stepBody":"","answerType":"string","variabilization":{},"choices":["Box Plot","Histogram","Bar Chart","Outlier","Standard Deviation"],"hints":{"DefaultPathway":[{"id":"add685ddesc-stat19a-h1","type":"hint","dependencies":[],"title":"Definitions","text":"Since we are looking for a plot, we can immediately rule out Outliers and Standard Deviations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add685ddesc-stat19a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Bar Chart"],"dependencies":["add685ddesc-stat19a-h1"],"title":"Solution","text":"Given our three charts, which most closely resembles our definition?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Box Plot","Histogram","Bar Chart","Outlier","Standard Deviation"]}]}}]},{"id":"add685ddesc-stat2","title":"Discrete vs Continuous","body":"Determine if the following variable is continuous or discrete:","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.8 Descriptive Statistics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"add685ddesc-stat2a","stepAnswer":["Continuous"],"problemType":"MultipleChoice","stepTitle":"The length of a dog\'s tail.","stepBody":"","answerType":"string","variabilization":{},"choices":["Discrete","Continuous","Both"],"hints":{"DefaultPathway":[{"id":"add685ddesc-stat2a-h1","type":"hint","dependencies":[],"title":"Definition of Discrete","text":"A discrete variable is a variable that can only take on a finite amount of values. For example, the sizes of pizzas at a certain restaurant would be a discrete variable, since they can only have lengths of 8\\", 12\\", and 16\\".","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add685ddesc-stat2a-h2","type":"hint","dependencies":["add685ddesc-stat2a-h1"],"title":"Definition of Continuous","text":"A continuous variable is a variable that can take on an infinite amount of values. For example, the temperature outside your front door is continuous since it can have almost any value. Whether its $$23.5$$, $$20.111...$$, or $$25$$ degrees outside, the temperature can take on an infinite amount of values since we can have an infinite amount of decimal points.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add685ddesc-stat2a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Continuous"],"dependencies":["add685ddesc-stat2a-h2"],"title":"Solution","text":"Knowing the definition of discrete and continuous variables, is the answer to our initial question continuous or discrete?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Discrete","Continuous","Both"]}]}}]},{"id":"add685ddesc-stat20","title":"What is this?","body":"Determine what the following definition is describing.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.8 Descriptive Statistics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"add685ddesc-stat20a","stepAnswer":["Standard Deviation"],"problemType":"MultipleChoice","stepTitle":"A measure of the amount of variation in a dataset.","stepBody":"","answerType":"string","variabilization":{},"choices":["Box Plot","Histogram","Bar Chart","Outlier","Standard Deviation"],"hints":{"DefaultPathway":[{"id":"add685ddesc-stat20a-h1","type":"hint","dependencies":[],"title":"Definitions","text":"We know that we are looking for a single value rather than a plot, therefore we can rule out Box Plot, Histogram, and Bar Chart.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add685ddesc-stat20a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Standard Deviation"],"dependencies":["add685ddesc-stat20a-h1"],"title":"Solution","text":"Given the option of Standard Deviation or Outlier, which more closely resembles our definition?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Box Plot","Histogram","Bar Chart","Outlier","Standard Deviation"]}]}}]},{"id":"add685ddesc-stat21","title":"Calculating Values","body":"Find the mean of the following dataset:","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.8 Descriptive Statistics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"add685ddesc-stat21a","stepAnswer":["$$11$$"],"problemType":"TextBox","stepTitle":"1,4,7,15,19,20","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$11$$","hints":{"DefaultPathway":[{"id":"add685ddesc-stat21a-h1","type":"hint","dependencies":[],"title":"Definitions","text":"Recall that the mean of a dataset is the sum of every value divided by the number of values in a dataset. For example, the mean of 1,2,3 is $$\\\\frac{6}{3}$$, or simply $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add685ddesc-stat21a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11$$"],"dependencies":["add685ddesc-stat21a-h1"],"title":"Solution","text":"Knowing our definition, what is our answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add685ddesc-stat22","title":"Calculating Values","body":"Find the median of the following dataset:","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.8 Descriptive Statistics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"add685ddesc-stat22a","stepAnswer":["$$15$$"],"problemType":"TextBox","stepTitle":"1,4,7,15,19,20,21","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$15$$","hints":{"DefaultPathway":[{"id":"add685ddesc-stat22a-h1","type":"hint","dependencies":[],"title":"Definitions","text":"Recall that the median of a dataset is the middle value. More specifically, if we have say $$5$$ values, our median would be the 3rd value in the sorted version of the data set.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add685ddesc-stat22a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["add685ddesc-stat22a-h1"],"title":"Solution","text":"Knowing our definition, what is our answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add685ddesc-stat23","title":"Calculating Values","body":"Find the mean of the following dataset:","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.8 Descriptive Statistics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"add685ddesc-stat23a","stepAnswer":["$$30$$"],"problemType":"TextBox","stepTitle":"15,19,20,30,66","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$30$$","hints":{"DefaultPathway":[{"id":"add685ddesc-stat23a-h1","type":"hint","dependencies":[],"title":"Definitions","text":"Recall that the median of a dataset is the middle value. More specifically, if we have say $$5$$ values, our median would be the 3rd value in the sorted version of the data set.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add685ddesc-stat23a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$30$$"],"dependencies":["add685ddesc-stat23a-h1"],"title":"Solution","text":"Knowing our definition, what is our answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add685ddesc-stat24","title":"Calculating Values","body":"Find the standard deviation of the following dataset (3 decimal points):","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.8 Descriptive Statistics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"add685ddesc-stat24a","stepAnswer":["$$18.665$$"],"problemType":"TextBox","stepTitle":"15,19,20,30,66","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$18.665$$","hints":{"DefaultPathway":[{"id":"add685ddesc-stat24a-h1","type":"hint","dependencies":[],"title":"Definitions","text":"Recall that the standard deviation of a dataset is the amount of spread a dataset has. More specifically, the standard deviation of a dataset is the sum each value of the dataset subtracted from the mean, then squared. Then you divide the sum by the total number of data points, and finally take the square root.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add685ddesc-stat24a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18.665$$"],"dependencies":["add685ddesc-stat24a-h1"],"title":"Solution","text":"Knowing our definition, what is our answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add685ddesc-stat25","title":"Calculating Values","body":"Find the standard deviation of the following dataset (3 decimal points):","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.8 Descriptive Statistics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"add685ddesc-stat25a","stepAnswer":["$$7.371$$"],"problemType":"TextBox","stepTitle":"1,4,7,15,19,20","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7.371$$","hints":{"DefaultPathway":[{"id":"add685ddesc-stat25a-h1","type":"hint","dependencies":[],"title":"Definitions","text":"Recall that the standard deviation of a dataset is the amount of spread a dataset has. More specifically, the standard deviation of a dataset is the sum each value of the dataset subtracted from the mean, then squared. Then you divide the sum by the total number of data points, and finally take the square root.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add685ddesc-stat25a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7.371$$"],"dependencies":["add685ddesc-stat25a-h1"],"title":"Solution","text":"Knowing our definition, what is our answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"add685ddesc-stat3","title":"Discrete vs Continuous","body":"Determine if the following variable is continuous or discrete:","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.8 Descriptive Statistics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"add685ddesc-stat3a","stepAnswer":["Discrete"],"problemType":"MultipleChoice","stepTitle":"The number of pixels on a computer screen.","stepBody":"","answerType":"string","variabilization":{},"choices":["Discrete","Continuous","Both"],"hints":{"DefaultPathway":[{"id":"add685ddesc-stat3a-h1","type":"hint","dependencies":[],"title":"Definition of Discrete","text":"A discrete variable is a variable that can only take on a finite amount of values. For example, the sizes of pizzas at a certain restaurant would be a discrete variable, since they can only have lengths of 8\\", 12\\", and 16\\".","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add685ddesc-stat3a-h2","type":"hint","dependencies":["add685ddesc-stat3a-h1"],"title":"Definition of Continuous","text":"A continuous variable is a variable that can take on an infinite amount of values. For example, the temperature outside your front door is continuous since it can have almost any value. Whether its $$23.5$$, $$20.111...$$, or $$25$$ degrees outside, the temperature can take on an infinite amount of values since we can have an infinite amount of decimal points.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add685ddesc-stat3a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Discrete"],"dependencies":["add685ddesc-stat3a-h2"],"title":"Solution","text":"Knowing the definition of discrete and continuous variables, is the answer to our initial question continuous or discrete?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Discrete","Continuous","Both"]}]}}]},{"id":"add685ddesc-stat4","title":"Discrete vs Continuous","body":"Determine if the following variable is continuous or discrete:","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.8 Descriptive Statistics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"add685ddesc-stat4a","stepAnswer":["Discrete"],"problemType":"MultipleChoice","stepTitle":"The amount of legs on a table.","stepBody":"","answerType":"string","variabilization":{},"choices":["Discrete","Continuous","Both"],"hints":{"DefaultPathway":[{"id":"add685ddesc-stat4a-h1","type":"hint","dependencies":[],"title":"Definition of Discrete","text":"A discrete variable is a variable that can only take on a finite amount of values. For example, the sizes of pizzas at a certain restaurant would be a discrete variable, since they can only have lengths of 8\\", 12\\", and 16\\".","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add685ddesc-stat4a-h2","type":"hint","dependencies":["add685ddesc-stat4a-h1"],"title":"Definition of Continuous","text":"A continuous variable is a variable that can take on an infinite amount of values. For example, the temperature outside your front door is continuous since it can have almost any value. Whether its $$23.5$$, $$20.111...$$, or $$25$$ degrees outside, the temperature can take on an infinite amount of values since we can have an infinite amount of decimal points.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add685ddesc-stat4a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Discrete"],"dependencies":["add685ddesc-stat4a-h2"],"title":"Solution","text":"Knowing the definition of discrete and continuous variables, is the answer to our initial question continuous or discrete?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Discrete","Continuous","Both"]}]}}]},{"id":"add685ddesc-stat5","title":"Discrete vs Continuous","body":"Determine if the following variable is continuous or discrete:","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.8 Descriptive Statistics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"add685ddesc-stat5a","stepAnswer":["Continuous"],"problemType":"MultipleChoice","stepTitle":"The volume of water in a cup.","stepBody":"","answerType":"string","variabilization":{},"choices":["Discrete","Continuous","Both"],"hints":{"DefaultPathway":[{"id":"add685ddesc-stat5a-h1","type":"hint","dependencies":[],"title":"Definition of Discrete","text":"A discrete variable is a variable that can only take on a finite amount of values. For example, the sizes of pizzas at a certain restaurant would be a discrete variable, since they can only have lengths of 8\\", 12\\", and 16\\".","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add685ddesc-stat5a-h2","type":"hint","dependencies":["add685ddesc-stat5a-h1"],"title":"Definition of Continuous","text":"A continuous variable is a variable that can take on an infinite amount of values. For example, the temperature outside your front door is continuous since it can have almost any value. Whether its $$23.5$$, $$20.111...$$, or $$25$$ degrees outside, the temperature can take on an infinite amount of values since we can have an infinite amount of decimal points.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add685ddesc-stat5a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Continuous"],"dependencies":["add685ddesc-stat5a-h2"],"title":"Solution","text":"Knowing the definition of discrete and continuous variables, is the answer to our initial question continuous or discrete?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Discrete","Continuous","Both"]}]}}]},{"id":"add685ddesc-stat6","title":"Plot Information","body":"Does the following plot usually take in continuous or discrete values?","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.8 Descriptive Statistics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"add685ddesc-stat6a","stepAnswer":["Continuous"],"problemType":"MultipleChoice","stepTitle":"Histogram","stepBody":"","answerType":"string","variabilization":{},"choices":["Discrete","Continuous","Both"],"hints":{"DefaultPathway":[{"id":"add685ddesc-stat6a-h1","type":"hint","dependencies":[],"title":"What is a histogram?","text":"A histogram is a visual representation of our data. The data is grouped into continuous number ranges, with each range corresponding to a vertical bar.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add685ddesc-stat6a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Continuous"],"dependencies":["add685ddesc-stat6a-h1"],"title":"Solution","text":"Knowing more about our plot, should our plot take in continuous or discrete values?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Discrete","Continuous","Both"]}]}}]},{"id":"add685ddesc-stat7","title":"Plot Information","body":"Does the following plot usually take in continuous or discrete values?","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.8 Descriptive Statistics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"add685ddesc-stat7a","stepAnswer":["Both"],"problemType":"MultipleChoice","stepTitle":"Box Plot","stepBody":"","answerType":"string","variabilization":{},"choices":["Discrete","Continuous","Both"],"hints":{"DefaultPathway":[{"id":"add685ddesc-stat7a-h1","type":"hint","dependencies":[],"title":"What is a box plot?","text":"A box plot is a graphic which demonstrates the locality, spread and skewness groups of numerical data through their quartiles. More specifically, a box plot tells us the median, 25% and 75% quartile ranges, and outliers among other things.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add685ddesc-stat7a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Both"],"dependencies":["add685ddesc-stat7a-h1"],"title":"Solution","text":"Knowing more about our plot, should our plot take in continuous or discrete values?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Discrete","Continuous","Both"]}]}}]},{"id":"add685ddesc-stat8","title":"Plot Information","body":"Does the following plot usually take in continuous or discrete values?","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.8 Descriptive Statistics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"add685ddesc-stat8a","stepAnswer":["Discrete"],"problemType":"MultipleChoice","stepTitle":"Bar Chart","stepBody":"","answerType":"string","variabilization":{},"choices":["Discrete","Continuous","Both"],"hints":{"DefaultPathway":[{"id":"add685ddesc-stat8a-h1","type":"hint","dependencies":[],"title":"What is a bar chart?","text":"A bar chart is a diagram in which the numerical values of variables are represented by the height or length of lines or rectangles of equal width. These lines are typically separated by white space, with each line representing exactly $$1$$ value from our data set.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add685ddesc-stat8a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Discrete"],"dependencies":["add685ddesc-stat8a-h1"],"title":"Solution","text":"Knowing more about our plot, should our plot take in continuous or discrete values?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Discrete","Continuous","Both"]}]}}]},{"id":"add685ddesc-stat9","title":"What is this?","body":"Determine what the following definition is describing.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.8 Descriptive Statistics","courseName":"OpenStax: Introductory Stats","steps":[{"id":"add685ddesc-stat9a","stepAnswer":["Minimum"],"problemType":"MultipleChoice","stepTitle":"The smallest quantity in a given data set.","stepBody":"","answerType":"string","variabilization":{},"choices":["Maximum","Minimum","Mean","Median","IQR","Quartile $$1$$","Quartile $$3$$"],"hints":{"DefaultPathway":[{"id":"add685ddesc-stat9a-h1","type":"hint","dependencies":[],"title":"Definitions","text":"We know that we are talking about a singular value with this definition, so we can immediately rule out the IQR, Quartile $$1$$, and Quartile $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"add685ddesc-stat9a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Minimum"],"dependencies":["add685ddesc-stat9a-h1"],"title":"Solution","text":"In addition, we can rule out the Mean and Median, since they represent middle values while we are looking for the name of the smallest quantity. Knowing this, what is our definition describing?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Maximum","Minimum","Mean","Median","IQR","Quartile $$1$$","Quartile $$3$$"]}]}}]},{"id":"aded0eclinear1","title":"Displaying in Interval Notation","body":"Write each inequality in interval notation (\\"inf\\" represents infinity)","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aded0eclinear1a","stepAnswer":["$$[-3$$, inf)"],"problemType":"MultipleChoice","stepTitle":"$$x \\\\geq (-3)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$[-3$$, inf)","$$(-3,0)$$","$$[-3$$, 0]","(-inf, $$-3]$$"],"hints":{"DefaultPathway":[{"id":"aded0eclinear1a-h1","type":"hint","dependencies":[],"title":"Writing the Left Endpoint","text":"Since the inequality includes a greater than or equal to symbol, $$ \\\\geq $$, we will need to use a bracket to denote the left endpoint.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aded0eclinear1a-h2","type":"hint","dependencies":["aded0eclinear1a-h1"],"title":"Writing the Right Endpoint","text":"There is no upper end to the solution to this inequality so we express it using the infinity symbol.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aded0eclinear1a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$[-3$$, inf)"],"dependencies":["aded0eclinear1a-h2"],"title":"Displaying Final Interval Notation","text":"What is the interval notation for the inequality?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$[-3$$, inf)","$$(-3,0)$$","$$[-3$$, 0]","(-inf, $$-3]$$"]}]}},{"id":"aded0eclinear1b","stepAnswer":["(-inf, $$2.5)$$"],"problemType":"MultipleChoice","stepTitle":"$$x<2.5$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"(-inf, $$2.5)$$","choices":["$$(2.5$$, inf)","[-inf, $$2.5)$$","(-inf, $$2.5)$$","(-inf, $$2.5]$$"],"hints":{"DefaultPathway":[{"id":"aded0eclinear1b-h1","type":"hint","dependencies":[],"title":"Writing the Right Endpoint","text":"Since the inequality includes a less than symbol, <, we will need to use a parantheses to denote the right endpoint.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aded0eclinear1b-h2","type":"hint","dependencies":["aded0eclinear1b-h1"],"title":"Writing the Left Endpoint","text":"There is no lower end to the solution to this inequality so we express it using the infinity symbol.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aded0eclinear1b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["(-inf, $$2.5)$$"],"dependencies":["aded0eclinear1b-h2"],"title":"Displaying Final Interval Notation","text":"What is the interval notation for the inequality?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(2.5$$, inf)","[-inf, $$2.5)$$","(-inf, $$2.5)$$","(-inf, $$2.5]$$"]}]}},{"id":"aded0eclinear1c","stepAnswer":["(-inf, -3/5]"],"problemType":"MultipleChoice","stepTitle":"$$x \\\\leq \\\\left(-\\\\frac{3}{5}\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"choices":["(-inf, -3/5)","(-inf, -3/5]","(-3/5, inf]","[-3/5, inf]"],"hints":{"DefaultPathway":[{"id":"aded0eclinear1c-h1","type":"hint","dependencies":[],"title":"Writing the Right Endpoint","text":"Since the inequality includes a less than or equal to symbol, $$ \\\\leq $$, we will need to use a bracket to denote the right endpoint.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aded0eclinear1c-h2","type":"hint","dependencies":["aded0eclinear1c-h1"],"title":"Writing the Left Endpoint","text":"There is no lower end to the solution to this inequality so we express it using the infinity symbol.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aded0eclinear1c-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["(-inf, -3/5]"],"dependencies":["aded0eclinear1c-h2"],"title":"Displaying Final Interval Notation","text":"What is the interval notation for the inequality?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["(-inf, -3/5)","(-inf, -3/5]","(-3/5, inf]","[-3/5, inf]"]}]}}]},{"id":"aded0eclinear10","title":"Displaying in Interval Notation","body":"Solve the inequality and write it in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aded0eclinear10a","stepAnswer":["(8, inf)"],"problemType":"MultipleChoice","stepTitle":"$$9c>72$$","stepBody":"","answerType":"string","variabilization":{},"choices":["(9, inf)","(8, inf)","(-inf, 72]","(-inf, 8]"],"hints":{"DefaultPathway":[{"id":"aded0eclinear10a-h1","type":"hint","dependencies":[],"title":"Using Divisioin 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Then, simplify to get $$c>8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aded0eclinear10a-h2","type":"hint","dependencies":["aded0eclinear10a-h1"],"title":"Writing the Left Endpoint","text":"Since the inequality includes a greater than symbol, >, we will need to use a bracket to denote the left endpoint.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aded0eclinear10a-h3","type":"hint","dependencies":["aded0eclinear10a-h2"],"title":"Writing the Right Endpoint","text":"There is no upper end to the solution to this inequality so we express it using the infinity symbol.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aded0eclinear10a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["(8, inf)"],"dependencies":["aded0eclinear10a-h3"],"title":"Displaying Final Interval Notation","text":"What is the interval notation for the inequality?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["(9, inf)","(8, inf)","(-inf, 72]","(-inf, 8]"]}]}}]},{"id":"aded0eclinear11","title":"Displaying in Interval Notation","body":"Solve the inequality and write it in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aded0eclinear11a","stepAnswer":["[64, inf)"],"problemType":"MultipleChoice","stepTitle":"$$24 \\\\leq \\\\frac{3}{8} m$$","stepBody":"","answerType":"string","variabilization":{},"choices":["[64, inf)","$$(-16$$, inf)","(-inf, 8]","(-inf, 24)"],"hints":{"DefaultPathway":[{"id":"aded0eclinear11a-h1","type":"hint","dependencies":[],"title":"Using Multiplication Property of Inequality","text":"Multiply $$\\\\frac{8}{3}$$ on both sides of the ienquality. Then, simplify to get $$m \\\\geq 64$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aded0eclinear11a-h2","type":"hint","dependencies":["aded0eclinear11a-h1"],"title":"Writing the Left Endpoint","text":"Since the inequality includes a greater than or equal to symbol, $$ \\\\geq $$, we will need to use a bracket to denote the left endpoint.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aded0eclinear11a-h3","type":"hint","dependencies":["aded0eclinear11a-h2"],"title":"Writing the Right Endpoint","text":"There is no upper end to the solution to this inequality so we express it using the infinity symbol.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aded0eclinear11a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["[64, inf)"],"dependencies":["aded0eclinear11a-h3"],"title":"Displaying Final Interval Notation","text":"What is the interval notation for the inequality?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["[64, inf)","$$(-16$$, inf)","(-inf, 8]","(-inf, 24)"]}]}}]},{"id":"aded0eclinear12","title":"Displaying in Interval Notation","body":"Solve the inequality and write it in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aded0eclinear12a","stepAnswer":["(-inf, 11/12]"],"problemType":"MultipleChoice","stepTitle":"$$r-\\\\frac{1}{3} \\\\leq \\\\frac{7}{12}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["(1/3, inf)","(-inf, 11/12]","(-inf, 1/4]","[5/12, inf)"],"hints":{"DefaultPathway":[{"id":"aded0eclinear12a-h1","type":"hint","dependencies":[],"title":"Using Addition Property of Inequality","text":"Add $$\\\\frac{1}{3}$$ to both sides of the inequality. Then, simplify to get $$x \\\\leq \\\\frac{11}{12}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aded0eclinear12a-h2","type":"hint","dependencies":["aded0eclinear12a-h1"],"title":"Writing the Right Endpoint","text":"Since the inequality includes a less than or equal to symbol, $$ \\\\leq $$, we will need to use a bracket to denote the right endpoint.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aded0eclinear12a-h3","type":"hint","dependencies":["aded0eclinear12a-h2"],"title":"Writing the Left Endpoint","text":"There is no lower end to the solution to this inequality so we express it using the infinity symbol.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aded0eclinear12a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["(-inf, 11/12]"],"dependencies":["aded0eclinear12a-h3"],"title":"Displaying Final Interval Notation","text":"What is the interval notation for the inequality?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["(1/3, inf)","(-inf, 11/12]","(-inf, 1/4]","[5/12, inf)"]}]}}]},{"id":"aded0eclinear13","title":"Displaying in Interval Notation","body":"Solve the inequality and write it in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aded0eclinear13a","stepAnswer":["(-inf, 5]"],"problemType":"MultipleChoice","stepTitle":"$$12d \\\\leq 60$$","stepBody":"","answerType":"string","variabilization":{},"choices":["(-inf, 7)","(12, inf)","(-inf, 5]","(-inf, 6)"],"hints":{"DefaultPathway":[{"id":"aded0eclinear13a-h1","type":"hint","dependencies":[],"title":"Using Divisioin Property of Inequality","text":"Divide $$12$$ on both sides of the inequality. Then, simplify to get $$d \\\\leq 5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aded0eclinear13a-h2","type":"hint","dependencies":["aded0eclinear13a-h1"],"title":"Writing the Right Endpoint","text":"Since the inequality includes a less than or equal to symbol, $$ \\\\leq $$, we will need to use a bracket to denote the right endpoint.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aded0eclinear13a-h3","type":"hint","dependencies":["aded0eclinear13a-h2"],"title":"Writing the Left Endpoint","text":"There is no lower end to the solution to this inequality so we express it using the infinity symbol.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aded0eclinear13a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["(-inf, 5]"],"dependencies":["aded0eclinear13a-h3"],"title":"Displaying Final Interval Notation","text":"What is the interval notation for the inequality?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["(-inf, 7)","(12, inf)","(-inf, 5]","(-inf, 6)"]}]}}]},{"id":"aded0eclinear14","title":"Displaying in Interval Notation","body":"Solve the inequality and write it in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aded0eclinear14a","stepAnswer":["$$(-18$$, inf)"],"problemType":"MultipleChoice","stepTitle":"$$-24<\\\\frac{4}{3} n$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$(-18$$, inf)","$$(-32$$, inf)","[6, inf)","[12, inf)"],"hints":{"DefaultPathway":[{"id":"aded0eclinear14a-h1","type":"hint","dependencies":[],"title":"Using Multiplication Property of Inequality","text":"Multiply $$\\\\frac{3}{4}$$ on both sides of the ienquality. 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Then, we can isolate the variable term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aded0eclinear16a-h2","type":"hint","dependencies":["aded0eclinear16a-h1"],"title":"Isolating the Variable Term","text":"$$a+\\\\frac{3}{4} \\\\geq \\\\frac{7}{10}$$. We can subtract by $$\\\\frac{3}{4}$$ on both sides to get $$a \\\\geq \\\\frac{-1}{20}$$ This is our final inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aded0eclinear17","title":"Solving Linear Inequalities","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aded0eclinear17a","stepAnswer":["$$b \\\\geq \\\\frac{-17}{24}$$"],"problemType":"TextBox","stepTitle":"Solve the inequality: $$b+\\\\frac{7}{8} \\\\geq \\\\frac{1}{6}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$b \\\\geq \\\\frac{-17}{24}$$","hints":{"DefaultPathway":[{"id":"aded0eclinear17a-h1","type":"hint","dependencies":[],"title":"General Procedure to Solve Inequalities","text":"To solve an inequality, we must first gather all constants on one side and all variable terms on the other. 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We can subtract by $$\\\\frac{7}{8}$$ on both sides to get $$b \\\\geq \\\\frac{-17}{24}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aded0eclinear18","title":"Solving Linear Inequalities","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aded0eclinear18a","stepAnswer":["$$f<\\\\frac{7}{30}$$"],"problemType":"TextBox","stepTitle":"Solve the inequality: $$f-\\\\frac{13}{20}<\\\\frac{-5}{12}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$f<\\\\frac{7}{30}$$","hints":{"DefaultPathway":[{"id":"aded0eclinear18a-h1","type":"hint","dependencies":[],"title":"General Procedure to Solve Inequalities","text":"To solve an inequality, we must first gather all constants on one side and all variable terms on the other. Then, we can isolate the variable term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aded0eclinear18a-h2","type":"hint","dependencies":["aded0eclinear18a-h1"],"title":"Isolating the Variable Term","text":"$$f-\\\\frac{13}{20}<\\\\frac{-5}{12}$$. We can add $$\\\\frac{13}{20}$$ on both sides to get $$f<\\\\frac{7}{30}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aded0eclinear19","title":"Solving Linear Inequalities","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aded0eclinear19a","stepAnswer":["$$g \\\\leq \\\\frac{23}{36}$$"],"problemType":"TextBox","stepTitle":"Solve the inequality: $$g-\\\\frac{11}{12} \\\\leq \\\\frac{-5}{18}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$g \\\\leq \\\\frac{23}{36}$$","hints":{"DefaultPathway":[{"id":"aded0eclinear19a-h1","type":"hint","dependencies":[],"title":"General Procedure to Solve Inequalities","text":"To solve an inequality, we must first gather all constants on one side and all variable terms on the other. 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We can add $$\\\\frac{11}{12}$$ to both sides of this equation to get $$g \\\\leq \\\\frac{23}{36}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aded0eclinear2","title":"Displaying in Interval Notation","body":"Write each inequality in interval notation (\\"inf\\" represents infinity)","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aded0eclinear2a","stepAnswer":["(2, inf)"],"problemType":"MultipleChoice","stepTitle":"$$x>2$$","stepBody":"","answerType":"string","variabilization":{},"choices":["(-inf, 2)","(-inf, 2]","(2, inf)","[2, inf)"],"hints":{"DefaultPathway":[{"id":"aded0eclinear2a-h1","type":"hint","dependencies":[],"title":"Writing the Left 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BY 4.0>","choices":["(-inf, 2)","(-inf, 2]","(2, inf)","[2, inf)"]}]}},{"id":"aded0eclinear2b","stepAnswer":["(-inf, $$-1.5]$$"],"problemType":"MultipleChoice","stepTitle":"$$x \\\\leq (-1.5)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"(-inf, $$-1.5]$$","choices":["(-inf, $$-1.5)$$","(-inf, $$-1.5]$$","$$(-1.5$$, inf)","$$[-1.5$$, inf)"],"hints":{"DefaultPathway":[{"id":"aded0eclinear2b-h1","type":"hint","dependencies":[],"title":"Writing the Right Endpoint","text":"Since the inequality includes a less than or equal to symbol, $$ \\\\leq $$, we will need to use a bracket to denote the right endpoint.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aded0eclinear2b-h2","type":"hint","dependencies":["aded0eclinear2b-h1"],"title":"Writing the Left Endpoint","text":"There is no lower end to the solution to this inequality so we express it using the infinity symbol.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aded0eclinear2b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["(-inf, $$-1.5]$$"],"dependencies":["aded0eclinear2b-h2"],"title":"Displaying Final Interval Notation","text":"What is the interval notation for the inequality?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["(-inf, $$-1.5)$$","(-inf, $$-1.5]$$","$$(-1.5$$, inf)","$$[-1.5$$, inf)"]}]}},{"id":"aded0eclinear2c","stepAnswer":["[3/4, inf)"],"problemType":"MultipleChoice","stepTitle":"$$x \\\\geq \\\\frac{3}{4}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["[3/4, inf)","(3/4, inf)","(-inf, 3/4]","(-inf, 3/4)"],"hints":{"DefaultPathway":[{"id":"aded0eclinear2c-h1","type":"hint","dependencies":[],"title":"Writing the Left 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["[3/4, inf)","(3/4, inf)","(-inf, 3/4]","(-inf, 3/4)"]}]}}]},{"id":"aded0eclinear20","title":"Solving Linear Inequalities","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aded0eclinear20a","stepAnswer":["$$u \\\\leq -13$$"],"problemType":"TextBox","stepTitle":"Solve the inequality: $$-5u \\\\geq 65$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$u \\\\leq -13$$","hints":{"DefaultPathway":[{"id":"aded0eclinear20a-h1","type":"hint","dependencies":[],"title":"General Procedure to Solve Inequalities","text":"To solve an inequality, we must first gather all constants on one side and all variable terms on the other. 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We must divide both sides by $$-5$$ to isolate u, which requires changing the sign. $$u \\\\leq -13$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aded0eclinear21","title":"Solving Linear Inequalities","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aded0eclinear21a","stepAnswer":["$$v \\\\geq -12$$"],"problemType":"TextBox","stepTitle":"Solve the inequality: $$-8v \\\\leq 96$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$v \\\\geq -12$$","hints":{"DefaultPathway":[{"id":"aded0eclinear21a-h1","type":"hint","dependencies":[],"title":"General Procedure to Solve Inequalities","text":"To solve an inequality, we must first gather all constants on one side and all variable terms on the other. 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We must divide both sides by $$-8$$ to isolate v, which requires changing the sign. $$v \\\\geq -12$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aded0eclinear22","title":"Solving Linear Inequalities","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aded0eclinear22a","stepAnswer":["$$c>-14$$"],"problemType":"TextBox","stepTitle":"Solve the inequality: $$-9c<126$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$c>-14$$","hints":{"DefaultPathway":[{"id":"aded0eclinear22a-h1","type":"hint","dependencies":[],"title":"General Procedure to Solve Inequalities","text":"To solve an inequality, we must first gather all constants on one side and all variable terms on the other. 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We must divide both sides by $$-9$$ to isolate c, which requires changing the sign. $$c>-14$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aded0eclinear23","title":"Solving Linear Inequalities","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aded0eclinear23a","stepAnswer":["$$d<-15$$"],"problemType":"TextBox","stepTitle":"Solve the inequality: $$-7d>105$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$d<-15$$","hints":{"DefaultPathway":[{"id":"aded0eclinear23a-h1","type":"hint","dependencies":[],"title":"General Procedure to Solve Inequalities","text":"To solve an inequality, we must first gather all constants on one side and all variable terms on the other. 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We must divide both sides by $$-7$$ to isolate $$d$$, which requires changing the sign. $$d<-15$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aded0eclinear24","title":"Solving Linear Inequalities","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aded0eclinear24a","stepAnswer":["$$x>9$$"],"problemType":"TextBox","stepTitle":"Solve the inequality: $$8x>72$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x>9$$","hints":{"DefaultPathway":[{"id":"aded0eclinear24a-h1","type":"hint","dependencies":[],"title":"General Procedure to Solve Inequalities","text":"To solve an inequality, we must first gather all constants on one side and all variable terms on the other. 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We must divide both sides by $$8$$ to isolate $$x$$. $$x>9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aded0eclinear25","title":"Solving Linear Inequalities","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aded0eclinear25a","stepAnswer":["$$y<8$$"],"problemType":"TextBox","stepTitle":"Solve the inequality: $$6y<48$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y<8$$","hints":{"DefaultPathway":[{"id":"aded0eclinear25a-h1","type":"hint","dependencies":[],"title":"General Procedure to Solve Inequalities","text":"To solve an inequality, we must first gather all constants on one side and all variable terms on the other. 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We must divide both sides by $$8$$ to isolate $$y$$. $$y<8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aded0eclinear26","title":"Solving Linear Inequalities","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aded0eclinear26a","stepAnswer":["$$t \\\\geq -3$$"],"problemType":"TextBox","stepTitle":"Solve the inequality: $$9t \\\\geq -27$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$t \\\\geq -3$$","hints":{"DefaultPathway":[{"id":"aded0eclinear26a-h1","type":"hint","dependencies":[],"title":"General Procedure to Solve Inequalities","text":"$$9t \\\\geq -27$$. To solve an inequality, we must first gather all constants on one side and all variable terms on the other. Then, we can isolate the variable term.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aded0eclinear26a-h2","type":"hint","dependencies":["aded0eclinear26a-h1"],"title":"Isolating the Variable Term","text":"We must divide both sides by $$9$$ to isolate $$t$$. $$t \\\\geq -3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aded0eclinear27","title":"Solving Linear Inequalities","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.5 Solve Linear Inequalities","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aded0eclinear27a","stepAnswer":["$$s<-4$$"],"problemType":"TextBox","stepTitle":"Solve the inequality: $$7s<-28$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$s<-4$$","hints":{"DefaultPathway":[{"id":"aded0eclinear27a-h1","type":"hint","dependencies":[],"title":"General Procedure to Solve Inequalities","text":"To solve an inequality, we must first gather all constants on one side and all variable terms on the other. 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Then, simplify to get $$z>-25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aded0eclinear9a-h2","type":"hint","dependencies":["aded0eclinear9a-h1"],"title":"Writing the Left Endpoint","text":"Since the inequality includes a greater than symbol, >, we will need to use a bracket to denote the left endpoint.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aded0eclinear9a-h3","type":"hint","dependencies":["aded0eclinear9a-h2"],"title":"Writing the Right Endpoint","text":"There is no upper end to the solution to this inequality so we express it using the infinity symbol.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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Algebra","steps":[{"id":"ae05e04inverse1a","stepAnswer":["$$\\\\frac{\\\\left(-2x\\\\right)}{x-1}$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{x}{x+2}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{\\\\left(-2x\\\\right)}{x-1}$$","choices":["$$\\\\frac{2x}{x+1}$$","$$\\\\frac{\\\\left(-2x\\\\right)}{x-1}$$","$$\\\\frac{x}{x-2}$$","$$\\\\frac{\\\\left(-2x\\\\right)}{x+1}$$"],"hints":{"DefaultPathway":[{"id":"ae05e04inverse1a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$y$$ for f(x)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse1a-h2","type":"hint","dependencies":["ae05e04inverse1a-h1"],"title":"Swap","text":"Swap $$y$$ and $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse1a-h3","type":"hint","dependencies":["ae05e04inverse1a-h2"],"title":"Multiply","text":"Multiply $$y+2$$ to both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse1a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x \\\\left(y+2\\\\right)=y$$"],"dependencies":["ae05e04inverse1a-h3"],"title":"Result","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x \\\\left(y+2\\\\right)=y$$","$$y \\\\left(y+2\\\\right)=x$$","$$x \\\\left(y+2\\\\right)=x$$"]},{"id":"ae05e04inverse1a-h5","type":"hint","dependencies":["ae05e04inverse1a-h4"],"title":"Expand","text":"Expand the left hand side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse1a-h6","type":"hint","dependencies":["ae05e04inverse1a-h5"],"title":"Subtract","text":"Subtract $$2x$$ from both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse1a-h7","type":"hint","dependencies":["ae05e04inverse1a-h6"],"title":"Subtract","text":"Subtract $$y$$ from both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse1a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x y-y=-2x$$"],"dependencies":["ae05e04inverse1a-h7"],"title":"Result","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x y-y=2x$$","$$x y+y=-2x$$","$$x y-y=-2x$$"]},{"id":"ae05e04inverse1a-h9","type":"hint","dependencies":["ae05e04inverse1a-h8"],"title":"Factor","text":"Factor out $$y$$ from the left hand side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse1a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y \\\\left(x-1\\\\right)=-2x$$"],"dependencies":["ae05e04inverse1a-h9"],"title":"Result","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y \\\\left(x-1\\\\right)=-2x$$","$$y \\\\left(x+1\\\\right)=-2x$$","$$y \\\\left(x-1\\\\right)=2x$$"]},{"id":"ae05e04inverse1a-h11","type":"hint","dependencies":["ae05e04inverse1a-h10"],"title":"Divide","text":"Divide $$x-1$$ from both sides to get the final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae05e04inverse10","title":"Intercepts from Slope-Intercept Form","body":"Find the intercepts of the function $$f(x)=5x+1$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.7 Inverse Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae05e04inverse10a","stepAnswer":["$$\\\\frac{-1}{5}$$"],"problemType":"TextBox","stepTitle":"X-Intercept","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-1}{5}$$","hints":{"DefaultPathway":[{"id":"ae05e04inverse10a-h1","type":"hint","dependencies":[],"title":"Plugging in $$0$$","text":"The x-intercept lies at the point of a function where the y-value is zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse10a-h2","type":"hint","dependencies":["ae05e04inverse10a-h1"],"title":"Plugging in $$0$$","text":"So, we have to set the equation to: $$0=5x+1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{5}$$"],"dependencies":["ae05e04inverse10a-h2"],"title":"Plugging in $$0$$","text":"What value of $$x$$ makes this equation true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse10a-h4","type":"hint","dependencies":["ae05e04inverse10a-h3"],"title":"Simplifying the Equation","text":"First, we subtract $$1$$ from both sides to get $$-1=5x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse10a-h5","type":"hint","dependencies":["ae05e04inverse10a-h4"],"title":"Simplifying the Equation","text":"Then, we divide both sides by $$5$$ to get $$x=\\\\frac{-1}{5}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ae05e04inverse10b","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"Y-Intercept","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"ae05e04inverse10b-h1","type":"hint","dependencies":[],"title":"Plugging in $$0$$","text":"Since this equation is in slope-intercept form (y=mx+b), we already know the intercept will be the constant (1), but let\'s check our answer algebreically. The y-intercept lies at the point of a function where the x-value is zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse10b-h2","type":"hint","dependencies":["ae05e04inverse10b-h1"],"title":"Plugging in $$0$$","text":"So, we have to set the equation to: $$f(0)=5x+1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse10b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ae05e04inverse10b-h2"],"title":"Plugging in $$0$$","text":"What is the value of f(0)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse10b-h4","type":"hint","dependencies":["ae05e04inverse10b-h3"],"title":"Simplifying the Equation","text":"When we plug in $$0$$ into the function, we get $$5\\\\times0+1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse10b-h5","type":"hint","dependencies":["ae05e04inverse10b-h4"],"title":"Simplifying the Equation","text":"Since $$5\\\\times0=0$$, we are left with $$0+1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse10b-h6","type":"hint","dependencies":["ae05e04inverse10b-h5"],"title":"Simplifying the Equation","text":"We add the constants: $$0+1=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae05e04inverse11","title":"Intercepts of Cubic Functions","body":"Find the intercepts of the function $$f(x)=x^3-27$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.7 Inverse Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae05e04inverse11a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"X-Intercept","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"ae05e04inverse11a-h1","type":"hint","dependencies":[],"title":"Plugging in $$0$$","text":"The x-intercept lies at the point of a function where the y-value is zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse11a-h2","type":"hint","dependencies":["ae05e04inverse11a-h1"],"title":"Plugging in $$0$$","text":"So, we have to set the equation to: $$0=x^3-27$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse11a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ae05e04inverse11a-h2"],"title":"Plugging in $$0$$","text":"What value of $$x$$ makes this equation true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse11a-h4","type":"hint","dependencies":["ae05e04inverse11a-h3"],"title":"Simplifying the Equation","text":"First, we add $$27$$ from both sides to get $$27=x^3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse11a-h5","type":"hint","dependencies":["ae05e04inverse11a-h4"],"title":"Simplifying the Equation","text":"Then, we take the cube root of both sides to get $$x=3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ae05e04inverse11b","stepAnswer":["$$-27$$"],"problemType":"TextBox","stepTitle":"Y-Intercept","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-27$$","hints":{"DefaultPathway":[{"id":"ae05e04inverse11b-h1","type":"hint","dependencies":[],"title":"Plugging in $$0$$","text":"The y-intercept lies at the point of a function where the x-value is zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse11b-h2","type":"hint","dependencies":["ae05e04inverse11b-h1"],"title":"Plugging in $$0$$","text":"So, we have to set the equation to: $$f(0)=x^3-27$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse11b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-27$$"],"dependencies":["ae05e04inverse11b-h2"],"title":"Plugging in $$0$$","text":"What is the value of f(0)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse11b-h4","type":"hint","dependencies":["ae05e04inverse11b-h3"],"title":"Simplifying the Equation","text":"When we plug in $$0$$ into the function, we get $$0^3-27$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse11b-h5","type":"hint","dependencies":["ae05e04inverse11b-h4"],"title":"Simplifying the Equation","text":"Since $$0^3=0$$, we are left with $$0-27$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse11b-h6","type":"hint","dependencies":["ae05e04inverse11b-h5"],"title":"Simplifying the Equation","text":"We add the constants: $$0-27=-27$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae05e04inverse12","title":"Intercepts from Graphs","body":"Find the intercepts of the function in the graph.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.7 Inverse Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae05e04inverse12a","stepAnswer":["$$x=0$$ and $$20$$"],"problemType":"MultipleChoice","stepTitle":"X-Intercepts","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=0$$ and $$20$$","choices":["$$x=0$$ and $$20$$","$$x=5$$ and $$40$$","$$x=10$$","There are no $$x-intercepts$$."],"hints":{"DefaultPathway":[{"id":"ae05e04inverse12a-h1","type":"hint","dependencies":[],"title":"Visualizing the X-Intercepts","text":"The x-intercept lies at the point of a function where the function touches the x-axis and y-value is zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse12a-h2","type":"hint","dependencies":["ae05e04inverse12a-h1"],"title":"Visualizing the X-Intercepts","text":"The graph of the function touches the x-axis when $$x$$ is equal to $$0$$ and $$20$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ae05e04inverse12b","stepAnswer":["$$x=0$$"],"problemType":"MultipleChoice","stepTitle":"Y-Intercepts","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=0$$","choices":["$$x=0$$ and $$20$$","$$x=0$$","$$x=10$$","There are no $$y-intercepts$$."],"hints":{"DefaultPathway":[{"id":"ae05e04inverse12b-h1","type":"hint","dependencies":[],"title":"Visualizing the Y-Intercepts","text":"The y-intercept lies at the point of a function where the function touches the y-axis and x-value is zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse12b-h2","type":"hint","dependencies":["ae05e04inverse12b-h1"],"title":"Visualizing the Y-Intercepts","text":"The graph of the function touches the y-axis when $$x$$ is equal to $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae05e04inverse13","title":"Intercepts of Piecewise Functions on Graphs","body":"Find the intercepts of the function in the graph.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.7 Inverse Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae05e04inverse13a","stepAnswer":["[0,10)"],"problemType":"MultipleChoice","stepTitle":"X-Intercepts","stepBody":"","answerType":"string","variabilization":{},"choices":["(0,10]","$$(-10,0)$$","[0,10)","$$[-10,0)$$"],"hints":{"DefaultPathway":[{"id":"ae05e04inverse13a-h1","type":"hint","dependencies":[],"title":"Visualizing the X-Intercepts","text":"The x-intercept lies at the point of a function where the function touches the x-axis and y-value is zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse13a-h2","type":"hint","dependencies":["ae05e04inverse13a-h1"],"title":"Visualizing the X-Intercepts","text":"The graph of the function touches the x-axis at all points from $$0$$ to $$10$$ except for $$10$$, since there is a hole in the graph.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ae05e04inverse13b","stepAnswer":["$$x=0$$"],"problemType":"MultipleChoice","stepTitle":"Y-Intercepts","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=0$$","choices":["$$x=0$$ and $$20$$","$$x=0$$","$$x=10$$","There are no $$y-intercepts$$."],"hints":{"DefaultPathway":[{"id":"ae05e04inverse13b-h1","type":"hint","dependencies":[],"title":"Visualizing the Y-Intercepts","text":"The y-intercept lies at the point of a function where the function touches the y-axis and x-value is zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse13b-h2","type":"hint","dependencies":["ae05e04inverse13b-h1"],"title":"Visualizing the Y-Intercepts","text":"The graph of the function touches the y-axis at the point $$(0,0)$$ when $$x$$ is equal to $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae05e04inverse14","title":"Interpreting a Graph","body":"Answer the questions based on the function in the graph.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.7 Inverse Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae05e04inverse14a","stepAnswer":["$$x=3$$"],"problemType":"MultipleChoice","stepTitle":"Find the y-intercept.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=3$$","choices":["$$x=2$$","$$x=3$$","$$x=1$$","$$x=0$$"],"hints":{"DefaultPathway":[{"id":"ae05e04inverse14a-h1","type":"hint","dependencies":[],"title":"Visualizing the Y-Intercepts","text":"The y-intercept lies at the point of a function at the value of f(0) the function touches the y-axis and $$f(x)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse14a-h2","type":"hint","dependencies":["ae05e04inverse14a-h1"],"title":"Visualizing the Y-Intercepts","text":"The graph of the function touches the y-axis when $$x$$ is equal to $$0$$ and $$y=3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ae05e04inverse14b","stepAnswer":["$$x=2$$"],"problemType":"MultipleChoice","stepTitle":"Find the x-intercept.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=2$$","choices":["$$x=2$$","$$x=3$$","$$x=1$$","$$x=0$$"],"hints":{"DefaultPathway":[{"id":"ae05e04inverse14b-h1","type":"hint","dependencies":[],"title":"Visualizing the X-Intercepts","text":"The x-intercept lies at the point of a function where the function touches the x-axis and $$f(x)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse14b-h2","type":"hint","dependencies":["ae05e04inverse14b-h1"],"title":"Visualizing the X-Intercepts","text":"The graph of the function touches the x-axis at $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae05e04inverse15","title":"Identifying an Inverse Function for a Given Input-Output Pair","body":"For a particular one-to-one function, $$f(2)=4$$ and $$f(5)=12$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.7 Inverse Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae05e04inverse15a","stepAnswer":["$$4f^{\\\\left(-1\\\\right)}=2$$ and $$12f^{\\\\left(-1\\\\right)}=5$$"],"problemType":"MultipleChoice","stepTitle":"What are the corresponding input and output values for the inverse function?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$4f^{\\\\left(-1\\\\right)}=2$$ and $$12f^{\\\\left(-1\\\\right)}=5$$","choices":["$$4f^{\\\\left(-1\\\\right)}=2$$ and $$12f^{\\\\left(-1\\\\right)}=5$$","$$2f^{\\\\left(-1\\\\right)}=12$$ and $$4f^{\\\\left(-1\\\\right)}=5$$","$$2f^{\\\\left(-1\\\\right)}=4$$ and $$5f^{\\\\left(-1\\\\right)}=12$$"],"hints":{"DefaultPathway":[{"id":"ae05e04inverse15a-h1","type":"hint","dependencies":[],"title":"Property of Inverse Functions","text":"The inverse function reverses the input and output quantities.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse15a-h2","type":"hint","dependencies":["ae05e04inverse15a-h1"],"title":"Reversing f(2) to find $$4f^{\\\\left(-1\\\\right)}$$","text":"$$f(2)=4$$ represents the point $$(2,4)$$. Reversing this point brings a point of the inverse function, $$(4,2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ae05e04inverse15a-h2"],"title":"Finding $$4f^{\\\\left(-1\\\\right)}$$","text":"$$4f^{\\\\left(-1\\\\right)}=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse15a-h4","type":"hint","dependencies":["ae05e04inverse15a-h3"],"title":"Reversing f(5) to find $$12f^{\\\\left(-1\\\\right)}$$","text":"$$f(5)=12$$ represents the point $$(5,12)$$. Reversing this point brings a point of the inverse function, $$(12,5)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse15a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["ae05e04inverse15a-h4"],"title":"Finding $$12f^{\\\\left(-1\\\\right)}$$","text":"$$f(-1)(12)=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae05e04inverse16","title":"Testing Inverse Relationships Algebraically","body":"Yes or No: Does $$g=f^{\\\\left(-1\\\\right)}$$?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.7 Inverse Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae05e04inverse16a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{1}{x+2}$$, $$g(x)=\\\\frac{1}{x}-2$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ae05e04inverse16a-h1","type":"hint","dependencies":[],"title":"Finding g(f(x))","text":"The first step is to find g(f(x)).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse16a-h2","type":"hint","dependencies":["ae05e04inverse16a-h1"],"title":"Calculating g(f(x))","text":"$$g(f(x))=\\\\frac{1}{\\\\frac{1}{x+2}}-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse16a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ae05e04inverse16a-h2"],"title":"Verifying the Value of g(f(x))","text":"Does $$g(f(x))=x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ae05e04inverse16a-h4","type":"hint","dependencies":["ae05e04inverse16a-h3"],"title":"Interpreting the Meaning of g(f(x))","text":"If $$g(f(x))=x$$, then $$g(x)=f^{\\\\left(-1\\\\right)} x$$ and $$f(x)=g^{\\\\left(-1\\\\right)} x$$. If g(f(x)) is not equal to $$x$$, then these statements are false.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae05e04inverse17","title":"Determining Inverse Relationships for Power Functions","body":"Yes or No: Does $$g=f^{\\\\left(-1\\\\right)}$$?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.7 Inverse Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae05e04inverse17a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=x^3$$ and $$g(x)=\\\\frac{1}{3} x$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ae05e04inverse17a-h1","type":"hint","dependencies":[],"title":"Finding f(g(x))","text":"The first step is to find f(g(x)).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse17a-h2","type":"hint","dependencies":["ae05e04inverse17a-h1"],"title":"Calculating f(g(x))","text":"$$f(g(x))=\\\\frac{x^3}{27}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse17a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ae05e04inverse17a-h2"],"title":"Verifying the Value of g(f(x))","text":"Does $$f(g(x))=x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ae05e04inverse17a-h4","type":"hint","dependencies":["ae05e04inverse17a-h3"],"title":"Interpreting the Meaning of g(f(x))","text":"If $$f(g(x))=x$$, then $$g(x)=f^{\\\\left(-1\\\\right)} x$$ and $$f(x)=g^{\\\\left(-1\\\\right)} x$$. If f(g(x)) is not equal to $$x$$, then these statements are false.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae05e04inverse18","title":"Testing Inverse Functions Algebraically","body":"Yes or No: Does $$g=f^{\\\\left(-1\\\\right)}$$?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.7 Inverse Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae05e04inverse18a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=x^3-4$$, $$g(x)=\\\\sqrt[3]{x+4}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ae05e04inverse18a-h1","type":"hint","dependencies":[],"title":"Finding g(f(x))","text":"The first step is to find g(f(x)).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse18a-h2","type":"hint","dependencies":["ae05e04inverse18a-h1"],"title":"Calculating g(f(x))","text":"$$g(f(x))=\\\\sqrt[3]{x^3-4+4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse18a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ae05e04inverse18a-h2"],"title":"Verifying the Value of g(f(x))","text":"Does $$g(f(x))=x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ae05e04inverse18a-h4","type":"hint","dependencies":["ae05e04inverse18a-h3"],"title":"Interpreting the Meaning of g(f(x))","text":"If $$g(f(x))=x$$, then $$g(x)=f^{\\\\left(-1\\\\right)} x$$ and $$f(x)=g^{\\\\left(-1\\\\right)} x$$. If g(f(x)) is not equal to $$x$$, then these statements are false.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae05e04inverse19","title":"Testing Inverse Functions Algebraically","body":"Yes or No: Does $$g=f^{\\\\left(-1\\\\right)}$$?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.7 Inverse Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae05e04inverse19a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$f(x)={\\\\left(x-1\\\\right)}^3$$, $$g(x)=\\\\sqrt[3]{x}+1$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ae05e04inverse19a-h1","type":"hint","dependencies":[],"title":"Finding g(f(x))","text":"The first step is to find g(f(x)).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse19a-h2","type":"hint","dependencies":["ae05e04inverse19a-h1"],"title":"Calculating g(f(x))","text":"$$g(f(x))=\\\\sqrt[3]{{\\\\left(x-1\\\\right)}^3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse19a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["ae05e04inverse19a-h2"],"title":"Verifying the Value of g(f(x))","text":"Does $$g(f(x))=x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ae05e04inverse19a-h4","type":"hint","dependencies":["ae05e04inverse19a-h3"],"title":"Interpreting the Meaning of g(f(x))","text":"If $$g(f(x))=x$$, then $$g(x)=f^{\\\\left(-1\\\\right)} x$$ and $$f(x)=g^{\\\\left(-1\\\\right)} x$$. If g(f(x)) is not equal to $$x$$, then these statements are false.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae05e04inverse2","title":"Inverse Functions","body":"Find the inverse form of the given function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.7 Inverse Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae05e04inverse2a","stepAnswer":["$$\\\\frac{\\\\left(-4x+3\\\\right)}{5x-2}$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{2x+3}{5x+4}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{\\\\left(-4x+3\\\\right)}{5x-2}$$","choices":["$$\\\\frac{2x+3}{5x-4}$$","$$\\\\frac{\\\\left(-4x+3\\\\right)}{5x-2}$$","$$\\\\frac{4x+3}{5x+2}$$","$$\\\\frac{\\\\left(-2x+3\\\\right)}{5x-2}$$"],"hints":{"DefaultPathway":[{"id":"ae05e04inverse2a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$y$$ for f(x)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse2a-h2","type":"hint","dependencies":["ae05e04inverse2a-h1"],"title":"Swap","text":"Swap $$y$$ and $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse2a-h3","type":"hint","dependencies":["ae05e04inverse2a-h2"],"title":"Multiply","text":"Multiply $$5y+4$$ to both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse2a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x \\\\left(5y+4\\\\right)=2y+3$$"],"dependencies":["ae05e04inverse2a-h3"],"title":"Result","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y \\\\left(5y+4\\\\right)=2y+3$$","$$x \\\\left(5y+4\\\\right)=2y+3$$","$$x \\\\left(5y+4\\\\right)=2x+3$$"]},{"id":"ae05e04inverse2a-h5","type":"hint","dependencies":["ae05e04inverse2a-h4"],"title":"Expand","text":"Expand the left hand side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse2a-h6","type":"hint","dependencies":["ae05e04inverse2a-h5"],"title":"Subtract","text":"Subtract $$4x$$ from both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse2a-h7","type":"hint","dependencies":["ae05e04inverse2a-h6"],"title":"Subtract","text":"Subtract $$2y$$ from both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse2a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$5xy-2y=-4x+3$$"],"dependencies":["ae05e04inverse2a-h7"],"title":"Result","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$5xy+2y=4x+3$$","$$5xy-2y=4x-9$$","$$5xy-2y=-4x+3$$"]},{"id":"ae05e04inverse2a-h9","type":"hint","dependencies":["ae05e04inverse2a-h8"],"title":"Factor","text":"Factor out $$y$$ from the left hand side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse2a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y \\\\left(5x-2\\\\right)=-4x+3$$"],"dependencies":["ae05e04inverse2a-h9"],"title":"Result","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y \\\\left(5x+2\\\\right)=-4x+3$$","$$y \\\\left(5x-2\\\\right)=-4x+3$$","$$y \\\\left(5x-2\\\\right)=4x-3$$"]},{"id":"ae05e04inverse2a-h11","type":"hint","dependencies":["ae05e04inverse2a-h10"],"title":"Divide","text":"Divide $$5x-2$$ from both sides to get the final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae05e04inverse20","title":"Explaining How to Find the Inverse of a Function Algebraically","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.7 Inverse Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae05e04inverse20a","stepAnswer":["Take the reciprocal of the function. $$1$$ divided by the function is the inverse function."],"problemType":"MultipleChoice","stepTitle":"How do you find the inverse of a function algebraically?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Take the reciprocal of the function. $$1$$ divided by the function is the inverse function.","choices":["Take the reciprocal of the function. $$1$$ divided by the function is the inverse function.","Set up an equation for $$y$$ in terms of $$x$$, then solve for $$x$$ in terms of $$y$$. Lastly, interpret the isolated $$x$$ as the inverse input, and the other side of the equation as the inverse output.","Multiply the function by $$\\\\frac{1}{x}$$, then switch the two sides of the equation. The left side is the inverse function input, and the right side the inverse function output."],"hints":{"DefaultPathway":[{"id":"ae05e04inverse20a-h1","type":"hint","dependencies":[],"title":"First Step","text":"The first step is to set up an equation for the function, for example $$y=4x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse20a-h2","type":"hint","dependencies":["ae05e04inverse20a-h1"],"title":"Second Step","text":"The second step is to solve the function in terms of $$x$$. For the example in the last hint, $$f(x)=4x$$, this would result in the equation $$x=\\\\frac{y}{4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse20a-h3","type":"hint","dependencies":["ae05e04inverse20a-h2"],"title":"Final Step","text":"The final step after solving for $$x$$ is to consider $$x$$ the output of the inverse function and the other side of the equation the ouput of the inverse function (after replacing $$y$$ with x.) Forthe example in the previous two hints, the output of the inverse function would thus be $$4x$$, and the input $$x$$. In equation form, $$f^{\\\\left(-1\\\\right)} x=\\\\frac{x}{4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae05e04inverse21","title":"Finding the Inverse","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.7 Inverse Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae05e04inverse21a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Is the function $$f(x)=a-x$$ its own inverse for all real numbers a?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ae05e04inverse21a-h1","type":"hint","dependencies":[],"title":"Setting up an Equation","text":"The first step is to set up an equation for the function, $$y=a-x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse21a-h2","type":"hint","dependencies":["ae05e04inverse21a-h1"],"title":"Solving the Equation in Terms of $$y$$","text":"The next step is to solve the equation in terms of $$x$$. $$y=a-x$$, therefore $$x=a-y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse21a-h3","type":"hint","dependencies":["ae05e04inverse21a-h2"],"title":"Interpreting the Inverse Based on the Equation","text":"After replacing $$y$$ with $$x$$ on the right side of the equation, we can see that $$f^{\\\\left(-1\\\\right)} x=a-x=a-y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae05e04inverse22","title":"Finding the Inverse","body":"Find the inverse form for the following function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.7 Inverse Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae05e04inverse22a","stepAnswer":["$$x-3$$"],"problemType":"TextBox","stepTitle":"$$f(x)=x+3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$x-3$$","hints":{"DefaultPathway":[{"id":"ae05e04inverse22a-h1","type":"hint","dependencies":[],"title":"Setting up an Equation","text":"The first step is to set up an equation for the function, $$y=x+3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse22a-h2","type":"hint","dependencies":["ae05e04inverse22a-h1"],"title":"Solving the Equation in Terms of $$y$$","text":"The next step is to solve the equation in terms of $$x$$. $$y=x+3$$, therefore $$x=y-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse22a-h3","type":"hint","dependencies":["ae05e04inverse22a-h2"],"title":"Interpreting the Inverse Based on the Equation","text":"After replacing $$y$$ with $$x$$ on the right side of the equation, we can see that $$f^{\\\\left(-1\\\\right)} x=x-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae05e04inverse23","title":"Finding the Inverse","body":"Find the inverse form for the following function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.7 Inverse Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae05e04inverse23a","stepAnswer":["$$2-x$$"],"problemType":"TextBox","stepTitle":"$$f(x)=2-x$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2-x$$","hints":{"DefaultPathway":[{"id":"ae05e04inverse23a-h1","type":"hint","dependencies":[],"title":"Setting up an Equation","text":"The first step is to set up an equation for the function, $$y=2-x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse23a-h2","type":"hint","dependencies":["ae05e04inverse23a-h1"],"title":"Solving the Equation in Terms of $$y$$","text":"The next step is to solve the equation in terms of $$x$$. $$y=2-x$$, therefore $$x=2-y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse23a-h3","type":"hint","dependencies":["ae05e04inverse23a-h2"],"title":"Interpreting the Inverse Based on the Equation","text":"After replacing $$y$$ with $$x$$ on the right side of the equation, we can see that $$f^{\\\\left(-1\\\\right)} x=2-x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae05e04inverse24","title":"Finding the Inverse","body":"Find the inverse form for the following function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.7 Inverse Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae05e04inverse24a","stepAnswer":["$$3-x$$"],"problemType":"TextBox","stepTitle":"$$f(x)=3-x$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3-x$$","hints":{"DefaultPathway":[{"id":"ae05e04inverse24a-h1","type":"hint","dependencies":[],"title":"Setting up an Equation","text":"The first step is to set up an equation for the function, $$y=3-x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse24a-h2","type":"hint","dependencies":["ae05e04inverse24a-h1"],"title":"Solving the Equation in Terms of $$y$$","text":"The next step is to solve the equation in terms of $$x$$. $$y=3-x$$, therefore $$x=3-y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse24a-h3","type":"hint","dependencies":["ae05e04inverse24a-h2"],"title":"Interpreting the Inverse Based on the Equation","text":"After replacing $$y$$ with $$x$$ on the right side of the equation, we can see that $$f^{\\\\left(-1\\\\right)} x=3-x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae05e04inverse3","title":"Inverse Functions","body":"Find the inverse form of the given function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.7 Inverse Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae05e04inverse3a","stepAnswer":["$$x^{\\\\frac{1}{2}}-7$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)={\\\\left(x+7\\\\right)}^2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x^{\\\\frac{1}{2}}-7$$","choices":["x**/2+7","$$x^{\\\\frac{1}{2}}+7$$","$$x^{\\\\frac{1}{2}}-7$$","$$x^{\\\\left(-\\\\frac{1}{2}\\\\right)}-7$$"],"hints":{"DefaultPathway":[{"id":"ae05e04inverse3a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$y$$ for f(x)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse3a-h2","type":"hint","dependencies":["ae05e04inverse3a-h1"],"title":"Swap","text":"Swap $$y$$ and $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse3a-h3","type":"hint","dependencies":["ae05e04inverse3a-h2"],"title":"Square Root","text":"Take the square root of both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse3a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x^{\\\\frac{1}{2}}=y+7$$"],"dependencies":["ae05e04inverse3a-h3"],"title":"Result","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x^2=y+7$$","$$x^{\\\\frac{1}{2}}=y-7$$","$$x^{\\\\frac{1}{2}}=y+7$$"]},{"id":"ae05e04inverse3a-h5","type":"hint","dependencies":["ae05e04inverse3a-h4"],"title":"Subtract","text":"Subtract $$7$$ from both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae05e04inverse4","title":"Inverse Functions","body":"Find the inverse form of the given function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.7 Inverse Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae05e04inverse4a","stepAnswer":["$$x^{\\\\frac{1}{2}}+6$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)={\\\\left(x-6\\\\right)}^2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x^{\\\\frac{1}{2}}+6$$","choices":["$$x^{\\\\frac{1}{2}}-6$$","$$x^{\\\\frac{1}{2}}+6$$","$$x^{\\\\left(-\\\\frac{1}{2}\\\\right)}+6$$","$$x^2+6$$"],"hints":{"DefaultPathway":[{"id":"ae05e04inverse4a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$y$$ for f(x)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse4a-h2","type":"hint","dependencies":["ae05e04inverse4a-h1"],"title":"Swap","text":"Swap $$y$$ and $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse4a-h3","type":"hint","dependencies":["ae05e04inverse4a-h2"],"title":"Square Root","text":"Take the square root of both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse4a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x^{\\\\frac{1}{2}}=y-6$$"],"dependencies":["ae05e04inverse4a-h3"],"title":"Result","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x^2=y-6$$","$$x^{\\\\frac{1}{2}}=y+6$$","$$x^{\\\\frac{1}{2}}=y-6$$"]},{"id":"ae05e04inverse4a-h5","type":"hint","dependencies":["ae05e04inverse4a-h4"],"title":"Add","text":"Add $$6$$ to both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae05e04inverse5","title":"Inverse Functions","body":"Find the inverse form of the given function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.7 Inverse Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae05e04inverse5a","stepAnswer":["$${\\\\left(x+5\\\\right)}^{\\\\frac{1}{2}}$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=x^2-5$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$${\\\\left(x+5\\\\right)}^{\\\\frac{1}{2}}$$","choices":["$${\\\\left(x+5\\\\right)}^2$$","$${\\\\left(x-5\\\\right)}^2$$","$${\\\\left(x+5\\\\right)}^{\\\\frac{1}{2}}$$","$${\\\\left(x-5\\\\right)}^{\\\\frac{1}{2}}$$"],"hints":{"DefaultPathway":[{"id":"ae05e04inverse5a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$y$$ for f(x)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse5a-h2","type":"hint","dependencies":["ae05e04inverse5a-h1"],"title":"Swap","text":"Swap $$y$$ and $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse5a-h3","type":"hint","dependencies":["ae05e04inverse5a-h2"],"title":"Add","text":"Add $$5$$ to both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse5a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x+5=y^2$$"],"dependencies":["ae05e04inverse5a-h3"],"title":"Result","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x+5=y^{\\\\frac{1}{2}}$$","$$x+5=y^2$$","$$x-5=y^2$$"]},{"id":"ae05e04inverse5a-h5","type":"hint","dependencies":["ae05e04inverse5a-h4"],"title":"Square Root","text":"Take the square root of both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae05e04inverse6","title":"Inverse Functions with Tables","body":"Use the values listed in the table to solve.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.7 Inverse Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae05e04inverse6a","stepAnswer":["$$7$$"],"problemType":"TextBox","stepTitle":"f(x) $$=$$ $$3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7$$","hints":{"DefaultPathway":[{"id":"ae05e04inverse6a-h1","type":"hint","dependencies":[],"title":"X-value","text":"This question is asking us what $$x$$ value needs to be inputted to get f(x) $$=$$ $$3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse6a-h2","type":"hint","dependencies":["ae05e04inverse6a-h1"],"title":"Table","text":"Look on the table to see which $$x$$ value corresponds to f(x) $$=$$ $$3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae05e04inverse7","title":"Inverse Functions","body":"Find the inverse form of the given function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.7 Inverse Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae05e04inverse7a","stepAnswer":["$$-2x+\\\\frac{3}{x}$$"],"problemType":"MultipleChoice","stepTitle":"f(x) $$=$$ $$\\\\frac{3}{x}-2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-2x+\\\\frac{3}{x}$$","choices":["$$-2x+\\\\frac{3}{x}$$","$$2x+\\\\frac{4}{x}$$","$$-2x-\\\\frac{3}{x}$$"],"hints":{"DefaultPathway":[{"id":"ae05e04inverse7a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute $$y$$ for f(x)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse7a-h2","type":"hint","dependencies":["ae05e04inverse7a-h1"],"title":"Swap","text":"Swap $$y$$ and $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse7a-h3","type":"hint","dependencies":["ae05e04inverse7a-h2"],"title":"Multiply","text":"Multiply $$y+2$$ to both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse7a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x \\\\left(y+2\\\\right)=3$$"],"dependencies":["ae05e04inverse7a-h3"],"title":"Result","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x \\\\left(y+2\\\\right)=3$$","$$x \\\\left(y+2\\\\right)=3x$$","$$y \\\\left(y+2\\\\right)=3$$"]},{"id":"ae05e04inverse7a-h5","type":"hint","dependencies":["ae05e04inverse7a-h4"],"title":"Expand","text":"Expand the left hand side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse7a-h6","type":"hint","dependencies":["ae05e04inverse7a-h5"],"title":"Subtract","text":"Subtract $$2x$$ from both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse7a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$xy=-2x+3$$"],"dependencies":["ae05e04inverse7a-h6"],"title":"Result","text":"What is the result?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x=-2x+3$$","$$xy=2x-3$$","$$xy=-2x+3$$"]},{"id":"ae05e04inverse7a-h8","type":"hint","dependencies":["ae05e04inverse7a-h7"],"title":"Divide","text":"Divide $$x$$ from both sides to get the final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae05e04inverse8","title":"Intercepts","body":"Find the intercepts of the function $$f(x)=\\\\sqrt{x}$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.7 Inverse Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae05e04inverse8a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"X-Intercept","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"ae05e04inverse8a-h1","type":"hint","dependencies":[],"title":"Plugging in $$0$$","text":"The x-intercept lies at the point of a function where the y-value is zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse8a-h2","type":"hint","dependencies":["ae05e04inverse8a-h1"],"title":"Plugging in $$0$$","text":"So, we have to set the equation to: $$0=\\\\sqrt{x}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["ae05e04inverse8a-h2"],"title":"Plugging in $$0$$","text":"What value of $$x$$ makes this equation true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse8a-h4","type":"hint","dependencies":["ae05e04inverse8a-h3"],"title":"Plugging in $$0$$","text":"The only number that has a square root of $$0$$ is $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ae05e04inverse8b","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"Y-Intercept","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"ae05e04inverse8b-h1","type":"hint","dependencies":[],"title":"Plugging in $$0$$","text":"The y-intercept lies at the point of a function where the x-value is zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse8b-h2","type":"hint","dependencies":["ae05e04inverse8b-h1"],"title":"Plugging in $$0$$","text":"So, we have to set the equation to: $$f(0)=\\\\sqrt{x}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse8b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["ae05e04inverse8b-h2"],"title":"Plugging in $$0$$","text":"What is the value of f(0)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae05e04inverse9","title":"Intercepts","body":"Find the intercepts of the function $$f(x)=\\\\sqrt[3]{3x+1}$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.7 Inverse Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae05e04inverse9a","stepAnswer":["$$\\\\frac{-1}{3}$$"],"problemType":"TextBox","stepTitle":"X-Intercept","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-1}{3}$$","hints":{"DefaultPathway":[{"id":"ae05e04inverse9a-h1","type":"hint","dependencies":[],"title":"Plugging in $$0$$","text":"The x-intercept lies at the point of a function where the y-value is zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse9a-h2","type":"hint","dependencies":["ae05e04inverse9a-h1"],"title":"Plugging in $$0$$","text":"So, we have to set the equation to: $$0=\\\\sqrt[3]{3x+1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{3}$$"],"dependencies":["ae05e04inverse9a-h2"],"title":"Plugging in $$0$$","text":"What value of $$x$$ makes this equation true?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse9a-h4","type":"hint","dependencies":["ae05e04inverse9a-h3"],"title":"Simplifying the Equation","text":"When we square both sides, we get that $$0=3x+1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse9a-h5","type":"hint","dependencies":["ae05e04inverse9a-h4"],"title":"Simplifying the Equation","text":"Then, we subtract $$1$$ from both sides to get $$-1=3x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse9a-h6","type":"hint","dependencies":["ae05e04inverse9a-h5"],"title":"Simplifying the Equation","text":"Finally, we divide both sides by $$3$$ to get $$x=\\\\frac{-1}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ae05e04inverse9b","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"Y-Intercept","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"ae05e04inverse9b-h1","type":"hint","dependencies":[],"title":"Plugging in $$0$$","text":"The y-intercept lies at the point of a function where the x-value is zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse9b-h2","type":"hint","dependencies":["ae05e04inverse9b-h1"],"title":"Plugging in $$0$$","text":"So, we have to set the equation to: $$f(0)=\\\\sqrt[3]{3x+1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse9b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ae05e04inverse9b-h2"],"title":"Plugging in $$0$$","text":"What is the value of f(0)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse9b-h4","type":"hint","dependencies":["ae05e04inverse9b-h3"],"title":"Simplifying the Equation","text":"When we plug in $$0$$ into the function, we get $$\\\\sqrt[3]{3\\\\times0+1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse9b-h5","type":"hint","dependencies":["ae05e04inverse9b-h4"],"title":"Simplifying the Equation","text":"Since $$3\\\\times0=0$$, we are left with $$\\\\sqrt[3]{1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae05e04inverse9b-h6","type":"hint","dependencies":["ae05e04inverse9b-h5"],"title":"Simplifying the Equation","text":"The cube root of $$1$$ is $$1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae0c19cmodels1","title":"Fitting a Regression Line to a Set of Data","body":"A regression was run to determine whether there is a relationship between the diameter of a tree ( $$x$$ , in inches) and the tree\u2019s age ( $$y$$ , in years). The results of the regression are given below. Use this to predict the age of a tree with diameter $$10$$ inches.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":" 4.3 Fitting Linear Models to Data","courseName":"OpenStax: College Algebra","steps":[{"id":"ae0c19cmodels1a","stepAnswer":["$$61.966$$"],"problemType":"TextBox","stepTitle":"$$y=ax+b$$, $$a=6.301$$, $$b=-1.044$$, $$r=-0.970$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$61.966$$","hints":{"DefaultPathway":[{"id":"ae0c19cmodels1a-h1","type":"hint","dependencies":[],"title":"Correlation Coefficient","text":"$$r$$ > $$0$$ suggests a positive (increasing) relationship. $$r$$ < $$0$$ suggests a negative (decreasing) relationship. 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The opposite is true when $$r$$ is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae0c19cmodels2a-h2","type":"hint","dependencies":["ae0c19cmodels2a-h1"],"title":"Strong and Weak Correlations","text":"When the value of $$r$$ is farther from $$0$$, there is a strong correlation. When the value of $$r$$ is closer to $$0$$, there is a weaker correlation and the data is more scattered.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae0c19cmodels2a-h3","type":"hint","dependencies":["ae0c19cmodels2a-h2"],"title":"Correlation Coefficient","text":"Combine both the sign of the correlation coefficient and the strength of the $$\\\\frac{correlation}{slope}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae0c19cmodels2a-h4","type":"hint","dependencies":["ae0c19cmodels2a-h3"],"title":"Correlation Coefficient","text":"The value of $$r$$ indicates a strong negative correlation coefficient which matches with Graph C.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae0c19cmodels3","title":"Drawing and Interpreting Scatter Plots","body":"Which scatter plot matches the specified correlation?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":" 4.3 Fitting Linear Models to Data","courseName":"OpenStax: College Algebra","steps":[{"id":"ae0c19cmodels3a","stepAnswer":["Graph B"],"problemType":"MultipleChoice","stepTitle":"$$r=-0.39$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Graph A","Graph B","Graph C","Graph D"],"hints":{"DefaultPathway":[{"id":"ae0c19cmodels3a-h1","type":"hint","dependencies":[],"title":"Negative and Positive Correlations","text":"When the value of $$r$$ is negative, it is a negative correlation. i.e the line has a decreasing slope. The opposite is true when $$r$$ is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae0c19cmodels3a-h2","type":"hint","dependencies":["ae0c19cmodels3a-h1"],"title":"Strong and Weak Correlations","text":"When the value of $$r$$ is farther from $$0$$, there is a strong correlation. When the value of $$r$$ is closer to $$0$$, there is a weaker correlation and the data is more scattered.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae0c19cmodels3a-h3","type":"hint","dependencies":["ae0c19cmodels3a-h2"],"title":"Correlation Coefficient","text":"Combine both the sign of the correlation coefficient and the strength of the $$\\\\frac{correlation}{slope}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae0c19cmodels3a-h4","type":"hint","dependencies":["ae0c19cmodels3a-h3"],"title":"Correlation Coefficient","text":"The value of $$r$$ indicates a weak negative correlation coefficient which matches with Graph B.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae0c19cmodels4","title":"Drawing and Interpreting Scatter Plots","body":"Which scatter plot matches the specified correlation?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":" 4.3 Fitting Linear Models to Data","courseName":"OpenStax: College Algebra","steps":[{"id":"ae0c19cmodels4a","stepAnswer":["Graph D"],"problemType":"MultipleChoice","stepTitle":"$$r=0.95$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Graph A","Graph B","Graph C","Graph D"],"hints":{"DefaultPathway":[{"id":"ae0c19cmodels4a-h1","type":"hint","dependencies":[],"title":"Negative and Positive Correlations","text":"When the value of $$r$$ is negative, it is a negative correlation. i.e the line has a decreasing slope. The opposite is true when $$r$$ is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae0c19cmodels4a-h2","type":"hint","dependencies":["ae0c19cmodels4a-h1"],"title":"Strong and Weak Correlations","text":"When the value of $$r$$ is farther from $$0$$, there is a strong correlation. When the value of $$r$$ is closer to $$0$$, there is a weaker correlation and the data is more scattered.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae0c19cmodels4a-h3","type":"hint","dependencies":["ae0c19cmodels4a-h2"],"title":"Correlation Coefficient","text":"Combine both the sign of the correlation coefficient and the strength of the $$\\\\frac{correlation}{slope}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae0c19cmodels4a-h4","type":"hint","dependencies":["ae0c19cmodels4a-h3"],"title":"Correlation Coefficient","text":"The value of $$r$$ indicates a strong positive correlation coefficient which matches with Graph D.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae0c19cmodels5","title":"Drawing and Interpreting Scatter Plots","body":"Which scatter plot matches the specified correlation?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":" 4.3 Fitting Linear Models to Data","courseName":"OpenStax: College Algebra","steps":[{"id":"ae0c19cmodels5a","stepAnswer":["Graph A"],"problemType":"MultipleChoice","stepTitle":"$$r=-0.26$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Graph A","Graph B","Graph C","Graph D"],"hints":{"DefaultPathway":[{"id":"ae0c19cmodels5a-h1","type":"hint","dependencies":[],"title":"Negative and Positive Correlations","text":"When the value of $$r$$ is negative, it is a negative correlation. i.e the line has a decreasing slope. The opposite is true when $$r$$ is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae0c19cmodels5a-h2","type":"hint","dependencies":["ae0c19cmodels5a-h1"],"title":"Strong and Weak Correlations","text":"When the value of $$r$$ is farther from $$0$$, there is a strong correlation. When the value of $$r$$ is closer to $$0$$, there is a weaker correlation and the data is more scattered.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae0c19cmodels5a-h3","type":"hint","dependencies":["ae0c19cmodels5a-h2"],"title":"Correlation Coefficient","text":"Combine both the sign of the correlation coefficient and the strength of the $$\\\\frac{correlation}{slope}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae0c19cmodels5a-h4","type":"hint","dependencies":["ae0c19cmodels5a-h3"],"title":"Correlation Coefficient","text":"The value of $$r$$ indicates a weaker negative correlation coefficient which matches with Graph A.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae0c19cmodels6","title":"Fitting a Regression Line to a Set of Data","body":"A regression was run to determine whether there is a relationship between hours of TV watched per day (x) and number of sit-ups a person can do (y). The results of the regression are given below. Use this to predict the number of sit-ups a person who watches $$11$$ hours of TV can do.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":" 4.3 Fitting Linear Models to Data","courseName":"OpenStax: College Algebra","steps":[{"id":"ae0c19cmodels6a","stepAnswer":["$$17.483$$"],"problemType":"TextBox","stepTitle":"$$y=ax+b$$, $$a=-1.341$$, $$b=32.234$$, $$r=-0.896$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$17.483$$","hints":{"DefaultPathway":[{"id":"ae0c19cmodels6a-h1","type":"hint","dependencies":[],"title":"Correlation Coefficient","text":"$$r$$ > $$0$$ suggests a positive (increasing) relationship. $$r$$ < $$0$$ suggests a negative (decreasing) relationship. In this problem, $$r$$ < $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae0c19cmodels6a-h2","type":"hint","dependencies":["ae0c19cmodels6a-h1"],"title":"Substitution","text":"Substitute the given values for a, $$b$$, and $$x$$ into $$y=ax+b$$ to get the value of $$y$$ when $$x=11$$ hours.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae0c19cmodels6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$17.483$$"],"dependencies":["ae0c19cmodels6a-h2"],"title":"Substitution","text":"What is $$y$$ when $$x=11$$, $$a=-1.341$$ and $$b=32.234$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae0c19cmodels7","title":"Drawing and Interpreting Scatter Plots","body":"Which scatter plot matches the data points?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":" 4.3 Fitting Linear Models to Data","courseName":"OpenStax: College Algebra","steps":[{"id":"ae0c19cmodels7a","stepAnswer":["Graph A"],"problemType":"MultipleChoice","stepTitle":"$$(16,46),(18,50),(20,54),(25,55),(30,62)$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Graph A","Graph B","Graph C","Graph D"],"hints":{"DefaultPathway":[{"id":"ae0c19cmodels7a-h1","type":"hint","dependencies":[],"title":"Plotting Points","text":"Plot the points on the graph using the $$x$$ and $$y$$ values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae0c19cmodels7a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph A"],"dependencies":["ae0c19cmodels7a-h1"],"title":"Plotting Points","text":"Which graph matches the given coordinate points?\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph B","Graph C","Graph D"]}]}}]},{"id":"ae0c19cmodels8","title":"Drawing and Interpreting Scatter Plots","body":"Which scatter plot matches the data points?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":" 4.3 Fitting Linear Models to Data","courseName":"OpenStax: College Algebra","steps":[{"id":"ae0c19cmodels8a","stepAnswer":["Graph C"],"problemType":"MultipleChoice","stepTitle":"$$(1,46),(2,50),(3,59),(4,75),(5,100),(6,136)$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Graph A","Graph B","Graph C","Graph D"],"hints":{"DefaultPathway":[{"id":"ae0c19cmodels8a-h1","type":"hint","dependencies":[],"title":"Plotting Points","text":"Plot the points on the graph using the $$x$$ and $$y$$ values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae0c19cmodels8a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph C"],"dependencies":["ae0c19cmodels8a-h1"],"title":"Plotting Points","text":"Which graph matches the given coordinate points?\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph B","Graph C","Graph D"]}]}}]},{"id":"ae0c19cmodels9","title":"Drawing and Interpreting Scatter Plots","body":"Which scatter plot matches the data points?","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College 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points?\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph B","Graph C","Graph D"]}]}}]},{"id":"ae173b9deriv1","title":"Derivatives as Rates of Change","body":"","variabilization":{},"oer":"https://openstax.org/details/books/physics <OpenStax: Physics>","license":0,"lesson":"3.4 Derivatives as Rates of Change","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ae173b9deriv1a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"If f(3) $$=$$ $$2$$ and f\'(3) $$=$$ $$5$$, estimate $$f(3.2)$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"ae173b9deriv1a-h1","type":"hint","dependencies":[],"title":"Understanding the Problem","text":"One application for derivatives is to estimate an unknown value of a function at a point by using a known value of a function at some given point, f(a), together with its rate of change at the given point, f\'(a). If f(x) is a function defined on an interval [a, a + h], then the amount of change of f(x) over the interval is the change in the $$y$$ values of the function over that interval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv1a-h2","type":"hint","dependencies":["ae173b9deriv1a-h1"],"title":"Solving for $$h$$","text":"The function at some given point, a, is f(a), and its rate of change at the given point is f\'(a). f(a + h) is the value that we wish to estimate. Solve for $$h$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ae173b9deriv1a-h2"],"title":"Determining the X-Value","text":"What is a, the x-value at which the function is defined?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ae173b9deriv1a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":[],"title":"f(a) and f\'(a)","text":"Given that f(a) is the function at point a, and f\'(a) is its rate of change at that point, what is a if f(3) and f\'(3)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"ae173b9deriv1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3.2$$"],"dependencies":["ae173b9deriv1a-h3"],"title":"The Estimated Value","text":"What is a + $$h$$, the x-value that we wish to estimate at?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ae173b9deriv1a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3.2$$"],"dependencies":[],"title":"f(a + h)","text":"f(a + h) is the y-value that we wish to estimate. Given that we are estimating $$f(3.2)$$, what is a + $$h$$, the x-value at which the estimated function is defined?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"ae173b9deriv1a-h5","type":"hint","dependencies":["ae173b9deriv1a-h4"],"title":"Determining the Amount of Change","text":"Now that you have found the values of a and a + $$h$$, use the following equation (a + h) - a to solve for $$h$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv1a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.2$$"],"dependencies":["ae173b9deriv1a-h5"],"title":"Plugging in to Find $$h$$","text":"What is $$h$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ae173b9deriv1a-h6-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.2$$"],"dependencies":[],"title":"Putting it Together","text":"If a $$=$$ $$3$$ and (a + h) $$=$$ $$3.2$$, what is $$h$$ using the equation $$h$$ $$=$$ (a + h) - a?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv1a-h6-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.2$$"],"dependencies":[],"title":"Simplifying $$h$$ in the Equation","text":"$$h$$ $$=$$ (a + h) - a $$=$$ $$3.2$$ - $$3$$. What is $$h$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"ae173b9deriv1a-h7","type":"hint","dependencies":["ae173b9deriv1a-h6"],"title":"Amount of Change Formula","text":"We can now solve for f(a + h) to get the amount of change formula: $$f{\\\\left(a+h\\\\right)}$$ \u2248 f(a) + f\'(a)h. Plug your values into this equation to solve for $$f(3.2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv1a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ae173b9deriv1a-h7"],"title":"Solving for the Estimated Function","text":"What is $$f(3.2)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv1a-h9","type":"hint","dependencies":["ae173b9deriv1a-h8"],"title":"Substituting Into the Amount of Change Equation","text":"Plug a $$=$$ $$3$$ and $$h$$ $$=$$ $$0.2$$ into the amount of change formula to get $$f(3.2)$$ $$=$$ f(3 + $$0.2)$$ \u2248 f(3) + $$f\'(3)(0.2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv1a-h10","type":"hint","dependencies":["ae173b9deriv1a-h9"],"title":"Plugging in Function Values","text":"In the problem, we are told that f(3) $$=$$ $$2$$ and f\'(3) $$=$$ $$5$$. Substituting these values into $$f(3.2)$$ $$=$$ f(3 + $$0.2)$$ \u2248 f(3) + $$f\'(3)(0.2)$$, we get $$2$$ + $$5(0.2)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv1a-h11","type":"hint","dependencies":["ae173b9deriv1a-h10"],"title":"Final Simplified Answer","text":"$$2$$ + $$5(0.2)$$ $$=$$ $$2$$ + $$1$$ $$=$$ $$3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ae173b9deriv10","title":"Derivatives as Rates of Change","body":"","variabilization":{},"oer":"https://openstax.org/details/books/physics <OpenStax: Physics>","license":0,"lesson":"3.4 Derivatives as Rates of Change","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ae173b9deriv10a","stepAnswer":["$$18000$$"],"problemType":"TextBox","stepTitle":"The population of a city is tripling every $$5$$ years. If its current population is 10,000, what will be its approximate population $$2$$ years from now?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$18000$$","hints":{"DefaultPathway":[{"id":"ae173b9deriv10a-h1","type":"hint","dependencies":[],"title":"Defining the Population","text":"Let P(t) be the population (in thousands) $$t$$ years from now.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["ae173b9deriv10a-h1"],"title":"The Current Population","text":"What is P(0)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$30000$$"],"dependencies":["ae173b9deriv10a-h2"],"title":"Tripling the Population","text":"Since the current population is 10,000 and P(t) is the population (in thousands) $$t$$ years from now, we know that P(0) $$=$$ $$10$$. Based on this information, what can we anticipate that the population is if it triples every $$5$$ years?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv10a-h4","type":"hint","dependencies":["ae173b9deriv10a-h3"],"title":"Estimating the Current Growth Rate","text":"The population of the city is tripling every $$5$$ years such that we can anticipate that the population is 30,000. In other words, P(5) $$=$$ $$30$$. Estimate P\'(0), the current growth rate, using P\'(0) \u2248 (P(t_2) - P(t_1))/(t_2 - t_1).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ae173b9deriv10a-h4"],"title":"Solving for P\'(0)","text":"What is P\'(0)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv10a-h6","type":"hint","dependencies":["ae173b9deriv10a-h5"],"title":"Defining $$t_1$$ and $$t_2$$","text":"P\'(0) \u2248 (P(t_2) - P(t_1))/(t_2 - t_1), where $$t_1$$ $$=$$ $$0$$ and $$t_2$$ $$=$$ $$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv10a-h7","type":"hint","dependencies":["ae173b9deriv10a-h6"],"title":"Plugging in the Times","text":"P\'(0) \u2248 (P(5) - P(0))/(5 - 0)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv10a-h8","type":"hint","dependencies":["ae173b9deriv10a-h7"],"title":"Pluggin in the Populations, in Thousands","text":"P\'(0) \u2248 (30 - 10)/(5 - 0)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv10a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ae173b9deriv10a-h8"],"title":"Simplification","text":"P\'(0) \u2248 $$\\\\frac{20}{5}$$. What is P\'(0)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv10a-h10","type":"hint","dependencies":["ae173b9deriv10a-h9"],"title":"Amount of Change Formula","text":"For small enough values of $$h$$, f\'(a) \u2248 f(a + h) - f(a)h. We can then solve for $$f{\\\\left(a+h\\\\right)}$$ to get the amount of change formula: f(a + h) \u2248 f(a) + f\'(a)h. Apply this equation to P(t) to estimate the population $$2$$ years from now.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv10a-h11","type":"hint","dependencies":["ae173b9deriv10a-h10"],"title":"Solving for P(2)","text":"P(2) \u2248 P(0) + (2)P\'(0)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv10a-h12","type":"hint","dependencies":["ae173b9deriv10a-h11"],"title":"Simplification","text":"P(2) \u2248 $$10$$ + 2(4)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv10a-h13","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18$$"],"dependencies":["ae173b9deriv10a-h12"],"title":"Finding P(2)","text":"P(2) \u2248 $$10$$ + $$8$$. What is P(2)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv10a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18000$$"],"dependencies":["ae173b9deriv10a-h13"],"title":"Final Answer","text":"P(2) $$=$$ $$18$$. Remember that P(t) is the population (in thousands) $$t$$ years from now. This means that P(2) is the population (in thousands) $$2$$ years from now. What is the approximate population $$2$$ years from now in thousands?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ae173b9deriv11","title":"Derivatives as Rates of Change","body":"Assume that the number of barbeque dinners that can be sold, $$x$$, can be related to the price charged, $$p$$, by the equation $$p(x)=9-0.03x$$, $$0$$ $$ \\\\leq $$ $$x$$ $$ \\\\leq $$ $$300$$. In this case, the revenue in dollars obtained by selling $$x$$ barbeque dinners is given by $$R(x)=x p\\\\left(x\\\\right)=x \\\\left(9-0.03x\\\\right)=-0.03x^2+9x$$ for $$0$$ $$ \\\\leq $$ $$x$$ $$ \\\\leq $$ $$300$$.","variabilization":{},"oer":"https://openstax.org/details/books/physics <OpenStax: Physics>","license":0,"lesson":"3.4 Derivatives as Rates of Change","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ae173b9deriv11a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"Use the marginal revenue function to estimate the revenue obtained from selling the 101st barbeque dinner.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"ae173b9deriv11a-h1","type":"hint","dependencies":[],"title":"Marginal Revenue Function","text":"MR(x) $$=$$ R\'(x), where R\'(x) is the derivative of R(x).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-0.06x$$ + $$9$$"],"dependencies":["ae173b9deriv11a-h1"],"title":"Solving for R\'(x)","text":"What is R\'(x)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv11a-h3","type":"hint","dependencies":["ae173b9deriv11a-h2"],"title":"Taking the Derivative of R\'(x)","text":"R(x) $$=$$ $$-0.03x^2$$ + $$9x$$\\\\nR\'(x) $$=$$ $$2\\\\left(-0.03\\\\right) x^{2-1}$$ + $$9$$\\\\nR\'(x) $$=$$ $$-0.06x$$ + $$9$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv11a-h4","type":"hint","dependencies":["ae173b9deriv11a-h3"],"title":"Estimating the Revenue of the 101st Dinner","text":"Now that we know that MR(x) $$=$$ R\'(x) $$=$$ $$-0.06x$$ + $$9$$, use R\'(100) to approximate R(101) - R(100), the revenue obtained from the sale of the 101st dinner.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ae173b9deriv11a-h4"],"title":"Solving for R\'(100)","text":"What is R\'(100)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv11a-h6","type":"hint","dependencies":["ae173b9deriv11a-h5"],"title":"Plugging in X","text":"Plug $$x$$ $$=$$ $$100$$ into R\'(x) $$=$$ $$-0.06x$$ + $$9$$ to solve for R\'(100).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv11a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ae173b9deriv11a-h6"],"title":"Final Calculations","text":"R\'(100) $$=$$ $$-0.06(100)$$ + $$9$$ $$=$$ $$-6$$ + $$9$$. What is R\'(100)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv11a-h8","type":"hint","dependencies":["ae173b9deriv11a-h7"],"title":"Interpretation of the Answer","text":"Since R\u2032(100) $$=$$ $$3$$, the revenue obtained from the sale of the 101st dinner is approximately $3.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ae173b9deriv12","title":"Derivatives as Rates of Change","body":"Assume that the number of barbeque dinners that can be sold, $$x$$, can be related to the price charged, $$p$$, by the equation $$p(x)=9-0.03x$$, $$0$$ $$ \\\\leq $$ $$x$$ $$ \\\\leq $$ $$300$$. In this case, the revenue in dollars obtained by selling $$x$$ barbeque dinners is given by $$R(x)=x p\\\\left(x\\\\right)=x \\\\left(9-0.03x\\\\right)=-0.03x^2+9x$$ for $$0$$ $$ \\\\leq $$ $$x$$ $$ \\\\leq $$ $$300$$.","variabilization":{},"oer":"https://openstax.org/details/books/physics <OpenStax: Physics>","license":0,"lesson":"3.4 Derivatives as Rates of Change","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ae173b9deriv12a","stepAnswer":["$$2.97$$"],"problemType":"TextBox","stepTitle":"The estimated revenue obtained from selling the 101st barbeque dinner is $3. What is the actual revenue? Compare the estimated revenue to the actual revenue obtained from the sale of this dinner.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2.97$$","hints":{"DefaultPathway":[{"id":"ae173b9deriv12a-h1","type":"hint","dependencies":[],"title":"Actual Revenue","text":"The actual revenue obtained from the sale of the 101st dinner is R(101) - R(100).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.97$$"],"dependencies":["ae173b9deriv12a-h1"],"title":"Solving for the Actural Revenue","text":"What is R(101) - R(100)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv12a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$602.97$$"],"dependencies":["ae173b9deriv12a-h2"],"title":"R(101)","text":"What is R(101)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv12a-h4","type":"hint","dependencies":["ae173b9deriv12a-h3"],"title":"Substituting $$x$$ $$=$$ $$101$$","text":"Plug $$x$$ $$=$$ $$101$$ into R(x) $$=$$ $$-0.03x^2$$ + $$9x$$ to solve for R(101).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv12a-h5","type":"hint","dependencies":["ae173b9deriv12a-h4"],"title":"Calculating R(101)","text":"R(x) $$=$$ $$-0.03x^2$$ + $$9x$$\\\\nR(101) $$=$$ $$-\\\\left({\\\\operatorname{0.03}\\\\left(101\\\\right)}^2\\\\right)$$ + 9(101)\\\\nR(101) $$=$$ $$-306.03$$ + $$909$$\\\\nR(101) $$=$$ $$602.97$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv12a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$600$$"],"dependencies":["ae173b9deriv12a-h5"],"title":"R(100)","text":"What is R(100)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv12a-h7","type":"hint","dependencies":["ae173b9deriv12a-h6"],"title":"Substituting $$x$$ $$=$$ $$100$$","text":"Plug $$x$$ $$=$$ $$100$$ into R(x) $$=$$ $$-0.03x^2$$ + $$9x$$ to solve for R(100).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv12a-h8","type":"hint","dependencies":["ae173b9deriv12a-h7"],"title":"Calculating R(100)","text":"R(x) $$=$$ $$-0.03x^2$$ + $$9x$$\\\\nR(100) $$=$$ $$-\\\\left({\\\\operatorname{0.03}\\\\left(100\\\\right)}^2\\\\right)$$ + 9(100)\\\\nR(100) $$=$$ $$-300$$ + $$900$$\\\\nR(100) $$=$$ $$600$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv12a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.97$$"],"dependencies":["ae173b9deriv12a-h8"],"title":"Subtracting the Calculations","text":"Now that you have determined that R(101) $$=$$ $$602.97$$ and R(100) $$=$$ $$600$$, what is R(101) - R(100)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv12a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.97$$"],"dependencies":["ae173b9deriv12a-h9"],"title":"Final Calculations","text":"R(101) - R(100) $$=$$ $$602.97$$ - $$600$$. What is the actual revenue obtained from the sale of the 101st dinner?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv12a-h11","type":"hint","dependencies":["ae173b9deriv12a-h10"],"title":"Interpretation of the Answer","text":"Since the estimated revenue obtained from the sale of the 101st dinner is approximately $3, and the actual revenue was $$\\\\$2.97$$, the marginal revenue is a fairly good estimate in this case and has the advantage of being easy to compute.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ae173b9deriv13","title":"Derivatives as Rates of Change","body":"The given function represents the position of a particle traveling along a horizontal line: s(t) $$=$$ $$2t^3$$ - $$15t^2$$ + $$36t$$ - $$10$$.","variabilization":{},"oer":"https://openstax.org/details/books/physics <OpenStax: Physics>","license":0,"lesson":"3.4 Derivatives as Rates of Change","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ae173b9deriv13a","stepAnswer":["$$6t^2$$ - $$30t$$ + $$36$$"],"problemType":"TextBox","stepTitle":"Find the velocity function.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6t^2$$ - $$30t$$ + $$36$$","hints":{"DefaultPathway":[{"id":"ae173b9deriv13a-h1","type":"hint","dependencies":[],"title":"Understanding the Problem","text":"One use for the derivative is to analyze motion along a line. Let s(t) be a function giving the position of an object at time $$t$$. The velocity of the object at time $$t$$ is given by v(t) $$=$$ s\'(t), the derivative of s(t). Solve for s\'(t).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6t^2$$ - $$30t$$ + $$36$$"],"dependencies":["ae173b9deriv13a-h1"],"title":"Solving for s\'(t)","text":"What is s\'(t)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv13a-h3","type":"hint","dependencies":["ae173b9deriv13a-h2"],"title":"Taking the Derivative of s(t)","text":"v(t) $$=$$ s\'(t)\\\\ns(t) $$=$$ $$2t^3$$ - $$15t^2$$ + $$36t$$ - $$10$$\\\\ns\'(t) $$=$$ $$3\\\\times2 t^{3-1}$$ - $$2\\\\times15 t^{2-1}$$ + $$36$$\\\\ns\'(t) $$=$$ $$6t^2$$ - $$30t$$ + $$36$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ae173b9deriv14","title":"Derivatives as Rates of Change","body":"The given function represents the position of a particle traveling along a horizontal line: s(t) $$=$$ $$2t^3$$ - $$15t^2$$ + $$36t$$ - $$10$$.","variabilization":{},"oer":"https://openstax.org/details/books/physics <OpenStax: Physics>","license":0,"lesson":"3.4 Derivatives as Rates of Change","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ae173b9deriv14a","stepAnswer":["$$12t$$ - $$30$$"],"problemType":"TextBox","stepTitle":"Find the acceleration function.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12t$$ - $$30$$","hints":{"DefaultPathway":[{"id":"ae173b9deriv14a-h1","type":"hint","dependencies":[],"title":"Acceleration Equation","text":"One use for the derivative is to analyze motion along a line. Let s(t) be a function giving the position of an object at time $$t$$. The acceleration of the object at time $$t$$ is given by a(t) $$=$$ v\'(t) $$=$$ s\'\'(t), where a(t) is the acceleration, v\'(t) is the derivative of the velocity, and s\'\'(t) is the double derivative of s(t). First, solve for s\'(t) or v(t).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6t^2$$ - $$30t$$ + $$36$$"],"dependencies":["ae173b9deriv14a-h1"],"title":"First Derivative: s\'(t)","text":"First, what is s\'(t)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv14a-h3","type":"hint","dependencies":["ae173b9deriv14a-h2"],"title":"Taking the First Derivative of s(t)","text":"s(t) $$=$$ $$2t^3$$ - $$15t^2$$ + $$36t$$ - $$10$$\\\\ns\'(t) $$=$$ $$3\\\\times2 t^{3-1}$$ - $$2\\\\times15 t^{2-1}$$ + $$36$$\\\\ns\'(t) $$=$$ $$6t^2$$ - $$30t$$ + $$36$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv14a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12t$$ - $$30$$"],"dependencies":["ae173b9deriv14a-h3"],"title":"Second Derivative: s\'\'(t)","text":"Now that you have calculated s\'(t), what is s\'\'(t)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv14a-h5","type":"hint","dependencies":["ae173b9deriv14a-h4"],"title":"Taking the Second Derivative of s(t)","text":"s\'(t) $$=$$ $$6t^2$$ - $$30t$$ + $$36$$\\\\ns\'\'(t) $$=$$ $$2\\\\times6 t^{2-1}$$ - $$30$$\\\\ns\'\'(t) $$=$$ $$12t$$ - $$30$$\\\\nThus, a(t) $$=$$ $$12t$$ - $$30$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ae173b9deriv15","title":"Derivatives as Rates of Change","body":"A model rocket is fired vertically upward from the ground. The distance s in feet that the rocket travels from the ground after $$t$$ seconds is given by s(t) $$=$$ $$-16t^2$$ + $$560t$$.","variabilization":{},"oer":"https://openstax.org/details/books/physics <OpenStax: Physics>","license":0,"lesson":"3.4 Derivatives as Rates of Change","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ae173b9deriv15a","stepAnswer":["$$464$$"],"problemType":"TextBox","stepTitle":"Find the velocity $$\\\\frac{ft}{s^2}$$ of the rocket $$3$$ seconds after being fired.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$464$$","hints":{"DefaultPathway":[{"id":"ae173b9deriv15a-h1","type":"hint","dependencies":[],"title":"Understanding the Problem","text":"One use for the derivative is to analyze motion along a line. Let s(t) be a function giving the position of an object at time $$t$$. The velocity of the object at time $$t$$ is given by v(t) $$=$$ s\'(t), the derivative of s(t). First, solve for s\'(t).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$464$$"],"dependencies":["ae173b9deriv15a-h1"],"title":"Solving for s\'(t)","text":"What is s\'(t)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv15a-h3","type":"hint","dependencies":["ae173b9deriv15a-h2"],"title":"Taking the Derivative of s(t)","text":"v(t) $$=$$ s\'(t)\\\\ns(t) $$=$$ $$-16t^2$$ + $$560t$$\\\\ns\'(t) $$=$$ $$2\\\\left(-16\\\\right) t^{2-1}$$ + $$560$$\\\\ns\'(t) $$=$$ $$-32t$$ + $$560$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv15a-h4","type":"hint","dependencies":["ae173b9deriv15a-h3"],"title":"Substituting Time","text":"Plug $$t$$ $$=$$ $$3$$ into s\'(t) $$=$$ $$-32t$$ + $$560$$ to solve for the velocity (ft/ s**2) of the rocket $$3$$ seconds after being fired.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv15a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$464$$"],"dependencies":["ae173b9deriv15a-h4"],"title":"Final Calculations","text":"s\'(3) $$=$$ $$-32(3)$$ + $$560$$ $$=$$ $$-96$$ + $$560$$. What is the velocity (ft/ s**2) of the rocket $$3$$ seconds after being fired?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ae173b9deriv16","title":"Derivatives as Rates of Change","body":"A model rocket is fired vertically upward from the ground. The distance s in feet that the rocket travels from the ground after $$t$$ seconds is given by s(t) $$=$$ $$-16t^2$$ + $$560t$$.","variabilization":{},"oer":"https://openstax.org/details/books/physics <OpenStax: Physics>","license":0,"lesson":"3.4 Derivatives as Rates of Change","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ae173b9deriv16a","stepAnswer":["$$-32$$"],"problemType":"TextBox","stepTitle":"Find the acceleration $$\\\\frac{ft}{s^2}$$ of the rocket $$3$$ seconds after being fired.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-32$$","hints":{"DefaultPathway":[{"id":"ae173b9deriv16a-h1","type":"hint","dependencies":[],"title":"Understanding the Problem","text":"The acceleration of the object at time $$t$$ is given by a(t) $$=$$ v\'(t) $$=$$ s\'\'(t), where a(t) is the acceleration, v\'(t) is the derivative of the velocity, and s\'\'(t) is the double derivative of s(t). First, solve for s\'(t) or v(t).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-32t$$ + $$560$$"],"dependencies":["ae173b9deriv16a-h1"],"title":"First Derivative: s\'(t)","text":"What is s\'(t)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv16a-h3","type":"hint","dependencies":["ae173b9deriv16a-h2"],"title":"Taking the First Derivative of s(t)","text":"v(t) $$=$$ s\'(t)\\\\ns(t) $$=$$ $$-16t^2$$ + $$560t$$\\\\ns\'(t) $$=$$ $$2\\\\left(-16\\\\right) t^{2-1}$$ + $$560$$\\\\ns\'(t) $$=$$ $$-32t$$ + $$560$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv16a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12t$$ - $$30$$"],"dependencies":["ae173b9deriv16a-h3"],"title":"Second Derivative: s\'\'(t)","text":"Now that you have calculated s\'(t), what is s\'\'(t)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv16a-h5","type":"hint","dependencies":["ae173b9deriv16a-h4"],"title":"Taking the Second Derivative of s(t)","text":"s\'(t) $$=$$ $$-32t$$ + $$560$$\\\\ns\'\'(t) $$=$$ $$1\\\\left(-32\\\\right) t^{1-1}$$\\\\ns\'\'(t) $$=$$ $$-32$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv16a-h6","type":"hint","dependencies":["ae173b9deriv16a-h5"],"title":"Substituting Time","text":"Normally, we would plug $$t$$ $$=$$ $$3$$ into a(t) $$=$$ $$-32$$, but because there is no $$t$$ value on the right side of the equation, this means that at any given time, the acceleration of the rocket is $$-32$$ $$\\\\frac{ft}{s^2}$$. Thus, the acceleration of the rocket $$3$$ seconds after being fired is $$-32$$ ft/ $$s^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ae173b9deriv17","title":"Derivatives as Rates of Change","body":"The position function s(t) $$=$$ $$t^2$$ - $$3t$$ - $$4$$ represents the position of the back of a car backing out of a driveway and then driving in a straight line, where s is in feet and $$t$$ is in seconds. In this case, s(t) $$=$$ $$0$$ represents the time at which the back of the car is at the garage door, so s(0) $$=$$ $$-4$$ is the starting position of the car, $$4$$ feet inside the garage.","variabilization":{},"oer":"https://openstax.org/details/books/physics <OpenStax: Physics>","license":0,"lesson":"3.4 Derivatives as Rates of Change","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ae173b9deriv17a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"Determine the velocity $$\\\\frac{ft}{s}$$ of the car when s(t) $$=$$ $$0$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"ae173b9deriv17a-h1","type":"hint","dependencies":[],"title":"Understanding the Problem","text":"One use for the derivative is to analyze motion along a line. Let s(t) be a function giving the position of an object at time $$t$$. The velocity of the object at time $$t$$ is given by v(t) $$=$$ s\'(t), the derivative of s(t). First, set s(t) to $$0$$ and solve for $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv17a-h2","type":"hint","dependencies":["ae173b9deriv17a-h1"],"title":"Setting s(t) to $$0$$","text":"The problem asks us to determine the velocity $$\\\\frac{ft}{s}$$ of the car when s(t) $$=$$ $$0$$. Solve for s(t) $$=$$ $$t^2$$ - $$3t$$ - $$4$$ $$=$$ $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv17a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ae173b9deriv17a-h2"],"title":"Solving for $$t$$","text":"What is $$t^2-3t-4=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv17a-h4","type":"hint","dependencies":["ae173b9deriv17a-h3"],"title":"Time When s(t) $$=$$ $$0$$","text":"$$t^2-3t-4=0$$\\\\n$$t=\\\\frac{3\\\\pm \\\\sqrt{3^2+4\\\\times4}}{2}$$\\\\n$$t=\\\\frac{3\\\\pm 5}{2}$$\\\\n$$t=\\\\frac{3+5}{2}=\\\\frac{8}{2}=4$$ and $$t=\\\\frac{3-5}{2}=\\\\frac{-2}{2}=-1$$\\\\nBecause time can only be positive, the time in which s(t) $$=$$ $$0$$ is $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv17a-h5","type":"hint","dependencies":["ae173b9deriv17a-h4"],"title":"Velocity Function","text":"Now that we have determined that $$t$$ $$=$$ $$4$$ when s(t) $$=$$ $$0$$, we can plug the time into the velocity function given by v(t) $$=$$ s\'(t), the derivative of s(t). Solve for the velocity function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv17a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2t$$ - $$3$$"],"dependencies":["ae173b9deriv17a-h5"],"title":"Calculating v(t)","text":"What is s\'(t)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv17a-h7","type":"hint","dependencies":["ae173b9deriv17a-h6"],"title":"Taking the Derivative of s(t)","text":"v(t) $$=$$ s\'(t)\\\\ns(t) $$=$$ $$t^2$$ - $$3t$$ - $$4$$\\\\ns\'(t) $$=$$ $$2t^{2-1}$$ - $$3$$\\\\ns\'(t) $$=$$ $$2t$$ - $$3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv17a-h8","type":"hint","dependencies":["ae173b9deriv17a-h7"],"title":"Plugging in Time","text":"Plug $$t$$ $$=$$ $$4$$ into s\'(t) $$=$$ $$2t$$ - $$3$$ to determine the velocity $$\\\\frac{ft}{s}$$ of the car when s(t) $$=$$ $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv17a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["ae173b9deriv17a-h8"],"title":"Final Calculations","text":"s\'(4) $$=$$ 2(4) - $$3$$ $$=$$ $$8$$ - $$3$$. What is the velocity $$\\\\frac{ft}{s}$$ of the car when s(t) $$=$$ 0?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ae173b9deriv18","title":"Derivatives as Rates of Change","body":"The position function s(t) $$=$$ $$t^2$$ - $$3t$$ - $$4$$ represents the position of the back of a car backing out of a driveway and then driving in a straight line, where s is in feet and $$t$$ is in seconds. In this case, s(t) $$=$$ $$0$$ represents the time at which the back of the car is at the garage door, so s(0) $$=$$ $$-4$$ is the starting position of the car, $$4$$ feet inside the garage.","variabilization":{},"oer":"https://openstax.org/details/books/physics <OpenStax: Physics>","license":0,"lesson":"3.4 Derivatives as Rates of Change","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ae173b9deriv18a","stepAnswer":["$$9$$"],"problemType":"TextBox","stepTitle":"Determine the velocity $$\\\\frac{ft}{s}$$ of the car when s(t) $$=$$ $$14$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9$$","hints":{"DefaultPathway":[{"id":"ae173b9deriv18a-h1","type":"hint","dependencies":[],"title":"Understanding the Problem","text":"One use for the derivative is to analyze motion along a line. Let s(t) be a function giving the position of an object at time $$t$$. The velocity of the object at time $$t$$ is given by v(t) $$=$$ s\'(t), the derivative of s(t). First, set s(t) to $$14$$ and solve for $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv18a-h2","type":"hint","dependencies":["ae173b9deriv18a-h1"],"title":"Setting s(t) to $$14$$","text":"The problem asks us to determine the velocity $$\\\\frac{ft}{s}$$ of the car when s(t) $$=$$ $$14$$. Solve for s(t) $$=$$ $$t^2$$ - $$3t$$ - $$4$$ $$=$$ $$14$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv18a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["ae173b9deriv18a-h2"],"title":"Solving for $$t$$","text":"What is $$t^2$$ - $$3t$$ - $$4$$ $$=$$ 14?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv18a-h4","type":"hint","dependencies":["ae173b9deriv18a-h3"],"title":"Time When s(t) $$=$$ $$14$$","text":"$$t^2-3t-4=14$$\\\\n$$t=\\\\frac{3\\\\pm \\\\sqrt{3^2+4\\\\times8}}{2}$$\\\\n$$t=\\\\frac{3\\\\pm 9}{2}$$\\\\n$$t=\\\\frac{3+9}{2}=\\\\frac{12}{2}=6$$ and $$t=\\\\frac{3-9}{2}=\\\\frac{-6}{2}=-3$$\\\\nBecause time can only be positive, the time in which $$s(t)=14$$ is $$6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv18a-h5","type":"hint","dependencies":["ae173b9deriv18a-h4"],"title":"Velocity Function","text":"Now that we have determined that $$t$$ $$=$$ $$6$$ when s(t) $$=$$ $$14$$, we can plug the time into the velocity function given by v(t) $$=$$ s\'(t), the derivative of s(t). Solve for the velocity function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv18a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2t$$ - $$3$$"],"dependencies":["ae173b9deriv18a-h5"],"title":"Calculating v(t)","text":"What is s\'(t)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv18a-h7","type":"hint","dependencies":["ae173b9deriv18a-h6"],"title":"Taking the Derivative of s(t)","text":"v(t) $$=$$ s\'(t)\\\\ns(t) $$=$$ $$t^2$$ - $$3t$$ - $$4$$\\\\ns\'(t) $$=$$ $$2t^{2-1}$$ - $$3$$\\\\ns\'(t) $$=$$ $$2t$$ - $$3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv18a-h8","type":"hint","dependencies":["ae173b9deriv18a-h7"],"title":"Plugging in Time","text":"Plug $$t$$ $$=$$ $$6$$ into s\'(t) $$=$$ $$2t$$ - $$3$$ to determine the velocity $$\\\\frac{ft}{s}$$ of the car when s(t) $$=$$ $$14$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv18a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["ae173b9deriv18a-h8"],"title":"Final Calculations","text":"s\'(6) $$=$$ 2(6) - $$3$$ $$=$$ $$12$$ - $$3$$. What is the velocity $$\\\\frac{ft}{s}$$ of the car when s(t) $$=$$ 14?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ae173b9deriv19","title":"Derivatives as Rates of Change","body":"A potato is launched vertically upward with an initial velocity of $$100$$ $$\\\\frac{ft}{s}$$ from a potato gun at the top of an 85-foot-tall building. The distance in feet that the potato travels from the ground after $$t$$ seconds is given by s(t) $$=$$ $$-16t^2$$ + $$100t$$ + $$85$$.","variabilization":{},"oer":"https://openstax.org/details/books/physics <OpenStax: Physics>","license":0,"lesson":"3.4 Derivatives as Rates of Change","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ae173b9deriv19a","stepAnswer":["$$84$$"],"problemType":"TextBox","stepTitle":"Find the velocity $$\\\\frac{ft}{s}$$ of the potato after $$0.5$$ s.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$84$$","hints":{"DefaultPathway":[{"id":"ae173b9deriv19a-h1","type":"hint","dependencies":[],"title":"Understanding the Problem","text":"One use for the derivative is to analyze motion along a line. Let s(t) be a function giving the position of an object at time $$t$$. The velocity of the object at time $$t$$ is given by v(t) $$=$$ s\'(t), the derivative of s(t). Solve for v(t).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-32t$$ + $$100$$"],"dependencies":["ae173b9deriv19a-h1"],"title":"Solving for v(t)","text":"What is s\'(t)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv19a-h3","type":"hint","dependencies":["ae173b9deriv19a-h2"],"title":"Taking the Derivative of s(t)","text":"s(t) $$=$$ $$-16t^2$$ + $$100t$$ + $$85$$\\\\ns\'(t) $$=$$ $$2\\\\left(-16\\\\right) t^{2-1}$$ + $$100$$\\\\ns\'(t) $$=$$ $$-32t$$ + $$100$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv19a-h4","type":"hint","dependencies":["ae173b9deriv19a-h3"],"title":"Plugging in Time","text":"Plug $$t$$ $$=$$ $$0.5$$ into v(t) $$=$$ $$-32t$$ + $$100$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv19a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$84$$"],"dependencies":["ae173b9deriv19a-h4"],"title":"Solving for $$v(0.5)$$","text":"What is $$v(0.5)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv19a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$84$$"],"dependencies":["ae173b9deriv19a-h5"],"title":"Calculating $$v(0.5)$$","text":"$$v(0.5)$$ $$=$$ $$-32(0.5)$$ + $$100$$ $$=$$ $$-16$$ + $$100$$. What is $$v(0.5)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ae173b9deriv2","title":"Derivatives as Rates of Change","body":"A ball is dropped from a height of $$64$$ feet. Its height above ground (in feet) $$t$$ seconds later is given by s(t) $$=$$ $$-16t^2$$ + $$64$$.","variabilization":{},"oer":"https://openstax.org/details/books/physics <OpenStax: Physics>","license":0,"lesson":"3.4 Derivatives as Rates of Change","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ae173b9deriv2a","stepAnswer":["$$-64$$"],"problemType":"TextBox","stepTitle":"What is the instantaneous velocity $$\\\\frac{ft}{s}$$ of the ball when it hits the ground?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-64$$","hints":{"DefaultPathway":[{"id":"ae173b9deriv2a-h1","type":"hint","dependencies":[],"title":"Understanding the Problem","text":"One use for the derivative is to analyze motion along a line. Let s(t) be a function giving the position of an object at time $$t$$. The instantaneous velocity of the object at time $$t$$ is given by v(t) $$=$$ s\'(t), the derivative of s(t).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv2a-h2","type":"hint","dependencies":["ae173b9deriv2a-h1"],"title":"Solving for Time When s(t) $$=$$ $$0$$","text":"The first step is determining how long it takes the ball to reach the ground. To do this, set s(t) $$=$$ $$0$$ and solve for $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ae173b9deriv2a-h2"],"title":"Determining $$t$$","text":"What is $$t$$, the time (in seconds) that it takes for the ball to reach the ground?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ae173b9deriv2a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Step by Step to Solving for $$t$$","text":"s(t) $$=$$ $$-16t^2$$ + $$64$$\\\\nSet s(t) to 0: $$-16t^2$$ + $$64$$ $$=$$ $$0$$\\\\nSubtract $$64$$ from both sides: $$-16t^2$$ $$=$$ $$-64$$\\\\nDivide $$-16$$ from both sides: $$\\\\frac{\\\\left(-16t^2\\\\right)}{-16}$$ $$=$$ $$\\\\frac{-64}{-16}$$ => $$t^2$$ $$=$$ $$4$$\\\\nTake the square root of both sides: $$\\\\sqrt{t^2}$$ $$=$$ $$\\\\sqrt{4}$$\\\\nWhat is $$t$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"ae173b9deriv2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-64$$"],"dependencies":["ae173b9deriv2a-h3"],"title":"Solving for the Instantaneous Velocity","text":"Now that we have determined that $$t$$ $$=$$ $$2$$ when the ball hits the ground, we can solve for the instantaneous velocity of the ball at v(2). If v(t) $$=$$ s\'(t), what is v(2)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv2a-h5","type":"hint","dependencies":["ae173b9deriv2a-h4"],"title":"Taking the Derivative","text":"Given s(t) $$=$$ $$-16t^2$$ + $$64$$, find s\'(t) by taking the derivative of s(t).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv2a-h6","type":"hint","dependencies":["ae173b9deriv2a-h5"],"title":"Solving for s\'(t)","text":"s(t) $$=$$ $$-16t^2$$ + $$64$$\\\\ns\'(t) $$=$$ 2(-16)t**(2-1) + $$0$$\\\\ns\'(t) $$=$$ $$-32t$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv2a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-64$$"],"dependencies":["ae173b9deriv2a-h6"],"title":"Finding v(2)","text":"Given that s\'(t) $$=$$ $$-32t$$ and v(t) $$=$$ s\'(t), what is v(2)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ae173b9deriv2a-h7-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-64$$"],"dependencies":[],"title":"Plugging Into the Equation","text":"Plug $$t$$ $$=$$ $$2$$ into s\'(t) $$=$$ $$-32t$$ to solve for v(2). What answer do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv2a-h7-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-64$$"],"dependencies":[],"title":"Steps to Obtaining the Answer","text":"v(t) $$=$$ s\'(t) $$=$$ $$-32t$$\\\\nv(2) $$=$$ s\'(2) $$=$$ $$-32(2)$$\\\\nWhat is v(2), the instantaneous velocity $$\\\\frac{ft}{s}$$ of the ball when it hits the ground?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}]}}]},{"id":"ae173b9deriv20","title":"Derivatives as Rates of Change","body":"A potato is launched vertically upward with an initial velocity of $$100$$ $$\\\\frac{ft}{s}$$ from a potato gun at the top of an 85-foot-tall building. The distance in feet that the potato travels from the ground after $$t$$ seconds is given by s(t) $$=$$ $$-16t^2$$ + $$100t$$ + $$85$$.","variabilization":{},"oer":"https://openstax.org/details/books/physics <OpenStax: Physics>","license":0,"lesson":"3.4 Derivatives as Rates of Change","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ae173b9deriv20a","stepAnswer":["$$-84$$"],"problemType":"TextBox","stepTitle":"Find the velocity $$\\\\frac{ft}{s}$$ of the potato after $$5.75$$ s.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-84$$","hints":{"DefaultPathway":[{"id":"ae173b9deriv20a-h1","type":"hint","dependencies":[],"title":"Understanding the Problem","text":"One use for the derivative is to analyze motion along a line. Let s(t) be a function giving the position of an object at time $$t$$. The velocity of the object at time $$t$$ is given by v(t) $$=$$ s\'(t), the derivative of s(t). Solve for v(t).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-32t$$ + $$100$$"],"dependencies":["ae173b9deriv20a-h1"],"title":"Solving for v(t)","text":"What is s\'(t)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv20a-h3","type":"hint","dependencies":["ae173b9deriv20a-h2"],"title":"Taking the Derivative of s(t)","text":"s(t) $$=$$ $$-16t^2$$ + $$100t$$ + $$85$$\\\\ns\'(t) $$=$$ $$2\\\\left(-16\\\\right) t^{2-1}$$ + $$100$$\\\\ns\'(t) $$=$$ $$-32t$$ + $$100$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv20a-h4","type":"hint","dependencies":["ae173b9deriv20a-h3"],"title":"Plugging in Time","text":"Plug $$t$$ $$=$$ $$5.75$$ into v(t) $$=$$ $$-32t$$ + $$100$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv20a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-84$$"],"dependencies":["ae173b9deriv20a-h4"],"title":"Solving for $$v(5.75)$$","text":"What is $$v(5.75)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ae173b9deriv3","title":"Derivatives as Rates of Change","body":"A ball is dropped from a height of $$64$$ feet. Its height above ground (in feet) $$t$$ seconds later is given by s(t) $$=$$ $$-16t^2$$ + $$64$$.","variabilization":{},"oer":"https://openstax.org/details/books/physics <OpenStax: Physics>","license":0,"lesson":"3.4 Derivatives as Rates of Change","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ae173b9deriv3a","stepAnswer":["$$-64$$"],"problemType":"TextBox","stepTitle":"What is the average velocity $$\\\\frac{ft}{s}$$ during its fall?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-64$$","hints":{"DefaultPathway":[{"id":"ae173b9deriv3a-h1","type":"hint","dependencies":[],"title":"Average Velocity Formula","text":"One use for the derivative is to analyze motion along a line. Let s(t) be a function giving the position of an object at time $$t$$. The average velocity of the object is given by $$v_{ave}$$ $$=$$ $$\\\\frac{\\\\Delta x}{\\\\Delta t}$$, where \u0394x is the displacement (total distance travelled) and \u0394t is the change in time.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv3a-h2","type":"hint","dependencies":["ae173b9deriv3a-h1"],"title":"Time When s(t) $$=$$ $$0$$","text":"The first step is determining how long it takes the ball to reach the ground. To do this, set s(t) $$=$$ $$0$$ and solve for $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["ae173b9deriv3a-h2"],"title":"Determining $$t$$","text":"What is $$t$$, the time (in seconds) that it takes for the ball to reach the ground?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ae173b9deriv3a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Step by Step for Solving $$t$$","text":"s(t) $$=$$ $$-16t^2$$ + $$64$$\\\\nSet s(t) to 0: $$-16t^2$$ + $$64$$ $$=$$ $$0$$\\\\nSubtract $$64$$ from both sides: $$-16t^2$$ $$=$$ $$-64$$\\\\nDivide $$-16$$ from both sides: $$\\\\frac{\\\\left(-16t^2\\\\right)}{-16}$$ $$=$$ $$\\\\frac{-64}{-16}$$ => $$t^2$$ $$=$$ $$4$$\\\\nTake the square root of both sides: $$\\\\sqrt{t^2}$$ $$=$$ $$\\\\sqrt{4}$$\\\\nWhat is $$t$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"ae173b9deriv3a-h4","type":"hint","dependencies":["ae173b9deriv3a-h3"],"title":"Solving for \u0394x","text":"Since we have determined that $$t$$ $$=$$ $$2$$ when the ball hits the ground, we know that \u0394t $$=$$ the end time - the start time $$=$$ $$2$$ - $$0$$ $$=$$ $$2$$. Now, we can determine \u0394x, the displacement, by solving for s(t $$=$$ $$2)-s(t$$ $$=$$ 0).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-64$$"],"dependencies":["ae173b9deriv3a-h4"],"title":"Finding the Total Distance Travelled","text":"What is \u0394x, the displacement?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv3a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["ae173b9deriv3a-h5"],"title":"Solving for s(t)","text":"The time it takes for the ball to reach the ground is $$t$$ $$=$$ $$2$$ seconds. What is s(2)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ae173b9deriv3a-h6-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":[],"title":"Determining s(2)","text":"s(2) $$=$$ $$-\\\\operatorname{16}\\\\left(2^2\\\\right)$$ + $$64$$ $$=$$ $$-16(4)$$ + $$64$$ $$=$$ $$-64$$ + $$64$$. What is s(2)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"ae173b9deriv3a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$64$$"],"dependencies":["ae173b9deriv3a-h6"],"title":"Solving for s(t)","text":"The initial time is $$t$$ $$=$$ $$0$$ seconds. What is s(0)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"ae173b9deriv3a-h7-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$64$$"],"dependencies":[],"title":"Determining s(0)","text":"s(0) $$=$$ $$-\\\\operatorname{16}\\\\left(0^2\\\\right)$$ + $$64$$ $$=$$ $$-16(0)$$ + $$64$$ $$=$$ $$0$$ + $$64$$. What is s(0)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"ae173b9deriv3a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-64$$"],"dependencies":["ae173b9deriv3a-h7"],"title":"Plugging in Values to Find \u0394x","text":"Given that s(2) $$=$$ $$0$$ and s(0) $$=$$ $$64$$, what is \u0394x, the displacement?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv3a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-64$$"],"dependencies":["ae173b9deriv3a-h8"],"title":"The Displacement","text":"\u0394x $$=$$ s(2) - s(0). What is $$0$$ - 64?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv3a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-32$$"],"dependencies":["ae173b9deriv3a-h9"],"title":"Solving for the Average Velocity","text":"$$v_{ave}$$ $$=$$ $$\\\\frac{\\\\Delta x}{\\\\Delta t}$$. What is the average velocity $$\\\\frac{ft}{s}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv3a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-32$$"],"dependencies":["ae173b9deriv3a-h10"],"title":"The Average Velocity","text":"$$v_{ave}$$ $$=$$ $$\\\\frac{\\\\Delta x}{\\\\Delta t}$$ $$=$$ $$\\\\frac{-64}{2}$$ $$=$$ ?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ae173b9deriv4","title":"Derivatives as Rates of Change","body":"A particle moves along a coordinate axis in the positive direction to the right. Its position at time $$t$$ is given by s(t) $$=$$ $$t^3$$ - $$4t$$ + $$2$$.","variabilization":{},"oer":"https://openstax.org/details/books/physics <OpenStax: Physics>","license":0,"lesson":"3.4 Derivatives as Rates of Change","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ae173b9deriv4a","stepAnswer":["Right to left"],"problemType":"MultipleChoice","stepTitle":"Is the particle moving from left to right or from right to left at time $$t$$ $$=$$ 1?","stepBody":"","answerType":"string","variabilization":{},"choices":["Left to right","Right to left","No direction"],"hints":{"DefaultPathway":[{"id":"ae173b9deriv4a-h1","type":"hint","dependencies":[],"title":"What to Solve For","text":"The direction of motion is determined by the velocity. Begin by finding v(t) where v(t) $$=$$ s\'(t), the derivative of s(t). A positive or negative sign of acceleration indicates if the velocity is increasing or decreasing. If v(t) > $$0$$, the particle is moving from left to right. If v(t) < $$0$$, the particle is moving from right to left. If v(t) $$=$$ $$0$$, there is no direction of motion.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3t^2$$ - $$4$$"],"dependencies":["ae173b9deriv4a-h1"],"title":"Solving for v(t)","text":"What is v(t), the velocity of the particle? Write out the full function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv4a-h3","type":"hint","dependencies":["ae173b9deriv4a-h2"],"title":"Taking the Derivative of s(t)","text":"To determine v(t), we must take the derivative of s(t) $$=$$ $$t^3$$ - $$4t$$ + $$2$$.\\\\ns(t) $$=$$ $$t^3$$ - $$4t$$ + $$2$$\\\\ns\'(t) $$=$$ $$3\\\\left(t^{3-1}\\\\right)$$ - $$4$$\\\\ns\'(t) $$=$$ $$3t^2$$ - $$4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["ae173b9deriv4a-h3"],"title":"Plugging $$t$$ $$=$$ $$1$$ Into v(t)","text":"v(t) $$=$$ $$3t^2$$ - $$4$$. What is v(1)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv4a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["ae173b9deriv4a-h4"],"title":"Solving for v(1)","text":"v(1) $$=$$ $$3\\\\left(1^2\\\\right)$$ - $$4$$ $$=$$ 3(1) - $$4$$ $$=$$ $$3$$ - $$4$$. What is v(1)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv4a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Less than"],"dependencies":["ae173b9deriv4a-h5"],"title":"Determining the Direction of Motion","text":"A positive or negative sign of acceleration indicates if the velocity is increasing or decreasing. If v(t) > $$0$$, the particle is moving from left to right. If v(t) < $$0$$, the particle is moving from right to left. If v(t) $$=$$ $$0$$, there is no direction of motion. Is v(t) equal to, greater than, or less than 0?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["Equal to","Less than","Greater than"]},{"id":"ae173b9deriv4a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Right to left"],"dependencies":["ae173b9deriv4a-h6"],"title":"Particle Direction","text":"v(1) $$=$$ $$-1$$ such that v(t) < $$0$$. What is the direction of motion of the particle?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["Left to right","Right to left","No direction"]}]}}]},{"id":"ae173b9deriv5","title":"Derivatives as Rates of Change","body":"A particle moves along a coordinate axis in the positive direction to the right. Its position at time $$t$$ is given by s(t) $$=$$ $$t^3$$ - $$4t$$ + $$2$$.","variabilization":{},"oer":"https://openstax.org/details/books/physics <OpenStax: Physics>","license":0,"lesson":"3.4 Derivatives as Rates of Change","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ae173b9deriv5a","stepAnswer":["Right to left"],"problemType":"MultipleChoice","stepTitle":"Is the particle speeding up or slowing down at time $$t$$ $$=$$ 1?","stepBody":"","answerType":"string","variabilization":{},"choices":["Left to right","Right to left","No direction"],"hints":{"DefaultPathway":[{"id":"ae173b9deriv5a-h1","type":"hint","dependencies":[],"title":"What to Solve For","text":"The speeding up or slowing down of an object is determined by the velocity and acceleration. Whenever the particle\'s velocity and acceleration have the same sign (positive or negative), the particle\'s speed is increasing. Likewise, when the particle\'s velocity and acceleration have opposing signs (one positive, one negative), the particle\'s speed is decreasing. Begin by finding the velocity (v(t) $$=$$ s\'(t), the derivative of s(t)) and the acceleration (a(t) $$=$$ v\'(t) $$=$$ s\'\'(t)).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3t^2$$ - $$4$$"],"dependencies":["ae173b9deriv5a-h1"],"title":"Solving for Velocity, v(t)","text":"What is v(t), the velocity of the particle? Write out the full function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv5a-h3","type":"hint","dependencies":["ae173b9deriv5a-h2"],"title":"Taking the Derivative of s(t)","text":"To determine v(t), we must take the derivative of s(t) $$=$$ $$t^3$$ - $$4t$$ + $$2$$.\\\\ns(t) $$=$$ $$t^3$$ - $$4t$$ + $$2$$\\\\ns\'(t) $$=$$ $$3\\\\left(t^{3-1}\\\\right)$$ - $$4$$\\\\ns\'(t) $$=$$ $$3t^2$$ - $$4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["ae173b9deriv5a-h3"],"title":"Plugging $$t$$ $$=$$ $$1$$ Into v(t)","text":"v(t) $$=$$ $$3t^2$$ - $$4$$. What is v(1)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["ae173b9deriv5a-h4"],"title":"Solving for v(1)","text":"v(1) $$=$$ $$3\\\\left(1^2\\\\right)$$ - $$4$$ $$=$$ 3(1) - $$4$$ $$=$$ $$3$$ - $$4$$. What is v(1)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv5a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Less than"],"dependencies":["ae173b9deriv5a-h5"],"title":"Determining the Direction of Motion of the Velocity","text":"A positive or negative sign of acceleration indicates if the velocity is increasing or decreasing. If v(t) > $$0$$, the particle is moving from left to right. If v(t) < $$0$$, the particle is moving from right to left. If v(t) $$=$$ $$0$$, there is no direction of motion. Is v(t) equal to, greater than, or less than 0?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["Equal to","Less than","Greater than"]},{"id":"ae173b9deriv5a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6t$$"],"dependencies":["ae173b9deriv5a-h6"],"title":"Solving for the Acceleration, a(t)","text":"What is a(t), the acceleration of the particle? Write out the full function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv5a-h8","type":"hint","dependencies":["ae173b9deriv5a-h7"],"title":"Taking the Derivative of v(t)","text":"To determine a(t), we must take the derivative of v(t) $$=$$ $$3t^2$$ - $$4$$.\\\\nv(t) $$=$$ $$3t^2$$ - $$4$$\\\\nv\'(t) $$=$$ $$2\\\\left(3\\\\right) t^{2-1}$$\\\\nv\'(t) $$=$$ $$6t$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv5a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["ae173b9deriv5a-h8"],"title":"Plugging $$t$$ $$=$$ $$1$$ Into a(t)","text":"a(t) $$=$$ $$6t$$. What is a(1)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv5a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["ae173b9deriv5a-h9"],"title":"Solving for a(1)","text":"a(1) $$=$$ $$6t$$ $$=$$ 6(1). What is a(1)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv5a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Greater than"],"dependencies":["ae173b9deriv5a-h10"],"title":"Determining the Direction of Motion of the Acceleration","text":"A positive or negative sign of acceleration indicates the rate at which the velocity is changing. If a(t) > $$0$$, the rate of the velocity is increasing. If a(t) < $$0$$, the rate of the velocity is decreasing. If a(t) $$=$$ $$0$$, there is no rate of change in the velocity. Is a(t) equal to, greater than, or less than 0?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["Equal to","Less than","Greater than"]},{"id":"ae173b9deriv5a-h12","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Different"],"dependencies":["ae173b9deriv5a-h11"],"title":"Same or Different Directions of Motion","text":"Are the directions of motion between the velocity and acceleration the same or different?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["Same","Different"]},{"id":"ae173b9deriv5a-h13","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Slowing down"],"dependencies":["ae173b9deriv5a-h12"],"title":"Speeding Up or Slowing Down","text":"Because v(1) < $$0$$ and a(1) > $$0$$, the velocity and acceleration are acting in opposite directions. The same direction of motion would indicate the speeding up of the particle whereas the opposite directions of motion would indicate the slowing down of the particle. Is the particle speeding up or slowing down at $$t$$ $$=$$ 1?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["Slowing down","Speeding up"]},{"id":"ae173b9deriv5a-h14","type":"hint","dependencies":["ae173b9deriv5a-h13"],"title":"Interpretation of the Problem","text":"Because v(1) < $$0$$ and a(1) > $$0$$, the velocity and acceleration are acting in opposite directions. In other words, the particle is being accelerated in the direction opposite the direction in which it is traveling, causing |v(t)| to decrease. The particle is slowing down.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ae173b9deriv6","title":"Derivatives as Rates of Change","body":"The position of a particle moving along a coordinate axis is given by s(t) $$=$$ $$t^3$$ - $$9t^2$$ + $$24t$$ + $$4$$, $$t$$ $$ \\\\geq $$ $$0$$.","variabilization":{},"oer":"https://openstax.org/details/books/physics <OpenStax: Physics>","license":0,"lesson":"3.4 Derivatives as Rates of Change","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ae173b9deriv6a","stepAnswer":["$$3t^2$$ - $$18t$$ + $$24$$"],"problemType":"TextBox","stepTitle":"Find v(t).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3t^2$$ - $$18t$$ + $$24$$","choices":["$$3t^2$$ $$-$$ $$18t$$ + $$24$$","Left to right","No direction","Right to left"],"hints":{"DefaultPathway":[{"id":"ae173b9deriv6a-h1","type":"hint","dependencies":[],"title":"v(t) Equation","text":"v(t) $$=$$ s\'(t), where s\'(t) is the derivative of s(t). Solve for s\'(t).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3t^2$$ - $$18t$$ + $$24$$"],"dependencies":["ae173b9deriv6a-h1"],"title":"Solving for s\'(t)","text":"What is s\'(t), the velocity of the particle?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv6a-h3","type":"hint","dependencies":["ae173b9deriv6a-h2"],"title":"Taking the Derivative of s(t)","text":"To determine v(t), we must take the derivative of s(t) $$=$$ $$t^3$$ - $$9t^2$$ + $$24t$$ + $$4$$, $$t$$ $$ \\\\geq $$ $$0$$.\\\\ns(t) $$=$$ $$t^3$$ - $$9t^2$$ + $$24t$$ + $$4$$\\\\ns\'(t) $$=$$ $$3t^{3-1}$$ - $$2\\\\left(9\\\\right) t^{2-1}$$ + $$24$$\\\\ns\'(t) $$=$$ $$3t^2$$ - $$18t$$ + $$24$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ae173b9deriv7","title":"Derivatives as Rates of Change","body":"The position of a particle moving along a coordinate axis is given by s(t) $$=$$ $$t^3$$ - $$9t^2$$ + $$24t$$ + $$4$$, $$t$$ $$ \\\\geq $$ $$0$$.","variabilization":{},"oer":"https://openstax.org/details/books/physics <OpenStax: Physics>","license":0,"lesson":"3.4 Derivatives as Rates of Change","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ae173b9deriv7a","stepAnswer":["2,4"],"problemType":"TextBox","stepTitle":"At what time(s) is the particle at rest? If your answer has multiple times, separate them by commas. For example, \\"1,2\\".","stepBody":"","answerType":"string","variabilization":{},"choices":["2,4","Left to right","No direction","Right to left"],"hints":{"DefaultPathway":[{"id":"ae173b9deriv7a-h1","type":"hint","dependencies":[],"title":"What to Solve For","text":"The particle is at rest when v(t) $$=$$ $$0$$. Set $$3t^2$$ - $$18t$$ + $$24$$ $$=$$ $$0$$ and solve for $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv7a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["2,4"],"dependencies":["ae173b9deriv7a-h1"],"title":"Time(s) the Particle is at Rest","text":"What is $$t$$? If your answer has multiple times, separate them by commas. For example, \\"1,2\\".","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv7a-h3","type":"hint","dependencies":["ae173b9deriv7a-h2"],"title":"Calculating $$3t^2$$ - $$18t$$ + $$24$$ $$=$$ $$0$$","text":"Factor the left-hand side of the equation to produce 3(t - 2)(t - 4) $$=$$ $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv7a-h4","type":"hint","dependencies":["ae173b9deriv7a-h3"],"title":"Simplifying the Equation","text":"When solving for $$t$$, we can simplify the equation 3(t - 2)(t - 4) $$=$$ $$0$$ to get $$t$$ - $$2$$ $$=$$ $$0$$ and $$t$$ - $$4$$ $$=$$ $$0$$ by dividing by $$3$$ on both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv7a-h5","type":"hint","dependencies":["ae173b9deriv7a-h4"],"title":"Final Answer","text":"We find that the particle is at rest at $$t$$ $$=$$ $$2$$ and $$t$$ $$=$$ $$4$$. Thus, your answer should be \\"2,4\\".","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ae173b9deriv8","title":"Derivatives as Rates of Change","body":"The position of a particle moving along a coordinate axis is given by s(t) $$=$$ $$t^3$$ - $$9t^2$$ + $$24t$$ + $$4$$, $$t$$ $$ \\\\geq $$ $$0$$.","variabilization":{},"oer":"https://openstax.org/details/books/physics <OpenStax: Physics>","license":0,"lesson":"3.4 Derivatives as Rates of Change","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ae173b9deriv8a","stepAnswer":["[0,2) U (4,+\u221e)"],"problemType":"MultipleChoice","stepTitle":"On what time intervals is the particle moving from left to right?","stepBody":"","answerType":"string","variabilization":{},"choices":["$$(0,2)$$ U (4,+\u221e)","[0,2) U (4,+\u221e)","$$[-\u221e$$, 0) U (2,+\u221e)","[0,1) U (4,+\u221e)"],"hints":{"DefaultPathway":[{"id":"ae173b9deriv8a-h1","type":"hint","dependencies":[],"title":"What to Look For","text":"The particle is moving from left to right when v(t) > $$0$$. The sign of v(t) determines the direction of the particle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv8a-h2","type":"hint","dependencies":["ae173b9deriv8a-h1"],"title":"Using a Number Line","text":"The number line shown presents the signs of v(t), where the numbers at the bottom represent the times.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv8a-h3","type":"hint","dependencies":["ae173b9deriv8a-h2"],"title":"Time Intervals","text":"To determine the time intervals that the particle is moving from left to right, we must locate where v(t) > $$0$$. In other words, we must find the times frames in which v(t) is positive.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv8a-h4","type":"hint","dependencies":["ae173b9deriv8a-h3"],"title":"Positive Sign Time Intervals","text":"In the number line, v(t) is shown to be positive from $$0$$ to $$2$$ and $$4$$ to $$\\\\infty$$.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv8a-h5","type":"hint","dependencies":["ae173b9deriv8a-h4"],"title":"Final Answer","text":"Now that we know that the time intervals in which the particle is moving left to right is $$0$$ to $$2$$ and $$4$$ to $$+\\\\infty$$, we must now construct this into a cohesive range statement. Square brackets mean that the end point is included, and round parentheses mean that it is excluded. Infinity symbols are always accompanied by round (exclusive) brackets, and U is used to combine intervals. Thus, the time interval is [0,2) U (4,+\u221e).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ae173b9deriv9","title":"Derivatives as Rates of Change","body":"The position of a particle moving along a coordinate axis is given by s(t) $$=$$ $$t^3$$ - $$9t^2$$ + $$24t$$ + $$4$$, $$t$$ $$ \\\\geq $$ $$0$$.","variabilization":{},"oer":"https://openstax.org/details/books/physics <OpenStax: Physics>","license":0,"lesson":"3.4 Derivatives as Rates of Change","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"ae173b9deriv9a","stepAnswer":["$$(2,4)$$"],"problemType":"MultipleChoice","stepTitle":"On what time intervals is the particle moving from right to left?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(2,4)$$","choices":["$$(2,4)$$","$$(0,4)$$","(4, +\u221e)","$$(0,2)$$"],"hints":{"DefaultPathway":[{"id":"ae173b9deriv9a-h1","type":"hint","dependencies":[],"title":"What to Look For","text":"The particle is moving from right to left when v(t) < $$0$$. The sign of v(t) determines the direction of the particle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv9a-h2","type":"hint","dependencies":["ae173b9deriv9a-h1"],"title":"Using a Number Line","text":"The number line shown presents the signs of v(t), where the numbers at the bottom represent the times.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv9a-h3","type":"hint","dependencies":["ae173b9deriv9a-h2"],"title":"Time Intervals","text":"To determine the time intervals that the particle is moving from right to left, we must locate where v(t) < $$0$$. In other words, we must find the times frames in which v(t) is negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv9a-h4","type":"hint","dependencies":["ae173b9deriv9a-h3"],"title":"Negative Sign Time Interval","text":"In the number line, v(t) is shown to be negative from $$2$$ to $$4$$.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"ae173b9deriv9a-h5","type":"hint","dependencies":["ae173b9deriv9a-h4"],"title":"Final Answer","text":"Now that we know that the time intervals in which the particle is moving right to left is $$2$$ to $$4$$, we must now construct this into a cohesive range statement. Square brackets mean that the end point is included, and round parentheses mean that it is excluded. Infinity symbols are always accompanied by round (exclusive) brackets, and U is used to combine intervals. Thus, the time interval is $$(2,4)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"ae25c01functions1","title":"Using the Vertical Line Test","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Graphs of Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae25c01functions1a","stepAnswer":["The graph is not a function."],"problemType":"MultipleChoice","stepTitle":"Determine whether the graph is the graph of a function.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["The graph is a function.","The graph is not a function."],"hints":{"DefaultPathway":[{"id":"ae25c01functions1a-h1","type":"hint","dependencies":[],"title":"Using the Vertical Line Test","text":"Since there exists a vertical line that intersects the graph at more than one point, the graph is not a function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae25c01functions10","title":"Finding the Domain of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Graphs of Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae25c01functions10a","stepAnswer":["(-inf,inf)"],"problemType":"TextBox","stepTitle":"Find the domain of the function in interval notation: $$f(x)=2x+5$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\infty)$$","hints":{"DefaultPathway":[{"id":"ae25c01functions10a-h1","type":"hint","dependencies":[],"title":"What is Domain?","text":"The domain is the set of x-values accepted by the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae25c01functions10a-h2","type":"hint","dependencies":["ae25c01functions10a-h1"],"title":"Finding the Domain","text":"Since this function accepts all values of $$x$$, its domain is $$(-\\\\infty,\\\\infty)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae25c01functions11","title":"Finding the Domain of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Graphs of Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae25c01functions11a","stepAnswer":["(-inf,inf)"],"problemType":"TextBox","stepTitle":"Find the domain of the function in interval notation: $$f(x)=-x-2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\infty)$$","hints":{"DefaultPathway":[{"id":"ae25c01functions11a-h1","type":"hint","dependencies":[],"title":"What is Domain?","text":"The domain is the set of x-values accepted by the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae25c01functions11a-h2","type":"hint","dependencies":["ae25c01functions11a-h1"],"title":"Finding the Domain","text":"Since this function accepts all values of $$x$$, its domain is $$(-\\\\infty,\\\\infty)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae25c01functions12","title":"Finding the Domain of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate 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$$(-\\\\infty,\\\\infty)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae25c01functions13","title":"Finding the Domain of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Graphs of Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae25c01functions13a","stepAnswer":["(-inf,inf)"],"problemType":"TextBox","stepTitle":"Find the domain of the function in interval notation: $$f(x)=-2x+2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\infty)$$","hints":{"DefaultPathway":[{"id":"ae25c01functions13a-h1","type":"hint","dependencies":[],"title":"What is Domain?","text":"The domain is the set of x-values accepted by the 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$$(-\\\\infty,\\\\infty)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae25c01functions2","title":"Using the Vertical Line Test","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Graphs of Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae25c01functions2a","stepAnswer":["The graph is a function."],"problemType":"MultipleChoice","stepTitle":"Determine whether the graph is the graph of a function.","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["The graph is a function.","The graph is not a function."],"hints":{"DefaultPathway":[{"id":"ae25c01functions2a-h1","type":"hint","dependencies":[],"title":"Using the Vertical Line Test","text":"Since there does not exist a 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Since the problem is about a mean, this is a test of a single population mean.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional10a-h2","type":"hint","dependencies":["ae2aa31Additional10a-h1"],"title":"Tail","text":"The 1% level of significance means that $$\\\\alpha=0.01$$. Based on the problem, $$H_0$$: $$p=0.50$$ and $$H_a$$: $$p \\\\neq 0.50$$. The words \\"is the same or different from\\" tell you this is a two-tailed test.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional10a-h3","type":"hint","dependencies":["ae2aa31Additional10a-h2"],"title":"Distribution","text":"Determine the distribution needed. The problem contains no mention of a mean. The information is given in terms of percentages. Use the distribution for P\u2032, the estimated proportion. \ud835\udc43\u2032~\ud835\udc41(\ud835\udc5d,sqrt((p * q)/n)). Fill in the equation with values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.5$$"],"dependencies":["ae2aa31Additional10a-h3"],"title":"Find $$p$$","text":"What is $$p$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.5$$"],"dependencies":["ae2aa31Additional10a-h3"],"title":"Calculate q","text":"$$q=1-p$$. What is q?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional10a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$100$$"],"dependencies":["ae2aa31Additional10a-h3"],"title":"Calculate $$n$$","text":"What is $$n$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional10a-h7","type":"hint","dependencies":["ae2aa31Additional10a-h4","ae2aa31Additional10a-h5","ae2aa31Additional10a-h6"],"title":"P-value","text":"p-value=P(p\u2032<0.47 or p\u2032>0.53)=0.5485","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional10a-h8","type":"hint","dependencies":["ae2aa31Additional10a-h7"],"title":"P-value","text":"$$p\u2032=0.53$$. Since the curve is symmetrical and the test is two-tailed, the p\u2032 for the left tail is equal to $$0.50-0.03=0.47$$ where $$\u03bc=p=0.50$$. $$(0.03$$ is the difference between $$0.53$$ and $$0.50.)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional10a-h9","type":"hint","dependencies":["ae2aa31Additional10a-h8"],"title":"Compare","text":"Compare \ud835\udefc and the $$p-value$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional10a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Less than"],"dependencies":["ae2aa31Additional10a-h9"],"title":"Compare","text":"Is \ud835\udefc less than or greater than the $$p-value$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Less than","Greater than"]},{"id":"ae2aa31Additional10a-h11","type":"hint","dependencies":["ae2aa31Additional10a-h10"],"title":"Interpret","text":"Since $$\u03b1<p-value$$, we cannot reject $$H_0$$. Find the conclusion that matches this.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae2aa31Additional11","title":"Full Hypothesis Test Examples","body":"Suppose a consumer group suspects that the proportion of households that have three cell phones is 30%. A cell phone company has reason to believe that the proportion is not 30%. Before they start a big advertising campaign, they conduct a hypothesis test. Their marketing people survey $$150$$ households with the result that $$43$$ of the households have three cell phones.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Additional Information and Full Hypothesis Test Examples","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae2aa31Additional11a","stepAnswer":["$$\\\\frac{43}{150}$$"],"problemType":"TextBox","stepTitle":"The value that helps determine the $$p-value$$ is p\u2032. Calculate p\u2032.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{43}{150}$$","hints":{"DefaultPathway":[{"id":"ae2aa31Additional11a-h1","type":"hint","dependencies":[],"title":"Set up","text":"Set up the Hypothesis Test. $$H_0$$: $$p=0.30$$ $$H_a$$: $$p \\\\neq 0.30$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional11a-h2","type":"hint","dependencies":["ae2aa31Additional11a-h1"],"title":"Distribution","text":"Determine the distribution needed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional11a-h3","type":"hint","dependencies":["ae2aa31Additional11a-h2"],"title":"Test","text":"The distribution for the hypothesis test is \ud835\udc43\'~\ud835\udc41(0.30,sqrt(((0.30)*(0.70))/150))","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional11a-h4","type":"hint","dependencies":["ae2aa31Additional11a-h3"],"title":"Calculate","text":"Calculate $$p\u2032=\\\\frac{x}{n}$$ where $$x$$ is the number of successes and $$n$$ is the total number in the sample.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$43$$"],"dependencies":["ae2aa31Additional11a-h4"],"title":"Calculate","text":"What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional11a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$150$$"],"dependencies":["ae2aa31Additional11a-h4"],"title":"Calculate","text":"What is $$n$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional11a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{43}{150}$$"],"dependencies":["ae2aa31Additional11a-h5","ae2aa31Additional11a-h6"],"title":"Calculate","text":"What is $$p\u2032=\\\\frac{x}{n}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae2aa31Additional12","title":"Full Hypothesis Test Examples","body":"Suppose a consumer group suspects that the proportion of households that have three cell phones is 30%. A cell phone company has reason to believe that the proportion is not 30%. Before they start a big advertising campaign, they conduct a hypothesis test. Their marketing people survey $$150$$ households with the result that $$43$$ of the households have three cell phones.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Additional Information and Full Hypothesis Test Examples","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae2aa31Additional12a","stepAnswer":["A success is having three cell phones in a household."],"problemType":"MultipleChoice","stepTitle":"What is a success for this problem?","stepBody":"","answerType":"string","variabilization":{},"choices":["A success is having three cell phones in a household.","A success is having two cell phones in a household.","A success is having four cell phones in a household.","A success is having five cell phones in a household."],"hints":{"DefaultPathway":[{"id":"ae2aa31Additional12a-h1","type":"hint","dependencies":[],"title":"Question","text":"The problem states that a consumer group suspects that the proportion of households that have three cell phones is 30%. Their marketing people survey $$150$$ households with the result that $$43$$ of the households have three cell phones. Interpret this.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional12a-h2","type":"hint","dependencies":["ae2aa31Additional12a-h1"],"title":"Meaning","text":"This means that $$3$$ cell phones in a household is a success as it aligns with what is stated in the problem.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae2aa31Additional13","title":"Full Hypothesis Test Examples","body":"Suppose a consumer group suspects that the proportion of households that have three cell phones is 30%. A cell phone company has reason to believe that the proportion is not 30%. Before they start a big advertising campaign, they conduct a hypothesis test. Their marketing people survey $$150$$ households with the result that $$43$$ of the households have three cell phones.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Additional Information and Full Hypothesis Test Examples","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae2aa31Additional13a","stepAnswer":["$$0.05$$"],"problemType":"TextBox","stepTitle":"What is the level of significance?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.05$$","hints":{"DefaultPathway":[{"id":"ae2aa31Additional13a-h1","type":"hint","dependencies":[],"title":"Preset \\\\alpha","text":"The level of significance is the preset \ud835\udefc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.05$$"],"dependencies":["ae2aa31Additional13a-h1"],"title":"Preset \\\\alpha","text":"What is the preset \ud835\udefc?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae2aa31Additional14","title":"Full Hypothesis Test Examples","body":"Suppose a consumer group suspects that the proportion of households that have three cell phones is 30%. A cell phone company has reason to believe that the proportion is not 30%. Before they start a big advertising campaign, they conduct a hypothesis test. Their marketing people survey $$150$$ households with the result that $$43$$ of the households have three cell phones.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Additional Information and Full Hypothesis Test Examples","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae2aa31Additional14a","stepAnswer":["$$0.7216$$"],"problemType":"TextBox","stepTitle":"Calculate the $$p-value$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.7216$$","hints":{"DefaultPathway":[{"id":"ae2aa31Additional14a-h1","type":"hint","dependencies":[],"title":"P-value","text":"p-value=P(p\u2032<0.30 or p\u2032>0.30).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.7216$$"],"dependencies":["ae2aa31Additional14a-h1"],"title":"P-value","text":"What is P(p\u2032<0.30 or p\u2032>0.30)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae2aa31Additional15","title":"Full Hypothesis Test Examples","body":"Suppose a consumer group suspects that the proportion of households that have three cell phones is 30%. A cell phone company has reason to believe that the proportion is not 30%. Before they start a big advertising campaign, they conduct a hypothesis test. Their marketing people survey $$150$$ households with the result that $$43$$ of the households have three cell phones.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Additional Information and Full Hypothesis Test Examples","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae2aa31Additional15a","stepAnswer":["The decision is do not reject $$H_0$$ because there is not sufficient evidence to conclude that the proportion of households that have three cell phones is not 30%."],"problemType":"MultipleChoice","stepTitle":"Make a decision. $$___$$ (Reject/Do not reject) $$H_0$$ because $$___$$ .","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"The decision is do not reject $$H_0$$ because there is not sufficient evidence to conclude that the proportion of households that have three cell phones is not 30%.","choices":["The decision is do not reject $$H_0$$ because there is not sufficient evidence to conclude that the proportion of households that have three cell phones is not 30%.","The decision is to reject $$H_0$$ because there is not sufficient evidence to conclude that the proportion of households that have three cell phones is not 30%.","The decision is do not reject $$H_0$$ because there is sufficient evidence to conclude that the proportion of households that have three cell phones is not 30%.","The decision is to reject $$H_0$$ because there is sufficient evidence to conclude that the proportion of households that have three cell phones is not 30%."],"hints":{"DefaultPathway":[{"id":"ae2aa31Additional15a-h1","type":"hint","dependencies":[],"title":"Compare","text":"Compare \ud835\udefc and the $$p-value$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional15a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Less than"],"dependencies":["ae2aa31Additional15a-h1"],"title":"Compare","text":"Is \ud835\udefc less than or greater than the $$p-value$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Less than","Greater than"]},{"id":"ae2aa31Additional15a-h3","type":"hint","dependencies":["ae2aa31Additional15a-h2"],"title":"Interpret","text":"Since $$\u03b1<p-value$$, we cannot reject $$H_0$$ because there is not enough evidence. Find the conclusion that matches this.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae2aa31Additional4","title":"Left-, Right-, and Two-Tailed Test","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Additional Information and Full Hypothesis Test Examples","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae2aa31Additional4a","stepAnswer":["$$Right-tailed$$"],"problemType":"MultipleChoice","stepTitle":"$$H_0$$: $$\u03bc \\\\leq 1$$, $$H_a$$: $$\u03bc>1$$. Assume the $$p-value$$ is $$0.1243$$. What type of test is this? Draw the picture of the $$p-value$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$Left-tailed$$","$$Right-tailed$$","$$Two-Tailed$$"],"hints":{"DefaultPathway":[{"id":"ae2aa31Additional4a-h1","type":"hint","dependencies":[],"title":"Tail","text":"$$H_a$$ tells you the whether the tail is left-, right, or two-tailed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional4a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Greater"],"dependencies":["ae2aa31Additional4a-h1"],"title":"Compare","text":"Is $$H_a$$ greater, less, or not equal to the mean?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Greater","Less","Not Equal"]},{"id":"ae2aa31Additional4a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Right"],"dependencies":["ae2aa31Additional4a-h2"],"title":"Interpret","text":"Does that mean we are looking to the left, right, or both sides of the mean?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Left","Right","Both Sides"]}]}}]},{"id":"ae2aa31Additional5","title":"Left-, Right-, and Two-Tailed Test","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Additional Information and Full Hypothesis Test Examples","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae2aa31Additional5a","stepAnswer":["$$Two-Tailed$$"],"problemType":"MultipleChoice","stepTitle":"$$H_0$$: $$p=0.5$$, $$H_a$$: $$p \\\\neq 0.5$$. Assume the $$p-value$$ is $$0.2564$$. What type of test is this?","stepBody":"","answerType":"string","variabilization":{},"choices":["$$Left-tailed$$","$$Right-tailed$$","$$Two-Tailed$$"],"hints":{"DefaultPathway":[{"id":"ae2aa31Additional5a-h1","type":"hint","dependencies":[],"title":"Tail","text":"$$H_a$$ tells you the whether the tail is left-, right, or two-tailed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional5a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Not equal"],"dependencies":["ae2aa31Additional5a-h1"],"title":"Compare","text":"Is $$H_a$$ greater, less, or not equal to the mean?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Greater","Less","Not Equal"]},{"id":"ae2aa31Additional5a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Both Sides"],"dependencies":["ae2aa31Additional5a-h2"],"title":"Interpret","text":"Does that mean we are looking to the left, right, or both sides of the mean?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Left","Right","Both Sides"]}]}}]},{"id":"ae2aa31Additional6","title":"Left-, Right-, and Two-Tailed Test","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Additional Information and Full Hypothesis Test Examples","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae2aa31Additional6a","stepAnswer":["$$Two-Tailed$$"],"problemType":"MultipleChoice","stepTitle":"$$H_0$$: $$p=50$$ and $$H_a$$: $$p \\\\neq 50$$. Assume the $$p-value$$ is $$0.2564$$. What type of test is this?","stepBody":"","answerType":"string","variabilization":{},"choices":["$$Left-tailed$$","$$Right-tailed$$","$$Two-Tailed$$"],"hints":{"DefaultPathway":[{"id":"ae2aa31Additional6a-h1","type":"hint","dependencies":[],"title":"Tail","text":"$$H_a$$ tells you the whether the tail is left-, right, or two-tailed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional6a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Not equal"],"dependencies":["ae2aa31Additional6a-h1"],"title":"Compare","text":"Is $$H_a$$ greater, less, or not equal to the mean?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Greater","Less","Not Equal"]},{"id":"ae2aa31Additional6a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Both Sides"],"dependencies":["ae2aa31Additional6a-h2"],"title":"Interpret","text":"Does that mean we are looking to the left, right, or both sides of the mean?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Left","Right","Both Sides"]}]}}]},{"id":"ae2aa31Additional7","title":"Full Hypothesis Test Examples","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Additional Information and Full Hypothesis Test Examples","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae2aa31Additional7a","stepAnswer":["At the 5% significance level, we conclude that Jeffrey swims faster using the new goggles. The sample data show there is sufficient evidence that Jeffrey\'s mean time to swim the $$25-yard$$ freestyle is less than $$16.43$$ seconds."],"problemType":"MultipleChoice","stepTitle":"Jeffrey, as an eight-year old, established a mean time of $$16.43$$ seconds for swimming the 25-yard freestyle, with a standard deviation of $$0.8$$ seconds. His dad, Frank, thought that Jeffrey could swim the 25-yard freestyle faster using goggles. Frank bought Jeffrey a new pair of expensive goggles and timed Jeffrey for $$15$$ 25-yard freestyle swims. For the $$15$$ swims, Jeffrey\'s mean time was $$16$$ seconds. Frank thought that the goggles helped Jeffrey to swim faster than the $$16.43$$ seconds. Conduct a hypothesis test using a preset $$\\\\alpha=0.05$$. Assume that the swim times for the 25-yard freestyle are normal.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"At the 5% significance level, we conclude that Jeffrey swims faster using the new goggles. The sample data show there is sufficient evidence that Jeffrey\'s mean time to swim the 25-yard freestyle is less than $$16.43$$ seconds.","choices":["At the 5% significance level, we conclude that Jeffrey swims faster using the new goggles. The sample data show there is sufficient evidence that Jeffrey\'s mean time to swim the $$25-yard$$ freestyle is less than $$16.43$$ seconds.","At the 5% significance level, we conclude that Jeffrey swims faster using the new goggles. The sample data show there is sufficient evidence that Jeffrey\'s mean time to swim the $$25-yard$$ freestyle is greater than $$16.43$$ seconds.","At the 5% significance level, we conclude that Jeffrey swims slower using the new goggles. The sample data show there is sufficient evidence that Jeffrey\'s mean time to swim the $$25-yard$$ freestyle is less than $$16.43$$ seconds.","At the 5% significance level, we conclude that Jeffrey swims slower using the new goggles. The sample data show there is sufficient evidence that Jeffrey\'s mean time to swim the $$25-yard$$ freestyle is greater than $$16.43$$ seconds."],"hints":{"DefaultPathway":[{"id":"ae2aa31Additional7a-h1","type":"hint","dependencies":[],"title":"Set up","text":"First, set up the hypothesis test. Since the problem is about a mean, this is a test of a single population mean.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional7a-h2","type":"hint","dependencies":["ae2aa31Additional7a-h1"],"title":"Tail","text":"Based on the problem, $$H_0$$: $$\u03bc=16.43$$ $$H_a$$: $$\u03bc<16.43$$. The \\"<\\" tells you this is left-tailed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional7a-h3","type":"hint","dependencies":["ae2aa31Additional7a-h2"],"title":"Distribution","text":"Determine the distribution needed. $$\ud835\udc4b~\ud835\udc41(\ud835\udf07,\\\\frac{\u03c3_X}{\\\\sqrt{n}})$$. Fill this in with values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16.43$$"],"dependencies":["ae2aa31Additional7a-h3"],"title":"Find \u03bc","text":"We know $$\u03c3=0.8$$ and $$n=15$$. What is \u03bc?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ae2aa31Additional7a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16.43$$"],"dependencies":[],"title":"Find \u03bc","text":"What is $$H_0$$? This is the value of \u03bc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ae2aa31Additional7a-h5","type":"hint","dependencies":["ae2aa31Additional7a-h4"],"title":"P-value","text":"Calculate the $$p-value$$ using the normal distribution for a mean. We need to know the sample mean for this.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional7a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["ae2aa31Additional7a-h5"],"title":"Mean","text":"What does the problem say is the sample mean?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional7a-h7","type":"hint","dependencies":["ae2aa31Additional7a-h6"],"title":"P-value","text":"$$p-value=P\\\\left(x<16\\\\right)=0.0187$$. The $$p-value$$ is the area to the left of the sample mean.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional7a-h8","type":"hint","dependencies":["ae2aa31Additional7a-h7"],"title":"Compare","text":"Compare \\\\alpha and the $$p-value$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional7a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Greater than"],"dependencies":["ae2aa31Additional7a-h8"],"title":"Compare","text":"Is \\\\alpha less than or greater than the $$p-value$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Less than","Greater than"]},{"id":"ae2aa31Additional7a-h10","type":"hint","dependencies":["ae2aa31Additional7a-h9"],"title":"Interpret","text":"Since \\\\alpha>p-value, we would reject $$H_0$$. This means that you reject $$\u03bc=16.43$$. In other words, you do not think Jeffrey swims the 25-yard freestyle in $$16.43$$ seconds but faster with the new goggles. Find the conclusion that matches with this idea.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae2aa31Additional8","title":"Full Hypothesis Test Examples","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Additional Information and Full Hypothesis Test Examples","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae2aa31Additional8a","stepAnswer":["At the $$2.5\\\\%$$ level of significance, from the sample data, there is not sufficient evidence to conclude that the true mean weight lifted is more than $$275$$ pounds."],"problemType":"MultipleChoice","stepTitle":"A college football coach records the mean weight that his players can bench press as $$275$$ pounds, with a standard deviation of $$55$$ pounds. Three of his players thought that the mean weight was more than that amount. They asked $$30$$ of their teammates for their estimated maximum lift on the bench press exercise. The data ranged from $$205$$ pounds to $$385$$ pounds. The actual different weights were (frequencies are in parentheses) 205(3); 215(3); 225(1); 241(2); 252(2); 265(2); 275(2); 313(2); 316(5); 338(2); 341(1); 345(2); 368(2); 385(1). Conduct a hypothesis test using a $$2.5\\\\%$$ level of significance to determine if the bench press mean is more than $$275$$ pounds.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"At the $$2.5\\\\%$$ level of significance, from the sample data, there is not sufficient evidence to conclude that the true mean weight lifted is more than $$275$$ pounds.","choices":["At the $$2.5\\\\%$$ level of significance, from the sample data, there is not sufficient evidence to conclude that the true mean weight lifted is more than $$275$$ pounds.","At the $$2.5\\\\%$$ level of significance, from the sample data, there is sufficient evidence to conclude that the true mean weight lifted is more than $$275$$ pounds.","At the $$2.5\\\\%$$ level of significance, from the sample data, there is not sufficient evidence to conclude that the true mean weight lifted is less than $$275$$ pounds.","At the $$2.5\\\\%$$ level of significance, from the sample data, there is sufficient evidence to conclude that the true mean weight lifted is less than $$275$$ pounds."],"hints":{"DefaultPathway":[{"id":"ae2aa31Additional8a-h1","type":"hint","dependencies":[],"title":"Set up","text":"First, set up the hypothesis test. Since the problem is about a mean, this is a test of a single population mean.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional8a-h2","type":"hint","dependencies":["ae2aa31Additional8a-h1"],"title":"Tail","text":"Based on the problem, $$H_0$$: $$\u03bc=275$$ and $$H_a$$: $$\u03bc>275$$. Therefore, this is a right-tailed test.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional8a-h3","type":"hint","dependencies":["ae2aa31Additional8a-h2"],"title":"Distribution","text":"Determine the distribution needed. \ud835\udf0e is known so the distribution is normal. $$\ud835\udc4b~\ud835\udc41(\ud835\udf07,\\\\frac{\u03c3_X}{\\\\sqrt{n}})$$. Fill this in with values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$275$$"],"dependencies":["ae2aa31Additional8a-h3"],"title":"Find \u03bc","text":"We know $$\u03c3=55$$ and $$n=30$$. What is \u03bc?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ae2aa31Additional8a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$275$$"],"dependencies":[],"title":"Find \u03bc","text":"What is $$H_0$$? This is the value of \u03bc.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ae2aa31Additional8a-h5","type":"hint","dependencies":["ae2aa31Additional8a-h4"],"title":"P-value","text":"Calculate the $$p-value$$ using the normal distribution for a mean. We need to know the sample mean for this.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional8a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$286.2$$"],"dependencies":["ae2aa31Additional8a-h5"],"title":"Mean","text":"What does the problem say is the sample mean?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional8a-h7","type":"hint","dependencies":["ae2aa31Additional8a-h6"],"title":"P-value","text":"$$p-value=P\\\\left(x>286.2\\\\right)=0.1323$$. The $$p-value$$ is the area to the right of the sample mean.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional8a-h8","type":"hint","dependencies":["ae2aa31Additional8a-h7"],"title":"Compare","text":"Compare \\\\alpha and the $$p-value$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional8a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Less than"],"dependencies":["ae2aa31Additional8a-h8"],"title":"Compare","text":"Is \\\\alpha less than or greater than the $$p-value$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Less than","Greater than"]},{"id":"ae2aa31Additional8a-h10","type":"hint","dependencies":["ae2aa31Additional8a-h9"],"title":"Interpret","text":"Since \\\\alpha<p-value, we would not reject $$H_0$$. Select the answer choice with a valid conclusion based on this.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae2aa31Additional9","title":"Full Hypothesis Test Examples","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Additional Information and Full Hypothesis Test Examples","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae2aa31Additional9a","stepAnswer":["At a 5% level of significance, the sample data show sufficient evidence that the mean (average) test score is greater than $$65$$, just as the math instructor thinks."],"problemType":"MultipleChoice","stepTitle":"Statistics students believe that the mean score on the first statistics test is $$65$$. A statistics instructor thinks the mean score is higher than $$65$$. He samples ten statistics students and obtains the scores 65; 65; 70; 67; 66; 63; 63; 68; 72; $$71$$. He performs a hypothesis test using a 5% level of significance. The data are assumed to be from a normal distribution.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"At a 5% level of significance, the sample data show sufficient evidence that the mean (average) test score is greater than $$65$$, just as the math instructor thinks.","choices":["At a 5% level of significance, the sample data show sufficient evidence that the mean (average) test score is greater than $$65$$, just as the math instructor thinks.","At a 5% level of significance, the sample data show insufficient evidence that the mean (average) test score is greater than $$65$$, just as the math instructor thinks.","At a 5% level of significance, the sample data show sufficient evidence that the mean (average) test score is less than $$65$$, just as the math instructor thinks.","At a 5% level of significance, the sample data show insufficient evidence that the mean (average) test score is less than $$65$$, just as the math instructor thinks."],"hints":{"DefaultPathway":[{"id":"ae2aa31Additional9a-h1","type":"hint","dependencies":[],"title":"Set up","text":"First, set up the hypothesis test. Since the problem is about a mean, this is a test of a single population mean.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional9a-h2","type":"hint","dependencies":["ae2aa31Additional9a-h1"],"title":"Tail","text":"A 5% level of significance means that $$\\\\alpha=0.05$$. Based on the problem, $$H_0$$: $$\u03bc=65$$ and $$H_a$$: $$\u03bc>65$$. Therefore, this is a right-tailed test.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional9a-h3","type":"hint","dependencies":["ae2aa31Additional9a-h2"],"title":"Distribution","text":"Determine the distribution needed. No population standard deviation given, and you are only given $$n=10$$ sample data values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional9a-h4","type":"hint","dependencies":["ae2aa31Additional9a-h3"],"title":"Distribution","text":"Use $$t_{df}$$. Therefore, the distribution for the test is $$t_9$$ where $$n=10$$ and $$df=10-1=9$$. Now, calculate the $$p-value$$ using the Student\'s $$t-distribution$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional9a-h5","type":"hint","dependencies":["ae2aa31Additional9a-h4"],"title":"P-value","text":"$$p-value=P\\\\left(x>67\\\\right)=0.0396$$ where the sample mean and sample standard deviation are calculated as $$67$$ and $$3.1972$$ from the data.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional9a-h6","type":"hint","dependencies":["ae2aa31Additional9a-h5"],"title":"Compare","text":"Compare \\\\alpha and the $$p-value$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Additional9a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Greater than"],"dependencies":["ae2aa31Additional9a-h6"],"title":"Compare","text":"Is \\\\alpha less than or greater than the $$p-value$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Less than","Greater than"]},{"id":"ae2aa31Additional9a-h8","type":"hint","dependencies":["ae2aa31Additional9a-h7"],"title":"Interpret","text":"Since \\\\alpha>p-value, we would reject $$H_0$$. This means you reject $$\u03bc=65$$. In other words, you believe the average test score is greater than $$65$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae2aa31Addtional1","title":"Left-, Right-, and Two-Tailed Test","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Additional Information and Full Hypothesis Test Examples","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae2aa31Addtional1a","stepAnswer":["$$Left-tailed$$"],"problemType":"MultipleChoice","stepTitle":"$$H_0$$: $$\u03bc=10$$, $$H_a$$: $$\u03bc<10$$. Assume the $$p-value$$ is $$0.0935$$. What type of test is this?","stepBody":"","answerType":"string","variabilization":{},"choices":["$$Left-tailed$$","$$Right-tailed$$","$$Two-Tailed$$"],"hints":{"DefaultPathway":[{"id":"ae2aa31Addtional1a-h1","type":"hint","dependencies":[],"title":"Tail","text":"$$H_a$$ tells you the whether the tail is left-, right, or two-tailed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Addtional1a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Less"],"dependencies":["ae2aa31Addtional1a-h1"],"title":"Compare","text":"Is $$H_a$$ greater, less, or not equal to the mean?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Greater","Less","Not Equal"]},{"id":"ae2aa31Addtional1a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Left"],"dependencies":["ae2aa31Addtional1a-h2"],"title":"Interpret","text":"Does that mean we are looking to the left, right, or both sides of the mean?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Left","Right","Both Sides"]}]}}]},{"id":"ae2aa31Addtional2","title":"Left-, Right-, and Two-Tailed Test","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Additional Information and Full Hypothesis Test Examples","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae2aa31Addtional2a","stepAnswer":["$$Left-tailed$$"],"problemType":"MultipleChoice","stepTitle":"Ho: $$\u03bc=5$$, $$H_a$$: $$\u03bc<5$$. What type of test is this?","stepBody":"","answerType":"string","variabilization":{},"choices":["$$Left-tailed$$","$$Right-tailed$$","$$Two-Tailed$$"],"hints":{"DefaultPathway":[{"id":"ae2aa31Addtional2a-h1","type":"hint","dependencies":[],"title":"Tail","text":"$$H_a$$ tells you the whether the tail is left-, right, or two-tailed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Addtional2a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Less"],"dependencies":["ae2aa31Addtional2a-h1"],"title":"Compare","text":"Is $$H_a$$ greater, less, or not equal to the mean?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Greater","Less","Not Equal"]},{"id":"ae2aa31Addtional2a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Left"],"dependencies":["ae2aa31Addtional2a-h2"],"title":"Interpret","text":"Does that mean we are looking to the left, right, or both sides of the mean?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Left","Right","Both Sides"]}]}}]},{"id":"ae2aa31Addtional3","title":"Left-, Right-, and Two-Tailed Test","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.5 Additional Information and Full Hypothesis Test Examples","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae2aa31Addtional3a","stepAnswer":["$$Right-tailed$$"],"problemType":"MultipleChoice","stepTitle":"Ho: $$H_0$$: $$p \\\\leq 0.2$$ and $$H_a$$: $$p>0.2$$ What type of test is this?","stepBody":"","answerType":"string","variabilization":{},"choices":["$$Left-tailed$$","$$Right-tailed$$","$$Two-Tailed$$"],"hints":{"DefaultPathway":[{"id":"ae2aa31Addtional3a-h1","type":"hint","dependencies":[],"title":"Tail","text":"$$H_a$$ tells you the whether the tail is left-, right, or two-tailed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae2aa31Addtional3a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Greater"],"dependencies":["ae2aa31Addtional3a-h1"],"title":"Compare","text":"Is $$H_a$$ greater, less, or not equal to the mean?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Greater","Less","Not Equal"]},{"id":"ae2aa31Addtional3a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Right"],"dependencies":["ae2aa31Addtional3a-h2"],"title":"Interpret","text":"Does that mean we are looking to the left, right, or both sides of the mean?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Left","Right","Both Sides"]}]}}]},{"id":"ae3c1a1exponential1","title":"Identifying Exponential Functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae3c1a1exponential1a","stepAnswer":["$$x^3$$"],"problemType":"MultipleChoice","stepTitle":"Which of the following equation are not exponential functions?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x^3$$","choices":["$$4^{3\\\\left(x-2\\\\right)}$$","$$x^3$$","$${\\\\left(\\\\frac{1}{3}\\\\right)}^x$$","$${\\\\left(-2\\\\right)}^x$$"],"hints":{"DefaultPathway":[{"id":"ae3c1a1exponential1a-h1","type":"hint","dependencies":[],"title":"Definition of Exponent","text":"What two components make up an exponential function?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential1a-h2","type":"hint","dependencies":["ae3c1a1exponential1a-h1"],"title":"Matching Definition with Examples","text":"Determine which choices don\'t meet up with the two components of the exponential function (constant as a base and independent variable as an exponent)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae3c1a1exponential10","title":"Evaluating Functinos with Base e","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae3c1a1exponential10a","stepAnswer":["$$0.60653$$"],"problemType":"TextBox","stepTitle":"Calculate $$e^{-0.5}$$. Round to five decimal places.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.60653$$","hints":{"DefaultPathway":[{"id":"ae3c1a1exponential10a-h1","type":"hint","dependencies":[],"title":"Locating $$e^x$$","text":"Press the button labeled $$e^x$$ on the calculator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential10a-h2","type":"hint","dependencies":["ae3c1a1exponential10a-h1"],"title":"Subsituting Values","text":"Input the value $$3.14$$ as the exponent and press enter.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae3c1a1exponential11","title":"Evaluating Exponential Function","body":"Evaluate the function. Round the answer to four decimal places, if necessary.","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"6.1 Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae3c1a1exponential11a","stepAnswer":["$$0.016$$"],"problemType":"TextBox","stepTitle":"$$f(x)=2\\\\times5^x$$, for $$f(-3)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.016$$","hints":{"DefaultPathway":[{"id":"ae3c1a1exponential11a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2\\\\times5^{\\\\left(-3\\\\right)}$$"],"dependencies":[],"title":"Using Substitution","text":"Substitute $$x=-3$$ into f(x). What is the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{125}$$"],"dependencies":["ae3c1a1exponential11a-h1"],"title":"Using Simplification","text":"Simplify the power. What is $$5^{\\\\left(-3\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential11a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{5}$$"],"dependencies":["ae3c1a1exponential11a-h2"],"title":"Using Simplification","text":"We can sequentially simplify the expression by first calculating $$5^{\\\\left(-1\\\\right)}$$, then later calculating that value to the power of $$3$$. What is $$5^{\\\\left(-1\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{125}$$"],"dependencies":["ae3c1a1exponential11a-h3"],"title":"Using Simplification","text":"What is $${\\\\left(\\\\frac{1}{5}\\\\right)}^3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.016$$"],"dependencies":["ae3c1a1exponential11a-h4"],"title":"Using Multiplication","text":"What is $$2\\\\frac{1}{125}$$? Express the value numerically. (Round to four decimal places if necessary)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae3c1a1exponential12","title":"Evaluating Exponential Function","body":"Evaluate the function. Round the answer to four decimal places, if necessary.","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"6.1 Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae3c1a1exponential12a","stepAnswer":["$$-4$$"],"problemType":"TextBox","stepTitle":"$$f(x)={\\\\left(-4\\\\right)}^{2x+3}$$, for $$f(-1)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-4$$","hints":{"DefaultPathway":[{"id":"ae3c1a1exponential12a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$${\\\\left(-4\\\\right)}^{2\\\\left(-1\\\\right)+3}$$"],"dependencies":[],"title":"Using Substitution","text":"Substitute $$x=-1$$ into f(x). What is the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ae3c1a1exponential12a-h1"],"title":"Add the Exponent","text":"What is the sum of the exponent?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential12a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["ae3c1a1exponential12a-h2"],"title":"Using Simplification","text":"Simplify the power. What is the value?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae3c1a1exponential13","title":"Evaluating Exponential Function","body":"Evaluate the function. Round the answer to four decimal places, if necessary.","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"6.1 Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae3c1a1exponential13a","stepAnswer":["$$20.0855$$"],"problemType":"TextBox","stepTitle":"$$f(x)=e^x$$, for f(3)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$20.0855$$","hints":{"DefaultPathway":[{"id":"ae3c1a1exponential13a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$e^3$$"],"dependencies":[],"title":"Using Substitution","text":"Substitute $$x=3$$ into f(x). What is the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential13a-h2","type":"hint","dependencies":["ae3c1a1exponential13a-h1"],"title":"Using Simplification","text":"Simplify the power. Use a calculator to obtain the numeric value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae3c1a1exponential14","title":"Evaluating Exponential Function","body":"Evaluate the function. Round the answer to four decimal places, if necessary.","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"6.1 Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae3c1a1exponential14a","stepAnswer":["$$-0.2707$$"],"problemType":"TextBox","stepTitle":"$$f(x)=-2e^{x-1}$$, for $$f(-1)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-0.2707$$","hints":{"DefaultPathway":[{"id":"ae3c1a1exponential14a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2e^{\\\\left(-1-1\\\\right)}$$"],"dependencies":[],"title":"Using Substitution","text":"Substitute $$x=-1$$ into f(x). What is the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["ae3c1a1exponential14a-h1"],"title":"Add the Exponent","text":"What is the sum of the exponent?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential14a-h3","type":"hint","dependencies":["ae3c1a1exponential14a-h2"],"title":"Using Simplification","text":"Simplify the power. Use a calculator to obtain the numeric value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential14a-h4","type":"hint","dependencies":["ae3c1a1exponential14a-h3"],"title":"Using Multiplication","text":"Multiply the terms in the expression together. Use a calculator to obtain the numeric value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae3c1a1exponential15","title":"Evaluating Exponential Function","body":"Evaluate the function. Round the answer to four decimal places, if necessary.","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"6.1 Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae3c1a1exponential15a","stepAnswer":["$$174.3$$"],"problemType":"TextBox","stepTitle":"$$f(x)=2.7\\\\times4^{\\\\left(-x+1\\\\right)}+1.5$$, for $$f(-2)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$174.3$$","hints":{"DefaultPathway":[{"id":"ae3c1a1exponential15a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2.7\\\\times4^{\\\\left(-\\\\left(-2\\\\right)+1\\\\right)}+1.5$$"],"dependencies":[],"title":"Using Substitution","text":"Substitute $$x=-2$$ into f(x). What is the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ae3c1a1exponential15a-h1"],"title":"Add the Exponent","text":"What is the sum of the exponent?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$64$$"],"dependencies":["ae3c1a1exponential15a-h2"],"title":"Using Simplification","text":"Simplify the power, $$4^3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$172.8$$"],"dependencies":["ae3c1a1exponential15a-h3"],"title":"Using Multiplication","text":"Multiply the terms in the expression, $$2.7\\\\times64$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential15a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$174.3$$"],"dependencies":["ae3c1a1exponential15a-h4"],"title":"Using Addition","text":"Add all the terms together, $$172.8+1.5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae3c1a1exponential16","title":"Evaluating Exponential Function","body":"Evaluate the function. Round the answer to four decimal places, if necessary.","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"6.1 Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae3c1a1exponential16a","stepAnswer":["$$483.8146$$"],"problemType":"TextBox","stepTitle":"$$f(x)=1.2e^{2x}-0.3$$, for f(3)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$483.8146$$","hints":{"DefaultPathway":[{"id":"ae3c1a1exponential16a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.2e^{2\\\\times3}-0.3$$"],"dependencies":[],"title":"Using Substitution","text":"Substitute $$x=3$$ into f(x). What is the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["ae3c1a1exponential16a-h1"],"title":"Add the Exponent","text":"What is the sum of the exponent?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential16a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$484.1146$$"],"dependencies":["ae3c1a1exponential16a-h2"],"title":"Using Multiplication","text":"Multiply the terms in the expression, $$1.2e^6$$. Use a calculator to obtain the numeric value. Round to four decimal places, if necessary.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential16a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$483.8146$$"],"dependencies":["ae3c1a1exponential16a-h3"],"title":"Using Addition","text":"Add all the terms together, $$484.1146-0.3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae3c1a1exponential17","title":"Evaluating Exponential Function","body":"Evaluate the function. Round the answer to four decimal places, if necessary.","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"6.1 Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae3c1a1exponential17a","stepAnswer":["$$1.3333$$"],"problemType":"TextBox","stepTitle":"$$f(x)=\\\\frac{-3}{2} 3^{\\\\left(-x\\\\right)}+\\\\frac{3}{2}$$, for f(2)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1.3333$$","hints":{"DefaultPathway":[{"id":"ae3c1a1exponential17a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-3}{2} 3^{\\\\left(-2\\\\right)}+\\\\frac{3}{2}$$"],"dependencies":[],"title":"Using Substitution","text":"Substitute $$x=2$$ into f(x). What is the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential17a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{9}$$"],"dependencies":["ae3c1a1exponential17a-h1"],"title":"Using Simplification","text":"Simplify the power, $$3^{\\\\left(-2\\\\right)}$$. Leave the answer as a fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential17a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{6}$$"],"dependencies":["ae3c1a1exponential17a-h2"],"title":"Using Multiplication","text":"Multiply the terms in the expression, $$\\\\left(-\\\\frac{3}{2}\\\\right) \\\\frac{1}{9}$$. Leave the answer as a fraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential17a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.3333$$"],"dependencies":["ae3c1a1exponential17a-h3"],"title":"Using Addition","text":"Add all the terms together, $$\\\\frac{-1}{6}+\\\\frac{3}{2}$$. Evaluate this value numerically. Round to four decimal places, if necessary.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae3c1a1exponential18","title":"Evaluating a Real-World Exponential Model","body":"","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"6.1 Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae3c1a1exponential18a","stepAnswer":["$$47622$$"],"problemType":"TextBox","stepTitle":"The fox population in a certain region has an annual growth rate of 9% per year. In the year $$2012$$, there were 23,900 fox counted in the area. What is the fox population predicted to be in the year 2020? Round down to the nearest whole number.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$47622$$","hints":{"DefaultPathway":[{"id":"ae3c1a1exponential18a-h1","type":"hint","dependencies":[],"title":"Exponential Growth","text":"A function that models exponential growth grows by a rate proportional to the amount present. For any real number $$x$$ and any positive real numbers a and $$b$$ such that $$b \\\\neq 1$$, an exponential growth function has the form $$f(x)=a b^x$$\\\\nwhere a is the initial or starting value of the function and $$b$$ is the growth factor or growth multiplier per unit $$x$$ .","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential18a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$23900{1.09}^x$$"],"dependencies":["ae3c1a1exponential18a-h1"],"title":"Defining the Exponential Model","text":"The function for exponential growth is given by $$f(x)=a b^x$$. What is exponential model, f(x), for this question?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential18a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$23900$$"],"dependencies":["ae3c1a1exponential18a-h2"],"title":"Defining the Exponential Model","text":"The function for exponential growth is given by $$f(x)=a b^x$$. What is the initial or starting value of the function, a?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential18a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.09$$"],"dependencies":["ae3c1a1exponential18a-h3"],"title":"Defining the Exponential Model","text":"The function for exponential growth is given by $$f(x)=a b^x$$. What is the growth factor or growth multiplier per unit $$x$$, $$b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential18a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["ae3c1a1exponential18a-h4"],"title":"Period of Growth","text":"How many years ahead are we trying to predict with our model?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential18a-h6","type":"hint","dependencies":["ae3c1a1exponential18a-h5"],"title":"Using Substitution","text":"Substitute $$x=8$$ into f(x) to evaluate the fox population in the year $$2020$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae3c1a1exponential19","title":"Calculating Continuous Decay","body":"","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"6.1 Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae3c1a1exponential19a","stepAnswer":["$$34.3$$"],"problemType":"TextBox","stepTitle":"A scientist begins with $$100$$ milligrams of a radioactive substance that decays exponentially. After $$35$$ hours, 50mg of the substance remains. How many milligrams will remain after $$54$$ hours? Round the answer to $$3$$ significant figures.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$34.3$$","hints":{"DefaultPathway":[{"id":"ae3c1a1exponential19a-h1","type":"hint","dependencies":[],"title":"The Continuous $$\\\\frac{Growth}{Decay}$$ Formula","text":"For all real numbers $$t$$, and all positive numbers a and $$r$$, continuous growth or decay is represented by the formula $$A(t)=a e^{r t}$$ where:\\\\na is the initial value,\\\\n$$r$$ is the continuous growth rate per unit time,\\\\nand $$t$$ is the elapsed time.\\\\n\\\\nIf $$r>0$$ , then the formula represents continuous growth. If $$r<0$$ , then the formula represents continuous decay.\\\\n\\\\nFor business applications, the continuous growth formula is called the continuous compounding formula and takes the form $$A(t)=P e^{r t}$$\\\\nwhere:\\\\nP is the principal or the initial invested,\\\\n$$r$$ is the growth or interest rate per unit time,\\\\nand $$t$$ is the period or term of the investment.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential19a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Continuous Decay"],"dependencies":["ae3c1a1exponential19a-h1"],"title":"Model","text":"What type of model is this?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Continuous Growth","Continuous Decay"]},{"id":"ae3c1a1exponential19a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Continuous growth rate per unit time, $$r$$"],"dependencies":["ae3c1a1exponential19a-h2"],"title":"Defining the Exponential Model","text":"Identify the terms of the model, $$A(t)=a e^{r t}$$, that are given and the terms that you would have to find. Which terms are not given?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Continuous growth rate per unit time, $$r$$","initial value, a"]},{"id":"ae3c1a1exponential19a-h4","type":"hint","dependencies":["ae3c1a1exponential19a-h3"],"title":"Continuous Growth Rate","text":"Find the continuous Growth Rate per unit time.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential19a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$50=100e^{35r}$$"],"dependencies":["ae3c1a1exponential19a-h4"],"title":"Continuous Growth Rate","text":"Substitute all known terms (a - initial values, $$t$$ - elapsed time, A(t) - model\'s estimate) provided in the question into the exponential decay model, $$A(t)=a e^{r t}$$ and find $$r$$, the continuous growth rate. What is the equation after substituting the values in?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$50=100e^{35r}$$","$$100=50e^r$$","$$50=a e^{35r}$$"]},{"id":"ae3c1a1exponential19a-h6","type":"hint","dependencies":["ae3c1a1exponential19a-h5"],"title":"Continuous Growth Rate","text":"Divide by $$100$$ on both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential19a-h7","type":"hint","dependencies":["ae3c1a1exponential19a-h6"],"title":"Continuous Growth Rate","text":"Take ln on both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential19a-h8","type":"hint","dependencies":["ae3c1a1exponential19a-h7"],"title":"Continuous Growth Rate","text":"Divide by $$35$$ on both sides. You would have successfully isolated $$r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential19a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-0.0198$$"],"dependencies":["ae3c1a1exponential19a-h8"],"title":"Continuous Growth Rate","text":"Evaluate $$r$$ numerically. Leave the answer in four decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential19a-h10","type":"hint","dependencies":["ae3c1a1exponential19a-h9"],"title":"Predicting the Weight","text":"Now that you have found all the terms for your exponential model, substitute $$t=54$$ to find the weight of the radioactive substance.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae3c1a1exponential2","title":"Identifying Exponential Functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae3c1a1exponential2a","stepAnswer":["$${0.875}^x$$"],"problemType":"MultipleChoice","stepTitle":"Which of the following equations represent exponential functions?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$${0.875}^x$$","choices":["$$2x^2-3x+1$$","$${0.875}^x$$","$$1.75x+2$$","$${1095.6}^{\\\\left(-2x\\\\right)}$$"],"hints":{"DefaultPathway":[{"id":"ae3c1a1exponential2a-h1","type":"hint","dependencies":[],"title":"Definition of Exponent","text":"What two components make up an exponential function?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential2a-h2","type":"hint","dependencies":["ae3c1a1exponential2a-h1"],"title":"Matching Definition with Examples","text":"Determine which choices don\'t meet up with the two components of the exponential function (constant as a base and independent variable as an exponent)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae3c1a1exponential20","title":"Calculating Continuous Growth","body":"","variabilization":{},"oer":"https://openstax.org/","license":0,"lesson":"6.1 Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae3c1a1exponential20a","stepAnswer":["$$1.38$$"],"problemType":"TextBox","stepTitle":"In the year $$1985$$, a house was valued at $110,000. By the year $$2005$$, the value had appreciated to $145,000. What was the percentage annual growth rate between $$1985$$ and 2005? Leave the answer in three significant figures.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1.38$$","hints":{"DefaultPathway":[{"id":"ae3c1a1exponential20a-h1","type":"hint","dependencies":[],"title":"The Continuous $$\\\\frac{Growth}{Decay}$$ Formula","text":"For all real numbers $$t$$, and all positive numbers a and $$r$$, continuous growth or decay is represented by the formula $$A(t)=a e^{r t}$$ where:\\\\na is the initial value,\\\\n$$r$$ is the continuous growth rate per unit time,\\\\nand $$t$$ is the elapsed time.\\\\n\\\\nIf $$r>0$$ , then the formula represents continuous growth. If $$r<0$$ , then the formula represents continuous decay.\\\\n\\\\nFor business applications, the continuous growth formula is called the continuous compounding formula and takes the form $$A(t)=P e^{r t}$$\\\\nwhere:\\\\nP is the principal or the initial invested,\\\\n$$r$$ is the growth or interest rate per unit time,\\\\nand $$t$$ is the period or term of the investment.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential20a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Continuous Growth"],"dependencies":["ae3c1a1exponential20a-h1"],"title":"Model","text":"What type of model is this?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Continuous Growth","Continuous Decay"]},{"id":"ae3c1a1exponential20a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Continuous growth rate per unit time, $$r$$"],"dependencies":["ae3c1a1exponential20a-h2"],"title":"Defining the Exponential Model","text":"Identify the terms of the model, $$A(t)=P e^{r t}$$, that are given and the terms that you would have to find. Which terms are not given?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Continuous growth rate per unit time, $$r$$","initial value, P"]},{"id":"ae3c1a1exponential20a-h4","type":"hint","dependencies":["ae3c1a1exponential20a-h3"],"title":"Continuous Growth Rate","text":"Find the continuous Growth Rate per unit time.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential20a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$145000=110000e^{20r}$$"],"dependencies":["ae3c1a1exponential20a-h4"],"title":"Continuous Growth Rate","text":"Substitute all known terms (P - initial values, $$t$$ - elapsed time, A(t) - model\'s estimate) provided in the question into the exponential decay model, $$A(t)=P e^{r t}$$ and find $$r$$, the continuous growth rate. What is the equation after substituting the values in?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$145000=110000e^{20r}$$","$$110000=145000e^{20r}$$","$$145000=35e^{110000r}$$"]},{"id":"ae3c1a1exponential20a-h6","type":"hint","dependencies":["ae3c1a1exponential20a-h5"],"title":"Continuous Growth Rate","text":"Divide by $$110000$$ on both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential20a-h7","type":"hint","dependencies":["ae3c1a1exponential20a-h6"],"title":"Continuous Growth Rate","text":"Take ln on both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential20a-h8","type":"hint","dependencies":["ae3c1a1exponential20a-h7"],"title":"Continuous Growth Rate","text":"Divide by $$20$$ on both sides. You would have successfully isolated $$r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential20a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.0138$$"],"dependencies":["ae3c1a1exponential20a-h8"],"title":"Continuous Growth Rate","text":"Evaluate $$r$$, the continuous growth rate, numerically. Leave the answer in three significant figures.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ae3c1a1exponential20b","stepAnswer":["$$155368.09$$"],"problemType":"TextBox","stepTitle":"Assume that the value continued to grow by the same percentage (Use the exact value that was found in the previous part). What was the value of the house in the year 2010? Round the answer to two decimal places.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$155368.09$$","hints":{"DefaultPathway":[{"id":"ae3c1a1exponential20b-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":[],"title":"Using Substitution","text":"How many years have elapsed since the first year, 1985?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential20b-h2","type":"hint","dependencies":["ae3c1a1exponential20b-h1"],"title":"Using Substitution","text":"Substitute $$t=25$$ into the exponential model that was found.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae3c1a1exponential21","title":"Exercise #1: Finding Exponential Formulas","body":"Find the formula for an exponential function that passes through the two points given. $$f(x)=$$? (Input the right side of the equation as your answer.)","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae3c1a1exponential21a","stepAnswer":["$$6\\\\times5^x$$"],"problemType":"TextBox","stepTitle":"$$(0,6)$$ and $$(3,750)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6\\\\times5^x$$","hints":{"DefaultPathway":[{"id":"ae3c1a1exponential21a-h1","type":"hint","dependencies":[],"title":"Finding Exponential Models Given a Point of Form (0,a)","text":"If one of the data points has the form (0,a), then a is the initial value. Using a, substitute the second point into the equation $$f(x)={a\\\\left(b\\\\right)}^x$$, and solve for $$b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential21a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["ae3c1a1exponential21a-h1"],"title":"Identifying \\"a\\" And Solving for \\"b\\"","text":"In this problem, the coordinate $$(0,6)$$ is given. Thus, following the previous hint, $$a=6$$. We can then plug in the other given point, $$(3,750)$$, to $$f(x)={a\\\\left(b\\\\right)}^x$$. This gives us $$750={6\\\\left(b\\\\right)}^3$$. What is $$b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential21a-h3","type":"hint","dependencies":["ae3c1a1exponential21a-h2"],"title":"Writing the Formula Knowing a and $$b$$","text":"Write the formula in the form $$f(x)={a\\\\left(b\\\\right)}^x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae3c1a1exponential22","title":"Exercise #2: Finding Exponential Formulas","body":"Find the formula for an exponential function that passes through the two points given. $$f(x)=$$? (Input the right side of the equation as your answer.)","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae3c1a1exponential22a","stepAnswer":["$$2000{\\\\left(\\\\frac{1}{\\\\sqrt{10}}\\\\right)}^x$$"],"problemType":"TextBox","stepTitle":"$$(0,2000)$$ and $$(2,20)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2000{\\\\left(\\\\frac{1}{\\\\sqrt{10}}\\\\right)}^x$$","hints":{"DefaultPathway":[{"id":"ae3c1a1exponential22a-h1","type":"hint","dependencies":[],"title":"Finding Exponential Models Given a Point of Form (0,a)","text":"If one of the data points has the form (0,a), then a is the initial value. Using a, substitute the second point into the equation $$f(x)={a\\\\left(b\\\\right)}^x$$, and solve for $$b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential22a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{\\\\sqrt{10}}$$"],"dependencies":["ae3c1a1exponential22a-h1"],"title":"Identifying \\"a\\" And Solving for \\"b\\"","text":"In this problem, the coordinate $$(0,2000)$$ is given. Thus, following the previous hint, $$a=2000$$. We can then plug in the other given point, $$(2,20)$$, to $$f(x)={a\\\\left(b\\\\right)}^x$$. This gives us $$200={\\\\operatorname{2000}\\\\left(b\\\\right)}^2$$. What is $$b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential22a-h3","type":"hint","dependencies":["ae3c1a1exponential22a-h2"],"title":"Writing the Formula Knowing a and $$b$$","text":"Write the formula in the form $$f(x)={a\\\\left(b\\\\right)}^x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae3c1a1exponential23","title":"Exercise #3: Finding Exponential Formulas","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae3c1a1exponential23a","stepAnswer":["$$3\\\\times2^x$$"],"problemType":"TextBox","stepTitle":"(-1, 3/2) and $$(3,24)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3\\\\times2^x$$","hints":{"DefaultPathway":[{"id":"ae3c1a1exponential23a-h1","type":"hint","dependencies":[],"title":"Finding Exponential Models Given No Points of Form (0,a)","text":"We can still find an exponential formula with two points that are not in the form (0,a). First, subsitute both points into two equations with the form $$f(x)={a\\\\left(b\\\\right)}^x$$. Solve the resulting system of two equations in two unknowns to find a and $$b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential23a-h2","type":"hint","dependencies":["ae3c1a1exponential23a-h1"],"title":"Using the First Data Point","text":"Plugging in the first data point, we get $$\\\\frac{3}{2}={a\\\\left(b\\\\right)}^{\\\\left(-1\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential23a-h3","type":"hint","dependencies":["ae3c1a1exponential23a-h2"],"title":"Using the Second Data Point","text":"Plugging in the second data point, we get $$24={a\\\\left(b\\\\right)}^3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential23a-h4","type":"hint","dependencies":["ae3c1a1exponential23a-h3"],"title":"Solving the System of Equations","text":"Then, by solving for a in the first equation, we get $$a=\\\\frac{3}{2} b$$. Next, subsitute this value for a in the second equation to solve for $$b$$. We get $$24=\\\\frac{3}{2} b b^3$$, which simplifies to $$24=\\\\frac{3}{2} b^4$$. Solve for $$b$$, and then use the numerical value of $$b$$ to find the numerical value of a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae3c1a1exponential24","title":"Exercise #4: Finding Exponential Formulas","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae3c1a1exponential24a","stepAnswer":["$$6{\\\\left(\\\\frac{1}{6}\\\\right)}^{\\\\frac{1}{5}+\\\\frac{2}{5} x}$$"],"problemType":"TextBox","stepTitle":"$$(-2,6)$$ and $$(3,1)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6{\\\\left(\\\\frac{1}{6}\\\\right)}^{\\\\frac{1}{5}+\\\\frac{2}{5} x}$$","hints":{"DefaultPathway":[{"id":"ae3c1a1exponential24a-h1","type":"hint","dependencies":[],"title":"Finding Exponential Models Given No Points of Form (0,a)","text":"We can still find an exponential formula with two points that are not in the form (0,a). First, subsitute both points into two equations with the form $$f(x)={a\\\\left(b\\\\right)}^x$$. Solve the resulting system of two equations in two unknowns to find a and $$b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential24a-h2","type":"hint","dependencies":["ae3c1a1exponential24a-h1"],"title":"Using the First Data Point","text":"Plugging in the first data point, we get $$6={a\\\\left(b\\\\right)}^{\\\\left(-2\\\\right)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential24a-h3","type":"hint","dependencies":["ae3c1a1exponential24a-h2"],"title":"Using the Second Data Point","text":"Plugging in the second data point, we get $$1={a\\\\left(b\\\\right)}^3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential24a-h4","type":"hint","dependencies":["ae3c1a1exponential24a-h3"],"title":"Solving the System of Equations","text":"Then, by solving for a in the first equation, we get $$a=6b^2$$. Next, subsitute this value for a in the second equation to solve for $$b$$. We get $$1=6b^2 b^3$$, which simplifies to $$1=6b^5$$. Solve for $$b$$, and then use the numerical value of $$b$$ to find the numerical value of a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae3c1a1exponential25","title":"Exercise #5: Finding Exponential Formulas","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae3c1a1exponential25a","stepAnswer":["$$\\\\frac{1}{8} 2^x$$"],"problemType":"TextBox","stepTitle":"$$(3,1)$$ and $$(5,4)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{8} 2^x$$","hints":{"DefaultPathway":[{"id":"ae3c1a1exponential25a-h1","type":"hint","dependencies":[],"title":"Finding Exponential Models Given No Points of Form (0,a)","text":"We can still find an exponential formula with two points that are not in the form (0,a). First, subsitute both points into two equations with the form $$f(x)={a\\\\left(b\\\\right)}^x$$. Solve the resulting system of two equations in two unknowns to find a and $$b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential25a-h2","type":"hint","dependencies":["ae3c1a1exponential25a-h1"],"title":"Using the First Data Point","text":"Plugging in the first data point, we get $$1={a\\\\left(b\\\\right)}^3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential25a-h3","type":"hint","dependencies":["ae3c1a1exponential25a-h2"],"title":"Using the Second Data Point","text":"Plugging in the second data point, we get $$4={a\\\\left(b\\\\right)}^5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential25a-h4","type":"hint","dependencies":["ae3c1a1exponential25a-h3"],"title":"Solving the System of Equations","text":"Then, by solving for a in the first equation, we get $$a=\\\\frac{1}{b^3}$$. Next, subsitute this value for a in the second equation to solve for $$b$$. We get $$4=\\\\frac{1}{b^3} b^5$$, which simplifies to $$4=b^2$$. Solve for $$b$$, and then use the numerical value of $$b$$ to find the numerical value of a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae3c1a1exponential26","title":"Exercise #1: Applying the Compound-Interest Formula","body":"After a certain number of years, the value of an investment account is represented by the equation $$A={\\\\operatorname{10250}\\\\left(1+\\\\frac{0.04}{12}\\\\right)}^{120}$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae3c1a1exponential26a","stepAnswer":["$$15281.03$$"],"problemType":"TextBox","stepTitle":"What is the value of the account? Round to the hundredths place.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$15281.03$$","hints":{"DefaultPathway":[{"id":"ae3c1a1exponential26a-h1","type":"hint","dependencies":[],"title":"Value of the Account in the Equation","text":"The value of the account is A.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae3c1a1exponential27","title":"Exercise #2: Applying the Compound-Interest Formula","body":"After a certain number of years, the value of an investment account is represented by the equation $$A={\\\\operatorname{10250}\\\\left(1+\\\\frac{0.04}{12}\\\\right)}^{120}$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae3c1a1exponential27a","stepAnswer":["$$10250$$"],"problemType":"TextBox","stepTitle":"What was the initial deposit made to the account?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10250$$","hints":{"DefaultPathway":[{"id":"ae3c1a1exponential27a-h1","type":"hint","dependencies":[],"title":"Identifying the Initial Deposit","text":"The initial deposit is the initial value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae3c1a1exponential28","title":"Exercise #3: Applying the Compound-Interest Formula","body":"After a certain number of years, the value of an investment account is represented by the equation $$A={\\\\operatorname{10250}\\\\left(1+\\\\frac{0.04}{12}\\\\right)}^{120}$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae3c1a1exponential28a","stepAnswer":["$$10$$"],"problemType":"TextBox","stepTitle":"How many years had the account been accumulating interest?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10$$","hints":{"DefaultPathway":[{"id":"ae3c1a1exponential28a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":[],"title":"Finding the Years","text":"In a compound-interest equation $$A(t)={P\\\\left(1+\\\\frac{r}{n}\\\\right)}^{nt}$$, $$n$$ is the number of compounding periods in one year and $$t$$ is measured in years. From the equation, we see that $$n=12$$ and $$nt=120$$. What is $$t$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae3c1a1exponential29","title":"Exercise #4: Applying the Compound-Interest Formula","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae3c1a1exponential29a","stepAnswer":["$$13268.58$$"],"problemType":"TextBox","stepTitle":"An account is opened with an initial deposite of $6500 and earns $$3.6$$ interest compounded semi-annually. What will the account be worth in $$20$$ years? Round to the hundredths place.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$13268.58$$","hints":{"DefaultPathway":[{"id":"ae3c1a1exponential29a-h1","type":"hint","dependencies":[],"title":"Compound-Interest Equation","text":"In a compound-interest equation $$A(t)={P\\\\left(1+\\\\frac{r}{n}\\\\right)}^{nt}$$, A(t) is the account value, $$t$$ is measured in years, P is the starting amount of the account, $$r$$ is the annual percentage rate expressed as a decimal, and $$n$$ is the number of compounding periods in one year.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential29a-h2","type":"hint","dependencies":["ae3c1a1exponential29a-h1"],"title":"Subsituting in Values","text":"By plugging the the values given by the problem into the compound interest equation, we get $$A(t)={\\\\operatorname{6500}\\\\left(1+\\\\frac{0.036}{2}\\\\right)}^{40}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae3c1a1exponential3","title":"Evaluating Exponential Functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae3c1a1exponential3a","stepAnswer":["$$135$$"],"problemType":"TextBox","stepTitle":"Let $$f(x)={5\\\\left(3\\\\right)}^{x+1}$$. Evaluate f(2) without a calculator.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$135$$","hints":{"DefaultPathway":[{"id":"ae3c1a1exponential3a-h1","type":"hint","dependencies":[],"title":"Using Substitution","text":"Substitute $$x=2$$ into the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential3a-h2","type":"hint","dependencies":["ae3c1a1exponential3a-h1"],"title":"Order of Operations","text":"Identify the highest order of operations that can be applied to the proble.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential3a-h3","type":"hint","dependencies":["ae3c1a1exponential3a-h2"],"title":"Simplifying the Equation","text":"Using your knowledge of the order of operations, simplify the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ae3c1a1exponential3a-h3"],"title":"Using Addition","text":"What is $$2+1$$ in the exponent?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$27$$"],"dependencies":["ae3c1a1exponential3a-h4"],"title":"Using Exponents","text":"$$3^3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential3a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$135$$"],"dependencies":["ae3c1a1exponential3a-h5"],"title":"Using Multiplication","text":"$$5\\\\times27$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae3c1a1exponential30","title":"Exercise #4: Applying the Compound-Interest Formula","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae3c1a1exponential30a","stepAnswer":["$$81.91$$"],"problemType":"TextBox","stepTitle":"An account is opened with an initial deposite of $6500 and earns $$3.6$$ interest compounded semi-annually. What will the account be worth in $$20$$ years? How much more would it have been worth if the account was compounded weekly? Only input the answer to the second question. Round to the hundredths place.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$81.91$$","hints":{"DefaultPathway":[{"id":"ae3c1a1exponential30a-h1","type":"hint","dependencies":[],"title":"Compound-Interest Equation","text":"In a compound-interest equation $$A(t)={P\\\\left(1+\\\\frac{r}{n}\\\\right)}^{nt}$$, A(t) is the account value, $$t$$ is measured in years, P is the starting amount of the account, $$r$$ is the annual percentage rate expressed as a decimal, and $$n$$ is the number of compounding periods in one year.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential30a-h2","type":"hint","dependencies":["ae3c1a1exponential30a-h1"],"title":"Solving for the Value of the Actual Account","text":"By plugging the the values given by the problem into the compound interest equation, we get $$A(t)={\\\\operatorname{6500}\\\\left(1+\\\\frac{0.036}{2}\\\\right)}^{40}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential30a-h3","type":"hint","dependencies":["ae3c1a1exponential30a-h2"],"title":"Solving for the Value of the Account if Compounded Weekly","text":"If the account was compounded weekly, it would be compounded $$52$$ times in a year. $$A(t)={\\\\operatorname{6500}\\\\left(1+\\\\frac{0.036}{52}\\\\right)}^{1040}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential30a-h4","type":"hint","dependencies":["ae3c1a1exponential30a-h3"],"title":"Finding the Difference","text":"The difference between the two accounts is $${\\\\operatorname{6500}\\\\left(1+\\\\frac{0.036}{52}\\\\right)}^{1040}-{\\\\operatorname{6500}\\\\left(1+\\\\frac{0.036}{2}\\\\right)}^{40}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae3c1a1exponential4","title":"Evaluating Exponential Functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae3c1a1exponential4a","stepAnswer":["$$5.5556$$"],"problemType":"TextBox","stepTitle":"Let $$f(x)={8\\\\left(1.2\\\\right)}^{x-5}$$. Evaluate f(3) using a calculator. Round to four decimal places.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5.5556$$","hints":{"DefaultPathway":[{"id":"ae3c1a1exponential4a-h1","type":"hint","dependencies":[],"title":"Using Substitution","text":"Substitue $$x=3$$ into the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential4a-h2","type":"hint","dependencies":["ae3c1a1exponential4a-h1"],"title":"Simplifying the Equation","text":"Using your knowledge of the order of operations, simplify the equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["ae3c1a1exponential4a-h2"],"title":"Subtracting Within the Exponent","text":"What is $$3-5$$ in the exponent?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.6944$$"],"dependencies":["ae3c1a1exponential4a-h3"],"title":"Using Exponents","text":"$${1.2}^{\\\\left(-2\\\\right)}$$ (Round to four decimal places but keep original number for calculations.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential4a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5.5556$$"],"dependencies":["ae3c1a1exponential4a-h4"],"title":"Using Multiplication","text":"$$8\\\\times0.6944$$ (Use original decimal instead of rounded four decimal places in calculation.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae3c1a1exponential5","title":"Evaluating Exponential Functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae3c1a1exponential5a","stepAnswer":["$$1.549$$"],"problemType":"TextBox","stepTitle":"At the beginning of this section, we learned that the population of India was about $$1.25$$ billion in the year $$2013$$, with an annual growth rate of about $$1.2\\\\%$$. This situation is represented by the growth function $$P(t)=1.25(1.012)t$$, where $$t$$ is the number of years since $$2013$$. To the nearest thousandth, what will the population of India be in 2031?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1.549$$","hints":{"DefaultPathway":[{"id":"ae3c1a1exponential5a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18$$"],"dependencies":[],"title":"Determining Years","text":"How many years after $$2013$$ till 2031?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential5a-h2","type":"hint","dependencies":["ae3c1a1exponential5a-h1"],"title":"Using Substitution","text":"Substitue $$t=18$$ into the population growth function","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential5a-h3","type":"hint","dependencies":["ae3c1a1exponential5a-h2"],"title":"Order of Operations","text":"Use order of operations to simplify the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.24$$"],"dependencies":["ae3c1a1exponential5a-h3"],"title":"Using Exponents","text":"$${1.012}^{18}$$ (Round to three decimal places but keep original number for calculations.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.549$$"],"dependencies":["ae3c1a1exponential5a-h4"],"title":"Using Multiplication","text":"$$1.25\\\\times1.2395$$ (Use original decimal instead of rounded three decimal places in calculation.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae3c1a1exponential6","title":"Finding Equations of Exponential Functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae3c1a1exponential6a","stepAnswer":["$$2.4492{0.6389}^2$$"],"problemType":"TextBox","stepTitle":"Find an exponential function that passes through the points $$(-2,6)$$ and $$(2,1)$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2.4492{0.6389}^2$$","hints":{"DefaultPathway":[{"id":"ae3c1a1exponential6a-h1","type":"hint","dependencies":[],"title":"Exponent Equation Form","text":"What is the equation form for exponential models?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential6a-h2","type":"hint","dependencies":["ae3c1a1exponential6a-h1"],"title":"Substitue Coordinate Values","text":"Substitute values into the exponential model equation $$f(x)={ab}^x$$ for each coordinate.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential6a-h3","type":"hint","dependencies":["ae3c1a1exponential6a-h2"],"title":"Rewrite","text":"Rewrite the equation $$6={ab}^{\\\\left(-2\\\\right)}$$ so that variable a is isolated","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential6a-h4","type":"hint","dependencies":["ae3c1a1exponential6a-h3"],"title":"Substitute Equation","text":"Substitute the equation for a from the previous hint into the other equation $$1={ab}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential6a-h5","type":"hint","dependencies":["ae3c1a1exponential6a-h4"],"title":"Simplifying the Function","text":"Use properties of exponents to isolate $$b$$ then evaluate remaining exponent","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae3c1a1exponential7","title":"Applying the Compoung-Interest Formula","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae3c1a1exponential7a","stepAnswer":["$$4045.05$$"],"problemType":"TextBox","stepTitle":"If we invest $3,000 in an investment account paying 3% interest compounded quarterly, how much will the account be worth in $$10$$ years?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4045.05$$","hints":{"DefaultPathway":[{"id":"ae3c1a1exponential7a-h1","type":"hint","dependencies":[],"title":"Compound Interest Formula","text":"The equation for calculating compound interest is $$A(t)={P\\\\left(1+\\\\frac{r}{n}\\\\right)}^{nt}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential7a-h2","type":"hint","dependencies":["ae3c1a1exponential7a-h1"],"title":"Identifying Variables","text":"Plug in the known values into the compound interest equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3000$$"],"dependencies":["ae3c1a1exponential7a-h2"],"title":"Investment Price","text":"What is the value of P?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.03$$"],"dependencies":["ae3c1a1exponential7a-h3"],"title":"Interest Rate","text":"What is the value of $$r$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential7a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["ae3c1a1exponential7a-h4"],"title":"Compounding Rate","text":"What is the value of $$n$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential7a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["ae3c1a1exponential7a-h5"],"title":"Solving for Time","text":"What is the value of $$t$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae3c1a1exponential7a-h7","type":"hint","dependencies":["ae3c1a1exponential7a-h6"],"title":"Solving the Equation","text":"Solve the equation using order of operations","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae3c1a1exponential8","title":"Applying the Compoung-Interest Formula","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.1 Exponential Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"ae3c1a1exponential8a","stepAnswer":["$$3644675.88$$"],"problemType":"TextBox","stepTitle":"An initial investment of $100,000 at 12% interest is compounded weekly (use $$52$$ weeks in a year). 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int21c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$24$$"],"dependencies":["ae45a45int21c-h1"],"title":"Rewriting the Expression","text":"What is $$9+15$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ae45a45int21d","stepAnswer":["$$-3$$"],"problemType":"TextBox","stepTitle":"$$-7-(-4)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-3$$","hints":{"DefaultPathway":[{"id":"ae45a45int21d-h1","type":"hint","dependencies":[],"title":"Rewriting the Expression","text":"Subtracting $$-4$$ is the same as adding $$4$$, since the two negative signs (one from the subtraction, the other from -4) cancel out into a +","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int21d-h2-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["ae45a45int21d-h2-s2"],"title":"Rewriting the Expression","text":"What is $$-7+3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int21d-h2-s4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["ae45a45int21d-h2-s3"],"title":"Rewriting the Expression","text":"What is $$-7+4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"ae45a45int22","title":"Add and Subtract Integers","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int22a-h3","type":"hint","dependencies":["ae45a45int22a-h2"],"title":"Solve the Expression","text":"Once the expression in the parantheses is solved into a number, subtract (from left to right because of the order of operations) to solve the expression","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int22a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$14$$"],"dependencies":["ae45a45int22a-h3"],"title":"Solve the Expression","text":"What is $$7-(-7)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int22a-h5","type":"hint","dependencies":["ae45a45int22a-h3"],"title":"Solve the Expression","text":"Subtracting $$-7$$ is the same as adding $$7$$, since the two negative signs (one from the subtraction, the other from -7) cancel out into a +","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ae45a45int22a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$14$$"],"dependencies":[],"title":"Solve the Expression","text":"What is $$7+7$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ae45a45int22a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["ae45a45int22a-h4"],"title":"Solve the Expression","text":"What is $$14-9$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae45a45int23","title":"Multiply and Divide Integers","body":"Multiply or 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int23a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["ae45a45int23a-h1"],"title":"Dividing with Negatives","text":"What is $$\\\\frac{100}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int23a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Negative"],"dependencies":["ae45a45int23a-h2"],"title":"Dividing with Negatives","text":"What is the sign of the numerator in the original problem? Negative or Positive?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Negative","Positive"]},{"id":"ae45a45int23a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Negative"],"dependencies":["ae45a45int23a-h3"],"title":"Dividing with Negatives","text":"What is the sign of the denominator in the original problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Negative","Positive"]},{"id":"ae45a45int23a-h5","type":"hint","dependencies":["ae45a45int23a-h4"],"title":"Signs","text":"Since the numerator and denominator had the the same sign in the original problem, the final answer is positive","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ae45a45int23c","stepAnswer":["$$-32$$"],"problemType":"TextBox","stepTitle":"$$4\\\\left(-8\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-32$$","hints":{"DefaultPathway":[{"id":"ae45a45int23c-h1","type":"hint","dependencies":[],"title":"Multiplying Numbers","text":"To multiply two numbers, you can multiply as if both numbers were positive, then make the final answer negative if the sign of each number is different","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int23c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$32$$"],"dependencies":["ae45a45int23c-h1"],"title":"Multiplying Numbers","text":"What is $$4\\\\times8$$?","variabilization":{},"oer":"https://OATutor.io 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answer will be negative","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ae45a45int23d","stepAnswer":["$$-9$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{-27}{3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-9$$","hints":{"DefaultPathway":[{"id":"ae45a45int23d-h1","type":"hint","dependencies":[],"title":"Dividing with Negatives","text":"To divide negative numbers, you can divide remove any negative signs and divide as if both the numerator and denominator were positive; then, if the signs of the numerator and denominator are different in the original problem (one is positive and the other is negative), then add a negative to your final answer","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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negative!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae45a45int25","title":"Simplify Expressions with Integers","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Integers","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae45a45int25a","stepAnswer":["$$49$$"],"problemType":"TextBox","stepTitle":"$${\\\\left(-7\\\\right)}^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$49$$","hints":{"DefaultPathway":[{"id":"ae45a45int25a-h1","type":"hint","dependencies":[],"title":"Exponents with Negatives","text":"The best approach to this is turning this into a multiplication problem: this is is equivalent to $$\\\\left(-7\\\\right) \\\\left(-7\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int25a-h2","type":"hint","dependencies":["ae45a45int25a-h1"],"title":"Multiplying Negative Numbers","text":"When multiplying two numbers, you can multiply as if they are both positive, then make the answer negative IF the signs of both numbers are different (ONLY ONE of the numbers is negative)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int25a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$49$$"],"dependencies":["ae45a45int25a-h2"],"title":"Multiplying Negative Numbers","text":"What is $$\\\\left(-7\\\\right) \\\\left(-7\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ae45a45int25a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$49$$"],"dependencies":[],"title":"Multiplying Negative Numbers","text":"What is $$7\\\\times7$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int25a-h3-s2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Negative"],"dependencies":["ae45a45int25a-h3-s1"],"title":"Multiplying Negative Numbers","text":"What is the sign of -7?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Negative","Positive"]}]},{"id":"ae45a45int25a-h4","type":"hint","dependencies":["ae45a45int25a-h3"],"title":"Signs","text":"Since the numbers have the same sign, the final answer will be positive","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ae45a45int25b","stepAnswer":["$$-49$$"],"problemType":"TextBox","stepTitle":"$$-\\\\left(7^2\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-49$$","hints":{"DefaultPathway":[{"id":"ae45a45int25b-h1","type":"hint","dependencies":[],"title":"Exponents with Negatives","text":"When doing a problem in the form $$-\\\\left(x^y\\\\right)$$, where $$x$$ and $$y$$ are some numbers, this is really saying $$-\\\\left(x^y\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int25b-h2","type":"hint","dependencies":["ae45a45int25b-h1"],"title":"Multiplying Negative Numbers","text":"When multiplying two numbers, you can multiply as if they are both positive, then make the answer negative IF the signs of both numbers are different (ONLY ONE of the numbers is 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Integers","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Integers","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae45a45int26a","stepAnswer":["$$9$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{8\\\\left(-9\\\\right)}{{\\\\left(-2\\\\right)}^3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9$$","hints":{"DefaultPathway":[{"id":"ae45a45int26a-h1","type":"hint","dependencies":[],"title":"Order of Operations","text":"Order of operations says that when multiplying or dividing, we must go from left to right","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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$$\\\\frac{\\\\left(-72\\\\right)}{\\\\left(-8\\\\right)}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ae45a45int26a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":[],"title":"Order of Operations","text":"What is $$\\\\frac{72}{8}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}},{"id":"ae45a45int26b","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{-30}{2}+\\\\left(-3\\\\right) \\\\left(-7\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"ae45a45int26b-h1","type":"hint","dependencies":[],"title":"Order of Operations","text":"First do all multiplying and dividing","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int26b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-15$$"],"dependencies":["ae45a45int26b-h1"],"title":"Order of Operations","text":"What is $$\\\\frac{-30}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int26b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$21$$"],"dependencies":["ae45a45int26b-h2"],"title":"Order of Operations","text":"What is (-3)*(-7)/","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int26b-h4","type":"hint","dependencies":["ae45a45int26b-h3"],"title":"Order of Operations","text":"Once all multiplication and division is done, we can add and subtract","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int26b-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["ae45a45int26b-h4"],"title":"Order of Operations","text":"What is $$-15+21$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae45a45int27","title":"Evaluate Variable Expressions with Integers","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Integers","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae45a45int27a","stepAnswer":["$$23$$"],"problemType":"TextBox","stepTitle":"Evaluate $$4x^2-2xy+3y^2$$ when $$x=2$$, 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\\\\left(-1\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int27a-h3-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ae45a45int27a-h3-s2"],"title":"Evaluation","text":"What is $$3{\\\\left(-1\\\\right)}^2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int27a-h3-s4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$23$$"],"dependencies":["ae45a45int27a-h3-s3"],"title":"Evaluation","text":"What is $$16-\\\\left(-4\\\\right)+3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}]}}]},{"id":"ae45a45int28","title":"Translate Phrases to Expressions with Integers","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Integers","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae45a45int28a","stepAnswer":["$$-1$$"],"problemType":"TextBox","stepTitle":"Translate and Simplify:","stepBody":"The sum of $$8$$ and $$-12$$, increased by $$3$$","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1$$","hints":{"DefaultPathway":[{"id":"ae45a45int28a-h1","type":"hint","dependencies":[],"title":"Terminology - Sum","text":"The word \\"sum\\" means that two numbers must be added","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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4.0>"},{"id":"ae45a45int28a-h4","type":"hint","dependencies":["ae45a45int28a-h3"],"title":"Terminology - Increasing","text":"The word \\"increasing\\" implies addition","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int28a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\left(-4+3\\\\right)$$"],"dependencies":["ae45a45int28a-h4"],"title":"Terminology - Increasing","text":"What is $$-4$$ increased by $$3$$, represented as an equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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By mid-afternoon, the temperature had dropped to $$-9$$ degrees. What was the difference in the morning and afternoon temperatures?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$20$$","hints":{"DefaultPathway":[{"id":"ae45a45int29a-h1","type":"hint","dependencies":[],"title":"Phrase","text":"First, write the problem as an algebraice phrase to solve. The problem is asking for the DIFFERENCE between $$11$$ degrees (the morning temperature) and $$-9$$ degrees (the afternoon temperature)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int29a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(11-(-9))$$"],"dependencies":["ae45a45int29a-h1"],"title":"Phrase","text":"What is the difference between $$11$$ degrees and $$-9$$ degrees, represented as an equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$11+\\\\left(-9\\\\right)$$","$$(11-(-9))$$","$$11\\\\left(-9\\\\right)$$","$$\\\\frac{11}{\\\\left(-9\\\\right)}$$"]},{"id":"ae45a45int29a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["ae45a45int29a-h2"],"title":"Phrase","text":"What does $$(11-(-9))$$ evaluate to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae45a45int3","title":"Simplify Expressions with Absolute Value","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Integers","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae45a45int3a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"$$10-3|9-3(3-1)|$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"ae45a45int3a-h1","type":"hint","dependencies":[],"title":"Order of Operation","text":"Start with the operation in the parenthetisis","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int3a-h2","type":"hint","dependencies":["ae45a45int3a-h1"],"title":"Simplify","text":"Simplify the value in the absolute value bars further by subtracting $$8$$ from $$8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["ae45a45int3a-h2"],"title":"Absolute Values Principle","text":"What is 3*|1|?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int3a-h4","type":"hint","dependencies":["ae45a45int3a-h3"],"title":"Absolute Values Principle","text":"The number within absolute sign will always be positive","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["ae45a45int3a-h4"],"title":"Magnitude of Number","text":"What is the magitude of 1?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae45a45int30","title":"Use Integers in Applications","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Integers","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae45a45int30a","stepAnswer":["$$9$$"],"problemType":"TextBox","stepTitle":"The temperature in Denver was $$-6$$ degrees at lunchtime. 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The problem is asking for the DIFFERENCE between $$-6$$ degrees (the luncthime temperature) and $$-15$$ degrees (the sunset temperature)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int30a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(-6-(-15))$$"],"dependencies":["ae45a45int30a-h1"],"title":"Phrase","text":"What is the difference between $$-6$$ degrees and $$-15$$ degrees, represented as an equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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Algebra","steps":[{"id":"ae45a45int4a","stepAnswer":["$$15$$"],"problemType":"TextBox","stepTitle":"$$34+\\\\left(-19\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$15$$","hints":{"DefaultPathway":[{"id":"ae45a45int4a-h1","type":"hint","dependencies":[],"title":"Principle of Signs","text":"Think of the negative sign as subtracting from the positive value","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int4a-h2","type":"hint","dependencies":["ae45a45int4a-h1"],"title":"Conversion","text":"Convert the expression to $$34-19$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae45a45int5","title":"Add and Subtract Integers","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Integers","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae45a45int5a","stepAnswer":["$$15$$"],"problemType":"TextBox","stepTitle":"24+3(-5+9)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$15$$","hints":{"DefaultPathway":[{"id":"ae45a45int5a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":[],"title":"Addition","text":"What is $$-5$$ + 9?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["ae45a45int5a-h1"],"title":"Multiplication","text":"What is $$3$$ * 4?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int5a-h3","type":"hint","dependencies":["ae45a45int5a-h2"],"title":"Add and Subtract Integers","text":"Now sum the numbers","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae45a45int6","title":"Add and Subtract Integers","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Integers","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae45a45int6a","stepAnswer":["$$29$$"],"problemType":"TextBox","stepTitle":"19+2(-3+8)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$29$$","hints":{"DefaultPathway":[{"id":"ae45a45int6a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":[],"title":"Addition","text":"What is $$-3$$ + 8?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["ae45a45int6a-h1"],"title":"Multiplication","text":"What is $$2$$ * 5?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int6a-h3","type":"hint","dependencies":["ae45a45int6a-h2"],"title":"Add and Subtract Integers","text":"Now sum the numbers","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae45a45int7","title":"Add and Subtract Integers","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Integers","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae45a45int7a","stepAnswer":["$$16$$"],"problemType":"TextBox","stepTitle":"$$44$$ - $$28$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16$$","hints":{"DefaultPathway":[{"id":"ae45a45int7a-h1","type":"hint","dependencies":[],"title":"Subtract $$28$$ from $$44$$","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"ae45a45int7b","stepAnswer":["$$16$$"],"problemType":"TextBox","stepTitle":"$$44$$ $$+\\\\left(-28\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16$$","hints":{"DefaultPathway":[{"id":"ae45a45int7b-h1","type":"hint","dependencies":[],"title":"Principle","text":"Think about adding $$-28$$ to $$44$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int7b-h2","type":"hint","dependencies":["ae45a45int7b-h1"],"title":"Conversion","text":"Convert the expression to $$44-28$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae45a45int8","title":"Add and Subtract Integers","body":"Simplify the following expression","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.2 Integers","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae45a45int8a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"$$(2-7)-(3-8)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"ae45a45int8a-h1","type":"hint","dependencies":[],"title":"Order of Operation","text":"Start with summing each of the values in the parenthese","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["ae45a45int8a-h1"],"title":"Subtraction","text":"What is $$2-7$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["ae45a45int8a-h2"],"title":"Subtraction","text":"What is $$3-8$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int8a-h4","type":"hint","dependencies":["ae45a45int8a-h3"],"title":"Principle","text":"Now, subtract the two values in the parenthetisis","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae45a45int8a-h5","type":"hint","dependencies":["ae45a45int8a-h4"],"title":"Conversion","text":"Convert the expression to $$-5+5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae647b3indep1","title":"Test of Independence","body":"In a volunteer group, adults $$21$$ and older volunteer from one to nine hours each week to spend time with a disabled senior citizen. The program recruits among community college students, four-year college students, and nonstudents. In Table $$11.15$$ is a sample of the adult volunteers and the number of hours they volunteer per week.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.3 Test of Independence","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae647b3indep1a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Is the number of hours volunteered independent of the type of volunteer?","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"ae647b3indep1a-h1","type":"hint","dependencies":[],"title":"Understanding the Question","text":"The observed table and the question at the end of the problem asks whether or not the number of hours volunteered is independent of the type of volunteer. This means that it is a test of independence. The two factors are number of hours volunteered and type of volunteer. This test is always right-tailed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae647b3indep1a-h2","type":"hint","dependencies":["ae647b3indep1a-h1"],"title":"Null and Alternative Hypotheses","text":"$$H_0$$: The number of hours volunteered is independent of the type of volunteer and $$H_a$$: The number of hours volunteered is dependent on the type of volunteer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae647b3indep1a-h3","type":"hint","dependencies":["ae647b3indep1a-h2"],"title":"Expected Frequency","text":"The calculation for the expected frequency for the top left cell is E $$=$$ ((row total)(column total))/total number surveyed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae647b3indep1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$90.57$$"],"dependencies":["ae647b3indep1a-h3"],"title":"Solving for E","text":"What is E $$=$$ ((row total)(column total))/total number surveyed? Round to the nearest hundredths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ae647b3indep1a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$255$$"],"dependencies":[],"title":"Row Total","text":"What is the row total for community college students?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae647b3indep1a-h4-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$298$$"],"dependencies":[],"title":"Column Total","text":"What is the column total for $$1-3$$ hours?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae647b3indep1a-h4-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$839$$"],"dependencies":[],"title":"Total Number Surveyed","text":"What is the total number surveyed (the sum of row totals or the sum of column totals)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae647b3indep1a-h4-s4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$90.57$$"],"dependencies":[],"title":"Plugging Into E","text":"What is E $$=$$ $$\\\\frac{255\\\\times298}{839}$$? Round to the nearest hundredths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ae647b3indep1a-h5","type":"hint","dependencies":["ae647b3indep1a-h4"],"title":"Test Statistic","text":"Calculate the test statistic: $$X^2$$ on your calculator or computer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae647b3indep1a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12.99$$"],"dependencies":["ae647b3indep1a-h5"],"title":"Solving $$X^2$$","text":"What is $$X^2$$? Round to the nearest hundredths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae647b3indep1a-h7","type":"hint","dependencies":["ae647b3indep1a-h6"],"title":"Probability Statement","text":"df $$=$$ (3 columns - 1)(3 rows - 1) $$=$$ (2)(2) $$=$$ $$4$$. Solve for $$p-value$$ $$=$$ P((X**2) > $$12.99)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae647b3indep1a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.0113$$"],"dependencies":["ae647b3indep1a-h7"],"title":"P-Value","text":"What is the $$p-value$$? Round your answer to four decimal places.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae647b3indep1a-h9","type":"hint","dependencies":["ae647b3indep1a-h8"],"title":"Comparing \ud835\udefc and the P-Value","text":"Since no \ud835\udefc is given, assume \ud835\udefc $$=$$ $$0.05$$. The $$p-value$$ $$=$$ $$0.0113$$. If \ud835\udefc > $$p-value$$, reject $$H_0$$. If \ud835\udefc < $$p-value$$, accept $$H_0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae647b3indep1a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["ae647b3indep1a-h9"],"title":"Greater Than or Less Than the P-Value","text":"Is \ud835\udefc > $$p-value$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"ae647b3indep1a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Reject"],"dependencies":["ae647b3indep1a-h10"],"title":"Reject or Accept $$H_0$$ Based on the P-Value","text":"Based on your previous answer, would you reject or accept $$H_0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Accept","Reject"]},{"id":"ae647b3indep1a-h12","type":"hint","dependencies":["ae647b3indep1a-h11"],"title":"Conclusion","text":"At a 5% level of significance, from the data, there is sufficient evidence to conclude that the number of hours volunteered and the type of volunteer are dependent on one another.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae647b3indep2","title":"Test of Independence","body":"De Anza College is interested in the relationship between anxiety level and the need to succeed in school. A random sample of $$400$$ students took a test that measured anxiety level and need to succeed in school. Table $$11.18$$ shows the results. De Anza College wants to know if anxiety level and need to succeed in school are independent events.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.3 Test of Independence","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae647b3indep2a","stepAnswer":["$$22$$"],"problemType":"TextBox","stepTitle":"How many high anxiety level students are expected to have a high need to succeed in school?","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$22$$","hints":{"DefaultPathway":[{"id":"ae647b3indep2a-h1","type":"hint","dependencies":[],"title":"Solving for E","text":"To answer this problem, solve for E $$=$$ ((row total)(column total))/total number surveyed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae647b3indep2a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$22.09$$"],"dependencies":["ae647b3indep2a-h1"],"title":"Applying Table to E","text":"What is E $$=$$ ((row total)(column total))/total number surveyed? Round to the nearest hundredths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ae647b3indep2a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$155$$"],"dependencies":[],"title":"Row Total","text":"What is the row total for high need to succeed?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae647b3indep2a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$57$$"],"dependencies":[],"title":"Column Total","text":"What is the column total for high anxiety?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae647b3indep2a-h2-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$400$$"],"dependencies":[],"title":"Total Number Surveyed","text":"What is the total number surveyed (sample size)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae647b3indep2a-h2-s4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$22.09$$"],"dependencies":[],"title":"Plugging Into E","text":"Plugging in the values, what is E $$=$$ $$\\\\frac{155\\\\times57}{400}$$? Round to the nearest hundredths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ae647b3indep2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$22$$"],"dependencies":["ae647b3indep2a-h2"],"title":"Rounding E","text":"Because we are calculating the expected number of students, your answer should be a whole number. What is E rounded down to nearest whole number?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae647b3indep3","title":"Test of Independence","body":"De Anza College is interested in the relationship between anxiety level and the need to succeed in school. A random sample of $$400$$ students took a test that measured anxiety level and need to succeed in school. Table $$11.18$$ shows the results. De Anza College wants to know if anxiety level and need to succeed in school are independent events.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"11.3 Test of Independence","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae647b3indep3a","stepAnswer":["$$8$$"],"problemType":"TextBox","stepTitle":"If the two variables are independent, how many students do you expect to have a low need to succeed in school and a med-low level of anxiety?","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8$$","hints":{"DefaultPathway":[{"id":"ae647b3indep3a-h1","type":"hint","dependencies":[],"title":"Solving for E","text":"To answer this problem, solve for E $$=$$ ((row total)(column total))/total number surveyed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae647b3indep3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8.19$$"],"dependencies":["ae647b3indep3a-h1"],"title":"Applying Table to E","text":"What is E $$=$$ ((row total)(column total))/total number surveyed? Round to the nearest hundredths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"ae647b3indep3a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$52$$"],"dependencies":[],"title":"Row Total","text":"What is the row total for low need to succeed?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae647b3indep3a-h2-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$63$$"],"dependencies":[],"title":"Column Total","text":"What is the column total for med-low anxiety?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae647b3indep3a-h2-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$400$$"],"dependencies":[],"title":"Total Number Surveyed","text":"What is the total number surveyed (sample size)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae647b3indep3a-h2-s4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8.19$$"],"dependencies":[],"title":"Plugging Into E","text":"Plugging in the values, what is E $$=$$ $$\\\\frac{52\\\\times63}{400}$$? Round to the nearest hundredths place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"ae647b3indep3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["ae647b3indep3a-h2"],"title":"Rounding E","text":"Because we are calculating the expected number of students, your answer should be a whole number. What is E rounded down to nearest whole number?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae6e5ceq1","title":"Finding the mean.","body":"Statistics are used to compare and sometimes identify authors. The following lists shows a simple random sample that compares the letter counts for three authors.\\\\nTerry: 7; 9; 3; 3; 3; 4; 1; 3; 2; $$2$$","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Skewness and the Mean, Median, and Mode","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae6e5ceq1a","stepAnswer":["$$3.7$$"],"problemType":"TextBox","stepTitle":"Calculate the mean for Terry.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3.7$$","hints":{"DefaultPathway":[{"id":"ae6e5ceq1a-h1","type":"hint","dependencies":[],"title":"Adding all the items.","text":"The first step would be to add all the letter counts of Terry.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae6e5ceq1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$37$$"],"dependencies":["ae6e5ceq1a-h1"],"title":"Adding all the items.","text":"What is the total number of letter counts for Terry?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae6e5ceq1a-h3","type":"hint","dependencies":["ae6e5ceq1a-h2"],"title":"Dividing by the number of values.","text":"The next step would be to divide that total by the number of entries. $$\\\\frac{37}{10}$$ $$=$$ $$3.7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae6e5ceq10","title":"Relationship of mean and median during skewness.","body":"Answer the following question.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Skewness and the Mean, Median, and Mode","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae6e5ceq10a","stepAnswer":["mean less than median"],"problemType":"MultipleChoice","stepTitle":"When the data are skewed left, what is the typical relationship between the mean and median?","stepBody":"","answerType":"string","variabilization":{},"choices":["mean less than median","mean greater than median","mean and median are close or the same"],"hints":{"DefaultPathway":[{"id":"ae6e5ceq10a-h1","type":"hint","dependencies":[],"title":"Thinking about the mean and median.","text":"Think about what the mean and median signify in a skewed left distribution. Having them mean less than median means that most of the data is in the right which is consistent with a skewed left distribution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae6e5ceq11","title":"Multiple modes.","body":"Answer the following question.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Skewness and the Mean, Median, and Mode","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae6e5ceq11a","stepAnswer":["Bimodal"],"problemType":"TextBox","stepTitle":"What word describes a distribution that has two modes?","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"ae6e5ceq11a-h1","type":"hint","dependencies":[],"title":"Knowing vocabulary.","text":"The technical term for a set of data with two modes is Bimodal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae6e5ceq12","title":"Seeing skewness.","body":"Answer the following question.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Skewness and the Mean, Median, and Mode","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae6e5ceq12a","stepAnswer":["Skewed Right"],"problemType":"MultipleChoice","stepTitle":"Describe the shape of this distribution.","stepBody":"","answerType":"string","variabilization":{},"choices":["Symmetrical","Skewed Left","Skewed Right"],"hints":{"DefaultPathway":[{"id":"ae6e5ceq12a-h1","type":"hint","dependencies":[],"title":"Looking at the distribution.","text":"First, look at where most of the data is concentrated. Here it seems as if the data is pulled out to the right so it is Skewed Right.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae6e5ceq13","title":"Relationship of mode and median during skewness.","body":"Answer the following question.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Skewness and the Mean, Median, and Mode","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae6e5ceq13a","stepAnswer":["mode less than median"],"problemType":"MultipleChoice","stepTitle":"Describe the relationship between the mode and the median of this distribution.","stepBody":"","answerType":"string","variabilization":{},"choices":["mode less than median","mode greater than median","mode and median are the same"],"hints":{"DefaultPathway":[{"id":"ae6e5ceq13a-h1","type":"hint","dependencies":[],"title":"Identifying the mode and median.","text":"The mode here is $$3$$ as it has the most entries. The median is higher than the mode as there are more than double the mode amount of entries.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae6e5ceq14","title":"Relationship of mean and median during skewness.","body":"Answer the following question.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Skewness and the Mean, Median, and Mode","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae6e5ceq14a","stepAnswer":["mean greater than median"],"problemType":"MultipleChoice","stepTitle":"Describe the relationship between the mean and the median of this distribution.","stepBody":"","answerType":"string","variabilization":{},"choices":["mean less than median","mean greater than median","mean and median are close or the same"],"hints":{"DefaultPathway":[{"id":"ae6e5ceq14a-h1","type":"hint","dependencies":[],"title":"Thinking about the mean and median.","text":"Think about what the mean and median signify in a skewed right distribution. Having them mean greater than median means that most of the data is in the left which is consistent with a skewed right distribution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae6e5ceq15","title":"Seeing skewness.","body":"Answer the following question.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Skewness and the Mean, Median, and Mode","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae6e5ceq15a","stepAnswer":["Symmetrical"],"problemType":"MultipleChoice","stepTitle":"Describe the shape of this distribution.","stepBody":"","answerType":"string","variabilization":{},"choices":["Symmetrical","Skewed Left","Skewed Right"],"hints":{"DefaultPathway":[{"id":"ae6e5ceq15a-h1","type":"hint","dependencies":[],"title":"Looking at the distribution.","text":"First, look at where most of the data is concentrated. Here it seems as if the data is in the center so it is Symmetrical.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae6e5ceq16","title":"Relationship of mode and median during skewness.","body":"Answer the following question.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Skewness and the Mean, Median, and Mode","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae6e5ceq16a","stepAnswer":["mode and median are the same"],"problemType":"MultipleChoice","stepTitle":"Describe the relationship between the mode and the median of this distribution.","stepBody":"","answerType":"string","variabilization":{},"choices":["mode less than median","mode greater than median","mode and median are the same"],"hints":{"DefaultPathway":[{"id":"ae6e5ceq16a-h1","type":"hint","dependencies":[],"title":"Identifying the mode and median.","text":"The mode here is $$5$$ as it has the most entries. The median is the same as the mode as it is a symmetrical distribution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae6e5ceq17","title":"Seeing skewness.","body":"Answer the following question.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Skewness and the Mean, Median, and Mode","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae6e5ceq17a","stepAnswer":["Skewed Left"],"problemType":"MultipleChoice","stepTitle":"Describe the shape of this distribution.","stepBody":"","answerType":"string","variabilization":{},"choices":["Symmetrical","Skewed Left","Skewed Right"],"hints":{"DefaultPathway":[{"id":"ae6e5ceq17a-h1","type":"hint","dependencies":[],"title":"Looking at the distribution.","text":"First, look at where most of the data is concentrated. Here it seems as if the data is pulled out to the left so it is Skewed Left.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae6e5ceq18","title":"Relationship of mode and median during skewness.","body":"Answer the following question.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Skewness and the Mean, Median, and Mode","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae6e5ceq18a","stepAnswer":["mode greater than median"],"problemType":"MultipleChoice","stepTitle":"Describe the relationship between the mode and the median of this distribution.","stepBody":"","answerType":"string","variabilization":{},"choices":["mode less than median","mode greater than median","mode and median are the same"],"hints":{"DefaultPathway":[{"id":"ae6e5ceq18a-h1","type":"hint","dependencies":[],"title":"Identifying the mode and median.","text":"The next step would be to divide that total by the number of entries. $$\\\\frac{37}{10}$$ $$=$$ $$3.7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae6e5ceq19","title":"Relationship of mean and median during skewness.","body":"Answer the following question.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Skewness and the Mean, Median, and Mode","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae6e5ceq19a","stepAnswer":["mean and median are close or the same"],"problemType":"MultipleChoice","stepTitle":"Describe the relationship between the mean and the median of this distribution.","stepBody":"","answerType":"string","variabilization":{},"choices":["mean less than median","mean greater than median","mean and median are close or the same"],"hints":{"DefaultPathway":[{"id":"ae6e5ceq19a-h1","type":"hint","dependencies":[],"title":"Finding the mean.","text":"To find the mean, add up all the instances and divide by the number of instances.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae6e5ceq19a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["ae6e5ceq19a-h1"],"title":"Finding the mean.","text":"What is the mean?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae6e5ceq19a-h3","type":"hint","dependencies":["ae6e5ceq19a-h2"],"title":"Finding the median.","text":"To find the median, find the midpoint.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae6e5ceq19a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["ae6e5ceq19a-h3"],"title":"Finding the median.","text":"What is the median?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae6e5ceq2","title":"Finding the median.","body":"Statistics are used to compare and sometimes identify authors. The following lists shows a simple random sample that compares the letter counts for three authors.\\\\nTerry: 7; 9; 3; 3; 3; 4; 1; 3; 2; $$2$$","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Skewness and the Mean, Median, and Mode","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae6e5ceq2a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"Calculate the median for Terry.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"ae6e5ceq2a-h1","type":"hint","dependencies":[],"title":"Order the numbers.","text":"The first step to finding the median would be to order the values in ascending order.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae6e5ceq2a-h2","type":"hint","dependencies":["ae6e5ceq2a-h1"],"title":"Finding the midpoint.","text":"The next step would be to find the midpoint. You do this by taking the total number of entries (10) and dividing by $$2$$. The mid point for an even number would be the average between entries $$5$$ and $$6$$. If there is an odd amount of entries you would round up and use that entry as the median. This gives us a median of $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae6e5ceq20","title":"Finding the greatest out of the mean, mode, median.","body":"Which is the greatest, the mean, the mode, or the median of the data set?","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Skewness and the Mean, Median, and Mode","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae6e5ceq20a","stepAnswer":["mean"],"problemType":"MultipleChoice","stepTitle":"11; 11; 12; 12; 12; 12; 13; 15; 17; 22; 22; $$22$$","stepBody":"","answerType":"string","variabilization":{},"choices":["mean","median","mode"],"hints":{"DefaultPathway":[{"id":"ae6e5ceq20a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15.1$$"],"dependencies":[],"title":"Finding the mean.","text":"What is the mean?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae6e5ceq20a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12.5$$"],"dependencies":["ae6e5ceq20a-h1"],"title":"Finding the median.","text":"What is the median?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae6e5ceq20a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["ae6e5ceq20a-h2"],"title":"Finding the mode.","text":"What is the mode?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae6e5ceq3","title":"Finding the mean.","body":"Statistics are used to compare and sometimes identify authors. The following lists shows a simple random sample that compares the letter counts for three authors.\\\\nDavis: 3; 3; 3; 4; 1; 4; 3; 2; 3; $$1$$","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Skewness and the Mean, Median, and Mode","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae6e5ceq3a","stepAnswer":["$$2.7$$"],"problemType":"TextBox","stepTitle":"Calculate the mean for Davis.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2.7$$","hints":{"DefaultPathway":[{"id":"ae6e5ceq3a-h1","type":"hint","dependencies":[],"title":"Adding all the items.","text":"The first step would be to add all the letter counts of Davis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae6e5ceq3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$27$$"],"dependencies":["ae6e5ceq3a-h1"],"title":"Adding all the items.","text":"What is the total number of letter counts for Davis?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae6e5ceq3a-h3","type":"hint","dependencies":["ae6e5ceq3a-h2"],"title":"Dividing by the number of values.","text":"The next step would be to divide that total by the number of entries. $$\\\\frac{27}{10}$$ $$=$$ $$2.7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae6e5ceq4","title":"Finding the median.","body":"Statistics are used to compare and sometimes identify authors. The following lists shows a simple random sample that compares the letter counts for three authors.\\\\nDavis: 3; 3; 3; 4; 1; 4; 3; 2; 3; $$1$$","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Skewness and the Mean, Median, and Mode","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae6e5ceq4a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"Calculate the median for Davis.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"ae6e5ceq4a-h1","type":"hint","dependencies":[],"title":"Order the numbers.","text":"The first step to finding the median would be to order the values in ascending order.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae6e5ceq4a-h2","type":"hint","dependencies":["ae6e5ceq4a-h1"],"title":"Finding the midpoint.","text":"The next step would be to find the midpoint. You do this by taking the total number of entries (10) and dividing by $$2$$. The mid point for an even number would be the average between entries $$5$$ and $$6$$. If there is an odd amount of entries you would round up and use that entry as the median. This gives us a median of $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae6e5ceq5","title":"Finding the mean.","body":"Statistics are used to compare and sometimes identify authors. The following lists shows a simple random sample that compares the letter counts for three authors.\\\\nMaris: 2; 3; 4; 4; 4; 6; 6; 6; 8; $$3$$","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Skewness and the Mean, Median, and Mode","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae6e5ceq5a","stepAnswer":["$$4.6$$"],"problemType":"TextBox","stepTitle":"Calculate the mean for Maris.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4.6$$","hints":{"DefaultPathway":[{"id":"ae6e5ceq5a-h1","type":"hint","dependencies":[],"title":"Adding all the items.","text":"The first step would be to add all the letter counts of Maris.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae6e5ceq5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$46$$"],"dependencies":["ae6e5ceq5a-h1"],"title":"Adding all the items.","text":"What is the total number of letter counts for Maris?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae6e5ceq5a-h3","type":"hint","dependencies":["ae6e5ceq5a-h2"],"title":"Dividing by the number of values.","text":"The next step would be to divide that total by the number of entries. $$\\\\frac{46}{10}$$ $$=$$ $$4.6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae6e5ceq6","title":"Finding the median.","body":"Statistics are used to compare and sometimes identify authors. The following lists shows a simple random sample that compares the letter counts for three authors.\\\\nMaris: 2; 3; 4; 4; 4; 6; 6; 6; 8; $$3$$","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Skewness and the Mean, Median, and Mode","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae6e5ceq6a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"Calculate the median for Maris.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"ae6e5ceq6a-h1","type":"hint","dependencies":[],"title":"Order the numbers.","text":"The first step to finding the median would be to order the values in ascending order.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae6e5ceq6a-h2","type":"hint","dependencies":["ae6e5ceq6a-h1"],"title":"Finding the midpoint.","text":"The next step would be to find the midpoint. You do this by taking the total number of entries (10) and dividing by $$2$$. The mid point for an even number would be the average between entries $$5$$ and $$6$$. If there is an odd amount of entries you would round up and use that entry as the median. This gives us a median of $$4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae6e5ceq7","title":"Stating skewness.","body":"State whether the data are symmetrical, skewed to the left, or skewed to the right.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.6 Skewness and the Mean, Median, and Mode","courseName":"OpenStax: Introductory Stats","steps":[{"id":"ae6e5ceq7a","stepAnswer":["Symmetrical"],"problemType":"MultipleChoice","stepTitle":"1; 1; 1; 2; 2; 2; 2; 3; 3; 3; 3; 3; 3; 3; 3; 4; 4; 4; 5; $$5$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Symmetrical","Skewed Left","Skewed Right"],"hints":{"DefaultPathway":[{"id":"ae6e5ceq7a-h1","type":"hint","dependencies":[],"title":"Find the mean.","text":"First, you want to find the mean. This is done by adding all the values together and dividing by the number of entries.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae6e5ceq7a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.85$$"],"dependencies":["ae6e5ceq7a-h1"],"title":"Find the mean.","text":"What is the mean of this data?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae6e5ceq7a-h3","type":"hint","dependencies":["ae6e5ceq7a-h2"],"title":"Find the median.","text":"Secondly you want to find the median. 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Having them close together means that most of the data is in the middle which is consistent with a symmetrical distribution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae8c2acComplex1","title":"Multiply Expression","body":"Multiply:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Use the Complex Number System","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae8c2acComplex1a","stepAnswer":["$$2x^2-x-15$$"],"problemType":"TextBox","stepTitle":"$$\\\\left(x-3\\\\right) \\\\left(2x+5\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2x^2-x-15$$","hints":{"DefaultPathway":[{"id":"ae8c2acComplex1a-h1","type":"hint","dependencies":[],"title":"Factoring Rules","text":"Remember the acronym, FOIL, which says to multiply the First, Outer, Inner, and Last terms to multiply the two expressions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8c2acComplex1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x^2$$"],"dependencies":["ae8c2acComplex1a-h1"],"title":"First Terms","text":"What will we get when we multiply our first terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8c2acComplex1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5x-6x$$"],"dependencies":["ae8c2acComplex1a-h2"],"title":"$$\\\\frac{Inner}{Outer}$$ Terms","text":"What will we get when we multiply our inner and outer terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8c2acComplex1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-15$$"],"dependencies":["ae8c2acComplex1a-h3"],"title":"Last Terms","text":"What will we get when we multiply our last terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8c2acComplex1a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x^2-x-15$$"],"dependencies":["ae8c2acComplex1a-h4"],"title":"Solution","text":"Combining our terms, what will our final expanded answer be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae8c2acComplex10","title":"Simplify Complex Expression","body":"Write each expression in terms of i if possible.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8c2acComplex10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["6i"],"dependencies":["ae8c2acComplex10a-h2"],"title":"Solution","text":"Now that we factored i, what will our final simplified expression be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae8c2acComplex11","title":"Add & Subtract Complex Numbers","body":"Add:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Use the Complex Number System","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae8c2acComplex11a","stepAnswer":["$$5i \\\\sqrt{3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{-12}+\\\\sqrt{-27}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5i \\\\sqrt{3}$$","hints":{"DefaultPathway":[{"id":"ae8c2acComplex11a-h1","type":"hint","dependencies":[],"title":"Factoring i","text":"If $$b$$ is a positive real number, then $$\\\\sqrt{-b}$$ $$=$$ $$i \\\\sqrt{b}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8c2acComplex11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$i \\\\sqrt{12}+i \\\\sqrt{27}$$"],"dependencies":["ae8c2acComplex11a-h1"],"title":"Factoring i","text":"What will our expression look like when we factor out i in both terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8c2acComplex11a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2i \\\\sqrt{3}+3i \\\\sqrt{3}$$"],"dependencies":["ae8c2acComplex11a-h2"],"title":"Simplifying Terms","text":"If we simplify our square roots, what will our expression be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8c2acComplex11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5i \\\\sqrt{3}$$"],"dependencies":["ae8c2acComplex11a-h3"],"title":"Solution","text":"Since we have common terms, we can get our solution by adding the terms together. What will our final solution be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae8c2acComplex12","title":"Add & Subtract Complex Numbers","body":"Add:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Use the Complex Number System","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae8c2acComplex12a","stepAnswer":["$$6i \\\\sqrt{2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{-8}+\\\\sqrt{-32}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6i \\\\sqrt{2}$$","hints":{"DefaultPathway":[{"id":"ae8c2acComplex12a-h1","type":"hint","dependencies":[],"title":"Factoring i","text":"If $$b$$ is a positive real number, then $$\\\\sqrt{-b}$$ $$=$$ $$i \\\\sqrt{b}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8c2acComplex12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$i \\\\sqrt{8}+i \\\\sqrt{32}$$"],"dependencies":["ae8c2acComplex12a-h1"],"title":"Factoring i","text":"What will our expression look like when we factor out i in both terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8c2acComplex12a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2i \\\\sqrt{2}+4i \\\\sqrt{2}$$"],"dependencies":["ae8c2acComplex12a-h2"],"title":"Simplifying Terms","text":"If we simplify our square roots, what will our expression be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8c2acComplex12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6i \\\\sqrt{2}$$"],"dependencies":["ae8c2acComplex12a-h3"],"title":"Solution","text":"Since we have common terms, we can get our solution by adding the terms together. What will our final solution be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae8c2acComplex13","title":"Add & Subtract Complex Numbers","body":"Add:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Use the Complex Number System","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae8c2acComplex13a","stepAnswer":["$$7i \\\\sqrt{3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{-27}+\\\\sqrt{-48}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7i \\\\sqrt{3}$$","hints":{"DefaultPathway":[{"id":"ae8c2acComplex13a-h1","type":"hint","dependencies":[],"title":"Factoring i","text":"If $$b$$ is a positive real number, then $$\\\\sqrt{-b}$$ $$=$$ $$i \\\\sqrt{b}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8c2acComplex13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$i \\\\sqrt{27}+i \\\\sqrt{48}$$"],"dependencies":["ae8c2acComplex13a-h1"],"title":"Factoring i","text":"What will our expression look like when we factor out i in both terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8c2acComplex13a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3i \\\\sqrt{3}+4i \\\\sqrt{3}$$"],"dependencies":["ae8c2acComplex13a-h2"],"title":"Simplifying Terms","text":"If we simplify our square roots, what will our expression be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8c2acComplex13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7i \\\\sqrt{3}$$"],"dependencies":["ae8c2acComplex13a-h3"],"title":"Solution","text":"Since we have common terms, we can get our solution by adding the terms together. What will our final solution be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae8c2acComplex14","title":"Add & Subtract Complex Numbers","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Use the Complex Number System","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae8c2acComplex14a","stepAnswer":["$$9+3i$$"],"problemType":"TextBox","stepTitle":"$$(4-3i)$$ + $$5+6i$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9+3i$$","hints":{"DefaultPathway":[{"id":"ae8c2acComplex14a-h1","type":"hint","dependencies":[],"title":"Adding Terms","text":"We can add our terms together by adding together \\"like\\" terms. For example, 3i + 3i $$=$$ 6i.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8c2acComplex14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9+3i$$"],"dependencies":["ae8c2acComplex14a-h1"],"title":"Solution","text":"What will our expression look like when we add like terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae8c2acComplex15","title":"Add & Subtract Complex Numbers","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Use the Complex Number System","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae8c2acComplex15a","stepAnswer":["$$-3-3i$$"],"problemType":"TextBox","stepTitle":"$$(2-5i)-(5-2i)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-3-3i$$","hints":{"DefaultPathway":[{"id":"ae8c2acComplex15a-h1","type":"hint","dependencies":[],"title":"Adding Terms","text":"We can add our terms together by adding together \\"like\\" terms. For example, 3i + 3i $$=$$ 6i.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8c2acComplex15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3-3i$$"],"dependencies":["ae8c2acComplex15a-h1"],"title":"Solution","text":"What will our expression look like when we add like terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae8c2acComplex16","title":"Add & Subtract Complex Numbers","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Use the Complex Number System","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae8c2acComplex16a","stepAnswer":["$$6+5i$$"],"problemType":"TextBox","stepTitle":"$$2+7i+4-2i$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6+5i$$","hints":{"DefaultPathway":[{"id":"ae8c2acComplex16a-h1","type":"hint","dependencies":[],"title":"Adding Terms","text":"We can add our terms together by adding together \\"like\\" terms. For example, 3i + 3i $$=$$ 6i.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8c2acComplex16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6+5i$$"],"dependencies":["ae8c2acComplex16a-h1"],"title":"Solution","text":"What will our expression look like when we add like terms?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae8c2acComplex17","title":"Add & Subtract Complex Numbers","body":"Simplify:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Use the Complex Number System","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae8c2acComplex17a","stepAnswer":["$$6-3i$$"],"problemType":"TextBox","stepTitle":"$$(8-4i)-(2-i)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6-3i$$","hints":{"DefaultPathway":[{"id":"ae8c2acComplex17a-h1","type":"hint","dependencies":[],"title":"Adding Terms","text":"We can add our terms together by adding together \\"like\\" terms. 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\\\\sqrt{5}$$"],"dependencies":["ae8c2acComplex8a-h1"],"title":"Factoring i","text":"What will our expression look like when we factor out i?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8c2acComplex8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$i \\\\sqrt{5}$$"],"dependencies":["ae8c2acComplex8a-h2"],"title":"Solution","text":"Now that we factored i, what will our final simplified expression be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae8c2acComplex9","title":"Simplify Complex Expression","body":"Write each expression in terms of i if possible.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.8 Use the Complex Number System","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae8c2acComplex9a","stepAnswer":["$$3i \\\\sqrt{2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{-18}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3i \\\\sqrt{2}$$","hints":{"DefaultPathway":[{"id":"ae8c2acComplex9a-h1","type":"hint","dependencies":[],"title":"Factoring i","text":"If $$b$$ is a positive real number, then $$\\\\sqrt{-b}$$ $$=$$ $$i \\\\sqrt{b}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8c2acComplex9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$i \\\\sqrt{18}$$"],"dependencies":["ae8c2acComplex9a-h1"],"title":"Factoring i","text":"What will our expression look like when we factor out i?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8c2acComplex9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3i \\\\sqrt{2}$$"],"dependencies":["ae8c2acComplex9a-h2"],"title":"Solution","text":"Now that we factored i, what will our final simplified expression be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae8dd20divradexp1","title":"Simplifying radical expressions.","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Divide Radical Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae8dd20divradexp1a","stepAnswer":["$$\\\\frac{2x}{3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\sqrt[2]{72x^3}}{\\\\sqrt[2]{162x}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2x}{3}$$","hints":{"DefaultPathway":[{"id":"ae8dd20divradexp1a-h1","type":"hint","dependencies":[],"title":"Radical Powers","text":"Are the radical powers the same? If they are, we can use the quotient property and simply put them underneath the same radical and divide.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8dd20divradexp1a-h2","type":"hint","dependencies":["ae8dd20divradexp1a-h1"],"title":"Quotient Property","text":"The quotient property is: $$\\\\frac{\\\\sqrt{x}}{\\\\sqrt{y}}$$ $$=$$ $$\\\\sqrt{\\\\frac{x}{y}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8dd20divradexp1a-h3","type":"hint","dependencies":["ae8dd20divradexp1a-h2"],"title":"Common Factors","text":"After we use the quotient property, we try to remove common factors and simplify.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae8dd20divradexp10","title":"Simplifying radical expressions.","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Divide Radical Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae8dd20divradexp10a","stepAnswer":["$$\\\\frac{-3p}{q^2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\sqrt[3]{-81p q^{-1}}}{\\\\sqrt[3]{3p^{-2} q^5}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-3p}{q^2}$$","hints":{"DefaultPathway":[{"id":"ae8dd20divradexp10a-h1","type":"hint","dependencies":[],"title":"Radical Powers","text":"Are the radical powers the same? If they are, we can use the quotient property and simply put them underneath the same radical and divide.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8dd20divradexp10a-h2","type":"hint","dependencies":["ae8dd20divradexp10a-h1"],"title":"Quotient Property","text":"The quotient property is: $$\\\\frac{\\\\sqrt{x}}{\\\\sqrt{y}}$$ $$=$$ $$\\\\sqrt{\\\\frac{x}{y}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8dd20divradexp10a-h3","type":"hint","dependencies":["ae8dd20divradexp10a-h2"],"title":"Common Factors","text":"After we use the quotient property, we try to remove common factors and simplify.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae8dd20divradexp11","title":"Simplifying radical expressions.","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Divide Radical Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae8dd20divradexp11a","stepAnswer":["$$3x y \\\\sqrt{2x}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\sqrt[2]{54x^5 y^3}}{\\\\sqrt[2]{3x^2 y}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3x y \\\\sqrt{2x}$$","hints":{"DefaultPathway":[{"id":"ae8dd20divradexp11a-h1","type":"hint","dependencies":[],"title":"Radical Powers","text":"Are the radical powers the same? If they are, we can use the quotient property and simply put them underneath the same radical and divide.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8dd20divradexp11a-h2","type":"hint","dependencies":["ae8dd20divradexp11a-h1"],"title":"Quotient Property","text":"The quotient property is: $$\\\\frac{\\\\sqrt{x}}{\\\\sqrt{y}}$$ $$=$$ $$\\\\sqrt{\\\\frac{x}{y}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8dd20divradexp11a-h3","type":"hint","dependencies":["ae8dd20divradexp11a-h2"],"title":"Common Factors","text":"After we use the quotient property, we try to remove common factors and simplify.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae8dd20divradexp12","title":"Simplifying radical expressions.","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Divide Radical Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae8dd20divradexp12a","stepAnswer":["$$4x y \\\\sqrt{2x}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\sqrt[2]{64x^4 y^5}}{\\\\sqrt[2]{2x y^3}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4x y \\\\sqrt{2x}$$","hints":{"DefaultPathway":[{"id":"ae8dd20divradexp12a-h1","type":"hint","dependencies":[],"title":"Radical Powers","text":"Are the radical powers the same? If they are, we can use the quotient property and simply put them underneath the same radical and divide.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8dd20divradexp12a-h2","type":"hint","dependencies":["ae8dd20divradexp12a-h1"],"title":"Quotient Property","text":"The quotient property is: $$\\\\frac{\\\\sqrt{x}}{\\\\sqrt{y}}$$ $$=$$ $$\\\\sqrt{\\\\frac{x}{y}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8dd20divradexp12a-h3","type":"hint","dependencies":["ae8dd20divradexp12a-h2"],"title":"Common Factors","text":"After we use the quotient property, we try to remove common factors and simplify.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae8dd20divradexp13","title":"Simplifying radical expressions.","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Divide Radical Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae8dd20divradexp13a","stepAnswer":["$$4x y \\\\sqrt{2x}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\sqrt[2]{96a^5 b^4}}{\\\\sqrt[2]{2a^3 b}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4x y \\\\sqrt{2x}$$","hints":{"DefaultPathway":[{"id":"ae8dd20divradexp13a-h1","type":"hint","dependencies":[],"title":"Radical Powers","text":"Are the radical powers the same? If they are, we can use the quotient property and simply put them underneath the same radical and divide.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8dd20divradexp13a-h2","type":"hint","dependencies":["ae8dd20divradexp13a-h1"],"title":"Quotient Property","text":"The quotient property is: $$\\\\frac{\\\\sqrt{x}}{\\\\sqrt{y}}$$ $$=$$ $$\\\\sqrt{\\\\frac{x}{y}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8dd20divradexp13a-h3","type":"hint","dependencies":["ae8dd20divradexp13a-h2"],"title":"Common Factors","text":"After we use the quotient property, we try to remove common factors and simplify.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae8dd20divradexp2","title":"Simplifying radical expressions.","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Divide Radical Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae8dd20divradexp2a","stepAnswer":["$$\\\\frac{2}{x}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\sqrt[3]{32x^2}}{\\\\sqrt[3]{4x^5}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2}{x}$$","hints":{"DefaultPathway":[{"id":"ae8dd20divradexp2a-h1","type":"hint","dependencies":[],"title":"Radical Powers","text":"Are the radical powers the same? If they are, we can use the quotient property and simply put them underneath the same radical and divide.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8dd20divradexp2a-h2","type":"hint","dependencies":["ae8dd20divradexp2a-h1"],"title":"Quotient Property","text":"The quotient property is: $$\\\\frac{\\\\sqrt{x}}{\\\\sqrt{y}}$$ $$=$$ $$\\\\sqrt{\\\\frac{x}{y}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8dd20divradexp2a-h3","type":"hint","dependencies":["ae8dd20divradexp2a-h2"],"title":"Common Factors","text":"After we use the quotient property, we try to remove common factors and simplify.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae8dd20divradexp3","title":"Simplifying radical expressions.","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Divide Radical Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae8dd20divradexp3a","stepAnswer":["$$\\\\frac{5s}{8}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\sqrt[2]{50s^3}}{\\\\sqrt[2]{128s}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5s}{8}$$","hints":{"DefaultPathway":[{"id":"ae8dd20divradexp3a-h1","type":"hint","dependencies":[],"title":"Radical Powers","text":"Are the radical powers the same? If they are, we can use the quotient property and simply put them underneath the same radical and divide.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8dd20divradexp3a-h2","type":"hint","dependencies":["ae8dd20divradexp3a-h1"],"title":"Quotient Property","text":"The quotient property is: $$\\\\frac{\\\\sqrt{x}}{\\\\sqrt{y}}$$ $$=$$ $$\\\\sqrt{\\\\frac{x}{y}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8dd20divradexp3a-h3","type":"hint","dependencies":["ae8dd20divradexp3a-h2"],"title":"Common Factors","text":"After we use the quotient property, we try to remove common factors and simplify.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae8dd20divradexp4","title":"Simplifying radical expressions.","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Divide Radical Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae8dd20divradexp4a","stepAnswer":["$$\\\\frac{2}{a}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\sqrt[3]{56a}}{\\\\sqrt[3]{7s^4}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2}{a}$$","hints":{"DefaultPathway":[{"id":"ae8dd20divradexp4a-h1","type":"hint","dependencies":[],"title":"Radical Powers","text":"Are the radical powers the same? If they are, we can use the quotient property and simply put them underneath the same radical and divide.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8dd20divradexp4a-h2","type":"hint","dependencies":["ae8dd20divradexp4a-h1"],"title":"Quotient Property","text":"The quotient property is: $$\\\\frac{\\\\sqrt{x}}{\\\\sqrt{y}}$$ $$=$$ $$\\\\sqrt{\\\\frac{x}{y}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8dd20divradexp4a-h3","type":"hint","dependencies":["ae8dd20divradexp4a-h2"],"title":"Common Factors","text":"After we use the quotient property, we try to remove common factors and simplify.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae8dd20divradexp5","title":"Simplifying radical expressions.","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Divide Radical Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae8dd20divradexp5a","stepAnswer":["$$\\\\frac{5q^2}{6}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\sqrt[2]{75q^5}}{\\\\sqrt[2]{108q}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5q^2}{6}$$","hints":{"DefaultPathway":[{"id":"ae8dd20divradexp5a-h1","type":"hint","dependencies":[],"title":"Radical Powers","text":"Are the radical powers the same? If they are, we can use the quotient property and simply put them underneath the same radical and divide.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8dd20divradexp5a-h2","type":"hint","dependencies":["ae8dd20divradexp5a-h1"],"title":"Quotient Property","text":"The quotient property is: $$\\\\frac{\\\\sqrt{x}}{\\\\sqrt{y}}$$ $$=$$ $$\\\\sqrt{\\\\frac{x}{y}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8dd20divradexp5a-h3","type":"hint","dependencies":["ae8dd20divradexp5a-h2"],"title":"Common Factors","text":"After we use the quotient property, we try to remove common factors and simplify.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae8dd20divradexp6","title":"Simplifying radical expressions.","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Divide Radical Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae8dd20divradexp6a","stepAnswer":["$$\\\\frac{2}{b}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\sqrt[3]{72b^2}}{\\\\sqrt[3]{9b^5}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2}{b}$$","hints":{"DefaultPathway":[{"id":"ae8dd20divradexp6a-h1","type":"hint","dependencies":[],"title":"Radical Powers","text":"Are the radical powers the same? If they are, we can use the quotient property and simply put them underneath the same radical and divide.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8dd20divradexp6a-h2","type":"hint","dependencies":["ae8dd20divradexp6a-h1"],"title":"Quotient Property","text":"The quotient property is: $$\\\\frac{\\\\sqrt{x}}{\\\\sqrt{y}}$$ $$=$$ $$\\\\sqrt{\\\\frac{x}{y}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8dd20divradexp6a-h3","type":"hint","dependencies":["ae8dd20divradexp6a-h2"],"title":"Common Factors","text":"After we use the quotient property, we try to remove common factors and simplify.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae8dd20divradexp7","title":"Simplifying radical expressions.","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Divide Radical Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae8dd20divradexp7a","stepAnswer":["$$\\\\frac{7b^2}{a}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\sqrt[2]{147a b^8}}{\\\\sqrt[2]{3a^3 b^4}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{7b^2}{a}$$","hints":{"DefaultPathway":[{"id":"ae8dd20divradexp7a-h1","type":"hint","dependencies":[],"title":"Radical Powers","text":"Are the radical powers the same? If they are, we can use the quotient property and simply put them underneath the same radical and divide.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8dd20divradexp7a-h2","type":"hint","dependencies":["ae8dd20divradexp7a-h1"],"title":"Quotient Property","text":"The quotient property is: $$\\\\frac{\\\\sqrt{x}}{\\\\sqrt{y}}$$ $$=$$ $$\\\\sqrt{\\\\frac{x}{y}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8dd20divradexp7a-h3","type":"hint","dependencies":["ae8dd20divradexp7a-h2"],"title":"Common Factors","text":"After we use the quotient property, we try to remove common factors and simplify.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae8dd20divradexp8","title":"Simplifying radical expressions.","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Divide Radical Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae8dd20divradexp8a","stepAnswer":["$$\\\\frac{-5m}{n^2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\sqrt[3]{-250m n^{-2}}}{\\\\sqrt[3]{2m^{-2} n^4}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{-5m}{n^2}$$","hints":{"DefaultPathway":[{"id":"ae8dd20divradexp8a-h1","type":"hint","dependencies":[],"title":"Radical Powers","text":"Are the radical powers the same? If they are, we can use the quotient property and simply put them underneath the same radical and divide.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8dd20divradexp8a-h2","type":"hint","dependencies":["ae8dd20divradexp8a-h1"],"title":"Quotient Property","text":"The quotient property is: $$\\\\frac{\\\\sqrt{x}}{\\\\sqrt{y}}$$ $$=$$ $$\\\\sqrt{\\\\frac{x}{y}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8dd20divradexp8a-h3","type":"hint","dependencies":["ae8dd20divradexp8a-h2"],"title":"Common Factors","text":"After we use the quotient property, we try to remove common factors and simplify.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae8dd20divradexp9","title":"Simplifying radical expressions.","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.5 Divide Radical Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"ae8dd20divradexp9a","stepAnswer":["$$\\\\frac{9x^2}{y^2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{\\\\sqrt[2]{162x^{10} y^2}}{\\\\sqrt[2]{2x^6 y^6}}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{9x^2}{y^2}$$","hints":{"DefaultPathway":[{"id":"ae8dd20divradexp9a-h1","type":"hint","dependencies":[],"title":"Radical Powers","text":"Are the radical powers the same? If they are, we can use the quotient property and simply put them underneath the same radical and divide.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8dd20divradexp9a-h2","type":"hint","dependencies":["ae8dd20divradexp9a-h1"],"title":"Quotient Property","text":"The quotient property is: $$\\\\frac{\\\\sqrt{x}}{\\\\sqrt{y}}$$ $$=$$ $$\\\\sqrt{\\\\frac{x}{y}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae8dd20divradexp9a-h3","type":"hint","dependencies":["ae8dd20divradexp9a-h2"],"title":"Common Factors","text":"After we use the quotient property, we try to remove common factors and simplify.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae9e419IneqApp1","title":"Emma\'s Rent","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Solve Applications with Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ae9e419IneqApp1a","stepAnswer":["$$1875$$"],"problemType":"TextBox","stepTitle":"Emma got a new job and will have to move. Her monthly income will be $5,265. To qualify to rent an apartment, Emma\u2019s monthly income must be at least three times as much as the rent. What is the highest rent Emma will qualify for?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1875$$","hints":{"DefaultPathway":[{"id":"ae9e419IneqApp1a-h1","type":"hint","dependencies":[],"title":"Creating an Inequality","text":"First, we must transfer the word problem into an inequality that represents Anna\'s rent (r).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp1a-h2","type":"hint","dependencies":["ae9e419IneqApp1a-h1"],"title":"Creating an Inequality","text":"We know that Emma\'s monthly income must be at least three times the rent. So, we have the inequality $$3r \\\\leq 5625$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp1a-h3","type":"hint","dependencies":["ae9e419IneqApp1a-h2"],"title":"Solving the Inequality","text":"Now, we must solve the inequality for $$r$$ by algebrically manipulating it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp1a-h4","type":"hint","dependencies":["ae9e419IneqApp1a-h3"],"title":"Solving the Inequality","text":"To do this, we can divide both sides by $$3$$ to isolate $$r$$. We now have $$r \\\\leq 1875$$ as our inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp1a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1875$$"],"dependencies":["ae9e419IneqApp1a-h4"],"title":"Creating a Sentence-Answer","text":"Since $$r$$ must be less than or equal to 1,875 dollars, what is the maximum rent?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae9e419IneqApp10","title":"Mona\'s Fun Zone Budget","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Solve Applications with Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ae9e419IneqApp10a","stepAnswer":["$$15$$"],"problemType":"TextBox","stepTitle":"Mona is planning her son\u2019s birthday party and has a budget of $285. The Fun Zone charges $19 per child. How many children can she have at the party and stay within her budget?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$15$$","hints":{"DefaultPathway":[{"id":"ae9e419IneqApp10a-h1","type":"hint","dependencies":[],"title":"Creating an Inequality","text":"First, we must transfer the word problem into an inequality that represents Mona\'s budget.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp10a-h2","type":"hint","dependencies":["ae9e419IneqApp10a-h1"],"title":"Creating an Inequality","text":"We know that Mona has a budget of $285 and the Fun Zone charges $19 per child. Let\'s call the number of child $$x$$, so we have the inequality $$19x \\\\leq 285$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp10a-h3","type":"hint","dependencies":["ae9e419IneqApp10a-h2"],"title":"Solving the Inequality","text":"Now, we must solve the inequality for $$x$$ by algebrically manipulating it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp10a-h4","type":"hint","dependencies":["ae9e419IneqApp10a-h3"],"title":"Solving the Inequality","text":"To do this, we can divide both sides by $$19$$ to isolate $$x$$. We now have $$x \\\\leq 15$$ as our inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp10a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["ae9e419IneqApp10a-h4"],"title":"Creating a Sentence-Answer","text":"Since $$x$$ must be less than or equal to $$15$$, what is the maximum number of children Mona can have at the party?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae9e419IneqApp11","title":"Water Taxi Load","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Solve Applications with Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ae9e419IneqApp11a","stepAnswer":["$$12$$"],"problemType":"TextBox","stepTitle":"A water taxi has a maximum load of 1,800 pounds. If the average weight of one person is $$150$$ pounds, how many people can safely ride in the water taxi?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12$$","hints":{"DefaultPathway":[{"id":"ae9e419IneqApp11a-h1","type":"hint","dependencies":[],"title":"Creating an Inequality","text":"First, we must transfer the word problem into an inequality that represents the load of the water taxi.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp11a-h2","type":"hint","dependencies":["ae9e419IneqApp11a-h1"],"title":"Creating an Inequality","text":"We know that the maximum load of the water taxi is 1,800 pounds and the average weight of one person is $$150$$ pounds. If there are $$x$$ people, then the total weight would be $$150x$$. Therefore, we can create the inequality $$150x \\\\leq 1800$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp11a-h3","type":"hint","dependencies":["ae9e419IneqApp11a-h2"],"title":"Solving the Inequality","text":"Now, we must solve the inequality for $$x$$ by algebrically manipulating it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp11a-h4","type":"hint","dependencies":["ae9e419IneqApp11a-h3"],"title":"Solving the Inequality","text":"To do this, we can divide both sides by $$150$$ to isolate $$x$$. We now have $$x \\\\leq 12$$ as our inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["ae9e419IneqApp11a-h4"],"title":"Creating a Sentence-Answer","text":"Since $$x$$ must be less than or equal to $$12$$, what is the maximum number of people that can safely ride in the water taxi?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae9e419IneqApp12","title":"Iced Drink with Gift Card","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Solve Applications with Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ae9e419IneqApp12a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"Arleen got a $20 gift card for the coffee shop. Her favorite iced drink costs $$\\\\$3.79$$. What is the maximum number of drinks she can buy with the gift card?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"ae9e419IneqApp12a-h1","type":"hint","dependencies":[],"title":"Creating an Inequality","text":"First, we must transfer the word problem into an inequality that represents the total cost of iced drink.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp12a-h2","type":"hint","dependencies":["ae9e419IneqApp12a-h1"],"title":"Creating an Inequality","text":"We know that Arleen can pay a maximum of $20 with her gift card and each iced drink costs $$\\\\$3.79$$. If we let $$x$$ represent the number of iced drinks she buy, then the total cost would be $$3.79x$$. Therefore, we have the inequality $$3.79x \\\\leq 20$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp12a-h3","type":"hint","dependencies":["ae9e419IneqApp12a-h2"],"title":"Solving the Inequality","text":"Now, we must solve the inequality for $$x$$ by algebrically manipulating it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp12a-h4","type":"hint","dependencies":["ae9e419IneqApp12a-h3"],"title":"Solving the Inequality","text":"To do this, we can divide both sides by $$3.79$$ to isolate $$x$$. We now have $$x \\\\leq 5.277$$ as our inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp12a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["ae9e419IneqApp12a-h4"],"title":"Creating a Sentence-Answer","text":"Since $$x$$ must be less than or equal to $$5.277$$ and $$x$$ has to be a whole number, what is the maximum number of drinks Arleen can buy with the gift card?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae9e419IneqApp13","title":"Selling Kitchen Aprons","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Solve Applications with Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ae9e419IneqApp13a","stepAnswer":["$$31$$"],"problemType":"TextBox","stepTitle":"Joni sells kitchen aprons online for $$\\\\$32.50$$ each. How many aprons must she sell next month if she wants to earn at least $1,000?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$31$$","hints":{"DefaultPathway":[{"id":"ae9e419IneqApp13a-h1","type":"hint","dependencies":[],"title":"Creating an Inequality","text":"First, we must transfer the word problem into an inequality that represents the total earning Joni makes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp13a-h2","type":"hint","dependencies":["ae9e419IneqApp13a-h1"],"title":"Creating an Inequality","text":"We know that Joni wants her earning to be at least $1000, and each kitchen apron can be sold for $$\\\\$32.50$$. If we let $$x$$ represent the number of kitchen aprons Joni sells, then her total earning would be $$32.5x$$. Therefore, we have the inequality $$32.5x \\\\geq 1000$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp13a-h3","type":"hint","dependencies":["ae9e419IneqApp13a-h2"],"title":"Solving the Inequality","text":"Now, we must solve the inequality for $$x$$ by algebrically manipulating it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp13a-h4","type":"hint","dependencies":["ae9e419IneqApp13a-h3"],"title":"Solving the Inequality","text":"To do this, we can divide both sides by $$32.5$$ to isolate $$x$$. We now have $$x \\\\geq 30.77$$ as our inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp13a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$31$$"],"dependencies":["ae9e419IneqApp13a-h4"],"title":"Creating a Sentence-Answer","text":"Since $$x$$ must be greater than or equal to $$30.77$$ and $$x$$ has to be a whole number, what is the minimum number of kitchen aprons Joni has to sell?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae9e419IneqApp14","title":"Keshad\'s Earning","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Solve Applications with Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ae9e419IneqApp14a","stepAnswer":["$$15000$$"],"problemType":"TextBox","stepTitle":"Keshad gets paid $2,400 per month plus 6% of his sales. His brother earns $3,300 per month. For what amount of total sales will Keshad\u2019s monthly pay be higher than his brother\u2019s monthly pay?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$15000$$","hints":{"DefaultPathway":[{"id":"ae9e419IneqApp14a-h1","type":"hint","dependencies":[],"title":"Creating an Inequality","text":"First, we must transfer the word problem into an inequality that represents the total earning Keshad makes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp14a-h2","type":"hint","dependencies":["ae9e419IneqApp14a-h1"],"title":"Creating an Inequality","text":"We know that Keshad wants his earning to be at least $3300, and he gets paid $2400 per month plus 6% (which is $$0.06$$ in decimals) of his sales. If we let $$x$$ represent his total sales, then his total earning is $$2400+0.06x$$. Therefore, we get the inequality $$2400+0.06x \\\\geq 3300$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp14a-h3","type":"hint","dependencies":["ae9e419IneqApp14a-h2"],"title":"Solving the Inequality","text":"Now, we must solve the inequality for $$x$$ by algebrically manipulating it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp14a-h4","type":"hint","dependencies":["ae9e419IneqApp14a-h3"],"title":"Solving the Inequality","text":"To isolate $$x$$, we can first subtract $$2400$$ from both sides and then divide both sides by $$0.06$$. We now have $$x \\\\geq 15000$$ as our inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp14a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15000$$"],"dependencies":["ae9e419IneqApp14a-h4"],"title":"Creating a Sentence-Answer","text":"Since $$x$$ must be greater than or equal to $$15000$$, what is the minimum sales Keshad has to make?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae9e419IneqApp15","title":"Andre\'s Earning","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Solve Applications with Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ae9e419IneqApp15a","stepAnswer":["$$400000$$"],"problemType":"TextBox","stepTitle":"Andre has been offered an entry-level job. The company offered him $48,000 per year plus $$3.5\\\\%$$ of his total sales. Andre knows that the average pay for this job is $62,000. What would Andre\u2019s total sales need to be for his pay to be at least as high as the average pay for this job?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$400000$$","hints":{"DefaultPathway":[{"id":"ae9e419IneqApp15a-h1","type":"hint","dependencies":[],"title":"Creating an Inequality","text":"First, we must transfer the word problem into an inequality that represents the total earning Andre makes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp15a-h2","type":"hint","dependencies":["ae9e419IneqApp15a-h1"],"title":"Creating an Inequality","text":"We know that Andre wants his earning to be at least $62,000, and he gets paid $48,000 per year plus $$3.5\\\\%$$ (which is $$0.035$$ in decimals) of his sales. If we let $$x$$ represent his total sales, then his total earning is $$48000+0.035x$$. Therefore, we get the inequality $$48000+0.035x \\\\geq 62000$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp15a-h3","type":"hint","dependencies":["ae9e419IneqApp15a-h2"],"title":"Solving the Inequality","text":"Now, we must solve the inequality for $$x$$ by algebrically manipulating it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp15a-h4","type":"hint","dependencies":["ae9e419IneqApp15a-h3"],"title":"Solving the Inequality","text":"To isolate $$x$$, we can first subtract $$48000$$ from both sides and then divide both sides by $$0.035$$. We now have $$x \\\\geq 400000$$ as our inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp15a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$400000$$"],"dependencies":["ae9e419IneqApp15a-h4"],"title":"Creating a Sentence-Answer","text":"Since $$x$$ must be greater than or equal to $$400000$$, what is the minimum sales Andre has to make?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae9e419IneqApp16","title":"Jake\'s Water Bill","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Solve Applications with Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ae9e419IneqApp16a","stepAnswer":["$$16$$"],"problemType":"TextBox","stepTitle":"Jake\u2019s water bill is $$\\\\$24.80$$ per month plus $$\\\\$2.20$$ per ccf (hundred cubic feet) of water. What is the maximum number of ccf Jake can use if he wants his bill to be no more than $60?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16$$","hints":{"DefaultPathway":[{"id":"ae9e419IneqApp16a-h1","type":"hint","dependencies":[],"title":"Creating an Inequality","text":"First, we must transfer the word problem into an inequality that represents the amount Jake pays for his water bill.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp16a-h2","type":"hint","dependencies":["ae9e419IneqApp16a-h1"],"title":"Creating an Inequality","text":"We know that Jake wants his water bill to be less than or equal to $60, and his bill consists of a fixed charge of $$\\\\$24.80$$ per month and an additional $$\\\\$2.20$$ per ccf of water used. If we let $$x$$ represent the number of ccf Jake uses, then the total amount of the water bill would be $$24.8+2.2x$$. Therefore, we get the inequality $$24.8+2.2x \\\\leq 60$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp16a-h3","type":"hint","dependencies":["ae9e419IneqApp16a-h2"],"title":"Solving the Inequality","text":"Now, we must solve the inequality for $$x$$ by algebrically manipulating it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp16a-h4","type":"hint","dependencies":["ae9e419IneqApp16a-h3"],"title":"Solving the Inequality","text":"To isolate $$x$$, we can subtract $$24.8$$ from both sides and then divide both sides by $$0.035$$. We now have $$x \\\\leq 16$$ as our inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp16a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":["ae9e419IneqApp16a-h4"],"title":"Creating a Sentence-Answer","text":"Since $$x$$ must be less than or equal to $$16$$, what is the maximum number of ccf Jake can use?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae9e419IneqApp17","title":"Marlon\'s TV Plan","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Solve Applications with Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ae9e419IneqApp17a","stepAnswer":["$$9$$"],"problemType":"TextBox","stepTitle":"Marlon\u2019s TV plan costs $$\\\\$49.99$$ per month plus $$\\\\$5.49$$ per first-run movie. How many first-run movies can he watch if he wants to keep his monthly bill to be a maximum of $100?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9$$","hints":{"DefaultPathway":[{"id":"ae9e419IneqApp17a-h1","type":"hint","dependencies":[],"title":"Creating an Inequality","text":"First, we must transfer the word problem into an inequality that represents the amount of Marlon\'s monthly TV bill.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp17a-h2","type":"hint","dependencies":["ae9e419IneqApp17a-h1"],"title":"Creating an Inequality","text":"We know that Marlon wants his monthly TV bill to be less than or equal to $100, and his bill consists of a fixed charge of $$\\\\$49.99$$ per month and an additional $$\\\\$5.49$$ for each first-run movie. If we let $$x$$ represent the number of first-run movies Marlon purchases, then the total amount of his TV bill would be $$49.99+5.49x$$. Therefore, we get the inequality $$49.99+5.49x \\\\leq 100$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp17a-h3","type":"hint","dependencies":["ae9e419IneqApp17a-h2"],"title":"Solving the Inequality","text":"Now, we must solve the inequality for $$x$$ by algebrically manipulating it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp17a-h4","type":"hint","dependencies":["ae9e419IneqApp17a-h3"],"title":"Solving the Inequality","text":"To isolate $$x$$, we can subtract $$49.99$$ from both sides and then divide both sides by $$5.49$$. We now have $$x \\\\leq 9.109$$ as our inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp17a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["ae9e419IneqApp17a-h4"],"title":"Creating a Sentence-Answer","text":"Since $$x$$ must be less than or equal to $$9.109$$ and $$x$$ has to be a whole number, what is the maximum number of first-run movies Jake can watch?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae9e419IneqApp18","title":"Moshde\'s Hairstyling Business","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Solve Applications with Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ae9e419IneqApp18a","stepAnswer":["$$48$$"],"problemType":"TextBox","stepTitle":"Moshde runs a hairstyling business from her house. She charges $45 for a haircut and style. Her monthly expenses are $960. She wants to be able to put at least $1,200 per month into her savings account order to open her own salon. How many \u201ccut & styles\u201d must she do to save at least $1,200 per month?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$48$$","hints":{"DefaultPathway":[{"id":"ae9e419IneqApp18a-h1","type":"hint","dependencies":[],"title":"Creating an Inequality","text":"First, we must transfer the word problem into an inequality that represents the amount of Moshde\'s monthly income.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp18a-h2","type":"hint","dependencies":["ae9e419IneqApp18a-h1"],"title":"Creating an Inequality","text":"We know that Moshde wants to save $1,200 each month. We also know that she spends $960 each month, and she earns $45 for each haircut and style. If we let $$x$$ represent the number of haircut and style Moshde does each month, then her total monthly income would be $$45x$$. Since all income are either spent or saved, we get the inequality $$45x \\\\geq 1200+960$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp18a-h3","type":"hint","dependencies":["ae9e419IneqApp18a-h2"],"title":"Solving the Inequality","text":"Now, we must solve the inequality for $$x$$ by algebrically manipulating it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp18a-h4","type":"hint","dependencies":["ae9e419IneqApp18a-h3"],"title":"Solving the Inequality","text":"To isolate $$x$$, we can simplify the right hand side and then divide both sides by $$45$$. We now have $$x \\\\geq 48$$ as our inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp18a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$48$$"],"dependencies":["ae9e419IneqApp18a-h4"],"title":"Creating a Sentence-Answer","text":"Since $$x$$ must be greater than or equal to $$48$$, what is the minimum number of haircut and style Moshde must do per month?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae9e419IneqApp19","title":"Katherine\'s Meals","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Solve Applications with Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ae9e419IneqApp19a","stepAnswer":["$$44$$"],"problemType":"TextBox","stepTitle":"Katherine is a personal chef. She charges $115 per four-person meal. Her monthly expenses are $3,150. How many four-person meals must she sell in order to make a profit of at least $1,900?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$44$$","hints":{"DefaultPathway":[{"id":"ae9e419IneqApp19a-h1","type":"hint","dependencies":[],"title":"Creating an Inequality","text":"First, we must transfer the word problem into an inequality that represents the amount of Katherin monthly income.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp19a-h2","type":"hint","dependencies":["ae9e419IneqApp19a-h1"],"title":"Creating an Inequality","text":"We know that Katherine wants to make a profit of at least $1,900 each month and has a monthly expense of $3,150. We also know that she charges $115 per four-person meal. If we let $$x$$ represent the number of four-person meals Katherine sells each month, then her total monthly income would be $$115x$$. Since profit is the difference between total income and the amount of expense, we get the inequality $$115x-3150 \\\\geq 1900$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp19a-h3","type":"hint","dependencies":["ae9e419IneqApp19a-h2"],"title":"Solving the Inequality","text":"Now, we must solve the inequality for $$x$$ by algebrically manipulating it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp19a-h4","type":"hint","dependencies":["ae9e419IneqApp19a-h3"],"title":"Solving the Inequality","text":"To isolate $$x$$, we can add $$3150$$ to both sides and then divide both sides by $$115$$. We now have $$x \\\\geq 43.913$$ as our inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp19a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$44$$"],"dependencies":["ae9e419IneqApp19a-h4"],"title":"Creating a Sentence-Answer","text":"Since $$x$$ must be greater than or equal to $$43.913$$ and $$x$$ has to be a whole number, what is the minimum number of four-person meals Katherine has to sell per month?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae9e419IneqApp2","title":"Dawn\'s Tablets","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Solve Applications with Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ae9e419IneqApp2a","stepAnswer":["$$15$$"],"problemType":"TextBox","stepTitle":"Dawn won a mini-grant of $4,000 to buy tablet computers for her classroom. The tablets she would like to buy cost $$\\\\$254.12$$ each, including tax and delivery. What is the maximum number of tablets Dawn can buy?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$15$$","hints":{"DefaultPathway":[{"id":"ae9e419IneqApp2a-h1","type":"hint","dependencies":[],"title":"Creating an Inequality","text":"First, we must transfer the word problem into an inequality that represents the number of tablets (n) Dawn can buy.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp2a-h2","type":"hint","dependencies":["ae9e419IneqApp2a-h1"],"title":"Creating an Inequality","text":"We know that $254.12*(the number of tablets) is no more than $4,000. So, we can create the inequality $$254.12n \\\\leq 4000$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp2a-h3","type":"hint","dependencies":["ae9e419IneqApp2a-h2"],"title":"Solving the Inequality","text":"Now we must solve the inequality by isolating $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp2a-h4","type":"hint","dependencies":["ae9e419IneqApp2a-h3"],"title":"Solving the Inequality","text":"To do this, we need to divide both sides by $$254.12$$. Now, we have $$n \\\\leq 15.74$$ as our inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp2a-h5","type":"hint","dependencies":["ae9e419IneqApp2a-h4"],"title":"Solving the Inequality","text":"Since the number of tablets can only be a whole number, we must round the inequality. Now, we have $$n \\\\leq 15$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["ae9e419IneqApp2a-h5"],"title":"Creating a Sentence-Answer","text":"Since $$n$$ must be less than or equal to $$15$$, what is the maximum number of tablets that Dawn can buy?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae9e419IneqApp20","title":"Trip to the State Convention","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Solve Applications with Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ae9e419IneqApp20a","stepAnswer":["$$49$$"],"problemType":"TextBox","stepTitle":"Five student government officers want to go to the state convention. It will cost them $110 for registration, $375 for transportation and food, and $42 per person for the hotel. There is $450 budgeted for the convention in the student government savings account. They can earn the rest of the money they need by having a car wash. If they charge $5 per car, how many cars must they wash in order to have enough money to pay for the trip?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$49$$","hints":{"DefaultPathway":[{"id":"ae9e419IneqApp20a-h1","type":"hint","dependencies":[],"title":"Sum Up Expenses","text":"The first thing we need to do is to solve for the total expenses that the five student government officers need to pay. Once we know the total expense, we can find out the additional amount of money that the officers need to make up for by washing cars.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp20a-h2","type":"hint","dependencies":["ae9e419IneqApp20a-h1"],"title":"Sum Up Expenses","text":"To find the total expense, we sum up the expense from registration ($110), transportation and food ($375), and hotel (5*$42).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp20a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$695$$"],"dependencies":["ae9e419IneqApp20a-h2"],"title":"Sum Up Expenses","text":"What is $$110+375+5\\\\times42$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp20a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$245$$"],"dependencies":["ae9e419IneqApp20a-h3"],"title":"Amount that Needs to be Earned","text":"We know that the five student government officers need $695 in total, but they only have $450 currently. What is the amount that they will need to earn by themselves?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp20a-h5","type":"hint","dependencies":["ae9e419IneqApp20a-h4"],"title":"Creating an Inequality","text":"We know that the five student government officers need to earn at least $245 by washing cars and they charge $5 per car. If we let $$x$$ represent the number of cars they wash, then the total earning from the car wash would be $$5x$$. Therefore, we get the inequality $$5x \\\\geq 245$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp20a-h6","type":"hint","dependencies":["ae9e419IneqApp20a-h5"],"title":"Solving the Inequality","text":"Now, we must solve the inequality for $$x$$ by algebrically manipulating it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp20a-h7","type":"hint","dependencies":["ae9e419IneqApp20a-h6"],"title":"Solving the Inequality","text":"To solve for $$x$$, we can divide both sides by $$5$$ to isolate $$x$$. We now have $$x \\\\geq 49$$ as our inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp20a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$49$$"],"dependencies":["ae9e419IneqApp20a-h7"],"title":"Creating a Sentence-Answer","text":"Since $$x$$ must be greater than or equal to $$49$$, what is the minimum number of cars the student government officers have to wash?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae9e419IneqApp21","title":"Renting Apartment","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Solve Applications with Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ae9e419IneqApp21a","stepAnswer":["$$9$$"],"problemType":"TextBox","stepTitle":"Alonzo works as a car detailer. He charges $175 per car. He is planning to move out of his parents\u2019 house and rent his first apartment. He will need to pay $120 for application fees, $950 for security deposit, and first and last months\u2019 rent at $1,140 per month. He has $1,810 in savings. How many cars must he detail to have enough money to rent the apartment?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9$$","hints":{"DefaultPathway":[{"id":"ae9e419IneqApp21a-h1","type":"hint","dependencies":[],"title":"Sum Up Expenses","text":"The first thing we need to do is to solve for the total expenses Alonzo has to pay. Once we know the total expense, we can find out the additional amount of money Alonzo needs to make by detailing cars.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp21a-h2","type":"hint","dependencies":["ae9e419IneqApp21a-h1"],"title":"Sum Up Expenses","text":"To find the total expense, we sum up the expense from paying the application fees ($120), security deposit ($950), and the rent for the first and last months (2*$1140).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp21a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3350$$"],"dependencies":["ae9e419IneqApp21a-h2"],"title":"Sum Up Expenses","text":"What is $$120+950+2\\\\times1140$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp21a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1540$$"],"dependencies":["ae9e419IneqApp21a-h3"],"title":"Amount that Needs to be Earned","text":"We know that Alonzo needs $3,350 in total, but he only has $1,810 currently. What is the additional amount that he will need to earn?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp21a-h5","type":"hint","dependencies":["ae9e419IneqApp21a-h4"],"title":"Creating an Inequality","text":"We know that Alonzo needs to earn at least $1,540 by detailing cars and he charges $175 per car. If we let $$x$$ represent the number of cars he details, then the total earning from the car detailing would be $$175x$$. Therefore, we get the inequality $$175x \\\\geq 1540$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp21a-h6","type":"hint","dependencies":["ae9e419IneqApp21a-h5"],"title":"Solving the Inequality","text":"Now, we must solve the inequality for $$x$$ by algebrically manipulating it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp21a-h7","type":"hint","dependencies":["ae9e419IneqApp21a-h6"],"title":"Solving the Inequality","text":"To solve for $$x$$, we can divide both sides by $$175$$ to isolate $$x$$. We now have $$x \\\\geq 8.8$$ as our inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp21a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["ae9e419IneqApp21a-h7"],"title":"Creating a Sentence-Answer","text":"Since $$x$$ must be greater than or equal to $$8.8$$ and $$x$$ has to be a whole number, what is the minimum number of cars Alonzo has to detail?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae9e419IneqApp22","title":"Picking College Classes","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Solve Applications with Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ae9e419IneqApp22a","stepAnswer":["$$13$$"],"problemType":"TextBox","stepTitle":"Marcela is registering for her college classes, which cost $105 per unit. How many units can she take to have a maximum cost of $1,365?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$13$$","hints":{"DefaultPathway":[{"id":"ae9e419IneqApp22a-h1","type":"hint","dependencies":[],"title":"Creating an Inequality","text":"First, we must transfer the word problem into an inequality that represents the total cost of classes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp22a-h2","type":"hint","dependencies":["ae9e419IneqApp22a-h1"],"title":"Creating an Inequality","text":"We know that the maximum cost Marcela can afford is $1,365, and each unit costs $105. If we let $$x$$ represent the number of units Marcela takes, then the total cost would be $$105x$$. Therefore, we have the inequality $$105x \\\\leq 1365$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp22a-h3","type":"hint","dependencies":["ae9e419IneqApp22a-h2"],"title":"Solving the Inequality","text":"Now, we must solve the inequality for $$x$$ by algebrically manipulating it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp22a-h4","type":"hint","dependencies":["ae9e419IneqApp22a-h3"],"title":"Solving the Inequality","text":"To do this, we can divide both sides by $$105$$ to isolate $$x$$. We now have $$x \\\\leq 13$$ as our inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp22a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$13$$"],"dependencies":["ae9e419IneqApp22a-h4"],"title":"Creating a Sentence-Answer","text":"Since $$x$$ must be less than or equal to $$13$$, what is the maximum number of units Marcela can take?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae9e419IneqApp23","title":"Ryan\'s Earning","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Solve Applications with Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ae9e419IneqApp23a","stepAnswer":["$$86$$"],"problemType":"TextBox","stepTitle":"Ryan charges his neighbors $$\\\\$17.50$$ to wash their car. How many cars must he wash next summer if his goal is to earn at least $1,500?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$86$$","hints":{"DefaultPathway":[{"id":"ae9e419IneqApp23a-h1","type":"hint","dependencies":[],"title":"Creating an Inequality","text":"First, we must transfer the word problem into an inequality that represents the total earning Ryan makes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp23a-h2","type":"hint","dependencies":["ae9e419IneqApp23a-h1"],"title":"Creating an Inequality","text":"We know that Ryan wants his earning to be at least $1,500, and he charges his neighbors $$\\\\$17.50$$ to wash their car. If we let $$x$$ represent the number of cars Ryan washes, then his total earning would be $$17.5x$$. Therefore, we have the inequality $$17.5x \\\\geq 1500$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp23a-h3","type":"hint","dependencies":["ae9e419IneqApp23a-h2"],"title":"Solving the Inequality","text":"Now, we must solve the inequality for $$x$$ by algebrically manipulating it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp23a-h4","type":"hint","dependencies":["ae9e419IneqApp23a-h3"],"title":"Solving the Inequality","text":"To do this, we can divide both sides by $$17.5$$ to isolate $$x$$. We now have $$x \\\\geq 85.714$$ as our inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp23a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$86$$"],"dependencies":["ae9e419IneqApp23a-h4"],"title":"Creating a Sentence-Answer","text":"Since $$x$$ must be greater than or equal to $$85.714$$ and $$x$$ has to be a whole number, what is the minimum number of cars Ryan needs to wash?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae9e419IneqApp24","title":"Kimuyen\'s Earning","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Solve Applications with Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ae9e419IneqApp24a","stepAnswer":["$$16875$$"],"problemType":"TextBox","stepTitle":"Kimuyen needs to earn $4,150 per month in order to pay all her expenses. Her job pays her $3,475 per month plus 4% of her total sales. What is the minimum Kimuyen\u2019s total sales must be in order for her to pay all her expenses?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16875$$","hints":{"DefaultPathway":[{"id":"ae9e419IneqApp24a-h1","type":"hint","dependencies":[],"title":"Creating an Inequality","text":"First, we must transfer the word problem into an inequality that represents the total earning Kimuyen makes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp24a-h2","type":"hint","dependencies":["ae9e419IneqApp24a-h1"],"title":"Creating an Inequality","text":"We know that Kimuyen wants her earning to be at least $4,150, and he gets paid $3,475 per month plus 4% (which is $$0.04$$ in decimals) of her total sales. If we let $$x$$ represent her total sales, then her total earning is $$3475+0.04x$$. Therefore, we get the inequality $$3475+0.04x \\\\geq 4150$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp24a-h3","type":"hint","dependencies":["ae9e419IneqApp24a-h2"],"title":"Solving the Inequality","text":"Now, we must solve the inequality for $$x$$ by algebrically manipulating it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp24a-h4","type":"hint","dependencies":["ae9e419IneqApp24a-h3"],"title":"Solving the Inequality","text":"To isolate $$x$$, we can first subtract $$3475$$ from both sides and then divide both sides by $$0.04$$. We now have $$x \\\\geq 16875$$ as our inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp24a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16875$$"],"dependencies":["ae9e419IneqApp24a-h4"],"title":"Creating a Sentence-Answer","text":"Since $$x$$ must be greater than or equal to $$16875$$, what is the minimum sales Kimuyen has to make?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae9e419IneqApp3","title":"Peter\'s Sales","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Solve Applications with Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ae9e419IneqApp3a","stepAnswer":["$$3541.67$$"],"problemType":"TextBox","stepTitle":"Pete works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3541.67$$","hints":{"DefaultPathway":[{"id":"ae9e419IneqApp3a-h1","type":"hint","dependencies":[],"title":"Creating an Inequality","text":"We know that $500+12% of Peter\'s sales must be greater than $925. If we let s equal the number of sales, then we can create an inequality representing a relationship between Peter\'s sales and his weekly pay.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp3a-h2","type":"hint","dependencies":["ae9e419IneqApp3a-h1"],"title":"Creating an Inequality","text":"We can represent the situation with the inequality $$500+0.12s>925$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp3a-h3","type":"hint","dependencies":["ae9e419IneqApp3a-h2"],"title":"Solving the Inequality","text":"We must now solve the inequality by isolating the variable s. This can be done by algebraically mainpulating the inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp3a-h4","type":"hint","dependencies":["ae9e419IneqApp3a-h3"],"title":"Solving the Inequality","text":"The first step is to subtract $$500$$ from both sides of the inequality. We now have $$0.12s>425$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp3a-h5","type":"hint","dependencies":["ae9e419IneqApp3a-h4"],"title":"Solving the Inequality","text":"Now, we must divide both sides of the inequality by $$0.12$$. Now, we have: $$s>3541.67$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp3a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3541.67$$"],"dependencies":["ae9e419IneqApp3a-h5"],"title":"Creating a Sentence-Answer","text":"Now all we have to do is describe the inequality with words. If Peter wants a weekly play greater than $925, what is the minimum sales he needs to make ($)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae9e419IneqApp4","title":"Boxes and Pallets","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Solve Applications with Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ae9e419IneqApp4a","stepAnswer":["$$20$$"],"problemType":"TextBox","stepTitle":"Alan is loading a pallet with boxes that each weighs $$45$$ pounds. The pallet can safely support no more than $$900$$ pounds. How many boxes can he safely load onto the pallet?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$20$$","hints":{"DefaultPathway":[{"id":"ae9e419IneqApp4a-h1","type":"hint","dependencies":[],"title":"Creating an Inequality","text":"First, we must translate the word problem into an inequality that represents the number of boxes one pallet can support.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp4a-h2","type":"hint","dependencies":["ae9e419IneqApp4a-h1"],"title":"Creating an Inequality","text":"Since we know that the boxes weight $$45$$ pounds each, and they cannot weigh more than $$900$$ pounds in total, let\'s let $$n$$ equal the number of boxes. Now, we have the inequality $$45n \\\\leq 900$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp4a-h3","type":"hint","dependencies":["ae9e419IneqApp4a-h2"],"title":"Solving the Inequality","text":"Now, we must isolate the variable $$n$$ in order to calculate the maximum number of boxes a pallet can support.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp4a-h4","type":"hint","dependencies":["ae9e419IneqApp4a-h3"],"title":"Solving the Inequality","text":"To isolate $$n$$, we must divide both sides by 45; we now have $$n \\\\leq 20$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp4a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["ae9e419IneqApp4a-h4"],"title":"Creating a Sentence-Answer","text":"Since we know that $$n$$ must be less than or equal to $$20$$, what is the maximum number of boxes one pallet can support?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae9e419IneqApp5","title":"Elevator Safety","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Solve Applications with Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ae9e419IneqApp5a","stepAnswer":["$$14$$"],"problemType":"TextBox","stepTitle":"The elevator in Yehire\u2019s apartment building has a sign that says the maximum weight is 2,100 pounds. If the average weight of one person is $$150$$ pounds, how many people can safely ride the elevator?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$14$$","hints":{"DefaultPathway":[{"id":"ae9e419IneqApp5a-h1","type":"hint","dependencies":[],"title":"Creating an Inequality","text":"First, we must transfer the word problen into an inequality that represents the maximum number of people that can ride in the elvator safely.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp5a-h2","type":"hint","dependencies":["ae9e419IneqApp5a-h1"],"title":"Creating an Inequality","text":"Since we know that the maximum weight one elevator can support is 2,100 pounds and that the average weight of one person is $$150$$ pounds, we can let $$n$$ equal the number of people in the elevator and create the following inequality: $$150n \\\\leq 2100$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp5a-h3","type":"hint","dependencies":["ae9e419IneqApp5a-h2"],"title":"Solving the Inequality","text":"Now, we must isolate $$n$$ so that we are able to calculate its value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp5a-h4","type":"hint","dependencies":["ae9e419IneqApp5a-h3"],"title":"Solving the Inequality","text":"To isolate $$n$$, we must divide both sides by $$150$$. We now have $$n \\\\leq 14$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$14$$"],"dependencies":["ae9e419IneqApp5a-h4"],"title":"Creating a Sentence-Answer","text":"Since $$n$$ must be less than or equal to $$14$$, what is the maximum number of people the elevator can safely support?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae9e419IneqApp6","title":"Juice Boxes","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Solve Applications with Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ae9e419IneqApp6a","stepAnswer":["$$7$$"],"problemType":"TextBox","stepTitle":"Angie has $20 to spend on juice boxes for her son\u2019s preschool picnic. Each pack of juice boxes costs $$\\\\$2.63$$. What is the maximum number of packs she can buy?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7$$","hints":{"DefaultPathway":[{"id":"ae9e419IneqApp6a-h1","type":"hint","dependencies":[],"title":"Creating an Inequality","text":"First, we must create an inequality that represents the maximum number of juice packs Angie can buy based on the amount of money she has.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp6a-h2","type":"hint","dependencies":["ae9e419IneqApp6a-h1"],"title":"Creating an Inequality","text":"We know that the total price of the juice boxes must be less than or equal to $$20$$. If we let $$n$$ equal the number of juice boxes, we have then inequality: $$2.63n \\\\leq 20$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp6a-h3","type":"hint","dependencies":["ae9e419IneqApp6a-h2"],"title":"Solving the Inequality","text":"Now, we must isolate the variable $$n$$ by algebraically manipulating the inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp6a-h4","type":"hint","dependencies":["ae9e419IneqApp6a-h3"],"title":"Solving the Inequality","text":"To isolate $$n$$, we must divide both sides by $$2.63$$. We now have the inequality $$n \\\\leq 7.60$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["ae9e419IneqApp6a-h4"],"title":"Creating a Sentence-Answer","text":"Since we know that $$n$$, the number of juice boxes, must be less than or equal to $$7.6$$, what is the maximum (whole) number of juice boxes Angie can buy?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae9e419IneqApp7","title":"Daniel\'s Party","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Solve Applications with Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ae9e419IneqApp7a","stepAnswer":["$$11$$"],"problemType":"TextBox","stepTitle":"Daniel wants to surprise his girlfriend with a birthday party at her favorite restaurant. It will cost $$\\\\$42.75$$ per person for dinner, including tip and tax. His budget for the party is $500. What is the maximum number of people Daniel can have at the party?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$11$$","hints":{"DefaultPathway":[{"id":"ae9e419IneqApp7a-h1","type":"hint","dependencies":[],"title":"Creating an Inequality","text":"We must first write an inequality to represent the number of people Daniel can have at his party (n) in terms of the amount of money he has ($500).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp7a-h2","type":"hint","dependencies":["ae9e419IneqApp7a-h1"],"title":"Creating an Inequality","text":"Since we know that the cost for dinner is $$\\\\$42.75$$ per person, 42.75*(the number of people) must be less than or equal to $500, which is the budget. This can be represented by the inequality $$42.75n \\\\leq 500$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp7a-h3","type":"hint","dependencies":["ae9e419IneqApp7a-h2"],"title":"Solving the Inequality","text":"We must now isolate the variable $$n$$ so that we can calculate the maximum number of people Daniel can have at the party.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp7a-h4","type":"hint","dependencies":["ae9e419IneqApp7a-h3"],"title":"Solving the Inequality","text":"To isolate the variable, we must divide both sides of the inequality by $$42.75$$. We now have $$n \\\\leq 11.695$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp7a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11$$"],"dependencies":["ae9e419IneqApp7a-h4"],"title":"Creating a Sentence-Answer","text":"Since we know that $$n$$, the number of people Daniel can have at his party, must be less than or equal to $$11.695$$, what is the maximum (whole) number of people Daniel can have at the party?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae9e419IneqApp8","title":"Tiffany\'s Salary","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Solve Applications with Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ae9e419IneqApp8a","stepAnswer":["$$4000000$$"],"problemType":"TextBox","stepTitle":"Tiffany just graduated from college and her new job will pay her $20,000 per year plus 2% of all sales. She wants to earn at least $100,000 per year. For what total sales will she be able to achieve her goal?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4000000$$","hints":{"DefaultPathway":[{"id":"ae9e419IneqApp8a-h1","type":"hint","dependencies":[],"title":"Creating an Inequality","text":"We must first create an inequality to represent Tiffany\'s salary with respect to her total number of sales, $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp8a-h2","type":"hint","dependencies":["ae9e419IneqApp8a-h1"],"title":"Creating an Inequality","text":"We know that $20,000 plus $$2$$ percent (or $$0.02)$$ of Tiffany\'s sales must be at least $100,000. This means that $$20000+0.02n \\\\geq 100000$$, where $$n$$ is the total number of Tiffany\'s sales.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp8a-h3","type":"hint","dependencies":["ae9e419IneqApp8a-h2"],"title":"Solving the Inequality","text":"To solve this inequality, we need to isolate the variable $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp8a-h4","type":"hint","dependencies":["ae9e419IneqApp8a-h3"],"title":"Solving the Inequality","text":"To start, we can subtract 20,000 from both sides of the inequality. We now have $$0.02n \\\\geq 80000$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp8a-h5","type":"hint","dependencies":["ae9e419IneqApp8a-h4"],"title":"Solving the Inequality","text":"We now need to divide both sides of the inequality by $$0.02$$. This means that $$n \\\\geq 4000000$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp8a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4000000$$"],"dependencies":["ae9e419IneqApp8a-h5"],"title":"Creating a Sentence-Answer","text":"Since we know that $$n$$, Tiffany\'s total sales, must be greater than or equal to $$4000000$$, what is the minimum sale she has to make?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"ae9e419IneqApp9","title":"Christian\'s Sales","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.6 Solve Applications with Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"ae9e419IneqApp9a","stepAnswer":["$$1200000$$"],"problemType":"TextBox","stepTitle":"Christian has been offered a new job that pays $24,000 a year plus 3% of sales. For what total sales would this new job pay more than his current job which pays $60,000?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1200000$$","hints":{"DefaultPathway":[{"id":"ae9e419IneqApp9a-h1","type":"hint","dependencies":[],"title":"Creating an Inequality","text":"We must first create an inequality to represent Christian\'s salary with respect to his total number of sales, $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp9a-h2","type":"hint","dependencies":["ae9e419IneqApp9a-h1"],"title":"Creating an Inequality","text":"We know that $24,000 plus $$3$$ percent (or $$0.03)$$ of Christian\'s sales must be greater than $60,000. This means that $$24000+0.03n>60000$$, where $$n$$ represents the total number of Christian\'s sales.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp9a-h3","type":"hint","dependencies":["ae9e419IneqApp9a-h2"],"title":"Solving the Inequality","text":"To solve this inequality, we need to isolate the variable $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp9a-h4","type":"hint","dependencies":["ae9e419IneqApp9a-h3"],"title":"Solving the Inequality","text":"To start, we can subtract 24,000 from both sides of the inequality. We now have $$0.03n>36000$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp9a-h5","type":"hint","dependencies":["ae9e419IneqApp9a-h4"],"title":"Solving the Inequality","text":"We now need to divide both sides of the inequality by $$0.03$$. This means that $$n>1200000$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"ae9e419IneqApp9a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1200000$$"],"dependencies":["ae9e419IneqApp9a-h5"],"title":"Creating a Sentence-Answer","text":"Since we know that $$n$$, Christian\'s total number of sales, must be greater than $$1200000$$, what is the minimum sale she has to make?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeae96dlogarithmic1","title":"Expanding Logarithms Using Product, Quotient, and Power Rules","body":"Expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.5 Logarithmic Properties","courseName":"OpenStax: College Algebra","steps":[{"id":"aeae96dlogarithmic1a","stepAnswer":["$$\\\\ln(3)+\\\\ln(a)+\\\\ln(b)+\\\\ln(5)+\\\\ln(c)$$"],"problemType":"TextBox","stepTitle":"$$\\\\ln(3a b 5c)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\ln(3)+\\\\ln(a)+\\\\ln(b)+\\\\ln(5)+\\\\ln(c)$$","hints":{"DefaultPathway":[{"id":"aeae96dlogarithmic1a-h1","type":"hint","dependencies":[],"title":"Product Rule for Logarithms","text":"The product rule for logarithms can be used to simplify a logarithm of a product by rewriting it as a sum of individual logarithms.\\\\n$$\\\\log_{b}\\\\left(M N\\\\right)=\\\\log_{b}\\\\left(M\\\\right)+\\\\log_{b}\\\\left(N\\\\right)$$ for $$b>0$$\\\\n\\\\nGiven the logarithm of a product, use the product rule of logarithms to write an equivalent sum of logarithms.\\\\n\\\\n1) Factor the argument completely, expressing each whole number factor as a product of primes.\\\\n2) Write the equivalent expression by summing the logarithms of each factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic1a-h2","type":"hint","dependencies":[],"title":"Quotient Rule for Logarithms","text":"The quotient rule for logarithms can be used to simplify a logarithm or a quotient by rewriting it as the difference of individual logarithms.\\\\n$$\\\\log_{b}\\\\left(\\\\frac{M}{N}\\\\right)=\\\\log_{b}\\\\left(M\\\\right)-\\\\log_{b}\\\\left(N\\\\right)$$\\\\n\\\\nGiven the logarithm of a quotient, use the quotient rule of logarithms to write an equivalent difference of logarithms.\\\\n\\\\n1) Express the argument in lowest terms by factoring the numerator and denominator and canceling common terms.\\\\n2) Write the equivalent expression by subtracting the logarithm of the denominator from the logarithm of the numerator.\\\\n3) Check to see that each term is fully expanded. If not, apply the product rule for logarithms to expand completely.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic1a-h3","type":"hint","dependencies":[],"title":"Power Rule for Logarithms","text":"The power rule for logarithms can be used to simplify the logarithm of a power by rewriting it as the product of the exponent times the logarithm of the base.\\\\n$$\\\\log_{b}\\\\left(M^n\\\\right)=n*\\\\log_{b}\\\\left(M\\\\right)$$\\\\n\\\\nGiven the logarithm of a power, use the power rule of logarithms to write an equivalent product of a factor and a logarithm.\\\\n\\\\n1) Express the argument as a power, if needed.\\\\n2) Write the equivalent expression by multiplying the exponent times the logarithm of the base.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic1a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Product Rule for Logarithms"],"dependencies":[],"title":"Identify the Relevant Rules","text":"Identify the relevant rules that can be used to expand the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Product Rule for Logarithms","Quotient Rule for Logarithms","Power Rule for Logarithms"]},{"id":"aeae96dlogarithmic1a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\ln(a)+\\\\ln(b)$$"],"dependencies":["aeae96dlogarithmic1a-h4"],"title":"Apply the Product Rule for Logarithms","text":"We can expand by applying the product rule to the entire expression. Consider expanding $$\\\\ln(a b)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeae96dlogarithmic10","title":"Changing Logarithmic Expressions to Expressions Involving Only Common Logs","body":"Rewrite each expression as an equivalent ratio of logs using the indicated base.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.5 Logarithmic Properties","courseName":"OpenStax: College Algebra","steps":[{"id":"aeae96dlogarithmic10a","stepAnswer":["$$\\\\frac{\\\\ln(55.875)}{\\\\ln(14)}$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{14}\\\\left(55.875\\\\right)$$ to base $$10$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{\\\\ln(55.875)}{\\\\ln(14)}$$","hints":{"DefaultPathway":[{"id":"aeae96dlogarithmic10a-h1","type":"hint","dependencies":[],"title":"Change-of-Base Formula","text":"The change-of-base formula can be used to evaluate a logarithm with any base.\\\\nFor any positive real numbers M,b, and $$n$$, where $$n \\\\neq 1$$ and $$b \\\\neq 1$$, $$\\\\log_{b}\\\\left(M\\\\right)=\\\\log_{n}\\\\left(M\\\\right)/\\\\log_{n}\\\\left(b\\\\right)$$.\\\\n\\\\n1) Determine the new base $$n$$, remembering that the common log, $$\\\\ln(x)$$, has base $$10$$, and the natural log, ln(x), has base e.\\\\n2) Rewrite the log as a quotient using the change-of-base formula\\\\na) The numerator of the quotient will be a logarithm with base $$n$$ and argument M.\\\\nb) The denominator of the quotient will be a logarithm with base $$n$$ and argument $$b$$.\\\\n","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["aeae96dlogarithmic10a-h1"],"title":"New Base","text":"What is the new base that we\'re changing to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic10a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\log_{b}\\\\left(M\\\\right)=\\\\frac{\\\\ln(M)}{\\\\ln(b)}$$"],"dependencies":["aeae96dlogarithmic10a-h2"],"title":"Change-of-Base","text":"What form does the quotient take after the change-of-base to the new base e?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\log_{b}\\\\left(M\\\\right)=\\\\frac{\\\\ln(M)}{\\\\ln(b)}$$","$$\\\\log_{b}\\\\left(M\\\\right)=\\\\frac{\\\\ln(M)}{\\\\ln(b)}$$"]},{"id":"aeae96dlogarithmic10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\ln(55.875)}{\\\\ln(14)}$$"],"dependencies":["aeae96dlogarithmic10a-h3"],"title":"Change-of-Base","text":"Replacing $$M=55.875$$ and $$b=14$$ in the question, what would the equivalent expression be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeae96dlogarithmic11","title":"Changing Logarithmic Expressions","body":"Suppose $$\\\\log_{5}\\\\left(6\\\\right)=a$$ and $$\\\\log_{5}\\\\left(11\\\\right)=b$$. Use the change-of-base formula along with properties of logarithms to rewrite each expression in terms of a and $$b$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.5 Logarithmic Properties","courseName":"OpenStax: College Algebra","steps":[{"id":"aeae96dlogarithmic11a","stepAnswer":["$$\\\\frac{b}{a}+\\\\frac{1}{a}$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{6}\\\\left(55\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{b}{a}+\\\\frac{1}{a}$$","hints":{"DefaultPathway":[{"id":"aeae96dlogarithmic11a-h1","type":"hint","dependencies":[],"title":"Change-of-Base Formula","text":"The change-of-base formula can be used to evaluate a logarithm with any base.\\\\nFor any positive real numbers M,b, and $$n$$, where $$n \\\\neq 1$$ and $$b \\\\neq 1$$, $$\\\\log_{b}\\\\left(M\\\\right)=\\\\log_{n}\\\\left(M\\\\right)/\\\\log_{n}\\\\left(b\\\\right)$$.\\\\n\\\\n1) Determine the new base $$n$$, remembering that the common log, $$\\\\ln(x)$$, has base $$10$$, and the natural log, ln(x), has base e.\\\\n2) Rewrite the log as a quotient using the change-of-base formula\\\\na) The numerator of the quotient will be a logarithm with base $$n$$ and argument M.\\\\nb) The denominator of the quotient will be a logarithm with base $$n$$ and argument $$b$$.\\\\n","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic11a-h2","type":"hint","dependencies":["aeae96dlogarithmic11a-h1"],"title":"New Base","text":"Observe that in the questions, both a and $$b$$ are of base $$5$$. Thus, we would want to use the change the base to $$5$$ so that we can express the logarithmic expressions in terms of a and $$b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic11a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\log_{5}\\\\left(55\\\\right)/\\\\log_{5}\\\\left(6\\\\right)$$"],"dependencies":["aeae96dlogarithmic11a-h2"],"title":"Change-of-Base","text":"What is the expression after changing the base to 5? Recall that for any positive real numbers M,b, and $$n$$, where $$n \\\\neq 1$$ and $$b \\\\neq 1$$, $$\\\\log_{b}\\\\left(M\\\\right)=\\\\log_{n}\\\\left(M\\\\right)/\\\\log_{n}\\\\left(b\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$(\\\\log_{5}\\\\left(11\\\\right)+\\\\log_{5}\\\\left(5\\\\right))/\\\\log_{5}\\\\left(6\\\\right)$$"],"dependencies":["aeae96dlogarithmic11a-h3"],"title":"Apply the Product Rule for Logarithms","text":"We observe that $$55$$ can be broken down into its prime factor of $$5$$ and $$11$$. Applying the product rule to the $$\\\\log_{5}\\\\left(55\\\\right)$$, what would the new expression be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic11a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$(b+\\\\log_{5}\\\\left(5\\\\right))/a$$"],"dependencies":["aeae96dlogarithmic11a-h4"],"title":"Substitution","text":"Substitute $$a=\\\\log_{5}\\\\left(6\\\\right)$$ and $$b=\\\\log_{5}\\\\left(11\\\\right)$$ into the expression. What is the expression now?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic11a-h6","type":"hint","dependencies":["aeae96dlogarithmic11a-h5"],"title":"Simplification","text":"Note that $$\\\\log_{5}\\\\left(5\\\\right)$$ can be simplified by the property that $$\\\\log_{b}\\\\left(b\\\\right)=1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeae96dlogarithmic12","title":"Changing Logarithmic Expressions","body":"Suppose $$\\\\log_{5}\\\\left(6\\\\right)=a$$ and $$\\\\log_{5}\\\\left(11\\\\right)=b$$. Use the change-of-base formula along with properties of logarithms to rewrite each expression in terms of a and $$b$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.5 Logarithmic Properties","courseName":"OpenStax: College Algebra","steps":[{"id":"aeae96dlogarithmic12a","stepAnswer":["$$\\\\frac{a}{b}-1$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{11}\\\\left(\\\\frac{6}{11}\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{a}{b}-1$$","hints":{"DefaultPathway":[{"id":"aeae96dlogarithmic12a-h1","type":"hint","dependencies":[],"title":"Change-of-Base Formula","text":"The change-of-base formula can be used to evaluate a logarithm with any base.\\\\nFor any positive real numbers M,b, and $$n$$, where $$n \\\\neq 1$$ and $$b \\\\neq 1$$, $$\\\\log_{b}\\\\left(M\\\\right)=\\\\log_{n}\\\\left(M\\\\right)/\\\\log_{n}\\\\left(b\\\\right)$$.\\\\n\\\\n1) Determine the new base $$n$$, remembering that the common log, $$\\\\ln(x)$$, has base $$10$$, and the natural log, ln(x), has base e.\\\\n2) Rewrite the log as a quotient using the change-of-base formula\\\\na) The numerator of the quotient will be a logarithm with base $$n$$ and argument M.\\\\nb) The denominator of the quotient will be a logarithm with base $$n$$ and argument $$b$$.\\\\n","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic12a-h2","type":"hint","dependencies":["aeae96dlogarithmic12a-h1"],"title":"New Base","text":"Observe that in the questions, both a and $$b$$ are of base $$5$$. Thus, we would want to use the change the base to $$5$$ so that we can express the logarithmic expressions in terms of a and $$b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic12a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\log_{5}\\\\left(\\\\frac{6}{11}\\\\right)/\\\\log_{5}\\\\left(11\\\\right)$$"],"dependencies":["aeae96dlogarithmic12a-h2"],"title":"Change-of-Base","text":"What is the expression after changing the base to 5? Recall that for any positive real numbers M,b, and $$n$$, where $$n \\\\neq 1$$ and $$b \\\\neq 1$$, $$\\\\log_{b}\\\\left(M\\\\right)=\\\\log_{n}\\\\left(M\\\\right)/\\\\log_{n}\\\\left(b\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$(\\\\log_{5}\\\\left(6\\\\right)-\\\\log_{5}\\\\left(11\\\\right))/\\\\log_{5}\\\\left(11\\\\right)$$"],"dependencies":["aeae96dlogarithmic12a-h3"],"title":"Apply the Quotient Rule for Logarithms","text":"We observe that the quotient rule can be applied to the quotient $$\\\\frac{6}{11}$$. Applying the quotient rule to the $$\\\\log_{5}\\\\left(\\\\frac{6}{11}\\\\right)$$, what would the new expression be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic12a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{a-b}{b}$$"],"dependencies":["aeae96dlogarithmic12a-h4"],"title":"Substitution","text":"Substitute $$a=\\\\log_{5}\\\\left(6\\\\right)$$ and $$b=\\\\log_{5}\\\\left(11\\\\right)$$ into the expression. What is the expression now?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeae96dlogarithmic13","title":"Evaluating Logarithmic Expressions with Properties of Logarithms","body":"Use properties of logarithms to evaluate the expression without using a calculator.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.5 Logarithmic Properties","courseName":"OpenStax: College Algebra","steps":[{"id":"aeae96dlogarithmic13a","stepAnswer":["$$-5$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{3}\\\\left(\\\\frac{1}{9}\\\\right)-3*\\\\log_{3}\\\\left(3\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-5$$","hints":{"DefaultPathway":[{"id":"aeae96dlogarithmic13a-h1","type":"hint","dependencies":[],"title":"Properties of Logarithms","text":"Some important properties of logarithms:\\\\n1) $$\\\\log_{b}\\\\left(1\\\\right)=0$$\\\\n2) $$\\\\log_{b}\\\\left(b\\\\right)=1$$\\\\n3) Inverse Property: $$\\\\log_{b}\\\\left(b^x\\\\right)=x$$, b**log{b}{x}=x,x>0\\\\n4) One-to-One Property: $$\\\\log_{b}\\\\left(M\\\\right)=\\\\log_{b}\\\\left(N\\\\right)$$ if and only if $$M=N$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\log_{3}\\\\left(1\\\\right)-\\\\log_{3}\\\\left(9\\\\right)-3*\\\\log_{3}\\\\left(3\\\\right)$$"],"dependencies":["aeae96dlogarithmic13a-h1"],"title":"Apply the Quotient Rule for Logarithms","text":"We observe that we can apply the quotient rule to the the logarithmic expression with the quotient, $$\\\\frac{1}{9}$$. What is the expression after applying the quotient rule?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic13a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\log_{3}\\\\left(1\\\\right)-2*\\\\log_{3}\\\\left(3\\\\right)-3*\\\\log_{3}\\\\left(3\\\\right)$$"],"dependencies":["aeae96dlogarithmic13a-h2"],"title":"Apply the Power Rule for Logarithms","text":"Note that $$9=3^2$$ and thus, we can use the power rule and write the equivalent expression by multiplying the exponent times the logarithm of the base. What is the expression now?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0-2\\\\times1-3\\\\times1$$"],"dependencies":["aeae96dlogarithmic13a-h3"],"title":"Use the Properties of Logarithms","text":"We can use the fact that $$\\\\log_{b}\\\\left(1\\\\right)=0$$ and $$\\\\log_{b}\\\\left(b\\\\right)=1$$ in our expression to simplify the expression. What is the simplified expression after applying the properties?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeae96dlogarithmic14","title":"Evaluating Logarithmic Expressions with Properties of Logarithms","body":"Use properties of logarithms to evaluate the expression without using a calculator.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.5 Logarithmic Properties","courseName":"OpenStax: College Algebra","steps":[{"id":"aeae96dlogarithmic14a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"$$6*\\\\log_{8}\\\\left(2\\\\right)+\\\\log_{8}\\\\left(64\\\\right)/(3*\\\\log_{8}\\\\left(4\\\\right))$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"aeae96dlogarithmic14a-h1","type":"hint","dependencies":[],"title":"Properties of Logarithms","text":"Some important properties of logarithms:\\\\n1) $$\\\\log_{b}\\\\left(1\\\\right)=0$$\\\\n2) $$\\\\log_{b}\\\\left(b\\\\right)=1$$\\\\n3) Inverse Property: $$\\\\log_{b}\\\\left(b^x\\\\right)=x$$, b**log{b}{x}=x,x>0\\\\n4) One-to-One Property: $$\\\\log_{b}\\\\left(M\\\\right)=\\\\log_{b}\\\\left(N\\\\right)$$ if and only if $$M=N$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic14a-h2","type":"hint","dependencies":["aeae96dlogarithmic14a-h1"],"title":"Exponent","text":"Since the expression is in base $$8$$, we can express the each term inside the logarithmic expressions with base $$8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic14a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["aeae96dlogarithmic14a-h2"],"title":"Exponent","text":"In $$6*\\\\log_{8}\\\\left(2\\\\right)$$, what is the exponent,b, of $$8$$ such that $$8^b=2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic14a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["aeae96dlogarithmic14a-h3"],"title":"Exponent","text":"In $$\\\\log_{8}\\\\left(64\\\\right)$$, what is the exponent,b, of $$8$$ such that $$8^b=64$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic14a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{3}$$"],"dependencies":["aeae96dlogarithmic14a-h4"],"title":"Exponent","text":"In $$3*\\\\log_{8}\\\\left(4\\\\right)$$, what is the exponent,b, of $$8$$ such that $$8^b=4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic14a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2*\\\\log_{8}\\\\left(8\\\\right)+2*\\\\log_{8}\\\\left(8\\\\right)/(3*(2/3)*\\\\log_{8}\\\\left(8\\\\right))$$"],"dependencies":["aeae96dlogarithmic14a-h5"],"title":"Apply the Power Rule for Logarithms","text":"Now that we have found all the exponent of each logarithmic expressions, write the equivalent expression by multiplying the exponent times the logarithm of the base. What is the expression now?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic14a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2\\\\times1+\\\\frac{2\\\\times1}{3\\\\frac{2}{3}}$$"],"dependencies":["aeae96dlogarithmic14a-h6"],"title":"Use the Properties of Logarithms","text":"We can use the fact that $$\\\\log_{b}\\\\left(b\\\\right)=1$$ in our expression to simplify the expression. What is the simplified expression after applying the properties?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeae96dlogarithmic15","title":"Using the Change-of-Base Formula with a Calculator","body":"Use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to five decimal places.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.5 Logarithmic Properties","courseName":"OpenStax: College Algebra","steps":[{"id":"aeae96dlogarithmic15a","stepAnswer":["$$2.00746$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{8}\\\\left(65\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2.00746$$","hints":{"DefaultPathway":[{"id":"aeae96dlogarithmic15a-h1","type":"hint","dependencies":[],"title":"Change-of-Base Formula","text":"The change-of-base formula can be used to evaluate a logarithm with any base.\\\\nFor any positive real numbers M,b, and $$n$$, where $$n \\\\neq 1$$ and $$b \\\\neq 1$$, $$\\\\log_{b}\\\\left(M\\\\right)=\\\\log_{n}\\\\left(M\\\\right)/\\\\log_{n}\\\\left(b\\\\right)$$.\\\\n\\\\n1) Determine the new base $$n$$, remembering that the common log, $$\\\\ln(x)$$, has base $$10$$, and the natural log, ln(x), has base e.\\\\n2) Rewrite the log as a quotient using the change-of-base formula\\\\na) The numerator of the quotient will be a logarithm with base $$n$$ and argument M.\\\\nb) The denominator of the quotient will be a logarithm with base $$n$$ and argument $$b$$.\\\\n","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["e"],"dependencies":["aeae96dlogarithmic15a-h1"],"title":"New Base","text":"What is the new base that we\'re changing to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic15a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\log_{b}\\\\left(M\\\\right)=\\\\ln{M}/ln(b)$$"],"dependencies":["aeae96dlogarithmic15a-h2"],"title":"Change-of-Base","text":"What form does the quotient take after the change-of-base to the new base e?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\log_{b}\\\\left(M\\\\right)=\\\\frac{\\\\ln(M)}{\\\\ln(b)}$$","$$\\\\log_{b}\\\\left(M\\\\right)=\\\\frac{\\\\ln(M)}{\\\\ln(b)}$$"]},{"id":"aeae96dlogarithmic15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\ln(65)}{\\\\ln(8)}$$"],"dependencies":["aeae96dlogarithmic15a-h3"],"title":"Change-of-Base","text":"Replacing $$M=65$$ and $$b=8$$ in the question, what would the equivalent expression be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic15a-h5","type":"hint","dependencies":["aeae96dlogarithmic15a-h4"],"title":"Using the Calculator","text":"Use the calculator to evaluate $$\\\\frac{\\\\ln(65)}{\\\\ln(8)}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeae96dlogarithmic16","title":"Using the Change-of-Base Formula with a Calculator","body":"Use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to five decimal places.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.5 Logarithmic Properties","courseName":"OpenStax: College Algebra","steps":[{"id":"aeae96dlogarithmic16a","stepAnswer":["$$-2.23266$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{1/2}\\\\left(4.7\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-2.23266$$","hints":{"DefaultPathway":[{"id":"aeae96dlogarithmic16a-h1","type":"hint","dependencies":[],"title":"Change-of-Base Formula","text":"The change-of-base formula can be used to evaluate a logarithm with any base.\\\\nFor any positive real numbers M,b, and $$n$$, where $$n \\\\neq 1$$ and $$b \\\\neq 1$$, $$\\\\log_{b}\\\\left(M\\\\right)=\\\\log_{n}\\\\left(M\\\\right)/\\\\log_{n}\\\\left(b\\\\right)$$.\\\\n\\\\n1) Determine the new base $$n$$, remembering that the common log, $$\\\\ln(x)$$, has base $$10$$, and the natural log, ln(x), has base e.\\\\n2) Rewrite the log as a quotient using the change-of-base formula\\\\na) The numerator of the quotient will be a logarithm with base $$n$$ and argument M.\\\\nb) The denominator of the quotient will be a logarithm with base $$n$$ and argument $$b$$.\\\\n","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["e"],"dependencies":["aeae96dlogarithmic16a-h1"],"title":"New Base","text":"What is the new base that we\'re changing to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic16a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\log_{b}\\\\left(M\\\\right)=\\\\ln{M}/ln(b)$$"],"dependencies":["aeae96dlogarithmic16a-h2"],"title":"Change-of-Base","text":"What form does the quotient take after the change-of-base to the new base e?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\log_{b}\\\\left(M\\\\right)=\\\\frac{\\\\ln(M)}{\\\\ln(b)}$$","$$\\\\log_{b}\\\\left(M\\\\right)=\\\\frac{\\\\ln(M)}{\\\\ln(b)}$$"]},{"id":"aeae96dlogarithmic16a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\ln(4.7)}{\\\\ln(\\\\frac{1}{2})}$$"],"dependencies":["aeae96dlogarithmic16a-h3"],"title":"Change-of-Base","text":"Replacing $$M=4.7$$ and $$b=\\\\frac{1}{2}$$ in the question, what would the equivalent expression be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic16a-h5","type":"hint","dependencies":["aeae96dlogarithmic16a-h4"],"title":"Using the Calculator","text":"Use the calculator to evaluate $$\\\\frac{\\\\ln(4.7)}{\\\\ln(\\\\frac{1}{2})}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeae96dlogarithmic2","title":"Expanding Logarithms Using Product, Quotient, and Power Rules","body":"Expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.5 Logarithmic Properties","courseName":"OpenStax: College Algebra","steps":[{"id":"aeae96dlogarithmic2a","stepAnswer":["$$\\\\log_{4}\\\\left(x\\\\right)-\\\\log_{4}\\\\left(z\\\\right)-\\\\log_{4}\\\\left(w\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{4}\\\\left(\\\\frac{\\\\frac{x}{z}}{w}\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\log_{4}\\\\left(x\\\\right)-\\\\log_{4}\\\\left(z\\\\right)-\\\\log_{4}\\\\left(w\\\\right)$$","hints":{"DefaultPathway":[{"id":"aeae96dlogarithmic2a-h1","type":"hint","dependencies":[],"title":"Product Rule for Logarithms","text":"The product rule for logarithms can be used to simplify a logarithm of a product by rewriting it as a sum of individual logarithms.\\\\n$$\\\\log_{b}\\\\left(M N\\\\right)=\\\\log_{b}\\\\left(M\\\\right)+\\\\log_{b}\\\\left(N\\\\right)$$ for $$b>0$$\\\\n\\\\nGiven the logarithm of a product, use the product rule of logarithms to write an equivalent sum of logarithms.\\\\n\\\\n1) Factor the argument completely, expressing each whole number factor as a product of primes.\\\\n2) Write the equivalent expression by summing the logarithms of each factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic2a-h2","type":"hint","dependencies":[],"title":"Quotient Rule for Logarithms","text":"The quotient rule for logarithms can be used to simplify a logarithm or a quotient by rewriting it as the difference of individual logarithms.\\\\n$$\\\\log_{b}\\\\left(\\\\frac{M}{N}\\\\right)=\\\\log_{b}\\\\left(M\\\\right)-\\\\log_{b}\\\\left(N\\\\right)$$\\\\n\\\\nGiven the logarithm of a quotient, use the quotient rule of logarithms to write an equivalent difference of logarithms.\\\\n\\\\n1) Express the argument in lowest terms by factoring the numerator and denominator and canceling common terms.\\\\n2) Write the equivalent expression by subtracting the logarithm of the denominator from the logarithm of the numerator.\\\\n3) Check to see that each term is fully expanded. If not, apply the product rule for logarithms to expand completely.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic2a-h3","type":"hint","dependencies":[],"title":"Power Rule for Logarithms","text":"The power rule for logarithms can be used to simplify the logarithm of a power by rewriting it as the product of the exponent times the logarithm of the base.\\\\n$$\\\\log_{b}\\\\left(M^n\\\\right)=n*\\\\log_{b}\\\\left(M\\\\right)$$\\\\n\\\\nGiven the logarithm of a power, use the power rule of logarithms to write an equivalent product of a factor and a logarithm.\\\\n\\\\n1) Express the argument as a power, if needed.\\\\n2) Write the equivalent expression by multiplying the exponent times the logarithm of the base.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic2a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Quotient Rule for Logarithms"],"dependencies":[],"title":"Identify the Relevant Rules","text":"Identify the relevant rules that can be used to expand the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Product Rule for Logarithms","Quotient Rule for Logarithms","Power Rule for Logarithms"]},{"id":"aeae96dlogarithmic2a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\log_{4}\\\\left(\\\\frac{x}{z}\\\\right)-\\\\log_{4}\\\\left(w\\\\right)$$"],"dependencies":["aeae96dlogarithmic2a-h4"],"title":"Apply the Quotient Rule for Logarithms","text":"Start by separating the numerator, $$\\\\frac{x}{z}$$ and denominator, w, of the fraction in the logarithmic expression using the quotient rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\log_{4}\\\\left(x\\\\right)-\\\\log_{4}\\\\left(z\\\\right)-\\\\log_{4}\\\\left(w\\\\right)$$"],"dependencies":["aeae96dlogarithmic2a-h5"],"title":"Apply the Quotient Rule for Logarithms","text":"Note that there is still a term with a fraction. Apply the quotient rule again on the term $$\\\\log_{4}\\\\left(\\\\frac{x}{z}\\\\right)$$. What is the fully expanded expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeae96dlogarithmic3","title":"Combining Logarithms Using Product, Quotient, and Power Rules","body":"Condense the expression to a single logarithm if possible.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.5 Logarithmic Properties","courseName":"OpenStax: College Algebra","steps":[{"id":"aeae96dlogarithmic3a","stepAnswer":["$$\\\\ln(\\\\frac{a}{d c})$$"],"problemType":"TextBox","stepTitle":"$$ln(a)-ln(d)-ln(c)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\ln(\\\\frac{a}{d c})$$","hints":{"DefaultPathway":[{"id":"aeae96dlogarithmic3a-h1","type":"hint","dependencies":[],"title":"Product Rule for Logarithms","text":"The product rule for logarithms can be used to simplify a logarithm of a product by rewriting it as a sum of individual logarithms.\\\\n$$\\\\log_{b}\\\\left(M N\\\\right)=\\\\log_{b}\\\\left(M\\\\right)+\\\\log_{b}\\\\left(N\\\\right)$$ for $$b>0$$\\\\n\\\\nGiven the logarithm of a product, use the product rule of logarithms to write an equivalent sum of logarithms.\\\\n\\\\n1) Factor the argument completely, expressing each whole number factor as a product of primes.\\\\n2) Write the equivalent expression by summing the logarithms of each factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic3a-h2","type":"hint","dependencies":[],"title":"Quotient Rule for Logarithms","text":"The quotient rule for logarithms can be used to simplify a logarithm or a quotient by rewriting it as the difference of individual logarithms.\\\\n$$\\\\log_{b}\\\\left(\\\\frac{M}{N}\\\\right)=\\\\log_{b}\\\\left(M\\\\right)-\\\\log_{b}\\\\left(N\\\\right)$$\\\\n\\\\nGiven the logarithm of a quotient, use the quotient rule of logarithms to write an equivalent difference of logarithms.\\\\n\\\\n1) Express the argument in lowest terms by factoring the numerator and denominator and canceling common terms.\\\\n2) Write the equivalent expression by subtracting the logarithm of the denominator from the logarithm of the numerator.\\\\n3) Check to see that each term is fully expanded. If not, apply the product rule for logarithms to expand completely.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic3a-h3","type":"hint","dependencies":[],"title":"Power Rule for Logarithms","text":"The power rule for logarithms can be used to simplify the logarithm of a power by rewriting it as the product of the exponent times the logarithm of the base.\\\\n$$\\\\log_{b}\\\\left(M^n\\\\right)=n*\\\\log_{b}\\\\left(M\\\\right)$$\\\\n\\\\nGiven the logarithm of a power, use the power rule of logarithms to write an equivalent product of a factor and a logarithm.\\\\n\\\\n1) Express the argument as a power, if needed.\\\\n2) Write the equivalent expression by multiplying the exponent times the logarithm of the base.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic3a-h4","type":"hint","dependencies":[],"title":"Condensing Logarithmic Expressions","text":"Given a sum, difference, or product of logarithms with the same base, write an equivalent expression as a single logarithm.\\\\n\\\\n1) Apply the power property first. Identify terms that are products of factors and a logarithm, and rewrite each as the logarithm of a power.\\\\n2) Next apply the product property. Rewrite sums of logarithms as the logarithm of a product.\\\\n3) Apply the quotient property last. Rewrite differences of logarithms as the logarithm of a quotient.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic3a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Quotient Rule for Logarithms"],"dependencies":[],"title":"Identify the Relevant Rules","text":"Identify the relevant rules that can be used to expand the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Product Rule for Logarithms","Quotient Rule for Logarithms","Power Rule for Logarithms"]},{"id":"aeae96dlogarithmic3a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\ln(\\\\frac{a}{d})$$"],"dependencies":["aeae96dlogarithmic3a-h5"],"title":"Apply the Quotient Rule for Logarithms","text":"Note that the logarithmic terms shares the same base, thus we can apply the quotient rule to the differences. We start with $$ln(a)-ln(d)$$. What does this expression condense to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic3a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\ln(\\\\frac{a}{d c})$$"],"dependencies":["aeae96dlogarithmic3a-h6"],"title":"Apply the Quotient Rule for Logarithms","text":"We can apply the quotient rule to the difference again. What is $$\\\\ln(\\\\frac{a}{d})-\\\\ln(c)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeae96dlogarithmic4","title":"Combining Logarithms Using Product, Quotient, and Power Rules","body":"Condense the expression to a single logarithm if possible.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.5 Logarithmic Properties","courseName":"OpenStax: College Algebra","steps":[{"id":"aeae96dlogarithmic4a","stepAnswer":["$$\\\\log_{b}\\\\left(7\\\\right)$$"],"problemType":"TextBox","stepTitle":"$$-\\\\log_{b}\\\\left(\\\\frac{1}{7}\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\log_{b}\\\\left(7\\\\right)$$","hints":{"DefaultPathway":[{"id":"aeae96dlogarithmic4a-h1","type":"hint","dependencies":[],"title":"Product Rule for Logarithms","text":"The product rule for logarithms can be used to simplify a logarithm of a product by rewriting it as a sum of individual logarithms.\\\\n$$\\\\log_{b}\\\\left(M N\\\\right)=\\\\log_{b}\\\\left(M\\\\right)+\\\\log_{b}\\\\left(N\\\\right)$$ for $$b>0$$\\\\n\\\\nGiven the logarithm of a product, use the product rule of logarithms to write an equivalent sum of logarithms.\\\\n\\\\n1) Factor the argument completely, expressing each whole number factor as a product of primes.\\\\n2) Write the equivalent expression by summing the logarithms of each factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic4a-h2","type":"hint","dependencies":[],"title":"Quotient Rule for Logarithms","text":"The quotient rule for logarithms can be used to simplify a logarithm or a quotient by rewriting it as the difference of individual logarithms.\\\\n$$\\\\log_{b}\\\\left(\\\\frac{M}{N}\\\\right)=\\\\log_{b}\\\\left(M\\\\right)-\\\\log_{b}\\\\left(N\\\\right)$$\\\\n\\\\nGiven the logarithm of a quotient, use the quotient rule of logarithms to write an equivalent difference of logarithms.\\\\n\\\\n1) Express the argument in lowest terms by factoring the numerator and denominator and canceling common terms.\\\\n2) Write the equivalent expression by subtracting the logarithm of the denominator from the logarithm of the numerator.\\\\n3) Check to see that each term is fully expanded. If not, apply the product rule for logarithms to expand completely.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic4a-h3","type":"hint","dependencies":[],"title":"Power Rule for Logarithms","text":"The power rule for logarithms can be used to simplify the logarithm of a power by rewriting it as the product of the exponent times the logarithm of the base.\\\\n$$\\\\log_{b}\\\\left(M^n\\\\right)=n*\\\\log_{b}\\\\left(M\\\\right)$$\\\\n\\\\nGiven the logarithm of a power, use the power rule of logarithms to write an equivalent product of a factor and a logarithm.\\\\n\\\\n1) Express the argument as a power, if needed.\\\\n2) Write the equivalent expression by multiplying the exponent times the logarithm of the base.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic4a-h4","type":"hint","dependencies":[],"title":"Condensing Logarithmic Expressions","text":"Given a sum, difference, or product of logarithms with the same base, write an equivalent expression as a single logarithm.\\\\n\\\\n1) Apply the power property first. Identify terms that are products of factors and a logarithm, and rewrite each as the logarithm of a power.\\\\n2) Next apply the product property. Rewrite sums of logarithms as the logarithm of a product.\\\\n3) Apply the quotient property last. Rewrite differences of logarithms as the logarithm of a quotient.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic4a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Power Rule for Logarithms"],"dependencies":[],"title":"Identify the Relevant Rules","text":"Identify the relevant rules that can be used to expand the expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Product Rule for Logarithms","Quotient Rule for Logarithms","Power Rule for Logarithms"]},{"id":"aeae96dlogarithmic4a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\log_{b}\\\\left({\\\\left(\\\\frac{1}{7}\\\\right)}^{\\\\left(-1\\\\right)}\\\\right)$$"],"dependencies":["aeae96dlogarithmic4a-h5"],"title":"Apply the Power Rule for Logarithms","text":"Because the logarithm of a power is the product of the exponent times the logarithm of the base, it follows that the product of a number and a logarithm can be written as a power. For the expression $$-\\\\log_{b}\\\\left(\\\\frac{1}{7}\\\\right)$$, we identify the factor, $$-1$$, as the exponent and the argument, $$\\\\frac{1}{7}$$, as the base. How can we rewrite the product as a logarithm of a power.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic4a-h7","type":"hint","dependencies":["aeae96dlogarithmic4a-h6"],"title":"Simplify","text":"Apply the exponent, $$-1$$, to the base, $$\\\\frac{1}{7}$$ to simplify the whole expression.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeae96dlogarithmic5","title":"Expanding Complex Logarithmic Expressions","body":"Expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.5 Logarithmic Properties","courseName":"OpenStax: College Algebra","steps":[{"id":"aeae96dlogarithmic5a","stepAnswer":["$$\\\\frac{3}{2} \\\\ln(y)-\\\\frac{1}{2} \\\\ln(1-y)$$"],"problemType":"TextBox","stepTitle":"$$\\\\ln(y \\\\sqrt{\\\\frac{y}{1-y}})$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3}{2} \\\\ln(y)-\\\\frac{1}{2} \\\\ln(1-y)$$","hints":{"DefaultPathway":[{"id":"aeae96dlogarithmic5a-h1","type":"hint","dependencies":[],"title":"Product Rule for Logarithms","text":"The product rule for logarithms can be used to simplify a logarithm of a product by rewriting it as a sum of individual logarithms.\\\\n$$\\\\log_{b}\\\\left(M N\\\\right)=\\\\log_{b}\\\\left(M\\\\right)+\\\\log_{b}\\\\left(N\\\\right)$$ for $$b>0$$\\\\n\\\\nGiven the logarithm of a product, use the product rule of logarithms to write an equivalent sum of logarithms.\\\\n\\\\n1) Factor the argument completely, expressing each whole number factor as a product of primes.\\\\n2) Write the equivalent expression by summing the logarithms of each factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic5a-h2","type":"hint","dependencies":[],"title":"Quotient Rule for Logarithms","text":"The quotient rule for logarithms can be used to simplify a logarithm or a quotient by rewriting it as the difference of individual logarithms.\\\\n$$\\\\log_{b}\\\\left(\\\\frac{M}{N}\\\\right)=\\\\log_{b}\\\\left(M\\\\right)-\\\\log_{b}\\\\left(N\\\\right)$$\\\\n\\\\nGiven the logarithm of a quotient, use the quotient rule of logarithms to write an equivalent difference of logarithms.\\\\n\\\\n1) Express the argument in lowest terms by factoring the numerator and denominator and canceling common terms.\\\\n2) Write the equivalent expression by subtracting the logarithm of the denominator from the logarithm of the numerator.\\\\n3) Check to see that each term is fully expanded. If not, apply the product rule for logarithms to expand completely.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic5a-h3","type":"hint","dependencies":[],"title":"Power Rule for Logarithms","text":"The power rule for logarithms can be used to simplify the logarithm of a power by rewriting it as the product of the exponent times the logarithm of the base.\\\\n$$\\\\log_{b}\\\\left(M^n\\\\right)=n*\\\\log_{b}\\\\left(M\\\\right)$$\\\\n\\\\nGiven the logarithm of a power, use the power rule of logarithms to write an equivalent product of a factor and a logarithm.\\\\n\\\\n1) Express the argument as a power, if needed.\\\\n2) Write the equivalent expression by multiplying the exponent times the logarithm of the base.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic5a-h4","type":"hint","dependencies":["aeae96dlogarithmic5a-h3"],"title":"Square Root","text":"Note that the square root applies to the fraction, $$\\\\frac{y}{1-y}$$. We can distribute the square root to the numerator and denominator to simplify future steps.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{1-y}$$"],"dependencies":["aeae96dlogarithmic5a-h4"],"title":"Apply the Quotient Rule for Logarithms","text":"We can start by identifying and separating the numerator and denominator of the quotient. What is the denominator in the logarithmic expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic5a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\ln(y \\\\sqrt{y})-\\\\ln(\\\\sqrt{1-y})$$"],"dependencies":["aeae96dlogarithmic5a-h5"],"title":"Apply the Quotient Rule for Logarithms","text":"After identifying the denominator, apply the quotient rule by subtracting the logarithm of the denominator from the logarithm of the numerator. What is the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic5a-h7","type":"hint","dependencies":["aeae96dlogarithmic5a-h6"],"title":"Exponent","text":"What are the exponent of each terms? Express them in terms of fractions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic5a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{2}$$"],"dependencies":["aeae96dlogarithmic5a-h7"],"title":"Exponent","text":"What is the exponent of $$y$$ in $$\\\\ln(y \\\\sqrt{y})$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic5a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["aeae96dlogarithmic5a-h8"],"title":"Exponent","text":"What is the exponent of $$(1-y)$$ in $$\\\\ln(\\\\sqrt{1-y})$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic5a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{2} \\\\ln(y)-\\\\frac{1}{2} \\\\ln(1-y)$$"],"dependencies":["aeae96dlogarithmic5a-h9"],"title":"Apply the Power Rule for Logarithms","text":"Write the equivalent expression by multiplying the exponent times the logarithm of the base. What is the final expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeae96dlogarithmic6","title":"Expanding Complex Logarithmic Expressions","body":"Expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.5 Logarithmic Properties","courseName":"OpenStax: College Algebra","steps":[{"id":"aeae96dlogarithmic6a","stepAnswer":["$$\\\\frac{8}{3} \\\\ln(x)+\\\\frac{14}{3} \\\\ln(y)$$"],"problemType":"TextBox","stepTitle":"$$\\\\ln(x^2 y^3 \\\\sqrt[3]{x^2 y^5})$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{8}{3} \\\\ln(x)+\\\\frac{14}{3} \\\\ln(y)$$","hints":{"DefaultPathway":[{"id":"aeae96dlogarithmic6a-h1","type":"hint","dependencies":[],"title":"Product Rule for Logarithms","text":"The product rule for logarithms can be used to simplify a logarithm of a product by rewriting it as a sum of individual logarithms.\\\\n$$\\\\log_{b}\\\\left(M N\\\\right)=\\\\log_{b}\\\\left(M\\\\right)+\\\\log_{b}\\\\left(N\\\\right)$$ for $$b>0$$\\\\n\\\\nGiven the logarithm of a product, use the product rule of logarithms to write an equivalent sum of logarithms.\\\\n\\\\n1) Factor the argument completely, expressing each whole number factor as a product of primes.\\\\n2) Write the equivalent expression by summing the logarithms of each factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic6a-h2","type":"hint","dependencies":[],"title":"Quotient Rule for Logarithms","text":"The quotient rule for logarithms can be used to simplify a logarithm or a quotient by rewriting it as the difference of individual logarithms.\\\\n$$\\\\log_{b}\\\\left(\\\\frac{M}{N}\\\\right)=\\\\log_{b}\\\\left(M\\\\right)-\\\\log_{b}\\\\left(N\\\\right)$$\\\\n\\\\nGiven the logarithm of a quotient, use the quotient rule of logarithms to write an equivalent difference of logarithms.\\\\n\\\\n1) Express the argument in lowest terms by factoring the numerator and denominator and canceling common terms.\\\\n2) Write the equivalent expression by subtracting the logarithm of the denominator from the logarithm of the numerator.\\\\n3) Check to see that each term is fully expanded. If not, apply the product rule for logarithms to expand completely.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic6a-h3","type":"hint","dependencies":[],"title":"Power Rule for Logarithms","text":"The power rule for logarithms can be used to simplify the logarithm of a power by rewriting it as the product of the exponent times the logarithm of the base.\\\\n$$\\\\log_{b}\\\\left(M^n\\\\right)=n*\\\\log_{b}\\\\left(M\\\\right)$$\\\\n\\\\nGiven the logarithm of a power, use the power rule of logarithms to write an equivalent product of a factor and a logarithm.\\\\n\\\\n1) Express the argument as a power, if needed.\\\\n2) Write the equivalent expression by multiplying the exponent times the logarithm of the base.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic6a-h4","type":"hint","dependencies":[],"title":"Exponent","text":"Start by simplifying the expression and finding the exponent for each variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic6a-h5","type":"hint","dependencies":["aeae96dlogarithmic6a-h4"],"title":"Exponent","text":"Distribute the cube root to the respective $$x$$ and $$y$$ terms in the expression, $$\\\\sqrt[3]{x^2 y^5}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic6a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{8}{3}$$"],"dependencies":["aeae96dlogarithmic6a-h5"],"title":"Exponent","text":"What is the exponent of $$x$$ after distributing the cube root and summing the exponents of all $$x$$ terms in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic6a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{14}{3}$$"],"dependencies":["aeae96dlogarithmic6a-h6"],"title":"Exponent","text":"What is the exponent of $$y$$ after distributing the cube root and summing the exponents of all $$y$$ terms in the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic6a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\ln(x^{\\\\frac{8}{3}})+\\\\ln(y^{\\\\frac{14}{3}})$$"],"dependencies":["aeae96dlogarithmic6a-h7"],"title":"Apply the Product Rule for Logarithms","text":"Write the equivalent expression of $$\\\\ln(x^{\\\\frac{8}{3}} y^{\\\\frac{14}{3}})$$ by summing the logarithms of each factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic6a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{8}{3} \\\\ln(x)+\\\\frac{14}{3} \\\\ln(y)$$"],"dependencies":["aeae96dlogarithmic6a-h8"],"title":"Apply the Power Rule for Logarithms","text":"Write the equivalent expression by multiplying the exponent times the logarithm of the base. What is the final expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeae96dlogarithmic7","title":"Condensing Complex Logarithmic Expressions","body":"Condense the expression to a single logarithm if possible.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.5 Logarithmic Properties","courseName":"OpenStax: College Algebra","steps":[{"id":"aeae96dlogarithmic7a","stepAnswer":["$$\\\\ln(\\\\frac{x z^3}{\\\\sqrt{y}})$$"],"problemType":"TextBox","stepTitle":"$$\\\\ln(x)-\\\\frac{1}{2} \\\\ln(y)+3\\\\ln(z)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\ln(\\\\frac{x z^3}{\\\\sqrt{y}})$$","hints":{"DefaultPathway":[{"id":"aeae96dlogarithmic7a-h1","type":"hint","dependencies":[],"title":"Product Rule for Logarithms","text":"The product rule for logarithms can be used to simplify a logarithm of a product by rewriting it as a sum of individual logarithms.\\\\n$$\\\\log_{b}\\\\left(M N\\\\right)=\\\\log_{b}\\\\left(M\\\\right)+\\\\log_{b}\\\\left(N\\\\right)$$ for $$b>0$$\\\\n\\\\nGiven the logarithm of a product, use the product rule of logarithms to write an equivalent sum of logarithms.\\\\n\\\\n1) Factor the argument completely, expressing each whole number factor as a product of primes.\\\\n2) Write the equivalent expression by summing the logarithms of each factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic7a-h2","type":"hint","dependencies":[],"title":"Quotient Rule for Logarithms","text":"The quotient rule for logarithms can be used to simplify a logarithm or a quotient by rewriting it as the difference of individual logarithms.\\\\n$$\\\\log_{b}\\\\left(\\\\frac{M}{N}\\\\right)=\\\\log_{b}\\\\left(M\\\\right)-\\\\log_{b}\\\\left(N\\\\right)$$\\\\n\\\\nGiven the logarithm of a quotient, use the quotient rule of logarithms to write an equivalent difference of logarithms.\\\\n\\\\n1) Express the argument in lowest terms by factoring the numerator and denominator and canceling common terms.\\\\n2) Write the equivalent expression by subtracting the logarithm of the denominator from the logarithm of the numerator.\\\\n3) Check to see that each term is fully expanded. If not, apply the product rule for logarithms to expand completely.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic7a-h3","type":"hint","dependencies":[],"title":"Power Rule for Logarithms","text":"The power rule for logarithms can be used to simplify the logarithm of a power by rewriting it as the product of the exponent times the logarithm of the base.\\\\n$$\\\\log_{b}\\\\left(M^n\\\\right)=n*\\\\log_{b}\\\\left(M\\\\right)$$\\\\n\\\\nGiven the logarithm of a power, use the power rule of logarithms to write an equivalent product of a factor and a logarithm.\\\\n\\\\n1) Express the argument as a power, if needed.\\\\n2) Write the equivalent expression by multiplying the exponent times the logarithm of the base.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic7a-h4","type":"hint","dependencies":[],"title":"Condensing Logarithmic Expressions","text":"Given a sum, difference, or product of logarithms with the same base, write an equivalent expression as a single logarithm.\\\\n\\\\n1) Apply the power property first. Identify terms that are products of factors and a logarithm, and rewrite each as the logarithm of a power.\\\\n2) Next apply the product property. Rewrite sums of logarithms as the logarithm of a product.\\\\n3) Apply the quotient property last. Rewrite differences of logarithms as the logarithm of a quotient.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic7a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\ln(x)-\\\\ln(\\\\sqrt{y})+\\\\ln(z^3)$$"],"dependencies":["aeae96dlogarithmic7a-h4"],"title":"Apply the Power Rule for Logarithms","text":"Apply the power property first. Identify terms that are products of factors and a logarithm, and rewrite each as the logarithm of a power. What is the expression after applying the power rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic7a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\ln(x z^3)-\\\\ln(\\\\sqrt{y})$$"],"dependencies":["aeae96dlogarithmic7a-h5"],"title":"Apply the Product Rule for Logarithms","text":"Rewrite sums of logarithms as the logarithm of a product. What is the expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic7a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\ln(\\\\frac{x z^3}{\\\\sqrt{y}})$$"],"dependencies":["aeae96dlogarithmic7a-h6"],"title":"Apply the Quotient Rule for Logarithms","text":"Rewrite differences of logarithms as the logarithm of a quotient. What is the final expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeae96dlogarithmic9","title":"Changing Logarithmic Expressions to Expressions Involving Only Natural Logs","body":"Rewrite each expression as an equivalent ratio of logs using the indicated base.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.5 Logarithmic Properties","courseName":"OpenStax: College Algebra","steps":[{"id":"aeae96dlogarithmic9a","stepAnswer":["$$\\\\frac{\\\\ln(15)}{\\\\ln(7)}$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{7}\\\\left(15\\\\right)$$ to base e","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{\\\\ln(15)}{\\\\ln(7)}$$","hints":{"DefaultPathway":[{"id":"aeae96dlogarithmic9a-h1","type":"hint","dependencies":[],"title":"Change-of-Base Formula","text":"The change-of-base formula can be used to evaluate a logarithm with any base.\\\\nFor any positive real numbers M,b, and $$n$$, where $$n \\\\neq 1$$ and $$b \\\\neq 1$$, $$\\\\log_{b}\\\\left(M\\\\right)=\\\\log_{n}\\\\left(M\\\\right)/\\\\log_{n}\\\\left(b\\\\right)$$.\\\\n\\\\n1) Determine the new base $$n$$, remembering that the common log, $$\\\\ln(x)$$, has base $$10$$, and the natural log, ln(x), has base e.\\\\n2) Rewrite the log as a quotient using the change-of-base formula\\\\na) The numerator of the quotient will be a logarithm with base $$n$$ and argument M.\\\\nb) The denominator of the quotient will be a logarithm with base $$n$$ and argument $$b$$.\\\\n","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["e"],"dependencies":["aeae96dlogarithmic9a-h1"],"title":"New Base","text":"What is the new base that we\'re changing to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogarithmic9a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\log_{b}\\\\left(M\\\\right)=\\\\frac{\\\\ln(M)}{\\\\ln(b)}$$"],"dependencies":["aeae96dlogarithmic9a-h2"],"title":"Change-of-Base","text":"What form does the quotient take after the change-of-base to the new base e?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\log_{b}\\\\left(M\\\\right)=\\\\frac{\\\\ln(M)}{\\\\ln(b)}$$","$$\\\\log_{b}\\\\left(M\\\\right)=\\\\frac{\\\\ln(M)}{\\\\ln(b)}$$"]},{"id":"aeae96dlogarithmic9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\ln(15)}{\\\\ln(7)}$$"],"dependencies":["aeae96dlogarithmic9a-h3"],"title":"Change-of-Base","text":"Replacing $$M=15$$ and $$b=7$$ in the question, what would the equivalent expression be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeae96dlogprop1","title":"Using the Product Rule for Logarithms","body":"Expand the logarithm.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.5 Logarithmic Properties","courseName":"OpenStax: College Algebra","steps":[{"id":"aeae96dlogprop1a","stepAnswer":["log{3}(2)+log{3}(3)+log{3}(5)+log{3}(x)+log{3}(3x+4)"],"problemType":"TextBox","stepTitle":"$$\\\\log_{3}\\\\left(30x\\\\left(3x+4\\\\right)\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"aeae96dlogprop1a-h1","type":"hint","dependencies":[],"title":"Factoring the Argument Completely","text":"$$\\\\log_{3}\\\\left(30x\\\\left(x+4\\\\right)\\\\right)=\\\\log_{3}\\\\left(2\\\\times3\\\\times5 x \\\\left(3x+4\\\\right)\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogprop1a-h2","type":"hint","dependencies":["aeae96dlogprop1a-h1"],"title":"Representing the Expression With Many Logarithms","text":"$$\\\\ln(ab)$$ is equivalent to $$\\\\ln(a)+\\\\ln(b)$$ by product rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeae96dlogprop10","title":"Using the Power Rule in Inverse","body":"Rewrite the logarithm.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.5 Logarithmic Properties","courseName":"OpenStax: College Algebra","steps":[{"id":"aeae96dlogprop10a","stepAnswer":["$$\\\\log_{3}\\\\left(16\\\\right)$$"],"problemType":"TextBox","stepTitle":"Rewrite $$2\\\\log_{3}\\\\left(4\\\\right)$$ using the inverse power rule such that it has a coefficient of $$1$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\log_{3}\\\\left(16\\\\right)$$","hints":{"DefaultPathway":[{"id":"aeae96dlogprop10a-h1","type":"hint","dependencies":[],"title":"Using the Inverse Power Rule to Expand","text":"We can use the power rule as follows. We know that the exponent that $$\\\\log_{b}\\\\left(M^n\\\\right)=n\\\\log_{b}\\\\left(M\\\\right)$$. In this case, $$M=4$$ and $$n=2$$. So, $$2\\\\log_{3}\\\\left(4\\\\right)=\\\\log_{3}\\\\left(4^2\\\\right)=\\\\log_{3}\\\\left(16\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeae96dlogprop11","title":"Expanding Complex Logarithms","body":"Expand the logarithm.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.5 Logarithmic Properties","courseName":"OpenStax: College Algebra","steps":[{"id":"aeae96dlogprop11a","stepAnswer":["$$4\\\\ln(x)+\\\\ln(y)-\\\\ln(7)$$"],"problemType":"TextBox","stepTitle":"Expand $$\\\\ln(\\\\frac{x^4 y}{7})$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4\\\\ln(x)+\\\\ln(y)-\\\\ln(7)$$","hints":{"DefaultPathway":[{"id":"aeae96dlogprop11a-h1","type":"hint","dependencies":[],"title":"Using the Quotient Rule for Logarithms","text":"Using the quotient rule, $$\\\\ln(\\\\frac{x^4 y}{7})=\\\\ln(x^4 y)-\\\\ln(7)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogprop11a-h2","type":"hint","dependencies":["aeae96dlogprop11a-h1"],"title":"Using the Product Rule for Logarithms","text":"Using the product rule, $$\\\\ln(x^4 y)-\\\\ln(7)=\\\\ln(x^4)+\\\\ln(y)-\\\\ln(7)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogprop11a-h3","type":"hint","dependencies":["aeae96dlogprop11a-h2"],"title":"Using the Power Rule","text":"Using the power rule, $$\\\\ln(x^4)+\\\\ln(y)-\\\\ln(7)=4\\\\ln(x)+\\\\ln(y)-\\\\ln(7)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeae96dlogprop12","title":"Expanding Complex Logarithms","body":"Expand the logarithm.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.5 Logarithmic Properties","courseName":"OpenStax: College Algebra","steps":[{"id":"aeae96dlogprop12a","stepAnswer":["$$2\\\\ln(x)+3\\\\ln(y)-4\\\\ln(z)$$"],"problemType":"TextBox","stepTitle":"Expand log((x**2*y**3)/z**4))","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2\\\\ln(x)+3\\\\ln(y)-4\\\\ln(z)$$","hints":{"DefaultPathway":[{"id":"aeae96dlogprop12a-h1","type":"hint","dependencies":[],"title":"Using the Quotient Rule for Logarithms","text":"Using the quotient rule, log((x**2*y**3)/z**4))=log(x**2*y**3)-log(z**4)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogprop12a-h2","type":"hint","dependencies":["aeae96dlogprop12a-h1"],"title":"Using the Product Rule for Logarithms","text":"Using the product rule, $$\\\\ln(x^2 y^3)-\\\\ln(z^4)=\\\\ln(x^2)+\\\\ln(y^3)-\\\\ln(z^4)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogprop12a-h3","type":"hint","dependencies":["aeae96dlogprop12a-h2"],"title":"Using the Power Rule","text":"Using the power rule, $$\\\\ln(x^2)+\\\\ln(y^3)-\\\\ln(z^4)=2\\\\ln(x)+3\\\\ln(y)-4\\\\ln(z)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeae96dlogprop13","title":"Using the Power Rule for Logarithms to Simplify the Logarithm of a Radical Expression","body":"Expand the logarithm.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.5 Logarithmic Properties","courseName":"OpenStax: College Algebra","steps":[{"id":"aeae96dlogprop13a","stepAnswer":["$$\\\\frac{1}{2} \\\\ln(x)$$"],"problemType":"TextBox","stepTitle":"Expand $$\\\\ln(\\\\sqrt{x})$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{2} \\\\ln(x)$$","hints":{"DefaultPathway":[{"id":"aeae96dlogprop13a-h1","type":"hint","dependencies":[],"title":"Rewriting the Radical with Exponentiation","text":"To enable the use of the power rule, we may rewrite $$\\\\sqrt{x}$$ as $$x^{\\\\frac{1}{2}}$$. This means that $$\\\\ln(\\\\sqrt{x})=\\\\ln(x^{\\\\frac{1}{2}})$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogprop13a-h2","type":"hint","dependencies":["aeae96dlogprop13a-h1"],"title":"Using the Power Rule to Simplify","text":"Using the power rule, $$\\\\ln(x^{\\\\frac{1}{2}})=\\\\frac{1}{2} \\\\ln(x)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeae96dlogprop14","title":"Using the Power Rule for Logarithms to Simplify the Logarithm of a Radical Expression","body":"Expand the logarithm.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.5 Logarithmic Properties","courseName":"OpenStax: College Algebra","steps":[{"id":"aeae96dlogprop14a","stepAnswer":["$$\\\\frac{2}{3} \\\\ln(x)$$"],"problemType":"TextBox","stepTitle":"Expand $$\\\\ln(x^{\\\\frac{2}{3}})$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2}{3} \\\\ln(x)$$","hints":{"DefaultPathway":[{"id":"aeae96dlogprop14a-h1","type":"hint","dependencies":[],"title":"Using the Power Rule to Simplify","text":"By Power rule, $$\\\\ln(a^b)=\\\\operatorname{blog}\\\\left(a\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeae96dlogprop15","title":"Condensing Logarithmic Expressions","body":"Condense the logarithm.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.5 Logarithmic Properties","courseName":"OpenStax: College Algebra","steps":[{"id":"aeae96dlogprop15a","stepAnswer":["$$\\\\log_{3}\\\\left(20\\\\right)$$"],"problemType":"TextBox","stepTitle":"Condense $$\\\\log_{3}\\\\left(5\\\\right)+\\\\log_{3}\\\\left(8\\\\right)-\\\\log_{3}\\\\left(2\\\\right)$$ into a single logarithm.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\log_{3}\\\\left(20\\\\right)$$","hints":{"DefaultPathway":[{"id":"aeae96dlogprop15a-h1","type":"hint","dependencies":[],"title":"Using the Power Rule","text":"Using the power rule, $$\\\\log_{3}\\\\left(5\\\\right)+\\\\log_{3}\\\\left(8\\\\right)-\\\\log_{3}\\\\left(2\\\\right)=\\\\log_{3}\\\\left(40\\\\right)-\\\\log_{3}\\\\left(2\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogprop15a-h2","type":"hint","dependencies":["aeae96dlogprop15a-h1"],"title":"Using the Quotient Rule","text":"Using the quotient rule, $$\\\\log_{3}\\\\left(40\\\\right)-\\\\log_{3}\\\\left(2\\\\right)=\\\\log_{3}\\\\left(20\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeae96dlogprop16","title":"Condensing Logarithmic Expressions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.5 Logarithmic Properties","courseName":"OpenStax: College Algebra","steps":[{"id":"aeae96dlogprop16a","stepAnswer":["$$\\\\ln(\\\\frac{15}{24})$$"],"problemType":"TextBox","stepTitle":"Condense $$\\\\ln(3)-\\\\ln(4)+\\\\ln(5)-\\\\ln(6)$$ into a single logarithm.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\ln(\\\\frac{15}{24})$$","hints":{"DefaultPathway":[{"id":"aeae96dlogprop16a-h1","type":"hint","dependencies":[],"title":"Using the Power Rule","text":"Using the power rule, $$\\\\ln(3)-\\\\ln(4)+\\\\ln(5)-\\\\ln(6)=\\\\ln(3)+\\\\ln(5)-\\\\ln(4)-\\\\ln(6)=\\\\ln(15)-\\\\ln(24)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogprop16a-h2","type":"hint","dependencies":["aeae96dlogprop16a-h1"],"title":"Using the Quotient Rule","text":"Using the quotient rule, $$\\\\ln(15)-\\\\ln(24)=\\\\ln(\\\\frac{15}{24})$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeae96dlogprop2","title":"Using the Product Rule for Logarithms","body":"Expand the logarithm.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.5 Logarithmic Properties","courseName":"OpenStax: College Algebra","steps":[{"id":"aeae96dlogprop2a","stepAnswer":["$$\\\\log_{b}\\\\left(8\\\\right)+\\\\log_{b}\\\\left(k\\\\right)$$"],"problemType":"TextBox","stepTitle":"Expand $$\\\\log_{b}\\\\left(8k\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\log_{b}\\\\left(8\\\\right)+\\\\log_{b}\\\\left(k\\\\right)$$","hints":{"DefaultPathway":[{"id":"aeae96dlogprop2a-h1","type":"hint","dependencies":[],"title":"Factoring the Argument Completely","text":"$$\\\\ln(ab)$$ is equivalent to $$\\\\ln(a)+\\\\ln(b)$$ by product rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeae96dlogprop3","title":"Using the Quotient Rule for Logarithms","body":"Expand the logarithm.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.5 Logarithmic Properties","courseName":"OpenStax: College Algebra","steps":[{"id":"aeae96dlogprop3a","stepAnswer":["$$(\\\\log_{2}\\\\left(3\\\\right)+\\\\log_{2}\\\\left(5\\\\right)+\\\\log_{2}\\\\left(x\\\\right)+\\\\log_{2}\\\\left(x-1\\\\right))-(\\\\log_{2}\\\\left(3x+4\\\\right)+\\\\log_{2}\\\\left(2-x\\\\right))$$"],"problemType":"TextBox","stepTitle":"Expand log{2}{(15x)(x-1))/((3x+4)(2-x)}","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$(\\\\log_{2}\\\\left(3\\\\right)+\\\\log_{2}\\\\left(5\\\\right)+\\\\log_{2}\\\\left(x\\\\right)+\\\\log_{2}\\\\left(x-1\\\\right))-(\\\\log_{2}\\\\left(3x+4\\\\right)+\\\\log_{2}\\\\left(2-x\\\\right))$$","hints":{"DefaultPathway":[{"id":"aeae96dlogprop3a-h1","type":"hint","dependencies":[],"title":"Factoring the Numerator and Denominator","text":"$$((15x)(x-1))=3\\\\times5 x \\\\left(x-1\\\\right)$$. $$\\\\left(3x+4\\\\right) \\\\left(2-x\\\\right)=\\\\left(3x+4\\\\right) \\\\left(2-x\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogprop3a-h2","type":"hint","dependencies":["aeae96dlogprop3a-h1"],"title":"Representing the Expression With Many Logarithms","text":"$$\\\\ln(\\\\frac{a}{b})$$ is equivalent to $$\\\\ln(a)-\\\\ln(b)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeae96dlogprop4","title":"Using the Quotient Rule for Logarithms","body":"Expand the logarithm.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.5 Logarithmic Properties","courseName":"OpenStax: College Algebra","steps":[{"id":"aeae96dlogprop4a","stepAnswer":["$$(\\\\log_{3}\\\\left(7\\\\right)+\\\\log_{3}\\\\left(x\\\\right)+\\\\log_{3}\\\\left(x+3\\\\right))-(\\\\log_{3}\\\\left(7\\\\right)+\\\\log_{3}\\\\left(x\\\\right)+\\\\log_{3}\\\\left(x-1\\\\right)+\\\\log_{3}\\\\left(x-2\\\\right))$$"],"problemType":"TextBox","stepTitle":"Expand $$\\\\log_{3}\\\\left(\\\\frac{7x^2+21x}{7x\\\\left(x-1\\\\right) \\\\left(x-2\\\\right)}\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$(\\\\log_{3}\\\\left(7\\\\right)+\\\\log_{3}\\\\left(x\\\\right)+\\\\log_{3}\\\\left(x+3\\\\right))-(\\\\log_{3}\\\\left(7\\\\right)+\\\\log_{3}\\\\left(x\\\\right)+\\\\log_{3}\\\\left(x-1\\\\right)+\\\\log_{3}\\\\left(x-2\\\\right))$$","hints":{"DefaultPathway":[{"id":"aeae96dlogprop4a-h1","type":"hint","dependencies":[],"title":"Factoring the Argument Completely","text":"$$7x^2+21x=7x\\\\left(x+3\\\\right)$$. $$(7x(x-1)(x-2))=7x \\\\left(x-1\\\\right) \\\\left(x-2\\\\right)$$. We can now use the product and quotient rules and expand.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogprop4a-h2","type":"hint","dependencies":["aeae96dlogprop4a-h1"],"title":"Representing the Expression With Many Logarithms","text":"Using the quotient and product rules, we can write the expression with many logarithms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeae96dlogprop5","title":"Using the Power Rule for Logarithms","body":"Expand the logarithm.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.5 Logarithmic Properties","courseName":"OpenStax: College Algebra","steps":[{"id":"aeae96dlogprop5a","stepAnswer":["$$5\\\\log_{2}\\\\left(x\\\\right)$$"],"problemType":"TextBox","stepTitle":"Expand $$\\\\log_{2}\\\\left(x^5\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5\\\\log_{2}\\\\left(x\\\\right)$$","hints":{"DefaultPathway":[{"id":"aeae96dlogprop5a-h1","type":"hint","dependencies":[],"title":"Using the Power Rule to Simplify","text":"We can use the power rule as follows. We know that the exponent that $$\\\\log_{b}\\\\left(M^n\\\\right)=n\\\\log_{b}\\\\left(M\\\\right)$$. In this case, $$M=x$$ and $$n=5$$. This means that $$\\\\log_{2}\\\\left(x^5\\\\right)=5\\\\log_{2}\\\\left(x\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeae96dlogprop6","title":"Using the Power Rule for Logarithms","body":"Expand the logarithm.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.5 Logarithmic Properties","courseName":"OpenStax: College Algebra","steps":[{"id":"aeae96dlogprop6a","stepAnswer":["2ln(x)"],"problemType":"TextBox","stepTitle":"Expand $$\\\\ln(x^2)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"aeae96dlogprop6a-h1","type":"hint","dependencies":[],"title":"Using the Power Rule to Simplify","text":"We can use the power rule as follows. We know that the exponent that $$\\\\log_{b}\\\\left(M^n\\\\right)=n\\\\log_{b}\\\\left(M\\\\right)$$. In this case, $$M=x$$ and $$n=2$$. This means that $$\\\\ln(x^2)=2ln(x)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeae96dlogprop7","title":"Rewriting an Expression as a Power before Using the Power Rule","body":"Rewrite the logarithm.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.5 Logarithmic Properties","courseName":"OpenStax: College Algebra","steps":[{"id":"aeae96dlogprop7a","stepAnswer":["$$2\\\\log_{3}\\\\left(5\\\\right)$$"],"problemType":"TextBox","stepTitle":"Rewrite $$\\\\log_{3}\\\\left(25\\\\right)$$ using the power rule.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2\\\\log_{3}\\\\left(5\\\\right)$$","hints":{"DefaultPathway":[{"id":"aeae96dlogprop7a-h1","type":"hint","dependencies":[],"title":"Rewriting the Log with Exponentiation","text":"Since we know that $$25=5^2$$, we can replace $$25$$ with $$5^2$$ in the log such that we now have $$\\\\log_{3}\\\\left(5^2\\\\right)$$. You may notice tthat this allows us to simplify with the power rule.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogprop7a-h2","type":"hint","dependencies":["aeae96dlogprop7a-h1"],"title":"Using the Power Rule to Simplify","text":"We can use the power rule as follows. We know that the exponent that $$\\\\log_{b}\\\\left(M^n\\\\right)=n\\\\log_{b}\\\\left(M\\\\right)$$. In this case, $$M=5$$ and $$n=2$$. This means that $$\\\\log_{3}\\\\left(5^2\\\\right)=2\\\\log_{3}\\\\left(5\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeae96dlogprop8","title":"Rewriting an Expression as a Power before Using the Power Rule","body":"Rewrite the logarithm.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.5 Logarithmic Properties","courseName":"OpenStax: College Algebra","steps":[{"id":"aeae96dlogprop8a","stepAnswer":["$$-2ln(x)$$"],"problemType":"TextBox","stepTitle":"Expand $$\\\\ln(\\\\frac{1}{x^2})$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-2ln(x)$$","hints":{"DefaultPathway":[{"id":"aeae96dlogprop8a-h1","type":"hint","dependencies":[],"title":"Rewriting the Log with Exponentiation","text":"We must notice that $$\\\\frac{1}{x^2}$$ can be written as $$x^{\\\\left(-2\\\\right)}$$, allowing us to use the power rule to simplify the log. We now have $$\\\\ln(x^{\\\\left(-2\\\\right)})$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeae96dlogprop8a-h2","type":"hint","dependencies":["aeae96dlogprop8a-h1"],"title":"Using the Power Rule to Simplify","text":"We can use the power rule as follows. We know that the exponent that $$\\\\log_{b}\\\\left(M^n\\\\right)=n\\\\log_{b}\\\\left(M\\\\right)$$. In this case, $$M=x$$ and $$n=-2$$. This means that $$\\\\ln(x^{\\\\left(-2\\\\right)})=-2ln(x)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeae96dlogprop9","title":"Using the Power Rule in Inverse","body":"Rewrite the logarithm.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.5 Logarithmic Properties","courseName":"OpenStax: College Algebra","steps":[{"id":"aeae96dlogprop9a","stepAnswer":["$$\\\\ln(x^4)$$"],"problemType":"TextBox","stepTitle":"Rewrite 4ln(x) using the inverse power rule such that it has a coefficient of $$1$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\ln(x^4)$$","hints":{"DefaultPathway":[{"id":"aeae96dlogprop9a-h1","type":"hint","dependencies":[],"title":"Using the Inverse Power Rule to Expand","text":"We can use the power rule as follows. We know that the exponent that $$\\\\log_{b}\\\\left(M^n\\\\right)=n\\\\log_{b}\\\\left(M\\\\right)$$. In this case, $$M=x$$ and $$n=4$$. So, $$4ln(x)=\\\\ln(x^4)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb11eedomain1","title":"Finding the Domain of a Function as a Set of Ordered Pairs","body":"Find the domain of the following function:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Domain and Range","courseName":"OpenStax: College Algebra","steps":[{"id":"aeb11eedomain1a","stepAnswer":["(2,3,4,5,6)"],"problemType":"MultipleChoice","stepTitle":"$$((2,10),(3,10),(4,20),(5,30),(6,40))$$","stepBody":"","answerType":"string","variabilization":{},"choices":["(4,5,6)","(10,20,30,40)","(2,3,4,5,6)","(10,10,20,30,40)"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain1a-h1","type":"hint","dependencies":[],"title":"Identifying Input Values","text":"The first step is to identify the input values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain1a-h2","type":"hint","dependencies":["aeb11eedomain1a-h1"],"title":"Input Value Definition","text":"An input value is the first coordinate in an ordered pair.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain1a-h3","type":"hint","dependencies":["aeb11eedomain1a-h2"],"title":"Identifying Restrictions","text":"There are no restrictions, since the ordered pairs are simply listed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain1a-h4","type":"hint","dependencies":["aeb11eedomain1a-h3"],"title":"Domain Definition","text":"The domain is the first set of coordinates in the ordered pairs.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb11eedomain10","title":"Finding the Domain and Range","body":"Find the domain and range of the function $$f(x)=2\\\\sqrt{x+4}$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Domain and Range","courseName":"OpenStax: College Algebra","steps":[{"id":"aeb11eedomain10a","stepAnswer":["$$x \\\\geq -4$$"],"problemType":"MultipleChoice","stepTitle":"Find the domain.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x \\\\geq -4$$","choices":["$$x \\\\leq -4$$","$$x \\\\geq -4$$","$$x \\\\geq 4$$","all real numbers"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain10a-h1","type":"hint","dependencies":[],"title":"Square Root Restrictions","text":"We cannot take the square root of a negative number, so the value inside the radical must be nonnegative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["aeb11eedomain10a-h1"],"title":"Solving the Inequality","text":"What value does $$x$$ have to be greater than or equal to for $$x+4 \\\\geq 0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aeb11eedomain10b","stepAnswer":["[0,inf)"],"problemType":"TextBox","stepTitle":"Find the range written in interval notation.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[0,\\\\infty)$$","hints":{"DefaultPathway":[{"id":"aeb11eedomain10b-h1","type":"hint","dependencies":[],"title":"Value of $$f(-4)$$","text":"We know that $$f(-4)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain10b-h2","type":"hint","dependencies":["aeb11eedomain10b-h1"],"title":"Trends of the Function Value","text":"The function value increases as $$x$$ increases without any upper limit.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain10b-h3","type":"hint","dependencies":["aeb11eedomain10b-h2"],"title":"Answer","text":"Therefore, we conclude that the range of f is $$[0,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb11eedomain11","title":"Writing a Piecewise Function","body":"A museum charges $5 per person for a guided tour with a group of $$1$$ to $$9$$ people or a fixed $50 fee for a group of $$10$$ or more people.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Domain and Range","courseName":"OpenStax: College Algebra","steps":[{"id":"aeb11eedomain11a","stepAnswer":["C(n) $$=$$ {5n if $$0<n<10$$, $$50$$ if $$n \\\\geq 10$$}"],"problemType":"MultipleChoice","stepTitle":"Write a function relating the number of people, $$n$$, to the cost, C.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"C(n) $$=$$ {5n if $$0<n<10$$, $$50$$ if $$n \\\\geq 10$$}","choices":["C(n) $$=$$ {5n}","C(n) $$=$$ {50n}","C(n) $$=$$ {n if $$0<n<10$$, $$50$$ if $$n \\\\geq 10$$}","C(n) $$=$$ {5n if $$0<n<10$$, $$50$$ if $$n \\\\geq 10$$}"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain11a-h1","type":"hint","dependencies":[],"title":"Number of Formulas Needed","text":"Two different formulas will be needed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain11a-h2","type":"hint","dependencies":["aeb11eedomain11a-h1"],"title":"$$0<n<10$$ Equation","text":"For $$n-values$$ under $$10$$, $$C=5n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain11a-h3","type":"hint","dependencies":["aeb11eedomain11a-h2"],"title":"$$n \\\\geq 10$$ Equation","text":"For values of $$n$$ that are $$10$$ or greater, $$C=50$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain11a-h4","type":"hint","dependencies":["aeb11eedomain11a-h3"],"title":"Answer","text":"C(n) $$=$$ {5n if $$0<n<10$$, $$50$$ if $$n \\\\geq 10$$}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb11eedomain12","title":"Working With a Piecewise Function","body":"A cell phone company uses the function C(g) $$=$$ {25 if $$0<g<2$$, $$25+\\\\operatorname{10}\\\\left(g-2\\\\right)$$ if $$g \\\\geq 2$$ to determine the cost, C, in dollars for g gigabytes of data transfer. Find the cost of using $$1.5$$ gigabytes of data and the cost of using $$4$$ gigabytes of data.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Domain and Range","courseName":"OpenStax: College Algebra","steps":[{"id":"aeb11eedomain12a","stepAnswer":["$$25$$"],"problemType":"TextBox","stepTitle":"Find the cost of using $$1.5$$ gigabytes of data","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$25$$","hints":{"DefaultPathway":[{"id":"aeb11eedomain12a-h1","type":"hint","dependencies":[],"title":"Finding Domain","text":"First, we look to see which part of the domain our input falls in. Because $$1.5$$ is less than $$2$$, we use the first formula.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["aeb11eedomain12a-h1"],"title":"Finding $$C(1.5)$$","text":"What is $$C(1.5)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aeb11eedomain12b","stepAnswer":["$$45$$"],"problemType":"TextBox","stepTitle":"Find the cost of using $$4$$ gigabytes of data","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$45$$","hints":{"DefaultPathway":[{"id":"aeb11eedomain12b-h1","type":"hint","dependencies":[],"title":"Choosing the Formula","text":"To find the cost of using $$4$$ gigabytes of data, C(4), we see that our input of $$4$$ is greater than $$2$$, so we use the second formula.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain12b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$45$$"],"dependencies":["aeb11eedomain12b-h1"],"title":"Finding C(4)","text":"What is C(4)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb11eedomain13","title":"Finding the Domain of a Function as a Set of Ordered Pairs","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Domain and Range","courseName":"OpenStax: College Algebra","steps":[{"id":"aeb11eedomain13a","stepAnswer":["{$$-5, 0, 5, 10, 15$$}"],"problemType":"MultipleChoice","stepTitle":"{$$(-5,4),(0,0),(5,-4),(10,-8),(15,-12)$$}","stepBody":"","answerType":"string","variabilization":{},"choices":["{$$4, 0, -4, -8, -12$$}","{$$-5, 0, 5, 10, 15$$}"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain13a-h1","type":"hint","dependencies":[],"title":"Input Values","text":"Identify the input values, or the first value in the coordinate","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb11eedomain14","title":"Finding the Domain of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Domain and Range","courseName":"OpenStax: College Algebra","steps":[{"id":"aeb11eedomain14a","stepAnswer":["all real numbers"],"problemType":"MultipleChoice","stepTitle":"Find the domain of the $$function:f(x)=5-x+x^3$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$x \\\\geq 0$$","$$x<0$$","$$x<5$$ and $$x>5$$","all real numbers"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain14a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":[],"title":"Restrictions","text":"Are there any restrictions on the values we can subsitute for $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"aeb11eedomain14a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["all real numbers"],"dependencies":["aeb11eedomain14a-h1"],"title":"Domain","text":"If there are no restrictions, then what is the domain?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x \\\\geq 0$$","$$x<0$$","$$x<5$$ and $$x>5$$","all real numbers"]}]}}]},{"id":"aeb11eedomain15","title":"Finding the Domain of a Function Involving a Denominator","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Domain and Range","courseName":"OpenStax: College Algebra","steps":[{"id":"aeb11eedomain15a","stepAnswer":["$$(-\\\\infty,\\\\frac{1}{2}) \\\\cup (\\\\frac{1}{2},\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"Find the domain of the function: $$f(x)=\\\\frac{1+4x}{2x-1}$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\frac{1}{2}) \\\\cup (\\\\frac{1}{2},\\\\infty)$$","choices":["$$(-\\\\infty,\\\\frac{1}{2})$$","$$(\\\\frac{1}{2},\\\\infty)$$","$$(-\\\\infty,\\\\frac{1}{2}) \\\\cup (\\\\frac{1}{2},\\\\infty)$$","$$(-\\\\infty,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain15a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":[],"title":"Denominator","text":"Set the denominator equal to zero and solve for $$x$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain15a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(-\\\\infty,\\\\frac{1}{2}) \\\\cup (\\\\frac{1}{2},\\\\infty)$$"],"dependencies":["aeb11eedomain15a-h1"],"title":"Domain","text":"When we exclude $$\\\\frac{1}{2}$$ from the domain, what is the domain?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(-\\\\infty,\\\\frac{1}{2})$$","$$(\\\\frac{1}{2},\\\\infty)$$","$$(-\\\\infty,\\\\frac{1}{2}) \\\\cup (\\\\frac{1}{2},\\\\infty)$$","$$(-\\\\infty,\\\\infty)$$"]}]}}]},{"id":"aeb11eedomain16","title":"Finding the Domain of a Function with an Even Root","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Domain and Range","courseName":"OpenStax: College Algebra","steps":[{"id":"aeb11eedomain16a","stepAnswer":["$$[\\\\frac{-5}{2},\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"Find the domain of the function $$f(x)=\\\\sqrt{5+2x}$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[\\\\frac{-5}{2},\\\\infty)$$","choices":["$$[\\\\frac{-5}{2},\\\\infty)$$","$$(\\\\frac{-5}{2},\\\\infty)$$","$$(-\\\\infty,\\\\frac{-5}{2}]$$","$$(-\\\\infty,\\\\frac{-5}{2})$$"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain16a-h1","type":"hint","dependencies":[],"title":"Define the Domain","text":"A square root function is undefined when the expression under the square root is negative. So, let\'s start by setting the expression under the sqare root to greater than or equal to zero. Then we know every value that makes this true is the domain of the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain16a-h2","type":"hint","dependencies":["aeb11eedomain16a-h1"],"title":"Inequality","text":"Set the value in the square root to being greater than or equal to $$0$$ and solve the inequality","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain16a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$[\\\\frac{-5}{2},\\\\infty)$$"],"dependencies":["aeb11eedomain16a-h2"],"title":"Domain","text":"What is the domain in interval notation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$[\\\\frac{-5}{2},\\\\infty)$$","$$(\\\\frac{-5}{2},\\\\infty)$$","$$(-\\\\infty,\\\\frac{-5}{2}]$$","$$(-\\\\infty,\\\\frac{-5}{2})$$"]}]}}]},{"id":"aeb11eedomain17","title":"Finding Domain and Range from a Graph","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Domain and Range","courseName":"OpenStax: College Algebra","steps":[{"id":"aeb11eedomain17a","stepAnswer":["[1950,2000]"],"problemType":"MultipleChoice","stepTitle":"Identify the domain of the function whose graph is shown in the figure using interval notation.","stepBody":"","answerType":"string","variabilization":{},"choices":["[1960,1990]","[1950,2000]","[47,90]","[47,77]"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain17a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1950$$"],"dependencies":[],"title":"Starting Point","text":"What year does the graph start?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain17a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2000$$"],"dependencies":["aeb11eedomain17a-h1"],"title":"Ending point","text":"What year does the graph end","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain17a-h3","type":"hint","dependencies":["aeb11eedomain17a-h2"],"title":"Interval Notation","text":"Write this in interval notation, inclusive of the years, and we get [1950,2000].","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aeb11eedomain17b","stepAnswer":["[47,90]"],"problemType":"MultipleChoice","stepTitle":"Identify the range of the function whose graph is shown in the figure using interval notation.","stepBody":"","answerType":"string","variabilization":{},"choices":["[1960,1990]","[1950,2000]","[47,90]","[47,77]"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain17b-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$47$$"],"dependencies":[],"title":"Starting Point","text":"What value is the lowest on the graph?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain17b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$90$$"],"dependencies":["aeb11eedomain17b-h1"],"title":"Ending point","text":"What value is the highest on the graph?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain17b-h3","type":"hint","dependencies":["aeb11eedomain17b-h2"],"title":"Interval Notation","text":"Write this in interval notation, inclusive of the number of endpoints, and we get [47,90].","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb11eedomain18","title":"Finding the Domain and Range","body":"Find the domain and range of $$f(x)=-\\\\sqrt{2-x}$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Domain and Range","courseName":"OpenStax: College Algebra","steps":[{"id":"aeb11eedomain18a","stepAnswer":["$$(-\\\\infty,2]$$"],"problemType":"MultipleChoice","stepTitle":"Find the domain of $$f(x)=-\\\\sqrt{2-x}$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,2]$$","choices":["$$(-\\\\infty,2)$$","$$(-\\\\infty,2]$$","$$[2,\\\\infty)$$","$$(-\\\\infty,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain18a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Set to $$0$$","text":"Set the expression in the square root to greater than or equal to $$0$$ and solve for $$x$$. What does $$x$$ has to be less than or equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain18a-h2","type":"hint","dependencies":["aeb11eedomain18a-h1"],"title":"Domain","text":"Now, write $$x \\\\leq 2$$ in interval notation, and we get $$(-\\\\infty,2]$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aeb11eedomain18b","stepAnswer":["$$(-\\\\infty,0]$$"],"problemType":"MultipleChoice","stepTitle":"Find the range of $$f(x)=-\\\\sqrt{2-x}$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,0]$$","choices":["$$(-\\\\infty,0]$$","$$[0,\\\\infty)$$","$$(-\\\\infty,2]$$","$$(-\\\\infty,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain18b-h1","type":"hint","dependencies":[],"title":"Plugging in $$x=2$$","text":"When $$x$$ is $$2$$, we get the maximum value of this equation, which is $$-\\\\sqrt{2-2}=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain18b-h2","type":"hint","dependencies":["aeb11eedomain18b-h1"],"title":"Range","text":"Therefore, the range of this function is less than or equal to $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain18b-h3","type":"hint","dependencies":["aeb11eedomain18b-h2"],"title":"Interval Notation","text":"The expression $$y \\\\leq 0$$ written in interval notation is $$(-\\\\infty,0]$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb11eedomain19","title":"Finding Domain","body":"Find the domain of the following function using interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Domain and Range","courseName":"OpenStax: College Algebra","steps":[{"id":"aeb11eedomain19a","stepAnswer":["$$(-\\\\infty,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=-2x(x-1)(x-2)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\infty)$$","choices":["$$(-\\\\infty,0)$$","$$(0,\\\\infty)$$","$$(-\\\\infty,1]$$","$$(-\\\\infty,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain19a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":[],"title":"Restrictions","text":"Are there any restrictions on the values of $$x$$ in this cubic function?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"aeb11eedomain19a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(-\\\\infty,\\\\infty)$$"],"dependencies":["aeb11eedomain19a-h1"],"title":"Domain","text":"If there are no restrictions, what is the domain?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(-\\\\infty,0)$$","$$(0,\\\\infty)$$","$$(-\\\\infty,1]$$","$$(-\\\\infty,\\\\infty)$$"]}]}}]},{"id":"aeb11eedomain2","title":"Finding the Domain of a Function","body":"Find the domain of the function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Domain and Range","courseName":"OpenStax: College Algebra","steps":[{"id":"aeb11eedomain2a","stepAnswer":["$$(-\\\\infty,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=x^2-1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\infty)$$","choices":["$$(-\\\\infty,\\\\infty)$$","$$(1,\\\\infty)$$","$$(0,\\\\infty)$$","$$(-\\\\infty,0)$$"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain2a-h1","type":"hint","dependencies":[],"title":"Identifying Restrictions on the Input","text":"The first step is to identify any restrictions on what $$x$$ can be by looking at the operations in the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain2a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["no"],"dependencies":["aeb11eedomain2a-h1"],"title":"Operations of the Function","text":"In the function, first, $$x$$ is squared. Then, $$1$$ is subtracted from $$x^2$$. Are there any limitations on the value of $$x$$ for the function to have a valid output?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["yes","no"]},{"id":"aeb11eedomain2a-h3","type":"hint","dependencies":["aeb11eedomain2a-h2"],"title":"Answer","text":"Any real number may be squared and then be lowered by one, so there are no restrictions on the domain of this function. The domain is the set of real numbers, negative infinity to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb11eedomain20","title":"Finding Domain","body":"Find the domain of the following function using interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Domain and Range","courseName":"OpenStax: College Algebra","steps":[{"id":"aeb11eedomain20a","stepAnswer":["$$(-\\\\infty,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=5-2x^2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\infty)$$","choices":["$$(-\\\\infty,5)$$","$$(2,\\\\infty)$$","$$(-\\\\infty,\\\\frac{5}{2}]$$","$$(-\\\\infty,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain20a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":[],"title":"Restrictions","text":"Are there any restrictions on the values of $$x$$ in this cubic function?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"aeb11eedomain20a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(-\\\\infty,\\\\infty)$$"],"dependencies":["aeb11eedomain20a-h1"],"title":"Domain","text":"If there are no restrictions, what is the domain?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(-\\\\infty,5)$$","$$(2,\\\\infty)$$","$$(-\\\\infty,\\\\frac{5}{2}]$$","$$(-\\\\infty,\\\\infty)$$"]}]}}]},{"id":"aeb11eedomain21","title":"Finding the Domain of a Function with an Even Root","body":"Find the domain of the following function using interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Domain and Range","courseName":"OpenStax: College Algebra","steps":[{"id":"aeb11eedomain21a","stepAnswer":["$$[2,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=3\\\\sqrt{x-2}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[2,\\\\infty)$$","choices":["$$[-2,\\\\infty)$$","$$(-2,\\\\infty)$$","$$[2,\\\\infty)$$","$$(-\\\\infty,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain21a-h1","type":"hint","dependencies":[],"title":"Define the Domain","text":"A square root function is undefined when the expression under the square root is negative. So, let\'s start by setting the expression under the sqare root to greater than or equal to zero. Then we know every value that makes this true is the domain of the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain21a-h2","type":"hint","dependencies":["aeb11eedomain21a-h1"],"title":"Inequality","text":"Set the value in the square root to being greater than equal to $$0$$ and solve the inequality. In other words, solve for $$x-2 \\\\geq 0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain21a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$[2,\\\\infty)$$"],"dependencies":["aeb11eedomain21a-h2"],"title":"Domain","text":"What is the domain in interval notation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$[-2,\\\\infty)$$","$$(-2,\\\\infty)$$","$$[2,\\\\infty)$$","$$(-\\\\infty,\\\\infty)$$"]}]}}]},{"id":"aeb11eedomain22","title":"Finding the Domain of a Function with an Even Root","body":"Find the domain of the following function using interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Domain and Range","courseName":"OpenStax: College Algebra","steps":[{"id":"aeb11eedomain22a","stepAnswer":["$$(-\\\\infty,3]$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=3-\\\\sqrt{6-2x}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,3]$$","choices":["$$(-\\\\infty,-3)$$","$$(-\\\\infty,3]$$","$$[3,\\\\infty)$$","$$(-\\\\infty,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain22a-h1","type":"hint","dependencies":[],"title":"Define the Domain","text":"A square root function is undefined when the expression under the square root is negative. So, let\'s start by setting the expression under the sqare root to greater than or equal to zero. Then we know every value that makes this true is the domain of the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain22a-h2","type":"hint","dependencies":["aeb11eedomain22a-h1"],"title":"Inequality","text":"Set the value in the square root to being greater than equal to $$0$$ and solve the inequality. In other words, solve for $$6-2x \\\\geq 0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain22a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(-\\\\infty,3]$$"],"dependencies":["aeb11eedomain22a-h2"],"title":"Domain","text":"What is the domain in interval notation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(-\\\\infty,-3)$$","$$(-\\\\infty,3]$$","$$[3,\\\\infty)$$","$$(-\\\\infty,\\\\infty)$$"]}]}}]},{"id":"aeb11eedomain23","title":"Find the domain of the function using interval notation.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Domain and Range","courseName":"OpenStax: College Algebra","steps":[{"id":"aeb11eedomain23a","stepAnswer":["$$(-\\\\infty,\\\\frac{4}{3}]$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\sqrt{4-3x}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\frac{4}{3}]$$","choices":["$$(-\\\\infty,\\\\infty)$$","$$(-\\\\infty,\\\\frac{-4}{3}]$$","$$(-\\\\infty,\\\\frac{4}{3}]$$","$$[\\\\frac{4}{3},\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain23a-h1","type":"hint","dependencies":[],"title":"Define the Domain","text":"A square root function is undefined when the expression under the square root is negative. So, let\'s start by setting the expression under the sqare root to greater than or equal to zero. Then we know every value that makes this true is the domain of the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain23a-h2","type":"hint","dependencies":["aeb11eedomain23a-h1"],"title":"Solving For the Domain","text":"Solving for $$4-3x \\\\geq 0$$, we get $$x \\\\leq \\\\frac{4}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain23a-h3","type":"hint","dependencies":["aeb11eedomain23a-h2"],"title":"Answer","text":"The domain of the function is $$(-\\\\infty,\\\\frac{4}{3}]$$, with the square bracket around $$\\\\frac{4}{3}$$ denoting the fact that $$\\\\frac{4}{3}$$ is a valid solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb11eedomain24","title":"Find the domain of the function using interval notation.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Domain and Range","courseName":"OpenStax: College Algebra","steps":[{"id":"aeb11eedomain24a","stepAnswer":["$$(-\\\\infty,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\sqrt{x^2+4}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\infty)$$","choices":["$$[2,\\\\infty)$$","$$(-\\\\infty,-2]$$","$$(2,\\\\infty)$$","$$(-\\\\infty,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain24a-h1","type":"hint","dependencies":[],"title":"Define the Domain","text":"A square root function is undefined when the expression under the square root is negative. So, let\'s start by setting the expression under the sqare root greater than or equal to zero. Then we know every value that makes this true is the domain of the function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain24a-h2","type":"hint","dependencies":["aeb11eedomain24a-h1"],"title":"Answer","text":"Since any integer squared is positive, and we are adding four to the squared number, there is no value of $$x$$ that makes the function undefined. So, the domain is $$(-\\\\infty,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb11eedomain25","title":"Find the domain of the function using interval notation.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Domain and Range","courseName":"OpenStax: College Algebra","steps":[{"id":"aeb11eedomain25a","stepAnswer":["$$(-\\\\infty,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)={\\\\left(1-2x\\\\right)}^{\\\\frac{1}{3}}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\infty)$$","choices":["$$[\\\\frac{1}{2},\\\\infty)$$","$$(-\\\\infty,\\\\frac{-1}{2}]$$","$$(-\\\\infty,\\\\frac{-1}{2}]$$","$$(-\\\\infty,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain25a-h1","type":"hint","dependencies":[],"title":"Answer","text":"Since cube roots are defined for positive, negative, and zero numbers, there are no values of $$x$$ that would make the function undefined. So, the domain is $$(-\\\\infty,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb11eedomain26","title":"Find the domain of the function using interval notation.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Domain and Range","courseName":"OpenStax: College Algebra","steps":[{"id":"aeb11eedomain26a","stepAnswer":["$$(-\\\\infty,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)={\\\\left(x-1\\\\right)}^{\\\\frac{1}{3}}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\infty)$$","choices":["$$(-\\\\infty,\\\\infty)$$","$$[1,\\\\infty)$$","$$(1,\\\\infty)$$","$$(-\\\\infty,-1]$$"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain26a-h2","type":"hint","dependencies":[],"title":"Answer","text":"Since cube roots are defined for positive, negative, and zero numbers, there are no values of $$x$$ that would make the function undefined. So, the domain is $$(-\\\\infty,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb11eedomain27","title":"Find the domain of the function using interval notation.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Domain and Range","courseName":"OpenStax: College Algebra","steps":[{"id":"aeb11eedomain27a","stepAnswer":["$$(-\\\\infty,6) \\\\cup (6,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{9}{x-6}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,6) \\\\cup (6,\\\\infty)$$","choices":["$$(-\\\\infty,\\\\infty)$$","$$(-\\\\infty,6)$$","$$(6,\\\\infty)$$","$$(-\\\\infty,6) \\\\cup (6,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain27a-h1","type":"hint","dependencies":[],"title":"Define the Domain","text":"A rational function is undefined when the denominator is equal to zero. So, let\'s start by setting the denominator equal to zero. Then we know every value other than that is a solution. (We can ignore the numerator since there is no value where it is undefined!)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain27a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":[],"title":"Solving For the Domain","text":"What is $$x$$ for $$x-6=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aeb11eedomain27a-h2-s1","type":"hint","dependencies":[],"title":"Solving For the Domain","text":"For $$x-6=0$$, add $$6$$ to both sides, and $$x=6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aeb11eedomain27a-h3","type":"hint","dependencies":["aeb11eedomain27a-h2"],"title":"Answer","text":"So, the function is defined for every value other than 6.The domain of the function is $$(-\\\\infty,6) \\\\cup (6,\\\\infty)$$, with the circle bracket around $$6$$ denoting the fact that $$6$$ is a not valid solution, and the U showing that both intervals are solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb11eedomain28","title":"Find the domain of the function using interval notation.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Domain and Range","courseName":"OpenStax: College Algebra","steps":[{"id":"aeb11eedomain28a","stepAnswer":["$$(-\\\\infty,\\\\frac{-1}{2}) \\\\cup (\\\\frac{-1}{2},\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{3x+1}{4x+2}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\frac{-1}{2}) \\\\cup (\\\\frac{-1}{2},\\\\infty)$$","choices":["$$(-\\\\infty,\\\\infty)$$","(-inf, -1/2)","(-1/2, inf)","$$(-\\\\infty,\\\\frac{-1}{2}) \\\\cup (\\\\frac{-1}{2},\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain28a-h1","type":"hint","dependencies":[],"title":"Define the Domain","text":"A rational function is undefined when the denominator is equal to zero. So, let\'s start by setting the denominator equal to zero. Then we know every value other than that is a solution. (We can ignore the numerator since there is no value where it is undefined!)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain28a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{2}$$"],"dependencies":["aeb11eedomain28a-h1"],"title":"Solving For the Domain","text":"What is $$x$$ for $$4x+2=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain28a-s1","type":"hint","dependencies":[],"title":"Solving For the Domain","text":"For $$4x+2=0$$, subtract $$2$$ from both sides, then divide both sides by $$4$$ to get $$x=\\\\frac{-1}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain28a-h3","type":"hint","dependencies":["aeb11eedomain28a-h2"],"title":"Answer","text":"So, the function is defined for every value other than -1/2.The domain of the function is $$(-\\\\infty,\\\\frac{-1}{2}) \\\\cup (\\\\frac{-1}{2},\\\\infty)$$, with the circle bracket around $$\\\\frac{-1}{2}$$ denoting the fact that $$\\\\frac{-1}{2}$$ is a not valid solution, and the U showing that both intervals are solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb11eedomain29","title":"Find the domain of the function using interval notation.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Domain and Range","courseName":"OpenStax: College Algebra","steps":[{"id":"aeb11eedomain29a","stepAnswer":["$$[-4,4) \\\\cup (4,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{\\\\sqrt{x+4}}{x-4}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[-4,4) \\\\cup (4,\\\\infty)$$","choices":["$$(-\\\\infty,\\\\infty)$$","$$[-4,4) \\\\cup (4,\\\\infty)$$","$$(-\\\\infty,4) \\\\cup (4,\\\\infty)$$","$$[-4,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain29a-h1","type":"hint","dependencies":[],"title":"Define the Domain","text":"A rational function is undefined when the denominator is equal to zero. So, let\'s start by setting the denominator equal to zero. In addition, this function also contains a square root in the numerator which is undefined when the expression under the square root is negative, so we can set the numerator greater than or equal to zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain29a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["aeb11eedomain29a-h1"],"title":"Solving For the Denominator","text":"What is $$x$$ for $$x-4=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aeb11eedomain29a-h2-s1","type":"hint","dependencies":[],"title":"Solving For the Denominator","text":"For $$x-4=0$$, add $$4$$ to both sides, and $$x=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aeb11eedomain29a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x \\\\geq -4$$"],"dependencies":["aeb11eedomain29a-h1"],"title":"Solving For the Numerator","text":"What is $$x$$ for $$\\\\sqrt{x+4} \\\\geq 0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x \\\\geq -4$$","$$x \\\\leq -4$$","$$x \\\\geq 4$$","$$x \\\\leq 4$$"],"subHints":[{"id":"aeb11eedomain29a-h3-s1","type":"hint","dependencies":[],"title":"Solving For the Numerator","text":"Set the expression under the numerator greater than or equal to $$0$$, and subtract $$4$$ to both sides. $$x \\\\geq -4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aeb11eedomain29a-h4","type":"hint","dependencies":["aeb11eedomain29a-h2","aeb11eedomain29a-h3"],"title":"Answer","text":"So, the function is defined for every value greater than or equal to $$-4$$ other than 4.The domain of the function is $$[-4,4) \\\\cup (4,\\\\infty)$$, with the circle brackets denoting the fact that $$4$$ is not a valid solution, and the U showing that values within the intervals are solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb11eedomain3","title":"Finding the Domain of a Function Involving a Denominator","body":"Find the domain of the function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Domain and Range","courseName":"OpenStax: College Algebra","steps":[{"id":"aeb11eedomain3a","stepAnswer":["all real numbers where $$x<2$$ or $$x>2$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{x+1}{2-x}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"all real numbers where $$x<2$$ or $$x>2$$","choices":["all real numbers where $$x$$ cannot be $$-1$$ or $$2$$","all real numbers where $$x<2$$ or $$x>2$$","all real numbers","all real numbers where $$x$$ cannot be $$-1$$"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain3a-h1","type":"hint","dependencies":[],"title":"Considering the Denominator","text":"When there is a denominator, we want to include only values of the input that do not force the denominator to be zero. Therefore, the first step is to set the denominator to $$0$$ and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["aeb11eedomain3a-h1"],"title":"Solving For When the Denominator is Zero","text":"$$2-x=0$$. $$x=$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain3a-h3","type":"hint","dependencies":["aeb11eedomain3a-h2"],"title":"Dealing With Inputs That Make the Denominator Zero","text":"Since the denominator is $$0$$ when $$x=2$$, $$2$$ is excluded from the domain.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain3a-h4","type":"hint","dependencies":["aeb11eedomain3a-h3"],"title":"Answer","text":"The answers are all real numbers where $$x<2$$ or $$x>2$$ as shown in the image. We can use a symbol known as the union, U, to combine the two sets. In interval notation, we write the solution:(-inf, $$2) \\\\cup (2,\\\\infty)$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb11eedomain30","title":"Find the domain of the function using interval notation.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Domain and Range","courseName":"OpenStax: College Algebra","steps":[{"id":"aeb11eedomain30a","stepAnswer":["$$(-\\\\infty,-11) \\\\cup (-11,2) \\\\cup (2,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{x-3}{x^2+9x-22}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,-11) \\\\cup (-11,2) \\\\cup (2,\\\\infty)$$","choices":["$$(-\\\\infty,\\\\infty)$$","$$(-\\\\infty,-11) \\\\cup (2,\\\\infty)$$","$$(-\\\\infty,-11) \\\\cup (-11,2) \\\\cup (2,\\\\infty)$$","$$(-\\\\infty,-11) \\\\cup (-11,2) \\\\cup (2,3) \\\\cup (3,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain30a-h1","type":"hint","dependencies":[],"title":"Define the Domain","text":"A rational function is undefined when the denominator is equal to zero. So, let\'s start by setting the denominator equal to zero. Then we know every value other than that is a solution. (We can ignore the numerator since there is no value where it is undefined!)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain30a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x=-11, 2$$"],"dependencies":["aeb11eedomain30a-h1"],"title":"Solving For the Domain","text":"What is $$x$$ for $$x^2+9x-22=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x=-2$$","$$x=-11$$","$$x=-2, 11$$","$$x=-11, 2$$"],"subHints":[{"id":"aeb11eedomain30a-h2-s1","type":"hint","dependencies":[],"title":"Solving For the Domain","text":"For $$x^2+9x-22=0$$, factor the equation to get $$\\\\left(x+11\\\\right) \\\\left(x-2\\\\right)$$. Set these two expressions equal to zero to get that $$x=-11$$ or $$x=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aeb11eedomain30a-h3","type":"hint","dependencies":["aeb11eedomain30a-h2"],"title":"Answer","text":"So, the function is defined for every value other than $$-11$$ and 2.The domain of the function is $$(-\\\\infty,-11) \\\\cup (-11,2) \\\\cup (2,\\\\infty)$$, with the circle brackets denoting the fact that $$-11$$ and $$2$$ are not valid solutions, and the U showing that values within the intervals are solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb11eedomain31","title":"Find the domain of the function using interval notation.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Domain and Range","courseName":"OpenStax: College Algebra","steps":[{"id":"aeb11eedomain31a","stepAnswer":["$$(-\\\\infty,-2) \\\\cup (-2,3) \\\\cup (3,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{1}{x^2-x-6}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,-2) \\\\cup (-2,3) \\\\cup (3,\\\\infty)$$","choices":["$$(-\\\\infty,\\\\infty)$$","$$(-\\\\infty,3)$$","$$(-2,3) \\\\cup (3,\\\\infty)$$","$$(-\\\\infty,-2) \\\\cup (-2,3) \\\\cup (3,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain31a-h1","type":"hint","dependencies":[],"title":"Define the Domain","text":"A rational function is undefined when the denominator is equal to zero. So, let\'s start by setting the denominator equal to zero. Then we know every value other than that is a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain31a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x=-2, 3$$"],"dependencies":["aeb11eedomain31a-h1"],"title":"Solving For the Domain","text":"What is $$x$$ for $$x^2-x-6=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x=1, 3$$","$$x=-2, 3$$","$$x=-3, 2$$","$$x=-1, -3$$"],"subHints":[{"id":"aeb11eedomain31a-h2-s1","type":"hint","dependencies":[],"title":"Solving For the Domain","text":"For $$x^2-x-6=0$$, factor the equation to get $$\\\\left(x-3\\\\right) \\\\left(x+2\\\\right)$$. Set these two expressions equal to zero to get that $$x=3$$ or $$x=-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aeb11eedomain31a-h3","type":"hint","dependencies":["aeb11eedomain31a-h2"],"title":"Answer","text":"So, the function is defined for every value other than $$-2$$ and 3.The domain of the function is $$(-\\\\infty,-2) \\\\cup (-2,3) \\\\cup (3,\\\\infty)$$, with the circle brackets denoting the fact that $$-2$$ and $$3$$ are not valid solutions, and the U showing that values within the intervals are solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb11eedomain32","title":"Find the domain of the function using interval notation.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Domain and Range","courseName":"OpenStax: College Algebra","steps":[{"id":"aeb11eedomain32a","stepAnswer":["$$(-\\\\infty,-3) \\\\cup (-3,5) \\\\cup (5,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\frac{2x^3-250}{x^2-2x-15}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,-3) \\\\cup (-3,5) \\\\cup (5,\\\\infty)$$","choices":["$$(-\\\\infty,\\\\infty)$$","$$(-\\\\infty,5) \\\\cup (5,\\\\infty)$$","$$(-\\\\infty,-3) \\\\cup (5,\\\\infty)$$","$$(-\\\\infty,-3) \\\\cup (-3,5) \\\\cup (5,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain32a-h1","type":"hint","dependencies":[],"title":"Define the Domain","text":"A rational function is undefined when the denominator is equal to zero. So, let\'s start by setting the denominator equal to zero. Then we know every value other than that is a solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain32a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x=-3, 5$$"],"dependencies":["aeb11eedomain32a-h1"],"title":"Solving For the Domain","text":"What is $$x$$ for $$x^2-2x-15=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x=3$$","$$x=3, 5$$","$$x=-3, -5$$","$$x=-3, 5$$"],"subHints":[{"id":"aeb11eedomain32a-h2-s1","type":"hint","dependencies":[],"title":"Solving For the Domain","text":"For $$x^2-2x-15=0$$, factor the equation to get $$\\\\left(x-5\\\\right) \\\\left(x+3\\\\right)$$. Set these two expressions equal to zero to get that $$x=5$$ or $$x=-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aeb11eedomain32a-h3","type":"hint","dependencies":["aeb11eedomain32a-h2"],"title":"Answer","text":"So, the function is defined for every value other than $$-3$$ and 5.The domain of the function is $$(-\\\\infty,-3) \\\\cup (-3,5) \\\\cup (5,\\\\infty)$$, with the circle brackets denoting the fact that $$-3$$ and $$5$$ are not valid solutions, and the U showing that values within the intervals are solutions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb11eedomain33","title":"Find the domain of the function using interval notation.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Domain and Range","courseName":"OpenStax: College Algebra","steps":[{"id":"aeb11eedomain33a","stepAnswer":["$$(3,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{5}{\\\\sqrt{x-3}}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(3,\\\\infty)$$","choices":["$$(-\\\\infty,\\\\infty)$$","$$(-\\\\infty,3)$$","$$(3,\\\\infty)$$","$$(-\\\\infty,3) \\\\cup (3,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain33a-h1","type":"hint","dependencies":[],"title":"Define the Domain","text":"A rational function is undefined when the denominator is equal to zero. Additionally, the denominator is a square root, so the expression under the square root cannot be negative. So, let\'s start by setting the denominator greater than zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain33a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["aeb11eedomain33a-h1"],"title":"Solving For the Domain","text":"What value does $$x$$ have to be greater than to make $$x-3>0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aeb11eedomain33a-h2-s1","type":"hint","dependencies":[],"title":"Solving For the Domain","text":"For $$x-3>0$$, add $$3$$ to both sides to get $$x>3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aeb11eedomain33a-h3","type":"hint","dependencies":["aeb11eedomain33a-h2"],"title":"Answer","text":"So, the function is defined for every value greater than 3.The domain of the function is $$(3,\\\\infty)$$, with the circle brackets denoting the fact that $$3$$ is not a valid solution.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb11eedomain4","title":"Finding the Domain of a Function With an Even Root","body":"Find the domain of the function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Domain and Range","courseName":"OpenStax: College Algebra","steps":[{"id":"aeb11eedomain4a","stepAnswer":["$$(-\\\\infty,7]$$"],"problemType":"MultipleChoice","stepTitle":"$$f(x)=\\\\sqrt{7-x}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,7]$$","choices":["$$(-\\\\infty,7]$$","$$(-\\\\infty,7)$$","$$[-7,\\\\infty)$$","$$(-7,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain4a-h1","type":"hint","dependencies":[],"title":"Dealing with Square Root","text":"When there is an even root in the formula, we exclude any real numbers that result in a negative number in the radicand.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain4a-h2","type":"hint","dependencies":["aeb11eedomain4a-h1"],"title":"Solving for $$x$$ When the Square Root is $$0$$","text":"The first step is to set the square root to greater than or equal to to zero and solve for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain4a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x \\\\leq 7$$"],"dependencies":["aeb11eedomain4a-h2"],"title":"Equation for When the Square Root is $$0$$","text":"$$(7-x) \\\\geq 0;$$ what is the inequality for $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x \\\\geq 7$$","$$x \\\\leq -7$$","$$x \\\\geq -7$$","$$x \\\\leq 7$$"]},{"id":"aeb11eedomain4a-h4","type":"hint","dependencies":["aeb11eedomain4a-h3"],"title":"Answer","text":"Now, we will exclude any number greater than $$7$$ from the domain. The answers are all real numbers less than or equal to $$7$$, or $$(-\\\\infty,7]$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb11eedomain5","title":"Describing Sets on the Real-Number Line","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Domain and Range","courseName":"OpenStax: College Algebra","steps":[{"id":"aeb11eedomain5a","stepAnswer":["$$1 \\\\leq x \\\\leq 3$$ or $$x>5$$"],"problemType":"MultipleChoice","stepTitle":"Describe the intervals of values in the image using inequality notation.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$1 \\\\leq x \\\\leq 3$$ or $$x>5$$","choices":["$$1<x<3$$ or $$x>5$$","$$1<x<3$$ or $$x \\\\geq 5$$","$$1 \\\\leq x \\\\leq 3$$ or $$x \\\\geq 5$$","$$1 \\\\leq x \\\\leq 3$$ or $$x>5$$"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain5a-h1","type":"hint","dependencies":[],"title":"Describing the Values","text":"To describe the values, $$x$$, included in the intervals shown, we would say, \u201cx is a real number greater than or equal to $$1$$ and less than or equal to $$3$$, or a real number greater than 5.\u201d","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain5a-h2","type":"hint","dependencies":["aeb11eedomain5a-h1"],"title":"Translate to Math Notation","text":"The above statement, translated into inequality notations, is $$1 \\\\leq x \\\\leq 3$$ or $$x>5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aeb11eedomain5b","stepAnswer":["{$$x|1 \\\\leq x \\\\leq 3$$ or $$x>5$$}"],"problemType":"MultipleChoice","stepTitle":"Describe the intervals of values in the image using set-builder notation.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"{$$x|1 \\\\leq x \\\\leq 3$$ or $$x>5$$}","choices":["{$$x|1<x<3$$ or $$x>5$$}","{$$x|1<x<3$$ or $$x \\\\geq 5$$}","{$$x|1 \\\\leq x \\\\leq 3$$ or $$x>5$$}","{$$x|1 \\\\leq x \\\\leq 3$$ or $$x \\\\geq 5$$}","{$$x|1 \\\\leq x \\\\leq 3$$ or $$x>5$$}"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain5b-h1","type":"hint","dependencies":[],"title":"Describing the Values","text":"To describe the values, $$x$$, included in the intervals shown, we would say, \u201cx is a real number greater than or equal to $$1$$ and less than or equal to $$3$$, or a real number greater than 5.\u201d","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain5b-h2","type":"hint","dependencies":[],"title":"Defining Set-Builder Notation","text":"The set-buidler notation takes the form {x|statement about x}, which is read as, \u201cthe set of all $$x$$ such that the statement about $$x$$ is true.\u201d","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain5b-h3","type":"hint","dependencies":["aeb11eedomain5b-h1","aeb11eedomain5b-h2"],"title":"Translate to Math Notation","text":"The above statement, translated into set-builder notation, is {$$x|1 \\\\leq x \\\\leq 3$$ or $$x>5$$.}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aeb11eedomain5c","stepAnswer":["$$[1,3] \\\\cup (5,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"Describe the intervals of values in the image using interval notation.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[1,3] \\\\cup (5,\\\\infty)$$","choices":["$$(1,3) \\\\cup [5,\\\\infty)$$","$$(1,3) \\\\cup (5,\\\\infty)$$","$$[1,3] \\\\cup [5,\\\\infty)$$","$$[1,3] \\\\cup (5,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain5c-h1","type":"hint","dependencies":[],"title":"Describing the Values","text":"To describe the values, $$x$$, included in the intervals shown, we would say, \u201cx is a real number greater than or equal to $$1$$ and less than or equal to $$3$$, or a real number greater than 5.\u201d","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain5c-h2","type":"hint","dependencies":["aeb11eedomain5c-h1"],"title":"Translate to Math Notation","text":"The above statement, translated into interval notations, is $$[1,3] \\\\cup (5,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb11eedomain6","title":"Finding Domain and Range from a Graph","body":"Find the domain and range of the function, f, shown in the image.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Domain and Range","courseName":"OpenStax: College Algebra","steps":[{"id":"aeb11eedomain6a","stepAnswer":["$$(-3,1]$$"],"problemType":"MultipleChoice","stepTitle":"Find the domain.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$[-3,1)$$","$$(-3,1]$$","$$[-4,0)$$","$$[-4,0]$$"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain6a-h1","type":"hint","dependencies":[],"title":"Horizontal Extent of Graph","text":"We can observe that the horizontal extent of the graph is $$-3$$ to $$1$$. $$-3$$ is not included, and $$1$$ and included.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain6a-h2","type":"hint","dependencies":["aeb11eedomain6a-h1"],"title":"Answer","text":"The domain of f is (-3,1].","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aeb11eedomain6b","stepAnswer":["$$[-4,0)$$"],"problemType":"MultipleChoice","stepTitle":"Find the range.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$(-3,1]$$","$$[-4,0)$$","$$[-4,0]$$","$$[-3,1)$$","$$[-4,0)$$"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain6b-h1","type":"hint","dependencies":[],"title":"Vertical Extent of Graph","text":"We can observe that the vertical extent of the graph is $$0$$ to $$-4$$. $$0$$ is not included, and $$-4$$ is included.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain6b-h2","type":"hint","dependencies":["aeb11eedomain6b-h1"],"title":"Answer","text":"The range is [-4,0).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb11eedomain7","title":"Finding Domain and Range from a Graph of Oil Production","body":"Find the domain and range of the function f whose graph is shown in the image.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Domain and Range","courseName":"OpenStax: College Algebra","steps":[{"id":"aeb11eedomain7a","stepAnswer":["$$1973 \\\\leq t \\\\leq 2008$$"],"problemType":"MultipleChoice","stepTitle":"Find the domain.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$1973 \\\\leq t \\\\leq 2008$$","choices":["$$1980 \\\\leq t \\\\leq 2000$$","$$1973 \\\\leq t \\\\leq 2008$$","$$180 \\\\leq t \\\\leq 2010$$","$$0 \\\\leq t \\\\leq 2200$$"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain7a-h1","type":"hint","dependencies":[],"title":"Representing the Horizontal Axis","text":"The input quantity along the horizontal axis is \u201cyears,\u201d which we represent with the variable $$t$$ for time.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain7a-h2","type":"hint","dependencies":["aeb11eedomain7a-h1"],"title":"Answer","text":"The graph may continue to the left and right beyond what is viewed, but based on the portion of the graph that is visible, we can determine the domain as $$1973 \\\\leq t \\\\leq 2008$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aeb11eedomain7b","stepAnswer":["$$180 \\\\leq b \\\\leq 2010$$"],"problemType":"MultipleChoice","stepTitle":"Find the range.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$180 \\\\leq b \\\\leq 2010$$","choices":["$$1980 \\\\leq b \\\\leq 2000$$","$$1973 \\\\leq b \\\\leq 2008$$","$$180 \\\\leq b \\\\leq 2010$$","$$0 \\\\leq b \\\\leq 2200$$"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain7b-h1","type":"hint","dependencies":[],"title":"Representing the Vertical Axis","text":"The output quantity is \u201cthousands of barrels of oil per day,\u201d which we represent with the variable $$b$$ for barrels.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain7b-h2","type":"hint","dependencies":["aeb11eedomain7b-h1"],"title":"Answer","text":"The graph may continue to the left and right beyond what is viewed, but based on the portion of the graph that is visible, we can determine the range as approximately $$180 \\\\leq b \\\\leq 2010$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb11eedomain8","title":"Finding the Domain and Range Using Toolkit Functions","body":"Find the domain and range of $$f(x)=2x^3-x$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Domain and Range","courseName":"OpenStax: College Algebra","steps":[{"id":"aeb11eedomain8a","stepAnswer":["(-inf,inf)"],"problemType":"TextBox","stepTitle":"Find the domain.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\infty)$$","hints":{"DefaultPathway":[{"id":"aeb11eedomain8a-h1","type":"hint","dependencies":[],"title":"Domain Restrictions","text":"There are no restrictions on the domain, as any real number may be cubed and then subtracted from the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain8a-h2","type":"hint","dependencies":["aeb11eedomain8a-h1"],"title":"Answer","text":"The domain is $$(-\\\\infty,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aeb11eedomain8b","stepAnswer":["(-inf,inf)"],"problemType":"TextBox","stepTitle":"Find the range","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\infty)$$","hints":{"DefaultPathway":[{"id":"aeb11eedomain8b-h1","type":"hint","dependencies":[],"title":"Range Restrictions","text":"Since there are no restrictions on the range, the range is $$(-\\\\infty,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb11eedomain9","title":"Finding the Domain and Range","body":"Find the domain and range of $$f(x)=\\\\frac{2}{x+1}$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.2 Domain and Range","courseName":"OpenStax: College Algebra","steps":[{"id":"aeb11eedomain9a","stepAnswer":["$$(-\\\\infty,-1) \\\\cup (-1,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"Find the domain.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,-1) \\\\cup (-1,\\\\infty)$$","choices":["$$(-\\\\infty,\\\\infty)$$","$$(-\\\\infty,-1)$$","$$(1,\\\\infty)$$","$$(-\\\\infty,-1) \\\\cup (-1,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain9a-h1","type":"hint","dependencies":[],"title":"Domain Restrictions","text":"We cannot evaluate the function at $$-1$$ because division by zero is undefined.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain9a-h2","type":"hint","dependencies":["aeb11eedomain9a-h1"],"title":"Answer","text":"The domain $$is(-\\\\infty,-1) \\\\cup (-1,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aeb11eedomain9b","stepAnswer":["$$(-\\\\infty,0) \\\\cup (0,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"Find the range.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,0) \\\\cup (0,\\\\infty)$$","choices":["$$(-\\\\infty,\\\\infty)$$","$$(-\\\\infty,0)$$","$$(0,\\\\infty)$$","$$(-\\\\infty,0) \\\\cup (0,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"aeb11eedomain9b-h1","type":"hint","dependencies":[],"title":"Range Restrictions","text":"The function is never $$0$$, so we exclude $$0$$ from the range.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb11eedomain9b-h2","type":"hint","dependencies":["aeb11eedomain9b-h1"],"title":"Answer","text":"The range is $$(-\\\\infty,0) \\\\cup (0,\\\\infty)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb975brationaldenom1","title":"Writing Equivalent Rational Expressions With a Given LCD","body":"Rewrite the expressions as rational expressions given their least common denominator.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add and Subtract Rational Expressions with Unlike Denominators","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aeb975brationaldenom1a","stepAnswer":["$$\\\\frac{6a-45}{\\\\left(a+9\\\\right) \\\\left(a+5\\\\right) \\\\left(a-9\\\\right)}$$, (5a**2 + 25a)/((a+9)(a+5)(a-9))"],"problemType":"MultipleChoice","stepTitle":"6/(a**2 + 14a + 45), $$\\\\frac{5a}{a^2-81}$$, LCD $$\\\\left(a+9\\\\right) \\\\left(a+5\\\\right) \\\\left(a-9\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{6a-45}{\\\\left(a+9\\\\right) \\\\left(a+5\\\\right) \\\\left(a-9\\\\right)}$$, (5a**2 + 25a)/((a+9)(a+5)(a-9))","choices":["$$\\\\frac{6a-40}{\\\\left(a+9\\\\right) \\\\left(a+5\\\\right) \\\\left(a-9\\\\right)}$$, (5a + 25)/((a+9)(a+5)(a-9))","$$\\\\frac{6a-45}{\\\\left(a+9\\\\right) \\\\left(a+5\\\\right) \\\\left(a-9\\\\right)}$$, (5a**2 + 25a)/((a+9)(a+5)(a-9))","(6a $$-$$ 60)/((a+9)(a+5)(a-9)), (5a**2 $$-$$ 25a)/((a+9)(a+5)(a-9))"],"hints":{"DefaultPathway":[{"id":"aeb975brationaldenom1a-h1","type":"hint","dependencies":[],"title":"Rewriting Denominators","text":"First, factor each denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom1a-h2","type":"hint","dependencies":["aeb975brationaldenom1a-h1"],"title":"Finding the Missing Factor","text":"Find the missing factor in the denominators that would give them the given LCD (least common denominator.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom1a-h3","type":"hint","dependencies":["aeb975brationaldenom1a-h2"],"title":"Multiplying each Expression","text":"Multiply each denominator by the \'missing\' factor and multiply each numerator by the same factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom1a-h4","type":"hint","dependencies":["aeb975brationaldenom1a-h3"],"title":"Simplifying the Numerator","text":"Simplify the numerators by multiplying out their factors. For example, $$6\\\\left(a+3\\\\right)$$ $$=$$ 6a + $$18$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb975brationaldenom10","title":"Adding Rational Expressions","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add and Subtract Rational Expressions with Unlike Denominators","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aeb975brationaldenom10a","stepAnswer":["$$\\\\frac{21y+8x}{30x^2 y^2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{7}{10x^2 y}$$ + $$\\\\frac{4}{15{xy}^2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{21y+8x}{30x^2 y^2}$$","hints":{"DefaultPathway":[{"id":"aeb975brationaldenom10a-h1","type":"hint","dependencies":[],"title":"Finding the LCD","text":"First, find the least common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom10a-h2","type":"hint","dependencies":["aeb975brationaldenom10a-h1"],"title":"Rewriting the Expression","text":"Rewrite each fraction as an equivalent fraction with the LCD.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom10a-h3","type":"hint","dependencies":["aeb975brationaldenom10a-h2"],"title":"Combining Terms","text":"Then, add the fractions and simplify if applicable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb975brationaldenom11","title":"Adding Rational Expressions","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add and Subtract Rational Expressions with Unlike Denominators","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aeb975brationaldenom11a","stepAnswer":["$$\\\\frac{3b+20a}{36a^3 b^3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{1}{12a^3 b^2}+\\\\frac{5}{9a^2 b^3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3b+20a}{36a^3 b^3}$$","hints":{"DefaultPathway":[{"id":"aeb975brationaldenom11a-h1","type":"hint","dependencies":[],"title":"Finding the LCD","text":"First, find the least common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom11a-h2","type":"hint","dependencies":["aeb975brationaldenom11a-h1"],"title":"Rewriting the Expression","text":"Rewrite each fraction as an equivalent fraction with the LCD.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom11a-h3","type":"hint","dependencies":["aeb975brationaldenom11a-h2"],"title":"Combining Terms","text":"Then, add the fractions and simplify if applicable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb975brationaldenom12","title":"Adding Rational Expressions","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add and Subtract Rational Expressions with Unlike Denominators","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aeb975brationaldenom12a","stepAnswer":["$$\\\\frac{4mn+7}{8m^2 n}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{1}{2m}+\\\\frac{7}{8m^2 n}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{4mn+7}{8m^2 n}$$","hints":{"DefaultPathway":[{"id":"aeb975brationaldenom12a-h1","type":"hint","dependencies":[],"title":"Finding the LCD","text":"First, find the least common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom12a-h2","type":"hint","dependencies":["aeb975brationaldenom12a-h1"],"title":"Rewriting the Expression","text":"Rewrite each fraction as an equivalent fraction with the LCD.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom12a-h3","type":"hint","dependencies":["aeb975brationaldenom12a-h2"],"title":"Combining Terms","text":"Then, add the fractions and simplify if applicable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb975brationaldenom13","title":"Adding Rational Expressions","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add and Subtract Rational Expressions with Unlike Denominators","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aeb975brationaldenom13a","stepAnswer":["$$\\\\frac{5r-7}{\\\\left(r+4\\\\right) \\\\left(r-7\\\\right)}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{3}{r+4}+\\\\frac{2}{r-5}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5r-7}{\\\\left(r+4\\\\right) \\\\left(r-7\\\\right)}$$","hints":{"DefaultPathway":[{"id":"aeb975brationaldenom13a-h1","type":"hint","dependencies":[],"title":"Finding the LCD","text":"First, find the least common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom13a-h2","type":"hint","dependencies":["aeb975brationaldenom13a-h1"],"title":"Rewriting the Expression","text":"Rewrite each fraction as an equivalent fraction with the LCD.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom13a-h3","type":"hint","dependencies":["aeb975brationaldenom13a-h2"],"title":"Combining Terms","text":"Then, add the fractions and simplify if applicable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb975brationaldenom14","title":"Adding Rational Expressions","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add and Subtract Rational Expressions with Unlike Denominators","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aeb975brationaldenom14a","stepAnswer":["$$\\\\frac{3\\\\left(n+6\\\\right)}{4\\\\left(n+1\\\\right) \\\\left(n-2\\\\right)}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{3}{4n+4}+\\\\frac{6}{n^2-n-2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3\\\\left(n+6\\\\right)}{4\\\\left(n+1\\\\right) \\\\left(n-2\\\\right)}$$","hints":{"DefaultPathway":[{"id":"aeb975brationaldenom14a-h1","type":"hint","dependencies":[],"title":"Finding the LCD","text":"First, find the least common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom14a-h2","type":"hint","dependencies":["aeb975brationaldenom14a-h1"],"title":"Rewriting the Expression","text":"Rewrite each fraction as an equivalent fraction with the LCD.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom14a-h3","type":"hint","dependencies":["aeb975brationaldenom14a-h2"],"title":"Combining Terms","text":"Then, add the fractions and simplify if applicable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb975brationaldenom15","title":"Adding Rational Expressions","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add and Subtract Rational Expressions with Unlike Denominators","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aeb975brationaldenom15a","stepAnswer":["$$\\\\frac{4\\\\left(8x+1\\\\right)}{10x-1}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2x+7}{10x-1}+3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{4\\\\left(8x+1\\\\right)}{10x-1}$$","hints":{"DefaultPathway":[{"id":"aeb975brationaldenom15a-h1","type":"hint","dependencies":[],"title":"Finding the LCD","text":"First, find the least common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom15a-h2","type":"hint","dependencies":["aeb975brationaldenom15a-h1"],"title":"Rewriting the Expression","text":"Rewrite each fraction as an equivalent fraction with the LCD.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom15a-h3","type":"hint","dependencies":["aeb975brationaldenom15a-h2"],"title":"Combining Terms","text":"Then, add the fractions and simplify if applicable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb975brationaldenom16","title":"Adding Rational Expressions with Different Denominators\\\\n","body":"Solve the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add and Subtract Rational Expressions with Unlike Denominators","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aeb975brationaldenom16a","stepAnswer":["$$\\\\frac{31}{36}$$"],"problemType":"TextBox","stepTitle":"Add $$\\\\frac{7}{12}$$ + $$\\\\frac{5}{18}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{31}{36}$$","hints":{"DefaultPathway":[{"id":"aeb975brationaldenom16a-h1","type":"hint","dependencies":[],"title":"Creating a Common Denominator","text":"We must first find the least common denominator (LCD). The LCD of $$12$$ and $$18$$ is $$36$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom16a-h2","type":"hint","dependencies":["aeb975brationaldenom16a-h1"],"title":"Rewriting Each Fraction","text":"With the LCD of $$36$$, we can rewrite the numerators. We have $$\\\\frac{21}{36}$$ + $$\\\\frac{10}{36}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom16a-h3","type":"hint","dependencies":["aeb975brationaldenom16a-h2"],"title":"Adding the Fractions","text":"We can now add these fractions to get $$\\\\frac{31}{36}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb975brationaldenom17","title":"Adding Rational Expressions with Different Denominators\\\\n","body":"Solve the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add and Subtract Rational Expressions with Unlike Denominators","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aeb975brationaldenom17a","stepAnswer":["$$\\\\frac{57}{60}$$"],"problemType":"TextBox","stepTitle":"Add $$\\\\frac{11}{30}+\\\\frac{7}{12}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{57}{60}$$","hints":{"DefaultPathway":[{"id":"aeb975brationaldenom17a-h1","type":"hint","dependencies":[],"title":"Creating a Common Denominator","text":"The LCD of $$30$$ and $$12$$ is $$60$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom17a-h2","type":"hint","dependencies":["aeb975brationaldenom17a-h1"],"title":"Rewriting Each Fraction","text":"We must rewrite each fraction so that $$60$$ is the common denominator. We can use a constant multipler to do this. We now have $$\\\\frac{22}{60}+\\\\frac{35}{60}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom17a-h3","type":"hint","dependencies":["aeb975brationaldenom17a-h2"],"title":"Adding the Fractions","text":"Adding the numerators, we get $$\\\\frac{57}{60}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb975brationaldenom18","title":"Adding Rational Expressions with Different Denominators\\\\n","body":"Solve the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add and Subtract Rational Expressions with Unlike Denominators","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aeb975brationaldenom18a","stepAnswer":["$$\\\\frac{33}{40}$$"],"problemType":"TextBox","stepTitle":"Add $$\\\\frac{3}{8}+\\\\frac{9}{20}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{33}{40}$$","hints":{"DefaultPathway":[{"id":"aeb975brationaldenom18a-h1","type":"hint","dependencies":[],"title":"Creating a Common Denominator","text":"We must find the LCD of $$8$$ and $$20$$ in order to add the fractions. The LCD is $$40$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom18a-h2","type":"hint","dependencies":["aeb975brationaldenom18a-h1"],"title":"Rewriting Each Fraction","text":"We must use $$40$$ as the LCD to create two equivalent fractions. We get $$\\\\frac{15}{40}+\\\\frac{18}{40}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom18a-h3","type":"hint","dependencies":["aeb975brationaldenom18a-h2"],"title":"Adding the Fractions","text":"Adding the numerators, $$15$$ and $$18$$, we get $$\\\\frac{33}{40}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb975brationaldenom19","title":"Adding Rational Expressions with Different Denominators\\\\n","body":"Solve the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add and Subtract Rational Expressions with Unlike Denominators","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aeb975brationaldenom19a","stepAnswer":["$$\\\\frac{16x+35y}{84x^2 y^2}$$"],"problemType":"TextBox","stepTitle":"Add $$\\\\frac{5}{12x^2 y}+\\\\frac{4}{21{xy}^2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{16x+35y}{84x^2 y^2}$$","hints":{"DefaultPathway":[{"id":"aeb975brationaldenom19a-h1","type":"hint","dependencies":[],"title":"Creating a Common Denominator","text":"We can create fractions with a common denominator by finding the LCD, which is $$84x^2 y^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom19a-h2","type":"hint","dependencies":["aeb975brationaldenom19a-h1"],"title":"Rewriting Each Fraction","text":"We can rewrite each fraction with a common scaler to get $$\\\\frac{35y}{84x^2 y^2}+\\\\frac{16x}{84x^2 y^2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom19a-h3","type":"hint","dependencies":["aeb975brationaldenom19a-h2"],"title":"Adding the Fractions","text":"Adding the fractions we get $$\\\\frac{16x+35y}{84x^2 y^2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb975brationaldenom2","title":"Writing Equivalent Rational Expressions With a Given LCD","body":"Rewrite the expressions as rational expressions given their least common denominator.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add and Subtract Rational Expressions with Unlike Denominators","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aeb975brationaldenom2a","stepAnswer":["$$\\\\frac{4b-20}{\\\\left(b+3\\\\right) \\\\left(b+3\\\\right) \\\\left(b-5\\\\right)}$$, $$\\\\frac{2b^2+6b}{\\\\left(b+3\\\\right) \\\\left(b+3\\\\right) \\\\left(b-5\\\\right)}$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{4}{b^2+6b+9}$$, $$\\\\frac{2b}{b^2-2b-15}$$, LCD: $$\\\\left(b+3\\\\right) \\\\left(b+3\\\\right) \\\\left(b-5\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{4b-20}{\\\\left(b+3\\\\right) \\\\left(b+3\\\\right) \\\\left(b-5\\\\right)}$$, $$\\\\frac{2b^2+6b}{\\\\left(b+3\\\\right) \\\\left(b+3\\\\right) \\\\left(b-5\\\\right)}$$","choices":["$$\\\\frac{4b-20}{\\\\left(b+3\\\\right) \\\\left(b+3\\\\right) \\\\left(b-5\\\\right)}$$, $$\\\\frac{2b^2+6b}{\\\\left(b+3\\\\right) \\\\left(b+3\\\\right) \\\\left(b-5\\\\right)}$$","$$\\\\frac{b-2}{\\\\left(b+3\\\\right) \\\\left(b+3\\\\right) \\\\left(b-5\\\\right)}$$, $$\\\\frac{2b+5}{\\\\left(b+3\\\\right) \\\\left(b+3\\\\right) \\\\left(b-5\\\\right)}$$"],"hints":{"DefaultPathway":[{"id":"aeb975brationaldenom2a-h1","type":"hint","dependencies":[],"title":"Rewriting Denominators","text":"First, factor each denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom2a-h2","type":"hint","dependencies":["aeb975brationaldenom2a-h1"],"title":"Finding the Missing Factor","text":"Find the missing factor in the denominators that would give them the given LCD (least common denominator.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom2a-h3","type":"hint","dependencies":["aeb975brationaldenom2a-h2"],"title":"Multiplying each Expression","text":"Multiply each denominator by the \'missing\' factor and multiply each numerator by the same factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom2a-h4","type":"hint","dependencies":["aeb975brationaldenom2a-h3"],"title":"Simplifying the Numerator","text":"Simplify the numerators by multiplying out their factors. For example, $$6\\\\left(a+3\\\\right)$$ $$=$$ 6a + $$18$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb975brationaldenom20","title":"Adding Rational Expressions with Different Denominators\\\\n","body":"Solve the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add and Subtract Rational Expressions with Unlike Denominators","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aeb975brationaldenom20a","stepAnswer":["$$\\\\frac{4b+5a}{30a^2 b^2}$$"],"problemType":"TextBox","stepTitle":"Add $$\\\\frac{2}{15a^2 b}+\\\\frac{5}{6{ab}^2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{4b+5a}{30a^2 b^2}$$","hints":{"DefaultPathway":[{"id":"aeb975brationaldenom20a-h1","type":"hint","dependencies":[],"title":"Creating a Common Denominator","text":"We must find the LCD in order to add the fractions. The LCM of $$15a^2 b$$ and $$6{ab}^2$$ is $$30a^2 b^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom20a-h2","type":"hint","dependencies":["aeb975brationaldenom20a-h1"],"title":"Rewriting Each Fraction","text":"We can rewrite each fraction with a common scaler to get $$\\\\frac{4b}{30a^2 b^2}+\\\\frac{25a}{30a^2 b^2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom20a-h3","type":"hint","dependencies":["aeb975brationaldenom20a-h2"],"title":"Adding the Fractions","text":"Adding these two fractions, we get $$\\\\frac{4b+5a}{30a^2 b^2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb975brationaldenom21","title":"Adding Rational Expressions with Different Denominators\\\\n","body":"Solve the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add and Subtract Rational Expressions with Unlike Denominators","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aeb975brationaldenom21a","stepAnswer":["$$\\\\frac{5d^2+6}{16{cd}^2}$$"],"problemType":"TextBox","stepTitle":"Add $$\\\\frac{5}{16c}+\\\\frac{3}{8{cd}^2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5d^2+6}{16{cd}^2}$$","hints":{"DefaultPathway":[{"id":"aeb975brationaldenom21a-h1","type":"hint","dependencies":[],"title":"Creating a Common Denominator","text":"We must find the LCD in order to add the fractions. The LCM of 16c and $$8{cd}^2$$ is $$16{cd}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom21a-h2","type":"hint","dependencies":["aeb975brationaldenom21a-h1"],"title":"Rewriting Each Fraction","text":"We can rewrite each fraction with a common scaler to get $$\\\\frac{5d^2}{16{cd}^2}+\\\\frac{6}{16{cd}^2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom21a-h3","type":"hint","dependencies":["aeb975brationaldenom21a-h2"],"title":"Adding the Fractions","text":"Adding these two fractions, we get $$\\\\frac{5d^2+6}{16{cd}^2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb975brationaldenom22","title":"Adding Rational Expressions with Different Denominators\\\\n","body":"Solve the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add and Subtract Rational Expressions with Unlike Denominators","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aeb975brationaldenom22a","stepAnswer":["$$\\\\frac{3\\\\left(x-2\\\\right)+2\\\\left(x-3\\\\right)}{\\\\left(x-3\\\\right) \\\\left(x-2\\\\right)}$$"],"problemType":"TextBox","stepTitle":"Add $$\\\\frac{3}{x-3}+2\\\\left(x-2\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{3\\\\left(x-2\\\\right)+2\\\\left(x-3\\\\right)}{\\\\left(x-3\\\\right) \\\\left(x-2\\\\right)}$$","hints":{"DefaultPathway":[{"id":"aeb975brationaldenom22a-h1","type":"hint","dependencies":[],"title":"Creating a Common Denominator","text":"We must find the LCD in order to be able to add the fractions. The LCM of $$(x-3)$$ and $$(x-2)$$ is $$(x-3)(x-2)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom22a-h2","type":"hint","dependencies":["aeb975brationaldenom22a-h1"],"title":"Rewriting Each Fraction","text":"We can rewrite each fraction with a common scaler to get $$\\\\frac{3\\\\left(x-2\\\\right)}{\\\\left(x-3\\\\right) \\\\left(x-2\\\\right)}+\\\\frac{2\\\\left(x-3\\\\right)}{\\\\left(x-2\\\\right) \\\\left(x-3\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom22a-h3","type":"hint","dependencies":["aeb975brationaldenom22a-h2"],"title":"Adding the Fractions","text":"Adding these two fractions, we get $$\\\\frac{3\\\\left(x-2\\\\right)+2\\\\left(x-3\\\\right)}{\\\\left(x-3\\\\right) \\\\left(x-2\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb975brationaldenom23","title":"Adding Rational Expressions with Different Denominators\\\\n","body":"Solve the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add and Subtract Rational Expressions with Unlike Denominators","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aeb975brationaldenom23a","stepAnswer":["$$\\\\frac{2\\\\left(x+3\\\\right)+5\\\\left(x-2\\\\right)}{\\\\left(x-2\\\\right) \\\\left(x+3\\\\right)}$$"],"problemType":"TextBox","stepTitle":"Add $$\\\\frac{2}{x-2}+\\\\frac{5}{x-3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2\\\\left(x+3\\\\right)+5\\\\left(x-2\\\\right)}{\\\\left(x-2\\\\right) \\\\left(x+3\\\\right)}$$","hints":{"DefaultPathway":[{"id":"aeb975brationaldenom23a-h1","type":"hint","dependencies":[],"title":"Creating a Common Denominator","text":"We must find the LCD in order to be able to add the fractions. The LCM of $$x+3$$ and $$(x-2)$$ is $$\\\\left(x+3\\\\right) \\\\left(x-2\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom23a-h2","type":"hint","dependencies":["aeb975brationaldenom23a-h1"],"title":"Rewriting Each Fraction","text":"We can rewrite each fraction with a common scaler to get $$\\\\frac{2\\\\left(x+3\\\\right)}{\\\\left(x-2\\\\right) \\\\left(x+3\\\\right)}+\\\\frac{5\\\\left(x-2\\\\right)}{\\\\left(x-2\\\\right) \\\\left(x+3\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom23a-h3","type":"hint","dependencies":["aeb975brationaldenom23a-h2"],"title":"Adding the Fractions","text":"Adding the fractions, we get $$\\\\frac{2\\\\left(x+3\\\\right)+5\\\\left(x-2\\\\right)}{\\\\left(x-2\\\\right) \\\\left(x+3\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb975brationaldenom24","title":"Adding Rational Expressions with Different Denominators\\\\n","body":"Solve the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add and Subtract Rational Expressions with Unlike Denominators","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aeb975brationaldenom24a","stepAnswer":["$$\\\\frac{4\\\\left(m+4\\\\right)+3\\\\left(m+3\\\\right)}{\\\\left(m+3\\\\right) \\\\left(m+4\\\\right)}$$"],"problemType":"TextBox","stepTitle":"Add $$\\\\frac{4}{m+3}+\\\\frac{3}{m+4}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{4\\\\left(m+4\\\\right)+3\\\\left(m+3\\\\right)}{\\\\left(m+3\\\\right) \\\\left(m+4\\\\right)}$$","hints":{"DefaultPathway":[{"id":"aeb975brationaldenom24a-h1","type":"hint","dependencies":[],"title":"Creating a Common Denominator","text":"We must find the LCD in order to be able to add the fractions. The LCM of $$m+3$$ and $$m+4$$ is $$\\\\left(m+3\\\\right) \\\\left(m+4\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom24a-h2","type":"hint","dependencies":["aeb975brationaldenom24a-h1"],"title":"Rewriting Each Fraction","text":"We can rewrite each fraction with a common scaler to get $$\\\\frac{4\\\\left(m+4\\\\right)}{\\\\left(m+3\\\\right) \\\\left(m+4\\\\right)}+\\\\frac{3\\\\left(m+3\\\\right)}{\\\\left(m+3\\\\right) \\\\left(m+4\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom24a-h3","type":"hint","dependencies":["aeb975brationaldenom24a-h2"],"title":"Adding the Fractions","text":"Adding the fractions, we get $$\\\\frac{4\\\\left(m+4\\\\right)+3\\\\left(m+3\\\\right)}{\\\\left(m+3\\\\right) \\\\left(m+4\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb975brationaldenom25","title":"Adding Rational Expressions with Different Denominators\\\\n","body":"Solve the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add and Subtract Rational Expressions with Unlike Denominators","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aeb975brationaldenom25a","stepAnswer":["$$\\\\frac{\\\\frac{2a\\\\left(4a^2-b^2\\\\right)+3a\\\\left(2ab+b^2\\\\right)}{4\\\\left(m+4\\\\right)+3\\\\left(m+3\\\\right)}}{\\\\left(m+3\\\\right) \\\\left(m+4\\\\right)}$$"],"problemType":"TextBox","stepTitle":"Add $$\\\\frac{2a}{2ab+b^2}+\\\\frac{3a}{4a^2-b^2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{\\\\frac{2a\\\\left(4a^2-b^2\\\\right)+3a\\\\left(2ab+b^2\\\\right)}{4\\\\left(m+4\\\\right)+3\\\\left(m+3\\\\right)}}{\\\\left(m+3\\\\right) \\\\left(m+4\\\\right)}$$","hints":{"DefaultPathway":[{"id":"aeb975brationaldenom25a-h1","type":"hint","dependencies":[],"title":"Creating a Common Denominator","text":"We must find the LCD in order to be able to add the fractions. The LCM of $$2ab+b^2$$ and (4a**2-b* *2) is (2ab+b**2)(4a**2-b* *2)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom25a-h2","type":"hint","dependencies":["aeb975brationaldenom25a-h1"],"title":"Rewriting Each Fraction","text":"We can rewrite each fraction with a common scaler to get $$\\\\frac{2a\\\\left(4a^2-b^2\\\\right)}{\\\\left(2ab+b^2\\\\right) \\\\left(4a^2-b^2\\\\right)}+\\\\frac{3a\\\\left(2ab+b^2\\\\right)}{\\\\left(2ab+b^2\\\\right) \\\\left(4a^2-b^2\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom25a-h3","type":"hint","dependencies":["aeb975brationaldenom25a-h2"],"title":"Adding the Fractions","text":"Adding the fractions, we get $$\\\\frac{\\\\frac{2a\\\\left(4a^2-b^2\\\\right)+3a\\\\left(2ab+b^2\\\\right)}{4\\\\left(m+4\\\\right)+3\\\\left(m+3\\\\right)}}{\\\\left(m+3\\\\right) \\\\left(m+4\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb975brationaldenom26","title":"Adding Rational Expressions with Different Denominators\\\\n","body":"Solve the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add and Subtract Rational Expressions with Unlike Denominators","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aeb975brationaldenom26a","stepAnswer":["$$\\\\frac{5x\\\\left(x^2-y^2\\\\right)+2x\\\\left(xy-y^2\\\\right)}{\\\\left(xy-y^2\\\\right) \\\\left(x^2-y^2\\\\right)}$$"],"problemType":"TextBox","stepTitle":"Add $$\\\\frac{5x}{xy-y^2}+\\\\frac{2x}{x^2-y^2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5x\\\\left(x^2-y^2\\\\right)+2x\\\\left(xy-y^2\\\\right)}{\\\\left(xy-y^2\\\\right) \\\\left(x^2-y^2\\\\right)}$$","hints":{"DefaultPathway":[{"id":"aeb975brationaldenom26a-h1","type":"hint","dependencies":[],"title":"Creating a Common Denominator","text":"We must find the LCD in order to be able to add the fractions. The LCM of $$xy-y^2$$ and $$x^2-y^2$$ is $$\\\\left(xy-y^2\\\\right) \\\\left(x^2-y^2\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom26a-h2","type":"hint","dependencies":["aeb975brationaldenom26a-h1"],"title":"Rewriting Each Fraction","text":"We can rewrite each fraction with a common scaler to get $$\\\\frac{5x\\\\left(x^2-y^2\\\\right)}{\\\\left(xy-y^2\\\\right) \\\\left(x^2-y^2\\\\right)}+\\\\frac{2x\\\\left(xy-y^2\\\\right)}{\\\\left(xy-y^2\\\\right) \\\\left(x^2-y^2\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom26a-h3","type":"hint","dependencies":["aeb975brationaldenom26a-h2"],"title":"Adding the Fractions","text":"Adding the fractions, we get $$\\\\frac{5x\\\\left(x^2-y^2\\\\right)+2x\\\\left(xy-y^2\\\\right)}{\\\\left(xy-y^2\\\\right) \\\\left(x^2-y^2\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb975brationaldenom27","title":"Adding Rational Expressions with Different Denominators\\\\n","body":"Solve the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add and Subtract Rational Expressions with Unlike Denominators","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aeb975brationaldenom27a","stepAnswer":["$$\\\\frac{7m+15}{2\\\\left(m+3\\\\right) \\\\left(m+1\\\\right)}$$"],"problemType":"TextBox","stepTitle":"Add $$\\\\frac{7}{2m+6}+\\\\frac{4}{m^2+4m+3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{7m+15}{2\\\\left(m+3\\\\right) \\\\left(m+1\\\\right)}$$","hints":{"DefaultPathway":[{"id":"aeb975brationaldenom27a-h1","type":"hint","dependencies":[],"title":"Creating a Common Denominator","text":"We must find the LCD in order to be able to add the fractions. The LCM of $$2m+6$$ and $$m^2+4m+3$$ is $$2\\\\left(m+3\\\\right) \\\\left(m+1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom27a-h2","type":"hint","dependencies":["aeb975brationaldenom27a-h1"],"title":"Rewriting Each Fraction","text":"We can rewrite each fraction with a common scaler to get $$\\\\frac{7\\\\left(m+1\\\\right)}{2\\\\left(m+3\\\\right) \\\\left(m+1\\\\right)}+\\\\frac{8}{2\\\\left(m+3\\\\right) \\\\left(m+1\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom27a-h3","type":"hint","dependencies":["aeb975brationaldenom27a-h2"],"title":"Adding the Fractions","text":"Adding the fractions, we get $$\\\\frac{7m+15}{2\\\\left(m+3\\\\right) \\\\left(m+1\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb975brationaldenom28","title":"Adding Rational Expressions with Different Denominators\\\\n","body":"Solve the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add and Subtract Rational Expressions with Unlike Denominators","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aeb975brationaldenom28a","stepAnswer":["$$\\\\frac{8x+24+3x^2-9x}{\\\\left(x+1\\\\right) \\\\left(x-3\\\\right) \\\\left(x+3\\\\right)}$$"],"problemType":"TextBox","stepTitle":"Add $$\\\\frac{8}{x^2-2x-3}+\\\\frac{3x}{x^2+4x+3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{8x+24+3x^2-9x}{\\\\left(x+1\\\\right) \\\\left(x-3\\\\right) \\\\left(x+3\\\\right)}$$","hints":{"DefaultPathway":[{"id":"aeb975brationaldenom28a-h1","type":"hint","dependencies":[],"title":"Creating a Common Denominator","text":"We must find the LCD in order to be able to add the fractions. The LCM of $$x^2-2x-3$$ and $$x^2+4x+3$$ is $$\\\\left(x+1\\\\right) \\\\left(x-3\\\\right) \\\\left(x+3\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom28a-h2","type":"hint","dependencies":["aeb975brationaldenom28a-h1"],"title":"Rewriting Each Fraction","text":"We can rewrite each fraction with a common scaler to get $$\\\\frac{8\\\\left(x+3\\\\right)}{\\\\left(x+1\\\\right) \\\\left(x-3\\\\right) \\\\left(x+3\\\\right)}$$ + $$\\\\frac{3x\\\\left(x-3\\\\right)}{\\\\left(x+1\\\\right) \\\\left(x+3\\\\right) \\\\left(x-3\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom28a-h3","type":"hint","dependencies":["aeb975brationaldenom28a-h2"],"title":"Adding the Fractions","text":"Adding the fractions, we get $$\\\\frac{8x+24+3x^2-9x}{\\\\left(x+1\\\\right) \\\\left(x-3\\\\right) \\\\left(x+3\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb975brationaldenom29","title":"Adding Rational Expressions with Different Denominators\\\\n","body":"Solve the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add and Subtract Rational Expressions with Unlike Denominators","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aeb975brationaldenom29a","stepAnswer":["$$\\\\frac{m+2+5m\\\\left(m-2\\\\right)}{\\\\left(m+1\\\\right) \\\\left(m-2\\\\right) \\\\left(m+2\\\\right)}$$"],"problemType":"TextBox","stepTitle":"Add $$\\\\frac{1}{m^2-m-2}$$ + $$\\\\frac{5m}{m^2+3m+2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{m+2+5m\\\\left(m-2\\\\right)}{\\\\left(m+1\\\\right) \\\\left(m-2\\\\right) \\\\left(m+2\\\\right)}$$","hints":{"DefaultPathway":[{"id":"aeb975brationaldenom29a-h1","type":"hint","dependencies":[],"title":"Creating a Common Denominator","text":"We must find the LCD in order to be able to add the fractions. The LCM of $$m^2-m-2$$ and $$m^2+3m+2$$ is $$\\\\left(m+1\\\\right) \\\\left(m-2\\\\right) \\\\left(m+2\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom29a-h2","type":"hint","dependencies":["aeb975brationaldenom29a-h1"],"title":"Rewriting Each Fraction","text":"We can rewrite each fraction with a common scaler to get $$\\\\frac{m+2}{\\\\left(m+1\\\\right) \\\\left(m-2\\\\right) \\\\left(m+1\\\\right)}$$ + $$\\\\frac{5m\\\\left(m-2\\\\right)}{\\\\left(m+1\\\\right) \\\\left(m-2\\\\right) \\\\left(m+1\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom29a-h3","type":"hint","dependencies":["aeb975brationaldenom29a-h2"],"title":"Adding the Fractions","text":"Adding the fractions, we get $$\\\\frac{m+2+5m\\\\left(m-2\\\\right)}{\\\\left(m+1\\\\right) \\\\left(m-2\\\\right) \\\\left(m+2\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb975brationaldenom3","title":"Writing Equivalent Rational Expressions With a Given LCD","body":"Rewrite the expressions as rational expressions given their least common denominator.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add and Subtract Rational Expressions with Unlike Denominators","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aeb975brationaldenom3a","stepAnswer":["$$\\\\frac{5c-40}{\\\\left(c-2\\\\right) \\\\left(c-2\\\\right) \\\\left(c-8\\\\right)}$$, $$\\\\frac{3c^2-6c}{\\\\left(c-2\\\\right) \\\\left(c-2\\\\right) \\\\left(c-8\\\\right)}$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{5}{c^2-4c+4}$$, $$\\\\frac{3c}{c^2-10c+16}$$, LCD: $$(c-2)(c-2)(c-8)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{5c-40}{\\\\left(c-2\\\\right) \\\\left(c-2\\\\right) \\\\left(c-8\\\\right)}$$, $$\\\\frac{3c^2-6c}{\\\\left(c-2\\\\right) \\\\left(c-2\\\\right) \\\\left(c-8\\\\right)}$$","choices":["$$\\\\frac{5c-4}{\\\\left(c-2\\\\right) \\\\left(c-2\\\\right) \\\\left(c-8\\\\right)}$$, $$\\\\frac{c^2-6c}{\\\\left(c-2\\\\right) \\\\left(c-2\\\\right) \\\\left(c-8\\\\right)}$$","$$\\\\frac{5c-40}{\\\\left(c-2\\\\right) \\\\left(c-2\\\\right) \\\\left(c-8\\\\right)}$$, $$\\\\frac{3c^2-6c}{\\\\left(c-2\\\\right) \\\\left(c-2\\\\right) \\\\left(c-8\\\\right)}$$","$$\\\\frac{5c-40}{\\\\left(c-2\\\\right) \\\\left(c-2\\\\right) \\\\left(c-8\\\\right)}$$, $$\\\\frac{4c^2-8c}{\\\\left(c-2\\\\right) \\\\left(c-2\\\\right) \\\\left(c-8\\\\right)}$$"],"hints":{"DefaultPathway":[{"id":"aeb975brationaldenom3a-h1","type":"hint","dependencies":[],"title":"Rewriting Denominators","text":"First, factor each denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom3a-h2","type":"hint","dependencies":["aeb975brationaldenom3a-h1"],"title":"Finding the Missing Factor","text":"Find the missing factor in the denominators that would give them the given LCD (least common denominator.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom3a-h3","type":"hint","dependencies":["aeb975brationaldenom3a-h2"],"title":"Multiplying each Expression","text":"Multiply each denominator by the \'missing\' factor and multiply each numerator by the same factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom3a-h4","type":"hint","dependencies":["aeb975brationaldenom3a-h3"],"title":"Simplifying the Numerator","text":"Simplify the numerators by multiplying out their factors. For example, $$6\\\\left(a+3\\\\right)$$ $$=$$ 6a + $$18$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb975brationaldenom30","title":"Adding Rational Expressions with Different Denominators\\\\n","body":"Solve the expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add and Subtract Rational Expressions with Unlike Denominators","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aeb975brationaldenom30a","stepAnswer":["$$\\\\frac{2n^2+12n-30}{\\\\left(n+2\\\\right) \\\\left(n-5\\\\right) \\\\left(n+3\\\\right)}$$"],"problemType":"TextBox","stepTitle":"Add $$\\\\frac{2n}{n^2-3n-10}$$ + $$\\\\frac{6}{n^2+5n+6}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2n^2+12n-30}{\\\\left(n+2\\\\right) \\\\left(n-5\\\\right) \\\\left(n+3\\\\right)}$$","hints":{"DefaultPathway":[{"id":"aeb975brationaldenom30a-h1","type":"hint","dependencies":[],"title":"Creating a Common Denominator","text":"We must find the LCD in order to be able to add the fractions. The LCM of $$n^2-3n-10$$ and $$n^2+5n+6$$ is $$\\\\left(n+2\\\\right) \\\\left(n-5\\\\right) \\\\left(n+3\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom30a-h2","type":"hint","dependencies":["aeb975brationaldenom30a-h1"],"title":"Rewriting Each Fraction","text":"We can rewrite each fraction with a common scaler to get $$\\\\frac{2n\\\\left(n+3\\\\right)}{\\\\left(n+2\\\\right) \\\\left(n-5\\\\right) \\\\left(n+3\\\\right)}$$ + $$\\\\frac{6\\\\left(n-5\\\\right)}{\\\\left(n+2\\\\right) \\\\left(n-5\\\\right) \\\\left(n+3\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom30a-h3","type":"hint","dependencies":["aeb975brationaldenom30a-h2"],"title":"Adding the Fractions","text":"Adding the fractions, we get $$\\\\frac{2n^2+12n-30}{\\\\left(n+2\\\\right) \\\\left(n-5\\\\right) \\\\left(n+3\\\\right)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb975brationaldenom4","title":"Writing Equivalent Rational Expressions With a Given LCD","body":"Rewrite the expressions as rational expressions given their least common denominator.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add and Subtract Rational Expressions with Unlike Denominators","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aeb975brationaldenom4a","stepAnswer":["$$\\\\frac{2d-12}{\\\\left(3d-1\\\\right) \\\\left(d+5\\\\right) \\\\left(d-6\\\\right)}$$, $$\\\\frac{5d^2+25d}{\\\\left(3d-1\\\\right) \\\\left(d+5\\\\right) \\\\left(d-6\\\\right)}$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{2}{3d^2+14d-5}$$, $$\\\\frac{5d}{3d^2-19d+6}$$ LCD: $$\\\\left(3d-1\\\\right) \\\\left(d+5\\\\right) \\\\left(d-6\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{2d-12}{\\\\left(3d-1\\\\right) \\\\left(d+5\\\\right) \\\\left(d-6\\\\right)}$$, $$\\\\frac{5d^2+25d}{\\\\left(3d-1\\\\right) \\\\left(d+5\\\\right) \\\\left(d-6\\\\right)}$$","choices":["$$\\\\frac{d-10}{\\\\left(3d-1\\\\right) \\\\left(d+5\\\\right) \\\\left(d-6\\\\right)}$$, $$\\\\frac{4d^2+16d}{\\\\left(3d-1\\\\right) \\\\left(d+5\\\\right) \\\\left(d-6\\\\right)}$$","$$\\\\frac{2d-12}{\\\\left(3d-1\\\\right) \\\\left(d+5\\\\right) \\\\left(d-6\\\\right)}$$, $$\\\\frac{5d^2+25d}{\\\\left(3d-1\\\\right) \\\\left(d+5\\\\right) \\\\left(d-6\\\\right)}$$","$$\\\\frac{d-6}{\\\\left(3d-1\\\\right) \\\\left(d+5\\\\right) \\\\left(d-6\\\\right)}$$, $$\\\\frac{3d^2+9d}{\\\\left(3d-1\\\\right) \\\\left(d+5\\\\right) \\\\left(d-6\\\\right)}$$"],"hints":{"DefaultPathway":[{"id":"aeb975brationaldenom4a-h1","type":"hint","dependencies":[],"title":"Rewriting Denominators","text":"First, factor each denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom4a-h2","type":"hint","dependencies":["aeb975brationaldenom4a-h1"],"title":"Finding the Missing Factor","text":"Find the missing factor in the denominators that would give them the given LCD (least common denominator.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom4a-h3","type":"hint","dependencies":["aeb975brationaldenom4a-h2"],"title":"Multiplying each Expression","text":"Multiply each denominator by the \'missing\' factor and multiply each numerator by the same factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom4a-h4","type":"hint","dependencies":["aeb975brationaldenom4a-h3"],"title":"Simplifying the Numerator","text":"Simplify the numerators by multiplying out their factors. For example, $$6\\\\left(a+3\\\\right)$$ $$=$$ 6a + $$18$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb975brationaldenom5","title":"Writing Equivalent Rational Expressions With a Given LCD","body":"Rewrite the expressions as rational expressions given their least common denominator.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add and Subtract Rational Expressions with Unlike Denominators","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aeb975brationaldenom5a","stepAnswer":["$$\\\\frac{3m+9}{\\\\left(5m+2\\\\right) \\\\left(m-1\\\\right) \\\\left(m+3\\\\right)}$$, $$\\\\frac{6m^2-6m}{\\\\left(5m+2\\\\right) \\\\left(m-1\\\\right) \\\\left(m+3\\\\right)}$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{3}{5m^2-3m-2}$$, $$\\\\frac{6m}{5m^2+17m+6}$$ LCD: $$\\\\left(5m+2\\\\right) \\\\left(m-1\\\\right) \\\\left(m+3\\\\right)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{3m+9}{\\\\left(5m+2\\\\right) \\\\left(m-1\\\\right) \\\\left(m+3\\\\right)}$$, $$\\\\frac{6m^2-6m}{\\\\left(5m+2\\\\right) \\\\left(m-1\\\\right) \\\\left(m+3\\\\right)}$$","choices":["$$\\\\frac{3m+9}{\\\\left(5m+2\\\\right) \\\\left(m-1\\\\right) \\\\left(m+3\\\\right)}$$, $$\\\\frac{6m^2-6m}{\\\\left(5m+2\\\\right) \\\\left(m-1\\\\right) \\\\left(m+3\\\\right)}$$","$$\\\\frac{5m+4}{\\\\left(5m+2\\\\right) \\\\left(m-1\\\\right) \\\\left(m+3\\\\right)}$$, $$\\\\frac{4m^2-8m}{\\\\left(5m+2\\\\right) \\\\left(m-1\\\\right) \\\\left(m+3\\\\right)}$$","$$\\\\frac{2m+6}{\\\\left(5m+2\\\\right) \\\\left(m-1\\\\right) \\\\left(m+3\\\\right)}$$, $$\\\\frac{6m^2-6m}{\\\\left(5m+2\\\\right) \\\\left(m-1\\\\right) \\\\left(m+3\\\\right)}$$"],"hints":{"DefaultPathway":[{"id":"aeb975brationaldenom5a-h1","type":"hint","dependencies":[],"title":"Rewriting Denominators","text":"First, factor each denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom5a-h2","type":"hint","dependencies":["aeb975brationaldenom5a-h1"],"title":"Finding the Missing Factor","text":"Find the missing factor in the denominators that would give them the given LCD (least common denominator.)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom5a-h3","type":"hint","dependencies":["aeb975brationaldenom5a-h2"],"title":"Multiplying each Expression","text":"Multiply each denominator by the \'missing\' factor and multiply each numerator by the same factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom5a-h4","type":"hint","dependencies":["aeb975brationaldenom5a-h3"],"title":"Simplifying the Numerator","text":"Simplify the numerators by multiplying out their factors. For example, $$6\\\\left(a+3\\\\right)$$ $$=$$ 6a + $$18$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb975brationaldenom6","title":"Adding Rational Expressions","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add and Subtract Rational Expressions with Unlike Denominators","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aeb975brationaldenom6a","stepAnswer":["$$\\\\frac{37}{72}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{5}{24}+\\\\frac{11}{36}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{37}{72}$$","hints":{"DefaultPathway":[{"id":"aeb975brationaldenom6a-h1","type":"hint","dependencies":[],"title":"Finding the LCD","text":"First, find the least common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom6a-h2","type":"hint","dependencies":["aeb975brationaldenom6a-h1"],"title":"Rewriting the Expression","text":"Rewrite each fraction as an equivalent fraction with the LCD.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom6a-h3","type":"hint","dependencies":["aeb975brationaldenom6a-h2"],"title":"Combining Terms","text":"Then, add the fractions and simplify if applicable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb975brationaldenom7","title":"Adding Rational Expressions","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add and Subtract Rational Expressions with Unlike Denominators","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aeb975brationaldenom7a","stepAnswer":["$$\\\\frac{47}{90}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{7}{30}+\\\\frac{13}{45}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{47}{90}$$","hints":{"DefaultPathway":[{"id":"aeb975brationaldenom7a-h1","type":"hint","dependencies":[],"title":"Finding the LCD","text":"First, find the least common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom7a-h2","type":"hint","dependencies":["aeb975brationaldenom7a-h1"],"title":"Rewriting the Expression","text":"Rewrite each fraction as an equivalent fraction with the LCD.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom7a-h3","type":"hint","dependencies":["aeb975brationaldenom7a-h2"],"title":"Combining Terms","text":"Then, add the fractions and simplify if applicable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb975brationaldenom8","title":"Adding Rational Expressions","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add and Subtract Rational Expressions with Unlike Denominators","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aeb975brationaldenom8a","stepAnswer":["$$\\\\frac{49}{60}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{9}{20}+\\\\frac{11}{30}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{49}{60}$$","hints":{"DefaultPathway":[{"id":"aeb975brationaldenom8a-h1","type":"hint","dependencies":[],"title":"Finding the LCD","text":"First, find the least common denominator.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom8a-h2","type":"hint","dependencies":["aeb975brationaldenom8a-h1"],"title":"Rewriting the Expression","text":"Rewrite each fraction as an equivalent fraction with the LCD.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeb975brationaldenom8a-h3","type":"hint","dependencies":["aeb975brationaldenom8a-h2"],"title":"Combining Terms","text":"Then, add the fractions and simplify if applicable.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeb975brationaldenom9","title":"Adding Rational Expressions","body":"Simplify the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.4 Add and Subtract Rational Expressions with Unlike Denominators","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"aeb975brationaldenom9a","stepAnswer":["$$\\\\frac{37}{54}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{8}{27}+\\\\frac{7}{18}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{37}{54}$$","hints":{"DefaultPathway":[{"id":"aeb975brationaldenom9a-h1","type":"hint","dependencies":[],"title":"Finding the 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{tan}^{-1\\\\left(\\\\frac{x}{3}\\\\right)}+C$$","hints":{"DefaultPathway":[{"id":"aed93bainv11a-h1","type":"hint","dependencies":[],"title":"Apply the Inverse Tangent Formula","text":"Use the formula for the inverse tangent.","variabilization":{},"oer":"","license":""},{"id":"aed93bainv11a-h2","type":"hint","dependencies":["aed93bainv11a-h1"],"title":"Inverse Tangent Formula","text":"$$\\\\int \\\\frac{1}{a^2+u^2} \\\\,du=\\\\frac{1}{a\\\\left({tan}^{-1}\\\\right)} \\\\frac{u}{a}+C$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"aed93bainv12","title":"Evaluating Indefinite Integrals","body":"Find each indefinite integral, using appropriate substitutions.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.7 Integrals Resulting in Inverse Trigonometric Functions","courseName":"OpenStax: Calculus Volume 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Then $$du=3dx$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aed93bainv2a-h2","type":"hint","dependencies":["aed93bainv2a-h1"],"title":"Apply $$a=2$$","text":"Apply the formula with $$a=2$$ then evaluate.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"aed93bainv20","title":"Computing Integrals","body":"Compute each definite integral.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.7 Integrals Resulting in Inverse Trigonometric Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aed93bainv20a","stepAnswer":["1/2ln(4/3)"],"problemType":"TextBox","stepTitle":"/int{tan(sin**-1)t/(sqrt(1-t**2),0,1/2,t}","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{1}{2} \\\\ln(\\\\frac{4}{3})$$","hints":{"DefaultPathway":[]}}]},{"id":"aed93bainv21","title":"Computing Integrals","body":"Compute each definite integral.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.7 Integrals Resulting in Inverse Trigonometric Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aed93bainv21a","stepAnswer":["1-2/sqrt(5)"],"problemType":"TextBox","stepTitle":"$$\\\\int_{0}^{\\\\frac{1}{2}} \\\\fracsin^1\\\\left({tan}^{-1}\\\\right) t+t^2} \\\\,dt$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$1-\\\\frac{2}{\\\\sqrt{5}}$$","hints":{"DefaultPathway":[]}}]},{"id":"aed93bainv22","title":"Computing Integrals","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.7 Integrals Resulting in Inverse Trigonometric Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aed93bainv22a","stepAnswer":["$$2{tan}^{-1\\\\left(A\\\\right)}$$ approaches pi as A approaches $$\\\\infty$$."],"problemType":"MultipleChoice","stepTitle":"For $$A>0$$, compute $$I(A)=\\\\int_{-A}^{A} \\\\frac{1}{1+t^2} \\\\,dt$$ and evaluate $$\\\\lim_{a\\\\to\\\\infty} I(A)$$, the area under the graph of $$\\\\frac{1}{1+t^2}$$ on $$[-\\\\infty,\\\\infty]$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2{tan}^{-1\\\\left(A\\\\right)}$$ approaches pi as A approaches $$\\\\infty$$.","choices":["$$2{tan}^{-1\\\\left(A\\\\right)}$$ approaches pi as A approaches $$\\\\infty$$.","$$2{cos}^{-1\\\\left(A\\\\right)}$$ approaches pi as A approaches $$\\\\infty$$.","$$2{sin}^{-1\\\\left(A\\\\right)}$$ approaches pi as A approaches $$\\\\infty$$."],"hints":{"DefaultPathway":[]}}]},{"id":"aed93bainv3","title":"Evaluating a Definite Integral","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.7 Integrals Resulting in Inverse Trigonometric Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aed93bainv3a","stepAnswer":["pi/3"],"problemType":"TextBox","stepTitle":"Evaluate the definite integral $$\\\\int_{0}^{\\\\frac{\\\\sqrt{3}}{2}} \\\\frac{1}{\\\\sqrt{1-u^2}} \\\\,du$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{\\\\pi}{3}$$","hints":{"DefaultPathway":[{"id":"aed93bainv3a-h1","type":"hint","dependencies":[],"title":"Inverse Sine Formula","text":"The format of the problem matches the inverse sine formula. 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function.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.7 Integrals Resulting in Inverse Trigonometric Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aed93bainv7a","stepAnswer":["pi/3"],"problemType":"TextBox","stepTitle":"$$\\\\int_{0}^{\\\\frac{\\\\sqrt{3}}{2}} \\\\frac{1}{\\\\sqrt{1-x^2}} \\\\,dx$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{\\\\pi}{3}$$","hints":{"DefaultPathway":[{"id":"aed93bainv7a-h1","type":"hint","dependencies":[],"title":"Apply the Inverse Sine Formula","text":"Use the formula for the inverse sine.","variabilization":{},"oer":"","license":""},{"id":"aed93bainv7a-h2","type":"hint","dependencies":["aed93bainv7a-h1"],"title":"Inverse Sine Formula","text":"$$\\\\int \\\\frac{1}{\\\\sqrt{a^2-u^2}} \\\\,du={sin}^{-1\\\\left(\\\\frac{u}{|a|}\\\\right)}+C$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"aed93bainv8","title":"Evaluating Integrals","body":"Evaluate the integral in terms of an inverse trigonometric function.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.7 Integrals Resulting in Inverse Trigonometric Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aed93bainv8a","stepAnswer":["-pi/12"],"problemType":"TextBox","stepTitle":"$$\\\\int_{\\\\sqrt{3}}^{1} \\\\frac{1}{1+x^2} \\\\,dx$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{-\\\\pi}{12}$$","hints":{"DefaultPathway":[{"id":"aed93bainv8a-h1","type":"hint","dependencies":[],"title":"Apply the Inverse Tangent Formula","text":"Use the formula for the inverse tangent.","variabilization":{},"oer":"","license":""},{"id":"aed93bainv8a-h2","type":"hint","dependencies":["aed93bainv8a-h1"],"title":"Inverse Tangent Formula","text":"$$\\\\int \\\\frac{1}{a^2+u^2} \\\\,du=\\\\frac{1}{a} {tan}^{-1} \\\\frac{u}{a}+C$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"aed93bainv9","title":"Evaluating Integrals","body":"Evaluate the integral in terms of an inverse trigonometric function.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5.7 Integrals Resulting in Inverse Trigonometric Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aed93bainv9a","stepAnswer":["pi/12"],"problemType":"TextBox","stepTitle":"$$\\\\int_{\\\\frac{2}{\\\\sqrt{3}}}^{\\\\sqrt{2}} \\\\frac{1}{|x| \\\\sqrt{x^2-1}} \\\\,dx$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{\\\\pi}{12}$$","hints":{"DefaultPathway":[{"id":"aed93bainv9a-h1","type":"hint","dependencies":[],"title":"Apply the Inverse Secant Formula","text":"Use the formula for the inverse secant.","variabilization":{},"oer":"","license":""},{"id":"aed93bainv9a-h2","type":"hint","dependencies":["aed93bainv9a-h1"],"title":"Inverse Secant Formula","text":"$$/int{1/(u*sqrt(u**2-a**2))=\\\\frac{1}{|a|} {sec}^{-1} \\\\frac{|u|}{a}+C$$","variabilization":{},"oer":"","license":""}]}}]},{"id":"aeec340Poisson1","title":"Poisson Distribution","body":"The average number of loaves of bread put on a shelf in a bakery in a half-hour period is $$12$$. Of interest is the number of loaves of bread put on the shelf in five minutes. The time interval of interest is five minutes.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Poisson Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aeec340Poisson1a","stepAnswer":["$$P(x=3), average$$ loaves:2"],"problemType":"MultipleChoice","stepTitle":"What is the mathematical statement for the probability that the number of loaves, selected randomly, put on the shelf in five minutes is three, and what is the average number of loaves at this time?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$P(x=3), average$$ loaves:2","choices":["$$P(x=3), average$$ loaves:2","$$P(x=4), average$$ loaves:2","$$P(x=6), average$$ loaves:4","$$P(x=7), average$$ loaves:4"],"hints":{"DefaultPathway":[{"id":"aeec340Poisson1a-h1","type":"hint","dependencies":[],"title":"Average Number","text":"Let X $$=$$ the number of loaves of bread put on the shelf in five minutes. Now, find the average number of loaves put on the shelf in five minutes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeec340Poisson1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["aeec340Poisson1a-h1"],"title":"Calculate Average Number","text":"If the average number of loaves put on the shelf in $$30$$ minutes $$(half-hour)$$ is $$12$$, what is the number of loaves put on the shelf in five minutes?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aeec340Poisson1a-h2-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":[],"title":"Calculate Average Number","text":"What is $$12\\\\frac{5}{30}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aeec340Poisson1a-h3","type":"hint","dependencies":["aeec340Poisson1a-h2"],"title":"Probability Statement","text":"We want to find the number of loaves is $$3$$. Find the associated probability statement.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeec340Poisson10","title":"Poisson Distribution","body":"According to Baydin, an email management company, an email user gets, on average, $$147$$ emails per day. Let X $$=$$ the number of emails an email user receives per day. The discrete random variable X takes on the values $$x$$ $$=$$ $$0$$, $$1$$, $$2$$ \u2026. The random variable X has a Poisson distribution: X ~ P(147). The mean is $$147$$ emails.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Poisson Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aeec340Poisson10a","stepAnswer":["$$12.1244$$"],"problemType":"TextBox","stepTitle":"What is the standard deviation? (Answer to four decimal places)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$12.1244$$","hints":{"DefaultPathway":[{"id":"aeec340Poisson10a-h1","type":"hint","dependencies":[],"title":"Standard deviation","text":"The standard deviation is the square root of the mean.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeec340Poisson10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12.1244$$"],"dependencies":["aeec340Poisson10a-h1"],"title":"Calculate","text":"What is $$\\\\sqrt{147}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeec340Poisson11","title":"Poisson Distribution","body":"Text message users receive or send an average of $$41.5$$ text messages per day.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Poisson Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aeec340Poisson11a","stepAnswer":["$$1.7292$$"],"problemType":"TextBox","stepTitle":"How many text messages does a text message user receive or send per hour? (Answer to four decimal places)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1.7292$$","hints":{"DefaultPathway":[{"id":"aeec340Poisson11a-h1","type":"hint","dependencies":[],"title":"Average Number","text":"Let X $$=$$ the number of texts that a user sends or receives in one hour. Find the average number of texts received per hour.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeec340Poisson11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.7292$$"],"dependencies":["aeec340Poisson11a-h1"],"title":"Average Number","text":"What is $$\\\\frac{41.5}{24}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeec340Poisson12","title":"Poisson Distribution","body":"Text message users receive or send an average of $$41.5$$ text messages per day.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Poisson Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aeec340Poisson12a","stepAnswer":["$$0.2653$$"],"problemType":"TextBox","stepTitle":"What is the probability that a text message user receives or sends two messages per hour? (Answer to four decimal places)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.2653$$","hints":{"DefaultPathway":[{"id":"aeec340Poisson12a-h1","type":"hint","dependencies":[],"title":"Average Number","text":"Let X $$=$$ the number of texts that a user sends or receives in one hour. Find the average number of texts received per hour.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeec340Poisson12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.7292$$"],"dependencies":["aeec340Poisson12a-h1"],"title":"Average Number","text":"What is $$\\\\frac{41.5}{24}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeec340Poisson12a-h3","type":"hint","dependencies":["aeec340Poisson12a-h2"],"title":"Probability","text":"We know the average number of texts received per hour is $$1.7292$$, and we are looking for the probability that a text message user receives or sends two messages per hour. Use this information for a calculator procedure.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeec340Poisson12a-h4","type":"hint","dependencies":["aeec340Poisson12a-h3"],"title":"Calculator","text":"$$P(x=2)=poissonpdf(1.7292$$, 2). Put this into the calculator to get the answer. poissonpdf is being used because we are finding the probability of an exact value, rather than a range.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeec340Poisson12a-h5","type":"hint","dependencies":["aeec340Poisson12a-h4"],"title":"Calculator","text":"Calculator steps: press 2nd DISTR, arrow down to poissonpdf. Press ENTER. Enter $$(1.7292$$, 2).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeec340Poisson13","title":"Poisson Distribution","body":"Text message users receive or send an average of $$41.5$$ text messages per day.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Poisson Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aeec340Poisson13a","stepAnswer":["$$0.2505$$"],"problemType":"TextBox","stepTitle":"What is the probability that a text message user receives or sends more than two messages per hour? (Answer to four decimal places)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.2505$$","hints":{"DefaultPathway":[{"id":"aeec340Poisson13a-h1","type":"hint","dependencies":[],"title":"Average Number","text":"Let X $$=$$ the number of texts that a user sends or receives in one hour. Find the average number of texts received per hour.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeec340Poisson13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.7292$$"],"dependencies":["aeec340Poisson13a-h1"],"title":"Average Number","text":"What is $$\\\\frac{41.5}{24}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeec340Poisson13a-h3","type":"hint","dependencies":["aeec340Poisson13a-h2"],"title":"Probability","text":"We know the average number of texts received per hour is $$1.7292$$, and we are looking for the probability that a text message user receives or sends two messages per hour. Use this information for a calculator procedure.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeec340Poisson13a-h4","type":"hint","dependencies":["aeec340Poisson13a-h3"],"title":"Calculator","text":"$$P\\\\left(x>2\\\\right)$$ $$=$$ $$1-P(x \\\\leq 2)$$ $$=$$ $$1-poissoncdf(1.7292$$, 2). Put this into the calculator to get the answer. poissoncdf is being used because we are finding the probability of a range of values, rather than an exact value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeec340Poisson13a-h5","type":"hint","dependencies":["aeec340Poisson13a-h4"],"title":"Calculator","text":"Calculator steps: press $$1$$ - and then press 2nd DISTR, arrow down to poissonpdf. Press ENTER. Enter $$(1.7292$$, 2).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeec340Poisson14","title":"On May $$13$$, $$2013$$, starting at 4:30 PM, the probability of low seismic activity for the next $$48$$ hours in Alaska was reported as about $$1.02\\\\%$$. Use this information for the next $$200$$ days to find the probability that there will be low seismic activity in ten of the next $$200$$ days.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Poisson Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aeec340Poisson14a","stepAnswer":["$$0.000039$$"],"problemType":"TextBox","stepTitle":"Use the binomial distribution to calculate the probability that there will be low seismic activity in ten of the next $$200$$ days. (Answer to four decimal places)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.000039$$","hints":{"DefaultPathway":[{"id":"aeec340Poisson14a-h1","type":"hint","dependencies":[],"title":"Calculator","text":"Let X $$=$$ the number of days with low seismic activity. We can use the binompdf procedure in the calculator. Find the values that should go in this procedure.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeec340Poisson14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$200$$"],"dependencies":["aeec340Poisson14a-h1"],"title":"Input","text":"We are using information for the next $$200$$ days. What is $$n$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeec340Poisson14a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.0102$$"],"dependencies":["aeec340Poisson14a-h2"],"title":"Input","text":"What would go into the second argument of the binompdf procedure? (Answer to four decimal places)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeec340Poisson14a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["aeec340Poisson14a-h3"],"title":"Input","text":"We are finding the probability that there will be low seismic activity in ten of the next $$200$$ days. What goes into the third argument of the binompdf procedure?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeec340Poisson14a-h5","type":"hint","dependencies":["aeec340Poisson14a-h4"],"title":"Calculator","text":"$$P(x=10)=binompdf(200$$, $$.0102$$, 10). Put this into the calculator to get the answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeec340Poisson14a-h6","type":"hint","dependencies":["aeec340Poisson14a-h5"],"title":"Calculator","text":"Calculator steps: press 2nd DISTR, arrow down to binompdf. Press ENTER. Enter (200, $$.0102$$, 10).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeec340Poisson15","title":"On May $$13$$, $$2013$$, starting at 4:30 PM, the probability of low seismic activity for the next $$48$$ hours in Alaska was reported as about $$1.02\\\\%$$. Use this information for the next $$200$$ days to find the probability that there will be low seismic activity in ten of the next $$200$$ days.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Poisson Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aeec340Poisson15a","stepAnswer":["$$0.000039$$"],"problemType":"TextBox","stepTitle":"Use the Poisson distribution to calculate the probability that there will be low seismic activity in ten of the next $$200$$ days. (Answer to six decimal places)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.000039$$","hints":{"DefaultPathway":[{"id":"aeec340Poisson15a-h1","type":"hint","dependencies":[],"title":"Calculator","text":"Let X $$=$$ the number of days with low seismic activity. We can use the poissonpdf procedure in the calculator. Find the values that should go in this procedure.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeec340Poisson15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.04$$"],"dependencies":["aeec340Poisson15a-h1"],"title":"Input","text":"We need to know the mean. The mean is given by np. What is the mean? (Answer to two decimal places)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeec340Poisson15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.04$$"],"dependencies":["aeec340Poisson15a-h2"],"title":"Input","text":"What is $$200\\\\times0.0102$$? (Answer to two decimal places)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeec340Poisson15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["aeec340Poisson15a-h3"],"title":"Input","text":"We are finding the probability that there will be low seismic activity in ten of the next $$200$$ days. What goes into the second argument of the poissonpdf procedure?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeec340Poisson15a-h5","type":"hint","dependencies":["aeec340Poisson15a-h4"],"title":"Calculator","text":"$$P(x=10)=poissonpdf(2.04$$, 10). Put this into the calculator to get the answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeec340Poisson15a-h6","type":"hint","dependencies":["aeec340Poisson15a-h5"],"title":"Calculator","text":"Calculator steps: press 2nd DISTR, arrow down to poissonpdf. Press ENTER. Enter $$(2.04$$, 10).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeec340Poisson16","title":"According to a recent poll by the Pew Internet Project, girls between the ages of $$14$$ and $$17$$ send an average of $$187$$ text messages each day. Let X $$=$$ the number of texts that a girl aged $$14$$ to $$17$$ sends per day. The discrete random variable X takes on the values $$x$$ $$=$$ $$0$$, $$1$$, $$2$$ \u2026. The random variable X has a Poisson distribution: X ~ P(187). The mean is $$187$$ text messages.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Poisson Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aeec340Poisson16a","stepAnswer":["$$13.6748$$"],"problemType":"TextBox","stepTitle":"What is the standard deviation? (Answer to four decimal places)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$13.6748$$","hints":{"DefaultPathway":[{"id":"aeec340Poisson16a-h1","type":"hint","dependencies":[],"title":"Standard deviation","text":"The standard deviation is the square root of the mean.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeec340Poisson16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$13.6748$$"],"dependencies":["aeec340Poisson16a-h1"],"title":"Calculate","text":"What is $$\\\\sqrt{187}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeec340Poisson2","title":"Poisson Distribution","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Poisson Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aeec340Poisson2a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"The average number of fish caught in an hour is eight. Of interest is the number of fish caught in $$15$$ minutes. The time interval of interest is $$15$$ minutes. What is the average number of fish caught in $$15$$ minutes?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"aeec340Poisson2a-h1","type":"hint","dependencies":[],"title":"Proportion","text":"Represent the time interval as a proportion.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeec340Poisson2a-h2","type":"hint","dependencies":["aeec340Poisson2a-h1"],"title":"Proportion","text":"Our proportion would be $$\\\\frac{15}{60}$$ since we want to find the average number of fish caught in $$15$$ minutes and we have information aboout $$60$$ minutes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeec340Poisson2a-h3","type":"hint","dependencies":["aeec340Poisson2a-h2"],"title":"Calculate","text":"Next, multiply this proportion with the average number of fish caught in an hour.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeec340Poisson2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["aeec340Poisson2a-h3"],"title":"Calculate","text":"What is $$8\\\\frac{15}{60}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeec340Poisson3","title":"Poisson Distribution","body":"A bank expects to receive six bad checks per day, on average. What is the probability of the bank getting fewer than five bad checks on any given day? Of interest is the number of checks the bank receives in one day, so the time interval of interest is one day. Let X $$=$$ the number of bad checks the bank receives in one day. If the bank expects to receive six bad checks per day then the average is six checks per day.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Poisson Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aeec340Poisson3a","stepAnswer":["$$P\\\\left(x<5\\\\right)$$"],"problemType":"MultipleChoice","stepTitle":"Write a mathematical statement for the probability question.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$P\\\\left(x<5\\\\right)$$","choices":["$$P\\\\left(x<5\\\\right)$$","$$P\\\\left(x>5\\\\right)$$","$$P\\\\left(x<6\\\\right)$$","$$P\\\\left(x>6\\\\right)$$"],"hints":{"DefaultPathway":[{"id":"aeec340Poisson3a-h1","type":"hint","dependencies":[],"title":"Probability","text":"We want to know the probability of the bank getting fewer than five bad checks on any given day. This means we are finding the probability that $$x<5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeec340Poisson4","title":"Poisson Distribution","body":"An electronics store expects to have ten returns per day on average. The manager wants to know the probability of the store getting fewer than eight returns on any given day.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Poisson Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aeec340Poisson4a","stepAnswer":["$$P\\\\left(x<8\\\\right)$$"],"problemType":"MultipleChoice","stepTitle":"State the probability question mathematically.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$P\\\\left(x<8\\\\right)$$","choices":["$$P\\\\left(x<6\\\\right)$$","$$P\\\\left(x>7\\\\right)$$","$$P\\\\left(x<8\\\\right)$$","$$P\\\\left(x<10\\\\right)$$"],"hints":{"DefaultPathway":[{"id":"aeec340Poisson4a-h1","type":"hint","dependencies":[],"title":"Probability","text":"We want to know the probability of the store getting fewer than eight returns on any given day. Let X $$=$$ the number of returns in one day. This means we are finding the probability that $$x<8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeec340Poisson5","title":"Poisson Distribution","body":"You notice that a news reporter says \\"uh,\\" on average, two times per broadcast. What is the probability that the news reporter says \\"uh\\" more than two times per broadcast. This is a Poisson problem because you are interested in knowing the number of times the news reporter says \\"uh\\" during a broadcast.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Poisson Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aeec340Poisson5a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"What is the interval of interest?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"aeec340Poisson5a-h1","type":"hint","dependencies":[],"title":"Interval","text":"We are focusing on the number of times \\"uh\\" is stated in one broadcast. Therefore, this would be the interval of interest.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aeec340Poisson5b","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"What is the average number of times the news reporter says \\"uh\\" during one broadcast?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"aeec340Poisson5b-h1","type":"hint","dependencies":[],"title":"Average","text":"This should be directly given in the problem. A news reporter says \\"uh,\\" on average, two times per broadcast.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeec340Poisson6","title":"Poisson Distribution","body":"You notice that a news reporter says \\"uh,\\" on average, two times per broadcast. What is the probability that the news reporter says \\"uh\\" more than two times per broadcast. This is a Poisson problem because you are interested in knowing the number of times the news reporter says \\"uh\\" during a broadcast.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Poisson Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aeec340Poisson6a","stepAnswer":["Let X $$=$$ the number of times the news reporter says \\"uh\\" during one broadcast. X can be any positive integer."],"problemType":"MultipleChoice","stepTitle":"Let X $$=$$ $$___$$ . What values does X take on?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Let X $$=$$ the number of times the news reporter says \\"uh\\" during one broadcast. X can be any positive integer.","choices":["Let X $$=$$ the number of times the news reporter says \\"uh\\" during one broadcast. X can be any positive integer.","Let X $$=$$ the number of times the news reporter says \\"uh\\" during two broadcasts. X can be any integer.","Let X $$=$$ the number of times the news reporter says \\"uh\\" during one broadcast. X can be any positive decimal.","Let X $$=$$ the number of times the news reporter says \\"uh\\" during two broadcasts. X can be any positive decimal."],"hints":{"DefaultPathway":[{"id":"aeec340Poisson6a-h1","type":"hint","dependencies":[],"title":"Define Variable","text":"X should be how many times \\"uh\\" is said. Find the time interval associated with it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeec340Poisson6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["aeec340Poisson6a-h1"],"title":"Time Interval","text":"What would the time interval be?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeec340Poisson6a-h3","type":"hint","dependencies":["aeec340Poisson6a-h2"],"title":"Define Variable","text":"Therefore, X should be how many times \\"uh\\" is said during one broadcast.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeec340Poisson6a-h4","type":"hint","dependencies":["aeec340Poisson6a-h3"],"title":"Define Variable Scope","text":"uh can only be said a positive number of times or $$0$$ times. \\"uh\\" can also only be said a whole number of times, not a decimal number of times.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeec340Poisson7","title":"Poisson Distribution","body":"You notice that a news reporter says \\"uh,\\" on average, two times per broadcast. What is the probability that the news reporter says \\"uh\\" more than two times per broadcast. This is a Poisson problem because you are interested in knowing the number of times the news reporter says \\"uh\\" during a broadcast.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Poisson Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aeec340Poisson7a","stepAnswer":["$$P\\\\left(x>2\\\\right)$$"],"problemType":"MultipleChoice","stepTitle":"The probability question is P( $$___$$ ).","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$P\\\\left(x>2\\\\right)$$","choices":["$$P\\\\left(x>2\\\\right)$$","$$P\\\\left(x<4\\\\right)$$","$$P\\\\left(x<5\\\\right)$$","$$P\\\\left(x>7\\\\right)$$"],"hints":{"DefaultPathway":[{"id":"aeec340Poisson7a-h1","type":"hint","dependencies":[],"title":"Probability","text":"We want to know the probability that the news reporter says \\"uh\\" more than two times per broadcast. Let X $$=$$ the number of times the news reporter says \\"uh\\" in one broadcast. This means we are finding the probability that $$x>2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeec340Poisson8","title":"Poisson Distribution","body":"According to Baydin, an email management company, an email user gets, on average, $$147$$ emails per day. Let X $$=$$ the number of emails an email user receives per day. The discrete random variable X takes on the values $$x$$ $$=$$ $$0$$, $$1$$, $$2$$ \u2026. The random variable X has a Poisson distribution: X ~ P(147). The mean is $$147$$ emails.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Poisson Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aeec340Poisson8a","stepAnswer":["$$0.018$$"],"problemType":"TextBox","stepTitle":"What is the probability that an email user receives exactly $$160$$ emails per day? (Answer to four decimal places)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.018$$","hints":{"DefaultPathway":[{"id":"aeec340Poisson8a-h1","type":"hint","dependencies":[],"title":"Probability","text":"We know the mean is $$147$$ emails, and we are looking for the probability that the email user receives exactly $$160$$ emails per day. Use this information for a calculator procedure.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeec340Poisson8a-h2","type":"hint","dependencies":["aeec340Poisson8a-h1"],"title":"Calculator","text":"$$P(x=160)=poissonpdf(147,160)$$. Put this into the calculator to get the answer. poissonpdf is being used because we are finding the probability of an exact value, rather than a range.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeec340Poisson8a-h3","type":"hint","dependencies":["aeec340Poisson8a-h2"],"title":"Calculator","text":"Calculator steps: press 2nd DISTR, arrow down to poissonpdf. Press ENTER. Enter $$(147,160)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aeec340Poisson9","title":"Poisson Distribution","body":"According to Baydin, an email management company, an email user gets, on average, $$147$$ emails per day. Let X $$=$$ the number of emails an email user receives per day. The discrete random variable X takes on the values $$x$$ $$=$$ $$0$$, $$1$$, $$2$$ \u2026. The random variable X has a Poisson distribution: X ~ P(147). The mean is $$147$$ emails.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.6 Poisson Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aeec340Poisson9a","stepAnswer":["$$0.8666$$"],"problemType":"TextBox","stepTitle":"What is the probability that an email user receives at most $$160$$ emails per day? (Answer to four decimal places)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.8666$$","hints":{"DefaultPathway":[{"id":"aeec340Poisson9a-h1","type":"hint","dependencies":[],"title":"Probability","text":"We know the mean is $$147$$ emails, and we are looking for the probability that the email user receives at most $$160$$ emails per day. Use this information for a calculator procedure.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeec340Poisson9a-h2","type":"hint","dependencies":["aeec340Poisson9a-h1"],"title":"Calculator","text":"$$P(x \\\\leq 160)=poissoncdf(147,160)$$. Put this into the calculator to get the answer. poissoncdf is being used because we are finding the probability of a range of values, rather than an exact value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aeec340Poisson9a-h3","type":"hint","dependencies":["aeec340Poisson9a-h2"],"title":"Calculator","text":"Calculator steps: press 2nd DISTR, arrow down to poissoncdf. Press ENTER. Enter $$(147,160)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aef8b03optimize10","title":"Minimizing Surface Area","body":"","variabilization":{},"oer":"https://openstax.org/books/calculus-volume-1/pages/4-7-applied-optimization-problems <Openstax: Applied Optimization Problems>","license":0,"lesson":"4.7 Applied Optimization Problems","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aef8b03optimize10a","stepAnswer":["108*sqrt(3,4)"],"problemType":"TextBox","stepTitle":"What is the minimum surface area for the previous problem?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$108\\\\sqrt[3]{4}$$","hints":{"DefaultPathway":[{"id":"aef8b03optimize10a-h1","type":"hint","dependencies":[],"title":"plug in $$x$$ and $$y$$","text":"Plug in the $$x$$ and $$y$$ values into the surface area formula to find the surface area.","variabilization":{},"oer":"","license":""},{"id":"aef8b03optimize10a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["108*sqrt(3,4)"],"dependencies":["aef8b03optimize10a-h1"],"title":"surface area","text":"If $$S=4xy+x^2$$, what does $$4\\\\left(6\\\\sqrt[3]{2}\\\\right) 3\\\\sqrt[3]{2}+{\\\\left(6\\\\sqrt[3]{2}\\\\right)}^2=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"aef8b03optimize13","title":"Maximizing the Area of a Cattle Pen","body":"","variabilization":{},"oer":"https://openstax.org/books/calculus-volume-1/pages/4-7-applied-optimization-problems <Openstax: Applied Optimization Problems>","license":0,"lesson":"4.7 Applied Optimization Problems","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aef8b03optimize13a","stepAnswer":["$$100$$ ft by $$100$$ ft"],"problemType":"MultipleChoice","stepTitle":"You have $$400$$ ft of fencing to construct a rectangular pen for cattle. What are the dimensions of the pen that maximizes the area?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$100$$ ft by $$100$$ ft","choices":["$$100$$ by $$50$$ ft","$$100$$ ft by $$100$$ ft","$$200$$ ft by $$20$$ ft"],"hints":{"DefaultPathway":[{"id":"aef8b03optimize13a-h1","type":"hint","dependencies":[],"title":"Constraint Equation","text":"First, write a constraint equation representing the perimeter in terms of its length(x) and width(y).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x+2y$$"],"dependencies":["aef8b03optimize13a-h1"],"title":"Perimeter","text":"What is the perimeter in terms of $$x$$ and $$y$$? $$P=400=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize13a-h3","type":"hint","dependencies":["aef8b03optimize13a-h2"],"title":"A(x)","text":"Write the area as a function of its length.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$200-x$$"],"dependencies":["aef8b03optimize13a-h3"],"title":"$$y$$","text":"First, solve for $$y$$ using the perimeter\'s equation. $$y=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize13a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-\\\\left(x^2\\\\right)+200$$"],"dependencies":["aef8b03optimize13a-h4"],"title":"subsitute $$y$$","text":"Rewrite $$A=xy$$ in terms of only $$x$$ by substituting $$y$$ with $$y=200-x$$. $$A(x)=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize13a-h6","type":"hint","dependencies":["aef8b03optimize13a-h5"],"title":"critical points","text":"Find the area function\'s critical point(s) using its derivative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize13a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2x+200$$"],"dependencies":["aef8b03optimize13a-h6"],"title":"derivative","text":"What is $$A\'(x)=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize13a-h8","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["100"],"dependencies":["aef8b03optimize13a-h7"],"title":"slope of zero","text":"When $$A\'(x)=0$$, $$x=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize13a-h9","type":"hint","dependencies":["aef8b03optimize13a-h8"],"title":"dimensions","text":"Lastly, find the dimension of the width.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize13a-h10","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["100"],"dependencies":["aef8b03optimize13a-h9"],"title":"width","text":"If $$x=100$$, what is $$y=200-x=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"aef8b03optimize14","title":"Maximizing the Area of a Pen for Hogs","body":"","variabilization":{},"oer":"https://openstax.org/books/calculus-volume-1/pages/4-7-applied-optimization-problems <Openstax: Applied Optimization Problems>","license":0,"lesson":"4.7 Applied Optimization Problems","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aef8b03optimize14a","stepAnswer":["$$100$$ ft by $$100$$ ft"],"problemType":"MultipleChoice","stepTitle":"You have $$800$$ ft of fencing to construct a rectangular pen for hogs. What are the dimensions of the pen that mamximizes the area?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$100$$ ft by $$100$$ ft","choices":["$$100$$ by $$50$$ ft","$$100$$ ft by $$100$$ ft","$$200$$ ft by $$20$$ ft"],"hints":{"DefaultPathway":[{"id":"aef8b03optimize14a-h1","type":"hint","dependencies":[],"title":"Constraint Equation","text":"First, write a constraint equation representing the perimeter in terms of its length(x) and width(y).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+2y$$"],"dependencies":["aef8b03optimize14a-h1"],"title":"Perimeter","text":"Let $$x$$ denote the side of the pen parallel to the river, and $$y$$ denote the side perpendicular to the river. What is the perimeter in terms of $$x$$ and $$y$$? $$P=800=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize14a-h3","type":"hint","dependencies":["aef8b03optimize14a-h2"],"title":"A(x)","text":"Write the area as a function of side $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize14a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$800-\\\\frac{1}{5} x$$"],"dependencies":["aef8b03optimize14a-h3"],"title":"$$y$$","text":"First, solve for $$y$$ using the perimeter\'s equation. $$y=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize14a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-\\\\left(\\\\frac{1}{2}\\\\right) x^2+400x$$"],"dependencies":["aef8b03optimize14a-h4"],"title":"subsitute $$y$$","text":"Rewrite $$A=xy$$ in terms of only $$x$$ by substituting $$y$$ with $$y=400-\\\\frac{1}{2} x$$. $$A(x)=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize14a-h6","type":"hint","dependencies":["aef8b03optimize14a-h5"],"title":"critical points","text":"Find the area function\'s critical point(s) using its derivative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize14a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-x+400$$"],"dependencies":["aef8b03optimize14a-h6"],"title":"derivative","text":"What is $$A\'(x)=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize14a-h8","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["400"],"dependencies":["aef8b03optimize14a-h7"],"title":"slope of zero","text":"When $$A\'(x)=0$$, $$x=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize14a-h9","type":"hint","dependencies":["aef8b03optimize14a-h8"],"title":"dimensions","text":"Lastly, find the dimension of the width.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize14a-h10","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["200"],"dependencies":["aef8b03optimize14a-h9"],"title":"$$y$$","text":"If $$x=400$$, what is $$y=400-\\\\frac{1}{2} x=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"aef8b03optimize15","title":"Minimizing Angled Wire Length","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/books/calculus-volume-1/pages/4-7-applied-optimization-problems <Openstax: Applied Optimization Problems>","license":0,"lesson":"4.7 Applied Optimization Problems","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aef8b03optimize15a","stepAnswer":["15"],"problemType":"TextBox","stepTitle":"Two poles are connected by a wire that is also connected to the ground. The first pole is 20ft tall and the second pole is 10ft tall. There is a distance of $$30$$ ft between the two poles. Where should the wire be ancored to the ground to minimize the amount of wire needed? $$x=___$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$15$$","hints":{"DefaultPathway":[{"id":"aef8b03optimize15a-h1","type":"hint","dependencies":[],"title":"Constraint Equation","text":"First, set up the constraint equation. In this case, it\'s the length of the wire in terms of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize15a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$30-x$$"],"dependencies":["aef8b03optimize15a-h1"],"title":"Pythagorean Theorem","text":"Use the Pythagorean theorem to determine the length of the wire, as the wire forms the hypotenuse of a right triangle. First, write the equation representing the distance between the wire that would be anchored and the $$20$$ ft tall pole.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$100+x^2$$"],"dependencies":["aef8b03optimize15a-h2"],"title":"legnth of the wire","text":"What is the length of the wire in which the sides are $$10$$ ft and $$x$$ ft?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2-60x+900$$"],"dependencies":["aef8b03optimize15a-h3"],"title":"legnth of the wire","text":"What is the length of the wire in which the sides are $$20$$ ft and $$(30-x)$$ ft?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize15a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x^2-60x+1400$$"],"dependencies":["aef8b03optimize15a-h4"],"title":"total lenght","text":"What is the total length of the wire in terms of $$x$$? $$L(x)=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize15a-h6","type":"hint","dependencies":["aef8b03optimize15a-h5"],"title":"critical point","text":"Find the critical point.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize15a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4x-60$$"],"dependencies":["aef8b03optimize15a-h6"],"title":"derivative","text":"First, find the derivative. $$L\'(x)=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize15a-h8","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["15"],"dependencies":["aef8b03optimize15a-h7"],"title":"slope of zero","text":"When $$L\'(x)=0$$, $$x=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"aef8b03optimize16","title":"Minimized Pulse","body":"","variabilization":{},"oer":"https://openstax.org/books/calculus-volume-1/pages/4-7-applied-optimization-problems <Openstax: Applied Optimization Problems>","license":0,"lesson":"4.7 Applied Optimization Problems","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aef8b03optimize16a","stepAnswer":["90"],"problemType":"TextBox","stepTitle":"A patient\'s pulse measure $$70$$ bpm, $$80$$ bpm, then $$120$$ bpm. To determine an accurate measurement of pulse, the doctor wants to know what value minimizes the expression $${\\\\left(x-70\\\\right)}^2+{\\\\left(x-80\\\\right)}^2+{\\\\left(x-120\\\\right)}^2$$? $$x=___$$?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$90$$","hints":{"DefaultPathway":[{"id":"aef8b03optimize16a-h1","type":"hint","dependencies":[],"title":"Find the derivative in order to find the critical point.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize16a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6x-540$$"],"dependencies":["aef8b03optimize16a-h1"],"title":"Derivative","text":"What is $$F\'(x)=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize16a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["90"],"dependencies":["aef8b03optimize16a-h2"],"title":"Derivative","text":"When $$F\'(x)=0$$, what is $$x=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"aef8b03optimize17","title":"Minimizing Fuel with Variable Constants","body":"","variabilization":{},"oer":"https://openstax.org/books/calculus-volume-1/pages/4-7-applied-optimization-problems <Openstax: Applied Optimization Problems>","license":0,"lesson":"4.7 Applied Optimization Problems","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aef8b03optimize17a","stepAnswer":["$$\\\\sqrt{\\\\frac{b}{a}}$$"],"problemType":"TextBox","stepTitle":"A truck uses gas as $$g(v)=av+\\\\frac{b}{v}$$ where v represents the speed of the truck and g represents the gallons of fuel per mile. At what speed is fuel consumption minimized?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\sqrt{\\\\frac{b}{a}}$$","hints":{"DefaultPathway":[{"id":"aef8b03optimize17a-h1","type":"hint","dependencies":[],"title":"Firstly, derive g(v) in order to find its critical point.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize17a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["a"],"dependencies":["aef8b03optimize17a-h1"],"title":"Derivative","text":"When deriving av from $$g(v)=av+\\\\frac{b}{v}$$, av becomes $$___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize17a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-b v^{-2}$$"],"dependencies":["aef8b03optimize17a-h2"],"title":"Derivative","text":"When deriving $$\\\\frac{b}{v}$$, also written as $$b v^{-1}$$ from $$g(v)=av+\\\\frac{b}{v}$$, av becomes $$___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize17a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$a-b v^{-2}$$"],"dependencies":["aef8b03optimize17a-h3"],"title":"Derivative","text":"What g\'(v)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize17a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{\\\\frac{b}{a}}$$"],"dependencies":["aef8b03optimize17a-h4"],"title":"Derivative","text":"When $$g\'(v)=0$$, $$v=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"aef8b03optimize2","title":"Maximizing the Volume of a Box","body":"","variabilization":{},"oer":"https://openstax.org/books/calculus-volume-1/pages/4-7-applied-optimization-problems <Openstax: Applied Optimization Problems>","license":0,"lesson":"4.7 Applied Optimization Problems","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aef8b03optimize2a","stepAnswer":["$$4.7$$"],"problemType":"TextBox","stepTitle":"An open-top box is to be made from a 24in. by 36in. piece of cardboard by removing a square from each corner of the box and folding up the flaps on each side. What size square should be cut out of each corner to get a box with the maximum volume? Round to the first decimal place.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4.7$$","hints":{"DefaultPathway":[{"id":"aef8b03optimize2a-h1","type":"hint","dependencies":[],"title":"The First Step","text":"The first step is to let $$x$$ be the side length of the square to be removed from each corner. Then, the remaining four flaps can be folded up to form an open-top box. Let V be the volume of the resulting box.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize2a-h2","type":"hint","dependencies":["aef8b03optimize2a-h1"],"title":"Find the equation","text":"The next step is to find the equation for the volume of the box.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize2a-h3","type":"hint","dependencies":["aef8b03optimize2a-h2"],"title":"Determine the Domain","text":"Next, determine the domain of consideration.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize2a-h4","type":"hint","dependencies":["aef8b03optimize2a-h3"],"title":"Find the $$x$$ value","text":"Lastly, find the $$x$$ value for the maximum.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"aef8b03optimize3","title":"Minimizing Travel Time","body":"\\\\n##figure2.gif","variabilization":{},"oer":"https://openstax.org/books/calculus-volume-1/pages/4-7-applied-optimization-problems <Openstax: Applied Optimization Problems>","license":0,"lesson":"4.7 Applied Optimization Problems","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aef8b03optimize3a","stepAnswer":["5.19"],"problemType":"TextBox","stepTitle":"Minimizing Travel Time","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$5.19$$","hints":{"DefaultPathway":[{"id":"aef8b03optimize3a-h1","type":"hint","dependencies":[],"title":"Diagram","text":"The first step is to set up a diagram and label all variables.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize3a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x+6-x$$"],"dependencies":["aef8b03optimize3a-h1"],"title":"shore\'s length","text":"Let $$x$$ be the distance running and let $$y$$ be the distance swimming. Let T be the time it takes to get from the cabin to the island. How can we rewrite the shore\'s length between the cabin and island using $$x$$?\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$x+6-x$$","$$6+x$$"]},{"id":"aef8b03optimize3a-h3","type":"hint","dependencies":["aef8b03optimize3a-h2"],"title":"Equations","text":"The second step is to set up equation for the total time spent traveling.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize3a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$T=RxD$$"],"dependencies":["aef8b03optimize3a-h3"],"title":"Distance","text":"To find the time spent traveling from the cabin to the island, add the tme spent running and the time spent swimming. Let $$D=distance$$, $$R=Rate$$, and $$T=Time$$. Which equation is correct?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$D=R+T$$","$$T=RxD$$","$$D=RxT$$"]},{"id":"aef8b03optimize3a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{x}{8}$$"],"dependencies":["aef8b03optimize3a-h4"],"title":"Time Spent Running","text":"Since $$Distance=RatexTime$$ can be written as $$Time=\\\\frac{Distance}{Running}$$, which equation represents the time spent running?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$8+x$$","$$x$$","$$\\\\frac{x}{8}$$"]},{"id":"aef8b03optimize3a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{y}{3}$$"],"dependencies":["aef8b03optimize3a-h5"],"title":"Time Spent Swimming","text":"Since $$Distance=RatexTime$$ can be written as $$Time=\\\\frac{Distance}{Running}$$, which equation represents the time spent swimming?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$y+3$$","$$\\\\frac{y}{3}$$","$$\\\\frac{y}{8}$$"]},{"id":"aef8b03optimize3a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{x}{8}+\\\\frac{y}{3}$$"],"dependencies":["aef8b03optimize3a-h6"],"title":"Total Time","text":"Now combine the two previous equations to represent the total time spent traveling. Which option represents this? In other words, $$T=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\frac{x}{8}+\\\\frac{y}{3}$$","$$\\\\frac{8}{x}+\\\\frac{y}{3}$$","$$\\\\frac{8}{x}+\\\\frac{3}{y}$$"]},{"id":"aef8b03optimize3a-h8","type":"hint","dependencies":["aef8b03optimize3a-h7"],"title":"Hypotenuse","text":"The third step is to set up the equation for the distance of $$y$$ miles that forms the hypotenuse of a right triangle with the legs of length of 2mi and $$(6-x)mi$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize3a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2^2+{\\\\left(6-x\\\\right)}^2=y^2$$"],"dependencies":["aef8b03optimize3a-h8"],"title":"pythagorean theorem","text":"If the Pythagorean theorem is $$a^2+b^2=c^2$$, which equation correctly represent the relevant variables and values?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$y^2+x^2=2^2$$","$${\\\\left(6-x\\\\right)}^2+x^2=y^2$$","$$2^2+{\\\\left(6-x\\\\right)}^2=y^2$$"]},{"id":"aef8b03optimize3a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\sqrt{{\\\\left(6-x\\\\right)}^2+4}$$"],"dependencies":["aef8b03optimize3a-h9"],"title":"isolate $$y$$","text":"Considering the previous equation, $$y=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["(6-x)**2+4)","$$\\\\sqrt{{\\\\left(6-x\\\\right)}^2+4}$$","$$\\\\sqrt{{\\\\left(4-x\\\\right)}^2+6}$$"]},{"id":"aef8b03optimize3a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{x}{8}+\\\\frac{{\\\\sqrt{6-x}}^2+4}{3}$$"],"dependencies":["aef8b03optimize3a-h10"],"title":"Total Time Spent Traveling","text":"If T(x) is the function for the total time spent traveling, $$T(x)=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\frac{x}{8}+\\\\frac{{\\\\sqrt{6-x}}^2+4}{3}$$","$$\\\\frac{{\\\\sqrt{6-x}}^2+4}{3}$$"]},{"id":"aef8b03optimize3a-h12","type":"hint","dependencies":["aef8b03optimize3a-h11"],"title":"subsitute $$y$$","text":"If $$T=\\\\frac{x}{8}+\\\\frac{y}{3}$$, subsitute $$y$$ with $$y=\\\\sqrt{{\\\\left(6-x\\\\right)}^2+4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize3a-h13","type":"hint","dependencies":["aef8b03optimize3a-h12"],"title":"Domain","text":"The third step is to find out the domain of consideration.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize3a-h14","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["[0,6]"],"dependencies":["aef8b03optimize3a-h13"],"title":"Shore length\'s domain","text":"If the shore length must be positive and less than $$6$$ miles, $$x$$ can only be within what two numbers?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["(0,6]","$$[-6,6]$$","[0,6]"]},{"id":"aef8b03optimize3a-h15","type":"hint","dependencies":["aef8b03optimize3a-h14"],"title":"Critical Point","text":"The final step is to find the critical point of interest.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize3a-h16","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1}{8}-\\\\frac{2\\\\frac{1}{2} \\\\left(6-x\\\\right) {\\\\left({\\\\left(6-x\\\\right)}^2+4\\\\right)}^{\\\\left(-\\\\frac{1}{2}\\\\right)}}{3}$$"],"dependencies":["aef8b03optimize3a-h15"],"title":"Derivative","text":"Since T(x) is a continuous function over a closed, bounded interval, it has a maximum and a minimum. In order to find the critical points, what is the derivative of T(x)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\frac{2\\\\frac{1}{2} \\\\left(6-x\\\\right) {\\\\left({\\\\left(6-x\\\\right)}^2+4\\\\right)}^{\\\\left(-\\\\frac{1}{2}\\\\right)}}{3}$$","$$\\\\frac{1}{8}-\\\\frac{2\\\\frac{1}{2} \\\\left(6-x\\\\right) {\\\\left({\\\\left(6-x\\\\right)}^2+4\\\\right)}^{\\\\left(-\\\\frac{1}{2}\\\\right)}}{3}$$"]},{"id":"aef8b03optimize3a-h17","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$6\\\\pm \\\\frac{6}{sqrt55}$$"],"dependencies":["aef8b03optimize3a-h16"],"title":"$$T\'(x)=0$$","text":"If $$T\'(x)=0$$, what is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$6\\\\pm \\\\frac{6}{sqrt55}$$","$$6+\\\\frac{6}{sqrt55}$$"]},{"id":"aef8b03optimize3a-h19","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$6-\\\\frac{6}{sqrt55}$$"],"dependencies":["aef8b03optimize3a-h17"],"title":"critial points within the domain","text":"Which of the following $$x$$ values are within the domain?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$6+\\\\frac{6}{sqrt55}$$","$$6-\\\\frac{6}{sqrt55}$$"]},{"id":"aef8b03optimize3a-h20","type":"hint","dependencies":["aef8b03optimize3a-h19"],"title":"Max or Min?","text":"Check if the critical point is a maximum or minimum by finding what T(x) equals at the endpoints.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize3a-h21","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["2.108"],"dependencies":["aef8b03optimize3a-h20"],"title":"left endpoint","text":"What does $$T(0)=___$$? Round to the third decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize3a-h22","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["1.417"],"dependencies":["aef8b03optimize3a-h21"],"title":"right endpoint","text":"What does $$T(6)=___$$? Round to the third decimal place.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize3a-h23","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["minimum"],"dependencies":["aef8b03optimize3a-h22"],"title":"justified critical point","text":"We find that $$T(0)\u22482.108h$$ and $$T(6)\u22481.417h$$, whereas T(6-6/sqrt55)\u22481.368h. Therefore, $$x=6-\\\\frac{6}{sqrt55}$$ is a $$___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["neither","minimum","maximum"]}]}}]},{"id":"aef8b03optimize4","title":"Maximizing Revenue","body":"","variabilization":{},"oer":"https://openstax.org/books/calculus-volume-1/pages/4-7-applied-optimization-problems <Openstax: Applied Optimization Problems>","license":0,"lesson":"4.7 Applied Optimization Problems","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aef8b03optimize4a","stepAnswer":["$$100$$"],"problemType":"TextBox","stepTitle":"Owners of a car rental company have determined that if they charge customers $$p$$ dollars per day to rent a car, where $$50 \\\\leq p \\\\leq 200$$, the number of cars $$n$$ they rent per day can be modeled by the linear function $$n(p)=1000-5p$$. If they charge $50 per day or less, they will rent all their cars. If they charge $200 per day or more, they will not rent any cars. Assuming the owners plan to charge customers between $50 per day and $200 per day to rent a car, how much should they charge to maximize their revenue?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$100$$","hints":{"DefaultPathway":[{"id":"aef8b03optimize4a-h1","type":"hint","dependencies":[],"title":"$$R=Revenue$$","text":"Let R be the revenue per day. First, make a function for R using the variables $$n$$ and $$p$$.","variabilization":{},"oer":"","license":""},{"id":"aef8b03optimize4a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$R=n p$$"],"dependencies":["aef8b03optimize4a-h1"],"title":"$$R=___$$?","text":"Which of the following represent R correctly?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$R=\\\\frac{n}{p}$$","$$R=n+p$$","$$R=n p$$"]},{"id":"aef8b03optimize4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5p^2+1000p$$"],"dependencies":["aef8b03optimize4a-h2"],"title":"$$R(x)=___$$?","text":"Subsitute $$n$$ in $$R=n p$$ with $$n(p)=1000-5p$$. Simplify that function","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize4a-h4","type":"hint","dependencies":["aef8b03optimize4a-h3"],"title":"Domain","text":"Secondly, find the domain of interest.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize4a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["[50,200]"],"dependencies":["aef8b03optimize4a-h4"],"title":"Domain","text":"If the owners plan to charge between $50 and $200 per day to rent a car, the domain is $$___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$(50,200)$$","(0,200]","[50,200]","$$-50, 200]$$"]},{"id":"aef8b03optimize4a-h6","type":"hint","dependencies":["aef8b03optimize4a-h5"],"title":"Critical Points","text":"Thirdly, find the critical points.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize4a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-10p+1000$$"],"dependencies":["aef8b03optimize4a-h6"],"title":"Derivative","text":"What is the derivative of R(x)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$-10p+1000$$","$$10p+1000$$","$$-100+10009$$"]},{"id":"aef8b03optimize4a-h8","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["100"],"dependencies":["aef8b03optimize4a-h7"],"title":"R\'(0)","text":"To find the critical point, set $$R\'(p)=0$$ and solve for $$p$$. $$P=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize4a-h9","type":"hint","dependencies":["aef8b03optimize4a-h8"],"title":"Max or Min","text":"Lastly, show if the critical point is a minimum or maximum.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize4a-h10","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["50000"],"dependencies":["aef8b03optimize4a-h9"],"title":"Critical Points","text":"What is $$R(100)=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize4a-h11","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["37500"],"dependencies":["aef8b03optimize4a-h10"],"title":"Left End Point","text":"What is $$R(50)=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize4a-h12","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["0"],"dependencies":["aef8b03optimize4a-h11"],"title":"Right End Point","text":"What is $$R(200)=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize4a-h13","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["maximum"],"dependencies":["aef8b03optimize4a-h12"],"title":"Max or Min","text":"Because R(100) is greater than R(50) and R(200), that means that $$p=100$$ is a $$___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["neither","maximum","minimum"]}]}}]},{"id":"aef8b03optimize5","title":"Maximizing Revenue","body":"","variabilization":{},"oer":"https://openstax.org/books/calculus-volume-1/pages/4-7-applied-optimization-problems <Openstax: Applied Optimization Problems>","license":0,"lesson":"4.7 Applied Optimization Problems","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aef8b03optimize5a","stepAnswer":["$$100$$"],"problemType":"TextBox","stepTitle":"A car rental company charges its customers $$p$$ dollars per day, where $$60 \\\\leq p \\\\leq 150$$. It has been found that the number of cars rented per day can be modeled by the linear function $$n(p)=750-5p$$. How much should the company charge each customer to maximize revenue?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$100$$","hints":{"DefaultPathway":[{"id":"aef8b03optimize5a-h1","type":"hint","dependencies":[],"title":"$$R=Revenue$$","text":"Let R be the revenue per day. First, make a function for R using the variables $$n$$ and $$p$$.","variabilization":{},"oer":"","license":""},{"id":"aef8b03optimize5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5p^2+750p$$"],"dependencies":["aef8b03optimize5a-h1"],"title":"$$R(x)=___$$?","text":"If $$R=n p$$, substitute $$n$$ with $$n(p)=750-5p$$. Simplify that function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize5a-h3","type":"hint","dependencies":["aef8b03optimize5a-h2"],"title":"Domain","text":"Secondly, find the domain of interest.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize5a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["[60,150]"],"dependencies":["aef8b03optimize5a-h3"],"title":"Domain","text":"Write $$60 \\\\leq p \\\\leq 150$$ in the correct mathematical connotation. Ex:[0,1000]","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$(60,150)$$","(0,150]","[60,150]","$$[-60,150]$$"]},{"id":"aef8b03optimize5a-h5","type":"hint","dependencies":["aef8b03optimize5a-h4"],"title":"Critical Points","text":"Thirdly, find the critical points.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize5a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-10p+750$$"],"dependencies":["aef8b03optimize5a-h5"],"title":"Derivative","text":"What is the derivative of R(x)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$-10p+1000$$","$$10p+1000$$","$$-10p+750$$"]},{"id":"aef8b03optimize5a-h7","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["75"],"dependencies":["aef8b03optimize5a-h6"],"title":"R\'(0)","text":"To find the critical point, set $$R\'(p)=0$$ and solve for $$p$$. $$P=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize5a-h8","type":"hint","dependencies":["aef8b03optimize5a-h7"],"title":"Max or Min","text":"Lastly, show if the critical point is a minimum or maximum.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize5a-h9","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["28152"],"dependencies":["aef8b03optimize5a-h8"],"title":"Critical Points","text":"What is $$R(75)=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize5a-h10","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["27000"],"dependencies":["aef8b03optimize5a-h9"],"title":"Left End Point","text":"What is $$R(60)=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize5a-h11","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["0"],"dependencies":["aef8b03optimize5a-h10"],"title":"Right End Point","text":"What is $$R(150)=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize5a-h12","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["maximum"],"dependencies":["aef8b03optimize5a-h11"],"title":"Max or Min","text":"Because R(100) is greater than R(50) and R(200), that means that $$p=100$$ is a $$___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["neither","maximum","minimum"]}]}}]},{"id":"aef8b03optimize6","title":"Domain of Consideration for a Rectangle in a circle","body":"","variabilization":{},"oer":"https://openstax.org/books/calculus-volume-1/pages/4-7-applied-optimization-problems <Openstax: Applied Optimization Problems>","license":0,"lesson":"4.7 Applied Optimization Problems","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aef8b03optimize6a","stepAnswer":["[0,1]"],"problemType":"MultipleChoice","stepTitle":"Modify the area function A if the rectangle is to be inscribed in the unit circle $$x^2+y^2=1$$. What is the domain of consideration? Write your answer in this format \\"[3, 7].\\"","stepBody":"","answerType":"string","variabilization":{},"choices":["[0,1]","[0,0]","[1,1]","[1,0]"],"hints":{"DefaultPathway":[{"id":"aef8b03optimize6a-h1","type":"hint","dependencies":[],"title":"Circle\'s Radius","text":"Find the radius of the circle equation.","variabilization":{},"oer":"","license":""},{"id":"aef8b03optimize6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["aef8b03optimize6a-h1"],"title":"c","text":"The value of c in circle equations of $$a^2+b^2=c^2$$ is the radius. For $$x^2+y^2=1$$, what is c and the radius equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize6a-h3","type":"hint","dependencies":["aef8b03optimize6a-h2"],"title":"End of the Domain","text":"At the end of the circle\'s radius is also the end of the domain.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["aef8b03optimize6a-h3"],"title":"Right endpoint","text":"If the end of the radius is at $$x=1$$ in the first quadrant, then the right end of the domain must also be $$___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"aef8b03optimize7","title":"Maximixing the Area of an Inscribed Rectangle","body":"","variabilization":{},"oer":"https://openstax.org/books/calculus-volume-1/pages/4-7-applied-optimization-problems <Openstax: Applied Optimization Problems>","license":0,"lesson":"4.7 Applied Optimization Problems","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aef8b03optimize7a","stepAnswer":["$$2sqrt2$$, $$sqrt2$$"],"problemType":"MultipleChoice","stepTitle":"A rectangle is to be inscribed in the ellipse $$\\\\frac{x^2}{4}=y^2=1$$. What should the dimensions of the rectangle be to maximize its area? Format your answer so that the longer dimension is first and the dimensions are separated with a space and comma. Ex: \\"5, 7\\"","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2sqrt2$$, $$sqrt2$$","choices":["$$2sqrt2$$, $$sqrt2$$","$$sqrt2$$, $$2sqrt2$$"],"hints":{"DefaultPathway":[{"id":"aef8b03optimize7a-h1","type":"hint","dependencies":[],"title":"A(x)","text":"First, write a function to represent that area of the rectangle using A(x).\\\\n##figure1.gif##","variabilization":{},"oer":"","license":""},{"id":"aef8b03optimize7a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["LW"],"dependencies":["aef8b03optimize7a-h1"],"title":"Area of a Rectangle","text":"Let the length (L) of the rectangle be the side of the rectangle parallel to the x-axis and the width (W) be the side parallel to the y-axis. What equation represents the area of the rectangle? In other words, $$A=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize7a-h3","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["2x"],"dependencies":["aef8b03optimize7a-h2"],"title":"Length","text":"Let (x,Y) be the corner of the rectangle in the first quadrant, as shown in the figure. Therefore, how can you rewrite L in terms of $$x$$? $$L=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize7a-h4","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["2y"],"dependencies":["aef8b03optimize7a-h3"],"title":"Width","text":"How can you rewrite W in terms of $$y$$? $$W=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize7a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{1-\\\\frac{x^2}{4}}$$"],"dependencies":["aef8b03optimize7a-h4"],"title":"Isolate $$y$$","text":"Since $$\\\\frac{x^2}{4}=y^2=1$$ and $$y>0$$, rewrite the equation so that $$y$$ is isolated. $$Y=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize7a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$A=2\\\\operatorname{xsqrt}\\\\left(4-x^2\\\\right)$$"],"dependencies":["aef8b03optimize7a-h5"],"title":"subsitute $$y$$","text":"Rewrite $$A=LW$$ in terms of $$x$$ by subsitutitng $$y$$ with $$y=\\\\sqrt{1-\\\\frac{x^2}{4}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$A=2x \\\\sqrt{4-x^2}$$","$$A=2\\\\sqrt{4-x^2}$$"]},{"id":"aef8b03optimize7a-h7","type":"hint","dependencies":["aef8b03optimize7a-h6"],"title":"Domain","text":"Next, find the domain of interest.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize7a-h8","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["[0, 2]"],"dependencies":["aef8b03optimize7a-h7"],"title":"First Quadrant Domain","text":"Looking at the first quadrant, the x-coordinate of the problem must satisfy what domain? Write your answer in the following example format: [3, 8].","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize7a-h9","type":"hint","dependencies":["aef8b03optimize7a-h8"],"title":"Critical Point","text":"Find the critical point within the domain.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize7a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{8-4x^2}{\\\\sqrt{4-x^2}}$$"],"dependencies":["aef8b03optimize7a-h9"],"title":"Derivative","text":"Firstly, find the derivative to find the critical point. What is $$A\'(x)=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\frac{8-4x^2}{\\\\sqrt{4-x^2}}$$","$$\\\\frac{4-4x^2}{\\\\sqrt{4-x^2}}$$"]},{"id":"aef8b03optimize7a-h11","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["2"],"dependencies":["aef8b03optimize7a-h10"],"title":"equal to zero","text":"Next, find where $$A\'(x)=0$$ to find the critical point. $$x^2=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize7a-h12","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$sqrt2$$"],"dependencies":["aef8b03optimize7a-h11"],"title":"crit point within the domain","text":"Since the interval of interest is [0,2], which critical point is within this domain?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$-sqrt2$$","$$sqrt2$$"]},{"id":"aef8b03optimize7a-h13","type":"hint","dependencies":["aef8b03optimize7a-h12"],"title":"Max or Min","text":"Justify that the critical point is a maximum.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize7a-h14","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["aef8b03optimize7a-h13"],"title":"Sole Critical Point?","text":"According to the figure, is $$sqrt2$$ the only critical point of A(x) in the interval [0,2]?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["Yes","No"]},{"id":"aef8b03optimize7a-h15","type":"hint","dependencies":["aef8b03optimize7a-h14"],"title":"L and W","text":"Lastly, find out the length L and width W.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize7a-h16","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1}{\\\\sqrt{2}}$$"],"dependencies":["aef8b03optimize7a-h15"],"title":"$$y$$","text":"If $$x=sqrt2$$, then $$y=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\frac{1}{\\\\sqrt{2}}$$","$$\\\\sqrt{2}$$"]},{"id":"aef8b03optimize7a-h17","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2\\\\sqrt{2}$$"],"dependencies":["aef8b03optimize7a-h16"],"title":"L","text":"If $$L=2x$$, $$L=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$2\\\\sqrt{2}$$","$$\\\\sqrt{2}$$"]},{"id":"aef8b03optimize7a-h18","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\sqrt{2}$$"],"dependencies":["aef8b03optimize7a-h17"],"title":"W","text":"If $$W=2y$$, $$W=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$2\\\\sqrt{2}$$","$$\\\\sqrt{2}$$"]}]}}]},{"id":"aef8b03optimize8","title":"Maximum Area of the Rectangle","body":"","variabilization":{},"oer":"https://openstax.org/books/calculus-volume-1/pages/4-7-applied-optimization-problems <Openstax: Applied Optimization Problems>","license":0,"lesson":"4.7 Applied Optimization Problems","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aef8b03optimize8a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"What is the maximum area for the rectangle of the previous problem?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"aef8b03optimize8a-h1","type":"hint","dependencies":[],"title":"Area","text":"Find the area of the rectangle by using the length and width from the previous problem.","variabilization":{},"oer":"","license":""},{"id":"aef8b03optimize8a-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["4"],"dependencies":["aef8b03optimize8a-h1"],"title":"Area of the Rectangle","text":"What is $$2sqrt2 sqrt2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"aef8b03optimize9","title":"Minimizing Surface Area","body":"\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/books/calculus-volume-1/pages/4-7-applied-optimization-problems <Openstax: Applied Optimization Problems>","license":0,"lesson":"4.7 Applied Optimization Problems","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"aef8b03optimize9a","stepAnswer":["$$108\\\\times4^{\\\\frac{1}{3}}$$"],"problemType":"MultipleChoice","stepTitle":"A rectangular box with a square base, an open top, and a volume of $$216$$ in**3 is to be constructed. What should the dimensions of the box be to minimize the surface area of the box?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$108\\\\times4^{\\\\frac{1}{3}}$$","choices":["$$108\\\\times4^{\\\\frac{1}{3}}$$","$${104}^{\\\\frac{1}{3}}$$"],"hints":{"DefaultPathway":[{"id":"aef8b03optimize9a-h1","type":"hint","dependencies":[],"title":"surface area","text":"Let S equal the surface area of the box. Find the equation that represents S.","variabilization":{},"oer":"","license":""},{"id":"aef8b03optimize9a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$4x y+x^2$$"],"dependencies":["aef8b03optimize9a-h1"],"title":"$$S=___$$?","text":"What is $$S=___$$ in terms of $$x$$ and $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$4x y+x^2$$","$$4x y+y^2$$"]},{"id":"aef8b03optimize9a-h3","type":"hint","dependencies":["aef8b03optimize9a-h2"],"title":"Volume","text":"Let V equal the volume of the box. Find the equation that represents $$V=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2 y$$"],"dependencies":["aef8b03optimize9a-h3"],"title":"$$V=___$$?","text":"In terms of the box\'s dimensions, what is V in terms of $$x$$ and $$y$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize9a-h5","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["216"],"dependencies":["aef8b03optimize9a-h4"],"title":"volume","text":"According to the information provided in the problem, $$V=216$$. Therefore, $$x^2 y$$ equals what numeric value?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize9a-h6","type":"hint","dependencies":["aef8b03optimize9a-h5"],"title":"S(x)","text":"Rewrite the surface area equation as a function in terms of $$x$$ only.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize9a-h7","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{216}{x^2}$$"],"dependencies":["aef8b03optimize9a-h6"],"title":"solving for $$y$$","text":"Solving for the constraint equation for $$y$$, $$y=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$\\\\frac{216}{x^2}$$","$$\\\\frac{216}{x}$$"]},{"id":"aef8b03optimize9a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{864}{x}+x^2$$"],"dependencies":["aef8b03optimize9a-h7"],"title":"subsitute $$y$$","text":"Substitute $$y$$ in $$4xy+x^2$$ with $$y=\\\\frac{216}{x^2}$$. Thus, we can rewrite $$S(x)=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize9a-h9","type":"hint","dependencies":["aef8b03optimize9a-h8"],"title":"domain","text":"Find the domain of interest.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize9a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["aef8b03optimize9a-h9"],"title":"left endpoint","text":"Because we need $$x^7=216$$, also written as $$y=\\\\frac{216}{x^2}$$, we cannot have $$x=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize9a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["positive"],"dependencies":["aef8b03optimize9a-h10"],"title":"right endpoint","text":"Fill in the blank. X is allowed to have any $$___$$ value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["positive","negative","positive and negative"]},{"id":"aef8b03optimize9a-h12","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(0,\\\\infty)$$"],"dependencies":["aef8b03optimize9a-h11"],"title":"domain in mathmatical connotation","text":"Write the domain in this format. Ex: \\"[3, 8)\\"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$(0,\\\\infty)$$","$$(-\\\\infty,\\\\infty)$$"]},{"id":"aef8b03optimize9a-h13","type":"hint","dependencies":["aef8b03optimize9a-h12"],"title":"critical point","text":"Find the x-value for when S(x) is at a minimum by finding the derivative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"aef8b03optimize9a-h14","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-\\\\left(\\\\frac{864}{x^2}\\\\right)+2x$$"],"dependencies":["aef8b03optimize9a-h13"],"title":"S\'(x)","text":"What is S\'(x)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$-\\\\left(\\\\frac{864}{x^2}\\\\right)+2x$$","$$-\\\\left(\\\\frac{864}{x^2}\\\\right)$$"]},{"id":"aef8b03optimize9a-h15","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$6\\\\sqrt[3]{2}$$"],"dependencies":["aef8b03optimize9a-h14"],"title":"$$0$$","text":"Set $$S\'(x)=0$$ to find the critical point and solve for $$x$$. $$X=___$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["$$6\\\\sqrt[3]{2}$$","$$6\\\\sqrt{2}$$"]},{"id":"aef8b03optimize9a-h16","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["True"],"dependencies":["aef8b03optimize9a-h15"],"title":"Max or Min","text":"$$x=6\\\\sqrt[3]{2}$$ is the only critical point of S and the absolute minimum. True or False?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["False","True"]}]}}]},{"id":"aefa26bnormal1","title":"Calculating the Confidence Interval","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 A Single Population Mean using the Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aefa26bnormal1a","stepAnswer":["$$(11.8, 18.2)$$"],"problemType":"MultipleChoice","stepTitle":"Suppose we have data from a sample. The sample mean is $$15$$, and the error bound for the mean is $$3.2$$. What is the confidence interval estimate for the population mean?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(11.8, 18.2)$$","choices":["$$(11.8, 18.2)$$","$$(11.2, 17.6)$$","$$(13.1, 19.4)$$","$$(13.4, 17.7)$$"],"hints":{"DefaultPathway":[{"id":"aefa26bnormal1a-h1","type":"hint","dependencies":[],"title":"Upper and Lower Limits","text":"The confidence interval estimate is the upper and lower bounds of the mean. Add the error to the mean to get the upper limit, and subtract the error from the mean to get the lower limit.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18.2$$"],"dependencies":["aefa26bnormal1a-h1"],"title":"Upper Limit","text":"What is $$15+3.2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11.8$$"],"dependencies":["aefa26bnormal1a-h1"],"title":"Lower Limit","text":"What is $$15-3.2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal1a-h4","type":"hint","dependencies":["aefa26bnormal1a-h2","aefa26bnormal1a-h3"],"title":"Combine","text":"Combine information about the upper and lower limits to get the answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aefa26bnormal10","title":"Changing the Confidence Level or Sample Size","body":"Suppose average pizza delivery times are normally distributed with an unknown population mean and a population standard deviation of six minutes. A random sample of $$28$$ pizza delivery restaurants is taken and has a sample mean delivery time of $$36$$ minutes.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 A Single Population Mean using the Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aefa26bnormal10a","stepAnswer":["$$(34.604, 37.396)$$"],"problemType":"MultipleChoice","stepTitle":"Assume the sample size is changed to $$50$$ restaurants with the same sample mean. Find a 90% confidence interval estimate for the population mean delivery time.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(34.604, 37.396)$$","choices":["$$(34.604, 37.396)$$","$$(32.137, 39.813)$$","$$(36.391, 38.211)$$","$$(35.924, 41.812)$$"],"hints":{"DefaultPathway":[{"id":"aefa26bnormal10a-h1","type":"hint","dependencies":[],"title":"Calculator","text":"Using the TI-83, 83+, $$84$$, 84+ calculator, Press STAT and arrow over to TESTS.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal10a-h2","type":"hint","dependencies":["aefa26bnormal10a-h1"],"title":"Calculator","text":"Arrow down to 7:ZInterval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal10a-h3","type":"hint","dependencies":["aefa26bnormal10a-h2"],"title":"Calculator","text":"Press ENTER. Arrow to Stats and press ENTER.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal10a-h4","type":"hint","dependencies":["aefa26bnormal10a-h3"],"title":"Calculator","text":"Arrow down and enter $$6$$ for \u03c3, $$36$$ for the mean, $$50$$ for $$n$$, and $$0.90$$ for the confidence level.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal10a-h5","type":"hint","dependencies":["aefa26bnormal10a-h4"],"title":"Calculator","text":"Arrow down to Calculate and press ENTER.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aefa26bnormal11","title":"Working Backwards to Find the Error Bound or Sample Mean","body":"Suppose we know that a confidence interval is $$(67.18, 68.82)$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 A Single Population Mean using the Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aefa26bnormal11a","stepAnswer":["$$0.82$$"],"problemType":"TextBox","stepTitle":"What is the error bound?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.82$$","hints":{"DefaultPathway":[{"id":"aefa26bnormal11a-h1","type":"hint","dependencies":[],"title":"Difference","text":"Calculate the difference between the upper and lower bounds.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.64$$"],"dependencies":["aefa26bnormal11a-h1"],"title":"Difference","text":"What is $$68.82-67.18$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal11a-h3","type":"hint","dependencies":["aefa26bnormal11a-h2"],"title":"Divide by $$2$$","text":"Divide this difference by $$2$$ since the error is on both sides of the mean.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.82$$"],"dependencies":["aefa26bnormal11a-h3"],"title":"Divide by $$2$$","text":"What is $$\\\\frac{1.64}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aefa26bnormal12","title":"Working Backwards to Find the Error Bound or Sample Mean","body":"Suppose we know that a confidence interval is $$(67.18, 68.82)$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 A Single Population Mean using the Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aefa26bnormal12a","stepAnswer":["$$68$$"],"problemType":"TextBox","stepTitle":"What is the sample mean?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$68$$","hints":{"DefaultPathway":[{"id":"aefa26bnormal12a-h1","type":"hint","dependencies":[],"title":"Sum","text":"Calculate the sum of the upper and lower bounds.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$136$$"],"dependencies":["aefa26bnormal12a-h1"],"title":"Sum","text":"What is $$68.82+67.18$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal12a-h3","type":"hint","dependencies":["aefa26bnormal12a-h2"],"title":"Divide by $$2$$","text":"Divide this sum by $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$68$$"],"dependencies":["aefa26bnormal12a-h3"],"title":"Divide by $$2$$","text":"What is $$\\\\frac{136}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aefa26bnormal13","title":"Working Backwards to Find the Error Bound or Sample Mean","body":"Suppose we know that a confidence interval is $$(42.12, 47.88)$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 A Single Population Mean using the Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aefa26bnormal13a","stepAnswer":["$$2.88$$"],"problemType":"TextBox","stepTitle":"What is the error bound?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2.88$$","hints":{"DefaultPathway":[{"id":"aefa26bnormal13a-h1","type":"hint","dependencies":[],"title":"Difference","text":"Calculate the difference between the upper and lower bounds.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal13a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5.76$$"],"dependencies":["aefa26bnormal13a-h1"],"title":"Difference","text":"What is $$47.88-42.12$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal13a-h3","type":"hint","dependencies":["aefa26bnormal13a-h2"],"title":"Divide by $$2$$","text":"Divide this difference by $$2$$ since the error is on both sides of the mean.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.88$$"],"dependencies":["aefa26bnormal13a-h3"],"title":"Divide by $$2$$","text":"What is $$\\\\frac{5.76}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aefa26bnormal14","title":"Working Backwards to Find the Error Bound or Sample Mean","body":"Suppose we know that a confidence interval is $$(42.12, 47.88)$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 A Single Population Mean using the Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aefa26bnormal14a","stepAnswer":["$$45$$"],"problemType":"TextBox","stepTitle":"What is the sample mean?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$45$$","hints":{"DefaultPathway":[{"id":"aefa26bnormal14a-h1","type":"hint","dependencies":[],"title":"Sum","text":"Calculate the sum of the upper and lower bounds.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal14a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$90$$"],"dependencies":["aefa26bnormal14a-h1"],"title":"Sum","text":"What is $$47.88+42.12$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal14a-h3","type":"hint","dependencies":["aefa26bnormal14a-h2"],"title":"Divide by $$2$$","text":"Divide this sum by $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal14a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$45$$"],"dependencies":["aefa26bnormal14a-h3"],"title":"Divide by $$2$$","text":"What is $$\\\\frac{90}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aefa26bnormal15","title":"Calculating the Sample Size $$n$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 A Single Population Mean using the Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aefa26bnormal15a","stepAnswer":["$$217$$"],"problemType":"TextBox","stepTitle":"The population standard deviation for the age of Foothill College students is $$15$$ years. If we want to be 95% confident that the sample mean age is within two years of the true population mean age of Foothill College students, how many randomly selected Foothill College students must be surveyed?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$217$$","hints":{"DefaultPathway":[{"id":"aefa26bnormal15a-h1","type":"hint","dependencies":[],"title":"Formula","text":"The formula for sample size is $$n=\\\\frac{z^2 {\\\\sigma}^2}{{EBM}^2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal15a-h2","type":"hint","dependencies":["aefa26bnormal15a-h1"],"title":"Find Missing Value","text":"We know the value of $$\u03c3=15$$ and $$EBM=2$$. We are missing $$z$$ so calculate its value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal15a-h3","type":"hint","dependencies":["aefa26bnormal15a-h2"],"title":"Z-value","text":"Since we want to be 95% confident, we will use z_0.025.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal15a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.96$$"],"dependencies":["aefa26bnormal15a-h3"],"title":"Calculator","text":"Use your calculator, a computer, or a probability table for the standard normal distribution to find z_0.025. What is its value?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal15a-h5","type":"hint","dependencies":["aefa26bnormal15a-h4"],"title":"Substitute","text":"Substitute all known values into the formula for sample size.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal15a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$216.09$$"],"dependencies":["aefa26bnormal15a-h5"],"title":"Calculate $$n$$","text":"What is $$n=\\\\frac{{1.96}^2 {15}^2}{2^2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal15a-h7","type":"hint","dependencies":["aefa26bnormal15a-h6"],"title":"Round $$n$$","text":"Round $$n$$ up to the next higher integer so the sample size is large enough.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal15a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$217$$"],"dependencies":["aefa26bnormal15a-h7"],"title":"Round $$n$$","text":"What is $$n$$ rounded up to the next integer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aefa26bnormal16","title":"Calculating the Sample Size $$n$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 A Single Population Mean using the Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aefa26bnormal16a","stepAnswer":["$$35$$"],"problemType":"TextBox","stepTitle":"The population standard deviation for the height of high school basketball players is three inches. If we want to be 95% confident that the sample mean height is within one inch of the true population mean height, how many randomly selected students must be surveyed?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$35$$","hints":{"DefaultPathway":[{"id":"aefa26bnormal16a-h1","type":"hint","dependencies":[],"title":"Formula","text":"The formula for sample size is $$n=\\\\frac{z^2 {\\\\sigma}^2}{{EBM}^2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal16a-h2","type":"hint","dependencies":["aefa26bnormal16a-h1"],"title":"Find Missing Value","text":"We know the value of $$\u03c3=3$$ and $$EBM=1$$. We are missing $$z$$ so calculate its value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal16a-h3","type":"hint","dependencies":["aefa26bnormal16a-h2"],"title":"Z-value","text":"Since we want to be 95% confident, we will use z_0.025.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal16a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.96$$"],"dependencies":["aefa26bnormal16a-h3"],"title":"Calculator","text":"Use your calculator, a computer, or a probability table for the standard normal distribution to find z_0.025. What is its value?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal16a-h5","type":"hint","dependencies":["aefa26bnormal16a-h4"],"title":"Substitute","text":"Substitute all known values into the formula for sample size.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal16a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$34.574$$"],"dependencies":["aefa26bnormal16a-h5"],"title":"Calculate $$n$$","text":"What is $$n=\\\\frac{{1.96}^2 3^2}{1^2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal16a-h7","type":"hint","dependencies":["aefa26bnormal16a-h6"],"title":"Round $$n$$","text":"Round $$n$$ up to the next higher integer so the sample size is large enough.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal16a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$35$$"],"dependencies":["aefa26bnormal16a-h7"],"title":"Round $$n$$","text":"What is $$n$$ rounded up to the next integer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aefa26bnormal2","title":"Calculating the Confidence Interval","body":"Suppose scores on exams in statistics are normally distributed with an unknown population mean and a population standard deviation of three points. A random sample of $$36$$ scores is taken and gives a sample mean (sample mean score) of $$68$$. Find a confidence interval estimate for the population mean exam score (the mean score on all exams).","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 A Single Population Mean using the Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aefa26bnormal2a","stepAnswer":["$$(67.1775, 68.8225)$$"],"problemType":"MultipleChoice","stepTitle":"Find a 90% confidence interval for the true (population) mean of statistics exam scores.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(67.1775, 68.8225)$$","choices":["$$(67.1775, 68.8225)$$","$$(62.3487, 87.1313)$$","$$(71.8111, 72.3434)$$","$$(66.2343, 69.1241)$$"],"hints":{"DefaultPathway":[{"id":"aefa26bnormal2a-h1","type":"hint","dependencies":[],"title":"Identify Calculations","text":"To find the confidence interval, the sample mean and error (EBM) are needed. We know the sample mean is $$68$$. Now, calculate the EBM.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal2a-h2","type":"hint","dependencies":["aefa26bnormal2a-h1"],"title":"Calculate the EBM","text":"The EBM is calculated through the formula: $$\\\\frac{z_\u03b1}{2} \\\\frac{\\\\sigma}{\\\\sqrt{n}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal2a-h3","type":"hint","dependencies":["aefa26bnormal2a-h2"],"title":"Calculate the EBM","text":"We know $$\u03c3=3$$ and $$n=36$$. The confidence level (CL) is 90%. Based on this, calculate the EBM.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.645$$"],"dependencies":["aefa26bnormal2a-h3"],"title":"Calculate $$\\\\frac{z_\u03b1}{2}$$","text":"What is $$\\\\frac{z_\u03b1}{2}$$? Remember, $$\\\\alpha=1-CL$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aefa26bnormal2a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1$$"],"dependencies":[],"title":"Calculate \ud835\udefc","text":"What is 1-CL?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal2a-h4-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.05$$"],"dependencies":[],"title":"Calculate $$\\\\frac{\u03b1}{2}$$","text":"What is $$\\\\frac{\u03b1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal2a-h4-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.05$$"],"dependencies":[],"title":"Right Area","text":"Now, we have z_0.05. What is the area to the right of z_0.05?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal2a-h4-s4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.95$$"],"dependencies":[],"title":"Left Area","text":"What is the area to the left of z_0.05?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal2a-h4-s5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.645$$"],"dependencies":[],"title":"Calculator","text":"What is $$invNorm(0.95, 0, 1)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal2a-h4-s6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.8225$$"],"dependencies":[],"title":"Calculate the EBM","text":"What is $$\\\\operatorname{1.645}\\\\left(\\\\frac{3}{\\\\sqrt{36}}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aefa26bnormal2a-h5","type":"hint","dependencies":["aefa26bnormal2a-h4"],"title":"Upper and Lower Limits","text":"The confidence interval estimate is the upper and lower bounds of the mean. Add the EBM to the mean to get the upper limit, and subtract the EBM from the mean to get the lower limit.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$68.8225$$"],"dependencies":["aefa26bnormal2a-h5"],"title":"Upper Limit","text":"What is $$68+0.8225$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal2a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$67.1775$$"],"dependencies":["aefa26bnormal2a-h5"],"title":"Lower Limit","text":"What is $$68-0.8225$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal2a-h8","type":"hint","dependencies":["aefa26bnormal2a-h6","aefa26bnormal2a-h7"],"title":"Combine","text":"Combine information about the upper and lower limits to get the answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aefa26bnormal3","title":"Calculating the Confidence Interval","body":"Suppose average pizza delivery times are normally distributed with an unknown population mean and a population standard deviation of six minutes. A random sample of $$28$$ pizza delivery restaurants is taken and has a sample mean delivery time of $$36$$ minutes.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 A Single Population Mean using the Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aefa26bnormal3a","stepAnswer":["$$(34.135, 37.865)$$"],"problemType":"MultipleChoice","stepTitle":"Find a 90% confidence interval estimate for the population mean delivery time.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(34.135, 37.865)$$","choices":["$$(34.135, 37.865)$$","$$(33.213, 38.912)$$","$$(33.121, 38.724)$$","$$(37.438, 39.124)$$"],"hints":{"DefaultPathway":[{"id":"aefa26bnormal3a-h1","type":"hint","dependencies":[],"title":"Calculator","text":"Using the TI-83, 83+, $$84$$, 84+ calculator, Press STAT and arrow over to TESTS.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal3a-h2","type":"hint","dependencies":["aefa26bnormal3a-h1"],"title":"Calculator","text":"Arrow down to 7:ZInterval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal3a-h3","type":"hint","dependencies":["aefa26bnormal3a-h2"],"title":"Calculator","text":"Press ENTER. Arrow to Stats and press ENTER.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal3a-h4","type":"hint","dependencies":["aefa26bnormal3a-h3"],"title":"Calculator","text":"Arrow down and enter $$6$$ for \u03c3, $$36$$ for the mean, $$28$$ for $$n$$, and $$0.90$$ for the confidence level.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal3a-h5","type":"hint","dependencies":["aefa26bnormal3a-h4"],"title":"Calculator","text":"Arrow down to Calculate and press ENTER.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal3a-h6","type":"hint","dependencies":["aefa26bnormal3a-h5"],"title":"Output","text":"The output is the answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aefa26bnormal4","title":"Calculating the Confidence Interval","body":"The Specific Absorption Rate (SAR) for a cell phone measures the amount of radio frequency (RF) energy absorbed by the user\u2019s body when using the handset. Every cell phone emits RF energy. Different phone models have different SAR measures. To receive certification from the Federal Communications Commission (FCC) for sale in the United States, the SAR level for a cell phone must be no more than $$1.6$$ watts per kilogram. Table $$8.1$$ shows the highest SAR level for a random selection of cell phone models as measured by the FCC.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 A Single Population Mean using the Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aefa26bnormal4a","stepAnswer":["$$(0.8809, 1.1671)$$"],"problemType":"MultipleChoice","stepTitle":"Find a 98% confidence interval for the true (population) mean of the Specific Absorption Rates (SARs) for cell phones. Assume that the population standard deviation is $$\u03c3=0.337$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0.8809, 1.1671)$$","choices":["$$(0.8809, 1.1671)$$","$$(0.8767, 1.1357)$$","$$(0.6757, 1.2657)$$","$$(0.7669, 1.296)$$"],"hints":{"DefaultPathway":[{"id":"aefa26bnormal4a-h1","type":"hint","dependencies":[],"title":"Sample Mean","text":"Find the sample mean.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.024$$"],"dependencies":["aefa26bnormal4a-h1"],"title":"Sample Mean Using SARs","text":"Add all the SARs and divide by the number of SARs to get the mean. What is this value?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal4a-h3","type":"hint","dependencies":["aefa26bnormal4a-h2"],"title":"Calculate the EBM","text":"Now, calculate the EBM.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal4a-h4","type":"hint","dependencies":["aefa26bnormal4a-h3"],"title":"Calculate the EBM","text":"The EBM is calculated through the formula: $$\\\\frac{z_\u03b1}{2} \\\\frac{\\\\sigma}{\\\\sqrt{n}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal4a-h5","type":"hint","dependencies":["aefa26bnormal4a-h4"],"title":"Calculate the EBM","text":"We know $$\u03c3=0.337$$. The confidence level (CL) is 98%, and there are $$30$$ entries in the table. Therefore, $$n=30$$. Based on this, calculate the EBM.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal4a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.236$$"],"dependencies":["aefa26bnormal4a-h5"],"title":"Calculate $$\\\\frac{z_\u03b1}{2}$$","text":"What is $$\\\\frac{z_\u03b1}{2}$$? Remember, $$\\\\alpha=1-CL$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aefa26bnormal4a-h6-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.02$$"],"dependencies":[],"title":"Calculate \ud835\udefc","text":"What is 1-CL?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal4a-h6-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.01$$"],"dependencies":[],"title":"Calculate $$\\\\frac{\u03b1}{2}$$","text":"What is $$\\\\frac{\u03b1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal4a-h6-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.01$$"],"dependencies":[],"title":"Calculator","text":"Now, we have z_0.01. Use your calculator, a computer, or a probability table for the standard normal distribution to find z_0.01. What is this value?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal4a-h6-s4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1431$$"],"dependencies":[],"title":"Calculate the EBM","text":"What is $$\\\\operatorname{2.236}\\\\left(\\\\frac{0.337}{\\\\sqrt{30}}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aefa26bnormal4a-h7","type":"hint","dependencies":["aefa26bnormal4a-h6"],"title":"Upper and Lower Limits","text":"The confidence interval estimate is the upper and lower bounds of the mean. Add the EBM to the mean to get the upper limit, and subtract the EBM from the mean to get the lower limit.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal4a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.1671$$"],"dependencies":["aefa26bnormal4a-h7"],"title":"Upper Limit","text":"What is $$1.024+0.1431$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal4a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.8809$$"],"dependencies":["aefa26bnormal4a-h7"],"title":"Lower Limit","text":"What is $$1.024-0.1431$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal4a-h10","type":"hint","dependencies":["aefa26bnormal4a-h8","aefa26bnormal4a-h9"],"title":"Combine","text":"Combine information about the upper and lower limits to get the answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aefa26bnormal5","title":"Calculating the Confidence Interval","body":"Table $$8.2$$ shows a different random sampling of $$20$$ cell phone models.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 A Single Population Mean using the Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aefa26bnormal5a","stepAnswer":["$$(0.804, 1.077)$$"],"problemType":"MultipleChoice","stepTitle":"Use this data to calculate a 93% confidence interval for the true mean SAR for cell phones certified for use in the United States. Assume that the population standard deviation is $$\u03c3=0.337$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0.804, 1.077)$$","choices":["$$(0.804, 1.077)$$","$$(0.782, 1.142)$$","$$(0.812, 1.162)$$","$$(0.623, 1.381)$$"],"hints":{"DefaultPathway":[{"id":"aefa26bnormal5a-h1","type":"hint","dependencies":[],"title":"Sample Mean","text":"Find the sample mean.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.9401$$"],"dependencies":["aefa26bnormal5a-h1"],"title":"Sample Mean Using SARs","text":"Add all the SARs and divide by the number of SARs to get the mean. What is this value?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal5a-h3","type":"hint","dependencies":["aefa26bnormal5a-h2"],"title":"Calculator","text":"Using the TI-83, 83+, $$84$$, 84+ calculator, Press STAT and arrow over to TESTS.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal5a-h4","type":"hint","dependencies":["aefa26bnormal5a-h3"],"title":"Calculator","text":"Arrow down to 7:ZInterval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal5a-h5","type":"hint","dependencies":["aefa26bnormal5a-h4"],"title":"Calculator","text":"Press ENTER. Arrow to Stats and press ENTER.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal5a-h6","type":"hint","dependencies":["aefa26bnormal5a-h5"],"title":"Calculator","text":"Arrow down and enter $$0.337$$ for \u03c3, $$0.9401$$ for the mean, $$20$$ for $$n$$, and $$0.93$$ for the confidence level.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal5a-h7","type":"hint","dependencies":["aefa26bnormal5a-h6"],"title":"Calculator","text":"Arrow down to Calculate and press ENTER.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal5a-h8","type":"hint","dependencies":["aefa26bnormal5a-h7"],"title":"Output","text":"The output is the answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aefa26bnormal6","title":"Changing the Confidence Level or Sample Size","body":"Suppose scores on exams in statistics are normally distributed with an unknown population mean and a population standard deviation of three points. A random sample of $$36$$ scores is taken and gives a sample mean (sample mean score) of $$68$$. Find a confidence interval estimate for the population mean exam score (the mean score on all exams).","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 A Single Population Mean using the Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aefa26bnormal6a","stepAnswer":["$$(67.02, 68.98)$$"],"problemType":"MultipleChoice","stepTitle":"Find a 95% confidence interval for the true (population) mean of statistics exam scores.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(67.02, 68.98)$$","choices":["$$(67.02, 68.98)$$","$$(66.43, 68.19)$$","$$(67.91, 69.84)$$","$$(66.18, 69.24)$$"],"hints":{"DefaultPathway":[{"id":"aefa26bnormal6a-h1","type":"hint","dependencies":[],"title":"Identify Calculations","text":"To find the confidence interval, the sample mean and error (EBM) are needed. We know the sample mean is $$68$$. Now, calculate the EBM.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal6a-h2","type":"hint","dependencies":["aefa26bnormal6a-h1"],"title":"Calculate the EBM","text":"The EBM is calculated through the formula: $$\\\\frac{z_\u03b1}{2} \\\\frac{\\\\sigma}{\\\\sqrt{n}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal6a-h3","type":"hint","dependencies":["aefa26bnormal6a-h2"],"title":"Calculate the EBM","text":"We know $$\u03c3=3$$ and $$n=36$$. The confidence level (CL) is 95%. Based on this, calculate the EBM.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.96$$"],"dependencies":["aefa26bnormal6a-h3"],"title":"Calculate $$\\\\frac{z_\u03b1}{2}$$","text":"What is $$\\\\frac{z_\u03b1}{2}$$? Remember, $$\\\\alpha=1-CL$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aefa26bnormal6a-h4-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.05$$"],"dependencies":[],"title":"Calculate \ud835\udefc","text":"What is 1-CL?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal6a-h4-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.025$$"],"dependencies":[],"title":"Calculate $$\\\\frac{\u03b1}{2}$$","text":"What is $$\\\\frac{\u03b1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal6a-h4-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.025$$"],"dependencies":[],"title":"Right Area","text":"Now, we have z_0.05. What is the area to the right of z_0.05?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal6a-h4-s4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.975$$"],"dependencies":[],"title":"Left Area","text":"What is the area to the left of z_0.05?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal6a-h4-s5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.96$$"],"dependencies":[],"title":"Calculator","text":"What is $$invNorm(0.975, 0, 1)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal6a-h4-s6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.98$$"],"dependencies":[],"title":"Calculate the EBM","text":"What is $$\\\\operatorname{1.96}\\\\left(\\\\frac{3}{\\\\sqrt{36}}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aefa26bnormal6a-h5","type":"hint","dependencies":["aefa26bnormal6a-h4"],"title":"Upper and Lower Limits","text":"The confidence interval estimate is the upper and lower bounds of the mean. Add the EBM to the mean to get the upper limit, and subtract the EBM from the mean to get the lower limit.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal6a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$67.02$$"],"dependencies":["aefa26bnormal6a-h5"],"title":"Upper Limit","text":"What is $$68+0.98$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal6a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$68.98$$"],"dependencies":["aefa26bnormal6a-h5"],"title":"Lower Limit","text":"What is $$68-0.98$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal6a-h8","type":"hint","dependencies":["aefa26bnormal6a-h6","aefa26bnormal6a-h7"],"title":"Combine","text":"Combine information about the upper and lower limits to get the answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aefa26bnormal7","title":"Changing the Confidence Level or Sample Size","body":"Suppose average pizza delivery times are normally distributed with an unknown population mean and a population standard deviation of six minutes. A random sample of $$28$$ pizza delivery restaurants is taken and has a sample mean delivery time of $$36$$ minutes.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 A Single Population Mean using the Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aefa26bnormal7a","stepAnswer":["$$(33.778, 38.222)$$"],"problemType":"MultipleChoice","stepTitle":"Find a 95% confidence interval estimate for the population mean delivery time.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(33.778, 38.222)$$","choices":["$$(33.778, 38.222)$$","$$(31.314, 37.904)$$","$$(32.371, 39.134)$$","$$(38.913, 44.133)$$"],"hints":{"DefaultPathway":[{"id":"aefa26bnormal7a-h1","type":"hint","dependencies":[],"title":"Calculator","text":"Using the TI-83, 83+, $$84$$, 84+ calculator, Press STAT and arrow over to TESTS.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal7a-h2","type":"hint","dependencies":["aefa26bnormal7a-h1"],"title":"Calculator","text":"Arrow down to 7:ZInterval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal7a-h3","type":"hint","dependencies":["aefa26bnormal7a-h2"],"title":"Calculator","text":"Press ENTER. Arrow to Stats and press ENTER.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal7a-h4","type":"hint","dependencies":["aefa26bnormal7a-h3"],"title":"Calculator","text":"Arrow down and enter $$6$$ for \u03c3, $$36$$ for the mean, $$28$$ for $$n$$, and $$0.95$$ for the confidence level.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal7a-h5","type":"hint","dependencies":["aefa26bnormal7a-h4"],"title":"Calculator","text":"Arrow down to Calculate and press ENTER.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal7a-h6","type":"hint","dependencies":["aefa26bnormal7a-h5"],"title":"Output","text":"The output is the answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aefa26bnormal8","title":"Changing the Confidence Level or Sample Size","body":"Suppose scores on exams in statistics are normally distributed with an unknown population mean and a population standard deviation of three points. A random sample of $$36$$ scores is taken and gives a sample mean (sample mean score) of $$68$$. The EBM is $$0.8225$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 A Single Population Mean using the Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aefa26bnormal8a","stepAnswer":["Decrease"],"problemType":"MultipleChoice","stepTitle":"Does the error bound increase or decrease if we increase the sample size and use $$n=100$$ instead of $$n=36$$?","stepBody":"","answerType":"string","variabilization":{},"choices":["Decrease","Increase"],"hints":{"DefaultPathway":[{"id":"aefa26bnormal8a-h1","type":"hint","dependencies":[],"title":"Calculate the EBM With $$n=100$$","text":"The EBM is calculated through the formula: $$\\\\frac{z_\u03b1}{2} \\\\frac{\\\\sigma}{\\\\sqrt{n}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal8a-h2","type":"hint","dependencies":["aefa26bnormal8a-h1"],"title":"Calculate the EBM With $$n=100$$","text":"We know $$\u03c3=3$$ and $$n=100$$. The confidence level (CL) is 90%. Based on this, calculate the EBM.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.645$$"],"dependencies":["aefa26bnormal8a-h2"],"title":"Calculate $$\\\\frac{z_\u03b1}{2}$$","text":"What is $$\\\\frac{z_\u03b1}{2}$$? Remember, $$\\\\alpha=1-CL$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aefa26bnormal8a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1$$"],"dependencies":[],"title":"Calculate \ud835\udefc","text":"What is 1-CL?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal8a-h3-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.05$$"],"dependencies":[],"title":"Calculate $$\\\\frac{\u03b1}{2}$$","text":"What is $$\\\\frac{\u03b1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal8a-h3-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.05$$"],"dependencies":[],"title":"Right Area","text":"Now, we have z_0.05. What is the area to the right of z_0.05?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal8a-h3-s4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.95$$"],"dependencies":[],"title":"Left Area","text":"What is the area to the left of z_0.05?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal8a-h3-s5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.645$$"],"dependencies":[],"title":"Calculator","text":"What is $$invNorm(0.95, 0, 1)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal8a-h3-s6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.4935$$"],"dependencies":[],"title":"Calculate the EBM","text":"What is $$\\\\operatorname{1.645}\\\\left(\\\\frac{3}{\\\\sqrt{100}}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aefa26bnormal8a-h4","type":"hint","dependencies":["aefa26bnormal8a-h3"],"title":"Compare","text":"Compare the EBMs of $$n=36$$ and $$n=100$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal8a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Less"],"dependencies":["aefa26bnormal8a-h4"],"title":"Compare","text":"Is $$0.4935$$ greater than or less than $$0.8225$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Greater","Less"]}]}},{"id":"aefa26bnormal8b","stepAnswer":["Narrower"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{Increase}{Decrease}$$","stepBody":"Does this $$\\\\frac{increase}{decrease}$$ make the confidence interval narrower or wider?","answerType":"string","variabilization":{},"choices":["Narrower","Wider"],"hints":{"DefaultPathway":[{"id":"aefa26bnormal8b-h1","type":"hint","dependencies":[],"title":"Number of Values in Confidence Interval","text":"Since the EBM decreased, there are less values in the confidence interval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aefa26bnormal9","title":"Changing the Confidence Level or Sample Size","body":"Suppose scores on exams in statistics are normally distributed with an unknown population mean and a population standard deviation of three points. A random sample of $$36$$ scores is taken and gives a sample mean (sample mean score) of $$68$$. The EBM is $$0.8225$$.","variabilization":{},"oer":"https://openstax.org/details/books/introductory-statistics <OpenStax: Introductory Statistics>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 A Single Population Mean using the Normal Distribution","courseName":"OpenStax: Introductory Stats","steps":[{"id":"aefa26bnormal9a","stepAnswer":["Increase"],"problemType":"MultipleChoice","stepTitle":"Does the error bound increase or decrease if we decrease the sample size to $$n=25$$ instead of $$n=36$$?","stepBody":"","answerType":"string","variabilization":{},"choices":["Decrease","Increase"],"hints":{"DefaultPathway":[{"id":"aefa26bnormal9a-h1","type":"hint","dependencies":[],"title":"Calculate the EBM With $$n=100$$","text":"The EBM is calculated through the formula: $$\\\\frac{z_\u03b1}{2} \\\\frac{\\\\sigma}{\\\\sqrt{n}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal9a-h2","type":"hint","dependencies":["aefa26bnormal9a-h1"],"title":"Calculate the EBM With $$n=100$$","text":"We know $$\u03c3=3$$ and $$n=100$$. The confidence level (CL) is 90%. Based on this, calculate the EBM.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.645$$"],"dependencies":["aefa26bnormal9a-h2"],"title":"Calculate $$\\\\frac{z_\u03b1}{2}$$","text":"What is $$\\\\frac{z_\u03b1}{2}$$? Remember, $$\\\\alpha=1-CL$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"aefa26bnormal9a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1$$"],"dependencies":[],"title":"Calculate \ud835\udefc","text":"What is 1-CL?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal9a-h3-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.05$$"],"dependencies":[],"title":"Calculate $$\\\\frac{\u03b1}{2}$$","text":"What is $$\\\\frac{\u03b1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal9a-h3-s3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.05$$"],"dependencies":[],"title":"Right Area","text":"Now, we have z_0.05. What is the area to the right of z_0.05?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal9a-h3-s4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.95$$"],"dependencies":[],"title":"Left Area","text":"What is the area to the left of z_0.05?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal9a-h3-s5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.645$$"],"dependencies":[],"title":"Calculator","text":"What is $$invNorm(0.95, 0, 1)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal9a-h3-s6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.987$$"],"dependencies":[],"title":"Calculate the EBM","text":"What is $$\\\\operatorname{1.645}\\\\left(\\\\frac{3}{\\\\sqrt{25}}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"aefa26bnormal9a-h4","type":"hint","dependencies":["aefa26bnormal9a-h3"],"title":"Compare","text":"Compare the EBMs of $$n=25$$ and $$n=100$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aefa26bnormal9a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Greater"],"dependencies":["aefa26bnormal9a-h4"],"title":"Compare","text":"Is $$0.987$$ greater than or less than $$0.8225$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Greater","Less"]}]}},{"id":"aefa26bnormal9b","stepAnswer":["Wider"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{Increase}{Decrease}$$","stepBody":"Does this $$\\\\frac{increase}{decrease}$$ make the confidence interval narrower or wider?","answerType":"string","variabilization":{},"choices":["Narrower","Wider"],"hints":{"DefaultPathway":[{"id":"aefa26bnormal9b-h1","type":"hint","dependencies":[],"title":"Number of Values in Confidence Interval","text":"Since the EBM increased, there are more values in the confidence interval.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af04890Rational1","title":"Determine the Values for Which a Rational Expression is Undefined","body":"Determine the value for which of the following rational expression is undefined:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Multiply and Divide Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af04890Rational1a","stepAnswer":["when $$c=0$$"],"problemType":"MultipleChoice","stepTitle":"a) $$\\\\frac{8a^2 b}{3c}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"when $$c=0$$","choices":["when $$a=b$$","when $$a=0$$","when $$b=0$$","when $$c=0$$"],"hints":{"DefaultPathway":[{"id":"af04890Rational1a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The rational will be undefined when the $$denominator=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational1a-h2","type":"hint","dependencies":["af04890Rational1a-h1"],"title":"Setting equal to $$0$$","text":"Set the $$denominator=0$$, $$3c=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af04890Rational1b","stepAnswer":["$$b=\\\\frac{-5}{2}$$"],"problemType":"MultipleChoice","stepTitle":"b) $$\\\\frac{4b-3}{2b+5}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$b=\\\\frac{-5}{2}$$","choices":["$$b=0$$","$$b=3$$","$$b=\\\\frac{-5}{2}$$"],"hints":{"DefaultPathway":[{"id":"af04890Rational1b-h1","type":"hint","dependencies":[],"title":"Principle","text":"The rational will be undefined when the $$denominator=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational1b-h2","type":"hint","dependencies":["af04890Rational1b-h1"],"title":"Setting equal to $$0$$","text":"Set the $$denominator=0$$, $$2b+5=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational1b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-5}{2}$$"],"dependencies":["af04890Rational1b-h2"],"title":"Finding values","text":"What is $$b$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af04890Rational1c","stepAnswer":["$$x=-2$$ or $$-3$$"],"problemType":"MultipleChoice","stepTitle":"c) $$\\\\frac{x+4}{x^2+5x+6}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=-2$$ or $$-3$$","choices":["$$x=-2$$ or $$-3$$","$$x=2$$ or $$3$$","$$x=2$$ or $$-3$$"],"hints":{"DefaultPathway":[{"id":"af04890Rational1c-h1","type":"hint","dependencies":[],"title":"Principle","text":"The rational will be undefined when the $$denominator=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational1c-h2","type":"hint","dependencies":["af04890Rational1c-h1"],"title":"Setting equal to $$0$$","text":"Set the $$denominator=0$$, $$x^2+5x+6=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational1c-h3","type":"hint","dependencies":["af04890Rational1c-h2"],"title":"Finding values","text":"Factor the denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af04890Rational10","title":"Multiply Rational Expressions","body":"Simplify the following rational expression:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Multiply and Divide Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af04890Rational10a","stepAnswer":["$$\\\\frac{5x \\\\left(x-2\\\\right)}{10x \\\\left(x+3\\\\right)}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{5x}{x^2+5x+6} \\\\frac{x^2-4}{10x}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5x \\\\left(x-2\\\\right)}{10x \\\\left(x+3\\\\right)}$$","hints":{"DefaultPathway":[{"id":"af04890Rational10a-h1","type":"hint","dependencies":[],"title":"Factoring the numerator","text":"Factor the numerator to $$\\\\left(x-2\\\\right) \\\\left(x+2\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational10a-h2","type":"hint","dependencies":["af04890Rational10a-h1"],"title":"Factoring the denominator","text":"Factor the denominator to $$\\\\left(x+2\\\\right) \\\\left(x+3\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational10a-h3","type":"hint","dependencies":["af04890Rational10a-h2"],"title":"Basic rule","text":"Multiply the rational while remaining in monomial form","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational10a-h4","type":"hint","dependencies":["af04890Rational10a-h3"],"title":"Dividing","text":"Divide by the common factor","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af04890Rational11","title":"Multiply Rational Expressions","body":"Simplify the following rational expression:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Multiply and Divide Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af04890Rational11a","stepAnswer":["$$\\\\frac{\\\\left(a-3\\\\right) \\\\left(a+5\\\\right)}{{\\\\left(a-5\\\\right)}^2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{3a^2-8a-3}{a^2-25} \\\\frac{a^2+10a+25}{3a^2-14a-5}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{\\\\left(a-3\\\\right) \\\\left(a+5\\\\right)}{{\\\\left(a-5\\\\right)}^2}$$","hints":{"DefaultPathway":[{"id":"af04890Rational11a-h1","type":"hint","dependencies":[],"title":"Factoring the numerator","text":"Factor the numerator to $$\\\\left(3a+1\\\\right) \\\\left(a-3\\\\right)$$ and $$\\\\left(a+5\\\\right) \\\\left(a+5\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational11a-h2","type":"hint","dependencies":["af04890Rational11a-h1"],"title":"Factoring the denominator","text":"Factor the denominator to $$\\\\left(a-5\\\\right) \\\\left(a+5\\\\right)$$ and $$\\\\left(3a+1\\\\right) \\\\left(a-5\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational11a-h3","type":"hint","dependencies":["af04890Rational11a-h2"],"title":"Basic rule","text":"Multiply the rational while remaining in monomial form","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational11a-h4","type":"hint","dependencies":["af04890Rational11a-h3"],"title":"Dividing","text":"Divide by the common factor","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af04890Rational12","title":"Multiply Rational Expressions","body":"Simplify the following rational expression:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Multiply and Divide Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af04890Rational12a","stepAnswer":["$$\\\\frac{2x-3}{\\\\left(x-3\\\\right) \\\\left(x-5\\\\right)}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2x^2+5x-12}{x^2-16} \\\\frac{x^2-8x+16}{2x^2-13x+15}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2x-3}{\\\\left(x-3\\\\right) \\\\left(x-5\\\\right)}$$","hints":{"DefaultPathway":[{"id":"af04890Rational12a-h1","type":"hint","dependencies":[],"title":"Factoring the numerator","text":"Factor the numerator to $$\\\\left(2x-3\\\\right) \\\\left(x+4\\\\right)$$ and $$(x-4)(x-4)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational12a-h2","type":"hint","dependencies":["af04890Rational12a-h1"],"title":"Factoring the denominator","text":"Factor the denominator to $$\\\\left(x+4\\\\right) \\\\left(x-4\\\\right)$$ and $$(x-3)(x-5)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational12a-h3","type":"hint","dependencies":["af04890Rational12a-h2"],"title":"Basic rule","text":"Multiply the rational while remaining in monomial form","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational12a-h4","type":"hint","dependencies":["af04890Rational12a-h3"],"title":"Dividing","text":"Divide by the common factor","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af04890Rational13","title":"Determine the domain of a rational function.","body":"Which values are not part of the domain of the following rational function:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Multiply and Divide Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af04890Rational13a","stepAnswer":["$$x=6$$ or $$-2$$"],"problemType":"MultipleChoice","stepTitle":"$$R(x)=\\\\frac{2x^2-14x}{4x^2-16x-48}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=6$$ or $$-2$$","choices":["$$x=6$$ or $$-2$$","$$x=6$$ or $$2$$","$$x=-6$$ or $$-2$$"],"hints":{"DefaultPathway":[{"id":"af04890Rational13a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The domain will be all real numbers except those values that make the denominator zero.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational13a-h2","type":"hint","dependencies":["af04890Rational13a-h1"],"title":"Setting equal to $$0$$","text":"Set the denominator to $$0$$, $$4x^2-16x-48=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational13a-h3","type":"hint","dependencies":["af04890Rational13a-h2"],"title":"Factoring","text":"Factor the denominator to $$4\\\\left(x-6\\\\right) \\\\left(x+2\\\\right)=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational13a-h4","type":"hint","dependencies":["af04890Rational13a-h3"],"title":"Dividing","text":"Divide by the common factor","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af04890Rational14","title":"Determine the domain of a rational function.","body":"Which values are not part of the domain of the following rational function:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Multiply and Divide Rational 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational14a-h3","type":"hint","dependencies":["af04890Rational14a-h2"],"title":"Factoring","text":"Factor the denominator to $$4\\\\left(x-1\\\\right) \\\\left(x+5\\\\right)=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational14a-h4","type":"hint","dependencies":["af04890Rational14a-h3"],"title":"Dividing","text":"Divide by the common factor","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af04890Rational15","title":"Multiply and Divide Rational Functions","body":"Calculate R(x)","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Multiply and Divide Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af04890Rational15a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"$$R(x)=f{\\\\left(x\\\\right)} g{\\\\left(x\\\\right)}$$ where $$f(x)=\\\\frac{2x-6}{x^2-8x+15}$$, $$g(x)=\\\\frac{x^2-25}{2x+10}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"af04890Rational15a-h1","type":"hint","dependencies":[],"title":"Factor the numerator","text":"Factor the numerator to $$2(x-3)$$ and $$\\\\left(x+5\\\\right) \\\\left(x-5\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational15a-h2","type":"hint","dependencies":["af04890Rational15a-h1"],"title":"Factor the denominator","text":"Factor the denominator to $$(x-3)(x-5)$$ and $$2\\\\left(x+5\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational15a-h3","type":"hint","dependencies":["af04890Rational15a-h2"],"title":"Expansion of the binomial","text":"Multiply the rational while remaining in monomial form","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational15a-h4","type":"hint","dependencies":["af04890Rational15a-h3"],"title":"Divisiding","text":"Divide by the common factor","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af04890Rational2","title":"Determine the Values for Which a Rational Expression is Undefined","body":"Determine the value for which of the following rational expression is undefined:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Multiply and Divide Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af04890Rational2a","stepAnswer":["when $$x=0$$"],"problemType":"MultipleChoice","stepTitle":"a) $$\\\\frac{3y^2}{8x}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"when $$x=0$$","choices":["when $$x=y$$","when $$x=0$$","when $$y=0$$"],"hints":{"DefaultPathway":[{"id":"af04890Rational2a-h1","type":"hint","dependencies":[],"title":"Principle","text":"The rational will be undefined when the $$denominator=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational2a-h2","type":"hint","dependencies":["af04890Rational2a-h1"],"title":"Setting equal to $$0$$","text":"Set the $$denominator=0$$, $$8x=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af04890Rational2b","stepAnswer":["$$n=\\\\frac{-1}{3}$$"],"problemType":"MultipleChoice","stepTitle":"b) $$\\\\frac{8n-5}{3n+1}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$n=\\\\frac{-1}{3}$$","choices":["$$b=\\\\frac{-1}{3}$$","$$b=0$$","$$b=\\\\frac{5}{8}$$","$$n=\\\\frac{-1}{3}$$"],"hints":{"DefaultPathway":[{"id":"af04890Rational2b-h1","type":"hint","dependencies":[],"title":"Principle","text":"The rational will be undefined when the $$denominator=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational2b-h2","type":"hint","dependencies":["af04890Rational2b-h1"],"title":"Setting equal to $$0$$","text":"Set the $$denominator=0$$, $$3n+1=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational2b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-1}{3}$$"],"dependencies":["af04890Rational2b-h2"],"title":"Finding values","text":"What is $$n$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af04890Rational2c","stepAnswer":["$$x=-1$$ or $$-3$$"],"problemType":"MultipleChoice","stepTitle":"c) $$\\\\frac{a+10}{a^2+4a+3}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=-1$$ or $$-3$$","choices":["$$x=1$$ or $$-3$$","$$x=1$$ or $$3$$","$$x=-1$$ or $$-3$$"],"hints":{"DefaultPathway":[{"id":"af04890Rational2c-h1","type":"hint","dependencies":[],"title":"Principle","text":"The rational will be undefined when the $$denominator=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational2c-h2","type":"hint","dependencies":["af04890Rational2c-h1"],"title":"Setting equal to $$0$$","text":"Set the $$denominator=0$$, $$a^2+4a+3=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational2c-h3","type":"hint","dependencies":["af04890Rational2c-h2"],"title":"Factoring","text":"Factor the denominator","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af04890Rational3","title":"Simplify Rational Expressions","body":"Simplify the following rational expression:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Multiply and Divide Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af04890Rational3a","stepAnswer":["$$\\\\frac{x+3}{x+6}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{x^2+5x+6}{x^2+8x+12}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{x+3}{x+6}$$","hints":{"DefaultPathway":[{"id":"af04890Rational3a-h1","type":"hint","dependencies":[],"title":"Factoring the numerator","text":"Factor the numerator to $$\\\\left(x+2\\\\right) \\\\left(x+3\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational3a-h2","type":"hint","dependencies":["af04890Rational3a-h1"],"title":"Factoring the denominator","text":"Factor the denominator to $$\\\\left(x+2\\\\right) \\\\left(x+6\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational3a-h3","type":"hint","dependencies":["af04890Rational3a-h2"],"title":"Dividing","text":"Divide by the common factor","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af04890Rational4","title":"Simplify Rational Expressions","body":"Simplify the following rational expression:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Multiply and Divide Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af04890Rational4a","stepAnswer":["$$\\\\frac{x+1}{x-1}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{x^2-x-2}{x^2-3x+2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{x+1}{x-1}$$","hints":{"DefaultPathway":[{"id":"af04890Rational4a-h1","type":"hint","dependencies":[],"title":"Factoring the numerator","text":"Factor the numerator to $$\\\\left(x-2\\\\right) \\\\left(x+1\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational4a-h2","type":"hint","dependencies":["af04890Rational4a-h1"],"title":"Factoring the denominator","text":"Factor the denominator to $$(x-2)(x-1)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational4a-h3","type":"hint","dependencies":["af04890Rational4a-h2"],"title":"Dividing","text":"Divide by the common factor","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af04890Rational5","title":"Simplify Rational Expressions","body":"Simplify the following rational expression:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Multiply and Divide Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af04890Rational5a","stepAnswer":["$$\\\\frac{a-2b}{2\\\\left(a+2b\\\\right)}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{3a^2-12ab+12b^2}{6a^2-24b^2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{a-2b}{2\\\\left(a+2b\\\\right)}$$","hints":{"DefaultPathway":[{"id":"af04890Rational5a-h1","type":"hint","dependencies":[],"title":"Factoring the numerator","text":"Factor the numerator to $$3(a-2b)(a-2b)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational5a-h2","type":"hint","dependencies":["af04890Rational5a-h1"],"title":"Factoring the denominator","text":"Factor the denominator to $$6\\\\left(a+2b\\\\right) \\\\left(a-2b\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational5a-h3","type":"hint","dependencies":["af04890Rational5a-h2"],"title":"Dividing","text":"Divide by the common factor","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af04890Rational6","title":"Simplify Rational Expressions","body":"Simplify the following rational expression:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Multiply and Divide Rational Expressions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af04890Rational6a","stepAnswer":["$$\\\\frac{\\\\frac{2}{3} \\\\left(x-3y\\\\right)}{x+3y}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2x^2-12xy+18y^2}{3x^2-27y^2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{\\\\frac{2}{3} \\\\left(x-3y\\\\right)}{x+3y}$$","hints":{"DefaultPathway":[{"id":"af04890Rational6a-h1","type":"hint","dependencies":[],"title":"Factoring the numerator","text":"Factor the numerator to $$2(x-3y)(x-3y)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational6a-h2","type":"hint","dependencies":["af04890Rational6a-h1"],"title":"Factoring the denominator","text":"Factor the denominator to $$3\\\\left(x-3y\\\\right) \\\\left(x+3y\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af04890Rational6a-h3","type":"hint","dependencies":["af04890Rational6a-h2"],"title":"Dividing","text":"Divide by the common factor","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af04890Rational7","title":"Simplify Rational Expressions","body":"Simplify the following rational expression:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.1 Multiply and Divide Rational Expressions","courseName":"OpenStax: Intermediate 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For example, $${10}^2=100$$, so $$100$$ is the square of $$10$$. 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Alternatively, $$\\\\sqrt{64}=8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af1a2a0roots19","title":"Finding the Square Root of a Number","body":"Solve the following expression.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Simplify and Use Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"af1a2a0roots19a","stepAnswer":["$$-11$$"],"problemType":"TextBox","stepTitle":"$$-\\\\sqrt{121}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-11$$","hints":{"DefaultPathway":[{"id":"af1a2a0roots19a-h1","type":"hint","dependencies":[],"title":"Square and Square Root of a Number","text":"If $$n^2=m$$, then $$m$$ is the square of $$n$$. 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What is the simplified expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$-11$$","$$12$$","$$-13$$","Not a real number"]}]}}]},{"id":"af1a2a0roots20","title":"Finding the Square Root of a Number","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Simplify and Use Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"af1a2a0roots20a","stepAnswer":["Undefined"],"problemType":"MultipleChoice","stepTitle":"$$\\\\sqrt{-121}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$-11$$","$$11$$","$$10$$","Undefined"],"hints":{"DefaultPathway":[{"id":"af1a2a0roots20a-h1","type":"hint","dependencies":[],"title":"Square Root of a Negative Number","text":"Since $$n^2=m$$ cannot be negative, a negative number does not have a square root.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af1a2a0roots21","title":"Finding the Square Root of a Number","body":"Solve the following expression. 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What is the simplified expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$-16$$","$$-12$$","$$-8$$","Not a real number"]}]}}]},{"id":"af1a2a0roots4","title":"Simplify and Use Square Roots","body":"Simplify the following exercises:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"9.1 Simplify and Use Square Roots","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"af1a2a0roots4a","stepAnswer":["Not a real number"],"problemType":"MultipleChoice","stepTitle":"$$\\\\sqrt{-196}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$-13$$","$$-14$$","$$-15$$","Not a real number"],"hints":{"DefaultPathway":[{"id":"af1a2a0roots4a-h1","type":"hint","dependencies":[],"title":"Identifying Square Root","text":"$$\\\\sqrt{-196}$$ is not a real number","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af1a2a0roots4b","stepAnswer":["$$-9$$"],"problemType":"MultipleChoice","stepTitle":"$$-\\\\sqrt{81}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-9$$","choices":["$$-9$$","$$-10$$","$$-11$$","Not a real number"],"hints":{"DefaultPathway":[{"id":"af1a2a0roots4b-h1","type":"hint","dependencies":[],"title":"Identifying Square Root","text":"$$9^2=81$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af1a2a0roots4b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-9$$"],"dependencies":["af1a2a0roots4b-h1"],"title":"Simplifying Expression","text":"The negative is in front of the radical sign. 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What is the simplified expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$-4$$","$$-7$$","$$-9$$","Not a real number"]}]}},{"id":"af1a2a0roots5b","stepAnswer":["Not a real number"],"problemType":"MultipleChoice","stepTitle":"$$\\\\sqrt{-121}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$-10$$","$$-11$$","$$-12$$","Not a real number"],"hints":{"DefaultPathway":[{"id":"af1a2a0roots5b-h1","type":"hint","dependencies":[],"title":"Identifying Square Root","text":"$$\\\\sqrt{-121}$$ is not a real number","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af1a2a0roots6","title":"Simplify and Use Square Roots","body":"Simplify the following exercises:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary 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$${15}^2=225$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af1a2a0roots8b-h2","type":"hint","dependencies":["af1a2a0roots8b-h1"],"title":"Order of Operations","text":"$$8+15$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af1a2a0roots8b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$23$$"],"dependencies":["af1a2a0roots8b-h2"],"title":"Simplify","text":"What is the simplified expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af1a2a0roots9","title":"Simplify and Use Square Roots","body":"Simplify the following exercises:","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary 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4.0>"}]}},{"id":"af1a2a0roots9b","stepAnswer":["y**8"],"problemType":"TextBox","stepTitle":"$$\\\\sqrt{y^{16}}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y^8$$","hints":{"DefaultPathway":[{"id":"af1a2a0roots9b-h1","type":"hint","dependencies":[],"title":"Identifying Square Root","text":"$${\\\\left(y^8\\\\right)}^2=y^{16}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af1a2a0roots9b-h2","type":"scaffold","problemType":"TextBox","answerType":"string","hintAnswer":["y**8"],"dependencies":["af1a2a0roots9b-h1"],"title":"Simplify","text":"What is the simplified expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af227bbMoApp1","title":"Modeling a Linear Equation to Solve an Unknown Number Problem","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.3 Models and Applications","courseName":"OpenStax: College Algebra","steps":[{"id":"af227bbMoApp1a","stepAnswer":["$$24$$ $$7$$"],"problemType":"MultipleChoice","stepTitle":"Find a linear equation to solve for the following unknown quantities","stepBody":"One number exceeds another number by $$17$$ and their sum is $$31$$. Find the larger of the two numbers.","answerType":"string","variabilization":{},"answerLatex":"$$24$$ $$7$$","choices":["$$26$$ $$5$$","$$25$$ $$6$$","$$24$$ $$7$$","$$23$$ $$8$$"],"hints":{"DefaultPathway":[{"id":"af227bbMoApp1a-h1","type":"hint","dependencies":[],"title":"Identify known quantities","text":"The known quantities is $$17$$ and $$31$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp1a-h2","type":"hint","dependencies":["af227bbMoApp1a-h1"],"title":"Determine unknown quantities","text":"There are two numbers which remain unknown. They are the target we want to find.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp1a-h3","type":"hint","dependencies":["af227bbMoApp1a-h2"],"title":"Assign a variable","text":"Because there is more than one unknown quantity, we should choose one varible to equal $$x$$, for example, the first number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp1a-h4","type":"hint","dependencies":["af227bbMoApp1a-h3"],"title":"Write the second number","text":"After choosing the first number, we need to write the second one in terms of $$x$$ (which equals to the first number).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp1a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x-17$$"],"dependencies":["af227bbMoApp1a-h4"],"title":"Write the second number","text":"When given the first number exceeds the second number by $$17$$, how can the second number be expressed?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp1a-h6","type":"hint","dependencies":["af227bbMoApp1a-h5"],"title":"Write an equation","text":"Then write an equation interpreting the words as mathematical operations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp1a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+x-17=31$$"],"dependencies":["af227bbMoApp1a-h6"],"title":"Translation to Math Operations","text":"What is the mathematical form of \\"their sum is 31\\"?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp1a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$24$$"],"dependencies":["af227bbMoApp1a-h7"],"title":"Simplify and Solve","text":"Solve the equation we write. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp1a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["af227bbMoApp1a-h8"],"title":"Calculate the other number","text":"What is $$x-17$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp1a-h10","type":"hint","dependencies":["af227bbMoApp1a-h9"],"title":"Explain the solution","text":"Since $$x$$ equals to the first number, $$x-17$$ equals to the second number, the first number is $$24$$ and the second number is $$7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af227bbMoApp10","title":"Solving a Perimeter Problem","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.3 Models and Applications","courseName":"OpenStax: College Algebra","steps":[{"id":"af227bbMoApp10a","stepAnswer":["28x27"],"problemType":"MultipleChoice","stepTitle":"Find the dimensions of a rectangle given that the perimeter is $$110$$ cm and the length is $$1$$ cm more than the width.","stepBody":"","answerType":"string","variabilization":{},"choices":["31x30","30x29","29x28","28x27"],"hints":{"DefaultPathway":[{"id":"af227bbMoApp10a-h1","type":"hint","dependencies":[],"title":"Identify known quantities","text":"There are two known quantities, $$110$$ cm and $$1$$ cm.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp10a-h2","type":"hint","dependencies":["af227bbMoApp10a-h1"],"title":"Determine unknown quantities","text":"There are two unknown quantities, the length L and width W of the rectangle.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp10a-h3","type":"hint","dependencies":["af227bbMoApp10a-h2"],"title":"Assign a variable","text":"In this question, there are more than one unknown quantities, we need to choose one, for example, the width of the rectangle to equal to $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+1$$"],"dependencies":["af227bbMoApp10a-h3"],"title":"Express other quantities","text":"What is the length of the rectangle in terms of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp10a-h5","type":"hint","dependencies":["af227bbMoApp10a-h4"],"title":"Using a formula","text":"The formula we can use in this problem is the perimeter formula $$P=2L+2W$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp10a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x+2\\\\left(x+1\\\\right)=110$$"],"dependencies":["af227bbMoApp10a-h5"],"title":"Translation to Math Operations","text":"What is the mathematical form of \\"the perimeter is $$110$$ cm\\"?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp10a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$27$$"],"dependencies":["af227bbMoApp10a-h6"],"title":"Simplify and Solve","text":"Solve the equation we write. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp10a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$28$$"],"dependencies":["af227bbMoApp10a-h7"],"title":"Calculate the other quantity","text":"What is $$x+1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp10a-h9","type":"hint","dependencies":["af227bbMoApp10a-h8"],"title":"Explain the solution","text":"The dimensions are $$L=28$$ cm and $$W=27$$ cm.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af227bbMoApp11","title":"Solving an Area Problem","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.3 Models and Applications","courseName":"OpenStax: College Algebra","steps":[{"id":"af227bbMoApp11a","stepAnswer":["$$135$$"],"problemType":"TextBox","stepTitle":"The perimeter of graph paper is $$48$$ in. The length is $$6$$ in. more than the width. Find the area of the graph paper.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$135$$","hints":{"DefaultPathway":[{"id":"af227bbMoApp11a-h1","type":"hint","dependencies":[],"title":"Identify known quantities","text":"There are two known quantities, $$48$$ in. and $$6$$ in.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp11a-h2","type":"hint","dependencies":["af227bbMoApp11a-h1"],"title":"Determine unknown quantities","text":"The unknown quantities are the dimensions and the area of the graph paper.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp11a-h3","type":"hint","dependencies":["af227bbMoApp11a-h2"],"title":"Assign a variable","text":"In this question, there are more than one unknown quantities, we need to choose one, for example, the width of the graph paper equal to $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+6$$"],"dependencies":["af227bbMoApp11a-h3"],"title":"Express other quantities","text":"What is the length of the graph paper in terms of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp11a-h5","type":"hint","dependencies":["af227bbMoApp11a-h4"],"title":"Using the perimeter formula","text":"First, we should use the perimeter formula $$P=2L+2W$$ to find the dimensions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp11a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x+2\\\\left(x+6\\\\right)=48$$"],"dependencies":["af227bbMoApp11a-h5"],"title":"Using the perimeter formula","text":"What is the mathematical form of \\"the perimeter of graph paper is $$48$$ in.\\"?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp11a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["af227bbMoApp11a-h6"],"title":"Simplify and Solve","text":"Solve the equation we write. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp11a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["af227bbMoApp11a-h7"],"title":"Calculate the other quantity","text":"What is $$x+6$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp11a-h9","type":"hint","dependencies":["af227bbMoApp11a-h8"],"title":"Using the area formula","text":"The standard formula for area is $$A=LW$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp11a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$135$$"],"dependencies":["af227bbMoApp11a-h9"],"title":"Substitute","text":"What is $$9\\\\times15$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp11a-h11","type":"hint","dependencies":["af227bbMoApp11a-h10"],"title":"Explain the solution","text":"The area of the graph paper is $${135}^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af227bbMoApp12","title":"Solving an Area Problem","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.3 Models and Applications","courseName":"OpenStax: College Algebra","steps":[{"id":"af227bbMoApp12a","stepAnswer":["$$250$$"],"problemType":"TextBox","stepTitle":"A game room has a perimeter of $$70$$ ft. The length is five more than twice the width. How many $${ft}^2$$ of new carpeting should be ordered?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$250$$","hints":{"DefaultPathway":[{"id":"af227bbMoApp12a-h1","type":"hint","dependencies":[],"title":"Identify known quantities","text":"There are three known quantities, $$70$$ ft, $$5$$, and $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp12a-h2","type":"hint","dependencies":["af227bbMoApp12a-h1"],"title":"Determine unknown quantities","text":"The unknown quantities are the dimensions and the area of the game room.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp12a-h3","type":"hint","dependencies":["af227bbMoApp12a-h2"],"title":"Assign a variable","text":"In this question, there are more than one unknown quantities, we need to choose one, for example, the width of the game room equal to $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x+5$$"],"dependencies":["af227bbMoApp12a-h3"],"title":"Express other quantities","text":"What is the length of the game room in terms of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp12a-h5","type":"hint","dependencies":["af227bbMoApp12a-h4"],"title":"Using the perimeter formula","text":"First, we should use the perimeter formula $$P=2L+2W$$ to find the dimensions.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp12a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x+2\\\\left(2x+5\\\\right)=70$$"],"dependencies":["af227bbMoApp12a-h5"],"title":"Using the perimeter formula","text":"What is the mathematical form of \\"a game room has a perimeter of $$70$$ ft\\"?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp12a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["af227bbMoApp12a-h6"],"title":"Simplify and Solve","text":"Solve the equation we write. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp12a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["af227bbMoApp12a-h7"],"title":"Calculate the other quantity","text":"What is $$2x+5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp12a-h9","type":"hint","dependencies":["af227bbMoApp12a-h8"],"title":"Using the area formula","text":"The standard formula for area is $$A=LW$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp12a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$250$$"],"dependencies":["af227bbMoApp12a-h9"],"title":"Substitute","text":"What is $$10\\\\times25$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp12a-h11","type":"hint","dependencies":["af227bbMoApp12a-h10"],"title":"Explain the solution","text":"The area of the game room is $$250$$ $${ft}^2$$, so $$250$$ ft ** $$2$$ of new carpeting should be ordered.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af227bbMoApp13","title":"Solving a Volume Problem","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.3 Models and Applications","courseName":"OpenStax: College Algebra","steps":[{"id":"af227bbMoApp13a","stepAnswer":["$$L=20$$ in. $$W=10$$ in. and $$H=8$$ in."],"problemType":"MultipleChoice","stepTitle":"Find the dimensions of a shipping box given that the length is twice the width, the height is $$8$$ inches, and the volume is 1,600**3.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$L=20$$ in. $$W=10$$ in. and $$H=8$$ in.","choices":["$$L=20$$ in. $$W=10$$ in. and $$H=8$$ in.","$$L=40$$ in. $$W=20$$ in. and $$H=8$$ in.","$$L=15$$ in. $$W=7.5$$ in. and $$H=8$$ in."],"hints":{"DefaultPathway":[{"id":"af227bbMoApp13a-h1","type":"hint","dependencies":[],"title":"Identify known quantities","text":"There are three known quantities, $$2$$, $$8$$ and 1,600.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp13a-h2","type":"hint","dependencies":["af227bbMoApp13a-h1"],"title":"Determine unknown quantities","text":"The unknown quantities are the dimensions (except the height) of the shipping box.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp13a-h3","type":"hint","dependencies":["af227bbMoApp13a-h2"],"title":"Assign a variable","text":"In this question, there are more than one unknown quantities, we need to choose one, for example, the width of the shipping box equal to $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x$$"],"dependencies":["af227bbMoApp13a-h3"],"title":"Express other quantities","text":"What the length of the shipping box in terms of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp13a-h5","type":"hint","dependencies":["af227bbMoApp13a-h4"],"title":"Using the volume formula","text":"The formula for the volume of a box is given as $$V=LWH$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp13a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x x\\\\times8=1600$$"],"dependencies":["af227bbMoApp13a-h5"],"title":"Using the volume formula","text":"What is the mathematical form of \\"the voulme is 1,600 in.**3\\"?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp13a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["af227bbMoApp13a-h6"],"title":"Simplify and Solve","text":"Solve the equation we write. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp13a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["af227bbMoApp13a-h7"],"title":"Calculate the other quantity","text":"What is $$2x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp13a-h9","type":"hint","dependencies":["af227bbMoApp13a-h8"],"title":"Explain the solution","text":"The length of the shipping box is $$20$$ in., the width is $$10$$ in. and the height is $$8$$ in.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af227bbMoApp2","title":"Modeling a Linear Equation to Solve an Unknown Number Problem","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.3 Models and Applications","courseName":"OpenStax: College Algebra","steps":[{"id":"af227bbMoApp2a","stepAnswer":["$$25$$ $$11$$"],"problemType":"MultipleChoice","stepTitle":"Find a linear equation to solve for the following unknown quantities","stepBody":"One number is three more than twice another number. If the sum of the two number is $$36$$, find the numbers.","answerType":"string","variabilization":{},"answerLatex":"$$25$$ $$11$$","choices":["$$19$$ $$8$$","$$21$$ $$9$$","$$23$$ $$10$$","$$25$$ $$11$$"],"hints":{"DefaultPathway":[{"id":"af227bbMoApp2a-h1","type":"hint","dependencies":[],"title":"Identify known quantities","text":"The known quantities is $$2$$ (twice), $$3$$ and $$36$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp2a-h2","type":"hint","dependencies":["af227bbMoApp2a-h1"],"title":"Determine unknown quantities","text":"There are two numbers which remain unknown. They are the target we want to find.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp2a-h3","type":"hint","dependencies":["af227bbMoApp2a-h2"],"title":"Assign a variable","text":"Because there is more than one unknown quantity, we should choose one varible to equal $$x$$, for example, the second number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp2a-h4","type":"hint","dependencies":["af227bbMoApp2a-h3"],"title":"Write the first number","text":"After choosing the second number, we need to write the first one in terms of $$x$$ (which equals to the second number).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp2a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x+3$$"],"dependencies":["af227bbMoApp2a-h4"],"title":"Write the first number","text":"When given the first number is three more than twice the second number, how can the first number be expressed?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp2a-h6","type":"hint","dependencies":["af227bbMoApp2a-h5"],"title":"Write an equation","text":"Then write an equation interpreting the words as mathematical operations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp2a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+2x+3=36$$"],"dependencies":["af227bbMoApp2a-h6"],"title":"Translation to Math Operations","text":"What is the mathematical form of \\"the sum of the two numbers is 36\\"?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp2a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$11$$"],"dependencies":["af227bbMoApp2a-h7"],"title":"Simplify and Solve","text":"Solve the equation we write. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp2a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["af227bbMoApp2a-h8"],"title":"Calculate the other number","text":"What is $$2x+3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp2a-h10","type":"hint","dependencies":["af227bbMoApp2a-h9"],"title":"Explain the solution","text":"Since $$x$$ equals to the second number, $$2x+3$$ equals to the second number, the first number is $$25$$ and the second number is $$11$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af227bbMoApp3","title":"Setting Up a Linear Equation to Solve a Real-World Application","body":"There are two cell phone companies that offer different packages. Company A charges a monthly service fee of $34 plus $.05/min talk-time. Company B charges a monthly service fee of $40 plus $.04/min talk-time.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.3 Models and Applications","courseName":"OpenStax: College Algebra","steps":[{"id":"af227bbMoApp3a","stepAnswer":["$$A=0.05x+34$$"],"problemType":"TextBox","stepTitle":"Write a linear equation that models the package offered by Company A (in the form of $$A=mx+b$$ )","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$A=0.05x+34$$","hints":{"DefaultPathway":[{"id":"af227bbMoApp3a-h1","type":"hint","dependencies":[],"title":"Identify known quantities","text":"The known quantities are $$34$$ and $$0.05$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp3a-h2","type":"hint","dependencies":["af227bbMoApp3a-h1"],"title":"Determine unknown quantities","text":"The unknown quantity in this problem is the talk-time.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp3a-h3","type":"hint","dependencies":["af227bbMoApp3a-h2"],"title":"Assign a variable","text":"Let $$x$$ equal the talk-time.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.05x+34$$"],"dependencies":["af227bbMoApp3a-h3"],"title":"Translation to Math Operations","text":"What is the mathematical form of \\"a monthly service fee of $34 plus $.05/min talk-time\\"?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af227bbMoApp3b","stepAnswer":["$$B=0.04x+40$$"],"problemType":"TextBox","stepTitle":"Write a linear equation that models the package offered by Company B (in the form of $$B=mx+b$$ )","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$B=0.04x+40$$","hints":{"DefaultPathway":[{"id":"af227bbMoApp3b-h1","type":"hint","dependencies":[],"title":"Identify known quantities","text":"The known quantities are $$40$$ and $$0.04$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp3b-h2","type":"hint","dependencies":["af227bbMoApp3b-h1"],"title":"Determine unknown quantities","text":"The unknown quantity in this problem is the talk-time.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp3b-h3","type":"hint","dependencies":["af227bbMoApp3b-h2"],"title":"Assign a variable","text":"Let $$x$$ equal the talk-time.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp3b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.04x+40$$"],"dependencies":["af227bbMoApp3b-h3"],"title":"Translation to Math Operations","text":"What is the mathematical form of \\"a monthly service fee of $40 plus $.04/min talk-time\\"?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af227bbMoApp3c","stepAnswer":["B"],"problemType":"MultipleChoice","stepTitle":"If the average number of minutes used each month is 1,160, which company offfers the better plan?","stepBody":"","answerType":"string","variabilization":{},"choices":["A","B"],"hints":{"DefaultPathway":[{"id":"af227bbMoApp3c-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$92$$"],"dependencies":[],"title":"Substitute","text":"What is $$0.05\\\\times1160+34$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp3c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$86.4$$"],"dependencies":["af227bbMoApp3c-h1"],"title":"Substitute","text":"What is $$0.04\\\\times1160+40$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp3c-h3","type":"hint","dependencies":["af227bbMoApp3c-h2"],"title":"Comparison","text":"If the average talk-time is 1,160 minutes, the plan form company A will cost $92 every month and the plan from company B will cost $$\\\\$86.4$$ every month, so company B offers the better plan.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af227bbMoApp3d","stepAnswer":["A"],"problemType":"MultipleChoice","stepTitle":"If the average number of miuntes used each month is $$420$$, which company offfers the better plan?","stepBody":"","answerType":"string","variabilization":{},"choices":["A","B"],"hints":{"DefaultPathway":[{"id":"af227bbMoApp3d-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$55$$"],"dependencies":[],"title":"Substitute","text":"What is $$0.05\\\\times420+34$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp3d-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$56.8$$"],"dependencies":["af227bbMoApp3d-h1"],"title":"Substitute","text":"What is $$0.04\\\\times420+40$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp3d-h3","type":"hint","dependencies":["af227bbMoApp3d-h2"],"title":"Comparison","text":"If the average talk-time is $$420$$ minutes, the plan from company A will cost $55 every month and the plan from company B will cost $$\\\\$56.8$$ every month, so company A offers the better plan.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af227bbMoApp3e","stepAnswer":["$$600$$"],"problemType":"TextBox","stepTitle":"How many minutes of talk-time would yield equal monthly statements from both companies?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$600$$","hints":{"DefaultPathway":[{"id":"af227bbMoApp3e-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.05x+34=0.04x+40$$"],"dependencies":[],"title":"Translation to Math Operations","text":"What is the mathematical form of \\"yield equal monthly statements from both companies\\"?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp3e-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$600$$"],"dependencies":["af227bbMoApp3e-h1"],"title":"Simplify and Solve","text":"Solve the equation we write. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af227bbMoApp4","title":"Find a linear equation to model this real-world application","body":"It costs ABC electronics company $$\\\\$2.50$$ per unit to produce a part used in a popular brand of desktop computers. The company has monthly operating expenses of $350 for utilities and $3,300 for salaries.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.3 Models and Applications","courseName":"OpenStax: College Algebra","steps":[{"id":"af227bbMoApp4a","stepAnswer":["$$2.5x+3650$$"],"problemType":"TextBox","stepTitle":"What are the company\'s monthly expenses, if the number parts produced is $$x$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2.5x+3650$$","hints":{"DefaultPathway":[{"id":"af227bbMoApp4a-h1","type":"hint","dependencies":[],"title":"Component of Total Expense","text":"The total expenses should be the sum of operating expenses, salaries and cost for producing the part.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp4a-h2","type":"hint","dependencies":["af227bbMoApp4a-h1"],"title":"Identify known quantities","text":"The known quantities are $$2.50$$, $$350$$ and 3,300.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp4a-h3","type":"hint","dependencies":["af227bbMoApp4a-h2"],"title":"Determine unknown quantities","text":"The unknown quantity in this problem is the number of this part produced.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp4a-h4","type":"hint","dependencies":["af227bbMoApp4a-h3"],"title":"Assign a variable","text":"Let $$x$$ be the number of parts produced.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp4a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.5x+350+3300$$"],"dependencies":["af227bbMoApp4a-h4"],"title":"Translation to Math Operations","text":"What is the mathematical form of the sum of this three components?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp4a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2.5x+3650$$"],"dependencies":["af227bbMoApp4a-h5"],"title":"Simplify","text":"Simplify the linear equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af227bbMoApp5","title":"For the following exercises, use the information to find a linear algebraic equation model to use to answer the question being asked.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.3 Models and Applications","courseName":"OpenStax: College Algebra","steps":[{"id":"af227bbMoApp5a","stepAnswer":["$$85$$ $$28$$"],"problemType":"MultipleChoice","stepTitle":"Mark and Don are planning to sell each of their marble collections at a garage sale","stepBody":"If Don has $$1$$ more than $$3$$ times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?","answerType":"string","variabilization":{},"answerLatex":"$$85$$ $$28$$","choices":["$$82$$ $$27$$","$$85$$ $$28$$","$$88$$ $$29$$","$$91$$ $$30$$"],"hints":{"DefaultPathway":[{"id":"af227bbMoApp5a-h1","type":"hint","dependencies":[],"title":"Identify known quantities","text":"The known quantities are $$1$$, $$3$$ and $$113$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp5a-h2","type":"hint","dependencies":["af227bbMoApp5a-h1"],"title":"Determine unknown quantities","text":"The number of marbles Don has and Mark has are two unknown quantities that we need to find.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp5a-h3","type":"hint","dependencies":["af227bbMoApp5a-h2"],"title":"Assign a variable","text":"Because there are more than $$1$$ unknown quantities, we should choose one of them, for example, the number of marbles Mark has, as $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp5a-h4","type":"hint","dependencies":["af227bbMoApp5a-h3"],"title":"Write the other quantity","text":"After choosing $$x$$, we should write the number of marbles Don has in terms of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x+1$$"],"dependencies":["af227bbMoApp5a-h4"],"title":"Write the other quantity","text":"When given Don has $$1$$ more than $$3$$ times the number of marbles Mark has, how many does Don have to sell?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp5a-h6","type":"hint","dependencies":["af227bbMoApp5a-h5"],"title":"Write an equation","text":"Then write an equation interpreting the words as mathematical operations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp5a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+3x+1=113$$"],"dependencies":["af227bbMoApp5a-h6"],"title":"Translation to Math Operations","text":"What is the mathematical form of \\"the total number of marbles is 113\\"?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp5a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$28$$"],"dependencies":["af227bbMoApp5a-h7"],"title":"Simplify and Solve","text":"Solve the equation we write. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp5a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$85$$"],"dependencies":["af227bbMoApp5a-h8"],"title":"Calculate another quantity","text":"What is $$3x+1$$ when $$x=28$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp5a-h10","type":"hint","dependencies":["af227bbMoApp5a-h9"],"title":"Explain the solution","text":"Don has to sell $$85$$ marbles and Mark has to sell $$28$$ marbles.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af227bbMoApp5b","stepAnswer":["$$46$$ $$23$$"],"problemType":"MultipleChoice","stepTitle":"Beth and Ann are joking that their combined ages equal Sam\'s age.","stepBody":"If Beth is twice Ann\'s age and Sam is $$69$$ yr old, what are Beth and Ann\'s ages?","answerType":"string","variabilization":{},"answerLatex":"$$46$$ $$23$$","choices":["$$48$$ $$24$$","$$50$$ $$25$$","$$46$$ $$23$$","$$44$$ $$24$$"],"hints":{"DefaultPathway":[{"id":"af227bbMoApp5b-h1","type":"hint","dependencies":[],"title":"Identify known quantities","text":"The known quantities are $$2$$ and $$69$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp5b-h2","type":"hint","dependencies":["af227bbMoApp5b-h1"],"title":"Determine unknown quantities","text":"Beth and Ann\'s ages are two unknown quantities.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp5b-h3","type":"hint","dependencies":["af227bbMoApp5b-h2"],"title":"Assign a variable","text":"Because there are more than $$1$$ unknown quantities, we should choose one of them, for example, Ann\'s age, as $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp5b-h4","type":"hint","dependencies":["af227bbMoApp5b-h3"],"title":"Write the other quantity","text":"After choosing $$x$$, we should write Beth\'s age in terms of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp5b-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x$$"],"dependencies":["af227bbMoApp5b-h4"],"title":"Write the other quantity","text":"When given Beth is twice Ann\'s age, what is Beth\'s age?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp5b-h6","type":"hint","dependencies":["af227bbMoApp5b-h5"],"title":"Write an equation","text":"Then write an equation interpreting the words as mathematical operations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp5b-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+2x=69$$"],"dependencies":["af227bbMoApp5b-h6"],"title":"Translation to Math Operations","text":"What is the mathematical form of \\"their combined ages equal Sam\'s age\\"?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp5b-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$23$$"],"dependencies":["af227bbMoApp5b-h7"],"title":"Simplify and Solve","text":"Solve the equation we write. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp5b-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$46$$"],"dependencies":["af227bbMoApp5b-h8"],"title":"Calculate another quantity","text":"What is $$2x$$ when $$x=23$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp5b-h10","type":"hint","dependencies":["af227bbMoApp5b-h9"],"title":"Explain the solution","text":"Beth is $$46$$ yr old and Ann is $$23$$ yr old.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af227bbMoApp5c","stepAnswer":["$$15$$ $$7$$"],"problemType":"MultipleChoice","stepTitle":"Ben originally filled out $$8$$ more applications than Henry. Then each boy filled out $$3$$ additional applications, bringing the total to $$28$$. How many applications did each boy originally fill out?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$15$$ $$7$$","choices":["$$15$$ $$8$$","$$16$$ $$9$$","$$15$$ $$7$$","$$14$$ $$8$$"],"hints":{"DefaultPathway":[{"id":"af227bbMoApp5c-h1","type":"hint","dependencies":[],"title":"Identify known quantities","text":"The known quantities are $$8$$, $$3$$ and $$28$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp5c-h2","type":"hint","dependencies":["af227bbMoApp5c-h1"],"title":"Determine unknown quantities","text":"There are four unknown quantities, the number of applications that Ben and Henry originally filled out and the number of applications that Ben and Henry each filled out in total.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp5c-h3","type":"hint","dependencies":["af227bbMoApp5c-h2"],"title":"Assign a variable","text":"Because there are more than $$1$$ unknown quantities, we should choose one of them, for example, the number of applications Henry originally filled out, as $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp5c-h4","type":"hint","dependencies":["af227bbMoApp5c-h3"],"title":"Write the other quantities","text":"After choosing $$x$$, we should write other three quantities in terms of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp5c-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+8$$"],"dependencies":["af227bbMoApp5c-h4"],"title":"Applications Ben originally filled out","text":"When Given Ben originally filled out $$8$$ more applications than Henry, how many applications did Ben originally fill out?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp5c-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+3$$"],"dependencies":["af227bbMoApp5c-h5"],"title":"Applications Henry filled out in total","text":"When given Henry filled out $$3$$ additional applications, how many applications did Henry fill out in total?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp5c-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+8+3$$"],"dependencies":["af227bbMoApp5c-h6"],"title":"Applications Ben filled out in total","text":"When given Ben filled out $$3$$ additional applications, how many applications did Ben fill out in total?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp5c-h8","type":"hint","dependencies":["af227bbMoApp5c-h7"],"title":"Write an equation","text":"Then write an equation interpreting the words as mathematical operations.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp5c-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+3+x+8+3=28$$"],"dependencies":["af227bbMoApp5c-h8"],"title":"Translation to Math Operations","text":"What is the mathematical form of \\"bringing the total to 28\\"?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp5c-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["af227bbMoApp5c-h9"],"title":"Simplify and Solve","text":"Solve the equation we write. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp5c-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["af227bbMoApp5c-h10"],"title":"Calculate the other target quantity","text":"What is $$x+8$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp5c-h12","type":"hint","dependencies":["af227bbMoApp5c-h11"],"title":"Explain the solution","text":"Henry originally filled out $$7$$ applications and Ben originally filled out $$15$$ applications.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af227bbMoApp6","title":"For the following exercises, use this scenario","body":"Two different telephone carries offer the following plans that a person is considering. Company A has a monthly fee of $20 and charges for $.05/min for calls. Company B has a monthly fee of $5 and charges $.10/min for calls.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.3 Models and Applications","courseName":"OpenStax: College Algebra","steps":[{"id":"af227bbMoApp6a","stepAnswer":["$$0.05x+20$$"],"problemType":"TextBox","stepTitle":"Find the model of the total cost of Company A\'s plan, using $$x$$ for the minutes","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.05x+20$$","hints":{"DefaultPathway":[{"id":"af227bbMoApp6a-h1","type":"hint","dependencies":[],"title":"Identify known quantities","text":"The known quantities are $$20$$ and $$0.05$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp6a-h2","type":"hint","dependencies":["af227bbMoApp6a-h1"],"title":"Determine unknown quantities","text":"The unknown quantity is the talk-time, which has been made $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.05x$$"],"dependencies":["af227bbMoApp6a-h2"],"title":"Fees charged for calls","text":"What is the fees charged for calls when calling $$x$$ minutes for a month?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp6a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.05x+20$$"],"dependencies":["af227bbMoApp6a-h3"],"title":"Total cost of this plan","text":"What is the total cost of a monthly fee and fees charged for calls?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af227bbMoApp6b","stepAnswer":["$$0.1x+5$$"],"problemType":"TextBox","stepTitle":"Find the model of the total cost of Company B\'s plan, using $$x$$ for the minutes","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.1x+5$$","hints":{"DefaultPathway":[{"id":"af227bbMoApp6b-h1","type":"hint","dependencies":[],"title":"Identify known quantities","text":"The known quantities are $$5$$ and $$0.1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp6b-h2","type":"hint","dependencies":["af227bbMoApp6b-h1"],"title":"Determine unknown quantities","text":"The unknown quantity is the talk-time, which has been made $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp6b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1x$$"],"dependencies":["af227bbMoApp6b-h2"],"title":"Fees charged for calls","text":"What is the fees charged for calls when calling $$x$$ minutes for a month?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp6b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.1x+5$$"],"dependencies":["af227bbMoApp6b-h3"],"title":"Total cost of this plan","text":"What is the total cost of a monthly fee and fees charged for calls?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af227bbMoApp6c","stepAnswer":["$$300$$"],"problemType":"TextBox","stepTitle":"Find out how many minutes of calling would make the two plans equal.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$300$$","hints":{"DefaultPathway":[{"id":"af227bbMoApp6c-h1","type":"hint","dependencies":[],"title":"Translation to Math Operations","text":"We have modeled the total cost of these two companies\' plans, so \\"make the two plans equal\\" can be translated to a equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp6c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.05x+20=0.1x+5$$"],"dependencies":["af227bbMoApp6c-h1"],"title":"Translation to Math Operations","text":"What is the equation that can model the question?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp6c-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$300$$"],"dependencies":["af227bbMoApp6c-h2"],"title":"Simplify and Solve","text":"Solve the equation. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp6c-h4","type":"hint","dependencies":["af227bbMoApp6c-h3"],"title":"Explaine the solution","text":"The solution of this equation is $$x=300$$, which means $$300$$ minutes of calling would make the two plans equal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af227bbMoApp6d","stepAnswer":["B"],"problemType":"MultipleChoice","stepTitle":"If the person makes a monthly average of $$200$$ min of calls, which plan should for the person choose?","stepBody":"","answerType":"string","variabilization":{},"choices":["A","B"],"hints":{"DefaultPathway":[{"id":"af227bbMoApp6d-h1","type":"hint","dependencies":[],"title":"Calculate the cost","text":"As the model of the total cost has been found in step $$1$$ and $$2$$, the total cost can be calculated when making a monthly average of $$200$$ min of calls.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp6d-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$30$$"],"dependencies":["af227bbMoApp6d-h1"],"title":"Substitute","text":"What is $$0.05\\\\times200+20$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp6d-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$25$$"],"dependencies":["af227bbMoApp6d-h2"],"title":"Substitute","text":"What is $$0.1\\\\times200+5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp6d-h4","type":"hint","dependencies":["af227bbMoApp6d-h3"],"title":"Comparison","text":"Next, we should compare the total cost to choose the right plan.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp6d-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$30$$"],"dependencies":["af227bbMoApp6d-h4"],"title":"Comparison","text":"Which is bigger, $$30$$ or 25?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$30$$","$$25$$"]},{"id":"af227bbMoApp6d-h6","type":"hint","dependencies":["af227bbMoApp6d-h5"],"title":"Explain the solution","text":"The person will spend $30 if he choose Company A\'s plan while $25 with Company B\'s plan, so he should choose Company B.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af227bbMoApp7","title":"Solving an Application Using a Formula","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.3 Models and Applications","courseName":"OpenStax: College Algebra","steps":[{"id":"af227bbMoApp7a","stepAnswer":["$$40$$"],"problemType":"TextBox","stepTitle":"It takes Andrew $$30$$ min to drive to work in the morning. He drives home using the same route, but it takes $$10$$ min longer, and he averages $$10$$ $$\\\\frac{mi}{h}$$ less than in the morning. How far does Andrew drive to work?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$40$$","hints":{"DefaultPathway":[{"id":"af227bbMoApp7a-h1","type":"hint","dependencies":[],"title":"Identify known quantities","text":"There are $$3$$ known quantities, $$30$$ min, $$10$$ min and $$10$$ $$\\\\frac{mi}{h}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp7a-h2","type":"hint","dependencies":["af227bbMoApp7a-h1"],"title":"Determine unknown quantities","text":"There are four unknown quantities, the speed Andrew drives in the morning, the speed he drives when he goes home, the distance he drives in the morning and that in the evening.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp7a-h3","type":"hint","dependencies":["af227bbMoApp7a-h2"],"title":"Assign a variable","text":"In this question, there are more than one unknown quantities, we need to choose one, for example, the speed Andrew drives in the morning and let it equal $$r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp7a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$r-10$$"],"dependencies":["af227bbMoApp7a-h3"],"title":"Express other quantities","text":"If the speed Andrew drives in the morning is $$r$$, what is the speed in the evening?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp7a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$r \\\\frac{1}{2}$$"],"dependencies":["af227bbMoApp7a-h4"],"title":"Express other quantities","text":"How far does Andrew drive to work in the morning in terms of $$r$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp7a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\left(r-10\\\\right) \\\\frac{2}{3}$$"],"dependencies":["af227bbMoApp7a-h5"],"title":"Express other quantities","text":"How far does Andrew drive home in the evening in terms of his speed in the evening?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp7a-h7","type":"hint","dependencies":["af227bbMoApp7a-h6"],"title":"Translation to Math Operations","text":"The only thing we can translated into a equation is the distance between Andrew\'s home and workplace remains unchanged in a day.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp7a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$r\\\\left(\\\\frac{1}{2}\\\\right)=\\\\left(r-10\\\\right) \\\\frac{2}{3}$$"],"dependencies":["af227bbMoApp7a-h7"],"title":"Translation to Math Operations","text":"What is the mathmatical form of the distance both trips covering is the same.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp7a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$40$$"],"dependencies":["af227bbMoApp7a-h8"],"title":"Simplify and Solve","text":"Solve the equation we write. What is $$r$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp7a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20$$"],"dependencies":["af227bbMoApp7a-h9"],"title":"Calculate the target quantity","text":"What is $$r \\\\frac{1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp7a-h11","type":"hint","dependencies":["af227bbMoApp7a-h10"],"title":"Explaine the solution","text":"Andrew dirve at the speed of $$40$$ $$\\\\frac{mi}{h}$$ in the morning, and it takes him $$30$$ mins to drive to work, so it is $$20$$ miles between his home and work.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af227bbMoApp8","title":"Solving an Application Using a Formula","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.3 Models and Applications","courseName":"OpenStax: College Algebra","steps":[{"id":"af227bbMoApp8a","stepAnswer":["$$45$$"],"problemType":"TextBox","stepTitle":"On Saturday morning, it took Jennifer $$3.6$$ $$h$$ to drive to her mother\'s house for the weekend. On Sunday evening, due to heavy traffic, it took Jennifer $$4$$ $$h$$ to return home. Her speed was $$5$$ $$\\\\frac{mi}{h}$$ slower on Sunday than on Saturday. What was her speed on Sunday in $$\\\\frac{mi}{h}$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$45$$","hints":{"DefaultPathway":[{"id":"af227bbMoApp8a-h1","type":"hint","dependencies":[],"title":"Identify known quantities","text":"There are three known quantities, $$3.6$$ $$h$$, $$4$$ $$h$$ and $$5$$ $$\\\\frac{mi}{h}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp8a-h2","type":"hint","dependencies":["af227bbMoApp8a-h1"],"title":"Determine unknown quantities","text":"There are four unknown quantities, the speed Jennifer drives on Saturday, the speed she drives on Sunday, the distance she drives on Saturday and Sunday.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp8a-h3","type":"hint","dependencies":["af227bbMoApp8a-h2"],"title":"Assign a variable","text":"In this question, there are more than one unknown quantities, we need to choose one, for example, the speed Jennifer drives on Saturday and let it equal $$r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp8a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$r-5$$"],"dependencies":["af227bbMoApp8a-h3"],"title":"Express other quantities","text":"If the speed Jennifer drives on Satuday is $$r$$ $$\\\\frac{mi}{h}$$ and her speed is $$5$$ $$\\\\frac{mi}{h}$$ slower on Sunday than on Saturday, what is her speed on Sunday?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp8a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3.6r$$"],"dependencies":["af227bbMoApp8a-h4"],"title":"Express other quantities","text":"How far does Jennifer drive on Saturday in terms of $$r$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp8a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4\\\\left(r-5\\\\right)$$"],"dependencies":["af227bbMoApp8a-h5"],"title":"Express other quantities","text":"How far does Jennifer drive on Sunday in terms of her speed on Sunday?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp8a-h7","type":"hint","dependencies":["af227bbMoApp8a-h6"],"title":"Translation to Math Operations","text":"The only thing we can translate into an equation is the distance between Jennifer and her mother\'s house remains unchanged.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp8a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3.6r=4\\\\left(r-5\\\\right)$$"],"dependencies":["af227bbMoApp8a-h7"],"title":"Translation to Math Operations","text":"What is the mathmatical form of the statement that the distance covered by both trips is the same?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp8a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$50$$"],"dependencies":["af227bbMoApp8a-h8"],"title":"Simplify and Solve","text":"Solve the equation we write. What is $$r$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp8a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$45$$"],"dependencies":["af227bbMoApp8a-h9"],"title":"Calculate the target quantity","text":"What is $$r-5$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp8a-h11","type":"hint","dependencies":["af227bbMoApp8a-h10"],"title":"Explain the solution","text":"Jennifer\' speed is $$50$$ $$\\\\frac{mi}{h}$$ on Saturday and $$45$$ $$\\\\frac{mi}{h}$$ on Sunday.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af227bbMoApp9","title":"Solving a Perimeter Problem","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.3 Models and Applications","courseName":"OpenStax: College Algebra","steps":[{"id":"af227bbMoApp9a","stepAnswer":["15x12"],"problemType":"MultipleChoice","stepTitle":"The perimeter of a rectangular outdoor patio is $$54$$ ft. The length is $$3$$ ft greater than the width. What are the dimensions of the ration?","stepBody":"","answerType":"string","variabilization":{},"choices":["14x13","15x12","16x11","17x10"],"hints":{"DefaultPathway":[{"id":"af227bbMoApp9a-h1","type":"hint","dependencies":[],"title":"Identify known quantities","text":"There are two known quantities, $$54$$ ft and $$3$$ ft.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp9a-h2","type":"hint","dependencies":["af227bbMoApp9a-h1"],"title":"Determine unknown quantities","text":"There are two unknown quantities, the length L and width W of the patio.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp9a-h3","type":"hint","dependencies":["af227bbMoApp9a-h2"],"title":"Assign a variable","text":"In this question, there are more than one unknown quantities, we need to choose one, for example, the width of the patio equal to $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp9a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x+3$$"],"dependencies":["af227bbMoApp9a-h3"],"title":"Express other quantities","text":"What the length of the patio in terms of $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp9a-h5","type":"hint","dependencies":["af227bbMoApp9a-h4"],"title":"Using a formula","text":"The formula we can use in this problem is the perimeter formula $$P=2L+2W$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp9a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2x+2\\\\left(x+3\\\\right)=54$$"],"dependencies":["af227bbMoApp9a-h5"],"title":"Translation to Math Operations","text":"What is the mathematical form of \\"the perimeter of a rectangular outdoor patio is $$54$$ ft\\"?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp9a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["af227bbMoApp9a-h6"],"title":"Simplify and Solve","text":"Solve the equation we write. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp9a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["af227bbMoApp9a-h7"],"title":"Calculate the other quantity","text":"What is $$x+3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af227bbMoApp9a-h9","type":"hint","dependencies":["af227bbMoApp9a-h8"],"title":"Explain the solution","text":"The dimensions are $$L=15$$ ft and $$W=12$$ ft.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af36e21FunNota1","title":"Determining If Menu Price Lists Are Functions","body":"The coffee shop menu, shown in Figure consists of items and their prices.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Functions and Function Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"af36e21FunNota1a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Is price a function of the item?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"af36e21FunNota1a-h1","type":"hint","dependencies":[],"title":"Identify the input values","text":"When considering price as a function, the input values are items, so the domain is {Plain Donut, Jelly Donut, Chocolate Donut}.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota1a-h2","type":"hint","dependencies":["af36e21FunNota1a-h1"],"title":"Identify the output values","text":"When considering price as a function, the output values are prices, so the range is {$$1.49$$, $$1.99$$.}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota1a-h3","type":"hint","dependencies":["af36e21FunNota1a-h2"],"title":"Classify the relation","text":"Because each input value in the domain leads to only one output value in the range, price is a function of the item.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota1b","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Is the item a function of the price?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"af36e21FunNota1b-h1","type":"hint","dependencies":[],"title":"Identify the input values","text":"When considering item as a function, the input values are prices, so the domain is {$$1.49$$, $$1.99$$.}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota1b-h2","type":"hint","dependencies":["af36e21FunNota1b-h1"],"title":"Identify the output values","text":"When considering item as a function, the output values are items, so the range is {Plain Donut, Jelly Donut, Chocolate Donut}.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota1b-h3","type":"hint","dependencies":["af36e21FunNota1b-h2"],"title":"Classify the relation","text":"Because the input value $$1.99$$ leads to two different output values Jelly Donut and Chocolate Donut, item is not a function of price.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af36e21FunNota10","title":"Solving Functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Functions and Function Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"af36e21FunNota10a","stepAnswer":["$$8$$"],"problemType":"TextBox","stepTitle":"Given the function $$g(m)=\\\\sqrt{m-4}$$, solve $$g(m)=2$$. What is $$m$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8$$","hints":{"DefaultPathway":[{"id":"af36e21FunNota10a-h1","type":"hint","dependencies":[],"title":"Find input value","text":"When we know an output value and want to determine the input values that would produce that output value, we set the output equal to the function\'s formula and solve for the input.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota10a-h2","type":"hint","dependencies":["af36e21FunNota10a-h1"],"title":"Substitution","text":"Substitute $$g(m)=2$$ into the original $$g(m)=\\\\sqrt{m-4}$$, and we get the equation $$2=\\\\sqrt{m-4}$$, which we are going to solve next.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota10a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["af36e21FunNota10a-h2"],"title":"Solving Equations","text":"Solve the equation $$2=\\\\sqrt{m-4}$$. What is $$m$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af36e21FunNota11","title":"Finding an Equation of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Functions and Function Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"af36e21FunNota11a","stepAnswer":["$$p=2-\\\\frac{1}{3} n$$"],"problemType":"TextBox","stepTitle":"Express the relationship $$2n+6p=12$$ as a function $$p=f(n)$$, if possible.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$p=2-\\\\frac{1}{3} n$$","hints":{"DefaultPathway":[{"id":"af36e21FunNota11a-h1","type":"hint","dependencies":[],"title":"Algebraic form","text":"To express the relationship in this form, we need to be able to write the relationship where $$p$$ is a fucntion of $$n$$, which means wirting it as $$p$$ $$=$$ [expression involving n].","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota11a-h2","type":"hint","dependencies":["af36e21FunNota11a-h1"],"title":"Subtraction","text":"Subtract $$2n$$ from both sides, and we get $$6p=12-2n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota11a-h3","type":"hint","dependencies":["af36e21FunNota11a-h2"],"title":"Division","text":"Divide both sides by $$6$$, and we get $$p=\\\\frac{12-2n}{6}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$p=2-\\\\frac{1}{3} n$$"],"dependencies":["af36e21FunNota11a-h3"],"title":"Simplification","text":"Simplify the expression. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af36e21FunNota12","title":"Expressing the Equation of a Circle as a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Functions and Function Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"af36e21FunNota12a","stepAnswer":["$$y=\\\\sqrt{1-x^2}$$ and $$-\\\\sqrt{1-x^2}$$, $$y$$ is not a function of $$x$$."],"problemType":"MultipleChoice","stepTitle":"Does the equation $$x^2+y^2=1$$ represent a function with $$x$$ as input and $$y$$ as output? If so, express the relationship as a function $$y=f(x)$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y=\\\\sqrt{1-x^2}$$ and $$-\\\\sqrt{1-x^2}$$, $$y$$ is not a function of $$x$$.","choices":["$$x=\\\\sqrt{1-y^2}$$","$$y=\\\\sqrt{1-x^2}$$","$$y=-\\\\sqrt{1-x^2}$$","$$y=\\\\sqrt{1-x^2}$$ and $$-\\\\sqrt{1-x^2}$$, $$y$$ is not a function of $$x$$."],"hints":{"DefaultPathway":[{"id":"af36e21FunNota12a-h1","type":"hint","dependencies":[],"title":"Algebraic form","text":"To express the relationship in this form, we need to be able to write the relationship where $$y$$ is a fucntion of $$x$$, which means wirting it as $$y$$ $$=$$ [expression involving x].","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota12a-h2","type":"hint","dependencies":["af36e21FunNota12a-h1"],"title":"Subtraction","text":"Subtract $$x^2$$ from both sides, and we get $$y^2=1-x^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota12a-h3","type":"hint","dependencies":["af36e21FunNota12a-h2"],"title":"Sqrt","text":"Solve $$y$$ in this equation, and we get $$y=\\\\sqrt{1-x^2}$$ and $$-\\\\sqrt{1-x^2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota12a-h4","type":"hint","dependencies":["af36e21FunNota12a-h3"],"title":"Explanation of the result","text":"We get two outputs corresponding to the same input, so this relationship cannot be represented as a single function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af36e21FunNota13","title":"Finding an Equation of a Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Functions and Function Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"af36e21FunNota13a","stepAnswer":["$$y=\\\\frac{1}{2} x^{\\\\frac{1}{3}}$$"],"problemType":"TextBox","stepTitle":"If $$x-8y^3=0$$, express $$y$$ as a function of $$x$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$y=\\\\frac{1}{2} x^{\\\\frac{1}{3}}$$","hints":{"DefaultPathway":[{"id":"af36e21FunNota13a-h1","type":"hint","dependencies":[],"title":"Algebraic form","text":"To express the relationship in this form, we need to be able to write the relationship where $$y$$ is a fucntion of $$x$$, which means wirting it as $$y$$ $$=$$ [expression involving x].","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota13a-h2","type":"hint","dependencies":["af36e21FunNota13a-h1"],"title":"Subtraction","text":"Subtract $$x$$ from both sides, and we get $$-8y^3=-x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota13a-h3","type":"hint","dependencies":["af36e21FunNota13a-h2"],"title":"Division","text":"Divide $$\\\\frac{-1}{8}$$ from both sides, and we get $$y^3$$ $$=$$ $$\\\\frac{x}{8}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota13a-h4","type":"hint","dependencies":["af36e21FunNota13a-h3"],"title":"Cubic root","text":"Extract the cubic root, and we get our final answer $$y=\\\\frac{1}{2} x^{\\\\frac{1}{3}}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af36e21FunNota14","title":"For the following exercises, determine whether the relation represents $$y$$ as a function of $$x$$.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Functions and Function Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"af36e21FunNota14a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$5x+2y=10$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"af36e21FunNota14a-h1","type":"hint","dependencies":[],"title":"Identify the input values","text":"When considering $$y$$ as a function of $$x$$, the input variable is $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota14a-h2","type":"hint","dependencies":["af36e21FunNota14a-h1"],"title":"Identify the output values","text":"When considering $$y$$ as a function of $$x$$, the output variable is $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota14a-h3","type":"hint","dependencies":["af36e21FunNota14a-h2"],"title":"Classify the relation","text":"Because each $$x$$ only leads to one particular $$y$$, $$y$$ is a function of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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$$x$$ only leads to one particular $$y$$, $$y$$ is a function of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota14c","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$x=y^2$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"af36e21FunNota14c-h1","type":"hint","dependencies":[],"title":"Identify the input values","text":"When considering $$y$$ as a function of $$x$$, the input variable is $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota14c-h2","type":"hint","dependencies":["af36e21FunNota14c-h1"],"title":"Identify the output values","text":"When considering $$y$$ as a function of $$x$$, the output variable is $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota14c-h3","type":"hint","dependencies":["af36e21FunNota14c-h2"],"title":"Classify the relation","text":"Because each $$x$$ leads to two different $$y$$ when $$x$$ is not $$0$$, $$y$$ is not a function of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota14d","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$3x^2+y=14$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"af36e21FunNota14d-h1","type":"hint","dependencies":[],"title":"Identify the input values","text":"When considering $$y$$ as a function of $$x$$, the input variable is $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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each $$x$$ only leads to one particular $$y$$, $$y$$ is a function of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota14g","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$y=\\\\frac{1}{x}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"af36e21FunNota14g-h1","type":"hint","dependencies":[],"title":"Identify the input values","text":"When considering $$y$$ as a function of $$x$$, the input variable is $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota14g-h2","type":"hint","dependencies":["af36e21FunNota14g-h1"],"title":"Identify the output values","text":"When considering $$y$$ as a function of $$x$$, the output variable is $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota14g-h3","type":"hint","dependencies":["af36e21FunNota14g-h2"],"title":"Classify the relation","text":"Because each $$x$$ in the domain (all real number except 0) only leads to one particular $$y$$, $$y$$ is a function of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota14h","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$x=\\\\frac{3y+1}{7y-1}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"af36e21FunNota14h-h1","type":"hint","dependencies":[],"title":"Identify the input values","text":"When considering $$y$$ as a function of $$x$$, the input variable is $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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Each $$x$$ only leads to a specific $$y$$. When $$x$$ equals $$\\\\frac{3}{7}$$, $$y$$ is not exist, so $$\\\\frac{3}{7}$$ is not in the domain.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota14i","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$x=\\\\sqrt{1-y^2}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"af36e21FunNota14i-h1","type":"hint","dependencies":[],"title":"Identify the input values","text":"When considering $$y$$ as a function of $$x$$, the input variable is $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota14i-h2","type":"hint","dependencies":["af36e21FunNota14i-h1"],"title":"Identify the output values","text":"When considering $$y$$ as a function of $$x$$, the output variable is $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota14i-h3","type":"hint","dependencies":["af36e21FunNota14i-h2"],"title":"Classify the relation","text":"Because each $$x$$ leads to two diffierent $$y$$ $$(y=\\\\sqrt{1-x^2}$$, y=-sqrt(1-x**2)), $$y$$ is not a function of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota14j","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$y=\\\\frac{3x+5}{7x-1}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"af36e21FunNota14j-h1","type":"hint","dependencies":[],"title":"Identify the input values","text":"When considering $$y$$ as a function of $$x$$, the input variable is $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota14j-h2","type":"hint","dependencies":["af36e21FunNota14j-h1"],"title":"Identify the output values","text":"When considering $$y$$ as a function of $$x$$, the output variable is $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota14j-h3","type":"hint","dependencies":["af36e21FunNota14j-h2"],"title":"Classify the relation","text":"Because each $$x$$ only leads to one particular $$y$$, $$y$$ is a function of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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each $$x$$ leads to two different $$y$$, $$y$$ is not a function of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota14l","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$2xy=1$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"af36e21FunNota14l-h1","type":"hint","dependencies":[],"title":"Identify the input values","text":"When considering $$y$$ as a function of $$x$$, the input variable is $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota14l-h2","type":"hint","dependencies":["af36e21FunNota14l-h1"],"title":"Identify the output values","text":"When considering $$y$$ as a function of $$x$$, the output variable is $$y$$.","variabilization":{},"oer":"https://OATutor.io 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considering $$y$$ as a function of $$x$$, the input variable is $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota14n-h2","type":"hint","dependencies":["af36e21FunNota14n-h1"],"title":"Identify the output values","text":"When considering $$y$$ as a function of $$x$$, the output variable is $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota14n-h3","type":"hint","dependencies":["af36e21FunNota14n-h2"],"title":"Classify the relation","text":"Because each $$x$$ only leads to one particular $$y$$, $$y$$ is a function of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota14o","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$y=\\\\sqrt{1-x^2}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"af36e21FunNota14o-h1","type":"hint","dependencies":[],"title":"Identify the input values","text":"When considering $$y$$ as a function of $$x$$, the input variable is $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota14o-h2","type":"hint","dependencies":["af36e21FunNota14o-h1"],"title":"Identify the output values","text":"When considering $$y$$ as a function of $$x$$, the output variable is $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota14o-h3","type":"hint","dependencies":["af36e21FunNota14o-h2"],"title":"Classify the relation","text":"Because each $$x$$ only leads to one particular $$y$$, $$y$$ is a function of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota14p","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$x=\\\\pm \\\\sqrt{1-y}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"af36e21FunNota14p-h1","type":"hint","dependencies":[],"title":"Identify the input values","text":"When considering $$y$$ as a function of $$x$$, the input variable is $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota14p-h2","type":"hint","dependencies":["af36e21FunNota14p-h1"],"title":"Identify the output values","text":"When considering $$y$$ as a function of $$x$$, the output variable is $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota14p-h3","type":"hint","dependencies":["af36e21FunNota14p-h2"],"title":"Classify the relation","text":"$$x=\\\\pm \\\\sqrt{1-y}$$ means $$y=1-x^2$$, so each $$x$$ can only lead to one particular $$y$$. Therefore, $$y$$ is a function of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota14q","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$y=\\\\pm \\\\sqrt{1-x}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"af36e21FunNota14q-h1","type":"hint","dependencies":[],"title":"Identify the input values","text":"When considering $$y$$ as a function of $$x$$, the input variable is $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota14q-h2","type":"hint","dependencies":["af36e21FunNota14q-h1"],"title":"Identify the output values","text":"When considering $$y$$ as a function of $$x$$, the output variable is $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota14q-h3","type":"hint","dependencies":["af36e21FunNota14q-h2"],"title":"Classify the relation","text":"When $$x$$ equals $$-1$$, we can get two different $$y$$ $$\\\\sqrt{2}$$ and $$-\\\\sqrt{2}$$, so $$y$$ is not a function of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota14r","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$y^2=x^2$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"af36e21FunNota14r-h1","type":"hint","dependencies":[],"title":"Identify the input values","text":"When considering $$y$$ as a function of $$x$$, the input variable is $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota14r-h2","type":"hint","dependencies":["af36e21FunNota14r-h1"],"title":"Identify the output values","text":"When considering $$y$$ as a function of $$x$$, the output variable is $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota14r-h3","type":"hint","dependencies":["af36e21FunNota14r-h2"],"title":"Classify the relation","text":"When $$x$$ equals $$1$$, we can get two different $$y$$ $$1$$ and $$-1$$, so $$y$$ is not a function of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota14s","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"$$y^3=x^2$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"af36e21FunNota14s-h1","type":"hint","dependencies":[],"title":"Identify the input values","text":"When considering $$y$$ as a function of $$x$$, the input variable is $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota14s-h2","type":"hint","dependencies":["af36e21FunNota14s-h1"],"title":"Identify the output values","text":"When considering $$y$$ as a function of $$x$$, the output variable is $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota14s-h3","type":"hint","dependencies":["af36e21FunNota14s-h2"],"title":"Classify the relation","text":"Because every real number has only one particular cube root, each $$x$$ can lead to one $$y$$ with the equation $$y^3=x^2$$. Therefore, $$y$$ is a function of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af36e21FunNota15","title":"Evaluate the function $$f(x)=2x-5$$ at the indicated values","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Functions and Function Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"af36e21FunNota15a","stepAnswer":["$$-11$$"],"problemType":"TextBox","stepTitle":"$$f(-3)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-11$$","hints":{"DefaultPathway":[{"id":"af36e21FunNota15a-h1","type":"hint","dependencies":[],"title":"Evaluating functions","text":"Given the equation for a function, we should replace the input variable in the equation with the value provided and then calculate the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota15a-h2","type":"hint","dependencies":["af36e21FunNota15a-h1"],"title":"Replacement","text":"Replace the variable $$x$$ with $$-3$$, and we get $$f(-3)=2\\\\left(-3\\\\right)-5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-11$$"],"dependencies":["af36e21FunNota15a-h2"],"title":"Calculation","text":"Calculate the expression $$2\\\\left(-3\\\\right)-5$$. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota15b","stepAnswer":["$$-1$$"],"problemType":"TextBox","stepTitle":"f(2)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-1$$","hints":{"DefaultPathway":[{"id":"af36e21FunNota15b-h1","type":"hint","dependencies":[],"title":"Evaluating functions","text":"Given the equation for a function, we should replace the input variable in the equation with the value provided and then calculate the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota15b-h2","type":"hint","dependencies":["af36e21FunNota15b-h1"],"title":"Replacement","text":"Replace the variable $$x$$ with $$2$$, and we get $$f(2)=2\\\\times2-5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota15b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["af36e21FunNota15b-h2"],"title":"Calculation","text":"Calculate the expression $$2\\\\times2-5$$. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota15c","stepAnswer":["-2a-5"],"problemType":"TextBox","stepTitle":"$$f(-a)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"af36e21FunNota15c-h1","type":"hint","dependencies":[],"title":"Evaluating functions","text":"Given the equation for a function, we should replace the input variable in the equation with the value provided and then calculate the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota15c-h2","type":"hint","dependencies":["af36e21FunNota15c-h1"],"title":"Replacement","text":"Replace the variable $$x$$ with -a, and we get $$f(-a)=2\\\\left(-a\\\\right)-5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota15c-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["-2a-5"],"dependencies":["af36e21FunNota15c-h2"],"title":"Simplification","text":"Simplify the expression $$2\\\\left(-a\\\\right)-5$$. What expression do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota15d","stepAnswer":["$$-2a+5$$"],"problemType":"TextBox","stepTitle":"$$-f(a)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-2a+5$$","hints":{"DefaultPathway":[{"id":"af36e21FunNota15d-h1","type":"hint","dependencies":[],"title":"Evaluating functions","text":"Given the equation for a function, we should replace the input variable in the equation with the value provided and then calculate the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota15d-h2","type":"hint","dependencies":["af36e21FunNota15d-h1"],"title":"Replacement","text":"Replace the variable $$x$$ with a, and we get $$-f(a)=-(2a-5)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota15d-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2a+5$$"],"dependencies":["af36e21FunNota15d-h2"],"title":"Simplification","text":"Simplify the expression $$-(2a-5)$$. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota15e","stepAnswer":["$$2a+2h-5$$"],"problemType":"TextBox","stepTitle":"$$f{\\\\left(a+h\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2a+2h-5$$","hints":{"DefaultPathway":[{"id":"af36e21FunNota15e-h1","type":"hint","dependencies":[],"title":"Evaluating functions","text":"Given the equation for a function, we should replace the input variable in the equation with the value provided and then calculate the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota15e-h2","type":"hint","dependencies":["af36e21FunNota15e-h1"],"title":"Replacement","text":"Replace the variable $$x$$ with $$a+h$$, and we get $$f{\\\\left(a+h\\\\right)}=2\\\\left(a+h\\\\right)-5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota15e-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2a+2h-5$$"],"dependencies":["af36e21FunNota15e-h2"],"title":"Simplification","text":"Simplify the expression $$2\\\\left(a+h\\\\right)-5$$. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af36e21FunNota16","title":"Evaluate the function $$f=-5x^2+2x-1$$ at the indicated values.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Functions and Function Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"af36e21FunNota16a","stepAnswer":["$$-52$$"],"problemType":"TextBox","stepTitle":"$$f(-3)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-52$$","hints":{"DefaultPathway":[{"id":"af36e21FunNota16a-h1","type":"hint","dependencies":[],"title":"Evaluating functions","text":"Given the equation for a function, we should replace the input variable in the equation with the value provided and then calculate the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota16a-h2","type":"hint","dependencies":["af36e21FunNota16a-h1"],"title":"Replacement","text":"Replace the variable $$x$$ with $$-3$$, and we get $$f(-3)=-5{\\\\left(-3\\\\right)}^2+2\\\\left(-3\\\\right)-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota16a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-52$$"],"dependencies":["af36e21FunNota16a-h2"],"title":"Calculation","text":"Calculate the expression $$-5{\\\\left(-3\\\\right)}^2+2\\\\left(-3\\\\right)-1$$. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota16b","stepAnswer":["$$-17$$"],"problemType":"TextBox","stepTitle":"f(2)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-17$$","hints":{"DefaultPathway":[{"id":"af36e21FunNota16b-h1","type":"hint","dependencies":[],"title":"Evaluating functions","text":"Given the equation for a function, we should replace the input variable in the equation with the value provided and then calculate the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota16b-h2","type":"hint","dependencies":["af36e21FunNota16b-h1"],"title":"Replacement","text":"Replace the variable $$x$$ with $$2$$, and we get $$f(2)=-5\\\\times2^2+2\\\\times2-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota16b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-17$$"],"dependencies":["af36e21FunNota16b-h2"],"title":"Calculation","text":"Calculate the expression $$-5\\\\times2^2+2\\\\times2-1$$. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota16c","stepAnswer":["$$-5a^2-2a-1$$"],"problemType":"TextBox","stepTitle":"$$f(-a)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-5a^2-2a-1$$","hints":{"DefaultPathway":[{"id":"af36e21FunNota16c-h1","type":"hint","dependencies":[],"title":"Evaluating functions","text":"Given the equation for a function, we should replace the input variable in the equation with the value provided and then calculate the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota16c-h2","type":"hint","dependencies":["af36e21FunNota16c-h1"],"title":"Replacement","text":"Replace the variable $$x$$ with -a, and we get $$f(-a)=-5{\\\\left(-a\\\\right)}^2+2\\\\left(-a\\\\right)-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota16c-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5a^2-2a-1$$"],"dependencies":["af36e21FunNota16c-h2"],"title":"Simplification","text":"Simplify the expression $$-5{\\\\left(-a\\\\right)}^2+2\\\\left(-a\\\\right)-1$$. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota16d","stepAnswer":["$$5a^2-2a+1$$"],"problemType":"TextBox","stepTitle":"$$-f(a)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5a^2-2a+1$$","hints":{"DefaultPathway":[{"id":"af36e21FunNota16d-h1","type":"hint","dependencies":[],"title":"Evaluating functions","text":"Given the equation for a function, we should replace the input variable in the equation with the value provided and then calculate the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota16d-h2","type":"hint","dependencies":["af36e21FunNota16d-h1"],"title":"Replacement","text":"Replace the variable $$x$$ with a, and we get $$f(a)=-5a^2+2a-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota16d-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5a^2-2a+1$$"],"dependencies":["af36e21FunNota16d-h2"],"title":"Simplification","text":"Simplify the expression $$-\\\\left(-5a^2+2a-1\\\\right)$$. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota16e","stepAnswer":["$$-5a^2-10ah-5h^2+2a+2h-1$$"],"problemType":"TextBox","stepTitle":"$$f{\\\\left(a+h\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-5a^2-10ah-5h^2+2a+2h-1$$","hints":{"DefaultPathway":[{"id":"af36e21FunNota16e-h1","type":"hint","dependencies":[],"title":"Evaluating functions","text":"Given the equation for a function, we should replace the input variable in the equation with the value provided and then calculate the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota16e-h2","type":"hint","dependencies":["af36e21FunNota16e-h1"],"title":"Replacement","text":"Replace the variable $$x$$ with $$a+h$$, and we get $$f{\\\\left(a+h\\\\right)}=-5{\\\\left(a+h\\\\right)}^2+2\\\\left(a+h\\\\right)-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota16e-h3","type":"hint","dependencies":["af36e21FunNota16e-h2"],"title":"Simplification","text":"Simplify the expression $$-5{\\\\left(a+h\\\\right)}^2+2\\\\left(a+h\\\\right)-1$$, and we get $$-5a^2-10ah-5h^2+2a+2h-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af36e21FunNota17","title":"Evaluation the function $$f=\\\\sqrt{2-x}+5$$ at the indicated values","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Functions and Function Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"af36e21FunNota17a","stepAnswer":["$$\\\\sqrt{5}+5$$"],"problemType":"TextBox","stepTitle":"$$f(-3)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\sqrt{5}+5$$","hints":{"DefaultPathway":[{"id":"af36e21FunNota17a-h1","type":"hint","dependencies":[],"title":"Evaluating functions","text":"Given the equation for a function, we should replace the input variable in the equation with the value provided and then calculate the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota17a-h2","type":"hint","dependencies":["af36e21FunNota17a-h1"],"title":"Replacement","text":"Replace the variable $$x$$ with $$-3$$, and we get $$f(-3)=\\\\sqrt{2-\\\\left(-3\\\\right)}+5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota17a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{5}+5$$"],"dependencies":["af36e21FunNota17a-h2"],"title":"Calculation","text":"Calculate the expression $$\\\\sqrt{2-\\\\left(-3\\\\right)}+5$$. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota17b","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"f(2)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"af36e21FunNota17b-h1","type":"hint","dependencies":[],"title":"Evaluating functions","text":"Given the equation for a function, we should replace the input variable in the equation with the value provided and then calculate the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota17b-h2","type":"hint","dependencies":["af36e21FunNota17b-h1"],"title":"Replacement","text":"Replace the variable $$x$$ with $$2$$, and we get $$f(2)=\\\\sqrt{2-2}+5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota17b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["af36e21FunNota17b-h2"],"title":"Calculation","text":"Calculate the expression sqrt(2 - 2)+5. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota17c","stepAnswer":["$$\\\\sqrt{2+a}+5$$"],"problemType":"TextBox","stepTitle":"$$f(-a)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\sqrt{2+a}+5$$","hints":{"DefaultPathway":[{"id":"af36e21FunNota17c-h1","type":"hint","dependencies":[],"title":"Evaluating functions","text":"Given the equation for a function, we should replace the input variable in the equation with the value provided and then calculate the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota17c-h2","type":"hint","dependencies":["af36e21FunNota17c-h1"],"title":"Replacement","text":"Replace the variable $$x$$ with -a, and we get $$f(-a)=\\\\sqrt{2-\\\\left(-a\\\\right)}+5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota17c-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{2+a}+5$$"],"dependencies":["af36e21FunNota17c-h2"],"title":"Simplification","text":"Simplify the expression $$\\\\sqrt{2-\\\\left(-a\\\\right)}+5$$. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota17d","stepAnswer":["$$-\\\\sqrt{2-a}+5$$"],"problemType":"TextBox","stepTitle":"$$-f(a)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-\\\\sqrt{2-a}+5$$","hints":{"DefaultPathway":[{"id":"af36e21FunNota17d-h1","type":"hint","dependencies":[],"title":"Evaluating functions","text":"Given the equation for a function, we should replace the input variable in the equation with the value provided and then calculate the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota17d-h2","type":"hint","dependencies":["af36e21FunNota17d-h1"],"title":"Replacement","text":"Replace the variable $$x$$ with a, and we get $$f(a)=\\\\sqrt{2-a}+5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota17d-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-\\\\sqrt{2-a}-5$$"],"dependencies":["af36e21FunNota17d-h2"],"title":"Simplification","text":"Simplify the expression $$-\\\\left(\\\\sqrt{2-a}+5\\\\right)$$. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota17e","stepAnswer":["$$\\\\sqrt{2-a-h}+5$$"],"problemType":"TextBox","stepTitle":"$$f{\\\\left(a+h\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\sqrt{2-a-h}+5$$","hints":{"DefaultPathway":[{"id":"af36e21FunNota17e-h1","type":"hint","dependencies":[],"title":"Evaluating functions","text":"Given the equation for a function, we should replace the input variable in the equation with the value provided and then calculate the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota17e-h2","type":"hint","dependencies":["af36e21FunNota17e-h1"],"title":"Replacement","text":"Replace the variable $$x$$ with a + $$h$$, and we get $$f{\\\\left(a+h\\\\right)}=\\\\sqrt{2-a+h}+5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota17e-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{2-a-h}+5$$"],"dependencies":["af36e21FunNota17e-h2"],"title":"Simplification","text":"Simplify the expression $$\\\\sqrt{2-a+h}+5$$. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af36e21FunNota18","title":"Evaluate the function $$f=\\\\frac{6x-1}{5x+2}$$ at the indicated values.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Functions and Function Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"af36e21FunNota18a","stepAnswer":["$$\\\\frac{19}{13}$$"],"problemType":"TextBox","stepTitle":"$$f(-3)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{19}{13}$$","hints":{"DefaultPathway":[{"id":"af36e21FunNota18a-h1","type":"hint","dependencies":[],"title":"Evaluating functions","text":"Given the equation for a function, we should replace the input variable in the equation with the value provided and then calculate the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota18a-h2","type":"hint","dependencies":["af36e21FunNota18a-h1"],"title":"Replacement","text":"Replace the variable $$x$$ with $$-3$$, so we get $$f(-3)=\\\\frac{6\\\\left(-3\\\\right)-1}{5\\\\left(-3\\\\right)+2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota18a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{19}{13}$$"],"dependencies":["af36e21FunNota18a-h2"],"title":"Calculation","text":"Calculate the expression $$\\\\frac{6\\\\left(-3\\\\right)-1}{5\\\\left(-3\\\\right)+2}$$. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota18b","stepAnswer":["$$\\\\frac{11}{12}$$"],"problemType":"TextBox","stepTitle":"f(2)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{11}{12}$$","hints":{"DefaultPathway":[{"id":"af36e21FunNota18b-h1","type":"hint","dependencies":[],"title":"Evaluating functions","text":"Given the equation for a function, we should replace the input variable in the equation with the value provided and then calculate the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota18b-h2","type":"hint","dependencies":["af36e21FunNota18b-h1"],"title":"Replacement","text":"Replace the variable $$x$$ with $$2$$, and we get $$f(2)=\\\\frac{6\\\\times2-1}{5\\\\times2+2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota18b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{11}{12}$$"],"dependencies":["af36e21FunNota18b-h2"],"title":"Calculation","text":"Calculate the expression $$\\\\frac{6\\\\times2-1}{5\\\\times2+2}$$. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota18c","stepAnswer":["$$\\\\frac{\\\\left(-6a-1\\\\right)}{\\\\left(-5a+2\\\\right)}$$"],"problemType":"TextBox","stepTitle":"$$f(-a)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{\\\\left(-6a-1\\\\right)}{\\\\left(-5a+2\\\\right)}$$","hints":{"DefaultPathway":[{"id":"af36e21FunNota18c-h1","type":"hint","dependencies":[],"title":"Evaluating functions","text":"Given the equation for a function, we should replace the input variable in the equation with the value provided and then calculate the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota18c-h2","type":"hint","dependencies":["af36e21FunNota18c-h1"],"title":"Replacement","text":"Replace the variable $$x$$ with -a, and we get $$f(-a)=\\\\frac{6\\\\left(-a\\\\right)-1}{5\\\\left(-a\\\\right)+2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota18c-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\left(-6a-1\\\\right)}{\\\\left(-5a+2\\\\right)}$$"],"dependencies":["af36e21FunNota18c-h2"],"title":"Simplification","text":"Simplify the expression $$\\\\frac{6\\\\left(-a\\\\right)-1}{5\\\\left(-a\\\\right)+2}$$. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota18d","stepAnswer":["$$\\\\frac{1-6a}{5a+2}$$"],"problemType":"TextBox","stepTitle":"$$-f(a)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1-6a}{5a+2}$$","hints":{"DefaultPathway":[{"id":"af36e21FunNota18d-h1","type":"hint","dependencies":[],"title":"Evaluating functions","text":"Given the equation for a function, we should replace the input variable in the equation with the value provided and then calculate the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota18d-h2","type":"hint","dependencies":["af36e21FunNota18d-h1"],"title":"Replacement","text":"Replace the variable $$x$$ with a, and we get $$f(a)=\\\\frac{6a-1}{5a+2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota18d-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1-6a}{5a+2}$$"],"dependencies":["af36e21FunNota18d-h2"],"title":"Simplification","text":"Simplify the expression $$-\\\\left(\\\\frac{6a-1}{5a+2}\\\\right)$$. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota18e","stepAnswer":["$$\\\\frac{6a+6h-1}{5a+5h+2}$$"],"problemType":"TextBox","stepTitle":"$$f{\\\\left(a+h\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{6a+6h-1}{5a+5h+2}$$","hints":{"DefaultPathway":[{"id":"af36e21FunNota18e-h1","type":"hint","dependencies":[],"title":"Evaluating functions","text":"Given the equation for a function, we should replace the input variable in the equation with the value provided and then calculate the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota18e-h2","type":"hint","dependencies":["af36e21FunNota18e-h1"],"title":"Replacement","text":"Replace the variable $$x$$ with a + $$h$$, and we get $$f{\\\\left(a+h\\\\right)}=\\\\frac{6\\\\left(a+h\\\\right)-1}{5\\\\left(a+h\\\\right)+2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota18e-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{6a+6h-1}{5a+5h+2}$$"],"dependencies":["af36e21FunNota18e-h2"],"title":"Simplification","text":"Simplify the expression $$\\\\frac{6\\\\left(a+h\\\\right)-1}{5\\\\left(a+h\\\\right)+2}$$. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af36e21FunNota19","title":"Evaluate the function $$f=|x-1|-|x+1|$$ at the indicated values","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Functions and Function Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"af36e21FunNota19a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"$$f(-3)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"af36e21FunNota19a-h1","type":"hint","dependencies":[],"title":"Evaluating functions","text":"Given the equation for a function, we should replace the input variable in the equation with the value provided and then calculate the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota19a-h2","type":"hint","dependencies":["af36e21FunNota19a-h1"],"title":"Replacement","text":"Replace the variable $$x$$ with $$-3$$, and we get $$f(-3)=|-3-1|-|-3+1|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota19a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["af36e21FunNota19a-h2"],"title":"Calculation","text":"Calculate the expression $$|-3-1|-|-3+1|$$. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota19b","stepAnswer":["$$-2$$"],"problemType":"TextBox","stepTitle":"f(2)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-2$$","hints":{"DefaultPathway":[{"id":"af36e21FunNota19b-h1","type":"hint","dependencies":[],"title":"Evaluating functions","text":"Given the equation for a function, we should replace the input variable in the equation with the value provided and then calculate the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota19b-h2","type":"hint","dependencies":["af36e21FunNota19b-h1"],"title":"Replacement","text":"Replace the variable $$x$$ with $$2$$, and we get $$f(2)=|2-1|-|2+1|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota19b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["af36e21FunNota19b-h2"],"title":"Calculation","text":"Calculate the expression $$|2-1|-|2+1|$$. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota19c","stepAnswer":["$$|-a-1|-|-a+1|$$"],"problemType":"TextBox","stepTitle":"$$f(-a)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$|-a-1|-|-a+1|$$","hints":{"DefaultPathway":[{"id":"af36e21FunNota19c-h1","type":"hint","dependencies":[],"title":"Evaluating functions","text":"Given the equation for a function, we should replace the input variable in the equation with the value provided and then calculate the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota19c-h2","type":"hint","dependencies":["af36e21FunNota19c-h1"],"title":"Replacement","text":"Replace the variable $$x$$ with -a, and we get $$f(-a)=|-a-1|-|-a+1|$$, which is our final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota19d","stepAnswer":["$$-|a-1|+|a+1|$$"],"problemType":"TextBox","stepTitle":"$$-f(a)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-|a-1|+|a+1|$$","hints":{"DefaultPathway":[{"id":"af36e21FunNota19d-h1","type":"hint","dependencies":[],"title":"Evaluating functions","text":"Given the equation for a function, we should replace the input variable in the equation with the value provided and then calculate the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota19d-h2","type":"hint","dependencies":["af36e21FunNota19d-h1"],"title":"Replacement","text":"Replace the variable $$x$$ with a, and we get $$f(a)=|a-1|-|a+1|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota19d-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-|a-1|+|a+1|$$"],"dependencies":["af36e21FunNota19d-h2"],"title":"Simplification","text":"Simplify $$-\\\\left(|a-1|-|a+1|\\\\right)$$. What do you get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota19e","stepAnswer":["$$|a+h-1|-|a+h+1|$$"],"problemType":"TextBox","stepTitle":"$$f{\\\\left(a+h\\\\right)}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$|a+h-1|-|a+h+1|$$","hints":{"DefaultPathway":[{"id":"af36e21FunNota19e-h1","type":"hint","dependencies":[],"title":"Evaluating functions","text":"Given the equation for a function, we should replace the input variable in the equation with the value provided and then calculate the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota19e-h2","type":"hint","dependencies":["af36e21FunNota19e-h1"],"title":"Replacement","text":"Replace the variable $$x$$ with a + $$h$$, and we get $$f{\\\\left(a+h\\\\right)}=|a+h-1|-|a+h+1|$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af36e21FunNota2","title":"Determining If Class Grade Rules Are Functions","body":"In a particular math class, the overall percent grade corresponds to a grade-point average. Table $$1$$ shows a possible rule for assigning grade points.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Functions and Function Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"af36e21FunNota2a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Is the percent grade a function of the grade-point average?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"af36e21FunNota2a-h1","type":"hint","dependencies":[],"title":"Identify the input values","text":"When considering percent grade as a function, the input values are grade-point averages, so the domain is {$$0.0$$, $$1.0$$, $$1.5$$, $$2.0$$, $$2.5$$, $$3.0$$, $$3.5$$, $$4.0$$.}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota2a-h2","type":"hint","dependencies":["af36e21FunNota2a-h1"],"title":"Identify the output values","text":"When considering percent grade as a function, the output values are percent grades, so the range is the set of all integers between $$0$$ and $$100$$ .","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota2a-h3","type":"hint","dependencies":["af36e21FunNota2a-h2"],"title":"Classify the relation","text":"Because the input value $$0.0$$ leads to $$56$$ different output values, percent grade is not a function of the grade-point average.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota2b","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Is grade-point average a function of the percent grade?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"af36e21FunNota2b-h1","type":"hint","dependencies":[],"title":"Identify the input values","text":"When considering grade-point average as a function, the input values are percent grades, so the domain is the set of all integers between $$0$$ and $$100$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota2b-h2","type":"hint","dependencies":["af36e21FunNota2b-h1"],"title":"Identify the output values","text":"When considering grade-point average as a function, the output values are grade-point averages, so the range is {$$0.0$$, $$1.0$$, $$1.5$$, $$2.0$$, $$2.5$$, $$3.0$$, $$3.5$$, $$4.0$$.}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota2b-h3","type":"hint","dependencies":["af36e21FunNota2b-h2"],"title":"Classify the relation","text":"Because each input value in the domain leads to only one output value in the range, grade-point average is a function of the percent grade.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af36e21FunNota20","title":"Determining whether a relationship is a one-to-one function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Functions and Function Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"af36e21FunNota20a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Is the area of a circle a function of its radius?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"af36e21FunNota20a-h1","type":"hint","dependencies":[],"title":"Identify the input values","text":"When considering area as a function of raidus, radius is the input values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota20a-h2","type":"hint","dependencies":["af36e21FunNota20a-h1"],"title":"Identify the output values","text":"When considering area as a function of raidus, area is the output values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota20a-h3","type":"hint","dependencies":["af36e21FunNota20a-h2"],"title":"Write the formula","text":"The area formula of a circle is $$A=\\\\pi r^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota20a-h4","type":"hint","dependencies":["af36e21FunNota20a-h3"],"title":"Classify the relation","text":"According to the formula $$A=\\\\pi r^2$$, each $$r$$ can lead to only one particular A, so the area of a circle is a function of its radius.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota20b","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"If yes, is the function one-to-one?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"af36e21FunNota20b-h1","type":"hint","dependencies":[],"title":"One-To-One Function","text":"A one-to-one function is a function in which each output value corresponds to exactly one input value. There are no repeated $$x-$$ or $$y-$$ values.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota20b-h2","type":"hint","dependencies":["af36e21FunNota20b-h1"],"title":"$$x-to-y$$","text":"From step one, each $$r$$ corresponds to only one A.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota20b-h3","type":"hint","dependencies":["af36e21FunNota20b-h2"],"title":"$$y-to-x$$","text":"When the area is A, the radius is $$\\\\sqrt{\\\\frac{A}{\\\\pi}}$$. So ecah A corresponds to exactly one $$r$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af36e21FunNota3","title":"Classify Relation","body":"For the following exercises, determine whether the relation represents a function.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Functions and Function Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"af36e21FunNota3a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"{(a, b), (c, d), (a, c)}","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"af36e21FunNota3a-h1","type":"hint","dependencies":[],"title":"Identify the input values","text":"The domain is {a, c}.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota3a-h2","type":"hint","dependencies":["af36e21FunNota3a-h1"],"title":"Identify the output values","text":"The range is {b, $$d$$, c}.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota3a-h3","type":"hint","dependencies":["af36e21FunNota3a-h2"],"title":"Classify the relation","text":"Because the input value a leads to two different output values $$b$$ and c, the relation does not represent a function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota3b","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"{(a, b), (b, c), (c, c)}","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"af36e21FunNota3b-h1","type":"hint","dependencies":[],"title":"Identify the input values","text":"The domain is {a, $$b$$, c}.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota3b-h2","type":"hint","dependencies":["af36e21FunNota3b-h1"],"title":"Identify the output values","text":"The range is {b, c}.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota3b-h3","type":"hint","dependencies":["af36e21FunNota3b-h2"],"title":"Classify the relation","text":"Because each input value in the domain leads to only one output value in the range, the relation represents a function.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af36e21FunNota4","title":"Using Function Notation for Days in a Month","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Functions and Function Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"af36e21FunNota4a","stepAnswer":["$$d=f(m)$$"],"problemType":"MultipleChoice","stepTitle":"Using function notation to represent a function whose input is the name of a month and output is the number of days in that month. Assume that the domain does not include leap years.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$d=f(m)$$","choices":["$$d=f(m)$$","$$m=f(d)$$"],"hints":{"DefaultPathway":[{"id":"af36e21FunNota4a-h1","type":"hint","dependencies":[],"title":"Function notation","text":"The number of days in a month is a function of the name of the month, so if we name the function f, we write days $$=$$ f(month) or $$d$$ $$=$$ f(m).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af36e21FunNota5","title":"Interpreting Function Notation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Functions and Function Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"af36e21FunNota5a","stepAnswer":["In the year $$2005$$ there are $$300$$ police officers in the town."],"problemType":"MultipleChoice","stepTitle":"A function N $$=$$ f(y) gives the number of police officers, N, in a town in year $$y$$. What does f(2005) $$=$$ $$300$$ represent?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"In the year $$2005$$ there are $$300$$ police officers in the town.","choices":["In the year $$2005$$ there are $$300$$ police officers in the town.","In the year $$2005$$ there are $$300$$ more police officers in the town than in the previous year.","In $$300$$ years there will be $$2005$$ police officers in the town."],"hints":{"DefaultPathway":[{"id":"af36e21FunNota5a-h1","type":"hint","dependencies":[],"title":"Function notation","text":"When we read f(2005) $$=$$ $$300$$, we see that the input year is $$2005$$. The value for the output, the number of police officers, is $$300$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af36e21FunNota6","title":"Evaluating Functions at Specific Values","body":"Evaluate $$f(x)=x^2+3x-4$$ at the following values of x:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Functions and Function Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"af36e21FunNota6a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"$$2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"af36e21FunNota6a-h1","type":"hint","dependencies":[],"title":"Evaluating functions","text":"Given the equation for a function, we should replace the input variable in the equation with the value provided and then calculate the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota6a-h2","type":"hint","dependencies":["af36e21FunNota6a-h1"],"title":"Replacement","text":"Replace the variable $$x$$ with the number $$2$$. We get the expression $$2^2+3\\\\times2-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota6a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["af36e21FunNota6a-h2"],"title":"Calculation","text":"Calculate the expression $$2^2+3\\\\times2-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota6b","stepAnswer":["$$a^2+3a-4$$"],"problemType":"TextBox","stepTitle":"a","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$a^2+3a-4$$","hints":{"DefaultPathway":[{"id":"af36e21FunNota6b-h1","type":"hint","dependencies":[],"title":"Evaluating functions","text":"Given the equation for a function, we should replace the input variable in the equation with the value provided and then calculate the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota6b-h2","type":"hint","dependencies":["af36e21FunNota6b-h1"],"title":"Replacement","text":"Replace the variable $$x$$ with the a. We get the expression $$a^2+3a-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af36e21FunNota6c","stepAnswer":["$$a^2+2ah+h^2+3a+3h-4$$"],"problemType":"MultipleChoice","stepTitle":"$$a+h$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$a^2+2ah+h^2+3a+3h-4$$","choices":["$$a^2+3a+3h-4$$","$$a^2+2ah+2h^2+6a+6h-4$$","$$a^2+2ah+h^2+3a$$","$$a^2+2ah+h^2+3a+3h-4$$"],"hints":{"DefaultPathway":[{"id":"af36e21FunNota6c-h1","type":"hint","dependencies":[],"title":"Evaluating functions","text":"Given the equation for a function, we should replace the input variable in the equation with the value provided and then calculate the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota6c-h2","type":"hint","dependencies":["af36e21FunNota6c-h1"],"title":"Replacement","text":"Replace the variable $$x$$ with the $$a+h$$. We get the expression $${\\\\left(a+h\\\\right)}^2+3\\\\left(a+h\\\\right)-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota6c-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$a^2+2ah+h^2+3a+3h-4$$"],"dependencies":["af36e21FunNota6c-h2"],"title":"Simplification","text":"Simplify the expression $${\\\\left(a+h\\\\right)}^2+3\\\\left(a+h\\\\right)-4$$. What do we get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$a^2+3a+3h-4$$","$$a^2+2ah+2h^2+6a+6h-4$$","$$a^2+2ah+h^2+3a$$","$$a^2+2ah+h^2+3a+3h-4$$"]}]}},{"id":"af36e21FunNota6d","stepAnswer":["$$2a+h+3$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{f{\\\\left(a+h\\\\right)}-f{\\\\left(a\\\\right)}}{h}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2a+h+3$$","hints":{"DefaultPathway":[{"id":"af36e21FunNota6d-h1","type":"hint","dependencies":[],"title":"Evaluating functions","text":"Given the equation for a function, we should replace the input variable in the equation with the value provided and then calculate the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota6d-h2","type":"hint","dependencies":["af36e21FunNota6d-h1"],"title":"Replacement","text":"Replace the term $$f{\\\\left(a+h\\\\right)}$$ and f(a) with the result in step $$2$$ and step $$3$$, namely, $$f{\\\\left(a+h\\\\right)}=a^2+2ah+h^2+3a+3h-4$$, and $$f(a)=a^2+3a-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota6d-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2a+h+3$$"],"dependencies":["af36e21FunNota6d-h2"],"title":"Simplification","text":"Simplify the expression after plugging in. What do we get?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af36e21FunNota7","title":"Evaluating functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Functions and Function Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"af36e21FunNota7a","stepAnswer":["$$24$$"],"problemType":"TextBox","stepTitle":"Given the function $$h(p)=p^2+2p$$, evaluate h(4).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$24$$","hints":{"DefaultPathway":[{"id":"af36e21FunNota7a-h1","type":"hint","dependencies":[],"title":"Evaluating functions","text":"Given the equation for a function, we should replace the input variable in the equation with the value provided and then calculate the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota7a-h2","type":"hint","dependencies":["af36e21FunNota7a-h1"],"title":"Replacement","text":"Replace the variable $$p$$ with $$4$$. We get the expression $$4^2+2\\\\times4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota7a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$24$$"],"dependencies":["af36e21FunNota7a-h2"],"title":"Simplification","text":"Evaluate the expression $$4^2+2\\\\times4$$. What is the answer?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af36e21FunNota8","title":"Evaluating functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Functions and Function Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"af36e21FunNota8a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"Given the function $$g(m)=\\\\sqrt{m-4}$$, evaluate g(5).","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"af36e21FunNota8a-h1","type":"hint","dependencies":[],"title":"Evaluating functions","text":"Given the equation for a function, we should replace the input variable in the equation with the value provided and then calculate the result.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota8a-h2","type":"hint","dependencies":["af36e21FunNota8a-h1"],"title":"Replacement","text":"Replace the variable $$m$$ with $$5$$. We get the expression $$\\\\sqrt{5-4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["af36e21FunNota8a-h2"],"title":"Simplification","text":"Evaluate the expression $$\\\\sqrt{5-4}$$. What is it equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af36e21FunNota9","title":"Solving Functions","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3.1 Functions and Function Notation","courseName":"OpenStax: College Algebra","steps":[{"id":"af36e21FunNota9a","stepAnswer":["$$-3, 1$$"],"problemType":"MultipleChoice","stepTitle":"Given the function $$h(p)=p^2+2p$$, solve for $$h(p)=3$$. What are the values for $$p$$?","stepBody":"","answerType":"string","variabilization":{},"choices":["$$-3, 1$$","$$-1, 3$$","$$-2, 2$$","No solution"],"hints":{"DefaultPathway":[{"id":"af36e21FunNota9a-h1","type":"hint","dependencies":[],"title":"Find input value","text":"When we know an output value and want to determine the input values that would produce that output value, we set the output equal to the function\'s formula and solve for the input.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota9a-h2","type":"hint","dependencies":["af36e21FunNota9a-h1"],"title":"Substitution","text":"Substitute $$h(p)=3$$ into the original $$h(p)=p^2+2p$$, and we get the equation $$3=p^2+2p$$, which we are going to solve next.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota9a-h3","type":"hint","dependencies":["af36e21FunNota9a-h2"],"title":"Factorization","text":"To solve $$3=p^2+2p$$, we can start by moving $$3$$ to the other side: $$p^2+2p-3=0$$. We can now factor the expression into $$\\\\left(p+3\\\\right) \\\\left(p-1\\\\right)=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af36e21FunNota9a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-3, 1$$"],"dependencies":["af36e21FunNota9a-h3"],"title":"Solving Equations","text":"Solve the equation $$\\\\left(p+3\\\\right) \\\\left(p-1\\\\right)=0$$. What is $$p$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$-3, 1$$","$$-1, 3$$","$$-2, 2$$","No solution"]}]}}]},{"id":"af4e405log1","title":"Converting to Logarithmic Form","body":"Convert the following to logarithmic form.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Evaluate and Graph Logarithmic Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af4e405log1a","stepAnswer":["$$3=\\\\log_{2}\\\\left(8\\\\right)$$"],"problemType":"MultipleChoice","stepTitle":"$$2^3=8$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$3=\\\\log_{2}\\\\left(8\\\\right)$$","choices":["$$3=\\\\log_{2}\\\\left(8\\\\right)$$","$$2=\\\\log_{3}\\\\left(8\\\\right)$$","$$8=\\\\log_{2}\\\\left(3\\\\right)$$","$$3=\\\\log_{8}\\\\left(2\\\\right)$$"],"hints":{"DefaultPathway":[{"id":"af4e405log1a-h1","type":"hint","dependencies":[],"title":"Identify the base and exponent","text":"To convert an exponential equation to logarithmic form, we need to identify the base of the exponential, a, and the exponent $$y$$ since $$y=\\\\log_{a}\\\\left(x\\\\right)$$ is equivalent to $$x=a^y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log1a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["af4e405log1a-h1"],"title":"Identify the base","text":"What is the base of the exponent?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["af4e405log1a-h2"],"title":"Identify the exponent","text":"What is the exponent?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log1a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3=\\\\log_{2}\\\\left(8\\\\right)$$"],"dependencies":["af4e405log1a-h3"],"title":"Determining Logarithmic Form","text":"Knowing the base and the exponent, what is the logarithmic form?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$3=\\\\log_{2}\\\\left(8\\\\right)$$","$$2=\\\\log_{3}\\\\left(8\\\\right)$$","$$8=\\\\log_{2}\\\\left(3\\\\right)$$","$$3=\\\\log_{8}\\\\left(2\\\\right)$$"]}]}},{"id":"af4e405log1b","stepAnswer":["$$\\\\frac{1}{2}=\\\\log_{5}\\\\left(\\\\sqrt{5}\\\\right)$$"],"problemType":"MultipleChoice","stepTitle":"$$5^{\\\\frac{1}{2}}=\\\\sqrt{5}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{1}{2}=\\\\log_{5}\\\\left(\\\\sqrt{5}\\\\right)$$","choices":["$$\\\\frac{1}{2}=\\\\log_{5}\\\\left(\\\\sqrt{5}\\\\right)$$","$$\\\\sqrt{5}=\\\\log_{5}\\\\left(\\\\frac{1}{2}\\\\right)$$","$$5=\\\\log_{sqrt(5)}\\\\left(\\\\frac{1}{2}\\\\right)$$","$$\\\\frac{1}{2}=\\\\log_{sqrt(5)}\\\\left(5\\\\right)$$"],"hints":{"DefaultPathway":[{"id":"af4e405log1b-h1","type":"hint","dependencies":[],"title":"Identify the base and exponent","text":"To convert an exponential equation to logarithmic form, we need to identify the base of the exponential, a, and the exponent $$y$$ since $$y=\\\\log_{a}\\\\left(x\\\\right)$$ is equivalent to $$x=a^y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log1b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["af4e405log1b-h1"],"title":"Identify the base","text":"What is the base of the exponent?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log1b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["af4e405log1b-h2"],"title":"Identify the exponent","text":"What is the exponent?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log1b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1}{2}=\\\\log_{5}\\\\left(\\\\sqrt{5}\\\\right)$$"],"dependencies":["af4e405log1b-h3"],"title":"Determining Logarithmic Form","text":"Knowing the base and the exponent, what is the logarithmic form?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{1}{2}=\\\\log_{5}\\\\left(\\\\sqrt{5}\\\\right)$$","$$\\\\sqrt{5}=\\\\log_{5}\\\\left(\\\\frac{1}{2}\\\\right)$$","$$5=\\\\log_{sqrt(5)}\\\\left(\\\\frac{1}{2}\\\\right)$$","$$\\\\frac{1}{2}=\\\\log_{sqrt(5)}\\\\left(5\\\\right)$$"]}]}},{"id":"af4e405log1c","stepAnswer":["$$4=\\\\log_{(1/2)}\\\\left(\\\\frac{1}{16}\\\\right)$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(\\\\frac{1}{2}\\\\right)}^4=\\\\frac{1}{16}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$4=\\\\log_{(1/2)}\\\\left(\\\\frac{1}{16}\\\\right)$$","choices":["$$4=\\\\log_{(1/2)}\\\\left(\\\\frac{1}{16}\\\\right)$$","$$\\\\frac{1}{2}=\\\\log_{4}\\\\left(\\\\frac{1}{16}\\\\right)$$","$$\\\\frac{1}{16}=\\\\log_{(1/2)}\\\\left(4\\\\right)$$","$$4=\\\\log_{(1/16)}\\\\left(\\\\frac{1}{12}\\\\right)$$"],"hints":{"DefaultPathway":[{"id":"af4e405log1c-h1","type":"hint","dependencies":[],"title":"Identify the base and exponent","text":"To convert an exponential equation to logarithmic form, we need to identify the base of the exponential, a, and the exponent $$y$$ since $$y=\\\\log_{a}\\\\left(x\\\\right)$$ is equivalent to $$x=a^y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log1c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["af4e405log1c-h1"],"title":"Identify the base","text":"What is the base of the exponent?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log1c-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["af4e405log1c-h2"],"title":"Identify the exponent","text":"What is the exponent?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log1c-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$4=\\\\log_{(1/2)}\\\\left(\\\\frac{1}{16}\\\\right)$$"],"dependencies":["af4e405log1c-h3"],"title":"Determining Logarithmic Form","text":"Knowing the base and the exponent, what is the logarithmic form?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$4=\\\\log_{(1/2)}\\\\left(\\\\frac{1}{16}\\\\right)$$","$$\\\\frac{1}{2}=\\\\log_{4}\\\\left(\\\\frac{1}{16}\\\\right)$$","$$\\\\frac{1}{16}=\\\\log_{(1/2)}\\\\left(4\\\\right)$$","$$4=\\\\log_{(1/16)}\\\\left(\\\\frac{1}{12}\\\\right)$$"]}]}}]},{"id":"af4e405log10","title":"Numerically Evaluating Logarithmic Expressions","body":"Find the exact value of each logarithm without using a calculator.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Evaluate and Graph Logarithmic Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af4e405log10a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{5}\\\\left(25\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"af4e405log10a-h1","type":"hint","dependencies":[],"title":"Convert to Exponential Form","text":"We know that we\'re trying to determine $$x$$ such that $$x=\\\\log_{5}\\\\left(25\\\\right)$$. Rewrite this expression first into exponential form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log10a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$5^x=25$$"],"dependencies":["af4e405log10a-h1"],"title":"Determing the Exponential Form","text":"What is the exponential form of $$\\\\log_{5}\\\\left(25\\\\right)=x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$5^x=25$$","$${25}^x=5$$","$$x^5=25$$","$$x^{25}=5$$"]},{"id":"af4e405log10a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$5^2$$"],"dependencies":["af4e405log10a-h2"],"title":"Rewrite $$25$$ in terms of $$5$$","text":"How can we rewrite $$25$$ in terms of powers of 5?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$5^2$$","$$5^4$$","$$5^5$$","$$5^3$$"]},{"id":"af4e405log10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["af4e405log10a-h3"],"title":"Solve for $$x$$","text":"Substitution $$5^2$$ into the exponential equation we found before, we get $$5^x=5^2$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af4e405log10b","stepAnswer":["$$\\\\frac{1}{2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{9}\\\\left(3\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{2}$$","hints":{"DefaultPathway":[{"id":"af4e405log10b-h1","type":"hint","dependencies":[],"title":"Convert to Exponential Form","text":"We know that we\'re trying to determine $$x$$ such that $$x=\\\\log_{9}\\\\left(3\\\\right)$$. Rewrite this expression first into exponential form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log10b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$9^x=3$$"],"dependencies":["af4e405log10b-h1"],"title":"Determing the Exponential Form","text":"What is the exponential form of $$\\\\log_{9}\\\\left(3\\\\right)=x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$9^x=3$$","$$3^x=9$$","$$x^9=3$$","$$x^3=9$$"]},{"id":"af4e405log10b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3^2$$"],"dependencies":["af4e405log10b-h2"],"title":"Rewrite $$9$$ in terms of $$3$$","text":"How can we rewrite $$9$$ in terms of powers of 3?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$3^2$$","$$3^3$$","$$3^4$$","$$3^5$$"]},{"id":"af4e405log10b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["af4e405log10b-h3"],"title":"Solve for $$x$$","text":"Substitution $$3^2$$ into the exponential equation we found before, we get $${\\\\left(3^2\\\\right)}^x=3^1$$. We can simplify this into $$3^{2x}=3^1$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af4e405log10c","stepAnswer":["$$-4$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{2}\\\\left(\\\\frac{1}{16}\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-4$$","hints":{"DefaultPathway":[{"id":"af4e405log10c-h1","type":"hint","dependencies":[],"title":"Convert to Exponential Form","text":"We know that we\'re trying to determine $$x$$ such that $$x=\\\\log_{2}\\\\left(\\\\frac{1}{16}\\\\right)$$. Rewrite this expression first into exponential form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log10c-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2^x=\\\\frac{1}{16}$$"],"dependencies":["af4e405log10c-h1"],"title":"Determing the Exponential Form","text":"What is the exponential form of $$\\\\log_{2}\\\\left(\\\\frac{1}{16}\\\\right)=x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2^x=\\\\frac{1}{16}$$","$${\\\\left(\\\\frac{1}{16}\\\\right)}^x=2$$","$$x^2=\\\\frac{1}{16}$$","$$x^{\\\\frac{1}{16}}=2$$"]},{"id":"af4e405log10c-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${\\\\left(\\\\frac{1}{2}\\\\right)}^4$$"],"dependencies":["af4e405log10c-h2"],"title":"Rewrite $$\\\\frac{1}{16}$$ in terms of $$\\\\frac{1}{2}$$","text":"Since $$2^4=16$$, we can actually rewrite $$\\\\frac{1}{16}$$ in terms of $$\\\\frac{1}{2}$$. What is $$\\\\frac{1}{16}$$ equal to in terms of $$\\\\frac{1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$${\\\\left(\\\\frac{1}{2}\\\\right)}^4$$","$$4^{\\\\frac{1}{2}}$$"]},{"id":"af4e405log10c-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2^{\\\\left(-4\\\\right)}$$"],"dependencies":["af4e405log10c-h3"],"title":"Simplifying the Exponential","text":"We note that $${\\\\left(\\\\frac{1}{x}\\\\right)}^y=x^{\\\\left(-y\\\\right)}$$. How can we rewrite $$\\\\frac{1}{16}={\\\\left(\\\\frac{1}{2}\\\\right)}^4$$ in terms of powers of 2?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2^{\\\\left(-4\\\\right)}$$","$${\\\\left(-4\\\\right)}^2$$","$${\\\\left(-16\\\\right)}^2$$","$$2^{\\\\left(-16\\\\right)}$$"]},{"id":"af4e405log10c-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["af4e405log10c-h4"],"title":"Solve for $$x$$","text":"Substitution $$2^{\\\\left(-4\\\\right)}$$ into the exponential equation we found before, we get $$2^x=2^{\\\\left(-4\\\\right)}$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af4e405log11","title":"Numerically Evaluating Logarithmic Expressions","body":"Find the exact value of each logarithm without using a calculator.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Evaluate and Graph Logarithmic Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af4e405log11a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{12}\\\\left(144\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"af4e405log11a-h1","type":"hint","dependencies":[],"title":"Convert to Exponential Form","text":"We know that we\'re trying to determine $$x$$ such that $$x=\\\\log_{12}\\\\left(144\\\\right)$$. Rewrite this expression first into exponential form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log11a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${12}^x=144$$"],"dependencies":["af4e405log11a-h1"],"title":"Determing the Exponential Form","text":"What is the exponential form of $$\\\\log_{12}\\\\left(144\\\\right)=x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$${12}^x=144$$","$${144}^x=12$$","$$x^{12}=144$$","$$x^{144}=12$$"]},{"id":"af4e405log11a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${12}^2$$"],"dependencies":["af4e405log11a-h2"],"title":"Rewrite $$144$$ in terms of $$12$$","text":"How can we rewrite $$144$$ in terms of powers of 12?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$${12}^2$$","$${12}^4$$","$${12}^5$$","$${12}^3$$"]},{"id":"af4e405log11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["af4e405log11a-h3"],"title":"Solve for $$x$$","text":"Substitution $${12}^2$$ into the exponential equation we found before, we get $${12}^x={12}^2$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af4e405log11b","stepAnswer":["$$\\\\frac{1}{2}$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{4}\\\\left(2\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{2}$$","hints":{"DefaultPathway":[{"id":"af4e405log11b-h1","type":"hint","dependencies":[],"title":"Convert to Exponential Form","text":"We know that we\'re trying to determine $$x$$ such that $$x=\\\\log_{4}\\\\left(2\\\\right)$$. Rewrite this expression first into exponential form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log11b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$4^x=2$$"],"dependencies":["af4e405log11b-h1"],"title":"Determing the Exponential Form","text":"What is the exponential form of $$\\\\log_{4}\\\\left(2\\\\right)=x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$4^x=2$$","$$2^x=4$$","$$x^4=2$$","$$x^2=4$$"]},{"id":"af4e405log11b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2^2$$"],"dependencies":["af4e405log11b-h2"],"title":"Rewrite $$4$$ in terms of $$2$$","text":"How can we rewrite $$4$$ in terms of powers of 2?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2^2$$","$$2^3$$","$$2^4$$","$$2^5$$"]},{"id":"af4e405log11b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["af4e405log11b-h3"],"title":"Solve for $$x$$","text":"Substitution $$2^2$$ into the exponential equation we found before, we get $${\\\\left(2^2\\\\right)}^x=2^1$$. We can simplify this into $$2^{2x}=2^1$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af4e405log11c","stepAnswer":["$$-5$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{2}\\\\left(\\\\frac{1}{32}\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-5$$","hints":{"DefaultPathway":[{"id":"af4e405log11c-h1","type":"hint","dependencies":[],"title":"Convert to Exponential Form","text":"We know that we\'re trying to determine $$x$$ such that $$x=\\\\log_{2}\\\\left(\\\\frac{1}{32}\\\\right)$$. Rewrite this expression first into exponential form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log11c-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2^x=\\\\frac{1}{32}$$"],"dependencies":["af4e405log11c-h1"],"title":"Determing the Exponential Form","text":"What is the exponential form of $$\\\\log_{2}\\\\left(\\\\frac{1}{32}\\\\right)=x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2^x=\\\\frac{1}{32}$$","$${\\\\left(\\\\frac{1}{32}\\\\right)}^x=2$$","$$x^2=\\\\frac{1}{32}$$","$$x^{\\\\frac{1}{32}}=2$$"]},{"id":"af4e405log11c-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${\\\\left(\\\\frac{1}{2}\\\\right)}^5$$"],"dependencies":["af4e405log11c-h2"],"title":"Rewrite $$\\\\frac{1}{16}$$ in terms of $$\\\\frac{1}{2}$$","text":"Since $$2^5=32$$, we can actually rewrite $$\\\\frac{1}{32}$$ in terms of $$\\\\frac{1}{2}$$. What is $$\\\\frac{1}{32}$$ equal to in terms of $$\\\\frac{1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$${\\\\left(\\\\frac{1}{2}\\\\right)}^5$$","$$5^{\\\\frac{1}{2}}$$"]},{"id":"af4e405log11c-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2^{\\\\left(-5\\\\right)}$$"],"dependencies":["af4e405log11c-h3"],"title":"Simplifying the Exponential","text":"We note that $${\\\\left(\\\\frac{1}{x}\\\\right)}^y=x^{\\\\left(-y\\\\right)}$$. How can we rewrite $$\\\\frac{1}{32}={\\\\left(\\\\frac{1}{2}\\\\right)}^5$$ in terms of powers of 2?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2^{\\\\left(-5\\\\right)}$$","$${\\\\left(-5\\\\right)}^2$$","$${\\\\left(-32\\\\right)}^2$$","$$2^{\\\\left(-32\\\\right)}$$"]},{"id":"af4e405log11c-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-5$$"],"dependencies":["af4e405log11c-h4"],"title":"Solve for $$x$$","text":"Substitution $$2^{\\\\left(-5\\\\right)}$$ into the exponential equation we found before, we get $$2^x=2^{\\\\left(-5\\\\right)}$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af4e405log12","title":"Numerically Evaluating Logarithmic Expressions","body":"Find the exact value of each logarithm without using a calculator.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Evaluate and Graph Logarithmic Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af4e405log12a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{9}\\\\left(81\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"af4e405log12a-h1","type":"hint","dependencies":[],"title":"Convert to Exponential Form","text":"We know that we\'re trying to determine $$x$$ such that $$x=\\\\log_{9}\\\\left(81\\\\right)$$. Rewrite this expression first into exponential form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log12a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$9^x=81$$"],"dependencies":["af4e405log12a-h1"],"title":"Determing the Exponential Form","text":"What is the exponential form of $$\\\\log_{9}\\\\left(81\\\\right)=x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$9^x=81$$","$${81}^x=9$$","$$x^9=81$$","$$x^{81}=9$$"]},{"id":"af4e405log12a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$9^2$$"],"dependencies":["af4e405log12a-h2"],"title":"Rewrite $$81$$ in terms of $$9$$","text":"How can we rewrite $$81$$ in terms of powers of 9?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$9^2$$","$$9^3$$","$$9^4$$","$$9^5$$"]},{"id":"af4e405log12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["af4e405log12a-h3"],"title":"Solve for $$x$$","text":"Substitution $$9^2$$ into the exponential equation we found before, we get $$9^x=9^2$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af4e405log12b","stepAnswer":["$$\\\\frac{1}{3}$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{8}\\\\left(2\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{1}{3}$$","hints":{"DefaultPathway":[{"id":"af4e405log12b-h1","type":"hint","dependencies":[],"title":"Convert to Exponential Form","text":"We know that we\'re trying to determine $$x$$ such that $$x=\\\\log_{8}\\\\left(2\\\\right)$$. Rewrite this expression first into exponential form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log12b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$8^x=2$$"],"dependencies":["af4e405log12b-h1"],"title":"Determing the Exponential Form","text":"What is the exponential form of $$\\\\log_{8}\\\\left(2\\\\right)=x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$8^x=2$$","$$2^x=8$$","$$x^8=2$$","$$x^2=8$$"]},{"id":"af4e405log12b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2^3$$"],"dependencies":["af4e405log12b-h2"],"title":"Rewrite $$8$$ in terms of $$2$$","text":"How can we rewrite $$8$$ in terms of powers of 2?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2^2$$","$$2^3$$","$$2^4$$","$$2^5$$"]},{"id":"af4e405log12b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["af4e405log12b-h3"],"title":"Solve for $$x$$","text":"Substitution $$2^3$$ into the exponential equation we found before, we get $${\\\\left(2^3\\\\right)}^x=2^1$$. We can simplify this into $$2^{3x}=2^1$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af4e405log12c","stepAnswer":["$$-2$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{3}\\\\left(\\\\frac{1}{9}\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-2$$","hints":{"DefaultPathway":[{"id":"af4e405log12c-h1","type":"hint","dependencies":[],"title":"Convert to Exponential Form","text":"We know that we\'re trying to determine $$x$$ such that $$x=\\\\log_{3}\\\\left(\\\\frac{1}{9}\\\\right)$$. Rewrite this expression first into exponential form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log12c-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3^x=\\\\frac{1}{9}$$"],"dependencies":["af4e405log12c-h1"],"title":"Determing the Exponential Form","text":"What is the exponential form of $$\\\\log_{3}\\\\left(\\\\frac{1}{9}\\\\right)=x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$3^x=\\\\frac{1}{9}$$","$${\\\\left(\\\\frac{1}{9}\\\\right)}^x=3$$","$$x^3=\\\\frac{1}{9}$$","$$x^{\\\\frac{1}{9}}=3$$"]},{"id":"af4e405log12c-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${\\\\left(\\\\frac{1}{3}\\\\right)}^2$$"],"dependencies":["af4e405log12c-h2"],"title":"Rewrite $$\\\\frac{1}{9}$$ in terms of $$\\\\frac{1}{3}$$","text":"Since $$3^2=9$$, we can actually rewrite $$\\\\frac{1}{9}$$ in terms of $$\\\\frac{1}{3}$$. What is $$\\\\frac{1}{9}$$ equal to in terms of $$\\\\frac{1}{3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$${\\\\left(\\\\frac{1}{3}\\\\right)}^2$$","$$2^{\\\\frac{1}{3}}$$"]},{"id":"af4e405log12c-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3^{\\\\left(-2\\\\right)}$$"],"dependencies":["af4e405log12c-h3"],"title":"Simplifying the Exponential","text":"We note that $${\\\\left(\\\\frac{1}{x}\\\\right)}^y=x^{\\\\left(-y\\\\right)}$$. How can we rewrite $$\\\\frac{1}{32}={\\\\left(\\\\frac{1}{2}\\\\right)}^5$$ in terms of powers of 2?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$3^{\\\\left(-2\\\\right)}$$","$${\\\\left(-2\\\\right)}^3$$","$${\\\\left(-9\\\\right)}^3$$","$$3^{\\\\left(-9\\\\right)}$$"]},{"id":"af4e405log12c-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["af4e405log12c-h4"],"title":"Solve for $$x$$","text":"Substitution $$3^{\\\\left(-2\\\\right)}$$ into the exponential equation we found before, we get $$3^x=3^{\\\\left(-2\\\\right)}$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af4e405log13","title":"Graphing Logarithmic Functions","body":"Choose the correct graph of the following logarithmic function.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Evaluate and Graph Logarithmic Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af4e405log13a","stepAnswer":["Graph B"],"problemType":"MultipleChoice","stepTitle":"$$y=\\\\log_{2}\\\\left(x\\\\right)$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Graph A","Graph B","Graph C","Graph D"],"hints":{"DefaultPathway":[{"id":"af4e405log13a-h1","type":"hint","dependencies":[],"title":"Convert to Exponential Form","text":"We know that we\'re trying to graph $$y=\\\\log_{2}\\\\left(x\\\\right)$$. Rewrite this expression first into exponential form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log13a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2^y=x$$"],"dependencies":["af4e405log13a-h1"],"title":"Determing the Exponential Form","text":"What is the exponential form of $$y=\\\\log_{2}\\\\left(x\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2^y=x$$","$$2^x=y$$","$$x^y=2$$","$$y^2=x$$"]},{"id":"af4e405log13a-h3","type":"hint","dependencies":["af4e405log13a-h2"],"title":"Determining Points of the Graph","text":"Next, we will use point plotting to determine which graph is correct. It will be easier to start with values of $$y$$ to get values of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log13a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{4}$$"],"dependencies":["af4e405log13a-h3"],"title":"Determining $$x$$ when $$y=-2$$","text":"We know $$x=2^y$$ so what is $$x$$ when $$y=-2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log13a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["af4e405log13a-h4"],"title":"Determining $$x$$ when $$y=-1$$","text":"We know $$x=2^y$$ so what is $$x$$ when $$y=-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log13a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["af4e405log13a-h5"],"title":"Determining $$x$$ when $$y=0$$","text":"We know $$x=2^y$$ so what is $$x$$ when $$y=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log13a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["af4e405log13a-h6"],"title":"Determining $$x$$ when $$y=1$$","text":"We know $$x=2^y$$ so what is $$x$$ when $$y=1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log13a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["af4e405log13a-h7"],"title":"Determining $$x$$ when $$y=2$$","text":"We know $$x=2^y$$ so what is $$x$$ when $$y=2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log13a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["af4e405log13a-h8"],"title":"Determining $$x$$ when $$y=3$$","text":"We know $$x=2^y$$ so what is $$x$$ when $$y=3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log13a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph B"],"dependencies":["af4e405log13a-h9"],"title":"Determining the Correct Graph","text":"Which of the graphs provided fits each of the $$6$$ plot points (x, y) that we found above where $$x$$ is the horizontal axis and $$y$$ is the vertical axis?\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph B","Graph C","Graph D"]}]}}]},{"id":"af4e405log14","title":"Graphing Logarithmic Functions","body":"Choose the correct graph of the following logarithmic function.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Evaluate and Graph Logarithmic Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af4e405log14a","stepAnswer":["Graph D"],"problemType":"MultipleChoice","stepTitle":"$$y=\\\\log_{3}\\\\left(x\\\\right)$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["Graph A","Graph B","Graph C","Graph D"],"hints":{"DefaultPathway":[{"id":"af4e405log14a-h1","type":"hint","dependencies":[],"title":"Convert to Exponential Form","text":"We know that we\'re trying to graph $$y=\\\\log_{3}\\\\left(x\\\\right)$$. Rewrite this expression first into exponential form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log14a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3^y=x$$"],"dependencies":["af4e405log14a-h1"],"title":"Determing the Exponential Form","text":"What is the exponential form of $$y=\\\\log_{3}\\\\left(x\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$3^y=x$$","$$3^x=y$$","$$x^y=3$$","$$y^3=x$$"]},{"id":"af4e405log14a-h3","type":"hint","dependencies":["af4e405log14a-h2"],"title":"Determining Points of the Graph","text":"Next, we will use point plotting to determine which graph is correct. It will be easier to start with values of $$y$$ to get values of $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log14a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{9}$$"],"dependencies":["af4e405log14a-h3"],"title":"Determining $$x$$ when $$y=-2$$","text":"We know $$x=3^y$$ so what is $$x$$ when $$y=-2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log14a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["af4e405log14a-h4"],"title":"Determining $$x$$ when $$y=-1$$","text":"We know $$x=3^y$$ so what is $$x$$ when $$y=-1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log14a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["af4e405log14a-h5"],"title":"Determining $$x$$ when $$y=0$$","text":"We know $$x=3^y$$ so what is $$x$$ when $$y=0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log14a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["af4e405log14a-h6"],"title":"Determining $$x$$ when $$y=1$$","text":"We know $$x=3^y$$ so what is $$x$$ when $$y=1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log14a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["af4e405log14a-h7"],"title":"Determining $$x$$ when $$y=2$$","text":"We know $$x=3^y$$ so what is $$x$$ when $$y=2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log14a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$27$$"],"dependencies":["af4e405log14a-h8"],"title":"Determining $$x$$ when $$y=3$$","text":"We know $$x=3^y$$ so what is $$x$$ when $$y=3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log14a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Graph D"],"dependencies":["af4e405log14a-h9"],"title":"Determining the Correct Graph","text":"Which of the graphs provided fits each of the $$6$$ plot points (x, y) that we found above where $$x$$ is the horizontal axis and $$y$$ is the vertical axis?\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Graph A","Graph B","Graph C","Graph D"]}]}}]},{"id":"af4e405log15","title":"Solving Logarithmic Equations","body":"Solve the following logarithmic equations.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Evaluate and Graph Logarithmic Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af4e405log15a","stepAnswer":["$$7$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{a}\\\\left(49\\\\right)=2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7$$","hints":{"DefaultPathway":[{"id":"af4e405log15a-h1","type":"hint","dependencies":[],"title":"Convert to Exponential Form","text":"We know that we\'re trying to solve $$\\\\log_{a}\\\\left(49\\\\right)=2$$. Rewrite this expression first into exponential form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log15a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$a^2=49$$"],"dependencies":["af4e405log15a-h1"],"title":"Determing the Exponential Form","text":"What is the exponential form of $$\\\\log_{a}\\\\left(49\\\\right)=2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$a^2=49$$","$${49}^2=a$$","$${49}^a=2$$","$$2^a=49$$"]},{"id":"af4e405log15a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["af4e405log15a-h2"],"title":"Solve for a","text":"What is the solution for a in the exponential equation $$a^2=49$$? Note that the domain of logarithmic equations is non-negative, so a must be non-negative!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af4e405log15b","stepAnswer":["$$e^3$$"],"problemType":"MultipleChoice","stepTitle":"$$ln(x)=3$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$e^3$$","choices":["$$e^3$$","$$3^e$$","$$1$$","e"],"hints":{"DefaultPathway":[{"id":"af4e405log15b-h1","type":"hint","dependencies":[],"title":"Convert to Exponential Form","text":"We know that we\'re trying to solve $$ln(x)=3$$. Rewrite this expression first into exponential form. Remember that the base for ln functions is e.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log15b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$e^3=x$$"],"dependencies":["af4e405log15b-h1"],"title":"Determing the Exponential Form","text":"What is the exponential form of $$ln(x)=3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$e^3=x$$","$$3^e=x$$","$$x^3=e$$","$$3^x=e$$"]},{"id":"af4e405log15b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$e^3$$"],"dependencies":["af4e405log15b-h2"],"title":"Solve for $$x$$","text":"What is the solution for $$x$$ in the exponential equation $$e^3=x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$e^3$$","$$3^e$$"]}]}}]},{"id":"af4e405log16","title":"Solving Logarithmic Equations","body":"Solve the following logarithmic equations.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Evaluate and Graph Logarithmic Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af4e405log16a","stepAnswer":["$$11$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{a}\\\\left(121\\\\right)=2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$11$$","hints":{"DefaultPathway":[{"id":"af4e405log16a-h1","type":"hint","dependencies":[],"title":"Convert to Exponential Form","text":"We know that we\'re trying to solve $$\\\\log_{a}\\\\left(121\\\\right)=2$$. 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Note that the domain of logarithmic equations is non-negative, so a must be non-negative!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af4e405log16b","stepAnswer":["$$e^7$$"],"problemType":"MultipleChoice","stepTitle":"$$ln(x)=7$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$e^7$$","choices":["$$1$$","$$3^e$$","e","$$e^3$$","$$e^7$$"],"hints":{"DefaultPathway":[{"id":"af4e405log16b-h1","type":"hint","dependencies":[],"title":"Convert to Exponential Form","text":"We know that we\'re trying to solve $$ln(x)=7$$. Rewrite this expression first into exponential form. Remember that the base for ln functions is e.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log16b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$e^7=x$$"],"dependencies":["af4e405log16b-h1"],"title":"Determing the Exponential Form","text":"What is the exponential form of $$ln(x)=7$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$e^7=x$$","$$7^e=x$$","$$x^7=e$$","$$7^x=e$$"]},{"id":"af4e405log16b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$e^7$$"],"dependencies":["af4e405log16b-h2"],"title":"Solve for $$x$$","text":"What is the solution for $$x$$ in the exponential equation $$e^7=x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$e^7$$","$$7^e$$"]}]}}]},{"id":"af4e405log17","title":"Solving Logarithmic Equations","body":"Solve the following logarithmic equations.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Evaluate and Graph Logarithmic Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af4e405log17a","stepAnswer":["$$4$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{a}\\\\left(64\\\\right)=3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4$$","hints":{"DefaultPathway":[{"id":"af4e405log17a-h1","type":"hint","dependencies":[],"title":"Convert to Exponential Form","text":"We know that we\'re trying to solve $$\\\\log_{a}\\\\left(64\\\\right)=3$$. Rewrite this expression first into exponential form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log17a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$a^3=64$$"],"dependencies":["af4e405log17a-h1"],"title":"Determing the Exponential Form","text":"What is the exponential form of $$\\\\log_{a}\\\\left(64\\\\right)=3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$a^3=64$$","$${64}^3=a$$","$${64}^a=3$$","$$3^a=64$$"]},{"id":"af4e405log17a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["af4e405log17a-h2"],"title":"Solve for a","text":"What is the solution for a in the exponential equation $$a^3=64$$? Note that the domain of logarithmic equations is non-negative, so a must be non-negative!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af4e405log17b","stepAnswer":["$$e^9$$"],"problemType":"MultipleChoice","stepTitle":"$$ln(x)=9$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$e^9$$","choices":["$$1$$","$$3^e$$","e","$$e^3$$","$$e^9$$"],"hints":{"DefaultPathway":[{"id":"af4e405log17b-h1","type":"hint","dependencies":[],"title":"Convert to Exponential Form","text":"We know that we\'re trying to solve $$ln(x)=9$$. Rewrite this expression first into exponential form. Remember that the base for ln functions is e.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log17b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$e^9=x$$"],"dependencies":["af4e405log17b-h1"],"title":"Determing the Exponential Form","text":"What is the exponential form of $$ln(x)=9$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$e^9=x$$","$$9^e=x$$","$$x^9=e$$","$$9^x=e$$"]},{"id":"af4e405log17b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$e^9$$"],"dependencies":["af4e405log17b-h2"],"title":"Solve for $$x$$","text":"What is the solution for $$x$$ in the exponential equation $$e^9=x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$e^9$$","$$9^e$$"]}]}}]},{"id":"af4e405log18","title":"Solving Algebraic Logarithmic Equations","body":"Solving the following logarithmic equations.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Evaluate and Graph Logarithmic Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af4e405log18a","stepAnswer":["$$7$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{2}\\\\left(3x-5\\\\right)=4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7$$","hints":{"DefaultPathway":[{"id":"af4e405log18a-h1","type":"hint","dependencies":[],"title":"Convert to Exponential Form","text":"We know that we\'re trying to solve $$\\\\log_{2}\\\\left(3x-5\\\\right)=4$$. 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Rewrite this expression first into exponential form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log18b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$e^4=e^{2x}$$"],"dependencies":["af4e405log18b-h1"],"title":"Determing the Exponential Form","text":"What is the exponential form of $$\\\\ln(e^{2x})=4$$? Remember that the base for ln functions is e.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$e^4=e^{2x}$$","$$2x^e=4^e$$","$$2x^e=e^4$$","$$e^{2x}=4^e$$"]},{"id":"af4e405log18b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["af4e405log18b-h2"],"title":"Solve for $$x$$","text":"Since the bases are the same, we can solve $$e^4=e^{2x}$$ as $$4=2x$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af4e405log19","title":"Solving Algebraic Logarithmic Equations","body":"Solving the following logarithmic equations.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Evaluate and Graph Logarithmic Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af4e405log19a","stepAnswer":["$$13$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{2}\\\\left(5x-1\\\\right)=6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$13$$","hints":{"DefaultPathway":[{"id":"af4e405log19a-h1","type":"hint","dependencies":[],"title":"Convert to Exponential Form","text":"We know that we\'re trying to solve $$\\\\log_{2}\\\\left(5x-1\\\\right)=6$$. Rewrite this expression first into exponential form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log19a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2^6=5x-1$$"],"dependencies":["af4e405log19a-h1"],"title":"Determing the Exponential Form","text":"What is the exponential form of $$\\\\log_{2}\\\\left(5x-1\\\\right)=6$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2^6=5x-1$$","$$6^2=5x-1$$","$${\\\\left(5x-1\\\\right)}^2=6$$","$${\\\\left(5x-1\\\\right)}^6=2$$"]},{"id":"af4e405log19a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$5x=65$$"],"dependencies":["af4e405log19a-h2"],"title":"Simplify the Expression","text":"Simplify the expression so all the terms with $$x$$ are on one side and all the constant $$(non-x)$$ terms are on the other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$5x=65$$","$$5x=64$$","$$5x=32$$","$$5x=35$$"],"subHints":[{"id":"af4e405log19a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$64$$"],"dependencies":[],"title":"Determine $$2^6$$","text":"What is $$2^6$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log19a-h3-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$65$$"],"dependencies":[],"title":"Determine $$64+1$$","text":"What is $$64+1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"af4e405log19a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$13$$"],"dependencies":["af4e405log19a-h3"],"title":"Solve for $$x$$","text":"What is the value of $$x$$ in $$5x=65$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af4e405log19b","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"$$\\\\ln(e^{3x})=6$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"af4e405log19b-h1","type":"hint","dependencies":[],"title":"Convert to Exponential Form","text":"We know that we\'re trying to solve $$\\\\ln(e^{3x})=6$$. Rewrite this expression first into exponential form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log19b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$e^6=e^{3x}$$"],"dependencies":["af4e405log19b-h1"],"title":"Determing the Exponential Form","text":"What is the exponential form of $$\\\\ln(e^{3x})=6$$? Remember that the base for ln functions is e.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$e^6=e^{3x}$$","$$3x^e=6^e$$","$$3x^e=e^6$$","$$e^{3x}=6^e$$"]},{"id":"af4e405log19b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["af4e405log19b-h2"],"title":"Solve for $$x$$","text":"Since the bases are the same, we can solve $$e^6=e^{3x}$$ as $$6=3x$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af4e405log2","title":"Converting to Logarithmic Form","body":"Convert the following to logarithmic form.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Evaluate and Graph Logarithmic Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af4e405log2a","stepAnswer":["$$2=\\\\log_{3}\\\\left(9\\\\right)$$"],"problemType":"MultipleChoice","stepTitle":"$$3^2=9$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2=\\\\log_{3}\\\\left(9\\\\right)$$","choices":["$$2=\\\\log_{3}\\\\left(9\\\\right)$$","$$3=\\\\log_{2}\\\\left(9\\\\right)$$","$$9=\\\\log_{3}\\\\left(2\\\\right)$$","$$2=\\\\log_{9}\\\\left(3\\\\right)$$"],"hints":{"DefaultPathway":[{"id":"af4e405log2a-h1","type":"hint","dependencies":[],"title":"Identify the base and exponent","text":"To convert an exponential equation to logarithmic form, we need to identify the base of the exponential, a, and the exponent $$y$$ since $$y=\\\\log_{a}\\\\left(x\\\\right)$$ is equivalent to $$x=a^y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 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form?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2=\\\\log_{3}\\\\left(9\\\\right)$$","$$3=\\\\log_{2}\\\\left(9\\\\right)$$","$$9=\\\\log_{3}\\\\left(2\\\\right)$$","$$2=\\\\log_{9}\\\\left(3\\\\right)$$"]}]}},{"id":"af4e405log2b","stepAnswer":["$$\\\\frac{1}{2}=\\\\log_{7}\\\\left(\\\\sqrt{7}\\\\right)$$"],"problemType":"MultipleChoice","stepTitle":"$$7^{\\\\frac{1}{2}}=\\\\sqrt{7}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{1}{2}=\\\\log_{7}\\\\left(\\\\sqrt{7}\\\\right)$$","choices":["(1/2)=log(7){sqrt(7)}","$$\\\\frac{1}{2}=\\\\log_{7}\\\\left(\\\\sqrt{7}\\\\right)$$","$$\\\\frac{1}{2}=\\\\log_{sqrt(7)}\\\\left(7\\\\right)$$","$$7=\\\\log_{sqrt(7)}\\\\left(\\\\frac{1}{2}\\\\right)$$","$$\\\\sqrt{7}=\\\\log_{7}\\\\left(\\\\frac{1}{2}\\\\right)$$"],"hints":{"DefaultPathway":[{"id":"af4e405log2b-h1","type":"hint","dependencies":[],"title":"Identify the base and exponent","text":"To convert an exponential equation to logarithmic form, we need to identify the base of the exponential, a, and the exponent $$y$$ since $$y=\\\\log_{a}\\\\left(x\\\\right)$$ is equivalent to $$x=a^y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log2b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["af4e405log2b-h1"],"title":"Identify the base","text":"What is the base of the exponent?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log2b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["af4e405log2b-h2"],"title":"Identify the exponent","text":"What is the exponent?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log2b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1}{2}=\\\\log_{7}\\\\left(\\\\sqrt{7}\\\\right)$$"],"dependencies":["af4e405log2b-h3"],"title":"Determining Logarithmic Form","text":"Knowing the base and the exponent, what is the logarithmic form?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{1}{2}=\\\\log_{7}\\\\left(\\\\sqrt{7}\\\\right)$$","$$\\\\sqrt{7}=\\\\log_{7}\\\\left(\\\\frac{1}{2}\\\\right)$$","$$7=\\\\log_{sqrt(7)}\\\\left(\\\\frac{1}{2}\\\\right)$$","$$\\\\frac{1}{2}=\\\\log_{sqrt(7)}\\\\left(7\\\\right)$$"]}]}},{"id":"af4e405log2c","stepAnswer":["$$x=\\\\log_{(1/3)}\\\\left(\\\\frac{1}{27}\\\\right)$$"],"problemType":"MultipleChoice","stepTitle":"$${\\\\left(\\\\frac{1}{3}\\\\right)}^x=\\\\frac{1}{27}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=\\\\log_{(1/3)}\\\\left(\\\\frac{1}{27}\\\\right)$$","choices":["$$x=\\\\log_{(1/3)}\\\\left(\\\\frac{1}{27}\\\\right)$$","$$\\\\frac{1}{3}=\\\\log_{x}\\\\left(\\\\frac{1}{27}\\\\right)$$","$$\\\\frac{1}{27}=\\\\log_{(1/3)}\\\\left(x\\\\right)$$","$$x=\\\\log_{(1/27)}\\\\left(\\\\frac{1}{3}\\\\right)$$"],"hints":{"DefaultPathway":[{"id":"af4e405log2c-h1","type":"hint","dependencies":[],"title":"Identify the base and exponent","text":"To convert an exponential equation to logarithmic form, we need to identify the base of the exponential, a, and the exponent $$y$$ since $$y=\\\\log_{a}\\\\left(x\\\\right)$$ is equivalent to $$x=a^y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log2c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["af4e405log2c-h1"],"title":"Identify the base","text":"What is the base of the exponent?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log2c-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x$$"],"dependencies":["af4e405log2c-h2"],"title":"Identify the exponent","text":"What is the exponent?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x$$","$$\\\\frac{1}{3}$$","$$\\\\frac{1}{27}$$"]},{"id":"af4e405log2c-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x=\\\\log_{(1/3)}\\\\left(\\\\frac{1}{27}\\\\right)$$"],"dependencies":["af4e405log2c-h3"],"title":"Determining Logarithmic Form","text":"Knowing the base and the exponent, what is the logarithmic form?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x=\\\\log_{(1/3)}\\\\left(\\\\frac{1}{27}\\\\right)$$","$$\\\\frac{1}{3}=\\\\log_{x}\\\\left(\\\\frac{1}{27}\\\\right)$$","$$\\\\frac{1}{27}=\\\\log_{(1/3)}\\\\left(x\\\\right)$$","$$x=\\\\log_{(1/27)}\\\\left(\\\\frac{1}{3}\\\\right)$$"]}]}}]},{"id":"af4e405log20","title":"Solving Algebraic Logarithmic Equations","body":"Solving the following logarithmic equations.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Evaluate and Graph Logarithmic Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af4e405log20a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{3}\\\\left(4x+3\\\\right)=3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"af4e405log20a-h1","type":"hint","dependencies":[],"title":"Convert to Exponential Form","text":"We know that we\'re trying to solve $$\\\\log_{3}\\\\left(4x+3\\\\right)=3$$. Rewrite this expression first into exponential form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log20a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3^3=4x+3$$"],"dependencies":["af4e405log20a-h1"],"title":"Determing the Exponential Form","text":"What is the exponential form of $$\\\\log_{3}\\\\left(4x+3\\\\right)=3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$3^3=4x+3$$","$$3^{4x+3}=3$$","$${\\\\left(4x+3\\\\right)}^3=3$$"]},{"id":"af4e405log20a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$4x=24$$"],"dependencies":["af4e405log20a-h2"],"title":"Simplify the Expression","text":"Simplify the expression so all the terms with $$x$$ are on one side and all the constant $$(non-x)$$ terms are on the other.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$4x=24$$","$$4x=27$$","$$4x=4$$","$$4x=8$$"],"subHints":[{"id":"af4e405log20a-h3-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$27$$"],"dependencies":[],"title":"Determine $$3^3$$","text":"What is $$3^3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log20a-h3-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$24$$"],"dependencies":[],"title":"Determine $$27-3$$","text":"What is $$27-3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"af4e405log20a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["af4e405log20a-h3"],"title":"Solve for $$x$$","text":"What is the value of $$x$$ in $$4x=24$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af4e405log20b","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"$$\\\\ln(e^{4x})=4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"af4e405log20b-h1","type":"hint","dependencies":[],"title":"Convert to Exponential Form","text":"We know that we\'re trying to solve $$\\\\ln(e^{4x})=4$$. Rewrite this expression first into exponential form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log20b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$e^4=e^{4x}$$"],"dependencies":["af4e405log20b-h1"],"title":"Determing the Exponential Form","text":"What is the exponential form of $$\\\\ln(e^{4x})=4$$? Remember that the base for ln functions is e.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$e^4=e^{4x}$$","$$4x^e=4^e$$","$$4x^e=e^4$$","$$e^{4x}=4^e$$"]},{"id":"af4e405log20b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["af4e405log20b-h2"],"title":"Solve for $$x$$","text":"Since the bases are the same, we can solve $$e^4=e^{4x}$$ as $$4=4x$$. What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af4e405log21","title":"Use Logarithmic Models in Applications","body":"Extended exposures to noice that measures $$85$$ dB (decibels) can cause permanent damage to the inner ear which will result in hearing loss. Use the formula: The loudness level, D, measured in decibels, of a sound of intensity, I, measured in watts per square inch is $$D=10*\\\\log_{10}\\\\left(\\\\frac{I}{{10}^{\\\\left(-12\\\\right)}}\\\\right)$$.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Evaluate and Graph Logarithmic Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af4e405log21a","stepAnswer":["$$100$$"],"problemType":"TextBox","stepTitle":"What is the decibel level of music coming through ear phones with intensity $${10}^{\\\\left(-2\\\\right)}$$ watts per square inch?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$100$$","hints":{"DefaultPathway":[{"id":"af4e405log21a-h1","type":"hint","dependencies":[],"title":"Decibel Level of Sound Formula","text":"We can use the decibel level of sound formula. The loudness level, D, measured in decibels, of a sound of intensity, I, measured in watts per square inch is $$D=10*\\\\log_{10}\\\\left(\\\\frac{I}{{10}^{\\\\left(-12\\\\right)}}\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log21a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$D=10*\\\\log_{10}\\\\left(\\\\frac{{10}^{\\\\left(-2\\\\right)}}{{10}^{\\\\left(-12\\\\right)}}\\\\right)$$"],"dependencies":["af4e405log21a-h1"],"title":"Plug in I","text":"Plug in the value for the intensity level, I.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$D=10*\\\\log_{10}\\\\left(\\\\frac{{10}^{\\\\left(-2\\\\right)}}{{10}^{\\\\left(-12\\\\right)}}\\\\right)$$","$$D=10*\\\\log_{10}\\\\left(\\\\frac{{10}^{\\\\left(-12\\\\right)}}{{10}^{\\\\left(-2\\\\right)}}\\\\right)$$","$${10}^{\\\\left(-12\\\\right)}=10*\\\\log_{10}\\\\left(\\\\frac{D}{{10}^{\\\\left(-12\\\\right)}}\\\\right)$$","$${10}^{\\\\left(-2\\\\right)}=10*\\\\log_{10}\\\\left(\\\\frac{{10}^{\\\\left(-12\\\\right)}}{D}\\\\right)$$"]},{"id":"af4e405log21a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${10}^{10}$$"],"dependencies":["af4e405log21a-h2"],"title":"Simplify the Logarithm","text":"We see log{10**(-2)/10**(-12)}. We can simplify the expression insid ethe logarithm, noting that $$\\\\frac{x^a}{x^b}=x^{a-b}$$. What is $$\\\\frac{{10}^{\\\\left(-2\\\\right)}}{{10}^{\\\\left(-12\\\\right)}}$$ simplified?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$${10}^{10}$$","$${10}^{\\\\left(-10\\\\right)}$$","$${10}^{\\\\left(-14\\\\right)}$$","$${10}^{14}$$"]},{"id":"af4e405log21a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["af4e405log21a-h3"],"title":"Simplify the Logarithm","text":"Now, we have $$D=10*\\\\log_{10}\\\\left({10}^{10}\\\\right)$$. What is $$\\\\log_{10}\\\\left({10}^{10}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","subHints":[{"id":"af4e405log21a-h4-s1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${10}^{10}={10}^x$$"],"dependencies":[],"title":"Convert to Exponential Form","text":"We know that we\'re trying to solve $$x=\\\\log_{10}\\\\left({10}^{10}\\\\right)$$. Rewrite this expression first into exponential form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$${10}^{10}={10}^x$$","$${10}^{10}=x^{10}$$"]},{"id":"af4e405log21a-h4-s2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":[],"title":"Solve for $$x$$","text":"Using the equation $${10}^{10}={10}^x$$, what is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"af4e405log21a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$100$$"],"dependencies":["af4e405log21a-h4"],"title":"Solve for D","text":"Now, we have $$D=10\\\\times10$$. What is D?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af4e405log23","title":"Convert to Logarithmic Form","body":"Convert the following exponential equation into logarithmic form.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Evaluate and Graph Logarithmic Functions","courseName":"OpenStax: Intermediate 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What is $$x$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af4e405log27b","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{5}\\\\left(1\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"af4e405log27b-h1","type":"hint","dependencies":[],"title":"Convert to Exponential Form","text":"We know that we\'re trying to determine $$x$$ such that x=log{}5{1}. 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Rewrite this expression first into exponential form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log30a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$a^2=81$$"],"dependencies":["af4e405log30a-h1"],"title":"Determing the Exponential Form","text":"What is the exponential form of $$\\\\log_{a}\\\\left(81\\\\right)=2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$a^2=81$$","$${81}^2=a$$","$${81}^a=2$$","$$2^a=81$$"]},{"id":"af4e405log30a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["af4e405log30a-h2"],"title":"Solve for a","text":"What is the solution for a in the exponential equation $$\\\\log_{a}\\\\left(81\\\\right)=2$$? Note that the domain of logarithmic equations is non-negative, so a must be non-negative!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af4e405log30b","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{a}\\\\left(27\\\\right)=3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"af4e405log30b-h1","type":"hint","dependencies":[],"title":"Convert to Exponential Form","text":"We know that we\'re trying to solve $$\\\\log_{a}\\\\left(27\\\\right)=3$$. Rewrite this expression first into exponential form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log30b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$a^3=27$$"],"dependencies":["af4e405log30b-h1"],"title":"Determing the Exponential Form","text":"What is the exponential form of $$\\\\log_{a}\\\\left(27\\\\right)=3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$a^3=27$$","$${27}^3=a$$","$${27}^a=3$$","$$3^a=27$$"]},{"id":"af4e405log30b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["af4e405log30b-h2"],"title":"Solve for a","text":"What is the solution for a in the exponential equation $$a^3=27$$? Note that the domain of logarithmic equations is non-negative, so a must be non-negative!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af4e405log4","title":"Converting to Exponential Form","body":"Convert the following to exponential form.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Evaluate and Graph Logarithmic Functions","courseName":"OpenStax: Intermediate 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To convert a logarithmic equation to exponential form, we need to identify the base of the exponential, a, and the exponent $$y$$ since $$y=\\\\log_{a}\\\\left(x\\\\right)$$ is equivalent to $$x=a^y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log7c-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${\\\\left(\\\\frac{1}{2}\\\\right)}^x=\\\\frac{1}{8}$$"],"dependencies":["af4e405log7c-h1"],"title":"Determining the Exponential Form","text":"What is the exponential form of the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$${\\\\left(\\\\frac{1}{2}\\\\right)}^x=\\\\frac{1}{8}$$","$${\\\\left(\\\\frac{1}{8}\\\\right)}^x=\\\\frac{1}{2}$$","$$x^{\\\\frac{1}{2}}=\\\\frac{1}{8}$$","$${\\\\left(\\\\frac{1}{8}\\\\right)}^{\\\\frac{1}{2}}=x$$"]},{"id":"af4e405log7c-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${\\\\left(\\\\frac{1}{2}\\\\right)}^3$$"],"dependencies":["af4e405log7c-h2"],"title":"Rewrite $$\\\\frac{1}{8}$$ with base $$\\\\frac{1}{2}$$","text":"Since $$2^3=8$$, we can actually rewrite $$\\\\frac{1}{8}$$ in terms of $$\\\\frac{1}{2}$$. What is $$\\\\frac{1}{8}$$ equal to in terms of $$\\\\frac{1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$${\\\\left(\\\\frac{1}{2}\\\\right)}^3$$","$$3^{\\\\frac{1}{2}}$$"]},{"id":"af4e405log7c-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["af4e405log7c-h3"],"title":"Solve for $$x$$","text":"Substituting $${\\\\left(\\\\frac{1}{2}\\\\right)}^3$$ into the exponential equation we found before, we get $${\\\\left(\\\\frac{1}{2}\\\\right)}^x={\\\\left(\\\\frac{1}{2}\\\\right)}^3$$. What is the solution for $$x$$ in this exponential equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af4e405log8","title":"Evaluate Logarithmic Functions","body":"Find the value of $$x$$ in the following logarithmic functions.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Evaluate and Graph Logarithmic Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af4e405log8a","stepAnswer":["$$8$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{x}\\\\left(64\\\\right)=2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8$$","hints":{"DefaultPathway":[{"id":"af4e405log8a-h1","type":"hint","dependencies":[],"title":"Convert to Exponential Form","text":"First, convert the expression to exponential form. To convert a logarithmic equation to exponential form, we need to identify the base of the exponential, a, and the exponent $$y$$ since $$y=\\\\log_{a}\\\\left(x\\\\right)$$ is equivalent to $$x=a^y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log8a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x^2=64$$"],"dependencies":["af4e405log8a-h1"],"title":"Determining the Exponential Form","text":"What is the exponential form of the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x^2=64$$","$${64}^2=x$$","$$x^{64}=2$$","$$2^x=64$$"]},{"id":"af4e405log8a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["af4e405log8a-h2"],"title":"Solve for $$x$$","text":"Knowing that the bases of a logarithmic function must be positive, what is the solution to the exponential equation you found above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af4e405log8b","stepAnswer":["$$125$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{5}\\\\left(x\\\\right)=3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$125$$","hints":{"DefaultPathway":[{"id":"af4e405log8b-h1","type":"hint","dependencies":[],"title":"Convert to Exponential Form","text":"First, convert the expression to exponential form. To convert a logarithmic equation to exponential form, we need to identify the base of the exponential, a, and the exponent $$y$$ since $$y=\\\\log_{a}\\\\left(x\\\\right)$$ is equivalent to $$x=a^y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log8b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$5^3=x$$"],"dependencies":["af4e405log8b-h1"],"title":"Determining the Exponential Form","text":"What is the exponential form of the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$5^3=x$$","$$3^5=x$$","$$5^x=3$$","$$3^x=5$$"]},{"id":"af4e405log8b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$125$$"],"dependencies":["af4e405log8b-h2"],"title":"Solve for $$x$$","text":"What is the solution for $$x$$ in the exponential equation you found above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af4e405log8c","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{(1/2)}\\\\left(\\\\frac{1}{4}\\\\right)=x$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"af4e405log8c-h1","type":"hint","dependencies":[],"title":"Convert to Exponential Form","text":"First, convert the expression to exponential form. To convert a logarithmic equation to exponential form, we need to identify the base of the exponential, a, and the exponent $$y$$ since $$y=\\\\log_{a}\\\\left(x\\\\right)$$ is equivalent to $$x=a^y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log8c-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${\\\\left(\\\\frac{1}{2}\\\\right)}^x=\\\\frac{1}{4}$$"],"dependencies":["af4e405log8c-h1"],"title":"Determining the Exponential Form","text":"What is the exponential form of the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$${\\\\left(\\\\frac{1}{2}\\\\right)}^x=\\\\frac{1}{4}$$","$${\\\\left(\\\\frac{1}{4}\\\\right)}^x=\\\\frac{1}{2}$$","$$x^{\\\\frac{1}{2}}=\\\\frac{1}{4}$$","$${\\\\left(\\\\frac{1}{4}\\\\right)}^{\\\\frac{1}{2}}=x$$"]},{"id":"af4e405log8c-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${\\\\left(\\\\frac{1}{2}\\\\right)}^2$$"],"dependencies":["af4e405log8c-h2"],"title":"Rewrite $$\\\\frac{1}{4}$$ with base $$\\\\frac{1}{2}$$","text":"Since $$2^2=4$$, we can actually rewrite $$\\\\frac{1}{4}$$ in terms of $$\\\\frac{1}{2}$$. What is $$\\\\frac{1}{4}$$ equal to in terms of $$\\\\frac{1}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$${\\\\left(\\\\frac{1}{2}\\\\right)}^2$$","$$2^{\\\\frac{1}{2}}$$"]},{"id":"af4e405log8c-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["af4e405log8c-h3"],"title":"Solve for $$x$$","text":"Substituting $${\\\\left(\\\\frac{1}{2}\\\\right)}^2$$ into the exponential equation we found before, we get $${\\\\left(\\\\frac{1}{2}\\\\right)}^x={\\\\left(\\\\frac{1}{2}\\\\right)}^2$$. What is the solution for $$x$$ in this exponential equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af4e405log9","title":"Evaluate Logarithmic Functions","body":"Find the value of $$x$$ in the following logarithmic functions.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.3 Evaluate and Graph Logarithmic Functions","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af4e405log9a","stepAnswer":["$$9$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{x}\\\\left(81\\\\right)=2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9$$","hints":{"DefaultPathway":[{"id":"af4e405log9a-h1","type":"hint","dependencies":[],"title":"Convert to Exponential Form","text":"First, convert the expression to exponential form. To convert a logarithmic equation to exponential form, we need to identify the base of the exponential, a, and the exponent $$y$$ since $$y=\\\\log_{a}\\\\left(x\\\\right)$$ is equivalent to $$x=a^y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log9a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x^2=81$$"],"dependencies":["af4e405log9a-h1"],"title":"Determining the Exponential Form","text":"What is the exponential form of the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x^2=81$$","$${81}^2=x$$","$$x^{81}=2$$","$$2^x=81$$"]},{"id":"af4e405log9a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["af4e405log9a-h2"],"title":"Solve for $$x$$","text":"Knowing that the baes of a logarithmic function must be positive, what is the solution to the exponential equation you found above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af4e405log9b","stepAnswer":["$$243$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{3}\\\\left(x\\\\right)=5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$243$$","hints":{"DefaultPathway":[{"id":"af4e405log9b-h1","type":"hint","dependencies":[],"title":"Convert to Exponential Form","text":"First, convert the expression to exponential form. To convert a logarithmic equation to exponential form, we need to identify the base of the exponential, a, and the exponent $$y$$ since $$y=\\\\log_{a}\\\\left(x\\\\right)$$ is equivalent to $$x=a^y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log9b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3^5=x$$"],"dependencies":["af4e405log9b-h1"],"title":"Determining the Exponential Form","text":"What is the exponential form of the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$5^3=x$$","$$3^5=x$$","$$5^x=3$$","$$3^x=5$$"]},{"id":"af4e405log9b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$243$$"],"dependencies":["af4e405log9b-h2"],"title":"Solve for $$x$$","text":"What is the solution for $$x$$ in the exponential equation you found above?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af4e405log9c","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"$$\\\\log_{(1/3)}\\\\left(\\\\frac{1}{27}\\\\right)=x$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"af4e405log9c-h1","type":"hint","dependencies":[],"title":"Convert to Exponential Form","text":"First, convert the expression to exponential form. To convert a logarithmic equation to exponential form, we need to identify the base of the exponential, a, and the exponent $$y$$ since $$y=\\\\log_{a}\\\\left(x\\\\right)$$ is equivalent to $$x=a^y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af4e405log9c-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${\\\\left(\\\\frac{1}{3}\\\\right)}^x=\\\\frac{1}{27}$$"],"dependencies":["af4e405log9c-h1"],"title":"Determining the Exponential Form","text":"What is the exponential form of the equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$${\\\\left(\\\\frac{1}{3}\\\\right)}^x=\\\\frac{1}{27}$$","$${\\\\left(\\\\frac{1}{27}\\\\right)}^x=\\\\frac{1}{3}$$","$$x^{\\\\frac{1}{3}}=\\\\frac{1}{27}$$","$${\\\\left(\\\\frac{1}{27}\\\\right)}^{\\\\frac{1}{3}}=x$$"]},{"id":"af4e405log9c-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$${\\\\left(\\\\frac{1}{3}\\\\right)}^3$$"],"dependencies":["af4e405log9c-h2"],"title":"Rewrite $$\\\\frac{1}{27}$$ with base $$\\\\frac{1}{3}$$","text":"Since $$3^3=27$$, we can actually rewrite $$\\\\frac{1}{27}$$ in terms of $$\\\\frac{1}{3}$$. What is $$\\\\frac{1}{27}$$ equal to in terms of $$\\\\frac{1}{3}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$${\\\\left(\\\\frac{1}{3}\\\\right)}^3$$","$$3^{\\\\frac{1}{3}}$$"]},{"id":"af4e405log9c-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["af4e405log9c-h3"],"title":"Solve for $$x$$","text":"Substituting $${\\\\left(\\\\frac{1}{3}\\\\right)}^3$$ into the exponential equation we found before, we get $${\\\\left(\\\\frac{1}{3}\\\\right)}^x={\\\\left(\\\\frac{1}{3}\\\\right)}^3$$. What is the solution for $$x$$ in this exponential equation?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af5fe5dSolveapp1","title":"Solve Proportions","body":"In the following exercises, solve each proportion.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Solve Applications with Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af5fe5dSolveapp1a","stepAnswer":["$$49$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{x}{56}=\\\\frac{7}{8}$$","stepBody":"Solve for $$x$$.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$49$$","hints":{"DefaultPathway":[{"id":"af5fe5dSolveapp1a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$56$$"],"dependencies":[],"title":"Find LCD","text":"What is the least common denominator of $$\\\\frac{x}{56}$$ and $$\\\\frac{7}{8}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af5fe5dSolveapp1a-h2","type":"hint","dependencies":["af5fe5dSolveapp1a-h1"],"title":"Multiply LCD Both Sides","text":"By multiplying both sides by LCD, we get $$x=49$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af5fe5dSolveapp1a-h3","type":"hint","dependencies":["af5fe5dSolveapp1a-h2"],"title":"Final Answer","text":"From the above steps, we get $$x=49$$ as the final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af5fe5dSolveapp10","title":"Solve Proportions","body":"In the following exercises, solve each proportion.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Solve Applications with Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af5fe5dSolveapp10a","stepAnswer":["$$\\\\frac{11}{7}$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{q-2}{2}=\\\\frac{2q-7}{18}$$","stepBody":"Solve for q.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{11}{7}$$","hints":{"DefaultPathway":[{"id":"af5fe5dSolveapp10a-h1","type":"hint","dependencies":[],"title":"Find LCD","text":"The least common denominator of \uff08q-2)/2 and $$\\\\frac{2q-7}{18}$$ is $$18$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af5fe5dSolveapp10a-h2","type":"hint","dependencies":["af5fe5dSolveapp10a-h1"],"title":"Multiply LCD Both Sides","text":"Through multiplying both sides by LCD, we get $$9(q-2)=2q-7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af5fe5dSolveapp10a-h3","type":"hint","dependencies":["af5fe5dSolveapp10a-h2"],"title":"Isolate q on One side","text":"We can further simplify $$9(q-2)=2q-7$$ to $$9q-18=2q-7$$. By subtracting 2q, we get $$7q-18=-7$$. We can add $$18$$ both sides and get $$7q=11$$. Lastly, we can divide both sides by $$7$$ and get $$q=\\\\frac{11}{7}$$ as the final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af5fe5dSolveapp11","title":"Solve Proportions","body":"In the following exercise, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Solve Applications with Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af5fe5dSolveapp11a","stepAnswer":["$$9$$"],"problemType":"TextBox","stepTitle":"Pediatricians prescribe $$5$$ milliliters (ml) of acetaminophen for every $$25$$ pounds of a child\u2019s weight. How many milliliters of acetaminophen will the doctor prescribe for Jocelyn, who weighs $$45$$ pounds?","stepBody":"Please enter you answer as number.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9$$","hints":{"DefaultPathway":[{"id":"af5fe5dSolveapp11a-h1","type":"hint","dependencies":[],"title":"Set Up Equation For the Given Condition","text":"Let $$x=ml$$ of acetaminophen that Jocelyn needs. We can translate into a proportion $$\\\\frac{ml}{pounds}=\\\\frac{ml}{pounds}$$ which gives $$\\\\frac{5}{25}=\\\\frac{x}{45}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af5fe5dSolveapp11a-h2","type":"hint","dependencies":["af5fe5dSolveapp11a-h1"],"title":"Solve Equation","text":"Solve for $$\\\\frac{5}{25}=\\\\frac{x}{45}$$. What is $$x$$? We can isolate $$x$$ by multiplying $$45$$ both sides. It gives $$x=9$$ as the final answer. Therefore, the doctor will prescribe $$9$$ milliliters of acetaminophen for Jocelyn.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af5fe5dSolveapp12","title":"Solve Proportions","body":"In the following exercise, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Solve Applications with Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af5fe5dSolveapp12a","stepAnswer":["$$325$$"],"problemType":"TextBox","stepTitle":"A veterinarian prescribed Sunny, a 65-pound dog, an antibacterial medicine in case an infection emerges","stepBody":"Please enter you answer as number.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$325$$","hints":{"DefaultPathway":[{"id":"af5fe5dSolveapp12a-h1","type":"hint","dependencies":[],"title":"Set Up Equation For the Given Condition","text":"Let $$x=mg$$ of medicine Sunny received. We can translate into a proportion $$\\\\frac{mg}{pounds}=\\\\frac{mg}{pounds}$$ which gives $$\\\\frac{5}{1}=\\\\frac{x}{65}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af5fe5dSolveapp12a-h2","type":"hint","dependencies":["af5fe5dSolveapp12a-h1"],"title":"Solve Equation","text":"Solve for $$\\\\frac{5}{1}=\\\\frac{x}{65}$$. What is $$x$$? We can isolate $$x$$ by multiplying $$65$$ both sides. It gives $$x=325$$ as the final answer. Therefore, the veterinarian prescribed 325mg medicine.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af5fe5dSolveapp13","title":"Solve Proportions","body":"In the following exercise, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Solve Applications with Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af5fe5dSolveapp13a","stepAnswer":["$$159$$"],"problemType":"TextBox","stepTitle":"A new energy drink advertises $$106$$ calories for $$8$$ ounces. How many calories are in $$12$$ ounces of the drink?","stepBody":"Please enter you answer as number.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$159$$","hints":{"DefaultPathway":[{"id":"af5fe5dSolveapp13a-h1","type":"hint","dependencies":[],"title":"Set Up Equation For the Given Condition","text":"Let $$x=calories$$ in $$12$$ ounces of the drink. We can translate into a proportion $$\\\\frac{calories}{ounces}=\\\\frac{calories}{ounces}$$ which gives $$\\\\frac{106}{8}=\\\\frac{x}{12}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af5fe5dSolveapp13a-h2","type":"hint","dependencies":["af5fe5dSolveapp13a-h1"],"title":"Solve Equation","text":"Solve for $$\\\\frac{106}{8}=\\\\frac{x}{12}$$. What is $$x$$? We can isolate $$x$$ by multiplying $$12$$ both sides. It gives $$x=159$$ as the final answer. Therefore, $$159$$ calories are in $$12$$ ounces of the drink.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af5fe5dSolveapp14","title":"Solve Proportions","body":"In the following exercise, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Solve Applications with Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af5fe5dSolveapp14a","stepAnswer":["$$400$$"],"problemType":"TextBox","stepTitle":"One 12-ounce can of soda has $$150$$ calories. If Josiah drinks the big 32-ounce size from the local mini-mart, how many calories does he get?","stepBody":"Please enter you answer as number.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$400$$","hints":{"DefaultPathway":[{"id":"af5fe5dSolveapp14a-h1","type":"hint","dependencies":[],"title":"Set Up Equation For the Given Condition","text":"Let $$x=calories$$ in $$32$$ ounces soda. We can translate into a proportion $$\\\\frac{calories}{ounces}=\\\\frac{calories}{ounces}$$ which gives $$\\\\frac{150}{12}=\\\\frac{x}{32}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af5fe5dSolveapp14a-h2","type":"hint","dependencies":["af5fe5dSolveapp14a-h1"],"title":"Solve Equation","text":"Solve for $$\\\\frac{150}{12}=\\\\frac{x}{32}$$. What is $$x$$? We can isolate $$x$$ by multiplying $$32$$ both sides. It gives $$x=400$$ as the final answer. Therefore, Josiah gets $$400$$ calories.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af5fe5dSolveapp15","title":"Solve Proportions","body":"In the following exercise, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Solve Applications with Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af5fe5dSolveapp15a","stepAnswer":["$$325$$"],"problemType":"TextBox","stepTitle":"Kyra is traveling to Canada and will change $250 US dollars into Canadian dollars. At the current exchange rate, $1 US is equal to $$\\\\$1.3$$ Canadian. How many Canadian dollars will she get for her trip?","stepBody":"Please enter you answer as number.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$325$$","hints":{"DefaultPathway":[{"id":"af5fe5dSolveapp15a-h1","type":"hint","dependencies":[],"title":"Set Up Equation For the Given Condition","text":"Let $$x=$$ Canadian that is equivalent to $$250$$ US dollars. We can translate into a proportion Canadian/US dollar=Canadian/US dollar which gives $$\\\\frac{1.3}{1}=\\\\frac{x}{250}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af5fe5dSolveapp15a-h2","type":"hint","dependencies":["af5fe5dSolveapp15a-h1"],"title":"Solve Equation","text":"Solve for $$\\\\frac{1.3}{1}=\\\\frac{x}{250}$$. What is $$x$$? We can isolate $$x$$ by multiplying $$250$$ both sides. It gives $$x=325$$ as the final answer. Therefore, Kyra will get $$325$$ Canadian for her trip.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af5fe5dSolveapp16","title":"Solve Proportions","body":"In the following exercise, solve.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Solve Applications with Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af5fe5dSolveapp16a","stepAnswer":["$$5530.5$$"],"problemType":"TextBox","stepTitle":". Maurice is traveling to Mexico and needs to exchange $450 into Mexican pesos. If each dollar is worth $$12.29$$ pesos, how many pesos will he get for his trip?","stepBody":"Please enter you answer as number.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5530.5$$","hints":{"DefaultPathway":[{"id":"af5fe5dSolveapp16a-h1","type":"hint","dependencies":[],"title":"Set Up Equation For the Given Condition","text":"Let $$x=$$ pesos that is equivalent to $$450$$ US dollars. We can translate into a proportion pesos/US dollar=pesos/US dollar which gives $$\\\\frac{12.29}{1}=\\\\frac{x}{450}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af5fe5dSolveapp16a-h2","type":"hint","dependencies":["af5fe5dSolveapp16a-h1"],"title":"Solve Equation","text":"Solve for $$\\\\frac{12.29}{1}=\\\\frac{x}{450}$$. What is $$x$$? We can isolate $$x$$ by multiplying $$450$$ both sides. It gives $$x=5530.5$$ as the final answer. Therefore, Maurice will get $$5530.5$$ pesos for his trip.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af5fe5dSolveapp17","title":"Solve Proportions","body":"Solve the following excercise","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Solve Applications with Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af5fe5dSolveapp17a","stepAnswer":["$$3$$"],"problemType":"TextBox","stepTitle":"Ronald needs a morning breakfast drink that will give him at least $$390$$ calories. Orange juice has $$130$$ calories in one cup. How many cups does he need to drink to reach his calorie goal?","stepBody":"Please enter you answer as number.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3$$","hints":{"DefaultPathway":[{"id":"af5fe5dSolveapp17a-h1","type":"hint","dependencies":[],"title":"Set Up Equation For the Given Condition","text":"Let $$x=number$$ of cups of orange juice that gives $$390$$ calories. We can translate into a proportion cups of orange $$\\\\frac{juice}{calories}=cups$$ of orange $$\\\\frac{juice}{calories}$$ which gives $$\\\\frac{1}{130}=\\\\frac{x}{390}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af5fe5dSolveapp17a-h2","type":"hint","dependencies":["af5fe5dSolveapp17a-h1"],"title":"Solve Equation","text":"Solve for $$\\\\frac{1}{130}=\\\\frac{x}{390}$$. What is $$x$$? We can isolate $$x$$ by multiplying $$390$$ both sides. It gives $$x=3$$ as the final answer. Therefore, Ronald needs to drink $$3$$ cups of orange juice to reach his calorie goal.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af5fe5dSolveapp18","title":"Solve Proportions","body":"Solve the following excercise","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Solve Applications with Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af5fe5dSolveapp18a","stepAnswer":["$$\\\\frac{640}{3}$$"],"problemType":"TextBox","stepTitle":"Sonya drinks a 32-ounce energy drink containing $$80$$ calories per $$12$$ oun\u2026","stepBody":"Please enter your answer as fraction.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{640}{3}$$","hints":{"DefaultPathway":[{"id":"af5fe5dSolveapp18a-h1","type":"hint","dependencies":[],"title":"Set Up Equation For the Given Condition","text":"Let $$x=calories$$ in 32-ounce energy drink. We can translate into a proportion cups of $$\\\\frac{calories}{ounces}=\\\\frac{calories}{ounces}$$ which gives $$\\\\frac{80}{12}=\\\\frac{x}{32}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af5fe5dSolveapp18a-h2","type":"hint","dependencies":["af5fe5dSolveapp18a-h1"],"title":"Solve Equation","text":"Solve for $$\\\\frac{80}{12}=\\\\frac{x}{32}$$. . What is $$x$$? We can isolate $$x$$ by multiplying $$32$$ both sides. It gives $$x=\\\\frac{640}{3}$$ as the final answer. Therefore, Sonya gets $$\\\\frac{640}{3}$$ calories from the drink.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af5fe5dSolveapp19","title":"Solve Proportions","body":"Solve the following excercise","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Solve Applications with Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af5fe5dSolveapp19a","stepAnswer":["$$\\\\frac{27}{8}$$"],"problemType":"TextBox","stepTitle":"Phil wants to fertilize his lawn. Each bag of fertilizer covers about 4,000 square feet of lawn. Phil\u2019s lawn is approximately 13,500 square feet. How many bags of fertilizer will he have to buy?","stepBody":"Please enter your answer as fraction.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{27}{8}$$","hints":{"DefaultPathway":[{"id":"af5fe5dSolveapp19a-h1","type":"hint","dependencies":[],"title":"Set Up Equation For the Given Condition","text":"Let $$x=bags$$ of fertilizer that cover $$13500$$ square feet of lawn. We can translate into a proportion cups of bags of $$\\\\frac{fertilizer}{areas}$$ of lawn in square feet $$=\\\\frac{fertilizer}{areas}$$ of lawn in square feet which gives $$\\\\frac{1}{4000}=\\\\frac{x}{13500}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af5fe5dSolveapp19a-h2","type":"hint","dependencies":["af5fe5dSolveapp19a-h1"],"title":"Solve Equation","text":"Solve for $$\\\\frac{1}{4000}=\\\\frac{x}{13500}$$. What is $$x$$? We can isolate $$x$$ by multiplying $$13500$$ both sides. It gives $$x=\\\\frac{27}{8}$$ as the final answer. Therefore, Phil needs to buy $$\\\\frac{27}{8}$$ bags of fertilizer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af5fe5dSolveapp2","title":"Solve Proportions","body":"In the following exercises, solve each proportion.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Solve Applications with Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af5fe5dSolveapp2a","stepAnswer":["$$7$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{56}{72}=\\\\frac{y}{9}$$","stepBody":"Solve for $$y$$.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7$$","hints":{"DefaultPathway":[{"id":"af5fe5dSolveapp2a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$72$$"],"dependencies":[],"title":"Find LCD","text":"What is the least common denominator of $$\\\\frac{56}{72}$$ and $$\\\\frac{y}{9}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af5fe5dSolveapp2a-h2","type":"hint","dependencies":["af5fe5dSolveapp2a-h1"],"title":"Multiply LCD Both Sides","text":"Through multiplying both sides by LCD, we get $$56=8y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af5fe5dSolveapp2a-h3","type":"hint","dependencies":["af5fe5dSolveapp2a-h2"],"title":"Isolate $$y$$ on One side","text":"We can divide by $$8$$ on both sides and get $$y=7$$ as the final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af5fe5dSolveapp20","title":"Solve Proportions","body":"Solve the following excercise","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Solve Applications with Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af5fe5dSolveapp20a","stepAnswer":["$$\\\\frac{5}{4}$$"],"problemType":"TextBox","stepTitle":"An oatmeal cookie recipe calls for $$\\\\frac{1}{2}$$ cup of butter to make $$4$$ dozen cookies. Hilda needs to make $$10$$ dozen cookies for the bake sale. How many cups of butter will she need?","stepBody":"Please enter your answer as fraction.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{5}{4}$$","hints":{"DefaultPathway":[{"id":"af5fe5dSolveapp20a-h1","type":"hint","dependencies":[],"title":"Set Up Equation For the Given Condition","text":"Let $$x=cups$$ of butter for making $$10$$ dozen cookies. We can translate into a proportion cups of $$\\\\frac{butter}{dozen}$$ of $$cookies=cups$$ of $$\\\\frac{butter}{dozen}$$ of cookies which gives $$\\\\frac{\\\\frac{1}{2}}{4}=\\\\frac{x}{10}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af5fe5dSolveapp20a-h2","type":"hint","dependencies":["af5fe5dSolveapp20a-h1"],"title":"Solve Equation","text":"Solve for $$\\\\frac{\\\\frac{1}{2}}{4}=\\\\frac{x}{10}$$. What is $$x$$? We can isolate $$x$$ by multiplying $$10$$ both sides. It gives $$x=\\\\frac{5}{4}$$ as the final answer. Therefore, Hilda needs $$\\\\frac{5}{4}$$ cups of butter.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af5fe5dSolveapp21","title":"Solve Proportions","body":"Solve the following excercise","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Solve Applications with Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af5fe5dSolveapp21a","stepAnswer":["$$\\\\frac{2}{3}$$"],"problemType":"TextBox","stepTitle":"A 2-foot-tall dog casts a 3-foot shadow at the same time a cat casts a one foot shadow. How tall is the cat in feet?","stepBody":"Please enter your answer as fraction.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{2}{3}$$","hints":{"DefaultPathway":[{"id":"af5fe5dSolveapp21a-h1","type":"hint","dependencies":[],"title":"Set Up Equation For the Given Condition","text":"Let $$x=actual$$ height of the cat. We can translate the given conditions into a proportion actual $$\\\\frac{height}{height}$$ of $$shadow=actual$$ $$\\\\frac{height}{height}$$ of shadow which gives $$\\\\frac{2}{3}=\\\\frac{x}{1}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af5fe5dSolveapp21a-h2","type":"hint","dependencies":["af5fe5dSolveapp21a-h1"],"title":"Solve Equation","text":"Solve for $$\\\\frac{2}{3}=\\\\frac{x}{1}$$. What is $$x$$? We can simplify the equation into $$x=\\\\frac{2}{3}$$ which gives the final answer. Therefore, the cat is $$\\\\frac{2}{3}$$ feet tall.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af5fe5dSolveapp22","title":"Solve Proportions","body":"Solve the following excercise","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Solve Applications with Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af5fe5dSolveapp22a","stepAnswer":["$$\\\\frac{403}{64}$$"],"problemType":"TextBox","stepTitle":"Larry and Tom were standing next to each other in the backyard when Tom challenged Larry to guess how tall he was. Larry knew his own height is $$6.5$$ feet and when they measured their shadows, Larry\u2019s shadow was $$8$$ feet and Tom\u2019s was $$7.75$$ feet long. What is Tom\u2019s height in feet?","stepBody":"Please enter your answer as fraction.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{403}{64}$$","hints":{"DefaultPathway":[{"id":"af5fe5dSolveapp22a-h1","type":"hint","dependencies":[],"title":"Set Up Equation For the Given Condition","text":"Let $$x=Tom\'s$$ actual height. We can translate the given conditions into a proportion actual $$\\\\frac{height}{height}$$ of $$shadow=actual$$ $$\\\\frac{height}{height}$$ of shadow which gives $$\\\\frac{6.5}{8}=\\\\frac{x}{7.75}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af5fe5dSolveapp22a-h2","type":"hint","dependencies":["af5fe5dSolveapp22a-h1"],"title":"Solve Equation","text":"Solve for $$\\\\frac{6.5}{8}=\\\\frac{x}{7.75}$$ . What is $$x$$? We can multiply both sides by $$7.75$$ which gets $$x=\\\\frac{6.5\\\\times7.75}{8}$$. We can simplify the fraction by multiplying $$1000$$ both numerator and denominator which gives $$\\\\frac{50375}{8000}$$. The greatest common factor of $$50375$$ and $$8000$$ is $$125$$. We can divide $$125$$ both numerator and denominator which get $$\\\\frac{\\\\frac{50375}{125}}{\\\\frac{8000}{125}}=\\\\frac{403}{64}$$. Therefore, Tom is $$\\\\frac{403}{64}$$ feet.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af5fe5dSolveapp23","title":"Solve Proportions","body":"Solve the following excercise","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Solve Applications with Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af5fe5dSolveapp23a","stepAnswer":["$$\\\\frac{742}{3}$$"],"problemType":"TextBox","stepTitle":"The tower portion of a windmill is $$212$$ feet tall. A six foot tall person standing next to the tower casts a seven-foot shadow. How long is the windmill\u2019s shadow in feet?","stepBody":"Please enter your answer as fraction.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{742}{3}$$","hints":{"DefaultPathway":[{"id":"af5fe5dSolveapp23a-h1","type":"hint","dependencies":[],"title":"Set Up Equation For the Given Condition","text":"Let $$x=length$$ of the windmill\'s shadow. We can translate the given conditions into a proportion length of $$\\\\frac{shadow}{actual}$$ $$height=length$$ of $$\\\\frac{shadow}{actual}$$ height which gives $$\\\\frac{7}{6}=\\\\frac{x}{212}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af5fe5dSolveapp23a-h2","type":"hint","dependencies":["af5fe5dSolveapp23a-h1"],"title":"Solve Equation","text":"Solve for $$\\\\frac{7}{6}=\\\\frac{x}{212}$$ . What is $$x$$? We can multiply both sides by $$212$$ which gets $$x=\\\\frac{7\\\\times212}{6}$$. The greatest common factor of $$7\\\\times212$$ and $$6$$ is $$2$$. We can divide $$2$$ both numerator and denominator which get $$\\\\frac{\\\\frac{1484}{2}}{\\\\frac{6}{2}}=\\\\frac{742}{3}$$. Therefore, the windmill shadow is $$\\\\frac{742}{3}$$ feet long.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af5fe5dSolveapp24","title":"Solve Proportions","body":"Solve the following excercise","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Solve Applications with Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af5fe5dSolveapp24a","stepAnswer":["$$366$$"],"problemType":"TextBox","stepTitle":"The height of the Statue of Liberty is $$305$$ feet. Nikia, who is standing next to the statue, casts a 6-foot shadow and she is $$5$$ feet tall. How long should the shadow of the statue be in feet?","stepBody":"Please enter your answer as fraction.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$366$$","hints":{"DefaultPathway":[{"id":"af5fe5dSolveapp24a-h1","type":"hint","dependencies":[],"title":"Set Up Equation For the Given Condition","text":"Let $$x=length$$ of shadow of Statue of Liberty. We can translate the given conditions into a proportion length of $$\\\\frac{shadow}{actual}$$ $$height=length$$ of $$\\\\frac{shadow}{actual}$$ height which gives $$\\\\frac{6}{5}=\\\\frac{x}{305}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af5fe5dSolveapp24a-h2","type":"hint","dependencies":["af5fe5dSolveapp24a-h1"],"title":"Solve Equation","text":"Solve for $$\\\\frac{6}{5}=\\\\frac{x}{305}$$ . What is $$x$$? We can multiply both sides by $$305$$ which gets $$x=366$$. Therefore, the shadow of the Statue of Liberty is $$366$$ feet long.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af5fe5dSolveapp25","title":"Solve Direct Variation Problems","body":"Solve the following excercise","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Solve Applications with Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af5fe5dSolveapp25a","stepAnswer":["y=14/3*x"],"problemType":"TextBox","stepTitle":"If $$y$$ varies directly as $$x$$ and $$y=14$$ when $$x=3$$. Find the equation that relates $$x$$ and $$y$$.","stepBody":"Please enter your answer as $$y=ax$$ where a is a real number that you need to find.","answerType":"string","variabilization":{},"answerLatex":"$$y=\\\\frac{14}{3} x$$","hints":{"DefaultPathway":[{"id":"af5fe5dSolveapp25a-h1","type":"hint","dependencies":[],"title":"Set Up Equation For the Given Condition","text":"Since $$y$$ varies directly as $$x$$, the ratio of $$y$$ to $$x$$ should be the same all the time. Therefore we get the proportion equation $$\\\\frac{y}{x}=\\\\frac{14}{3}$$. We can simplify the equation by multiplying $$x$$ both sides which gives $$y=\\\\frac{14}{3} x$$ as the final asnwer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af5fe5dSolveapp3","title":"Solve Proportions","body":"In the following exercises, solve each proportion.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Solve Applications with Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af5fe5dSolveapp3a","stepAnswer":["$$-11$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{98}{154}=\\\\frac{-7}{p}$$","stepBody":"Solve for $$p$$.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-11$$","hints":{"DefaultPathway":[{"id":"af5fe5dSolveapp3a-h1","type":"hint","dependencies":[],"title":"Find LCD","text":"The least common denominator of $$\\\\frac{98}{154}$$ and $$\\\\frac{-7}{p}$$ is $$154p$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af5fe5dSolveapp3a-h2","type":"hint","dependencies":["af5fe5dSolveapp3a-h1"],"title":"Multiply LCD Both Sides","text":"Through multiplying both sides by LCD, we get $$98p=-7\\\\times154$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af5fe5dSolveapp3a-h3","type":"hint","dependencies":["af5fe5dSolveapp3a-h2"],"title":"Isolate $$p$$ on One side","text":"We can divide by $$98$$ on both sides and get $$p=-11$$ as the final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af5fe5dSolveapp4","title":"Solve Proportions","body":"In the following exercises, solve each proportion.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Solve Applications with Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af5fe5dSolveapp4a","stepAnswer":["$$-13$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{72}{156}=\\\\frac{-6}{q}$$","stepBody":"Solve for q.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-13$$","hints":{"DefaultPathway":[{"id":"af5fe5dSolveapp4a-h1","type":"hint","dependencies":[],"title":"Find LCD","text":"The least common denominator of $$\\\\frac{72}{156}$$ and $$\\\\frac{-6}{q}$$ is 156q.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af5fe5dSolveapp4a-h2","type":"hint","dependencies":["af5fe5dSolveapp4a-h1"],"title":"Multiply LCD Both Sides","text":"Through multiplying both sides by LCD, we get $$72q=-6\\\\times156$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af5fe5dSolveapp4a-h3","type":"hint","dependencies":["af5fe5dSolveapp4a-h2"],"title":"Isolate q on One side","text":"We can divide by $$72$$ on both sides and get $$q=-13$$ as the final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af5fe5dSolveapp5","title":"Solve Proportions","body":"In the following exercises, solve each proportion.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Solve Applications with Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af5fe5dSolveapp5a","stepAnswer":["$$16$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{a}{a+12}=\\\\frac{4}{7}$$","stepBody":"Solve for a.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16$$","hints":{"DefaultPathway":[{"id":"af5fe5dSolveapp5a-h1","type":"hint","dependencies":[],"title":"Find LCD","text":"The least common denominator of $$\\\\frac{a}{a+12}$$ and $$\\\\frac{4}{7}$$ is $$7\\\\left(a+12\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af5fe5dSolveapp5a-h2","type":"hint","dependencies":["af5fe5dSolveapp5a-h1"],"title":"Multiply LCD Both Sides","text":"Through multiplying both sides by LCD, we get $$7a=4\\\\left(a+12\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af5fe5dSolveapp5a-h3","type":"hint","dependencies":["af5fe5dSolveapp5a-h2"],"title":"Isolate a on One side","text":"We can further simplify $$7a=4\\\\left(a+12\\\\right)$$ to $$7a=4a+48$$. By subtracting 4a both sides, we get $$3a=48$$. Then we divide by $$3$$ both sides of the equation and get $$a=16$$ as the final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af5fe5dSolveapp6","title":"Solve Proportions","body":"In the following exercises, solve each proportion.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Solve Applications with Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af5fe5dSolveapp6a","stepAnswer":["$$88$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{b}{b-16}=\\\\frac{11}{9}$$","stepBody":"Solve for $$b$$.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$88$$","hints":{"DefaultPathway":[{"id":"af5fe5dSolveapp6a-h1","type":"hint","dependencies":[],"title":"Find LCD","text":"The least common denominator of $$\\\\frac{b}{b-16}$$ and $$\\\\frac{11}{9}$$ is $$9(b-16)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af5fe5dSolveapp6a-h2","type":"hint","dependencies":["af5fe5dSolveapp6a-h1"],"title":"Multiply LCD Both Sides","text":"Through multiplying both sides by LCD, we get $$9b=11(b-16)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af5fe5dSolveapp6a-h3","type":"hint","dependencies":["af5fe5dSolveapp6a-h2"],"title":"Isolate $$b$$ on One side","text":"We can further simplify $$9b=11(b-16)$$ to $$9b=11b-176$$. By subtracting $$11b$$ both sides, we get $$-2b=-176$$. Then we divide by $$-2$$ both sides of the equation and get $$b=88$$ as the final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af5fe5dSolveapp7","title":"Solve Proportions","body":"In the following exercises, solve each proportion.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Solve Applications with Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af5fe5dSolveapp7a","stepAnswer":["$$60$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{m+90}{25}=\\\\frac{m+30}{15}$$","stepBody":"Solve for $$m$$.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$60$$","hints":{"DefaultPathway":[{"id":"af5fe5dSolveapp7a-h1","type":"hint","dependencies":[],"title":"Find LCD","text":"The least common denominator of $$\\\\frac{m+90}{25}$$ and $$\\\\frac{m+30}{15}$$ is $$75$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af5fe5dSolveapp7a-h2","type":"hint","dependencies":["af5fe5dSolveapp7a-h1"],"title":"Multiply LCD Both Sides","text":"Through multiplying both sides by LCD, we get $$3\\\\left(m+90\\\\right)=5\\\\left(m+30\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af5fe5dSolveapp7a-h3","type":"hint","dependencies":["af5fe5dSolveapp7a-h2"],"title":"Isolate $$m$$ on One side","text":"We can further simplify $$3\\\\left(m+90\\\\right)=5\\\\left(m+30\\\\right)$$ to $$3m+270=5m+150$$. By subtracting $$3m$$ and $$150$$ both sides, we get $$120=2m$$. Then we divide both sides of the equation by $$2$$ and get $$m=60$$ as the final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af5fe5dSolveapp8","title":"Solve Proportions","body":"In the following exercises, solve each proportion.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Solve Applications with Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af5fe5dSolveapp8a","stepAnswer":["$$10$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{n+10}{4}=\\\\frac{40-n}{6}$$","stepBody":"Solve for $$n$$.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10$$","hints":{"DefaultPathway":[{"id":"af5fe5dSolveapp8a-h1","type":"hint","dependencies":[],"title":"Find LCD","text":"The least common denominator of $$\\\\frac{n+10}{4}$$ and $$\\\\frac{40-n}{6}$$ is $$12$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af5fe5dSolveapp8a-h2","type":"hint","dependencies":["af5fe5dSolveapp8a-h1"],"title":"Multiply LCD Both Sides","text":"Through multiplying both sides by LCD, we get $$3\\\\left(n+10\\\\right)=2(40-n)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af5fe5dSolveapp8a-h3","type":"hint","dependencies":["af5fe5dSolveapp8a-h2"],"title":"Isolate $$n$$ on One side","text":"We can further simplify $$3\\\\left(n+10\\\\right)=2(40-n)$$ to $$3n+30=80-2n$$. By subtracting $$30$$, we get $$3n=50-2n$$. We can add $$2n$$ both sides isolate $$n$$ on one side and get $$5n=50$$. Lastly, we can divide both sides by $$5$$ and get $$n=10$$ as the final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af5fe5dSolveapp9","title":"Solve Proportions","body":"In the following exercises, solve each proportion.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"7.5 Solve Applications with Rational Equations","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"af5fe5dSolveapp9a","stepAnswer":["$$30$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{2p+4}{8}=\\\\frac{p+18}{6}$$","stepBody":"Solve for $$p$$.","answerType":"arithmetic","variabilization":{},"answerLatex":"$$30$$","hints":{"DefaultPathway":[{"id":"af5fe5dSolveapp9a-h1","type":"hint","dependencies":[],"title":"Find LCD","text":"The least common denominator of $$\\\\frac{2p+4}{8}$$ and $$\\\\frac{p+18}{6}$$ is $$24$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af5fe5dSolveapp9a-h2","type":"hint","dependencies":["af5fe5dSolveapp9a-h1"],"title":"Multiply LCD Both Sides","text":"Through multiplying both sides by LCD, we get $$3\\\\left(2p+4\\\\right)=4\\\\left(p+18\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af5fe5dSolveapp9a-h3","type":"hint","dependencies":["af5fe5dSolveapp9a-h2"],"title":"Isolate $$p$$ on One side","text":"We can further simplify $$3\\\\left(2p+4\\\\right)=4\\\\left(p+18\\\\right)$$ to $$6p+12=4p+72$$. By subtracting $$12$$, we get $$6p=4p+60$$. We can subtract $$4p$$ both sides and get $$2p=60$$. Lastly, we can divide both sides by $$2$$ and get $$p=30$$ as the final answer.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af60b4fellipse1","title":"Writing Equations of Ellipses in Standard Form","body":"Find the standard form equation of the ellipse with the following coordinates.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 The Ellipse","courseName":"OpenStax: College Algebra","steps":[{"id":"af60b4fellipse1a","stepAnswer":["$$\\\\frac{x^2}{64}+\\\\frac{y^2}{39}=1$$"],"problemType":"TextBox","stepTitle":"Vertices $$(8,0)$$ and $$(-8,0)$$ and foci $$(5,0)$$ and $$(-5,0)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{x^2}{64}+\\\\frac{y^2}{39}=1$$","hints":{"DefaultPathway":[{"id":"af60b4fellipse1a-h1","type":"hint","dependencies":[],"title":"Vertice and Foci","text":"Consider whether the major axis lies on the $$x-$$ or y-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af60b4fellipse1a-h2","type":"hint","dependencies":["af60b4fellipse1a-h1"],"title":"Standard Equation Form","text":"The form of the standard equation is $$c^2=a^2-b^2$$, given $$a=vertices$$ and $$c=foci$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af60b4fellipse1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["af60b4fellipse1a-h2"],"title":"Solving for a","text":"What is a equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af60b4fellipse1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["af60b4fellipse1a-h3"],"title":"Solving for c","text":"What is c equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af60b4fellipse1a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$39$$"],"dependencies":["af60b4fellipse1a-h4"],"title":"Solving for $$b$$","text":"What is $$b^2$$ equal to using the relationship between a, $$b$$, and c?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af60b4fellipse1a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{x^2}{64}+\\\\frac{y^2}{39}=1$$"],"dependencies":["af60b4fellipse1a-h5"],"title":"Plugging in the Values","text":"Plug in the values for a and $$b$$ into the standard form equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af60b4fellipse10","title":"The Ellipse","body":"For the following exercise, determine whether the given equations represent ellipses. If yes, write in standard form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 The Ellipse","courseName":"OpenStax: College Algebra","steps":[{"id":"af60b4fellipse10a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$2x^2+y=4$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"af60b4fellipse10a-h1","type":"hint","dependencies":[],"title":"Consider whether the equation contains the same variable exponents as the standard equation form of an ellipse.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af60b4fellipse10a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["af60b4fellipse10a-h1"],"title":"Is it possible to write the equation in standard form?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"af60b4fellipse11","title":"The Ellipse","body":"For the following exercise, determine whether the given equations represent ellipses. If yes, write in standard form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 The Ellipse","courseName":"OpenStax: College Algebra","steps":[{"id":"af60b4fellipse11a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"$$4x^2-y^2=4$$","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"af60b4fellipse11a-h1","type":"hint","dependencies":[],"title":"Does the equation have the same signs as the standard equation form of an ellipse?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af60b4fellipse11a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["af60b4fellipse11a-h1"],"title":"Is it possible to write the equation in standard form?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]}]}}]},{"id":"af60b4fellipse13","title":"The Ellipse","body":"$$x^2+9y^2=1$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 The Ellipse","courseName":"OpenStax: College Algebra","steps":[{"id":"af60b4fellipse13a","stepAnswer":["$$\\\\frac{x^2}{1^2}+\\\\frac{y^2}{{\\\\left(\\\\frac{1}{3}\\\\right)}^2}=1$$"],"problemType":"TextBox","stepTitle":"Write the equation of an ellipse in standard form","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{x^2}{1^2}+\\\\frac{y^2}{{\\\\left(\\\\frac{1}{3}\\\\right)}^2}=1$$","hints":{"DefaultPathway":[{"id":"af60b4fellipse13a-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{x^2}{1^2}+\\\\frac{y^2}{{\\\\left(\\\\frac{1}{3}\\\\right)}^2}=1$$"],"dependencies":[],"title":"Use algebra to rewrite the equation in standard form.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af60b4fellipse13b","stepAnswer":["$$(1,0)$$ and $$(-1,0)$$"],"problemType":"MultipleChoice","stepTitle":"Identify the end points of the major axis","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(1,0)$$ and $$(-1,0)$$","choices":["$$(1,0)$$ and $$(-1,0)$$","$$(2,0)$$, and $$(-2,0)$$","$$(3,0)$$ and $$(-3,0)$$"],"hints":{"DefaultPathway":[{"id":"af60b4fellipse13b-h1","type":"hint","dependencies":[],"title":"Major Axis","text":"Does the major axis lie on the $$x-$$ or y-axis?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af60b4fellipse13b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(1,0)$$ and $$(-1,0)$$"],"dependencies":["af60b4fellipse13b-h1"],"title":"Finding Major Axis End Point","text":"Given that the major axis is on the x-axis, use the information from the standard form to find the endpoints on the major axis","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(1,0)$$ and $$(-1,0)$$","$$(2,0)$$ and $$(-2,0)$$","$$(3,0)$$ and $$(-3,0)$$"]}]}},{"id":"af60b4fellipse13c","stepAnswer":["$$(0,\\\\frac{1}{3})$$ and $$(0,\\\\frac{-1}{3})$$"],"problemType":"MultipleChoice","stepTitle":"Identify the end points of the minor axis","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,\\\\frac{1}{3})$$ and $$(0,\\\\frac{-1}{3})$$","choices":["$$(0,\\\\frac{1}{3})$$ and $$(0,\\\\frac{-1}{3})$$","$$(0,\\\\frac{1}{2})$$ and $$(0,\\\\frac{-1}{2})$$","$$(0,\\\\frac{1}{4})$$ and $$(0,\\\\frac{-1}{4})$$"],"hints":{"DefaultPathway":[{"id":"af60b4fellipse13c-h1","type":"hint","dependencies":[],"title":"Minor Axis","text":"Does the minor axis lie on the $$x-$$ or y-axis?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af60b4fellipse13c-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(0,\\\\frac{1}{3})$$ and $$(0,\\\\frac{-1}{3})$$"],"dependencies":["af60b4fellipse13c-h1"],"title":"Finding Minor Axis End Point","text":"Given that the minor axis is on the y-axis, use the information from the standard form to find the endpoints on the minor axis","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(0,\\\\frac{1}{3})$$ and $$(0,\\\\frac{-1}{3})$$","$$(0,\\\\frac{1}{4})$$ and $$(0,\\\\frac{-1}{4})$$","$$(0,\\\\frac{1}{5})$$ and $$(0,\\\\frac{-1}{5})$$"]}]}},{"id":"af60b4fellipse13d","stepAnswer":["$$(\\\\frac{2\\\\sqrt{2}}{3},0)$$ and $$(\\\\frac{-2\\\\sqrt{2}}{3},0)$$"],"problemType":"MultipleChoice","stepTitle":"Identify the Foci","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(\\\\frac{2\\\\sqrt{2}}{3},0)$$ and $$(\\\\frac{-2\\\\sqrt{2}}{3},0)$$","choices":["$$(\\\\frac{2\\\\sqrt{2}}{3},0)$$ and $$(\\\\frac{-2\\\\sqrt{2}}{3},0)$$","$$(\\\\frac{3\\\\sqrt{2}}{3},0)$$ and $$(\\\\frac{-3\\\\sqrt{2}}{3},0)$$"],"hints":{"DefaultPathway":[{"id":"af60b4fellipse13d-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(\\\\frac{2\\\\sqrt{2}}{3},0)$$ and $$(\\\\frac{\\\\left(-2\\\\sqrt{2}\\\\right)}{3},0)$$"],"dependencies":[],"title":"Finding the Foci","text":"Given that we know $$a^2$$ and $$b^2$$, use the equation $$c^2=a^2-b^2$$ to determine the coordinates of the foci","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(\\\\frac{2\\\\sqrt{2}}{3},0)$$ and $$(\\\\frac{\\\\left(-2\\\\sqrt{2}\\\\right)}{3},0)$$","$$(\\\\frac{3\\\\sqrt{2}}{3},0)$$ and $$(\\\\frac{\\\\left(-3\\\\sqrt{2}\\\\right)}{3},0)$$"]}]}}]},{"id":"af60b4fellipse14","title":"The Ellipse","body":"$$\\\\frac{x^2}{100}+\\\\frac{y^2}{62}=1$$","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 The Ellipse","courseName":"OpenStax: College Algebra","steps":[{"id":"af60b4fellipse14a","stepAnswer":["$$\\\\frac{x^2}{{10}^2}+\\\\frac{y^2}{8^2}=1$$"],"problemType":"TextBox","stepTitle":"Write the equation of an ellipse in standard form","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{x^2}{{10}^2}+\\\\frac{y^2}{8^2}=1$$","hints":{"DefaultPathway":[{"id":"af60b4fellipse14a-h1","type":"hint","dependencies":[],"title":"Matching Standard Form","text":"Change the denominator so that it\'s in a squared format.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af60b4fellipse14b","stepAnswer":["$$(10,0)$$ and $$(-10,0)$$"],"problemType":"MultipleChoice","stepTitle":"Identify the end points of the major axis","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(10,0)$$ and $$(-10,0)$$","choices":["$$(10,0)$$ and $$(-10,0)$$","$$(9,0)$$ and $$(-9,0)$$","$$(8,0)$$ and $$(-8,0)$$"],"hints":{"DefaultPathway":[{"id":"af60b4fellipse14b-h1","type":"hint","dependencies":[],"title":"Major Axis","text":"Does the major axis lie on the $$x-$$ or y-axis?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af60b4fellipse14b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(10,0)$$ and $$(-10,0)$$"],"dependencies":["af60b4fellipse14b-h1"],"title":"Finding Major Axis End Point","text":"Given that the major axis is on the x-axis, use the information from the standard form to find the endpoints on the major axis","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(10,0)$$ and $$(-10,0)$$","$$(9,0)$$ and $$(-9,0)$$","$$(8,0)$$ and $$(-8,0)$$"]}]}},{"id":"af60b4fellipse14c","stepAnswer":["$$(0,8)$$ and $$(0,-8)$$"],"problemType":"TextBox","stepTitle":"Identify the end points of the minor axis","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,8)$$ and $$(0,-8)$$","choices":["$$(0,8)$$ and $$(0,-8)$$","$$(0,7)$$ and $$(0,-7)$$"],"hints":{"DefaultPathway":[{"id":"af60b4fellipse14c-h1","type":"hint","dependencies":[],"title":"Minor Axis","text":"Does the minor axis lie on the $$x-$$ or y-axis?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af60b4fellipse14c-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(0,8)$$ and $$(0,-8)$$"],"dependencies":["af60b4fellipse14c-h1"],"title":"Finding Minor Axis End Point","text":"Given that the minor axis is on the y-axis, use the information from the standard form to find the endpoints on the minor axis","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(0,8)$$ and $$(0,-8)$$","$$(0,7)$$ and $$(0,-7)$$"]}]}},{"id":"af60b4fellipse14d","stepAnswer":["$$(6,0)$$ and $$(-6,0)$$"],"problemType":"MultipleChoice","stepTitle":"Identify the foci","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(6,0)$$ and $$(-6,0)$$","choices":["$$(6,0)$$ and $$(-6,0)$$","$$(0,8)$$ and $$(0,-8)$$","$$(0,7)$$ and $$(0,-7)$$"],"hints":{"DefaultPathway":[{"id":"af60b4fellipse14d-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(6,0)$$ and $$(-6,0)$$"],"dependencies":[],"title":"Finding the Foci","text":"Given that we know $$a^2$$ and $$b^2$$, use the equation $$c^2=a^2-b^2$$ to determine the coordinates of the foci","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(6,0)$$ and $$(-6,0)$$","$$(0,8)$$ and $$(0,-8)$$","$$(0,7)$$ and $$(0,-7)$$"]}]}}]},{"id":"af60b4fellipse15","title":"Finding Ellipse Characteristics","body":"Find the major axes endpoints, minor axes endpoints, and foci endpoints of the ellipse:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 The Ellipse","courseName":"OpenStax: College Algebra","steps":[{"id":"af60b4fellipse15a","stepAnswer":["(11,-1),(7,-1),(2,-5),(2,3),(2+sqrt(65),-1),(2-sqrt(65),-1)"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{{\\\\left(x-2\\\\right)}^2}{81}+\\\\frac{{\\\\left(y+1\\\\right)}^2}{16}=1$$","stepBody":"","answerType":"string","variabilization":{},"choices":["(11,-1),(7,-1),(2,-5),(2,3),(2+sqrt(65),-1),(2-sqrt(65),-1)","(11,-2),(7,-1),(2,-3),(2,3),(2+sqrt(63),-1),(2-sqrt(65),-1)","(10,-1),(6,-1),(2,-4),(2,3),(1+sqrt(65),-1),(2-sqrt(65),-1)"],"hints":{"DefaultPathway":[{"id":"af60b4fellipse15a-h1","type":"hint","dependencies":[],"title":"Axes Endpoints","text":"The axes endpoints can be found with $$b$$ and a. $$a=9$$ and $$b=4$$ according to the equation. Since this is a horizontal ellipse, the major axes endpoints are $$(11,-1)$$ and $$(-7,-1)$$. The minor axes endpoints are $$(2,-5)$$ and $$(2,3)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af60b4fellipse15a-h2","type":"hint","dependencies":["af60b4fellipse15a-h1"],"title":"Foci Endpoints","text":"The foci endpoints can be found with c, which is equal to $$\\\\sqrt{a^2-b^2}=\\\\sqrt{65}$$. Since this is a horizontal ellipse, the foci are $$(2+\\\\sqrt{65},-1)$$ and $$(2-\\\\sqrt{65},-1)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af60b4fellipse16","title":"Finding Ellipse Characteristics","body":"Find the major axes endpoints, minor axes endpoints, and foci endpoints of the ellipse:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 The Ellipse","courseName":"OpenStax: College Algebra","steps":[{"id":"af60b4fellipse16a","stepAnswer":["(-5,10),(-5,4),(-3,7),(-7,7),(-5,7+sqrt(5)),(-5,7-sqrt(5))"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{{\\\\left(x+5\\\\right)}^2}{4}+\\\\frac{{\\\\left(y-7\\\\right)}^2}{9}=1$$","stepBody":"","answerType":"string","variabilization":{},"choices":["(-5,10),(-5,4),(-3,7),(-7,7),(-5,7+sqrt(5)),(-5,7-sqrt(5))","(11,-1),(7,-1),(2,-5),(2,3),(2+sqrt(65),-1),(2-sqrt(65),-1)","(11,-2),(7,-1),(2,-3),(2,3),(2+sqrt(63),-1),(2-sqrt(65),-1)","(10,-1),(6,-1),(2,-4),(2,3),(1+sqrt(65),-1),(2-sqrt(65),-1)"],"hints":{"DefaultPathway":[{"id":"af60b4fellipse16a-h1","type":"hint","dependencies":[],"title":"Axes Endpoints","text":"According to the equation, $$a=3$$ and $$b=2$$. This means that the major axes endpoints are: $$(-5,10),(-5,4)$$. The minor axes endpoints are: $$(-3,7),(-7,7)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af60b4fellipse16a-h2","type":"hint","dependencies":["af60b4fellipse16a-h1"],"title":"Foci Endpoints","text":"Since $$a=3$$ and $$b=2$$, we can use $$c=\\\\sqrt{a^2-b^2}$$ to get $$c=\\\\sqrt{5}$$. This means that the foci are (-5,7+sqrt(5)),(-5,7-sqrt(5))","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af60b4fellipse17","title":"Finding Ellipse Characteristics","body":"Find the major axes endpoints, minor axes endpoints, and foci endpoints of the ellipse:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 The Ellipse","courseName":"OpenStax: College Algebra","steps":[{"id":"af60b4fellipse17a","stepAnswer":["DNE"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{{\\\\left(x-7\\\\right)}^2}{49}+\\\\frac{{\\\\left(y-7\\\\right)}^2}{49}=1$$. If the equation is not one of an ellipse, enter DNE.","stepBody":"","answerType":"string","variabilization":{},"choices":["(-5,10),(-5,4),(-3,7),(-7,7),(-5,7+sqrt(5)),(-5,7-sqrt(5))","(11,-1),(7,-1),(2,-5),(2,3),(2+sqrt(65),-1),(2-sqrt(65),-1)","(11,-2),(7,-1),(2,-3),(2,3),(2+sqrt(63),-1),(2-sqrt(65),-1)","(10,-1),(6,-1),(2,-4),(2,3),(1+sqrt(65),-1),(2-sqrt(65),-1)","DNE"],"hints":{"DefaultPathway":[{"id":"af60b4fellipse17a-h1","type":"hint","dependencies":[],"title":"Not an Ellipse","text":"Since $$a=b$$, this is a circle of radius $$7$$. This means that there are no $$\\\\frac{major}{minor}$$ axes or foci. DNE.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af60b4fellipse18","title":"Finding Ellipse Characteristics","body":"Find the foci of the ellipse.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 The Ellipse","courseName":"OpenStax: College Algebra","steps":[{"id":"af60b4fellipse18a","stepAnswer":["(-3,-1-sqrt(11)),(-3,-1+sqrt(11))"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{{\\\\left(x+3\\\\right)}^2}{25}+\\\\frac{{\\\\left(y+1\\\\right)}^2}{36}=1$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["(-3,-1-sqrt(11)),(-3,-1+sqrt(11))","(-2,-1-sqrt(11)),(-2,-1+sqrt(11))","(-1,-1-sqrt(11)),(-1,-1+sqrt(11))"],"hints":{"DefaultPathway":[{"id":"af60b4fellipse18a-h1","type":"hint","dependencies":[],"title":"Finding the Foci with c","text":"We know that $$a=36$$ and $$b=25$$. So, we can solve for c using the equation $$c=\\\\sqrt{a^2-b^2}$$, leading to $$\\\\sqrt{11}$$. So, the foci are (-3,-1-sqrt(11)),(-3,-1+sqrt(11))","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af60b4fellipse19","title":"Finding Ellipse Characteristics","body":"Find the foci of the ellipse.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 The Ellipse","courseName":"OpenStax: College Algebra","steps":[{"id":"af60b4fellipse19a","stepAnswer":["$$(-1-\\\\sqrt{96},2)$$, $$(-1+\\\\sqrt{96},2)$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{{\\\\left(x+1\\\\right)}^2}{100}+\\\\frac{{\\\\left(y-2\\\\right)}^2}{4}=1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-1-\\\\sqrt{96},2)$$, $$(-1+\\\\sqrt{96},2)$$","choices":["$$(-3,-1-\\\\sqrt{11})$$, $$(-3,-1+\\\\sqrt{11})$$","$$(-2,-1-\\\\sqrt{11})$$, $$(-2,-1+\\\\sqrt{11})$$","$$(-1-\\\\sqrt{96},2)$$, $$(-1+\\\\sqrt{96},2)$$"],"hints":{"DefaultPathway":[{"id":"af60b4fellipse19a-h1","type":"hint","dependencies":[],"title":"Finding the Foci with c","text":"Since we know that $$a=10$$ and $$b=2$$, we can solve for c using the equation $$c=\\\\sqrt{a^2-b^2}$$, which gives us $$c=\\\\sqrt{96}$$. Since this is a horizontal ellipse, the foci are (-1-sqrt(96),2),(-1+sqrt(96),2)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af60b4fellipse2","title":"Writing Equations of Ellipses in Standard Form","body":"Find the standard form equation of the ellipse with the following coordinates","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 The Ellipse","courseName":"OpenStax: College Algebra","steps":[{"id":"af60b4fellipse2a","stepAnswer":["$$\\\\frac{x^2}{1}+\\\\frac{y^2}{16}=1$$"],"problemType":"TextBox","stepTitle":"Vertices $$(0,4)$$ and $$(0,-4)$$, and foci $$(0,\\\\sqrt{15})$$, and $$(0,-\\\\sqrt{15})$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{x^2}{1}+\\\\frac{y^2}{16}=1$$","hints":{"DefaultPathway":[{"id":"af60b4fellipse2a-h1","type":"hint","dependencies":[],"title":"Vertice and Foci","text":"Consider whether the major axis lies on the $$x-$$ or y-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af60b4fellipse2a-h2","type":"hint","dependencies":["af60b4fellipse2a-h1"],"title":"Standard Equation Form","text":"The form of the standard equation is $$c^2=a^2-b^2$$, given $$a=vertices$$ and $$c=foci$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af60b4fellipse2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["af60b4fellipse2a-h2"],"title":"Solving for a","text":"What is a equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af60b4fellipse2a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{15}$$"],"dependencies":["af60b4fellipse2a-h3"],"title":"Solving for c","text":"What is c equal to?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af60b4fellipse2a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["af60b4fellipse2a-h4"],"title":"Solving for $$b$$","text":"What is $$b^2$$ equal to using the relationship between a, $$b$$, and c?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af60b4fellipse2a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{x^2}{1}+\\\\frac{y^2}{16}=1$$"],"dependencies":["af60b4fellipse2a-h5"],"title":"Plugging in the Values","text":"Plug in the values for a and $$b$$ into the standard form equation.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af60b4fellipse20","title":"Finding Ellipse Characteristics","body":"Find the foci of the ellipse. If the equation is not of an ellipse, enter DNE.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 The Ellipse","courseName":"OpenStax: College Algebra","steps":[{"id":"af60b4fellipse20a","stepAnswer":["DNE"],"problemType":"MultipleChoice","stepTitle":"$$x^2+y^2=1$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$(-3,-1-\\\\sqrt{11})$$, $$(-3,-1+\\\\sqrt{11})$$","$$(-2,-1-\\\\sqrt{11})$$, $$(-2,-1+\\\\sqrt{11})$$","$$(-1-\\\\sqrt{96},2)$$, $$(-1+\\\\sqrt{96},2)$$","DNE"],"hints":{"DefaultPathway":[{"id":"af60b4fellipse20a-h1","type":"hint","dependencies":[],"title":"Equation is a Circle","text":"Since $$a=b$$ in this equation, it represents a circle. Thus, the answer is DNE.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af60b4fellipse21","title":"Finding the Center of an Ellipse","body":"Find the center of the following ellipse:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 The Ellipse","courseName":"OpenStax: College Algebra","steps":[{"id":"af60b4fellipse21a","stepAnswer":["$$(0,0)$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{x^2}{25}+\\\\frac{y^2}{36}=1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,0)$$","choices":["$$(0,0)$$","$$(1,1)$$","$$(2,2)$$"],"hints":{"DefaultPathway":[{"id":"af60b4fellipse21a-h1","type":"hint","dependencies":[],"title":"We will notice that there is nothing being added or subtracted to either $$x$$ or $$y$$, meaning that the ellipse is centered at the origin.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af60b4fellipse22","title":"Finding the Center of an Ellipse","body":"Find the center of the following ellipse:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 The Ellipse","courseName":"OpenStax: College Algebra","steps":[{"id":"af60b4fellipse22a","stepAnswer":["$$(0,0)$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{x^2}{16}+\\\\frac{y^2}{9}=1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,0)$$","choices":["$$(0,0)$$","$$(1,1)$$","$$(2,2)$$"],"hints":{"DefaultPathway":[{"id":"af60b4fellipse22a-h1","type":"hint","dependencies":[],"title":"We will notice that there is nothing being added or subtracted to either $$x$$ or $$y$$, meaning that the ellipse is centered at the origin.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af60b4fellipse23","title":"Finding the Center of an Ellipse","body":"Find the center of the following ellipse:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 The Ellipse","courseName":"OpenStax: College Algebra","steps":[{"id":"af60b4fellipse23a","stepAnswer":["$$(2,4)$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{{\\\\left(x-2\\\\right)}^2}{64}+\\\\frac{{\\\\left(y-4\\\\right)}^2}{16}=1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(2,4)$$","choices":["$$(0,0)$$","$$(1,1)$$","$$(2,4)$$"],"hints":{"DefaultPathway":[{"id":"af60b4fellipse23a-h1","type":"hint","dependencies":[],"title":"We will notice that there $$2$$ is being subtracted from the $$x$$ term and $$4$$ is being subtracted from the $$y$$ term. The center is $$(2,4)$$","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af60b4fellipse24","title":"Finding the Center of an Ellipse","body":"Find the center of the following ellipse:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 The Ellipse","courseName":"OpenStax: College Algebra","steps":[{"id":"af60b4fellipse24a","stepAnswer":["$$(-3,3)$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{{\\\\left(x+3\\\\right)}^2}{9}+\\\\frac{{\\\\left(y-3\\\\right)}^2}{9}=1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-3,3)$$","choices":["$$(-1,1)$$","$$(-3,3)$$","$$(-5,5)$$"],"hints":{"DefaultPathway":[{"id":"af60b4fellipse24a-h1","type":"hint","dependencies":[],"title":"We must notice that $$3$$ is being added to the $$x$$ term and $$3$$ is being subtracted from the $$y$$ term. Thus, the center is $$(-3,3)$$","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af60b4fellipse25","title":"Finding the Center of an Ellipse","body":"Find the center of the following ellipse:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 The Ellipse","courseName":"OpenStax: College Algebra","steps":[{"id":"af60b4fellipse25a","stepAnswer":["$$(0,-1)$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{x^2}{2}+\\\\frac{{\\\\left(y+1\\\\right)}^2}{5}=1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,-1)$$","choices":["$$(0,-1)$$","$$(0,-2)$$","$$(0,-3)$$"],"hints":{"DefaultPathway":[{"id":"af60b4fellipse25a-h1","type":"hint","dependencies":[],"title":"It\'s important to notice that nothing is being subtracted or added to the $$x$$ term while $$1$$ is being added to the $$y$$ term. This means that the center is $$(0,-1)$$","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af60b4fellipse26","title":"Finding the Area of an Ellipse","body":"Find the area of the ellipse:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 The Ellipse","courseName":"OpenStax: College Algebra","steps":[{"id":"af60b4fellipse26a","stepAnswer":["$$12\\\\pi$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{{\\\\left(x-3\\\\right)}^2}{9}+\\\\frac{{\\\\left(y-3\\\\right)}^2}{16}=1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$12\\\\pi$$","choices":["$$12\\\\pi$$","$$11\\\\pi$$","$$10\\\\pi$$"],"hints":{"DefaultPathway":[{"id":"af60b4fellipse26a-h1","type":"hint","dependencies":[],"title":"Using the Area Formula","text":"The area of an ellipse is given by $$a b \\\\pi$$. Since $$a=4$$ and $$b=3$$, the area of the ellipse is $$12\\\\pi$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af60b4fellipse27","title":"Finding the Area of an Ellipse","body":"Find the area of the ellipse:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 The Ellipse","courseName":"OpenStax: College Algebra","steps":[{"id":"af60b4fellipse27a","stepAnswer":["$$24\\\\pi$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{{\\\\left(x+6\\\\right)}^2}{16}+\\\\frac{{\\\\left(y-6\\\\right)}^2}{36}=1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$24\\\\pi$$","choices":["$$24\\\\pi$$","$$10\\\\pi$$","$$5\\\\pi$$"],"hints":{"DefaultPathway":[{"id":"af60b4fellipse27a-h1","type":"hint","dependencies":[],"title":"Using the Area Formula","text":"The area of an ellipse is given by $$a b \\\\pi$$. Since $$a=6$$ and $$b=4$$, the area of the ellipse is $$24\\\\pi$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af60b4fellipse28","title":"Finding the Area of an Ellipse","body":"Find the area of the ellipse:","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 The Ellipse","courseName":"OpenStax: College Algebra","steps":[{"id":"af60b4fellipse28a","stepAnswer":["$$2\\\\sqrt{5} \\\\pi$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{{\\\\left(x+1\\\\right)}^2}{4}+\\\\frac{{\\\\left(y-2\\\\right)}^2}{5}=1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2\\\\sqrt{5} \\\\pi$$","choices":["$$2\\\\sqrt{3} \\\\pi$$","$$2\\\\sqrt{4} \\\\pi$$","$$2\\\\sqrt{5} \\\\pi$$"],"hints":{"DefaultPathway":[{"id":"af60b4fellipse28a-h1","type":"hint","dependencies":[],"title":"Using the Area Formula","text":"The area of an ellipse is given by $$a b \\\\pi$$. Since $$a=\\\\sqrt{5}$$ and $$b=2$$, the area of the ellipse is $$2\\\\sqrt{5} \\\\pi$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af60b4fellipse29","title":"Finding the Area of an Ellipse","body":"Find the area of the ellipse.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 The Ellipse","courseName":"OpenStax: College Algebra","steps":[{"id":"af60b4fellipse29a","stepAnswer":["$$9\\\\pi$$"],"problemType":"MultipleChoice","stepTitle":"$$9x^2-54x+9y^2-54y+81=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$9\\\\pi$$","choices":["$$4\\\\pi$$","$$9\\\\pi$$","$$12\\\\pi$$"],"hints":{"DefaultPathway":[{"id":"af60b4fellipse29a-h1","type":"hint","dependencies":[],"title":"Putting the Ellipse Equation into Standard Form","text":"We must put this equation into standard form in order to identify a and $$b$$. $$9\\\\left(x^2-6x\\\\right)+9\\\\left(y^2-6y\\\\right)=-81$$. $${9\\\\left(x-3\\\\right)}^2+{9\\\\left(y-3\\\\right)}^2=-81+81+81$$. $${\\\\left(x-3\\\\right)}^2+{\\\\left(y-3\\\\right)}^2=\\\\frac{81}{9}$$. $${\\\\left(x-3\\\\right)}^2+{\\\\left(y-3\\\\right)}^2=9$$. $$\\\\frac{{\\\\left(x-3\\\\right)}^2}{9}+\\\\frac{{\\\\left(y-3\\\\right)}^2}{9}=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af60b4fellipse29a-h2","type":"hint","dependencies":["af60b4fellipse29a-h1"],"title":"Finding the Area of the Circle","text":"Since both a and $$b=3$$, we know this is a circle. The radius of the circle is $$3$$, so the area is $$9\\\\pi$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af60b4fellipse3","title":"Writing the Equation of an Ellipse Centered at a Point Other Than the Origin","body":"Find the standard form equation of the ellipse with the following coordinates","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 The Ellipse","courseName":"OpenStax: College Algebra","steps":[{"id":"af60b4fellipse3a","stepAnswer":["$$\\\\frac{{\\\\left(x+2\\\\right)}^2}{9}+\\\\frac{{\\\\left(y+3\\\\right)}^2}{25}=1$$"],"problemType":"TextBox","stepTitle":"Vertices $$(-2,-8)$$ and $$(-2,2)$$ and foci $$(-2,-7)$$ and $$(-2,1)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{{\\\\left(x+2\\\\right)}^2}{9}+\\\\frac{{\\\\left(y+3\\\\right)}^2}{25}=1$$","hints":{"DefaultPathway":[{"id":"af60b4fellipse3a-h1","type":"hint","dependencies":[],"title":"Vertice and Foci","text":"Consider whether the major axis lies on the $$x-$$ or y-axis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af60b4fellipse3a-h2","type":"hint","dependencies":["af60b4fellipse3a-h1"],"title":"Standard Equation Form","text":"The form of the standard equation is $$c^2=a^2-b^2$$, given $$a=vertices$$ and $$c=foci$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af60b4fellipse3a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(-2,-8)$$ and $$(-2,2)$$"],"dependencies":["af60b4fellipse3a-h2"],"title":"Midpoint Formula","text":"Using the midpoint formula, identify the coordinate for the point halfway between the vertices","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(-2,-8)$$ and $$(-2,2)$$","$$(-1,-6)$$ and $$(-2,1)$$","$$(-1,-8)$$ and $$(-2,0)$$"]},{"id":"af60b4fellipse3a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$5$$"],"dependencies":["af60b4fellipse3a-h3"],"title":"Solving for a","text":"Solve for a by finding the distance between the y-coordinates of the vertices.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af60b4fellipse3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["af60b4fellipse3a-h4"],"title":"Solving for c","text":"Solve for c using the known information on the foci.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af60b4fellipse3a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["af60b4fellipse3a-h5"],"title":"Solving for $$b$$","text":"Solve for $$b$$ using the equation $$c^2=a^2-b^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af60b4fellipse3a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{{\\\\left(x+2\\\\right)}^2}{9}+\\\\frac{{\\\\left(y+3\\\\right)}^2}{25}=1$$"],"dependencies":["af60b4fellipse3a-h6"],"title":"Plugging in the Values","text":"Plug in the values for $$h$$, k, a, and $$b$$ into the standard form equation for an ellipse.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af60b4fellipse4","title":"Graphing Ellipses Centered at the Origin","body":"Identify the center, vertices, co-vertices, and foci of the following equation of an ellipse (list coordinates in the same order).","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 The Ellipse","courseName":"OpenStax: College Algebra","steps":[{"id":"af60b4fellipse4a","stepAnswer":["$$(0,0),(0,5)$$ and $$(0,-5),(3,0)$$ and $$(-3,0),(0,4)$$ and $$(0,-4)$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{x^2}{9}+y^2+25=1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,0),(0,5)$$ and $$(0,-5),(3,0)$$ and $$(-3,0),(0,4)$$ and $$(0,-4)$$","choices":["$$(0,0),(0,5)$$ and $$(0,-5),(3,0)$$ and $$(-3,0),(0,4)$$ and $$(0,-4)$$","$$(0,0),(0,1)$$ and $$(0,-1),(3,0)$$ and $$(-3,0),(0,4)$$ and $$(0,-4)$$","$$(0,0),(0,2)$$ and $$(0,-2),(3,0)$$ and $$(-3,0),(0,4)$$ and $$(0,-4)$$"],"hints":{"DefaultPathway":[{"id":"af60b4fellipse4a-h1","type":"hint","dependencies":[],"title":"Vertice and Foci","text":"Does the major axis lie on the $$x-$$ or y-axis?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af60b4fellipse4a-h2","type":"hint","dependencies":["af60b4fellipse4a-h1"],"title":"Standard Equation Form","text":"What is the standard equation form of this ellipse?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af60b4fellipse4a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(0,0)$$"],"dependencies":["af60b4fellipse4a-h2"],"title":"Center of Ellipse","text":"Where is the center of the ellipse?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(0,0)$$","$$(1,1)$$","$$(2,2)$$","$$(3,3)$$"]},{"id":"af60b4fellipse4a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(0,5)$$ and $$(0,-5)$$"],"dependencies":["af60b4fellipse4a-h3"],"title":"Vertices of Ellipse","text":"Given the value of a is known, what are the coordinates for the vertices?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(0,4)$$ and $$(0,-4)$$","$$(0,5)$$ and $$(0,-5)$$","$$(0,6)$$ and $$(0,-6)$$","$$(0,7)$$ and $$(0,-7)$$"]},{"id":"af60b4fellipse4a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(3,0)$$ and $$(-3,0)$$"],"dependencies":["af60b4fellipse4a-h4"],"title":"Co-vertices of Ellipse","text":"Given the value of $$b$$ is known, what are the coordinates for the co-vertices?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(3,0)$$ and $$(-3,0)$$","$$(0,4)$$ and $$(0,-4)$$","$$(0,5)$$ and $$(0,-5)$$"]},{"id":"af60b4fellipse4a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(0,4)$$ and $$(0,-4)$$"],"dependencies":["af60b4fellipse4a-h5"],"title":"Foci of Ellipse","text":"Using the equation $$c^2=a^2-b^2$$, solve for c and obtain the coordinates of the foci.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(3,0)$$ and $$(-3,0)$$","$$(0,4)$$ and $$(0,-4)$$","$$(0,5)$$ and $$(0,-5)$$"]}]}}]},{"id":"af60b4fellipse5","title":"Graphing Ellipses Centered at the Origin","body":"Rewrite the following equation in standard form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 The Ellipse","courseName":"OpenStax: College Algebra","steps":[{"id":"af60b4fellipse5a","stepAnswer":["$$\\\\frac{x^2}{25}+\\\\frac{y^2}{4}=1$$"],"problemType":"TextBox","stepTitle":"$$4x^2+25y^2=100$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{x^2}{25}+\\\\frac{y^2}{4}=1$$","hints":{"DefaultPathway":[{"id":"af60b4fellipse5a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{x^2}{25}+\\\\frac{y^2}{4}=1$$"],"dependencies":[],"title":"Rewriting in Standard Form","text":"Use algebra to rewrite the equation in standard form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af60b4fellipse5b","stepAnswer":["$$(0,0),(5,0)$$ and $$(-5,0),(0,2)$$ and (0,-2),(sqrt(21),0) and $$(-\\\\sqrt{21},0)$$"],"problemType":"MultipleChoice","stepTitle":"Then identify and label the center, vertices, co-vertices, and foci (list coordinates in the same order)","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(0,0),(5,0)$$ and $$(-5,0),(0,2)$$ and (0,-2),(sqrt(21),0) and $$(-\\\\sqrt{21},0)$$","choices":["$$(0,0),(5,0)$$ and $$(-5,0),(0,2)$$ and (0,-2),(sqrt(21),0) and $$(-\\\\sqrt{21},0)$$","$$(0,1),(1,0)$$ and $$(-5,0),(0,2)$$ and (0,-2),(sqrt(2),0) and $$(-\\\\sqrt{2},0)$$"],"hints":{"DefaultPathway":[{"id":"af60b4fellipse5b-h1","type":"hint","dependencies":[],"title":"Vertice and Foci","text":"Does the major axis lie on the $$x-$$ or y-axis?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af60b4fellipse5b-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(0,0)$$"],"dependencies":["af60b4fellipse5b-h1"],"title":"Center of Ellipse","text":"Where is the center of the ellipse?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(0,0)$$","$$(0,1)$$","$$(0,2)$$"]},{"id":"af60b4fellipse5b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(5,0)$$ and $$(-5,0)$$"],"dependencies":["af60b4fellipse5b-h2"],"title":"Vertices of Ellipse","text":"Given the value of a is known, what are the coordinates for the vertices?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(3,0)$$ and $$(-3,0)$$","$$(0,4)$$ and $$(0,-4)$$","$$(5,0)$$ and $$(-5,0)$$"]},{"id":"af60b4fellipse5b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(0,2)$$ and $$(0,-2)$$"],"dependencies":["af60b4fellipse5b-h3"],"title":"Co-vertices of Ellipse","text":"Given the value of $$b$$ is known, what are the coordinates for the co-vertices?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(0,2)$$ and $$(0,-2)$$","$$(0,2)$$ and $$(0,-3)$$","$$(0,4)$$ and $$(0,-4)$$"]},{"id":"af60b4fellipse5b-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(0,\\\\sqrt{21})$$ and $$(0,-\\\\sqrt{21})$$"],"dependencies":["af60b4fellipse5b-h4"],"title":"Foci of Ellipse","text":"Using the equation $$c^2=a^2-b^2$$ solve for c and obtain the coordinates of the foci","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(0,\\\\sqrt{21})$$ and $$(0,-\\\\sqrt{21})$$","$$(0,\\\\sqrt{2})$$ and $$(0,-\\\\sqrt{2})$$"]}]}}]},{"id":"af60b4fellipse6","title":"Graphing Ellipses Not Centered at the Origin","body":"Identify the center, vertices, co-vertices, and foci of the following equation of an ellipse (list coordinates in the same order)","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 The Ellipse","courseName":"OpenStax: College 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af60b4fellipse6a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(-2,5)$$"],"dependencies":["af60b4fellipse6a-h2"],"title":"Center of Ellipse","text":"Given the format of the standard equation reveals the value of $$h$$ and k, what are the coordinates for the center of the ellipse?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(-2,5)$$","$$(-1,5)$$","$$(-3,5)$$"]},{"id":"af60b4fellipse6a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["(-2,5~sqrt(9))"],"dependencies":["af60b4fellipse6a-h3"],"title":"Vertices of Ellipse","text":"Given we know the value of a and the center coordinates, what are the coordinates for the vertices?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["(-2,5~sqrt(9))","(-1,5~sqrt(9))","(-3,5~sqrt(9))"]},{"id":"af60b4fellipse6a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\left(-2\\\\pm \\\\sqrt{4}, 5\\\\right)$$"],"dependencies":["af60b4fellipse6a-h4"],"title":"Co-vertices of Ellipse","text":"Given we know the value of $$b$$ and the center coordinates, what are the coordinates for the vertices?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\left(-1\\\\pm \\\\sqrt{4}, 5\\\\right)$$","$$\\\\left(-2\\\\pm \\\\sqrt{4}, 5\\\\right)$$","$$\\\\left(-3\\\\pm \\\\sqrt{4}, 5\\\\right)$$"]},{"id":"af60b4fellipse6a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["(-2,5~sqrt(5))"],"dependencies":["af60b4fellipse6a-h5"],"title":"Foci of Ellipse","text":"Using the equation $$c^2=a^2-b^2$$ solve for c and obtain the coordinates of the foci","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["(-1,5~sqrt(5))","(-2,5~sqrt(5))","(-3,5~sqrt(5))"]}]}}]},{"id":"af60b4fellipse7","title":"Graphing Ellipses Centered at the Origin","body":"Rewrite the following equation in standard form.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 The Ellipse","courseName":"OpenStax: College Algebra","steps":[{"id":"af60b4fellipse7a","stepAnswer":["$$\\\\frac{{\\\\left(x-5\\\\right)}^2}{9}+\\\\frac{{\\\\left(y+2\\\\right)}^2}{4}=1$$"],"problemType":"TextBox","stepTitle":"$$\\\\frac{{\\\\left(x-5\\\\right)}^2}{9}+\\\\frac{{\\\\left(y+2\\\\right)}^2}{4}=1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{{\\\\left(x-5\\\\right)}^2}{9}+\\\\frac{{\\\\left(y+2\\\\right)}^2}{4}=1$$","hints":{"DefaultPathway":[{"id":"af60b4fellipse7a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{{\\\\left(x-5\\\\right)}^2}{9}+\\\\frac{{\\\\left(y+2\\\\right)}^2}{4}=1$$"],"dependencies":[],"title":"Rewriting in Standard Form","text":"Use algebra to rewrite the equation in standard form.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af60b4fellipse7b","stepAnswer":["(5,-2),(5~sqrt(9),-2),(5,-2~sqrt(4)),(5,-2~sqrt(4))"],"problemType":"MultipleChoice","stepTitle":"Then identify and label the center, vertices, co-vertices, and foci (list coordinates in the same order).","stepBody":"","answerType":"string","variabilization":{},"choices":["(5,-2),(5~sqrt(9),-2),(5,-2~sqrt(4)),(5,-2~sqrt(4))","(5,-1),(3~sqrt(9),-2),(4,-2~sqrt(4)),(5,-2~sqrt(4))"],"hints":{"DefaultPathway":[{"id":"af60b4fellipse7b-h2","type":"hint","dependencies":["af60b4fellipse7a-h1"],"title":"Vertice and Foci","text":"Does the major axis lie on the $$x-$$ or y-axis?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af60b4fellipse7b-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(5,-2)$$"],"dependencies":["af60b4fellipse7b-h2"],"title":"Center of Ellipse","text":"Given the format of the standard equation reveals the value of $$h$$ and k, what are the coordinates for the center of the ellipse?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$(5,-2)$$","$$(-2,5)$$"]},{"id":"af60b4fellipse7b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["(5~sqrt(9),-2)"],"dependencies":["af60b4fellipse7b-h3"],"title":"Vertices of Ellipse","text":"Given we know the value of a and the center coordinates, what are the coordinates for the vertices?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["(5~sqrt(9),-2)","(-2,5~sqrt(9))"]},{"id":"af60b4fellipse7b-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$5-2\\\\pm \\\\sqrt{4}$$"],"dependencies":["af60b4fellipse7b-h4"],"title":"Co-vertices of Ellipse","text":"Given we know the value of $$b$$ and the center coordinates, what are the coordinates for the vertices?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$5-2\\\\pm \\\\sqrt{4}$$","(2~sqrt(4),5)"]},{"id":"af60b4fellipse7b-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$5-2\\\\pm \\\\sqrt{4}$$"],"dependencies":["af60b4fellipse7b-h5"],"title":"Foci of Ellipse","text":"Using the equation $$c^2=a^2-b^2$$ solve for c and obtain the coordinates of the foci","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$5-2\\\\pm \\\\sqrt{4}$$","(2~sqrt(4),5)"]}]}}]},{"id":"af60b4fellipse8","title":"The Ellipse","body":"For the following exercise, determine whether the given equations represent ellipses. If yes, write in standard form. If no, enter the number $$0$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 The Ellipse","courseName":"OpenStax: College Algebra","steps":[{"id":"af60b4fellipse8a","stepAnswer":["$$\\\\frac{x^2}{3^2}+\\\\frac{y^2}{2^2}=1$$"],"problemType":"TextBox","stepTitle":"$$4x^2+9y^2=36$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{x^2}{3^2}+\\\\frac{y^2}{2^2}=1$$","hints":{"DefaultPathway":[{"id":"af60b4fellipse8a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":[],"title":"Does the equation contain the same variable exponents as the standard equation form of an ellipse?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"af60b4fellipse8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{x^2}{3^2}+\\\\frac{y^2}{2^2}=1$$"],"dependencies":["af60b4fellipse8a-h1"],"title":"Use algebra to rewrite the equation in standard form","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af60b4fellipse9","title":"The Ellipse","body":"For the following exercise, determine whether the given equations represent ellipses. If yes, write in standard form. If no, enter the number $$0$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"8.1 The Ellipse","courseName":"OpenStax: College Algebra","steps":[{"id":"af60b4fellipse9a","stepAnswer":["$$\\\\frac{x^2}{{\\\\left(\\\\frac{1}{2}\\\\right)}^2}+\\\\frac{y^2}{{\\\\left(\\\\frac{1}{3}\\\\right)}^2}$$"],"problemType":"TextBox","stepTitle":"$$4x^2+9y^2=1$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{x^2}{{\\\\left(\\\\frac{1}{2}\\\\right)}^2}+\\\\frac{y^2}{{\\\\left(\\\\frac{1}{3}\\\\right)}^2}$$","hints":{"DefaultPathway":[{"id":"af60b4fellipse9a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":[],"title":"Does the equation contain the same variable exponents as the standard equation form of an ellipse?","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["Yes","No"]},{"id":"af60b4fellipse9a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{x^2}{{\\\\left(\\\\frac{1}{2}\\\\right)}^2}+\\\\frac{y^2}{{\\\\left(\\\\frac{1}{3}\\\\right)}^2}$$"],"dependencies":["af60b4fellipse9a-h1"],"title":"Use algebra to rewrite the equation in standard form.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af69facexpolog1","title":"Modeling Exponential Growth and Decay","body":"The half-life of carbon-14 is 5,730 years.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Exponential and Logarithmic Models","courseName":"OpenStax: College Algebra","steps":[{"id":"af69facexpolog1a","stepAnswer":["$$f(t)=A_0 e^{\\\\frac{\\\\ln(0.5) t}{5730}}$$"],"problemType":"MultipleChoice","stepTitle":"Express the amount of carbon-14 remaining as a function of time, $$t$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$f(t)=A_0 e^{\\\\frac{\\\\ln(0.5) t}{5730}}$$","choices":["$$f(t)=A_0 e^{\\\\frac{\\\\ln(5) t}{5630}}$$","$$f(t)=A_0 e^{\\\\frac{\\\\ln(0.5) t}{5730}}$$","$$f(t)=A_0 e^{\\\\frac{\\\\ln(0.5) t}{5830}}$$"],"hints":{"DefaultPathway":[{"id":"af69facexpolog1a-h1","type":"hint","dependencies":[],"title":"Identifying the Formula","text":"What is the continuous growth formula?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog1a-h2","type":"hint","dependencies":["af69facexpolog1a-h1"],"title":"Solve for k","text":"Analyze the formula with substituted values and take the necessary steps to solve for k","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog1a-h3","type":"hint","dependencies":["af69facexpolog1a-h2"],"title":"Substitute Known Values","text":"Substitute the half life for $$t$$ and and $$0.5A_0$$ for f(t)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog1a-h4","type":"hint","dependencies":["af69facexpolog1a-h3"],"title":"Eliminate Variable","text":"Divide both side by $$A_0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog1a-h5","type":"hint","dependencies":["af69facexpolog1a-h4"],"title":"Using Natural Log","text":"Take the natural log of both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog1a-h6","type":"hint","dependencies":["af69facexpolog1a-h5"],"title":"Finding k","text":"Divide by the coefficient of k on both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog1a-h7","type":"hint","dependencies":["af69facexpolog1a-h6"],"title":"Final Formula","text":"Substitute final value for k as the rate in the continuous growth formula","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af69facexpolog10","title":"Use Newton\'s Law of Cooling","body":"A pitcher of water at $$40$$ degrees Fahrenheit is placed into a $$70$$ degree room. One hour later, the temperature has risen to $$45$$ degrees.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Exponential and Logarithmic Models","courseName":"OpenStax: College Algebra","steps":[{"id":"af69facexpolog10a","stepAnswer":["$$362$$"],"problemType":"TextBox","stepTitle":"How long will it take for the temperature to rise to $$60$$ degrees? (round to the nearest whole number in minutes)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$362$$","hints":{"DefaultPathway":[{"id":"af69facexpolog10a-h1","type":"hint","dependencies":[],"title":"Identify Formula","text":"What is the formula for Newton\'s Law of Cooling?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$70$$"],"dependencies":["af69facexpolog10a-h1"],"title":"Substituting Known Values","text":"What will be substituting for $$T_s$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog10a-h3","type":"hint","dependencies":["af69facexpolog10a-h2"],"title":"Solve for A","text":"Plug in the initial temperature value and set the time as $$0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog10a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-30$$"],"dependencies":["af69facexpolog10a-h3"],"title":"Solve for A","text":"What is A?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog10a-h5","type":"hint","dependencies":["af69facexpolog10a-h4"],"title":"Solve for K","text":"Another value given in the problem is 45\xb0F at $$60$$ minutes. Plug in those values to solve for K.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog10a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-0.00304$$"],"dependencies":["af69facexpolog10a-h5"],"title":"Solve for K","text":"What is K? (round to the nearest ten-thousandths place)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog10a-h7","type":"hint","dependencies":["af69facexpolog10a-h6"],"title":"Final Formula","text":"After solve for K and A, plug in those values into the original Newton\'s cooling formula.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog10a-h8","type":"hint","dependencies":["af69facexpolog10a-h7"],"title":"Solve for T","text":"To find the time it takes for the cake to cool to $$60$$ degrees, plug in $$60$$ for T(t) and solve for $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog10a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$362$$"],"dependencies":["af69facexpolog10a-h8"],"title":"Solve for T","text":"What is T? (round to nearest whole number)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af69facexpolog11","title":"Temperature Decrease","body":"The temperature of an object in degrees Fahrenheit after $$t$$ minutes is represented by the equation $$T(t)=68e^{\\\\left(-1\\\\times0.0174 t\\\\right)}+@{C}$$.","variabilization":{"C":["70","71","72","73"],"ans":["136","137","138","139"]},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Exponential and Logarithmic Models","courseName":"OpenStax: College Algebra","steps":[{"id":"af69facexpolog11a","stepAnswer":["@{ans}"],"problemType":"TextBox","stepTitle":"What is the temperature of the object after $$1.5$$ hours to the nearest degree?","stepBody":"","answerType":"arithmetic","variabilization":{"C":["70","71","72","73"],"ans":["136","137","138","139"]},"hints":{"DefaultPathway":[{"id":"af69facexpolog11a-h1","type":"hint","dependencies":[],"title":"Plugging in $$1.5$$ as T","text":"To solve this problem, we simply have to plug in $$1.5$$ for the variable $$t$$. This will give us a value T(t) that represents our answer.","variabilization":{"C":["70","71","72","73"],"ans":["136","137","138","139"]},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af69facexpolog12","title":"Interpreting Logistic Growth Models","body":"Use the logistic growth model $$f(x)=\\\\frac{@{numerator}}{1+8e^{\\\\left(-2x\\\\right)}}$$","variabilization":{"numerator":["150","160","170"],"ans":["17","18","19"]},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Exponential and Logarithmic Models","courseName":"OpenStax: College Algebra","steps":[{"id":"af69facexpolog12a","stepAnswer":["@{ans}"],"problemType":"TextBox","stepTitle":"Find f(0). Round to the nearest whole number.","stepBody":"","answerType":"arithmetic","variabilization":{"numerator":["150","160","170"],"ans":["17","18","19"]},"hints":{"DefaultPathway":[{"id":"af69facexpolog12a-h1","type":"hint","dependencies":[],"title":"Plugging in $$0$$","text":"Plugging in $$0$$ for $$x$$ in the function f(x) will give us the answer. This means that $$f(0)=@{ans}$$.","variabilization":{"numerator":["150","160","170"],"ans":["17","18","19"]},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af69facexpolog13","title":"Interpreting Logistic Growth Models","body":"Use the logistic growth model $$f(x)=\\\\frac{@{numerator}}{1+8e^{\\\\left(-2x\\\\right)}}$$.","variabilization":{"numerator":["150","160","170"],"ans":["150","160","170"]},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Exponential and Logarithmic Models","courseName":"OpenStax: College Algebra","steps":[{"id":"af69facexpolog13a","stepAnswer":["@{ans}"],"problemType":"TextBox","stepTitle":"Find f(4). Round to the nearest whole number.","stepBody":"","answerType":"arithmetic","variabilization":{"numerator":["150","160","170"],"ans":["150","160","170"]},"hints":{"DefaultPathway":[{"id":"af69facexpolog13a-h1","type":"hint","dependencies":[],"title":"Plugging in $$0$$","text":"Plugging in $$4$$ for $$x$$ in the function f(x) will give us the answer. This means that $$f(4)=@{ans}$$.","variabilization":{"numerator":["150","160","170"],"ans":["150","160","170"]},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af69facexpolog14","title":"Interpreting Logistic Growth Models","body":"For the following problem, use the logistic growth model $$f(x)=\\\\frac{@{numerator}}{1+8e^{\\\\left(-2x\\\\right)}}$$.","variabilization":{"numerator":["150","160","170"]},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Exponential and Logarithmic Models","courseName":"OpenStax: College Algebra","steps":[{"id":"af69facexpolog14a","stepAnswer":["@{numerator}"],"problemType":"TextBox","stepTitle":"Find the carrying capacity.","stepBody":"","answerType":"arithmetic","variabilization":{"numerator":["150","160","170"]},"hints":{"DefaultPathway":[{"id":"af69facexpolog14a-h1","type":"hint","dependencies":[],"title":"The Numerator is the Carrying Capacity","text":"Since this is a logistic growth model, we know that the numerator is the carrying capacity. This means that our answer is @{numerator}","variabilization":{"numerator":["150","160","170"]},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af69facexpolog15","title":"Fish Population","body":"$$P(t)=\\\\frac{1000}{1+9e^{@{exponentt}}}$$ represents the population of fish as a function of time.","variabilization":{"exponent":["-0.6","-0.8","-0.9"]},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Exponential and Logarithmic Models","courseName":"OpenStax: College Algebra","steps":[{"id":"af69facexpolog15a","stepAnswer":["$$100$$"],"problemType":"TextBox","stepTitle":"What is the initial population?","stepBody":"","answerType":"arithmetic","variabilization":{"exponent":["-0.6","-0.8","-0.9"]},"answerLatex":"$$100$$","hints":{"DefaultPathway":[{"id":"af69facexpolog15a-h1","type":"hint","dependencies":[],"title":"Plugging in $$0$$ for $$t$$","text":"We must plug in $$0$$ for $$t$$, as this represents the initial population. This leaves us with $$\\\\frac{1000}{10}$$ which simplifies to $$100$$.","variabilization":{"exponent":["-0.6","-0.8","-0.9"]},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af69facexpolog16","title":"Fish Population","body":"$$P(t)=\\\\frac{1000}{1+9e^{-0.6 t}}$$ represents the population of fish as a function of time, $$t$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Exponential and Logarithmic Models","courseName":"OpenStax: College Algebra","steps":[{"id":"af69facexpolog16a","stepAnswer":["$$1.4$$"],"problemType":"TextBox","stepTitle":"What is the doubling time to the nearest tenth?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1.4$$","hints":{"DefaultPathway":[{"id":"af69facexpolog16a-h1","type":"hint","dependencies":[],"title":"Setting up an Equation","text":"Since we know that the initial population of fish is $$100$$, we simply set the function equal to $$200$$. $$200=\\\\frac{1000}{1+9e^{-0.6 t}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog16a-h2","type":"hint","dependencies":["af69facexpolog16a-h1"],"title":"Solving for $$t$$","text":"Now, we can simply solve for $$t$$. $$0.2=\\\\frac{1}{1+9e^{-0.6 t}}$$. $$t=1.4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af69facexpolog17","title":"Fish Population","body":"$$P(t)=\\\\frac{1000}{1+9e^{-0.6 t}}$$ represents the population of fish as a function of time passed, $$t$$ (in years.)","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Exponential and Logarithmic Models","courseName":"OpenStax: College Algebra","steps":[{"id":"af69facexpolog17a","stepAnswer":["$$269$$"],"problemType":"TextBox","stepTitle":"What will the fish population be after $$2$$ years to the nearest whole number?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$269$$","hints":{"DefaultPathway":[{"id":"af69facexpolog17a-h1","type":"hint","dependencies":[],"title":"Plugging in $$2$$ for $$t$$","text":"In order to solve this problem, we can simply plug in $$2$$ for $$t$$ and put the function P(2) into a calculator. This leaves us with $$P(2)=269$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af69facexpolog18","title":"Fish Population","body":"$$P(t)=\\\\frac{1000}{1+9e^{-0.6 t}}$$ represents the population of fish.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Exponential and Logarithmic Models","courseName":"OpenStax: College Algebra","steps":[{"id":"af69facexpolog18a","stepAnswer":["$$7.3$$"],"problemType":"TextBox","stepTitle":"To the nearest tenth, how long will the fish population take to reach 900?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$7.3$$","hints":{"DefaultPathway":[{"id":"af69facexpolog18a-h1","type":"hint","dependencies":[],"title":"Setting up an Equation","text":"We can simply set P(t) to $$900$$ and then solve for $$t$$. $$900=\\\\frac{1000}{1+9e^{-0.6 t}}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog18a-h2","type":"hint","dependencies":["af69facexpolog18a-h1"],"title":"Solving the Equation","text":"Now, we must solve for $$t$$ by isolating the variable. $$0.9=\\\\frac{1}{1+9e^{-0.6 t}}$$. This means that $$t=7.3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af69facexpolog19","title":"Fish Population","body":"$$P(t)=\\\\frac{@{numerator}}{1+9e^{-0.6 t}}$$ represents the population of fish as a function of time.","variabilization":{"numerator":["1000","2000","3000","4000"]},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Exponential and Logarithmic Models","courseName":"OpenStax: College Algebra","steps":[{"id":"af69facexpolog19a","stepAnswer":["@{numerator}"],"problemType":"TextBox","stepTitle":"What is the carrying capacity?","stepBody":"","answerType":"arithmetic","variabilization":{"numerator":["1000","2000","3000","4000"]},"hints":{"DefaultPathway":[{"id":"af69facexpolog19a-h1","type":"hint","dependencies":[],"title":"Carrying Capacity is the Numerator","text":"To find the carrying capacity, we can simply look at the numerator since it is equal to the carrying capacity. This means that the carrying capacity is @{numerator}","variabilization":{"numerator":["1000","2000","3000","4000"]},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af69facexpolog2","title":"Modeling Exponential Growth and Decay","body":"The half-life of plutonium-244 is 80,000,000 years.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Exponential and Logarithmic Models","courseName":"OpenStax: College Algebra","steps":[{"id":"af69facexpolog2a","stepAnswer":["$$f(t)=A_0 e^{\\\\frac{\\\\ln(0.5) t}{800000000}}$$"],"problemType":"MultipleChoice","stepTitle":"Find a function that gives the amount of plutonium-244 remaining as a function of time, measured in years.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$f(t)=A_0 e^{\\\\frac{\\\\ln(0.5) t}{800000000}}$$","choices":["$$f(t)=A_0 e^{\\\\frac{\\\\ln(0.6) t}{800000000}}$$","$$f(t)=A_0 e^{\\\\frac{\\\\ln(0.5) t}{800000000}}$$","$$f(t)=A_0 e^{\\\\frac{\\\\ln(0.6) t}{700000000}}$$"],"hints":{"DefaultPathway":[{"id":"af69facexpolog2a-h1","type":"hint","dependencies":[],"title":"Identifying the Formula","text":"What is the continuous growth formula?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog2a-h2","type":"hint","dependencies":["af69facexpolog2a-h1"],"title":"Solve for k","text":"Analyze the formula with substituted values and take the necessary steps to solve for k","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog2a-h3","type":"hint","dependencies":["af69facexpolog2a-h2"],"title":"Substitute Known Values","text":"Substitute the half life for $$t$$ and and $$0.5A_0$$ for f(t)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog2a-h4","type":"hint","dependencies":["af69facexpolog2a-h3"],"title":"Eliminate Variable","text":"Divide both side by $$A_0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog2a-h5","type":"hint","dependencies":["af69facexpolog2a-h4"],"title":"Using Natural Log","text":"Take the natural log of both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog2a-h6","type":"hint","dependencies":["af69facexpolog2a-h5"],"title":"Finding k","text":"Divide by the coefficient of k on both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog2a-h7","type":"hint","dependencies":["af69facexpolog2a-h6"],"title":"Final Formula","text":"Substitute final value for k as the rate in the continuous growth formula","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af69facexpolog20","title":"Y-Intercept of Logistic Growth Model","body":"For the following problem, use the logistic growth model $$f(x)=\\\\frac{c}{1+{ae}^{\\\\left(-rx\\\\right)}}$$.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Exponential and Logarithmic Models","courseName":"OpenStax: College Algebra","steps":[{"id":"af69facexpolog20a","stepAnswer":["$$\\\\frac{c}{1+a}$$"],"problemType":"TextBox","stepTitle":"What is the y-intercept?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{c}{1+a}$$","hints":{"DefaultPathway":[{"id":"af69facexpolog20a-h1","type":"hint","dependencies":[],"title":"Solving for f(x) when $$x=0$$","text":"We know that a function\'s y-intercept is the y-value at which $$x=0$$. This means that we can just plug in $$0$$ for $$x$$ and simplify to find f(0) in terms of c and a. $$f(0)=\\\\frac{c}{1+{ae}^0}=\\\\frac{c}{1+a}$$. The y-intercept is $$\\\\frac{c}{1+a}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af69facexpolog21","title":"Finding the Function that Describes Radioactive Decay","body":"A scientist begins with $$250$$ grams of a radioactive substance. After $$250$$ minutes, the sample has decayed to $$32$$ grams.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Exponential and Logarithmic Models","courseName":"OpenStax: College Algebra","steps":[{"id":"af69facexpolog21a","stepAnswer":["$$250e^{-0.00822 t}$$"],"problemType":"TextBox","stepTitle":"Rounding to five decimal places, write an exponential equation representing this situation.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$250e^{-0.00822 t}$$","hints":{"DefaultPathway":[{"id":"af69facexpolog21a-h1","type":"hint","dependencies":[],"title":"Exponential Growth and Decay","text":"We are able to use the formula $$y=A_0 e^{k t}$$ where $$A_0$$ is equal to the value at time zero, e is Euler\u2019s constant, and k is a positive constant that determines the rate of growth or decay to model exponential growth or decay respectively. In the case of an exponential growth, the k value is positive while in the case of an exponential decay, the k value is negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog21a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$250$$"],"dependencies":["af69facexpolog21a-h1"],"title":"Initial Value","text":"What is the value associated with time zero, $$A_0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog21a-h3","type":"hint","dependencies":["af69facexpolog21a-h2"],"title":"Rate of $$\\\\frac{Growth}{Decay}$$","text":"We want to find the rate of growth of decay, k, so that we can obtain the expression. The model for exponential decay is $$y=A_0 e^{k t}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog21a-h4","type":"hint","dependencies":["af69facexpolog21a-h3"],"title":"Rate of $$\\\\frac{Growth}{Decay}$$","text":"Divide by $$A_0$$ on both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog21a-h5","type":"hint","dependencies":["af69facexpolog21a-h4"],"title":"Rate of $$\\\\frac{Growth}{Decay}$$","text":"Take the natural log of both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog21a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\ln(\\\\frac{y}{A_0})}{t}$$"],"dependencies":["af69facexpolog21a-h5"],"title":"Rate of $$\\\\frac{Growth}{Decay}$$","text":"Divide by the time variable. We now obtain an expression for the rate of decay, k. What is k? Express in terms of $$y$$, $$A_0$$ and $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog21a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-0.00822$$"],"dependencies":["af69facexpolog21a-h6"],"title":"Rate of $$\\\\frac{Growth}{Decay}$$","text":"Given that the sample decayed to $$32$$ grams after $$250$$ minutes, we can substitute $$t=250$$ and $$y=32$$ into the expression to solve for k. What is k? Round to five decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog21a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$250e^{-0.00822 t}$$"],"dependencies":["af69facexpolog21a-h7"],"title":"Exponential Decay Model","text":"Now that we have found $$A_0$$ and k, we can substitute the values into the exponential equation for this question, $$y=A_0 e^{k t}$$. What is the exponential model?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af69facexpolog21b","stepAnswer":["$$84$$"],"problemType":"TextBox","stepTitle":"To the nearest minute, what is the half-life of this substance?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$84$$","hints":{"DefaultPathway":[{"id":"af69facexpolog21b-h1","type":"hint","dependencies":[],"title":"Solving for $$t$$","text":"We can rearrange the equation that we obtained such that $$t$$ is the subject.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog21b-h2","type":"hint","dependencies":["af69facexpolog21b-h1"],"title":"Solving for $$t$$","text":"Divide by $$A_0$$ on both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog21b-h3","type":"hint","dependencies":["af69facexpolog21b-h2"],"title":"Solving for $$t$$","text":"Take the natural log of both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog21b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\ln(\\\\frac{y}{A_0})}{k}$$"],"dependencies":["af69facexpolog21b-h3"],"title":"Solving for $$t$$","text":"Divide by the rate of decay, k. We now obtain an expression for time, $$t$$. What is $$t$$? Express in terms of $$y$$, $$A_0$$ and k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog21b-h5","type":"hint","dependencies":["af69facexpolog21b-h4"],"title":"Half-life","text":"Observe that the expression, $$\\\\frac{y}{A_0}$$ within the natural log expression is the proportion of the object\'s mass left. At half-life, this would be 50% or $$0.5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog21b-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$84$$"],"dependencies":["af69facexpolog21b-h5"],"title":"Substitution","text":"We can substitute $$\\\\frac{y}{A_0}=0.5$$ and k that was previously found to solve for $$t$$. What is $$t$$? Round to the nearest minute.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af69facexpolog22","title":"Finding the Function that Describes Radioactive Decay","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Exponential and Logarithmic Models","courseName":"OpenStax: College Algebra","steps":[{"id":"af69facexpolog22a","stepAnswer":["$$-0.0004$$"],"problemType":"TextBox","stepTitle":"The half-life of Radium-226 is $$1590$$ years. What is the annual decay rate? Express the decimal result to four decimal places.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-0.0004$$","hints":{"DefaultPathway":[{"id":"af69facexpolog22a-h1","type":"hint","dependencies":[],"title":"Exponential Growth and Decay","text":"We are able to use the formula $$y=A_0 e^{k t}$$ where $$A_0$$ is equal to the value at time zero, e is Euler\u2019s constant, and k is a positive constant that determines the rate of growth or decay to model exponential growth or decay respectively. In the case of an exponential growth, the k value is positive while in the case of an exponential decay, the k value is negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog22a-h2","type":"hint","dependencies":["af69facexpolog22a-h1"],"title":"Rate of $$\\\\frac{Growth}{Decay}$$","text":"We want to find the rate of decay, k, so that we can obtain the expression. The model for exponential decay is $$y=A_0 e^{k t}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog22a-h3","type":"hint","dependencies":["af69facexpolog22a-h2"],"title":"Rate of $$\\\\frac{Growth}{Decay}$$","text":"Divide by $$A_0$$ on both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog22a-h4","type":"hint","dependencies":["af69facexpolog22a-h3"],"title":"Rate of $$\\\\frac{Growth}{Decay}$$","text":"Take the natural log of both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog22a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\ln(\\\\frac{y}{A_0})}{t}$$"],"dependencies":["af69facexpolog22a-h4"],"title":"Rate of $$\\\\frac{Growth}{Decay}$$","text":"Divide by the time variable. We now obtain an expression for the rate of decay, k. What is k? Express in terms of $$y$$, $$A_0$$ and $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog22a-h6","type":"hint","dependencies":["af69facexpolog22a-h5"],"title":"Half-life","text":"Observe that the expression, $$\\\\frac{y}{A_0}$$ within the natural log expression is the proportion of the object\'s mass left. At half-life, this would be 50% or $$0.5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog22a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-0.0004$$"],"dependencies":["af69facexpolog22a-h6"],"title":"Substitution","text":"We can substitute $$\\\\frac{y}{A_0}=0.5$$ and the half-life time, $$t=1590$$ that is given. What is k? Round to four decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af69facexpolog23","title":"Finding the Function that Describes Radioactive Decay","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Exponential and Logarithmic Models","courseName":"OpenStax: College Algebra","steps":[{"id":"af69facexpolog23a","stepAnswer":["$$-0.0666$$"],"problemType":"TextBox","stepTitle":"The half-life of Erbium-165 is $$10.4$$ hours. What is the hourly decay rate? Express the decimal result to four decimal places.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$-0.0666$$","hints":{"DefaultPathway":[{"id":"af69facexpolog23a-h1","type":"hint","dependencies":[],"title":"Exponential Growth and Decay","text":"We are able to use the formula $$y=A_0 e^{k t}$$ where $$A_0$$ is equal to the value at time zero, e is Euler\u2019s constant, and k is a positive constant that determines the rate of growth or decay to model exponential growth or decay respectively. In the case of an exponential growth, the k value is positive while in the case of an exponential decay, the k value is negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog23a-h2","type":"hint","dependencies":["af69facexpolog23a-h1"],"title":"Rate of $$\\\\frac{Growth}{Decay}$$","text":"We want to find the rate of decay, k, so that we can obtain the expression. The model for exponential decay is $$y=A_0 e^{k t}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog23a-h3","type":"hint","dependencies":["af69facexpolog23a-h2"],"title":"Rate of $$\\\\frac{Growth}{Decay}$$","text":"Divide by $$A_0$$ on both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog23a-h4","type":"hint","dependencies":["af69facexpolog23a-h3"],"title":"Rate of $$\\\\frac{Growth}{Decay}$$","text":"Take the natural log of both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog23a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\ln(\\\\frac{y}{A_0})}{t}$$"],"dependencies":["af69facexpolog23a-h4"],"title":"Rate of $$\\\\frac{Growth}{Decay}$$","text":"Divide by the time variable. We now obtain an expression for the rate of decay, k. What is k? Express in terms of $$y$$, $$A_0$$ and $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog23a-h6","type":"hint","dependencies":["af69facexpolog23a-h5"],"title":"Half-life","text":"Observe that the expression, $$\\\\frac{y}{A_0}$$ within the natural log expression is the proportion of the object\'s mass left. At half-life, this would be 50% or $$0.5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog23a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-0.0666$$"],"dependencies":["af69facexpolog23a-h6"],"title":"Substitution","text":"We can substitute $$\\\\frac{y}{A_0}=0.5$$ and the half-life time, $$t=10.4$$ that is given. What is k? Round to four decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af69facexpolog24","title":"Finding the Function that Describes Radioactive Decay","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Exponential and Logarithmic Models","courseName":"OpenStax: College Algebra","steps":[{"id":"af69facexpolog24a","stepAnswer":["$$4223$$"],"problemType":"TextBox","stepTitle":"A wooden artifact from an archeological dig contains $$60$$ percent of the carbon-14 that is present in living trees. To the nearest year, about how many years old is the artifact? (The half-life of carbon-14 is $$5730$$ years.)","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4223$$","hints":{"DefaultPathway":[{"id":"af69facexpolog24a-h1","type":"hint","dependencies":[],"title":"Exponential Growth and Decay","text":"We are able to use the formula $$y=A_0 e^{k t}$$ where $$A_0$$ is equal to the value at time zero, e is Euler\u2019s constant, and k is a positive constant that determines the rate of growth or decay to model exponential growth or decay respectively. In the case of an exponential growth, the k value is positive while in the case of an exponential decay, the k value is negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog24a-h2","type":"hint","dependencies":["af69facexpolog24a-h1"],"title":"Rate of $$\\\\frac{Growth}{Decay}$$","text":"We want to find the rate of decay, k, so that we can obtain the expression. The model for exponential decay is $$y=A_0 e^{k t}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog24a-h3","type":"hint","dependencies":["af69facexpolog24a-h2"],"title":"Rate of $$\\\\frac{Growth}{Decay}$$","text":"Divide by $$A_0$$ on both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog24a-h4","type":"hint","dependencies":["af69facexpolog24a-h3"],"title":"Rate of $$\\\\frac{Growth}{Decay}$$","text":"Take the natural log of both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog24a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\ln(\\\\frac{y}{A_0})}{t}$$"],"dependencies":["af69facexpolog24a-h4"],"title":"Rate of $$\\\\frac{Growth}{Decay}$$","text":"Divide by the time variable. We now obtain an expression for the rate of decay, k. What is k? Express in terms of $$y$$, $$A_0$$ and $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog24a-h6","type":"hint","dependencies":["af69facexpolog24a-h5"],"title":"Half-life","text":"Observe that the expression, $$\\\\frac{y}{A_0}$$ within the natural log expression is the proportion of the object\'s mass left. At half-life, this would be 50% or $$0.5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog24a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-0.00012$$"],"dependencies":["af69facexpolog24a-h6"],"title":"Substitution","text":"We can substitute $$\\\\frac{y}{A_0}=0.5$$ and the half-life time, $$t=5730$$ that is given. What is k? Round to five decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog24a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\ln(\\\\frac{y}{A_0})}{k}$$"],"dependencies":["af69facexpolog24a-h7"],"title":"Solving for $$t$$","text":"We can rearrange the equation that we obtained such that $$t$$ is the subject. Express $$t$$ in terms of y,A_0,k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog24a-h9","type":"hint","dependencies":["af69facexpolog24a-h8"],"title":"Proportionality","text":"Observe that the expression, $$\\\\frac{y}{A_0}$$ within the natural log expression is the proportion of the object\'s mass left. We are interested to find how long it took to reach 60% of the carbon-14 present in living trees. Thus, we can let $$\\\\frac{y}{A_0}=0.6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog24a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4223$$"],"dependencies":["af69facexpolog24a-h9"],"title":"Substitution","text":"We can substitute $$\\\\frac{y}{A_0}=0.6$$ and the rate of decay,k that was previously found. What is $$t$$? Round to nearest year.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af69facexpolog25","title":"Finding a Function That Describes Exponential Growth","body":"A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was $$1350$$ bacteria.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Exponential and Logarithmic Models","courseName":"OpenStax: College Algebra","steps":[{"id":"af69facexpolog25a","stepAnswer":["$$1350e^{0.03466t}$$"],"problemType":"TextBox","stepTitle":"Rounding to five decimal places, write an exponential equation representing this situation.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1350e^{0.03466t}$$","hints":{"DefaultPathway":[{"id":"af69facexpolog25a-h1","type":"hint","dependencies":[],"title":"Exponential Growth and Decay","text":"We are able to use the formula $$y=A_0 e^{k t}$$ where $$A_0$$ is equal to the value at time zero, e is Euler\u2019s constant, and k is a positive constant that determines the rate of growth or decay to model exponential growth or decay respectively. In the case of an exponential growth, the k value is positive while in the case of an exponential decay, the k value is negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog25a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1350$$"],"dependencies":["af69facexpolog25a-h1"],"title":"Initial Value","text":"What is the value associated with time zero, $$A_0$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog25a-h3","type":"hint","dependencies":["af69facexpolog25a-h2"],"title":"Rate of $$\\\\frac{Growth}{Decay}$$","text":"We want to find the rate of growth, k, so that we can obtain the expression. The model for exponential growth is $$y=A_0 e^{k t}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog25a-h4","type":"hint","dependencies":["af69facexpolog25a-h3"],"title":"Rate of $$\\\\frac{Growth}{Decay}$$","text":"Divide by $$A_0$$ on both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog25a-h5","type":"hint","dependencies":["af69facexpolog25a-h4"],"title":"Rate of $$\\\\frac{Growth}{Decay}$$","text":"Take the natural log of both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog25a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\ln(\\\\frac{y}{A_0})}{t}$$"],"dependencies":["af69facexpolog25a-h5"],"title":"Rate of $$\\\\frac{Growth}{Decay}$$","text":"Divide by the time variable. We now obtain an expression for the rate of decay, k. What is k? Express in terms of $$y$$, $$A_0$$ and $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog25a-h7","type":"hint","dependencies":["af69facexpolog25a-h6"],"title":"Proportionality","text":"Observe that the expression, $$\\\\frac{y}{A_0}$$ within the natural log expression is the proportion of bacteria count after time $$t$$. We were given that the bacteria count doubled, thus $$\\\\frac{y}{A_0}=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog25a-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.03466$$"],"dependencies":["af69facexpolog25a-h7"],"title":"Substitution","text":"We can substitute $$\\\\frac{y}{A_0}=2$$ and $$t=20$$ to find k. What is k? Round to five decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog25a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1350e^{0.03466t}$$"],"dependencies":["af69facexpolog25a-h8"],"title":"Exponential Growth Model","text":"Now that we\'ve found $$A_0$$ and k, substitute them into the growth model, $$y=A_0 e^{k t}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af69facexpolog25b","stepAnswer":["$$691200$$"],"problemType":"TextBox","stepTitle":"To the nearest whole number, what is the population size after $$3$$ hours?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$691200$$","hints":{"DefaultPathway":[{"id":"af69facexpolog25b-h1","type":"hint","dependencies":[],"title":"Substitution","text":"How many minutes are there in three hours? Substitute this in to the model using the exact k value to obtain the population count.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af69facexpolog26","title":"Finding a Function That Describes Exponential Growth","body":"For the following exercises, use this scenario: A biologist recorded a count of $$360$$ bacteria present in a culture after $$5$$ minutes and $$1000$$ bacteria present after $$20$$ minutes.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Exponential and Logarithmic Models","courseName":"OpenStax: College Algebra","steps":[{"id":"af69facexpolog26a","stepAnswer":["$$256$$"],"problemType":"TextBox","stepTitle":"To the nearest whole number, what was the initial population in the culture?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$256$$","hints":{"DefaultPathway":[{"id":"af69facexpolog26a-h1","type":"hint","dependencies":[],"title":"Exponential Growth and Decay","text":"We are able to use the formula $$y=A_0 e^{k t}$$ where $$A_0$$ is equal to the value at time zero, e is Euler\u2019s constant, and k is a positive constant that determines the rate of growth or decay to model exponential growth or decay respectively. In the case of an exponential growth, the k value is positive while in the case of an exponential decay, the k value is negative.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog26a-h2","type":"hint","dependencies":["af69facexpolog26a-h1"],"title":"Solving for $$A_0$$","text":"We were given that the count was $$360$$ at $$5$$ minutes and the count was $$1000$$ at $$20$$ minutes. We can express this as two equations that can be used to solve for the initial population, $$A_0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog26a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$360=A_0 e^{5k}$$"],"dependencies":["af69facexpolog26a-h2"],"title":"Solving for $$A_0$$","text":"Find the first equation. Substitute $$y=360$$ and $$t=5$$ into the exponential growth model, $$y=A_0 e^{k t}$$. What is the expression now?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$360=A_0 e^{5k}$$","$$y=360e^{5k}$$","$$360=A_0 e^{5t}$$"]},{"id":"af69facexpolog26a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$1000=A_0 e^{20k}$$"],"dependencies":["af69facexpolog26a-h3"],"title":"Solving for $$A_0$$","text":"Find the second equation. Substitute $$y=1000$$ and $$t=20$$ into the exponential growth model, $$y=A_0 e^{k t}$$. What is the expression now?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$1000=A_0 e^{20k}$$","$$y=1000e^{20k}$$","$$1000=A_0 e^{20t}$$"]},{"id":"af69facexpolog26a-h5","type":"hint","dependencies":["af69facexpolog26a-h4"],"title":"Solving for $$A_0$$","text":"If we let $$g=e^{5k}$$, then in the first equation, $$360=g A_0$$ and $$1000=g^4 A_0$$. We can use this information to solve for $$A_0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog26a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{\\\\frac{31000}{360}}$$"],"dependencies":["af69facexpolog26a-h5"],"title":"Solving for $$A_0$$","text":"Solve for g. We can do so by dividing the second equation by the first equation to obtain $$g^3$$. Then, cube root the value to obtain g. What is g?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog26a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$256$$"],"dependencies":["af69facexpolog26a-h6"],"title":"Solving for $$A_0$$","text":"Substitute $$g=\\\\sqrt[3]{\\\\frac{1000}{360}}$$ into the first equation to obtain $$A_0$$. Round to the nearest whole number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af69facexpolog26b","stepAnswer":["$$256e^{0.06811t}$$"],"problemType":"TextBox","stepTitle":"Rounding to six decimal places, write an exponential equation representing this situation.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$256e^{0.06811t}$$","hints":{"DefaultPathway":[{"id":"af69facexpolog26b-h1","type":"hint","dependencies":[],"title":"Solving for Rate of Growth, k","text":"We previously solve for g where $$g=e^{5k}$$. We can use this equation to obtain k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog26b-h2","type":"hint","dependencies":["af69facexpolog26b-h1"],"title":"Solving for Rate of Growth, k","text":"Take natural log on both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog26b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0.06811$$"],"dependencies":["af69facexpolog26b-h2"],"title":"Solving for Rate of Growth, k","text":"Divide by $$5$$ on both sides. What is k? Round to six decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog26b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$256e^{0.06811t}$$"],"dependencies":["af69facexpolog26b-h3"],"title":"Exponential Growth Model","text":"Now that we\'ve found $$A_0$$ and k, substitute them into the growth model, $$y=A_0 e^{k t}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af69facexpolog26c","stepAnswer":["$$10$$"],"problemType":"TextBox","stepTitle":"To the nearest minute, how long did it take the population to double?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10$$","hints":{"DefaultPathway":[{"id":"af69facexpolog26c-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2A_0$$"],"dependencies":[],"title":"Population Count","text":"The population doubles when $$y$$ is equal to? Express in terms of $$A_0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog26c-h2","type":"hint","dependencies":["af69facexpolog26c-h1"],"title":"Doubling Time","text":"We can substitute $$y=2A_0$$ and solve for $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog26c-h3","type":"hint","dependencies":["af69facexpolog26c-h2"],"title":"Doubling Time","text":"Divide by $$A_0$$ on both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog26c-h4","type":"hint","dependencies":["af69facexpolog26c-h3"],"title":"Doubling Time","text":"Take natural log on both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog26c-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["af69facexpolog26c-h4"],"title":"Doubling Time","text":"Divide by k on both sides. This should have isolated $$t$$. Substitute $$A_0$$ and k that was previously found to solve for $$t$$. What is $$t$$? Round to the nearest minute","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af69facexpolog27","title":"Using Newton\'s Law of Cooling","body":"For the following exercises, use this scenario: A pot of warm soup with an internal temperature of 100\xb0 Fahrenheit was taken off the stove to cool in a 69\xb0 F room. After fifteen minutes, the internal temperature of the soup was 95\xb0 F.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Exponential and Logarithmic Models","courseName":"OpenStax: College Algebra","steps":[{"id":"af69facexpolog27a","stepAnswer":["$$31e^{-0.011726 t}+69$$"],"problemType":"TextBox","stepTitle":"Use Newton\u2019s Law of Cooling to write a formula that models this situation.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$31e^{-0.011726 t}+69$$","hints":{"DefaultPathway":[{"id":"af69facexpolog27a-h1","type":"hint","dependencies":[],"title":"Newton\'s Law of Cooling","text":"The temperature of an object, T, in surrounding air with temperature $$T_s$$ will behave according to the formula $$T(t)=A e^{k t}+T_s$$ where\\\\n$$t$$ is time,\\\\nA is the difference between the initial temperature of the object and the surroundings,\\\\nk is a constant, the continuous rate of cooling of the object.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog27a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$69$$"],"dependencies":["af69facexpolog27a-h1"],"title":"Surrounding Temperature","text":"What is the surrounding temperature, $$T_s$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog27a-h3","type":"hint","dependencies":["af69facexpolog27a-h2"],"title":"Solving for A","text":"Find the difference between the initial temperature of the object and the surroundings.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog27a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$100$$"],"dependencies":["af69facexpolog27a-h3"],"title":"Solving for A","text":"What is the initial temperature of the soup?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog27a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$31$$"],"dependencies":["af69facexpolog27a-h4"],"title":"Solving for A","text":"What is the difference between the initial temperature of the soup and the surroundings?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog27a-h6","type":"hint","dependencies":["af69facexpolog27a-h5"],"title":"Continuous rate of Cooling, k","text":"We can substitute the A and $$T_s$$ value that was previously found to solve for k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog27a-h7","type":"hint","dependencies":["af69facexpolog27a-h6"],"title":"Continuous rate of Cooling, k","text":"Subtract $$T_s$$ from both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog27a-h8","type":"hint","dependencies":["af69facexpolog27a-h7"],"title":"Continuous rate of Cooling, k","text":"Divide by A.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog27a-h9","type":"hint","dependencies":["af69facexpolog27a-h8"],"title":"Continuous rate of Cooling, k","text":"Take natural log on both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog27a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\ln(\\\\frac{T\\\\left(t\\\\right)-T_s}{A})}{t}$$"],"dependencies":["af69facexpolog27a-h9"],"title":"Continuous rate of Cooling, k","text":"Divide by time, $$t$$ to isolate k. What is the expression k equals to? Express in terms of T(t), $$T_s$$, A, $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog27a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-0.011726$$"],"dependencies":["af69facexpolog27a-h10"],"title":"Continuous rate of Cooling, k","text":"It was given that after $$15$$ minutes, the internal temperature was 95\xb0 F. Substitute these values into the expression, along with A and $$T_s$$ that was previously found. What is k? Round to six decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog27a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$31e^{-0.011726 t}+69$$"],"dependencies":["af69facexpolog27a-h11"],"title":"Newton\'s Law of Cooling Model","text":"Substitute all the values found into $$T(t)=A e^{k t}+T_s$$. What is T(t)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af69facexpolog27b","stepAnswer":["$$88$$"],"problemType":"TextBox","stepTitle":"To the nearest minute, how long will it take the soup to cool to 80\xb0 F?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$88$$","hints":{"DefaultPathway":[{"id":"af69facexpolog27b-h1","type":"hint","dependencies":[],"title":"Solving for $$t$$","text":"Substitute $$T(t)=80$$ to solve for $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog27b-h2","type":"hint","dependencies":["af69facexpolog27b-h1"],"title":"Solving for $$t$$","text":"Subtract $$69$$ from both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog27b-h3","type":"hint","dependencies":["af69facexpolog27b-h2"],"title":"Solving for $$t$$","text":"Divide by $$31$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog27b-h4","type":"hint","dependencies":["af69facexpolog27b-h3"],"title":"Solving for $$t$$","text":"Take natural log on both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog27b-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$88$$"],"dependencies":["af69facexpolog27b-h4"],"title":"Solving for $$t$$","text":"Divide by continuous rate of cooling, $$k=-0.011726$$. What is $$t$$? Round to the nearest minute.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af69facexpolog27c","stepAnswer":["$$74$$"],"problemType":"TextBox","stepTitle":"To the nearest degree, what will the temperature be after $$2$$ and a half hours?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$74$$","hints":{"DefaultPathway":[{"id":"af69facexpolog27c-h1","type":"hint","dependencies":[],"title":"Solving for T(t)","text":"Substitute $$t=150$$ to solve for T(t).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af69facexpolog28","title":"Using Newton\'s Law of Cooling","body":"For the following exercises, use this scenario: A turkey is taken out of the oven with an internal temperature of 165\xb0F and is allowed to cool in a 75\xb0F room. After half an hour, the internal temperature of the turkey is 145\xb0F.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Exponential and Logarithmic Models","courseName":"OpenStax: College Algebra","steps":[{"id":"af69facexpolog28a","stepAnswer":["$$90e^{-0.008377 t}+75$$"],"problemType":"TextBox","stepTitle":"Write a formula that models this situation.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$90e^{-0.008377 t}+75$$","hints":{"DefaultPathway":[{"id":"af69facexpolog28a-h1","type":"hint","dependencies":[],"title":"Newton\'s Law of Cooling","text":"The temperature of an object, T, in surrounding air with temperature $$T_s$$ will behave according to the formula $$T(t)=A e^{k t}+T_s$$ where\\\\n$$t$$ is time,\\\\nA is the difference between the initial temperature of the object and the surroundings,\\\\nk is a constant, the continuous rate of cooling of the object.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog28a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$75$$"],"dependencies":["af69facexpolog28a-h1"],"title":"Surrounding Temperature","text":"What is the surrounding temperature, $$T_s$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog28a-h3","type":"hint","dependencies":["af69facexpolog28a-h2"],"title":"Solving for A","text":"Find the difference between the initial temperature of the object and the surroundings.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog28a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$165$$"],"dependencies":["af69facexpolog28a-h3"],"title":"Solving for A","text":"What is the initial temperature of the turkey?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog28a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$90$$"],"dependencies":["af69facexpolog28a-h4"],"title":"Solving for A","text":"What is the difference between the initial temperature of the turkey and the surroundings?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog28a-h6","type":"hint","dependencies":["af69facexpolog28a-h5"],"title":"Continuous rate of Cooling, k","text":"We can substitute the A and $$T_s$$ value that was previously found to solve for k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog28a-h7","type":"hint","dependencies":["af69facexpolog28a-h6"],"title":"Continuous rate of Cooling, k","text":"Subtract $$T_s$$ from both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog28a-h8","type":"hint","dependencies":["af69facexpolog28a-h7"],"title":"Continuous rate of Cooling, k","text":"Divide by A.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog28a-h9","type":"hint","dependencies":["af69facexpolog28a-h8"],"title":"Continuous rate of Cooling, k","text":"Take natural log on both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog28a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{\\\\ln(\\\\frac{T\\\\left(t\\\\right)-T_s}{A})}{t}$$"],"dependencies":["af69facexpolog28a-h9"],"title":"Continuous rate of Cooling, k","text":"Divide by time, $$t$$ to isolate k. What is the expression k equals to? Express in terms of T(t), $$T_s$$, A, $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog28a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-0.008377$$"],"dependencies":["af69facexpolog28a-h10"],"title":"Continuous rate of Cooling, k","text":"It was given that after $$30$$ minutes, the internal temperature was 145\xb0 F. Substitute these values into the expression, along with A and $$T_s$$ that was previously found. What is k? Round to six decimal places.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog28a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$90e^{-0.008377 t}+75$$"],"dependencies":["af69facexpolog28a-h11"],"title":"Newton\'s Law of Cooling Model","text":"Substitute all the values found into $$T(t)=A e^{k t}+T_s$$. What is T(t)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af69facexpolog28b","stepAnswer":["$$134$$"],"problemType":"TextBox","stepTitle":"To the nearest degree, what will the temperature be after $$50$$ minutes?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$134$$","hints":{"DefaultPathway":[{"id":"af69facexpolog28b-h1","type":"hint","dependencies":[],"title":"Solving for T(t)","text":"Substitute $$t=50$$ to solve for T(t).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af69facexpolog28c","stepAnswer":["$$113$$"],"problemType":"TextBox","stepTitle":"To the nearest minute, how long will it take the turkey to cool to 110\xb0 F?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$113$$","hints":{"DefaultPathway":[{"id":"af69facexpolog28c-h1","type":"hint","dependencies":[],"title":"Solving for $$t$$","text":"Substitute $$T(t)=110$$ to solve for $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog28c-h2","type":"hint","dependencies":["af69facexpolog28c-h1"],"title":"Solving for $$t$$","text":"Subtract $$75$$ from both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog28c-h3","type":"hint","dependencies":["af69facexpolog28c-h2"],"title":"Solving for $$t$$","text":"Divide by $$90$$ from both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog28c-h4","type":"hint","dependencies":["af69facexpolog28c-h3"],"title":"Solving for $$t$$","text":"Take natural log on both sides.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog28c-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$113$$"],"dependencies":["af69facexpolog28c-h4"],"title":"Solving for $$t$$","text":"Divide by continuous rate of cooling, $$k=-0.008377$$. What is $$t$$? Round to the nearest minute.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af69facexpolog29","title":"Logarithmic Application","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Exponential and Logarithmic Models","courseName":"OpenStax: College Algebra","steps":[{"id":"af69facexpolog29a","stepAnswer":["$$5.82$$"],"problemType":"TextBox","stepTitle":"Recall the formula for calculating the magnitude of an earthquake, $$M=\\\\frac{2}{3} \\\\ln(\\\\frac{S}{S_0})$$. One earthquake has magnitude $$3.9$$ on the MMS scale. If a second earthquake has $$750$$ times as much energy as the first, find the magnitude of the second quake. Round to the nearest hundredth.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5.82$$","hints":{"DefaultPathway":[{"id":"af69facexpolog29a-h1","type":"hint","dependencies":[],"title":"Understanding the Model","text":"We were given that the model is for measuring magnitude of earthquake is $$M=\\\\frac{2}{3} \\\\ln(\\\\frac{S}{S_0})$$. In the model, M is the magnitude of the earthquake, $$S_0$$ is the initial seismic moment that led to the occurence of earthquake. S is the seismic moment associated with an earthquake. If we denote the seismic moment of the first earthquake as $$S_1$$. In the context of the question, since the second earthquake has $$750$$ times as much energy as the first, we can interpret its seismic moment as $$750S_1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog29a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{3} \\\\ln(\\\\frac{750S_1}{S_0})$$"],"dependencies":["af69facexpolog29a-h1"],"title":"Logarithmic Model","text":"Given that the equation for the first earthquake is $$M=\\\\frac{2}{3} \\\\ln(\\\\frac{S_1}{S_0})=3.9$$. What is M equals to for the second earthquake? Express in terms of $$S_1$$ and $$S_0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog29a-h3","type":"hint","dependencies":["af69facexpolog29a-h2"],"title":"Product Rule for Logarithms","text":"The product rule for logarithms can be used to simplify a logarithm of a product by rewriting it as a sum of individual logarithms.\\\\n$$\\\\log_{b}\\\\left(M N\\\\right)=\\\\log_{b}\\\\left(M\\\\right)+\\\\log_{b}\\\\left(N\\\\right)for$$ $$b>0$$\\\\n\\\\nGiven the logarithm of a product, use the product rule of logarithms to write an equivalent sum of logarithms.\\\\n\\\\n1) Factor the argument completely, expressing each whole number factor as a product of primes.\\\\n2) Write the equivalent expression by summing the logarithms of each factor.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog29a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2}{3} \\\\ln(\\\\frac{S_1}{S_0})+\\\\frac{2}{3} \\\\ln(750)$$"],"dependencies":["af69facexpolog29a-h3"],"title":"Apply the Product Rule for Logarithms","text":"We can use the product rule to separate the factors, $$750$$ and $$\\\\frac{S_1}{S_0}$$, from within the logarithmic expression of the equation of second earthquake. Write the equivalent expression by summing the logarithms of each factor. What is the new expression?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog29a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\ln(M)+\\\\ln(N)$$"],"dependencies":["af69facexpolog29a-h4"],"title":"Apply the Product Rule for Logarithms","text":"What is the expression after applying product rule to $$\\\\ln(M N)$$? Express in terms of M and N.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog29a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$a \\\\ln(M)+a \\\\ln(N)$$"],"dependencies":["af69facexpolog29a-h5"],"title":"Apply the Product Rule for Logarithms","text":"What is the expression after applying product rule to $$a \\\\ln(M N)$$? Express in terms of a, M and N. (In our question, what values / variables does a, M and N refer to?)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog29a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1.92$$"],"dependencies":["af69facexpolog29a-h6"],"title":"Simplification","text":"What is $$\\\\frac{2}{3} \\\\ln(750)$$? Round to the nearest hundredth.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog29a-h8","type":"hint","dependencies":["af69facexpolog29a-h7"],"title":"Substitution","text":"We were previously given in the question that $$M=\\\\frac{2}{3} \\\\ln(\\\\frac{S_1}{S_0})=3.9$$. We can substitute this as well as the simplification that was previously done into $$M=\\\\frac{2}{3} \\\\ln(\\\\frac{S_1}{S_0})+\\\\frac{2}{3} \\\\ln(750)$$ to find the magnitude of the second earthquake.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af69facexpolog3","title":"Modeling Exponential Growth and Decay","body":"A bone fragment is found that contains 20% of its original carbon-14.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Exponential and Logarithmic Models","courseName":"OpenStax: College Algebra","steps":[{"id":"af69facexpolog3a","stepAnswer":["$$13301$$"],"problemType":"TextBox","stepTitle":"To the nearest year, how old is the bone?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$13301$$","hints":{"DefaultPathway":[{"id":"af69facexpolog3a-h1","type":"hint","dependencies":[],"title":"Identifying the Formula","text":"What is the continuous growth formula?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog3a-h2","type":"hint","dependencies":["af69facexpolog3a-h1"],"title":"Substitute Known Values","text":"Substitute $$0.2A_0$$ for f(t) and $$0.2$$ for $$r$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog3a-h3","type":"hint","dependencies":["af69facexpolog3a-h2"],"title":"Solve for $$t$$","text":"Rearrange the equation using algebraic methods to solve for $$t$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af69facexpolog30","title":"Using the Logistic-Growth Model","body":"For the following exercises, use this scenario: The equation $$N(t)=\\\\frac{500}{1+49e^{-0.7 t}}$$ models the number of people in a town who have heard a rumor after $$t$$ days.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Exponential and Logarithmic Models","courseName":"OpenStax: College Algebra","steps":[{"id":"af69facexpolog30a","stepAnswer":["$$10$$"],"problemType":"TextBox","stepTitle":"How many people started the rumor?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10$$","hints":{"DefaultPathway":[{"id":"af69facexpolog30a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":[],"title":"Initial Value","text":"What day did the rumor start?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog30a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$10$$"],"dependencies":["af69facexpolog30a-h1"],"title":"Substitution","text":"Substituting $$t=0$$, what is N(0)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af69facexpolog30b","stepAnswer":["$$71$$"],"problemType":"TextBox","stepTitle":"To the nearest whole number, how many people will have heard the rumor after $$3$$ days?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$71$$","hints":{"DefaultPathway":[{"id":"af69facexpolog30b-h1","type":"hint","dependencies":[],"title":"Substitution","text":"Substituting $$t=3$$, what is N(3)?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"af69facexpolog30c","stepAnswer":["$$500$$"],"problemType":"TextBox","stepTitle":"As $$t$$ increase without bound, what value does N(t) approach?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$500$$","hints":{"DefaultPathway":[{"id":"af69facexpolog30c-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":[],"title":"Limits","text":"As $$t$$ approaches $$\\\\infty$$, what does the expression $$e^{-0.7 t}$$ approach?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog30c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["af69facexpolog30c-h1"],"title":"Simplification","text":"Given that $$e^{-0.7 t}$$ approaches $$0$$ as $$t$$ tends to $$\\\\infty$$, what does $$49e^{-0.7 t}$$ approach?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog30c-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$500$$"],"dependencies":["af69facexpolog30c-h2"],"title":"Population Size","text":"After the previous simplification, what does the N(t) approach as $$t$$ tends to $$\\\\infty$$? We observe that as $$t$$ tends to $$\\\\infty$$, everyone in the town would hear of the rumor, thus N(t) which models the number of people in town that heard the rumor would tells us how many people there are in the town.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af69facexpolog4","title":"Modeling Exponential Growth and Decay","body":"Cesium-137 has a half-life of about $$30$$ years.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Exponential and Logarithmic Models","courseName":"OpenStax: College Algebra","steps":[{"id":"af69facexpolog4a","stepAnswer":["More"],"problemType":"MultipleChoice","stepTitle":"If we begin with $$200$$ mg of cesium-137, will it take more or less than $$230$$ years until only $$1$$ milligram 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Growth has slowed to a doubling time of approximately three years.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Exponential and Logarithmic Models","courseName":"OpenStax: College Algebra","steps":[{"id":"af69facexpolog6a","stepAnswer":["$$A_0 e^{\\\\frac{\\\\ln(2) t}{3}}$$"],"problemType":"MultipleChoice","stepTitle":"Find the new function that takes that longer doubling time into account.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$A_0 e^{\\\\frac{\\\\ln(2) t}{3}}$$","choices":["$$A_0 e^{\\\\frac{\\\\ln(4) t}{3}}$$","$$A_0 e^{\\\\frac{\\\\ln(3) t}{3}}$$","$$A_0 e^{\\\\frac{\\\\ln(2) t}{3}}$$"],"hints":{"DefaultPathway":[{"id":"af69facexpolog6a-h1","type":"hint","dependencies":[],"title":"Identifying the Formula","text":"What is the doubline time formula?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog6a-s1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":[],"title":"Substituting Known Values","text":"For this problem, what value will we be substituting for $$t$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog6a-h2","type":"hint","dependencies":["af69facexpolog6a-h1"],"title":"Solve for k","text":"Multiply by k and divide by $$2$$ to isolate k","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog6a-h3","type":"hint","dependencies":["af69facexpolog6a-h2"],"title":"Final Function","text":"Substitute k into the continuous growth formula","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af69facexpolog7","title":"Expressing an Exponential Model in Base e","body":"Changing to base e","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Exponential and Logarithmic Models","courseName":"OpenStax: College Algebra","steps":[{"id":"af69facexpolog7a","stepAnswer":["$$2.5e^{\\\\ln(3.1) x}$$"],"problemType":"TextBox","stepTitle":"Change the function $$y={\\\\operatorname{2.5}\\\\left(3.1\\\\right)}^x$$ so that this same functino is written in the form $$y=A_0 e^{kx}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2.5e^{\\\\ln(3.1) x}$$","hints":{"DefaultPathway":[{"id":"af69facexpolog7a-h1","type":"hint","dependencies":[],"title":"Inserting Exponential and Inverse","text":"Rewrite $$y={ab}^x$$ as $$y=a e^{\\\\ln(b^x)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog7a-h2","type":"hint","dependencies":["af69facexpolog7a-h1"],"title":"Law of Logs","text":"Use the power rule of logarithms to rewrite $$y$$ as $$y=a e^{x \\\\ln(b)}=a e^{\\\\ln(b) x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af69facexpolog8","title":"Expressing an Exponential Model in Base e","body":"Changing to base e","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Exponential and Logarithmic Models","courseName":"OpenStax: College Algebra","steps":[{"id":"af69facexpolog8a","stepAnswer":["$$3e^{\\\\ln(0.5) x}$$"],"problemType":"TextBox","stepTitle":"Change the function $$y={3\\\\left(0.5\\\\right)}^x$$ to one having e as the base","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3e^{\\\\ln(0.5) x}$$","hints":{"DefaultPathway":[{"id":"af69facexpolog8a-h1","type":"hint","dependencies":[],"title":"Inserting Exponential and Inverse","text":"Rewrite $$y={ab}^x$$ as $$y=a e^{\\\\ln(b^x)}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog8a-h2","type":"hint","dependencies":["af69facexpolog8a-h1"],"title":"Law of Logs","text":"Use the power rule of logarithms to rewrite $$y$$ as $$y=a e^{x \\\\ln(b)}=a e^{\\\\ln(b) x}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af69facexpolog9","title":"Use Newton\'s Law of Cooling","body":"A cheesecake is taken out of the oven with an ideal internal temperature of 165\xb0F, and is placed into a 35\xb0F refrigerator. After $$10$$ minutes, the cheesecake has cooled to 150\xb0F.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"6.7 Exponential and Logarithmic Models","courseName":"OpenStax: College Algebra","steps":[{"id":"af69facexpolog9a","stepAnswer":["$$107$$"],"problemType":"TextBox","stepTitle":"If we must wait until the cheesecake has cooled to 70\xb0F before we eat it, how long will we have to wait? 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Plug in those values to solve for K.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog9a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-0.0123$$"],"dependencies":["af69facexpolog9a-h5"],"title":"Solve for K","text":"What is K? (round to the nearest ten-thousandths place)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog9a-h7","type":"hint","dependencies":["af69facexpolog9a-h6"],"title":"Final Formula","text":"After solve for K and A, plug in those values into the original Newton\'s cooling formula.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog9a-h8","type":"hint","dependencies":["af69facexpolog9a-h7"],"title":"Solve for T","text":"To find the time it takes for the cake to cool to $$70$$ degrees, plug in $$70$$ for T(t) and solve for $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"af69facexpolog9a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$106.68$$"],"dependencies":["af69facexpolog9a-h8"],"title":"Solve for T","text":"What is T? (round to nearest hundreths place)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"af9dcc3physical1","title":"Calculating Mass from Linear Density","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.5 Physical Applications","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"af9dcc3physical1a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"Consider a thin rod oriented on the x-axis over the interval $$[\\\\frac{\\\\pi}{2},pi]$$. If the density of the rod is given by $$p(x)=sinx$$, what is the mass of the rod?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"af9dcc3physical1a-h1","type":"hint","dependencies":[],"title":"Mass-Density Formula","text":"Given a thin rod oriented along the x-axis over the interval [a,b], let p(x) denote a linear density function giving the density of the rod at a point $$x$$ in the interval. 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Then the mass of the rod is given by $$m=\\\\int_{a}^{b} p(x) \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical10a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3x+2$$"],"dependencies":["af9dcc3physical10a-h1"],"title":"Set up the integral","text":"What is the density of the rod?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical10a-h3","type":"hint","dependencies":["af9dcc3physical10a-h2"],"title":"Set up the integral","text":"Think about integral over the length of antenna, and $$x$$ is starting from $$0$$. so Interval would be [0,3]","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical10a-h4","type":"hint","dependencies":["af9dcc3physical10a-h3"],"title":"Set up the integral","text":"$$M=\\\\int_{0}^{3} 3x+2 \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical10a-h5","type":"hint","dependencies":["af9dcc3physical10a-h4"],"title":"Find the integral","text":"(3/2)*(x**2)+ $$2x$$ as the limits going from $$x=0$$ to $$x=3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical10a-h6","type":"hint","dependencies":["af9dcc3physical10a-h5"],"title":"Evaluate","text":"Solve $$\\\\frac{3}{2} 3^2+2\\\\times3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"af9dcc3physical11","title":"Find the mass of the one-dimensional object.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.5 Physical Applications","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"af9dcc3physical11a","stepAnswer":["$$\\\\frac{20}{3}$$"],"problemType":"TextBox","stepTitle":"A wire that is $$2$$ ft long (starting at $$x=0)$$ and has a density function of $$\u03c1(x)=x^2+2x$$ $$\\\\frac{lb}{ft}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{20}{3}$$","hints":{"DefaultPathway":[{"id":"af9dcc3physical11a-h1","type":"hint","dependencies":[],"title":"Mass-Density Formula","text":"Given a thin rod oriented along the x-axis over the interval [a,b], let p(x) denote a linear density function giving the density of the rod at a point $$x$$ in the interval. Then the mass of the rod is given by $$m=\\\\int_{a}^{b} p(x) \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$x^2+2x$$"],"dependencies":["af9dcc3physical11a-h1"],"title":"Set up the integral","text":"What is the density of the rod?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical11a-h3","type":"hint","dependencies":["af9dcc3physical11a-h2"],"title":"Set up the integral","text":"Think about integral over the length of antenna and $$x$$ is startinig from $$0$$. so Interval would be [0,2]","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical11a-h4","type":"hint","dependencies":["af9dcc3physical11a-h3"],"title":"Set up the integral","text":"$$M=$$ $$\\\\int_{0}^{2} x^2+2x \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical11a-h5","type":"hint","dependencies":["af9dcc3physical11a-h4"],"title":"Simplify","text":"We can break the integral into two integrals here : $$\\\\int_{0}^{2} x^2+2x \\\\,dx=\\\\int_{0}^{2} x^2 \\\\,dx+\\\\int_{0}^{2} 2x \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical11a-h6","type":"hint","dependencies":["af9dcc3physical11a-h5"],"title":"Find the integral","text":"$$\\\\frac{x^3}{3}$$ as the limits going from $$x=0$$ to $$x=2$$ plus $$x^2$$ as the limit going from $$x=0$$ to $$x=2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical11a-h7","type":"hint","dependencies":["af9dcc3physical11a-h6"],"title":"Evaluate","text":"Solve $$\\\\frac{2^3}{3}+2^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"af9dcc3physical12","title":"Find the mass of the two-dimensional object that is centered at the origin.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.5 Physical Applications","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"af9dcc3physical12a","stepAnswer":["$$\\\\frac{332\\\\pi}{15}$$"],"problemType":"TextBox","stepTitle":"An oversized hockey puck of radius $$2$$ in. with density function $$\u03c1(x)=x^3-2x+5$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{332\\\\pi}{15}$$","hints":{"DefaultPathway":[{"id":"af9dcc3physical12a-h1","type":"hint","dependencies":[],"title":"Mass-Density Formula","text":"Let p(x) be an integrable function representing the radial density of a disk of radius $$r$$. Then the mass of the disk is given by $$m=\\\\int_{0}^{r} 2\\\\pi p\\\\left(x\\\\right) \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical12a-h2","type":"hint","dependencies":["af9dcc3physical12a-h1"],"title":"Set up the integral","text":"$$m=\\\\int_{0}^{2} 2\\\\pi x \\\\left(x^3-2x+5\\\\right) \\\\,dx=m=\\\\int_{0}^{2} 2\\\\pi \\\\left(x^4-2\\\\left(x^2\\\\right)+5x\\\\right) \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical12a-h3","type":"hint","dependencies":["af9dcc3physical12a-h2"],"title":"Find the integral","text":"$$2\\\\pi$$ times $$\\\\frac{x^5}{5}-\\\\frac{2x^3}{3}+\\\\frac{5x^2}{2}$$ as the limits going from $$x=0$$ to $$x=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical12a-h4","type":"hint","dependencies":["af9dcc3physical12a-h3"],"title":"Evaluate","text":"$$2\\\\pi \\\\left(\\\\frac{2^5}{5}-\\\\frac{2x^3}{3}+\\\\frac{5\\\\times2^2}{2}\\\\right)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"af9dcc3physical13","title":"Find the mass of the two-dimensional object that is centered at the origin.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.5 Physical Applications","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"af9dcc3physical13a","stepAnswer":["$$100\\\\pi$$"],"problemType":"TextBox","stepTitle":"A plate of radius $$10$$ in. with density function $$\u03c1(x)=1+cos\\\\left(\\\\pi x\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$100\\\\pi$$","hints":{"DefaultPathway":[{"id":"af9dcc3physical13a-h1","type":"hint","dependencies":[],"title":"Mass-Density Formula","text":"Let p(x) be an integrable function representing the radial density of a disk of radius $$r$$. Then the mass of the disk is given by $$m=\\\\int_{0}^{r} 2\\\\pi p\\\\left(x\\\\right) \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical13a-h2","type":"hint","dependencies":["af9dcc3physical13a-h1"],"title":"Set up the integral","text":"m=/int{2*pi*x*(1+cos(pi*x),0,10,x}=2*pi*/int{x,0,10,x}+2*pi*/int{cos(pi*x),0,10,x}","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical13a-h3","type":"hint","dependencies":["af9dcc3physical13a-h2"],"title":"Find the integral","text":"We can evaluate the integral separately: $$2*pi*\\\\int_{0}^{10} x \\\\,dx$$ and $$2*pi*\\\\int_{0}^{10} cos\\\\left(\\\\pi x\\\\right) \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical13a-h4","type":"hint","dependencies":["af9dcc3physical13a-h3"],"title":"Evaluate first integral","text":"$$2*pi*\\\\int_{0}^{10} x \\\\,dx$$ is $$\\\\frac{2\\\\pi x^2}{2}$$ with the limit from $$x=0$$ to $$x=10$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical13a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$100\\\\pi$$"],"dependencies":["af9dcc3physical13a-h4"],"title":"Solve first integral","text":"What is $$\\\\frac{2\\\\pi {10}^2}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical13a-h6","type":"hint","dependencies":["af9dcc3physical13a-h5"],"title":"Evaluate second integral","text":"$$2*pi*\\\\int_{0}^{10} cos\\\\left(\\\\pi x\\\\right) \\\\,dx$$ is $$-\\\\left(\\\\frac{x}{\\\\pi}\\\\right) sin\\\\left(\\\\pi x\\\\right)$$ with the limit from $$x=0$$ to $$x=10+\\\\frac{1}{\\\\pi} \\\\left(-\\\\frac{1}{\\\\pi} cos\\\\left(\\\\pi x\\\\right)\\\\right)$$ with the limit from $$x=0$$ to $$x=10$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical13a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["af9dcc3physical13a-h6"],"title":"Solve second integral","text":"What is $$\\\\left(-\\\\frac{10}{\\\\pi} sin\\\\left(10\\\\pi\\\\right)-0\\\\right)+\\\\frac{1}{\\\\pi} \\\\left(-\\\\frac{1}{\\\\pi} cos\\\\left(10\\\\pi\\\\right)-\\\\left(-\\\\frac{1}{\\\\pi} cos\\\\left(0\\\\right)\\\\right)\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical13a-h8","type":"hint","dependencies":["af9dcc3physical13a-h7"],"title":"Add the first and second integral","text":"Solve $$100\\\\pi+0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"af9dcc3physical14","title":"Find the work","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.5 Physical Applications","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"af9dcc3physical14a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"A spring has a natural length of $$10$$ cm. It takes $$2$$ J to stretch the spring to $$15$$ cm. How much work would it take to stretch the spring from $$15$$ cm to $$20$$ cm?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"af9dcc3physical14a-h1","type":"hint","dependencies":[],"title":"cm to $$m$$","text":"We need to covert all values with cm into $$m$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical14a-h2","type":"hint","dependencies":["af9dcc3physical14a-h1"],"title":"Find the constant spring k","text":"From $$W=\\\\int_{a}^{b} F(x) \\\\,dx$$, where F(x) is $$k x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical14a-h3","type":"hint","dependencies":["af9dcc3physical14a-h2"],"title":"Find the interval","text":"Because natural length is $$0.1m$$, when $$x=0$$, the interval will end at $$x=$$ (distance - $$0.1)m$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical14a-h4","type":"hint","dependencies":["af9dcc3physical14a-h3"],"title":"Find k: Plug information into formula","text":"$$2=\\\\int_{0}^{0.05} k x \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical14a-h5","type":"hint","dependencies":["af9dcc3physical14a-h4"],"title":"Evaluate integral","text":"$$2=k \\\\frac{x^2}{2}$$ with the limit from $$x=0$$ to $$x=0.05$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical14a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1600$$"],"dependencies":["af9dcc3physical14a-h5"],"title":"Fint the constant spring k","text":"What is k when $$2=$$ $$k \\\\frac{{0.05}^2}{2}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical14a-h7","type":"hint","dependencies":["af9dcc3physical14a-h6"],"title":"Define F(x)","text":"$$F(x)=1600x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical14a-h8","type":"hint","dependencies":["af9dcc3physical14a-h7"],"title":"Find work to streach the string from 15cm to 20xm","text":"$$W=\\\\int_{0.05}^{0.1} 1600x \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical14a-h9","type":"hint","dependencies":["af9dcc3physical14a-h8"],"title":"Evaluate integral","text":"Solve $$\\\\frac{1600x^2}{2}$$ with the limit from $$x=0.05$$ to $$x=$$ $$0.1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical14a-h10","type":"hint","dependencies":["af9dcc3physical14a-h9"],"title":"Solve","text":"1600*((0.1**2)/2 - $$\\\\frac{{0.05}^2}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"af9dcc3physical15","title":"Find the natural length of spring","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.5 Physical Applications","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"af9dcc3physical15a","stepAnswer":["$$36$$"],"problemType":"TextBox","stepTitle":"A force of $$F=20x-x^3$$ N stretches a nonlinear spring by $$x$$ meters. What work is required to stretch the spring from $$x=0$$ to $$x=2$$ $$m$$? Answer in $$___$$ J.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$36$$","hints":{"DefaultPathway":[{"id":"af9dcc3physical15a-h1","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$20x-x^3$$"],"dependencies":[],"title":"Define F(x)","text":"What is the force F(x) applied to stretch the spring?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical15a-h2","type":"hint","dependencies":["af9dcc3physical15a-h1"],"title":"Set up the integral","text":"$$W=\\\\int_{a}^{b} F(x) \\\\,dx=\\\\int_{0}^{2} 20x-x^3 \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical15a-h3","type":"hint","dependencies":["af9dcc3physical15a-h2"],"title":"Find the integral","text":"$$\\\\frac{20x^2}{2}-\\\\frac{x^4}{4}$$ as the limits going from $$x=0$$ to $$x=2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical15a-h4","type":"hint","dependencies":["af9dcc3physical15a-h3"],"title":"Evaluate","text":"Solve (20*(2**2)/2-(2**4)/2.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"af9dcc3physical2","title":"Calculating Mass from Radial Density","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.5 Physical Applications","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"af9dcc3physical2a","stepAnswer":["$$\\\\frac{128\\\\pi}{5}$$"],"problemType":"TextBox","stepTitle":"Let $$p(x)=\\\\sqrt{x}$$ represent the radial density of a disk. Calculate the mass of a disk of radius $$4$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{128\\\\pi}{5}$$","hints":{"DefaultPathway":[{"id":"af9dcc3physical2a-h1","type":"hint","dependencies":[],"title":"Mass-Density Formula","text":"Let p(x) be an integrable function representing the radial density of a disk of radius $$r$$. Then the mass of the disk is given by $$m=\\\\int_{0}^{r} 2\\\\pi p\\\\left(x\\\\right) \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical2a-h2","type":"hint","dependencies":["af9dcc3physical2a-h1"],"title":"Set up the integral","text":"$$m=\\\\int_{0}^{4} 2\\\\pi x \\\\sqrt{x} \\\\,dx=m=\\\\int_{0}^{4} 2\\\\pi x^{\\\\frac{3}{2}} \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical2a-h3","type":"hint","dependencies":["af9dcc3physical2a-h2"],"title":"Find the integral","text":"$$2\\\\pi \\\\frac{2}{5} x^{\\\\frac{5}{2}}$$ as the limits going from $$x=0$$ to $$x=4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical2a-h4","type":"hint","dependencies":["af9dcc3physical2a-h3"],"title":"Evaluate","text":"$$\\\\frac{32\\\\times4 \\\\pi}{5}=\\\\frac{128\\\\pi}{5}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"af9dcc3physical3","title":"The Work Required to Stretch or Compress a Spring","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.5 Physical Applications","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"af9dcc3physical3a","stepAnswer":["$$6.25$$"],"problemType":"TextBox","stepTitle":"Suppose it takes a force of $$10$$ N (in the negative direction) to compress a spring $$0.2$$ $$m$$ from the equilibrium position. How much work is done to stretch the spring $$0.5$$ $$m$$ from the equilibrium position?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6.25$$","hints":{"DefaultPathway":[{"id":"af9dcc3physical3a-h1","type":"hint","dependencies":[],"title":"Find the spring constant","text":"Since $$x=-0.2$$ and $$F(x)=-10$$ are given, we have $$-10=k \\\\left(-0.2\\\\right)$$ then $$k=50$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical3a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$50x$$"],"dependencies":["af9dcc3physical3a-h1"],"title":"Define F(x)","text":"What is the force F(x) applied to stretch the spring?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical3a-h3","type":"hint","dependencies":["af9dcc3physical3a-h2"],"title":"Set up the integral","text":"$$W=\\\\int_{a}^{b} F(x) \\\\,dx=\\\\int_{0}^{0.5} 50x \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical3a-h4","type":"hint","dependencies":["af9dcc3physical3a-h3"],"title":"Find the integral","text":"$$25x^2$$ as the limits going from $$x=0$$ to $$x=0.5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"af9dcc3physical3a-h5","type":"hint","dependencies":["af9dcc3physical3a-h4"],"title":"Evaluate","text":"$$25{0.5}^2=6.25$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"af9dcc3physical4","title":"Calculating Mass from Linear Density","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6.5 Physical Applications","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"af9dcc3physical4a","stepAnswer":["$$\\\\frac{70}{3}$$"],"problemType":"TextBox","stepTitle":"Consider a thin rod oriented on the x-axis over the interval [1,3]. If the density of the rod is given by $$\u03c1(x)=2\\\\left(x^2\\\\right)+3$$, what is the mass of the rod?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{70}{3}$$","hints":{"DefaultPathway":[{"id":"af9dcc3physical4a-h1","type":"hint","dependencies":[],"title":"Mass-Density Formula","text":"Given a thin rod oriented along the x-axis over the interval [a,b], let p(x) denote a linear density function giving the density of the rod at a point $$x$$ in the interval. 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Remember to include the positive and negative case.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afa0cfequadraticeq3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["afa0cfequadraticeq3a-h2"],"title":"Square Root Property","text":"What is the greatest square in $$\\\\sqrt{50}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afa0cfequadraticeq30","title":"Solving Equations with the Square Root Property","body":"Solve the equation for the value of the variable.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afa0cfequadraticeq30a","stepAnswer":["$$3\\\\sqrt{2}$$ or $$-3\\\\sqrt{2}$$"],"problemType":"MultipleChoice","stepTitle":"$$a^2-18=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$3\\\\sqrt{2}$$ or $$-3\\\\sqrt{2}$$","choices":["$$5\\\\sqrt{2}$$ or $$-5\\\\sqrt{2}$$","$$3\\\\sqrt{2}$$ or $$-3\\\\sqrt{2}$$","$$2\\\\sqrt{2}$$ or $$-2\\\\sqrt{2}$$"],"hints":{"DefaultPathway":[{"id":"afa0cfequadraticeq30a-h1","type":"hint","dependencies":[],"title":"Square Root Property","text":"If $$x^2=k$$, and $$k \\\\geq 0$$, then $$x=\\\\sqrt{k}$$ or $$x=-\\\\sqrt{k}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afa0cfequadraticeq4","title":"Square Root Property","body":"Use the Square Root Property to solve for the variable.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afa0cfequadraticeq4a","stepAnswer":["$$3\\\\sqrt{3}$$"],"problemType":"TextBox","stepTitle":"$$x^2$$ $$-27$$ $$=$$ $$0$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3\\\\sqrt{3}$$","hints":{"DefaultPathway":[{"id":"afa0cfequadraticeq4a-h1","type":"hint","dependencies":[],"title":"Square Root Property","text":"Solve for $$x^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afa0cfequadraticeq4a-h2","type":"hint","dependencies":["afa0cfequadraticeq4a-h1"],"title":"Square Root Property","text":"Square Root both sides of the equation. Remember to include the positive and negative case.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afa0cfequadraticeq4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["afa0cfequadraticeq4a-h2"],"title":"Square Root Property","text":"What is the greatest square in $$\\\\sqrt{27}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afa0cfequadraticeq5","title":"Square Root Property","body":"Use the Square Root Property to solve for the variable.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Elementary 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$$-7$$"],"dependencies":["afa0cfequadraticeq5a-h2"],"title":"Quadratics","text":"What is result from square rooting both sides?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$8$$ and $$-8$$","$$7$$ and $$-7$$","$$6$$ and $$-6$$"]}]}}]},{"id":"afa0cfequadraticeq6","title":"Square Root Property","body":"Use the Square Root Property to solve for the variable.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Elementary 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$$-6$$"],"dependencies":["afa0cfequadraticeq6a-h2"],"title":"Quadratics","text":"What is result from square rooting both sides?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$8$$ and $$-8$$","$$7$$ and $$-7$$","$$6$$ and $$-6$$"]}]}}]},{"id":"afa0cfequadraticeq7","title":"Square Root Property","body":"Use the Square Root Property to solve for the variable.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afa0cfequadraticeq7a","stepAnswer":["No real solution"],"problemType":"MultipleChoice","stepTitle":"$$c^2+12=0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$8$$ and $$-8$$","$$7$$ and $$-7$$","$$6$$ and $$-6$$","No real solution"],"hints":{"DefaultPathway":[{"id":"afa0cfequadraticeq7a-h1","type":"hint","dependencies":[],"title":"Quadratics","text":"Solve for $$c^2$$ by subtracting $$12$$ from both sides of the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afa0cfequadraticeq7a-h2","type":"hint","dependencies":["afa0cfequadraticeq7a-h1"],"title":"Quadratics","text":"Notice that solving for c would result in a square root of a negative number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afa0cfequadraticeq8","title":"Square Root Property","body":"Use the Square Root Property to solve for the variable.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afa0cfequadraticeq8a","stepAnswer":["No real solution"],"problemType":"MultipleChoice","stepTitle":"$$d^2$$ + $$81$$ $$=$$ $$0$$","stepBody":"","answerType":"string","variabilization":{},"choices":["$$8$$ and $$-8$$","$$7$$ and $$-7$$","$$6$$ and $$-6$$","No real solution"],"hints":{"DefaultPathway":[{"id":"afa0cfequadraticeq8a-h1","type":"hint","dependencies":[],"title":"Quadratics","text":"Solve for $$d^2$$ by subtracting $$81$$ from both sides of the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afa0cfequadraticeq8a-h2","type":"hint","dependencies":["afa0cfequadraticeq8a-h1"],"title":"Square Root Property","text":"Notice that solving for $$d$$ would result in a square root of a negative number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afa0cfequadraticeq9","title":"Square Root Property","body":"Use the Square Root Property to solve for the variable.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"10.1 Solve Quadratic Equations Using the Square Root Property","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afa0cfequadraticeq9a","stepAnswer":["$$-2\\\\sqrt{10}$$ and $$2\\\\sqrt{10}$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{1}{2} x^2+4=24$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$-2\\\\sqrt{10}$$ and $$2\\\\sqrt{10}$$","choices":["$$-2\\\\sqrt{7}$$ and $$2\\\\sqrt{7}$$","$$-2\\\\sqrt{10}$$ and $$2\\\\sqrt{10}$$","$$-\\\\sqrt{10}$$ and $$\\\\sqrt{10}$$"],"hints":{"DefaultPathway":[{"id":"afa0cfequadraticeq9a-h1","type":"hint","dependencies":[],"title":"Quadratics","text":"Solve for $$x^2$$ by subtracting $$4$$ from both sides and multiplying by $$2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afa0cfequadraticeq9a-h2","type":"hint","dependencies":["afa0cfequadraticeq9a-h1"],"title":"Square Root Property","text":"Square root both sides of the equation. Remember the positive and negative case.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afaf721inequalities1","title":"Interval Notation Practice","body":"Graph the following inequalities on the number line, and choose the correct interval domain as the answer.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afaf721inequalities1a","stepAnswer":["$$[-3,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$x \\\\geq -3$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[-3,\\\\infty)$$","choices":["$$[-3,\\\\infty)$$","$$[3,\\\\infty)$$","$$(-\\\\infty,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities1a-h1","type":"hint","dependencies":[],"title":"Interpreting the Problem","text":"The first step is to interpret the problem. $$ \\\\geq $$ means \\"greater or equal than,\\" so we know the valid domain is all numbers greater than or equal to $$-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities1a-h2","type":"hint","dependencies":["afaf721inequalities1a-h1"],"title":"Graphing the Solution","text":"On a number line, shade to the right of $$3$$, and put a bracket at three. The answer should look like the image attached to this hint.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities1a-h3","type":"hint","dependencies":["afaf721inequalities1a-h2"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, brackets are used when the left or right sides of the interval are included in the solution. For example, the expression $$\\"x \\\\leq -1\\"$$ includes all numbers less than or equal to $$-1$$. Since $$-1$$ is included in the solution, the interval notation is $$(-\\\\infty,-1]$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities1a-h4","type":"hint","dependencies":["afaf721inequalities1a-h3"],"title":"Example of Interval Notation","text":"The attached image shows the inequality, number line, and interval notation of $$x \\\\leq 1$$.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"afaf721inequalities1b","stepAnswer":["(-inf,2.5)"],"problemType":"MultipleChoice","stepTitle":"$$x<2.5$$","stepBody":"","answerType":"string","variabilization":{},"choices":["(-inf,2.5]","[-inf,2.5)","(-inf,2.5)"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities1b-h1","type":"hint","dependencies":[],"title":"Interpreting the Problem","text":"The first step is to interpret the problem. < means \\"less than,\\" so we know the valid domain is all numbers less than $$2.5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities1b-h2","type":"hint","dependencies":["afaf721inequalities1b-h1"],"title":"Graphing the Solution","text":"On a number line, shade to the left of $$2.5$$, and put a parentheses at $$2.5$$. The answer should look like the image attached to this hint.\\\\n##figure3.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities1b-h3","type":"hint","dependencies":["afaf721inequalities1b-h2"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, parentheses show that the endpoint of the inequality is not included. For example, the expression \\"x>3\\" includes all numbers greater than $$3$$. Since $$3$$ is not included in the solution, the interval notation is $$(3,\\\\infty)$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities1b-h4","type":"hint","dependencies":["afaf721inequalities1b-h3"],"title":"Example of Interval Notation","text":"The attached image shows the inequality, number line, and interval notation of $$x>3$$.\\\\n##figure4.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"afaf721inequalities1c","stepAnswer":["$$(-\\\\infty,\\\\frac{-3}{5}]$$"],"problemType":"MultipleChoice","stepTitle":"$$x \\\\leq \\\\frac{-3}{5}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\frac{-3}{5}]$$","choices":["$$(-\\\\infty,\\\\frac{-3}{5}]$$","$$[-\\\\infty,\\\\frac{-3}{5}]$$","$$(-\\\\infty,\\\\frac{3}{5})$$"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities1c-h1","type":"hint","dependencies":[],"title":"Interpreting the Problem","text":"The first step is to interpret the problem. $$ \\\\leq $$ means \\"less than or equal to,\\" so we know the valid domain is all numbers less than or equal to $$\\\\frac{-3}{5}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities1c-h2","type":"hint","dependencies":["afaf721inequalities1c-h1"],"title":"Graphing the Solution","text":"On a number line, shade to the left of $$\\\\frac{-3}{5}$$, and put a bracket at $$\\\\frac{-3}{5}$$. The answer should look like the image attached to this hint.\\\\n##figure5.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities1c-h3","type":"hint","dependencies":["afaf721inequalities1c-h2"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, brackets are used when the left or right sides of the interval are included in the solution. For example, the expression $$\\"x \\\\leq -1\\"$$ includes all numbers less than or equal to $$-1$$. Since $$-1$$ is included in the solution, the interval notation is $$(-\\\\infty,-1]$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities1c-h4","type":"hint","dependencies":["afaf721inequalities1c-h3"],"title":"Example of Interval Notation","text":"The attached image shows the inequality, number line, and interval notation of $$x \\\\leq 1$$.\\\\n##figure6.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afaf721inequalities10","title":"Solve the inequality, graph the solution on the number line, and write the solution in interval notation.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afaf721inequalities10a","stepAnswer":["$$(-4,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$-8q<32$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-4,\\\\infty)$$","choices":["$$(-4,\\\\infty)$$","$$(-\\\\infty,-4)$$","$$(-\\\\infty,4)$$"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities10a-h1","type":"hint","dependencies":[],"title":"Isolating q","text":"First, to isolate q, we can divide both sides of the inequality by $$-8$$. Since $$-8<0$$, we need to flip the inequality sign. Hence, we get $$q>-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities10a-h2","type":"hint","dependencies":["afaf721inequalities10a-h1"],"title":"Graphing on the Number Line","text":"Next, graph the solution on the number line. To do this, shade to the right of $$-4$$ and put a parenthesis at $$-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities10a-h3","type":"hint","dependencies":["afaf721inequalities10a-h2"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, parentheses show that the endpoint of the inequality is not included. For example, the expression \\"x>3\\" includes all numbers greater than $$3$$. Since $$3$$ is not included in the solution, the interval notation is $$(3,\\\\infty)$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities10a-h4","type":"hint","dependencies":["afaf721inequalities10a-h3"],"title":"Example of Interval Notation","text":"The attached image shows the inequality, number line, and interval notation of $$x>3$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afaf721inequalities11","title":"Solve the inequality, graph the solution on the number line, and write the solution in interval notation.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afaf721inequalities11a","stepAnswer":["$$[10,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$-7r \\\\leq -70$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[10,\\\\infty)$$","choices":["$$(-\\\\infty,10]$$","$$(-\\\\infty,-10]$$","$$[10,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities11a-h1","type":"hint","dependencies":[],"title":"Isolating $$r$$","text":"First, to isolate $$r$$, we can divide both sides of the inequality by $$-7$$. Since $$-7<0$$, we need to flip the inequality sign. Hence, we get $$r \\\\geq 10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities11a-h2","type":"hint","dependencies":["afaf721inequalities11a-h1"],"title":"Graphing on the Number Line","text":"Next, graph the solution on the number line. To do this, shade to the right of $$10$$ and put a bracket at $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities11a-h3","type":"hint","dependencies":["afaf721inequalities11a-h2"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, brackets are used when the left or right sides of the interval are included in the solution. For example, the expression $$\\"x \\\\leq -1\\"$$ includes all numbers less than or equal to $$-1$$. Since $$-1$$ is included in the solution, the interval notation is $$(-\\\\infty,-1]$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities11a-h4","type":"hint","dependencies":["afaf721inequalities11a-h3"],"title":"Example of Interval Notation","text":"The attached image shows the inequality, number line, and interval notation of $$x \\\\leq 1$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afaf721inequalities12","title":"Solve the inequality, graph the solution on the number line, and write the solution in interval notation.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afaf721inequalities12a","stepAnswer":["$$(-25,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$-20<\\\\frac{4}{5} u$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-25,\\\\infty)$$","choices":["$$(-25,\\\\infty)$$","$$(-16,\\\\infty)$$","$$(-\\\\infty,-25)$$"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities12a-h1","type":"hint","dependencies":[],"title":"Isolating u","text":"First, to isolate u, we can multiply both sides of the inequality by $$\\\\frac{5}{4}$$. Since $$\\\\frac{5}{4}>0$$, we don\'t need to flip the inequality sign. Hence, we get $$u>-25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities12a-h2","type":"hint","dependencies":["afaf721inequalities12a-h1"],"title":"Graphing on the Number Line","text":"Next, graph the solution on the number line. To do this, shade to the right of $$-25$$ and put a parenthesis at $$-25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities12a-h3","type":"hint","dependencies":["afaf721inequalities12a-h2"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, parentheses show that the endpoint of the inequality is not included. For example, the expression \\"x>3\\" includes all numbers greater than $$3$$. Since $$3$$ is not included in the solution, the interval notation is $$(3,\\\\infty)$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities12a-h4","type":"hint","dependencies":["afaf721inequalities12a-h3"],"title":"Example of Interval Notation","text":"The attached image shows the inequality, number line, and interval notation of $$x>3$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afaf721inequalities13","title":"Solve the inequality, graph the solution on the number line, and write the solution in interval notation.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afaf721inequalities13a","stepAnswer":["$$[64,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$24 \\\\leq \\\\frac{3}{8} m$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[64,\\\\infty)$$","choices":["$$[64,\\\\infty)$$","$$(-\\\\infty,64]$$","$$(-\\\\infty,9]$$"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities13a-h1","type":"hint","dependencies":[],"title":"Isolating $$m$$","text":"First, to isolate $$m$$, we can multiply both sides of the inequality by $$\\\\frac{8}{3}$$. Since $$\\\\frac{8}{3}>0$$, we don\'t need to flip the inequality sign. Hence, we get $$m \\\\geq 64$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities13a-h2","type":"hint","dependencies":["afaf721inequalities13a-h1"],"title":"Graphing on the Number Line","text":"Next, graph the solution on the number line. To do this, shade to the right of $$64$$ and put a bracket at $$64$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities13a-h3","type":"hint","dependencies":["afaf721inequalities13a-h2"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, brackets are used when the left or right sides of the interval are included in the solution. For example, the expression $$\\"x \\\\leq -1\\"$$ includes all numbers less than or equal to $$-1$$. Since $$-1$$ is included in the solution, the interval notation is $$(-\\\\infty,-1]$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities13a-h4","type":"hint","dependencies":["afaf721inequalities13a-h3"],"title":"Example of Interval Notation","text":"The attached image shows the inequality, number line, and interval notation of $$x \\\\leq 1$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afaf721inequalities14","title":"Solve the inequality, graph the solution on the number line, and write the solution in interval notation.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afaf721inequalities14a","stepAnswer":["$$(-\\\\infty,-16]$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{-1}{2} t \\\\geq 8$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,-16]$$","choices":["$$(-\\\\infty,-4]$$","$$(-\\\\infty,-16]$$","$$[-16,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities14a-h1","type":"hint","dependencies":[],"title":"Isolating $$t$$","text":"First, to isolate $$t$$, we can multiply both sides of the inequality by $$-2$$. Since $$-2<0$$, we need to flip the inequality sign. Hence, we get $$t \\\\leq -16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities14a-h2","type":"hint","dependencies":["afaf721inequalities14a-h1"],"title":"Graphing on the Number Line","text":"Next, graph the solution on the number line. To do this, shade to the left of $$-16$$ and put a bracket at $$-16$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities14a-h3","type":"hint","dependencies":["afaf721inequalities14a-h2"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, brackets are used when the left or right sides of the interval are included in the solution. For example, the expression $$\\"x \\\\leq -1\\"$$ includes all numbers less than or equal to $$-1$$. Since $$-1$$ is included in the solution, the interval notation is $$(-\\\\infty,-1]$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities14a-h4","type":"hint","dependencies":["afaf721inequalities14a-h3"],"title":"Example of Interval Notation","text":"The attached image shows the inequality, number line, and interval notation of $$x \\\\leq 1$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afaf721inequalities15","title":"Solve the inequality, graph the solution on the number line, and write the solution in interval notation.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afaf721inequalities15a","stepAnswer":["$$(-\\\\infty,-64]$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{-1}{4} y \\\\geq 16$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,-64]$$","choices":["$$(-\\\\infty,-4)$$","$$(-\\\\infty,-64]$$","$$[-64,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities15a-h1","type":"hint","dependencies":[],"title":"Isolating $$y$$","text":"First, to isolate $$y$$, we can multiply both sides of the inequality by $$-4$$. Since $$-4<0$$, we need to flip the inequality sign. Hence, we get $$y \\\\leq -64$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities15a-h2","type":"hint","dependencies":["afaf721inequalities15a-h1"],"title":"Graphing on the Number Line","text":"Next, graph the solution on the number line. To do this, shade to the left of $$-64$$ and put a bracket at $$-64$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities15a-h3","type":"hint","dependencies":["afaf721inequalities15a-h2"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, brackets are used when the left or right sides of the interval are included in the solution. For example, the expression $$\\"x \\\\leq -1\\"$$ includes all numbers less than or equal to $$-1$$. Since $$-1$$ is included in the solution, the interval notation is $$(-\\\\infty,-1]$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities15a-h4","type":"hint","dependencies":["afaf721inequalities15a-h3"],"title":"Example of Interval Notation","text":"The attached image shows the inequality, number line, and interval notation of $$x \\\\leq 1$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afaf721inequalities16","title":"Solve Inequality","body":"In the following exercise, solve the inequality, and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afaf721inequalities16a","stepAnswer":["$$(-\\\\infty,107]$$"],"problemType":"MultipleChoice","stepTitle":"$$m-45 \\\\leq 62$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,107]$$","choices":["$$(-\\\\infty,107]$$","$$(107,\\\\infty)$$","$$(-\\\\infty,17]$$"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities16a-h1","type":"hint","dependencies":[],"title":"Addition Property of Inequality","text":"First, we can add $$45$$ to both sides of the inequality. This gives us $$m-45+45 \\\\leq 62+45$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities16a-h2","type":"hint","dependencies":["afaf721inequalities16a-h1"],"title":"Simplify","text":"Then, simplify both sides of the inequality. This gives us $$m \\\\leq 107$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities16a-h3","type":"hint","dependencies":["afaf721inequalities16a-h2"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, brackets are used when the left or right sides of the interval are included in the solution. For example, the expression $$\\"x \\\\leq -1\\"$$ includes all numbers less than or equal to $$-1$$. Since $$-1$$ is included in the solution, the interval notation is $$(-\\\\infty,-1]$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afaf721inequalities17","title":"Solve Inequality","body":"In the following exercise, solve the inequality, and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afaf721inequalities17a","stepAnswer":["$$(-9,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$v+12>3$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-9,\\\\infty)$$","choices":["$$[-9,\\\\infty)$$","$$(-9,\\\\infty)$$","$$(-\\\\infty,-9]$$","$$(-\\\\infty,-9)$$"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities17a-h1","type":"hint","dependencies":[],"title":"Subtraction Property of Inequality","text":"First, we can subtract $$12$$ from both sides of the inequality. This gives us $$v+12-12>3-12$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities17a-h2","type":"hint","dependencies":["afaf721inequalities17a-h1"],"title":"Simplify","text":"Then, simplify both sides of the inequality. This gives us $$v>-9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities17a-h3","type":"hint","dependencies":["afaf721inequalities17a-h2"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, parentheses show that the endpoint of the inequality is not included. For example, the expression \\"x>3\\" includes all numbers greater than $$3$$. Since $$3$$ is not included in the solution, the interval notation is $$(3,\\\\infty)$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afaf721inequalities18","title":"Solve Inequality","body":"In the following exercise, solve the inequality, and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afaf721inequalities18a","stepAnswer":["$$[\\\\frac{-1}{20},\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$a+\\\\frac{3}{4} \\\\geq \\\\frac{7}{10}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[\\\\frac{-1}{20},\\\\infty)$$","choices":["$$[\\\\frac{-1}{20},\\\\infty)$$","$$[\\\\frac{1}{20},\\\\infty)$$","$$(-\\\\infty,\\\\frac{-1}{20}]$$"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities18a-h1","type":"hint","dependencies":[],"title":"Subtraction Property of Inequality","text":"First, we can subtract $$\\\\frac{3}{4}$$ from both sides of the inequality. This gives us $$a+\\\\frac{3}{4}-\\\\frac{3}{4}=\\\\frac{7}{10}-\\\\frac{3}{4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities18a-h2","type":"hint","dependencies":["afaf721inequalities18a-h1"],"title":"Simplify","text":"Then, simplify both sides of the inequality. This gives us $$a \\\\geq \\\\frac{-1}{20}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities18a-h3","type":"hint","dependencies":["afaf721inequalities18a-h2"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, brackets are used when the left or right sides of the interval are included in the solution. For example, the expression $$\\"x \\\\leq -1\\"$$ includes all numbers less than or equal to $$-1$$. Since $$-1$$ is included in the solution, the interval notation is $$(-\\\\infty,-1]$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afaf721inequalities19","title":"Solve Inequality","body":"In the following exercise, solve the inequality, and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afaf721inequalities19a","stepAnswer":["$$[\\\\frac{-17}{24},\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$b+\\\\frac{7}{8} \\\\geq \\\\frac{1}{6}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[\\\\frac{-17}{24},\\\\infty)$$","choices":["$$[\\\\frac{-17}{24},\\\\infty)$$","$$(\\\\frac{17}{24},\\\\infty)$$","$$(-\\\\infty,\\\\frac{-17}{24}]$$"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities19a-h1","type":"hint","dependencies":[],"title":"Subtraction Property of Inequality","text":"First, we can subtract $$\\\\frac{7}{8}$$ from both sides of the inequality. This gives us $$b+\\\\frac{7}{8}-\\\\frac{7}{8} \\\\geq \\\\frac{1}{6}-\\\\frac{7}{8}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities19a-h2","type":"hint","dependencies":["afaf721inequalities19a-h1"],"title":"Simplify","text":"Then, simplify both sides of the inequality. This gives us $$b \\\\geq \\\\frac{-17}{24}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities19a-h3","type":"hint","dependencies":["afaf721inequalities19a-h2"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, brackets are used when the left or right sides of the interval are included in the solution. For example, the expression $$\\"x \\\\leq -1\\"$$ includes all numbers less than or equal to $$-1$$. Since $$-1$$ is included in the solution, the interval notation is $$(-\\\\infty,-1]$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afaf721inequalities2","title":"Interval Notation Practice","body":"Graph the following inequalities on the number line, and choose the correct interval domain as the answer.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afaf721inequalities2a","stepAnswer":["$$(2,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$x>2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(2,\\\\infty)$$","choices":["$$[2,\\\\infty)$$","$$(2,\\\\infty)$$","$$(2,\\\\infty]$$"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities2a-h1","type":"hint","dependencies":[],"title":"Interpreting the Problem","text":"The first step is to interpret the problem. > means \\"greater than,\\" so we know the valid domain is all numbers greater than $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities2a-h2","type":"hint","dependencies":["afaf721inequalities2a-h1"],"title":"Graphing the Solution","text":"On a number line, shade to the right of $$2$$, and put a parentheses at $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities2a-h3","type":"hint","dependencies":["afaf721inequalities2a-h2"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, parentheses show that the endpoint of the inequality is not included. For example, the expression \\"x>3\\" includes all numbers greater than $$3$$. Since $$3$$ is not included in the solution, the interval notation is $$(3,\\\\infty)$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities2a-h4","type":"hint","dependencies":["afaf721inequalities2a-h3"],"title":"Example of Interval Notation","text":"The attached image shows the inequality, number line, and interval notation of $$x>3$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"afaf721inequalities2b","stepAnswer":["(-inf,-1.5]"],"problemType":"MultipleChoice","stepTitle":"$$x \\\\leq -1.5$$","stepBody":"","answerType":"string","variabilization":{},"choices":["(-inf,-1.5]","(-inf,1.5)","$$[-\\\\infty,3]$$"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities2b-h1","type":"hint","dependencies":[],"title":"Interpreting the Problem","text":"The first step is to interpret the problem. $$ \\\\leq $$ means \\"less than or equal to,\\" so we know the valid domain is all numbers less than or equal to $$-1.5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities2b-h2","type":"hint","dependencies":["afaf721inequalities2b-h1"],"title":"Graphing the Solution","text":"On a number line, shade to the left of $$-1.5$$, and put a bracket at $$-1.5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities2b-h3","type":"hint","dependencies":["afaf721inequalities2b-h2"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, brackets are used when the left or right sides of the interval are included in the solution. For example, the expression $$\\"x \\\\leq -1\\"$$ includes all numbers less than or equal to $$-1$$. Since $$-1$$ is included in the solution, the interval notation is $$(-\\\\infty,-1]$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities2b-h4","type":"hint","dependencies":["afaf721inequalities2b-h3"],"title":"Example of Interval Notation","text":"The attached image shows the inequality, number line, and interval notation of $$x \\\\leq 1$$.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"afaf721inequalities2c","stepAnswer":["$$(-\\\\infty,\\\\frac{3}{4}]$$"],"problemType":"MultipleChoice","stepTitle":"$$x \\\\geq \\\\frac{3}{4}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\frac{3}{4}]$$","choices":["$$(-\\\\infty,\\\\frac{3}{4}]$$","$$(-\\\\infty,\\\\frac{3}{4})$$","$$[-\\\\infty,3]$$"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities2c-h1","type":"hint","dependencies":[],"title":"Interpreting the Problem","text":"The first step is to interpret the problem. $$ \\\\geq $$ means \\"greater or equal than,\\" so we know the valid domain is all numbers greater than or equal to $$\\\\frac{3}{4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities2c-h2","type":"hint","dependencies":["afaf721inequalities2c-h1"],"title":"Graphing the Solution","text":"On a number line, shade to the right of $$\\\\frac{3}{4}$$, and put a bracket at $$\\\\frac{3}{4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities2c-h3","type":"hint","dependencies":["afaf721inequalities2c-h2"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, brackets are used when the left or right sides of the interval are included in the solution. For example, the expression $$\\"x \\\\leq -1\\"$$ includes all numbers less than or equal to $$-1$$. Since $$-1$$ is included in the solution, the interval notation is $$(-\\\\infty,-1]$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities2c-h4","type":"hint","dependencies":["afaf721inequalities2c-h3"],"title":"Example of Interval Notation","text":"The attached image shows the inequality, number line, and interval notation of $$x \\\\leq 1$$.\\\\n##figure3.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afaf721inequalities20","title":"Solve Inequality","body":"In the following exercise, solve the inequality, and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afaf721inequalities20a","stepAnswer":["$$(-\\\\infty,\\\\frac{23}{36})$$"],"problemType":"MultipleChoice","stepTitle":"$$g-\\\\frac{11}{12}<\\\\frac{-5}{18}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\frac{23}{36})$$","choices":["$$(-\\\\infty,\\\\frac{25}{36})$$","$$(-\\\\infty,\\\\frac{23}{36})$$","$$(\\\\frac{22}{36},\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities20a-h1","type":"hint","dependencies":[],"title":"Addition Property of Inequality","text":"First, we can add $$\\\\frac{11}{12}$$ to both sides of the inequality. This gives us $$g-\\\\frac{11}{12}+\\\\frac{11}{12}<\\\\frac{-5}{18}+\\\\frac{11}{12}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities20a-h2","type":"hint","dependencies":["afaf721inequalities20a-h1"],"title":"Simplify","text":"Then, simplify both sides of the inequality. This gives us $$g<\\\\frac{23}{36}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities20a-h3","type":"hint","dependencies":["afaf721inequalities20a-h2"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, parentheses show that the endpoint of the inequality is not included. For example, the expression \\"x>3\\" includes all numbers greater than $$3$$. Since $$3$$ is not included in the solution, the interval notation is $$(3,\\\\infty)$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afaf721inequalities21","title":"Solve Inequality","body":"In the following exercise, solve the inequality, and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afaf721inequalities21a","stepAnswer":["$$(-\\\\infty,8]$$"],"problemType":"MultipleChoice","stepTitle":"$$7r \\\\leq 56$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,8]$$","choices":["$$(-\\\\infty,8]$$","$$(-\\\\infty,8)$$","$$[8,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities21a-h1","type":"hint","dependencies":[],"title":"Division Property of Inequality","text":"First, to isolate $$r$$, we can divide both sides of the inequality by $$7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities21a-h2","type":"hint","dependencies":["afaf721inequalities21a-h1"],"title":"Inequality Sign","text":"Since we divided by a positive number, the inequality sign stays the same. Hence, we have $$r \\\\leq 8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities21a-h3","type":"hint","dependencies":["afaf721inequalities21a-h2"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, brackets are used when the left or right sides of the interval are included in the solution. For example, the expression $$\\"x \\\\leq -1\\"$$ includes all numbers less than or equal to $$-1$$. Since $$-1$$ is included in the solution, the interval notation is $$(-\\\\infty,-1]$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afaf721inequalities22","title":"Solve Inequality","body":"In the following exercise, solve the inequality, and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afaf721inequalities22a","stepAnswer":["$$(-14,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$-9c<126$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-14,\\\\infty)$$","choices":["$$(-\\\\infty,14)$$","$$(-\\\\infty,14]$$","$$(-14,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities22a-h1","type":"hint","dependencies":[],"title":"Division Property of Inequality","text":"First, to isolate c, we can divide both sides of the inequality by $$-9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities22a-h2","type":"hint","dependencies":["afaf721inequalities22a-h1"],"title":"Inequality Sign","text":"Since we divided by a negative number, the inequality reverses. Hence, we have $$c>-14$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities22a-h3","type":"hint","dependencies":["afaf721inequalities22a-h2"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, parentheses show that the endpoint of the inequality is not included. For example, the expression \\"x>3\\" includes all numbers greater than $$3$$. Since $$3$$ is not included in the solution, the interval notation is $$(3,\\\\infty)$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afaf721inequalities23","title":"Solve Inequality","body":"In the following exercise, solve the inequality, and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afaf721inequalities23a","stepAnswer":["$$(-\\\\infty,-15)$$"],"problemType":"MultipleChoice","stepTitle":"$$-7d>105$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,-15)$$","choices":["$$(-\\\\infty,-15)$$","$$(-\\\\infty,15]$$","$$(-15,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities23a-h1","type":"hint","dependencies":[],"title":"Division Property of Inequality","text":"First, to isolate $$d$$, we can divide both sides of the inequality by $$-7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities23a-h2","type":"hint","dependencies":["afaf721inequalities23a-h1"],"title":"Inequality Sign","text":"Since we divided by a negative number, the inequality reverses. Hence, we have $$d<-15$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities23a-h3","type":"hint","dependencies":["afaf721inequalities23a-h2"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, parentheses show that the endpoint of the inequality is not included. For example, the expression \\"x>3\\" includes all numbers greater than $$3$$. Since $$3$$ is not included in the solution, the interval notation is $$(3,\\\\infty)$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afaf721inequalities24","title":"Solve Inequality","body":"In the following exercise, solve the inequality, and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afaf721inequalities24a","stepAnswer":["$$(64,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$40<\\\\frac{5}{8} k$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(64,\\\\infty)$$","choices":["$$(25,\\\\infty)$$","$$(-\\\\infty,64)$$","$$(64,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities24a-h1","type":"hint","dependencies":[],"title":"Multiplication Property of Inequality","text":"First, to isolate $$d$$, we can multiply both sides of the inequality by $$\\\\frac{8}{5}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities24a-h2","type":"hint","dependencies":["afaf721inequalities24a-h1"],"title":"Inequality Sign","text":"Since we multiplied by a positive number, the inequality stays the same. Hence, we have $$k>64$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities24a-h3","type":"hint","dependencies":["afaf721inequalities24a-h2"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, parentheses show that the endpoint of the inequality is not included. For example, the expression \\"x>3\\" includes all numbers greater than $$3$$. Since $$3$$ is not included in the solution, the interval notation is $$(3,\\\\infty)$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afaf721inequalities25","title":"Solve Inequality","body":"In the following exercise, solve the inequality, and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afaf721inequalities25a","stepAnswer":["$$[-27,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{a}{-3} \\\\leq 9$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[-27,\\\\infty)$$","choices":["$$[-27,\\\\infty)$$","$$[-3,\\\\infty)$$","$$(-\\\\infty,-27)$$"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities25a-h1","type":"hint","dependencies":[],"title":"Multiplication Property of Inequality","text":"First, to isolate a, we can multiply both sides of the inequality by $$-3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities25a-h2","type":"hint","dependencies":["afaf721inequalities25a-h1"],"title":"Inequality Sign","text":"Since we multiplied by a negative number, the inequality reverses. Hence, we have $$a \\\\geq -27$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities25a-h3","type":"hint","dependencies":["afaf721inequalities25a-h2"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, brackets are used when the left or right sides of the interval are included in the solution. For example, the expression $$\\"x \\\\leq -1\\"$$ includes all numbers less than or equal to $$-1$$. Since $$-1$$ is included in the solution, the interval notation is $$(-\\\\infty,-1]$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afaf721inequalities26","title":"Solve Inequality","body":"In the following exercise, solve the inequality, and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afaf721inequalities26a","stepAnswer":["$$(-\\\\infty,8]$$"],"problemType":"MultipleChoice","stepTitle":"$$4v \\\\geq 9v-40$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,8]$$","choices":["$$(-\\\\infty,-8]$$","$$(-\\\\infty,8]$$","$$(8,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities26a-h1","type":"hint","dependencies":[],"title":"Isolate","text":"We want to isolate the variable on one side of the inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities26a-h2","type":"hint","dependencies":["afaf721inequalities26a-h1"],"title":"Subtraction Property of Inequality","text":"To isolate the variable, we can subtract 9v from both sides of the inequality. This gives us $$4v-9v \\\\geq 9v-40-9v$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities26a-h3","type":"hint","dependencies":["afaf721inequalities26a-h2"],"title":"Simplify","text":"Simplifying the inequality gives us $$-5v \\\\geq -40$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities26a-h4","type":"hint","dependencies":["afaf721inequalities26a-h3"],"title":"Division Property of Inequality","text":"Then, we can divide both sides of the inequality by $$-5$$, which gives us v on the left hand side and $$8$$ on the right hand side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities26a-h5","type":"hint","dependencies":["afaf721inequalities26a-h4"],"title":"Inequality Sign","text":"Since we divided by a negative number, the inequality reverses. Hence, we have $$v \\\\leq 8$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities26a-h6","type":"hint","dependencies":["afaf721inequalities26a-h5"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, brackets are used when the left or right sides of the interval are included in the solution. For example, the expression $$\\"x \\\\leq -1\\"$$ includes all numbers less than or equal to $$-1$$. Since $$-1$$ is included in the solution, the interval notation is $$(-\\\\infty,-1]$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afaf721inequalities27","title":"Solve Inequality","body":"In the following exercise, solve the inequality, and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afaf721inequalities27a","stepAnswer":["$$[7,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$5u \\\\leq 8u-21$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[7,\\\\infty)$$","choices":["$$(-\\\\infty,-7]$$","$$(-\\\\infty,7]$$","$$[7,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities27a-h1","type":"hint","dependencies":[],"title":"Isolate","text":"We want to isolate the variable on one side of the inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities27a-h2","type":"hint","dependencies":["afaf721inequalities27a-h1"],"title":"Subtraction Property of Inequality","text":"To isolate the variable, we can subtract 8u from both sides of the inequality. This gives us $$5u-8u \\\\leq 8u-21-8u$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities27a-h3","type":"hint","dependencies":["afaf721inequalities27a-h2"],"title":"Simplify","text":"Simplifying the inequality gives us $$-3u \\\\leq -21$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities27a-h4","type":"hint","dependencies":["afaf721inequalities27a-h3"],"title":"Division Property of Inequality","text":"Then, we can divide both sides of the inequality by $$-3$$, which gives us u on the left hand side and $$7$$ on the right hand side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities27a-h5","type":"hint","dependencies":["afaf721inequalities27a-h4"],"title":"Inequality Sign","text":"Since we divided by a negative number, the inequality reverses. Hence, we have $$v \\\\geq 7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities27a-h6","type":"hint","dependencies":["afaf721inequalities27a-h5"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, brackets are used when the left or right sides of the interval are included in the solution. For example, the expression $$\\"x \\\\leq -1\\"$$ includes all numbers less than or equal to $$-1$$. Since $$-1$$ is included in the solution, the interval notation is $$(-\\\\infty,-1]$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afaf721inequalities28","title":"Solve Inequality","body":"In the following exercise, solve the inequality, and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afaf721inequalities28a","stepAnswer":["$$(-\\\\infty,\\\\frac{18}{5})$$"],"problemType":"MultipleChoice","stepTitle":"$$9p>14p-18$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\frac{18}{5})$$","choices":["$$(-\\\\infty,\\\\frac{18}{5})$$","$$(-\\\\infty,\\\\frac{-18}{5})$$","$$(-\\\\infty,\\\\frac{5}{18})$$"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities28a-h1","type":"hint","dependencies":[],"title":"Isolate","text":"We want to isolate the variable on one side of the inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities28a-h2","type":"hint","dependencies":["afaf721inequalities28a-h1"],"title":"Subtraction Property of Inequality","text":"To isolate the variable, we can subtract $$14p$$ from both sides of the inequality. This gives us $$9p-14p>14p-18-14p$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities28a-h3","type":"hint","dependencies":["afaf721inequalities28a-h2"],"title":"Simplify","text":"Simplifying the inequality gives us $$-5p>-18$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities28a-h4","type":"hint","dependencies":["afaf721inequalities28a-h3"],"title":"Division Property of Inequality","text":"Then, we can divide both sides of the inequality by $$-5$$, which gives us $$p$$ on the left hand side and $$\\\\frac{18}{5}$$ on the right hand side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities28a-h5","type":"hint","dependencies":["afaf721inequalities28a-h4"],"title":"Inequality Sign","text":"Since we divided by a negative number, the inequality reverses. Hence, we have $$p<\\\\frac{18}{5}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities28a-h6","type":"hint","dependencies":["afaf721inequalities28a-h5"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, parentheses show that the endpoint of the inequality is not included. For example, the expression \\"x>3\\" includes all numbers greater than $$3$$. Since $$3$$ is not included in the solution, the interval notation is $$(3,\\\\infty)$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afaf721inequalities29","title":"Solve Inequality","body":"In the following exercise, solve the inequality, and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afaf721inequalities29a","stepAnswer":["$$(-9,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$12x+3\\\\left(x+7\\\\right)>10x-24$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-9,\\\\infty)$$","choices":["$$(-\\\\infty,-9)$$","$$(-\\\\infty,-9]$$","$$(-9,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities29a-h1","type":"hint","dependencies":[],"title":"Simplify","text":"First, we need to use the Distributive Property to get rid of the parentheses, and then combine like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities29a-h2","type":"hint","dependencies":["afaf721inequalities29a-h1"],"title":"Simplified Inequality","text":"The simplified inequality is $$15x+21>10x-24$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities29a-h3","type":"hint","dependencies":["afaf721inequalities29a-h2"],"title":"Isolate","text":"We want to isolate the variable on one side of the inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities29a-h4","type":"hint","dependencies":["afaf721inequalities29a-h3"],"title":"Subtraction Property of Inequality","text":"To isolate the variable, we can subtract $$10x$$ and $$21$$ from both sides of the inequality. This gives us $$15x+21-10x-21>10x-24-10x-21$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities29a-h5","type":"hint","dependencies":["afaf721inequalities29a-h4"],"title":"Simplify","text":"Simplifying the inequality gives us $$5x>-45$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities29a-h6","type":"hint","dependencies":["afaf721inequalities29a-h5"],"title":"Division Property of Inequality","text":"Then, we can divide both sides of the inequality by $$5$$, which gives us $$x$$ on the left hand side and $$-9$$ on the right hand side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities29a-h7","type":"hint","dependencies":["afaf721inequalities29a-h6"],"title":"Inequality Sign","text":"Since we divided by a postive number, the inequality stays the same. Hence, we have $$x>-9$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities29a-h8","type":"hint","dependencies":["afaf721inequalities29a-h7"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, parentheses show that the endpoint of the inequality is not included. For example, the expression \\"x>3\\" includes all numbers greater than $$3$$. Since $$3$$ is not included in the solution, the interval notation is $$(3,\\\\infty)$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afaf721inequalities3","title":"Interval Notation Practice","body":"Graph the following inequalities on the number line, and choose the correct interval domain as the answer.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afaf721inequalities3a","stepAnswer":["$$(-\\\\infty,-4]$$"],"problemType":"MultipleChoice","stepTitle":"$$x \\\\leq -4$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,-4]$$","choices":["$$(-\\\\infty,-4]$$","$$[-\\\\infty,4)$$","$$[-\\\\infty,-4)$$"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities3a-h1","type":"hint","dependencies":[],"title":"Interpreting the Problem","text":"The first step is to interpret the problem. $$ \\\\leq $$ means \\"less than or equal to,\\" so we know the valid domain is all numbers less than $$-4$$, including $$-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities3a-h2","type":"hint","dependencies":["afaf721inequalities3a-h1"],"title":"Graphing the Solution","text":"On a number line, shade to the left of $$-4$$, and put a bracket at $$-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities3a-h3","type":"hint","dependencies":["afaf721inequalities3a-h2"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, brackets are used when the left or right sides of the interval are included in the solution. For example, the expression $$\\"x \\\\leq -1\\"$$ includes all numbers less than or equal to $$-1$$. Since $$-1$$ is included in the solution, the interval notation is $$(-\\\\infty,-1]$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities3a-h4","type":"hint","dependencies":["afaf721inequalities3a-h3"],"title":"Example of Interval Notation","text":"The attached image shows the inequality, number line, and interval notation of $$x \\\\leq 1$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"afaf721inequalities3b","stepAnswer":["[0.5,inf)"],"problemType":"MultipleChoice","stepTitle":"$$x \\\\geq 0.5$$","stepBody":"","answerType":"string","variabilization":{},"choices":["(0.5,inf)","[0.5,inf)","(-inf,0.5]"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities3b-h1","type":"hint","dependencies":[],"title":"Interpreting the Problem","text":"The first step is to interpret the problem. $$ \\\\geq $$ means \\"greater or equal than,\\" so we know the valid domain is all numbers greater than or equal to $$0.5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities3b-h2","type":"hint","dependencies":["afaf721inequalities3b-h1"],"title":"Graphing the Solution","text":"On a number line, shade to the right of $$0.5$$, and put a bracket at $$0.5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities3b-h3","type":"hint","dependencies":["afaf721inequalities3b-h2"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, brackets are used when the left or right sides of the interval are included in the solution. For example, the expression $$\\"x \\\\leq -1\\"$$ includes all numbers less than or equal to $$-1$$. Since $$-1$$ is included in the solution, the interval notation is $$(-\\\\infty,-1]$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities3b-h4","type":"hint","dependencies":["afaf721inequalities3b-h3"],"title":"Example of Interval Notation","text":"The attached image shows the inequality, number line, and interval notation of $$x \\\\leq 1$$.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"afaf721inequalities3c","stepAnswer":["$$(\\\\infty,\\\\frac{-2}{3})$$"],"problemType":"MultipleChoice","stepTitle":"$$x<\\\\frac{-2}{3}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(\\\\infty,\\\\frac{-2}{3})$$","choices":["$$(\\\\infty,\\\\frac{-2}{3})$$","$$(-\\\\infty,\\\\frac{-2}{3}]$$","$$(\\\\frac{-2}{3},\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities3c-h1","type":"hint","dependencies":[],"title":"Interpreting the Problem","text":"The first step is to interpret the problem. < means \\"less than,\\" so we know the valid domain is all numbers less than $$\\\\frac{-2}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities3c-h2","type":"hint","dependencies":["afaf721inequalities3c-h1"],"title":"Graphing the Solution","text":"On a number line, shade to the left of $$\\\\frac{-2}{3}$$, and put a parentheses at $$\\\\frac{-2}{3}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities3c-h3","type":"hint","dependencies":["afaf721inequalities3c-h2"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, parentheses show that the endpoint of the inequality is not included. For example, the expression \\"x>3\\" includes all numbers greater than $$3$$. Since $$3$$ is not included in the solution, the interval notation is $$(3,\\\\infty)$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities3c-h4","type":"hint","dependencies":["afaf721inequalities3c-h3"],"title":"Example of Interval Notation","text":"The attached image shows the inequality, number line, and interval notation of $$x>3$$.\\\\n##figure3.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afaf721inequalities30","title":"Solve Inequality","body":"In the following exercise, solve the inequality, and write the solution in interval notation.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afaf721inequalities30a","stepAnswer":["$$[3,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$6h-4(h-1) \\\\leq 7h-11$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[3,\\\\infty)$$","choices":["$$[3,\\\\infty)$$","$$[-3,\\\\infty)$$","$$(-\\\\infty,3]$$"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities30a-h1","type":"hint","dependencies":[],"title":"Simplify","text":"First, we need to use the Distributive Property to get rid of the parentheses, and then combine like terms.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities30a-h2","type":"hint","dependencies":["afaf721inequalities30a-h1"],"title":"Simplified Inequality","text":"The simplified inequality is $$2h+4 \\\\leq 7h-11$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities30a-h3","type":"hint","dependencies":["afaf721inequalities30a-h2"],"title":"Isolate","text":"We want to isolate the variable on one side of the inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities30a-h4","type":"hint","dependencies":["afaf721inequalities30a-h3"],"title":"Subtraction Property of Inequality","text":"To isolate the variable, we can subtract $$7h$$ and $$4$$ from both sides of the inequality. This gives us $$2h+4-7h-4 \\\\leq 7h-11-7h-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities30a-h5","type":"hint","dependencies":["afaf721inequalities30a-h4"],"title":"Simplify","text":"Simplifying the inequality gives us $$-5h \\\\leq -15$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities30a-h6","type":"hint","dependencies":["afaf721inequalities30a-h5"],"title":"Division Property of Inequality","text":"Then, we can divide both sides of the inequality by $$-5$$, which gives us $$h$$ on the left hand side and $$3$$ on the right hand side.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities30a-h7","type":"hint","dependencies":["afaf721inequalities30a-h6"],"title":"Inequality Sign","text":"Since we divided by a negative number, the inequality reverses. Hence, we have $$h \\\\geq 3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities30a-h8","type":"hint","dependencies":["afaf721inequalities30a-h7"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, brackets are used when the left or right sides of the interval are included in the solution. For example, the expression $$\\"x \\\\leq -1\\"$$ includes all numbers less than or equal to $$-1$$. Since $$-1$$ is included in the solution, the interval notation is $$(-\\\\infty,-1]$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afaf721inequalities4","title":"Solve the Inequality","body":"Graph the solution on the number line, and choose the correct interval notation as the answer.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afaf721inequalities4a","stepAnswer":["$$(-\\\\infty,\\\\frac{9}{8}]$$"],"problemType":"MultipleChoice","stepTitle":"$$n-\\\\frac{1}{2} \\\\leq \\\\frac{5}{8}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\frac{9}{8}]$$","choices":["$$(\\\\frac{9}{8},\\\\infty)$$","$$[\\\\frac{9}{8},\\\\infty)$$","$$(-\\\\infty,\\\\frac{9}{8}]$$"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities4a-h1","type":"hint","dependencies":[],"title":"First Step: Addition","text":"First, we can add $$\\\\frac{1}{2}$$ to both sides of the inequality. This gives us $$n-\\\\frac{1}{2}+\\\\frac{1}{2} \\\\leq \\\\frac{5}{8}+\\\\frac{1}{2}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities4a-h2","type":"hint","dependencies":["afaf721inequalities4a-h1"],"title":"Second Step: Simplify","text":"Then, simplify both sides of the inequality. This gives us $$n \\\\leq \\\\frac{9}{8}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities4a-h3","type":"hint","dependencies":["afaf721inequalities4a-h2"],"title":"Third Step: Graphing on the Number Line","text":"Thirdly, graph the solution on the number line, as shown in the image.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities4a-h4","type":"hint","dependencies":["afaf721inequalities4a-h3"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, brackets are used when the left or right sides of the interval are included in the solution. For example, the expression $$\\"x \\\\leq -1\\"$$ includes all numbers less than or equal to $$-1$$. Since $$-1$$ is included in the solution, the interval notation is $$(-\\\\infty,-1]$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities4a-h5","type":"hint","dependencies":["afaf721inequalities4a-h4"],"title":"Example of Interval Notation","text":"The attached image shows the inequality, number line, and interval notation of $$x \\\\leq 1$$.\\\\n##figure2.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afaf721inequalities5","title":"Solve the inequality, graph the solution on the number line, and choose the correct interval notation as the answer.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afaf721inequalities5a","stepAnswer":["$$[\\\\frac{11}{12},\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"$$p-\\\\frac{3}{4} \\\\geq \\\\frac{1}{6}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$[\\\\frac{11}{12},\\\\infty)$$","choices":["$$[\\\\frac{11}{12},\\\\infty)$$","$$(\\\\frac{11}{12},\\\\infty)$$","$$(-\\\\infty,\\\\frac{11}{12}]$$"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities5a-h1","type":"hint","dependencies":[],"title":"Isolating $$p$$","text":"First, add $$\\\\frac{3}{4}$$ to both sides to isolate $$p$$ on one side of the inequality. This gives us $$p-\\\\frac{3}{4}+\\\\frac{3}{4} \\\\geq \\\\frac{1}{6}+\\\\frac{3}{4}$$, which simplifies to $$p \\\\geq \\\\frac{11}{12}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities5a-h2","type":"hint","dependencies":["afaf721inequalities5a-h1"],"title":"Graphing the Solution","text":"Next, graph the solution on the number line. To do this, shade to the right of $$\\\\frac{-11}{12}$$ and put a bracket at $$\\\\frac{-11}{12}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities5a-h3","type":"hint","dependencies":["afaf721inequalities5a-h2"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, brackets are used when the left or right sides of the interval are included in the solution. For example, the expression $$\\"x \\\\leq -1\\"$$ includes all numbers less than or equal to $$-1$$. Since $$-1$$ is included in the solution, the interval notation is $$(-\\\\infty,-1]$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities5a-h4","type":"hint","dependencies":["afaf721inequalities5a-h3"],"title":"Example of Interval Notation","text":"The attached image shows the inequality, number line, and interval notation of $$x \\\\leq 1$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afaf721inequalities6","title":"Solve the inequality, graph the solution on the number line, and choose the correct interval notation as the answer.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afaf721inequalities6a","stepAnswer":["$$(-\\\\infty,\\\\frac{11}{12}]$$"],"problemType":"MultipleChoice","stepTitle":"$$r-\\\\frac{1}{3} \\\\leq \\\\frac{7}{12}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\frac{11}{12}]$$","choices":["$$(-\\\\infty,\\\\frac{11}{12})$$","$$(-\\\\infty,\\\\frac{11}{12}]$$","$$(\\\\frac{11}{12},\\\\infty]$$"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities6a-h1","type":"hint","dependencies":[],"title":"Isolating $$r$$","text":"First, we can add $$\\\\frac{1}{3}$$ to both sides to isolate $$r$$ on one side of the inequality. This gives us $$r-\\\\frac{1}{3}+\\\\frac{1}{3} \\\\leq \\\\frac{7}{12}+\\\\frac{1}{3}$$, which simplifies to $$r \\\\leq \\\\frac{11}{12}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities6a-h2","type":"hint","dependencies":["afaf721inequalities6a-h1"],"title":"Graphing the Solution","text":"Next, graph the solution on the number line. To do this, shade to the left of $$\\\\frac{-11}{12}$$ and put a bracket at $$\\\\frac{-11}{12}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities6a-h3","type":"hint","dependencies":["afaf721inequalities6a-h2"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, brackets are used when the left or right sides of the interval are included in the solution. For example, the expression $$\\"x \\\\leq -1\\"$$ includes all numbers less than or equal to $$-1$$. Since $$-1$$ is included in the solution, the interval notation is $$(-\\\\infty,-1]$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities6a-h4","type":"hint","dependencies":["afaf721inequalities6a-h3"],"title":"Example of Interval Notation","text":"The attached image shows the inequality, number line, and interval notation of $$x \\\\leq 1$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afaf721inequalities7","title":"Solve the inequality, graph the solution on the number line, and write the solution in interval notation.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afaf721inequalities7a","stepAnswer":["$$(-\\\\infty,6)$$"],"problemType":"MultipleChoice","stepTitle":"7y<\u200b\u200b42","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,6)$$","choices":["$$(-\\\\infty,6]$$","$$(-\\\\infty,6)$$","$$(6,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities7a-h1","type":"hint","dependencies":[],"title":"Isolating $$y$$","text":"First, to isolate $$y$$, we can divide both sides of the inequality by $$7$$. Since $$7>0$$, the inequality sign stays the same. Hence, we get $$y<6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities7a-h2","type":"hint","dependencies":["afaf721inequalities7a-h1"],"title":"Graphing on the Number Line","text":"Next, graph the solution on the number line. To do this, shade to the left of $$6$$ and put a parenthesis at $$6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities7a-h3","type":"hint","dependencies":["afaf721inequalities7a-h2"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, parentheses show that the endpoint of the inequality is not included. For example, the expression \\"x>3\\" includes all numbers greater than $$3$$. Since $$3$$ is not included in the solution, the interval notation is $$(3,\\\\infty)$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities7a-h4","type":"hint","dependencies":["afaf721inequalities7a-h3"],"title":"Example of Interval Notation","text":"The attached image shows the inequality, number line, and interval notation of $$x>3$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afaf721inequalities8","title":"Solve the inequality, graph the solution on the number line, and write the solution in interval notation.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afaf721inequalities8a","stepAnswer":["$$(-\\\\infty,5]$$"],"problemType":"MultipleChoice","stepTitle":"$$12d \\\\leq 60$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,5]$$","choices":["$$(-\\\\infty,5]$$","$$(-\\\\infty,5)$$","$$[5,\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities8a-h1","type":"hint","dependencies":[],"title":"Isolating $$d$$","text":"First, to isolate $$d$$, we can divide both sides of the inequality by $$12$$. Since $$12>0$$, the inequality sign stays the same. Hense, we get $$d \\\\leq 5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities8a-h2","type":"hint","dependencies":["afaf721inequalities8a-h1"],"title":"Graphing on the Number Line","text":"Next, graph the solution on the number line. To do this, shade to the left of $$5$$ and put a bracket at $$5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities8a-h3","type":"hint","dependencies":["afaf721inequalities8a-h2"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, brackets are used when the left or right sides of the interval are included in the solution. For example, the expression $$\\"x \\\\leq -1\\"$$ includes all numbers less than or equal to $$-1$$. Since $$-1$$ is included in the solution, the interval notation is $$(-\\\\infty,-1]$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities8a-h4","type":"hint","dependencies":["afaf721inequalities8a-h3"],"title":"Example of Interval Notation","text":"The attached image shows the inequality, number line, and interval notation of $$x \\\\leq 1$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afaf721inequalities9","title":"Solve the inequality, graph the solution on the number line, and write the solution in interval notation.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.7 Solve Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afaf721inequalities9a","stepAnswer":["$$(-\\\\infty,-5]$$"],"problemType":"MultipleChoice","stepTitle":"$$-10a \\\\geq 50$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,-5]$$","choices":["$$[-5,\\\\infty)$$","$$(-\\\\infty,-5]$$","$$(-\\\\infty,5]$$"],"hints":{"DefaultPathway":[{"id":"afaf721inequalities9a-h1","type":"hint","dependencies":[],"title":"Isolating a","text":"First, to isolate a, we can divide both sides of the inequality by $$-10$$. Since $$-10<0$$, we need to flip the inequality sign. Hence, we get $$a \\\\leq -5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities9a-h2","type":"hint","dependencies":["afaf721inequalities9a-h1"],"title":"Graphing on the Number Line","text":"Next, graph the solution on the number line. To do this, shade to the left of $$-5$$ and put a bracket at $$-5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities9a-h3","type":"hint","dependencies":["afaf721inequalities9a-h2"],"title":"Interval Notation: Parentheses and Brackets","text":"In interval notation, brackets are used when the left or right sides of the interval are included in the solution. For example, the expression $$\\"x \\\\leq -1\\"$$ includes all numbers less than or equal to $$-1$$. Since $$-1$$ is included in the solution, the interval notation is $$(-\\\\infty,-1]$$. Since $$\\\\infty$$ represents \\"infinity\\" and is not an actual number, we always put a parenthesis next to $$\\\\infty$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afaf721inequalities9a-h4","type":"hint","dependencies":["afaf721inequalities9a-h3"],"title":"Example of Interval Notation","text":"The attached image shows the inequality, number line, and interval notation of $$x \\\\leq 1$$.\\\\n##figure1.gif##","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afb3ccdlang1","title":"Find Factors, Prime Factorizations, and Least Common Multiples","body":"Select the options with numbers that are all factors of $$5625$$.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Use the Language of Algebra","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"afb3ccdlang1a","stepAnswer":["$$3$$"],"problemType":"MultipleChoice","stepTitle":"Is $$5625$$ divisible by $$2$$, $$3$$, $$4$$, $$5$$ and $$10$$, or 6?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$3$$","choices":["$$2$$","$$3$$","$$5$$ and $$10$$","$$6$$"],"hints":{"DefaultPathway":[{"id":"afb3ccdlang1a-h1","type":"hint","dependencies":[],"title":"Checking for Divisibility","text":"Is $$5625$$ divisible by 2?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang1a-h2","type":"hint","dependencies":["afb3ccdlang1a-h1"],"title":"Checking for Divisibility","text":"Does it end in $$0$$, $$2$$, $$4$$, $$6$$ or 8? No. $$5626$$ is not divisible by $$2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang1a-h3","type":"hint","dependencies":["afb3ccdlang1a-h2"],"title":"Checking for Divisibility","text":"Is $$5625$$ divisible by 3?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang1a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$18$$"],"dependencies":["afb3ccdlang1a-h3"],"title":"Checking for Divisibility","text":"What is the sum of the digits?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang1a-h5","type":"hint","dependencies":["afb3ccdlang1a-h4"],"title":"Checking for Divisibility","text":"$$5626$$ is divisible by $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang1a-h6","type":"hint","dependencies":["afb3ccdlang1a-h5"],"title":"Checking for Divisibility","text":"Is $$5625$$ divisible by $$5$$ or 10?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang1a-h7","type":"hint","dependencies":["afb3ccdlang1a-h6"],"title":"Checking for Divisibility","text":"The last digit is $$5$$, it\'s divisible by $$5$$ not by $$10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang1a-h8","type":"hint","dependencies":["afb3ccdlang1a-h7"],"title":"Checking for Divisibility","text":"Is $$5625$$ divisible by 6?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang1a-h9","type":"hint","dependencies":["afb3ccdlang1a-h8"],"title":"Checking for Divisibility","text":"$$5626$$ is not divisible by $$2$$, so it\'s not divisible by $$6$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afb3ccdlang10","title":"Translate an English Phrase to an Algebraic Expression","body":"June has dimes and quarters in her purse. The number of dimes is seven less than four times the number of quarters. Let q represent the number of quarters.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Use the Language of Algebra","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"afb3ccdlang10a","stepAnswer":["$$4q-7$$"],"problemType":"TextBox","stepTitle":"Write an expression for the number of dimes.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$4q-7$$","hints":{"DefaultPathway":[{"id":"afb3ccdlang10a-h1","type":"hint","dependencies":[],"title":"Write a phrase","text":"Write a phrase about the number of dimes: seven less than four times the number of quarters","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang10a-h2","type":"hint","dependencies":["afb3ccdlang10a-h1"],"title":"Substitution","text":"Then, we substitute q for the number of quarters","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang10a-h3","type":"hint","dependencies":["afb3ccdlang10a-h2"],"title":"Translate","text":"Translate $$4$$ times q: $$7$$ less than $$4q$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang10a-h4","type":"hint","dependencies":["afb3ccdlang10a-h3"],"title":"Translate","text":"Translate the phrase into algebr: $$4q$$ - $$7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afb3ccdlang11","title":"Simplify Expressions Using the Order of Operations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Use the Language of Algebra","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"afb3ccdlang11a","stepAnswer":["$$58$$"],"problemType":"TextBox","stepTitle":"Simplify: $$2+8\\\\left(6+1\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$58$$","hints":{"DefaultPathway":[{"id":"afb3ccdlang11a-h1","type":"hint","dependencies":[],"title":"Parentheses","text":"Since there are parentheses, we do the calculations inside the parentheses first.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$7$$"],"dependencies":["afb3ccdlang11a-h1"],"title":"Parentheses","text":"what is $$6+1$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang11a-h3","type":"hint","dependencies":["afb3ccdlang11a-h2"],"title":"$$\\\\frac{Multiplication}{Division}$$","text":"Then, we calculate the division and multiplication.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$56$$"],"dependencies":["afb3ccdlang11a-h3"],"title":"$$\\\\frac{Multiplication}{Division}$$","text":"what is $$8\\\\times7$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang11a-h5","type":"hint","dependencies":["afb3ccdlang11a-h4"],"title":"$$\\\\frac{Addition}{Subtraction}$$","text":"The last step is to do the addition and subtraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang11a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$58$$"],"dependencies":["afb3ccdlang11a-h5"],"title":"$$\\\\frac{Addition}{Subtraction}$$","text":"What is $$2+56$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afb3ccdlang12","title":"Simplify Expressions Using the Order of Operations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Use the Language of Algebra","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"afb3ccdlang12a","stepAnswer":["$$58$$"],"problemType":"TextBox","stepTitle":"Simplify: $$4+6\\\\left(3+6\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$58$$","hints":{"DefaultPathway":[{"id":"afb3ccdlang12a-h1","type":"hint","dependencies":[],"title":"Parentheses","text":"Since there are parentheses, we do the calculations inside the parentheses first.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang12a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$9$$"],"dependencies":["afb3ccdlang12a-h1"],"title":"Parentheses","text":"what is $$3+6$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang12a-h3","type":"hint","dependencies":["afb3ccdlang12a-h2"],"title":"$$\\\\frac{Multiplication}{Division}$$","text":"Then, we calculate the division and multiplication.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang12a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$54$$"],"dependencies":["afb3ccdlang12a-h3"],"title":"$$\\\\frac{Multiplication}{Division}$$","text":"what is $$6\\\\times9$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang12a-h5","type":"hint","dependencies":["afb3ccdlang12a-h4"],"title":"$$\\\\frac{Addition}{Subtraction}$$","text":"The last step is to do the addition and subtraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang12a-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$58$$"],"dependencies":["afb3ccdlang12a-h5"],"title":"$$\\\\frac{Addition}{Subtraction}$$","text":"What is $$4+54$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afb3ccdlang13","title":"Identify and Combine Like Terms","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Use the Language of Algebra","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"afb3ccdlang13a","stepAnswer":["$$10x+6$$"],"problemType":"TextBox","stepTitle":"Simplify: $$7x+2+3x+4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10x+6$$","hints":{"DefaultPathway":[{"id":"afb3ccdlang13a-h1","type":"hint","dependencies":[],"title":"Like terms","text":"First, we identify the like terms (terms that have the same exponents).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang13a-h2","type":"hint","dependencies":["afb3ccdlang13a-h1"],"title":"Combination","text":"Then, we combine the like terms (adding the coefficient of the like terms), and get $$10x+6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afb3ccdlang14","title":"Identify and Combine Like Terms","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Use the Language of Algebra","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"afb3ccdlang14a","stepAnswer":["$$10y+1$$"],"problemType":"TextBox","stepTitle":"Simplify: $$8y+5+2y-4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10y+1$$","hints":{"DefaultPathway":[{"id":"afb3ccdlang14a-h1","type":"hint","dependencies":[],"title":"Like terms","text":"First, we identify the like terms (terms that have the same exponents).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang14a-h2","type":"hint","dependencies":["afb3ccdlang14a-h1"],"title":"Combination","text":"Then, we combine the like terms (adding the coefficient of the like terms), and get $$10y+1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afb3ccdlang15","title":"Identify and Combine Like Terms","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Use the Language of Algebra","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"afb3ccdlang15a","stepAnswer":["$$22a+1$$"],"problemType":"TextBox","stepTitle":"Simplify: $$10a+7+5a-2+7a-4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$22a+1$$","hints":{"DefaultPathway":[{"id":"afb3ccdlang15a-h1","type":"hint","dependencies":[],"title":"Like terms","text":"First, we identify the like terms (terms that have the same exponents).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang15a-h2","type":"hint","dependencies":["afb3ccdlang15a-h1"],"title":"Combination","text":"Then, we combine the like terms (adding the coefficient of the like terms), and get $$22a+1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afb3ccdlang16","title":"Simplifying Expressions","body":"Simplify the given expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Use the Language of Algebra","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"afb3ccdlang16a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"Simplify $$2^3$$ - $$\\\\frac{12}{9-5}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"afb3ccdlang16a-h1","type":"hint","dependencies":[],"title":"Use PEMDAS","text":"We must simplify operations in the following order: parentheses, exponents, $$\\\\frac{multiplication}{division}$$ $$(left-to-right)$$, $$\\\\frac{addition}{subttraction}$$ $$(left-to-right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang16a-h2","type":"hint","dependencies":["afb3ccdlang16a-h1"],"title":"Simplifying the Terms","text":"$$2^3-\\\\frac{12}{9-5}=2^3-\\\\frac{12}{4}=8-\\\\frac{12}{4}=8-3=5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afb3ccdlang17","title":"Simplifying Expressions","body":"Simplify the given expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Use the Language of Algebra","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"afb3ccdlang17a","stepAnswer":["$$6$$"],"problemType":"TextBox","stepTitle":"Simplify $$3^2-\\\\frac{18}{11-5}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$6$$","hints":{"DefaultPathway":[{"id":"afb3ccdlang17a-h1","type":"hint","dependencies":[],"title":"Use PEMDAS","text":"We must simplify operations in the following order: parentheses, exponents, $$\\\\frac{multiplication}{division}$$ $$(left-to-right)$$, $$\\\\frac{addition}{subttraction}$$ $$(left-to-right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang17a-h2","type":"hint","dependencies":["afb3ccdlang17a-h1"],"title":"Simplifying the Terms","text":"$$3^2-\\\\frac{18}{11-5}=3^2-\\\\frac{18}{6}=9-\\\\frac{18}{6}=9-3=6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afb3ccdlang18","title":"Simplifying Expressions","body":"Simplify the given expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Use the Language of Algebra","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"afb3ccdlang18a","stepAnswer":["$$58$$"],"problemType":"TextBox","stepTitle":"Simplify $$2+8\\\\left(6+1\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$58$$","hints":{"DefaultPathway":[{"id":"afb3ccdlang18a-h1","type":"hint","dependencies":[],"title":"Use PEMDAS","text":"We must simplify operations in the following order: parentheses, exponents, $$\\\\frac{multiplication}{division}$$ $$(left-to-right)$$, $$\\\\frac{addition}{subttraction}$$ $$(left-to-right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang18a-h2","type":"hint","dependencies":["afb3ccdlang18a-h1"],"title":"Simplifying the Terms","text":"$$2+8\\\\left(6+1\\\\right)=2+8\\\\left(7\\\\right)=2+56=58$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afb3ccdlang19","title":"Simplifying Expressions","body":"Simplify the given expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Use the Language of Algebra","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"afb3ccdlang19a","stepAnswer":["$$58$$"],"problemType":"TextBox","stepTitle":"Simplify $$4+6\\\\left(3+6\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$58$$","hints":{"DefaultPathway":[{"id":"afb3ccdlang19a-h1","type":"hint","dependencies":[],"title":"Use PEMDAS","text":"We must simplify operations in the following order: parentheses, exponents, $$\\\\frac{multiplication}{division}$$ $$(left-to-right)$$, $$\\\\frac{addition}{subttraction}$$ $$(left-to-right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang19a-h2","type":"hint","dependencies":["afb3ccdlang19a-h1"],"title":"Simplifying the Terms","text":"$$4+6\\\\left(3+6\\\\right)=4+6\\\\left(9\\\\right)=4+54=58$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afb3ccdlang2","title":"Find Factors, Prime Factorizations, and Least Common Multiples","body":"List the factors in increasing order, with commas without spaces dividing them. An example of the input format is: \\"2,2,4,5\\" without the quotes.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Use the Language of Algebra","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"afb3ccdlang2a","stepAnswer":["2,2,2,2,3"],"problemType":"TextBox","stepTitle":"Please factor $$48$$.","stepBody":"","answerType":"string","variabilization":{},"hints":{"DefaultPathway":[{"id":"afb3ccdlang2a-h1","type":"hint","dependencies":[],"title":"First, we need to find two factors whose product is the given number, which is $$2\\\\times24$$.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang2a-h2","type":"hint","dependencies":["afb3ccdlang2a-h1"],"title":"Since $$2$$ is a prime number, we are done with this. We then turn to $$24$$, and factor $$24$$.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang2a-h3","type":"hint","dependencies":["afb3ccdlang2a-h2"],"title":"Then, we find two factors whose product is $$24$$, which is $$2\\\\times12$$.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang2a-h4","type":"hint","dependencies":["afb3ccdlang2a-h3"],"title":"Since $$2$$ is a prime number, we are done with this. We then turn to $$12$$, and factor $$12$$.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang2a-h5","type":"hint","dependencies":["afb3ccdlang2a-h4"],"title":"Then, we find two factors whose product is $$12$$, which is $$2\\\\times6$$.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang2a-h6","type":"hint","dependencies":["afb3ccdlang2a-h5"],"title":"Since $$2$$ is a prime number, we are done with this. We then turn to $$6$$, and factor $$6$$.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang2a-h7","type":"hint","dependencies":["afb3ccdlang2a-h6"],"title":"Then, we find two factors whose product is $$6$$, which is $$2\\\\times3$$.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afb3ccdlang20","title":"Simplifying Expressions","body":"Simplify the given expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Use the Language of Algebra","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"afb3ccdlang20a","stepAnswer":["$$29$$"],"problemType":"TextBox","stepTitle":"Simplify $$\\\\frac{20}{4}+6\\\\left(5-1\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$29$$","hints":{"DefaultPathway":[{"id":"afb3ccdlang20a-h1","type":"hint","dependencies":[],"title":"Use PEMDAS","text":"We must simplify operations in the following order: parentheses, exponents, $$\\\\frac{multiplication}{division}$$ $$(left-to-right)$$, $$\\\\frac{addition}{subttraction}$$ $$(left-to-right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang20a-h2","type":"hint","dependencies":["afb3ccdlang20a-h1"],"title":"Simplifying the Terms","text":"$$\\\\frac{20}{4}+6\\\\left(5-1\\\\right)=\\\\frac{20}{4}+6\\\\left(4\\\\right)=5+6\\\\times4=5+24=29$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afb3ccdlang21","title":"Simplifying Expressions","body":"Simplify the given expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Use the Language of Algebra","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"afb3ccdlang21a","stepAnswer":["$$31$$"],"problemType":"TextBox","stepTitle":"Simplify $$\\\\frac{33}{3}+4\\\\left(7-2\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$31$$","hints":{"DefaultPathway":[{"id":"afb3ccdlang21a-h1","type":"hint","dependencies":[],"title":"Use PEMDAS","text":"We must simplify operations in the following order: parentheses, exponents, $$\\\\frac{multiplication}{division}$$ $$(left-to-right)$$, $$\\\\frac{addition}{subttraction}$$ $$(left-to-right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang21a-h2","type":"hint","dependencies":["afb3ccdlang21a-h1"],"title":"Simplifying the Terms","text":"$$\\\\frac{33}{3}+4\\\\left(7-2\\\\right)=\\\\frac{33}{3}+4\\\\left(5\\\\right)=11+4\\\\times5=11+20=31$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afb3ccdlang22","title":"Simplifying Expressions","body":"Simplify the given expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Use the Language of Algebra","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"afb3ccdlang22a","stepAnswer":["$$149$$"],"problemType":"TextBox","stepTitle":"Simplify $$3\\\\left(1+9\\\\times6\\\\right)-4^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$149$$","hints":{"DefaultPathway":[{"id":"afb3ccdlang22a-h1","type":"hint","dependencies":[],"title":"Use PEMDAS","text":"We must simplify operations in the following order: parentheses, exponents, $$\\\\frac{multiplication}{division}$$ $$(left-to-right)$$, $$\\\\frac{addition}{subttraction}$$ $$(left-to-right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang22a-h2","type":"hint","dependencies":["afb3ccdlang22a-h1"],"title":"Simplifying the Terms","text":"$$3\\\\left(1+9\\\\times6\\\\right)-4^2=3\\\\left(1+54\\\\right)-4^2=3\\\\left(55\\\\right)-4^2=3(55)-16=165-16=149$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afb3ccdlang23","title":"Simplifying Expressions","body":"Simplify the given expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Use the Language of Algebra","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"afb3ccdlang23a","stepAnswer":["$$121$$"],"problemType":"TextBox","stepTitle":"Simplify $$5\\\\left(2+8\\\\times4\\\\right)-7^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$121$$","hints":{"DefaultPathway":[{"id":"afb3ccdlang23a-h1","type":"hint","dependencies":[],"title":"Use PEMDAS","text":"We must simplify operations in the following order: parentheses, exponents, $$\\\\frac{multiplication}{division}$$ $$(left-to-right)$$, $$\\\\frac{addition}{subttraction}$$ $$(left-to-right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang23a-h2","type":"hint","dependencies":["afb3ccdlang23a-h1"],"title":"Simplifying the Terms","text":"$$5\\\\left(2+8\\\\times4\\\\right)-7^2=5\\\\left(2+32\\\\right)-7^2=5\\\\left(36\\\\right)-7^2=5\\\\times36-49=121$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afb3ccdlang24","title":"Simplifying Expressions","body":"Simplify the given expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Use the Language of Algebra","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"afb3ccdlang24a","stepAnswer":["$$50$$"],"problemType":"TextBox","stepTitle":"Simplify $$2\\\\left(1+3\\\\left(10-2\\\\right)\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$50$$","hints":{"DefaultPathway":[{"id":"afb3ccdlang24a-h1","type":"hint","dependencies":[],"title":"Use PEMDAS","text":"We must simplify operations in the following order: parentheses, exponents, $$\\\\frac{multiplication}{division}$$ $$(left-to-right)$$, $$\\\\frac{addition}{subttraction}$$ $$(left-to-right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang24a-h2","type":"hint","dependencies":["afb3ccdlang24a-h1"],"title":"Simplifying the Terms","text":"$$2\\\\left(1+3\\\\left(8\\\\right)\\\\right)=2\\\\left(1+24\\\\right)=2(25)=50$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afb3ccdlang25","title":"Simplifying Expressions","body":"Simplify the given expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Use the Language of Algebra","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"afb3ccdlang25a","stepAnswer":["$$30$$"],"problemType":"TextBox","stepTitle":"Simplify $$5\\\\left(2+4\\\\left(3-2\\\\right)\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$30$$","hints":{"DefaultPathway":[{"id":"afb3ccdlang25a-h1","type":"hint","dependencies":[],"title":"Use PEMDAS","text":"We must simplify operations in the following order: parentheses, exponents, $$\\\\frac{multiplication}{division}$$ $$(left-to-right)$$, $$\\\\frac{addition}{subttraction}$$ $$(left-to-right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang25a-h2","type":"hint","dependencies":["afb3ccdlang25a-h1"],"title":"Simplifying the Terms","text":"$$5\\\\left(2+4\\\\left(3-2\\\\right)\\\\right)=5\\\\left(2+4\\\\left(1\\\\right)\\\\right)=5\\\\times6=30$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afb3ccdlang26","title":"Simplifying Expressions","body":"Simplify the given expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Use the Language of Algebra","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"afb3ccdlang26a","stepAnswer":["$$5$$"],"problemType":"TextBox","stepTitle":"Simplify $$8+2\\\\left(7-2\\\\left(5-3\\\\right)\\\\right)-3^2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$5$$","hints":{"DefaultPathway":[{"id":"afb3ccdlang26a-h1","type":"hint","dependencies":[],"title":"Use PEMDAS","text":"We must simplify operations in the following order: parentheses, exponents, $$\\\\frac{multiplication}{division}$$ $$(left-to-right)$$, $$\\\\frac{addition}{subttraction}$$ $$(left-to-right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang26a-h2","type":"hint","dependencies":["afb3ccdlang26a-h1"],"title":"Simplifying the Terms","text":"$$8+2\\\\left(7-2\\\\left(5-3\\\\right)\\\\right)-3^2=8+2\\\\left(7-2\\\\left(2\\\\right)\\\\right)-3^2=8+2\\\\left(7-4\\\\right)-3^2=8+2\\\\left(3\\\\right)-3^2=8+2\\\\left(3\\\\right)-9=8+6-9=14-9=5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afb3ccdlang27","title":"Simplifying Expressions","body":"Simplify the given expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Use the Language of Algebra","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"afb3ccdlang27a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"Simplify $$10+3\\\\left(6-2\\\\left(4-2\\\\right)\\\\right)-2^4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"afb3ccdlang27a-h1","type":"hint","dependencies":[],"title":"Use PEMDAS","text":"We must simplify operations in the following order: parentheses, exponents, $$\\\\frac{multiplication}{division}$$ $$(left-to-right)$$, $$\\\\frac{addition}{subttraction}$$ $$(left-to-right)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang27a-h2","type":"hint","dependencies":["afb3ccdlang27a-h1"],"title":"Simplifying the Terms","text":"$$10+3\\\\left(6-2\\\\left(4-2\\\\right)\\\\right)-2^4=10+3\\\\left(6-2\\\\left(2\\\\right)\\\\right)-2^4=10+3\\\\left(6-4\\\\right)-2^4=10+3\\\\left(2\\\\right)-2^4=10+3\\\\left(2\\\\right)-16=10+6-16=0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afb3ccdlang28","title":"Evaluating Expressions","body":"Solve for the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Use the Language of Algebra","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"afb3ccdlang28a","stepAnswer":["$$64$$"],"problemType":"TextBox","stepTitle":"What is $$x^6$$ when $$x=2$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$64$$","hints":{"DefaultPathway":[{"id":"afb3ccdlang28a-h1","type":"hint","dependencies":[],"title":"Plugging In For $$x$$","text":"if $$x=2$$ in $$x^6$$, then we have $$2^6$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang28a-h2","type":"hint","dependencies":["afb3ccdlang28a-h1"],"title":"Simplifying the Expression","text":"$$2^6=64$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afb3ccdlang29","title":"Evaluating Expressions","body":"Solve for the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Use the Language of Algebra","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"afb3ccdlang29a","stepAnswer":["$$243$$"],"problemType":"TextBox","stepTitle":"What is $$x^5$$ when $$x=3$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$243$$","hints":{"DefaultPathway":[{"id":"afb3ccdlang29a-h1","type":"hint","dependencies":[],"title":"Plugging In For $$x$$","text":"if $$x=3$$ in $$x^5$$, then we have $$3^5$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang29a-h2","type":"hint","dependencies":["afb3ccdlang29a-h1"],"title":"Simplifying the Expression","text":"$$3^5=243$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afb3ccdlang3","title":"Find Factors, Prime Factorizations, and Least Common Multiples","body":"Find the least common multiple (LCM) of the numbers using the prime factor method.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Use the Language of Algebra","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"afb3ccdlang3a","stepAnswer":["$$36$$"],"problemType":"TextBox","stepTitle":"$$12$$ and $$18$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$36$$","hints":{"DefaultPathway":[{"id":"afb3ccdlang3a-h1","type":"hint","dependencies":[],"title":"First, we find the prime factors of both number: $$2\\\\times3\\\\times3$$ $$=18$$, $$2\\\\times2\\\\times3=12$$.","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang3a-h2","type":"hint","dependencies":["afb3ccdlang3a-h1"],"title":"Then, we multiple the factors: $$2\\\\times2\\\\times3\\\\times3$$ $$=$$ $$36$$","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afb3ccdlang30","title":"Evaluating Expressions","body":"Solve for the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Use the Language of Algebra","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"afb3ccdlang30a","stepAnswer":["$$21$$"],"problemType":"TextBox","stepTitle":"What is $$x^2+3xy-7y^2$$ wheen $$x=4, y=1$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$21$$","hints":{"DefaultPathway":[{"id":"afb3ccdlang30a-h1","type":"hint","dependencies":[],"title":"Plugging In For $$x$$ and $$y$$","text":"Plugging in $$x=4$$ and $$y=1$$, we get $$4^2+13\\\\left(4\\\\right)-{7\\\\left(1\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang30a-h2","type":"hint","dependencies":["afb3ccdlang30a-h1"],"title":"Simplifying the Expression","text":"Simplifying the given expression using PEMDAS, we get $$21$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afb3ccdlang31","title":"Evaluating Expressions","body":"Solve for the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Use the Language of Algebra","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"afb3ccdlang31a","stepAnswer":["$$36$$"],"problemType":"TextBox","stepTitle":"What is $$6x^2+3xy-9y^2$$ when $$x=3, y=2$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$36$$","hints":{"DefaultPathway":[{"id":"afb3ccdlang31a-h1","type":"hint","dependencies":[],"title":"Plugging In For $$x$$ and $$y$$","text":"Plugging in $$x=3, y=2$$, we get $${6\\\\left(3\\\\right)}^2+23\\\\left(3\\\\right)-{9\\\\left(2\\\\right)}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang31a-h2","type":"hint","dependencies":["afb3ccdlang31a-h1"],"title":"Simplifying the Expression","text":"$${6\\\\left(3\\\\right)}^2+23\\\\left(3\\\\right)-{9\\\\left(2\\\\right)}^2=54+18-36=36$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afb3ccdlang32","title":"Evaluating Expressions","body":"Solve for the value of the expression.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Use the Language of Algebra","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"afb3ccdlang32a","stepAnswer":["$$73$$"],"problemType":"TextBox","stepTitle":"What is $$a^2+b^2$$ when $$a=3, b=8$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$73$$","hints":{"DefaultPathway":[{"id":"afb3ccdlang32a-h1","type":"hint","dependencies":[],"title":"Plugging In For a and $$b$$","text":"Plugging in $$a=3, b=8$$, we get $$3^2+8^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang32a-h2","type":"hint","dependencies":["afb3ccdlang32a-h1"],"title":"Simplifying the Expression","text":"$$3^2+8^2=9+64=53$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afb3ccdlang4","title":"Simplify Expressions Using the Order of Operations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Use the Language of Algebra","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"afb3ccdlang4a","stepAnswer":["$$15$$"],"problemType":"TextBox","stepTitle":"Simplify: $$\\\\frac{18}{6}+4\\\\left(5-2\\\\right)$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$15$$","hints":{"DefaultPathway":[{"id":"afb3ccdlang4a-h1","type":"hint","dependencies":[],"title":"Parentheses","text":"Since there are parentheses, we do the calculations inside the parentheses first.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang4a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["afb3ccdlang4a-h1"],"title":"Parentheses","text":"what is $$5$$ -2?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang4a-h3","type":"hint","dependencies":["afb3ccdlang4a-h2"],"title":"$$\\\\frac{Multiplication}{Division}$$","text":"Then, we calculate the division and multiplication.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["afb3ccdlang4a-h3"],"title":"$$\\\\frac{Multiplication}{Division}$$","text":"what is $$\\\\frac{18}{6}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang4a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$12$$"],"dependencies":["afb3ccdlang4a-h4"],"title":"$$\\\\frac{Multiplication}{Division}$$","text":"What is $$4\\\\times3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang4a-h6","type":"hint","dependencies":["afb3ccdlang4a-h5"],"title":"$$\\\\frac{Addition}{Subtraction}$$","text":"The last step is to do the addition and subtraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang4a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["afb3ccdlang4a-h6"],"title":"$$\\\\frac{Addition}{Subtraction}$$","text":"What is $$3+12$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afb3ccdlang5","title":"Simplify Expressions Using the Order of Operations","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Use the Language of Algebra","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"afb3ccdlang5a","stepAnswer":["$$13$$"],"problemType":"TextBox","stepTitle":"Simplify: 5+2**3+3*[6-3*(4-2)].","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$13$$","hints":{"DefaultPathway":[{"id":"afb3ccdlang5a-h1","type":"hint","dependencies":[],"title":"Parentheses","text":"Since there are parentheses, we do the calculations inside the parentheses first.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang5a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["afb3ccdlang5a-h1"],"title":"Parentheses","text":"What is $$4-2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang5a-h3","type":"hint","dependencies":["afb3ccdlang5a-h2"],"title":"Parentheses","text":"Since there are still parentheses, we do the calculations inside the parentheses. There are multiplication and subtraction inside the parentheses, we do the multiplication first.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang5a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["afb3ccdlang5a-h3"],"title":"Parentheses","text":"What is $$3\\\\times2$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang5a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["afb3ccdlang5a-h4"],"title":"Parentheses","text":"What is $$6-6$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang5a-h6","type":"hint","dependencies":["afb3ccdlang5a-h5"],"title":"Exponents","text":"Outsdie the parentheses, there are exponents, we do the calculation for it.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang5a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8$$"],"dependencies":["afb3ccdlang5a-h6"],"title":"Exponents","text":"What is $$2^3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang5a-h8","type":"hint","dependencies":["afb3ccdlang5a-h7"],"title":"$$\\\\frac{Addition}{Subtraction}$$","text":"The last step is to do the addition and subtraction.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang5a-h9","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$13$$"],"dependencies":["afb3ccdlang5a-h8"],"title":"$$\\\\frac{Addition}{Subtraction}$$","text":"What is $$5$$ + $$8$$ + 0?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afb3ccdlang6","title":"Evaluate an Expression","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Use the Language of Algebra","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"afb3ccdlang6a","stepAnswer":["$$16$$"],"problemType":"TextBox","stepTitle":"When $$x$$ $$=$$ $$4$$, evaluate $$x^2$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$16$$","hints":{"DefaultPathway":[{"id":"afb3ccdlang6a-h1","type":"hint","dependencies":[],"title":"Exponents","text":"First, we replace $$x$$ with $$4$$, so we will be evaluating $$4^2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$16$$"],"dependencies":[],"title":"Exponents","text":"What is $$4\\\\times4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"afb3ccdlang6b","stepAnswer":["$$81$$"],"problemType":"TextBox","stepTitle":"When $$x$$ $$=$$ $$4$$, evaluate $$3^x$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$81$$","hints":{"DefaultPathway":[{"id":"afb3ccdlang6b-h1","type":"hint","dependencies":[],"title":"Exponents","text":"First, we replace $$x$$ with $$4$$, so we will be evaluating $$3^4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang6b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$81$$"],"dependencies":[],"title":"Exponents","text":"What is $$3\\\\times3\\\\times3\\\\times3$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"afb3ccdlang6c","stepAnswer":["$$52$$"],"problemType":"TextBox","stepTitle":"When $$x$$ $$=$$ $$4$$, evaluate $$2x^2+3x+8$$?","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$52$$","hints":{"DefaultPathway":[{"id":"afb3ccdlang6c-h1","type":"hint","dependencies":[],"title":"Replace","text":"First, we replace all the $$x$$ with $$4$$, so we will be evaluating $$2\\\\times4^2$$ + $$3\\\\times4$$ $$+8$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang6c-h2","type":"hint","dependencies":["afb3ccdlang6c-h1"],"title":"Order of Operations","text":"Then, we follow the order of operations to calculate.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang6c-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$52$$"],"dependencies":["afb3ccdlang6c-h2"],"title":"Order of Operations","text":"What is $$32$$ + $$12$$ + 8?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afb3ccdlang7","title":"Identify and Combine Like Terms","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Use the Language of Algebra","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"afb3ccdlang7a","stepAnswer":["$$3x^2+7x+12$$"],"problemType":"TextBox","stepTitle":"Simplify: $$2x^2+3x+7+x^2+4x+5$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$3x^2+7x+12$$","hints":{"DefaultPathway":[{"id":"afb3ccdlang7a-h1","type":"hint","dependencies":[],"title":"Like terms","text":"First, we identify the like terms (terms that have the same exponents).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang7a-h2","type":"hint","dependencies":["afb3ccdlang7a-h1"],"title":"Combination","text":"Then, we combine the like terms (adding the coefficient of the like terms), and get $$3^2+7x+12$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afb3ccdlang8","title":"Translate an English Phrase to an Algebraic Expression","body":"","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Use the Language of Algebra","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"afb3ccdlang8a","stepAnswer":["$$8\\\\left(x+y\\\\right)$$"],"problemType":"TextBox","stepTitle":"Translate the English phrase into an algebraic expression: eight times the sum of $$x$$ and $$y$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8\\\\left(x+y\\\\right)$$","hints":{"DefaultPathway":[{"id":"afb3ccdlang8a-h1","type":"hint","dependencies":[],"title":"Math Operations","text":"Times tell us to multiply and sum tells us to add.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang8a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8\\\\left(x+y\\\\right)$$"],"dependencies":["afb3ccdlang8a-h1"],"title":"Math Operations","text":"Because we are multiplying $$8$$ times the sum, we need parentheses around the sum of $$x$$ and $$y$$, $$x+y$$. This forces us to determine the sum first.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"afb3ccdlang8b","stepAnswer":["$$8x+y$$"],"problemType":"TextBox","stepTitle":"Translate the English phrase into an algebraic expression: the sum of eight times $$x$$ and $$y$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$8x+y$$","hints":{"DefaultPathway":[{"id":"afb3ccdlang8b-h1","type":"hint","dependencies":[],"title":"Math Operations","text":"Times tell us to multiply and sum tells us to add.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang8b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$8\\\\left(x+y\\\\right)$$"],"dependencies":["afb3ccdlang8b-h1"],"title":"Math Operations","text":"To take a sum, we look for the words of and and to see what is being added. Here we are taking the sum of eight times $$x$$ and $$y$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afb3ccdlang9","title":"Translate an English Phrase to an Algebraic Expression","body":"The length of a rectangle is $$14$$ less than the width. Let w represent the width of the rectangle.","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"1.1 Use the Language of Algebra","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"afb3ccdlang9a","stepAnswer":["w-14"],"problemType":"TextBox","stepTitle":"Write an expression for the length of the rectangle.","stepBody":"","answerType":"arithmetic","variabilization":{},"hints":{"DefaultPathway":[{"id":"afb3ccdlang9a-h1","type":"hint","dependencies":[],"title":"Write a phrase","text":"Write a phrase about the length of the rectangl: $$14$$ less than the width","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang9a-h2","type":"hint","dependencies":["afb3ccdlang9a-h1"],"title":"Substitution","text":"Then, we substitute w for \\"the width\\".","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang9a-h3","type":"hint","dependencies":["afb3ccdlang9a-h2"],"title":"Subtration","text":"Rewrite less than as subtracted from.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afb3ccdlang9a-h4","type":"hint","dependencies":["afb3ccdlang9a-h3"],"title":"Translate","text":"Translate the phrase into algebra: w - $$14$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afbe5ccpower1","title":"Identifying the End Behavior of a Power Function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.2 Power Functions and 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<OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbe5ccpower18a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["afbe5ccpower18a-h3"],"title":"Finding the Degree","text":"What is the degree?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afbe5ccpower19","title":"Determing the Leading Coefficient and Degree","body":"For the following exercises, find the degree and leading coefficient for the given polynomial.","variabilization":{},"oer":"https://openstax.org/details/books/college-algebra-2e <OpenStax: College Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"5.2 Power Functions and Polynomial Functions","courseName":"OpenStax: College Algebra","steps":[{"id":"afbe5ccpower19a","stepAnswer":["Leading Coefficient: $$-3$$, Degree: 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found by evaluating f(0)","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbe5ccpower9a-h2","type":"hint","dependencies":["afbe5ccpower9a-h1"],"title":"f(0) $$=$$ $$-4(0)(0$$ + 3)(0 - 4) $$=$$ $$0$$","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbe5ccpower9a-h3","type":"hint","dependencies":["afbe5ccpower9a-h2"],"title":"The y-intercept is $$(0,0)$$","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbe5ccpower9a-h4","type":"hint","dependencies":["afbe5ccpower9a-h3"],"title":"The x-intercepts are found by determining the zeros of het function","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbe5ccpower9a-h5","type":"hint","dependencies":["afbe5ccpower9a-h4"],"title":"$$0$$ $$=$$ -4x(x + 3)(x - 4). $$x$$ $$=$$ $$0$$ or $$x$$ $$=$$ $$-3$$ or $$x$$ $$=$$ $$4$$","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbe5ccpower9a-h6","type":"hint","dependencies":["afbe5ccpower9a-h5"],"title":"The x-intercepts are $$(0,0)$$, $$(-3,0)$$, and $$(4,0)$$","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbe5ccpower9a-h7","type":"hint","dependencies":["afbe5ccpower9a-h6"],"title":"The degree is $$3$$ so the graph has at most $$2$$ turning points","text":"","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afbeae7inequalities1","title":"Determining Solutions","body":"Determine whether the given ordered pair is a valid solution.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphs of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afbeae7inequalities1a","stepAnswer":["The ordered pair is not a solution."],"problemType":"MultipleChoice","stepTitle":"Determine whether $$(0,0)$$ is a solution to the inequality $$y>x+4$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["The ordered pair is a solution.IThe ordered pair is not a solution.","The ordered pair is not a solution."],"hints":{"DefaultPathway":[{"id":"afbeae7inequalities1a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute the $$x$$ and $$y$$ values of the ordered pair into the inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities1a-h2","type":"hint","dependencies":["afbeae7inequalities1a-h1"],"title":"Simplify","text":"Simplify the equation so that there is a single digit on each side of the inequality symbol.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities1a-h3","type":"hint","dependencies":["afbeae7inequalities1a-h2"],"title":"Interpret","text":"The equation in this case simplifies to $$0>4$$, which is not a true statement. Therefore, $$(0,0)$$ is not a solution to the inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afbeae7inequalities10","title":"Identifying Graphed Inequalities","body":"Choose the correct inequality shown by the graph.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphs of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afbeae7inequalities10a","stepAnswer":["$$x+2y \\\\leq -2$$"],"problemType":"MultipleChoice","stepTitle":"The boundary line shown is $$x+2y=-2$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x+2y \\\\leq -2$$","choices":["$$x+2y \\\\leq -2$$","$$x+2y \\\\geq -2Ix+2y \\\\leq -2Ix+2y>-2Ix+2y<-2$$"],"hints":{"DefaultPathway":[{"id":"afbeae7inequalities10a-h1","type":"hint","dependencies":[],"title":"Test","text":"We need to test the inequality to determine the signage. Plug the point $$(0,0)$$ into the equation twice, once with the equation being $$x+2y>-2$$, and once with $$x+2y<-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities10a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x+2y<-2$$"],"dependencies":["afbeae7inequalities10a-h1"],"title":"Find the correct sign","text":"In this case, when substituting $$(0,0)$$ into the equation, does $$x+2y>-2$$ result in a true statement, or does $$x+2y<-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x+2y<-2Ix+2y>-2$$"],"subHints":[{"id":"afbeae7inequalities10a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$x+2y<-2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"afbeae7inequalities10a-h3","type":"hint","dependencies":["afbeae7inequalities10a-h2"],"title":"Determine line type","text":"If the line is solid, the sign will include the values of the line. In other words, the sign will be greater than or equal to, or less than or equal to. If the line is dotted, the sign will simply be > or <.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities10a-h4","type":"hint","dependencies":["afbeae7inequalities10a-h3"],"title":"Answer","text":"Therefore, the equation for the inequality is $$x+2y \\\\leq -2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afbeae7inequalities11","title":"Identifying Graphed Inequalities","body":"Choose the correct inequality shown by the graph.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphs of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afbeae7inequalities11a","stepAnswer":["$$2x-y<4$$"],"problemType":"MultipleChoice","stepTitle":"The boundary line shown is $$2x-y=4$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2x-y<4$$","choices":["$$2x-y<4$$","$$2x-y \\\\geq 4I2x-y \\\\leq 4I2 x-y>4I2 x-y<4$$"],"hints":{"DefaultPathway":[{"id":"afbeae7inequalities11a-h1","type":"hint","dependencies":[],"title":"Test","text":"We need to test the inequality to determine the signage. Plug the point $$(0,0)$$ into the equation twice, once with the equation being $$2x-y>4$$, and once with $$2x-y<4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities11a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2x-y<4$$"],"dependencies":["afbeae7inequalities11a-h1"],"title":"Find the correct sign","text":"In this case, when substituting $$(0,0)$$ into the equation, does $$2x-y<4$$ result in a true statement, or does $$2x-y<4$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$2x-y<4I2 x-y>4$$"],"subHints":[{"id":"afbeae7inequalities11a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$2x-y<4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"afbeae7inequalities11a-h3","type":"hint","dependencies":["afbeae7inequalities11a-h2"],"title":"Determine line type","text":"If the line is solid, the sign will include the values of the line. In other words, the sign will be greater than or equal to, or less than or equal to. If the line is dotted, the sign will simply be > or <.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities11a-h4","type":"hint","dependencies":["afbeae7inequalities11a-h3"],"title":"Answer","text":"Therefore, the equation for the inequality is $$2x-y<4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afbeae7inequalities12","title":"Identifying Graphed Inequalities","body":"Choose the correct inequality shown by the graph.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphs of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afbeae7inequalities12a","stepAnswer":["$$4x-3y>12$$"],"problemType":"MultipleChoice","stepTitle":"The boundary line shown is $$4x-3y=12$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$4x-3y>12$$","choices":["$$4x-3y>12$$","$$4x-3y \\\\geq 12I4x-3y \\\\leq 12I4 x-3y>12I4 x-3y<12$$"],"hints":{"DefaultPathway":[{"id":"afbeae7inequalities12a-h1","type":"hint","dependencies":[],"title":"Test","text":"We need to test the inequality to determine the signage. Plug the point $$(0,0)$$ into the equation twice, once with the equation being $$4x-3y>12$$, and once with $$4x-3y<12$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities12a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$4x-3y>12$$"],"dependencies":["afbeae7inequalities12a-h1"],"title":"Find the correct sign","text":"In this case, when substituting $$(0,0)$$ into the equation, does $$4x+3y>12$$ result in a true statement, or does $$4x+3y<12$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$4x-3y<12I4 x-3y>12$$"],"subHints":[{"id":"afbeae7inequalities12a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$4x-3y<12$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"afbeae7inequalities12a-h3","type":"hint","dependencies":["afbeae7inequalities12a-h2"],"title":"Determine line type","text":"If the line is solid, the sign will include the values of the line. In other words, the sign will be greater than or equal to, or less than or equal to. If the line is dotted, the sign will simply be > or <.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities12a-h4","type":"hint","dependencies":["afbeae7inequalities12a-h3"],"title":"Answer","text":"Therefore, the equation for the inequality is $$4x-3y<12$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afbeae7inequalities13","title":"Determining Solutions","body":"Determine whether the given ordered pair is a valid solution.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphs of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afbeae7inequalities13a","stepAnswer":["The ordered pair is a solution."],"problemType":"MultipleChoice","stepTitle":"Determine whether $$(0,1)$$ is a solution to the inequality $$y>x-1$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["The ordered pair is a solution.","The ordered pair is a solution.IThe ordered pair is not a solution."],"hints":{"DefaultPathway":[{"id":"afbeae7inequalities13a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute the $$x$$ and $$y$$ values of the ordered pair into the inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities13a-h2","type":"hint","dependencies":["afbeae7inequalities13a-h1"],"title":"Simplify","text":"Simplify the equation so that there is a single digit on each side of the inequality symbol.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities13a-h3","type":"hint","dependencies":["afbeae7inequalities13a-h2"],"title":"Interpret","text":"The equation in this case simplifies to $$1>-1$$, which is a true statement. Therefore, $$(0,1)$$ is a solution to the inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afbeae7inequalities14","title":"Determining Solutions","body":"Determine whether the given ordered pair is a valid solution.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphs of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afbeae7inequalities14a","stepAnswer":["The ordered pair is a solution."],"problemType":"MultipleChoice","stepTitle":"Determine whether $$(-4,-1)$$ is a solution to the inequality $$y>x-1$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["The ordered pair is a solution.","The ordered pair is a solution.IThe ordered pair is not a solution."],"hints":{"DefaultPathway":[{"id":"afbeae7inequalities14a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute the $$x$$ and $$y$$ values of the ordered pair into the inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities14a-h2","type":"hint","dependencies":["afbeae7inequalities14a-h1"],"title":"Simplify","text":"Simplify the equation so that there is a single digit on each side of the inequality symbol.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities14a-h3","type":"hint","dependencies":["afbeae7inequalities14a-h2"],"title":"Interpret","text":"The equation in this case simplifies to $$-1>-5$$, which is a true statement. Therefore, $$(-4,-1)$$ is a solution to the inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afbeae7inequalities15","title":"Determining Solutions","body":"Determine whether the given ordered pair is a valid solution.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphs of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afbeae7inequalities15a","stepAnswer":["The ordered pair is not a solution."],"problemType":"MultipleChoice","stepTitle":"Determine whether $$(4,2)$$ is a solution to the inequality $$y>x-1$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["The ordered pair is a solution.IThe ordered pair is not a solution.","The ordered pair is not a solution."],"hints":{"DefaultPathway":[{"id":"afbeae7inequalities15a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute the $$x$$ and $$y$$ values of the ordered pair into the inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities15a-h2","type":"hint","dependencies":["afbeae7inequalities15a-h1"],"title":"Simplify","text":"Simplify the equation so that there is a single digit on each side of the inequality symbol.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities15a-h3","type":"hint","dependencies":["afbeae7inequalities15a-h2"],"title":"Interpret","text":"The equation in this case simplifies to $$2>3$$, which is not a true statement. Therefore, $$(4,2)$$ is not a solution to the inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afbeae7inequalities16","title":"Graph Linear Inequalities","body":"Graph the linear inequality.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphs of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afbeae7inequalities16a","stepAnswer":["A"],"problemType":"MultipleChoice","stepTitle":"$$y>\\\\frac{2}{3} x-1$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"afbeae7inequalities16a-h1","type":"hint","dependencies":[],"title":"y-intercept","text":"By plugging in the value $$x=0$$, we can conclude that the line will cross over $$(0,-1)$$ as the y-intercept","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities16a-h2","type":"hint","dependencies":["afbeae7inequalities16a-h1"],"title":"Slope","text":"Given the equation, we can conclude that the slope of the equation is $$\\\\frac{2}{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities16a-h3","type":"hint","dependencies":["afbeae7inequalities16a-h2"],"title":"Line","text":"Because the equation does not include an \\"equal to\\" sign, it is a dotted line and the solution will not be on that line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities16a-h4","type":"hint","dependencies":["afbeae7inequalities16a-h3"],"title":"Plug in","text":"Plug in any coordinate value that is either on top or under the line but not on the line. If the value is accepted by the equation, then shade the side of the graph the point is on. (Ex: If point is above line and works, shade the area above line vice versa)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities16a-h5","type":"hint","dependencies":["afbeae7inequalities16a-h4"],"title":"Match","text":"Match your graph with the graph in the answer choices","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afbeae7inequalities17","title":"Graph Linear Inequalities","body":"Graph the linear inequality.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphs of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afbeae7inequalities17a","stepAnswer":["B"],"problemType":"MultipleChoice","stepTitle":"$$y<\\\\frac{3}{5} x+2$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"afbeae7inequalities17a-h1","type":"hint","dependencies":[],"title":"y-intercept","text":"By plugging in the value $$x=0$$, we can conclude that the line will cross over $$(0,2)$$ as the y-intercept","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities17a-h2","type":"hint","dependencies":["afbeae7inequalities17a-h1"],"title":"Slope","text":"Given the equation, we can conclude that the slope of the equation is $$\\\\frac{3}{5}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities17a-h3","type":"hint","dependencies":["afbeae7inequalities17a-h2"],"title":"Line","text":"Because the equation does not include an \\"equal to\\" sign, it is a dotted line and the solution will not be on that line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities17a-h4","type":"hint","dependencies":["afbeae7inequalities17a-h3"],"title":"Plug in","text":"Plug in any coordinate value that is either on top or under the line but not on the line. If the value is accepted by the equation, then shade the side of the graph the point is on. (Ex: If point is above line and works, shade the area above line vice versa)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities17a-h5","type":"hint","dependencies":["afbeae7inequalities17a-h4"],"title":"Match","text":"Match your graph with the graph in the answer choices","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afbeae7inequalities18","title":"Graph Linear Inequalities","body":"Graph the linear inequality.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphs of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afbeae7inequalities18a","stepAnswer":["A"],"problemType":"MultipleChoice","stepTitle":"$$y \\\\leq \\\\left(-\\\\frac{1}{2}\\\\right) x+4$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"afbeae7inequalities18a-h1","type":"hint","dependencies":[],"title":"y-intercept","text":"By plugging in the value $$x=0$$, we can conclude that the line will cross over $$(0,4)$$ as the y-intercept","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities18a-h2","type":"hint","dependencies":["afbeae7inequalities18a-h1"],"title":"Slope","text":"Given the equation, we can conclude that the slope of the equation is $$\\\\frac{-1}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities18a-h3","type":"hint","dependencies":["afbeae7inequalities18a-h2"],"title":"Line","text":"Because the equation includes an \\"equal to\\" sign, it is a straight, non-dotted line, and the solution can be on the line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities18a-h4","type":"hint","dependencies":["afbeae7inequalities18a-h3"],"title":"Plug in","text":"Plug in any coordinate value that is either on top or under the line but not on the line. If the value is accepted by the equation, then shade the side of the graph the point is on. (Ex: If point is above line and works, shade the area above line vice versa)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities18a-h5","type":"hint","dependencies":["afbeae7inequalities18a-h4"],"title":"Match","text":"Match your graph with the graph in the answer choices","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afbeae7inequalities19","title":"Graph Linear Inequalities","body":"Graph the linear inequality.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphs of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afbeae7inequalities19a","stepAnswer":["D"],"problemType":"MultipleChoice","stepTitle":"$$y \\\\geq \\\\left(-\\\\frac{1}{3}\\\\right) x-2$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"afbeae7inequalities19a-h1","type":"hint","dependencies":[],"title":"y-intercept","text":"By plugging in the value $$x=0$$, we can conclude that the line will cross over $$(0,-2)$$ as the y-intercept","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities19a-h2","type":"hint","dependencies":["afbeae7inequalities19a-h1"],"title":"Slope","text":"Given the equation, we can conclude that the slope of the equation is $$\\\\frac{-1}{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities19a-h3","type":"hint","dependencies":["afbeae7inequalities19a-h2"],"title":"Line","text":"Because the equation includes an \\"equal to\\" sign, it is a straight, non-dotted line, and the solution can be on the line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities19a-h4","type":"hint","dependencies":["afbeae7inequalities19a-h3"],"title":"Plug in","text":"Plug in any coordinate value that is either on top or under the line but not on the line. If the value is accepted by the equation, then shade the side of the graph the point is on. (Ex: If point is above line and works, shade the area above line vice versa)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities19a-h5","type":"hint","dependencies":["afbeae7inequalities19a-h4"],"title":"Match","text":"Match your graph with the graph in the answer choices","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afbeae7inequalities2","title":"Determining Solutions","body":"Determine whether the given ordered pair is a valid solution.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphs of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afbeae7inequalities2a","stepAnswer":["The ordered pair is a solution."],"problemType":"MultipleChoice","stepTitle":"Determine whether $$(1,6)$$ is a solution to the inequality $$y>x+4$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["The ordered pair is a solution.","The ordered pair is a solution.IThe ordered pair is not a solution."],"hints":{"DefaultPathway":[{"id":"afbeae7inequalities2a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute the $$x$$ and $$y$$ values of the ordered pair into the inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities2a-h2","type":"hint","dependencies":["afbeae7inequalities2a-h1"],"title":"Simplify","text":"Simplify the equation so that there is a single digit on each side of the inequality symbol.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities2a-h3","type":"hint","dependencies":["afbeae7inequalities2a-h2"],"title":"Interpret","text":"The equation in this case simplifies to $$6>5$$, which is a true statement. Therefore, $$(1,6)$$ is a solution to the inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afbeae7inequalities20","title":"Graph Linear Inequalities","body":"Graph the linear inequality.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphs of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afbeae7inequalities20a","stepAnswer":["B"],"problemType":"MultipleChoice","stepTitle":"$$x-y \\\\leq 3$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"afbeae7inequalities20a-h1","type":"hint","dependencies":[],"title":"Convert","text":"$$x-y \\\\leq 3$$ can be rewritten as $$y \\\\geq x-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities20a-h2","type":"hint","dependencies":["afbeae7inequalities20a-h1"],"title":"y-intercept","text":"By plugging in the value $$x=0$$, we can conclude that the line will cross over $$(0,-3)$$ as the y-intercept","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities20a-h3","type":"hint","dependencies":["afbeae7inequalities20a-h2"],"title":"Slope","text":"Given the equation, we can conclude that the slope of the equation is $$1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities20a-h4","type":"hint","dependencies":["afbeae7inequalities20a-h3"],"title":"Line","text":"Because the equation includes an \\"equal to\\" sign, it is a straight, non-dotted line, and the solution can be on the line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities20a-h5","type":"hint","dependencies":["afbeae7inequalities20a-h4"],"title":"Plug in","text":"Plug in any coordinate value that is either on top or under the line but not on the line. If the value is accepted by the equation, then shade the side of the graph the point is on. (Ex: If point is above line and works, shade the area above line vice versa)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities20a-h6","type":"hint","dependencies":["afbeae7inequalities20a-h5"],"title":"Match","text":"Match your graph with the graph in the answer choices","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afbeae7inequalities21","title":"Graph Linear Inequalities","body":"Graph the linear inequality.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphs of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afbeae7inequalities21a","stepAnswer":["B"],"problemType":"MultipleChoice","stepTitle":"$$x-y \\\\geq -2$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"afbeae7inequalities21a-h1","type":"hint","dependencies":[],"title":"Convert","text":"$$x-y \\\\geq -2$$ can be rewritten as $$y \\\\leq x+2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities21a-h2","type":"hint","dependencies":["afbeae7inequalities21a-h1"],"title":"y-intercept","text":"By plugging in the value $$x=0$$, we can conclude that the line will cross over $$(0,2)$$ as the y-intercept","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities21a-h3","type":"hint","dependencies":["afbeae7inequalities21a-h2"],"title":"Slope","text":"Given the equation, we can conclude that the slope of the equation is $$1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities21a-h4","type":"hint","dependencies":["afbeae7inequalities21a-h3"],"title":"Line","text":"Because the equation includes an \\"equal to\\" sign, it is a straight, non-dotted line, and the solution can be on the line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities21a-h5","type":"hint","dependencies":["afbeae7inequalities21a-h4"],"title":"Plug in","text":"Plug in any coordinate value that is either on top or under the line but not on the line. If the value is accepted by the equation, then shade the side of the graph the point is on. (Ex: If point is above line and works, shade the area above line vice versa)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities21a-h6","type":"hint","dependencies":["afbeae7inequalities21a-h5"],"title":"Match","text":"Match your graph with the graph in the answer choices","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afbeae7inequalities22","title":"Graph Linear Inequalities","body":"Graph the linear inequality.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphs of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afbeae7inequalities22a","stepAnswer":["A"],"problemType":"MultipleChoice","stepTitle":"$$4x+y>-4$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"afbeae7inequalities22a-h1","type":"hint","dependencies":[],"title":"Convert","text":"The equation can be rewritten as $$y>-4x-4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities22a-h2","type":"hint","dependencies":["afbeae7inequalities22a-h1"],"title":"y-intercept","text":"By plugging in the value $$x=0$$, we can conclude that the line will cross over $$(0,-4)$$ as the y-intercept","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities22a-h3","type":"hint","dependencies":["afbeae7inequalities22a-h2"],"title":"Slope","text":"Given the equation, we can conclude that the slope of the equation is $$-4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities22a-h4","type":"hint","dependencies":["afbeae7inequalities22a-h3"],"title":"Line","text":"Because the equation does not include an \\"equal to\\" sign, it is a dotted line and the solution will not be on that line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities22a-h5","type":"hint","dependencies":["afbeae7inequalities22a-h4"],"title":"Plug in","text":"Plug in any coordinate value that is either on top or under the line but not on the line. If the value is accepted by the equation, then shade the side of the graph the point is on. (Ex: If point is above line and works, shade the area above line vice versa)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities22a-h6","type":"hint","dependencies":["afbeae7inequalities22a-h5"],"title":"Match","text":"Match your graph with the graph in the answer choices","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afbeae7inequalities23","title":"Graph Linear Inequalities","body":"Graph the linear inequality.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphs of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afbeae7inequalities23a","stepAnswer":["A"],"problemType":"MultipleChoice","stepTitle":"$$x+5y<-5$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"afbeae7inequalities23a-h1","type":"hint","dependencies":[],"title":"Convert","text":"The equation can be rewritten as $$y<\\\\left(-\\\\frac{1}{5}\\\\right) x-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities23a-h2","type":"hint","dependencies":["afbeae7inequalities23a-h1"],"title":"y-intercept","text":"By plugging in the value $$x=0$$, we can conclude that the line will cross over $$(0,-1)$$ as the y-intercept","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities23a-h3","type":"hint","dependencies":["afbeae7inequalities23a-h2"],"title":"Slope","text":"Given the equation, we can conclude that the slope of the equation is $$\\\\frac{-1}{5}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities23a-h4","type":"hint","dependencies":["afbeae7inequalities23a-h3"],"title":"Line","text":"Because the equation does not include an \\"equal to\\" sign, it is a dotted line and the solution will not be on that line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities23a-h5","type":"hint","dependencies":["afbeae7inequalities23a-h4"],"title":"Plug in","text":"Plug in any coordinate value that is either on top or under the line but not on the line. If the value is accepted by the equation, then shade the side of the graph the point is on. (Ex: If point is above line and works, shade the area above line vice versa)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities23a-h6","type":"hint","dependencies":["afbeae7inequalities23a-h5"],"title":"Match","text":"Match your graph with the graph in the answer choices","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afbeae7inequalities24","title":"Graph Linear Inequalities","body":"Graph the linear inequality.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphs of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afbeae7inequalities24a","stepAnswer":["B"],"problemType":"MultipleChoice","stepTitle":"$$3x+2y \\\\geq -6$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"afbeae7inequalities24a-h1","type":"hint","dependencies":[],"title":"Convert","text":"The equation can be rewritten as $$y \\\\geq \\\\left(-\\\\frac{3}{2}\\\\right) x-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities24a-h2","type":"hint","dependencies":["afbeae7inequalities24a-h1"],"title":"y-intercept","text":"By plugging in the value $$x=0$$, we can conclude that the line will cross over $$(0,-3)$$ as the y-intercept","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities24a-h3","type":"hint","dependencies":["afbeae7inequalities24a-h2"],"title":"Slope","text":"Given the equation, we can conclude that the slope of the equation is $$\\\\frac{-3}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities24a-h4","type":"hint","dependencies":["afbeae7inequalities24a-h3"],"title":"Line","text":"Because the equation includes an \\"equal to\\" sign, it is a straight, non-dotted line, and the solution can be on the line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities24a-h5","type":"hint","dependencies":["afbeae7inequalities24a-h4"],"title":"Plug in","text":"Plug in any coordinate value that is either on top or under the line but not on the line. If the value is accepted by the equation, then shade the side of the graph the point is on. (Ex: If point is above line and works, shade the area above line vice versa)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities24a-h6","type":"hint","dependencies":["afbeae7inequalities24a-h5"],"title":"Match","text":"Match your graph with the graph in the answer choices","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afbeae7inequalities25","title":"Graph Linear Inequalities","body":"Graph the linear inequality.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphs of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afbeae7inequalities25a","stepAnswer":["A"],"problemType":"MultipleChoice","stepTitle":"$$4x+2y \\\\geq -8$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"afbeae7inequalities25a-h1","type":"hint","dependencies":[],"title":"Convert","text":"The equation can be rewritten as $$y \\\\geq -2x-4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities25a-h2","type":"hint","dependencies":["afbeae7inequalities25a-h1"],"title":"y-intercept","text":"By plugging in the value $$x=0$$, we can conclude that the line will cross over $$(0,-4)$$ as the y-intercept","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities25a-h3","type":"hint","dependencies":["afbeae7inequalities25a-h2"],"title":"Slope","text":"Given the equation, we can conclude that the slope of the equation is $$-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities25a-h4","type":"hint","dependencies":["afbeae7inequalities25a-h3"],"title":"Line","text":"Because the equation includes an \\"equal to\\" sign, it is a straight, non-dotted line, and the solution can be on the line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities25a-h5","type":"hint","dependencies":["afbeae7inequalities25a-h4"],"title":"Plug in","text":"Plug in any coordinate value that is either on top or under the line but not on the line. If the value is accepted by the equation, then shade the side of the graph the point is on. (Ex: If point is above line and works, shade the area above line vice versa)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities25a-h6","type":"hint","dependencies":["afbeae7inequalities25a-h5"],"title":"Match","text":"Match your graph with the graph in the answer choices","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afbeae7inequalities26","title":"Graph Linear Inequalities","body":"Graph the linear inequality.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphs of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afbeae7inequalities26a","stepAnswer":["D"],"problemType":"MultipleChoice","stepTitle":"$$y>4x$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"afbeae7inequalities26a-h1","type":"hint","dependencies":[],"title":"y-intercept","text":"By plugging in the value $$x=0$$, we can conclude that the line will cross over $$(0,0)$$ as the y-intercept","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities26a-h2","type":"hint","dependencies":["afbeae7inequalities26a-h1"],"title":"Slope","text":"Given the equation, we can conclude that the slope of the equation is $$4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities26a-h3","type":"hint","dependencies":["afbeae7inequalities26a-h2"],"title":"Line","text":"Because the equation does not include an \\"equal to\\" sign, it is a dotted line and the solution will not be on that line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities26a-h4","type":"hint","dependencies":["afbeae7inequalities26a-h3"],"title":"Plug in","text":"Plug in any coordinate value that is either on top or under the line but not on the line. If the value is accepted by the equation, then shade the side of the graph the point is on. (Ex: If point is above line and works, shade the area above line vice versa)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities26a-h5","type":"hint","dependencies":["afbeae7inequalities26a-h4"],"title":"Match","text":"Match your graph with the graph in the answer choices","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afbeae7inequalities27","title":"Graph Linear Inequalities","body":"Graph the linear inequality.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphs of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afbeae7inequalities27a","stepAnswer":["C"],"problemType":"MultipleChoice","stepTitle":"$$y>x$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"afbeae7inequalities27a-h1","type":"hint","dependencies":[],"title":"y-intercept","text":"By plugging in the value $$x=0$$, we can conclude that the line will cross over $$(0,0)$$ as the y-intercept","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities27a-h2","type":"hint","dependencies":["afbeae7inequalities27a-h1"],"title":"Slope","text":"Given the equation, we can conclude that the slope of the equation is $$1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities27a-h3","type":"hint","dependencies":["afbeae7inequalities27a-h2"],"title":"Line","text":"Because the equation does not include an \\"equal to\\" sign, it is a dotted line and the solution will not be on that line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities27a-h4","type":"hint","dependencies":["afbeae7inequalities27a-h3"],"title":"Plug in","text":"Plug in any coordinate value that is either on top or under the line but not on the line. If the value is accepted by the equation, then shade the side of the graph the point is on. (Ex: If point is above line and works, shade the area above line vice versa)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities27a-h5","type":"hint","dependencies":["afbeae7inequalities27a-h4"],"title":"Match","text":"Match your graph with the graph in the answer choices","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afbeae7inequalities28","title":"Graph Linear Inequalities","body":"Graph the linear inequality.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphs of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afbeae7inequalities28a","stepAnswer":["A"],"problemType":"MultipleChoice","stepTitle":"$$y \\\\leq -x$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"afbeae7inequalities28a-h1","type":"hint","dependencies":[],"title":"y-intercept","text":"By plugging in the value $$x=0$$, we can conclude that the line will cross over $$(0,0)$$ as the y-intercept","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities28a-h2","type":"hint","dependencies":["afbeae7inequalities28a-h1"],"title":"Slope","text":"Given the equation, we can conclude that the slope of the equation is $$-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities28a-h3","type":"hint","dependencies":["afbeae7inequalities28a-h2"],"title":"Line","text":"Because the equation includes an \\"equal to\\" sign, it is a straight, non-dotted line, and the solution can be on the line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities28a-h4","type":"hint","dependencies":["afbeae7inequalities28a-h3"],"title":"Plug in","text":"Plug in any coordinate value that is either on top or under the line but not on the line. If the value is accepted by the equation, then shade the side of the graph the point is on. (Ex: If point is above line and works, shade the area above line vice versa)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities28a-h5","type":"hint","dependencies":["afbeae7inequalities28a-h4"],"title":"Match","text":"Match your graph with the graph in the answer choices","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afbeae7inequalities29","title":"Graph Linear Inequalities","body":"Graph the linear inequality.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphs of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afbeae7inequalities29a","stepAnswer":["C"],"problemType":"MultipleChoice","stepTitle":"$$y \\\\leq -3x$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"afbeae7inequalities29a-h1","type":"hint","dependencies":[],"title":"y-intercept","text":"By plugging in the value $$x=0$$, we can conclude that the line will cross over $$(0,0)$$ as the y-intercept","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities29a-h2","type":"hint","dependencies":["afbeae7inequalities29a-h1"],"title":"Slope","text":"Given the equation, we can conclude that the slope of the equation is $$-3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities29a-h3","type":"hint","dependencies":["afbeae7inequalities29a-h2"],"title":"Line","text":"Because the equation includes an \\"equal to\\" sign, it is a straight, non-dotted line, and the solution can be on the line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities29a-h4","type":"hint","dependencies":["afbeae7inequalities29a-h3"],"title":"Plug in","text":"Plug in any coordinate value that is either on top or under the line but not on the line. If the value is accepted by the equation, then shade the side of the graph the point is on. (Ex: If point is above line and works, shade the area above line vice versa)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities29a-h5","type":"hint","dependencies":["afbeae7inequalities29a-h4"],"title":"Match","text":"Match your graph with the graph in the answer choices","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afbeae7inequalities3","title":"Determining Solutions","body":"Determine whether the given ordered pair is a valid solution.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphs of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afbeae7inequalities3a","stepAnswer":["The ordered pair is not a solution."],"problemType":"MultipleChoice","stepTitle":"Determine whether $$(2,6)$$ is a solution to the inequality $$y>x+4$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["The ordered pair is a solution.IThe ordered pair is not a solution.","The ordered pair is not a solution."],"hints":{"DefaultPathway":[{"id":"afbeae7inequalities3a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute the $$x$$ and $$y$$ values of the ordered pair into the inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities3a-h2","type":"hint","dependencies":["afbeae7inequalities3a-h1"],"title":"Simplify","text":"Simplify the equation so that there is a single digit on each side of the inequality symbol.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities3a-h3","type":"hint","dependencies":["afbeae7inequalities3a-h2"],"title":"Interpret","text":"The equation in this case simplifies to $$6>6$$, which is not a true statement. Therefore, $$(2,6)$$ is not a solution to the inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afbeae7inequalities30","title":"Graph Linear Inequalities","body":"Graph the linear inequality.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphs of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afbeae7inequalities30a","stepAnswer":["C"],"problemType":"MultipleChoice","stepTitle":"$$y \\\\geq 2$$","stepBody":"##figure1.gif## ","answerType":"string","variabilization":{},"choices":["A","B","C","D"],"hints":{"DefaultPathway":[{"id":"afbeae7inequalities30a-h1","type":"hint","dependencies":[],"title":"y-intercept","text":"By plugging in the value $$x=0$$, we can conclude that the line will cross over $$(0,2)$$ as the y-intercept","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities30a-h2","type":"hint","dependencies":["afbeae7inequalities30a-h1"],"title":"Slope","text":"Given the equation, we can conclude that the slope of the equation is $$0$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities30a-h3","type":"hint","dependencies":["afbeae7inequalities30a-h2"],"title":"Line","text":"Because the equation includes an \\"equal to\\" sign, it is a straight, non-dotted line, and the solution can be on the line","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities30a-h4","type":"hint","dependencies":["afbeae7inequalities30a-h3"],"title":"Plug in","text":"Plug in any coordinate value that is either on top or under the line but not on the line. If the value is accepted by the equation, then shade the side of the graph the point is on. (Ex: If point is above line and works, shade the area above line vice versa)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities30a-h5","type":"hint","dependencies":["afbeae7inequalities30a-h4"],"title":"Match","text":"Match your graph with the graph in the answer choices","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afbeae7inequalities4","title":"Determining Solutions","body":"Determine whether the given ordered pair is a valid solution.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphs of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afbeae7inequalities4a","stepAnswer":["The ordered pair is not a solution."],"problemType":"MultipleChoice","stepTitle":"Determine whether $$(-5,-15)$$ is a solution to the inequality $$y>x+4$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["The ordered pair is a solution.IThe ordered pair is not a solution.","The ordered pair is not a solution."],"hints":{"DefaultPathway":[{"id":"afbeae7inequalities4a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute the $$x$$ and $$y$$ values of the ordered pair into the inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities4a-h2","type":"hint","dependencies":["afbeae7inequalities4a-h1"],"title":"Simplify","text":"Simplify the equation so that there is a single digit on each side of the inequality symbol.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities4a-h3","type":"hint","dependencies":["afbeae7inequalities4a-h2"],"title":"Interpret","text":"The equation in this case simplifies to $$-15>-1$$, which is not a true statement. Therefore, $$(-5,-15)$$ is not a solution to the inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afbeae7inequalities5","title":"Determining Solutions","body":"Determine whether the given ordered pair is a valid solution.","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphs of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afbeae7inequalities5a","stepAnswer":["The ordered pair is not a solution."],"problemType":"MultipleChoice","stepTitle":"Determine whether $$(-8,12)$$ is a solution to the inequality $$y>x+4$$.","stepBody":"","answerType":"string","variabilization":{},"choices":["The ordered pair is a solution.IThe ordered pair is not a solution.","The ordered pair is not a solution."],"hints":{"DefaultPathway":[{"id":"afbeae7inequalities5a-h1","type":"hint","dependencies":[],"title":"Substitute","text":"Substitute the $$x$$ and $$y$$ values of the ordered pair into the inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities5a-h2","type":"hint","dependencies":["afbeae7inequalities5a-h1"],"title":"Simplify","text":"Simplify the equation so that there is a single digit on each side of the inequality symbol.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities5a-h3","type":"hint","dependencies":["afbeae7inequalities5a-h2"],"title":"Interpret","text":"The equation in this case simplifies to $$12>-4$$, which is not a true statement. Therefore, $$(1,6)$$ is not a solution to the inequality.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afbeae7inequalities6","title":"Identifying Graphed Inequalities","body":"Choose the correct inequality shown by the graph.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphs of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afbeae7inequalities6a","stepAnswer":["$$y \\\\geq 2x-1$$"],"problemType":"MultipleChoice","stepTitle":"The boundary line shown is $$y=2x-1$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y \\\\geq 2x-1$$","choices":["$$y \\\\geq 2x-1$$","$$y \\\\geq 2x-1Iy \\\\leq 2x-1Iy>2x-1Iy<2x-1$$"],"hints":{"DefaultPathway":[{"id":"afbeae7inequalities6a-h1","type":"hint","dependencies":[],"title":"Test","text":"We need to test the inequality to determine the signage. Plug the point $$(0,0)$$ into the equation twice, once with the equation being $$y>2x-1$$, and once with $$y<2x-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities6a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y>2x-1$$"],"dependencies":["afbeae7inequalities6a-h1"],"title":"Find the correct sign","text":"In this case, when substituting $$(0,0)$$ into the equation, does $$y>2x-1$$ result in a true statement, or does $$y<2x-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y<2x-1Iy>2x-1$$"],"subHints":[{"id":"afbeae7inequalities6a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$y>2x-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"afbeae7inequalities6a-h3","type":"hint","dependencies":["afbeae7inequalities6a-h2"],"title":"Determine line type","text":"If the line is solid, the sign will include the values of the line. In other words, the sign will be greater than or equal to, or less than or equal to. If the line is dotted, the sign will simply be > or <.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities6a-h4","type":"hint","dependencies":["afbeae7inequalities6a-h3"],"title":"Answer","text":"Therefore, the equation for the inequality is $$y \\\\geq 2x-1$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afbeae7inequalities7","title":"Identifying Graphed Inequalities","body":"Choose the correct inequality shown by the graph.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphs of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afbeae7inequalities7a","stepAnswer":["$$y<2x-4$$"],"problemType":"MultipleChoice","stepTitle":"The boundary line shown is $$y=2x-4$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y<2x-4$$","choices":["$$y<2x-4$$","$$y \\\\geq 2x-4Iy \\\\leq 2x-4Iy>2x-4Iy<2x-4$$"],"hints":{"DefaultPathway":[{"id":"afbeae7inequalities7a-h1","type":"hint","dependencies":[],"title":"Test","text":"We need to test the inequality to determine the signage. Plug the point $$(0,0)$$ into the equation twice, once with the equation being $$y>2x-4$$, and once with $$y<2x-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities7a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y<2x-4$$"],"dependencies":["afbeae7inequalities7a-h1"],"title":"Find the correct sign","text":"In this case, when substituting $$(0,0)$$ into the equation, does $$y>2x-4$$ result in a true statement, or does $$y<2x-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y<2x-4Iy>2x-4$$"],"subHints":[{"id":"afbeae7inequalities7a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$y<2x-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"afbeae7inequalities7a-h3","type":"hint","dependencies":["afbeae7inequalities7a-h2"],"title":"Determine line type","text":"If the line is solid, the sign will include the values of the line. In other words, the sign will be greater than or equal to, or less than or equal to. If the line is dotted, the sign will simply be > or <.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities7a-h4","type":"hint","dependencies":["afbeae7inequalities7a-h3"],"title":"Answer","text":"Therefore, the equation for the inequality is $$y<2x-4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afbeae7inequalities8","title":"Identifying Graphed Inequalities","body":"Choose the correct inequality shown by the graph.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphs of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afbeae7inequalities8a","stepAnswer":["$$y \\\\leq \\\\frac{-x}{3}-2$$"],"problemType":"MultipleChoice","stepTitle":"The boundary line shown is $$y=\\\\frac{-x}{3}-2$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$y \\\\leq \\\\frac{-x}{3}-2$$","choices":["$$y \\\\leq \\\\frac{-x}{3}-2$$","$$y \\\\geq \\\\frac{-x}{3}-2Iy \\\\leq \\\\frac{-x}{3}-2Iy>\\\\frac{-x}{3}-2Iy<\\\\frac{-x}{3}-2$$"],"hints":{"DefaultPathway":[{"id":"afbeae7inequalities8a-h1","type":"hint","dependencies":[],"title":"Test","text":"We need to test the inequality to determine the signage. Plug the point $$(0,0)$$ into the equation twice, once with the equation being $$y>2x-4$$, and once with $$y<2x-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities8a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$y<\\\\frac{-x}{3}-2$$"],"dependencies":["afbeae7inequalities8a-h1"],"title":"Find the correct sign","text":"In this case, when substituting $$(0,0)$$ into the equation, does $$y>2x-4$$ result in a true statement, or does $$y<2x-4$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$y<\\\\frac{-x}{3}-2Iy>\\\\frac{-x}{3}-2$$"],"subHints":[{"id":"afbeae7inequalities8a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$y<\\\\frac{-x}{3}-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"afbeae7inequalities8a-h3","type":"hint","dependencies":["afbeae7inequalities8a-h2"],"title":"Determine line type","text":"If the line is solid, the sign will include the values of the line. In other words, the sign will be greater than or equal to, or less than or equal to. If the line is dotted, the sign will simply be > or <.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities8a-h4","type":"hint","dependencies":["afbeae7inequalities8a-h3"],"title":"Answer","text":"Therefore, the equation for the inequality is $$y \\\\leq \\\\frac{-x}{3}-2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afbeae7inequalities9","title":"Identifying Graphed Inequalities","body":"Choose the correct inequality shown by the graph.\\\\n##figure1.gif","variabilization":{},"oer":"https://openstax.org/details/books/elementary-algebra-2e <OpenStax: Elementary Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"4.7 Graphs of Linear Inequalities","courseName":"OpenStax: Elementary Algebra","steps":[{"id":"afbeae7inequalities9a","stepAnswer":["$$x+y \\\\geq 3$$"],"problemType":"MultipleChoice","stepTitle":"The boundary line shown is $$x+y=3$$.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x+y \\\\geq 3$$","choices":["$$x+y \\\\geq 3$$","$$x+y \\\\geq 3Ix+y \\\\leq 3Ix+y>3Ix+y<3$$"],"hints":{"DefaultPathway":[{"id":"afbeae7inequalities9a-h1","type":"hint","dependencies":[],"title":"Test","text":"We need to test the inequality to determine the signage. Plug the point $$(0,0)$$ into the equation twice, once with the equation being $$x+y>3$$, and once with $$x+y<3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities9a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x+y>3$$"],"dependencies":["afbeae7inequalities9a-h1"],"title":"Find the correct sign","text":"In this case, when substituting $$(0,0)$$ into the equation, does $$x+y>3$$ result in a true statement, or does $$x+y<3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$x+y<3Ix+y>3$$"],"subHints":[{"id":"afbeae7inequalities9a-h2-s1","type":"hint","dependencies":[],"title":"Answer","text":"The answer is $$x+y>3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]},{"id":"afbeae7inequalities9a-h3","type":"hint","dependencies":["afbeae7inequalities9a-h2"],"title":"Determine line type","text":"If the line is solid, the sign will include the values of the line. In other words, the sign will be greater than or equal to, or less than or equal to. If the line is dotted, the sign will simply be > or <.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"afbeae7inequalities9a-h4","type":"hint","dependencies":["afbeae7inequalities9a-h3"],"title":"Answer","text":"Therefore, the equation for the inequality is $$x+y \\\\geq 3$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"afec02cEL1","title":"Evaluating an Exponential Function","body":"Evaluate the given exponential functions as indicated, accurate to two significant figures after the decimal.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.5 Exponential and Logarithmic Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"afec02cEL1a","stepAnswer":["$$2.24$$"],"problemType":"TextBox","stepTitle":"$$f(x)=5^x$$ where $$x=\\\\frac{1}{2}$$.","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2.24$$","hints":{"DefaultPathway":[{"id":"afec02cEL1a-h1","type":"hint","dependencies":[],"title":"Interpret the Exponent","text":"Substitute $$x=\\\\frac{1}{2}$$ into f(x)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"afec02cEL1a-h2","type":"hint","dependencies":["afec02cEL1a-h1"],"title":"Exponent","text":"Note that an exponent of $$\\\\frac{1}{2}$$ is equivalent to taking the square root of the base number.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"afec02cEL10","title":"Determine the domain of the exponential function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.5 Exponential and Logarithmic Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"afec02cEL10a","stepAnswer":["All real numbers"],"problemType":"MultipleChoice","stepTitle":"f(x) $$=$$ $$e^{-x}-1$$","stepBody":"","answerType":"string","variabilization":{},"choices":["All real numbers","Not all real numbers"],"hints":{"DefaultPathway":[{"id":"afec02cEL10a-h1","type":"hint","dependencies":[],"title":"Domain","text":"Determine whether or not there are values of $$x$$ that make the exponential function equal to a number that is not real.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"afec02cEL10a-h2","type":"hint","dependencies":["afec02cEL10a-h1"],"title":"Domain","text":"If there are any, the domain doesn\'t include all real numbers. If there aren\'t any, the domain includes all real numbers","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"afec02cEL10a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["afec02cEL10a-h2"],"title":"Check the Domain","text":"Does the domain include all real numbers?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["Yes","No"]},{"id":"afec02cEL10a-h4","type":"hint","dependencies":["afec02cEL10a-h3"],"title":"Conclusion","text":"Therefore, the domain of the exponential function includes all real numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"afec02cEL11","title":"Determine the vertical asymptote of the logarithmic function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.5 Exponential and Logarithmic Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"afec02cEL11a","stepAnswer":["$$x=1$$"],"problemType":"MultipleChoice","stepTitle":"f(x) $$=$$ $$ln(x-1)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=1$$","choices":["$$x=-1$$","$$x=0$$","$$x=1$$"],"hints":{"DefaultPathway":[{"id":"afec02cEL11a-h1","type":"hint","dependencies":[],"title":"Vertical Asymptote","text":"Determine what x-value makes the logarithmic functon equal to $$0$$. The x-value will be the value of the vertical asymptote.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"afec02cEL11a-h2","type":"hint","dependencies":["afec02cEL11a-h1"],"title":"Vertical Asymptote","text":"When $$x=1$$, the logarithmic functon equal to $$0$$. Therefore, the vertical asymptote is $$x=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"afec02cEL12","title":"Determine the vertical asymptote of the logarithmic function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.5 Exponential and Logarithmic Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"afec02cEL12a","stepAnswer":["$$x=0$$"],"problemType":"MultipleChoice","stepTitle":"f(x) $$=$$ $$1-ln(x)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=0$$","choices":["$$x=-1$$","$$x=0$$","$$x=1$$"],"hints":{"DefaultPathway":[{"id":"afec02cEL12a-h1","type":"hint","dependencies":[],"title":"Vertical Asymptote","text":"Determine what x-value makes the logarithmic functon equal to $$0$$. The x-value will be the value of the vertical asymptote.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"afec02cEL12a-h2","type":"hint","dependencies":["afec02cEL12a-h1"],"title":"Vertical Asymptote","text":"When $$x=0$$, the logarithmic functon equal to $$0$$. Therefore, the vertical asymptote is $$x=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"afec02cEL13","title":"Determine the vertical asymptote of the logarithmic function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.5 Exponential and Logarithmic Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"afec02cEL13a","stepAnswer":["$$x=-1$$"],"problemType":"MultipleChoice","stepTitle":"f(x) $$=$$ $$\\\\ln(x+1)$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x=-1$$","choices":["$$x=-1$$","$$x=0$$","$$x=1$$"],"hints":{"DefaultPathway":[{"id":"afec02cEL13a-h1","type":"hint","dependencies":[],"title":"Vertical Asymptote","text":"Determine what x-value makes the logarithmic functon equal to $$0$$. The x-value will be the value of the vertical asymptote.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"afec02cEL13a-h2","type":"hint","dependencies":["afec02cEL13a-h1"],"title":"Vertical Asymptote","text":"When $$x=-1$$, the logarithmic functon equal to $$0$$. Therefore, the vertical asymptote is $$x=-1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"afec02cEL14","title":"Solve the exponential equation","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.5 Exponential and Logarithmic Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"afec02cEL14a","stepAnswer":["$$\\\\frac{\\\\ln(15)}{3}$$"],"problemType":"MultipleChoice","stepTitle":"$$e^{3x}-15=0$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{\\\\ln(15)}{3}$$","choices":["$$\\\\frac{15}{\\\\ln(3)}$$","$$\\\\frac{\\\\ln(15)}{\\\\ln(3)}$$","$$\\\\frac{\\\\ln(15)}{3}$$"],"hints":{"DefaultPathway":[{"id":"afec02cEL14a-h1","type":"hint","dependencies":[],"title":"Determining $$x$$","text":"To determine the value of $$x$$, add $$15$$ to both sides","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"afec02cEL14a-h2","type":"hint","dependencies":["afec02cEL14a-h1"],"title":"Determining $$x$$","text":"To get rid of e, apply ln to both sides, leaving only $$3x$$ on the left side of the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"afec02cEL14a-h3","type":"hint","dependencies":["afec02cEL14a-h2"],"title":"Simplification","text":"Simplify the equation so only $$x$$ remains on the left hand side of the equation","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"afec02cEL15","title":"Determine the value of the horizontal asymptote of the exponential function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.5 Exponential and Logarithmic Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"afec02cEL15a","stepAnswer":["$$2$$"],"problemType":"TextBox","stepTitle":"$$f(x)=e^x+2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2$$","hints":{"DefaultPathway":[{"id":"afec02cEL15a-h1","type":"hint","dependencies":[],"title":"Horizontal Asymptote","text":"Determine what value $$y$$ approaches as $$x$$ approaches infinity or negative infinity to find the horizontal asymptote.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"afec02cEL15a-h2","type":"hint","dependencies":["afec02cEL15a-h1"],"title":"Horizontal Asymptote","text":"As $$x$$ apporaches negative $$\\\\infty$$, $$e^x$$ approaches $$0$$. Therefore, the horizontal asymptote is $$y=0+2=2$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"afec02cEL16","title":"Determine the value of the horizontal asymptote of the exponential function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.5 Exponential and Logarithmic Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"afec02cEL16a","stepAnswer":["$$0$$"],"problemType":"TextBox","stepTitle":"$$f(x)=3^{x+1}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0$$","hints":{"DefaultPathway":[{"id":"afec02cEL16a-h1","type":"hint","dependencies":[],"title":"Horizontal Asymptote","text":"To find the horizontal asymptote, determine where the $$y$$ value approaches as the $$x$$ value goes to infinity or negative infinity","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"afec02cEL16a-h2","type":"hint","dependencies":["afec02cEL16a-h1"],"title":"Horizontal Asymptote","text":"As $$x$$ apporaches negative $$\\\\infty$$, $$3^{x+1}$$ approaches $$0$$. Therefore, the horizontal asymptote is $$y=0$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"afec02cEL17","title":"Determine the value of the horizontal asymptote of the exponential function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.5 Exponential and Logarithmic Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"afec02cEL17a","stepAnswer":["$$1$$"],"problemType":"TextBox","stepTitle":"$$f(x)=1-2^{-x}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$1$$","hints":{"DefaultPathway":[{"id":"afec02cEL17a-h1","type":"hint","dependencies":[],"title":"Horizontal Asymptote","text":"To find the horizontal asymptote, determine where the $$y$$ value approaches as the $$x$$ value goes to infinity or negative infinity","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"afec02cEL17a-h2","type":"hint","dependencies":["afec02cEL17a-h1"],"title":"Horizontal Asymptote","text":"As $$x$$ apporaches $$\\\\infty$$, $$2^{\\\\left(-x\\\\right)}$$ approaches $$0$$. Therefore, the horizontal asymptote is $$y=1-0=1$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"afec02cEL2","title":"Evaluating an Exponential Function","body":"Evaluate the given exponential functions as indicated, accurate to two significant figures after the decimal.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.5 Exponential and Logarithmic Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"afec02cEL2a","stepAnswer":["$$125$$"],"problemType":"TextBox","stepTitle":"$$f(x)=5^x$$ where $$x=3$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$125$$","hints":{"DefaultPathway":[{"id":"afec02cEL2a-h1","type":"hint","dependencies":[],"title":"Interpret the Exponent","text":"Substitute $$x=3$$ into f(x)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"afec02cEL3","title":"Evaluating an Exponential Function","body":"Evaluate the given exponential functions as indicated, accurate to two significant figures after the decimal.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.5 Exponential and Logarithmic Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"afec02cEL3a","stepAnswer":["$$9.74$$"],"problemType":"TextBox","stepTitle":"$$f(x)=5^x$$ where $$x=\\\\sqrt{2}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$9.74$$","hints":{"DefaultPathway":[{"id":"afec02cEL3a-h1","type":"hint","dependencies":[],"title":"Interpret the Exponent","text":"Substitute $$x=\\\\sqrt{2}$$ into f(x)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"afec02cEL4","title":"Evaluating an Exponential Function","body":"Evaluate the given exponential functions as indicated, accurate to two significant figures after the decimal.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.5 Exponential and Logarithmic Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"afec02cEL4a","stepAnswer":["$$0.01$$"],"problemType":"TextBox","stepTitle":"$$f(x)={10}^x$$ where $$x=-2$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$0.01$$","hints":{"DefaultPathway":[{"id":"afec02cEL4a-h1","type":"hint","dependencies":[],"title":"Interpret the Exponent","text":"Substitute $$x=-2$$ into f(x)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"afec02cEL5","title":"Evaluating an Exponential Function","body":"Evaluate the given exponential functions as indicated, accurate to two significant figures after the decimal.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.5 Exponential and Logarithmic Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"afec02cEL5a","stepAnswer":["$$10000$$"],"problemType":"TextBox","stepTitle":"$$f(x)={10}^x$$ where $$x=4$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$10000$$","hints":{"DefaultPathway":[{"id":"afec02cEL5a-h1","type":"hint","dependencies":[],"title":"Interpret the Exponent","text":"Substitute $$x=4$$ into f(x)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"afec02cEL6","title":"Evaluating an Exponential Function","body":"Evaluate the given exponential functions as indicated, accurate to two significant figures after the decimal.","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.5 Exponential and Logarithmic Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"afec02cEL6a","stepAnswer":["$$46.42$$"],"problemType":"TextBox","stepTitle":"$$f(x)={10}^x$$ where $$x=\\\\frac{5}{3}$$","stepBody":"","answerType":"arithmetic","variabilization":{},"answerLatex":"$$46.42$$","hints":{"DefaultPathway":[{"id":"afec02cEL6a-h1","type":"hint","dependencies":[],"title":"Interpret the Exponent","text":"Substitute $$x=\\\\frac{5}{3}$$ into f(x)","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"afec02cEL7","title":"Determine the domain of the exponential function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.5 Exponential and Logarithmic Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"afec02cEL7a","stepAnswer":["All real numbers"],"problemType":"MultipleChoice","stepTitle":"f(x) $$=$$ $$e^x+2$$","stepBody":"","answerType":"string","variabilization":{},"choices":["All real numbers","Not all real numbers"],"hints":{"DefaultPathway":[{"id":"afec02cEL7a-h1","type":"hint","dependencies":[],"title":"Domain","text":"Determine whether or not there are values of $$x$$ that make the exponential function equal to a number that is not real.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"afec02cEL7a-h2","type":"hint","dependencies":["afec02cEL7a-h1"],"title":"Domain","text":"If there are any, the domain doesn\'t include all real numbers. If there aren\'t any, the domain includes all real numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"afec02cEL7a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["afec02cEL7a-h2"],"title":"Check the Domain","text":"Does the domain include all real numbers?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["Yes","No"]},{"id":"afec02cEL7a-h4","type":"hint","dependencies":["afec02cEL7a-h3"],"title":"Conclusion","text":"Therefore, the domain of the exponential function includes all real numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"afec02cEL8","title":"Determine the domain of the exponential function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.5 Exponential and Logarithmic Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"afec02cEL8a","stepAnswer":["All real numbers"],"problemType":"MultipleChoice","stepTitle":"f(x) $$=$$ $$3^{x+1}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["All real numbers","Not all real numbers"],"hints":{"DefaultPathway":[{"id":"afec02cEL8a-h1","type":"hint","dependencies":[],"title":"Domain","text":"Determine whether or not there are values of $$x$$ that make the exponential function equal to a number that is not real.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"afec02cEL8a-h2","type":"hint","dependencies":["afec02cEL8a-h1"],"title":"Domain","text":"If there are any, the domain doesn\'t include all real numbers. If there aren\'t any, the domain includes all real numbers","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"afec02cEL8a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["afec02cEL8a-h2"],"title":"Check the Domain","text":"Does the domain include all real numbers?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["Yes","No"]},{"id":"afec02cEL8a-h4","type":"hint","dependencies":["afec02cEL8a-h3"],"title":"Conclusion","text":"Therefore, the domain of the exponential function includes all real numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"afec02cEL9","title":"Determine the domain of the exponential function","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"1.5 Exponential and Logarithmic Functions","courseName":"OpenStax: Calculus Volume 1","steps":[{"id":"afec02cEL9a","stepAnswer":["All real numbers"],"problemType":"MultipleChoice","stepTitle":"f(x) $$=$$ $$1-2^{-x}$$","stepBody":"","answerType":"string","variabilization":{},"choices":["All real numbers","Not all real numbers"],"hints":{"DefaultPathway":[{"id":"afec02cEL9a-h1","type":"hint","dependencies":[],"title":"Domain","text":"Determine whether or not there are values of $$x$$ that make the exponential function equal to a number that is not real.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"afec02cEL9a-h2","type":"hint","dependencies":["afec02cEL9a-h1"],"title":"Domain","text":"If there are any, the domain doesn\'t include all real numbers. If there aren\'t any, the domain includes all real numbers","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"afec02cEL9a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["afec02cEL9a-h2"],"title":"Check the Domain","text":"Does the domain include all real numbers?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["Yes","No"]},{"id":"afec02cEL9a-h4","type":"hint","dependencies":["afec02cEL9a-h3"],"title":"Conclusion","text":"Therefore, the domain of the exponential function includes all real numbers.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"aff0960mixture16","title":"Solve the Coin Word Problems","body":"Solve the following problem:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Solve Mixture and Uniform Motion Applications","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aff0960mixture16a","stepAnswer":["$$59$$ nickels and $$7$$ pennies"],"problemType":"MultipleChoice","stepTitle":"Jesse has $$\\\\$3.02$$ worth of pennies and nickels in his piggy bank. The number of nickels is three more than eight times the number of pennies. How many nickels and how many pennies does Jesse have?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$59$$ nickels and $$7$$ pennies","choices":["$$59$$ nickels and $$7$$ pennies","$$60$$ nickels and $$8$$ pennies","$$7$$ nickels and $$59$$ pennies8 nickels and $$60$$ dimes"],"hints":{"DefaultPathway":[{"id":"aff0960mixture16a-h1","type":"hint","dependencies":[],"title":"Variables","text":"Represent the number of each type of coin using variables. The number of nickels is three more than eight times the number of pennies, so you can represent nickels as $$8p+3$$, and write pennies simply as $$p$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture16a-h2","type":"hint","dependencies":["aff0960mixture16a-h1"],"title":"Multiplication","text":"Multiple the number of each coin type(in variable form) by its respective value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture16a-h3","type":"hint","dependencies":["aff0960mixture16a-h2"],"title":"Conversion","text":"Now, translate the expressions that were multiplied by their coin values into a single equation. This will be $$\\\\operatorname{0.05}\\\\left(8p+3\\\\right)+0.01p=3.02$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture16a-h4","type":"hint","dependencies":["aff0960mixture16a-h3"],"title":"Calculation","text":"Solve the equation for $$p$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture16a-h5","type":"hint","dependencies":["aff0960mixture16a-h4"],"title":"Substitution","text":"Plug $$p$$ into the original relationship between the coins to find $$8p+3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture16a-h6","type":"hint","dependencies":["aff0960mixture16a-h5"],"title":"Calculation","text":"The answer is $$59$$ nickels and $$7$$ pennies.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aff0960mixture16b","stepAnswer":["$$9$$ nickels and $$16$$ dimes"],"problemType":"MultipleChoice","stepTitle":"Michaela has $$\\\\$2.05$$ in dimes and nickels in her change purse. She has seven more dimes than nickels. How many coins of each type does she have?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$9$$ nickels and $$16$$ dimes","choices":["$$9$$ nickels and $$16$$ dimes","$$10$$ nickels and $$23$$ dimes","$$2$$ nickels and $$21$$ dimes","$$3$$ nickels and $$7$$ dimes"],"hints":{"DefaultPathway":[{"id":"aff0960mixture16b-h1","type":"hint","dependencies":[],"title":"Variables","text":"Represent the number of each type of coin using variables. She has seven more dimes than nickels, so you can represent dimes as $$n+7$$, and write nickels simply as $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture16b-h2","type":"hint","dependencies":["aff0960mixture16b-h1"],"title":"Multiplication","text":"Multiple the number of each coin type(in variable form) by its respective value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture16b-h3","type":"hint","dependencies":["aff0960mixture16b-h2"],"title":"Conversion","text":"Now, translate the expressions that were multiplied by their coin values into a single equation. This will be $$\\\\operatorname{0.05}\\\\left(n+7\\\\right)+0.1n=2.05$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture16b-h4","type":"hint","dependencies":["aff0960mixture16b-h3"],"title":"Calculation","text":"Solve the equation for $$n$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture16b-h5","type":"hint","dependencies":["aff0960mixture16b-h4"],"title":"Substitution","text":"Plug $$n$$ into the original relationship between the coins to find $$n+7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture16b-h6","type":"hint","dependencies":["aff0960mixture16b-h5"],"title":"Calculation","text":"The answer is $$9$$ nickels and $$16$$ dimes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aff0960mixture17","title":"Solve the Coin Word Problems","body":"Solve the following problem:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Solve Mixture and Uniform Motion Applications","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aff0960mixture17a","stepAnswer":["$$41$$ nickels and $$18$$ quarters"],"problemType":"MultipleChoice","stepTitle":"Jesse has $$\\\\$6.55$$ worth of quarters and nickels in his pocket. The number of nickels is five more than two times the number of quarters. How many nickels and how many quarters does Jesse have?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$41$$ nickels and $$18$$ quarters","choices":["$$41$$ nickels and $$18$$ quarters","$$40$$ nickels and $$17$$ quarters","$$18$$ nickels and $$41$$ quarters","$$36$$ nickels and $$10$$ quarters"],"hints":{"DefaultPathway":[{"id":"aff0960mixture17a-h1","type":"hint","dependencies":[],"title":"Variables","text":"Represent the number of each type of coin using variables. The number of nickels is five more than two times the number of quarters, so you can represent as $$2q+5$$, and write pennies simply as q.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture17a-h2","type":"hint","dependencies":["aff0960mixture17a-h1"],"title":"Multiplication","text":"Multiple the number of each coin type(in variable form) by its respective value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture17a-h3","type":"hint","dependencies":["aff0960mixture17a-h2"],"title":"Conversion","text":"Now, translate the expressions that were multiplied by their coin values into a single equation. This will be $$\\\\operatorname{0.05}\\\\left(2q+5\\\\right)+0.25q=6.55$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture17a-h4","type":"hint","dependencies":["aff0960mixture17a-h3"],"title":"Calculation","text":"Solve the equation for q.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture17a-h5","type":"hint","dependencies":["aff0960mixture17a-h4"],"title":"Substitution","text":"Plug q into the original relationship between the coins to find $$2q+5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture17a-h6","type":"hint","dependencies":["aff0960mixture17a-h5"],"title":"Calculation","text":"The answer is $$41$$ nickels and $$18$$ quarters.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aff0960mixture17b","stepAnswer":["$$10$$ $10 bills and $$5$$ $5 bills"],"problemType":"MultipleChoice","stepTitle":"In a cash drawer there is $125 in $5 and $10 bills. The number of $10 bills is twice the number of $5 bills. How many of each type of bill is in the drawer?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$10$$ $10 bills and $$5$$ $5 bills","choices":["$$10$$ $10 bills and $$5$$ $5 bills","$$2$$ $10 bills and $$7$$ $5 bills","$$10$$ $12 bills and $$1$$ $5 bill","$$10$$ $8 bills and $$7$$ $5 bills"],"hints":{"DefaultPathway":[{"id":"aff0960mixture17b-h1","type":"hint","dependencies":[],"title":"Variables","text":"Represent the number of each type of cash using variables. The number of $10 bills is twice the number of $5 bills, so you can represent $10 as 2f, and write five dollar bills as f.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture17b-h2","type":"hint","dependencies":["aff0960mixture17b-h1"],"title":"Multiplication","text":"Multiple the number of each cash type(in variable form) by its respective value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture17b-h3","type":"hint","dependencies":["aff0960mixture17b-h2"],"title":"Conversion","text":"Now, translate the expressions that were multiplied by their cash values into a single equation. This will be $$125=20f+5f$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture17b-h4","type":"hint","dependencies":["aff0960mixture17b-h3"],"title":"Calculation","text":"Solve the equation for f.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture17b-h5","type":"hint","dependencies":["aff0960mixture17b-h4"],"title":"Substitution","text":"Plug f into the original relationship between the cash to find 2f.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture17b-h6","type":"hint","dependencies":["aff0960mixture17b-h5"],"title":"Calculation","text":"The answer is $$10$$ $10 bills and $$5$$ $5 bills.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aff0960mixture18","title":"Solve Ticket and Stamp Word Problems","body":"Solve the following problem:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Solve Mixture and Uniform Motion Applications","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aff0960mixture18a","stepAnswer":["$$25$$ $$49-cent$$ stamps and $$10$$ $$35-cent$$ stamps."],"problemType":"MultipleChoice","stepTitle":"Danny paid $$\\\\$15.75$$ for stamps. The number of 49-cent stamps was five less than three times the number of 35-cent stamps. How many 49-cent stamps and how many 35-cent stamps did Danny buy?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$25$$ 49-cent stamps and $$10$$ 35-cent stamps.","choices":["$$25$$ $5 bills and $$7$$ $10 bills","$$24$$ $$49-cent$$ stamps and $$8$$ $$35-cent$$ stamps.","$$25$$ $$49-cent$$ stamps and $$10$$ $$35-cent$$ stamps.","$$7$$ $5 bills and $$23$$ $10 bills"],"hints":{"DefaultPathway":[{"id":"aff0960mixture18a-h1","type":"hint","dependencies":[],"title":"Variables","text":"Represent the number of each type of cash using variables. The number of 49-cent stamps was five less than three times the number of 35-cent stamps, so you can represent 49-cent stamp as $$3t-5(where$$ $$t$$ is 35-cent stamp).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture18a-h2","type":"hint","dependencies":["aff0960mixture18a-h1"],"title":"Multiplication","text":"Multiply the number of each stamp type(in variable form) by its respective value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture18a-h3","type":"hint","dependencies":["aff0960mixture18a-h2"],"title":"Conversion","text":"Now, translate the expressions that were multiplied by their cash values into a single equation. This will be $$0.49\\\\left(3t-5\\\\right)+0.35t=15.75$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture18a-h4","type":"hint","dependencies":["aff0960mixture18a-h3"],"title":"Calculation","text":"Solve the equation for $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture18a-h5","type":"hint","dependencies":["aff0960mixture18a-h4"],"title":"Substitution","text":"Plug $$t$$ into the original relationship to find $$3t-5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture18a-h6","type":"hint","dependencies":["aff0960mixture18a-h5"],"title":"Calculation","text":"The answer is $$25$$ 49-cent stamps and $$10$$ 35-cent stamps.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aff0960mixture18b","stepAnswer":["$$21$$ $5 bills and $$7$$ $10 bills"],"problemType":"MultipleChoice","stepTitle":"Sumanta has $175 in $5 and $10 bills in his drawer. The number of $5 bills is three times the number of $10 bills. How many of each are in the drawer?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$21$$ $5 bills and $$7$$ $10 bills","choices":["$$21$$ $5 bills and $$7$$ $10 bills","$$15$$ $5 bills and $$20$$ $10 bills","$$3$$ $5 bills and $$7$$ $10 bills","$$7$$ $5 bills and $$21$$ $10 bills"],"hints":{"DefaultPathway":[{"id":"aff0960mixture18b-h1","type":"hint","dependencies":[],"title":"Variables","text":"Represent the number of each type of cash using variables. The number of $5 bills is three times the number of $10 bills, so you can represent this as 3t(where $$t$$ is ten dollar bills).","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture18b-h2","type":"hint","dependencies":["aff0960mixture18b-h1"],"title":"Multiplication","text":"Multiply the number of each cash type(in variable form) by its respective value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture18b-h3","type":"hint","dependencies":["aff0960mixture18b-h2"],"title":"Conversion","text":"Now, translate the expressions that were multiplied by their cash values into a single equation. This will be $$175=15t+10t$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture18b-h4","type":"hint","dependencies":["aff0960mixture18b-h3"],"title":"Calculation","text":"Solve the equation for $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture18b-h5","type":"hint","dependencies":["aff0960mixture18b-h4"],"title":"Substitution","text":"Plug $$t$$ into the original relationship between the cash to find $$3t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture18b-h6","type":"hint","dependencies":["aff0960mixture18b-h5"],"title":"Calculation","text":"The answer is $$21$$ $5 bills and $$7$$ $10 bills.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aff0960mixture19","title":"Solve Ticket and Stamp Word Problems","body":"Solve the following problem:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Solve Mixture and Uniform Motion Applications","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aff0960mixture19a","stepAnswer":["$$26$$ passengers paying $32 and $$14$$ passengers paying $26"],"problemType":"MultipleChoice","stepTitle":"A whale-watching ship had $$40$$ paying passengers on board. The total revenue collected from tickets was $1,196. Full-fare passengers paid $32 each and reduced-fare passengers paid $26 each. How many full-fare passengers and how many reduced-fare passengers were on the ship?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$26$$ passengers paying $32 and $$14$$ passengers paying $26","choices":["$$26$$ passengers paying $32 and $$14$$ passengers paying $26","$$25$$ passengers paying $32 and $$14$$ passengers paying $26","$$26$$ passengers paying $32 and $$10$$ passengers paying $26","$$35$$ passengers paying $32 and $$14$$ passengers paying $26"],"hints":{"DefaultPathway":[{"id":"aff0960mixture19a-h1","type":"hint","dependencies":[],"title":"Variables","text":"Represent the number of each type of passengers using variables, $32 as $$x$$ and $26 as $$(40-x)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture19a-h2","type":"hint","dependencies":["aff0960mixture19a-h1"],"title":"Multiplication","text":"Multiply the number of each ticket type(in variable form) by its respective value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture19a-h3","type":"hint","dependencies":["aff0960mixture19a-h2"],"title":"Conversion","text":"Now, translate the expressions that were multiplied by their monetary values into a single equation. This will be $$\\\\operatorname{32}\\\\left(x\\\\right)+\\\\operatorname{26}\\\\left(40-x\\\\right)=1196$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture19a-h4","type":"hint","dependencies":["aff0960mixture19a-h3"],"title":"Calculation","text":"Solve the equation for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture19a-h5","type":"hint","dependencies":["aff0960mixture19a-h4"],"title":"Substitution","text":"Plug $$x$$ into the original relationship to find $$40-x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture19a-h6","type":"hint","dependencies":["aff0960mixture19a-h5"],"title":"Calculation","text":"The answer is $$26$$ passengers paying $32 and $$14$$ passengers paying $26.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aff0960mixture19b","stepAnswer":["$$63$$ dimes and $$20$$ quarters"],"problemType":"MultipleChoice","stepTitle":"Chi has $$\\\\$11.30$$ in dimes and quarters. The number of dimes is three more than three times the number of quarters. How many of each are there?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$63$$ dimes and $$20$$ quarters","choices":["$$63$$ dimes and $$20$$ quarters","$$42$$ dimes and $$30$$ quarters","$$27$$ dimes and $$34$$ quarters","$$89$$ dimes and $$52$$ quarters"],"hints":{"DefaultPathway":[{"id":"aff0960mixture19b-h1","type":"hint","dependencies":[],"title":"Variables","text":"Represent the number of each type of coin using variables. The number of dimes is three more than three times the number of quarters, which you can represent as $$3+3q$$, and you can represent quarters simply as q.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture19b-h2","type":"hint","dependencies":["aff0960mixture19b-h1"],"title":"Multiplication","text":"Multiply the number of each coin type(in variable form) by its respective value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture19b-h3","type":"hint","dependencies":["aff0960mixture19b-h2"],"title":"Conversion","text":"Now, translate the expressions that were multiplied by their monetary values into a single equation. This will be $$\\\\operatorname{0.1}\\\\left(3+3q\\\\right)+0.25q=11.30$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture19b-h4","type":"hint","dependencies":["aff0960mixture19b-h3"],"title":"Calculation","text":"Solve the equation for q.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture19b-h5","type":"hint","dependencies":["aff0960mixture19b-h4"],"title":"Substitution","text":"Plug q into the original relationship between the coins to find $$d$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture19b-h6","type":"hint","dependencies":["aff0960mixture19b-h5"],"title":"Calculation","text":"The answer is $$63$$ dimes and $$20$$ quarters.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aff0960mixture20","title":"Solve Mixture Word Problems","body":"Solve the following problem:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Solve Mixture and Uniform Motion Applications","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aff0960mixture20a","stepAnswer":["$$10$$ pounds of raisins and $$15$$ pounds of nuts"],"problemType":"MultipleChoice","stepTitle":"Henning is mixing raisins and nuts to make $$25$$ pounds of trail mix. Raisins cost $$\\\\$4.50$$ a pound and nuts cost $8 a pound. If Henning wants his cost for the trail mix to be $$\\\\$6.60$$ a pound, how many pounds of raisins and how many pounds of nuts should he use?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$10$$ pounds of raisins and $$15$$ pounds of nuts","choices":["$$15$$ pounds of raisins and $$15$$ pounds of nuts","$$10$$ pounds of raisins and $$20$$ pounds of nuts","$$10$$ pounds of raisins and $$15$$ pounds of nuts","$$10$$ pounds of raisins and $$25$$ pounds of nuts"],"hints":{"DefaultPathway":[{"id":"aff0960mixture20a-h1","type":"hint","dependencies":[],"title":"Variables","text":"Represent the number of each type of passengers using variables, raisin as $$x$$ and nut as $$(25-x)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture20a-h2","type":"hint","dependencies":["aff0960mixture20a-h1"],"title":"Multiplication","text":"Multiply the number of each grain type(in variable form) by its respective value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture20a-h3","type":"hint","dependencies":["aff0960mixture20a-h2"],"title":"Conversion","text":"Now, translate the expressions that were multiplied by their monetary values into a single equation. This will be $$\\\\operatorname{4.5}\\\\left(x\\\\right)+8\\\\left(25-x\\\\right)=6.6\\\\times25$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture20a-h4","type":"hint","dependencies":["aff0960mixture20a-h3"],"title":"Calculation","text":"Solve the equation for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture20a-h5","type":"hint","dependencies":["aff0960mixture20a-h4"],"title":"Substitution","text":"Plug $$x$$ into the original relationship to find $$25-x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture20a-h6","type":"hint","dependencies":["aff0960mixture20a-h5"],"title":"Calculation","text":"The answer is $$10$$ pounds of raisins and $$15$$ pounds of nuts.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aff0960mixture20b","stepAnswer":["$$7$$ dimes and $$36$$ quarters"],"problemType":"MultipleChoice","stepTitle":"Alison has $$\\\\$9.70$$ in dimes and quarters. The number of quarters is eight more than four times the number of dimes. How many of each coin does she have?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$7$$ dimes and $$36$$ quarters","choices":["$$7$$ dimes and $$36$$ quarters","$$42$$ dimes and $$30$$ quarters","$$27$$ dimes and $$34$$ quarters","$$89$$ dimes and $$52$$ quarters"],"hints":{"DefaultPathway":[{"id":"aff0960mixture20b-h1","type":"hint","dependencies":[],"title":"Variables","text":"Represent the number of each type of coin using variables.The number of quarters is eight more than four times the number of dimes, which you can represent as $$8+4d$$, and you can represent dimes as $$d$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture20b-h2","type":"hint","dependencies":["aff0960mixture20b-h1"],"title":"Multiplication","text":"Multiply the number of each coin type(in variable form) by its respective value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture20b-h3","type":"hint","dependencies":["aff0960mixture20b-h2"],"title":"Conversion","text":"Now, translate the expressions that were multiplied by their monetary values into a single equation. This will be $$\\\\operatorname{0.25}\\\\left(8+4d\\\\right)+0.1d=9.70$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture20b-h4","type":"hint","dependencies":["aff0960mixture20b-h3"],"title":"Calculation","text":"Solve the equation for $$d$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture20b-h5","type":"hint","dependencies":["aff0960mixture20b-h4"],"title":"Substitution","text":"Plug $$d$$ into the original relationship between the coins to find q.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture20b-h6","type":"hint","dependencies":["aff0960mixture20b-h5"],"title":"Calculation","text":"The answer is $$7$$ dimes and $$36$$ quarters.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aff0960mixture21","title":"Solve Mixture Word Problems","body":"Solve the following problem:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Solve Mixture and Uniform Motion Applications","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aff0960mixture21a","stepAnswer":["$$5$$ pounds of cereal squares and $$25$$ pounds of nuts"],"problemType":"MultipleChoice","stepTitle":"Orlando is mixing nuts and cereal squares to make a party mix. Nuts sell for $7 a pound and cereal squares sell for $4 a pound. Orlando wants to make $$30$$ pounds of party mix at a cost of $$\\\\$6.50$$ a pound, how many pounds of nuts and how many pounds of cereal squares should he use?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$5$$ pounds of cereal squares and $$25$$ pounds of nuts","choices":["$$5$$ pounds of cereal squares and $$25$$ pounds of nuts","$$15$$ pounds of cereal squares and $$25$$ pounds of nuts","$$5$$ pounds of cereal squares and $$35$$ pounds of nuts","$$5$$ pounds of cereal squares and $$30$$ pounds of nuts"],"hints":{"DefaultPathway":[{"id":"aff0960mixture21a-h1","type":"hint","dependencies":[],"title":"Variables","text":"Represent the number of each type of passengers using variables, cereal square as $$x$$ and nut as $$(30-x)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture21a-h2","type":"hint","dependencies":["aff0960mixture21a-h1"],"title":"Multiplication","text":"Multiply the number of each grain type(in variable form) by its respective value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture21a-h3","type":"hint","dependencies":["aff0960mixture21a-h2"],"title":"Conversion","text":"Now, translate the expressions that were multiplied by their monetary values into a single equation. This will be $$4\\\\left(x\\\\right)+7\\\\left(30-x\\\\right)=6.5\\\\times30$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture21a-h4","type":"hint","dependencies":["aff0960mixture21a-h3"],"title":"Calculation","text":"Solve the equation for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture21a-h5","type":"hint","dependencies":["aff0960mixture21a-h4"],"title":"Substitution","text":"Plug $$x$$ into the original relationship to find $$30-x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture21a-h6","type":"hint","dependencies":["aff0960mixture21a-h5"],"title":"Calculation","text":"The answer is $$5$$ pounds of cereal squares and $$25$$ pounds of nuts.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aff0960mixture21b","stepAnswer":["$$330$$ day passes and $$367$$ tournament passes"],"problemType":"MultipleChoice","stepTitle":"The first day of a water polo tournament the total value of tickets sold was $17,610. One-day passes sold for $20 and tournament passes sold for $30. The number of tournament passes sold was $$37$$ more than the number of day passes sold. How many day passes and how many tournament passes were sold?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$330$$ day passes and $$367$$ tournament passes","choices":["$$330$$ day passes and $$367$$ tournament passes","$$254$$ day passes and $$437$$ tournament passes","$$620$$ day passes and $$35$$ tournament passes","$$200$$ day passes and $$342$$ tournament passes"],"hints":{"DefaultPathway":[{"id":"aff0960mixture21b-h1","type":"hint","dependencies":[],"title":"Variables","text":"Represent the number of each type of pass using variables.The number of tournament passes sold was $$37$$ more than the number of day passes sold, which you can represent as $$d+37$$, and you can represent day passes simply as $$d$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture21b-h2","type":"hint","dependencies":["aff0960mixture21b-h1"],"title":"Multiplication","text":"Multiply the number of each pass type(in variable form) by its respective value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture21b-h3","type":"hint","dependencies":["aff0960mixture21b-h2"],"title":"Conversion","text":"Now, translate the expressions that were multiplied by their monetary values into a single equation. This will be $$\\\\operatorname{30}\\\\left(d+7\\\\right)+20d=17610$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture21b-h4","type":"hint","dependencies":["aff0960mixture21b-h3"],"title":"Calculation","text":"Solve the equation for $$d$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture21b-h5","type":"hint","dependencies":["aff0960mixture21b-h4"],"title":"Substitution","text":"Plug $$d$$ into the original relationship between the passes to find $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture21b-h6","type":"hint","dependencies":["aff0960mixture21b-h5"],"title":"Calculation","text":"The answer is $$330$$ day passes and $$367$$ tournament passes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aff0960mixture22","title":"Solve Mixture Word Problems","body":"Solve the following problem:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Solve Mixture and Uniform Motion Applications","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aff0960mixture22a","stepAnswer":["$$21$$ gallons of fruit juice and $$7$$ gallons of soda"],"problemType":"MultipleChoice","stepTitle":"Becca wants to mix fruit juice and soda to make a punch. She can buy fruit juice for $3 a gallon and soda for $4 a gallon. If she wants to make $$28$$ gallons of punch at a cost of $$\\\\$3.25$$ a gallon, how many gallons of fruit juice and how many gallons of soda should she buy?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$21$$ gallons of fruit juice and $$7$$ gallons of soda","choices":["$$21$$ gallons of fruit juice and $$8$$ gallons of soda","$$21$$ gallons of fruit juice and $$7$$ gallons of soda","$$25$$ gallons of fruit juice and $$7$$ gallons of soda","$$24$$ gallons of fruit juice and $$8$$ gallons of soda"],"hints":{"DefaultPathway":[{"id":"aff0960mixture22a-h1","type":"hint","dependencies":[],"title":"Variables","text":"Represent the number of each type of passengers using variables, fruit punch as $$x$$ and soda as $$(28-x)$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture22a-h2","type":"hint","dependencies":["aff0960mixture22a-h1"],"title":"Multiplication","text":"Multiply the number of each drink type(in variable form) by its respective value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture22a-h3","type":"hint","dependencies":["aff0960mixture22a-h2"],"title":"Conversion","text":"Now, translate the expressions that were multiplied by their monetary values into a single equation. This will be $$3\\\\left(x\\\\right)+4\\\\left(28-x\\\\right)=3.25\\\\times28$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture22a-h4","type":"hint","dependencies":["aff0960mixture22a-h3"],"title":"Calculation","text":"Solve the equation for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture22a-h5","type":"hint","dependencies":["aff0960mixture22a-h4"],"title":"Substitution","text":"Plug $$x$$ into the original relationship to find $$28-x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture22a-h6","type":"hint","dependencies":["aff0960mixture22a-h5"],"title":"Calculation","text":"The answer is $$21$$ gallons of fruit juice and $$7$$ gallons of soda.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aff0960mixture22b","stepAnswer":["$$112$$ adult passes and $$199$$ tournament passes"],"problemType":"MultipleChoice","stepTitle":"At the movie theater, the total value of tickets sold was $$\\\\$2, 612.50$$. Adult tickets sold for $10 each and $$\\\\frac{senior}{child}$$ tickets sold for $$\\\\$7.50$$ each. The number of $$\\\\frac{senior}{child}$$ tickets sold was $$25$$ less than twice the number of adult tickets sold. How many $$\\\\frac{senior}{child}$$ tickets and how many adult tickets were sold?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$112$$ adult passes and $$199$$ tournament passes","choices":["$$112$$ adult passes and $$199$$ tournament passes","$$123$$ adult passes and $$209$$ tournament passes","$$72$$ adult passes and $$239$$ tournament passes","$$22$$ adult passes and $$75$$ tournament passes"],"hints":{"DefaultPathway":[{"id":"aff0960mixture22b-h1","type":"hint","dependencies":[],"title":"Variables","text":"Represent the number of each type of ticket using variables.The number of $$\\\\frac{senior}{child}$$ tickets sold was $$25$$ less than twice the number of adult tickets sold, so you can represent $$\\\\frac{child}{senior}$$ tickets(c) as 2a-25, and you can represent adult passes simply as a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture22b-h2","type":"hint","dependencies":["aff0960mixture22b-h1"],"title":"Multiplication","text":"Multiply the number of each ticket type(in variable form) by its respective value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture22b-h3","type":"hint","dependencies":["aff0960mixture22b-h2"],"title":"Conversion","text":"Now, translate the expressions that were multiplied by their monetary values into a single equation. This will be $$10a+\\\\operatorname{7.5}\\\\left(2a-25\\\\right)=2612.5$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture22b-h4","type":"hint","dependencies":["aff0960mixture22b-h3"],"title":"Calculation","text":"Solve the equation for a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture22b-h5","type":"hint","dependencies":["aff0960mixture22b-h4"],"title":"Substitution","text":"Plug a into the original relationship between the passes to find c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture22b-h6","type":"hint","dependencies":["aff0960mixture22b-h5"],"title":"Calculation","text":"The answer is $$112$$ adult passes and $$199$$ tournament passes.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aff0960mixture23","title":"Solve A Uniform Motion Application","body":"Solve the following problem:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Solve Mixture and Uniform Motion Applications","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aff0960mixture23a","stepAnswer":["$$21$$ mph for Wayne and $$28$$ mph for Dennis"],"problemType":"MultipleChoice","stepTitle":"Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis\u2019s speed is seven miles per hour faster than Wayne\u2019s speed, so it takes Wayne two hours to ride to the beach while it takes Dennis $$1.5$$ hours for the ride. Find the speed of both bikers.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$21$$ mph for Wayne and $$28$$ mph for Dennis","choices":["$$21$$ mph for Wayne and $$28$$ mph for Dennis","$$24$$ mph for Wayne and $$31$$ mph for Dennis","$$18$$ mph for Wayne and $$25$$ mph for Dennis","$$25$$ mph for Wayne and $$32$$ mph for Dennis"],"hints":{"DefaultPathway":[{"id":"aff0960mixture23a-h1","type":"hint","dependencies":[],"title":"Variables","text":"Represent the speeds using variables, Wayne\'s speed as $$x$$, Dennis\'s speed as $$x+7$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture23a-h2","type":"hint","dependencies":["aff0960mixture23a-h1"],"title":"Multiplication","text":"Multiply the speed(in variable form) by its respective value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture23a-h3","type":"hint","dependencies":["aff0960mixture23a-h2"],"title":"Conversion","text":"Now, translate the expressions that were multiplied by their monetary values into a single equation. This will be $$\\\\operatorname{1.5}\\\\left(x+7\\\\right)=2(x)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture23a-h4","type":"hint","dependencies":["aff0960mixture23a-h3"],"title":"Calculation","text":"Solve the equation for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture23a-h5","type":"hint","dependencies":["aff0960mixture23a-h4"],"title":"Substitution","text":"Plug $$x$$ into the original relationship find $$x+7$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture23a-h6","type":"hint","dependencies":["aff0960mixture23a-h5"],"title":"Calculation","text":"The answer is $$21$$ mph for Wayne and $$28$$ mph for Dennis.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aff0960mixture23b","stepAnswer":["$$40$$ postcards and $$100$$ stamps"],"problemType":"MultipleChoice","stepTitle":"Julie went to the post office and bought both $$\\\\$0.41$$ stamps and $$\\\\$0.26$$ postcards. She spent $$\\\\$51.40$$. The number of stamps was $$20$$ more than twice the number of postcards. How many of each did she buy?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$40$$ postcards and $$100$$ stamps","choices":["$$40$$ postcards and $$100$$ stamps","$$20$$ postcards and $$37$$ stamps","$$58$$ postcards and $$44$$ stamps","$$21$$ postcards and $$50$$ stamps"],"hints":{"DefaultPathway":[{"id":"aff0960mixture23b-h1","type":"hint","dependencies":[],"title":"Variables","text":"Represent the number of stamps and postcards using variables.The number of stamps was $$20$$ more than twice the number of postcards, so you can represent stamps(s) as $$2p+20$$, and you can represent postcards simply as $$p$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture23b-h2","type":"hint","dependencies":["aff0960mixture23b-h1"],"title":"Multiplication","text":"Multiply the number of postcards and stamps(in variable form) by its respective value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture23b-h3","type":"hint","dependencies":["aff0960mixture23b-h2"],"title":"Conversion","text":"Now, translate the expressions that were multiplied by their monetary values into a single equation. This will be $$\\\\operatorname{0.41}\\\\left(2p+20\\\\right)+0.26p=51.40$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture23b-h4","type":"hint","dependencies":["aff0960mixture23b-h3"],"title":"Calculation","text":"Solve the equation for $$p$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture23b-h5","type":"hint","dependencies":["aff0960mixture23b-h4"],"title":"Substitution","text":"Plug $$p$$ into the original relationship between the postcards and stamps to find s.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture23b-h6","type":"hint","dependencies":["aff0960mixture23b-h5"],"title":"Calculation","text":"The answer is $$40$$ postcards and $$100$$ stamps.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aff0960mixture24","title":"Solve A Uniform Motion Application","body":"Solve the following problem:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Solve Mixture and Uniform Motion Applications","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aff0960mixture24a","stepAnswer":["$$48$$ mph for local train and $$60$$ mph for express train"],"problemType":"MultipleChoice","stepTitle":"An express train and a local train leave Pittsburgh to travel to Washington, D.C. The express train can make the trip in four hours and the local train takes five hours for the trip. The speed of the express train is $$12$$ miles per hour faster than the speed of the local train. Find the speed of both trains.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$48$$ mph for local train and $$60$$ mph for express train","choices":["$$48$$ mph for local train and $$65$$ mph for express train","$$50$$ mph for local train and $$60$$ mph for express train","$$48$$ mph for local train and $$60$$ mph for express train"],"hints":{"DefaultPathway":[{"id":"aff0960mixture24a-h1","type":"hint","dependencies":[],"title":"Variables","text":"Represent the speeds using variables, the speed of local train as $$x$$, the speed of express train as $$x+12$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture24a-h2","type":"hint","dependencies":["aff0960mixture24a-h1"],"title":"Multiplication","text":"Multiply the speed(in variable form) by its respective value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture24a-h3","type":"hint","dependencies":["aff0960mixture24a-h2"],"title":"Conversion","text":"Now, translate the expressions that were multiplied by their monetary values into a single equation. This will be $$4\\\\left(x+12\\\\right)=5(x)$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture24a-h4","type":"hint","dependencies":["aff0960mixture24a-h3"],"title":"Calculation","text":"Solve the equation for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture24a-h5","type":"hint","dependencies":["aff0960mixture24a-h4"],"title":"Substitution","text":"Plug $$x$$ into the original relationship find $$x+12$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture24a-h6","type":"hint","dependencies":["aff0960mixture24a-h5"],"title":"Calculation","text":"The answer is $$48$$ mph for local train and $$60$$ mph for express train.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aff0960mixture24b","stepAnswer":["$$15$$ $10 shares and $$5$$ $12 shares"],"problemType":"MultipleChoice","stepTitle":"Hilda has $210 worth of $10 and $12 stock shares. The number of $10 shares is five more than twice the number of $12 shares. How many of each type of share does she have?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$15$$ $10 shares and $$5$$ $12 shares","choices":["$$15$$ $10 shares and $$5$$ $12 shares","$$22$$ $10 shares and $$4$$ $12 shares","$$21$$ $10 shares and $$3$$ $12 shares","$$8$$ $10 shares and $$12$$ $12 shares"],"hints":{"DefaultPathway":[{"id":"aff0960mixture24b-h1","type":"hint","dependencies":[],"title":"Variables","text":"Represent the number of shares using variables.The number of $10 shares is five more than twice the number of $12 shares, so you can represent $10 shares(t) as $$2l+5$$, and you can represent $12 shares simply as L.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture24b-h2","type":"hint","dependencies":["aff0960mixture24b-h1"],"title":"Multiplication","text":"Multiply the number of shares(in variable form) by its respective value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture24b-h3","type":"hint","dependencies":["aff0960mixture24b-h2"],"title":"Conversion","text":"Now, translate the expressions that were multiplied by their monetary values into a single equation. This will be $$\\\\operatorname{10}\\\\left(2l+5\\\\right)+12l=210$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture24b-h4","type":"hint","dependencies":["aff0960mixture24b-h3"],"title":"Calculation","text":"Solve the equation for l.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture24b-h5","type":"hint","dependencies":["aff0960mixture24b-h4"],"title":"Substitution","text":"Plug l into the original relationship between the shares to find $$t$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture24b-h6","type":"hint","dependencies":["aff0960mixture24b-h5"],"title":"Calculation","text":"The answer is $$15$$ $10 shares and $$5$$ $12 shares.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aff0960mixture25","title":"Solve A Uniform Motion Application","body":"Solve the following problem:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Solve Mixture and Uniform Motion Applications","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aff0960mixture25a","stepAnswer":["$$65$$ mph for Carina and $$50$$ mph for her brother"],"problemType":"MultipleChoice","stepTitle":"Carina is driving from her home in Anaheim to Berkeley on the same day her brother is driving from Berkeley to Anaheim, so they decide to meet for lunch along the way in Buttonwillow. The distance from Anaheim to Berkeley is $$395$$ miles. It takes Carina three hours to get to Buttonwillow, while her brother drives four hours to get there. Carina\u2019s average speed is $$15$$ miles per hour faster than her brother\u2019s average speed. Find Carina\u2019s and her brother\u2019s average speeds.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$65$$ mph for Carina and $$50$$ mph for her brother","choices":["$$65$$ mph for Carina and $$50$$ mph for her brother","$$80$$ mph for Carina and $$45$$ mph for her brother","$$60$$ mph for Carina and $$55$$ mph for her brother"],"hints":{"DefaultPathway":[{"id":"aff0960mixture25a-h1","type":"hint","dependencies":[],"title":"Variables","text":"Represent the speeds using variables, Carina\'s speed as $$x+15$$, the speed of her brother as $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture25a-h2","type":"hint","dependencies":["aff0960mixture25a-h1"],"title":"Multiplication","text":"Multiply the speed(in variable form) by its respective value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture25a-h3","type":"hint","dependencies":["aff0960mixture25a-h2"],"title":"Conversion","text":"Now, translate the expressions that were multiplied by their monetary values into a single equation. This will be $$3\\\\left(x+15\\\\right)+4\\\\left(x\\\\right)=395$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture25a-h4","type":"hint","dependencies":["aff0960mixture25a-h3"],"title":"Calculation","text":"Solve the equation for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture25a-h5","type":"hint","dependencies":["aff0960mixture25a-h4"],"title":"Substitution","text":"Plug $$x$$ into the original relationship find $$x+15$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture25a-h6","type":"hint","dependencies":["aff0960mixture25a-h5"],"title":"Calculation","text":"The answer is $$65$$ mph for Carina and $$50$$ mph for her brother.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aff0960mixture25b","stepAnswer":["$$34$$ general and $$61$$ youth"],"problemType":"MultipleChoice","stepTitle":"The ice rink sold $$95$$ tickets for the afternoon skating session, for a total of $828. General admission tickets cost $10 each and youth tickets cost $8 each. How many general admission tickets and how many youth tickets were sold?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$34$$ general and $$61$$ youth","choices":["$$34$$ general and $$61$$ youth","$$20$$ general and $$30$$ youth","$$32$$ general and $$42$$ youth","$$77$$ general and $$63$$ youth"],"hints":{"DefaultPathway":[{"id":"aff0960mixture25b-h1","type":"hint","dependencies":[],"title":"Variables","text":"Represent the number of each type of ticket using variables.The ice rink sold $$95$$ tickets for the afternoon skating session, so you can represent general admissions tickets(g) as 95-c, and you can represent child tickets simply as c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture25b-h2","type":"hint","dependencies":["aff0960mixture25b-h1"],"title":"Multiplication","text":"Multiply the number of each ticket type(in variable form) by its respective value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture25b-h3","type":"hint","dependencies":["aff0960mixture25b-h2"],"title":"Conversion","text":"Now, translate the expressions that were multiplied by their monetary values into a single equation. This will be $$\\\\operatorname{10}\\\\left(95-c\\\\right)+8c=828$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture25b-h4","type":"hint","dependencies":["aff0960mixture25b-h3"],"title":"Calculation","text":"Solve the equation for c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture25b-h5","type":"hint","dependencies":["aff0960mixture25b-h4"],"title":"Substitution","text":"Plug c into the original relationship between the passes to find g.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture25b-h6","type":"hint","dependencies":["aff0960mixture25b-h5"],"title":"Calculation","text":"The answer is $$34$$ general and $$61$$ youth.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aff0960mixture26","title":"Solve A Uniform Motion Application","body":"Solve the following problem:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Solve Mixture and Uniform Motion Applications","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aff0960mixture26a","stepAnswer":["$$50$$ mph for Christopher and $$40$$ mph for his parents"],"problemType":"MultipleChoice","stepTitle":"Christopher and his parents live $$115$$ miles apart. They met at a restaurant between their homes to celebrate his mother\u2019s birthday. Christopher drove one and a half hours while his parents drove one hour to get to the restaurant. Christopher\u2019s average speed was ten miles per hour faster than his parents\u2019 average speed. What were the average speeds of Christopher and of his parents as they drove to the restaurant?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$50$$ mph for Christopher and $$40$$ mph for his parents","choices":["$$50$$ mph for Christopher and $$45$$ mph for his parents","$$50$$ mph for Christopher and $$40$$ mph for his parents","$$55$$ mph for Christopher and $$40$$ mph for his parents","$$50$$ mph for Christopher and $$30$$ mph for his parents"],"hints":{"DefaultPathway":[{"id":"aff0960mixture26a-h1","type":"hint","dependencies":[],"title":"Variables","text":"Represent the speeds using variables, Christopher\'s speed as $$x+10$$, the speed of his parents as $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture26a-h2","type":"hint","dependencies":["aff0960mixture26a-h1"],"title":"Multiplication","text":"Multiply the speed(in variable form) by its respective value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture26a-h3","type":"hint","dependencies":["aff0960mixture26a-h2"],"title":"Conversion","text":"Now, translate the expressions that were multiplied by their monetary values into a single equation. This will be $$\\\\operatorname{1.5}\\\\left(x+10\\\\right)+1\\\\left(x\\\\right)=115$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture26a-h4","type":"hint","dependencies":["aff0960mixture26a-h3"],"title":"Calculation","text":"Solve the equation for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture26a-h5","type":"hint","dependencies":["aff0960mixture26a-h4"],"title":"Substitution","text":"Plug $$x$$ into the original relationship find $$x+10$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture26a-h6","type":"hint","dependencies":["aff0960mixture26a-h5"],"title":"Calculation","text":"The answer is $$50$$ mph for Christopher and $$40$$ mph for his parents.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aff0960mixture26b","stepAnswer":["$$114$$ general and $$246$$ student"],"problemType":"MultipleChoice","stepTitle":"The box office sold $$360$$ tickets to a concert at the college. The total receipts were $4,170. General admission tickets cost $15 and student tickets cost $10. How many of each kind of ticket was sold?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$114$$ general and $$246$$ student","choices":["$$114$$ general and $$246$$ student","$$72$$ general and $$333$$ student","$$24$$ general and $$156$$ student","$$189$$ general and $$112$$ student"],"hints":{"DefaultPathway":[{"id":"aff0960mixture26b-h1","type":"hint","dependencies":[],"title":"Variables","text":"Represent the number of each type of ticket using variables.The box office sold $$360$$ tickets, so you can represent general admission tickets as $$(360-s)$$, and you can represent student tickets simply as s.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture26b-h2","type":"hint","dependencies":["aff0960mixture26b-h1"],"title":"Multiplication","text":"Multiply the number of each ticket type(in variable form) by its respective value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture26b-h3","type":"hint","dependencies":["aff0960mixture26b-h2"],"title":"Conversion","text":"Now, translate the expressions that were multiplied by their monetary values into a single equation. This will be $$\\\\operatorname{15}\\\\left(360-s\\\\right)+10s=4170$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture26b-h4","type":"hint","dependencies":["aff0960mixture26b-h3"],"title":"Calculation","text":"Solve the equation for s.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture26b-h5","type":"hint","dependencies":["aff0960mixture26b-h4"],"title":"Substitution","text":"Plug s into the original relationship between the passes to find g.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture26b-h6","type":"hint","dependencies":["aff0960mixture26b-h5"],"title":"Calculation","text":"The answer is $$114$$ general and $$246$$ student.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aff0960mixture27","title":"Solve A Uniform Motion Application","body":"Solve the following problem:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Solve Mixture and Uniform Motion Applications","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aff0960mixture27a","stepAnswer":["$$2.5$$ hours"],"problemType":"MultipleChoice","stepTitle":"Two truck drivers leave a rest area on the interstate at the same time. One truck travels east and the other one travels west. The truck traveling west travels at $$70$$ mph and the truck traveling east has an average speed of $$60$$ mph. How long will they travel before they are $$325$$ miles apart?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2.5$$ hours","choices":["$$3.5$$ hours","$$2.5$$ hours","$$3$$ hours","$$2$$ hours"],"hints":{"DefaultPathway":[{"id":"aff0960mixture27a-h1","type":"hint","dependencies":[],"title":"Variables","text":"Assume the traveling time as $$t$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture27a-h2","type":"hint","dependencies":["aff0960mixture27a-h1"],"title":"Multiplication","text":"Multiply the time (in variable form) by its respective value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture27a-h3","type":"hint","dependencies":["aff0960mixture27a-h2"],"title":"Conversion","text":"Now, translate the expressions that were multiplied by their monetary values into a single equation. This will be $$\\\\operatorname{70}\\\\left(t\\\\right)+\\\\operatorname{60}\\\\left(t\\\\right)=325$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture27a-h4","type":"hint","dependencies":["aff0960mixture27a-h3"],"title":"Calculation","text":"Solve the equation for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture27a-h5","type":"hint","dependencies":["aff0960mixture27a-h4"],"title":"Calculation","text":"The answer is $$2.5$$ hours.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aff0960mixture27b","stepAnswer":["$$4$$ pounds of macadamia nuts and $$8$$ pounds almonds"],"problemType":"MultipleChoice","stepTitle":"Macario is making $$12$$ pounds of nut mixture with macadamia nuts and almonds. Macadamia nuts cost $9 per pound and almonds cost $$\\\\$5.25$$ per pound. How many pounds of macadamia nuts and how many pounds of almonds should Macario use for the mixture to cost $$\\\\$6.50$$ per pound to make?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$4$$ pounds of macadamia nuts and $$8$$ pounds almonds","choices":["$$4$$ pounds of macadamia nuts and $$8$$ pounds almonds","$$2$$ pounds of macadamia nuts and $$9$$ pounds almonds","$$3$$ pounds of macadamia nuts and $$6$$ pounds almonds","$$12$$ pounds of macadamia nuts and $$3$$ pounds almonds"],"hints":{"DefaultPathway":[{"id":"aff0960mixture27b-h1","type":"hint","dependencies":[],"title":"Variables","text":"Represent the number of each type of nut using variables.Macario is making $$12$$ pounds of nut mixture with macadamia nuts and almonds, so you can represent pounds of macadamia(m) as $$(120-a)$$, and you can represent pounds of almonds simply as a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture27b-h2","type":"hint","dependencies":["aff0960mixture27b-h1"],"title":"Multiplication","text":"Multiply the number of each nut type(in variable form) by its respective value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture27b-h3","type":"hint","dependencies":["aff0960mixture27b-h2"],"title":"Conversion","text":"Now, translate the expressions that were multiplied by their monetary values into a single equation. This will be $$12(6.5)=9\\\\left(120-a\\\\right)+5.25a$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture27b-h4","type":"hint","dependencies":["aff0960mixture27b-h3"],"title":"Calculation","text":"Solve the equation for a.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture27b-h5","type":"hint","dependencies":["aff0960mixture27b-h4"],"title":"Substitution","text":"Plug a into the original relationship between the nuts to find $$m$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture27b-h6","type":"hint","dependencies":["aff0960mixture27b-h5"],"title":"Calculation","text":"The answer is $$4$$ pounds of macadamia nuts and $$8$$ pounds almonds.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aff0960mixture28","title":"Solve A Uniform Motion Application","body":"Solve the following problem:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Solve Mixture and Uniform Motion Applications","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aff0960mixture28a","stepAnswer":["$$3$$ hours"],"problemType":"MultipleChoice","stepTitle":"Pierre and Monique leave their home in Portland at the same time. Pierre drives north on the turnpike at a speed of $$75$$ miles per hour while Monique drives south at a speed of $$68$$ miles per hour. How long will it take them to be $$429$$ miles apart?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$3$$ hours","choices":["$$3.5$$ hours","$$2.5$$ hours","$$3$$ hours","$$2$$ hours"],"hints":{"DefaultPathway":[{"id":"aff0960mixture28a-h1","type":"hint","dependencies":[],"title":"Variables","text":"Assume the traveling time as $$t$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture28a-h2","type":"hint","dependencies":["aff0960mixture28a-h1"],"title":"Multiplication","text":"Multiply the time (in variable form) by its respective value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture28a-h3","type":"hint","dependencies":["aff0960mixture28a-h2"],"title":"Conversion","text":"Now, translate the expressions that were multiplied by their monetary values into a single equation. This will be $$\\\\operatorname{75}\\\\left(t\\\\right)+\\\\operatorname{68}\\\\left(t\\\\right)=429$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture28a-h4","type":"hint","dependencies":["aff0960mixture28a-h3"],"title":"Calculation","text":"Solve the equation for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture28a-h5","type":"hint","dependencies":["aff0960mixture28a-h4"],"title":"Calculation","text":"The answer is $$3$$ hours.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aff0960mixture28b","stepAnswer":["$$3.6$$ lbs Bermuda seed and $$5.4$$ lbs Fescue seed"],"problemType":"MultipleChoice","stepTitle":"Riley is planning to plant a lawn in his yard. He will need nine pounds of grass seed. He wants to mix Bermuda seed that costs $$\\\\$4.80$$ per pound with Fescue seed that costs $$\\\\$3.50$$ per pound. How much of each seed should he buy so that the overall cost will be $$\\\\$4.02$$ per pound?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$3.6$$ lbs Bermuda seed and $$5.4$$ lbs Fescue seed","choices":["$$3.6$$ lbs Bermuda seed and $$5.4$$ lbs Fescue seed","$$4.3$$ lbs Bermuda seed and $$4.7$$ lbs Fescue seed","$$2$$ lbs Bermuda seed and $$3.4$$ lbs Fescue seed","$$1.2$$ lbs Bermuda seed and $$7.3$$ lbs Fescue seed"],"hints":{"DefaultPathway":[{"id":"aff0960mixture28b-h1","type":"hint","dependencies":[],"title":"Variables","text":"Represent the number of each type of ticket using variables. He will need nine pounds of grass seed, so you can represent pounds of Bermuda seed(b) as $$(9-f)$$, and you can represent pounds of Fescue seed simply as f.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture28b-h2","type":"hint","dependencies":["aff0960mixture28b-h1"],"title":"Multiplication","text":"Multiply the number of each seed type(in variable form) by its respective value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture28b-h3","type":"hint","dependencies":["aff0960mixture28b-h2"],"title":"Conversion","text":"Now, translate the expressions that were multiplied by their monetary values into a single equation. This will be $$9(4.02)=\\\\operatorname{4.8}\\\\left(9-f\\\\right)+3.5f$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture28b-h4","type":"hint","dependencies":["aff0960mixture28b-h3"],"title":"Calculation","text":"Solve the equation for f.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture28b-h5","type":"hint","dependencies":["aff0960mixture28b-h4"],"title":"Substitution","text":"Plug f into the original relationship between the nuts to find $$b$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture28b-h6","type":"hint","dependencies":["aff0960mixture28b-h5"],"title":"Calculation","text":"The answer is $$3.6$$ lbs Bermuda seed and $$5.4$$ lbs Fescue seed.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aff0960mixture29","title":"Solve A Uniform Motion Application","body":"Solve the following problem:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Solve Mixture and Uniform Motion Applications","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aff0960mixture29a","stepAnswer":["$$65$$ mph for Carina and $$50$$ mph for her brother"],"problemType":"MultipleChoice","stepTitle":"When Naoko walks to school, it takes her $$30$$ minutes. If she rides her bike, it takes her $$15$$ minutes. Her speed is three miles per hour faster when she rides her bike than when she walks. What is her speed walking and her speed riding her bike?","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$65$$ mph for Carina and $$50$$ mph for her brother","choices":["$$2$$ mph for walking and $$5$$ mph for bike","$$3$$ mph for walking and $$6$$ mph for bike","$$4$$ mph for walking and $$7$$ mph for bike","$$6$$ mph for walking and $$9$$ mph for bike","$$65$$ mph for Carina and $$50$$ mph for her brother"],"hints":{"DefaultPathway":[{"id":"aff0960mixture29a-h1","type":"hint","dependencies":[],"title":"Variables","text":"Represent the speeds using variables, the speed of riding bike as $$x+3$$, her walking speed as $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture29a-h2","type":"hint","dependencies":["aff0960mixture29a-h1"],"title":"Multiplication","text":"Multiply the speed(in variable form) by its respective value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture29a-h3","type":"hint","dependencies":["aff0960mixture29a-h2"],"title":"Conversion","text":"Now, translate the expressions that were multiplied by their monetary values into a single equation. This will be $$\\\\frac{1}{4} \\\\left(x+3\\\\right)=\\\\frac{1}{2} x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture29a-h4","type":"hint","dependencies":["aff0960mixture29a-h3"],"title":"Calculation","text":"Solve the equation for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture29a-h5","type":"hint","dependencies":["aff0960mixture29a-h4"],"title":"Substitution","text":"Plug $$x$$ into the original relationship find $$x+3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture29a-h6","type":"hint","dependencies":["aff0960mixture29a-h5"],"title":"Calculation","text":"The answer is $$3$$ mph for walking and $$6$$ mph for bike.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aff0960mixture29b","stepAnswer":["Kathy\'s speed: $$5$$ mph, Cheryl\'s speed: $$3$$ mph"],"problemType":"MultipleChoice","stepTitle":"Kathy and Cheryl are walking in a fundraiser. Kathy completes the course in $$4.8$$ hours and Cheryl completes the course in eight hours. Kathy walks two miles per hour faster than Cheryl. Find Kathy\u2019s speed and Cheryl\u2019s speed.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Kathy\'s speed: $$5$$ mph, Cheryl\'s speed: $$3$$ mph","choices":["Kathy\'s speed: $$5$$ mph, Cheryl\'s speed: $$3$$ mph","Kathy\'s speed: $$3$$ mph, Cheryl\'s speed: $$5$$ mph","Kathy\'s speed: $$2$$ mph, Cheryl\'s speed: $$7$$ mph","Kathy\'s speed: $$4$$ mph, Cheryl\'s speed: $$6$$ mph"],"hints":{"DefaultPathway":[{"id":"aff0960mixture29b-h1","type":"hint","dependencies":[],"title":"Variables","text":"Represent their speed using variables. Kathy walks two miles per hour faster than Cheryl, so you can represent Kathy(k) as $$2+c$$, and you can represent Cheryl simply as c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture29b-h2","type":"hint","dependencies":["aff0960mixture29b-h1"],"title":"Multiplication","text":"Multiply the representations of walkers(in variable form) by how long they take.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture29b-h3","type":"hint","dependencies":["aff0960mixture29b-h2"],"title":"Conversion","text":"Now, translate the expressions that were multiplied by their monetary values into a single equation. This will be $$\\\\operatorname{4.8}\\\\left(2+c\\\\right)=8c$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture29b-h4","type":"hint","dependencies":["aff0960mixture29b-h3"],"title":"Calculation","text":"Solve the equation for c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture29b-h5","type":"hint","dependencies":["aff0960mixture29b-h4"],"title":"Substitution","text":"Plug c into the original relationship between the walkers to find k.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture29b-h6","type":"hint","dependencies":["aff0960mixture29b-h5"],"title":"Calculation","text":"Kathy walks at $$5$$ mph while Cheryl walks at $$3$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aff0960mixture30","title":"Solve A Uniform Motion Application","body":"Solve the following problem:","variabilization":{},"oer":"https://openstax.org/details/books/intermediate-algebra-2e <OpenStax: Intermediate Algebra>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"2.4 Solve Mixture and Uniform Motion Applications","courseName":"OpenStax: Intermediate Algebra","steps":[{"id":"aff0960mixture30a","stepAnswer":["$$10$$ mph for running and $$16$$ mph for biking"],"problemType":"MultipleChoice","stepTitle":"Cruz is training to compete in a triathlon. He left his house at 6:00 and ran until 7:30. Then he rode his bike until 9:45. He covered a total distance of $$51$$ miles. His speed when biking was $$1.6$$ times his speed when running. Find Cruz\u2019s biking and running speeds.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$10$$ mph for running and $$16$$ mph for biking","choices":["$$12$$ mph for running and $$16$$ mph for biking","$$12$$ mph for running and $$19$$ mph for biking","$$10$$ mph for running and $$16$$ mph for biking","$$15$$ mph for running and $$27$$ mph for biking"],"hints":{"DefaultPathway":[{"id":"aff0960mixture30a-h1","type":"hint","dependencies":[],"title":"Variables","text":"Represent the speeds using variables, biking speed as $$1.6x$$, running speed as $$x$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture30a-h2","type":"hint","dependencies":["aff0960mixture30a-h1"],"title":"Multiplication","text":"Multiply the speed(in variable form) by its respective value.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture30a-h3","type":"hint","dependencies":["aff0960mixture30a-h2"],"title":"Conversion","text":"Now, translate the expressions that were multiplied by their monetary values into a single equation. This will be $$1.5x+2.25\\\\times1.6x=51$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture30a-h4","type":"hint","dependencies":["aff0960mixture30a-h3"],"title":"Calculation","text":"Solve the equation for $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture30a-h5","type":"hint","dependencies":["aff0960mixture30a-h4"],"title":"Substitution","text":"Plug $$x$$ into the original relationship find $$1.6x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture30a-h6","type":"hint","dependencies":["aff0960mixture30a-h5"],"title":"Calculation","text":"The answer is $$10$$ mph for running and $$16$$ mph for biking.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}},{"id":"aff0960mixture30b","stepAnswer":["Commercial $$550$$ mph, private plane $$340$$ mph"],"problemType":"MultipleChoice","stepTitle":"A commercial jet and a private airplane fly from Denver to Phoenix. It takes the commercial jet $$1.6$$ hours for the flight, and it takes the private airplane $$2.6$$ hours. The speed of the commercial jet is $$210$$ miles per hour faster than the speed of the private airplane. Find the speed of both airplanes to the nearest $$10$$ mph.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"Commercial $$550$$ mph, private plane $$340$$ mph","choices":["Commercial $$550$$ mph, private plane $$340$$ mph","Commercial $$230$$ mph, private plane $$120$$ mph","Commercial $$600$$ mph, private plane $$320$$ mph","Commercial $$70$$ mph, private plane $$345$$ mph"],"hints":{"DefaultPathway":[{"id":"aff0960mixture30b-h1","type":"hint","dependencies":[],"title":"Variables","text":"Represent their speed using variables. The speed of the commercial jet is $$210$$ miles per hour faster than the speed of the private airplane, so you can represent the commercial jet(c) as $$210+p$$, and you can represent the private jet simply as $$p$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture30b-h2","type":"hint","dependencies":["aff0960mixture30b-h1"],"title":"Multiplication","text":"Multiply the representations of the planes(in variable form) by how long they take.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture30b-h3","type":"hint","dependencies":["aff0960mixture30b-h2"],"title":"Conversion","text":"Now, translate the expressions that were multiplied by their time values into a single equation. This will be $$\\\\operatorname{1.6}\\\\left(p+210\\\\right)=2.6p$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture30b-h4","type":"hint","dependencies":["aff0960mixture30b-h3"],"title":"Calculation","text":"Solve the equation for $$p$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture30b-h5","type":"hint","dependencies":["aff0960mixture30b-h4"],"title":"Substitution","text":"Plug $$p$$ into the original relationship between the walkers to find c.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"aff0960mixture30b-h6","type":"hint","dependencies":["aff0960mixture30b-h5"],"title":"Calculation","text":"The commercial jet travels at $$550$$ mph, and the private plane at $$340$$ mph.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"aff8a70precal6","title":"Algebra with Absolute Values","body":"These problems are harder, often highlighting an important subtlety","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Algebra with Absolute Values","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"aff8a70precal6a","stepAnswer":["$$(-\\\\infty,\\\\frac{-5}{2})$$ $$\\\\cup$$ $$(\\\\frac{1}{2},\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"Find all values of $$x$$ such that $$|\\\\frac{3}{x+1}|<2$$. Express your answer in interval notation.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\frac{-5}{2})$$ $$\\\\cup$$ $$(\\\\frac{1}{2},\\\\infty)$$","choices":["$$(-\\\\infty,\\\\frac{-5}{2})$$ $$\\\\cup$$ $$(\\\\frac{1}{2},\\\\infty)$$","$$(-\\\\infty,\\\\frac{-5}{2})$$ $$\\\\cup$$ $$(\\\\frac{-1}{2},\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"aff8a70precal6a-h1","type":"hint","dependencies":[],"title":"Can\'t divide by $$0$$","text":"Note that the denominator cannot be $$0$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precal6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["aff8a70precal6a-h1"],"title":"Can\'t divide by $$0$$","text":"What value $$x$$ cannot equal if $$x+1 \\\\neq 0$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precal6a-h3","type":"hint","dependencies":["aff8a70precal6a-h2"],"title":"Property","text":"Note that $$|\\\\frac{a}{b}|=\\\\frac{|a|}{|b|}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precal6a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["aff8a70precal6a-h3"],"title":"Property","text":"Is $$|\\\\frac{3}{x+1}|$$ equaled to $$\\\\frac{|3|}{|x+1|}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"aff8a70precal6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["aff8a70precal6a-h4"],"title":"Absolute value","text":"What is the value of $$|3|$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precal6a-h6","type":"hint","dependencies":["aff8a70precal6a-h5"],"title":"Absolute value","text":"For $$x \\\\neq -1$$, $$\\\\frac{3}{|x+1|}<2$$, which means $$2|x+1|>3$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precal6a-h7","type":"hint","dependencies":["aff8a70precal6a-h6"],"title":"Absolute value","text":"Here, we use the fact that for $$b>0$$, $$a<c$$, we can imply $$b a<b c$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precal6a-h8","type":"hint","dependencies":["aff8a70precal6a-h7"],"title":"Simplification","text":"$$2|x+1|>3$$, so we have $$|x+1|>\\\\frac{3}{2}$$ if we divide both side by $$2$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precal6a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x+1>\\\\frac{3}{2}$$"],"dependencies":["aff8a70precal6a-h8"],"title":"Remove the absolute value","text":"If $$x+1 \\\\geq 0$$, what is $$|x+1|>\\\\frac{3}{2}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$x+1>\\\\frac{3}{2}$$","$$x+1>\\\\frac{-3}{2}$$","$$x+1<\\\\frac{-3}{2}$$"]},{"id":"aff8a70precal6a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x+1<\\\\frac{-3}{2}$$"],"dependencies":["aff8a70precal6a-h9"],"title":"Remove the absolute value","text":"If $$x+1<0$$, what is $$|x+1|>\\\\frac{3}{2}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$x+1>\\\\frac{3}{2}$$","$$x+1>\\\\frac{-3}{2}$$","$$x+1<\\\\frac{-3}{2}$$"]},{"id":"aff8a70precal6a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["aff8a70precal6a-h10"],"title":"Simplification","text":"$$x+1>\\\\frac{3}{2}$$, x>?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{1}{2}$$","$$\\\\frac{-1}{2}$$"]},{"id":"aff8a70precal6a-h12","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{-5}{2}$$"],"dependencies":["aff8a70precal6a-h11"],"title":"Simplification","text":"$$x+1<\\\\frac{-3}{2}$$, x<?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{-5}{2}$$","$$\\\\frac{5}{2}$$"]},{"id":"aff8a70precal6a-h13","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(-\\\\infty,\\\\frac{-5}{2})$$ $$\\\\cup$$ $$(\\\\frac{1}{2},\\\\infty)$$"],"dependencies":["aff8a70precal6a-h12"],"title":"Interval Notation","text":"Write $$x>\\\\frac{1}{2}$$ or $$x<\\\\frac{-5}{2}$$ in interval notation. What is the answer?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$(-\\\\infty,\\\\frac{-5}{2})$$ $$\\\\cup$$ $$(\\\\frac{1}{2},\\\\infty)$$","$$(-\\\\infty,\\\\frac{-5}{2})$$ $$\\\\cup$$ $$(\\\\frac{-1}{2},\\\\infty)$$"],"subHints":[{"id":"aff8a70precal6a-h13-s1","type":"hint","dependencies":[],"title":"Interval Notation","text":"Remember that $$x \\\\neq -1$$","variabilization":{},"oer":"https://OATutor.io","license":""}]}]}}]},{"id":"aff8a70precal7","title":"Algebra with Absolute Values","body":"These questions are challenging, requiring mastery of each concept and their interrelations.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Algebra with Absolute Values","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"aff8a70precal7a","stepAnswer":["$$(\\\\frac{-1}{2},\\\\frac{1}{2})$$"],"problemType":"MultipleChoice","stepTitle":"Find all values of $$x$$ such that $$|\\\\frac{5}{x^2+1}|>4$$. Express your answer in interval notation.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(\\\\frac{-1}{2},\\\\frac{1}{2})$$","choices":["$$(\\\\frac{-1}{2},\\\\frac{1}{2})$$","$$\\\\frac{\\\\pm 1}{2}$$","$$(-1,1)$$","$$\\\\pm 1$$"],"hints":{"DefaultPathway":[{"id":"aff8a70precal7a-h1","type":"hint","dependencies":[],"title":"Remove the absolute value","text":"Note $$x^2 \\\\geq 0$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precal7a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["aff8a70precal7a-h1"],"title":"Remove the absolute value","text":"Is $$x^2+1 \\\\geq 0$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"aff8a70precal7a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["aff8a70precal7a-h2"],"title":"Remove the absolute value","text":"$$|x^2+1|$$ $$=x^2+1$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"aff8a70precal7a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{5}{x^2+1}$$"],"dependencies":["aff8a70precal7a-h3"],"title":"Remove the absolute value","text":"What does $$|\\\\frac{5}{x^2+1}|$$ equal to?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{5}{x^2+1}$$","$$\\\\frac{5}{\\\\left(-x^2-1\\\\right)}$$"],"subHints":[{"id":"aff8a70precal7a-h4-s1","type":"hint","dependencies":[],"title":"Remove the absolute value","text":"Use the property $$|\\\\frac{a}{b}|=\\\\frac{|a|}{|b|}$$ to simplify.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"aff8a70precal7a-h5","type":"hint","dependencies":["aff8a70precal7a-h4"],"title":"Simplification","text":"For $$b>0$$, $$a>c$$, then $$b a>b c$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precal7a-h6","type":"hint","dependencies":["aff8a70precal7a-h5"],"title":"Simplification","text":"$$\\\\frac{5}{x^2+1}>4$$ is equivalent to $$5>4\\\\left(x^2+1\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precal7a-h7","type":"hint","dependencies":["aff8a70precal7a-h6"],"title":"Simplification","text":"$$5>4\\\\left(x^2+1\\\\right)$$ is equivalent to $$x^2+1<\\\\frac{5}{4}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precal7a-h8","type":"hint","dependencies":["aff8a70precal7a-h7"],"title":"Simplification","text":"$$x^2+1<\\\\frac{5}{4}$$ is equivalent to $$x^2<\\\\frac{1}{4}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precal7a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["aff8a70precal7a-h8"],"title":"Simplification","text":"Is $$x^2<\\\\frac{1}{4}$$ equivalent to $${|x|}^2<\\\\frac{1}{4}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"],"subHints":[{"id":"aff8a70precal7a-h9-s1","type":"hint","dependencies":[],"title":"Simplification","text":"$$x^2={\\\\left(-x\\\\right)}^2$$, so $$x^2={|x|}^2$$","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"aff8a70precal7a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["aff8a70precal7a-h9"],"title":"Simplification","text":"Is $${|x|}^2<\\\\frac{1}{4}$$ equivalent to $$|x|<\\\\frac{1}{2}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"],"subHints":[{"id":"aff8a70precal7a-h10-s1","type":"hint","dependencies":[],"title":"Simplification","text":"For a,b>0, $$a<b$$ is equivalent to $$\\\\sqrt{a}<\\\\sqrt{b}$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"aff8a70precal7a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x<\\\\frac{1}{2}$$"],"dependencies":["aff8a70precal7a-h10"],"title":"Simplification","text":"For $$|x|<\\\\frac{1}{2}$$, if $$x \\\\geq 0$$, which one is correct?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$x<\\\\frac{1}{2}$$","$$x>\\\\frac{1}{2}$$","$$x<\\\\frac{-1}{2}$$","$$x>\\\\frac{-1}{2}$$"]},{"id":"aff8a70precal7a-h12","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x>\\\\frac{-1}{2}$$"],"dependencies":["aff8a70precal7a-h11"],"title":"Simplification","text":"For $$|x|<\\\\frac{1}{2}$$, if $$x<0$$, which one is correct?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$x<\\\\frac{1}{2}$$","$$x>\\\\frac{1}{2}$$","$$x<\\\\frac{-1}{2}$$","$$x>\\\\frac{-1}{2}$$"]},{"id":"aff8a70precal7a-h13","type":"hint","dependencies":["aff8a70precal7a-h12"],"title":"Interval Notation","text":"Write $$x<\\\\frac{1}{2}$$ if $$x \\\\geq 0$$ and $$x>\\\\frac{-1}{2}$$ if $$x<0$$ in interval notation.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precal7a-h14","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(\\\\frac{-1}{2},\\\\frac{1}{2})$$"],"dependencies":["aff8a70precal7a-h13"],"title":"Interval Notation","text":"What is the interval notation of $$\\\\frac{-1}{2}<x<\\\\frac{1}{2}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$(\\\\frac{-1}{2},\\\\frac{1}{2})$$","$$\\\\frac{\\\\pm 1}{2}$$","$$(-1,1)$$","$$\\\\pm 1$$"]}]}}]},{"id":"aff8a70precal8","title":"Algebra with Absolute Values","body":"These problems are harder, often highlighting an important subtlety","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Algebra with Absolute Values","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"aff8a70precal8a","stepAnswer":["$$(-\\\\infty,-4)$$ $$\\\\cup$$ $$(0,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"Find all values of $$x$$ such that $$|\\\\frac{{\\\\left(x+2\\\\right)}^3}{2}|>|2x+4|$$. Express your answer in interval notation.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,-4)$$ $$\\\\cup$$ $$(0,\\\\infty)$$","choices":["$$(-\\\\infty,-4)$$ $$\\\\cup$$ $$(0,\\\\infty)$$","$$(-4,0)$$"],"hints":{"DefaultPathway":[{"id":"aff8a70precal8a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":[],"title":"LHS simplification","text":"Is $$|\\\\frac{{\\\\left(x+2\\\\right)}^3}{2}|$$ equivalent to $$\\\\frac{|{\\\\left(x+2\\\\right)}^3|}{|2|}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"],"subHints":[{"id":"aff8a70precal8a-h1-s1","type":"hint","dependencies":[],"title":"LHS simplification","text":"Use the property $$|\\\\frac{a}{b}|=\\\\frac{|a|}{|b|}$$ to simplify.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"aff8a70precal8a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["aff8a70precal8a-h1"],"title":"LHS simplification","text":"Is $$\\\\frac{|{\\\\left(x+2\\\\right)}^3|}{|2|}$$ equivalent to $$\\\\frac{{|x+2|}^3}{2}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"],"subHints":[{"id":"aff8a70precal8a-h2-s1","type":"hint","dependencies":[],"title":"LHS simplification","text":"Use the property $$|a b|=|a| |b|$$","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"aff8a70precal8a-h3","type":"hint","dependencies":["aff8a70precal8a-h2"],"title":"RHS simplification","text":"$$|2x+4|=2|x+2|$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precal8a-h4","type":"hint","dependencies":["aff8a70precal8a-h3"],"title":"RHS simplification","text":"Use the property $$|a b|=|a| |b|$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precal8a-h5","type":"hint","dependencies":["aff8a70precal8a-h4"],"title":"Special Case","text":"Note that $$x=-2$$ gives $$LHS=0$$ and $$RHS=0$$, so $$x$$ cannot be $$-2$$, which means $$x+2$$ cannot be $$0$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precal8a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["aff8a70precal8a-h5"],"title":"Special Case","text":"Is $$x+2 \\\\neq 0$$ equivalent to $$|x+2|>0$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"],"subHints":[{"id":"aff8a70precal8a-h6-s1","type":"hint","dependencies":[],"title":"Special Case","text":"$$|a|=0$$ is equivalent to $$a=0$$","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"aff8a70precal8a-h7","type":"hint","dependencies":["aff8a70precal8a-h6"],"title":"Simplification","text":"For $$x \\\\neq 2$$, we now have $$\\\\frac{{|x+2|}^3}{2}>2|x+2|$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precal8a-h8","type":"hint","dependencies":["aff8a70precal8a-h7"],"title":"Simplification","text":"Multiply by $$2$$ on both sides, so we get $${|x+2|}^3>4|x+2|$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precal8a-h9","type":"hint","dependencies":["aff8a70precal8a-h8"],"title":"Simplification","text":"$${|x+2|}^3>4|x+2|$$ is equivalent to $${|x+2|}^2>4$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precal8a-h10","type":"hint","dependencies":["aff8a70precal8a-h9"],"title":"Simplification","text":"For $$b>0$$, $$a b>c b$$ is equivalent to $$a>c$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precal8a-h11","type":"hint","dependencies":["aff8a70precal8a-h10"],"title":"Simplification","text":"$${|x+2|}^2>4$$ is equivalent to $$|x+2|>2$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precal8a-h12","type":"hint","dependencies":["aff8a70precal8a-h11"],"title":"Simplification","text":"For a,b > $$0$$, $$a>b$$ is equivalent to $$\\\\sqrt{a}>\\\\sqrt{b}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precal8a-h13","type":"hint","dependencies":["aff8a70precal8a-h12"],"title":"Simplification","text":"$$|x+2|>2$$ is equivalent to $$x+2>2$$ or $$x+2<-2$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precal8a-h14","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$0$$"],"dependencies":["aff8a70precal8a-h13"],"title":"Simplification","text":"$$x+2>2$$, x>?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precal8a-h15","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-4$$"],"dependencies":["aff8a70precal8a-h14"],"title":"Simplification","text":"$$x+2<-2$$, x<?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precal8a-h16","type":"hint","dependencies":["aff8a70precal8a-h15"],"title":"Interval Notation","text":"Write $$x>0$$ or $$x<4$$ in interval notation.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precal8a-h17","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(-\\\\infty,-4)$$ $$\\\\cup$$ $$(0,\\\\infty)$$"],"dependencies":["aff8a70precal8a-h16"],"title":"Interval Notation","text":"What is the answer in interval notation?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$(-\\\\infty,-4)$$ $$\\\\cup$$ $$(0,\\\\infty)$$","$$(-4,0)$$"]}]}}]},{"id":"aff8a70precal9","title":"Algebra with Absolute Values","body":"These problems are harder, often highlighting an important subtlety","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Algebra with Absolute Values","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"aff8a70precal9a","stepAnswer":["Yes"],"problemType":"MultipleChoice","stepTitle":"Using the triangle inequality, show that for any two numbers A and B, $$|A-B| \\\\geq |A|-|B|$$. Can we prove this?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"aff8a70precal9a-h1","type":"hint","dependencies":[],"title":"Proof","text":"$$|A|=|A-B+B|$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precal9a-h2","type":"hint","dependencies":["aff8a70precal9a-h1"],"title":"Triangle Inequality","text":"$$|A-B+B| \\\\leq |A-B|+|B|$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precal9a-h3","type":"hint","dependencies":["aff8a70precal9a-h2"],"title":"Triangle Inequality","text":"Use the triangle inequality $$|x+y| \\\\leq |x|+|y|$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precal9a-h4","type":"hint","dependencies":["aff8a70precal9a-h3"],"title":"Triangle Inequality","text":"Now, we have $$|A|=|A-B+B| \\\\leq |A-B|+|B|$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precal9a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["aff8a70precal9a-h4"],"title":"Triangle Inequality","text":"Can we prove $$|A-B| \\\\geq |A|-|B|$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"],"subHints":[{"id":"aff8a70precal9a-h5-s1","type":"hint","dependencies":[],"title":"Triangle Inequality","text":"$$|A|-|B|=|A-B+B| \\\\leq |A-B|$$, so $$|A|-|B| \\\\leq |A-B|$$. Proved.","variabilization":{},"oer":"https://OATutor.io","license":""}]}]}}]},{"id":"aff8a70precala1","title":"Algebra with Absolute Values","body":"These questions test your knowledge of the core concepts.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Algebra with Absolute Values","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"aff8a70precala1a","stepAnswer":["$$\\\\frac{5}{2}$$ or $$\\\\frac{3}{2}$$"],"problemType":"MultipleChoice","stepTitle":"Find all values of $$x$$ such that $$|\\\\frac{-3}{2-x}|=6$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{5}{2}$$ or $$\\\\frac{3}{2}$$","choices":["$$\\\\frac{5}{2}$$ or $$\\\\frac{3}{2}$$","$$\\\\frac{5}{2}$$","$$\\\\frac{3}{2}$$"],"hints":{"DefaultPathway":[{"id":"aff8a70precala1a-h1","type":"hint","dependencies":[],"title":"The property of the absolute value","text":"$$|\\\\frac{a}{b}|=\\\\frac{|a|}{|b|}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precala1a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{|-3|}{|2-x|}$$"],"dependencies":["aff8a70precala1a-h1"],"title":"The property of the absolute value","text":"How to change $$|\\\\frac{-3}{2-x}|$$ using the given property?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{|-3|}{|2-x|}$$","$$\\\\frac{|-3|}{2-x}$$"]},{"id":"aff8a70precala1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["aff8a70precala1a-h2"],"title":"Simplify","text":"What is $$|-3|$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precala1a-h4","type":"hint","dependencies":["aff8a70precala1a-h3"],"title":"Simplify","text":"Solve the equation $$\\\\frac{3}{|2-x|}=6$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precala1a-h5","type":"hint","dependencies":["aff8a70precala1a-h4"],"title":"Simplify","text":"Note that $$\\\\frac{a}{b}=c$$ equals to $$b=\\\\frac{a}{c}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precala1a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$|2-x|=\\\\frac{3}{6}$$"],"dependencies":["aff8a70precala1a-h5"],"title":"Simplify","text":"What is the simplified equation using the given hint?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$|2-x|=\\\\frac{3}{6}$$","$$|2-x|=\\\\frac{6}{3}$$"]},{"id":"aff8a70precala1a-h7","type":"hint","dependencies":["aff8a70precala1a-h6"],"title":"Simplify","text":"Note that $$|2-x|=|x-2|$$, $$\\\\frac{3}{6}=\\\\frac{1}{2}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precala1a-h8","type":"hint","dependencies":["aff8a70precala1a-h7"],"title":"Simplify","text":"$$|x-2|=\\\\frac{1}{2}$$ can be simplified to $$x-2=\\\\pm \\\\left(\\\\frac{1}{2}\\\\right)$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precala1a-h9","type":"hint","dependencies":["aff8a70precala1a-h8"],"title":"Simplify","text":"Solve $$x$$ for $$x-2=\\\\pm \\\\left(\\\\frac{1}{2}\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precala1a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{5}{2}$$"],"dependencies":["aff8a70precala1a-h9"],"title":"Solve for $$x$$","text":"What is the $$x$$ value when $$x-2=\\\\frac{1}{2}$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precala1a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{3}{2}$$"],"dependencies":["aff8a70precala1a-h10"],"title":"Solve for $$x$$","text":"What is the $$x$$ value when $$x-2=-\\\\left(\\\\frac{1}{2}\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"aff8a70precala2","title":"Algebra with Absolute Values","body":"These questions test your knowledge of the core concepts.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Algebra with Absolute Values","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"aff8a70precala2a","stepAnswer":["$$[-5,4]$$"],"problemType":"MultipleChoice","stepTitle":"Find all values of $$x$$ such that $$|\\\\frac{2x+1}{3}| \\\\leq 3$$. Express your answers in interval notation.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$[-5,4]$$","$$[-5,-4]$$","[5,4]","$$[5,-4]$$"],"hints":{"DefaultPathway":[{"id":"aff8a70precala2a-h1","type":"hint","dependencies":[],"title":"The property of the absolute value","text":"Use the property $$|\\\\frac{a}{b}|=\\\\frac{|a|}{|b|}$$ to simplify $$|\\\\frac{2x+1}{3}|$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precala2a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{|2x+1|}{|3|}$$"],"dependencies":["aff8a70precala2a-h1"],"title":"The property of the absolute value","text":"How to change $$|\\\\frac{2x+1}{3}|$$ using the given property?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{|2x+1|}{|3|}$$","$$\\\\frac{2x+1}{|3|}$$"]},{"id":"aff8a70precala2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["aff8a70precala2a-h2"],"title":"Simplify","text":"What is $$|3|$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precala2a-h4","type":"hint","dependencies":["aff8a70precala2a-h3"],"title":"Simplify","text":"Now we have $$\\\\frac{|2x+1|}{3} \\\\leq 3$$. Think about how we can simplify in the next step.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precala2a-h5","type":"hint","dependencies":["aff8a70precala2a-h4"],"title":"Simplify","text":"Note that $$\\\\frac{a}{b}=c$$ equals to $$b=\\\\frac{a}{c}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precala2a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$|2x+1| \\\\leq 9$$"],"dependencies":["aff8a70precala2a-h5"],"title":"Simplify","text":"What is the simplified equation using the given hint?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$|2x+1| \\\\leq 9$$","$$|2x+1| \\\\geq 9$$","$$|2x+1| \\\\leq -9$$","$$|2x+1| \\\\geq -9$$"]},{"id":"aff8a70precala2a-h7","type":"hint","dependencies":["aff8a70precala2a-h6"],"title":"Simplify","text":"$$|2x+1| \\\\leq 9$$ is equivalent to $$2x+1 \\\\leq 9$$ or $$2x+1 \\\\geq -9$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precala2a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x \\\\leq 4$$"],"dependencies":["aff8a70precala2a-h7"],"title":"Solve for $$x$$","text":"What is the $$x$$ value for $$2x+1 \\\\leq 9$$","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$x \\\\leq 4$$","$$x \\\\geq 4$$","$$x \\\\leq -4$$","$$x \\\\geq -4$$"]},{"id":"aff8a70precala2a-h9","type":"hint","dependencies":["aff8a70precala2a-h8"],"title":"Explanation","text":"$$2x \\\\leq 8$$, so $$x \\\\leq 4$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precala2a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x \\\\geq -5$$"],"dependencies":["aff8a70precala2a-h9"],"title":"Solve for $$x$$","text":"What is the $$x$$ value for $$2x+1 \\\\geq -9$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$x \\\\geq -5$$","$$x \\\\leq -5$$","$$x \\\\geq 5$$","$$x \\\\leq 5$$"]},{"id":"aff8a70precala2a-h11","type":"hint","dependencies":["aff8a70precala2a-h10"],"title":"Explanation","text":"$$2x \\\\geq -10$$, so $$x \\\\geq -5$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precala2a-h12","type":"hint","dependencies":["aff8a70precala2a-h11"],"title":"In interval notation","text":"Write $$x \\\\leq 4$$ and $$x \\\\geq -5$$ in interval notation","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precala2a-h13","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$[-5,4]$$"],"dependencies":["aff8a70precala2a-h12"],"title":"In interval notation","text":"What is $$x \\\\leq 4$$ and $$x \\\\geq -5$$ in interval notation?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$[-5,4]$$","$$[-5,-4]$$","[4,5]","$$[-4,5]$$"]}]}}]},{"id":"aff8a70precala3","title":"Algebra with Absolute Values","body":"These questions test your knowledge of the core concepts.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Algebra with Absolute Values","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"aff8a70precala3a","stepAnswer":["$$15$$"],"problemType":"TextBox","stepTitle":"Find a number $$K>0$$ with the following property: If $$x$$ is in the interval $$(-2,1)$$, then it is guaranteed that $$|2x^2-3x+1|<K$$.","stepBody":"Hint: How large can $$|x|$$ be?","answerType":"arithmetic","variabilization":{},"answerLatex":"$$15$$","hints":{"DefaultPathway":[{"id":"aff8a70precala3a-h1","type":"hint","dependencies":[],"title":"Triangle Inequality","text":"Use triangle inequality: $$|a+b| \\\\leq |a|+|b|$$ to simplify $$|2x^2-3x+1|$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precala3a-h2","type":"hint","dependencies":["aff8a70precala3a-h1"],"title":"Triangle Inequality","text":"$$|2x^2-3x+1|$$ $$ \\\\leq |2x^2|+|-3x+1|$$ using triangle inequality","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precala3a-h3","type":"hint","dependencies":["aff8a70precala3a-h2"],"title":"Triangle Inequality","text":"$$|2x^2|+|-3x+1| \\\\leq |2x^2|+|-3x|+|1|$$ using triangle inequality","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precala3a-h4","type":"hint","dependencies":["aff8a70precala3a-h3"],"title":"Absolute value of a product","text":"Use $$|a b|=|a| |b|$$ to simplify $$|2x^2|+|-3x|+|1|$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precala3a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2{|x|}^2$$"],"dependencies":["aff8a70precala3a-h4"],"title":"Absolute value of a product","text":"What is the result of $$|2x^2|$$ using the given hint?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$2{|x|}^2$$","$$4{|x|}^2$$"]},{"id":"aff8a70precala3a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3|x|$$"],"dependencies":["aff8a70precala3a-h5"],"title":"Absolute value of a product","text":"What is the result of $$|-3x|$$ using the given hint?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$3|x|$$","$$\\\\left(-3\\\\right) |x|$$"]},{"id":"aff8a70precala3a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["aff8a70precala3a-h6"],"title":"Simplify","text":"What is $$|1|$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precala3a-h8","type":"hint","dependencies":["aff8a70precala3a-h7"],"title":"$$|x|$$","text":"Now we have $$|2x^2-3x+1| \\\\leq 2{|x|}^2+3|x|+1$$. Think about how large $$|x|$$ can be.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precala3a-h9","type":"hint","dependencies":["aff8a70precala3a-h8"],"title":"$$|x|$$","text":"Since $$x$$ is in $$(-2,1)$$, $$|x|<2$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precala3a-h10","type":"hint","dependencies":["aff8a70precala3a-h9"],"title":"Substitution","text":"Substitute $$x=2$$ into $$2{|x|}^2+3|x|+1$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precala3a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$15$$"],"dependencies":["aff8a70precala3a-h10"],"title":"Substitution","text":"What is $$2{|2|}^2+3|2|+1$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precala3a-h12","type":"hint","dependencies":["aff8a70precala3a-h11"],"title":"Substitution","text":"$$2\\\\times2^2+3\\\\times2+1=8+6+1=15$$. So $$K=15$$ guarantees this.","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"aff8a70precalb4","title":"Algebra with Absolute Values","body":"These problems are harder, often highlighting an important subtlety. Use the piecewise definition of the absolute value to rewrite the following expressions in piecewise notation, without absolute value bars. Some are done for you, to demonstrate.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Algebra with Absolute Values","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"aff8a70precalb4a","stepAnswer":["$$2x-2$$ if $$x \\\\geq 1$$ and $$\\\\left(-2\\\\right) x+2$$ if $$x<1$$"],"problemType":"MultipleChoice","stepTitle":"$$2|x-1|$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2x-2$$ if $$x \\\\geq 1$$ and $$\\\\left(-2\\\\right) x+2$$ if $$x<1$$","choices":["$$2x-2$$ if $$x \\\\geq 1$$ and $$\\\\left(-2\\\\right) x+2$$ if $$x<1$$","$$\\\\left(-2\\\\right) x+2$$ if $$x \\\\geq 1$$ and $$2x-2$$ if $$x<1$$"],"hints":{"DefaultPathway":[{"id":"aff8a70precalb4a-h1","type":"hint","dependencies":[],"title":"Demonstration","text":"We can put $$2$$ inside the absolute value, which is $$|2\\\\left(x-1\\\\right)|$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalb4a-h2","type":"hint","dependencies":["aff8a70precalb4a-h1"],"title":"Demonstration","text":"Two cases: $$2\\\\left(x-1\\\\right)$$ if $$x-1 \\\\geq 0$$ and $$\\\\left(-2\\\\right) \\\\left(x-1\\\\right)$$ if $$x-1<0$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalb4a-h3","type":"hint","dependencies":["aff8a70precalb4a-h2"],"title":"Demonstration","text":"Simplify: $$2x-2$$ if $$x \\\\geq 1$$ and $$\\\\left(-2\\\\right) x+2$$ if $$x<1$$","variabilization":{},"oer":"https://OATutor.io","license":""}]}},{"id":"aff8a70precalb4b","stepAnswer":["$$x^4$$ if $$x \\\\geq 0$$ and $$-\\\\left(x^4\\\\right)$$ if $$x<0$$"],"problemType":"MultipleChoice","stepTitle":"$$x^3 |x|$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x^4$$ if $$x \\\\geq 0$$ and $$-\\\\left(x^4\\\\right)$$ if $$x<0$$","choices":["$$x^4$$ if $$x \\\\geq 0$$ and $$-\\\\left(x^4\\\\right)$$ if $$x<0$$","$$-\\\\left(x^4\\\\right)$$ if $$x \\\\geq 0$$ and $$x^4$$ if $$x<0$$"],"hints":{"DefaultPathway":[{"id":"aff8a70precalb4b-h1","type":"hint","dependencies":[],"title":"Demonstration","text":"Two cases: $$x^3 x$$ if $$x \\\\geq 0$$ and $$x^3 \\\\left(-x\\\\right)$$ if $$x<0$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalb4b-h2","type":"hint","dependencies":["aff8a70precalb4b-h1"],"title":"Demonstration","text":"Simplify: $$x^4$$ if $$x \\\\geq 0$$ and $$-\\\\left(x^4\\\\right)$$ if $$x<0$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalb4b-h3","type":"hint","dependencies":["aff8a70precalb4b-h2"],"title":"Demonstration","text":"Note we couldn\'t easily move $$x^3$$ into the absolute value, as it is sometimes positive and sometimes negative.","variabilization":{},"oer":"https://OATutor.io","license":""}]}},{"id":"aff8a70precalb4c","stepAnswer":["$$12x-15$$ if $$x \\\\geq \\\\frac{5}{4}$$ and $$\\\\left(-12\\\\right) x+15$$ if $$x<\\\\frac{5}{4}$$"],"problemType":"MultipleChoice","stepTitle":"$$3|4x-5|$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$12x-15$$ if $$x \\\\geq \\\\frac{5}{4}$$ and $$\\\\left(-12\\\\right) x+15$$ if $$x<\\\\frac{5}{4}$$","choices":["$$12x-15$$ if $$x \\\\geq \\\\frac{5}{4}$$ and $$\\\\left(-12\\\\right) x+15$$ if $$x<\\\\frac{5}{4}$$","$$\\\\left(-12\\\\right) x+15$$ if $$x \\\\geq \\\\frac{5}{4}$$ and $$12x-15$$ if $$x<\\\\frac{5}{4}$$"],"hints":{"DefaultPathway":[{"id":"aff8a70precalb4c-h1","type":"hint","dependencies":[],"title":"Property","text":"Note we can use the property that $$|a| |b|=|a b|$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalb4c-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["aff8a70precalb4c-h1"],"title":"Property","text":"Can we change to $$|3\\\\left(4x-5\\\\right)|$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"aff8a70precalb4c-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$4x-5 \\\\geq 0$$"],"dependencies":["aff8a70precalb4c-h2"],"title":"Two cases","text":"Under what condition we can remove the absolute value and change to $$3\\\\left(4x-5\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$4x-5 \\\\geq 0$$","$$4x-5<0$$"]},{"id":"aff8a70precalb4c-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$4x-5<0$$"],"dependencies":["aff8a70precalb4c-h3"],"title":"Two cases","text":"Under what condition we can remove the absolute value and change to $$\\\\left(-3\\\\right) \\\\left(4x-5\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$4x-5 \\\\geq 0$$","$$4x-5<0$$"]},{"id":"aff8a70precalb4c-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$12x-15$$"],"dependencies":["aff8a70precalb4c-h4"],"title":"Simplify","text":"What is the simplified version of $$3\\\\left(4x-5\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$12x-15$$","$$\\\\left(-12\\\\right) x+15$$"]},{"id":"aff8a70precalb4c-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\left(-12\\\\right) x+15$$"],"dependencies":["aff8a70precalb4c-h5"],"title":"Simplify","text":"What is the simplified version of $$\\\\left(-3\\\\right) \\\\left(4x-5\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$12x-15$$","$$\\\\left(-12\\\\right) x+15$$"]},{"id":"aff8a70precalb4c-h7","type":"hint","dependencies":["aff8a70precalb4c-h6"],"title":"Simplify","text":"$$4x-5 \\\\geq 0$$, then $$4x \\\\geq 5$$, so $$x \\\\geq \\\\frac{5}{4}$$. $$4x-5<0$$, then $$4x<5$$, so $$x<\\\\frac{5}{4}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalb4c-h8","type":"hint","dependencies":["aff8a70precalb4c-h7"],"title":"Answer","text":"$$12x-15$$ if $$x \\\\geq \\\\frac{5}{4}$$ and $$\\\\left(-12\\\\right) x+15$$ if $$x<\\\\frac{5}{4}$$","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"aff8a70precalb41","title":"Algebra with Absolute Values","body":"These problems are harder, often highlighting an important subtlety. Use the piecewise definition of the absolute value to rewrite the following expressions in piecewise notation, without absolute value bars. Some are done for you, to demonstrate.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Algebra with Absolute Values","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"aff8a70precalb41a","stepAnswer":["$$x^6+x$$ if $$x \\\\geq 0$$ and $$-\\\\left(x^6\\\\right)-x$$ if $$x<0$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\left(x^5+1\\\\right) |x|$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$x^6+x$$ if $$x \\\\geq 0$$ and $$-\\\\left(x^6\\\\right)-x$$ if $$x<0$$","choices":["$$x^6+x$$ if $$x \\\\geq 0$$ and $$-\\\\left(x^6\\\\right)-x$$ if $$x<0$$","$$-\\\\left(x^6\\\\right)+x$$ if $$x \\\\geq 0$$ and $$x^6-x$$ if $$x<0$$"],"hints":{"DefaultPathway":[{"id":"aff8a70precalb41a-h1","type":"hint","dependencies":[],"title":"Two cases","text":"$$\\\\left(x^5+1\\\\right) x$$ if $$x \\\\geq 0$$ and $$\\\\left(x^5+1\\\\right) \\\\left(-x\\\\right)$$ if $$x<0$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalb41a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x^6+x$$"],"dependencies":["aff8a70precalb41a-h1"],"title":"Simplify","text":"What is the simplified version of $$\\\\left(x^5+1\\\\right) x$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$x^6+x$$","$$-\\\\left(x^6\\\\right)-x$$"]},{"id":"aff8a70precalb41a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-\\\\left(x^6\\\\right)-x$$"],"dependencies":["aff8a70precalb41a-h2"],"title":"Simplify","text":"What is the simplified version of $$\\\\left(x^5+1\\\\right) \\\\left(-x\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$x^6+x$$","$$-\\\\left(x^6\\\\right)-x$$"]}]}},{"id":"aff8a70precalb41b","stepAnswer":["$$1$$ if $$x>1$$ and $$-1$$ if $$x<1$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{|x-1|}{x-1}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$1$$ if $$x>1$$ and $$-1$$ if $$x<1$$","choices":["$$1$$ if $$x>1$$ and $$-1$$ if $$x<1$$","$$1$$ if $$x \\\\geq 1$$ and $$-1$$ if $$x<1$$","$$-1$$ if $$x>1$$ and $$1$$ if $$x<1$$"],"hints":{"DefaultPathway":[{"id":"aff8a70precalb41b-h1","type":"hint","dependencies":[],"title":"Can\'t divide by $$0$$","text":"Note that the denominator cannot be $$0$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalb41b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["aff8a70precalb41b-h1"],"title":"Can\'t divide by $$0$$","text":"What value $$x$$ cannot equal if $$x-1 \\\\neq 0$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalb41b-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{x-1}{x-1}$$"],"dependencies":["aff8a70precalb41b-h2"],"title":"Remove the absolute value","text":"If $$x-1>0$$, what should $$\\\\frac{|x-1|}{x-1}$$ be?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalb41b-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{-\\\\left(x-1\\\\right)}{x-1}$$"],"dependencies":["aff8a70precalb41b-h3"],"title":"Remove the absolute value","text":"If $$x-1<0$$, what should $$\\\\frac{|x-1|}{x-1}$$ be?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{-\\\\left(x-1\\\\right)}{x-1}$$","$$\\\\frac{-\\\\left(x-1\\\\right)}{-\\\\left(x-1\\\\right)}$$","$$\\\\frac{x-1}{x-1}$$"]},{"id":"aff8a70precalb41b-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["aff8a70precalb41b-h4"],"title":"Simplify","text":"What is $$\\\\frac{x-1}{x-1}$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalb41b-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-1$$"],"dependencies":["aff8a70precalb41b-h5"],"title":"Simplify","text":"What is $$\\\\frac{-\\\\left(x-1\\\\right)}{x-1}$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalb41b-h7","type":"hint","dependencies":["aff8a70precalb41b-h6"],"title":"Answer","text":"$$1$$ if $$x>1$$ and $$-1$$ if $$x<1$$","variabilization":{},"oer":"https://OATutor.io","license":""}]}},{"id":"aff8a70precalb41c","stepAnswer":["$$3$$ if $$x>-2$$ and $$-3$$ if $$x<-2$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{|3x+6|}{x+2}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$3$$ if $$x>-2$$ and $$-3$$ if $$x<-2$$","choices":["$$3$$ if $$x>-2$$ and $$-3$$ if $$x<-2$$","$$3$$ if $$x \\\\geq -2$$ and $$-3$$ if $$x<-2$$","$$-3$$ if $$x>-2$$ and $$3$$ if $$x<-2$$"],"hints":{"DefaultPathway":[{"id":"aff8a70precalb41c-h1","type":"hint","dependencies":[],"title":"Can\'t divide by $$0$$","text":"Note that the denominator cannot be $$0$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalb41c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["aff8a70precalb41c-h1"],"title":"Can\'t divide by $$0$$","text":"What value $$x$$ cannot equal if $$x+2 \\\\neq 0$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalb41c-h3","type":"hint","dependencies":["aff8a70precalb41c-h2"],"title":"Property","text":"Note we can use the property that $$|a b|=|a| |b|$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalb41c-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["aff8a70precalb41c-h3"],"title":"Property","text":"Can we simplify $$\\\\frac{|3x+6|}{x+2}$$ to $$\\\\frac{3|x+2|}{x+2}$$ ?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"aff8a70precalb41c-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{3\\\\left(x+2\\\\right)}{x+2}$$"],"dependencies":["aff8a70precalb41c-h4"],"title":"Remove the absolute value","text":"If $$x+2>0$$, what should $$\\\\frac{3|x+2|}{x+2}$$ be?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{3\\\\left(x+2\\\\right)}{x+2}$$","$$\\\\frac{\\\\left(-3\\\\right) \\\\left(x+2\\\\right)}{x+2}$$"]},{"id":"aff8a70precalb41c-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{\\\\left(-3\\\\right) \\\\left(x+2\\\\right)}{x+2}$$"],"dependencies":["aff8a70precalb41c-h5"],"title":"Remove the absolute value","text":"If $$x+2<0$$, what should $$\\\\frac{3|x+2|}{x+2}$$ be?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{3\\\\left(x+2\\\\right)}{x+2}$$","$$\\\\frac{\\\\left(-3\\\\right) \\\\left(x+2\\\\right)}{x+2}$$"]},{"id":"aff8a70precalb41c-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["aff8a70precalb41c-h6"],"title":"Simplify","text":"What is $$\\\\frac{3\\\\left(x+2\\\\right)}{x+2}$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalb41c-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["aff8a70precalb41c-h7"],"title":"Simplify","text":"What is $$\\\\frac{\\\\left(-3\\\\right) \\\\left(x+2\\\\right)}{x+2}$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalb41c-h9","type":"hint","dependencies":["aff8a70precalb41c-h8"],"title":"Answer","text":"$$3$$ if $$x>-2$$ and $$-3$$ if $$x<-2$$","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"aff8a70precalb5","title":"Algebra with Absolute Values","body":"These problems are harder, often highlighting an important subtlety","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Algebra with Absolute Values","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"aff8a70precalb5a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Find all values of $$x$$ such that $$\\\\left|\\\\frac{2**{x}+\\\\sqrt[3]{x}}{\\\\log_{5}(x**{2}+1)}\\\\right|=-2.$$ Is it possible to find the $$x$$ values?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"aff8a70precalb5a-h1","type":"hint","dependencies":[],"title":"Absolute value","text":"Note that for all $$y$$, $$|y| \\\\geq 0$$ always.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalb5a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$0>-2$$"],"dependencies":["aff8a70precalb5a-h1"],"title":"Absolute value","text":"Which one is correct? $$0>-2$$ or $$0<-2$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$0>-2$$","$$0<-2$$"]},{"id":"aff8a70precalb5a-h3","type":"hint","dependencies":["aff8a70precalb5a-h2"],"title":"Absolute value","text":"Since $$0>-2$$, it is impossible for $$|y|=-2$$ where $$y$$ can be any values.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalb5a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["aff8a70precalb5a-h3"],"title":"Absolute value","text":"Thus, does such an $$x$$ value exist in solving the equation?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]}]}}]},{"id":"aff8a70precalnew1","title":"Algebra with Absolute Values","body":"These questions test your knowledge of the core concepts.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Algebra with Absolute Values","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"aff8a70precalnew1a","stepAnswer":["$$4$$ or $$2$$"],"problemType":"MultipleChoice","stepTitle":"Find all values of $$x$$ such that $$|\\\\frac{-6}{3-x}|=6$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$4$$ or $$2$$","choices":["$$4$$ or $$2$$","$$4$$","$$2$$"],"hints":{"DefaultPathway":[{"id":"aff8a70precalnew1a-h1","type":"hint","dependencies":[],"title":"The property of the absolute value","text":"$$|\\\\frac{a}{b}|=\\\\frac{|a|}{|b|}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew1a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{|-6|}{|3-x|}$$"],"dependencies":["aff8a70precalnew1a-h1"],"title":"The property of the absolute value","text":"How to change $$|\\\\frac{-6}{3-x}|$$ using the given property?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{|-6|}{|3-x|}$$","$$\\\\frac{|-6|}{3-x}$$"]},{"id":"aff8a70precalnew1a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$6$$"],"dependencies":["aff8a70precalnew1a-h2"],"title":"Simplify","text":"What is $$|-6|$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew1a-h4","type":"hint","dependencies":["aff8a70precalnew1a-h3"],"title":"Simplify","text":"Solve the equation $$\\\\frac{6}{|3-x|}=6$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew1a-h5","type":"hint","dependencies":["aff8a70precalnew1a-h4"],"title":"Simplify","text":"Note that $$\\\\frac{a}{b}=c$$ equals to $$b=\\\\frac{a}{c}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew1a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$|3-x|=\\\\frac{6}{6}$$"],"dependencies":["aff8a70precalnew1a-h5"],"title":"Simplify","text":"What is the simplified equation using the given hint?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$|3-x|=\\\\frac{6}{6}$$","$$|3-x|=\\\\frac{-6}{6}$$"]},{"id":"aff8a70precalnew1a-h7","type":"hint","dependencies":["aff8a70precalnew1a-h6"],"title":"Simplify","text":"Note that $$|3-x|=|x-3|$$ due to the absolute\'s property. $$\\\\frac{6}{6}=1$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew1a-h8","type":"hint","dependencies":["aff8a70precalnew1a-h7"],"title":"Simplify","text":"$$|x-3|=1$$ can be simplified to $$x-3=\\\\pm 1$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew1a-h9","type":"hint","dependencies":["aff8a70precalnew1a-h8"],"title":"Simplify","text":"Solve $$x$$ for $$x-3=\\\\pm 1$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew1a-h10","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["aff8a70precalnew1a-h9"],"title":"Solve for $$x$$","text":"What is the $$x$$ value when $$x-3=1$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew1a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["aff8a70precalnew1a-h10"],"title":"Solve for $$x$$","text":"What is the $$x$$ value when $$x-3=-1$$?","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"aff8a70precalnew2","title":"Algebra with Absolute Values","body":"These questions test your knowledge of the core concepts.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Algebra with Absolute Values","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"aff8a70precalnew2a","stepAnswer":["$$[-2,1]$$"],"problemType":"MultipleChoice","stepTitle":"Find all values of $$x$$ such that $$|\\\\frac{4x+2}{3}| \\\\leq 2$$. Express your answers in interval notation.","stepBody":"","answerType":"string","variabilization":{},"choices":["$$[-2,-1]$$","$$[-2,1]$$","[1,2]","[2,1]"],"hints":{"DefaultPathway":[{"id":"aff8a70precalnew2a-h1","type":"hint","dependencies":[],"title":"The property of the absolute value","text":"Use the property $$|\\\\frac{a}{b}|=\\\\frac{|a|}{|b|}$$ to simplify $$|\\\\frac{4x+2}{3}|$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew2a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{|4x+2|}{|3|}$$"],"dependencies":["aff8a70precalnew2a-h1"],"title":"The property of the absolute value","text":"How to change $$|\\\\frac{4x+2}{3}|$$ using the given property?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{|4x+2|}{|3|}$$","$$\\\\frac{4x+2}{|3|}$$"]},{"id":"aff8a70precalnew2a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["aff8a70precalnew2a-h2"],"title":"Simplify","text":"What is $$|3|$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew2a-h4","type":"hint","dependencies":["aff8a70precalnew2a-h3"],"title":"Simplify","text":"Now we have $$\\\\frac{|4x+2|}{3} \\\\leq 2$$. Think about how we can simplify in the next step.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew2a-h5","type":"hint","dependencies":["aff8a70precalnew2a-h4"],"title":"Simplify","text":"Note that $$\\\\frac{a}{b}=c$$ equals to $$b=\\\\frac{a}{c}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew2a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$|4x+2| \\\\leq 6$$"],"dependencies":["aff8a70precalnew2a-h5"],"title":"Simplify","text":"What is the simplified equation using the given hint?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$|4x+2| \\\\leq 6$$","$$|4x+2| \\\\geq 6$$","$$|4x+2| \\\\leq -6$$","$$|4x+2| \\\\geq -6$$"]},{"id":"aff8a70precalnew2a-h7","type":"hint","dependencies":["aff8a70precalnew2a-h6"],"title":"Simplify","text":"$$|4x+2| \\\\leq 6$$ is equivalent to $$4x+2 \\\\leq 6$$ or $$4x+2 \\\\geq -6$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew2a-h8","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x \\\\leq 1$$"],"dependencies":["aff8a70precalnew2a-h7"],"title":"Solve for $$x$$","text":"What is the $$x$$ value for $$4x+2 \\\\leq 6$$","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$x \\\\leq 1$$","$$x \\\\geq 1$$","$$x \\\\leq -1$$","$$x \\\\geq -1$$"],"subHints":[{"id":"aff8a70precalnew2a-h8-s1","type":"hint","dependencies":[],"title":"Explanation","text":"$$4x \\\\leq 4$$, so $$x \\\\leq 1$$","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"aff8a70precalnew2a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x \\\\geq -2$$"],"dependencies":["aff8a70precalnew2a-h8"],"title":"Solve for $$x$$","text":"What is the $$x$$ value for $$4x+2 \\\\geq -6$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$x \\\\geq -2$$","$$x \\\\leq -2$$","$$x \\\\geq 2$$","$$x \\\\leq 2$$"],"subHints":[{"id":"aff8a70precalnew2a-h9-s1","type":"hint","dependencies":[],"title":"Explanation","text":"$$4x \\\\geq -8$$, so $$x \\\\geq -2$$","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"aff8a70precalnew2a-h10","type":"hint","dependencies":["aff8a70precalnew2a-h9"],"title":"In interval notation","text":"Write $$x \\\\leq 1$$ and $$x \\\\geq -2$$ in interval notation","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew2a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$[-2,1]$$"],"dependencies":["aff8a70precalnew2a-h10"],"title":"In interval notation","text":"What is $$x \\\\leq 1and$$ $$x \\\\geq -2$$ in interval notation?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$[-2,-1]$$","$$[-2,1]$$","[1,2]","[2,1]"]}]}}]},{"id":"aff8a70precalnew3","title":"Algebra with Absolute Values","body":"These questions test your knowledge of the core concepts.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Algebra with Absolute Values","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"aff8a70precalnew3a","stepAnswer":["$$28$$"],"problemType":"TextBox","stepTitle":"Find a number $$K>0$$ with the following property: If $$x$$ is in the interval $$(-2,3)$$, then it is guaranteed that $$|-2x^2-3x+1|<K$$.","stepBody":"Hint: How large can $$|x|$$ be?","answerType":"arithmetic","variabilization":{},"answerLatex":"$$28$$","hints":{"DefaultPathway":[{"id":"aff8a70precalnew3a-h1","type":"hint","dependencies":[],"title":"Triangle Inequality","text":"Use triangle inequality: $$|a+b| \\\\leq |a|+|b|$$ to simplify $$|-2x^2-3x+1|$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew3a-h2","type":"hint","dependencies":["aff8a70precalnew3a-h1"],"title":"Triangle Inequality","text":"$$|-2x^2-3x+1|$$ $$ \\\\leq |-2x^2|+|-3x+1|$$ using triangle inequality","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew3a-h3","type":"hint","dependencies":["aff8a70precalnew3a-h2"],"title":"Triangle Inequality","text":"$$|-2x^2|+|-3x+1| \\\\leq |-2x^2|+|-3x|+|1|$$ using triangle inequality","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew3a-h4","type":"hint","dependencies":["aff8a70precalnew3a-h3"],"title":"Absolute value of a product","text":"Use $$|a b|=|a| |b|$$ to simplify $$|-2x^2|+|-3x|+|1|$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew3a-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2{|x|}^2$$"],"dependencies":["aff8a70precalnew3a-h4"],"title":"Absolute value of a product","text":"What is the result of $$|-2x^2|$$ using the given hint?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$2{|x|}^2$$","$$4{|x|}^2$$"]},{"id":"aff8a70precalnew3a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3|x|$$"],"dependencies":["aff8a70precalnew3a-h5"],"title":"Absolute value of a product","text":"What is the result of $$|-3x|$$ using the given hint?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$3|x|$$","$$\\\\left(-3\\\\right) |x|$$"]},{"id":"aff8a70precalnew3a-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["aff8a70precalnew3a-h6"],"title":"Simplify","text":"What is $$|1|$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew3a-h8","type":"hint","dependencies":["aff8a70precalnew3a-h7"],"title":"$$|x|$$","text":"Now we have $$|-2x^2-3x+1| \\\\leq 2{|x|}^2+3|x|+1$$. Think about how large $$|x|$$ can be.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew3a-h9","type":"hint","dependencies":["aff8a70precalnew3a-h8"],"title":"$$|x|$$","text":"Since $$x$$ is in $$(-2,3)$$, $$|x|<3$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew3a-h10","type":"hint","dependencies":["aff8a70precalnew3a-h9"],"title":"Substitution","text":"Substitute $$x=3$$ into $$2{|x|}^2+3|x|+1$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew3a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$28$$"],"dependencies":["aff8a70precalnew3a-h10"],"title":"Substitution","text":"What is $$2{|3|}^2+3|3|+1$$","variabilization":{},"oer":"https://OATutor.io","license":"","subHints":[{"id":"aff8a70precalnew3a-h11-s1","type":"hint","dependencies":[],"title":"Substitution","text":"$$2\\\\times3^2+3\\\\times3+1=18+9+1=28$$. So $$K=28$$ guarantees this.","variabilization":{},"oer":"https://OATutor.io","license":""}]}]}}]},{"id":"aff8a70precalnew4","title":"Algebra with Absolute Values","body":"These problems are harder, often highlighting an important subtlety. Use the piecewise definition of the absolute value to rewrite the following expressions in piecewise notation, without absolute value bars. Some are done for you, to demonstrate.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Algebra with Absolute Values","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"aff8a70precalnew4a","stepAnswer":["$$3x-6$$ if $$x \\\\geq 2$$ and $$\\\\left(-3\\\\right) x+6$$ if $$x<2$$"],"problemType":"MultipleChoice","stepTitle":"$$3|x-2|$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$3x-6$$ if $$x \\\\geq 2$$ and $$\\\\left(-3\\\\right) x+6$$ if $$x<2$$","choices":["$$3x-6$$ if $$x \\\\geq 2$$ and $$\\\\left(-3\\\\right) x+6$$ if $$x<2$$","$$\\\\left(-3\\\\right) x+6$$ if $$x \\\\geq 2$$ and $$3x-6$$ if $$x<2$$"],"hints":{"DefaultPathway":[{"id":"aff8a70precalnew4a-h1","type":"hint","dependencies":[],"title":"Demonstration","text":"We can put $$3$$ inside the absolute value, which is $$|3\\\\left(x-2\\\\right)|$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew4a-h2","type":"hint","dependencies":["aff8a70precalnew4a-h1"],"title":"Demonstration","text":"Two cases: $$3\\\\left(x-2\\\\right)$$ if $$x-2 \\\\geq 0$$ and $$\\\\left(-3\\\\right) \\\\left(x-2\\\\right)$$ if $$x-2<0$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew4a-h3","type":"hint","dependencies":["aff8a70precalnew4a-h2"],"title":"Demonstration","text":"Simplify: $$3x-6$$ if $$x \\\\geq 2$$ and $$\\\\left(-3\\\\right) x+6$$ if $$x<2$$","variabilization":{},"oer":"https://OATutor.io","license":""}]}},{"id":"aff8a70precalnew4b","stepAnswer":["$$2x^4$$ if $$x \\\\geq 0$$ and $$-2x^4$$ if $$x<0$$"],"problemType":"MultipleChoice","stepTitle":"$$x^3 |2x|$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2x^4$$ if $$x \\\\geq 0$$ and $$-2x^4$$ if $$x<0$$","choices":["$$2x^4$$ if $$x \\\\geq 0$$ and $$-2x^4$$ if $$x<0$$","$$-2x^4$$ if $$x \\\\geq 0$$ and $$2x^4$$ if $$x<0$$"],"hints":{"DefaultPathway":[{"id":"aff8a70precalnew4b-h1","type":"hint","dependencies":[],"title":"Demonstration","text":"Two cases: $$2x^3 x$$ if $$x \\\\geq 0$$ and $$x^3 \\\\left(-2x\\\\right)$$ if $$x<0$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew4b-h2","type":"hint","dependencies":["aff8a70precalnew4b-h1"],"title":"Demonstration","text":"Simplify: $$2x^4$$ if $$x \\\\geq 0$$ and $$-2x^4$$ if $$x<0$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew4b-h3","type":"hint","dependencies":["aff8a70precalnew4b-h2"],"title":"Demonstration","text":"Note we couldn\'t easily move $$x^3$$ into the absolute value, as it is sometimes positive and sometimes negative.","variabilization":{},"oer":"https://OATutor.io","license":""}]}},{"id":"aff8a70precalnew4c","stepAnswer":["$$12x-24$$ if $$x \\\\geq 2$$ and $$\\\\left(-12\\\\right) x+24$$ if $$x<2$$"],"problemType":"MultipleChoice","stepTitle":"$$4|3x-6|$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$12x-24$$ if $$x \\\\geq 2$$ and $$\\\\left(-12\\\\right) x+24$$ if $$x<2$$","choices":["$$12x-24$$ if $$x \\\\geq 2$$ and $$\\\\left(-12\\\\right) x+24$$ if $$x<2$$","$$-12x+24$$ if $$x \\\\geq 2$$ and $$12x-24$$ if $$x<2$$"],"hints":{"DefaultPathway":[{"id":"aff8a70precalnew4c-h1","type":"hint","dependencies":[],"title":"Property","text":"Note we can use the property that $$|a| |b|=|a b|$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew4c-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["aff8a70precalnew4c-h1"],"title":"Property","text":"Can we change to $$|4\\\\left(3x-6\\\\right)|$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"aff8a70precalnew4c-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3x-6 \\\\geq 0$$"],"dependencies":["aff8a70precalnew4c-h2"],"title":"Two cases","text":"Under what condition we can remove the absolute value and change to $$4\\\\left(3x-6\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$3x-6 \\\\geq 0$$","$$3x-6<0$$"]},{"id":"aff8a70precalnew4c-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3x-6<0$$"],"dependencies":["aff8a70precalnew4c-h3"],"title":"Two cases","text":"Under what condition we can remove the absolute value and change to $$\\\\left(-3\\\\right) \\\\left(4x-5\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$3x-6 \\\\geq 0$$","$$3x-6<0$$"]},{"id":"aff8a70precalnew4c-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$12x-24$$"],"dependencies":["aff8a70precalnew4c-h4"],"title":"Simplify","text":"What is the simplified version of $$4\\\\left(3x-6\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$12x-24$$","$$\\\\left(-12\\\\right) x+24$$"]},{"id":"aff8a70precalnew4c-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\left(-12\\\\right) x+24$$"],"dependencies":["aff8a70precalnew4c-h5"],"title":"Simplify","text":"What is the simplified version of $$\\\\left(-4\\\\right) \\\\left(3x-6\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$12x-24$$","$$\\\\left(-12\\\\right) x+24$$"]},{"id":"aff8a70precalnew4c-h7","type":"hint","dependencies":["aff8a70precalnew4c-h6"],"title":"Simplify","text":"$$3x-6 \\\\geq 0$$, then $$3x \\\\geq 6$$, so $$x \\\\geq 2$$, then $$3x<6$$, so $$x<2$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew4c-h8","type":"hint","dependencies":["aff8a70precalnew4c-h7"],"title":"Answer","text":"$$12x-24$$ if $$x \\\\geq 2$$ and $$\\\\left(-12\\\\right) x+24$$ if $$x<2$$","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"aff8a70precalnew41","title":"Algebra with Absolute Values","body":"These problems are harder, often highlighting an important subtlety. Use the piecewise definition of the absolute value to rewrite the following expressions in piecewise notation, without absolute value bars. Some are done for you, to demonstrate.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Algebra with Absolute Values","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"aff8a70precalnew41a","stepAnswer":["$$2x^6+2x$$ if $$x \\\\geq 0$$ and $$-2x^6-2x$$ if $$x<0$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\left(2x^5+2\\\\right) |x|$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2x^6+2x$$ if $$x \\\\geq 0$$ and $$-2x^6-2x$$ if $$x<0$$","choices":["$$2x^6+2x$$ if $$x \\\\geq 0$$ and $$-2x^6-2x$$ if $$x<0$$","$$-2x^6+2x$$ if $$x \\\\geq 0$$ and $$2x^6-2x$$ if $$x<0$$"],"hints":{"DefaultPathway":[{"id":"aff8a70precalnew41a-h1","type":"hint","dependencies":[],"title":"Two cases","text":"$$\\\\left(2x^5+2\\\\right) x$$ if $$x \\\\geq 0$$ and $$\\\\left(2x^5+2\\\\right) \\\\left(-x\\\\right)$$ if $$x<0$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew41a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$2x^6+2x$$"],"dependencies":["aff8a70precalnew41a-h1"],"title":"Simplify","text":"What is the simplified version of $$\\\\left(2x^5+2\\\\right) x$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$2x^6+2x$$","$$-2x^6-2x$$"]},{"id":"aff8a70precalnew41a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-2x^6-2x$$"],"dependencies":["aff8a70precalnew41a-h2"],"title":"Simplify","text":"What is the simplified version of $$\\\\left(2x^5+2\\\\right) \\\\left(-x\\\\right)$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$2x^6+2x$$","$$-2x^6-2x$$"]}]}},{"id":"aff8a70precalnew41b","stepAnswer":["$$2$$ if $$x>1$$ and $$-2$$ if $$x<1$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{|2x-2|}{x-1}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$2$$ if $$x>1$$ and $$-2$$ if $$x<1$$","choices":["$$2$$ if $$x>1$$ and $$-2$$ if $$x<1$$","$$2$$ if $$x \\\\geq 1$$ and $$-2$$ if $$x<1$$","$$-2$$ if $$x>1$$ and $$2$$ if $$x<1$$"],"hints":{"DefaultPathway":[{"id":"aff8a70precalnew41b-h1","type":"hint","dependencies":[],"title":"Can\'t divide by $$0$$","text":"Note that the denominator cannot be $$0$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew41b-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$1$$"],"dependencies":["aff8a70precalnew41b-h1"],"title":"Can\'t divide by $$0$$","text":"What value $$x$$ cannot equal if $$x-1 \\\\neq 0$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew41b-h3","type":"hint","dependencies":["aff8a70precalnew41b-h2"],"title":"Remove the absolute value","text":"We will discuss cases when $$2x-2>0$$ or $$2x-2<0$$, which is simplified to $$x>1$$ or $$x<1$$. The reason why we do not have $$x=1$$ is because $$x$$ cannot be $$1$$ as the denominator cannot be $$0$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew41b-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{2x-2}{x-1}$$"],"dependencies":["aff8a70precalnew41b-h3"],"title":"Remove the absolute value","text":"If $$x>1$$, what should $$\\\\frac{|2x-2|}{x-1}$$ be?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew41b-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{-\\\\left(2x-2\\\\right)}{x-1}$$"],"dependencies":["aff8a70precalnew41b-h4"],"title":"Remove the absolute value","text":"If $$x<1$$, what should $$\\\\frac{|x-1|}{x-1}$$ be?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{-\\\\left(2x-2\\\\right)}{x-1}$$","$$\\\\frac{-\\\\left(2x-2\\\\right)}{-\\\\left(x-1\\\\right)}$$"]},{"id":"aff8a70precalnew41b-h6","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$2$$"],"dependencies":["aff8a70precalnew41b-h5"],"title":"Simplify","text":"What is $$\\\\frac{2x-2}{x-1}$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew41b-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["aff8a70precalnew41b-h6"],"title":"Simplify","text":"What is $$\\\\frac{-\\\\left(2x-2\\\\right)}{x-1}$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew41b-h8","type":"hint","dependencies":["aff8a70precalnew41b-h7"],"title":"Answer","text":"$$2$$ if x>1and $$-2$$ if $$x<1$$","variabilization":{},"oer":"https://OATutor.io","license":""}]}},{"id":"aff8a70precalnew41c","stepAnswer":["$$3$$ if $$x>-3$$ and $$-3$$ if $$x<-3$$"],"problemType":"MultipleChoice","stepTitle":"$$\\\\frac{|3x+9|}{x+3}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$3$$ if $$x>-3$$ and $$-3$$ if $$x<-3$$","choices":["$$3$$ if $$x>-3$$ and $$-3$$ if $$x<-3$$","$$3$$ if $$x \\\\geq -3$$ and $$-3$$ if $$x<-3$$","$$-3$$ if $$x>-3$$ and $$3$$ if $$x<-3$$"],"hints":{"DefaultPathway":[{"id":"aff8a70precalnew41c-h1","type":"hint","dependencies":[],"title":"Can\'t divide by $$0$$","text":"Note that the denominator cannot be $$0$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew41c-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["aff8a70precalnew41c-h1"],"title":"Can\'t divide by $$0$$","text":"What value $$x$$ cannot equal if $$x+3 \\\\neq 0$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew41c-h3","type":"hint","dependencies":["aff8a70precalnew41c-h2"],"title":"Property","text":"Note we can use the property that $$|a b|=|a| |b|$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew41c-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["aff8a70precalnew41c-h3"],"title":"Property","text":"Can we simplify $$\\\\frac{|3x+9|}{x+3}$$ to $$\\\\frac{3|x+3|}{x+3}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"aff8a70precalnew41c-h5","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{3\\\\left(x+3\\\\right)}{x+3}$$"],"dependencies":["aff8a70precalnew41c-h4"],"title":"Remove the absolute value","text":"If $$x+3>0$$, what should $$\\\\frac{3|x+3|}{x+3}$$ be?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{3\\\\left(x+3\\\\right)}{x+3}$$","$$\\\\frac{\\\\left(-3\\\\right) \\\\left(x+3\\\\right)}{x+3}$$"]},{"id":"aff8a70precalnew41c-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{\\\\left(-3\\\\right) \\\\left(x+3\\\\right)}{x+3}$$"],"dependencies":["aff8a70precalnew41c-h5"],"title":"Remove the absolute value","text":"If $$x+3<0$$, what should $$\\\\frac{3|x+3|}{x+3}$$ be?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{3\\\\left(x+3\\\\right)}{x+3}$$","$$\\\\frac{\\\\left(-3\\\\right) \\\\left(x+3\\\\right)}{x+3}$$"]},{"id":"aff8a70precalnew41c-h7","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$3$$"],"dependencies":["aff8a70precalnew41c-h6"],"title":"Simplify","text":"What is $$\\\\frac{3\\\\left(x+3\\\\right)}{x+3}$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew41c-h8","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-3$$"],"dependencies":["aff8a70precalnew41c-h7"],"title":"Simplify","text":"What is $$\\\\frac{\\\\left(-3\\\\right) \\\\left(x+3\\\\right)}{x+3}$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew41c-h9","type":"hint","dependencies":["aff8a70precalnew41c-h8"],"title":"Answer","text":"$$3$$ if $$x>-3$$ and $$-3$$ if $$x<-3$$","variabilization":{},"oer":"https://OATutor.io","license":""}]}}]},{"id":"aff8a70precalnew5","title":"Algebra with Absolute Values","body":"These problems are harder, often highlighting an important subtlety","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Algebra with Absolute Values","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"aff8a70precalnew5a","stepAnswer":["No"],"problemType":"MultipleChoice","stepTitle":"Find all values of $$x$$ such that $$\\\\left|\\\\frac{x**{10}+\\\\sqrt[3]{x}+100}{\\\\log_{6}(x**2+20)}\\\\right|=-1.$$ Is it possible to find the $$x$$ values?","stepBody":"","answerType":"string","variabilization":{},"choices":["Yes","No"],"hints":{"DefaultPathway":[{"id":"aff8a70precalnew5a-h1","type":"hint","dependencies":[],"title":"Absolute value","text":"Note that for all $$y$$, $$|y| \\\\geq 0$$ always.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew5a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$0>-1$$"],"dependencies":["aff8a70precalnew5a-h1"],"title":"Absolute value","text":"Which one is correct? $$0>-1$$ or $$0<-1$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$0>-1$$","$$0<-1$$"]},{"id":"aff8a70precalnew5a-h3","type":"hint","dependencies":["aff8a70precalnew5a-h2"],"title":"Absolute value","text":"Since $$0>-1$$, it is impossible for $$|y|=-1$$ where $$y$$ can be any values.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew5a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["aff8a70precalnew5a-h3"],"title":"Absolute value","text":"Thus, does such an $$x$$ value exist in solving the equation?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]}]}}]},{"id":"aff8a70precalnew6","title":"Algebra with Absolute Values","body":"These problems are harder, often highlighting an important subtlety","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Algebra with Absolute Values","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"aff8a70precalnew6a","stepAnswer":["$$(-\\\\infty,\\\\frac{-10}{3})$$ $$\\\\cup$$ $$(\\\\frac{-2}{3},\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"Find all values of $$x$$ such that $$|\\\\frac{4}{x+2}|<3$$. Express your answer in interval notation.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,\\\\frac{-10}{3})$$ $$\\\\cup$$ $$(\\\\frac{-2}{3},\\\\infty)$$","choices":["$$(-\\\\infty,\\\\frac{-10}{3})$$ $$\\\\cup$$ $$(\\\\frac{-2}{3},\\\\infty)$$","$$(-\\\\infty,\\\\frac{-10}{3})$$ $$\\\\cup$$ $$(\\\\frac{2}{3},\\\\infty)$$"],"hints":{"DefaultPathway":[{"id":"aff8a70precalnew6a-h1","type":"hint","dependencies":[],"title":"Can\'t divide by $$0$$","text":"Note that the denominator cannot be $$0$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew6a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$-2$$"],"dependencies":["aff8a70precalnew6a-h1"],"title":"Can\'t divide by $$0$$","text":"What value $$x$$ cannot equal if $$x+2 \\\\neq 0$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew6a-h3","type":"hint","dependencies":["aff8a70precalnew6a-h2"],"title":"Property","text":"Note that $$|\\\\frac{a}{b}|=\\\\frac{|a|}{|b|}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew6a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["aff8a70precalnew6a-h3"],"title":"Property","text":"Is $$|\\\\frac{4}{x+2}|$$ equaled to $$\\\\frac{|4|}{|x+2|}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"aff8a70precalnew6a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$4$$"],"dependencies":["aff8a70precalnew6a-h4"],"title":"Absolute value","text":"What is the value of $$|4|$$?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew6a-h6","type":"hint","dependencies":["aff8a70precalnew6a-h5"],"title":"Absolute value","text":"For $$x \\\\neq -2$$, we have $$\\\\frac{4}{|x+2|}<3$$, which means $$3|x+2|>4$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew6a-h7","type":"hint","dependencies":["aff8a70precalnew6a-h6"],"title":"Absolute value","text":"Here, we use the fact that for $$b>0$$, $$a<c$$, we can imply $$b a<b c$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew6a-h8","type":"hint","dependencies":["aff8a70precalnew6a-h7"],"title":"Simplification","text":"$$3|x+2|>4$$, so we have $$|x+2|>\\\\frac{4}{3}$$ if we divide both side by $$3$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew6a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x+2>\\\\frac{4}{3}$$"],"dependencies":["aff8a70precalnew6a-h8"],"title":"Remove the absolute value","text":"If $$x+2 \\\\geq 0$$, what is $$|x+2|>\\\\frac{4}{3}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$x+2>\\\\frac{4}{3}$$","$$x+2>\\\\frac{-4}{3}$$","$$x+2<\\\\frac{-4}{3}$$","$$x+2<\\\\frac{4}{3}$$"]},{"id":"aff8a70precalnew6a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x+2<\\\\frac{-4}{3}$$"],"dependencies":["aff8a70precalnew6a-h9"],"title":"Remove the absolute value","text":"If $$x+2<0$$, what is $$|x+2|>\\\\frac{4}{3}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$x+2>\\\\frac{4}{3}$$","$$x+2>\\\\frac{-4}{3}$$","$$x+2<\\\\frac{-4}{3}$$","$$x+2<\\\\frac{4}{3}$$"]},{"id":"aff8a70precalnew6a-h11","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-2}{3}$$"],"dependencies":["aff8a70precalnew6a-h10"],"title":"Simplification","text":"$$x+2>\\\\frac{4}{3}$$, x>?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew6a-h12","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{-10}{3}$$"],"dependencies":["aff8a70precalnew6a-h11"],"title":"Simplification","text":"$$x+2<\\\\frac{-4}{3}$$, x<?","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew6a-h13","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(-\\\\infty,\\\\frac{-10}{3})$$ $$\\\\cup$$ $$(\\\\frac{-2}{3},\\\\infty)$$"],"dependencies":["aff8a70precalnew6a-h12"],"title":"Interval Notation","text":"Write $$x>\\\\frac{-2}{3}$$ or $$x<\\\\frac{-10}{3}$$ in interval notation. What is the answer?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$(-\\\\infty,\\\\frac{-10}{3})$$ $$\\\\cup$$ $$(\\\\frac{-2}{3},\\\\infty)$$","$$(-\\\\infty,\\\\frac{-10}{3})$$ $$\\\\cup$$ $$(\\\\frac{2}{3},\\\\infty)$$"],"subHints":[{"id":"aff8a70precalnew6a-h13-s1","type":"hint","dependencies":[],"title":"Interval Notation","text":"Remember that $$x \\\\neq -2$$","variabilization":{},"oer":"https://OATutor.io","license":""}]}]}}]},{"id":"aff8a70precalnew7","title":"Algebra with Absolute Values","body":"These questions are challenging, requiring mastery of each concept and their interrelations.","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Algebra with Absolute Values","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"aff8a70precalnew7a","stepAnswer":["$$(-1,1)$$"],"problemType":"MultipleChoice","stepTitle":"Find all values of $$x$$ such that $$|\\\\frac{4}{x^2+1}|>2$$. Express your answer in interval notation.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-1,1)$$","choices":["$$(-1,1)$$","$$\\\\pm 1$$","$$(\\\\frac{-1}{2},\\\\frac{1}{2})$$","$$\\\\frac{\\\\pm 1}{2}$$"],"hints":{"DefaultPathway":[{"id":"aff8a70precalnew7a-h1","type":"hint","dependencies":[],"title":"Remove the absolute value","text":"Note $$x^2 \\\\geq 0$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew7a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["aff8a70precalnew7a-h1"],"title":"Remove the absolute value","text":"Is $$x^2+1 \\\\geq 0$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"aff8a70precalnew7a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["aff8a70precalnew7a-h2"],"title":"Remove the absolute value","text":"$$|x^2+1|$$ $$=x^2+1$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"aff8a70precalnew7a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{4}{x^2+1}$$"],"dependencies":["aff8a70precalnew7a-h3"],"title":"Remove the absolute value","text":"What does $$|\\\\frac{4}{x^2+1}|$$ equal to?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\frac{4}{x^2+1}$$","$$\\\\frac{4}{\\\\left(-x^2-1\\\\right)}$$"],"subHints":[{"id":"aff8a70precalnew7a-h4-s1","type":"hint","dependencies":[],"title":"Remove the absolute value","text":"Use the property $$|\\\\frac{a}{b}|=\\\\frac{|a|}{|b|}$$ to simplify.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"aff8a70precalnew7a-h5","type":"hint","dependencies":["aff8a70precalnew7a-h4"],"title":"Simplification","text":"For $$b>0$$, $$a>c$$, then $$b a>b c$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew7a-h6","type":"hint","dependencies":["aff8a70precalnew7a-h5"],"title":"Simplification","text":"$$\\\\frac{4}{x^2+1}>2$$ is equivalent to $$4>2\\\\left(x^2+1\\\\right)$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew7a-h7","type":"hint","dependencies":["aff8a70precalnew7a-h6"],"title":"Simplification","text":"$$4>2\\\\left(x^2+1\\\\right)$$ is equivalent to $$x^2+1<2$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew7a-h8","type":"hint","dependencies":["aff8a70precalnew7a-h7"],"title":"Simplification","text":"$$x^2+1<2$$ is equivalent to $$x^2<1$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew7a-h9","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["aff8a70precalnew7a-h8"],"title":"Simplification","text":"Is $$x^2<1$$ equivalent to $${|x|}^2<1$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"],"subHints":[{"id":"aff8a70precalnew7a-h9-s1","type":"hint","dependencies":[],"title":"Simplification","text":"$$x^2={\\\\left(-x\\\\right)}^2$$, so $$x^2={|x|}^2$$","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"aff8a70precalnew7a-h10","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["aff8a70precalnew7a-h9"],"title":"Simplification","text":"Is $${|x|}^2<1$$ equivalent to $$|x|<1$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"],"subHints":[{"id":"aff8a70precalnew7a-h10-s1","type":"hint","dependencies":[],"title":"Simplification","text":"For a,b>0, $$a<b$$ is equivalent to $$\\\\sqrt{a}<\\\\sqrt{b}$$.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"aff8a70precalnew7a-h11","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x<1$$"],"dependencies":["aff8a70precalnew7a-h10"],"title":"Simplification","text":"For $$|x|<1$$, if $$x \\\\geq 0$$, which one is correct?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$x<1$$","$$x>1$$","$$x<-1$$","$$x>-1$$"]},{"id":"aff8a70precalnew7a-h12","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$x>-1$$"],"dependencies":["aff8a70precalnew7a-h11"],"title":"Simplification","text":"For $$|x|<1$$, if $$x<0$$, which one is correct?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$x<1$$","$$x>1$$","$$x<-1$$","$$x>-1$$"]},{"id":"aff8a70precalnew7a-h13","type":"hint","dependencies":["aff8a70precalnew7a-h12"],"title":"Interval Notation","text":"Write $$x<1$$ if $$x \\\\geq 0$$ and $$x>-1$$ if $$x<0$$ in interval notation.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew7a-h14","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(-1,1)$$"],"dependencies":["aff8a70precalnew7a-h13"],"title":"Interval Notation","text":"What is the interval notation of $$-1<x<1$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$(-1,1)$$","$$\\\\pm 1$$","$$(\\\\frac{-1}{2},\\\\frac{1}{2})$$","$$\\\\frac{\\\\pm 1}{2}$$"]}]}}]},{"id":"aff8a70precalnew8","title":"Algebra with Absolute Values","body":"These problems are harder, often highlighting an important subtlety","variabilization":{},"oer":"https://OATutor.io","license":0,"lesson":"Algebra with Absolute Values","courseName":"Pre-Calculus Essentials (UC Berkeley Math 1B)","steps":[{"id":"aff8a70precalnew8a","stepAnswer":["$$(-\\\\infty,-\\\\sqrt{6}-2)$$ $$\\\\cup$$ $$(\\\\sqrt{6}-2,\\\\infty)$$"],"problemType":"MultipleChoice","stepTitle":"Find all values of $$x$$ such that $$|\\\\frac{{\\\\left(x+2\\\\right)}^3}{2}|>|3x+6|$$. Express your answer in interval notation.","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$(-\\\\infty,-\\\\sqrt{6}-2)$$ $$\\\\cup$$ $$(\\\\sqrt{6}-2,\\\\infty)$$","choices":["$$(-\\\\infty,-\\\\sqrt{6}-2)$$ $$\\\\cup$$ $$(\\\\sqrt{6}-2,\\\\infty)$$","$$(-\\\\sqrt{6}-2,\\\\sqrt{6}-2)$$"],"hints":{"DefaultPathway":[{"id":"aff8a70precalnew8a-h1","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":[],"title":"LHS simplification","text":"Is $$|\\\\frac{{\\\\left(x+2\\\\right)}^3}{2}|$$ equivalent to $$\\\\frac{|{\\\\left(x+2\\\\right)}^3|}{|2|}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"],"subHints":[{"id":"aff8a70precalnew8a-h1-s1","type":"hint","dependencies":[],"title":"LHS simplification","text":"Use the property $$|\\\\frac{a}{b}|=\\\\frac{|a|}{|b|}$$ to simplify.","variabilization":{},"oer":"https://OATutor.io","license":""}]},{"id":"aff8a70precalnew8a-h2","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["Yes"],"dependencies":["aff8a70precalnew8a-h1"],"title":"LHS simplification","text":"Is $$\\\\frac{|{\\\\left(x+2\\\\right)}^3|}{|2|}$$ equivalent to $$\\\\frac{{|x+2|}^3}{2}$$?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"aff8a70precalnew8a-h3","type":"hint","dependencies":["aff8a70precalnew8a-h2"],"title":"LHS simplification","text":"Use the property $$|a b|=|a| |b|$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew8a-h4","type":"hint","dependencies":["aff8a70precalnew8a-h3"],"title":"RHS simplification","text":"$$|3x+6|=3|x+2|$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew8a-h5","type":"hint","dependencies":["aff8a70precalnew8a-h4"],"title":"RHS simplification","text":"Use the property $$|a b|=|a| |b|$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew8a-h6","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["No"],"dependencies":["aff8a70precalnew8a-h5"],"title":"Special Case","text":"Is there any value $$x$$ cannot be?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["Yes","No"]},{"id":"aff8a70precalnew8a-h7","type":"hint","dependencies":["aff8a70precalnew8a-h6"],"title":"Simplification","text":"We now have $$\\\\frac{{|x+2|}^3}{2}>3|x+2|$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew8a-h8","type":"hint","dependencies":["aff8a70precalnew8a-h7"],"title":"Simplification","text":"Multiply by $$2$$ on both sides, so we get $${|x+2|}^3>6|x+2|$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew8a-h9","type":"hint","dependencies":["aff8a70precalnew8a-h8"],"title":"Simplification","text":"$${|x+2|}^3>6|x+2|$$ is equivalent to $${|x+2|}^2>6$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew8a-h10","type":"hint","dependencies":["aff8a70precalnew8a-h9"],"title":"Simplification","text":"For $$b>0$$, $$a b>c b$$ is equivalent to $$a>c$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew8a-h11","type":"hint","dependencies":["aff8a70precalnew8a-h10"],"title":"Simplification","text":"$${|x+2|}^2>6$$ is equivalent to $$|x+2|>\\\\sqrt{6}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew8a-h12","type":"hint","dependencies":["aff8a70precalnew8a-h11"],"title":"Simplification","text":"For a,b>0, $$a>b$$ is equivalent to $$\\\\sqrt{a}>\\\\sqrt{b}$$.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew8a-h13","type":"hint","dependencies":["aff8a70precalnew8a-h12"],"title":"Simplification","text":"$$|x+2|>\\\\sqrt{6}$$ is equivalent to $$x+2>\\\\sqrt{6}$$ or $$x+2<-\\\\sqrt{6}$$","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew8a-h14","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\sqrt{6}-2$$"],"dependencies":["aff8a70precalnew8a-h13"],"title":"Simplification","text":"$$x+2>\\\\sqrt{6}$$, x>?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\sqrt{6}-2$$","$$-\\\\sqrt{6}-2$$"]},{"id":"aff8a70precalnew8a-h15","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-\\\\sqrt{6}-2$$"],"dependencies":["aff8a70precalnew8a-h14"],"title":"Simplification","text":"$$x+2<\\\\left(-\\\\sqrt{6}\\\\right)$$, x<?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$\\\\sqrt{6}-2$$","$$-\\\\sqrt{6}-2$$"]},{"id":"aff8a70precalnew8a-h16","type":"hint","dependencies":["aff8a70precalnew8a-h15"],"title":"Interval Notation","text":"Write $$x>\\\\sqrt{6}-2$$ or $$x<-\\\\sqrt{6}-2$$ in interval notation.","variabilization":{},"oer":"https://OATutor.io","license":""},{"id":"aff8a70precalnew8a-h17","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$(-\\\\infty,-\\\\sqrt{6}-2)$$ $$\\\\cup$$ $$(\\\\sqrt{6}-2,\\\\infty)$$"],"dependencies":["aff8a70precalnew8a-h16"],"title":"Interval Notation","text":"What is the answer in interval notation?","variabilization":{},"oer":"https://OATutor.io","license":"","choices":["$$(-\\\\infty,-\\\\sqrt{6}-2)$$ $$\\\\cup$$ $$(\\\\sqrt{6}-2,\\\\infty)$$","$$(-\\\\sqrt{6}-2,\\\\sqrt{6}-2)$$"]}]}}]},{"id":"swedish_a1cc0dcareas1","title":"Att hitta arean av ett omr\xe5de mellan tv\xe5 kurvor $$1$$","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"5 Area av ett omr\xe5de mellan tv\xe5 kurvor","courseName":"Matematik 4","steps":[{"id":"swedish_a1cc0dcareas1a","stepAnswer":["$$\\\\frac{57}{4}$$"],"problemType":"TextBox","stepTitle":"Om R \xe4r omr\xe5det som begr\xe4nsas ovan av grafen f\xf6r funktionen $$f(x)=x+4$$ och nedan av grafen f\xf6r funktionen $$g(x)=3-\\\\frac{x}{2}$$ \xf6ver intervallet [1,4], best\xe4m arean av omr\xe5det R.","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{57}{4}$$","hints":{"DefaultPathway":[{"id":"swedish_a1cc0dcareas1a-h1","type":"hint","dependencies":[],"title":"Identifiera kurvorna","text":"F\xf6rst m\xe5ste du identifiera de tv\xe5 kurvorna som definierar omr\xe5det vars area du vill hitta. I detta fall \xe4r f(x) den \xf6vre kurvan och g(x) den nedre kurvan","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"swedish_a1cc0dcareas1a-h2","type":"hint","dependencies":["swedish_a1cc0dcareas1a-h1"],"title":"St\xe4ll upp integralen","text":"$$A=\\\\int (f(x)-g(x) \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"swedish_a1cc0dcareas1a-h3","type":"hint","dependencies":["swedish_a1cc0dcareas1a-h2"],"title":"St\xe4ll upp integralen","text":"Med $$f(x)=x+4$$ och $$g(x)=3-\\\\frac{x}{2}$$ \xf6ver det givna intervallet [1,4], kan vi fullborda integraluttrycket som $$A=\\\\int_{1}^{4} x+4-3-\\\\frac{x}{2} \\\\,dx=\\\\int_{1}^{4} \\\\frac{3x}{2}+1 \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"swedish_a1cc0dcareas1a-h4","type":"hint","dependencies":["swedish_a1cc0dcareas1a-h3"],"title":"Best\xe4m integralen","text":"$$\\\\int_{1}^{4} \\\\frac{3x}{2}+1 \\\\,dx=\\\\frac{3x^2}{4}+x$$ n\xe4r gr\xe4nserna g\xe5r fr\xe5n $$x=1$$ till $$x=4$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"swedish_a1cc0dcareas1a-h5","type":"hint","dependencies":["swedish_a1cc0dcareas1a-h4"],"title":"Utv\xe4rdera","text":"$$16-\\\\frac{7}{4}=\\\\frac{57}{4}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"swedish_a1cc0dcareas10","title":"F\xf6r f\xf6ljande \xf6vningar, skissa graferna till funktionerna och skugga omr\xe5det mellan kurvorna. 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Om n\xf6dv\xe4ndigt, dela upp omr\xe5det i delomr\xe5den f\xf6r att best\xe4mma dess totala area.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4 Areor","courseName":"Matematik 4","steps":[{"id":"swedish_a1cc0dcareas11a","stepAnswer":["$$\\\\frac{15}{2}$$"],"problemType":"TextBox","stepTitle":"$$y=x^3$$ och $$y=x^2-2x$$ \xf6ver $$x=[-1,1]$$","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{15}{2}$$","hints":{"DefaultPathway":[{"id":"swedish_a1cc0dcareas11a-h1","type":"hint","dependencies":[],"title":"Separera intervallet","text":"I det h\xe4r problemet kan arean av omr\xe5det mellan $$2$$ kurvor delas upp i $$2$$ delomr\xe5den som [-1,0] och [0,1].","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"swedish_a1cc0dcareas11a-h2","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{19}{12}$$"],"dependencies":["swedish_a1cc0dcareas11a-h1"],"title":"Area fr\xe5n $$-1$$ till $$0$$","text":"Vad \xe4r arean av intervallet fr\xe5n $$-1$$ till 0?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"swedish_a1cc0dcareas11a-h3","type":"hint","dependencies":["swedish_a1cc0dcareas11a-h2"],"title":"Area fr\xe5n $$-1$$ till $$0$$","text":"$$\\\\int_{-1}^{0} x^2-2x-x^3 \\\\,dx=0-\\\\frac{-1^3}{3}-(-1)^2-\\\\frac{(-1)^4}{4}=\\\\frac{19}{12}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"swedish_a1cc0dcareas11a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{11}{12}$$"],"dependencies":["swedish_a1cc0dcareas11a-h3"],"title":"Area fr\xe5n $$0$$ till $$1$$","text":"Vad \xe4r arean av intervallet fr\xe5n $$0$$ till 1?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"swedish_a1cc0dcareas11a-h5","type":"hint","dependencies":["swedish_a1cc0dcareas11a-h4"],"title":"Area fr\xe5n $$0$$ till $$1$$","text":"$$\\\\int_{0}^{1} x^3-x^2+2x \\\\,dx=\\\\frac{1^4}{4}-\\\\frac{1^3}{3}+1^2=\\\\frac{11}{12}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"swedish_a1cc0dcareas11a-h6","type":"hint","dependencies":["swedish_a1cc0dcareas11a-h5"],"title":"Arean av hela omr\xe5det","text":"$$\\\\frac{19}{12}+\\\\frac{11}{12}=\\\\frac{5}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"swedish_a1cc0dcareas3","title":"Om R \xe4r omr\xe5det begr\xe4nsat ovanf\xf6r av grafen av funktionen $$f(x)=sinx$$ och nedanf\xf6r av grafen av funktionen $$g(x)=cosx$$ \xf6ver intervallet [0,pi], best\xe4m arean av omr\xe5det R.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6 Omr\xe5den av sammansatta regioner","courseName":"Matematik 4","steps":[{"id":"swedish_a1cc0dcareas3a","stepAnswer":["$$2\\\\sqrt{2}$$"],"problemType":"TextBox","stepTitle":"Identifiera kurvorna","stepBody":"F\xf6rst m\xe5ste du identifiera de tv\xe5 kurvorna som definierar omr\xe5det vars area du vill hitta. I detta fall \xe4r f(x) den \xf6vre kurvan och g(x) den nedre kurvan. Notera att fr\xe5n $$0$$ till $$\\\\frac{\\\\pi}{4}$$ \xe4r grafen g(x) en \xf6vre kurva. Men fr\xe5n $$\\\\frac{\\\\pi}{4}$$ till pi \xe4r f(x) en \xf6vre kurva och g(x) en nedre kurva.##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$2\\\\sqrt{2}$$","hints":{"DefaultPathway":[{"id":"swedish_a1cc0dcareas3a-h1","type":"hint","dependencies":[],"title":"St\xe4ll upp integralen","text":"$$A=\\\\int_{a}^{b} |f(x)-g(x)| \\\\,dx=\\\\int_{0}^{\\\\frac{\\\\pi}{4}} cosx-sinx \\\\,dx+\\\\int_{\\\\frac{\\\\pi}{4}}^{pi} sinx-cosx \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"swedish_a1cc0dcareas3a-h2","type":"hint","dependencies":["swedish_a1cc0dcareas3a-h1"],"title":"Integrera fr\xe5n $$0$$ till $$\\\\frac{\\\\pi}{4}$$","text":"$$\\\\int_{0}^{\\\\frac{\\\\pi}{4}} cosx-sinx \\\\,dx=sinx+cox$$ med $$x$$ g\xe5r fr\xe5n $$0$$ till $$\\\\frac{\\\\pi}{4}$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"swedish_a1cc0dcareas3a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{2}-1$$"],"dependencies":["swedish_a1cc0dcareas3a-h2"],"title":"Utv\xe4rdera","text":"Vad \xe4r arean av ett omr\xe5de begr\xe4nsat fr\xe5n $$x=0$$ till $$x=\\\\frac{\\\\pi}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"swedish_a1cc0dcareas3a-h4","type":"hint","dependencies":["swedish_a1cc0dcareas3a-h3"],"title":"Integrera fr\xe5n $$\\\\frac{\\\\pi}{4}$$ till pi","text":"$$\\\\int_{0}^{\\\\frac{\\\\pi}{4}} -cosx-sinx \\\\,dx=sinx+cox$$ med $$x$$ g\xe5r fr\xe5n $$\\\\frac{\\\\pi}{4}$$ till pi","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"swedish_a1cc0dcareas3a-h5","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\sqrt{2}+1$$"],"dependencies":["swedish_a1cc0dcareas3a-h4"],"title":"Utv\xe4rdera","text":"Vad \xe4r arean av ett omr\xe5de begr\xe4nsat fr\xe5n $$x=0$$ till $$x=\\\\frac{\\\\pi}{4}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"swedish_a1cc0dcareas3a-h6","type":"hint","dependencies":["swedish_a1cc0dcareas3a-h5"],"title":"Arean av hela omr\xe5det","text":"$$\\\\sqrt{2}-1+\\\\sqrt{2}+1=2\\\\sqrt{2}$$ $${units}^2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"swedish_a1cc0dcareas4","title":"Att hitta arean av en komplex region","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"6 Omr\xe5den av sammansatta regioner","courseName":"Matematik 4","steps":[{"id":"swedish_a1cc0dcareas4a","stepAnswer":["$$\\\\frac{5}{6}$$"],"problemType":"TextBox","stepTitle":"Betrakta omr\xe5det som visas i Figur $$6.7$$. 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Efter att ha l\xf6st f\xf6r $$x$$, f\xe5r vi $$x=1$$ d\xe4r graferna sk\xe4r varandra.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"swedish_a1cc0dcareas4a-h2","type":"hint","dependencies":["swedish_a1cc0dcareas4a-h1"],"title":"Separera intervallet","text":"Eftersom arean fr\xe5n $$0$$ till $$1$$ och arean fr\xe5n $$1$$ till $$2$$ inte \xe4r desamma, m\xe5ste vi integrera var och en av dem separat.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"swedish_a1cc0dcareas4a-h3","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{3}$$"],"dependencies":["swedish_a1cc0dcareas4a-h2"],"title":"Area fr\xe5n $$0$$ till $$1$$","text":"Vad \xe4r arean av intervallet fr\xe5n $$0$$ till 1?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"swedish_a1cc0dcareas4a-h3-s1","type":"hint","dependencies":[],"title":"Area fr\xe5n $$0$$ till $$1$$","text":"$$\\\\int_{0}^{1} x^2 \\\\,dx=\\\\frac{1^3}{3}-0=\\\\frac{1}{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"swedish_a1cc0dcareas4a-h4","type":"scaffold","problemType":"TextBox","answerType":"arithmetic","hintAnswer":["$$\\\\frac{1}{2}$$"],"dependencies":["swedish_a1cc0dcareas4a-h3"],"title":"Area fr\xe5n $$1$$ till $$2$$","text":"Vad \xe4r arean av intervallet fr\xe5n $$1$$ till 2?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","subHints":[{"id":"swedish_a1cc0dcareas4a-h4-s1","type":"hint","dependencies":[],"title":"Area fr\xe5n $$1$$ till $$2$$","text":"$$\\\\int_{1}^{2} 2-x \\\\,dx=2\\\\times2-\\\\frac{2^2}{2}-2\\\\times1-\\\\frac{1^2}{2}=\\\\frac{1}{2}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]},{"id":"swedish_a1cc0dcareas4a-h5","type":"hint","dependencies":["swedish_a1cc0dcareas4a-h4"],"title":"Adderar areorna","text":"Genom att addera dessa omr\xe5den tillsammans, erh\xe5ller vi $$A=\\\\frac{1}{3}+\\\\frac{1}{2}=\\\\frac{5}{6}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"swedish_a1cc0dcareas6","title":"F\xf6r f\xf6ljande \xf6vningar, best\xe4m arean av omr\xe5det mellan de tv\xe5 kurvorna i den givna figuren genom att integrera \xf6ver x-axeln.","body":"","variabilization":{},"oer":"https://openstax.org/details/books/calculus-volume-1 <OpenStax: Calculus Volume 1>","license":0,"lesson":"4 Areor","courseName":"Matematik 4","steps":[{"id":"swedish_a1cc0dcareas6a","stepAnswer":["$$\\\\frac{32}{3}$$"],"problemType":"TextBox","stepTitle":"$$y=x^2-3$$ och $$y=1$$","stepBody":"##figure1.gif## ","answerType":"arithmetic","variabilization":{},"answerLatex":"$$\\\\frac{32}{3}$$","hints":{"DefaultPathway":[{"id":"swedish_a1cc0dcareas6a-h1","type":"hint","dependencies":[],"title":"Best\xe4m gr\xe4nsv\xe4rdena","text":"F\xf6r att st\xe4lla upp de best\xe4mda integralerna av $$2$$-funktionerna m\xe5ste vi hitta gr\xe4nserna d\xe4r de sk\xe4r varandra genom att s\xe4tta dem lika och l\xf6sa f\xf6r $$x$$.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"swedish_a1cc0dcareas6a-h2","type":"hint","dependencies":["swedish_a1cc0dcareas6a-h1"],"title":"Best\xe4m gr\xe4nsv\xe4rdena","text":"$$x^2-3=1$$ sedan $$x=2$$ och $$x=-2$$. S\xe5 vi erh\xe5ller $$x=-2$$ som en nedre gr\xe4ns och $$x=2$$ som en \xf6vre gr\xe4ns.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"swedish_a1cc0dcareas6a-h3","type":"hint","dependencies":["swedish_a1cc0dcareas6a-h2"],"title":"Definiera den \xf6vre grafen","text":"Arean av A ges av $$int{|f(x)-g(x)|, a, b, x$$, eftersom f(x) \xe4r en \xf6vre graf och g(x) \xe4r den nedre.","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"swedish_a1cc0dcareas6a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["g(x)"],"dependencies":["swedish_a1cc0dcareas6a-h3"],"title":"Definiera den \xf6vre grafen","text":"Vad \xe4r den \xf6vre grafen i detta problem?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"","choices":["g(x)","f(x)"]},{"id":"swedish_a1cc0dcareas6a-h5","type":"hint","dependencies":["swedish_a1cc0dcareas6a-h4"],"title":"St\xe4ll upp integralen","text":"$$\\\\int_{-2}^{2} 1-x^2+3 \\\\,dx$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"swedish_a1cc0dcareas6a-h6","type":"hint","dependencies":["swedish_a1cc0dcareas6a-h5"],"title":"Ber\xe4kna integralen","text":"$$4x-\\\\frac{x^3}{3}$$ n\xe4r gr\xe4nserna g\xe5r fr\xe5n $$x=-2$$ till $$x=2$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""},{"id":"swedish_a1cc0dcareas6a-h7","type":"hint","dependencies":["swedish_a1cc0dcareas6a-h6"],"title":"Utv\xe4rdera","text":"$$4\\\\times2-\\\\frac{2^3}{3}-4\\\\left(-2\\\\right)-\\\\frac{{\\\\left(-2\\\\right)}^3}{3}=\\\\frac{32}{3}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":""}]}}]},{"id":"swedish_a2d77b8chainrule1","title":"\xd6vningar p\xe5 kedjeregeln","body":"Givet $$y=f(u)$$ och $$u=g(x)$$, best\xe4m $$\\\\frac{dy}{dx}$$ i termer av $$x!$$","variabilization":{},"oer":"https://openstax.org/books/college-physics-2e/pages/3-problems-exercises <OpenStax: College Physics 2e>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3 Kedjeregeln","courseName":"Matematik 4","steps":[{"id":"swedish_a2d77b8chainrule1a","stepAnswer":["$$12x$$"],"problemType":"MultipleChoice","stepTitle":"$$y=3u-6$$, $$u=2x^2$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$12x$$","choices":["$$12x$$","$$4x$$","$$16x$$","$$8x$$"],"hints":{"DefaultPathway":[{"id":"swedish_a2d77b8chainrule1a-h1","type":"hint","dependencies":[],"title":"Kedjeregeln","text":"Kom ih\xe5g, kedjeregeln i Leibniz notation s\xe4ger oss att $$\\\\frac{dy}{dx}=\\\\frac{dy}{du} \\\\frac{du}{dx}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"swedish_a2d77b8chainrule1a-h2","type":"hint","dependencies":["swedish_a2d77b8chainrule1a-h1"],"title":"Best\xe4m derivatan","text":"F\xf6rst, f\xf6rs\xf6k att hitta derivatan av $$y$$ med avseende p\xe5 u. Best\xe4m sedan derivatan av u med avseende p\xe5 $$x$$. Att multiplicera de tv\xe5 borde ge det r\xe4tta svaret!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"swedish_a2d77b8chainrule1a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$3$$"],"dependencies":["swedish_a2d77b8chainrule1a-h2"],"title":"Best\xe4m $$\\\\frac{dy}{du}$$","text":"Vad \xe4r $$\\\\frac{dy}{du}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$3$$","$$\\\\frac{3}{2} u^2$$","3u","$$3u^2$$"]},{"id":"swedish_a2d77b8chainrule1a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$4x$$"],"dependencies":["swedish_a2d77b8chainrule1a-h3"],"title":"Best\xe4m $$\\\\frac{du}{dx}$$","text":"Vad \xe4r $$\\\\frac{du}{dx}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$4x$$","$$\\\\frac{2}{3} x^3$$","$$4x^2$$","$$2x^3$$"]},{"id":"swedish_a2d77b8chainrule1a-h5","type":"hint","dependencies":["swedish_a2d77b8chainrule1a-h4"],"title":"S\xe4tt in","text":"Gl\xf6m inte att s\xe4tta in vad vi har f\xf6r u i termer av $$x$$ f\xf6r att ge v\xe5rt slutgiltiga svar i termer av $$x!$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"swedish_a2d77b8chainrule11","title":"$$y={\\\\left(5-2x\\\\right)}^{\\\\left(-2\\\\right)}$$","body":"","variabilization":{},"oer":"https://openstax.org/books/college-physics-2e/pages/3-problems-exercises <OpenStax: College Physics 2e>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3 Kedjeregeln","courseName":"Matematik 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Sedan, best\xe4m derivatan av u med avseende p\xe5 $$x$$. Att multiplicera de tv\xe5 borde ge r\xe4tt svar!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"swedish_a2d77b8chainrule2a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$18u^2$$"],"dependencies":["swedish_a2d77b8chainrule2a-h2"],"title":"Best\xe4m $$\\\\frac{dy}{du}$$","text":"Vad \xe4r $$\\\\frac{dy}{du}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$18u^2$$","$$18u^3$$","18u","$$18$$"]},{"id":"swedish_a2d77b8chainrule2a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$7$$"],"dependencies":["swedish_a2d77b8chainrule2a-h3"],"title":"Best\xe4m $$\\\\frac{du}{dx}$$","text":"Vad \xe4r $$\\\\frac{du}{dx}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$7$$","$$\\\\frac{7}{2} x^2$$","$$7x$$","$$7x^2$$"]},{"id":"swedish_a2d77b8chainrule2a-h5","type":"hint","dependencies":["swedish_a2d77b8chainrule2a-h4"],"title":"S\xe4tt in","text":"Gl\xf6m inte att s\xe4tta in vad vi har f\xf6r u i termer av $$x$$ f\xf6r att ge v\xe5rt slutgiltiga svar i termer av $$x!$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"swedish_a2d77b8chainrule3","title":"\xd6vningar p\xe5 kedjeregeln","body":"Givet $$y=f(u)$$ och $$u=g(x)$$, best\xe4m $$\\\\frac{dy}{dx}$$ i termer av $$x!$$","variabilization":{},"oer":"https://openstax.org/books/college-physics-2e/pages/3-problems-exercises <OpenStax: College Physics 2e>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3 Kedjeregeln","courseName":"Matematik 4","steps":[{"id":"swedish_a2d77b8chainrule3a","stepAnswer":["$$5cos\\\\left(5x-1\\\\right)$$"],"problemType":"MultipleChoice","stepTitle":"$$y=sin(u)$$, $$u=5x-1$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$5cos\\\\left(5x-1\\\\right)$$","choices":["$$5cos\\\\left(5x-1\\\\right)$$","$$5cos\\\\left(u\\\\right)$$","$$5sin\\\\left(u\\\\right)$$","$$5sin\\\\left(5x-1\\\\right)$$"],"hints":{"DefaultPathway":[{"id":"swedish_a2d77b8chainrule3a-h1","type":"hint","dependencies":[],"title":"Kedjeregeln","text":"Kom ih\xe5g, kedjeregeln i Leibniz notation s\xe4ger oss att $$\\\\frac{dy}{dx}=\\\\frac{dy}{du} \\\\frac{du}{dx}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"swedish_a2d77b8chainrule3a-h2","type":"hint","dependencies":["swedish_a2d77b8chainrule3a-h1"],"title":"Best\xe4m derivatan","text":"F\xf6rst, f\xf6rs\xf6k att hitta derivatan av $$y$$ med avseende p\xe5 u. D\xe4refter, best\xe4m derivatan av u med avseende p\xe5 $$x$$. Att multiplicera de tv\xe5 borde ge r\xe4tt svar!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"swedish_a2d77b8chainrule3a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["cos(u)"],"dependencies":["swedish_a2d77b8chainrule3a-h2"],"title":"Best\xe4m $$\\\\frac{dy}{du}$$","text":"Vad \xe4r $$\\\\frac{dy}{du}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["cos(u)","$$-cos(u)$$","sin(u)","$$-sin(u)$$"]},{"id":"swedish_a2d77b8chainrule3a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$5$$"],"dependencies":["swedish_a2d77b8chainrule3a-h3"],"title":"Best\xe4m $$\\\\frac{du}{dx}$$","text":"Vad \xe4r $$\\\\frac{du}{dx}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$5$$","$$\\\\frac{5}{2} x^2$$","$$5x$$","$$5x^2$$"]},{"id":"swedish_a2d77b8chainrule3a-h5","type":"hint","dependencies":["swedish_a2d77b8chainrule3a-h4"],"title":"Anslut","text":"Gl\xf6m inte att s\xe4tta in vad vi har f\xf6r u i termer av $$x$$ f\xf6r att ge v\xe5rt slutgiltiga svar i termer av $$x!$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"swedish_a2d77b8chainrule4","title":"\xd6vningar p\xe5 kedjeregeln","body":"Givet $$y=f(u)$$ och $$u=g(x)$$, best\xe4m $$\\\\frac{dy}{dx}$$ i termer av $$x!$$","variabilization":{},"oer":"https://openstax.org/books/college-physics-2e/pages/3-problems-exercises <OpenStax: College Physics 2e>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3 Kedjeregeln","courseName":"Matematik 4","steps":[{"id":"swedish_a2d77b8chainrule4a","stepAnswer":["$$\\\\frac{\\\\sin\\\\left(-\\\\frac{x}{8}\\\\right)}{8}$$"],"problemType":"MultipleChoice","stepTitle":"$$y=cos(u)$$, $$u=\\\\frac{-x}{8}$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{\\\\operatorname{sin}\\\\left(\\\\frac{x}{8}\\\\right)}{8}$$","choices":["$$\\\\frac{\\\\sin\\\\left(-\\\\frac{x}{8}\\\\right)}{8}$$","$$-\\\\frac{\\\\sin\\\\left(-\\\\frac{x}{8}\\\\right)}{8}$$","$$\\\\sin\\\\left(-\\\\frac{x}{8}\\\\right)$$","$$-\\\\sin\\\\left(-\\\\frac{x}{8}\\\\right)$$"],"hints":{"DefaultPathway":[{"id":"swedish_a2d77b8chainrule4a-h1","type":"hint","dependencies":[],"title":"Kedjeregeln","text":"Kom ih\xe5g, kedjeregeln i Leibniz notation s\xe4ger oss att $$\\\\frac{dy}{dx}=\\\\frac{dy}{du} \\\\frac{du}{dx}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"swedish_a2d77b8chainrule4a-h2","type":"hint","dependencies":["swedish_a2d77b8chainrule4a-h1"],"title":"Best\xe4m derivatan","text":"F\xf6rst, f\xf6rs\xf6k att hitta derivatan av $$y$$ med avseende p\xe5 u. Best\xe4m sedan derivatan av u med avseende p\xe5 $$x$$. Att multiplicera de tv\xe5 borde ge r\xe4tt svar!","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"swedish_a2d77b8chainrule4a-h3","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$-sin(u)$$"],"dependencies":["swedish_a2d77b8chainrule4a-h2"],"title":"Best\xe4m $$\\\\frac{dy}{du}$$","text":"Vad \xe4r $$\\\\frac{dy}{du}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["sin(u)","$$-sin(u)$$","cos(u)","$$-cos(u)$$"]},{"id":"swedish_a2d77b8chainrule4a-h4","type":"scaffold","problemType":"MultipleChoice","answerType":"string","hintAnswer":["$$\\\\frac{-1}{8}$$"],"dependencies":["swedish_a2d77b8chainrule4a-h3"],"title":"Best\xe4m $$\\\\frac{du}{dx}$$","text":"Vad \xe4r $$\\\\frac{du}{dx}$$?","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","choices":["$$\\\\frac{-1}{8}$$","$$\\\\frac{-x}{8}$$","$$\\\\frac{-\\\\left(x^2\\\\right)}{8}$$","$$\\\\frac{-\\\\left(x^3\\\\right)}{8}$$"]},{"id":"swedish_a2d77b8chainrule4a-h5","type":"hint","dependencies":["swedish_a2d77b8chainrule4a-h4"],"title":"Anslut","text":"Gl\xf6m inte att s\xe4tta in vad vi har f\xf6r u i termer av $$x$$ f\xf6r att ge v\xe5rt slutgiltiga svar i termer av $$x!$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"}]}}]},{"id":"swedish_a2d77b8chainrule6","title":"\xd6vningar p\xe5 kedjeregeln","body":"Givet $$y=f(u)$$ och $$u=g(x)$$, best\xe4m $$\\\\frac{dy}{dx}$$ i termer av $$x!$$","variabilization":{},"oer":"https://openstax.org/books/college-physics-2e/pages/3-problems-exercises <OpenStax: College Physics 2e>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>","lesson":"3 Kedjeregeln","courseName":"Matematik 4","steps":[{"id":"swedish_a2d77b8chainrule6a","stepAnswer":["$$\\\\frac{4x-12}{\\\\sqrt{4x^2-24x+3}}$$"],"problemType":"MultipleChoice","stepTitle":"$$y=\\\\sqrt{4u+3}$$, $$u=x^2-6x$$","stepBody":"","answerType":"string","variabilization":{},"answerLatex":"$$\\\\frac{4x-12}{\\\\sqrt{4x^2-24x+3}}$$","choices":["$$\\\\frac{4x-12}{\\\\sqrt{4x^2-24x+3}}$$","$$\\\\frac{1}{\\\\sqrt{4x^2-24x+3}}$$","$$\\\\frac{4x-12}{\\\\sqrt{4u+3}}$$","$$\\\\frac{1}{\\\\sqrt{4u+3}}$$"],"hints":{"DefaultPathway":[{"id":"swedish_a2d77b8chainrule6a-h1","type":"hint","dependencies":[],"title":"Kedjeregeln","text":"Kom ih\xe5g, kedjeregeln i Leibniz notation s\xe4ger oss att $$\\\\frac{dy}{dx}=\\\\frac{dy}{du} \\\\frac{du}{dx}$$","variabilization":{},"oer":"https://OATutor.io <OATutor>","license":"https://creativecommons.org/licenses/by/4.0/ <CC BY 4.0>"},{"id":"swedish_a2d77b8chainrule6a-h2","type":"hint","dependencies":["swedish_a2d77b8chainrule6a-h1"],"title":"Best\xe4m derivatan","text":"F\xf6rst, f\xf6rs\xf6k att hitta derivatan av $$y$$ med avseende p\xe5 u. Best\xe4m sedan derivatan av u med avseende p\xe5 $$x$$. 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t?Number(t.string):null}},{key:"toColor",value:function(e){var t,i;if(null===e)return null;var a=this.evaluate(e);return void 0===a?null:"string"in a?null!=(i=null==(t=this.colorMap)?void 0:t.call(this,a.string))?i:a.string:null}},{key:"toBackgroundColor",value:function(e){var t,i;if(null===e)return null;var a=this.evaluate(e);return void 0===a?null:"string"in a?null!=(i=null==(t=this.backgroundColorMap)?void 0:t.call(this,a.string))?i:a.string:null}}]),e}(),Qu=["body","above","below","superscript","subscript"];function Ku(e){return"string"==typeof e&&Qu.includes(e)}function Zu(e){return void 0!==e&&Array.isArray(e)&&2===e.length}var Ju=function(){function e(t){var i,a,o,n,r,s,l;(0,h.Z)(this,e),this.type=t.type,"string"==typeof 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0!==this.verbatimLatex&&(e.verbatimLatex=this.verbatimLatex),this.subsupPlacement&&(e.subsupPlacement=this.subsupPlacement),this.explicitSubsupPlacement&&(e.explicitSubsupPlacement=!0),this.isFunction&&(e.isFunction=!0),this.displayContainsHighlight&&(e.displayContainsHighlight=!0),this.isExtensibleSymbol&&(e.isExtensibleSymbol=!0),this.skipBoundary&&(e.skipBoundary=!0),this.captureSelection&&(e.captureSelection=!0),this.args&&(e.args=function(e){return e.map((function(e){return null===e?"<null>":Array.isArray(e)&&e[0]instanceof Ju?{atoms:e.map((function(e){return e.toJson()}))}:"object"==typeof e&&"group"in e?{group:e.group.map((function(e){return e.toJson()}))}:e}))}(this.args)),this._branches)for(var t=0,i=Object.keys(this._branches);t<i.length;t++){var a=i[t];this._branches[a]&&(e[a]=this._branches[a].filter((function(e){return"first"!==e.type})).map((function(e){return e.toJson()})))}return"mord"===e.type&&2===Object.keys(e).length&&"value"in 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Tu.serialize(this.body,ks(Is({},e),{defaultMode:null!=(t=e.defaultMode)?t:this.latexMode}))}},{key:"aboveToLatex",value:function(e){return Tu.serialize(this.above,e)}},{key:"belowToLatex",value:function(e){return Tu.serialize(this.below,e)}},{key:"supsubToLatex",value:function(e){var t="";if(e=ks(Is({},e),{defaultMode:"math"}),void 0!==this.branch("subscript")){var i=Tu.serialize(this.subscript,e);0===i.length?t+="_{}":1===i.length&&/^[0-9]$/.test(i)?t+="_".concat(i):t+="_{".concat(i,"}")}if(void 0!==this.branch("superscript")){var a=Tu.serialize(this.superscript,e);0===a.length?t+="^{}":1===a.length?"\u2032"===a?t+="^\\prime ":"\u2033"===a?t+="^\\doubleprime ":/^[0-9]$/.test(a)?t+="^".concat(a):t+="^{".concat(a,"}"):t+="^{".concat(a,"}")}return t}},{key:"treeDepth",get:function(){for(var e=1,t=this.parent;t;)t=t.parent,e+=1;return e}},{key:"inCaptureSelection",get:function(){for(var 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:host([disabled]:focus), :host([disabled]:focus-within) { outline: none; opacity:  .5; }\n    :host(:focus), :host(:focus-within) {\n      outline: Highlight auto 1px;    /* For Firefox */\n      outline: -webkit-focus-ring-color auto 1px;\n    }\n    :host([readonly]:focus), :host([readonly]:focus-within),\n    :host([read-only]:focus), :host([read-only]:focus-within) {\n      outline: none;\n    }\n    @media (hover: none) and (pointer: coarse) {\n      :host(:not(:focus)) :first-child { pointer-events: none !important; }\n    }";break;case"core":t=".ML__container {\n  min-height: auto !important;\n  --_hue: var(--hue, 212);\n  --_placeholder-color: var(--placeholder-color, hsl(var(--_hue), 40%, 49%));\n  --_placeholder-opacity: var(--placeholder-opacity, 0.4);\n  --_text-font-family: var(--text-font-family, system-ui, -apple-system, BlinkMacSystemFont, 'Segoe UI', 'Roboto', 'Oxygen', 'Ubuntu', 'Cantarell', 'Fira Sans', 'Droid Sans', 'Helvetica Neue', sans-serif);\n}\n.ML__sr-only {\n  position: absolute;\n  width: 1px;\n  height: 1px;\n  margin: -1px;\n  padding: 0;\n  overflow: hidden;\n  clip: rect(0, 0, 0, 0);\n  clip-path: inset(50%);\n  white-space: nowrap;\n  border: 0;\n}\n.ML__is-inline {\n  display: inline-block;\n}\n.ML__base {\n  visibility: inherit;\n  display: inline-block;\n  position: relative;\n  cursor: text;\n  padding: 0;\n  margin: 0;\n  box-sizing: content-box;\n  border: 0;\n  outline: 0;\n  vertical-align: baseline;\n  font-weight: inherit;\n  font-family: inherit;\n  font-style: inherit;\n  text-decoration: none;\n  width: min-content;\n}\n.ML__strut,\n.ML__strut--bottom {\n  display: inline-block;\n  min-height: 0.5em;\n}\n.ML__small-delim {\n  font-family: KaTeX_Main;\n}\n/* Text mode */\n.ML__text {\n  font-family: var(--_text-font-family);\n  white-space: pre;\n}\n/* Use cmr for 'math upright' */\n.ML__cmr {\n  font-family: KaTeX_Main;\n  font-style: normal;\n}\n.ML__mathit {\n  font-family: KaTeX_Math;\n  /* The KaTeX_Math font is italic by default, so the font-style below is only \n     useful when a fallback font is used\n  */\n  font-style: italic;\n}\n.ML__mathbf {\n  font-family: KaTeX_Main;\n  font-weight: bold;\n}\n/* Lowercase greek symbols should stick to math font when \\mathbf is applied \n   to match TeX idiosyncratic behavior */\n.lcGreek.ML__mathbf {\n  font-family: KaTeX_Math;\n  font-weight: normal;\n}\n.ML__mathbfit {\n  font-family: KaTeX_Math;\n  font-weight: bold;\n  font-style: italic;\n}\n.ML__ams {\n  font-family: KaTeX_AMS;\n}\n/* Blackboard */\n.ML__bb {\n  font-family: KaTeX_AMS;\n}\n.ML__cal {\n  font-family: KaTeX_Caligraphic;\n}\n.ML__frak {\n  font-family: KaTeX_Fraktur;\n}\n.ML__tt {\n  font-family: KaTeX_Typewriter;\n}\n.ML__script {\n  font-family: KaTeX_Script;\n}\n.ML__sans {\n  font-family: KaTeX_SansSerif;\n}\n.ML__series_ul {\n  font-weight: 100;\n}\n.ML__series_el {\n  font-weight: 100;\n}\n.ML__series_l {\n  font-weight: 200;\n}\n.ML__series_sl {\n  font-weight: 300;\n}\n.ML__series_sb {\n  font-weight: 500;\n}\n.ML__bold,\n.ML__boldsymbol {\n  font-weight: 700;\n}\n.ML__series_eb {\n  font-weight: 800;\n}\n.ML__series_ub {\n  font-weight: 900;\n}\n.ML__series_uc {\n  font-stretch: ultra-condensed;\n}\n.ML__series_ec {\n  font-stretch: extra-condensed;\n}\n.ML__series_c {\n  font-stretch: condensed;\n}\n.ML__series_sc {\n  font-stretch: semi-condensed;\n}\n.ML__series_sx {\n  font-stretch: semi-expanded;\n}\n.ML__series_x {\n  font-stretch: expanded;\n}\n.ML__series_ex {\n  font-stretch: extra-expanded;\n}\n.ML__series_ux {\n  font-stretch: ultra-expanded;\n}\n.ML__it {\n  font-style: italic;\n}\n.ML__shape_ol {\n  -webkit-text-stroke: 1px black;\n  text-stroke: 1px black;\n  color: transparent;\n}\n.ML__shape_sc {\n  font-variant: small-caps;\n}\n.ML__shape_sl {\n  font-style: oblique;\n}\n/* First level emphasis */\n.ML__emph {\n  color: #bc2612;\n}\n/* Second level emphasis */\n.ML__emph .ML__emph {\n  color: #0c7f99;\n}\n.ML__highlight {\n  color: #007cb2;\n  background: #edd1b0;\n}\n.ML__center {\n  text-align: center;\n}\n.ML__label_padding {\n  padding: 0 0.5em;\n}\n.ML__frac-line {\n  width: 100%;\n  min-height: 1px;\n}\n.ML__frac-line:after {\n  content: '';\n  display: block;\n  margin-top: max(-1px, -0.04em);\n  min-height: max(1px, 0.04em);\n  /* Ensure the line is visible when printing even if \"turn off background images\" is on*/\n  -webkit-print-color-adjust: exact;\n  print-color-adjust: exact;\n  /* There's a bug since Chrome 62 where \n      sub-pixel border lines don't draw at some zoom \n      levels (110%, 90%). \n      Setting the min-height used to work around it, but that workaround\n      broke in Chrome 84 or so.\n      Setting the background (and the min-height) seems to work for now.\n      */\n  background: currentColor;\n  box-sizing: content-box;\n  /* Vuetify sets the box-sizing to inherit \n            causes the fraction line to not draw at all sizes (see #26) */\n  /* On some versions of Firefox on Windows, the line fails to \n            draw at some zoom levels, but setting the transform triggers\n            the hardware accelerated path, which works */\n  transform: translate(0, 0);\n}\n.ML__sqrt {\n  display: inline-block;\n}\n.ML__sqrt-sign {\n  display: inline-block;\n  position: relative;\n}\n.ML__sqrt-line {\n  display: inline-block;\n  height: max(1px, 0.04em);\n  width: 100%;\n}\n.ML__sqrt-line:before {\n  content: '';\n  display: block;\n  margin-top: min(-1px, -0.04em);\n  min-height: max(1px, 0.04em);\n  /* Ensure the line is visible when printing even if \"turn off background images\" is on*/\n  -webkit-print-color-adjust: exact;\n  print-color-adjust: exact;\n  background: currentColor;\n  /* On some versions of Firefox on Windows, the line fails to \n            draw at some zoom levels, but setting the transform triggers\n            the hardware accelerated path, which works */\n  transform: translate(0, 0);\n}\n.ML__sqrt-line:after {\n  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width: 100%;\n  left: 0;\n  overflow: hidden;\n}\n.ML__latex .stretchy:before,\n.ML__latex .stretchy:after {\n  content: '';\n}\n.ML__latex .stretchy svg {\n  display: block;\n  position: absolute;\n  width: 100%;\n  height: inherit;\n  fill: currentColor;\n  stroke: currentColor;\n  fill-rule: nonzero;\n  fill-opacity: 1;\n  stroke-width: 1;\n  stroke-linecap: butt;\n  stroke-linejoin: miter;\n  stroke-miterlimit: 4;\n  stroke-dasharray: none;\n  stroke-dashoffset: 0;\n  stroke-opacity: 1;\n}\n.ML__latex .slice-1-of-2 {\n  display: inline-flex;\n  position: absolute;\n  left: 0;\n  width: 50.2%;\n  overflow: hidden;\n}\n.ML__latex .slice-2-of-2 {\n  display: inline-flex;\n  position: absolute;\n  right: 0;\n  width: 50.2%;\n  overflow: hidden;\n}\n.ML__latex .slice-1-of-3 {\n  display: inline-flex;\n  position: absolute;\n  left: 0;\n  width: 25.1%;\n  overflow: hidden;\n}\n.ML__latex .slice-2-of-3 {\n  display: inline-flex;\n  position: absolute;\n  left: 25%;\n  width: 50%;\n  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95%));\n  --_smart-fence-color: var(--smart-fence-color, currentColor);\n  --_smart-fence-opacity: var(--smart-fence-opacity, 0.5);\n  --_latex-color: var(--latex-color, hsl(var(--_hue), 80%, 40%));\n  --_correct-color: var(--correct-color, #10a000);\n  --_incorrect-color: var(--incorrect-color, #a01b00);\n  --_composition-background-color: var(--composition-background-color, #fff1c2);\n  --_composition-text-color: var(--composition-text-color, black);\n  --_composition-underline-color: var(--composition-underline-color, transparent);\n}\n/* This is the actual field content (formula) */\n.ML__content {\n  display: flex;\n  align-items: center;\n  align-self: center;\n  position: relative;\n  overflow: hidden;\n  padding: 2px 3px 2px 1px;\n  width: 100%;\n}\n.ML__virtual-keyboard-toggle,\n.ML__menu-toggle {\n  box-sizing: border-box;\n  display: flex;\n  align-self: center;\n  align-items: center;\n  flex-shrink: 0;\n  flex-direction: column;\n  justify-content: center;\n  width: 34px;\n  height: 34px;\n  padding: 0;\n  margin-right: 4px;\n  cursor: pointer;\n  /* Avoid some weird blinking with :hover */\n  border-radius: 8px;\n  border: 1px solid transparent;\n  transition: background 0.2s cubic-bezier(0.64, 0.09, 0.08, 1);\n  color: hsl(var(--_hue), 40%, 50%);\n  fill: currentColor;\n  background: transparent;\n}\n.ML__virtual-keyboard-toggle:hover,\n.ML__menu-toggle:hover {\n  background: hsla(0, 0%, 70%, 0.3);\n  color: #333;\n  fill: currentColor;\n}\n.ML__virtual-keyboard-toggle > span,\n.ML__menu-toggle > span {\n  display: flex;\n  align-self: center;\n  align-items: center;\n}\n/* The invisible element used to capture keyboard events. We're just trying\n really hard to make sure it doesn't show. */\n.ML__keyboard-sink {\n  display: inline-block;\n  resize: none;\n  outline: none;\n  border: none;\n  /* Need these for Microsoft Edge */\n  position: fixed;\n  clip: rect(0 0 0 0);\n  /* Need this to prevent iOS Safari from auto-zooming */\n  font-size: 1em;\n  font-family: KaTeX_Main;\n  line-height: 0.5;\n  /* On Chromium, if this is 0, no keyboard events are received */\n}\n[part=\"placeholder\"] {\n  color: var(--neutral-400);\n}\n.ML__composition {\n  background: var(--_composition-background-color);\n  color: var(--_composition-text-color);\n  text-decoration: underline var(--_composition-underline-color);\n}\n.ML__caret::after {\n  content: '';\n  visibility: hidden;\n  width: 0;\n  display: inline-block;\n  height: 0.76em;\n  --_caret-width: clamp(2px, 0.08em, 10px);\n  border: none;\n  border-radius: calc(var(--_caret-width) / 2);\n  border-right: var(--_caret-width) solid var(--_caret-color);\n  margin-right: calc(-1 * var(--_caret-width));\n  position: relative;\n  left: -0.045em;\n  bottom: -0.05em;\n  animation: ML__caret-blink 1.05s step-end forwards infinite;\n}\n.ML__text-caret::after {\n  content: '';\n  visibility: hidden;\n  width: 0;\n  display: inline-block;\n  height: 0.76em;\n  --_caret-width: clamp(2px, 0.08em, 10px);\n  border: none;\n  border-radius: calc(var(--_caret-width) / 2);\n  border-right: var(--_caret-width) solid var(--_caret-color);\n  margin-right: calc(-1 * var(--_caret-width));\n  position: relative;\n  left: -0.045em;\n  bottom: -0.05em;\n  animation: ML__caret-blink 1.05s step-end forwards infinite;\n}\n.ML__latex-caret::after {\n  content: '';\n  visibility: hidden;\n  --_caret-width: clamp(2px, 0.08em, 10px);\n  border: none;\n  border-radius: calc(var(--_caret-width) / 2);\n  border-right: var(--_caret-width) solid var(--_latex-color);\n  margin-right: calc(-1 * var(--_caret-width));\n  position: relative;\n  left: -0.019em;\n  animation: ML__caret-blink 1.05s step-end forwards infinite;\n}\n.ML__focused .ML__latex-caret::after,\n.ML__focused .ML__text-caret::after,\n.ML__focused .ML__caret::after {\n  visibility: visible;\n}\n.ML__focused .ML__text {\n  background: var(--_text-highlight-background-color);\n}\n/* When using smartFence, the anticipated closing fence is displayed\nwith this style */\n.ML__smart-fence__close {\n  opacity: var(--_smart-fence-opacity);\n  color: var(--_smart-fence-color);\n}\n.ML__selected,\n.ML__focused .ML__selected .ML__contains-caret,\n.ML__focused .ML__selected .ML__smart-fence__close,\n.ML__focused .ML__selected .ML__placeholder {\n  color: var(--_selection-color);\n  opacity: 1;\n}\n.ML__selection {\n  box-sizing: border-box;\n  background: var(--_selection-background-color) !important;\n}\n.ML__contains-caret.ML__close,\n.ML__contains-caret.ML__open,\n.ML__contains-caret > .ML__close,\n.ML__contains-caret > .ML__open,\n.ML__contains-caret .ML__sqrt-sign,\n.ML__contains-caret .ML__sqrt-line {\n  color: var(--_caret-color);\n}\n.ML__contains-highlight {\n  box-sizing: border-box;\n  background: transparent;\n}\n.ML__focused .ML__contains-highlight {\n  background: var(--_contains-highlight-background-color);\n}\n.ML__raw-latex {\n  font-family: 'Berkeley Mono', 'IBM Plex Mono', 'Source Code Pro', Consolas, 'Roboto Mono', Menlo, 'Bitstream Vera Sans Mono', 'DejaVu Sans Mono', Monaco, Courier, monospace;\n  font-weight: 400;\n  font-size: 0.8em;\n  letter-spacing: -0.05em;\n  color: var(--_latex-color);\n}\n.ML__suggestion {\n  color: var(--neutral-500);\n}\n.ML__virtual-keyboard-toggle.is-visible.is-pressed:hover {\n  background: hsl(var(--_hue), 25%, 35%);\n  color: #fafafa;\n  fill: currentColor;\n}\n.ML__virtual-keyboard-toggle:focus {\n  outline: none;\n  border-radius: 8px;\n  border: 2px solid hsl(var(--_hue), 40%, 50%);\n}\n.ML__virtual-keyboard-toggle.is-pressed,\n.ML__virtual-keyboard-toggle.is-active:hover,\n.ML__virtual-keyboard-toggle.is-active {\n  background: hsl(var(--_hue), 25%, 35%);\n  color: #fafafa;\n  fill: currentColor;\n}\n/* Add an attribute 'data-tooltip' to automatically show a\n   tooltip over a element on hover.\n*/\n[data-tooltip] {\n  position: relative;\n}\n[data-tooltip]::after {\n  content: attr(data-tooltip);\n  position: absolute;\n  display: block;\n  z-index: 2;\n  pointer-events: none;\n  right: auto;\n  top: calc(-100% - 4px);\n  width: max-content;\n  max-width: 200px;\n  padding: 8px 8px;\n  border-radius: 4px;\n  background: #616161;\n  color: #fff;\n  box-shadow: 0 2px 2px 0 rgba(0, 0, 0, 0.14), 0 1px 5px 0 rgba(0, 0, 0, 0.12), 0 3px 1px -2px rgba(0, 0, 0, 0.2);\n  text-align: center;\n  font-family: system-ui, -apple-system, BlinkMacSystemFont, 'Segoe UI', 'Roboto', 'Oxygen', 'Ubuntu', 'Cantarell', 'Fira Sans', 'Droid Sans', 'Helvetica Neue', sans-serif;\n  font-style: normal;\n  font-weight: 400;\n  font-size: 13px;\n  /* Phone */\n  opacity: 0;\n  transform: scale(0.5);\n}\n@media only screen and (max-width: 767px) {\n  [data-tooltip]::after {\n    padding: 8px 16px;\n    font-size: 16px;\n  }\n}\nmenu [data-tooltip]::after {\n  left: 100%;\n  top: 0%;\n}\nmenu .ML__base {\n  cursor: default;\n}\n/** Don't display if we're tracking, i.e. have the pointer down */\n.tracking [data-tooltip]:hover::after {\n  /* Use visibility, not display. Display will remove the after from the DOM, and the override below will not work */\n  visibility: hidden;\n}\n/** But do display if tracking and inside a menu */\n.tracking menu li[data-tooltip]:hover::after,\n[data-tooltip]:hover::after {\n  visibility: visible;\n  opacity: 1;\n  transform: scale(1);\n  transition-property: opacity, scale;\n  transition-duration: 0.15s;\n  transition-delay: 1s;\n  transition-timing-function: cubic-bezier(0.4, 0, 1, 1);\n}\n.ML__prompt {\n  border-radius: 2px;\n}\n.ML__editablePromptBox {\n  outline: 1px solid #acacac;\n  border-radius: 2px;\n  z-index: -1;\n}\n.ML__focusedPromptBox {\n  outline: highlight auto 1px;\n}\n.ML__lockedPromptBox {\n  background-color: rgba(142, 142, 141, 0.4);\n  z-index: -1;\n}\n.ML__correctPromptBox {\n  outline: 1px solid var(--_correct-color);\n  box-shadow: 0 0 5px var(--_correct-color);\n}\n.ML__incorrectPromptBox {\n  outline: 1px solid var(--_incorrect-color);\n  box-shadow: 0 0 5px var(--_incorrect-color);\n}\n.variant-submenu {\n  display: flex;\n  flex-direction: column;\n  padding: 8px;\n}\n.variant-submenu [part=menu-item] {\n  font-size: 2rem;\n  text-align: center;\n  margin: 0;\n}\n.insert-matrix-submenu {\n  display: grid;\n  padding: 8px;\n  align-content: center;\n  justify-content: center;\n  grid-template-columns: repeat(5, minmax(0, 1fr));\n}\n.insert-matrix-submenu [part=menu-item] {\n  font-size: 21px;\n  border: none;\n  border-radius: 0;\n  line-height: 21px;\n  text-align: center;\n  padding: 0;\n  margin: 0;\n}\n.border-submenu [part=menu-item] {\n  font-size: 2rem;\n  line-height: 1.2;\n  text-align: center;\n}\n.swatches-submenu {\n  --_swatch-size: 2rem;\n  --_columns: 4;\n  display: flex;\n  flex-flow: wrap;\n  padding: 8px;\n  max-width: calc(var(--_columns) * (var(--_swatch-size) + 18px) + 16px);\n  box-sizing: border-box;\n}\n.menu-swatch {\n  display: flex;\n  align-items: center;\n  justify-content: center;\n  box-sizing: border-box;\n  width: fit-content;\n  height: fit-content;\n  margin: 2px;\n  padding: 0;\n  background: var(--neutral-200);\n}\n.menu-swatch > .label {\n  padding: 0;\n  margin: 0;\n  line-height: 0;\n}\n.menu-swatch > .label > span {\n  display: inline-block;\n  margin: 6px;\n  min-width: var(--_swatch-size);\n  min-height: var(--_swatch-size);\n  border-radius: 50%;\n}\n.menu-swatch.active {\n  background: var(--neutral-100);\n  scale: 1.4;\n}\n.menu-swatch.active > .label > span {\n  border-radius: 2px;\n}\n.menu-swatch .ui-checkmark,\n.menu-swatch .ui-mixedmark {\n  position: absolute;\n  margin: 0;\n  padding: 0;\n  color: white;\n}\n.menu-swatch.dark-contrast .ui-checkmark,\n.menu-swatch.dark-contrast .ui-mixedmark {\n  color: #000;\n}\n";break;case"environment-popover":t="#mathlive-environment-popover.is-visible {\n  visibility: visible;\n}\n#mathlive-environment-popover {\n  --_environment-panel-height: var(--environment-panel-height, 70px);\n  --_accent-color: var(--accent-color, #aaa);\n  --_background: var(--environment-panel-background, #fff);\n  --_button-background: var(--environment-panel-button-background, white);\n  --_button-background-hover: var(--environment-panel-button-background-hover, #f5f5f7);\n  --_button-background-active: var(--environment-panel-button-background-active, #f5f5f7);\n  --_button-text: var(--environment-panel-button-text, #e3e4e8);\n  position: absolute;\n  width: calc(var(--_environment-panel-height) * 2);\n  height: var(--_environment-panel-height);\n  border-radius: 4px;\n  border: 1.5px solid var(--_accent-color);\n  background-color: var(--_background);\n  box-shadow: 0 0 30px 0 var(--environment-shadow, rgba(0, 0, 0, 0.4));\n  pointer-events: all;\n  visibility: hidden;\n}\n#mathlive-environment-popover .MLEP__array-buttons {\n  height: calc(var(--_environment-panel-height) * 5/4);\n  width: calc(var(--_environment-panel-height) * 5/4);\n  margin-left: calc(0px - var(--_environment-panel-height) * 0.16);\n  margin-top: calc(0px - var(--_environment-panel-height) * 0.19);\n}\n#mathlive-environment-popover .MLEP__array-buttons .font {\n  fill: white;\n}\n#mathlive-environment-popover .MLEP__array-buttons circle {\n  fill: #7f7f7f;\n  transition: fill 300ms;\n}\n#mathlive-environment-popover .MLEP__array-buttons .MLEP__array-insert-background {\n  fill-opacity: 1;\n  fill: var(--_background);\n  stroke: var(--_accent-color);\n  stroke-width: 3px;\n}\n#mathlive-environment-popover .MLEP__array-buttons line {\n  stroke: var(--_accent-color);\n  stroke-opacity: 0;\n  stroke-width: 40;\n  pointer-events: none;\n  transition: stroke-opacity 300ms;\n  stroke-linecap: round;\n}\n#mathlive-environment-popover .MLEP__array-buttons g[data-command]:hover circle {\n  fill: var(--_accent-color);\n}\n#mathlive-environment-popover .MLEP__array-buttons g[data-command]:hover line {\n  stroke-opacity: 1;\n}\n#mathlive-environment-popover .MLEP__environment-delimiter-controls {\n  height: 100%;\n  width: 50%;\n}\n#mathlive-environment-popover .MLEP__environment-delimiter-controls .MLEP__array-delimiter-options {\n  width: var(--_environment-panel-height);\n  height: var(--_environment-panel-height);\n  display: flex;\n  flex-wrap: wrap;\n  flex-direction: row;\n  justify-content: space-around;\n}\n#mathlive-environment-popover .MLEP__environment-delimiter-controls .MLEP__array-delimiter-options svg {\n  pointer-events: all;\n  margin-top: 2px;\n  width: calc(var(--_environment-panel-height) / 3 * 28 / 24);\n  height: calc(var(--_environment-panel-height) / 3 - 2px);\n  border-radius: calc(var(--_environment-panel-height) / 25);\n  background-color: var(--_button-background);\n}\n#mathlive-environment-popover .MLEP__environment-delimiter-controls .MLEP__array-delimiter-options svg:hover {\n  background-color: var(--_button-background-hover);\n}\n#mathlive-environment-popover .MLEP__environment-delimiter-controls .MLEP__array-delimiter-options svg path,\n#mathlive-environment-popover .MLEP__environment-delimiter-controls .MLEP__array-delimiter-options svg line {\n  stroke: var(--_button-text);\n  stroke-width: 2;\n  stroke-linecap: round;\n}\n#mathlive-environment-popover .MLEP__environment-delimiter-controls .MLEP__array-delimiter-options svg rect,\n#mathlive-environment-popover .MLEP__environment-delimiter-controls .MLEP__array-delimiter-options svg path {\n  fill-opacity: 0;\n}\n#mathlive-environment-popover .MLEP__environment-delimiter-controls .MLEP__array-delimiter-options svg.active {\n  pointer-events: none;\n  background-color: var(--_button-background-active);\n}\n#mathlive-environment-popover .MLEP__environment-delimiter-controls .MLEP__array-delimiter-options svg.active path,\n#mathlive-environment-popover .MLEP__environment-delimiter-controls .MLEP__array-delimiter-options svg.active line {\n  stroke: var(--_accent-color);\n}\n#mathlive-environment-popover .MLEP__environment-delimiter-controls .MLEP__array-delimiter-options svg.active circle {\n  fill: var(--_accent-color);\n}\n";break;case"suggestion-popover":t="/* The element that display info while in latex mode */\n#mathlive-suggestion-popover {\n  background-color: rgba(97, 97, 97);\n  color: #fff;\n  text-align: center;\n  border-radius: 8px;\n  position: fixed;\n  z-index: 1;\n  display: none;\n  flex-direction: column;\n  justify-content: center;\n  box-shadow: 0 14px 28px rgba(0, 0, 0, 0.25), 0 10px 10px rgba(0, 0, 0, 0.22);\n}\n#mathlive-suggestion-popover.top-tip::after {\n  content: '';\n  position: absolute;\n  top: -15px;\n  left: calc(50% - 15px);\n  width: 0;\n  height: 0;\n  border-left: 15px solid transparent;\n  border-right: 15px solid transparent;\n  border-bottom: 15px solid rgba(97, 97, 97);\n  font-size: 1rem;\n}\n#mathlive-suggestion-popover.bottom-tip::after {\n  content: '';\n  position: absolute;\n  bottom: -15px;\n  left: calc(50% - 15px);\n  width: 0;\n  height: 0;\n  border-left: 15px solid transparent;\n  border-right: 15px solid transparent;\n  border-top: 15px solid rgba(97, 97, 97);\n  font-size: 1rem;\n}\n#mathlive-suggestion-popover.is-animated {\n  transition: all 0.2s cubic-bezier(0.64, 0.09, 0.08, 1);\n  animation: ML__fade-in cubic-bezier(0, 0, 0.2, 1) 0.15s;\n}\n#mathlive-suggestion-popover.is-visible {\n  display: flex;\n}\n@keyframes ML__fade-in {\n  from {\n    opacity: 0;\n  }\n  to {\n    opacity: 1;\n  }\n}\n/* The wrapper class for the entire content of the popover panel */\n#mathlive-suggestion-popover ul {\n  display: flex;\n  flex-flow: column;\n  list-style: none;\n  margin: 0;\n  padding: 0;\n  align-items: flex-start;\n  max-height: 400px;\n  overflow-y: auto;\n}\n#mathlive-suggestion-popover li {\n  display: flex;\n  flex-direction: row;\n  justify-content: space-between;\n  margin: 8px;\n  padding: 8px;\n  width: calc(100% - 16px - 16px);\n  column-gap: 1em;\n  border-radius: 8px;\n  cursor: pointer;\n  /* Since the content can be clicked on, provide feedback on hover */\n}\n#mathlive-suggestion-popover li a {\n  color: #5ea6fd;\n  padding-top: 0.3em;\n  margin-top: 0.4em;\n  display: block;\n}\n#mathlive-suggestion-popover li a:hover {\n  color: #5ea6fd;\n  text-decoration: underline;\n}\n#mathlive-suggestion-popover li:hover,\n#mathlive-suggestion-popover li.is-pressed,\n#mathlive-suggestion-popover li.is-active {\n  background: rgba(255, 255, 255, 0.1);\n}\n/* The command inside a popover (inside a #mathlive-suggestion-popover) */\n.ML__popover__command {\n  font-size: 1.6rem;\n  font-family: KaTeX_Main;\n}\n.ML__popover__current {\n  background: #5ea6fd;\n  color: #fff;\n}\n.ML__popover__latex {\n  font-family: 'IBM Plex Mono', 'Source Code Pro', Consolas, 'Roboto Mono', Menlo, 'Bitstream Vera Sans Mono', 'DejaVu Sans Mono', Monaco, Courier, monospace;\n  align-self: center;\n}\n/* The keyboard shortcuts for a symbol as displayed in the popover */\n.ML__popover__keybinding {\n  font-family: system-ui, -apple-system, BlinkMacSystemFont, 'Segoe UI', 'Roboto', 'Oxygen', 'Ubuntu', 'Cantarell', 'Fira Sans', 'Droid Sans', 'Helvetica Neue', sans-serif;\n  font-size: 0.8em;\n  opacity: 0.7;\n}\n/* Style for the character that joins the modifiers of a keyboard shortcut \n(usually a \"+\" sign)*/\n.ML__shortcut-join {\n  opacity: 0.5;\n}\n";break;case"keystroke-caption":t="/* The element that displays the keys as the user type them */\n#mathlive-keystroke-caption-panel {\n  visibility: hidden;\n  /*min-width: 160px;*/\n  /*background-color: rgba(97, 97, 200, .95);*/\n  background: var(--secondary, hsl(var(--_hue), 19%, 26%));\n  border-color: var(--secondary-border, hsl(0, 0%, 91%));\n  box-shadow: 0 3px 6px rgba(0, 0, 0, 0.16), 0 3px 6px rgba(0, 0, 0, 0.23);\n  text-align: center;\n  border-radius: 6px;\n  padding: 16px;\n  position: absolute;\n  z-index: 1;\n  display: flex;\n  flex-direction: row-reverse;\n  justify-content: center;\n  --keystroke: white;\n  --on-keystroke: #555;\n  --keystroke-border: #f7f7f7;\n}\n@media (prefers-color-scheme: dark) {\n  body:not([theme='light']) #mathlive-keystroke-caption-panel {\n    --keystroke: hsl(var(--_hue), 50%, 30%);\n    --on-keystroke: hsl(0, 0%, 98%);\n    --keystroke-border: hsl(var(--_hue), 50%, 25%);\n  }\n}\nbody[theme='dark'] #mathlive-keystroke-caption-panel {\n  --keystroke: hsl(var(--_hue), 50%, 30%);\n  --on-keystroke: hsl(0, 0%, 98%);\n  --keystroke-border: hsl(var(--_hue), 50%, 25%);\n}\n#mathlive-keystroke-caption-panel > span {\n  min-width: 14px;\n  /*height: 8px;*/\n  margin: 0 8px 0 0;\n  padding: 4px;\n  background-color: var(--keystroke);\n  color: var(--on-keystroke);\n  fill: currentColor;\n  font-family: system-ui, -apple-system, BlinkMacSystemFont, 'Segoe UI', 'Roboto', 'Oxygen', 'Ubuntu', 'Cantarell', 'Fira Sans', 'Droid Sans', 'Helvetica Neue', sans-serif;\n  font-size: 1em;\n  border-radius: 6px;\n  border: 2px solid var(--keystroke-border);\n  /*box-shadow: 0 7px 14px rgba(0,0,0,0.25), 0 5px 5px rgba(0,0,0,0.22);*/\n}\n";break;case"virtual-keyboard":t=".ML__keyboard {\n  --_keyboard-height: 0;\n  --_keyboard-zindex: var(--keyboard-zindex, 105);\n  --_accent-color: var(--keyboard-accent-color, #0c75d8);\n  --_background: var(--keyboard-background, #cacfd7);\n  --_border: var(--keyboard-border, #ddd);\n  --_padding-horizontal: var(--keyboard-padding-horizontal, 0px);\n  --_padding-top: var(--keyboard-padding-top, 5px);\n  --_padding-bottom: var(--keyboard-padding-bottom, 0px);\n  --_row-padding-left: var(--keyboard-row-padding-left, 0px);\n  --_row-padding-right: var(--keyboard-row-padding-right, 0px);\n  --_toolbar-text: var(--keyboard-toolbar-text, #2c2e2f);\n  --_toolbar-text-active: var(--keyboard-toolbar-text-active, var(--_accent-color));\n  --_toolbar-background: var(--keyboard-toolbar-background, transparent);\n  --_toolbar-background-hover: var(--keyboard-toolbar-background-hover, #eee);\n  --_toolbar-background-selected: var(--keyboard-toolbar-background-selected, transparent);\n  --_toolbar-font-size: var(--keyboard-toolbar-font-size, '135%');\n  --_horizontal-rule: var(--keyboard-horizontal-rule, 1px solid #fff);\n  --_keycap-background: var(--keycap-background, white);\n  --_keycap-background-hover: var(--keycap-background-hover, #f5f5f7);\n  --_keycap-background-active: var(--keycap-background-active, var(--_accent-color));\n  --_keycap-background-pressed: var(--keycap-background-pressed, var(--_accent-color));\n  --_keycap-border: var(--keycap-border, #e5e6e9);\n  --_keycap-border-bottom: var(--keycap-border-bottom, #8d8f92);\n  --_keycap-text: var(--keycap-text, #000);\n  --_keycap-text-active: var(--keycap-text-active, #fff);\n  --_keycap-text-hover: var(--keycap-text-hover, var(--_keycap-text));\n  --_keycap-text-pressed: var(--keycap-text-pressed, #fff);\n  --_keycap-shift-text: var(--keycap-shift-text, var(--_accent-color));\n  --_keycap-primary-background: var(--keycap-primary-background, var(--_accent-color));\n  --_keycap-primary-text: var(--keycap-primary-text, #ddd);\n  --_keycap-primary-background-hover: var(--keycap-primary-background-hover, #0d80f2);\n  --_keycap-secondary-background: var(--keycap-secondary-background, #a0a9b8);\n  --_keycap-secondary-background-hover: var(--keycap-secondary-background-hover, #7d8795);\n  --_keycap-secondary-text: var(--keycap-secondary-text, #060707);\n  --_keycap-secondary-border: var(--keycap-secondary-border, #c5c9d0);\n  --_keycap-secondary-border-bottom: var(--keycap-secondary-border-bottom, #989da6);\n  --_keycap-height: var(--keycap-height, 60px);\n  /* Keycap width (incl. margin) */\n  --_keycap-max-width: var(--keycap-max-width, 100px);\n  --_keycap-gap: var(--keycap-gap, 8px);\n  --_keycap-font-size: var(--keycap-font-size, clamp(16px, 4cqw, 24px));\n  --_keycap-small-font-size: var(--keycap-small-font-size, calc(var(--keycap-font-size) * 0.8));\n  --_keycap-extra-small-font-size: var(--keycap-extra-small-font-size, calc(var(--keycap-font-size) / 1.42));\n  --_variant-panel-background: var(--variant-panel-background, #fff);\n  --_variant-keycap-text: var(--variant-keycap-text, var(--_keycap-text));\n  --_variant-keycap-text-active: var(--variant-keycap-text-active, var(--_keycap-text-active));\n  --_variant-keycap-background-active: var(--variant-keycap-background-active, var(--_accent-color));\n  --_variant-keycap-length: var(--variant-keycap-length, 70px);\n  --_variant-keycap-font-size: var(--variant-keycap-font-size, 30px);\n  --_variant-keycap-aside-font-size: var(--variant-keycap-aside-font-size, 12px);\n  --_keycap-shift-font-size: var(--keycap-shift-font-size, 16px);\n  --_keycap-shift-color: var(--keycap-shift-color, var(--_accent-color));\n  --_box-placeholder-color: var(--box-placeholder-color, var(--_accent-color));\n  --_box-placeholder-pressed-color: var(--box-placeholder-pressed-color, var(--keycap-text-pressed));\n}\n.is-math-mode .MLK__rows .if-text-mode,\n.is-text-mode .MLK__rows .if-math-mode {\n  display: none;\n}\n.if-can-undo,\n.if-can-redo,\n.if-can-copy,\n.if-can-cut,\n.if-can-paste {\n  opacity: 0.4;\n  pointer-events: none;\n}\n.can-undo .if-can-undo,\n.can-redo .if-can-redo,\n.can-copy .if-can-copy,\n.can-cut .if-can-cut,\n.can-paste .if-can-paste {\n  opacity: 1;\n  pointer-events: all;\n}\nbody > .ML__keyboard {\n  position: fixed;\n  --_padding-bottom: calc(var(--keyboard-padding-bottom, 0px) + env(safe-area-inset-bottom, 0));\n}\nbody > .ML__keyboard.is-visible > .MLK__backdrop {\n  box-shadow: 0 -5px 6px rgba(0, 0, 0, 0.08);\n  border-top: 1px solid var(--_border);\n}\nbody > .ML__keyboard.backdrop-is-transparent.is-visible > .MLK__backdrop {\n  box-shadow: none;\n  border: none;\n}\nbody > .ML__keyboard.is-visible.animate > .MLK__backdrop {\n  transition: 0.28s cubic-bezier(0, 0, 0.2, 1);\n  transition-property: transform, opacity;\n  transition-timing-function: cubic-bezier(0.4, 0, 1, 1);\n}\n.ML__keyboard {\n  position: relative;\n  overflow: hidden;\n  top: 0;\n  left: 0;\n  height: 100%;\n  width: 100%;\n  z-index: var(--_keyboard-zindex);\n  box-sizing: border-box;\n  outline: none;\n  border: none;\n  margin: 0;\n  padding: 0;\n  line-height: 1;\n  overflow-wrap: unset;\n  text-align: left;\n  vertical-align: baseline;\n  cursor: auto;\n  white-space: pre;\n  box-shadow: none;\n  opacity: 1;\n  transform: none;\n  pointer-events: none;\n}\n.ML__keyboard :where(div) {\n  box-sizing: border-box;\n  outline: none;\n  border: none;\n  margin: 0;\n  padding: 0;\n  line-height: 1;\n  overflow-wrap: unset;\n  text-align: left;\n  vertical-align: baseline;\n  cursor: auto;\n  white-space: pre;\n  box-shadow: none;\n  transform: none;\n}\n.MLK__backdrop {\n  position: absolute;\n  bottom: calc(-1 * var(--_keyboard-height));\n  width: 100%;\n  height: var(--_keyboard-height);\n  box-sizing: border-box;\n  padding-top: var(--_padding-top);\n  padding-bottom: var(--_padding-bottom);\n  padding-left: var(--_padding-horizontal);\n  padding-right: var(--_padding-horizontal);\n  opacity: 0;\n  visibility: hidden;\n  transform: translate(0, 0);\n  background: var(--_background);\n}\n.backdrop-is-transparent .MLK__backdrop {\n  background: transparent;\n}\n/* If a custom layout has a custom container/backdrop\n  (backdrop-is-transparent), make sure to let pointer event go through. */\n.backdrop-is-transparent .MLK__plate {\n  background: transparent;\n  pointer-events: none;\n}\n/* If a custom layout has a custom container/backdrop, make sure to \n   allow pointer events on it. */\n.backdrop-is-transparent .MLK__layer > div > div {\n  pointer-events: all;\n}\n.ML__keyboard.is-visible > .MLK__backdrop {\n  transform: translate(0, calc(-1 * var(--_keyboard-height)));\n  opacity: 1;\n  visibility: visible;\n}\n.caps-lock-indicator {\n  display: none;\n  width: 8px;\n  height: 8px;\n  background: #0cbc0c;\n  box-shadow: inset 0 0 4px 0 #13ca13, 0 0 4px 0 #a9ef48;\n  border-radius: 8px;\n  right: 8px;\n  top: 8px;\n  position: absolute;\n}\n.ML__keyboard.is-caps-lock .caps-lock-indicator {\n  display: block;\n}\n.ML__keyboard.is-caps-lock .shift {\n  background: var(--_keycap-background-active);\n  color: var(--_keycap-text-active);\n}\n.MLK__plate {\n  position: absolute;\n  top: var(--_padding-top);\n  left: var(--_padding-horizontal);\n  width: calc(100% - 2 * var(--_padding-horizontal));\n  margin: 0;\n  padding: 0;\n  box-sizing: border-box;\n  container-type: inline-size;\n  touch-action: none;\n  -webkit-user-select: none;\n  user-select: none;\n  pointer-events: all;\n  font-family: system-ui, -apple-system, BlinkMacSystemFont, 'Segoe UI', 'Roboto', 'Oxygen', 'Ubuntu', 'Cantarell', 'Fira Sans', 'Droid Sans', 'Helvetica Neue', sans-serif;\n  font-size: 16px;\n  /* Size of toolbar labels */\n  font-weight: 400;\n  text-shadow: none;\n}\n.ML__box-placeholder {\n  color: var(--_box-placeholder-color);\n}\n.MLK__tex {\n  font-family: KaTeX_Main, KaTeX_Math, 'Cambria Math', 'Asana Math', OpenSymbol, Symbola, STIX, Times, serif !important;\n}\n.MLK__tex-math {\n  font-family: KaTeX_Math, KaTeX_Main, 'Cambria Math', 'Asana Math', OpenSymbol, Symbola, STIX, Times, serif !important;\n  font-style: italic;\n}\n.MLK__layer {\n  display: none;\n  outline: none;\n}\n.MLK__layer.is-visible {\n  display: flex;\n  flex-flow: column;\n}\n/* Keyboard layouts are made or rows of keys... */\n.MLK__rows {\n  --_keycap-width: min(var(--_keycap-max-width), 10cqw);\n  display: flex;\n  flex-flow: column;\n  align-items: center;\n  border-collapse: separate;\n  clear: both;\n  border: 0;\n  margin: 0;\n  margin-bottom: var(--_keycap-gap);\n  gap: var(--_keycap-gap);\n  /* If the styling include, e.g., some shadows, they will be\n  cut off by the overflow. In that case, set the padding to \n  compensate. */\n  padding-left: var(--_row-padding-left);\n  padding-right: var(--_row-padding-right);\n  overflow: visible;\n  touch-action: none;\n}\n.MLK__rows > .MLK__row {\n  display: flex;\n  flex-flow: row;\n  justify-content: center;\n  width: 100%;\n  gap: var(--_keycap-gap);\n  margin: 0;\n  padding: 0;\n  /* For the alignment of the text on some modifiers (e.g. shift) */\n  /* Extra spacing between two adjacent keys */\n}\n.MLK__rows > .MLK__row .tex {\n  font-family: KaTeX_Math, KaTeX_Main, 'Cambria Math', 'Asana Math', OpenSymbol, Symbola, STIX, Times, serif !important;\n}\n.MLK__rows > .MLK__row .tex-math {\n  font-family: KaTeX_Math, 'Cambria Math', 'Asana Math', OpenSymbol, Symbola, STIX, Times, serif !important;\n}\n.MLK__rows > .MLK__row .big-op {\n  font-size: calc(1.25 * var(--_keycap-font-size));\n}\n.MLK__rows > .MLK__row .small {\n  font-size: var(--_keycap-small-font-size);\n}\n.MLK__rows > .MLK__row .bottom {\n  justify-content: flex-end;\n}\n.MLK__rows > .MLK__row .left {\n  align-items: flex-start;\n  padding-left: 12px;\n}\n.MLK__rows > .MLK__row .right {\n  align-items: flex-end;\n  padding-right: 12px;\n}\n.MLK__rows > .MLK__row .w0 {\n  width: 0;\n}\n.MLK__rows > .MLK__row .w5 {\n  width: calc(0.5 * var(--_keycap-width) - var(--_keycap-gap));\n}\n.MLK__rows > .MLK__row .w15 {\n  width: calc(1.5 * var(--_keycap-width) - var(--_keycap-gap));\n}\n.MLK__rows > .MLK__row .w20 {\n  width: calc(2 * var(--_keycap-width) - var(--_keycap-gap));\n}\n.MLK__rows > .MLK__row .w40 {\n  width: calc(4 * var(--_keycap-width) - var(--_keycap-gap));\n}\n.MLK__rows > .MLK__row .w50 {\n  width: calc(5 * var(--_keycap-width) - var(--_keycap-gap));\n}\n.MLK__rows > .MLK__row .MLK__keycap.w50 {\n  font-size: 80%;\n  padding-top: 10px;\n  font-weight: 100;\n}\n.MLK__rows > .MLK__row .separator {\n  background: transparent;\n  border: none;\n  pointer-events: none;\n}\n.MLK__rows > .MLK__row .horizontal-rule {\n  height: 6px;\n  margin-top: 3px;\n  margin-bottom: 0;\n  width: 100%;\n  border-radius: 0;\n  border-top: var(--_horizontal-rule);\n}\n.MLK__rows > .MLK__row .ghost {\n  background: var(--_toolbar-background);\n  border: none;\n  color: var(--_toolbar-text);\n}\n.MLK__rows > .MLK__row .ghost:hover {\n  background: var(--_toolbar-background-hover);\n}\n.MLK__rows > .MLK__row .bigfnbutton {\n  font-size: var(--_keycap-extra-small-font-size);\n}\n.MLK__rows > .MLK__row .shift,\n.MLK__rows > .MLK__row .action {\n  color: var(--_keycap-secondary-text);\n  background: var(--_keycap-secondary-background);\n  border-color: var(--_keycap-secondary-border);\n  border-bottom-color: var(--_keycap-secondary-border-bottom);\n  line-height: 0.8;\n  font-size: min(1rem, var(--_keycap-small-font-size));\n  font-weight: 600;\n  padding: 8px 12px 8px 12px;\n}\n.MLK__rows > .MLK__row .shift:hover,\n.MLK__rows > .MLK__row .action:hover {\n  background: var(--_keycap-secondary-background-hover);\n}\n.MLK__rows > .MLK__row .action.primary {\n  background: var(--_keycap-primary-background);\n  color: var(--_keycap-primary-text);\n}\n.MLK__rows > .MLK__row .action.primary:hover {\n  background: var(--_keycap-primary-background-hover);\n  color: var(--_keycap-primary-text);\n}\n.MLK__rows > .MLK__row .shift.selected,\n.MLK__rows > .MLK__row .action.selected {\n  color: var(--_toolbar-text-active);\n}\n.MLK__rows > .MLK__row .shift.selected.is-pressed,\n.MLK__rows > .MLK__row .action.selected.is-pressed,\n.MLK__rows > .MLK__row .shift.selected.is-active,\n.MLK__rows > .MLK__row .action.selected.is-active {\n  color: white;\n}\n.MLK__rows > .MLK__row .warning {\n  background: #cd0030;\n  color: white;\n}\n.MLK__rows > .MLK__row .warning svg.svg-glyph {\n  width: 24px;\n  height: 24px;\n  min-height: 24px;\n}\n/** A regular keycap\n * Use the :where() pseudo-class to give it a very low specifity, \n * so that it can be overriden by custom style.\n */\n:where(.MLK__rows > .MLK__row  div) {\n  display: flex;\n  flex-flow: column;\n  align-items: center;\n  justify-content: space-evenly;\n  width: calc(var(--_keycap-width) - var(--_keycap-gap));\n  height: var(--_keycap-height);\n  box-sizing: border-box;\n  padding: 0;\n  vertical-align: top;\n  text-align: center;\n  float: left;\n  color: var(--_keycap-text);\n  fill: currentColor;\n  font-size: var(--_keycap-font-size);\n  background: var(--_keycap-background);\n  border: 1px solid var(--_keycap-border);\n  border-bottom-color: var(--_keycap-border-bottom);\n  border-radius: 6px;\n  cursor: pointer;\n  touch-action: none;\n  /* Keys with a variants panel */\n  position: relative;\n  overflow: hidden;\n  -webkit-user-select: none;\n  user-select: none;\n  -webkit-tap-highlight-color: transparent;\n}\n:where(.MLK__rows > .MLK__row  div):hover {\n  overflow: visible;\n  background: var(--_keycap-background-hover);\n}\n:where(.MLK__rows > .MLK__row  div) .ML__latex {\n  pointer-events: none;\n  touch-action: none;\n}\n:where(.MLK__rows > .MLK__row  div) svg.svg-glyph {\n  margin: 8px 0;\n  width: 20px;\n  height: 20px;\n  min-height: 20px;\n}\n:where(.MLK__rows > .MLK__row  div) svg.svg-glyph-lg {\n  margin: 8px 0;\n  width: 24px;\n  height: 24px;\n  min-height: 24px;\n}\n:where(.MLK__rows > .MLK__row  div).MLK__tex-math {\n  font-size: 25px;\n}\n:where(.MLK__rows > .MLK__row  div).is-pressed {\n  background: var(--_keycap-background-pressed);\n  color: var(--_keycap-text-pressed);\n  --_box-placeholder-color: var(--_box-placeholder-pressed-color);\n}\n:where(.MLK__rows > .MLK__row  div).MLK__keycap.is-active,\n:where(.MLK__rows > .MLK__row  div).action.is-active,\n:where(.MLK__rows > .MLK__row  div).MLK__keycap.is-pressed,\n:where(.MLK__rows > .MLK__row  div).action.is-pressed {\n  z-index: calc(var(--_keyboard-zindex) - 5);\n}\n:where(.MLK__rows > .MLK__row  div).MLK__keycap.is-active aside,\n:where(.MLK__rows > .MLK__row  div).action.is-active aside,\n:where(.MLK__rows > .MLK__row  div).MLK__keycap.is-pressed aside,\n:where(.MLK__rows > .MLK__row  div).action.is-pressed aside {\n  display: none;\n}\n:where(.MLK__rows > .MLK__row  div).MLK__keycap.is-active .MLK__shift,\n:where(.MLK__rows > .MLK__row  div).action.is-active .MLK__shift,\n:where(.MLK__rows > .MLK__row  div).MLK__keycap.is-pressed .MLK__shift,\n:where(.MLK__rows > .MLK__row  div).action.is-pressed .MLK__shift {\n  display: none;\n}\n:where(.MLK__rows > .MLK__row  div).shift.is-pressed,\n:where(.MLK__rows > .MLK__row  div).MLK__keycap.is-pressed,\n:where(.MLK__rows > .MLK__row  div).action.is-pressed {\n  background: var(--_keycap-background-pressed);\n  color: var(--_keycap-text-pressed);\n}\n:where(.MLK__rows > .MLK__row  div).shift.is-active,\n:where(.MLK__rows > .MLK__row  div).MLK__keycap.is-active,\n:where(.MLK__rows > .MLK__row  div).action.is-active {\n  background: var(--_keycap-background-active);\n  color: var(--_keycap-text-active);\n  --_box-placeholder-color: var(--_box-placeholder-pressed-color);\n}\n:where(.MLK__rows > .MLK__row  div) small {\n  color: var(--_keycap-secondary-text);\n}\n:where(.MLK__rows > .MLK__row  div) aside {\n  font-family: system-ui, -apple-system, BlinkMacSystemFont, 'Segoe UI', 'Roboto', 'Oxygen', 'Ubuntu', 'Cantarell', 'Fira Sans', 'Droid Sans', 'Helvetica Neue', sans-serif;\n  font-size: 10px;\n  line-height: 10px;\n  color: var(--_keycap-secondary-text);\n}\n/* Add an attribute 'data-tooltip' to display a tooltip on hover.\nNote there are a different set of tooltip rules for the keyboard toggle\n(it's in a different CSS tree) */\n.ML__keyboard [data-tooltip] {\n  position: relative;\n}\n.ML__keyboard [data-tooltip]::after {\n  position: absolute;\n  display: inline-table;\n  content: attr(data-tooltip);\n  top: inherit;\n  bottom: 100%;\n  width: max-content;\n  max-width: 200px;\n  padding: 8px 8px;\n  background: #616161;\n  color: #fff;\n  text-align: center;\n  z-index: 2;\n  box-shadow: 0 2px 2px 0 rgba(0, 0, 0, 0.14), 0 1px 5px 0 rgba(0, 0, 0, 0.12), 0 3px 1px -2px rgba(0, 0, 0, 0.2);\n  border-radius: 2px;\n  font-family: system-ui, -apple-system, BlinkMacSystemFont, 'Segoe UI', 'Roboto', 'Oxygen', 'Ubuntu', 'Cantarell', 'Fira Sans', 'Droid Sans', 'Helvetica Neue', sans-serif;\n  font-weight: 400;\n  font-size: 12px;\n  transition: all 0.15s cubic-bezier(0.4, 0, 1, 1) 1s;\n  opacity: 0;\n  transform: scale(0.5);\n}\n.ML__keyboard [data-tooltip]:hover {\n  position: relative;\n}\n.ML__keyboard [data-tooltip]:hover::after {\n  opacity: 1;\n  transform: scale(1);\n}\n.MLK__toolbar {\n  align-self: center;\n  display: flex;\n  flex-flow: row;\n  justify-content: space-between;\n  width: 100%;\n  max-width: 996px;\n  min-height: 32px;\n  /* Icons for undo/redo, etc. */\n}\n.MLK__toolbar svg {\n  height: 20px;\n  width: 20px;\n}\n.MLK__toolbar > .left {\n  position: relative;\n  display: flex;\n  justify-content: flex-start;\n  flex-flow: row;\n}\n.MLK__toolbar > .right {\n  display: flex;\n  justify-content: flex-end;\n  flex-flow: row;\n}\n.MLK__toolbar > div > div {\n  /* \"button\" in the toolbar */\n  display: flex;\n  align-items: center;\n  justify-content: center;\n  color: var(--_toolbar-text);\n  fill: currentColor;\n  background: var(--_toolbar-background);\n  font-size: var(--_toolbar-font-size);\n  padding: 4px 15px;\n  cursor: pointer;\n  width: max-content;\n  min-width: 42px;\n  min-height: 34px;\n  border: none;\n  padding-left: 10px;\n  padding-right: 10px;\n  padding-bottom: 8px;\n  padding-top: 8px;\n  margin-top: 0;\n  margin-bottom: 4px;\n  margin-left: 4px;\n  margin-right: 4px;\n  border-radius: 8px;\n  box-shadow: none;\n  border-bottom: 2px solid transparent;\n}\n.MLK__toolbar > div > div:not(.disabled):not(.selected):hover {\n  background: var(--_toolbar-background-hover);\n}\n.MLK__toolbar > div > div.disabled svg,\n.MLK__toolbar > div > div.disabled:hover svg,\n.MLK__toolbar > div > div.disabled.is-pressed svg {\n  color: var(--_toolbar-text);\n  opacity: 0.2;\n}\n.MLK__toolbar > div > div:hover,\n.MLK__toolbar > div > div:active,\n.MLK__toolbar > div > div.is-pressed,\n.MLK__toolbar > div > div.is-active {\n  color: var(--_toolbar-text-active);\n}\n.MLK__toolbar > div > div.selected {\n  color: var(--_toolbar-text-active);\n  background: var(--_toolbar-background-selected);\n  border-radius: 0;\n  border-bottom-color: var(--_toolbar-text-active);\n  padding-bottom: 4px;\n  margin-bottom: 8px;\n}\n/* This is the element that displays variants on press+hold */\n.MLK__variant-panel {\n  visibility: hidden;\n  position: fixed;\n  display: flex;\n  flex-flow: row wrap-reverse;\n  justify-content: center;\n  align-content: center;\n  margin: 0;\n  padding: 0;\n  bottom: auto;\n  top: 0;\n  box-sizing: content-box;\n  transform: none;\n  z-index: calc(var(--_keyboard-zindex) + 1);\n  touch-action: none;\n  max-width: 350px;\n  background: var(--_variant-panel-background);\n  text-align: center;\n  border-radius: 6px;\n  padding: 6px;\n  box-shadow: 0 14px 28px rgba(0, 0, 0, 0.25), 0 10px 10px rgba(0, 0, 0, 0.22);\n  transition: none;\n}\n.MLK__variant-panel.is-visible {\n  visibility: visible;\n}\n.MLK__variant-panel.compact {\n  --_variant-keycap-length: var(--variant-keycap-length, 50px);\n  --_variant-keycap-font-size: var(--variant-keycap-font-size, 24px);\n  --_variant-keycap-aside-font-size: var(--variant-keycap-aside-font-size, 10px);\n}\n.MLK__variant-panel .item {\n  display: flex;\n  flex-flow: column;\n  align-items: center;\n  justify-content: center;\n  font-size: var(--_variant-keycap-font-size);\n  height: var(--_variant-keycap-length);\n  width: var(--_variant-keycap-length);\n  margin: 0;\n  box-sizing: border-box;\n  border-radius: 5px;\n  border: 1px solid transparent;\n  background: transparent;\n  pointer-events: all;\n  cursor: pointer;\n  color: var(--_variant-keycap-text);\n  fill: currentColor;\n}\n@media (max-height: 412px) {\n  .MLK__variant-panel .item {\n    --_variant-keycap-font-size: var(--variant-keycap-font-size, 24px);\n    --_variant-keycap-length: var(--variant-keycap-length, 50px);\n  }\n}\n.MLK__variant-panel .item .ML__latex {\n  pointer-events: none;\n}\n.MLK__variant-panel .item.is-active {\n  background: var(--_variant-keycap-background-active);\n  color: var(--_variant-keycap-text-active);\n}\n.MLK__variant-panel .item.is-pressed {\n  background: var(--_variant-keycap-background-pressed);\n  color: var(--_variant-keycap-text-pressed);\n}\n.MLK__variant-panel .item.small {\n  font-size: var(--_keycap-small-font-size);\n}\n.MLK__variant-panel .item.swatch-button {\n  box-sizing: border-box;\n  background: #fbfbfb;\n}\n.MLK__variant-panel .item.swatch-button > span {\n  display: inline-block;\n  margin: 6px;\n  width: calc(100% - 12px);\n  height: calc(100% - 12px);\n  border-radius: 50%;\n}\n.MLK__variant-panel .item.swatch-button:hover {\n  background: #f0f0f0;\n}\n.MLK__variant-panel .item.swatch-button:hover > span {\n  border-radius: 2px;\n}\n.MLK__variant-panel .item.box > div,\n.MLK__variant-panel .item.box > span {\n  border: 1px dashed rgba(0, 0, 0, 0.24);\n}\n.MLK__variant-panel .item .warning {\n  min-height: 60px;\n  min-width: 60px;\n  background: #cd0030;\n  color: white;\n  padding: 5px;\n  display: flex;\n  align-items: center;\n  justify-content: center;\n  border-radius: 5px;\n}\n.MLK__variant-panel .item .warning.is-pressed,\n.MLK__variant-panel .item .warning.is-active {\n  background: red;\n}\n.MLK__variant-panel .item .warning svg.svg-glyph {\n  width: 50px;\n  height: 50px;\n}\n.MLK__variant-panel .item aside {\n  font-size: var(--_variant-keycap-aside-font-size);\n  line-height: 12px;\n  opacity: 0.78;\n  padding-top: 2px;\n}\n.MLK__keycap {\n  position: relative;\n}\n.MLK__shift {\n  display: block;\n  position: absolute;\n  right: 4px;\n  top: 4px;\n  font-size: var(--_keycap-shift-font-size);\n  color: var(--_keycap-shift-color);\n}\n.hide-shift .MLK__shift {\n  display: none;\n}\n@media (max-width: 414px) {\n  .MLK__variant-panel {\n    max-width: 350px;\n    --_variant-keycap-font-size: var(--variant-keycap-font-size, 24px);\n    --_variant-keycap-length: var(--variant-keycap-length, 50px);\n  }\n}\n/* @xs breakpoint: iPhone 5 */\n@container (max-width: 414px) {\n  .MLK__rows {\n    --_keycap-gap: max(var(--_keycap-gap, 2px), 2px);\n    --_keycap-height: max(var(--_keycap-height), 42px);\n    --_keycap-width: min(min(var(--_keycap-max-width), 10cqw), 62px);\n  }\n  .MLK__toolbar > div > div {\n    font-size: 100%;\n    margin-left: 2px;\n    margin-right: 2px;\n  }\n  .MLK__rows .shift,\n  .MLK__rows .action {\n    font-size: 65%;\n  }\n  .MLK__rows .warning svg.svg-glyph {\n    width: 14px;\n    height: 14px;\n    min-height: 14px;\n  }\n}\n@container (max-width: 744px) {\n  .MLK__rows {\n    --_keycap-gap: max(var(--keycap-gap, 2px), 2px);\n    --_keycap-height: max(var(--keycap-height, 52px), 52px);\n    --_keycap-width: min(min(var(--_keycap-max-width), 10cqw), 62px);\n  }\n  .MLK__toolbar > div > div {\n    padding-left: 0;\n    padding-right: 0;\n  }\n  .MLK__tooltip::after {\n    padding: 8px 16px;\n    font-size: 16px;\n  }\n  .MLK__rows > .MLK__row > div.fnbutton {\n    font-size: 16px;\n  }\n  .MLK__rows > .MLK__row > div.bigfnbutton {\n    font-size: calc(var(--_keycap-extra-small-font-size) / 1.55);\n  }\n  .MLK__rows > .MLK__row > div.small {\n    font-size: 13px;\n  }\n  .MLK__rows > .MLK__row > div > aside {\n    display: none;\n  }\n  .MLK__shift {\n    display: none;\n  }\n}\n/* Medium breakpoint: larger phones */\n@container (max-width: 768px) {\n  .MLK__rows {\n    --_keycap-height: max(var(--keycap-height, 42px), 42px);\n  }\n  .MLK__rows > .MLK__row > div > small {\n    font-size: 14px;\n  }\n}\n@media (max-height: 768px) {\n  .MLK__rows {\n    --_keycap-height: max(var(--keycap-height, 42px), 42px);\n  }\n  .MLK__rows > .MLK__row > div > small {\n    font-size: 14px;\n  }\n}\n@container (max-width: 1444px) {\n  .MLK__rows .if-wide {\n    display: none;\n  }\n}\n@media (prefers-color-scheme: dark) {\n  .ML__keyboard {\n    --_accent-color: var(--keyboard-accent-color, #0b5c9c);\n    --_background: var(--keyboard-background, #151515);\n    --_border: var(--keyboard-border, transparent);\n    --_toolbar-text: var(--keyboard-toolbar-text, #e3e4e8);\n    --_toolbar-background-hover: var(--keyboard-toolbar-background-hover, #303030);\n    --keyboard-toolbar-background-hover: #303030;\n    --_horizontal-rule: var(--keyboard-horizontal-rule, 1px solid #303030);\n    --_keycap-background: var(--keycap-background, #1f2022);\n    --_keycap-background-hover: var(--keycap-background-hover, #2f3032);\n    --_keycap-border: var(--_keycap-border, transparent);\n    --_keycap-border-bottom: var(--_keycap-border-bottom, transparent);\n    --_keycap-text: var(--keycap-text, #e3e4e8);\n    --_keycap-secondary-background: var(--keycap-secondary-background, #3d4144);\n    --_keycap-secondary-background-hover: var(--keycap-secondary-background-hover, #4d5154);\n    --_keycap-secondary-text: var(--keycap-secondary-text, #e7ebee);\n    --keycap-secondary-border: transparent;\n    --keycap-secondary-border-bottom: transparent;\n    --_keycap-secondary-border: var(--keycap-secondary-border, transparent);\n    --_keycap-secondary-border-bottom: var(--keycap-secondary-border-bottom, transparent);\n    --_variant-panel-background: var(--variant-panel-background, #303030);\n    --_variant-keycap-text-active: var(--variant-keycap-text-active, #fff);\n  }\n}\n/* Same as the media query, but with a class */\n[theme='dark'] .ML__keyboard {\n  --_accent-color: var(--keyboard-accent-color, #0b5c9c);\n  --_background: var(--keyboard-background, #151515);\n  --_border: var(--keyboard-border, transparent);\n  --_toolbar-text: var(--keyboard-toolbar-text, #e3e4e8);\n  --_toolbar-background-hover: var(--keyboard-toolbar-background-hover, #303030);\n  --keyboard-toolbar-background-hover: #303030;\n  --_horizontal-rule: var(--keyboard-horizontal-rule, 1px solid #303030);\n  --_keycap-background: var(--keycap-background, #1f2022);\n  --_keycap-background-hover: var(--keycap-background-hover, #2f3032);\n  --_keycap-border: var(--_keycap-border, transparent);\n  --_keycap-border-bottom: var(--_keycap-border-bottom, transparent);\n  --_keycap-text: var(--keycap-text, #e3e4e8);\n  --_keycap-secondary-background: var(--keycap-secondary-background, #3d4144);\n  --_keycap-secondary-background-hover: var(--keycap-secondary-background-hover, #4d5154);\n  --_keycap-secondary-text: var(--keycap-secondary-text, #e7ebee);\n  --keycap-secondary-border: transparent;\n  --keycap-secondary-border-bottom: transparent;\n  --_keycap-secondary-border: var(--keycap-secondary-border, transparent);\n  --_keycap-secondary-border-bottom: var(--keycap-secondary-border-bottom, transparent);\n  --_variant-panel-background: var(--variant-panel-background, #303030);\n  --_variant-keycap-text-active: var(--variant-keycap-text-active, #fff);\n}\n[theme='light'] .ML__keyboard {\n  --_accent-color: var(--keyboard-accent-color, #0c75d8);\n  --_background: var(--keyboard-background, #cacfd7);\n  --_border: var(--keyboard-border, #ddd);\n  --_toolbar-text: var(--keyboard-toolbar-text, #2c2e2f);\n  --_toolbar-background: var(--keyboard-toolbar-background, transparent);\n  --_toolbar-background-hover: var(--keyboard-toolbar-background-hover, #eee);\n  --_toolbar-background-selected: var(--keyboard-toolbar-background-selected, transparent);\n  --_horizontal-rule: var(--keyboard-horizontal-rule, 1px solid #fff);\n  --_keycap-background: var(--keycap-background, white);\n  --_keycap-background-hover: var(--keycap-background-hover, #f5f5f7);\n  --_keycap-background-active: var(--keycap-background-active, var(--_accent-color));\n  --_keycap-background-pressed: var(--keycap-background-pressed, var(--_accent-color));\n  --_keycap-border: var(--_keycap-border, #e5e6e9);\n  --_keycap-border-bottom: var(--_keycap-border-bottom, #8d8f92);\n  --_keycap-text: var(--keycap-text, #000);\n  --_keycap-text-active: var(--keycap-text-active, #fff);\n  --_keycap-text-hover: var(--keycap-text-hover, var(--_keycap-text));\n  --_keycap-text-pressed: var(--keycap-text-pressed, #fff);\n  --_keycap-shift-text: var(--keycap-shift-text, var(--_accent-color));\n  --_keycap-secondary-background: var(--keycap-secondary-background, #a0a9b8);\n  --_keycap-secondary-background-hover: var(--keycap-secondary-background-hover, #7d8795);\n  --_keycap-secondary-text: var(--keycap-secondary-text, #060707);\n  --_keycap-secondary-border: var(--keycap-secondary-border, #c5c9d0);\n  --_keycap-secondary-border-bottom: var(--keycap-secondary-border-bottom, #989da6);\n  --_variant-panel-background: var(--variant-panel-background, #fff);\n  --_variant-keycap-text: var(--variant-keycap-textvar, var(--_keycap-text));\n  --_variant-keycap-text-active: var(--variant-keycap-text-active, var(--_keycap-text-active));\n  --_variant-keycap-background-active: var(--variant-keycap-background-active, var(--_accent-color));\n}\n";break;case"ui":t=":host {\n  --primary-color: #5898ff;\n  --primary-color-dimmed: #c0c0f0;\n  --primary-color-dark: var(--blue-500);\n  --primary-color-light: var(--blue-100);\n  --primary-color-reverse: #ffffff;\n  --secondary-color: #ff8a65;\n  --secondary-color-dimmed: #f0d5c5;\n  --secondary-color-dark: var(--orange-500);\n  --secondary-color-light: var(--orange-100);\n  --secondary-color-reverse: #ffffff;\n  --link-color: #5898ff;\n  --link-color-dimmed: #c5c5c5;\n  --link-color-dark: #121212;\n  --link-color-light: #e2e2e2;\n  --link-color-reverse: #ffffff;\n  --semantic-blue: var(--blue-700);\n  --semantic-red: var(--red-400);\n  --semantic-orange: var(--orange-400);\n  --semantic-green: var(--green-700);\n  --neutral-100: #f5f5f5;\n  --neutral-200: #eeeeee;\n  --neutral-300: #e0e0e0;\n  --neutral-400: #bdbdbd;\n  --neutral-500: #9e9e9e;\n  --neutral-600: #757575;\n  --neutral-700: #616161;\n  --neutral-800: #424242;\n  --neutral-900: #212121;\n  --red-25: #fff5f5;\n  --red-50: #ffefee;\n  --red-100: #ffd6d5;\n  --red-200: #ffb3b4;\n  --red-300: #ff8f91;\n  --red-400: #ff6f71;\n  --red-500: #ff4f52;\n  --red-600: #f93f42;\n  --red-700: #e33539;\n  --red-800: #c92c30;\n  --red-900: #a71f23;\n  --orange-25: #fff7f2;\n  --orange-50: #fff8f2;\n  --orange-100: #ffebd9;\n  --orange-200: #ffdcb8;\n  --orange-300: #ffc694;\n  --orange-400: #ffa96d;\n  --orange-500: #ff8f3f;\n  --orange-600: #f57e33;\n  --orange-700: #d86e2a;\n  --orange-800: #b95e22;\n  --orange-900: #8f4217;\n  --brown-25: #fffaf5;\n  --brown-50: #f7f2e8;\n  --brown-100: #e7d8c9;\n  --brown-200: #d1b7a5;\n  --brown-300: #b78a76;\n  --brown-400: #9e604c;\n  --brown-500: #844027;\n  --brown-600: #733721;\n  --brown-700: #622c1d;\n  --brown-800: #4f2218;\n  --brown-900: #3f1913;\n  --yellow-25: #fffdf5;\n  --yellow-50: #fff9e7;\n  --yellow-100: #fff2c4;\n  --yellow-200: #ffe59e;\n  --yellow-300: #ffd875;\n  --yellow-400: #ffc53d;\n  --yellow-500: #ffb900;\n  --yellow-600: #e6a100;\n  --yellow-700: #bf8f00;\n  --yellow-800: #997d00;\n  --yellow-900: #7d6700;\n  --lime-25: #f9f9e8;\n  --lime-50: #f5f9e8;\n  --lime-100: #eaf2c6;\n  --lime-200: #d9e6a1;\n  --lime-300: #c5d377;\n  --lime-400: #b1c24c;\n  --lime-500: #92b215;\n  --lime-600: #7ea612;\n  --lime-700: #6d8f0f;\n  --lime-800: #5c6d0b;\n  --lime-900: #3f4a05;\n  --green-25: #f2f9e8;\n  --green-50: #e4f7e7;\n  --green-100: #bceac4;\n  --green-200: #90dd9d;\n  --green-300: #64cf75;\n  --green-400: #42c458;\n  --green-500: #21ba3a;\n  --green-600: #1da736;\n  --green-700: #199533;\n  --green-800: #137c2e;\n  --green-900: #0b5726;\n  --teal-25: #e8f9f9;\n  --teal-50: #e3f9f9;\n  --teal-100: #b9f1f1;\n  --teal-200: #8be7e7;\n  --teal-300: #5ddddd;\n  --teal-400: #3ad6d6;\n  --teal-500: #17cfcf;\n  --teal-600: #14b7b7;\n  --teal-700: #129f9f;\n  --teal-800: #0e7f7f;\n  --teal-900: #094e4e;\n  --cyan-25: #e8f9fd;\n  --cyan-50: #e3f4fd;\n  --cyan-100: #b8e5f9;\n  --cyan-200: #89d3f6;\n  --cyan-300: #5ac1f2;\n  --cyan-400: #36b4ef;\n  --cyan-500: #13a7ec;\n  --cyan-600: #1197d3;\n  --cyan-700: #1088ba;\n  --cyan-800: #0e7398;\n  --cyan-900: #0a5366;\n  --blue-25: #e8f5fd;\n  --blue-50: #e2f0fd;\n  --blue-100: #b6d9fb;\n  --blue-200: #86c0f9;\n  --blue-300: #56a6f6;\n  --blue-400: #3193f4;\n  --blue-500: #0d80f2;\n  --blue-600: #0c75d8;\n  --blue-700: #0c6abe;\n  --blue-800: #0b5c9c;\n  --blue-900: #094668;\n  --indigo-25: #f0e8fd;\n  --indigo-50: #ede7f9;\n  --indigo-100: #d1c2f0;\n  --indigo-200: #b399e6;\n  --indigo-300: #9470db;\n  --indigo-400: #7d52d4;\n  --indigo-500: #6300ff;\n  --indigo-600: #5b2fb6;\n  --indigo-700: #502b9f;\n  --indigo-800: #422581;\n  --indigo-900: #2c1d54;\n  --purple-25: #f9e8fd;\n  --purple-50: #f4e3fc;\n  --purple-100: #e3baf8;\n  --purple-200: #d18cf3;\n  --purple-300: #be5eee;\n  --purple-400: #b03cea;\n  --purple-500: #a219e6;\n  --purple-600: #9118cd;\n  --purple-700: #8116b3;\n  --purple-800: #6b1492;\n  --purple-900: #49115f;\n  --magenta-25: #fde8fd;\n  --magenta-50: #fde9f3;\n  --magenta-100: #f9c8e0;\n  --magenta-200: #f5a3cc;\n  --magenta-300: #f17eb8;\n  --magenta-400: #ee63a8;\n  --magenta-500: #eb4799;\n  --magenta-600: #d83f88;\n  --magenta-700: #c53777;\n  --magenta-800: #ac2d60;\n  --magenta-900: #861d3d;\n}\n@media (prefers-color-scheme: dark) {\n  :host {\n    --semantic-blue: var(--blue-700);\n    --semantic-red: var(--red-400);\n    --semantic-orange: var(--orange-400);\n    --semantic-green: var(--green-700);\n    --semantic-bg-blue: var(--blue-25);\n    --semantic-bg-red: var(--red-25);\n    --semantic-bg-orange: var(--orange-25);\n    --semantic-bg-green: var(--green-25);\n    --neutral-100: #121212;\n    --neutral-200: #424242;\n    --neutral-300: #616161;\n    --neutral-400: #757575;\n    --neutral-500: #9e9e9e;\n    --neutral-600: #bdbdbd;\n    --neutral-700: #e0e0e0;\n    --neutral-800: #eeeeee;\n    --neutral-900: #f5f5f5;\n  }\n}\n/* @media (prefers-color-scheme: dark) {\n  :host {\n      --label-color: #fff;\n      --active-label-color: #000;\n      --menu-bg: #525252;\n      --active-bg: #5898ff;\n      --active-bg-dimmed: #5c5c5c;\n  }\n} */\n:host {\n  --ui-font-family: 'Inter', system-ui, -apple-system, BlinkMacSystemFont,\n    'Segoe UI', Helvetica, Arial, sans-serif, 'Apple Color Emoji',\n    'Segoe UI Emoji', 'Segoe UI Symbol';\n  --ui-font-size: 14px;\n  --ui-line-height: 1.5;\n  --ui-letter-spacing: 0.007em;\n  --mono-font-family: 'Berkeley Mono', 'JetBrains Mono', 'IBM Plex Mono',\n    'Source Code Pro', Menlo, Monaco, 'Courier New', monospace;\n  --ui-layer-1: var(--neutral-100);\n  --ui-layer-2: var(--neutral-200);\n  --ui-layer-3: var(--neutral-300);\n  --ui-layer-4: var(--neutral-400);\n  --ui-layer-5: var(--neutral-500);\n  --ui-layer-6: var(--neutral-600);\n  --ui-border-color: var(--primary-color);\n  --ui-border-radius: 4px;\n  --ui-text: var(--neutral-900);\n  --ui-text-secondary: var(--neutral-700);\n  --ui-text-placeholder: var(--neutral-500);\n  --ui-text-muted: var(--neutral-300);\n  /** A field is a UI element in which a user can type data, for\n  * example an input or textarea element.\n  */\n  --ui-field-bg: var(--neutral-100);\n  --ui-field-bg-hover: var(--neutral-100);\n  --ui-field-bg-disabled: var(--neutral-300);\n  --ui-field-bg-invalid: var(--red-100);\n  --ui-field-bg-focus: var(--neutral-100);\n  --ui-field-border: 0.5px solid var(--border-color);\n  --ui-field-border-hover: 0.5px solid var(--border-color);\n  --ui-field-border-disabled: 0.5px solid var(--border-color);\n  --ui-field-border-invalid: 0.5px solid var(--border-color);\n  --ui-field-border-focus: 0.5px solid var(--border-color);\n  --ui-menu-bg: var(--neutral-100);\n  --ui-menu-text: var(--neutral-900);\n  --ui-menu-bg-hover: var(--neutral-200);\n  --ui-menu-text-hover: var(--neutral-900);\n  /** The `active` state is used for the state of menu items\n  * when they are selected.\n  */\n  --ui-menu-bg-active: var(--primary-color);\n  --ui-menu-text-active: var(--primary-color-reverse);\n  /** The `active-muted` set is used for the state of\n  * submenus when they are open.\n  */\n  --ui-menu-bg-active-muted: var(--neutral-300);\n  --ui-menu-text-active-muted: var(--neutral-900);\n  /* --ui-menu-shadow: 0 1px 2px 0 rgba(60, 64, 67, 0.302),\n0 2px 6px 2px rgba(60, 64, 67, 0.149); */\n  --ui-menu-shadow: 0 0 2px rgba(0, 0, 0, 0.5), 0 0 20px rgba(0, 0, 0, 0.2);\n  --ui-menu-divider: 0.5px solid #c7c7c7;\n  /* var(--neutral-300); */\n  --ui-menu-z-index: 10000;\n  --page-bg: var(--neutral-100);\n  --content-bg: var(--neutral-200);\n}\n@media (prefers-color-scheme: dark) {\n  :host {\n    --ui-menu-bg: var(--neutral-200);\n  }\n}\n/* PingFang SC is a macOS font. 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e.silenceNotifications||(a=null==(i=null==(t=e.mathfield.host)?void 0:t.dispatchEvent(new CustomEvent("move-out",{detail:{direction:"upward"},cancelable:!0,bubbles:!0,composed:!0})))||i),e.announce(a?"plonk":"line"),a},r=e.at(e.position),s=r;s&&"below"!==s.parentBranch&&!(Array.isArray(s.parentBranch)&&s.parent instanceof nb);)s=s.parent;if(Array.isArray(null==s?void 0:s.parentBranch)&&s.parent instanceof nb){var l=s.parent;if(s.parentBranch[0]<1)return n();var c=s.parentBranch[0]-1,p=l.array[c][s.parentBranch[1]];if(!p.some((function(e){return"prompt"===e.type&&!e.captureSelection}))&&e.mathfield.hasEditablePrompts)return n();mg(e,r,p,o,"up")}else{if(!s)return n();var h=null!=(a=s.parent.branch("above"))?a:s.parent.createBranch("above");if(!h.some((function(e){return"prompt"===e.type&&e.placeholderId}))&&e.mathfield.hasEditablePrompts)return n();mg(e,r,h,o,"up")}return e.mathfield.stopCoalescingUndo(),!0}(e,i);if("downward"===t)return function(e,t){var i,a,o=null!=(i=null==t?void 0:t.extend)&&i;o||e.collapseSelection("forward");for(var n=function(){var t,i,a=!0;return e.silenceNotifications||(a=null==(i=null==(t=e.mathfield.host)?void 0:t.dispatchEvent(new CustomEvent("move-out",{detail:{direction:"downward"},cancelable:!0,bubbles:!0,composed:!0})))||i),e.announce(a?"plonk":"line"),a},r=e.at(e.position),s=r;s&&"above"!==s.parentBranch&&!(Array.isArray(s.parentBranch)&&s.parent instanceof nb);)s=s.parent;if(Array.isArray(null==s?void 0:s.parentBranch)&&s.parent instanceof nb){var l=s.parent;if(s.parentBranch[0]+1>l.array.length-1)return n();var c=s.parentBranch[0]+1,p=l.array[c][s.parentBranch[1]];if(!p.some((function(e){return"prompt"===e.type&&!e.captureSelection}))&&e.mathfield.hasEditablePrompts)return n();mg(e,r,p,o,"down")}else{if(!s)return n();var h=null!=(a=s.parent.branch("below"))?a:s.parent.createBranch("below");if(!h.some((function(e){return"prompt"===e.type}))&&e.mathfield.hasEditablePrompts)return n();mg(e,r,h,o,"down")}return!0}(e,i);if(i.extend){var p=dg(e,e.position,t);p<0&&(p=0),p>e.lastOffset&&(p=e.lastOffset);var h=e.setSelection(e.anchor,p);return e.mathfield.stopCoalescingUndo(),h}if(e.selectionIsPlaceholder){e.collapseSelection(t);var $=$g(e,t);return e.mathfield.stopCoalescingUndo(),$}var d=e.position,u=d;if(e.collapseSelection(t)?ug(e,d=e.position)||(d=dg(e,d,t)):d=dg(e,d,t),d<0||d>e.lastOffset){var m=!0;return e.silenceNotifications||(m=null==(o=null==(a=e.mathfield.host)?void 0:a.dispatchEvent(new CustomEvent("move-out",{detail:{direction:t},cancelable:!0,bubbles:!0,composed:!0})))||o),m&&e.announce("plonk"),m}return e.setPositionHandlingPlaceholder(d),e.mathfield.stopCoalescingUndo(),e.announce("move",u),!0}function dg(e,t,i){return(t+="forward"===i?1:-1)<0||t>e.lastOffset||ug(e,t)?t:dg(e,t,i)}function ug(e,t){for(var i,a=e.at(t),o=a.parent;o&&!o.inCaptureSelection;)o=o.parent;return!(null!=o&&o.inCaptureSelection||null!=(i=a.parent)&&i.skipBoundary&&(!a.isFirstSibling&&a.isLastSibling||"first"===a.type)||e.mathfield.hasEditablePrompts&&!a.parentPrompt)}function mg(e,t,i,a,o){var n=e.mathfield.hasEditablePrompts,s=n?i.filter((function(e){return"prompt"===e.type&&!e.captureSelection})):i,l=Bm(e.mathfield.getHTMLElement(t)).right,c=e.offsetOf(function(e,t,i){for(var a=1/0,o=0;o<t.length;o++){var n=Bm(e.getHTMLElement(t[o])).right,r=Math.abs(i-n);if(!(r<=a))break;a=r}return t[o-1]}(e.mathfield,s,l))-(n?1:0);if(a){var p,h=(0,r.Z)(e.selection.ranges[0],2),$=h[0],d=h[1];p=c<("up"===o?$:d)?{ranges:[[c,d]],direction:"backward"}:{ranges:[[$,c]],direction:"forward"},e.setSelection(p)}else e.setPositionHandlingPlaceholder(c);e.announce("move ".concat(o))}function bg(e){var t=e.position,i=e.at(t).parent;return null!=i&&i.parent?(e.position=e.offsetOf(i),e.mathfield.stopCoalescingUndo(),e.announce("move",t),!0):(e.announce("plonk"),!1)}function yg(e){var t;if(e.collapseSelection(),function(e){for(var t=0,i=e.at(e.position),a=!1;i;)(!i.hasEmptyBranch("superscript")||!i.hasEmptyBranch("subscript"))&&(t+=1),i.hasEmptyBranch("superscript")?i.hasEmptyBranch("subscript")||(a=!1):a=!0,i=i.parent;return a?t:0}(e)>=e.mathfield.options.scriptDepth[1])return e.announce("plonk"),!1;var i=e.at(e.position);return void 0===i.subsupPlacement&&("subsup"!==(null==(t=i.rightSibling)?void 0:t.type)&&i.parent.addChildAfter(new fb({style:i.computedStyle}),i),i=i.rightSibling),i.createBranch("superscript"),e.setSelection(e.getSiblingsRange(e.offsetOf(i.superscript[0]))),!0}function fg(e){var t;if(e.collapseSelection(),function(e){for(var t=0,i=e.at(e.position),a=!1;i;)(!i.hasEmptyBranch("superscript")||!i.hasEmptyBranch("subscript"))&&(t+=1),i.hasEmptyBranch("superscript")?i.hasEmptyBranch("subscript")||(a=!0):a=!1,i=i.parent;return a?t:0}(e)>=e.mathfield.options.scriptDepth[0])return e.announce("plonk"),!1;var i=e.at(e.position);return void 0===i.subsupPlacement&&("subsup"!==(null==(t=e.at(e.position+1))?void 0:t.type)&&i.parent.addChildAfter(new fb({style:e.at(e.position).computedStyle}),i),i=e.at(e.position+1)),i.createBranch("subscript"),e.setSelection(e.getSiblingsRange(e.offsetOf(i.subscript[0]))),!0}function gg(){function e(e){return!(!function(e){return!(e.disabled||"hidden"===e.type&&"INPUT"===e.tagName.toUpperCase()||function(e){if(!Ps()||e===document.activeElement||e.contains(document.activeElement))return!1;if("hidden"===getComputedStyle(e).visibility)return!0;var t=e.getBoundingClientRect();if(0===t.width||0===t.height)return!0;for(;e;){if("none"===getComputedStyle(e).display)return!0;e=e.parentElement}return!1}(e))}(e)||function(e){return"INPUT"===e.tagName.toUpperCase()&&"radio"===e.type&&!function(e){var t;if(!e.name)return!0;var i=function(e,t){var i,a=C(e);try{for(a.s();!(i=a.n()).done;){var o=i.value;if(o.checked&&o.form===t)return o}}catch(n){a.e(n)}finally{a.f()}return null}((null!=(t=e.form)?t:e.ownerDocument).querySelectorAll('input[type="radio"][name="'+e.name+'"]'),e.form);return!i||i===e}(e)}(e)||t(e)<0)}function t(e){var t,i=Number.parseInt(null!=(t=e.getAttribute("tabindex"))?t:"NaN",10);return Number.isNaN(i)?"true"===e.contentEditable||("AUDIO"===e.nodeName||"VIDEO"===e.nodeName)&&null===e.getAttribute("tabindex")?0:e.tabIndex:i}return Ps()?function(i){var a=[],o=[];return(0,Z.Z)(i.querySelectorAll('input, select, textarea, a[href], button,\n        [tabindex], audio[controls], video[controls],\n        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e.announce("move",i),e.mathfield.stopCoalescingUndo(),!0}function Tg(e,t){var i,a,o=!(arguments.length>2&&void 0!==arguments[2])||arguments[2],n="forward"===t?1:-1;"placeholder"===e.at(e.anchor).type&&$g(e,t);var r,s=e.at(e.anchor).parentPrompt;r=s?"forward"===t?e.offsetOf(s)+1:e.offsetOf(s.leftSibling):Math.max(e.position+n,0);var l=Ag(e,r,t);if(!l||"forward"===t&&e.offsetOf(l)<r||"backward"===t&&e.offsetOf(l)>r){if(!o||null!=(a=null==(i=e.mathfield.host)?void 0:i.dispatchEvent(new CustomEvent("move-out",{detail:{direction:t},cancelable:!0,bubbles:!0,composed:!0})))&&!a)return e.announce("plonk"),!1;var c=gg();if(!document.activeElement||c.length<=1)return e.announce("plonk"),!1;var p=c.indexOf(document.activeElement)+n;return p<0&&(p=c.length-1),p>=c.length&&(p=0),c[p].focus(),e.mathfield.stopCoalescingUndo(),!0}return vg(e,l),!0}function Ag(e){var t=arguments.length>1&&void 0!==arguments[1]?arguments[1]:0,i=arguments.length>2&&void 0!==arguments[2]?arguments[2]:"forward";return 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d=Cy(t),u=[].concat((0,Z.Z)(null!=(a=null==(i=$[$.length-1])?void 0:i.keystrokes)?a:[]),[d]);$.push({state:n.getState(),keystrokes:u,leftSiblings:qg(e)});for(var m=0,b="";!c&&m<u.length;)p=$.length-(u.length-m),b=u.slice(m).join(""),!(c=sm($[p].leftSiblings,b,e.options.inlineShortcuts))&&/^[a-zA-Z][a-zA-Z0-9]+?([_\^][a-zA-Z0-9\*\+\-]+?)?$/.test(b)&&(c=e.options.onInlineShortcut(e,b)),m+=1;e.flushInlineShortcutBuffer({defer:!0})}else e.flushInlineShortcutBuffer();e.options.smartMode&&(c?e.switchMode("math"):function(e,t,i){var a=e.model;if("latex"===a.mode||!a.at(a.position).isLastSibling||!i||!Ty(i))return!1;var o=Cy(i);if(!a.selectionIsCollapsed)return!("text"!==e.model.mode||!/[/_^]/.test(o));var n=function(e){for(var t="",i=e.position,a=!1;!a;){var o=e.at(i);(a=!(o&&("text"===o.mode||"math"===o.mode&&o.type&&/mord|mpunct/.test(o.type))))||(t=o.value+t),i-=1}return t}(a)+o;if("text"===e.model.mode){if("Esc"===t||/[/\\]/.test(o))return!0;if(/[\^_]/.test(o))return/(^|\s)[a-zA-Z][^_]$/.test(n)&&Og(a,1),!0;var r={")":"(","}":"{","]":"["}[o],s=a.at(a.position).parent;if(r&&s instanceof db&&s.leftDelim===r)return!0;if(/(^|[^a-zA-Z])(a|I) $/.test(n))return!1;if(/[\$\xA2-\xA5\u0E3F\u20A1\u20A4\u20A7-\u20A9\u20AC\u20B1\u20B9\u20BA]/.test(o))return!0;if(/(^|[^a-zA-Z'\u2019])[a-zA-Z] $/.test(n))return Og(a,1),!1;if(/\D\.[^\d\s]$/.test(n)){Og(a,1);var l=a.at(a.position);return l.value="\u22c5",l.style.variant="normal",l.command="\\cdot",l.verbatimLatex=void 0,a.contentDidChange({data:"\\cdot",inputType:"insertText"}),!0}if(/(^|\s)[a-zA-Z][^a-zA-Z]$/.test(n)||/\.\d$/.test(n)||/\([\d+\-.]$/.test(n))return Og(a,1),!0;if(/\([a-z][,;]$/.test(n))return Og(a,2),!0;if(/[\d+\-=><*|]$/.test(o))return Sg(a),!0}else{if("[Space]"===t)return Cg(a,void 0,(function(e){return/[a-z][:,;.]$/.test(e.value)})),!0;if(/[a-zA-Z]{3,}$/.test(n)&&!/(dxd|abc|xyz|uvw)$/.test(n))return 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a=hy(py(Oy(i))),o=hy({key:i.key,shift:i.shiftKey,alt:i.altKey,ctrl:i.ctrlKey,meta:i.metaKey||i.ctrlKey&&/macos|ios/.test(Ms()),cmd:!1,win:!1}),n=e.length-1;n>=0;n--)if((e[n].key===a||e[n].key===o)&&(!e[n].ifMode||e[n].ifMode===t))return e[n].command;return""}(e.keybindings,n.mode,t)),!h&&("[Enter]"===s||"[Return]"===s)){var y=!1;return n.contentWillChange({inputType:"insertLineBreak"})&&(e.host&&(y=!e.host.dispatchEvent(new Event("change",{bubbles:!0,composed:!0}))),y||t.preventDefault&&(t.preventDefault(),t.stopPropagation()),n.contentDidChange({inputType:"insertLineBreak"})),y}if((!h||"[Space]"===s)&&"math"===n.mode){if("[Space]"===s){if(e.adoptStyle="none",e.flushInlineShortcutBuffer(),e.options.mathModeSpace)return ym.insert(n,e.options.mathModeSpace,{format:"latex",mode:"math"}),e.snapshot("insert-space"),h="",e.dirty=!0,e.scrollIntoView(),t.preventDefault&&(t.preventDefault(),t.stopPropagation()),!1;var f=n.at(n.position+1),g=n.at(n.position-1);if("text"===(null==f?void 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T={"(":"(","{":"\\lbrace","[":"\\lbrack"}[v],A={"(":")","{":"\\rbrace","[":"\\rbrack"}[v],w=am(n.selection),O=(0,r.Z)(w,2),S=O[0],B=O[1];return e.snapshot(),n.position=B,ym.insert(n,A,{format:"latex"}),n.position=S,ym.insert(n,T,{format:"latex"}),n.setSelection(S+1,B+1),n.contentDidChange({data:v,inputType:"insertText"}),e.snapshot("insert-fence"),e.dirty=!0,e.scrollIntoView(),t.preventDefault&&t.preventDefault(),!1}}}return!0}var _=n.at(Math.max(n.position,n.anchor)),z=_.parent;if("moveAfterParent"===h&&"leftright"===(null==z?void 0:z.type)&&_.isLastSibling&&e.options.smartFence&&kg(n,".",e.defaultStyle)&&(h="",of(e)),e.keyboardDelegate.cancelComposition(),h)e.executeCommand(h);else if(c){l=e.effectiveStyle;n.setState($[p].state);var q,I=C(u=$[$.length-1].keystrokes);try{for(I.s();!(q=I.n()).done;){m=q.value;ym.insert(n,m,{silenceNotifications:!0,style:l})}}catch(k){I.e(k)}finally{I.f()}e.snapshot("insert-shortcut"),n.setState($[p].state),n.deferNotifications({content:!0,selection:!0,data:c,type:"insertText"},(function(){return ym.insert(n,c,{format:"latex",style:l}),Sg(e.model),c.endsWith(" ")&&(e.switchMode("text"),ym.insert(n," ",{style:l,mode:"text"})),e.snapshot(),n.selectionIsCollapsed||e.flushInlineShortcutBuffer(),!0})),e.dirty=!0,n.announce("replacement")}return e.scrollIntoView(),t.preventDefault&&t.preventDefault(),!1}function zg(e,t,i){var a=e.model;if(e.isSelectionEditable){null!=i||(i={}),i.focus&&e.focus(),i.feedback&&window.MathfieldElement.playSound("keypress"),"string"==typeof i.mode&&(e.switchMode(i.mode),e.snapshot());var o=uu(t),n=window.mathVirtualKeyboard;if(null!=n&&n.isShifted&&(o="string"==typeof o?o.toUpperCase():o.map((function(e){return e.toUpperCase()}))),i.simulateKeystroke){var 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db){v.leftDelim=n,e.mathfield.snapshot();var T=e.extractAtoms([e.position,f-1]);return T=T.filter((function(e){return"first"!==e.type})),v.addChildren(T,v.parentBranch),e.position+=1,e.contentDidChange({data:n,inputType:"insertText"}),e.mathfield.snapshot("insert-fence"),!0}if(o instanceof db&&("?"===o.leftDelim||"."===o.leftDelim)&&Dg(n,o.rightDelim)){o.isDirty=!0,o.leftDelim=n,e.mathfield.snapshot();var A,w=C(e.extractAtoms([e.offsetOf(a.firstSibling),e.position]));try{for(w.s();!(A=w.n()).done;){var O=A.value;o.parent.addChildBefore(O,o)}}catch(E){w.e(E)}finally{w.f()}return e.contentDidChange({data:n,inputType:"insertText"}),e.mathfield.snapshot("insert-fence"),!0}if(!(o instanceof db&&"|"===o.leftDelim)){if(e.mathfield.snapshot(),ym.insert(e,"\\left".concat(n,"\\right?"),{format:"latex",style:i}),"first"!==a.lastSibling.type){var S=e.offsetOf(a.lastSibling),B=e.extractAtoms([e.position,S]);e.at(e.position).body=B,e.position-=1}return 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e.mathfield.snapshot(),Y.rightDelim=n,Y.addChildren(e.extractAtoms([k,e.position]),Y.parentBranch),e.contentDidChange({data:n,inputType:"insertText"}),e.mathfield.snapshot("insert-fence"),!0;if(o instanceof db&&("?"===o.rightDelim||"."===o.rightDelim)&&Ng(o.leftDelim,n))return e.mathfield.snapshot(),o.isDirty=!0,o.rightDelim=n,o.parent.addChildren(e.extractAtoms([e.position,e.offsetOf(a.lastSibling)]),o.parentBranch),e.position=e.offsetOf(o),e.contentDidChange({data:n,inputType:"insertText"}),e.mathfield.snapshot("insert-fence"),!0;var P=o.parent;return!(!(P instanceof db)||"?"!==P.rightDelim&&"."!==P.rightDelim||!e.at(e.position).isLastSibling)&&(e.position=e.offsetOf(P),kg(e,n,i))}return!1}function Ng(e,t){return!e||(["(","\\lparen","{","\\{","\\lbrace","[","\\lbrack"].includes(e)?[")","\\rparen","}","\\}","\\rbrace","]","\\rbrack"].includes(t):Ym[e]===t)}function 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e.selectionIsCollapsed&&0===e.position?(e.announce("plonk"),!1):(e.position=0,e.mathfield.stopCoalescingUndo(),!0)},moveToMathfieldEnd:function(e){return e.selectionIsCollapsed&&e.position===e.lastOffset?(e.announce("plonk"),!1):(e.position=e.lastOffset,e.mathfield.stopCoalescingUndo(),!0)},moveToSuperscript:yg,moveToSubscript:fg},{target:"model",changeSelection:!0}),cf({moveToNextPlaceholder:function(e){return Tg(e,"forward")},moveToPreviousPlaceholder:function(e){return Tg(e,"backward")}},{target:"model",changeSelection:!0,audioFeedback:"return"}),cf({undo:function(e){return e.undo(),!0},redo:function(e){return e.redo(),!0},scrollIntoView:function(e){return e.scrollIntoView(),!0},scrollToStart:function(e){return e.field.scroll(0,0),!0},scrollToEnd:function(e){var t=e.field.getBoundingClientRect();return e.field.scroll(t.left-window.scrollX,0),!0},toggleKeystrokeCaption:function(e){return 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0))}},{key:"scheduleOperation",value:function(e){var t=this;this.cancelDelayedOperation();var a=i.SUBMENU_DELAY;a<=0?e():this.hysteresisTimer=setTimeout((function(){t.hysteresisTimer=0,e()}),a)}},{key:"cancelDelayedOperation",value:function(){this.hysteresisTimer&&(clearTimeout(this.hysteresisTimer),this.hysteresisTimer=0)}}]),i}(wx);Cx.SUBMENU_DELAY=120;var Ox=Cx;function Sx(e){return!fx(e)&&(!!("function"==typeof e.label||"function"==typeof e.ariaLabel||"function"==typeof e.tooltip||(yx(e)||bx(e))&&("function"==typeof e.enabled||"function"==typeof e.visible)||yx(e)&&"function"==typeof e.checked)||!!bx(e)&&e.submenu.some(Sx))}var Bx=(0,$.Z)((function e(){(0,h.Z)(this,e)}));function _x(e){return new Promise((function(t,i){var a=gy(e);a||t(!1);for(var o=a,n=setTimeout((function(){r.abort(),t(function(e,t){return Math.hypot(t.x-e.x,t.y-e.y)}(o,a)<Bx.MAX_DISTANCE)}),Bx.DELAY),r=new AbortController,s=r.signal,l=0,c=["pointermove","pointerup","pointercancel"];l<c.length;l++){var 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e.deferNotifications({content:!0,selection:!0,type:i},(function(){var t,i,a=c.removeBranch(p);if(1===a.length&&"placeholder"===a[0].type)null==(i=c.parent)||i.removeChild(c),e.position=Math.max(0,h);else{var o=c.parent.addChildrenAfter(a,c);null==(t=c.parent)||t.removeChild(c),e.position=e.offsetOf(o)}}))}}return e.deferNotifications({content:!0,selection:!0,type:i},(function(){return e.deleteAtoms(t)}))}cf({deleteAll:function(e){return e.contentWillChange({inputType:"deleteContent"})&&_v(e,[0,-1],"deleteContent")},deleteForward:function(e){return function(e){return!(!e.mathfield.isSelectionEditable||!e.contentWillChange({inputType:"deleteContentForward"}))&&(e.selectionIsCollapsed?e.deferNotifications({content:!0,selection:!0,type:"deleteContentForward"},(function(){var 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l.abort()},cancelComposition:function(){n&&(e.blur(),requestAnimationFrame((function(){return e.focus({preventScroll:!0})})))},blur:function(){"function"==typeof e.blur&&e.blur()},focus:function(){!r&&"function"==typeof e.focus&&e.focus({preventScroll:!0})},hasFocus:function(){return Ay()===e},setAriaLabel:function(t){return e.setAttribute("aria-label",t)},setValue:function(t){var i;e.textContent=t,e.style.left="-1000px",null==(i=window.getSelection())||i.selectAllChildren(e)},moveTo:function(t,i){e.style.top="".concat(i,"px"),e.style.left="".concat(t,"px")}}}(this.element.querySelector(".ML__keyboard-sink"),this.element,this),window.addEventListener("resize",this,{signal:s}),document.addEventListener("scroll",this,{signal:s}),this.resizeObserver=new ResizeObserver((function(){return of(r)})),this.resizeObserver.observe(this.field),window.mathVirtualKeyboard.addEventListener("virtual-keyboard-toggle",this,{signal:s}),$y&&!Ls.locale.startsWith($y.locale)&&function(e){$y=uy.find((function(t){return e.startsWith(t.locale)}))}(Ls.locale),"ready"!==Jy&&document.fonts.ready.then((function(){return sf(r)})),t.querySelector("[part=container]").style.removeProperty("visibility"),this.undoManager.startRecording(),this.undoManager.snapshot("set-value")}else console.error("%cMathLive 0.98.3: Something went wrong and the mathfield could not be created.%c\nIf you are using Vue, this may be because you are using the runtime-only build of Vue. 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t.computedStyle}},{key:"connectToVirtualKeyboard",value:function(){this.connectedToVirtualKeyboard||(this.connectedToVirtualKeyboard=!0,window.addEventListener("message",this,{signal:this.eventController.signal}),window.mathVirtualKeyboard.connect(),window.mathVirtualKeyboard.visible&&window.mathVirtualKeyboard.update(ex(this)),$x(this))}},{key:"disconnectFromVirtualKeyboard",value:function(){this.connectedToVirtualKeyboard&&(window.removeEventListener("message",this),window.mathVirtualKeyboard.disconnect(),this.connectedToVirtualKeyboard=!1,hx())}},{key:"showMenu",value:function(e){var t,i,a=null!=(i=null!=(t=null==e?void 0:e.location)?t:vm(this.field))?i:void 0,o=null==e?void 0:e.modifiers,n=this.element.querySelector("[part=container]");return this._menu.show({target:n,location:a,modifiers:o})}},{key:"colorMap",get:function(){var e=this;return function(t){var i,a,o;return null!=(o=null==(a=(i=e.options).colorMap)?void 0:a.call(i,t))?o:Qs(t)}}},{key:"backgroundColorMap",get:function(){var e=this;return function(t){var i,a,o,n,r,s;return null!=(s=null!=(r=null==(a=(i=e.options).backgroundColorMap)?void 0:a.call(i,t))?r:null==(n=(o=e.options).colorMap)?void 0:n.call(o,t))?s:Ks(t)}}},{key:"smartFence",get:function(){var e;return null!=(e=this.options.smartFence)&&e}},{key:"readOnly",get:function(){var e;return null!=(e=this.options.readOnly)&&e}},{key:"disabled",get:function(){var e,t;return null!=(t=null==(e=this.host)?void 0:e.disabled)&&t}},{key:"contentEditable",get:function(){var e;return"false"!==(null==(e=this.host)?void 0:e.getAttribute("contenteditable"))}},{key:"userSelect",get:function(){if(!this.host)return"";var e=getComputedStyle(this.host);return e.getPropertyValue("user-select")||e.getPropertyValue("-webkit-user-select")}},{key:"hasEditableContent",get:function(){return!(this.disabled||!this.contentEditable)&&(!this.readOnly||this.hasEditablePrompts)}},{key:"hasEditablePrompts",get:function(){return this.readOnly&&!this.disabled&&this.contentEditable&&void 0!==this.model.findAtom((function(e){return"prompt"===e.type&&!e.locked}))}},{key:"isSelectionEditable",get:function(){if(this.disabled||!this.contentEditable)return!1;if(!this.readOnly)return!0;var e=this.model.at(this.model.anchor),t=this.model.at(this.model.position),i=Ju.commonAncestor(e,t);return!!("prompt"===(null==i?void 0:i.type)||null!=i&&i.parentPrompt)}},{key:"letterShapeStyle",get:function(){var e;return null!=(e=this.options.letterShapeStyle)?e:"tex"}},{key:"minFontScale",get:function(){return this.options.minFontScale}},{key:"selectionStyle",get:function(){if(this.model.selectionIsCollapsed)return this.effectiveStyle;var e=this.model.getAtoms(this.model.selection);if(0===e.length)return{};var t,i=Is({},e[0].style),a=C(e);try{for(a.s();!(t=a.n()).done;)for(var o=t.value,n=0,s=Object.entries(o.style);n<s.length;n++){var l=(0,r.Z)(s[n],2),c=l[0],p=l[1];i[c]!==p&&delete i[c]}}catch(h){a.e(h)}finally{a.f()}return i}},{key:"queryStyle",value:function(e){var t=Yg(this,e);"verbatimColor"in t&&delete t.verbatimColor,"verbatimBackgroundColor"in t&&delete t.verbatimBackgroundColor;var i=Object.keys(t).length;if(0===i)return"all";if(i>1){for(var o=0,n=Object.keys(t);o<n.length;o++){var r=n[o],s=this.queryStyle((0,a.Z)({},r,t[r]));if("none"===s)return"none";if("some"===s)return"some"}return"all"}var l=Object.keys(t)[0],c=t[l];if(this.model.selectionIsCollapsed)return this.effectiveStyle[l]===c?"all":"none";var p=this.model.getAtoms(this.model.selection,{includeChildren:!0}),h=p.length;if(0===h)return"none";var $,d=0,u=C(p);try{for(u.s();!($=u.n()).done;){var m=$.value;"first"!==m.type?m.style[l]===c&&(d+=1):h-=1}}catch(b){u.e(b)}finally{u.f()}return 0===d?"none":d===h?"all":"some"}},{key:"keybindings",get:function(){var e,t;if(this._keybindings)return this._keybindings;var i=function(e,t){var i,a=[],o=[],n=C(e);try{for(n.s();!(i=n.n()).done;){var r=i.value;try{if("continue"===function(){var e=_y(r,t);if(!e)return"continue";var i=o.find((function(t){return t.key===e.key&&t.ifMode===e.ifMode}));if(i)throw new Error("Ambiguous key binding ".concat(r.key," (").concat(zy(r.command),") matches ").concat(i.key," (").concat(zy(i.command),") with the ").concat(t.displayName," keyboard layout"));o.push(e)}())continue}catch(s){s instanceof Error&&a.push(s.message)}}}catch(l){n.e(l)}finally{n.f()}return[o,a]}(this.options.keybindings,null!=(e=yy())?e:function(){switch(my()){case"apple":return sy;case"windows":return ly;case"linux":return cy}return sy}()),a=(0,r.Z)(i,2),o=a[0],n=a[1];return(null==(t=yy())?void 0:t.score)>0&&(this._keybindings=o,n.length>0&&console.error("MathLive 0.98.3: Invalid keybindings for current keyboard layout",n)),o}},{key:"menu",get:function(){return this._menu}},{key:"setOptions",value:function(e){var t;this.options=Is(Is({},this.options),Wf(e)),this._keybindings=void 0,"inline-math"===this.options.defaultMode?this.element.classList.add("ML__is-inline"):this.element.classList.remove("ML__is-inline");var i=this.options.defaultMode;"inline-math"===i&&(i="math"),(null==(t=this.model.root.firstChild)?void 0:t.mode)!==i&&(this.model.root.firstChild.mode=i),this.options.readOnly&&this.hasFocus()&&window.mathVirtualKeyboard.visible&&this.executeCommand("hideVirtualKeyboard");var a=Ju.serialize([this.model.root],{expandMacro:!1,defaultMode:this.options.defaultMode});("macros"in e||this.model.getValue()!==a)&&ym.insert(this.model,a,{insertionMode:"replaceAll",selectionMode:"after",format:"latex",silenceNotifications:!0,mode:"math"}),("value"in e||"macros"in e||"registers"in e||"colorMap"in e||"backgroundColorMap"in e||"letterShapeStyle"in e||"minFontScale"in e||"readOnly"in e||"placeholderSymbol"in e)&&of(this)}},{key:"getOptions",value:function(e){return Gf(this.options,e)}},{key:"getOption",value:function(e){return Gf(this.options,e)}},{key:"handleEvent",value:function(){var e=p(l().mark((function e(t){var i,a,o,n=this;return l().wrap((function(e){for(;;)switch(e.prev=e.next){case 0:if(!xm(this)){e.next=53;break}if(!df(t)){e.next=14;break}if(Sm(t.origin,null!=(i=this.options.originValidator)?i:"none")){e.next=4;break}throw new DOMException("Message from unknown origin (".concat(t.origin,") cannot be handled"),"SecurityError");case 4:if("execute-command"!==(a=t.data.action)){e.next=12;break}if("virtual-keyboard"!==hf(o=t.data.command)){e.next=9;break}return e.abrupt("return");case 9:this.executeCommand(o),e.next=13;break;case 12:"update-state"===a||("focus"===a?this.focus({preventScroll:!0}):"blur"===a&&this.blur());case 13:return e.abrupt("return");case 14:e.t0=t.type,e.next="focus"===e.t0?17:"blur"===e.t0?19:"mousedown"===e.t0?21:"pointerdown"===e.t0?23:"contextmenu"===e.t0?35:"virtual-keyboard-toggle"===e.t0?44:"resize"===e.t0?46:"scroll"===e.t0?48:"wheel"===e.t0?50:52;break;case 17:return this.onFocus(),e.abrupt("break",53);case 19:return this.onBlur(),e.abrupt("break",53);case 21:return"none"!==this.userSelect&&Wg(this,t),e.abrupt("break",53);case 23:if(e.t1="none"!==this.userSelect,!e.t1){e.next=34;break}if(Wg(this,t),e.t2=!1===t.shiftKey,!e.t2){e.next=31;break}return e.next=30,zx(t,this.element.querySelector("[part=container]"),this._menu);case 30:e.t2=e.sent;case 31:if(e.t3=e.t2,!e.t3){e.next=34;break}Lg.stop();case 34:return e.abrupt("break",53);case 35:if(e.t4="none"!==this.userSelect&&!1===t.shiftKey,!e.t4){e.next=40;break}return e.next=39,zx(t,this.element.querySelector("[part=container]"),this._menu);case 39:e.t4=e.sent;case 40:if(e.t5=e.t4,!e.t5){e.next=43;break}Lg.stop();case 43:return e.abrupt("break",53);case 44:return this.hasFocus()&&$x(this),e.abrupt("break",53);case 46:case 48:return this.geometryChangeTimer&&cancelAnimationFrame(this.geometryChangeTimer),this.geometryChangeTimer=requestAnimationFrame((function(){return xm(n)&&n.onGeometryChange()})),e.abrupt("break",53);case 50:return this.onWheel(t),e.abrupt("break",53);case 52:console.warn("Unexpected event type",t.type);case 53:case"end":return e.stop()}}),e,this)})));return function(t){return e.apply(this,arguments)}}()},{key:"dispose",value:function(){xm(this)&&(Ls.unsubscribe(this._l10Subscription),this.keyboardDelegate.dispose(),this.keyboardDelegate=void 0,this.eventController.abort(),this.eventController=void 0,this.resizeObserver.disconnect(),window.mathVirtualKeyboard.removeEventListener("virtual-keyboard-toggle",this),this.disconnectFromVirtualKeyboard(),this.model.dispose(),delete this.element.mathfield,this.element=void 0,this.host=void 0,this.field=void 0,this.ariaLiveText=void 0,document.getElementById("mathlive-keystroke-caption-panel")&&(Ry("mathlive-keystroke-caption-panel"),Nm("core"),Nm("keystroke-caption")),document.getElementById("mathlive-suggestion-popover")&&(Ry("mathlive-suggestion-popover"),Nm("suggestion-popover"),Nm("core")),document.getElementById("mathlive-environment-popover")&&(Ry("mathlive-environment-popover"),Nm("environment-popover"),Nm("core")))}},{key:"flushInlineShortcutBuffer",value:function(e){var t=this;if(null!=e||(e={defer:!1}),!e.defer)return this.inlineShortcutBuffer=[],clearTimeout(this.inlineShortcutBufferFlushTimer),void(this.inlineShortcutBufferFlushTimer=0);this.options.inlineShortcutTimeout>0&&(clearTimeout(this.inlineShortcutBufferFlushTimer),this.inlineShortcutBufferFlushTimer=setTimeout((function(){return t.flushInlineShortcutBuffer()}),this.options.inlineShortcutTimeout))}},{key:"executeCommand",value:function(e){var t=this;return"virtual-keyboard"===hf(e)?(this.focus({preventScroll:!0}),window.mathVirtualKeyboard.executeCommand(e),requestAnimationFrame((function(){return window.mathVirtualKeyboard.update(ex(t))})),!1):function(e,t){var i;if(!t)return!1;var a,o=[],n=!1,r=!1;Ys(t)?(a=t[0],o=t.slice(1)):a=t,a=a.replace(/-\w/g,(function(e){return e[1].toUpperCase()}));var s=lf[a],l=null==s?void 0:s.target;if("model"===l){var c;if(!e.isSelectionEditable&&null!=s&&s.changeContent)return e.model.announce("plonk"),!1;/^(delete|add)/.test(a)&&("deleteBackward"!==a&&e.flushInlineShortcutBuffer(),e.snapshot(a)),/^complete/.test(a)||Dy(e),(c=lf[a]).fn.apply(c,[e.model].concat((0,Z.Z)(o))),Yy(e),r=!0,n=!0}else if("virtual-keyboard"===l)r=null!=(i=window.mathVirtualKeyboard.executeCommand(t))&&i,n=!0;else{if(!lf[a])throw new Error('Unknown command "'.concat(a,'"'));var p;if(!e.isSelectionEditable&&null!=s&&s.changeContent)return e.model.announce("plonk"),!1;/^(undo|redo)/.test(a)&&e.flushInlineShortcutBuffer(),r=(p=lf[a]).fn.apply(p,[e].concat((0,Z.Z)(o))),n=!0}return"virtual-keyboard"!==l&&(!e.model.selectionIsCollapsed||null!=s&&s.changeSelection&&"deleteBackward"!==t)&&(e.flushInlineShortcutBuffer(),null!=s&&s.changeContent||e.stopCoalescingUndo(),e.defaultStyle={}),r&&of(e),n}(this,e)}},{key:"errors",get:function(){return Tb(this.model.getValue(),{context:this.context})}},{key:"getValue",value:function(e,t,i){return this.model.getValue(e,t,i)}},{key:"setValue",value:function(e,t){var i;void 0===(t=null!=t?t:{mode:"math"}).insertionMode&&(t.insertionMode="replaceAll"),(void 0===t.format||"auto"===t.format)&&(t.format="latex"),(void 0===t.mode||"auto"===t.mode)&&(t.mode=null!=(i=function(e,t){var i,a=e.at(t);if(a){i=a.mode;for(var o=a.parent;!i&&o;)o&&(i=o.mode),o=o.parent}return i}(this.model,this.model.position))?i:"math");var a=this.undoManager.canUndo();ym.insert(this.model,e,t)&&(of(this),a||this.undoManager.reset(),this.undoManager.snapshot("set-value"))}},{key:"expression",get:function(){var e=window.MathfieldElement.computeEngine;return e?e.box(e.parse(this.model.getValue("latex-unstyled"))):(console.error("MathLive 0.98.3:  no compute engine available. Make sure the Compute Engine library is loaded."),null)}},{key:"scrollIntoView",value:function(){var e;if(this.element){if(this.host)if(this.options.onScrollIntoView)this.options.onScrollIntoView(this);else if(this.host.scrollIntoView({block:"nearest",inline:"nearest"}),window.mathVirtualKeyboard.visible&&window.mathVirtualKeyboard.container===window.document.body){var t=window.mathVirtualKeyboard.boundingRect,i=this.host.getBoundingClientRect();i.bottom>t.top&&(null==(e=window.document.scrollingElement)||e.scrollBy(0,i.bottom-t.top+8))}this.dirty&&rf(this,{interactive:!0});var a=this.field.getBoundingClientRect(),o=null;if(this.model.selectionIsCollapsed)o=vm(this.field);else{var n=Om(this);if(n.length>0){var r,s=-1/0,l=-1/0,c=C(n);try{for(c.s();!(r=c.n()).done;){var p=r.value;p.right>s&&(s=p.right),p.top<l&&(l=p.top)}}catch(b){c.e(b)}finally{c.f()}o={x:s+a.left-this.field.scrollLeft,y:l+a.top-this.field.scrollTop,height:0}}}if(this.host&&o){var h=this.host.getBoundingClientRect(),$=o.y,d=this.host.scrollTop;$<h.top?d=$-h.top+this.host.scrollTop:$>h.bottom&&(d=$-h.bottom+this.host.scrollTop+o.height),this.host.scroll({top:d,left:0})}if(o){var u=o.x-window.scrollX,m=this.field.scrollLeft;u<a.left?m=u-a.left+this.field.scrollLeft-20:u>a.right&&(m=u-a.right+this.field.scrollLeft+20),this.field.scroll({top:this.field.scrollTop,left:m})}}}},{key:"insert",value:function(e,t){return!("string"!=typeof e||0===e.length&&("insertBefore"===(null==t?void 0:t.insertionMode)||"insertAfter"===(null==t?void 0:t.insertionMode))||0===e.length&&this.model.selectionIsCollapsed)&&(this.flushInlineShortcutBuffer(),(t=null!=t?t:{mode:"math"}).focus&&this.focus(),t.feedback&&(window.MathfieldElement.keypressVibration&&Es()&&navigator.vibrate(3),window.MathfieldElement.playSound("keypress")),"\\\\"===e?ag(this.model):"&"===e?og(this.model):this.model.selectionIsCollapsed?ym.insert(this.model,e,Is({style:this.model.at(this.model.position).computedStyle},t)):ym.insert(this.model,e,t),this.snapshot("insert-".concat(this.model.at(this.model.position).type)),of(this),t.scrollIntoView&&this.scrollIntoView(),!0)}},{key:"switchMode",value:function(e){var t,i=this,a=arguments.length>1&&void 0!==arguments[1]?arguments[1]:"",o=arguments.length>2&&void 0!==arguments[2]?arguments[2]:"";if(this.model.mode!==e&&this.hasEditableContent&&this.contentEditable&&!this.disabled){var n=this.model.mode;if(this.model.mode=e,null!=(t=this.host)&&t.dispatchEvent(new Event("mode-change",{bubbles:!0,composed:!0,cancelable:!0}))){var r=this.model.mode,s=this.model;s.deferNotifications({content:!!o||!!a,selection:!0,type:"insertText"},(function(){var t=!1;if(i.flushInlineShortcutBuffer(),i.stopCoalescingUndo(),a&&"latex"!==e){var n=vb(a,{context:i.context,parseMode:e});s.collapseSelection("forward");var l=s.at(s.position);s.position=s.offsetOf(l.parent.addChildrenAfter(n,l)),t=!0}if(i.model.mode=e,"latex"===e){var c=s.selectionIsCollapsed;Ey(i,"accept");var p,h=s.at(s.position);if(c)p="\\";else{var $=am(s.selection);p=i.model.getValue($,"latex");var d=i.model.extractAtoms($);1===d.length&&"placeholder"===d[0].type&&(p=a,c=!0),h=s.at($[0])}var u=new kb(p);h.parent.addChildAfter(u,h),c?s.position=s.offsetOf(u.lastChild):s.setSelection(s.offsetOf(u.firstChild),s.offsetOf(u.lastChild))}else ky(s).forEach((function(e){e.isError=!1}));if(o){var m=vb(o,{context:i.context,parseMode:r});s.collapseSelection("forward");var b=s.at(s.position);s.position=s.offsetOf(b.parent.addChildrenAfter(m,b)),t=!0}return of(i),i.undoManager.snapshot("latex"===e?"insert-latex":"insert"),t})),this.model.mode=e,window.mathVirtualKeyboard.update(ex(this))}else this.model.mode=n}}},{key:"hasFocus",value:function(){return!this.blurred}},{key:"focus",value:function(e){this.hasFocus()||(this.keyboardDelegate.focus(),this.connectToVirtualKeyboard(),this.onFocus(),this.model.announce("line")),null!=e&&e.preventScroll||this.scrollIntoView()}},{key:"blur",value:function(){this.disconnectFromVirtualKeyboard(),this.hasFocus()&&this.keyboardDelegate.blur()}},{key:"select",value:function(){this.model.selection={ranges:[[0,this.model.lastOffset]]},this.focus()}},{key:"applyStyle",value:function(e){var t,i,a=this,o=arguments.length>1&&void 0!==arguments[1]?arguments[1]:{},n={operation:"set",silenceNotifications:!1};nm(o)?n.range=o:("toggle"===o.operation&&(n.operation="toggle"),n.range=o.range,n.silenceNotifications=null!=(t=o.silenceNotifications)&&t);var r=Yg(this,e),s=null!=(i=n.operation)?i:"set";if(void 0===n.range&&this.model.selectionIsCollapsed){if("set"===s)return void(this.defaultStyle=Is(Is({},this.defaultStyle),r));for(var l=Is({},this.defaultStyle),c=0,p=Object.keys(r);c<p.length;c++){var h=p[c];l[h]===r[h]?("color"===h&&delete l.verbatimColor,"backgroundColor"===h&&delete l.verbatimBackgroundColor,delete l[h]):l[h]=r[h]}this.defaultStyle=l}else this.model.deferNotifications({content:!n.silenceNotifications,type:"insertText"},(function(){if(void 0===n.range){var e,t=C(a.model.selection.ranges);try{for(t.s();!(e=t.n()).done;){var i=e.value;jf(a.model,i,r,{operation:s})}}catch(o){t.e(o)}finally{t.f()}}else jf(a.model,n.range,r,{operation:s})})),of(this)}},{key:"toggleContextMenu",value:function(){var e,t=this;return!!this._menu.visible&&("open"===this._menu.state?(this._menu.hide(),!0):(this._menu.show({target:this.element.querySelector("[part=container]"),location:null!=(e=this.getCaretPoint())?e:void 0,onDismiss:function(){var e;return null==(e=t.element)?void 0:e.focus()}}),!0))}},{key:"getCaretPoint",value:function(){var e=vm(this.field);return e?{x:e.x,y:e.y}:null}},{key:"setCaretPoint",value:function(e,t){var i=Hg(this,e,t,{bias:0});if(i<0)return!1;var a=this.model.position;return this.model.position=i,this.model.announce("move",a),of(this),!0}},{key:"getPrompt",value:function(e){var t=this.model.findAtom((function(t){return"prompt"===t.type&&t.placeholderId===e}));return t}},{key:"getPromptValue",value:function(e,t){var i=this.getPrompt(e);if(!i)return"";var a=this.model.offsetOf(i.firstChild),o=this.model.offsetOf(i.lastChild);return this.model.getValue(a,o,t)}},{key:"getPrompts",value:function(e){return this.model.getAllAtoms().filter((function(t){return"prompt"===t.type&&(!e||!(e.id&&t.placeholderId!==e.id||e.locked&&t.locked!==e.locked||"undefined"===e.correctness&&t.correctness||e.correctness&&t.correctness!==e.correctness))})).map((function(e){return e.placeholderId}))}},{key:"setPromptValue",value:function(e,t,i){if(void 0!==t){var a=this.getPrompt(e);if(!a)return void console.error("MathLive 0.98.3: unknown prompt ".concat(e));var o=this.model.getBranchRange(this.model.offsetOf(a),"body");this.model.setSelection(o),this.insert(t,ks(Is({},i),{insertionMode:"replaceSelection"}))}null!=i&&i.silenceNotifications&&(this.valueOnFocus=this.getValue()),of(this)}},{key:"setPromptState",value:function(e,t,i){var a=this.getPrompt(e);a?("undefined"===t?a.correctness=void 0:"string"==typeof t&&(a.correctness=t),"boolean"==typeof i&&(a.locked=i,a.captureSelection=i),of(this)):console.error("MathLive 0.98.3: unknown prompt ".concat(e))}},{key:"getPromptState",value:function(e){var t=this.getPrompt(e);return t?[t.correctness,t.locked]:(console.error("MathLive 0.98.3: unknown prompt ".concat(e)),[void 0,!0])}},{key:"getPromptRange",value:function(e){var t=this.getPrompt(e);return t?this.model.getBranchRange(this.model.offsetOf(t),"body"):(console.error("MathLive 0.98.3: unknown prompt ".concat(e)),[0,0])}},{key:"canUndo",value:function(){return this.undoManager.canUndo()}},{key:"canRedo",value:function(){return this.undoManager.canRedo()}},{key:"popUndoStack",value:function(){this.undoManager.pop()}},{key:"snapshot",value:function(e){var t;this.undoManager.snapshot(e)&&(window.mathVirtualKeyboard.visible&&window.mathVirtualKeyboard.update(ex(this)),null==(t=this.host)||t.dispatchEvent(new CustomEvent("undo-state-change",{bubbles:!0,composed:!0,detail:{type:"snapshot"}})))}},{key:"stopCoalescingUndo",value:function(){this.undoManager.stopCoalescing(this.model.selection)}},{key:"stopRecording",value:function(){this.undoManager.stopRecording()}},{key:"startRecording",value:function(){this.undoManager.startRecording()}},{key:"undo",value:function(){var e;this.undoManager.undo()&&(window.mathVirtualKeyboard.visible&&window.mathVirtualKeyboard.update(ex(this)),null==(e=this.host)||e.dispatchEvent(new CustomEvent("undo-state-change",{bubbles:!0,composed:!0,detail:{type:"undo"}})))}},{key:"redo",value:function(){var e;this.undoManager.redo()&&(window.mathVirtualKeyboard.visible&&window.mathVirtualKeyboard.update(ex(this)),null==(e=this.host)||e.dispatchEvent(new CustomEvent("undo-state-change",{bubbles:!0,composed:!0,detail:{type:"undo"}})))}},{key:"resetUndo",value:function(){var e;null==(e=this.undoManager)||e.reset()}},{key:"onSelectionDidChange",value:function(){var e,t,i=this.model;this.keyboardDelegate.setValue(i.getValue(this.model.selection,"latex-expanded"));var a=i.at(i.position),o=null!=(e=a.mode)?e:Uf(this.options);this.model.mode!==o&&("latex"===this.model.mode?(Ey(this,"accept",{mode:o}),i.position=i.offsetOf(a)):this.switchMode(o)),null==(t=this.host)||t.dispatchEvent(new Event("selection-change",{bubbles:!0,composed:!0})),window.mathVirtualKeyboard.visible&&window.mathVirtualKeyboard.update(ex(this)),$x(this)}},{key:"onContentWillChange",value:function(e){var t,i,a;return null==(a=null==(i=this.host)?void 0:i.dispatchEvent(new InputEvent("beforeinput",ks(Is({},e),{data:e.data?e.data:null!=(t=e.inputType)?t:"",cancelable:!0,bubbles:!0,composed:!0}))))||a}},{key:"onFocus",value:function(){this.focusBlurInProgress||!this.blurred||(this.focusBlurInProgress=!0,this.blurred=!1,this.keyboardDelegate.focus(),this.stopCoalescingUndo(),rf(this,{interactive:!0}),this.valueOnFocus=this.model.getValue(),this.hasEditablePrompts&&!this.model.at(this.model.anchor).parentPrompt&&this.executeCommand("moveToNextPlaceholder"),this.focusBlurInProgress=!1)}},{key:"onBlur",value:function(){var e,t,i,a=this;if(!this.focusBlurInProgress&&!this.blurred){this.focusBlurInProgress=!0,this.stopCoalescingUndo(),this.blurred=!0,this.ariaLiveText.textContent="",Gy(this),this.model.getValue()!==this.valueOnFocus&&(null==(e=this.host)||e.dispatchEvent(new Event("change",{bubbles:!0,composed:!0}))),this.disconnectFromVirtualKeyboard(),null==(t=this.host)||t.dispatchEvent(new Event("blur",{bubbles:!1,composed:!0})),null==(i=this.host)||i.dispatchEvent(new UIEvent("focusout",{bubbles:!0,composed:!0})),of(this),this.focusBlurInProgress=!1,hx();var o=new AbortController,n=o.signal;document.addEventListener("visibilitychange",(function(){"hidden"===document.visibilityState&&document.addEventListener("visibilitychange",(function(){xm(a)&&"visible"===document.visibilityState&&a.focus({preventScroll:!0})}),{once:!0,signal:n})}),{once:!0,signal:n}),setTimeout((function(){return o.abort()}),100)}}},{key:"onInput",value:function(e){zg(this,e)}},{key:"onKeystroke",value:function(e){return _g(this,e)}},{key:"onCompositionStart",value:function(e){var t=this;this.model.deleteAtoms(am(this.model.selection));var i=vm(this.field);i&&requestAnimationFrame((function(){rf(t),t.keyboardDelegate.moveTo(i.x,i.y-i.height)}))}},{key:"onCompositionUpdate",value:function(e){(function(e,t){var i=e.at(e.position);if("composition"===i.type)i.value=t;else{var a=i.caret;i.caret=void 0;var o=new pb(t,{mode:i.mode});o.caret=a,i.parent.addChildAfter(o,i),e.position+=1}})(this.model,e),of(this)}},{key:"onCompositionEnd",value:function(e){(function(e){var t=e.at(e.position);"composition"===t.type&&(t.parent.removeChild(t),e.position-=1)})(this.model),zg(this,e,{simulateKeystroke:!0})}},{key:"onCut",value:function(e){this.isSelectionEditable?this.model.contentWillChange({inputType:"deleteByCut"})&&(this.stopCoalescingUndo(),ym.onCopy(this,e),_v(this.model,am(this.model.selection),"deleteByCut"),this.snapshot("cut"),of(this)):this.model.announce("plonk")}},{key:"onCopy",value:function(e){ym.onCopy(this,e)}},{key:"onPaste",value:function(e){var t=this.isSelectionEditable;return t&&(t=ym.onPaste(this.model.at(this.model.position).mode,this,e.clipboardData)),t||this.model.announce("plonk"),e.preventDefault(),e.stopPropagation(),t}},{key:"onGeometryChange",value:function(){this._menu.hide(),Wy(this),$x(this)}},{key:"onWheel",value:function(e){var t=5*e.deltaX;if(Number.isFinite(t)&&0!==t){var i=this.field;t<0&&0===i.scrollLeft||t>0&&i.offsetWidth+i.scrollLeft>=i.scrollWidth||(i.scrollBy({top:0,left:t}),e.preventDefault(),e.stopPropagation())}}},{key:"getHTMLElement",value:function(e){for(var t=e;!t.id&&t.hasChildren;)t=e.children[0];return this.field.querySelector('[data-atom-id="'.concat(t.id,'"]'))}},{key:"context",get:function(){var e,t,i=this;return{registers:null!=(e=this.options.registers)?e:{},smartFence:this.smartFence,letterShapeStyle:this.letterShapeStyle,minFontScale:this.minFontScale,placeholderSymbol:null!=(t=this.options.placeholderSymbol)?t:"\u25a2",colorMap:function(e){return i.colorMap(e)},backgroundColorMap:function(e){return i.backgroundColorMap(e)},getMacro:function(e){return Sl(e,i.options.macros)},atomIdsSettings:{seed:"random",groupNumbers:!1}}}}]),e}();function kv(e){if(e&&(e.classList.remove("ML__highlight"),e.children)){var t,i=C(e.children);try{for(i.s();!(t=i.n()).done;){kv(t.value)}}catch(a){i.e(a)}finally{i.f()}}}function Nv(e,t){var i;e&&(t&&(null==(i=e.dataset)?void 0:i.atomId)!==t?(e.classList.remove("ML__highlight"),e.children&&e.children.length>0&&(0,Z.Z)(e.children).forEach((function(e){e instanceof HTMLElement&&Nv(e,t)}))):(e.classList.add("ML__highlight"),e.children&&e.children.length>0&&(0,Z.Z)(e.children).forEach((function(e){e instanceof HTMLElement&&Nv(e)}))))}Ps()||console.error('MathLive 0.98.3: this version of the MathLive library is for use in the browser. A subset of the API is available on the server side in the "mathlive-ssr" library. If using server side rendering (with React for example) you may want to do a dynamic import of the MathLive library inside a `useEffect()` call.');var Dv=new WeakMap,Yv={letterShapeStyle:"mf.letterShapeStyle = ...",horizontalSpacingScale:'Removed. Use `"thinmuskip"`, `"medmuskip"`, and `"thickmuskip"` registers ',macros:"mf.macros = ...",registers:"mf.registers = ...",backgroundColorMap:"mf.backgroundColorMap = ...",colorMap:"mf.colorMap = ...",enablePopover:"mf.popoverPolicy = ...",mathModeSpace:"mf.mathModeSpace = ...",placeholderSymbol:"mf.placeholderSymbol = ...",readOnly:"mf.readOnly = ...",removeExtraneousParentheses:"mf.removeExtraneousParentheses = ...",scriptDepth:"mf.scriptDepth = ...",smartFence:"mf.smartFence = ...",smartMode:"mf.smartMode = ...",smartSuperscript:"mf.smartSuperscript = ...",inlineShortcutTimeout:"mf.inlineShortcutTimeout = ...",inlineShortcuts:"mf.inlineShortcuts = ...",keybindings:"mf.keybindings = ...",virtualKeyboardMode:"mf.mathVirtualKeyboardPolicy = ...",customVirtualKeyboardLayers:"mathVirtualKeyboard.layers = ...",customVirtualKeyboards:"mathVirtualKeyboard.layouts = ...",keypressSound:"mathVirtualKeyboard.keypressSound = ...",keypressVibration:"mathVirtualKeyboard.keypressVibration = ...",plonkSound:"mathVirtualKeyboard.plonkSound = ...",virtualKeyboardContainer:"mathVirtualKeyboard.container = ...",virtualKeyboardLayout:"mathVirtualKeyboard.alphabeticLayout = ...",virtualKeyboardTheme:"No longer supported",virtualKeyboardToggleGlyph:"No longer supported",virtualKeyboardToolbar:"mathVirtualKeyboard.editToolbar = ...",virtualKeyboards:"Use `mathVirtualKeyboard.layouts`",speechEngine:"`MathfieldElement.speechEngine`",speechEngineRate:"`MathfieldElement.speechEngineRate`",speechEngineVoice:"`MathfieldElement.speechEngineVoice`",textToSpeechMarkup:"`MathfieldElement.textToSpeechMarkup`",textToSpeechRules:"`MathfieldElement.textToSpeechRules`",textToSpeechRulesOptions:"`MathfieldElement.textToSpeechRulesOptions`",readAloudHook:"`MathfieldElement.readAloudHook`",speakHook:"`MathfieldElement.speakHook`",computeEngine:"`MathfieldElement.computeEngine`",fontsDirectory:"`MathfieldElement.fontsDirectory`",soundsDirectory:"`MathfieldElement.soundsDirectory`",createHTML:"`MathfieldElement.createHTML`",onExport:"`MathfieldElement.onExport`",onInlineShortcut:"`MathfieldElement.onInlineShortcut`",onScrollIntoView:"`MathfieldElement.onScrollIntoView`",locale:"MathfieldElement.locale = ...",strings:"MathfieldElement.strings = ...",decimalSeparator:"MathfieldElement.decimalSeparator = ...",fractionNavigationOrder:"MathfieldElement.fractionNavigationOrder = ..."},Pv=function(e){m(i,e);var t=g(i);function i(e){var a;if((0,h.Z)(this,i),a=t.call(this),e){for(var o=[],n=0,r=Object.keys(e);n<r.length;n++){var s=r[n];if(Yv[s])if(Yv[s].startsWith("mf."))if(Yv[s].startsWith("mf.".concat(s)))o.push("Option `".concat(s,"` cannot be used as a constructor option. Use ").concat(Yv[s]));else{var l=Yv[s].match(/([a-zA-Z]+) =/);o.push("Option `".concat(s,"` has been renamed `").concat(l[1],"`"))}else o.push("Option `".concat(s,"` cannot be used as a constructor option. Use ").concat(Yv[s]))}if(o.length>0){console.group("%cMathLive 0.98.3: %cInvalid Options","color:#12b; font-size: 1.1rem","color:#db1111; font-size: 1.1rem"),console.warn("Some of the options passed to `new MathfieldElement(...)` are invalid. \n          See https://cortexjs.io/mathlive/changelog/ for details.");var c,p=C(o);try{for(p.s();!(c=p.n()).done;){var $=c.value;console.warn($)}}catch(u){p.e(u)}finally{p.f()}console.groupEnd()}}if(Lv()&&(a._internals=a.attachInternals(),a._internals.role="math",a._internals.ariaLabel="math input field",a._internals.ariaMultiLine="false"),a.attachShadow({mode:"open",delegatesFocus:!0}),a.shadowRoot&&"adoptedStyleSheets"in a.shadowRoot){a.shadowRoot.adoptedStyleSheets=[Im("core"),Im("mathfield"),Im("mathfield-element"),Im("ui"),Im("menu")],a.shadowRoot.appendChild(document.createElement("span"));var d=document.createElement("slot");d.style.display="none",a.shadowRoot.appendChild(d)}else a.shadowRoot.innerHTML="<style>"+qm("core")+qm("mathfield")+qm("mathfield-element")+qm("ui")+qm("menu")+'</style><span></span><slot style="display:none"></slot>';return e&&a._setOptions(e),a}return(0,$.Z)(i,[{key:"fontsDirectory",get:function(){throw new Error("Use MathfieldElement.fontsDirectory instead")},set:function(e){throw new Error("Use MathfieldElement.fontsDirectory instead")}},{key:"soundsDirectory",get:function(){throw new Error("Use MathfieldElement.soundsDirectory instead")},set:function(e){throw new Error("Use MathfieldElement.soundsDirectory instead")}},{key:"locale",get:function(){throw new Error("Use MathfieldElement.locale instead")},set:function(e){throw new Error("Use MathfieldElement.locale instead")}},{key:"strings",get:function(){throw new Error("Use MathfieldElement.strings instead")},set:function(e){throw new Error("Use MathfieldElement.strings instead")}},{key:"decimalSeparator",get:function(){throw new Error("Use MathfieldElement.decimalSeparator instead")},set:function(e){throw new Error("Use MathfieldElement.decimalSeparator instead")}},{key:"computeEngine",get:function(){throw new Error("Use MathfieldElement.computeEngine instead")},set:function(e){throw new Error("Use MathfieldElement.computeEngine instead")}},{key:"showMenu",value:function(e){var t,i;return null!=(i=null==(t=this._mathfield)?void 0:t.showMenu(e))&&i}},{key:"mathVirtualKeyboard",get:function(){throw new Error("The `mathVirtualKeyboard` property is not available on the MathfieldElement. Use `window.mathVirtualKeyboard` instead.")}},{key:"onPointerDown",value:function(){var e=this;window.addEventListener("pointerup",(function(t){var i;t.target===e&&(null==(i=e._mathfield)||!i.disabled)&&e.dispatchEvent(new MouseEvent("click",{altKey:t.altKey,button:t.button,buttons:t.buttons,clientX:t.clientX,clientY:t.clientY,ctrlKey:t.ctrlKey,metaKey:t.metaKey,movementX:t.movementX,movementY:t.movementY,relatedTarget:t.relatedTarget,screenX:t.screenX,screenY:t.screenY,shiftKey:t.shiftKey}))}),{once:!0})}},{key:"getPromptValue",value:function(e,t){var i,a;return null!=(a=null==(i=this._mathfield)?void 0:i.getPromptValue(e,t))?a:""}},{key:"setPromptValue",value:function(e,t,i){var a;null==(a=this._mathfield)||a.setPromptValue(e,t,i)}},{key:"getPromptRange",value:function(e){var t,i;return null!=(i=null==(t=this._mathfield)?void 0:t.getPromptRange(e))?i:null}},{key:"getPrompts",value:function(e){var t,i;return null!=(i=null==(t=this._mathfield)?void 0:t.getPrompts(e))?i:[]}},{key:"form",get:function(){var e;return null==(e=this._internals)?void 0:e.form}},{key:"name",get:function(){var e;return null!=(e=this.getAttribute("name"))?e:""}},{key:"type",get:function(){return this.localName}},{key:"mode",get:function(){var e,t;return null!=(t=null==(e=this._mathfield)?void 0:e.model.mode)?t:"text"===this.defaultMode?"text":"math"},set:function(e){var t;null==(t=this._mathfield)||t.switchMode(e)}},{key:"expression",get:function(){if(this._mathfield)return window[Symbol.for("io.cortexjs.compute-engine")]?this._mathfield.expression:(console.error('MathLive 0.98.3: The CortexJS Compute Engine library is not available.\n        \n        Load the library, for example with:\n        \n        import "https://unpkg.com/@cortex-js/compute-engine?module"'),null)},set:function(e){var t,a;if(this._mathfield){var o=null!=(a=null==(t=i.computeEngine)?void 0:t.box(e).latex)?a:null;null!==o&&this._mathfield.setValue(o),window[Symbol.for("io.cortexjs.compute-engine")]||console.error('MathLive 0.98.3: The CortexJS Compute Engine library is not available.\n        \n        Load the library, for example with:\n        \n        import "https://unpkg.com/@cortex-js/compute-engine?module"')}}},{key:"errors",get:function(){var e,t;return null!=(t=null==(e=this._mathfield)?void 0:e.errors)?t:[]}},{key:"_getOptions",value:function(e){return this._mathfield?Gf(this._mathfield.options,e):Dv.has(this)?Is({},Gf(Is(Is({},Vf()),Wf(Dv.get(this).options)),e)):null}},{key:"getOptions",value:function(e){return console.warn("%cMathLive 0.98.3: %cDeprecated Usage%c\n      `mf.getOptions()` is deprecated. Read the property directly on the mathfield instead.\n      See https://cortexjs.io/mathlive/changelog/ for details.","color:#12b; font-size: 1.1rem","color:#db1111; font-size: 1.1rem","color: inherit, font-size: 1rem"),this._mathfield?Gf(this._mathfield.options,e):Dv.has(this)?Gf(Is(Is({},Vf()),Wf(Dv.get(this).options)),e):null}},{key:"reflectAttributes",value:function(){var e=this,t=Vf(),a=this._getOptions();Object.keys(i.optionsAttributes).forEach((function(o){var n=Mv(o);"on/off"===i.optionsAttributes[o]?t[n]!==a[n]?e.setAttribute(o,a[n]?"on":"off"):e.removeAttribute(o):t[n]!==a[n]&&("boolean"===i.optionsAttributes[o]?a[n]?e.setAttribute(o,""):e.removeAttribute(o):("string"==typeof a[n]||"number"==typeof a[n])&&e.setAttribute(o,a[n].toString()))}))}},{key:"getOption",value:function(e){return console.warn("%cMathLive 0.98.3: %cDeprecated Usage%c\n      `mf.getOption()` is deprecated. Read the property directly on the mathfield instead.\n      See https://cortexjs.io/mathlive/changelog/ for details.","color:#12b; font-size: 1.1rem","color:#db1111; font-size: 1.1rem","color: inherit, font-size: 1rem"),this._getOptions([e])[e]}},{key:"_getOption",value:function(e){return this._getOptions([e])[e]}},{key:"_setOptions",value:function(e){if(this._mathfield)this._mathfield.setOptions(e);else if(Dv.has(this)){var t=Is(Is({},Dv.get(this).options),e);Dv.set(this,ks(Is({},Dv.get(this)),{selection:{ranges:t.readOnly?[[0,0]]:[[0,-1]]},options:t}))}else Dv.set(this,{value:void 0,selection:{ranges:[[0,0]]},options:e,menuItems:void 0});this.reflectAttributes()}},{key:"setOptions",value:function(e){console.group("%cMathLive 0.98.3: %cDeprecated Usage","color:#12b; font-size: 1.1rem","color:#db1111; font-size: 1.1rem"),console.warn(" `mf.setOptions()` is deprecated. 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i=Dv.get(this).options;Dv.set(this,{value:e,selection:{ranges:i.readOnly?[[0,0]]:[[0,-1]],direction:"forward"},options:i,menuItems:void 0})}else{var a=Rv(this);Dv.set(this,{value:e,selection:{ranges:a.readOnly?[[0,0]]:[[0,-1]],direction:"forward"},options:a,menuItems:void 0})}}},{key:"hasFocus",value:function(){var e,t;return null!=(t=null==(e=this._mathfield)?void 0:e.hasFocus())&&t}},{key:"focus",value:function(){var e;null==(e=this._mathfield)||e.focus()}},{key:"blur",value:function(){var e;null==(e=this._mathfield)||e.blur()}},{key:"select",value:function(){var e;null==(e=this._mathfield)||e.select()}},{key:"insert",value:function(e,t){var i,a;return null!=(a=null==(i=this._mathfield)?void 0:i.insert(e,t))&&a}},{key:"applyStyle",value:function(e,t){var i;return null==(i=this._mathfield)?void 0:i.applyStyle(e,t)}},{key:"queryStyle",value:function(e){var t,i;return null!=(i=null==(t=this._mathfield)?void 0:t.queryStyle(e))?i:"none"}},{key:"caretPoint",get:function(){var e,t;return 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